Confectionery and Chocolate Engineering Principles and Applications

Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

To the memory of my parents Ferenc Mohos and Viktória Tevesz

Confectionery and Chocolate Engineering Principles and Applications

Professor Ferenc Á. Mohos, PhD Chairman Codex Alimentarius Hungaricus Confectionery Products Working Committee

A John Wiley & Sons, Ltd., Publication

This edition first published 2010 © 2010 Ferenc Á. Mohos Blackwell Publishing was acquired by John Wiley & Sons in February 2007. Blackwell’s publishing programme has been merged with Wiley’s global Scientific, Technical, and Medical business to form Wiley-Blackwell. Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom Editorial offices 9600 Garsington Road, Oxford, OX4 2DQ, United Kingdom 2121 State Avenue, Ames, Iowa 50014-8300, USA For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com/ wiley-blackwell. The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Library of Congress Cataloging-in-Publication Data Mohos, Ferenc Á. Confectionery and chocolate engineering : principles and applications / Ferenc Á. Mohos. p. cm. Includes bibliographical references and index. ISBN 978-1-4051-9470-9 (hardback : alk. paper) 1. Confectionery. 2. Chocolate. 3. Chemistry, Technical. 4. Food–Analysis. I. Title. TX783.M58 2010 641.8′6–dc22 2009042943 A catalogue record for this book is available from the British Library. Set in 10 on 12 pt Times NR Monotype by Toppan Best-set Premedia Limited Printed in Singapore 1

2010

Contents

Preface Acknowledgements Part I

Theoretical introduction

Chapter 1 Principles of food engineering 1.1 Introduction 1.1.1 The peculiarities of food engineering 1.1.2 The hierarchical and semi-hierarchical structure of materials 1.1.3 Application of the Damköhler equations in food engineering 1.2 The Damköhler equations 1.3 Investigation of the Damköhler equations by means of similarity theory 1.3.1 Dimensionless numbers 1.3.2 Degrees of freedom of an operational unit 1.3.3 Polynomials as solutions of the Damköhler equations 1.4 Analogies 1.4.1 The Reynolds analogy 1.4.2 The Colburn analogy 1.4.3 Similarity and analogy 1.5 Dimensional analysis 1.6 The Buckingham Π theorem Further reading Chapter 2 Characterization of substances used in the confectionery industry 2.1 Qualitative characterization of substances 2.1.1 Principle of characterization 2.1.2 Structural formulae of confectionery products 2.1.3 Classification of confectionery products according to their characteristic phase conditions 2.1.4 Phase transitions – a bridge between sugar sweets and chocolate 2.2 Quantitative characterization of confectionery products 2.2.1 Composition of chocolates and compounds 2.2.2 Composition of sugar confectionery 2.2.3 Composition of biscuits, crackers and wafers 2.3 Preparation of recipes 2.3.1 Recipes and net/gross material consumption 2.3.2 Planning of material consumption

xviii xxi 1 3 3 3 5 6 6 8 8 11 12 13 13 15 16 16 17 18 19 19 19 20 27 28 29 29 35 43 45 45 48

vi

Contents

Chapter 3 Engineering properties of foods 3.1 Introduction 3.2 Density 3.2.1 Solids and powdered solids 3.2.2 Particle density 3.2.3 Bulk density and porosity 3.2.4 Loose bulk density 3.2.5 Dispersions of various kinds, and solutions 3.3 Fundamental functions of thermodynamics 3.3.1 Internal energy 3.3.2 Enthalpy 3.3.3 Specific heat capacity calculations 3.4 Latent heat and heat of reaction 3.4.1 Latent heat and free enthalpy 3.4.2 Phase transitions 3.5 Thermal conductivity 3.5.1 First Fourier equation 3.5.2 Heterogeneous materials 3.5.3 Liquid foods 3.5.4 Liquids containing suspended particles 3.5.5 Gases 3.6 Thermal diffusivity and Prandtl number 3.6.1 Second Fourier equation 3.6.2 Liquids and gases 3.6.3 Prandtl number 3.7 Mass diffusivity and Schmidt number 3.7.1 Law of mass diffusion (Fick’s first law) 3.7.2 Mutual mass diffusion 3.7.3 Mass diffusion in liquids 3.7.4 Temperature dependence of diffusion 3.7.5 Mass diffusion in complex solid foodstuffs 3.7.6 Schmidt number 3.8 Dielectric properties 3.8.1 Radio frequency and microwave heating 3.8.2 Power absorption – the Lambert–Beer law 3.8.3 Microwave and radio frequency generators 3.8.4 Analytical applications 3.9 Electrical conductivity 3.9.1 Ohm’s law 3.9.2 Electrical conductivity of metals and electrolytes; the Wiedemann–Franz law and Faraday’s law 3.9.3 Electrical conductivity of materials used in confectionery 3.9.4 Ohmic heating technology 3.10 Infrared absorption properties 3.11 Physical characteristics of food powders 3.11.1 Classification of food powders 3.11.2 Surface activity 3.11.3 Effect of moisture content and anticaking agents

52 53 53 54 54 55 55 56 56 56 58 58 62 62 63 66 66 67 67 68 68 69 69 69 70 71 71 72 72 73 74 75 76 76 77 78 81 81 81 82 83 83 85 86 86 87 87

Contents

3.11.4 3.11.5 3.11.6 3.11.7 3.11.8 3.11.9 3.11.10 Further reading

Mechanical strength, dust formation and explosibility index Compressibility Angle of repose Flowability Caking Effect of anticaking agents Segregation

Chapter 4 The rheology of foods and sweets 4.1 Rheology: its importance in the confectionery industry 4.2 Stress and strain 4.2.1 Stress tensor 4.2.2 Cauchy strain, Hencky strain and deformation tensor 4.2.3 Dilatational and deviatoric tensors; tensor invariants 4.2.4 Constitutive equations 4.3 Solid behaviour 4.3.1 Rigid body 4.3.2 Elastic body (or Hookean body/model) 4.3.3 Linear elastic and nonlinear elastic materials 4.3.4 Texture of chocolate 4.4 Fluid behaviour 4.4.1 Ideal fluids and Pascal bodies 4.4.2 Fluid behaviour in steady shear flow 4.4.3 Extensional flow 4.4.4 Viscoelastic functions 4.4.5 Oscillatory testing 4.4.6 Electrorheology 4.5 Viscosity of solutions 4.6 Viscosity of emulsions 4.6.1 Viscosity of dilute emulsions 4.6.2 Viscosity of concentrated emulsions 4.6.3 Rheological properties of flocculated emulsions 4.7 Viscosity of suspensions 4.8 Rheological properties of gels 4.8.1 Fractal structure of gels 4.8.2 Scaling behaviour of the elastic properties of colloidal gels 4.8.3 Classification of gels with respect to the nature of the structural elements 4.9 Rheological properties of sweets 4.9.1 Chocolate mass 4.9.2 Truffle mass 4.9.3 Praline mass 4.9.4 Fondant mass 4.9.5 Dessert masses 4.9.6 Nut brittle (croquante) masses 4.9.7 Whipped masses

vii 88 89 91 91 92 95 95 96 97 98 98 98 100 103 104 105 105 105 107 108 109 109 109 126 132 141 144 144 146 146 147 148 149 151 151 152 153 156 156 162 163 163 164 165 166

viii

Contents

4.10

Rheological properties of wheat flour doughs 4.10.1 Complex rheological models for describing food systems 4.10.2 Special testing methods for the rheological study of doughs 4.10.3 Studies of the consistency of dough Further reading

166 166 170 172 175

Chapter 5 Introduction to food colloids 5.1 The colloidal state 5.1.1 Colloids in the confectionery industry 5.1.2 The colloidal region 5.1.3 The various types of colloidal systems 5.2 Formation of colloids 5.2.1 Microphases 5.2.2 Macromolecules 5.2.3 Micelles 5.2.4 Disperse (or non-cohesive) and cohesive systems 5.2.5 Energy conditions for colloid formation 5.3 Properties of macromolecular colloids 5.3.1 Structural types 5.3.2 Interactions between dissolved macromolecules 5.3.3 Structural changes in solid polymers 5.4 Properties of colloids of association 5.4.1 Types of colloids of association 5.4.2 Parameters influencing the structure of micelles and the value of cM 5.5 Properties of interfaces 5.5.1 Boundary layer and surface energy 5.5.2 Formation of boundary layer: adsorption 5.5.3 Dependence of interfacial energy on surface morphology 5.5.4 Phenomena when phases are in contact 5.5.5 Adsorption on the free surface of a liquid 5.6 Electrical properties of interfaces 5.6.1 The electric double layer and electrokinetic phenomena 5.6.2 Structure of the electric double layer 5.7 Theory of colloidal stability: the DLVO theory 5.8 Stability and changes of colloids and coarse dispersions 5.8.1 Stability of emulsions 5.8.2 Two-phase emulsions 5.8.3 Three-phase emulsions 5.8.4 Two liquid phases plus a solid phase 5.8.5 Emulsifying properties of food proteins 5.8.6 Emulsion droplet size data and the kinetics of emulsification 5.8.7 Bancroft’s rule for the type of emulsion 5.8.8 HLB value and stabilization of emulsions 5.8.9 Emulsifiers used in the confectionery industry 5.9 Emulsion instability 5.9.1 Mechanisms of destabilization 5.9.2 Flocculation 5.9.3 Sedimentation (creaming)

176 177 177 177 179 179 179 180 180 180 181 182 182 184 184 188 188 190 190 190 190 191 193 196 198 198 199 200 203 203 205 205 205 207 207 209 210 211 212 212 213 215

Contents

ix

5.9.4 Coalescence 5.9.5 Ostwald ripening in emulsions 5.10 Phase inversion 5.11 Foams 5.11.1 Transient and metastable (permanent) foams 5.11.2 Expansion ratio and dispersity 5.11.3 Disproportionation 5.11.4 Foam stability: coefficient of stability and lifetime histogram 5.11.5 Stability of polyhedral foams 5.11.6 Thinning of foam films and foam drainage 5.11.7 Methods of improving foam stability Further reading

219 220 221 222 222 224 225 229 230 230 231 233

Part II

235

Physical operations

Chapter 6 Comminution 6.1 Changes during size reduction 6.1.1 Comminution of non-cellular and cellular substances 6.1.2 Grinding and crushing 6.1.3 Dry and wet grinding 6.2 Rittinger’s ‘surface’ theory 6.3 Kick’s ‘volume’ theory 6.4 The third, or Bond, theory 6.5 Energy requirement for comminution 6.5.1 Work index 6.5.2 Differential equation for the energy requirement for comminution 6.6 Particle size distribution of ground products 6.6.1 Particle size 6.6.2 Screening 6.6.3 Sedimentation analysis 6.6.4 Electrical-sensing-zone method of particle size distribution determination (Coulter method) 6.7 Particle size distributions 6.7.1 Rosin–Rammler (RR) distribution 6.7.2 Normal distribution (Gaussian distribution, N distribution) 6.7.3 Log-normal (LN) distribution (Kolmogorov distribution) 6.7.4 Gates–Gaudin–Schumann (GGS) distribution 6.8 Kinetics of grinding 6.9 Comminution by five-roll refiners 6.9.1 Effect of a five-roll refiner on particles 6.9.2 Volume and mass flow in a five-roll refiner 6.10 Grinding by a melangeur 6.11 Comminution by a stirred ball mill 6.11.1 Kinetics of comminution in a stirred ball mill 6.11.2 Power requirement of a stirred ball mill 6.11.3 Residence time distribution in a stirred ball mill Further reading

237 238 238 238 239 239 240 241 241 241 241 242 242 243 245 245 245 245 246 246 247 247 248 248 251 253 256 257 257 259 261

x

Contents

Chapter 7 Mixing/kneading 7.1 Technical solutions to the problem of mixing 7.2 Power characteristics of a stirrer 7.3 Mixing-time characteristics of a stirrer 7.4 Representative shear rate and viscosity for mixing 7.5 Calculation of the Reynolds number for mixing 7.6 Mixing of powders 7.6.1 Degree of heterogeneity of a mixture 7.6.2 Scaling up of agitated centrifugal mixers 7.6.3 Mixing time for powders 7.6.4 Power consumption 7.7 Mixing of fluids of high viscosity 7.8 Effect of impeller speed on heat and mass transfer 7.8.1 Heat transfer 7.8.2 Mass transfer 7.9 Mixing by blade mixers 7.10 Mixing rolls 7.11 Mixing of two liquids Further reading

263 263 264 266 266 266 267 267 271 272 273 274 275 275 275 276 277 277 278

Chapter 8 Solutions 8.1 Preparation of aqueous solutions of carbohydrates 8.1.1 Mass balance 8.1.2 Parameters characterizing carbohydrate solutions 8.2 Solubility of sucrose in water 8.2.1 Solubility number of sucrose 8.3 Aqueous solutions of sucrose and glucose syrup 8.3.1 Syrup ratio 8.4 Aqueous sucrose solutions containing invert sugar 8.5 Solubility of sucrose in the presence of starch syrup and invert sugar 8.6 Rate of dissolution Further reading

279 279 279 280 282 282 283 283 285 285 286 288

Chapter 9 Evaporation 9.1 Theoretical background – Raoult’s law 9.2 Boiling point of sucrose/water solutions at atmospheric pressure 9.3 Application of a modification of Raoult’s law to calculate the boiling point of carbohydrate/water solutions at decreased pressure 9.3.1 Sucrose/water solutions 9.3.2 Dextrose/water solutions 9.3.3 Starch syrup/water solutions 9.3.4 Invert sugar solutions 9.3.5 Approximate formulae for the elevation of the boiling point of aqueous sugar solutions 9.4 Vapour pressure formulae for carbohydrate/water solutions 9.4.1 Vapour pressure formulae 9.4.2 Antoine’s rule 9.4.3 Trouton’s rule

289 289 291 291 291 292 292 292 292 295 295 297 299

Contents

xi

9.4.4 Ramsay–Young rule 9.4.5 Dühring’s rule 9.5 Practical tests for controlling the boiling points of sucrose solutions 9.6 Modelling of an industrial cooking process for chewy candy 9.6.1 Modelling of evaporation stage 9.6.2 Modelling of drying stage Further reading

301 302 303 304 305 307 307

Chapter 10 Crystallization 10.1 Introduction 10.2 Crystallization from solution 10.2.1 Nucleation 10.2.2 Supersaturation 10.2.3 Thermodynamic driving force for crystallization 10.2.4 Metastable state of a supersaturated solution 10.2.5 Nucleation kinetics 10.2.6 Thermal history of the solution 10.2.7 Secondary nucleation 10.2.8 Crystal growth 10.2.9 Theories of crystal growth 10.2.10 Effect of temperature on growth rate 10.2.11 Dependence of growth rate on the hydrodynamic conditions 10.2.12 Modelling of fondant manufacture based on the diffusion theory 10.3 Crystallization from melts 10.3.1 Polymer crystallization 10.3.2 Spherulite nucleation, spherulite growth and crystal thickening 10.3.3 Melting of polymers 10.3.4 Isothermal crystallization 10.3.5 Non-isothermal crystallization 10.3.6 Secondary crystallization 10.4 Crystal size distributions 10.4.1 Normal distribution 10.4.2 Log-normal distribution 10.4.3 Gamma distribution 10.4.4 Histograms and population balance 10.5 Batch crystallization 10.6 Isothermal and non-isothermal recrystallization 10.6.1 Ostwald ripening 10.6.2 Recrystallization under the effect of temperature or concentration fluctuations 10.6.3 Ageing 10.7 Methods for studying the supermolecular structure of fat melts 10.7.1 Cooling/solidification curve 10.7.2 Solid fat content 10.7.3 Dilatation: Solid fat index 10.7.4 Differential scanning calorimetry, differential thermal analysis and low-resolution NMR methods

309 310 310 310 311 312 313 315 317 318 319 322 323 324 326 329 329 330 333 334 345 346 346 346 346 347 347 349 350 350 351 351 351 351 352 353 354

xii

Contents

10.8 10.9

Crystallization of glycerol esters: Polymorphism Crystallization of cocoa butter 10.9.1 Polymorphism of cocoa butter 10.9.2 Tempering of cocoa butter and chocolate mass 10.9.3 Shaping (moulding) and cooling of cocoa butter and chocolate 10.9.4 Sugar blooming and dew point temperature 10.9.5 Crystallization during storage of chocolate products 10.9.6 Bloom inhibition 10.9.7 Tempering of cocoa powder 10.10 Crystallization of fat masses 10.10.1 Fat masses and their applications 10.10.2 Cocoa butter equivalents and improvers 10.10.3 Fats for compounds and coatings 10.10.4 Cocoa butter replacers 10.10.5 Cocoa butter substitutes 10.10.6 Filling fats 10.10.7 Fats for ice cream coatings and ice dippings/toppings 10.11 Crystallization of confectionery fats with a high trans-fat portion 10.11.1 Coating fats and coatings 10.11.2 Filling fats and fillings 10.11.3 Future trends in the manufacture of trans-free special confectionery fats 10.12 Modelling of chocolate cooling processes and tempering 10.12.1 Franke model for the cooling of chocolate coatings 10.12.2 Modelling the temperature distribution in cooling chocolate moulds 10.12.3 Modelling of chocolate tempering process Further reading

355 359 359 360 365 367 368 370 371 371 371 372 374 376 378 379 381 382 383 383

Chapter 11 Gelling, emulsifying, stabilizing and foam formation 11.1 Hydrocolloids used in confectionery 11.2 Agar 11.2.1 Isolation of agar 11.2.2 Types of agar 11.2.3 Solution properties 11.2.4 Gel properties 11.2.5 Setting point of sol and melting point of gel 11.2.6 Syneresis of an agar gel 11.2.7 Technology of manufacturing agar gels 11.3 Alginates 11.3.1 Isolation and structure of alginates 11.3.2 Mechanism of gelation 11.3.3 Preparation of a gel 11.3.4 Fields of application 11.4 Carrageenans 11.4.1 Isolation and structure of carrageenans 11.4.2 Solution properties

394 395 395 395 396 396 397 398 398 399 400 400 401 401 402 402 402 403

384 385 385 386 390 392

Contents

11.4.3 Depolymerization of carrageenan 11.4.4 Gel formation and hysteresis 11.4.5 Setting temperature and syneresis 11.4.6 Specific interactions 11.4.7 Utilization 11.5 Furcellaran 11.6 Gum arabic 11.7 Gum tragacanth 11.8 Guaran gum 11.9 Locust bean gum 11.10 Pectin 11.10.1 Isolation and composition of pectin 11.10.2 High-methoxyl (HM) pectins 11.10.3 Low-methoxyl (LM) pectins 11.10.4 Low-methoxyl (LM) amidated pectins 11.10.5 Gelling mechanisms 11.10.6 Technology of manufacturing pectin jellies 11.11 Starch 11.11.1 Occurrence and composition of starch 11.11.2 Modified starches 11.11.3 Utilization in the confectionery industry 11.12 Xanthan gum 11.13 Gelatin 11.13.1 Occurrence and composition of gelatin 11.13.2 Solubility 11.13.3 Gel formation 11.13.4 Viscosity 11.13.5 Amphoteric properties 11.13.6 Surface-active/protective-colloid properties and utilization 11.13.7 Methods of dissolution 11.13.8 Stability of gelatin solutions 11.13.9 Confectionery applications 11.14 Egg proteins 11.14.1 Fields of application 11.14.2 Structure 11.14.3 Egg-white gels 11.14.4 Egg-white foams 11.14.5 Egg-yolk gels 11.14.6 Whole-egg gels 11.15 Foam formation 11.15.1 Fields of application 11.15.2 Velocity of bubble rise 11.15.3 Whipping 11.15.4 Continuous industrial aeration 11.15.5 Industrial foaming methods 11.15.6 In situ generation of foam Further reading

xiii 404 405 405 405 406 407 407 408 408 409 409 409 410 411 411 411 412 413 413 414 414 416 416 416 417 417 418 418 419 420 421 421 422 422 422 423 424 424 425 425 425 426 429 430 432 432 433

xiv

Contents

Chapter 12 Transport 12.1 Types of transport 12.2 Calculation of flow rate of non-Newtonian fluids 12.3 Transporting dessert masses in long pipes 12.4 Changes in pipe direction 12.5 Laminar unsteady flow 12.6 Transport of flour and sugar by air flow 12.6.1 Physical parameters of air 12.6.2 Air flow in a tube 12.6.3 Flow properties of transported powders 12.6.4 Power requirement of air flow 12.6.5 Measurement of a pneumatic system Further reading

434 434 434 436 437 438 438 438 438 439 441 442 444

Chapter 13 Pressing 13.1 Applications of pressing in the confectionery industry 13.2 Theory of pressing 13.3 Cocoa liquor pressing Further reading

445 445 445 448 449

Chapter 14 Extrusion 14.1 Flow through a converging die 14.1.1 Theoretical principles of the dimensioning of extruders 14.1.2 Pressure loss in the shaping of pastes 14.1.3 Design of converging die 14.2 Feeders used for shaping confectionery pastes 14.2.1 Screw feeders 14.2.2 Cog-wheel feeders 14.2.3 Screw mixers and extruders 14.3 Extrusion cooking 14.4 Roller extrusion 14.4.1 Roller extrusion of biscuit doughs 14.4.2 Feeding by roller extrusion Further reading

451 451 451 455 456 459 459 460 461 464 465 465 467 467

Chapter 15 15.1

Particle agglomeration: Instantization and tabletting Theoretical background 15.1.1 Processes resulting from particle agglomeration 15.1.2 Solidity of a granule 15.1.3 Capillary attractive forces in the case of liquid bridges 15.1.4 Capillary attractive forces in the case of no liquid bridges 15.1.5 Solidity of a granule in the case of dry granulation 15.1.6 Water sorption properties of particles 15.1.7 Effect of electrostatic forces on the solidity of a granule 15.1.8 Effect of crystal bridges on the solidity of a granule 15.1.9 Comparison of the various attractive forces affecting granulation 15.1.10 Effect of surface roughness on the attractive forces

469 469 469 472 472 473 474 475 477 478 479 479

Contents

15.2

xv

Processes of agglomeration 15.2.1 Agglomeration in the confectionery industry 15.2.2 Agglomeration from liquid phase 15.2.3 Agglomeration of powders: Tabletting or dry granulation 15.3 Granulation by fluidization 15.3.1 Instantization by granulation: Wetting of particles 15.3.2 Processes of fluidization 15.4 Tabletting 15.4.1 Tablets as sweets 15.4.2 Types of tabletting 15.4.3 Compression, consolidation and compaction 15.4.4 Characteristics of the compaction process 15.4.5 Quality properties of tablets Further reading

481 481 481 482 482 482 483 484 484 485 486 488 492 492

Part III

493

Chemical and complex operations: Stability of sweets

Chapter 16

Chemical operations (inversion and caramelization), ripening and complex operations 16.1 Inversion 16.1.1 Hydrolysis of sucrose by the effect of acids 16.1.2 A specific type of acidic inversion: Inversion by cream of tartar 16.1.3 Enzymatic inversion 16.2 Caramelization 16.2.1 Maillard reaction 16.2.2 Sugar melting 16.3 Alkalization of cocoa material 16.3.1 Purposes and methods of alkalization 16.3.2 German process 16.4 Ripening 16.4.1 Ripening processes of diffusion 16.4.2 Chemical and enzymatic reactions during ripening 16.5 Complex operations 16.5.1 Complexity of the operations used in the confectionery industry 16.5.2 Conching 16.5.3 New trends in the manufacture of chocolate 16.5.4 Modelling the structure of dough Further reading Chapter 17 Water activity, shelf life and storage 17.1 Water activity 17.1.1 Definition of water activity 17.1.2 Adsorption/desorption of water 17.1.3 Measurement of water activity 17.1.4 Factors lowering water activity 17.1.5 Sorption isotherms

495 495 495 498 499 502 502 504 505 505 506 507 507 509 510 510 510 521 522 523 525 525 525 527 527 533 534

xvi

Contents

17.1.6 17.1.7

Hygroscopicity of confectionery products Calculation of equilibrium relative humidity of confectionery products 17.2 Shelf life and storage 17.2.1 Definition of shelf life 17.2.2 Role of light and atmospheric oxygen 17.2.3 Role of temperature 17.2.4 Role of water activity 17.2.5 Role of enzymatic activity 17.2.6 Concept of mould-free shelf life 17.3 Storage scheduling Further reading

538 541 541 541 541 541 542 542 547 548

Chapter 18 Stability of food systems 18.1 Common use of the concept of food stability 18.2 Stability theories: types of stability 18.2.1 Orbital stability and Lyapunov stability 18.2.2 Asymptotic and marginal (or Lyapunov) stability 18.2.3 Local and global stability 18.3 Shelf life as a case of marginal stability 18.4 Stability matrix of a food system 18.4.1 Linear models 18.4.2 Nonlinear models

550 550 550 550 551 552 552 553 553 554

Part IV

555

Appendices

535

Appendix 1 Data on engineering properties of materials used and made by the confectionery industry A1.1 Carbohydrates A1.2 Oils and fats A1.3 Raw materials, semi-finished products and finished products

557 557 566 567

Appendix 2 Solutions of sucrose, corn syrup and other monosaccharides and disaccharides

579

Appendix 3 Survey of fluid models A3.1 Decomposition method for calculation of flow rate of rheological models A3.1.1 Principle of the decomposition method A3.1.2 Bingham model A3.1.3 Casson model (n = 1/2) A3.1.4 Peek, McLean and Williamson model A3.1.5 Reiner–Philippoff model A3.1.6 Reiner model A3.1.7 Rabinowitsch, Eisenschitz, Steiger and Ory model A3.1.8 Oldroyd model A3.1.9 Weissenberg model A3.1.10 Ellis model

582 582 582 583 585 586 587 587 588 589 590 591

Contents

xvii

A3.1.11 Meter model A3.1.12 Herschel–Bulkley–Porst–Markowitsch–Houwink (HBPMH) (or generalized Ostwald–deWaele) model A3.1.13 Ostwald–de Waele model A3.1.14 Williamson model A3.2 Calculation of the friction coefficient ξ of non-Newtonian fluids in the laminar region A3.3 Generalization of the Casson model A3.3.1 Theoretical background to the exponent n A3.3.2 Theoretical foundation of the Bingham model A3.4 Determination of the exponent n of the flow curve of a generalized Casson fluid A3.5 Dependence of shear rate on the exponent n in the case of a generalized Casson fluid A3.6 Calculation of the flow rate for a generalized Casson fluid A3.7 Lemma on the exponent in the generalized Casson equation Further reading

591

600 601 603 605

Appendix 4 Fractals A4.1 Irregular forms – fractal geometry A4.2 Box-counting dimension A4.3 Particle-counting method A4.4 Fractal backbone dimension Further reading

606 606 606 607 608 608

Appendix 5 Introduction to structure theory A5.1 General features of structure theory A5.2 Attributes and structure: A qualitative description A5.3 Hierarchical structures A5.4 Structure of measures: A quantitative description A5.5 Equations of conservation and balance A5.6 Algebraic structure of chemical changes A5.7 The technological triangle: External technological structure A5.8 Conserved substantial fragments

609 609 610 611 611 612 614 614 615

Appendix 6 Technological lay-outs Further reading

617 629

References Index

630 668

592 594 595 596 597 597 598 598

Preface

The purpose of this book is to describe features of the unit operations in confectionery manufacturing. The approach adopted here might be considered as a novelty in the confectionery literature. The choice of the subject might perhaps seem surprising, owing to the fact that the word ‘confectionery’ is usually associated with handicraft instead of engineering. It must be acknowledged that the attractiveness of confectionery can be partly attributed to the coexistence of handicraft and engineering in this field. Nevertheless, large-scale industry has also had a dominant presence in this field for about a century. The traditional confectionery literature focuses on technology. The present work is based on a different approach, where, by building on the scientific background of chemical engineering, it is intended to offer a theoretical approach to practical aspects of the confectionery and chocolate industry. However, one of the main aims is to demonstrate that the structural description of materials used in chemical engineering must be complemented by taking account of the hierarchical structure of the cellular materials that are the typical objects of food engineering. By characterizing the unit operations of confectionery manufacture, without daring to overestimate the eventual future exploitation of the possibilities offered by this book, I intend to inspire the development of new solutions both in technology and machinery, including the intensification of operations, the application of new materials, and new and modern applications of traditional raw materials. I have studied unit operations in the confectionery industry since the 1960s. During my university years I began dealing with the rheological properties of molten chocolate (the Casson equation, rheopexy etc.). This was an attractive and fruitful experience for me. Later on, I worked for the Research Laboratory of the Confectionery Industry for three years. Altogether I spent – on and off – half a century in this field, working on product development, production, quality control/assurance, purchasing and trading. These tasks, related mainly to sugar confectionery and chocolate, convinced me that a uniform attitude is essential for understanding the wide-ranging topics of confectionery and chocolate manufacture. As a young chemical engineer, I also started lecturing undergraduate and graduate students. Having gathered experience in education (compiling lectures etc.), I found that this conviction was further confirmed. In the late 1960s my attention was firmly focused on the unit operations in this industry, and I tried to utilize and build on the results produced by the Hungarian school of chemical engineering (M. Korach (Maurizio Cora), P. Benedek, A. László and T. Blickle). Benedek and László discussed the topics of chemical engineering, placing the Damköhler equations in the centre of the theory, similarly to the way in which electricity is based on the Maxwell equations. Blickle and the mathematician Seitz developed structure theory and applied it to chemical engineering. Structure theory exploits the tools of abstract

Preface

xix

algebra to analyse the structures of system properties, materials, machinery, technological changes etc. It is a useful method for defining concepts and studying their relations. The outcome of these studies is well reflected in several books and university lectures published by me, and serves as the theoretical background for the present book as well. Chapter 1 introduces the Damköhler equations as a framework for chemical engineering. This chapter outlines the reasons why this framework is suitable for studying the unit operations of the confectionery industry in spite of the cellular structure of the materials. In Chapter 2, the structural characterization of raw materials and products is discussed by means of structure theory. This chapter also demonstrates in detail methods for preparing confectionery recipes taking compositional requirements into account. Chapter 3 and Appendices 1 and 2 all deal with the engineering properties of the materials used in confectionery. Heat and mass transfer are not discussed individually but are included in other chapters. Rheology is essential to confectionery engineering. Therefore, a relatively large part of the book (Chapter 4) discusses the rheological properties of both Newtonian and non-Newtonian fluids, along with elasticity, plasticity, extensional viscosity etc. Non-Newtonian flow, especially that of Casson fluids, is discussed in Chapter 12 and Appendix 3. Some relevant topics in colloid chemistry are discussed in Chapters 5 and 11. In this context, the basics of fractal geometry cannot be ignored; thus, Appendix 4 offers an outline thereof. Comminution plays an important role in this field, as new procedures and machines related to comminution enable new chocolate technologies to be developed. Chapters 7, 8 and 9 discuss the operations of mixing, as well as the topics of solutions of carbohydrates in water and the evaporation of these solution. These chapters provide confirmation that the Dühring rule, the Ramsay–Young rule etc. are also valid for these operations. Crystallization (Chapter 10) from aqueous solutions (candies) and fat melts (chocolate and compounds) is a typical operation in confectionery practice, and thus I highlight its dominant characteristics. In Chapter 13, pressing is briefly discussed. Extrusion (Chapter 14) and agglomeration (Chapter 15) are typical operations that manifest the wide-ranging nature of the confectionery industry. Chapter 16 deals with inversion, the Maillard reaction and such complex operations as conching, and also new trends in chocolate manufacture and (tangentially) baking. Chapter 17 deals with the issues of water activity and shelf life. A separate chapter (18) is devoted to food stability. The real meaning of such an approach is that from the start of production to the consumer’s table the kinetics of the changes in the raw materials and products must be taken into consideration. Furthermore, in the light of this attitude, the concept of ‘food stability’ must be defined more exactly by using the concepts of stability theory. For the sake of completeness, Appendix 6 contains some technological outlines. I intended to avoid the mistake of ‘he who grasps much holds little’ (successfully? who knows?); therefore, I have not been so bold as to discuss such operations – however essential – as fermentation, baking and panning, about which I have very little or no practical knowledge. Similarly, I did not want to provide a review of the entire circle of relevant references. Thus the substance that I grasped turned out to be great but rather difficult, and I wish I could say that I have coped with it. Here the gentle reader is requested to send me their remarks and comments for a new edition hopefully to be published in the future.

xx

Preface

My most pleasant obligation is to express my warmest thanks to all the colleagues who helped my work. First of all, I have to mention the names of my professors, R. Lásztity (Technical University of Budapest) and T. Blickle (University of Chemical Engineering, Veszprém), who were my mentors in my PhD work, and Professor J. Varga (Technical University of Budapest), my first instructor in ‘chocolate science’. I am grateful to Professor S. Szántó and Professor L. Maczelka (Research Laboratory of the Confectionery Industry), who consulted me very much as a young colleague on the topics of this field. I highly appreciate the encouragement obtained from Mr M. Halbritter, the former President of the Association of Hungarian Confectionery Manufacturers, Professor Gy. Karlovics (Corvinus University of Budapest and Bunge Laboratories, Poland), Professor A. Fekete (Corvinus University of Budapest), Professor A. Salgó (Technical University of Budapest), Professor G. Szabo (Rector, Szeged University of Sciences), Professor A. Véha (Dean, Szeged University of Sciences) and Professor E. Gyimes (Szeged University of Sciences). I am also indebted to Professor C. Alamprese (Università degli Studi di Milano, Italy), Ms P. Alexandre, a senior expert at CAOBISCO, Brussels, Belgium, Professor R. Scherer (Fachhochschule Fulda, Germany), Professor H.-D. Tscheuschner and Professor K. Franke (Dresden University of Technology, Germany), moreover, to D. Meekison for his valuable help provided in copyediting. Last but not least, I wish to express my deep and cordial thanks to my family: to my daughter Viktória for correcting my poor English, and to my wife Irén, who with infinite patience has tolerated my whimsicality and the permanent and sometimes shocking disorder around me, and (despite all this) assured me a normal way of life. F.Á.M., Budapest

Acknowledgements

The author gratefully acknowledges the permission granted to reproduce the copyright material in this book: Akadémiai Kiadó, Budapest (Fig. 14.1); Archer Daniels Midland Co. (ADM), IL, USA (Fig. 17.5); Carle & Montanari SpA, Milan (Figs 6.3, 6.5 and 6.6; Table 6.4); Elsevier Science Ltd, The Netherlands (Figs 9.1, 9.2, 10.1, 10.6(a)–(d), 10.7, 10.24, 10.25, 10.26, 10.27, 10.28, 10.29, 10.30 and 11.6; Tables 3.8 and 3.9); Professor K. Kerti, Budapest (Table 10.3); Professor R. Lásztity, Budapest (Figs 4.26 and 4.27); Springer Science and Business Media, The Netherlands (Figs 3.1, 3.2 and 3.3; Tables 3.1, 3.2, 3.19, 3.20, 17.2, 17.3 and 17.8; Section 17.1.6); Professor J.F. Steffe, Michigan, USA (Figs 4.5, 4.11, 4.13, 4.15, 4.16, 4.17, 4.18 and 4.21; Table 4.1); P. Székely, Budapest (Figs 16.1 and 16.2; Tables 16.3 and 16.4); Wiley-VCH Verlag GmbH & Co KGaA, Germany, and Mrs Liselotte Rumpf, Karlsruhe (Figs 15.1, 15.2, 15.3 and 15.4; Table 15.1). Every effort has been made to trace copyright holders and to obtain their permission for the use of copyright material. The publisher apologizes for any errors or omissions in the above list and would be grateful if notified of any corrections that should be incorporated in future reprints or editions of this book.

Part I

Theoretical introduction

Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

Chapter 1

Principles of food engineering

Contents 1.1 Introduction 1.1.1 The peculiarities of food engineering 1.1.2 The hierarchical and semi-hierarchical structure of materials 1.1.3 Application of the Damköhler equations in food engineering 1.2 The Damköhler equations 1.3 Investigation of the Damköhler equations by means of similarity theory 1.3.1 Dimensionless numbers 1.3.2 Degree of freedom of an operational unit 1.3.3 Polynomials as solutions of the Damköhler equations 1.4 Analogies 1.4.1 The Reynolds analogy 1.4.2 The Colburn analogy 1.4.3 Similarity and analogy 1.5 Dimensional analysis 1.6 The Buckingham Π theorem Further reading

1.1 1.1.1

3 3 5 6 6 8 8 11 12 13 13 15 16 16 17 18

Introduction The peculiarities of food engineering

Food engineering is based to a great extent on the results of chemical engineering. However, the differences in overall structure between chemicals and foods, that is, the fact that the majority of foods are of cellular structure, result in at least three important differences in the operations of food engineering – the same is valid for biochemical engineering. (1) Chemical engineering applies the Gibbs theory of multicomponent chemical systems, the principal relationships of which are based on chemical equilibrium, for example the Gibbs phase rule. Although the supposition of equilibrium is only an approximation, it frequently works, and provides good results. In the case of cellular substances, however, the conditions of equilibrium do not apply in general, because the cell walls function as semi-permeable membranes, which makes equilibrium practically possible only in aqueous media and for long-lasting processes. Consequently, the Gibbs phase Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

4

Confectionery and Chocolate Engineering: Principles and Applications

rule cannot be a basis for determining the degrees of freedom of food engineering systems in general. For further details, see Section 1.3.2. (2) Another problem is that cellular substances prove to be chemically very complex after their cellular structure has been destroyed. In the Gibbs theory, the number of components in a multicomponent system is limited and well defined, not infinite. The number of components in a food system can be practically infinite or hard to define; in addition, this number depends on the operational conditions. Certainly, we can choose a limited set of components for the purpose of a study – and this is the usual way – but this choice will not guarantee that exclusively those components will participate in the operation considered. Therefore, interpretation of the degrees of freedom in food engineering systems causes difficulties and is often impossible, because the number and types of participants (chemical compounds, cell fragments, crystalline substances, etc.) in food operations are hard to estimate: many chemical and physical changes may take place simultaneously, and a small change in the conditions (temperature, pH, etc.) may generate other types of chemical or physical changes. If we compare this situation with a complicated heterogeneous catalytic chemical process with many components, it is evident that in food engineering we struggle with complex tasks that are not easier, only different. Evidently, comminution plays a decisive role in connection with these peculiarities. However, in the absence of comminution, these two peculiarities – the existence of intact cell wall as barriers to equilibrium and the very high number of operational participants – may appear together as well; for example, in the roasting of cocoa beans, the development of flavours takes place inside unbroken cells. In such cases, cytological aspects (depot fat, mitochondria, etc.) become dominant because the cell itself works as a small chemical plant, the heat and mass transfer of which cannot be influenced by traditional (e.g. fluid-mechanical) means. This problem is characteristic of biochemical engineering. (3) The third peculiarity, which is a consequence of the cellular structure, is that the operational ‘participants’ in food engineering may be not only chemical compounds, chemical radicals and other molecular groups but also fragments of comminuted cells. In the case of chemical compounds/radicals, etc., although the set of these participants can be infinitely diverse, the blocks from which they are built are well defined (atoms), the set of atoms is limited and the rules according to the participants are built are clear and well defined. In the case of cellular fragments, none of this can be said. They can, admittedly, be classified; however, any such classification must be fitted to a given task, without any possibility of application to a broader range of technological problems. This is a natural consequence of the fact that the fragments generated by comminution, in their infinite diversity, do not manifest such conspicuous qualitative characteristics as chemicals; nevertheless, they can be distinguished because slight differences in their properties which occur by accident because of their microstructure may become important. This situation may be understood as the difference between discrete and continuous properties of substances: while chemical systems consist of atoms and combinations of them, to which stoichiometry can be applied, the systems of food engineering can not be built up from such well-defined elements. This stoichiometry means that welldefined amounts by mass (atomic masses or molecular masses) may be multiplied by

Principles of food engineering

5

integers in order to get the mass fluxes in a reaction. However, in the recipes that are used for describing the compositions of foods, the mass fluxes are treated as continuous variables, contrary to the idea of stoichiometry.

1.1.2

The hierarchical and semi-hierarchical structure of materials

Although foods also consist of atoms in the final analysis, it is characteristic of food engineering that it does not go to an elementary decomposition of the entire raw material; however, a certain part of the raw material will be chemically modified, another part will be modified at the level of cells (by comminution), etc. The structures of materials are hierarchical, where the levels of the hierarchy are joined by the containing relation, which is reflexive, associative and transitive (but not commutative): A → B means that B contains A, i.e. ‘→’ is the symbol for the containing relation. The meaning of the reflexive, associative and transitive properties is: • Reflexive: A contains itself. • Associative: if A → (B → C), then (A → B) → C. • Transitive: if A → B → C, then A → C (the property is inheritable). The transitive property is particularly important: if A = atom, B = organelle and C = cell (considered as levels), then the transitive relation means that if an organelle (at level B) contains an atom (at level A) and if a cell (at level C) contains this organelle (at level B), then that cell (at level C) contains the atom in question (at level A) as well. The hierarchical structure of materials is illustrated in Fig. 1.1. For the sake of completeness, Fig. 1.1 includes the hierarchical levels of tissue, organs and organisms, which

Atom

Group of atoms Food engineering and biochemical engineering

Chemical compound

Cellular organelle

Cell

Tissue, organ, organism

Fig. 1.1

Hierarchical structure of materials.

Chemical engineering

6

Confectionery and Chocolate Engineering: Principles and Applications

are of interest when one is choosing ripened fruit, meat from a carcase, etc. In a sense, the level of the organism is the boundary of the field of food (and biochemical) engineering. This hierarchical structure is characteristic of cellular materials only when they are in an intact, unbroken state. Comminution may disrupt this structure; for example if cellular fragments are dispersed in an aqueous solution, and these fragments may themselves contain aqueous solutions as natural ingredients, then these relations can be represented by A1 → C → A 2 where A1 represents the natural ingredients of a cell (an aqueous solution), C represents the cellular material and A2 represents the aqueous solution in which the cellular material is dispersed. Evidently, in this case the hierarchical levels are mixed, although they still exist to some extent. Therefore, for such cases of bulk materials, the term ‘semihierarchical structure’ seems more appropriate. If we allow that the degrees of freedom cannot be regarded as the primary point of view, a more important, in fact crucial, question is whether the set of chemical and/or physical changes that occur in an operation can be defined at all. The answer is difficult, and one must take into consideration the fact that an exact determination of this set is not possible in the majority of cases. Instead, an approximate procedure must be followed that defines the decisive changes and, moreover, the number and types of participants. In the most favourable cases, this procedure provides the result (i.e. product) needed.

1.1.3

Application of the Damköhler equations in food engineering

In spite of the differences discussed above, the Damköhler equations, which describe the conservation of the fluxes of mass, components, heat and momentum, can provide a mathematical framework from the field of chemical engineering that can be applied to the tasks in food engineering (and biochemical engineering), with a limitation relating to the fluxes of components. The essence of this limitation is that the entire set of components cannot be defined in any given case. This limitation has to be taken into account by defining both the chemical components studied and their important reactions. The conservation law of component fluxes does hold approximately for this partial system. The correctness of the approximation may be improved if this partial set approaches the entire set of components. For example, if we consider the baking of biscuit dough, it is impossible to define all the chemical reactions taking place and all the components participating in them; therefore, the conservation equations for the components cannot be exact, because of the disturbing effect of by-reactions. However, what counts as a by-reaction? This uncertainty is a source of inaccuracy. The conservation equations for mass, heat and momentum flux can be used without any restrictions for studying physical (and mechanical) operations, since they concern bulk materials.

1.2

The Damköhler equations

This chapter principally follows the ideas of Benedek and László (1964). Some further important publications (although not a comprehensive list) that are relevant are Charm

Principles of food engineering

7

(1971), Pawlowski (1971), Schümmer (1972), Meenakshi Sundaram and Nath (1974), Loncin and Merson (1979), Stephan and Mitrovic (1984), Zlokarnik (1985), Mahiout and Vogelpohl (1986), Hallström et al. (1988), Stichlmair (1991), VDI-Wärmeatlas (1991), Zogg (1993), Chopey (1994), Stiess (1995), Perry (1998), Hall (1999), Sandler (1999), McCabe et al. (2001), Zlokarnik (2006) and Dobre (2007). According to Damköhler, chemical-technological systems can be described by equations of the following type: convection + conduction + transfer + source = local change

(1.1)

In detail, div [ Γ v ] − div [δ grad Γ ] + ωε ΔΓ + G = −

∂Γ ∂t

(1.2)

where v = linear velocity (in units of m/s); Γ is a symbol for mass, a component, heat or momentum; δ = generalized coefficient of convection (m2/s); ω = transfer surface area per unit volume (m2/m3); ε = generalized coefficient of transfer; G = flux of source; and t = time (s). Such equations can be set up for fluxes of mass, components, heat and momentum. The Damköhler equations play a role in chemical and food engineering similar to that of the Maxwell equations in electrodynamics. The application of the Damköhler equations to food-technological systems is presented in Chapter 2. Let us consider these equations one by one. Flux of mass: div [ ρ v ] − [ D grad ρ ] + ωβ ′Δρ + G = −

∂ρ ∂t

(1.3)

where v = linear velocity (m/s), ρ = density (kg/m3), β′ = mass transfer coefficient (m/s), D = self-diffusion coefficient (m2/s) and G = source of mass flux (kg/m3s). Flux of a component:

div[ci v] − div[D grad ci] + wb Δci + nir = –∂ ci /∂ t (1.4)

Fick’s 2nd law where ci = concentration of the i-th component (mol/m3), D = diffusion coefficient (m2/s), β = component transfer coefficient (m/s), νi = degree of reaction for the i-th component and r = velocity of reaction [(mol/(m3 s)]. Flux of heat:

div[rcpTv] − div[l grad T ] + wa ΔT + nir ΔH = –∂ (rcpT )/∂ t (1.5)

Fourier’s 2nd law

Newton’s law of cooling

where cp = specific heat (p = constant) [J/(kg K)], T = temperature (K), λ = thermal conductivity (W/m K), ΔH = heat of reaction (J/mol) and α = heat transfer coefficient [J/(m2 s K)].

8

Confectionery and Chocolate Engineering: Principles and Applications

The flux of momentum is described by the Navier–Stokes law, Div {ρ v ⋅ v} − Div {η Grad v} + ωγ Δv + grad p = −

∂ [ρv] ∂t

(1.6)

where Div = tensor divergence, Grad = tensor gradient, · is the symbol for a dyadic product, η = dynamic viscosity [kg/(ms)], γ = (f′ρv/2) = coefficient of momentum transfer [kg/(m2 s)] and p = pressure [kg/(ms2)] Equations (1.3)–(1.6) are called the Damköhler equation system. In general, the Damköhler equations cannot be solved by analytical means. In some simpler cases, described below, however, there are analytical solutions. For further details see Grassmann (1967), Charm (1971), Loncin and Merson (1979), Hallström et al. (1988) and Banks (1994).

1.3

Investigation of the Damköhler equations by means of similarity theory

1.3.1

Dimensionless numbers

Let us suppose that a set of Damköhler equations called ‘Form 1’ are valid for a technological system called ‘System 1’, and a set of equations ‘Form 2’ are valid for ‘System 2’. It is known from experience that if similar phenomena take place in the two systems, then this similarity of phenomena can be expressed by a relationship denoted by ‘∼’, as in ‘Form 1 ∼ Form 2’. Similarity theory deals with the description of this relationship. The simplest characteristics of this similarity are the ratios of two geometric sizes, two concentrations, etc. These are called simplex values. 1.3.1.1

Complex values

The first perception of such a relationship is probably connected with the name of Reynolds, who made the observation, in relation to the flow of fluids, that System 1 and System 2 are similar if the ratios of momentum convection to momentum conduction in these systems are equal to each other. Let us consider Eqn (1.1), convection + conduction + transfer + source = local change

(1.1)

for momentum flux. Since the terms for convection, conduction, etc. on the left-hand side evidently have the same dimensions in the equation, their ratios are dimensionless. One of the most important dimensionless quantities is the ratio of momentum convection to momentum conduction, which is called the Reynolds number, denoted by Re. Re = Dvρ/η, where D is a geometric quantity characteristic of the system and v is a linear velocity, v=

Q R2 π

(1.7)

Principles of food engineering

9

where Q = volumetric flow rate (m3/s) and R = radius of tube (m). For conduits of non-circular cross-section, the definition of the equivalent diameter De is De =

area of stream cross-section wetted perimeter

(1.8)

The value of De for a tube is 4D2π/4Dπ = D (the inner diameter of the tube), and for a conduit of square section it is 4a2/4a = a (the side of the square). For heat transfer, the total length of the heat-transferring perimeter is calculated instead of the wetted perimeter (e.g. in the case of part of a tube). It has been shown that several different types of flow can be characterized by their Reynolds numbers: Re < about 2300: laminar flow; Re > 2300 to Re < 10 000: transient flow; Re > 10 000: turbulent flow. This means, for example, that if for System 1 the Reynolds number Re(1) is 1000 and for System 2 the Reynolds number Re(2) is 1000, then the flow shows the same (laminar) properties in both systems. Moreover, all systems in which the Reynolds numbers are the same show the same flow properties. In order to understand the role of the Reynolds number, let us interpret the form of Eqn (1.6) as momentum convection + momentum conduction = local change of momentum If Re = 1, this means for the momentum part that convection = 50% and conduction = 50%; if Re = 3, then convection = 75% and conduction = 25%; and if Re = 99, then convection = 99% and conduction = 1%. It is difficult to overestimate the importance of Reynolds’ idea of similarity, because this has become the basis of modelling. One can investigate phenomena first with a small model, which is relatively cheap and can be made quickly, and then the size of the model can be increased on the basis of the results. Modelling and increasing the size (scaling-up) are everyday practice in shipbuilding, and in the design of chemical and food machinery, etc. If, for a given system, D, ρ and η are constant, the type of flow depends on the linear velocity (v) if only convection and conduction take place. Using similar considerations, many other dimensionless numbers can be derived from the Damköhler equations; some of these are presented in Tables 1.1 and 1.2. From Table 1.1, we have the following, for example: • In Eqn (1.4), the ratio of convection to conduction is the Peclet number for component transfer (Pe′), Pe ′ =

div [ci v ] vd = div [ D grad ci ] D

10

Confectionery and Chocolate Engineering: Principles and Applications

Table 1.1

Derivation of dimensionless numbers.

Flux

Convection/ conduction

Transfer/ convection

Source/ convection

Component (Eqn 1.4) Heat (Eqn 1.5) Momentum (Eqn 1.6)

Pe′ Pe Re

St′ St f′/2

Da(I) Da(III) Eu or Fa

Table 1.2

Another way of deriving dimensionless numbers.

Flux

Convection/ conduction

Transfer/ conduction

Source/ convection

Component (Eqn 4) Heat (Eqn 5) Momentum (Eqn 6)

Pe′ Pe Re

Nu′ Nu A (no name)

Da(I) Da(III) Eu or Fa

• In Eqn (1.6), the ratio of the momentum source to the momentum convection is the Euler number (Eu), Eu =

Δp grad p = Div {ρ v ⋅ v} ρv 2

Another way of deriving dimensionless numbers is illustrated in Table 1.2. In the third column of this table, the ratio of transfer to conduction is represented instead of the ratio of transfer to convection, and in this way another system of dimensionless numbers (i.e. variables) is derived. Note that: • If the source is a force due to a stress, equal to Δp d 2, then the Euler number is obtained. • If the source is a gravitational force, equal to ρgd 3, then the Fanning number is obtained. The dimensionless numbers in Tables 1.1 and 1.2 are as follows: Pe′ = vd/D, the Peclet number for component transfer. Pe = vd/a, the Peclet number for heat transfer (a = temperature conduction coefficient or heat diffusion coefficient). St′ = β/v, the Stanton number for component transfer (β = component transfer coefficient). St = α /ρcpv, the Stanton number for heat transfer (α = heat transfer coefficient). γ = f ′ρv/2, the momentum transfer coefficient ( f ′/2 = γ /ρv). Da(I) = νird/civ, the first Damköhler number; this is the component flux produced by chemical reaction divided by the convective component flux. Da(III) = νi ΔH rd/ρcPv ΔT, the third Damköhler number; this is the heat flux produced by chemical reaction divided by the convective heat flux.

Principles of food engineering

11

Eu = Δp/ρv2, the Euler number; this is the stress force divided by the inertial force. Fa = gd/v2, the Fanning number; this is the gravitational force divided by the inertial force. Nu′ = βd/D, the Nusselt number for component transfer (D = diffusion coefficient). Nu = αd/λ, the Nusselt number for heat transfer (λ = thermal conductivity). Following van Krevelen’s treatment (1956), 3 × 3 = 9 independent dimensionless numbers can be derived in this way from three equations (‘rows’) and four types of phenomena (‘columns’, namely convection, conduction, transfer and sources), and three rates can be produced from these numbers. With the help of such matrices of nine elements (see Tables 1.1 and 1.2), other dimensionless numbers can also be obtained, which play an important role in chemical and food engineering. For example, values of efficiency can be derived in this way: Pr = Pe/Re = ν/a, the Prandtl number; Sc = Pe′/Re = ν/D, the Schmidt number; Le = Sc/Pr = a/D, the Lewis number.

1.3.2

Degrees of freedom of an operational unit

The number of degrees of freedom of an operational unit is a generalization of corresponding concept in the Gibbs phase rule. The question of how to determine the number of degrees of freedom of an operational unit was first put by Gilliland and Reed (1942); further references are Morse (1951), Benedek (1960) and Szolcsányi (1960). For multiphase systems, the Gibbs classical theory, as is well known, prescribes the equality of the chemical potentials for each component in each phase in equilibrium. If μkf (where k = 1, 2, … , K, and f = 1, 2, … , F) denotes the chemical potential of the k-th component in the f-th phase, then the following holds in equilibrium: • For the f-th phase, when there are K components,

μ1f = μ2f = … = μ Kf , i.e. F(K − 1) equations. • For the k-th component, when there are F phases,

μ1k = μk2 = … = μkF , i.e. K(F − 1) equations. In equilibrium, the additional variables which are to be fixed are T and p. Consequently, in equilibrium, the number of variables (ϕ) which can be freely chosen is

ϕ = F ( K − 1) − K ( F − 1) + 2 = K − F + 2

(1.9)

This is the Gibbs phase rule, which is essential for studying multiphase systems. Even in the extreme case where the solubility of a component in a solvent is practically zero, the phase rule can nevertheless be applied by considering the fact that the chemical

12

Confectionery and Chocolate Engineering: Principles and Applications

potential of this component is sufficient for equilibrium in spite of its very small concentration. The generalization that we need in order to obtain ϕ for an operational unit is given by

ϕ = L−M

(1.10)

where ϕ is the number of degrees of freedom, L is the total number of variables describing the system and M is the number of independent relations between variables. In the simplest case, that of a simple stationary operational unit with an isolated wall, if the number of input phases is F and the number of output phases is F′, then the total number of variables is L = (F + F ′ ) (K + 2) where K is the number of components. (To describe a homogeneous phase, (K + 2) data points are needed.) Let us now consider the constraints. There are constraints derived from the conservation laws for every component and also for energy and momentum, which means (K + 2) constraints for every phase. The number of constraints for equilibrium between two phases is (K + 2), which means (F ′ − 1)(K + 2) constraints for the output phases. Consequently, the total number of constraints is M = ( K + 2 ) + ( F ′ − 1) ( K + 2 ) and, finally,

ϕ = F (K + 2)

(1.11)

However, in the case of cellular substances the conditions of equilibrium typically do not apply; moreover, the number of components can usually not be determined. Therefore, the Gibbs phase rule cannot be used for food-technological systems except in special cases where exclusively chemical changes are taking place in the system studied. This uncertainty relating to the degrees of freedom is an essential characteristic of food engineering.

1.3.3

Polynomials as solutions of the Damköhler equations

A solution of the Damköhler equation system can be approximated by the product

Π 1a = Π 2b × Π 3c × … × Π id × …

(1.12)

where the Πi are dimensionless numbers created from the terms of the Damköhler equations and a, b, c, d, … are exponents, which can be positive or negative integers or fractions. It is to be noted that Eqn (1.12) assumes that the solution is provided by ‘monomials’ (and not by ‘binomials’ as in, for example, Π 1a = Π 2b × Π 3c × … × Π id × … ) – this assumption is not valid in every case.

Principles of food engineering

13

The principal idea represented by Eqn (1.12) is that convergent polynomial series, for example a Taylor series, can approximate well almost any algebraic expression, and thus also a solution of the Damköhler equations. But it is not unimportant how many terms are taken into account. There are algebraic expressions which cannot be approximated by a monomial, because they are not a product of terms but a sum of terms. However, the general idea is correct, and formulas created from the dimensionless numbers Πi according to Eqn (1.12) provide good approximations of monomial or binomial form. (Trinomials are practically never used.) How can this practical tool be used? Let us consider a simple example. A warm fluid flows in a tube which heats the environment; for example, this might be the heating system of a house. If heat radiation is negligible, the Nusselt, Reynolds and Peclet numbers for the simultaneous transfer of momentum and heat should be taken into account (see Table 1.2). Since the appropriate dimensionless numbers created from the terms of the Damköhler equations are Nu for heat (convection/conduction), Re for momentum (convection/conduction), Pe for heat (convection/conduction) or Pr = Pe/Re, therefore neglecting the gravitational force, we obtain the following function f: Nu = f ( Re, Pr )

(1.13)

which is an expression of Eqn (1.12) for the case above. Equation (1.12) is one of the most often applied relationships in chemical and food engineering. Its usual form is Nu = CRe a × Pr b

(1.14)

which has the same monomial form as Eqn (1.12). Many handbooks give instructions for determining the values of the exponents a and b and the constant C, depending upon the boundary conditions. Let us consider the physical ideas on which this approach is based.

1.4 1.4.1

Analogies The Reynolds analogy

An analogy can be set up between mechanisms as follows: momentum transfer ↔ heat transfer; momentum transfer ↔ component transfer; component transfer ↔ heat transfer. This analogy can be translated into the mathematical formalism of the transfer processes.

14

Confectionery and Chocolate Engineering: Principles and Applications

From physical considerations, Reynolds expected that the momentum flux (Jp) and the heat flux (Jq) would be related to each other, i.e. if Jq =

α × A Δ ( ρc pT ) cp ρ

(1.15)

γ × A Δ (ρv ) ρ

(1.16)

then Jq =

In other words, the moving particles transport their heat content also. Then he supposed that

α γ = cp ρ ρ

(1.17)

or, in another form, f′ α = = St cp ρ 2

(1.18)

If the flux of a component is Ji = β F Δci

(1.19)

then Reynolds’ supposition can be extended to this third kind of flux as follows: St = St ′ = f ′ 2

(1.20)

where St is the Stanton number for heat transfer (St = α/cpρ), St′ is the Stanton number for component transfer (St′ = β/v), f ′/2 = γ /ρv and γ is the momentum transfer coefficient. If the Reynolds analogy formulated in Eqn (1.20) is valid, then if we know one of the three coefficients α, β or γ, the other two can be calculated from this equation. This fact would very much facilitate practical work, since much experimental work would be unnecessary. But proof of the validity of the Reynolds analogy is limited to the case of strong turbulence. In contrast to the Reynolds analogy, a≠v≠D

(1.21)

Pr ≠ Sc ≠ Le

(1.22)

i.e.

Equation (1.17) is valid only for turbulent flow of gases. In the case of gases,

Principles of food engineering

Pr ≈ 0.7 − 1

15 (1.23)

is always valid.

1.4.2

The Colburn analogy

Colburn introduced a new complex dimensionless number, and this made it possible to maintain the form of the Reynolds analogy: J q = St Pr 2 3

(1.24)

Ji = St ′ Sc 2 3

(1.25)

and Jp =

f′ 2

(1.26)

Finally, formally similarly to the Reynolds analogy, St Pr 2 3 = St ′ Sc 2 3 =

f′ 2

(1.27)

The Colburn analogy formulated in Eqn (1.27) essentially keeps Reynolds’ principal idea about the coupling of the momentum (mass) and thermal flows, and gives an expression that describes the processes better. Equation (1.27) is the basis of the majority of calculations in chemical engineering. In view of the essential role of Eqn (1.27), it is worth looking at its structure: St =

Nu α = Re Pr ρc pv

St ′ =

Nu ′ β = Re Sc v

f′ γ = 2 ρv The numbers Pr and Sc are parameters of the fluid: Pr =

v a

Sc =

v D

Additional material parameters are needed for calculations, namely α, ρ and cp. If v is known, f ′and β can be calculated.

16

Confectionery and Chocolate Engineering: Principles and Applications

This theoretical framework (see Eqns 1.13, 1.14 and 1.27) can be modified if, for example, a buoyancy force plays an important role – in such a case the Grashof number, which is the ratio of the buoyancy force to the viscous force, appears in the calculation. A detailed discussion of such cases would, however, be beyond the scope of this book. A similar limitation applies to cases where the source term is related to a chemical reaction: chemical operations in general are not the subject of this book. A more detailed discussion of these topics can be found in the references given in Section 1.2.

1.4.3

Similarity and analogy

Similarity and analogy are quite different concepts in chemical and food engineering, although they are more or less synonyms in common usage. Therefore it is necessary to give definitions of these concepts which emphasize the differences in our understanding of them in the present context. Similarity refers to the properties of machines or media. Similarity means that the geometric and/or mechanical properties of two machines or streaming media can be described by the same mathematical formulae (i.e. by the same dimensionless numbers), that our picture of the flux (e.g. laminar or turbulent) is similar in two media, etc. Similarity is the basis of scaling-up. Analogy refers to transfer mechanisms. Analogy means that the mechanisms of momentum, heat and component transfer are related to each other by the way that components are transferred by momentum and, moreover, components transfer heat energy (except in the case of heat radiation). This fact explains the important role of the Reynolds number, which refers to momentum transfer.

1.5

Dimensional analysis

This is a simple mathematical tool for creating relationships between physical variables using the rule that physical expressions should be homogeneous from the point of view of dimensions. Dimensional analysis is applied in various fields of science because, on the one hand, dimensionally homogeneous practical expressions can be derived for the description of phenomena, and on the other hand, the number of variables can be reduced with its help. The word ‘physical’ relates here not only to the phenomena studied in physics but also to phenomena studied in any branch of science (economics, biology, etc.), since homogeneity of equations is a principal requirement for the interpretation of such mathematical operations as addition and multiplication. Homogeneity means also that the equation remains unchanged if the system of the fundamental units used changes (e.g. between the SI and ‘Anglo-Saxon’ systems). Dimensional analysis contracts physical variables into dimensionless groups, which become new variables; by this process, the number of variables is decreased. Having fewer variables is a great advantage. For example, if instead of six variables, only three variables need to be experimentally studied, and supposing that five points have to be measured for every variable, then instead of 56 = 15 625 only 53 = 125 points need to be measured in laboratory experiments. Example 1.1 shows how this method works.

Principles of food engineering

17

Example 1.1 The Hagen–Poiseuille equation, Δp =

32ηvL D2

(1.28)

can be rewritten by dividing both sides by ρv2: Δp ⎛ 32 ⎞ ⎛ L ⎞ = ρv 2 ⎜⎝ ρvD η ⎟⎠ ⎝ D ⎠

(1.29)

32 Re = Eu ( D L )

(1.30)

i.e.

The number of independent variables is six (L, D, Δp, v, ρ, η) in Eqn (1.28); however, the number of dimensionless independent variables in Eqn (1.30) is three (Re, Eu and the D/L simplex) – the second equation is easier to study. The steps of dimensional analysis are: • A system of equations is set up; the number of equations is equal to the number of fundamental units (m, s, kg, K, etc.). These equations express the dimensional homogeneity of the relationship studied. • Since the number of independent variables (i) is more than the number of fundamental units (u) in the general case, i − u = d dimensionless variables can be chosen independently; the usual notation for them is Πj, following Buckingham. Evidently, a simple linear algebraic method results in the relationship that is sought. Applying the methods of dimensional analysis can be very fruitful because complicated problems may turn out to be easily solved. There is a well-developed theory of dimensional analysis which applies abundantly the results of linear algebra and computerization; see Huntley (1952), Barenblatt (1987), Szirtes (1998, 2006) and Zlokarnik (1991). However, even this theory leads to cases in which these approaches must be used cautiously (e.g. if the solution is not a monomial but a binomial) or the results provided are not useful.

1.6

The Buckingham Π theorem

The principle of dimensional analysis was probably first expressed by Buckingham; therefore it is known as the Buckingham Π theorem. According to the formulation of Loncin and Merson (1979), if n independent variables occur in a phenomenon and if n′ fundamental units are necessary to express these variables, every relation between these n variables can be reduced to a relation between n − n′ dimensionless variables. Dimensional analysis and the approximation given by Eqn (1.12) lead to the same formula, which demonstrates their common mathematical background. The question can

18

Confectionery and Chocolate Engineering: Principles and Applications

be put of why the approximation according to Eqn (1.12) was discussed independently of dimensional analysis. The very purpose of such an approach is that the method of creating dimensionless numbers and the use of dimensional analysis as a research tool are not inherently linked, although it goes without saying that dimensional analysis leads to dimensionless variables. While the derivation of dimensionless numbers from the Damköhler equations relates to a special range of transfer phenomena that are crucial from our point of view, dimensional analysis is a general method that is not limited to chemical engineering.

Further reading Baker, W.E., Westine, P.S. and Dodge, F.T. (1991) Similarity Methods in Engineering Dynamics: Theory and Practice of Scale Modeling, Fundamental Studies in Engineering, Vol. 12. Elsevier, Amsterdam. Brimbenet, J.-J., Schunbert, H. and Trystram, G. (2007) Advances in research in food process engineering as presented at ICEF 9. J Food Sci 78: 390–404. Couper, J.R. (ed.) (2005) Chemical Process Equipment: Selection and Design. Elsevier, Boston, MA. Dobre, T.G. and Marcano, J.G.S. (2007) Chemical Engineering: Modelling, Simulation and Similitude. Wiley-VCH, Weinheim. Earle, R.L. and Earle, M.D. (1983) Unit Operations in Food Processing: The Web Edition. http:// www.nzifst.org.nz/unitoperations Fito, P., LeMaguer, M., Betoret, N. and Fito, P.J. (2007) Advanced food engineering to model real foods and processes: The “SAFES” methodology. J Food Eng 83: 173–185. Ghoshdastidar, P.S. (2005) Heat Transfer, 2nd edn. Oxford University Press, Oxford. Grassmann, P., Widmer, F. and Sinn, H. (1997) Einführung in die thermische Verfahrenstechnik, 3. vollst. überarb. Aufl. de Gruyter, Berlin. Heldmann, D.R. and Lund, D.B. (1992) Handbook of Food Engineering, Food Science and Technology, No. 51. Marcel Dekker, New York. Lienhard, J.H., IV and Lienhard, J.H., V. (2005) A Heat Transfer Textbook, 3rd edn. Phlogiston Press, Cambridge, MA. Rohsenow, W.M., Hartnett, J.P. and Cho, Y.I. (eds) (1998) Handbook of Heat Transfer, 3rd edn, McGraw-Hill Handbooks. McGraw-Hill, New York. Sedov, L.I. (1982) Similarity and Dimensional Methods in Mechanics. Mir, Moscow. Singh, R.P. and Heldman, D.R. (2001) Introduction to Food Engineering. Academic Press, San Diego, CA. Szu˝cs, E. (1980) Similitude and Modelling. Elsevier Scientific, Amsterdam. Toledo, R.T. (1991) Fundamentals of Food Process Engineering. Van Nostrand Reinhold, New York. Tscheuschner, H.D. (1996) Grundzüge der Lebensmitteltechnik. Behr’s, Hamburg. Uicker, J.J., Pennock, G.R. and Shigley, J.E. (2003) Theory of Machines and Mechanisms, 3rd edn. Oxford University Press, New York. Valentas, K.J., Rotstein, E. and Singh, R.P. (1997) Handbook of Food Engineering Practice. CRC Prentice Hall, Boca Raton, FL. Vauck, W.R.A. (1974) Grundoperationen chemischer Verfahrenstechnik. Steinkopff, Dresden. VDI-GVC (2006) VDI-Wärmeatlas. Springer, Berlin. Watson, E.L. and Harper, J.C. (1988) Elements of Food Engineering, 2nd edn. Van Nostrand Reinhold, New York.

Chapter 2

Characterization of substances used in the confectionery industry

Contents 2.1 Qualitative characterization of substances 2.1.1 Principle of characterization 2.1.2 Structural formulae of confectionery products 2.1.3 Classification of confectionery products according to their characteristic phase conditions 2.1.4 Phase transitions – a bridge between sugar sweets and chocolate 2.2 Quantitative characterization of confectionery products 2.2.1 Composition of chocolates and compounds 2.2.2 Composition of sugar confectionery 2.2.3 Composition of biscuits, crackers and wafers 2.3 Preparation of recipes 2.3.1 Recipes and net/gross material consumption 2.3.2 Planning of material consumption

2.1 2.1.1

19 19 20 27 28 29 29 35 43 45 45 48

Qualitative characterization of substances Principle of characterization

The characterization of the substances used in the confectionery industry is based on two suppositions: (1) The substances are partly of colloidal and partly of cellular nature. (2) From a technological point of view, their properties are essentially determined by the hydrophilic or hydrophobic characteristics of their ingredients. These substances are complex colloidal systems, that is, organic substances of mostly natural origin which consist of various simple colloidal systems with a hierarchical or quasi-hierarchical structure. Let us consider the example of the hierarchical structure of a food represented in Fig. 2.1. The left-hand part shows, in outline, the structure of a substance: a solution containing solids and oil droplets. The right-hand part shows a structural formula using an oriented graph consisting of vertices and arrows. The vertices of the graph are symbols representing the components from which the substance is theoretically constructed. The arrows relate to the ‘containing relation’, and are directed from the contained symbol to the containing Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

20

Confectionery and Chocolate Engineering: Principles and Applications

Solution

Oil droplets

Solution Water

Solids Dissolved substance Emulsifier Solids

Oil droplets

Fig. 2.1 Hierarchical structure of foods. Example: an aqueous solution contains solid particles and oil droplets coupled by an emulsifier to the aqueous phase.

symbol; for example, dissolved substances are contained by water. Such a diagram can be regarded as a primitive formula of the given substance which, to some extent, imitates the structural formulae of the simplest chemical compounds. A ‘quasi-hierarchical’ attribute is more expressive, since there can be cross-relations as well; see the position of ‘emulsifier’. The structure shown in Fig. 2.1 is less complex than this, however. Although this way of representing structural relations is very simple, it can express the hydrophilic/hydrophobic behaviour of a system. Evidently, from an external viewpoint, this system behaves like a hydrophilic system, as does, for example, milk cream (as opposed to milk butter); that is, it is an oil-in-water (O/W) system. The materials studied often have a cellular structure. The cell walls hinder the free transport of material to a great extent, and therefore the actual material flows are determined by the particle size, since comminution more or less destroys the cell walls. This effect can be important in the case of cocoa mass because the amount of free cocoa butter equals the total cocoa butter content only if all the cocoa cells are cut up. This characterization of substances is not capable of reflecting those properties which need to be explored by microstructural studies, for example the polymorphism of lactose in milk powder and the fine structure of proteins.

2.1.2

Structural formulae of confectionery products

Structural formulae of various confectionery products obtained by the application of structure theory (see Appendix 5) are shown in Figs 2.2–2.16. The substances named in these figures may be considered as ‘conserved substantial fragments’ (referred to from now on simply as ‘fragments’). The set of fragments is tailored to the technological system studied. Let us consider chocolate (Fig. 2.2). Although the usual ingredients of milk chocolate are sugar powder, cocoa mass, cocoa butter, milk powder and lecithin, it is expedient to use the following fragments to describe the manufacture of milk chocolate: sugar (powder), cocoa butter, fat-free cocoa, water and lecithin. This is because these fragments determine such essential properties of chocolate as viscosity and taste. The recipe for a chocolate product must obey some restrictions on the ratios of these fragments because, on the one hand, there are definitive prescriptions laid down by authorities (see e.g. European Union (2000)) and, on the other hand, there are certain practical rules of thumb concerning the fragments that provide a starting point for preparing recipes:

Characterization of substances used in the confectionery industry

Water

Sugar

e (Lecithin)

d

Cocoa butter

Crystallization

d

d

Dried milk

Fat-free cocoa d Milk fat

Fig. 2.2

Structural formula of chocolate. d = dispersion; e = emulsion.

Sugar

Glucose syrup

s

s Water s Acid, flavour, colour

Fig. 2.3

Structural formula of hard-boiled candy. s = solution.

Cream of tartar

Sugar s (Inversion + retarded crystallization)

s

Water s Acid, flavour, colour Fig. 2.4

Structural formula of crystallized hard-boiled candy. s = solution.

21

22

Confectionery and Chocolate Engineering: Principles and Applications

Glucose syrup

Sugar

s

s (cry in the case of fudge) Water

e

s

(lecithin)

Acid, flavour, colour

Fat s Milk

Fig. 2.5

Structural formula of toffee/fudge. s = solution; e = emulsion; cry = crystallization.

Glucose syrup

Sugar s + cry

s

Water

s Acid, flavour, colour

Fig. 2.6

Structural formula of fondant. s = solution; cry = crystallization.

Sugar

Glucose syrup

s

s

Water

s

Acid, flavour, colour Fig. 2.7

sw + s

Gelling agent

Structural formula of jelly. s = solution; sw = swelling.

Characterization of substances used in the confectionery industry

23

Melted sugar

d

d

Cut nuts Fig. 2.8

Structural formula of nut brittle (croquant). d = dispersion.

Sugar

d

d

Cut almonds

Fig. 2.9

Structural formula of marzipan (or of persipan, with apricot stones). d = dispersion.

Air

Sugar

Glucose syrup

s

s

f

Water s

Acid, flavour, colour Fig. 2.10

• • • • •

Foaming sw + s Foaming agent

Structural formula of confectionery foams. s = solution; sw = swelling; f = foaming.

content of cocoa butter, 30–38 m/m%; content of sugar, 30–50 m/m% (depending on the kind of chocolate, i.e. dark or milk); content of milk dry matter (milk fat + fat-free milk solids), 15–25 m/m%; content of milk fat, minimum 3.5 m/m%; content of lecithin, 0.3–0.5 m/m%.

24

Confectionery and Chocolate Engineering: Principles and Applications

Crystallized sugar

Glucose syrup

d

d

Water s

sw + s

d

Acid, flavour, colour

Binder

Lubricant (tablets) Fig. 2.11

Structural formula of granules, tablets and lozenges. s = solution; d = dispersion; sw = swelling.

Glazing layer Colouring layer Coating ... n Coating 2 Coating 1 Bonding layer

Centre

Fig. 2.12

Structural formula of dragées.

Example 2.1 Let us consider a milk chocolate with the following parameters (in m/m%): • • • • •

sugar content, c. 40–44; total fat content, 31–33; cocoa mass content, 12–16 (cocoa butter 50% of this); lecithin content, 0.4; whole milk powder, 20–24 (milk fat 26% of this). The calculation of the recipe is an iterative task.

Characterization of substances used in the confectionery industry

25

Flour Starch

Sugar

Gluten sw + s

s

g+s Water

s

e

Leavening/ yeast

Fig. 2.13

Fat/ margarine

Structural formula of dough. s = solution; e = emulsion; g = gelling; sw = swelling.

Sugar/ glucose syrup

Fat/ margarine

d

d

Flour

Gluten Starch d

Leavening

Fig. 2.14

Structural formula of biscuits and crackers. d = dispersion.

The procedure for the calculation is: • Calculate Total 1, which contains all the ingredients without cocoa butter (e.g. 79.2 in Version 1). • Calculate the amount of cocoa butter required to make up the total to 100 (20.8 in Version 1). • Calculate the fat content of the ingredients (Total 1) without cocoa butter (12.72 in Version 1). • Add the amount of cocoa butter calculated previously (in Version 1, 20.8 + 12.72 = 33.52 – the value is too high).

26

Confectionery and Chocolate Engineering: Principles and Applications

Flour Sugar Starch Gluten d sw + s g+s Water e

e

(+ lecithin)

Milk/ eggs

Fat s Leavening

Fig. 2.15

Structural formula of wafers. s = solution; d = dispersion; e = emulsion; g = gelling; sw = swelling.

Sugar

Fat

e + cry

Air

f

s + cry

Dried milk

Fig. 2.16

e + cry

Water sw + s

Stabilizer, thickener

Structural formula of ice cream. s = solution; e = emulsion; sw = swelling; f = foaming;

Note that the milk fat content is higher than 3.5 m/m% in every case. Moreover, no chemical reactions are taken into consideration. Consequently, the elements of set A (see Appendix 5) are sufficient for preparing the recipe. However, when the Maillard reaction that takes place during conching is to be studied, a ‘deeper’ analysis of the participant substances is necessary; that is, the elements of set B must be determined, for example the lysine content of the milk protein, the reducing sugar content of the sugar powder, water, etc.

Characterization of substances used in the confectionery industry

27

Table 2.1 Calculation of a milk chocolate recipe (all values in m/m%). Raw materials

Version 1

Fat, Version 1

Version 2

Fat, Version 2

Version 3

Fat, Version 3

Lecithin Sugar Whole milk powder Cocoa mass Water content TOTAL 1 Cocoa butter TOTAL 2 Comments

0.4 42 22

0.4 0 5.72

0.4 43 23

0.4 0 5.98

0.4 43 23

0.4 0 5.98

14 0.8 79.2 20.8 100

7 0 12.72 20.8 33.52 Too high

14 0.8 81.2 18.8 100

7 0 12.98 18.8 31.78

15 0.8 82.2 17.8 100

7.5 0 13.38 17.8 31.18

Fair

Good

Table 2.2 Cartesian product of phases.a 1×1 Fat melts

1×2 (W/O) Emulsions

1×3 Chocolate, compounds

2×1 (O/W) Toffee, fudge, ice cream

2×2 Hard-/soft-boiled candies

2×3 Jellies, foams, wafers

3×1 Cocoa/chocolate powders, pudding powders

3×2 Dragées, tablets, lozenges

3×3 Biscuits, crackers

1 = hydrophobic phase; 2 = hydrophilic phase; 3 = solids (hydrophilic)

a

2.1.3

Classification of confectionery products according to their characteristic phase conditions

In colloids and coarse dispersions, various phases are present (see Chapter 5). Since the gaseous phase is of minor importance in the majority of confectionery products, the basis of classification is the hydrophilic/hydrophobic character, which applies to both the liquid and the solid phase. Table 2.2 (Mohos 1982) represents a classification of confectionery products with the help of a 3 × 3 Cartesian product, which represents a combination of hydrophobic solutions (1), hydrophilic solutions (2) and (hydrophilic) solids (3). The gaseous phase is not represented, but can be taken into account as a possible combination in particular cases. The first factor in an element of this Cartesian product represents the dominant or continuous phase, and the second factor represents the contained phase; for example, 1 × 2 means a water-in-oil (W/O) emulsion (e.g. milk butter or margarine), and 2 × 1 means an O/W emulsion (e.g. toffee, fudge or ice cream). It should be emphasized that this classification is a simplification in the following senses: • There is not one single classification that is appropriate in all cases, and other classifications which take the phase conditions into account in more detail may give a more differentiated picture of the important properties.

28

Confectionery and Chocolate Engineering: Principles and Applications

• Table 2.2 contains only some large groups of finished confectionery products that are characteristic of each element (i × j) of the product; however, all materials used or made in the confectionery industry can be classified into one or other of these elements. • The classification of products containing flour (biscuits, wafers, crackers etc.) is very haphazard because of the complexity of their structure. • The elements (3 × 1), (3 × 2) and (3 × 3) can hardly be regarded as different; the only difference is that the hydrophobicity decreases from cocoa/chocolate powders to biscuits and crackers containing flour. However, cases showing the opposite trend in the hydrophobicity are very frequent (e.g. cocoa powder with 8% cocoa butter content compared with cakes with 30% fat content). • Chocolate and compounds are actually W/O emulsions [see element (1 × 2)], but the water content is in practice less than 1 m/m%. • There are likely to be other appropriate classifications that are not based on combinations of hydrophilic/hydrophobic/solid/liquid phases. Despite these objections and contradictions, this classification correctly expresses the hydrophobic/hydrophilic properties of the materials used and/or made in the confectionery industry because these properties play an essential role in the technologies used and in the shelf life of the substances (i.e. raw materials, semi-finished products and finished products).

2.1.4

Phase transitions – a bridge between sugar sweets and chocolate

To study the phase conditions of chocolate, Mohos (1982) produced so-called crystal chocolate in the following way: Recipe for Experiments 1, 2 and 3 (laboratory scale) (in g): sugar, 58.5; water, 19.5; cocoa mass, 18.0; cocoa butter, 16 (sum = 112.0). Recipe for Experiment 4 (plant scale) (in kg): sugar, 50.0; water, 16.7; cocoa mass, 15.5; cocoa butter, 13.7 (sum = 95.9). The results are presented in Table 2.3. Three steps may be distinguished in the experiments: Step 1: At a water content of about 10%, the cocoa butter phase separates. (The consistency of the mass is similar to that of sugar sweets.) Step 2: At about 100 min (water content ≈ 5.2%), a phase inversion (O/W → W/O) starts, and this lasts up to a water content of about 1.38% (235 min). In the final period, the crystallization of sugar and the comminution of sugar crystals by the rubbing effects of conching start. Step 3: The consistency of crystal chocolate is developed. A plot of water percentage versus time can be approximated by the function wt = (w0 − w∞ ) exp ( − ki t ) + w∞

(2.1)

Characterization of substances used in the confectionery industry

Table 2.3

29

Manufacture of crystal chocolate: experimental results.

Experiment 1 Time (min)

Experiment 2 Water %

0 30 45 90 165

17.4 3.85 1.57 0.52 0.43

Air temperature (°C) Air RH (%)a Air velocity (m/s)

43 35 2

Time (min) 0 30 45 90 165

Experiment 3 Water % 17.4 3.73 1.46 0.33 0.22

Time (min) 0 25 60 120 180

43 35 2

Experiment 4 Water %

Time (min)

17.4 7.31 1.54 0.35 0.34

34 38 2

0 20 50 70 100 120 140 160 190 235 265 295 325

Water % 17.4 10.7 9.74 9.2 5.2 6.21 4.67 3.53 2.68 1.75 1.38 0.88 0.72 72 (input) 20 (input) 22.3–25.1

RH = relative humidity.

a

where t = time of conching/drying (min), w0 = initial water content (%), w∞ = water content after long drying (≈0.3%), ki = velocity constant of drying (min−1) and i is the number of the experiment. For the experiments above, k1 = 9.83 × 10−3, k2 = 8.4 × 10−3, k3 = 7.78 × 10−3 and k4 = 2.17 × 10−3. At the end of production, the size of the sugar crystals is similar to that in a fondant mass (c. 5–30 μm); however, after a short time the larger crystals are in the majority because of Ostwald ripening, similarly to the changes that occur in fondant. A noteworthy phenomenon. The two methods of (1) comminution by mill and (2) solution + crystallization provide similar results. However, while comminution is not followed by Ostwald ripening, the operations of solution + crystallization are. Just the same phenomenon can be observed when a ripened fondant is rekneaded and then shaped. While the structure of the centres of ripened fondant hardly changes in storage, the centres of unripened fondant are easily dried, etc.; that is, their structure is more changeable and less stable. All of this emphasizes the importance of Ostwald ripening (see Sections 5.9.5, 10.6.1 and 16.4).

2.2 2.2.1

Quantitative characterization of confectionery products Composition of chocolates and compounds

Quantitative relations can be given which characterize the composition of chocolates and compounds [see the (1 × 3) element of the Cartesian product in Table 2.2]; the latter contain special fats instead of cocoa butter as the dispersing phase. Dark chocolate and milk chocolate are typical examples of these product groups.

30

Confectionery and Chocolate Engineering: Principles and Applications

2.2.1.1

Composition of dark chocolate

If the proportions of the ingredients (in %) are S, sugar, B, cocoa butter, M, cocoa mass, L, lecithin, then S + M + B + L = 100

(2.2)

The cocoa content (C) is C =M +B

(2.3)

Taking into account the consistency requirements, the total fat content (F) must be between 30 and 40%, i.e. F = L + cM M + B = 30−40

(2.4)

where cM is the cocoa butter content (mass concentration) of the cocoa mass (c. 0.50–0.56). The usual value of S for dark chocolate is 30–50%, and the usual value of L is 0.3–0.5%. On the basis of these relations, many chocolate recipes can be prepared, as shown in Table 2.4. Because of price considerations, the total fat content is chosen to be nearer to 30% than to 40% (usually, F = 30–33). A more detailed picture of the fragments is not needed in general for preparing a recipe for chocolate; for example, the water content does not usually play any role, since only the cocoa mass has a relatively high water content (1–2 m/m%), which is decreased during conching. The water content of sugar, cocoa butter (particularly if it is deodorized) and lecithin can be neglected. Also, the water content of cocoa mass can be made low if it is refined by a special film evaporator (e.g. the Petzomat, from Petzholdt), which can be regarded as a pre-conching machine.

Table 2.4

Recipes for dark chocolate.

Raw materials Sugar Cocoa mass (50%) Cocoa butter Lecithin TOTAL Cocoa content

Version 1

Fat, Version 1

36 60 3.7 0.3 100 63.7

0 30 3.7 0.3 34

Version 2 30 70 0 0 100 70

Fat, Version 2

Version 3

Fat, Version 3

0 35 0 0 35

39.6 46 14 0.4 100 60

0 23 14 0.4 37.4

Characterization of substances used in the confectionery industry

2.2.1.2

31

Composition of milk chocolate

The following equation (in %) is valid for a milk chocolate: S + M + B + L + W + b = 100

(2.5)

where S, M, B and L have meanings similar to those above, W is the percentage of whole milk powder and b is the percentage of (dry) milk fat (about 1 m/m% of the water content). The use of dry milk fat is optional. Equation (2.3) is valid for the cocoa content. The usual value of S for milk chocolate is 40–45%, and the usual value of L is 0.3–0.5%. Taking into account the consistency requirements, the total fat content must be between 30 and 40%, i.e. F = L + cMM + B + WcW + b = 30−40

(2.6)

where cW is the milk fat content (mass concentration) of whole milk powder (c. 0.26–0.27). An additional requirement related to the consistency is the ratio R = cocoa butter/ non-cocoa-butter fats (mass/mass) because non-cocoa-butter fats soften the consistency and, in extreme cases, make it too soft for correct shaping of the chocolate. One principal requirement for milk chocolate, which is laid down by authority (European Union 2000), is that the milk fat content should be at least 3.5 m/m%. (In tropical countries, a value of 2.5 m/m% is accepted because of the hot climate.) The usual values of milk fat content are in the range 3.5–6%, and the usual values of total fat content are in the range 30–40%; consequently, the value of R + 1 can theoretically vary as follows: 40 30 ≈ 11.4 ≥ R + 1 ≥ =5 3.5 6 i.e. 10.4 ≥ R ≥ 4 However, a ratio R = 4 is not available, since the consistency would be very soft. Instead, the practical minimum value is given by R + 1 = 30/3.5 = 8.57, i.e. about R = 7.6. On the other hand, an intense milky taste is an important quality requirement too, and therefore increasing the dry milk content is an understandable ambition of producers. Another way to produce milk chocolate with an intensely milky taste is to use special milk preparations, for example condensed sugared milk (milk crumb) or chococrumb (see Chapter 16), where the Maillard reaction is used. An essential quality requirement is a suitably high value of the fat-free cocoa content, which gives the product its cocoa taste. The practical value is at least 3–4 m/m% for compounds and at least 5–6 m/m% for milk chocolate. However, for compounds, cocoa powder of low cocoa butter content (10–12 m/m%) has to be used because the fats used in compounds are not compatible with cocoa butter or are only partly compatible. For a

32

Confectionery and Chocolate Engineering: Principles and Applications

milk chocolate this minimum value of fat-free cocoa content means that the percentage of cocoa mass must be at least 10–12 m/m% (assuming that the cocoa butter content of cocoa mass is about 50 m/m%).

Example 2.2 On the basis of these relations, a chocolate recipe can easily be prepared as shown in Table 2.4. Let us look at the percentages in Version 3. The amount of non-cocoa-butter fats is 5.98 + 0.4 = 6.38, and the total fat content is 31.18. From these data, the lesson is that the consistency will be too soft because R + 1 = 31.18/6.38 = 4.887, i.e. R is approximately 3.9! We shall present the steps of a calculation of a milk chocolate recipe. Let S = 40, W = 20 (26 m/m% of milk fat) and L = 0.4. If M (cocoa mass) = 12, we then do the following calculation:

Ingredients Sugar Lecithin Whole milk powder Cocoa mass TOTAL

Total fat 40.0 0.4 20.0 12.0 72.4

0.0 0.4 5.2 6.0 11.6

If the balance of these ingredients is made up by cocoa butter (100 − 72.4 = 27.6), then the total fat content will be 27.6 + 11.6 = 39.2% – too high! If M = 14 and S = 43, then we do the following calculation:

Ingredients Sugar Lecithin Whole milk powder Cocoa mass TOTAL

Total fat 43.0 0.4 20.0 14.0 77.4

0.0 0.4 5.2 7.0 12.6

If the balance of these ingredients is made up by cocoa butter (100 − 77.4 = 22.6), then the total fat content will be 22.6 + 12.6 = 35.2% – this is acceptable. Taking the price of cocoa butter into account, this is an important alteration. In the above recipe, R + 1 = 35.2/(5.2 + 0.4) = 6.28, i.e. R = 5.28. The usual way of reducing the proportion of non-cocoa-butter fat is to use whole and skimmed milk powder together as follows. The amount of whole milk powder is calculated according to the minimum requirement of 3.5% milk fat, i.e. 3.5%/0.26 ≈ 13.5%. This amount is then made up to 20%, i.e. the amount of skimmed milk powder is 6.5%.

Characterization of substances used in the confectionery industry

33

The calculation is modified as follows:

Ingredients Sugar Lecithin Whole milk powder Skimmed milk powder Cocoa mass TOTAL Cocoa butter TOTAL

Total fat 43.0 0.4 13.5 6.5 14.0 77.4 100 − 77.4 = 22.6 100

0.0 0.4 3.5 0.0 7.0 10.9 22.6 33.5

In this recipe R + 1 = 33.5/(3.5 + 0.4) ≈ 8.6, i.e. R ≈ 7.6. Note that in this example, a blend of two kinds of milk powder has been used; the average milk fat content of this blend is 3.5/20 = 17.5% (instead of 26%).

2.2.1.3

Preparation of Gianduja recipes

The relevant European Union directive (European Union 2000) defines Gianduja chocolate as a blend of dark or milk chocolate and hazelnut paste (and pieces); both dark and milk Gianduja chocolate are defined in detail. The minimum and maximum amounts of hazelnut are 20% and 40%, respectively, for dark Gianduja and 15% and /40% for milk Gianduja. The recipes for both types of Gianduja chocolate are actually very simple.

Example 2.3 Seventy-five per cent dark chocolate is mixed with 25% hazelnut paste or 70% milk chocolate is mixed with 30% hazelnut paste. Since shelled hazelnuts have an oil content of about 40–60% and hazelnut oil has a very low cold point (−18°C), the hazelnut paste softens the consistency of the product to a great extent. If milk chocolate of the composition calculated above is used in a proportion of 70% and the assumed oil content of the hazelnuts is 50%, then the distribution of the various oils/fats will be: 70% milk chocolate: 0.7 × (22.6 + 7)% cocoa butter + 0.7 × 3.9% (lecithin + milk fat); 30% hazelnut paste: 0.5 × 30% hazelnut oil. In summary, this Gianduja product contains 20.72% cocoa butter + 2.73% (lecithin + milk fat) + 15% hazelnut oil (total fat content 38.45%), and therefore R +1 =

38.45 38.45 = ≈ 2.17, i.e. R ≈ 1.17. 38.45 − 20.72 17.73

34

Confectionery and Chocolate Engineering: Principles and Applications

In order to avoid a consistency that is too soft, the hazelnuts are used partly as paste and partly as tiny pieces. The hazelnut oil remains in the cells in the latter, and therefore this portion of hazelnut oil does not soften the consistency of the chocolate . For example, the above composition can be modified so that 70% milk chocolate is mixed with 15% hazelnut paste and 15% chopped hazelnuts. The milk Gianduja mass will have the following composition and fat/oil distribution: 70 kg milk chocolate: 20.72 kg cocoa butter + 2.73 kg (lecithin + milk fat); 15 kg hazelnut paste: 7.5 kg hazelnut oil. The distribution of the various fats in this milk Gianduja mass will be (in %) 20.72/0.85 = 24.38% cocoa butter; 2.73/0.85 = 3.21% lecithin + milk fat; 7.5/0.85 = 8.82% hazelnut oil; TOTAL: 36.41% oils/fats. For this solution, R + 1 = 36.41/(36.41 − 24.38) = 36.41/12.03 ≈ 3.03, i.e. R ≈ 2.03. Evidently, the softness of the consistency has been moderated. For the sake of completeness, let us calculate a recipe for a compound that is similar to milk chocolate. The corresponding formula (in %) is S + P + V + L + m = 100

(2.7)

where S refers to sugar, P to cocoa powder, V to special vegetable fat, L to lecithin and m to whole or skimmed milk powder. Taking the consistency requirements into account, the total fat content (F) must be between 30 and 40%, i.e. F = L + cm m + V + PcP = 30−40

(2.8)

where cm is the milk fat content of whole or skimmed milk powder (m/m) and cP is the cocoa butter content of cocoa powder (m/m). The further requirements concerning compounds are similar to those for chocolate. Example 2.4 Let us take an example in which a blend of milk powder of 15% milk fat content and cocoa powder of 10% cocoa butter content is used: Ingredients Sugar Lecithin Milk powder blend Cocoa powder TOTAL Special vegetable fat TOTAL

Total fat 43.0 0.4 20.0 6.0 69.4 100 − 69.4 = 30.6 100

0.0 0.4 3.0 0.6 4.0 30.6 34.6

Characterization of substances used in the confectionery industry

35

Comment: From the point of view of cocoa taste, 6% cocoa powder (10% cocoa butter content) is equivalent to 2 × 6% × 0.9 = 10.8% cocoa mass (50% cocoa butter content) since the fat-free cocoa content of both is 6% × 0.9 = 5.4%. (This would be acceptable for milk chocolate as well.) If the cocoa powder content is less than 3%, the taste of the product is not characteristic of cocoa.

2.2.2

Composition of sugar confectionery

The composition of the various types of sugar confectionery is principally determined by the water content and the syrup ratio (SR) in the product (see Chapters 8 and 9 for further details). The syrup ratio is the ratio of the starch syrup dry content to the sugar content, expressed in the form 100 : X or 100/X, where for each 100 kg of sugar there is X kg of starch syrup dry content. Example 2.5 If SR = 100 : 50, this means that in the prepared solution there are dissolved 100 kg of sugar and 50 kg of starch syrup dry content. Assuming the usual dry content of starch syrup of 80 m/m%, 100 kg of sugar and 50 kg/0.8 = 62.5 kg of (wet) starch syrup should be blended. In addition to the water content, the reducing sugar content plays an important role in determining the properties of sugar confectionery. The reducing content of a sugar/starch syrup solution, derived from the dextrose content of the syrup, can be calculated using the formula R = (1 − W ) ×

DE SR + 1

(2.9)

where R is the reducing sugar content of the solution (%), W is the concentration of water in the solution, DE is the dextrose equivalent of the starch syrup (%) and SR is the syrup ratio. The other important source of the reducing content of carbohydrate solutions is inversion, which produces the reducing sugar glucose (also known as dextrose) by hydrolysis of sucrose (also known as saccharose) under the action of catalysts (acids or the enzyme invertase): sucrose + H2 O = glucose + fructose (Water is chemically built into the dry content during inversion: 342 g sucrose + 18 g water = 180 g glucose + 180 g fructose, i.e. a 5% increase in dry content.) The reducing sugar content of carbohydrate solutions and sugar confectionery can easily be determined. Titrimetric or iodometric methods are the methods mostly used for the determination of reducing sugar content, and do not require sophisticated, expensive laboratory equipment. However, what is measured by these iodometric methods? According to Erdey (1958), iodometric methods (the Fehling/Bertrand and Fehling/ Schoorl-Regenbogen methods) may be used for the quantitative determination of glucose, fructose, invert sugar, sucrose (after inversion), maltose, galactose, mannose, arabinose,

36

Confectionery and Chocolate Engineering: Principles and Applications

xylose and mannose by use of a table containing the corresponding data for reduced Cooper measuring solution (0.1 N) versus the kind of sugar measured (in mg). (The determination is not strictly stoichiometric.) Aldoses may be oxidized easily; the oxidation of ketoses (e.g. fructose) takes place only in more strongly oxidizing media, but the alkaline medium that is that is typically used in these methods of sugar determination is favourable for oxidation of all the various sugars; for further details see Bruckner (1961). Colorimetric methods are also widely used for determining reducing sugar content (e.g. in investigations of human blood; see Section 16.1.1). Why does the reducing sugar content of carbohydrate solutions play such an important role in confectionery practice? The reducing sugar content, together with the water content, determines the following: • the crystallization of sucrose; • water adsorption on the surface of the product, i.e. the hygroscopic properties of the surface; • the consistency of the product. The ability of sucrose to crystallize is an important property from two contradictory points of view: • Certain products are of crystalline structure (e.g. crystalline drops, fondant and fudge). • There are types of sugar confectionery (e.g. drops, toffees, jellies and marshmallows) which must not be of crystalline structure. During their production, the crystallization of sucrose must be hindered by glucose syrup, invert sugar etc. The hygroscopic properties of the surface of sugar confectionery may have unintended consequences. Packaging materials can defend sugar confectionery against water adsorption, which would make the surface sticky. The water permeability of packaging materials can be adjusted to the given task. However, if the product is left unpacked for some time, stickiness becomes a serious problem. Experience shows that when the reducing sugar content of a sugar mass is more than 16 m/m%, the mass becomes stickier and stickier. The syrup ratio for a reducing sugar content of 16% (and a water concentration W of 0.02 and DE = 40%) can be obtained from Eqn (2.9): 16 = (1 − W ) × DE (SR + 1) = 0.98 × 40 (SR + 1) From this equation, SR = 1.45, i.e. 100 kg of sugar and 100/1.45 ≈ 69 kg of glucose syrup dry content (c. 69/0.8 ≈ 86.25 kg ‘wet’ glucose syrup) should be mixed to produce a solution of 16% reducing sugar content. This ratio is economic, since the dry content of glucose syrup is always a little cheaper than sugar. However, Eqn (2.9) does not take into account the inversion of sucrose, which is caused by the acid content (sulphuric and hydrochloric acid) of glucose syrup derived from the acidic conversion of starch. Although the acid content of glucose syrup remaining after the conversion of starch is neutralized, and the pH of glucose syrups is about 4.5–5.5, hydrolysis caused by the residual acid must not be ignored. An additional reason for increasing the reducing sugar content is the presence of other acidic agents in candies, above all the various flavouring acids (citric, malic, lactic and tartaric acids).

Characterization of substances used in the confectionery industry

37

The inversion abilities of various acids are rather different, and cannot be exactly characterized by a single parameter, because inversion is catalysed by hydrogen ions: i.e. the process of inversion is strongly dependent on the conditions in the acidic medium (the kind of acid, the concentration, etc.) (for more details, see Section 16.1 and Chapter 17). Sokolovsky (1958) discussed in detail the hygroscopicity of sugar masses and their ingredients under various conditions of production and storage. In confectionery practice, the typical hygroscopic substances are fructose, invert sugar, sorbitol and glycerol. The orders of hygroscopicity and of solubility are the same: glucose < sucrose < invert sugar < fructose On the basis of the considerations above, the conclusion is that the reducing sugar content itself cannot characterize the hygroscopicity of sugar masses. Instead, the kinds of sugar (monosaccharides and disaccharides) that the reducing sugar content is composed of are decisive: the value of 16% is a rough threshold only, and reducing sugar contents of 16% derived exclusively from glucose and derived partly from glucose and partly from fructose have entirely different effects. An increase in the reducing sugar content makes the consistency of candies softer, although an exact description of the circumstances that influence the consistency has to be limited to individual cases. Taking into account the effects of water content and reducing sugar content, Fig. 2.17 shows the intervals which can be regarded as optimal for various sugar confectioneries; see Mohos (1975). Naturally, these intervals are experimentally determined, and are not derived from any scientific law. Moreover, their boundaries are not strictly fixed, and this statement relates to the dividing line between the amorphous and crystalline regions too. Nevertheless, Fig. 2.17 should be informative for the preparation of recipes for sugar confectionery.

15 V

Reducing suqar (%)

I IV/a IV/b

Crystalline

I: drops II: crystalline drops III: fondant IV/a: toffee IV/b: fudge V: jellies

II III

5

0

Fig. 2.17

Amorphous

10

5

10

15 20 Water content (%)

Reducing sugar vs water content in sugar confectionery.

38

Confectionery and Chocolate Engineering: Principles and Applications

A typical instance of the crucial role of reducing sugar content is provided by the technology for crystalline drops (or ‘grained drops’), the characteristic region for which is denoted by ‘II’ in Fig. 2.17. Two kinds of technology are possible: inversion of sucrose by cream of tartar (also known as cremor tartari or potassium hydrogen tartrate), and the use of sucrose + glucose syrup. Before flavouring, colouring and pulling, a sugar mass made by either of these technologies has to have the following composition: 4% water; 3% glycerol; 6% reducing sugar. The recipe for the cream of tartar technology is: 92–93 kg sugar; 3 kg glycerol; c. 0.2 kg cream of tartar; 25–30 kg water for dissolution. YIELD: c. 100 kg sugar mass. In the case of the recipe for the glucose syrup technology, we assume that the parameters of the glucose syrup are DE = 40% and dry content = 80%, i.e. 100 kg of glucose syrup contains 40 kg × 0.8 = 32 kg of reducing sugar. Therefore, 6 kg of reducing sugar is contained in 6 kg/0.23 = 18.75 kg of (wet) glucose syrup, the dry content of which is 18.74 kg × 0.8 = 15 kg. Compared with the cream of tartar technology, the amount of sugar is decreased by 15 kg, and the amount is water is decreased by 3–4 kg (= 18.75 − 15). The recipe is: 77–78 kg sugar; 18.75 kg (wet) glucose syrup; 3 kg glycerol; 23–27 kg water for dissolution. YIELD: c. 100 kg sugar mass. In both technologies, first atmospheric and then vacuum evaporation are necessary, and there must be strictly no mixing or moving of the solution. Moreover, an essential requirement is that the dissolution of the sugar must be perfect, i.e. no sugar crystals must remain undissolved, otherwise crystallization of sucrose will start during the evaporation. To avoid such a mistake, sufficient water must be used for dissolution. But these two technologies are very different. The cream of tartar technology is based on the inversion effect of cream of tartar, a process which is strongly time-dependent; consequently, the durations of the two evaporation steps have definite limits imposed on them. A slow evaporation results in more reducing sugar than necessary, and the crystallization in the end product will occur late or be impossible. The other ‘sensitive point’ of this technology is that sugar always contains Ca2+ and Mg2+ ions, which form salts with cream of tartar, and the Ca salt is insoluble.

Characterization of substances used in the confectionery industry

39

Example 2.6 A simple calculation shows that this consumption of cream of tartar by calcium and magnesium ions may be considerable. The molecular mass of cream of tartar (KHC4H4O6) is 188, i.e. 188 g of cream of tartar reacts with 40 g of calcium or 24.3 g of magnesium. The average calcium content of sugar per kilogram is c. 0.15 g, and the corresponding value for magnesium is c. 0.025 g. This means that 90 kg sugar contains c. 13.5 g Ca and 2.25 g Mg, which react with 13.5 g × (188 40 ) + 2.25 g × (188 24.3) = 80.87 g cream of tartar. If c. 200 g of cream of tartar is used in the batch, the decrease in the amount of it because of the effect of Ca2+ and Mg2+ is c. 40%. (Naturally, these data are indicative only.) This consumption is the reason why we give only an approximate amount of cream of tartar (c. 0.2 kg) in the recipe. This means that the amount of cream of tartar has to be adjusted to the sugar used. However, the quality of the product made by the cream of tartar technology is much better: the sucrose crystals are of small (5–9 μm) and very homogeneous size, whereas the product made by the glucose syrup technology has a consistency somewhat similar to that of starch sugar made from potatoes. (If potato starch is converted by acid in aqueous solution, the evaporated reaction mixture, containing c. 80 m/m% dextrose, can be sold as a cheap product. In former years, this process was done in the kitchen at home as well.) But the glucose syrup technology is practically insensitive to the duration of the evaporation steps. An improved variation of the glucose syrup technology which eliminates the consistency properties of the end product that remind consumers of starch sugar uses liquid sugar instead of glucose syrup as follows: 18.75 kg of wet glucose syrup (dry content = 80%, DE = 40%) contains about 15 kg of dry content and 6 kg of reducing sugar, and 9 kg sugar + 6 kg liquid sugar dry content is equivalent to 15 kg syrup dry content. Taking into account the usual parameters of liquid sugars (dry content = 75%, and fructose : glucose = 55 : 45), this means a blend of 9 kg sugar + 6 kg/0.75 = 8 kg liquid sugar. The acid residues in both glucose syrup and liquid sugar cause unwanted inversion, and therefore the acid content has to be rather low. (In both technologies, glycerol is added because its hygroscopic effect accelerates the crystallization of the product.) Both technologies are very sensitive to the reducing sugar and water content parameters from the point of view of both pulling and crystallization of sucrose in the product – these latter operations are very closely connected with each other. Sokolovsky (1951) studied the effect of pulling on the density and water content of sugar masses made for the production of grained drops; Fig. 2.18 has been compiled from this study. It can be seen that in the pulling operation, the density of the sugar mass first decreases, and later – after about 7 min – the density starts to increase again, which shows that its tubular structure is becoming more and more broken. In the pulling operation, the water content increases linearly up to about the seventh minute, then a drying process starts, and after about 10 min of pulling the initial water content is restored. Both of these phenomena show that there is an optimum pulling time (c. 6–7 min); after this, pulling will be disadvantageous. In fact, Sokolovsky measured a value of about 2.6 m/m% for the water content of the sugar mass, whereas this parameter had a minimum of 4 m/m% in

40

Confectionery and Chocolate Engineering: Principles and Applications

Increase in water content 1.0

1.5 0.5

1.0 0.9

Fig. 2.18

Increase of water content (%)

Density of sugar mass (kg/L)

Density

0

5

10 Duration of pulling (min)

0

Pulling of sugar mass [according to Sokolovsky (1951)].

the present author’s studies in the Research Laboratory of the Hungarian Confectionery Industry in 1969 (unpublished). However, the present author also found that if the reducing sugar content is lower than 6 m/m%, crystallization of the sugar mass is liable to start dramatically during the pulling operation. This may have the result that the entire amount of sucrose is crystallized in only a few minutes, and the latent heat of crystallization is liberated. Meanwhile, the sugar mass transforms into large crystalline pieces, and falls from the pulling arms. Because of the huge amount of liberated latent heat, the bulk sugar mass gets very hot, almost ‘glowing’. (This may happen in both technologies.) Example 2.7 In order to estimate the warming effect of this crystallization, let us do a calculation. First, as an approximation, the effect of the size of the batch (mass, surface area etc.) can be neglected. The specific heat capacity of sucrose is about 1.42 kJ/kg, and its latent heat of solution (positive) or of crystallization (negative) is about 18.7 kJ/kg (see Chapter 3). For a unit weight of sugar mass, the following equation is valid when the temperature t is close to 40°C:

(t − 40 ) × 1.42 − 18.7 = 0 → t = 53.16°C (The liberation of latent heat – negative enthalpy – means that the system loses heat, i.e. it is warmed; this is a thermodynamic convention.) The temperature t increases to a value of 53.16°C because of the latent heat of crystallization. This calculation supposes a homogeneous distribution of heat during crystallization; however, thermal inhomogeneities cause strong overheating, which can easily be observed as mentioned above.

Characterization of substances used in the confectionery industry

41

On the other hand, as the reducing sugar content approaches 10 m/m%, the crystallization of sucrose in the product becomes slower and slower, and at about a value of 12% crystallization becomes impossible. Since it is rather difficult to obtain exactly 6 m/m% of reducing sugar, a range of 6–8 m/m% is recommended. Before packaging, storage of the end product for 1 or 2 days in a hot (c. 40°C), wet (c. 80% RH) room is desirable because crystallization is to a certain extent stimulated by humidity and heat. The phenomenon discussed above raises the question of stability; for more details, see Chapter 18. It is important to discuss the technology for grained drops because this product represents an extreme case in sugar confectionery. Of all of the grained products (e.g. fondant), grained drops have the lowest water content and reducing sugar content. Some other examples of the effect of reducing sugar content are: • If the reducing sugar content of a jelly is 12–14 m/m%, graining is very probable (this is unambiguously a fault). • If filled hard-boiled sugar confectionery is being produced, and the sugar mass used is ‘dry’ enough (reducing sugar content c. 12–13 m/m%), the filling, if it is an aqueous solution (e.g. a fruit filling), can induce crystallization of the sugar mass cover. The final product will be soft and crisp. This may be the aim in some cases, but otherwise it qualifies as a fault. It should be emphasized that the reducing sugar content and the water content cannot on their own characterize completely the production conditions and the properties of the products; nevertheless, the relations presented in Fig. 2.17 should provide a useful orientation for preparing recipes and for making judgements about how a product will be behave if its composition is known. Let us consider some recipes in order to show how Fig. 2.17 can be applied for preparing recipes. In these recipes, the glucose syrup has the parameters dry content = 80%; DE = 40% 2.2.2.1

Drops

The parameters of the end product are: water content, 2 m/m%; reducing sugar content, 15%. 100 kg of ‘wet’ glucose syrup has 32 kg of reducing sugar and 80 kg of dry content. Therefore, 15% of reducing sugar is contained in 100 kg × (15/32) = 46.9 kg of glucose syrup, which contains 37.5 kg dry content (+11.4 kg water). The recipe is: 46.9 kg glucose syrup; 60.5 kg sugar (= 98 − 37.5); c. 30 kg water (for dissolution, which will be evaporated). YIELD: 100 kg sugar mass (2 m/m% water content). 2.2.2.2

Agar-agar jelly

The parameters of an agar-agar jelly are: water content, 23 m/m%; reducing sugar content, 12 m/m%; agar-agar content, 1.25 kg. The amount of glucose syrup is 12 kg /0.32 = 37.5 kg, which has a dry content (obtained by multiplying by 0.8) of 30 kg. The dry content of the

42

Confectionery and Chocolate Engineering: Principles and Applications

jelly (100 kg − 23 kg = 77 kg) consists of 30 kg dry glucose syrup + 1.25 kg agaragar + 45.75 kg sugar. The recipe is: 1.25 kg agar-agar mixed with 10 kg sugar is added to: 35.75 kg sugar +20 kg water for dissolution. The solution is cooked to c. 106°C. At this temperature, 37.5 kg of glucose syrup is added to the solution, which is then boiled again. Finally, the agar solution may be flavoured, coloured, dosed with starch powder, etc. 2.2.2.3

Fudge

The parameters are: Water content (W), 10 m/m%. Reducing sugar content (R), 10 m/m%. Milk dry content (M), 10 m/m%. The fondant content is to be 20 m/m%, which has the following parameters: Water content, 9 m/m%. Reducing sugar content, 7 m/m%. The task is to prepare a mass I, which does not contain fondant, and to prepare a mass II of fondant; these are then to be mixed in the proportion 4 : 1. Recipe for mass I: because the parameters of the fudge relate to the end product, the parameters of mass I are calculated as follows: W × 0.8 + 9% × 0.2 = 10%, i.e. W = 10.25% R × 0.8 + 7 × 0.2 = 10%, i.e. R = 10.75% M × 0.8 = 10%, i.e. M = 12.5% Because the dry content of the condensed milk used is 70%, the amount of condensed milk for mass I must be 12.5 kg 0.7 = 17.9 kg The amount of glucose syrup for mass I must be 10.75 kg/0.32 = 33.6 kg, which has a dry content of 33.6 kg × 0.8 = 26.9 kg. The dry content of mass I (100 kg − 10.25 kg = 89.75 kg) consists of: 12.5 kg condensed milk dry content; 26.9 kg glucose syrup dry content; 50.35 kg sugar. 17.9 kg condensed milk + 33.6 kg glucose syrup + 50.35 kg sugar are dissolved in c. 30 kg water, and this solution is boiled to c. 122°C. Recipe for mass II: if the reducing sugar content is 7%, then the amount of glucose syrup must be

Characterization of substances used in the confectionery industry

43

7 kg 0.32 = 21.9 kg The dry content of this is 21.9 kg × 0.8 = 17.5 kg Because the total dry content of the fondant mass II is 91%, the amount of sugar is

(91 − 17.5) kg = 73.5 kg Therefore, 73.5 kg sugar and 21.9 kg glucose syrup are dissolved in c. 25 kg water, and boiled to c. 124°C. Finally, 80 kg of mass I and 20 kg of mass II are mixed. Comment: The exact values of the water contents above can be established only by measuring the boiling points of the solutions. These examples of agar jelly and fudge may also be informative for calculating recipes for sugar confectioneries (e.g. marshmallow) that contain ingredients than other than sugar and glucose syrup.

2.2.3

Composition of biscuits, crackers and wafers

It is almost impossible to compile a comprehensive survey of the composition of the various biscuits, crackers etc. that exist. Table 2.5 presents some typical composition

Table 2.5 Composition of various confectionery products containing flour. Product type

Sugar (%)

Fat (%)

Honey cake

30–38 ×

Weisse Lebkuchen

25

10 ×

Elise-Makronen

33–35

22–30 ×

Nusslebkuchen

48–50

18 ×

16

17 ××

Makronen Short biscuits/Weichkeks for cutting for shaping for dosing Mürbteig (1 : 2 : 3) Spekulatius

45–48

13 ×

24

15

20 16 17 13–15 30

17 10 22 27 15 ×

56 60 53 40–45 40

7× 14 × 8× 19 15 ××

Hard/sweet biscuits Patience Zwieback Ladyfingers Wafers

10–12 40 × 7 33 1

10 0 6 0 8

70 37 70 33 28

8–10 23 17 34 × 63

0

Flour (%)

Water (%)

9.5–15 ××

50–57

40

25 ××

7–13

30 ××

Comments × Sugar/honey ×× Wheat/rice × Oil-containing seeds ×× Water/egg × Almond ×× Egg white/fruit × Oil kernels ×× Egg white/fruit × Egg white × Egg × Milk × Egg × Almond ×× Egg × Partly caramelized × Egg

44

Confectionery and Chocolate Engineering: Principles and Applications

Table 2.6 Hydrophilic character and presence of gluten skeleton for products containing flour.a Product type

Hydrophilic character

Gluten content

Honey cake Weisse Lebkuchen Elise-Makronen-Lebkuchen Nusslebkuchen Makronen Short biscuits/Weichkeks for cutting for shaping for dosing Mürbteig (1 : 2 : 3) Spekulatius Hard/sweet biscuits Patience Zwieback Ladyfingers Wafers

S M M M W

Yes No No No No

W W W W W S S S S S

No No No No No No No Yes Yes Yes

S = strong, M = medium, W = weak

a

values of various confectionery products containing flour. Table 2.6 presents the relationship between the hydrophilic character and the presence of a gluten skeleton for various products containing flour. We emphasize that the products listed in Tables 2.5 and 2.6 correspond roughly to German practice, see Ölsamen und daraus hergestellete Massen und Süsswaren (1995), and the data are merely for information. Nevertheless, these tables illustrate the fact that, out of the factors related to colloidal characteristics, the most important factors influencing the properties of these products are: • the strength of their hydrophilic character; and • the presence or absence of a developed gluten skeleton. Some additional comments are: • All these products are of hydrophilic character, i.e. none of these products, although they contain greater or lesser amounts of fat, are hydrophobic. This hydrophilic character means an O/W colloidal nature. • The development of a gluten skeleton requires a rather high proportion of water (or honey or milk, etc.) in the composition. An example recipe for honey cake according to the values in Table 2.6 is the following: 100 kg of sugar is dissolved in 40 kg of water, the solution is boiled to 106–107°C (85°R), and finally 185 kg of wheat flour is added in portions while the mixture is continuously mixed (Meyer 1949). Some other relevant references are Kengis (1951); Les codes d’usages en confiserie (1965); Gutterson (1969); Schwartz (1974); Verordening (1979); Meiners et al. (1984);

Characterization of substances used in the confectionery industry

45

Williams (1964); Richtlinie für Zuckerwaren (1992); Földes and Ravasz (1998); Manley (1998); Minifie (1999); Biscuits et gateaux, Répertoire des dénominations et recueil des usages (2001); Édesipari termékek (2003); and Real Decreto 2419/1978, 1124/1982, 1787/1982, 1137/1984 and 1810/1991.

2.3 2.3.1

Preparation of recipes Recipes and net/gross material consumption

From a practical point of view, the concepts of ‘recipe’, ‘net material consumption’ and ‘gross material consumption’ can be distinguished as follows: Recipe. This is a description of the procedure by which a product or semi-product is to be made. It itemizes the amounts of raw materials to be used, and the technological parameters, and it may also refer to the method of shaping. Net material consumption (NMC). This is defined as ‘composition of product by weight percentage’. The NMC is an important part of the product specification which does not contain any technological parameters. It is the basis for calculating the percentage composition. Gross material consumption (GMC). The GMC is the total amount (in kg) of raw materials used for manufacturing a unit (e.g. 100 kg or 1 ton) of end product. It does not contain any technological parameters. The GMC is the basis for calculation of the raw material demand of a product, material provision, stockpiling, pricing etc. At this point, it is reasonable to explain two other concepts related to the NMC, which are important in trading: Product specification. This contains product parameters, including the percentage composition, which every item of the product must comply with. These parameters may be either continuous (e.g. the value of a mass) or discrete (e.g. a number of pieces). Certificate. This refers to a certain lot, and therefore the number of the lot is the basis for identification. Certificates are issued by the quality assurance department of the producer. The parameters in the certificate, which are always concrete (i.e. discrete) values that have been measured or determined, must comply with the product specification. What is the relationship between these concepts? A recipe is the salient point for preparing a product composition (i.e. the NMC) and the GMC.

Example 2.8 Let us consider an example of the preparation of the NMC and GMC for a milk chocolate in a way which takes into account the steps of the technology, i.e. starting from batch recipes. In Example 2.2, the recipe was prepared as follows:

46

Confectionery and Chocolate Engineering: Principles and Applications

Ingredients Sugar Lecithin Whole milk powder Skimmed milk powder Cocoa mass TOTAL Cocoa butter TOTAL

Total fat 43.0 0.4 13.5 6.5 14.0 77.4 100 − 77.4 = 22.6 100

0.0 0.4 3.5 0.0 7.0 10.9 22.6 33.5

It can be seen that the recipe contains the amounts of raw materials in kilograms. If these amounts of raw materials are mixed in a closed tank, then the numbers of kilograms can be replaced by percentages because there is no loss and no growth, i.e. statements such as ‘the sugar content is 43 m/m%’ or ‘the fat content of the mass is 33.5 m/m%’ are true. However, this recipe must be related to a complete technological process, during which loss or growth cannot be avoided. A brief description of the technology is: • • • •

A mass of c. 27 m/m% fat content is made for refining. ‘Dry conching’ (c. 29 m/m% fat content)’ is carried out in a conche machine. Cocoa butter is then added (‘wet conching’, with c. 31 m/m% of cocoa butter). Finally, a 33.5 m/m% fat content is achieved by adding cocoa butter and lecithin.

Although continuous kneaders are exclusively used in modern plant, we shall do the calculation for a batch (c. 250 kg) technology because it is more instructive. The starting point is the recipe above, which can be regarded a ‘plan of the product’. Evidently, the addition of cocoa butter is necessary, since without it the fat content would be (10.9 − 0.4)/(77.4 − 0.4) = 13.64% (0.4 kg of lecithin is added at the end of conching). The first fraction of cocoa butter added is x kg, where 10.5 + x = 0.27 → x = 14.1 kg 77.0 + x (It is assumed that the blended mass can be refined.) The second fraction of cocoa butter added is y kg, where 10.5 + 14.1 + y = 0.29 → y = 2.6 kg 77 + 14.1 + y This mass is then ‘dry’ conched. The third fraction of cocoa butter added, in order to begin the ‘wet’ conching, is z kg, where 10.5 + 14.1 + 2.6 + z = 0.31 → z = 2.7 kg 77 + 14.1 + 2.6 + z The last fraction of cocoa butter added is w kg (0.4 kg of lecithin is also added to the mass in this step), where

Characterization of substances used in the confectionery industry

47

10.5 + 14.1 + 2.6 + 2.7 + 0.4 + w = 0.335 → w = 3.2 kg 77 + 14.1 + 2.6 + 2.7 + 0.4 + w [For control, x + y + z + w = (14.1 + 2.6 + 2.7 + 3.2) kg = 22.6 kg; see the recipe above.] What is the water content of this chocolate mass? The following calculation takes the water content of the raw materials into account also: Ingredients Sugar Lecithin Whole milk powder Skimmed milk powder Cocoa mass Cocoa butter TOTAL

43.0 0.4 13.5 6.5 14 22.6 100

Water content (%)

Water (kg)

0.0 0.0 4.0 4.0 1.5 0.0

0.0 0.0 0.54 0.26 0.21 0.0 1.01

Evidently, the water content of the end product that would be measured by a laboratory measurement would be different, for example 0.85 m/m%. What should be done in this case? It may be assumed that a certain amount of water will be evaporated, i.e. the yield is (100 − 1.01) kg/0.9915 = 99.84 kg [where (100 − 0.85)% = 0.9915], and the amount of water evaporated is 0.16 kg. Therefore, all amounts of raw materials should be increased by a factor of 100/99.84 = 1.0016, i.e. instead 43.0 kg of sugar, 43.0 kg × 1.0016 = 43.07 kg, is calculated, and similarly for other ingredients. The sources of possible loss include evaporation (e.g. in the case of sugar confectionery), smearing and shape defects (broken centres, etc.). On the one hand, all of these losses are dependent on the level of the technology used, and on the other hand, they are a question of economic efficiency: if the losses are too high, a more advanced technology should be used because it is too expensive to use the existing technology. A simple rule is that for every production line and for every product made by that line, a certain amount of loss may be accepted. For high-tech, continuous machinery producing chocolate mass, the evaporation of water is the single source of loss; however, for a batch technology, about 100.1–100.2 kg of input results in 100 kg of end product. Let us assume a value of 100.2 kg, so the material consumption will be as follows (i.e. with a multiplying factor of 1.002): Ingredients Sugar Lecithin Whole milk powder Skimmed milk powder Cocoa mass Cocoa butter TOTAL LOSS YIELD

43.09 0.40 13.53 6.51 14.03 22.64 100.20 0.20 100 end product (0.85 m/m% water content)

This is the typical form of the GMC of a milk chocolate.

48

Confectionery and Chocolate Engineering: Principles and Applications

Since we are assuming a batch technology, the blending of raw materials follows the size of the batch (250 kg), i.e. all of the amounts of raw materials and the additional amounts of cocoa butter are to be multiplied by 2.5 (43.09 kg × 2.5, etc.). The exact values of the sugar content, fat content etc., are checked by analytical methods, and these are included in the product specification. They are the basis of the product composition (i.e. the NMC). It can be seen clearly that losses play an important role in the operation of a plant: they cause costs that are mostly unnecessary, and environmental pollution as well. If 0.2 kg of chocolate is lost per 100 kg of end product, for 10 tons of chocolate there is 20 kg of waste!

2.3.2

Planning of material consumption

The use of the GMC is illustrated here with an example. Example 2.9 Suppose that we need to produce five products, namely three kinds of chocolate (A, B and C), and an unfilled (D) and a filled (E) hard-boiled sugar confectionery, using ten kinds of raw materials. A general scheme for planning the material consumption of the production processes can be given with the help of matrices, as follows: The material matrix M10×5 contains the gross material consumption (per unit of product); in this example, the ten rows relate to the raw materials, and the five columns relate to the different products. The production matrix P5×1 (a column matrix) contains the amounts to be made; in this example, the five rows relate to the amounts of the five products. The G-matrix G10×1 shows the possible consumption for every material according to the gross material consumption; in this example, the ten rows relate to the raw materials and the one column shows summarized amounts of the raw materials which can be used for the different products. In the form of a matrix product, G10×1 = M10×5 × P5×1

(2.10)

where ‘×’ indicates matrix multiplication. In the general case, the form of Eqn (2.10) is Gn×1 = M n× m × Pm ×1

(2.11)

where the n rows of Mn×m relate to n kinds of raw material and the m columns relate to m products, i.e. this matrix contains the GMCs of m products, and each GMC relates to n raw materials. In Pm×1, the m rows relate to the m products and show the amounts to be produced; and in Gn×1, the n rows relate to the demands for the n raw materials if m products are made. The corresponding GMCs for 100 kg of end product are represented by the material matrix M (Table 2.7).

49

Characterization of substances used in the confectionery industry

Table 2.7 Material matrix for 100 kg of end product. Raw materials

A (chocolate)

B (chocolate)

C (chocolate)

D (drops)

E (drops)

40 56 3.8 0 0.4 0.01 0 0 0

43 51.2 5.6 0 0.4 0.01 0 0 0

45.8 0 28 26 0.4 0.01 0 0 0

61.32 0 0 0 0 0 45.24 21.46 0.895

55.1 0 0 0 0 0 64.2 17.2 0.795

Sugar Cocoa mass Cocoa butter Cocoa powder Lecithin Vanillin crystals Glucose syrup Water Citric acid hydrate Fruit

0

0

0

0

4

Table 2.8 Production matrix, showing the amounts (in tonnes and in units of 100 kg) to be produced for each product. Product A B C D E

Amount (tonnes)

100 kg

50 60 40 90 80

500 600 400 900 800

Table 2.9 G-matrix, showing the total raw material demand of the production process, for each raw material. Raw materials Sugar Cocoa mass Cocoa butter Cocoa powder Lecithin Vanillin crystals Glucose syrup Water Citric acid hydrate Fruit

G=M×P 163 388 58 720 16 460 10 400 600 15 92 076 33 074 1441.5 3200

Comment: It is not recommended to manufacture chocolate without cocoa mass (see ‘C’)! The production matrix (P) (Table 2.8) shows the amounts to be produced. The demands for raw materials are represented by the G-matrix (Table 2.9). Evaluation of efficiency The opening inventory matrix O10×1 (a column matrix) contains ten rows concerning the raw materials.

50

Confectionery and Chocolate Engineering: Principles and Applications

The drawing matrix F10×1 (a column matrix) contains ten rows concerning the raw materials. The closing inventory matrix Z10×1 (a column matrix) contains ten rows concerning the raw materials. The consumption matrix C10×1 (a column matrix) contains the actual amounts of raw materials consumed. The G-matrix shows the obligations (soll values in German), and the column matrix C shows the actual values consumed (ist values in German). In the form of a matrix equation, C10×1 = O10×1 + F10×1 − Z10×1

(2.12)

The differences are calculated using the difference matrix: The difference matrix D10×1 (a column matrix) shows the differences between the prescribed amounts (G) and the actual amounts consumed for each of the raw materials (in this example there are ten rows, since ten raw materials are used). In the form of a matrix equation, D10×1 = C10×1 − G10×1

(2.13)

Or, in percentages, Efficiency ( + − ) = (C10×1 − G10×1 ) × 100% G10×1

(2.14)

A positive value of the efficiency indicates a surplus of consumption relative to the amount expected. Details of a calculation are shown in Table 2.10 for the present example. This calculation may be followed by a calculation of a value which, in the general case, has the form

Table 2.10

Calculation of efficiency. Opening inventory

Drawing

Closing inventory

G

O

F

Z

C=O−Z+F

(C − G)/G

163 388 58 720 16 460 10 400 600 15 92 076 33 074 1441.5 3200

4210 153 539 105 128 11 24 321 21 004 103 235

162 000 58 000 16 000 11 000 550 20 75 000 20 000 1500 4109

1751 −747 288 553 29 17 7462 6973 119.5 1150

164 459 58 900 16 251 10 552 649 14 91 859 34 031 1483.5 3194

0.6554949 0.3065395 −1.2697448 1.4615385 8.1666667 −6.6666667 −0.2356749 2.8935115 2.9136316 −0.1875

Raw materials

Sugar Cocoa mass Cocoa butter Cocoa powder Lecithin Vanillin crystals Glucose syrup Water Citric acid hydrate Fruit

Consumption

Efficiency (%)

Characterization of substances used in the confectionery industry

Table 2.11 Price ( /kg) T

V

0.5 2 4 1 1 10 0.5 0.001 1.5 4

51

Values of the matrix V T and the product V1×n * Dn×1. Difference D=C−G 1071 180 −209 152 49 −1 −217 957 42 −6 TOTAL

V1× n ∗ Dn×1 = ∑ vi di , i = 1, 2, … , n

Materials V*D 535.5 360 −836 152 49 −10 −108.5 0.957 63 −24

Sugar Cocoa mass Cocoa butter Cocoa powder Lecithin Vanillin crystals Glucose syrup Water Citric acid hydrate Fruit 181.957 = Σvidi

(2.15)

where V1×n is a row matrix with n rows and one column, and contains the unit prices of n materials, and * indicates a scalar product. The difference matrix Dn×1 = Cn×1 − Gn×1 shows the difference in consumption for each of the n raw materials. Since V1×n is a row matrix and Dn×1 is a column matrix, their product is a scalar (a number) – in this case, for example, it may be expressed in euros or US dollars. [The product of a row matrix with a column matrix (in this order!) is a scalar.] The values of the matrix VT and the product V1×n * Dn×1 for the present example are presented in Table 2.11. These mean a cost surplus of about 182. (The prices in the matrix V are intended to serve as examples only.)

Chapter 3

Engineering properties of foods

Contents 3.1 3.2

3.3

3.4

3.5

3.6

3.7

3.8

Introduction Density 3.2.1 Solids and powdered solids 3.2.2 Particle density 3.2.3 Bulk density and porosity 3.2.4 Loose bulk density 3.2.5 Dispersions of various kinds, and solutions Fundamental functions of thermodynamics 3.3.1 Internal energy 3.3.2 Enthalpy 3.3.3 Specific heat capacity calculations Latent heat and heat of reaction 3.4.1 Latent heat and free enthalpy 3.4.2 Phase transitions Thermal conductivity 3.5.1 First Fourier equation 3.5.2 Heterogeneous materials 3.5.3 Liquid foods 3.5.4 Liquids containing suspended particles 3.5.5 Gases Thermal diffusivity and Prandtl number 3.6.1 Second Fourier equation 3.6.2 Liquids and gases 3.6.3 Prandtl number Mass diffusivity and Schmidt number 3.7.1 Law of mass diffusion (Fick’s first law) 3.7.2 Mutual mass diffusion 3.7.3 Mass diffusion in liquids 3.7.4 Temperature dependence of diffusion 3.7.5 Mass diffusion in complex solid foodstuffs 3.7.6 Schmidt number Dielectric properties 3.8.1 Radio frequency and microwave heating 3.8.2 Power absorption – the Lambert–Beer law 3.8.3 Microwave and radio frequency generators 3.8.4 Analytical applications

Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

53 53 54 54 55 55 56 56 56 58 58 62 62 63 66 66 67 67 68 68 69 69 69 70 71 71 72 72 73 74 75 76 76 77 78 81

Engineering properties of foods

3.9

Electrical conductivity 3.9.1 Ohm’s law 3.9.2 Electrical conductivity of metals and electrolytes; the Wiedemann–Franz law and Faraday’s law 3.9.3 Electrical conductivity of materials used in confectionery 3.9.4 Ohmic heating technology 3.10 Infrared absorption properties 3.11 Physical characteristics of food powders 3.11.1 Classification of food powders 3.11.2 Surface activity 3.11.3 Effect of moisture content and anticaking agents 3.11.4 Mechanical strength, dust formation and explosibility index 3.11.5 Compressibility 3.11.6 Angle of repose 3.11.7 Flowability 3.11.8 Caking 3.11.9 Effect of anticaking agents 3.11.10 Segregation Further reading

3.1

53 81 81 82 83 83 85 86 86 87 87 88 89 91 91 92 95 95 96

Introduction

It is important to study the changes in physical properties of foodstuffs during unit operations because some properties (often called engineering properties) influence heat and mass transport; such properties include, for instance, the density, specific heat capacity and thermal conductivity/diffusivity; see Hallström et al. (1988) or Bálint (2001). Knowledge of the physical properties of foods is required to perform the various engineering calculations that are involved in the design of food-producing machinery and storage and refrigeration equipment, and for estimating process times for the refrigeration, freezing, heating and drying of foods. The thermal properties of foods are strongly dependent upon chemical composition and temperature, and there are a multitude of food and raw-material items available. It is difficult to generate a database of experimentally determined physical properties for all possible conditions and compositions of foods. The most viable option is to predict the physical properties of foods using mathematical models that account for the effects of chemical composition and temperature. These properties of foods have been discussed in detail by, among others, Rha (1975), Loncin and Merson (1979, pp. 24–30), Szczesniak (1983), Hallström et al. (1988, Chapter 2) and Fricke and Becker (2001). Appendix 1 gives data on the materials used in and made by the confectionery industry, classified according to the kinds of materials.

3.2

Density

The definition of the density ρ is

ρ=

mass ( kg ) volume ( m3 )

(3.1)

54

Confectionery and Chocolate Engineering: Principles and Applications

In the simplest case, the structure of the substance is homogeneous and continuous, in which case this definition does not need any further expansion. However, the structures of foods can be of very different kinds, such as solids, powdered solids, solutions, and dispersions of solids, fluids or gases, and therefore the composition is not homogeneous. Since the density of a food is dependent on its composition, several different definitions of density are applicable. In spite of all this, most foods and particles have a similar solid density of about 1.4–1.5 kg/m3, depending on the moisture content. Let us consider the various kinds of density.

3.2.1

Solids and powdered solids

The solid density is the density of the solid material from which the particles of a material are made, disregarding any internal pores. (This definition is valid for all substances and foods, whether they are porous or not, of course.) This similarity in density is due mainly to the similar densities of the main ingredients (Table 3.1). Notable exceptions are salt-based and fat-rich powders, whose density may vary considerably according to composition.

3.2.2

Particle density

Another density is the particle density, which is defined as follows: Particle density =

actual mass of particles actual volume of particles

(3.2)

This parameter takes account of the existence of internal pores and therefore can be considered as a measure of the true density of the particles. (This parameter is more relevant to situations where the relationship between particle weight and interparticle forces is of concern.) However, this parameter does not provide any information regarding the shapes of the internal pores and their positions in the particle structure. The latter can have distinctly different forms, whose character depends on the type of process and the conditions under which the particles were formed. Table 3.1 Densities of some ingredients [reproduced from Peleg (1983), with kind permission of Springer Science and Business Media]. Material Glucose Sucrose Starch Cellulose Protein (globular) Fat Salt Citric acid Water

Density (kg/m3) 1.56 1.59 1.5 1.27–1.61 1.4 0.9–0.95 2.16 1.54 1

Engineering properties of foods

3.2.3

55

Bulk density and porosity

The bulk (or apparent) density is the mass of particles that occupies a unit volume of a bed. It is usually determined by weighing a container of known volume and dividing the net weight of the powder by the container’s volume. The porosity is the fraction of the volume not occupied by particles or solid material, and therefore can be expressed as either Total porosity = 1 −

bulk density solid density

(3.3)

or Interparticle porosity = 1 −

bulk density particle density

(3.4)

Because powders are compressible, their bulk density is usually given with an additional specifier, i.e. as the loose bulk density (as poured), the tapped bulk density (after vibration) or the compact density (after compression). Another way to express the bulk density is in the form of a fraction of the solid density of the particles, which is sometimes referred to as the ‘theoretical density’. This expression, as well as the use of porosity instead of density, enables and facilitates a unified treatment and meaningful comparison of powders that may have considerably different solid or particle densities.

3.2.4

Loose bulk density

Approximate values of the loose bulk density of a variety of food powders are listed in Table 3.2, which shows that with very few exceptions food powders have apparent densities in the range 0.3–0.8 g/cm3. As previously mentioned, the solid density of most food powders is about l.4 and therefore these values are an indication that food powders have high porosity (i.e. 40–80%), which can be internal, external or both. There are many published theoretical and experimental studies of porosity as a function of the particle size, distribution and shape. Mostly they pertain to free-flowing powders or models (e.g. steel shot and metal powders), where porosity can be treated as primarily due to geometrical and statistical factors only; see Gray (1968) and McGeary (1967–1970). Even though the porosity can vary considerably in such cases, depending on such factors as the concentration of fines, it is still evident that the exceedingly low density of food powders cannot be explained by geometrical considerations only. Most food powders are known to be cohesive (Carr 1976), which means that their attractive interparticle forces are significant relative to the weight of the particles. Since the bulk density of food powders depends on the combined effects of interrelated factors, namely the intensity of the attractive interparticle forces, the particle size and the number of contact points (Rumpf 1961), a change in any one of the characteristics of a powder may result in a significant change in the bulk density, with a magnitude that cannot always be anticipated; for details, see Peleg et al. (1982).

56

Confectionery and Chocolate Engineering: Principles and Applications

Table 3.2 Approximate bulk density and moisture content of various food powders [reproduced from Peleg (1983, Table 10.1), with kind permission of Springer Science and Business Media]. Powder Baby formula Cocoa Coffee (ground and roasted) Coffee (instant) Coffee creamer Cornmeal Cornstarch Egg (whole) Gelatin (ground) Microcrystalline cellulose Milk Oatmeal Onion (powdered) Salt (granulated) Salt (powdered) Soy protein (precipitated) Sugar (granulated) Sugar (powdered) Wheat flour Wheat (whole) Whey Yeast (active dry baker’s) Yeast (active dry wine)

Bulk density (kg/m3)

Moisture content (%)

0.40 0.48a 0.33c 0.33c 0.47 0.66a 0.56a 0.34a 0.68 0.68 0.61a 0.43a 0.51 0.96a 0.95 0.28 0.80 0.48 0.48 0.801 0.56 0.52 0.82

2.5 3–5b 7c 2.5c 3 12b 12b 2–4b 12 6 2–4 8b 1–4 0.2b 0.2b 2–3 0.52 0.52 12b 12b 4.5b 8 8

a

Data from Carr (1976). Data from Watt and Merrill (1975). Data from Schwartzberg (1982).

b c

3.2.5

Dispersions of various kinds, and solutions

Most solid foodstuffs contain gas, which is in general a mixture of air and water vapour. The gas is contained in capillaries, which can be open or completely closed. If the diameter of some of the pores is less than 10−7 m, the material is said to be capillary porous. Tables 3.7 and 3.8 later in this chapter show data and power series for approximate calculation of the densities of the principal components of foods, as a function of temperature. The basis of the calculations is the fact that density is an additive property:

ρ = ∑ xi ρi

(3.5)

where ρ is the density of a substance that is a mixture of n components, labelled by i = 1, 2, … , n; ρi is the density of the i-th component (in kg/m3), and xi is the mass fraction of the i-th component (in kg/kg).

3.3 3.3.1

Fundamental functions of thermodynamics Internal energy

(In this section, capital letters denote molar quantities, except for T (temperature).) The internal energy U is the total microscopic energy of a system. It is related to the molecular structure and the degree of molecular activity in the system. In this respect, it

Engineering properties of foods

57

Table 3.3 Some fundamental functions of thermodynamics. Fundamental function

Molar relationship

Enthalpy Free energy Free enthalpy Energy Enthalpy

H = U + pV = TS + μN F = U − TS = −pV + μN G = U − TS + pV = μN U = F + TS H = G + TS

can be considered as the sum of the potential and kinetic energy of the system, but at the molecular scale such as the energy used for chemical bonds between atoms or molecules. It is defined by U = TS − pV + μ N

(3.6)

where S = molar entropy, p = pressure, V = molar volume, μ = chemical potential and N = molar number. From this relationship, the so-called fundamental functions of thermodynamics can be derived; the functions most often used are summarized in Table 3.3. The enthalpy (ΔH) is of great practical importance since it equals the change of internal energy of the system plus the work provided to its surroundings. It can therefore be used to calculate the heat absorbed or released during a reversible isobaric (constant pressure) reaction. The usual name of the function F is the Helmholtz free energy, and the Gibbs energy (or free enthalpy, G) is the maximum quantity of energy that can be released as nonexpansion (process initiating) work from a closed system during an isothermal and isobaric reaction. To reach this maximum it has to be a reversible (quasistatic) process. In food practice, the importance of non-molar quantities cannot be overestimated, because the molar amounts of foods are usually unknown. Therefore, in the following discussion we shall often ignore the convention of using capital letters to denote molar quantities only. We emphasize that thermodynamic relationships that are valid in general are valid also for molar quantities, but the converse of this statement cannot be assumed to be true. The molar specific heat capacity at constant volume (CV ≡ (∂U/∂T)V) and at constant pressure (Cp) can be derived (if p = constant) from the equation H = U + pV → dH = dU (T ; V ) + p dV ⎛ dH ⎞ ≡ C = ⎛ ∂U ⎞ + ⎛ ∂U ⎞ p ⎝ dT ⎠ p ⎝ ∂T ⎠ V ⎝ ∂V ⎠ T

(3.7)

⎛ ∂ V ⎞ + p ⎛ ∂V ⎞ ⎝ ∂T ⎠ p ⎝ ∂T ⎠ p

⎡ ∂U ⎞ ⎤ ∂V ⎞ + p⎥ ⎛ = CV + ⎢⎛ ⎣ ⎝ ∂V ⎠ T ⎦ ⎝ ∂T ⎠ p

(3.8)

For an ideal gas, pV = RT, i.e. (∂U/∂V)T = 0 and (∂V/∂T)p = R/p, where R = 8.31434 J/ mol K, the molar gas constant. Consequently, for an ideal gas, C p = CV + R

(3.9)

For real gases, the appropriate alternative gas law (the van der Waals equation, etc.) is the basis of calculations. The relationship

58

Confectionery and Chocolate Engineering: Principles and Applications

c p > cV

(3.10)

is a general rule for all substances, where cp and cV are the specific heat capacities per unit mass at constant pressure and volume, respectively. For gases, cp and cV are of the same magnitude. The difference between the two types of specific heat capacity for homogeneous solids and liquids can be calculated on the basis of the second law of thermodynamics: C p − CV = T ( ∂p ∂T )V ( ∂V ∂T ) p = TV ° α 2 χ

(3.11)

where V° is the volume for a given standard condition, α = (1/V°)(∂V/∂T)p is the coefficient of thermal expansion (>0) and χ = −(1/V°)(∂V/∂p)T is the coefficient of compressibility (>0). This difference between the two types of specific heat capacity corresponds to the work of volume, and for solids and liquids this is only a small percentage of Cp (or cp), in contrast to the case of gases, for which the work of volume can be important. The relationships between the two types of specific heat capacity are important because a practical determination of cV cannot be carried out: heating at constant volume is not possible. In practice, the specific heat capacity at constant pressure is always used.

3.3.2

Enthalpy

The enthalpy can be calculated as an integral of the specific heat capacity over a given interval of temperature: h=

T2

∫ c p dT

(3.12)

T1

Normally, cp is almost constant within the temperature region of interest (in which T is usually measured in °C, and usually −20°C < T < 100°C), and therefore the above equation may be approximated as h = c p (T2 − T1 )

(3.13)

This supposition of constant specific heat capacity is reflected in Tables 3.7 and 3.8 and Eqn (3.19), which give average values of the density, specific heat capacity and thermal conductivity for various food components (water, carbohydrate, protein, fat, air, ice and inorganic minerals). If the composition of a foodstuff is known, an approximate calculation of these material parameters can be done.

3.3.3

Specific heat capacity calculations

The specific heat capacities of foodstuffs depend very much on the composition. The specific heat capacity of water is 4.18 kJ/kg K, while that of the solid constituents is much lower, 1–2 kJ/kg K (Table 3.4). Heldman (1975) suggested the following formula, together with numerical values, to estimate the specific heat capacities of foodstuffs based on composition: c p = ∑ xi c pi

(3.14)

59

Engineering properties of foods

Table 3.4 Thermal properties of food constituents (reproduced with kind permission of Springer Science and Business Media).

Component

Mass concentration (kg/kg)

Water Carbohydrate Protein Fat Air Ice Inorganic minerals

cw cc cpr cf ca ci cm

Density (kg/m3)

Specific heat capacity (kJ/kg K)

Thermal conductivity (W/m K)

1000 1550 1380 930 1.24 917 2400

4.182 1.42 1.55 1.67 1.00 2.11 0.84

0.60 0.58 0.20 0.18 0.025

Table 3.5 Calculation of specific heat capacity of milk chocolate and dark chocolate.a Calculation using the data in Table 3.4 Milk chocolate {8.301}

Water Ash Protein Fat Carbohydrate

Dark chocolate {8.302}

cp

Ingredients (g/kg)

cp (J/g K)

Ingredients (g/kg)

cp (J/g K)

4.182 0.84 1.55 1.67 1.42

8 19 86 348 539

33.456 15.96 133.3 581.16 765.38

10 12 62 310 606

41.82 10.08 96.1 517.7 860.52

Total for composition

1529.256 J/kg K

1526.22 J/kg K

Calculation using Eqns (3.17) and (3.18)

cp = 1.67 + 2.5xw (Eqn 3.17) cp = 1.40 + 3.2xw (Eqn 3.18)

Milk chocolate {8.301}

Dark chocolate {8.302}

1.69 kJ/kg K 1.423 kJ/kg K

1.695 kJ/kg K 1.432 kJ/kg K

a

Source of compositions: Livsmedelstabeller – Energi och vissa Näringsämmen (1978). The numbers in braces { } are the product numbers used in that publication.

where xi is the mass concentration of the i-th constituent, i.e. the proportion by mass. Equation (3.14) expresses the fact that specific heat capacity is an additive property, but calculations based on the data in Table 3.4 do not differentiate between the various kinds of carbohydrates, fats, etc. Thus the fact, for example, that aqueous solutions of 80 m/m% sucrose and 80 m/m% corn syrup are different cannot be taken into account. However, if a change of phase or some other type of transformation takes place, the enthalpy of the system changes because of latent heat. In this case a so-called apparent specific heat capacity is measured; see Section 3.4.2 below. Some calculations performed using Table 3.4 are summarized in Tables 3.5–3.7. An approximate expression for foodstuffs containing mainly water is c p = 4.18xw

(3.15)

c p = 4.18xw + 2 xd

(3.16)

or

60

Confectionery and Chocolate Engineering: Principles and Applications

Table 3.6

Calculation of specific heat capacity of orange marmalade and almond paste.a

Calculation using the data in Table 3.4 Orange marmalade {8.100}

Water Ash Protein Fat Carbohydrate

Almond paste {8.350}

cp

Ingredients (g/kg)

cp (J/g K)

Ingredients (g/kg)

cp (J/g K)

4.182 0.84 1.55 1.67 1.42

408 2 0 0 590

1706.256 1.68 0 0 837.8

90 13 98 229 570

376.38 10.92 151.9 382.43 809.4

Total for composition

2545.736 J/kg K

1731.03 J/kg K

Orange marmalade {8.100}

Almond paste {8.350}

2.69 kJ/kg K 2.706 kJ/kg K

1.895 kJ/kg K 1.688 kJ/kg K

Calculation using Eqns (3.17) and (3.18)

cp = 1.67 + 2.5xw (Eqn 3.17) cp = 1.40 + 3.2xw (Eqn 3.18) a

Source of compositions: Livsmedelstabeller – Energi och vissa Näringsämmen (1978). The numbers in braces { } are the product numbers used in that publication.

Table 3.7

Calculation of specific heat capacity of cocoa nibs and sweets.a

Calculation using the data in Table 3.4 Cocoa nibs cp Water Ash Protein Fat Carbohydrate

Ingredients (g/kg)

4.182 0.84 1.55 1.67 1.42

Total for composition

27.5 34 23.5 525 390

Sweets cp (J/g K)

Ingredients (g/kg)

115.005 28.56 36.425 876.75 553.8

100 0 0 0 900

1610.54 J/kg K

cp (J/g K) 418.2 0 0 0 1278 1696.2 J/kg K

Calculation using Eqns (3.17) and (3.18)

cp = 1.67 + 2.5xw (Eqn 3.17) cp = 1.40 + 3.2xw (Eqn 3.18)

Cocoa nibs

Sweets

1.739 kJ/kg K 1.488 kJ/kg K

1.92 kJ/kg K 1.72 kJ/kg K

Source of compositions: Minifie (1999, pp. 21–22) for cocoa nibs. The composition for sweets is a typical composition of soft confectionery provided by the present author.

where xd is the dry matter content of the material (xd = 1 − xw). For fish and meat, with xw < 0.25, and for fruit and vegetables, with xw > 0.50, the following formula was suggested by Andersen and Risum (1982): c p = 1.67 + 2.5xw

(3.17)

For sorghum and other cereals with a low water content, the following equation may be used:

Engineering properties of foods

c p = 1.40 + 3.2 xw

61 (3.18)

Let us calculate some specific heat capacity values using Table 3.4 and Eqns (3.17) and (3.18). The specific heat capacity, similarly to the density, thermal conductivity and thermal diffusivity, is usually expressed as a series in T (in K or °C); see Tables 3.8 and 3.9. The values for the specific heat capacity of milk chocolate (No. 8.301; see Table 3.5) shown in Table 3.10 were calculated according to Tables 3.8 and 3.9. If, instead, we calculate the specific heat capacity for milk chocolate of the same composition by the methods described earlier, where the temperature is not taken into account, the results are: using the data in Table 3.4, 1529 J/kg K; using Eqn (3.17), 1690 J/kg K; using Eqn (3.18), 1423 J/kg K. For unfrozen foods, the following simple approximation for the specific heat capacity was given by Chen (1985): c ( J kg K ) = 4190 − 2300w (s ) − 628w (s )

2

(3.19)

where w(s) is the mass fraction of the solids in the food. Using this equation, the specific heat capacities (in J/kg K) shown in Table 3.10 were obtained for the various foods listed in Tables 3.5–3.7. Table 3.8 Thermal-property equations for food components (−40°C ≤ t ≤ 150°C) [reproduced with permission from Choi and Okos (1986); Copyright Elsevier]. Thermal property and food component Thermal conductivity [W/(m K)] Protein Fat Carbohydrate Fibre Ash Thermal diffusivity (m2/s) Protein Fat Carbohydrate Fibre Ash Density (kg/m3) Protein Fat Carbohydrate Fibre Ash Specific heat capacity [J/(kg K )] Protein Fat Carbohydrate Fibre Ash

Thermal-property model k = 1.7881 × 10−1 + k = 1.8071 × 10−1 + k = 2.0141 × 10−1 + k = 1.8331 × 10−1 + k = 3.2962 × 10−1 +

1.1958 2.7604 1.3874 1.2497 1.4011

× × × × ×

l0−3 t − 2.7178 × 10−6 t2 10−3 t − 1.7749 × l0−7 t2 10−3 t − 4.3312 × 10−6 t2 10−3 t − 3.1683 × 10−6 t2 10−3 t − 2.9069 × 10−6 t20

a = 6.8714 × 10−8 + a = 9.8777 × 10−8 − a = 8.0842 × 10−8 + a = 7.3976 × 10−8 + a = 1.2461 × 10−7 +

4.7578 1.2569 5.3052 5.1902 3.7321

× × × × ×

10−10 t − 10−10 t − 10−10 t − 10−10 t − 10−10 t −

ρ = 1.3299 × 103 ρ = 9.2559 × 102 ρ = 1.5991 × 103 ρ = 1.3115 × 103 ρ = 2.4238 × 103

5.1840 4.1757 3.1046 3.6589 2.8063

× × × × ×

10−1 t 10−1 t 10−1 t 10−1 t 10−1 t

cp cp cp cp cp

= = = = =

2.0082 1.9842 1.5488 1.8459 1.0926

× × × × ×

103 103 103 103 103

− − − − −

+ 1.2089t + 1.4733t + 1.9625t + 1.8306t + 1.8896t

− − − − −

1.3129 4.8008 5.9399 4.6509 3.6817

1.4646 3.8286 2.3218 2.2202 1.2244

× × × × ×

10−3 t2 10−3 t2 10−3 t2 10−3 t2 10−3t2

× × × × ×

10−12 t2 10−14 t2 10−12 t2 10−12t2 10−12 t2

62

Confectionery and Chocolate Engineering: Principles and Applications

Table 3.9 Thermal-property equations for water and ice (−40°C ≤ t ≤ 150°C) [reproduced with permission from Choi and Okos (1986); Copyright Elsevier]. Thermal property and food component Water Thermal conductivity [W/(m K)] Thermal diffusivity (m2/s) Density (kg/m3) Specific heat capacity [J/(kg K)]a Specific heat capacity [J/(kg K)]b Ice Thermal conductivity [W/(m K)] Thermal diffusivity (m2/s) Density (kg/m3) Specific heat capacity [J/(kg K)] a b

Thermal-property model kw = 5.7109 × 10−1 + 1.7625 × 10−3 t − 6.7036 × 10−6 t2 aw = 1.3168 × 10−7 + 6.2477 × 10−10 t − 2.4022 × 10−12 t2 ρw = 9.9718 × 102 + 3.1439 × 10−3 t − 3.7574 × 10−3 t2 cp = 4.0817 × 103 − 5.3062t + 9.9516 × 10−1 t2 cp = 4.1762 × 103 − 9.0864 × 10−2 t + 5.4731 × 10−3 t2 kice = 2.2196 − 6.2489 × 10−3 t + 1.0154 × 10−4 t2 aice = 1.1756 × 10−6 − 6.0833 × 10−9 t + 9.5037 × 10−11 t2 ρice = 9.1689 × 102 − 1.3071 × 10−1 t cp ice = 2.0623 × 103 + 6.0769t

For the temperature range −40°C to 0°C. For the temperature range 0–150°C.

Table 3.10 Specific heat capacity of milk chocolate No. 8.301 calculated according to the thermal-property model of Choi and Okos (1986). C(t) (J/kg K)

Temperature (°C) 0 10 20 35

1752.179 1768.779 1784.388 1805.942

Table 3.11 Specific heat capacity (J/kg K) of several materials used or produced by the confectionery industry.

3.4 3.4.1

Product

w(s)

c (Chen)

Milk chocolate Bitter chocolate Orange marmalade Almond paste Cocoa nibs Sweets

0.992 0.958 0.592 0.91 0.973 0.9

1295.352 1434.451 2698.106 1623.757 1373.607 1662.188

Latent heat and heat of reaction Latent heat and free enthalpy

A definition of the heat of transformation can be given by studying the general stoichiometric relationship of a transformation,

∑ rA M A → ∑ rB M B

(3.20)

⎛ ∂H ⎞ LAB = ⎜ = ( ∑ rB H B − ∑ rA H A ) ⎝ ∂ξ ⎟⎠ p,T

(3.21)

or

Engineering properties of foods

63

where LAB is the thermal effect of the transformation under isobaric and isothermal conditions when a substance A transforms to a substance B, ξ(0 → 1) is the degree of change, rA and rB are the stoichiometric values in the stoichiometric equation for the relationship, and HA and HB are the molar enthalpies of A and B, respectively. Equation (3.21) can be regarded as the definition of the heat of a phase transition. If the system studied consists of n components of chemical activity μ, such that N μ = ∑ Niμi , i = 1, 2, … , n

(3.22)

where N is the number of molecules and μi is the chemical activity (in J/kg) of the i-th type of molecule, of number Ni (and where N = Σ Ni), then the free enthalpy function g (in J/kg) introduced by Gibbs is g = h − Ts = N μ

(3.23)

where s is the entropy (J/kg K) and T is the temperature (K). Under isothermal and isobaric conditions (i.e. T = constant and p = constant), Δg = Δμ

(3.24)

where Δμ is the change in the chemical activity of the system. The name ‘free enthalpy’ (see Eqn 3.23) can be understood since G = H − Ts. H is the total enthalpy and Ts is the enthalpy ‘bound’ to the system. The difference between these two terms is therefore the free (available) enthalpy of the system under isolated conditions, which can be released as a result of chemical reactions. The ‘bound’ energy relies in the arrangement of molecules and can be released when the structure of the material is compromised as during a phase change in chocolate when fat crystals melt. It can be derived from the potential functions of thermodynamics that ⎛ ∂s ⎞ ⎜⎝ ⎟ = λi T ∂ηi ⎠ T ,η

(3.25)

where ηi is the concentration of the substance the latent heat (λi) of which is involved in a phase transition, η is the concentration of all other substances in the system and T is the temperature of the phase transition. The term ‘latent heat’, although old-fashioned, is reasonable since during a phase transition the temperature remains constant (an ‘isothermal condition’), i.e. heat absorption is not accompanied by an increase in temperature, because during the phase transition a structural transformation (e.g. ice → water at 0°C) takes place. According to the convention used in thermodynamics, a process is endothermic (LAB is positive) if the system absorbs heat; in the opposite case it is exothermic, i.e. the system loses (produces) heat; see Lund (1983).

3.4.2

Phase transitions

In the traditional classification scheme of Ehrenfest [see Fényes et al. (1971)], phase transitions are divided into two broad categories. First-order phase transitions are those that involve a latent heat. During such a transition, a system either absorbs or releases a fixed (and typically large) amount of energy. Because energy cannot be instantaneously transferred between the system and its

64

Confectionery and Chocolate Engineering: Principles and Applications

environment, first-order transitions are associated with ‘mixed-phase regimes’, in which some parts of the system have completed the transition and others have not. This phenomenon is familiar to anyone who has boiled a pan of water: the water does not instantly turn into gas, but forms a turbulent mixture of water and water vapour bubbles. Mixedphase systems are difficult to study, because their dynamics are violent and hard to control. According to IUPAC (1997), a first-order phase transition is a transition in which the molar Gibbs energies (G, the free enthalpy) or molar Helmholtz energies (F, the free energy) of the two phases (or the chemical potentials of all components in the two phases) are equal at the transition temperature, but their first derivatives with respect to temperature and pressure are discontinuous at the transition point, i.e. ⎛ ∂G ⎞ = ⎛ ∂ λ i ⎞ = λ i → ∞ ⎝ ∂T ⎠ p ⎝ ∂T ⎠ p 0 where T is constant, i.e. ∂T = 0. At this temperature, dissimilar phases coexist and can be transformed into one another by a change in a field variable such as the pressure, temperature, magnetic field or electric field. The second class of phase transitions is that of continuous phase transitions, also called second-order phase transitions. These have no associated latent heat. Examples are the ferromagnetic transition and the superfluid transition – but these are not interesting from the point of view of our study. Several transitions are known to be infinite-order phase transitions. For further details, see Fényes (1971). The topic of phase transitions is a current research area in physics and mathematics. The definition of the heat of transformation in Eqn (3.21) can also be written in the form

∑ rA H A + ΔH = ∑ rB HB

(3.26)

where ΔH is the change of enthalpy in a chemical reaction or the latent heat if Eqn (3.26) relates only to a phase transition. In practice, the process takes place over a temperature range from t1 to t2, and one or more components of the substance may partially melt or evaporate, etc., i.e. phase transitions may occur to some extent without any chemical reaction. For such a process, the enthalpy balance is

∑ ai cai = ∑ b j cbj

(3.27)

where ai and bj are the amounts of the output and input substances, respectively, and cai and cbj are the specific heat capacities of the output and input substances, respectively. In such cases the specific heat capacity of the output substances cbi apparently incorporates latent heats too. Therefore, such specific heat capacities should be called ‘apparent specific heat capacities’. A deeper thermal analysis of the process needs a determination of the latent heat separately. Let us consider the types of phase transitions that are important from the viewpoint of confectionery manufacture. These are, in general, evaporation/condensation, melting/ solidification, and modification of the crystal structure of fats and oils (e.g. cocoa butter) and of lactose.

Engineering properties of foods

65

In confectionery practice, the solubility of carbohydrates (sucrose, etc.) in water and the solid-phase content of the fats used also play an important role. 3.4.2.1

Solution–evaporation–crystallization

The heat of vaporization of water is 2256 kJ/kg at l00°C and 101.3 kPa. Other volatile substances in food are normally of minor importance when one is calculating the heat of vaporization. For liquid foods, the boiling point is somewhat higher than 100°C, depending on the concentration of solids. For well-defined solutions, the elevation of the boiling point is proportional to the molar concentration of the solute. This topic will be discussed in detail for the solutions used in confectionery manufacture in Chapters 8 and 9. Table 3.12 shows the (positive) latent heat of solution of some carbohydrates and polyalcohols (bulk sweeteners) that are important in confectionery practice. Their cooling effect is a consequence of the positivity of the latent heat of solution. For further details relating to bulk sweeteners, see Albert et al. (1980). In the mouth, these carbohydrates exert a cooling effect according to the following equation, which treats the mouth as an adiabatic (closed) system: Δh + c p ΔT = 0

(3.28)

where Δh is the latent heat (in J/kg) of the carbohydrate to be dissolved in the mouth, cp is the specific heat capacity of saliva and ΔT is the change in temperature due to the effect of the dissolution of the carbohydrate. Because Δh > 0, ΔT < 0 (cp > 0). The cooling effect is strongest in the case of xylitol; the smallest effect, which can be imperceptible, is exerted by sucrose. Although the supposition of an adiabatic system is only a rough approximation, it characterizes the conditions correctly. Another important consequence of the positive latent heat of these substances is that the process of dissolution can be made to take place more effectively by warming. The latent heat of crystallization is equal to the latent heat of solution, but the sign is negative. This means that the crystallization of these carbohydrates is an exothermic process, and crystallization can be induced by cooling; this is done, for example, in the manufacture of fondant mass. The exothermic nature of the crystallization of sucrose can be observed well during the operation of pulling when satin bonbons of grained structure are produced: the arms of the pulling machine must cooled to prevent sticking of the invert sugar content of the sugar mass. Moreover, if the reducing sugar content of the sugar mass is too low Table 3.12 Latent heat of some carbohydrates used in confectionery practice. Substance

Latent heat (kJ/kg)

Sucrose Fructose Sorbitol Xylitol Maltitol Isomalt Lactitol

18.7 48.6 82.3 125.3 46.8 39.3 26.2

66

Confectionery and Chocolate Engineering: Principles and Applications

Table 3.13

Temperatures and enthalpies (ΔH) of exothermic reactions of carbohydrates during pyrolysis.

Carbohydrate

Onset temperature (°C)

Peak temperature (°C)

Enthalpy (kJ/kg)

170–200 190–220 160–200

195–230 215–245 200–245

620–780 600–800 630–720

Monosaccharides Disaccharides Polysaccharides

(below c. 4%), a very rapid crystallization starts, and as a consequence the sugar mass transforms into large crystals while very rapidly growing warm. For further details, see Chapter 10. 3.4.2.2

Chemical reactions

If a chemical reaction takes place, then ΔH relates to the change of enthalpy called the ‘heat of reaction’. In confectionery practice, the pyrolysis of carbohydrates plays an important role, for example in the melting of sugar. Raemy and Schweizer (1982, 1986) have carried out extensive calorimetric investigations of the thermal degradation of a range of sugars and polysaccharides. Under the experimental conditions employed in their studies, the decomposition reactions yielded exothermic transitions; temperature and enthalpy values are given in Table 3.13. For example, in the case of sugar, pyrolysis starts at 190°C and culminates at 215°C. The pyrolysis of sucrose will be discussed in detail in Chapter 16. Intense exothermic effects were observed by Raemy (1981), Raemy and Lambelet (1982) and Raemy and Loliger (1982) with foods of high carbohydrate content (>60%), such as coffee and chicory products and a range of cereals and oilseeds; the roasting and carbonization of these materials were linked to the pyrolytic exothermic events. Concerning the heat of reaction, we should mention that the energy content of foods from the point of view of nutrition is equal to their heat of combustion. This nutritional topic, however, is beyond the scope of this book.

3.5 3.5.1

Thermal conductivity First Fourier equation

In a steady-state situation, the rates of heat transfer in every section of a rod conducting heat are equal. Considering an element of the rod of differential length dx, the rate of heat transfer through this element is given by the first Fourier equation, dQ dT = −λ A dt dy

(3.29)

where y is the distance in the direction of heat transfer (m), Q is the amount of heat transferred (J) (dQ/dt is in units of W), t is the time (s), A is the area at right angles to the direction of heat transfer (m2), λ is the thermal conductivity of the material (W/m K) and T is the temperature (K).

67

Engineering properties of foods

The steady-state situation is equivalent to dQ dT = constant and =0 dt dt

3.5.2

(3.30)

Heterogeneous materials

In food and confectionery practice, the materials processed are mainly heterogeneous, although there are a few exceptions, for example crystalline sucrose, salt and anhydrous citric acid. The simplest case is of a food consisting of two components. If the thermal conductivities in a two-component material are λ1 and λ2, the total ‘apparent’ or ‘effective’ conductivity depends on the heat flow direction. If the flow of heat is parallel to the layers of material, the thermal conductivity is given by

λ = λ1 (1 − x ) + λ2 x

(3.31)

where x is the volume fraction of material 2. On the other hand, if the heat flow is perpendicular to the layers of material, the thermal conductivity is given by 1 1− x x = + λ⊥ λ1 λ2

(3.32)

or

λ⊥ =

λ1λ2 xλ1 + (1 − x ) λ2

(3.33)

Evidently, Eqns (3.31) and (3.32) follow Kirchhoff’s laws. If the materials are not oriented in layers as assumed above but instead are completely random, the conductivity will have a value between λ and λ⊥. For further details, see Hallström et al. (1988) and Fricke and Becker (2001).

3.5.3

Liquid foods

The thermal conductivities of liquids are generally modelled by equations as follows:

λ = λ0 + BT

(3.34)

λ = λ0 + BT + CT 2

(3.35)

or

Further expressions describing the thermal conductivity as a function of temperature and also of concentration and constituents can be found, for example, in the reviews by Cuevas and Cheryan (1978) and Fricke and Becker (2001).

68

Confectionery and Chocolate Engineering: Principles and Applications

According to Loncin and Merson (1979), the thermal-conductivity value for starch (0.15 W/m K) is a good average value for carbohydrates and proteins in the compact state; this conductivity is clearly lower when these products occur in a porous or fibrous form containing air.

3.5.4

Liquids containing suspended particles

Maxwell gave an equation for calculating the thermal conductivity of a composite medium consisting of a liquid containing suspended particles:

λS = λ L

[2λL + λP − 2xV (λL − λP )] 2λL + λP + xV ( λ L − λ P )

(3.36)

where λS is the thermal conductivity of the whole suspension, λL is the thermal conductivity of the liquid suspending medium, λP is the thermal conductivity of the dispersed particles and xV is the volume fraction of the suspended particles. The distance between the particles must be large compared with the particle radius, i.e. xV must be small. For xV greater than 0.1, modifications have to be made to Eqn (3.36). Eucken (1940) and, later, Levy (1981) introduced a modified version of the Maxwell equation, cited by Fricke and Becker (2001).

3.5.5

Gases

The area of the thermal properties of gases is one where theoretical and empirical methods of calculation are in relatively good agreement. The following relationship can be derived theoretically for pure gases:

λM ⎛ 9 ⎞ R + CV = ⎝ 4⎠ η

(3.37)

where λ is the thermal conductivity [in cal/(cm K s)], η is the dynamic viscosity of the gas (in poise = 0.1 Pa s), M is the molar mass of the gas (in g/mol), CV is the molar specific heat capacity at constant volume (in cal/mol K) and R is the molar gas constant = 1.98 cal/ mol K. Many modifications of this relationship are in use. The dependence of the thermal conductivity on temperature can be expressed by the simple empirical formula

λ1 ⎛ T2 ⎞ =⎜ ⎟ λ2 ⎝ T1 ⎠

1,786

(3.38)

where the indices 1 and 2 relate to the two different temperatures T1 and T2. More complicated formulae are in use for the dependence on pressure: For gas mixtures, a calculation of the weighted average

λ = ∑ xi λi or the weighted average of the reciprocals of the thermal conductivities

(3.39)

Engineering properties of foods

1 x =∑ i λ λi

69

(3.40)

gives a relatively good result. A detailed discussion of the topic has been given by Szolcsányi (1975).

3.6 3.6.1

Thermal diffusivity and Prandtl number Second Fourier equation

The thermal diffusivity is defined by the second Fourier equation, which refers to unsteadystate conditions, i.e. dT/dt ≠ 0: dT λ ∂ 2T = dt c p ρ ∂y2

(3.41)

λ = a, the thermal diffusivity ( m 2 s ) cp ρ

(3.42)

where λ is the thermal conductivity of the material (W/m K), cp is the specific heat capacity of the material [J/(kg K)] and ρ is the density of the material (kg/m3). The name ‘thermal diffusivity’ suggests a similarity to the mass diffusivity D, which is defined by the second Fick equation, which in turn is formally similar to the second Fourier equation. However, this is more than a formal similarity: the mechanisms of heat and mass diffusion are not only similar but also coupled – the various analogies used in chemical engineering are founded on this idea, probably first proposed by Reynolds (see Chapter 1).

3.6.2

Liquids and gases

The thermal diffusivity a is dependent on the thermal conductivity λ, the specific heat capacity cp and the density ρ of the material in which thermal diffusion takes place. This fact illustrates why it can be difficult to give a simple method for exactly calculating the thermal diffusivity of a material, since all the latter three properties themselves are difficult to calculate. No general method exists for calculating the thermal conductivity of liquids. Numerous measurement methods have been published; for details see Loncin and Merson (1979) and Section 3.5 above. Generally, the thermal conductivity of a weakly polar liquid diminishes slightly when the temperature is raised; however, that of a strongly polar liquid increases. This increase is most significant in the case of water at temperatures between 0 and 150°C. For solid or liquid products containing at least 40% water and for temperatures between 0 and 100°C, Riedel (1969) showed that the thermal diffusivity is a weighted average of that of water at the same temperature and that of the dry protein, lipid or carbohydrate material, for which he obtained a mean experimental value of 0.0885 × 10−6 m2/s. Thus, a = 0.0885 × 10 −6 ( awater − 0.0885 × 10 −6 ) xwater

(3.43)

70

Confectionery and Chocolate Engineering: Principles and Applications

where a is the thermal diffusivity of the product, awater is the thermal diffusivity of liquid water at the given temperature and xwater is the mass fraction of water. When the temperature of a product rich in water is less than 0°C, the properties depend essentially on the proportion of frozen water, the thermal conductivity of ice being distinctly above that of water and of dry material.

3.6.3

Prandtl number

The Prandtl number Pr is important in heat and mass transfer. Recall the following numbers from Chapter 1: Reynolds number: Re = 2Rvρ/η; Prandtl number: Pr = Pe/Re = ν/a; Schmidt number: Sc = Pe′/Re = ν/D; Lewis number: Le = Sc/Pr = a/D; where R is the characteristic radius of a tube, ν = η/ρ is the kinematic viscosity (m2/s), D is the diffusion coefficient (m2/s) and a is the thermal diffusivity (m2/s). The Prandtl and Schmidt numbers are material parameters of a fluid. Although Pr = ν/a, i.e. it can be calculated as a ratio of two empirically measured quantities, there are other relationships for calculating it too. For gases, the following ‘rules’ are recognized: monatomic gases: Pr = 0.67 ± 5%; non-polar gases: Pr = 0.79 ± 15%; polar gases with linear molecules: Pr = 0.73 ± 15%; strongly polar gases: Pr = 0.86 ± 8%; water vapour and ammonia: Pr ≈ 1. The dependence of Pr on the pressure can be calculated by taking account of the specific heat capacity cp as a function of p and T, and also the coefficient of compressibility Z (Szolcsányi 1972; 1975, Chapter 2). For liquids, according to Denbigh (1946), the Prandtl number can be calculated from the latent heat of evaporation (ΔH)ev, or from the change of entropy (ΔS)ev of evaporation and the normal boiling point Tn at a given temperature (T) (in K): ⎡ ( ΔH )ev ⎤ − 1.8 log Pr = 0.2 ⎢ ⎣ RT ⎥⎦

(3.44)

or log Pr =

0.10 [( ΔS )ev (Tn )] T − 1.8

(3.45)

where R = 1.98 cal/mol K, (ΔH)ev is in cal and (ΔS)ev is in cal/K. Note that the relationships described in Eqns (3.44) and (3.45) are valid for liquids consisting of chemically homogeneous substances. Therefore – disregarding a few exceptions – they cannot be applied to confectionery practice, in which the majority of solutions

Engineering properties of foods

71

are aqueous solutions of carbohydrates, and when they are evaporated pure water will be evaporated, and not the ‘dry content’. Nevertheless, the enthalpy and entropy of evaporation of water and the Prandtl numbers of such solutions are practically independent of each other. The Riedel equation (Eqn 3.43) facilitates estimation of the Prandtl numbers of foods with a water content of at least 40 m/m% because the calculation is simplified to a measurement of the dynamic viscosity. From Eqn (3.43), the thermal diffusivities a of foods lie in the following range: 0.0885 × 10 −6 m 2 s < a < 0.143 × 10 −6 m 2 s

(3.46)

where 0.143 × 10−6 m2/s is the thermal diffusivity of water at 20°C. Since the densities ρ of foods lie in the range 500 kg m3 < ρ < 1500 kg m3

(3.47)

(and the density can be calculated relatively well), the calculation of the Prandtl number Pr = η/(aρ) is influenced mostly by the accuracy of the viscosity value. If we calculate with the mean values a = 0.11 × 10−6 m2/s and ρ = 1000 kg/m3, then the Prandtl number may be estimated in the following way: Pr =

η η ( Pa s ) = −3 aρ 0.11 × 10 [( m 2 s )( kg m3 )]

(3.48)

where Pa s = (m2/s)(kg/m3). Example 3.1 Let us calculate the Prandtl number of a chocolate mass at 50°C using the parameters given by Rapoport and Tarchova (1939) (valid for the interval 30–70°C):

ρ = (1320 − 0.5t ) ( kg m3 ) a × 108 = 2.7778 ( 4 + 0.017t ) ( m 2 s ) if η = 3 Pa s (an average value). At 50°C, ρ = 1295 kg/m3 and a = 1.347 × 10−7 m2/s; that is, Pr =

3 = 1.72 × 10 4 1295 × 1.347 × 10 −7

In Table A1.24 in Appendix 1, a = 0.1244 × 106 m2/s at 35°C for chocolate according to Antokolskaia (1964).

3.7 3.7.1

Mass diffusivity and Schmidt number Law of mass diffusion (Fick’s first law)

The various transport phenomena may be divided into pressure diffusion, thermal diffusion, forced diffusion and ordinary diffusion. Ordinary diffusion is the net transport of

72

Confectionery and Chocolate Engineering: Principles and Applications

liquid or solid without any movement of the fluid. Ordinary diffusion is caused by a concentration gradient and is proportional to this gradient according to Fick’s first law (or the first Fick equation), mA′ = − DA ρA

dcA dx

(3.49)

where mA′ [kg/(m2 s)] is the mass flux of substance A per unit area (m2) through a section perpendicular to the direction of flow (the prime means a time derivative), DA (m2/s) is the diffusion coefficient of A, ρA (kg/m3) is the density of A, cA is the concentration of A (kg/kg or mol/mol, etc.) and x (m) is the coordinate in the direction of flow.

3.7.2

Mutual mass diffusion

In the case of a binary system, mutual diffusion takes place: substance A diffuses into substance B while B diffuses into A. For constant ρA, the net flux of A may be written as mA′ = − DA ρA

dcA + cA ( mA′ + mB′ ) dx

(3.50)

In several cases the diffusion rate of one component is zero, i.e. mB′ = 0. This is, for instance, the case in convection drying, as vapour leaves the material surface and diffuses through a boundary layer of stagnant air. With mB′ = 0, Eqn (3.50) above gives mA′ = − DA ρA (1 − cA )

−1

dcA dx

(3.51)

In this example A is a gas phase (vapour), and the equation may be rewritten as mA′ = −

M WW DA P dpW RT P − pW dx

(3.52)

where MWW = molar weight of water, R = molar gas constant, T = absolute temperature (K), P = vapour pressure of pure water at T and pW = partial vapour pressure of water in the gas phase at T.

3.7.3

Mass diffusion in liquids

The diffusion coefficients of liquids are, in most practical systems, heavily dependent on the concentration of solids in solution. According to Besskow (1953), the diffusion coefficients of fluids are mostly of magnitude 10−3–10−4 cm2/min, i.e. 10−9–10−10 m2/s. According to Bruin (1979), the scale of the self-diffusivity of water is 10−9 (m2/s), and that of the diffusivity of water in some materials (cellophane, gelatin, starch, maltodextrin, coffee extract and amylopectin) is in the range 5 × 10−9–10−14 (m2/s) and depends on the water concentration: lower diffusion coefficients occur for lower concentrations.

Engineering properties of foods

3.7.4

73

Temperature dependence of diffusion

The temperature dependence may be described by means of an Arrhenius-like equation, B D = D0 exp ⎛ − ⎞ ⎝ T⎠

(3.53)

where D0 and B are constants. According to Einstein, D = kT f

(3.54)

where D is the empirical diffusion constant (m2/s), k is the Boltzmann constant, equal to 1.38062 × 10−23 J/K) (= R/N, where N is the Avogadro number), T is the absolute temperature (K) and f is the drag coefficient of the fluid (kg/s). In the laminar flow region, the following equation due to Stokes is valid for colloids: f = 6π rη

(3.55)

where r is the radius of a particle of spherical shape (m) and η is the dynamic viscosity of the fluid [in units of Pa s = kg/(m s)]. The empirical diffusion constant can therefore be determined from the Stokes–Einstein equation, D=

kT 6π rη

(3.56)

if r and η are known. Because the activation energy in the Arrhenius equation is the same for both diffusion and viscosity, the temperature dependence of the phenomena can be regarded as the same. Consequently, the product Dη is approximately constant, since the linear dependence in the Stokes–Einstein equation can be neglected in comparison with the exponential dependence in the Arrhenius equation (Erdey-Grúz and Schay 1962). In other words, if the viscosity increases, then the diffusion coefficient decreases, and vice versa. The following models were suggested by van der Lijn (1976) for sugar solutions: maltose: log D = −7.870 − 9.40 ( x + 0.194 ) (548 − T ) T

(3.57)

sucrose: log D = −8.209 − 17.8 ( x + 0.121) ( 447 − T ) T

(3.58)

glucose: log D = −8.405 − 15.9 ( x + 0.417 ) (397 − T ) T

(3.59)

where x is the mole fraction of solids (mol/mol); T is in K and D is in m2/s. (In these equations, the logarithmic form is merely a simplification for computation; log D has no physical meaning.) Stokes’s law relates to the drag on particles of spherical shape, and therefore it cannot be expected to be valid for molecules of dissolved substances, the shapes of which are generally not spherical. Nevertheless the Stokes–Einstein equation gives acceptable values for the diffusion coefficients of dissolved molecules. The reason for this is likely to be that

74

Confectionery and Chocolate Engineering: Principles and Applications

rotation of the molecules is induced so much at the usual temperatures that the conditions of spherical symmetry are met (Erdey-Grúz and Schay (1962). Example 3.2 Let us calculate the diffusion coefficient of sucrose in a 20 m/m% solution at 303 K from Eqn (3.58), using the Stokes–Einstein equation and estimating the radius of the sucrose molecule on the basis of the molar volume. From Eqn (3.58), if x = 0.2 and T = 303 K, log D = −10.924, i.e. D = 1.19 × 10−11 m2/s. Since the molar weight of sucrose is 342 g, the volume of one molecule is 342 4r 3π = 35.05 × 10 −29 m 3 = 23 3 6.02 × 10 cm 3 where r is the radius of the molecule assuming a spherical shape, i.e. r = 4.37 × 10 −10 m From the Stokes–Einstein equation, D=

kT = 3.386 × 10 −10 m 2 s 6π rη

taking into account the fact that the dynamic viscosity of this sugar solution (20 m/m%) is 1.5 × 10−3 Pa s (Junk and Pancoast 1973). According to Rohsetzer (1986), the diffusion coefficient D of sucrose can be estimated as 2.5 × 10−10 m2/s.

3.7.5

Mass diffusion in complex solid foodstuffs

Regarding the mass transfer properties of solid foodstuffs, two types of material are often distinguished: • laminates, consisting of layers with different properties sandwiched together; • particulates, in which discrete particles of one phase are dispersed in a continuum of another (Holliday 1963). For a slab consisting of a laminate in which the layers are perpendicular to the direction of flow, the mass diffusivity may be calculated (Bruin and Luyben 1980) according to 1 ⎛ 1 ⎞ = ∑⎜ , i = 1, 2, … , n ⎝ Di σ i ⎟⎠ D

(3.60)

where σi represents the solubility coefficient of the i-th laminate. Several examples of systems made up of one continuous polymer phase and one dispersed phase have been described in the literature; see Holliday (1963). Mathematically, the diffusion of particles in these materials may, essentially, be treated like the phenomena

Engineering properties of foods

75

of electrical and thermal conduction. Here, a two-phase system is considered where a number of particles (spheres) of material A are embedded in a continuous medium of material B. The diffusion coefficients of the two materials, DA and DB, are assumed to be constant. The effective mass diffusion coefficient of this medium is calculated according to

γ ( DA − DB ) D − DB = D + 2 DB DA − 2 DB

(3.61)

It is assumed that any interaction between the spheres is negligible (following Maxwell), and γ is an empirical parameter.

3.7.6

Schmidt number

If the diffusivity D and kinematic viscosity ν are known, the Schmidt number Sc can be calculated from Sc =

v D

(3.62)

where ν = η/ρ is the kinematic viscosity. The Schmidt number plays a role in mass transfer similar to that of the Prandtl number in heat transfer; the role of D in mass transfer is the same as that of a (the thermal diffusivity) in heat transfer: Prandtl number: Pr = Pe/Re = ν/a; Schmidt number: Sc = Pe′/Re = ν/D. These two numbers are closely connected to each other in accordance with the Colburn analogy (see Eqn (1.24) in Chapter 1): St Pr 2 3 = St ′Sc 2 3 =

f′ 2

Example 3.3 Let us calculate the Schmidt number for an aqueous sucrose solution for which D = 2.5 × 10−10 m2/s (Rohrsetzer 1986), η = 10−3 kg/m s and ρ = 1050 kg/m3. The Schmidt number is Sc =

η 10 −3 = = 3.81 × 103 ρD 1050 × 2.5 × 10 −10

The scale of the Schmidt number is actually determined by the dynamic viscosity and the diffusion coefficient, since the value of the density is always between relatively narrow limits [see Eqn (3.47)]. On the other hand, when the temperature increases, the viscosity decreases and the diffusion coefficient increases, and consequently their ratio – and also the Schmidt number – decreases, which is not compensated by the decrease in density.

76

Confectionery and Chocolate Engineering: Principles and Applications

3.8

Dielectric properties

3.8.1

Radio frequency and microwave heating

The dielectric properties of foods are important if radio frequency (often abbreviated to ‘RF’) or microwave heating is used. (Other names of this type of heating are ‘capacitive dielectric heating’ and ‘capacitance heating’.) In dielectric heating, the foodstuff interacts with electromagnetic waves oscillating at frequencies between 3 and 300 000 MHz. High-frequency heating is usually carried out at radio frequencies between 13.9 and 27 MHz and microwave heating at between 915 and 2450 MHz. According to Manley (1998), there is an increased interest in the use of microwaves and radio-frequency energy to enhance baking speed and efficiency. APV Baker has been offering microwave applications within standard ovens to heat both dough pieces and biscuits later in the bake period (to encourage more rapid drying). Sasib Bakery offers a radio-frequency application to speed drying in the later parts of a conventional oven … RF-ovens are available in 25…85 kW modules. RF units have an overall efficiency of between 65 and 72% in terms of conversion of mains electrical consumption to transfer of RF energy to the product … Microwave energy is used in the first zone to heat the dough piece rapidly, in the middle zones to control leavening gas production and in later zones to increase the rate of moisture removal. Microwave energy must be used in combination with conventional heating as this determines the colouration and flavour development.

The amount of heat generated in dielectric heating depends on the dielectric properties of the food such as the dielectric constant (ε) and the loss angle (δ). These are both highly dependent on the food composition, the temperature and the radiation frequency (or wavelength). Dielectric heating generates heat directly inside the material exposed to the electromagnetic waves. The conversion of electrical energy to heat results from dielectric losses in the electrically non-conducting material, which is usually a poor thermal conductor. Dielectric heating depends on the interaction between polar groups in the molecules of a non-conductive material and an alternating electric field. The atomic carriers of the charges in such fluid and solid materials are not able to move when an electric field E is imposed; instead, they can only be slightly displaced from their initial positions. The effective force is proportional to the electric field strength, and because of the displacement, negative and positive surface charges arise at sites on the boundary. This phenomenon is quantified by the polarization P, which is related to the electric field by the following equation: P = ( ε r′ − 1) ε 0 E = D − ε 0 E

(3.63)

where P is the polarization vector (C/m2), ε r′ is the dielectric constant (>1), ε0 is the dielectric constant of the vacuum (F/m), E is the electric field vector (non-alternating) (V/m) and D is the dielectric displacement vector (C/m2). D is defined in a vacuum by the Maxwell equation D = ε0E

(3.64)

Engineering properties of foods

77

and ε0 is defined by the Maxwell equations as c = ( μ0 ε 0 )

−1 2

(3.65)

where c is the speed of light in vacuum = 2.99792458 × 108 m/s and μ0 is the magnetic permeability of the vacuum, defined as equal to 4π × 10−7 N/A2 = 4π × 10−7 H/m (A = ampere and H = henry). Consequently, ε0 = 8.854187817 × 10−12 F/m. If E is an alternating electric field, the dielectric constant becomes complex:

ε # = ε r′ − jε r′′

(3.66)

where j is the complex unit vector (= −1). The dielectric loss factor ε r′′ expresses the degree to which an externally applied electric field will be converted to heat:

δ = ε r′′ ε r′

(3.67)

where δ is the dielectric loss, or loss tangent, and ε r′′ is the dielectric loss factor. Both ε r′ and ε r′′ are dependent on the frequency of the alternating current and also on the temperature. The loss tangent contains contributions from both dielectric relaxation and electrical resistive heating, which dominates at lower frequencies. If the surface area of the plates of a capacitor is S (m2), the distance between them is d (m) and the dielectric constant is ε r′ , then its capacitance is C (F) = ε 0 (F m ) ε r′ (S d ) ( m 2 m )

(3.68)

If f (Hz) is the frequency of an alternating voltage U (V) and δ is the loss tangent, the power P is P = 2π fCU 2 tan δ

(3.69a)

or P ( W m 3 ) = 2πε0 ε r′ f (U d ) tan δ ≈ 55.603 × 10 −12 ε ′ f (U d ) tan δ 2

(3.69b)

or P (W cm 3 ) ≈ 55.603 × 10 −14 ε r′ f (U d ) tan δ

(3.70)

if d is in cm.

3.8.2

Power absorption – the Lambert–Beer law

The Lambert–Beer law for power absorption gives N (W ) = N 0 exp ( −2α z )

(3.71)

where N and N0 are the attenuated and the generated power, respectively, z is the penetration depth (m) and α is the attenuation factor (1/m), i.e.

78

Confectionery and Chocolate Engineering: Principles and Applications

{

}

α = ( 2π λ ) ( ε r′ 2 ) (1 + tan2 δ ) − 1

12

(3.72)

where λ = c/f is the wavelength (m) (where c ≈ 3 × 108 m/s is the velocity of light). The wavelengths for frequencies of 27.12, 915 and 2450 MHz are λ27.12 = (3 × 108 m/s)/(27.12 × 106 s−1) = 11.06 m; λ915 = (3 × 108 m/s)/(915 × 106 s−1) = 0.328 m; λ2450 = (3 × 108 m/s)/(2450 × 106 s−1) = 0.122 m. If z = ze = 1/2α, then N = N0/e, where e = 2.71… is the base of natural logarithms. The penetration depth z can be calculated from Eqn (3.72):

λ ⎛ ε′ ⎞ z =⎛ 0 ⎞⎜ ⎝ 2π ⎠ ⎝ ε ′′ ⎟⎠

(3.73)

where λ0 is the vacuum wavelength, ε′ is the real part of the complex dielectric constant and ε″ is the imaginary part of the complex dielectric constant. Microwave energy at 915 MHz penetrates more deeply than that at 2450 MHz for the same material because λ(915 MHz) = (3 × 108 m/s)/(915 × 106 s−1) = 0.328 m and λ(2450 MHz) = (3 × 108 m/s)/(2450 × 106 s−1) = 0.122 m, and the penetration depth is proportional to the vacuum wavelength. At the extremes of the frequency range for dielectric heating (3 MHz and 30 GHz), the corresponding penetration depths are 100 m and 0.01 m, respectively. The above relationship also shows that infrared radiation, for practical purposes, acts only on the surfaces of bodies because its wavelength is much shorter (e.g. λ = 3 × 10−6 m at f = 1014 Hz) than that of radio frequency radiation. Therefore infrared radiation from incandescent heaters is very often used to provide intense heating for product colouring. The relationship described in Eqn (3.72) is still valid, although both ε r′ and tan δ are dependent on the frequency. In industrial microwave heating units, the most commonly used frequencies are 915 MHz and 2.45 GHz. A frequency of 5.8 GHz is increasingly being used for special applications. The most commonly used frequencies for radio frequency heating are 13.56 MHz and 27.12 MHz. For further details, see Stammer and Schlünder (1992) and Grüneberg et al. (1993).

3.8.3 3.8.3.1

Microwave and radio frequency generators Microwave generators (magnetrons)

The type of microwave generator most frequently used is the magnetron. Magnetrons were developed in the 1950s for radar applications and have been used for microwave heating since the discovery of this application of high-frequency waves.

