Katholieke Universiteit Leuven Faculteit Bio-ingenieurswetenschappen
DISSERTATIONES DE AGRICULTURA Doctoraatsproefschrift Nr. 796 aan de Faculteit Bio-ingenieurswetenschappen van de K.U.Leuven
DATA BASED MECHANISTIC MODELLING OF THREE DIMENSIONAL TEMPERATURE DISTRIBUTION IN VENTILATED ROOMS FILLED WITH BIOLOGICAL PRODUCTS
Promotor:
Proefschrift voorgedragen tot het
Prof. Daniel Berckmans, K.U. Leuven
behalen van de graad van Doctor in de Toegepaste
Leden van de Examencommissie:
Biologische Wetenschappen
Prof. Marc Hendrickx, KU Leuven
Door
Prof. Erik Vranken, KU Leuven
Võ Tấn Thành
Prof. Jean Lebrun, Liege University Dr. Ha Thanh Toan, Cantho University
May 2008
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Acknowledgment Firstly, I would like to express my sincere gratefulness to my advisor, Prof. Daniel Berckmans for his great suggestions, endless support and encouragement, and professional guidance throughout my PhD study.
My sincere thanks to Prof. Erik Vranken for his daily help during my working time in Leuven. I would like to thank my committee members, Prof. Rony Geers, Prof. Marc Hendrickx, Prof. Jean Lebrun and Dr. Ha Thanh Toan for their valuable suggestions and kind support during my research work. Special thanks to Mr. Jean-Lou Lemaire and Mr. Ludo Happaerts for their assistance in building the experimental installation and guidance to collect the equipments during my research.
I would like to thank all my friends and colleagues at M3-BIORES for helping me during the time I was in Leuven.
I would also like to thank the Belgian government for giving me an opportunity to pursue my study. My specially thanks to VLIR through the Institutional University Cooperation program with Cantho University for their financial support to this study. Special thanks to the Vietnamese student community at Catholic University of Leuven for their friendship and great time we have spent together.
Lastly, the deepest appreciation goes to my dear parents and my lovely wife in Vietnam for their unconditional love, encouragement, and support.
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Abstract In this PhD research, a data based mechanistic modelling approach was developed for real time monitoring (non-invasive measurement method) and online adaptive control of threedimensional temperature distribution in the room filled with biological products.
An empty room and room filled with obstacles and a room filled with biological products were selected for the model development. During the experiments, step inputs on an air inlet temperature were applied while airspace, obstacles and biological product temperature were recorded. The continuous-time simplified refined instrument variable algorithm (SRVIC) in the CAPTAIN Toolbox for Matlab was used to identify the model parameters and select the best model order.
In a first step was analysed wherever the data based mechanistic method, as developed for an empty room, could be applied in a ventilated room filled with obstacles. An empty room and a room filled with plastic balls as simulative products were used, and 36 thermocouples were installed in two vertical planes in a process room. The results showed that the temperature uniformity index in the empty and in the room filled with balls was dependent on the ventilation rates. High ventilation rates in combination with high acceptable temperature differences resulted in a high temperature uniformity index. The temperature uniformity index in both cases reached 100% with a ventilation rate of 240 m3 h-1 and an acceptable temperature difference of 1oC, due to adequate mixing. By adopting the mechanistic phase from an empty room to a room filled with obstacles, to model the dynamic response of airspace from inlet air temperature was shown that data based mechanistic modelling (DBM) approach worked in a room filled with obstacles.
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In the second step, the data based mechanistic modelling was further developed for a ventilated room filled with 480 balls. The dynamic response of airspace temperature and air inside the balls after steps on the air inlet temperature were used to find a best fitted model based on the coefficient of determination ( RT2 ), Young Information Criterion (YIC) and standard error (SE). A third order transfer function from the dynamic response of airspace temperature from steps inlet air temperature was found as the best fitted model in correspondence with the general theory of heat transfer from airspace to the balls. Physically meaningful parameters were found from the fitted model: there were the local volumetric of fresh air concentration (β1), and α2 value which was related to a local overall heat transfer coefficient from the airspace to the air inside the balls, it is the same structure of “cooling rate” term in Newton’s law for cooling processes. The local volumetric of fresh air concentration values (β1) in a room filled with obstacles represented an airflow pattern as in an empty room, and α2 was explained as the amount of energy which transfers from airspace to the air inside the ball, and it can be used to predict the temperature of air inside the ball.
In the third step, the data based mechanic model was developed for a room filled with solid material with different shapes as biological products. The dynamic response of airspace and product temperature from steps on inlet air temperature was recorded. The temperature uniformity index of airspace and the bulk of potatoes were calculated at different acceptable temperature differences and ventilation rates. The calculated temperature uniformity index for airspace varied from 2.7 to 66%, and the bulk of potatoes from 8.0 to 66%. The cumulative energy in the potatoes during the process is not only dependent on airspace temperature but also on the air velocity. A third order transfer function was obtained from the dynamic response of airspace temperature from step on inlet air temperature. This equation explained a heat transfer process in the room filled with biological products. Two important physically vi
meaningful parameters found from the model parameters are a local volumetric of fresh air concentration value (β1) and a local ‘cooling rate’ value (α2). The local cooling rate value in the model explained the rate of energy transfer to the product from the airspace. The local cooling rate value contains both a temperature difference (between airspace and product) and the airspace velocity. This is an important model parameter which existing in the model could be used for online prediction of product temperature and real time control of the 3D temperature distribution in a room filled with biological products.
The last step of this research was the evaluation of the model. Two methods to calculate the local cooling rate value to estimate the product temperature from the DBM model were found in the second and third steps. Results of local cooling rate values with several kinds of material and several levels of ventilation rates showed that the differences of the local cooling rates between the two methods at the different ventilation rates were 22 to 26%. A new method to calculate a cooling rate or heat transfer coefficient was found in this step, it was calculated under unsteady conditions with the information of the dynamic response of the product from airspace temperature that can be used to real time calculation of product temperature during heating/cooling applications.
Finally, the data based mechanistic model approach was successfully applied to model the temperature in a room filled with biological products and it extends for the 3D temperature distribution by using a well-mixed zone approach. Physically meaningful parameters were found such as a local volumetric of fresh air concentration and a local ‘cooling rate’. Their existence in a model are a key factor to explain an energy transfer within a fluid flow, a rate of energy transfer from fluid flow to products and a fluid flow velocity around products during heating/cooling processes. A local cooling rate is a physically meaningful parameter
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in the model, which can be used in a control algorithm to predict the product temperature, and be applied for real time controlling of 3D temperature uniformity index in heating and cooling applications.
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Table of content Acknowledgment Abstract Table of Content List of tables List of figures List of symbols and abbreviations
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Chapter 1 Introduction and Objectives ............................................................................... 1 1.1 Introduction ................................................................................................................ 3 1.2 Objectives ................................................................................................................. 16 1.3 Brief outline of next chapters ................................................................................... 18 Chapter 2 Materials and Methods..................................................................................... 21 2.1 Introduction .............................................................................................................. 23 2.2 Technical descriptions .............................................................................................. 23 2.3 Materials and methods .............................................................................................. 34 2.3.1. Material ............................................................................................................. 34 2.3.2. Methods ............................................................................................................ 35 Chapter 3 Modelling of Three-dimensional Air Temperature Distributions in Porous Media ........................................................................................................................................ 45 3.1 Introduction .............................................................................................................. 47 3.2 Materials and methods .............................................................................................. 48 3.2.1. Laboratory test room ........................................................................................ 48 3.2.2. Experiments ...................................................................................................... 48 3.2.3. Data-based mechanistic modelling approach ................................................... 52 3.3 Results and discussions ............................................................................................ 59 3.3.1. Uniformity of air temperature in the empty test installation ............................ 59 3.3.2. Data-based phase in the empty room................................................................ 63 3.3.3. Uniformity of air temperature in the test installation filled with obstacles ...... 66 3.3.4. Data-based mechanistic modelling approach in the obstacles room ................ 68 3.3.5. Comparison between the empty test installation and the test installation filled with obstacles ................................................................................................................... 72 3.4 Conclusions .............................................................................................................. 74 Chapter 4 Data Based Mechanistic Modelling for Control of Three Dimensional Temperature distribution in a Ventilated Room Filled with Obstacles ................................... 75 4.1 Introduction .............................................................................................................. 77 4.2 Materials and methods .............................................................................................. 78 4.2.1. Laboratory test room ........................................................................................ 78 4.2.2. Experiments ...................................................................................................... 79 4.2.3. Data based mechanistic modelling approach.................................................... 81 4.2.4. Data based identification and estimation .......................................................... 81 4.2.5. Mechanistic phase in the ventilated porous media ........................................... 82 4.3 Results and discussions ............................................................................................ 89 4.3.1. The third order transfer function in ventilated porous media ........................... 89 4.3.2. Physical meanings of estimated parameters ..................................................... 90 4.4 Conclusions .............................................................................................................. 96 Chapter 5 Data Based Mechanistic Modelling of Three Dimensional Temperature Distribution in Ventilated Rooms Filled with Biological Material .......................................... 97 5.1 Introduction .............................................................................................................. 99