Engineering properties of foods

79

Magnetrons are produced with output powers ranging from 200 W to 60 kW or even higher. The majority of magnetrons are produced with output powers between 800 W and 1200 W for household microwave ovens. Magnetrons of very low power are commonly used in medical applications, and magnetrons of high power are used for industrial heating and in research. Owing to the mass production of magnetrons with a power of about 800–1200 W, the price of such magnetrons is comparatively low. Therefore, these magnetrons are also used for industrial heating applications. During operation, magnetrons must be cooled to prevent overheating. Magnetrons with a power of up to about 2 kW are usually air cooled, while those with a higher power are usually water cooled, requiring water recirculation units. Those magnetrons also require the use of special protection equipment against reflected power that could overheat and destroy the magnetron. Low-power magnetrons are more robust and can be operated without protection equipment. There are many other types of microwave generators, such as klystrons and travelling wave tubes. However, none of these generator types is used for industrial microwave heating as the costs are too high compared with magnetrons. 3.8.3.2

Radio frequency generators

Radio frequency waves are usually generated by tube or semiconductor generators. Tube generators use a vacuum tube to generate the high-frequency waves. Semiconductor generators are a comparatively new development for industrial heating and have only limited output power. Tube generators can have an output power of several 100 kW. In a radio frequency heating unit, a high-frequency field is generated between two or more electrodes. The shapes of the electrodes determine the shape of the generated field. Although many electrode shapes are possible, two types of electrodes are most commonly used, namely rod electrodes and plate electrodes. For heating, the lossy dielectric material is placed between or over the electrodes. The electric field strength is determined by the applied voltage and the distance between the electrodes. The minimum technically feasible distance between the electrodes is determined by the applied voltage. With increasing distance between the electrodes, the voltage required to maintain the electric field strength also increases. The maximum distance between the electrodes and thus the maximum product thickness are determined by the necessity to avoid arcing between the electrodes. Tables 3.14–3.16 show the dielectric constants of some substances that are of interest in the confectionery industry. For the dielectric constant of cocoa butter, see Fincke (1965). These data can be regarded as merely indicative, since both εr and tan δ are dependent on the frequency used. Table 3.17 shows the temperature and frequency dependence of tan δ of water. Table 3.14 Dielectric properties of water.

Ice Water (at 25°C)

Relative dielectric constant (ε r′ )

Relative dielectric loss constant (ε ″)

Loss tangent (tan δ)

3.2 78

0.0029 12.48

0.0009 0.16 (= 12.48/78)

80

Confectionery and Chocolate Engineering: Principles and Applications

Table 3.15 Relative dielectric constants of some materials used in confectionery practice at room temperature (indicative values). Substance

Relative dielectric constant

Air (dry) Cereals Margarine, liquid Sorbitol Soy beans Starch Sucrose Syrup

1.000536 3–5 2.8–3.2 33.5 2.8 3–5 3.3 50

Table 3.16 Relative dielectric constants of some vegetable oils at 20°C and their dependence on temperature (Kiss 1988). Vegetable oil Arachis Cottonseed Linseed Sunflower

ε(rel), 20°C

Range (°C)

(1/ε)(Δε/Δt)

3.051 3.149 3.192 3.11

0–100 0–100 0–100 0–100

−0.00313 −0.00366 −0.00385 −0.0034

Table 3.17 Temperature and frequency dependence of tan δ of water (Jeppson 1964). Temperature (°C) 15 55 95

900 MHz

2450 MHz

0.07 0.03 0.02

0.17 0.07 0.04

Although a considerable body of published information exists on the dielectric properties of many foodstuffs, because of the various factors describing the electric field and the food that affect these properties, precise values for a particular product under a specific set of conditions can be obtained only by actual measurements. The benefit of dielectric heating is that as the water content of the material heated decreases, both the relative dielectric constant εr and the loss tangent tan δ decrease. Consequently, the heat loading on material that is drying becomes less and less. This process prevents excessive heating of the material, i.e. this operation can be described as ‘gentle’ or ‘considerate’. Dielectric drying lines, which consist of a transport band and a heating oven above it, are made for industrial purposes (Vauck and Müller 1994). The usual technical data of such a drying line are: length of band, 16 m; width of band, 0.7 m; length of heating oven, 5 m; voltage, 7 kV; frequency, 15–19 MHz; throughput, up to 400 kg/h; power consumption, 20 kW; source of radiant energy, tube generators; efficiency, 30–50%; specific energy demand, c. 0.2–2 kWh/kg. The power consumption can, however, reach 350 kW. The main fields of application of dielectric drying lines are in the chemical industry and in biotechnology. Since the process is relatively expensive, its use in the confectionery industry is very limited at present.

Engineering properties of foods

3.8.4

81

Analytical applications

The dielectric properties of materials can give valuable information about their composition. There is a relatively direct relationship between the dielectric properties and the water content of a material – this is the basis of many electronic moisture meters. The data provided by these measurements must be compared with data obtained by ‘absolute’ methods; for example, the calibration of measurements of the water content of cocoa powder using dielectric properties may be done with results obtained by the Karl Fischer method. For confectionery applications, see Minifie (1970); for near infrared reflectance/ transmittance (NIR/NIT) investigations of cocoa and chocolate products, see Kaffka et al. (1982a), Bollinger et al. (1999) and Schulz (2004); for investigations of proteins using NIR/NIT and the Kjeldahl method, see Horváth et al. (1985); for investigations of water content, see Kaffka et al. (1990); for investigations of oil, protein, water and fibre content, see Kaffka et al. (1982b), El-Rafey et al. (1988) and Bázár (2008); and for investigations of wavelength optimization using the Polar Qualification System (PQS), see Kaffka and Seregély (2002).

3.9

Electrical conductivity

3.9.1

Ohm’s law

With foods that are conductors (e.g. sugar, whole egg, salt and dried milk), the electrical conductivity is significantly dependent on the frequency of the electromagnetic field. Most foods, however, are poor conductors and their conductivity is essentially independent of the electromagnetic field. The electrical conductivity is defined by Ohm’s law for direct current (DC), E = IR

(3.74)

where E = voltage (V), I = current (A) and R = resistance (Ω). The electrical conductivity G is the inverse of the resistance: G=

1 R

(3.75)

It is measured in units of siemens (S), where 1S=

s3 A 2 A 1 = = 2 kg m V Ω

The specific resistance γ is defined as the resistance of a line of 1 mm2 cross-section and 1 m length, i.e. its units are mm 2 m [R ][ A] =Ω = 10 −6 Ω m = 10 −6 l m S [] specific resistance ( Ω m ) = 1 specific conductivity (S m )

(3.76)

82

Confectionery and Chocolate Engineering: Principles and Applications

For alternating current (AC), the electrical resistance can be expressed in units of 1/S = Ω too, but in this case it is called the ‘impedance’ Z, and is of complex value: Z = R + jω L +

1 1 ⎞ = R + j ⎛ωL − ⎝ ωC ⎠ jωC

(3.77)

where ω = 2πf, and R is the real (‘ohmic’) part of the complex impedance Z. If |Z| is the absolute value (in 1/S = Ω) of the complex number Z, then its reciprocal (in S) is the absolute value of the complex conductivity. Materials can be classified by their conductivity into three classes: • Conductors. Metals have a high conductivity. Electrolytes have either a medium conductivity (good electrolytes, e.g. aqueous solutions of mineral acids) or a low conductivity (poor electrolytes, e.g. dilute aqueous solutions and water itself). • Insulators, such as glass or a vacuum, have a low conductivity. • Semiconductors: their conductivity is generally intermediate, but varies widely under different conditions, such as exposure of the material to electric fields or to light of certain frequencies.

3.9.2

Electrical conductivity of metals and electrolytes; the Wiedemann–Franz law and Faraday’s law

The various atomic mechanisms of electrical conduction result in differences in the properties of matter. Electrical conduction is closely connected to the movement of electrons. In metals there is a very mobile electron cloud, which moves easily under the effect of an electric field. Therefore, metals are excellent conductors of both electrons and heat. (The high surface reflection of metals is also a result of their mobile electrons.) If the temperature increases, the movement of the electrons becomes more and more difficult, and consequently the conductivity decreases. In contrast, if the temperature of an electrolyte increases, the ions become more and more mobile, and consequently its conductivity increases. There is an important relationship between the thermal and electrical conductivities of metals called the Wiedemann–Franz law. The ratio of the thermal conductivity to the electrical conductivity of a metal is proportional to the temperature. This relationship is based upon the fact that heat transport and electrical transport both involve the free electrons in the metal. The thermal conductivity increases with the average particle velocity, since that increases the forward transport of energy. However, the electrical conductivity decreases with increasing particle velocity because collisions divert the electrons from the forward transport of charge. This means that the ratio of the thermal to the electrical conductivity depends upon the average velocity squared, which is proportional to the temperature. According to the Wiedemann–Franz law, L=

λ = 2.45 × 10 −8 W Ω K 2 GT

(3.78)

where L is the Lorenz number, λ is the thermal conductivity and G is the electrical conductivity.

83

Engineering properties of foods

The Lorenz number is practically independent of the temperature, and lies in the range 2.3–3.2 for many metals. The extension of the Wiedemann–Franz law to other kinds of materials is questionable. For electrolytes, Faraday’s law states that m=

Q M 1 QM = qn N F n

(3.79)

where m is the mass of the substance produced at an electrode (in g), Q is the total electric charge that has passed through the solution (in coulombs), q is the electron charge = 1.602 × 10−19 coulombs/electron, n is the valence of the substance as an ion in solution (electrons/ion), F is Faraday’s constant = 96.485 C/mol, M is the molar mass of the substance (g/mol) and N is the Avogadro number = 6.022 × 1023 ions/mole.

3.9.3

Electrical conductivity of materials used in confectionery

The raw materials, the products and the semi-products of the confectionery industry can be roughly separated into two groups: hydrophilic materials, which contain a hydrophilic continuous phase; hydrophobic (or lipophilic) materials, which contain a hydrophobic continuous phase. A hydrophilic phase can be regarded as a more or less concentrated aqueous solution in which hydrophilic/lipophilic substances are dispersed. The base of a lipophilic phase is a vegetable oil or fat in which the other ingredients are dissolved or dispersed. The electrical conductivity of such materials is determined by the following facts: • Hydrophilic materials are electrolytes, mostly poor electrolytes, the electrical conductivity of which is dependent on the water activity. • Hydrophobic materials (e.g. chocolate) are either good or poor insulators, depending upon the amount of free ions, free fatty acids etc. Some typical conductivity values are given in Table 3.18 for both aqueous solutions and fats/oils (see the entry for ‘paraffin’). Since the ohmic conductivity of foods is low in general, measurement of the complex (ohmic + inductive + capacitive) conductivity is mostly used, because this makes many-sided studies of their properties possible.

3.9.4

Ohmic heating technology

The study of the electrical properties of electrolytes (Kohlrausch’s rule, etc.) is beyond the scope of the present work, and this topic may not seem important from the point of view of present practice in the confectionery industry. Nevertheless, we must take into account the fact that ohmic heating technology is developing; see Fine (2007). With ohmic heating, the food material, which serves as an electrical resistor, is heated by passing electricity through it. At an atomic level, this use of electricity – or Joule heating – is the result of moving electrons colliding with atoms in the conductor,

84

Confectionery and Chocolate Engineering: Principles and Applications

Table 3.18 Some typical electrical conductivities of materials. Electrical conductivity Metals Silver Copper Aluminium

63.01 × 106 S/m at 20°C (630 100 S/cm; the highest electrical conductivity of any metal) 59.6 × 106 S/m (20°C) 37.8 × 106 S/m (20°C)

Electrolytes Sea water Drinking water Ultrapure water Glycerol Ethanol Sulphuric acid (30 m/m%, aqueous)

5 S/m 0.0005–0.05 S/m 5.5 × 10−6 S/m 2.2 × 10−3 S/m (0°C); 12.3 × 10−3 S/m (21.3°C) 3 × 10−4 S/m 74 S/m

Insulators Paraffin Quartz

10−16 S/m 5 × 10−15 S/m

Semiconductors Germanium Silicon Graphite Selenium

1.1236 × 106 S/m (0°C) 1.725 × 106 S/m 12 S/m (0°C) 1.2 × 10−7 S/m

whereupon momentum is transferred to the atoms, increasing their kinetic energy. This electrical energy is dissipated as heat, which results in rapid, uniform heating throughout the product, producing a potentially far higher-quality product than its canned counterpart. The heat generation is effective throughout the entire volume of the product and depends on the food’s electrical properties (mainly the electrical conductivity). Unlike radiative techniques (e.g. microwave heating), ohmic heating is not limited by the penetration of waves; rather, heat is generated uniformly throughout the product exposed to the electric field, if the conductivities of different parts of the product are the same. As a fluid represents an electrical resistance to a current, it can be heated rapidly, and increases of 2°C are possible within 1 s. The heating rate, however, is dependent on the current used, together with the product’s physical chemistry and electrical properties. The conductivity is also an important factor. Conductivity values change with both the temperature (as the temperature increases, the conductivity increases and results in a gradual improvement in the heating process over time) and the frequency of the current if AC is used. The heating rate is also dependent on other parameters, such as the electric field distribution, and the size, shape and orientation of particulates in liquid foods. A high solids content is desirable for effective ohmic heating because it often results in faster heating. Ohmic heating is also a more efficient treatment for high-viscosity products and particulate foods (with particle sizes of up to 4 cm) than are conventional heating techniques, which require time for heat penetration to occur to the centre of the material and in which particles heat up more slowly than the fluid phase of the food. In addition, the lack of mechanical action makes ohmic heating suitable for use with sensitive products. In the past, one drawback of ohmic heating was that electrolytic reactions could take place at the surface of the electrodes, leading to burning of the product and corrosion if

Engineering properties of foods

85

the electrodes were made of common food-grade metals. The major electrolytic effect was dissolution of the metallic electrodes, which could cause product contamination. To overcome this drawback, ohmic suppliers now use more resistant electrodes (such as electrodes made of pure carbon), use AC instead of DC and increase the frequency of the electric supply (no corrosion takes place at high frequencies, especially at high current densities of 3500 A/m2). More significantly, owing to the variation in the performance of electrical resistance heating from one product to another, the main disadvantage of ohmic heating is that its application varies from product to product. Despite this, a large number of potential future applications exist for ohmic heating, including use in blanching, evaporation, dehydration, fermentation and extraction. The development of non-acid sterilized food products is now closely tracking the development of innovative aseptic packaging systems. The prospects for ohmic heating in the confectionery industry cannot yet be judged; however, the diversity of sweets makes possible the application of up-to-date technologies. Furthermore, electricity is an environmentally friendly source of energy, and therefore its use will be intensified.

3.10

Infrared absorption properties

The infrared absorption properties of foodstuffs are not easily described. Birth (1976) discussed how light interacts with food materials and described the important principles of normal surface reflection, body reflection and light scattering. Surface reflection takes place, as the name implies, at the surface of a material and is about 4% for most organic components. In the case of body reflection, the light enters the material, becomes diffuse owing to scattering, and undergoes some absorption. The remaining light leaves the material close to where it entered. Normal surface reflection produces the gloss or shine observed on polished surfaces, while body reflection produces the colours and patterns that constitute most of the information we obtain visually. For materials with a rough surface, both the surface and the body reflection will be diffuse. Scattering is the mechanism which redirects the radiant energy from its original direction of propagation. The optical characteristics of various media were also discussed theoretically by Krust et al. (1962) and by Ginzburg (1969) using data from studies by Bolshakov et al. (1976) of the optical characteristics of various materials and products. These studies demonstrated the necessity for taking account of scattered radiation during measurement. As a first step, equations for short-wave radiation were determined using experimental transmission values for depths greater than 8 mm for bread in a regression analysis (with no long-wave radiation remaining). As halogen lamps (λmax = 1.12 μm) were used during these experiments, about 33% of the total radiation, calculated from Planck’s equation, was absorbed in this wavelength range, and therefore qout = 0.33 exp ( −1.6 x ) qin

(3.80)

where qin is the input radiation energy flux, qout is the output radiation energy flux and x is the penetration depth.

86

Confectionery and Chocolate Engineering: Principles and Applications

The equations for the long-wave (63% of the total radiation) penetration curves were then calculated by subtracting the calculated transmission values above from the experimental transmission values for the total radiation: qout = 63 exp ( −6.6 x ) qin

(3.81)

The total penetration curves describing the experimentally measured transmission values are therefore given by qout = 33 exp ( −1.6 x ) + 63 exp ( −6.6 x ) qin

(3.82)

A summation of the coefficients gives 96% total absorption for the radiation. This is in agreement with the theoretical value, as the surface reflection for most organic materials is about 4%.

3.11 3.11.1

Physical characteristics of food powders Classification of food powders

Food powders are a large group of different kinds of powders that have little in common, except for being used as (or in) foods. In the confectionery industry, the most important raw materials in the powder state are sucrose, wheat flour, milk powder, soy flour, gelatin, pectin, agar-agar and starch; among the finished products, cocoa powder and various pudding powders should be mentioned. Many mixtures in the powder state are made during production too. Even this incomplete enumeration indicates the importance of this topic. The classification criteria for food powders may, therefore, vary for the purpose of convenience or according to any particular practical application. Peleg (1983) provided a classification of powders by: • • • • • •

use (flour, sweeteners etc.); major chemical component (starchy, sugar etc.); process (ground, spray-dried etc.); size (fine, coarse); moisture sorption pattern (extremely hygroscopic, moderately hygroscopic etc.); flowability (free-flowing, cohesive etc.).

This classification demonstrates the difficulties of treating food powders as a group at a level of generalization that will not make the analysis too vague and consequently impractical. Furthermore, some of the more interesting and potentially useful criteria, for example hygroscopicity and flowability, are not easy to quantify because they represent the combined effect of different sorts of physical and physicochemical phenomena. The composition and properties of many food powders may vary to different degrees and may also change with time. Therefore, it is not uncommon that a free-flowing powder, for example, may become sticky during storage, or that a relatively non-hygroscopic powder

Engineering properties of foods

87

(e.g. salt) may becomes highly hygroscopic in the presence of impurities. Realizing these problems, and with the understanding that exceptions to the discussion are not only possible but also sometimes unavoidable, this chapter is an attempt to evaluate the factors that determine or influence the physical properties of food powders, with special emphasis on their specific or unique characteristics. The physical properties of powders are usually characterized at two levels, that of the individual particles and that of the bulk powder. Although it is self-evident that the bulk properties are primarily influenced by the properties of the particles, the relationship between the two is by no means simple and involves external factors such as the system geometry and the mechanical and thermal history of the powder. The bulk properties of fine powders, always interdependent, are determined by the physicochemical properties of the material (e.g. composition and moisture content), the geometry, size and surface characteristics of the individual particles, and the history of the system as a whole. The shape of the container can affect flowability, and the powder density usually increases as a result of vibration, for example. Numerical values assigned to such properties therefore ought to be regarded as useful only under the conditions under which they were determined, or as indicators of the order of magnitude only.

3.11.2

Surface activity

Since the phenomenon of water vapour sorption in food has been extensively studied and discussed in the literature, it need not be discussed here. Less information is available on the capacity of many kinds of food surfaces to adsorb fine solid particles or to interact with other particles and equipment surfaces. These interactions are not limited to particles of the same or similar chemical species, although there is evidence to suggest that surface affinity can differ considerably between materials (e.g. in the case of certain anticakingagent–powder systems). The mechanisms by which particle surfaces interact are also of several different kinds, including liquid bridging by surface moisture or melted fat, electrostatic charge (as in dust), molecular forces and the surface energy of crystalline materials. Detailed theoretical discussions and mathematical analyses of such interactions and their implications in powder technology have been published by Rumpf (1961), Pietsch (1969) and Zimon (1969).

3.11.3

Effect of moisture content and anticaking agents

In general, moisture sorption is associated with increased cohesiveness, due mainly to interparticle liquid bridges. Therefore, especially in the case of hygroscopic food powders, a higher moisture content ought to result in a lowering of the loose bulk density, as indeed is the case for powdered sugar and salt, for example. It should be mentioned, however, that this decrease will only be detected in freshly sieved or flowing powders, where these same interparticle forces are not allowed to cause caking of the mass (see below). Another notable exception to this trend is in the case of fine powders that are very cohesive even in their dry form (e.g. baby formula and coffee creamer). In such cases it appears that the bed array has reached its maximum ‘openness’ at a low moisture content, and therefore a further lowering of the density becomes impossible. It is also worth remembering that excessive moisture levels, especially in powders containing soluble crystalline compounds (such as sugars or salt), may result in liquefaction of the

88

Confectionery and Chocolate Engineering: Principles and Applications

Table 3.19 Effect of moisture content on the mechanical characteristics of selected food powders [reproduced from Peleg (1983), with permission of Springer Science and Business Media]. Loose bulk density Compressibility (g/cm3) (value of b)

Cohesion (g/cm2)

Angle of internal friction (deg)a

Reference

Powder

Moisture (%)

Powdered salt (100/200 mesh)

Dry 0.6

1.26 0.78

0.02 0.12

0 50

40 36

Moreyra and Peleg (1981)

Powdered sucrose (60/80 mesh)

Dry 0.1

0.62 0.5

0.152 0.185

10 14

39 37

Peleg and Mannheim (1973)

Starch

Dry 18.5

0.81 0.69

0.12 0.15

6 13

33 30

Peleg (1971)

Baker’s yeast

8.4 13

0.52 0.49

0.08 0.26

14 TCb

42 TCb

Dobbs et al. (1982)

a b

Determined by a Jenike Flow Factor Tester at consolidation levels of 0.2–0.5 kg/cm3. ‘TC’ indicates that the powder was too cohesive for measurement by the Flow Factor Tester.

powder, and consequently in an increase in its density. At this stage the powder most probably has already lost its utility, and therefore this phenomenon has little practical importance. Anticaking agents (or flow conditioners) are supposed to reduce interparticle forces and, as such, they are expected to increase the bulk density of powders (Peleg and Mannheim 1973). It has been observed, though, that there may be an optimal concentration beyond which the effect will diminish (Nash et al. 1965) or will be practically unaffected by the conditioner concentration (Hollenbach et al. 1982). It can also be observed that for a noticeable effect on the bulk density (i.e. an increase of the order of 10% or more), the agent and the host particles must have surface affinity. Otherwise, the particles of the agent may segregate and, instead of reducing the interparticle forces, will only fill the interparticle space. It seems, however, that there is very little information on the exact nature of these surface interactions and the mechanism by which they affect the bed structure. Examples of the effects of moisture and anticaking agents on the bulk properties of selected food powders are given in Tables 3.19 and 3.20 (Peleg 1983). The value of b in these tables is the constant in the equation

ρB = a + b log σ n

(3.83)

where ρB is the bulk density, σN is the applied stress and a and b are constants.

3.11.4

Mechanical strength, dust formation and explosibility index

Many solid food materials, especially when dry, are brittle and fragile. Their hardness on the Mohs scale is of the order of 1–2 (Carr 1976). Since, particularly for small objects, surface or shape irregularities are normally associated with mechanical weakness (due to stress concentrations, for example), dry food particles have a tendency to wear down or disintegrate. Mechanical attrition of food powders usually occurs during handling and processing, when the particles are subjected

Engineering properties of foods

89

Table 3.20 Effect of anticaking agents on the bulk density and compressibility of selected food powders [reproduced from Peleg (1983), with permission of Springer Science and Business Media]. Concentration (%)

Loose bulk density (g/cm3)

Compressibility (value of b)

None Ca stearate Silicon dioxide Ca3(PO4)2

0 0.5 0.5 0.5

0.7 0.87a 0.75a 0.76*

0.066 0.039a 0.052a 0.044a

Hollenbach et al. (1982)

Gelatin (powdered)

None Al silicate

0 1

0.68 0.7

0 0.016

Peleg (1971)

Cornstarch

None Ca stearate Silicon dioxide Ca3(PO4)2

0 1 1 1

0.62 0.59 0.67 0.61

0.109 0.099 0.077a 0.062a

Hollenbach et al. (1982)

Soy protein

None Ca stearate Silicon dioxide Ca3(PO4)2

0 1 1 1

0.27 0.27 0.27 0.31a

0.04 0.041 0.036 0.024a

Hollenbach et al. (1982)

Powder

Agent

Sucrose

Reference

a

Significant change relative to the untreated powder.

to impact and frictional forces. The result is frequently a dust problem that may also develop into a dust explosion hazard. The incidence of dust explosions depends mainly on the dust particle size, the dust-toair ratio and the availability of a triggering spark. Carr (1976) listed potentially explosive agricultural dusts and ranked them according to their explosibility in the following descending order: starch (50), sugar (13.2), grain (9.2), wheat flour (3.8), wheat (2.5), skimmed milk (1.4), cocoa (1.4), coffee (<<0.1). (The numbers in parentheses are the ‘explosibility index’, where a severe hazard is denoted by an index of ≥10, a strong hazard by 1–10, a moderate hazard by 0.1–1 and a weak hazard by <0.1.) Other implications of dust in the food industry include human health and safety, plant and equipment maintenance, and material loss. When the size of the particles produced through mechanical attrition is larger than that of dust particles, the process can still influence the product’s bulk density or cause a segregation problem that may affect the product’s appearance, as in the case of instant coffee.

3.11.5

Compressibility

Powders can be compacted in two ways: by tapping, i.e. by the application of vibration (which may occur during transport or handling, during storage in tall bins, as a result of vibrating the powder in order to increase the weight in a given container, or as a result of vibrating the dosing equipment for the purpose of tableting, as in the case of some soups and candies), and by mechanical compression. The stresses that develop in storage are usually a few orders of magnitude smaller than those applied in tableting and similar forming operations.

90

Confectionery and Chocolate Engineering: Principles and Applications

3.11.5.1

Tapping

According to Sone (1972), for a variety of food powders, the relationship between the volume reduction fraction γn and the number of taps n is given by

γn =

V0 − Vn abn = V0 1 + bn

(3.84)

or, in linearized form, n 1 n = + γ n ab a

(3.85)

where V0 is the initial volume, Vn is the volume after n taps, and a and b are constants. The constant a in this equation represents the asymptotic level of the volume change or, in other words, the level that will be obtained after a large number of taps or a long time of vibration: lim γ n = a

x →∞

(3.86)

The constant b represents the rate at which the compaction is achieved, i.e. 1/b is the number of vibrations necessary to reach half of the asymptotic change. The intersection with the ordinate is 1/ab. Consequently, both constants can be determined from Eqn (3.85). 3.11.5.2

Compaction under compressive load

According to Gray (1968) the main modes, or mechanisms, by which a powder bed deforms are: • spatial rearrangement of particles (without deformation of the particles); • filling of interparticle voids by deformation (mainly plastic) of particles; • filling of voids by fragmentation of particles. The relative contribution of each mechanism depends mainly on the properties of the particles (e.g. size, shape and hardness), the magnitude of the applied pressure and the stress distribution within the compacted specimen. Therefore, the compressibility pattern of a given powder may be totally different in different load ranges. At the low end of the pressure range, i.e. for a normal stress of the order of up to 10–50 N/cm2, the first mechanism seems to be dominant, and, with few exceptions, such as brittle coffee agglomerates and fatty powders, most of the change in density is due to structural rearrangement of the bed. The original structure of both cohesive and non-cohesive powders is greatly influenced by the presence of interparticle forces. Therefore, it is expected that an open structure supported by such forces will readily collapse under small stresses, resulting in a high apparent compressibility in such a stress range. This has indeed been observed in a large variety of food powders (Peleg and Mannheim 1973; Moreyra and Peleg 1981; Peleg et al. 1982; Peleg 1983). The compressibility under small loads is a sensitive index of the cohesiveness of a powder (Carr 1976) and can be used to detect potential flow problems. In this low stress

Engineering properties of foods

91

range, the compressive deformability of food powders can be described by Eqn (3.83) above. The constant b in that equation has been called the ‘compressibility’. Its values for selected food powders are listed in Tables 3.19 and 3.20 (Peleg 1983). It should be mentioned that Eqn (3.83) is dimensionally not correct.

3.11.6

Angle of repose

The angle of repose (Fig. 3.1) is an indispensable parameter in the design of systems for the processing, storage and conveying of powders. Its actual magnitude, however, depends on the way in which the powder heap is formed (e.g. the impact velocity), and therefore published values of the angle are not always comparable (Brown and Richards 1970). For cohesive powders, the measurement of this angle is sometimes difficult because of the irregular shapes that heaps can assume. The angle of repose is sometimes confused with the angle of internal friction. Although its magnitude is certainly influenced by frictional forces (especially in free-flowing powders), it is also affected by interparticle attractive forces – a factor that becomes dominant in wet and cohesive powders. Mainly for this reason, the angle of repose (regardless of how it is measured) can be used as a rough indicator of flowability. According to Carr (1976), angles of repose of up to about 35° indicate free flowability, 35–45° some cohesiveness, 45–55° cohesiveness (loss of free flowability), and 55° and above very high cohesiveness and very limited flowability, if at all.

3.11.7

Flowability

Despite a few superficial similarities, the flowability of liquids and of powders are different physical phenomena. The main distinctions are as follows: (1) The flow rate of powders (Fig. 3.2), in contrast to that of liquids, is practically independent of the height h (i.e. the head) of the powder above the aperture if h is more than about 2.5 times the aperture diameter D (Brown and Richards 1970). (2) Powders can resist appreciable shear stresses. Once compacted, under their own weight or by external pressure, they can also form mechanically stable structures (e.g. arches) that will halt flow altogether despite the existence of a head.

Fall from fixed height

Bed rupture

Rotary drum (instantaneous angle)

α α

α

Fig. 3.1 Angle of repose α. Reproduced from Peleg and Bagley (1983, p. 311) with permission of Springer Science and Business Media.

92

Confectionery and Chocolate Engineering: Principles and Applications

POWDER If h > 2.5D Q ≈ constant

LIQUID Q = const. D2h1/2

h

Dp

Q θ

D

D Q

Fig. 3.2 Flow rate of powders and of liquids. D = aperture diameter; h = head height above aperture; Dp = typical size of granules; Q = volume flux of stored material. Reproduced from Peleg and Bagley (1983, p. 313) with permission of Springer Science and Business Media.

For these reasons, treatment of a powder’s gravitational flowability must be based on theories of solid mechanics (Jenike 1964), and not on hydrodynamics. The general principle on which such analyses are based is that, from a physical point of view, powder flow is equivalent to failure of a solid in shear. In ideal free-flowing powders or granular materials, the resistance to flow is primarily through friction, and therefore flowability evaluation is fairly straightforward. In cohesive powders (as in the case of most food powders), interparticle forces are enhanced by compaction (through an increase in the number of contact points, for example), with the result that the ‘compact’ can develop appreciable mechanical strength. Therefore, under even a small pressure, many food powders may cause serious flow problems. For cohesive powders, the system geometry (e.g. the bin angle and aperture diameter) plays a decisive role in establishing the flow regime, i.e. mass or funnel flow, and its rate and stability. A detailed description and analysis of the methodology involved can be found in, for example, the work of Jenike (1964). In the case of cohesive food powders, internal friction has very little influence on flowability. Most food powders have an angle of internal friction of the order of 30–45° (Peleg 1983), and it usually decreases slightly as moisture is absorbed. This is mainly because of a reduction in the surface roughness of the particles through dissolution and lubrication. It is obvious that this reduction in internal friction does not result in improved flowability. Further details are given in Charm (1971, p. 113–114). A comprehensive discussion of topics such as the flow of powders and the storage of powders in silos has been given by Schulze and Schwedes (1993).

3.11.8

Caking

Many food powders, especially those containing soluble components or fats, tend to agglomerate spontaneously when exposed to a moist atmosphere or elevated storage temperatures. This phenomenon can result in anything from small, soft aggregates that break easily to rock-hard lumps of variable size or solidification of the whole powder

Engineering properties of foods

Moisture sorption or heating Wet

Dry Surface

93

Equilibration

Continued sorption or melting Drying or cooling Attraction Fusion Caked (hard) Rewetting

Free flowing

Cohesive Cohesive (sticky) (sticky)

Caked (sticky)

Continued drying or cooling

Drying or cooling Caked (hard) Fig. 3.3 Schematic representation of the most common caking mechanisms in food powders. Reproduced from Peleg and Bagley (1983, p. 315) with permission of Springer Science and Business Media.

mass. In most cases, the process is initiated by the formation of liquid bridges between particles that can later solidify by drying or cooling. This mechanism, known as ‘humidity caking’ (Fig. 3.3), was schematically described by Burak (1966). Incidentally, the attraction stage shown in Fig. 3.3 is not a hypothetical stage. It can actually be observed that moisture absorption is accompanied by shrinkage of the powder. 3.11.8.1

Source of liquid bridges in food products

The source of the liquid bridges in food powders is sticky or liquefied particle surfaces, produced mainly by the following effects: • Melting of fats: this mechanism has practically no importance in confectionery practice. • Moisture sorption, accidental wetting or moisture condensation, which causes dissolution of the surface and/or the presence of a liquid film around the particles. • Liquefaction of the surface itself as a result of the temperature at which amorphous sugars become thermoplastic being exceeded, without the addition of external moisture. [A detailed study of the physicochemical aspects of this phenomenon was reported by To and Flink (1978) and Flink (1983)]. This temperature, also known as the ‘sticky point’, has a strong dependency on the powder’s moisture content. The sticky point is especially low for fruit juice powders, and this is the main reason for their physical instability. This temperature can easily be exceeded in other sugar-rich powders (e.g. onion and certain spices) during grinding (and therefore refrigerated mills are recommended in such cases) and during storage when the temperature or moisture is not tightly controlled; • Liberation of absorbed moisture when amorphous sugars crystallize. A notable example is the crystallization of lactose in milk powders (Berlin et al. 1968), one of the main reasons for a variety of instantization processes; see also Flink (1983).

94

Confectionery and Chocolate Engineering: Principles and Applications

According to Kargin (1957), glucose glass sorbs water only on the surface, owing to the low diffusivity of water in the glass. As the surface hydrates, a saturated solution forms and the glass softens, giving higher water diffusion. In tightly packed sugar glasses, water sorption is a surface phenomenon, and on the usual weight basis, sorption will be higher the higher the specific area. Lees (1968) described the graining of boiled sweets, which is a result of recrystallization of amorphous sugar. According to Lees, seeding studies at low humidity indicated that the induction period could be related to the formation of nuclei. The course of recrystallization was measured by the loss of water from the sample and by X-ray diffraction. For further details, see Lees (1968). Makower and Dye (1956) studied the recrystallization of spray-dried sucrose occurring during storage at various relative humidities. An induction period prior to the start of recrystallization, which depended on the relative humidity, was noted. Guilbot and Drapron (1969) gave moisture sorption isotherms for amorphous and crystalline sugars. Crystalline sugars begin to sorb after a minimum level of water activity aw is reached, while amorphous sugars sorb at any water activity. Crystalline maltose first starts sorbing water at aw = 0.25 and immediately forms the monohydrate. Amorphous maltose sorbs from aw greater than zero and appears to recrystallize at aw = 0.52, at which time it loses water down to the monohydrate level. Tests on a series of amorphous sugars showed all of them to have approximately the same isotherm, at least up to their recrystallization points, probably indicating that the same sites (–OH groups) are available for uptake of water for all sugars. As these sugars recrystallized, they lost water to become their respective hydrates, although an anhydrous sugar would lose all the previously sorbed water. At high enough aw, the sugars eventually go into solution. Guilbot and Drapron also observed that the temperatures for solid–liquid transitions of amorphous sugars were much lower than the melting temperatures of the respective crystalline sugars. The similarity of this transition temperature (collapse temperature) for the various amorphous sugars tested indicated that their bonding energy and degree of bonding were similar in the amorphous state. Sloan and Labuza (1974) also presented many sorption isotherms, including those for amorphous and crystalline sucrose. Amorphous sucrose begins sorbing water at aw = 0.10, with a major rise in moisture content at aw = 0.30. Crystalline sucrose remains essentially dry (i.e. with a moisture content of about 0) until about aw = 0.84, and then begins to absorb considerably, meeting the amorphous-sucrose sorption curve at about aw = 0.90. The isotherms do not reflect any recrystallization effect, which probably means that the time was too short for recrystallization to occur. Roth (1976, 1977) demonstrated that mechanical crushing of sugar crystals produces an amorphous surface capable of recrystallization after sorbing water. Measurements showed that the amorphous layer on the surface comprised about 2% of the sugar mass. Recrystallization of sucrose was noted at aw = 0.42 at 20°C, at which point there was caking and liquefaction due to recrystallization. Subsequent isotherms for the recrystallized sucrose were normal for the crystalline material and were repeatable, indicating no further change after the recrystallization had been completed. Simatos and Blond (1975) noted that freeze-dried sucrose adsorbed water at high rates, while a glass produced from the melt was slow to sorb water, this being due to the difference in specific surface areas of the two samples. Differential thermal analysis of freezedried sucrose showed a glass transition at 45°C, devitrification at 91°C and melting of a crystalline phase at 180°C. Glassy sucrose produced from the melt had a glass transition

Engineering properties of foods

95

at about 60°C and a shallow, broad melting peak at about 160°C. Simatos and Blond concluded that freeze-dried sucrose contains sucrose nuclei that crystallize upon rewarming, these crystals then melting at the normal melting point. X-ray diffraction did not indicate a crystalline structure, but electron diffraction revealed some organization in freeze-dried sucrose. The size of the structured regions must be small, and it could be questioned whether these are crystal nuclei, small crystals, or parts of organized regions in the glass. The onset and progression of the caking phenomenon do not necessitate liquefaction of whole particles or the whole powder bed. It is enough that only part of the surface becomes wet to initiate agglomeration. Furthermore, the intensity and the spread of the phenomenon within a given bed depend on the moisture absorption rate, the rate of diffusion of moisture into the interior of the particles, and rate of penetration of moisture into the bed. The hardness of the aggregates will depend on the material – crystalline, glassy or fatty – and on the temperature history of the particles, including the temperature range and the frequency of fluctuations. It can also happen that because of insufficient drying or because of mixing of ingredients with different moisture contents, caking will occur only after moisture equilibration inside the package. The meanings of the terms ‘caking tendency’ and ‘caking intensity’ are fairly vague. Despite this limitation, however, it can be shown that most powders that are known to be cohesive (in terms of bulk properties and flowability) also tend to cake readily, especially if under static pressure (Peleg and Mannheim 1973).

3.11.9

Effect of anticaking agents

Anticaking agents (also known as flow conditioners, glidants and free-flowing agents) are very fine powders (particle size 1–4 μm) of an inert or fairly inert chemical substance that are added to powders with a much larger particle size in order to inhibit caking and improve flowability. In studies of sucrose and onion powders, Peleg and Mannheim (1973) showed that such agents were effective (in both roles) in only a limited relative humidity range. The explanation is that coverage of the host particle surfaces by particles of the agent is sufficient to reduce interparticle attraction and perhaps to interfere with the continuity of liquid bridges. The presence of agent particles is not sufficient, however, to cover moisture sorption sites. Therefore, if moisture sorption is not disturbed, liquid bridges will eventually be formed and caking will occur as if the agent were not present. Additional aspects of the use of anticaking agents in food and other powders are discussed by, for example, Nash et al. (1965), Burak (1966), Peleg and Mannheim (1973), Carr (1976) and Hollenbach et al. (1982).

3.11.10

Segregation

Segregation of particles occurs when particles with different properties are distributed preferentially in different parts of a bed. The main reasons for segregation are differences in particle size, density, shape and resilience. However, in practice, differences in particle size are by far the most important factor. Segregation usually occurs when free-flowing powders having a significant range of particle sizes are exposed to vibration or other types of mechanical motion. Under such conditions, the smaller particles migrate to the bottom of the bed so that their concentration decreases as a function of height in the bed. The phenomenon is not limited to mixtures of particles of different types. It can and does occur

96

Confectionery and Chocolate Engineering: Principles and Applications

in chemically uniform powders whenever significant size differences exist. The segregation phenomenon is particularly noticeable when the powder contains a considerable amount of fines (e.g. colourants in drink powders and fines at the bottom of an instant-coffee jar). In the case of cohesive powders, segregation of fines is less likely to occur. The reason is that in such powders the fines usually adhere to the surface of the larger particles to form what are known as ‘ordered mixtures’ (Yeung and Hersey 1979; Egermann 1980). For further details, see Molerus (1993) and Mak and Kelly (1976).

Further reading Barton, A.F.M. (1991) CRC Handbook of Solubility Parameters and Other Cohesion Parameters, 2nd edn. CRC Press, Boca Raton, FL. Beaton, C.F. and Hewitt, G.F. (1989) Physical Property Data for the Design Engineer. Hemisphere Publishing, New York. Boethling, R.S. and Mackay, D. (2000) Handbook of Property Estimation Methods for Chemicals, Environmental and Health Sciences. Lewis Publishers, CRC Press LLC, Boca Raton, FL. Cropper, W.H. (1998) Mathematical Computer Programs for Physical Chemistry. Springer, New York. Eszterle, M. (1993) Molecular structure and specific volume of pure sucrose solutions. Zuckerindustrie 118 (6): 459–464. Figura, L.O. and Teixeira, A.A. (2007) Food Physics: Physical Properties, Measurement and Applications. Springer, New York. Francis, B., Hastings, W.R. and Jeans, P.A. (1962) Pilot Scale High Frequency Biscuit Baking with Particular Reference to the Checking of Hard Sweet Biscuits. BBIRA, Report 63. Hayes, G.D. (1987) Food Engineering Data Handbook. Longman Science and Technology, Harlow. Hewitt, G.F. (ed.) (1992) Handbook of Heat Exchanger Design. Begell House, New York. Kress-Rogers, E. and Brimelow, C.J.B. (2001) Instrumentation and Sensors for the Food Industry. CRC Press, Boca Raton, FL. Krizhanoskiy, I.S., Leppo, R.M., Chernatyin, G.A. (1968) Calculation of the heat-, energy- and water resources in the confectionery industry (in Russian). Khlebopekar-naya i Konditerskaya Promyshlennost 12 (5): 25–28. Lawson, R., Miller, A.R. and Thacker, D. (1986) Heat Transfer in Biscuit Baking, Part I: The Effects of Radiant Energy on Semi-sweet Biscuits. C&CFRA (FMBRA), Report 132. Lewis, M.J. (1996) Physical Properties of Foods and Food Processing Systems. Woodhead Publishing, Cambridge. Lienhard, J.H., IV and Lienhard, J.H., V (2005) A Heat Transfer Textbook, 3rd edn. Phlogiston Press, Cambridge, MA. Lyman, W.J., Reehl, W.F. and Rosenblatt, D.H. (1982) Handbook of Chemical Property Estimation Methods: Environmental Behavior of Organic Compounds. McGraw-Hill, New York. Murra, F., Zhang, L. and Lyng, J.G. (2009) Radio frequency treatment of foods: Review of recent advances. J Food Eng 91 (4): 297–508. Povey, M.J.W. and Mason, T.J. (1998) Ultrasound in Food Processing. Blackie Academic & Professional, London. Raznjavic, K. (1996) Handbook of Thermodynamic Tables, 2nd edn. Begell House, New York. Reid, R.C. and Sherwood, T.K. (1966) The Properties of Gases and Liquids – Their Estimation and Correlations. McGraw Hill, New York, NY. Rohsenow, W.M., Hartnett, J.P. and Cho, Y.I. (eds) (1998) Handbook of Heat Transfer, 3rd edn. McGraw-Hill, New York. Strayfield International Ltd (1986) An array of applications are evolving for radiofrequency drying. Food Eng Int 11: 11.

Chapter 4

The rheology of foods and sweets

Contents 4.1 4.2

4.3

4.4

4.5 4.6

4.7 4.8

4.9

Rheology: its importance in the confectionery industry Stress and strain 4.2.1 Stress tensor 4.2.2 Cauchy strain, Hencky strain and deformation tensor 4.2.3 Dilatational and deviatoric tensors; tensor invariants 4.2.4 Constitutive equations Solid behaviour 4.3.1 Rigid body 4.3.2 Elastic body (or Hookean body/model) 4.3.3 Linear elastic and nonlinear elastic materials 4.3.4 Texture of chocolate Fluid behaviour 4.4.1 Ideal fluids and Pascal bodies 4.4.2 Fluid behaviour in steady shear flow 4.4.3 Extensional flow 4.4.4 Viscoelastic functions 4.4.5 Oscillatory testing 4.4.6 Electrorheology Viscosity of solutions Viscosity of emulsions 4.6.1 Viscosity of dilute emulsions 4.6.2 Viscosity of concentrated emulsions 4.6.3 Rheological properties of flocculated emulsions Viscosity of suspensions Rheological properties of gels 4.8.1 Fractal structure of gels 4.8.2 Scaling behaviour of the elastic properties of colloidal gels 4.8.3 Classification of gels with respect to the nature of the structural elements Rheological properties of sweets 4.9.1 Chocolate mass 4.9.2 Truffle mass 4.9.3 Praline mass 4.9.4 Fondant mass

Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

98 98 98 100 103 104 105 105 105 107 108 109 109 109 126 132 141 144 144 146 146 147 148 149 151 151 152 153 156 156 162 163 163

98

Confectionery and Chocolate Engineering: Principles and Applications

4.9.5 Dessert masses 4.9.6 Nut brittle (croquante) masses 4.9.7 Whipped masses 4.10 Rheological properties of wheat flour doughs 4.10.1 Complex rheological models for describing food systems 4.10.2 Special testing methods for rheological study of doughs 4.10.3 Studies of the consistency of dough Further reading

4.1

164 165 166 166 166 170 172 175

Rheology: its importance in the confectionery industry

The rheological properties of sweets play an important role both in engineering calculations for the manufacture of sweets and in the evaluation of the quality of sweets. Heat and mass transfer processes are strongly influenced by the rheological properties (viscosity, yield stress etc.) of sweets. On the other hand, the textural characteristics of sweets are of paramount importance. A study of the rheological properties of sweets demands a thorough knowledge of rheology because sweets, as a consequence of their great variety, are typical examples of fluids that require complex rheological models to represent them.

4.2

Stress and strain

Rheology is the science of the flow and deformation of matter, and therefore a good understanding of stress and strain is an essential prerequisite for a study of rheology.

4.2.1

Stress tensor

Stress, defined as a force per unit area and usually expressed in pascals (Pa ≡ N/m2), may be tensile, compressive (= ‘negative tensile’) or shear. Nine separate quantities are required to completely describe the state of stress in a material. A small elementary cube (Fig. 4.1) may be considered in terms of Cartesian coordinates (x, y and z or, in another notation, x1, x2 and x3). Stresses are indicated by σij, where the first subscript refers to the normal to the face upon which the force acts, and the second subscript refers to a direction tangential to the face if i ≠ j; if i = j, both superscripts refer to the normal direction. Therefore, σ11 is a normal (tensile or compressive) stress acting in the plane perpendicular to x1, in the direction of x1, and σ23 is a shear stress acting in the plane perpendicular to x2, in the direction of x3. Normal stresses are considered to be positive where they act outward (acting to create a tensile stress) and negative where they act inward (acting to create a compressive stress). Stress components may be summarized as a stress tensor written in the form of a matrix, ⎡σ 11 σ 12 σ 13 ⎤ t = σ ij = ⎢σ 21 σ 22 σ 23 ⎥ ⎥ ⎢ ⎢⎣σ 31 σ 32 σ 33 ⎥⎦

(4.1)

The rheology of foods and sweets

99

szz tyz txz

tzy tzx syy tyx

txy

sxx

z

y

x Fig. 4.1

Typical stresses on a cube.

Another notation for a stress tensor is ⎡σ 11 τ12 τ13 ⎤ Φ ij = ⎢ τ 21 σ 22 τ 23 ⎥ ⎥ ⎢ ⎢⎣ τ 31 τ 32 σ 33 ⎥⎦

(4.2)

This latter notation emphasizes the difference between the normal stresses (σii) and the shear stresses (τij, i ≠ j). Thus, the usual notation for a yield stress is τ0 because it is a kind of shear stress – in general, σ is used for the notation of both normal and shear stresses; however, τ is used for notating shear stresses exclusively. Also, a common notation is σii = σi, so, for example, σ11 = σ1, which relates to the diagonal elements of the tensor. The basic laws of mechanics can be used, by considering the moments about the axes under equilibrium conditions, to prove that the stress matrix (σij or Φij) is symmetrical:

σ ij = σ ji

(4.3)

i.e.

σ 12 = σ 21 σ 31 = σ 31 σ 32 = σ 23 meaning that here are only six (6 = 9 − 3) independent components in the stress tensor represented in Eqn (4.1).

100

Confectionery and Chocolate Engineering: Principles and Applications

4.2.2

Cauchy strain, Hencky strain and deformation tensor

Strain is a consequence of a stress in the normal direction (i.e. a tensile or compressive stress). Let us consider a specimen of initial length L0 which is extended to a length L = L0 + δL. Since this deformation is small, i.e. infinitesimal (δL), it may be thought of in terms of the Cauchy strain (also called the engineering strain),

εC =

δL L = −1 L0 L0

(4.4)

However, for large deformations, the Hencky strain (also called the true strain) is commonly used; this is determined by evaluating an integral from L0 to L:

εH =

L

⎛ ⎞ = ln ⎜ ⎟ ∫ ⎝ L0 ⎠ L0 L dL

L

(4.5)

The choice of which strain measure to use is largely a matter of convenience, and one can be calculated from the other:

ε H = ln (1 + ε C )

(4.6)

Since the Taylor series of ln(1 + εC) is

ε H = ln (1 + ε C ) = ε C −

(ε C )2 2

+

(ε C )3 3

, etc. ( if 0 ≤ ε C < 1)

the Cauchy and Hencky strains are approximately equal at small strains: εH ≈ εC. Evidently, εH and εC are dimensionless ratios. The Hencky (true) strain is important in food engineering. Another type of deformation commonly found in rheology is simple shear (or simple shear strain) as a consequence of a shear stress. This can be illustrated by considering a rectangular bar of height h. The lower surface is stationary and the upper surface is displaced linearly by an amount equal to δL. Each element is subject to the same level of deformation, so the size of the element is not relevant. The angle of shear, γ, may be calculated as tg γ =

δL h

(4.7)

For small deformations, the angle of shear (in radians) is equal to the shear strain (also symbolized by γ), i.e. tan γ = γ. A related strain tensor can also be expressed in matrix form, ⎡ ε11 ε12 εij = ⎢ε 21 ε 22 ⎢ ⎣⎢ ε31 ε32

ε13 ⎤ ε 23 ⎥ ⎥ ε33 ⎥⎦

(4.8)

The rheology of foods and sweets

B

u(x +

r)

B′

r′

r A

101

u(x)

A′

x

0 Fig. 4.2

Interpretation of the strain tensor.

which includes the strains in both the normal direction (the diagonal positions in the matrix) and the tangential direction. For an interpretation of the strain tensor, let us consider Fig. 4.2. In a deformation experiment, for example a tensile test, AB changes to A′B′. In a small vicinity of the point A, i.e. for small deformations, a Taylor expansion can be used: u ( x + r ) = u ( x ) + D ( x ) r or

u (x + r) − u (x) du = D (x) = r dr

where D(x) is a derivative tensor (the gradient tensor or strain tensor). In detail,

D (x) =

⎡( ∂ux ∂x ) ( ∂ux ∂y ) ( ∂ux ∂z ) ⎤ du = [ uij ] = ⎢( ∂uy ∂x ) ( ∂uy ∂y ) ( ∂uy ∂z )⎥ ⎥ ⎢ dx ⎢⎣ ( ∂uz ∂x ) ( ∂uz ∂y ) ( ∂uz ∂z ) ⎥⎦

For small deformations, the symmetric deformation tensor can be used: E = εij = (1 2 ) ( D + DT ) or, in detail,

γ 12 2 γ 13 2 ⎤ ⎡ ε11 ⎢ E = εij = γ 21 2 ε 22 γ 23 2 ⎥ ⎥ ⎢ ⎢⎣γ 31 2 γ 32 2 ε 33 ⎥⎦

(4.9)

where DT is the transposed matrix of D, i.e. (D)ij = (DT)ji, and, for the mixed entries, γ ij = γ ji (i ≠ j). The following motion function is important for studying deformations: x = a (t ) + R (t ) D (t ) X

(4.10)

102

Confectionery and Chocolate Engineering: Principles and Applications

where t is the time coordinate, x = x(X, t) is a vector describing the motion of a point P0, a(t) is a vector describing translation, R(t) is a tensor describing rotation, D(t) is a tensor describing deformation without rotation and X is the material (or Lagrange) coordinates (in three dimensions). If there is no rotation, the following holds: D=E+I

(4.11)

where I is the unit tensor, ⎡1 0 0 ⎤ I = ⎢0 1 0 ⎥ ⎥ ⎢ ⎢⎣0 0 1 ⎥⎦

(4.12)

If there is no translation, it follows from Eqn (4.10) that x = RD = dR

(4.13)

where d is another form of the deformation tensor. The equation RD = dR is a result of the polar decomposition theorem of Cauchy; for further details, see Verhás (1985, Appendix F5). The product RD means rotation + deformation, and dR means deformation + rotation; in general, RD ≠ DR for products of matrices (the matrix product is not commutative in general). According to the polar decomposition theorem, every tensor T can be decomposed into a product of an isometric tensor M and a symmetric tensor S or R: T = MS = RM, where the isometric property is defined by the equation v = (Mr)2 = r2, and v and r are vectors (‘isometric’ means ‘measure-conservative’). Equation (4.10) describes the motion around a point P0. However, in order to study the change of volume around the point P0, let us consider the equation dV = jdV0 = det R det D dV0

(4.14)

where j is the Jacobi determinant (0 < j < ∞): ⎡ ( ∂x1 ( ∂X1 ) ( ∂x1 ( ∂X 2 ) ( ∂x1 ( ∂X 3 ) ⎤ j = ⎢( ∂x2 ( ∂X1 ) ( ∂x2 ( ∂X 2 ) ( ∂x2 ( ∂X 3 )⎥ ⎥ ⎢ ⎢⎣( ∂x3 ( ∂X1 ) ( ∂x3 ( ∂X 2 ) ( ∂x3 ( ∂X 3 ) ⎥⎦

(4.15)

A constant value of j represents the fact that the material cannot be annihilated. Since det R = 1, the following holds from Eqn (4.14): dV = det D dV0

(4.16)

Let us define the factor λV = (det D)1/3; we can then write ⎛ 1 ⎞ ⎛ 1 ⎞ D 0 = ⎜ ⎟ D and d 0 = ⎜ ⎟ d ⎝ λV ⎠ ⎝ λV ⎠

(4.17)

The rheology of foods and sweets

4.2.3

103

Dilatational and deviatoric tensors; tensor invariants

Both the stress and the strain tensors can be broken down into two tensors: the dilatational and the deviatoric tensor. If we introduce the notation q = (V′ − V)/V for the specific volume change, the following holds: q ≈ ε11 + ε 22 + ε 33 = tr ε ij

(4.18)

The average specific strain can be expressed as

ε* =

tr ε ij 3

(4.19)

For a pure volume change, DDil = ε*I

(4.20)

where DDil is the dilatational tensor and I is the unit tensor. For a pure deformation, DDev = εij − ε*I

(4.21)

In detail,

DDev

⎡ε11 − ε γ 13 2 ⎤ * γ 12 2 ⎢ ⎥ = εij − ε*I = ⎢ γ 21 2 ε 22 − ε* γ 23 2 ⎥ ⎢ ⎥ γ 32 2 ε33 − ε* ⎥⎦ ⎢⎣ γ 31 2

(4.22)

This decomposition is not done simply for its own sake: Hamann (1983) noted that it may be important in the study of material failure because some foods, particularly those that are incompressible, may not be sensitive to volume-changing stresses but may be very sensitive to shape-changing stresses. For example, butter and gelled egg white, which are nearly incompressible, are unaffected by hydrostatic pressure (the deviatoric part is practically zero); however, if they are deformed in terms of shape (the deviatoric part is not zero), they eventually break. Shear stresses are deviatoric, and if we have the condition called ‘pure shear’, the overall effect is a change in shape with negligible change in volume. A decomposition of the stress tensor similar to Eqn (4.21) can also be given: TDil = σ *I TDev = σ ij − σ *I 4.2.3.1

(4.23) (4.24)

Scalar invariants of a tensor

In the theory of tensors of second rank, the eigenvalue problem is important; this means the solution of the equation

104

Confectionery and Chocolate Engineering: Principles and Applications

AI = λ I or ( A − λ I ) = 0

(4.25)

where A = [Aij] is a tensor of second rank, I is the unit tensor, 0 is the zero tensor and λ is a scalar. By calculating det (A − λI) = 0, the following equation can be obtained:

λ 3 − S1λ 2 + S2 λ − S3 = 0

(4.26)

where the scalar invariants of the tensor A are S1 = A11 + A22 + A33 = tr A

(4.26a)

S2 = ( A11A22 − A12 A21 ) + ( A11A33 − A13 A31 ) + ( A22 A33 − A23 A32 )

(4.26b)

S3 = A11A22 A33 = det A

(4.26c)

Some further important relationships are S1 = λ1 + λ2 + λ3 1 1 (tr A )2 − (tr A 2 ) 2 2 1 1 1 3 3 2 S3 = λ11λ 22 λ33 = (tr A ) − (tr A ) tr A + (tr A ) 3 2 6 S2 = λ1λ2 + λ1λ3 + λ 2 λ3 =

(4.27a)

where λ1, λ2 and λ3 are the solutions of Eqn (4.24), called the eigenvalues of the tensor A. The eigenvalues can be used to represent the normal stresses of the tensor A in the form ⎡λ1 0 A = ⎢ 0 λ2 ⎢ ⎢⎣ 0 0

0⎤ 0⎥ ⎥ λ3 ⎦⎥

(4.27b)

The normal stresses and the scalar invariants are important in the theory of plasticity. It is relevant to note that in the case of a symmetric tensor the characteristic equation (Eqn 4.26) always has three real roots; furthermore, the eigenvectors corresponding to them are orthogonal to each other.

4.2.4

Constitutive equations

Equations that show a relationship between stress (σij) and strain (εij) are called rheological equations of state or constitutive equations. In the case of complex materials, these equations may include other variables such as time, temperature and pressure. The word rheogram refers to a graph of a rheological relationship, for example strain vs stress (for solids) or shear rate vs shear stress (for fluids; see later). Rheometry investigates the rheological properties of materials under special conditions for which the rheological equations can be applied in a relatively simplified, possibly scalar, form, for example a flow curve (shear stress vs shear rate).

The rheology of foods and sweets

4.3

105

Solid behaviour

4.3.1

Rigid body

The definitive equation for a rigid (or Euclidean) body is d=I

(4.28)

i.e., for a rigid body, no deformation takes place during its movement. (Euclidean geometry can be interpreted as a physical theory dealing with the motion of rigid bodies.) In other words, the condition for deformation is d − I ≠ 0.

4.3.2

Elastic body (or Hookean body/model)

When a force is applied to a solid material and the resulting stress vs strain curve is a straight line through the origin, the material obeys Hooke’s law. This relationship may be stated for shear stress and shear strain as

σ 12 = μγ

(4.29)

where μ (in Pa) is the shear modulus. Hookean materials do not flow, and are linearly elastic. The stress remains constant until the strain is removed, and the material then returns to its original shape. Sometimes shape recovery, though complete, is delayed by atomic-level processes. This time-dependent, or delayed, elastic behaviour is known as elasticity. Hooke’s law can be used to describe the behaviour of many solids (e.g. steel, eggshell, dry pasta and hard candy) when they are subjected to small strains, typically such that γ12 < 0.01. Large strains often produce brittle fracture or nonlinear behaviour. The behaviour of a Hookean solid may be investigated by studying the uniaxial compression of a cylindrical sample (Fig. 4.3). If a material is compressed so that it experiences a change in length and radius, such that h0 → (h0 − δh) and R0 → (R0 + δR), then the normal stress and strain may be calculated as follows (assuming that the vertical direction in the figure is denoted by x2):

σ 22 = εC =

F πR02

δh h0

(4.30) (4.31)

where the absolute value of δh is used. From Eqns (4.30) and (4.31), Young’s modulus, E (Pa), also called the modulus of elasticity, can be determined: E=

σ 22 εC

(4.32)

106

Confectionery and Chocolate Engineering: Principles and Applications

F δh < 0 R0

ho δR > 0

Fig. 4.3

Uniaxial compression of a cylindrical sample.

If the deformations are large, the Hencky strain εH should be used to calculate the strain, and the stress calculation should be adjusted for the increase in radius caused by the compression:

σ 22 =

F 2 π ( R0 + δR )

(4.33)

In the case of axial compression, if the deformation is large, not only the normal strain in the x2 direction but also the tangential shear strains in the x1 and x3 directions should be taken into account. A critical assumption in these calculations is that the sample remains cylindrical in shape. For this reason, lubricated contact surfaces are often recommended when materials such as food gels are tested. Young’s modulus may also be determined by flexural testing of beams; for a detailed description, see Steffe (1996). In addition to Young’s modulus, Poisson’s ratio ν can be defined from compression data: v=

lateral strain δR R0 = axial strain δh h0

(4.34)

where the absolute value of δh is used. Poisson’s ratio may range from 0 to 0.5. Typically, ν varies from 0.0 for rigid-like materials containing large amounts of air to near 0.5 for liquid-like materials. Values from 0.2 to 0.5 are common for biological materials, with 0.5 representing an incompressible substance such as potato flesh. Tissues with a high level of cellular gas, such as apple flesh, have values closer to 0.2. Metals usually have values of Poisson’s ratio between 0.25 and 0.35; see Steffe (1996, p. 9). If a material is subjected to a uniform change in external pressure, it may experience a volume change. These quantities are used to define the bulk modulus K: K ( Pa ) = change in pressure volumetric strain

(4.35)

The rheology of foods and sweets

107

i.e. the pressure causes a change in volume that is related to the original value of the volume. The bulk modulus of dough is approximately 106 Pa, while the value for steel is 1011 Pa. Another common property, the bulk compressibility, is defined as the reciprocal of the bulk modulus. The definitive equation for a Hookean body is t = ⎣( d − I )

(4.36)

where ⎣ is a tensor function. If our study is limited to isotropic media, the following wellknown form of Hooke’s law is obtained: t = 2 μ ( d − I ) + λ tr ( d − I ) I

(4.37)

where μ and λ are the Lamé parameters. If tr(d − I) = tr E ≈ 0, i.e. the volume change is small, then Eqn (4.37) is simplified; see Eqn (4.29). In the case of incompressibility, div E = tr E = 0

(4.38)

(Definitions: modulus = stress/strain; compliance = strain/stress, i.e. modulus = 1/ compliance.) The following elastic moduli are used for characterizing homogeneous isotropic materials: E = Young’s modulus, μ = shear modulus, ν = Poisson’s ratio, K = bulk modulus (which is an extension of Young’s modulus to three dimensions), λ = first Lamé parameter and M = P-wave modulus (uninteresting from our point of view). When two material constants describing the behaviour of a Hookean solid are known, the others can be calculated using theoretical relationships. For example, the first Lamé parameter can be calculated from

λ=K−

2μ 3

(4.39)

Numerous experimental techniques applicable to food materials may be used to determine Hookean material constants. The methods include testing in tension, in compression and in torsion; see Dally and Riley (1965), Polakowski and Kipling (1966) and Mohsenin (1986). The Hookean properties of some typical materials have been published; see Lewis (1987) [cited by Steffe (1996, Appendix 6.5)].

4.3.3

Linear elastic and nonlinear elastic materials

Both linear elastic materials and nonlinear elastic materials (such as rubber) return to their original shape when the stress is removed. Food may be solid in nature but not Hookean. A comparison of curves for linear elastic (Hookean), elastoplastic and nonlinear elastic materials (Fig. 4.4) shows a number of similarities and differences.

Confectionery and Chocolate Engineering: Principles and Applications

Fluid (H oo kea n)

Yield stress s0

Ela stic

Shear stress s12

108

tic

las

re nea

nli

No

Deformation g Fig. 4.4

Deformation curves for linear elastic (Hookean), elastoplastic and nonlinear elastic materials.

An elastoplastic material has a Hookean-type behaviour below the yield stress (σ0) but flows to produce permanent deformation above that value. Margarine and butter, at room temperature, may behave as elastoplastic substances. The method of investigation of this type of material, as a solid or as a fluid, depends on whether the shear stress is above or below σ0. It is worth noting, however, that typical elastic materials, such as steel and rubber, also have a yield stress, although it has a very high value and the elastic (linear) section in Fig. 4.4 is much steeper than in the case of foods. Furthermore, to fully distinguish fluid-like from solid-like behaviour, the characteristic time of the material and the characteristic time of the deformation process involved must be considered simultaneously. The Deborah number, proposed by Reiner, has been defined to address this issue (see Steffe 1996, p. 332). Food rheologists also find the failure behaviour of solid food (particularly brittle materials and firm gels) to be very useful because such data sometimes correlate well with the conclusions of human sensory panels (Kawanari et al. 1981; Hamann 1983; Montejano et al. 1985). A typical characteristic of brittle solids is that they break when given a small deformation. Hamann (1983) summarized an evaluation of the structural failure of solid foods in compression, torsion and sandwich shear modes. The jagged force–deformation relationships of crunchy materials may offer alternative texture classification criteria for brittle or crunchy foods (Ulbricht et al. 1995; Peleg and Normand 1995).

4.3.4

Texture of chocolate

Tscheuschner (2008) investigated the texture of chocolate with an Instron-type instrument (from Kögel u.a.). If the height of a sample body of cylindrical form is H0 and the reduction under the effect of compression is ΔH, the compression can be defined by the ratio K = ΔH/H0. If the cross-sectional area of the probe body is A and the force of compression is F, then the pressure of compression can be defined by P = F/A. If Fm is the maximum force needed for breaking the sample, then the solidity with respect to biting can be defined by Pm = Fm/A. By studying plots of P vs K, it can be observed that the curves, starting from the origin, go through a maximum value which provides a measure of the biting solidity for the given conditions (recipe, temperature etc.); the value of P finally decreases. The typical range

The rheology of foods and sweets

109

of P is 0–2.5 × 106 Pa, and that of K is 0–30%. If the velocity of compression is increased, the maximum of P (Pm) is displaced to higher values of K. According to Tscheuschner’s study, there is a linear relationship between the sensory biting solidity IB and the instrumentally measured solidity Pm, which has the form IB = 1.03Pm + 0.91. Moreover, he determined that, at constant values of compression and temperature, the decrease in the height h(t) of the sample as a function of the time t of melting gives a linear plot of h(t) vs t. If the duration of melting is fixed (e.g. 20 min), then the values hA for this time interval are characteristic of the melting of the chocolate. When this measured behaviour of chocolate was compared with the sensory behaviour (characterized by IM) in the mouth, a linear relationship was found, IM = 0.9hA + 0.14. Both IB and IM may provide a basis for classifications of sensory quality. For further details, see Tscheuschner (2008).

4.4

Fluid behaviour

4.4.1

Ideal fluids and Pascal bodies

The model of ideal fluids is based on Pascal’s theorem, according to which no shear stress evolves during the movement of ideal fluids. Consequently, the Cauchy stress tensor t is simplified to a spherical tensor t = − pI

(4.40)

where p is the scalar pressure and I is the unit tensor. The other principal equation expresses the incompressibility of ideal fluids. If the equation of continuity is dρ + ρ div v = 0 dt

(4.41)

where ρ is the fluid density and v is the velocity vector of the fluid, then the following holds for an incompressible fluid: div v = 0 and

dρ = 0 → ρ = ρ (t ) = constant dt

(4.42)

Equations (4.40) and (4.42) determine the flow properties of ideal fluids, or Pascal bodies.

4.4.2 4.4.2.1

Fluid behaviour in steady shear flow Simple steady shear flow; Newtonian fluids

To interpret the deformation rate, let us consider the time derivative of the deformation tensor E = εij = (1/2)(D + DT):

110

Confectionery and Chocolate Engineering: Principles and Applications

V (t, x ) =

dE d ( u r ) 1 = = ( D + DT ) dt dt 2

(4.43)

where D = dv/dx, i.e. the velocity gradient. For isotropic fluids, the constitutive equation of a Newtonian body (or model) can be given as t = 2η (Grad v ) + (ηV div v − p ) I S

(4.44)

where t is the stress tensor (Pa), η is the shear viscosity (Pa s), (Grad v)S is the symmetric part of the velocity gradient tensor (s−1), ηV is the volumetric viscosity (Pa s), div v is the divergence of the velocity (s−1) and p is the static pressure (Pa). For a Newtonian model, the velocity gradient tensor can be given in the form ⎡0 ε ′ 0 ⎤ Grad v = ⎢0 0 0 ⎥ ⎥ ⎢ ⎢⎣0 0 0 ⎥⎦

(4.45)

where ε′ ≡ dε/dt is the shear rate. Every asymmetric tensor can be constructed from a symmetric and an antisymmetric tensor; when this is done for Grad v, we obtain

ε ′ 2 0⎤ ⎡ 0 ε ′ 2 0⎤ ⎡0 ε ′ 0 ⎤ ⎡ 0 ⎢0 0 0 ⎥ = ⎢ε ′ 2 0 0⎥ 0 0 ⎥ + ⎢ −ε ′ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢⎣0 0 0 ⎥⎦ ⎢⎣ 0 0 0 ⎥⎦ 0 0 ⎥⎦ ⎢⎣ 0 symmetric tenso or

antisymmetric tensor

In Eqns (4.36) and (4.43), only the symmetric tensor is taken into account. If the fluid is incompressible (div v = 0), the static pressure can also be neglected (p = 0), and then the stress tensor has the form ⎡ 0 ηε ′ 0 ⎤ t = ⎢ηε ′ 0 0 ⎥ ⎥ ⎢ ⎢⎣ 0 0 0 ⎥⎦

(4.46)

If only a single component differs from zero, the well-known equation of a flow curve is obtained,

τ12 = ηε ′

(4.47)

Let us consider Cauchy’s equation of motion, which is an application of Newton’s second law to continua:

ρ

dv = Div t + ρf dt

(4.48)

The rheology of foods and sweets

111

Area Force

u

Velocity h

x2

profile x1

Fig. 4.5

u=0

Velocity profile of fluid between parallel plates [reproduced from Steffe (1996), with permission].

On the left side of Eqn (4.48), we have (mass × acceleration)/volume; on the right side, we have forces acting on the surface/volume (Div t) + forces acting in the volume/volume. If the value of t in Eqn (4.44) is substituted into Eqn (4.48), we obtain

ρ

dv − ρf − grad p − η div grad v = 0 dt

(4.49)

assuming that the fluid is incompressible (div v = 0, i.e. ηV can be neglected). Equations (4.41) and (4.49) are the Navier–Stokes equations in a simpler form. Fluids may be studied by subjecting them to continuous shearing at a constant rate. Ideally, this can be accomplished using two parallel plates with a fluid in the gap between them (Fig. 4.5). The lower plate is fixed and the upper one moves at a constant velocity u, which can be thought of as an incremental change in position L divided by a small interval of time t, i.e. δL/δt. A force per unit area on the plate is required for motion, resulting in a shear stress σ21 on the upper plate, which, conceptually, could also be considered to be a layer of fluid. The flow described above is steady simple shear, and the shear rate (also called the strain rate) is defined as the rate of change of strain:

γ′≡

dγ d δL⎞ u = ⎛ = ( velocity gradient ) dt dt ⎝ h ⎠ h

(4.50)

Other symbols for the shear rate are ∂v/∂r (where r is the radius of a tube) and D. This experimental arrangement is a realization of Newton’s original definition of viscosity, which may be applied to streamline (laminar) flow between parallel plates:

η = η (γ ′ ) =

σ 21 γ′

(4.51)

where σ21 is an element of the stress tensor. In steady, simple shear flow the coordinate system may be oriented parallel to the direction of flow so that the stress tensor given by Eqns (4.1) and (4.4) reduces to ⎡σ 11 σ 12 0 ⎤ σ ij = ⎢σ 21 σ 22 0 ⎥ ⎥ ⎢ 0 σ 33 ⎥⎦ ⎣⎢ 0

(4.52)

112

Confectionery and Chocolate Engineering: Principles and Applications

Methods of applying this definition for different types of viscometer are described by, for example, Steffe (1996, Chapter 1). Spreading (of creams or spreads) and brushing operations are frequently found in the food industry. In this case, the maximum shear rate can be estimated from the velocity u of the brush or knife divided by the thickness z of the coating:

γ max ′ =

u z

(4.53a)

The idea of ‘maximum speed divided by gap size’ can be useful for estimating the shear rates found in particular applications such as brush coating (Steffe 1996). The shear rate induced by mixing can be calculated by the Metzner–Otto method (Metzner and Otto 1957) according to the formula

γ ′ = Kn

(4.53b)

where n is the revolution rate (in s−1) and K is the Metzner–Otto constant. For details of the determination of K, see Windhab (1993). Simple shear flow is called viscosimetric flow. It includes: • • • •

axial flow in a tube; rotational flow between concentric cylinders; rotational flow between a cone and a plate; torsional flow (and also rotational flow) between parallel plates.

In viscosimetric flow, three shear-rate-dependent material functions, collectively called viscometric functions, are needed to completely establish the state of stress in the fluid, since they contain all the elements of the stress tensor given by Eqns (4.45) and (4.53). These may be described as: the viscosity function,

η = η (γ ′ ) =

σ 21 γ′

(σ 21 = σ12 )

(4.54)

the first normal stress coefficient,

Φ1 = Φ1 (γ ′ ) =

σ 11 − σ 22 N1 = (γ ′ )2 (γ ′ )2

(4.55)

and the second normal stress coefficient,

Φ 2 = Φ 2 (γ ′ ) =

σ 22 − σ 33 N2 = (γ ′ )2 (γ ′ )2

(4.56)

The first normal-stress difference N1 = σ11 − σ22 is always positive and is assumed to be approximately 10 times greater than the second normal-stress difference N2 = σ22 − σ33.

113

The rheology of foods and sweets

Table 4.1 Values of steady shear and normal-stress differencesa [from Dickie and Kokini (1983), cited by Steffe (1996), with kind permission of Professor James F. Steffe]. Product

K (Pan)

n

K′ (Pa sm)

Apple butter Panned frosting Honey Ketchup Marshmallow cream Mayonnaise Mustard Peanut butter Stick butter Stick margarine Squeezable margarine Tub margarine Whipped butter Whipped cream

222.90 355.84 15.39 29.10 563.10 100.13 35.05 501.13 199.29 297.58 8.68 106.68 312.30 422.30

0.145 0.117 0.989 0.136 0.379 0.131 0.196 0.065 0.085 0.074 0.124 0.077 0.05? 0.05

156.03 816.11

0.566 0.244

39.47 185.45 256.40 65.69 3785.00 3403.00 3010.13 15.70 177.20 110.76 363.70

0.258 0.127 −0.048 0.136 0.175 0.393 0.299 0.168 0.358 0.476 0.418

m

The constants K and n are those of the Ostwald–de Waele model, σ = K(γ ′)n; N = σ11 − σ22 = K′(γ ′)m.

a

Table 4.2 Shear rates typical of familiar materials and processes. Process/situation Extrusion, pipe flow Dough sheeting, dip coating Mixing/stirring

Shear rate (s−1) 1–1000 10–100 10–1000

Measurement of N2 is difficult; fortunately, the assumption that N2 = 0 is usually satisfactory. The ratio N1/σ12, known as the recoverable shear (or recoverable elastic strain), has proven to be a useful parameter in modelling die swell phenomena in polymers (Tanner 1988). Some data on the N1 values of fluid foods have been published (Table 4.1). If a fluid is ideally Newtonian, • η(γ ′) is a constant, and equal to the Newtonian viscosity; and • the first and second normal-stress differences (N1 and N2) are zero. As γ ′ approaches zero, elastic fluids tend to display Newtonian behaviour. Table 4.2 shows several shear-rate regions that are important in food processing. 4.4.2.2

Classification of rheological behaviour

Classifying fluids is a valuable way to conceptualize fluid behaviour. However, we do not mean to imply that the types of behaviour noted in Figs 4.6–4.8 are mutually exclusive. Every type of model is characterized by two plots:

114

Confectionery and Chocolate Engineering: Principles and Applications

Hookean body

e

e

Non-Hookean body

Reversible e

t e

t e

t e

t

t

t

(Ideal) Fig. 4.6

Types of elastic deformation.

e

Plastoelastic

e

t ∂e/∂t

Viscoplastic (ideal Bingham body) e

t ∂e/∂t

t Fig. 4.7

Plastic (non-elastic)

t ∂e/∂t

t

t

Types of plastic deformation.

• deformation (ε) vs time (t); and • deformation (ε) vs stress (τ) for elastic deformation (Fig. 4.6); in addition, there is a plot of shear rate (∂ε/∂t) vs shear stress (τ) (called the flow curve) for plastic (Fig. 4.7) and viscous (Fig. 4.8) deformation. All three types of rheological behaviour (elastic, plastic and viscous) are ideally manifested in the so-called ideal bodies (the ideal Hookean, ideal Bingham and ideal Newtonian bodies), for which both plots are linear. The plastic models have the typical characteristic that in a plot of ∂ε/∂t vs τ, the curve (or, in the case of an ideal Bingham body, the line) does not start from the origin but from a value τ0, called the yield stress. Among the plastic and viscous models, there are types which also have elastic properties under the usual technological circumstances. It is, however, worth emphasizing that under extreme conditions (e.g. a sudden, very strong stress), even such typical viscous media as water behave as elastic rather than viscous materials.

The rheology of foods and sweets

Viscoelastic e

e

Viscous (non-elastic)

e

t ∂e/∂t

t

Fig. 4.8

Ideal (Newton body)

t

∂e/∂t

115

t ∂e/∂t

t

t

Types of viscous deformation.

A further classification of the various models emphasizes the time- independent/ dependent or shear-thinning/thickening behaviour of a fluid, and the ‘mixing’ of these behaviours; for example, a material showing elastic behaviour (such as dough) may simultaneously show shear thinning and time-dependent behaviour. Other factors, such as ageing, may also influence the rheological behaviour. In the following sections, mathematical models will be presented which describe the rheological behaviour of various types of fluid. 4.4.2.3

Mathematical models for inelastic fluids

Time-independent material functions for viscous and plastic fluids The elastic behaviour of many fluid foods is small or can be neglected (materials such as dough are an exception), leaving the viscosity and plasticity function as the main area of interest. The ideal Newton model can be described by the equation

τ = ηD

(4.57)

where τ is the shear stress (Pa), η is the dynamic viscosity (Pa s) = constant and D is the shear rate (s−1). In the case of generalized Newton models, η depends on the shear rate or shear stress. Fluids that have a yield stress τ0 ≥ 0 are called Bingham fluids, in general. The flow curve can be described by the equation

τ = ηp1γ ′ + τ 0

(4.58)

where ηpl is the plastic (or Bingham) viscosity (Pa s), which is constant in the case of an ideal Bingham body. In the case of generalized Bingham fluids, ηpl is not constant but is dependent on the shear rate or shear stress.

116

Confectionery and Chocolate Engineering: Principles and Applications

Bingham fluids (ideal) (ideal)

Shear stress (Pa)

ing

nn thi

r-

ea

Sh Yield

Newtonian fluids

ing

nn

stress

i -th ar

g

e

Sh

thic ear-

in ken

Sh

0

0

Shear rate (s–1)

Fig. 4.9 Flow curves for typical time-independent fluids. The flow curves of Newtonian and Bingham fluids are straight only in the ideal case; otherwise, they are convex or concave.

Figure 4.9 demonstrates the flow curves of Newtonian and Bingham fluids; both types may show either shear-thinning or shear-thickening behaviour. The flow curves of Newtonian and Bingham fluids are straight only in the ideal case; in other cases, they are convex or concave. One of the mathematical models used to describe the behaviour of certain types of generalized Newtonian fluids and generalized Bingham fluids is the Herschel–Bulkley model [or Herschel–Bulkley–Porst–Markowitsch–Houwink (HBPMH) or generalized Ostwald–de Waele model],

τ = KD n + τ 0

(4.59)

where K ( > 0) is the consistency coefficient, n (> 0) is the flow behaviour index and τ0 (≥ 0) is the yield stress. One special case of the Herschel–Bulkley model is the Ostwald–de Waele (or power-law) model, where τ0 = 0, i.e. the Ostwald–de Waele fluid is a type of Newtonian fluid. The Herschel–Bulkley model is appropriate to many fluid foods. K is commonly called the viscosity (η) or the plastic viscosity (ηPl). Shear-thinning and shear-thickening behaviour of time-independent Newtonian and Bingham fluids can be distinguished by the following relations (where the shear rate is denoted here by D): • ∂η/∂D > 0 corresponds to shear-thickening behaviour, or dilatancy – there is a convex flow curve of τ vs D; • ∂η/∂D < 0 corresponds to shear-thinning, or pseudo-plastic, behaviour – there is a concave flow curve. For further details, see Langer and Werner (1981), Ellenberger et al. (1984), Tebel and Zehner (1985), Henzler (1988), Grüneberg and Wilk (1992), Schmerwitz (1992), Schnabel and Reher (1992) and Steffe (1996, Appendix 2).

The rheology of foods and sweets

117

Apparent viscosity The apparent viscosity has a precise definition. It is, as in Eqn (4.51), the shear stress divided by the shear rate:

ηapp = ηapp ( D ) =

τ 21 ⎛ τ ⎞ = D ⎝ D⎠

(4.60)

(Below, we shall sometimes use the notation τ21 ≡ τ.) For Herschel–Bulkley fluids, the apparent viscosity is determined in a like manner from Eqns (4.59) and (4.60). Therefore

ηapp =

τ τ = KD n−1 + 0 D D

(4.61)

During flow, materials that show shear-thinning behaviour may exhibit three distinct regions: The lower Newtonian region, where the apparent viscosity η0, called the limiting viscosity at zero shear rate, can be regarded as constant as the shear rate is varied. The lower Newtonian region may be relevant in problems involving low shear rates, such as those related to the sedimentation of fine particles in fluids. The middle region, where the apparent viscosity η changes with shear rate (decreasing for shear-thinning fluids) and the power-law equation is a suitable model for the phenomenon. The middle region is most often examined when the performance of foodprocessing equipment is considered. The upper Newtonian region, where the slope of the curve η∞, called the limiting viscosity at infinite shear rate, can again be regarded as constant as the shear rate is varied. When the flow of an Ostwald–de Waele fluid in a tube is studied, the so-called consistency variables are defined: P ( Pa ) =

ΔpR 2L

and V (s −1 ) =

4Q R3 π

(4.62)

where R (m) and L (m) are the radius and length, respectively, of the tube, Δp (Pa) is the pressure difference between the two ends of tube and Q (m3/s) is the flow rate. The usual V vs P plot is of the form presented in Fig. 4.10, where tan α =

1 η0

and tan β =

1 η∞

(4.63)

Time-dependent material functions of plastic and viscous fluids Time-dependent materials are considered to be inelastic with a viscosity function which depends on time. The response of the substance to a stress is instantaneous, and the timedependent behaviour is due to changes in the structure of the material itself. In contrast, the time-dependent effects found in viscoelastic materials arise because the response of the stress to an applied strain is not instantaneous and is not associated with

118

Confectionery and Chocolate Engineering: Principles and Applications

V

b

a P Fig. 4.10 Shear-thinning behaviour of an Ostwald–de Waele fluid, represented by a plot of the consistency variables.

Table 4.3

Terminology used for fluids with time-dependent behaviour.

Change

Time-thinning

Time-thickening

Reversible Irreversible

Thixotropy Rheomalaxis (rheodestruction)

Rheopexy Rheoretrogradation

a structural change in the material. Also, the timescale of thixotropy may be quite different from the timescale associated with viscoelasticity. The most characteristic effects are usually observed in situations involving short process times. Real materials may be both time-dependent and viscoelastic. Materials with time-dependent characteristics may exhibit either a decreasing or an increasing shear stress (and apparent viscosity) with time at a fixed rate of shear. Both phenomena can be described by the following relations (where the shear rate D is constant): • (∂τ/∂t)D > 0 for fluids with time-thickening behaviour; and • (∂τ/∂t)D < 0 for fluids with time-thinning behaviour. Table 4.3 shows the terminology that is used for fluids with time-dependent behaviour. For example, thixotropy may be observed when the rotor of a rotational viscosimeter turns at a constant angular velocity (D = shear rate ∼ dω/dt) while the measured values relating to the shear stress decrease continuously. Irreversible thixotropy, called rheomalaxis (or rheodestruction), is common in food products, and may be a factor in evaluating the yield stress as well as in the general flow behaviour of a material. (‘Antithixotropy’ and ‘negative thixotropy’ are synonyms for ‘rheopexy’.) The thixotropy of many fluid foods may be described in terms of the sol–gel transition phenomenon. This terminology could apply, for example, to starch-thickened baby food or to yogurt. After being manufactured and placed in a container, these foods slowly develop a three-dimensional network and may be described as gels. When they are subjected to shear (by standard rheological testing or by mixing with a spoon), the structure is broken down (the gel → sol transition) and the material reaches a minimum thickness, where it exists in the sol state. In foods that show reversibility, the network is rebuilt and

The rheology of foods and sweets

119

the gel state is reobtained. Irreversible materials remain in the sol state. When a material is subjected to a constant shear rate, the shear stress will decay over time. During a rest period, the material may completely recover, partially recover or not recover any of its original structure, leading to a high, medium or low torque response, respectively, in the sample. Thixotropic behaviour is common in the confectionery industry; for example, before dosing, filling masses are mixed through in order to obtain the correct viscosity. The structural changes caused by mixing can be linked, for example, to recrystallization, and then breaking and/or solution of large crystals, in the case of fondant used as an ingredient. Torque decay data may be used to model irreversible thixotropy by adding a structural decay parameter λ to the Herschel–Bulkley model to account for breakdown (Tiu and Boger 1974):

τ = λ {KD n + τ 0 }

(4.64)

where λ = λ(t), the structural parameter, is a function of time. λ = 1 before the onset of shearing (t = 0), and an equilibrium value λE < λ is obtained after complete breakdown as a result of shearing, which means irreversibility. The decay of the structural parameter with time may be assumed to obey a secondorder equation, dλ 2 = − k1 ( λ − λE ) dt

(4.65)

where k1 is the rate constant, which is a function of shear rate. From Eqns (4.52), (4.53), (4.61) and (4.64),

λ=

τ KD + τ 0 n

=

ηD ≡ ηA KD n + τ 0

(4.66)

where A ≡ D/{KDn + τ0} = constant (since D and τ0 are constant). Taking into account Eqns (4.65) and (4.66), we can write dλ dη 2 2 =A = − k1 ( λ − λE ) = − k1A2 (η − ηE ) dt dt

(4.67)

where λ(t) → η(t) and λE → ηE. After integrating the differential equations (4.53) and (4.67) with respect to η, the result is 1 1 = + Bt η − ηE η0 − ηE

(4.68)

where η0 is the initial value of the apparent viscosity calculated from the initial shear stress and shear rate (for t = 0 and λ = 1), and B = k1A.

120

Confectionery and Chocolate Engineering: Principles and Applications

Table 4.4 Viscosity data. η (Pa s) 10 9 8.3

Time (s) 0 10 20

Using Eqn (4.68), a plot of 1/(η − ηE) vs t, at a particular shear rate, can be made to obtain B. This is done for numerous shear rates and the resulting information is used to determine the relation between B and γ ′ and the relation between k1 and γ ′. This method supposes also that K (= viscosity) is constant with time at a given shear rate, i.e. the change of the ratio denoted by A can be neglected. Example 4.1 The data for viscosity as a function of mixing time listed in Table 4.4 were obtained. By applying Eqn (4.68), ηE and B (Pa s2)−1 are calculated as follows: Eqn I: Eqn II: Eqn II − Eqn I:

1 1 = + 10 B 9 − ηE 10 − ηE 1 1 = + 20 B 8.3 − ηE 10 − ηE 1 1 1 1 − = 10 B = − 8.3 − ηE 9 − ηE 9 − ηE 10 − ηE

i.e. 1 1 2 + = 8.3 − ηE 10 − ηE 9 − ηE → ηE = 4.33 Pa s and, with substitution from Eqn I, B = 0.00377… (Pa s2)−1. 4.4.2.4

Yield stress phenomena

An important characteristic of Bingham plastic materials is the presence of a yield stress σ0 (another common notation is τ0), which represents a finite stress required to achieve flow. Below the yield stress, the material exhibits solid-like characteristics: it stores energy at small strains, and does not level out under the influence of gravity to form a flat surface. This characteristic is very important in process design and quality assessment for materials such as butter, yogurt and spreads, and also for dipping in chocolate or any other fatty mass. The yield stress is a practical, but idealized, concept. Thickness of a falling film The yield stress is important in the covering of centres, for example covering with chocolate mass. Let us consider a wall which has an angle α to the vertical. The shear stress gradient affecting an infinitesimally thick layer of chocolate mass is

The rheology of foods and sweets

dτ yz = ρ g cos α dy

121

(4.69)

where z is a coordinate directed along the wall, z ⊥ y, ρ is the density of the falling film and g is the gravitational constant (9.81 m/s2). To integrate Eqns (4.25) and (4.69) between the boundaries of the film, we take τ = τ0 from 0 to a, and then a variable value of τ from a to y. Although the flow curve most commonly used for chocolate mass is the Casson equation (see later), we use here the general form for a Bingham fluid for simplicity (see Eqn (4.58), dτ = dτ0 (= 0) + η(dvyz/ dy) dy, which holds also for a Casson body: a+ y

y

dv

y

dv

yz yz ∫ dτ = τ 0 + ∫ η dy dy = aρ g cos α + ∫ η dy dy 0 a a

(4.70)

The flow starts where τ = τ0 (i.e. y = a), which means the following in the case of a vertical wall parallel to the z direction (since z ⊥ y, α = 0, and therefore cos α = 1):

τ 0 = ρ ga

(4.71)

where a is the thickness of the film. For further details, see Szolcsányi (1972, pp. 191–194) and Lásztity (1987a, p. 267). Example 4.2 The yield stress (or yield value) of couverture chocolate is σ0 = 10 Pa; ρ = 1.2 × 103 kg/m3. The thickness of chocolate cover will be (see Eqn 4.60) a=

σ0 10 = ( m ) = 0.849 mm. ρ g 1.2 × 103 × 9.81

The yield stress τ0 may be defined as the minimum stress required to initiate flow. Although the existence of a yield stress has been challenged, there is little doubt from a practical standpoint that σ0 is an engineering reality which may strongly influence process engineering calculations. There are many ways to evaluate the yield stress for fluid-like substances (Steffe 1996, p. 35). Cheng (1986) has written an excellent review of the yield stress problem, and has also described a concept of static and dynamic yield stresses that has great practical value in the rheological testing of fluid foods. Many foods, such as starch-thickened baby food (Steffe and Ford 1985), thicken during storage and exhibit irreversible thixotropic behaviour when stirred before consumption. Chemical changes (e.g. starch retrogradation) cause a weak gel structure to form in the material during storage. This structure is sensitive and is easily disrupted by movement of the fluid. The yield stress measured on an undisturbed sample is the static yield stress. The yield stress of a completely broken-down sample, often determined from extrapolation of the equilibrium flow curve, is the dynamic yield stress (Fig. 4.11). The static yield stress may be significantly higher than the dynamic yield stress. If the material recovers its structure during a short period of time (which is uncommon in

Confectionery and Chocolate Engineering: Principles and Applications

Shear stress

122

Static yield stress Equilibrium flow curve

Dynamic yield stress

Shear rate Fig. 4.11

Static and dynamic yield stresses [reproduced from Steffe (1996, Fig. 1.20, p. 38), with permission].

food products), then a rate parameter may be used to fully describe the rheological behaviour. The idea of a static and a dynamic yield stress can be explained by assuming that there are two types of structure in a thixotropic fluid (Cheng 1986): • One structure is insensitive to shear rate and serves to define the dynamic yield stress associated with the equilibrium flow curve. • A second structure, the weak structure, forms over a certain period of time when the sample is at rest. Combined, the two structures cause a resistance to flow which determines the static yield stress. Yoo et al. (1995) defined a new dimensionless number, the yield number: Yield number =

static yield stress dynamic yield stress

(4.72)

An important issue in the measurement of yield stress, particularly from a quality control standpoint, is reproducibility of the experimental data. This is critical when one is comparing the overall characteristics of products made on different production lines or in different plants. Bingham fluids Bingham fluids have a yield stress, and in certain circumstances they behave both like solids (elastic and/or plastic behaviour if τ ≤ yield stress) and like fluids (if τ > yield stress). Various theoretical descriptions of the solid → plastic → fluid transitions make use of a threshold value denoted by f, which characterizes the plastic state. This value is called the yield stress (see above). f is usually expressed in terms of the scalar invariants of the stress tensor T: f = f(t1, t2, t3). In the plastic state,

The rheology of foods and sweets

f (t1, t2, t3 ) ≥ 0

123

(4.73)

It is known from experiment that f is independent (or nearly independent) of the first scalar invariant (t1 = tr T), i.e. the deviatoric stress (the shearing effect) essentially determines the value of f: f = f(t2, t3). If, also, the effect of t3 can be neglected, the Huber–von Mises criterion is obtained: 2 (σ F ) =0 3 2

f = TDev ⋅ TDev −

(4.74)

where · denotes the scalar product of the deviatoric tensor TDev with itself, and σF is a material property. Equation (4.74), written in detail, is 2 2 2 f = (σ y − σ z ) + (σ z − σ x ) + (σ x − σ y ) + 6 (σ xy + σ yz + σ zx )− 2

2

2

2 (σ F ) =0 3 2

(4.75)

Taking the Huber–von Mises criterion (Eqns 4.74 and 4.75) into account, the constitutive equation of a generalized Bingham fluid can be written as ⎤ 23 ⎛ 1 ⎞⎡ V = ⎜ ⎟ ⎢1 − σ F TDev ⎝ 2η ⎠ ⎣ TDev ⋅ TDev ⎥⎦

(4.76)

where V is the shear rate tensor and η is the dynamic viscosity. Tresca’s criterion can be easily represented by means of Mohr circles (Fig. 4.12). The abscissa is the stress σ in the normal direction, the ordinate is the shear stress τ, and j1 < l2 < l3 are the eigenvalues of the stress tensor T. Tresca’s criterion is defined by the equation

τ cr =

l3 − l1 2

(4.77)

i.e. the critical value of the shear stress is equal to the radius of largest semicircle (see Fig. 4.12).

t

tcrit

s1 Fig. 4.12

s2

Representation of Tresca’s criterion by means of Mohr circles.

s3

s

124

Confectionery and Chocolate Engineering: Principles and Applications

Example 4.3 A stress tensor is given by its elements (in Pa) as follows: ⎡4 5 0 ⎤ σ ij = ⎢5 4 0 ⎥ ⎥ ⎢ ⎢⎣ 0 0 4 ⎥⎦ The characteristic equation of the matrix σij is l3 − 12 × l2 + 23 × l + 36 = 0; the roots (i.e. the normal stresses) are l1 = −1, l2 = 4 and l3 = 9. According to Eqns (4.66) and (4.77),

τ cr =

4.4.2.5

l3 − l1 9 + 1 = = 5 Pa 2 2 Dependence of dynamic viscosity on temperature and pressure

Dependence on temperature The following relationship can be derived on the basis of Eyring’s theory for pure liquid chemical substances: U ⎞ η = A exp ⎛ ⎝ RT ⎠

(4.78)

where A is a constant, U is the activation energy of the viscosity, R is the universal gas constant and T is the temperature (K). The activation energy of diffusion is equal to that of viscosity, and hence U ⎞ kT η = A exp ⎛ = ⎝ RT ⎠ Dδ

(4.79)

where D is the diffusion constant, k is the Boltzmann constant and δ is the distance between neighbouring layers of the fluid. Equation (4.78) gives good agreement with empirical results close to the normal boiling point; however, in a broad region, a plot of ln η vs 1/T is not linear. The following rule can be applied to estimate U: U=

ΔH evap 2.45

(4.80)

where ΔHevap is the latent heat of evaporation. For further details, see Liszi (1975). For the temperature region t = 28–100°C, the viscosity vs temperature function of (crystal-free) cocoa butter has been given by Tscheuschner (1993a) as

η = 5.7 × 10 −7 exp ⎛ ⎝

3533.7 ⎞ (Pa s ) T ⎠

(4.81)

The rheology of foods and sweets

125

If we consider an unknown viscosity η at any temperature T and a reference viscosity ηr at a reference temperature Tr, the constant A can be eliminated from Eqn (4.78), and the resulting equation can be written in logarithmic form: E ⎛1 1⎞ ⎛η⎞ ln ⎜ ⎟ = ⎛ ⎞ ⎜ − ⎟ ⎝ ⎝ ηr ⎠ R ⎠ ⎝ T Tr ⎠

(4.82)

Steffe (1996, Appendices 6.14) gives data on activation energies for fruit juices and various egg products (Ostwald–de Waele fluids); the range of the values is 1.2– 14.2 kcal/g mol at 50°C (1 kcal = 4.1868 kJ). Dependence on pressure The activation energy U for viscosity and diffusion consists of two parts: one is the energy that is needed for the creation of a new ‘hole’, and the other is the potential that has to be passed through by a molecule in order to reach a neighbouring hole. If an external pressure P acts on a liquid, then additional work is needed because not only the cohesive forces but also this pressure has to be compensated. If the volume of a hole v0 is regarded as independent of the external pressure P, then an additional amount of work Pv0 will be added to the activation energy U; consequently, the following relation will be valid for the pressure dependence: Pv η = η0 exp ⎛ 0 ⎞ ⎝ kT ⎠

(4.83)

where η0 is the dynamic viscosity of the fluid at P ≈ 0 (a very small pressure). See also Stephan and Lucas (1979) and Lucas (1981). The effects of shear rate and temperature can be combined into a single expression (Harper and El Sahrigi 1965): E ⎞ η = f (T ; γ ′ ) = KT exp ⎛ (γ ′ )m −1 ⎝ RT ⎠

(4.84)

where m is the average value of the flow behaviour index based on all temperatures, and KT is a constant at a given temperature T. The effects of temperature and concentration C on the apparent viscosity at a constant shear rate can be combined into a single relationship (Vitali and Rao 1984; Castaldo et al. 1990): E ⎞ η = f (T ; C ) = C B KTC exp ⎛ ⎝ RT ⎠

(4.85)

where KTC is a constant at a given temperature T and concentration C, and B is an exponent. The three constants KTC, E and B must be determined from experimental data. The effects of shear rate, temperature and concentration (or moisture content) can also be combined into a single expression (Mackey et al. 1989): E m −1 η = f (T ; γ ′; C ) = KTγ ′ ;C exp ⎛ + BC ⎞ (γ ′ ) ⎝ RT ⎠

(4.86)

126

Confectionery and Chocolate Engineering: Principles and Applications

where the influence of shear rate is given in terms of a power-law function. The parameters KTγ ′;C, m, E and BC cannot be given an exact physical interpretation because the sequence of steps used in determining them influences the magnitude of the constants. The parameters of the equation may be determined using stepwise regression analysis with the assumption that interaction effects (such as the temperature dependence of m and BC) can be neglected. The rheological behaviour of fluid foods is complex and is influenced by numerous factors. The time–temperature history and strain history may be added to Eqn (4.43) to form a more comprehensive equation; see Dolan et al. (1989), Mackey et al. (1989), Morgan (1989) and Dolan and Steffe (1990). This is applicable to protein- and starchbased dough and slurry systems.

4.4.3

Extensional flow

4.4.3.1

Shear-free flow

Viscometric flow may be defined as that type of flow which is indistinguishable from steady simple shear flow. Pure extensional flow, which yields an extensional viscosity, does not involve shearing and is sometimes referred to as ‘shear-free’ flow. (In the published literature, ‘elongational viscosity’ and ‘Trouton viscosity’ are common synonyms for ‘extensional viscosity’.) Many food-processing operations involve extensional deformation, and the molecular orientation caused by extension, compared with shear, can produce unique food products and behaviours. The reason why shear and extensional flow have a different influence on rotational behaviour is that flow fields orient long molecules of high molecular weight. In shear flow, the preferred orientation corresponds to the direction of flow; however, the presence of a differential velocity across the flow field encourages molecules to rotate, thereby reducing the degree of stretching induced in molecular chains. However, in extensional flow, the situation is very different. The preferred molecular orientation is in the direction of the flow field because there are no competing forces to cause rotation. Hence, extensional flow will induce the maximum possible stretching of the molecules, producing a tension in the chains that may result in a large resistance to deformation (compared with the case of shear flow). Stiffer molecules are oriented more quickly in an extensional flow field. This phenomenon may be a factor in the choice of the thickening agent for pancake syrup: stringiness can be reduced, while maintaining thickness, when stiffer molecules are selected as additives. Reduced stringiness leads to what can be called a clean ‘cut-off’ after syrup is poured from a bottle. An example of a stiff molecule is the rod-like biopolymer xanthan; this can be compared with sodium alginate and carboxymethylcellulose, which exhibit a random coil-type conformation in solution (Padmanabhan and Bhattacharya 1993, 1994). Extensional flow is an important aspect of food process engineering and is prevalent in many operations, such as dough processing. Sheet stretching and extrudate drawing provide good examples of extensional flow (Fig. 4.13). The converging flow into dies such as those found in single- and twin-screw extruders involves a combination of shear and extensional flow; the extensional component of deformation is illustrated in Fig. 4.13. In the analysis of converging flow in a die, the pressure drop across the die can be separated into shear and extensional components.

The rheology of foods and sweets

Sheet stretching

127

Extruder die

Calendering Gravity induced sagging

Squeezing

Bubble growth Fig. 4.13

Various ways of shaping dough using extensional flow.

Converging flow may also be observed when a fluid is sucked into a pipe or a straw, and when food is spread with a knife. One of the most common examples of extensional flow is seen when a filled candy bar or a fruit-filled pastry is pulled apart. Extensional deformation is also present in calendering (Fig. 4.13), a standard operation performed with dough sheeting. Gravity-induced sagging (see Fig. 4.13 again) also involves extensional deformation. This may be observed in the cut-off apparatus associated with fruit-filling systems for pastry products. Extensional flow in this situation is undesirable because it may contribute to inconsistent levels of filling or an unsightly product appearance due to smeared filling. Bubble growth from the production of carbon dioxide during dough fermentation, extrudate expansion caused by vaporization of water, and squeezing to achieve spreading of a product also involve extensional deformation. Extensional flow is also a factor in die swell and in mixing, particularly dough mixing with ribbon blenders (Steffe 1996). This shows that extensional viscosity plays an important role in many fields of confectionery practice. 4.4.3.2

Extensional viscosity

Although extensional viscosity is clearly a factor in food processing, the use of this rheological property in the engineering design of processes and equipment is still at an early

128

Confectionery and Chocolate Engineering: Principles and Applications

stage of development. Extensional flow is also an important factor in the human perception of texture with regard to the mouthfeel and swallowing of fluid foods and fluid drugs. Extensional viscosity has been measured for various food products. First of all, Trouton’s investigations (Trouton 1906) and Leighton et al. (1934) should be mentioned. Extensional viscosity plays an essential function in extrusion and in calendering, which are important operations in both the food and the plastic industry. Rheological measurements of food doughs have certainly been carried out for many years in the food industry. The farinograph, amylograph and Brabender torque rheometer are some of the instruments that have been used to provide an indication of the deformation characteristics of doughs. Data from the Chopin Alveograph, a common doughtesting device in which a spherical bubble of material is formed by inflating a sheet, can be interpreted in terms of a biaxial extensional viscosity (Launay and Buré 1977; Faridi and Rasper 1987). Doughs have also been evaluated by subjecting them to uniaxial extension (de Bruijne et al. 1990). However, although these instruments have proved to be quite useful for obtaining qualitative information about the properties of doughs, they do not give these properties in quantities defined in engineering or scientific units. This is because the flow field created in these devices is usually so complicated that basic material properties cannot be obtained (Baird 1983). 4.4.3.3

Types of extensional flow

There are three basic types of extensional flow (Fig. 4.14): • During uniaxial extension, material is stretched in one direction (x1) with a corresponding size reduction in the other two directions (x2 and x3); for example, a cube is stretched into a prism of square cross-section (x2 = x3). This is truly uniaxial, because the extension occurs only in one direction, and there is a contraction in the other two directions.

Uniaxial extension

Planar extension

Biaxial extension Fig. 4.14

The basic types of extensional flow.

The rheology of foods and sweets

129

• In planar extension, material is stretched in the x1 direction with a corresponding decrease in thickness (in the x2 direction), while the width (in the x3 direction) remains unchanged. • Biaxial extension appears similar to uniaxial compression, but it is usually thought of as a flow which produces a radial tensile stress, for example when a column of circular cross-section is compressed. Uniaxial extension For a material of constant density in uniaxial extension, the velocity distribution in Cartesian coordinates, described using the Hencky strain rate, is u1 = ε H′ x1 ( extension )

(4.87)

u2 = −

ε H′ x2 2

(contraction)

(4.88)

u3 = −

ε H′ x3 2

(contraction)

(4.89)

where ε H′ > 0 is the Hencky shear rate. If the direction of stretching is x1, then a size reduction (equal to half of the stretching) takes place in the x2 and x3 directions. Pure extensional flow does not involve shear deformation; therefore, all the shear stress terms are zero:

σ 12 = σ 13 = σ 12 = 0

(4.90)

The stress is also axisymmetric:

σ 22 = σ 33

(4.91)

This results in one normal-stress difference that can be used to define the tensile extensional viscosity:

ηE =

σ 11 − σ 22 σ11 − σ 33 = ε H′ ε H′

(4.92)

Materials are said to be tension-thinning (or extensionally thinning) if ηE decreases with increasing ε H′ . They are tension-thickening (or extensionally thickening) if ηE increases with increasing ε H′ . These terms are analogous to the shear-thinning and shear-thickening used previously to describe changes in the apparent viscosity with shear rate. Biaxial extension The velocity distribution produced by a uniaxial compression that causes a biaxial extensional flow can be expressed in Cartesian coordinates as u1 = ε B′ x1 ( extension ) u2 = −2ε B′ x2

(contraction)

(4.93) (4.94)

130

Confectionery and Chocolate Engineering: Principles and Applications

u3 = ε B′ x3

(extension)

(4.95)

Since ε H′ = 2ε B′ ( > 0 ) , biaxial extension can actually be viewed as a form of tensile deformation. Uniaxial compression, however, should not be viewed as being simply the opposite of uniaxial tension, because the tendency of molecules to orient themselves is stronger in tension than in compression (Steffe 1996). Axial symmetry allows the above equations to be rewritten in cylindrical coordinates (z = axial direction, r = radial direction and θ = angle of rotation): uz = −2ε B′ z ( compression)

(4.96)

ur = ε B′ r ( extension )

(4.97)

u0 = 0 ( no rotation )

(4.98)

The biaxial extensional viscosity is defined in terms of the normal-stress difference and the strain rate:

σ 11 − σ 22 σ 11 − σ 33 = ε B′ ε B′ σ − σ rr 2 (σ zz − σ rr ) = zz = ε B′ ε H′

ηB =

(4.99)

where σzz and σrr are the corresponding diagonal elements of the stress tensor in cylindrical coordinates. Planar extension In planar extension, the velocity distribution is u1 = ε H′ x1 ( extension ) u2 = −ε H′ x2

(contraction)

u3 = 0 ( no change )

(4.100) (4.101) (4.102)

This type of flow produces two distinct stress differences, (σ11 − σ22) and (σ11 − σ33). The planar extensional viscosity is defined in terms of the more easily measured of these two stress differences:

ηP =

σ 11 − σ 22 ε H′

(4.103)

It is difficult to generate planar extensional flow, and experimental tests of this type are less common than those involving tensile or biaxial flow. 4.4.3.4

Relation between extensional and shear viscosities

The following limiting relationships between the extensional and shear viscosities can be expected for non-Newtonian fluids at small strains (Walters 1975; Petrie 1979; Dealy 1994):

The rheology of foods and sweets

131

lim ηE ( ε H′ ) = 3 lim η (γ ′ ) → ηE = 3η

(4.104)

lim ηB ( ε B′ ) = 6 lim η (γ ′ ) → ηB = 6η

(4.105)

lim ηP ( ε H′ ) = 4 lim η (γ ′ ) → ηP = 4η

(4.106)

εH ′ →0

γ ′→0

ε B′ →0

γ ′→0

εH ′ →0

γ ′→0

The values relating to the special case of Newtonian fluids are indicated by the sign ‘→’, where η is the Newtonian viscosity in steady shear flow. These latter three equations can be used to verify the operation of extensional viscometers. Clearly, however, a Newtonian fluid must be extremely viscous to maintain its shape and give the solid-like appearance required in many extensional flow tests (Steffe 1996). Reliable relationships for non-Newtonian fluids at large strains have not been developed. Trouton (1906) established a mathematical relationship between tensile extensional viscosity and shear viscosity. Data for extensional and shear viscosities are often compared using a dimensionless ratio known as the Trouton number Tr, where Tr =

extensional viscosity shear viscosity

(4.107)

Since the extensional and shear viscosities are functions of different strain rates, a conventional method of comparison is needed to remove ambiguity. Based on a consideration of viscoelastic and inelastic fluid behaviour, Jones et al. (1987) proposed the following conventions for computing Trouton numbers for uniaxial and planar extensional flow: Truniaxial = Trplanar =

ηE ( ε H′ ) η 3ε H′

(

ηP ( ε H′ ) η ( 2ε H′ )

)

(4.108) (4.109)

meaning that shear viscosities (η) are calculated at shear rates equal to 3ε H′ for uniaxial extension and 2ε H′ for planar extension. Using similar considerations, Huang and Kokini (1993) proposed the following convention for the case of biaxial extension: Trbiaxial =

η

ηB ( ε B′ ) 12 ε B′

(

)

(4.110)

The Trouton ratio for a Newtonian fluid may be determined from Eqns (4.96)–(4.98). Any departures from these numbers are due to viscoelastic material behaviour. Experimental results may produce considerably higher values. For example, Peck et al. (2006) measured the following Trouton numbers in roller extrusion of biscuit doughs for uniaxial extension as a function of die entry angle: for short doughs, 45–124 (die entry angle 60°) and 51–141 (45°); and for hard doughs, 89–108 (60°) and 100–133 (45°). These Trouton numbers were calculated on the basis of power-law parameters, obtained using the Gibson equation (see Chapter 14).

132

Confectionery and Chocolate Engineering: Principles and Applications

For further studies, see Steffe (1996, Chapter 4); for further discussion of simple extension, see Leblans and Scholtens (1986); and for methods of measuring extensional viscosity, see Cheremisinoff (1988, pp. 991–1059).

4.4.4 4.4.4.1

Viscoelastic functions Viscoelastic phenomena

Viscoelastic fluids simultaneously exhibit obvious fluid-like (viscous) and solid-like (elastic) behaviour. The manifestations of this behaviour due to a high elastic component can be very strong and can create difficult problems in process engineering design. These problems are particularly prevalent in the plastics-processing industry but are also present in the processing of foods such as doughs, particularly those containing large quantities of wheat protein. Figures 4.15–4.18 illustrate several viscoelastic phenomena (Steffe 1996). During mixing or agitation, a viscoelastic fluid may climb an impeller shaft in a phenomenon known as the Weissenberg effect. This can be observed in the home mixing of cake or chocolate brownie batter. When a Newtonian fluid emerges from a long, round tube into the air, the emerging jet will normally contract. The normal-stress differences present in a viscoelastic fluid, however, may cause jet expansion. This behaviour contributes to the challenge of designing extruder dies to produce the desired shape of many pet, snack and cereal foods. In addition, highly elastic fluids may exhibit a tubeless siphon effect. This phenomenon is well known in the confectionery industry in relation to dosing fillings that show elastic Impeller

Impeller

A

B

Fig. 4.15 The Weissenberg effect: a viscoelastic fluid may climb an impeller shaft. (A) Newtonian fluid and (B) viscoelastic fluid [reproduced from Steffe (1996), with permission].

Vacuum

A

B

Vacuum

Fig. 4.16 The tubeless siphon effect. Under the effect of a vacuum, a viscoelastic fluid may pull a fibre. (A) Newtonian fluid and (B) viscoelastic fluid [reproduced from Steffe (1996), with permission].

The rheology of foods and sweets

A

133

B

Fig. 4.17 A viscoelastic fluid may produce jet expansion. (A) Newtonian fluid and (B) viscoelastic fluid [reproduced from Steffe (1996), with permission].

A

B

STOP

Recoil Fig. 4.18 Recoil phenomenon of a viscoelastic fluid. (A) Newtonian fluid and (B) viscoelastic fluid [reproduced from Steffe (1996), with permission].

behaviour (although they are not actually sucked up). A drop of filling does not separate from the dosing head, and thus the head ‘pulls a fibre’ from the drop. Various fillings with an aqueous base frequently contain a gelling agent in order to fix the water content; however, this gives a certain amount of elasticity to the filling. The fibre pulled by the dosing head causes problems because, for example, a chocolate cover cannot perfectly close a praline. A recoil phenomenon, where tensile forces in a fluid cause particles to move backwards (snap back) when the flow is stopped, may also be observed in viscoelastic fluids. A summary of the behaviour of viscoelastic polymer solutions in various flow situations has been given by Boger and Walters (1993). 4.4.4.2

Importance of large deformations in food rheology

In process engineering, data on viscoelasticity may be very helpful in understanding various problems. When materials are tested in the linear range, material functions do not depend on the magnitude of the stress, the magnitude of the deforming strain or the rate of application of the strain. If the behaviour is linear, an applied stress will produce a proportional strain response. The linear range of testing is determined from experimental data. Testing can easily enter the nonlinear range if excessive strains (usually > 1%) or high deformation rates are applied to a sample.

134

Confectionery and Chocolate Engineering: Principles and Applications

The importance of large-deformation (nonlinear) behaviour in food rheology, however, should not be overlooked. Many processes, such as mastication and swallowing, are accomplished only with very large deformations. The collection of viscoelastic data relevant to this type of problem involves testing in the nonlinear range of behaviour. These data may be useful in attacking practical problems; however, from a fundamental standpoint, they can only be used for comparative purposes, because the theoretical complexity of nonlinear viscoelasticity makes it impractical for most applications. Elastic behaviour may be evaluated using viscometric methods to determine the normal-stress differences found in steady shear flow. Alternatively, viscoelastic material functions may be determined from experiments involving the application of unsteadystate deformations. Generally, these dynamic testing techniques may be divided into two major categories: transient and oscillatory. Transient methods include tests of start-up flow, cessation of steady shear flow, step strain, creep and recoil. In oscillatory testing, samples are deformed by the application of a harmonically varying strain, which is usually applied over a simple shear field. In a creep test, the material is subjected to a constant stress and the corresponding strain is measured as a function of time, γ(t) (in %/100%). The data are often plotted in terms of the shear creep compliance J ( m 2 N ) = J (t ) =

γ σ constant

(4.111)

versus time. In a step-strain test, commonly called a stress relaxation test, a constant strain is applied to the test sample and the changing stress over time, σ(t), is measured. The data are commonly presented in terms of a shear stress relaxation modulus G = G (t ) =

σ γ constant

(4.112)

versus time. Data from creep and stress relaxation tests can also be described in terms of mechanical (spring and dashpot) analogues; see Polakowski and Kipling (1966), Sherman (1970), Mohsenin (1986) and Barnes et al. (1989). 4.4.4.3

Mechanical analogues for describing viscoelastic behaviour

Massless mechanical models, composed of springs and dashpots, are useful for conceptualizing rheological behaviour. A spring is considered to be an ideal solid element obeying Hooke’s law,

σ = Gy

(4.113)

and a dashpot is considered to be an ideal fluid element obeying Newton’s law,

σ = μγ ′

(4.114)

The rheology of foods and sweets

135

Hooke

Newton

Fig. 4.19

Maxwell model (series connection).

Hooke Newton

Fig. 4.20

Kelvin model (parallel connection).

where σ is the stress (Pa), G is the elasticity modulus (Pa), γ is the strain (a ratio, i.e. a dimensionless number), γ ′ is the shear rate (s−1) and μ is the dynamic viscosity (Pa s). Springs and dashpots can be connected in various ways to portray the behaviour of viscoelastic materials; however, the combination of elements is not unique, because many different combinations can be used to model the same set of experimental data. The most common mechanical analogues of rheological behaviour are the Maxwell (Fig. 4.19) and Kelvin (or Kelvin–Voigt) models (Fig. 4.20). Stress relaxation – the Maxwell model A wide range of behaviour may be observed in stress relaxation tests. No relaxation would be observed in an ideal elastic material, while an ideal viscous substance would relax instantaneously. Viscoelastic materials relax gradually, with the end point depending on the molecular structure of the material being tested: the stress in a viscoelastic solid decays to an equilibrium stress (σE > 0), but the residual stress in a viscoelastic liquid is zero. Stress relaxation data are commonly presented in terms of a stress relaxation modulus (see Eqn 4.113) G = f (t ) =

σ γ constant

(4.115)

If the material is perfectly elastic, the relaxation modulus is equal to the shear modulus. The Maxwell model (Fig. 4.19), which contains a Hookean spring in series with a Newtonian dashpot, has frequently been used to interpret stress relaxation data for viscoelastic liquids, particularly polymeric liquids. The total shear strain in a Maxwell fluid element is equal to the sum of the strain in the spring and the dashpot:

γ = γ spring + γ dashpot

(4.116)

136

Confectionery and Chocolate Engineering: Principles and Applications

By differentiating Eqn (4.116), and taking into account Eqns (4.77) and (4.79), the following equation is obtained: dy 1 dσ σ =γ′= + dt G dt μ

(4.117)

or

σ + λrel

dσ = μγ ′ dt

(4.118)

where the relaxation time (also called the characteristic time) is defined as

λrel =

μ G

(4.119)

Although an exact definition of λrel is difficult, it can be thought of as the time it takes a macromolecule to be stretched out when deformed. The above equations have been presented in terms of shear deformation. If testing is conducted in uniaxial tension or compression, then the relaxation time can be thought of in terms of an extensional viscosity ηEx and Young’s modulus E. The Maxwell model is useful in understanding stress relaxation data. Consider a stepstrain (stress relaxation) experiment where there is a sudden application of a constant shear strain γ0. When the strain is constant, the shear rate is equal to zero (γ ′ = 0) and Eqn (4.118) becomes

σ + λrel

dσ =0 dt

(4.120)

This equation may be integrated using the initial condition that σ = σ0 at t = 0; after evaluating the integral, ⎛ t ⎞ σ = σ 0 exp ⎜ − ⎝ λrel ⎟⎠

(4.121)

Equation (4.121) describes the gradual relaxation of the stress (from σ0 to zero) after the application of a sudden strain. The relationship provides a means of determining the relaxation time: λrel is the time it takes for the stress to decay to 1/e (approximately 36.8%) of its initial value. Experimental data show that the Maxwell model does not account for the stress relaxation behaviour of many viscoelastic materials, because it does not include an equilibrium stress σE. This problem may be addressed for numerous foods by constructing a model that consists of a combination of various elements in series or parallel coupling. This concept can be generalized to determine a relaxation spectrum for a viscoelastic material (Ferry 1980). Peleg and Normand (1983) noted two major problems in collecting stress relaxation data for foods:

The rheology of foods and sweets

137

(1) When subjected to large deformations, foods usually exhibit nonlinear viscoelastic behaviour. (2) Natural instability and biological activity make it difficult to determine equilibrium mechanical parameters. To overcome these difficulties, Peleg and Normand suggested that stress relaxation data should be calculated as a normalized stress (a normalized force term is also acceptable) and fitted to the following linear equation:

σ 0t = k1 + k2t σ0 − σ

(4.122)

where σ0 is the initial stress, σ is the decreasing stress at time t, and kl (s) and k2 are constants. The reciprocal of k1 represents the initial decay rate (s−1), and k2, which is dimensionless, is a hypothetical value of the asymptotic normalized force. Creep and recovery – the Kelvin model In a creep test, an instantaneous stress is applied to the sample and the change in strain (called the creep) is observed over time. When the stress is released, some recovery may be observed as the material attempts to return to the original shape. These tests can be particularly useful in studying the behaviour in constant-stress environments such as those found in levelling, sedimentation and coating applications, where gravity is the driving force. Creep experiments can also be conducted in uniaxial tension or compression. Viscoelastic materials (e.g. bread dough) can exhibit a nonlinear response to strain and, owing to their ability to recover some structure as a result of storing energy, show a permanent deformation less than the total deformation applied to the sample. This strain recovery, or creep recovery, is also known as ‘recoil’ and may be investigated in terms of a recoil function (Dealy 1994). The starting point for developing a mechanical analogue describing creep behaviour is the Kelvin model (Fig. 4.20), which contains a spring connected in parallel with a dashpot. When this system is subjected to shear strain, the spring and dashpot are strained equally; see Eqn (4.116). The total shear stress (σ) caused by the deformation is the sum of the individual shear stresses which, using Eqns (4.113) and (4.114), can be written as

σ = Gy + μγ ′

(4.123)

Differentiating Eqn (4.123) with respect to time (where G is constant) yields 1 dσ dγ ′ = γ ′ + λret G dt dt

(4.124)

where the retardation time λret = μ/G is unique for any substance, and dγ /dt = γ ′ (see Eqn 4.117). If the material were a Hookean solid, the retardation time would be zero and the maximum strain would be obtained immediately on the application of the stress: the achievement of the maximum strain in a viscoelastic material is delayed (or retarded). The retardation time can be thought of in terms of the extensional viscosity ηEx and Young’s modulus E if the testing is conducted in uniaxial tension or compression.

138

Confectionery and Chocolate Engineering: Principles and Applications

In creep, where the material is allowed to flow after being subjected to a constant shear stress, i.e. dσ/dt = 0, the solution to Eqn (4.124) is

σ ⎡ ⎛ t ⎞⎤ γ = f (t ) = ⎛ 0 ⎞ ⎢1 − exp ⎜ − ⎝ G ⎠⎣ ⎝ λret ⎟⎠ ⎥⎦

(4.125)

showing that the initial strain is zero (σ0 = 0 at t = 0). Equations (4.93) and (4.125) predict a strain that asymptotically approaches the maximum strain (σ0/G) associated with the spring. λret is the time taken for the delayed strain to reach approximately 63.2% (1 − 1/e) of the final value. Materials with a large retardation time reach their full deformation slowly. The Kelvin model (Fig. 4.20) shows excellent elastic retardation (Fig. 4.21), but is not general enough to model creep in many biological materials. The solution to this problem is to use the Burgers model (Fig. 4.22), which is a Kelvin and a Maxwell model placed in series. t

t0 t1

t

g g∞ g1

t1 Fig. 4.21

t

Response of a Kelvin fluid to a constant stress τ0.

s Maxwell G0 m0

G1

Kelvin

m1

s Fig. 4.22

The Burgers model: a Kelvin and a Maxwell model coupled in series.

The rheology of foods and sweets

139

Data following this mechanical analogue show an initial elastic response due to the free spring, a retarded elastic behaviour related to the parallel spring–dashpot combination, and Newtonian-type flow after long periods of time due to the free dashpot:

γ = f (t ) =

σ0 ⎛ σ0 ⎞ ⎡ ⎛ t ⎞ ⎤ σ 0t + + ⎜ ⎟ 1 − exp ⎜ − ⎝ λret ⎟⎠ ⎥⎦ μ0 G ⎝ G1 ⎠ ⎢⎣

(4.126)

where λret = μ1/Gl, the retardation time of the Kelvin portion of the model. The Burgers model can also be expressed in terms of the creep compliance by dividing Eqn (4.126) by the constant stress σ0: J [1 Pa ] =

1 ⎛ 1 ⎞⎡ γ ⎛ t ⎞⎤ t + = J (t ) = + ⎜ ⎟ ⎢1 − exp ⎜ − ⎝ λret ⎟⎠ ⎥⎦ μ0 G ⎝ G1 ⎠ ⎣ σ0

(4.127)

where J0 = 1/G (t = 0) is the instantaneous compliance, J1 is the retarded compliance, λret is the retardation time (= μ1/Gl) of the Kelvin component and μ0 is the Newtonian viscosity of the free dashpot. The sum of J0 and J1 is called the steady-state compliance. When creep experiments are conducted, controlled-stress rheometers allow the strain recovered when the constant stress is removed to be measured. The complete creep and recovery curve may be expressed using the Burgers model. When calculated as a compliance, the creep is given by Eqn (4.127) for 0 < t < t1, where t1 is the time when the constant stress is removed. At the beginning of creep (t = 0), there is an instantaneous change J = J0 in the compliance (where J0 = 1/G) due to the spring in the Maxwell portion of the model. Then, the Kelvin component produces an exponential change (if t < 0, then 0 < (1/G1) [1 − exp(−t/λret)] → 1/G1) in the compliance related to the retardation time. After sufficient time has passed, the independent dashpot generates a purely viscous response (t/μ0) since the other additive terms (1/G + 1/G1) do not change any more for practical purposes. If necessary, additional Kelvin elements can be added to the Burgers model to represent the experimental data better. Mathematically, this idea can be described by the equation ⎛ t ⎞ t J = J (t ) = J 0 + Σ i Ji exp ⎜ − + , i = 1, 2, … , m ⎝ λret ⎟⎠ μ0

(4.128)

where m is the total number of Kelvin elements in the model, each having a unique retarded compliance and retardation time. A simple linearized model has been suggested by Peleg (1980) to characterize the creep of biological materials: t = k1 + k2t J

(4.129)

where t is the time (s), J is the compliance function (1/Pa), and k1 (Pa s) and k2 (Pa) are constants. 4.4.4.4

Analogy between rheological models and electrical networks

On the basis of the Kirchhoff laws, an analogy (denoted by ‘∼’) can be established between rheological and electrical networks in the following way:

140

Confectionery and Chocolate Engineering: Principles and Applications

Hooke model ~ electrical resistance i.e. Hooke’s law ~ Ohm’s law:

(4.130)

σ = Gγ ∼ I = (1/R)U σ ∼ I (electrical circuit) G ∼ 1/R (R = electrical resistance) γ ∼ U (electrical potential) Newton model ~ electrical capacitor, i.e.

(4.131)

i.e. σ = μγ ′ ∼ I = C(dU/dt) γ∼U γ ′ = dγ /dt ∼ dU/dt μ∼C This analogy facilitates the analysis of rheological models. For further details, see Foster (1924, 1932), Guillemin (1935, 1950) and Verhás (1985). 4.4.4.5

Series and parallel coupling of models; relaxation functions

Verhás (1985) showed that for series and parallel coupling, respectively, the following simple relationships hold: M1 ( k1 ) SM2 ( k2 )  M1,2 ( k1,2 ) , where 1 k1,2 = 1 k1 + 1 k2

(4.132)

and M1 ( k1 ) PM2 ( k2 )  M1,2 ( k1,2 ) , where k1,2 = k1 + k2

(4.133)

In the above, M denotes the model (Hooke, Newton or coupled); ki is a characteristic of the model (e.g. G for the Hooke model or η for the Newton model); i (= 1 or 2) is an index related to the model; S and P denote serial and parallel coupling, respectively; and 䉴 indicates the resultant model after coupling. For example, for serially coupled Newton models, N1 (η1 ) SN 2 (η2 )  N1,2 (η1,2 ) where 1/η1,2 = 1/η1 + 1/η2. Evidently, for serial coupling, σ = ∑ σ i , and for parallel coupling, γ = ∑ yi . In addition, Verhás (1985) showed that: • If two Kelvin models or two Maxwell models, with equal relaxation times, are coupled in parallel or in series, respectively, the resultant Kelvin model or Maxwell model also has this common relaxation time.

The rheology of foods and sweets

141

• For any Maxwell model, a Kelvin model can be found which is equivalent to it, and vice versa. This generalization of rheological models leads to the notion of a relaxation function,

σ (t ) = γ 0Y (t )

(4.134)

where σ(t) is the stress (Pa) as a function of time t, γ 0 is the deformation when t = 0, and Ψ(t) is the relaxation function (Pa). The clear meaning of Ψ(t) is the stress remaining in the body after time t, after a sudden deformation of unit size. A detailed survey of mechanical models of food has been given by Tscheuschner (1993b).

4.4.5

Oscillatory testing

In oscillatory instruments, samples are subjected to a harmonically varying stress or strain. This testing procedure is the most common dynamic method for studying the viscoelastic behaviour of food. The results are very sensitive to chemical composition and physical structure, so they are useful in a variety of applications, including evaluation of gel strength, monitoring starch gelatinization, studying glass transition phenomena, observing protein coagulation or denaturation, evaluating curd formation in dairy products, studying the melting of cheese, studying texture development in bakery and meat products, shelf-life testing, and correlation of rheological properties with human sensory perception. Food scientists have found oscillatory testing instruments to be particularly valuable tools for product development work. Oscillatory testing may be conducted in tension, bulk compression or shear. Typical commercial instruments operate in the shear deformation mode, and this is the predominant testing method used for food. A shear strain may be generated using parallel-plate, cone-and-plate or concentric-cylinder fixtures. Dynamic testing instruments may be divided into two general categories: controlled-rate instruments, where the deformation (strain) is fixed and the stress is measured, and controlled-stress instruments, where the stress amplitude is fixed and the deformation is measured. Both produce similar results. The emphasis in this section is on fluid and semi-solid foods. In oscillatory tests, materials are subjected to a deformation (in controlled-rate instruments) or a stress (in controlled-stress instruments) which varies harmonically with time. Sinusoidal simple shear is typical. To illustrate the concept, consider two rectangular plates oriented parallel to each other (Fig. 4.23). The lower plate is fixed and the upper plate is allowed to move back and forth in a horizontal direction. Assume that the sample being tested is located between the plates of a controlled-rate device. Suppose the strain in the material between the plates is a function of time defined by

γ = γ 0 sin (ωt )

(4.135)

where γ 0 is the amplitude of the strain, equal to L/h when the motion of the upper plate is L sin(ωt); ω = 2πν is the angular frequency expressed in rad/s, and ν is the frequency expressed in hertz (cycles/s).

142

Confectionery and Chocolate Engineering: Principles and Applications

L sin(w t) L

Oscillating plate

h

Stationary plate Fig. 4.23

Oscillatory strain between rectangular plates [reproduced from Steffe (1996), with permission].

For example, if the two plates in Fig. 4.23 are separated by a distance of h = 1.5 mm and the upper plate is moved by L = 0.3 mm from the centre line, then the maximum strain amplitude may be calculated as 0.2 or 20% (γ 0 = L/h = 0.3/1.5 = 0.2). This can be regarded as a large deformation. Using a sine wave for the strain input results in a periodic shear rate, found by taking the derivative of Eqns (4.100) and (4.135): dγ = γ ′ = γ 0ω cos (ωt ) dt

(4.136)

For a small strain amplitude (in the linear viscoelastic region, σ ∼ γ), the following shear stress is produced by the strain input:

σ = σ 0 sin (ωt + δ )

(4.137)

where σ0 is the amplitude of the shear stress (not to be confused with the yield stress symbolized by σ0 or τ0 in earlier sections), and δ is the phase lag or phase shift (also called the mechanical loss angle) relative to the strain. The time period associated with the phase lag is equal to δ/ω. Dividing both sides of Eqn (4.137) by γ 0 yields

σ ⎛ σ0 ⎞ = sin (ωt + δ ) γ 0 ⎜⎝ γ 0 ⎟⎠

(4.138)

The results of small-amplitude oscillatory tests can be described by plots of the amplitude ratio σ0/γ 0 and the phase shift δ as frequency-dependent functions. However, the shear stress output produced by a sinusoidal strain input is usually written as G ′′ ⎞ σ = G ′γ + ⎛ γ′ ⎝ ω ⎠

(4.139)

where G′ is the shear storage modulus and G″ is the shear loss modulus. In addition, these two moduli can be expressed as ⎛σ ⎞ G ′ = ⎜ 0 ⎟ cos δ ⎝ γ0 ⎠

(4.140)

The rheology of foods and sweets

143

and ⎛σ ⎞ G ′′ = ⎜ 0 ⎟ sin δ ⎝ γ0 ⎠

(4.141)

G′γ 0 may be interpreted as the component of the stress in phase with the strain; G″γ 0 may be interpreted as the component of the stress 90° out of phase with the strain. Some additional frequency-dependent material functions are: The complex modulus G*, G* =

σ0 = G ′ 2 + G ′′ 2 γ0

(4.142)

The absolute value of the viscosity, η*,

η* =

G* = η ′ 2 + η ′′ 2 ω

(4.143)

the components of which are the dynamic viscosity η′,

η′ =

G ′′ ω

(4.144)

and the complex viscosity η″,

η ′′ =

G′ ω

(4.145)

Using Eqns (4.104) and (4.106), Eqns (4.99) and (4.101) can be expressed as

σ = G ′γ + η ′γ ′

(4.146)

which represents the material behaviour well because it clearly indicates the elastic (G ′γ ) and viscous (η′γ ′) nature of the substance. The tangent of the phase shift or phase angle (tan δ) is also a function of frequency: tan δ =

G ′′ G′

(4.147)

This parameter is directly related to the energy lost per cycle divided by the energy stored per cycle. Values of tan δ for typical food systems (dilute solutions, concentrated solutions and gels) have been given by Steffe (1996, pp. 325–326). The Maxwell model of a fluid is often used to interpret data from the dynamic testing of polymeric liquids. If the strain input is harmonic (see Eqns (4.135) and (4.136)), Eqn (4.136) can be substituted into Eqn (4.120), and the resulting differential equation can be solved to produce a number of frequency-dependent rheological functions for Maxwell fluids:

144

Confectionery and Chocolate Engineering: Principles and Applications

2 Gω 2 λrel 2 1 + ω 2 λrel

G′ = G ′′ =

η′ =

(4.148)

Gωλrel 2 1 + ω 2 λrel

(4.149)

η 2 1 + ω 2 λrel

tan δ =

(4.150)

G ′′ 1 = G ′ ωλrel

(4.151)

where λrel is the relaxation time of the Maxwell fluid and is equal to μ/G. Looking at experimental data may allow the material constants of the Maxwell model to be evaluated from the asymptotes: as ω goes to zero, η′ goes to η; and as ω goes to infinity, G′ goes to G. For further details of the dynamic testing of foods, see, for example, Sherman (1983), Stastna et al. (1986) and van Vliet (1999).

4.4.6

Electrorheology

Steffe (1996, Section 1.11) discussed the topic of electrorheology, sometimes called the Winslow effect (Winslow 1947), which refers to changes in the rheological behaviour due to the imposition an electric field on a material. Electrorheological fluids are dispersions of solid particles, typically 0.1–100 μm in diameter, in an insulating (non-conducting) oil. An example, milk chocolate, was discussed by Steffe (1996). At low shear rates, in the absence of an electric filed, the particles are randomly distributed, and many electrorheological fluids show near-Newtonian behaviour. With the application of an electric field, the particles become polarized, causing particle alignment across the electrode gap and creating an enhanced, fibre-like structure. The application of a voltage causes some materials to develop high yield stresses, which can be so high that flow ceases, effectively transforming the material from a liquid to a solid. The dielectric properties of chocolate are well known, although barely studied, and are worthy of more interest.

4.5

Viscosity of solutions

When a polymer is dissolved in a solvent, there is a noticeable increase in the (dynamic) viscosity of the resulting solution. The viscosities of pure solvents and solutions can be measured, and various values calculated from the resulting data: Relative viscosity = ηrel =

ηsolution ηsolvent

Specific viscosity = ηsp = ηrel − 1 Reduced viscosity = ηred =

ηsp C

(4.152) (4.153) (4.154)

The rheology of foods and sweets

145

η Inherent viscosity = ηinh = ln ⎛ red ⎞ ⎝ C ⎠

(4.155)

⎛ ηsp ⎞ Intrinsic viscosity = ηint = lim ⎜ ⎟ ≡ [η ] C→0 ⎝ C ⎠

(4.156)

where C is the volume or mass concentration of the solution. Using these concepts for the well-known Einstein equation,

ηsolution = ηsolvent (1 + 2.5F )

(4.157)

where Φ is the volume concentration of solid spheres in a solution, the value of which must be small, we can derive

ηrel = 1 + 2.5F

(4.158)

ηsp = 2.5F

(4.159)

ηred = 2.5 = ηint

(4.160)

If Φ is small enough, ln ηrel ≈ ηsp

(4.161)

If Φ is measured in g/100 ml, and the swelling of the polymer is taken into account by a factor s, then from the Einstein equation,

[η ] = 0.025 sF

(4.162)

The Einstein equation can be used to describe the viscosity properties of emulsions too (see later). The intrinsic viscosity has great practical value in molecular-weight determinations of high polymers, using the equation

[η ] = KM a

(4.163)

where M is the molecular weight of the polymer, a = 0.7–1 (according to Staudinger, a = 1), and K is a constant characterizing the monomer of the polymer and the solvent. If a ≠ 1 (the general case), Eqn (4.163) is called the Mark–Houwink equation. For details, see Erdey-Grúz and Schay (1954), Sun (2004) and Section 5.3.1 of this book. The viscosities of solutions are useful in understanding the behaviour of some biopolymers, including aqueous solutions of locust bean gum, guar gum and carboxymethylcellulose (Rao 1986). The intrinsic viscosities of numerous protein solutions have been summarized by Rha and Pradipasena (1986). For further details, see Krieger (1983) and Sun (2004).

146

Confectionery and Chocolate Engineering: Principles and Applications

Example 4.4 From a practical viewpoint, a 20 m/m% aqueous sucrose solution can be regarded as dilute, but not according to the physical chemistry! According to Junk and Pancoast (1973), the dynamic viscosity η of such a sucrose solution is 1.957 cP = 1.957 × 10−3 Pa s, and its density is 1.06655 g/ml (at 20°C). Let us calculate the viscosity of this solution with the help of Eqn (4.158), given that at 20°C the viscosity of water η0 is 1.002 × 10−3 Pa s and the density of water ρ0 is 0.998 g/ ml. In this case the volume of 100 g of sucrose solution is (100/1.06655) ml = 93.76 ml, and the volume of 80 g of water is (80/1.002) ml = 79.84 ml. Consequently, the volume ratio of sucrose is

Φ=

93.76 − 79.84 = 0.1485 93.76

According to the Einstein equation (4.157),

ηsolution = ηwater (1 + 2.5Φ ) = 1.002 × 10 −3 Pa s (1 + 2.5 × 0.1485) = 1.374 × 10 −3 Pa s The difference is (1.957 − 1.374)/1.957 = 29.8%.

4.6 4.6.1

Viscosity of emulsions Viscosity of dilute emulsions

Very dilute emulsions exhibit a Newtonian viscosity, and this is often defined in terms of the viscosity η0 of the continuous phase and the droplet volume fraction Φ by using the equation proposed by Einstein (Eqn 4.158). Equation (4.159) is valid provided: • • • •

the droplets behave as solid, rigid spheres; they are large with respect to the size of the molecules of the continuous phase; there is no hydrodynamic interaction between the droplets; and slippage does not occur at the oil/water interface.

The increase in viscosity above the value η0 results from energy dissipation when droplets of an immiscible liquid are introduced into the continuous phase, and the flow pattern of the latter phase is then modified in the vicinity of the droplets. The limitations on the applicability of Eqn (4.158) are often satisfied by very dilute emulsions, particularly if the droplet size does not exceed a few microns and the droplets are enveloped by an elastic or viscoelastic film of adsorbed emulsifier molecules. However, when the adsorbed emulsifier film is fluid, as with ionic emulsifiers, Eqns (4.122) and (4.159) have to be modified to allow the transmission of normal and tangential components of stress across the interface and into the droplets (Taylor 1932). This produces fluid circulation within the droplets and reduces the distortion of the flow pattern in the continuous phase around the droplets. Equations (4.121) and (4.122) now become

ηrel = 1 +

2.5 {ηi + ( 2 5) η0 } Φ ηi + η0

(4.164)

The rheology of foods and sweets

Phase

hrel

inversion

fcrit Newtonian flow Fig. 4.24

147

f

Non-Newtonian flow

Phase inversion of an emulsion (relative viscosity vs volume concentration of solid).

where ηi is the viscosity of the liquid forming the drops. When ηi >> η0, Eqns (4.128) and (4.164) reduce to Eqns (4.122) and (4.158), but in all other situations Eqn (4.164) gives a lower ηrel. The validity of Eqn (4.164) has been confirmed by viscosity studies with a large number of O/W emulsions.

4.6.2

Viscosity of concentrated emulsions

For very dilute emulsions, ηrel increases linearly as Φ increases. For more concentrated emulsions, Φ exerts a greater influence and the viscosity changes from Newtonian to nonNewtonian. The non-Newtonian character is initially pseudoplastic, but in very concentrated systems it may become plastic and exhibit viscoelasticity. Very often the influence of ηrel is as portrayed in Fig. 4.24, with ηrel increasing almost exponentially to a maximum value just prior to emulsion inversion, where a critical value of Φ (Φcrit) is exceeded. A phase inversion experiment was discussed in Section 2.1.4 (see also Mohos 1982). When Φ increases beyond the limit of validity of Eqns (4.122) and (4.158), the distorted flow patterns around the droplets draw closer together and eventually overlap. The resulting hydrodynamic interaction leads to an increased ηrel. This effect has been represented in many different forms, but they usually reduce to a power series in Φ,

ηrel = 1 + 2.5Φ + bΦ 2 + cΦ 3 + …

(4.165)

provided the droplets behave as discrete spheres; b and c are constants. Many different values of b, between 0 and 10 for O/W and W/O emulsions, have been quoted in the literature, but there are very few values for c. The hydrodynamic interaction between spherical droplets on opposite sides of a hypothetical spherical enclosure and separated by a distance f can be defined by a coefficient λ, where

λ=

1− D 2 f D 2f

(4.166)

and D is the droplet diameter. This equation is valid provided D/2f lies between 0.5 and 1. Therefore, the hydrodynamic interaction depends both on the size of the droplets and

148

Confectionery and Chocolate Engineering: Principles and Applications

on the distance between them. The latter will also be influenced by droplet size in that, at constant Φ, the value of f will decrease as D decreases. The sharp increase in viscosity which is observed in more concentrated emulsions can be explained by applying lubrication theory to calculate the viscous dissipation of energy (Frankel and Acrivos 1967). The viscosity of concentrated emulsions at high shear rates such that the droplets are completely deflocculated can often be satisfactorily described by the relation

η∞ 2.5Φ ⎞ = exp ⎛ ⎝ 1 − kΦ ⎠ η0

(4.167)

where k depends on the hydrodynamic interaction between droplets and increases as the droplet size decreases (Saunders 1961). This equation has the same form as that proposed by Mooney (1951), with k now being described as a geometric crowding factor such that 1.35 < k < 1.91. When expanded, Eqn (4.167) gives a power series (a geometric series with a quotient kΦ) in Φ similar to Eqn (4.165). Practical emulsions are never monodisperse with respect to droplet size, and the characteristics of the size distribution influence the rheological properties. The model represented by Eqn (4.167) can be extended to emulsions containing an i-modal size distribution, and the product of the relative viscosities of the various size fractions in the continuous phase at the same volume concentration gives the resultant relative viscosity: ⎛ 2.5Φ i ⎞ ηrel = Πi exp ⎜ ⎝ 1 − kiΦ i ⎟⎠

(4.168)

The viscosity of an emulsion can also be related to the droplet size distribution by an alternative relation (Djakovic’ et al. 1976),

η = SK − B

(4.169)

where K is the rate of change of viscosity with respect to the specific surface S, so that K=

dη dS

(4.170)

and B is a constant.

4.6.3

Rheological properties of flocculated emulsions

Following preparation, the droplets in emulsions flocculate, and the size of the aggregates so formed increases with storage time. These aggregates immobilize liquid from the continuous phase within the voids between the droplets, so that when the emulsion is examined in a viscometer at low shear rates such that the aggregate structure is not seriously damaged, an anomalously high viscosity is exhibited. When the shear rate is increased, the aggregate size is progressively reduced, as is the volume of the continuous phase immobilized. The effect of aggregation on viscosity can be demonstrated by a simple

The rheology of foods and sweets

149

procedure in which an emulsion is first stirred vigorously or subjected to a high shear rate and is then examined at a low shear rate (Mooney 1946; Sherman 1967). In emulsions stabilized by emulsifiers with not too high a molecular weight, van der Waals attraction forces are primarily responsible for the bonds between droplets in the aggregates. This gives rise to viscoelastic properties in the near-stationary state. When a small shear stress is applied to the emulsion, the resulting time-dependent strain leads to creep compliance–time behaviour; see the earlier discussion of the Kelvin fluid. The values of the various parameters decrease as the mean droplet size increases, with the precise influence of droplet size varying from one parameter to another. It is noteworthy that in the case of emulsions with a relatively small mean droplet size, small changes in the mean size can produce substantial changes in the magnitudes of the viscoelastic parameters. For further details see Sherman (1983, pp. 405–437). When freshly prepared emulsions are stored at ambient temperature, the droplets flocculate and coalesce for some time before there is visible separation of the disperse phase. At the same time, the rheological properties alter significantly, provided no other processes are associated with storage. Measurements made at high shear rates on W/O and O/W emulsions with medium to high concentrations of disperse phase indicate a sharp decrease of viscosity with storage time. This is associated with an increasing mean droplet size. The kinetics of droplet coalescence are defined by Nt = N 0 exp ( −Ct )

(4.171)

where Nt and N0 are the numbers of droplets per millilitre of emulsion at time t and initially, respectively, and C is the rate of droplet coalescence. For further details, see Tscheuschner (1993b).

4.7

Viscosity of suspensions

According to Scott Blair (1969), Hatschek proposed the following equation for the viscosity of concentrated suspensions:

ηsusp =

1 13 1 − (Φ K )

(4.172)

where Φ is the ratio of volume suspended to total volume and K is the voluminosity factor, which takes into account the swelling due to the solvent attaching to the suspended phase and increasing the volume of the particles. Roscoe (1952) proposed the following equation for a suspension of uniformly sized particles in high concentration,

ηsus =

ηsolv (1 − 1.35Φ )2.5

and, for a suspension of diversely sized particles in high concentration,

(4.173)

150

Confectionery and Chocolate Engineering: Principles and Applications

ηsus =

ηsolv (1 − Φ )2.5

(4.174)

Oldroyd (1959) dealt with various cases of deformation of disperse systems: slow, steady rates; variable, small rates; and finite rates. In addition, he discussed various forms of the Einstein equation (Eqn 4.158) and differential equations describing more complicated rheological properties. Example 4.5 Let us calculate the approximate value of Φ for the fat-free suspended fraction of a chocolate that contains about 35 m/m% cocoa butter, and estimate the (dynamic) viscosity of this chocolate mass if the viscosity of cocoa butter ηc.butter is about 0.03 Pa s. The density of chocolate is 1.235 g/ml and that of cocoa butter is 0.91 g/ml. 100 g chocolate has a volume of 100/1.235 = 80.97 ml. 35 g cocoa butter has a volume of 35/0.91 = 38.46 ml. If we assume that no voluminosity needs to be taken into account, the volume of fat-free components is the difference between the above volumes:

Φ=

80.97 − 38.46 = 0.525 80.97

Using Eqn (4.173),

ηchocol. =

ηc.butter

(1 − 1.35Φ )2.5

=

0.03 = 0.655 Pa s 0.0458

Using Eqns (4.139) and (4.174),

ηchocol. =

ηc.butter 0.03 = = 0.193 Pa s 2.5 0.1555 (1 − Φ )

Both of these results are much less than the real value of the viscosity of chocolate, which has a magnitude of about 2 Pa s at least. For further details, see Tscheuschner (1993b). Example 4.6 Let us calculate the voluminosity K of the fat-free components of chocolate according to Eqn (4.172) if the viscosity of the chocolate mass is 2.1 Pa s. 2.1 =

0.03 13 1 − (Φ K )

(0.525K )1 3 =

69 70

The rheology of foods and sweets

151

and 3

1 ⎞ ⎛ 69 ⎞ K =⎛ = 1.824 ⎝ 0.525 ⎠ ⎝ 70 ⎠

4.8 4.8.1

Rheological properties of gels Fractal structure of gels

A large variety of intermediate and finished products in the confectionery industry are gels. Their mechanical properties, as determined at small deformations, vary widely from very soft and deformable to rather stiff, as can be easily experienced by hand. The smalldeformation properties are frequently studied, although their practical importance is limited. The most important reason for studying them is that they can account for certain aspects of the undisturbed structure of the gel, if the experiments are done well. In this context, structure is defined as the spatial distribution of the relevant structural elements (building blocks) of the network and the interaction forces between them. The measurement of mechanical properties is especially suitable for investigating the structure of materials, because they can take account of both the spatial distribution of the structural elements and the interaction forces between them, in contrast to most other methods. However, this also makes their interpretation more complicated. A ‘small deformation’ is defined as a relative deformation (strain) so small that applying it does not affect the structure of the material studied. A characteristic of gels is that they consist of a continuous solid-like network in a continuous liquid phase over the timescale considered. The latter aspect implies that certain products can be considered as gels over short times but as liquids over long times. At intermediate timescales, their reaction to an applied stress will be partly elastic and partly viscous, i.e. they behave viscoelastically. So the dependence on time is an important characteristic of the small-deformation properties of gels. A new mathematical tool, fractal geometry, was used in the 1990s for studying the networks of gel structures (and of fats). Fractal geometry is useful for describing many of the irregular and fragmented patterns found in nature. The shapes of these patterns are not lines, planes or three-dimensional objects, and therefore cannot be described using Euclidean geometry. Fractal geometry is concerned with the geometric scaling relationships and symmetries associated with fractal objects, which is the name of a new family of geometrical shapes. The creator of fractal theory was Mandelbrot (1983). An important characteristic of a perfect fractal object is that it is self-similar at all levels of magnification. A fractal system can display statistical self-similarity rather than exact self-similarity, where the microstructure is similar over a certain range of magnification (Meakin 1988). The principles of fractal geometry can also be used to describe a disordered distribution of mass, including particles in a colloidal gel and fat crystal networks. In this case, the patterns are statistically self-similar at different scales of observation, and the relationship of the radius R to the mass M is given by M (R ) ~ R D

(4.175)

152

Confectionery and Chocolate Engineering: Principles and Applications

where D is the mass fractal dimension (more frequently referred to simply as the fractal dimension); see Meakin (1988), Vreeker et al. (1992), Marangoni and Rousseau (1996) and Narine and Marangoni (1999a). If Eqn (4.175) were describing a two-dimensional Euclidean object such as a square, then the value of D would be 2. However, a fractal object may be something intermediate between a line and a plane or between a plane and a cube. Therefore, the fractal dimension may be ‘fract(ion)al’: 1 < D < 2 or 2 < D < 3. A short summary of the concept of fractals and the determination of various fractal dimensions is given in Appendix 4 in order to facilitate understanding of the applications of this concept in engineering. After the introduction of the concept of fractals, many studies were carried out of the structures of polymer and colloidal aggregates. Scaling theory has been used to explain the elastic properties of protein gels (de Gennes 1979; Bremer et al. 1989a,b; Vreeker et al. 1992; Stading et al. 1993). Colloidal aggregates have been shown to be fractal structures both rheologically and optically (Weitz and Oliveira 1984; Brown and Ball 1985; Sonntag and Russel 1987; Buscall et al. 1988; Ball 1989; Uriev and Ladyzhinsky 1996).

4.8.2

Scaling behaviour of the elastic properties of colloidal gels

A power-law relationship between the elastic modulus and the solids volume fraction has been established from work with colloidal aggregates (Brown and Ball 1985; Sonntag and Russel 1987; Buscall et al. 1988; Ball 1989; Shih et al. 1990). The scaling behaviour of the elastic properties of colloidal gels was studied by Shih et al. (1990), who developed a scaling theory based on treating the structure of the gel network as a collection of flocs that are fractal objects, closely packed together throughout the sample. Two regimes, the strong-link and the weak-link regimes, were identified based on the strength of the links between the flocs relative to the strength of the links within the flocs themselves. When a network is composed of very large flocs, which occurs at low particle concentrations (low solid fat content, SFC), the links between the flocs are stronger than the flocs themselves. 4.8.2.1

Determination of the elastic constant (shear modulus)

Consider a network to which an external force f is applied in the x direction, causing a deformation. Across a cross-section A perpendicular to x there are N strands or chains per unit area bearing the stress σ, each exerting a reaction force −(df/dx) Δx, where Δx is the distance over which the relevant structural elements of the network have moved with respect to each other. This gives df σ = −N ⎛ ⎞ Δ x ⎝ dx ⎠

(4.176)

There is no restriction on the nature of the strands, on the elements that the strands are constructed from, or on the nature of the interaction forces involved. If the measurements are done in the so-called linear region, which is normally the case for small deformation experiments, df/dx is constant. Generally, f can be expressed as −dF/dx, where dF is the change in the Gibbs energy (free enthalpy) when the elements are moved apart by a distance dx, so we can write

The rheology of foods and sweets

⎛ d2 f ⎞ σ = N ⎜ 2 ⎟ Δx ⎝ dx ⎠

153

(4.177)

The local deformation Δx can be related to the macroscopic shear strain γ by a characteristic length C determined by the geometric structure of the network: Δx = γ C

(4.178)

In general, C has a tensor character. As the shear modulus G is given by σ/γ, ⎛ d2 F ⎞ G = NC ⎜ 2 ⎟ ⎝ dx ⎠

(4.179)

At constant temperature, we have dF = dH − T dS for the free entropy, where H is the enthalpy and S is the entropy, which results in the equation G=

NC d ( dH − T dS ) dx 2

(4.180)

According to Shih et al. (1990), in the strong-link regime (low SFC), the elastic constant is related to the solids volume fraction by G ~ Φ (3 − x ) (3 − D )

(4.181)

where x is the so-called backbone fractal dimension, which usually lies between 1 and 1.3 (Shih et al. 1990), and D is the fractal dimension. When a network is composed of very small flocs formed at a high particle concentration (high SFC), the links between the flocs are weaker than the flocs themselves. In this weak-link regime (high SFC), the elastic constant is related to the solids volume fraction by G ~ Φ (3 − 2 ) ( 3 − D )

(4.182)

Equations (4.181) and (4.182) show that in the weak-link regime the elastic constant of the system increases more slowly with particle concentration than in the strong-link regime.

4.8.3

Classification of gels with respect to the nature of the structural elements

According to the discussion by van Vliet (1999), several types of gels can be distinguished on the basis of the nature of the structural elements. Such a classification is irrespective of the energy content of the bonds or of their relaxation times. 4.8.3.1

Gels formed from flexible macromolecules

Flory (1953) derived the classical equation for the shear modulus of flexible macromolecules,

154

Confectionery and Chocolate Engineering: Principles and Applications

G = nkT

(4.183)

where G is the shear modulus (N/m2), n is the number of elastically effective chains per unit volume (m−3), k = 1.38062 × 10−23 J/K is the Boltzmann constant, and T is the temperature (K). A chain is defined as a part of a macromolecule extending from one cross-link to the next along the primary molecule. The cross-links represent the fixed points of the structure in the sense that the chain ends meeting there have to move together, irrespective of the motion of the cross-link. In terms of Eqn (4.183), the quantity kT stems from the second derivative of the Gibbs energy, and n stems from NC. The enthalpic contribution (see Eqn 4.178) may be neglected, as the contour length L of the chain between cross-links is much longer than the root-mean-square end-to-end distance 〈r2〉1/2 of a free chain of length L (Treloar 1975). Equation (4.183) has been shown to hold for many gels composed of synthetic polymers. For food-grade macromolecules, it holds for gelatin gels under the conditions that exist in food, and also for heat-set ovalbumin gels in 6 M urea. Even if Eqn (4.183) holds, however, the relation between the shear modulus and the concentration of macromolecules is less straightforward than we would expect from it, ⎛ c ⎞ G =⎜ RT ⎝ M c ⎟⎠

(4.184)

where c is the concentration of the gelling substance (w/w), Mc is the average molecular weight of the chain between two cross-links and R is the gas constant. Equation (4.182) cannot be used for the determination of molecular weight, and the probable explanation of this fact is that a proportion of the molecules are not involved in gel formation and/or that a proportion of the various chains are elastically ineffective. The proportion that are elastically ineffective will depend on the history (the cooling regime) and concentration. The storage modulus G′ of gelatin gels (see Eqns 4.140 and 4.141) has been found to be proportional to the concentration squared for concentrations above 2% (te Nijenhuis 1981). In the case of polysaccharide gels, the macromolecular chains are rather stiff. This means that the requirement L > 〈r2〉1/2 does not hold, especially for highly cross-linked gels. 4.8.3.2

Gels formed from hard particles

Our understanding of the relation between the structure of gels of hard particles and their small-deformation properties has been greatly enhanced by the introduction of the idea that clusters with a fractal structure are formed during aggregation (Bremer et al. 1989a,b, 1990; van Vliet 1999). The number of primary particles NP in a cluster with a fractal structure scales with the radius R: NP ⎛ R ⎞ =⎜ ⎟ N 0 ⎝ aeff ⎠

D

(4.185)

The rheology of foods and sweets

155

where D is the fractal dimension (D < 3), aeff is the radius of the effective building blocks forming the fractal cluster, and N0 is the number of primary particles forming such a building block. The size of the clusters scales with R3, and so the volume fraction of particles Φc in the cluster decreases with increasing radius. At a certain radius Rg, the average Φc will equal the volume fraction Φ of primary particles in the system; the clusters will then fill the total volume and a gel will be formed, with r2

= aeff Φ 1 (D −3)

12

(4.186)

where Rg is a measure of the average cluster radius at the moment the gel is formed. In fact, this quantity gives an upper cut-off length, i.e. the largest length scale at which the fractal regime exists. So, given the value of Φ, at least one additional parameter besides D needs to be known for a full characterization of the fractal clusters forming the gel network – namely Rg, aeff or N0. Gels built from fractal clusters with the same value of D but a different value of aeff will exhibit different structures at the same magnification, and hence a different permeability (van Vliet et al. 1997).

4.8.3.3

Gels formed from flexible particles

After aggregation of protein particles, such as casein, the particles start to fuse and the interaction between the original casein micelles becomes just as stiff as the rest of the casein particles. In such a case it is inappropriate to speak any more of an interaction energy between particles. An alternative is to assign a modulus to the protein chain (van Vliet 1999). For a cylindrical chain of aggregated particles of length L and radius a, where the stiffness of the particles is the same as that of the bonds between them, assuming a linear regime, the Young’s modulus E is given by E=

f ⎞⎛ L ⎞ ⎛ f ⎞⎛ x ⎞ σ =⎛ = ΔL L ⎝ πa 2 ⎠ ⎝ ΔL ⎠ ⎝ πa 2 ⎠ ⎜⎝ Δ x ⎟⎠

(4.187)

Since ⎛ d2 F ⎞ f = ⎜ 2 ⎟ Δx ⎝ dx ⎠

(4.188)

therefore 1 ⎛ d2 F ⎞ E =⎛ ⎞⎜ 2 ⎟ ⎝ πa ⎠ ⎝ dx ⎠

(4.189)

For the shear modulus of the gel, the relations G = NCπaE

(4.190)

156

Confectionery and Chocolate Engineering: Principles and Applications

where C = a/26 (Bremer and van Vliet 1991; Bremer et al. 1990), and G ~ ( π 26 ) EΦ 2 (3− D)

(4.191)

are obtained.

4.9

Rheological properties of sweets

Machikhin and Machikhin (1987) have given a comprehensive survey of the rheological properties of various sweets.

4.9.1 4.9.1.1

Chocolate mass Fluid models for describing the flow properties of chocolate mass

Rheological measurements of the properties of chocolate started in the 1950s. Some notable publications are those of Fincke (1956a), Kleinert (1957), Heiss and Bartusch (1957a,b), Steiner (1959b), Heimannn and Fincke (1962a–d) and Duck (1965). These researchers clarified some essential points about the measurement of the rheological properties of various types of chocolate mass (e.g. the importance of the preparation of the mass for measurement) and the types of flow curves of chocolate (e.g. the generalized Bingham model). These investigations also clarified the effects of cocoa butter, temperature, water content and addition of lecithin on the viscosity and yield stress of molten chocolate. Kleinert (1954a,b,c) studied the rheological properties of chocolate couvertures with a Drage viscometer, and determined that the Bingham model,

τ = τ 0 + ηPl D

(4.192)

where τ is the shear stress (Pa s), τ0 is the yield stress (Pa s), ηPl is the Bingham plastic viscosity (Pa s) and D is the shear rate (velocity gradient) (s−1), was useful. Thus a yield stress could be measured, and the Bingham plastic viscosity was approximately linear. Koch (1959) worked with a falling-ball viscometer of the Koch type. The viscosity data measured for chocolate couvertures were in the region of 15–50 Pa s; the values for couvertures manufactured for hollow figures were in the region of 25–26 Pa s, and those for chocolate bars were in the region of 25–55 Pa s. Steiner (1959b) studied the hypothesis of Bingham behaviour for chocolate at higher shear rates. His data for the yield stress were in the range 9–38 Pa, and the data for the Bingham plastic viscosity were in the range 1–3.4 Pa s, depending on the types of viscosimeter used. Fincke (1956a,b) and Heiss and Bartusch (1956) did not support the concept of the Bingham model for chocolate mass; however, they performed their investigations at low shear rate values. These investigators also reported evidence of thixotropic and sometimes rheopexic behaviour, as a result of making viscosity measurements first with increasing and then with decreasing rates of shear. Also, Mohos (1966) observed thixotropy and rheopexy in milk chocolate (water content, 1.23%; fat content, 38.4%; equipment,

The rheology of foods and sweets

157

falling-ball rheoviscometer of Höppler type). Rheopexic behaviour was characteristic of milk chocolate at higher temperatures (above 60°C). Fincke (1956a,b), Heiss and Bartusch (1956) and Steiner (1959a,b) determined that the Casson model (Casson 1959),

τ = τ 0 CA + ηCA D

(4.193)

where τ is the shear stress (Pa s), τ0 CA is the Casson yield stress (Pa s), ηCA is the Casson viscosity (Pa s) (independent of D) and D is the shear rate (s−1), can be used to describe the rheological properties of chocolate mass. It is important to note that the Casson model is the only one which is based on a physical picture, and not merely an empirical formula as are the other fluid models. For this physical picture, see Appendix 3. Later, Heinz (1959) and Heimann and Fincke (1962a–d) obtained the best fit of Eqn (4.193) for milk chocolate when the exponent was equal to 2/3 instead of 1/2, i.e. 3 23 τ 2 3 = τ 02 CA + ηCA D2 3

(4.193a)

These investigators had mentioned earlier that the exponent n was sometimes in the range 1/2 ≤ n ≤ 1. The fundamental results of these investigations were summarized in the classical publication of Fincke (1965). The IOCCC method (the use of which is presently suspended because it is under checking) uses the Casson equation, which provides the best fit (standard deviation < 3%) in the shear rate range D = 5–60 s−1 (Tscheuschner and Finke 1988a). In a relatively broad range of shear rate (D = 0.90–45 s−1), a good fit is obtained for dark chocolate at t = 50°C if n = 0.77 (Tscheuschner 1993a). According to the studies of Mohos (1966b, 1967a,b), a general equation

τ n = (τ 0 CA n ) + (ηCA n )n D n n

(4.194)

where K0 = (τ0 CA n)n and K1 = (ηCA n)n are constant, can be used for the description of the rheological properties of milk chocolate, where 1/2 ≤ n ≤ 1 in general. However, in some cases where the milk proteins have been strongly denatured owing to the effect of increased temperature (> 60°C) during conching or as a result of being pumped through too hot a tube, the relation n > 1 may be valid. Mohos (1967a,b) showed that an easy modification of the theoretical framework applied in the Casson model results in the general formula given in Eqn (4.194). For further details, see Appendix 3. It is to be emphasized, however, that the exponents n = 2/3 and 1/2 ≤ n ≤ 1, etc. derive from an effort to obtain a linear relationship, although the theoretical background may be interpreted in terms of the fractal nature of the material as well. In connection with this, the units of τn and (τ0 CA n)n are Pa, those of (ηCA n)n are Pa s and those of Dn are s−1. Table 4.5 presents an evaluation of a flow curve according to two different flow models (the Bingham model and the Casson model with n = ½). It is to be emphasized that the values of 19.62 Pa and 1.94 Pa s in the table are the Bingham yield stress and the Bingham (or plastic) viscosity, respectively. Similarly, the values of 9.18 Pa and 1.09 Pa s are the Casson yield stress and the Casson viscosity, respectively. These values are conceptually different, since in one case they relate to a Bingham fluid and in the other case they relate to a Casson fluid.

158 Table 4.5

Confectionery and Chocolate Engineering: Principles and Applications

Flow curve of a plain chocolate evaluated as a Bingham fluid and as a Casson (n = ½) fluid.

Shear stress (Pa)

Shear rate (s−1)

Shear stress0.5

Shear rate0.5

25.3 27.5 30.8 33.9 38.1 43.4 49.9 57.7 67.6 79.7

3.52 4.47 5.7 7.28 9.29 11.9 15.1 19.3 24.7 31.5

5.029911 5.244044 5.549775 5.822371 6.17252 6.587868 7.063993 7.596052 8.221922 8.927486

1.876166 2.114237 2.387467 2.698148 3.04795 3.449638 3.885872 4.393177 4.969909 5.612486

Intercept Slope Correlation

Yield stress Plastic viscosity Correlation r

19.62 Pa 1.94 Pa s 0.999138

Casson: Casson:

9.18 Pa 1.09 Pa s 0.999714

= Intercept2 = Slope2

3.029152 1.043251 0.999714

The most serious objection against the Casson equation is that the Casson viscosity ηCA and the Casson yield stress τ0 CA are not equal to the measured viscosity and yield stress values, respectively. The difference between ηCA and η is understandable: because of linearization, ηCA is independent of the shear rate but η is dependent on the shear rate. On the other hand, the yield stress is always a result of extrapolation, and in the linear plot of the Casson equation the extrapolation is easier. The Tscheuschner equation (Eqn 4.201 below) is tailored to the special properties of dark and milk chocolate; therefore, it provides the best fit to the flow curves. For further details, see Tscheuschner (1993a). 4.9.1.2

Effect of solids content and cocoa butter on the viscosity of chocolate

Steiner (1959a) refers to Habbard (1956), who determined that not all of the cocoa butter is available as a medium for dispersion of the solid particles. Part of it is probably absorbed on the surface of the particles present, and will not influence the viscosity. According to Habbard, a general relation exists over a wide range of concentrations: −k 1− Φ ⎞ ηPl = η0 ⎛ ⎝ 1− v ⎠

(4.195)

where ηPl is the plastic viscosity of chocolate (Pa s), η0 is the dynamic viscosity of cocoa butter when serving as a dispersion medium (Pa s), Φ is the volumetric proportion of solids (V/V), v is the volumetric proportion of voids in the packed solids (V/V) determined by centrifugation, and k is an exponent (constant). Habbard proposed that the value of k could be determined for a single pair of ηPl and η0 if c and v were measured, and thus the corresponding values of ηPl and η0 could be calculated for other concentrations. Although the Habbard formula has not achieved widespread use, its principal idea is in agreement with that represented by Eqns (4.172)–(4.174), and it provides a correct picture of the relationship between viscosity and solids content for chocolate. The Habbard formula is based on the very simple idea expressed by the equation

The rheology of foods and sweets

159

Table 4.6 Variation of plastic viscosity of chocolate as a function of cocoa butter content (CB), c and v, assuming that the viscosity of cocoa butter is 0.03 Pa s; see the Habbard formula (Eqn 4.195).

CB (%)

C

32 35 40

0.57 0.53 0.46

ηPl (v = 0.05, k = 5.73) (Pa s)

ηPl (v = 0.08, k = 5.44) (Pa s)

ηPl (v = 0.1, k = 5.26) (Pa s)

5.39 3 1.29

5.43 3 1.25

5.53 3 1.24

bΦ + cF = 1

(4.196)

where b = −1/(1 − v) = 1/(v − 1) is the volume ratio of solids, c is the volume ratio of liquids and F is the volume of free liquids (i.e. fats; cocoa butter in the case of chocolate). The constant b takes into account also the immobilized fat, which is either absorbed on the particle surfaces or not melted. From the Habbard formula, a modified form of the Einstein equation can be derived,

η∞ = η0 (1 + bΦ )k

(4.197)

Bartusch (1974) proposed the Eilers–Maron equation, ⎡ 1 − dΦ ⎤ η∞ = η0 ⎢ ⎥ ⎣ 1 − βVd ⎦

2

(4.198)

(where d and β are constant), which can also be regarded as a variant of the Habbard equation. Example 4.7 Let us consider a chocolate mass of plastic viscosity ηPl = 3 Pa s, and assume that η0 = 0.03 Pa s for cocoa butter, that the volumetric proportion of solids Φ for 35 m/m% cocoa butter is 0.525, as in Example 4.5, and that v = 0.05. From Eqn (4.195), 3 = 0.03[1 − 0.525/(1 − 0.05)]−k, and therefore k ≈ 5.73. Table 4.6 shows the results for ηPl for various values of cocoa butter content (CB) and v; the values of Φ were calculated as in Example 4.5. According to Steiner (1959a,b), the fat (mainly cocoa butter) content, which may vary between c. 32 and 40 m/m%, influences the viscosity very effectively because over this range the apparent viscosity may be reduced by a factor of 10 relative to its original value. In Example 4.7, the viscosities at the ends of this range are 5.39 Pa s (32% cocoa butter) and 1.29 Pa s (40% cocoa butter), which means a reduction by a factor of 4.18. The effect of solids depends not only on the amount but also on the quality of the solids. In milk chocolate, these solids may be sugar, fat-free dry cocoa cells and fat-free dried milk, all of which are likely to affect the properties in various ways. The size distribution of the solids plays an especially great role. A basic requirement of chocolate quality is that the largest particles must be smaller than 20 μm. However, if the proportion of

160

Confectionery and Chocolate Engineering: Principles and Applications

very small (<5 μm) particles is too high, their huge specific surface adsorbs a considerable amount of cocoa butter and, as a result, the proportion of dispersed phase will be decreased, which causes a strong increase in viscosity. Evidently, comminution has to be done under strict control in chocolate manufacture. Tscheuschner and Finke (1988a) determined that liquid, crystal-free cocoa butter in the shear rate range from 0 to 500 s−1 and at temperatures between 35°C and 100°C is a Newtonian fluid. For the temperature function of the viscosity of cocoa butter, the Frenkel–Eyring equation (Eqn 4.78) U ⎞ η = A exp ⎛ ⎝ RT ⎠ holds in the form 3533.7 ⎞ η = 5.7 × 10 −7 exp ⎛ (Pa s ) ⎝ RT ⎠

(4.199)

where R = 8.32 J/kmol. According to Tscheuschner and Finke (1988b), the rheological properties of cocoa butter–cocoa solid matter dispersions with a constant particle size distribution depend on the solids volume fraction Φ, the temperature, the moisture content, the lecithin content and the shear rate. Besides the structure-related viscosity ηstr(D1) (see below), a yield stress τ0 occurs at solids volume fractions Φ > 0.21. The value of the yield stress increases with increasing Φ. Additionally, weak thixotropic flow properties can be observed. However, the Casson model is not suitable, because at Φ < 0.2 there is no yield; rather, there is a viscous flow of the structure. Even at higher Φ values, the Casson model is fairly inaccurate in the shear rate range D < 5 s−1 (Tscheuschner and Finke 1988a). Tscheuschner (1993) discussed several suspensions of cocoa butter, among them chocolate. For cocoa butter/cocoa solids, cocoa butter/sugar and cocoa butter/milk powder suspensions, he recommended

ηsp = ηrel − 1 = [η ]Φ + AΦ n + BΦ n

(4.200)

where ηrel = η∞/ηCB is the relative viscosity of the suspension, expressed as the ratio of the viscosity of the suspension (in equilibrium, D → ∞) to the viscosity of clear cocoa butter; Φ is the volume concentration of solid spheres in suspension; and A and B are constants. Equation (4.200) can be regarded as a variant of the Einstein equation (Eqn 4.157). Tscheuschner (1989, 1993c) developed a flow model with four parameters for molten chocolate, −n

⎛ D⎞ τ = η∞ D + τ 0 + ηstr ( D1 ) ⎜ ⎟ D ⎝ D1 ⎠

(4.201)

or, for the viscosity ηS of chocolate,

ηS ( D ) = η∞ +

τ0 ⎛ D⎞ + ηstr ( D1 ) ⎜ ⎟ ⎝ D1 ⎠ D

−n

(4.202)

The rheology of foods and sweets

161

where ηstr(D) is called the structural parameter, and depends on D. The units of D and D1 are s−1. If D = D1, Eqn (4.202) can be written

ηS ( D1 ) = η∞ +

τ0 + ηstr ( D1 ) D1

(4.203)

If the value of D is increased (D → ∞), the additive terms τ0/D and ηstr(D) disappear (see Eqn 4.202) because the high shear rate destroys the structure of the suspension and, as a result, Eqn (4.203) simplifies to

ηS ( D ) = η∞ = constant

(4.204)

The viscosity η∞ refers to the state where D > 200 s−1. In a study of 19 chocolate masses by Tscheuschner (1993a), the (modified) Eqns (4.201) and (4.202) provided a better fit than the Casson equation did. For the relations between the various fluid models (the Newton model, the Casson model with d = 1/2 and n = 2/3, the Heinz model with a general value of n, and the Herschel–Bulkley model) applied to chocolate, see Tscheuschner (1999). 4.9.1.3

Effect of lecithin on the viscosity of chocolate

A substance which radically affects the viscosity of chocolate is lecithin, which is mostly of soya origin. Most of the reduction occurs with the first 0.2–0.3%, and there is little further gain beyond additions of 0.5%. Roughly, 0.3% of commercial lecithin is equivalent to a replacement of 4–5% cocoa butter. Excessive quantities of lecithin, however, have been reported as leading to an increase in viscosity (Liebig 1953). Here it should be mentioned that control of the yield stress σ0 is important in the shaping of chocolate products, both in the case of the covering of centres (see Eqn 4.71) and in the case of shell moulding and the shaping of figures by use of a spinner. A food additive widely applied for this purpose is polyglycerol polyricinoleate (PGPR, E476), which strongly increases the yield stress. A combination of lecithin and polyglycerol polyricinoleate in a ratio of 3 : 1 to 4 : 1 is used, at a maximum content of 0.5%. Tscheuschner (1993a, 2008), in connection with the agglomeration phenomena that take place in chocolate during conching, stated that an addition of 0.2% lecithin somewhat improves the rheological properties because its amphoteric molecules cover the hydrophilic particle surfaces, leading to a decrease in the interfacial free energy. But the effect of lecithin alone is not sufficient to hinder the increase in yield stress at high water content. In this case a mix of lecithin and PGPR (in the ratio 7 : 3) gives better results. The effect of PGPR can be explained by the binding of the PGPR molecules to the heterogeneous surfaces of milk powder particles, rather than to the hydrophilic surfaces of sugar particles. It is to be stressed that the value of 0.2% for the lecithin content seems to be optimal, because when both lower and higher amounts are added, the resulting reduction in the viscosity of chocolate is weaker. An excellent summary of the emulsifiers used in chocolate has been given by Minifie (1999, Chapter 4). The effect of emulsifiers has a close connection with the water content of the chocolate. As a general rule, the water content must be below 0.4% – above this value, the viscosity

162

Confectionery and Chocolate Engineering: Principles and Applications

starts to increase rapidly. The effect of emulsifiers is to disperse the water content of the chocolate. The undesirable effect of humidity on chocolate manifests itself strongly above 0.4%, which is equivalent to a water content of the milk powder of about 2.5%. This value of 0.4% is a critical threshold for particle aggregation. It was demonstrated by Tscheuschner (2002) that if the initial water content of a chocolate suspension is low enough, a long conching time is not necessary, because it cannot improve the rheological properties. This perception is fundamental from the point of view of reducing the conching time and energy consumption. The key point is that it is necessary to decrease the water content of the milk powder, which can be done, for example, by some previous preparation of the milk powder. 4.9.1.4

Effect of temperature on the viscosity of chocolate

The relation between the viscosity of melted chocolate and temperature has been studied by several researchers (Stanley 1941; Kleinert 1954d; Fincke and Heinz 1956), and appears to follow an exponential law (Eqn 4.78). Fincke and Heinz (1956), using a Rotovisco viscometer, plotted log η against 1/T and obtained a straight line up to 80 or 90°C in the case of plain chocolate. The magnitude of the effect of temperature was of the order of a 2–3% decrease per 1°C, and was similar to the effect for cocoa butter alone. The temperature coefficient appeared to increase slightly with shear rate. For milk chocolate, the logarithmic relationship did not hold above about 60°C, owing to changes consequent upon heat treatment. According to Heimann and Fincke (1962c,d), the critical temperature region starts at about 60°C, where the Maillard reaction between the milk protein and the sugars in milk chocolate becomes more and more intense, and this causes a definite increase in both the viscosity and the yield stress. According to the experiments of Mohos (1982), the progress of the Maillard reaction, characterized by the hydroxymethylfurfurol (HMF) content of the milk chocolate, is a function of temperature; for further details, see Section 16.2.1. 4.9.1.5

Pressure dependence of the flow curve of chocolate mass

Machikhin (1968) investigated the pressure dependence of the flow curves of chocolate masses at 44°C with a rotoviscometer. According to this study, chocolate masses were shown to be Bingham fluids. The exponential relationship in Eqn (4.72) proved valid for the viscosity but less valid for the yield stress. For more details, see Machikhin and Machikhin (1987, pp. 135–151).

4.9.2

Truffle mass

Machikhin et al. (1976) investigated ‘Extra’ truffle mass (with a fat content of 43.3% and a water content of 7%) with a Rheotest-2 rotational viscometer. In the manufacturing process, chocolate mass and milk butter (both molten) were whipped for c. 30–40 min. Evaluation of the flow curve according to the Ostwald–de Waele model gave the results

τ = (306 − 7.05t ) D 0.37 and

(4.205)

The rheology of foods and sweets

η = (306 − 7.05t ) D −0.63

163

(4.206)

where τ is the shear stress (Pa); η is the dynamic viscosity (Pa s); D is the shear rate (s−1), which ranged from 0.5 to 218.7 s−1; and t is the temperature (°C), which ranged from 25 to 34°C. The ‘power law index’ n is 0.37 and the ‘consistency index’ K is equal to 306 − 7.05t.

4.9.3

Praline mass

Birfeld (1970) and Birfeld and Machikhin (1970) investigated the viscosity of various praline masses, using rotational viscometers of types RV-8 and RM-1. The result of the evaluation followed the Ostwald–de Waele model: four praline masses gave the results η (kPa s) = 1.62D−0.49, 1.44D−0.51, 1.08D−0.41 and 0.79D−0.62, where D is in s−1. According to Machikhin and Machikhin (1987, Chapter 3), the flow curves of various praline masses can be described by the Bingham model (Eqn 4.192). However, in the region D > 6–7 s−1, the following formula can be used: b η = ηPl exp ⎛ ⎞ ⎝ D⎠

(4.207)

where b is a constant, and if D → ∞, then η → ηPl.

4.9.4

Fondant mass

Nikiforov et al. (1964) found that the viscosity of various fondant masses can be described by flow curves of the Ostwald–de Waele type. In the region D = 0.4–0.3 s−1,

η ( kPa s ) = 8.2 D(0.53−1)

(4.208)

and in the region D = 0.3–10 s−1,

η ( kPa s ) = 5.2 D(0.123−1)

(4.209)

Nikiforov et al. (1964) investigated the temperature dependence of the viscosity of the fondant masses by evaluating the flow curves in a linear form (ln η vs ln D). The following relations were obtained: at 16°C: η = 6.92 D −1.073

(4.210)

at 20°C: η = 4.77 D −0.771

(4.211)

at 24°C: η = 3.65D

(4.212)

−0.79

at 28°C: η = 2.85D −0.814

(4.213)

at 32°C: η = 1.98D

(4.214)

−0.695

The temperature dependence of the viscosity of fondant mass can be described by the relationship

164

Confectionery and Chocolate Engineering: Principles and Applications

Table 4.7

The values of the constants n, A, B, a and b in Eqn (4.216).

N 0.5 0.55

A

B

a

b

28525 12900

−875 −400

21323 9524

−667 −282

η = (12.113 − 0.345 × t ) D n−1

4.9.5

(4.215)

Dessert masses

4.9.5.1

Dessert masses containing fondant

Flow curves of various dessert masses containing fondant were given by Koryachkhin (1975), who proposed the following formula according to the Herschel–Bulkley model:

η ( Pa s ) =

A + Bt + ( a + bt ) D n−1 D

(4.216)

where η (Pa s) is the structure viscosity, D (s−1) is the stress rate, A + Bt (Pa) is the yield stress, a + bt (Pa sn−1) is the plastic viscosity and t (°C) is the temperature. The values of n, A, B, a and b are shown in Table 4.7. The conditions of validity of Eqn (4.216) are D = 2 − 140 s −1, t = 28 − 30°C

(4.217)

Marshalkin et al. (1970) and Marshalkin and Karpin (1971) investigated the structure viscosity of dessert masses containing fondant. They recommended the following formula: log η = ( 4.19 − 0.036t ) (1 − 0.223 log D )

(4.218)

where t (°C) is the temperature (10–70°C), η (Pa s) is the structure viscosity and D (s−1) is the shear rate. (Here, ‘log’ means the logarithm to base 10.) Maksimov (1976) and Maksimov and Machikhin (1976) investigated the rheological properties of dessert masses containing fondant with a rotoviscometer of type RV-8 in such a way that the external rotating cylinder was vibrated at a frequency f (Hz), and the gap between the two cylinders was regarded as the amplitude of the strain γ. In this arrangement, the shear rate can be calculated as dγ dt = 2 πfγ 0

(4.219)

where γ 0 is the maximum amplitude (i.e. the gap). The following relationship was recommended:

η=a

dγ dt

(4.220)

The rheology of foods and sweets

165

where η is the viscosity (kPa s), a is a constant that decreases if the temperature is increased (with values in the range 948–33 500), and b is a constant that increases if the temperature is increased (with values in the range −0.594 to −1.465). As Eqns (4.174) and (4.186) show, these masses follow the Ostwald–de Waele model, i.e. a ↔ K (the consistency coefficient) and b ↔ n (the index of the power law or fluid model). Under the effect of vibration, K is decreases, while n increases up to 1. Consequently, the behaviour of these masses approaches that of a Newtonian fluid. This effect of vibration is very evident in the region of small shear rates, 0–20 s−1. Because the usual shear rates used in shaping are about 10 s−1, this fact is important from a technological viewpoint. 4.9.5.2

Rheological model of dessert fillings

For modelling the extrusion of dessert fillings, Kot and Gligalo (1969) recommended a complex model consisting of two Hooke elements, a Newton element and a St Venant (plastic) element (Fig. 4.25). According to Kot and Gligalo’s tests, the rheological properties of dessert fillings are strongly influenced by the amount and the size distribution of crystals; for example, if the amount of crystals of maximum size 0.05 mm is less than 50 m/m%, the filling can be regarded as a Bingham fluid.

4.9.6

Nut brittle (croquante) masses

Maksimov et al. (1973) proposed an Ostwald–de Waele model for nut brittle masses at various temperatures (Table 4.8),

t G1

h

P0 G2 t

Fig. 4.25

Fluid model given by Kot and Gligalo (1969) for extrusion of dessert fillings.

Table 4.8 n and K values at various temperatures for Maksimov et al.’s model. Temperature (°C) n (flow index) K (consistency coefficient)

90 0.46 1730

110 0.37 1050

130 0.44 350

166

Confectionery and Chocolate Engineering: Principles and Applications

Table 4.9 Viscosity of whipped masses as a function of whipping time. Time (s) 180 210 240 270 300

Viscosity (Pa s) 10.34 18.27 30 42.5 50

η ( Pa s ) = KD n−1

(4.221)

For further details, see Machikhin and Machikhin (1987, Section 3.1.1).

4.9.7

Whipped masses

Goguyeva (1965) measured the viscosity of whipped masses as a function of the duration of whipping (Table 4.9). A plot of viscosity vs time is approximately linear, but both the initial and the terminal parts of it definitely differ from linearity.

4.10 4.10.1

Rheological properties of wheat flour doughs Complex rheological models for describing food systems

As Scott Blair (1975) writes, systematic work on the rheology of flour dough was done by Kosutány and Hankóczy in the first decades of the 20th century. Kosutány (1907) described an apparatus designed by Rejto˝. Strips of dough, rectangular in cross-section, were stretched on a series of low-friction metal rollers. Hankóczy (1920) was a pioneer in developing a method for measuring the work done during the kneading of dough in a mechanical mixer, a method later commercialized by Brabender in Germany and still used today. In connection with the methods for measuring the rheological properties of dough, Baird (1983) gave a review of measurements with devices that provided material properties in engineering/scientific units. In order to obtain the well-defined flows described above that lead to material functions, certain test geometries must be employed. The material functions defined for shear flow are most directly obtained with a cone-and-plate rheometer but can also be determined with a plate–plate rheometer. The cone-and-plate geometry leads to a uniform shear rate throughout the sample. Hence, from torque and normal-thrust measurements, η and the first normal-stress coefficient can be obtained as a function of γ ′ as follows (see Eqn 4.55):

Φ1 = Φ1 (γ ′ ) =

σ 11 − σ 22 N1 = (γ ′ )2 (γ ′ )2

(4.222)

In a plate–plate rheometer, γ ′ varies with the radial position, which requires extra calculations to obtain η and N1 as a function of γ ′. These test geometries can also be used to

167

The rheology of foods and sweets

carry out transient shear experiments, although the cone-and-plate geometry is the preferred geometry. The dynamic viscosity (η) in a steady shear flow can be obtained at higher shear rates using a capillary rheometer rather than a rotary rheometer. However, the shear rate also varies with radial position and is a function of the viscosity. The wall shear rate can be obtained using a procedure that corrects for the non-parabolic velocity profile. Unfortunately, there is no established way to obtain N1 from a capillary rheometer at present. Two companies, Rheometrics Inc. and Sangamo Ltd, manufacture rotary rheometers suited to carrying out various shear flow experiments. The extensional (or elongational) viscosity is obtained most often by extending the end of a cylindrical specimen exponentially with time, which leads to values of ε′ independent of the position in the sample, or by extending the end at a constant rate, which requires a knowledge of the diameter profile to calculate ε′. By definition,

σ 33 − σ 11 = −η ∧ ( ε ′ ) ε ′

(4.223)

where η^(ε′) is the extensional viscosity as function of ε′, and ε′ is the extension rate. Methods for generating both biaxial extension and planar extension are also available. An instrument for carrying out various unidirectional elongational flow experiments is manufactured by Rheometrics. For some special testing methods, see Section 4.10.2 Some models for describing the properties of doughs are presented below, making use of the results of measurements which provide quantities with engineering significance. 4.10.1.1

Bread dough

Bread provides an instructive example of the investigation of the rheological properties of foods containing flour as the principal component. A rheological model of the soft part of bread given by Lásztity (1987b) is presented in Figs 4.26 and 4.27. Figure 4.26 shows the case if the stress P is less than P0 (the yield stress). Figure 4.27 shows the case if P > P0. This model can be characterized by five parameters: G1 and G2 for the Hooke elements, μ1 and μ2 for the Newton elements, and P0 (stress yield) for the St Venant plasticity model. G1 can be calculated from the elastic deformation, by applying Hooke’s law (Fig. 4.26): G1 =

P ε

(elastic deformation)

(4.224)

The following equation applies to the Kelvin model characterized by G2 and μ1: ⎛ G t⎞⎤ ⎛ P ⎞⎡ ε ( deformation ) = ⎜ ⎟ ⎢1 − exp ⎜ − 2 ⎟ ⎥ ⎝ G2 ⎠ ⎣ ⎝ μ1 ⎠ ⎦

(4.225)

The retarded elastic deformation (see Fig. 4.26 or 4.27) is equal to P/G2, where P is the stress applied. G2 can be calculated by applying Hooke’s law, G2 =

p (retarded elastic deformation) ε

(4.226)

168

Confectionery and Chocolate Engineering: Principles and Applications

Hooke

G1 m1

G2 St Venant

P0

Newton

m2 a Deformation

b

c

e

d

P < P0

Elastic Total Retarded Elastic

t(0)

t(1)

Time

Fig. 4.26 Model for the soft part of bread (1). Consecutive phases of the deformation process are indicated by a–e. a, initial phase at t = t(0); b, elastic deformation; c, retarded elastic deformation; d, after t = t(1) (the stress has ceased), recovery of the first phase of elastic deformation (phase ‘b’); e, recovery of phase ‘c’. [Reproduced from Lásztity (1987b), with permission.]

At a given t = τ, the appropriate deformation ε = ε(τ) can be read from the curve (Fig. 4.26), and μ1 can be calculated from Eqns (4.191) and (4.222):

μ1 = −

G2 τ ln [1 − ε (τ )G2 p ]

(4.227)

Determination of P0 (the yield stress) and μ2 can be carried out if p is changed and the flow curve (P vs dε/dt) is evaluated according to the model dε P = P0 + μ2 ⎛ ⎞ ⎝ dt ⎠

(4.228)

P0 is obtained by interpolation of the curve of P to obtain the value at which dε/dt = 0. Glücklich and Shelef (1962) recommended a complex model consisting of eight elements for wheat doughs (Fig. 4.28). 4.10.1.2

Pretzel dough

Machikhin (1975) determined the temperature dependence of the viscosity of sweet pretzel doughs as follows:

The rheology of foods and sweets

a

b

c

e

d

169

f

Deformation

Elastic

Total elastic Total

Retarded elastic Lasting deformation t(0)

t(1)

Time

Fig. 4.27 Soft part of bread (2). Consecutive phases of the deformation process are indicated by a–f. a–c, as in Fig. 4.26; d, plastic deformation (represented by a St Venant model); e, recovery of phase ‘b’ (elastic deformation); f, recovery of phase ‘c’ (retarded elastic deformation). [Reproduced from Lásztity (1987b), with permission.]

t G1

G2

P0,1

h1

G3

h2

P0,3

t Fig. 4.28

Glücklich/Shelef model for wheat dough.

P0,2

170

Confectionery and Chocolate Engineering: Principles and Applications

t Hooke G1

Hooke G2 St Venant t0

Newton h

t Fig. 4.29

Shvedov model.

A η = 10 exp ⎡⎢⎛ ⎞ B ⎤⎥ ⎝ ⎣ t⎠ ⎦

(4.229)

where η is the viscosity (Pa s), t is the temperature (30–60°C), and A and B are constants dependent on the shear rate, the water content of the dough and the overpressure (between 0 and 1.5 MPa). The values of A and B were in the range 0.967–1.062 and 0.106–0.182, respectively, if the water content was 32% and the shear rate was in the range 0.1–0.35 s−1 (at 1.47 MPa), and 1.118–1.061 and 0.198–0.227, respectively, if the water content was 30.4% and the shear rate was in the range 0.1–1.0 s−1 (again at 1.47 MPa). Machikhin (1975) recommended the Shvedov model (Fig. 4.29) for characterizing sweet pretzel doughs. This consists of a Newton, a St Venant and two Hooke elements; consequently, the pretzel mass has a yield stress τ0 (see the St Venant element). The following differential equation applies to the Shvedov model (Machikhin 1975): dτ ⎛ 1 1 ⎞ τ − τ0 D=⎛ ⎞⎜ + ⎟ + ⎝ dt ⎠ ⎝ G1 G2 ⎠ η

(4.230)

where D is the shear rate, τ is the shear stress, τ0 is the stress yield, and G1 and G2 are the elasticity moduli of the Hooke elements. For further details, see Machikhin and Machikhin (1987, pp. 121–125).

4.10.2

Special testing methods for the rheological study of doughs

Szczesniak (1963a) classified texture-measuring instruments into three groups: • fundamental tests, in which properties such as Young’s modulus and viscosity are measured; • empirical tests that measure properties that are usually poorly defined but which have been shown by practical experience to be related to textural quality in some way;

The rheology of foods and sweets

171

• imitative tests that measure various properties under conditions similar to those to which the food is subjected during mastication. Bourne (1975) has given a review of texture measurements and instruments for that purpose. Food doughs are defined here to be low-moisture mixtures of water and wheat, corn, oat, semolina or soya flour or mixtures of these flours. Other ingredients can also be added, such as flavourings and oils. Rheological measurements of food doughs have certainly been carried out for many years in the food industry. The various special testing methods for the rheological study of doughs will not be discussed here in detail, but some references are given below. 4.10.2.1

Farinograph

Two of the most widely used physical dough-testing instruments for wheat quality evaluation studies are the farinograph, designed by Hankóczy (1920), and the mixograph, designed by Swanson and Working (1933). The technique is to combine flour and water (and other ingredients), and to record the torque required to mix the resulting dough. This record provides a quantitative measure related to the rheological properties of the dough. 4.10.2.2

Brabender farinograph

The most complete and comprehensive source of information dealing with the farinograph is the third edition of The Farinograph Handbook by D’Appolonia and Kunerth (1984). The mixing action is brought about by two sigma-type blades which rotate at speeds with a ratio of 3 : 2. The type of mixing created by this type of blade is different from that caused by the pin-type mixer in the mixograph. The temperature during mixing is controlled by the use of temperature-controlled water circulating in a jacket surrounding the bowl in which the dough is mixed. For the evaluation of the results, see Farinograph-E Worldwide Standard for Testing Flour Quality (1997).

4.10.2.3

Brabender extensigraph

This is used for measuring the resistance (in Brabender units, B.U.) of dough as a function of extensibility (in cm). Measurements made with a Brabender extensigraph can be regarded as supplements to measurements made with a farinograph. If a plot of σ0 − σ vs time is prepared, a curve is obtained that is very similar to that obtained with an extensigraph for wheat flours (Buschuk 1985), which follows the Maxwell model (Eqn 4.84).

4.10.2.4

Mixograph

The mixograph is a small, high-speed recording dough mixer originally designed by Swanson and Working (1933) to provide a method of measuring quality, as far as quality is related to gluten structure. The device measures the rate of development of the dough,

172

Confectionery and Chocolate Engineering: Principles and Applications

the maximum resistance of the dough to mixing and the duration of the resistance to mechanical overmixing. The first systematic statistical study relating characteristics measured by mixograph to the results of baking was conducted by Johnson et al. (1943). 4.10.2.5

Alveograph

The alveograph, designed by Chopin (1927), is a rheological technique designed for the routine testing of wheat flours. A dough prepared from the flour under standard conditions of water addition and mixing is made into a sheet and cut into a circular test piece, which, after a period of resting, is subjected to biaxial extension by inflating it into the shape of a bubble until it ruptures. The pressure in the bubble is measured with a manometer and recorded on a chart as a function of time. For further detail, see Hlynka and Barth (1955a,b), Bloksma (1957), Chopin (1962), Scott Blair and Potel (1973) and Rasper and Hardy (1985). 4.10.2.6

Texturometer

Many of the instrumental tests described in the literature relate to the characterization of ‘as is’ baked goods. In the mouth, the products are not only disintegrated mechanically by mastication, but are also mixed with saliva, which softens and hydrates the structure. The imitative tests applicable to baked goods are mainly of at type simulating mastication. A typical example here is the General Foods texturometer, composed of mechanical jaws, a strain gauge and a recording system (Friedman et al. 1963). Excellent correlations with sensory ratings have been reported by Szczesniak (1963a,b). Further references include Brandt (1962) and Tanaka (1975). A similar test may be performed using a universal testing machine such as an Instron tester (Bourne 1968). The method is known as instrumental texture profiling analysis (TPA). 4.10.2.7

Penetrometer

The penetrometer is a simple instrument used commonly to assess the strength of baked goods. A probe or indenter is generally used, and the depth of penetration at a definite time after loading with a constant weight is recorded. The greater the penetration, the more tender the product. The theory and application of puncture testing were described by Bourne (1979). The penetrometer also is widely used for measuring the textural properties of jellies, whipped sweets, fondant products etc. For further details, see Babb (1965), Funk et al. (1969), Morandini et al. (1972), Smejkalova (1974) and Choishner et al. (1983).

4.10.3

Studies of the consistency of dough

Miller (1985), after many studies with several penetrometers, including an Instron Universal Tester, developed a method of measuring the consistency of short doughs using a Stevens LFRA Texture Analyzer. This analysis generated a standard relationship between the consistency of a dough and the water level of the recipe which was applicable to all 18 flours tested. This relationship is

The rheology of foods and sweets

W0 =

Wt 1 − 0.71 log (C 234 g )

173

(4.231)

where W0 is the optimum recipe water level (% by flour wt), Wt is the recipe water level (% by flour wt), C is the consistency (g), and W0 is the optimum recipe water level (% by flour wt) which produces a dough consistency of 234 g. Under standardized test conditions, the dough consistency measurements correlated well with both the weight of dough samples and the weight of biscuit pieces. The dough consistency also correlated reasonably well with biscuit thickness. This method has potential applications in problem solving and in laboratory matching of production doughs. Nyikolajev (1967), Nyikolajev and Mityukova (1976) and Nyikolajev et al. (1976, 1978) investigated the tensile strength of doughs made with an aqueous sugar solution and a yeast suspension using a plastometer of type KP-3. The following relationship between the tensile strength of the dough and the amount of thinning solution was given:

τ 0 = A − Bc

(4.232)

where c is the amount of aqueous sugar solution or suspension of yeast added, and A and B are constants. According to Rebindyer (1958), the following relationship can be applied to data obtained with a plastometer:

τ0 =

K (α ) P h2

(4.233)

where τ0 is the tensile strength (Pa), K(α) = (1/π) cos2 (α/2)cgt (α/2) is a constant depending on the angle of the measuring cone, P is the force (N) exerted by the measuring cone and h is the sinking depth (m) of the measuring cone. Mazur and Dyatlov (1972) investigated the tensile strength of various doughs containing fat, sugar and yeast with a plastometer of type KP-3. Based on their results, the following relationship was recommended:

τ 0 = aθ b

(4.234)

where τ0 is the tensile strength (kPa); θ is the rising time (min); a is a constant, with values in the range 3.6–4.7 × 10−4, and b is a constant, with values in the range 0.7–1.1. The values of the constants depend on the fat and sugar contents of the dough. 4.10.3.1

Compressibility of doughs

The compressibility of doughs is an important parameter in the shaping of doughs by compression. Volarovich and Nyikiforova (1968) investigated the compressibility of biscuit masses. The decrease in volume was measured. The following relationship could be applied for biscuit masses: pV a = b ×10c

(4.235)

174

Confectionery and Chocolate Engineering: Principles and Applications

where p is the pressure (MPa); V is the volume of dough (cm3) at a pressure p; and a, b and c are constants depending on the volume concentration of air in the dough. 4.10.3.2

Doughs without yeast

Blagoveshchenskaya (1975) obtained a relationship for the viscosity of doughs without yeast,

η = c + e aW a

(4.236)

where η is the viscosity (Pa s); W is the water content (%); and a, b, c and e are constants. If the temperature dependence of the viscosity of such doughs is investigated at a given shear rate, a minimum value of the viscosity is found, and the temperature at which this minimum value occurs becomes higher as the water content of the dough is made lower. 4.10.3.3

Tadzhik girdle cakes

Libkin et al. (1978) investigated the flow curve of Tadzhik girdle cakes, and the following relationship was determined:

η=

a Dn

(4.237)

where η is the viscosity at 20°C (Pa s) and D is the shear rate (s−1). The dimension of the constant a depends on the value of n; for example if n = 0.668, then a = 5426 Pa s(1−0.668). The Tadzhik girdle cakes followed the Ostwald–de Waele model. For the temperature dependence of the viscosity of Tadzhik girdle cakes, the following relationship gave a good approximation:

η = a + bt + ct 2

(4.238)

where η is the viscosity (Pa s); t is the temperature (20–34°C); and a, b and c are constants depending on the experimental conditions and the shear rate. 4.10.3.4

Biscuit doughs

Machikhin and Machikhin (1987) recommended the following relationship for the viscosity vs shear rate curve (measured with an RM-1 rotoviscometer) of biscuit doughs:

η = 1 ( a + bD )

(4.239)

where η is the viscosity (Pa s); D is the shear rate (s−1); a is a constant [(Pa s)−1], with values between 0.018 and 0.08; and b is a constant (Pa−1), with values between 0.062 and 0.636. Different viscosity values were obtained when the components of the recipe (fat, flour, yeast, and aqueous solutions of sugar and salt) were changed. For further details, see Machikhin and Machikhin (1987, Chapter 3).

The rheology of foods and sweets

175

Manohar and Rao (1997) studied the effect of mixing time and additives on the rheological characteristics of dough and on the quality of the resulting biscuits. They found that an increased mixing time influenced the rheological characteristics by increasing the compliance, elastic recovery, cohesiveness, adhesiveness and stickiness, and by reducing the extrusion time, the apparent biaxial extensional viscosity and the consistency hardness. Doughs mixed for 180 s gave biscuits of superior quality compared with those made from a dough mixed for either 90 s or 300 s. The incorporation of cysteine or dithioerythriol, particularly in doughs from medium strong wheat flour, resulted in biscuits with greater spread and crispness.

Further reading Cakebread, S.H. (1971) Physical properties of confectionery ingredients – Viscosity of carbohydrate solutions. Confect Prod 37 (11): 662–665. Cakebread, S.H. (1971) Physical properties of confectionery ingredients – Viscosity of high boilings. Mixtures of high solids content of high temperatures. Confect Prod 37 (12): 705–709. Eszterle, M. (1990) Viscosity and molecular structure of pure sucrose solutions. Zuckerindustrie 115 (4): 263–267. Figura, L.O. and Teixeira, A.A. (2007) Food Physics: Physical Properties, Measurement and Applications. Springer. Kress-Rogers, E. and Brimelow, C.J.B. (2001) Instrumentation and Sensors for the Food Industry. CRC Press, Boca Raton, FL. Launey, B. and Bure, J. (1974) Stress relaxation in wheat flour dough following a finite period of shearing. 1. Qualitative study. Cereal Chem 51 (2): 151. Lewis, M.J. (1996) Physical Properties of Foods and Food Processing Systems. Woodhead Publishing, Cambridge. Manley, D.J.R. (1981) Dough mixing and its effect on biscuit forming. Cake and Biscuit Alliance Technologists’ Conference. Marangoni, A.G. and Narine, S.S. (2004) Fat Crystal Networks. Marcel Dekker, New York. Miller, A.R. (1984) Rotary Moulded Short-Dough Biscuits, Part V: The Use of Penetrometers in Measuring the Consistency of Short Doughs. FMBRA Report 120. Narine, S.S. and Marangoni, A.G. (2002) Physical Properties of Lipids. Marcel Dekker, New York. Scholey, J., et al. (1975) Physical Properties of Bakery Jams. BFMIRA Report 217. Scholey, J. and Vane-Wright, R. (1973) Physical Properties of Bakery Jams: An Investigation into Methods of Measurement. BFMIRA Technology Circular 540. Steele, I.W. (1977) Measurement of biscuit dough consistency. FMBRA Bull 2, 50. Steele, I.W. (1977) The search for consistency in biscuit doughs. Baking Ind J 9 (3): 21. Thacker, D. and Miller, A.R. (1979) Process variables in the manufacture of rotary moulded Lincoln biscuits. Cake and Biscuit Alliance Technologists’ Conference. VDI-GVC (2006) VDI-Wärmeatlas. Springer, Berlin. Verkroost, J.A. (1979) Some Effects on Recipe Variations on Physical Properties of Baker Jams. BFMIRA Report 297. Wade, P. (1965) Investigation of the Mixing Process for Hard Sweet Biscuit Doughs, Part I, Comparison of Large and Small Scale Doughs. BBIRA Report 76. Wade, P. and Davis, R.I. (1964) Energy Requirement for the Mixing of Biscuit Doughs under Industrial Conditions. BBIRA Report 71.

Chapter 5

Introduction to food colloids

Contents 5.1

5.2

5.3

5.4

5.5

5.6

5.7 5.8

The colloidal state 5.1.1 Colloids in the confectionery industry 5.1.2 The colloidal region 5.1.3 The various types of colloidal systems Formation of colloids 5.2.1 Microphases 5.2.2 Macromolecules 5.2.3 Micelles 5.2.4 Disperse (or non-cohesive) and cohesive systems 5.2.5 Energy conditions of colloid formation Properties of macromolecular colloids 5.3.1 Structural types 5.3.2 Interactions between dissolved macromolecules 5.3.3 Structural changes in solid polymers Properties of colloids of association 5.4.1 Types of colloids of association 5.4.2 Parameters influencing the structure of micelles and the value of cM Properties of interfaces 5.5.1 Boundary layer and surface energy 5.5.2 Formation of boundary layer: adsorption 5.5.3 Dependence of interfacial energy on surface morphology 5.5.4 Phenomena when phases are in contact 5.5.5 Adsorption on the free surface of a liquid Electrical properties of interfaces 5.6.1 The electric double layer and electrokinetic phenomena 5.6.2 Structure of the electric double layer Theory of colloidal stability: the DLVO theory Stability and changes of colloids and coarse dispersions 5.8.1 Stability of emulsions 5.8.2 Two-phase emulsions 5.8.3 Three-phase emulsions 5.8.4 Two liquid phases plus a solid phase 5.8.5 Emulsifying properties of food proteins

Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

177 177 177 179 179 179 180 180 180 181 182 182 184 184 188 188 190 190 190 190 191 193 196 198 198 199 200 203 203 205 205 205 207

Introduction to food colloids

5.8.6 Emulsion droplet size data and the kinetics of emulsification 5.8.7 Bancroft’s rule for the type of emulsion 5.8.8 HLB value and stabilization of emulsions 5.8.9 Emulsifiers used in the confectionery industry 5.9 Emulsion instability 5.9.1 Mechanisms of destabilization 5.9.2 Flocculation 5.9.3 Sedimentation (creaming) 5.9.4 Coalescence 5.9.5 Ostwald ripening in emulsions 5.10 Phase inversion 5.11 Foams 5.11.1 Transient and metastable (permanent) foams 5.11.2 Expansion ratio and dispersity 5.11.3 Disproportionation 5.11.4 Foam stability: coefficient of stability and lifetime histogram 5.11.5 Stability of polyhedral foams 5.11.6 Thinning of foam films and foam drainage 5.11.7 Methods of improving foam stability Further reading

5.1 5.1.1

177 207 209 210 211 212 212 213 215 219 220 221 222 222 224 225 229 230 230 231 233

The colloidal state Colloids in the confectionery industry

The materials used in the chocolate, confectionery and biscuit industries are very diverse from the point of view of their structural complexity. Among them there are both chemical compounds consisting of small molecules (water, baking salts, monosaccharides etc.) and substances consisting of giant macromolecules (starch, cellulose derivatives, proteins of vegetable and animal origin etc.); however, between these structural extremities, many more materials can be found. Disregarding the substances consisting of small molecules (M < c. 500 Da), all other substances used or produced by these industries behave like colloids, and this statement holds even for a concentrated solution of sucrose! During comminution, substances that are originally of cellular structure are dispersed into particles with a large surface area. These particles have peculiar properties that are characteristic of the group of materials called ‘colloids’. Therefore a study of food colloids is essential for understanding the engineering aspects of food production.

5.1.2

The colloidal region

During comminution, smaller and smaller particles are generated. At the beginning of the process, the surface of these particles does not play an important role in the bulk properties of the substance, because the proportion of the mass on the surface is small. However, as the degree of comminution increases, this proportion becomes increasingly dominant in the bulk properties. There is a size region between 500 μm and 1 μm in which interfacial phenomena determine the bulk properties of substances. This is the colloidal region, a characteristic of which is that the material parameters change continuously between

178

Confectionery and Chocolate Engineering: Principles and Applications

Interface

Substance A

Substance B

Fig. 5.1 In the colloidal region, the material parameters do not change discontinuously (dotted line), but continuously (continuous line).

1 μm

500 μm

Coarse disperse systems

COLLOIDS (microscopic disperse systems)

Submicroscopic disperse systems

Generated by Deformation

Lamination (e.g. soap layer on water surface) Pulling fibres (e.g. pulling of a fibre of jelly, or silk pulled by a silkworm) Fig. 5.2

Dispersion

Corpuscles Sheets Fibres

The colloidal region: generation of colloids by deformation and dispersion.

particles even though the local coordinates of the particles are discontinuous (Fig. 5.1). Figure 5.2 shows the position of the colloidal region, and how to generate colloids in practice. The principal difference between the deformation and dispersion methods is that one dimension of the colloids generated by deformation is macroscopic; for example, the length of a silk fibre can be large enough to be considered macroscopic even though its diameter (two dimensions) is tiny. The colloids generated by dispersion (i.e. comminution) have particles of colloidal size in all three dimensions. In connection with this characteristic size region, colloidal solutions have particular optical properties which make possible the determination of the size and the molecular weight of the colloidal particles. In the colloidal size region, two subregions can be distinguished according to the size d of the dissolved colloidal molecules: • the Rayleigh region, d < 1/λ, where λ is the wavelength of visible light; and • the Debye region, 1/λ < d < λ.

Introduction to food colloids

179

Dispersion medium Gaseous

Fluid

Solid

Colloidal gas dispersion

Solid gas dispersion (solid foam if coarse)

Fluid

Aerosol or colloidal fog

Emulsion

Xerosol (fluid inclusions if coarse)

Solid

Aerosol or colloidal smoke

Suspension

Xerosol or aerosol

Dispersed phase

Gaseous

Fig. 5.3

The various types of colloidal systems, considered as combinations of states.

These regions are different with respect to the absorption and dispersion of light. Many colloidal solutions show optical anisotropy as well. Although the optical properties of colloidal solutions are important in food chemistry and engineering, this topic does not play an essential role from our viewpoint.

5.1.3

The various types of colloidal systems

Figure 5.3 shows the possible combinations of states that can give colloidal systems; of these, emulsions and suspensions are of particular interest.

5.2

Formation of colloids

The colloidal region may be approached either • from large sizes, i.e. by deformation or dispersion; or • from molecular sizes: molecules → microphases → macromolecules → micelles → disperse and cohesive systems.

5.2.1

Microphases

When the concentration of a molecule exceeds its solubility, the solution becomes saturated, and a new phase of agglomerates is then formed from the molecules, which has a surface with a physical meaning. The consequences of phase formation are that: • the surface has an interfacial energy; • neighbouring molecules will adhere to this new surface; • adhering molecules on the surface can react with each other.

180

Confectionery and Chocolate Engineering: Principles and Applications

The size region where this surface emerges can be regarded as the minimum value of the colloidal range of size, and is about 1 nm (10−9 m). The upper boundary of this region is about 1 μm (10−6 m), which is the characteristic size of so-called microphases. The structure of microphases can be crystalline or amorphous, solid or liquid – even similar to that of macrophases.

5.2.2

Macromolecules

Another method of formation of particles of large mass is the coupling of small molecules to create macromolecules by means of covalent bonds. The characteristic parts of a macromolecule are the primer molecules (or monomers) and the segments; the latter are welldifferentiated parts of the chain of the macromolecule. The emergence of segments means new qualitative behaviour. The range of size of macromolecules that can be formed by this method is about 104–107 Da. Giant macromolecules formed by the effect of covalent bonds, which are of infinite mass in the chemical sense, cannot be regarded as colloids, because they do not have the ability to perform thermal motion.

5.2.3

Micelles

Micelles are associations of molecules. Their typical size range is about 102–103 Da. The molecules that form micelles have a polar and a non-polar part – such molecules are called ‘amphipathic’. Micelles containing 50–100 molecules can be formed only in more concentrated solutions, and they are in equilibrium with the free molecules, and of uniform globular form. Micelles of laminar structure or of very large size can be formed in very concentrated solutions, and their size is inhomogeneous.

5.2.4

Disperse (or non-cohesive) and cohesive systems

If there are no attractive forces between the colloidal particles or these forces are too weak to overcome the energy of thermal motion, then the particles are independent of each other, and the system formed by them is fluid: 0 ≤ U coh ≤ kT

(5.1)

where Ucoh is the attractive energy, kT is the energy of thermal motion, k is the Boltzmann constant and T is the absolute temperature This type of system is called disperse or non-cohesive. Such systems can be regarded as stable systems from the point of view of thermodynamics, which fact is a consequence of the relationship expressed in Eqn (5.1). The classes of disperse systems are: • • • •

macromolecular colloidal solutions, e.g. dissolved proteins; colloidal solutions of association, e.g. detergents; colloid dispersions, or sols, e.g. sugar particles finely dispersed in fat; coarse dispersions, e.g. sugar particles coarsely dispersed in fat. For cohesive systems, kT < U coh

(5.2)

Introduction to food colloids

181

Two types of cohesive systems can be differentiated according to the strength of the attractive forces between the particles: (1) If the attractive forces are relatively weak, the system is a gel or agglomerate (or heap). Its characteristic feature is a stable shape, although the system becomes fluid under the effect of even a weak force. If the system is thinned (Ucoh is decreased) or warmed (kT is increased), then the above relationship may change to kT > U coh

(5.3)

and the system transforms into a disperse (incoherent) system. For this reason, such systems are called reversible cohesive systems. The classes that these systems are usually divided into are: • polymer gels, e.g. gelatin gel; • micelle gels, e.g. soaps; • colloidal aggregates (referred to as gels if they are a mixture of a liquid medium and solid particles), e.g. finely crystallized fats; • coarse aggregates, e.g. monodisaccharides coarsely crystallized from aqueous solution with starch syrup (fondant). (2) If the attractive forces are of chemical nature, the system does not contain individual particles any more, and both the medium and the ‘particles’ in it form unbroken, continuous networks. Both phases become deformed to a great extent; therefore such systems were called deformed systems by Buzágh (1937). The important representatives of these systems are: • chemical gels, e.g. pectin gels; • solid–gas xerogels, e.g. activated charcoal; • solid–liquid xerogels, e.g. porous catalysts in a liquid medium. Some typical values of interaction energies are: • chemical bonds: 80–800 kJ/mol; • hydrogen bonds: 8–40 kJ/mol; • dispersion bonds: 1–8 kJ/mol. The relatively strong hydrogen bonds are characteristic of pectin jellies, which are chemical gels. If the molecular polarity is decreased, the interaction energy decreases in parallel: • Polar molecule/polar molecule: for example, for water (M = 18), the internal molar heat of evaporation is 44 kJ/mol, so the cohesive-energy density is 44/18 kJ/g = 2.444 kJ/g. • Non-polar molecule/non-polar molecule: for example, for pentane (M = 72), the internal molar heat of evaporation is 25 kJ/mol, so the cohesive-energy density is 25/72 kJ/g = 0.3472 kJ/g; • Polar molecule/non-polar molecule: the values of the cohesive-energy density are between those above.

5.2.5

Energy conditions for colloid formation

The formation of a colloid is governed by the change of (Gibbs) free enthalpy, ΔG = ΔW − TΔS

(5.4)

182

Confectionery and Chocolate Engineering: Principles and Applications

taking into consideration the fact that ΔS is always positive, and ΔW = Wm − m + Wp − p − 2Wp − m

(5.5)

where ΔG is the change of free enthalpy, ΔW is the change of interaction energy, Wm–m is the interaction energy between molecules of the medium, Wm–p is the interaction energy between a molecule of the medium and a molecule of the particles, Wp–p is the interaction energy between molecules of the particles, T is the absolute temperature and ΔS is the change of entropy. The condition for the formation of a colloid is ΔG < 0

(5.6)

Evidently, an increase in temperature helps in the formation of colloids. Moreover, if ΔW ≤ 0

(5.7)

then the formation of a colloid will certainly occur spontaneously. This means that Wm − m + Wp − p ≤ 2Wp − m

(5.8)

If ΔW >> 0, then the formation of a colloid is impossible. Solutions of macromolecules and colloidal solutions of association may be formed if ΔG < 0, so these are stable. Colloidal dispersions cannot be formed spontaneously, because of the high value of the interfacial energy of the particles – if, nevertheless, they are formed, they are unstable. In the case of microphases, the attractive forces between the particles are strong. However, if a protective layer at the interface between the phases hinders the aggregation of particles, the separation of the particles can be maintained for some time. After a shorter or longer delay, however, such a system becomes heterogeneous, directly or indirectly via a cohesive system, because this transformation is accompanied by a decrease of interfacial energy. When the attractive energy Ucoh between the particles overcomes the kinetic energy derived from thermal motion (see Eqn 5.2), the energy barrier can no longer hinder collisions between particles, and a network, i.e. a cohesive system, will be formed.

5.3 5.3.1

Properties of macromolecular colloids Structural types

The properties of macromolecular colloids are determined by the monomers and the segments. A segment consisting of monomer molecules is capable of microscopic Brownian movement but does not participate in the macroscopic Brownian movement of the whole polymer. The structural types of macromolecules may be classified as follows: • Macromolecules built up from monomers by covalent bonds: – homopolymers: the structural element is a single type of monomer (e.g. amylose, amylopectin and cellulose);

Introduction to food colloids

183

– copolymers: these are made up from structural elements that are more than one type of monomer (e.g. substituted compounds of a monomer, such as methylated/amylated glucose, and alginic acid). • Macromolecules (called polyelectrolytes) containing dissociating groups; typical representatives are proteins containing carboxyl, hydroxyl, methyl and amyl groups. The coil volume of these macromolecules is influenced by pH as well, and is minimum at the isoelectric point. The skeleton of a macromolecule may be of the following types: • • • •

linear, e.g. amylose, agarose, alginic acid, carrageenans and cellulose; branched, e.g. amylopectin; globular, e.g. casein; network, e.g. gelatin. The flexibility of a chain is dependent on:

• chemical structure (bond angles and rotation); • solvation. The structure of a dissolved linear polymer is loose and coil-like, and is permeated by the solvent. The volume of the coil is larger in a better solvent since the solvation is higher. The average chain-terminal distance h is a characteristic quantity for linear macromolecular colloids. If it is supposed that there is no interaction between the segments, the shortest average chain-terminal distance (h0) is obtained in a θ-solvent. This is the worst solvent which still dissolves the polymer. Solvation means loosening of the coil, a measure of which is the expansion factor α, for which the following is valid: h2 = α 2 h02

(5.9)

The conditions for solution of a polymer are given by the following equations: Wm − m + Wp − p = 2Wp − m

(athermic process )

(5.10)

Wm − m + Wp − p < 2Wp − m

(exothermic process )

(5.11)

Wm − m + Wp − p < 2Wp − m

(endothermic process )

(5.12)

Many polymer solutions are thermodynamically stable systems. Another important parameter which characterizes the interaction between a polymer and a solvent is the interaction parameter χ: For good solvents, α > 1 and 0 < χ < 0.5. For a theta solvent, α = 1 and χ = 0.5. For unsuitable solvents, α < 1 and χ > 0.5. The interaction parameter can be determined by measuring the specific osmotic pressure π/c of the polymer solution as a function of polymer concentration:

π c = RT ⎛

1 + Ac + Bc 2 + …⎞ ⎝M ⎠

(5.13)

184

Confectionery and Chocolate Engineering: Principles and Applications

where M is the molar mass of the polymer (the intercept of the curve of π/c vs c is RT/M); R is the universal gas constant; T is the absolute temperature; and A is calculated from the slope of the linear section of the curve, which is equal to RTA. In addition, we can write A = k ( 0.5 − χ )

(5.14)

From Eqn (5.14), χ can be calculated. The value of k (not to be confused with the Boltzmann constant k) is dependent on the type of solvent. The expansion factor can be determined from the Mark–Houwink equation (see Eqn 4.163) and a modification of this equation by Flory and Fox. The interaction parameter can be determined from the Stockmayer–Fixman equation. These equations are related to the intrinsic viscosity of the solution. For further details, see Sun (2004).

5.3.2

Interactions between dissolved macromolecules

When the concentration of a polymer solution is increased, the segments may permeate through neighbouring segments, and the movement of neighbouring molecules will be more and more hindered. The viscosity is increased, and the stretching of the coils becomes more and more elastic. The elastic and viscosity properties of the concentrated solution become characteristic of such concentrated solutions. The effect of a precipitant on dilute polymer solutions can be expressed by saying that the attractive forces between segments become stronger and, as a result, the coils shrink and globules are formed. Finally, as a result of the effect of attraction, the globules are united into flocs, or coacervates. However, the effect of a precipitant on concentrated polymer solutions is different: the attractive forces between segments that are permeating each other stimulate the formation of a network of segments. As a result, a cohesive system, i.e. a polymer gel, is formed. In protein solutions, the interactions are strongest at the isoelectric point, at which isolabile proteins are precipitated from a sufficiently concentrated solution, and a protein gel is formed.

5.3.3

Structural changes in solid polymers

When a polymer is cooled, the linear molecules settle into bundles, micelles are formed from the bundles and, finally, crystallites are formed from the micelles. Amorphous regions remain among the crystallites; the crystallites produce solidity, and the amorphous fraction provided flexibility to the structure. Under the effect of a solvent, a solid polymer becomes swollen (in the case of foods, the only solvent of interest is water). The swelling of the macromolecules in a chemical network is limited. The swelling of the macromolecules in a physical network is limited if the solvent is worse than a θ-solvent; if the solvent is better than a θ-solvent, the swelling is unlimited, and the polymer becomes dissolved. 5.3.3.1

Velocity of swelling

The velocity of wetting is the amount of solvent absorbed per unit time, which can be described by (Gábor 1987, p. 35)

Introduction to food colloids

185

Table 5.1 Calculation of rate constant for the swelling of stringy agar according to Eqn (5.16).a Time (h)

W (m/m%)

36 − W

16/(36 − W)

ln

Slope

20 25 28 30 31

16 11 8 6 5

1 1.454545 2 2.666667 3.2

0 0.374693 0.693147 0.980829 1.163151

0.293244

0 1 2 3 4 W = water content.

a

m (t ) = m∞ − ( m∞ − m0 ) exp ( − kt )

(5.15)

or ln

m∞ − m0 = kt m∞ − m (t )

(5.16)

where m(t) is the mass of the polymer plus the swelling solvent (e.g. water) (kg), m∞ is the value of m(t) if the time t is large (t → ∞), m0 is the initial value of m(t) (at t = 0) and k is the rate constant for swelling (with dimensions of 1/time). In the case of unlimited swelling, the value of m∞ is not exactly defined. The value of k is dependent on the surface of the swelling macromolecule and the temperature of the solvent. Example 5.1 Stringy agar is soaked in cold water (its water content can easily be measured during the swelling). The first column in Table 5.1 shows the time points at which the water content was measured, and the second column shows the measured values. We assume that m∞ = 36 m/m%. The result is k = 0.293 … h−1. Syneresis can be understood as the reverse process, during which a gel loses water according to the equation m (t ) = m∞ + ( m0 − m∞ ) exp ( − kt )

(5.17)

where k is the rate constant for syneresis, and m∞ refers to a very dry state. 5.3.3.2

Capillary rise and flow dynamics

The driving force for liquid penetration into a capillary is given by the Laplace equation, Δp =

2γ LV cos θ r

(5.18)

186

Confectionery and Chocolate Engineering: Principles and Applications

where Δp is the driving force (sucking effect); γLV is the liquid/vapour tension (>0); θ is the so-called apparent contact angle, which determines the curvature of the meniscus; and r is the radius of the capillary. In order to interpret the results of capillary penetration experiments, theoretical models are required. The simplest is the well-known Washburn equation (Washburn 1921). Washburn showed that the velocity v = dh/dt of the liquid–air meniscus along the tube drops very quickly to such a value that the conditions of laminar flow assumed in the Hagen–Poiseuille equation are established, so that dv r 2 π dh r 4 πΔp = = dt dt 8ηh i.e. dh r 2 Δp = dt 8ηh

(5.19)

where h is the height of the liquid front, t is the time of penetration and η is the dynamic viscosity of the liquid. If the value of Δp is substituted from Eqn (5.18) into Eqn (5.19), the following differential equation is obtained: dh rγ LV cos θ = dt 4ηh

(5.20)

After integration (from h = 0 to h and from t = 0 to t), h2 =

trγ LV cos θ 2η

(5.21)

In many cases, it has been found experimentally that the Washburn law (Eqn 5.21) is dimensionally applicable for liquids penetrating a porous medium, i.e. h ∼ t1/2. However, this type of Washburn law is valid only for the short-time regime. In the long-time limit, the penetration slows down and shows an exponential relaxation towards the equilibrium height, h∞: ⎛ − ρ gr 2t ⎞ ⎤ ⎡ h (t ) = h∞ ⎢1 − exp ⎜ ⎝ 8ηh∞ ⎟⎠ ⎥⎦ ⎣

(5.22)

and h∞ =

2γ LV cos θ ρ gr

(5.23)

where ρ is the density of the liquid, g is the gravitational acceleration, and h∞ is defined by the balance of capillary and hydrostatic pressures.

187

Introduction to food colloids

For details of the wettability of porous solids, see Li and Neumann (1996) and Grundke (2002). Example 5.2 Water penetrates into a capillary of radius r = 0.1 mm = 10−4 m. The viscosity of water is η = 10−3 Pa s, the interfacial tension of water is γLV = 73 × 10−3 N/m and the apparent contact angle is zero, i.e. cos θ = 1. According to Eqn (5.21), trγ LV cos θ t × 10 −4 m × 73 × 10 −3 ( N m ) × 1 = 2η 2 × 10 −3 Pa s 73 = t × 10 −4 × ⎛ ⎞ m 2 s ⎝ 2⎠

h2 =

If t = 1 s, the distance of penetration is h = (73 2 ) = 6.04… × 10 −2 m = 6.04… cm. The penetration height at equilibrium (see Eqn 5.23) is 2γ LV cos θ ρ gr 2 × 73 × 10 −3 ( N m ) × 1 = 3 10 ( kg m 3 ) × 9.81 ( m s 2 ) × 10 −4 m = 14.88… cm

h∞ =

5.3.3.3

Swelling pressure

During the solvation of macromolecules, molecules of the solvent penetrate into the inside of the macromolecules. If the energy of solvation exceeds the energy associated with the binding forces in a network of a macromolecules, the binding points of the network become loose, and dissolution of the macromolecules will start. This process may be accelerated by a rise in temperature. The swelling pressure is the pressure difference between the gel phase and the pure solvent. It can be calculated from the equation

π SW =

RT ln aS VS

(5.24)

where R is the universal gas constant, T is the temperature, aS is the activity of the solvent in the gel phase and VS is the partial molar volume of the solvent in the gel phase. The accelerating effect of a temperature rise can be read directly from Eqn (5.24). The volume of a macromolecule is increased as a result of solvation; however, the increase is not equal to the volume of solvent added, because intermolecular changes take place during the process, for example contraction, amorphous/crystalline state transitions and similar structural changes. 5.3.3.4

Effect of heat on amorphous polymers

Figure 5.4 shows the effect of a constant strain (tension) on an amorphous polymers as the temperature is increased. At Tgl, the polymer ceases to have a glassy consistency, and

188

Confectionery and Chocolate Engineering: Principles and Applications

Strain

Fluid

Elastic

Glassy Tgl

Tm

Tfl

Temperature Fig. 5.4

The effect of strain on an amorphous polymer as the temperature is increased.

it then behaves elastically. With a further increase in temperature, the polymer melts at Tm, and above Tfl it behaves like a fluid, i.e. in the range Tgl < Tm < Tfl. This behaviour is characteristic of fats. This phenomenon can be observed when a sugar mass is being shaped into drops, although the temperature sequence is reversed. When the sugar mass is fluid at c. 120– 100°C, flavouring and colouring can be done; then, it is shaped into a sugar rope, which is elastic at c. 40–35°C; and, finally, the glassy consistency of the drops is achieved on cooling (at about 16°C).

5.4 5.4.1

Properties of colloids of association Types of colloids of association

Organic substances which contain both polar and non-polar groups are capable of forming associations. For associations of large size, a sufficiently large solubility, a suitable temperature, a large molecular mass and a special molecular structure (that of an amphipathic compound consisting of 30–100 atoms) are needed. If the polar and non-polar groups are well separated spatially, a so-called critical micelle-forming concentration cM can be determined, and if the concentration c of a colloidal solution is higher than cM, then globular micelles of almost homogeneous size are formed. However, in highly concentrated solutions (c >> cM), large micelles with a sheet structure are formed and, finally, the aggregation of these large micelles produces a micelle gel. If the colloid molecules contain larger numbers of polar and non-polar groups (e.g. the majority of non-ionic surfactants are of this kind), the micelles become less regular in shape, and also the value of cM is less sharp in this case. If the concentration in an aqueous system is increased, a weak network of non-polar bonds is formed, which is highly viscous. Amphipathic compounds have an important property: they can be adsorbed at interfaces with a large polarity difference (e.g. air–water and oil–water interfaces) and, as a

Introduction to food colloids

189

result, the polarity of the interface is modified, the wetting ability is influenced, and colloids and coarse dispersions may be stabilized. The compounds with a molecular mass of about 300–3000 Da which form micelles contain both polar and non-polar groups, and the balance of these groups is an important characteristic of such compounds: the HLB (hydrophile–lipophile balance) number has been defined to characterize their micelle-forming behaviour. For details, see Section 5.8.8. Colloidal solutions containing small micelles are in equilibrium in the thermodynamic sense. For a reaction of the type T  nT  ( T )n the following equation is valid in equilibrium: K=

[( T )n ] = [T ]

n

c (q n) [c (1 − q )]n

(5.25)

where T is the symbol for a surfactant (amphipathic) molecule, the square brackets [ ] denote a molar concentration, q is the molecular ratio of the surfactant in a micelle, n is the number of molecules of the surfactant in the micelle, c is the molar concentration and K is the equilibrium constant The thermodynamic force for micelle formation is mainly the interaction between the water molecules and the surrounding non-polar groups (denoted by A) of the amphipathic molecules. The formation of a colloidal solution is determined by the Gibbs free-enthalpy change ΔG = ΔW − T ΔS, where ΔW is the enthalpy change by virtue of the interactions (H2O–H2O, A–H2O and A–A) between the water molecules and the non-polar groups. The distribution of colloidal particles in a solvent is always accompanied by an increase in entropy, i.e. ΔS > 0. Consequently, the sign and value of ΔW determine the sign of ΔG, since 2 A + 2H2 O  2 A−H2 O i.e. ΔW = −2WA −H2 O + WH2 O −H2 O + WA − A

(5.26)

where WA–H2O is the interaction energy between non-polar groups and water molecules, WH2O–H2O is the interaction energy between water molecules, and WA–A is the interaction energy between non-polar groups. If ΔW ≤ 0, the formation of a colloidal solution (→) occurs spontaneously; if ΔW >> 0, the formation of a colloidal solution is hindered, although micelle formation (←) is possible. A compact monomolecular interfacial layer is formed by amphipathic compounds at a relatively low critical micelle-forming concentration cM, and therefore these substances are called surface-active compounds. It can be seen that an increase in temperature decreases ΔG (i.e. provides easier solubilization) until the increase of ΔW, which also depends on the temperature, compensates this effect.

190

Confectionery and Chocolate Engineering: Principles and Applications

5.4.2

Parameters influencing the structure of micelles and the value of cM

In water the non-polar groups, and in a non-polar solvent the polar groups, are associated. Micelles of regular structure (globular or sheet-like) are formed in ionic surfactants. Large, sheet-like micelles are formed in concentrated solutions (c >> cM). Chain-like micelles may be formed in organic solvents from ionic surfactants and in water from non-ionic surfactants. In aqueous solutions, the value of cM increases as the non-polar part of the amphipathic molecule becomes larger; for example, for paraffin derivatives (Cn–), if n increases then cM decreases. Concentrated surfactant solutions with a network structure are viscous sols, and in the case of a network made up of large micelles, the solution forms a gel, i.e. it becomes solid. Amphipathic substances decrease the interfacial tension of water because the exterior side of the interfacial layer is formed by the non-polar groups. This decrease of interfacial tension continues up to the point where c = cM is approached. The specific molar electrical conductivity Λ decreases steeply at c = cM since a certain proportion of the molecules in a micelle do not dissociate. Some non-polar substances that are insoluble in water become soluble in surfactant solutions if c > cM because these substances become enriched in the non-polar parts of the micelles or because mixed micelles form. This phenomenon is called solubilization.

5.5 5.5.1

Properties of interfaces Boundary layer and surface energy

The properties of colloids and of coarse dispersions made up of microphases are determined by the structure of the boundary layer between the two phases. The thickness of the boundary layer is usually about 1–2 nm, and the parameters change over this distance, rather than discontinuously. However, when the liquid also contains dissolved substances, these substances may press on the molecules of solvent from the interface, and adsorb at it. This phenomenon is called adsorption. The thickness of the boundary layer can be increased to c. 100 nm in the case of adsorbed macromolecules. The atoms or molecules on the free surface of a phase are not bonded in the direction of the free side. The forces resulting from this lack of bonding are manifested in surface and interfacial tension and in surface and interfacial surplus energy. (In the case of a gas–liquid boundary, we speak of ‘surface surplus energy’, and in the case of a liquid– liquid boundary layer, ‘interface surplus energy’.) The energy of a free surface is equal to the work needed to create a unit area of surface in a reversible way. In the case of chemically pure substances, the interfacial tension and the interfacial surplus energy are numerically equal, and may be measured in units of J/m2.

5.5.2

Formation of boundary layer: adsorption

When two immiscible phases come into contact, their atoms or molecules bond partially to the field of the other phase. Two cases can be distinguished:

Introduction to food colloids

191

(1) Two phases of chemically pure substances come into contact. The original surfaces disappear, and a new boundary layer is formed. As a result, the interfacial energy is always less than the sum of the energies of the original surfaces. The usual interfacial phenomena are adhesion, wetting and an exothermic thermal effect. (2) Two phases of solutions come into contact – this is a more complicated case. The chemical composition of the boundary layer usually differs from the compositions of the two phases that constitute it because the concentration conditions in the boundary layer are determined by the requirement that the decrease in the (Gibbs) free enthalpy should be a maximum. For example, if the dissolved substance is an amphipathic compound, its concentration in the boundary layer will be higher than that in the solution because its polar groups will be oriented towards the water and its non-polar groups will be oriented towards the non-polar phase. As a result, the amphipathic compound will be enriched in the boundary layer. The concentrations of components in the boundary layer cannot be directly measured in general, and therefore the usual way is to determine the decrease in the concentration in the interior of the solution after adsorption, and then to calculate the interfacial surplus (nσ, in units of mol). The Gibbs interfacial concentration is defined by

Γσ =

nσ AS

( mol

m2 )

(5.27)

where AS is the surface area of the boundary layer (m2). If the surface area of the boundary layer is unknown, the interfacial surplus can be related to the amount mS of one of the phases (e.g. the mass of the adsorbent). In this case, this parameter is called the specific adsorbed amount, mσ =

nσ mS

( mol kg )

(5.28)

The adsorption isotherm is defined as the plot of Γ σ vs c or Γ σ vs p at constant temperature, where c is the (decreased) concentration of an adsorbed substance inside the solution after adsorption and p is the (decreased) partial tension of an adsorbed gas after adsorption. (In these isotherms, mσ can also be used, if necessary, instead of Γ σ.) The thermodynamic force for adsorption can be rather strong when the boundary is established by a contact of the type polar ↔ (polar + non-polar) ↔ non-polar groups, where ‘↔’ means ‘contact’ and ‘(polar + non-polar)’ relates to the amphipathic substance, which, so to speak, joins together the polar and non-polar phases, which are otherwise immiscible. Ions of ordinary electrolytes may be adsorbed on a polar surface if the atoms or polar groups of the surface are chemically similar to those ions. In this case the adsorbed ions are located near to the surface, and their counter-ions a little further away. An electric double layer is formed in this way.

5.5.3

Dependence of interfacial energy on surface morphology

It is easy to demonstrate that liquid drops are thermodynamically unstable.

192

Confectionery and Chocolate Engineering: Principles and Applications

It is well known that the vapour pressure of a curved liquid surface differs from that of a planar surface. The relationship between these pressures can be determined from the capillary elevation or depression. If the vapour pressure above a planar liquid surface is p0, and is pr in a closed capillary, where the surface is curved, then, according to the barometer formula (which is a consequence of the Boltzmann distribution), −Mgh ⎞ pr = p0 exp ⎛ ⎝ RT ⎠

(5.29)

where M is the molar mass of the liquid, g is the gravitational acceleration (9.81 m s−2), h is the capillary elevation (positive) or depression (negative), R is the universal gas constant and T is the temperature (in K). However, an expression for h can be obtained from the equilibrium of the weight of the liquid column in the capillary and the interfacial force acting on the wall of the capillary:

( interfacial force ) 2π rγ = r 2π hgρ ( weight of liquid )

(5.30)

i.e. h=

2γ rgρ

(5.31)

where r is the radius of the capillary, γ is the interfacial tension of the liquid (this is always positive!) and ρ is the density of the liquid. If the surface of the liquid is concave or convex, r is positive or negative, respectively. Taking this expression for h into account, the relative change of vapour pressure can be obtained from the barometer formula as follows (using ex ≈ 1 + x + …, if x → 0): ⎛ −2γ M ⎞ ⎛ 2γ M ⎞ pr = p0 exp ⎜ ≈ p0 ⎜1 − ⎟ ⎝ rρ RT ⎠ ⎝ rρ RT ⎟⎠

(5.32)

Δp pr − p0 2γ M = =− p0 p0 rρ RT

(5.33)

Since the surface of a drop is always convex, i.e. r is negative, the vapour pressure of a drop of liquid is always higher than that of the pure liquid. Consequently, diffusion of vapour occurs from smaller drops to larger ones, and from the larger drops to the planar surface of the pure liquid. (The continuous liquid can be regarded as being in a stable state from the point of view of thermodynamics.) This picture is valid for pure liquids only, and if two (or more) liquids are dispersed in each other, diffusion of this kind between them can be decreased or inhibited – this is the aim of the techniques of emulsification. A similar relationship can be developed for the solubility of small crystals, and explains why nuclei are necessary before crystals can form, even from a supersaturated solution. It explains why larger crystals grow at the expense of smaller ones, a fact that is made use

Introduction to food colloids

193

Table 5.2 Surface energy compared with interaction energy (approximate values). Substance

Type of bond

n-hexane Water Mercury

Dispersion Hydrogen Molecular

Surface energy (mJ/m2)

Interaction energy (kJ/mol)

18 73 480

1–8 8–40 80–800

of in the chemical industry and is known as Ostwald ripening; see Section 10.6.1. Consequently, the surface of a solid is always heterogeneous from the point of view of energy: on the peaks and edges, the free energies of atoms or molecules are higher than on a planar surface.

5.5.4

Phenomena when phases are in contact

The surface energy of chemically pure substances in contact with their own vapour is proportional to the strength of the bonds between their atoms or molecules. This fact is demonstrated in Table 5.2. The surface energy of polar, chemically pure liquids is roughly proportional to their polarity. The surface energy of amphipathic substances is less than expected because the non-polar groups in them are oriented towards the vapour phase. The surface tension of non-polar homologous substances is roughly proportional to their molar mass because the dispersive forces are stronger if the molecule is larger. When a solid and a liquid substance (condensed phases) plus their vapour are in contact, the total surface energy is less than the sum of the surface energies of the separate phases. The energy liberated is called the adhesion energy Wa: Wa = −γ SV − γ LV + γ SL

(5.34)

where γ SV is the interfacial tension of the solid, related to the solid–vapour interface; γ LV is the interfacial tension of the liquid, related to the liquid–vapour interface; and γ SL is the interfacial tension of the solid–liquid interface. When the temperature is increased, the interfacial tension always decreases; this observation has led to some important perceptions. According to Ramsay,

γ = const. (Tcr′ − T )

(5.35)

where γ is the interfacial tension of the liquid, Tcr′ is a characteristic temperature value, usually less than the critical temperature of the liquid by 4–6 K, and T is the temperature. If a drop of liquid is placed on the surface of a solid, the drop will spread to an extent that depends on the relevant surface energies: all three phases (solid, liquid and gas) attempt to decrease their surface area because of their surface energy. The extent of spreading can be described by Young’s equation (illustrated in Fig. 5.5), cos θ =

γ SV − γ SL γ LV

(5.36)

194

Confectionery and Chocolate Engineering: Principles and Applications

g LV

Liquid surface Vapour phase

g SV Solid surface

q

g

SL

g LV cos q

Fig. 5.5 Illustration of Young’s equation. γ SV = solid–vapour interfacial energy, γ SL = solid–liquid interfacial energy and γ LV = liquid–vapour interfacial energy.

where θ is the contact angle, which is defined as the angle formed at the junction of the three phases. Solid particles are preferentially wetted by the liquid phase if cos θ is positive (θ < 90°), i.e. γ SV > γ SL. The balance of surface tensions, considered as vectors, is obtained as follows:

γ SV = γ SL + γ LV cos θ

(5.37)

In the case of solid particles absorbed at an oil–water interface, Young’s equation may be written in the form cos θ =

γ PO − γ PW γ OW

(5.38)

where O denotes oil, P denotes the solid particles and W denotes water. Solid particles are preferentially wetted by an aqueous phase if cos θ is positive (θ < 90°), i.e. γ SV > γ SL (see Eqn 5.36). 5.5.4.1

Mercury porosimetry

A contrasting example is provided by mercury, which has a very high surface tension (480 mN/m) (see Table 5.2) and does not wet solids (θ = 140°). This specific property of mercury is used for the determination of pore distributions because mercury can fill the volume of pores without any gaps owing to its extremely high surface tension. Mercury intrusion porosimetry requires the sample to be placed in a special filling device that allows the sample to be evacuated, followed by the introduction of liquid mercury. The size of the envelope of the mercury is then measured as a function of increasing applied pressure. The basis of evaluation is the Laplace equation (Eqn 5.18), where Δp is the pressure difference acting on a fluid of surface tension γLV if the contact angle is θ. If Δp is positive, the surface tension pulls the fluid up from a dish into a capillary – this is the case for fluids (e.g. water) that wet the solid. This is called capillary rise or positive capillarity. It should be mentioned that the interfacial tension of a surface is always positive!

Introduction to food colloids

195

Since in the case of mercury cos 140° = −0.7660, Δp is negative, i.e. a capillary depression (negative capillarity) will result. When the pores of a material are filled with mercury, a measurement of this counter-pressure can be used to determine the size of the pores. In mercury porosimetry, the sample is first evacuated and then surrounded with mercury and, finally, pressure is applied to force mercury into the void spaces while the amount of mercury intruded is monitored. Data for the intruded volume of mercury versus applied pressure are obtained, and the pressures are converted to pore sizes using Eqn (5.18). The greater the applied pressure, the smaller the pores entered by the mercury. This method is typically used over the range of pore sizes from 300 μm to 0.0035 μm. The difference between water and mercury is well manifested by the fact that water is easily adsorbed by filling the pores independently of their size, i.e. such a method is suitable for measuring the total volume of pores. However, mercury is suitable for measuring the distribution of pore sizes according to Eqn (5.18). Because of increased concern over the use of mercury, several non-mercury intrusion techniques have been developed. For further details, see, for example, Brouwer et al. (2002) for applications to investigations of cocoa, and for chocolate products, see Loisel et al. (1997). Example 5.3 Let us use Eqn (5.18) to calculate the pressure p of mercury that has to be applied if the radius r of the pores is 5, 10 or 15 μm. p (5 μm ) =

2γ cos θ 0.766 = 2 × 480 × 10 −3 ( N m ) × ≈ 147 × 103 Pa r 5 × 10 −6 m

147 ⎞ p (10 μm ) = ⎛ × 103 Pa ⎝ 2 ⎠ 147 ⎞ p (15 μm ) = ⎛ × 103 Pa ⎝ 3 ⎠ 5.5.4.2

Location of particles on a water–oil interface

According to Dickinson (1992, p. 33), for casein micelles (protein particles) in homogenized milk, reasonable values for the interfacial tensions are γ PO = 10 mN/m, γ PW = 0 mN/m and γ OW = 20 mN/m. (1 dyn/cm = 1 mN/m = 1 mJ/m2.) Substituting these values into Eqn (5.38) gives a contact angle of the order of 60°, which means that the casein micelles are located predominantly on the outside of the milk fat globules. The smaller the contact angle, the more effective the wetting is, and the larger the spreading of drops is. For total spreading,

θ = 0 and cos θ = 1

(5.39)

This means that the vector component γ OW cos θ = γ OW causes a drop to spread entirely over a surface. The spreading coefficient S is defined as S = γ SV − γ SL − γ LV

(5.40)

196

Confectionery and Chocolate Engineering: Principles and Applications

For total spreading, S ≥0

5.5.5

(5.41)

Adsorption on the free surface of a liquid

On the free surface of a liquid, the interfacial tension is dependent on the concentration of dissolved substances, and the change of the interfacial tension relative to that of the solvent is caused by adsorption of the dissolved substance on the boundary layer between the liquid and the vapour. Let us investigate the change of the (Gibbs) free enthalpy. The free enthalpy g (J/mol) is the sum of the chemical potentials of the components and the surface energy, g = μ1n1 + μ2 n2 + γ F

(5.42)

or, as a total differential, dg = μ1dn1 + μ2 dn2 + dμ1 n1 + dμ2 n2 + γ dF + dγ F

(5.43)

where μ1n1 is the chemical potential × number of moles of the solvent (J), μ2n2 is the chemical potential × number of dissolved moles of the dissolved substance (J), and γ (J/ m2) is the interfacial tension of a free surface of area F (m2). Since dg = − sdT + vdp + d ( μ1 n1 + μ2 n2 )

(5.44)

is the isobaric (p = constant) reversible work, where s is the entropy (J/K), then if T and p are constant, the isobaric reversible work which establishes a new surface of area dF is equal to γ dF, and as a result, dg = μ1dn1 + μ2 dn2 + γ dF

(5.45)

Considering Eqn (5.43), the following equation holds for part of the total differential dg: dμ1 n1 + dμ2 n2 + dγ F = 0

(5.46)

(the Gibbs–Duhem equation). This is because n1 and n2 do not depend on μ1 and μ2, respectively, and F does not depend on γ. The Gibbs–Duhem equation is valid also for the interior of the homogeneous phase (where molar quantities are denoted by ‘°’: dμ1 n1° + dμ2 n2° = 0

(5.47)

The chemical potential μ1 of the solvent can be eliminated from Eqn (5.46): ⎡ ⎛ n2° ⎞ ⎤ ⎢ n2 − n1 ⎜ ⎟ ⎥ dμ 2 + Fdγ = 0 ⎝ n1° ⎠ ⎥⎦ ⎢⎣

(5.48)

Introduction to food colloids

197

Note that the expression in the square brackets [ ] in Eqn (5.48) is equal to the surplus of the dissolved substance on the surface of area F, i.e. [ n2 − n1 ( n2° /n1° )] ≡ Γ2 F

(the Gibbs interfacial concentration)

(5.49)

or, in another form, −dy a ⎛ dγ ⎞ = Γ2 = −⎛ 2 ⎞ ⎜ ⎟ ⎝ dμ 2 RT ⎠ ⎝ da2 ⎠

(5.50)

where a2 is the chemical activity of the dissolved substance. Shishkowsky derived a relationship for amphipathic (‘capillary-active’) substances: Δγ = γ ° − γ = A ln (1 + Bc )

(5.51)

where γ° is the interfacial tension of the pure solvent, γ is the interfacial tension of the solution, A and B are constants, and c is the concentration of the dissolved substance. After differentiation of Eqn (5.51) with respect to c, −dγ A ⎡ Bc ⎤ = Γ2 = dc RT ⎢⎣ 1 + Bc ⎥⎦

(5.52)

This type of equation was studied for the first time by Langmuir, although in that case it was related to kinetic topics, and it is called the Langmuir isotherm. If c → 0 (initial section of the isotherm), the slope of the isotherm is AB/RT. If c → ∞ (the region of high concentration),

Γ∞ =

A RT

(5.53)

It should be mentioned that in the case of capillary-active substances, Γ∞ cannot be reached for several reasons. In order to understand this discrepancy, we must take into account the fact that instead of the concentration c, the chemical activity a determines the behaviour of the system. The constant B can be regarded as a measure of capillarity, for which an empirical rule was given by Traube in the case of homologous series: Bn +1 ≈ 3.4 Bn

(5.54)

which shows that capillarity becomes stronger as the molecular mass increases. If an amphipathic substance of concentration c that is the n-th member of a homologous series produces a decrease Δγ in the interfacial tension, then the (n + 1)-th member of this series produces the same decrease with a solution of concentration c/3 [(1/3) × 3.4) ≈ 1]. It can be easily seen, from Eqn (5.52), that

198

{

Confectionery and Chocolate Engineering: Principles and Applications

} {

A ⎡ Bc ⎤ RT ⎢⎣ 1 + Bc ⎥⎦

≈

n +1

}

A ⎡ Bc ⎤ RT ⎢⎣ 1 + Bc ⎥⎦

n

Bn +1cn +1 Bn cn ≈ 1 + Bn +1cn +1 1 + Bn cn Bn +1 c c ≈ 3.4 ≈ n → cn +1 ≈ n Bn cn +1 3 Water-soluble macromolecular substances are mostly of amphipathic structure; consequently, the more non-polar the surface of a solid or fluid is, the better they can be adsorbed from aqueous solution. It is characteristic of polymers that only some fraction of their monomers are coupled to the surface; the other monomers remain in the solution in the form of coils. If the amount of adsorbed polymer is increased, the length of these coils becomes larger. Langmuir showed that fatty acids, alcohols and esters with long carbon chains form a monomolecular layer (or film) on the surface of water, and that this film strongly decreases the surface tension of the water. To decrease the surface tension from 73 × 10−3 to 50 × 10−3 N/m, the thickness of this monomolecular layer must be about 20 (20 × 10−10 m). This thickness is practically equal to the length of a paraffin chain (for n = 16–18) if the chain is maximally stretched. This means that the polar part of the chain is oriented towards the water, and the non-polar part is oriented towards the vapour phase. The decrease in the surface tension of water caused by a small amount of oil may result in moderation of the bubbling of aqueous solutions or calming of the surging of the sea around a ship – the latter phenomenon was first described by Benjamin Franklin. Bubbling may be very disadvantageous when one is warming or evaporating carbohydrate solutions, and a small amount of oil or fat is widely used also in the confectionery industry as a bubbling inhibitor.

5.6 5.6.1

Electrical properties of interfaces The electric double layer and electrokinetic phenomena

If two layers are in contact or move relative to each other, there exists an electric potential difference between them. This potential difference appears as an electric double layer at the interface. The reason for this potential difference may be that the electric charges of the layers are not equal or that there is relative movement between them. The latter induces electrokinetic phenomena. The charges at interfaces may be derived from specially adsorbed ions, adsorbed ionic surfactants, adsorbed polyelectrolytes or interfacial dissociation. The size and sign (positive or negative) of the interfacial charge are dependent on pH if the charge is derived from adsorbed polyelectrolytes or the surface is an inorganic substance. It should be emphasized, however, that the electric charges of protein molecules derive from dissociation of their acidic or basic groups and not from the electric double layer. At the isoelectric point (iep), the positive and negative charges compensate each other. In the case of a surplus of positive charges (a cation surplus in the solution), pH > iep; in the opposite case, pH < iep. Therefore, the movement of charges is determined by their mobility, which depends on the net charge difference.

Introduction to food colloids

199

The various electrokinetic phenomena are: Electrophoresis: small suspended or colloidal particles move under the effect of an electric field to a positive or negative electrode. Electro-osmosis: the movement of a fluid through capillaries or pores in a solid under the effect of an electric field. Streaming potentials (the reverse of electro-osmosis): these are induced when a fluid is forced through capillaries or pores in a solids. Sedimentation potentials (or electrophoretic potentials) (the reverse of electrophoresis): settling particles are charged by the effect of their zeta potential, and their movement creates a potential difference. The phenomenon of electrophoresis is very important in colloids because their stability is definitely dependent on their zeta potential. The effect of electrolytes on the stability of colloids is exerted mainly via a change in the surface charge of the colloidal particles at the isoelectric point; see, for example, the discussion of gelatin in Section 11.13.5.

5.6.2

Structure of the electric double layer

The presence of an electric double layer is demonstrated by the fact that there is always a certain amount of liquid, adsorbed on the solid surface, that remains fixed to the particles when particles move in a liquid or when a liquid moves relative to particles. The various conditions that can occur in the electric double layer are illustrated in Fig. 5.6. In concentrated electrolytes, the entire double layer moves together with the particles. In this case the double layer is similar to a planar capacitor (a Helmholtz-like double layer), and the potential ε is a linear function of position x (Fig. 5.6(a)). Moreover, there is no electrokinetic potential difference between the particles and the solution because both parts of the electric double layer move together.

Potential

Potential d

e

Potential

d

d

e

e y

z x Helmholtz-like electric double layer (a)

x Diffuse electric double layer (b)

x y

z (c)

Fig. 5.6 Electric double layer: (a) in concentrated electrolyte; (b) in dilute electrolyte; (c) when one ion adsorbs strongly.

200

Confectionery and Chocolate Engineering: Principles and Applications

However, in dilute electrolytes, the double layer penetrates deeply into the interior of the solution (Fig. 5.6(b)), and its structure consists of two parts, a planar capacitor and a diffuse part, which are separated by the ‘splitting plane’. The electrode potential ε vs position x is an exponential function in this case, if ε0 < 25 mV:

ε = ε 0 exp ( −κ x )

(5.55)

where κ is a parameter related to the characteristic thickness of the ion atmosphere and the strength of the ions, and x is the distance from the solid surface. If x = 0, then ε = ε0. If x = 1/κ, then ε = ε0/e, i.e. 1/κ is the fictive or characteristic thickness of the electric double layer (in Fig. 5.6(b), 1/κ = d is the thickness of the splitting plane); 1/κ is the position where the density of electron charge has its maximum. In equilibrium, the splitting plane separates the electrode potential ε into two parts:

ε = ψ +ζ

(5.56)

where ψ is the potential difference in the layer adsorbed at the surface of the solid particle (ψ0 = ε0 is the surface potential), and ζ is the electrokinetic or zeta potential derived from the relative movement of the solid and liquid phases. If one of the adsorbing ions adsorbs strongly at the solid surface, then the case shown in Fig. 5.6(c) is also possible, i.e. ε < ψ. The more dilute the solution, the larger the diffuse part of the electric double layer. If the ion concentration is increased, the value of the zeta potential can decrease to zero. If the dispersion forces are very strong, an increase in the ion concentration can cause the zeta potential to decrease below zero, and then, from a negative value (‘trans-charging’ in the case of ions of three or more valencies), it starts to increase back to zero as in Fig. 5.6(c). The diffuse part of the electric double layer can be described by the Debye–Hückel theory, if we suppose that the Helmholtz-like part of the electric double layer is absent. The usual value of the zeta potential is less than 0.1 V; its sign (positive or negative) is dependent on the qualitative nature of the solid and liquid, and also on the concentration of the liquid. Electrolytes change the zeta potential in both size and sign in a complicated way. Amphipathic substances may greatly influence the zeta potential if they are strongly adsorbed, by changing the structure of the electric double layer.

5.7

Theory of colloidal stability: the DLVO theory

The theory of the stability of sols, i.e. the DLVO (Derjaguin–Landau–Verwey–Overbeek) theory, gives the potential describing the interaction between two globular particles as the sum of a repulsive potential VR and an attractive potential VA. In simple cases, the form of these potentials is ⎛ ε aψ 02 ⎞ VR = ⎜ ln (1 + eκ H ) ⎝ 2 ⎟⎠

(5.57)

if ψ0 < 25 mV (small surface potential) and κa > 10 (medium to thick double layer), where ψ0 is the permittivity of the medium, a is the radius of the particles, ψ0 is the surface

201

Introduction to food colloids

potential, κ is a parameter (see Eqn (5.55) above) and H is the distance between the two globular particles, and VA =

A12 a 12 H

(5.58)

if H << a, where A12 is the complex Hamaker constant. The repulsive effect may be derived from the steric potential of adsorbed amphetic substances (macromolecules or surfactants) or from coils that protrude into the interior of the solution to a distance dp. In this case the repulsive potential contains two additional parts: • If H < 2dp, then the osmotic pressure is increased as a result of penetration of the coils of the macromolecules into each other; this is the mixing part VR(M). • If H < dp, the repulsive effect is strengthened as a result of compression of the polymer layer; this is the volume restriction part VR(V). Consequently, in the general case the entire (repulsive + attractive) potential (VR+A) is given by VR = VR(E) + VR(M) + VR(V )

(5.59)

and VR+ A = VR(E) + VR(M) + VR(V ) − VA

(5.60)

Repulsion

Potential V

VR+A

VR

Potential barrier(VM) H

Attraction

HM

Fig. 5.7

Primary minimum(Vm)

Secondary minimum VA

Attractive (VA), repulsive (VR) and resultant (VR+A) potentials according to the DLVO theory.

202

Confectionery and Chocolate Engineering: Principles and Applications

where VR(E) is the repulsive potential of the electric double layer (Fig. 5.7). Figure 5.7 shows the attractive (VA), repulsive (VR) and resultant (VR+A) potentials according to the DLVO theory. The condition for flocculation is Δg = Δh − TΔs > 0

(5.61)

where Δg is the change of free (Gibbs) enthalpy, Δh is the change of enthalpy, T is the temperature and Δs is the change of entropy of the system. During the approach of particles to each other, as the coils penetrate into each other, the entropy decreases continuously (Δs < 0) because the solution becomes more concentrated. However, this decrease becomes more important when volume restriction starts, since volume restriction causes an additional decrease of entropy: as a result, the free movement of the chains becomes more and more limited. This means a decrease in the configurational entropy as well. As the coils penetrate into each other, the segment–segment interaction becomes stronger and the segment–solution interaction changes weaker. The balance of these interactions determines the sign of the enthalpy change Δh. Among the attractive forces, the dispersive (van der Waals) forces are the most important; these are the sum of all the forces between the atoms of the particles. Therefore the distance over which they are effective is the largest. Very near to the particle (i.e. when H is very small), the attractive force exceeds the repulsion (see the primary minimum Vm in Fig. 5.7). In contrast, the distance over which the repulsive effects act is relatively small, and its terminal point is at the secondary minimum in Fig. 5.7. The depth of the secondary minimum is about kT (the energy of thermal movement). Between the two minima there is a potential barrier VM, at which the repulsive effect is the strongest. If a particle can get over the potential barrier (i.e. the particles get nearer to each other), the attractive effect starts to intensify and, according to the DLVO theory, will result in the direct contact of particles. If VM > 10kT, the system is stable, since only some particles have sufficient thermal energy to get over the potential barrier. So no flocculation takes place. However, if the effective distance of the repulsive effect is small, there is no potential barrier (VM = 0), and every collision between particles will result in flocculation. However, the situation is in fact more complicated than this because of secondary processes if VM < 10kT. If an electrolyte is added to a sol stabilized by an electric double layer in increasing amounts, flocculation (or ‘coagulation’ as it is called in the case of electrolytes) will start, the rate of which increases until the potential barrier VM equals zero. In the case of a thick double layer (H > H″) (Fig. 5.8), V(H″) + VM is not too large, and the particles can get back, for example because of the effect of thermal energy, into a state in which exclusively the attractive forces affect them if the electrolyte is subsequently extracted from the system. Consequently, the process Sol → flocculation → ( repeptization ) → sol is reversible. Figure 5.8 shows the potential conditions for flocculation according to the DLVO theory. If H = H′ > H″, i.e. the particles approach too near to each other, V(H′) + VM is too large, and repeptization is impossible. If the electrolyte concentration is very high,

Introduction to food colloids

203

VR+A

Potential barrier(VM) H′ H″

HM

Primary minimum(Vm)

Fig. 5.8

Potential conditions for flocculation according to the DLVO theory.

the polymer goes into the theta state, and then will be precipitated. But this process is essentially distinct from flocculation. The approach of particles to each other may be caused by cooling, ultrafiltration and other means, not only by flocculation. The most readily apparent merit of the DLVO theory is the explanation of the destabilizing action of neutral salts. If the concentration is increased, the potential drops faster with distance. At very short distances, the attractive potential is always dominant. At medium distances, where the maximum is found, the reduction of the repulsive potential may be considerable. In this way, the compression of the double layer may cause the energy barrier to disappear. The mechanism gives an explanation of the fact that the addition of a salt which does not adsorb nevertheless causes flocculation. Divalent ions are about 50 times as efficient as monovalent ones in destabilizing a suspension. This is known as the Schulze–Hardy rule. However, emulsions generally contain two liquids and the electric double layer extends into both phases. This fact makes the treatment of emulsions more complicated. While the DLVO theory can be applied to describe the conditions beyond the potential barrier, i.e. where H > HM and VM = V(HM), the investigation of repeptization relates to the region where H < HM.

5.8 5.8.1

Stability and changes of colloids and coarse dispersions Stability of emulsions

Food emulsions cover an extremely wide area in practice. One finds ‘semi-solid’ varieties such as margarine, butter, many confectionery fillings and creams, and liquid varieties such as milk, sauces, dressings, and various beverages. In addition, the concept of food emulsions also covers an array of products that contain both solids (suspensions) and/or gases in addition to two liquid phases (e.g. ice cream).

204

Confectionery and Chocolate Engineering: Principles and Applications

Condensation + stabilization

DISPERSION (sol)

Dispersion + stabilization

Flocculation

Solution, evaporation

Crystallization, coalescence

HOMOGENEOUS SYSTEMS

HETEROGENEOUS SYSTEMS

Solution, evaporation

Crystallization, coalescence

Peptization Condensation + aggregation Fig. 5.9

HEAP (gel)

Dispergation + aggregation

Stability of colloids and coarse dispersions, and transformations between them.

Dispersions are systems that contain microphases dispersed in a medium. Their position is between homogeneous and heterogeneous systems (Fig. 5.9). In contrast to dissolved macromolecules, these microphases have a surface in a physical sense, and therefore they have a surface energy as well; consequently, they are thermodynamically unstable. The stability of food emulsions is a field which offers a large spectrum of scientifically interesting phenomena that are only incompletely understood. The three most common dispersed phases in food colloids are liquid water (or an aqueous solution), liquid oil (or partly crystalline fat) and gaseous air (or carbon dioxide). It is natural, therefore, to think of many food colloids as being primarily emulsions or foams rather than colloidal dispersions. Nevertheless, despite the foremost importance of emulsions and foams, there are two good reasons why colloidal dispersions should also be interesting. There is a pragmatic reason for studying particulate dispersions. It concerns the behaviour of casein micelles, the ubiquitous dispersed particles found in milk and in most other dairy colloids. Interactions between casein micelles in different states of dispersion determine the colloidal stability of milk, as well as the formation, structure and rheology of dairy products such as cheese and yogurt. Milk derivatives also play an important role in various aspects of the confectionery industry. As will be seen later, a rigid dividing line cannot be drawn between emulsions and suspensions. As mentioned above, emulsions are unstable systems. This is easily understood, since Eqns (5.29) and (5.33), as a consequence of the Boltzmann distribution, refer also to emul-

Introduction to food colloids

205

sions: p means ‘solubility’ in this case. That is, the smaller particles tend to become associated into larger particles; see the discussion of Ostwald ripening in Section 5.9.5. Therefore, the technical aim of achieving ‘stable emulsions’ has, scientifically, to be limited to control of the kinetics of the processes that lead to the breakdown of emulsions. The technologist has two main tools available for this purpose: (1) the use of mechanical devices to disperse the system and (2) the addition of stabilizing chemical additives or natural compounds (low-molecular-weight emulsifiers and polymers) to keep it dispersed.

5.8.2

Two-phase emulsions

In a two-phase emulsion, one liquid is dispersed in another in the form of large droplets (≥0.3 μm). The emulsion is called an oil-in-water (O/W) emulsion if the continuous phase is water; the opposite arrangement is called a water-in-oil (W/O) emulsion. In some cases the dispersed droplets themselves are emulsions; for example, the dispersed phase (L1) may now be the continuous phase for droplets of the original continuous phase (L2). An emulsion of this kind is called a multiple emulsion and is denoted by ‘W/O/W’ or ‘O/W/O’ depending on the nature of the continuous phase. For further details see John (1970, 1972), Bauckhage (1973), Mersmann and Grossmann (1980), Koglin et al. (1981), Pörtner and Werner (1989) and Pedrocchi and Widmer (1989).

5.8.3

Three-phase emulsions

Most of the ‘emulsions’ encountered in food systems are more complicated than the systems of two liquids described above. It is not feasible to describe all the variations of solid, gel, liquid and gas dispersions found in food emulsions. We shall mention only three examples here, each illustrating a property that cannot be achieved in a two-phase emulsion. In the first, the presence of a third liquid facilitates emulsification to form emulsions with small droplets; in the second, small solid particles stabilize an emulsion; and in the third, an emulsifier forms a liquid crystal, incorporating part of the aqueous and oil phases. For further details, see Kriechbaumer and Marr (1983) and Friberg and ElNokaly (1983).

5.8.4

Two liquid phases plus a solid phase

The mechanism of stabilization by solid particles is of importance in food emulsions, considering the fact that the most common food emulsifiers, the monoglycerides, show crystallization during their use, forming particles at the interface. It is a well-known fact (King and Mukerjee 1938, Schulman and Leja 1954) that the wetting conditions of the two liquids on the solid particles are the key factor in the stabilization mechanism. The particles will stabilize the emulsion if they are located at the interface between the two liquids (see Fig. 5.7), where they serve as a mechanical barrier to prevent coalescence of the droplets. If they are electrically charged in a continuous aqueous phase, the stabilization against flocculation will also be enhanced by the electric double layer. However, the focus in this section will be on the mechanical action against coalescence. The protection against coalescence is based on the wetting energy needed to expel the particles from the interface into the dispersed droplets. This energy depends on the contact

206

Confectionery and Chocolate Engineering: Principles and Applications

Oil phase g = 90°

g < 90°

g > 90°

Solid particle Fig. 5.10

Water droplet on an interface between a solid particle and an oil phase. γ = contact angle.

angles between the liquids and the solid. It is obvious from Fig. 5.10 that a particle with a contact angle of 90° will give the most stable emulsion. The energy that is necessary to force a sphere into the most strongly wetting phase is ΔE = πr 2γ O W (1 − cos θ )

2

(5.62)

where ΔE is the energy required to expel a spherical particle of radius r from the interface into a phase with which its contact angle is θ, and γO/W is the interfacial tension between the oil and water phases. A contact angle of 75° gives only half of the energy compared with a sphere with a contact angle of 90°, and the energy is almost zero at 30°: cos 90° = 0 → (1 − cos 90°) = 1 2

cos 75° ≈ 0.26 → 0.742 ≈ 0.55 cos 30° ≈ 0.87 → 0.132 ≈ 0.0169 These values clearly demonstrate that the contact angle must be close to 90°. An angle greater than 90° (cos θ < 0) will give an even higher energy in Eqn (5.62). Unfortunately, a contact angle greater than 90° has been shown to give less stability in practice; the solid particles are now squeezed into the continuous phase during flocculation. The wetting energies involved in Eqn (5.62) are sufficient to stabilize an emulsion. Preliminary calculations by Davies (1964) showed the wetting energy to be significantly greater than the van der Waals attraction potential at optimal distances between the droplet surfaces. However, the wetting energy is strongly reduced with increasing distance and serves, in a way, to prevent flocculation of the droplets. Hence, this form of stabilization is most useful for systems with high-internal-ratio emulsions. A detailed procedure to optimize the contact angle was described by Friberg et al. (1990). In this procedure, the solid particles must be small (≤ 0.1 μm) and are assumed to be heavier than the aqueous phase, which in turn is assumed to be heavier than the oil. Optimal results are obtained by bringing the oil, with all added components, into contact with the aqueous phase, which also contains all its components, to obtain equilibrium before the experiment. The separated oil phase is placed in a vessel and a drop of the aqueous phase is placed on a powder consisting of the solid particles. The shape of the drop decides the next steps taken. Three different cases may emerge, as shown in Fig. 5.10.

Introduction to food colloids

207

• A contact angle θ of 90° between the aqueous phase and the solid material means that the solid particles are optimally useful for stabilizing the emulsion, and no further action is needed. • If θ < 90°, the interfacial free energy between the oil and the solid material is too high. In this case an oil-soluble surface-active agent must be added, which should bind strongly to the solid surface. • If θ > 90°, a surfactant is added to the aqueous phase, and the same adjustments as in the second case are made.

5.8.5

Emulsifying properties of food proteins

The tests used for the evaluation of proteins as emulsifiers are more or less empirical. The most popular is measurement of the emulsifying capacity (EC), where the maximum amount of fat emulsified by a protein dispersion just prior to the inversion point is determined. The EC method was originally developed by Swift et al. (1961), and it has been used widely, though modified in certain respects. Comparisons between results from different laboratories are difficult to make because this type of investigation is very much influenced by the conditions of measurement. The emulsifying activity index (EAI), as developed by Pearce and Kinsella (1978), is a rough estimate of the particle size of the emulsion, based on the interfacial area (calculated via turbidity) per unit of protein. Dagorn-Scaviner et al. (1987) have compared the EC and EAI methods by studying the emulsifying properties of some food proteins (bovine serum albumin (BSA) and casein, among others). The ranking order of the proteins, BSA being the best, was the same, irrespective of the method used. However, a ‘proper’ characterization of the emulsifying power of a protein requires as full a description as possible of the protein-stabilized emulsion formed.

5.8.6

Emulsion droplet size data and the kinetics of emulsification

The size distribution of the droplets is a most important parameter for characterizing any emulsion. Two emulsions may have the same average droplet diameter yet exhibit quite dissimilar behaviour because of differences in the distribution of diameters. Stability and resistance to creaming, rheology, chemical reactivity, and physiological efficiency are but a few of the phenomena influenced by both relative size and size distribution. Thus the evaluation of an emulsion for size can involve measurements of its droplet number, length (diameter), area, volume and mass (Tornberg et al. 1990). 5.8.6.1

Distribution functions and oil droplet size distribution

The distribution functions most often used to characterize the size distribution of emulsion droplets are the normal and log-normal distributions, the modified log-probability distribution, the Espenscheid and Kerker distributions, the Matijevic distribution (another modification of the logarithmic function), the gamma and Weibull distributions, and the Nukiyama–Tanasawa distribution (which is bimodal and is used in the case of atomization of a liquid). For details, see Bürkholz (1973), Kurzhals and Reuter (1973) and Orr (1983).

208

Confectionery and Chocolate Engineering: Principles and Applications

Emulsification is a dynamic process involving the disruption and recombination or coalescence (called recoalescence) of fat globules. Coalescence is the joining of small droplets together into larger ones. Thus, the final droplet size distribution will be governed by the detailed conditions of the balance between disruption and coalescence during emulsification. Inertial and viscous forces can deform and disrupt globules. Viscous forces generate velocity differences, and inertial forces give rise to pressure gradients within the liquids. To achieve disruption of globules, they have to be deformed to such an extent as to oppose the Laplace pressure within the globule, Δp =

2γ r

(5.63)

where γ is the interfacial tension and r is the radius of the globule (see Eqn 5.18). Therefore, pressure or velocity gradients of the order of 2γ /r2 (applied over a distance r) have to be formed. These high velocity and pressure gradients needed are produced by intense agitation, but unfortunately most of the energy of this agitation is dissipated as heat. Some kind of average of the oil droplet size distribution is given by a droplet size determination, but one needs to be aware of the type of average being calculated. The n-th moment of the frequency distribution of the globule diameter can be used as an auxiliary parameter, Sn = ∑ i ni din

(5.64)

For example, S0 is the total number of droplets per unit volume. The number average diameter d10 is given by S1/S0, and the volume/surface average diameter d32 is equal to S3/S2, called the Sauter mean diameter. The latter average diameter is related to the specific surface area A of the emulsion by the formula A = 6Φ/d32, where Φ is the volume fraction of the dispersed phase. According to Eqn (5.64), S0 = ∑ ni = total number of droplets per unit volume; S1 = ∑ ndi and S1/S0 = d10 = number average diameter; S2 = ∑ ni di2 , S3 = ∑ ni di3 and the Sauter mean diameter is d32 = S3/S2. If it is assumed that the shape of the particles is spherical (and that d is the diameter), then V ( volume ) =

d 3π S 6 , S (surface area ) = d 2π , i.e. = 6 V d

If the volume concentration of the dispersed phase in the emulsion is Φ, where 0 ≤ Φ ≤ 1, then 6Φ ⎛S⎞ =A= ⎝ V ⎠ phase d32

(5.65)

Introduction to food colloids

209

For further details, see Tornberg et al. (1990). The moments of a distribution are the coefficients of the Taylor series of the generating function of the distribution (Alexits and Fenyő 1955; Gnedenko 1988). 5.8.6.2

Kinetics of emulsification

Using an intuitive approach, several researchers have proposed that the time for d32 to reach its equilibrium value d32(∞) could be described by analogy to reaction kinetics: dZ dU = −aZ b

(5.66)

where Z = d32(t)/d32(∞) − 1, U = Nt, N is the impeller speed (in revolutions per second), t is the time and d32(t) is the Sauter mean diameter at the time t. The terms a and b are analogous to a reaction rate constant and reaction order, respectively. Equation (5.66) can be written dZ dt = −cZ b

(5.66a)

where c = aN is a constant. An implicit assumption is that the entire droplet size distribution evolves similarly. For b = 1, d32(t) decays exponentially. Hong and Lee (1985) found that this was the case for stirred-tank systems undergoing simultaneous breakage and coalescence (0.05 < Φ < 0.2). For details, see Treiber and Kiefer (1976), Koglin et al. (1981), Heusch (1983), Armbruster et al. (1991) and Leng and Calabrese (2003). For studies of the kinetics of dispersion, see Zielinski et al. (1974), Becker et al. (1981), Herndl and Mershmann (1982), Kneuele (1983), Ebert (1983), Volt and Mershmann (1985), Zehner (1986), Latzen and Molerus (1987), Kraume and Zehner (1988), Kipke (1992) and Gyenis (1992).

5.8.7

Bancroft’s rule for the type of emulsion

Bancroft’s rule tells us that the type of emulsion is dictated by the emulsifier and that the emulsifier should be soluble in the continuous phase. This empirical observation can be rationalized by considering the interfacial tension at the oil–surfactant and water–surfactant interfaces. There are some exceptions to Bancroft’s rule, but it is a very useful rule of thumb for most systems. If one translates Bancroft’s rule to the ‘HLB language (see Section 5.8.8), it means: For O/W emulsions, one should use emulsifying agents that are more soluble in water than in oil (high-HLB surfactants). For W/O emulsions, one should use emulsifying agents that are more soluble in oil than in water (low-HLB surfactants). The facts that lecithin (HLB ≈ 4, i.e. a low value) is soluble in cocoa butter and that it has a very strong viscosity-lowering effect are in accordance with Bancroft’s rule, since cocoa butter is the continuous phase in chocolate.

210

Confectionery and Chocolate Engineering: Principles and Applications

5.8.8

HLB value and stabilization of emulsions

The main factor in the stabilizing action of surfactants is their tendency to adsorb at the interface instead of being dissolved in one of the liquid phases, i.e. their properties must be balanced between hydrophilic and lipophilic characteristics. The methods for selecting a surfactant are of two principal kinds. In the first, the surfactant per se is characterized by a value for the balance in question, and each W/O combination will have its specific value for the optimal surfactant. The second kind of method considers the combination of the surfactant with the oil and the water, and the whole system is characterized by a number. The best-known system of the first kind is based on the HLB number, introduced by Griffin (1949). This number is based on the relative percentage of hydrophilic to hydrophobic groups in the surfactant molecule. The original method for determining the HLB number requires a long and laborious experimental procedure (Griffin 1954). However, for certain types of non-ionic surfactant, namely polyoxyethylene derivatives of fatty alcohols R(CH2CH2O)xOH and polyhydric alcohol fatty acid esters, the HLB number may be calculated using the following expression: S HLB = 20 ⎛1 − ⎞ ⎝ A⎠

(5.67)

where S is the saponification number of the ester and A is the acid number of the acid. But for many fatty acid esters (e.g. lanolin and beeswax), it is difficult to determine S accurately. In this case, Griffin gave the following expression: HLB =

E+P 5

(5.68)

where E is the weight percentage of the oxyethylene content and P is the weight percentage of the polyhydric alcohol content. In surfactants where only ethylene oxide is used as the hydrophilic portion, the HLB number is simply 5. For a summary of the HLB number ranges required for various systems, see Atlas Chemical Industries (1963). Davies (1959) divided the structure of emulsifiers into component groups, each of which can be assigned a number (positive or negative) that contributes to the total HLB number. The HLB number can then be calculated using the relation HLB = 7 + E ( hydrophilic-group number ) − F ( lipophilic-group number )

(5.69)

HLB numbers are approximately additive; for example, a combination of several surfactants will act as one surfactant that has the weighted average (by mass) of the HLB numbers. If, for example, the optimal HLB value is 12, this can be implemented by using a mixture of a solubilizer (HLB = 18, concentration c1 = 0.4 m/m) and an emulsifier (HLB = 8, concentration c2 = 0.6 m/m): HLB ( mixture ) = 0.4 × 18 + 0.6 × 8 = 12 To produce emulsions of high stability, mixtures of surfactants are used that consist of surfactants of low and high HLB value because the optimal HLB value is dependent

Introduction to food colloids

211

on the material properties of the phases to be emulsified. If a surfactant soluble in both water and oil is used, a thick adsorption layer can be established. Complex emulsions can also be produced in this way; for example, water droplets of size 1–2 μm can be emulsified in oil droplets of size 10–12 μm which, in turn, are dispersed in water. HLB numbers are usually assigned to emulsifying agents without taking into consideration the properties of other components in the emulsion. Marszall and Van Valkenburg (1982) argued that the HLB value is based on only the molecular structure of the emulsifier and does not take into consideration all the factors that affect the performance of an emulsifier, such as the type of oil, the temperature, and the additives in the oil and water phases. With these facts in mind, Marszall and Van Valkenburg (1982) argued for the term ‘effective HLB value’. This is a performance value that takes the above factors into account. Shinoda and Arai (1964) introduced the concept of the HLB temperature, or phase inversion temperature (PIT), which is a characteristic property of an emulsion with a surfactant present. The PIT of an emulsion is the temperature at which the hydrophilic and lipophilic properties of a non-ionic surfactant are balanced. At higher temperatures emulsions are of W/O type, but change to an O/W type at lower temperatures. A correlation exists between the HLB number and the HLB temperature (Shinoda and Sagitani 1978), and one can determine the HLB number from the HLB temperature of a surfactant using a calibration curve (Shinoda and Friberg 1986). The simplest method to determine the PIT of an emulsion is by direct visual observation (Shinoda and Arai 1964). A more sensitive method is to follow the conductivity of the emulsion as a function of temperature. Parkinson and Sherman (1977) suggested the use of the measured PIT value as a rapid method for evaluating emulsion stability. The HLB number and HLB temperature provide a tool for designing energy-efficient methods of emulsification. For more detail, see Ludwig (1969) and Friberg et al. (1990).

5.8.9

Emulsifiers used in the confectionery industry

Lecithin (E 322). This is the most frequently used emulsifier, and is a natural mixture of phosphatidylcholine (PC), phosphatidylethanolamine and other phospholipids. The standard lecithin (mainly of soybean origin) is a hydrophobic mixture dominated by the properties of phosphatidylethanolamine (effective HLB about 4). It is typically used in margarines and spreads as a hydrophobic emulsifier and in chocolates as a viscosity regulator; it is also used as a wetting additive in powders. PC-enhanced lecithin (E 322). The PC concentration is increased by selective extraction of the non-PC components of the lecithin. It is more hydrophilic than the native mixture, and is used in applications where more hydrophilic properties are required. The product has less taste and a purer character than the original material. Hydrolysed lecithin (E 322). This is also more hydrophilic than standard lecithin. It is dispersible in water, and is used in applications where the continuous phase is water, such as mayonnaise and dressings. Distilled monoglycerides (MGs) (E 471). These are about 90% MGs, with a fatty acid composition depending on the fat base, and are slightly on the lipophilic side (HLB about 5). They are used, for example, in the margarine industry as a lipophilic emulsifier, in the baking industry as an additive to retard the staling of bread, and in whipped toppings.

212

Confectionery and Chocolate Engineering: Principles and Applications

Monoglycerides/diglycerides (MGs/DGs) (E 471). These are typically 40% MGs and 60% DGs. They are more lipophilic than distilled monoglycerides (HLB less than 5) and are used as emulsion destabilizers in the ice cream industry. Modified monoglycerides (E 472). (Lactylated, acetylated etc.) These are used in baked products, whipped toppings, and frozen desserts and cakes. Polyglycerol esters (E 475). These are hydrophobic emulsifiers (HLB typically less than 4) and are used in the chocolate industry in combination with lecithin as viscosity regulators. Polyglycerol polyricinoleate (PGPR) is a derivative of ricinic (castor) oil. Lecithin has a radical decreasing effect on the viscosity of chocolate mass; however, PGPR decreases the shear yield (also called the yield stress, τ0) of chocolate mass. The usual combination is 0.25–0.3 m/m% lecithin + c. 0.1 m/m% PGPR (calculated relative to 100% chocolate). Sorbitan esters (E 491). Sorbitan stearate (solid) and sorbitan oleate (liquid). These are lipophilic emulsifiers (HLB about 4) and are used in emulsions in a wide range of products. Polysorbates (E 433). Polysorbate 80 (oleate, liquid) is a hydrophilic emulsifier (HLB typically about 12–16), used in frozen desserts and dressings.

5.9 5.9.1

Emulsion instability Mechanisms of destabilization

Four main mechanisms of emulsion destabilization can be identified: (1) Creaming, which is separation caused by the upward motion of emulsion droplets that have a lower density than the surrounding medium. (2) Flocculation, which is the aggregation of droplets. Flocculation takes place when the kinetic energy during collisions brings droplets over the repulsive force barrier and into a region where attractive forces operate and cause the droplets to attach to each other. (3) Coalescence, which means that two droplets, when they collide, lose their identity and form a single larger one. (4) Ostwald ripening, which is caused by diffusional transport from small droplets to larger ones. The reason for this process is that the chemical potential of the liquid in the droplets decreases as the droplet radius increases. This is analogous to the radius dependence of the vapour pressure above water droplets and above water condensed in capillaries. The concentration of droplets in an emulsion and the droplet size are key parameters in determining the timescale of the instability process. Basically, coalescence is dominant at high concentrations (above 10–50%), flocculation at low concentrations and for small droplets (below 5% and 1 μm in size), and creaming at low concentrations and for large droplets (below 10–50% and above 2–5 μm in size). At intermediate concentrations and sizes, each instability mechanism has to be considered in more detail to identify which one predominates. This information is important because different instability mechanisms are influenced differently by emulsion parameters such as concentration, particle size, type of emulsifier and viscosity.

Introduction to food colloids

213

The first stages in the destabilization of an emulsion are flocculation and sedimentation (or creaming), where two droplets adhere to each other after they have collided. First, the number of droplets is reduced and, second, the enlarged mass of the droplets makes sedimentation faster. These two processes are instrumental in destabilization and depend on each other. The quantitative relationships are well established. Flocculation and creaming are followed by coalescence, in which two adhering droplets become one larger droplet. Ostwald ripening is a relatively long-lasting process.

5.9.2

Flocculation

The induction of flocculation by polymers is a well-known process. A suspension that is stable over time may suddenly start sedimentation when a small concentration of a polymer is added to the solution. In the case where the interactions responsible for this process are confined to polymer–surface interactions, the validity of the bridging theory of La Mer and Healy (1963) has been demonstrated in extensive investigations by Fleer et al. (1971). The mechanism consists of adsorption, on the uncovered surface of a second particle, of a polar group of a polymer that is already attached to an initial particle. A polymer is adsorbed at an interface so as to form ‘trains’ (the molecules on the liquid surface are joined up like the carriages of a train), ‘loops’ (as if the carriages are jammed) and ‘tails’ (a string of carriages stands on its end). The results of Fleer et al. (1971) show convincingly that flocculation will take place when the loops or tails reach the uncovered surface of a particle on which the polymer groups are strongly adsorbed. The stabilization and flocculation of a suspension by polymers are, however, dependent not only on the interactions between polymer groups and the surfaces of the particles but also on polymer–polymer and polymer–solvent interactions. When two particles approach each other, interpenetration of segments from different polymer molecules occurs, and compression also takes place if the distance between the particles is sufficiently small. Thus, the approach of the particles alters the free energy of the system, and the choice between flocculation and stabilization is indicated by the sign of the change of the total (Gibbs) free enthalpy, H a ΔGtot = ΔGpi − ⎛ ⎞ ⎛ ⎞ ⎝ 12 ⎠ ⎝ d ⎠

(5.70)

where Gtot is the change of the total free enthalpy, Gpi is the change in free enthalpy due to the interaction of polymers, H is the Hamaker constant, a is the radius of a spherical particle, d is the distance between the surfaces of the particles, and the last term is the van der Waals interaction. Repulsive forces due to the electric double layer are neglected here. Since the distance is comparatively large, the first term in Eqn (5.70) is dominant, and the following discussion is limited to this term. The change in free enthalpy is divided into an enthalpic and an entropic term: ΔGpi = ΔH − TΔS

(5.71)

The enthalpic term H might be said to reflect the change of the molecular interaction from a mainly polymer–solvent one to a more pronounced polymer–polymer one on interpenetration. The entropic term T ΔS describes the change in the order of the system

214

Confectionery and Chocolate Engineering: Principles and Applications

when solvent molecules are replaced by polymer segments in the interaction zone. Also, compression of the chains leads to a reduction in the magnitude of the entropic term. A decrease in temperature causes a reduction in the size of the entropic term, and destabilization might occur. A good solvent for the polymer will bring about stabilization, since the interpenetration of the polymer chains will give rise to an increase in the free enthalpy, and ΔGpi (and also ΔGtot) will be positive. Changing the solvent by addition of a non-solvent may lead to the creation of conditions where the polymer will not react to the presence of the solvent but will instead behave as if in a vacuum. When these conditions are obtained, the suspension will flocculate, but, by addition of a good solvent, the suspension can be spontaneously redispersed. 5.9.2.1

Flocculation kinetics

If no distance-dependent forces act on the droplets, the number of collisions and the flocculation depend on the diffusion of droplets only. This is called Brownian flocculation. The flocculation rate is described by the number of the original particles which disappear per unit time and volume, given by the Smoluchowski equation, dn 8 kTn2 = −16 πDan2 = − ⎛ ⎞ ⎝ 3⎠ η dt

(5.72)

where dn/dt is the flocculation rate (s m3)−1, a is the droplet radius (m), D is the diffusion coefficient for one droplet (m2/s), n is the number of particles per unit volume (m−3), k = 1.38062 × 10−23 J/K is the Boltzmann constant, T is the absolute temperature (K) and η is the viscosity of the continuous medium (Pa s). The second equation in Eqn (5.72) is obtained from the first one by using the Einstein equation, D=

kT f

(5.73)

and Stokes’ law for friction in a fluid, f = 6 πηa → D =

kT 6 πηa

(5.74)

where f (kg/s) is the friction coefficient for a droplet. The rate of destabilization is easier to understand in terms of the half-life (the time required for the number of droplets to be reduced to one half of its original value), t1 2 = 5.9.2.2

3η 8kTn

(5.75)

Stability from viscosity increase

An increase in the viscosity of the continuous phase adds to the kinetic stability, and this is a fact that is intuitively evident. However, the effect is smaller than intuition might lead

Introduction to food colloids

215

us to believe and, without a concurrent energy barrier, viscosity as such has only a small effect on stabilization. Creaming and flocculation induce emulsion instability at a rate that depends on the droplet size. When the diameter of the particles is reduced by a factor of two, the particle concentration (number of droplets per unit volume) increases by a factor of 23. Therefore, for a fixed volume concentration, the flocculation rate increases rapidly with decreasing particle size. The Smoluchowski equation assumes that hydrodynamic interactions are unimportant, and that the system is dilute (concentrations below 1%). However, in practice, hydrodynamic interactions are of importance in collision events. Furthermore, in technologically important systems the concentration is often considerably larger than 1%, which means that the flocculation rate is lower than that predicted by the Smoluchowski formula. The reason is that particles shield each other (compare the discussion of hindered settling in Section 5.9.3.1). Concerning shear-induced flocculation, see Zeichner and Schowalter (1979), and for gravity-induced flocculation, see Reddy et al. (1981) and Bergenstahl and Claesson (1990).

5.9.3

Sedimentation (creaming)

A droplet moves in a gravitational field. Its movement is slowed by the frictional force from the surrounding medium. For small droplets, Brownian movement (diffusion) is also important. The first two forces and Brownian movement (gravitation − friction ± diffusion)) determine the sedimentation rate and sedimentation equilibrium. 5.9.3.1

Sedimentation rate

The settling rate v, given by Stokes’ law (see Eqn 5.74), is the predominant cause of emulsion instability when the size of the droplets is above around 2–5 μm: ⎛ 4a3π ⎞ Gravitation ⎜ g Δρ = vf = v 6 πηa (friction ) ⎝ 3 ⎟⎠ v=

2a 2 g Δρ 9η

(5.76) (5.77)

where a is the particle radius (m), Δρ is the density difference (kg/m3), g = 9.81 (m/s2) is the gravitational acceleration and η is the viscosity of the continuous phase (Pa s). Equation (5.76) means that, initially, gravitation accelerates the particle and then its velocity reaches v (see Eqn 5.77). From this moment, gravitation and friction compensate each other, and the particle maintains its velocity according to Newton’s first law since the effect of diffusion can be neglected. Greenwald, cited by Gábor (1987, p. 134), gave a relation for the settling rate w in a bulk liquid, the particle size distribution of which is characterized by a general value ri: ⎡ 2 ( r )2 g Δρ ⎤ ⎡ 4 ( ri )3 π ⎤ ⎛ 1 ⎞ ⎡ 8π ( ri )2 gΔρ ⎤ = ∑⎢ w = ∑⎢ i ∑⎢ ⎥ ⎥ ⎥ 9η ⎦ ⎣ 3 ⎦ ⎝V ⎠ ⎣ ⎣ 27ηV ⎦

(5.77a)

where V is the volume of liquid. Evidently, Eqn (5.77a) is a variant of Stokes’ law (Eqn 5.76).

216

Confectionery and Chocolate Engineering: Principles and Applications

Stokes’ law holds under dilute conditions Φ < 2%. In concentrated systems, a smaller settling rate is observed experimentally than that calculated from Stokes’ law. Stokes’ law can be applied to the settling of solid particles in suspensions as well. For more concentrated dispersions, Buscall et al. (1982) suggested ⎛ Φ⎞ v ( effective ) = v (Stokes ) × ⎜1 − ⎟ ⎝ p⎠

5p

(5.78)

where Φ is the volume fraction of the dispersed phase and p is an empirical variable, approximately equal to the final volume of the dispersed phase in the sediment or cream layer. The reason for the decrease in velocity is that the emulsion droplets get in each other’s way and hinder each other’s movement. This phenomenon, which is common in practice, is called hindered settling. This is, for instance, the reason why cream is a stable emulsion, whereas unhomogenized milk (which is a less concentrated emulsion) is unstable towards creaming.

5.9.3.2

Sedimentation equilibrium

Sedimentation equilibrium can be illustrated by the number of droplets per unit volume at two levels in a container after equilibrium has been reached. The Boltzmann distribution gives directly the ratio g Δh ΔρV ⎛n ⎞ ln ⎜ 1 ⎟ = − ⎝ n2 ⎠ kT

(5.79)

where nl and n2 are the numbers of droplets per unit volume at two levels, g = 9.81 m/s2 is the gravitational acceleration, Δh (m) is the difference in height between the two levels, Δρ (kg/m3) is the density difference between the dispersed liquid and the continuous liquid, V (m3) is the volume of one droplet, k (J/K) is the Boltzmann constant, and T (K) is the temperature. The numerator of the right-hand side is the difference of potential energy in the gravitational field, which is related to the distribution as a function of height. This is really the barometer formula (Eqn 5.29). Example 5.4 Equation (5.77) is the basis of Andreasen’s pipette method of particle size analysis by sedimentation; see Andreasen (1935), Koglin (1972) and Thomas (2006). With the substitution a = d/2, where d is the particle size, the equation d2 =

18ηv g Δρ

(5.77b)

provides the appropriate relation; the measured parameters are h and t (the height and the time of sedimentation, respectively), and v = h/t. Cocoa powder is settling in water at 20°C; its particle size is 50 μm. What is the distance by which it sediments after a settling time of 10 s? (The viscosity of water is 0.001 Pa s.) The density of cocoa powder is 1232 (kg/m3), i.e. Δρ ≈ 232 (kg/m3). From Eqn (5.77b),

Introduction to food colloids

25 × 10 −10 = 18 × 10 −3 ×

217

v → v = 315.88 μm s → h ≈ 3.158 mm ( per 10 s ) 9.81 × 232

If h and t are known, d can be calculated. 5.9.3.3

Structure of aggregates, gels and sediments

The stability behaviour of concentrated dispersions is much less well understood than that of dilute systems. The complex interplay of Brownian motion, colloidal forces and hydrodynamic interactions means that the theory of aggregation is far more complicated than for dilute sols, where a description in terms of pair interactions is generally adequate. Aggregates produced by the irreversible coagulation of colloidal particles are not closely packed. They have an open disordered structure and are examples of fractals. It is the sticking together of the particles (and clusters of particles) under the influence of Brownian motion which bestows upon the structure its fractal character, and, indeed, the irregular trajectory of a Brownian particle is itself a fractal object. See Appendix 4 for a discussion of fractals. While the sticking together of individual particles one at a time is a reasonable model for deposition or sedimentation in dilute systems, the aggregates formed in the later stages of Brownian coagulation in the absence of an external field occur by cluster–cluster aggregation and not by particle–cluster aggregation. Very large simulated structures formed by diffusion-limited cluster–cluster aggregation in two or three dimensions are found to be self-similar, i.e. at length scales appreciably larger than the particle radius a, the structure is scale invariant. The fractal dimension D is defined by R ∼ N p1 D a

(N p → ∞ )

(5.80)

where R is the radius of gyration of an aggregate composed of Np particles. In three dimensions (d = 3), the fractal dimension for diffusion-limited coagulation is 1.78 ± 0.01. Denser aggregates are formed if particles are able to alter their relative positions immediately after collision. This is expressed by a sticking probability PS, which is less than unity. In the limit PS → 0, we reach the situation known as reaction-limited cluster–cluster aggregation. This also leads to self-similar structures, but with a larger fractal dimension D ≈ 2.05 in the limit of very large clusters (Np → ∞) at very low total particle volume fractions (Φ → 0). At finite particle volume fractions, the end result of the coagulation process is a sediment (or cream) or a particle gel. That is, the large aggregates either settle under gravity to form a low-density porous sediment, or join together to fill all the available space with a particle gel network. The size of the aggregates making up the network depends strongly on the particle concentration. The number Nc of particles in a close-packed aggregate of radius R is given by R Nc = ⎛ ⎞ ⎝ a⎠

3

and so, from Eqn (5.80), we have

(5.81)

218

Φf =

Confectionery and Chocolate Engineering: Principles and Applications

N p ⎛ R ⎞ 1 ( D − 3) = Nc ⎝ a ⎠

(5.82)

The condition for gelation is that the average volume fraction Φf of the fractal aggregate is equal to the overall volume fraction Φ:

Φf = Φ

(5.83)

The critical aggregate radius Rc at which a gel is formed is therefore Rc = aΦ 1 (D −3)

(5.84)

The strong dependence of Rc on the volume fraction Φ leads to a strong dependence on Φ of the properties of the particle gel, both mechanical (e.g. rigidity) and structural (e.g. porosity). In a low-density sediment or particle gel formed from irreversibly aggregated spherical colloidal particles, three different spatial scales of structure may be identified (Dickinson 1992): • short-range order from packing and excluded-volume effects; • medium-range disorder associated with the fractal characteristics of the diffusioncontrolled aggregation process; and • long-range uniformity in the case of a material that is macroscopically homogeneous. Food particle gels may be produced by the aggregation of casein micelles (in cheese and yogurt) or fat crystals (in margarine). The fractal dimensions of casein particle gels produced by renneting or acidification tend to lie in the range 2.2 < D < 2.4 depending on the experimental conditions. However, the network structure of casein particle gels is more complicated than that of idealized models because of the polydispersity and heterogeneity of the aggregating particles, and also because of macromolecular rearrangements within the network, which continue to occur after gelation. The relationship between the structure and the rheology of casein particle gels depends in a complicated way on the conditions of aggregation of the colloidal particles. It is clear that the concepts of fractal geometry are a useful tool for unravelling this important aspect of the processing of food colloids. 5.9.3.4

Polymer gels and particle gels

Nevertheless, even though casein gels are composed of polymers, their properties are quite different from those of true polymer gels such as gelatin or alginate. In contrast to a polymer gel, whose elasticity is mainly of entropic origin, the rheology of a particle gel is related to energetic (enthalpic) aspects such as the bending energies of network connections and the breaking energies of linkages (Dickinson 1992). The three main physical factors affecting the rheology of a particle gel are: • the volume fraction of particles; • the deformability of the particles and their linkages; and • the fractal dimension of the network.

Introduction to food colloids

5.9.3.5

219

Interparticle interactions

The interparticle interactions are determined mainly by the properties of the surfaces of the droplets in the emulsion, which in food emulsions are coated with various surfaceactive molecules, in most cases of biological origin. Surface forces. The surface forces are all of the static forces that act between particles and depend on the separation of the particles. These forces are influenced by the properties of both the particles and the separating medium. The term ‘surface forces’ is used because the chemical composition of the outermost layer of the particles influences the range and magnitude of the forces much more than does the bulk composition. The types of forces most commonly observed are van der Waals, electrostatic double-layer, hydration, hydrophobic and steric forces. Only van der Waals and double-layer forces are taken into account in the DLVO theory of colloidal stability. Electrostatic double-layer forces. Typical food emulsions coated with proteins or hydrocolloids have small surface charge densities, corresponding to low zeta potentials, normally between −1 and −20 mV. In long-shelf-life emulsions, such as mayonnaise, the electrolyte concentration is also rather high, which reduces the range of the electrostatic repulsion. Consequently, in many food emulsions double-layer forces are not very important, and the DLVO theory can rarely explain emulsion stability or solve stability problems in food systems. Van der Waals forces. These are repulsive when the dielectric function of the medium is between those of two interacting particles; for example, repulsive van der Waals forces act between an air bubble and an oil droplet in aqueous solutions according to Hough and White (1980). The magnitude and sign of the van der Waals force are determined by the dielectric properties of the particles and of the surrounding medium, and it is in principle easy to calculate the non-retarded van der Waals force from dielectric data; see Hough and White (1980). However, in practice it is often the case that the dielectric properties of the media are unknown.

5.9.4

Coalescence

How rapidly two droplets coalesce depends on the stability of the thin film separating them. When the dispersed phase is a liquid, coalescence follows rapidly once the separating film ruptures. Numerous studies have dealt with the coalescence of a single droplet with a planar interface created by a settled, coalesced layer. These studies involve measurement of the time that elapses from the arrival of a droplet at the interface to coalescence. Many factors influence the waiting time, or film drainage time, including the age of the interface. The times can be correlated using film drainage theory. The simplest model for film drainage assumes that the conditions affecting the drainage rate are time invariant. By analogy to squeezing flow between parallel discs (the lubrication approximation), the rate at which the film thins is given by dh = − k1h3 dt

(5.85)

where h (m) is the thickness of the film, k1 (s m2)−1 is a constant and t is the time (s). The interface is assumed to be mobile but motionless. The initial separation distance is h0, and

220

Confectionery and Chocolate Engineering: Principles and Applications

h is the separation distance after time t. The constant k1 accounts for all the factors that determine the drainage time. After integration of Eqn (5.85), 1 1 − = k1t h2 h02

(5.86)

Estimation of the initial film thickness h0 is not critical, since the initial thinning is fast. After a short time, h−2 >> h0−2, allowing evaluation of the drainage rate constant k1 from precise measurements of film thickness versus time. Estimates of the film thickness at rupture from 25 to 500 Å have been reported. Studies involving mass transfer from droplets show that in the presence of mass transfer, coalescence times are much shorter. In the case of a collision between two droplets with equal diameters d, the leading edges of the two deformable droplets become flattened on collision. This deformation creates a parallel, disc-like geometry. Therefore, the dynamics of film drainage can be represented as a squeezing flow between two discs of radius R (m), separated by distance h (m), that approach each other owing to a force F (N). The excess pressure in the film must be of the order of the Young–Laplace pressure. These suppositions lead to the following drainage rate (compare Eqn 5.85): dh 32 πσ 2 h3 =− dt 3ηd 2 F

(5.87)

where η is the viscosity (Pa s) of the continuous phase and σ is the interfacial tension (N/m). Equation (5.87) shows that the film drainage rate is inversely proportional to the approach force, again demonstrating that coalescence is promoted by gentle collisions. Integration of Eqn (5.87) with the initial condition h = h0 at t = 0 and the final condition h = hc at t = τ leads to 1⎞ ⎛ 3ηd 2 F ⎞ ⎛ 1 τ =⎜ − ⎝ 64 πσ 2 ⎟⎠ ⎜⎝ hc2 h02 ⎟⎠

(5.88)

where τ is the time (s) required for film rupture and hc (m) is the critical thickness required for film rupture. The initial distance h0 is usually much greater than hc, so that 1 1 1 − ≈ hc2 h02 hc2

(5.89)

Coalescence occurs only if the contact time tc is greater than τ. For further detail, see Leng and Calabrese (2003).

5.9.5

Ostwald ripening in emulsions

Ostwald ripening is caused by diffusional transport from small droplets to larger ones. The reason for this process is that the chemical potential of the liquid in a droplet decreases as the droplet radius increases. This is analogous to the radius dependence of the vapour pressure above water droplets and above water condensed in capillaries (see

Introduction to food colloids

221

Eqn 5.29). A similar observation can be made in connection with the solubility of crystals: particles that are smaller than a critical particle radius will disappear owing to their higher solubility, and larger particles will grow owing to their lower solubility. The Ostwald– Thomson equation gives the solubility of crystals of different sizes (Gábor 1987, p. 95): 2γ SL M ⎛L ⎞ RT ln ⎜ r ⎟ = ⎝ L∞ ⎠ r ( dS − d L )

(5.90)

where R is the universal gas constant (8.31434 J/mol K), T is the temperature (K), Lr is the solubility of a crystal of radius r m, L∞ is the solubility of a crystal of infinite radius (i.e. that of a planar surface), γSL is the solid–liquid interfacial tension (N/m), M is the molar mass of the crystalline substance (kg/kmol), dS is the density of the crystalline substance (kg/m3) and dL is the density of the solution (kg/m3). The Ostwald–Thomson equation can be derived directly from the Boltzmann distribution, which is valid for equilibrium (see also Eqn (5.79) for sedimentation): U ⎛N ⎞ ln ⎜ 1 ⎟ = − ⎝ N2 ⎠ kT

(5.91)

where N1 and N2 are the numbers of particles per unit volume at positions 1 and 2, respectively, and U is the potential-energy difference between these positions. A model for Ostwald ripening in emulsions has been developed by Yarranton and Masliyah (1997); see also Matz (1984), Pawlowski et al. (1986), and Sections 10.6.1 and 16.4.

5.10

Phase inversion

The conversion of an O/W emulsion into a W/O emulsion or vice versa is called emulsion phase inversion (or simply ‘inversion’). In food systems, the process of emulsion phase inversion does not usually occur spontaneously: large amounts of mechanical energy are frequently required. This is because inversion is not a single physical process like creaming, coalescence or flocculation, but is a composite phenomenon, possibly involving all three of these primary processes, as well as involving one or more complex intermediate colloidal states (foams, multiple emulsions, bicontinuous structures etc.). The most important parameter in the description of emulsion phase inversion is the volume fraction Φ of the dispersed phase. Experimentally, as Φ increases at constant emulsifier concentration, there is a systematic increase in the viscosity of the emulsion until, at a certain critical volume fraction Φc, there is a sudden drop in viscosity corresponding to a sudden change in volume fraction from a high value Φc to a low value 1 − Φc at the inversion point. Another easily measurable quantity showing a sharp change at the inversion point is the electrical conductivity: high for O/W systems but low for W/O systems. Assuming that oil or water droplets in emulsions can be regarded as nondeformable spheres, Ostwald suggested that Φc should be taken as 0.74, corresponding to the maximum ordered packing fraction for identical hard spheres. Both emulsion types are possible for 1 − Φc ≤ Φc ≤ Φ, but only one type exists outside this range. In practice, some emulsions do invert near Φ ≈ 0.74, but many others do not. This is due, in part, to

222

Confectionery and Chocolate Engineering: Principles and Applications

the polydispersity and deformability of real emulsion droplets. In addition, Φc depends substantially on the chemical nature of the emulsifier, a factor ignored completely in the geometrical approach of Ostwald. Irrespective of O/W ratio, an emulsifier with a low HLB number will be unlikely to stabilize an O/W emulsion, and one with a high HLB number will not readily stabilize a W/O emulsion. The universal features of emulsion phase inversion in a system consisting of oil + water + emulsifier are neatly described by a mathematical approach called catastrophe theory. The four reasons why emulsion phase inversion may be regarded mathematically as a catastrophic event are: (1) Emulsion morphologies are bimodal; i.e. for a large range of volume fractions, they can exist indefinitely in one of two stable states (O/W or W/O), but not as something in between. (Note, in this respect, that these latter states are thermodynamically unstable; no minima of the Gibbs free enthalpy g correspond to them.) (2) In accordance with the abrupt change in morphology, inversion involves a sudden jump in physical properties such as viscosity and electrical conductivity. (3) The system exhibits hysteresis, i.e. the morphology depends on the experimental path or the previous history of the emulsion. (4) Two emulsions prepared only slightly differently from the same amounts of oil + water + emulsifier may show divergent behaviour. In the mathematical theory of phase inversion (Dickinson 1981; 1992, pp. 100–115), the Gibbs free enthalpy g is expressed as a fourth-degree polynomial of a variable s, and the three roots of the equation dg/ds = 0, where the derivative dg/ds is a third-degree polynomial, provide the characteristic quantities of the states O/W and W/O and the transition. Such a form of the Gibbs free enthalpy as a function of fourth order is, however, a pure mathematical construct with no underlying physical justification apart from its success in describing the observed phenomenon (Dickinson 1992). See Section 2.1.4 for details of a phase inversion experiment by Mohos (1982).

5.11 5.11.1

Foams Transient and metastable (permanent) foams

A foam is a coarse dispersion of gas bubbles in a liquid (or, sometimes, solid) continuous phase (usually water). It is a colloidal state in the sense that the thin films separating adjacent gas cells in a foam are typically of colloidal dimensions. Two easily recognizable structures in foam systems can be described. A kugelschaum, or sphere foam, is produced in freshly prepared systems, and consists of small, roughly spherical bubbles separated by thick films of viscous liquid. The foam may be considered as a temporary dilute dispersion of bubbles in the liquid. But on ageing the structure gradually changes, and the bubbles transform into polyhedral gas cells with thin, flat walls. A few terms are useful for distinguishing qualitatively between different kinds of liquid foams. In structural terms, a bubbly foam (e.g. ice cream), in which the amount of gas incorporated is low enough for the bubbles to retain their roughly spherical shape, is substantially different from a polyhedral foam (e.g. beer foam), in which the gas-to-liquid ratio is so large that the bubbles are pressed against one another in a honeycomb-type

Introduction to food colloids

223

structure. In kinetic terms, it is convenient to distinguish between an unstable transient foam (e.g. champagne bubbles), whose lifetime is measured in seconds or minutes, and a metastable permanent foam (e.g. meringue), whose lifetime is measured in days. In the confectionery industry, it is an essential requirement to produce permanent foams. Pure water does not foam. Gas bubbles introduced beneath the liquid surface burst as fast as the liquid can drain away from them. The stability of a transient foam may be estimated by noting the persistence time of the bubbles. In the context of foods, the most important gas used to make bubbles, apart from air, is carbon dioxide. This gas has the advantage of being non-toxic and natural, being produced in situ during breadmaking, biscuit/cake production (by yeast or baking powder) and beer fermentation (by yeast). Nitrogen gas, however, gives a more stable head than carbon dioxide because it diffuses more slowly from small bubbles to large ones owing to its lower solubility in water (see Example 5.5 below). Permanent food foams are stabilized by macromolecules or by particles. As the gas dissolves into the aqueous phase from a bubble, the surface area of the bubble decreases, and, since there is negligible desorption of adsorbed macromolecules or particles, there is a decrease in surface tension, which stabilizes the Laplace pressure difference across the film, and so the bubble shrinks no further. The adsorbed macromolecules most commonly used to stabilize food foams are egg-white proteins and milk proteins. Many dairy colloids, such as whipped cream and ice cream, are emulsions as well as foams; they are primarily stabilized not by adsorbed protein films but by a matrix of partially aggregated fat globules at the air–water interface. The major food ingredients apart from fat that have a significant effect on the structure and stability of food foams are starch and egg yolk (in baked products) and sugar (in confectionery). Foams and emulsions have much in common in terms of their colloidal stability, but there are several important differences from the physical point of view. Gas bubbles are about 103 times as large as emulsion droplets, mainly because gases are much more soluble in water than oil is in water or water is in oil. The surface tension of a gas bubble is several times the interfacial tension of an emulsion droplet. A bubbly foam has a strong tendency to cream because, in addition to the large bubble size, the density difference between the phases is more than ten times that in an emulsion. In addition, gas bubbles are about 105 times as compressible as emulsion droplets, and they are more easily deformed because of their large size. The combination of fast creaming and the deformability of bubbles rapidly converts a bubbly foam into a polyhedral foam, unless the aqueous phase is a high-viscosity liquid or a gel-like solid. In addition, liquid foams are much more susceptible than emulsions to disturbing influences, for example evaporation, dust, draughts, temperature gradients, vibration and the addition of foam-breaking chemicals. This is mainly because of the much larger dimensions of the liquid films in foams than of those in emulsions. On top of it all, the disproportionation of bubbles in foams is much faster than the Ostwald ripening of emulsion droplets. The overall effect of all these physical factors is that small bubbles are hard to make and tend to disappear rapidly, and a foam with large bubbles is susceptible to fast drainage and rupture. Stability is best achieved with ‘insoluble’ adsorbed layers of coagulated protein (e.g. egg white in meringue) or immobile particles (e.g. fat globules in whipped cream), or by converting the liquid foam into a solid foam (as in the baking of a cake). Conversion of a bubbly foam into a creamed layer happens in a matter of minutes if the liquid has a viscosity of the order of that of water. Creaming may be prevented by

224

Confectionery and Chocolate Engineering: Principles and Applications

the addition of a hydrocolloid that gives a yield stress in excess of c. 10 Pa. This stress is so low as not to be perceptible during normal handling. This means that bubbles can be kept in suspension under quiescent conditions by a weak gel network in a liquid medium that behaves like a low-viscosity aqueous solution under normal conditions of pouring, mixing and drinking.

5.11.2

Expansion ratio and dispersity

The expansion ratio n is expressed by the equation n=

VF VG + VL V = = 1+ G VL VL VL

(5.92)

where VF is the volume of the foam, VL is the volume of the liquid content of the foam and VG is the volume of the gas content of the foam. The foam density ρF can also be used to characterize the expansion ratio of a foam:

ρF =

( mG + mL ) VF

=

ρGVG + ρLVL VF

(5.93)

where mG is the mass of gas in the foam, mL is the mass of liquid in the foam and ρL is the density of liquid in the foam. Since the foams of interest are usually aqueous solutions, ρG/ρL = ρAIR/ρWATER ≈ 1/1000, and therefore Eqn (5.93) can be simplified to

ρF =

ρGVG + ρLVL ρLVL ρL ρ ≈ = →n≈ L VF VF n ρF

(5.93a)

In confectionery practice, the values of ρF are in the range 0.25–0.8 and those of ρL are in the range 1.1–1.2 (aqueous solutions of carbohydrates); consequently, from Eqn (5.93), the values of n vary over a range of about 1.5–5. The dispersity of gas emulsions and polyhedral foams is a very important parameter, which determines many of their properties and the processes occurring in them (diffusion transfer, drainage etc.), and therefore their technological characteristics and areas of application. The procedure usually followed in order to obtain detailed information about the bubble size distribution involves grouping bubbles into fractions by size, and counting the number of bubbles Ni and determining the radius of the bubbles Ri in each fraction. Thus, it is possible to evaluate: • the bubble radius RV, averaged by volume: 3 RV3 = ⎛ ⎞ v = ⎝ 4π ⎠

∑ Ri3 Ni ∑ Ni

where v is the average bubble volume; • the bubble radius RA, averaged by area:

(5.94)

Introduction to food colloids

RA2 =

A = 4π

∑ Ri2 ∑ Ni

225

(5.95)

where A is the average surface area; and • the bubble radius RL, averaged by length:

RL =

∑ Ri Ni ∑ Ni

(5.96)

The results of dispersion analysis make it possible to calculate also the specific surface area,

εG =

3∑ Ri2 Ni ∑ Ri3 Ni

(5.97)

and the average radius by volume and surface area, RAV =

∑ Ri3 Ni ∑ Ri2 Ni

(5.98)

The distribution function is represented graphically both as integral and as differential distribution curves. All of the universal distribution functions (gamma, Gaussian, Maxwell, Pearson, Boltzmann, binomial, Poisson etc.) are used for evaluation of the size distribution of foams. For details, see Exerowa and Kruglyakov (1998, pp. 25–30).

5.11.3

Disproportionation

5.11.3.1

The Plateau border

The diffusion of gas from small bubbles into large bubbles is referred to as disproportionation. In the absence of a stabilizing film of polymer molecules or particles, disproportionation occurs remarkably quickly. Overall, it is probably the most important type of instability in foams. The driving force for disproportionation is the Laplace pressure difference of a curved bubble surface, which results in a higher pressure in a small bubble than in a large one [see Fig. 5.11, which depicts a Plateau border (Section 5.11.5)]: ⎛1 1⎞ Δp = γ ⎜ + ⎟ ⎝ r1 r2 ⎠

(5.99)

where γ is the surface tension, and r1 and r2 are the radii of the two bubbles of different size. As the solubility of gases increases with pressure (Henry’s law), more gas dissolves near the small bubble than near the large one, and so the latter grows at the expense of the former. Assuming that gas transport takes place by diffusion (obeying Fick’s law) through the continuous phase (Dickinson 1992, p. 126),

226

Confectionery and Chocolate Engineering: Principles and Applications

P1

P2 = P0 + g /R P1 R P1

P1 r

P1 = P0 + g /r

Fig. 5.11

A Plateau border.

1 1 dV dr Aγ = A ∼ − A ( P1 − P2 ) = − Aγ ⎛ − ⎞ ∼ − ⎝ r R⎠ dt dt r i.e. dr γ ∼ − → r dr ∼ −γ dt dt r

(5.100)

where ‘∼’ means ‘proportional to’, V = Ar is the volume of the bubble, A is the surface area of the bubble, P1 − P2 is the Laplace pressure difference and γ is the surface tension. After integration of Eqn (5.100), the change in radius r of a small bubble with respect to time t is given by ⎛ 4RTDG L S∞γ ⎞ r 2 = r02 − ⎜ ⎟⎠ t ⎝ pλ

(5.101)

where r0 is the bubble radius at t = 0, DG/L is the diffusion coefficient of the gas in the liquid, S∞ is the solubility of the gas at a planar interface (r → ∞), p is the pressure and λ is the distance over which gas diffuses from the small bubble to one with an infinite radius of curvature. A plot of r vs t for Eqn (5.101) is a parabola (Fig. 5.12). Example 5.5 Prins (1987) published the following data relating to the parameters in Eqn (5.101): r0 = 125 × 10−6 m γ = 39 × 10−3 N/m λ = 10−5 m Dg/l(C) = 1.77 × 10−9 m2/s (diffusion coefficient of CO2 in the liquid) Dg/l(N) = 1.99 × 10−9 m2/s (diffusion coefficient of N2 in the liquid) S∞(C) = 3.9 × 10−4 mol/N m (solubility of CO2 at a planar interface if r → ∞) S∞(N) = 6.9 × 10−6 mol/N m (solubility of N2 at a planar interface if r → ∞) P = 105 N/m2 λ = 10−5 m

Introduction to food colloids

227

Food foams

Bubble radius r

Parabola (theoretical)

Inflection Fig. 5.12

Time t

Shrinkage of bubbles.

Let us calculate (at T = 293 K) the time t needed for r → 0 (see Eqn 5.101). For carbon dioxide, 1252 × 10 −12 = 4 × 8.31 ( N m mol K ) × 293 K × 39 × 10 −3 N m × 1.77 × 10 −9 ( m 2 s ) × 3.9 × 10 −4 mol N m × t [105 ( N m 2 ) × 10 −5 m ] For nitrogen, 1252 × 10 −12 = 4 × 8.31 ( N m mol K ) × 293 K × 39 × 10 −3 N m × 1.99 × 10 −9 ( m 2 s ) × 6.9 × 10 −6 mol N m × t [105 ( N m 2 ) × 10 −5 m ] After some calculations, t (carbon dioxide) = 4.11 × 10−11 s/[1.77 × 10−9 × 3.9 × 10−4] = 59.5 s t (nitrogen) = 4.11 × 10−11 s/[1.99 × 10−9 × 6.9 × 10−6] = 2993 s Obviously, in Example 5.5 the difference between the times that are necessary for the disappearance of the bubbles is derived mainly from the difference between the solubilities S∞(C) and S∞(N) of the gases. In food foams, however, with surface-active species present at the air–water interface, the situation is more complicated because the surface tension of a shrinking bubble is less than the equilibrium value. Therefore, the surface tension decreases during shrinkage, and the disproportionation process may stop completely, i.e. r may stop decreasing. In this case a different type of curve for r = r(t) is obtained; see Fig. 5.12, which has an inflection point P. 5.11.3.2

Surface dilational viscosity and surface dilational modulus

The change in the surface tension γ with decreasing bubble size can be expressed in terms of the surface dilational viscosity ηd, defined by

228

Confectionery and Chocolate Engineering: Principles and Applications

ηd =

Δγ d ln A dt

(5.102)

where Δγ (N/m) is the increase (or decrease) in surface tension compared with the equilibrium value, and d ln A/dt (s−1) is the relative rate at which the area is changing; d ln A is dimensionless. The surface dilational viscosity ηd measures the ability of a liquid surface to resist an external disturbance, such as an increase in surface area A or a shrinking stress exerted on the surface by a streaming liquid. The related surface dilational modulus (or Gibbs coefficient) εd is defined by

εd =

dγ d ln A

(5.103)

Taking account of ηd when r is large makes little difference, but it can have a considerable slowing-down effect when r is small. If A = r2, then d ln A = 2 d ln r. Consequently, if r is large, ηd is small, and if r is small, ηd is large, i.e. the slowing-down effect of ηd is strong. When the dilational surface rheology of the bubble surface has a purely elastic component, the disproportionation process stops completely. The stability condition is

εd =

dγ 1 ⎛ dγ ⎞ γ =⎛ ⎞ ≥ d ln A ⎝ 2 ⎠ ⎜⎝ d ln r ⎟⎠ 2

(5.104)

where εd is the surface dilational modulus and γ is the surface tension in equilibrium. For small-molecule surfactants, εd is effectively zero on the timescale of bubble shrinkage because the surfactant dissolves into the bulk phase as r becomes smaller to restore the equilibrium adsorption condition. 5.11.3.3

Gibbs adsorption equation

Materials that adsorb strongly at an interface, and therefore cause a substantial lowering of the surface tension at low concentrations, are called surfactants. For a surfactant solution, it is usually a very good approximation to take Γ2 as the absolute surface concentration in the Gibbs adsorption equation 1 ⎞ dγ Γ2 = −⎛ ⎝ RT ⎠ d ln ( x2 f2 )

(5.105)

where x2 and f2 are the concentration and the activity coefficient (unity for an ideal solution), respectively, of the solute. Because a small-molecule surfactant dissolves into the bulk phase as r becomes small, the absolute surface concentration Γ2 of it becomes zero at the interface. This means that foams made with surfactants that form a simple monolayer are unstable with respect to bubble collapse by disproportionation. However, the situation is different with an adsorbed protein film at the interface, especially when the protein is susceptible to surface coagulation, as is the case with the egg-white protein ovalbumin. There is no desorption over the timescale for bubble

Introduction to food colloids

229

shrinkage, and as r decreases, d becomes large enough to satisfy Eqn (5.104): hence the foam is stable towards disproportionation. The condition for stability is also satisfied if the bubble surface becomes packed with hydrophilic solid particles (as in the case of the fat globules in whipped dairy foams) or if a small-molecule surfactant is present in sufficient concentration to form an elastic liquid-crystalline gel phase around the bubbles. For details of the manufacture of foams, see Henzler (1980), Beyer von Morgenstern and Mersmann (1982), Stein (1987a,b, 1988) and Brauer et al. (1989).

5.11.4

Foam stability: coefficient of stability and lifetime histogram

All foams are thermodynamically unstable owing to their high interfacial free energy, which decreases on rupture or drainage. A detailed description of test methods for foam stability, together with literature references, has been given by Pugh (2002). Various methods are employed to estimate and compare foam stability with respect to destruction of a foam column. Most often, these methods can be reduced to determination of the lifetime of a foam column (or part of it) up to its complete disappearance. In connection with the stability of flotation foams (froths), the concept of a coefficient of stability B has been developed: B=

t VF

(5.106)

where t is the time for foam destruction and VF is the volume of the foam. When a foam decays in a gravitational field, the capillary pressure in its upper parts is reduced owing to the diminishing height of the foam column. Hence, the times of decay of the various local layers are different, and the total lifetime of the whole foam column is an integral that takes account of the effect of the local pressure and the total height H0 of the foam column: t=

H0

dt

∫ dH dt 0

(5.107)

The function t vs H is often obtained in the form of a histogram of the distribution of the lifetimes of local foam layers of, for example, 2 cm thickness. To compare the stability of foams made using various surfactants or with different surfactant concentrations, it is advisable to measure the foam lifetime at constant pressure, tP, in the Plateau borders. The quantity tP is a much better-defined indicator of foam stability since the pressure in the borders throughout the height of the foam column remains constant during its destruction. This parameter is also much more sensitive to the kind of surfactant, the electrolyte concentration and the presence of other additives than the lifetime of the foam in a gravitational field. Drainage strongly affects foam collapse. The higher the drainage rate, the more rapidly the equilibrium state is reached and, therefore, stability can be reduced. That is why a correlation is often observed between the rate of drainage and foam stability: the slower the rate of drainage, the longer the foam lifetime is.

230

Confectionery and Chocolate Engineering: Principles and Applications

5.11.5

Stability of polyhedral foams

In a polyhedral foam, the liquid films (lamellae) between the bubbles are thin and flat. In order to satisfy the condition of mechanical equilibrium, the films meet each other at an angle of 120°. The meeting point is called a Plateau border. Owing to the curvature of the interface, the pressure in a Plateau border is lower than that in a bubble by an amount Δp =

γ r1

(5.108)

where r1 is the radius of curvature of the Plateau border surface. Equation (5.108) takes into account the fact that in polyhedral foams the lamellae are thin, and the other radius (r2) of curvature of the Plateau border surface has become practically infinite, i.e. 1/r2 ≈ 0; see Eqn (5.99). As in the case of coalescence in emulsions, the stability of a polyhedral foam depends on two distinct processes: film drainage and film rupture.

5.11.6 5.11.6.1

Thinning of foam films and foam drainage Thinning

The lifetime Δt of a thinning foam film can be estimated from the relation Δt =

hCR

dh

∫ h0 w

(5.109)

where h0 is the initial thickness of the film, hCR is the critical thickness at which the film ruptures and w = dh/dt is the rate of thinning. Under certain conditions, the hydrodynamics of films in foams is very well described by the lubrication theory of Reynolds (the Stefan–Reynolds relation): −

dh 2 h3 Δp = dt 3ηr 2

(5.110)

where h is the thickness between two solid circular plates of radius r, as a function of the time t; Δp is the pressure drop between the capillary pressure of the meniscus pσ and the disjoining pressure of the film; and η is the dynamic viscosity of the solution. Equation (5.110) leads to d ( h −2 ) 4 Δp = 3ηr 2 dt

(5.111)

Manev et al. (1974) showed the following relationship to be useful: h = h0 exp ( − kt ) where k (s−1) is a constant, experimentally determined.

(5.112)

Introduction to food colloids

5.11.6.2

231

Foam drainage; foam syneresis

At the moment of formation, the liquid content of a foam is usually considerably larger than that in hydrostatic equilibrium. For this reason, liquid starts draining out of a foam even during generation of the foam. The ‘excess’ liquid in the films drains into the Plateau borders, and then flows down through them from the upper to the lower layers of the foam following the direction of gravity until the gradient of the capillary pressure balances the gravitational force (dpσ/dL = ρg, where L is a coordinate in the direction opposite to gravity). Simultaneously with drainage from the films into the borders, the liquid begins to flow out from the foam when the pressure in the lower layers exceeds the external pressure. By analogy with gel syneresis, the outflow of liquid from a foam was called ‘foam syneresis’ by Arbuzov and Grebenshchikov (1937). The main driving force for drainage is gravity, which acts directly on the liquid in a non-horizontal film, and indirectly through suction acting on the Plateau borders. The rate of drainage is determined not only by the hydrodynamic characteristics of the foam (the shape and size of the borders, the viscosity of the liquid phase, the pressure gradient, the mobility of the liquid–gas interface, etc.) but also by the rate of internal collapse of the foam (both films and borders) and the breakdown of the foam column. The outflow of liquid from the foam represents the last stage of a process that includes film thinning and rupture, and outflow through the borders and films.

5.11.6.3

Film rupture: the role of particular materials and mechanical disruption

Film rupture is a stochastic process. In practice, the most important mechanism of film rupture is that involving particular materials in the film. If a hydrophobic particle is large enough to touch both surfaces, the Laplace pressure in the film adjacent to the particle may become positive. This will lead to flow of liquid away from the extraneous particle, eventually resulting in the liquid breaking contact with the particle (i.e. film rupture). Another type of contaminant particle is one that spreads its contents over the air–water interface. This spreading causes adjacent liquid in the film to move in the same direction. This movement of liquid induces local thinning of the film, which enhances the probability of rupture. This type of mechanism is believed to be responsible for the destabilizing effect of fatty particles in aqueous foams. Examples of this phenomenon are the poor foaming behaviour of whole milk as compared with skimmed milk, and the detrimental effect of a small amount of egg yolk on the foaming of egg white. In addition to contamination by particles, other forms of disturbance which may induce film rupture are mechanical disruption (stirring, shaking etc.) and evaporation. At the top of a foam exposed to the external atmosphere, evaporation of water may reduce stability by reducing the film thickness to the value at which there is spontaneous hole formation. There are various empirical and semi-empirical equations that are used in the quantitative description of the drainage process; their application, however, is usually limited to short time intervals and narrow ranges of foam expansion ratio.

5.11.7

Methods of improving foam stability

Several methods have been developed for improving foam stability:

232

Confectionery and Chocolate Engineering: Principles and Applications

(1) Stability increase caused by an increase in bulk viscosity. As a general rule, the drainage rate of a foam may be decreased by increasing the bulk viscosity of the liquid from which the foam is prepared. For many food foams, drainage can easily be halted by formation of a hydrous gel, and the lamellae can be stabilized at relatively large thicknesses (∼1 μm). The more viscous the liquid, the slower is the drainage between layers. (2) Stability increase caused by an increase in surface viscosity. An alternative method to slow down the foam drainage kinetics is to increase the surface viscosity by packing a high concentration of surfactants or particles into the surface, for example by adding relatively high-molar-mass polymers, proteins or polysaccharides, or certain types of particles (e.g. castor sugar). In addition, high cohesive forces in the surface films can be achieved by using mixed surfactant systems. (3) An absorbed surfactant film can control the viscosity of the surface layer. The experimentally measurable parameters that characterize the mechanical–dynamical properties of monolayers are the surface elasticity and surface viscosity. The surface viscosity reflects the speed of the relaxation processes which restore equilibrium in a system after a stress has been imposed on it. The surface viscosity (for simple ‘Newtonian surface flow’) is defined by the equation dv τ S = ηS ⎛ ⎞ ⎝ dr ⎠ S

(5.113)

where τS is the surface shear stress (N/m), ηS is the surface shear viscosity (N s/m) and (dv/dr)S is the surface shear rate [m/(s m)]. The surface viscosity is also a measure of the energy dissipation in the surface layer. In contrast, the surface elasticity is a measure of the energy stored in the surface layer as a result of an external stress. The surface elasticity (for simple ‘Hookean surface elasticity’) is defined by dL ⎞ τ S = GS ⎛ ⎝ dr ⎠ S

(5.114)

where τS is the surface shear stress (N/m), GS is the surface shear modulus (N/m) and (dL/dr)S is the surface shear deformation (m/m). (4) Gibbs–Marangoni effect. This is caused by adsorbed surfactants, and heals thinning surfaces and prevents the drainage of thin films that leads to rupture. For thick lamellae, under dynamic conditions, the Gibbs–Marangoni effect becomes important and operates on both expanding and contracting films. The Gibbs–Marangoni effect is the transfer of mass on or in a liquid layer due to differences in surface tension. Since a liquid with a high surface tension pulls more strongly on the surrounding liquid than one with a low surface tension, the presence of a gradient in the surface tension will naturally cause the liquid to flow away from regions of low surface tension. This surface tension gradient may be caused by a concentration gradient or a temperature gradient, because surface tension is a function of temperature. The Gibbs–Marangoni effect tends to oppose any rapid displacement of a surface, and may, at fairly high surfactant concentrations, provide a temporary restoring or stabilizing force to ‘dangerous’ thin films that can easily rupture. The Gibbs–Marangoni effect is superimposed on the Gibbs elasticity, so that the effective restoring force is a function of the rate of extension, as well as of the thickness.

Introduction to food colloids

233

(5) Stabilization of films by a combination of surfactants (mixed films). In many cases it has been found that the use of a combination of surfactants gives slower drainage and improved foam stability through interfacial cohesion. There are several possible explanations for the enhanced stability, including: • A non-ionic surfactant causes a reduction in the critical micelle concentration of a solution of an anionic surfactant. • Although the anionic surfactant should not be too strongly absorbed, a lowering of the surface tension is expected to occur for the combination of a non-ionic and an anionic surfactant. • An increase in surface viscosity and drainage is expected to occur when a combination of surfactants is used. In many cases, a gelatinous surface layer is believed to be formed, which gives a low gas permeability.

Further reading Adamson, A.W. and Gast, A.P. (1997) Physical Chemistry of Surfaces, 6th edn. Wiley, New York. Anon. (1985) The Dairy Handbook. Alfa-Laval AB, Lund. Burt, D.J. and Thacker, D. (1981) Use of emulsifiers in short dough biscuits. Food Trade Rev 47: 344. CABATEC (1991) Dairy Ingredients in the Baking and Confectionery Industries, An audio-visual open learning module, Ref. C6. Biscuit, Cake, Chocolate and Confectionery Alliance, London. Friberg, S. E., Larsson, K. and Sjöblom, J. (2003) Food Emulsions. Marcel Dekker, New York. Hlynka, I. (ed.) (1964) Wheat Chemistry and Technology. American Association of Cereal Chemists. Holmberg, K. (ed.) (2002) Handbook of Applied Surface and Colloid Chemistry. Wiley, Chichester. Horn, J.D. (1970) Emulsifiers – A Practical Appraisal, British Chapter, A.S.B.E. Conference, November. Hutchinson, P.E. (1978) Emulsifiers in cookies: Yesterday, today and tomorrow. 53rd Annual Biscuit and Cracker Manufacturers Association Technologists’ Conference. Hutchinson, P.E. et al. (1977) Effect of emulsifiers on texture of cookies. J Food Sci 42: 2. Kattenerg, H.R. (De Zaan) Cocoa-powder for the dairy industry. US Patent 4.704.292. Kulp, K. (ed.) (1994) Cookie Chemistry and Technology. American Institute of Baking, Kansas. Marangoni, A.G. and Narine, S.S. (2004) Fat Crystal Networks. Marcel Dekker, New York. Narine, S.S. and Marangoni, A.G. (2002) Physical Properties of Lipids. Marcel Dekker, New York. Payens, T.A.J. (1979) Casein micelles: The colloid chemical approach. J Dairy Res 46: 291–306. Pocius, A.V. (2002) Adhesion and Adhesives Technology: An Introduction, 2nd edn. Hanser, Munich. Tamime, A.Y. and Robinson, R.K. (1999) Yoghurt: Science and Technology. CRC Press/Woodhead, Cambridge.

Part II

Physical operations

Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

Chapter 6

Comminution

Contents 6.1

Changes during size reduction 6.1.1 Comminution of non-cellular and cellular substances 6.1.2 Grinding and crushing 6.1.3 Dry and wet grinding 6.2 Rittinger’s ‘surface’ theory 6.3 Kick’s ‘volume’ theory 6.4 The third, or Bond, theory 6.5 Energy requirement for comminution 6.5.1 Work index 6.5.2 Differential equation for the energy requirement for comminution 6.6 Particle size distribution of ground products 6.6.1 Particle size 6.6.2 Screening 6.6.3 Sedimentation analysis 6.6.4 Electrical-sensing-zone method of particle size distribution determination (Coulter principle) 6.7 Particle size distributions 6.7.1 Rosin–Rammler (RR) distribution 6.7.2 Normal distribution (Gaussian distribution, N distribution) 6.7.3 Log-normal (LN) distribution (Kolmogorov distribution) 6.7.4 Gates–Gaudin–Schumann (GGS) distribution 6.8 Kinetics of grinding 6.9 Comminution by five-roll refiners 6.9.1 Effect of a five-roll refiner on particles 6.9.2 Volume and mass flow in a five-roll refiner 6.10 Grinding by a melangeur 6.11 Comminution by a stirred ball mill 6.11.1 Kinetics of comminution in a stirred ball mill 6.11.2 Power requirement of a stirred ball mill 6.11.3 Residence time distribution in a stirred ball mill Further reading Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

238 238 238 239 239 240 241 241 241 241 242 242 243 245 245 245 245 246 246 247 247 248 248 251 253 256 257 257 259 261

238

6.1 6.1.1

Confectionery and Chocolate Engineering: Principles and Applications

Changes during size reduction Comminution of non-cellular and cellular substances

The role of comminution in food engineering has a peculiarity which is closely connected to the cellular structure of the substances that are comminuted. Taking into consideration the hierarchical structure of the materials, there is an important difference between the properties of unbroken and broken cells. Greater or lesser amounts of substances flow out of cells as a result of comminution, making it possible for many chemical and physical changes to take place that were hindered by the original cellular structure. The degree of comminution (given, for example, by the particle size distribution) is a factor that determines the ratio of materials in unbroken and in broken cells. The materials that are comminuted in food production may be grouped as follows: • Non-cellular substances, for example – sucrose (sugar), which is ground as a powder itself and together with other ingredients (cocoa derivatives, milk powder etc.); – milk powder, which is ground together with other ingredients. • Cellular substances, for example cocoa nibs and various nuts (almonds, hazelnuts etc.). The degree of comminution has to be high (the particle size has to be very small) in the cases of cocoa mass and pastes of roasted almonds or hazelnuts. In other words, the ratio of free substance to bound substance must be high; for example, the entire content of cocoa butter should be free. However, if marzipan is being prepared, the almonds ground together with sugar must not be finely comminuted, to prevent the marzipan paste ‘leaking’ almond oil, i.e. the ratio of free to bound almond oil must be low.

6.1.2

Grinding and crushing

Size reduction, or comminution, is an operation in which the particle size of a bulk material is decreased, using various techniques and types of equipment. The changes that take place during size reduction can be grouped as follows: • Reduction of large, irregularly shaped solid particles to smaller sizes. • Creation of new free surfaces. • Changes in the number and size of the particles and in the surface area of the mass. (These changes are connected with changes in the bonding in the crystal lattice or, in general, changes in the structure of the material.) The first type of process seems self-evident and is related to the coarse stage, i.e. crushing. The second type was highlighted by the theory of Rittinger, which is more than 100 years old but is still useful today. This type of process leads to new contact surfaces, which are highly important in chemical reaction kinetics, and is characteristic of grinding. The changes of the third type, which characterize the very fine stage of grinding, are more complex: besides the phenomena of size reduction and surface area increase, agglomeration also occurs and possibly prevails.

Comminution

239

The materials used in the confectionery industry also have very different structures from the point of view of comminution, for example crystalline (e.g. sugars), cellular or amorphous (e.g. melted and cooled sugar) – every case has to be studied separately. Grinding, which is essential from the point of view of confectionery manufacture, is the fine phase of comminution. In the coarse stage, called crushing, particles with sizes of millimetres to centimetres are produced, and this operation works down to millimetre sizes. However, grinding produces particles of micron size, and its processes are more complicated.

6.1.3

Dry and wet grinding

The types of comminution can be distinguished according to whether or not the material to be ground is suspended in a continuous medium. If not, as for example in the case of the grinding of sugar, it is called dry grinding. The processing of cocoa into cocoa mass or chocolate provides a typical example of wet grinding: the continuous medium in which grinding takes place is cocoa butter or some other kind of vegetable fat. The roll refiners and pearl mills used for these purposes are continuously operated machines.

6.2

Rittinger’s ‘surface’ theory

Rittinger’s ‘surface’ theory, dating from 1867, deals with comminution by an imaginary process of slicing. The material, assumed to be homogeneous, in the form of a cube xl m in size, is sliced in the three principal directions by parallel planes with a spacing of x2 = x1/ν, producing ν 3 smaller cubes. The ratio ν = x1/x2 is called the ‘reduction ratio’. According to the principle of Rittinger’s theory, each individual slicing operation requires the same amount of energy, i.e. the energy requirement for comminution is proportional to the area of the newly created surfaces. We start with a cube of edge x1, which has an (initial) surface area 6x12. After slicing, we get a final surface area of 6τ 3 x22, the increase in surface area being equal to ΔS = 6ν 3 x22 − 6x12

(6.1)

According to Rittinger’s theory, the specific energy requirement w is the ratio of the energy requirement W (J) to the volume V (m3) of material comminuted: w=

W cΔS = V V

1⎞ ⎛ 6ν 3 x 2 6x 2 ⎞ ⎛ 1 w = c1⎜ 3 32 − 31 ⎟ = c ⎜ − ⎟ ⎝ x2 x1 ⎠ ⎝ ν x2 x1 ⎠

(6.2) (6.3)

After some algebraic transformation, ⎛ c⎞ w = ⎜ ⎟ (ν − 1) ⎝ x1 ⎠

(6.4)

240

Confectionery and Chocolate Engineering: Principles and Applications

Since in common cases ν >> 1, we can write ⎛ c⎞ w =⎜ ⎟ν ⎝ x1 ⎠

(6.5)

wR x2 = cR

(6.6)

i.e.

where c (J/m2) is a constant; its dimension is equal to that of the surface tension of a liquid. Equation (6.6) expresses the essence of the Rittinger theory: the reduction of the particle size and the specific energy requirement for the size reduction process are inversely proportional to each other. The Rittinger formula calculates only the effect of the breaking of molecular bonds and neglects the work of elastic deformation preceding the fracture.

6.3

Kick’s ‘volume’ theory

Kick’s ‘volume’ theory, dating from 1885, takes into consideration also the work of elastic deformation. Accordingly, the infinitesimal energy requirement for the fracture of a cube of size x m is dW = x 2σ d ( xλ )

(6.7)

where σ (Pa) is the fracture stress, x2σ (N) is the force of fracture, and λ (m/m) = σ/E is the specific deformation (assuming the validity of Hooke’s law; E (Pa) is Young’s modulus). From Eqn (6.7), we obtain dW = x 2σ (σ E ) dx

(6.8)

After integrating Eqn (6.7) between the initial and final states (characterized by x1 and x2, respectively), we find that the energy requirement is W=

( x13 − x23 ) σ 2 3E

=

x23(ν 3 − 1) σ 2 x23ν 3σ 2 x13σ 2 ≈ = 3E 3E 3E

(6.9)

i.e. w=

W σ2 = x23 3E

(6.10)

The conspicuous defect of Eqn (6.10) is that it does not include the reduction ratio, i.e. the energy requirement for comminution according to Eqn (6.10) is independent of the size reduction; it depends exclusively on the volume x13 of the material to be fractured.

Comminution

6.4

241

The third, or Bond, theory

The so-called third theory, also known as the Bond theory, aims to solve this contradiction by calculating the energy requirement of the first fracture according to the Kick (volume) theory but the energy requirement of subsequent fractures according to the Rittinger (surface) theory. The specific energy requirement for comminution according to the Bond theory is 1 ⎞ ⎛ 1 wB = cB ⎜ − ⎟ ⎝ x2 x1 ⎠

(6.11)

or, expressed in terms of the reduction ratio, ⎛ c ⎞ wB = ⎜ B ⎟ ⎝ x1 ⎠

(

)

c ν −1 ≈ ⎛ B ⎞ ν ⎝ x⎠

(6.12)

Equation (6.12) is empirical, but Bond demonstrated its suitability for practical calculations by the results of very many tests. This can be considered to be a result of the fact that the Kick theory has been proved suitable for crushing (large particles, the first step of comminution), and the Rittinger theory has been proved suitable for grinding (small particles, the later steps of comminution).

6.5 6.5.1

Energy requirement for comminution Work index

A ‘work index’ can be defined by means of Eqn (6.12). The total energy Wi required to comminute 1 short ton (= 907.2 kg) of material to a particle size of 100 μm is the ‘work index’, which can be calculated from test results for a size reduction from x1 to x2: Wi 1 100 = W 1 x2 − 1 x1

(6.13)

If the work index is determined for a material, then the energy requirement W for any comminution of that material from xp to xr can be calculated: ⎛ 1 1 ⎞ W = 10Wi ⎜ − ⎝ xr xp ⎟⎠

(6.14)

However, work indices have not yet been determined for the materials used in confectionery manufacture.

6.5.2

Differential equation for the energy requirement for comminution

All of these three theories of comminution can easily be formulated as a single differential equation,

242

Confectionery and Chocolate Engineering: Principles and Applications

Table 6.1 Material behaviour with respect to hardness. a: very soft

b: soft

c: medium-hard

d: hard

A: fibrous B: elastic C: plastic D: tough E: brittle

dW c =− n dx x

(6.15)

where W is the energy requirement for comminution; x is the particle size, considered as the independent variable characterizing the comminution process; and c is a constant, the dimension of which is dependent upon the value of n. n depends on the theory: if n = 1, the equation represents the Kick theory; if n = 2, the Rittinger theory; and if n = 1.5, the Bond (third) theory. Tarján (1981, p. 258) provided a set of codes of matrix type to indicate material behaviour, as shown in Table 6.1. According to this classification, fresh fruit and sugar beet are classified as Aa, sugar cane as Ab and sugar as Eb.

6.6 6.6.1

Particle size distribution of ground products Particle size

In Chapter 5, the particle size was used as a concept without any definition, even though homogeneity of size cannot be assumed in bulk grinding, i.e. a set of particle sizes is to be studied in reality. For this reason, the particle size distribution is the proper tool for studying ground products. The particle size concerns a single particle; the particle size distribution shows how the sizes of the particles are distributed in a bulk sample of particles. Consequently, the particle size distribution concerns a given bulk sample. A single particle is usually an irregular geometric body with several different possible measures, and behind these measures one can imagine various techniques of determination. The size of a particle is defined by the method of determination of the particle size. In general, the masses of particles larger than about 40–60 μm can be determined by screening; below this range, determination is generally done by sedimentation, microscopy or other techniques. Various recording apparatuses have been developed to eliminate manual work; these use, for example, gravimetric registration of settling, observation of the change of transparency of suspensions (turbidimetry), microscopic particle counters with digital marking, electrical-resistance changes and light scattering in laser beams. A description of these methods of particle size determination is beyond the scope of this book. Allen (1981, p. 104) has given a detailed overview of these methods with many definitions of particle size. Let us consider the most important methods used in the confectionery industry.

Comminution

6.6.2

243

Screening

From a theoretical and practical point of view, screening plays an essential role in comminution because the basic concepts of comminution are defined by reference to screening. If a particle falls on a screen, there can be two outcomes of this event: the particle passes or it does not. This decision of a given testing sieve divides the bulk sample into two parts: (1) The passing part is labelled D; if the size of the openings of the sieve is d (m) and the size of a tested particle is x (m), and if the relationship x≤d

(6.16)

is valid, then that particle belongs to the passing part. (2) The retained part, retained on the sieve, is labelled R (‘residue’); it can be characterized by the relationship x>d

(6.17)

The screening method makes decisions of this kind with a series of sieves; if n sieves are used, n points are obtained for both D and R. The usual form in which D(d) is presented is as a ratio (%) with respect to the whole sample as a function of the size of the sieve. Data on the testing sieves of particular interest to the confectionery industry are given in Table 6.2. IOCCC (1990a) recommends a water-sieving method for the determination of the sieve residue of cocoa powder and cocoa mass, which uses plate sieves with square apertures (holes) of size 75 μm × 75 μm or 125 μm × 125 μm. In practice, continuous functions R and D that approximate the real conditions to a greater or lesser extent are used. These functions can be differentiated. So, for example, D′(x) gives the frequency of particles of size x in the sample. The maximum of the frequency is regarded as a characteristic value of the sample: particles of that size are the most frequent in the sample. Table 6.3 shows data for the particle size distribution of a chocolate mass, obtained by the Coulter counter method (see Section 6.6.4). It can be

Table 6.2 Testing-sieve data. US sieve number 100 120 140 170 200 230 270 325 400

Tyler sieve number 100 115 150 170 200 250 270 325 400

Opening [mm (inches)] 0.149 0.125 0.105 0.088 0.074 0.063 0.053 0.044 0.037

(0.0059) (0.0049) (0.0041) (0.0035) (0.0029) (0.0025) (0.0021) (0.0017) (0.0015)

244

Confectionery and Chocolate Engineering: Principles and Applications

Table 6.3 Particle size distribution of a chocolate mass measured by the Coulter counter method. x (μm) 0.9 1.1 1.3 1.5 1.8 2.2 2.6 3.1 3.7 4.3 5 6 7.5 9 10.5 12.5 15 18 21 25 30 36 43 51

Cumulative (%)

Differential (%)

3.52 5.69 8.09 10.57 14.73 20.29 25.81 32.42 39.66 46.03 52.37 59.75 68.02 74.08 78.72 83.5 88.02 92 94.8 97.18 98.74 99.54 99.87 100

3.52 2.17 2.4 2.48 4.16 5.56 5.52 6.61 7.24 6.37 6.34 7.38 8.27 6.06 4.64 4.78 4.52 3.98 2.8 2.38 1.56 0.8 0.33 0.13

10

× 10%

8 6 4 2 0 0

10

20

30

Size (μm) Fig. 6.1

Particle size distribution of chocolate (differential curve).

seen from the first and second columns of the table that the (cumulative) percentage of particles less than 25 μm in size is 97.18%, i.e. D(x ≤ 25) = 97.18%. Figure 6.1 shows the differential curve, and Fig. 6.2 the cumulative curve. The maximum of the frequency in Fig. 6.1 occurs at about 13 μm.

Comminution

245

100

%

80 60 40 20 0 0

20

40

60

Size (μm) Fig. 6.2

Particle size distribution of chocolate (cumulative curve).

6.6.3

Sedimentation analysis

The other classical method is the sedimentation method, which is based on Stokes’ law; this is valid in the laminar region (Re = duρ/η < 4, where d = particle size, u = velocity of sedimentation, ρ = density of medium and η = dynamic viscosity of medium). For further details, see Section 5.9.3. To accelerate the speed of sedimentation in the size range below 10 μm, sedimentometers functioning in a centrifugal field have been developed (Németh and Horányi 1970).

6.6.4

Electrical-sensing-zone method of particle size distribution determination (Coulter method)

The Coulter technique is a method of determining the number and size of particles suspended in an electrolyte by causing them to pass through a small orifice, on either side of which there is an immersed electrode. The changes in resistance as particles pass through the orifice generate voltage pulses whose amplitudes are proportional to the volumes of the particles. The pulses are amplified, sized and counted, and the size distribution of the suspended phase may be determined from the data derived. The technique was originally applied to blood cell counting, and then to counting of bacterial cells and the measurement of cell volume distributions as well as number counting. Modified instruments were soon developed to size and count particles. An excellent description of the methods used in the confectionery industry to determine particle size distributions can be found in, for example, Minifie (1999, pp. 825–843). For further details, see Allen (1981).

6.7 6.7.1

Particle size distributions Rosin–Rammler (RR) distribution

The distribution function most often used in Europe is that of Rosin and Rammler. In the transcription given by Bennett, this is

246

Confectionery and Chocolate Engineering: Principles and Applications

⎡ ⎛ d ⎞n⎤ 1 − D ( d ) = R ( d ) = (100% ) exp ⎢ − ⎜ ⎟ ⎥ ⎣ ⎝ d0 ⎠ ⎦

(6.18)

where d is the particle size, D(d) is the cumulative passing function, R(d) is the cumulative residue function, d0 is the mode and n is the uniformity coefficient. If d = d0, then R (%) = 100/e, i.e. the particle size belongs to the residue in 100%/e = 36.8% of the sample (and D(p) = 63.2%), where e = 2.718, the Euler constant and base of natural logarithms. The standard deviation σ(RR) of the RR distribution is inversely proportional to the uniformity coefficient: π ⎞ ⎛ 1 ⎞ 1.282 σ ( RR ) = ⎛ ≈ ⎝ 6 ⎠ ⎝ n⎠ n

(6.19)

Equation (6.19) explains the meaning of the uniformity coefficient n.

6.7.2

Normal distribution (Gaussian distribution, N distribution)

The normal distribution does not play an important role in the context of grinding, since this distribution characterizes mainly natural processes, for example growth, crystallization and sublimation. However, it is mentioned here for the sake of completeness.

6.7.3

Log-normal (LN) distribution (Kolmogorov distribution)

This distribution is frequently used in the context of grinding. Kolmogorov (1937) demonstrated that particle sizes produced by crushing obey the LN distribution law; see Gnedenko (1988, p. 193). If the variable used to characterize the particle size d is x = log d

(6.20)

where ‘log’ means the logarithmic function to base 10, and the logarithmic standard deviation is

σ log = log d2 − log d1

(6.21)

then the equation for the normal distribution can be applied unchanged as follows: D ( d ) = Φ ( log d )

(6.22)

where Φ is the error integral (i.e. the Gaussian function). The frequency function of the LN distribution is ⎧ log ( d d0 log )2 ⎫ D ′( d ) = (5.772d0 logσ log ) exp ⎨ − ⎬ 2 2σ log ⎩ ⎭

(6.23)

Whereas the mean value d0 in the normal distribution is an algebraic mean, it is a geometric mean (d0 log) in the LN distribution.

Comminution

6.7.4

247

Gates–Gaudin–Schumann (GGS) distribution

This is used mostly in the USA. It can be expressed as ⎛d⎞ D (d ) = ⎜ ⎟ ⎝ d0 ⎠

m

(6.24)

or ⎛d⎞ R (d ) = 1 − ⎜ ⎟ ⎝ d0 ⎠

m

(6.25)

where d0 is the characteristic size of the particles and m is a parameter of homogeneity. For further details, see Beke (1981), Allen (1981) and Tarján (1981).

6.8

Kinetics of grinding

The decrease with time t of the weight percentage s of the oversize fraction coarser than a limiting particle size x is given to a first approximation by the following differential equation (Tarján 1981): ds = −cs, dt

(6.26)

where c (s−1) is a constant. Equation (6.26) states that the rate of grinding varies with the amount of oversize fraction remaining. The solution to Eqn (6.26) is s = exp ( −ct ) , s0

(6.27)

where s0 is the oversize fraction at the instant t = 0. If the grindability of the material is not constant during the entire grinding process (e.g. if it decreases owing to the gradual elimination of defects in crystals), then the equation ds = −csz dt

(6.28)

holds, where the relative grindability z is a function of the decrease of s (Razumov 1968). Experiments have shown the actual process to be well represented by the function s = exp ( − Kt n ) s0

(6.29)

which is analogous in structure to the Rosin–Rammler (RR) function (Eqn 6.18). (The units of K are s−n.)

248

Confectionery and Chocolate Engineering: Principles and Applications

The analogy to the RR function means that plotting log s against t on semi-logarithmic graph paper (as in the case of log R vs d) gives a straight line. Differentiation with respect to time of the expression for the oversize fraction s (Eqn 6.29) yields the equation ds = − snKt n −1 dt

(6.30)

This implies that for n = 1 the rate of grinding is exponential, i.e. it follows the first-order kinetics of chemical reactions. The grinding rate decreases if n < 1 and increases if n > 1, compared with first-order kinetics. The constants K and n measured for different materials are contravariant: a high value of K (easy grindability) entails a low value of n. The parameter K is a sensitive function of the grain size x. For small sizes, K is proportional to xm, and m is the slope of the Rosin–Rammler distribution. The variation of n with x is slight for ball mills; 0.7 ≤ n ≤ 1.3.

6.9

Comminution by five-roll refiners

Five-roll (and also three-roll) refiners are traditionally used in the fine grinding of chocolate, but they are used also in the manufacture of marzipan, compounds and various filling masses.

6.9.1

Effect of a five-roll refiner on particles

The effect is double: • comminution by stresses caused by the radial forces exerted by the rollers; and • comminution by shear caused by frictional forces, which are tangential to the matt surfaces of the rollers. The usual arrangement of the five rollers is shown in Fig. 6.3. During the refining process, there is very intense heat production; therefore cooling of the rollers by water (at c. 15–18°C) is necessary. A peculiarity of five-roll refining is that a special consistency of the product is necessary, which can be achieved with a fat content of about 25–27%. If the fat content is lower, the first roller pair ‘refuses’ the mass; if it is higher, the rollers merely ‘lick’ the mass and do not pick it up. 6.9.1.1

Angle of pulling-in

Let us consider Fig. 6.4 in order to study the quantitative relations in a roll refiner. The half-angle of pulling-in, ϕ, can be determined from the relationship cos ϕ =

AO D 2 + b 2 D + b = = MO D 2 + d 2 D + d

(6.31a)

i.e. D=

d cos ϕ − b 1 − cos ϕ

(6.31b)

Comminution

249

R5 g(4–5)

R4 g(3–4)

R3 g(2–3)

R2 R1 g(1–2)

g(1–2)

g(2–3)

g(3–4)

g(4–5)

Fig. 6.3 Five-roll refiner. R = roller [reproduced from Bertini (1996), by kind permission of Carle & Montanari SpA].

Particle

d/2 M D/2 j O

jj A b

Cylinders Fig. 6.4

Angle of pulling-in and distance of rubbing in a roll refiner.

250

Confectionery and Chocolate Engineering: Principles and Applications

where D is the cylinder diameter (m), b is the space between the cylinders (m) and d is the particle diameter (m). If ϕ is given, this determines the minimum value of the cylinder diameter. The usual values of ϕ for various states of the surfaces of the cylinders are: • lustrous, polished: 11°; • unpolished: 15°; • rough, unpolished: 17°. 6.9.1.2

Distance of rubbing

According to Eqn (6.31b), cylinders of larger diameter are necessary only when large particles are to be rolled. If the values of d and b are small, the minimum cylinder diameter D can also be small, as Example 6.1 shows. Example 6.1 Let d = 0.05 mm and b = 0.025 mm; cos 17° = 0.9563. Then, D=

( 0.05 × 0.9563 − 0.02 ) × 10 −3 1 − 0.9563

( mm ) ≈ 0.6365 … mm

This is obviously a very low value. The essential reason for using refiners with largediameter rollers in chocolate manufacture is to make the distance of rubbing, 2s, in these machines as large as is required, and this makes it possible to approach the necessary fineness of the particles in chocolate, which is an essential quality requirement. From Fig. 6.4, the half-distance of rubbing can be calculated: 2

D d D b 2 2 2 s 2 = ( AM ) = ( MO ) − ( AO ) = ⎛ + ⎞ − ⎛ + ⎞ ⎝ 2 2 ⎠ ⎝ 2 2⎠

2

(6.32)

and

( d 2 − b2 ) D s = ⎛ ⎞ (d − b) + ⎝ 2⎠ 4 Since D >> d and D >> b, we can write D s ≈ ⎛ ⎞ (d − b) ⎝ 2⎠

(6.33)

s∼ D

(6.34)

i.e.

Example 6.2 Let us calculate the distances of rubbing for two sizes of cylinders, D(1) = 0.35 m and D(2) = 0.45 m; in both cases, d = 0.030 mm and b = 0.020 mm.

Comminution

251

0.35 ⎞ s (1) = ⎛ ( 0.030 − 0.020 ) × 10 −6 ≈ 0.0418 mm ⎝ 2 ⎠ 0.45 ⎞ s (2 ) = ⎛ ( 0.030 − 0.020 ) × 10 −6 ≈ 0.0474 mm ⎝ 2 ⎠ Therefore, the diameter of the rollers in modern refiners is a minimum of 300 mm.

6.9.2

Volume and mass flow in a five-roll refiner

The calculation of the power requirement of a five-roll refiner seems to be an unsolved problem. However, a relationship between the revolution rates of the rollers can be determined on the basis of continuity if it is supposed that the volume (or mass) flow remains unchanged during the refining process; namely, that the moisture loss accompanying the work of the rollers is practically negligible. This is represented in Fig. 6.3; see Bertini (1996). The volume flow dV/dt between two rollers can be expressed as dV g (i − (i + 1)) πD (i + 1) L (i + 1) n (i + 1) ( m3 s ) = dt 60

(6.35)

where g(i–(i + 1)) is the gap (m) between rollers i and (i + 1), πD(i + 1) L(i + 1) is the surface area (m2) of roller (i + 1), and n(i + 1) is the revolution rate (min−1) of roller (i + 1). Evidently, for rollers 1 and 2, for example, the following relation is valid: g (1− 2 ) v ( 2 ) = g ( 2 −3) v (3) = g (3− 4 ) v ( 4 ) = g ( 4 −5) v (5)

(6.36)

where, for example, v(3) = D(3)n(3) since the lengths of the rollers are equal. Figure 6.5 shows the effect of an increase in the velocity of roller 1 on productivity. This figure shows the particle fineness and the yield as a function of the percentage

2320

Five-roll refiner, 1800 mm Milk chocolate(25.4% fat)

2030 1740

30

1450 1160

20

Yield (kg/h)

Particle size (μm)

40

870 580 10 0

20

40

60

80

100

Increase (%) in velocity of roller 1 Fig. 6.5 Particle size and yield as functions of the velocity increase of roller 1 [reproduced from Bertini (1996), by kind permission of Carle & Montanari SpA].

252

Confectionery and Chocolate Engineering: Principles and Applications

Type 2 35 Type 1

Particle size (μm)

30

25

Type 3

20 Upper lines: with pre-refining Lower lines: without pre-refining 15

10 500

1000

1500 2000 Yield (kg/h)

2500

3000

Fig. 6.6 Particle size versus productivity for a five-roll refiner with and without pre-refining [reproduced from Bertini (1996), by kind permission of Carle & Montanari SpA].

Table 6.4 Relationship between length of rollers and productivity for Carle & Montanari machines [reproduced from Bertini (1996), by kind permission of Carle & Montanari SpA). Type HF 513 HF 518 HF 525

Length of rollers (mm)

Productivity (kg/h)

1300 1800 2500

700–850 1000–1200 1400–1600

increase in the velocity of a type HFE five-roll refiner. Roller 2 remains in a fixed position, which roller 1 is drawn near to. If the input created by roller 1 becomes more intense, the productivity increases, but the particle fineness becomes poorer (Bertini 1996). The usual solution is to use pre-refining with a two-roll refiner; its effect is represented in Fig. 6.6, which shows results for five-roll refiners of three different types. As both Figs 6.5 and 6.6 show, linearity seems to be a good approximation. The relationship between the length of the rollers and the productivity is presented in Table 6.4, assuming that the characteristic particle size of the product is 20 μm. According to the manufacturer’s technical information, the specific energy requirement for modern machines of the kind described here varies between 30 and 70 kWh/t. Example 6.3 In a five-roll refiner, the parameters of the rollers are L = 1.8 m and D(5) = 0.4. The productivity is 1500 kg/h, and the specific weight of the refined product is 1200 kg/m3. Let us calculate the revolution rate of roller 5 if the gap between rollers 4 and 5 is 0.020 mm (= 2 × 105 m).

Comminution

253

Evidently, dm ρ dV ρ g ( 4 −5)D (5) πLn (5) = = dt dt 60 1500 kg h =

1500 kg = 0.42 kg s 3600 s

0.42 kg s = 1200 kg m3 × 2 × 10 −5 m × 0.4 m × 3.14 × 1.8 m ×

n (5 ) 60

n (5) = 464.43 min = 7.741 s If g(1–2) = 6 × 10−5 m, then n(2) = (464.43/min) × 2/6 = 154.81 min−1 = 2.58 s−1.

6.10

Grinding by a melangeur

The melangeur cannot be regarded as a true machine for comminution; it is actually a mixer, as suggested by its name. However, it is perhaps one of the most versatile machines in the confectionery industry and is almost indispensable in small plants because it can be used, although not very effectively, for mixing, comminution and conching. Let us consider the forces on a particle in a melangeur (Fig. 6.7). At the point A, a force F acts on the particle. This force can be analysed into a tangential force t and a radial force r, i.e. F= t+r

(6.37)

j

Rotating cylinder D/2

q ϕ p

r

a

Forces on a particle in a melangeur.

d

jm

F

Rotating table

Fig. 6.7

n r

t

Rotating table

A

Rotating cylinder

254

Confectionery and Chocolate Engineering: Principles and Applications

These forces are made up as follows: t = p+q

(6.38)

r= m+n

(6.39)

The forces q and m are compensated by a rotating table (or plate). However, the condition for comminution is p>n

(6.40)

Since p = t cos ϕ and n = r sin ϕ, this requirement can be expressed as t cos ϕ > r sin ϕ

(6.41)

t > tan ϕ r

(6.42)

i.e.

where ϕ is the angle of pulling-in; its usual value for a melangeur is about 25–30°. Since t = tan ρ r

(6.43)

where ρ is the angle of friction, it follows from Eqn (6.42) that tan ρ > tan ϕ , i.e. ρ > ϕ

(6.44)

On the basis of geometric considerations (see Fig. 6.7), the following equation for ϕ is valid: D−d = cos ϕ D+d

(6.45)

i.e. d=

D (1 − cos ϕ ) 1 + cos ϕ

(6.46)

where d is the size of the largest particle that can be pulled in by the melangeur. Example 6.4 If D = 0.6 m, ϕ = 30° and cos ϕ = 0.8660, then d = 0.6 × 0.0718 ≈ 0.04309 m = 43.09 mm.

Comminution

255

In the optimal case the ratio D/d is about 40, which determines the value of D if d is given. The rolling of the cylinders (rollers) is slip-free at their centre line, but at the edges of the rollers the slipping effect is strong, and this causes both mixing and some comminution (see Fig. 6.7). The peripheral velocity vm of the table at the centre line of a cylinder is a n vm ( m s ) = 2 π ⎛ r + ⎞ ⎝ 2 ⎠ 60

(6.47)

where n is the revolution rate (min−1) of the table, r is the distance (m) of the inner edge of the cylinder from the centre of the rotating table, and a is the length (m) of the surface of the cylinder. It can be assumed that there is no slip at the centre line, i.e. vm = peripheral velocity of cylinder

(6.48)

The peripheral velocity vi of the table at the inner edge of a cylinder is vi ( m s ) =

2 πrn 60

(6.49)

and the peripheral velocity ve of the table at the external edge of a cylinder is ve ( m s ) =

2π (r + a ) n 60

(6.50)

The maximum slip relative to vm is w = ve − vm = vm − vi =

ve − vi a n = π⎛ ⎞ ⎝ 2 ⎠ 60 2

(6.51)

If P is the pulling force (N) at the centre line of a cylinder, K is the weight (N) of a cylinder and μ is the friction coefficient, then the power balance (W) is W = Pvm = Kμw

(6.52)

i.e. P (vo + vi ) a n a n = P2π ⎛ r + ⎞ = Kμπ ⎛ ⎞ ⎝ ⎝ 2 ⎠ 60 2 2 ⎠ 60

(6.53)

and P=

Kμ a 2 ( 2r + a )

(6.54)

Since a melangeur has two cylinders, the power required to overcome friction for the two cylinders is

256

Confectionery and Chocolate Engineering: Principles and Applications

N = 2W = 2Pvm =

Kμπan 60

(6.55)

Example 6.5 If the surfaces of the rollers have a length a = 0.9 m, and n = 12 min−1, K = 3 × 104 N and μ = 0.35, then from Eqn (6.55), N=

Kμπan 12 = 3 × 10 4 × 0.35 × 3.14 × 0.9 × = 5.93 kW 60 60

It should be emphasized that N is the power requirement of friction only; the total power requirement of a melangeur contains additionally the power consumption for moving the machine. To determine n (the revolution rate of the table), we suppose that the frictional force mgμ on a particle must exceed the centrifugal force originating from the revolution of the cylinder: mgμ >

mvm2 ρ

(6.56)

where ρ = r + a/2 is the radius of the cylinder from the centre of the table at the centre line. The value of n is then obtained from a n gμρ > vm = 2 π ⎛ r + ⎞ ⎝ 2 ⎠ 60

(6.57)

i.e. 30 gμρ >n π (r + a 2 )

(6.58)

Example 6.6 The parameters of a melangeur are r = 0.3 m, ρ = 0.8 m, a = 0.9 m and μ = 0.35. Let us calculate the maximum revolution rate n (min−1) of the table. From Eqn (6.58) (using the fact that π ≈ g ), n < 30 ×

6.11

0.35 × 0.8 = 21.17 min −1 ≈ 1 3 s −1 0.3 + 0.45

Comminution by a stirred ball mill

Stirred (or agitated) ball (or pearl) mills became indispensable machines for comminution in cocoa processing and chocolate manufacture in the second half of the 20th century. From the beginning it was clear that stirred ball mills were suitable for refining cocoa mass (cocoa liquor) if the cocoa nibs were pre-ground. The question was whether they could refine milk chocolate, taking into consideration the heat sensitivity of milk protein

Comminution

257

and the particular properties of sucrose and lactose. Extensive research work has addressed these questions, examples of which are Anonymous (1971, 1981, 1995), Niediek (1973, 1978), Bauermeister (1978), Goryacheva et al. (1979), Shlamas et al. (1984), Lucisano et al. (2006) and Alamprese et al. (2007). For further details, see Hoepffner and Patat (1973), Hörner and Patat (1975), Kirchner and Aigner (1979), Freiermuth and Kirchner (1981, 1983), Bühler (1982), Kipke (1982), Stehr and Schwedes (1983), Rolf and Vongluekiet (1983), Kersting (1984), Kirchner and Leluschko (1986), Weit and Schwedes, (1986, 1988), Stehr (1989), Ulfik (1991) and Bunge and Schwedes (1992). A frequently cited review of the mathematical modelling of grinding kinetics was given by Austin and Bhatia (1971/1972).

6.11.1

Kinetics of comminution in a stirred ball mill

A simple kinetics of batch comminution in the micrometre and submicrometre range (1–40 μm) was presented by Strazisar and Runovc (1996) on the basis of experiments where barite, magnetite, dolomite and calcite in aqueous suspension were ground by a stirred ball mill, a vibratory mill and a planetary mill. These authors interpreted the results of the grinding experiments using a cumulative log-normal distribution Q (d ) = (

{

}

1 1 ) ∫ exp ⎡⎢ − ⎛ ⎞ x 2 ⎤⎥ dx ⎝ ⎣ 2⎠ ⎦ 2π

(6.59)

where d is the particle size, x = (1/s) ln (d/d50) is a variable and s = ln (d84/d50) is the standard deviation. The dependence of d50 on the grinding time can be approximated by an exponential equation, t d50(t ) = d50( ∞ ) + {d50( 0 ) − d50( ∞ )} exp ⎛ − ⎞ ⎝ τ⎠

(6.60)

where d50(t) is the median particle size at time t, d50(0) is the median particle size of the feed material, d50(∞) is the expected median particle size after a long grinding time and τ is the characteristic time. For further details, see Dück et al. (2003). Example 6.7 The particle size of the input cocoa mass into a stirred ball mill is d50(0) = 120 μm; after t = 60 s, the particle size of the output is d50(t) = 50 μm; and the probable value of d50(∞) is 50 μm. Let us calculate the value of the characteristic time τ of this stirred ball mill. From Eqn (6.60), 50 − 10 ⎞ −60 s ln ⎛ = −1.0116 = → τ = 59.312 s. ⎝ 120 − 10 ⎠ τ

6.11.2

Power requirement of a stirred ball mill

According to Stiess (1994, Vol. 2, pp. 307–309), the power requirement of a stirred ball mill can be expressed by means of the dimensionless Newton number

258

Confectionery and Chocolate Engineering: Principles and Applications

Ne = P ρS n3 d 5

(6.61)

where P is the power requirement (W), ρS is the density of the solid being comminuted (kg/m3), n is the rotation rate of the mixer (s−1) and d is the diameter of the mixer (m). Since stirred ball mills work in the turbulent region, Ne is constant. The effective energy requirement can be calculated using the formula EV =

P − P0 (dV dt ) cV

(6.62)

where EV is the energy requirement per unit volume (J/m3), P0 is the power requirement for free running (W), dV/dt is the volume flow of the suspension (m3/s) and cV is the volume concentration of solids in suspension. We may also write Em =

EV ρS

(6.63)

where ρS (kg/m3) is the density of the material being ground. In the turbulent region, Ne = ξ is independent of the Reynolds number and the viscosity of the fluid, and the usual value of Ne is approximately 0.1–5 (Stiess 1995, Vol. 1, p. 231). The value of Ne is more than unity if there are objects bumping in the mixing space. Weit and Schwedes (1986) gave a relationship between EV and the mean particle size x, x=

655 μm EV0.84 1

(6.64)

x=

223 μm EV0.84 2

(6.65)

or

where x (μm) is a weighted mean particle size, calculated from the volume distribution as x = ∫ xq(x) dx between the boundaries x(min) and x(max), and EV 1 (J/cm3) and EV 2 (kW h/ m3) are values of the volumetric energy requirement. Comment: Equations (6.64) and (6.65) hold (and are equivalent) over a rather broad range of parameters of the process (e.g. concentration of solid material, volume flow and particle size). Note that 1 kW h = 3.6 × 106 J, 1 m3 = 106 cm3

and

655 μm ≈ 223 μm. 3.60.84

Example 6.8 The power requirement of a pearl mill for refining cocoa mass is to be calculated for a mean particle size after milling of 65.5 μm; P0 is 60% of the total power requirement, dV/ dt = 0.72 m3/h = 2 × 10−4 m3/s and ρS = 1100 kg/m3.

Comminution

259

To calculate cV, we take the following data into account: the concentration of cocoa solids in cocoa liquor is 45 m/m% and its density ρS is 1100 kg/m3, and the density of cocoa butter ρB is 850 kg/m3. The volume (in litres) of 1 kg of cocoa liquor is V = 0.45/1.1 + 0.55/0.85 = 0.409 + 0.647 = 1.056 L. Consequently, 0.409 ( V V ) = 0.387 1.056

cV =

From Eqn (6.64), 65.5 μm =

655 μm → EV 1 = 101 0.84 = 12.6 × 106 J m3 EV0.84 1

From Eqn (6.62), EV =

P − P0

(dV dt ) cV

P (1 − 0.6 ) = 12.6 × 106 J m3 × (2 × 10 −4 m3 s ) × 0.387 = 0.97524 kW P=

0.97524 kW = 2.4381 kW 0.4

Em =

6.11.3

EV 12.6 × 106 J m3 = = 11.45 kJ kg . ρS 1100 kg m3

Residence time distribution in a stirred ball mill

In a continuously operated stirred ball mill, a two-phase mixture consisting of a dispersed solid and a continuous liquid phase is transported through a fixed cylinder equipped with a stirring device. This formal similarity to a ball mill used for batch processing leads us to choose an axial transport model for describing the transport through the ball mill. The following treatment follows the method of Stehr (1984). The differential equation for the model of axial dispersion, with normalized variables, is ∂C ∂C ⎛ 1 ⎞ ∂2C =− + ∂τ ∂X ⎝ Pe ⎠ ∂X 2

(6.66)

where C = c/c0 is the normalizevd concentration of a tracer, c is the tracer concentration at time t, c0 is the maximum concentration of the tracer, τ = t/tm is the normalized time variable, t is the time, tm is the mean residence time, X = x/L is the relative position in the mill, x is the distance from the input, L is the length of the grinding chamber, Pe = νmL/D is the Peclet number, νm is the mean axial transport velocity and D is the axial dispersion coefficient. The solid fraction of the feed suspension can itself be considered to be a tracer. The first term on the right-hand side of the differential equation represents convection and the second term represents diffusion, both related to the mean axial transport velocity.

260

Confectionery and Chocolate Engineering: Principles and Applications

To solve the differential equation, a close–close system is assumed, although the ideal boundary conditions of a close–close system are valid only if no dispersion occurs in the inlet and outlet regions. With regard to the substantial difference between the crosssectional areas of the pipes attached to the inlet and outlet and the grinding chamber, it can be assumed that the dispersion in the pipes, and hence at the boundaries of the system, is negligible. Consequently, the model of a close–close system, where both convection and dispersion occur, applies to the stirred ball mill. Molerus (1966) pointed out that only for the above conditions does the concentration cimp(t) measured at the outlet yield the residence time density distribution E(t) of an injected tracer of quantity q as follows: dV ⎞ cimp(t ) E (t ) = ⎛ ⎝ dt ⎠ q

(6.67)

where dV/dt is the flow rate. In addition, Molerus pointed out that the transport coefficients in the differential equation can be determined using the first and second moments of the residence time density distribution E(t):

{

ν L 2 ⎛ 2D ⎞ ⎛ D⎞ σ 2 = μ (2) − {μ (1) } = ⎜ 3 ⎟ l − ⎜ ⎟ ⎡⎢1 − exp ⎛ − m ⎞ ⎤⎥ ⎝ D ⎠⎦ ⎝ νm ⎠ ⎝ νm ⎠ ⎣

}

(6.68)

where σ 2 is the variance. The zeroth moment is the normalization; μ(1) = ∫ tE(t) dt = tm is the first moment, defining the mean value of E(t); and μ(2) = ∫ t2E(t)dt is the second moment. (All of these integrals are evaluated from 0 to ∞.) Hence, the mean axial transport velocity can be defined by

νm =

L L = (1) tm μ

(6.69)

From response data, E(t), μ(1) = tm, μ(2), σ 2 and D can be calculated by iteration from Eqn (6.68). For details, see Levenspiel (1972). Stehr (1984) determined a relationship between the stirrer parameters and the Peclet number, referring to the residence time distribution of single-phase flow in rotating-disc contactors: Pe =

νmL 6 = D 1 + 1.33 × 10 −3 dRS n ν m

(6.70)

where dRS is the diameter of the discs in the grinding chamber and n is the number of discs. In the investigations of Stehr (1984), the range of the dimensionless number dRSn/νm was 419–6534, i.e. the range of Pe was 3.854–0.619. Example 6.9 Let us calculate the residence time distribution in a rotating-disc contactor for the following parameters: L = 1.5 m, νm = 0.01 m/s, n = 14 and dRS = 0.4 m. From Eqn (6.69),

Comminution

ν m = 0.01 =

261

1.5 → μ (1) = 150 s μ (1)

From Eqn (6.70),

νmL 6 = D 1 + 1.33 × 10 −3 dRS n ν m 6 = ≈ 3.44 1 + 1.33 × 10 −3 × 0.4 × 14 0.01

Pe =

Pe =

1.5 νmL = 3.44 = 0.01 × → D = 4.36 × 10 −3 m 2 s . D D

From Eqn (6.68),

{

}

ν L D ⎡ 2 ⎛ 2D ⎞ σ 2 = μ (2) − {μ (1) } = ⎜ 3 ⎟ l − 1 − exp ⎛ − m ⎞ ⎤⎥ ⎝ D ⎠⎦ ⎝ νm ⎠ ν m ⎣⎢ ⎛ 2 × 4.36 × 10 −3 ⎞ =⎜ ⎟⎠ × {1 − 0.436 × [1 − exp ( −3.44 )] ⎝ 10 −6 = 8.72 × 10 −3{1 − 0.436 × (1 − 0.033)} = 3.68 × 10 −3 s2 6.11.3.1

Effect of comminution behaviour

Up to now, studies of stirred ball mills have concerned their use as stirrers. In order to determine the grinding effect of a stirred ball mill, it can be assumed that the result of comminution for a differential amount of material is a function of residence time only. Hence, the particle size distribution of the product obtained in the continuous mode is calculable, using the residence time distribution of the total amount of material. In addition, the results of batch-grinding experiments are required. In principle, the calculation can be accomplished using the equation Qcont (S ) = ∫ Qbatch (t; S ) × E (t ) dt

(6.71)

where Qcont is the cumulative mass percentage finer than the stated size S in continuous mode, and Qbatch is the corresponding quantity for the batch mode. Equation (6.71) expresses the essence of this calculation method: the cumulative mass percentage finer than a given particle size is constructed from the cumulative particle size distribution data for the batch mode and the residence time distribution. For details, see Stehr (1984), and for the modelling of comminution in a stirred ball mill, see Bernhardt et al. (1999) and Schwedes (2003).

Further reading Atiemo-Obeng, V.A., Penney, W.R. and Armenante, P. (2003) Solid–liquid mixing, in Handbook of Industrial Mixing: Science and Practice, ed. Paul, E., Atiemo-Obeng, V.A. and Kresta, S.M., pp. 543–584. Wiley Interscience.

262

Confectionery and Chocolate Engineering: Principles and Applications

Bauermeister (Probat Group). Technical brochures. Beckett, S.T. (ed.) (1988) Industrial Chocolate Manufacture and Use. Van Nostrand Reinhold, New York. Beckett, S.T. (2000) The Science of Chocolate. Royal Society of Chemistry, Cambridge. Dück, J., Schaaff, F. and Neesse, T. (2003) Modellierung der Mahlattrition in Rührwerkskugelmühlen. Aufbereitungs Technik 44 (3): 8–18. FrymaKoruma. Technical brochures. IOCCC Analytical Method 38 (1990, reprint 1997) Determination of the Sieve Residue of Cocoa Powder and of Cocoa Mass (Water Sieving Method). CAOBISCO, Brussels. Kempf, N.W. (1964) The Technology of Chocolate. Manufacturing Confectioner Publishing Co., Glen Rock, NJ. Minifie, B.W. (1999) Chocolate, Cocoa and Confectionery: Science and Technology, 3rd edn. Aspen Publications, Gaithersburg, MD. NETZSCH. Technical brochures. Nopens, I. and Biggs, C.A. (2005) Advances in Population Balance Modelling. Elsevier, Oxford. Posner, E.S. and Hibbs, A.N. (2005) Wheat Flour Milling, 2nd edn. American Association of Cereal Chemists, St Paul, MN. Pratt, C.D. (ed.) (1970) Twenty Years of Confectionery and Chocolate Progress. AVI Publishing, Westport, CT. Remény, K. (1974) The Theory of Grindability and the Comminution of Binary Mixtures. Akadémia Kiadó, Budapest. Wieland, H. (1972) Cocoa and Chocolate Processing. Noyes Data Corp., Park Ridge, NJ.

Chapter 7

Mixing/kneading

Contents 7.1 7.2 7.3 7.4 7.5 7.6

Technical solutions to the problem of mixing Power characteristics of a stirrer Mixing-time characteristics of a stirrer Representative shear rate and viscosity for mixing Calculation of the Reynolds number for mixing Mixing of powders 7.6.1 Degree of heterogeneity of a mixture 7.6.2 Scaling up of agitated centrifugal mixers 7.6.3 Mixing time for powders 7.6.4 Power consumption 7.7 Mixing of fluids of high viscosity 7.8 Effect of impeller speed on heat and mass transfer 7.8.1 Heat transfer 7.8.2 Mass transfer 7.9 Mixing by blade mixers 7.10 Mixing rolls 7.11 Mixing of two liquids Further reading

7.1

263 264 266 266 266 267 267 271 272 273 274 275 275 275 276 277 277 278

Technical solutions to the problem of mixing

Mixing is any process that increases the randomness of the distribution of two or more materials with different properties. In practice, mixing can take place between solids, between liquids, between solids and liquids, or between other combinations. In certain cases, gases, particularly air, can be incorporated, either intentionally or accidentally. Mixing can be obtained by any one of the following techniques: • The ingredients are placed in a vessel that is rotated or tumbled, subjecting the ingredients to a variety of motions. The blending of solids in double-cone, vee, tumbler and mushroom-type mixers provides good examples. • The ingredients are placed in a vessel in which an arm or an agitator stirs the mass. Examples are provided by stirred tanks for making liquid mixtures, kneading machines for mixing bread or cookie doughs, mixers for solids or pasty materials, and ribbon and screw mixers. Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

264

Confectionery and Chocolate Engineering: Principles and Applications

• The mixture of ingredients is pumped through an orifice, valve, nozzle or other static device, which causes mixing. Examples are provided by the homogenization of milk and the blending of pastes with ‘static’ mixers. In all cases, the operation can be carried out in discrete batches or continuously by feeding the separate (or partially pre-mixed) ingredients to the mixer and continuously withdrawing the mixed product. The mixing is caused by splitting of the flow at the beginning of each mixing element, by changes in flow direction caused by alternate right- and left-handed helices, and by acceleration and deceleration of a fluid as boundary layers are built up and destroyed at the beginning and end of each element.