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5.2 Material and method ............................................................................................... 101 5.2.1. Laboratory test room ...................................................................................... 101 5.2.2. Data based mechanistic modelling approach.................................................. 103 5.3 Results and discussion ............................................................................................ 110 5.3.1. Temperature distribution in airspace and potatoes during experiment........... 110 5.3.2. Uniformity of air and potato temperature in experiment ................................ 113 5.3.3. Data based phase ............................................................................................ 114 5.3.4. Mechanistic phase........................................................................................... 115 5.3.5. Analysis and suggestion of the order of transfer function .............................. 117 5.4 Conclusions ............................................................................................................ 119 Chapter 6 Validation of a Data Based Mechanistic Model for Online Calculation of the Cooling Rate during Cooling Processes ................................................................................. 121 6.1 Introduction ............................................................................................................ 123 6.2 Material and method ............................................................................................... 124 6.2.1. Laboratory test room ...................................................................................... 124 6.2.2. Experiment ..................................................................................................... 125 6.3 Data based mechanistic model approach ................................................................ 125 6.3.1. Data based phase ............................................................................................ 125 6.3.2. Mechanistic phase........................................................................................... 125 6.4 Results and discussions .......................................................................................... 128 6.4.1. The third order transfer function for the heat transfer in a room filled with products 128 6.4.2. Cooling rate from the first order transfer function ......................................... 129 6.4.3. Cooling rate as found from the estimated parameters of the third order transfer function 130 6.4.4. Comparison of ‘Cooling rate’ values obtained by two methods .................... 130 6.4.5. Calculation of a heat transfer coefficient from ‘Cooling rate’ with known physical properties of material ....................................................................................... 131 6.4.6. An example for biological products ............................................................... 132 6.5 Conclusions ............................................................................................................ 133 Chapter 7 General conclusions and Suggestions for future research ............................. 135 7.1 General conclusions................................................................................................ 137 7.2 Suggestions for future research .............................................................................. 142 References Appendices
137 148 -Appendix A -Appendix B -Appendix C
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List of tables Table 1-1 Researches related to the improvement of product temperature during heat treatments ........................................................................................................................... 9 Table 2-1 Physical characteristics of materials used in this research ....................................... 34 Table 2-2 Comparison between mechanistic, data based and data based mechanistic modelling .......................................................................................................................................... 39 Table 3-1 Overview of the dynamic experiments .................................................................... 51 Table 3-2 The model parameter estimates with for sensor position (1) and (32) in the empty room with ventilation rate is 280 m3 h−1........................................................................... 63 Table 3-3 The model parameter estimates with for sensor position (1) and (32) (in the obstacle room with ventilation rate is 280 m3 h−1) ......................................................................... 70 Table 4-1 Overview of the dynamic experiments .................................................................... 81 Table 4-2 Third order model parameter associated statistical measures for step up experimental data at 280 m3 h-1 inlet flow rate ................................................................. 89 Table 4-3 First order model parameters in a relationship between temperature in WMZ1 ( ti (t ) ) and material temperature ( tm (t ) ) with associates statistics measure for step up experiment at 280 m3 h-1................................................................................................... 92 Table 4-4 Calculated parameters at position (33) with ventilation rate from 220 to 280 m3 h-1 .......................................................................................................................................... 95 Table 5-1 Overview of the dynamic experiments .................................................................. 103 Table 5-2 Temperature uniformity index as a function of acceptable temperature difference and ventilation rates in the airspace................................................................................ 113 Table 5-3 Temperature uniformity index as a function of acceptable temperature difference and ventilation rates in bulk of potatoes ......................................................................... 113 Table 5-4 Estimated parameters from the third order of transfer function at position (6) and (30) with ventilation rate of 280 m3 h-1 as an example of calculating method ............... 115 Table 5-5 α2 values from numerator of the third order transfer function at ventilation rate 200 m3 h-1 ............................................................................................................................... 116 Table 5-6 Analysis and suggestion of the order of transfer function in several processes..... 118 Table 6-1 Estimated parameters from the third order transfer function at the different ventilation rates .............................................................................................................. 128 Table 6-2 Estimated parameters from the first order transfer function .................................. 129 Table 6-3 Cooling rate values at the different ventilation rates (1st order TF) ....................... 129 Table 6-4 Roots of second order equation .............................................................................. 130 Table 6-5 Comparison cooling rate values from two methods at different ventilation rates 130 Table 6-6 Estimated bulk heat transfer coefficient from known physical characteristics of material and ‘Cooling rate’ term from model parameters at different ventilation rates . 131 Table 6-7 Estimated parameters from Data based mechanistic model for forced air-cooling of apples at ventilation rate 20 m3 h-1 ................................................................................. 132