7.2

Power characteristics of a stirrer

According to dimensional analysis, the power consumption P of mixing can be calculated using a function Ne = f(Re), where P = Ne ( Newton number ) ρ d 5 n3

(7.1)

d 2 n d 2 nρ = = Re ( Reynolds number for mixing ) ν η

(7.2)

Here ρ is the density of the fluid (kg/m3), d is the diameter of the mixing element (m), n is the characteristic velocity of mixing (s−1), ν is the kinematic viscosity of the fluid (m2/s) and η is the dynamic viscosity of the fluid (kg/m s). The determination of the function Ne = f(Re) is an experimental task. If Re < 20, then Ne × Re = constant. This is in the laminar flow region. If Re > 50 (for a vessel with baffles) or Re > 5 × 104 (for an unbaffled vessel), then Ne = constant. This is in the turbulent flow region. Stiess (1995, Vol. 1, pp. 228–231), gave a general characterization of the relationship between Ne and Re for mixing. Three regions can be distinguished: (1) In the laminar region (Re ≈ 10–50), Ne =

KI Re

(7.3)

where KI ≈ 50–150 depending on the type of mixer. (2) In the transitional region (Re ≈ 150–1000), Ne =

K II Re m

(7.4)

where 0 < m < 1. The boundaries of the transitional region and the value of KII vary to a great extent depending on the type of mixer.

Mixing/kneading

265

(3) In the turbulent region (Re > 1000 in general), Ne = K III = constant ≈ 0.1− 0.5

(7.5)

For a baffled mixer, Ne is higher than for an unbaffled one. The number Ne and the Euler number Eu are closely connected with each other (sometimes the Newton number is called the modified Euler number): Eu = pressure force/inertial force; Ne = drag force/centrifugal force. The influence of baffles is nil in the laminar flow region but extremely strong at Re > 5 × 104. The installation of baffles under otherwise unchanged operating conditions increases the stirrer power. Reher (1969) dealt in detail with the power requirements of mixing for blade, turbine and spiral impellers, taking into account the effect of the geometric conditions as well. His results can be summarized by the formula v

H D Ne = CRe u ⎛ ⎞ ⎛ ⎞ ⎝ D⎠ ⎝ d ⎠

w

(7.6)

where Ne = P/n3d 5ρ; Re = ρnd 2/ηrep (where ηrep is the representative viscosity; see Section 7.4); u, v and w are exponents; H is the height of the fluid level (m); D is the inner diameter of the tank (m); and d is the diameter of the impeller (m). For a blade impeller, C = 82.8; u = −1 if 2.3 × 10−5 < Re < 5; v = 0.19 if 0.7 < H/D < 1.25; w = 0.685 if 1.363 < D/d < 2.00. For a turbine impeller, C = 36.4; u = −1 if 1.4 × 10−5 < Re < 7; v = 0.43 if 0.9 < H/D < 1.3; w = 0.52 if 1.743 < D/d < 2.56. For a spiral impeller, C = 48.7; u = −1 if 1.2 × 10−4 < Re < 1; v = 0.59 if 0.7 < H/D < 1.25; w = 0.423 if 1.82 < D/d < 3.33. For further types of impeller, see Reher (1970).

266

7.3

Confectionery and Chocolate Engineering: Principles and Applications

Mixing-time characteristics of a stirrer

Zlokarnik (1991) obtained the following formula using dimensional analysis: nθ = f ( Re, Sc )

(7.7)

where θ is the mixing time (s), Sc = ν/D is the Schmidt number, nθ is the characteristic time of mixing (dimensionless) and the function f is to be determined experimentally.

7.4

Representative shear rate and viscosity for mixing

For ideal Newtonian fluids, the viscosity η is independent of the shear rate D; therefore, the calculation of the Reynolds number is simple (Re = dvρ/η) since the shear rate, which is a function of v, has no effect on η. A ‘representative viscosity’ ηrep is used if the viscosity η depends on the shear rate or stress. At the working point, the coordinates of the flow curve τ = f(η; D) are τw and Dw, and, by definition (Fig. 7.1),

ηrep =

τw Dw

(7.8)

For real Newtonian fluids, the Reynolds number is defined by Eqn (7.2) using the representative viscosity ηrep instead of η (Riquart 1975).

7.5

Calculation of the Reynolds number for mixing

For the mixing of ideal Newtonian fluids, the Reynolds number can be calculated from Eqn (7.2). For Ostwald–de Waele fluids, the corresponding flow curve is

τ = KD m

(7.9)

where K is a constant (kg m−1 s−(2+m)) and m is the flow index. The representative viscosity at the working point is

Shear stress t

t

Flow curve

Working point tw α Dw

Fig. 7.1

Definition of representative viscosity.

Shear rate

D

267

Mixing/kneading

ηrep =

τ KD m = = KD m −1 D D

(7.10)

The value of D (the shear rate) for mixing is calculated using the formula D = ks n

(7.11)

where n is the revolution rate and ks is a constant value that is characteristic of the type of impeller used. For the leaf impeller, disc impeller and propeller mixer, ks = 11

(7.12)

in the case of viscoelastic fluids, and ks =

22 s 2 s2 − 1

(7.13)

in the case of dilatant fluids, where s = dt/dm, dt is the inner diameter of the tank and dm is the width of the impeller. For the anchor impeller, ks = 9.5 +

9s 2 s2 − l

(7.14)

For the spiral impeller, ks = 4π

(7.15)

In the case of viscoelastic fluids, 0.512 ≤ m (Reher 1970).

Example 7.1 Let us calculate the value of Re for a viscoelastic fluid mixed using a spiral impeller. According to Eqns (7.11) and (7.15), D = ksn = 4πn. According to Eqn (7.10), ηrep = KDm − 1 = K(4πn)m − 1. 1− m According to Eqn (7.2), Re = nd m2 ρ ηrep = nd m2 ρ ( 4 πn ) K.

7.6 7.6.1

Mixing of powders Degree of heterogeneity of a mixture

A peculiarity of the mixing of powders is that it can be more complicated to achieve the required homogeneity of the mixture than is the case for the mixing of gases or liquids with each other or the mixing of liquids with solids. Therefore, the degree of homogeneity and the rate of mixing are important technological parameters in the mixing of powders.

268

Confectionery and Chocolate Engineering: Principles and Applications

Among the most objective procedures for estimating the degree of homogeneity, and among the simplest, are those of Hixson and Tenney (1935) and Coulson and Maitra (1950), which consist of collecting a given number of samples from time to time, and determining by inspection the number or fraction of each set of samples that ‘appears’ to be homogeneous. Very often, however, it is preferable to analyse random samples of a predetermined constant mass. Whatever the procedure used, the degree of heterogeneity is expressed conveniently by either the variance or the standard deviation. A thorough review of the topic of the mixing of solids has been given by Muzzio et al. (2003). 7.6.1.1

Homogeneity – effectiveness of mixing

The effectiveness of mixing can be expressed by the duration of mixing necessary to reach a given homogeneity. The usual degrees of homogeneity are 75%, 90% and 95%, where, for example, 75% means that the difference from the final value of the concentration, etc. is ±25% (and 90% and 95% mean differences of ±10% and ±5%, respectively); τ75 means the time (s) taken for a homogeneity of 75% to be reached. According to Hoogendoorn and Den Hartog (1967), the product nτ75 definitely decreases if the Reynolds number is increased. Also, the use of a leading tube strongly improves the effectiveness of mixing, and it holds in general that nτ 75 ≈ 140

(7.16)

where n is the rotation rate. The true variance is obtained from the expression

σ 2 = ( N − 1)−1∑ ( x − xm )2

(7.17)

where xm is the true value of the mean concentration of a constituent, and x is each of the values obtained by analysing each of the N samples. If the true value of the mean concentration is not known, the experimental variance is used, i.e. s 2 = ( N − 1)

−1

∑ ( x − xe )

2

(7.18)

where xe is the arithmetic mean of the results obtained. An evident measure of homogeneity is M = 1−

σ xm

or M = 1 −

s xe

(7.19)

If M = 0.50, the mixture is not yet homogeneous. If V = s/xe = 0.95, the homogeneity may be sufficient for a given purpose. Here V is the variance coefficient, i.e. M +V = 1

(7.20)

Precautions must be taken that the samples are collected completely at random; during sampling, one must also endeavour not to enhance the mixing, which would distort the results obtained for subsequent samples.

Mixing/kneading

269

It is important to note that there is a fundamental difference between a uniform distribution, corresponding to a theoretical variance of zero, and a random distribution, for which the variance tends towards a final limiting value, designated by σ r2 , which can always be determined experimentally. For liquids or gases, where mixing takes place at a molecular level, a random distribution leads very nearly to a uniform distribution for any sample of the size that is normally encountered in analysis. In summary, the quality of mixing can be determined by: • estimating the proportion of samples that ‘appear’ homogeneous; • determining the variance or standard deviation by analysis of samples of fixed size; or • determining the mass of the sample which must be taken in order to have a chosen standard deviation. In addition, the degree of heterogeneity can be estimated by other procedures, such as by determination of the amount of contact area between phases, or determination of the droplet size in the case of emulsions. For further details, see Hiby (1979) and Söderman and Laine (1990).

7.6.1.2

Range prescription

The simplest way to characterize the homogeneity of a component i is to prescribe a range within which the concentration xi of this component will be found when the mixing is stopped, i.e. xi min ≤ xi , j ≤ xi max

(7.21)

If n samples (labelled by j = 1, … , n) are taken at different points in time, with concentrations xi,j, all n samples have to meet this requirement. 7.6.1.3

Prescription of ratio of components

The standard deviation itself cannot characterize the ratio p/q of two components (of concentrations p and q). If we take n samples at different points in time (labelled by j = 1, 2, …), the ratios of them will be pi/qi where i = 1, 2, … , n, and the mean value Xj of these ratios is given by 1 ⎛p⎞ X j = ⎛ ⎞ ∑ ⎜ i ⎟ , i = 1, 2, … , n and j = 1, 2, … ⎝ n ⎠ ⎝ qi ⎠ j

(7.22)

where j is the serial number the sample. If the quality requirement is that the difference between the planned value of p/q and Xj cannot be more than w%, then mixing has to be carried on until 1−

Xj < w 100 p q

where p/q is the prescription in the recipe (the planned value).

(7.23)

270 7.6.1.4

Confectionery and Chocolate Engineering: Principles and Applications

Rate of mixing

The driving force for mixing is the degree of heterogeneity, σ 2 − σ r2. The rate of mixing depends on this driving force and on a rate coefficient k, which is characteristic of the equipment and the material being mixed. An equation proposed by Oyama (see Weidenbaum 1958) is dσ 2 = − k (σ 2 − σ r2 ) dt

(7.24)

or, integrated, ⎛ σ 2 − σ r2 ⎞ ln ⎜ 02 = kt ⎝ σ − σ r2 ⎟⎠

(7.25)

where σ 02 is the initial variance. By definition, the rate coefficient k (s−1) is constant for a given product in a given apparatus. It must be measured experimentally, for example as a function of the rate of rotation. 7.6.1.5

Separation during mixing of powders

The results of mixing can sometimes be influenced by factors that are difficult to foresee. For example, during the mixing of certain powders, stratification can take place, where the larger particles come to the surface or separation of particles of different density occurs if the mixing is prolonged. In certain cases, especially for pastes, the addition of very small quantities of additives such as surface-active agents can radically modify the rate coefficient. There is a simple model that takes separation during mixing into consideration. The differential equation in this model is dM = A (1 − M ) − BΦ dt

(7.26)

where M is the measure of homogeneity according to Eqn (7.19), Φ is the potential of separation (≤1, possibly negative; see Eqn (7.29)), A is the coefficient of mixing and B is the coefficient of separation. The following relation applies: M = 1−Φ2

(7.27)

If the heavier component is the upper one (Φ is positive),

{

At M = 1 − (1 − r ) exp ⎛ − ⎞ + r ⎝ 2 ⎠

}

2

(7.28)

If the heavier component is the lower one (Φ is negative),

{

At M = 1 − r − (1 + r ) exp ⎛ − ⎞ ⎝ 2 ⎠

}

2

(7.29)

Mixing/kneading

271

where t is the duration of mixing (s) and r = A/B. Equations (7.28) and (7.29) are the integrals of Eqn (7.26). The actual circumstances in mixing processes are slightly different, and, according to Rose (1959a,b), this can be taken into account by a internal degree of efficiency χ, which characterizes the equipment: if t → ∞, M → M ( equivalent ) = χ (1 − r 2 )

(7.30)

For further details, see Sommer (1975). Example 7.2 Let us calculate the velocity constant k of mixing supposing that Eqns (7.16) and (7.25) are valid for the mixing process used; in addition, σ0/xm = 0.8, σr/xm = 0.02, τ75 ↔ σ75/xm = 0.25, and n = 0.5 s−1: ⎛ σ 2 − σ r2 ⎞ ⎛ 0.82 − 0.02 2 ⎞ ln ⎜ 02 = ln ⎜ = 2.332 = kτ 75 2⎟ ⎝ σ − σr ⎠ ⎝ 0.252 − 0.02 2 ⎟⎠ From Eqn (7.16), nτ 75 ≈ 140, τ 75 ≈ 140 0.5 = 280 s → k = 2.332 280 s = 8.33 × 10 −3 s −1 In addition, we can calculate the value of τ95 according to Eqn (7.25): ⎛ σ 2 − σ r2 ⎞ ⎛ 0.82 − 0.02 2 ⎞ −3 ln ⎜ 02 ln = ⎜⎝ 0.052 − 0.02 2 ⎟⎠ = 5.719 = 8.33 × 10 × τ 95 → τ 95 = 686.6 s ⎝ σ − σ r2 ⎟⎠

7.6.2

Scaling up of agitated centrifugal mixers

It is nearly impossible to formulate generalized scaling-up equations for the mixing of solids. However, extensive experimental investigations conducted by Scheuber et al. (1980), Merz and Holzmüller (1981) and Müller (1982) have resulted in the following useful criteria. Two regions are demarcated by a Froude number of 3. The improvement in mixing coefficient for a given mixer at Fr > 3 is dramatic. The coefficient of mixing M proposed by Müller is a parameter used in his semiempirical one-dimensional model of horizontal mixers. This mixing coefficient determines how quickly concentration equalization will occur in a mixer. A large mixing coefficient will result in a short mixing time for a given quality of mix. The mixing coefficient is assumed to remain constant at all points in the mixer for the duration of the mixing process. It should be noted that M depends on the type of mixer, the geometry of the internal components and the operating conditions, but it does not depend on the properties of the components of the mixture (e.g. size or density). If Fr < 3, then

M = constant D2 n

(7.31)

If Fr > 3, then

M ≈ Fr 2 D2 n

(7.32)

272

Confectionery and Chocolate Engineering: Principles and Applications

where D (m) is the diameter of the mixer, n (rpm) is the revolution rate, and Fr, the Froude number, is defined as Fr =

v2 Rω 2 = gR g

(7.33)

where v (m/s) is the peripheral velocity of the mixing element (plough, paddle etc.), R (m) is the mixer radius (= D/2), ω (rad/s) is the angular velocity of the agitators and n = 30ω/π. Two common approaches are used for scaling these mixers, assuming geometric similarity and the same quality of mixing: (1) Keep the peripheral speed constant between the pilot mixer and the full-scale mixer, i.e. n ( pilot ) R ( full scale ) = n ( full scale ) R ( pilot )

(7.34)

(2) Keep the Froude number constant between the pilot mixer and the full-scale mixer, i.e. n ( pilot ) = n ( full scale )

R ( full scale ) R ( pilot )

(7.35)

Both of these approaches are used by mixer equipment manufacturers, and this suggests that more research and development are required to increase our understanding of the mixing processes of solids. It is common practice to use a Froude number of 7 for mixing non-friable materials. For friable materials, the effect of breakage caused by agitator impact must be evaluated. Attrition is nonlinear with impact velocity, whereas it is linear with mixing time. Therefore, an optimum can be found through experimentation.

7.6.3

Mixing time for powders

Rumpf and Müller (1962) have shown experimentally that the mixing coefficient can also be related to the mixer length L if the mixer diameter D is kept constant: Mt = constant L2

(7.36)

where M is the mixing coefficient, t (s) is the mixing time and L (m) is the mixer length. For Froude numbers below 3 and for geometrically similar mixers operating at the same peripheral speed of the agitator, the mixing time increases linearly with the mixer diameter: 2

L D t∼⎛ ⎞ ⎝ D⎠ v

(7.37)

Mixing/kneading

273

At higher Froude numbers (>3), the mixing time is linear with the volume (not the diameter) of the mixer. The effect of the agitator speed v is significant in this range: 2

L D3 t∼⎛ ⎞ 5 ⎝ D⎠ v

7.6.4

(7.38)

Power consumption

A relationship between power consumption and Froude number for agitated centrifugal mixers was given by Müller (1982). The power consumption is expressed in a non-dimensional form using the Newton number Ne: Ne =

P ρs(1 − ε ) D5 n3( L D )

(7.39)

where ρs is the density of the mixture (kg/m3), ε is the voidage of the packed bed and n (s−1) is the rotation rate. The Froude number Fr has the form Rω 2 g

Fr =

where R is the radius of the mixer, ω = 2πn is the angular velocity and g is the gravitational acceleration. For Fr < 1, where the acceleration forces are relatively small, and the material is not fluidized or under plastic shear, the following relationship holds: Ne ∼

1 Fr

(7.40)

At higher Froude numbers, the configuration of paddles/agitators (i.e. the roughness and shape) and the size of the particles have a significant influence on the shape of the curve of log Ne vs log Fr: the linear region in the plot becomes curved. Example 7.3 A mixture of sugar and cocoa powder is being homogenized in a vessel, where D = 1 m, L = 1 m, ε = 0.3, ρs = 1350 kg/m3 and n = 90 min−1. What is the approximate power requirement?

ω= Fr =

nπ = 3π, n = 1.5 s −1 30 Rω 2 9π 2 = 0.5 × ≈ 4.5 ( > 3) g 9.81

 Ne ≈ 4 ( Muller 1982 ) L P = 4 ρs(1 − ε )D5 n3⎛ ⎞ = 4 × 1350 × 0.7 × 1.53 = 12757.5 W ≈ 12.6 kW ⎝ D⎠

274

7.7

Confectionery and Chocolate Engineering: Principles and Applications

Mixing of fluids of high viscosity

According to Schmidt (1968), the flow pattern in the case of a propeller mixer changes with the Reynolds number. If the viscosity is low and the Reynolds number is high (Re > 104), a propeller mixer can work efficiently, since the mixer generates an axial movement of the liquid, and the flow pattern is determined by the mass forces. If the viscosity is higher and the Reynolds number is lower (103 < Re < 50), the flow rays generated by the propeller spread out in a radial direction, since the effect of the viscous forces is increased. At high viscosity and low Reynolds number (Re < 20, in the laminar flow region), the axial flow disappears entirely. Under these conditions the shape of the mixer has no importance, but the size of it is important because this determines the volume of flow moved. When a fluid of high viscosity is mixed, places can often be found in the tank where the flow is very slow or does not develop, because the energy dissipation caused by the viscous forces rapidly consumes the kinetic energy of the mixer within a short distance from the mixer. According to Schmidt (1968), the effective distance of a mixer can be expressed by the formula R ≈C

P η

(7.41)

where R is the effective distance of the mixer measured from the axle. In the case of high viscosity, mixers work in the laminar flow region, and the active region of mixing is decreased in size. Therefore, the geometric shape of the mixing region and the mixer must be tailored to a specific objective: the basis of achieving mixing efficiency is to increase the velocity difference between the parts that are unmoved and moved. In this case mixers have to work on the principle of volume displacement. For this reason, the mixer often fills almost the whole volume of the tank: this ensures that the effective distance of the mixer reaches every point of the tank. There are batch and continuous agitators with various structural shapes, and in both types there can be one or more mixers in the place where mixing is done. An important characteristic of the mixing fluids of high viscosity is that the power requirement per unit volume is higher than in the case of lower viscosity. Further references for the making of suspensions and dispersions are Brauer and Mewes (1973), Zielinski et al. (1974), Kale et al. (1974), Nagel and Kürten (1976), Staudinger and Moser (1976), Einenkel (1979), Kipke (1979, 1985, 1992), Mersmann and Grossmann (1980), Becker et al. (1981), Koglin et al. (1981), Herndl and Mershmann (1982), Kneuele (1983), Ebert (1983), Bertrand (1985), Zehner (1986), Geisler et al. (1988), Xanthopoulos and Stamatoudis (1988), Kraume and Zehner (1988, 1990), Pörtner and Werner (1989), Brauer et al. (1989), Markopoulos et al. (1990), Fleischli and Streiff (1990) and Pörtner et al. (1991). Further references for helical screw agitators and for static and continuous mixers are Chapman and Holland (1965), Henzler (1979), Riedel (1979), Pahl (1985), Gyenis (1992) and Sarghini and Masi (2008). Further references for the topic of the ‘just-suspended speed’ in stirred tanks are Zwietering (1958), Baldi et al. (1978), Volt and Mershmann (1985), Davies (1986), Latzen and Molerus (1987), Mak (1992), Atiemo-Obeng et al. (2003), Ibrahim and Nienow (1994) and Joosten et al. (1977).

Mixing/kneading

275

Further references for the residence time distribution are Schönemann and Hein (1993) and Schönemann et al. (1993).

7.8 7.8.1

Effect of impeller speed on heat and mass transfer Heat transfer

Detailed discussions with a rich list of references have been given by Gaddis and Vogelpohl (1991) and Sprehe et al. (1999) The principal relationship for heat transfer is ⎛η ⎞ Nu = C Re a Pr b ⎜ FL ⎟ ⎝ ηW ⎠

c

(7.42)

where Nu = αD/λ (Nusselt number), α is the heat transfer coefficient (W/m2 K), D is the inner diameter of the vessel (m), λ is the thermal conductivity of the fluid at the temperature in the centre (W/m K), Re = d 2nρ/ηFL (Reynolds number), d is the diameter of the impeller (m), ηFL is the dynamic viscosity of the fluid at the temperature in the centre (Pa s), n is the rotation rate (s−1), ρ is the density of the fluid at the temperature in the centre (kg/m3), Pr = νFL/a, νFL is the kinematic viscosity of the fluid at the temperature in the centre (m2/s), a is the thermal diffusivity of the fluid at the temperature in the centre (m2/s), ηW is the dynamic viscosity of the fluid at the temperature of the wall, and a, b and c are exponents depending on the conditions of mixing (construction of mixer, type of fluid etc.). The actual form of the relationship in Eqn (7.42) can be strongly influenced by the type of impeller and fluid. Further discussion is beyond the scope of this book. For further references, see Pawlowski and Zlokarnik (1972), Poggemann et al. (1979), Kahilainen et al. (1979), Schulz (1979), Yüce and Schlegel (1990) and Fingrhut (1991).

7.8.2

Mass transfer

The diffusional mass transfer rate is affected primarily by the impact of agitation on the hydrodynamic environment near the surfaces of the particles, in particular on the thickness of the diffusional boundary layer surrounding the solid particles. The hydrodynamic environment near a particle surface depends on the properties of the fluid and of the particles. In addition, the diffusivity DA also influences the diffusional mass transfer. In general, the specific impact of agitation must be determined experimentally for each system. The correlations discussed below are presented to provide a guide to and some insight into the expected effects of various variables on solid–liquid mass transfer. Referring to solid–liquid mass transfer, several correlations for the quantity kSL that appears in the Sherwood number have been reported in the literature. The following Froessling-type equation, developed by Nienow and Miles (1978), is based on the theory of the slip velocity between a liquid and a solid particle: Sh = 2 + 0.44 Re1 2 Sc 0.38

(7.43)

where Sh = kSLd/DA is the Sherwood number (where DA is in m2/s), d is the characteristic size of the particles (m), Re = ρLV(s)d/ηL is the Reynolds number, V(s) is the settling

276

Confectionery and Chocolate Engineering: Principles and Applications

velocity or slip velocity (m/s), ηL is the viscosity (Pa s) of the fluid, Sc = ηL/ρLDA = νL/DA is the Schmidt number and νL is the kinematic viscosity (m2/s) of the fluid. This has proven useful for estimating kSL and for establishing the effect of the properties of the solid and fluid and the effect of the agitation parameters. The Froessling correlation is not applicable to solid–liquid systems where the settling velocity or slip velocity is small, i.e. where V(s) << 0.0005 m/s. For further details, see Nienow (1975), Baldi (1978), Doriaswarmy and Sharma (1984) and Davies (1986).

7.9

Mixing by blade mixers

Blade mixers are used with various deformable or plastic solids and high-consistency pastes to achieve a kneading and mixing action accompanied by heating or cooling. They are used for mixing the components of chocolate, soft sugar confectioneries containing fondant mass, chewing/bubble gum, etc. The process involves compressing the fluid mass flat, folding it over on itself and then compressing it again. The material is usually torn apart, and high shear is produced between the moving and stationary fluid elements. The mixing is usually performed by two Z-shaped heavy blades rotating in opposite directions at different speeds on parallel horizontal shafts. The following formula was derived from dimensional analysis for highway mixers: ⎛γ ⎞ N = 150 × d 4.56 × n2.78 × ⎜ ⎟ ⎝ g⎠

0.78

× ηa0.22

(7.44)

where d is the diameter of the circle traced out by the blade (m), n is the rotational velocity of the blade (rot/s) and ηa is the apparent viscosity of the mixture (Pa s). Example 7.4 Let us calculate the approximate power consumption of a blade mixer with the parameters d = 0.5 m, n = 0.5 s−1 and ηa = 30 Pa s. We substitute these parameters into Eqn (7.44), and obtain N = 87.8 kW. For approximate calculations of power, Kharkhuta et al. (1968) recommended the following formulae: If Q < 1400 kg, then N = 0.035Q ( kW )

(7.45)

where Q is the mass of the mix (kg). If Q > 1400 kg, then N = 30 + 0.01Q ( kW )

(7.46)

277

Mixing/kneading

For details of the measurement of the performance of blade mixers, see Cheremisinoff (1988, pp. 788–790). Example 7.5 Let us calculate the approximate power consumption for a batch of size Q = 250 kg. According to Eqn (7.45), N = 8.75 kW.

7.10

Mixing rolls

Mixing rolls subject pastes and deformable solids to intense shear by passing them between smooth or corrugated metal rolls that revolve at different speeds. These machines are widely used in the cocoa, confectionery and biscuit industries. A typical area of application is in five-roll refiners, which mix and comminute chocolate paste at the same time. The material enters the mixing rolls in the form of lumps, powder or friable laminated material. As a result of rotation, adhesion and friction, the material is entrained into the gap between the rolls, and upon discharge it sticks to one of the rolls, depending on their temperature difference and velocities. The rolls are temperature controlled. The rolling process is also influenced by the gap between the rolls. Both the shearing action and the entrainment of material into the gap are very important in the mixing process and in transporting the material through the unit. The hydrodynamic theory of mixing rolls was originally developed by von Kármán (1925). For a detailed discussion of the mechanism governing the rolling process, see Cheremisinoff (1988), who cites work by Bernhard (1962), Soroka and Soroka (1965), Bekin and Nemytkov (1966) and Lukach et al. (1967). The power requirements of mixing rolls can be calculated on the basis of similarity theory, and the type of formula obtained is N = Kγω D a Lb hc f d B 3

(7.47)

where K is a constant, γ is the specific weight, ω is the angular velocity, D is the roll diameter, L is the length of the roll, h is the minimum gap between rolls (cm), f is the friction and B is the batch weight. Equation (7.47) was obtained in experiments on various types of plastic with the parameters h = 0.6–2.6 mm, v1 = 6.28–18 m/s, f = 1–3, L = 150– 1050 mm and D = 200–400 mm. The exponents obtained were a = 2 and 2.3, b = 0.6, c = 0.1, and d = –0.2. No data have been found by the author for the mixing rolls used in the confectionery industry. For further details, see Section 14.2.3.

7.11

Mixing of two liquids

This is a typical task in the manufacture of emulsions. This topic was discussed in Section 5.8.7. Holmes et al. (1964) determined a formula called the Holmes–Voncken–Dekker formula for baffled turbine-stirred machines, which are frequently used for producing emulsions: nτ ( d D ) ≈ constant 2

(7.48)

278

Confectionery and Chocolate Engineering: Principles and Applications

where τ is the mixing time (s) and n is the revolution rate (s−1). For geometrically similar machines, d/D = constant, i.e. nτ ≈ constant

(7.49)

It is worth mentioning the formal similarity of Eqns (7.16) and (7.49): the former refers to powders, the latter to emulsions.

Further reading Azzopardi, B. (ed.) (2000) Fluid Mixing. Special issue, Chem Eng Res Des 78 (A3). Baldyga, J. and Bourne, J.R. (1999) Turbulent Mixing and Chemical Reactions. Wiley, Chichester. CABATEC (1992) Biscuit Mixing. Audio-visual open learning module, Ref. S10. The Biscuit, Cake, Chocolate and Confectionery Alliance, London. Gassis, E.S. and Vogelpohl, A. (1991) Wärmeübergang in Rührbehältern. In: VDI-Wärmeatlas, 6th edn. Ma 1. Kempf, N.W. (1964) The Technology of Chocolate. Manufacturing Confectioner Publishing, Glen Rock, NJ. Levins, D.M. and Glastonbury, J. (1972) Application of Kolmogoroff’s theory to particle-liquid mass transfer in agitated vessels. Chem Eng Sci 27: 537–542. Lienhard, J.H., IV and Lienhard, J.H., V (2005) A Heat Transfer Textbook, 3rd edn. Phlogiston Press, Cambridge, MA. Manley, D.J.R. (1981) Dough mixing and its effect on biscuit forming. Cake and Biscuit Alliance Technologists’ Conference. Manley, D.J.R. (1998) Biscuit, Cookie and Cracker Manufacturing Manuals. Vol. 3, Biscuit Dough Piece Forming. Woodhead Publishing, Cambridge. Makins, A.H. (1974) The evolution of sheeters and laminators. Baking Ind J, Oct., pp. 28–29. Muller, W. (1982) Mixing of solids; methods and present state of design. Germ Chem Eng 5: 263–277. NETZSCH. Technical brochures. Portner, R., Langer, G. and Werner, U. (1991) Zur dimensionsanalytischen Beschreibung von Mischprozessen in geruhrten Newtonschen, strukturviskosen und viskoelastischen Flussigkeiten. Chemie-Ing Techn 63: 172. Povey, M.J.W. and Mason, T.J. (1998) Ultrasound in Food Processing. Blackie Academic & Professional, London. Pratt, C.D. (ed.) (1970) Twenty Years of Confectionery and Chocolate Progress. AVI Publishing, Westport, CT. Rohsenow, W.M., Hartnett, J.P. and Cho, Y.I. (eds) (1998) Handbook of Heat Transfer, 3rd edn. McGraw-Hill Handbooks. McGraw-Hill, New York. Schonemann, E., Mandel K.-M. and Hein, J. (1993) Das Mischzeitverhalten bei sehr geringen 1 Konzentrationen. Chemie-Ing Techn 65: 68. Sollich. Technical brochures. Tanguy, P.A., Bertrand, J. and Xuereb, C. (2005) Innovative studies in industrial mixing processes. Chem Eng Sci 60 (8–9): 2099. Wade, P. (1965) Investigation of the Mixing Process for Hard Sweet Biscuit Doughs. Part I, Comparison of Large and Small Scale Doughs. BBIRA Report 76. Wade, P. and Davis, R.I. (1964) Energy Requirement for the Mixing of Biscuit Doughs under Industrial Conditions. BBIRA Report 71. Werner & Pfleiderer. Technical brochures. Wieland, H. (1972) Cocoa and Chocolate Processing. Noyes Data Corporation, Park Ridge, NJ. Yianneskis, M. (ed.) (2006) Fluid Mixing, 8th International Conference. Special issue, Chem. Eng. Sci. 61(9).

Chapter 8

Solutions

Contents 8.1 Preparation of aqueous solutions of carbohydrates 8.1.1 Mass balance 8.1.2 Parameters characterizing carbohydrate solutions 8.2 Solubility of sucrose in water 8.2.1 Solubility number of sucrose 8.3 Aqueous solutions of sucrose and glucose syrup 8.3.1 Syrup ratio 8.4 Aqueous sucrose solutions containing invert sugar 8.5 Solubility of sucrose in the presence of starch syrup and invert sugar 8.6 Rate of dissolution Further reading

8.1 8.1.1

279 279 280 282 282 283 283 285 285 286 288

Preparation of aqueous solutions of carbohydrates Mass balance

The first step in the manufacture of sugar confectionery is the preparation of aqueous solutions of carbohydrates such as sucrose, starch syrup and invert syrup. It is useful to study the mass balance for the case of two components, Aa + Bb = x ( A + B )

(8.1)

where a is the concentration (m/m) of component A (mass A kg), b is the concentration (m/m) of component B (mass B kg) (m/m) and x is the resultant concentration of the mixture. For example, if sugar (A) is dissolved in water (B), then the sugar concentration of pure sugar is a = 1 and the (sugar) concentration of water is b = 0, i.e. A × 1 + B × 0 = x (A + B ) → x =

A A+ B

(8.2)

For a multicomponent mixture where the components are labelled by an index i,

∑ mi ci = x ∑ mi Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

(8.3) Ferenc Á. Mohos

280

Confectionery and Chocolate Engineering: Principles and Applications

Sometimes the equation

(a − x ) A = ( x − b) B

(8.4)

is more practical for calculations because it relates to differences in percentages. An important property of Eqns (8.1) and (8.4) is that a < x < b or a > x > b

(8.5)

i.e. the value of x is between those of a and b. The value of x can exceed the values of a and b because of evaporation. Therefore, a more general mass balance is Aa + Bb = x ( A + B − V )

(8.6)

where V is the mass of vapour (kg) that is extracted by evaporation from the mixture. Equation (8.6) makes a common treatment of the operations of dissolution and evaporation possible.

8.1.2

Parameters characterizing carbohydrate solutions

Various types of concentrations are used. (1) Mass ratio concentration: c = mass of dissolved material (kg ) 100 kg of solution

(8.7)

This is the most frequently used concentration; it is usually expressed as a mass ratio (m/m) or as a mass percentage (m/m%). The latter is also referred to as degrees Brix (see later). (2) The volume concentration v is used only rarely, but it is used for solutions containing alcohol: v = volume of dissolved material (L ) 100 L of solution

(8.8)

(3) Mixed concentration: C = mass of dissolved material (kg ) 100 L of solution

(8.9)

The use of this type of concentration is complicated because the density of the solution is dependent on its solid content. (4) The Raoult concentration or ‘molality’ (not ‘molarity’!) is defined as m = number of dissolved moles 1000 g of solvent

(8.10)

It is used in connection with the elevation of the boiling point (see Chapter 9) and the depression of the freezing point of solutions, although the latter plays hardly any role in the confectionery industry.

Solutions

281

(5) Degrees Baumé. The number Bé of degrees Baumé (symbol °Bé) is a kind of concentration, although it has a close connection to the density of the solution (denoted by d and expressed in g/cm3), measured at 20°C and related to the density of water at 4°C. For solutions heavier than water, i.e. if d > 1 g/cm3, then d ( g cm3 ) =

145 145 − Be´

(8.11)

For solutions lighter than water, i.e. if d < 1 g/cm3, then d ( g cm3 ) =

140 130 + Be´

(8.12)

See Examples 8.1 and 8.2 below. (6) Degrees Brix. The number Bx of degrees Brix (symbol °Bx) is a measurement of the mass ratio of dissolved sucrose to the mass of aqueous sugar solution; for example, a 25°Bx solution contains 25 g of sucrose per 100 g of solution. The number of degrees Brix can be approximated as Bx = 261.3 −

261.3 1 = 261.3 ⎛ 1 − ⎞ ⎝ d⎠ d

(8.13)

where d (> 1) is the density (g/cm3) of the sugar solution measured by a refractometer at 20°C. After some algebraic transformation, we obtain the following result from Eqns (8.11) and (8.13) for ‘heavy’ solutions: d=

261.3 145 = 261.3 − Bx 145 − Be´

i.e. Bx =

261.3 Be´ = 1.8021 Be´ 145

(8.14)

It should be emphasized that Eqn (8.14) is only an approximation, but it can sometimes be used for engineering purposes. The exact relations between degrees Brix and Baumé (at 68°F = 20°C) are given in Appendix 2 (Table A2.1); for a detailed scale (at intervals of 0.5°B), see Meiners et al. (1983, Vol. 1/I, p. 11).

Example 8.1 Sugar solutions are ‘heavy’. If Bé = 18°, then d=

145 = 1.1417 g cm 3 145 − 18

282

Confectionery and Chocolate Engineering: Principles and Applications

Example 8.2 If the concentration of an alcoholic solution (‘light’) is 18°Bé, then d=

8.2

140 = 0.8459 g cm 3 148

Solubility of sucrose in water

An essential point in relation to the preparation of sugar solutions is the solubility of sugar in water at given temperatures. Table A2.2 gives the solubility of sugar and the density of saturated sugar solutions as a function of temperature (Antokolskaia 1964). According to Junk and Pancoast (1973), the concentration of a saturated sucrose/water solution as a function of temperature (in the interval 0–100°C) can be approximated by the following formula (at atmospheric pressure): c = 64.397 + 0.07251t + 0.0020569t 2 − 0.000009035t3

(8.15a)

where c is the amount of dissolved sucrose (g) per 100 g of saturated solution and t is the temperature (°C) of the solution. Vavrinecz (1955a,b) proposed the following formula: c = 64.347 + 0.10236t + 0.001424t 2 − 0.000006020t 3

(8.15b)

Example 8.3 From Table A2.2, a saturated aqueous solution contains 260.4 g of sucrose per 100 g of water at 50°C. According to Eqn (8.15a), when t = 50°C, c = 74.294125 g/100 g saturated solution, i.e. 25.705875 g of water dissolves 74.294125 g of sucrose = 289.0161 g sucrose/100 g water. The difference shows the incorrectness of Eqn (8.15a): 289.0161/260.4 = 1.1099 … , i.e. an error of ≈ 11%! Under similar conditions, Eqn (8.15b) results in c = 73.9745 g of sucrose in (100 − 73.9745) = 26.0255 g of water, i.e. 284.2385 g sucrose/100 g water (instead of 260.4 g). The error is a little less: 284.2385/260.4 = 1.0915 … (≈ 9.2%).

8.2.1

Solubility number of sucrose

For saturated sucrose solutions, the ‘solubility number’, denoted by σ, is defined as

σ=

mass of dissolved substance mass of solvent

(8.16)

Table A2.3 shows the solubility number of sucrose as a function of temperature (Sokolovsky 1958). For further details, see Maczelka (1962).

Solutions

8.3

283

Aqueous solutions of sucrose and glucose syrup

Starch syrup is usually characterized by two parameters: the dry content D and the socalled ‘dextrose equivalent’ DE – the reducing sugar content of the dry content of the starch syrup expressed in terms of dextrose. A basic task in the manufacture of sugar confectionery is the preparation of sucrose–starch syrup solutions of given dry content and given reducing sugar content. Example 8.4 The usual parameters of starch syrup are D = 80% (m/m) and DE = 40% (m/m), which means that starch syrup contains 20% (m/m) water + 80% (m/m) dry content, and this dry content consists of 80% × 0.40 = 32% reducing sugars and 48% non-reducing sugars (carbohydrates): 100 kg starch syrup = 20 kg water + 32 kg reducing sugars + 48 kg nonreducing sugars. If the dry content is to be calculated, the mass balance is Aa + BD = x ( A + B )

(8.17)

where A is the amount of sugar of concentration a and B is the amount of starch syrup of dry content D. Example 8.5 50 kg of a sugar solution with a sucrose content a = 75% and 40 kg of starch syrup with D = 82% are mixed. The resultant dry content of the solution is 50 × 0.75 + 40 × 0.82 = x (50 + 40 ) → x = 75.33 … % Let us calculate the reducing sugar content of this solution. The reducing sugar content of sucrose is practically zero: a = 0. The reducing sugar content of starch syrup is determined by the value of DE. In this case we assume that DE = 43, so 50 × 0 + 40 × 0.82 × 0.43 = xred (50 + 40 ) → xred = 15.671 … %

8.3.1

Syrup ratio

A parameter commonly used to characterize sugar–starch syrup solutions is the syrup ratio, which is expressed as follows, by definition: Syrup ratio (SR ) = 100 kg sugar : X kg starch syrup dry cont ent

(8.18)

Example 8.6 If SR = 100 : 50, then 100 kg sugar and 50 kg starch syrup dry content are dissolved in the solution prepared.

284

Confectionery and Chocolate Engineering: Principles and Applications

The value of X used with advanced machinery (about 50–60) is higher than that used with traditional machinery (about 30–40). The syrup ratio and water content of the solution unambiguously determine how to prepare an aqueous sucrose–corn syrup solution. The water content can be prescribed in two ways: by the concentration (m/m%) or by the amount (kg). (If another type of concentration is not specified, it should be assumed that it is in mass per cent.)

Example 8.7 If the water content of a solution is 20%, SR = 100 : 60 and DE = 38%, then 100 kg of solution consists of the following components: 20 kg water 80 kg dry content, which is divided according to SR as 80 × 100/160 = 50 kg sugar 80 × 60/160 = 30 kg starch syrup dry content The reducing sugar content of the solution is 30 kg × 0.38 = 11.4%. It should be mentioned that, in addition to the 30 kg dry content of the starch syrup, the starch syrup has a water content; if D = 80%, this water content is 30 kg × [(1/0.8) − 1] = 7.5 kg, which is included in the 20 kg of water.

Example 8.8 If the amount of water W is 15 kg and the dry content is 90 kg, the solution consists of the following components (assuming SR = 100 : 60 again): 15 kg water 90 kg dry content, which is divided according to SR as 90 × 100/160 = 56.25 kg sugar 90 × 60/160 = 33.75 kg starch syrup dry content The reducing sugar content of the solution is 33.75 kg × 0.38/(90 + 15) = 12.214%. (The water content included in the starch syrup is 33.75 kg × [(1/0.82) − 1] = 7.41 kg water if D = 82%.)

Because the reducing sugar content of sugar–starch syrup solutions is an important parameter in the technology of sugar confectionery, the following general formula for its calculation is useful: R = (1 − W ) ×

DE SR + 1

(8.19)

where R is the reducing sugar content of the solution (%), W is the concentration of water in the solution, DE is the dextrose equivalent of the starch syrup (%) and SR is the syrup ratio. For the previous two examples, we have the following results:

Solutions

285

Example 8.7: if W = 0.20 and SR = 100 : 60 = 1.66, R = 0.8 ×

38% = 11.42% 2.66

Example 8.8: if W = 15/(15+90) = 0.14029, R = (1 − 0.1429) ×

8.4

38% = 12.214% 2.66

Aqueous sucrose solutions containing invert sugar

A similar calculation needs to be done if invert sugar solution is mixed with sugar solution or starch syrup. Invert sugar solutions can also be characterized by two technological parameters: the dry content and the reducing sugar content. Invert sugar is formed when sucrose is chemically split by acid (‘inverted’): sucrose ( in presence of acids or invertase ) → glucose + fructose As a general rule, the inversion of sucrose is mostly regarded as undesirable in the confectionery industry because the resulting fructose makes the product sticky. However, there are special applications of invert sugar solutions in which the strong hygroscopic property of fructose is exploited for conservation of the water content against drying. In these cases, the correct way to use invert sugar is to add invert sugar solution to the system, rather than to invert the sucrose content of the system, because the control of inversion is difficult. For details, see Sections 2.2.2 and 16.1. The main sources of invert sugar solution are ‘fluid sugars’. These ‘fluid sugars’ are prepared from acidic or enzymatic conversion of starch, which results in glucose, and then the glucose is partly (or entirely) transformed to fructose by enzymatic catalysis. In an invert sugar solution prepared from sucrose, the proportion of glucose to fructose is always 50 : 50; however, in fluid sugars this proportion can change according to the target use. The calculations of the dry content and reducing sugar content of solutions containing invert sugar are similar to those presented above for starch syrup.

8.5

Solubility of sucrose in the presence of starch syrup and invert sugar

In the presence of both starch syrup and invert sugar, the solubility of sucrose is decreased; however, the total dry content of the saturated solution is higher than when sucrose alone is dissolved. This fact makes it possible to produce sugar solutions of high dry content and, in the end, to produce sugar confectionery. Table A2.4 gives solubility data for sucrose–starch syrup–water solutions (Sokolovsky 1958, p. 16) and Table A2.5 gives data for sucrose–invert sugar–water solutions (Sokolovsky 1958, p. 17). It can be seen that the presence of invert sugar reduces the solubility of sucrose less than the presence of starch syrup does, and, in addition, it increases

286

Confectionery and Chocolate Engineering: Principles and Applications

the total soluble dry content of the solution far more than starch syrup does. For example, at 50°C the solubility of sucrose is 260.36 g/100 g water, which changes under the effect of starch syrup to 176.56 g sucrose + 188.56 g starch syrup dry content (both values per 100 g water); however, under the effect of invert sugar, the values are 196.43 g sucrose + 253.2 g invert sugar (both values again per 100 g water). It is evident from the solubility data that the amount of water required to dissolve sugar ingredients is more than the water content of sugar confectionery, which ranges from 1.5 m/m% (for hard-boiled bonbons) up to 22 m/m%. Consequently, the evaporation of surplus water is necessary. For example, at 20°C, 100 g of water dissolves 257.89 g dry content (154.82 g sucrose + 103.07 g starch syrup dry content); however, the minimum water content of the saturated solution is about 27.94% at room temperature. To achieve evaporation, the solution is warmed, and at higher temperatures the solubility of the dry components is increased. A fundamental condition for successful evaporation is that while the solution is becoming more and more concentrated, no component should start to crystallize. Since the rate of evaporation is speeded up by mixing, the crystallization of sucrose is a real danger if the syrup ratio is not high enough.

8.6

Rate of dissolution

The dissolution of solid substances is a process of diffusion. The process of dissolution can be accelerated by intensive mixing, which, on the one hand, disperses the solid particles in the solvent, and on the other hand, causes a turbulent flow that reduces the thickness of the laminar boundary layer on the surface of the particles, through which the movement of the soluble material into the solvent is relatively slow. The differential equation for the dissolution process is dS = kA (cSAT − c ) dt

(8.20)

where S is the mass of dissolved substance (kg), t is the time (s), k is the mass transfer coefficient (m/s), A is the surface area of the soluble substance (m2), c is the concentration of the solution (kg/m3) and cSAT is the concentration of the saturated solution (kg/m3). If it may be assumed that S = Zc, where Z (m3) is a constant, then kA ⎞ ⎛ c −c ⎞ ln ⎜ SAT 0 ⎟ = ⎛ Δt ⎝ cSAT − cTERM ⎠ ⎝ Z ⎠

(8.21)

where c0 is the concentration of the dissolved substance when t = 0, cTERM is the concentration of the dissolved substance when t = Δt, Δt is the duration of the process and A/Z is the specific surface area (1/m). The difficult question is the need to suppose that A is constant, because the surface area of a substance that is dissolving will not remain constant. The mass transfer coefficient k can be calculated according to the Colburn–Chilton analogy (see Section 1.4.2): Sh = CRe a Sc b

(8.22)

Solutions

287

Table 8.1 Constants in Eqn (8.22) according to type of impeller (from Fejes 1970).

Type of mixer

Characteristic lengths

Propeller Turbine, flat blade Turbine, oblique blade

D D d

d d d

C

a

b

Flow region

0.66 3.3 0.625

0.667 0.55 0.62

0.3 0.3 0.5

4 × 104 < Re < 18 × 104 2.3 × 104 < Re < 11 × 104 7500 < Re < 6.7 × 105

where Sh is the Sherwood number (the Nusselt number for mass transfer), Re is the Reynolds number, Sc is the Schmidt number = ν/CD, a and b are exponents, ν is the kinematic viscosity of the solvent (m2/s), and CD is the diffusion coefficient of the solid substance (m2/s). The Sherwood and Reynolds numbers can be calculated in various ways: If the characteristic length (m) is the diameter d of an impeller, Shi =

kd d 2n , Rei = ν CD

If the characteristic length is the diameter D of a tank, Sht =

kD D2 n , Ret = ν CD

If the characteristic length is the mean equivalent diameter dp of the particles, Shp =

kd p dpν , Rep = ν C

where v is the peripheral velocity of the impeller (m/s). Fejes (1970, p. 67) has reviewed studies of the various types of agitators for different flow regions; see Table 8.1.

Example 8.9 100 g of sugar is to be dissolved in 100 g of water at 20°C; the particle size is 0.2 mm, and the particles are assumed to be homogeneously of cubic form. The diameter d of the impeller of the mixer is 0.2 m, n = 1/s and the diffusion constant of sucrose is 2.5 × 10−10 m2/s (Rohrsetzer 1986, p. 15). We choose an impeller of the oblique-blade turbine type. From Table 8.1, C = 0.625, a = 0.62 and b = 0.5, and hence Rei =

d 2n 1 = 0.22 × −6 = 40 000 10 ν

Sc = 10 −6 ( 2.5 × 10 −10 ) = 4000

288

Confectionery and Chocolate Engineering: Principles and Applications

Shi = kd CD = k × 0.2 ( 2.5 × 10 −10 ) = 0.625 × ( 40 000 )

0.62

× 4000 → k = 3.52 × 10 −5 m s

The surface area of a cubic particle is 6 × (2 × 10−4 m)2 = 1.2 × 10−5 m2. If the density of sugar is 1500 kg/m3, then the volume of 100 g of sugar is 66.67 × 10−6 m3. The volume of a cubic particle is 8 × 10−12 m3. Consequently, 100 g of sugar consists of 66.67 × 10−6 m3/8 × 10−12 m3 = 8.33 × 106 particles, the total surface area of which is A = 1.2 × 10−5 m2 × 8.33 × 106 = 100 m2. In Eqn (8.21), c0 = 0 and cTERM = 100 g sugar/100 g water; however, in Eqns (8.20) and (8.21), the concentrations must be given in kg/m3. Since cTERM = 100/200 = 50°Bx, we obtain d = cTERM = 1.2367 g/cm3 = 1236.7 kg/m3 from Eqn (8.13). From Table A2.2, cSAT = 1332.7 kg/m3 at 20°C. From Eqn (8.21), ln [1332.7 (1332.7 − 1236.7 )] = ( kA Z ) Δt 2.631 = ( kA Z ) Δt = [3.52 × 10 −5 m s × 100 m 2 m 3 ] Δt → Δt = 747.44 s ≈ 12.45 min In practice, dissolving is accelerated by warming the solution.

Further reading AVP Baker. Technical brochures, Alikonis, J.J. (1979) Candy Technology. AVI Publishing, Westport, CT. Cakebread, S.H. (1975) Sugar and Chocolate Confectionery. Oxford University Press, Oxford. Lees, R. (1980) A Basic Course in Confectionery. Specialized Publications Ltd, Surbiton. Lienhard, J.H., IV and Lienhard, J.H., V (2005) A Heat Transfer Textbook, 3rd edn. Phlogiston Press, Cambridge, MA. Meiners, A. and Joike, H. (1969) Handbook for the Sugar Confectionery Industry. SilesiaEssenzenfabrik, Gerhard Hanke K.G. Norf, Germany. Pratt, C.D. (ed.) (1970) Twenty Years of Confectionery and Chocolate Progress. AVI Publishing, Westport, CT. Robert Bosch/Hamac. Technical brochures Schwartz, M.E. (1974) Confections and Candy Technology, Food Technology Review, 12. Noyes, Park Ridge, NJ. Sullivan, E.T. and Sullivan, M.C. (1983) The Complete Wilton Book of Candy. Wilton Enterprises Inc., Woodridge, IL. Ter Braak. Technical brochures.

Chapter 9

Evaporation

Contents 9.1 9.2 9.3

Theoretical background – Raoult’s law Boiling point of sucrose/water solutions at atmospheric pressure Application of a modification of Raoult’s law to calculate the boiling point of carbohydrate/water solutions at decreased pressure 9.3.1 Sucrose/water solutions 9.3.2 Dextrose/water solutions 9.3.3 Starch syrup/water solutions 9.3.4 Invert sugar solutions 9.3.5 Approximate formulae for the elevation of the boiling point of aqueous sugar solutions 9.4 Vapour pressure formulae for carbohydrate/water solutions 9.4.1 Vapour pressure formulae 9.4.2 Antoine’s rule 9.4.3 Trouton’s rule 9.4.4 Ramsay–Young rule 9.4.5 Dühring’s rule 9.5 Practical tests for controlling the boiling points of sucrose solutions 9.6 Modelling of an industrial cooking process for chewy candy 9.6.1 Modelling of evaporation stage 9.6.2 Modelling of drying stage Further reading

9.1

289 291 291 291 292 292 292 292 295 295 297 299 301 302 303 304 305 307 307

Theoretical background – Raoult’s law

For the theoretical background to the various formulae for the boiling-point elevation and vapour pressure of aqueous solutions of carbohydrates such as sucrose, dextrose, starch syrup and invert sugar, we need to study Raoult’s law. This law states that the elevation Δtb of the boiling point of a solution is given by Δtb = mΔtm ;b

(9.1)

where m, the so-called Raoult concentration or ‘molality’, is the number of dissolved moles per 1000 g of solvent, and Δtm;b is the molar elevation of the boiling point (in units of K/mol), given by Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

290

Confectionery and Chocolate Engineering: Principles and Applications

Δtm;b =

RTb2 1000 Lb

(9.2)

i.e. Δtb =

mRTb2 1000 Lb

(9.3)

where R, the universal gas constant, is equal to 8.31434 J/mol K; Tb (K) is the boiling point of the solvent at the given pressure; and Lb (J/1000 g solvent) is the molar latent heat of vaporization of the solvent at the given pressure (Erdey-Grúz and Schay 1954, Vol. 2, p. 51; Lengyel et al. 1960, p. 73). According to Raoult’s law, which is strictly valid only for dilute solutions, if one gram of a substance is dissolved in 1000 g of solvent, then Δtb g = Δtm;b M

(9.4)

where M is the molar mass of the dissolved substance. In other words, the elevation of the boiling point is dependent on the number m of moles of the dissolved substance (see Eqn 9.3); moreover, it is dependent on the qualitative nature of the solvent, defined by the molar elevation of the boiling point of the solvent (Eqn 9.2), but independent of the nature of the dissolved substance. In the case of aqueous solutions of carbohydrates, the solvent is water, and for water, Δtm;b = 0.52 K. For dilute aqueous solutions, the appropriate form of Eqn (9.4) is Δtb g = 0.52 K M

(9.5)

at atmospheric pressure (750 mmHg). Example 9.1 Let us calculate the molar elevation of the boiling point of water at atmospheric pressure using Eqn (9.2). The molar heat of vaporization of water is Lb = 9710 cal/18 g = 539.44 cal/ g = 2258.53 J/g, and Tb = 373.1 K (Erdey-Grúz and Schay 1962, Vol. 1, p. 715). We obtain 8.31434 J mol K × (373.1 K ) RTb2 = = 0.511245 K mol 1000 Lb 1000 g × 2258.53 J g 2

Δtm ;b =

In practice, the value Δtm;b = 0.52 K/mol is used. It should be emphasized that at higher concentrations, Raoult’s law is not valid; for example, if 1 mole (180 g) of dextrose is dissolved in 1000 g of water (so that the concentration s is 180/1180 = 15.25%), the elevation of the boiling point should be 0.52°C. However, according to Bukharov’s measurements (see Table 9.2), this value is merely ≈ 0.35°C. For 1 mole (342 g) of sucrose (s = 342/1342 = 25.48%), the actual elevation of the boiling point is ≈ 0.45 K according to Bukharov’s measurements (see Table 9.2 and Sokolovsky 1958, p. 20). That is, these solutions cannot be regarded as dilute.

Evaporation

9.2

291

Boiling point of sucrose/water solutions at atmospheric pressure

Sokolovsky (1958, p. 19) gives a simple formula for the boiling point of sucrose/water solutions: T ( s ) = 100 +

2.33s 1− s

(9.6)

where T(s) is the boiling point (°C) of a sucrose/water solution at atmospheric pressure, and s is the concentration of sucrose in the solution (m/m or m/m%). Equation (9.6) can be regarded a modification of Raoult’s law. Table 9.1 shows the relation between measured values of the boiling point and the corresponding values calculated on the basis of Eqn (9.6). For higher concentrations of sucrose, Eqn (9.6) cannot be correct, because it becomes divergent when s → 1. Exclusively for the interval 15% > W > 2% of the water content W (m/m%), a better approximation has been obtained by the present author using the formula T ( s ) = 146.7 − 4.2W + 0.138W 2

9.3 9.3.1

(9.7)

Application of a modification of Raoult’s law to calculate the boiling point of carbohydrate/water solutions at decreased pressure Sucrose/water solutions

Values of the boiling-point elevation of sucrose/water solutions at decreased pressures have been given by Sokolovsky (1958, p. 20), following Bukharov (1935); see Table 9.2.

Table 9.1 Boiling points of aqueous sucrose solutions of various concentrations (m/m%), measured by Bukharov (1935) and calculated according to Eqn (9.6). Boiling point (°C) Sugar concentration (%) 10 20 30 40 50 60 70 80 90

Measured

Calculated

100.1 100.3 100.6 101 101.8 103 105.5 109.4 119.6

100.2589 100.5825 100.9986 101.5533 102.33 103.495 105.4367 109.32 120.97

292

Confectionery and Chocolate Engineering: Principles and Applications

Table 9.2 Elevation of boiling point (°C) of aqueous sucrose solutions as a function of concentration at various pressures (Sokolovsky 1958). p (105 Pa) 0.12279

0.19916

0.31157

0.47393

0.70096

1.01325

t (°C) Concentration (m/m%) 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 a

50

60

70

80

90

100

0.05 0.1 0.17 0.26 0.39 0.52 0.69 0.8 1.02 1.32 1.7 2.3 2.8 3.65 5.05 6.8 a

0.05 0.1 0.18 0.27 0.4 0.54 0.71 0.85 1.1 1.4 1.82 2.45 3 3.9 5.4 7.3 10a

0.05 0.11 0.18 0.28 0.42 0.55 0.73 0.9 1.18 1.52 1.94 2.6 3.2 4.18 5.8 7.85 10.75 16a

0.06 0.11 0.19 0.28 0.43 0.57 0.76 0.95 1.25 1.61 2.06 2.75 3.4 4.46 6.2 8.35 11.5 17.2

0.06 0.12 0.19 0.29 0.44 0.58 0.78 1 1.32 1.72 2.18 2.9 3.6 4.75 6.6 8.9 12.25 18.4

0.06 0.12 0.2 0.3 0.45 0.6 0.8 1.05 1.4 1.8 2.3 3.05 3.8 5.05 7 9.4 13 19.6

a a

a

Data uncertain or unknown.

9.3.2

Dextrose/water solutions

Values of the boiling-point elevation of aqueous solutions of dextrose at decreased pressures have been summarized by Sokolovsky (1958, p. 41); see Table 9.3.

9.3.3

Starch syrup/water solutions

The elevation of the boiling point of aqueous starch syrup solutions as a function of concentration at various pressures, according to Bukharov (1935), is given in Table 9.4.

9.3.4

Invert sugar solutions

Sokolovsky (1951, p. 18) has also published data on the elevation of the boiling point of aqueous invert sugar solutions (Table 9.5).

9.3.5

Approximate formulae for the elevation of the boiling point of aqueous sugar solutions

Equation (9.6), T(s) = 100 + 2.33s/(1 − s), which refers to atmospheric pressure, means that if s = 0.5 (50%) then the elevation of the boiling point (in °C) is equal to the factor

Evaporation

293

Table 9.3 Elevation of boiling point (°C) of dextrose solutions as a function of concentration at various pressures (Sokolovsky 1958). p (105 Pa) 0.12279

0.19916

0.31157

0.47393

0.70096

1.01325

t (°C) Concentration (m/m%) 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

50

60

70

80

90

100

0.08 0.16 0.25 0.39 0.51 0.62 0.78 1.04 1.45 1.98 2.7 3.63 4.73 6.04 7.47 9.29 12.01 19.14

0.08 0.17 0.26 0.41 0.55 0.66 0.84 1.11 1.55 2.12 2.9 3.9 5.07 6.47 8.02 9.98 13.6 20.5

0.09 0.18 0.28 0.44 0.59 0.7 0.9 1.2 1.66 2.28 3.1 4.17 5.43 6.93 8.58 10.69 14.69 21.08

0.1 0.19 0.3 0.48 0.63 0.75 0.96 1.28 1.78 2.42 3.3 4.45 5.89 7.4 9.17 11.42 15.59 23.62

0.11 0.21 0.32 0.51 0.67 0.8 1.02 1.36 1.9 2.59 3.59 4.75 6.19 7.9 9.79 12.17 16.65 25.27

0.11 0.22 0.35 0.55 0.7 0.85 1.05 1.45 2 2.75 3.75 5.05 6.6 8.4 10.45 13 17.75 27

2.33. This consideration helps us to construct approximate formulae similar to Eqn (9.6). A detailed calculation shows that this factor is dependent on the pressure applied. The resulting approximate formulae are: T ( s ) = Ts =0 +

(1.1 + 0.012Ts =0 ) s 1− s

T ( d ) = Td =0 + T (st ) = Td =0 + T (i ) = Ti =0 +

(1.2 + 0.0154Td =0 ) d 1− d

(0.63 + 0.0077Ty=0 ) y 1− y

(1.47 + 0.0188Ts =0 ) i 1− i

(9.8) (9.9) (9.10) (9.11)

where Ts =0 = Td=0 = Ty=0 = Ti=0 is the boiling point (°C) of pure water at the given pressure (decreased or atmospheric), and s, d, y and i are the concentrations (m/m%) of sucrose, dextrose, starch and invert sugar, respectively.

294

Confectionery and Chocolate Engineering: Principles and Applications

Table 9.4 Elevation of boiling point (°C) of aqueous starch syrup solutions as a function of concentration at various pressures (Sokolovsky 1958). p (105 Pa) 0.12279

0.19916

0.31157

0.47393

0.70096

1.01325

t (°C) Concentration (m/m%) 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 92 94 96

50

60

70

80

90

100

0.04 0.06 0.1 0.14 0.18 0.27 0.33 0.405 0.57 0.74 1.02 1.4 1.94 2.62 3.49 4.62 6.44 9.73 12.11 16.18 24.92

0.04 0.06 0.1 0.15 0.19 0.28 0.35 0.48 0.61 0.79 1.09 1.51 2.07 2.81 3.74 4.96 6.97 10.45 13.01 18.07 26.82

0.04 0.07 0.11 0.16 0.21 0.31 0.37 0.52 0.66 0.85 1.17 1.62 2.23 3.12 4 5.3 7.4 11.2 13.94 18.85 28.84

0.04 0.07 0.11 0.18 0.22 0.33 0.4 0.55 0.7 0.91 1.25 1.74 2.38 3.21 4.28 5.66 7.89 11.97 14.91 19.96 30.85

0.04 0.08 0.13 0.19 0.24 0.35 0.44 0.59 0.75 0.97 1.33 1.89 2.53 3.43 4.55 6.05 8.43 12.79 15.91 21.33 33.28

0.04 0.08 0.15 0.2 0.26 0.38 0.5 0.63 0.8 1.03 1.4 1.95 2.7 3.65 4.85 6.45 9 13.6 17 22.75 40

Table 9.5 Elevation of boiling point (°C) of aqueous invert sugar solutions as a function of concentration at various pressures (Sokolovsky 1951). p (105 Pa) 0.12279

0.31157

0.70096

1.01325

t (°C) Concentration (m/m%) 70 75 80 85 90 a

50

70

90

100

5.8 7.53 9.66 12.83 17.42

6.8 8.65 11.17 14.76 20.09

a

9.86 12.54 16.94 22.99

8.1 10.5 13.5 18 24.55

Data uncertain or unknown.

Evaporation

295

Example 9.2 The boiling point of pure water at atmospheric pressure is 100°C. Therefore, according to Eqn (9.8), for sucrose solutions, T ( s ) = Ts = 0 +

(1.1 + 0.012Ts = 0 ) s 1− s

= 100 +

(1.1 + 1.2 ) s 1− s

which agrees with Eqn (9.6).

Example 9.3 If the boiling point of pure water is 50°C, then, according to Eqn (9.9), for dextrose solutions, T ( d ) = Td = 0 +

(1.2 + 0.0154Td = 0 ) d 1− d

= 50 +

(1.2 + 0.0154 × 50 ) d 1− d

= 50 +

1.97d 1− d

Let d = 0.7; then T(d) = 50 + 1.97 × 0.7/0.3 = 54.6°C. The value in Table 9.3 is 56.04°C.

More detailed data for aqueous solutions of various monosaccharides and disaccharides can be found in the handbook by Junk and Pancoast (1973).

9.4 9.4.1

Vapour pressure formulae for carbohydrate/water solutions Vapour pressure formulae

The vapour pressure function most often used has the form log p = −

A +B T

(9.12)

where p is the vapour pressure, T is the boiling point of the liquid (K), and A and B are constants. For pure water (Erdey-Grúz and Schay 1962, p. 708), log pw = −

10313 2253.715 + 8.9296 = − + 8.9296 4.576Tw Tw

(9.13)

where pw is the vapour pressure of water (mmHg); the subscript ‘w’ denotes water vapour. Let us now write Tw − 273 = Ts=0 = Td=0 = Td=0; that is, we transcribe Eqns (9.8)–(9.11) into formulae using Tw, which is expressed in kelvin. After some simple algebraic transformations, the following equations are obtained: T ( s ) = Tw (1 + 0.012S ) − 2.176S

(9.14)

296

Confectionery and Chocolate Engineering: Principles and Applications

T ( d ) = Tw (1 + 0.0154S ) − 3.0042S

(9.15)

T (st ) = Tw (1 + 0.0077S ) − 1.4721S

(9.16)

T (i ) = Tw (1 + 0.0188S ) − 3.6624S

(9.17)

where, for the sake of simplicity, S denotes the ratios of concentrations s/(1 − s), d/(1 − d), y/(1 − y) and i/(1 − i), and ‘st’ refers to aqueous starch syrup solutions. Expressions for Tw can be obtained from Eqns (9.14)–(9.17); for example, from Eqn (9.14), we obtain Tw =

T ( s ) + 2.176S 1 + 0.012S

When these expressions are substituted into Eqn (9.13), the following pressure functions are obtained: For aqueous sucrose solutions: log pw = −

2253.715 (1 + 0.012S ) + 8.9296 T ( s ) + 2.176S

(9.18)

For aqueous dextrose solutions: log pw = −

2253.715 (1 + 0.0154S ) + 8.9296 T ( d ) + 3.0042S

(9.19)

For aqueous starch syrup solutions: log pw = −

2253.715 (1 + 0.0077S ) + 8.9296 T (st ) + 1.4721S

(9.20)

For aqueous invert sugar solutions: log pw = −

2253.715 (1 + 0.0188S ) + 8.9296 T (i ) + 3.6624S

(9.21)

In the above equations, pw is in mmHg. The correct form is actually ‘log(pw/mmHg)’ because the argument of the log function must be dimensionless. Example 9.4 An aqueous sucrose solution has a concentration s = 0.3, i.e. S = 0.3/0.7 = 0.4286. The boiling point of this solution (measured value) is 60°C + 0.54°C = 333.54 K = T(s) (see Table 9.2). What is the vapour pressure at this temperature? According to Eqn (9.18), log pw = −

2253.715 (1 + 0.012 × 0.4286 ) + 8.9296 333.54 + 2.176 × 0.4286

Evaporation

297

From this, pw = 147.23 mmHg = 0.19629 × 105 Pa. The correct value is 0.19916 × 105 Pa (see Table 9.2). Example 9.5 An aqueous starch syrup solution has a concentration s = 0.7, i.e. S = 0.7/0.3 = 2.33. The boiling point of this solution (measured value) is 80°C + 3.21°C = 356.21 K = T(st) (see Table 9.4). What is the vapour pressure at this temperature? According to Eqn (9.20), log pw = −

2253.715 (1 + 0.0077 × 2.33) + 8.9296 356.21 + 1.4721 × 2.33

From this, pw = 355.3 mmHg = 0.473694 × 105 Pa. The correct value is 0.47393 × 105 Pa (see Table 9.4). Example 9.6 An aqueous dextrose solution is evaporated at pw = 250 mmHg and 75°C = 348 K. What is the equilibrium concentration (S and s) in these circumstances? Using Eqn (9.19), log 250 = −

2253.715 (1 + 0.0154 × S ) + 8.9296 348 + 3.0042 × S

By solving this equation, we obtain S = 1.2806 = s/(1 − s), and s = 0.562 (m/m). It should be emphasized that all of these equations concerning the elevation of boiling point and the vapour pressure of carbohydrate solutions are only theoretical, although they are based on the laboratory measurements of Bukharov. Moreover, the concentration interval in which they work acceptably is only about c = 0.3–0.7, and the data referring to aqueous invert sugar solutions are insufficient. Consequently, they must be regarded as being for information only and must not replace trials.

9.4.2

Antoine’s rule

Equations (9.14)–(9.17) can be transcribed into a general form. For example, T ( s ) = Tw (1 + 0.012S ) − 2.176S

(9.14)

can be written as T ( s ) − Tw = Δtb = Tw × 0.012S − 2.176S = S (Tw × 0.012 − 2.176 ) i.e. Δtb = S (Tw a − b ) where a and b are constants (Table 9.6).

(9.22)

298

Confectionery and Chocolate Engineering: Principles and Applications

Table 9.6 Constants in Eqns (9.22) and (9.23). Carbohydrate

a

b (K)

Tr (K) = b/a

Sucrose Dextrose Starch syrup Invert sugar

0.012 0.0154 0.0077 0.0188

2.176 3.0042 1.4721 3.6624

181.330 195.078 191.182 194.809

Equation (9.22) can be transformed to ⎛T ⎞ Δtb = Ti − Tw = Sb ⎜ w − 1⎟ ⎝ Tr ⎠

(9.23)

where Tr ≡ b/a and Ti is the boiling point of any of the sugars discussed above. A consequence of Eqn (9.23) is that Tw ≥ Tr. From Eqns (9.13) and (9.23), an equation for the vapour pressure of carbohydrate solutions can be constructed: ⎛ Sb ⎞ log pw = − 2253.715 ⎜1 + i ⎟ (Ti + Sbi ) + 8.9296 ⎝ Tri ⎠

(9.24)

where i refers to the type of carbohydrate dissolved and Ti is the boiling point of the solution. There is a well-known general relationship for the vapour pressure of solutions, called Antoine’s rule: log pw = −

A +B T +C

(9.25)

where A, B and C are constants; in most cases C = −43 K. Equation (9.24) can be regarded as an equation of Antoine type, where A = 2253.715(1 + Sbi/Tri), B = 8.9296 and C = Sbi. C is dependent on the composition (i.e. on S) and is positive. For further details, see Elliot and Lira (1999). Example 9.7 Let us calculate the Antoine equation for a starch syrup/water solution in which the concentration of starch syrup is s = 75%. We have S=

0.75 =3 0.25

From Table 9.6, bi = 1.4721 K and Tr = 191.182 K. 3 × 1.4721⎞ ⎛ Sb ⎞ A = 2253.715 ⎜1 + i ⎟ = 2253.715 ⎛1 + = 2314.565 ⎝ ⎝ Tri ⎠ 191.182 ⎠

Evaporation

299

B = 8.9296 C = Sbi = 3 × 1.4721 K = 4.4163 K log pw = −

9.4.3

A 2314.565 +B =− + 8.9296 T +C T + 4.4163

Trouton’s rule

Equation (9.2) can be expressed with the help of the Trouton constant: Δtm =

mRTb2 18RTb = m× 1000 Lb 1000 Tr

(9.26)

where Tr =

Lb Tb

is the Trouton constant, which is equal to 26.0 cal/K (= 108.784 J/K) for water (ErdeyGrúz and Schay 1962, p. 715). S and m are proportional to each other: m=

1000S M

(9.27)

where M is the molar mass (in grams) of the carbohydrate. For example, if M/2 = 342 g/2 = 171 g of sucrose is dissolved in 1000 g of water, then m = 1/2. Since S = 171/1000 = 0.174, we obtain from Eqn (9.27) the result m = 1000 × 0.171/342 = 1/2. These facts can be explained as a loose relationship between Eqn (9.22) and Raoult’s law: Δtb =

mRTw2 18RTw = m× = S (Tw a − b ) 1000 Tr 1000Lw

(9.28)

An important question can be asked: How to calculate the elevation of the boiling point of aqueous solutions of carbohydrate mixtures? A possible interpretation of this question is ‘how to calculate the value of S?’ The starting supposition is that the values of the constants a and b have taken into account the ‘molar masses’ of the carbohydrates in question. Thus, for the values of both a and b, the following sequence of magnitudes can be observed: starch syrup < sucrose < dextrose < invert sugar which agrees with the sequence of molality of these carbohydrates if their masses are the same. There are no data on the DE value of the starch syrup used in Bukharov’s investigations. If an estimated value DE ≈ 40%, which is usual in the confectionery industry,

300 Table 9.7

Confectionery and Chocolate Engineering: Principles and Applications

Elevation of boiling point for aqueous carbohydrate solutions, for S = 1.a Elevation of boiling point (K) if S = 1

Pressure (105 Pa)

Boiling point Tw (K)

Sucrose

Dextrose

Starch syrup

Invert sugar

323 333 343 353 363 373

1.700071 1.820073 1.940076 2.060078 2.18008 2.300082

1.969998 2.123998 2.277998 2.431998 2.585998 2.739998

1.014998 1.091998 1.168997 1.245997 1.322997 1.399997

2.409985 2.597984 2.785984 2.973983 3.161983 3.349982

0.12279 0.19916 0.31157 0.47393 0.70096 1.01325 a

A maximum of two decimal places is sufficient.

is assumed, then the dissolved dry substance of the starch syrup consists roughly of 40% reducing sugar (expressed as dextrose) + 60% other components, which are dissolved as well. Consequently, 100 kg of dextrose, when dissolved, produces roughly double the number of molecules that 100 kg of starch syrup does. This can be observed in the values of a and b. Therefore the simple mass ratios can presumably be used to approximate the real conditions by a type of equation of mixture: Δtb = S (Tw a − b ) =

∑ s j (Tw a j − b j ) = ∑ s j ( Δtb j ) 1− s

1− s

(9.29)

where Δtbi is the elevation of the boiling point for each carbohydrate if S = 1; s = Σsj is the total solid content dissolved (m/m); j = 1, 2, … , n; and aj and bj are constants for each particular type of substance. In order to facilitate the calculation, Table 9.7 gives the values of Δtbi for the carbohydrates in question.

Example 9.8 At p = 0.19916 × 105 Pa (Tw = 60°C = 333 K), the distribution of the dissolved solid content is (concentrations in m/m) sucrose = 0.45; starch syrup = 0.20; invert sugar = 0.05. The elevation of the boiling point is to be calculated (s = 0.70). Using Eqn (9.29), Δtb = {0.45 (333 × 0.012 − 2.176 ) + 0.2 (333 × 0.0077 − 1.4721) + 0.05 (333 × 0.0188 − 3.6624 )} 0.3 = 3.891 K

(According to Bukharov (Table 9.2), the measured value is 3.9 K for a sucrose solution of s = 0.7.) The calculation can easily be done for this example by using the values in Table 9.7: Δtb = ( 0.45 × 1.82 + 0.2 × 1.092 + 0.05 × 2.6 ) 0.3 = 3.8915 K

301

Evaporation

9.4.4

Ramsay–Young rule

The Ramsay–Young rule makes it possible to calculate the boiling point of a solution at a pressure p that differs from atmospheric pressure (1 bar): ⎡ T ( p) ⎤ ⎡ T ( p) ⎤ =⎢ ⎢ T (1 bar ) ⎥ ⎥ ⎣ ⎦ water ⎣ T (1 bar ) ⎦solution

(9.30)

where T(p) is the boiling point (in K) of water at a (decreased) pressure p, T(1 bar) is the boiling point of water at atmospheric pressure, T(p) is the boiling point of the solution at the pressure p and T(1 bar) is the boiling point of the solution at atmospheric pressure. For pure water, the vapour pressure function is log pw = −

10313 2253.715 + 8.9296 = − + 8.9296 4.576Tw Tw

(9.13)

For aqueous solutions of the carbohydrates in question, the vapour pressure function is log pw = −

2253.715 (1 + Sbi Tri ) + 8.9296 Ti + Sbi

(9.24)

The Ramsay–Young rule concerns the same pressures, i.e. the equality of Eqns (9.13) and (9.24), and therefore

(pure water )

1 1 + Sbi Tri = Tw Ti + Sbi

(solution)

(9.31)

Since Eqn (9.31) refers to any pressure p, it can be written for atmospheric pressure:

(pure water ) 1 373 =

1 + Sbi Tri Ti 0 + Sbi

(solution)

(9.32)

where Ti 0 is the boiling point of the solution at atmospheric pressure. From Eqns (9.31) and (9.32), Tw 373 =

Ti + Sbi Ti 0 + Sbi

(9.33)

which is a modified form of the Ramsay–Young rule.

Example 9.9 Let us apply the Ramsay–Young rule to an aqueous dextrose solution with s = 60% at the pressures 0.12279 × 105 Pa and 0.31157 × 105 Pa. From Table 9.6, bi = 3.0042 K; S = 0.6/0.4 = 3/2, and therefore Sbi = 6.634. For this solution, from Table 9.3,

302

Confectionery and Chocolate Engineering: Principles and Applications

Δtb = 5.05 K at atmospheric pressure. Δtb = 3.63 K at 0.12279 × 105 Pa. Δtb = 4.17 K at 0.31157 × 105 Pa. At 0.12279 × 105 Pa, the boiling point of pure water is 50°C = 323 K. Supposing that the Ramsay–Young rule is valid, we write 323 323 + 3.63 + 6.634 = 0.866, = 0.866 373 373 + 5.05 + 6.634 At 0.31157 × 105 Pa, the boiling point of pure water is 70°C = 343 K: 343 343 + 4.17 + 6.634 = 0.9196, = 0.9204 373 373 + 5.05 + 6.634 In both cases, the two ratios for the same pressure can be regarded as equal (e.g. 0.9196 ≈ 0.9204); our supposition was correct.

9.4.5

Dühring’s rule

According to Dühring’s rule for two substances 1 and 2, the boiling points are in a linear relationship: T1 ( p ) = k1T2 ( p ) + k2

(9.34)

where T1(p) is the boiling point (K) of substance 1 at pressure p, T2(p) is the boiling point (K) of substance 2 at pressure p, and k1 and k2 are constants. The Ramsay–Young rule can be regarded as a special case of Dühring’s rule: in the former case, k2 = 0. In general, Dühring’s rule can be assumed to be valid for less familiar substances. Equation (9.31) provides a starting point for applying Dühring’s rule to the aqueous carbohydrate solutions previously studied:

(pure water )

1 1 + Sbi Tri = Tw Ti + Sbi

(solution)

(9.35)

i.e. 1 + Sbi Tri 1 + Sb j Trj = Ti + Sbi T j + Sb j

(9.36)

where i and j refer to two different carbohydrates. The equation Ti + Sbi 1 + Sbi Tri = =K T j + Sb j 1 + Sb j Trj

(9.37)

303

Evaporation

or Ti = KT j + S ( Kb j − bi )

(9.38)

where K is a constant, agrees with Dühring’s rule as expressed in Eqn (9.34).

9.5

Practical tests for controlling the boiling points of sucrose solutions

In the production of hard-boiled and low-boiled sugar sweets by confectioners, the end point of boiling can be checked without special equipment, using experiments based on assessment of the sugar mass by the confectioner’s own senses (Table 9.8).

Example 9.10 Conversion between the various temperature scales is a common task. A temperature of 32°F is assigned to the melting point of ice, and 212°F to the boiling point of water, so that the temperature interval between these points is divided into 180 parts; 32°F = 0°C and 212°F = 100°C. The following equation makes it very simple to convert temperatures from one scale to another by the help of the Sena formula:

(t − 273) K 5 = t°C 5 = t°R 4 = (t − 32 ) °F 9

(9.39)

Table 9.8 Sugar-boiling tests used in practice.a Hungarian Szirup Gyenge szál Erös szál Szál Gyöngy Kis gyöngy Nagy gyöngy Gyenge pflúg Pflúg (fújás) Erös pflúg Gyenge golyó Golyó Kemény golyó Törés Karamel Cukor (nyilt lángon)

Germanb

Englishc Crystal syrup

Schwach. Faden Starker Faden Thread Pearl Kleine Perlen Grosse Perlen Schwach. Pflug Blow/soufflé Starker Pflug Soft ball Ballen Bruch Karamel Bonbons/Feuer

Hard ball Crack Caramel

Boiling point (°C) 104 105 107.5 108 110 110 111 112.5 113 116 118 122.5 123 131 150 156

For ‘doctor solution’ (Läuter-Lösung in German), a boiling point of 102.5°C = 82°R is recommended by Földes and Ravasz (1998). b For German test names, see Besselich (1951). c For English test names, see Meiners et al.(1983). a

304

Confectionery and Chocolate Engineering: Principles and Applications

Here, absolute zero (0 K) is calculated to be equal to −273°C (the exact value is −273.16°C). For example, if t(K) = 573 K, then

(573 − 273) K 5 = t°C 5 → 573 K=300°C = t°R 4 → 573 K = 300°R × ( 4 5) = 240°R = (t − 32 ) °F 9 → (573 − 273) 5 = (t − 32 ) 9 → t = 572°F → 573 K = 572°F

(As a check, we can calculate t°F if t°C = 300°C: 300°C 5 = (t − 32 ) °F 9 → t = 572°F If t°F = 0, what is t°C? t°C 5 = ( 0 − 32 ) 9 → t°C = 32 × (5 9) °C = −17.77°C Comments: • The equality 573 K = 572°F does not mean that the two scales are similar; see above. • The scale in which the magnitude of a degree is the same as in the Fahrenheit scale but where the temperature is counted from absolute zero is called the Rankine scale. In this scale, a temperature of 459.67° corresponds to 0°F, 491.67° to the freezing point of water and 671.67° = (491.67 + 180)° to the boiling point of water. (Since Δt°C = 1.8 Δt°F, −273.16°C (= 0 K) = 1.8 × 273.16 °Rankine = 491.688 °Rankine – the difference derives from the value of absolute zero.) • The Réaumur scale (°R) is used in old recipes (0°R = 0°C; 80°R = 100°C).

9.6

Modelling of an industrial cooking process for chewy candy

This section presents a study by Oliveira et al. (2008) of the use of a hybrid modelling approach that consists of a phenomenological model of the evaporation step and an artificial neural network to model the vacuum-drying step of a cooking process for chewy candy. Figure 9.1 presents a schematic illustration of a ‘classical’ candy-cooking machine. An aqueous sugar/starch syrup solution (with a boiling point of about 106°C) is pumped through a stainless steel coil located inside a vapour chamber. The temperature of the vapour chamber is controlled by a control loop composed of a PT100 sensor, a PID controller and a control valve. The candy solution enters the expansion chamber at atmospheric pressure, where the evaporated water is removed through an outlet. The resulting paste (candy mass) is accumulated in this chamber and then transferred to the vacuum chamber through an outlet by a tempering piston. The air inlet valve is then opened, allowing the candy mass to flow into the reservoir for the cooling step. The cooking process must be done to meet the requirement of a total solids concentration of around 98%; it is conducted in a temperature range of 125–132°C.

Evaporation

305

8

12 13

7

4

6 10 3

9 11

5

2

1

Fig. 9.1 Schematic diagram of an industrial candy-cooking machine: 1, feed pump; 2, vapour chamber; 3, stainless steel coil; 4, vapour input valve; 5, vapour purge; 6, expansion chamber; 7, tempering piston; 8, outlet; 9, vacuum chamber; 10, air inlet valve; 11, paste reservoir; 12, PT100 sensor; 13, temperature controller [reproduced from Oliveira et al. (2008), by kind permission of Elsevier].

9.6.1

Modelling of evaporation stage

The following assumptions were made: • homogeneity of composition and temperature inside the steel coil; • constant amount of contents in the steel coil; • thermodynamic equilibrium of the system. The global balance for the evaporation stage is given by dM = qb′ − q ′ − qv′ dt

(9.40)

where dM/dt (kg/s) is the mass flow, M (kg) is the total mass in the evaporation stage, qb′ (kg/s) is the feed flow, q′ (kg/s) is the concentrate product flow and qv′ (kg/s) is the evaporate flow. The total mass in the system was assumed to be constant, i.e. dM/dt = 0, and thus qb′ = q ′ + qv′

(9.41)

The mass balance for the total solids is given by dX ⎞ = X b qb′ − Xq ′ M⎛ ⎝ dt ⎠

(9.42)

306

Confectionery and Chocolate Engineering: Principles and Applications

where X (expressed as a ratio) is the total solids concentration and Xb is the solids concentration of the feed. The energy balance is given by dH ⎞ M⎛ = H b qb′ − XHq ′ − H v qv′ + Q ′ ⎝ dt ⎠

(9.43)

where H (kJ/kg) is the enthalpy of the concentrated product, Hb (kJ/kg) is the enthalpy of the feed, Hv (kJ/kg) is the enthalpy of the evaporate, and Q′ (kJ/s) is the heat exchange through the wall, which is given by Q ′ = κΔTML

(9.44)

where κ (kJ/s K) is the overall heat transfer coefficient and ΔTML (°C) is the logarithmic mean of the temperature. The overall heat transfer coefficient of the cooker used was determined from a steadystate evaporation model and from steady-state data for the industrial process, using the following equation:

κ=

XHq ′ + H v qv′ − H b qb′ ΔTML

(9.45)

where qv′ was computed from Eqn (9.42). The enthalpy of the candy solution was defined by H = C P ( X )T

(9.46)

H b = CP ( X b )Tb

(9.47)

where the subscript ‘b’ denotes the feed. To calculate the specific heat capacity of the candy solution as a function of the percentage composition mi of the system, the following equation (Singh and Heldman 1993) was used: CP = 1.424 mc + 1.549mp + 1.675mf + 0.837 ma + 4.187 mw

(9.48)

where c = carbohydrate, p = protein, f = fat, a = ash and w = water. The enthalpy of the saturated vapour Hv was given by Hv = 2509.2888 + 1.6747Tv

(9.49)

and the enthalpy of the condensate Hcond was given by H cond = 4.187Tcond

(9.50)

where Tv and Tcond are the corresponding temperatures. To correlate the boiling-point elevations ΔTeb measured, the ‘boiling point rise’ (BPR) model of Capriste and Lozano (1988) was adopted: ΔTeb = aX b ecX P d

(9.51)

Evaporation

307

140 This work (BPR data) Adjusted BPR model Industrial process Lees and Jackson (1999)

Syrup boiling temperature (°C)

135 130 125 120 115 110 105 100 95 90 30

40

50 80 60 70 Total solids concentration (wt %)

90

100

Fig. 9.2 Boiling point of candy solution as a function of the total solids concentration at 100 kPa (= 1 bar). BPR = boiling point rise [reproduced from Oliveira et al. (2008), by kind permission of Elsevier].

where a = 0.4846 × 10−2, b = −1.0718, c = 8.5714, d = 0.09689, e = 2.71828 … (the base of natural logarithms) and P (bar) is the absolute pressure of the system. The parameters a, b, c and d were obtained by fitting the measured boiling-point elevation data for the candy solution (R = 0.9882; mean absolute error 0.56°C). Figure 9.2 shows values of the boiling point versus total solids concentration for the candy solution at 1 bar pressure. The experimental data measured in this work are in good agreement with values presented in the literature (Lees and Jackson 1999) and with data obtained from industrial processes, despite differences in the compositions of the solutions used. It is worth noting, in addition, that a plot of the boiling-point elevation data for the solution versus boilingpoint elevation data for water was linear for solids concentrations of 0.6, 0.7, 0.745 and 0.8 (at atmospheric pressure), and thus the model used was capable of satisfactorily representing the experimental data on boiling-point elevation for chewy candy solutions.

9.6.2

Modelling of drying stage

It should be noted that while evaporation is a continuous process with this type of machine, the drying stage is a batch operation. A multilayer perceptron artificial neural network (ANN) with one hidden layer was used to represent the variation of moisture content in the vacuum chamber. The inputs to the ANN were the total solids concentration X in the vacuum chamber feed, the absolute pressure P of the system and the process time t; the moisture content of the candy mass at the vacuum chamber outlet was the output of the ANN. For details, see Oliveira et al. (2008).

Further reading Alikonis, J.J. (1979) Candy Technology. AVI Publishing, Westport, CT. AVP Baker. Technical brochures.

308

Confectionery and Chocolate Engineering: Principles and Applications

Andreasen, G. (1972) Are traditional sugar boiling techniques really the best way. Confect Prod 38 (12): 641–656. Cakebread, S.H. (1972) Confectionery ingredients – Vapour pressures of carbohydrate solutions II. Confect Prod 38 (9): 486–492, 496. Cakebread, S.H. (1972) Confectionery ingredients – Vapour pressures of carbohydrate solutions III. Confect Prod 38 (10): 524–526, 550. Cakebread, S.H. (1975) Sugar and Chocolate Confectionery. Oxford University Press, Oxford. Lees, R. (1972) High boiled sweets – Simple in composition but physical structure is complex. Confect Prod 38 (9): 456–457. Lees, R. (1972) High boiled sweets – Products should not grain nor become sticky. Confect Prod 38 (9): 484. Lees, R. (1980) A Basic Course in Confectionery. Specialized Publications Ltd, Surbiton. Lienhard, J.H., IV and Lienhard, J.H., V (2005) A Heat Transfer Textbook, 3rd edn. Phlogiston Press, Cambridge, MA. Meiners, A. and Joike, H. (1969) Handbook for the Sugar Confectionery Industry. SilesiaEssenzenfabrik, Gerhard Hanke K.G. Norf, Germany. Pratt, C.D. (ed.) (1970) Twenty Years of Confectionery and Chocolate Progress. AVI Publishing, Westport, CT. Robert Bosch/Hamac. Technical brochures. Rohsenow, W.M., Hartnett, J.P. and Cho, Y.I. (eds) (1998) Handbook of Heat Transfer, 3rd edn, McGraw-Hill Handbooks. McGraw-Hill, New York, Chapter 15. Schwartz, M.E. (1974) Confections and Candy Technology, Food Technology Review, 12. Noyes, Park Ridge, NJ. Sullivan, E.T. and Sullivan, M.C. (1983) The Complete Wilton Book of Candy. Wilton Enterprises Inc., Woodridge, IL. Ter Braak. Technical brochures. VDI-GVC (2006). VDI-Wärmeatlas. Springer, Berlin.

Chapter 10

Crystallization

Contents 10.1 10.2

10.3

10.4

10.5 10.6

10.7

Introduction Crystallization from solution 10.2.1 Nucleation 10.2.2 Supersaturation 10.2.3 Thermodynamic driving force for crystallization 10.2.4 Metastable state of a supersaturated solution 10.2.5 Nucleation kinetics 10.2.6 Thermal history of the solution 10.2.7 Secondary nucleation 10.2.8 Crystal growth 10.2.9 Theories of crystal growth 10.2.10 Effect of temperature on growth rate 10.2.11 Dependence of growth rate on the hydrodynamic conditions 10.2.12 Modelling of fondant manufacture based on the diffusion theory Crystallization from melts 10.3.1 Polymer crystallization 10.3.2 Spherulite nucleation, spherulite growth and crystal thickening 10.3.3 Melting of polymers 10.3.4 Isothermal crystallization 10.3.5 Non-isothermal crystallization 10.3.6 Secondary crystallization Crystal size distributions 10.4.1 Normal distribution 10.4.2 Log-normal distribution 10.4.3 Gamma distribution 10.4.4 and population balance Batch crystallization Isothermal and non-isothermal recrystallization 10.6.1 Ostwald ripening 10.6.2 Recrystallization under the effect of temperature or concentration fluctuations 10.6.3 Ageing Methods for studying the supermolecular structure of fat melts 10.7.1 Cooling/solidification curve 10.7.2 Solid fat content

Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

310 310 310 311 312 313 315 317 318 319 322 323 324 326 329 329 330 333 334 345 346 346 346 346 347 347 349 350 350 351 351 351 351 352

310

Confectionery and Chocolate Engineering: Principles and Applications

10.7.3 10.7.4

Dilatation: Solid fat index Differential scanning calorimetry, differential thermal analysis and low-resolution NMR methods 10.8 Crystallization of glycerol esters: Polymorphism 10.9 Crystallization of cocoa butter 10.9.1 Polymorphism of cocoa butter 10.9.2 Tempering of cocoa butter and chocolate mass 10.9.3 Shaping (moulding) and cooling of cocoa butter and chocolate 10.9.4 Sugar blooming and dew point temperature 10.9.5 Crystallization during storage of chocolate products 10.9.6 Bloom inhibition 10.9.7 Tempering of cocoa powder 10.10 Crystallization of fat masses 10.10.1 Fat masses and their applications 10.10.2 Cocoa butter equivalents and improvers 10.10.3 Fats for compounds and coatings 10.10.4 Cocoa butter replacers 10.10.5 Cocoa butter substitutes 10.10.6 Filling fats 10.10.7 Fats for ice cream coatings and ice dippings/toppings 10.11 Crystallization of confectionery fats with a high trans-fat portion 10.11.1 Coating fats and coatings 10.11.2 Filling fats and fillings 10.11.3 Future trends in the manufacture of trans-free special confectionery fats 10.12 Modelling of chocolate cooling processes and tempering 10.12.1 Franke model for the cooling of chocolate coatings 10.12.2 Modelling the temperature distribution in cooling chocolate moulds 10.12.3 Modelling of chocolate tempering process Further reading

10.1

353 354 355 359 359 360 365 367 368 370 371 371 371 372 374 376 378 379 381 382 383 383 384 385 385 386 390 392

Introduction

Crystallization is a process in which an ordered solid phase is precipitated from a gaseous, liquid or solid phase. The liquid phase may be either a melt or a solution; both cases occur in confectionery practice. Crystallization from molten fat is characteristic of chocolate and similar products, the continuous phase of which is a molten fat. On the other hand, crystallization from solution is characteristic of various types of candies, primarily fondant and some hard-boiled sugar confectioneries.

10.2 10.2.1

Crystallization from solution Nucleation

A solid phase is precipitated from a solution if the chemical potential of the solid phase is less that that of the dissolved components to be precipitated from the solution. A solution

Crystallization

311

in which the chemical potential of a dissolved component is the same as that of the corresponding solid phase is in equilibrium with this solid phase under the given conditions and is termed a saturated solution. In order to for crystallization to proceed, this equilibrium concentration must be exceeded as the result of some method for producing supersaturation: cooling the solution and evaporation of the solvent are both used in confectionery technology. These methods can be carried out both continuously and batchwise. In both of these methods, the concentration of the solution (i.e. the chemical potential of the component) is somewhat greater than that corresponding to equilibrium. This excess concentration or chemical potential, which is actually the driving force for crystallization, is termed the supersaturation. If the supersaturation is obtained by cooling, then the difference between the temperature corresponding exactly to saturation and the actual temperature of the solution is termed the supercooling. Provided the supersaturation is not too great, the rate of formation of new crystal nuclei is negligible and the state of the solution corresponds to a metastable region: new crystals are formed only to a limited extent, and crystals already present grow. If the supersaturation is increased further, then the maximum permissible supersaturation is attained, which defines the boundary of the metastable region. When this boundary is exceeded, the rate of nucleation increases rapidly and the crystallization process becomes uncontrolled. Thus, it is expedient to control the crystallization process so that the state of the solution is characterized by a point lying inside the metastable region, which is limited on one side by the above-mentioned boundary and on the other side by the solubility curve. According to Nyvlt et al. (1985), the kinetics of crystallization can be divided into two stages: formation of crystal nuclei (or nucleation), and crystal growth proper. Both of these stages occur simultaneously in a crystallizer, but they will be considered separately in the study of crystallization processes presented below. It is usual to subdivide the formation of crystal nuclei according to the following scheme, depending on the mechanism involved: (1) primary nucleation (in the absence of solid particles), which may be either homogeneous or heterogeneous (catalytically initiated by a foreign surface); (2) secondary nucleation, which can be classified further into apparent, true and contact secondary nucleation. A basic criterion for this distinction is the presence or absence of a solid phase. Secondary nucleation is contingent on the presence of crystals.

10.2.2

Supersaturation

To study the nucleation and growth of crystals as a function of the driving force for crystallization, we define the concept of supersaturation as follows: Δc = c − ceq

(10.1)

We can also use the relative supersaturation, s=

c − ceq Δc = ceq ceq

(10.2)

312

Confectionery and Chocolate Engineering: Principles and Applications

or the supersaturation ratio, S=

c = s +1 ceq

(10.3)

where c is the concentration of the dissolved substance in the supersaturated solution, and ceq is the concentration in the saturated solution. The numerical values of Δc, s and S are dependent on the choice of units in which the concentration of the substance in solution is given. This dependence is demonstrated in the following example. Example 10.1 In an example given by Mullin (1973), the concentration of a saturated aqueous solution of sugar is ceq = 2040 kg sugar/1000 kg water, and the concentration of a supersaturated solution is c = 2450 kg sugar/1000 kg water (both values at 20°C). Taking these concentrations into account, we obtain s=

Δc = ( 2450 − 2040 ) kg sugar 1000 kg water = 410 kg sugar 1000 kg water ceq

s=

Δc = 410 kg sugar 2040 kg sugar = 0.2001, and S = s + 1 = 1.2001 ceq

If we work with concentrations in kg sugar/kg solution, then c = 2450 kg sugar 3450 kg solution = 0.7101 kg sugar kg solution , and ceq = 2040 kg sugar 3040 kg solution = 0.6711 kg sugar kg solution Δc = ( 0.7101 − 0.6711) kg kg solution = 0.039 kg sugar kg solution s = Δc ceq = 0.039 kg sugar 0.6711 kg sugar = 0.058, and S = s + 1 = 1.058 If we work with molarities x, where M(sugar) = 342 and M(water) = 18, xeq = x=

2040 342 = 0.097 2040 342 + 1000 18

2450 342 = 0.114 2450 342 + 1000 18

Δx = 0.114 − 0.097 = 0.017 s=

10.2.3

Δx 0.017 = = 0.175, and S = s + 1 = 1.175 xeq 0.097

Thermodynamic driving force for crystallization

The driving force for crystallization is expressed thermodynamically by the difference between the chemical potentials of a crystalline substance 1 in the supersaturated solution (state ′) and in the saturated solution (state ″):

Crystallization

Δμ1 = μ1′ − μ1′′

313 (10.4)

The expression for the chemical potential of a substance i in a solution is

μi = μi° + RT ln ai

(10.5)

where μi° is the chemical potential in the standard state, R is the gas constant, T is the temperature (K), ai = xiξi is the chemical activity of substance i in the solution, xi is the molarity (molar concentration) of substance i in the solution and ξi is the activity coefficient of substance i in the solution. Taking Eqn (10.5) into account, the form of Eqn (10.4) is Δμ1 = ln (a1 aeq ) = ln Sa RT

(10.6)

where a1 and aeq are the chemical activities of substance 1 in the supersaturated and the saturated state, respectively, and Sa is the thermodynamic supersaturation ratio calculated from the chemical activities. (Carrying on from Example 10.1, if Sa = 1.175, then from Eqn (10.6), Δμ1 = ln (a1 aeq ) = ln Sa = 0.1613 RT and Δμ1 = 0.1613RT = 0.1613 × 8.31434 ( J mol K ) × 293 K = 392.94 J mol ) A simplification of Eqn (10.6) can be used, Δμ ≈s RT

(10.7)

where s is calculated as a ratio of molarities, if the following assumptions are fulfilled: • the ratio of the activity coefficients ξi/ξi.eq is equal to 1; • dissociation of the substance in the solution can be neglected; • ln(1 + s) ≈ s in the whole supersaturation region. In theoretical studies, the driving force for crystallization must always be expressed by means of the exact expression Δμ/RT and not simply on the basis of concentrations (for example, ln 1.175 = 0.1613 ≠ 0.175).

10.2.4

Metastable state of a supersaturated solution

The phase diagram of a two-component solid phase is given in Fig. 10.1. The lower limit of the metastable zone is the solubility of the substance, and the upper limit is the metastable boundary of the solution. The liquid system contains a substance with a positive

314

Confectionery and Chocolate Engineering: Principles and Applications

Region of spontaneous nucleation

Equilibrium comcentration C

c3

c2

No spontaneous nucleation C

Metastability limit

c1

B

A′ A

A″

B´

Solubility

T1

T2

T3

Temperature T Fig. 10.1

Metastable zone [reproduced from Nyvlt et al. (1985), by kind permission of Elsevier].

temperature coefficient of solubility in the temperature interval considered, i.e. dceq/dT > 0. According to Ting and McCabe (1934), the metastable zone is separated into two parts, separated by the dashed line in Fig. 10.1. In the region between the dotted line and the upper boundary of the metastable zone, spontaneous nucleation is possible. The position of the metastability boundary is expressed by the maximum attainable supercooling, ΔTmax = T2 − T1

(10.8)

which corresponds to the maximum attainable supersaturation, ⎛ dceq ⎞ Δcmax = ceq.T (2) − ceq.T (1) ≈ ΔTmax ⎜ ⎝ dT ⎟⎠

(10.9)

Preparation of a solution by the route A′ → A → B represents the polythermal method. Isothermal preparation of a supersaturated solution begins at point A″ and involves evaporation of solvent at a constant temperature up to the saturation point A, and then proceeds through the metastable region up to point C at the metastability boundary. After this limit is passed, the solution is in a labile state, and the solid phase is immediately and spontaneously precipitated. Since the lines of solubility and the metastability boundary are not parallel in general (see later), ceq.T (2) − ceq.T (1) ≈ ceq.T (3) − ceq.T (2)

(10.10)

i.e. the value of Δcmax obtained by the polythermal method does not agree with that obtained by isothermal preparation.

Crystallization

315

The width of the metastable zone can be measured either using the isothermal method or using the Nyvlt polythermal method; details can be found in Nyvlt et al. (1985, pp. 47–65).

10.2.5

Nucleation kinetics

The initial concept of nucleation is the formation and decomposition of clusters of molecules of the dissolved substance – aggregates – as a result of local fluctuations. For each value of the supersaturation of the solution, a critical cluster size can be determined – the critical nucleus – which is in equilibrium with the surrounding medium, and has the same probability of growth as of disintegration. If an aggregate is smaller than the critical size, then the probability of its decomposition is large, whereas clusters larger than the critical size grow spontaneously. In order for a stable nucleus to be formed in a solution, a certain degree of supersaturation must be exceeded. The solubility of small particles depends on their size L according to a relationship given by Ostwald and Freundlich, ln

cL 2σ slM = c∞ RTρc L

(10.11)

where cL is the solubility of small crystals of size L; c∞ is the solubility of large crystals, i.e. c∞ = ceq, and cL/c∞ ≥ 1; σsl is the specific surface energy of the solid–liquid surface (J/ m2); M is the molar mass of the substance dissolved (kg/mol); R = 8.31434 J/mol K is the gas constant; T is the temperature (K); and ρc is the density of the substance dissolved (kg/m3). The fact that small particles are more readily soluble can be easily shown: ∂ ln (cL c∞ ) 2σ M = − sl (1 L2 ) RTρc ∂L c 2σ M ∂cL = − L sl (1 L2 ) < 0 RTρc ∂L i.e. cL decreases if L is increased, and vice versa. This phenomenon is known as Ostwald ripening. For some values of σsl, see Nyvlt et al. (1985, pp. 71–73, 309). For example, the value of σsl for KCl is 35 J/m2, that for cholesterol is 17 J/m2 and that for BaSO4 is 116 J/m2. The usual values are between 20 and 200 J/m2 (the values for sparingly soluble substances are in the region of 100 or higher).

Example 10.2 Let us calculate the ratio of the solubilities of crystals of radii L10 = 10 μm and L1 = 1 μm (the corresponding concentrations are denoted by c10 and c1, respectively), supposing that all other parameters in Eqn (10.11) are unchanged. With such a supposition, Eqn (10.11) can be written as ln(c10/c∞) = K/10 and ln(c1/c∞) = K, where K is a dimensionless constant. After some algebraic manipulations,

316

Confectionery and Chocolate Engineering: Principles and Applications

⎛ c10 ⎞ ⎜⎝ ⎟⎠ c∞

10

=

c1 c∞

9

c ⎛c ⎞ and ⎜ 10 ⎟ = 1 ⎝ c∞ ⎠ c10

and, since c10/c∞ > 1, therefore c1/c10 >> 1.

The classical theory of nucleation states that clusters of particles are formed in solution according to the following scheme: a + a ↔ a (2) a ( 2 ) + a ↔ a (3 ) a (i − 1) + a ↔ a (i ) As soon as these clusters attain a critical size corresponding to the relationship given in Eqn (10.10), the intermolecular forces between the particles within a cluster begin to predominate over the effect of the surrounding particles, and the cluster becomes stable. Taking this mechanism into consideration, a relationship can be given for the rate of nucleus formation: dN ΔG ⎤ = k ′ exp ⎡⎢ − dt ⎣ kT ⎥⎦

(10.12)

where dN/dt is the rate of increase of the number of nuclei (s−1), k′ is a constant (s−1), k = 1.38062 × 10−23 J/K (the Boltzmann constant), T is the temperature (K) and G is the Gibbs free enthalpy (J). According to the classical expression of Nielsen (1964, 1969), dN K′ ⎞ ⎛ = Ω exp ⎜ − 3 ⎝ T log 2 S ⎟⎠ dt

(10.13)

where Ω is a pre-exponential factor (s−1), K′ is a constant (K3), T is the temperature (K) and S is the supersaturation ratio (calculated from the chemical activity). It follows from Eqn (10.13) that the width of the metastable region decreases with increasing saturation temperature of the equilibrium solution. An empirical relationship has been proposed by Tobvin and Krasnova (1949, 1951) and by Akhumov (1960) for the dependence of Smax on temperature: B Smax = 1 + A exp ⎛ ⎞ ⎝T ⎠

(10.14)

where A and B are constants. During cluster formation, N particles (atoms, molecules or ions) of the given substance in the bulk of the original phase are transferred from the original phase 1 into the final phase 2. This process is accompanied by a change in the Gibbs free enthalpy of ΔG = − Δμ N + G1 ( N )

(10.15)

Crystallization

317

where Δμ is the free enthalpy difference of a single species in the phase considered, and G1(N) is the free enthalpy, which depends on the formation of an interphase boundary and on the translational and rotational movement of the cluster. Although there are other approaches to the kinetics of nucleation, it should be emphasized that a correct application of the classical theory leads in most cases to an interpretation of any given experiment with equal success. Moreover, it seems that the rate of homogeneous nucleation in melts is generally described well by the classical theory.

10.2.6

Thermal history of the solution

Of the many factors affecting the width of the metastable zone, the thermal history of the solution is particularly interesting. It has long been known that solutions that have been maintained at a temperature sufficiently higher than the equilibrium temperature for several hours have broader metastable zones or slower nucleation than solutions whose temperature has not increased much above the equilibrium temperature. Theoretical considerations have demonstrated that the experimental data can be explained by assuming that there is a change in the mean subcritical cluster size, produced by a deviation from its steady-state value, and that the rate of change of this deviation can be described by a first-order kinetic equation, dN ( n ) = const. ( N − N eq ) dt

(10.16)

ln ( N − N eq ) = −const. t + C

(10.17)

− or

where N is the number of characteristic cluster sizes at time t, Neq is the equilibrium number of characteristic cluster sizes corresponding to the temperature of the solution, t is the time of overheating, C is an integration constant, n is an arbitrary degree of aggregation and N(n) is the number of characteristic sizes of clusters which have an aggregation degree n. (N − Neq) is proportional to the width of the metastable zone, and it decreases if t (the time of overheating) is increased. For details, see Nyvlt and Pekárek (1980) and Nyvlt et al. (1985, pp. 85–94). 10.2.6.1

Influence of mechanical action on the metastable zone

Unstirred solutions have broader metastable zones than have stirred solutions (Mullin and Osman 1973, Garside et al. 1972). According to the theory of local isotropy, regions with an isotropic character are formed even in very strongly stirred solutions, with a size corresponding to the intensity of stirring. The overall volume of the solution can be divided into a large number of elementary volumes. The number of elementary volumes in an isotropic region depends on a quantity w characterizing the intensity of stirring. The greater the effect of stirring, the greater are the supersaturation and the temperature of the solution. This effect can be described by the following equation: dcn = k (c0 − cn ) or cn − cmin = (c0 − cn )[1 − exp ( − kw )] dw

(10.18)

318

Confectionery and Chocolate Engineering: Principles and Applications

where w is the intensity of stirring, cmin is the minimum value of the concentration in the vicinity of a cluster provided that the stirring does not lead to exchange of elementary volumes, c0 is the average concentration in the solution, cn is the decreased concentration of the substance in the isotropic region around a cluster and k is a constant. 10.2.6.2

Effect of viscosity of solution on the width of the metastable zone

In very viscous solutions, the nucleation rate and thus also the width of the metastable zone are a function of the viscosity of the solution (Mullin and Leci 1969, Pacák and Sláma 1979). Above a certain critical viscosity, the nucleation rate in the solution decreases with increasing viscosity, even if the supersaturation of the solution increases at the same time.

10.2.7

Secondary nucleation

The mechanisms of nucleation resulting from the presence of crystals in a supersaturated solution are generally termed secondary nucleation. These mechanisms can be separated into three groups, according to Botsaris and Denk (1970) and Nyvlt (1973b, 1978); these groups differ in the source of crystal nuclei. (1) Apparent secondary nucleation. The types of apparent secondary nucleation are: • Seeding with crystal dust (dust breeding), which occurs when a supersaturated solution is seeded with untreated crystals. • Polycrystalline breeding: here, it is necessary for the crystal growth to occur at such high supersaturation values that the crystals do not grow regularly but form polycrystalline aggregates. • Macroabrasion can become important during intense stirring of suspensions. In the case of the attrition mechanism of macroabrasion, the rate of nucleation does not depend markedly on the supersaturation (Asselbergs and De Jong 1972). According to Nyvlt (1981a), the rate of this type of nucleation is dependent on the number of nuclei formed by macroabrasion, the mean retention time of the solution and the rate constant of macroabrasion; the latter is largely determined by the hardness of the crystals and the quality of the crystal surface. (2) True secondary nucleation. It is difficult to distinguish between the various mechanisms of true secondary nucleation (see below), but in some instances it is possible: • formation of nuclei from the solid phase, i.e. from a seed crystal; • formation of nuclei from a dissolved substance in solution; • formation of nuclei from a transition phase at the crystal surface. The kinetics of true secondary nucleation can be described by a modification of the Becker–Döring equation in the form (Nyvlt 1981b) ⎡ ⎤ dN N C = k ′ exp ⎢ − 2 ⎥ dt ⎣ ln (w weq ) ⎦

(10.19a)

where dNN/dt (s−1) is the number of nuclei coming into existence by true secondary nucleation per unit time, k′ (s−1) is the rate constant of true secondary nucleation, C

Crystallization

319

is a constant, ‘ln’ denotes the natural logarithm, w is the concentration of the solution, weq is the solubility and W is a reference concentration close to the saturation point. An approximate form of Eqn (10.19a) was given by Nyvlt (1972), dN N dt = kN Δw n

(10.19b)

where kN is the rate constant, Δw = W − weq, W is a reference concentration close to the saturation point, n = (Δw/W)N′ is an exponent and N′ is the number of particles forming a crystal nucleus. This is the power law that is widely used for describing the kinetics of nucleation in addition to Eqns (10.12) and (10.13). (3) Contact nucleation. This mechanism predominates in stirred crystallizers. It occurs when a crystal is contacted with a glass rod or various other materials, and even this contact induces nucleation. In contrast to simple abrasion, which can also occur in undersaturated solutions, this mechanism is always connected with growth of the seed crystal, where visible crystal damage need not occur.

10.2.8

Crystal growth

With the development of industrial crystallization, ever greater attention has been devoted to questions connected with the growth rate of crystals. The rate of growth can be characterized in several ways, the most obvious way being the linear growth rate dL/dt, which expresses the rate of change of a characteristic crystal dimension L with time. Other characteristic parameters concern the surface area and the volume of the crystals: in the case of regular (model) geometric bodies, the relationships between these parameters can easily be expressed (Nyvlt 1981a). The most marked and obvious property of crystals is their shape, which differs for different substances. Explanations for this variability have been sought in the energy conditions in the crystal lattice, leading to different rates of growth for different individual planes.

10.2.8.1

ΔL law: Constant growth of crystals

Constant growth was described by McCabe (1929), who postulated a ‘ΔL law’, whereby crystallographically equivalent faces of similar crystals would grow at the same rate, i.e. dL/dt = constant: dL ⎞ L (t ) = L0 + ⎛ Δt = L0 + ΔL ⎝ dt ⎠

(10.20)

where L0 (m) is the original size of the crystal; dL/dt (m/s) is a constant, equal to the linear rate of growth of the size; Δt (s) is the duration of crystal growth; and ΔL (m) is the increment of crystal size during a time interval Δt. The implicit meaning of the relationship described by Eqn (10.20) is that dL/dt is independent of L0. Moreover, since the weight of a crystal is proportional to the cube of its size, if the ΔL law holds, then the mass distribution does not change during crystallization. Consequently, the weight ratio R for bulk crystal growth can be calculated from

320

Confectionery and Chocolate Engineering: Principles and Applications

Table 10.1

Data for demonstration of the ΔL law in Example 10.3.

w(i)

L(0) (mm)

L(1)

L(2)

S(0)

S(1)

S(2)

0.1 0.27 0.43 0.15 0.05 1

0.248 0.183 0.119 0.078 0.059

0.36 0.295 0.231 0.19 0.171

0.383 0.318 0.254 0.213 0.194

0.001525 0.001655 0.000725 7.12 × 10−5 1.03 × 10−5 0.003986

0.004666 0.006932 0.0053 0.001029 0.00025 0.018176 R(1) = 4.559978

0.005618 0.008683 0.007046 0.00145 0.000365 0.023162 R(2) = 5.810684

∑ wi ( L0,i + ΔL ) 3 ∑ wi ( L0,i )

3

R=

(10.21)

where wi is the mass ratio of the i-th crystal fraction, L0,i is the original size of the i-th crystal fraction and ΔL is the increment during Δt. McCabe’s ΔL law is important also because, in a sense, all other theories of crystal growth treat it as a starting point, the deviations from which are to be explained. Example 10.3 shows how the ΔL law works in practice.

Example 10.3 In Table 10.1, w(i) is the mass ratio (unchanged after t1 and t2 minutes); i = 1, 2, 3, 4, 5 labels the various fractions; L(0), L(1) and L(2) are the sizes of the crystals at times t0, t1 and t2, respectively; S(0) is the denominator of Eqn (10.21); S(1) and S(2) are the numerator of Eqn (10.21); R(1) = S(1)/S(0); and R(2) = S(2)/S(0). For every fraction, ΔL (t1 ) = L (1) − L ( 0 ) = 0.112 mm ΔL (t2 ) = L ( 2 ) − L ( 0 ) = 0.135 mm The constant linear rate of crystal growth means that

[ L (1) − L (0 )]t1 = [ L (2 ) − L (0 )]t2 = [ L (2 ) − L (1)] (t1 − t2 ) The actual values of t2 and t1 are uninteresting from our point of view. However, we can calculate the constant linear growth rate of the crystals using, for example, the values t1 = 100.00 min and t2 = 120.54 min. Then, dL 0.112 0.135 0.135 − 0.112 = = ≈ 0.00112 mm min = 100 120.54 120.54 − 100.00 dt The original bulk weight of the crystals has grown by a factor of R(1) = 4.56 times after t1 minutes and R(2) = 5.81 times after t2 minutes. Procedure of calculation above The ratios of the various fractions remain unchanged during the crystallization.

321

Crystallization

The differences between the values in columns L(1) and L(0) are equal to 0.112 mm (e.g. 0.36 − 0.248 = 0.171 − 0.059 = 0.112). The differences between the values in columns L(2) and L(0) are equal to 0.135 mm (e.g. 0.318 − 0.183 = 0.213 − 0.078 = 0.135). These equal differences correspond to ΔL. Let us now calculate the values of S(0), S(1) and S(2) for fraction 3, with w(i) = 0.43, according to Eqn (10.21): S ( 0 ) = 0.43 × 0.1193 = 0.000725 S (1) = 0.43 × 0.2313 = 0.0053 S ( 2 ) = 0.43 × 0.2543 = 0.007046

∑ S (0) = 0.003986, ∑ S (1) = 0.018176, ∑ S (2) = 0.023162 R (1) =

S (1) 0.018176 = = 4.559978 S ( 0 ) 0.003986

and S ( 2 ) 0.023162 = = 5.810684 S ( 0 ) 0.003986

R (2) =

In Eqn (10.21), it is supposed that the weight of a crystal is proportional to the cube of its size; the proportionality factor is simplified by the formation of a ratio. For further details, see Nordeng and Sibley (1996), Kile et al. (2000) and Kile and Eberl (2003). 10.2.8.2

Deviations from the ΔL law: Size-proportionate growth of crystals

However, it has been found in a number of experimental studies that substantial deviations from the ΔL law occur in some systems; for example, in a stirred suspension of crystals, large crystals mostly grow faster than small crystals. The deviations may be consequences of Ostwald ripening (the solubility of crystals is dependent on crystal size), differences in diffusion rate for crystals of different size and dependence of the surface integration mechanism on the crystal size. Proportionate (size-dependent) growth, evidenced in both natural and synthetic crystal systems, appears to account better for observed crystal size distributions (CSDs). Simple mathematical arguments that favour proportionate rather than constant growth for most natural systems have been presented by Eberl et al. (2002). Proportionate growth can be approximated by (Kile and Eberl 2003) X j +1 = X j + k j X j

(10.22)

Proportionate growth has also been approximated, contrary to Eqn (10.20), as dr = kr dt and ascribed (where k is constant) to:

(10.23)

322

Confectionery and Chocolate Engineering: Principles and Applications

• an accelerated solution velocity around larger crystals; • a greater density of dislocation defects on the surfaces of larger crystals; and • effects of lattice strain as a function of crystal size. In both Eqn (10.21) and Eqn (10.22), kj may contain inherent randomness. According to the law of proportionate effect, kj is replaced by a random number εj in an equation similar to Eqn (10.22), where εj usually varies between 0 and 1. Such randomness is required to produce a log-normal CSD, which is one of the most commonly observed CSD shapes. Constant growth can be distinguished from proportionate growth by the effects that the growth mechanisms have on the shapes of CSDs. Constant growth maintains the absolute size differences between crystals as the mean size increases because such growth can be described by adding the same layer thickness to each crystal per unit time. For example, if one crystal is 2 mm smaller than another at the beginning of growth, this 2 mm size difference will be maintained throughout the growth process. Proportionate growth, however, maintains the relative size differences between crystals because growth is modelled by multiplying each size by a constant. In other words, if one crystal is twice the size of another at the beginning of proportionate growth, it will remain twice the size as growth proceeds. There is some dispute as to whether only proportionate growth can generate and maintain a log-normal CSD, and whether only proportionate growth can maintain the theoretical shape of the universal steady-state curve expected from Ostwald ripening (see later) after ripening has ceased.

10.2.9

Theories of crystal growth

The theories of crystal growth can, in principle, be divided into two broad categories: theories dealing with crystal growth from a purely thermodynamic point of view, and theories dealing with the actual kinetics of crystal growth that attempt to describe the effects of external parameters (such as concentration, temperature and pressure) on the final crystal shape and also determine the effect of these parameters on the rate of growth of the individual crystal faces. Since we are dealing in this book with industrial crystallization processes and are focusing on the technological questions of confectionery production, only the boundary layer theory and the diffusion layer model will be discussed here. 10.2.9.1

Boundary layer theory

Volmer (1939) determined experimentally the existence of a boundary layer between the mother phase and the crystal, which adheres strongly to the crystal surface and in which the structural species (molecules, atoms or ions) move. A complicated potential surface can be assigned to the crystal surface, where the valleys correspond to possible resting positions of adsorbed particles, and the peaks are a measure of the potential energy that the adsorbed species must overcome to change their position. This energy is lower that that necessary to leave the Volmer layer and for a transition into the mother phase. The coefficient D of surface self-diffusion of an adsorbed species was given by Taylor and Langmuir (1933) by the equation D=

d2 4tp

(10.24)

Crystallization

323

where d is the average distance between adsorbed species in the boundary layer and tp is the mean period of time spent by a particle in a position corresponding to a potential valley. Volmer’s discovery of the existence of the boundary layer and of surface diffusion of species within this layer was especially important for the development of all modern theories of crystal growth, and from the point of view of practical applications. 10.2.9.2

Diffusion layer model

The diffusion theory of crystal growth is one of the oldest theories in this field. According to this model, the crystallization process is separated into the following steps: (1) (2) (3) (4)

transfer of the substance to the diffusion layer; diffusion of the substance through the diffusion layer; incorporation of particles of the substance into the crystal lattice; removal of heat released during crystal growth from the crystal into the mother phase. The diffusion rate can generally be described by Fick’s first law in the form dm DA (c − ck ) = dt δ

(10.25)

where dm/dt is the amount of substance diffusing per unit time through an area A, D is the diffusion coefficient, c is the concentration of the substance in the mother phase (c > ck > ceq), ck > ceq is the concentration of the substance at the crystal surface, inside the Volmer layer, and δ is the thickness of the Volmer layer. According to the investigations of Nyvlt and Václavu (1972) and Garside and Mullin (1968), Eqn (10.25) can be written in the form dm g = kG A (c − ceq ) dt

(10.26)

where kG includes the ratio D/δ and is a formal rate constant, and g is an exponent (= 1–2).

10.2.10

Effect of temperature on growth rate

It is known that a rise in temperature promotes diffusion. For a broad interval of temperature, the usual form of the diffusion coefficient D is B D = AT n exp ⎛ − ⎞ ⎝ T⎠

(10.27)

where n, A and B are positive constants, and T is the temperature (K) (Liszi 1975, p. 286). It can be shown from Eqn (10.27) that ∂D B = [ nAT n−1 + ABT n−2 ] exp ⎛ − ⎞ > 0, ⎝ T⎠ ∂T

324

Confectionery and Chocolate Engineering: Principles and Applications

i.e. if the temperature is raised, D is increased. For a smaller interval of temperature, a linear approximation holds: D = D0 (T0 + aΔT )

(10.28)

where a (> 1) is a constant. The expression for the growth rate in Eqn (10.26) contains the concentration difference c − ceq, which is also a function of temperature, and this usually decreases under the effect of a temperature rise. But this decrease is compensated by the increase in the diffusion coefficient D. Consequently, a rise in temperature speeds up the growth rate. This effect is important from the point of view of confectionery technology as well: in the production of grained sweets, the pulling operation starts a slow crystallization process, which must not be completed during shaping but must be completed during storage before the product leaves the plant. Therefore, before packaging, an overnight relaxation is needed in a warm room at about 45–50°C to complete the crystallization.

10.2.11

Dependence of growth rate on the hydrodynamic conditions

McCabe and Stevens (1951) demonstrated that the mean linear rate of crystal growth depends on the relative velocity u of the liquid and solid phases according to the relationship ⎛ dL ⎞ ⎝ dt ⎠

−1

1⎞ ρ ⎛ 1 =⎜ + ⎟ c ⎝ kd ki ⎠ Δc

(10.29)

where L is the linear size of the crystal, kd (kg/m2 s) is the rate constant of diffusion, ki (kg/m2 s) is the rate constant of incorporation of particles into the crystal lattice, ρc is the density of a solution of concentration c, and Δc is the concentration difference, which serves as the driving force. If u → ∞ (i.e. for very intense stirring) and ki << kd → ∞, then ⎛ dL ⎞ ⎝ dt ⎠

−1

⎛ 1⎞ ρ =⎜ ⎟ c ⎝ ki ⎠ Δc

(10.30)

If u → 0 (the rate of crystal growth is diffusion controlled) and kd << ki, then ⎛ dL ⎞ ⎝ dt ⎠

−1

⎛ 1⎞ρ =⎜ ⎟ c ⎝ kd ⎠ Δc

(10.31)

According to Karpinski (1980), the rate of crystal growth can be expressed as a function of dimensionless criteria in the usual form of a Froessling correlation (see the discussion of the Colburn–Chilton analogy in Section 1.4.2): Sh = a + bRe c Sc d

(10.32)

where a is a constant (a = 2 for a sphere, 2 6 for a tetrahedron and 2 2 for an octahedron; if Sh > 30, then a = 0 can be used), b is a constant (0.33–0.79), c is a constant (0.5–0.6) and d = 1/3. The most commonly used form of Eqn (10.32) is

325

Crystallization

Sh = 2 + bRec Sc1 3

(10.33)

The Sherwood number is defined as Sh =

kd L ρs D

(10.34)

where ρs is the density of the solution. It should be mentioned that in this case an equivalent rate constant ke would be more correct than kd, where 1 1 1 = + ke kd ki

(10.35)

because no distinction can be made between kd and ki during intense stirring. The Reynolds number is defined as follows: For a single crystal: Re =

uLρs η

(10.36)

where η is the dynamic viscosity of the supersaturated solution. For crystals growing in a fluidized bed: Re =

uLρs ηε

(10.37)

where ε is the porosity of the fluidized bed. For a rotating disc: Re =

ω d 2 ρs η

(10.38)

where d is the diameter of the rotating disc and ω is the frequency of rotation. For growth in a stirred suspension: Re =

L4 3M ′1 3 ρs η

(10.39)

where M′ is the energy dissipated by the stirrer per unit amount of suspension, i.e. M ′ ( kg m 2 s3 ) =

W msusp

326

Confectionery and Chocolate Engineering: Principles and Applications

The Schmidt number is defined as Sc =

10.2.12

η ρs D

(10.40)

Modelling of fondant manufacture based on the diffusion theory

We illustrate the calculation of the growth rate constant ke with an example. Example 10.4 Fondant mass is produced with a continuous crystallizer (‘fondant gun’), which is a typical MSMPR (mixed suspension, mixed product removal) crystallizer. It has a capacity of 360 kg/h, and the power consumption W of its motor that produces dissipation energy is 3 kW. The characteristic size L of the (sugar) crystals in the fondant is 20 × 10−6 m. The density ρs of the supersaturated sugar/glucose syrup solution used is 1.4 × 103 kg/m3. The (mean) dynamic viscosity η of the solution is 30 Pa s. The diffusion coefficient D of the sugar crystals is 2.5 × 10−10 m2/s. Let us calculate the growth rate constant ke by using the equation Sh = 2 + 0.6 Re1 2Sc1 3 During 1 s, 360 kg/3600 = 0.1 kg of fondant solution is produced with consumption of 3 kJ of energy; consequently, M′ = 3 kW/0.1 kg = 3 × 104 W/kg. Re =

L4 3M ′1 3 ρs 1.4 × 103 43 13 = ( 20 × 10 −6 ) (3 × 10 4 ) × = 7.7 × 10 −4 30 η

Sc =

η 30 = = 8.57 × 107 ρs D 1.4 × 103 × 2.5 × 10 −10

Since Sc > 30, the Sherwood number can be calculated: Sh = 0.6 Re1 2 Sc1 3 = 0.6 × (7.7 × 10 −4 ) (8.57 × 107 ) = 0.6 × ( 2.77 × 10 −2 ) ( 4.34 × 102 ) = 7.21 12

Sh =

13

ke ( 20 × 10 −6 ) ke L = 7.21 = ρs D (1.4 × 103 )(2.5 × 10−10 )

ke = 7.21 × 1.4 × 103 ×

2.5 × 10 −10 = 0.126 kg m 2 s 20 × 10 −6

Calculation of growth rate of the sugar crystals The recipe for the fondant mass is as follows: sugar, 80 kg; glucose syrup, 13 kg (80% dry content = 10.4 kg, dextrose = 40%); yield, 100 kg fondant; water content, 9.6 kg (%).

Crystallization

327

The boiling point of the sugar solution is 120°C; at the end of cooling and crystallization, the temperature is 30°C. The concentrations at the end of cooking are: sugar, 80 kg; glucose syrup dry content, 10.4 kg, which is distributed into two parts: • ‘dextrose’, 10.4 kg × 0.40 = 4.16 kg; • dextrins, 6.24 kg. The molar concentration of sugar is 80 342 = 0.2951 80 342 + 4.16 180 + 9.6 18 + 6.24 2000 (The molar mass of the dextrins has been calculated with M = 2000; however, their molar ratio can actually be neglected.) At 30°C, the solubility of sugar is 218.14 kg/100 kg water; therefore, the total amount of dissolved substance (sugar + glucose syrup) can be calculated from 225 kg/100 kg water (exact values are not at our disposal). The amount y (kg) of the dissolved phase at 30°C can be calculated: y 225 + 100 = , i.e. y = 31.2 kg 100 9.6 Consequently, the amount of dissolved phase is 31.2 kg, and the amount of crystallized sugar is (100 − 31.2) kg = 68.8 kg. The concentrations in the dissolved phase are: sugar, (80 − 68.8) kg = 11.2 kg; ‘dextrose’ (assuming that it is not crystallized), 4.16 kg; dextrins, 6.24 kg; water, 9.6 kg. The molar concentration of sugar in the equilibrium phase is xe =

11.2 342 = 0.0557 11.2 342 + 4.16 180 + 9.6 18 + 6.24 2000

and Δc = x − xe = 0.2951 − 0.0557 = 0.2394 From Eqn (10.29), ⎛ dL ⎞ ⎝ dt ⎠

−1

1⎞ ρ ⎛ 1 =⎜ + ⎟ c ⎝ kd ki ⎠ Δc

dL ⎞ or ⎛ ⎝ dt ⎠

−1

⎛ 1⎞ρ =⎜ ⎟ c ⎝ ke ⎠ Δc

Confectionery and Chocolate Engineering: Principles and Applications

Sucrose molar concentration x

328

0.2951

Boundary of supersaturation

A

Saturation 0.0557

C

30

Fig. 10.2

B3 B3 B1

120 Temperature (°C)

Work curve for fondant crystallization.

or, with the substitution 1/kd + 1/ki = 1/ke, 0.2394 dL ke Δc = = 0.126 × = 2.15 × 10 −5 [ m s ] = 0.215 μm s 1.4 × 103 dt ρc The mean crystal size develops during (20/0.215) s = 93.02 s. Work curve of the fondant crystallizer and calculation of retention time of the sugar crystals Figure 10.2 shows the A–B–C work curve; if the amount of cooling at the beginning of the process is small, the appropriate work curve is A–B1–C and the point B1 is in the metastable region above the dotted line, where spontaneous crystallization is possible. The points B2 and B3 are located in the region where strong crystallization starts. Let us calculate the retention time tr of a crystal particle in the crystallizer, which has a length H = 2.5 m and an inner diameter d = 0.15 m, with a free opening of 50%. The free volume of the fondant gun is 0.152 × 3.14 × 0.5 × 2.5 m3 = 0.0221 m3. The volume flow rate of the fondant mass is (0.1 kg/s)/(1.4 × 103 kg/m3) = 0.71 × 10−4 m3/s. The retention time is defined as tr =

volume of crystallizer volume flow rate

(10.41)

In this case tr = 0.0221 m3/0.71 × 10−4 m3/s ≈ 311 s. This calculation illustrates a process in which, as a result of strong cooling at the gun wall, the sugar content is crystallized, but is then redissolved because the stirrer mixes the developed sugar crystals with supersaturated, insufficiently cooled solution from the axle of the stirrer. This crystallization–solution process is repeated c. 3–4 times (311/93.02 ≈ 3.34) before the ready fondant mass has left the fondant gun.

Crystallization

10.3 10.3.1

329

Crystallization from melts Polymer crystallization

Polymer crystallization controls the macroscopic structure of polymer materials, and thereby determines the properties of the final product. The morphology of polymer crystals is different from that of crystals consisting of small molecules, mainly because of the difference between the connectivity of the chains in a polymer and the absence of such connectivity in assemblies of simple molecules. This affects not only the equilibrium crystal structures but also the kinetics of crystal growth. In this context, fats and oils can be regarded as polymers. The essential fatty acids of cocoa butter, which are typical of the kinds of fat used by the chocolate industry, are: • palmitic acid (P), CH3−(CH2)14−COOH, molecular mass 256; • stearic acid (St), CH3−(CH2)16−COOH, molecular mass 284; • oleic acid (O), CH3−(CH2)7−CH−CH−(CH2)7−COOH, molecular mass 282. The molecular mass of glycerol (C3H8O3) is 92. The molecular mass of the triacylglycerol (TAG) P−O−St is P + O + St + Glycerol − 3H2 O = 860 (In the following, if a TAG is given as, for example, P−O−St, ‘O’ means oleic acid, not oxygen.) The triacylglycerols of cocoa butter and other special confectionery vegetable fats can be regarded as medium-to-large molecules. When a system is cooled from the equilibrium melting temperature Tm to a lower crystallization temperature, polymer crystals can form two-dimensional (2D) lamellar structures in both the melt and the solution via the stages of nucleation, growth of lamellae and aggregative growth of spherulites. The formation of a three-dimensional (3D) crystal structure from a disordered state begins with nucleation and involves the creation of a stable nucleus from a disordered polymer melt or solution. Depending on whether any second phase, such as foreign particles or the surface of another polymer, is present in the system, nucleation is classified as either homogeneous (primary nucleation) or heterogeneous (secondary nucleation). In primary nucleation, the creation of a stable nucleus by intermolecular forces orders the chains into a parallel array. As the temperature falls below the melting temperature Tm, the molecules tend to move towards their lowest-energy conformation, with stiffer chain segments, and this favours the formation of ordered chains and thus nuclei. Since it facilitates the formation of stable nuclei, secondary nucleation is also involved at the beginning of crystallization through heterogeneous nucleation agents, such as dust particles. Following nucleation, crystals grow by the deposition of chain segments on the surface of the nuclei. This growth is controlled by a small diffusion coefficient at low temperatures and by thermal redispersion of chains at the crystal–melt interface at high temperatures. Thus crystallization can occur only in a range of temperatures between the glass transition temperature Tg and the melting point Tm, which is always higher than Tg.

330

Confectionery and Chocolate Engineering: Principles and Applications

Cluster

Particle

Crystallite

Fig. 10.3

Fat crystal network.

As a consequence of their long-chain nature, subsequent entanglements and their particular crystal structure, polymers crystallized in the bulk state are never totally crystalline, and a certain fraction of the polymer is amorphous. Polymers fail to achieve complete crystallinity because polymer chains cannot completely disentangle and align properly during a finite period of cooling. Lamellar structures can be formed, but a single polymer chain can pass through several lamellae, with the result that some segments of the polymer chains are crystallized into lamellae and some other parts of the polymer chains are in an amorphous state between adjacent lamellae. Fat crystal networks are composed of branched, interlinked particles that form a three-dimensional network, the voids of which are filled by liquid fat. The particles, which are aggregates of crystallites, form clusters. The clusters pack in a regular, homogeneous manner and represent the largest structural building block of the fat crystal network. Figure 10.3 shows the hierarchical structure of the concepts of crystallite < particle < cluster.

10.3.2

Spherulite nucleation, spherulite growth and crystal thickening

In this section, the 3D formation of spherulites will be described, based on random nucleation. The derivation of formulae for other cases, for example needle-like growth, is similar. The kinetics of crystallization depend on both diffusion of the polymer and nucleation. The stages of spherulite nucleation and growth are nucleation, growth of spherulites and crystal thickening. Nucleation can occur at any temperature T below the melting temperature Tm when the chemical potential of a monomer in the amorphous state (ga) and in the crystalline state (gc) are the same, i.e. ga(Tm) = gc(Tm). The main driving force for nucleation (Δgm) upon cooling is the difference between the chemical potentials of a monomer in the crystalline state and in the amorphous state, Δgm = ga − gc

(10.42)

Crystallization

331

For slight cooling, ⎛ ΔH m ⎞ Δgm ≈ ΔSm (Tm − T ) = ⎜ (T − T ) ⎝ Tm ⎟⎠ m

(10.43)

where ΔSm is the entropy of fusion (per unit volume) per monomer, and ΔHm is the enthalpy (heat) of fusion (per unit volume) per monomer. The formation of an interface between the amorphous and crystalline phases changes the Gibbs free enthalpy: ΔG = ΔgV + ∑ i Aiσ i

(10.44)

where V is the (total) volume of the nucleus, Ai is the i-th part of the surface area of the nucleus and σi is the free enthalpy associated with Ai. In the case of a spherical nucleus of radius r, 4π ΔG = ⎛ ⎞ r 3 Δg + 4 πr 2σ ⎝ 3 ⎠

(10.45)

Minimizing the free enthalpy with respect to r, we obtain ∂ ( ΔG ) 2σ = 0 → r° = − ∂r Δg

(10.46)

where r is the critical nucleus size. Taking Eqn (10.43) into account, we obtain r° = −

2σTm ΔH m (Tm − T )

(10.47)

Since ΔHm is negative, the critical radius increases with a decrease of the degree of cooling Tm − T. The free enthalpy barrier for nucleation is obtained by substituting the value of the critical radius into Eqn (10.44): 4π 2σ 3 2 2 2 σ ΔG ° = ⎛ ⎞ ( r°) Δg + 4 π ( r°) σ = 4 π ( r°) ⎛ − + σ ⎞ = 4 π ( r° ) ⎝ 3 ⎠ ⎝ 3 ⎠ 3 16 πσ 3Tm2 = 2 2 3 ( ΔH m ) (Tm − T )

(10.48)

It can be seen that ΔG° is proportional to Tm2/(Tm − T)2. The Arrhenius equation can be used in this case: ⎤ ⎡ ΔG ° ⎞ −σ 3Tm2 N = exp ⎛ − = C exp ⎢ 2 2 ⎥ ⎝ kT ⎠ ⎣ 3 ( ΔH m ) (Tm − T ) ⎦

(10.49)

332

Confectionery and Chocolate Engineering: Principles and Applications

where N is the nucleation constant per unit volume and time [in units of nuclei/ (m3 s) = 1/(m3 s)], k is the Boltzmann constant, T is the temperature to which the system is overcooled and C = ln (16π). Two types of nucleation can be observed: homogeneous and heterogeneous. The characteristics of homogeneous nucleation are: • Polymer chains can aggregate spontaneously below the melting point. • The distribution of the nuclei is random. • The generation of nuclei is usually a first-order function of time: ΔP = N Δt

(10.50)

where ΔP is the number of nuclei generated during a time Δt. • The size of the growing units is given by K Φ =⎛ ⎞ ⎝N⎠

34

(10.51)

where Φ is the final average volume of the crystallized units (m3), N is as above and K is the rate constant for the growth of the radius (m/s). For homogeneous nucleation, the relationship between the nucleation rate and temperature is, according to Turnbull and Fischer, E ΔG ° ⎞ N = N 0 exp ⎛ − d − ⎝ kT kT ⎠

(10.52)

Nucleation constant N

where N0 is a material constant [nuclei/(m3 s)] and Ed is the activation energy at the surface of the nucleus. Because ΔG° is proportional to Tm2/(Tm − T)2, the nucleation rate increases with increasing overcooling (i.e. with decreasing temperature); see Eqn (10.52). However, at low temperatures Ed/kT becomes dominant, and since the activation energy Ed decreases in proportion to temperature, nucleation slows down (Fig. 10.4).

Ed/kT dominant

ΔG/kT dominant

Melting point Tm (Overcooling)

Temperature T

Fig. 10.4 Homogeneous nucleation: with increasing overcooling (decreasing temperature), first the term ΔG°/kT becomes dominant, which increases the nucleation rate, and later the term Ed/kT becomes dominant and, as a result, the nucleation rate is decreased.

Crystallization

333

The characteristics of heterogeneous nucleation are: • Heterogeneous nuclei start from impurities. • Nuclei form simultaneously as soon as the sample reaches the crystallization temperature. • The time dependence of nucleation is a zero-order function of time (i.e. independent of time). • The size of the growing units is given by

Φ=

V∞ 1 ≈ N ′V0 N ′

(10.53)

where V∞ is the volume of the system when t → ∞, V0 is the volume of the system when t = 0 and N′ is the number of nuclei per unit volume. The crystal growth of low-molecular-mass materials is described by ⎤ ⎡ A B N = exp ⎢ − − m ⎥ ⎣ T T (Tm − T ) ⎦

(10.54)

where m = 1 if the formation of nuclei is 2D. For a spiral-form crystal where growth occurs with the help of a screw dislocation, N = C (Tm − T )

2

(10.55)

where A, B and C are material constants. For details, see Bodor (1991, p. 214). After the spherulites start to touch each other, a further decrease in the Gibbs free enthalpy can be achieved only by crystal thickening, which is very slow; typically, dfc t ~ log ⎛ ⎞ ⎝K⎠ dt

(10.56)

where fc is the crystallinity ratio, t is the time (s) and K (s) is a constant.

10.3.3

Melting of polymers

High-molecular-mass crystalline materials do not melt at a single, well-defined temperature but over a fairly wide temperature interval. The Thomson equation is valid for the melting point of these materials: 2σ e ⎤ ⎡ Tm = T0.m ⎢1 − ⎥ H Δ ⎣ mD ⎦

(10.57)

where Tm is the melting point for lamellae of thickness D, T0.m is the melting point for an infinite crystal, σe is the surface energy of the basal plane, involving chain folding, and ΔHm is the melting enthalpy per unit volume.

334

Confectionery and Chocolate Engineering: Principles and Applications

Since fats are mixtures of different triglycerides, and every triglyceride has its own typical melting point, the melting process means the successive melting of fractions of different melting point as the temperature increases. From a morphological point of view, melting is not simply the inverse process of crystallization. Crystallization means nucleation and growth, while melting occurs simultaneously at all crystallized parts of spherulites. At interfaces, where incompatible impurities are concentrated, melting proceeds faster.

10.3.4

Isothermal crystallization

10.3.4.1

Kolmogorov–Avrami heuristic phase transition theory

Kolmogorov (1937) and Avrami (1939, 1940, 1941) developed a description of the overall kinetics of phase transitions known as the Kolmogorov–Avrami equation. This equation can be applied to several types of phase transition, from crystallization to cosmological problems. From the point of view of crystallization, the Kolmogorov–Avrami equation can be demonstrated as follows. Early stages of crystallization: Primary crystallization If spherulite nucleation and growth proceed for x minutes, during a given time interval dx, Nm0 dx ρL

(10.58)

nuclei are formed, where N is the formation (rate) constant for nucleation, i.e. the number of nuclei per unit volume and time [nuclei/(m3 s)]; m0 is the mass (kg) of the crystallizing material at t = 0; and ρL is the density of the liquid phase (kg/m3). The nucleation rate is then w=

Nm0 ρL

(10.59)

where w is the growth rate of the number of spherulites (nuclei/s). Moreover, if ρS is the density of the crystallizing solid phase (kg/m3), r is the radius of a crystallizing spherulite (m) and K is the rate constant for the growth of the radius (m/s), then r = Kx

(10.60)

expresses the size of a spherulite formed after time t, and 4 4 3 mS = ⎛ ⎞ r 3 πρS = ⎛ ⎞ ( Kx ) πρS ⎝ 3⎠ ⎝ 3⎠

(10.61)

expresses the total mass of a solid spherulite formed after time t. The mass growth rate of a spherulite is dmS ⎛ Nm0 ⎞ ⎛ 4 ⎞ = wmS = ⎜ ( Kx )3 πρS ⎟ ⎝ ⎠ ⎝ ⎠ dt 3 ρL

(10.62)

Crystallization

335

or ⎛ Nm0 ⎞ ⎛ 4 ⎞ dmS = ⎜ ( Kx )3 πρSdt ⎝ ρL ⎟⎠ ⎝ 3 ⎠

(10.63)

After integration of Eqn (10.63) from x = 0 to x = t, the following is obtained: mS NK 3t 4 πρS = m0 3ρ L

(10.64)

If mL is the mass of the liquid phase, then m0 = mS + mL

(10.65)

and mL m NK 3t 4 πρS = 1− S = 1− 3ρL m0 m0

(10.66)

where mL m = 1− S m0 m0

(10.67)

is the proportionality ratio that shows the relative amount of liquid phase compared with the initial mass of the crystallizing material. Description of the overall crystallization process, including collisions During the crystallization process, the proportionality ratio changes from one (at the beginning of crystallization) to zero (at the end of crystallization). The Kolmogorov– Avrami theory takes the probability of collisions into account by a modification of Eqn (10.63) with the proportionality ratio, dmS′ =

dmS ⎛ Nm0 ⎞ ⎛ 4 ⎞ = ( Kt )3 πρS dt 1 − mS m0 ⎜⎝ ρL ⎟⎠ ⎝ 3 ⎠

(10.68)

or dmS ⎛ Nm0 ⎞ ⎛ 4 ⎞ =⎜ ( Kt )3 πρS dt 1 − mS m0 ⎝ ρL ⎟⎠ ⎝ 3 ⎠

(10.69)

After integration of Eqn (10.69) from t = 0 to t = t, the Kolmogorov–Avrami equation is obtained: NK 3t 4 πρS ⎛ m − mS ⎞ ln ⎜ 0 =− ⎟ ⎝ m0 ⎠ 3ρ L

(10.70)

336

Confectionery and Chocolate Engineering: Principles and Applications

Equilibrium solid

100

Crystallization (%)

80 60 Induction time 40 Maximum growth rate 20 0

Fig. 10.5

0

10

20

30

40

50

60 Time (s)

70

Shape of the Kolmogorov–Avrami equation.

or mL = exp ( − zt 4 ) m0

(10.71)

where z=

NK 3 πρS 3ρ L

(10.72)

From Eqn (10.70), the so-called crystallinity ratio f is given by f (t ) =

mS = 1 − exp ( − zt 4 ) m0

(10.73)

which is a better-known form of the Kolmogorov–Avrami equation. In the general case, f (t ) =

mS = 1 − exp ( − zt n ) m0

(10.74)

where n is the Avrami exponent. Figure 10.5 shows the typical sigmoid shape of the Kolmogorov–Avrami equation, which starts with an induction period (or induction time), then continues with a quasilinear segment corresponding to the maximum growth rate, and finally ends with equilibrium. The induction period is defined by the time at which mS/m0 (as given in Eqn (10.74) differs appreciably from unity. The half-time, which is also characteristic of crystallization, is defined by ln 2 ⎞ t1 2 = ⎛ ⎝ z ⎠

12

(10.75)

Crystallization

337

Table 10.2 Avrami exponent n for different growth and nucleation mechanisms. Nucleation mechanism Growth mechanism

Random addition, n = 1

Instantaneous addition, n = 0

3+1=4 2+1=3 1+1=2

3+0=3 2+0=2 1+0=1

Spherulitic, n = 3 Disc-like, n = 2 Rod-like, n = 1

The value of the exponent n in the Avrami equation (the Avrami exponent) can vary between 1 and 4 depending on the nucleation and growth mechanisms, as shown in Table 10.2. In reality, 100% crystallization is never achieved. Therefore, a correction to Eqn (10.74) is needed: 1−

f (t ) = exp ( − zt n ) fmax

(10.76)

where fmax is the maximum crystalline fraction that can be achieved. For isothermal crystallization, data obtained from differential scanning calorimetry (DSC) can be evaluated by use of the relation t

1 ⎞ f (t )T = ⎛ ⎝ ΔH ⎠

∫ (dHC dt ) dt

0 t∞

(10.77)

∫ (dHC dt ) dt 0

For the sake of completeness, we also give here the corresponding relation for evaluating DSC data for non-isothermal crystallization, in which the variable is the temperature T instead of the time t: T

∫ (dHC dT ) dT

1 ⎞ T0 f (T ) = ⎛ ⎝ ΔH ⎠ T∞

(10.78)

∫ (dHC dT ) dT

T0

where HC is the enthalpy of crystallization, and ΔH is the total enthalpy of crystallization (i.e. for 100% crystallization). In the case of mL/m0 = 0.5, t1/2 = ln 2/z can easily be determined from a plot of f(t) vs t; see Eqn (10.77). The reasons for deviations from the Avrami equation can be: • simultaneous appearance of different growth mechanisms (see Table 10.2); • the influence of impurities on crystal growth; • the density of the growing phase is not uniform (it is higher in the internal region), so that

338

Confectionery and Chocolate Engineering: Principles and Applications

mL = exp ( − zt 4 At − m ) m0

(10.79)

where the factor At−m takes the time dependence of ρS into account; • the molecular-mass distribution can influence the kinetics of crystallization. Analysis of dilatometry data using the Avrami equation Let the volume of the (total) crystallizing mass m0 be Vt at a given instant of time, and let the final volume be V∞ = m0/ρS when crystallization has ended. Then mL mS + ρL ρS

(10.80)

m0 = mS + mL

(10.81)

Vt = Since

therefore Vt =

mL m0 − mL m0 1⎞ ⎛ 1 + = + mL ⎜ − ⎟ ⎝ ρL ρS ⎠ ρL ρS ρS

(10.82)

Taking into consideration that V∞ =

m0 V V , ρL = 0 , ρS = ∞ m0 m0 ρS

(10.83)

Eqn (10.82) becomes ⎛m ⎞ Vt = V∞ + ⎜ L ⎟ (V0 − V∞ ) ⎝ m0 ⎠

(10.84)

mL Vt − V∞ h −h = = exp ( − zt n ) ≈ t ∞ m0 V0 − V∞ h0 − h∞

(10.85)

or

where htD = Vt is the volume of the crystallizing material (at t = 0 and at t → ∞), D is the diameter of the tube of the dilatometer (considered as constant) and ht is the height of the surface of the crystallizing material at time t. From Eqn (10.85), exp ( − zt n ) ≈

ht − h∞ = 1− Xt h0 − h∞

where Xt is the relative crystallinity.

(10.86)

339

Crystallization

The evaluation of dilatometry data according to the Avrami equation can be done using the double logarithm of Eqn (10.85),

{

}

⎡h −h ⎤ ln − ln ⎢ t ∞ ⎥ = ln {− [(1 − X t )]} = ln z + n ln t ⎣ h0 − h∞ ⎦

(10.87)

or by using Eqn (10.76), f ⎞⎫ ⎧ ⎛ ln ⎨− ln ⎜1 − ⎬ = ln z + n ln t ⎝ fmax ⎟⎠ ⎭ ⎩

(10.88)

The linearity of Eqn (10.88) forms the basis of the evaluation. Kerti (2000) applied the Kolmogorov–Avrami equation in the form log {− ln (1 − x )} = log k + n log t

(10.89)

to distinguish between various special vegetable fats. Her results are shown in Table 10.3, where tX is the time for which log {− ln (1 − x )} = log z + n log tX = 0

(10.90)

i.e. − ln (1 − x ) = 1 → x =

e −1 ≈ 63% e

(10.91)

(The base of the logarithm ‘log’ is 10.) At the time tX defined by Kerti using Eqn (10.89), the crystallinity (the value of x) is about 63%. This characteristic time can be used to distinguish the group of cocoa butter and cocoa butter equivalents from the group of cocoa butter replacers and cocoa butter substitutes (Kerti 2000). Unfortunately, the application of the Kolmogorov–Avrami equation in the lipid crystallization literature is inconsistent. Three different fits of the Avrami model have produced significantly different values for the Avrami exponent and constant (Narine et al. 2006). Some researchers suggest that only a portion of the crystallization curve should be fitted with the model, thereby ignoring important information about the entire crystallization process. It has also been suggested that there are a number of line segments within a typical dataset that can each be fitted with the Kolmogorov–Avrami model, and

Table 10.3 Distinction between various confectionery vegetable fats using the Kolmogorov–Avrami equation (rounded values) (reproduced from Kerti 2000, with permission). Parameter N log k tX

Cocoa butter

Cocoa butter equivalent

Cocoa butter replacer

Cocoa butter substitute

6.42 −22.725 3.54

4.572 −17.478 3.823

2.282 −5.52 2.419

2.979 −8.127 2.728

340

Confectionery and Chocolate Engineering: Principles and Applications

researchers have arbitrarily chosen one segment to fit with the model, without any justification. In fact, the crystallization kinetics of most lipid systems are not characterized by the conditions that are assumed to be valid in the Kolmogorov–Avrami model. In order to solve this problem, a modification of the original Komogorov–Avrami model was developed by Narine et al. (2006), the essence of which is the application of the Kolmogorov–Avrami equation to consecutive segments of the curve of solid fat content (SFC) versus time: F1 (t ) = 1 − exp ( − A1t m(1) ) F1∞

(10.92)

where F1(t) is the absolute crystallinity at time t, F1∞ is the crystallinity at some time when either the growth rate or the nucleation conditions change, and A1 and m(1) are the Avrami constant and exponent applicable to the nucleation, growth and dimensionality of the crystallizing lipid over the segment of time where such conditions are constant. In this manner, the Kolmogorov–Avrami equation for step i is given by Fi (t ) m (i ) = 1 − exp ⎡⎣ − Ai (t − τ i ) ⎤⎦ F1∞

(10.93)

The total absolute crystallinity is the sum of the individual absolute crystallinities. Marangoni (1998) emphasized that a modification which uses the form F (t ) m = 1 − exp ⎡⎣ − ( At ) ⎤⎦ F∞

(10.94)

does not solve the problem of fitting but exacerbates it, since such a modification as Eqn (10.94) would transform the Avrami constant from a complex constant for a k-th-order process to a first-order rate constant with units of t−1. However, crystallization of fats from the melt is not a kinetically first-order process. Several experimental techniques can be used to follow the isothermal crystallization of fats as a function of time. In an isothermal DSC experiment, the relative amount of material crystallized as a function of time is calculated by integration of the isothermal DSC curve. The area enclosed by the baseline and the exothermic peak corresponds to the heat of crystallization ΔH. The relative amount of crystallized material is given by Eqn (10.77) or (10.78). In the pNMR (pulsed nuclear magnetic resonance) technique, the solid fat content is measured directly. The samples are first melted to destroy any memory effect and then transferred to a thermostatted water bath at the crystallization temperature. SFC readings are taken at appropriate time intervals. Wright et al. (2001) compared several different techniques used in lipid crystallization studies and concluded that pNMR was the best method to characterize the overall crystallization process. For further details of these techniques, see Foubert et al. (2003).

10.3.4.2

Gompertz model

The Gompertz model was used by Kolek et al. (2000), who claimed that there were several analogies between crystallization of fats and bacterial growth: the reproduction

Crystallization

341

of bacteria is comparable to the nucleation and growth of crystals, and the consumption of nutrients is comparable to the decrease in supersaturation. Kolek et al. (2000) and Vanhoutte (2002) fitted their crystallization curves to a reparametrized Gompertz equation as deduced by Zwietering et al. (1990):

{

}

μe S (t ) = a exp ⎡⎢ − exp ⎛ ⎞ ( λ − t ) + 1 ⎤⎥ ⎝ a⎠ ⎣ ⎦

(10.95)

where S(t) (%) is the SFC curve as a function of time t, a (%) is the value of S(t) when t approaches infinity, μ (%/s) is the maximum crystallization rate, e (= 2.7182818) is the base of natural logarithms and λ (s) is a parameter proportional to the induction time. 10.3.4.3

Aggregation and flocculation models

Berg and Brimberg (1983) noted that the course of fat crystallization is similar to that of aggregation and flocculation of colloids: solid fat is formed by aggregation of dispersed particles, and fat crystals also grow by aggregation. Prior to the main phase, an induction period exists, where the following equations apply: Aggregation:C − C0 = − k1 (t − t0 )

2

(10.96)

Flocculation: ln (C C0 ) = − k3 (t − t0 )

2

(10.97)

where C is the concentration of particles in the liquid phase at time t; C0 and t0 are the initial values of C and t, respectively; the ki are rate constants; and C − C0 = S(t) is the amount of solid fat. For the main phase, the following equations were used: Aggregation:C − C0 = − k2 t − t0

(10.98)

Flocculation: ln (C C0 ) = − k4 t − t0

(10.99)

10.3.4.4

Foubert model

The model of Foubert et al. (2002) was, in contrast to the above models, originally written in the form of a differential equation; however, an algebraic solution assuming isothermal conditions was also developed. The variable chosen is h, which is the amount of remaining crystallizable fat: h (t ) =

a − f (t ) a

(10.100)

where f(t) is the amount of crystallization at time t, and a is the maximum amount of crystallization. The variable h(t) is related to the remaining supersaturation and thus decreases – in contrast to f(t) – in a sigmoidal way with time. In this model, crystallization is represented as if it were a combination of a first-order forward reaction and a reverse reaction of order n with rate constants Ki for each of the reactions:

342

Confectionery and Chocolate Engineering: Principles and Applications

dh = K n h n − K1h dt

(10.101)

Extensive parameter estimation studies revealed that the approximation Kn = K1 is acceptable, and therefore the model can be simplified to dh = K (hn − h) dt

(10.102)

and h (0) =

a − f (0) a

(10.103)

Since the physical interpretation of a parameter called the ‘induction time’ is more straightforward that that of the parameter h(0), and since the induction time can be more easily extracted from a crystallization curve, the function τ(x) is introduced instead of h(0); this is defined as the time needed to obtain x % crystallization. Thus the integrated form of Eqns (10.102) and (10.103) is h = ⎡⎣1 + {(1 − x )(

1− n )

− 1} exp {− (1 − n ) K (t − τ )}⎤⎦

1 (1− n )

(10.104)

Figure 10.6(a) shows a visual comparison of the fit between the Avrami, Gompertz and Foubert models. It can be seen that the Foubert model shows a better fit than the two other models. The adequacy of the various models for describing isothermal fat crystallization was tested statistically by Foubert et al. (2002). This study revealed that the Gompertz and Foubert models always perform better than the Avrami model, and that the Foubert model performs better than the Gompertz model in the majority of cases. Modelling of two-step isothermal crystallization (Foubert et al. 2006a) Since fats are complex mixtures of triglycerides, their crystallization can lead to the formation of many crystal types, owing either to polymorphism or to concomitant growth of several crystal types. This may lead to crystallization curves in which two steps can be identified [Fig. 10.6(b)]. The assumptions used to build a model of crystallization were based on the presence of an isosbestic point [Fig. 10.6(b)], indicating that the first step involves crystallization from the melt to the α phase, and the second step involves a polymorphic transformation from α to β′ without direct crystallization from the melt into β′. The datasets shown in Fig. 10.6(b) were acquired by means of time-resolved X-ray diffraction (tr-XRD). The scattering patterns at small angles (SAXS) represent the long spacings. To develop the two-step model, the Foubert model was reformulated (Foubert et al. 2006a) (see Eqn 10.100) as df a− f ⎤ = K ( a − f ) − aK ⎡⎢ dt ⎣ a ⎥⎦

n

and this equation formed the basis of the proposed two-step model.

(10.105)

Released crystallization heat (J/g)

Crystallization

343

70 60 50 40 30 20 10 0

0

0.2

0.4

Date

0.6

0.8

Avrami

1 1.2 Time (h)

1.4

1.6

Gompertz (a)

0.18

1.8

2

Foubert

β′

0.16

Peak intensity

0.14

1.66h 1.55h 1.44h 1.33h 1.22h 1.11h 1.00h 0.89h 0.78h 0.67h 0.56h

Isosbestic point 0.12 0.1 0.08 0.06 0.04 0.02 0 0.016

0.56h 0.67h 0.78h 0.89h 1.00h 1.11h 1.22h 1.33h 1.44h 1.55h 1.66h 0.018

α

0.02

0.022

0.024

0.026

0.028

0.03

s (Å ) –1

(b)

Fig. 10.6 (a) Visual comparison of fit between the Avrami, Gompertz and Foubert models (isothermal crystallization of cocoa butter as measured by means of DSC). (b) Isothermal crystallization of cocoa butter at 20°C: SAXS (small-angle X-ray scattering) diffraction patterns as a function of time. Time span 2, from 0.56 h onwards; s (horizontal axis) is the wavenumber of the X-rays. In time span 1 (0 h to 10.56 h), the formation of the α modification is practically entirely completed, but the formation of β′ is still at an early stage. In time span 2, the formation of the β′ modification takes place. (c) Example of a two-step process and a fit obtained by combining two Foubert equations. (d) Example of crystallization curves obtained with a fractional model using the following parameter values: Kα = 6/h, Kβ′ = 3/h, nα = 100, nβ′ = 4, τα = 0.01 h and τβ′ = 0.5 h [(a) and (c) reproduced from Foubert et al. (2003), and (b) and (d) from Foubert et al. (2006), by kind permission of Elsevier].

344

Confectionery and Chocolate Engineering: Principles and Applications

Released crystallization heat (J/g)

70 60 50 40 30 20 10 0 0

0.5

1

1.5

2

2.5

3

3.5

Time (h) Combination of two Foubert equations

Data

(c)

1.2

frβ′

Crystallized fraction

1 0.8 0.6 0.4 0.2

frα

0 0

1

2

3

4

Time (h) (d)

Fig. 10.6

Continued

The change in the fractions of α and β′ crystals, frα and frβ′, as a function of time can be written as a function of the rate rα of formation of α crystals from the melt and the rate rβ′ of transformation of α to β′ crystals; see Eqn (10.105), where a = 1: n(α )

rα = K α [1 − ( frα + frβ′ )] − K α [1 − ( frα + frβ′ )] n(β′ )

rβ′ = K β′ (1 − frβ′ ) − K β′ [1 − (1 − frβ′ )]

(10.106) (10.107)

In addition, the following equations hold: dfrα = rα − rβ′ dr

(10.108)

Crystallization

drβ′ = rβ′ dr

345

(10.109)

The initial values of frα + frβ′ were calculated for x = 0.01. Figure 10.6(c) shows the crystallization curves of the α and β′ modifications as a function of time. Figure 10.6(d) shows several crystallization curves obtained with a fractional model by Foubert et al. (2006a). For further details of kinetic formulae, see Smith (2005).

10.3.5

Non-isothermal crystallization

In the study of non-isothermal crystallization, the energy released during the crystallization process is measured as a function of the temperature T by means of the DSC technique. This method was developed by Jeziorny (1978). The relative crystallinity, X(t), is given by X (t ) =

ΔH T ΔH C

(10.110)

where ΔHT (J) is the enthalpy of crystallization released during a temperature change T − T0, and ΔHC is the overall enthalpy of crystallization, which is equal to the area enclosed by the crystallization peak in a plot of H versus T obtained from the DSC data. The crystallization time t can be calculated from the relation 1 t = ⎛ ⎞ T − T0 ⎝Φ⎠

(10.111)

where Φ is the heating or cooling rate (K/s), T is an arbitrary temperature (K) and T0 is the onset temperature (K). The basis of data evaluation may be the Avrami equation, f (t ) =

mS = 1 − exp ( − zt 4 ) m0

(10.73)

where mS/m0 = X(t); see Eqns (10.109) and (10.110). Another approach was proposed by Ozawa (1971), who used a modified Avrami equation of the form K (T ) ⎤ 1− X (t ) = exp ⎡⎢ − ⎣ Φ m ⎥⎦

(10.112)

where K(T) (K/min) is a cooling/heating function, Φ is the heating or cooling rate (K/s) and m is the Ozawa exponent, which depends on the dimensionality of the crystal growth (Ziru 1997). K(T) and m can be determined after linearization of Eqn (10.112) in the usual way (Yuxian 1998). Joson et al. (2003) studied both of these approaches, and the Avrami equation in the original form provided better fitting. Non-isothermal crystallization needs further study. The modelling approaches used for isothermal crystallization (e.g. the Avrami equation) may provide starting points; however, the variables and parameters of the equations describing the isothermal case cannot be

346

Confectionery and Chocolate Engineering: Principles and Applications

used unchanged. It should be emphasized that from the engineering point of view, it seems essential to consider the non-isothermal case.

10.3.6

Secondary crystallization

Crystallization does not always end as predicted by the Kolmogorov–Avrami equation, which can be applied to primary crystallization only. A secondary stage of crystallization can proceed after the first stage, and this process can last for a considerable period of time. The crystallinity vs time relationship in the case of secondary crystallization can be given by the formula t − t0 ⎤ x (t ) = C + D ln ⎡⎢ ⎣ E ⎥⎦

(10.113)

where x(t) (< 100%) is the crystalline mass fraction at time t; C, D and E (s) are constants; and t0 is the time point at the beginning of the secondary crystallization process.

10.4 10.4.1

Crystal size distributions Normal distribution

In the study of crystallization processes, a great deal of attention is devoted to the size distribution of the particles of the product formed. It has been found that the sieve spectrum may be a useful diagnostic indicator of the operation of a crystallizer, and that the data from a sieve analysis can be used to evaluate a number of fundamental kinetic parameters that characterize the crystallization process and can be used to guide the design of machinery. The best-known distribution function (Randolph and Larson 1971) is the normal distribution, defined as follows: f (L) =

⎡ ( L − L50 )2 ⎤ 1 exp ⎢ − 12 2σ 2 ⎥⎦ σ ( 2τ ) ⎣

(10.114)

where σ is the scatter and L50 is the crystal size corresponding to an oversize fraction of 50%. The normal distribution is frequently used, especially to express the distribution of particle sizes of poorly soluble substances.

10.4.2

Log-normal distribution

Crystal size distributions in the larger size region are mostly better expressed by an empirical log-normal distribution (Randolph and Larson 1971, Nyvlt and Cipová 1979): f (L) =

1 ⎡ log 2 ( L L50 ) ⎤ exp ⎢ − 12 2 log 2 σ ⎥⎦ log σ ( 2τ ) ⎣

(10.115)

Crystallization

347

For both the normal and the log-normal distribution functions, the distribution of the number of particles is expressed by the relationship N ( L ) = 0.5 + 0.5 erf ( x )

(10.116)

where for the normal distribution, x=

L − L50 21 2 σ

(10.117)

and for the log-normal distribution, x=

log ( L L50 ) 21 2 log σ

(10.118)

As has been demonstrated by Nyvlt and Cipová (1979), even the log-normal distribution cannot express the experimental data in corresponding coordinates as a straight line; the plot is curved, especially in the region below oversize fractions of 10% and above 90%.

10.4.3

Gamma distribution

The gamma distribution function, f (L) =

La′ exp ( a ′L b ) Γ ( a ′ + 1) ( b a ′ )(a′+1)

(10.119)

is very useful for expressing the CSDs of crystallization products, as for a perfectly stirred continuous crystallizer, its form can be derived theoretically and the values of the parameters a′ and b are related directly to the crystallization process: a′ = 3

(10.120)

dL ⎞ b = 3tr ⎛ ⎝ dt ⎠

(10.121)

See Randolph and Larson (1971). The Rosin–Rammler–Sperling distribution is less useful for describing CSDs (Nyvlt and Cipová 1979).

10.4.4

Histograms and population balance

Histograms of N vs L are important if prepared for equidistant values of the sieve aperture. If the population density of the crystals n(L) can be expressed as a function of their sizes, then a complete description of the crystal size distribution can be obtained by integration from L = 0 to L = ∞ to obtain the following expressions:

348

Confectionery and Chocolate Engineering: Principles and Applications

N = ∫ n ( L ) dL

(10.122)

where N is the number of crystals; L = ∫ n ( L ) L dL

(10.123)

where L is a summary of the crystal size, or the visible crystal size; A = β ∫ n ( L ) L2 dL

(10.124)

where A is the surface area of the crystals and is β a surface shape factor; and m = αρc ∫ n ( L ) L3 dL

(10.125)

where m is the concentration of the crystal mass in a suspension, β is the surface shape factor and ρc is the density of a solution of concentration c. These integrals (Eqns 10.122–10.125) represent the zeroth, first, second and third moments of the distribution. In order to describe a distribution function n(L) for a given model of a crystallizer, the population density must be introduced. 10.4.4.1

Population balance

The balance of the number of particles has the general form Accumulation = input − output To describe this balance, the following assumptions are made: • • • • •

perfect mixing of the suspension in the crystallizer; steady state (constant crystal population density at all times); samples of the suspension correspond to the suspension in the crystallizer; the feed is constant, and does not contain any crystals; the volume of the suspension is constant.

Under these conditions, the balance simplifies to the form dn ∂ ⎡ ⎛ dL ⎞ ⎤ n ( dV dt ) =0 = n + dt ∂L ⎢⎣ ⎝ dt ⎠ ⎥⎦ V

(10.126)

If the crystal growth is controlled by the McCabe ΔL law, ∂ ⎛ dL ⎞ =0 ∂L ⎝ dt ⎠

(10.127)

and at steady state (dn/dt = 0), dn ⎛ dn ⎞ ⎛ dL ⎞ n = + =0 dt ⎝ dL ⎠ ⎝ dt ⎠ tr

(10.128)

Crystallization

349

Batch crystallizer

3 3′

log n

log n

Continuous crystallizer

1

1′ 2

L (a)

2′ L

(b)

Fig. 10.7 Plots of log n vs L for (a) a continuous and (b) a batch crystallizer [reproduced from Nyvlt et al. (1985), by kind permission of Elsevier].

where tr = V/(dV/dt). From Eqn (10.128), we obtain dn n =0 + dL ( dL dt ) tr dn

dL

∫ n = ∫ (dL dt ) t r

(10.129) (10.130)

After integration of these two equations between the limits n and n0, and 0 and L, respectively, we obtain L ⎧ ⎫ n = n0 exp ⎨− ⎬ ⎩ ( dL dt ) tr ⎭

(10.131)

For an MSMPR (mixed suspension, mixed product removal) crystallizer, i.e. a continuous crystallizer, a plot of log n vs L theoretically gives a straight line [Fig. 10.7(a)]. The following reasons can be given for deviations from linearity (Canning 1971; Nyvlt 1973a): • • • • • •

unintentional dissolution of small crystals; separate dissolution of small crystals; deviation from the McCabe ΔL law; internal classification; sampling of a classified product; splitting of the crystals into particles with comparable sizes.

10.5

Batch crystallization

An unseeded batch crystallizer has no input or output. The crystal population density balance (Eqn 10.126) is thus reduced to the form (Randolph and Larson 1971) ∂ ⎡ ⎛ dL ⎞ ⎤ ∂ n V + ( nV ) = 0 ∂L ⎣⎢ ⎝ dt ⎠ ⎦⎥ ∂t

(10.132)

350

Confectionery and Chocolate Engineering: Principles and Applications

If the population density n is related to the overall working volume V, then ∂w ∂ ⎡ ⎛ dL ⎞ ⎤ + w =0 ∂t ∂L ⎢⎣ ⎝ dt ⎠ ⎥⎦

(10.133)

where w = nV. The integration of this partial differential equation is easy if the initial distribution w = w(0; L) is known; however, this assumption is not fulfilled in most cases. For details, see Baliga (1970), Mucsai (1971), Randolph and Larson (1971), Wey and Estrin (1973), Blickle and Halász (1973, 1977) and Nyvlt et al. (1985, p. 228). Figure 10.7(b) shows a plot of log n vs L for a batch crystallizer. The curves 1 → 1′ mean the primary distribution, the curves 2 → 2′ the secondary distribution and the curves 3 → 3′ the resultant distribution, which is the sum of the previous two. The symbol ‘→’ here represents a time shift of the curves. The primary curve (1) has a maximum, which causes a maximum in the resultant curve (3) also.

10.6

Isothermal and non-isothermal recrystallization

Recrystallization is a process involving a change in the size and/or shape of crystals by a mechanism of surface diffusion of the solid or mass transport through the liquid phase. Isothermal recrystallization (or Ostwald ripening) and non-isothermal recrystallization can be distinguished depending on the conditions.

10.6.1

Ostwald ripening

The reasons for the change in the CSD of crystals in contact with the mother liquor follow from the expression for the change in the overall Gibbs free enthalpy of the system, dG = −S dT − V dP + ∑ μi dni + σ sl dA

(10.134)

For a system at equilibrium, G is a minimum, and if dT = 0, dP = 0 and dni = 0, then the surface area A of the interface must be as small as possible (i.e. a minimum). It can thus be expected that spontaneous processes leading to a decrease in the surface area of the solid phase will occur in the heterogeneous system considered here. A decrease in the surface area of the solid phase means dissolution of the small particles. The use of modern experimental techniques has justified Gibbs’s original ideas. According to the Gibbs–Thomson equation, the activity a(r) of a solid substance in a solution and the concentration a0 of the solution in equilibrium with the solid particle are a function of the particle size r: ⎛ a ( r ) ⎞ βvσ sl ln ⎜ = ⎝ a0 ⎟⎠ rkT

(10.135)

where β is the surface shape factor, v is the particle volume, σsl is the surface tension at the solid–liquid interface, k is the Boltzmann constant and T (K) is the temperature. From the Gibbs–Thomson equation, it can be seen that if r1 < r2 < r3, then a(r1) > a(r2) > a(r3), i.e. the smaller the particles are, the greater is the corresponding equilibrium concentration

Crystallization

351

and vice versa. Moreover, the greater the degree of polydispersity in the system, the more readily and rapidly ripening occurs (Glasner 1973). Evidently, the Ostwald–Freundlich equation (Eqn 10.11) and the Gibbs–Thomson equation (Eqn 10.135) are equivalent; both can be derived from the fact that the Gibbs free enthalpy is a minimum in equilibrium. For more details, see Section 16.4.1.

10.6.2

Recrystallization under the effect of temperature or concentration fluctuations

A special case of recrystallization is involved in systems containing medium-size particles. These particles are ‘too large’ for Ostwald ripening, i.e. their changes in solubility as a result of the differences between crystal sizes are negligible according to the Gibbs– Thomson equation. The consequence is that recrystallization of these particles occurs because of the effect of fluctuations in temperature or concentration.

10.6.3

Ageing

Two mechanisms are classified into this category: (1) Recrystallization of primary particles in forms such as needles, dendrites and thin plates into more compact shapes by surface diffusion or mass transport through the liquid phase. (2) Transformation of metastable crystal modifications into stable modifications by dissolving and recrystallization. This process is essential to fat crystallization.

10.7 10.7.1

Methods for studying the supermolecular structure of fat melts Cooling/solidification curve

For the correct moulding of chocolate, it is essential that the crystallization behaviour of the cocoa butter used should be critically examined before the cocoa butter is added to the chocolate mixture. In this respect, the cooling/solidification curve may give valuable information. Since phase transitions are accompanied by thermal effects, these may be indicated by changes in the slope of the cooling/solidification curve. Various fats, in particular cocoa butter, and mixtures of fats give characteristic cooling/solidification curves. The cooling/solidification curve is a time–temperature curve measured during the cooling of cocoa butter or other fats until crystallization occurs under the specified conditions of the test. The steps of the method are immersion of a Shukoff flask filled with a given quantity of molten cocoa butter at a specified depth in an ice–water mixture at 0°C, and recording, during cooling of the cocoa butter under precisely specified conditions, the temperature of the fat at regular time intervals. A plot of temperature versus time is then constructed. The method is illustrated in Fig. 10.8. In Fig. 10.8, the cooling curves of soybean oil and of cocoa butter and other special confectionery fats are presented together, schematically. The first point (temperature value) at which the curves start to diverge is the ‘prime stay point’, then the curve of cocoa butter or other special fat passes through a temperature minimum and a temperature

Confectionery and Chocolate Engineering: Principles and Applications

Temperature (°C)

352

Prime stay point

T(max)

Cocoa butter or other special fat for confectionery

T(min) Soybean oil t(min)

t(max)

Cooling time (min)

Fig. 10.8 Cooling curves of fats, determined with a Shukoff flask. The crystallization is characterized by [T(max) − T(min)]/[t(max) − t(min)].

maximum, and finally the temperature starts to decrease again. A ratio formed from the temperature and time values at the extremities gives a measure of crystallization. The reason for the increase in the temperature of cocoa butter during the cooling is that crystallization processes have taken place in the cocoa butter, and the latent heat of these processes is negative, i.e. they are exothermic (heat-producing) (IOCCC Analytical Method 31 1988, IUPAC 1986).

10.7.2

Solid fat content

The cooling curve indicates an exothermic process but does not give information about the proportion in the solid state in cocoa butter or fats as a function of temperature. Fats are a mixture of different triglycerides, and therefore there is a characteristic temperature region in which the various triglyceride fractions become solid. In this region, the proportion of solid phase increases from 0% to 100% as a function of temperature. A curve of the solid fat content curve of cocoa butter is presented in Fig. 10.9. A characteristic feature of this curve is a certain hardness in the temperature interval 25–30°C; however, at the temperature of the human body (36.5°C) the SFC is practically zero (total melting). There are several techniques for determination of the SFC. NMR (pulsed or continuous-wave) is the only one that makes a direct measurement of SFC; other techniques, such as dilatation and thermal analysis, utilize related properties (changes in specific volume and changes in enthalpy, respectively). Pulsed NMR is the preferred technique and the predominant one in use.

Crystallization

353

100 Hardness

Solid fat content (%)

80

60

40 Heat resistance 20 Waxiness 0 20

25

30

35

40

Temperature (°C) Fig. 10.9

Solid-fat-content curve of cocoa butter.

e

id lin

Liqu Specific volume

x

ne

id li

Sol

Temperature Fig. 10.10

10.7.3

Interpretation of solid-fat-index curve (dilatation curve).

Dilatation: Solid fat index

Dilatation is a classic, simple technique (Fig. 10.10). The change in specific volume (dilatation) gives a guide to the relative proportions of solid fats and liquid oils in a semi-solid sample. The results should be stated as a solid fat index (SFI), which is expressed in mm3/25 g or ml/kg. The SFI is merely an empirical index and an arbitrary approximation of an absolute value, which should never be used to state the percentage of solids.

354

Confectionery and Chocolate Engineering: Principles and Applications

Enthalpy ΔH

Temperature 100

Liquid fat (%)

50

0 Fig. 10.11

10.7.4

Temperature

The melting peak – the change of enthalpy during the melting of fats.

Differential scanning calorimetry, differential thermal analysis and low-resolution NMR methods

Thermal analysis methods such as differential scanning calorimetry (DSC) and differential thermal analysis (DTA) are used to determine the melting properties of a fat, and are based on changes in enthalpy during a temperature–time programme (Fig. 10.11). A plot of enthalpy (H) vs temperature (T) shows a steep fall and then a rise at the temperature of a first-order phase transition (melting) (see Section 3.4.2). This type of determination also relates to merely a function of the percentage of solids, since the melting enthalpy differs from triglyceride to triglyceride and varies considerably between different polymorphs. Integration of the ‘melting peak’ always gives results between 0% and 100% liquid fat and should preferably be stated in the form of an SFI. For further details, see Lund (1983). Low-resolution NMR (pulsed or continuous-wave) is the only technique yet devised that gives a direct measurement of SFC. A charge in motion generates a magnetic field. Protons have ‘spin’ and thus act as tiny bar magnets that tend to align themselves in the direction of a powerful, steady magnetic field in the NMR instrument and precess at a specific angular frequency. After energy is introduced by means of a strong radio frequency pulse, the angle of precession is changed by 90°. This induces a signal in the receiver coil of the instrument, proportional to the number of protons in the solid or liquid phase. When the energy input ceases, certain relaxation processes start and the system returns to its initial state. In SFC determination the so-called ‘spin–spin’ relaxation is significant. In this case, the energy is distributed among protons in the same molecule, and different relaxation times occur in different nuclear environments. On this basis, protons in the solid phase relax much faster than those in the liquid phase, because of the denser and more rigid ‘distribution net’ in the solid. A schematic magnetization decay obtained from a fat sample is shown in Fig. 10.12. The SFC can be calculated by either a direct or an indirect method. In the (simpler) direct method, the calculation employs signals from both the solid fat (S′) and the liquid fat (L) using the formula

Crystallization

355

90° pulse Instrument 'dead time'

So lid

S

y ca

de

S′

Signal

Liquid decay

L

0

L

10

70

105

Time (μs) Fig. 10.12

Principle of solid-fat-content determination by NMR.

Solids ( % ) = 100 ×

f × S′ S = 100 × f × S′ + L S+L

(10.136)

where f = S′/S is a factor that takes into account the ‘dead time’ of the instrument. (The indirect method, by which the f-factor is excluded, is recommended where the accuracy of the results is very important.) The concept of the SFC plays an important role in understanding the crystallization of the fats used in cocoa, chocolate and confectionery manufacture. A conclusion of the above-mentioned studies is that the temperature region of 25–40°C needs particular attention from the point of view of determining the specific heat capacity and enthalpy of substances that contain cocoa butter or special fats. For details of the methods used, see Wunderlich (1990), Dean (1995) and Pungor (1995). For further details, see Bodor (1991, Chapter V), Karlshamns Oils & Fats Academy (1991), Minifie (1999, pp. 848–855) and McGauley (2001).

10.8

Crystallization of glycerol esters: Polymorphism

To study the behaviour of cocoa butter, let us look at the crystallization characteristics of glycerol esters. The triacylglycerol composition of a fat is one of its most important parameters because it governs the physical properties and the polymorphic behaviour of the fat. Polymorphism is defined as the ability of a TAG molecule to crystallize in different molecular packing arrangements (polymorphs or polymorphic forms), corresponding to different unit cell structures, which are typically characterized by X-ray diffraction spectroscopy. Two types of polymorphism exist: enantiotropy and monotropy. Enantiotropic polymorphism is characterized by a greater number of stable crystal forms in a given

356

Confectionery and Chocolate Engineering: Principles and Applications

Table 10.4 Crystal polymorphs. Polymorph

Unit cell

Short spacing(s) (Å)

α β′ β

Hexagonal Orthorhombic Triclinic

4.15 3.8 and 4.2 (both strong) Multiple peaks + one strong (4.6)

temperature range, i.e. the transformations between crystal forms are reversible. In monotropic polymorphism, which is characteristic of cocoa butter as well, only one stable crystal form exists, and the transformation of other crystal forms to the stable form is irreversible. The Gay-Lussac–Ostwald step rule states that if more than one modification can occur, then the most stable modification, which has the lowest free enthalpy, never comes into existence first; instead, the spontaneous decrease of free enthalpy always takes place stepby-step. Although the Gay-Lussac–Ostwald step rule is not a strict natural law – there are exceptions – it mostly provides good guidance about the direction of transitions between modifications. Since Chapman’s study (Chapman 1971), fat polymorphs have been delineated into three main forms, denoted by α, β′ and β, and variations within these main types. The main crystal characteristics of the various polymorphs are summarized in Table 10.4. Schenk and Peschar (2004) discussed the structure of chocolate and the polymorphism of cocoa butter. For further details, see also Larsson (1997). When a melt of a simple TAG is cooled quickly, it solidifies in its lowest-melting form (α), with perpendicular alkyl chains in its unit cell (the angle of tilt is 90°). When heated slowly, this melts but, if held just above this melting point, it will resolidify in the β′ crystalline form. In the same way, a more stable β form can be obtained from the β′ form. The β form has the highest melting point and can be obtained directly by crystallization from solvent. The β′ and β forms have tilted alkyl chains, which permit more efficient packing of the TAGs in the crystal lattice. Glycerol esters with only one type of acyl chain are easy to make and have been thoroughly studied. The results have provided useful guidance, but such molecules are not generally significant components of natural fats (except perhaps after complete hydrogenation). With mixed saturated TAGs such as PStP (P = palmitic acid and St = stearic acid), the β form is only obtained with difficulty, and such compounds usually exist in their β′ form. Among TAGs with saturated (S) and unsaturated (U) acyl chains, symmetrical compounds (SUS and USU) have higher-melting (more stable) β forms – this applies to cocoa butter as well as for other fats. The main TAG components of cocoa butter, according to Jovanovic et al. (1995) are POSt, 16.5–41.2%; StOSt, 22.6–28.8%; POP, 12.0–18.4%. However, the unsymmetrical compounds (USS and UUS) have stable β′ forms. A schematic picture of the ‘chair’ shape of the TAGs and the average composition of cocoa butter are presented in Fig. 10.13. For the determination of mono-oleo disaturated symmetrical triglycerides (SOS) in the oils and fats used in chocolate and in sugar confectionery products, see IOCCC Analytical Method 35 (1990a). For the determination of the composition of the fatty acids in the 2-position of glycerides in the oils and fats used in chocolate and in sugar confectionery products, see IOCCC Analytical Method 41 (1990b).

Crystallization

H2C

P

H2C

P

H2C

St

HC

O

HC

O

HC

O

P

H2C

St

H2C

H2C

POSt 40%

POP 16%

St StOSt 26%

O

P or St Double bond (cis–trans rotation) Typical 'chair' shape of TAGs

Fig. 10.13

357

P or St

Average composition of cocoa butter and the ‘chair’ shape of triacylgylcerols (TAGs).

Elaidic acid (trans) HO O O (Rotation) HO

Fig. 10.14

Oleic acid (cis)

Oleic and elaidic acids – cis–trans isomers.

It should be mentioned that oleic acid and elaidic acid are cis–trans isomers of each other; their schematic geometries are represented in Fig. 10.14. There are important differences between their physical characteristics; for example, the melting point of oleic acid is 16°C, and that of elaidic acid is 51–52°C. In practice, the olefin carboxylic acids that occur as natural components of fats are always of cis structure (Bruckner 1961, p. 564). However, as a consequence of hydrogenation, a certain degree of cis → trans transition takes place, and this is disadvantageous from the point of view of nutrition. Cocoa butter contains oleic acid (the cis isomer). Figure 10.15 shows the double-chain-length (DCL) and triple-chain-length (TCL) arrangements and the short and long spacings of tilted dimers of triglycerides provided by X-ray diffraction. The stable β form generally crystallizes in a double-chain-length

358

Confectionery and Chocolate Engineering: Principles and Applications

Double-chain-length

Triple-chain-length

Long spacing

Long spacing

Short spacing

Tilt Short spacing

Fig. 10.15 Double- and triple-chain-length arrangements and short and long spacings of tilted dimers of triglycerides.

arrangement (β2), but if one acyl group is very different from the others in either chain length or degree of unsaturation, the crystals assume a triple-chain-length arrangement (β3), since this allows more efficient packing of the alkyl chains and head groups. The crystals of this form have the short spacing expected of a β crystalline form but the long spacing is about 50% longer than usual. In the DCL arrangement, the molecules align themselves (like tuning forks) with two chains in an extended line (to give the double chain length) and a third chain parallel to these (see Fig. 10.15 and Table 10.4). Some mixed glycerol esters, which have a TCL form when crystallized on their own, give high-melting (well-packed) mixed crystals with an appropriate second glycerol ester. The methyl groups at the top and bottom of each triacylglycerol layer do not usually lie in a straight line, but form a boundary with a structure depending on the lengths of the various acyl groups. This is called the ‘methyl terrace’. The molecules tilt with respect to their methyl end-planes to give the best fit of the upper methyl terrace of one row of glycerol esters with the lower methyl terrace of the next row of esters. There may be several β2 modifications, differing in the slope of the methyl terrace and in the angle of tilt. Crystallization occurs in two stages: nucleation and growth/thickening. A crystal nucleus is the smallest crystal that can exist in a solution and is dependent on concentration and temperature. Spontaneous (homogeneous) nucleation rarely occurs in fats. Instead, heterogeneous nucleation occurs on solid particles (dust etc.) or on the walls of the container. Once crystals are formed, fragments may drop off and either redissolve or act as nuclei for further crystals. The nucleation rates for the various polymorphs are in the order α > β′ > β so that α and β′ crystals are more readily formed in the first instance, even though the β polymorph is the most stable and is favoured thermodynamically. Crystal nuclei grow by incorporation of other molecules from the adjacent liquid layer at a rate depending on the amount of supercooling and the viscosity of the melt. Figure 10.16 shows the change of the free enthalpy G of several modifications as a function of the temperature T. The free enthalpies of two modifications labelled ‘1’ and ‘2’ are G1 = U1,0 − kT ln Ω1 and G2 = U 2,0 − kT ln Ω 2

(10.137)

Free enthalpy G

Crystallization

U2,0

G2

U1,0

Transition point T0

359

T

G1 G1 G2 Fig. 10.16

Change of free enthalpy of fat crystal modifications as a function of temperature.

where U1,0 and U2,0 are the internal energies of the modifications 1 and 2 at absolute zero temperature, k is the Boltzmann constant, and ln Ω1 and ln Ω2 are the partition functions of the modifications 1 and 2. ΔU0 is the latent heat of the transition at absolute zero temperature, and is defined by U 2,0 − U1,0 = ΔU 0 > 0

(10.138)

Since the bonding relationships between the lattice elements of the various modifications are different, the partition functions are also different and, consequently, the curves of G1 and G2 are not parallel to each other. Generally, if the bonding energy is lower, then the vibration frequencies are also lower and the series of energy levels is denser; therefore, Ω2 increases more quickly with temperature than Ω1 does, and G1 and G2 intersect at a transition point T0. Since U1,0 has been chosen as the zero energy level in Fig. 10.16, the stable modification is represented at temperatures less than T0 by the curve G1, and by the curve G2 at temperatures higher than T0. The change between the modifications takes place through transformation of the crystal network. The condition for such a transformation is the existence of sufficient mobility of the crystals, which is rather limited in general; however, the mobility increases quickly with any temperature rise. This fact explains how the origin of unstable modifications and their existence for a long time are possible. Moreover, it points towards the importance of temperature conditions in storage.

10.9 10.9.1

Crystallization of cocoa butter Polymorphism of cocoa butter

In the confectionery industry, crystallization of cocoa butter (alone or in chocolate) is carried out in two steps: (1) pre-crystallization, or tempering; (2) crystallization by cooling (moulding etc.) and in storage.

360

Confectionery and Chocolate Engineering: Principles and Applications

Since cocoa butter has six crystal modifications, the purpose of pre-crystallization is to produce the necessary amount of crystal seeds of the least unstable modification, β (V) (see below). Here, ‘least unstable’ means that this modification remains unchanged over several months. To produce the stable modification β (VI) directly needs sophisticated technology, and such technology is not yet in everyday use, but investigations aimed at solving this problem are in progress. If the correct technology is used, the proportion of the β (V) modification generated by tempering is about 1–5%, and the proportion generated by cooling is about 45–60%. Crystallization is finished in storage, when the proportion of crystals of the β (V) modification increases to 60–80%. Several authors have discussed the crystalline forms of cocoa butter polymorphs (Duck 1964, Wille and Lutton 1966, Huyghebeart and Hendrickx 1971, Lovegren et al. 1976, Dimick and Davis 1986, Jovanovic et al. 1995). For the determination of the melting points of cocoa butter, see IOCCC Analytical Method 4 (1961). The data on the melting points are rather different for the various crystal modifications. In the confectionery industry, the data provided by Wille and Lutton (1966) are perhaps the most often used, although in a ‘mixed’ form (Greek letter + numbering): γ = I, 17.3°C; α1 = II, 23.3°C; α2 = III, 25.5°C; β′ = IV, 27.3°C; β (V), 33.8°C; β (VI), 36.3°C. The idea of polymorphic crystalline forms of cocoa butter – as well as of other fats – refers not to the external microscopic or macroscopic geometrical appearance of the fat crystals but to the internal structure of the crystals at a molecular level, i.e. the packing of the triglycerides in the molecular crystal lattice. Figure 10.17 represents the transition β′ (IV) → β (V) of the crystal modifications of cocoa butter. The transition β′ → β is stimulated by shearing, which is caused by strong mixing of chocolate mass. The characteristic feature of the α (II and III) modifications is that the TAGs start to align along the axis of the fatty acids and a chair-type arrangement is formed. (The modification I is designated by γ in the literature.) The β′ (IV) modification is more compact, its consistency is harder and two chairs form one bond (DCL arrangement). The characteristic feature of the β (V) modification is a compact structure in which three chairs form one bond (TCL arrangement). In the modification β (VI), which is the stable one and evolves over weeks or months, the consistency becomes more compact through the development of a curved tuning-fork shape of the parts of the TAGs where the oleic acid groups are located.

10.9.2

Tempering of cocoa butter and chocolate mass

The crystallization of cocoa butter or chocolate mass, usually containing about 28–38% cocoa butter, means the solidification of the material in such a way that the cocoa butter is crystallized in the form of the β (V) modification. The series of operations starts with tempering, the next operation is the shaping of the cocoa butter or chocolate mass and, finally, the operation of cooling finishes this series.

Crystallization

361

β′(IV)

Shearing H

H

β(V), triclinic

H

H H

H H H

H

H

H

H

H

H

H

H

H

H H H

H H H

H H

H H

H H

H H

H H H β′(IV), orthorhombic

H H

Carbon atoms Fig. 10.17

Polymorphic transformation of cocoa butter modifications β′ (IV) → β (V) on shearing.

In the following descriptions, the tempering of cocoa butter and of chocolate mass are presented together. However, there is an important difference: since the contraction of cocoa butter in a chocolate mass is proportional to the volume ratio of cocoa butter, the contraction of a cocoa butter bar is about three times higher than that of a chocolate bar, assuming that they are of the same volume. Therefore, the moulding of cocoa butter bars, which is a relatively rare task, needs more cautious cooling because the bars can crack. The risk of such a phenomenon is less in the case of the moulding of chocolate mass. 10.9.2.1

Tempering

From the point of view of the technology, the control of the transitions α (III) → β′ (IV) → β (V) plays an essential role. This is the tempering operation. At the end of tempering, all of the β′ (IV) modification has to be melted and, at the same time, tempering must provide a seed concentration of the β (V) modification of 0.1–1.15% of the cocoa butter mass according to Loisel et al. (1997). Jewell (1972), however, reported that larger amounts of seeds, 2–5% of the cocoa butter, were needed for good temper. According to Lonchampt and Hartel (2004), this difference may be due to differences in seed size, which affects the number of seed crystals. Von Drachenfels et al. (1962) specified the importance

362

Confectionery and Chocolate Engineering: Principles and Applications

of crystal size. The smaller and more regular the size of the seed crystals, the glossier the chocolate and the greater its bloom resistance. On the other hand, if the crystal size is too large, the crystals tend to recrystallize during storage. It was mentioned above that the transitions from modification I to modification VI are increasingly slow. At the beginning of the cooling of cocoa butter, the γ (I) and α (II and III) modifications occur but they change rapidly to the β′ (IV) and β (V) modifications. For details, see Ziegleder (1988). Since the crystallization of cocoa butter follows monotropic polymorphism, the direction of the changes is exclusively γ (I) → α (II) → α (III) → β′ (IV) → β (V) → β (VI). Moreover, under the usual conditions all the modifications can be crystallized directly from molten cocoa butter except for β (VI), which crystallizes slowly from the β (V) modification (Fig. 10.18). The stable form β (VI) cannot be produced directly from melted chocolate except by the addition of β (VI) cocoa butter seeds and under very well-controlled conditions (Giddey and Clerc 1961, van Langevelde et al. 2001). It should be emphasized that the target of tempering is to bring about the β (V) modification, which is unstable, although its transition to the stable β (VI) modification is very slow: it needs weeks or months. During these monotropic changes the Gibbs free enthalpy decreases continuously; its minimum is reached in the β (VI) modification.

Molten chocolate (∼50°C)

γ (I) Quick transformation

ling

α1 (II)

Coo

Quick transformation α2 (III) Quick transformation β′ (IV) + β (V)

e siv g en Int earin sh

g Warmin

β (V) crystallized

Cooli ng

β′ (IV)molten β (V) seeds remain Fig. 10.18

Monotropic (one-way) changes of cocoa butter modifications during tempering and cooling.

Crystallization

363

However, if the tempering results in a majority of crystals of the β′ (IV) modification, the transition β′ (IV) → β (V) will take place in the chocolate product within hours or days, and the consequence of such a transition will be the appearance of fat bloom on the surface of the chocolate product. This is a severe quality defect, called blooming. Taking into account all the considerations above, the principle of the tempering process is to produce the β′ (IV) and β (V) modifications, and then to melt the β′ (IV) modification while the β (V) modification is retained. Although the β (V) modification can be produced directly from a molten chocolate mass, such a direct method cannot exclude the development of crystals of the β′ (IV) modification. A warming period is necessary in the tempering operation which destroys the crystals of the β′ (IV) modification – this is the way to avoid fat bloom. Figure 10.19 shows the temperature profile of a correct tempering operation for chocolate mass, which consists of three steps: two steps of cooling and one step of warming between them. The traditional tempering machine is similarly partitioned in the direction of advance of the chocolate mass. It is evident that a simple conical double-jacketed chocolate tank with a mixer is hardly suitable for performing tempering correctly, because it is difficult to carry out the warming phase. Strong mixing of the chocolate mass during tempering promotes the development of crystals of the β (V) modification by the shearing effect. The measurement of tempering, for which the ‘temperimeter’ is a practical instrument, provides important technological parameters. This instrument includes a small vessel, which the tempered chocolate mass is poured into. The vessel is placed in an ice–water bath, and the temperature of the chocolate mass is measured as a function of time. The resulting temperature vs time plots are represented in Fig. 10.20. The curve for a well-tempered chocolate mass is characterized by a horizontal line: the amount of crystals of the β (V) modification is sufficient, and in the time interval

Temperature (°C)

∼ 50

∼ 32 30–31 ∼ 27 20

β′ (IV) + β (V) Warming Cooling

Fig. 10.19

β′ (IV) molten + β (V) Warming Time

Temperature profile in the chocolate tempering process.

β (V) crystallization

Cooling

364

Confectionery and Chocolate Engineering: Principles and Applications

Temperature (°C)

Undertempered

Well-tempered Overtempered

Time Fig. 10.20

Typical temperimeter curves.

represented by this line the latent heat generated by crystallization (an exothermic effect) and the cooling effect of the bath (an endothermic effect) are in balance. Consequently, the temperature does not change in this interval. When the chocolate mass is undertempered, too many crystals of the β′ (IV) modification develop, which rapidly transform to the β (V) modification. Consequently, the latent heat dissipated by their crystallization exceeds the cooling effect of the bath. Therefore, an increase in temperature occurs. When the chocolate mass is overtempered, too many crystals of the β (V) modification develop, which melt too slowly to compensate the cooling effect of the bath. Consequently, the temperature decreases continuously. In many publications, bloom in chocolate is often described as a process involving the migration by capillary action of a liquid fat to the surface (Kleinert 1962). Loisel et al. (1997) considered chocolate as a porous material and were able to determine, by mercury porosimetry, the porosity volume of well-tempered dark chocolate [β (V)], undertempered chocolate [β (IV)] and overtempered chocolate [a mixture of β (V) and β (VI)]. The volume of air bubbles due to the process was determined by X-ray radiography to be less than 0.1% of the sample volume. The porosity of normal chocolate was about 1% of the total volume, and this increased to 2% for the undertempered chocolate and 4% for the overtempered chocolate. The results did not allow determination of the precise pore diameter, but suggested that the chocolate did not have open, interconnected pores with a mean diameter larger than 0.1 μm at the surface. Moreover, it seems that the pores were filled by the liquid fraction of cocoa butter at room temperature. As a result, it is better to talk about empty cavities rather than pores. Khan et al. (2003) highlighted the presence of pores at the surface of milk chocolate by scanning the surface with an atomic force microscope. These authors estimated the concentration of pores to be thousands/cm2; the pores, 1–2.5 μm in depth, were randomly distributed on the surface. As mentioned previously, a preferable method of crystallization from melts is to add crystal seeds of the stable modification to the molten substance, which start an overall crystallization in the stable modification. This is the principle of the Seedmaster tempering machine manufactured by Bindler, in which crystals of the stable β (VI) modification are produced by intensive shearing (‘Seedmaster cryst’), and the pre-tempered chocolate mass is seeded by these stable crystals in the Seedmaster mix.

Crystallization

365

Warm side Cool side

Pre-warming of moulds

Cooling tunnel

Dosing Vibration

Demoulding

Wall

Fig. 10.21

Moulding of chocolate.

Beside cocoa butter, several types of chocolate may contain milk fat (milk chocolate) and/or oils derived from added almonds or hazelnuts (dark and milk chocolate) if these nuts are refined together with the chocolate mass. Since the properties of these fats/oils are essentially different from those of cocoa butter, they can exert an important effect on the crystallization of cocoa butter in chocolate. As a rule, it can be stated that in the case of milk fat, almond oil or /hazelnut oil the end point of cooling will be ∼26°C instead of 27°C, and the end point of warming will be 29–31°C instead of 30–32°C. The decrease in temperature that is to be used is dependent on the amount of these fats/oils. For further details, see Kniel (2000) and McGauley (2001).

10.9.3

Shaping (moulding) and cooling of cocoa butter and chocolate

Tempering is followed by shaping and cooling. In order to avoid sudden cooling of the well-tempered chocolate mass, the moulds (metal or plastic) are pre-tempered, i.e. prewarmed to a temperature of about 30°C. Figure 10.21 shows the usual method of moulding; the various kinds of moulding machines, which are not discussed here, are a subject of confectionery technology. On entering the cooling machine, the temperature of the chocolate mass is a little lower than at the dosing stage because there is some cooling in the vibration section as well. Figure 10.22 shows three stages in the cooling tunnel, although the tunnel is not actually divided into three stages: each stage means about one third of the length of the tunnel. The values of the temperature of the chocolate mass and cooling air are for information only; the unit mass and the shape of the moulded product are crucial factors in determining these values. In Fig. 10.22, the temperature profiles of the cooling in the case of cocoa butter/cocoa butter equivalent (CBE) fats and in the case of cocoa butter replacer (CBR)/

366

Confectionery and Chocolate Engineering: Principles and Applications

30°C

12°C

16°C

16°C

Cooling of cocoa butter and CBE

40°C

8°C

8°C

16°C

Cooling of CBR and CBS Air flow

Product flow

Fig. 10.22 Temperature profile of cooling process. CBE = cocoa butter equivalent, CBR = cocoa butter replacer, CBS cocoa butter substitute.

cocoa butter substitute (CBS) fats are represented in parallel in order to stress the differences between these two types of fats. In the first stage, a moderate cooling is recommended in order to avoid the development of a solid crust on the surface of the chocolate that could hinder heat transfer and the correct crystallization into the β (V) modification. In any case, the second stage of cooling is essential from the point of view of crystallization because the majority of the crystal seeds are formed in this stage. The latent heat of crystallization of cocoa butter, which is on average −1.88 kJ/kg, is released mainly in this stage, and as a result of the temperature of the chocolate is slightly increased. This stage is characterized by the growth of crystal seeds. If the tempering is not correct, few crystal seeds develop on slow cooling. However, if the cooling is too fast, unstable seeds develop, and the surface of the product acquires a reddish tint, which is soon followed by fat bloom. As a result of crystallization of the majority of the cocoa butter in chocolate, a contraction takes place, equal to about 9.3% (V/V%) relative to the cocoa butter and about 3% (V/V%) relative to the chocolate product if we take into account the fact that the proportion of cocoa butter in chocolate is about 30 m/m%. This contraction makes demoulding possible. If the tempering is not correct, the consequence is a smaller contraction. In extreme cases, the demoulding is not perfect: the surface of the product is not bright, the consistency (Brucheigenschaft) is not ‘cracking’, etc.

Crystallization

10.9.4

367

Sugar blooming and dew point temperature

In the third stage, the chocolate leaves the cooling tunnel, the temperature of which increases continuously. On leaving the cooling machine, the moulded chocolate has a temperature of about 16°C, which must not be lower than the dew point of the external air in the room, otherwise water from the air will condense as dew on the surface of the chocolate and dissolve the sugar content of the chocolate surface. Later, this sugar solution will dry and its solid content will remain as ‘sugar bloom’. The two types (sugar and fat) of blooming can be easily differentiated either microscopically or by fingering (in the case of sugar bloom, the surface is coarse). Barenbrug (1974) used the Magnus–Tetens formula for the saturated vapour pressure of water pw.s, ⎛ aTd ⎞ pw.s = 0.6105 exp ⎜ (kPa ) ⎝ b + Td ⎟⎠

(10.139)

where Td is the dew point temperature (°C), a = 17.27 and b = 237.7°C. It is known that if T > Td, then pw.s(T) > pw.s(Td). Moreover, if the vapour at the temperature T is not saturated, then the relative humidity (RH) can be defined by the relationship pw.s (Td ) = RHpw.s (T )

(10.140)

From Eqn (10.139), aT ⎞ ⎛ aTd ⎞ 0.6105 exp ⎜ = RH × 0.6105 exp ⎛ ⎝ b +T ⎠ ⎝ b + Td ⎟⎠

(10.141)

aTd aT = + ln ( RH) ( = α ) b + Td b + T

(10.142)

From Eqn (10.142), the dew point temperature can be calculated: Td =

bα a −α

(10.143)

where

α=

aT + ln ( RH) b +T

(10.144)

α is a function of T and RH is the relative humidity of the air at temperature T; see Eqn (10.140). If 0°C < T < 60°C, 0.01 < RH < 1.0 and 0°C < Td < 50°C, then the uncertainty in the calculated dew point temperature is ±0.4°C. For more details, see Koninklijk Nederlands Meterologisch Instituut (2000).

368

Confectionery and Chocolate Engineering: Principles and Applications

Example 10.5 Assume that the temperature at the end of the cooling tunnel is 16°C. Let us calculate the dew point temperatures for the cases RH = 0.4, RH = 0.6 and RH = 0.7. From Eqn (10.144),

α ( 0.4 ) = 17.27 ×

16 + ln 0.4 = 0.173 237.7 + 16

α ( 0.6 ) = 17.27 ×

16 + ln 0.6 = 0.578 237.7 + 16

α ( 0.7 ) = 17.27 ×

16 + ln 0.7 = 0.732 237.7 + 16

From Eqn (10.143), Td ( 0.4 ) = 237.7 ×

0.173 = 2.4°C 17.27 − 0.173

Td ( 0.6 ) = 237.7 ×

0.578 = 8.23°C 17.27 − 0.578

Td ( 0.7 ) = 237.7 ×

0.732 = 10.52°C 17.27 − 0.732

This means, for example, if a chocolate bar leaves the cooling machine at 10°C and the external air parameters are 16°C and 70% RH, then vapour will condense on its surface (10°C < 10.52°C).

10.9.5

Crystallization during storage of chocolate products

The crystallization of cocoa butter finishes during the storage of chocolate products. This after-crystallization is dependent on the previous steps of crystallization, the conditions of storage and the behaviour of the chocolate product. The problem of formation of fat bloom during storage originates mainly from: • Too high a ratio of after-crystallization. According to Kniel (2000), post-crystallization means that there is still formation of crystals after the cooling process. In optimal cases, approximately 20% of the cocoa butter crystallizes during the cooling step, and the fraction of crystallized cocoa butter reaches about 45–60% in the first few hours after cooling. Then this fraction slowly increases to 56–80% during storage, i.e. the fraction of post-crystallization is about 20%. However, if the residence time of the chocolate in the cooler is too short or the temperature of the cooler is too low, the post-crystallization can reach 40% (i.e. after cooling, the cystallized fraction is too low, less than 45% instead of 45–60%). In this case the post-crystallization occurs slowly under uncontrolled conditions. The consequence is an uneven structure with large crystals, and a high fat bloom risk.

Crystallization

369

• Tempering was not correct. During tempering, the β (IV) modification might not be melted, and the remaining seeds of it may cause fat bloom. • Fractionation of triglycerides. In addition to symmetric triglycerides, cocoa butter also contains the asymmetric triglycerides POO and StOO. If crystallization is too quick, the symmetric triglycerides migrate to the solid crystals, and the asymmetric ones to the melt. Consequently, the random distribution of triglycerides in the melted cocoa butter will no longer exist: the crystals will be enriched in symmetric triglycerides (and the melt will contain more asymmetric triglycerides), and this promotes fat blooming. Fat bloom contains mainly symmetric triglycerides. • Fat migration. It is important to remember that even a seemingly solid product such as chocolate at room temperature contains considerable amounts of liquid oil. The solid fat content of cocoa butter at 25°C is approximately 80–85%, the remainder of the fat consisting of low-melting triglycerides that do not crystallize at this temperature. These liquid triglycerides are trapped in a matrix consisting of solid fats and normally move only by slow diffusion processes. However, if the temperature is raised to 30°C, the amount of liquid increases to approximately 50% and the solid matrix becomes much less efficient as a migration barrier. At this concentration of solid material, the distance between particles is so large that continuous liquid channels may form and the movement of liquid triglycerides becomes rapid. The liquid also dissolves some of the more high-melting triglycerides in the cocoa butter such as POP (a symmetric triglyceride) and transports them to the chocolate surface, where they can recrystallize on existing POP-rich crystals. When the crystals have increased sufficiently in size, they can be observed as fat bloom, and even before that as a visible dulling of the surface. The problem is of course accentuated in a composite product such as a filled praline. The filling is normally quite fluid owing to the desired sensory characteristics, and the oil content of the filling may be as high as 50%. When in contact with a chocolate shell, the oil gradually enters the solid chocolate and dissolves some of the high-melting cocoa butter. The triglycerides of the soft fat contained in a filling or in pieces of nuts (hazelnut, almond etc.) migrate much more quickly than the triglycerides of cocoa butter do (e.g. OOO migrates four times quicker than POSt). Therefore, these fast-moving triglycerides ‘push out’ cocoa butter to the surface of the product, and the consequence of this phenomenon is fat blooming. The above problems of fat bloom can develop separately and together as well. However, fat blooming may be delayed for years by correct use of technology and correct storage conditions. The transition β (V) → β (VI) is relatively quick in dark chocolate, but can be slowed down by adding 1–2% of milk fat because milk fat consists of many different triglycerides, and this seems to hinder fat blooming. For details of the properties and polymorphism of milk fat, see Timms (1984), Campbell and Pavlasek (1987), Breitschuh and Windhab (1998) and ten Grotenhuis et al. (1999). When we study the rates of various phenomena as a function of storage temperature, which plays an essential role in after-crystallization, it is surprising that for chocolate containing soft fats the temperature interval of 20–22°C is not beneficial, because crystallization has not yet ceased in this interval but migration is speeded up. The combination of these two effects results in a maximum in the rate of fat blooming. (The fat-blooming curve is a sum of the curves due to these two effects.)

370

Confectionery and Chocolate Engineering: Principles and Applications

10.9.6

Bloom inhibition

Lonchampt and Hartel (2004) have divided the factors having an effect on fat blooming into two groups: compositional factors and factors associated with the processing method. 10.9.6.1

Compositional factors

In general, the higher the solid content and the lower the liquid fraction, the more resistant a chocolate is to bloom. However, because of organoleptic consequences, it is possible to increase the melting point of chocolate by only 1°C (Arishima and McBrayer 2002). Several different ways have been used to increase the solid fat content, including the use of a stearine (high-melting) fraction of cocoa butter (after fractionation), and adding specific TAGs to chocolate, namely StOSt, POP or asymmetric TAGs such as StStO or PPO, to impede the β (V) → β (VI) transition. Milk fat has long been known to have an anti-bloom effect when blended with cocoa butter in chocolate. (However, it is also known to enhance bloom when used with compound coatings; see below.) The anti-bloom effect of milk fat and emulsifiers was discussed by Lonchampt and Hartel (2004) in detail. It should be mentioned that the effect of emulsifiers on the improvement of bloom resistance is strongly dependent on the type of emulsifier. 10.9.6.2

Processing factors

The key processing factor is tempering, which must be different for dark and milk chocolate, as previously mentioned. As already discussed, cooling that is either too slow or too quick can induce bloom. Rapid cooling produces small cracks and pores on the chocolate surface, enhancing bloom (Kleinert 1962). Rapid cooling may also promote the formation of unstable polymorphs in regions that have cooled too quickly. Proper cooling of both types of chocolate (and compound coatings) is needed to protect against early bloom formation. Warm treatment prior to storage It has been found that a brief period of warming to 32–35°C protects chocolate against bloom formation. Following his earlier work, Kleinert (1962) investigated the possibility of exposing chocolate to a brief warm-temperature hold to prevent bloom formation. A minimum ‘treatment’ time of 80 min at 32–35°C was sufficient to protect the chocolate against bloom for more than 1 year, although a similar hold at temperatures from 28 to 31°C did not prevent the chocolate from blooming. Minifie (1989) also noted a similar treatment (32.2°C for 2 h); however, he described also a second treatment using a lower temperature for a longer period of time. Treatment for 2 days at 26.7–29.4°C for dark chocolate and 22.8–25°C for milk chocolate also inhibited bloom formation. However, this later method decreased the final gloss. After warm treatment, the chocolate was in the β (VI) form. For details, see Lonchampt and Hartel (2004, p. 264). Storage conditions The inhibition of storage bloom is maximum when the chocolate is stored at 18°C or below, without any temperature fluctuations. Chocolate can be stored frozen for a very long time. However, even though the ideal storage conditions that prevent bloom are well known, it is impossible to control the temperature after the chocolate leaves the plant.

Crystallization

10.9.7

371

Tempering of cocoa powder

Because cocoa powder contains 8–24% of cocoa butter – the usual values of cocoa butter content are 8–10% or 10–12% if produced for industrial purposes, and 16–24% if produced for household use – this fraction of cocoa butter is tempered as well in order to hinder fat blooming. Fat blooming in cocoa powder has been studied less than in chocolate, although modern powder-cooling and stabilizing systems can solve this problem. The colour of cocoa powder is an essential quality property: a deep red/brown colour is preferred. According to Fincke (1965), two types of colour can be distinguished. The inherent colour derives from the flavonoid substances of fat-free cocoa cells, which can be enhanced by alkalization; see Section 16.3. The inherent colour plays an important role in cocoa drinks because the flavonoids are dissolved and provide the colour of the drink. The outer colour is a result of the correct tempering of the cocoa butter content. Alkalization and tempering have a synergistic effect from the point of view of the outer colour, which is regarded by customers as essential. A simple experiment shows how the outer colour depends on tempering. If we warm some cocoa powder up to about 45°C while mixing it, and then cool it below about 20°C, the result is a deep red/brown outer colour, which is likely to become greyish in a short time. This phenomenon is more evident when the cocoa butter content is higher: this gives both a deeper red/brown colour and faster blooming. The tempering of cocoa powder has two steps: (1) In pre-tempering, before grinding, the cakes of cocoa powder are tempered at 43–45°C in order to melt the total cocoa butter content. (2) In the grinding machine, the cocoa powder is pulverized in a fluidized bed, which ensures uniform, gradual cooling first down to 20–24°C, and then down to 16–18°C. The DSC curve of a well-tempered cocoa powder shows that the highest peak of heat absorption is at 34°C, which is related to the β (V) and β (VI) modifications. A peak at 28–29°C is evidence of poor tempering, i.e. the existence of the very unstable β (IV) modification.

10.10 10.10.1

Crystallization of fat masses Fat masses and their applications

In the following discussion, the expression ‘fat mass’ relates to products and semi-products of the confectionery industry which consist of about 28–38 m/m% of non-cocoa butter vegetable fats, which forming the continuous dispersion phase of these products. The usual names for such products are compounds (or surrogates), which are similar to chocolate; coatings, which substitute for couverture chocolate in cheap products; and creams/ fillings, which have a soft consistency relative to compounds and coatings, the consistency of which is similar to that of chocolate. In the following ‘vegetable fat’ always means non-cocoa butter vegetable fats. Briefly, the crystallization of fat masses is determined by the properties of the vegetable fat that they contain. Since compounds and coatings have, if possible, a

372

Confectionery and Chocolate Engineering: Principles and Applications

similar consistency to chocolate, the vegetable fat contained in them also has a similar consistency to chocolate. Namely, the consistency of the product and the consistency of the vegetable fat are in the closest possible relationship allowed by the properties of the crystallization. As a consequence of the above reasoning, on the one hand, the vegetable fats used for producing compounds and coatings ‘imitate’ the crystallization (consistency) properties of cocoa butter; on the other hand, the vegetable fats used for producing creams and fillings ‘imitate’ the crystallization (consistency) properties of milk fat, milk butter or milk cream. Although this is a simplification, and there are differences of greater or lesser extent between this picture and the true situation, it essentially expresses the fact that fat masses provide a cheaper solution than products made with cocoa butter or milk cream. The vegetable fats other than cocoa butter that are used in the confectionery industry are discussed in the following sections.

10.10.2

Cocoa butter equivalents and improvers

Cocoa butter equivalents (CBEs) and cocoa butter improvers (CBIs) are designed to be completely miscible with cocoa butter. The range comprises in part products that are nearly identical to cocoa butter, but also products that can be used to alter the properties of chocolate to make it more heat-resistant or slightly softer, to mention just a few examples. CBEs are composed of the triglycerides POP, POS and SOS in order to mimic the properties of cocoa butter. POP is obtained from palm oil by fractionation and is then blended with fats rich in POS and SOS. A typical source of SOS is shea nut oil, while illipe fat contains POS and SOS. The typical triglyceride compositions of the components used in CBEs are shown in Table 10.5 Other sources of oils and fats used in CBEs are sal, mango and kokum. Today, CBElike products can also be manufactured through enzymatic interesterification of more abundant raw materials; however, these products are not approved for use in chocolate in the European Union (see later). The melting points of the β polymorphs of POP, POS and SOS are 37°C, 37°C and 43°C, respectively. This implies that the higher the SOS content of the fat the harder it will be, although an excessively high content may increase the viscosity of the tempered chocolate mass. If POP is the main component, the fat will temper slowly and be soft. Since POP is normally the cheapest component, there must be an optimum cost/benefit, depending on the application of the fat.

Table 10.5 Typical triglyceride compositions (average in m/m%) of components used in CBEs. CBE raw material Palm mid fraction Illipe butter Shea stearine Cocoa butter (as reference)

POP

POSt

StOSt

Others

65 10 1 17

13 36 8 39

2 42 69 26

20 12 22 18

Crystallization

373

There is another dimension to fat quality: the purity of the fractions or the concentration of the required symmetric triglycerides that are used in the final CBE. This will also be reflected in the performance, as well as the price of the fat. Generally, CBEs show the same properties in chocolate as cocoa butter does with regard to crystallization, texture and eating properties. The shelf life can be prolonged and the bloom stability improved by the addition of a good-quality CBE to the chocolate formula. With a normal CBE, no modification of the manufacturing process is necessary, although slight adjustments of temperature during tempering may be required in the case of the softer and harder CBEs. A good-quality CBE can be mixed with cocoa butter in any proportion without changing the melting sequence. A CBE will react in the same way as cocoa butter when milk fat or other softening fats such as nut oils are mixed in. Owing to their homogeneous triglyceride composition, both CBEs and cocoa butter crystallize in a highly ordered structure, and they do so more easily than does a fat with a large number of different fatty acids. This fact is responsible for the hardness and uniquely well-defined melting behaviour of CBEs. A CBE based mainly on POP results in a softer chocolate product than does a SOSrich CBE. CBEs rich in POP are also more difficult to temper, requiring tempering temperatures some degrees lower than for cocoa butter. The properties and the tempering temperatures for chocolate based on these POP-rich CBEs are very dependent on the level of CBEs used. Softer CBEs are normally used at a lower level or may also replace part of the milk fat if that is used. The entire chocolate formula must be considered in order to find the desired properties and the most cost-efficient product. In products with a very high milk fat content, the softening effect of the milk fat can be compensated for by using a hard CBE. A soft cocoa butter can be hardened with a hard CBE as well. These hard CBEs, sometimes called cocoa butter improvers, make it possible to improve the shelf life of cocoa butter-based chocolate in hot climates. The production process for chocolates containing hard CBEs requires a slightly higher tempering temperature. There is a special group of vegetable fats called ‘cocoa butter equivalents’ defined by the European Union Directive 2000/36/EC relating to cocoa and chocolate products. According to the Preamble, ‘(5) The addition to chocolate products of vegetable fats other than cocoa butter, up to a maximum of 5%, is permitted in certain Member States’. Moreover (Annex II), ‘In conformity with the criteria laid down by this directive, the following vegetable fats, obtained from the plants listed below, may be used [Table 10.6]. Furthermore, as an exception to the above, Member States may allow the use of coconut

Table 10.6 Vegetable fats that may be used in chocolate to a maximum of 5% according to the European Union Directive 2000/36/EC. Common name of vegetable fat

Scientific name of plant from which the listed fat is obtained

Illipe, Borneo tallow or Tengkawang Palm oil Sal Shea Kokum gurgi Mango kernel fat

Shorea spp. Elaeis guineensis, Elaeis olifera Shorea robusta Butyrospermum parkii Garcinia indica Mangifera indica

374

Confectionery and Chocolate Engineering: Principles and Applications

oil for the following purpose: in chocolate used for the manufacture of ice cream and similar frozen products.’

10.10.3

Fats for compounds and coatings

Cocoa butter, CBEs and other fats in the tempering group are based mainly on speciality raw materials. This places them in a high price range compared with fats based on the major vegetable oils. By means of fat modification techniques such as hydrogenation, fractionation and interesterification, it is possible to produce fats with melting profiles similar to cocoa butter at significantly lower cost. These fats have a completely different composition and crystallization pattern, but the sensory properties of the end product still resemble those of chocolate. There are two types of so-called ‘non-tempering fats’: cocoa butter replacers (non-lauric) and cocoa butter substitutes (lauric). 10.10.3.1

Crystallization properties of non-tempering confectionery fats

In contrast to cocoa butter and other fats based on triglycerides with an unsaturated fatty acid in the mid position on the glycerol backbone, hydrogenated and lauric fats do not require tempering in the same way as cocoa butter does to reach a stable crystalline form. It should be emphasized that these non-tempering vegetable fats also require controlled cooling conditions, but these are different from the tempering process used for cocoa butter and chocolate. In this type of fat, the triglycerides tend to crystallize in a double-layer structure, since the saturated and trans-unsaturated fatty acids have a higher degree of molecular similarity. The formation of a double-layer (DCL) structure is rapid compared with the triplelayer (TCL) formation characteristic of cocoa butter. Most hydrogenated and lauric fats are reasonably stable in the β′ form, although the thermodynamically favoured state is the β form. 10.10.3.2

Crystallization dynamics of non-tempering fats

Hydrogenated fats comprise mixtures of saturated (palmitic and stearic), cis-unsaturated (oleic) and trans-unsaturated (elaidic) fatty acids. The molecular structures of elaidic and stearic acid are quite similar (they are cis–trans isomers), and these fatty acids are very compatible with each other. Elaidic acid has a melting point that is intermediate between those of stearic and oleic acids, and this fact also determines the melting points of the triglycerides where elaidic acid is present. The mixtures of longer-chain (C-16 to C-18) and medium-chain (C-12 and C-14) saturated fatty acids in combination with oleic acid that occur in ‘lauric’ fats (coconut oil and palm kernel oil) are also very compatible with each other (but not with cocoa butter!), especially when the unsaturated fatty acid is present in one of the outer glycerol positions. The crystallization process proceeds via nucleation and crystal growth. Hydrogenated fats based on long-chain fatty acids nucleate readily, whereas lauric fats may need a higher degree of supercooling to start crystallization. Once nucleated, however, the lower molecular weight of the lauric fats tends to give a higher crystal growth rate compared with the larger palmitic/stearic/elaidic-based systems. The exact composition of the fat determines the overall crystallization kinetics. In general, the higher the solid fat content, the more rapid is the crystallization (nucleation + growth). The shape of the melting curve (flat or

Crystallization

375

steep) is also a good indication of the crystallization rate: the steeper the melting curve, the faster the setting. Effect of additives and matrix on crystallization dynamics The product matrix and the presence of additives strongly affect the overall crystallization rates of non-tempering fats. Nucleation rates are strongly influenced by the presence of particles such as those of sugars, cocoa and milk solids. Surface-active components, either naturally present in the system or added as functional additives, influence the crystallization rate. It is well known that diglycerides, lecithin and sorbitan esters influence the crystallization kinetics. Diglycerides and sorbitan triesters slow down the overall crystallization by reducing either the nucleation rate or the crystal growth. On the other hand, sorbitan monoesters, as well as monoglycerides and other more polar emulsifiers, may act as nucleating agents and thus increase the overall crystallization rate. The non-tempering systems are sensitive to differences in cooling conditions. If a nontempering fat is cooled rapidly to very low temperatures, it can crystallize in an unstable α form, especially if the fat has a high melting point and a steep melting profile. To avoid subsequent fat bloom, such systems should be either crystallized or annealed at higher temperatures to obtain a controlled transformation to the desired β′ form before packing and storage. Fat bloom and fat bloom inhibition in non-tempering systems The desired crystal form in non-tempering systems is normally the β′ (beta-prime) form. β forms can be used on the condition that the processing conditions and the composition are optimized to meet melting-profile and crystallization-rate criteria. Generally, β′ crystals are thermodynamically unstable, although some individual triglycerides exist that do not show any β form. Similarly to the case of cocoa butter and the Sat–Unsat–Sat type of triglycerides, β′ crystals may exist in at least two modifications, with different melting points and structural features. The β′ forms are numbered in order of decreasing stability: β′-1, β′-2 etc. In a confectionery fat based on hydrogenated fats, the conversion of β′-2 to either β′-1 or β crystals may constitute a driving force for fat bloom formation. It can also be argued, by analogy with the cocoa butter case, that these crystal form transformations are not the cause of but rather a consequence of fat bloom formation. In general, fat bloom is a crystal growth process that requires the triglycerides in the bloom to be mobile at the time of formation, either by a melting process or by solubilization in a liquid phase. These crystal growth processes may be associated with a phase transformation process, but not necessarily so (Liedefelt 2002, p. 129). With this in mind, any measure taken to minimize the solubilization and transport of solubilized triglycerides will enhance the bloom stability in non-tempering systems. By optimizing the triglyceride composition so that only co-crystallizing triglyceride species are present in the system, the solubilization of unstable compositions can be inhibited. Similarly, adding components that slow down the crystal growth process, such as diglycerides, will also slow down bloom formation. The presence of non-co-crystallizing fractions, for example low-melting trisaturated triglycerides such as trilaurin, is sometimes the cause of fat bloom, especially in lauric fats. The transport of solubilized triglycerides may of course be minimized by lowering the amount of liquid phase in the system. This will also bring about an undesired increase in

376

Confectionery and Chocolate Engineering: Principles and Applications

the melting point of the fat, however. The use of lower-melting triglyceride fractions to bind the liquid phase is sometimes possible.

10.10.4

Cocoa butter replacers

Compounds and coatings based on cocoa butter replacers (CBRs) do not need tempering like cocoa butter. Upon cooling they crystallize in the β′ form, which in practice is the stable modification for these fats. CBRs can be produced from a number of different raw materials, such as soybean oil, rapeseed oil, palm oil, cottonseed oil and sunflower oil. The production of this type of fat involves special hydrogenation and fractionation techniques. The main fatty acids are the saturated palmitic and stearic acids, together with the monounsaturated oleic acid and its trans isomer. Thus, the fatty acid chain lengths are C-16 and C-18. The use of different raw materials or combinations of them, in conjunction with the flexibility of the processes, allows a considerable range of compositions and provides a means of customizing a wide range of fats. CBRs may contain small amounts of additives such as sorbitan tristearate to help stabilize the β′ form. This improves both the gloss retention and the initial gloss of compound coatings. There are two main reasons for using CBRs instead of cocoa butter: the price is lower, and production is simplified since the tempering step (in the sophisticated form that is necessary for cocoa butter and chocolate) can be omitted. Setting times are adequate for modern high-speed equipment, even though they are somewhat longer than for cocoa butter substitute (CBS) compounds. The recommended cooling conditions are represented in Fig. 10.23. The melting properties of CBR compounds do not quite match the standards of chocolate or CBS compounds in moulded products. For coating applications, they offer several advantages, such as good heat stability and simplified manufacturing procedures. In general, CBR coatings have very good initial gloss and gloss retention. In some applications, the fact that they are non-lauric and hence cannot develop a soapy flavour makes a CBRs a preferred choice over CBSs. Finally, the possibility to use some cocoa liquor together with a CBR (see below) is a way to improve the flavour. 10.10.4.1

Compatibility with cocoa butter

It is obvious that the triglycerides of cocoa butter are different from those of CBRs. Since the fatty acids of both of these alternatives are C-16 and C-18, however, there is a certain degree of compatibility, and up to 20% cocoa butter (20% of the fat phase) can be tolerated before eutectic effects become too severe for practical use. Blends of cocoa butter and CBRs show typical eutectic behaviour. Within limits, this can be used to improve the melting and sensory properties of CBR-based compounds. In the 10–20% range, the addition of cocoa butter has a controlled softening effect that is more pronounced at higher temperatures. This good miscibility enables manufacturers to use all types of cocoa powder or a certain amount of cocoa liquor, which ensures that the final product can be given a rich chocolate flavour and good flavour release. The drawback of including cocoa liquor is that the setting time is prolonged and gloss retention reduced to some extent. The use of higher levels of cocoa butter than 20% is not recommended, since the eutectic effects will be too severe.

Crystallization

377

80

70

Fats for ice cream

Solid fat content (%)

60

50 Filling fats

40

Fats for toppings

30

20

10

0 10

Fig. 10.23

15

20 25 30 Temperature (°C)

35 37

Solid-fat-content curves for filling fats and fats for coating of ice cream.

As a final remark, it should be mentioned that accidental admixtures, such as might occur when a manufacturer switches from chocolate to compound production on a line, will not have severe negative effects on product quality. 10.10.4.2

Milk fat and CBR blends

CBRs work well with milk fat. Milk fat/CBR blends exhibit no eutectic effects, resulting in a predictable linear decrease in solid fat content with increasing content of milk fat. The practical limit may be 20%, since the product may be too soft at higher levels. Nut oils such as peanut, hazelnut and almond oils act in a manner analogous to milk fat in this respect. The presence of milk fat extends setting time but may improve gloss stability. CBRs may be used in solid and filled moulded items, but are also excellent for all kinds of coating applications. Since CBRs tolerate relatively high amounts of cocoa butter (20%), cocoa liquor may be used in the formulation to give a good, full cocoa flavour to the product. CBRs, being non-tempering systems, have low viscosity, fast crystallization and some elasticity, which makes them suitable for coating. The high gloss and good gloss retention give the product an appealing appearance.

378

Confectionery and Chocolate Engineering: Principles and Applications

Within the CBR range there are also products with a high solids content, well suited for use in warmer climates. 10.10.4.3

Moulding and coating

It is important to use appropriate cooling conditions for all CBRs used for moulding. To achieve good contraction, as well as good gloss and shelf life, CBRs benefit from so-called shock cooling. When a CBR is crystallized in a cooling tunnel, the temperature should be around 6–8°C. The temperature should be increased to around 16°C at the end of the tunnel to match the ambient temperature without risk of condensation. When CBRs are used in coating applications, the cooling parameters are of importance for achieving the best gloss and shelf life. A low initial temperature in the cooling tunnel is recommended, similarly to the requirements for the crystallization of moulded items.

10.10.5

Cocoa butter substitutes

Compounds and coatings based on cocoa butter substitutes (CBSs) are widely used in both the chocolate and the bakery industries. Like CBRs, CBSs do not need tempering like cocoa butter, since they crystallize spontaneously in the β′ form, which is stable in practice. CBSs are based on lauric fats, i.e. fats that contain a high percentage of lauric acid in their fatty acid composition. The main raw materials in this group are coconut and palm kernel oil, with palm kernel oil being the preference for CBS manufacture. The production of CBSs involves special hydrogenation and fractionation techniques. Lauric acid (C-12) makes up approximately 50% of the fatty acid composition, with myristic acid (C-14) as the second most abundant. In addition, smaller amounts of the longer-chain palmitic and stearic acids are present. Sorbitan tristearate has sometimes been added to CBSs to help improve the stability of the β′ crystals. Such addition of sorbitan tristearate improves both initial gloss and gloss retention. To a large extent, the use of CBSs in compounds is driven by the same motives as the use of CBRs: lower costs and simplified production procedures. A CBS similarly needs no tempering, and high crystallization rates allow a high throughput in the cooling tunnel. For recommended cooling conditions, see Fig. 10.22. CBS-based compounds generally have better melting properties than CBRs, almost on a par with chocolate. As can be seen below, however, CBSs are less compatible with cocoa butter. In some applications, when the water content of a product is more than c. 3% or when there is an effect of lipase originating from moulds, the fact that lauric fats may develop a soapy flavour makes CBRs a preferred choice over CBSs. Fats with a high content of short-chain fatty acids have a lower viscosity than fats based on longer-chain fatty acids. This means that for a given fat content and a fixed set of emulsifiers, CBS coatings and compounds have a lower viscosity than CBR coatings and compounds. CBSs and cocoa butter have completely different fatty acid and triglyceride profiles. Hence, eutectics are apparent even at low levels of cocoa butter. In contrast to cocoa butter/CBR blends, the strongest effects are seen at intermediate temperatures, from 20°C to 30°C. At higher temperatures (> 33–35°C), only small effects are observed. This means that cocoa liquor cannot be used in CBS-based compounds and

Crystallization

379

coatings. Cocoa powders typically contain 10–12% cocoa butter. A typical content of cocoa powder in compounds is 15%, which gives a CBS/cocoa butter ratio of approximately 95/5, as a short calculation shows. Let the cocoa butter content of a product be 15% × 0.11 = 1.65%. If the total fat content in the product is 33.3%, which is a commonly used value, then the amount of CBS is 31.65% (= 33.3% − 1.65%), and the CBS/cocoa butter ratio is 31.65/1.65 = 94.95/4.95 ≈ 95/5. Going beyond this will result in excessive softening of the product. In a recipe for a compound such as milk chocolate, a cocoa powder content of at least 4–5% is needed; however, in a recipe for a compound such as dark chocolate, 15% of cocoa powder, as an upper value, is sufficient for the appropriate taste. Consequently, this limited compatibility of CBSs with cocoa butter does not tighten the field of applications. CBS fats are excellent for both tablet and shell moulding, especially the types based on fractionated and hydrogenated components. Products containing fractionated CBSs display very fast crystallization and excellent contraction properties. The resulting tablet has a good snap and melts very quickly. CBS fats are highly suitable for a wide range of coating purposes, and are often used for this purpose in the bakery industry to coat various pastry products. They are appreciated for their convenience in that they do not need tempering, and set very quickly on the coated item. It is easy to produce a thin, even layer of coating. CBS fats also leave a good gloss on the surface. Thanks to their low viscosity and good flow properties, coatings based on CBSs are very good to use for hollow-figure moulding. In addition, CBS-based coatings give a good contraction that facilitates demoulding. Other suitable application areas for CBS fats are pan-coated products and lentils.

10.10.6

Filling fats

Although there are many differences between the various types of fillings, the essential similarity is that the suspension phase is fat, and the crystallization properties of this fat determines the consistency of the filling. It is not only the filling fat itself that affects the final properties of the filling. Besides adding to the flavour sensation, other components will contribute to building up the fat phase, for example cocoa powder or liquor, milk powder, nuts and possibly others as well. Fat bloom is often the main factor that limits the shelf life of moulded or coated products. Bloom may occur as a result of migration of fat from the filling to the surrounding chocolate layer. The migration process is very difficult to avoid, since it is driven by the fact that the filling contains more liquid fat than the coating. In an attempt to reach equilibrium, this liquid fat will migrate into the coating and recrystallize on the surface of the product, with fat bloom formation as a consequence. Instead of trying to hinder this migration, more recent product developments in filling fats have focused on gaining control over the effects of the migration (Liedefelt et al. 2002, p. 137). 10.10.6.1

Fat-based confectionery fillings

The fat-based confectionery fillings include various praline fillings, such as nougat, truffles, yogurt fillings and chocolate spreads. The filling can also be a centre for a coated product or a filling in a bakery product. In these products, fat is the continuous phase,

380

Confectionery and Chocolate Engineering: Principles and Applications

constituting some 30–40% of the product. The other ingredients are sugar, milk powder, cocoa products and, often, some kind of nuts. Depending on the choice of fat for the filling, almost any kind of eating sensation from very hard and cool-melting to soft and creamy can be achieved. Figure 10.22 shows curves of the solid fat content versus temperature for filling fats and fats for ice cream. The region covered by the SFC curves of filling fats is peculiarly broad, which shows the great variety of demands to be met. 10.10.6.2

Firm fillings

The firm fillings include praline and nougat fillings, for example. The fats used in these products consist of fractionated non-lauric components with very steep melting curves. They are rather similar to cocoa butter in their composition, and are therefore highly suitable for use in fillings with high amounts of chocolate in the recipe. The similarity to cocoa butter also means that these filling fats are stable in the β crystal form and, as such, normally require tempering. If tempering is not possible, good melting properties can be achieved in other ways. There are hydrogenated and fractionated non-lauric fats that are stable in the β′ form. With these types of fats, good melting can be achieved in combination with simple production steps, since tempering can be excluded. Lauric fats can also give a cool-melting sensation to a filled product. However, lauric fats are not suitable for mixing with cocoa butter; therefore, the cocoa powder that provides the cocoa flavour of a filling must be of low cocoa butter content (8–12%) when lauric fats are employed. Hydrogenated non-lauric fats have widespread use in the confectionery industry. They are suitable for praline fillings, as well as for wafer fillings and sugar confectionery. They generally show a fast crystallization pattern and are thus convenient to use in most production units. In addition, they are stable in the β′ form and give products with high migration stability. 10.10.6.3

Soft fillings and ‘chocolate’ spreads

For soft, creamy fillings, very soft fats are needed. Oil migration is even more difficult to control in such cases. The same type of fats are suitable for chocolate spreads. In this application as well, a soft and creamy product is required. The product must also be spreadable over a wide temperature range, from refrigerator to room temperature, and have a good shelf life. It should have a shiny and attractive appearance and, above all, should not separate so that liquid oil shows on the surface of the product. 10.10.6.4

Aerated fillings

Generally, all kinds of fat-based fillings may be aerated. There are two ways of aerating a filling: either by melting the fat and aerating during crystallization, or by starting from a solid pre-crystallized fat and aerating it. A fat suitable for whipping must be able to retain air bubbles during aeration. This result is obtained through small β′ fat crystals that create a network around the air bubbles. To stabilize the air bubbles, a certain

Crystallization

381

amount of solid fat is necessary, and thus it is not possible to aerate fats that are completely liquid. The method of starting from a pre-crystallized fat is often used for manufacturing wafer and biscuit fillings. These normally consist of fat, sugar and a flavouring. First the fat is tempered to room temperature, after which it is mixed with the other ingredients and whipped. It is also possible to use a continuous aeration process. In this case the filling is melted, and subsequently aerated during crystallization. It is thus important to have good control over the temperature gradient during the process to obtain optimal aeration, as well as pumpability of the cream. 10.10.6.5

Sugar-based confectionery fillings

The fat content of sugar-based fillings is normally less than 20%, and the continuous phase is a sugar and water solution emulsified with fat. The most important types of sugar-based fillings for confectionery are toffee/fudge and fondant. Toffee/fudge Toffee and fudge consist of oil, sugar, water and milk ingredients. The heating of sugar and milk proteins results in a Maillard reaction (see Section 16.2.1), a process that gives the toffee its characteristic flavour and colour. The hardness and texture of the final toffee/fudge are mainly determined by the water content, which in turn is controlled by the boiling temperature. The higher the temperature, the lower the water content and the harder the final product. In toffee/fudge applications, the fat works as a smoothing and shortening agent, making the product less sticky. Another very important function of the fat is as a flavour carrier, which means that it needs to have very good flavour and flavour stability. Because of the processing conditions, the fat must also tolerate high temperatures and a high water content, without the risk of oxidation, hydrolytic reactions or the formation of off flavours. Fondant When a sugar solution is cooled under constant agitation, small crystals are formed, transforming the solution into a white, sticky dough or mass – a fondant. Here as well, the water content is the main factor that determines the hardness of the product. A low water content allows a fondant to be moulded in starch moulds or formed into various shapes. With a higher water content, the fondant is more liquid and can be used as a fondant cream filling. The fat’s role in a fondant is similar to its role in toffee. It can smooth the texture and give the fondant extra creaminess, but most of all it acts as a flavour carrier. In this case the fat might be added to the sugar solution at more moderate temperatures. It still needs to tolerate a high water content and be very stable against oxidation, however, since air will be mixed into the fondant mass during processing.

10.10.7 10.10.7.1

Fats for ice cream coatings and ice dippings/toppings Coatings

Both chocolate and compounds are used in the coating of ice cream. For chocolate, the normal chocolate legislation is applicable, i.e. if vegetable fat is allowed, the same rules

382

Confectionery and Chocolate Engineering: Principles and Applications

apply to its use in ice cream coatings. In the European Union, coconut oil is added to the list of permitted vegetable fats, as mentioned previously. Coatings are in common use today in the production of ice cream. Since the fat content of a coating varies between 55% and 70%, its properties are derived mainly from the fat used. Besides economic considerations, vegetable fats have an advantage over cocoa butter in ice cream coatings as they are better equipped to meet the special demands made. In particular, fast crystallization is critical, and non-transparent but thin layers are required. Ice cream coatings should have a melting point below 30°C, otherwise the coating will not melt during eating, since the mouth temperature falls a few degrees when ice cream is being eaten. A coating must have good snap, and then melt rapidly and totally to give a good mouthfeel. The SFC curve (Fig. 10.23) is a good indicator in this respect. With a view to simplifying the production process and ensuring that the product is completely covered with a thin, non-transparent layer, ice cream coatings should have a low viscosity. If this is not the case, ‘bleeding’ will occur, with the ice cream appearing in white spots on the coating. Not only is this unattractive, but it may also cause wrappers to stick. Moreover, coatings must also have a somewhat elastic structure to prevent them from breaking or chipping off the ice cream during eating. A coating should have a rapid rate of solidification, so that production output can be kept at a high level. A fast flavour release is particularly essential in coatings. Since the fat is the carrier of the flavour, it should be chosen with care to guarantee fast flavour release. The types of fats used in ice cream coatings are mainly lauric fats, such as coconut oil. These fats have an advantage over cocoa butter in their high crystallization rates – an important property in this field. To prevent the coating from cracking, other, softer vegetable fats are often added to the lauric fats to increase the elasticity. There are also nonlauric fats available for ice cream coatings. These are particularly suitable in cases where the same equipment is used for the production of both chocolate and ice cream coatings. The risk of contamination between a lauric and a non-lauric fat is then eliminated. Whenever a thicker layer of coating on ice cream is desired, a non-lauric fat is also a good alternative. 10.10.7.2

Toppings

Ice dippings and ice toppings are other types of ice cream coatings, applied at the point of sale or at home. These should be liquid at room temperature and solidify almost instantly when in contact with cold ice cream. As the product may be kept at room temperature for a long period of time, stability against oxidation and rancidity is important. The quality of an ice dipping or ice topping is largely determined by the properties of the fat system. The product should have a low viscosity and be liquid at room temperature. Since cooling is only by contact with the ice cream, a rapidly crystallizing fat is needed. The melting profile of the fat should also be designed to ensure good flavour release.

10.11

Crystallization of confectionery fats with a high trans-fat portion

There are health concerns about trans fatty acids formed by hydrogenation, related to the coating fats and filling fats used in confectionery.

Crystallization

10.11.1

383

Coating fats and coatings

Foubert et al. (2006b) investigated a trans-containing coating fat (TCF) with 36% trans fatty acid content and a trans-free coating fat (TFCF) with 0.4% trans fatty acid content of lauric type. (In both samples, the trans fatty acid was of C 18:1 type.) The isothermal crystallization behaviour was as follows: • TCF: modified β′ at 17°C and direct β′ at 23°C. • TFCF: mediated β′ and direct β′ at 23°C. • The crystallization rate was higher for the TFCF. The behaviour in storage was as follows: • • • •

TFC: increasing hardness due to continued crystallization. TFCF: increasing hardness due to changes in microstructure and sintering. No significant difference in hardness. However, there were large differences in the hardness of coatings due to various triglyceride interactions and other components in the coating matrix.

10.11.2

Filling fats and fillings

Vereecken et al. (2007) investigated a trans-containing filling fat (TCFF) with 10% trans fatty acid content of lauric type and a trans-free filling fat (TFFF) with 0.2% trans fatty acid content. (In both samples, the trans fatty acid was of C 18:1 type.) The isothermal crystallization behaviour was as follows: • TCFF: at 15–20°C, limited post-crystallization, direct β′, no sandiness. • TFFF: at 8–10°C, direct α; at 12–17.5°C, α-mediated β′; at 20°C, direct β′. • The crystallization rate was higher for the TCFF. The behaviour in storage was as follows: • TCFF: decreasing hardness. • TFFF: decreasing hardness. • Both TCFF and TFFF: decreasing hardness; no significant difference in hardness after one week of storage; reduction in differences in crystal size and shape. • The filling matrix had an important influence. Pajin et al. (2007) investigated the influence of filling fat type on praline products with nougat fillings. Of three samples, two (S1 and S2) were practically lauric acid-free with a high trans fatty acid content (S1 = 38.08%; S2= 33.68%); the third sample had a lauric acid content of 11.28% and a relatively low trans fatty acid content (6.99%). The crystallization kinetics were observed by means of the change in the SFC, under static conditions at a 20°C crystallization temperature, using the NMR technique and a modified Gompertz model (see Eqn 10.95). The results showed that the amount of solid phase formed in S1 in the course of crystallization was 2–2.5 times larger than that in S2 and S3. In addition, the maximum rate of crystallization in S1 was about twice that in the two other samples. However, there were large differences in the induction periods: the induction period for

384

Confectionery and Chocolate Engineering: Principles and Applications

S2 was four times higher than that for S2, and the induction period of S3 was practically negligible, i.e. crystallization of S3 started promptly at 20°C. The investigation was complemented by a rheometric study. Measurement of SFC and viscosity are two approaches to determining the suitability of a filling fat to be used in a praline. Both methods provide valuable information and, particularly, viscosity build-up is very important from the point of view of production on a factory scale.

10.11.3

Future trends in the manufacture of trans-free special confectionery fats

Karlovits et al. (2006) discussed the possibilities of manufacturing filling confectionery fats of low trans fatty acid (TFA) content. They cited the conclusions of the European Food Safety Authority of 1 September 2004, which can be summarized as follows: • Both TFAs and SAFAs (saturated fatty acids) are risk factors for high blood pressure and coronary heart disease. • At equivalent dietary levels, the effects of TFAs on the heart may be greater than those of SAFAs; however, the intake of TFAs is c. 10 times less than that of SAFAs, taking into account the dietary recommendations in many European countries. • The relationship between TFA intake and cancer, Type 2 diabetes and allergies is weak and inconsistent. • At present, TFAs from natural sources and those formed during food processing cannot be distinguished analytically. Karlovits et al. (2006) highlighted a contradiction: if two blends of the same SFC are produced with a low trans fat content and a high trans fat content, the following equation holds for the sum SAFA + TRANS:

(SAFA + TRANS)low trans fat ≥ (SAFA + TRANS)high trans fat

(10.145)

That is, it is easier to produce a blend with high-trans-fat components than with lowtrans-fat components if the sum SAFA + TRANS and the SFC are specified. However, the traditional possibilities (an increase in the ratio of fully hardened vegetable oils, lauric oils, palm oil fractions etc.) decrease the TRANS content but increase the sum SAFA + TRANS. According to Karlovits et al. (2006), a non-traditional possibility may be to develop a new generation of confectionery fats which consist of liquid oil, fat replacers, gelling/ texture-building agents, emulsifiers and antioxidants. However, a series of new filling fats has already been presented (the ‘Kruszwica’ range): all-purpose fats, a medium-trans range and a low-trans range, the latter of which is non-lauric and GMO-free. Health concerns about trans fatty acids formed by hydrogenation have led to the use of interesterification, fractionation, and blending of saturated and polyunsaturated oils as an alternative method to hydrogenation. These alternatives are costly and do not easily produce the desirable physical and chemical properties of oils suitable for broad ranges of food products. Therefore, these alternative methods cannot easily replace the hydrogenation of vegetable oils. Hydrogenation is still a viable choice for food manufacture if trans fatty acids can be substantially reduced during the hydrogenation process (Jang et al. 2006). New hydrogenation processes such as electrocatalytic hydrogenation,

Crystallization

385

precious-metal catalyst hydrogenation and supercritical-fluid-state hydrogenation have shown promising results for the reduction of trans fatty acids below the level of 8%. These hydrogenation techniques would be viable alternatives for replacing the conventional Ni catalyst hydrogenation to produce hydrogenated products with low trans fatty acids. However, further research needs to be done on the economic feasibility of new hydrogenation processes and the reusability of precious-metal catalysts.

10.12 10.12.1

Modelling of chocolate cooling processes and tempering Franke model for the cooling of chocolate coatings

Franke (1998) proposed a one-dimensional, unsteady-state heat transfer model for the crystallization of chocolate coatings of coated products using a linear term q′/cp to account for latent-heat evolution: ∂T q′ = ∇ (a ∇ T ) + ∂t cp

(10.146)

where ∂T/∂t (K/s) is the change of temperature T as a function of time t, a (m2/s) is the thermal diffusivity of the chocolate mass, q′ (kJ/kg s) is the specific heat of crystallization per unit time, cp (kJ/kg K) is the specific heat capacity of the chocolate and ∇ = ∂( )/∂xi, where i = 1, 2, 3. The form of the function T is T = T(t, x), where the x direction is perpendicular to the chocolate surface. Obviously, Eqn (10.146) is Fourier’s second equation for thermal conduction plus an external heat source originating from crystallization of cocoa butter. This equation is combined with appropriate heat-transfer-coefficient boundary conditions for convective heat transfer from the surrounding air (Newton’s law of cooling): ∂Q = α F (Tair − Tsurface ) ∂t

(10.147)

where α (kJ/m2 K s) is the heat transfer coefficient and F (m2) is the surface area of the cooling chocolate. The integral of Eqn (10.147) leads to a function T = T(t) of the following form: T (t ) = T0 exp ( − kt )

(10.148)

where T0 and k are determined from the boundary conditions. An essential part of the Franke model is the supposition that q′ can be decomposed into a product of two variables, q ′ = f1 (Qm ) f2 (T )

(10.149)

where f1(Qm) (kJ/kg s) is the dependence of the rate of crystallization on the undercooling and Qm (kJ/kg) is the specific heat of crystallization per unit time that has already been released during crystallization, and

386

Confectionery and Chocolate Engineering: Principles and Applications

f1 f1m m1

m2

Qmax

Qm

Qend

Fig. 10.24 Modelling of heat flux as a function of released crystallization heat Qm [reproduced from Franke (1998), by kind permission of Elsevier].

f2 (T ) = fT (Tcryst − T )

(10.150)

where fT is a coefficient (0.00043/K, experimentally determined), and Tcryst (°C) is the temperature of crystallization (= 26.5°C, experimentally determined for plain chocolate). Consequently, one of the initial conditions is the following relationship derived from Eqn (10.150): if Tcryst ≤ T , then q ′ = 0

(10.151)

The function f2(T) is linear, as Eqn (10.150) shows. The modelling of f1(Qm) is represented in Fig. 10.24. The released heat of crystallization is proportional to f1. The evolution of the heat flux of crystallization (q′ vs t) is presented in Fig. 10.26 below, which is a mapping of the f1 vs Qm plot modified by the effect of f2. Calculated temperature plots for different layers were obtained from the first term of Eqn (10.146), which describes the penetration of the cooling effect with respect to both time t and depth x. On the surface, x = 0. Equations (10.146)–(10.150) provide a complete description of the cooling process of chocolate couverture; solutions can be obtained by numerical integration by computer. The model is able to fit the general course of temperature variation, including the plateau phase caused by the release of heat of crystallization. A plot of T(t)x is presented in Fig. 10.25 to illustrate this fact. Simulations of the cooling of coated cookies under different cooling conditions using the model showed possibilities for optimizing the process with respect to the expected surface gloss and hardness. Figure 10.26 presents a plot of the specific crystallization rate versus cooling time. Temperature plots for different layers of the chocolate coating during cooling by forced convection were calculated.

10.12.2

Modelling the temperature distribution in cooling chocolate moulds

Tewkesbury et al. (2000) developed a computational model using the alternative approach of the effective heat capacity, in which latent heat is included in the specific-heat term:

Crystallization

387

26 Calculated values Measured values

Temperature (°C)

25 24 23 22 21 20 0

1

2

3 Cooling time (min)

4

5

6

Fig. 10.25 Cooling curve T(t) for chocolate crystallization obtained from measured and calculated values [reproduced from Franke (1998), by kind permission of Elsevier].

Specific crystallization rate (kJ/kg/min)

35

0.4 mm (bottom) 30

1.2 mm (middle) 2.0 mm (surface)

25 20 15 10 5 0 0

60

120

180

240

300

Cooling time (s) Fig. 10.26 Specific crystallization rate as a function of cooling time. Calculated temperature plot in different layers of the chocolate coating during cooling by forced convection [reproduced from Franke (1998), by kind permission of Elsevier].

ρc p eff

∂T = ∇ (λ ∇ T ) ∂t

(10.152)

where ρ (kg/m3) is the density of the chocolate, cp eff (kJ/kg K) is the effective heat capacity, which is a function of temperature and thermal history, and λ (kJ/m s) is the coefficient of thermal conduction. This approach takes as its starting point the fact that chocolate displays a range of melting temperatures instead of a single melting point. In the classical Maxwell–Stefan

388

Confectionery and Chocolate Engineering: Principles and Applications

Polycarbonate mould

A

Chocolate

C1 C2

C3

4 mm 4 mm 4 mm

Thermocouple leads to data logger

2 mm B

4 mm

Ø 60 mm Fig. 10.27 Vertical cross-section of a mould with thermocouple positions [reproduced from Tewkesbury et al. (2000), by kind permission of Elsevier].

formulation of phase transition problems, solidification occurs at a well-defined front that moves through the material (Taylor et al. 1993, Viskanta et al. 1997). This is appropriate for simple materials such as water and elemental metals but is inappropriate for chocolate, which is a multicomponent mixture of triglycerides that display a spread of melting points. The effective specific heat capacity of tempered chocolate can be obtained from DSC experiments at various cooling rates. Crystallization is a kinetic process, so the effective heat capacity of chocolate is dependent on the thermal history of the sample and on the cooling rate (Stapley et al. 1999). A plan view of a mould with thermocouple positions used by Tewkesbury et al. (2000) is shown in Fig. 10.27. Figure 10.28 presents several cooling curves for (a) untempered and (b) tempered chocolate with a nominal air-cooling rate of 2°C/min. These experiments proved, as Stapley et al. (1999) showed, that the slower the material is cooled, the higher the temperature at which latent heat is evolved, and also the greater the amount of latent heat. Two methods were used to model the latent-heat release: • Use of cp vs T data from a single dataset corresponding to the nominal cooling rate of the cooling tunnel, i.e. working with a function cp = cp(T)w (where w is the cooling rate). • Use of a set of cp vs T curves over a range of cooling rates, i.e. working with a function cp = cp(T; w), which is a 3D surface plot. The effective specific heat capacity is high in the region of melting (20–30°C), as the peak in Fig. 10.29 shows. To solve the differential equation (10.152) taking the effective heat capacity functions cp = cp(T)w and cp = cp(T; w) into account, a simple moving-average technique was applied, which takes a window of N data points and finds the average ordinate. Good agreement was found between the model and experiment for cooling rates of 1 and 2°C/min, but they diverged at higher and lower cooling rates. Simulations that used a specific dataset for a single cooling rate alone failed to predict the temperature at which crystallization occurred. The program was thus altered to allow the specific-heat-capacity data for chocolate to be calculated as a function of both temperature and cooling rate, and the resulting datasets were used in the simulations. This fitted the experimentally measured mould temperatures well within a cooling-rate window of 0.5–2°C.

Crystallization

389

35

Temperature (°C)

30 C3 25 A

C2

20 B

C1

15 10 0

200

400

(a)

1000

800

600

1200

1400

Time (s) 30 28

Temperature (°C)

26

C3 A

24 22

B

20

C2

18 C1

16 14 12 10 0

500

(b)

1000 Time (s)

1500

2000

8 7 6 5 4

8 10

(°C)

0

–10

10

°C/m

ling

30 20 Temp eratu re

Coo

50 20

rate (

2 0 2 4 6

in)

3

Effective heat capa city (kJ/kg K)

Fig. 10.28 Cooling curves for (a) untempered and (b) tempered chocolate with a nominal air-cooling rate of 2°C/min [reproduced from Tewkesbury et al. (2000), by kind permission of Elsevier].

Fig. 10.29 Plot of effective heat capacity as a function of temperature and cooling rate [reproduced from Tewkesbury et al. (2000), by kind permission of Elsevier].

390

Confectionery and Chocolate Engineering: Principles and Applications

10.12.3

Modelling of chocolate tempering process

Debaste et al. (2008) developed a model that enhanced our understanding and control of the chocolate tempering process, and aimed to predict the temperature field during melting and crystallization. The heat transfer problem was simplified by using an effective thermal conductivity to model the mixing obtained with a newly designed stirrer. The heat conduction equation was solved using Comsol. The essential attributes of this model are: • A mechanical stirrer was designed to simulate manual mixing in a controlled and reproducible manner. • The convective terms in the heat balance equation were neglected, and an effectivethermal-conductivity approach was used to take into account the enhancement of heat transport by the mixing process. • A sink term was added to the heat balance equation to take into account the additional cooling arising from the latent heat of melting of the solid pieces used as crystallization seeds. This term was written in the form of a kinetic equation, whose parameters were identified by differential scanning calorimetry and by melting experiments carried out under adiabatic conditions. One should note that the term ‘kinetic’ is used in a broad sense in thermal analysis: it covers the study and modelling of the rate(s) of change of measured quantities (Várhegyi 2007). • The predicted transient temperature profiles were validated against the results of tempering experiments, using the new stirring device. The mathematical model consists of the following three equations. The first equation is

ρc p

∂T = ∇ ( λeff ∇ T ) + Q ∂t

(10.153)

where ρ and cp are the density and specific heat capacity, respectively, of chocolate; λeff (kJ/m s) is a lumped thermal-conductivity parameter taking into account the enhancement of heat transport by the mixing process of tempering; and Q is expresses the heat originating from crystallization The initial condition (the second equation) is expressed as T (t = 0 ) = T0

(10.154)

where the temperature T0 is uniform within the whole tempering bowl. The heat flux boundary condition (the third equation) at the wall of the bowl and on the free surface is given as a sum of a convective flux and a radiative flux: ∂T ⎞ 4 −k ⎛ = htot (T − Text ) + σε (T 4 − Text ) ⎝ ∂t ⎠ wall

(10.155)

where the emissivity ε and the convective heat transfer coefficient htot depend on the nature and positions of the surfaces considered; σ (W/m2 K4) is the Stefan–Boltzmann constant. (The units of k are W s/m2 K.) Figure 10.30 shows a comparison of the convective heat flux with the radiative heat flux as a function of temperature in the case of a grey (ε = 0.93)

Crystallization

391

Ratio of convective to radiative flux

1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0

20

40 60 Temperature (°C)

80

100

Fig. 10.30 Convective heat flux divided by radiative heat flux as a function of temperature; ε = 0.93, ambient air temperature = 20°C [reproduced from Debaste et al. (2008), by kind permission of Elsevier].

body in ambient air at 20°C. Obviously, the radiative heat flux is too important to be neglected in the model. The appropriate relationship for calculating the global heat transfer coefficient htot at the wall is 1 1 d wall 1 = + + htot hin λ wall hout

(10.156)

where the corresponding heat transfer coefficients (W/m2 K) are hin (on the inner wall) and hout (the resistance of the outer wall); dwall is the thickness of the wall (m) and λwall is the conductivity of the wall (W/m K). The heat resistance of the inner wall hin was calculated from a relationship of the type Nu = f(Re; Pr) referred to the anchor stirrers used. When the bowl was immersed in water, the resistance of the outer wall could be neglected, and when the bowl was in air this resistance was calculated from a relationship of the type Nu = g(Ra; Pr), where Ra is the Rayleigh number. For details, see Debaste et al. (2008). If the total observed heat flow used for melting is F = F(t), then the degree of conversion is given by the fraction of the heat flow that has been consumed by the melting,

α=

∫ F (t ) dt (t: 0 → t ) ∫ F (t ) dt (t: 0 → ∞ )

(10.157)

It was proposed in this model to describe the kinetic model of melting under nonisothermal conditions by the following equation (Chen et al. 2007): dα Ea ⎛ ⎞ = A0 exp ⎜ − (1 − α )n ⎝ R (T − Tref ) ⎟⎠ dt

(10.158)

392

Confectionery and Chocolate Engineering: Principles and Applications

where Tref = 0°C. The parameters A0, Ea and n were fitted to DSC results by a nonlinear multivariable least-squares technique (dα/dt as a function of α and T) (Wasan 1970). The values of the kinetic parameters of Eqn (10.158) were log A0 = 5.34, Ea = 1001.4 cal/mol and n = 2.8. The curves of α = α(t) (see Eqn 10.158) were of the usual S-shape. The model gives an accurate prediction of the cooling rate and the temperature field within a mass of melted chocolate seeded with small solid grains and left at ambient temperature. It can be used to identify the criteria for good tempering. It could be observed that the initial temperature of the seeds was not be a critical parameter, whereas the ambient temperature, not surprisingly, had a large influence in the case of cooling in ambient air. The opinion of Debaste et al. (2008) is that, in its present state of development, this model is unable to correlate the prediction of the evolution of temperature with time to the quality of tempering. These authors’ ongoing studies are focusing on the development of a shrinking-core model to get a better description of the kinetics of the melting of seeds, which is related to the value of Q (see Eqn 10.153). In addition, to complete the model, the nucleation that takes place later in the second stage of the tempering process will be studied.

Further reading Alikonis, J.J. (1979) Candy Technology. AVI Publishing, Westport, CT. Beckett, S.T. (ed.) (1988) Industrial Chocolate Manufacture and Use. Van Nostrand Reinhold, New York. Beckett, S.T. (2000) The Science of Chocolate. Royal Society of Chemistry, Cambridge. da Silva Martins, P.M. (2006) Modelling crystal growth from pure and impure solutions: A case study on sucrose. Doctoral thesis, University of Porto, Portugal. De Graef, V., Van Puyvelde, P., Goderis, B. and Dewettinck, K. (2009) Influence of shear flow on polymorphic behavior and microstructural development during palm oil crystallization. Eur J Lipid Sci Technol 111: 290–302. De Graef, V., Goderis, B., Van Puyvelde, P., Foubert, I. and Dewettinck, K. (2008) Development of a rheological method to characterize palm oil crystallizing under shear. Eur J Lipid Sci Technol 110: 521–529. De Graef, V., Foubert, I., Smith, K.W., Cain, F.W. and Dewettinck, K. (2007) Crystallization behavior and texture of trans-containing and trans-free palm oil based confectionery fats. J Agric Food Chem 55(25): 10258. De Graef, V., Dewettinck, K., Verbeken, D. and Foubert, I. (2006) Rheological behavior of crystallizing palm oil. Eur J Lipid Sci Technol 108: 864–870. Friberg, S.E., Larsson, K. and Sjöblom, J. (2003) Food Emulsions. Marcel Dekker, New York. Kempf, N.W. (1964) The Technology of Chocolate. The Manufacturing Confectioner Publishing Co., Glen Rock, NJ. Lakatos, B.L. and Blickle, T. (1995) Nonlinear dynamics of isothermal CMSMPR crystallizers: A simulation study. Computers Chem Eng 11 (Suppl 1): 501–506. Lees, R. (1980) A Basic Course in Confectionery. Specialized Publications Ltd, Surbiton. Marangoni, A.G. and Narine, S.S. (2004) Fat Crystal Networks. Marcel Dekker, New York. Meiners, A. and Joike, H. (1969) Handbook for the Sugar Confectionery Industry. SilesiaEssenzenfabrik, Gerhard Hanke K.G. Norf, Germany. Minifie, B.W. (1970) Chemical analysis and its application to candy technology. Confect Prod 36 (7): 423–426, 449. Minife, B.W. (1970) Chemical analysis and its application to candy technology – The analysis of fats. Confect Prod 36 (9): 554–555.

Crystallization

393

Minife, B.W. (1970) Chemical analysis and its application to candy technology – The analysis of fats. Confect Prod 36 (10): 615–616. Minife, B.W. (1970) Chemical analysis and its application to candy technology – The analysis of fats. Confect Prod 36 (12): 746–747, 770. Minifie, B.W. (1999) Chocolate, Cocoa and Confectionery, Science and Technology, 3rd edn. Aspen Publications, Gaithersburg, MD. Narine, S.S. and Marangoni, A.G. (2002) Physical Properties of Lipids. Marcel Dekker, New York. Pratt, C.D. (ed.) (1970) Twenty Years of Confectionery and Chocolate Progress. AVI Publishing, Westport, CT. Rojkowski, Z. (1977) New empiracal kinetic equation of size dependent crystal growth and its use. Krist u Technik 12 (11): 1121–1128. Rojkowski, Z.H. (1993) Crystal growth rate models and similarity of population balances for sizedependent growth rate and for constant growth rate dispersion. Chem Eng Sci 48 (8): 1475–1485. da Silva Martins, P.M. (2006) Modelling Crystal Growth from Pure and Impure Solutions – A Case Study on Sucrose. Doctorial Thesis. University of Porto, Portugal. Sullivan, E.T. and Sullivan, M.C. (1983) The Complete Wilton Book of Candy. Wilton Enterprise, Inc., Woodridge, IL. Vereecken, J., Foubert, I., Smith, K.W. and Dewettinck, K. (2007) Relationship between crystallization behavior, microstructure, and macroscopic properties in trans-containing and trans-free filling fats and fillings. J Agric Food Chem 55 (19):7793. Widlak, N. (1999) Physical Properties of Fats, Oils, and Emulsifiers. American Oil Chemists Society. Wieland, H. (1972) Cocoa and Chocolate Processing. Noyes Data Corp., Park Ridge, NJ.

Chapter 11

Gelling, emulsifying, stabilizing and foam formation

Contents 11.1 11.2

Hydrocolloids used in confectionery Agar 11.2.1 Isolation of agar 11.2.2 Types of agar 11.2.3 Solution properties 11.2.4 Gel properties 11.2.5 Setting point of sol and melting point of gel 11.2.6 Syneresis of an agar gel 11.2.7 Technology of manufacturing agar gels 11.3 Alginates 11.3.1 Isolation and structure of alginates 11.3.2 Mechanism of gelation 11.3.3 Preparation of a gel 11.3.4 Fields of application 11.4 Carrageenans 11.4.1 Isolation and structure of carrageenans 11.4.2 Solution properties 11.4.3 Depolymerization of carrageenan 11.4.4 Gel formation and hysteresis 11.4.5 Setting temperature and syneresis 11.4.6 Specific interactions 11.4.7 Utilization 11.5 Furcellaran 11.6 Gum arabic 11.7 Gum tragacanth 11.8 Guaran gum 11.9 Locust bean gum 11.10 Pectin 11.10.1 Isolation and composition of pectin 11.10.2 High-methoxyl (HM) pectins 11.10.3 Low-methoxyl (LM) pectins 11.10.4 Low-methoxyl (LM) amidated pectins 11.10.5 Gelling mechanisms 11.10.6 Technology of manufacturing pectin jellies 11.11 Starch Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

395 395 395 396 396 397 398 398 399 400 400 401 401 402 402 402 403 404 405 405 405 406 407 407 408 408 409 409 409 410 411 411 411 412 413

Gelling, emulsifying, stabilizing and foam formation

11.11.1 Occurrence and composition of starch 11.11.2 Modified starches 11.11.3 Utilization in the confectionery industry 11.12 Xanthan gum 11.13 Gelatin 11.13.1 Occurrence and composition of gelatin 11.13.2 Solubility 11.13.3 Gel formation 11.13.4 Viscosity 11.13.5 Amphoteric properties 11.13.6 Surface-active/protective-colloid properties and utilization 11.13.7 Methods of dissolution 11.13.8 Stability of gelatin solutions 11.13.9 Confectionery applications 11.14 Egg proteins 11.14.1 Fields of application 11.14.2 Structure 11.14.3 Egg-white gels 11.14.4 Egg-white foams 11.14.5 Egg-yolk gels 11.14.6 Whole-egg gels 11.15 Foam formation 11.15.1 Fields of application 11.15.2 Velocity of bubble rise 11.15.3 Whipping 11.15.4 Continuous industrial aeration 11.15.5 Industrial foaming methods 11.15.6 In situ generation of foam Further reading

11.1

395 413 414 414 416 416 416 417 417 418 418 419 420 421 421 422 422 422 423 424 424 425 425 425 426 429 430 432 432 433

Hydrocolloids used in confectionery

Jellies and foams (e.g. marshmallows) are popular types of confectionery. These products are made from hydrocolloids. Consequently, gelling and foaming are important operations. Emulsifying and stabilizing are essential from the point of view of emulsions (see Section 5.8). The technological role of hydrocolloids in the confectionery industry is very complex and can be difficult to categorize because of the wide range of effects exerted. In the following sections, the most characteristic properties of the hydrocolloids used in confectionery are discussed.

11.2 11.2.1

Agar Isolation of agar

Agar is a gelatinous product isolated from seaweed (red algae class, Rhodophyceae, e.g. Gelidium spp., Pterocladia spp. and Gracilaria spp.) by a hot-water extraction process. Purification is possible by congealing the gel.

396

Confectionery and Chocolate Engineering: Principles and Applications

Agar is a heterogeneous complex mixture of related polysaccharides having the same backbone chain structure. The main components of the chain are β-D-galactopyranose and 3.6-anhydro-α-L-galactopyranose, which alternate through 1 → 4 and 1 → 3 linkages. The chains are esterified to a low extent with sulphuric acid. The sulphate content differentiates the agarose fraction (the main gelling component of agar), in which close to every tenth galactose unit of the chain is esterified, and the agaropectin fraction, which has a higher sulphate esterification degree and, in addition, has pyruvic acid bound in ketal form [4.6-(1-carboxyethylidene)-D-galactose]. The ratio of the two polymers can vary greatly. Uronic acid, when present, does not exceed 1%.

11.2.2

Types of agar

Bar-style agar. The weight of one piece is 7.5 g on average. Owing to its honeycomb-like structure, the bulk density is about 0.030–0.036 g/cm3. Stringy agar. The normal length of an agar string is 28–36 cm, although there are no definite required dimensions. Agar flakes and powdered agar. ‘Flakes’ (or coarse powder) are normally produced by a freezing process, while most of the ‘powdered agar’ (or fine powder) is processed by a pressing dehydration method. The Japanese agricultural standard for powdered agar (officially, ‘special type agar’) is given in Table 11.1.

11.2.3

Solution properties

Agar is not soluble in cold water but is soluble in boiling water. Merely heating the water, if it is kept at a temperature below boiling point, does not bring about perfect dissolution. When bar-style agar, stringy agar or agar flakes are used, soaking in cold water beforehand, preferably overnight, greatly assists in full dissolution. Even when so-called ‘quickly dissolvable agar’ is used, at least 5–10 min soaking is recommended. Two ranges of viscosity of agar sols can be distinguished (see Matsuhashi 1990): (1) Low concentrations, c (%) = 0.06–0.2 and ηrel = 1.2–3.5. For this region, log ηrel = Kc

(11.1)

where K is a constant (0.9–1.22, depending on the type of agar).

Table 11.1 Grades specified by Japanese agricultural standard for powdered agar [reproduced from Matsuhashi (1990), by kind permission of Springer Science and Business Media].

Gel strength (g/cm3) Insoluble in hot water (%) Crude protein (%) Crude ash (%) Moisture (%)

Superior

1st

2nd

3rd

≥ 600 < 0.5 < 0.5 <4 < 22

≥ 350 <2 < 1.5 <4 < 22

≥ 250 <3 <2 <4 < 22

≥ 150 <4 <3 <4 < 22

Gelling, emulsifying, stabilizing and foam formation

397

(2) High concentrations, c (%) = 0.8–4.1. Gel setting occurs upon cooling. Agar gels are typical dissolution gels because gelation is started by the effect of cooling a solution of agar. Agar is a most potent gelling agent, as gelation is perceptible even at 0.04%. The setting and stability of the gel are affected by the agar concentration and its average molecular weight. A 1.5% solution sets to a gel at 32–39°C, but does not ‘melt’ (transition from the gel state to the sol state) below 60–97°C. The great difference between the gelling and melting temperatures, due to hysteresis, is a distinct and unique feature of agar.

11.2.4

Gel properties

Agar can be regarded as a prototype and model for all gelling systems (Rees 1969). Typically, a 1% concentration in water is good enough to make a rigid gel suitable for most applications. Upon cooling to about 30–40°C, the sol sets to a firm gel that must be heated to about 85–95°C before it will ‘melt’, i.e. soften. Thus, the hysteresis of the thermally reversible changes between the gel and sol phases of agar is significant (Indovina et al. 1979), because the sol can set at room temperature, and the resultant gel is stable even during the hottest summer season. For the preparation of a gel, is essential to boil the agar at natural pH in order to make a ‘true gel’; merely mixing the agar with water will make only a ‘false gel’. 11.2.4.1

Dynamic properties of agar gels

The concept of ‘gel strength’ varies between countries. Matsuhashi (1990, p. 23), recommended and briefly described the methods of determination of gel strength used by Marine Colloids and by Meer Corporation. The gel strength (as used by Marine Colloids) in the case of agarose is defined as that force, expressed in grams per square centimetre, which must be applied to a 1% agarose gel to cause it to fracture. ‘Apparent gel strength’ means a value at a non-defined concentration (Matsuhashi 1970). Matsuhashi found a linear relationship between the apparent gel strength Y and the concentration of the gel X over a wide range of concentration for a number of agars: Y = aX − b

(11.2)

where a and b are constants. The value of X ≥ b/a should theoretically correspond to the minimum concentration (in per cent) of agar that forms a gel (Y ≥ 0). The concept of ‘gel strength’ was established by Tanii (1957), who measured the rigidity coefficient T/θ and the breaking (or cracking) load TC of agar gels using an improved torsion dynamometer (by Sheppard), where T is the twisting moment (in grams), θ is the angle of twist in degrees, and TC is the breaking load (in grams) that causes a crack of about 2 mm in a moulded gel. By using numerical relationships between the physical properties and the concentration of a gel, the properties of any sampled gel can be expressed relative to those of standard agar as follows: Cr, related to the rigidity coefficient; Cc, related to the cracking load; Cr.s, related to the rigidity coefficient of a 1% gel.

398

Confectionery and Chocolate Engineering: Principles and Applications

The gel strength (or the gel properties in general) of agar is most stable at about or a little above pH 7.0. Therefore, if any kind of acidic substance needs to be added to a gel preparation, it should be added after the agar is dissolved and at a temperature lower than the boiling point of water.

11.2.5

Setting point of sol and melting point of gel

The setting point (or gelling point) of a sol is the temperature of gelation (= setting). The melting point is the temperature at which the gel → sol change takes place. It should be emphasized that melting of a gel does is not the same as the melting of a crystalline substance, although both types of melting are a solid → liquid change. For methods of measuring the melting point, see Matsuhashi (1990, p. 28). There is a relationship between gel strength and melting point. In general, as the gel strength increases, the melting point also increases. However, there are exceptions to the general rule, for example Ceramium agars. At low concentrations, the melting point of most agar gels is dependent on concentration; at high concentrations, the melting point becomes independent of concentration. The actual melting point of agar depends on the type and the seaweed source. 11.2.5.1

Mechanism of gelation

It is the agarose component of agar that is responsible for gelation. At elevated temperatures, agarose exists as a disordered ‘random coil’, but on cooling it forms strong gels at a low polymer concentration. The gelation process involves the adoption of an ordered double-helix state (Ablett et al. 1978; Morris and Norton 1983).

11.2.6

Syneresis of an agar gel

An agar gel in a closed vessel will eliminate a certain amount of water at its surface as time passes. This phenomenon is called syneresis and results in volumetric shrinkage of the gel. The basic factors that influence the syneresis of agar gels are: Concentration of agar in gel. The amount of synerized water Δw(t) after a time t is approximately inversely proportional (∼) to the square root of the agar concentration X for most practical concentrations (Tanii 1959): Δw (t ) ~

1 X

(11.3)

Holding time of gel. At 30°C, syneresis almost reaches an equilibrium state in 72 h; 144 h is recognized as the time to reach practical equilibrium. See Chapter 5, Eqn (5.17), m (t ) = m∞ + ( m0 − m∞ ) exp ( − kt )

(11.4)

where k is the rate constant of syneresis, and m(t), m∞ and m0 are the weight of gel at time t, after very long storage and at the beginning of storage, respectively. The amount of synerized water is Δw(t) = m0 − m(t).

399

Gelling, emulsifying, stabilizing and foam formation

Apparent gel strength. The higher the apparent gel strength, the lower the syneresis: Δw (t ) ~

1 Y

(11.5)

This relation is a consequence of the relation expressed by Eqn (11.2). Rigidity coefficient. For a variety of agars, the amount of synerized water is proportional to the rigidity coefficient Cr.s: Δw (t ) ~ Cr.s

(11.6)

Pressurization. For most agar gels, the application of pressure increases syneresis; however, exceptions have been reported as well: Δw (t ) ~ Δp

(11.7)

Total sulphate content. The amount of synerized water is generally inversely proportional to the total sulphate content: Δw (t ) ~

11.2.7

1 (SO4 )

(11.8)

Technology of manufacturing agar gels

The effect of added sucrose on the rheological properties of 1% agar gel was investigated by Nakahama (1966) and Isozaki et al. (1976) using sugar solutions in the range 0–75% and a concentration of agar of exactly 1%. Agar is widely used, for example, in preparing nutritive media in microbiology. Its application in the food industry is based on its main properties: it is essentially indigestible, forms heat-resistant gels and has emulsifying and stabilizing activity. Agar is added as a stabilizer to sherbets (frozen desserts of fruit juice, sugar, water or milk) and ice creams (at about 0.1%), often in combination with gum tragacanth or locust (carob) bean gum or gelatin. Agar has a role in vegetarian diets (meat substitute products), desserts and pre-treated instant cereal products. An amount of 0.1–1% stabilizes yogurt, some cheeses, candies and bakery products (pastry fillings). Agar (or pectin) dissolved in water is added to the fondant filling of pralines to hinder the separation of a thin aqueous sugar solution from the fondant mass under gravity; otherwise, the separated thin solution easily finds slits in the chocolate coating. The manufacture starts with soaking of the agar, and then the agar is dissolved in boiling water together with sugar. The solution is cooked to a boiling point of about 106°C, and then starch syrup (and possibly invert syrup) is added to this solution, which is boiled to 106°C again. After cooling to about 60°C, flavouring and colouring agents are added, and finally the resulting sol is cast in starch or Teflon moulds and allowed to gel at room temperature overnight (Fig. 11.1). The dry content of the cast gel is about 75%. Agar is one of the most favoured gelling agents (besides pectin and gelatin) by the confectionery industry for manufacturing jellies and jelly centres (for panning etc.).

400

Confectionery and Chocolate Engineering: Principles and Applications

Starch syrup

Sugar Agar

Soaking

Water

Dissolution

Evapo ration, 106°C

Mixing

Water Colouring, flavouring

Fig. 11.1

11.3 11.3.1 11.3.1.1

Cooling to about 60°C

Dosing

Schematic layout of agar jelly production.

Alginates Isolation and structure of alginates Isolation

Alginates occur in all brown algae (Phaeophyceae) as a skeletal component of their cell walls. The major source of industrial production is the giant kelp, Macrocystis pyrifera. Some species of Laminaria, Ascophyllum and Sargassum are also used. Algae are extracted with alkali. The polysaccharide is usually precipitated from the extract with acid or calcium salts. 11.3.1.2

Structure

The building of alginate are β-D-mannuronic and α-L-guluronic acid, joined by 1 → 4 linkages. The ratio of the two sugars (mannuronic/guluronic acids) is generally 1.5, with some deviations depending on the source. Alginates extracted from Laminaria hyperborea have ratios of 0.4–1.0. Partial hydrolysis of alginate yields chain fragments that consist predominantly of either mannuronic or guluronic acid, and also fragments where the two uronic acid residues alternate in a 1 : 1 ratio. Owing to the differing monomer (mannuronic or guluronic acid) ratios, alginates are commonly referred to as being either ‘high-M’ or ‘high-G’ alginates. The molecular weights M of alginates are in the range 32–200 kDa. This corresponds to a degree of polymerization n = 180–930. The pK values (the negative logarithm of the dissociation constant) of the carboxyl groups are 3.4–4.4. Alginates are water-soluble in the form of alkali, magnesium, ammonium and amine salts. The viscosity of alginate solutions is influenced by the molecular weight and the counter-ion of the salt. In the absence of divalent and trivalent cations or in the presence of a chelating agent, the viscosity is low (‘long flow’ property). However, with a rise in the levels of multivalent cations (e.g. calcium) there is a parallel rise in viscosity (‘short flow’). Thus, the viscosity can be adjusted as desired. As a general rule of thumb, doubling the alginate concentration increases the viscosity of the solution by a factor of 10. Freezing and thawing of a sodium alginate solution containing Ca2+ ions can result in a further rise in viscosity. A 1% solution, depending on the type of alginate, can have a

Gelling, emulsifying, stabilizing and foam formation

401

viscosity range of 20–20 000 cPa s. The viscosity is unaffected in a pH range of 4.5–10. It rises at a pH below 4.5, reaching a maximum at pH 3–3.5. Gels, fibres or films are formed by adding Ca2+ or acids to sodium alginate solutions. A slow reaction is needed for uniform gel formation. This is achieved by use of a mixture of sodium alginate, calcium phosphate and glucono-δ-lactone, or a mixture of sodium alginate and calcium sulphate. Propylene glycol alginate is a derivative of economic importance. This ester is obtained by the reaction of propylene oxide with partially neutralized alginic acid. It is soluble down to pH 2 and, in the presence of Ca2+ ions, forms soft, elastic, less brittle and syneresis-free gels.

11.3.2

Mechanism of gelation

By far the most important gels, particularly in food applications, are the gels formed with calcium ions. It is considered that calcium alginate gels are formed by simple ionic bringing of two carboxyl groups on adjacent polymer chains by calcium ions. The gel-forming ability is related mainly to the content of guluronic acid: two contiguous, diaxially linked G (guluronic) residues form a cavity that acts as a binding site for calcium ions. Long sequences of such sites form cross-links with similar sequences in other alginate molecules, giving rise to junctions in the gel network. This is often referred to as the ‘egg-box’ model of alginate gelation. The proportion of G blocks is the main structural feature relevant to gel formation.

11.3.3

Preparation of a gel

Alginates are capable of producing gels of differing strengths and textures suitable for a wide range of applications, but correct formulation requires the consideration of the fact that high-G alginates form strong, brittle gels with a tendency to synerize, whereas high-M alginates form weaker, more elastic gels that are less prone to syneresis. The term calcium conversion refers to the ratio of calcium ions to sodium in an alginate. A molar ratio of 0.5 (where, theoretically, there is sufficient calcium to totally replace the sodium) is expressed as a calcium conversion of 100%. For further details of gelation and syneresis, see Sime (1990, pp. 65–66). Alginate gels can be prepared by either diffusion or internal setting. Diffusion setting is the simplest technique and, as the name implies, the gel is set by allowing calcium ions to diffuse into an alginate solution. Since the diffusion process is slow, it can only be utilized effectively to produce thin strips or small spheres, or to provide a thin gelled coating on the surface of a food product. The most common source of calcium ions for diffusion is calcium chloride. Diffusion setting is ideally suited to the preparation of fruits with an outer skin and a liquid centre such as blackcurrants and blueberries. An alginate solution and a fruit purée mix are used, but they are kept separate and fed through nozzles consisting of two coaxial tubes. This process is sometimes referred to as ‘co-extrusion’. In internal or bulk setting, which is normally carried out at room temperature, the calcium is released under controlled conditions from within the system. Calcium sulphate, calcium carbonate and calcium hydrogen orthophosphate are the most common sources of calcium. The rate at which the calcium is made available to the alginate molecules

402

Confectionery and Chocolate Engineering: Principles and Applications

depends primarily on pH, and the amount, particle size and solubility characteristics of the calcium salt. A small particle size and low pH favour rapid release of calcium. Sequestrants are used in alginate solutions either to prevent the alginate from reacting with polyvalent ions (and other contaminants) in the solution or to sequester the calcium inherent in the alginate. The viscosity of an alginate solution is dependent on the molecular weight and the level of residual calcium. Sequestrants can be used to establish how much of the ‘viscosity’ is due to the presence of this residual calcium. In almost all situations in internal setting, the calcium release during mixing of the ingredients is so rapid that a calcium sequestrant is required to control the reaction by competing with the alginate for the calcium ions. Typical food-approved sequestrants are sodium hexametaphosphate, tetrasodium pyrophosphate and sodium citrate. Disodium hydrogen orthophosphate, although it has little affinity for calcium at pH values below 5, is sometimes used to remove calcium ions (as insoluble calcium phosphate) from tap water. Removal of the calcium permits more efficient hydration and solubilization of the alginate.

11.3.4

Fields of application

Alginate is a powerful thickening, stabilizing and gel-forming agent. At a level of 0.25– 0.5%, it improves and stabilizes the consistency of fillings for baked products (e.g. cakes and pies) and of salad dressings, and prevents formation of larger ice crystals in ice creams during storage. Furthermore, alginates are used in a variety of gel products (e.g. cold instant puddings, fruit gels, dessert gels and chocolate milks). Shear-reversible alginate gels are ideal for the production of layered desserts. The gel can be prepared aseptically in bulk and then pumped into individual containers. As the gel is deposited cold, and regains a large proportion of its gel strength almost immediately after the shearing action is removed, other layers can be deposited on top without delay. For further details, see Sime (1990, pp. 53–78)

11.4 11.4.1 11.4.1.1

Carrageenans Isolation and structure of carrageenans Isolation

Red seaweeds (Rhodophyceae) produce two types of galactans: agar and agar-like polysaccharides, composed of D-galactose and 3.6-anhydro-L-galactose residues; and carrageenan and related polysaccharides, composed of D-galactose and 3.6-anhydro-D-galactose, which are partially sulphated in the form of 2-, 4- and 6-sulphates and 2.6-disulphates. Galactose residues are linked alternately by 1 → 3 and 1 → 4 linkages. Carrageenans are isolated from Chondrus (Chondrus crispus, Irish moss), Eucheuma, Gigartina, Gloiopeltis and Iridaea species by hot-water extraction under mild alkaline conditions, followed by drying or isolate precipitation. 11.4.1.2

Structure

Carrageenans are a complex mixture of various polysaccharides. They can be separated by fractional precipitation with potassium ions. The two major fractions are κ-carrageenan

Gelling, emulsifying, stabilizing and foam formation

403

(the gelling and K+-insoluble fraction) and λ-carrageenan (non-gelling, K+-soluble), and there also is a smaller fraction ι-carrageenan (iota). κ-Carrageenan is composed of D-galactose, 3.6-anhydro-D-galactose and ester-bound sulphate in a molar ratio of 6 : 5 : 7. The galactose residues are essentially fully sulphated in position 4, whereas the anhydrogalactose residues can be sulphated in position 2 or substituted by α-D-galactose-6-sulphate or 2.6-disulphate. λ-Carrageenan is characterized by a higher sulphate content, which favours a zigzag, ribbon-shaped conformation. The molecular weights of the κ- and λ-carrageenans are 200–800 kDa. The water solubility becomes higher as the carrageenan sulphate content becomes higher and as the content of anhydrosugar residue becomes lower.

11.4.2

Solution properties

Carrageenans typically form highly viscous aqueous solutions. This is due to their linear macromolecular structure and polyelectrolytic nature. The viscosity of the solution depends on the type of carrageenan, the molecular weight, the temperature, the ions present and the carrageenan concentration. Carrageenan powders tend to pick up water very quickly, and the resultant viscous coating of the particles can lead to clumping, which impedes dissolution. Therefore, efficient powder dispersal is essential. To achieve this, one should: • Always use cold water (or milk) to disperse the carrageenan, and then heat to dissolve. • Use a high-shear mixer, which creates a vortex in the solution (without cavitation, which would entrap air) and add the carrageenan slowly into the vortex. • Add a diluent such as sugar to the carrageenan, at least 3 parts sugar to 1 part of carrageenan. • Use a ‘solution retardant’ such as liquid sugar, alcohol or glycerine as an initial dispersant since carrageenan will not dissolve in them. All carrageenans are soluble in hot water (> 75°C). Solutions of up to 10% of commercial carrageenan can normally be handled by conventional mixing equipment. The sodium salts of κ and ι are soluble in cold water, while salts of other cations such as Ca2+ and K+ do not dissolve completely but exhibit a varying degree of swelling. λ is fully soluble in cold water. 11.4.2.1

Viscosity of carrageenan solutions

Commercial carrageenans are generally available in viscosities ranging from about 5 to 800 mPa s when measured at 75°C and 1.5% concentration. Carrageenans with viscosities less than 100 mPa s have flow properties very close to Newtonian. The degree of deviation from Newtonian flow increases with concentration and molecular weight of the carrageenan. The solutions exhibit pseudoplastic (shear-thinning) flow properties and are sufficiently shear-dependent that it is necessary to specify the shear rate used for making a viscosity measurement. The pseudoplastic behaviour of carrageenans corresponds to a power-law (Ostwald–de Waele) equation (Guiseley et al. 1980):

404

Confectionery and Chocolate Engineering: Principles and Applications

τ = η = kD n−1 D

(11.9)

where τ (Pa) is the shear stress, D (s−1) is the shear rate, η is the apparent viscosity, n ≤ 1 is the flow behaviour index and k (Pa sn) is a constant. For a wide range of shear rates (two or more decades), a better fit to the experimental data can be achieved by adding a quadratic term:

τ 2 log η = log ⎛ ⎞ = log a + ( n − 1) log D + b ( log D ) ⎝ D⎠

(11.10)

where a > 0 and b ≤ 0 are constants. The viscosity increases nearly exponentially with concentration. This behaviour is typical of linear polymers carrying charged groups and is a consequence of the increase with concentration of the interaction between polymer chains. Salts lower the viscosity of carrageenan solutions by reducing electrostatic repulsion among the sulphate groups. This behaviour, likewise, is normal for ionic macromolecules. At high enough salt concentrations, however, κ- and ι-carrageenan solutions may gel, with an increase in apparent viscosity. The viscosity decreases with temperature and the change is exponential. The change is reversible provided that heating is done at or near the stability optimum at c. pH 9, and is not prolonged to the point where significant thermal degradation occurs. On cooling, however, the gelling types of carrageenan show an abrupt increase in apparent viscosity when the gelling point is reached, provided that the counter-ions (notably K+ and Ca2+) that promote gelation are present. The viscosity increases with molecular weight in accordance with the Mark–Houwink equation,

[η ] = KM α

(11.11)

where [η] = lim(c → 0) (ηsp/c) = lim(c → 0) (ln ηr/c) is the intrinsic viscosity, M is the average molecular weight (since carrageenans are polydisperse), and K and α are constants.

11.4.3

Depolymerization of carrageenan

Loss of molecular weight may occur through depolymerization of carrageenan. Carrageenans are particularly susceptible to depolymerization through acid-catalysed hydrolysis. This is related to the 3.6-anhydride content. Cleavage occurs preferentially at the 1.3 glycosidic linkages and is promoted by the strained ring system of the anhydride. Sulphation of the 1.4-linked units at C-2 appears to mitigate attack by acid, degradation rates for ι-carrageenan being roughly one-half those for κ-carrageenan under similar conditions. Carrageenans in the gel state are more stable to acid than are those in the sol state. The rate of acid-catalysed depolymerization is proportional to the hydrogen ion activity. Carrageenans are relatively resistant to alkaline degradation, though this does occur through ‘peeling’ reactions at high pH. As a consequence, carrageenans have maximum stability slightly on the alkaline side, at about pH 9.

Gelling, emulsifying, stabilizing and foam formation

11.4.4

405

Gel formation and hysteresis

The gelling-type (κ- and ι-) carrageenans and furcellaran require heat to bring them into solution. On cooling, the hot solutions set to gels. The anionic carrageenans require specific counter-ions, notably potassium and calcium, to be present for gelation to occur. According to Rees (1972), the molecular chains of carrageenans in aqueous jellies associate into double helices. Evidence for double-helix formation has been found from X-ray diffraction patterns of fibres of ι-carrageenan (Arnott et al. 1974; Bryce et al. 1974, 1982). In the case of ι- and κ-carrageenan, only when potassium or other gel-promoting cations are present can the domains aggregate into a three-dimensional network. The radius of the hydrated cation appears to be a factor; bulky cations, such as tetramethylammonium, do not induce gelation. The sol–gel transition temperatures of carrageenans typically exhibit hysteresis, the melting temperature being higher than the gelling temperature. This is a function of charge density. Hysteresis is most pronounced for furcellaran and κ-carrageenan (15–27°C), and least (2–5°C) for ι-carrageenan, which has the highest half-ester sulphate content of the gelling carrageenans. For further details, see Rees et al. (1969).

11.4.5

Setting temperature and syneresis

The setting temperatures of carrageenans depend primarily on the concentration of gelling cations present, and are relatively insensitive to the carrageenan concentration. Although both K+ and Ca2+ can cause both κ- and ι-carrageenan to gel, the K+ concentration is the factor controlling the sol–gel transition for κ, whereas the Ca2+ concentration controls the transition for ι. The transition temperature data are well fitted by functions of the form Tg = a + b c

(11.12)

where a and b are constants, Tg is the gelling temperature, and c is the concentration of gelling cation. This linear dependence on the square root of the ion concentration indicates that Coulombic effects, rather than salting-out effects, control the sol → gel transition. The melting temperatures are similarly related to the cation concentration. κ-Carrageenan and furcellaran gels are relatively rigid and subject to syneresis in the presence of calcium ions. If potassium is the sole counter-ion, the gels are compliant, resembling those of ι-carrageenan. ι-Carrageenan by itself yields compliant gels with very little tendency to undergo syneresis.

11.4.6

Specific interactions

Electrostatic interactions can occur between the negatively charged carrageenans and positively charged sites on proteins. The commercially important interaction of carrageenans with milk proteins is an example of this. It has been established that at least one aspect of the ‘milk reactivity’ of carrageenans is a gelation phenomenon involving a highly specific interaction between carrageenan and κ-casein. In order for it to manifest itself, two events must occur:

406

Confectionery and Chocolate Engineering: Principles and Applications

(1) Carrageenan and κ-casein interact to form a complex. (2) The resulting complex aggregates into a three-dimensional gel network. Gelation is overtly manifested in such applications as blancmange-type puddings, whereas the suspension of cocoa in seemingly fluid chocolate milk occurs only below a well-defined sol–gel transition temperature (Payens 1972). Although carrageenans may interact with other casein fractions (αs1 and β) in milk, the binding is much weaker than that with κ-casein, and does not result in gel formation. The presence of the other fractions, such as occurs in whole casein, in fact decreases the sharpness of the sol → gel transition. Specific interactions with κ-casein can occur at pH levels above the isoelectric point of the protein. (Non-specific interactions can, of course, occur below this point.) The former type of interaction is a most convenient one from a commercial standpoint as it permits the carrageenan to be functional without disturbing the integrity of the casein micelles and of the milk as a whole. This interaction has been ascribed to electrostatic attraction between the negatively charged sulphate groups of the carrageenan and a predominantly positively charged region in the peptide chain of κ-casein (Snoeren et al. 1975). Although this association of κ-casein with carrageenan occurs for all carrageenan types, it is primarily the gel-forming types (κ- and ι-carrageenan) that exhibit a sol–gel transition with temperature. This is in accord with the practical observation that κ- and ι-carrageenan are effective for the suspension of cocoa in chocolate milk, whereas λcarrageenan is ineffective. This observation confirms that gelation is required, and suggests that the formation of the gel network may be a function solely of the carrageenan component of the carrageenan–casein complex. Moreover, suspension of cocoa does not occur if the carrageenan has a low molecular weight. For further details, see Stanley (1990, pp. 96–104).

11.4.7

Utilization

The utilization of carrageenan in food processing, principally in dairy products, is based on the ability of the polymer to gel, to increase the viscosity of solutions and to stabilize emulsions and various dispersions. A level as low as c. 0.03% in chocolate milk prevents fat droplet separation and stabilizes the suspension of cocoa particles. Chocolate milk typically contains 1% cocoa, 6% sugar and from 0.025% to 0.035% carrageenan. Vanillin is added as a flavour modifier. Most commonly, these ingredients are sold to the dairy as a blended ‘dairy powder’ to be incorporated into the milk during pasteurization. On cooling, the interaction between κ-casein and carrageenan produces the delicate gel structure necessary to keep the cocoa suspended and to give the drink a rich mouthfeel. κ-Carrageenan is used when the drink is pasteurized; ι and λ are also functional, though less economical. Chocolate syrups are used instead of powders by some dairies, as well as by consumers. Here the syrup is dispersed into cold milk at a ratio of about 1 part syrup to 10–12 parts milk. As in other cold-processed applications, λ-carrageenan is used. Cooked puddings and pie fillings have traditionally been starch-based. However, κcarrageenan can be used to provide a more uniform set. For further details, see Guiseley et al. (1980).

Gelling, emulsifying, stabilizing and foam formation

407

Carrageenans are also used to stabilize ice cream and to clarify beverages. Ice cream and sherbet have been defined as partly frozen foams, containing a gas (air) dispersed as small cells in a partially frozen continuous aqueous phase. Stanley (1990) has given a detailed survey of the applications of carrageenan in the food industry.

11.5

Furcellaran

Furcellaran (Danish agar) is produced from red seaweed (algae Furcellaria fastigiata). Production began in 1943 when Europe was cut off from its agar suppliers. After alkali pre-treatment of the algae, the polysaccharide is isolated using hot water. The extract is then concentrated under vacuum and seeded with 1–1.5% KCl solution. The separated gel threads are concentrated further by freezing, the excess water is removed by centrifugation or pressing and, lastly, the polysaccharide is dried. The product is a potassium salt and contains, in addition, 8–15% occluded KCl. Furcellaran is composed of D-galactose (46–53%), 3.6-anhydro-D-galactose (30–33%) and sulphated portions of both of these sugars (16–20%). The structure of furcellaran is similar to that of κ-carrageenan. The essential difference is that κ-carrageenan has one sulphate ester residue per two sugar residues, while furcellaran has one sulphate ester residue per three to four sugar residues. The sugar sulphates identified are D-galactose-2sulphate, -4-sulphate and -6-sulphate, and 3.6-anhydro-D-galactose-2-sulphate. Branching of the polysaccharide chain cannot be excluded. Furcellaran forms thermally reversible aqueous gels by a mechanism involving double-helix formation, similarly to κ-carrageenan. The gelling ability is affected by the polysaccharide polymerization degree, the amount of 3.6-anhydro-D-galactose and the radius of the cations present. K+ and NH+4 form very stable, strong gels. Ca2+ has a lesser effect, while Na+ prevents gel setting. Addition of sugar affects the gel texture, which goes from a brittle to a more elastic texture. Furcelleran provides good gels with milk, and therefore is used as an additive in puddings. It is also suitable for cake fillings and icings. In the presence of sucrose, it gels rapidly and retains good stability, even against food-grade acids. Furcelleran has an advantage over pectin in marmalades since it allows stable gel setting at sugar concentrations even below 50–60%. The required amount of polysaccharide is 0.2–0.5%, depending on the marmalade’s sugar content and the desired gel strength. To keep the extent of hydrolysis low, a cold aqueous 2–3% solution of furcellaran is mixed into a hot, cooked slurry of fruit and sugar.

11.6

Gum arabic

Gum arabic is a tree exudate of various Acacia species, primarily Acacia senegal, and is obtained as a result of tree bark injury. It is collected as air-dried droplets with diameters from 2 to 7 cm. The annual yield per tree averages 0.9–2.0 kg. The major producer is Sudan, which produces 50 000–60 000 tonnes per annum, followed by several other African countries. Gum arabic has been known since the time of ancient Egypt as ‘kami’, an adhesive for pigmented paints.

408

Confectionery and Chocolate Engineering: Principles and Applications

Gum arabic is a mixture of closely related polysaccharides, with an average molecularweight range of 260–1160 kDa. The main structural units in the exudate of A. senegal are L-arabinose, L-rhamnose, D-galactose and D-glucuronic acid. The proportions vary significantly with Acacia species. Gum arabic has a major core chain built from β-Dgalactopyranosyl residues linked by 1→3 bonds, in part carrying side chains attached at position 6. Gum arabic occurs as a neutral or weakly acidic salt. The counter-ions are Ca2+, Mg2+ and K+. Gum arabic is very soluble in water, and solutions of up to 50% gum can be prepared. The viscosity of the solution starts to rise steeply only at high concentrations (above 30 m/m%). This property is unlike that of many other polysaccharides, which provide highly viscous solutions even at low concentrations (about 4–5 m/m%). Gum arabic is used as an emulsifier and stabilizer, for example in baked products. It retards sugar crystallization and fat separation in confectionery products and the formation of large ice crystals in ice creams, and can be used as a foam stabilizer in beverages. Gum arabic is also applied as a flavour fixative in the production of encapsulated, powdered aroma concentrates. For example, essential oils may be emulsified with gum arabic solution and then spray dried. In this process, the polysaccharide forms a film surrounding the oil droplets, which then protects the oil against oxidation and other changes.

11.7

Gum tragacanth

Gum tragacanth is a tasteless and odourless plant exudate collected from Astragalus species shrubs grown in the Middle East (Iran, Syria and Turkey). Gum tragacanth consists of a water-soluble fraction, the so-called tragacanthic acid, and an insoluble swelling component, bassorin. The soluble fraction is a complex mixture of various acidic polysaccharides. Their basic units are D-galacturonic acid, D-galactose, L-fucose, D-xylose and L-arabinose. Most of the carboxyl groups in the insoluble fraction are esterified with methano1; their hydrolysis yields tragacanthic acid. Its molecular weight is about 840 kDa. The molecules are highly elongated (450 × 1.9 nm) in aqueous solution and are responsible for the high viscosity of the solution. The viscosity decreases strongly with increasing shear rate. Gum tragacanth is used as a thickening agent and a stabilizer in salad dressings (0.4–1.2%) and in fillings and icings in baked goods. As an additive in ice creams (0.5%), it provides a soft texture.

11.8

Guaran gum

Guar flour is obtained from the seed endosperm of the leguminous plant Cyamopsis tetragonoloba. The seed is decoated and the germ removed. In addition to the polysaccharide guaran, guar flour contains 10–15% moisture, 5–6% protein, 2.5% crude fibre and 0.5–0.8% ash. The plant is cultivated for forage in India, Pakistan and the US (Texas). Guaran gum consists of a chain of β-D-mannopyranosyl units joined by 1 → 4 linkages. Every second residue has a side chain, a D-galactopyranosyl residue that is bound to

Gelling, emulsifying, stabilizing and foam formation

409

the main chain by an α(1 → 6) linkage. Guaran gum forms highly viscous solutions, the viscosity of which decreases sharply as the shear rate becomes higher than 8–10 rpm. Guaran gum is used as a thickening agent and a stabilizer in salad dressings and ice creams (application level 0.3%). (In addition to the food industry, it is widely used in the paper, cosmetic and pharmaceutical industries.)

11.9

Locust bean gum

The locust bean (also known as carob bean or St. John’s bread) is from an evergreen cultivated in the Mediterranean area since ancient times. Its long, edible, fleshy seed pods are also used as fodder. The dried seeds were called carat by the Arabs and served as a unit of weight (approximately 200 mg). Even today, the carat is used as a unit of weight for precious stones, diamonds and pearls, and as a measure of the purity of gold (1 carat = 1/24 part of pure gold). The locust bean seeds consist of 30–33% hull material, 23–25% germ and 42–46% endosperm. The seeds are milled and the endosperm is separated and utilized like the guar flour described above. The commercial flour contains 88% galactomannoglucan, 5% other polysaccharides, 6% protein and 1% ash. The main locust bean polysaccharide is similar to that of guaran gum: a linear chain of 1 → 4-linked β-D-mannopyranosyl units, with α-D-galactopyranosyl residues joined 1 → 6 as side chains. The ratio of mannose to galactose is 3 : 6; this indicates that, instead of every second mannose residue, as in guaran gum, only every fourth to fifth is substituted at the C-6 position with a galactose molecule. The molecular weight of this galactomannan is close to 310 kDa. The physical properties correspond to those of guar gum, except that the viscosity of the solution is not as high (it is similar to that of tragacanth). Locust bean flour is used as a thickener, binder and stabilizer in meat canning, salad dressings, sausages, soft cheeses and ice creams. It also improves the water-binding capacity of dough, especially when flour of low gluten content is used.

11.10 11.10.1

Pectin Isolation and composition of pectin

Pectin is widely distributed in plants. It is produced commercially from the peel of citrus fruits and from apple pomace (crushed and pressed residue). It is 20–40% of the dry matter content in citrus fruit peel and 10–20% in apple pomace. Extraction is achieved at pH 1.5–3 and 60–100°C. The process is carefully controlled to avoid hydrolysis of glycosidic and ester linkages. The extract is concentrated to a liquid pectin product or is dried by spray or drum drying to a powdered product. Purified preparations are obtained by precipitation of pectin with ions that form insoluble pectin salts (e.g. Al3+), followed by washing with acidified alcohol to remove the added ions, or by alcoholic precipitation using isopropranol and ethanol. Pectin is a chain-like polymer consisting of α-D-galacturonic structural units joined by 1 → 4 linkages. The main chain, however, is one-tenth rhamnose residues. In segments in which rhamnose is enriched, rhamnose units may be in adjacent or alternate positions. Pectin also contains small amounts of D-galactan and arabinan in its extended side chains

410

Confectionery and Chocolate Engineering: Principles and Applications

and, to a lesser extent, fucose and xylose sugars in its short side chains (1–3 unit chains). These short side chains are not regarded as typical pectin constituents. The carboxyl groups of galacturonic acid along the main chain are esterified to a variable extent with methanol, while the OH groups in the 2- and 3-positions may be acetylated to a small extent. The stability of pectin is highest at pH 3–4. For further details, see Rolin and De Vries (1990). At a pH of about 3, and also at higher pH in the presence of Ca2+ ions, pectin forms a thermally reversible gel. The gel-forming ability, under comparable conditions, is directly proportional to the molecular weight and inversely proportional to the degree of esterification (see below). For gel formation: • Low-ester pectins require very low pH values and/or calcium ions, but they set to a gel in the presence of a relatively low sugar content. • High-ester pectins require an increasing amount of sugar with increasing degree of esterification. • The gel-setting time for high-ester pectins is longer than that for pectin products of low degree of esterification. Commercial pectins are divided into high-ester pectins and low-ester pectins according to the degree of esterification (DE), which is the percentage of galacturonic acid subunits that are methyl esterified.

11.10.2

High-methoxyl (HM) pectins

High-ester pectins (DE > 50%), also known as high-methoxyl (HM) pectins, intended for gel-making are further subdivided into: • rapid-set, DE 70–75%; • medium-rapid-set, DE 65–69%; • slow-set, DE 60–64%. The degree of esterification correlates with the setting rate and texture of the gel under otherwise similar conditions, which means that very high-methoxyl pectins jellify quicker (or, for example, at higher temperatures) than less high-methoxyl pectins. Therefore, slow-set pectin is a commonly used raw material for confectionery manufacture because flavouring, colouring and dosing need a certain time before setting. Citrus pectins with the same degree of esterification have a slightly higher setting temperature and are somewhat more elastic than apple pectins. 11.10.2.1

Gel strength standardization

Almost all high-ester pectins are standardized to USA-SAG grade 150. This designation means that 1 part of the pectin is able to turn 150 parts of sucrose into a jelly prepared under standard conditions and with standard properties as follows: • refractometer-soluble solids, 65%; • pH = 2.20–2.40; • gel strength, 23.5% SAG.

Gelling, emulsifying, stabilizing and foam formation

411

This method was implemented by the 1959 IFT Committee for Pectin Standardization, and a detailed procedure is available (Final Report of the IFT Committee 1959).

11.10.3

Low-methoxyl (LM) pectins

Pectins with a proportion of methoxylated polygalacturonic acid units (DE) less than 50%, known as low-methoxyl (LM) pectins, can jellify with Ca2+ ions. LM pectins thus form gels not only with sugar and acids, but also with less soluble solids containing calcium ions. The volume of pectin, type of pectin, content of dry soluble solids, pH range and concentration of buffer salts and calcium ions contained in the environment are decisive for the gel strength. A well-matched balance between the pectin and calcium concentrations leads to an optimal texture. If the calcium optimum is exceeded, on the other hand, this will produce a brittle gel with a tendency towards syneresis or, eventually, calcium pectinate formation. Since gel setting with LM pectins is also possible with a low soluble-solids content and at a high pH value, this opens up numerous possibilities of applications in dietetic and dairy products.

11.10.4

Low-methoxyl (LM) amidated pectins

The degree of amidation (DA) is the percentage of galacturonic acid subunits that are amidated. Amidated pectins are low-methoxyl pectins that have been demethoxylated with ammonia instead of acid. During demethoxylation, some of the ester groups are replaced by amide groups, which modifies the gelling properties in comparison with aciddemethoxylated pectins. LM amidated pectins, just like non-amidated pectins, require Ca2+ ions for gelling; however, they can set sufficiently well with only minor amounts present. Furthermore, the gel-setting temperature of amidated pectins is much less influenced by the calcium dosage.

11.10.5

Gelling mechanisms

A combination of hydrogen bonding and hydrophobic interactions is the mechanism accepted to be responsible for gel formation with high-ester pectin. The ester groups are the hydrophobic parts of the pectin molecules. An energy contribution is associated with the contact between these hydrophobic areas and water; the hydrophobic areas will tend to aggregate in order to minimize the area of the contacting surface, in analogy to the coalescence of oil drops in water. Hydrogen bonds that form between adjacent galacturonan chains contribute even more to the energy decrease associated with the formation of junction zones (Oakenfull and Scott 1984). It has been suggested that the rigidity of the pectin molecule is positively correlated with the DE and the concentration of sugars in the solution, and that this is also an important factor that plays a role in gel formation with high-ester pectin (Michel et al. 1984; Plashchina et al. 1985). The ‘egg-box’ model has been widely accepted as an explanation of the gelation of low-ester pectin in the presence of calcium ions. Pectins and pectates are configured in helices with three subunits per turn. However, the junction zones are organized so that

412

Confectionery and Chocolate Engineering: Principles and Applications

the pectin chains are in the form of helices with only two subunits per turn. These twofold helix structures are joined by calcium ions bridging two opposing carboxyl groups. The model forces us to make the assumption that the helix structure changes from twofold to threefold when the gel is dried to a powder. Since pectin can set into a gel, it is widely used in marmalade and jelly production. An example of standard conditions to form a stable gel is pectin content < 1%, sucrose 58–75% and pH 2.8–3.5. In low-sugar products, low-ester pectin is used in the presence of Ca2+ ions. The use of pectin as a stabilizer for beverages (to emulsify the essential-oil components) and ice creams is also of importance. For further details, see Rolin and De Vries (1990).

11.10.6 11.10.6.1

Technology of manufacturing pectin jellies Preparation of high-ester pectin gels

The conditions necessary for gelation are a fairly low pH and a high soluble-solids concentration. A critical temperature exists above which gelation will not take place, even though all other conditions for gelation are fulfilled. If the temperature is reduced below this limit, gelation commences after some time. The gelation temperature depends on the combination of pectin type and the composition of the batch; it can, to a great extent, be varied at will by selecting an appropriate pectin type. It is not usually possible to melt a high-ester pectin gel once it has solidified. If a gel batch is stirred or poured while gelation is in progress (it may still look liquid!), structures that have already formed may be broken. They will not reform, and the corresponding part of the pectin will not be utilized in the final gel structure. This phenomenon is known as pre-gelation. Pre-gelation is very similar to the situation of poor dissolution or distribution of the pectin. The results are similar: the gel is weaker than expected, and gelation may, in severe cases, appear to be absent. The cause of failure is also basically the same in the two cases: only part of the pectin is utilized. Two important differences between the gelation of low-ester pectin and of high-ester pectin must be mentioned. The difference between the setting and melting temperatures of low-ester pectin gels is modest, and it is possible in many cases to remelt a low-ester pectin gel. This is in contrast to the situation for high-ester pectin gels. A low-ester pectin gel solidifies almost immediately when the gelling conditions are reached; a high-ester pectin gel system shows a time lag. The pectin jellies used in confectionery are produced from high-methoxy, slow-set pectins with a typical degree of esterification of 60–64%. Since pectin gels are so-called chemo-gels, i.e. gelation is induced by a decrease in pH, an aqueous sugar solution containing pectin is boiled to about 106°C in a slightly alkaline medium (pH = 7.5–8) containing the sequestrant (buffer) di/trisodium citrate. After starch syrup has been added, the solution is coloured and flavoured, and dosing can be started at about 90–95°C: under the effect of the flavouring acid (in general, a water–citric acid solution in the ratio 1 : 1), gelation starts slowly and must not end until dosing is in progress. The synchronization of dosing and gelation demands the use of slow-set pectin. Figure 11.2 shows a schematic layout for pectin jelly technology. The technology is sensitive to the hardness of the water used; therefore, the amount and pH of the buffer are fitted to the circumstances of manufacture. The duration of gelation is relatively short, 2–3 h, which makes the technology very productive. The water content of the end product

Gelling, emulsifying, stabilizing and foam formation

Mixing

Sugar powder

Fig. 11.2

Starch syrup

Sugar

Pectin powder

Dissolution

Water

413

Evapo ration 106°C

Mixing

Buffer Colouring, flavouring

Cooling to about 90–95°C

Dosing

Schematic layout of pectin jelly technology.

is about 21–23%, which, depending on the conditions of storage, may decrease slowly to about 15% because of the effect of syneresis in about 6 months. Pectin jellies are favoured because, perhaps, they are mostly similar to the natural fruits, which also contain pectin; moreover, their texture is such that they are easily split, and the split jelly layers are translucent and glassy. These properties of a splitting texture and translucency are the very quality requirements of confectionery jellies. 11.10.6.2

Preparation of low-ester pectin gels

Low-ester pectin gels may be prepared by the same sequence of events as that described above for high-ester pectin gels, but it is not necessary to achieve as high a solids content or as low a pH in order to induce gelling conditions. The gelation is dependent on calcium, which is in most cases provided by the fruit material added. In analogy to high-ester pectin gels, the gel temperature may be varied by appropriate selection of the combination of pectin type and the properties of the medium. The same considerations regarding dispersion of the pectin prior to gelation and the possibility of pre-gelation apply to low-ester pectin gels as to high-ester pectin gels.

11.11 11.11.1

Starch Occurrence and composition of starch

Starch is widely distributed in various plant organs as a storage carbohydrate. Starch granules are formed from two glucans, amylose and amylopectin. Most starches contain 20–39% amylose. New corn cultivars (amylomaize) have been developed that contain 50–80% amylose. Normal starch granules contain 70–80% amylopectin, while some corn cultivars and millet, referred to as waxy maize or waxy millet, contain almost 100% amylopectin. Amylopectin, when heated in water, forms a transparent, highly viscous solution, which is ropy, sticky and coherent. Unlike the case with amylose, there is no tendency to retrogradation. There are no staling or ageing phenomena and no gelling, except at very

414

Confectionery and Chocolate Engineering: Principles and Applications

high concentrations. However, there is a rapid viscosity drop in acidic media and when the solution is autoclaved or subjected to a strong mechanical shear force. Starch is an important thickening and binding agent and is used extensively in the production of puddings, soups, sauces, salad dressings, diet food preparations for infants, pastry fillings, mayonnaise etc. Amylose films can be used for food packaging, as edible wrapping or tubing, as exemplified by a variety of instant coffee and tea products. The uses of amylopectin are also diverse. It is used to a large extent as a thickener and stabilizer and as an adhesive and binding agent.

11.11.2

Modified starches

The properties of starch and of amylose and amylopectin can be improved or ‘tailored’ by physical and chemical methods to fit or adjust the properties to a particular application or food product: • Pregelatinized starch. Heating of a starch suspension above its gelatinization temperature, followed by suspension drying, provides a starch product that is soluble in cold water and that gels. These products are used in instant foods (e.g. puddings) and as baking aids. • Thin-boiling starch. Partial acidic hydrolysis yields a starch product that is not very soluble in cold water but is readily soluble in boiling water. The solution has a lower viscosity than the untreated starch, and remains fluid after cooling. Retrogradation is low. These starches are utilized as thickeners and as protective films. • Starch ethers. When a 30–40% starch suspension is reacted with ethylene oxide or propylene oxide in the presence of hydroxides of alkali and/or alkaline earth metals (pH 11–13), hydroxyethyl or hydroxypropyl derivatives are obtained. (These derivatives are also obtained in reactions with the corresponding epichlorohydrins.) The degree of substitution can he controlled over a wide range by adjusting the process parameters. These products are utilized as thickeners for refrigerated foods and heat-sterilized canned foods. Reaction of starch with monochloracetic acid in an alkaline solution yields carboxymethyl starch. This product swells instantly, even in cold water and in ethanol. Dispersions of 1–3% carboxymethyl starch have an ointment-like consistency, whereas 3–4% dispersions provide a gel-like consistency. Such products are of interest as thickeners and gel-forming agents. • Starch esters. Starch monophosphate esters are produced by dry heating of starch with alkaline orthophosphate or alkaline tripolyphosphate at 120–175°C. They are used as thickeners and stabilizers, among other things, in bakery products.

11.11.3

Utilization in the confectionery industry

Modified starches are used in a number of confectionery applications: • Acid-hydrolysed starches are used in wine gums, jellies, jelly beans and gelatin replacements. The specific properties used are hot-water solubility, medium/low viscosity, fast setting and high gel strength. • Oxidized starches are used in wine gums, jellies and hard gums, and for pan coating and replacement of gum arabic. The specific properties used are hot-water solubility, medium/low viscosity, slow setting, good clarity and good film formation.

Gelling, emulsifying, stabilizing and foam formation

415

• Dextrinized starches are used in hard gums, pan coating, blazing, and replacement of gum arabic. The specific properties used are hot- and cold-water solubility, low viscosity, slow setting, good clarity and good film formation. • Enzyme-hydrolysed starches are used in hard gums, pan coating, and replacement of gum arabic. The specific properties used are cold-water solubility, low viscosity, slow set-back and good clarity. • Acetylated starches are used in jellies, hard gums, and replacement of gelatin and gum arabic. The specific properties used are hot-water solubility and stability against retrogradation. • Hydroxypropylated starches are used in hard gums, pan coating, and replacement of gum arabic. The specific properties used are hot-water solubility and stability against retrogradation. The fields of application of several types of starch, according to function, are: • • • • • • • • • • • • •

wine gums: gelling; jelly pastilles: gelling; pan coating: adhesion and film strength; hard gums: texturing; extruded liquorice: texturing; marshmallows: foam strengthening; nuts: glazing; jelly beans: glazing; caramels: stiffness; wafers: binding; lozenges: binding; gums, jellies and centres: moulding; chewing gum: dusting.

A schematic layout of a starch jelly technology is shown in Fig. 11.3. Cooking can be done also by an extruder. Finally, a typical Rahat Lakoum recipe is as follows: 5 kg thin-boiling starch is dissolved in 30 kg water, the solution is boiled for 2 min, and then 30 kg sugar, 8 kg starch

Instant gelatin + gum arabic

Sugar

Dissolution

Starch

Water

Fig. 11.3

Starch syrup

Precooking

Liquorice extract

Schematic layout of a starch jelly technology (liquorice).

Vacuum cooking

Cooling

Shaping

416

Confectionery and Chocolate Engineering: Principles and Applications

syrup and 2 kg invert syrup (70%) are added. This solution is boiled to 78–80% dry content, and then flavoured with citric acid and rose oil on a cooling table.

11.12

Xanthan gum

Xanthan gum, an extracellular polysaccharide from Xantomonas campestris and some related microorganisms, is produced on a nutritive medium containing glucose, NH4Cl, and a mixture of amino acids and minerals. The polysaccharide is recovered from the medium by isopropanol precipitation in the presence of KCl. Xanthan gum is a heteroglycan consisting of D-glucose, D-mannose and D-glucuronic acid in a molar ratio of 2.8 : 2 : 2. Some sugar residues are acetylated, while some are present as ketals, formed by pyruvate [4.6-O-(1-carboxyethylidine)-D-glucopyranose]. The molecule consists of a ‘backbone’ made of 1.4-linked (β-glucopyranosyl) residues. The molecular weight of xanthan gum is > 106 Da. In spite of this high value, it is quite soluble in water. The highly viscous solution exhibits a pseudoplastic behaviour. The viscosity is, to a great extent, independent of temperature. Solutions, emulsions and gels, in the presence of xanthan gums, acquire a high freeze–thaw stability. The practical importance of xanthan gum is based on its emulsion-stabilizing and particle-suspending abilities (it is applied to turbidity problems, and essential-oil emulsions in beverages). Owing to its high thermal stability, it is useful as a thickening agent in food canning. The addition of xanthan gum to starch gels substantially improves their freeze–thaw stability. The properties of xanthan gum can also be utilized in instant puddings: a mixture of locust bean flour, sodium pyrophosphate and milk powder with xanthan gum as an additive provides instant jelly after reconstitution in water. The pseudoplastic thixotropic properties, i.e. a high viscosity in the absence of a shear force and a drop in viscosity to a fluid state under a shear force, due to intermolecular association of single-stranded xanthan gum molecules, are of interest for the production of salad dressings.

11.13 11.13.1

Gelatin Occurrence and composition of gelatin

Collagen constitutes 20–25% of the total protein in mammals. The high content of glycine and proline and the occurrence of 4-hydroxyproline and 5-hydroxylysine are characteristic features. Collagen also contains carbohydrates (glucose and galactose). The collagen chain consists of three polypeptide chains and, in the most prevalent species (collagen type I), two chains are the same, while the third chain differs. The three peptide chains, each of which has a helical structure, occur together as a triple-stranded helix. Collagen swells but does not solubilize. One characteristic of the intact collagen fibre is that it shrinks when heated (on cooking or roasting). The shrinkage temperature TS is different for different species of collagen. For fish collagen, the value of TS is 45°C and for mammals, 60–60°C. When native or intact collagen is heated to T > TS, the triple-stranded helix is destroyed to a great extent, depending on the cross-links. The disrupted structure now exists as random coils, which are soluble in water and are called gelatin.

Gelling, emulsifying, stabilizing and foam formation

417

Gelatin, one of the most versatile food substances, is the only natural protein of commercial importance that is capable of producing clear, thermoreversible gels in water at near to body temperature. Composition. The drying loss (water content) is typically not more than 15 m/m%. Protein, 84%; ash, 1%. The density is about 1300 kg/m3; the pH value ranges from 4.0 to 9.0 depending on type. The thermoreversibility of gelatin gels, coupled with their typical elastic texture, meltin-the-mouth characteristics and excellent flavour release, gives gelatin-based food products their unique textural and sensory properties. Gelatin’s unique characteristics give it wide application in the food industry, where its functional properties are used to gel, thicken, stabilize, emulsify, bind, form films and allow aeration. Being a protein, it is also a very useful nutritive component. Gelatin is extracted from selected beef or pork skin material obtained from approved abattoirs and meatworks. This collagenous material may be subjected to an alkaline pretreatment followed by hot-water extraction, yielding Type B gelatin, with an isoelectric point (IEP) of approximately 5.0. Gelatin obtained from pork skin material using an acid pre-treatment, yielding Type A gelatin, has an isoelectric point between 7.0 and 9.0. These two types of gelatin are entirely interchangeable in most food product applications. 11.13.1.1

Structure and amino acid composition of gelatin

The raw materials, type of pre-treatment and gelatin extraction conditions during manufacture all affect the molecular distribution of the gelatin polypeptides obtained. Commercial gelatins are a heterogeneous protein mixture of polypeptide chains. Gelatin molecules are quite large, with a molecular weight ranging from a few thousand up to several hundred thousand daltons. The molecular-weight distribution of gelatin has a great bearing on its physical properties and particularly affects the values of the viscosity and gel strength.

11.13.2

Solubility

Gelatin is relatively insoluble in cold water but hydrates readily in warm water. When added to cold water, gelatin granules swell into discrete swollen particles, absorbing 5–10 times their weight in water. Raising the temperature above 40°C dissolves the swollen gelatin particles, forming a solution, which gels upon cooling to the setting point. The extent to which gelatin granules swell in cold water is dependent on pH, with the maximum swelling occurring at pH values away from gelatin’s isoelectric point. The rate of dissolution is affected by factors such as temperature, concentration and particle size. Gelatin is insoluble in alcohol and most other organic solvents.

11.13.3

Gel formation

Gelatin forms thermoreversible gels that convert to a solution as the temperature is increased to 30–35°C. Upon cooling of the gelatin solution to the setting point, a gel structure is reformed, i.e. gelatin gel, similarly to agar gel, is a typical dissolution gel. This solution-to-gel conversion process is reversible and can be repeated many times (sol–gel– sol, etc.). Gelatin gels, therefore, produce a ‘melt-in-the-mouth’ characteristic giving excellent flavour release, which is a desirable property in a number of foods.

418

Confectionery and Chocolate Engineering: Principles and Applications

Gelatin has the ability, at all pH levels found in food systems, to form gels that do not undergo syneresis. At the onset of gelation, there is a tremendous increase in viscosity until a gel has completely formed. 11.13.3.1

Strength of gelatin gels

Gelatins are graded and marked according to their Bloom strength, which is a measurement of the gel strength determined according to international standards and methodologies (Bloom 1925; GME 2004; ISO 9665). For further methods, see Johnston-Banks (1990). The strength of gelatin gels depends upon the concentration and the intrinsic strength of the gelatin used; the intrinsic strength is a function of both the structure and the molecular weight. This means that for a type of gelatin specified by its Bloom grade, the gel strength is proportional to the concentration (m/m%) of gelatin. The gel strength is largely independent of pH over a wide range above a value of about 5.0 but is sensitive to an acidic medium. This is important in acidic food systems such as those found in certain confectionery products, water-based dessert gels and cultured dairy products. Moreover, the gel strength increases with time as the gel matures, reaching an equilibrium after approximately 18 h of maturation. Other factors affecting gel rigidity include temperature, thermal history and the presence and concentration of electrolytes, non-electrolytes and other ingredients. Unlike most polysaccharide gelling agents, the formation of gelatin gels does not require the presence of other reagents such as sucrose, salts and divalent cations and is not dependent on the pH.

11.13.4

Viscosity

The viscosity characteristics displayed by a gelatin of given Bloom grade are primarily related to the molecular-weight distribution of the gelatin molecules. The viscosity of gelatin solutions increases strongly with increasing concentration and with decreasing temperature. In salt-free solutions, the minimum viscosity occurs at the pH of the iso-ionic point of the gelatin. Changes in the molecular shape and charge distribution result in changes in viscosity at different pH values. The viscosity of gelatin plays a significant role in certain food systems. Examples of this include applications in starch-moulded confectionery, where the high working speeds demanded by modern processing equipment require a gelatin with a low viscosity to prevent the formation of ‘tails’, together with a rapid distribution in the moulds. The viscosity also affects the gel properties, including the setting and melting points. Highviscosity gelatin gives gels with higher melting and setting temperatures and produces a quicker setting rate than gelatins of lower viscosity. Gelatins of higher viscosity are preferred for stabilizing certain emulsions.

11.13.5

Amphoteric properties

Gelatin, like other proteins, displays amphoteric characteristics, having both acidic (carboxyl) and basic (amino and guanidine) amino acid groups.

Gelling, emulsifying, stabilizing and foam formation

419

pe Ty

3

5

pIA 7

9

pH 11

pe Ty

Electric charge Negative Positive

A

pIB

B

Fig. 11.4 Electric charge of a Type A and a Type B gelatin molecule as a function of pH of solution. pI = isoionic point.

At a unique pH, gelatin has an equal number of negative and positive charges on the molecule. In the case of salt-free solutions, the pH at which there is no net charge on the molecule is referred to as the iso-ionic point (pI). In the case of gelatin solutions that contain salts or other electrolytes, the pH where the net charge on the molecule is zero and no movement occurs in an electric field is called the isoelectric point. For deionized solutions, the pI and the IEP are virtually identical. Type A gelatin has a pI of pH 7.0–9.0, whereas Type B gelatin has a pI of 4.8–5.2 (see Section 11.13.1). The ionic character of gelatin as a function of pH is illustrated in Fig. 11.4. Irrespective of the pH, Type A gelatin is positively charged in all food systems, whereas Type B gelatin can be either positively charged or negatively charged. The pI controls the charge density and net charge on the gelatin molecule as a function of the pH of the food system in which the gelatin is employed. The charge upon the gelatin and its intensity determine whether the gelatin remains compatible with the other substances present as ingredients. Negatively charged polysaccharides such as xanthan gum, carrageenan and alginates will coacervate with gelatins that are positively charged if the conditions are favourable, resulting in removal of some of the gelatin from the system and a consequent loss in functionality. When negatively charged polysaccharides are combined with gelatin, as is done in certain applications, Type B gelatin should be used to ensure compatibility and to avoid a loss in functionality.

11.13.6

Surface-active/protective-colloid properties and utilization

Gelatin, like virtually all proteins, possesses surface-active properties and will adsorb at air–water and oil–water interfaces, making it capable of assisting in the formation and stability of foams and emulsions such those found in marshmallow, toffee and mousse (and also mayonnaise and certain meat products). The emulsifying and emulsion-stabilizing ability of gelatin is linked to its ability to generate a film around the dispersed phase. A gelatin content increases the viscosity of the aqueous phase in oil-in-water (O/W) emulsions, assisting in emulsion stability. The surface-active properties of gelatins are influenced by the pH of the food system in which they are employed because of the influence of the pH on the charge balance.

420

Confectionery and Chocolate Engineering: Principles and Applications

Gelatin acts as a protective colloid in supersaturated solutions such as are found in ice cream, icings and marshmallow, where it is able to restrict the growth of ice and sugar crystals and therefore gives the desired mouthfeel and shelf-life stability to the product. Consequently, gelatin is used as a functional ingredient in many food product applications. Use is made of its multifunctional properties, together with its ease of dissolution and neutral flavour, to achieve food products of desirable texture and mouthfeel and to provide shelf-life stability. Gelatin is used in food systems as a gelling agent, thickener, film former, protective colloid, adhesive agent, stabilizer, emulsifier, foaming/whipping agent and beverage fining agent. It is important to recognize that in many commercial food product applications, more than one functional property of gelatin is utilized at any one time.

11.13.7

Methods of dissolution

The following methods are those most often used for hydrating gelatin in solution. 11.13.7.1

Cold-water swelling methods

Although they take longer to carry out, cold-water swelling methods are advantageous as the solutions are produced free from entrained air, which is an essential requirement. (1) Gelatin may be swollen in cold water for a specific time depending on the particle size. The swollen particles are then hydrated by heating in a jacketed vessel under gentle agitation to 60°C. (2) Gelatin may be swollen in cold water, and then directly added to warm liquids such as sugar syrups, in which it fully hydrates. 11.13.7.2

Hot-water dissolution methods

Hot-water dissolution provides a rapid way to obtain gelatin solutions of high concentration and is extensively used, particularly in the confectionery industry, where concentrations of up to 40% w/w are often prepared. Gelatin is added to hot water at 90°C in a vortex created by mechanical agitation. Once dispersion has taken place, the agitation speed should be reduced to avoid air entrainment. Dual-speed agitators are recommended for this purpose. If air entrainment occurs, standing the solution at 60°C will allow air bubbles to escape and the solution will clear. For preparing concentrated solutions, coarse-mesh gelatins should be used as these disperse easily without lumping problems. Johnston-Banks (1990, p. 253) recommends 16–28 mesh BSI (18 mesh BSI = 0.85 mm; 30 mesh BSI = 0.5 mm). To obtain rapid hydration in the preparation of concentrated gelatin solutions, the water temperature used should be from 75°C to 90°C. Lower temperatures, i.e. 60–75°C, may be used but this significantly increases the time needed to hydrate the gelatin particles. The method used to prepare gelatin solutions depends on a number of factors, which include the end product to be manufactured, the particle size of the gelatin, the concentration of gelatin required, the time available, equipment available (degree of agitation, type of stirrer etc.) and the viscosity of the gelatin. At a given concentration, high-viscosity gelatins are more difficult to hydrate than gelatins of lower viscosity.

Gelling, emulsifying, stabilizing and foam formation

11.13.7.3

421

General precautions

(1) To prevent extensive thermal hydrolysis, never boil gelatin solutions. (2) To avoid clumping problems, always add gelatin to water and not water to gelatin. (3) To prevent unnecessary hydrolysis, never dissolve gelatin in the presence of acids or fruit juices. (4) Care should be taken to ensure that complete dissolution of the gelatin has occurred. In many instances, failure of gelatin to perform properly results from incomplete hydration of the gelatin, with a consequential deficient concentration in the final product. (5) If the water/gelatin mixture is agitated, no air must be introduced into the mixture, which easily forms unwanted foam. For further details, see Johnston-Banks (1990).

11.13.8

Stability of gelatin solutions

Once in solution, gelatin can suffer a loss in gel strength and viscosity when exposed to elevated temperatures for any prolonged period of time. This loss can be magnified under acidic conditions arising from the presence of fruit juices and organic acids. As with other proteins, the presence of proteolytic enzymes can also degrade gelatin solutions. The loss of gel strength and viscosity is a result of hydrolysis of the gelatin molecules. The rate of hydrolysis is a function of temperature, time and the pH of the solution. To avoid unnecessary hydrolysis, the addition of acid to a product formulation should be delayed until the latest possible time in the process.

11.13.9

Confectionery applications

The main areas of application of gelatin in confectionery are summarized in Table 11.2. A schematic layout of the technology for gelatin jellies is shown in Fig. 11.5.

Table 11.2 Main areas of application of gelatin in confectionery. Bloom grade Recommended: Type A Marshmallow (extruded) Marshmallow (deposited) Angel Kiss Recommended: Type A or B Soft-boiled sweets Jellies Dragées Tablets

200–280 120–240 100–200 100–200 180–260 100–180 100–180

422

Confectionery and Chocolate Engineering: Principles and Applications

Sugar

Water

Dissolution

Starch syrup

Water

Soaking

Cooking (113–121°C)

Cooling (100°C)

Mixing

Gelatin Fig. 11.5

Schematic layout of technology for gelatin jellies.

11.14

Egg proteins

11.14.1

Cooling

Colouring, flavouring

Shaping

Fields of application

Eggs are capable of performing various useful functions in foods, including foaming, coagulating or gelling, and emulsifying. All liquid components of eggs have the capacity to form gels upon heating. This important property is a major factor in the daily consumption of eggs, in which they are scrambled, fried, poached, hard-boiled and used in the preparation of omelettes, quiches, soufflés, custards and cakes. Egg white is used in candy to incorporate air, control the growth of sugar crystals, provide structural support through thermal coagulation and prevent syneresis (Cotterill et al. 1963; Baldwin 1986). The main fields of application are in roasted gingerbreads (geröstete Nusslebkuchen in German), macaroons (Makronen in German) and meringues. Although egg yolk gels or sets in any food in which it is cooked, it has little application outside traditional egg cookery, such as in custards, cakes and related foods in which the flavour, texture and colour depend on the yolk.

11.14.2

Structure

The liquid portion of shell eggs consists of about 67–70% albumen and 30–33% yolk. For detailed information on the chemistry and composition of eggs and egg proteins, the reader is referred to reviews of these topics: Cook (1968), Shenstone (1968), Baker (1968), Cotterill and Geiger (1977), Osuga and Feeney (1977), Powrie and Nakai (1986) and Cook and Briggs (1986). Egg albumen is made up of thick and thin white, with the thick white comprising about 60% of the albumen in a fresh egg (Shenstone 1968). The thin white has no apparent structural organization. The thick white has been described as a weak gel, with its rigidity attributed to ovomucin, a high-molecular-weight glycoprotein that is organized into fibres. The thick-white gel is broken down to thin white during the normal ageing of eggs. This is termed ‘egg white thinning’, and in this process the pH of the albumen rises from

Gelling, emulsifying, stabilizing and foam formation

423

about 7.6 to 9.3 owing to a loss of CO2 through the pores of the shell. Thick white is also easily broken down by mechanical means, such as blending and homogenization. Egg albumen can be regarded as a solution of globular proteins containing ovomucin fibres (Powrie and Nakai 1986). Egg yolk, on the other hand, is a complex structural system in which several types of particles are suspended in a protein solution or ‘plasma’ (Robinson 1979; Powrie and Nakai 1986). These particles include spheres (4–150 μm diameter), granules (0.3–2 μm), profiles or low-density lipoproteins (25 nm), and myelin figures (56–130 nm in length). Further references are Grodzinski (1951), Romanoff and Romanoff (1961), Bellairs (1961) and Bellairs et al. (1972).

11.14.3

Egg-white gels

Egg white has been preferred over egg yolk and whole egg for its gelling abilities in food systems for several reasons. The foaming ability of egg white broadens its range of gelling applications. The preparation of egg gels or of food gels containing eggs consists of three steps: (1) The egg product is brought into a liquid state by rehydrating a powder or thawing a frozen product as necessary. (2) Next, any desired ingredients are incorporated, such as starches, salts, sugars, buffers or acids. (3) Finally, the liquid is poured into the desired type of container, heated to the desired temperature and cooled. Common methods of heating include immersion in a water bath, steam injection, baking and frying. Most research on egg gels has been conducted on gels heated by immersion in a water bath. This is the usual method also in the confectionery industry. It is important to understand the factors controlling the gelation of albumen in order to maximize its function in any given food product. The quality of egg white gels is greatly affected by pH and ionic conditions, similarly to the gels of most globular proteins. The gel strength and cohesiveness are at their minimum at the pH value where the net charge of the protein is minimum, in the range of pH 6–7 for egg white. The behaviour of albumen gels with respect to pH is of critical importance in food gels. When egg white is gelled alone, the normal pH is 9, and gels of excellent quality are obtained. However, when egg white is added to other food systems, the pH is lowered and egg white functions much less effectively in gelling. This fact must not be overlooked if egg white is to be used successfully in combination with other foods. Further references are Meyer and Hood (1973), Nakamura et al. (1978), Hegg et al. (1979), Beveridge et al. (1980), Egelandsdal (1980), Hickson et al. (1980, 1982), Holt et al. (1984), Diehl and Gardner (1984), Gossett et al. (1984), Woodward and Cotterill (1986a) and Powrie and Nakai (1986). Since heat is the driving force behind protein denaturation and gelation, the temperature of heating is of major importance in forming gels. Egg albumen begins to lose fluidity at about 60°C with the denaturation of conalbumin (Hickson et al. 1982; Montejano et al. 1984; Baldwin 1986). The hardness of the gel increases logarithmically with protein concentration (Beveridge et al. 1980; Woodward and Cotterill 1986a). As the protein concentration is increased,

424

Confectionery and Chocolate Engineering: Principles and Applications

egg white has an enhanced tendency towards aggregation, resulting in increased fracturability of the gel (Woodward and Cotterill 1986). One inherent property of protein gels is an ability to bind or trap water. In albumen gels, water-binding ability is influenced by the same factors that control gel hardness. Water-binding ability is enhanced by increasing the pH, heating temperature and protein concentration (Woodward and Cotterill 1986).

11.14.4

Egg-white foams

Egg white (egg albumen) is an excellent foaming material because of the special functional properties of its constituent proteins (Dickinson 1992). The main constituent proteins are (in m/m%) ovalbumin (53–55), conalbumin (12–14) and ovomucoid (10–12). The effectiveness of egg white as a foaming agent in meringues and cake batter arises from successful teamwork between the various constituent proteins and glycoproteins in chicken egg albumen. Not only is egg white effective in stabilizing the liquid polyhedral foam, but it is involved also in converting the liquid foam into a solid polyhedral foam during baking. A reasonably acceptable meringue can be made with just the major component, ovalbumin, although the resulting texture is coarser than that of whole-egg-white meringue, and it takes much longer to do the beating. Ovalbumin is easily denatured by heat, and its foaming properties are enhanced by prior heat treatment. Egg albumen with the ovalbumin completely removed gives satisfactory foamability and short-term stability, but the foam collapses after baking. In whole egg white, it appears that the main functional role of the ovalbumin (and conalbumin) is to convert the liquid foam into a more permanent solid foam as the protein becomes thermally denatured during baking. The presence of highly surface-active globulins is essential for forming a meringue (or a cake batter) with small bubbles and a smooth texture. The glycoprotein ovomucoid is largely responsible for the high viscosity of egg albumen, which assists foam stability by retarding foam drainage. In contrast to ovalbumin, ovomucoid is extremely stable to heat. The complexing of lysozyme (c. 3.5 m/m%) with other egg-white proteins, especially ovomucin (c. 1.5 m/m%), leads to a reduction in foaming capacity, but once formed, the complex increases the film strength and hence enhances foam stabilization. Lysozyme alone is a rather poor foaming agent. Apart from egg white, two other ingredients are usually included in recipes for meringue dishes and cakes: sugar and lemon juice. Sugar is added slowly with continued beating, but only after the egg white has first been beaten into a stable foam. Caster sugar should be used in preference to ordinary granulated sugar, since the crystals of the latter are too large to dissolve properly in the aqueous phase of the walls surrounding the air bubbles. The purpose of the lemon juice is to reduce the pH towards the isoelectric points of the acidic egg-white proteins. This leads to enhanced surface rheological properties and hence better foam stability. All the surface rheological parameters reach maximum values at the isoelectric point.

11.14.5

Egg-yolk gels

Egg yolk contains about 16% protein, including livetins, phosvitin and various lipoproteins. In native egg yolk, the proteins are arranged structurally in yolk spheres (up to 150 μm in diameter) and granules (0.3–1.7 μm in diameter). When yolk is heated in a shell

Gelling, emulsifying, stabilizing and foam formation

425

egg, the spherical structures are heat-set independently of one another, producing the familiar coarse, mealy texture of a hard-cooked egg yolk (Woodward and Cotterill 1987a). However, stirring of raw yolk easily disrupts the spheres, resulting in a protein solution that is capable of forming firm gels with a texture similar to that of albumen (Hawley 1970; Woodward and Cotterill 1987a). Egg-yolk gels have minimum hardness at pH 6, and increase in hardness as the pH is increased (Woodward and Cotterill 1987b). The effects of pH are not as pronounced in yolk as in albumen gels, however. Yolk gelation appears to be initiated at about 70°C, and the gel strength increases with increasing temperature. Egg-yolk gels have a much greater capacity for binding water than do egg-white gels. Water-binding capacity is increased with increasing pH, protein concentration and heating temperature (Woodward and Cotterill 1987b). For the role of low-density lipoprotein (LDL) in the gelation of egg yolk, see Nakamura et al. (1982), Meyer and Hood (1973) and Kojima and Nakamura (1985).

11.14.6

Whole-egg gels

Whole egg has many properties in common with egg white, since it is composed of approximately 67% albumen and 33% yolk. Beveridge and Ko (1984) found that wholeegg gels were very similar to egg-white gels. The hardness increased linearly from a minimum at pH 5 to a maximum at pH 9. The hardness also increased as the temperature was increased from 77 to 90°C and as the time of heating was increased from 5 to 25 min. Dilution of whole egg caused a logarithmic decrease in hardness. Whole egg has a greater gelling capacity than either albumen or yolk, based on the interaction of the yolk and albumen components (Woodward 1984; Woodward and Cotterill 1986b). Perhaps the most important interaction is the binding of iron from the yolk by conalbumin, which results in an increase of its denaturation temperature from 61°C in egg white to 78°C in whole egg. The stability of proteins in heated whole egg is enhanced with increasing pH and the addition of sucrose. The optimum heating temperature for maximum water binding is about 80°C; excessively high temperatures or long heating times result in overcoagulation and syneresis (Baldwin 1986, p. 345). Non-fat milk solids, starches and food gums have been used to improve water binding in whole-egg gels (Bengsston 1967; Ziegler et al. 1971; O’Brien et al. 1982).

11.15 11.15.1

Foam formation Fields of application

Some typical confectionery foams are listed in Table 11.3. 11.15.1.1

Ice cream

Ice cream is a partly frozen foam containing emulsified fat (c. 10 m/m%) and an air content of about 50% by volume. The role of the air cells is to give ice cream its soft, light texture. Without air, the frozen emulsion would be too cold for the mouth and too rich for the stomach. The main ingredients of an ice cream mix are (milk) fat, sugar (c. 15 m/m%), non-fat milk solids (including lactose) (c. 10 m/m%) and water. In addition, commercial products

426

Confectionery and Chocolate Engineering: Principles and Applications

Table 11.3 Foamed confectionery products. Product Angel Kiss Foamed wafer filling Marshmallow Montelimar Fondant/foamed caramel Frappé Meringue

Density (kg/L) 0.2 0.2 0.25–0.4 > 0.65 0.85–1.2 0.4–0.55 0.6–0.8

contain emulsifiers, stabilizers, colours and flavours. The ingredients are mixed, pasteurized, homogenized and rapidly cooled. After the oil-in-water emulsion has been aged, it is passed into a chamber whose temperature is low enough to partly freeze the mixture, and at the same time air is beaten in. The technological objective of the freezing/aeration stage is the formation of a stable foam through partial destabilization of the emulsion. Beating without freezing does not give a stable foam. The temperature of the ice cream mix falls by about 10°C during its short residence time (c. 20 s) in a scraped-surface heat exchanger. Air is incorporated intensively as the mix is frozen against the cylinder wall. The ice crystals that begin to form at the wall are vigorously scraped off by the blades of the rotor, and the solid foam leaves the freezer at a temperature of c. −5°C. An acceptable texture is achieved with an overrun (see Section 11.15.2) of 100%. Too little foam makes the product appear wet, hard and excessively cold. Too much foam gives a texture that is dry and crumbly. The air cells should be of the order of 100 μm in size. If the air cells are too large, the ice cream melts too quickly. On the other hand, if the air cells are too small, the foam becomes too stable, and an undesirable ‘head’ is left on melting. Ice cream has a complex colloidal structure, being both a foam and an emulsion. The solid foam is held together partially by emulsified fat, and partially by a network of small ice crystals dispersed in a sweetened aqueous macromolecular solution. The role of the emulsifier (e.g. glycerol monostearate) is to assist in the controlled destabilization of the emulsion in the freezer. Polymorphic changes in the solidified fat during ageing of the ice cream mix cause distortion of the initial spherical shape of the globules, which, in combination with the partially loosened protein film, cause shear-induced clumping in the freezer. In whipped cream, the fat content is c. 35–40 m/m%, which is enough to stabilize the foam with a fat globule network alone. The additional stabilization of the air bubbles of ice cream is due to the ice crystals (size ∼ 50 μm) and the highly viscous aqueous phase, the viscosity of which is often increased by addition of polysaccharide stabilizers (e.g. carrageenan or locust bean gum) Ice cream is one of the most complex food colloids; not only a foam, but also an emulsion, a dispersion and a gel. However, the majority of confectionery colloids are similarly complex systems.

11.15.2

Velocity of bubble rise

At the onset of formation by injection methods, a foam represents a gas emulsion. The rate of its transformation into a polyhedral foam depends on the velocity of bubble rise

Gelling, emulsifying, stabilizing and foam formation

427

and the consequent drainage of the ‘excess’ liquid from the foam thus formed. The size of the bubbles, the gas volume fraction (gas concentration in the liquid) and the surfactant concentration determine the velocity of bubble rise. Bubble rise has a destructive effect on foam stability. In the absence of a surfactant, the velocity at which individual gas bubbles rise is expressed by an equation due to Hadamar and Rybczynski (Dukhin et al. 1986),

η + η1 ρ gR 2 2 u = ⎛ ⎞ ( ρ2 − ρ1 ) g ( R 2 η2 ) 2 ≈ 2 ⎝ 3⎠ 2η2 + 3η1 3η2

(11.13)

where ‘1’ refers to the gas phase, ‘2’ refers to the liquid phase, ρ is the density, η is the dynamic viscosity and R is the radius of the spherical bubbles. As is known, the velocity of ascent (or descent) of solid particles is expressed by Stokes’ law, 2 ρ gR 2 uS = ⎛ ⎞ 2 ⎝ 9 ⎠ η2

(11.14)

From Eqns (11.13) and (11.14), u 3 = uS 2

(11.15)

i.e. the rise of bubbles is 1.5 times faster than that of solid particles. The regime of gas bubble rise depends significantly on the hydrodynamic conditions, i.e. on the Reynolds number Re, where Re =

2 ρ2 uR η2

(11.16)

From Eqns (11.13) and (11.16), u2 =

RgRe 6

(11.17)

Bubbles with a diameter smaller than 0.01 cm = 100 μm (Re ≤ 0.5) rise as solid particles, and obey Stokes’ law. When Re > 0.5, a deviation from Stokes’ law is observed but the spherical shape of the bubbles is retained up to Re values close to 1500. When Re > 200–300, the Example 11.1 Let us calculate the velocity of bubble rise according to Eqn (11.17) if Re = 0.5 and R = 10−4 m: u2 =

RgRe 0.5 2 ≈ 0.8 × 10 −4 ( m s ) → u ≈ 0.9 cm s ≈ 10 −4 m × 10 ( m s 2 ) × 6 6

Let us calculate the dynamic viscosity η2 of the liquid phase if ρ2 = 1.02 × 103 kg/m3:

428

Confectionery and Chocolate Engineering: Principles and Applications

Re = 0.5 =

2 ρ2 uR 10 −4 = 2 × 1.02 × 103 × 0.9 × 10 −2 × η2 η2

10 −4 ⎞ ⎛ −3 → η2 = ⎜ 2 × 1.02 × 103 × 0.9 × 10 −2 × ⎟ Pa s ≈ 3.6 × 10 Pa s ⎝ 0.5 ⎠ Comment: This is a rather low value of dynamic viscosity. From Eqn (11.13), it is evident that the dynamic viscosity and the bubble rise are in an inverse relationship. For example, if η2 = 3.6 × 10−1 Pa s (which is a common value in practice), then u ≈ 0.9 × 10−2 cm/s, i.e. the bubble rise will be 10 times slower. velocity of bubble rise in the absence of a surfactant satisfies the following equation (Levich 1962): u=

ρ2 gR 2 9η2

(11.18)

The following relation was derived from results on the velocity of rise of large (R = 2– 10 mm) bubbles (Harrison and Leung 1962): u = 0.792 g1 2 (VB )

16

(11.19)

where VB is the bubble volume. For further details, see Exerowa and Kruglyakov (1998, Chapter 1). Example 11.2 Let us calculate the velocity of rise of a bubble of radius R = 5 mm according to Eqn (11.19): u = 0.792 × 10 m ( s −1 )

(

)

16

3.14 ⎤ 3 m ⎡⎢ 4 × (5 × 10 −3 ) × 3 ⎥⎦ ⎣

≈ 22.3 cm s

A rather high velocity! When an ensemble of bubbles rises, the collective velocity U depends also on the volume fraction ϕ of the dispersed gas. In the absence of a surfactant, the collective velocity of bubble rise in the Stokes hydrodynamic regime of bubble movement (Re ≤ 0.5), as given by Galor and Walso (1968) and Rulev (1970), is ⎛ ρ gR 2 ⎞ U =⎜ 2 (1 − ϕ1 3 ) ⎝ 3η2 ⎟⎠

(11.20)

In the regime 100 ≤ Re ≤ 1500, the corresponding equation is U=

ρ2 gR 2 (1 − ϕ )2 9 (1 − ϕ 1 2 )

(11.21)

Gelling, emulsifying, stabilizing and foam formation

U/u

1.0

429

1: Stokes regime (with surfactant) 2: Stokes regime (no surfactant) 3: high Reynolds number (no surfactant)

0.5

1 0 0

2

3

0.5 j

1.0

Fig. 11.6 Bubble rise ratio U/u as a function of volume fraction ϕ in various regions of Reynolds number [reproduced from Exerowa and Kruglyakov (1998), by kind permission of Elsevier].

For the regime 0.5 ≤ Re ≤ 100, no analytical expression for the collective velocity U has been obtained. Surfactants exert a significant influence on the collective velocity of rise. The ratio between the collective velocity of rise and the velocity of a single bubble (U/u) is given in Fig. 11.6 (Rulev 1970). In the presence of a surfactant (curve 1), the dependence of U and ϕ is more pronounced. In relation to foam formation, the amount of air incorporated is described by the ‘overrun’ or ‘expansion ratio’. This is the gas-to-liquid ratio in an aerated product expressed as a percentage on a volume basis. So, a 200% overrun means that 1 L of liquid has taken up 2 L of gas to form 3 L of foam, i.e.

ϕ=

2 2 = 1+ 2 3

11.15.3

Whipping

There are three main ways of making food foams: • whipping (beating); • bubble formation at an orifice; and • bubble generation in situ. Whipping (or beating) is the traditional way of making a foam from egg white or cream. Air is entrapped in the viscous liquid in the form of large bubbles, which are then broken down into smaller bubbles by mechanical action. In principle, there is an unlimited amount of air available for incorporation; but, in practice, the overrun is limited by the fact that no new air can be introduced once the whipping rod is covered by foam. The initial bubble formation takes place in the region immediately behind the moving rod where the local hydrostatic pressure is low. The newly formed bubbles become elongated by the stress in the liquid, and break into a number of smaller bubbles. For inducing this kind of bubble deformation and rupture, elongational motion of the liquid is more effective than shear flow. Agitation during whipping causes local pressure fluctuations,

430

Confectionery and Chocolate Engineering: Principles and Applications

Density of foam (kg/L)

1.0

Rapid whipping

0.5

Slow whipping

0

Fig. 11.7

5

10 Time of whipping (min)

Density of foam as a function of time of whipping (values for information only).

Density of foam (kg/L)

0.4

0.3

0.2

p

yru

s rch

a St 0.1

40

50

ar ug

se

cro

Su

60

s rt

ve In

70

Sugar dry content (m/m%)

Fig. 11.8

Density of foam as a function of sugar dry content (values for information only).

which in turn induce surface tension fluctuations due to rapid changes in bubble volume and surface area. Beyond a certain whipping speed, the surface tension fluctuations become so intense that the liquid films between the pulsating bubbles rupture, and hence the foam collapses. This is the explanation for the sharp maximum in the plot of foam volume against whipping speed observed experimentally with kitchen mixers. Figures 11.7–11.9 show the dependence of the foam density on various technological parameters.

11.15.4

Continuous industrial aeration

This method is often applied by introducing the required gas volume into the liquid by generating bubbles at an orifice, and then reducing them in size later on in the process with a mixer or whipping rod.

Gelling, emulsifying, stabilizing and foam formation

431

Density of foam (kg/L)

1.4 1.2 1.0

0.5%

0.8 0.6 0.4 0.2 0

4.0%

2

4 6 8 Time of whipping (min)

1.0% 2.0% 10

Fig. 11.9 Density of foam as a function of time of whipping with various concentrations of whipping agent (values for information only).

For a bubble emerging from a single orifice, there are two forces acting: the buoyancy (or Archimedes) force, ⎛ 4 πr 3 ⎞ FA = ⎜ ρg ⎝ 3 ⎟⎠

(11.22)

and the surface tension force at the orifice, FS = 2πRγ

(11.23)

where r is the radius of the bubble, ρ is the density of the liquid, g is the gravitational acceleration, R is the radius of the orifice and γ is the surface tension of the liquid. (The density ρg of the gas can be neglected. For example, the density of air at 20°C is c. 29 g/24 L = 1.208 kg/m3; however, the liquids used in food production have densities in the region of 103 kg/m3 and, consequently, ρ ≈ ρ − ρg.) From Eqns (11.22) and (11.23), ⎛ 3γ R ⎞ r=⎜ ⎝ 2 ρ g ⎟⎠

13

(11.24)

In the presence of a surface-active material, the surface tension γ of the growing bubble is higher that the equilibrium tension owing to the finite dilational viscosity. This results in the development of larger bubbles; see Eqn (11.24). Example 11.3 The parameters in an industrial aeration process are R = 5 × 10−3 m, γ = 2 × 10−2 N/m and ρ = 1.01 × 103 kg/m3. Let us calculate the radius r of the bubbles according to Eqn (11.23): r3 =

3 × 5 × 10 −3 m × 2 × 10 −2 ( N m ) = 15.1 × 10 −9 m 3 2 × 1.01 × 103 kg m3 × 8.91 ( m s 2 )

and r = 2.47 mm.

432

Confectionery and Chocolate Engineering: Principles and Applications

Ingredients

Whipping agent

Water

Filtered air

Whipping

Dissolution

Colouring, flavouring

Shaping

(a)

Water Whipping agent

Sugar

Mixing

Whipping

Dissolution

Filtered air

Colouring, flavouring

Other ingredients

Shaping

(b) Fig. 11.10

11.15.5

Foaming by (a) one-step method and (b) two-step method.

Industrial foaming methods

Figure 11.10 shows schematic layouts for foaming by one- and two-step methods.

11.15.6

In situ generation of foam

An important example of this is the in situ generation of carbon dioxide bubbles by yeast cells or by baking powder during the baking of cakes (and bread). When baking powder is used, vapour (and ammonia) is also generated. The baking powders usually used are represented by the following chemical reactions: 2 NaHCO3 → Na 2HCO3 + H2 O + CO2

(11.25)

( NH4 )2 CO3 → 2NH3 + H2 O + CO2

(11.26)

NaHCO3 + KHC 4H 4 O6 → KNaC 4H4 O6 + H2 O + CO2

(11.27)

2 NaHCO3 + Na 2H2 P2 O7 → Na 2 P2 O7 + 2H2 O + 2CO2

(11.28)

8NaHCO3 + 3CaH4 ( PO 4 )2 → Ca3 ( PO 4 )2 + 4Na 2HPO 4 + 8H2 O + 8CO2

(11.29)

NaHCO3 + NH 4Cl → NaCl + NH3 + H2 O + CO2

(11.30)

The list of permitted components in baking powder is legally regulated. For further details of foams and foam films, see Exerowa and Kruglyakov (1998). Although the generation of foams by yeast cells is a general method of preparing several types of dough, this type of biochemical operation is beyond the scope of this book.

Gelling, emulsifying, stabilizing and foam formation

433

Further reading Anon. (1985) The Dairy Handbook. Alfa-Laval AB, Lund. AVEBE. Technical brochures/specifications. Boomgaard, T.V.D., van Vliet, T. and van Hooydonk, A.M.C. (1987) Physical stability of chocolate milk. Int J Food Sci Techn 22: 279–291. CABATEC (1991) Dairy Ingredients in the Baking and Confectionery Industries. An audio-visual open learning module, Ref. C6. The Biscuit, Cake, Chocolate and Confectionery Alliance, London. Copenhagen Pectin A/S. Technical brochures/specifications. DGF Stoess. Technical brochures/specifications. Friberg, S.E., Larsson, K. and Sjöblom, J. (2003) Food Emulsions. Marcel Dekker, New York. Goldsmith, S.M. and Toledo, R.T. (1985) Studies on egg albumin using nuclear magnetic resonance. J Food Sci 50: 59. Gossett, P. W. and Baker, R. C. (1983) Effect of pH and succinylation on the water retention properties of coagulated, frozen, and thawed egg albumen. J Food Sci 48: 1391. Harris, P. (ed) (1990) Food Gels. Elsevier Applied Science, London. Ingleton, J.F. (1971) Agar agar in confectionery jellies. Confect Prod 37 (9): 544. IPPA (International Pectin Producers Association). Technical brochures/specifications. Lees, R. (1972) Use of alginates for the manufacture of confectionery. Confect Prod 38 (8): 416–418. Lucas Meyer. Technical brochures/specifications. Mellema, M., van Opheusden, J.H.J. and van Vliet, T. (1999) In: Dickinson, J.M. and Patino, R. (eds) Food Emulsions and Foams. Interfaces, Interactions and Stability, Special Publication, No. 227. Royal Society of Chemistry, Cambridge, p. 176. Muir, D.D. (1984) The chemistry of milk powders. Proc IFST 17(1). National Starch. Technical brochures/specifications. Perry, R.H. (1972) Turkish delight: Halwa, Chalwa, Rahat, Lokhum, Baslogke. Confect Prod 38 (6): 317–318, 326. Scholey, J. et al. (1975) Physical Properties of Bakery Jams. BFMIRA Report 217. Scholey, J. and Vane-Wright, R. (1973) Physical Properties of Bakery Jams – An Investigation into Methods of Measurement. BFMIRA Technical Circular 540. Snoeren, Th.H.M. (1976) Kappa-carrageenan, a Study on its Physico-chemical Properties, Sol-Gel Transition and Interaction with Milk Protein. PhD Thesis. Wageningen, The Netherlands. Snoeren, Th.H.M., Both, P. and Schmidt, D.G. (1976) An electro-microscopic study of carrageenan and its interaction with kappa-casein. Neth Milk Dairy J 30: 132–141. Verkroost, J.A. (1979) Some Effects on Recipe Variations on Physical Properties of Baker Jams. BFMIRA Report 297.

Chapter 12

Transport

Contents 12.1 12.2 12.3 12.4 12.5 12.6

Types of transport Calculation of flow rate of non-Newtonian fluids Transporting dessert masses in long pipes Changes in pipe direction Laminar unsteady flow Transport of flour and sugar by air flow 12.6.1 Physical parameters of air 12.6.2 Air flow in a tube 12.6.3 Flow properties of transported powders 12.6.4 Power requirement of air flow 12.6.5 Measurement of a pneumatic system Further reading

12.1

434 434 436 437 438 438 438 438 439 441 442 444

Types of transport

Two types of transport will be studied here: (1) Flow of non-Newtonian fluids in a tube of circular cross-section, with remarks on cases where the cross-section is different. The fluids are chocolate mass, fondant mass and praline mass. (2) Transport of flour with air; although this operation is common in the milling industry, it is of great importance in the confectionery industry as well.

12.2

Calculation of flow rate of non-Newtonian fluids

Taking into account the fact that the flow curve of a non-Newtonian fluid is of the form

τ = τ 0 + ηPl D

(12.1)

the flow rate Q = Q(Δp) can be calculated using the following integral (the Rabinowitsch– Mooney equation): Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

Transport

Q=

435

πR3 τ R 2 ∫ Dτ dτ τ 3R τ 0

(12.2)

rΔp 2L

(12.3)

where

τ=

and R is the radius of the tube (r is used as a variant), L is the length of the tube, τ0 = r0 Δp/2L is the yield stress, r0 is defined by Eqn (12.3), Δp is the pressure difference between the two ends of the tube and ηPl is the plastic (dynamic) viscosity of the fluid. Sections A3.1 and A3.2 show the flow rate for the most commonly used fluid models, calculated on the basis of the Rabinowitsch–Mooney equation. Mohos developed a method that decomposes the flow rate Q for the most frequently used models of non-Newtonian fluids into a product. This product consists of a factor K, which is proportional to the pressure difference Δp, and a factor M(Bu) containing Δp but not proportional to it. Values of M as a function of Bu (the Buckingham number) can be tabulated according to this method. For further details, see Appendix 3. The physical basis of the decomposition is that, on the one hand, the volumetric rate Q=

R 4 πΔp 8L η

has a dimension m3/s. On the other hand, since the plastic viscosity or Casson viscosity ηPl and the other types of viscosity (except the surface viscosity) have the same dimension as that of the dynamic viscosity (kg/m s), the following dimensionally correct equation can be written:

[Q ] =

R 4 πΔp R 4 πΔp R 4 πΔp R 4 πΔp = N Pl × = N CA × = NX × 8Lη 8LηPl 8LηCA 8LηX

where the symbol [ ] means the dimension, and the N-values are pure numbers. The Nvalues are as follows: NPl = 1 − (4/3)Bu + (1/3)Bu4, a number depending on the Buckingham number Bu, where Bu = τ0/τ; τ = R Δp/2L, τ0 is the yield stress, and R and L are the radius and length of the tube, respectively. NCA is a number referring to a Casson fluid (and the Casson viscosity). NX is a number referring to an arbitrary fluid, etc.; see Sections 3.1 and 3.2. Because the Casson equation in both the original (n = ½) and the generalized form is mostly used for linearization of the flow curves of molten chocolate masses, this is dealt with in detail in Sections 3.3–3.7. Examples of calculations are given in Appendix 3. The examples below present the evaluation of flow curves according to the Bingham, Casson and Ostwald–de Waele models. These flow models are the ones most frequently used in the confectionery industry.

436

Confectionery and Chocolate Engineering: Principles and Applications

Example 12.1 Table 12.1 Evaluation of the flow curve of a milk chocolate according to the Bingham (τ; D) and Casson (τ0.5; D0.5) models. D(s−1)

t (Pa)

τ0.5

D0.5

23 26.1 53.2 75 91 139

0.16 0.36 2.42 4.4 7 13.7

4.795832 5.108816 7.293833 8.660254 9.539392 11.78983

0.4 0.6 1.555635 2.097618 2.645751 3.701351

Slope 8.469394

Constant 28.30303

Slope 2.141655

Constant 3.938165

Table 12.1 shows a flow curve evaluated according to the Bingham (n = 1) and Casson (n = 0.5) models. The results are: According to the Bingham model, τ = 8.47 (Pa) + 28.3 (Pa s) D Correlation coefficient: 0.988481. According to the Casson model, τ1/2 = 2.14 (Pa)1/2 + 3.94 (Pa s)1/2 D1/2 Correlation coefficient: 0.998945. It can be seen that the evaluation according to the Casson model provides a slightly better correlation than that according to the Bingham model. This example shows that there is no exclusively acceptable method of flow curve evaluation. Example 12.2 Table 12.2

Evaluation of the flow curve of a truffle mass according to the Ostwald–de Waele model.

τ (Pa)

D (s−1)

ln τ

ln D

Exponent

ln (intercept)

73 169 435 606 708

0.5 5 50 150 200

4.290459 4.290459 6.075346 6.40688 6.562444

−0.69315 1.609438 3.912023 5.01063 5.298317

0.378806

4.546189 Intercept 94.27241 r = 0.9994

Table 12.2 shows the flow curve of a truffle mass and its evaluation (Machikhin et al. 1976). The resulting flow curve is described by

τ = 94.27 ( Pa s 0.379 ) D 0.379

12.3

Transporting dessert masses in long pipes

Lapitov and Filatov (1963) developed a calculation method for viscoelastic masses, in fact for dessert masses, which starts from the Buckingham equation written in the following form by Mohos:

Transport

{

4 1 ⎡ R 4 πΔp ⎤ Q=⎢ 1 − ⎛ ⎞ Bu + ⎛ ⎞ Bu 4 ⎥ ⎝ ⎠ ⎝ L 8 η 3 3⎠ ⎣ Pl ⎦

}

437

(12.4)

where Bu = τ0/τ. Lapitov and Filatov (1963) worked with the ratio d0/d (= Bu) of the diameters of the undeformed material and of the tube. However, they calculated with Bu2 instead of Bu4, and also assumed that Bu > 0.75. After these simplifications, the following linear function was applied: 1 − (4/3)Bu + (1/3)Bu4 ≈ 0.32 (1 − Bu). The Buckingham equation (Eqn 12.4) was written

λ=

200 Re ′

(12.5)

λ=

2 Δp LρV 2

(12.6)

where λ is the generalized friction coefficient; Re′ = Vdρ/η is the generalized Reynolds number; η = ηPl + dτ0/25V; V = 4Q/πd2 (d = diameter of pipe), i.e the average linear velocity of the mass; L is the length of the pipe; and Δp is the pressure difference. According to experimental results, instead of Eqns (12.5) and (12.6), the following can be used:

λ′ =

200 (Re ′ )0.92

D 0.08 Δp = 91.6 × η 0.92 × V 1.08 × 1.92 L d

(12.7) (12.8)

The interval of validity of these equations is given by V = 0.01–0.2 m/s, τ0 = 100–1000 Pa, ηPl = 10–200 Pa s and d = 0.03–0.08 m. For these experiments, the calculated values varied from λ′ = 2590 and Re′ = 0.082 to λ′ = 164 200 and Re′ = 0.000734 as the diameter of the tube was varied from 42 mm to 69 mm.

12.4

Changes in pipe direction

If the direction of the pipe changes, the length of its track has to be taken into account when L is calculated. Machikhin and Birfeld (1969) gave a method of calculation that takes into account sudden changes in the pipe diameter, which follows the method of Bagley (1957): it calculates with a fictive length of pipe L′ = N × L, where the value of N is about 6–8, and needs to be experimentally determined. If the pipe has a perpendicular part, then two additional pressure differences need to be calculated to obtain Δp in Eqn (12.8): a hydrostatic pressure difference Δph, Δph = ρ gH

(12.9)

and a pressure difference ΔpV depending on the average linear velocity V of the mass, ΔpV = KV ρH

(12.10)

438

Confectionery and Chocolate Engineering: Principles and Applications

KV = 1.08 + 0.563 × V 0.5

(12.11)

where ρ is the density of the mass, g is the gravitational acceleration and H is the length of the perpendicular part of the pipe.

12.5

Laminar unsteady flow

A detailed discussion of laminar unsteady flow in long pipes was presented by Letelier and Céspedes (1986). These authors published a mathematical tool called ‘OneDimensional Analysis’, by means of which the equations of motion can be easily dealt with; it was used for discussion of viscoelastic flow (Colemann–Noll and Maxwell fluids) and power-law flow (Ostwald–de Waele fluids). A description of this method is beyond the scope of this book.

12.6

Transport of flour and sugar by air flow

12.6.1

Physical parameters of air

The density of air as a function of temperature and pressure is given by the equation ⎛ p ⎞ 273° d = dN ⎜ ⎝ pN ⎟⎠ 273° + t

(12.12)

where dN = 1.295 kg/m3 is the density of air in the normal state, pN = 101 337 N/m2 is the normal atmospheric pressure, p is the pressure of the air and t (°C) is the temperature of the air. Some physical parameters of air are given in Table 12.3.

12.6.2

Air flow in a tube

The pressure drop in a straight tube with a rough inner surface can be expressed as L dv 2 Δp ( Pa ) = f ⎛ ⎞ G ⎝ D⎠ 2

(12.13)

where f is the air friction coefficient, L is the length of the tube (m), D is the diameter of the tube (m), d is the density of the air (kg/m3) and vG is the velocity of the air (m/s). The friction coefficient f is a function of the Reynolds number of the air (see Fig. 12.1), Table 12.3

Physical parameters of air at normal atmospheric pressure (Egry 1973).

Temperature (°C) Density (kg/m3) Viscosity (kg/(m s)) × 106 Kinematic viscosity (m2/s) × 106

0

20

40

60

80

1.295 17.17 13.3

1.207 18.15 15.1

1.13 19.1 16.9

1.06 20 18.9

1 20.9 20.9

439

Transport

f 0.03 Rough 0.02 Smooth 0.01

0

0

100

200

300

400

500 Re × 10−3

Fig. 12.1 Air friction coefficient as a function of Reynolds number for smooth and rough tube surfaces (Egry 1973).

Re =

DvG νG

(12.14)

where νG is the kinematic viscosity of air (m2/s). Figure 12.1 shows the air friction coefficient as a function of the Reynolds number for smooth and rough tube surfaces (see Egry 1973). The values of f for a smooth surface should be used because, during transport, the surface will become smooth within a short time. Example 12.3 For a tube with L = 30 m and D = 0.12 m, and t = 25°C, vG = 22 m/s and p = 0.95 × 105 Pa, let us calculate the pressure drop at the end of the tube. The density of the air is 0.95 273 ⎛ p ⎞ 273° d = dN ⎜ = 1.295 × × = 1.112 kg m3 ⎟ ⎝ pN ⎠ 273° + t 1.01337 298 From Table 12.3, a good approximation for the kinematic viscosity νG (at 25°C) is 15.5 × 10−6 m2/s. Thus Re = 0.12 × 22/(15.5 × 10−6) = 170 323. From Fig. 12.1, f ≈ 0.017, and therefore L dv 2 30 222 Δp ( Pa ) = f ⎛ ⎞ G = 0.017 × × 1.112 × = 1143.7 Pa ⎝ D⎠ 2 0.12 2 The absolute pressure at the end of the tube is (95 000 − 1143.7) Pa = 93856.3 Pa.

12.6.3

Flow properties of transported powders

Materials transported in a powder state cannot be characterized by a single particle size value but must be characterized by a particle size distribution; however, for the purpose of calculation, a so-called ‘reduced particle size’ x is used:

440

Confectionery and Chocolate Engineering: Principles and Applications

⎛ 6M ⎞ x (m) = ⎜ ⎝ πdM ⎟⎠

13

(12.15)

where M is the mass of one particle (or grain) (kg), and dM is the density of the material transported in the powder state (kg/m3). This calculation of the reduced particle size assumes a globular shape of the particles, which is a good approximation if x < 0.1 mm. The ‘reduced cross-section’ of a particle is defined as 13

⎡ ⎛ 3M ⎞ 2 ⎤ A (m2 ) = ⎢π ⎜ ⎟ ⎥ ⎣ ⎝ 4d M ⎠ ⎦

(12.16)

From Eqns (12.15) and (12.16), a practical relation can be obtained: x2 π ≈ 0.78x 2 82 3

A=

(12.17)

The air flow acts on a particle with a force F (N), equal to F=

CE Adw 2 2

(12.18)

where CE is the hydrodynamic friction coefficient of the particles, which depends on the Reynolds number ReM referred to the material transported in the powder state; vM is the velocity of the transported material (m/s); and w = vG − vM is the difference between the velocities of the air and the particles (m/s). The usual definition of the slip s is s=

w v = 1− M vG vG

(12.19)

The definition of ReM is ReM =

xw νG

(12.20)

Values of CE are shown in Fig. 12.2, which presents the hydrodynamic friction coefficient of globular particles as a function of the Reynolds number (Egry 1973). For the limiting velocity c of a floating particle, hydrodynamic friction force = weight of particle i.e. CE Adc 2 ⎛ 4 ⎞ ⎛ x ⎞ = πd g ⎝ 3⎠ ⎝ 2⎠ M 2 and

(12.21)

Transport

441

CE 103 102 101 100 10−1 10−2 10−2 10−1 100 101 102 103 104 105 106 107 ReM Fig. 12.2 1973).

Hydrodynamic friction coefficient CE of globular particles as a function of Reynolds number (Egry

4 d ⎛ xg ⎞ c = ⎛ ⎞⎛ M⎞⎜ ⎟ ⎝ 3 ⎠ ⎝ d ⎠ ⎝ CE ⎠

(12.22)

where g is the gravitational acceleration (= 9.81 m/s2). Example 12.4 The size of a particle is x = 0.05 mm = 5 × 10−5 m, dM = 400 kg/m3, w = 25 m/s, t = 25°C, νG (at 25°C) = 15.5 × 10−6 m2/s and d = 1.112 kg/m3. Let us calculate ReM and the limiting velocity c of a floating particle. ReM =

xw 25 = 5 × 10 −5 × = 80.65 15.5 × 10 −6 νG

From Fig. 12.2, CE ≈ 2: 4 d ⎛ xg ⎞ 4 400 ⎞ ⎛ 5 × 10 −5 × 9.81⎞ c = ⎛ ⎞⎛ M⎞⎜ ⎟ = ⎛ ⎞⎛ ⎟⎠ ≈ 0.343 m s ⎝ 3 ⎠ ⎝ d ⎠ ⎝ CE ⎠ ⎝ 3 ⎠ ⎝ 1.112 ⎠ ⎝⎜ 2

12.6.4

Power requirement of air flow

If ΔpΣ < 105 Pa, air can be regarded as incompressible, and then P=

VΔpΣ η

(12.23)

where P is the power requirement of the air flow (in W = N m/s), ΔpΣ is the total pressure drop of the transporting air flow (N/m2), V is the volume velocity of the air (m3/s) and η is the efficiency of the pneumatic machinery. At higher values of ΔpΣ, the power requirement is calculated by taking isothermal compression into account:

442

Confectionery and Chocolate Engineering: Principles and Applications

pV 1 1 ln ( p1 p2 ) η

P=

(12.24)

where p1 and p2 are the initial and final pressure, respectively (p1 < p2), and V1 is the volume velocity of the air (m3/s) at p1.

12.6.5

Measurement of a pneumatic system

According to Pápai (1965), the principle of measurement is the summation of the partial pressure drops along the pneumatic pipe. First, the air volume velocity V is determined from the following equation if the material flow QM (kg/s) and the mixing ratio r are given (r = QM/QAIR): VG =

QM rd

(12.25)

The recommended mixing ratio r for flour is in the range 1–4. From this equation, the calculated cross-section A of the pipe can be determined: A=

VG vG

(12.26)

In the general case a pneumatic pipe consists of a vertical section of length L and diameter D, a curved section, a cyclone, a pipe section behind the cyclone, a ventilator and a powder filter. The following equation for the total pressure drop Δp3 is valid: Δp3 ( N m 2 ) = Δp1 + Δp2

(12.27)

where Δp1 is the pressure drop of the pipe and Δp2 is the pressure drop arising from the effect of the cyclone up to the end of the powder filter. The partial pressure drops are Δp1 = Δp10 + Δp11

(12.28)

where Δp10 is the pressure drop of the pipe containing only air (‘vacant run’), and Δp11 is the pressure drop of the pipe containing air + material. Here, f ( L + LE ) ⎤ dvG2 Δp10 = ⎡⎢ζ IN + ⎥⎦ 2 D ⎣

(12.29)

where ζIN = 1.2–2 is the input air friction coefficient, LE = 1.5LC is the equivalent length of the curved section if the ratio of the radius R to the diameter D is high (≥ 10), and LC is the length of the curved section, and Δp11 = Δp111 + Δp112 + Δp113 + Δp114 + Δp115

(12.30)

Transport

443

In detail, Δp111 = ζACQMvM/A is the pressure drop derived from acceleration of material (where ζAC = 1.2–1.4, depending on the method of input; when a sucking head is used, ζAC = 1.7–1.9); Δp112 = kHLHQMvG/A is the pressure drop in the horizontal section of the pipe; Δp113 = HgQM/vMA is the pressure drop derived from lifting through a height H; Δp114 = kFLFQMvM/A is the pressure drop derived from collisions of particles in the vertical straight section of the pipe (for collisions in the vertical section, kF = 0.05–0.12 m−1; for collisions in the horizontal section, kF = 0.04–0.1 m−1); Δp115 = kCYLCYQMvM/A is the pressure drop in the curved section. Also, Δp2 = Δp21 + Δp22 + Δp23

(12.31)

where Δp21 = ζCYdv2G/2 is the pressure drop in the cyclone; ζCY = 2–8 is the friction coefficient of the cyclone; Δp22 = Δp10 is the pressure drop in the pipe section between the cyclone (+ ventilator) and the filter; this is practically equal to the pressure drop for a vacant run; Δp23 is the pressure drop in the filter, which depends on its type.

Total pressure drop

Figure 12.3 shows how to choose a suitable ventilator, taking into account the minimum volume velocity of the air (Vg min) on the basis of the characteristics of the ventilator. The formulae for the various partial pressure drops show that the total pressure drop is a quadratic function of the velocity (and volume velocity) of the air. The continuous line in Fig. 12.3 represents the values of the volume velocity of air in the run. A ventilator of type 1 is not suitable, because it cannot produce a sufficient pressure difference to provide Vg min.

Characteristic of ventilator 2

1

Vg min Fig. 12.3

Choice of ventilator (Egry 1973).

Vg

444

Confectionery and Chocolate Engineering: Principles and Applications

For further details, see Leschonski (1975), Bohnet (1983), Schmidt et al. (1992), Tomas (1992), Wilms (1993) and Schulze (1993).

Further reading BEPEX/HUTT. Technical brochures. Boger, D.V., Crochet, M.J. and Keiller, R.A. (1992) On viscoelastic flows through abrupt contractions. J Non-Newtonian Fluid Mechanics 44: 267–279. Carle & Montanari. Technical brochures. NETZSCH-Mohnopumpen. Technical brochures. Rohsenow, W.M., Hartnett, J.P. and Cho, Y.I. (eds) (1998) Handbook of Heat Transfer, 3rd edn. McGraw-Hill Handbooks. McGraw-Hill, New York. Sollich. Technical brochures. Tomay, T. (ed) (1973) Gabonaipari kézikönyv, Technológiai gépek és berendezések (Manual of Grain Processors, Technological Machines and Equipments of the Grain Silos, in Hungarian). Mezo˝gazdasági Kiadó, Budapest, p. 942–954.

Chapter 13

Pressing

Contents 13.1 Applications of pressing in the confectionery industry 13.2 Theory of pressing 13.3 Cocoa liquor pressing Further reading

13.1

445 445 448 449

Applications of pressing in the confectionery industry

Pressing of cocoa mass (cocoa liquor) is used for producing cocoa butter and cocoa powder, which are essential in chocolate manufacture. (The machinery for pressing cocoa mass can also be used for decreasing the fat content of hazelnut paste, which is made from roasted, unshelled, comminuted hazelnuts.)

13.2

Theory of pressing

Pressing can be regarded as filtration under the effect of excess pressure – in filtration, the driving force for separation originates from the hydraulic pressure of the slurry. The Ruth equation can be used to describe the evolution of the pressing process over time: w = dV dt =

Δp η α (1− ε ) hC + β

(13.1)

where V (m) is the volume of filtrate VB (m3) per unit area A (m2) of filter, t (s) is the time, w (m/s) is the rate of pressing, Δp (Pa) is the pressure applied, η (Pa s) is the dynamic viscosity of the filtrate, ε (dimensionless) is the porosity of the cake, hC (m) is the height of the cake, α (m/m3 = m−2) is the average flow resistance of the cake per unit volume and β (m−1) is the average flow resistance of the filter used. The total volume VSL of slurry can be imagined as consisting of two parts: VSL = VC + VB Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

(13.2) Ferenc Á. Mohos

446

Confectionery and Chocolate Engineering: Principles and Applications

where VC is the volume of cake and VB is the volume of filtrate. The corresponding heights are: hSL: height of slurry at the beginning of pressing; hC: height of cake as a function of t; hB: height of filtrate as a function of t. If s is the volume concentration of the suspension, i.e. s=

VC VSL

then VB =

(1 − s )VC s

(13.3)

and V=

VB (1 − s ) (1 − ε ) hC = A s

(13.4)

Differentiating both sides, w=

dV [(1 − s ) (1 − ε )] ⎛ dhC ⎞ = ⎝ dt ⎠ s dt

(13.5)

If we consider the case of a non-compressible cake, i.e. ε = 0, the Ruth equation (Eqn 13.1) can be written in a simpler form, w=

dV Δp η = dt [α yV + β ]

(13.6)

where y = VC/VB and yV = VC/A. There are two usual methods for integration of the differential equation (13.6). Case 1. Suppose that w = dV/dt = constant, and Δp increases up to its maximum value (Δp)max. If Δp = (Δp)max, then V = VB/A = Vmax, and, from Eqn (13.6), ⎛ 1 ⎞ ⎡ ( Δp )max ⎤ Vmax = ⎜ − β⎥ ⎝ α y ⎟⎠ ⎢⎣ wη ⎦

(13.7)

The duration of pressing τ, during which Δp increases to (Δp)max, is, from Eqn (13.7),

τ=

Vmax ⎛ 1 ⎞ ⎡ ( Δp )max ⎤ =⎜ − β⎥ ⎝ α yw ⎟⎠ ⎢⎣ wη w ⎦

(13.8)

Pressing

447

Case 2. Suppose that Δp is constant. From Eqn (13.6), we obtain V = 0 at the beginning of pressing, i.e. dV ⎞ Δp w (t = 0 ) = ⎛ = ⎝ dt ⎠ 0 ηβ

(13.9)

Furthermore, Eqn (13.6) can be written in the form ⎛ dV ⎞ α yV + β = Δp [ ] ⎝ dt ⎠ η or ⎛ dV ⎞ (α yV ) + ⎛ dV ⎞ β = Δp ⎝ dt ⎠ ⎝ dt ⎠ η After separating the variables, we obtain ⎛ Δp ⎞ dVα yV + dVβ = ⎜ ⎟ dt ⎝ η⎠ After integration (V = 0 → V = V when t = 0 → t = τ), 2 Δp ⎛ 2β ⎞ V2 +⎜ V =τ ⎝ α y ⎟⎠ ηα y

(13.10)

V 2 + 2CV = Kτ

(13.11)

or

where C (m) = β/αy and K (m2/s) = 2 Δp/ηαy. From the quadratic equation (13.11), the value of V = V(τ) can easily be calculated: V=

−2C + 4C + 4 Kτ 2

(13.12)

The empirical constants of pressing can also be determined from Eqn (13.11): 1 2C τ V =V ⎛ ⎞ + ⎝K⎠ K

(13.13)

In a plot of τ/V vs V, the intercept is 2C/K and the slope is 1/K. Therefore ⎛ 1 ⎞ ⎛ 2C ⎞ intercept × ⎜ = K = 2C ⎝ slope ⎟⎠ ⎝ K ⎠ where 1/slope = K.

(13.14)

448

Confectionery and Chocolate Engineering: Principles and Applications

13.3

Cocoa liquor pressing

Cocoa powder is manufactured by hydraulic pressing of finely ground cocoa liquor, which must have been made from well-winnowed, high-grade cocoa beans. Hydraulic presses can automatically and accurately obtain the required fat content in the cocoa cake. In modern horizontal presses, 12–14 pots are mounted in a horizontal frame and each pot is provided with a metal filter screen supported on plates in the pot. The press, when closed, is filled automatically with hot cocoa mass under pressure, and a proportion of the free cocoa butter is removed during the filling operation. A higher pressure (up to 400–450 bar with older machines and 800–850 bar with modern machines) is then applied. The ultimate fat content of the cocoa cake is controlled by the time cycle, the weight of cocoa butter and the distance of travel of the ram. The pressing resistance of the cake is strongly dependent on the size distribution and moisture content of the cocoa liquor that is being pressed. The moisture content is critical; the optimum interval is 0.8–1.5 m/m% but the recommendations of the manufacturer of the press are always decisive. For efficient pressing, cocoa liquor has to be coarser than for chocolate: 98–99% through a 200-mesh (74 μm) sieve. In modern practice, the best results are obtained from pressing at 95–105°C, since – as the Ruth equation (Eqn 13.1) shows – if the dynamic viscosity decreases, the pressure difference needed for pressing decreases also: dV/dt ∼ Δp/η. However, higher temperatures than this range result in inferior, strongly flavoured cocoa butter without any improvement in yield. The output is strongly dependent on the fat content of the cocoa cake: c. 3 tonne/h for 24% and 0.5–0.7 tonne/h for 10–12%. Example 13.1 A horizontal cocoa-pressing machine has 12 pots of 15 kg cocoa mass capacity; the total capacity is 12 × 15 kg = 180 kg. The cocoa mass, which has a cocoa butter content of 54 m/m%, is pressed to a cocoa butter content of 16 m/m%. The temperature of pressing is 80°C. The pressing resistance can be calculated assuming that the resistance of the filter (β) can be neglected. The time required to press a charge is τ = 600 s (= 10 min). Let us calculate the amount of cocoa press cake and cocoa butter. The mass balance of cocoa butter is 0.54 = 0.16 x + 1 − x → x =

1 − 0.54 ≈ 0.55 1 − 0.16

i.e. the proportion of press cake is x ≈ 0.55 and the proportion of cocoa butter pressed is 1 − x ≈ 0.45. For 180 kg of cocoa mass, the amount of press cake is 180 kg × 0.55 = 99 kg and that of cocoa butter pressed is 81 kg. Consequently, the output is 6 × 99 kg = 594 kg of cocoa cake/h. The diameter d of a pot is 0.45 m; therefore, the distance of the ram from the filter screen at the start of pressing is (assuming a density of cocoa liquor of 984 kg/m3 at 80°C)

Pressing

H = 15 kg ×

449

4 ≈ 0.096 m 0.45 × 3.14 m 2 × 984 kg m 3 2

According to Rapoport and Sosnovsky (1951, pp. 171–179), the density of cocoa mass is 1075 kg/m3, the density of cocoa butter is 872 kg/m3 (both at 80°C) and the dynamic viscosity (centipoise) as a function of temperature T = 35–80°C can be described by logη = 2.2 − 0.0145T i.e. at 80°C, η (cocoa butter) ≈ 10.96 cP ≈ 0.011 Pa s. At the end of pressing, the height of the cocoa cake is hC = 15 kg × 0.55 ×

4 ≈ 0.048 m 0.452 × 3.14 m 2 × 1075 kg m3

At the end of pressing, the distance moved by the ram is hB = 15 kg × 0.45 ×

4 ≈ 0.049 0.452 × 3.14 m 2 × 872 kg m3

(It can be seen that H ≈ 0.096 m ≈ hC + hB = 0.097 m.) Since hB = Vmax = 0.097 m, w=

0.097 m hB = 600 s 600 s

Since β ≈ 0, and the pressure increases continuously up to (Δp)max = 400 × 105 Pa, Case 1 is applicable here. Applying Eqn (13.8),

τ=

Vmax ⎛ 1 ⎞ ⎡ ( Δp )max ⎤ =⎜ ⎝ α yw ⎟⎠ ⎢⎣ wη − β ⎥⎦ w

(13.8)

Since y = VC/VB = hC/hB = 0.048/0.049 ≈ 1, η = 0.011 Pa s and β = 2 × 107/m (an assumed value), Eqn (13.8) can be written as 400 × 105 Pa ⎛ 600 s ⎞ ⎡ ⎤ 600 s = ⎜ × − 2 × 107 m ⎥ ⎟ ⎢ ⎝ α 0.097 m ⎠ ⎣ 600 s ( 0.097 m × 0.011 Pa s ) ⎦ and α = (1/0.097)(2.474 × 1011 − 2 × 107) ≈ 2.55 × 1012 m−2. The usual ranges of values are α = 1012–1014 m−2 and β = (1.6–6.5) × 107 m−1. See Fábry (1995, p. 146).

Further reading Bauermeister (Probat Group). Technical brochures. Beckett, S.T. (ed.) (1988) Industrial Chocolate Manufacture and Use. Van Nostrand Reinhold, New York.

450

Confectionery and Chocolate Engineering: Principles and Applications

Beckett, S.T. (2000) The Science of Chocolate. Royal Society of Chemistry, Cambridge. Cakebread, S.H. (1975) Sugar and Chocolate Confectionery. Oxford University Press, Oxford. Carle & Montanari. Technical brochures. Meursing, E.H. (1983) Cocoa Powders for Industrial Processing, 3rd edn. Cacaofabriek de Zaan, Koog aan de Zaan, The Netherlands. Minifie, B.W. (1999) Chocolate, Cocoa and Confectionery, Science and Technology, 3rd edn. Aspen Publications, Gaithersburg, MD. Nemeth, J. and Horanyi, R. (1970) Untersuchungen uber die Teilchengrosse als Kennwert der Leistung schneller Klarzentrifugen. Periodica Polytechnica Ch XIV (2): 183–193. Pratt, C.D. (ed.) (1970) Twenty Years of Confectionery and Chocolate Progress. AVI Publishing, Westport, CT. Wieland, H. (1972) Cocoa and Chocolate Processing. Noyes Data Corp., Park Ridge, NJ.

Chapter 14

Extrusion

Contents 14.1 Flow through a converging die 14.1.1 Theoretical principles of the dimensioning of extruders 14.1.2 Pressure loss in the shaping of pastes 14.1.3 Design of converging die 14.2 Feeders used for shaping confectionery pastes 14.2.1 Screw feeders 14.2.2 Cog-wheel feeders 14.2.3 Screw mixers and extruders 14.3 Extrusion cooking 14.4 Roller extrusion 14.4.1 Roller extrusion of biscuit doughs 14.4.2 Feeding by roller extrusion Further reading

14.1 14.1.1

451 451 455 456 459 459 460 461 464 465 465 467 467

Flow through a converging die Theoretical principles of the dimensioning of extruders

First, it should be emphasized that both shear and extensional viscosity play an important role in flow through a converging die. The extensional viscosity ηE must not be confused with the volumetric viscosity ηV, which latter appears in the constitutive (Navier–Stokes) equation of isotropic compressible Newtonian fluids, t = 2η (Grad v ) + (ηv div v − p ) d s

(14.1)

where t is the deformation tensor, v is the velocity vector, δ is the unit tensor, p is the pressure (scalar), η is the shear (or dynamic) viscosity and the index ‘s’ refers to the symmetric part of the tensor Grad v. If the fluid is incompressible, div v = 0

(14.2)

the term containing the volumetric viscosity becomes zero, and Eqn (14.1) can be simplified: Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

452

Confectionery and Chocolate Engineering: Principles and Applications

t = 2η (Grad v ) − pdd s

(14.3)

Now, the volumetric viscosity is not taken into account. In the extensional deformation of incompressible bodies, the shape of the body is deformed but its volume is unchanged. However, flow through a converging die may occur with both incompressible and compressible fluids (div v ≠ 0). In Chapter 15, the case of incompressible fluids will be discussed exclusively. However, it should be mentioned that among the fluids used in the confectionery industry (chocolate, fat masses, cremes based on fondant, etc.), there are some types that are compressible (see Section 15.4.3). The literature on confectionery technology hardly discusses how to dimension extruders for shaping various masses made from fondant, fat compounds, etc. The reason for this deficiency is likely to be the fact that, on the one hand, an exact mathematical discussion of the questions that arise is possible only in some simple cases, and on the other hand, the solutions obtained in those cases are not sufficiently accurate. As a consequence, methods based on empirical values have been in widespread use. In the following, an attempt is made to summarize certain parts of this branch of science in a substantially simplified form, and to make it applicable in practice. Steffe (1996, Chapter 4) discussed in detail the extensional type of flow, which is of great importance in food process engineering. He described many rheological methods for determination of the extensional viscosity of several foods, and also the appropriate rheological calculations for obtaining relationships between the extensional flow rate and the extensional viscosity. Discussions concerning rheometry are beyond the scope of this book. Steffe (1996, Section 4.4) discussed calculation methods related to flow through a converging die. The following description briefly presents these methods. 14.1.1.1

Cogswell–Gibson method

Both Cogswell (1972, 1978, 1981) and Gibson (1988) assumed that the entry pressure drop δPen across an area of converging flow, from a circular barrel of radius Rb into a capillary die of radius R, was made up of two components, one related to shear flow (δPen.S) and one related to extensional flow (δPen.E): δPen = δPen.S + δPen.S

(14.4)

Cogswell (1981) experimentally determined the total entry pressure loss δPen by the procedure of Bagley (1957). In a Bagley plot, the pressure drop δP is plotted versus L/d at a constant value of the volume flow rate Q (m3/s), i.e. L δP vs ⎛ ⎞ ⎝d⎠

where Q = constant

(14.5)

or at a constant value of the shear rate D (s−1) = 4Q/πR3, where d = 2R is the diameter and R is the radius of the tube, i.e. L (δP ) vs ⎛ ⎞ where D = constant ⎝d⎠

(14.6)

Extrusion

453

In these plots, the intercept (L/d = 0) gives the entry pressure drop δPen at a given value of Q (or D). Visibly, the Bagley plot provides data on the effect of the geometry of the device; therefore, this effect cannot be ignored when the total entry pressure drop is calculated. The pressure drop due to shear flow associated with convergence is calculated from 3n n 3n + 1 ⎤ ⎡ 2 K ⎤ ⎡ ⎛ R ⎞ ⎤ δPen.S = Q n ⎡⎢ 1 − ⎢ ⎜ ⎟ ⎥ ⎣ 4 n ⎥⎦ ⎢⎣ 3n tan α ⎥⎦ ⎣ ⎝ Rb ⎠ ⎦

(14.7)

where α is the acute angle between the wall of the barrel and the axis of the tube. The shear viscosity parameters K and n are calculated for Ostwald–de Waele fluids from

τ = KD n

(14.8)

The pressure drop due to extensional flow associated with convergence is calculated from 3m m 2 K E ⎞ ⎛ tan α ⎞ ⎡ ⎛ R ⎞ ⎤ δPen.E = Q m ⎛ 1− ⎜ ⎟ ⎥ ⎢ ⎝ 3m ⎠ ⎝ 2 ⎠ ⎣ ⎝ Rb ⎠ ⎦

(14.9)

The extensional-viscosity parameters KE and m are calculated for Ostwald–de Waele fluids from

τ E = KE D m

(14.10)

(The Cogswell equations and the Gibson equations relate to Ostwald–de Waele fluids.) The steps of the calculation are: (1) From the Bagley plot, δPen is determined. (2) Then δPen.S is calculated with the help of K and n, which are determined from shear rheological data. (3) From Eqn (14.4), δPen − δPen.S = δPen.E is calculated. (4) Finally, from Eqn (14.9), the parameters KE and m are calculated by regression analysis of ln (δPen.E) vs ln Q: ⎧ 2 K E ⎞ ⎛ tan α ⎞ m ⎡ ⎛ R ⎞ 3 m ⎤ ⎫ ln (δPen.E ) = m ln Q + ln ⎨⎛ ⎢1 − ⎜ ⎟ ⎥ ⎬ ⎩⎝ 3m ⎠ ⎝ 2 ⎠ ⎣ ⎝ Rb ⎠ ⎦ ⎭

(14.11)

From the slope, m can be calculated (Q is known), and from the intercept, ln{ }; KE is the only unknown, and thus it also can be calculated. (Step 4 can be regarded a special method of rheometry for determination of the extensional-viscosity parameters.) This method can be successfully applied up to die angles α ≈ 45°. In the region 10° < α < 45°, both shear and extensional flow are present in some proportions. If α < 10°, shear flow is dominant. However, if α > 45°, materials form their own convergence pattern (Cogswell 1981); this is a characteristic feature for food fluids, which have a relatively high yield stress. A special case is α = 90°, for which Cogswell (1972) provided simple relations.

454

Confectionery and Chocolate Engineering: Principles and Applications

14.1.1.2

Gibson method

The Gibson method uses another approach. The pressure drop due to shear flow associated with convergence is calculated from 3n n 3n + 1 ⎤ ⎡ 2 K sin3 n α ⎤ ⎡ ⎛ R ⎞ ⎤ 1 δPen.S = Q n ⎡⎢ − ⎜ ⎟ ⎥ ⎢ ⎣ 4 n ⎥⎦ ⎢⎣ 3nα 3 n +1 ⎥⎦ ⎣ ⎝ Rb ⎠ ⎦

(14.12)

where α is in radians. The pressure drop due to extensional flow associated with convergence is calculated from ⎧ 2 ⎞ ⎡ (sin α ) (1 + cos α ) ⎤ m ⎡ ⎛ R ⎞ 3 m Φ ⎤ ⎫ δPen.E = Q m K E ⎨⎛ ⎢ ⎥⎦ ⎢1 − ⎜⎝ Rb ⎟⎠ + 4 m ⎥ ⎬ 4 ⎩⎝ 3m ⎠ ⎣ ⎦⎭ ⎣

(14.13)

where Φ = f(m, α) is a tabulated integral. Using these expressions, the calculation steps of the Gibson method are similar to those above. The linearized form of Eqn (14.13) provides the value of m from the slope. Since m and α are known, the value of Φ is given by the tabulated integral and, consequently, δPen.E can be calculated in the region 0 < α < 90°. The Gibson method is suitable for determining the extensional viscosity for the full range of die angles up to 90°. 14.1.1.3

Empirical method

Steffe (1996) presented also an empirical method, according to which the extensional viscosity may be estimated using a standard material. Assuming that the shear contribution to the pressure loss is small, an average extensional viscosity can be calculated. From Eqn (14.10),

ηE = K E D m −1

(14.14)

Assuming that m = 1, then

ηE = K E

(14.15)

From Eqns (14.13) and (14.15), a simpler form for δPen is obtained: ⎛ Dπ R3 ⎞ ηEC1 δPen.S = ⎜ ⎝ 4 ⎟⎠

(14.16)

That is, the average extensional viscosity is given by

ηE =

CδPen D

where D = 4Q/πR3 is the apparent wall shear rate, and C is a constant.

(14.17)

Extrusion

455

It is assumed that the dimensionless constant C is a function of the system geometry but independent of the strain rate and the rheological properties of the sample. The numerical value of C could be estimated using a standard Newtonian material with a known value of ηE.

14.1.2

Pressure loss in the shaping of pastes

Machikhin and Machikhin (1987, Chapter 6) discussed the case of flow in short tubes for which the ratio of the length L to the diameter D is less than 10. This is exactly the case in which a mass is shaped by extrusion. The total decrease in pressure Δpt can be expressed as Δpt = − Δph + ∑ Δpvis + ∑ Δpg + ΔpE + Δpkin + Δpin + Δprel

(14.18)

Let us consider the various terms in this sum: Δph: this is the hydrostatic pressure, which helps the flow in machines with a horizontal structure (the minus sign shows this fact). Σ Δpvis: this term takes into account the viscous losses involved in front of the matrices and in the matrix channels. Σ Δpg: this term originates from the sudden geometrical change of the surface and tubes of the extruder. ΔpE: this term takes into account the pressure loss originating from extensional deformation during the shaping process: ΔPE = ηE ε E′

(14.19)

where ηE is the extensional viscosity (Pa s) and ε′VE is the extensional deformation rate (s−1). Δpkin: this term takes into account the pressure loss associated with increasing the kinetic energy of the mass that is being shaped. Δpin: this term takes into account the pressure loss associated with accelerating the mass in a non-steady state, i.e. the pressure loss related to the inertia of the mass. Δprel: this term takes into account the pressure loss originating from pressure relaxation in the mass: Δprel = p0 − f ( p0 , t )

(14.20)

where p0 is the initial pressure (at t = 0). The determination of Δprel is difficult, and therefore the approximation p0 = Δpt/2 is generally used, which assumes that p0 = Δpt/2 decreases to Δprel when the relaxation is stopped. As a result, the following equation is obtained from Eqn. (14.20): Δpt Δp = ∑ ∑ Δp − f ⎛ t , t⎞ ⎝ 2 2 ⎠

(14.21)

where Σ ΣΔp denotes the known terms on the right-hand side of Eqn (14.18). Equation (14.21) can be solved for Δpt/2 if the function f is known.

456

Confectionery and Chocolate Engineering: Principles and Applications

For viscoplastic Bingham fluids, the pressure loss associated with shaping (Δp) can be calculated for tubes of circular cross-section: 8Lτ 0 ⎛ 2 Lηpl ⎞ 4Q Δp = ⎜ + C1 ⎟ ⎛ 3 ⎞ + + C2 ⎝ R ⎠ ⎝ R π⎠ 3R

(14.22)

where L is the length of the tube (m), ηPl is the plastic viscosity of the fluid that is being shaped (Pa s), R is the radius of the tube (m), C1 is a material constant (Pa s), Q is the flow rate (m3/s), τ0 is the yield stress of the fluid (Pa) and C2 is a material constant (Pa). Machikhin and Machikhin (1987, Chapter 6) cited some products for which the values of the constants are in the region 77 ≤ C1 (Pa s) ≤ 2570 and 7850 ≤ C2 (Pa) ≤ 63 700. Example 14.1 L = 0.4 m, R = 0.03 m, ηPl = 15 Pa s, τ0 = 5 Pa, Q = 10−3 m3/s, C1 = 2000 Pa s and C2 = 50 000 Pa. From Eqn (14.22), 1.5 4 × 10 −3 ⎡ ⎤ 8 × 0.4 × 5 Δp = ⎛ 2 × 0.4 × + 2000⎞ ⎢ 3 −6 ⎥ + 3 × 0.3 + 50 000 ⎝ ⎠ 0.03 ⎣ 3 × 10 × 3.14 ⎦ ≈ 1.46 × 105 Pa = 1.46 bar To determine the material constants, the following relationship is taken into consideration: ∂ ( Δp ) ⎛ 2 Lηpl ⎞ 4 =⎜ + C1 ⎟ 3 ⎝ R ⎠Rπ ∂Q Consequently, from a Bagley plot of Δp vs Q, ⎛ 2 Lηpl ⎞ 4 Slope = ⎜ + C1 ⎟ 3 ⎝ R ⎠Rπ

(14.23)

and Intercept =

8L τ 0 + C2 3R

(14.24)

The unknown material constants C1 and C2 can be calculated from the slope and intercept, respectively.

14.1.3

Design of converging die

The rheological properties of plastics used for moulding are similar to those of the fluids used in the confectionery industry (e.g. Bingham or Ostwald–de Waele fluids); consequently, knowledge from this field can be applied in the area of confectionery. Sors et al. (1981) analysed three cases; for simplicity, circular sections were considered:

Extrusion

457

(1) a section with a uniformly converging diameter (convex velocity diagram); (2) a section where the velocity increases in proportion to the distance travelled (linear velocity diagram); (3) a section where there is a uniformly accelerating flow (concave velocity diagram). Any other profile can be referred back to a circular section by introducing the hydraulic radius rh =

2T K

(14.25)

where rh is the hydraulic radius, T is the area of the cross-section and K is the circumference of the cross-section. However, the following considerations need to be taken into account: (1) If identical inlet and outlet cross-sections are assumed in all three cases, then, naturally, the inlet and outlet velocities will be identical too. (2) In the case of a profile with a uniformly converging diameter, the flow velocity, starting from the inlet cross-section, will first increase at a lower rate, and then, on approaching the outlet, this rate will rapidly increase. (3) If the flow velocity increases in proportion to the distance travelled, the diameter of the profile will decrease at a fast rate in the vicinity of the inlet cross-section, but will hardly decrease at all in the vicinity of the outlet cross-section. Thus, the profile is ‘trumpet-shaped’. (4) In the case of a uniformly accelerating flow, the flow velocity will increase rapidly at the inlet cross-section, but the rate of increase will be less in the vicinity of the outlet cross-section. The section will converge steeply at the inlet, while its convergence will be flat in the vicinity of the outlet section. The longitudinal section is trumpet-shaped in this case too. Since the friction that arises in a flowing melt increases in proportion to the velocity squared, it can be stated that a profile with a uniformly decreasing diameter is very unfavourable, whereas the two trumpet shapes are much more favourable. Taking these aspects into account, the case of a uniformly accelerating flow will be considered here. Trumpet-shaped longitudinal sections can be plotted very simply (Fig. 14.1). If the inlet diameter is D, the outlet diameter is d (n = D/d) and the diameter at a distance z from the diameter D is denoted by y, then y = D−

(D − d ) z L

(14.26)

where z = ZL. It is known that the velocity of a uniformly accelerating motion at a crosssection at a distance of z = ZL from the inlet cross-section is

νz = ν 0 + 2az

(14.27)

where a (a constant) is the acceleration. Without giving further details, we state that a function f = f(n; z; d) can be obtained, so that

458

Confectionery and Chocolate Engineering: Principles and Applications

Length of extruder

60°

4 3

2 D/2 = 4.5d 1

P 1

2

3

4

5

6

7

0

d

0.1 0.2

1/y 2

0.3 0.4 0.5

Flow velocity (n = 9)

0.6 0.7 0.8 0.9 1.0

0

0.5

Z

1.0

Fig. 14.1 Plot of adaptor zone profile and flow velocity [reproduced from Sors et al. (1981), by kind permission of Akadémiai Kiadó, Budapest].

y=

nd 1 + ( n2 − 1) Z

(

= Kd

(14.28)

)

and K = n 1 + ( n2 − 1) Z is constant if n and Z are fixed. First the value of n = D/d is determined (in the example illustrated in Fig. 14.1, n = 9) and measured at a certain point on the y axis of a coordinate system (see Fig. 14.1). From this point, taking the viscosity of the melt into consideration, a straight line is drawn at an angle of 120–150° to the positive direction of the x axis (i.e. 60–30° to the positive direction of the y axis). (An angle of 150° is used with higher viscosities and 120° at lower viscosities, according to Sors et al. (1981).) Starting from the intersection point of this straight line and the straight line y = 1 (= d), both straight sections are divided into an equal number of parts (up to seven parts in the direction of the +x axis), and the points with opposite numbers are connected: x axis 6 5 4

y axis 1 2 3

Extrusion

459

These connecting lines cover a curve of second order. The lower diagram in Fig. 14.1 shows the value of 1/y2 as a function of Z if n is fixed (in the present example, n = 9). For more details, see Sors et al. (1981, Section 2.2) and Sors and Balázs (1989). The latter gives a collection of examples and designs. Machikhin and Machikhin (1987, pp. 219–221) dealt with dimensioning in the case of a fluid of Ostwald–de Waele type as well, and with cases where the extrusion die was of quadrangular section and the shape of the extruder was conical. This discussion started from the fact that, according to Prager and Hodge (1956), the pressure acting in the direction of the z axis can be calculated from π Δp = ⎛1 + ⎞ τ 0 ⎝ 2⎠

(14.29)

The surface of the velocity distribution is a rotational paraboloid in a cylinder of radius R − δ, where the width of the laminar boundary layer is

δ=

ηpl u τ0

(14.30)

and u is the linear velocity of the mass at the surface of the boundary layer. This method assumes rotation-free flow, and is intended to calculate conditions such that the linear velocity of the mass at the surface of the boundary layer and inside the die are the same. The cross-section of the die proposed is a rotational parabola of third degree. However, while the trumpet profile proposed by Sors et al. (1981) is biconcave, Machikhin and Machikhin (1972, pp. 219–221) proposed a biconvex profile.

14.2 14.2.1

Feeders used for shaping confectionery pastes Screw feeders

Several types of feeders are frequently used in the confectionery operations of shaping by extrusion, transportation and dosage of masses. Machikhin and Machikhin (1972, pp. 221–229) discussed these questions in detail. Neglecting the complicated hydrodynamic calculations, the results can be summarized as follows. The flow rate of a spiral feeder can be calculated from ⎡ π 2 D 2 nH sin ϕ cos ϕ ⎤ ⎡ πDH 3 sin2 ϕ ⎤ ⎡ p2 − p1 ⎤ Q=⎢ Fdψ − ⎢ ⎥ ⎥ ⎢⎣ L ⎥⎦ Fa 2 12 η ⎣ ⎦ ⎣ ⎦

(14.31)

where Q is the flow rate or transport output (m3/s), n is the rate of revolution (s−1), D is the outside diameter of the screw (m), H is the depth of the thread (m), ϕ is the ascent angle of the thread (radians), Fd is a correction coefficient depending on the shape of the thread, Ψ is a correction coefficient taking into account the non-Newtonian behaviour of the fluid, η = ηPl + τ0 /γ ′ is the dynamic viscosity of the fluid (Pa s) (where γ ′ is the shear rate), p2 is the pressure in the screw (Pa), p1 is the pressure outside (Pa), L is the length of the screw and Fa is a correction coefficient related to the shape of the flow.

460

Confectionery and Chocolate Engineering: Principles and Applications

The corresponding values of H/W, Fd, Fa and Ψ have been tabulated (see Machikhin and Machikhin 1972, p. 219, Table 93). The ratio H/W relates to tubes of square crosssection (where H = height and W = width). Example 14.2 Let us calculate the flow rate if the parameters of the screw are D = 0.1 m, H = 0.05 m, ϕ = 20°, η = 10 Pa s, p2 − p1 = 3 × 103 Pa, L = 0.4 m, n = 3 s−1, Fa = 0.74, Fd = 0.78 and Ψ = 0.697. ⎡ π 2 D 2 nH sin ϕ cos ϕ ⎤ ⎡ πDH 3 sin2 ϕ ⎤ ⎡ p2 − p1 ⎤ ψ Q=⎢ F − d ⎥ ⎢ ⎥ ⎢⎣ L ⎥⎦ Fa 2 12η ⎣ ⎦ ⎣ ⎦ 2 −2 −2 = 3.14 × 10 × 3 × 5 × 10 × 0.342 × 0.9397 × 0.6972 2 − (3.14 × 10−2 × 125) × 10−6 × 0.3422 × 3 × 103 × 0.74 (12 × 10 × 0.4) = (16.564 − 2.123) × 10 −4 m 3 s = 1.4441 × 10 −3 m3 s

For a twin screw, the flow output can be calculated from the formula Q = n ( 2 π − α ) ( πD tan ϕ − b ) ( D − H ) H −

(2π − α ) Dδ 3 tan ϕ ( p2 − p1 ) 12ηb

(14.32)

where b is the width of the screw in the axial direction (m), δ is the size of the gap (m) and α is the central angle, for which cos α = 1−

2H H2 + D − H 2 ( D − H )2

(14.33)

Example 14.3 Let us calculate the output of a twin screw which, in addition to those given in Example 14.2, has the parameters b = 5 × 10−3 m, δ = 10−3 m (tan 20° = 0.364) and cos α = 1 − 2 × 0.05/0.05 + 0.052/(2 × 0.052) = −0.5 → α = 120° = 2π/3: Q = 3 × ( 2 π − 2 π 3) (π × 0.1 × 0.364 − 5 × 10 −3 ) × 0.05 × 0.05 − ( 2 π − 2 π 3) × 0.1 × 10 −9 × 0.364 × 10 −3 (12 × 10 × 5 × 10 −3 ) = 3.42 × 10 −3 m 3 s

14.2.2

Cog-wheel feeders

The benefits of a cog-wheel feeder are uniform feeding, a high pressure and simple construction. For the materials used in the food industry, the vacuum created by the feeder is insufficient to supply the material into the space between the cogs, and therefore a constrained additional supply is applied (Machikhin and Birfeld 1969). The usual revolution rate is 5–20 min−1. The efficiency coefficient is defined by the formula

ϕ = 1−

Qg + Qf Qt

(14.34)

461

Extrusion

where Qg is the loss associated with the gap (kg/min), Qf is the loss associated with the feeder (kg/min) and Qt is the theoretical output of the feeder (kg/min). For dessert masses, the dependence of the efficiency on the revolution rate n (min−1) is given by the equation

ϕ = 0.8 + 0.017 ( min )2 n − 0.0009 ( min ) n2

(14.35)

The theoretical output can be calculated from a formula given by Yudyin (1964): Qt = 2 πρbnm 2 ( z + sin2 α ) × 10 −6

(14.36)

where ρ is the density of the mass (kg/m3), b is the width of the cogs (mm), n is the revolution rate (min−1), m is the modulus (mm), z is the number of cogs and α is the coupling angle of the cogs. Example 14.4 The parameters of a cog-wheel feeder are ρ = 1200 kg/m3 (the density of the mass), b = 30 mm, n = 8 min−1, m = 3 mm, z = 15 and α = 20° (sin 20° = 0.3420). The output (see Eqn 14.36) is Qt ( kg min ) = 2 × 3.14 × 1200 × 30 × 8 × 32 × (15 + 0.34202 ) × 10 −6 = 738.21 kg min The optimal region of the feeding efficiency ϕ is n = 8–15 min−1; the maximum of ϕ can be obtained when n = 9.5 min−1. For more details, see Machikhin and Machikhin (1987, Chapter 6).

14.2.3

Screw mixers and extruders

Screw-type mixers, or extruders, of which there are numerous design configurations, are widely employed for extrusion of various confectionery products consisting of fondant mass, fats, milk powder, soy meal, etc. These products contain some proportion of molten phase, and from this point of view both fats and fondant mass behave similarly: they tend to melt under the effect of heat or pressure, and the resulting molten phase forms the liquid phase of the products in question. (The word ‘fondant’ means ‘melting’, although the phenomenon that takes place in fondant mass as a result of the effect of heat is actually solution.) The main application area of extruders in the confectionery industry is shaping of rope in order to produce centres for coated products. Mixing and material extrusion can be accomplished in single or double rotating-screw machines. In single-screw mixers, the mixing quality is determined by the total shear deformation for a given material volume. The total shear deformation may be increased by decreasing the height of the screw channel, by varying the angle of pitch of the helix, and/or by increasing the countercurrent or sinking through an increase in the head resistance. There are two types of flow in the channel of a single-screw mixer: longitudinal (along the helical axis of the channel) and transverse (circulatory). Bernhard (1962) and Lukach

462

Confectionery and Chocolate Engineering: Principles and Applications

et al. (1967) recommended Moore’s equation, among others, for determining the shear strains. The shear strain parallel to the screw axis (in the z direction) is

{

L B Dz = ⎛ ⎞ A − ⎝ h⎠ 1− a

}

(14.37)

where L is the length of the screw, h is the height of the screw channel, ψ is the helix angle, a is the ratio of the countercurrent rate to the rate of forced flow, and A and B are dimensionless functions of the ratio y/h of the base and height of the thread channel. The shear strain perpendicular to the screw axis is L ⎧ A B tan ψ ⎫ Dx = ⎛ ⎞ ⎨ − ⎬ ⎝ h ⎠ ⎩ tan ψ 1− a ⎭

(14.38)

By integrating these equations over the cross-sectional area of the flow, an average value of the shear strain is obtained. Twin-screw mixers are more effective because they process material by the action of intermeshing screws. The material streamlines are interrupted when they pass through a zone of low velocity gradient with low mixing effects. However, overall, the bulk of the material volume enclosed between the wall of the casing and the screw surfaces undergoes a highly efficient form of mixing. A successful modification of the screw mixer has working members comprising one screw and an interrupted helix (teeth and gaps are located on the helix). The material between the meshing teeth is thus subjected to longitudinal, axial and radial shear. Another variation is the ‘KO kneader’, which can handle light and medium-stiff pastes. In this design, a worm runs along a horizontal casing fitted with teeth. The worm is interrupted at regular intervals by gaps. The shaft not only turns but also reciprocates in the axial direction, so that the teeth periodically clean the gaps on the casing. This periodic reciprocation minimizes the material flow in one direction only. However, the intensive shearing strain in this machine is periodic, and the gaps tend to reduce the effective shear stress. Large surface areas of material are exposed as a result of the shear stresses generated by large torques. This limits the viscosity or rigidity of material to be mixed. For the theoretical principles of design, the work of Cheremisinoff (1988, pp. 828–857) can be recommended, from which the calculation of dosing-zone capacity and head resistance presented here has been taken. The volumetric flow capacity through the head is directly proportional to the pressure drop and inversely proportional to the material’s dynamic viscosity: Q=

KΔp η

(14.39)

where K (with dimensions of volume) is a coefficient depending on the head geometry. On the other hand, the effective material flow is the difference between the flow injection and the countercurrent flow + leakage: ⎛ Δp ⎞ Q = α n − ⎜ ⎟ (β + γ ) ⎝ η⎠

(14.40)

Extrusion

463

where n is the rotation rate (rot/s), α is the coefficient of the injection flow, β is the coefficient of the countercurrent flow and γ is the coefficient of leakage. These coefficients (with dimensions of volume) can be determined for screws of both variable pitch and variable depth. Combining Eqn (14.29) with Eqn (14.30), we obtain Q=

α Kn K +β +γ

(14.41)

The capacity of a screw mixer can be readily determined from information about K for various screw-mixer heads. For each head, there is an optimum screw-channel geometry that provides the maximum capacity. For details of the design of heads and the optimization of the working conditions of an extruder, see Cheremisinoff (1988, pp. 840–843). Example 14.5 Let us calculate the values of K, α, β and γ if Δp = 5 × 105 Pa, η = 50 Pa s, Q = 10 × 10−6 m3/s, n = 4 s−1, β = 0.1K and γ = 0.02K. According to Eqn (14.40), Q = 10 × 10 −6 m 3 s = K × 5 × 105 Pa 50 Pa s i.e. K = 10−9 m3 → β = 0.1 × 10−9 m3 and γ = 0.02 × 10−9 m3. According to Eqn (14.41), Q = 10 × 10 −6 m 3 s =

α nK K (1 + 0.1 + 0.02 )

or

α n = 1.12 × 10 −5 m3 s → α = 0.28 × 10 −5 m3 14.2.3.1

Modelling of single- and twin-screw mixers

Silin (1964) applied a similar theory to develop design correlations for screw mixers. The study of the flow was based on the following dimensionless relation: Eu = CRe n (l d )

m

(14.42)

where Eu = Δp/ρv2 (the Euler number); Re = dvρ/η (the Reynolds number); Δp is the pressure drop in the channel; ρ is the density of the material melt; v is the average velocity of the material; η is the dynamic viscosity of the material; l and d are the length and inner diameter, respectively, of the screw; and C is a constant. Further references are Janssen and Smith (1975), Spreckley (1987), Treiber (1988) and Klasen and Mewes (1991).

464

Confectionery and Chocolate Engineering: Principles and Applications

14.3

Extrusion cooking

Extrusion cooking is a process of forcing a material to flow under a variety of conditions through a shaped hole (die) at a predetermined rate to achieve various products (Dziezak 1989). Extrusion cooking of foods has been practised for over 50 years. Initially, the role of an extruder was limited to mixing and forming macaroni and ready-to-eat (RTE) cereal pellets. Today, the food extruder is considered as a high-temperature–short-time bioreactor that transforms raw ingredients into modified intermediate and finished products (Harper (1989). Extrusion cooking technology is used today for the production of pasta, breakfast cereals, breadcrumbs, biscuits, crackers, croutons, baby foods, snack foods, confectionery items, chewing gum, texturized vegetable protein (TVP), modified starch, pet foods, dried soups and dry beverage mixes (Linko et al. 1983). For the benefits of extrusion cooking, see Wiedemann and Strobel (1987). Three major types of extruders are used in the food industry: piston extruders, rollertype extruders and screw extruders (Thorz 1986). A piston extruder can consist of a single piston or a set of pistons that deposit a precise amount of product onto conveyers or trays. Piston extruders are primarily used for forming product shapes and are used in both confectionery and bakery production facilities. One example of the function of a piston extruder is where cake, cookie or muffin dough is deposited onto a sheet with the use of a wire cutter, or into individual cups in an already shaped pan, and is then conveyed to an oven for baking. Another example is in the depositing of fillings into doughnuts, cupcakes and chocolate-type products. Roller extruders are used to form the shape of a product. A roller extruder consists of two counter-rotating rollers that turn at similar or differential speeds. This process is also referred to as ‘calendering’ in the dough industry. The roller surfaces can be smooth to create a long thin strip or can be perforated to form the dough into shaped products. The roller extruder can be altered to control the width of the layer of product moving between the rollers. Products such as crackers and hard cookies can be formed by creating the desired shape within the rollers and conveying the dough between the rollers. The dough is forced into the pattern on the roller and is then conveyed to an oven for baking. Excess dough can be collected and reused. Products such as graham crackers and saltines are created using smooth roller systems to form thin layers. Screw extruders utilize single, twin or multiple screws rotating within a metal cabinet called the barrel. The screws convey the material forward and through a small orifice called a die, which can take many shapes and sizes. Several external parameters such as screw speed and configuration, the temperature of the barrel, the size and shape of the die, and the length of the barrel affect the properties of the final product. The first food application of extrusion occurred in the 1800s in the production of ground sausage and meats, stuffed into natural casings. The pasta press was introduced in 1935 for forming and shaping pasta dough. Screw extruders providing both cooking and forming capabilities came in around 1950 for the production of animal feeds. Because of the demand for pre-cooked cereals and starches in the 1960s, large machines were required. These larger cooker extruders led to new applications in RTE cereals and snack products, as well as expanding the drypet-food market. Pre-cooked infant foods were also developed. Improvements to the cooker extruder in the 1970s led to the development of soft, moist pet foods and co-extrusion; the use of two extruders, one for cooking and another for forming, was

Extrusion

465

developed. The 1980s saw expanded use of the twin-screw extruder owing to its versatility and productivity (Harper 1989). For further details, see Dziezak (1989) and Gray and Chinnaswamy (1995).

14.4 14.4.1

Roller extrusion Roller extrusion of biscuit doughs

Biscuit doughs are dense solid–liquid pastes that exhibit complex rheological behaviour. The sheeting of doughs using roll mills is a prominent operation in the production of biscuits. Rolling is well understood in the context of many operations: the body of literature describing the production of metal sheets is considerable (e.g. Orowan 1943), and the rolling of polymer melts, called calendering, has also been studied extensively. The roller extrusion of food materials such as doughs, however, is a broad area in which current practice still frequently relies on approximations and pilot plant testing. Castell-Perez (1992) and Rao (1992) described the sizeable gap that exists between the material models of practical use in the analysis of forming processes and the observations of the complex history-dependent response of the materials themselves. First, Levine et al. (2002) developed a model describing the two-dimensional calendering of finite-width sheets. The main objective of this work was to take into account both the lengthwise and the widthwise flow occurring during the processing of viscous polymeric materials, the rheology of which can be characterized by flow curves with a power law. The model appropriately describes two-dimensional calendering, and predicts reasonably well the sideways deformation of the calendered material. Results show that as narrower and thicker sheets are fed to the rollers, the sideways spread of the sheet increases. The same effect is observed with larger-diameter rollers. Furthermore, as the feed becomes narrower, the maximum pressure exerted on the material, as well as the forces developed and the power consumed per unit width, decreases. Although the model does not describe the real flaking process, which is characterized by unsteady-state conditions, it could be used as a first approximation to solve this problem. For the mathematical details, see Levine et al. (2002). Peck et al. (2006) studied the roller extrusion of short and hard biscuit doughs. Short doughs are relatively crumbly materials because of their high percentage of fat (25–30 wt% of the flour or more) and are poorly cohesive and fairly inelastic. Hard doughs are more elastic and glutinous because they contain less fat and are mixed more aggressively to develop the three-dimensional network of the dough. In this work, Peck et al. (2006) studied the analysis of hot rolling of metals given by Orowan (1943), which was used to describe the rolling behaviour of stiff ceramic pastes with moderate success. Peck et al. determined that this analysis could not be used for calendering, because as the rolled sheet exits the nip (the narrowest point between the rollers), the thickness of a hot metal sheet does not change, unlike that of a biscuit sheet, which increases. In their experiments, the measured exit sheet thickness was slightly larger than the estimated nip separation (e.g. by up to 6%), which could be the result of elastic effects. Although the standard plasticity model for sheet metal was modified to include strain rate dependence, the behaviour of neither of the doughs could be adequately described by this model. In response to the problems encountered in characterizing these materials, Benbow and Bridgwater (1993) developed an approximate method of material characterization via

466

Confectionery and Chocolate Engineering: Principles and Applications

capillary extrusion. In this approach, the total pressure drop required to extrude a soft solid is composed of two terms. The first term represents the work due to the paste undergoing quasi-plastic deformation associated with the contraction. The second term represents the work due to the effects of stresses in the region of the die surface. The Benbow–Bridgwater approach therefore describes the deformation behaviour of a soft solid in terms of two distinct properties of the material, namely the bulk deformational response and the wall slip response. The Benbow–Bridgwater analysis assumes that shearing of the dough occurs only at the die wall. However, internal shearing may be expected as the wall yield stress approaches the shear yield stress of the dough. Horrobin (1999) provided a relation for the uniaxial yield stress for the system, σy ≈ 0.82σ0 (assuming that the extrusion pressure does not depend on the extrudate velocity), where y is perpendicular to the direction x of the sheet velocity. Applying a von Mises yield criterion, the plug flow assumption would be therefore be valid if ⎛ 0.82 ⎞ σ > τ + βV n 0 ⎝ 3 ⎠ 0

(14.43)

(the die wall shear stress term), where β is a parameter in the second term of the Benbow– Bridgwater equation, V is the mean velocity of the material and n is the index in the second term of the Benbow–Bridgwater equation, used with V to introduce a strain rate dependence of the yield and shear stresses. If a value of σ0 = 0.1 MPa is employed, then internal shearing is unlikely, even at the highest extrudate velocity. However, the wall yield stress term rises to about 80% of the bulk shear yield stress; therefore, it may be concluded that internal shearing could be having an influence, and so an alternative characterization approach, such as that of capillary analysis, must be considered. A power-law-fluid fit makes a corresponding calculation for the die entry region during extrusion possible. The key principle behind this analysis is the separation of the die entry pressure drop into two components, the first due to the simple shear viscosity and the second to the extensional viscosity (Steffe 1996), by using the Gibson equations (see Eqns 14.13 and 14.14). For the mathematical details, see the references cited above. The rheological characteristics of the two doughs were quantified by a power-law model. The results indicated that the doughs were not ideally suited to a quasi-plastic analysis. The power-law parameters varied noticeably between the doughs, but both were strongly shear-thinning (power-law shear indices of 0.25 and 0.5) with large extensional viscosities, as the relatively high Trouton numbers (45–141) also show. It was ascertained that the underpredictions of the power-law model were in contrast to those provided by the modified plasticity approach. In the die entry region of the ram extrusion tests, the material undergoes a large reduction in cross-sectional area (by over 98%), and so its extensional response governs the extrusion behaviour. Therefore, the plasticity model predicts larger values than those predicted by the power-law-based fluid mechanics model. The rolling of the two doughs was modelled by Peck et al. (2006) using the analysis of Levine (1996) incorporating a power-law rheology. The model predictions were too low. The reasons can be summed up thus: as the high extensional viscosities of the doughs acted to increase the measured roller torque and separating force, in line with observations, the deviation of the materials from a power-law-based description became evident

Extrusion

467

from the impossibility of generating power-law parameters to satisfy both the torque and the force comparisons simultaneously. The characterization of doughs requires further work for full use to be made of a more complete model of the operation. In addition, a finite element solution such as that developed by Levine et al. (2002), which includes shear and extensional deformation terms, seems to be an area that would benefit from further work.

14.4.2

Feeding by roller extrusion

Feeding by roller extrusion was discussed by Machikhin and Machikhin (1987, Chapter 6). For calendering, Ardichvili gave the following relation for the mass flow Q considered as the velocity of a plane (cited by Machikhin and Machikhin 1987, Section 6.4): dp h3 Q (m2 s) = ν h − ⎛ ⎞ ⎝ dx ⎠ 12η

(14.44)

where v (m/s) is the circumferential velocity of the cylinders, h (m) is the gap between the cylinders, dp/dx (Pa/m) is the pressure gradient created by the cylinders at right angles to the direction of movement of the plane, and η (Pa s) is the dynamic viscosity of the material rolled. The feeding capacity is given by Q ( m 2 s ) = ν hmin K =

vhminQact Qther

(14.45)

where hmin (m) is the minimum gap, Qact (m2/s) is the actual capacity and Qther (m2/s) is the theoretical capacity. When the machine is fed with bread dough, the following relations between the driving moment M (N m/s) and the other parameters can be used: M vs hmin : M = a1hmin exp ( −a2 hmin )

(14.46)

M vs n : M = b1 + b2 n

(14.47)

M vs L : M = c1 + c2 L

(14.48)

where a1, a2, b1, b2, c1 and c2 are coefficients of appropriate dimension, which are experimentally determined; n (s−1) is the rotation rate; and L (m) is the width of the matrix.

Further reading Almond, N. et al. (1991) Biscuit, Cookies and Crackers. Elsevier Applied Science, London. BEPEX/HUTT. Technical brochures. Berman, G.K., Machikhin, Y.A. and Lunyin, L.N. (1972) Flow of visco-plastic food mass in extruder (in Russian). Khlebopekarnaya i Konditerskaya Promyshlennost 3: 18–20. Biscuit and Cracker Manufacturers Association (1970) The Biscuit and Cracker Handbook. Biscuit and Cracker Manufacturers Association, Chicago, IL.

468

Confectionery and Chocolate Engineering: Principles and Applications

Bloksma, A.H. (1990) Dough structure, dough rheology and baking quality. Cereal Foods World 35 (2): 237. Ellis, P.E. (ed.) (1990) Cookie and Cracker Manufacturing. Biscuit and Cracker Manufacturers Association, Washington, DC. Faridi, F. (ed.) (1994) The Science of Cookie and Cracker Production. Chapman and Hall, New York. Georgopoulos, T., Larsson, H. and Eliasson, A-Ch. (2004) A comparison of the rheological properties of wheat flour dough and its gluten prepared by ultracentrifugation. Food Hydrocolloids 18 (1): 143. Hasegawa, T. and Nakamura, H. (1991) Experimental study of the elongational stress of dilute polymer solutions in orifice flows. J Non-Newtonian Fluid Mechan 38: 159–181. Kulp, K. (ed.) (1994) Cookie Chemistry and Technology. American Institute of Baking, Kansas. Letang, C., Piau, M and Verdier, C. (1999) Characterization off wheat flour-water doughs. Part I: Rheometry and microstructure. J Food Eng 41 (2): 121. Manley, D. (1998) Biscuit, Cookie and Cracker Manufacturing Manuals (Vol. 1: Ingredients; Vol. 2: Biscuit Doughs; Vol. 3: Biscuit Dough Piece Forming; Vol. 4: Baking and Cooling of Biscuits; Vol. 5: Secondary Processing in Biscuit Manufacturing; Vol. 6: Biscuit Packaging and Storage). Woodhead, Cambridge. Smith, W.H. (1972) Biscuit, Crackers and Cookies. Applied Science Publishers, London. Sollich. Technical brochures. Wade, P. (1988) Biscuit, Cookies and Crackers. Elsevier Applied Science, London. Whiteley, P.R. (1971) Biscuit Manufacture. Applied Science Publishers, London.

Chapter 15

Particle agglomeration: Instantization and tabletting

Contents 15.1 Theoretical background 15.1.1 Processes resulting from particle agglomeration 15.1.2 Solidity of a granule 15.1.3 Capillary attractive forces in the case of liquid bridges 15.1.4 Capillary attractive forces in the case of no liquid bridges 15.1.5 Solidity of a granule in the case of dry granulation 15.1.6 Water sorption properties of particles 15.1.7 Effect of electrostatic forces on the solidity of a granule 15.1.8 Effect of crystal bridges on the solidity of a granule 15.1.9 Comparison of the various attractive forces affecting granulation 15.1.10 Effect of surface roughness on the attractive forces 15.2 Processes of agglomeration 15.2.1 Agglomeration in the confectionery industry 15.2.2 Agglomeration from liquid phase 15.2.3 Agglomeration of powders: Tabletting or dry granulation 15.3 Granulation by fluidization 15.3.1 Instantization by granulation: Wetting of particles 15.3.2 Processes of fluidization 15.4 Tabletting 15.4.1 Tablets as sweets 15.4.2 Types of tabletting 15.4.3 Compression, consolidation and compaction 15.4.4 Characteristics of the compaction process 15.4.5 Quality properties of tablets Further reading

15.1 15.1.1

469 469 472 472 473 474 475 477 478 479 479 481 481 481 482 482 482 483 484 484 485 486 488 492 492

Theoretical background Processes resulting from particle agglomeration

In studying particle agglomeration, Rumpf (1974) performed fundamental investigations. The present discussion of the theoretical background is to a great extent based on his results. Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

470

Confectionery and Chocolate Engineering: Principles and Applications

Table 15.1 Binding mechanisms and technical operations of dry and wet agglomeration [reproduced from Rumpf (1974) by kind permission of Wileyhyphen;VCH Verlag GmbH amp; Co. KGaA and courtesy of Mrs Liselotte Rumpf]. Binding mechanism

Operation

Van der Waals forces Electrostatic forces Fluid bridges, capillary forces, binders

Agglomeration in dispersion or in bulk Building up agglomerate in dispersion or in bulk Shaping by extrusion Shaping rope by extrusion Compression by cylinders Tabletting Drying Heating (plastifying, sintering etc.)

Bridges of solid particles Mechanical interlocking

Dry

Wet

× ×

× × × ×

× × × × ×

× ×

Particle agglomeration (or aggregation) is the direct mutual attraction between particles (e.g. atoms or molecules) via van der Waals forces or chemical bonding. When particles in fluid collide, there is a chance that they will attach to each other and become a larger particle. There are three major physical mechanisms of formation of aggregates: Brownian motion, fluid motion forced by shear and differential settling forced by gravitation. Table 15.1 summarizes the binding mechanisms and technical operations for dry and wet agglomeration. The practical ways of forming agglomerates are the result of formation of solid bridges: (1) (2) (3) (4) (5)

by by by by by

sintering: molecular diffusion caused by thermal effects; chemical reactions; melting in the case of thermoplastic materials; additives (e.g. limestone in ore briquettes); crystallization of soluble materials.

If an elastic material of unit cross-sectional area (1 m2) is subjected to a tensile force and as a result it breaks, two new surfaces will be created. If both sides of the broken material are of the same composition, then the work of cohesion is defined as Wcoh (N m m 2 ) = 2γ

(15.1)

where γ (N/m) is the surface tension of the material. In the situation in which two dissimilar materials of unit cross-sectional area are in intimate contact, there are intermolecular forces present that are lost when the materials are separated, i.e. an interfacial energy may have been present before the materials were split apart. As this energy is lost after the two surfaces are separated, we must subtract it from the energy used to create the two new surfaces according to the Dupré equation, Wadh (N m m 2 ) = γ 1 + γ 2 − γ 12

(15.2)

where Wadh (N m/m2) is the work of adhesion, γ1 and γ2 are the surface tensions of the two materials created by splitting, and γ12 is the interfacial tension (surface energy) of the material before splitting.

Particle agglomeration: Instantization and tabletting

15.1.1.1

471

Capillary forces in freely moving fluid surfaces

First, wet and dry aggregation must be distinguished. Three cases of wet aggregation were discussed separately by Rumpf (1958a,b): (1) fluid bridges exist between the solid particles; (2) capillary forces due to fluid surfaces in the holes between the particles cause aggregation; (3) fluid droplets saturated with solids attempt to unite in order to provide stabilization. Let P be a point on a curved surface of a freely moving fluid, the surface tension of which is γ, and let the surface at this point be characterized by two circles of radii R1 and R2 which are in planes perpendicular to each other. The pressure difference at point P is defined by the Laplace equation, 1 ⎞ ⎛ 1 Δp = γ ⎜ + ⎝ R1 R2 ⎟⎠

(15.3)

If both radii are oriented towards the inside of the curved (concave) surface, their signs are positive and the force generated is oriented towards the inside of the surface. If the signs are different because the orientations of the radii are opposite (i.e. we have a saddle surface), then Δp is decreased. In Case 1 above, the capillary forces are oriented towards the fluid surface at the solid– fluid–gas interface, and this additionally involves a decreased pressure in the network of fluid bridges. Both of these effects cause an attraction between the solid particles. In Case 2, the holes between the solid particles are saturated with fluid, and the capillary forces of the fluid affect the surface of the granulate (the entire assembly of particles) only. However, since the fluid surface is concave (viewed from the gas side) around the particles, attractive forces of a certain strength bond the particles together. In Case 3, the fluid surface is convex. Consequently, and in contrast to Case 2, the binding forces are lost, and the surface tension is the only force holding the droplets together. As a result of the effect of this surface tension, two fluid droplets are inclined to unite. 15.1.1.2

Cohesion and adhesion without freely moving fluid surfaces

(1) Bonds by viscous binder. Since no equilibrium of the forces defined by the Laplace equation can be involved in the case of viscous fluids, their effect is not important. However, the binding forces of binders exceed that defined by the Laplace equation; additionally, some binders harden later. (2) Bonds by adsorption layers. An adsorption layer of water can be established by a high pressure (e.g. during briquetting), which is not thicker than 30 Å (30 × 10−10 m). The attraction between such layers is rather strong. 15.1.1.3

No material bridges between solid particles

(1) Molecular forces: the effective distance of covalent forces is very short; therefore, they play practically no role in aggregation. However, van der Waals forces, with a range of c. 100 Å (=10−8 m), have some effect.

472

Confectionery and Chocolate Engineering: Principles and Applications

(2) The effect of electrostatic forces must not be entirely neglected in studying aggregation. 15.1.1.4

Mechanical interlocking

The surfaces of solid particles are not smooth, and are full of ‘lock and key’ sites at which they can be linked together. Adhesion stimulates such couplings.

15.1.2

Solidity of a granule

The solidity of a granule is characterized by a tensile strength σt related to the mean crosssectional area of a granulate consisting of particles. It is supposed that: • The number of particles in the cross-section is high. • The particle size distribution in both the cross-sectional area and the entire granulate is the same. • A representative mean binding force of effective size H may characterize the solidity of the granule. The following calculations relate to globular particles of various size, which the granule is assumed to be composed of. According to Rumpf (1974),

σt =

(1 − ε ) kH πd

2

≈

2H d2

(15.4)

where σt is the tensile strength; is the fraction of hole volume in a granule (0 < ε < 1); k is the coordination number, indicating how many neighbouring particles are in contact with the particle studied; H is a representative binding force; and d is the diameter of a globular particle. Equation (15.4) can be applied to the most frequently encountered values of these parameters.

15.1.3

Capillary attractive forces in the case of liquid bridges

According to Batel (1956), a good approximation can be obtained from the formula H ≈ ( 2.2 − 2.7 ) γ d

(15.5)

and

σt =

2.2 (1 − ε ) kγ πd

(15.6)

On the basis of this, Rumpf (1974) recommended

σt ≈

4.4γ d

(15.7)

Particle agglomeration: Instantization and tabletting

473

b

d

H

d

d 3.0

H

2.5 a H/dg

2.0

0.1

1.5

0.05

1.0 10−4

0.5

−5

0

5×10 0

0.005 0.0005 0.05

0.01

0.001 0.10

0.15

0.20

a/d Fig. 15.1 Effect of distance between particles. The curves are labelled with values of Vliq/Vsol; see text for details [reproduced from Rumpf (1974) by kind permission of Wiley-VCH Verlag GmbH & Co. KGaA, and courtesy of Mrs Liselotte Rumpf].

Example 15.1 In the case of water, γ = 72 dyn/cm = 72 × 10−5 N/(10−2 m) = 72 × 10−3 N/m. If d = 1 μm = 10−6 m, then σt ≈ 4.4 × 72 × 103 N/m = 3.168 × 105 N/m – this is a considerable value. Figure 15.1 demonstrates the effect of the distance between the particles. In a model with two spherical particles of diameter d separated by a distance a, the central angle of the bridge is 2β and the contact angle is δ if the surface tension of the fluid is γ. The surfaces of the particles are assumed to be entirely smooth. The variables in the plot are two dimensionless numbers, H/dγ plotted versus a/d. Along each of the curves, Vliq/Vsol is constant (where Vliq is the volume of the liquid bridge and Vsol is the volume of the solid particles). The intercept on the H/dγ axis relates to the values when a/d → 0; this gives the maximum value of H. In the case of contact between the particles, a = a0 ≈ 4 Å and H = Hmax (for particles with a smooth surface).

15.1.4

Capillary attractive forces in the case of no liquid bridges

The surface of the solid particles is saturated with fluid but it retains its concave form. The decreased pressure binding the particles together is determined by pt =

γ m

(15.8)

where m = ε/[S(1 − ε)] is the mean hydraulic radius. Here, S is the specific surface area of the particles per unit volume. For spheres of radius r, S=

4 πr 2 3 6 = = 4 πr 3 3 r d

(15.9)

474

Confectionery and Chocolate Engineering: Principles and Applications

As a result, pt =

6γ (1 − ε ) εd

(15.10)

However, the tensile strength is dependent on the proportion of solid matter:

σ t = ε pt = γ (1− ε ) S

(15.11)

For globular particles,

σt =

6γ (1 − ε ) d

(15.12)

A comparison of the two effects represented by Eqns (15.6) and (15.11) can be made as follows:

ρ = σt

(fluid bridges ) Sdπ = σ t (saturation ) 2.2 k

(15.13)

Since in general kε ≈ 3.1 (constant),

ρ

6 πε ≈ 2.75ε 3.1 × 2.2

(15.14)

Consequently, this ratio is proportional to the volume of holes relative to the particles.

15.1.5

Solidity of a granule in the case of dry granulation

Whereas in wet granulation the range of the particle size distribution is about 1–100 μm and the characteristic size (60–80%) is about 60 μm, in dry granulation this range is that of very fine powders (particle sizes < 1 μm), and the applied pressure may be rather high. For such small particles, van der Waals forces have to be taken into account. Hamaker (1937) determined a formula for the van der Waals forces on the basis of the London–Heitler theory, H=

Ad 24a 2

(15.15)

where A (J) is a constant, d is the diameter of the particles (spheres) and a is the distance between the two spheres. Equation (15.15) has been proved valid for the range a < 1000 Å. For the range a > 2000 Å, the theory of Casimir and Polder (1948) and Lifschitz (1955) gave the equation H=

Bd 36a3

where B (J m) is a constant.

(15.16)

Particle agglomeration: Instantization and tabletting

475

10

Tensile solidity st (105 Pa)

5 d=

0.0

1μ

1

m

0.5 d=

0.1 μm

0.1 0.05

d= 0.01 0.1

2

1μ m 3

4

5

6 7 8 9 10

a (distance between particles) (10−9 m) Fig. 15.2 Van der Waals forces. d = particle diameter [reproduced from Rumpf (1958a) by kind permission of Wiley-VCH Verlag GmbH & Co. KGaA, and courtesy of Mrs Liselotte Rumpf].

Krupp (1967) gave a formula for the van der Waals forces H vdW

Ld 8πa 2

(15.17)

where L denotes the Lifschitz–van der Waals constants. Figure 15.2 presents the theoretical tensile solidity of granules as a function of the distance between two spheres a and the diameter of the spheres d (Rumpf 1958a). From Fig. 15.2, it is evident that in dry granulation, high solidity can be achieved only by the use of high pressure.

15.1.6

Water sorption properties of particles

It is observed that dry granulation can be performed more easily in a wet than in a dry environment, i.e. if the water content w of the bulk material consisting of particles is below a typical value wlow, dry granulation is hard to perform, if it can be performed at all. However, if wlow < w < wup, where wup is an upper limit, dry granulation can easily be performed. The explanation is evident: the adsorbed water is located in the cavities of the rough surfaces of the particles, and as a result the shape of the particles becomes more or less spherical. The effect is double: the van der Waals forces produce an additional attraction between the spheres, and the distance a between two spheres is reduced, which increases the attraction (see Eqn 15.15). The thickness of a monomolecular water layer (slow) is about 3 Å (3 × 10−10 m), and the upper limit of the layer thickness (sup) is about 30 Å; namely, if the water layer is thicker, the case of a freely moving fluid has to be considered (i.e. slow → wlow and sup → wup). Rumpf (1974) gave an equation for the water content w of a particle with a rough surface:

476

Confectionery and Chocolate Engineering: Principles and Applications

6 w = ⎛ ⎞ sq ⎝d⎠

(15.18)

where w is the water content per unit volume of solid, s is the thickness of the water layer (m), q is the roughness factor and d is the particle diameter (m). For a better understanding, we note that 6/d = S/V, i.e. the specific surface area of a particle of volume V; the product (6/d)s gives the volume of the water layer on a particle, which is then corrected by factor q for roughness. Example 15.2 Let us calculate the thickness of a water layer if the characteristic particle size is d = 5 × 10−6 m, the water content of a sucrose powder is 0.02 m/m% (c. 40% relative humidity), the density of sucrose is about 1600 kg/m3 and q = 1.2. w = 0.02 kg water kg sucrose = 0.02 × 1.6 m3 water m 3 sucrose = 0.02 × 1.6 From Eqn (15.18), 6 6 0.02 × 1.6 = ⎛ ⎞ sq = ⎛ × 10 −6 ⎞ s × 1.2 → s ≈ 2.22 × 10 −8 m = 222 Å ⎝d⎠ ⎝5 ⎠ [A relative-humidity curve for sucrose was published by Junk and Pancoast (1973, p. 14)]. If d = 1 × 10−6 m, then w = 0.02 × 1.6 = 6 × 106 × 1.2 × s → s ≈ 44.4 Å This value is close to the limit of total coverage by a water layer. It should be noted that finely powered sucrose can be relatively easily dry-compressed. The tensile solidity of a granule can be calculated (Rumpf 1974) as follows. The area A on which the cohesive tension acts is approximately s A = π ⎛ max ⎞ ⎝ 2d ⎠

(15.19)

where smax is the maximum value of the distance between two spherical particles. Then the cohesive force is s H = σ coh A = π ⎛ max ⎞ σ coh ⎝ 2d ⎠

(15.20)

and the theoretical tensile solidity is

σ theo,t =

2H s = σ coh π ⎛ max ⎞ ⎝ d d ⎠

where σcoh is the cohesive tension.

(15.21)

Particle agglomeration: Instantization and tabletting

477

The concept of a theoretical tensile solidity supposes that there is no distance between the particles, i.e. a = 0, which makes Eqns (15.15), (15.16) and (15.17) divergent. The size of the theoretical tensile solidity is probably rather high (> 108 Pa), but because of defective sites it is actually much lower. The size of smax is about 30 Å.

15.1.7

Effect of electrostatic forces on the solidity of a granule

A regular arrangement of positive and negative charges may not be assumed in granulation; consequently, the following calculations concern a maximum value of the electrostatic forces. However, these effects cannot be entirely neglected; this means that the charges of different sign that are certainly formed during granulation will not neutralize each other entirely. This is taken into account by a factor of 0.2905 and also by assuming that neighbouring particles have a repulsion that decreases the attraction between the two particles studied. According to Coulomb’s law, QQ 2a H = − ⎛ 1 2 2 ⎞ ⎛1 − ⎞ ⎝ L ⎠⎝ d ⎠

(15.22)

where L is the distance between the centres of the charges Q1 and Q2, of spherical shape. If a << d, then L ≈ d, and H ≈−

Q1Q2 d2

(15.23)

If −Q1 = Q2 = ϕπd 2

(15.24)

where ϕ is the surface density of charge, then H ≈ (ϕπd )

2

(15.25)

In a network of particles, H eff = 0.2905H

(15.26)

As a result,

σt =

2 H eff ≈ 5.7ϕ 2 d2

(15.27)

Rumpf (1974) proposed, for the maximum surface density of charge, that

ϕ max ≈ 1.6

(

)

dyn cm ≈ 0.506

By using Eqn (15.27),

(

N

)

m

(15.28)

478

Confectionery and Chocolate Engineering: Principles and Applications

σ t.max ≈ 1.459 N m 2 which is a very low value. Bodenstedt (1952) measured the highest electric charging in powders of cereals (explosion danger!), although he estimated the value of ϕmax to be about five times lower than the value estimated by Rumpf (1974). Rumpf (1974) calculated the maximum distance between two particles (dmax) within which the weight of a particle is compensated (or exceeded) by electrostatic attraction: Q1Q2

(a + d )

2

=

(ϕπ 2 d )2

⎛ d3 ⎞ = π ⎜ ⎟ ρg ⎝ 6⎠ (a + d ) 2

(15.29)

where ρ = 1000 kg/m3 for the sake of simplicity. From Eqn (15.29), K d −d = a

(15.30)

where K = 6πϕ2/1000 g. From Eqn. (15.30), ∂a ⎛ 1 ⎞ = ∂d ⎝ 2 ⎠

K −1 = 0 d

(The second derivative shows that this is a maximum.) It can be stated that the electrostatic force exceeds the weight of a particle within approximately 123 μm under the conditions assumed. Since the electrostatic attractive forces are not sensitive to the surface roughness of particles (see later), their additional effect in accelerating the agglomeration process is important in dry systems.

15.1.8

Effect of crystal bridges on the solidity of a granule

Rumpf (1974) calculated this effect using the equation ⎛ ρ ⎞ σ t ≈ xf ⎜ ⎟ (1 − ε ) σ X ⎝ ρX ⎠

(15.31)

where x is the concentration of the dissolved substance X from which the granulate is built up (m/m), f is the water content of the crystallized substance X before drying (m/m), ρ is the density of substance X, ρx is the density of substance X inside at crystal bridge, ε is the hole volume ratio in the granulate and σX is the tensile solidity of substance X inside the crystal bridge. An approximate calculation shows that σt << σX. If ρ/ρX ≈ 1, x = 0.3, f = 0.02 and (1 − ε) = 0.65, then

σ t ≈ 0.3 × 0.02 × 0.65σ X ≈ 0.0039σ X

(15.32)

If the order of σX is 105 Pa (= 1 bar), then σt ≈ 390 Pa. In agglomeration, the strongest attractive binding effect is exerted by fluid bridges.

Particle agglomeration: Instantization and tabletting

479

1000

Tensile solidity st (105 Pa)

100

D:

I : Briquetting, material interlocking

Si

nt

er

s theo,t 10

g

C

:C

3

B:

ap

illa

va

n

1 A:

0.1

in

ry

de

II : Crystallized salts fo r

ce

rW aa

s

ls

va

n

de

rW aa

ls

0.01 0.01

0.1

10 100 1 d (size of particle) (10-6 m)

1000

Fig. 15.3 Theoretical solidity of a granule [reproduced from Rumpf (1958a) by kind permission of Wiley-VCH Verlag GmbH & Co. KGaA, and courtesy of Mrs Liselotte Rumpf].

15.1.9

Comparison of the various attractive forces affecting granulation

On the basis of the theoretical considerations previously discussed, Rumpf (1958a) summarized the values of the theoretical solidity of granules (Fig. 15.3). The upper part of this figure (‘I. Briquetting, material interlocking’) (σt ≥ 3 × 105 Pa) shows the range of forces that operate in briquetting and sintering. In this range, the particle size plays a relatively small role. The ranges shown by oblique lines concern the various types of forces between the particles: A: range of very weak van der Waals forces, for which w < wlow, where w is the water content (see Section 15.1.6); B: range of van der Waals forces, for which wlow < w < wup; C: range of capillary forces (δ = 0, see Fig. 15.1); D: range of sintering, briquetting and material interlocking. These values, although they are based on theoretical considerations, have proved basically correct.

15.1.10

Effect of surface roughness on the attractive forces

Let us consider a particle of diameter d, on the surface of which there are hemispheres of radius r to represent surface roughness. The distance between two particles is a, and the minimum distance is a0 = 4 Å. Figure 15.4 presents the maximum values of the forces acting between smooth-surfaced spherical particles of diameter d. Hmax is calculated assuming a0 = 4 Å. The parameters relating to Fig. 15.4 are:

480

Confectionery and Chocolate Engineering: Principles and Applications

1

es idg

id

br

0.01

e nd

s

ce

als

for

a rW

rs

cto

Va

u nd

o

lc

ca

i ctr

ln on -c

e

El

tri ca

Hmax (10-5 N)

u Liq

on du cto rs

0.1

El ec

0.001

0.0001 0.1

1

10

100

d (diameter) (10-6 m) Fig. 15.4 Comparison of forces acting between particles [reproduced from Rumpf (1974) by kind permission of Wiley-VCH Verlag GmbH & Co. KGaA, and courtesy of Mrs Liselotte Rumpf].

Liquid bridges: β = 20°, γwater = 0.072 N/m, δ = 0 (see Fig. 15.1). Van der Waals forces: L = 5 eV (see Eqn 15.17). Electrical conductors: surface potential U = 0.5 V. Electrical non-conductors: density of elementary charge ψ = 102 e/μm2 = 1014 e/m2, which corresponds to an electric field of about 20 000 V/cm (close to the dielectric breakdown field of air). Evidently, in the range below 100 μm, the effect of the electrostatic attractive forces can be neglected compared with the van der Waals forces and the effect of liquid bridges if the particle surfaces are smooth. The roughness of the particle surface changes this situation: because the effective distance of the van der Waals forces is short, they are much more sensitive to surface roughness (r). If we study the change of H as a function of r, a minimum can be observed in the interval 0.01–0.1 μm. This fact may be exploited in practice: agglomeration may be hindered by addition of a powder of high particle fineness. Liquid bridges are much less sensitive to surface roughness. If β (see Fig. 15.1) is not too small, for example β ≈ 20°, the small protrusions contribute to the distance between the particles. The smaller the angle β, the larger the distance. If β ≈ 2.5°, which a value of r ≈ 0.1 μm corresponds to, the liquid bridges will be torn apart, and they will remain between particles less than 0.1 μm in size only. If β < 2.5°, capillary condensation becomes increasingly important. If β ≈ 1° (the corresponding value of r is ≈ 0.05 μm), the capillary condensation on the peaks of the protrusions is pronounced, and the forces of the liquid bridges exceed the van der Waals forces. The electrostatic forces, in the case of both conductors and non-conductors, are hardly influenced by surface protrusions. Assuming a density of elementary charge

Particle agglomeration: Instantization and tabletting

481

ψ = 102 e/μm2 = 1014 e/m2, the electrostatic attractive forces between spherical particles of diameter 50 μm are stronger than the van der Waals forces over the entire interval of r. For further references, see Schubert (1973, 1979), Zorli (1988) and Borho et al. (1991).

15.2 15.2.1

Processes of agglomeration Agglomeration in the confectionery industry

It is reasonable to study the various types of agglomeration (frequently called ‘aggregation’) taking the continuous phase into consideration. In the following discussions, the term agglomeration is used when the product is not powder-like, i.e. its size is of the order of centimetres. The term granulation is used when the product consists of small granules, i.e. powder-like products with pieces having a size of the order of millimetres or smaller. In this understanding, the initial operations of both agglomeration and granulation are mostly comminution and mixing (blending); the actual operation in which agglomeration or granulation is performed is shaping, and is characteristic of each particular type of product. Chapters 6 and 7 deal in detail with comminution and mixing, respectively; the majority of sweets are a blend of comminuted components.

15.2.2

Agglomeration from liquid phase

• Fat crystallization. The continuous phase of the dispersion, also containing another ingredient (sucrose, cocoa, etc.), is molten fat; agglomeration is a result of crystallization of the fat in moulds. It is similar to sintering, but it takes place as a result of cooling. Various chocolate and compound products are manufactured in this way. This topic is discussed in Chapter 10. • Sucrose crystallization. Sucrose is crystallized from aqueous solution, and other soluble components, mainly glucose and/or invert syrup, control the crystallization. All fondant products are produced by this method. The ‘sintering’ effect is induced by cooling, similarly to the previous case. This topic is discussed in Chapter 10. • Agglomeration under the effect of gelling/foaming agents. An aqueous solution of sucrose, glucose syrup and gelling or foaming agents (plus other ingredients) solidifies under the effect of attractive forces between the molecules of the gelling/foaming agent and dissolved carbohydrates. Confectionery jellies and foams are manufactured in this way. Since solutions of carbohydrates are produced in these operations, ‘comminution’ means dissolving these carbohydrates in water. This topic is discussed in Chapter 11. • Agglomeration by baking. All confectionery biscuits, wafers, cakes, etc. are produced in this way. The attractive forces in this type of agglomeration are essentially of chemical nature. (Some similarity between baking and sintering may perhaps be mentioned in this context.). • Granulation from liquid phase (wet granulation). This is the typical method of manufacturing various instant products, for example cocoa drink powders. The fluidized-bed method is typically used for this purpose. In addition, wet granulation frequently provides semi-products for tabletting as well. During wet granulation, particle development can occur in two ways:

482

Confectionery and Chocolate Engineering: Principles and Applications

–crystallization of sugars, etc. on solid seeds, which may be soluble or insoluble in water; –solidification of material without any preliminary seeding; this process generally takes place in other steps, during which crystals are formed as seeding kernels (e.g. crystallization of lactose during the powdering of milk).

15.2.3

Agglomeration of powders: Tabletting or dry granulation

In tabletting, a bulk powder is compressed without any wetting process; therefore, this operation is called dry granulation. In order to make a powder suitable for tabletting, previous handling may be necessary, during which the powder is wetted by an aqueous solution of binding agents (carbohydrates and/or gelling agents). The bulk material produced by wet granulation is then compressed. Because of its simplicity and high efficiency, dry granulation is a preferred method. Wet and dry granulation will be discussed further below, focusing on their use in confectionery practice. Further references are Schilp (1975), Hermann (1979), Leuenberger et al. (1980, 1981), Stahl (1983), Pahl (1985), Dötsch and Sommer (1985), Schmidt (1989), Heinz et al. (1989), Koch (1991) and Bakele (1992).

15.3 15.3.1

Granulation by fluidization Instantization by granulation: Wetting of particles

Instant cocoa and fruit powders are popular confectionery products that are manufactured by granulation. The essence of instantization is that capillary forces help in the wetting and disintegration of particles in contact with the aqueous medium (milk, water, etc.). Under the usual conditions, ‘non-instant’ particles float on the surface of a drink because the surface tension of the liquid compensates for gravitation, and although the soluble ingredients (sugars, etc.) slowly start to dissolve, this effect is not sufficient for penetration of the particles into the liquid. However, in the case of instant powders, the liquid phase penetrates into the capillaries of the particles, and this induces the immersion and disintegration of the particles. For the theoretical basis, see Section 5.3.3.2 (Capillary rise and flow dynamics, Eqns 5.19–5.23). Example 5.2 is related to instantization. Using a Schmidt–Enslin instrument (Pfalzer et al. 1973), the process of liquid penetration can be investigated, as a result of which plots of h2 vs t (lines) are obtained: h2 =

trγ LV cos θ 2η

(5.21)

The steeper the line, the quicker the penetration is. A commonly used quality parameter for wettability is the ratio h2/t (mm2/s) at given values of t (e.g. 10 s) and of θ (the apparent contact angle). If the line of a plot is broken, this indicates that the particles have two systems of cavities, which have different hydraulic radii: the external value rEx and the internal value rIn. The air penetration method (Carman 1956) is suitable for determination of the external hydraulic radius.

Particle agglomeration: Instantization and tabletting

483

According to Pfalzer et al. (1974), the wettability of porous solids is primarily determined by the external system of cavities: the larger the radius of this system, the quicker the penetration, i.e. the wettability of fine particles is poor and, consequently, the fine portion of the bulk powder becomes separated and re-agglomerated. The larger particles, about 2–3 mm in size, have excellent wettability because they contain large cavities. For details of the process of wetting and dissolution in instant foods, see Dörnyei (1981), and for the determination of the wettability of instant cocoa powder in water, see IOCCC Analytical Method 27 (1988b).

15.3.2

Processes of fluidization

A discussion of the theoretical background of fluidization is beyond the scope of this book, and the following references are recommended: Leva (1959), Davidson and Harrison (1971), Kunii and Levenspiel (1991) and Steinmetz (2001). Ormós (1975) summarized the various methods of granulation in a fluidized bed: • Partial melting of the particle surfaces, and then agglomeration and cooling (US Patent 3 514 510). • Preliminary wetting of particles and then drying, with no addition of solids (Henszelman and Blickle 1959). • Wetting of particles by vapour plus addition of a binding agent, and then agglomeration and drying (Hungarian Patent 152 386). • Increase in the size of the particle crust by addition of a solution or suspension of another ingredient, and solidification of particles by drying or cooling (Wurster 1960). • Wetting the particle surfaces with a solution of a binding agent, and then solidification by drying, cooling or chemical reaction (Contini and Atasoy 1966). For further details, see Pintauro (1972), especially for chocolate drink powders (pp. 46– 78), for natural sugars (e.g. brown sugars and fondant mixtures) and synthetic sweeteners (pp. 102–139), for soluble coffee and soluble tea (pp. 141–172) and for flour and cake mixes (pp. 175–219). Granulation takes place in the space around a grid. Fluidization is carried out by filtered, heated air coming from under the grid and by vibration. Dosing of the powder provides the kernels of the granules, which are wetted by a solution of binder sprayed from the top of a tower. The end product is transported out by air flow. Running is possible both in a charged and in a stationary state. (In view of the many kinds of machines, the details of the technological calculations are beyond the scope of this book.) The usual arrangement of a fluidized bed is shown in Fig. 15.5. For cocoa drink powders, Dörnyei (1981) proposed a ratio 2 : 1 of sucrose to cocoa powder; some deviations from this value are possible, but this value can be taken as a good guide. For further recipes and details of the technology and machinery, see Pintauro (1972). For further details on cocoa powder agglomeration, see Omobuwajo et al. (2000), Vu et al. (2003) and Mércia de Freitas and Caetano da Silva Lannes (2007). Further references include Schügerl (1974), Hein et al. (1982), Furchner et al. (1990), Schubert (1990), Uhlemann (1990), Bauckhage (1990), Troetsch (1991), Weber (1993), Ratti and Mujumdar (1995) and Ruiz-López et al. (2008).

484

Confectionery and Chocolate Engineering: Principles and Applications

Air Solution

Air (filtered, heated)

Spraying

Dosing of powder

Granulation

Product outlet

Grid Air for outlet of product

Air (filtered, heated)

Vibration Fig. 15.5

15.4 15.4.1

Aggregation in a fluidized bed.

Tabletting Tablets as sweets

Tablets can be regarded as hard sugar confectionery. The fact that boiling is not used in their manufacture makes tablet manufacture attractive and relatively cheap. An additional advantage is the high efficiency of tabletting machines. The surface of a tablet may be either coated, i.e. the tablet serves as a centre (‘corpus’) for a coated product, or uncoated. The most commonly used raw materials for tabletting are (flavoured) sucrose, dextrose, sorbitol and a mixture of various vitamins for use with these carbohydrates (e.g. enrichment with ascorbic acid, citric acid etc.).

Particle agglomeration: Instantization and tabletting

15.4.2

485

Types of tabletting

During tabletting, granular substances are compressed. Two types of tabletting can be distinguished: • In direct tabletting/compression, all the raw materials are blended and directly compressed after having been loaded into the tabletting machine. • In indirect tabletting, the raw materials are blended and mixed with surfactant agents, which wet the surface of the components and increase the interfacial forces, i.e. adhesive forces. This wetting process, which results in a granulate, is usually followed by a drying period, and finally the compression of the mixture. This is called wet granulation. Another method of indirect tabletting is dry granulation. 15.4.2.1

Manufacture of tablets by wet granulation

Granulation is still the most frequently used method of preparing a tabletting mixture. There are at least four different variations of the procedure: (1) granulation of the active substance (+ filler) with a binder solution; (2) granulation of a mixture of the active substance (+ filler) and a binder with the pure solvent. (3) granulation of a mixture of the active substance (+ filler) and a portion of the binder with a solution of the remaining binder; (4) granulation of the active substance (+ filler) with a solution of a portion of the binder followed by dry addition of the remaining binder to the finished granulate. A number of factors dictate which of the above methods is used. With many formulations, method 1 gives tablets with a shorter disintegration time and quicker release of the active substance than method 2 (Wan and Lim 1990). Also, in many cases, method 1 gives somewhat harder tablets than method 2. Method 3 is useful if method 1 cannot be used, as when the tabletting mixture lacks the capacity for the quantity of liquid required for the total amount of binder. If the disintegration time of a tablet presents a problem, it is worth trying method 4, mixing in about a third of the binder together with a lubricant and, last of all, the disintegrant. Methods 2 and 3 have proved best for active substances of high solubility, as the quantity of liquid can be kept small to avoid clogging the granulating screens. The capacity of the powder mixture to bind liquid is one of the parameters determined in wet granulation. Every powder mixture to be granulated has a different binding capacity. Water is nowadays the most commonly used solvent. Sometimes, if water cannot be used, as with effervescent tablets, active ingredients that are prone to hydrolysis, etc., ethanol or isopropanol is used as the solvent, although fluidized-bed granulation is preferred. However, this field is beyond the scope of this text. 15.4.2.2

Manufacture of tablets by dry granulation

Dry granulation is less widely used than wet granulation as a method for preparing tabletting mixtures. The best-known dry granulation technique is compaction. It is the method of choice whenever wet granulation cannot be used for reasons of stability and when the physical properties of any ingredient do not allow direct compression.

486

Confectionery and Chocolate Engineering: Principles and Applications

15.4.2.3

Direct compression

Direct compression is becoming ever more widely used in the confectionery industry because it is an easy technique. For good tabletting properties, the mixture of substances must fulfil a number of physical requirements. It must be free-flowing, it must not be prone to electrostatic charging, its crystals must not be too brittle and its compression characteristics must result in tablets of adequate hardness (Bühler 1993). 15.4.2.4

Types of machines for granulation

Two types of machine are used in the confectionery industry: • machines that perform building-up: these are the traditional panning machines; • machines that perform briquetting: these are stamping apparatuses (rotary tablet presses). Compaction by stamping machines will be discussed below; see also Landillon et al. (2008) and Djuric et al. (2009).

15.4.3

Compression, consolidation and compaction

These three concepts have a close connection with each other. Compression: the reduction in the bulk volume of a material as a result of the removal of the gaseous phase (air) by applied pressure. Consolidation: this involves an increase in the mechanical strength of a material resulting from particle–particle interactions. Compaction: the compression and consolidation of a two-phase (solid + gas) system due to an applied force. The stages of compaction are: (1) (2) (3) (4) (5)

particle rearrangement, or interparticle slippage; deformation of particles; bonding, or cold welding; deformation of the solid body; elastic recovery, or expansion of the mass as a whole.

15.4.3.1

Particle rearrangement

• This occurs at low pressures. • It reduces the relative volume of the powder bed. • Small particles flow into voids between larger particles, leading to a closer packing arrangement. As the pressure increases, relative particle movement becomes impossible, inducing deformation.

Particle agglomeration: Instantization and tabletting

15.4.3.2

487

Deformation mechanisms of materials

• Elastic deformation: the shape of the tablet is recovered. • Plastic deformation: no recovery of shape. • Brittle fragmentation: during application of the force, the tablet becomes fragmented, and after removal of the force the fragments retain the shape of the tablet. 15.4.3.3

Bonding

Solid bridges form directly between particles in the absence of any binding elements or additives. Intermolecular and electrostatic forces project beyond the particle surface as small discrete fields with very short-range order. Under the effect of surface tension and capillary suction, four types of contact can be established: direct contact, discontinuous contact, contact while saturated by fluid and contact while enclosed by fluid (the proportion of binding fluid increases in this sequence). For the compressive stress in the case of direct contact and globular particles of equal size,

σD =

3.1[(1 − ε ) ε ] γ x (1 + tan β 2 )

(15.33)

where σD is the tensile stress, ε is the ratio of void volume (0 < ε < 1), γ = 72 dyn/cm (= 72 × 10−3 N/m) is the surface tension of water, x is the diameter of a particle and β is the central angle of the fluid bridge. Example 15.3 ε = 0.4, x = 10−3 m, β = 60°, σD = ? From Eqn. (15.33), 0.6 ⎞ 72 × 10 −3 σ D = 3.1⎛ × −3 = 212.3 Pa ⎝ 0.4 ⎠ 10 × 1.577 In the case of saturated contact and globular particles of equal size, the following equation can be applied: 1− ε ⎤ γ σ S = 6 ⎡⎢ ⎣ d ⎦⎥ where d is the diameter of the granulate. See Tóth (1973, p. 717). Example 15.4 ε = 0.4, d = 10−3 m, σS = ? From Eqn (15.34), 1 − 0.4 σ S = 6 ⎡⎢ −3 ⎤⎥ × 72 × 10 −3 = 259.2 Pa ⎣ 10 ⎦ Mechanical interlocking is shape-dependent.

(15.34)

488

Confectionery and Chocolate Engineering: Principles and Applications

15.4.3.4

Deformation of the solid body

As the pressure increases, the bonded solid is consolidated towards a limiting density by plastic and/or elastic deformation. 15.4.3.5

Recovery

The compact is ejected, allowing radial and axial recovery. The elastic character of the material tends to cause the compact to revert to its original shape.

15.4.4

Characteristics of the compaction process

The steps of the compression cycle are ejection, compression, adjustment of weight and die filling. The pressure–volume relationships for this process are shown in Fig. 15.6. The curve A–B–A represents a quasi-static process in which the granules slip on each other and form a compact structure. Then the shapes of the granules are deformed; the energy consumption in this period of interfacial connections between the granules is high. However, the curve A–B–C represents a realistic process in which the tablet dilates (relaxation and recovery). In industrial practice, tabletting is rather quick, and therefore, the curve A–B–C describes the actual conditions, and dilatation must always be taken into account. A typical compression profile is shown in Fig. 15.7. Figure 15.8 shows typical crushingstrength vs pressure plots, comparing single and double compression. 15.4.4.1

Lubricant selection

Pressure

Figure 15.9 shows plots of ejection force vs compression pressure for use in choosing a lubricant. The three curves relate to three different lubricants. For any given value of compression pressure, the curves provide three different values of the ejection force, and the lubricant that gives the minimum value may be chosen.

A

B

C Shift of piston

Fig. 15.6

Plot of pressure vs volume in tabletting.

Particle agglomeration: Instantization and tabletting

489

Force (kN)

Displacement (mm)

15

0

0 0

Consolidation

0.10 Time (s)

Effective contact time

Fig. 15.7

Compression profile: plot of displacement vs time and force vs time.

2 Crushing strength (MPa)

Double compressing

Single compressing

0 0

300 Pressure (MPa)

Fig. 15.8 Plots of crushing strength vs pressure for single and double compression.

Ejection force (N)

1000

0 0

400 Compression pressure (MPa)

Fig. 15.9

Lubricant selection: plots of ejection force vs compression pressure for three different lubricants.

490

Confectionery and Chocolate Engineering: Principles and Applications

15.4.4.2

Side-pressure coefficient

An essential characteristic of the tabletting process is the side-pressure coefficient: side-pressure coefficient (ξ ) =

side pressure tabletting pressure

(15.35)

According to Pascal’s law, the pressure in a spreading fluid is the same in every direction. Consequently, for fluids, ξ = 1; for entirely rigid solids, ξ = 0. The pressure in a bulk sample of granules cannot spread unchanged in every direction since, on the one hand, the directions are not equivalent, and on the other hand, the equalization of pressure differences is hindered; therefore reversibility, which is an assumption of Pascal’s law, is an unacceptable approximation. According to Lunyin et al. (1976), for the tabletting of tea,

ξ = 1.5 × 10 −4W 2 + (1.7 × 10 −3 + 5 × 10 −5W ) p + 0.1415

(15.36)

where W is the water content (%) of the tea leaves (values 6–10.7%) and p is the tabletting pressure (MPa) (values 10–250 MPa). Example 15.5 Let us calculate the value of ξ if W = 10% and p = 20 MPa, and if W = 7% and p = 150 MPa. From Eqn (15.36),

ξ (W = 10%, p = 20 MPa ) = 0.1965 and

ξ (W = 7%, p = 150 MPa ) = 0.3596 For instant tea and coffee, Nazarov et al. (1973) published the following results: If p < 20 MPa, then ξ ≤ = 0.05. If p = 20–200 MPa, then ξ = 0.05–0.26 for instant tea and 0.05–0.21 for instant coffee. For various praline masses (at p = 0–2 MPa and 36–46°C), Nazarov et al. (1973) found that, at constant temperature t, if p is increased, ξ also increases and approaches 1. If both t and p are increased, then ξ increases rapidly. For example, ξ = 0.9 when t = 46°C and p = 0.15 MPa and when t = 38°C and p = 1 MPa. 15.4.4.3

Volume modulus

An important characteristic of materials for tabletting is the volume modulus ⎛ ΔV ⎞ EV = − p ⎜ ⎝ V0 ⎟⎠

(15.37)

where p is the pressure (MPa), ΔV is the decrease of volume under the effect of the pressure p (m3) and V0 is the initial volume of the material tabletted (m3).

491

Particle agglomeration: Instantization and tabletting

15.4.4.4

Change of density under the effect of compression

According to Nazarov et al. (1973), the Balshin equation is widely applied to describe the change of density under the effect of compression:

( ρ ρmax )m = p pmax

(15.38a)

where ρ is the density of the material compressed (kg/m3); ρmax is the maximum density of the material (kg/m3), related to the maximum pressure (pmax) of compression; p is the pressure of compression (MPa); and pmax (MPa) is the maximum pressure. Taking into consideration the fact that ρmax = M/Vmin and ρ = M/V (where Vmin is the volume of the sample when the pressure p is maximum, i.e. pmax, and V is the volume when the pressure is p), Eqn (15.38a) is an equation of state of solid matter, pmax (Vmin ) = pV m m

(15.38b)

where m is a constant. The logarithmic form of Eqn (15.38a) can easily be used, m log10 ρ = − log10 p + log10 pmax

(15.38c)

Equation (15.38a, b or c) can be applied to describe the density change of refined sugar (Krivcun et al. 1974), tea leaves (Lunyin et al. 1976) and instant tea and coffee (Nazarov et al. 1973) during tabletting. Khvedelidze (cited by Lunyin et al. 1976) studied the tabletting of tea, and determined that in the tabletting mould the density associated with the maximum compactness that can be reached is about 340–350 kg/m3. However, in a quasi-static compression the final (maximum) density is 1485 kg/m3. An approximate equation describing the change of density as a function of pressure and water content is

ρ=

1485 ( 6.49 − 0.49W + p ) 28.46 − 2.13W + p

(15.39)

where ρ is the density of the material compressed (kg/m3), p is the pressure in the tabletting process (MPa) and W is the water content (%). For instant green tea and instant coffee, the following empirical equation was recommended by Khvedelidze:

ρ = A+

p −10 m + np

(15.40)

where ρ is the density (kg/m3), p is the pressure in the tabletting process (MPa) (values 20–200 MPa), and A, m and n are constants (their values are shown in Table 15.2). Table 15.2 Values of constants for instant green tea and instant coffee. Product Instant green tea Instant coffee

A (kg/m3)

m

N

716 729

0.066 0.117

0.002 0.001

492

Confectionery and Chocolate Engineering: Principles and Applications

For recipes for tabletting see, for example, Bühler (1982) and Johnson (1974), and for compression of snack foods, see Mazumder et al. (2007).

15.4.5

Quality properties of tablets

The quality properties of tablets are viewed somewhat differently in food production than in pharmaceutical production. The common aspects are content uniformity, crushing strength, friability, weight, and shape uniformity. For effervescent mouthwash tablets, such parameters as disintegration time and dissolution time play an important role too. For references, see Pfalzer et al. (1973) and Koch and Sommer (1993).

Further reading Hemati, M., Flamant, G., Steinmetz, D. And Gauthier, D. (2001) Special issue – Euro fluidization III. Powder Technol 120 (1–2), 1. Nopens, I., Biggs, C.A. De Clercq, B. and Govoreanu, R. (2006) Advances in population balance modelling. Special issue, Chem Eng Sci 61 (1): 63–74. Pocius, A.V. (2002) Adhesion and Adhesives Technology: An Introduction, 2nd edn. Hanser, Munich. Salman, A.D. and Hounslow, M.J. (2005) Granulation across the length scales. Second International Workshop on Granulation. Chem Eng Sci 60 (14). Salman, A.D. and Hounslow, M.J. (2007) Fluidized bed applications. Chem Eng Sci 62 (1–2).

Part III

Chemical and complex operations: Stability of sweets

Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

Chapter 16

Chemical operations (inversion and caramelization), ripening and complex operations

Contents 16.1 Inversion 16.1.1 Hydrolysis of sucrose by the effect of acids 16.1.2 A specific type of acidic inversion: Inversion by cream of tartar 16.1.3 Enzymatic inversion 16.2 Caramelization 16.2.1 Maillard reaction 16.2.2 Sugar melting 16.3 Alkalization of cocoa material 16.3.1 Purposes and methods of alkalization 16.3.2 German process 16.4 Ripening 16.4.1 Ripening processes of diffusion 16.4.2 Chemical and enzymatic reactions during ripening 16.5 Complex operations 16.5.1 Complexity of the operations used in the confectionery industry 16.5.2 Conching 16.5.3 New trends in the manufacture of chocolate 16.5.4 Modelling the structure of dough Further reading

16.1 16.1.1

495 495 498 499 502 502 504 505 505 506 507 507 509 510 510 510 521 522 523

Inversion Hydrolysis of sucrose by the effect of acids

The hydrolysis of sucrose, or ‘inversion’, takes place according to the following chemical reaction: C12H22 O11 (sucrose ) + H2 O ( water ) = C 6H12 O6 (glucose ) + C 6H12O6 (fructose )

(16.1)

Comment: During this reaction, 342 g dry content of sucrose becomes 360 g dry content, i.e. 18 g water (5 m/m%) is chemically absorbed and accounts for the increase. Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

496

Confectionery and Chocolate Engineering: Principles and Applications

Table 16.1 Catalytic ability of various acids in inversion, according to Ostwald. Acid

Inversion ability (%)

HBr HCl HNO3 H2SO4 H2SO3 Oxalic acid H3PO4 Citric acid Maleic acid Lactic acid Acetic acid

111.4 100 100 53.6 30.4 18.6 6.2 1.72 1.27 1.07 0.5

This is a famous reaction because, in 1850, Wilhelmy worked out the principles of the kinetics of chemical reactions by studying it; see Erdey-Grúz and Schay (1954, p. 401). Inversion is stoichiometrically a bimolecular reaction but it is kinetically first-order since it takes place in a dilute aqueous solution (the water is in excess, i.e. the concentration of water remains practically unchanged). Consequently, Eqn (15.1) can be written in symbols as AB → A + B

(16.2)

and the corresponding kinetic equation is −

d [ AB] = k [ AB] dt

(16.3)

where k (s−1) is a kinetic constant. If the yield is denoted by x, Eqn (16.3) can be written as dx = k (a − x ) dt

(16.4)

or, in integrated form, 1 a ⎞ k = ⎛ ⎞ ln ⎛ ⎝ t⎠ ⎝ a − x⎠

(16.5)

where a is the initial concentration of sucrose. In inversion, the hydroxonium ion functions as a catalyst. The catalytic ability of various acids may be characterized by a scale (Table 16.1). Comment: The inversion ability is defined as a ratio of kinetic constants related to hydrochloric acid (HCl) as 100% (Sokolovsky 1958, p. 160). In acid-catalysed reactions of first order, the reaction constant k is a sum in general: k = k0 + ∑ ki [ ion ]i

(16.6)

Chemical operations (inversion and caramelization), ripening and complex operations

497

Table 16.2 Relationship between the inversion ability and the dissociation of inorganic and organic acids.a Sources of dissociation constants (at 20°C): Erdey (1958, p. 37) and Kaltofen et al. (1957, p. 270). Acid HNO3 H2SO4 H2SO3 Oxalic H3PO4 Citric Lactic Acetic

Inversion ability (%)

Log (inversion ability)

pK

100 53.6 30.4 18.6 6.2 1.72 1.07 0.5

2 1.72916479 1.48287358 1.26951294 0.79239169 0.23552845 0.02938378 −0.30103

1.2 1.52 1.77 1.42 1.96 3.1 3.82 4.73

Intercept: 2.414717. Slope: −0.61885. Correlation: r = −0.9436.

a

where k0 is a constant, and ki is the rate constant of the i-th ion, of concentration [ion]i. However, in inversion, k depends exclusively on the activity of the hydroxonium ion (H3O+); thus Eqn (16.6) is valid for inversion in the form k = kH+ [H3O + ]

(16.7)

The activity of hydroxonium ions can be determined on the basis of this relationship (due to Ostwald); see Erdey-Grúz and Schay (1954, p. 536). Table 16.2 shows the relationship between the logarithm to base 10 of the inversion ability and pK (where pK is the negative logarithm to base 10 of the dissociation constant). The relatively good correlation (r = −0.9436) of a plot of the data in Table 16.2 is in accordance with the equation Δ log k = α Δ log K acid = α Δ pK acid

(16.8)

where α is a constant. See Erdey-Grúz and Schay (1954, p. 538). Commercially prepared acid-catalysed solutions are neutralized when the desired level of inversion is reached. Neutralization, however, needs to be done carefully; if the pH exceeds 7, the colour of the solution rapidly becomes brown because of the development of so-called reversion products. It is known that sucrose rotates the plane of polarization of light to the right: its specific rotation is + 66.67°. Under the effect of acid as a catalyst, glucose, which rotates the plane to the right (specific rotation + 52.74°), and fructose, which rotates it to the left (specific rotation −93.78°), are formed, and because fructose rotates the plane more strongly to the left than glucose does to the right (and the amount of sucrose causing a rotation to the right is decreasing by splitting), after a certain degree of splitting the solution will rotate the plane to the left by an angle

[α ]D =

−93.78 + 52.74 = −20.52° 2

All of these rotation angles above relate to the D-line of light. The reason for the division by 2 is that two molecules (glucose and fructose) are formed in equimolar amounts

498

Confectionery and Chocolate Engineering: Principles and Applications

(Maczelka 1962, p. 37). As a result, the initial rotation to the right is inverted. Therefore this process is called ‘inversion’ and the product is called invert sugar. [Another name for glucose is dextrose (from dexter, Latin for ‘right’), and another name for fructose is laevulose (from laevus, Latin for ‘left’); these are related to the particular optical properties of these compounds.] The determination of the reducing sugar content of carbohydrate solutions is a routine task in the confectionery industry because the reducing sugar content has proved to be an important quality and consistency parameter for many sweets. Both iodometric (the Fehling, Schoorl–Regenbogen and Bertrand methods) and polarimetric methods are used for measuring the reducing content. It should be mentioned that the cupric → cuprous reduction, which is the basis of these methods, is not a stoichiometrically correct reaction since partial oxidation of the sugar takes place at the same time. However, this problem can be eliminated by fixing the conditions of the analysis. High-performance liquid chromatography (HPLC) methods provide a fingerprint of the sugar components. Colorimetric methods are widely applied for the determination of glucose in human blood. The Somogyi–Nelson micromethod (Morris 1958) is based on the blue colour (‘molybdenum blue’) produced by reaction of cuprous oxide (Cu2O) and arsenic molybdenate. The cuprous oxide is formed from CuSO4 (from Seignette salt) by the reducing effect of the reducing sugar content. 16.1.1.1

Dependence of acidic inversion on temperature

According to the Arrhenius law, acidic inversion is catalysed by temperature rise as well: ΔH ⎞ k = A exp ⎛ − ⎝ RT ⎠

(16.9)

where A (s−1) is a frequency factor (not entirely independent of the temperature T); ΔH (J) is the activation energy, which is always positive for thermal reactions; R = 8.31434 J/ (mol K) (the molar gas constant); and T (K) is the absolute temperature (see Erdey-Grúz and Schay 1954, p. 440). It is known that raising the temperature by 10 K increases the rate of first-order reactions by two to three times (Lengyel et al. 1960, p. 186). This fact is important in practice because the majority of sugar confectioneries contains flavouring acids (citric, lactic, tartaric or, by accident, acetic acid) and the temperature of the process is rather high (60–100°C). However, the consequences of inversion may be both advantageous and disadvantageous, depending upon the properties of the end product; see Chapter 8. Invert sugar (and fructose) has a lower water activity than has sucrose, and as a consequence, it improves the shelf life of certain products by inhibiting drying, but can make the product surface and consistency sticky. To briefly summarize, control of the inversion of the sucrose content is an essential technological task.

16.1.2

A specific type of acidic inversion: Inversion by cream of tartar

Besides the edible acids, potassium bitartrate or cream of tartar (C4H5KO6; M = 188.18) is also used in hard-boiled, pulled and grained sugar confectionery as an inversion cata-

Chemical operations (inversion and caramelization), ripening and complex operations

499

lyst. The pH value of a saturated aqueous solution of cream of tartar is 3.577 at 25°C; a solution of 0.01 mol/kg has a pH value of 3.639 at 25°C. (For comparison, a 0.1 N aqueous solution of carbonic acid has a pH of 3.73, i.e. it has slightly weaker acidity than a potassium bitartrate solution of concentration 0.01 mol/kg.) Potassium bitartrate (KHTa) is formed in wine, from a reaction between the bitartrate ion (HTa−) from tartaric acid (H2Ta) and the potassium ion (K+) found in grapes, especially grape skins: K + + HTa −  KHTa

(16.10)

Its solubility in water is low: 5.7 g/L at 20°C. However, the solubility of calcium tartrate is lower: 0.1 g/L. Consequently, in the presence of calcium ions, a proportion of the dissociated bitartrate ions will be precipitated: HTa − → H+ + Ta 2−

(16.11)

Ca 2+ + Ta 2− → Ca-Ta ↓

(16.12)

1 kg of sucrose contains about 100 mg of free calcium ions, which consumes 100 mg ×

188.18 = 469.6 mg cream of tartar 40.08

Assuming that a charge of 100 kg of sucrose needs 0.2 kg of cream of tartar for inversion, the free calcium ions consume 100 × 469.6 mg = 46.96 g of cream of tartar, which is a quarter of the total amount of cream of tartar used. Taking into consideration the fact that the free calcium content of sucrose items is variable, the technology of inversion by cream of tartar is always accompanied by some uncertainty.

16.1.3

Enzymatic inversion

Sucrose can be hydrolysed by the enzymes α-glucosidase and β-fructofuranosidase, because the glucose part has a glucoside carbon atom of α-configuration and the fructose part is of β-configuration. The enzyme invertase (E 1103) that is of industrial importance is produced from yeast, and is β-fructofuranosidase. (The invertase produced from moulds is α-glucosidase.) The hydrolysis of sucrose is similar to an esterification during which a trisaccharide is formed (Sumner and Somers 1953): Sucrose + enzyme ↔ fructosido-enzyme + glucose Fructosido-enzyme + sucrose ↔ trisaccharide + enzyme Trisaccharide + enzyme + H2 O ↔ fructose + sucrose + enzyme Sucrose + enzyme ↔ etc. (‘da capo al fine’) The enzyme activity is dependent on the concentration of the substrate, in this case sucrose. The activity of invertase increases up to a sucrose concentration of 5 m/m%; at higher concentrations it decreases (Höber, cited by Tolnay 1963). The optimal pH interval

500

Confectionery and Chocolate Engineering: Principles and Applications

16 0.05%

Invert sugar (%)

14

(deliquesced) 0.025%

12 10

0.01%

8 Control: 0% 6 5

0

5

10

20 30 Storage (days)

45

Fig. 16.1 Effect of invertase on the reducing sugar content of fondant products during storage at 20°C and 50% relative humidity [reproduced from Székely (1966), by kind permission of Mr Péter Székely].

of yeast invertase is 4.5–5.0 and that for honey invertase is 5.5–6.6 (Bergmeyer 1962). The optimal temperature for the function of invertase is 55°C, according to Bersin (see Tolnay 1963). However, it can work well at room temperature, which means that invertase added to a product continues to cause hydrolysis during storage. (Invert sugar is produced in situ.) This property is used to soften fondant and marzipan products. With regard to denaturation, the critical temperature of the function of invertase is about 69°C (Görög 1964). For further details, see Combes and Monsan (1982). Székely (1966) studied the application of invertase (Naarden, powder No. 5, activity 5086 Sumner invertase units (SU)/g enzyme) for softening fondant products covered with a sucrose crystal layer (‘kandis’) at a level of 6.2 m/m%. Because of heat sensitivity, the invertase preparation was mixed with the fondant mass and flavours at a temperature below 60°C. The Somogyi–Nelson method was applied to determine the invert sugar (reducing sugar) content; see Morris (1958). Figure 16.1 shows a storage experiment during which the change of the reducing sugar content of the fondant was investigated as a function of storage time at 20°C at a relative humidity of 50%, with different invertase concentrations (0.05%, 0.025% and 0.01%) and compared with a control (0%). Similar experiments were performed with the same samples at 20°C and a relative humidity of 70%, and at 30°C. Székely (1966) studied, using a Höppler Consistometer (Rod No. 27, diameter 4 mm) and the same samples, the effect of invertase on the softening of fondant. Figure 16.2 shows the penetration of the consistometer rod into the fondant product. (The conditions of storage were 20°C and relative humidity 50%.) The corresponding penetration vs consistency relationship is shown in Table 16.3 (Székely 1966). Székely (1966) determined the liquid phase content of the same samples during storage at 30°C. (The temperature of determination was 22°C.) The results are presented in Table 16.4. The liquid phase content refers to the centre, without the ‘kandis’ layer (6.2 m/m%). Under the effect of invertase, the liquid phase ratio is slightly increased. The liquid phase content was determined by centrifugal separation; see Ravasz (1964). According to Székely (1966), the reducing sugar (invert sugar) content must not exceed 14–15%, or the fondant product will be liquefied. An invertase concentration of 0.01 m/m%

Chemical operations (inversion and caramelization), ripening and complex operations

501

9 0.05%

Penetration (mm)

8

(deliquesced)

7 0.025%

6 5

0.01%

4 3 2 Control: 0%

1 0 0

10

20 Storage (days)

35

45

Fig. 16.2 Change of penetration under the effect of invertase during storage at 20°C and 50% relative humidity [reproduced from Székely (1966), by kind permission of Mr Péter Székely].

Table 16.3 Penetration vs consistency relationship for stored fondant products that contain invertase [reproduced from Székely (1966), with kind permission of Mr Péter Székely]. Penetration (mm)

Consistency

0.3–0.45 0.45–1.00 1.00–5.00 5.00–8.00 8.00–9.00

Semi-hard Soft Very soft Creamy Semi-fluid

Table 16.4 Variation of liquid phase content during storage as a function of invertase content [reproduced from Székely (1966), with kind permission of Mr Péter Székely]. Invertase content (%) Storage (days)

0.05

0.025

0.01

0

1 5 15 25 35 45

75.8 77.7 79 80.1 81.2 82

75.2 77 78.5 79.6 80.7 81.2

75.7 76.8 77.6 78.7 79.4 80

75.5 75.7 76.4 77.1 78 78.9

502

Confectionery and Chocolate Engineering: Principles and Applications

(5000 SU units/g) can be recommended for softening the consistency. A fat content of 14 m/m% does not disturb the effect of invertase in fondant.

16.2 16.2.1

Caramelization Maillard reaction

In the confectionery industry, the concept of ‘caramelization’ is applied in two different ways: the Maillard reaction (non-enzymatic browning) and sugar melting. The Maillard reaction, also called the non-enzymatic browning reaction or amino– carbonyl reaction, is a reaction between compounds containing amino groups and compounds containing reducing groups such as aldehydes and ketones. The amino compounds in foods are mostly free or protein-bound amino acids, and the reducing compounds are reducing sugars. The Maillard reaction plays a central role in food science. The reaction is very common and occurs in almost all foods and other systems containing amino acids and sugars that undergo heat treatment or are subjected to long storage times. Owing to the complexity of the reaction, it has both positive and negative effects; for example, the formation of flavour is in most cases positive, while a loss of essential amino acids is a negative consequence of the reaction. Although intensively studied, the chemistry of the reaction is not fully understood. Various authors (Hodge 1953; Ellis 1959; Reynolds 1963, 1965, 1969; Feeney et al. 1975; Mauron 1981) have reviewed the chemical nature of this reaction. The classical scheme worked out by Hodge (1953) is still valid in its description of the main stages of the reaction. The initial stage is the condensation of the amino and the carbonyl groups, which can undergo the ‘Amadori rearrangement’ to give amino-deoxy compounds, often called ‘Amadori compounds’. The Amadori compounds can then react further, depending on the reaction conditions. These reactions are mainly of three types: dehydration, degradation and fission. The last stage in the Maillard reaction is the formation of brown polymers from precursors originating in the previous stages. The reaction pathways are greatly influenced by the reaction conditions. The reactants are obviously of great importance. Temperature, water activity, pH and duration of the reaction are also very important. The formation of the brown colour is one of the most easily detectable changes caused by the Maillard reaction. The coloured compounds are of high molecular weight, but their structure is basically unknown. The formation of flavour is another easily detectable change caused by the Maillard reaction; see Fors (1983), Lingert (1990) and Schieberle (1990). The flavoured compounds are formed mainly by condensation of the degradation products. As amino acids are involved in the Maillard reaction, these are lost during the reaction. The principal amino acid lost in foods is lysine. As lysine is often the limiting amino acid in proteins, the Maillard reaction can lower the nutritional quality of proteins (Mauron 1981). The Maillard reaction can lead to increased toughness of the food as a consequence of polymerization and cross-linking of proteins (Labuza et al. 1977) In the confectionery industry, the Maillard reaction plays an essential role in manufacturing fudge, toffee, butterscotch (commonly called caramel), milk chocolate, milk crumb and choco crumb. Fondant mass is often enriched with sugared condensed milk in order to produce a speciality fondant with a beige colour and caramel flavor – the

Chemical operations (inversion and caramelization), ripening and complex operations

503

Maillard reaction is similarly important in this process. This type of fondant mass can be regarded as a kind of caramel too. Also, sugared condensed milk itself provides a classical example of the Maillard reaction, which is accompanied by reactions of the milk fat during evaporation. Naturally, the Maillard reaction takes place in all cases where proteins and reducing sugars are heated to elevated temperature, for example in the roasting of cocoa, nuts and coffee, in the alkalization of cocoa nibs or mass, and in baking. However, the typical operation is a controlled Maillard reaction in the manufacture of various types of caramels and milk chocolate. A common feature of these examples is that some kind of fat (mostly butter or cocoa butter) is present among the reactants. It seems that a fundamental condition for the typical caramel flavour is the presence of some fat – this is proved by experience in confectionery practice. Tressl (1990) has given the principal pathways for the thermal generation of aromas in foodstuffs and a list of references concerning the Maillard reaction. The proteins and carbohydrates are transformed into smaller compounds (amino acids, and monosaccharides and disaccharides, respectively), and then, during many reactions called early and advanced Maillard reactions (the Amadori transformation, Strecker degradation etc.), they are terminated by N- or S-containing heterocycles. The lipids are transformed into fatty acids and then peroxidized and degraded, and finally the resultant lipid-derived products are transformed into N- or S-containing heterocycles. The products of lipid degradation play an important role in the advanced Maillard reactions. This fact is important from the point of view of producing fudges and toffees; namely, the lipid medium (milk butter) stimulates the generation of typical milky flavour components. On the other hand, many of the flavour components are lipid-soluble (hazelnut, coconut etc.); therefore, these types of flavour are not intense in lipid-poor media. The same applies also to milk powders and products: the butter-poor products are flavour-poor as well. From the point of view of our objective, two technological parameters are important if the product composition is fixed: the temperature and the duration of the effect of heat. In the production of various types of caramel products, the temperature range is about 120–130°C (for butterscotch, c. 145–152°C). The effect of the heat used in evaporation is sufficient to generate the typical caramel flavour as well. In the production of milk chocolate, the temperature of conching is about 45–60°C, which is relatively low; however, the effect of heat lasts c. 12–20 h in modern conching machine. In the production of milk crumb or choco crumb, the temperature is higher but the duration of the effect of heat is shorter. Milk proteins are essential to the generation of caramel flavour, particularly the whey protein, which is present in an amount of 14–24% (while the casein proportion is about 76–86%), because the Maillard products of lysine seem to be the most important factors in the flavour, and the lysine content is higher (10–12%) in whey protein than in casein (about 8.4%); see Gasztonyi (1979, p. 152). For this reason, whey is a favourite raw material for fudges and toffees, and it is used also in chocolate production, although whey must not be estimated as milk content according to EU regulations. Care is needed when whey is used for chocolate since the microbiological state of whey is often objectionable, and the temperature of conching is not sufficient to totally destroy its microflora. Mohos (1982) studied the hydroxymethylfurfurol (HMF) content of caramelized milk powders made by a roll dryer (Vonda type). The pasteurized milk used, of pH 6.4–6.9, contained 10% sucrose or dextrose and 20% milk fat. This solution was condensed up to

504

Confectionery and Chocolate Engineering: Principles and Applications

40–41% dry content. When this condensed solution was digested for 1.5–2 h, the flavour of the end product (milk powder) was excellent. (A longer digestion is not preferable, because the viscosity of the solution becomes too high for drying.) The optimal HMF content of caramelized milk powders can be regarded as 15–25 mg/kg. The HMF content of milk chocolate made from such milk powders is about 10–15 mg/kg, which shows that a certain part of the final HMF content is also generated also during the chocolate production. If the average ratio of milk powder in milk chocolate is about 20 m/m%, this results in about 2–3 mg/kg of HMF (10/5–15/5); the other part (8–12 mg/kg) develops during the production of the milk chocolate. Many brands of milk chocolate were investigated, and an HMF value of 10–15 mg/kg was determined in general. The HMF value is one of the characteristic properties of the Maillard reaction and the chemical reactions taking place in molten sucrose. For its determination, see Örsi (1962) and Mohos and Lengyel (1981). The principle of this method is that HMF results in a colour reaction with a solution of aniline and barbituric acid in glacial acetic acid.

16.2.2

Sugar melting

The Maillard reaction is not the only non-enzymatic browning reaction. The caramelization reaction is in many ways similar to the Maillard reaction, except that only sugar compounds take part in the caramelization process. Brown-coloured products with a typical caramel aroma are obtained by melting sugar or heating sugar syrup in the presence of acid and/or alkaline catalysts. The process can be directed more towards aroma formation or more towards accumulation of brown pigment. The heating of sucrose syrup in a buffered solution enhances molecular fragmentation and, thereby, formation of aroma substances. Primarily dihydrofuranoses, cyclopentanolones, cyclohexenolones and pyrones are formed. On the other hand, heating glucose syrup with sulphuric acid in the presence of ammonia provides intensely coloured polymers (‘sucre couleur’, E 150a, b, c, d). The stability and solubility of these polymers are enhanced by bisulphite anions. For details of the reactions involved, see, for example, Belitz and Grosch (1987). These reactions provide the basis for industrial processes for ‘colour production’. In the confectionery industry, a simple method is applied for producing colour: the intensity of the brown colour of the melted sugar is tailored according to demand by dissolving it in water, and this aqueous solution is the ‘colour’ used for colouring and flavouring. The other field of application is the manufacture of ‘grillage’ (Krokant in German) products. Sugar is melted, and then cut nuts, honey or glucose syrup is mixed with it. The mixture is cooled and may be shaped, etc. The melting point of sugar is about 175–180°C, depending on the amount of nonsucrose ingredients. The temperature range of technological importance is about 200– 210°C. Mohos (1982) and Mohos et al. (1981) studied the operation of sugar melting. Three steps may be distinguished: (1) The temperature range is about 207–210°C; the colour of the molten sugar is strawyellow, and its HMF content is about 1.5–2 g/kg. This type of molten sugar is suitable for covering the so-called ‘Dobos’ fancy cake (Dobos-tart, named after C. József Dobos, Hungarian confectioner, 1847–1927) because its flavour is very mild. This temperature range is suitable for producing various grillage products as well because

Chemical operations (inversion and caramelization), ripening and complex operations

505

HMF (mg/kg)

HMF (mg/kg)

3000 5000 4000 3000 2000

2000

1000

1000 0

0 0

100

150

200

205

210

215

Temperature (°C)

Temperature (°C)

(a)

(b)

Fig. 16.3 Melting of sugar: variation of hydroxymethylfurfurol (HMF) content of molten sugar with temperature (a) under laboratory conditions and (b) by Sucromelt 3400 machine (O. Hänsel).

there is no danger of burning them. The blending of cut nuts facilitates keeping within this range, since the increase in temperature is arrested for a short time by the addition of the nuts, which are at room temperature. (2) The typical temperature range is 210–215°C; the HMF content of the molten sugar is about 2–3 g/kg, the colour becomes browner and the molten sucrose becomes more and more difficult to manage. This product is suitable for manufacturing colour solutions. (3) The HMF content of the molten sugar is more than 3 g/kg; the colour becomes deep brown and, finally, black. Figure 16.3 shows the variation of the HMF content of molten sugar as a function of the temperature, under laboratory conditions and in the product produced by the Sucromelt 3400 machine (O. Hänsel); the output of this machine is 80–180 kg molten sucrose/h. Below 210°C, the dependence can be regarded as quasi-linear, however, when the temperature rises above 210°C, HMF production is accelerated. Since the temperature is only one of the important parameters in sucrose melting, in addition to the duration/intensity of heating, etc., the HMF content of sucrose can be regarded as a reliable characteristic of caramelization. For further results, see Quintas et al. (2007a,b).

16.3 16.3.1

Alkalization of cocoa material Purposes and methods of alkalization

The cocoa substance (cocoa nibs, the cotyledon) contains many acidic components, which have a harsh flavour; see Rohan and Stewart (1963, 1964a,b, 1965a,b, 1966a,b) and Bonar et al. (1968). Also, flavonoids play an important role in the development of cocoa powder. The purposes of alkalization, first applied by Van Houten in 1828, are:

Acids

Flavons

Cellulose

Confectionery and Chocolate Engineering: Principles and Applications

Hemicellulose

506

Cocoa

butter

(Free)

(Bound)

Neutralization Swelling and partial solubilization

Condensation

Diffusion

Aqueous alkali

Fig. 16.4

Changes during alkalization (German process).

• neutralization of acidic components; • to bring about chemical changes in flavonoids in order to develop the deep brown colour of cocoa powder; • to bring about swelling and solubilization of the cellulose and hemicellulose components of the nibs to improve the flotation properties of the cocoa particles in solution; • to increase the proportion of free cocoa butter by opening the cells of the nibs. Alkalization has to be finished by evaporation in order to reduce the moisture content of the cocoa mass below 1%, which is a principal requirement for pressing. The most frequently applied methods of alkalization (often called ‘preparation’) are: • the Dutch process: alkalization of nibs; • the German process: alkalization of cocoa liquor (cocoa mass).

16.3.2

German process

Alkalization is discussed in detail in classical publications, for example Fincke (1965) and Minifie (1999). Here only the German process will be briefly described (Fig. 16.4), in order to emphasize an interesting aspect: phase inversion. In alkalization according by any method, the pH is increased to about 7–8 (in extreme cases, up to 9.5) by using aqueous alkali, mostly a solution of potassium carbonate (potash) or bicarbonate. The characteristic concentrations are as follows: 100 kg of cocoa mass is stirred with 10 kg of water containing about 1–3 kg of the potassium compound. (If more potash is added, alkalization is terminated by addition of citric acid in order to lower the pH to neutral.) The potash solution is continuously stirred into the cocoa liquor for about 8 h at c. 110–115°C to allow the continuous evaporation of water. In contrast, if the potash solution is added to cocoa liquor in one block, the cocoa liquor, of W/O type emulsion, is inverted into an O/W type emulsion, and as a consequence, the rheological properties change drastically: the power requirement of the stirrer is strongly increased. Also, the mixture takes on a mud-like appearance and is violet–brown in colour.

Chemical operations (inversion and caramelization), ripening and complex operations

507

However, this method of alkalization is radical, and the cocoa liquor–potash solution mixture needs very robust machinery for kneading and evaporation.

16.4

Ripening

In food production, the collective designation for most of the enzymatic processes that take place is ‘ripening’. However, in confectionery manufacture the concept of ‘ripening’ is used also for processes that occur as a result of diffusion. A typical example is Ostwald ripening, which can be manifested as coagulation of very small particles in an emulsion or dissolution of crystals of very small size. From the point of view of thermodynamics, ripening is a process of stabilization, i.e. the first step is to bring about an unstable state, which then changes spontaneously into a stable state in a certain time (e.g. during storage). Some processes of ripening are practically complete before the product leaves the factory; however, many carry on in the consumers’ home, for example transcrystallization of cocoa butter. These phenomena point to the fact that a definition of food stability is necessary that is in conformity with the understanding of thermodynamics and takes into account the fact that in certain cases the technological aim is to create an unstable state; see Chapter 18.

16.4.1

Ripening processes of diffusion

Fondant ripening is a typical case of Ostwald ripening. A well-known experience is that if fondant mass is allowed to rest overnight before shaping, its consistency will be entirely changed to its advantage (e.g. such a fondant, when blended with other ingredients, is not so apt to dry). However, modern technologies operate continuously and do not allow the new fondant mass to rest. Fondant ripening was studied by Maczelka and Gyo˝rbíróné (1958), who determined that the optimal size interval of the sucrose crystals was 5–20 μm; crystals larger than 30 μm can be perceptible in the mouth. Karácsony and Pentz (1955a,b) determined that very small crystals (c. 1 μm) were dissolved after leaving the fondant machine, and about 24 h was necessary for development of a stable state in the crystalline bulk of the fondant mass. Consequently, calculations of the liquid phase based on the chemical composition of the fondant (Hinton 1931) can be regarded as an approximation only, because the solubility values are dependent on the crystal size; however, this effect cannot be calculated. For crusted liqueur bonbons/pralines, an aqueous sucrose solution of boiling point 112.5°C is made (schwach pflug, ‘weak blow’), and then ethyl alcohol and flavouring (rum, brandy etc.) are blended with it; finally, this flavoured alcoholic solution is cast into moulds made of starch, and the surface of the centres is dusted with starch. After 10–12 h, the palettes are turned. The entire period of crystallization takes about 24 h, during which a continuous sucrose crust, including the diluted liqueur, is formed. After the centres have been carefully undusted, they may be packed or covered by chocolate. Traces of acid are the ‘killer’ of this technology because inversion hinders the crystallization of sucrose.

508

Confectionery and Chocolate Engineering: Principles and Applications

The following types of grained sweets can be distinguished: • Hard-boiled sweets grained by humidity of filling. The trick of the technology is that crystallization in a sugar mass needs a certain water content and a suitably low reducing sugar content (a ‘dry sugar mass’ of about 13–14%). In the case of hard-boiled sugar bonbons with an aqueous filling (e.g. fruit jellies), this condition means that the water content of the filling can start the crystallization (‘graining’) of the sucrose in the sugar mass. As a result, the sugar mass becomes entirely grained and slightly whitened: the consistency of a hard-boiled sugar bonbon becomes soft. • Hard-boiled sweets grained by pulling (forced crystallization). In Section 2.2.2, the technology of crystalline grained drops was discussed. Both the cream of tartar and the glucose syrup technology result in a sugar mass that is suited to crystallization because of its low reducing sugar content. When the sugar mass is pulled, a forced crystallization starts in the sugar mass, which takes some days. The effect of pulling is wide-ranging: small, thin tubes are formed by it. At the beginning, the volume of the pulled mass increases (the density of the sugar mass decreases from 1.5 to c. 0.9 in 6–7 min), later the small tubes break, which causes an increase in the density of the sugar mass; see Sokolovsky (1951, p. 103) and Fig. 2.18. The pulling induces, if the composition of the sugar mass is correct, a slow crystallization that extends step-by-step over the entire volume of the product. The warm, wet air of the store room promotes crystallization of the unpacked product. If the product is packed promptly after leaving the cooling tunnel, crystallization will be slower. • Sweets grained by addition of fondant (fudge). In fudge, crystallization originates from sucrose crystals added to the fondant mass. The process of crystallization starts quickly but continues slowly for about a half an hour after the product has left the production line. The structure is entirely homogeneous because of the thorough mixing. • Sweets grained by spontaneous crystallization (grained caramels). Crystallization is generated by the correct high ratio of sugar to glucose syrup (i.e. a low reducing sugar content). The usual method of shaping such grained caramels is casting into moulds of starch dust. The crystalline structure of the centres develops slowly during solidification in the moulds (3–4 h), and it has a certain inhomogeneity: the external part of the centres is more solid than the internal part, which is slightly creamy. This peculiar consistency is preferred. 16.4.1.1

Dissolution of fondant in cherry liqueur pralines

The preparation of the fruit involves preserving washed sour cherries in an aqueous solution of ethanol of c. 70 V/V% until the end of diffusion (4–6 weeks) at 12–15°C. For the production of pralines, the cherries are placed in chocolate shells and then covered by fondant mass. The cooled, solid fondant layer is then covered with chocolate. The praline pieces need intensive cooling for demoulding; therefore, a slow, step-by-step warming-up is necessary to avoid cracking them. It is reasonable to couple this warmingup (6–8°C → 15–16°C) period with the ripening of the pieces, which takes approximately overnight. During ripening, the change of temperature induces high tensions in the praline pieces because the dilatations of the chocolate shell and the centre are different (Zs. SzabóForrás, Confectionery Research Laboratory, Budapest, 1970, personal communication).

Chemical operations (inversion and caramelization), ripening and complex operations

509

In addition, during this period the osmotic-pressure differences between the fondant mass and the cherries start to be balanced. Consequently, the fondant is dissolved by the aqueous alcoholic solution, which diffuses from the cherry into it. The effect of diffusion is that the filling becomes thinner and thinner, and the solution can easily bubble through cracks. The high tensions, the cracking and the thinning of the filling together may cause considerable problems if ripening is neglected and the pieces are packed promptly after demoulding. 16.4.1.2

Humidity balance between cookies and fillings (toughening)

A rule of thumb may be applied in this relationship (here the term ‘cookie’ refers to any sweets made of flour): the cookie and the filling have to be contrary from the point of view of hygroscopicity, i.e. a cookie with hydrophilic properties (e.g. wafers and biscuits of low fat content) needs a fatty filling, and a cookie with hydrophobic properties (e.g. linzer and others of high fat content) needs a hydrophilic filling (e.g. fruit jelly or jam). If the cookie is hydrophilic, the use of a fatty filling hinders the toughening of the cookie. If the cookie is hydrophobic (‘short pastry’), the water content of the filling migrates to a certain extent into the cookie, which makes its consistency short. (The effect of air humidity is similar.) Thus, a cookie with hydrophilic properties must be protected from humidity (= fatty filling), while a cookie with hydrophobic properties needs some humidity for an optimal consistency (= aqueous filling). 16.4.1.3

Ripening of dough before shaping

Relaxation is recommended because beneficial processes take place in the dough, which are driven by diffusion in general. Naturally, in the case of a yeast dough, ripening means both fermentation and diffusion. The fermentation of doughs could deserve an entire chapter in this book because of its importance, but this would go beyond the scope of this book. 16.4.1.4

Tempering of chocolate mass and polymorphism

The aim of the tempering of chocolate mass is to generate the β (V) modification of cocoa butter, the transcrystallization of which into the β (VI) modification is sufficiently slow to produce a product of ‘stable quality’. At the same time, during tempering, the crystals of the very unstable β (IV) modification are melted. Cooling and storage have to provide the correct conditions to prolong the β (V) → β (VI) transcrystallization of cocoa butter as long as possible. Consequently, tempering generates an unstable state, the stabilization of which is controlled both in production and in storage. Polymorphism is discussed in Section 10.9.1.

16.4.2

Chemical and enzymatic reactions during ripening

Only acidic and enzymatic inversion have been mentioned here, because a detailed discussion of the processes that take place in baking, in the development of flavour, etc. is beyond the scope of this book.

510

Confectionery and Chocolate Engineering: Principles and Applications

16.5

Complex operations

16.5.1

Complexity of the operations used in the confectionery industry

Foods are systems consisting of many ingredients, and the ingredients themselves are complex systems as well. Consequently, any operation exerts a wide range of effects on them. This complexity of operations means an essential difference compared with chemical operations. The various fields of confectionery engineering are typical from this point of view: many raw materials are blended, and many operations are used to process them. In Sections 16.5.2 and 16.5.4, structure theory (see Appendix 5) is used to take this complexity of operations into account when conching and dough preparation are discussed.

16.5.2

Conching

Conching is one of the most studied fields of chocolate manufacture. The first steps in studying the conching process were taken by Zipperer (1924), Fincke (1936), Aasted (1941), Taubert (1954), Heiss (1955) and Rohan (1965). Fincke (1965), Minifie (1999) and Marshalkin (1978) discussed the conching process in detail. According to Fincke (1956b) and Acker and Diemair (1956), the fat-free cocoa material has a strong anti-oxidative effect, which protects cocoa butter from oxidation. Bartusch and Mohr (1966) determined that a decrease in the water and acid contents of chocolate mass took place together during conching. Although the tendencies of both processes may be described by a decreasing concentration vs time curve, this curve actually consists of concentration minima and maxima, which become fewer and fewer as time progresses. The reason for this phenomenon may be explained by briquetting of the chocolate particles during the refining process, which is a consequence of the pressing performed by the rollers. During conching, the small briquettes are comminuted step-by-step, and humidity and acid vapours are evaporated from the surface of the briquettes. But the regeneration of the evaporation surface is not continuous, and this causes fluctuations of the concentrations of about 0.1%. Sommer (1973) studied the change of the concentration deviation in the process of homogenization of a 1 : 1 mixture of cocoa butter and sugar powder. Tscheuschner et al. (1992, 1993) experimentally demonstrated that dry conching may lead to aggregation or de-aggregation of chocolate particles, depending on the temperature, the water content and the shear rate generated by the conching machine. These phenomena are highly influenced by the types of sugars and milk solids. A high temperature and a low-to-medium intensity of shearing promote aggregation, whereas intensive shearing and long conching diminish the size of the aggregated particles in chocolate. The aggregation of particles during dry conching of milk chocolate is a complex process in which amorphous lactose and the humidity play an essential role. For details of the physico-chemical relations of the polymorphism of lactose, see King (1965). In the initial step of aggregation, the free humidity, sugar and other hydrophilic ingredients are in contact. As a result, a soft aggregate is formed, which may be dissolved by the effect of a high shear rate. In contrast, the aggregate may solidify when the humidity decreases. For this reason, it is important that the humidity of the chocolate, which became free during roll-refining, is rapidly decreased at the beginning of conching. Free

Chemical operations (inversion and caramelization), ripening and complex operations

511

humidity is absorbed by amorphous lactose, and is needed for lactose crystallization. In the process of lactose crystallization, there is free water at the start. If this water is not quickly eliminated, it dissolves a small portion of the sugar content, which is sufficient to start aggregation. The process of lactose crystallization is dependent on temperature, and it starts below about 30% relative humidity if the temperature is higher than about 60°C. For further details, see Mohos (1979), Tscheuschner et al. (1992, 1993), Winkler and Tscheuschner (1998), Scheruhn et al. (2000), Tscheuschner (2000), Franke et al. (2001, 2002), Franke and Heinzelmann (2006) and Gray (2006). In brief, the technological procedure consists of the following steps. The chocolate mass is refined by five rollers; a total fat content of c. 26–27 m/m% is necessary for refining. After refining, the chocolate, of powder consistency, is transported to the conche machines, where it is liquefied by adding 1–2% cocoa butter and kneading strongly. According to practical observations, intensive kneading of chocolate mass of high viscosity is very beneficial from the point of view of quality. This period is called ‘dry conching’ and takes some hours. Dry conching is followed by ‘wet conching’, in which a 1–2% amount of cocoa butter is added to the chocolate mass again. Finally, half an hour before the end of the process, c. 0.3–0.4% of lecithin is added to the chocolate mass to emulsify the residual water content of the mass in the suspending cocoa butter phase. The use of lecithin is very financially advantageous because it causes a decrease in viscosity equivalent to that caused by about 7% cocoa butter (Fincke 1965). Thus, the final cocoa butter content of the chocolate mass for bars can be relatively low, about 30–33% (for couverture, the value is higher, 35–38%). When conching was first used, the duration was very long, c. 3 days for dark chocolate and 1 day for milk chocolate, because the conche pot (Langreiber in German) was unable to provide sufficiently intensive kneading and aeration for the removal of the water and acid content. Later, as a result of many developments, the alternating motion of the rollers agitating the chocolate in the conche pot was replaced by a rotating motion, and several different solutions have also been developed for intensifying the aeration (e.g. the use of vacuum, and blowing-in of warm, dry air). In the last 30–40 years, compact conching solutions have been developed, the common property of which is that refining, kneading and aeration are coupled together either in one machine or in a closed transport circle of machines. Although opinions on the quality of the chocolate produced with such compact machines differ, they have many advantages (saving of space, simplicity of operation, cheapness, low repair costs etc.). In addition, these solutions have contributed to the demystification of conching. 16.5.2.1

Batch conching

Mohos (1982) developed a qualitative model (the triangle model) of batch conching by the application of structure theory as shown in Fig. 16.5. The quantitative model (Fig. 16.6) consists of the following parts (the conserved substantial fragments that the calculation refers to are given in parentheses). The set of conserved substantial fragments: (1) (2) (3) (4) (5)

dry, fat-free cocoa material (7 kg); cocoa butter (dry) (26 kg); sucrose (dry) + milk dry material (62 kg); oxygen; water (1 kg).

512

Confectionery and Chocolate Engineering: Principles and Applications

Aroma development (Maillard reaction, oxidation etc.)

Steam distillation (decrease of acid content)

Change of flow properties by kneading and emulsification Fig. 16.5

Triangle model of conching.

a4 = | 1 | 4 | 5 |

a3 = | 4 | 5 | Gas phase Liquid phase

Chocolate mass

a1 = | 1 | 2 | 3 | 5 |

Chocolate mass

|1|2|3|4|5| Conched

a2 = | 1 | 2 | 3 | 5 |

a5 = | 4 | 5 | a6 = | 3 |

Fig. 16.6 Quantitative conching model based on conserved substantial fragments. For example, a3 = |4|5| means a material flux consisting of oxygen and water.

The set of material fluxes: a1, the chocolate mass at the start of conching; a2, the dry ingredients of the chocolate mass conched (without Maillard product or lecithin); a3, the humidity and oxygen of the air entering the conche machine; a4, the humidity and oxygen of the air leaving the conche machine, enriched by acid content from the cocoa material; a5, the water content and amount of oxygen bound in the chocolate mass by oxidation; a6, M (the product of the Maillard reaction).

513

Chemical operations (inversion and caramelization), ripening and complex operations

The chocolate mass that is conched is a union of three material fluxes: a2, a5 and a6. The set of technological changes (where t is the time coordinate): evaporation of acids, s = s(t): the amount of acid evaporated; Maillard reaction, M = M(t): the amount of Maillard reaction product; oxidation, y = y(t): the amount of oxygen that reacts with the cocoa material; evaporation of water, w = w(t): the amount of water evaporated. The set of phases: G = gaseous; L = liquid. We can express the value of a2 as follows, taking into account the set of technological changes: a2 = (7 − s − y ) + 26 + (62 − M ) + (1 − v )

(16.13)

Let us study the elements of the set of technological changes in order to obtain the value of every material flux at the end of conching. Evaporation of acids. According to Bartusch and Mohr (1966), the changes of the water and acid contents of the cocoa mass parallel each other, i.e. we may write s0 − st = k1 (w0 − wt )

(16.14)

where s0 and st are the acid contents of a2 when t = 0 and t = t, respectively, w0 and wt are the water contents of a2 when t = 0 and t = t, respectively, and k1 is the rate constant (1/ time). Maillard reaction. For the rate of the Maillard reaction, it is assumed that Mt − M 0 = k2 (w0 − wt )

(16.15)

where M0 and Mt are the amounts of the conserved substantial fragment (3) in a2 that participate in the Maillard reaction when t = 0 and t = t, respectively. Oxidation. For the rate of oxidation, it is assumed that yt − y0 = k3 (w0 − wt )

(16.16)

where y0 and yt are the amounts of (1) in a2 that are oxidized when t = 0 and t = t, respectively. We introduce the simplifying notation (w0 − wt) ≡ w. It may be assumed that ΔY = k4 ( y0 − yt ) = k3 k4w where ΔY is the loss of oxygen in the input air. Thus, a5 = ΔY + ( y0 − yt ) = k3w (1 + k4 )

(16.17)

a3 = L + k5 L = L (1 + k5 )

(16.18)

514

Confectionery and Chocolate Engineering: Principles and Applications

where L is the oxygen content of the input air, and k5L is the water content of the input air if p and V are constant. The value of k5 is 10.76 × 10−3 kg/0.2314 kg dry air at t = 35°C and relative humidity 30%. (The increase in the proportion of nitrogen in the air that accompanies the decrease in the amount of oxygen is neglected.) a4 = L (1 + k5 ) + w (1 + k1 − k3 k4 )

(16.19)

a6 = (M 0 − Mt ) − k2w

(16.20)

The value of a2 at the end of conching (see Eqn 16.13) is a2 = (7 − k1w − k3w ) + 26 + (62 − k2w ) + (1 − w ) = 96 − (1 + k1 + k2 + k3 ) w

(16.21)

Evidently, the task has been simplified to a drying task because 1 + k1 + k2 + k3 = constant

(16.22)

A drying process may be most easily modelled by a first-order kinetics: d (w0 − wt ) = k0 (wt − w∞ ) dt

(16.23)

where w∞ is the water content of the chocolate mass after very long conching (≈0.3%). (The value of w0 is ≈ 1.3–1.5%). After integration, wt = (w0 − w∞ ) exp ( − k0t ) + w∞

(16.24)

The halving time can be obtained from

τ1 2 =

ln 2 k0

(16.25)

The value of the halving time τ1/2 can be regarded as a measure of the effectiveness of a conching machine. In order to obtain comparable results for conche machine effectiveness, either the value of w∞ should be given together with τ1/2 (e.g. τ1/2 = 8 h; w∞ = 0.3%) or a value of w∞ should be defined (standardized) by the industry. The kinetic constant k0 and, therefore, also the value of the halving time are dependent on the amount of charge, i.e. the way to correctly characterize a conche machine is as τ1/2 = X hours (referring to Y tonnes of chocolate). Since Y tonnes means a characteristic property of the construction of the machine, every amount of charge has its own characteristic halving time. Example 16.1 In a conching machine of volume 5 m3, w0 = 1.4% and wt = 0.8% when t = 6 h and w∞ = 0.3%. (This is a very effective conche.) Let us calculate the halving time τ1/2. From Eqn (16.24), 0.8 − 0.3 wt − w∞ = exp ( − k0 × 6 ) = 1.4 − 0.3 w0 − w∞

Chemical operations (inversion and caramelization), ripening and complex operations

515

i.e. exp ( k0 × 6 ) =

1.1 = 2.2 0.5

k0 × 6 = ln 2.2 = 0.7885 … k0 = 0.1314 … and

τ1 2 =

ln 2 = 5.274 h. 0.13142

Checking: the half-value is (1.4 + 0.3)%/2 = 0.85%, i.e. 1.4% → 0.85% ↔ ( 0 → 5.274 …) h 0.85% → 0.8% ↔ (5.274 … → 6 ) h If conching is continued, the water content after 2 × τ1/2 ( 2 × 5.274 h = 10.548 h) will be (0.85 + 0.3)%/2 = 0.575%. An approximate calculation of the values of rate constants. Rohan and Stewart (1963, 1964a,b, 1965a,b, 1966a,b) and Bonar et al. (1968) studied the change of acid content of chocolate during conching. They determined that the loss of acid content of cocoa mass during conching for 48 h was about 0.14%. If the loss of water content is approximated by a value of 0.8%, and the ratio of cocoa mass in milk chocolate is R, then, from Eqn (16.14), the value of k1 is k1 = R

Δs 0.14% acid =R× = R × 0.175% acid % water Δw 0.8% water

If the value of R is 0.12, then k1 = 0.12 × 0.175 g acid g water = 0.021 g acid g water Mohos (1982) studied the change of the hydroxymethylfurfurol content of milk chocolate during conching, and the following equation was obtained: HMF = [ 2.25 + 0.674 (T − 45°C ) °C + 0.1723 t h ]( mg kg milk chocolate )

(16.26)

where T is the temperature of conching (°C) (interval studied 45–60°C), and t is the duration of conching (h) (interval studied 0–20 h). (The values of r were 0.9954 at 45°C, 0.9673 at 50°C, 0.9384 at 55°C and 0.9781 at 60°C.) Evidently, the quantitative description of conching sketched above can be regarded as only approximate, because Eqn (16.24) is linear in t; however, Eqn (16.15) leads to an exponential function of t. Nevertheless, this linear kinetics is uncommon, and therefore first-order kinetics will be applied in the following treatment; further studies are necessary to clear up this question.

516

Confectionery and Chocolate Engineering: Principles and Applications

Regarding the exactness of such an approximation, the following Taylor series can be considered: C 2 C exp ( kt ) = C + Ckt + ⎛ ⎞ ( kt ) + … ⎝ 2⎠ where C and k are constants, and t is the time. In the present case, C = [ 2.25 + 0.674 (T − 45°C ) °C ]( mg kg milk chocolate ) k=

0.1723 (1 h ) 2.25 + 0.674 (T − 45°C ) °C

The greatest error is for T = 45°C because at this temperature the value of k is highest (the rounded values are k = 0.0766 and C = 2.25). For convergence, kt < 1 is needed, i.e. t < (1/0.0766) h → t < 13.055 h. Consequently, if an exponential function is applied instead of a linear function (Eqn 16.26), as time passes, this approximation becomes more and more inaccurate, and when t ≥ 13.055 h, it is not usable at all. According to Mohos (1982), in dry conching of milk chocolate the temperature must not be higher than 60°C; the optimum is about 55°C, and this period is recommended until the HMF content reaches 10–15 mg/kg. For storage of milk chocolate, a maximum temperature of 45°C is recommended. Rheological measurements (Mohos 1966a) show that above 45°C the viscosity of milk chocolate starts to increase and the mass shows the phenomenon of rheopexy, which is related to structural changes. Rheopexy is greater in milk chocolates produced with milk crumb than with milk powder. According to Gray (2006), the viscosity of chocolate is high below 40°C, although such temperatures are essential for masses using monohydrate sugar alcohols (and, according to Mohos, also dextrose monohydrate). Above 60°C, white chocolate will darken and its flavour will be affected. Above 75°C, milk chocolate may caramelize, depending upon its recipe – this is often desirable. Above 85°C, milk chocolate can start to burn, which introduces a bitter note. Again, this may be desirable. The temperature range 50–100°C is suitable for dark chocolates. At the end of conching, the mass must be cooled to 40– 45°C for storage or immediate use. White chocolate should be stored at the lower end of this range. For estimation of an average value of k2, we assume that T = 50°C, t = 20 h and the loss of water content is 0.7%. Then, ΔM = 2.25 + 0.674 (50 − 5) + 0.1723 × 20 = 9.066 mg HMF kg milk chocolate k2 = −

ΔM = −9.066 mg HMF 7 g water = −1.291 mg HMF g water Δw

The negative sign means that while the water content of the chocolate mass is decreasing, the HMF content of it is increasing. In Eqn (16.21), k2 and k3 are of negative sign because an increase in HMF content (and in the amount of oxidation product) means a loss similar to evaporation of water and volatile acids; however, from the viewpoint of kinetics the rate constants k0 and k1 are negative, and k2 and k3 are positive.

Chemical operations (inversion and caramelization), ripening and complex operations

517

Lipscomb (1954) determined that 100 g of cocoa mass absorbs about 12.3 mg of oxygen in 16 h, i.e. 1 kg of milk chocolate of 12% cocoa mass content absorbs 0.123 g × 0.1 = 0.01476 g of oxygen. If the loss of water content is approximated as 0.6%, then k3 = −

Δy = −14.76 mg oxygen 6 g water = −2.46 mg oxygen g water Δw

The results may be summarized by a matrix equation: ⎡ w ⎤ ⎡ − k0 ⎢ ⎥ ⎢ ⎛ d⎞⎢ s ⎥=⎢ 0 ⎝ dt ⎠ ⎢M ⎥ ⎢ 0 ⎢ ⎥ ⎢ ⎣y⎦ ⎣ 0

0 − k0 k1 0 0

0 0 k0 k2 0

0 ⎤ ⎡w⎤ 0 ⎥ ⎢s ⎥ ⎥×⎢ ⎥ 0 ⎥ ⎢M ⎥ ⎥ ⎢ ⎥ k0 k3 ⎦ ⎣ y ⎦

(16.27)

The stability matrix (see Section 18.4) is regular (its determinant is not zero), but its two positive elements (k0k2 and k0k3) show that the Maillard reaction and the oxidation cannot be controlled. These processes are theoretically unstable: the increase in the amounts of Maillard and oxidized products is unlimited in an infinite time, although these possibilities have no practical importance. If linear kinetics are supposed in the cases of the Maillard reaction and oxidation, then the correct form of Eqn (16.27) is dw = − k0w, dt

ds = − k0 k1s, dt

dM = k0 k2 , dt

dy = k0 k3 dt

(16.28)

Also in this case, the system is unstable, although the increase of M and y is slower. The development of aroma in chocolate is a very complicated process, and the present discussion has been limited to modelling the decrease of the volatile acid content and the increase of the Maillard reaction product. Besides these factors, many others need to be taken into account, but a discussion of them is beyond the scope of this book. Similarly, the change of consistency during conching is an important topic (see Chapter 4). 16.5.2.2

Modelling a continuous conching process with high shear rate

Franke and Tscheuschner (1991) developed a model of a high-shear-rate conching process, applying the Danckwerts one-dimensional dispersion model (Danckwerts 1953). It is assumed that the materials move axially at a mean rate w in the conching equipment, which is a double-mantled tube of length L, and the chocolate mass and the cooling water flow in countercurrent on the two sides of the mantle. To this motion a turbulence produced by inserts and by an air feed is added, which is shown by the axial diffusion coefficient Dax: ∂m ∂ ( Dax ∂m ∂x ) ∂m = −w + ∂t ∂x ∂x

(16.29)

where x is the coordinate in the axial direction (m), 0 ≤ x ≤ L; w is the mean speed (m/s), assumed to be constant in the steady state; dm/dt is the rate of change of the quantity of

518

Confectionery and Chocolate Engineering: Principles and Applications

chocolate mass in a considerable space (kg/s); −w ∂m/∂x is the convective transport at a mean rate w; Dax is the axial diffusion remixing (m2/s); and ∂(Dax ∂m/∂x)/∂x is the mass transport by axial diffusion. The model assumes linearity of Dax(x) as a function of x: Dax ( x ) = Dax.0 + xDax.1

(16.30)

where Dax.0 and Dax.1 are constant. By using this assumption, the equation of continuity (Eqn 16.29) is dm ∂m ∂2 m = ( Dax.1 − w ) + ( Dax.0 + xDax.1 ) 2 dt ∂x ∂x

(16.31)

The heat balance can be written by applying the following assumptions: (1) (2) (3) (4)

Heat conduction between layers within the chocolate mass is neglected. The possible state transitions of cocoa butter are neglected. Heat exchange with the environment of the conching equipment is neglected. Steady-state conditions are assumed, i.e. the heat balance for a chocolate layer of width dx is zero: − dH back ′ − dH ch ′ − dH w′ + dH D′ = 0

(16.32)

where dH back ′ is the enthalpy change due to diffusion back mixing (W), dH ch ′ is the convection enthalpy change of the chocolate mass (W), dH w′ is the heat exchange by the cooling water (W) and dH D′ is the heat dissipated from the power input by the shearing devices (W). The primes ′ mean derivatives with respect to time, i.e. d( )/dt. The terms of Eqn (16.32) can be expressed as ⎛ ∂ 2Tch ⎞ ⎛ Dax.0 + xDax.1 ⎞ dH back dx mch ′ =⎜ ′ c p.ch ⎜ ⎟ ⎝ Dax.1 − w ⎠ ⎝ ∂x 2 ⎟⎠

(16.33a)

The model uses an approximation Dax ( x ) ⎤ ⎛ ∂ 2Tch ⎞ dx −dH back m c ′ ′ = ⎡⎢ ch p.ch ⎜ ⎝ ∂x 2 ⎟⎠ ⎣ w ⎥⎦

(16.33b)

Moreover, dH ch ′ = mch ′ c p.ch dTch′

(16.34)

where mch ′ is the mass flow of chocolate (kg/s), cp.ch is the specific heat capacity of the chocolate (J/kg K) and dTch is the temperature change of the chocolate mass (K). dH w′ = ( d π dx ) kw (Tch − Tw )

(16.35)

where d π dx is the area of the heat transfer surface (m2), d is the inner diameter of the tube (m), kw is the (resultant) heat transfer coefficient between the chocolate mass and the

Chemical operations (inversion and caramelization), ripening and complex operations

519

cooling water (W/m2 K), and Tch and Tw are the temperatures of the chocolate mass and cooling water, respectively (K). dH D′ = PD ( x ) dx

(16.36)

where PD(x) is the position-dependent power dissipated per unit length input by the shear elements of the conching equipment. Substituting Eqns (16.33)–(16.36) into Eqn (16.32), a common linear second-order differential equation results, which describes the temperature profile (Tch vs x) of the chocolate mass along the length of the conching equipment: 2 ⎡ Dax ( x ) ⎤ m ′ c ⎛ ∂ Tch ⎞ + m ′ c ⎛ dTch′ ⎞ + ( d π ) k (T − T ) − P ( x ) = 0 ch p.ch ⎜ ch p.ch w ch w D ⎢⎣ w ⎥⎦ ⎝ dt ⎠ ⎝ ∂x 2 ⎟⎠

(16.37)

In Eqn (16.37), Tw is unknown, and can be calculated from the energy balance around a thin layer of the cooling jacket of the conching equipment. To calculate the temperature profile of the cooling water: (1) A steady-state process of cooling is assumed, i.e. dH w′ dt = 0 (the enthalpy flow of the cooling water is constant). (2) Back mixing within the cooling water is neglected. These assumptions result in the equation dT mw′ c p.w ⎛ ch ⎞ + ( d π ) kw (Tch − Tw ) − ( dout π ) ken (Tw − Ten ) = 0 ⎝ dx ⎠

(16.38)

where mw′ is the mass flow of cooling water (kg/s), cp.w is the specific heat capacity of the chocolate (J/kg K), dout is the outside diameter of the cooling jacket (m), ken is the heat transfer coefficient between the cooling water and the environment (W/m2 K), and Ten is the temperature of the environment (K). The system of coupled differential equations (16.37) and (16.38) represents the Franke– Tscheuschner process model. The variables included are: process variables: mch ′ , mw′ ; a geometric variable: x; thermodynamic variables: cp.ch, cp.w, kw, ken transport variables: w, Dax.0, Dax.1 These variables can be determined for any particular conching equipment in order to calculate the process model. The variable that cannot be simply determined is the dissipated power per unit length, PD(x), because both dissipation and diffusion take place in the chocolate mass in parallel. It is assumed that the total power required, Ptotal(x), can be separated into two parts: Ptotal ( x ) = PD ( x ) + PD ( x )str

(16.39)

where PD(x)str is the power used for the structural modification of the chocolate mass.

520

Confectionery and Chocolate Engineering: Principles and Applications

By using the notation for the ratio PD(x)/Ptotal(x) ≡ 1 − zstr(x), we assume the linearity of zstr(x): zstr ( x ) = zstr.0 + xzstr.1

(16.40)

where zstr.0 is a constant (the initial value at x = 0), and zstr.1 is a constant (the variation of zstr(x) along the length of the equipment). The linear model represented by Eqn (16.40) has been validated by the fact that there is good agreement between real measured temperatures and the temperatures calculated from the process model using the parameter zstr(x). By varying the chocolate mass flow mch ′ , a plot of w vs mch ′ is constructed, which can ′ ) ; furthermore, a plot of Dax.0 be approximated by a quadratic function f, i.e. w = f ( mch vs mch ′ is constructed, which can be approximated by a quadratic function g, i.e. Dax.0 = g ( mch ′ ). To describe the variation of the power input with time, P(t), an exponential approximation is applied: t P ( x ) = P0 exp ⎛ − ⎞ ⎝ τ⎠

(16.41)

where P0 is the starting value of the power input (W), and τ is a time constant (s) describing the decrease of power input with increasing shearing time. When plots of P0 vs T and τ vs T for different amounts of chocolate mass are constructed, the types of approximate equation obtained are P0 = − A + Bm − C ( mT )

(16.42)

τ = D − Em − FT + G (T )

(16.43)

where m is the amount of chocolate in the container (kg), T is the temperature (K), and A (W), B (W/kg), C (W/kg K), D (s), E (s/kg), F (s/K) and G (s/kg K) are positive constants. According to Franke and Tscheuschner’s experiments, the fraction of the power input required to modify the structure of the chocolate mass varies from 60% to 85%, and with increasing throughput, the fraction used for this modification of the chocolate mass decreases while the dissipative fraction increases, i.e. the efficiency of the equipment deteriorates. No significant correlation between the conching temperature and the percentage of structural modification was identified. For further details, see Franke and Tscheuschner (1991). 16.5.2.3

Conching degree

The ‘conching degree’ (CD) can be defined qualitatively as a noticeable effect of conching on the taste of chocolate. A high CD means a harmonious, intense chocolate flavour in sensory terms. The CD can be measured (Ziegleder et al. 2003, Ziegleder 2004), and can be calculated as the reciprocal values of chosen markers (benzaldehyde and tetramethylpyrazine). These volatile marker compounds are obtained by fractional short-time steam distillation, and then identified and quantified by the GC-MS technique. The analytical values and the sensory results are in good correlation.

521

Chemical operations (inversion and caramelization), ripening and complex operations

16.5.3

New trends in the manufacture of chocolate

In the second half of the last century, many studies aimed at simplifying chocolate manufacture focused on the following targets: • to cut the duration of the conching process; • to improve the efficiency of certain unit operations, e.g. comminution and aeration of the chocolate mass; • to cut the costs and space requirements of the machinery (compact plant). The most easily damaged parts of the machinery are eight- and five-roll refiners. Substitution of eight-roll refiners was the easiest task: the comminution of cocoa nibs to cocoa mass can be performed by a disc mill, an agitated ball mill, a beater blade mill or a combination of these. A logical idea was to substitute a five-roll refiner by an agitated ball mill, which is a relatively cheap machine with a small space requirement and high efficiency. At the beginning of modern chocolate production, the melangeur was used as a machine for homogenization – and for conching too! This tradition offered a practical solution to the problem of finding a machine that could unify homogenization and conching. The development of the conching process resulted in many types of conche machines that used intensive circular motion instead of alternating motion (as in the old longitudinal conche, the Langreiber), and intensive aeration. The typical conche machine of today is a circular container with a rotating kneader, plus strong ventilation. Table 16.5 lists the relationships between the raw materials processed and the unit operations in chocolate manufacture. It is evident that the sugar, milk dry content and cocoa nibs need to be comminuted; however, there is a question of which combinations of them would be the most efficient or whether separate comminution would be preferred. This question, among others, was discussed by Simon (1969a,b) and Ubezio (1976). Processing using separate unit operations is one of the trends. For comminution of cocoa nibs by a combination of a disc mill, agitated ball mill and beater blade mill, see Niediek (1973, 1978), Bauermeister (1978), Samans (1978), Goryacheva et al. (1979), Bertini (1989), Taylor and Zumbe (2000) and Alamprese et al. (2007). For evaporation of cocoa mass and cocoa butter by a thin-layer evaporator (for ‘pre-conching’), see Schmitt (1974, 1986, 1988), Heemskerk and Komen (1987) and Bäucker (1974). Separate production of milk crumb (milk + sugar) or choco crumb (milk + sugar + cocoa mass) in order to focus on the Maillard reaction is another trend. Both types of crumb are produced by evaporation, condensation, kneading and drying to c. 4% of final water

Table 16.5

Relationships between raw materials and unit operations carried out in chocolate manufacture.

Sugar Milk dry content Cocoa nibs Cocoa butter Lecithin

Homogenization

Comminution

Evaporation

Maillard

Rheology

Yes Yes Yes Yes Yes

Yes Yes Yes No No

No Yes Yes Yes No

Yes Yes No No No

Yes Yes Yes Yes Yes

522

Confectionery and Chocolate Engineering: Principles and Applications

content. For further details, see Bouwman-Timmermans and Siebenga (1995), Aguilera and Stanley (1999), Minifie (1999, pp. 144–148), McCarthy et al. (2000), Beckett (2003), Lim and Barigou (2004), Edmondson (2005) and Aguilera (2005). Universal conching machines achieve all operations at the same time in the same space. Continuous kneading machines have been well proven for homogenization, and, consequently, agitated ball mills seemed to be an obvious solution for coupling homogenization and comminution. An extreme case of such a solution is when all the unit operations of chocolate manufacture are implemented in a single machine of special construction: comminution is performed by a tailored device using a knife, as in the MacIntyre and Lloveras machines. All the other unit operations are automatically done during comminution. For further details, see Riedel (1991) for the MacIntyre technology, and see the Lloveras technical documentation for the Lloveras Universal Refiner Conche. Some of the various non-traditional chocolate technologies are discussed below; however, it is nearly impossible to achieve completeness, and so instead we cite the following references: • LSCP (Lindt & Sprüngli Chocolate Process): see Kleinert (1971, 1973) and Coccolo (1973); • Wiener process: see Karoblene et al. (1981) and Tadema (1988); • EMA plant: see Heemskerk and Komen (1987); • Konticonche: see Tscheuschner et al. (1981a,b) and Tscheuschner and Winkler (1997). In the first step (preparation of cocoa mass) of the Tscheuschner chocolate process, a portion of the sugar (1–2%), dissolved in water, is added to the cocoa mass, which is warmed to 80°C in order to induce the Maillard reaction. Then the water content of the cocoa mass is evaporated to a value of 0.3% by insufflation of conditioned air. The time and temperature conditions of preparation are controlled and fitted to the quality requirements. In the second step, this prepared cocoa mass is refined together with sugar. The agglomeration in the refining is small after such a preparation and, as a result of intensive shear during conching for 10–20 min, the agglomerated particles are reduced in size. Because a decrease in moisture content and a sufficient amount of Maillard reaction, which are the usual requirements in the traditional conching process, have already been provided in the preparation step, the conching time may be drastically reduced. A similar preparation of milk powder is done before using any of the following: • the NETZSCH process: see Harbs and Jung 2005; • the Carle & Montanari system: see Anonymous 1985; • the BFMIRA process: see BFMIRA 1970. Further references are Perry (1968), Dunning and Dannert (1970), Hofmann and Tscheuschner (1976), Billon (1984), Ritschel et al. (1985), Ley (1988), Aguilar et al. (1995), Ziegler and Aguilar (1995, 2003), Müntener (1997, 2007), Kuster (2000), Jolly et al. (2003) and Blakemore (2006).

16.5.4

Modelling the structure of dough

One of the most general and important operations is dough preparation. This discussion is confined to a study of this field by means of structure theory. The starting point is a triangle model of the structure of dough (Fig. 16.7). The structure can

Chemical operations (inversion and caramelization), ripening and complex operations

Inhibitio

n Swelling Gelation

Yeast Mono-, dior oligobaking powder

ilic

Poly-

Hy dro

ning) brow

Fig. 16.7

Proteins (gluten)

Hy dro p

ibit

ion

pm en t

Inh

ev elo sd Ga

Swelling gelation foaming

hil ic

Carbohydrates

enzy mafic

rd re a

(non-

Mailla

Inhi bitio n

ction

ph

Dissolution

523

Emulsifiers (egg, lecithin)

Lipids Emulsifying

Triangle model of the structure of dough: model of processes in aqueous medium.

be modelled by three elements: (mono-, di-, oligo- and poly-) carbohydrates, proteins and lipids. The set of technological changes may be constructed from these elements as follows: • • • • • • • •

homogenization (kneading or solution); emulsifying (by using egg or lecithin, etc.); swelling of gluten; gelation of polycarbohydrates and/or proteins; foaming of polycarbohydrates and/or proteins; gas development under the action of chemical agents or fermentation by yeast; change of consistency; Maillard reaction, of small extent.

This enumeration cannot aim for completeness. The set of conserved substantial elements is determined by the set of technological changes; the same refers to the set of elements of machinery as well.

Further reading Aasted. Technical brochures. Almond, N. et al. (1991) Biscuit, Cookies and Crackers. Elsevier Applied Science, London. Bauermeister (Probat Group). Technical brochures.

524

Confectionery and Chocolate Engineering: Principles and Applications

Beckett, S.T. (ed.) (1988) Industrial Chocolate Manufacture and Use. Van Nostrand Reinhold, New York. Beckett, S.T. (2000) The Science of Chocolate. Royal Society of Chemistry, Cambridge. Bessiêre, Y. and Thomas, A.F. (1990) Flavour Science and Technology. 6th Weurman Symposium. Wiley, Chichester. Biscuit and Cracker Manufacturers Association (1970) The Biscuit and Cracker Handbook. Biscuit and Cracker Manufacturers Association, Chicago. Cakebread, S.H. (1975) Sugar and Chocolate Confectionery. Oxford University Press, Oxford. Eiarsson, H. (1987) Antibacterial Maillard reaction products. Doctoral Thesis, University of Göteborg. Ellis, P.E. (ed.) (1990) Cookie and Cracker Manufacturing. Biscuit and Cracker Manufacturers Association, Washington, DC. Faridi, F. (ed.) (1994) The Science of Cookie and Cracker Production. Chapman and Hall, New York. Gaonkar, A.G. and Anulkumar, G. (1995). Ingredient Interactions: Effects on Food Quality. Dekker, New York. Hodge, D.G. and Wade, P. (1968) Investigation of the Baking of Semi Sweet Biscuits, Part I. C&CFRA (FMBRA) Report 14. Kempf, N.W. (1964) The Technology of Chocolate. The Manufacturing Confectioner Publishing Co., Glen Rock, NJ. Kulp, K. (ed.) (1994) Cookie Chemistry and Technology. American Institute of Baking, Kansas. Lehmann. Technical brochures. MacFarlane, I. (1989) Measurement of Oven Conditions and Types of Ovens Used in the Biscuit Industry, Biscuit Seminar. ZDS Solingen, Germany. Manley, D. (1998) Biscuit, Cookie and Cracker Manufacturing Manuals (Vol. 1: Ingredients; Vol. 2: Biscuit Doughs; Vol. 3: Biscuit Dough Piece Forming; Vol. 4: Baking and Cooling of Biscuits; Vol. 5: Secondary Processing in Biscuit Manufacturing; Vol. 6: Biscuit Packaging and Storage). Woodhead, Cambridge. Meursing, E.H. (1983) Cocoa Powders for Industrial Processing, 3rd edn. Cacaofabriek de Zaan, Koog aan de Zaan. NETZSCH. Technical brochures. Pratt, C.D. (ed.) (1970) Twenty Years of Confectionery and Chocolate Progress. AVI Publishing, Westport, CT. Smith, W.H. (1972) Biscuit, Crackers and Cookies. Applied Science Publishers, London. Sollich. Technical brochures. Wade, P. (1988) Biscuit, Cookies and Crackers. Elsevier Applied Science, London. Wade, P. and Bold, E.R. (1968) Investigation of the Baking of Semi Sweet Biscuits. Part I, C&CFRA (FMBRA) Report 14. Whiteley, P.R. (1971) Biscuit Manufacture. Applied Science Publishers, London Wieland, H. (1972) Cocoa and Chocolate Processing. Noyes Data Corp., Park Ridge, NJ.

Chapter 17

Water activity, shelf life and storage

Contents 17.1 Water 17.1.1 17.1.2 17.1.3 17.1.4 17.1.5 17.1.6 17.1.7

activity Definition of water activity Adsorption/desorption of water Measurement of water activity Factors lowering water activity Sorption isotherms Hygroscopicity of confectionery products Calculation of equilibrium relative humidity of confectionery products 17.2 Shelf life and storage 17.2.1 Definition of shelf life 17.2.2 Role of light and atmospheric oxygen 17.2.3 Role of temperature 17.2.4 Role of water activity 17.2.5 Role of enzymatic activity 17.2.6 Concept of mould-free shelf life 17.3 Storage scheduling Further reading

17.1 17.1.1

525 525 527 527 533 534 535 538 541 541 541 541 541 542 542 547 548

Water activity Definition of water activity

The term ‘water activity’ (aw) describes the (equilibrium) amount of water available for hydration of materials; a value of unity indicates pure water, and zero indicates the total absence of ‘free’ water molecules; the addition of solutes always lowers the water activity. The water activity aw is the equilibrium vapour pressure that the water in a food exerts (pw) divided by the vapour pressure of pure water (p0). The water activity can also be defined as the relative humidity of air (RH%) at which a food, if held, would neither gain nor lose moisture. aw = γ w xw =

pw RH% = 100 p0

(at a given temperature )

Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

(17.1)

526

Confectionery and Chocolate Engineering: Principles and Applications

Non-hygroscopic

Relative humidity (%)

100

Typical S-shaped profile (e.g. foodstuffs, paper)

50

Hygroscopic

0 0

10

20

30

Water content (m/m%) Fig. 17.1

Typical water sorption isotherms.

where γw is the activity coefficient of water, xw is the mole fraction of water in the aqueous fraction, p is the partial pressure of water above the material and p0 is the partial pressure of pure water at the same temperature. The water activity is strongly dependent on temperature. The value of the water activity coefficient is often calculated, but it is only a rough approximation in most cases and holds only in a narrow region of concentration. For a given food, different amounts of water are held or bound at different water activities. The relationship is called the water sorption isotherm, since it defines the moisture content m in equilibrium with different values of aw at constant temperature. Figure 17.1 shows typical water sorption isotherms. Equation (17.1) can be written in another form: aw =

(RH% )air 100

=

(ERH% )food 100

(17.2)

where (ERH%)food is the equilibrium relative humidity of the food. The value of ERH% is, by definition, equal to the water activity of the food and is defined by the relative humidity of the air in equilibrium. The water activity aw usually increases with temperature and pressure. For small temperature increases (T1 → T2) at low aw, an often used relationship is ln a (T1 ) ⎛ ΔH ⎞ ⎛ 1 1 ⎞ = − ⎝ R ⎠ ⎜⎝ T1 T2 ⎟⎠ a (T2 )

(17.3)

where a(T1) and a(T2) are the water activities at temperatures T1 and T2 (K), respectively, ΔH is the enthalpy change (J/mol) (e.g. the enthalpy change of absorption or mixing), and R = 8.31434 J/mol K is the universal gas constant. Differences in water activity may cause water migration between food components.

Water activity, shelf life and storage

17.1.2

527

Adsorption/desorption of water

In order to understand the essential role of the concept of water activity and ERH%, let us consider the physical phenomena taking place in a food. Two extreme cases can be differentiated (where RH is a characteristic of the air in the room, and aw = ERH is a characteristic of the food): (1) Mass of air >> mass of food. This is the case when an unpacked substance (e.g. food) is stored in a large room with free space (e.g. a storehouse). The phenomena are: if RH > aw → adsorption (the food becomes wet); if RH < aw → desorption (the food becomes dry); if RH = aw → equilibrium, i.e. RH = ERH. It should be emphasized that during both adsorption and desorption, the food changes; the original state of the food is preserved only when the RH of the air is equal to the ERH, which is an essential characteristic of the food. (2) Mass of air << mass of food. This case is characteristic of a packed food: the air space above the product is tiny, and thus the process RH (of air space) → ERH (of product) is characteristic of this state. This state will be disturbed (slowly or quickly) by migration of air (if it is possible at all) through the packaging material if the RH of the outside air is very much different from the RH of the inside air. Evidently, in such a case the water activity (= ERH) of the food determines the microbiological, physicochemical etc. processes taking place in the food during storage as well. If the ERH of the food is high enough to reach the region where microbiological deterioration is probable, a packaging that does not permit continuous water loss from the product may soon cause microbiological defects (e.g. mould). (At the same time, this water loss from the product must remain above a minimum water content defined by the requirement for a fair quality until the shelf life expires.)

17.1.3

Measurement of water activity

Many techniques, both direct and indirect, are available for the measurement of water activity. The oldest technique, that of the hair hygrometer, was in fact reported by Leonardo da Vinci. A hair, being basically protein, will absorb water and change in both weight and length as the humidity increases. The hair can be tied to a pointer to indicate the degree of saturation of the vapour space in terms of relative humidity. The device is very inaccurate below 30% and above 80% RH but is used (usually with synthetic fibres) in home air-conditioning units for humidity control. It is not accurate enough for research purposes. A very accurate technique, which cannot be applied to measurement of aw of small samples, is wet bulb psychrometry. The method involves spinning two thermometers, one of which is immersed in a wet wick, in the vapour space. Based on the properties of the air, the relative humidity can be determined. It is still used in the measurement of the RH in large warehouses, such as storehouses where there is a large air volume. The method is fast but cannot be applied to small samples. However, a dew point device employing the same principle can be used. In this case a surface is cooled and the temperature at which water condenses is measured optically. This can then be related to aw. The instruments available have an accuracy of ± 3% RH. In relation to this topic, see Section 10.9.4 and Example 10.5.

528

Confectionery and Chocolate Engineering: Principles and Applications

Table 17.1 Evaluation of relative humidity of air at various temperatures by measuring the dew point. Dew point (°C) Temperature (°C) 15 20 25 30

6

8

10

12

0.548848 0.400467 0.295728 0.22087

0.629508 0.459321 0.339189 0.25333

0.720425 0.525658 0.388176 0.289917

0.822693 0.600278 0.44328 0.331072

Example 17.1 Let us prepare a table using Eqns (10.143) and (10.144): Td =

bα a −α

α=

aT + ln ( RH ) b +T

if T (temperature of air, °C) = 15, 20, 25 and 30, Td (the measured dew point, °C) = 6, 8, 10 and 12, a = 17.27 and b = 237.7 K. The procedure of calculation is as follows. From Eqn (10.143), Td → α =

17.27Td →α 237.7 + Td

From Eqn (10.144),

(α ; T ) → ln RH = α −

17.27T → RH 237.7 + T

The results are shown in Table 17.1. For example, if the dew point of air at 20°C is 8°C, then the RH of the air is 0.459.

One of the best direct measurements of aw is the direct measurement of the vapour pressure of water in the space surrounding the food by manometric techniques. In addition, electrical devices, for example the Sina-Scope, should be mentioned; of these, the most often used devices are relative-humidity sensors based on electrical resistance, with an accuracy of ± 0.5 RH%. For measurements of high-aW systems (low solute concentration), the measurement of freezing-point depression provides an suitable method; it is used by microbiologists for studies of microbiological growth as a function of aw. This method is based on Raoult’s law being valid for dilute solutions, according to which the freezing-point depression behaves as follows:

Water activity, shelf life and storage

529

• it is equal to Δt = −1.86 K if 1 mol of substance is dissolved in water (this is the so-called ‘molal’ freezing-point depression); and • it is proportional to the Raoult concentration (‘molality’) of the dissolved substance. If the Raoult concentration of the solute (denoted by a subscript 2) is m2, then the freezing-point depression is Δtf = −1.86m2 ( K )

(17.4)

If A2 g of solute is dissolved in A1 g of water, then the numbers of moles are n1 =

A1 M1

and n2 =

A2 M2

and the Raoult concentration of the solute is m2 =

n2 A2 = 1000 1000M 2

(17.5)

Supposing that the activity coefficient of water γ1 = 1, then n1 n1 + n2

aw = γ 1x1 = x1 =

(17.6)

To simplify the calculations, we write aw = x1 =

1000 18

(1000 18) + m2

=

55.56 55.56 + m2

(17.7)

and m2 (the Raoult concentration) is calculated from the freezing-point depression; see Eqn (17.4). The value of the freezing-point depression (Δtm.f) for a given solvent is determined by the Clausius–Clapeyron equation, R (Tf ) 1000 Lf

2

Δtm.f =

(17.8)

where Tf is the freezing point of the pure solvent (K); Lf is the latent heat of freezing/ melting of the solvent; and R = Nk is the molar gas constant, equal to 8.31434 J/mol K (N = 6.022 × 10−23/mol is the Avogadro number and k = 1.38062 × 10−23 J/K is the Boltzmann constant). For example, Δtm.f for benzene is 5.12 K. If Tev is the boiling point of the pure solvent (K) and Lev is the latent heat of boiling of the solvent, then the Clausius–Clapeyron equation gives the elevation of boiling point for the given solvent: R (Tev ) 1000 Lev 2

Δtm.ev =

(17.9)

530

Confectionery and Chocolate Engineering: Principles and Applications

Both the freezing-point depression and the boiling-point elevation depend exclusively on the type of solvent and not on the dissolved substance. Example 17.2 Let us calculate the theoretical values of Δtm.f and Δtm.ev according to the Clausius– Clapeyron equation. R = 8.31434 J mol K Freezing-point depression: Tf = 273 K and Lf = 334.9 J/g: Δtm.f =

8.314 × 2732 = 1.85 K ( measured value: 1.86 K ) 1000 × 334.9

Boiling-point elevation: Tev = 373 K and Lev = 2260.87 J/g: Δtm.ev =

8.314 × 3732 = 0.51 K ( measured value: 0.52 K ) 1000 × 2260.87

Example 17.3 Let us calculate the water activity of an aqueous sucrose solution of concentration 20 m/m%, although it is questionable whether this solution could be regarded as dilute. If 20 g of sucrose is dissolved in 80 g of water, then (1000/80) × 20 g of sucrose is dissolved in 1000 g of water. Since M1 = 342 for sucrose, m2 =

1000 20 × = 0.731 80 342

and Δtf = −1.86m2 = −1.36 K According to the above method of measuring the water activity, the freezing-point depression is measured, and from its value (assumed here to be Δtf = −1.36 K) the water concentration of the sucrose solution (20 m/m% → x1) is deduced: x1 =

n1 80 18 = = 0.9871 n1 + n2 80 18 + 20 342

Values of the water activity of aqueous sucrose solutions of various concentrations, according to Labuza (1975), are shown in Table 17.2. It can be seen that the measured (0.993) and calculated (0.9871) values are somewhat different. If the sucrose concentration is increased, the difference increases. In general, the molar mass of the solute is not known and the Raoult concentration of it must be calculated from the freezing-point depression caused by the solute; then Eqn (17.7) is used.

Water activity, shelf life and storage

531

Table 17.2 Water activity of sucrose solutions [reproduced from Labuza (1975), by kind permission of Springer Science and Business Media]. Sucrose (m/m%) 5 10 20 40 60

Water activity aw = ERH 0.999 0.994 0.993 0.96 0.89

Example 17.4 The water concentration of a culture medium is 90 m/m% and the measured freezing-point depression of it is −1.2 K. Let us calculate the water activity of this solution. m2 =

1. 2 = 0.645 1.86

This means that in 1000 g of water, the number of moles of the solute is 0.645. Consequently, the mole fraction of water (see Eqn 17.7) is aw = x1 =

1000 18 55.556 = = 0.9885 1000 18 + m2 55.555 + 0.645

Moreover, by using Eqn (17.5), an average molar mass for the culture medium can be calculated, taking into account the fact that A2 = 10 g: m2 = 0.645 =

A2 10g = 1000M 2 1000M 2

(17.10)

and from Eqn (17.10), M2 = 645 g. Measurement of the freezing-point depression is a common tool for determining the molar mass of a solute. An indirect method uses a chamber system in which aw can be directly controlled by some means. The moisture content after equilibrium is then measured rather than the water activity, as this can be done by a much simpler technique. The basic technique is the use of a saturated salt solution slurry. In a desiccator, various salts at saturation give a definitive value of aw, which varies little with temperature. Table 17.3 lists some of the more common salt systems; see Labuza (1975, p. 204). For further details, see Labuza (1975). Table 17.4 shows calculated water activity data for three saturated salt solutions. The calculation for LiCl.H2O is M=

1000 82.8 × = 19.53 100 42.4

532

Confectionery and Chocolate Engineering: Principles and Applications

Table 17.3 Equilibrium relative humidity (ERH) of salts used in chamber system [reproduced from Labuza (1975), by kind permission of Springer Science and Business Media]. Water activity aw = ERH

Salt

<0.001 0.11 0.33 0.49 0.55 0.65 0.76 0.82 0.88 0.94 0.99

Desiccant LiCl.H2O MgCl2 K2CO3 Mg(NO3)2 NaNO3 NaCl CdCl2 K2CrO4 KNO3 Na2HPO4

Table 17.4 Calculated water activity data for three saturated salt solutions. Solubility data taken from Kaltofen et al.(1957). Salt

Solubility at 20°C (g/100 g water)

Molar mass

Water activity aw, calculated

82.82 88.27 31.66

42.4 85 101.1

0.7399 0.843 0.947

LiCl.H2O NaNO3 KNO3

aw =

55.56 = 0.7399 55.56 + 19.53

If it is (rightly) supposed that LiCl is entirely dissociated, then xw =

55.56 = 0.587 55.56 + 2 × 19.53

Furthermore, the measured water activity coefficient is γw = 0.19, and thus aw = 0.19 × 0.587 = 0.11 which is the actual water activity of a saturated LiCl solution. It can be seen from Tables 17.3 and 17.4 that the measured and calculated values differ strongly for LiCl.H2O and NaNO3, which proves that the assumptions γw = 1 and aw ≈ xw cannot be accepted. In the confectionery industry, the minimum concentrations of carbohydrates used are about 60 m/m% because only more concentrated solutions are stable against microbiological deterioration. As the concentration of carbohydrates increases, the accuracy of Eqn (17.7), based on Raoult’s law, becomes unreliable.

Water activity, shelf life and storage

17.1.4

533

Factors lowering water activity

The basic factor that lowers aw is association of solutes with water to form a hydration shell. Depending on the amount present, the availability of the vapour pressure of water is decreased from x1 = n1/n1 = 1, according to Eqn (17.6), to aw = γ 1x1 ≈ x1 =

n1 ( <1 if n2 > 0 ) n1 + n2

Most solutes, especially as the concentration is increased, behave non-ideally and bind or structure water more than predicted. This is true for solutes (e.g. carbohydrates and hydrocolloids) of importance in the confectionery industry as well. With any solid system, a second important factor that lowers aw is the capillary effect. According to the Kelvin equation, the vapour pressure or activity of a liquid present in a capillary is reduced as the radius of the capillary decreases: aw = exp ⎛ − ⎝

2γ V cos θ ⎞ rRT ⎠

(17.11)

where aw is the water activity if the liquid is water, γ is the surface tension of the liquid in the capillary, V is the molar volume of liquid, θ is the contact angle, r is the radius of the capillary, R is the universal gas constant and T is the temperature (K). The capillary effect becomes important only in capillaries of radius less than 0.1 μm, as the next calculation shows. For simplicity, let us approximate Eqn (17.11) in the form aw = 1 −

2γ V cos θ rRT

Substituting the values γ = 73 dyn/cm, V = 18 cm3/mol, cos θ = 1, r = 10−5 cm, R = 8.31 erg/ mol and T = 300 K, we obtain aw = 1 − 0.01 = 0.99. If r = 10−6 cm, then aw = 1 − 0.1 = 0.9, etc. The exact values must be calculated according to Eqn (17.11) because the approximation applied becomes incorrect for higher values of the exponent. Bluestein and Labuza (cited in Labuza 1971) have shown that most of the capillaries in foods are larger than 10 μm but that as water is removed, the water present in small capillaries (r < 0.1 μm) comprises a significant amount of the total water. Thus, these capillaries do partially control the lowering of aw. The third factor responsible for lowering aw can account partially for the fact that hysteresis occurs (Fig. 17.2). A different path is followed depending on whether or not the isotherm is approached by adsorption or desorption. The most important factor controlling the vapour pressure is the interaction of water with solid surfaces and with high-molecular-weight colloidal systems. Water molecules usually interact with the polar groups on surfaces and are held very tightly. The energy required to remove these water molecules is greater than the energy required to vaporize a water molecule from the surface of pure water. Many aspects of a food system cause a lowering of the availability or activity of water. Many people have tried to quantify the degree to which water is bound or free in a food;

534

Confectionery and Chocolate Engineering: Principles and Applications

25

Less strongly bound water layer and capillary absorbed water

Strongly bound monolayer

Solvent and free water

Water content (%)

20

Desorption

15

10

5

Adsorption A

0 0.0

0.1

0.2

0.3

B 0.4

0.5

0.6

0.7

0.8

0.9

1.0

Water activity (aw) Fig. 17.2 Adsorption/desorption of water on the surface of foods – data for information only. The middle region ‘Less strongly bound water layer and capillary absorbed water’ has boundaries at about A = 0.33 and B = 0.75.

however, as shown by Labuza et al. (1970) and Labuza (1971), water even below aw = 1 still has the properties of bulk water: this water can still dissolve solutes, act as a medium for their mobilization, allow reactions to occur within the structure itself and be available as a reactant for reactions such as hydrolysis.

17.1.5

Sorption isotherms

Many theories have been developed to describe the shape of sorption isotherms from both a theoretical and a mathematical standpoint. The mostly commonly used equations are the following: • the Kelvin equation (Eqn 17.11); • the BET isotherm: 1 aw (c − 1) aw = + m (1 − aw ) m0c m0c

(17.12)

where m is the moisture content at given aw, m0 is the monolayer value and c is a constant; • the Harkins–Jura isotherm: ln aw = B − Am −2

(17.13)

• the Henderson isotherm: ln (1− aw ) = Am B

(17.14)

Water activity, shelf life and storage

535

• the Kuhn isotherm: m=

A +B ln (1 aw )

(17.15)

• the Labuza isotherm: m = Baw

(17.16)

• the Oswin isotherm: ⎛ a ⎞ m = A⎜ w ⎟ ⎝ 1 − aw ⎠

B

(17.17)

• a statistical equation:

aw =

A+ m B+m

(17.18)

where A and B are constants. For details, see Labuza (1975). The profile of a sorption isotherm is characteristic of the hygroscopicity of a product, the sorption behaviour of a product being dependent on temperature. Sandoval and Barreiro (2002) studied the water sorption isotherms of non-fermented cocoa beans at 25°C, 30°C and 35°C in the water activity region aw = 0.08–0.94. The best fit was shown by the BET equation when aw < 0.50 (r = 0.986) and by the Harkins–Jura model when aw ≥ 0.50 (r = −0.986). An average value of the sorption enthalpy of 1.81 ± 0.10 kcal/g mol was obtained for the temperature range studied.

17.1.6

Hygroscopicity of confectionery products

The fact that sucrose is a typical non-hygroscopic material under certain conditions plays an important role in products covered by sucrose crystals, for example candied and crystallized confectioneries. The sucrose crystals may effectively restrict the porosity on the surface of the centre, through which the migration of moisture in or out is strongly hindered until the ambient air humidity reaches the region RH = 50–70%. Above RH = 70% sucrose starts to become hygroscopic, and consequently the product may become wet. [Below RH = 50%, the product slowly dries; see later and Minifie (1999, Group 5 of confectioneries).] 17.1.6.1

Non-hygroscopic materials (ERH < 60%)

Among the non-hygroscopic materials, sucrose is the first that we shall mention. The isotherm of sucrose (at room temperature) is shown in Fig. 17.3.

536

Confectionery and Chocolate Engineering: Principles and Applications

100 90

Relative humidity (%)

80 70 60

Intensive wetting of sucrose Critical region

50 40

Practically no wetting of sucrose

30 20 10 0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Water content (m/m%) Fig. 17.3

Isotherm of sucrose.

The water content of sucrose must not exceed 0.1 m/m%, which relates to an ERH ≈ 80% (!) as the isotherm of sucrose shows. For refined sucrose, the prescription is usually maximum of 0.08 m/m% (↔ ERH ≈ 75%). When the sucrose is stored in a silo, the upper limit of the water content is 0.02–0.04 m/m%. The higher values are tolerable for storing sucrose in sacks because the moisture slowly evaporates through the sack material, although the released water may stick the sucrose particles together, and the consequence of such a process is the development of a crust on the sack material (e.g. paper). However, in a silo, the moisture cannot leave. Therefore, wetting zones are formed in the bulk of the sucrose, which promote the compression of hard lumps of sucrose. When the water content increases, osmophilic mould fungi on the surface of the sucrose particles may produce invert sugar, which causes intensive sticking (caking) of the sucrose particles. Accordingly, the condition of the air in silos is essential. The isotherm of sucrose shows that the maximum relative humidity of the air in a storehouse for sucrose must not exceed 70% (↔ water content ≈ 0.03 m/m%), otherwise the sucrose stock starts to get wet. However, the region of RH = 0.6–0.7% may also be critical because of traces of several impurities. In this region, bulk sugar is liable to agglomeration under the effect of own weight both in the sack and in the silo. The relatively non-hygroscopic behaviour of sucrose plays an essential role in fondant products covered by a sucrose crust (Kandisschicht): up to RH = 70%, the sucrose crust provides a barrier against the diffusion of humidity in and out, which blocks both drying and wetting of the product. If this crust is damaged (cracked), the protective effect is lost. According to Csajághy (2001), the water content of sucrose is dependent on the size of the sucrose crystals as well. There is an (average) optimal crystal size of sucrose (region

537

Moisture content

Water activity, shelf life and storage

I

0.3

II

0.5

III

0.7

0.9

1.1

Size of sucrose crystals (mm) Fig. 17.4

Moisture content of sucrose as a function of size of sucrose crystals.

II) ≈ 0.5–0.77 mm. In this region, the curve of water content vs crystal size has a minimum (Fig. 17.4). The explanation of this phenomenon is that: • The surface area of very small crystals (region I) is very high, and this surface absorbs a rather large amount of water. • If the sucrose particles are too large (region III), many large conglomerates come into being, which may enclose a large amount of water. Fats and oils are other non-hygroscopic materials that should be mentioned. If the continuous dispersion phase is a lipid, ‘outwardly’ the product behaves non-hygroscopically. The best example of this is sweets covered by chocolate (or fat mass): the relationship between the product and its environment is determined by the non-hygroscopic behaviour of chocolate (if the chocolate coating is not damaged). 17.1.6.2

Less hygroscopic materials (ERH = 40–60%)

Many confectionery products can be classified into this group. The essential matter is to ensure that the product in question has properties that guarantee that its ERH remains in this relatively narrow range. The risk is double: if the ERH sinks below 40% (too much invert substance), the product will be sticky; however, if the ERH rises above 60% (too little syrup and too much sucrose), the product will be hard and dry, and possibly grained. Figure 17.5 shows an isotherm for cocoa powder at room temperature (Meursing 1976). It can be seen why the critical value of 8 m/m% is the upper limit prescribed in product specifications: this moisture content corresponds to ERH = 60% and, at higher ERH values, microbiological deterioration will start. Cocoa powder is not a hygroscopic material, since its moisture content does not rise steeply in the region RH = 20–60% as the relative humidity of the air increases (Fig. 17.5). 17.1.6.3

Hygroscopic materials (ERH < 40%)

As the appropriate isotherm shows, even low values of the relative humidity of the air may cause a rapid increase in the moisture content of a strongly hygroscopic substance. Typical

538

Confectionery and Chocolate Engineering: Principles and Applications

20

Moisture content (m/m%)

18 16 14 12 10 8 6 4 2 0 0

10 20 30 40 50 60 70 80 90 100 ERH (%)

Fig. 17.5 Isotherm of cocoa powder at room temperature. ERH = equilibrium relative humidity. Note that this figure is for information only [reproduced from Meursing (1976) by kind permission of ADM, see The Cocoa Manual (1998) ADM Cocoa]. Table 17.5 Approximate equilibrium relative humidity (ERH) for confectionery products [reproduced from Minifie (1999), by kind permission of Springer Science and Business Media]. Class of confectionery products or raw materials

ERH (%)

1. 2. 3. 4. 5. 6. 7.

15–25 25–30 35–50 50–60 60–56 65–75 See text

Water cookies (freshly baked), roast nuts, fresh cereals (corn flakes) Hard candy, aerated high boilings, hard toffee, butterscotch Hard caramels, hard nougats (ungrained), soft cookies, milk crumb Gums, pastilles, low-moisture jellies, liquorice pastes, soft caramels Soft fondants, marzipan, pastes, fudge Soft marshmallows, Turkish delight, some fondants, fruit jellies, soft nougats (grained) Chocolate coatings, compound coatings, lozenge pastes, dragées

hygroscopic substances in the confectionery industry are fructose, invert sugar, sorbitol and glycerine. The order of hygroscopicity and solubility is the same: glucose < sucrose < invert sugar < fructose, i.e. the hygroscopicity and solubility rise in parallel. Minifie (1999) has given approximate ERH intervals for several confectionery products (Table 17.5). Group 7 needs some comment. The hygroscopicity of these products, the moisture content of which is mostly about 1 m/m%, is dependent on their ingredients. Dark chocolate will start to absorb moisture at RH = c. 85%, and milk chocolate at c. 78%. The water activity of normal chocolate is about 0.1–0.2. – these low values accord with the fact that the continuous dispersion phase of chocolate is cocoa butter; see Section 17.1.6.1.

17.1.7

Calculation of equilibrium relative humidity of confectionery products

The basic assumption of this calculation is that the dry content of a substance can be divided into two parts: sucrose and others. The non-sucrose part of the dry substance is

Water activity, shelf life and storage

539

taken into account as an ‘additional value of sucrose’ sA, and the values for several types of substances are calculated by the use of special factors. According to Grover (1947), the additional value of sucrose sA is calculated from sA = 1.994 − 0.339 ( g + i ) + 0.038 ( g + i )

2

(17.19)

where sA is the content of non-sucrose ingredients (g/g of water), g is the content of glucose (g/g of water) and i is the content of invert substances (g/g of water). The special factors for calculating sA for various kinds of non-sucrose substances are: sucrose = 1; invert sugar, gelatine, milk protein, sorbitol = 1.3; glucose, glucose dry content, pectin and other jellifying materials = 0.8; fruit acids = 2.5. It can be seen that the typical hygroscopic substances are taken into account with a factor greater than 1. The non-hygroscopic substances that are less hygroscopic than sucrose are calculated with a factor less than 1. The next step in the calculation is sT = s + sA

(17.20)

where s is the content of sucrose in formula g/g of water, and sT is the content of calculated total sucrose (g/g of water). Finally, aW (= ERH/100%) is calculated according to Eqn (17.1). Völker (1968) studied the applicability of the Grover equation (Eqn 17.19) in detail. Cakebread (1969a,b, 1970a,b, 1971a–g, 1972a–c) studied the ERH properties of confectionery products and modified the Grover equation, since, according to Eqn (17.19), the solubility should increase after a decrease if sT increases: sA =

1.994 1 + 0.1775 ( g + i )

(17.21)

Applying Raoult’s law, Money and Born (1951) suggested the following equation for calculating ERH: ERH =

100 1 + 0.27N

where N is the number of moles dissolved in 100 g of water. Example 17.5 Let us consider hard-boiled drops of the following composition (%): water, 1.5; sucrose, 65.6; glucose syrup, 35.0; citric acid, 0.9. Using the Grover equation (Eqn 17.19),

(17.22)

540

Confectionery and Chocolate Engineering: Principles and Applications

g = 35 ×

0. 8 2.5 = 18.67, i = 0.9 × = 1.2, g + i = 19.87 1.5 1.5

sA = 1.994 − 0.339 ( g + i ) + 0.038 ( g + i ) = 1.994 − 0.339 × 19.87 − 0.038 × 19.872 = 10.25 2

sT = s + sA =

65.6 + 10.25 = 53.98 1.5

The (modified) Raoult concentration of sucrose is m2 =

1000 53.98 × = 105.2 ( moles of sucrose 1000 g of water ) 1.5 342

According to Eqn. (17.7), aw =

55.56 55.56 = = 0.3456 55.56 + m2 55.56 + 105.2

ERH = 34.56%. Using the Cakebread equation (Eqn 17.21), sA =

1.994 1.994 = = 0.44 1 + 0.1775 ( g + i ) 1 + 0.1775 × 19.87

sT = s + sA =

65.6 + 0.44 = 44.17 1.5

m2 =

1000 44.17 × = 86.1 1.5 342

aw =

55.56 55.56 = = 0.392 55.56 + m2 55.56 + 86.1

ERH = 39.2%. Let us calculate ERH using the Money–Born equation (Eqn 17.22): ERH =

100 1 + 0.27N

(17.22)

where N is the number of moles dissolved in 100 g of water. According to Grover, N = 10.52 and ERH = 100/(1 + 0.27 × 10.52) = 26.04%. According to Cakebread, N = 8.61 and ERH = 100/(1 + 0.27 × 8.61) = 30.12%. The results of these calculations are summarized in Table 17.6. According to Minifie (1999), the ERH of hard-boiled drops is 25–30%, with which the values calculated above according to the Money–Born equation are in good agreement. Minifie (1999) gave methods for determining the water activity of confectionery products and raw materials. Further references are Mansvelt (1962), Jansen (1969), Lees (1972) and Thieme (1972).

Water activity, shelf life and storage

541

Table 17.6 Calculation of equilibrium relative humidity (ERH) for hardboiled drops according to several models. Principle Grover equation Cakebread equation Money–Born equation (According to Grover) (According to Cakebread)

17.2 17.2.1

N

ERH (%) 34.56 39.2

10.5 8.61

26.04 30.12

Shelf life and storage Definition of shelf life

The shelf life is the period of time that starts at the moment a food has been made and up to end of which the quality of the food remains faultless. Shelf life covers all parts of the period, such as storage in the factory, in trading, in storehouses and finally in the home of the consumer. The shelf life and the stability of foods are concepts closely connected to each other. The shelf life is defined by the concept of marginal (or Lyapunov) stability (see Appendix 6 for details). In order to understand the optimal conditions of storage, let us consider the factors influencing shelf life.

17.2.2

Role of light and atmospheric oxygen

Light and oxygen promote destabilization reactions of fats/oils (autoxidation), proteins and sugars, and consequently the conditions of storage and the properties of the packaging have to be chosen to isolate foods from the effect of light and oxygen.

17.2.3

Role of temperature

There is an optimal interval of temperature for every food, which is dependent on its specific quality properties. For example, in the case of chocolate, the microbiological aspects are usually not a priority but the melting properties of the cocoa butter in it (polymorphism etc.) must be taken into account. In contrast, marzipan is sensitive to macrobiological (yeast) infections, and therefore the temperature of storage is important from this point of view. This means that a rise in temperature may cause problems. On the other hand, a decrease in temperature may also be damaging: if dew condenses from the ambient air onto the surface of the food, moulding may start. Thus the effect of temperature on the humidity of air is an important aspect of storage. Finally, it should be mentioned that temperature differences in food induce water migration and the equalization of mechanical stresses.

17.2.4

Role of water activity

The main types of deterioration related to the water activity of a medium are: • In the interval of water activity aw = 0.2–0.4, lipid oxidation is relatively slow; at lower and higher values it is stimulated.

542

Confectionery and Chocolate Engineering: Principles and Applications

• Non-enzymatic browning starts at about aw = 0.2 and reaches its maximum at about aw = 0.8. • Enzymatic activity starts at about aw = 0.3, mould growth starts at about aw = 0.75, yeast growth starts at aw = 0.8 and bacterial growth starts at aw = 0.9 – all these four types are stimulated by an increase in aw.

17.2.5

Role of enzymatic activity

Enzymatic activity may be important in products containing lauric acid (e.g. coconut confectioneries, and compounds made with hydrogenated coconut oil or palm kernel oil) because of saponification. It is well known that if the moisture content of coconut products exceeds 3–4 m/m%, the risk of saponification becomes high. Enzymatic activity is closely connected with the microbiological state of the raw materials and with infections during production. The measures prescribed by good manufacturing practice and hazard analysis of critical control points, etc. should be taken into account to prevent such problems. An everyday problem is the defence of products against moulding if the water activity exceeds aw = 0.75. Conditions in which this high value of water activity can develop may originate from: • Ambient air (supposing that the packaging does not defend the product against wetting): – if the climate is hot and wet; – if the ambient temperature is low (air humidity is condensed on the packaging). • The packaged product: – if the water activity of the product is higher than 0.75, the RH of the small air space above the product becomes equal to this high value since the packaging does not allow the humidity to escape through it (the product becomes ‘stuffy’); – if the product continuously releases its water content because of syneresis, which is a common phenomenon in, for example, jellies. If the packaging material is not capable of permitting continuous water diffusion of appropriate extent, the product will be ‘stuffy’. Both phenomena become more pronounced when the product is cooled. Non-enzymatic browning is important in confectionery practice; see Chapter 16. In conclusion, it can be established that packaging plays an important role in shelf life.

17.2.6

Concept of mould-free shelf life

The concept of the mould-free shelf life (MFSL) is connected to the important role of moulding, which may reduce the shelf life of food. A method has been worked out that takes into account the permeability of the packaging material, and the length of time during which the water activity of the food reaches a critical value can be calculated in this way. A unique use of isotherm equations and a storage stability map has been made by a research group from MIT for the prediction of the shelf life of foods stored in semipermeable films; see Mizrahi et al. (1970a,b), Simon et al. (1971), Labuza et al. (1972a,b) Quast and Karel (1972) and Quast et al. (1972). In these studies, it was shown how the reaction kinetics of deterioration as determined under steady-state conditions could be applied to a condition where moisture was being slowly transported into a package.

Water activity, shelf life and storage

543

Let us take the case of moisture adsorption in a package of a food (Labuza 1975, pp. 214–217). One limiting acceptability factor is the critical water activity mcrt at which mould will grow, or at which the product will lose its crispness (e.g. for wafers) or the surface will become sticky (e.g. for sugar bonbons). The rate at which a package of food picks up water is modelled by a first-order reaction: dw ⎛ k ⎞ = A ( Pout − Pe ) dt ⎝ x ⎠

(17.23)

where t is the duration of storage (hours, days or months), w is the weight of water gained by the package (kg), k is the permeability of the package to water (kg/m h), x is the film thickness (m), A is the film area (m2), Pout is the outside relative water vapour pressure (%/100%), assumed constant, and Pe is the relative vapour pressure of water in equilibrium with the food (%/100%). If the assumption is made that the limiting resistance to water vapour flow is the film, then the water in the package should equilibrate rapidly, i.e. Pe ≈ Pt, where Pt is the instantaneous relative vapour pressure of water in equilibrium with the food (%/100%) as a function of the time t. Thus, Pt can be determined from the isotherm. In the simplest case, a linear isotherm is assumed: mout − mt = C ( Pout − Pt )

(17.24)

where mout is the outside equilibrium moisture content (%/100%), mt is the instantaneous moisture content of the food (at time t) (%/100%) and C is the constant of the linear isotherm. If wS is the weight of dry solids in the food, then

(dw dt ) wS

=

dm ⎛ k ⎞ ⎛ A ⎞ ⎛ k ⎞ ⎛ A ⎞ ( mout − mt ) = ⎜⎝ ⎟⎠ ( Pout − Pe ) = ⎝ ⎠ ⎜⎝ ⎟⎠ ⎝ ⎠ dt x wS x wS C

(17.25)

i.e. dmt ⎛ k ⎞ ⎛ A ⎞ ( mout − mt ) = = K ( mout − mt ) ⎜ ⎟ dt ⎝ x ⎠ ⎝ wS ⎠ C

(17.26)

where w/wS = mt, and K = (k/x)(A/wS)/C is a constant. After integrating Eqn (17.26) from t = 0 to t = tcrt, 1 ⎡ m − m0 ⎤ tcrt = ⎛ ⎞ ln ⎢ out ⎝ K ⎠ ⎣ mout − mcrt ⎥⎦

(17.27)

where tcrt is the shelf life of the food investigated (hours or days), m0 is the initial moisture content of the food (at t = 0) and mcrt is the critical water activity in the food. The procedure for determination of the shelf life is demonstrated in Fig. 17.6. If the outside humidity is lower than the instantaneous water activity of the food, the water content of the food will decrease, i.e. the food will dry. In this case Eqn (17.23) can be applied in the form

544

Confectionery and Chocolate Engineering: Principles and Applications

Outside relative humidity

Water activity

mout

Critical water

mcrt

activity Changing water activity of food

m0

0 tcrt

t

(shelf life)

Time (h)

0

Fig. 17.6

Determination of shelf life of foods.

dw k = − ⎛ ⎞ A ( Pe − Pout ) ⎝ x⎠ dt

(17.28)

The corresponding differential equation is dmt = − K ( mt − mout ) dt

(17.29)

The change in the water activity of the food is shown by a curve similar to that in Fig. 17.6 but reflected in the line mout, i.e. in this case mout < m0. The integration of Eqn (17.29) (from t = 0 to t = tcrt) gives 1 ⎡ m − mout ⎤ tcrt = ⎛ ⎞ ln ⎢ 0 ⎝ K ⎠ ⎣ mcrt − mout ⎥⎦

(17.30)

In both cases (see Eqns 17.27 and 17.30), mt = mout when t → ∞, i.e. after an infinite time the water activity of the product reaches the water activity (relative humidity) of the environment. Note that in Eqns (17.26) and (17.29) the coefficient of mt is negative – this fact makes possible the stabilization of both hydration (Eqn 17.36) and dehydration (Eqn 17.39) of the product. Example 17.6 A filled wafer (m0 = 3.0%) is packed and stored under the following conditions: the air humidity is Pout = mout × 80% and the critical water activity of the product is mcrt = 5.0%. The size of the package is 210 × 100 × 15 mm, and the solid content of the product is 150 g (the total mass of the package is 160 g). The thickness of the packaging film is x = 2.5 × 10−5 m. The permeability of the film is k = 3.75 × 10−6 kg/(m × month).

545

Water activity, shelf life and storage

Two points (%) of the linear isotherm of the wafer m (%) = CP (%) are (m = 8.0, P = 90) and (m = 2.0, P = 20), from which C = (8.0 − 2.0)/(90 − 20) = 6/70. The equation of the linear isotherm is 6 m − 2 = ⎛ ⎞ ( P − 20 ) ⎝ 70 ⎠ 6 m = ⎛ ⎞ ( P − 20 ) + 2 ⎝ 70 ⎠ and therefore 6 6 mout = ⎛ ⎞ ( Pout − 20 ) + 2 = ⎛ ⎞ (80 − 20 ) + 2 = 7.143% ⎝ 70 ⎠ ⎝ 70 ⎠ 1 ⎡ m − m0 ⎤ tcrt = ⎛ ⎞ ln ⎢ out ⎝ K ⎠ ⎣ mout − mcrt ⎥⎦

(17.30)

where K = (k/x)(A/wS)/C. From the geometry of the product, A = 2 ( 0.21 × 0.1 + 0.21 × 0.015 + 0.1 × 0.015) m 2 = 0.026 m 2 7.143 − 3.0 ⎤ ⎡ m − m0 ⎤ ln ⎢ out = ln ⎡⎢ = 0.6592 ⎥ ⎣ 7.143 − 5.0 ⎥⎦ ⎣ mout − mcrt ⎦ K=

( k x )( A wS ) ⎛ 3.75 × 10 −6 ⎞ ⎛ 0.026 ⎞ ⎛ 70 ⎞ =⎜ = 0.3033 month −5 ⎟ C

⎝ 2.5 × 10

⎠ ⎝ 0.15 ⎠ ⎝ 6 ⎠

0.6592 ⎞ tcrt = ⎛ months = 2.17 months ⎝ 03033 ⎠ From the isotherm, the value of Pcrt corresponding to mcrt can also be calculated: 6 5 = ⎛ ⎞ ( Pcrt − 20 ) + 2 → Pcrt = 55% ⎝ 70 ⎠ This value of air humidity is lower than the boundary of the risk of moulding (RH = c. 75%) but is necessary for preserving the crispness of the wafer. This example demonstrates that a high humidity in the storehouse cuts the shelf life of a product.

Another formula for calculating the mould-free shelf life assumes that during storage the water activity of the food does not change and, principally, the lower the water activity is, the longer the mould-free shelf life will be: log10 MFSL ( days ) = 7.91 − 8.1aw

(17.31)

546

Confectionery and Chocolate Engineering: Principles and Applications

Table 17.7 Calculated values of mould-free shelf life (MFSL). aw

log10(MFSL)

MFSL (days)

0.2 0.5 0.7 0.9

6.29 3.86 2.24 0.62

1949844.6 7244.3596 173.78008 4.1686938

Table 17.8 Water vapour permeability (WVP) of some packaging materials (reproduced from Minifie (1999), by kind permission of Springer Science and Business Media).a Material Cellulose film 300 P (30 g/m2) Cellulose film 300 MS (33 g/m2) Cellulose film 330 XS (33 g/m2) Polyamide 6 (Capran) Polyamide 11 (Rilsan) EVA (ethyl vinyl alcohol) PVC (plasticized) PVC (non-plasticized) Polyester Polyethylene LD Polyethylene MD Polypropylene (bioriented) PVDC (extruded) Aluminium foil (25 μm) Aluminium foil (12 μm) Aluminium foil (9 μm)

WVP [g/(m2 × 24 h)] 1000 18 5 220 55 36 35 35 22 22 6 8 5 0.05 1.5 4

Gauge of all materials: 25 μm unless otherwise specified.

a

at 21°C. Table 17.7 shows the values of MFSL calculated for some values of water activity from Eqn (17.31). It can be seen that an increase in water activity radically (exponentially) decreases the value of the probable MFSL. Minifie (1999) has given data on the water vapour permeability (WVP) of some packaging materials (Table 17.8). The relationship between the WVP given by Minifie and k is WVP [ g ( m 2 × 24 h )] = 30 WVP × 10 −3 [ kg ( m 2 × month )] = k (25 × 10 −6 m ) → k [ kg ( m × month )] = 30 × 25 × WVP × 10 −9 = 0.75 × WVP × 10 −6. For example, if WVP = 5, then k = 3.75 × 10−6 kg/(m × month).

Water activity, shelf life and storage

547

The actual shelf life of confectionery products should be experimentally determined in each case – none of the theoretical and empirical equations, although they are indicative, predicts with sufficient accuracy the behaviour of products with respect to flavour/offflavour, odour, brightness, crispness etc. For more details, see Minifie (1999) and Man and Jones (2000).

17.3

Storage scheduling

Fuzzy logic seems to be suitable for describing the properties of a stored product that has a definite value of shelf life. During the time interval up to the end of the shelf life, the quality properties of the product are changing, and although this period can be partitioned with the help of qualitative characteristics, picking out the quantitative thresholds of such parts may be open to dispute. The essence of the matter is that every lot with the same shelf life is a little different, and during storage the differences are likely to become more evident. However, a trader has no means to check them to decide how to manage the lot in terms of shelf life. Fuzzy logic can handle this vagueness phenomenon. The model presented in Fig. 17.7 is proposed. In the example shown in the figure, the shelf life of a product is 12 months, which is partitioned into four parts: Fresh, Good, Medium and Fair. These ‘labels’ qualitatively characterize the product over 12 months. The membership function m(xi) measures the truth related to the given quality property, e.g. m(x1) measures the change of freshness, which is maximum (= 1) at the start (month = 0) and is minimum (= 0) at the end of the third month. The corresponding membership functions of the model are: For the interval ‘Fresh’: m ( x1 ) = 1 −

x1 3

if 0 ≤ x1 ≤ 3

(17.32)

For the interval ‘Good’: m ( x2 ) =

x2 − 2 2

if 2 ≤ x2 ≤ 4

(17.33)

Membership

Shelf life

x1

Fresh

x2

x3

x4

Good

Medium

Fair x5

0 0

Fig. 17.7

3

6 Months

Model for storage scheduling by fuzzy logic.

9

Stale

1

12

548

Confectionery and Chocolate Engineering: Principles and Applications

and m ( x2 ) =

6 − x2 2

if 4 ≤ x2 ≤ 6

(17.34)

For the interval ‘Medium’: m ( x3 ) =

x3 − 5 2

if 5 ≤ x3 ≤ 7

(17.35)

7 − x3 2

if 7 ≤ x3 ≤ 9

(17.36)

and m ( x3 ) =

For the interval ‘Fair’: m ( x4 ) =

x4 − 8 2

if 8 ≤ x4 ≤ 10

(17.37)

and m ( x4 ) =

10 − x4 2

if 10 ≤ x4 ≤ 12

(17.38)

For the interval ‘Stale’: m ( x5 ) =

x5 − 11 if 11 ≤ x5 ≤ 12 2

(17.39)

Both the shelf life and the partitioning of it must be adapted to the given task. The fuzzy approach provides doubly covered time intervals ({2; 3}, {5; 6}, {8; 9}, {11; 12} in the present example), which show the manager that a new period will be starting soon. The simplest way of applying this approach is to generate a signal by a clock in the store documentation system that links a colour to each period. For example, the numbers showing the amount of products ‘Fresh’ may be green (and ‘Good’ may be represented by blue, ‘Medium’ by yellow, ‘Fair’ by orange and ‘Stale’ by red), and the doubly covered time intervals may be linked to intermediate colours (green → blue etc.).

Further reading Buck, D.F. and Edwards, M.K. (1997) Anti-oxidants to prolong shelf life. Food Tech Int 2: 29–33. Cooper, R.M., Knight, R.A., Robb, J. and Seiler, D.L.A. (1968) The Equilibrium Relative Humidity of Baked Products with Particular Reference to the Shelf Life of Cakes. FMBRA Report 19. Duckworth, D. (1975) Water Relations of Foods. Academic Press, London.

Water activity, shelf life and storage

549

ERH Calc, A Versatile Computer Program for Calculating Water Activity and Predicting Mould-free Shelf Life for Perishable Products. Campden & Chorleywood Food Research Association, UK. Labuza, T.P. (1984) Moisture sorption: Practical aspects of isotherm measurement and use. American Association of Cereal Chemists, St Paul, MN. Leistner, L. (1970) Bedeutung des pH-Wertes, des Redox-potentiales und der Wasseraktivitaet fuer die Praxis des Fleischwarenherstellung. Arch Lebensmittelhyg 21: 121–126; Einfluss der Wasseraktivitaet auf Vermehrungsfaehigkeit von Mikroorganismen. Arch Lebensmittelhyg 21: 264–267. Manley, D. (1998) Biscuit, Cookie and Cracker Manufacturing Manuals (Vol. 1: Ingredients; Vol. 2: Biscuit Doughs; Vol. 3: Biscuit Dough Piece Forming; Vol. 4: Baking and Cooling of Biscuits; Vol. 5: Secondary Processing in Biscuit Manufacturing; Vol. 6: Biscuit Packaging and Storage). Woodhead Publishing, Cambridge. Oswin, C.R. (1983) Package Life Theory and Practice. The Institute of Packaging, UK. Institute of Food Science and Technology (1993) Shelf-Life of Foods – Guidelines for Its Determination and Prediction. Institute of Food Science and Technology, London. Reade, M.G. (1970) Hygrometry of confectionery products. Ingredients and methods for testing. Confect Prod 36 (2): 94–99, 116. Reade, M.G. (1970) Hygrometry of cocoa and chocolate products. Confect Prod 36 (8): 475–478, 494. Reade, M.G. (1970) Hygrometry of cocoa and chocolate products. Confect Prod 36 (9): 557–562. Reade, M.G. (1970) Hygrometry of cocoa and chocolate products. Confect Prod 36 (10): 619–625. Sieler, D.A.L. (1978) The Microflora of Cake and Its Ingredients. Cake and Biscuit Alliance Technologists’ Conference. Sieler, D.A.L. (1984) Preservation of bakery products. Proc Inst Food Sci Technol 17 (1): 31. Wedzicha, B.L. and Quine, D.E.C. (1983) Sorption of water vapor by wafer biscuits. Lebensm Wiss Techn 16: 115. Wolf, W., Speiss, W.E.L. and Jung, G. (1989) Sorption Isotherms and Water Activity of Food Materials. Science and Technology Publishers, London.

Chapter 18

Stability of food systems

Contents 18.1 Common use of the concept of food stability 18.2 Stability theories: Types of stability 18.2.1 Orbital stability and Lyapunov stability 18.2.2 Asymptotic and marginal (or Lyapunov) stability 18.2.3 Local and global stability 18.3 Shelf life as a case of marginal stability 18.4 Stability matrix of a food system 18.4.1 Linear models 18.4.2 Nonlinear models

18.1

550 550 550 551 552 552 553 553 554

Common use of the concept of food stability

The use of the concept of ‘stability’ in food science does not entirely agree with its use in physics and chemistry because the word ‘stable’ is used to mean ‘reliable’, ‘correct’ or ‘sound’. However, one of the essential aims of the technology is to produce an appropriately unstable food structure; for example, fresh bread is rather unstable but dry bread is close to a stable state. Another provocative example is that the processes of deterioration lead to stable states that are not wanted. In addition to deterioration, several types of ageing (Ostwald ripening, browning phenomena, hydrolysis, inversion etc.) can be mentioned as well. These are well-known facts. On the other hand, the concept of a ‘stable state’ has a very real meaning in relation to food processing and storage: it relates to a well-defined system of conditions, among them that the food product must be preserved. One of the essential technological targets is to establish the characteristics of a food that meet the requirements of food safety and food quality (laid down in the specification of the product) during a specified period of storage (called the ‘shelf life’). How is the concept of stability used in food engineering?

18.2 18.2.1

Stability theories: types of stability Orbital stability and Lyapunov stability

Stability theory was founded by Lyapunov (1892) and Poincaré at the end of the 19th century. Poincaré dealt with orbital stability, concerning the cyclic motion of orbits (the Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

Stability of food systems

551

‘three-body problem’). His investigations led to the foundation of perturbation theory. Since orbital stability seems to be uninteresting from our point of view, however, it is not discussed in the following. Lyapunov worked out the mathematical principles of stability and methods for deciding whether a system is stable or unstable. His theory includes definitions of the types of stability, their characteristics (‘Lyapunov numbers’), and functions for deciding the existence of stability. These Lyapunov functions simplify the description of the behaviour of a system in a similar way to the potential function for a central force field.

18.2.2

Asymptotic and marginal (or Lyapunov) stability

Let us characterize a system by a point P on a trajectory x that has n parameters as follows: ⎡ x1 ⎤ ⎢x ⎥ 2 P1× n( x ) = ⎢ ⎥ ⎢⎥ ⎢ ⎥ ⎣ xn ⎦ i.e. P(x) is described by a column vector x of n dimensions, where the parameters x are functions of the time t (P has 1 column and n rows, as the usual designation 1 × n shows.) The evolution of this system can be described by the trajectory P(x; t): P ( xi ; t = 0 ) → P ( xi ; t = tf )

(18.1)

where the xi are the variables of the substantial changes (i = 1, 2, … , n); in the case of a food, the variables x can be physical, chemical or microbiological characteristics, characteristics related to consistency, etc.; t = 0 is the initial value of t, and t = tf is the final value of t when the process is stopped. To define the various types of stability, let us consider two circles of radii r and R (r < R), with a common centre. The initial point of the trajectory P(xi; t = 0) is located inside a circle of radius r. On the basis of the behaviour of this trajectory, the following types of stability can be distinguished. 18.2.2.1

Asymptotic stability

• The trajectory remains inside the circle of radius R; and • its end is inside the circle of radius r. A special case is exponential stability, in which the trajectory reaches the equilibrium state fastest, out of all the various asymptotic trajectories. In many cases the system can be described by a linear model (see below), the elements of which determine the stable or unstable behaviour. If all these elements are negative, the system is exponentially stable. Because nonlinear models are mostly very complicated, a linear approximation is used if possible. (This approximation is used for modelling the Maillard reaction that takes place in batch conching; see Section 16.5.2.) Therefore, exponential stability is a highlighted topic of stability theory.

552

Confectionery and Chocolate Engineering: Principles and Applications

18.2.2.2

Marginal (or Lyapunov) stability

• The trajectory remains inside the circle of radius R; but • it does not return into the circle of radius r. The term ‘Lyapunov stability’ is applied in all problems that are not connected to cyclic motion, and this type of stability is deduced in all cases where the trajectory does not leave the circle of radius R. Essentially, asymptotic stability can be regarded as a subconcept of marginal (or Lyapunov) stability. 18.2.2.3

Instability

• The trajectory leaves the circle of radius R. 18.2.2.4

Blowing up

This is a special type of instability that refers to the case in which the trajectory very quickly diverges to infinity. In linear systems, instability is equivalent to blowing up because unstable poles always lead to exponential growth of the system states. However, for nonlinear systems, blowing up is only one form of instability.

18.2.3

Local and global stability

If asymptotic (or exponential) stability holds for any initial state, the equilibrium point is said to be globally asymptotically (or exponentially) stable. Marginal stability is a type of local stability that refers to a given time interval only. The distinction between global and local stability needs tools for analysis of the trajectory function, which is a function of many variables in general. For further details, see Slotine and Li (1991). It is important to point out the fact that in many engineering applications, marginal (Lyapunov) stability is not enough; for example, in mechanical systems, asymptotic stability may be an essential requirement. However, in food science, marginal stability seems to be essential, as a type of local stability.

18.3

Shelf life as a case of marginal stability

The concept of shelf life can be defined (Mohos 1990) by a time value tsh as follows. There are vectors Pmin and Pmax prescribed by the product specification so that for every xi = xi(t) (where i = 1, 2, … , n), Pmin ≤ P ( x; t ) ≤ Pmax

(18.2)

in the time interval t = 0 → t = tsh. Let us consider how this definition relates to the different types of stability. The concepts of stability expressed above and the definition in Eqn (18.2) can be related as follows:

Stability of food systems

553

Radius of circle

Unstable Marginally stable R Asymptotically stable

x(t) r 0 0

Fig. 18.1

Shelf life

Time t

Shelf life defined as a kind of marginal (Lyapunov) stability.

P ( x; r )min ≤ P ( x; t ) ≤ P ( x; R )max

(18.3)

in the time interval t = 0 → t = tsh, where P(x; r)min symbolizes the circle of radius r and P(x; R)max symbolizes the circle of radius R. The evolution of x(t) can be imagined as an advance in a tube, the cross-section of which is of radius R and the length of which is defined by the coordinate t (Fig. 18.1). The shelf life is the time interval in which x(t) does not leave the region {P(x; r)min; P(x; R)max}, taking account of every variable. This definition is characteristic of marginal (Lyapunov) stability. It should be mentioned that asymptotic stability is not suitable for defining food stability; for example, dried jelly is asymptotically stable but defective.

18.4 18.4.1

Stability matrix of a food system Linear models

In the simplest case, the evolution of a system can be described by one variable in a linear form, i.e. dx/dt = kx, where k is constant, and P(x; t) is an integral of this differential equation. If k is negative, we have asymptotic stability; if k is positive, we have instability. The form of linear systems is similar: d ( x ) = Kx dt

(18.4)

where x is a column vector of variables with n rows, and K is an (n × n) matrix of constants (the ‘stability matrix’). The investigation of the stability of linear systems can be done with the help of the K matrix. The case of nonlinear systems is much more difficult. For further details, see Borbély (1961) and Slotine and Li (1991).

554

18.4.2

Confectionery and Chocolate Engineering: Principles and Applications

Nonlinear models

The stability matrix of a linear model is a practical formulation of the fact that: • There are n linear differential equations. • The system characterized by these equations as uniform whole can be handled by mathematical tools from the point of view of stability. In this case, every equation has a solution of the same form, and the only difference is manifested in the positive or negative value of the elements of the stability matrix. If the differential equations describing the system’s kinetics are not linear, such a general form of solution cannot be given: every case has to be studied one by one. A stability matrix can be also constructed for nonlinear models. However, it consists of qualitative relations only: if A and B are model variables, the related element of the stability matrix rA,B expresses merely the fact that the model contains also the differential equation dA/dt = F(B). Notwithstanding, the stability matrix expresses a requirement for considering the system as a uniform whole with respect to stability, although the mathematical possibilities do not provide a uniform solution. In this narrow sense, the stability matrix of a food system refers to parameters (variables) of the following types: • • • •

physical (P); chemical (C); sensory (S); and microbiological (M).

Here, P, C, S and M are sets of individual parameters, for example • • • •

P: viscosity (p1), elasticity (p2), interfacial tension (p3), etc.; C: water content (c1), pH (c2), reducing sugar content (c3), etc.; S: odour (s1), colour (s2), stickiness (s3), fracture properties (s4), etc.; M: numbers of cells of various bacteria (m1), total number of cells (m2), etc.

Such methods for the investigation of food stability are not in widespread use. An attempt has been presented using the example of conching; see Section 16.5.2.

Part IV

Appendices

Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

Appendix 1

Data on engineering properties of materials used and made by the confectionery industry

Contents A1.1 A1.2 A1.3

A1.1

Carbohydrates Oils and fats Raw materials, semi-finished products and finished products

557 566 567

Carbohydrates

Table A1.1

Thermophysical data for sugar (Antokolskaia 1964).

Type

T (°C)

ρ (kg/m3)

λ (W/m degree)

c (W s/kg degree)

a × 106 (m2/s)

Cube

−5 15 35 – – 15 35 60

– 1600 – 900 660 1198 1188 1160 1538.4 (Sokolovsky 1959) 1571.4 (Sokolovsky 1959) 1500 (Sokolovsky (1959)

0.157 0.153 0.145 0.139 0.139 0.336 0.334 0.353

1340 1361 1390 712 879 3207 3199 3408

0.1347 0.1280 0.1194 0.2390 0.2360 0.0875 0.0833 0.0894

Castor1 Icing2 Invert

Dextrose3 Dextrose hydrate Maltose 1

0.1% water content. 0.3% water content. 3 Water-free. 2

For further details, see Desent and Bouscher (1961). Table A1.2 1964).

Thermal conductivity of crystalline sugars (Antokolskaia

λ (W/m degree)

Type Powder Raffinated Light pressed Castor Raw

Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

0.069 0.081 0.093 0.58 0.17

Ferenc Á. Mohos

558

Confectionery and Chocolate Engineering: Principles and Applications

Table A1.3 Specific and molar heat capacity of crystalline sugars (Antokolskaia 1964). T (°C) 0 20 30 40 50 60 70 80 90

c (W s/kg degree)

C (W s/mole degree)

1088.57 1214.17 1256.04 1323.03 1356.52 1419.32 1469.57 1532.37 1578.42

385 415 435 452 464 485 502 523 540

604 749 427 174 734 668 416 350 097

Table A1.4 Melting point of sugars (Antokolskaia 1964). Sugar

Melting point (°C)

Dextrose Maltose Sucrose Raw sugar Powdered sugar Fructose

146 108 170–188 170–188 170–188 104

Table A1.5 Density of saturated sugar solutions as a function of temperature (Antokolskaia 1964). T (°C) 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Sugar content (g/100 cm3 water)

Density (kg/m3)

179.2 184.7 190.5 197.0 203.9 211.4 219.9 228.4 238.1 248.7 260.4 273.1 287.3 302.9 320.5 339.9 362.1 386.8 415.7 448.6 487.2

1314.0 1319.2 1323.5 1328.0 1331.9 1342.7 1342.7 1348.0 1353.5 1359.2 1365.1 1371.2 1377.5 1384.0 1390.8 1397.7 1404.9 1412.2 1419.9 1427.7 1435.9

Appendix 1 Data on engineering properties

559

Table A1.6 Concentration of saturated sugar/water solutions as a function of temperature (Sokolovsky 1958). t (°C) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

%

t (°C)

%

t (°C)

%

64.18 64.31 64.45 64.59 64.73 64.87 65.01 65.15 65.29 65.48 65.58 65.73 65.88 66.03 66.18 66.33 66.48 66.63 66.78 66.93 67.09 67.25 67.41 67.57 67.73 67.89 68.05 68.21 68.37 68.53 68.7 68.87 69.04 69.21

34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67

69.38 69.55 69.72 69.89 70.06 70.24 70.42 70.6 70.78 70.96 71.14 71.32 71.5 71.68 71.87 72.06 72.25 72.44 72.63 72.82 73.01 73.2 73.39 73.58 73.78 73.98 74.18 74.38 74.58 74.78 74.98 75.18 75.38 75.59

68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

75.8 76.01 76.22 76.43 76.54 76.85 77.06 77.27 77.48 77.7 77.92 78.14 78.36 78.58 78.8 79.02 79.24 79.46 79.69 79.92 80.15 80.38 80.61 80.84 81 81.3 81.53 81.77 82.01 82.25 82.49 82.73 82.97

Table A1.7 Thermophysical data for sucrose solutions with various boiling points (Antokolskaia 1964). Concentration (m/m%)

T (°C)

λ (W/m degree)

c (W s/kg degree)

η (N s/m2) × 106

0 10 20 30 40 50 60

100.0 100.2 100.4 100.7 101.2 102.0 103.5

0.682 0.645 0.642 0.570 0.531 0.493 0.456

4187 4120 3864 3626 3358 3256 2939

282.4 366.7 452.1 619.7 960.1 1765.2 3341.1

560

Confectionery and Chocolate Engineering: Principles and Applications

Table A1.8 Thermophysical characteristics of sucrose–water solutions (Sokolovsky 1959). Concentration (m/m%) 20

30

40

50

60

801

T (°C)

λ (W/m degree)

c (W s/kg degree)

ν × 106 (m2/s)

a (Prandtl)

50 60 70 80 50 60 70 80 50 60 70 80 50 60 70 80 50 60 70 80 15

0.5700 0.5809 0.5893 0.5965 0.5368 0.5956 0.5536 0.5604 0.502 0.510 0.518 0.524 0.468 0.475 0.482 0.488 0.433 0.440 0.447 0.452 0.326

3760 3775 3790 3805 3546 3568 3591 3614 3333 3363 3393 3423 3119 3157 3195 3232 2906 2951 2996 3041 1361

0.9065 0.7605 0.6420 0.5610 1.282 1.082 0.9063 0.7750 2.140 1.701 1.389 1.153 4.173 3.148 2.442 1.956 11.02 7.63 5.54 4.15 0.1280

6.38 5.26 4.37 3.76 9.71 7.84 6.49 5.48 16.52 12.97 10.48 8.62 33.82 25.30 19.47 15.50 93.90 64.75 46.82 34.98

1

Tonn (1961). Comment: Since the Prandtl number Pr is equal to ν/a, a = ν/Pr. For example (in the first row), a = (0.9065 × 10−6/6.38) (m2/s) = 0.1421 × 10−6 m2/s.

For further details, see Tonn (1961). Table A1.9 Thermal conductivity (W/m K) of sucrose–water solutions (Antokolskaia 1964). Temperature (°C) Concentration (m/m%) 0 10 20 30 40 50 60

0

20

30

40

0.583 0.551 0.520 0.488 0.457 0.391 0.384

0.599 0.566 0.535 0.501 0.470 0.437 0.405

0.614 0.581 0.548 0.514 0.480 0.449 0.415

0.628 0.594 0.560 0.526 0.492 0.458 0.419

50

60

70

80

0.641 0.607 0.572 0.536 0.502 0.481 0.434

0.652 0.617 0.538 0.547 0.512 0.477 0.441

0.663 0.628 0.592 0.555 0.519 0.484 0.449

0.672 0.636 0.600 0.563 0.526 0.491 0.455

Temperature (°C)

0 10 20 30 40 50 60

Appendix 1 Data on engineering properties

561

Specific heat capacity of aqueous sugar solutions (Sokolovsky 1958, p. 32): cP = 1 − ( 0.6 − 0.0018t ) S where cP is the specific heat capacity of the aqueous sugar solution (kcal/kg = 4.1868 kJ/ kg), t is the temperature (°C) and S is the sugar concentration (m/m). Thermal conductivity of aqueous sugar solutions (Sokolovsky 1958, p. 32):

λ = λ W (1 − 10 −5 × KS ) where λ is the thermal conductivity of the aqueous sugar solution at 20°C (kcal/m h K = 1.163 W/m K); λW is the thermal conductivity of water at 20°C; K is a constant, with a value of 556 at 20°C; and S is the sugar concentration (m/m). The thermal conductivity of an aqueous sugar solution of concentration 80 m/m% (at 20°C) is 0.28 kcal/m h K = 0.32564 W/m K. Thermal diffusivity of crystalline sugar (Sokolovsky 1958, p. 32): a = 4.93 × 10 −4 m 2 h = 4.93 × 10 −4 × 2.778 × 10 −4 m 2 s = 1.3696 × 10 −7 m 2 s

Table A1.10 Boiling point of sucrose–water solutions (Antokolskaia 1964). Concentration (m/m%) 10 20 30 40 50 60 70 80 90

Boiling point (°C) 100.1 100.3 100.6 101.0 101.8 103.0 105.5 109.4 119.6

An approximate formula for the boiling point of sugar–water solutions is (Sokolovsky 1958, p. 19) T (°C ) = 100°C + 2.33 (S W ) where T is the boiling point, S is the concentration of sugar (m/m%) in the solution and W is the concentration of water (m/m%) in the solution. For example, if S = 20% and W = 80%, then T (°C) = 100 + 2.33(20/80) = 100.5825°C (in Table A1.10, 100.3°C). If S = 90% and W = 10%, then T (°C) = 100 + 2.33(90/10) = 120.97°C (in Table A1.10, 119.6°C). For example, if the pressure is 92.51 mmHg (= 92.51 mmHg × 133.2 Pa/mmHg = 12322.332 Pa), then the boiling point of water is 50°C, and an aqueous sugar solution of 70 m/m% concentration has a boiling point t = (50 + 3.65)°C.

562

Confectionery and Chocolate Engineering: Principles and Applications

Table A1.11 Elevation of boiling point (°C) of sugar solutions as a function of concentration at various pressures (according to Bukharov; cited by Sokolovsky 1958, p. 20). Pressure (mmHg = 133.2 Pa) 92.51

149.38

233.7

355.1

525.76

760

Sugar (%)

50°C

60°C

70°C

80°C

90°C

100°C

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

0.05 0.10 0.17 0.26 0.39 0.52 0.69 0.80 1.02 1.32 1.70 2.30 2.80 3.65 5.05 (6.80)

0.05 0.10 0.18 0.27 0.40 0.54 0.71 0.85 1.10 1.40 1.82 2.15 3.00 3.90 5.40 7.30 (10.00)

0.05 0.11 0.18 0.28 0.42 0.55 0.73 0.90 1.18 1.52 1.94 2.60 3.20 4.18 5.80 7.85 10.75 (16.00)

0.06 0.11 0.19 0.28 0.43 0.57 0.76 0.95 1.25 1.61 2.06 2.75 3.40 4.46 6.20 8.35 11.50 17.20

0.06 0.12 0.19 0.29 0.41 0.58 0.78 1.00 1.32 1.72 2.18 2.90 3.60 4.75 6.60 8.90 12.25 18.40

0.06 0.12 0.20 0.30 0.45 0.60 0.80 1.05 1.40 1.80 2.30 3.05 3.80 5.05 7.00 9.40 13.00 19.60

Table A1.12 Solubility of sugar in water in the presence of glucose syrup (Sokolovsky 1958, p. 16).

20°C

50°C

70°C

A

B

C

D

E

67.09 57.51 51.23 4.51 43.26 72.25 62.97 55.05 51.03 46.81 44.47 37.96 76.22 67.43 60.6 55.14 52.7 49.69

0 10.56 17.74 21.76 28.8 0 10.05 18.26 24 28.86 32.02 40.54 0 9.92 17.55 24.95 28.1 32.16

203 180.2 165.09 163.16 154.82 260.36 233.39 208.16 204.37 193.19 189.15 176.56 322.83 207.7 277.35 276.95 274.48 273.77

0 33.1 57.17 73.19 103.07 0 37.25 69.01 96.12 119.52 136.2 188.56 0 43.7 80.32 125.31 146.35 177.19

203 213.3 222.26 236.35 257.89 260.36 270.64 277.17 300.49 312.71 325.35 365.12 322.83 341.49 357.67 402.26 420.83 450.96

A = concentration of sugar (m/m%) in 100 g of solution. B = concentration of glucose syrup (m/m%) in 100 g of solution. C = sugar (g) per 100 g of water. D = glucose syrup dry content (g) per 100 g of water. E = C + D.

Appendix 1 Data on engineering properties

Table A1.13 Solubility of sugar in water in the presence of invert sugar (Sokolovsky 1958, p. 17).

23.1°C

30°C

50°C

A

B

C

D

E

67.59 57.84 47.31 28.66 68.11 56.32 50.97 49.91 48.95 46.36 39.23 32.06 31.85 26.03 21.18 20.59 72.22 62.81 53.8 46.2 35.75

0 11.9 25.39 36.9 – 14.94 21.86 23.21 24.46 28.01 37.48 47.02 47.62 56.37 63.68 64.47 – 11.42 22.65 32.32 46.05

208.55 191.14 173.3 158.18 213.58 195.96 187.6 185.68 184.09 180.88 168.43 153.25 155.13 147.9 139.89 137.82 260.36 243.73 228.45 215.08 196.43

0 39.32 93 150.98 – 51.98 80.46 86.34 91.99 109.29 160.93 224.76 231.95 320.28 420.61 431.52 – 44.31 96.17 150.46 253.2

208.55 230.46 266.3 309.16 213.58 247.94 268.06 272.02 276.08 290.17 329.36 378.01 387.08 468.18 560.5 569.34 260.36 288.04 324.62 365.54 449.45

A = concentration of sugar (m/m%) in 100 g of solution. B = concentration of glucose syrup (m/m%) in 100 g of solution. C = sugar (g) per 100 g of water. D = glucose syrup dry content (g) per 100 g of water. E = C + D.

Table A1.14 Boiling point of aqueous glucose syrup solutions at atmospheric pressure (Sokolovsky 1958, p. 40). G (%) 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 G (%) = glucose syrup dry content (m/m%).

Boiling point (°C) 100.55 100.7 100.85 101.05 101.45 102 102.75 103.75 105.05 106.6 108.4 110.45 113 117.75 127

563

564

Confectionery and Chocolate Engineering: Principles and Applications

Table A1.15 Elevation of boiling point of glucose solutions as a function of concentration at various pressures (according to Bukharov; cited by Sokolovsky 1958). Pressure (mmHg) 92.51

149.8

233.7

355.1

525.76

760

Glucose (%)

50°C

60°C

70°C

80°C

90°C

100°C

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

0.08 0.16 0.25 0.39 0.51 0.62 0.78 1.04 1.15 1.98 2.7 3.63 4.73 6.04 7.47 9.29 12.01 19.14

0.03 0.17 0.26 0.41 0.55 0.66 0.84 1.11 1.55 2.12 2.9 3.9 5.07 6.47 8.02 9.98 13.6 20.5

0.09 0.18 0.28 0.41 0.59 0.70 0.9 1.2 1.66 2.28 3.1 1.17 5.43 6.93 8.58 10.69 14.69 21.08

0.1 0.19 0.30 0.48 0.63 0.75 0.96 1.28 1.78 2.42 3.3 4.45 5.89 7.4 9.17 11.42 15.59 23.62

0.11 0.21 0.32 0.51 0.67 0.80 1.02 1.36 1.9 2.59 3.59 4.75 6.19 7.9 9.79 12.17 16.65 25.27

0.11 0.22 0.35 0.55 0.7 0.85 1.05 1.45 2 2.75 3.75 5.05 6.6 8.l0 10.45 13 17.75 27

1 mmHg = 133.2 Pa.

Table A1.16 Dynamic viscosity (N s/m2 = Pa s) of sucrose–water solutions (Antokolskaia 1964). Temperature (°C) Concentration (m/m%) 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84

20

30

40

50

60

70

80

90

0.0572 0.0678 0.0809 0.0970 0.1171 0.1428 0.1759 0.2190 0.2780 0.3540 0.4600 0.6737

0.0331 0.0386 0.0454 0.0634 0.0636 0.0760 0.0916 0.1113 0.1388 0.1697 0.2141 0.2739 0.3561 0.4695 0.6310 0.8640 1.2140

0.0206 0.0238 0.0275 0.0320 0.0374 0.0441 0.0522 0.0622 0.0747 0.0906 0.1111 0.1381 0.1737 0.2220 0.2885 0.3805 0.5130 0.7010 0.9800 1.430 2.160

0.0137 0.0156 0.0178 0.0204 0.0235 0.0272 0.0317 0.0372 0.0440 0.0524 0.0631 0.0768 0.0945 0.1178 0.1487 0.1900 0.2463 0.3234 0.4330 0.5930 0.8320 1.2000 1.8000

0.0095 0.0107 0.0121 0.0137 0.0156 0.0172 0.0206 0.0239 0.0279 0.0328 0.0388 0.0463 0.0559 0.0682 0.0841 0.1050 0.1330 0.1707 0.2216 0.2927 0.3939 0.5460 0.7700 1.2500 1.7000

0.0069 0.0077 0.0086 0.0098 0.0108 0.0122 0.0139 0.0159 0.0183 0.0212 0.0248 0.0292 0.0348 0.0418 0.0507 0.0620 0.0768 0.0951 0.1215 0.1562 0.2044 0.2722 0.3727 0.5190 0.7400

0.0053 0.0058 0.0064 0.0071 0.0079 0.0088 0.0099 0.0112 0.0127 0.0145 0.0167 0.0194 0.0227 0.0268 0.0319 0.0384 0.0466 0.0572 0.0711 0.0896 0.1152 0.1505 0.2000 0.2800 0.3760

0.0830 0.0940 0.1110 0.1420 0.1860

565

Appendix 1 Data on engineering properties

Table A1.17 Dynamic viscosity of saturated dextrose–water solutions (Sokolovsky 1959). T (°C)

Concentration of dextrose (m/m%)

Dextrose dissolved in 100 g water

Viscosity (N s/m2)

47.72 54.64 61.83 70.91 74.73 78.23 81.83 84.63

91.60 120.46 162.14 243.80 295.00 359.94 436.31 552.77

0.0183 0.0187 0.0224 0.0509 0.0662 0.0784 0.1040

20 30 40 50 60 70 80 90

Table A1.18 Interfacial tension of sugar solutions at 20°C (Antokolskaia 1964). Concentration (g/100 g)

σ (N/m)

Concentration (g/100 g)

σ (N/m)

0 6.8 10.0 13.1 20.5 22.2

0.0727 0.0731 0.07335 0.0736 0.0745 0.0749

29.8 31.0 40.7 47.5 51.2 62.7

0.0760 0.0762 0.0771 0.0780 0.0787 0.0796

Table A1.19 Dynamic viscosity (N s/m2) of honey as a function of temperature (°C) and water content (m/m%) (Antokolskaia 1964). Temperature (°C) Water content (%) 14 16 19 24

10

20

30

40

50

60

70

80

– – 28.0 4.52

59.2 22.8 6.5 1.3

14.4 5.9 2.7 0.5

4.6 2.1 1.0 0.4

1.5 0.9 0.5 0.2

1.0 0.6 0.3 0.1

0.70 0.30 0.20 0.05

0.25 0.15 0.10 0.03

Table A1.20 Thermophysical characteristics of starch syrup used for caramel production (Schriftsammlung der Arbeiten des WKNU 1950). Dry content (m/m%) 80

85

88

92

t (°C)

ρ (kg/m3)

λ (W/m K)

cP (W s/kg K)

a × 106 (m2/s)

20 40 60 80 20 40 60 80 20 40 60 80 20 40 60 80

1420 1395 1370 1340 1450 1425 1400 1370 1480 1455 1430 1400 1520 1490 1460 1430

0.326 0.314 0.314 0.302 0.314 0.302 0.302 0.291 0.314 0.302 0.302 0.291 0.314 0.302 0.291 0.279

1967.8 2009.7 2051.6 2093.4 1884.1 1925.9 1967.8 2008.7 1842.2 1884.1 1925.9 1967.8 1758.5 1800.3 1884.1 1925.9

0.1167 0.1139 0.1111 0.1055 0.1167 0.1111 0.1083 0.1055 0.1167 0.1111 0.1083 0.1055 0.1167 0.1111 0.1055 0.1028

566

Confectionery and Chocolate Engineering: Principles and Applications

According to Sokolovsky (1958, p. 103), the specific heat capacity of starch (cP′ ) can be calculated from cP′ = [ acP + f (100% − a )] 100% where cP = 0.2697 cal/g °C (and f = 1) or cP = 1.1284 kJ/kg °C (and f = 4.1839) is the specific heat capacity of the dry content of starch, and a (%) is the water content of the starch. [This relation is based on the fact that the specific heat capacity of water is about 1 cal/ (grade g) = 4.1839 kJ/(grade kg) and the values of the specific heat capacity are additive according to the ratio of ingredients.] The values of the specific heat capacity of commercial starch products are: cP′ (cal/g)

Temperature (°C) 0–20 21–42 43–62

Table A1.21 1959).

0.2765 0.2978 0.3061

Dynamic viscosity of aqueous solutions of invert sugar as a function of temperature (Sokolovsky

Total dry content (m/m%)

Invert sugar content (m/m%)

t (°C)

η (Pa s)

t (°C)

η (Pa s)

73.65 79.35 78.97 73.65 79.35 78.97

30.0 10.1 20.3 40.2 30.7 30.2

0.663 5.344 23.714 0.304 2.076 7.752

50.0 45.1 45.0 70.5 60.0 70.0

0.140 0.431 1.648 0.060 0.145 0.157

74.00 78.84 81.75 74.00 79.84 81.75

A1.2

Oils and fats

Table A1.22

Temperature dependence of the thermal parameters of margarine (Derneko and Schafchid 1959).

T

For household use, λ (W/m degree)

Cream margarine, a × 106 (m2/s)

20 22 24 26 27 29 32

– 0.165 – 0.172 – 0.173 0.179

0.0736 0.0717 0.0714 – 0.0711 0.0678 0.0667

An approximation for cream margarine in the temperature range 20–32°C is a × 108 = 2.7778 [2.65 − 0.02 (T − 20 )] ( m 2 s ) where T is in degrees Celsius.

Appendix 1 Data on engineering properties

567

Table A1.23 Thermal properties of vegetable oils and fats at 20°C (Kiss 1988). Thermal conductivity λ (106 × W/m K) Sunflower oil Soy oil Sunflower oil (hydrogenated)

0.1675 0.1763 0.1675

Thermal diffusivity a (10 × m2/s) Sunflower oil Soy oil

0.944 1.056

Specific heat capacity cP (kJ/(kg °C) Sunflower oil (raffinated) Soy oil Sunflower oil (hydrogenated)

1.7752 1.8149 2.1311

Specific heat capacity cP(kJ/(kg °C) of hydrogenated vegetable oils cP = cP.20[1 + α(t − 20)] where cP.20 = 2.147 kJ/kg °C and α × 103 = 1.380 K−1 Density ρ (kg/m3) Arachis oil Sunflower oil Soy oil

913.7 918.9 919.4

Dielectric constant (permittivity) εt (F/m) as a function of temperature (t = 0–100°C) Arachis oil εt = 3.051 − 0.00313(t − 20) Sunflower oil εt = 3.11 − 0.0034(t − 20)

A1.3

Raw materials, semi-finished products and finished products

Table A1.24 Thermophysical properties of cocoa butter, cocoa mass and chocolate (Antokolskaia 1964). ρ (kg/m3)

λ (W/m K)

927 910 895 880

0.291 0.325 0.372 0.430

2512 2512 2512 2512

0.1250 0.1444 0.1867 0.1944

Chocolate 0 10 20 35

1235 – – –

0.214 0.223 0.233 0.246

1482.1 1854.7 2122.7 1603.5

0.1172 0.0975 0.0889 0.1244

Cocoa mass 10 30 50 70

1110 1100 1090 1080

0.372 0.360 0.349 0.337

2837.7 2637.7 2637.7 2837.7

0.1279 0.1250 0.1194 0.1197

t (°C) Cocoa butter 10 30 50 70

cp (W s/kg K)

a (106 m2/s)

568

Confectionery and Chocolate Engineering: Principles and Applications

For cocoa mass,

λ = 1.163 ( 0.325 − 0.005t ) W m K a × 108 = 2.7778 ( 4.65 − 0.005t ) m 2 s For chocolate mass (Rapoport and Tarchova 1939), for the temperature range 30–70°C, ρ = (1320 − 0.5t) kg/m3 λ = 1.163(0.2 + 0.0007t) W/m K cp = 1674.7 W s/kg K a × 108 = 2.7778(4 + 0.017t) m2/s Table A1.25

Chemical properties of cocoa butter.

Property

Value

Acid degree Acid value Saponification value Iodine value Peroxide value (mEqO2/kg)

1.6–6.0 (exceptionally up to 8) 0.9–3.4 (exceptionally up to 4.5) 192–197 32–42 (exceptionally up to 45) <2

For further details, see Fincke (1965). Table A1.26

Physical properties of cocoa butter.

Property

Value

Melting point (complete fusion) of modifications γ form α form β (IV) form β (V) form β (VI) form Heat of melting, mean value [β (IV) form] Specific heat capacity, solid and liquid Density in liquid state at 30°C

16–18°C 21–24°C 27–29°C 33–35°C 36–37°C 150.68 kJ/kg 2.093 kJ/kg 901 kg/m3

For further details, see Fincke (1965). Table A1.27 Dynamic viscosity (Pa s) of cocoa butter as a function of temperature (°C) (Sokolovsky 1959). t (°C)

η (Pa s)

t (°C)

η (Pa s)

t (°C)

η (Pa s)

35 40 45

0.0520 0.0383 0.0349

50 55 60

0.0278 0.0249 0.0206

65 70 75

0.0192 0.0158 0.0154

Appendix 1 Data on engineering properties

569

Table A1.28 Variation of the dynamic viscosity (Pa s) of chocolate as a function of the conching time (Antokolskaia 1964). Conching time (h) 0 Chocolate, t = 50–60°C ‘Sport’, t = 32°C ‘Prima’, t = 32°C ‘Extra’, t = 32°C Chocolate, t = 32°C (Water content %)

80 35 25.5 30.5 7.6 1.19

0.5 – – – 27 – –

1

2

3

4

5

24

48

72

96

45 15 15.5 25 – –

37.5 12 14.5 21 – –

35 12 – – – –

33.5 12 14 18.5 – –

23 – – 15 – –

– – 12 11.4 5.00 0.96

– 11 11 11 4.65 –

– 11 – – 5.3 0.91

– – 11 – –

Table A1.29 Thermophysical characteristics of some raw materials used in the confectionery industry (Danilova 1961). Material Cocoa beans

Citric acid

Melange

Honey

t (°C)

ρ (kg/m3)

λ (W/(m K)

cP (W s/kg K)

a × 106 (m2/s)

20 50 70 110

560 0.099 0.093 0.093

0.105 2260.9 2260.9 2260.9

2260.9 0.0777 0.0763 0.0750

0.0819

15 30 50

1542 – 0.174

0.179 0.177 1873.3

1394.2 1381.6 0.1411

0.1436 0.1430

−10 5 15 25

952 – 1015 1010

0.209 0.456 0.463 0.467

4438 3810 3747.2 3642.5

0.3249 0.1167 0.1222 0.1277

−5 15 20 35

1010 1435 1345 1345

0.654 0.349

1821.3 2306.9 2428.3 2993.6

0.1250 0.1055 0.1055 0.0867

– 0.370

According to Danilova (1961), the data for honey can be calculated in the temperature interval 5–35°C as follows: ρ = 1442 − 0.4t (kg/m3) λ = 1.163(0.29 + 0.00075t) (W/m K) cP = 4186.8(0.54 + 0.0035t) (W s/kg K) a × 108 = 2.778(4.3 − 0.033t) (m2/s) For further details, see Kältetechnik (1960).

570

Confectionery and Chocolate Engineering: Principles and Applications

Table A1.30 Thermophysical properties of sweets and chocolate. ρ (kg/m3)

Product Pralines with fruit juice1 Wafer Zwieback ‘Sport’ cake Chocolate

– – – 1315

λ (W/m K)

a × 106 (m2/s)

c (W s/kg K)

0.1 − 7.4 × 10−4(t − 40) 1.163(0.046 + 0.0015t) 4186.8(0.35 + 0.00033t) 2.7778(7.7 − 0.28t) – 4186.8(0.572 − 0.0007t) 2.7778(3.35 + 0.0047t) 1.163(0.098 + 0.0001t) 4186.8(0.575 − 0.0022t) 2.7778(2.8 + 0.121t) 1591–1675 1.163(0.2 + 0.0007t) 2.7778(4 + 0.017t)

t = temperature in °C. Latent heat of melting: 124.604 kW s/kg (Danilova et al.1961). 1 Antokolskaia (1964).

For chocolate mass, the following relationships were given by Rapoport and Tarchova (1939) for the temperature range 30–70°C: ρ (kg/m3) = 1320 − 0.5t λ (W/m K) = 1.163(0.2 + 0.0007t) c (W s/kg K) = 1674.7 (= 0.4 kcal/kg °C ) a × 108 (m2/s) = 2.7778 (4 + 0.017t) where t (°C) is the temperature. For the viscosity of chocolate, see Section 4.9.1 and Appendix 3.

Table A1.31 Hardness of various chocolate brands (Sokolovsky 1959). Brand Prima Goldetikett Sport Vanille Extra Cream Soy

Hardness (N/m2) 5981 5680 3130 3230 3430 1860 1270

Appendix 1 Data on engineering properties

Table A1.32

571

Thermophysical properties of sweets (Danilova et al. 1961).

Product

t (°C)

ρ (kg/m3)

λ (W/m K)

c (W s/kg K)

a × 106 (m2/s)

Wafer Zwieback

30 15 25 30 5 30 45 1224 25 50 85 14 25 35 – 25 50 85 – 25 45 65 85 25 45 25 45 65 85 0 10 15 20 27 35 40 15 22 30

– – – – 1370 – – 0.283 1411 – – 1360 – – 1204 940 – – 519 642 – – – 705 – – – – – – – 14751 – – – – 538 – – 520 648 950 – – – 1270 1260 1250 1240 1150 – – –

0.106 0.123 0.123 0.124 0.436 0.436 0.436 1704 0.384 0.372 0.360 0.372 0.366 0.359 0.250 0.212 0.215 0.222 0.099 0.116 0.120 0.122 0.124 0.128 0.128 0.116 0.120 0.122 0.124 0.062 0.063 0.064 0.064 0.065 0.066 0.066 0.080 0.081 0.085 0.099 0.087 0.196 0.206 0.200 0.213 0.244 0.256 0.267 0.267 0.219 0.209 – –

– 2365.5 2302.7 2302.7 2080.8 2281.8 2453.5 0.1353 2951.9 2834.5 2784.2 1808.7 1800.3 1779.4 1410.9 2101.8 2491.2 3328.5 1724.9 2181.3 1997.1 1687.3 1624.5 2177.1 1967.8 2177.1 1997.1 1687.3 1624.5 1226.7 1377.5 1557.5 1988.7 1821.3 1423.5 1285.3 1791.9 1771.0 1825.4 1984.5 1930.1 1976.2 2265.1 2265.1 2499.5 1674.7 1674.7 1674.7 1591.0 2344.6 1992.9 1381.6 1967.8

0.1900 0.0944 0.0967 0.0972 0.1528 0.1394 0.1297

Filled drops

Romaschka (drops) Jelly (marmalade)

Marzipan

Pralines Pralines with fruit juice filling ‘Lakton’ ‘Cream’

Cake ‘Unsere Marke’ ‘Sport’

Cocoa powder

Honey cake

Honey cake ‘Minze’ ‘Sachsen’ Halawa

Chocolates ‘Gold anchor’

‘Soy beans’ ‘Sport’ ‘Extra’ (milk) 1

– 0 26 40 60 10 30 50 70 18 35 15 15

Akhindinov and Polakova (1962). According to Akhindinov and Polakova (1962), for cocoa powder ρ = 857–1475 (kg/m3).

0.0922 0.0944 0.0950 0.1514 0.1500 0.1486 0.1475 0.1075 0.0919 0.0708 0.0867 0.0830 – – – 0.0833 0.0917 0.0830 0.0933 0.1128 0.1219 0.1067 0.0967 0.0864 0.0680 0.0753 0.0983 0.1094 0.0833 0.0855 0.0864 0.0928 0.0808 0.1047 0.0958 0.0930 0.0894 0.1139 0.1122 0.1278 0.1361 0.0980 0.0911 – –

572

Confectionery and Chocolate Engineering: Principles and Applications

Table A1.33 Density of sugar confectionery products (Antokolskaia 1964). Product ‘Bon-Bon’ drops with chocolate–hazelnut filling (35%) Drops with berry filling Drops Fruit jelly bonbon

Density (kg/m3) 1350–1355 1430–1435 1220–1225 1359

Table A1.34 Dynamic viscosity (Pa s) of various fondant masses as a function of temperature (°C) (Antokolskaia 1964). Temperature (°C) Fondant mass for ‘Riviera’ (water content 10.5%; invert sugar 6.3%) 70 68 67 66 65 62 60 Fondant mass for ‘Happy Childhood’ (water content 10.5%; invert sugar 5.8%) 79 77 75 74 72 Fondant mass for ‘Shio-Shio-San’ (water content 9.0%; invert sugar 6.0%) 80 79 77 76 75 74 70 Toffee mass ‘Kis-Kis’ (water content 17%) 60 70 80 90 100 Toffee mass ‘Kis-Kis’ (water content 9%) 60 80 85 90 100 Toffee mass ‘Kis-Kis’ (water content 8%) 70 85 90 Toffee mass ‘Gold Key’ (water content 19%) 60 70 80 90 100

Viscosity (Pa s)

5.2 5.4 6.0 8.0 8.6 10.4 11.6 16.0 20.0 23.0 36.0 44.0 8.0–9.0 13.0 18.4 21.6 25.2 32.0 50.0 2.48 1.15 0.67 0.36 0.26 559.3 116.6 64.4 43.0 33.7 487.8 122.8 43.0 3.91 2.18 1.13 0.62 0.38

Appendix 1 Data on engineering properties

Table A1.35

573

Thermophysical properties of semi-products used for cakes (Antokolskaia 1964). ρ (kg/m3)

λ (W/m K)

c (W s/kg K)

a × 106 (m2/s)

25 40 60 85

1400 – – –

0.291 0.291 0.291 0.291

2239.9 2260.9 2281.8 2311.1

0.0928 0.0919 0.0911 0.0900

20 40 60 80

1600 1570 1540 1500

0.314 0.291 0.267 0.256

1381.6 1465.4 1832.8 1718.6

0.1444 0.1278 0.1055 0.1000

20 40 60 80

1550 1520 1490 1460

0.314 0.302 0.291 0.279

1716.8 1758.5 1842.2 1884.1

0.1187 0.1111 0.1055 0.1000

20–60

1392 1005 1397 1397

0.373 0.173 0.352 0.327

1737.5 1490.5 1632.8 1511.4

0.1542 0.1058 0.1480 0.1564

t (°C) Toffee

Hard-boiled sugar mass Water content 2%

Water content 3–5%

Fondant With nuts 20 With cream

For hard-boiled sugar mass, in the temperature range 15–85°C,

λ = 1.163 ( 0.265 + 0.0005t ) ( W m K ) cP = 4186.8 ( 0.43 + 0.0025t ) ( W s kg K ) a × 108 = 2.7778 ( 4.55 − 0.021t ) ( m 2 s ) On the basis of the figures published by Antokolskaia (1964), the following approximate formulae can be used in the temperature range 0–80°C. For fondant mass,

λ = 0.43 + (t − 10 ) 8.3 × 10 −4 ( W m K ) cP = 1778 + 15.7 (t − 10 ) ( W s kg K ) For fondant filling, a × 106 = 0.16 − 0.001(t − 20 ) ( m 2 s ) For hard-boiled sugar mass, cP = 1100 + 6.67 (t − 10 ) ( W s kg K ) a × 106 = 0.1 − 6.67 × 10 −4 (t − 50 ) ( m 2 s )

574

Confectionery and Chocolate Engineering: Principles and Applications

Table A1.36 Dynamic viscosity (Pa s) of hard-boiled sugar masses with different molasses contents as a function of temperature (°C) (Nachschlagewerk des Konditors 1958, Sokolovsky et al. 1958). Water (%)

Temperature (°C)

Dynamic viscosity (Pa s)

Molasses (%)

1.91

135 125 115 105 95 90 85 80 75

18.4 31.0 79.7 240 950 2020 4680 11 800 31 700

15

1.84

135 125 115 105 95 90 85 80

–

25

135 125 115 105 95 90 85 80 75

– –

135 125 115 105 95 90 85 80 75

– –

135 125 115 105 95 90 85 80

– –

2.48

2.3

2.7

37.2 95.8 325 1400 3030 7320 17 400 25

55.0 154.7 690.0 1562.3 4470 12 100 39 900 35

100.3 382 2000 4820 11 500 30 060 95 600

110 390 2400 5000 11 700 35 000

50

Appendix 1 Data on engineering properties

575

Table A1.37 Density and dynamic viscosity of biscuit dough as a function of temperature (Antokolskaia 1964). t (°C)

ρ (kg/m3)

η (Pa s)

Kneading under pressure (water content 38%) 18 880 19 880 20 880

11.5 9.5 7.9

Kneading at atmospheric pressure (water content 37%) 20 1034 21 1034 22 1032 23 1032 24 1027 25 1027 26 1024

7.4 6.5 5.9 5.5 4.9 4.3 4.0

Table A1.38 Density and thermophysical properties of wafer batter and of biscuit and cracker doughs (Antokolskaia 1964). t (°C)

ρ (kg/m3)

λ (W/m K)

cP (W s/kg K)

a × 106 (m2/s)

Wafer batter 15 25 40 60 85

– – – 1100 –

0.477 0.477 0.483 0.483 0.488

3621.6 3600.6 3600.6 3558.8 3600.6

0.1205 0.1250 0.1267 0.1244 –

Zwieback dough 15 22 40

1165 – –

0.331 0.338 0.348

2625.1 2713.1 2847.0

0.1083 0.1069 0.1047

Semi-sweet biscuit dough 20 – 26 1222–1330 36 –

0.401 0.409 0.420

2909.8 29515.9 2997.7

0.1036 0.1039 0.1050

Cracker dough (laminated) 15 1295 22 – 30 – 40 –

0.326 0.328 0.329 0.335

2352.9 2323.7 2260.9 2219.0

0.1069 0.1092 0.1125 0.1167

Sweet biscuit dough 15 1280 22 – 30 –

0.338 0.340 0.338

2491.1 2512.1 2533.0

0.1061 0.1056 0.1044

Hard–sweet biscuit dough 15 1330 24 – 30 –

0.385 0.407 0.430

2658.8 2888.9 3181.9

0.1083 0.1055 0.1014

576

Confectionery and Chocolate Engineering: Principles and Applications

For Zwieback dough,

λ = 1.163 ( 0.275 + 0.005t ) ( W m K ) cP = 4186.8 ( 0.62 + 0.0012t ) ( W s kg K ) According to Danilova et al. (1961), for short dough (sheeted),

λ = 1.163 ( 0.28 + 0.00014t ) ( W m K ) cP = 4186.8 ( 0.58 − 0.0013t ) ( W s kg K ) a × 108 = 2.7778 (3.65 + 0.0143t ) ( m 2 s )

Table A1.39 Dynamic viscosity of sweet biscuit dough as a function of time of kneading (Antokolskaia 1964). η (MPa s)

Time of kneading (min) 10 20 30 40 60 80 90

0.38 0.36 0.26 0.17 0.60 0.69 1.1

Table A1.40 Viscosity (MPa s) of the cracker dough for ‘Krokett’ and the sweet biscuit dough for ‘Sachar’ as a function of the amount of gluten of different qualities (Antokolskaia 1964). Viscosity (MPa s)

Amount of gluten (%)

Cracker dough (39°C) (water content 24–25%)

Sweet biscuit dough (25–26°C) (water content 17–19%)

Weak

17 20 33

0.86 0.95 1.00

1.00 2.0 2.60

Medium

19 22 32

0.36 0.70 0.96

0.10 0.80 0.85

Strong

19 22 24 34

0.30 0.25 0.25 0.25

0.80 0.70 0.65 0.60

Elasticity of gluten

Appendix 1 Data on engineering properties

Table A1.41 Dynamic viscosity of honey-cake dough (water content 20%) as a function of time of kneading at 30°C (Antokolskaia 1964). Time of kneading (min)

Viscosity (MPa s)

15 30 45 60

0.25 0.26 0.30 0.29

Table A1.42 Dynamic viscosity of honey-cake dough as a function of temperature (Antokolskaia 1964). Temperature (°C)

Viscosity (MPa s)

20 30 40

0.55–0.93 0.27 0.26

Table A1.43 Dynamic viscosity of honey-cake dough as a function of water content (Antokolskaia 1964). Brand name

Water content (%)

Viscosity (MPa s)

20.0 20.8 21.0 22.6 23.0 21.0 23.0

0.40 0.25 0.18 0.16 0.14 0.30 0.20

Moskauer Batoni

Honey

Table A1.44 Density and dynamic viscosity of wafer batter as a function of water content at 20°C (two types, with different elasticity values of gluten) (Antokolskaia 1964). Density (kg/m3)

Viscosity (Pa s)

Gluten content of flour 27% 60.0 62.0 64.0 65.0

1136–1140 1154 1142–1144 1133–1137

4.2–4.6 1.1–1.2 1.15–1.20 0.83–0.84

Gluten content of flour 32% 60.6 62.0 64.0 65.0 66.0

– – – – –

Water content (%)

12.4 9.5 3.3 2.3 1.1

577

578

Confectionery and Chocolate Engineering: Principles and Applications

Table A1.45 Dynamic viscosity of wafer batter (water content 62.3– 62.6%) as a function of temperature (Antokolskaia 1964). Temperature (°C)

Viscosity (Pa s)

15 20 25 30

1.80 1.51 1.40 1.08

Table A1.46 Thermal conductivity of raw materials (Nachschlagewerk des Konditors 1958). Raw material Wheat flour

Crystalline sugar Sugar powder Maize starch Cows’ milk Condensed milk Sodium carbonate Salt Ammonium carbonate Butter

Butter, fried Margarine Honey Starch syrup Invert syrup Egg Crystalline vanillin Flavourings

Water content (m/m%)

λ (cal/g °C)

Dry content 13.0 13.5 14.0 14.5 15.0 15.5 0.1–0.2 0.5 13.0 87.5 30.0 1.0 2.0 – 14.2 13.6 13.5 0.1 15.0 18.0 20.0 33.5 74.0 0.5–1.0 −

0.340 0.426 0.429 0.432 0.436 0.439 0.442 0.301 0.308 0.423 0.940 0.630 0.539 0.220 0.610 0.688 0.574 0.557 0.521 0.500 0.450 0.700 0.600 0.760 0.310 0.540

Appendix 2

Solutions of sucrose, corn syrup and other monosaccharides and disaccharides

Table A2.1 Comparison of Brix and Baumé concentrations of aqueous sucrose solutions at 20°C (68°F). Degrees Brix

Degrees Baumé

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95

2.8 5.6 8.3 11.1 13.8 16.6 19.3 22 24.6 27.5 30.2 32.5 35 37.6 40.3 42.5 44.9 47.4 49.5

Comment: Linear interpolation provides fair results:

Degrees Brix 10 15 12.5 60 65 62.5 80 85 82.5

Degrees Baumé

Interpolated values

5.57 8.34 6.96 32.49 35.04 33.77 42.47 44.86 43.67

6.955

33.765

43.665

For a detailed scale (per 0.5 degree Brix), see Meiners et al. (1983, Vol. 1/I, p. 11). Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

580

Confectionery and Chocolate Engineering: Principles and Applications

Table A2.2 Sugar content and density of saturated aqueous sucrose solutions as a function of temperature (Antokolskaya et al. 1964). Temperature (°C)

Sugar (g/100 g water)

Density (kg/m3)

179.2 184.7 190.5 197 203.9 211.4 219.9 228.4 238.1 248.7 260.4 273.1 287.3 302.9 320.5 339.9 362.1 386.8 415.7 448.6 487.2

1314 1319.2 1323.5 1328 1332.7 1342.7 1345.7 1348 1353.5 1359.2 1365.1 1371.2 1377.5 1384 1390.8 1397.7 1404.9 1412.2 1419.9 1427.7 1435.9

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Table A2.3 Solubility number of sucrose as a function of temperature (Sokolovsky 1958, p. 15). t (°C) 0 10 20 30 40 50 60 70 80 90 100

(m/m%)

100σ

64.18 65.58 67.09 68.7 70.42 72.25 74.18 76.22 78.36 80.84 82.97

179.1736 190.5288 203.859 219.4888 238.0663 260.3604 287.2967 320.5214 362.1072 421.9207 487.1991

For further details, see Maczelka (1962, p. 24). Melting-point values of sucrose (depending on purity): 185–190°C. Mean specific heat capacity of sucrose (in the range 0–100°C): cP = 0.2712 + 0.00103t ( kcal kg °C )

(1 cal = 4.1868 J ) . For further details, see Maczelka (1962, p. 27).

Appendix 2 Solutions of sucrose, corn syrup and other monosaccharides and disaccharides

581

Table A2.4 Solubility of sugar in water in the presence of glucose syrup (Sokolovsky 1958, p. 16). 100 g solution consists of

100 g water dissolves

Sugar (g)

Syrup (g)

Sugar (g)

Syrup (g)

Total dry weight (g)

20

67.09 57.54 51.23 48.51 43.26

0 10.56 17.74 21.76 28.8

203 180.2 165.09 163.16 154.82

0 33.1 57.17 73.19 103.07

203 213.3 222.26 236.35 257.89

50

72.25 62.97 55.65 51.03 46.81 44.47 37.96

0 10.05 18.26 24 28.86 32.02 40.54

260.36 233.39 208.16 204.37 193.19 189.15 176.56

0 37.25 69.01 96.12 119.52 136.2 188.56

260.36 270.64 277.17 300.49 312.71 325.35 365.12

70

76.22 67.43 60.6 55.14 52.7 49.69

0 9.92 17.55 24.95 28.1 32.16

322.83 297.79 277.35 276.95 274.48 273.77

0 43.7 80.32 125.31 146.35 177.19

322.83 341.49 357.67 402.26 420.83 450.96

Temperature (°C)

Table A2.5 Solubility of sugar in water in the presence of invert syrup (Sokolovsky 1958, p. 17). 100 g solution consists of

100 g water dissolves

Sugar (g)

Invert sugar (g)

Sugar (g)

23.1

67.59 57.84 47.31 38.66

0 11.9 25.39 36.9

208.55 191.14 173.3 158.18

0 39.32 93 150.98

208.55 230.46 266.3 309.16

30

68.11 56.32 50.97 49.91 48.95 46.36 39.23 32.06 31.85 26.03 21.18 20.59

0 14.94 21.86 23.21 24.46 28.01 37.48 47.02 47.62 56.37 63.68 64.47

213.58 195.96 187.6 185.68 184.09 180.88 168.43 153.25 155.13 147.9 139.89 137.82

0 51.98 80.46 86.34 91.99 109.29 160.93 224.76 231.95 320.28 420.61 431.52

213.58 247.94 268.06 272.02 276.08 290.17 329.36 378.01 387.08 468.18 560.5 569.34

50

72.22 62.81 53.8 46.2 36.75

0 11.42 22.65 32.32 46.05

260.36 243.73 228.45 215.08 196.43

0 44.31 96.17 150.46 253.2

260.36 288.04 324.62 365.54 449.63

Temperature (°C)

Invert sugar (g)

Total dry weight (g)

Appendix 3

Survey of fluid models

Contents A3.1

Decomposition method for calculation of flow rate of rheological models A3.1.1 Principle of the decomposition method A3.1.2 Bingham model A3.1.3 Casson model (n = 1/2) A3.1.4 Peek, McLean and Williamson model A3.1.5 Reiner–Philippoff model A3.1.6 Reiner model A3.1.7 Rabinowitsch, Eisenschitz, Steiger and Ory model A3.1.8 Oldroyd model A3.1.9 Weissenberg model A3.1.10 Ellis model A3.1.11 Meter model A3.1.12 Herschel–Bulkley–Porst–Markowitsch–Houwink (HBPMH) (or generalized Ostwald–deWaele) model A3.1.13 Ostwald–de Waele model A3.1.14 Williamson model A3.2 Calculation of the friction coefficient ξ of non-Newtonian fluids in the laminar region A3.3 Generalization of the Casson model A3.3.1 Theoretical background to the exponent n A3.3.2 Theoretical foundation of the Bingham model A3.4 Determination of the exponent n of the flow curve of a generalized Casson fluid A3.5 Dependence of shear rate on the exponent n in the case of a generalized Casson fluid A3.6 Calculation of the flow rate for a generalized Casson fluid A3.7 Lemma on the exponent in the generalized Casson equation Further reading

A3.1 A3.1.1

582 582 583 585 586 587 587 588 589 590 591 591 592 594 595 596 597 597 598 598 600 601 603 605

Decomposition method for calculation of flow rate of rheological models Principle of the decomposition method

Reher et al. (1969) gave a family of flow curves in the form of a genealogy and gave formulae for the volume flow rate of the various rheological types of fluids studied below. Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

Appendix 3 Survey of fluid models

583

The expressions for the flow rate are generally rather complicated, and thus it is difficult to recognize and use the essential physical relations. Below, a simplification of these formulae is presented that will be beneficial also from the point of view of memorizing them. The principle of the method is that the expression for the flow rate Q is decomposed into two factors, namely K and M: Q ( m3 s ) = K × M

(A3.1)

where K (m3/s) = R4π Δp/8Lη* is the flow rate of an ideal Newtonian fluid of viscosity η* that contains Δp proportionally, and M is a dimensionless number that contains Δp but not proportionally; R is the radius of a cylindrical tube, L is the length of the tube, η* is the apparent dynamic viscosity of the fluid, and Δp is the pressure difference between the two ends of the tube. M (0 ≤ M ≤ 1) can be regarded as the degree of efficiency relative to the flow rate of an ideal Newtonian fluid: M = Q ( non-Newtonian ) Q ( ideal Newtonian )

(A3.2)

In the case of fluids of plastic nature, there is a yield stress τ0, which is the value of the shear stress corresponding to D = 0 (where the shear stress is τ = R Δp/2L). Therefore, the flow of plastic fluids may be characterized by a dimensionless number, now designated by Bu, the Buckingham number: Bu = τ0/τ. For plastic fluids, M is a function of Bu. Taking into account the fact that τ = R Δp/2L, Eqn (A3.1) is modified in this case to ⎛ R 3 πτ 0 ⎞ ⎛ 1 ⎞ ⎛ R 3 πτ 0 ⎞ Q=⎜ ×M = ⎜ ×F ⎟ ⎝ 4η* ⎠ ⎝ Bu ⎠ ⎝ 4η* ⎟⎠

(A3.3)

where F = (1/Bu) × M (Bu > 0). Both M and F can be tabulated as functions of Bu = τ0/τ = 2Lτ0/R Δp, enabling one to solve the following problems (where τ0 is known in both cases): • Δp is known and Q is to be calculated: M is used; • Q is fixed and Δp is to be calculated: F is used. Since in the case of non-Newtonian fluids Q and Δp are not proportional, the decomposition method decomposes Q into a factor K containing Δp proportionally and a factor M(Bu) containing Δp not proportionally, and tabulates the values of M (and F) as a function of Bu. As a result, by calculating the value of Bu, all the usual calculations can be done – the other parameters (R, L, η*) are proportional to Q. In certain fluid models, the role of Bu in M is replaced by similar parameters, which are always dimensionless (see later).

A3.1.2

Bingham model

The decomposition method can be presented by use of a simple example connected with the Bingham fluid. From the Buckingham–Reiner equation (where D is the shear rate),

τ = ηpl D + τ 0 the following equation can be obtained with some algebraic manipulation:

584

Confectionery and Chocolate Engineering: Principles and Applications

1 ⎛ R 4 π Δp ⎞ ⎛ 4 1 − Bu + Bu 4 ⎞ Q=⎜ ⎠ ⎝ 8LηPl ⎟⎠ ⎝ 3 3

(A3.4)

where Bu =

τ0 tR

(A3.5)

is the Buckingham number. Consequently, we can write Q = K ×M where K=

R 4 π Δp 8LηPl

( the Newtonian part )

(A3.6)

and M ( Bu ) = 1 −

4 1 Bu + Bu 4 3 3

(A3.7)

which is a dimensionless number. M and F = M/Bu can be tabulated as functions of Bu (Table A3.1). If Bu = 1 (i.e. no flow), then M(Bu) = 0, which shows that M functions as a degree of efficiency. As mentioned in Section 12.3, Lapitov and Filatov (1963) performed such a decomposition of the Buckingham–Reiner equation in practice. Table A3.1 Factors M and F of the Bingham model. Bu 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.0025 0.003 0.0035 0.004 0.002 0.0021

M (Bingham)

F (Bingham)

0.8667 0.733867 0.6027 0.4752 0.354167 0.2432 0.1467 0.069867 0.0187 0.996667 0.996 0.995333 0.994667 0.997333 0.9972

8.667 3.669333 2.009 1.188 0.708333 0.405333 0.209571 0.087333 0.020778 398.6667 332 284.381 248.6667 498.6667 474.8571

(In the following calculations, SI units are used exclusively.)

Appendix 3 Survey of fluid models

585

Example A3.1 Suppose τ0 = 0.5 Pa, ηPl = 3 Pa s, R = 2 × 10−2 m, L = 10 m and Δp = 2 × 105 Pa. Then τR = R Δp/2L = 2 × 102 Pa and Bu = τ0/τR = 0.0025. From Table A3.1, M (Bingham) = 0.996667. On the basis of the data in Table A3.1, K ≈ 0.4186 × 10−3 m3/s and Q ≈ 0.996667 × 0.4 186 × 10−3 m3/s.

Example A3.2 The ‘inverse’ problem can also be solved by using this method. We want to increase the flow rate to Q = 0.5 × 10−3 m3/s while all the other parameters remain unchanged. The increased value of Δp is calculated from Q = 0.5 × 10 −3 =

R 3 πτ 0 3.14 ⎞ × F ( Bingham ) = ⎛ × ( 2 × 10 −6 ) × F ( Bingham m) ⎝ 2 ⎠ 4ηPl

From this equation, F (Bingham) ≈ 477.7. The corresponding Buckingham number Bu (see Table A3.1) is approximately 0.0021 = 0.5/τR, and τR = 238 Pa = R Δp/2L = 2 × 10−2 Δp/20 and Δp = 2.38 × 105 Pa.

(Preparing a detailed table of M and F values fitted to a given task is very easy using a computer; Table A3.1 is an example only.)

A3.1.3

Casson model (n = ½)

The usual form of the flow curve is

τ = τ 0.CA + ηCA

∂w ∂r

(A3.8)

where τ is the shear stress (Pa = N/m2), τ0.CA is the Casson yield stress (Pa), ηCA is the Casson (dynamic) viscosity (Pa s) and ∂w/∂r is the shear rate (s−1). The flow rate is 4 1 ⎛ Δp πR 4 ⎞ ⎛ 16 1 − Ca + Ca 2 − Ca8 ⎞ Q=⎜ ⎠ ⎝ 8LηCA ⎟⎠ ⎝ 7 3 21

(

(A3.9)

)

where Ca = τ 0.CA τ R = r0 R = Bu , the Casson number. (The Casson number Ca is not to be confused with the Cauchy number, also denoted by Ca.) 16 4 1 M (Ca ) = ⎛1 − Ca + Ca 2 − Ca8 ⎞ ⎝ ⎠ 7 3 21 If Ca = 1 (i.e. no flow), then M = 0.

(A3.10)

586

Confectionery and Chocolate Engineering: Principles and Applications

A3.1.4

Peek, McLean and Williamson model

The flow curve of this model is

τ = η∞ D +

(η0 − η∞ ) D 1+ τ τm

(A3.11)

where τ is the shear stress, η0 and η∞ (constants) are viscosities (at t = 0 and t → ∞, respectively), τm is the constant shear stress, and D is the shear rate. This flow curve is characterized by three constants above.

A3.1.4.1

Evaluation of the constants

Three points (designated 1, 2 and 3) are needed to estimate the three constants. After some algebraic transformations, for point 1, we obtain an equation (I) τ(1)/D(1) (1 + τ (1)/τm) = η0 + η∞τ(1)/τm. We obtain similar equations, designated by (II) and (III), for points 2 and 3, respectively. Then

(I ) − (II ) =

η∞[τ (1) − τ ( 2 )] (η0 is eliminated ) τm

(I ) − (II ) τ (1) − τ (2 ) = (η∞ τ m is eliminated ) (II ) − (III ) τ (1) − τ (2 ) From this equation, the τm contained in the left-hand side can be estimated. Then the linear equations (I) and (II) contain only two unknowns (η0 and η∞), which can be estimated easily. An easier method is to determine η0 and η∞ from a plot of η vs D by extrapolations; for η0, we take lim D → 0, and for η∞, we take lim D → ∞. After interpolation, τm can be calculated from equation (I). A3.1.4.2

Decomposition

Using the substitutions τ = R Δp/2L and t = (η∞/η0)(τ/τm), the flow rate is ⎛ pR 4 Δp ⎞ ⎧ ⎡ η0 − η8 ⎤ ⎛ 1⎞ ⎡ 4 2 4 ⎛ 4 ⎞ ⎫ Q ( Δp ) = ⎜ − + − ln (t + 1)⎤⎥ ⎬ ⎨1 − ⎝ 8Lη∞ ⎟⎠ ⎩ ⎢⎣ η∞ ⎥⎦ ⎝ t ⎠ ⎣⎢ 3 t t 2 ⎝ t 3 ⎠ ⎦⎭

(A3.12)

The expression 1 4 2 4 4 M (t ) = ⎛ ⎞ ⎡⎢ − + 2 − ⎛ 3 ⎞ ln (t + 1)⎤⎥ ⎝ t⎠ ⎣3 t t ⎝t ⎠ ⎦ can be tabulated.

(A3.13)

Appendix 3 Survey of fluid models

A3.1.5

587

Reiner–Philippoff model

The flow curve is

τ = η∞ D +

(η0 − η∞ ) D

(A3.14)

1 + [τ τ m ]2

where the notation is the same as for the Peek, McLean and Williamson model. A3.1.5.1

Evaluation of the constants

As for the Peek, McLean and Williamson model (see above). A3.1.5.2

Decomposition

By using the substitutions τ = R Δp/2L and t = (τ τ m ) (η∞ η0 ) , the flow rate is Q=

πR 3τ 4η∞

⎧ ⎡ 2 (η0 − η∞ ) ⎤ ⎛ 1 ⎞ ⎡ ⎛ 1 ⎞ ⎤⎫ ⎨ ⎢1 − ⎥ × ⎝ t 2 ⎠ × ⎢⎣1 + ⎝ t 2 ⎠ ln (t + 1)⎥⎦ ⎬ η0 ⎩⎣ ⎭ ⎦

(A3.15)

The expressions ⎡ 2 (η0 − η∞ ) ⎤ ⎛ 1 ⎞ ⎡ ⎛ 1 ⎞ M ( K ; η∞; η0 ) = ⎢1 − × 2 × ⎢1 + 2 ln (t + 1)⎤⎥ ⎥ η0 ⎦ ⎣ ⎦ ⎝t ⎠ ⎣ ⎝t ⎠

(A3.16a)

and 1 1 M ′(t ) = ⎛ 2 ⎞ × ⎡⎢1 + ⎛ 2 ⎞ ln (t + 1)⎤⎥ ⎝t ⎠ ⎣ ⎝t ⎠ ⎦

}

(A3.16b)

can be tabulated. Comment: It is more practical to tabulate the values of M if it is dependent on a single parameter, for example t. This is recommended in the cases discussed later also.

A3.1.6

Reiner model

The flow curve is

τ = η∞ D +

(η0 − η∞ ) τ η0 exp (τ c )

(A3.17)

where c is a constant, and the other notation is the same as for the Peek, McLean and Williamson model (c > 0). A3.1.6.1

Evaluation of the constants

From a plot of η vs D, η0 and η∞ can be determined by extrapolations, taking lim D → 0 and lim D → ∞, respectively. With the notation

588

k=

Confectionery and Chocolate Engineering: Principles and Applications

η0 − η∞ η0

the following equations are obtained for two points 1 and 2:

τ (2 ) ⎤ τ ( 2 ) − η∞ D ( 2 ) = kτ ( 2 ) exp ⎡⎢ ⎣ c ⎦⎥ τ (1) ⎤ τ (1) − η∞ D (1) = kτ (1) exp ⎡⎢ ⎣ c ⎥⎦ By dividing the first equation by the second, c can be calculated:

τ ( 2 ) − τ (1) ⎤ [τ (2 ) − η∞ D (2 )] τ (1) = exp ⎡⎢ c ⎣ ⎦⎥ [τ (1) − η∞ D (1)] τ (2 ) ln ( left-hand side ) =

A3.1.6.2

τ ( 2 ) − τ (1) c

Decomposition

By using the substitution τ/c = K = R Δp/2Lc, the flow rate is 8 ⎞ ⎛ η∞ ⎞ ⎡ 3 K 3 3 3 ⎫ ⎛ πR 4 Δ p ⎞ ⎧ ⎛ 1− Q=⎜ − exp ( − K ) ⎛ + + + 2 ⎞ ⎤⎥ ⎬ ⎨ 1− ⎝ 2 2 K K ⎠ ⎦⎭ ⎝ 8Lη∞ ⎟⎠ ⎩⎝ K 2 ⎠ ⎜⎝ η0 ⎟⎠ ⎣⎢ K 2

(A3.18)

where K > 0. The expression 8 ⎛ η ⎞ 3 K 3 3 3 M ( K ; η∞; η0 ) = ⎛1 − 2 ⎞ ⎜1 − ∞ ⎟ ⎡⎢ 2 − exp ( − K ) ⎛ + + + 2 ⎞ ⎤⎥ ⎝ K ⎠ ⎝ η0 ⎠ ⎣ K ⎝ 2 2 K K ⎠⎦

(A3.19)

is a dimensionless part that can be tabulated if c, η∞ and η0 are given. (Also, M′(K) = (1 − 8/K2)[3/K2 − exp(−K)(K/2 + 3/2 + 3/K + 3/K2)] can be tabulated.)

A3.1.7

Rabinowitsch, Eisenschitz, Steiger and Ory model

The flow curve is D = a1τ + a2τ 3 The units of [a1] and [a2] are

[ a1 ] = 1 Pa s = 1 dynamic viscosity [ a2 ] = 1 ( Pa 2 × Pa s )

(A3.20)

Appendix 3 Survey of fluid models

A3.1.7.1

589

Determination of the constants

For two points 1 and 2, two equations are obtained, D (1) = a1τ (1) + a2τ 3(1) D ( 2 ) = a1τ ( 2 ) + a2τ 3( 2 ) from which the constants can be calculated. The volume flow rate is a R 2 Δp 2 ⎞ ⎛ πR 4 Δp ⎞ ⎛ Q=⎜ a1 + 2 2 ⎟ ⎟ ⎜ ⎝ 8L ⎠ ⎝ ⎠ 6L

(A3.21)

This formula can be easily calculated, and thus decomposition is not necessary.

A3.1.8

Oldroyd model

The flow curve is

τ′ τ = η0 ⎛ D − + λ2 D ′⎞ ⎝ ⎠ N

(A3.22)

where the prime ′ denotes differentiation with respect to the radius, and N and λ2 are constants.

A3.1.8.1

Decomposition

If we assume that τ = 0, the volume flow rate is 2 ⎛ π λ Δp ⎞ ⎧ ⎡ ⎞ ⎤ R 4 R3 ⎫ ⎛ R⎞⎛ R Q=⎜ 2 + λ2 R − λ22 ⎟ + λ22 ⎥ + + ⎨λ2 ⎢exp ⎜⎝ ⎟⎠ ⎜ − ⎬ ⎟ ⎠ ⎝ Lη0 ⎠ ⎩ ⎣ 6 ⎭ λ2 ⎝ 2 ⎦ 8λ2

(A3.23)

Using the substitution R/λ2 = u (where u ≠ 0), the flow rate is 1 1 1 1 1 1⎤ ⎛ πR 4 Δp ⎞ Q=⎜ × 8 ⎡⎢e u ⎛ − 2 + 3 − 4 ⎞ + 4 + + ⎟ ⎝ ⎠ ⎝ 8Lη0 ⎠ 2u 6u 8 ⎥⎦ u u u ⎣

(A3.24)

The expression 1 1 1 1 1 1⎤ M ( u ) = 8 ⎡⎢e u ⎛ − 2 + 3 − 4 ⎞ + 4 + + u u ⎠ u 6u 8 ⎦⎥ ⎣ ⎝ 2u can be tabulated.

(A3.25)

590

Confectionery and Chocolate Engineering: Principles and Applications

A3.1.9

Weissenberg model

The flow curve is

τ=

η0 D cosh τ G 2

(

)

(A3.26)

where D is the shear rate, and η0 and G are constants. A3.1.9.1

Calculation of the constants

Using the series cosh x = 1 +

x2 x 4 + + 2! 4!

for two points 1 and 2, the following equations are obtained: 2

4

⎡τ (1) G 2 ⎤⎦ ⎡τ (1) G 2 ⎤⎦ η D (1) +⎣ + = 0 1+ ⎣ 2 4! τ (1) and 2

4

⎡τ ( 2 ) G 2 ⎤⎦ ⎡τ ( 2 ) G 2 ⎤⎦ η D (2 ) +⎣ + = 0 1+ ⎣ 2 4! τ (2 ) By dividing the first equation by the second, an equation is obtained for 1/G2, which is linear if the series is stopped after the second term and quadratic if it is stopped after the third term. So the value of G can be calculated, and then also η0.

A3.1.9.2

Decomposition

(

)

Using the substitutions τ G 2 = k and τ = R Δp/2L, the flow rate is 24 ⎤ ⎡ πR 3τ ⎤ ⎡⎛ 8 ⎞ ⎛ 1 ⎛ 8⎞ ⎛ 3 3 ⎞ 2⎞ Q=⎢ ⎥ ⎢⎣⎝ k ⎠ ⎝ 3 + 3k ⎠ sinh k − ⎝ k ⎠ ⎝ k + k 3 ⎠ cosh k + k 4 ⎥⎦ 4 η ⎣ 0 ⎦

(A3.27)

The expression 8 1 8 3 3 24 M ( k ) = ⎡⎢⎛ ⎞ ⎛ + 3k 2 ⎞ sinh k − ⎛ ⎞ ⎛ + 3 ⎞ cosh k + 4 ⎥⎤ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ k k k k ⎦ ⎣ k 3

(A3.28)

can be tabulated. It should be mentioned that the range in which Eqns (A3.27) and (A3.28) are valid is 1.1433 < k < 1.1511. This applies also to the undecomposed expression for Q.

Appendix 3 Survey of fluid models

A3.1.10

591

Ellis model

The flow curve is

τ=

η0 D α −1 1 + (τ τ 1 2 )

(A3.29)

where η0 is a constant, τ1/2 is the ‘half-value’ of τ used (= R Δp/4L), and α is a constant. A3.1.10.1

Determination of the constants

From Eqn (A3.29),

τ1 2 =

η0 D [τ1 2 ] 2

η0 can be calculated. (The value of D when τ = τ1/2 is D[τ1/2].) From the equation

τ (1) =

η0 D (1) α −1 1 + {τ (1) τ1 2 }

α can be calculated. A3.1.10.2

Decomposition

Using the substitutions τ = R Δp/2L and t = (τ/τ1/2)α−1, the flow rate is ⎛ πR 3τ ⎞ ⎛ 4tα −1 ⎞ 1+ Q=⎜ ⎝ 4η0 ⎟⎠ ⎜⎝ α + 3 ⎟⎠

(A3.30)

The expression M (t ) = 1 +

4tα −1 α +3

(A3.31)

can be tabulated.

A3.1.11

Meter model

The flow curve is

τ=

η0 D {1 + (τ τ1 2 ) ad [(α − 1)(η∞ η0 )]} 1 + (τ τ 1 2 ) ad (α − 1)

(A3.32)

where η0 is a constant, τ1/2 is the ‘half-value’ of τ (= R Δp/4L) used, and α is a constant.

592

Confectionery and Chocolate Engineering: Principles and Applications

A3.1.11.1

Decomposition

We assume that η∞ << η0, and with the substitutions τ = R Δp/2L, s = (τ/τ1/2)α−1, the flow rate is 2 4s 2 s ⎤ ⎛ η∞ ⎞ 2 ⎡ 2 4 s ⎤ ⎫⎪ ⎛ πR 3τ ⎞ ⎧⎪ ⎛ η∞ ⎞ ⎡ 4 + + + 1 Q=⎜ + − s s ⎢ ⎨ ⎬ ⎜ ⎟ ⎜ ⎟ ⎟ ⎢ ⎥ ⎝ 4η0 ⎠ ⎩⎪ α + 3 ⎝ η0 ⎠ ⎣ α + 3 α + 1 ⎦ ⎝ η0 ⎠ ⎣ α + 1 3α + 1 ⎥⎦ ⎭⎪

(A3.33)

The expr4ession 2

M = 1+

4s 4 2 s ⎤ ⎛ η∞ ⎞ 2 ⎡ 2 4s ⎤ ⎛η ⎞ + + +⎜ ⎟ s ⎢ − ⎜ ∞ ⎟ s ⎡⎢ ⎥ α + 3 ⎝ η0 ⎠ ⎣ α + 3 α + 1 ⎦ ⎝ η0 ⎠ ⎣ α + 1 3α + 1 ⎥⎦

(A3.34)

can be tabulated. Equation (A3.34) can be simplified by taking into account the fact that η∞ << η0 and neglecting the second-order terms: M ( s; α ; η∞ ; η0 ) = 1 +

4s 4 2s ⎤ ⎛ η∞ ⎞ 4 s ⎛ η∞ ⎞ s 2 ⎛η ⎞ + = + − − − ⎜ ∞ ⎟ s ⎡⎢ 1 1 2 ⎜⎝ ⎟⎠ ⎜ ⎟ ⎝ η0 ⎠ α + 3 η0 α + 1 α + 3 ⎝ η0 ⎠ ⎣ α + 3 α + 1 ⎦⎥ (A3.35)

If the value of α is close to 1, the term containing s2 can be neglected: ⎛ η ⎞ 4s M ( s; α ; η∞; η0 ) ≈ 1 + ⎜1 − ∞ ⎟ ⎝ η0 ⎠ α + 3

(A3.36)

which is a very simple expression.

A3.1.12

Herschel–Bulkley–Porst–Markowitsch–Houwink (HBPMH) (or generalized Ostwald–deWaele) model

The HBPMH model is a power-law model that takes into account a yield stress τ0:

τ = τ 0 + kD n A3.1.12.1

Determination of the constants

Three points are needed for evaluating the flow curve,

τ1 = τ 0 + k ( D1 )n, τ 2 = τ 0 + k ( D2 )n, τ 3 = τ 0 + k ( D3 )n Then the following differences are taken: ⎡⎛ D ⎞ n ⎤ τ 2 − τ1 = k ( D2 )n − k ( D1 )n = k ( D1 )n ⎢⎜ 2 ⎟ − 1⎥ ⎣⎝ D1 ⎠ ⎦

(A3.37)

Appendix 3 Survey of fluid models

593

⎡⎛ D ⎞ n ⎤ τ 3 − τ 2 = k ( D3 )n − k ( D2 )n = k ( D2 )n ⎢⎜ 3 ⎟ − 1⎥ ⎣⎝ D2 ⎠ ⎦ If the values of D are chosen such that D2 D3 = D1 D2 (e.g. D1 = 2, D2 = 3 and D3 = 4.5 s−1), then

τ 3 − τ 2 ⎛ D2 ⎞ =⎜ ⎟ τ 2 − τ1 ⎝ D1 ⎠

n

(A3.37a)

and n can be calculated. If n is known, k and τ0 can be taken from a plot of τ vs Dn (see Eqn A3.37); k = slope and τ0 = intercept. It is expedient to improve the correlation by iterating the value of n, because the above calculation is based on three points only. Example A3.3 Since the HBPMH model is often applied in the food industry, an example is given here to demonstrate the evaluation described above. In Table A3.2, column 2 was calculated according to the formula τ = 5.82 + 4.93D0.63. Column 3 shows rounded values of τ, towards which the evaluation is directed. By using Eqn (A3.37a) for D = 20, 30 and 45 s−1, we obtain n τ 3 − τ 2 49.17 − 39.4 30 = = 1.29 = ⎛ ⎞ = 1.5n ⎝ 20 ⎠ τ 2 − τ1 39.4 − 31.83

n=

ln 1.29 = 0.6296 ln 1.5

Column 4 shows the values of D0.6296, and from a plot of τ vs D0.6296 we obtain τ0 = 5.810 … and k = 3.946, i.e. we have successfully restored the original formula. Table A3.2 Values of D and τ used to demonstrate the evaluation of τ0 and k. D (s−1) 10 15 20 25 30 35 40 45 50

τ (Pa)

τ (Pa)

D0.6296

22.62723 27.5187 31.83029 35.75632 39.40015 42.82489 46.07261 49.17309 52.1484

22.63 27.52 31.83 35.76 39.4 42.82 46.07 49.17 52.14

4.261868 5.501322 6.593689 7.588274 8.511293 9.378756 10.20133 10.98658 11.74009

Intercept = 5.810417 Slope = 3.946409

594

Confectionery and Chocolate Engineering: Principles and Applications

A3.1.12.2

Calculation of flow rate and decomposition

Reher et al. (1969) did not discuss this generalized case. Therefore, the calculation of the flow rate is briefly presented here. The flow rate Q = Q(Δp) can be calculated from the following integral (the Rabinowitsch– Mooney equation): Q=

πR 3 τ R3

τR

2 ∫ Dτ dτ

τ0

where

τ=

r Δp 2L

From the flow curve given in Eqn (A3.37), we obtain dτ = knD n −1 dD and the boundaries of integration are τ0 ↔ D = 0 and τR ↔ D. For simplicity we write τR = τ, and thus the integral that is to be done is Q=

πR 3 τ R3

τR

∫

Dτ 2 dτ =

τ0

πR 3 τ3

D

n 2 n −1 ∫ D (τ 0 + kD ) knD dD

(A3.38)

0

Using the notation Bu = τ0/τ (the Buckingham number), after some algebraic transformation we obtain 1n ⎡ Bu 2 2 Bu (1 − Bu ) (1 − Bu )2 ⎤ R Δp ⎞ 1 − Bu ) ⎢ + + Q = nπR 3⎛ ( ⎝ 2 Lk ⎠ 2n + 1 3n + 1 ⎥⎦ ⎣ n +1

(A3.39)

The expression ⎡ Bu 2 2 Bu (1 − Bu ) (1 − Bu )2 ⎤ + + M ( Bu ) = (1 − Bu ) ⎢ 2n + 1 3n + 1 ⎥⎦ ⎣ n +1 is a dimensionless number that may be tabulated. The decomposed flow rate is R Δp ⎞ Q = nπR 3⎛ ⎝ 2 Lk ⎠

1n

× M ( Bu )

(A3.39a)

If Bu = 1, then Q = 0 (the flow does not start).

A3.1.13

Ostwald–de Waele model

If Bu = 0 (i.e. τ0 = 0) in Eqn (A3.39), the Ostwald–de Waele model is obtained, for which the flow rate can be calculated:

Appendix 3 Survey of fluid models

nπR 3( R Δp 2 L ) 3n + 1

595

1n

Q=

(A3.40)

Evidently, the number n/(3n + 1) does not need to be tabulated. If n = 1, i.e. k = η, then from Eqn (A3.40), πR 4 R Δp 8ηL

Q=

which is the Hagen–Poiseuille equation. The flow curve of the Ostwald–de Waele model can be easily linearized by plotting τ = kDn in the form ln τ = ln k + n ln D, from which the intercept, equal to ln k, and the slope, equal to n, can be obtained.

A3.1.14

Williamson model

The flow curve is

τ=

fD + η∞ D s+D

(A3.41)

where f, s and η∞ are constants. The formula for the volume flow rate Q is very complicated (Reher et al 1969) and is hardly suitable for practical purposes. Therefore we consider the following extreme cases: lim

D →∞

τ = η∞ D

and lim

D→ 0

τ f = D s

At low values of D, we have η (= τ/D) > η∞, and η decreases from f/s to η∞. The slope of the curve is dependent on the above three constants as follows: f = η∞ s ( K + 1)

(A3.42)

where K is a number. An integral mean of the viscosity can be estimated for this initial region of the flow curve:

η* =

1 ⎛ f ⎞ K dk ⎛ f ⎞ = η∞ ln ⎜ = η∞ ln ( K + 1) ∫ ⎝ ⎠ ⎝ sη∞ ⎟⎠ K +1 s 0 k +1

(A3.43)

The approximate flow curve is obtained as ⎡ ⎛ f ⎞⎤ τ = (η* + η∞ )D = η∞ ⎢ln ⎜ ⎟ ⎥ D = η∞[ ln ( K + 1)] D ⎣ ⎝ sη∞ ⎠ ⎦

(A3.44)

This is a flow curve of Newton type, for which the Hagen–Poiseuille equation can be used. For the velocity profiles of all the flow curves described above, see Reher et al. (1969).

596

Confectionery and Chocolate Engineering: Principles and Applications

Calculation of the friction coefficient ξ of non-Newtonian fluids in the laminar region

A3.2

The decomposition method gives the flow rate in the form ⎡ R 4 π Δp ⎤ Q = K ×M = ⎢ ⎥×M ⎣ 8Lη* ⎦

(A3.45)

In the laminar region, a modified dynamic viscosity can be defined as

ηM =

η* M

(A3.46)

Using this, the friction coefficient can be expressed as

ξ=

64 Re

(A3.47)

where Re =

Dvρ DvρM = ηM η*

(A3.48)

where 0 < M < 1. Thus,

ξ=

64η* DvρM

(A3.49)

In the laminar region, which is important in the case of non-Newtonian fluids, the friction coefficient ξ can be derived from the Hagen–Poiseuille equation for the flow rate. The friction coefficient is defined by the equation L ρv 2 Δp = ξ ⎛ ⎞ ⎝ D⎠ 2

(A3.50)

Δp ⎛ ξ ⎞ ρv 2 = L ⎝ D⎠ 2

(A3.51)

or

where Δp is the pressure difference inducing the flow, ξ is the friction coefficient defined in Eqn (A3.47), L is the length of the circular tube, D is the diameter of the tube, ρ is the density of the fluid and v is the linear velocity of the fluid.

Appendix 3 Survey of fluid models

A3.3 A3.3.1

597

Generalization of the Casson model Theoretical background to the exponent n

Casson (1959) developed a theoretical model that is based on the supposition that in suspensions, the particles form rod-like agglomerates. These rods are cylindrical, their half-length is L and their cross-sectional radius is r, and the value of J = L/r is a function of shear rate. On the basis of mechanical considerations following from those assumptions, Casson derived a linear relationship between J and y = (η0D)−1/2 when J >> 1: J = α + β y = α + β (η0 D )

−1 2

(A3.52)

where η0 is the dynamic viscosity of the dispersion medium, D is the shear rate, α and β are constants, and A = aα − 1 12

1 B = ⎛ ⎞ aβ c ⎝ D⎠ where a is a constant related to the spatial orientation of the rods and c is the concentration of the dispersed particles. Starting from this model, it was derived that 12

aβ c ⎞ ⎡ η0 ⎤ ⎡(1 − c )− A 2 − 1⎤⎦ τ1 2 = ⎢ D1 2 + ⎛ −A ⎥ ⎝ A ⎠⎣ ⎣ (1 − c ) ⎦

(A3.53)

which is the usual form of the Casson equation,

τ 1 2 = K1D1 2 + K 0

(A3.54)

where K1 = η 2CA (the so-called Casson viscosity) and K0 = τ 20.CA (the so-called Casson yield stress) are constants. Heinz (1959) and Heimann and Fincke (1962c,d) determined that for milk chocolate, an exponent of 2/3 gives a better fit than Eqn (A3.54). According to the studies of Mohos (1966a), a general equation

τ n = K1D n + K 0

(A3.55)

can be applied to describe the rheological properties of milk chocolate, where 1/2 ≤ n ≤ 1. However, in some cases where the milk proteins have been strongly denatured owing to the effect of increased temperature (>60°C) during conching or transportation, n > 1. Mohos (1967b) demonstrated that Eqn (A3.55) can be derived from the assumption J = α + β y = α + β (η0 D )

−n

instead of Eqn (A3.57), and this relation leads to

(A3.56)

598

Confectionery and Chocolate Engineering: Principles and Applications

aβc ⎞ ⎡(1 − c )− An − 1⎤⎦ τ n = ⎡⎣η0n(1 − c )− An ⎤⎦ D n + ⎛ ⎝ A ⎠⎣

(A3.57)

which is equivalent to Eqn (A3.55).

A3.3.2

Theoretical foundation of the Bingham model

The above generalization of the Casson model gives a theoretical foundation for the Bingham model as well if n = 1, i.e. the Bingham model can be regarded as a special case of the Casson model, the plastic viscosity of which is given by the equation

ηPL = η0(1 − c )− A

(A3.58)

where A = aα − 1 > 1 (see above). This equation shows that the plastic viscosity expressed in this form is independent of the shear rate, but it is dependent on the concentration c of the dispersed particles and proportional to the viscosity η0 of the dispersing medium. It should be mentioned, in addition, that the original Casson model (n = 1/2) can be regarded as a special type of Bingham model. The square of the Casson flow curve is

τ = τ 0.CA + 2 DηCAτ 0.CA + ηCA D

(A3.59)

which is the flow curve of a special Bingham fluid, for which

τ = τ 0.CA + ηPL D

(A3.60)

where 1 ηPL = ⎛ ⎞ 2 DηCAτ 0.CA + ηCA ⎝ D⎠

(A3.61)

It is worth emphasizing that all the various fluid models, except for the Casson model, are merely mathematical formulae that best fit the flow curves; they cannot be related to the structure of the fluid modelled.

A3.4

Determination of the exponent n of the flow curve of a generalized Casson fluid

The search for an analytical method for determination of the exponent n has not been successful; therefore an iterative method is recommended. The crucial question is the value of n, because if it is known, then K0 and K1 can be determined by linearization. Since τ > D, if n < 1 then it can be shown that ∂τ/∂D increases monotonically; consequently, the function τ = τ(D) is convex. In contrast, if n > 1, then ∂τ/∂D decreases monotonically, and the shape of the function is concave. On this basis, the correct region in which to search for n may be chosen.

Appendix 3 Survey of fluid models

599

Example A3.4 We seek the exponent n (= 0.83) of the following flow curve:

τ 0.83 = 3 + 4D 0.83 i.e. τ0 (Pa) = 31/0.83 {(Pa0.83)}1/0.83 and η (Pa s) = 41/0.83 {(Pa s0.83)}1/0.83. The measured values of D and τ are given in Table A3.3. Table A3.3 Measured values of D (s−1) and τ (Pa), and their values as a result of exact linearization with n = 0.83. D

τ = (τ0.83)1/0.83

D0.83

τ0.83

0.5 1.3 5 10 20 40 60

7.375706 12.20296 33.01793 60.35124 114.3895 221.737 328.7452

0.56253 1.24329 3.80316 6.76083 12.0186 21.3653 29.9134

5.25012 7.97317 18.2127 30.0433 51.0745 88.4614 122.654

It should be emphasized that the measured (known) values are: τ

D 0.5 1.3 etc.

7.375 12.20

but the linearity has to apply to the relationship between τ0.83 and D0.83, not to these measured values. Table A3.4 shows a process of iteration that divides the region in question into halves in a reasonable way, for example n n n n n

= = = = =

0.875 = (1 + 0.75)/2; 0.9375 = (1 + 0.875)/2; 0.8125 = (0.75 + 0.875)/2; 0.84375 = (0.8125 + 0.875)/2; 0.828125 = (0.8125 + 0.84375)/2.

Table A3.4 Steps of iteration: the values of exponent vs standard deviation and the results (n = 0.828 … , slope = K1, intercept = K0). Values of n

SD Slope Intercept

1

0.5

0.75

0.875

0.9375

0.8125

0.84375

0.828125

0.776468

0.14705

0.121122

0.1158565

0.370818

0.036868

0.029122

0.006258 3.989727 2.976559

600

Confectionery and Chocolate Engineering: Principles and Applications

The process of iteration is (τ1 vs D1), (τ0.5 vs D0.5), (τ0.75 vs D0.75), (τ0.875 vs D0.875), etc. The basis of the decision is the value of the standard deviation

SD =

[ Σ ( x − x ) ( y − y )]2 ⎤ 2 1 ⎡ ⎢Σ ( y − y) − ⎥ 2 N − 2 ⎢⎣ Σ (x − x) ⎥⎦

(A3.62)

which is to be minimized (where N is the number of elements in the sample, and the underlining means the sample average). The values of the slope and the intercept are shown in Table A3.4 for the sake of completeness in the case of n = 0.828175 only – they are uninteresting in the other cases.

A3.5

Dependence of shear rate on the exponent n in the case of a generalized Casson fluid

Let us investigate the dependence of flow rate on the value of the exponent n in the case of a generalized Casson fluid. This means an investigation of the sign of the derivative ∂Q ⎛ ∂Q ⎞ ⎛ ∂ τ ⎞ ⎛ ∂D ⎞ = ∂n ⎝ ∂ τ ⎠ ⎝ ∂D ⎠ ⎝ ∂n ⎠

(A3.63)

(the chain rule). Since ∂Q/∂τ is a derivative of the Hagen–Poiseuille equation, it is positive. From the flow curve τn = K1Dn + K0 (see Eqn A3.55), ∂τ ⎞ τ n −1⎛ = K1D n −1 ⎝ ∂D ⎠ i.e. D ∂τ = K1⎛ ⎞ ⎝ ∂D τ⎠

n −1

(A3.64)

which is positive. From the flow curve (Eqn A3.55) of a generalized Casson fluid, D is defined by the equation D=

(τ n − τ 0n )1 n η

1n ⎛ 1⎞ = ⎜ ⎟ (r n − 1) ⎝ η⎠

(A3.65)

where K0 = τ0n and K1 = ηn; moreover,

τ = r ( >1) τ0

(A3.66)

601

Appendix 3 Survey of fluid models

It can be proved also that the expression

(r n − 1)1 n

(A3.67)

monotonically increases if n (> 0) increases, and consequently ∂D/∂n is positive (see Eqn A3.7). As a result, ∂Q/∂n is positive also, i.e. the higher the exponent n in the flow curve of a generalized Casson fluid, the higher its flow rate. A numerical example presented later (Table A3.5) will demonstrate this result. Human blood is a proven Casson fluid, with n = 1/2; see, for example, Charm and Kurland (1965) and Lee et al. (2007). We can ask whether any method (e.g. consumption of lecithin) exists to increase the exponent n = 1/2 in order to improve the flow of blood, that is, to increase the flow rate of blood in the arteries.

A3.6

Calculation of the flow rate for a generalized Casson fluid

Taking into account the fact that the flow curve of a Casson-type fluid is τn = K0 + K1Dn (Eqn A3.55), the flow rate Q = Q(Δp) can be calculated from the following integral (the Rabinowitsch–Mooney equation): Q=

πR 3 τ R3

τR

2 ∫ Dτ dτ

(A3.68)

τ0

where τ = r Δp/2L, R is the radius of the tube (r is used as a variant), L is the length of the tube, τ0 = r0 Δp/2L is the yield stress, Δp is the pressure difference between the two ends of the tube, K0 = (τ0)1/n is a constant, K1 = (ηn)1/n is a constant and ηn is the dynamic viscosity of a generalized Casson fluid with exponent n. From the flow curve (A3.55), we obtain 1n ⎛ 1⎞ D = ⎜ ⎟ (τ n − τ 0n ) ⎝ η⎠

(A3.69)

However, this equation makes direct integration of the Rabinowitsch–Mooney equation (A3.68) impossible. If the expression for D is changed into the form 1n

τ n⎤ ⎛ 1⎞ ⎡ D = ⎜ ⎟ τ ⎢1 − ⎛ 0 ⎞ ⎥ ⎝ η⎠ ⎣ ⎝ τ ⎠ ⎦

(A3.70)

then Newton’s generalized binomial theorem can be applied (Taylor and Zafiratos 1991, p. 524; Filep 1997), which results in an infinite power series:

(1 − x n )1 n = 1 −

{

} {

1 n−0 n 1 n−0 x + 1 1

}{ } {

1 n − 1 2n 1 n − 0 x + 2 1

}{ }{ 1 n −1 2

}

1 n − 2 3n x + 3 (A3.71)

602

Confectionery and Chocolate Engineering: Principles and Applications

where n = p/q and p and q are integers, and 0 ≤ x ≤ 1. Under these conditions, the series (Eqn A3.70) is convergent. Comment: An arbitrary m = r/t (where r and t are integers) can be used instead of 1/n in the series in Eqn (A3.70); moreover, m can be complex as well. In this case the coefficients of the power series are ⎡ m − 0 ⎤ , ⎡ m − 0 ⎤ ⎡ m − 1 ⎤ , ⎡ m − 0 ⎤ ⎡ m − 1 ⎤ ⎡ m − 2 ⎤ , etc. ⎢⎣ 1 ⎥⎦ ⎢⎣ 1 ⎥⎦ ⎢⎣ 2 ⎥⎦ ⎢⎣ 1 ⎥⎦ ⎢⎣ 2 ⎥⎦ ⎢⎣ 3 ⎥⎦

(A3.72)

If the expression for D(τ) is replaced by its infinite series in the integral in Eqn (A3.68), assuming that x = τ0/τ (see Eqn A3.70), an approximate value of the flow rate can be calculated. The condition for convergence 0 ≤ x = τ0/τ = Bu ≤ 1 (where Bu is the Buckingham number) is fulfilled. (The smaller the value of x, the quicker the convergence.) The result of integration for any value of n is ⎛ π Δp R 4 ⎞ Q=⎜ ×M ⎝ 8ηn L ⎟⎠

(A3.73)

where K1 = (ηn)1/n is a constant, and ηn is the dynamic viscosity of a generalized Casson fluid with exponent n. For specific values of n, we have the following results: Bingham model, n = 1 (for the sake of completeness): 4 1 M = 1 − ⎛ ⎞ Bu + ⎛ ⎞ Bu 4 ⎝ 3⎠ ⎝ 3⎠

(A3.7)

Casson-type model, n = 4/5: 25 25 ⎞ 25 ⎞ 12 5 ⎛ 11 345 ⎞ ⎡ 4⎤ M = ⎢1 − ⎛ ⎞ Bu 4 5 + ⎛ Bu8 5 − ⎛ Bu + ⎜ ⎟⎠ Bu ⎥ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ 16 105 256 107 520 ⎣ ⎦

(A3.74)

Casson-type (or Heinz) model, n = 2/3: 9 9 1 9 ⎞ 267 ⎞ M = ⎡⎢1 − ⎛ ⎞ Bu 2 3 + ⎛ ⎞ Bu 4 3 − ⎛ ⎞ Bu3 + ⎛ Bu8 3 + ⎛ Bu 4 ⎤⎥ ⎝ 16 ⎠ ⎝ 4⎠ ⎝ 128 ⎠ ⎝ 640 ⎠ ⎣ ⎝ 5⎠ ⎦

(A3.75)

Casson model, n = 1/2: 16 4 1 M = ⎛1 − Ca + Ca 2 − Ca8 ⎞ , Ca 2 = Bu ⎝ ⎠ 7 3 21

(A3.10)

Table A3.5 shows the values of M for the various models. It can be observed that if Bu > 0.5, the power series for n = 4/5 and 2/3 is not useful in practice as an approximation, because the convergence will be very slow. Additional terms would be required for a more exact approximation. It can be observed, moreover, that higher values of n are associated with higher values of Q, and that the effectiveness of Casson fluids (n = 1/2) is very poor. Consequently, an increase in the exponent n may be of great practical importance.

Appendix 3 Survey of fluid models

603

Table A3.5 Values of M in four types of generalized Casson model (n = 1, 4/5, 2/3, 1/2) as a function of the Buckingham number. Bu

Bingham

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.986667 0.973333 0.96 0.946668 0.933335 0.920004 0.906675 0.893347 0.880022 0.8667 0.733867 0.6027 0.4752 0.354167 0.2432 0.1467 0.069867 0.0187 M(Bu)

A3.7

n = 4/5 0.9609 0.932112 0.906331 0.882359 0.859668 0.837962 0.817052 0.796808 0.777136 0.757963 0.585081 0.433738 0.296128 0.169214 0.051818 −0.05622 −0.15439 −0.24165 M(4/5)

n = 2/3 (Heinz) 0.91778779 0.87059359 0.83164581 0.79737306 0.76628203 0.73756517 0.71072242 0.68541649 0.6614061 0.63851092 0.45008597 0.30600754 0.18956824 0.09534568 0.02236004 −0.0278088 −0.0517485 −0.0445527 M(2/3)

n = 1/2 (Casson) 0.784761904 0.703417845 0.644102634 0.596190354 0.555565117 0.520116013 0.488589033 0.460167088 0.43428259 0.410522249 0.244387972 0.147677011 0.086501641 0.047446405 0.023321899 0.009534225 0.002756897 0.000338176 M(Ca)

Lemma on the exponent in the generalized Casson equation

We have to prove that the expression

(r n − 1)1 n

(A3.67)

monotonically increases as (0 <) n increases. It is sufficient to prove that ⎛ d ⎞ ⎡ ex − 1 1 x ⎤ ( ) ⎦ ⎝ dx ⎠ ⎣

(A3.76)

is positive when x increases. As the expression (ex − 1)1/x is a general power function, we have to start from the definition of a general power function zα: z α = e [ exp (α ln z )]

(A3.77)

604

Confectionery and Chocolate Engineering: Principles and Applications

where α can be an irrational expression as well. In the case discussed, z = ex − 1 and

α=

1 x

we have (d/dx)zα = e[exp(α ln z)][(dα/dx) ln z + (α/z)(dα/dx)] ⎛ d ⎞ z α = e exp (α ln z ) ⎡⎛ dα ⎞ ln z + ⎛ α ⎞ ⎛ dα ⎞ ⎤ [ ] ⎢⎝ ⎠ ⎝ dx ⎠ ⎝ z ⎠ ⎝ dx ⎠ ⎥⎦ ⎣ dx x ⎤ ex 1 x ⎡ ln ( e − 1) = ( e x − 1) ⎢ − + ⎥ 2 x x x ( e − 1) ⎦ ⎣

(A3.78)

and −

ln ( e x − 1) ex + >0 x2 x ( e x − 1)

(A3.79)

since ln ( e x − 1) ex > x ex − 1

(A3.80)

In detail, ex >1 ex − 1 ln ( e x − 1) < ln e x = x →

(A3.81) ln ( e x − 1) <1 x

(A3.82)

Finally, because all the multiplying factors of the equation x ln n ( e x − 1) ⎤ ⎡ ln ( e x − 1) ⎤ ex 1 x⎛ 1 ⎞ ⎡ e x 1 − = − + e ( ) ⎥ ⎥ ⎝ x ⎠ ⎢⎣ e x − 1 x2 x ( e x − 1) ⎦ x ⎦ ⎣

( e x − 1)1 x ⎢ −

(A3.83)

are positive, it can be stated that the derivative function (d/dx)[(ex − 1)1/x] is positive. Thus, if x is increased, the value of (ex − 1)1/x increases as well. If, instead of ex, the investigation of the sign is directed towards rn, then the derivative (Eqn A3.78) will contain also a multiplying factor ln r, which is positive since r > 1.

Appendix 3 Survey of fluid models

605

Further reading Rohsenow, W.M., Hartnett, J.P. and Cho, Y.I. (eds) (1998) Handbook of Heat Transfer, 3rd edn. McGraw-Hill, New York. Steffe, J.F. (1996) Rheological Methods in Food Process Engineering, 2nd edn. Freeman Press, East Lansing, MI. Tscheuschner, H.-D. (1993) Schokolade, Süsswaren. In: Weipert, D., Tscheuschner, H.D. and Windhab, E. (eds) Rheologie der Lebensmittel. Behr’s Verlag, Hamburg. Tscheuschner, H.-D. (1993) Rheologische Eigenschaften von Lebensmittelsystemen. In: Weipert, D., Tscheuschner, H.D. and Windhab, E. (eds) Rheologie der Lebensmittel. Behr’s Verlag, Hamburg. VDI-GVC (2006) VDI-Wärmeatlas. Springer, Berlin.

Appendix 4

Fractals

Contents A4.1 Irregular forms – fractal geometry A4.2 Box-counting dimension A4.3 Particle-counting method A4.4 Fractal backbone dimension Further reading

A4.1

606 606 607 608 608

Irregular forms – fractal geometry

It is well known that Mandelbrot studied the length of the coast of Britain using maps of various scales and with compasses set to span various distances: smaller scales gave longer results. If the compass setting was 500 km, then the length obtained was 2600 km; if the compass setting was 17 km, then the length obtained was 8640 km. A Portuguese encyclopedia gave a larger value for the length of the Portuguese/Spanish border than a Spanish one did, since Portugal is a smaller country than Spain and the map used in Portugal for the measurement had more detail. The study of such irregular shapes led Mandelbrot to establish a new geometry called ‘fractal geometry’ (the word ‘fractal’ refers to the Latin word fractus, ‘broken’) (Mandelbrot 1977, 1983; Peitgen and Richter 1986; Peitgen and Jürgens 1990; Peitgen et al. 1991; Schröder 1991). Fractals have fine structure at arbitrarily small scales and are too irregular to be easily described in traditional Euclidean geometric language. Many such objects can be found in nature: crystals, electrochemically deposited zinc metal leaves (with a dendritic growth pattern), the arteries and veins of a kidney, landscapes, etc. At the turn of the 19th and 20th century, mathematicians came up with some 10 different notions of dimensions, which are all related, and are all special forms of Mandelbrot’s ‘fractal dimensions’. Of these notions of dimensions, the ‘box-counting dimension’ has the most applications in science.

A4.2

Box-counting dimension

The structure to be studied is put onto a grid with a mesh size C, and the number of grid boxes that contain some of the structure is counted. This gives a number, say N, which is dependent on C, of course. Then we plot the logarithms, and find Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

Appendix 4 Fractals

(a) Fig. A4.1

607

(b)

(a) and (b) show two shapes on a grid for evaluation of the box-counting dimension.

Table A4.1 Evaluation of the two shapes shown in Fig. A4.1 according to Eqn (A4.1). Shape (a) N

1/C

ln N

ln (1/C)

Slope = D(b)

4 15 50 141

2 4 8 16

1.386294 2.70805 3.912023 4.94876

0.693147 1.386294 2.079442 2.772589

1.715562

N

1/C

ln N

ln(1/C)

Slope = D(b)

4 16 54 158

2 4 8 16

1.386294 2.772589 3.988984 5.062595

0.693147 1.386294 2.079442 2.772589

1.766623

Shape (b)

log N ∼ log (1 C )

D(b)

(A4.1)

where D(b) is the box-counting dimension, ≤ 2 for a plane. (Comment: In a plane, the box-counting dimension can never exceed 2; at the same time, however, the self-similarity dimension can do so. The reason for this discrepancy is that in the case of curves that have overlapping parts, the box-counting dimension does not take these overlapping parts into account.) Figure A4.1 shows two shapes on a grid, and Table A4.1 presents the evaluation of them.

A4.3

Particle-counting method

The particle-counting method was applied to determine the fractal dimension of a structure consisting of fat crystal networks by Narine and Marangoni (1999b). The images of

608

Confectionery and Chocolate Engineering: Principles and Applications

fat networks that are acquired from polarized light microscopy are not suitable for analysis by the traditional methods of fractal-dimension determination. The reason is that such images are subsets of two-dimensional (2D) space, but represent a subset of a threedimensional (3D) network. Therefore, the number of particles present in a 3D portion of the sample is counted by first representing all the particles present in that portion of the sample in the plane of the image. Those particles which do not appear in the picture owing to geometrical shadowing are missed, but the number of these can be rendered negligible by making the thickness of the sample very small. In order to calculate the value of D (with d = 3, i.e. in space), the number of microstructural elements N(R) projected onto a square area of side length R is counted, the square being drawn in the focal plane of the image: N = R σ , N >> 1 and N = cR D

(A4.2)

where N(R) is the number of microstructural elements in the image, R is the linear size of the fractal, σ is the linear size of one particle (microstructural element), d is the usual topological dimension (in the following, always equal to 1 for a line, 2 for a plane and 3 for space), D is the fractal dimension and c is a constant.

A4.4

Fractal backbone dimension

According to Narine and Marangoni (1999b), the fractal backbone dimension x of a network may be thought of as an indicator of the spatial distribution of microstructural elements in chains, and the elements in the chains constitute a microstructure. These chains are arbitrary in terms of the fact that a microstructural element may belong to any chain. In a 2D system such as the screen of a microscope, the following formula holds: N ∼ (R σ )

x

(A4.3)

where R is the length of an area enveloping the fractal chain, and x is the fractal backbone dimension (the chemical length exponent or tortuosity). For details of the microscopic method of determination, see Narine and Marangoni (1999b).

Further reading Rothschild, W.G. (1998) Fractals in Chemistry. Wiley-Interscience, New York. Stauffer, D. and Stanley, H.E. (1996) From Newton to Mandelbrot: A Primer in Theoretical Physics with Fractals for the Personal Computer, 2nd edn. Springer, Berlin.

Appendix 5

Introduction to structure theory

Contents A5.1 A5.2 A5.3 A5.4 A5.5 A5.6 A5.7 A5.8

A5.1

General features of structure theory Attributes and structure: A qualitative description Hierarchical structures Structure of measures: A quantitative description Equations of conservation and balance Algebraic structure of chemical changes The technological triangle: External technological structure Conserved substantial fragments

609 610 611 611 612 614 614 615

General features of structure theory

Structure theory was developed by Blickle and Seitz (Seitz and Blickle 1974, Blickle and Seitz 1975, Blickle 1978). Further references are Szép and Seitz (1975) and Seitz et al. (1975, 1976). Structure theory deals with the attributes of a system and their relations. The attributes have both qualitative and quantitative characteristics. In chemical engineering, attributes may reasonably be distinguished into the following categories: • substantial attributes (e.g. atoms or ions, and other chemical or physical properties such as density and energy); • attributes of machinery (e.g. type of tank, tube or distillation device); • attributes of technological changes (dissociation, double decomposition, substitution, distillation, etc.). These three types of attributes form a so-called technological triangle (Mohos 1982; see Fig. A5.1). An attribute can be regarded as a set of attribute elements, and the structure of this attribute is modelled by an internal product of its elements, i.e. the relations between these elements. The relations between different attributes are modelled by an external product of attributes. This method of system theory entirely follows the algebra of relations. A typical example of the technological triangle is the following. To choose the machinery for a technological task, catalogues are studied that describe the attributes (parameters) of the machinery offered, the properties of the substances (substantial attributes) Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

610

Confectionery and Chocolate Engineering: Principles and Applications

M Machinery

S Substances

Fig. A5.1

C Changes

Technological triangle.

that may be produced with it, and the technological changes (attributes of operations) that may be implemented with it. This procedure may be executed electronically today using the Internet. When structure theory was developed by Blickle and Seitz in the 1970s, a tailor-made computer was developed for this purpose in the Mu˝szaki Kémiai Kutató Intézet (Research Institute of Industrial Chemistry), Veszprém. In chemical engineering, these attributes and relations are discovered in one way or another, and then systematization is done by the use of system theory. This method of systematization provides an example that may be applied also in the confectionery industry – and perhaps in food science as well.

A5.2

Attributes and structure: A qualitative description

The elements ai of the ‘attribute set’ A characterize a part W of four-dimensional space (x, y, z and t), where A ∋ ai and W ∋ [xi; yi; zi; ti], which latter is a point of W. Γ(k; i) is a function with values 1 and 0 that refers to set A, where k refers to the part [xk; yk; zk; tk] of W and i refers to the element ai of set A. If there exists the attribute combination {ai; aj} in the point k, then

Γ ( k; i ) = Γ ( k; j ) = 1 = Γ ( k; i ∧ j )

(A5.1)

Otherwise Γ(k; i ∧ j) = 0 (if this combination does not exist), where ∧ means logical disjunction. Let us consider the product set of A of finite cardinality, P(A) ∋ A. In addition, let us consider the equivalency relation ε, which decomposes the set A into equivalency classes designated A1, A2, … , Am. Finally, let us consider the set P of possibilities, the elements Ln of which are the possible combinations of attributes P ∋ Ln, where n = 1, 2, … , r. If P ∋ Lp; P ∋ Lq and Lp ⊃ Lq, then P ∋ Lp, Lq

(A5.2)

i.e. a part of a possible attribute combination is possible too. Moreover, for ∀Pp ∋ Lq and ∀Ap, Lq ∩ Ap = 1 or 0 i.e. any possible combination can contain only one element of an attribute class.

(A5.3)

Appendix 5 Introduction to structure theory

611

Let us define a function F: P(A) → {1; 0} such that F(Lp) = 1 if P ∋ Lp, otherwise F(Lp) = 0. The construction [A; ε; F] is called a structure. Example A5.1(Blickle and Seitz 1975, p. 35) The various types of phase systems may be characterized by combinations bi = f j ∧ f j ∧ gk

(A5.4)

where i = 1, 2, 3, 4, 5; f is the phase, which may be solid (j = 1), liquid (j = 2) or gaseous (j = 3); and g is the homogeneous relation (k = 1) or heterogeneous relation (k = 2). In detail, b1 b2 b3 b4 b5

= = = = =

f1 f2 f2 f3 f3

A5.3

∧ ∧ ∧ ∧ ∧

f1 ∧ (g1 ∨ g2): homogeneous solid phase or a mixture of powders; f2 ∧ g1: homogeneous liquid phase; (f1 ∨ f2 ∨ f3) ∧ g2: solid, liquid or gaseous substance dispersed in liquid; f3 ∧ g1: homogeneous gaseous phase; (f1 ∨ f2) ∧ g2: solid or liquid substance dispersed in gas.

Hierarchical structures

Among the various types of structures, the hierarchical structures are especially of interest. As can be seen, the equivalency relation ε decomposes the set A into equivalency classes Ai (i = 1, 2, … , m), and at the same time, the space belonging to set A is decomposed into partial spaces Si (i = 1, 2, … , m), where Si belongs to Ai. This means that certain relations can exist between two different attribute classes that contain only one common element; for example, a certain oxygen atom (an element of the attribute class ‘atom’) belongs exclusively to only one molecule of water (an element of the attribute class ‘molecule’). Naturally, there are other types of relations, but such an exclusivity characterizes the socalled arranging relation denoted by →; for example, a → b means a is contained by b. The characteristics of an arranging relation are: • it is reflexive: a → a; • it is non-commutative: if a → b, then b → / a (b is not contained by a); • it is transitive: if a → b and b → c, then a → c. The arranging relation transmits a certain attribute of a to b; for example, an atom (a) is contained by a molecule (b), which is contained by a phase (c). Evidently, this atom is contained also by the phase in question. The arranging relation transmits an attribute by heredity; for example, a family tree is built up from arranging relations between different generations.

A5.4

Structure of measures: A quantitative description

The attribute combinations that are possible (Γ = 1) may be characterized by measures as well; for example, combinations of atoms (‘compounds’) have a molar mass, molar volume, melting point, etc.

612

Confectionery and Chocolate Engineering: Principles and Applications

The set function μ(Ωi) is called a measure defined on the set Ω, where Ω ∋ Ωi, if: • its values are non-negative sums; and • it is additive, i.e. for any decomposition of it, it is valid that

Ωi = Ωi ,1 ∪ Ωi ,2 ∪, … , ∪ Ωi ,n

and Ω ∋ Ωi

(A5.5)

A ratio of measures of the same type referring to different attribute classes is called a homogeneous measure. For example, if the type of measure is the mass, and the attribute classes are ‘atom’ (hydrogen) and ‘molecule’ (water), then m(H2) = 2 g and m(H2O) = 18 g; their ratio is 18/2 (or 2/18, or the ratio of oxygen is (18 − 2)/18), i.e. measures of the same type refer to different types of attribute. In contrast, a heterogeneous measure can also be defined, for example the volume (cm3) of hydrogen in 18 g of water – this measure is of dimension (cm3/g), i.e. heterogeneous measures are expressed in ‘mixed’ concentration units. An important example is the atomic mass, which is a ratio of the mass and number of atoms of the same type, i.e. different types of measure refer to the same type of attribute.

A5.5

Equations of conservation and balance

If Zi(Pj; t) is the measure of possibility Pj and t2

IN OUT ∫ [ Zi ( Pj ; t ) − Zi ( Pj ; t )] dt + [ Zi ( Pj ; t1 ) − Zi ( Pj ; t2 )] = 0

(A5.6)

t1

or, with a simpler notation, D [ Zi Pj ] = 0

(A5.7)

then it is said that the measure Zi(Pj; t) of possibility Pj is conserved, and D is the operator of conservation. If Zi(Pj; t) can be differentiated with respect to t (time), this equation can be written in the form ZiIN( Pj ; t ) − ZiOUT( Pj ; t ) −

dZi ( Pj ; t ) =0 dt

(A5.8)

It is easy to see that D {[ Zi Pj ] + [ Zi Pk ]} = D {[ Zi Pj ]} + D {[ Zi Pk ]}

(A5.9)

and D {const.[ Zi Pj ]} = const. D {[ Zi Pj ]}

(A5.10)

Let ak j be an element of the attribute class of Ak, where j = 1, 2, … , w. If for all these elements of Ak it is valid that

Appendix 5 Introduction to structure theory

D ⎡⎣∑ j Zi ( ak j )⎤⎦ = 0

613

(A5.11)

then Ak is called a ‘conservative attribute class’. For example, the chemical elements are a conservative attribute class in chemical engineering (where, for example, the splitting of atoms does not belong to the attribute class of changes in the system!). The stoichiometric equations are based on this fact: M×C = A

(A5.12)

where M is the stoichiometric matrix of the reaction, C is a column matrix of compounds, consisting of the numbers of molecules, and A is a column matrix of atoms, consisting of the numbers of atoms. Since the atom attribute is conservative, D ( M × C) = D ( A) = 0

(A5.13)

where the operator D is used on every row of the product M × C.

Example A5.2 Let us consider the mixing of sodium hydroxide and sulphuric acid, 2NaOH + H2SO4:

Na O H S

NaOH 1 1 1 0

H2SO 4 ⎡1 0 ⎢1 4 =M=⎢ ⎢1 2 ⎢ ⎣0 4

⎡1 × 2 + 0 × 1⎤ ⎡2 ⎤ ⎢1 × 2 + 4 × 1⎥ ⎢6 ⎥ ⎥=⎢ ⎥ A=⎢ ⎢1 × 2 + 2 × 1⎥ ⎢ 4 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣0 × 2 + 1 × 1⎦ ⎣ 1 ⎦

0⎤ 4⎥ 2 NaOH ⎥ and C = ⎡ ⎤ ⎢ ⎥ 2⎥ ⎣1 ⎦ H2SO 4 ⎥ 4⎦

Na O H S

In chemical engineering, Z is some kind of specific measure (mass, component, heat or momentum flux per unit volume) and the equations of conservation are the Damköhler equations referring to those specific measures. (The use of the symbol D is not accidental.) Therefore, the most commonly applied form of the equations of conservation is D (Z ) =

dZ + div [ v ( Z ) − K grad ( Z )] dt

(A5.14)

For a batch process, D (Z ) =

dZ =p dt

(A5.15)

614

Confectionery and Chocolate Engineering: Principles and Applications

For a continuous process with zero mixing (K = 0), D ( Z ) = vx

dZ =q dx

(A5.16)

where p and q are variables referring to sources, sinks and transfer, vx is the linear velocity of the component characterized by the parameter Z in the direction x, and K is the factor of mixing.

A5.6

Algebraic structure of chemical changes

Structure theory defines chemical changes as a mapping h, the kernel of which is the structure of the input materials and the image is the structure of the output materials. In Example A5.2, the kernel is 2NaOH + H2SO4, and the image is Na2SO4 + 2 H2O; h is an ionic exchange reaction. In a technological system, the set H of changes (H ∋ hi, where i = 1, 2, … , n) may be defined. Moreover, two algebraic operations between these changes can be defined: 䊝: series coupling; 䊟: parallel coupling. These refer to material streams separated in space, according to the original work of Blickle and Seitz (1975). It can be shown (Blickle and Seitz 1975) that the set H and these two algebraic operations establish an algebraic structure (H; 䊝; 䊟), which is a non-distributive lattice (Birkhoff 1948). Using this algebraic structure, chemical changes (operations) can be modelled. For details, see Blickle and Seitz (1975).

A5.7

The technological triangle: External technological structure

A technological system consists of three parts (Fig. A5.1): • machinery (M); • substances (raw materials etc.) (S); • changes (or operations) (C) executed by the machinery on the substances. These parts of a technological system may be regarded as three vertices of a triangle (Mohos 1982). Applying the usual definition, it may be said that the relations between the elements of the machinery (or of the substances or changes) can be expressed by internal (Cartesian) products; however, the relations between the above three parts are external relations. ‘Bundles’ (denoted by —) of external relations, which can be regarded as the sides of the triangle, link the vertices, i.e. M—S, S—C and C—M. The usual form of a technical data sheet reflects some ‘collection’ of the attribute elements of a machine, and is constructed on the basis of technical and marketing considerations. A factory producing many different machines is obliged to construct a database of

Appendix 5 Introduction to structure theory

615

these sheets on uniform principles. Such a database contains references to the attributes of substances and the attributes of the changes (operations) that can be executed on substances. Consequently, the database can be used for solving a given problem in the chemical industry if two ‘points’ of the triangle (the raw materials and the operations of the processes) are known; the third ‘point’ (the suitable machinery) can then be chosen. Another case is when the machinery and the operations executed by this machinery are given. In this case, the substances (the third ‘point’) the processing of which these two ‘points’ make possible can be chosen (for example, a particular unit may suitable for nitration of benzene, etc. but unsuitable for hydrogenation of oils, etc. at high pressure).

A5.8

Conserved substantial fragments

A great advantage of chemistry is that its structural description is based on atoms, groups of atoms, and molecules, because the atoms are conserved substantial attributes. However, foods are typically of cellular structure; therefore, atoms and molecules alone are not sufficient for characterization of the structure of foods. Since there is no uniform description of the structure of the materials, and also the definition of the technological changes is contingent on the given task and must be fitted to it, it can be said in the spirit of structure theory that both the kernel and the image of the mapping are missing, compared with the sound definitions of chemistry. Structure theory aims to surmount these difficulties by defining a conserved substantial fragment (CSF) (Mohos 1982). Atoms are conserved in chemical reactions; similarly, the CSFs are conserved in the technological changes of a given food technology. Evidently, the CSFs are bound by this definition to individual cases, and although their role is as ‘the atoms of food’, they cannot be summarized in any periodic system à la Mendeleyev because their conservative property is valid in a given food technology only. The concept of a conserved substantial fragment is defined as follows: (1) The set of the conserved substantial fragments of the system studied (C) is a union of the sets A and B: C = A∪ B

(A5.17)

(2) The elements of the set A are those CSFs which are brought about by physical operations exclusively, and also have different properties from each other. It is supposed that the chemical changes taking place in these fragments can be neglected from the point of view of our study. (3) The elements of the set B are chemical atoms or groups of atoms, and, in practice, only these fragments participate in the chemical operations performed in the technological system. (4) The fragments defined in this way behave as conserved substantial parts, i.e. their quality remains unchanged during the appropriate operations. Consequently, they can be regarded as ‘quasi-chemical components’, for which the following conservation equation is valid (the second Damköhler equation, for the flux of a component; see Chapter 1):

616

Confectionery and Chocolate Engineering: Principles and Applications

div [ci v ] − div [ D grad ci ] + ωβ Δci + νi r = − ∂ci ∂t

(1.4)

where ci is the concentration of component i [(mol/m3) for the elements of set B; (kg/ m3) for the e1lements of set A], D is the diffusion coefficient (m2/s), β is the component transfer coefficient (m/s), νi is the degree of reaction for component i, and r is the velocity of reaction (mol/m3 s). The ‘source’ term νir in the second Damköhler equation is worthy of attention: if the elements of set A are the materials, then νi is the degree and r is the velocity of physical transformation; this transformation may be, for example, the decomposition of cells by breaking up. In this case, the units of r may be, for example, kg cocoa mass/m3 s. What is the meaning of the concept of a ‘conserved substantial fragment’ in the case presented in Fig. 2.1 in Chapter 2? The elements of set C are [solids; emulsifier; water; oil; dissolved substances]. If no chemical reaction takes place then set B is empty, i.e. C = A. If there are some chemical reactions between the dissolved substances, for example b1, b2 and b3, which may be atoms, groups of atoms etc., then B = [b1; b2; b3] and A = [solids; emulsifier; water; oil; other dissolved substances]. In addition, C = A U B. The concept of a CSF can be regarded as a theoretical basis for recipe calculations as well. Confectionery products are mostly blends of components, and the role of chemical reactions (inversion, the Maillard reaction, the reactions of flavonoids in cocoa beans, etc.) is relatively small from the point of view of mass changes. Naturally, however, if the chemical reactions are the subject of study, the blend character of the product has a secondary importance. The concept of a CSF is far from the exactness of the concept of atoms or groups of atoms, even though the set of groups of atoms is not a closed system but is well defined by the chemistry. Regarding the algebraic structure (H; 䊝; 䊟), series and parallel coupling can be applied as operations with the modification that the material flows are not, by any means, separated in space. This modification is not contrary to the thinking of structure theory, and is very useful in food engineering, in which many different changes take place either in parallel (parallel coupling) or after each other (series coupling) in the same space. A classical example is provided by conching.

Appendix 6

Technological lay-outs

Too small/large beans

Fibre, stone metal etc.

Storage of cocoa beans

Raw-bean cleaning, sorting

Storage of cocoa liquor, homogenization

Fig. A6.1

Roasting, cooling

Shell, germ

Winnowing, separation

Pre-grinding of cocoa nibs

Fine grinding

Production of cocoa liquor (Traditional method).

Fibre, stone, metal etc.

Storage of cocoa beans

Too small/large beans

Raw-bean cleaning, sorting

Shell, germ

Thermal shock, cooling

Winnowing, separation

Roasting of cocoa nibs Storage of cocoa liquor, homogenization Fig. A6.2

Fine grinding

Production of cocoa liquor (Method of nib roasting).

Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

Pre-grinding of cocoa nib

618

Confectionery and Chocolate Engineering: Principles and Applications

Fibre, stone, metal etc.

Too small/large beans

Raw-bean cleaning, sorting

Storage of cocoa beans

Shell, germ Pasteurization + thermal shock, cooling

Winnowing, separation

Pre-grinding of cocoa nibs Storage of cocoa liquor, homogenization Fig. A6.3

Roasting of cocoa liquor

Fine grinding

Production of cocoa liquor (Method of roasting cocoa liquor).

Sugar powder

Cocoa liquor

Cocoa butter Refining (five-roll refiners)

Homogenization of ingredients

Wet conching

Vapour, acids

Hazelnut paste

Milk powder

Cocoa butter

Dry conching

Cocoa butter

Lecithin

Emulsification Storage of chocolate mass, homogenization

Fig. A6.4

Production of milk chocolate mass (Traditional method).

Tempering 3 (≈30–32°C)

Tempering 2 (≈27°C)

Warm side Moulding (dosing)

Vibration

Warming the moulds (≈30°C)

Fig. A6.5

Production of chocolate bar.

Tempering 1 (≈32°C)

Storage of chocolate mass (≈50°C)

Cool side Cooling 1 (chocolate ≈16°C)

Cooling 2 (chocolate ≈9°C)

Demoulding

Cooling 3 (chocolate ≈12°C)

Packaging of bars

Appendix 6 Technological lay-outs

Storage of cocoa liquor

Vapour + odours

Pressing (separation)

Air

Packaging of cocoa powder

For moulding

Deodorization by film evaporation

“Preparation” (chemical reaction +evaporation)

Acidic solution for neutralization

For chocolate

Cocoa butter

Aqueous alkali solution

Vapour

Cooling the cake

Air

Tempering

Kibbling the cake

Pulverization

Cool air

Tempered air Fig. A6.6

Production of cocoa powder and cocoa butter (Alkalization of cocoa liquor).

Vapour

Aqueous alkali solution

“Nib alkalization” (roasting, chemical reaction + evaporation) Acidic solution for neutralization Shell, germ Fig. A6.7

Winnowing, separation

Pre-roasting the beans

Cocoa butter

Cocoa cake

Pressing (separation)

Fine grinding

Pre-grinding of cocoa nibs

Raw-bean cleaning, sorting

Production of cocoa powder and cocoa butter (Dutch Process).

Storage of cocoa beans

619

620

Confectionery and Chocolate Engineering: Principles and Applications

Tempering chocolate for shell

Warm side

Vibrating, flipping

Depositing

Tempering chocolate for bottoming

Spinning

Precooling, scraping

Vibration

Cooling the filling

Dosing, filling

Cool side

Turning back, cooling

Nut + fat cream or cherry + fondant cream

Rim heating Final cooling Bottoming

Praline production.

Sugar silo

Glucose syrup

Predissolving

Dissolving

Atmospheric evaporation

Vacuum evaporation

Water tank Sugar solution or wax

Packaging the drops Fig. A6.9

Sorting

Packaging of pralines

Tempering the moulds

Moulds

Fig. A6.8

Demoulding

Drops manufacture.

Warm air

Surface treatment

Warm tempering (decoration)

Cool tempering (colouring, flavouring)

Cooling

Filling

Shaping the rope + dosing of filling

Cutting the rope

Cream of tartar

Sugar silo

Atmospheric evaporation

Dissolving

Water tank

Vacuum evaporation

Glycerine

Cool tempering (colouring, flavouring)

Pulling

Warm vapour

Surface wetting in warm room

Cutting the rope

Cooling

Warm tempering (decoration)

Rope shaping + filling

Packaging the sweets

Filling

Fig. A6.10 Production of grained sweets with cremor tartari. By addition of glucose syrup instead of cremor tartari a simpler technology can be installed which results a similar quality of product.

Glucose syrup

Sugar silo

Atmospheric evaporation

Shaping the starch moulds

Dissolving

Water tank

Packaging the covered fondant product

Cooling

Dosing into starch (colouring, flavouring)

Mogul system Conditioning the starch Air

Covering with chocolate curtain

Crystallization

Solidification of fondant centres

Demoulding

Covering machine Bottoming

Arranging the centres on the covering machine

Couverture chocolate, tempered Fig. A6.11

Production of fondant centres by Mogul, and covering by chocolate.

622

Confectionery and Chocolate Engineering: Principles and Applications

Sugar silo

Puffer storage

Cooling (≈ 28–32°C)

Loading the candy pots

Storage of centres

Fig. A6.12

Water tank

Crystallization on the surface of centres

Flooding the candy pots

Dissolving

Evaporation (≈ 105–107°C)

Circle of candy solution

Draining the candy pots

Circle of candy pots

Packaging the centres

Drying the centres

Covering the fondant centres by candy layer (‘candis layer’, traditional).

Sugar silo

Milk (milk powder etc.)

Water

Dissolving

Glucose syrup

Molten butter

Emulsification

Evaporation

Emulsifier Caramelization (evaporation) Cooling

Shaping, e.g. by Mogul

Fig. A6.13

Homogenization (colouring, flavouring)

Manufacture of toffee and fudge (by adding fondant mass).

Fondant mass

Appendix 6 Technological lay-outs

Sugar silo

Gelling agent

Puffer (pectin gel)

Glucose syrup

Homogenization + boiling

Evaporation (≈ 106°C)

Dissolving

Water Colouring, flavouring (sol → gel transition of pectin gel)

Shaping, e.g. by Mogul

Fig. A6.14

Jelly manufacture (general scheme). For further details see Chapter 11.

Sugar silo

Water

Dissolving

Foaming agent

Glucose syrup

Evaporation

Other ingredients Fig. A6.15

Cooling (sol → gel transition)

Manufacture of sweet foams.

Air

Foaming

Shaping, e.g. by Mogul or extruder

623

624

Confectionery and Chocolate Engineering: Principles and Applications

Sugar

Water

Glucose syrup

Honey, brown sugar, molasses etc.

Emulsifier

Butter or fat

Dissolving Dissolving

Flavours

Dissolving

Evaporation Pasteurization, homogenization

Butter

Fondant gun Foaming (continuous) Extruder (homogenization)

Fig. A6.16

Covering with chocolate and/or packaging

Shaping by cutting

Montelimar manufacture (continuous).

Sugar

Honey Nuts

Sugar melting (≈ 200–210°C)

Glucose syrup

Homogenization

Covering with chocolate or packaging

Fig. A6.17

Pasteurization, homogenization

Croquante manufacture.

Rolling

Cutting

Appendix 6 Technological lay-outs

Blanching (≈ 80°C, 10–20 min)

Sorting almonds

Warm water

Fig. A6.18

Homogenization

Bitter almonds (≈ 5%)

Shaping by cutting or extruder

Cooling

Impurities

Sugar (crystalline)

Shells

Grinding by rolling

Cooking (≈ 30–40 min)

Sorting almonds

Shelling

Marzipan manufacture (traditional).

Wetting

Dragée centres Castor/icing sugar

Castor/icing sugar

Drying

Sugar solution

Sugar solution

Air Wetting

Repeated Air

Drying

Drying

Colouring, flavouring

Air (cold/warm)

Glazing

Air Fig. A6.19

Packaging

Drying

Dragee manufacture (cold/warm method).

Dragée centres, e.g. hazelnuts

‘Wetting’

Tempered chocolate Fig. A6.20

Manufacture of chocolate dragee.

Hardening

Repeated

Air

Packaging

625

626

Confectionery and Chocolate Engineering: Principles and Applications

Icing sugar

Homogenization

Packaging

Homogenization and granulation

Lubricant

Drying Homogenization

Colouring, flavouring

Binding agent, e.g. gelatin solution

Lubricant

Indirect method

Direct method

Homogenization

Tabletting

Packaging

Manufacture of tablets.

Chicle gum

Pre-warming (c. 60°C)

Colouring, flavouring (special)

Z-kneading

Glucose syrup

Packaging

Extrusion

Rolling

Icing sugar Shaping, e.g. into balls

Fig. A6.23

Drying

Manufacture of lozenges.

Icing sugar/ glucose

Fig. A6.22

Cutting or punching out

Colouring, flavouring

Binding agent, e.g. gelatin solution Fig. A6.21

Rolling

Surface treatment

Manufacture of chewing and bubble gums.

Cutting

Packaging

Appendix 6 Technological lay-outs

Z-kneading (c. 35°C)

Flour

Icing sugar

Relaxation

Baking powder

Water

Sheeting and gauging

Cutting

Baking (c. 180°C)

Baking (c. 200°C)

Cooling

Surface treatment

Special operations

Baking (c. 160°C)

Packaging

Cracker production

Oil spraying

Manufacture of semi-sweet biscuits and crackers.

Z-kneading (c. 20°C)

Flour

Icing sugar

Fig. A6.25

Lamination

Scrap return

Vegetable fat

Fig. A6.24

Sheeting

Vegetable fat

Baking powder

Baking (c. 150°C)

Baking (c. 175°C)

Cooling

Surface treatment

Manufacture of short-dough biscuits.

Shaping by extrusion + cutting or moulding

Baking (c. 175°C)

Packaging

627

628

Confectionery and Chocolate Engineering: Principles and Applications

Baking powder

Rye flour

Honey

Water

Baking (c. 200°C) Surface treatment (covering by chocolate or candy crust)

Cooling

Sheeting and gauging

Baking (c. 160°C)

Packaging

Manufacture of Lebkuchen (‘honey cakes’).

Sugar solution

Emulsification

Oil/fat Salt

Suspending

Flour

Filtering

Lecithin Milk/milk derivatives

Cream preparation

Scrap return

Fig. A6.27

Scrap

Re-kneading

Shaping by rotary moulding or punching out

Spices

Baking (c. 180°C)

Fig. A6.26

Relaxation (c. 2 weeks)

Z-kneading (c. 60°C)

Wheat flour

Wafer manufacture.

Egg/egg derivatives Spreading/ layering

Cooling

Buffer storage

Cooling

Dosing/baking

Cutting

Packaging

Appendix 6 Technological lay-outs

629

Further reading Alikonis, J.J. (1979) Candy Technology. AVI, Westport, CT. Almond, N., Wade, P., Gordon, M.H. and Reardon, P. (1991) Biscuit, Cookies and Crackers. Elsevier Applied Science, London. Beckett, S.T. (editor) (1988) Industrial Chocolate Manufacture and Use. Van Nostrand Reinhold, New York. Biscuit and Cracker Manufacturers Association (1970) The Biscuit and Cracker Handbook. Chicago. Cakebread, S.H. (1975) Sugar and Chocolate Confectionery. Oxford University Press, Oxford. Ellis, P.E. (ed.) (1990) Cookie and Cracker Manufacturing. Biscuit and Cracker Manufacturers Association, Washington, DC. Fabry, Y. and Bryselbout, P.H. (1985) Guide Technologique de la Confiserie Industrielle. SEPAIC, Paris. Faridi, F. (ed.) (1994) The Science of Cookie and Cracker Production. Chapman and Hall, New York. Gutterson, M., Noyes Development Corporation (1969) Baked Goods Production Processes. NDC, London, Food Processing Review 9. Kempf, N.W. (1964) The Technology of Chocolate. The Manufacturing Confectioner Publishing Co., Glen Rock, NJ. Kulp, K. (ed.) (1994) Cookie Chemistry and Technology. American Institute of Baking, Kansas. Lees, R. (1980) A Basic Course in Confectionery. Specialized Publications Ltd., Surbiton, UK. Manley, D. (1998) Biscuit, Cookie and Cracker Manufacturing Manual. (Vol 1: Ingredients, Vol 2: Biscuit doughs, Vol 3: Biscuit dough piece forming, Vol 4: Baking and cooling of biscuits, Vol 5: Secondary processing in biscuit manufacturing, Vol 6: Biscuit packaging and storage.) Woodhead Publishing, Cambridge, UK. Meiners, A. and Joike, H. (1969) Handbook for the Sugar Confectionery Industry. SilesiaEssenzenfabrik, Gerhard Hanke K.G. Norf., West Germany. Meursing, E.H. (1983) Cocoa Powders for Industrial Processing, 3rd edn. Cacaofabriek de Zaan, Koog aan de Zaan, The Netherlands. Pratt, C.D., de Vadetzsky, E., Landwill, K.E. et al. (1970) Twenty Years of Confectionery and Chocolate Progress. AVI, Westport, CT. Schwartz, M. E. (1974) Confections and Candy Technology. Noyes, Park Ridge, N.J., Food Technology Review 12. Smith, W.H. (1972) Biscuit, Crackers and Cookies. Applied Science, London. Sullivan, E.T., Sullivan, M.C. (1983) The Complete Wilton Book of Candy. Wilton Enterprises, Woodridge, IL. Wade, P. (1988) Biscuit, Cookies and Crackers. Elsevier Applied Science, London. Whiteley, P.R. (1971) Biscuit Manufacture. Applied Science, London. Wieland, H. (1972) Cocoa and Chocolate Processing. Noyes Data Corp., Park Ridge, NJ.

References

Aasted, K. (1941) Studien über den Conchierprozess. Dissertation, Berlin. Ablett, S., Lillford, P.J., Baghdadi, S.M.A. and Derbyshire, W. (1978) J Colloid Interface Sci 67: 355. Acker, L. and Diemair, W. (1956) Zeitschrift für die Süsswarenwirtschaft 9: 1187. Aguilar, C.A., Dimick, P.S., Hollender, R. and Ziegler, G.R. (1995) Flavor modification of milk chocolate by conching in a twin-screw, co-rotating, continuous mixer. J Sensory Stud 10 (4): 369–380. Aguilera, J.M. (2005) Why food microstructure? J Food Eng 67: 3–11. Aguilera, J.M. and Stanley, D.W. (1999) Microstrutural Principles of Food Processing and Engineering, 2nd edn. Aspen Publication, Maryland. Akhindinov, J.N. and Polakova, N.D. (1962) Untersuchung der Viskosität von Futtermelassenaus kubanischen Zuckerfabriken. Zuckerindustrie 2: 27. Akhumov, E.I. (1960) Issledovanie Peresyshkhennyh Vodnuh Rastvorov Solei, no 42. Tr. Vses. nauchno-isled. Inst. Metalurgii, Leningrad. Alamprese, C., Datei, L. and Semeraro, Q. (2007) Optimization of processing parameters of a ball mill refiner for chocolate. J Food Eng 83 (4): 629–636. Albert, R., Strätz, A. and Vollheim, G. (1980) Die katalitische Herstellung von Zuckeralkoholon und deren Verwendung. Chemie-Ing Techn 52: 582. Alexits, G. and Feny , I. (1955) Matematika Vegyészeknek (Mathematics for Chemists, in Hungarian). Tankönyvkiadó, Budapest, p. 326–329. Allen, T. (1981) Particle Size Measurement. Chapman and Hall, London. Andersen, P.E. and Risum, J. (1982) Levnedsmiddeltechnologien, Bind 1. Polytechnish Forlag, Lyngby. Andreasen, A.H.M. (1935) Angewandte Chemie 20: 283. Anonymous (1971) Grinding – equipment review. Chem Process Eng 52 (1): 39–42. Anonymous (1981) Refining of cocoa mass. CCB Rev Choc Confect Bakery 6 (1): 12. Anonymous (1985) Conching and the PDAT – Carle & Montanari system. Confect Prod 51 (11): 626–627. Anonymous (1995) Optimizing the milling process by a combination of milling techniques. Zucker Süsswarenwirtschaft 48 (10): 420–421. Antokolskaia, H.J. (1964) Nachschlagewerk über Rohstoffe, Halbfabrikate und Fertigwaren der Konditorwarenherstellung. Verlag, Lebensmittelindustrie, Moscow. Arbuzov, N. and Grebenshchikov, B.N. (1937) Zh fiz khim 10: 32. Arishima, T. and McBrayer, T. (2002) Application of specialty fats and oils. Manuf Confect 82: 65–76. Armbruster, H., Karbstein, H. and Schubert, H. (1991) Herstellung von Emulsionen unter Berücksichtigung der Grenzflächenbesetzungskinetik des Emulgators. Chemie-Ing Techn 63: 266. Arnott, S., Scott, W.E., Rees, D.A. and McNab, C.G.A. (1974) ι-Carrageenan: molecular structure and packing of polysaccharide double helices in oriented fibres of divalent cation salts. J Mol Biol 90: 253–267. Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Ferenc Á. Mohos

References

631

Asselbergs, C.J. and De Jong, E.J. (1972) Secondary Nucleation, Discussion of the Working Party on Crystallization. European Federation of Chemical Engineering, Boekelo. Atiemo-Obeng, V.A., Penney, W.R. and Armenante, P. (2003) Solid–liquid mixing. In: Paul, E., Atiemo-Obeng, V.A. and Kresta, S.M. (eds) Handbook of Industrial Mixing: Science and Practice. Wiley-Interscience, a John Wiley & Sons Publication, New York, NY, p. 543–584. Atlas Chemical Industries (1963) The Atlas HLB System, 2nd edn. Atlas Chemical Industries, Wilmington, Delaware. Austin, L.G. (1971/1972) A review: Introduction to the mathematical description of grinding as a rate process. Powder Technol 5: 1–17. Austin, L.G. and Bathia, V.K. (1971/1972) Experimental methods for grinding studies in laboratory mills. Powder Technol 5: 261–266. Avrami, M. (1939) Kinetics of phase change. I. General theory. J Chem Phys 7: 1103–1112. Avrami, M. (1940) Kinetics of phase change. II. Transformation – time relations for random distribution of nuclei. J Chem Phys 8: 212–224. Avrami, M. (1941) Kinetics of phase change. III. Granulation, phase change and microstructure. J Chem Phys 9: 177–184. Babb, A.T.S. (1965) A recording instrument for rapid evaluation of compressibility of bakery goods. J Sci Fd Agr 16 (11): 670–679. Bagley, E.B. (1957) End corrections in the capillary flow of polyethylene. J Appl Phys 28: 624–627. Baird, D.G. (1983) Food dough rheology. In: Peleg, M. and Bagley, E.B. (eds) Physical Properties of Foods. AVI Publishing Co. Inc., Westport, CT, p. 343–350. Bakele, W. (1992). Grundlagen, Methoden und Technik der Trockengranulierung. Chemie-Ing Techn 64: 273. Baker, C.M.A. (1968) The proteins of egg white. In: Carter, T.C. (ed) Egg Quality: A Study of the Hen’s Egg. Oliver and Boyd, Edinburgh, p. 67. Baker, W.E., Westine, P.S. and Dodge, F.T. (1991) Similarity Methods in Engineering Dynamics: Theory and Practice of Scale Modelling. Fundamental Studies in Engineering, vol 12. Elsevier, Amsterdam. Baldi, G., Conti, R. and Alaria, E. (1978). Complete suspension of particles in mechanically agitated vessels. Chem Eng Sci 33: 21. Baldwin, R. E. (1986) Functional properties of eggs in food. In: Stadelman, W.J. and Cotterill, O.J. (eds) Egg Science and Teclrnologv, 3rd edn. AVI Publishing Co. Inc., Westport, CT, p. 345. Baliga, J.B. (1970) Crystal Nucleation and Growth Kinetics in Batch Evaporative Crystallization. PhD Thesis. Iowa State University, Ames. Bálint, Á. (2001) Prediction of physical properties of foods for unit operatuions. Period Polytechnica Ser Chem Eng 45 (1): 35–40. Ball, R.C. (1989) Fractal colloidal aggregates: Consolidation and elasticity. Physica D 38: 13–15. Banks, R.B. (1994) Growth and Diffusion Phenomena. Springer Verlag, Berlin. Barenblatt, G.L. (1987) Dimensional Analysis. Gordon and Breach, New York, NY. Barenbrug, A.W.T. (1974) Psychometry and Psychometric Charts, 3rd edn. Cape and Transvaal Printers Ltd, Cape Town. Barnes, H.A., Hutton, J.F. and Walters, K. (1989) An Introduction to Rheology. Elsevier Science Publishgin Co. Inc, New York, NY. Bartusch, W. (1974) Gordian 11: 346. Bartusch, W. and Mohr, W. (1966) Über die Vorgänge beim Conchieren von Schokolademassen I: Pysikalische und chemische Veränderungen bei dunklen Massen. Fette Seifen Anstrichmittel 68: 635. Batel, W. (1956) Forsch.-Ber. Wirtschafts- u.Verkehrmin (Research Reports of the Ministry of Economics and Communication), Nr. 262. Nordrh.-Westf., Dusseldorf. Bäucker, A. (1974) Thin layer processing of intermediate products of chocolate manufacture. Int Rev Sugar Confect 27 (7): 290–292.

632

Confectionery and Chocolate Engineering: Principles and Applications

Bauckhage, K. (1973) Viskosität von Emulsionen, I–IV. Chemie-Ing Techn 45: 1001, 1087, 1181 and 1273. Bauckhage, K. (1990) Das Zerstäuben als Grundverfahren. Chemie-Ing Techn 62: 613. Bauermeister, J. (1978) Grinding of cocoa mass. Industrie Alimentari 17 (5): 424–428. Bázár, G. (2008) NIR classification of frying fat samples by means of Polar Qualification System. 14th International Diffuse Reflectance Conference, 3–8 August, Chambersburg, PA. Becker, U., Langer, G. and Werner, U. (1981) Einfluss physikochemischer Stoffeigenschaften auf das Fliessverhalten konzentrierter Suspensionen. Chemie-Ing Techn 53: 568. Beckett, S.T. (2000) The Science of Chocolate. Royal Society of Chemistry, Cambridge. Beckett, S.T. (2003) Is the taste of British milk chocolate different? Int J Dairy Technol 56 (3): 139–142. Beke, B. (1981) The Process of Fine Grinding. Martinus Nijhoff/Dr W. Junk Publ., The Hague. Bekin, N.G. and Nemytkov, V.A. (1966) Kauchiek i Resina 10: 31. Belitz, H.-D. and Grosch, W. (1987) Food Chemistry. Springer Verlag, Berlin, p. 213–219. Bellairs, R. (1961) The structure of the yolk of the hen’s egg as studied by electron microscopy. I. The yolk of the unincubated egg. J Biophys Biochem Cytol 11: 207. Bellairs, R., Backhouse, M. and Evans, R.J. (1972) A correlated chemical and morphological study of egg yolk and its constituents. Micron 3: 328. Benbow, J.J. and Bridgwater, J. (1993) Paste Flow and Extrusion. Clarendon Press, Oxford. Benedek, P. (1960) Stacionárius mu˝veleti egységek szabadsági foka (Degree of Freedom of Stationary Operational Units, in Hungarian). Doctorial Thesis. M. Ásványolaj és Földgáz Kísérleti Intézet, Publ. Nr. 218. Benedek, P. and László, A. (1964) A vegyészmérnöki tudomány alapjai (The Principles of Chemical Engineering). Mu˝szaki Könyvkiadó, Budapest, Chapters 5–7. (Grundlagen des Chemieingenieurwesens, Deutsche Verl. für Grundstoffindustrie, Leipzig, 1965, in German.) Bengsston, N. (1967) Ultrafast freezing of cooked egg white. Food Technol 21: 1259. Berg, T.G.O. and Brimberg, U.I. (1983) Fette Seifen Anstrichmittel 85: 142. Bergenstahl, B.A. and Claesson, P.M. (1990) Surface forces in emulsions. In: Larsson, K. and Friberg, S.E. (eds) Food Emulsions, 2nd edn. Marcel Dekker, New York, NY. Bergmeyer, H.U. (1962) Methoden der Enzymatischen Analyse. Verlag Chemie, Weinheim. Berlin, E., Anderson, B.A. and Pallansch, M.J. (1968) Comparison of water vapor absortion by milk powder components. J Dairy Sci 51: 1912–1915. Bernhard, E. (1962) Pererabotka Termoplaszticheskikh Materialov. M. Goshkhi-nisdat, USSR. Bernhardt, C., Reinsch, E. and Husemann, K. (1999) The influence of suspension properties on ultra-fine grinding in stirred ball mills. Powder Technol 105: 357–361. Bertini, A. (1989) Cocoa liquor roasting. Confect Manufacture Marketing 26 (7,8): 39, 41, 42. Bertini, A. (1996) Walzen auf Walzwerken ist immer noch am besten. Carle & Montanari Nachrichten (Technical Bulletins of Carle & Montanari SpA) 16, Mai. Bertrand, J. (1985) Design of impellers for agitation of highly viscous non-Newtonian fluids. Chemie-Ing Techn 57: 334. Besselich, N. (1951) Die Bonbon-Fabrikation. Verlag der Konditor-Zeitung Trier, 6. neubearb./ erweit. Auflage von Riedel, H.R., Klepsch, G., p. 57. Besskow, S.D. (1953) Technisch-chemische Berechnungen. VEB Verlag Technik, Berlin, p. 238. Beveridge, T. and Ko, S. (1984) Firmness of heat-induced whole egg coagulum. Poultry Sci 63: 1372. Beveridge, T., Arntfield, S., Ko, S. and Chung, J.K.L. (1980) Firmness of heat induced albumen coagulum. Poultry Sci 59: 1229. BFMIRA (1970) No conching. New chocolate processing plant. BFMIRA demonstrate pilot system. Confect Product 36 (3): 143–145. Beyer von Morgenstern, J. and Mersmann, A. (1982) Begasen hochviskoser Flüssigkeiten. ChemieIng Techn 54: 684.

References

633

Billon, M. (1984) Conching chocolate in a cooker-extruder. Revue des Industries de la Biscuiterie Biscotterie Boulangerie Chocolaterie Confiserie 79: 12–14. Birfeld, A.A. (1970) Candidate Thesis. Technical University of Food Industries, Moscow. Birfeld, A.A. and Machikhin, Y.A. (1970) Viscosity of praline masses (in Russian). Khlebopekarnaya i Konditerskaya Promyshlennost 6: 12–13. Birkhoff, G. (1948) Lattice Theory, vol 25. American Mathematical Society Colloquim Publications, New York, NY. Birth, G.S. (1976) Quality Detection in Foods. American Society of Agricultural Engineers, Michigan. Biscuits et gateaux, Répertoire des dénominations et recueil des usages (2001). Syndicat National de la Biscuiterie Francaise, France. Blagoveshchenskaya, M.M. (1975) Izvestiya Vuzov. Candidate Thesis. Technical University of Food Industries, Moscow. Blakemore, I. (2006) Playing forward to spin. Food Process 75 (11): 12. Blickle, T. (1978) Mu˝szaki Kémiai Rendszerek Modellezése a Szerkezetelmélet Felhasználásával (Modelling by Application of Structure Theory in Chemical Engineering, in Hungarian). Veszprém University of Chemical Industries (today: University of Pannonia), Veszprém. Blickle, T. and Halász, S. (1973) A study of the relation between crystal growth kinetics and particle size distribution of crystals produced from solution. Krist Tech 8 (6): 679–687. Blickle, T. and Halász, S. (1977) The balance equation set of crystallization. In: Blickle, T. (ed) Mathematical Models of Mass- and Heat Transfer Systems (in Hungarian). Mu˝szaki Könyvkiadó, Budapest, p. 123–154. Blickle, T. and Seitz, K. (1975) A Modern Algebrai Módszerek Felhasználása a Mu˝szaki Kémiában (Application of Modern Algebraic Methods in Chemical Engineering, in Hungarian). Mu˝szaki Könyvkiadó, Budapest. Bloksma, A.H. (1957) A calculation of the shape of the alveographs of some rheological model substances. Cereal Chem 34: 126–136. Bloom, O.T. (1925) US Patent 1.540.979, June 9th. Bodenstedt, E. (1952) Dissertation. University of Bonn, Bonn. Bodor, G. (1991) Structural Investigation of Polymers. Akadémiai Kiadó, Budapest/Ellis Horwood, Chichester, p. 214. Boger, D.V. and Walters, K. (1993) Rheological Phenomena in Focus. Elsevier Science Publishers, New York, NY. Bohnet, M. (1983) Fortschritte bei der Auslegung pneumatischen Förderanlagen, Vortrag auf dem Jahrestreffen der Verfahrens-Ingenieure, 29 Sept bis 1 Okt. 1982, Basel. Chemie-Ing Techn 55 (7): 524–539. Bollinger, S., Zeng and Windhab, E.J. (1999) In-line measurement of cocoa butter and chocolate by emans of infrared spectroscopy. J Am Oil Chem Soc 76: 659. Bolshakov, A.S., Borekov, V.G., Kasulin, G.N., Rogov, F.A., Skryabin, U.P. and Zhukov, N.N. (1976) Paper 35. 22nd European Meeting of Meat Research Workers. Bonar, A, Rohan, T.A. and Stewart, T. (1968) Revue de la Confiserie 1: 8. Borbély, S. (1961) Ordinary differential equations. 79. Stability criteria (in Hungarian). In: Sályi, I. (ed) ‘Pattantyús’ Gépész-és villamosmérnökök (Manual for Mechanical and Electrical Engineers), vol 2. Müszaki Könyvkiadó, Budapest, pp. 309–311. Borho, K., Polke, R., Wintermantel, K., Schubert, H. and Sommer, K. (1991) Produkteigenschaften und Verfahrenstechnik. Chemie-Ing Techn 63: 792. Botsaris, G.D. and Denk, E.G. (1970) Ann Rev Ind Eng Chem I: 337. Bourne, M.C. (1968) Texture profile of ripening pears. J Food Sci 33: 223–226. Bourne, M.C. (1975) Texture measurements in vegetable. In: Rha, Ch.D. (ed) Theory, Determination and Control of Physical Properties of Food Materials. Reisel Publishing Co., Dordrecht, p. 133–162.

634

Confectionery and Chocolate Engineering: Principles and Applications

Bourne, M.C. (1979) Theory and application of the puncture test in food texture measurement. In: Sherman, P. (ed) Food Texture and Rheology. Academic Press, New York, NY, p. 95–142. Bouwman-Timmermans, M. and Siebenga, A. (1995) Chocolate crumb – dairy ingredients for milk chocolate. Manufact Confect 74–79. Brandt, M.A., Skinner, E.Z. and Coleman, J.A. (1962) Texture profile method. J Food Sci 28 (4): 404–409. Brauer, H. and Mewes, D. (1973) Einfluss von Strombechern auf die Rührleistung. Chemie-Ing Techn 45: 461. Brauer, H., Dylag, M. and Talaga, J. (1989) Zur Fluiddynamik zur gerührten Gas/Feststoff/ Flüssigkeits-Systemen. Chemie-Ing Techn 61: 978. Breitschuh, B. and Windhab, E. (1998) Parameters influencing the use of vegetable fats in chocolate. J Am Chem Soc 75: 897–904. Bremer, L.G.B. and van Vliet, T. (1991) Rheol Acta 30: 98. Bremer, L.G.B., van Vliet, T. and van den Tempel, M. (1989) J Chem Soc Faraday Trans 85: 3359. Bremer, L., van Vliet, T. and Walstra, P. (1989) Theoretical and experimental study of the fractal nature of the structure of casein gels. J Chem Soc Faraday Trans 85: 3359–3372. Bremer, L.G.B., Bijsterbosch, B.H., Schrijvers, R.,van Vliet, T. and Walstra, P. (1990) Colloids Surf 51: 159. Brouwer, S., Groen, J.C., Pérez-Ramírez, J. and Peffer, L.A.A. (2002) Incorporation of the appropriate contact angle in mercury intrusion porosimetry. In: IX. Workshop über die Charakterisierung von feinteiligen und porösen Festkörpern, 14–15 November, Abstract Serie 1. Brown, W.D. and Ball, R.C. (1985) Computer simulation of chemically limited aggregation. J Phys A Math Gen 1985: L 18: 517–521. Brown, R.L. and Richards, J.C. (1970) Principles of Powder Mechanics. Pergamon Press, Oxford. Bruckner, Gy. (1961) Szerves kémia I-2.köt. Aminosavak, peptidek, fehérjék, szénhidrátok (Kucsman, Á. és Kajtár, M. közremu˝ködésével) (Organic Chemistry, vol 1 and 2. Aminoacids, Peptides, Proteins, Carbohydrates, Co-workers: Kucsman, Á, Kajtár, M., in Hungarian). Tankönyvkiadó, Budapest, p. 944–947, 998–1000. Bruin, S. (1979) Preconcentration and drying of food materials. Postgraduate Course, Lund, Sweden. Bruin, S. and Luyben, K.Ch.A.M. (1980) Drying of food materials: a review on recent developments. In: Mujumdar, A.S. (ed) Advances in Drying, vol 1. Hemisphere Publishing Corporation, Washington, DC. Bryce, T.A., Mc.Kinnon, A., Morris, E.R., Rees, D.A. and Thom, D. (1974) Chain conformations in the sol–gel transitions, and their characterisation by spectroscopic methods. J Chem Faraday Disc 57: 221–229. Bryce, T.A., Clark, A.H., Rees, D.A. and Reid, D.S. (1982) Concentration dependence of the orderdisorder transition of carrageeenans. Further confirmatory evidence for the double helix in solution. Eur J Biochem 122: 63–69. Buck, D.F. and Edwards, M.K. (1997) Anti-oxidants to prolong shelf-life. Food Tech Int 2: 29–33. Bühler, G. (1982) Fryma-CoBall-Mill, eine neuartige Nasskugelmühle mit hoher Energiedichte. Chemie-Ing Techn 54: 371. Bühler, V. (1993) Kollidon, Polyvinylpyrrolidone for the Pharmaceutical Industry. BASF AG, Feinchemie Ludwigshaven. Bukharov, P.S. (1935) Tablici povysheniya temperatur kipeniya (Tables of Rise of Boiling Point, in Russian). NKPP SSSR, Glavkonditer Zhurn.Tekhnika i tekhnologiya konditerskovo proizvodstva. Bunge, F. and Schwedes, J. (1992) Energiebedarf beim Zellaufschluss in Rührwerkskugelmühlen. Chemie-Ing Techn 64: 844. Burak, N. (1966) Chemicals for improving the flow properties of powders. Chem Ind 21: 844–850.

References

635

Bürkholz, A. (1973) Messmethoden zur Tropfengrössebestimmung. Chemie-Ing Techn 45: 1. Burt, D.J. and Thacker, D. (1981) Use of emulsifiers in short dough biscuits. Food Trade Rev 47: 344. Buscall, R., Goodwin, J.W., Ottewill, L.H. and Tadros, Th.F. (1982) The settling of particles through Newtonian and non-Newtonian media. J Colloid Intertface Sci 82: 78–86. Buscall, R., Mills, D.A., Goodwin, J.W. and Lawson, D.W. (1988) Scaling behaviour of the rheology of aggregate network formed from colloidal particles. J Chem Soc Faraday Trans 84: 4249–4260. Buschuk, W. (1985) Rheology: Theory and application to wheat flour doughs. In: Faridi, H. (ed) Rheology of Wheat Products. American Association of Cereal Chemists, St. Paul, MN, p. 1–26. von Buzágh, A. (1937) Colloid Systems. Technical Press, London. Cakebread, S.H. (1969a) Amount of sucrose left in solution. Calculation of ERH; Compositions of creamy toffee batches. Confect Prod 35 (9): 573–578. Cakebread, S.H. (1969b) Calculating the ERH – The Money and Born equation. Confect Prod 35 (10): 651–655. Cakebread, S.H. (1970a) Confectionery ingredients – Composition and properties, Part VII. Confect Prod 36 (7): 436–437. Cakebread, S.H. (1970b) Confectionery ingredients – Composition and properties, Part III – Sugars. Confect Prod 36 (8): 481–484. Cakebread, S.H. (1971a) Confectionery ingredients – Composition and properties, Part VII – The individual carbohydrates: Sucrose. Confect Prod 37 (2): 81–89. Cakebread, S.H. (1971b) Confectionery ingredients – Composition and properties, Part VII – The individual carbohydrates: Dextrose. Confect Prod 37 (3): 140–147. Cakebread, S.H. (1971c) Confectionery ingredients – Composition and properties, Part VII – The individual carbohydrates: Laevulose and invert sugar. Confect Prod 37 (4): 214–218. Cakebread, S.H. (1971d) Physical properties of confectionery ingredients – Supersaturation. Confect Prod 37 (7): 407–412. Cakebread, S.H. (1971e) Physical properties of confectionery ingredients – Supersaturation. Confect Prod 37 (8): 461–464, 470. Cakebread, S.H. (1972a) Physical properties of confectionery ingredients. Confect Prod 38 (1): 16–18. Cakebread, S.H. (1972b) Grained confectionery – Kinds of grain: crystallization: solubility. Confect Prod 38 (2): 78–83. Cakebread, S.H. (1972c) Grained confectionery – Viscosity: shelf-life: effects of manipulation. Confect Prod 38 (3): 132–149. Campbell, B. and Pavlasek, S. (1987) Dairy products as ingredients in chocolate and confections. Food Technol 74: 78–85. Canning, T.F. (1971) Chem Eng Prog Symp Ser 67 (110): 74. Capriste, G.H. and Lozano, J.E. (1988) Effect of confentration and pressure of the boiling point rise of apple juice and related sugar solutions. J Food Sci 53 (3): 865. Carman, P.C. (1956) Flow of Gases Through Porous Media. Academic Press Inc., London. Carr, R.L. (1976) Powder and granule properties and mechanics. In: Marchello, J.M and Gomezplata, A. (eds) Gas–Solids Handling in the Processing Industries. Marcel Dekker, Inc., New York, NY, p. 13–88. Casimir, H.B.G. and Polder, D. (1948) The influence of retardation on the London-van der Waals forces. Physic Rev 73: 360. Casson, N. (1959) A flow equation for pigment-oil suspensions of the printing ink. In: Mill, C.C. (ed) Rheology of Disperse Systems. Pergamon Press, London, p. 84–104. Castaldo, D., Palmieri, L., Lo Voi, A. and Costabile, R. (1990) Flow properties of Babaco (Carica Pentagona) purees and concentrates. J Texture Stud 21: 253–264.

636

Confectionery and Chocolate Engineering: Principles and Applications

Castell-Perez, M.E. (1992) Viscoelastic properties of dough. In: Rao, M.A. and Steffe, J.F. (eds) Viscoelastic Properties of Foods. Elsevier Science Publisher, London, p. 77–102. Chapman, G. (1971) Cocoa butter and confectionery fats. Studies using programmed X-ray diffraction and differential scanning calirometry. J Am Oil Chem Soc 48: 824–830. Chapman–Holland (1965) A study of turbine and helical screw agitators in liquid mixing. Trans Instn Chem Eng 43: T131–T140. Charm, S.E. (1971) The Fundamentals of Food Engineering. AVI Publishing Co, Inc., Westport, CT. Charm, S. and Kurland, G. (1965) Viscometry of human blood for shear rates of 0–100,000 s−1. Nature 206: 617–618. Chen, C.S. (1985) Thermodynamical analysis of the freezing and thawing of foods: enthalpy and apparent specific heat. J Food Sci 50 (4): 1158–1162. Chen, G., Li, C., Kuo, Y.L. and Yen, Y.W. (2007) A DSC study on the kinetics of disproportionation reaction of (hfac)Cul(COD). Thermochimica Acta 456: 89–93. Cheng, D.C-H. (1986) Yield stress: a time-dependent property and how to measure it. Rheol Acta 25: 542–554. Cheremisinoff, N.P. (1988) Principles of polymer mixing and extrusion. In: Cheremisinoff, N.P. (ed) Encyclopedia of Fluid Mechanics, vol 7 Rheology and Non-Newtionian Flows. Gulf Publishing Co., Houston, TX, p. 761–866. Choi, Y. and Okos, M.R. (1986) Effects of temperature and composition on the thermal properties of foods. Food Eng Process Appl 1: p: 93–101. Choishner (Tscheuschner), Kh.D., Brindrikh, U., Puchkova, L.I. and Tarasova, L.P. (1983) Investigations of texture of sweet biscuits (in Russian). Khlobepekarnaya i Konditerskaya Promyshlennost 7: 27–28. Chopey, N.P. (ed) (1994) Handbook of Chemical Engineering Calculations. McGraw-Hill, New York, NY. Chopin, M. (1927) Determination of baking value of wheat by means of specific energy of deformation of dough. Cereal Chem 4: 1–13. Chopin, M. (1962) Sur l’utilization du rapport P/L dans l’essai des farines avec l’alvéographe. Bull de l’École Francaise de Meunerie 189: 139–141. Coccolo, E. (1973) New methods of chocolate manufacture. Industrie Alimentari 12 (7, 8): 75–84. Cogswell, F.N. (1972) Converging flow of polymer melts in extrusion dies. Polymer Eng Sci 12 (1): 64–73. Cogswell, F.N. (1978) Converging flow and streching flow: a compilation. J Non-Newtonian Fluid Mech 4: 23–38. Cogswell, F.N. (1981) Polymer Melt Rheology. Halsted Press, New York, NY. Combes, D. and Monsan, P. (1982) Hydrolyse continue de solutions concentrées de saccharose par l’invertase immobilisée. In: Dupuy, P. (ed) Utilisation des Enzymes en Technologie Alimentaire. Symposium International, Versailles, p. 177–188. Contini, S. and Atasoy, K. (1966) Pharm Ind 28: 144. Cook, W.H. (1968) Macromolecular components of egg yo1k. In: Carter, T.C. (ed) Egg Quality: A Study of the Hen’s Egg. Oliver and Boyd, Edinburgh, p. 109. Cook, F. and Briggs, G. M. (1986) The nutritive value of eggs. In: Stadelman, W.J. and O. J. Cotterill, O.J. (eds) Egg Science und Technology, 3rd edn. AVI Publishing Co. Inc., Westport, CT, p. 141. Cotterill, O. J. and Geiger, G. S. (1977) Egg product yield trends from shell eggs. Poultry Sci 56: 1027. Cotterill, O.J., Amick, G.M., Kluge, B.A. and Rinard, V.C. (1963) Some factors affecting the performance of egg white in divinity of candy. Poultry Sci 42: 218. Coulson, J.M. and Maitra, N.K. (1950) Mixing of solid particles. Ind Chem 26 (301): 55–60. Csajághy, K. (2001) Cukoripari technológia (Technology of Candy Industry, in Hungarian), vol 2. Agrárszakoktatási Intézet, Budapest, p. 271.

References

637

Cuevas, R. and Cheryan, M. (1978) Thermal conductivity of liquid foods: a review. J Food Process Eng 2: 223. Dagorn-Scaviner, C., Guegen, J. and Lefebvre, J. (1987) Emulsifying properties of pea globulins as related to their adsorption. J Food Sci 52: 335–341. Dally, J.W. and Riley, W.F. (1965) Experimental Stress Analysis. McGraw-Hill Book Co., New York, NY. Danckwerts, P.V. (1953) Continuous flow system, distribution of residence times. Chem Eng Sci (Lond) 3: 1. Danilova, G.N. (ed) (1961) Sammlung von aufgaben und Berechnungen zur Wärmeübertragung. Verlag Gostorgizdat, Moscow. D’Appolonia, B.L. and Kunerth, W.H. (eds) (1984) The Farinograph Handbook. American Association of Cereal Chemists, St. Paul, MN. Davidson, J.F. and Harrison, D. (1971) Fluidization. Academic Press, New York, NY. Davies, J.T. (1959) Proceedings of the 2nd International Congress, Surface Activity, vol 1. Butterworths, London, p. 426. Davies, J.T. (1964) Rec Progr Surf Sci 2: 129. Davies, J.T. (1986) Particle suspension and mass transfer rates in agitated vessels. Chem Eng Process 20 (4): 175–181. Dealy, J.M. (1994) Official nomenclature for material functions describing the response to a viscoelastic fluid to various shearing and extensional deformations. J Rheol 38: 179– 191. Dean, J. (1995) The Analytical Chemistry Handbook. McGraw Hill, Inc., New York, NY: 15.1–15.5. Debaste, F., Kegelaers, Y., Liégeots, S., Ben Amor, H. and Halloin, V. (2008) Contribution to the modelling of chocolate tempering process. J Food Eng 88: 568. Denbigh, K.G. (1946) A method of estimating the Prandtl number of liquids. J Soc Chem Ind 64: 61. Derneko, A.P. and Schafchid, M.M. (1959) Bestimmung der Temperaturko-effizienten einiger Lebensmittel. Sammlung von Studentenarbeiten LTJCPa, p. 9. Desent, G.M. and Bouscher, T.A. (1961) Ausrüstung und Taktstrassen für die Speiseherstellung. Verlag Gostorizdat, Moscow. Dickie, A.M. and Kokini, J.L. (1983) An improved model for food thickness from non-Newtonian fluid mechanics in the mouth. J Food Sci 48: 57–61, 65. Dickinson, E. (1981) Interpretation of emulsion phase inversion as a cusp catastrophe. J Colloid Interface Sci 84: 284. Dickinson, E. (1992) An Introduction to Food Colloids. Oxford Science Publications, Oxford University Press, Oxford. Diehl, K.C. and Gardner, F.A. (1984) Correlation of the rheological behaviour of egg albumen to temperature, pH and NaCl concentration. J Food Sci 49: 137. Dimick, S.P. and Davis, R.T. (1986) Solidification of cocoa butter. Manuf Confect 66 (6): 123–128. Djakovic’, L, Dokic, P., Radivojevic’, P. and Kler, V. (1976) Investigation of the dependance of rheological characteristics on the parameters of particle size distribution at O/W emulsions. Colloid Polymer Sci 254: 907. Djuric, D., Van Melkebeke, Kleinebudde, B.P., Remon, J.P.C and Vervaet, C. (2009) Comparison of two twin-screw extruders for continuous granulation. Eur J Pharm Biopharm 71 (1): 155. Dobbs, A.J., Peleg, M., Mudgett, R.E. and Rufner, R. (1982) Some physical characteristics of active dry yeast. Powder Technol 32: 75–81. Dobre, T.G. (2007) Chemical Engineering: Modelling, Simulation and Similitude. Wiley-VCH, Weinheim. Dolan, K.D. and Steffe, J.F. (1990) Modeling the rheological behaviour of gelatinizing starch solutions using mixer viscometric data. J Texture Stud 21: 265–294.

638

Confectionery and Chocolate Engineering: Principles and Applications

Dolan, K.D., Steffe, J.F. and Morgan, R.G. (1989) Back extrusion abd simulation of viscosity development during starch gelatinization. J Food Proc Eng 11: 79–101. Doriaswarmy, L.K. and Sharma, M.M. (1984) Heterogeneous Reactions: Analysis, Examples and Reactor Design, vol 2. Fluid–Fluid–Solid Reactions. Wiley, New York, NY, p. 233–316. Dörnyei, J. (1981) Pillanatoldódó élelmiszerek gyártása (Manufacture of Instant Foods, in Hungarian). Mez gazdasági Kiadó, Budapest, p. 68–75. Dötsch, W. and Sommer, K. (1985) Agglomerationskinetik der Tellergranulation. Chemie-Ing Techn 57: 871. von Drachenfels, H., Kleinert, J. and Hanssen, E. (1962) A new method of preventing fat bloom. Rev Int Choc XVII: 409–410. Duck, W. (1964) Manuf Confect 44: 67. Duck, W. (1965) Candy Ind Confect J 11: 124. Dukhin, S.S., Rulev, N.N. and Dimitrov, D.S. (1986) Koagulyatiya i dinamika tonkikh plenok (in Russian). Naukova Dumka, Kiev. Dunning, H.N. and Dannert, R.D. (1970) Chocolate conching. United States Patent 3 544 328. Dziezak, J.D. (1989) Single and twin screw extruders in food processing. Food Technol 43 (4): 164–174. Eberl, D.D., Kile, D.E. and Drits, V.A. (2002) On geological interpretations of crystal size distributions: Constant versus proportionate growth. Am Minerol 87: 1235–1241. Ebert, F. (1983) Zur Bewegung feiner Partikeln, die turbulent strömenden Medien suspendiert sind. Chem-Ing Techn 55: 931. Édesipari termékek (2003) (Confectionery Products, Waffles and Biscuits) Codex Alimentarius Hungaricus 2–84, directive, Hungary. Edmondson, P. (2005) Why is microstructure important in food systems? New Food 4: 36. Egelandsdal, B. (1980) Heat-induced gelling in solutions of ovalbumin. J Food Sci 45: 570. Egermann, H. (1980) Effects of mixing on mixing homogeneity, Part I, Ordered adhesion-random adhesion. Powder Technol 27: 183–188. Egry, L. (1973) Légáramos szállítóberendezések (air flow transport machines). In: Tomay, L. (ed) Gabonaipari kézikönyv, Technológiai gépek és berendezések (Manual of Grain Processors, Technological Machines and Equipments of the Grain Silos, in Hungarian). Mezo˝gazdasági Kiadó, Budapest, p. 937–1003. Einenkel, W.-D. (1979) Fluiddynamik des Suspendierens. Chemie-Ing Techn 51: 697. Ellenberger, J., Hamersma, P.J. and Fortuin, J.M.H. (1984) Drei-Parameter-Modell zur Beschreibung der Fliesskurven von viskoelastischen Flüssigkeiten. Chemie-Ing Techn 56: 783. Elliot, J.R. and Lira, C.T. (1999) Introductory Chemical Engineering Thermodynamics. Prentice Hall, Englewood Cliffs, New Jersey, p. 635. Ellis, G.P. (1959) The Maillard reaction. In: Wolform, M.L. (ed) Advances in Carbohydrate Chemistry, no 14. Academic Press, New York, NY, p. 63–134. El-Rafey, H.H.A., Perédi, J., Kaffka, K., Náday, B. and Balogh, A. (1988) Possibilities of application of NIR technique in the analysis of of oilseeds and their derivatives – examination of used frying oils. Fat Sci Technol 90 (5): 175. Erdey, L. (1958) Bevezetés a kémiai analízisbe II. rész: Térfogatos analízis, 6. kiadás (An Introduction to Chemical Analysis, Part II: Volumetric Analysis, 6. edn, in Hungarian). Takönyvkiadó, Budapest. Erdey-Grúz, T. and Schay, G. (1954) Elméleti fizikai kémia, vol 2 (Theoretical Physical Chemistry, vol 2, in Hungarian). Tankönyvkiadó, Budapest, p. 202–203. Erdey-Grúz, T. and Schay, G. (1962) Elméleti fizikai kémia, vol 1 (Theoretical Physical Chemistry, vol 1, in Hungarian). Tankönyvkiadó, Budapest. Eszterle, M. (1990) Viscosity and molecular structure of pure sucrose solutions. Zuckerindustrie 115 (4): 263–267. Eszterle, M. (1993) Molecular structure and specific volume of pure sucrose solutions. Zuckerindustrie 118 (6): 459–464.

References

639

Eucken, A. (1940) Allgemeine Gesetzmässigkeiten für das Wärmeleitvermögen verschiedener Stoffarten und Aggregatzustände, Forschung auf dem Gebiete des Ingeneurwesens, Ausgabe A, 11 (1), p. 6. European Union (2000) Council Directive 2000/36/EC: Cocoa and Chocolate Products. European Union. Exerowa, D. and Kruglyakov, P.M. (1998) Foam and Foam Films – Theory, Experiment, Application. Elsevier Science B.V., Amsterdam. Fábry, Gy. (1995) Élelmiszeripari eljárások és berendezések (Food processes and machinery, in Hungarian). Mez gazda, Budapest. Faridi, H. and Rasper, V.F. (1987) The Alveograph Handbook. American Association of Cereal Chemists, St. Paul, MN. Farinograph-E Worldwide Standard for Testing Flour Quality (1997) Wheat Flour – Physical Characteristics of Doughs – Part 1: Determination of Water Absorption and Rheological Properties using a Farinograph. ISO 5530-1. Feeney, R.E., Blankenhorn, G. and Dixon, H.B.F. (1975) Carbonyl-amine reactions in protein chemistry. In: Anfinsen, C.B., Edsall, J.T. and Richards, F.M. (eds) Advances in Protein Chemistry. Academic Press, New York, NY, p. 135–203. Fejes, G. (1970) Ipari kevero˝berendezések (Industrial Mixing Machines, in Hungarian). Mu˝szaki Könyvkiadó, Budapest. Fényes, I. (ed) (1971) Modern fizikai kisenciklopédia (Advaced Concise Encyclo-paedia of Physics, in Hungarian) Gondolat Könyvkiadó, Budapest, p. 691. Ferry, J.D. (1980) Viscoelastic Properties of Polymers, 3rd edn. John Wiley and Sons, New York. Filep, L. (1997) A tudományok királyno˝je – A matematika fejlo˝dése (The Queen of Sciences – The Development of Mathematics, in Hungarian). Typotex Kft, Budapest, Bessenyei Kiadó, Nyíregyháza, p. 170–171. Final Report of the IFT Committee (1959) Pectin standardization. Food Technol 13: 496. Fincke, H. (1936) Handbuch der Kakaoerzeugnisse, 1. Auflage. Springer Verlag, Berlin. Fincke, A. (1956a) Zucker und Süsswarenwirtschaft 9: 629, 677, 725. Fincke, A. (1956b) Zeitschrift für die Süsswarenwirtschaft 9: 561. Fincke, A. (1965) Handbuch der Kakaoerzeugnisse, 2nd edn. Springer Verlag, Berlin. Fincke, A. and Heinz, W. (1956) Untersuchungen zur Rheologie der Schokoladen. Fette Seifen 58: 905. Fine, F. (2007) Ohmic heating technology. Food Beverage Int Sept: 43. Fingrhut, H. (1991) Die Wärmeaustauschfläche am Rührbehälter. Chemie-Ing Techn 63: 142. Fleer, G.J., Koopal, L.K. and Lyklema, J. (1971) Polymer adsorption and its effect on the stability of hydrophobic colloids. I. Characterization of polyvinylalcohol adsoprtion on sivel iodide. Kolloid ZUZ Polymere 250: 689. Fleischli, M. and Streiff, F.A. (1990) Neue Erkenntnisse zum Mischen und Dispergieren von Flüssigkeiten mit grossen Viskositätsunterschieden in statischen Mischern. Chemie-Ing Techn 62: 650. Flink, M. (1983) Structure and structure transitions in dried carbohydrate materials. In: Peleg, M. and Bagley, E.B. (eds) Physical Properties of Foods. AVI Publishing Co., Inc., Westport, CT, p. 473–522. Flory, J.D. (1953) Viscoelastic Properties of Polymer Chemistry. Cornell University Press, Ithaca. Földes, J. and Ravasz, L. (1998) Cukrászat, 3rd edn (Confectioner’s Business, in Hungarian), Útmutató Kiadó, Budapest, p. 88. Fors, S. (1983) Sensory properties of volatile Maillard reaction products and related compounds: A literature review. In: Waller, G.R. and Feather, M.S. (eds) The Maillard Reaction in Foods and Nutrition. ACS Symposium Series 215, American Chemical Society, Washington, DC, p. 185–286.

640

Confectionery and Chocolate Engineering: Principles and Applications

Foster, R.M. (1924) A resistance theorem. Bell System Techl J April: 259–267. Foster, R.M. (1932) Geometrical circuits of electrical networks. Trans AIEE 51: 309–317. Foubert, I., Vanrolleghem, P.A., Vanhoutte, B. and Dewettink, K. (2002) Dynamic mathematical model of the crystallization kinetics of fats. Food Res Int 35: 945. Foubert, I., Dewettink, K. and Vanrolleghem, P.A. (2003) Modelling of the crystallization kinetics of fats. Trends Food Sci Technol 14: 79. Foubert, I., Dewettink, K., Janssen, G. and Vanrolleghem, P.A. (2006a) Modelling two-step isothermal fat crystallization. J Food Eng 75: 551. Foubert, I., Vereecken, J., Smith, K.W. and Dewettinck, K. (2006b) Relationship between crystallization behaviour, microstructure and macroscopic properties in trans containing and trans free coating fats and coatings. J Agricult Food Chem 54: 7256–7262. Franke, K. (1998) Modelling the conching kinetics of chocolate coatings with respect to final product quality. J Food Eng 36: 371. Franke, K. and Tscheuschner, H.D. (1991) Modelling of the high shear rate conching process for chocolate. J Food Eng 14: 103–115. Franke, K. and Heinzelmann, K. (2006) Optimierung der Eigenschaften von Sprühmilchpulver für die Anwendung in Schokolade. Süsswaren 51 (6): 19–22. Franke, K., Heinzelmann, K. and Tscheuschner, H.-D. (2001) Structure formation processes during chocolate manufacturing, Critical – the solid sugar. Zucker Süsswarenwirtschaft 54 (12): 18–22. Franke, K., Heinzelmann, K. and Tscheuschner, H.-D. (2002) Einfluss von Feuchte und Emulgator auf Struktur und Rheologie von Zucker-Kakaobutter-Dispersionen nach der Feinvermahlung. Chemie-Ing Technik 74 (11): 1633–1636. Frankel, N.A. and Acrivos, A. (1967) On the viscosity of a concentrated suspension of solid spheres. Chem Eng Sci 22: 847. Freiermuth, D. and Kirchner, K. (1981) Verteilung der Kugelenergie in Kugelmühlen in Abhängigkeit von den Mahlparametern. Chemie-Ing Techn 53: 384. Freiermuth, D. and Kirchner, K. (1983) Ein Beitrag zur experimentellen Ermittlung der Kugelumlaufdauer in Kugelmühlen. Chemie-Ing Techn 55: 60. Friberg, S.E. and El-Nokaly, M. (1983): Multilayer emulsions. In: Peleg, M. and Bagley, E.B. (eds) Physical Properties of Foods. AVI Publishing Co. Inc. Westport, CT, p. 145–156. Friberg, S.E., Goubran, R.F. and Kayali, I.H. (1990) Emulsion stability. In: Larsson, K. and Friberg, S.E. (eds) Food Emulsion, 2nd edn. Marcel Dekker, New York, NY, p. 1–40. Fricke, B.A. and Becker, B.R. (2001) Evaluation of thermophysical property models for foods. Int J Refraction 7 (4): 311–330 (Table 1), 312 (Table 2). Friedman, H.H., Whitney, J.E. and Szczesniak, A.S. (1963) The texturometer – A new instrument for objective texture measurement. J Food Sci 28: 390–396. Funk, K., Zabik, M.E. and Egidsily, D.A. (1969) Objective measurement for baked products. J Home Econ 61: 119–123. Furchner, B., Schwechten, D. and Hackl, H. (1990) Die kontinuierliche Fliessbett-SprühGranulation. Chemie-Ing Techn 62: 220. Gábor, Mné. (ed) (1987) Az élelmiszer-elo˝állítás kolloidikai alapjai (The Colloid Chemical Fundamentals of Food Production, in Hungarian). Mezo˝gazdasági Kiadó, Budapest. Gaddis, E.S. and Vogelpohl, A. (1991) Wärmeübergang in Rührbehältern. VDI-Wämeatlas 6. Auflage, Ma 1. Galor, B. and Walso, S. (1968) Chem Eng 23: 1431. Garside, J. and Mullin, J.W. (1968) Trans Inst Chem Engrs 46, T11. Garside, J., Gaska, J. and Mullin, J.W. (1972) Crystal growth rate studies with potassium sulfate in a fluidized bed crystallizer. J Crystal Growth 13/14: 510. Gasztonyi, K. (ed) (1979) Az élelmiszerkémia alapjai (The Fundamentals of Food Chemistry, in Hungarian). Mezo˝gazdasági Kiadó, Budapest, p. 152. Geisler, R., Mersmann, A. and Voit, H. (1988) Makro- und Mikromischen im Rührkessel. ChemieIng Techn 60: 947.

References

641

de Gennes, P.G. (1979) Scaling Concepts in Polymer Physics. Cornell University Press, Ithaca, NY. Gibson, A.G. (1988) Converging dies. In: Collyer, A.A. and Clegg, D.W. (eds) Rheological Measurements. Elsevier Applied Science, New York, NY, p. 49–92. Giddey, C. and Clerc, E. (1961) Polymorphism of cocoa butter and its importance in the chocolate industry. Rev Int Choc 16: 548–554. Gilliland, E.R. and Reed, C.E. (1942) Degrees of freedom in multicomponent absorption and rectification columns Ind Eng Chem 34: 551. Ginzburg, A.S. (1969) Application of Infra-Red Radiation in Food Processing. Lenard Hill Books, London. Glasner, A. (1973) The mechanism of crystallization: a revision of concepts. Materials Res Bull 8 (4): 413–421. Glücklich, J. and Shelef, L. (1962) An investigation into the rheological properties of flour dough. Studies in shear and compression. Cereal Chem 39 (3): 242–255. GME (2004) Gelatine Manufacturers of Europe Monograph 2004. Food Technology Co. Ltd, West Sussex. Gnedenko, B.V. (1988) The Theory of Probability. Mir Publishers, Moscow, Chapter 7. Goguyeva, M.P. (1965) Data on the structure-mechanic properties of whipped masses (in Russian). Khlebopekarnaya i Konditerskaya Promyshlennost’ 10: 20–24. Görög, J. (1964) Ipari mikrobiológia és enzimológia (Industrial Microbiology and Enzymology, in Hungarian) Lectures at the Technical University of Budapest. Goryacheva, G.N., Kleshko, G.M., Osetrova, E.V., Simutenko, V.V. and Khisyametdinova, R.Ya. (1979) Specific features of 2-stage grinding of cacao beans (in Russian). Khlebopekarnaya i Konditerskaya Promyshlennost' 10: 30–31. Gossett, P.W., Rizvi, S.S.H. and Baker, R.C. (1984) Quantitative analysis of gelation in egg protein systems. Food Technol 38 (5): 67. Grassmann, P. (1967) Einführung in die thermische Verfahrenstechnik. De Gruyter, Berlin. Gray, W.A. (1968) The Packing of Solid Particles. Chapman and Hall Ltd., London. Gray, M. (2006) What changes occur in chocolate during conching? New Food 1: 35–40. Gray, D.R. and Chinnaswamy, R. (1995) The role of extrusion in food processing. In: Gaonkar, A.G. (ed) Food Processing – Recent Developments. Elsevier, New York, NY, p. 241. Griffin, W.C. (1949) Classification of surface-active agents by HLB. J Soc Cosmet Chem 1: 311. Griffin, W.C. (1954) Calculation of HLB of non-ionic surfactants. J Soc Cosmet Chem 5: 249. Grodzinski, Z. (1951) The yolk’s spheres of the hen’s egg as osmometers. Biol Rev 26: 253. ten Grotenhuis, E., van Aken, G.A., van Malssen, K.F. and Schenk, H. (1999) Polymorphism of milk fat studied by differential scanning calorimetry and real-time X-ray powder diffraction, J Am Oil Chem Soc 76: 1031–1039. Grover, D.W. (1947) J Soc Chem Ind 66: 201. Grundke, K. (2002) Wetting, spreading and penetration. In: Holmber, K. (ed) Handbook od Applied Surface and Colloid Chemistry, vol 2. Wiley, Chichester, p. 138–140. Grüneberg, M. and Wilk, M. (1992) Rheologisch bedingte Einflussfaktoren auf die Verarbeitungseigenschaften vorkristallisierter Schokoladenmassen. Chemie-Ing Techn 64: 668. Grüneberg, M., Persch, C. and Schubert, M. (1993) Modelle zur Berechnung der MikrowellenErwärmung von Dielektrika am Beispiel von Lebensmitteln, Chemie-Ing Techn 65: 1083. Guilbot, A. and Drapron, R. (1969) Evolution of the state of organization and affinity for water of various carbohydrates as a function of relative humidity (in French). Bull Int Inst Froid Annexe 9: 191–197. Guillemin, E.A. (1935) Communication Networks I–II. John Wiley and Sons Inc., New York, NY. Guillemin, E.A. (1950) Mathematics of Circuit Analysis. John Wiley and Sons Inc., New York, NY. Guiseley, K.B., Stanley, N.F. and Whitehouse, P.A. (1980) Carrageenan. In: Davidson, R.L. (ed) Handbook of Water-Soluble Gums and Resins. McGraw-Hill, New York, NY, p. 5–30. Gutterson, M. (1969) Confectionery Products, Manufacturing Processes. NDC, London. Gyenis, J. (1992) Strömungseigenschaften von Schüttgütern im statischen Mischrohr. Chemie-Ing Techn 64: 306.

642

Confectionery and Chocolate Engineering: Principles and Applications

Hall, A. (1999) Special topic issue – Process and product development. Chem Eng Design 77 (3):173–174. Hallström, B., Skjöldebrand, Ch. and Trägárdh, Ch. (1988) Heat Transfer and Food Products. Elsevier Applied Science, London, p. 5 (Table 1.1). Hamaker, H.C. (1937) The London–van der Waals attraction between spherical particles. Physica 4: 1038. Hamann, D.D. (1983) Structural failure in solid foods. In: Peleg, M. and Gabley, E.B. (ed) Physical Properties of Foods. AVI Publishing Co. Inc, Westport, CT, p. 351–384. Hankóczy, E.V. (1920) Z ges Getreidewesen 12: 57. Habbard, E.H. (1956) Chem Ind: 491. Harbs, T. and Jung, S. (2005) Chocolate made easy – ChocoEasy™ – The compact plant for the production of chocolate. Confect Prod 5: 1–4. Harper, J.M. (1989) Food extruders and their applications. In: Mercier, C., Linko, P. and Harper, J.M. (eds) Extrusion Cooking. American Association of Cereal Chemistry, St. Paul, MN, p. 1–14. Harper, J.C. and El Sahrigi, A.F. (1965) Viscometric behavior of tomato concentrates. J Food Sci 30: 470–476. Harrison, D. and Leung, L. (1962) Trans Inst Chem Eng 40: 146. Hawley, R.L. (1970) Egg product. US Patent 3 510 315. Heemskerk, R. and Komen, G. (1987) The engineering of chocolate. Confect Manuf Market 24 (3): 19, 21. Hegg, P.O., Martens, H. and Lofgvist, B. (1979) Effects of pH and neutral salts on the formation and quality of thermal aggregates of ovalbumin. A study on thermal aggregation and denaturation. J Sci Food Agric 30: 981. Heimann, W. and Fincke, A. (1962a) Beiträge zur Rheometrie der Schokoladen. Zeitschrift für Lebensmittel-Untersuchung und Forschung 117 (2): 93–103. Heimann, W. and Fincke, A. (1962b) Beiträge zur Rheometrie der Schokoladen, II, Mitteilung: messung der Fleissgrenze und ihre Berechnung aus der Casson-Gleichung. Zeitschrift für Lebensmittel-Untersuchung und Forschung 117 (3): 225–230. Heimann, W. and Fincke, A. (1962c) Beiträge zur Rheometrie der Schokoladen, II, Mitteilung: Anwendung einer modifizierten Casson-Gleichung auf Milchschiokoladen und Kakaomassen. Zeitschrift für Lebensmittel-Untersuchung und Forschung 117 (4): 297–301. Heimann, W. and Fincke, A. (1962d) Beiträge zur Rheometrie der Schokoladen, II, Mitteilung: Temperaturabhängigkeit des fliessverhaltens geschmolzener Schokolade. Zeitschrift für Lebensmittel-Untersuchung und Forschung 117 (4): 301–306. Hein, J.-C., Rafflenbeul, R. and Beckmann, M. (1982) Fortschritte in der Zerstäubungs-trocknungsTechnologie. Chemie-Ing Techn 54: 787. Heinz, W. (1959) The Casson flow equation: Its validity for suspension of paints. Materialprüfung 1: 311. Heinz, R., Hofmann, F., Petersen, H. and Polke, R. (1989) Produktkenndaten zur Charakterisierung des Verdichtungsverhaltens für die Auslegung von Walzenpressen. Chemie-Ing Techn 61: 424. Heiss, R. (1955). Übersicht über die neue Forschungs- und Entwicklungsarbeiten auf dem Gebiet der Schokoladeherstellung. Zucker Süsswarenwirtschaft 8: 14. Heiss, R. and Bartusch, W. (1956) Über die rheologischen Eigenschaften von dunklen SchokoladeMassen. Fette Seifen Anstrichmittel 58: 868. Heiss, R. and Bartusch, W. (1957a) Int Fachschrift Schokolade-Industrie 8: 312. Heiss, R. and Bartusch, W. (1957b) Int Fachschrift Schokolade-Industrie 9: 350. Heldman, D.R. (1975) Food Process Engineering. AVI Publishing Co. Inc., Westport, CT. Henszelman, F. and Blickle, T. (1959) Veszprém Vegyipari Eljárások és Mu˝veletek Konferenciája (Proceedings of the Conference of Chemical Processes and Operations), p. 185. Henzler, H.J. (1979) Eignung von kontinuierlich durchströmten Mischern zum Homogenisieren. Chemie-Ing Techn 51: 1.

References

643

Henzler, H.-J. (1980) Begasen höherviskoser Flüssigkeiten. Chemie-Ing Techn 52: 643. Henzler, J. (1988) Rheologische Stoffeigenschaften – Erklärung, Messung, Erfassung und Bedeutung. Chemie-Ing Techn 60: 1. Hermann, W. (1979) Verfahren und Kosten der Agglomeration. Chemie-Ing Techn 51: 277. Herndl, G. and Mershmann, A. (1982) Fluiddynamik und Stoffübergang in gerührten Suspensionen, Chemie-Ing Techn 54: 258. Heusch, R. (1983) Elementarprozesse bei der Herstellung und Stabilisierung von Emulsionen. Chemie-Ing Techn 55: 608–616. Hiby, J.W. (1979) Definition und Messung der Mischgüte in flüssigen Gemischen. Chemie-Ing Techn 51: 704. Hickson, D.W., Dill, C.W., Morgan, R.G., Suter, D.A. and Carpenter, Z.L. (1980) A comparison of heat-induced gel strengths of bovine plasma and egg albumen proteins. J Anim Sci 51: 69. Hickson, D.W., Alford, E.S., Gardner, F.A., Diehl, K., Sanders, J.O. and Dill, C.W. (1982) Changes in heat-induced rheological properties during cold storage of egg albumen. J Food Sci 47: 1908. Hinton, G.L. (1931) Res Records 29: 1931. Hixson, A.W. and Tenney, A.H. (1935) Trans Am Inst Chem Eng 31: 113–127. Hlynka, I. and Barth, F.W. (1955a) Chopin alveograph studies I. Dough resistance at constant sample deformation. Cereal Chem 32: 463–471. Hlynka, I. and Barth, F.W. (1955b) Chopin alveograph studies II. Structural relaxation in dough. Cereal Chem 32: 472–480. Hodge, J.E. (1953) Dehydrated foods, chemistry of browning reactions in model system. J Agric Food Chem 1: 928–943. Hoepffner, L. and Patat, F. (1973) Untersuchunger zur Optimierung von Kugelmühlen. Chemie-Ing Techn 45: 961. Hofmann, T. and Tscheuschner, H.-D. (1976) Scaling-up the high-shear process for finishing of chocolate mass. Lebensmittel-Industrie 23 (11): 499–502. Hollenbach, A.M., Peleg, M. and Rufner, R. (1982) Effect of four anticaking agents on the bulk characteristics of ground sugar. J Food Sci 47: 538–544. Holliday, L. (1963) Chem Ind 18: 794. Holmes–Voncken–Dekker (1964) Fluid flow in turbine-stirred, baffled tanks. Chem Eng Sci 19 (3): 201–213. Holt, D.L., Watson, M.A., Dill, C.W., Alford, E.S., Edwards, R.L., Diehl, K.C. and Gardner, F.A. (1984) Correlation of the rheological behavior of egg albumen to temperature, pH, and NaCI concentration. J Food Sci 49: 137. Hong, P.O. and Lee, J.M. (1985) Changes of average drop sizes during initial period of liquid–liquid dispersions in agitated vessels. Ind Eng Chem Process Des Dev 24: 868–872. Hoogendoorn–Den Hartog (1967) Model studies on mixers in the viscous flow region. Chem Eng Sci 22 (12): 1689–1699. Hörner, B. and Patat, F. (1975) Zur Probleme der Optimierung von Kugelmühlen. Chemie-Ing Techn 47: 25. Horrobin, D.J. (1999) Theoretical Aspects of Paste Extrusion. PhD Thesis. University of Cambridge, Cambridge. Horváth, L., Norris, K. and Horváth-Mosonyi, M. (1985) Acta Alimentaria 14 (2): 113. Hough, D.B. and White, L.R. (1980) The calculation of Hamaker constants from Lifshitz theory with application of wetting phenomena. Adv Colloid Interface Sci 14: 3. Huang, H. and Kokini, J.L. (1993) Measurement of biaxial extensional viscosity and wheat flour doughs. J Rheol 37: 879–891. Huntley, H.E. (1952) Dimensional Analysis. MacDonald & Co., London. Huyghebeart, A. and Hendrickx, H. (1971) Polymorphism of cocoa butter, shown by differential scanning calorimetry. Lebensm-Wiss Technol 4: 59.

644

Confectionery and Chocolate Engineering: Principles and Applications

Ibrahim, S.B. and Nienow, A.W. (1994) The effect of viscosity on mixing pattern and solid suspension in stirred vessels. Inst Chem Eng Symp Ser 136: 25–36. Indovina, P.L., Tettamanti, E., Micciancio-Giammarinaro, M.S. and Palma, M.U. (1979) Thermal hysteresis and reversibility of gel-sol transition in agarose-water systems. J Chem Phys 70: 2841. IOCCC Analytical Method 4 (1961, reprint 1996) Determination of the Melting Points of Cocoa Butter. CAOBISCO, Brussels. IOCCCAnalytical Method 31 (1988a, Reprint 1996) Determination of the Cooling/Solidification Curve of Cocoa Butter and of Other Fats Used in Chocolate and Confectionery Products. CAOBISCO, Brussels. IOCCC Analytical Method 27 (1988b, reprint 1996) Determination of the Wettability of Instant Cocoa Powder in Water. CAOBISCO, Brussels. IOCCC Analytical Method 35 (1990a, Reprint1996) Determination of Mono-Oleo Disaturated Symmetrical Triglycerides (SOS) in Oils and Fats used in Chocolate and in Sugar Confectionery Products by Thin-Layer Chromatography and Gas-Liquid Chromatography. CAOBISCO, Brussels. IOCCC Analytical Method 41 (1990b, reprint 1996) Determination of the Composition of the Fatty Acids in the 2-Position of Glycerides in Oils and Fats Used in Chocolate and in Sugar Confectionery Products. CAOBISCO, Brussels. Isozaki, H., Akabane, H. and Nakahama, N. (1976) Viscoelasticity of hydrogel of agar-agar; analysis of creep and stress relaxation. J Agric Chem Japan 50: 265. IUPAC Standard Methods for the Analysis of Oils, Fats & Derivatives. (1986) No. 2.132 Determination of the Cooling Curve of Fats, 7th edn. Blackwell Scientific Publications, Oxford. IUPAC (1997) IUPAC Compendium of Chemical Technology, 2nd edn. IUPAC. Jang, E.S., Jung, M.Y. and Min, D.B. (2006) Hydrogenation for low trans and high conjugated fatty acids. Compr Rev Food Sci Food Safety 1: 22. Jansen, F. (1969) Candy Ind 3: 33–41. Janssen, L.P.B.M. and Smith, J.M. (1975) Der Stand des Wissens auf dem Gebiet des ZweiSchnecken-Extruders. Chemie-Ing Techn 47: 445. Jenike, A.W. (1964) Storage and Flow of Solids. Bulletin No. 123, Utah Engin. Exp. Stn., University of Utah, Salt Lake City, UT. Jeppson, M.R. (1964) Consider microwaves. Food Eng 36 (11): 49–52. Jewell, G. (1972) Some observations on bloom on chocolate. Rev Int Choc 27: 161–162. Jeziorny, A. (1978) Polymers 19: 1142. John, K. (1970) Zum Fliessverhalten von Emulsionen I. Chemie-Ing Techn 42: 132. John, K. (1972) Zum Fliessverhalten von Emulsionen I. Chemie-Ing Techn 44: 622. Johnson, J.C. (1974) Tablet Manufacture. Noyes Data Corporation, Park Ridge, NJ, p. 246–251. Johnson, J.A., Swanson, C.O. and Bayfield, E.G. (1943) The correlation of mixograms with baking results. Cereal Chem 20: 625. Johnston-Banks, F.A. (1990) Gelatine. In: Harris, P. (ed) Food Gels. Elsevier Applied Science, London, p. 233–290. Jolly, M.S., Blackburn, S. and Beckett, S.T. (2003) Energy reduction during chocolate conching using a reciprocating multihole extruder. Food J Eng 59 (2, 3): 137–142. Jones, D.M., Walters, K. and Williams, P.R. (1987) On the extensional viscosity of mobile polymer systems. Rheol Acta 26: 20–30. Joosten, G.E.H., Schilder, J.G.M. and Broere, A.M. (1977) The suspension of floating solids in stirred vessels. Trans Inst Chem Eng 55: 220. Joson, J.F., Davila, L.T. and Domingo, Z.B. (2003) Kinetics of non-isothermal crystallization of coconut-based cholesteryl ester: Avrami and Ozawa approaches. Science Diliman (University of Philippines Diliman) 15 (1): 51–56. Jovanovic, O., Karlovic, Dj. and Jakovlevic, J. (1995) Chocolate pre-crystallization: a review. Acta Alimentaria 24 (3): 225–239.

References

645

Junk, W.R and Pancoast, H.M. (1973) Handbook of Sugars for Processors, Chemists and Technologist. AVI Publishing Co. Inc., Westport, CT, Table 4.15. Kaffka, K. J., Norris, K.H., Kulcsár, F. and Draskovits, I. (1982a) Attempts to determine fat, protein and carbohydrate content in cocoa powder by the NIR technique. Acta Alimentaria 11: 271. Kaffka, K. J., Norris, K.H., Perédi, J. and Balogh, A. (1982b) Attempts to determine oil, protein, water and fiber content in sunflower seeds by the NIR technique. Acta Alimentaria 11 (3): 253. Kaffka, K.J., Horváth, L., Kulcsár, F. and Váradi, M. (1990) Investigation of the state of water in fibrous foodstuffs by near infrared spectroscopy. Acta Alimentaria 19 (2): 125. Kaffka, K.J. and Seregély, Z. (2002) PQS (Polar Qualification System) The new data reduction and product qualification method. Acta Alimentaria 31 (1): 3. Kahilainen, H., Kurki-Suonio, I. and Laine, J. (1979) Konvektiver Wärmeübergang in einer begasten, nicht-Newtonschen Flüssigkeit. Chemie-Ing Techn 51: 1143. Kale, D.D., Mashelkar, R.A. and Ulbrecht, J. (1974) High-speed agitation of non-Newtonian fluids: Influence of elasticity and fluid intertia. Chemie-Ing Techn 46: 69. Kältetechnik (Enzyklopädisches Nachschlagewerk), Band 2 (1960). Gostorgizdat, Moscow. Kaltofen, R., Pagels, I., Schumann, K., Ziemann, J. and Otto, S. (1957) Tabellenbuch Chemie, Volk und Wissen. Volkseigener Verlag, Berlin, p. 270. Karácsony, D. and Pentz, L. (1955a) Theoretical and practical issues of candying and storage of fondant bonbons (in Hungarian). Élelmezési Ipar 9: 45–52. Karácsony, D. and Pentz, L. (1955b) Theoretical and practical study of the circumstances of candying and storage of fondant bonbons (in Hungarian). Élelmezési Ipar 9: 236. Kargin, V.A. (1957) Absorptive properties of glasslike polymers. J Polymer Sci 23: 47–55. Karlovits, G., Martine, J. and Barczak, L. (2006) Trends in production of confectionery filling fats, prensetation. XIV International Science Conference ‘Progress in Technology of Vegetable Fats’, Witaszyce, Poland. Karlshamns Oils & Fats Academy (1991) Cocoa Butter and Alternatives. Karlshamns Oils & Fats Academy. von Kármán, Th. (1925) Beitrag zur Theorie des Walzvorganges. Zeitschrift für Angewandte Mathematik und Mechanik, vol 5. Karoblene, S.A., Kondakova, I.A. and Ermakova, T.P. (1981) Degree of dispersion of chocolate mass obtained by different processes (in Russian). Khlebopekarnaya i Konditerskaya Promyshlennost 10: 28–29. Karpinski, P. (1980) Crystallization as a mass transfer phenomenon. Chem Eng Sci 35: 2321–2324. Kawanari, M., Hamann, D.D., Swartzel, K.R. and Hansen, A.P. (1981) Rheological and texture studies of butter. J Texture Stud 12: 483–505. Kengis, R.P. (1951) Prigatovleniye muchnikh konditerskikh izdeliy (Manufacture of Confectioneries from Flour, in Russian). Gostorgizdat. Kersting, F.-J. (1984) Zur Lösung des Differenzialgleichungssystems der Zerkleinerung. Chem-Ing Techn 56 (12): 937. Kerti, K. (2000) Isotherm DSC method for distinction of various special vegetable fats used for confectionery purposes (in Hungarian). Édesipar 4: 10. Khan, R.S., Hodge, S.M. and Rousseau, D. (2003) Morphology of surface pores in milk chocolate. 94th American Oil Chemistry Society Annual Meeting and Expo May 4–7, Kansas City, MI. Kile, D.E. and Eberl, D.D. (2003) On the origin of size-dependent and size-independent crystal growth: Influence of advection and diffusion. Am Minerol 88: 1514–1521. Kile, D.E., Eberl, D.D., Hoch, A.R. and Reddy, MM. (2000) An assessment of calcite crystal growth mechanisms based on crystal size distributions. Geochimica et Cosmochimica Acta 64: 2937–2950. King, N. (1965) The physical structure of dried milk. Dairy Sci Abstr 27: 91. King, A. and Mukerjee, L.N. (1938) J Soc Chem Ind 57: 431.

646

Confectionery and Chocolate Engineering: Principles and Applications

Kipke, K. (1979) Rühren von dünnflüssigen und mittelviskosen Medien. Chemie-Ing Techn 51: 430. Kipke, K. (1982) Offene Probleeme in der Rührtechnik. Chemie-Ing Techn 54: 416. Kipke, K. (1985) Auslegung für Industrie-Rührwerken. Chemie-Ing Techn 57: 813. Kipke, K. (1992) Suspendieren in einem 21-m3-Behälter. Chemie-Ing Techn 64: 624. Kirchner, K. and Aigner, M. (1979) Beitrag zur Auslegung von Kugelmühlen aufgrund von Modellversuchen. Chemie-Ing Techn 51: 820. Kirchner, K. and Leluschko, J. (1986) Untersuchungen zum Einfluss der Flüssigkeit bei der Nassmahlung in Kugelmühlen. Chemie-Ing Techn 58: 417. Kiss, B. (ed) (1988) Növényolajipari és Háztartás-vegyipari Táblázatok (Technical Data for the Vegetable Oil Processing and Household Chemical Industries; in Hungarian). Mezo˝gazdasági Könyvkiadó, Budapest, p. 100. Klasen, C.-J. and Mewes, D. (1991) Strangpressen partikelförmiger Feststoffe. Chemie-Ing Techn 63: 768. Kleinert, J. (1954a) Zucker Süsswarenwirtschaft, 7: 469. Kleinert, J. (1954b) Int Choc Revue 9: 225. Kleinert, J. (1954c) Manufact Confect 34 (12): 19. Kleinert, J. (1954d) Zucker Süsswarenwirtschaft 7: 511. Kleinert, J. (1957) Int Fachschrift Schokolade-Industrie 4: 130. Kleinert, J. (1962) Studies on the formation of fat bloom and methods of delaying it. Rev Int Choc 16: 201–219. Kleinert, J. (1971) Chocolate manufacture by the LSCP process. Int Choc Rev 26 (1): 2–8. Kleinert, J. (1973) Method for the manufacture of a chocolate composition, particularly milk chocolate composition. Swiss Patent 532 365. Kneuele, F. (1983) Zur Massstabsübertragung beim Suspendieren im Rührgefäss. Chemie-Ing Techn 55: 275. Kniel, K. (2000) ZDS-Practical Course in Chocolate Confectionery 2000. ZDS-Seminar PEO-10, Soligen. Koch, J. (1959) Multi-point viscosity determination. Manuf Confect 39 (10): 23–27. Koch, T. (1991) Formgebung durch Agglomerieren. Chemie-Ing Techn 63: 1170. Koch, T. and Sommer, K. (1993) Prüfmethoden für Agglomerate. Chemie-Ing Techn 65: 935. Koglin, B. (1972) Experimentelle Untersuchungen zur Sedimentation vonTeilchenkomplexen in Suspensionen. Chemie-Ing Techn 44: 515. Koglin, B., Pawlowski, J. and Schnöring, H. (1981) Kontinuierliches Emulgieren mit Rotor/Stator Maschinen: Einfluss der volumenbezogenen Dispergierleistung und der Verweilzeit auf die Emulsionfeinheit. Chemie-Ing Techn 53: 641. Kojima, E. and Nakamura, R. (1985) Heat gelling properties of hen’s egg yolk low density lipoprotein (LDL) in the presence of other protein. J Food Sci 50: 63. Kolek, W., Walstra, P. and Van Vliet, T. (2000) Crystallization kinetics of fully hydrogenated palm oil in sunflower oil mixtures. J Am Oil Chem Soc 77: 389–398. Kolmogorov, A. (1937) Bull Acad Sci URSS (Sci Mat Nat ) 5: 355. Koninklijk Nederlands Meterologisch Instituut (KNMI) (2000) Handbook for Meterological Observation. KNMI, Chapter 5. Koryachkhin, V.P. (1975) Candidate Thesis. University of Food Technologies, Moscow. Kosutány, T. (1907) Der ungarische Weizen und das ungarisches Mehl. Verlag Molnárok Lapja, Budapest. Kot, Yu.D. and Gligalo, E.M. (1969) Isvestiya Vusov. Pishchevaya Tyechnologiya 1: 69–71. Kraume, M. and Zehner, P. (1988) Suspendieren im Rührbehälter – Vergleich unterschiedlicher Berechnungsgleichungen. Chemie-Ing Techn 60: 822. Kraume, M. and Zehner, P. (1990) Homogenisieren in gerührten Fest/Flüssig-Systemen. ChemieIng Techn 62: 306. van Krevelen, D.W. (1956) Steinkohlveredlung und Verfahrenstechnik. Brennstoff-Chemie 37 (5/6): 65–70.

References

647

Kriechbaumer, A. and Marr, R. (1983) Herstellung, Stabilität und Spaltung Multipler Emulsionen. Chemie-Ing Tech 55: 700–707. Krieger, I.M. (1983) Rheology of emulsions and dispersions. In: Peleg, M. and Bagley, E.B. (eds) Physical Properties of Foods. AVI Publishing Co. Inc., Westport, CT, 385–398. Krivcun, L.V., Karyakina, A.B. and Gildebrandt, N.A. (1974) The effect of swelling corn starch on the structure-mechanical properties of foam (in Russian). Khlebopekar-naya i Konditerskaya Promyshlennost’ 11: 39–41. Krupp, H. (1967) Particle adhesion theory and experiment. Adv Colloid Interface Sci 1: 111. Krust, P.W., McGlauchlin, L.D. and McQuistan, R.B. (1962) Elements of Infra-Red Technology. John Wiley and Sons, New York, NY. Kunii, D. and Levenspiel, C. (1991) Fludization Engineering, 2nd edn. John Wiley & Sons, Inc., New York, NY. Kurzhals, H.-A. And Reuter, H. (1973) Fettkügelchen-Grössenverteilung in homogenisierter Milch und anderen Öl-in-Wasser-Emulsionen. Chemie-Ing Techn 45: 491. Kuster, W. (2000) Two-stage chocolate refining – an important step for optimising the conching process. Eur Food Drink Rev Spring: 21–27. Labuza, T.P. (1971) Properties of water as related to the keeping quality of foods. In: Proceedings of the 3rd International Congress on Food Science and Technology, p. 618. Labuza, T.P. (1975) Absorption phenomena in foods: Theoretical and practical aspects. In: Rha, Ch. (ed) Theory, Determination and Control of Physical Properties of Food materials. D. Reidel Publishing Co., Dordrecht, p. 205. Labuza, T.P., Tannenbaum, S.R. and Karel, M. (1970) Water content and stability of low mositure and intermediate moisture foods. Food Tech 24: 543. Labuza, T.P., Mizrahi, S. and Karel, M. (1972a) Mathematical models for optimization of flexible film packaging of foods for storage. Trans A.S.A.E 15: 150. Labuza, T.P., McNally, L., Gallagher, D., Hawkes, J. and Hurtado, F. (1972b) Stability of intermediate moisture foods. J Food Sci 37: 154. Labuza, T.P., Warren, R.M. and Warmbier, H.C. (1977) The physical aspects with respect to water and non-enzymatic browning. In: Friedman, M. (ed) Protein crosslinking. Nutritional and Medical Consequences. Advances in Experimental Medicine and Biology, 86B. Plenum Press, New York, 379–418. La Mer, V.K. and Healy, Th.W. (1963) Adsorption-flocculation reactions of macromolecules at the solid–liquid interface. Rev Pure Appl Chem 13: 112. Landillon V., Cassan D., Morel M.-H. and Cuq, B. (2008) J Food Eng 86 (3): 178. Langer, G. and Werner, U. (1981) Messung der Viskosität von Suspensionen. Chemie-Ing Techn 53: 132. van Langevelde, A., van Malssen, K., Peschar, R. and Schenk, H. (2001) Effect of temperature on recrystallization behavior of cocoa butter. J Am Oil Chem Soc 78: 919–925. Lapitov, E.K. and Filatov, B.S. (1963) Kolloidnaya Zhurnal 125: 43–49. Larsson, K. (1997) Molecular organization in lipids. In: Friberg, S. and Larsson, K. (eds) Food Emulsions, 3rd edn. Marcel Dekker Inc., New York, NY, p. 111–140. Lásztity, R. (1987a) Complex colloid systems in food industries. In: Gabor, Mné, (ed) Colloid Chemical Principles of Food Manufacture. Mezo˝gazdasági Kiadó, Budapest, 246–269. Lásztity, R. (1987b) Reológiai vizsgálati módszerek (Methods of rheological measurements). In: Lásztity, R. and Törley, D. (eds) Az élelmiszer-analítika elméleti alapjai, Part 1 (The Theoretical Principles of Food Analysis, in Hungarian, Part 1). Mezo˝gazdasági Könyvkiadó, Budapest, p. 262–264. Latzen, W. and Molerus, O. (1987) Mindestrührerdrehzahlen beim Suspendieren von Feststoffen. Chemie-Ing Techn 59: 236. Launay, B. and Buré, J. (1977) Use of the Chopin Alveographe as a rheological tool. II. Dough properties in biaxial extension. Cereal Chem 54: 1152–1158. Leblans, P.J.R. and Scholtens, B.J.R. (1986) Constitutive analysis of the nonlinear viscoplastic properties in simple extension of polymeric fluids and networks: a comparison. In: Cheremisinoff,

648

Confectionery and Chocolate Engineering: Principles and Applications

N.P. (ed) Encyclopedia of Fluid Mechanics, vol 7 Rheology and Non-Newtonian Flows. Gulf Publishing Co., Houston, TX, p. 555–579. Lees, R. (1968) Published books of historic significance to the British sweet and chocolate industry. Confect Prod 34 (1): 30–33. (continued in each volume until 1969 12) Lees, R. (1972) High boiled sweets – Suggested data sheet for remedying faults. Confect Prod 9: 456–484; 11: 582–583. Lees, R. and Jackson, E.B. (1999) Sugar Confectionery and Chocolate Manufacture. St. Edmundsbury Press Ltd., Bury St. Edmunds, p. 284. Lee, B.K., Alexy, T., Wendy, R.B. and Meiselman, H.J. (2007) Red blood cell aggregation via Myrenne aggregometer and yield shear stress. Biorheology 44 (1): 29–53. Leighton, A., Leviton, A. and Williams, O.E. (1934) The apparent viscosity of ice cream. J Dairy Sci 17: 639–650. Leng, D.E. and Calabrese, R.V. (2003) Immiscible liquid – liquid systems. In: Paul, E.L., AtiemoObeng, V.A. and Kresta, S.M. (eds) Handbook of Industrial Mixing. Wiley-Interscience, a John Wiley & Sons, Inc. Publication, Chichester, p. 639–754. Lengyel, S., Proszt, J. and Szarvas, P. (1960) Általános és szervetlen kémia (General and Inorganic Chemistry, in Hungarian), Tankönyvkiadó, Budapest. Leschonski, K. (1975) Das Klassieren disperser Feststoffe in gasförmigen Medien. Chemie-Ing Techn 47: 708. Les codes d’usages en confiserie (1965) Chambre Syndicale nationale de la confiserie, France. Letelier, M.F.S. and Céspedes, J.F.B. (1986) Laminar unsteady flow of elementary non-Newtonian fluids in long pipes. In: Cheremisinoff, N.P. (ed) Encyclopedia of Fluid Mechanics, vol 7 Rheology and Non-Newtonian Flows. Gulf Publishing Co., Houston, TX, p. 55–88. Leuenberger, H., Bier, H.P., Sucker, H. (1980) Theorie and Praxix der Bestimmung des Granulierflüssigkeitsbedarfes beim konventionellen Granulieren. Chemie-Ing Techn 52: 609. Leuenberger, H., Hiestand, E.N. and Sucker, H. (1981) Ein Beitrag zur Theorie der Pulverkompression. Chemie-Ing Techn 53: 45. Leva, M. (1959) Fluidization. McGraw-Hill Book Co. Inc., New York, NY. Levenspiel, O. (1972) Chemical Reaction Engineering. Wiley, New York, NY, p. 253. Levich, V.G. (1962) Physicochemical Hydrodynamics. Prentice Hall, Englewood Cliffs, NJ. Levine, L. (1996) Model for the sheeting of dough between rolls operating at different speeds. Cereal Foods World 41 (8): 690. Levine, L., Corvalan, C.M., Campanella, O.H. and Okos, M.R. (2002) A model describing the two dimensional calender of finite width sheets. Chem Eng Sci 57: 643. Levy, F.L. (1981) A modified Maxwell-Eucken equation for calculating the thermal conductivity of two-component solutions or mixtures. Int J Refrigeration 4: 223–225. Lewis, M.J. (1987) Physical Properties of Foods and Food Processing Systems. Ellis Horwood, Chichester. Ley, D. (1988) The Frisse approach to conching. Confect Prod 53 (8): 605–608. Li, D. and Neumann, A.W. (1996) Thermodynamic status of contact angles. In: Neumann, A.W. and Spelt, J.K. (eds) Applied Surface Thermodynamics. Surfactant Science Series, Vol.63. Marcel Dekker, New York, NY, p. 109–168. Libkin, A.A., Rjabov, O.A. and Machikhin, S.A. et al. (1978) Rheological properties of doughs of Tadzhik girdle-cakes (in Russian). Khlobepekarnaya i Konditerskaya Promyshlennost 5: 24–26. Liebig, A.N. (1953) Manufact Confect 33 (9): 47. Liedefelt, J.-O. (ed) (2002) Handbook of Vegetable Oils and Fats. Karlshamns AB. Lifschitz, E.M. (1955) Exp Theoret Physik USSR 29: 94. van der Lijn, J. (1976) Simulation of Heat and Mass Transfer in Spray Drying. Doctoral Thesis. Wageningen. Lim, K.S. and Barigou, M. (2004) X-ray micro-computed tomography of cellular food products. Food Res Int 37: 1001–1012.

References

649

Lingert, H. (1990) In: Finot, P.A., Aeschbacher, H.U., Hurrell, R.F. and Liardon, R. (eds) The Maillard Reaction in Food Processing, Human Nutrition and Physiology. Birkhäuser Verlag, Basel, p. 171. Linko, P., Linko, Y.Y. and Olkku, J. (1983) Extrusion cooking and bioconversions. J Food Eng 2 (4): 243–257. Lipscomb, A.G. (1954) Ungelöste Probleme bei der Herstellung von Nahrungsmitteln: Schokolade. Fette u Seifen 56: 803–809. Liszi, J. (1975) Vezetéses transzport együtthatók (Conductive transport coefficients). In: Szolcsányi, P. (ed) Vegyészmérnöki számítások termodinamikai alapjai (Thermodynamic Bases of Calculations in Chemical Engineering, in Hungarian). Mu˝szaki Könyvkiadó, Budapest, p. 237–257. Livsmedelstabeller – Energi och vissa Näringsämmen (1978) Statens Livsmedelsverk. Liber Tryck, Stockholm. Loisel, C., Lecq, G., Keller, G. and Ollivon, M. (1997) Fat bloom and chocolate structure studied by mercury porosimetry. J Food Sci 62: 781–788. Lonchampt, P. and Hartel, R.W. (2004) Fat bloom in chocolate and compound coatings. Eur J Lipid Sci Technol 106: 241–274. Loncin, M. and Merson, R.L. (1979) Food Engineering, Principles and Selected Applications. Academic Press, New York, NY. Lovegren, N., Gray, M.S. and Feuge, R. (1976) Effect of liquid fat on melting point and polymorphic behavior of cocoa butter and a cocoa butter fraction. J Am Chem Soc 53: 83. Lucas, K. (1981) Die Druckabhängigkeit der Viskosität der Flüssigkeiten – eine einfache Abschätzung. Chemie-Ing Techn 53: 959. Lucisano, M., Casiraghi, E. and Mariotti, M. (2006) Influence of formulation and processing variables on ball mill refining of milk chocolate. Eur Food Res Technol 223 (6): 797–802. Ludwig, K.G. (1969) Modern emulsifiers for retarding the formation of fat bloom in chocolate. Fette Seifen Anstrichmittel 71 (8): 672–678. Lukach, Y.E., Ryabinin, D.D. and Metlov, B.V. (1967). Valvovye Mashinydlia Pereraboki Plastmass i Resinovykh Smesey. M. Mashinostroenie, USSR. Lund, D.B. (1983) Applications of differential scanning calorimetry in foods. In: Peleg, M. and Bagley, E.B. (eds) Physical Properties of Foods. AVI Publishing Co. Inc, Westport, CT, p. 125–144. Lunyin, O.G., Sokolov, Y.D. and Berkovich, M.A. (1976) Supposed velocity gradient in flow of plum cream “Charlotte” (in Russian). Khlebepekarnaya i Konditerskaya Promyshlennost’ 1976 8: 31–33. Lyapunov, M.A. (1892) The General Problem of Motion Stability. Princeton University Press, Princeton, 1947. Machikhin, Yu.A. (1968) Konditerskaya Promyshlennost’, CNIITEI Pishcheprom 1: 13–15. Machikhin, S.A. (1975) Doctorial Thesis. Technical University of the Food Industries in Moscow. Machikhin, Y.A. and Birfeld, A.A. (1969) Mekhanizaciya perekachki pralinovykh mass, CNIITEI Pishcheprom, Moscow. Machikhin, Y.A. and Machikhin, S.A. (1987) Élelmiszeripari termékek reológiája (Rheology of Confectionery Products, in Hungarian). Mezo˝gazdasági Kiadó, Budapest, Chapters 3–7. (Inzheniernaya reologiya pishchevykh matierialov, Izdatielstvo Lyogkaya i pishchevaya promyshlennost’, Moscow, 1981.) Machikhin, Y.A., Chuvakhin, S.V. and Karpin, R.A. (1976) Structure-mechanic properties of Truffle masses (in Russian). Khlebopekarnayai i Konditerskaya Promyshlennost’ 9, p. 34–35. Mackey, K.L., Ofoli, R.Y., Morgan, R.G. and Steffe, J.F. (1989) Rheological Modeling of potato flour during extrusion cooking. J Food Proc Eng 11: 1–11. Maczelka, L. (1962) Édesipari anyagismeret (Raw Materials of Confectionery Production, in Hungarian). Mu˝szaki Könyvkiadó, Budapest, p. 38. Maczelka, L. and Gyo˝rbíróné, E. (1958) Édesipar IX–X: 191.

650

Confectionery and Chocolate Engineering: Principles and Applications

Mahiout, S. and Vogelpohl, A. (1986) Stoffübergang in hochviskosen Gemischen. Chemie-Ing Techn 58: 62. Mak, A.T.C. (1992) Solid–Liquid Mixing in a Mechanically Agitated Vessel. PhD Dissertation. University College, London. Mak, F.K. and Kelly, F.H.C. (1976) Some physical and chemical properties of sucrose surfaces. Zeitschrift für die Zuckerindustrie 26 (11): 713–718. Makins, A.H. (1974) The evolution of sheeters and laminators. Baking Ind J Oct: 28–29. Makower, B. and Dye.W.B. (1956) Equilibrium moisture content and crystallization of amorphous sucrose and glucose. J Agric Food Chem 4: 72–77. Maksimov, A.S. (1976) Candidate Thesis. Technical University of the Food Industries, Moscow. Maksimov, A.S. and Machikhin, Yu.A. (1976) Isvestiya Vuzov. Pishchevaya Tekhnologiya 6: 108–111. Maksimov, A.S., Machikhin, Yu.A., Selekhov, V.A. and Shapiro, A.G. (1973) Konditerskaya Promyshlennost’ 7 CNIITEI pishcheprom, p. 15–19. Man, D. and Jones, A. (2000) Shelf-life Evaluation of Foods, Part II (Chocolate Confectionery), 2nd edn. Food Preservation Technology Books from C.H.I.P.S. Mandelbrot, B.B. (1977) Fractals: Form, Chance, and Dimension. W.H. Freeman and Co., San Francisco, CA. Mandelbrot, B.B. (1983) The Fractal Geometry of Nature. W.H. Freeman and Co., New York, NY. Manev, E., Schedulko, A. and Exerowa, D. (1974) Colloid Polym Sci 252: 586. Manley, D. (1998a) Biscuit, Cookie and Cracker Manufacturing Manuals, 3. Biscuit Dough Piece Forming. Woodhead Publishing Ltd., Cambridge. Manohar, R.S. and Rao, P.H. (1997) Effect of mixing period and additives on the rheological characteristics of dough and quality of biscuits. J Cereal Sci 25: 197–206. Mansvelt, J.M. (1962) Kakao Zucker 6: 280–286. Marangoni, A.G. (1998) On the use and misuse of the Avrami equation in characterization of the kinetics of fat crystallization. J Am Oil Chem Soc 75 (10): 1465. Marangoni, A.G. and Rousseau, D. (1996) Is platic fat rheology govered by the fractal nature of the fat crystal network. J Am Oil Chem.Soc 73: 1265–1271. Markopoulos, J., Kalwourzis, A. and Raptopoulos, W. (1990) Fluid dynamics and suspension characteristics of double impeller agitated vessels. Chemie-Ing Techn 62: 659. Marshalkin, G.A. (1978) Tekhnologiya konditerskikh izdeliy. Pishchevaya Prom., Moscow. Marshalkin, G.A. and Karpin, B.A. (1971) Calculation of effective viscosity of fondant masse (in Russian). Khlebioejarnaya i Konditerskaya Promyshlennost’ 8: 12–14. Marshalkin, G.A., Jegorov, V.A. and Karghin, B. (1970) Adhesion and rheological properties of fondant mass (in Russian). Khlebioejarnaya i Konditerskaya Promyshlennost 12: 14–16. Marszall, L. and Van Valkenburg, J.W. (1982) The effect of glycols on the hydrophile–lipophile balance and the micelle formation of nonionic surfactants. JAOCS 59 (2): 84. Matsuhashi, T. (1970) Regression line of ’apparent jelly strength’ of agar gel on agar concentration 1; 2. J Jap Soc Food Sci 17: 29, 32. Matsuhashi, T. (1990) Agar. In: Harris, P. (ed) Food Gels. Elsevier Applied Science, London, p. 19. Matz, G. (1984) Ostwald-Reifung – in neuer Sicht. Chemie-Ing Techn 56: 886. Mauron, J. (1981) The Maillard reaction in food; A critical review from the nutritional standpoint. In: Eriksson, C. (ed) Progress in Food and Nutrition Science. Pergamon Press, Oxford, p. 5–35. Mazumder, P., Roopan B.S. and Bhattacharya, S. (2007) Textural attributes of a model snack food at different moisture content. J Food Eng 79 (2): 511. Mazur, P.Y. and Dyatlov, V.A. (1972) The effect of fat and sucrose on the surface tension during the rising of dough (Russian). Khlebioejarnaya i Konditerskaya Promyshlennost’ 12: 22–25. McCabe, W.L. (1929) Crystal growth in aqueous solutions. Ind Eng Chem 21: 30–33. McCabe, W.L. and Stevens, J. (1951) Rate of growth of crystals in aqueous solutions. Chem Eng Prog 47:168.

References

651

McCabe, W.L., Smith, J.C. and Harriott, P. (2001) Unit Operatons of Chemical Engineering, 6th edn. McGraw-Hill, Boston, MA. McCarthy, M., Walton, J. and McCarthy, K. (2000) Magnetic resonance imaging – analysis for confectionery products and processes. 54th PMCA Production Conference. McGauley, S.E. (2001) The Relationship Between Polymorphism, Crystallization Kinetics and Microstructure of Statically Crystallized Cocoa Butter. Masters Thesis. The University of Guelph, Guelph. McGeary, R.K. (1967–1970) Mechanical packing of spherical particles. In: Hausner, H.H., Roll, K.H. and Johnson, P.K. (eds) Perspectives of Power Metallurgy, vol 2 Vibratory Compacting. Plenum Press, New York, NY. Meakin, P. (1988) Fractal aggregates. Adv Colloid Interface Sci 28: 249–331. Meenakshi Sundaram, K. and Nath, G. (1974) Heat transfor to an Ellis model fluid flowing in a circular pipe. Chemie-Ing Techn 46: 863. Meiners, A., Kreiten, K. and Joike, H. (1983) Silesia Confiserie Manual No. 3. Silesia-Essenzenfabrik Gerhard Hanke K.G., Neuss. Mércia de Freitas, E. and Caetano da Silva Lannes, S. (2007) Use of texture analysis to determine compaction force of powders. J Food Eng 80: 568. Mersmann, A. and Grossmann, H. (1980) Dispergieren im flüssigen Zweiphasensystem. Chemie-Ing Techn 52: 621. Merz, A. and Holzmüller, R. (1981) Radionuclide investigations into the influence of mixing chamber geometry on the mass transport in a continuous plowshar mixer. Proceedings of the Powtech Conference on Mixing of Particulate Solids. Institution of Chemical Engineers, p. S 1/D/1–11. Metzner, A.B. and Otto, R.E. (1957) Agitation of non-Newtonian fluids. AICHE J 3 (1): 3–10. Meursing, E.H. (1976) Cocoa Powders for Industrial Processing, 2nd edn. Cacaofabriek de Zaan, Koog aan de Zaan. Meyer, A. (1949) Der Zuckerbäcker. Manu Verlag, Augsburg, p. 17. Meyer, R. and Hood, L.F. (1973) The effect of pH and heat on the ultrastructure of thick and thin hen’s egg albumen. Poultry Sci 52: 1814. Michel, E., Thibault, J.-F. and Doublier, J.-L. (1984) Viscometric and potentiometric study of high-methoxyl pectins in the presence of sucrose. Carbohydrate Polymers 4: 283 Miller, A.R. (1985) The use of a penetrometer to measure the consistency of short doughs. In: Faridi, H. (ed) Rheology of Wheat Products. The American Association of Cereal Chemists, St. Paul, MN, p. 117–132. Minifie, B.W. (1970a) Chemical analysis and its application to candy technology. Confect Prod 36 (7) 423–426, 449. Minife, B.W. (1970b) Chemical analysis and its application to candy technology – The analysis of fats. Confect Prod 36 (9): 554–555. Minife, B.W. (1970c) Chemical analysis and its application to candy technology – The analysis of fats. Confect Prod 36 (10): 615–616. Minife, B.W. (1970d) Chemical analysis and its application to candy technology – The analysis of fats. Confect Prod 36 (12): 746–747, 770. Minifie, B.W. (1989) Bloom, microbiological and other spoilage problems. Choc Cocoa Confect 69: 495–518. Minifie, B.W. (1999) Chocolate, Cocoa and Confectionery, Science and Technology, 3rd edn. An Aspen Publication, Maryland. Mizrahi, S., Labuza, T.P. and Karel, M. (1970a) Computer aided prediction of food storage stability: Extent of browning in dehydrated cabbage. J Food Sci 35: 799. Mizrahi, S., Labuza, T.P. and Karel, M. (1970b) Feasibility of accelerated storage tests for browning of cabbage. J Food Sci 35: 804. Mohos, F.Á. (1966a) Rheological Studies of Milk Chocolate. Theses of Dipl. Technical University of Budapest, Budapest.

652

Confectionery and Chocolate Engineering: Principles and Applications

Mohos, F.Á. (1966b) Rheological studies of chocolate masses, particularly in point of view of the Casson equation, Part I (in Hungarian). Édesipar 6: 168–173. Mohos, F.Á. (1967a) Rheological studies of chocolate masses, particularly in point of view of the Casson equation, Part II (in Hungarian). Édesipar 3: 77–79. Mohos, F.Á. (1967b) Rheological studies of chocolate masses, particularly in point of view of the Casson equation, Part III (in Hungarian). Édesipar 6: 174–176. Mohos, F.Á. (1975) Édesipari termékek gyártása (The Manufacture of Confectionery Products, in Hungarian). Mezo˝gazdasági Kiadó, Budapest, 1975, p. 135. Mohos, F.Á. (1979) A konsolás folyamán végbemeno˝ változásokról (On the Changes Taking Place in Conching, in Hungarian). Édesipar 6: 176–180. Mohos, F.Á. (1982) A kémiai technológiák szerkezetelméletének alkalmazása az édesipar gyakorlatban (The Application of Structure Theory of Chemical Engineering in the Confectionery Practice, in Hungarian). Candidate Thesis. The Hungarian Academy of Sciences, Budapest. Mohos, F.Á. (1990) Édesipari technológia (Confectionery Technology, in Hungarian). Lectures. University of Horticulture and Food Technologies, Budapest, p. 266. Mohos, F.Á. and Lengyel, Sné (1981) Vizsgálat tejcsokoládé hidroxi-metil-furfurol (HMF) tartalmára (Study on the Hydroxy-methyl-furfurol Content /HMF/ of Milk Chocolate, in Hungarian). Édesipar 2: 108–110. Mohos, F.Á., Örsi, F., Kaszás, J., Horváth, Zs., Orsolics, E., Grabska, D. and Demjén, Z. (1981) Karamellizációval kapcsolatos vizsgálatok az édesiparban (Studies on Caramelization in the Confectionery Industry, in Hungarian). Édesipar 2: 44–48. Mohsenin, H (1986) Physical Properties of Plant and Animal Materials, vol 1, 2nd edn. Gondon and Breach Science Publishers, New York, NY. Molerus, O. (1966) Chemie-Ing Techn 38 (2): 137. Molerus, O. (1993) Verhalten feinkörniger Schüttgüter. Chemie-Ing Techn 65 (6): 710–718. Montejano, J.G., Hamann, D.D., Ball, H.R., Jr. and Lanier, T.C. (1984) Thermally induced gelation of native and modified egg white rheological changes during processing; final strengths and microstructures. J Food Sci 49: 1249. Montejano, J.G., Hamann, D.D. and Lanier, T.C. (1985). Comparison of two instrumental methods with sensory texture of protein gels. J Texture Stud 16: 403–424. Money, R.W and Born, R. (1951) Equilibrium humidity of sugar solutions. J Sci Food Agric 2 (4): 180–185. Mooney, M. (1946) J Colloid Sci 1: 195. Mooney, M. (1951) The viscosity of a concentrated suspension of spherical particles. J Colloid Sci 6: 162–170. Morandini, W., Engle, H. and Wassermann, L. (1972) Measurement of crumb strength of Madeira cakes and yeast-raised baking products. Getreide Mehl Brot 26 (3): 68–75. Moreyra, R. and Peleg, M. (1981) Effect of equilibrium water activity on the bulk properties of selected food powders. J Food Sci 46: 1918–1922. Morgan, R.G. (1989) Modeling the Effects of Temperature–Time History, Temperature, Shear Rate and Moisture on the Viscosity of Defatted Soy Flour. PhD Dissertation. Texas A&M University, College Station, TX. Morris, B.J. (1958) The Chemical Analysis of Foods and Food Products. Academic Press, New York, NY. Morris, E.R. and Norton, I.T. (1983) Polysaccharide aggregation in solutions and gels. In: WynJones, E. and Gormally, J. (eds) Aggregation Processes in Solution. Elsevier, Amsterdam, 549–593. Morse, P.L. (1951) Degrees of freedom for steady state flow systems. Ind Eng Chem 43: 1863. Mucsai, L. (1971) Crystallization (in Hungarian). Mu˝szaki Könyvkiadó, Budapest. Mullin, J.W. (1973) Chem Eng (London) 274: 316. Mullin, J.W. and Leci, C.L. (1969) Some nucleation characteristics of aqueous citric acid solutions. J Crystal Growth 5: 75.

References

653

Mullin, J.W. and Osman, M.M. (1973) The nucleation and precipitation of nickel ammonium sulphate crystals from aqueous solution. Krist Technik 8 (4): 471–481. Müntener, K. (1997) Process for refining of chocolate. German Federal Republic Patent Application DE 196 23 206 A1. Müntener, K. (2007) Process for refining of chocolate mass. German Federal Republic Patent DE 196 23 206 B4. Murra, F., Zhang, L. and Lyng, J.G. (2009) Radio frequency treatment of foods. Review of recent advances. J Food Eng 91 (4): 497–508. Muzzio, F.J., Alexander, A., Goodridge, Ch., Shen, E. and Shinbrot, T. (2003) Solids mixing. In: Paul, E., Atiemo-Obeng, V.A. and Kresta, S.M. (eds) Handbook of Industrial Mixing: Science and Practice. Wiley-Interscience, a John Wiley & Sons, Inc. Publication, New York, NY. Nachschlagewerk des Konditors (1958) Verlag Lebensmittelindustrie, Moscow. Nagel, O. and Kürten, H. (1976) Untersuchungen zum Dispergieren im turbulenten Scherfeld. Chemie-Ing Techn 48: 513. Nakahama, N. (1966) Rheological studies of the agar-agar gel. J Home Econ Japan 17: 197. Nakamura, R., Sugiyama, H. and Sato, Y. (1978) Factors contributing to the heat-induced aggregation of ovalbumin. Agric Biol Chem 42: 819. Nakamura, R., Fukano, T. and Taniguchi, M. (1982) Heat-induced gelation on hen’s egg yolk low density protein (LDL) dispersion. J Food Sci 47: 1449. Narine, S.S. and Marangoni, G. (1999a) Fractal nature of fat crystal networks. Phys Rev E 59: 1908–1920. Narine, S.S. and Marangoni, A.G. (1999b) Relating structure of fat crystal networks to mechanical properties: a review. Food Res Int 32: 227–248. Narine, S.S., Humphrey, K. and Bouzidi, L. (2006) Modification of the Avrami model for application to the kinetics of the melt crystallization of lipids. J Am Oil Chem Soc 83 (11): 913. Nash, J.H., Leiter, R. and Johnson, A.P. (1965) Effect of antiagglomerant agents on physical properties of finely devided solids. Ind Eng Chem (Prod Res Div) 4 (2): 140–145. Nazarov, N.I., Shebershnyeva, N.N., Kalinyin, Yu.V. and Mizunov, A.I. (1973) Measurement of structure-mechanic properties of maccaroni dough before extrusion (in Russian). Khlebopekarnaya Konditerskaya Promyshlennost’ 10: 22–24. Niediek, E.A. (1973) Comminution of cocoa kernels. II. Structural investigations. Int Choc Rev 28 (4): 82, 84, 86–88, 90–92, 94–95. Niediek, E. A. (1978) Optimal process for fine grinding of cocoa masses. Int Rev Sugar Confect 31, (3): 13–18. Nielsen, A.E. (1964) Kinetics of Precipitation. Pergamon Press, Oxford. Nielsen, A.E. (1969) Ultraviolet spectra and structure of complexes of pyridine 1-oxide and oxygen acids. Krist Tech 4: 17. Nienow, A.W. (1975) Agitated vessel particle-liquid mass transfer: a comparison between theories and data. Chem Eng J 9: 153. Nienow, A.W. and Miles, D. (1978) The effect of impeller/tank configurations on fluid-particle mass transfer. Chem Eng J 15: 13. te Nijenhuis, K. (1981) Colloid Polym Sci 259: 1017. Nikiforov, V.N., Teplova, R.V., Zobova, R.G. and Lyagoda, G.A. (1964) Khimicheskiye i fizicheskiye kharakteristiki irisa i nachinok karameli (Chemical and Physical Properties of Iris and Caramel Masses). CINTI Pishcheprom, Moscow. Nopens, I, Biggs, C.A., De Clercq, B. and Govoreanu, R. (2006) Advances in population balance modelling. Special issue. Chem Eng Sci 61 (1):63–74. Nordeng, S.H. and Silbey, D.F. (1996) A crystal growth rate equation for ancient dolomites: Evidence for millimeter-scale flux-limited growth J Sedimentary Res 66: 477–481. Nyvlt, J. (1972) Collect Czechosl Chem Commun 37: 3155. Nyvlt, J. (1973a) A simple analysis of the work of crystallizers. Krist Tech 8 (5): 595–601. Nyvlt, J. (1973b) Chem Prumysl 23: 417.

654

Confectionery and Chocolate Engineering: Principles and Applications

Nyvlt, J. (1978) Industrial Crystallization – The Present State of the Art. Verlag Chemie, Weinheim. Nyvlt, J. (1981a) Collect Czechosl Chem Commun 46: 79. Nyvlt, J. (1981b) Chem Prumysl 33: 333. Nyvlt, J. and Václavu, V. (1972) Collect Czechosl Chem Commun 37: 3664. Nyvlt, J. and Cipová, H. (1979) Chem Prumysl 29: 230. Nyvlt, J. and Pekárek, V. (1980) Z Physik Chem (Neue Folge) 122: 199. Nyvlt, J., Söhnel, O., Matuchová, M. and Broul, M. (1985) The Kinetics of Industrial Crystallization. Elsevier, New York, NY. Oakenfull, D. and Scott, A. (1984) Hydrophobic interaction in the gelation of high methoxyl pectins. J Food Sci 49: 1093. O'Brien, S.W., Baker, R.C., Hood, L.F. and Liboff, M. (1982) Water-holding capacity and textural acceptability of precooked, frozen, whole-egg omelettes. J Food Sci 47: 412. Oldroyd, J.G. (1959) Complicated rheological properties. In: Mill, C.C. (ed) Rheology of Disperse Systems. Pergamon Press, London, p. 1–15. Oliveira, G.A., Noqueira, M.R.C., Brusamarello, C.Z., Corazza, M.L. and Corazza, F.C. (2008) A hybrid approach to modeling of an industrial cooking process of chewy candy. J Food Eng 89 (3): 251. Omobuwajo, T.O., Busari, O.T. and Osemwegie, A.A. (2000) Thermal agglomeration of chocolate drink powder. J Food Eng 46 (2): 73–81. Orowan, E. (1943) The calculation of roll pressure in hot and cold flat rolling. Proc Instit Mechn Engineers 150: 140. Osuga, D.T. and Feeney, R.E. (1977) Egg proteins. In: Whitaker, J.R. and Tannenbaum, S.R. (eds) Food Proteins. AVI Publishing Co. Inc., Westport, CT, p. 193. Ölsamen und daraus hergestellte Massen und Süsswaren (1995) In: Schriftenreihe des Bundes für Lebensmittelrecht und Lebensmittelkunde e.V. Bundesverband der Deutschen Süsswarenindustrie e.V., Germany. Ormós, Z. (1975) Granulálás fluidizált rétegben. Magyar Kémikusok Lapja XXX (10): 511–519. Orr, C. (1983) Emulsion droplet size data. In: Becher, P. (ed) Encyclopedia of Emulsion Technology, vol I. Marcel Dekker, Inc., New York, NY, p. 369. Örsi, F. (1962) Adatok a hidroximetilfurforul kolorimetriás meghatározásához (Data to Colorimetric Determination of Hydroxy-methyl-furfurol, in Hungarian). Department of Food Chemistry, Technical University of Budapest, Budapest. Ozawa, T. (1971) Kinetics of non-isothermal crystallization. Polymers 12: 150. Pacák, P. and Sláma, I. (1979) Soln Chem 8: 529. Padmanabhan, M. and Bhattacharya, M. (1993) Planar extensional viscosity of corn meal dough, J Food Eng 18: 389–411. Padmanabhan, M. and Bhattacharya, M. (1994) In-line measurements of rheological properties of polymer melts. Rheol Acta 33: 71–87. Pahl, M.H. (1985) Mischen in Schneckenmaschinen. Chemie-Ing Techn 57: Teil 1, 421, Teil 2, 506. Pajin, B., Karlovic’, D., Omorjan, R., Sovilj, V. and Antic’, D. (2007) Influence of filling fat type on praline products with nougat filling. Eur J Lipid Sci Technol 109: 1203. Pápai, L. (1965) Anyagszállítás Légáramban és Folyadékáramban (Transport by Air and Fluid, in Hungarian). Mérnöktovábbképzo˝ Intézet, Budapest. Parkinson, J.C. and Sherman, P. (1977) Colloid Polym.Sci 255: 122. Pawlowski, J. (1971) Die Aenlichkeitstheorie in der Physikalisch-technischen Forschung. Springer Verlag, Berlin. Pawlowski, J. and Zlokarnik, M. (1972) Optimieren von Rührern für eine maximale Ableitung von Reaktionswärme. Chemie-Ing Techn 44: 982. Pawlowski, J., Kalkert, N. and Müller, P.-F. (1986) Mathematische Modelierungvon Keimbildung, Keimschwund und Umlösung von Kristallen. Chemie-Ing Techn 58: 602. Payens, T.A.J. (1972) Light scattering of protein reactivity of polysaccharides, especially of carrageenans. J Dairy Sci 55: 141–150.

References

655

Pearce, K.N. and Kinsella, K. (1978) Emulsifying properties of proteins: evaluation turbidimetric technique. J Agric Food Chem 26: 716–723. Peck, M.C., Rough, S.L., Barnes, J. and Wilson, D.I. (2006) Roller extrusion of biscuit doughs. J Food Eng 74 (4): 431. Pedrocchi, L. and Widmer, F. (1989) Emulsionsherstellung im turbulenten Scherfeld. Chemie-Ing Techn 61: 82. Peitgen, H.-O. and Richter, P.H. (1986) The Beauty of Fractals. Springer Verlag, Berlin. Peitgen, H.-O. and Jürgens, H. (1990) Fraktale: Gezähmtes Chaos. Carls Friedrich von Siemens Stiftung, Munich. Peitgen, H.-O., Jürgens, H. and Saupe, D. (1991) Fractals for the Classroom, Part One: Introduction to Fractals and Chaos. Springer-Verlag/National Council of Teachers of Mathematics, Berlin. Peleg, M. (1971) Measurements of Cohesiveness and Flow Properties of Food Powders. DSc Thesis. Technion, Haifa. Peleg, M. (1980) Linearization of relaxation and creep curves of solid biological materials. J Rheol 24: 451–463. Peleg, M. (1983) Physical characteristics of food powders. In: Peleg, M. and Bagley, E.B. (eds) Physical Properties of Foods. AVI Publishing Co. Inc., Westport, CT, p. 295–296, Table 10.1. Peleg, M. and Bagley, E.B. (eds) (1983) Physical Properties of Foods. AVI Publishing Co. Inc., Westport, CT. Peleg, M. and Mannheim,C.H. (1973) Effect of conditions on the flow properties of powdered sucre. Powder Technol 7: 45–50. Peleg, M. and Normand, M.D. (1995) Stiffness assessment from jagged force-deformation relationships. J Texture Stud 26: 353–370. Peleg, M. and Normand, M.D. (1983) Comparison of two methods for stress relaxation data presentation of solid foods. Rheol Acta 22: 108–113. Peleg, M., Moreyra, R. and Scoville, E. (1982) Rheological characteristics of food powders. AIChE Symp Series 78: 138–143. Perry, F. (1968) Chocolate conching in the double overthrow conche. Confect Product 34 (12): 774–777. Perry, R.H. (1998) Perry’s Chemical Engineers’ Handbook. McGraw-Hill, New York, NY. Petrie, C.J.S. (1979) Extensional Flows. Pittman, London. Pfalzer, L., Bartusch, W. and Heiss, R. (1973) Untersuchungen über die physikalischen Eigenschaften agglomerierter Pulver. Chem-Ing Technik 45 (8): 510–516. Pfalzer, L., Bartusch, W. and Heiss, R. (1974) Die Benetzungseigenschaften agglomerierter Pulver in Abhängigkeit von ihrer Struktur. Gordian, 3. Pietsch, W.B. (1969) Adhesion and agglomeration solids during storage flow and handling, Trans ASME Ser B5: 435–449. Pintauro, N. (1972) Agglomeration Processes in Food Manufacture. Noyes Data Corp., Park Ridge, NJ. Plashchina, I.G., Semenova, M.G., Braudo, E.E. and Tolstoguzov, V.B. (1985) Structural studies of the solutions of anionic polysaccharides, IV. Study of pectin solutions by light-scattering. Carbohydrate Polymers 5: 159. Poggemann, R., Steiff, A. and Weinspach, P.-M. (1979) Wärmeübergang in Rührkesseln mit einphasigen Flüssigkeiten. Chemie-Ing Techn 51: 948. Polakowski, N.H. and Kipling, E.J. (1966) Strength and Structure of Engineering Materials. PrenticeHall Inc., Englewood Cliffs, NJ. Pörtner, R. and Werner, U. (1989) Untersuchungen zur Homogenisierung strukturviskoser und viskoelastischer Flüssigkeiten in Rührbehältern. Chemie-Ing Techn 61: 72. Powrie, W.D. and Nakai, S. (1986) The chemistry of eggs and egg products. In: Stadelman W.J. and Cotterill, O.J. (eds) Egg Science und Technology, 3rd edn. AVI Publishing Co. Inc., Westport, CT, p. 97. Prager, V. and Hodge, F. (1956) Tyeoriya Idealno Plastyicheskih. Inostrannaya Lityeratura, Moscow.

656

Confectionery and Chocolate Engineering: Principles and Applications

Prins, A. (1987) In: Dickinson, E (ed) Food Emulsion and Foams. Royal Society of Chemistry, London. Pugh, R.J. (2002) Foams and foaming. In: Holmberg, K. (ed) Handbook of Applied Surface and Colloid Chemistry, vol 2. J. Wiley & Sons, Chichester, p. 31. Pungor, E. (1985) A Practical Guide to Instrumental Analysis. CRC Press, Boca Raton, FL, p. 181–191. Quast, D.G. and Karel, M. (1972) Computer simulation of storage life of foods undergoing spoilage by two interacting mechanisms. J Food Sci 37: 679. Quast, D.G., Karel, M. and Rand, W.M. (1972) Development of mathematical model for oxidation of chips as a function of oxygen pressure, extent of oxidation and equilibrium relative humidity. J Food Sci 37: 673. Quintas, M.A.C., Brandao, T.R.S. and Silva, C.L.M. (2007a) Modelling colour changes during the caramelisation reaction. J Food Eng 83 (4): 483–488. Quintas, M., Guimaraes, C., Baylina, J., Brabdao, T.R.S. and Silva, C.L.M. (2007b) Innovative Food Sci Emerging Technol 8 (2): 306. Raemy, A. (1981) Differential thermal analysis and heat flow calorimetry of coffee and chicory products. Thermochim Acta 43: 229–236. Raemy, A. and Lambelet, P. (1982) A calorimetric study of self-heating in coffee and chicory. J Food Technol 17: 451–460. Raemy, A. and Loliger, J. (1982) Thermal behavior of cereals studied by heat flow calorimetry. J Cereal Chem 59: 189–191. Raemy, A. and Schweizer, T. (1982) Proc Journées de Genéve Calorimetrie et Analyse Thermique 13: III.11.70. Raemy, A. and Schweizer, T. (1986) Nestlé Res News, 1984/1985. Nestec, Vevey, p. 180. Randolph, A.D. and Larson, M.A. (1971) Theory of Particulate Processes. Academic Press, New York, NY. Rao, M.A. (1986) Rheological properties of fluid foods. In: Rao, M.A. and Rizvi, S.S.H. (eds.) Engineering Properties of Foods. Marcel Dekker, Inc., New York, NY, p. 1–48. Rao, M.A. (1992) Classification, description and measurement of viscoelastic properties of solid food. In. Rao, M.A. and Steffe, J.F. (eds) Viscoelastic Properties of Foods. Elsevier Science Publisher, London, p. 3–47. Rapoport, A.L. and Sosnovsky, L.B. (1951) Tekhnologiya konditerskovo proizvodstva. Pishchepromizdat, Moscow, p. 171–179. Rapoport, A.L. and Tarchova, A.V. (1939) Thermische Konstanten von Konditorwaren. Arbeiten des ZKNII. Rasper, V.F. and Hardy, K.M. (1985) Constant water vs. constant consistency techniques in alveography of soft wheat flours. In: Faridi, H. (ed) Rheology of Wheat Products. The American Association of Cereal Chemists, Inc., St. Paul, MN. Ratti, C. and Mujumdar, A.S. (1995) Simulation of packed bed drying of. foodstuffs with airflow reversal. J Food Eng 26 (3): 259. Ravasz, L. (1964) Candidate Thesis. Hungarian Academy of Sciences, Budapest. Razumov, K.A. (1968) Some Regularities of Ball Mill Grinding. IMPV, Leningrad, A-1. Real Decreto 2419 (1978) de 19 de mayo, por el que se aprueba la Reglamentación Técnico-Sanitaria para la elaboración, circulación y comercio de productos de confiteria-pasteleria, bolleria y repostaria, B.O. del E. – Núm.224, p. 23699, Spain. Real Decreto 1124 (1982) de 30 de abril, por el que se aprueba la Reglamentación Técnico-Sanitaria para la Elaboración, Fabricación, Circulación y Comercio de Galletas, B.O. del E. – Núm.133, p. 15069, Spain. Real Decreto 1787 (1982) de 14 de mayo, por el que se aprueba la Reglamentación Técnico-Sanitaria para la elaboración y venta de turrones y mazapanes, B.O. del E. – Núm.183, p. 20918, Spain. Real Decreto 1137 (1984) de 28 de marzo, por el que se aprueba la Reglamentación TécnicoSanitaria para la Fabricación, Circulación y Comercio del Pan y Panes Especiales, B.O. del E. – Núm.146, p. 17901, Spain.

References

657

Real Decreto 1810 (1991) de 13 de diciembre, por el que se aprueba la Reglamentación TécnicoSanitaria para la elaboración, circulación y comercialización de caramelos, chocles, confites y golosinas, B.O. del E. – Núm.308, p. 41513, Spain. Rebindyer, P.A. (1958) Fiziko-khimicheskaya mekhanyika, novaya oblasty nauki (Physico-Chemical Mechanics, a New Area of Science). Znanyiya, Moscow. Reddy, S.R., Melik, D.H. and Fogler, H.S. (1981) Emulsion stability-theoretical studies on simultaneous flocculation and creaming. J Colloid Interface Sci 82: 116–127. Rees, D.A. (1969) Structure, conformation and mechanism in the formation of polysaccharide gels and networks. In: Wolform, M.L. and Tipson, R.S. (eds) Advances in Carbohydrate Chemistry and Biochemistry. Academic Press, New York, NY, p. 267. Rees, D.A. (1972) Mechanism of gelation in polysaccharide systems. In: Gelation and Gelling Agents, B.F.M.I.R.A. Symposium Proceedings, No. 13, London, p. 7–12. Rees, D.A., Steele, I.W. and Williamson, F.B. (1969) Conformational analysis of polysaccharides. III. The relation between stereochemistry and properties of soma natural polysaccharides. J Polymer Sci Part C 28: 261–276. Reher, E.O. (1969) Chem Techn 21 (1): 14–22. Reher, E.-O. (1970) Chem Techn 22 (3): 136–140. Reher, E.-O., Haroske, D. and Köhler, K. (1969) Strömungen nicht-Newtonscher Flüssigkeiten, 1. Mitteilung: Eine Analyse der nicht-Newtonscher Reibungsgesetze und deren Anwendnung für die Rohrströmung. Chem Techn Teil I 21 (3): 137–143; Teil II 21(5): 281–284. Reynolds, T.M. (1963) Chemistry of nonenzymic browning, I. Adv Food Res 12: 1–52. Reynolds, T.M. (1965) Chemistry of nonenzymic browning, II. Adv Food Res 14: 167–283. Reynolds, T.M. (1969) Nonenzymic browning sugar–amine interaction. In: Schultz, H.W., Cain, R.F. and Wrolstad, R.W. (eds) Carbohydrates and Their Role. AVI Publications Ltd. Inc., Westport, CT, p. 219–252. Rha, C. (1975) Thermal properties of food materials. In: Rha., Ch. and Reidel, R. (eds) Theory, Determination and Control of Physical Properties of Food Materials. D. Riedel Publishing, Dordrecht, p. 311. Rha, C. and Pradipasena, P. (1986) Viscosity of proteins. In: Mitchell, J.R. and Leward, D.A. (eds) Functional Properties of Food Macromolecules. Elsevier Science Publishing Co., Inc., New York, NY. Richtlinie für Zuckerwaren – Leitsätze für Feine Backwaren – Leitsätze für Ölsamen und daraus hergestellte Massen und Süsswaren (1992). Bund für Lebensmittelrecht und Lebensmittelkunde e.V. Deutscher Bäcker-Verlag GmbH, Bochum. Riedel, L. (1969) Thermal conductivity measurements on water-rich foods (in German). Kältetechnik 21: 315–316. Riedel, H. R. (1979) Continuous mixing of chocolate masses. Kakao Zucker 31 (4): 70–72. Riedel, H.R. (1991) Simplified chocolate production – Four processes on a single machine. Confect Prod 57 (7): 528, 534. Riquart, H.-P. (1975) Bestimmung der effektiven Viskosität einer Dispersion aus dem Widerstandgesetz der Partikeln. Chemie-Ing Techn 47: 822. Ritschel, G., Berkes, K. and Hoerig, J. (1985) Continuous manufacture of chocolate mass. German Democratic Republic Patent DD 223 626. Robinson, D.S. (1979) The domestic hen’s egg. In: Vaughn, J.G. (ed). Food Microscopy. Academic Press, New York, NY, p. 313. Rohan, T.A. (1965) Forschungen auf dem Gebiet des Schokoladenaromas. Gordian: 1555. Rohan, T.A. and Stewart, T. (1963) Revue de la Confiserie 8: 799. Rohan, T.A. and Stewart, T. (1964a) Revue de la Confiserie 4: 456. Rohan, T.A. and Stewart, T. (1964b) Revue de la Confiserie 11: 502. Rohan, T.A. and Stewart, T. (1965a) Revue de la Confiserie 4: 416. Rohan, T.A. and Stewart, T. (1965b) Revue de la Confiserie 12: 408. Rohan, T.A. and Stewart, T. (1966a) Revue de la Confiserie 2: 202. Rohan, T.A. and Stewart, T. (1966b) Revue de la Confiserie 2: 206.

658

Confectionery and Chocolate Engineering: Principles and Applications

Rohrsetzer, S. (1986) Kolloidika, Mikrófázisok, micellák, makromolekulák (Colloid Chemistry – Microphases, Micelles and Macromolecules, in Hungarian). Tanikönyvkiadó, Budapest. Rolf, L. and Vongluekiet, Th. (1983) Ermittlung der Energieverteilungen in Kugelmühlen. ChemieIng Techn 55: 800. Rolin, C. and De Vries, J. (1990) Pectin. In: Harris (ed) Food Gels. Elsevier Applied Science, London, p. 401–434. Romanoff, A.L. and Romanoff, A.J. (1961) The Avian Egg. Wiley, New York, NY, p. 1949. Roscoe, R. (1952) The viscosity of suspensions of rigid spheres. Br J Appl Phys 3: 267–269. Rose, H.E. (1959a) Eine neue Formel für das Trockenmischen von Pulvern. Chemie-Ing Techn 3 (3): 192–194. Rose, H.E. (1959b) A suggested equation relating to the mixing of powders and its application to the study of performance of certain types of machines. Trans Instn Chem Engrs 37 (2): 47–56. Roth, D. (1976) Production of Amorphous State During Grinding and Recrystallization as a Cause of Agglomeration of Powdered Sucrose and Processes to Prevent This. Dissertation, University of Karlsruhe, Karlsruhe. Roth, D. (1977) The water vapor absorption behavior of icing sugar (in German). Zucker 30: 274–284. Ruiz-López, I.I., Martínez-Sánchez, C.E., Cobos-Vivlado, R. and Herman-Lara, E. (2008) Mathematical modeling and simulation of batch drying of foods in fixed beds with airflow reversal. J Food Eng 89 (3): 310. Rulev, N.N. (1970) Kolloid Zh 39: 80. Rumpf, H. (1958a) Grundlagen und Methoden des Granulierens, 1. und 2. Teile. Chemie-Ing Techn 30 (3): 144–158. Rumpf, H. (1958b) Grundlagen und Methoden des Granulierens, 3.Teil. Chemie-Ing Techn 30 (5): 329–336. Rumpf, H. (1961) The strength of granules and agglomerates. In: Knepper, W.A. (ed) Agglomeration. Industrial Publications, New York, NY. Rumpf, H. (1974) Die Wissenschaft des Granulierens. Chemie-Ing Techn 46 (1): 1–11. Rumpf, H. and Müller, W. (1962) An investigation into mixing of powders in centrifugal mixers. In: Proceedings of the Symposium on the Handling of Solids. Institution of Chemical Engineers, p. 38–46. Samans, H. (1978) Chocolate technology, with improvement of individual stages of processing. Optimization of grinding. Kakao Zucker 30 (5): 134, 136–137. Sandler, S.I. (1999) Chemical and Engineering Thermodynamics. Wiley, New York, NY. Sandoval, A.J. and Barreiro, J.A. (2002) Water adsorption isotherms of non-fermented cocoa beans (Theobroma cacao). J Food Eng 51 (2): 119–123. Sarghini, F. and Masi, P. (2008) Optimal shape design of a static mixer for food processing. Italian Food Technol 53: 5. Saunders, F.L. (1961) J Colloid Sci 16: 13. Schenk, H and Peschar, R. (2004) Understanding the structure of chocolate. Radiation Physics Chem 71: 729–735. Scheruhn, E., Franke, K. and Tscheuschner, H.D. (2000) Einfluss der Milchpulverart auf das Verarbeitungsverhalten und die Fliesseigenschaften von Milchschokoladenmasse, Teil I: Anmischen der Rohstoffe und fliesseigenschaften der conchierten Schokoladenmasse. Zucker Süsswarenwirtschaft 53 (4): 131–136. Scheuber, G., Alt, Cr. and Leucke, R. (1980) Untersuchung des Mischungverlaufs in Feststoffmischern unterschiedlicher Grösse. Aufberet-Techn 21 (2): 57–68. Schieberle, P. (1990) In: Finot, P.A., Aeschbacher, H.U., Hurrell, R.F. and Liardon, R. (eds) The Maillard Reaction in Food Processing, Human Nutrition and Physiology. Birkhäuser Verlag, Basel, p. 187. Schilp, R. (1975) Zur Technologie der Pastengranulierung. Chemie-Ing Techn 47: 374.

References

659

Schmerwitz, H. (1992) Temperaturabhängigkeit und Mischungsverhalten von Flüssigkeiten, welche dem Potenzgesetz folgen. Chemie-Ing Techn 64: 62. Schmidt, P. (1968) Rührgeräte zum Mischen zäher Flüssigkeiten. Aufbereitungstechnik 9 (9): 438–441. Schmidt, P.C. (1989) Instrumentierungsmöglichkeiten an Rundlauf-tablettenpressent. Chemie-Ing Techn 61: 115. Schmidt, E., Wadenpohl, Ch. and Löffler, F. (1992) Mathematische Beschreibung von Agglomerationsvorgängen im Zykloneinlauf. Chemie-Ing Techn 64: 76. Schmitt, A. (1974) The Petzomat. An original machine for processing thin layers of cocoa pastes. Revue des Fabricants de Confiserie Chocolaterie Confiturerie Biscuiterie 49 (3): 39–51. Schmitt, A. (1986) Refining of cocoa mass with the PIV unit. Süsswaren 30 (1, 2): 46–48. Schmitt, A. (1988) Spray/thin layer roasting of cocoa mass. Süsswaren 32 (12): 470–486. Schnabel, R. and Reher, E.-O. (1992) Stand der Rheometrie und Anwendung neuer Signalformen für die Stoffkennzeichnung nicht-Newtonscher Fluide. Chemie-Ing Techn 64: 864. Schönemann, E. and Hein, J. (1993) Ideales und reales Mischzeitverhalten im Rührkessel. ChemieIng Techn 65: 1357. Schröder, M. (1991) Fractals, Chaos, Power Laws. Freeman, New York, NY. Schubert, H. (1973) Kapillardruck und Zugfestigkeit von feuchten Haufwerken aus körnigen Stoffen. Chemie-Ing Techn 45: 396. Schubert, H. (1979) Grundlagen des Agglomerierens. Chemie-Ing Techn 51: 266. Schubert, H. (1990) Instantisieren pulverförmiger Lebensmittel. Chemie-Ing Techn 62: 892. Schügerl, K. (1974) Fliessbett-Technik. Chemie-Ing Techn 46: 525. Schulman, J.H. and Leja, I. (1954) Control of contact angles at the oil–water–solid interfaces. Trans Faraday Soc 50: 598–606. Schulz, N. (1979) Mischen und Wärmeaustausch in hochkonsistenten Medien, Chemie-Ing Techn 51: 693. Schulz, H. (2004) Analysis of coffee, tea, cocoa, tobocco, spices, medicinal and aromatic plants, and related products. In: Roberts, C.A., Workman Jr., J. and Reeves, J.B. (eds) Near-Infrared Spectroscopy in Agriculture. American Society of Agronomy, Inc. Madison, WI, p. 345–376. Schriftsammlung der Arveiten des WKNU (1950) Ausgabe V–VI. Verlag Lebensmittelindustrie, Moscow. Schulze, D. (1993) Austragorgane und Austraghilfen. Chemie-Ing Techn 65: 48. Schulze, D. and Schwedes, J. (1993) Fliessverhalten von Schüttgütern. In: Weipert, D., Tscheuschner, H.D. and Windhab, E. (eds) Rheologie der Lebensmittel. Behr’s Verlag, Hamburg, p. 257–302. Schümmer, P. (1972) Gesetzmässigkeiten der Modelübertragung bei rheologisch komplexem Stoffverhalten. Chemie-Ing Techn 44: 1057. Schwartz, M.E. (1974) Confections and Candy Technology, Noyes, Park Ridge, NJ. Schwedes, J. (2003) Fließverhalten von Schüttgütern in Bunkern. Aufbereitungs Technik 44: 8–18. Scott Blair, G.W. (1969) Elementary Rheology. Academic Press, London. Scott Blair, G.W. (1975) Survey of the rheological studies of food materials. In: Rha, Ch. (ed) Theory, Determination and Control of Physical Properties of Food Materials. D. Reidel Publishing Co., Dordrecht, p. 3–5. Scott Blair, G.W. and Potel, P. (1973) A preliminary study of the physical significance of certain properties measured by the Chopin extensimeter for testing flour doughs. Cereal Chem 14: 257–262. Seitz, K. and Blickle, T. (1974) The Structure of Systems. MKKE (Karl Marx Economic University), Budapest. Seitz, K., Blickle, T. and Grega, G. (1975) On the Semi-groups of Type with a Special Emphasis on Certain Applications in Chemical Engineering. MKKE (Karl Marx Economic University), Budapest. Seitz, K., Blickle, T. and Balázs, J. (1976) The Study of Special Equations Over the Partial Algebraic System S(F+, XF+, 0, Ð). MKKE (Karl Marx Economic University), Budapest.

660

Confectionery and Chocolate Engineering: Principles and Applications

Shenstone, F.S. (1968) The gross composition, chemistry and physico-chemical basis of organization of the yolk and the white. In: Carter, T.C. (ed) Egg Quality: A Study of the Hen’s Egg. Oliver and Boyd, Edinburgh, p. 26. Sherman, P. (1967) Rheological changes in emulsions on aging. III. At very low rates of shear. J Colloid Interface Sci 24: 97. Sherman, P. (1970) Industrial Rheology. Academic Press, New York, NY. Sherman, P. (1983) Rheological properties of emulsions. In: Becher, P. (ed) Encyclopedia of Emulsion Technology I. Marcel Dekker, Inc., New York, NY, p. 405–437. Shih, W.-H., Kim, S.W.Y.S.-I., Liu, J. and Aksay, A. (1990) Scaling behaviour of the elastic properties of colloid gels. Phys Rev A42: 4772–4778. Shinoda, K. and Arai, H. (1964) The correlation between phase inversion temperature in emulsuion and cloud point in solution of nonionic emulsifier. J Phys Chem 68 (12): 3485. Shinoda, K. and Friberg, S. (1986) Emulsion and Stabilization. Wiley, New York, NY, p. 63. Shinoda, K. and Sagitani, H. (1978) Emulsifier selection in water/oil type emulsions by the hydrophile-lipophile balance-temperature system. J Colloid Interface Sci 64: 68–71 Shlamas, M.A., Stravinskas, Yu.K., Goryacheva, G.N. and Chernyshev, V.V. (1984) System to prepare chocolate masses. USSR Patent 1 114 389 A. Silin, V.A. (1964) Modelirovanie Chervyachnyk Mashin. sb. Trudov Ukrniplastmash. Oborudovanie dlia Pererabotki Polymerov, Kiev, Tekhnika, USSR. Simatos, D. and Blond, G. (1975) The porous texture of freeze dried products. In: Goldblith, L.R. and Rothmayr, W.W. (ed) Freeze Drying and Advanced Food Technology. Academic Press, New York, NY, p. 401–412. Sime, W.J. (1990) Alginates. In: Harris, P. (ed) Food Gels. Elsevier Applied Science, London, p. 53–78. Simon, E.J. (1969a) Comminution during chocolate manufacture – separate or joint milling? Int Choc Rev 24 (4): 140–155. Simon, E.J. (1969b) Particle size diminution in chocolate production – separate or combined grinding. Int Choc Rev 24 (10): 413–421. Simon, I.B., Labuza, T.P. and Karel, M. (1971) Computer aided prediction of food storage stability: oxidation. J Food Sci 36: 208. Singh, R.P. and Heldman, D.R. (1993) Introduction to Food Engineering, 2nd edn. Academic Press, London, p. 140. Sloan, A.E. and Labuza, T.P. (1974) Investigating alternative humectants for use in food. Food Prod Dev 8 (9): 75, 78, 80, 82, 84, 88. Slotine, J-J. and Li, W. (1991) Applied Nonlinear Control. Prentice-Hall International, Inc. Englewood Cliffs, NJ, p. 33. Smejkalova, Z. (1974) Measurements of rheological properties of bakery products. MlynskoPekarensky Primyshl 20 (6): 174–178. Smith, K.W. (2005) Fat Crystallization – Fundamentals. Unilever Research, Colworth. Snoeren, Th.H.M., Payens, T.A.J., Jeunink, J. and Both, P. (1975) Electrostatic interaction between κ-carrageenan and κ-casein. Milchwissenschaft 30: 393–396. Söderman, J. and Laine, J. (1990) Calculation of degree of mixing and energy in agitated vessels as a function of system geometry and fluids. Chemie-Ing Techn 62: 244. Sokolovsky, A.L. (1951) Fiziko-khimicheskiye osnovy proizvodstva karameli (Physico-chemical Principles of Production of Sugar Confectioneries, in Russian). Pishchepromizdat, Moscow, p. 24–69, 82–135. Sokolovsky, A.L. (1958) Spravochnik konditera, Chast I: Cyrye i tekhnologiya konditerskovo promyshlennost’ (Confectionery Handbook, Part I: Raw materials and Technology of Confectionery Production, in Russian). Pishchepromizdat, Moscow, p. 160. Sokolovsky, A.L. (ed) (1959) Technologie der Konditorwarenherstellung. Verlag Lebensmittelindustrie, Moscow. Sommer, K. (1973) Revue de la Confiserie 1: 5.

References

661

Sommer, K. (1975) Mechanismen des Pulvermischens. Chemie-Ing Techn 47: 305. Sone, T. (1972) Consistency of Foodstuffs. D. Riedel Publishing Co., Dordrecht. Sonntag, R.C. and Russel, W.B. (1987) Elastic properties of flocculated networks. J Colloid Interface Sci 116: 485–488. Soroka, B.S. and Soroka, B.A. (1965) Khimichestoie Mashinostroienie 2: 15–19. Sors, L. and Balázs, I. (1989) Design of Plastic Moulds and Dies. Akadémiai Kiadó, Budapest, Elsevier Science Publishers, Amsterdam. Sors, L., Bardócz, L. and Radnóti, I. (1981) Plastic Molds and Dies. Van Nostrand Reinhold Co., New York, NY and Akadémiai Kiadó, Budapest. Spreckley, N. (1987) Shaping up with extrusion. Food Flavourings Ingredients Packaging Processing 9 (9): 51, 53, 55. Sprehe, M., Gaddus, E. and Vogelpohl, A. (1999) On the mass transfer in an impinging-stream reactor. Chem Eng Techn 21 (1): 19–21. Stading, M., Langton, M. and Hermansson, A. (1993) Microstructure and rheological behaviour of particulate β-lactoglobulin gels. Food Hydrocolloids 7: 195–212. Stahl, P.H. (1983) Trocknen von Tablettengranulaten mittels ablufttemperatur-geregelter Gleichgewichtseinstellung. Chemie-Ing Techn 55: 221. Stammer, A. and Schlünder, E.-U. (1992) Mikrowellen-Trocknung – Grundlegende Untersuchungen und Scale-up. Chemie-Ing Techn 64: 986. Stanley, J. (1941) Ind Eng Chem (anal. edition) 13: 398. Stanley, N.F. (1990) Carrageenans. In: Harris, P. (ed) Food Gels. Elsevier Applied Science, London, p. 79–119. Stapley, A.G.F., Tewkesbury, H. and Fryer, P.J. (1999) The effects of shear and temperature history on the crystallization of chocolate. J Am Oil Chem Soc 76: 677. Stastna, J., De Kee, D. and Powley, M.B. (1986) Modeling complex viscosity as a response function. In: Cheremisinoff, N.P. (ed) Encyclopedia of Fluid Mechanics, vol 7 Rheology and Non-Newtonian Flows. Gulf Publishing Co., Houston, TX, p. 581–610. Staudinger, G. and Moser, F. (1976) ‘100 proz Suspension’ von Feststoffen im Rührgefä – Definition und Berechnung. Chemie-Ing Techn 48: 1071. Steffe, J.F. (1996) Rheological Methods in Food Process Engineering, 2nd edn. Freeman Press, MI. Steffe, J.F. and Ford, E.W. (1985) Rheological techniques to evaluate the shelf-stability of starchthickened, strained apricots. J Texture Stud 16: 179–192. Stehr, N. and Schwedes, J. (1983) Zerkleinerungstechnische Untersuchungen an einer Rührwerkskugelmühle. Chemie-Ing Techn 55: 233. Stehr, N. (1984) Residence time distribution in a stirred ball mill and their effect on comminution. Chem Eng Process 18: 73–83. Stehr, N. (1989) Möglichkeiten zur Verhinderung der Mahlkörper-Verpressung in kontinuierlich betriebenen Rührwerksmühlen. Chemie-Ing Techn 61: 422. Stein, W.A. (1987a) Mischzeiten beim Rühren von begasten viskosen Flüssigkeiten,Teil 1. ChemieIng Techn 59: 750. Stein, W.A. (1987b) Mischzeiten beim Rühren von begasten viskosen Flüssigkeiten,Teil 2. ChemieIng Techn 59: 886. Stein, W.A. (1988) Mischzeiten beim Rühren von begasten viskosen Flüssigkeiten,Teil 3. ChemieIng Techn 60: 972. Steiner, E.H. (1959a) Revue International de la Chocolaterie, p. 302 Steiner, E.H. (1959b) The rheology of molten chocolate. In: Mill, C.C. (ed) Rheology of Disperse Systems. Pergamon Press, London, p. 167–180. Steinmetz, D. (2001) Euro Fluidization III. Elsevier, Oxford. Stephan, K. and Lucas, K. (1979) Viscosity of Dense Fluids. Plenum Press, New York, NY. Stephan, K. and Mitrovic, J. (1984) Massnahmen zur Intensivierung des Wärmeübergangs. ChemieIng Techn 56: 427.

662

Confectionery and Chocolate Engineering: Principles and Applications

Stichlmair, J. (1991) Anwendung der Aehnlichkeitsgesetze bei vollständiger und partieller Aehnlichkeit. Chemie-Ing Techn 63: 38. Stiess, M. (1995) Mechanische Verfahrenstechnik I–II. Springer Verlag, Berlin. Strayfield International Ltd. (1986) An array of applications are evolving for radiofrequency drying. Food Eng Int 11: 11. Strazisar, J. and Runovc, F. (1996) Kinetics of comminution in micro- and sub-micrometer ranges. Int J Miner Process 44–45: 673–682. Sumner, J.B. and Somers, G.F. (1953) Chemistry and Methods of Enzymes. Academic Press, New York, NY. Sun, S.F. (2004) Physical Chemistry of Macromolecules: Basic Principles and Issues, 2nd edn. John Wiley & Sons, Inc., Chichester, Chapter 8. Swanson, C.O. and Working, E.B. (1933) Testing quality of flour by the recording dough mixer. Cereal Chem 10: 1. Swift, C.E., Lockett, C. and Fryar, A.J. (1961) Comminuted meat emulsions – the capacity of meats for emulsifying fat. Food Techn 15: 468. Szczesniak, A.S. (1963a) Classification of textural characteristics. J Food Sci 28: 385–389. Szczesniak, A.S. (1963b) Objective measurements of food texture. J Food Sci 28: 410–420. Szczesniak, A.S. (1983) Physical properties of foods. What they are and their relation to other food properties. In: Peleg, M. and Bagley, E.B. (eds) Physical Properties of Foods. AVI Publishing Co. Inc., Westport, CT, 1–42. Székely, P. (1966) Az invertáz édesipari alkalmazásának néhány kérdése (A Study on the Application of Invertase in the Confectionery Practice, in Hungarian) Diploma Theses. Department of Food Chemistry, Technical University of Budapest, Budapest. Szép, J. and Seitz, K. (1975) A rendszerelméleti és rendszerszerkezeti kutatások újabb eredményei (New results in system theory and structure theory, in Hungarian). In: Proceedings of Technical Chemistry Conference, I. Veszprém Academic Committee of Hungarian Academy of Sciences, p. 26–28. Szirtes, T. (1998) Applied Dimensional Analysis and Modeling. McGraw-Hill Co., Inc., New York, NY. Szirtes, T. (2006) Dimenzióanalízis és alkalmazott modellelmélet (Dimensional Analysis and Applied Model Theory, in Hungarian). Typotex, Budapest. Szolcsányi, P. (1960) Die Rolle der Freiheitsgrade in der Thermodynamik der Vorgänge. Z. der Techn. Hochschule für Chemie, Leuna-Merseburg, 3: 33. Szolcsányi, P. (ed) (1972) Transzportfolyamatok (Transport Processes, in Hungarian). Tankönyvkiadó, Budapest. Szolcsányi, P. (ed) (1975) Thermodynamical Bases of the Calculations in Chemical Engineering (in Hungarian). Muszaki Könyvkiadó, Budapest. Tadema, J.C. (1988) Chocolate making by the smaller manufacturer. Confect Manuf Market 25 (3): 6–7. Tamime, A.Y. and Robinson, R.K. (1999) Yoghurt – Science and Technology. CRC Press, Woodhead Publishing Ltd., Cambridge. Tanaka, M. (1975) General foods texturometer applications to food texture research in Japan. J Food Texture Stud 6: 101–106. Tanii, K. (1957) Study on agar. Bull Tohoku Reg Fish Res Lab 9: 1. Tanii, K. (1959) Study on agar. Bull Tohoku Reg Fish Res Lab 15: 67. Tanner, R.I. (1988) Recoverable elastic strain and swelling ratio. In: Collyer, A.A. and Clegg, D.W. (eds) Rheological Measurements. Elsevier Applied Science, New York, NY, 93–118. Tarján, G. (1981) Mineral Processing. Akadémiai Kiadó, Budapest. Taubert, A. (1954) Die geschmacklichen Veränderungen beim Conchieren, Zucker Süsswarenwirtschaft. Taylor, G.I. (1932) The viscosity of a fluid containing small drops of another fluid. Proc Roy Soc (London) A138: 41.

References

663

Taylor, J.B. and Langmuir, I.D. (1933) The evaporation of atoms, ions and electrons from caesium films on tungsten. Phys Rev 44: 423–458. Taylor, R. and Krishna, R. (1993) Multicomponent Mass Transfer. Series in Chemical Engineering, Part I/2.1 – 2.3. Wiley, New York, NY. Taylor, J.R. and Zafiratos, Ch.D. (1991) Modern Physics for Scientists and Engineers. Prentice-Hall International Editions, New York, NY. Taylor, C. and Zumbe, A. (2000) Processing overkill? Confectionery April: 12–13, 15, 17– 18. Tebel, K.H. and Zehner, P. (1985) Ein Konzept der repräsentativen Viskosität für Mischprozesse. Chemie-Ing Techn 57: 49. Tewkesbury, H., Stapley, A.G.F. and Fryer, P.J. (2000) Modelling temperature distributions in cooling chocolate moulds. Chem Eng Sci 55: 3123. The Cocoa Manual (1998) ADM Cocoa, Thieme, E. (1972) Gordian 4: 132–135. Thomas, W.D.E. (2006) A sedimentation method for the determination of the effective particle size distribution of DDT dispersible powders. J Sci Food Agr 7: 270–276. Thorz, M.S. (1986) Extruding–cooling and cutting technology as applied to the processing of confectionery products. Manuf Confect 66 (11): 57–61. Timms, R. (1984) Phase behavior of fats and their mixtures. Prog Lipid Res 4: 23–38. Ting, H.H. and McCabe, W.L. (1934) Supersaturation and crystal formation in seeded solutions. Ind Eng Chem 26: 1201. Tiu, C. and Boger, D.V. (1974) Complete rheological characterization of time-dependent food products. J Texture Stud 5: 329–338. To, E.C. and Flink, J.M. (1978) ‘Collapse’ a structural transition in freeze-dried carbohydrates. J Food Technol 13: 555–565, 567–581, 583–594. Tobvin, M.V. and Krasnova, S.I. (1949) Zhur Fiz Khim 23:963. Tobvin, M.V. and Krasnova, S.I. (1951) Zhur Fiz Khim 25: 161. Tolnay, P. (1963) Ipari enzimológia (Industrial Enzymology, in Hungarian). Mu˝szaki Kiadó, Budapest. Tomas, J. (1992) Modellierung der Zeitverfestigungen von Schüttgütern – Probleme und erste Lösungsansätze. Chemie-Ing Techn 64: 746. Tonn, H. (1961) Zeitschrift für die Zuckerindustrie, p. 6 Tornberg, E., Olsson, A. and Persson, K. (1990) The structural and interfacial properties of food proteins in relation to their function in emulsions. In: Larsson, K. and Friberg, S.E. Food Emulsions, 2nd edn. Marcel Dekker, New York, NY, 247–326. Tóth, Á. (1973) A keveréktakarmány-gyártás gépei (The machinery of mixed feed manufacture). In: Tomay, L. (ed) Gabonaipari kézikönyv, Technológiai gépek és berendezések (Manual of Grain Processors, Technological Machines and Equipments of the Grain Silos, in Hungarian). Mezo˝gazdasági Kiadó, Budapest, p. 625–758. Treiber, A. (1988) Extrusionsaufbereitung scherempfindlicher Nahrungsmittel-produkte. ChemieIng Techn 60: 699. Treiber, A. and Kiefer, P. (1976) Kavitation und Turbulenz als Zerkleinerungs-mechanismen bei der Homogenisation von O/W-Emulsionen. Chemie-Ing Techn 48: 259. Treloar, L.R.G. (1975) The Physics of Rubber Elasticity, 3rd edn. Clarendon Press, Oxford. Tressl, R. (1990) Processed flavors – Scope and limitations. In: Bessiére, Y. and Thomas, A.F. (eds) Flavour Science and Technology. John Wiley & Sons, New York, NY, p. 88–104. Troetsch, H.A. (1991) Zerstäubung von Flüssigkeiten. Chemie-Ing Techn 63: 1152. Trouton, F.T. (1906) On the coefficient of viscous traction and its relation to that of viscosity. Proc Roy Soc London Series A LXXVII: 475–500. Tscheuschner, H.-D. (1989) Rheologie, Rheometrie und instrumentelle Texturanalyse von Schokoladenmassen und Schokolade. Quakenbrück/BRD, Industriekurs Rheologie/rheometrie (Juni).

664

Confectionery and Chocolate Engineering: Principles and Applications

Tscheuschner, H.-D. (1993a) Schokolade, Süsswaren. In: Weipert, D., Tscheuschner, H.D. and Windhab, E. (eds). Rheologie der Lebensmittel. Behr’s Verlag, Hamburg, p. 431–470. Tscheuschner, H.-D. (1993b) Rheologische Eigenschaften von Lebensmittelsystemen. In: Weipert, D., Tscheuschner, H.D. and Windhab, E. (eds). Rheologie der Lebensmittel. Behr’s Verlag, Hamburg, p. 101–172. Tscheuschner, H.-D. (1993c) Zeitschrift für Süsswarenwirtschaft 3: 136. Tscheuschner, H.-D. (1994) Rheological and processing properties of fluid chocolate (in German). Rheology June: 83. Tscheuschner, H.-D. (1999) Confectionery Science II. In: Ziegler, G.R. (ed) Proceedings of the International Symposium Nov 14–16. Penn State University, p. 75. Tscheuschner, H.-D. (2000) Grundlagen dr Schokoladenherstellung. Süsswaren 3: 24. Tscheuschner, H.-D. (2002) Strukturbildungsprozesse bei der Schokoladen-herstellung. Feststoffkomponente Milchpulver als Feuchtelieferant. Zeitschrift Süsswarenwirtschaft 6: 16. Tscheuschner, H.-D. (2008) Einfluss von Rezepturbestandteilen auf die Rheologie von Schokoladenmassen. Fachseminar Lebensmittel-Rheologie, Nov. 13, Karlsruhe. Tscheuschner, H.-D. and Finke, A. (1988a) Rheologische Untersuchungen von Kakaobutter und ihren Dispersionen mit Kakaofeststoff und Zucker, Teil 1. Zeitschrift Süsswarenwirtschaft 7/8: 244. Tscheuschner, H.-D. and Finke, A. (1988b) Rheologische Untersuchungen von Kakaobutter und ihren Dispersionen mit Kakaofeststoff und Zucker, Teil 2. Zeitschrift Süsswarenwirtschaft 11. Tscheuschner, H.-D. and Winkler, Th. (1997) Alternative Verfahrensführung des conchirprozesses. LTV (Quakenbrück) 42 (2): 72. Tscheuschner, H.-D., Schebiella, G. and Foerster, H. (1981a) Continuous intensive conching of chocolate masses with the Konticonche Type 420. Kakao Zucker 33 (5): 122, 124, 126. Tscheuschner, H.D., Suess, H.W., Schebiella, G., Foerster, H. and Herricht, R. (1981b) Drastic rationalization of the conching process by means of the microelectronically controlled Konticonche Type 420. Lebensmittelindustrie 28 (12): 546–548. Tscheuschner, H.-D., Franke, K. and Hubald, B. (1992) Einfluss von Prozessbedingungen des conchierens auf die Griessbildung von Schokoladen-massen. Zucker Süsswarenwirtschaft 45 (11): 428–433. Tscheuschner, H.-D., Franke, K. and Hunbald, B. (1993) Milchpulver und Lactose. Einfluss stofflich bedingter Faktoren auf die Griessbildung von Schokoladenmassen beim Conchieren. Zeitschrift Süsswarenwirtschaft 12: 580–583. Ubezio, P. (1976) Manufacture of chocolate by the separate grinding process. Industrie Alimentari 15 (9): 137–140. Uhlemann, H. (1990) Kontinuierliche Wirbelschicht-Sprühgranulation. Chemie-Ing Techn 62: 822. Ulbricht, D., Normand, M.D. and Peleg, M. (1995) Creating typical jagged force-deformation relationships from the irregular and irreducible compression data of crunchy foods. J Sci Food Agric 67: 453–459. Ulfik, B. (1991) Zerkleinerungsmaschinen. Chemie-Ing Techn 63: 1054. Uriev, N.B. and Ladyzhinsky, I.Y. (1996) Fractal models in rheology of colloid gels. Colloids Surfaces A: Physicochem Eng Aspects 108: 1–11. Vanhoutte, B. (2002) Milk Fat Crystallization: Fractionation and Texturisation. PhD Thesis. Ghent University, Ghent. Várhegyi, G. (2007) Aims and methods in non-siothermal reaction kinetics. J Anal Appl Pyrolysis 79: 278. Vauck, W.R.A. and Müller, H.A. (1994) Grundoperationen chemischer Verfahrens-technik. Deztscher Verlag für Grundstoffindustrie, Leipzig, p. 489. VDI-Wärmeatlas (1991). VDE-Verlag, Düsseldorf (enlarged editions frequently published). Vavrinecz, G. (1955a) Contributions to the physical chemistry of saccharose, I. Solubility of saccharose to pure water from 0°C to 100°C (in Hungarian). Élelmezési ipar 9: 270–275.

References

665

Vavrinecz, G. (1955b) Contributions to the physical chemistry of saccharose, II. Phase-diagram of the saccharose–water system (in Hungarian). Élelmezési ipar 9: 300–304. Vereecken, J., Foubert, I., Smith, K.W. and Dewettinck, K. (2007) Relationship between crystallization behaviour, microstructure and macroscopic properties in trans containing and trans free filling fats and fillings. J Agric Food Chem 55: 7793–7801. Verhás, J. (1985) Termodinamika és reológia (in Hungarian). Mu˝szaki Könyvkiadó, Budapest, p. 55–67. Verordening van 8 maart (1979) Vb. Bo.van 14 mei 1979, afl.18, nr.G.Z.P. 2, van het Produktschap voor Granen, Zaden en Peulvruchten, Warenwet, P.O.B.-voorschriften (E-14 b), The Netherlands. Vitali, A.A. and Rao,M.A. (1984) Flow properties of low pulp concentrated orange juice: effect of temperature and concentration. J Food Sci 49: 882–888. van Vliet, T. (1999) Factors determining small deformation behaviour of gels. In: Dickinson, J.M. and Patino, R. (eds) Food Emulsions and Foams. Interfaces, Interactions and Stability, Special Publication No. 227. Royal Society of Chemistry, Cambridge, p. 3–13. van Vliet, T., Lucey, J.A., Grolle, K. and Walstra, P. (1997) Rearrangement in acid-induced casein gels during and after gel formation. In: Dickinson, E. and Bergenstahl, B. (eds) Food Colloids, Proteins, Lipids and Polysaccharides. Royal Society of Chemistry, Cambridge, p. 335. Viskanta, R., Bianchi, M.V.A., Crister, J.K. and Gao, D. (1997) Solidification processes of solutions. Cryobiology 34: 348. Volarovich, M.P. and Nyikiforova, V.N. (1968) Structure-mechanical properties of candy mass and its effect on the distribution of candy filling (in Russian) Khlebopekarnaya i Konditerskaya Promyshlennost’ 5: 16–18. Volmer, M. (1939) Kinetik der Phasenbildung. Steinkopff, Dresden. Volt, H. and Mershmann, A. (1985) Allgemeingültige Aussage zut Mindest-Rührdrehzahl beim Suspendieren. Chemie-Ing Techn 57: 692. Völker, H.H. (1968) Gordian 10: 444–447. Vreeker, R., Hoekstra, L.L., Boer, D.C. and Agterof, W.G.M. (1992) The fractal nature of fat crystal networks. Collids Surf 65: 185–189. Vu, T.O., Galet, L., Fages, J. and Oulahna, D. (2003) Improving the dispersion kinetics of a cocoa powder by size enlargement. Powder Techn 130 (1–3): 400–405. Walters, K. (1975) Rheometry. Chapman and Hall, London. Wan, L.S.C. and Lim, K.S. (1990) Sci Pract Pharm 6 (8): 567–573. Wasan, T. (1970) Parametric Estimation. McGraw-Hill, New York, NY. Washburn, E.W. (1921) Penetration of liquids into capillaries. Phys Rev Ser 17 (2): 273–283. Watt, B.K. and Merrill, A.L. (1975) Composition of foods. In: Agriculture Handbook No.8 .US Department of Agriculture, Washington, DC. Weber, M. (1993) Anwendung des Dispersionsmodells auf Mischvorgänge in Wirbelschichten und turbulenten Rohrströmungen. Chemie-Ing Tech 65: 1485. Weidenbaum, S.S. (1958) Mixing of solids. Adv Chem Eng 2: 209–324. Weit, H. and Schwedes, J. (1986) Massstabsvergrösserung von Rührwerkskugelmühlen. Chemie-Ing Techn 58: 818. Weit, H. and Schwedes, J. (1988) Messstabvergrösserung bezüglich der Leistungs-aufnahme bei Rührwerkskugelmühlen. Chemie-Ing Techn 60: 318. Weitz, D.A. and Oliveira, M. (1984) Fractal structures formed by kinetic aggregation of aqueous gold colloids. Phys Rev Lett 52: 1433–1436. Wey, J.S. and Estrin, J. (1973) The growth and nucleation of ice in a batch Couette flow crystallizer. Ind Eng Chem Process Des Develop 18: 421. Wiedemann, W. and Strobel, E. (1987) Verfahrenstechnische und Wirtschaftliche Vorteile der Kochextrusion im Vergleich zu herkömmlichen Verfahren der Lebensmittel-Industrie. ChemieIng Techn 59: 35. Wille, L.R. and Lutton, S.E. (1966) Polymorphism of cocoa butter. J Am Oil Chem Soc 43: 491–496.

666

Confectionery and Chocolate Engineering: Principles and Applications

Williams, C.T. (1964) Chocolate and Confectionery. Leonard Hill, London. Wilms, H. (1993) Verfahrenstechnische und statische Aspekte der Siloaus-legung. Chemie-Ing Techn 65: 284. Windhab, E. (1993) Allegemeine rheologische Messprinzipe, Geräte und Methoden. Ausgewählte Beispiele der Ingenieurtechnischen Anwendung der Rheologie. In: Weipert, D., Tscheuschner, H.D. and Windhab, E. (eds) Rheologie der Lebensmittel. Behr’s Verlag, Hamburg, p. 173– 256. Winkler, T. and Tscheuschner, H.-D. (1998) Feuchte der Kakaofeststoffe fördert Agglomeration. Zeitschrift Süsswarenwirtschaft 3: 119. Winslow, W.M. (1947) US Patent Specification 2417850. Woodward, S.A. (1984) Texture and Microstructure of Heat-Formed Egg White, Egg Yolk, and Whole Egg Gels. PhD Dissertation. University of Missouri, Columbia, MI. Woodward, S.A. and Cotterill, O.J. (1986a) Texture and microstructure of heat-formed egg white gels. J Food Sci 51: 333. Woodward, S.A. and Cotterill, O.J. (1986b) Texture and microstructure of heat-formed whole egg gels. In: IFT86 Program and Abstracts. Institute of Food Technologists, Chicago, IL, p. 159. Woodward, S.A. and Cotterill, O.J. (1987a) Texture and microstructure of cooked whole egg yolks and heat-formed gels of stirred egg yolk. J Food Sci 52: 63. Woodward, S.A. and Cotterill, O.J. (1987b) Texture profile analysis, expressed serum, and microstructure of heat-formed egg yolk gels. J Food Sci 52: 68. Wright, A.J., Narine, S.S. and Marangoni, A.G. (2001) Comparison of experimental techniques used in lipid crystallization studies. In: Widlak, N., Hartel, R. and Narine, S. (eds) Crystallization and Solidifaction Properties of Lipids. AOCS Press, Illinois, p. 120. Wunderlich, B. (1990) Thermal Analysis. Academic Press, New York, NY, p.137–140. Wurster, D.E. (1960) Preparation of compressed tablet granulations by the the air-suspension technique II. J Pharm Sci 49: 82–84. Xanthopoulos, C. and Stamatoudis, M. (1988) Effect of impeller and vessel size on impeller power number in closed vessels for Reynolds numbers above 75000. Chemie-Ing Techn 60: 560. Yarranton, H.W. and Masliyah, J.H. (1997) Numerical simulation of Ostwald ripening in emulsions. J Colloid Interface Sci 196: 157–169. Yeung, C.C. and Hersey, J.A. (1979) Ordered powder mixing of coarse and fine particulate systems. Powder Technol 22: 127–131. Yoo, B., Rao, M.A. and Steffe, J.F. (1995) Yield stress of food dispersions with the vane method at controlled shear rate ands shear stress. J Texture Stud 26: 1–10. Yüce, S. and Schlegel, D. (1990) Wärmeübergang an nicht-Newtonische Substanzen in statischen Mischern. Chemie-Ing Techn 62: 418. Yudyin, Ye.M. (1964) Shestyerennye nasosi (Cog-Wheelers). Mashino-stroyenniye, Moscow. Yuxian, A. (1998) J Polym Sci 36 B: 1305 Zehner, P. (1986) Fluiddynamik beim Suspendieren im Rührbehälter. Chemie-Ing Techn 58: 830. Zeichner, G.R. and Schowalter, W.R. (1979) Effects of hydrodynamics and colloidal forces on the coagulation of dispersions. J Colloid Interface Sci 71: 237. Ziegleder G. (1988) Kristallisation von Kakobutter unter statischen und dynamischen Bedingungen (DSC, Rheometrie). Süsswaren 32 (12): 287–293. Ziegleder, G. (2004) Conching. Advanced chocolate technology. Bühler Seminar, Uzwill, Sept. Ziegleder, G., Balimann, G., Mikle, H. and Zaki, H. (2003) New knowledge about conching. Part III: Sensory analysis and conclusion. Süsswaren 48 (5): 14–16. Ziegler, G. and Aguilar, C. (1995) Research probes conching in a continuous processor. Candy Industry 160 (5): 36–41. Ziegler, G.R. and Aguilar, C.A. (2003) Residence time distribution in a co-rotating, twin-screw continuous mixer by the step change method. J Food Eng 59 (2–3): 161. Ziegler, H.F. Jr., Seeley, R.D. and Holland, R.L. (1971) Frozen egg mixture. US Patent 3 565 638.

References

667

Zielinski, M., Niediek, E. A. and Sommer, K. (1974) Effect of kneading and stirring on flow behaviour of highly concentrated suspensions. 1. Mixtures of cocoa butter and a solid component. Gordian 74 (4): 135–138, 141. Zimon, A.D. (1969) Adhesion of Dust and Powders. Plenum Press, New York, NY. Zipperer, P. (1924) Die Schokoladefabrikation, 4. Auflage, Verlag M. Krayn, Berlin. Ziru, H. (1997) A kinetics study of crystallization from discotic mesophases. Liquid Crystals 23 (3): 317. Zlokarnik, M. (1985) Modellübertragung bei partiellen Aehnlichkeit. Chemie-Ing Techn 57: 420. Zlokarnik, M. (1991) Dimensional Analysis and Scale-up in Chemical Engineering. Springer Verlag, Berlin. Zlokarnik, M. (2006) Scale-up in chemical engineering. Wiley-VCH, Weinheim. Zogg, M. (1993) Einführung in die Mechanische Verfahrenstechnik, 2. Aufl. B.G. Teubner, Stuttgart. Zorli, U. (1988) Hafttechnik – Grundlagen und Anwendungen. Chemie-Ing Techn 60: 162. Zwietering, T.N. (1958) Suspending of solid particles in liquid by agitator. Chem Eng Sci 8: 244. Zwietering, M.H., Jongenburger, I., Rombouts, F.M. and Van’t Riet, K. (1990) Modelling of the bacterial growth curve. Appl Environ Microbiol 56: 1875.

Index

ability of sucrose to crystallize, 36 acid-hydrolyzed, 414 activation energy of diffusion, 124 additive property, 56 adsorption, 190 aerated fillings, 380 after-crystallization, 368 agar, 395–9 ageing, 351 agglomeration, 469, 470, 481 by baking, 481 from liquid phase, 481 in the confectionery industry, 481 of powders, 482 particle, 469 aggregation, 470, 481 aggregation, wet, 471, 474, 485 aggregative growth of sphelurites, 329 agitated ball mill, power demand of, 257 air flow in a tube, 438 air flow, power demand of, 441 air friction coefficient, 438 algebraic structure (H; 䊝; 䊟), 615, 617 algebraic structure of chemical changes, 615 alginate gel, diffusion setting of an, 401 alginate gel, internal (or bulk) setting of an, 401 alginate gel, preparation of, 401 alginate solutions, viscosity of, 400 alginates, 400–2 alginates, shear-reversible, 402 alkalization (often called ‘preparation’), 503, 505, 506 alternative motion of the rollers, 511 Confectionery and Chocolate Engineering: Principles and Applications © 2010 Ferenc Á. Mohos. ISBN: 978-1-405-19470-9

Alveograph, 172 Amadori compounds/rearrangement/ transformation, 502, 503 amino acids, 502 amorphous sugars, 93, 94, 510, 511 amphoteric properties of gelatin, 418 analogy between the rheological models and the electrical networks, 139 analysis dilatometry data using the Avrami equation, 338 Andreasen’s pipette method, 216 angle of pulling-in, 248, 254 angle of repose, 53, 91 anticaking agents, 53, 87, 88, 95 Antoine’s rule, 297, 298 approximate ERH for confectionery products, according to Minifie, 538 aqueous solutions of carbohydrates, solution of, 279 Archimedes (or buoyancy) force, 431 Ardichvili formula, 467 aroma development, 512, 516 arranging relation, 611 Arrhenius equation, 73, 331, 498 arterial and veins of a kidney, 606 asymmetric TAGs, 370 attenuation factor, 77 attractive interparticle forces, 55, 200, 479 attributes of a system, 609 attributes of machinery, 609 attributes of technological changes, 609 attritube combination, 610 average chain-terminal distance, 183 Avrami exponent, 337 axial diffusion coefficient, 517 Ferenc Á. Mohos

Index

bacterial growth, 542 Bagley method/plot, 437, 452, 453 baking powder, 223, 432 Balshin equation, 491 Bancroft’s Rule, 209 barometer formula, 192, 216 batch conching, 511 batch crystallization, 349 batch process, 614 Batel formula, 472, Baumé degree, 281 Becker–Döring–Nyvlt equation, 318 Benbow–Bridgewater method, 465 Bertrand method, 498 BFMIRA process, 532 biaxial extension, 131 binder solution (for granulation), 485 binding mechanisms, 470 Bingham fluids, 115, 120, 122, 156, 157, 163, 435, 436, 582, 583, 584, 598, 602 binomial, 13 biscuit doughs, 174, 575 Bloom grade/strength, 418 bloom inhibition, 370 blowing up, 552 boiling point, 65 boiling point of aqueous glucose syrup solutions at atmospheric pressure, 563 boiling point of sucrose/water solutions, 291, 561 Boltzmann distribution, 192, 216, 221 Bond’s (the ‘third’) theory, 241 bonding (or cold welding), 486, 487 bound energy, 63 boundary layer, 190, 322 boundary layer, formation of, 190 box-counting dimension, 606 Brabender extensigraph/farinograph, 171 bread dough, 167 briquetting, 486, 510 Brix degree, 280 Brownian coagulation, 217 Brownian flocculation, 214 browning phenomena, 550 Brucheigenschaft, 366 Brunauer–Emmett–Teller (BET) isotherm, 534, 535

669

bubble growth, 127 bubble rise, velocity of, 426 bubbling of aqueous solutions, 198 Buckingham equation, 436 Buckingham number, 435, 582, 584, 594, 602, 603 Buckingham π-theorem, 3, 17 Buckingham–Reiner equation, 582, 583 bulk compressibility, 107 bulk density, 52, 55, 87 bulk modulus, 106 bulk powder, 87 buoyancy (or Archimedes) force, 431 Burgers model, 138, 139 butterscotch, 502, 503 cake, 424, 570, 571 Cakebread equation, 539, 540 caking, 53, 87, 92 calculation of boiling point of carbohydrates/water solutions at decreased pressures, 291 ERH of confectionery products, 538–41 flow rate and decomposition, 594 flow rate of a generalized Casson fluid, 601 flow rate of rheological models, 582 friction coefficient of non-Newtonian fluids in laminar region, 596 Reynolds number for mixing, 266 specific heat capacity of cocoa nib and sweets, 60, specific heat capacity of milk and dark chocolate, 59 specific heat of orange marmelade, almond paste, 60 calendering, 127, 128 candy-cooking machine, 304 capacitance heating, 76 capillary forces, 471, 472, 473 capillary effect, 533 capillary elevation (rise) or depression, 185, 192, 194 capillary rheometer, 167 Capriste–Lozano ‘boiling point rise’ (BPR) model, 306 caramel; caramelization, 495, 502, 503, 504, 565

670

Index

carbohydrate solutions, parameters characterizing, 280 carbohydrates, 557 Carle & Montanari system, 532 Carman’s air penetration method, 482 carrageenen, 402–6 Cartesian product of phases, 27 Casimir–Polder theory, 474 Casson body/equation/fluid/ model (n = ½), 121, 157, 158, 435, 436, 585, 597, 598 Casson number, 585 Casson type (or Heinz) model, n = 2/3 602 Casson type model, n = 4/5 , 602 Casson viscosity, 157, 158, 435, 585, 597 Casson yield stress, 157, 158, 585, 597 catalytic ability of various acids in version, according to Ostwald, 496 Cauchy number, 585 Cauchy strain, 97, 100 Cauchy’s equation of motion, 110 cell fragments, 4 cells of the nibs, 506 cellular fragments, 4 cellular structure, 3, 20 cellular substances, 12 centre (‘corpus’), 484 chair shape of triacylglycerols TAGs, 357 chamber system, 531 change of consistency, 533 changes in pipe direction, 437 chelating agent, 400 chemical properties of cocoa butter, 568 chemical activity, 63 chemical length exponent (or tortuosity), 608 chemical operations, 495 chemical potential, 57, 312, 313 chemical reactions, 66 chemical variables, 554 chewing and bubble gums, manufacture of, 626 choco crumb, 502 chococrumb, 31 chocolate, 29, 156, 567–71 production, 618,

manufacture of dragee, 625 spreads, 380 choice of ventilator, 443 cis-trans isomers, 357 classification of confectionery products according to their characteristic phase conditions, 19 food powders, 53, 86 gels, 97, 153 rheological behavior, 113 sensory quality, 109 Clausius–Clapeyron equation, 529 closing inventory matrix, 50 cluster-cluster aggregation, 217 coacervates, 184 coagulation, 202 coalescence, 212, 219 coarse disperse systems, 27, 178 coating, 112 coatings, 112, 374, 376, 381 cocoa liquor, production of (traditional method), 617 cocoa beans cocoa beans, 569 cocoa butter, 567, 568, equivalent (CBE) fats, 365, 372, 373 improver (CBI) fats, 372, replacer (CBR) fats, 366, 376, 378 substitute (CBS) fats, 366, 376, 378 cocoa cells, 20 cocoa liquor (or mass), 448, 567 cocoa nib (cotyledon), 506 cocoa powder, 448, 571 cocoa powder and cocoa butter, production of (alkalization of cocoa liquor), 619 cocoa powder and cocoa butter, production of (Dutch Process), 619 cocoa powder, outer colour of, 371 cocoa press cake, 448 coconut oil, saponification of, 542 coefficient of compressibility, 58 coefficient of stability, 229 coefficient of thermal expansion, 58 Cogswell–Gibson method, 452, 453 cohesive systems, 180, 181 cohesiveness of a powder, 90

Index

Colburn (or -Chilton) analogy, 3, 15, 75, 286, 324 cold welding (or bonding), 486 cold-water swelling methods (for gelatin), 420 collagen, 416 collapse temperature, 94 colloidal state, 19, 27, 177, 179, 188 colloids in the confectionery industry, 177 combination of a disc mill, agitated ball and beater blade mill, 531 comminution, energy requirement for, 241, 242 comminution, separate, 531 comminution, 237–61 compact density (after compression), 55 compaction, 90, 486, 488 comparison of Brix and Baumé concentrations of aqueous sucrose solutions at 20°C, 579 compass setting, 606 compatibility with cocoa butter, 376, 378 complex conductivity, 82 complex dielectric constant, 78 complex modulus, 143 complex operations, 495, 510 complex rheological models for describing food systems, 166 complexity of operations used in the confectionery industry, 510 compliance, 107 composition of biscuits, crackers, wafers, 19, 43 chocolates and compounds, 19 confectionery products containing flour, 43 gelatine, 416, 417 starch, 413 sugar confectionery, 19 sugar confectionery, 35 compounds, 376 compressibility, 53, 89 compressibility of doughs, 173 compression, 482–92 compression, change of density under the effect of, 491 compressive stress, 98

671

concentration of saturated sugar/water solutions as a function of temperature, 559 concept of mould-free shelf life (MFSL), 542 conche pot (Langreiber in German), 511, 531 conching, 26, 503, 510, 617 conching degree (CD), according to Ziegleder, 520 conching time, 569, conching, dry, 511 condensation, 64, condensed milk, 31, 578 condition for flocculation, 202 conduction, 7 conductors, 82 cone-and-plate-rheometer, 166 confectionery foams, fields of application, 425 configurational entropy, 202 conservation and balance, equation of, 612 conservative attribute class, 613 conserved substantial fragments (CFS), 20, 511, 533, 616 consistency of candies, 37 consistency of dough, 172 consistency of the product, 36 consistency variables, 117 consolidation, 486 constitutive equations, 97, 104 contact angle, 195 contact nucleation, 319 containing relation, 5, 19 continuous industrial aeration, 430 continuous phase phase transitions, 64 continuous process, 615 continuous sucrose crust, 507 continuous water loss from the product, 527 contraction, 129 convection, 7 converging flow, 126 conversion between the various temperature scales, 303, 304 conversion of sucrose, 36 cooling effect, 65

672

Index

cooling/solidification curve, 351 Coulomb’s law, 477 Coulter technique, 245 counting method, 607 cow’s milk, 578 cracker dough (laminated), 575 creaming, 212 creep, 137, 138, 139 critical aggregate radius, 218 critical cluster size, 315 critical micelle-forming concentration (cM), 188 critical volume fraction (phase inversion) Φc, 221 critical water activity of the food, 543 croquante, manufacture of, 624 cross-section, reduced, 440 crushing, 238 crusted liqueur bonbons/pralines, 507 crystal chocolate, 28 growth, 319, 326 modifications of cocoa butter, 360, 361 size distribution (CSD), 346 thickening, 330 crystalline drops, 38 crystalline sucrose, 94, 578 crystalline vanillin, 578 crystallization, 65, 309–92, 470, 481, 507 driving force for, 311, 312 during storage of chocolate products, 368 dynamics of non-tempering fats, 374 from melts, 130, 329 from solution, 310 non-isothermal, 337, 345, 350 of cocoa butter, 356, 359 of confectionery fats with a high trans-fat portion, 382 of fat masses, 371 of glycerol esters, 355 of sucrose, 36 secondary, 346 stimulated by wet and heat, 41 cyclohexanolones, 504 cyclopentanolones, 504 cylindrical tube, 583

Damköhler number, first, 10 Damköhler number, third, 10 Damköhler equations, 3, 6–8, 614 Danckwerts one-dimensional dispersion model, 517 data on engineering properties of materials used and made by the confectionery industry, 557 Deborah number, 108 Debye region, 178 Debye–Hückel theory, 200 decomposition method, 435, 582–603 definition of a structure, 611 definitive water activity of various salt at saturation, 531 deformation, 109, 178, 487 deformed systems (Buzágh), 181 degree of amidation (DA), 411 degree of efficiency, 583 degree of esterification, 410 degree of freedom, 3, 4, 11 degree of heterogeneity of a mixture, 267 deliberation of latent heat – negative enthalpy, 40 dendritic growth pattern, 606 density, 52, 53, 567 and dynamic viscosity of biscuit dough as a function of temperature, 575, and dynamic viscosity of wafer batter as a function of water content at 20°C, 577 and thermophysical properties of wafer batter and of biscuit and cracker doughs, 575 of sugar confectionery products, 572 of cocoa liquor, 449 of foam, 431 of saturated sugar solutions as a function of temperature, 558 dependence of dynamic viscosity on temperature and pressure, 124 dependence of growth rate on the hydrodynamic conditions, 324 dependence of shear rate on the n exponent in case of a generalized Casson fluid, 600 depolymerization of carrageenan, 404 depot fat, 4

Index

design of converging die, 456 dessert fillings/masses, rheological model of, 164, 165 dessert masses containing fondant, 164 destabilization (mechanisms of), 212 destabilization action of neutral salts, 203 deterioration, 541 deviations from the ΔL law: sizeproportionate growth of crystals, 321 dew point device, 527 dew point temperature, 367 dextrose equivalent (DE), 35, 284 dextrose/water solutions, 292, dielectric constant (permittivity), 567 dielectric constant, 76 dielectric constant of cocoa butter, 79 dielectric displacement vector, 76 dielectric loss factor, 77 dielectric properties, 52, 76 dielectric properties of water, 79 difference matrix, 50 differential canning calorimetry (DSC), 354 differential thermal analysis (DTA), 354 diffusion layer model, 322, 508 diffusion, temperature dependence of, 52, 73 diffusional transport, 220 dihydrofuranoses, 504 dilatancy, 116 dilatation, 353, 508 dimensional analysis, 3, 16 dimensioning of extruders, 451 dimensionless numbers, 3, 8, 583 direc tabletting/compression, 485, 486 disperse (or non-cohesive) systems, 180 dispersion, 52, 56, 178 dispersity, 224 disproportionation, 223, 225 dissolution, rate of, 286 distance of rubbing, 249, 250 distilled monoglycerides (MGs) (E 471), 211 distribution functions, 207 distribution of pore sizes, 195 DLVO theory, 200 Dobos fancy cake (‘Dobos tart’), 504 double-chain-length (DCL), 357, 374

673

dough processing, 126 doughs without yeast, 174 drag on particles of spherical shape, 73 dragee, manufacture (cold/warm method) of, 625 drainage, 220, 229 drawing matrix, 50 driving force for liquid penetration, 185 droplet coalescence, kinetics of, 149 drops, 571, 572, 620 dry sugar mass, 508 drying stage modelling, 307 drying, velocity constant, 29, 514 DSC curve of a well-tempered cocoa powder, 371 Dupré (or Young-) equation, 470 duration of pressing, 446 dust explosion, 89 dust formation, 53, 88 Dutch process: alkalization of nibs, 506 Dühring’s rule, 302, 303 dynamic viscosity, 8, 115, 143, 167, of a generalized Casson fluid with exponent n, 602 aqueous solutions of invert sugar as a function of temperature, 566 cocoa butter as a function of temperature, 568 hard-boiled sugar masses with different molasses content as a function of temperature, 574 honey as a function of temperature and water content, 565 honey-cake dough as a function of water content, 577 honey-cake dough as a function of time of kneading at 30°C, 577 saturated dextrose–water solutions, 565 sucrose-water solutions, 564 sweet biscuit dough as a function of time of kneading, 576 various fondant masses as a function of temperature, 572 wafer batter as a function of temperature, 578 dynamic yield stress, 121

674

Index

effect of heat on amorphous polymers, 187 impeller speed on heat and mass transfer, 275 invertase on the reducing sugar content of fondant products, 500 lecithine on the viscosity of chocolate, 161 solids content and cocoa butter on the viscosity of chocolate, 158 temperature on the growth rate, 323 temperature on the viscosity of chocolate, 162 effective distance of a mixer, 274 effectiveness of mixing, 268 egg, 421–5 Eilers–Maron equation, 159 Einstein equation of diffusion, 214 Einstein equation of viscosity, 73, 145, 159 elastic body (or Hookean body/model), 97, 106 elastic constant (shear modulus), 152 elastoplastic material, 108 elecrical capacitor, 140 electric double-layer, 198, 199 electric field vector, 76 electric properties of the interfaces, 198 electrical conductivity, 53, 81 electrical conductivity of materials used in confectionery, 83 electrical conductors, 480 electrical resistance, 140 electrical sensing zone method of particle size distribution determination, 245 Coulter principle, 245 electrode potential, 200 electrodynamics, 7 electrokinetic (or zeta) potential, 200 electrokinetic phenomena, 198 electrolytes, 83 electron charge, 83 electro-osmosis, 199 electrophoresis, 199 electrorheology, 97, 144 electrostatic double-layer forces, 219 electrostatic forces, 472, 477 elevation of boiling point, 65, 529

elevation of the boiling point (molar), 289, 290 Ellis model, 591 elongational (or extensional) viscosity, 126, 167 EMA plant, 532 emulsification, kinetics of, 192, 207, 209 emulsifiers used in chocolate, 161 emulsifiers used in the confectionery industry, 211 emulsifying, 395, 533 emulsifying activity index (EAI), 207 emulsifying capacity (EC), 207 emulsifying properties of food proteins, 207 emulsion droplets size data, 207 emulsion, 203–21 enantriotropy, 355 endothermic process, 63 energy conditions of colloid formation, 181 engineering properties of foods, 52 engineering strain, 100 enthalpy function, 57, 58, 63, 196 enthalpy of the candy solution, 306 enthalpy of the saturated vapour, 306 entropy, 63, 196 environmental pollution, 48 enzymatic activity, 542 enzymatic inversion, 499, enzyme-hydrolysed starch, 415 equation of mixture, 279 equilibrium relative humidity (ERH), 526, 532, 537, 538 equivalent diameter, 9 Espenscheid–Kerker distribution, 207 essential fatty acids of cocoa butter, 329 Euler number, 11, 265, 463 European Union Directive 2000/36/EC relating to cocoa and chocolate products, 373 eutectic effects, 376 evaluation of efficiency, 49 evaporation, 64, 289, 513 existence of stability, 551 exothermic process, 63 exothermic reactions of carbohydrates during pyrolysis, 66

Index

expansion ratio (or overrun), 224, 429 expansion ratio, 224 explosibility index, 53, 88 exponent n of the flow curve of a generalized Casson fluid, determination of the, 598 extensional (or elongational) flow, 97, 126, 127, 167, 451, 454 extensional deformation of incompressible bodies, 452 external technological structure, 615 extruders, 126, 127, 451, 464 extruders, screw, 461, 464 Eyring’s theory, 124 falling film (thickness of a), 121 Fanning number, 11 Faraday’s constant, 83 Faraday’s law, 53, 82 farinograph, 171 fat bloom, 363, 369, 375 fat crystal network, 330, 607 fat migration, 369 fat-free cocoa content, 31 fats for ice cream, 377, 381 feeders, cog-wheel, 460 feeders, screw, 459 feeding by roller extrusion, 467 Fehling method, 498 fermentation by yeast, 223, 533 ferromagnetic transition, 64 Fick equation, second, 69 Fick’s 2nd law, 7 Fick’s law, 225 Fick’s law of mass diffusion, 71 Fick’s first law, 71, 72, 323 filler (for granulation), 485 filling fats, 377, 379, 383 fillings, 380, 383 fillings, soft, 380 film draining, 219, 230, 231 film-evaporator (e.g. Petzomat from Petzholdt), 30 first normal stress coefficient, 112 first-order phase transitions, 63 flavonoids, 505, 506 flavorings, 578

675

flocculated emulsions, rheological properties of, 148 flocculation, 212, 213, 214 Flory–Fox equation, 184 flow of non-Newtonian fluids, 434 flow conditioners, 88, 95 flow dynamics in a capillary, 185 flow properties of transported powders, 439 flow properties, change, 512 flow rate of non-Newtonian fluids, 434 flow through a converging die, 451 flowability, 53, 92 flui model of human blood, 601 fluid behavior, 97, 109 fluid bridges, 471 fluid models for describing the flow properties of chocolate mass, 156 fluid models, 582 fluidization, 482, 484 fluids, ideal, 97, 109, 583 flux of a component, 7 flux of heat, 7 flux of mass, 6, 7 flux of momentum, 8 foam, 222,395, 425–33 drainage, 230, 231, 424 films, thinning of, 230 formation in the confectionery industry, 425 stability, 229 syneresis, 231 foaming methods, industrial, 432 foaming of polycarbohydrates and/or proteins, 533 fondant, 163, 381, 461, 502, 503, 536, 572, 573 centres, production of by candy layer convering (‘candis layer’, traditional), 622 centres, production of by Mogul, and covering by chocolate, 621 in cherry liqueur pralines, dissolution of, 508 ripening, 507 food stability, 550 food powders, classification of, 53, 86

676

Index

food powders, physical characteristics of, 53 forced crystallization, 508 Foubert model, 341, Fourier equation, first, 52, 66 Fourier equation, second, 52, 69 Fourier’s 2nd law, 7 fractal dimension, 151,152, 217, 606, 608 fractal structure of gels, 97, 151 fractionation of triglycerides, 369 fragmentation of particles, 90 fragments of cells comminuted, 4 Franke model for modelling the cooling of chocolate coatings, 385 Franke–Tscheuschner model, 517–20 free energy (or Helmholtz), 57 free energy, 63 free enthalpy (Gibbs or free energy), 52, 57, 63, 189, 196, 202, 213, 316, 331, 333, 350, 358, 362 free water molecules, 525 free-flowing, 55, 95 freezing-point depression, 528, 529, 531 friction coefficient, 596, Froessling correlation, 275, 324 Froude number for mixing, 271, 272, 273 fructose, 35 fruit jelly, 572 fundamental functions of thermodynamics, 56 furcellaran, 407 fuzzy logic, 537 galactose, 35 Galor–Walso–Rulev equation for collective velocity of bubble rise, 428 gamma distribution, 207, 347 gas development, 533 Gates–Gaudin–Schumann (GGS) distribution, 247 Gaussian distribution, 246 Gay-Lussac–Ostwald step rule, 356 gel → sol transition, 118 gel strength, 397, 399 gel, holding time of, 398 gelatin, 416 –21 gelation of polycarbohydrates and/or proteins, 533

gelling, 395 gels, classification of, 97, 153 general precautions when hydrating gelatine in solution, 421 generalization of the Casson model, 597 generalized Casson fluid, 600, 601 generalized Newton models, 115 generalized Ostwald–de Waele model, 116 generating function of the distribution, 209 German Process: alkalization of cocoa liquor (cocoa mass), 506 Gianduja recipes, 33 Gibbs adsorption equation, 228 Gibbs elasticity, 232 Gibbs interfacial concentration, 191, 197 Gibbs phase rule, 3, 11 Gibbs–Duhem equation, 196 Gibbs–Marangoni effect, 232 Gibbs–Thomson equation, 350 Gibson method/equation, 131, 454, 466 glassy consistency, 188 glassy sucrose, 94 glidants, 95 global balance for evaporation stage, 305 glowing, 40 glucose dry content, 539 gluten skeleton, 44 glycerol, 39 G-matrix, 48 Gompertz model, 340, good gloss retention, 377 graham crackers, 464 grained products (e.g. fondant, drops), 38, 41, 94, 508 grained sweets, production of with cremor tartari, 621 granulation, 481, 482 granulation from liquid phase (wet granulation), 481 granulation, dry, 474, 482, 485 Grashof number, 16 gravity-induced flocculation, 215 Greenwald formula, 215, 216 grillage (or Krokant in German), 504 grinding, 238, 253 grinding, dry, 239

Index

grinding, kinetics of, 247 gross material consumption, 45 Grover equation, 539 growth of lamellae, 329 guaran gum, 408 guluronic acid, 401 gum arabic, 407 gum tragacanth, 408 Habbard formula, 158, 159 Hadamard–Rybczynski equation for velocity of bubble rise, 427 Hagen–Poiseuille equation, 185, 595, 596, 600 hair hygrometer, 527 halawa, 571 halving time of conching, 514 Hamaker constant (complex), 201, 213 Hamaker formula, 474 hard-boiled sugar mass, 573 hard-boiled sweets grained by humidity of filling, 508 hard-boiled sweets grained by pulling (forced crystallization), 508 hardness of various chocolate brands, 570 hard-sweet biscuit dough, 575 Harkins–Jura isotherm, 534, 535 Harrison–Leung equation for velocity of bubble rise, 428 hazelnut paste, 33 heat absorption, 63 heat of a phase transition, 63 heat of combustion, 66 heat of reaction, 52, 62, 66 heat transfer in mixing, 275 Heinz model, 161 Helmholtz-like double layer, 199 Hencky strain, 97, 100 Henderson isotherm, 534 Henry’s law, 225 Herschel–Bulkley–Porst–Markowitsch– Houwink (HBPMH) (or generalized Ostwald–deWaele) model, 592 Herschel–Bulkley–Porst–Markowitsch– Houwink (HBPMH) model, 116, 117 heterogeneous materials, 52, 67 heterogeneous measure, 612 heterogeneous nucleation, 332

677

heterogeneous relation, 611 hierarchical structure, 3, 5, 6, 19, 611 high gloss, 377 high tensions in the praline pieces, 508 hindered settling, 216 histograms, 347 HLB temperature, 211 HLB-value, 209, 210 Holmes–Voncken–Dekker formula, 277 homogeneity of equations, 16 homogeneous measure, 612 homogeneous relation, 611 homogeneous nucleation, 332 homogenization (kneading or solution), 533 honey, 565, 569, 571, 577, 578 Hooke body/element/model, 140, 165 Hooke’s law, 105, 134, 140 Hookean surface elasticity, 232 horizontal cocoa-pressing machine of 12 pots, 448 Huber–von Mises criterion, 123 humidity balance between cookies and fillings, 509 humidity caking, 93 hydrocolloids applied for confectioneries, 395 hydrodynamic friction coefficient of the particles, 440 hydrolysis, 550 hydrolysis of sucrose by the effect of acids, 495 hydrophilic or hydrofobic characteristics, 19 hydroxymethylfurfurol (HMF), 503, 504, 515, 516 hygroscopic materials, 537 hygroscopic properties of the surface, 36 hygroscopic property of fructose, 285 hygroscopicity of confectionery products, 535 hygroscopicity of sugar masses, 37 hysteresis, 405, 533 ice cream, 425 image of a mapping, 615, 616 impedance, 82 improving foam stability, 231

678

Index

in situ generation of foam, 432 infections during production, 542 infinite-order phase transitions, 64, infrared absorption properties, 53, 85 inherent colour of cocoa powder, 371 instability, 552 insulators, 82 intensifying the aeration, 511 intensity of shearing, 510 interactions between dissolved macromolecules, 183, 184 interfaces, 190, 191 interfacial tension of sugar solutions at 20°C, 565 internal energy (molar), 57 internal energy, 56 internal pores, 54 internal product of elements, 609 interparticle forces, 92 interparticle porosity, 55 interparticle voids, 90 intrinsic viscosity, 145 inversion, 35, 285, 495, 550 abilities of various acids, 37 by cream of tartar, 498 of sucrose by cream of tartar, 38 inversion point, 221 invert substances, 539 invert sugar, 35, 292, 566, 578 invertase, 35 irregular forms, 606 isobaric reversible work, 196 isoelectric point (IEP), 184, 417 iso-ionic point (pI) of the gelatine, 418 isotherm of cocoa powder at room temperature, 538 isotherm of sucrose, 536 isothermal crystallization, 334, 337, 350 isothermal preparation of a supersaturated solution, 314 isotropic compressible Newtonian fluids, 451 jellies, 395, 571 jellifying materials, 539 jelly, manufacture of (general scheme), 623 jet expansion, 132, 133

junction zones, 411 Junk–Pancoast formula, 282 just-suspended speed, 274 Kandisschicht (or sucrose crust), 536 Karlovits formula, 384 Kelvin (or Kelvin–Voigt) models, 135, 137, 138, 139, 140, 141 Kelvin equation, 533, 534 kernel of a mapping, 615, 616 Kick’s ‘volume’ theory, 240 kinetic and potential energy, 56 Kirchhoff’s laws, 67 kneading, 263 kneading of chocolate mass of high viscosity, 511 Kohlrausch’s rule, 83 KO-kneader, 462 Kolmogorov distribution, 246 Kolmogorov–Avrami heuristic phase transition theory, 334–40 Konticonche, 532 Krokant (or grillage), 504 Krupp formula, 475 kugelschaum or sphere form, 222 Kuhn isotherm, 535 La Mer–Healy bridging theory, 213 Labuza isotherm, 535 lactose crystallization, 511 Lambert–Beer law, 52, 77 Lamé parameters, 107, laminar flow, 9, 185, 438 lamination, 178 Langmuir isotherm, 197 Langreiber (or conche pot), 511, 531 Laplace equation, 185, 194, 208, 471 large deformations, 133 latent heat, 52, 62, 63, 65, 124 Lebkuchen (‘honey cakes’), manufacture of, 628 lecithin (E 322), 211, 511 lemma on the exponent in the generalized Casson equation, 603 length of coast of Britain, 606 Leonardo da Vinci, 537 less hygroscopic materials, 537 Levine model, 465

Index

Lewis number, 11, 70 lifetime histogram, 229 Lifschitz equation, 474 Lifschitz–van der Waals constant, 475 Lindt & Sprüngli Chocolate Process (LSCP), 532 linear-elastic materials, 97, 107 lipid oxidation, 541 liquid bridges, 93 liquids containing suspended particles, 68 locust bean gum, 409, log-normal (LN) distribution (Kolmogorov distribution), 246, 346 London–Heitler theory, 474 long spacing, 358 loops, 213 loose bulk density, 52, 55 Lorenz number, 82 low-resolution NMR methods, 354 lozenges, manufacture of, 626 lubricant selection for compaction, 488 lubrication theory, 230 Lyapunov stability/function/number, 550, 551 lysine, 503 lysozyme, 424 Machikhin–Birfeld method, 437 Machikhin–Machikhin method, 454–5, 459 machines for granulation, 486 macroabrasion, 318 macrobiological state of the raw materials, 542 macromolecular colloids, properties of, 182 Magnus–Tetens formula, 367–8 Maillard reaction, 162 Maillard reaction, 26, 31, 162, 381, 502, 512, 516, 551 margarine, 566, 578 marginal (of Lyapunov) stability, 541, 551, 552 Mark–Houwink equation, 145, 184, 404 marzipan, 571, manufacture (traditional) of, 625 mass diffusion, 52, 69, 74 mass flux, 5

679

mass fractal dimension, 152 mass ratio concentration, 280 mass transfer in mixing, 275 mastication, 134 material behaviour with respect to hardness (Tarján’s classification), 242 material matrix, 48 mathematical models for inelastic fluids, 115 Matijevic distribution, 207 Matsuhashi formula, 397 maximum attainable supersaturation, 314 Maxwell equation of thermal conductivity, 68 Maxwell equations of electrodynamics, 7, 76 Maxwell model, 134, 135, 136, 138,140, 141, 143, 144 Maxwell–Stefan formulation of phase transition, 387 McCabe’s ΔL law, 319, 320, 348 McCabe–Stevens relationship, 324 measurement of the water vapour pressure, 528 measurement of a pneumatic system, 442 mechanical analogues, 134 mechanical interlocking, 472, 487 mechanical strength, 53, 88 medium-rapid-set pectin, 410 melange, 569 melangeur, power requirement of friction in a, 256 melting, 64 melting of polymers, 333 melting of sugar, 505 melt-in-the-mouth characteristics, 417 membership function, 537 mercury porosimetry, 194 meringue, 223, 424 metastable (permanent) foams, 222 metastable state of a supersaturated solution, 313, 314 metastable zone, 313, 314, 317, 318 Meter model, 591 methods for hydrating gelatin in solution, 420 Metzner–Otto method, 112 micelles, 180, 189

680

Index

micelles, parameters influencing the structure of, 190 microbiological defects (e.g. mould), 527 microbiological variables, 554 microphases, 179 microscopic disperse systems, 178 microwave and radio frequency generators, 52, 76, 78 milk chocolate, 29, 502, 597 milk chocolate, production of (traditional method), 618 milk crumb, 31, 502 milk crumb or choco crumb, separate production of, 531 milk fat and CBR blends, 377 milky taste, 31 mitochondria, 4 mixed concentration, 280 mixed-phase regimes, 64 mixers, screw, 461 mixers, single-screw, 461 mixers, twin-screw, 462 mixing, 263–78 by blade mixers, 276 of fluids of high viscosity, 274 of powders, 267 of two liquids, 277 rate of, 270 rolls, 277 time for powders, 272 time characteristics of a stirrer, 266 Mixograph, 171 model for storage scheduling by fuzzy logic, 537 model of axial dispersion, 259 modelling of the structure of dough, 532 a continuous conching process with high shear rate, 517–20 an industrial cooking process for chewy candy, 304 chocolate cooling processes and tempering, 385 chocolate tempering process, 390 evaporation stage, 305 fondant manufacture based on the diffusion theory, 326–8 single- and twin-screw mixers, 463

temperature distributions in cooling chocolate moulds, 386 the structure of dough, 522 two-step isothermal crystallization, 342–5 modification of Raoult’s law, 291 modification of the crystal structure, 64 modified dynamic viscosity, 596 Mohr circles, 123 Mohs scale, 88 molar volume, 57 elevation of the boiling point, 289, 290 internal energy, 57 specific heat capacity at constant pressure, 57 specific heat capacity at constant volume, 57 molecular forces, 471 moments of the residence time density distribution, 260 momentum conduction, 8 momentum convection, 8 momentum flux, 8 momentum transfer coefficient, 14 Money–Born equation, 539, 540 monomial, 13 monomolecular water layer, 47 monotropy, 355, 362 Montelimar of sweet foams, manufacture of, 623 Montelimar, manufacture (continuous) of, 624 Moore’s equation, 462 morphology of surface, 191 mould, 527, 542 moulding of chocolate, 365 mouth as an adiabatic (closed) system, 65 MSMPR (mixed suspension, mixed product removal) crystallizer, 349 mud-like appearance of cocoa mass, 506 multilayer perception artificial neural network (ANN), 307 multiphase systems, 11 mutual mass diffusion, 52, 72 Müller formula, 273

Index

Navier–Stokes law, 8, 111, 451 near infrared reflectance/transmittance (NIR/NIT) investigations, 81 net material consumption, 45 NETZSCH process, 532 new trends in the manufacture of chocolate, 531 Newton element/model, 140, 165 Newton number, 264, 257, 273 Newton’s second law, 110 Newton’s generalized binomial theorem, 601 Newton’s first law, 215 Newton’s law of cooling, 7, 385 Newton’s law of viscosity, 134 Newtonian fluids, 109 Newtonian region, lower/middle/upper, 117 Newtonian surface flow, 232 Nielsen formula, 316 Nienow–Miles equation, 275 non-commutative characteristic, 611 non-distributive lattice, 615 non-enzymatic browning, 502, 542 non-hygroscopic material, 535, 539 non-linear elastic materials, 97, 107 non-molar quantities, 57 non-Newtonian fluid, 583, 596 normal distribution (Gaussian distribution; N distribution), 246, 346 normal stress coefficient, second, 112 normal-stress difference, 130 n-th moment of the frequency distribution, 208 nuclear magnetic resonance (NMR) spectroscopy, 354 nucleation , secondary, 311, 318, 329 nucleation, 310, 315 nucleation, true secondary, 318 Nukiyama–Tanasawa distribution, 207 Nusselt number, 11, 13, 275, 287 nut brittle (croquante) masses, 165 Nyvlt formula, 319 Nyvlt polythermal method, 315 O/W type emulsion, 27, 506 Ohm’s law, 53, 81, 140 ohmic heating technology, 53, 83

681

oil droplet size distribution, 207 Oldroyd model, 589 opening inventory matrix, 49 operator of conservation, 613 orbital stability, 550 orders of hygroscopicity and of solubility, 37 ordinary diffusion, 72 organelle, 5 oscillatory testing, 97, 134, 141 osmotic pressure, 201 Ostwald–de Waele fluids/model, 116, 162, 163, 165, 174, 226, 403, 435, 436, 453, 456, 459, 594, 595 Ostwald ripening, 29, 193, 212, 213, 220, 221, 223, 315, 321, 350, 507, 550 Ostwald–Freundlich relationship, 315 Ostwald–Thomson equation, 221 Oswin isotherm, 535 ovalbumin, 424 overheating, 40 overrun (or expansion ratio), 429 ovomucin, 424 oxidation, 513, 516 Oyama equation, 270 Ozawa equation/exponent, 345 packaging material, 527 palm kernel oil, saponification of, 542 panning machines, 486 parallel coupling of changes, 615, 617 particle agglomeration, 469 counting method, 607 density, 52, 54 rearrangement, 486 size, 242 size distribution of chocolate mass, 244, 245 size distribution of ground products, 242 partitioning, 538 Pascal bodies, 97, 109 Pascal’s law, 490 Peck model, 466 Peclet number, 9, 13, 259, 260 pectin, 409–12 apple/citrus, 410

682

Index

gel strength standardization of, 410 high methoxyl (HM), 410 low methoxyl (LM), 411 low methoxyl (LM) amidated, 411 Peek, McLean and Williamson model, 586 penetration depth, 77 penetrometer, 172 periodic shear rate, 142 periodic system á la Mendeleyev, 616 permittivity of the medium, 200 phase inversion (or HLB) temperature (PIT), 211 phase inversion of an emulsion, 147, 221 phase lag (or phase shift), 142 phase transition, 19, 28, 52, 63 phenomena when phases are in contact, 193 physical properties of cocoa butter, 568 physical parameters of air, 438 physical variables, 554 piston extruder, 464 Planck’s equation, 85 planning of material consumption, 19, 48 plastifying, 470 plastometer, 173 Plateau border, 225, 229, 230 plate-plate rheometer, 166 Poisson’s ratio, 106 polarization vector, 76 polarized light microscopy, 608 polycrystalline breading, 318 polyglycerol polyricinoleate (PGPR), 161, 212 polyglycerol esters (E, 472) polyhedral foams, stability of, 230 polyhedral gas cells, 222 polymer crystallization, 329 polymer gels, 218 polymorphism, 355, 509 polymorphism of cocoa butter, 356, 359 polymorphism of lactose, 20, 510 polynomials, 12 polysorbates (E 433), 212 polythermal preparation of a supersaturated solution, 314 population balance, 347, 348 porosity, 52, 55

porosity of chocolate determined by X-ray radiography, 364 potash solution, 506 potential barrier, 201 potential function, 57 powdered solids, 54 power absorption, 52, 77 power characteristic of a stirrer, 264 power-law fluid/model (or Ostwald–de Waele), 116, 466 practical test for controlling the boiling points of sucrose solutions, 303 praline, 163, 570, 571 praline, production of, 620 Prandtl number, 11, 52, 69, 70, 71, 75, 275, 560 Prandtl number of a chocolate mass at 50°C, 71 pre-conching of cocoa mass and cocoa butter by thin-layer evaporator, 531 pre-conching machine, 30 pressing, 445–9 pressing in the confectionery industry, 445 pressing, hydraulic, 448 pressure loss of in the shaping of pastes, 455 pressure-dependence of the flow curve of chocolate mass, 16 pretzel dough, 168 primary nucleation, 311, 329 principle of characterization, 19 product specification, 45 production matrix, 48 protection against coalescence, 205 pseudo-plastic behaviour, 116 pulling, 39, 65, 133, 508 pyrolysis of sucrose, 66 pyrones, 504 qualitative characterization of substances, 19 qualitative description, 610 quasi-chemical components, 616 Rabinowitsch, Eisenschitz, Steiger and Ory model, 588 Rabinowitsch–Mooney equation, 434, 594, 601

Index

radio frequency and microwave heating, 52, 76, 78 Rahat Lakoum recipe, 415 Ramsay’s formula, 193 Ramsay–Young rule, 301 Randolph–Larson equation, 349 Raoult concentration (or molality), 280, 289, 529, 530, 532, 540 Raoult’s law, 289, 290, 299, 528, 539 raw materials, semi-finished products and finished products, 567 Rayleigh region, 178 reactions of flavonoids in cocoa bean recipes, 19, 34, 38, 41, 42, 45, 617 recoil phenomenon, 133, 137 recoverable shear, 113, 137 recovery after compaction, 488 recrystallization of amorphous sugar, 94 recrystallization under the effect of temperature or concentration fluctuations, 351 reducing sugar content, 26, 35, 37, 284, 498, 502 refining, kneading and aeration are coupled together, 511 reflexive characteristic, 611 Reher’s formula, 265 Reiner model, 587 Reiner–Philippoff model, 587 relation between extensional and shear viscosities, 130 relative dielectric constants of some materials used in confectionery practice, 80, relative humidity of air (RH), 525, 526, 528, 535, 536, 537 relaxation functions, 136, 140, 141, 509 repeptization, 202 repulsive potential, 200 residual acid, 36 resultant potential, 201 retardation time, 137 reversion products, 497 Reynolds analogy, 3, 13, 14 Reynolds number, 8, 70, 264, 274, 275, 287, 324, 325, 427, 439, 440, 441, 463 rheological behavior, classification of, 113 rheological properties of gels, 97, 151

683

rheomalaxis (rheodestruction), 118 rheopexy, 118, 156 rheoretrogradation, 118 Riedel equation, 71 rigid (or Euclidean) body, 97, 105 rigidity coefficient, 397, 399 ripening, 495, 507, 509 (see also Ostwald) Rittinger’s ‘surface’ theory, 239 roasting, 66, 503 robust machinery of alkalization for kneading and evaporation, 506 rod-like agglomerates, 597 roller extrusion, 464, 465 roll-refining, 510 Rosin–Rammler distribution, 245, 247, 248 rotating motion, 511 rotoviscometer, 162, 174 Rumpf equation for the maximum distance between two charged particles, 478 Rumpf equation for the water content of a particle, 475 Rumpf equation for the tensile strength, 472 Rumpf equation for the tensile solidity of a granule, 476 Rumpf–Müller formula, 272 Ruth equation, 445, 446, 448 saddle surface, 471 sagging (gravity induced), 127 saltines, 464 saponification number of the ester, 210 satin bonbons of grained structure, 65 saturated fatty acids (SAFA), 384 saturated salt solution slurry, 531 Sauter mean diameter, 208, 209 scaling behaviour of the elastic properties of colloidal gels, 152 scaling up of agitated centrifugal mixers, 271 scattering, 85 Schmidt formula, 274, Schmidt number, 11, 52, 70, 71, 75, 266, 276, 287, 324, 326 Schoorl–Regenbogen method, 498 Schulze–Hardy rule, 203

684

Index

schwach pflug/weak blow ≈ 112,5°C, 507 screening, 243 second principal law of thermodynamics, 58 secondary nucleation, 318 sedimentation (creaming), 199, 212, 215, 216, 245 seeding with crystal dust, 318 segregation, 53, 89, 95 Seignette salt, 498, self-similarity, 151, 607 semiconductors, 82 semi-hierarchic structure, 3, 6 semi-permeable membranes, 3 semi-sweet biscuit dough, 575 semi-sweet biscuits and crackers, manufacture of, 627 sensory quality, classifications of, 109 sensory variables, 554 separation during mixing of powders, 270 sequestrants, 402 serial and parallel coupling of models, 140, 615, 617 shaping (moulding) and cooling of cocoa butter and chocolate, 365 shaping by extrusion, 470 shaping dough, 127 shaping of rope, 461, 452, 470 shear creep compliance, 134 shear loss modulus, 142 shear modulus, 105, 142 shear rate, 111, 115 shear stress, 98, 115, 134, 135 shear, viscosity, shear, 110 shear yield, 212 shear-free flow, 126 shear-induced flocculation, 215 shear-reversible alginates, 402 shear-thickening fluids, 116 shear-thinning fluids, 116 sheet stretching, 126 shelf life, 28, 544–6, 552, 553 shell eggs, 422 Sherwood number, 275, 287, 324, 325, 326 Shishkowsky’s relationship, 197 short spacing, 358 short-dough (sheeted), 576 short-dough biscuits, manufacture of, 627

shrinkage of the powder, 93 shrinkage temperature, 416 Shukoff flask, 351 Shvedov model, 170 side-pressure coefficient, 490 Silin equation, 463 similarity and analogy, 3, 16 simplex values, 8 sintering, 470, 481 size of sucrose crystal vs. moisture content of sucrose, 537 small strain amplitude, 142 Smoluchowski equation, 214, 215 Sokolovsky’s formula for boiling point of sugar-water solutions, 291, 561 solid → plastic → fluid transition, 122 solid behavior, 97, 105 solid bridges, 470 solid density, 54 solid fat content (SFC), 352, 353 solid-fat-index (SFI) curve (or dilatation curve), 353 solidification, 64 solidity of granule, 472, 474 solubility number of sucrose, 282, 580 solubility of sucrose, 192, 282, 285, 562, 563, 581 solubility of carbohydrates, 65 solubilization, 189, 506 solutions of sucrose, corn syrup and other monosaccharides and disaccharides, 579 Somogyi–Nelson micromethod, 498, 500 sorbitan esters (E 491), 212 sorption isotherms, 534 Sors’ method, 456–9 special confectioney vegetable fats, 329 special milk-preparations, 31 specific and molar heat capacity of crystalline sugars, 558 specific conductivity, 81 specific heat capacity, 52, 57, 64, 580 calculations, 58 of aqueous sugar solutions, 561 of commercial starch products, 566 of several materials used or produced by the confectionery industry, 62 specific molar electrical conductivity, 190

Index

specific resistance, 81 spherulites, 330 splitting plane, 200 spreading, 112, 195 squeezing, 127 St. Venant (plastic) element, 165 stability matrix of a food system, 553, 554 stability from viscosity increase, 214 stability theories, 550–4 stabilization, 507 stable modification of cocoa butter crystal β (VI), 360 stamping apparatuses, 486 Stanton number, 10, 14 starch, 413, 414, 566, 578 static yield stress, 121 statistical isotherm equation, 535 statistical self-similarity, 151 steady shear flow, 97, 109 steady-state evaporation model, 306 steam distillation, 512 Stefan–Reynolds equation, 230 Steffe’s empirical method, 454 sticking probability, 217 stirred ball mill, residence time distribution in a, 259 Stockmayer–Fixman equation, 184 stoichiometric equations, 613 stoichiometric matrix of a reaction, 614 Stokes law, 73, 214, 215, 427 Stokes–Einstein law, 73 storage, 541–8 storage conditions for chocolate, 370 storage scheduling, 537 strain, 98, 97, 111 streaming potentials (the reserve of electro-osmosis), 199 streamline (laminar) flow, 111 Strecker degradation, 503 strength of gelatine gels, 418 stress, 98, 97, 135, 136 strongly hygroscopic substance, 537 structural changes in solid polimers, 184 structural energy, 56 structural formulae of confectionery products, 19, 20 structural types of macromolecules, 182

685

structure of aggregates, gels and sediments, 217 structure of measures, 611 structure theory, 20, 510, 609–16 submicroscopic disperse systems, 178 substancial attributes, 609 sucre couleur (E 150a, b, c, d), 504 sucrose crust (or Kandisschicht in German), 536 sucrose, solution of, 283, 285 sucrose/water solutions, 291 sugar blooming, 367 sugar content and density of saturated aqueous sucrose solutions ias a function of temperature, 580 sugar melting, 504 sugar-based confectionery fillings, 381 sugared condensed milk, 502, 503 sugars, melting point of, 558, 580 supercooling, 311 superfluid transition, 64 supermolecular structure of fat melts, 351 supersaturation, 311, 312 surface activity, 53, 87 surface and interface surplus energy, 190 surface dilational modulus (or Gibbs coefficient), 227, 228 surface potential, 200 surface protrusions, 480 surface roughness, 479 surface tension, 470, surface tension force, 431 surface-active compounds, 189 surfactant (amphipathic) molecule, 189 suspending cocoa butter phase, 511 swallowing, 134 sweet biscuit dough, 575 sweets grained by addition of fondant (fudge), 508 sweets grained by spontaneous crystallization (grained caramels), 508 sweets, rheological properties of, 97, 156 swell phenomena, 113, 187, 533, 506 swelling, velocity of, 184 syneresis, 185, 398, 405 syrup ratio (SR), 35, 283

686

Index

tablets as sweets, 470, 483–92, 626 tabletting, indirect, 485 Tadzik girdle cakes, 174 tails, 213 tapping, 55, 89 Taylor–Langmuir equation, 322 technical data sheet, 615 technological triangle, 609–10, 615 tempering, 360, 361, 364, 509 tempering of cocoa powder, 371 tensile stress, 98 tensor antisymmetric/asymetric/symmetric, 110 deformation, 97, 100, 101 deviatoric, 97, 103 dilatational, 97, 103 divergence, 8 eigenvalues/eigenvectors of a, 104 gradient, 8 invariants, 97, 103 spherical, 109 strain, 99 stress, 98, 99, 97 testing methods for rheological study of doughs, 170 testing-sieve data, 263 texture of chocolate, 97 texturometer, 172 theoretical density, 55 theoretical foundation of the Bingham model, 598 theoretical tensile solidity, 476 theories of crystal growth, 322 theory of colloidal stability, 200 thermal conductivity, 52, 68, 557, 560, 561, 567, 578 thermal diffusivity, 52, 69, 561, 567 thermal effect of transformation in isobar and isotherm conditions, 63 thermal history of the solution, 317 thermal-property equations for food components (−40°C ≤ t ≤ 150°C), 61 thermal-property equations for water and ice (−40°C ≤ t ≤ 150°C), 62 thermal physical characteristics of cocoa butter, cocoa mass and chocolate, 567 semi-products used for cakes, 573

some raw materials used in the confectionery industry, 569 starch syrup used for caramel production, 565 sucrose-water solutions, 559, 560 sugar, 557 sweets and chocolate, 570, 571 vegetable oils and fats at 20°C, 567 thickening fluids, 116 thinning fluids, 116 thixotropy, 118, 156 Thomson equation, 333 three-body problem, 551 tilt, 358 time-independent material functions of viscous and plastic fluids, 115, 117 titrimetric or iodometric methods, 35 Tobvin–Krasnova–Akhumov formula, 316 toffee/fudge, 381, 502, 503, 573 toffee/fudge, manufacture of (by adding fondant mass), 622 toppings, 377, 382, torque decay, 119 total volume of pores, 195 toughening, 509 traces of acid as ‘killer’ of technology, 507 trains, 213 trans fatty acid (TFA), 384 trans-containing coating fat (TCF), 383 transcrystallization of cocoa butter, 507, 509 transformations between colloids and coarse dispersions, 204 trans-free coating fat (TFCF), 383 transient flow, 9 transient foams, 222 transient method, 134 transitive characteristic, 611 transitive property, 5 transport, 434–8 Traube’s rule, 197 Tresca’s criterion, 123 triacylglycerol (TAG), 329, 355 triangle model of batch conching, 511–7 triangle model of the structure of dough, 533 triggering spark, 89 triglycerides of the soft fat, 369

Index

triple-chain-length (TCL), 357, 360, 374 triple-stranded helix, 416 Trouton constant, 299 Trouton number, 131, 466 Trouton viscosity, 126 Trouton’s rule, 299 truffle mass, 162, 436 Tscheuschner chocolate process, 532 Tscheuschner equation, 158 Tscheuschner’s formula of four parameters, 160 tubeless syphon effect, 132 Turnbull–Fischer formula, 332 typical electrical conductivities of materials, 84 ultimate fat content of the cocoa cake, 448 unbroken cells, 4, universal conching machines, 532 van der Waals attraction potential, 206, van der Waals forces, 219, 470, 471, 474, 475, 479, 480 van der Waals interaction, 213, vapor pressure formulae, 295, 525 variation of the dynamic viscosity of chocolate as a function of the conching time, 569 Vavrinecz formula, 282 velocity gradient tensor, 110, 111 violet-brown in colour, 506 viscoelastic functions, 97, 132 viscosimetric flow /functions, 112 viscosity apparent, 117 bulk, 232 complex, 143 dynamic, 8, 115, 143, 167 inherent, 145 limiting, 117 of solutions, 97, 144, 146, 147 of suspensions, 97, 149 of the cracker dough as a function of the amount of gluten of different qualities, 576 plastic, 116, 583 Volmer layer, 322, 323

687

volume and mass flow in a five-roll refiner, 251 volume modulus, 490 von Kármán’s hydrodynamic theory, 277 von Mises yield criterion, 466 W/O type emulsion, 27, 506 wafer, 570, 571, 575, 628 warm treatment prior to storage, 370 Washburn equation, 186 water activity, 525, 527, 531 water adsorption of the packed food, rate of, 543 water adsorption on the surface of product, 36 water migration, 526 water permeability of packaging materials, 36 water sorption isotherm, 526 water sorption properties of particles, 475 water vapour permeability (WVP) of some packaging material, according to Minifie, 546 weak blow/schwach pflug, 112,5°C, 507 Weibull distribution, 207 Weissenberg effect, 132 Weissenberg model, 590 Weit–Schwedes relationship, 258 wet bulb psychrometry, 527 wet conching, 511 wet grinding, 239, wettability, 187, 189 483, wetting, 205, 482, 485, 536 wheat flour doughs, rheological properties of, 97, 166 whipped masses, 166 whipping, 429 whole-egg gels, 425, Wiedemann–Franz law, 53, 82 Wiener process, 532 Williamson model, 595 Winslow effect, 144 work index, 241 work of adhesion, 470 work of cohesion, 470

688

Index

work of volume, 58 working point, 266

Young’s modulus, 105, 137 Young–Laplace pressure, 220, 223, 225, 226

xanthan gum, 416

zwieback, 570, 571, 575, 576

yeast growth, 542 yield number, 122 yield stress phenomena, 120 Young (or -Dupre) equation, 193

β (V) modification of cocoa butter crystal, 360, 362, 363 β′ (beta-prime) form, 375 ΔL law: constant growth of crystals, 319