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List of figures Fig. 1-1 Scheme of model predictive control of 3D temperature distribution in a room filled with biological product ..................................................................................................... 14 Fig. 1-2 Energy exchanges in the Well Mixed Zone (WMZ) .................................................. 16 Fig. 2-1 First of the laboratory test installation ........................................................................ 25 Fig. 2-2 Slot of air inlet ............................................................................................................ 26 Fig. 2-3 Air outlet systems ....................................................................................................... 26 Fig. 2-4 Air outlet tube ............................................................................................................. 26 Fig. 2-5 Holes on the air outlet system ..................................................................................... 26 Fig. 2-6 Control of the ventilation rate through the test room.................................................. 27 Fig. 2-7 Cooling unit (a) and heat exchanger (b) ..................................................................... 28 Fig. 2-8 Numbered sensors in the test installation.................................................................... 28 Fig. 2-9 Room filled with balls................................................................................................. 30 Fig. 2-10 Room filled with boxes of potatoes .......................................................................... 31 Fig. 2-11 Position of balls in the test room and measuring air temperature inside the ball ..... 31 Fig. 2-12 Product temperature measurements ......................................................................... 31 Fig. 2-13 Test installation for validating the obtained model................................................... 33 Fig. 2-14 Air extracted system ................................................................................................ 33 Fig. 2-15 Researching methods to control 3D product bulk temperature in heating/cooling applications ....................................................................................................................... 36 Fig. 2-16 Step up (a) and step down (b) experiments:.............................................................. 36 Fig. 2-17 Step up inlet air temperature and dynamic temperature changes in airspace and potatoes at ventilation rate 280 m3 h-1 .............................................................................. 37 Fig. 2-18 Data-based mechanistic (DBM) modelling technique (Young et al., 2002)............. 40 Fig. 2-19 Diagrammatic representation of the continuous time SRIVC algorithm .................. 41 Fig. 3-1 Step up (a) and step down (b) experiments (c) The measured dynamic response of airspace and buffer temperature from the inlet air temperature for a ventilation rate 280m-3 h-1.......................................................................................................................... 50 Fig. 3-2 Three-dimensional view of the test installation. (1) Air inlet, (2) Air outlet, ............. 52 Fig. 3-3 Schematic representation of the well-mixed zone concept: ........................................ 54 Fig. 3-4 Block diagram of the feedback connected the second order transfer function: .......... 58 Fig. 3-5 (a) Visualisation of measured three-dimensional temperature distribution in the empty room under the following steady-state conditions: ventilation rate 120 m3 h−1, air supply temperature 17°C and internal heat production 300W; (b) visualisation of the 3D zone in the empty room with acceptable temperature difference 0.2°C and average temperature in a well mixed zone 23.1±0.2 oC................................................................. 61 Fig. 3-6 Temperature uniformity index (Itemp) of the empty room: (a) low ventilation rates (120–160 m3 h−1); (b) high ventilation rates (240–280 m3 h−1) with an acceptable temperature difference from 0.2 to 1°C ............................................................................ 62 Fig. 3-7 Partial contours of parameter β1 in the empty chamber: (a) front plane at the ventilation rate is 120 m3 h−1; (b) front plane at the ventilation rate is 280 m3 h−1 .......... 64 Fig. 3-8 Visualisation of the airflow pattern with smoke: (a) 120 m3 h−1; (b) 280 m3 h− ........ 65 Fig. 3-9 (a) Visualization of measured three-dimensional airspace temperature distribution in obstacle room under the following steady state conditions: ventilation rate 120 m3 h−1, air supply temperature 17 °C and internal heat production 300 W; (b) Visualisation of the 3D zone in the obstacle room with acceptable temperature difference 0.2°C, a well mixed zone 23.6±0.2 oC ................................................................................................... 67 Fig. 3-10 Temperature uniformity index of airspace (Itemp) of the obstacle chamber at (a) low ventilation rates (120–160 m3 h−1); (b) high ventilation rates (240–280 m3 h−1) ............. 68 xiii
Fig. 3-11 (a) The output of the first-order, second-order transfer function model compared with the measured airspace temperature response at the sensor position (1); (b) at the sensor position (32) in the obstacle chamber at ventilation rate 280 m3 s−1; , first order; , second order; noise line, experiment data ...................................................................... 69 Fig. 3-12 Residual plots of the first, second and third order model at the sensor position (32) in the obstacle room with the ventilation rate 280 m3 h−1 ................................................ 71 Fig. 3-13 Partial contours of parameter β1 in the obstacle chamber at the front plane with the ventilation rate 280 m3 h−1 ................................................................................................ 72 Fig. 3-14 Comparison between the temperature uniformity index of airspace (Itemp) for two values of the acceptable temperature difference 0.2°C (a) and 1°C (b); .......................... 73 Fig. 4-1 Numbered sensors in the test chamber........................................................................ 79 Fig. 4-2 Step up (a), step down (b) of dynamic experiments,(c) The measured dynamic response of airspace and air inside ball temperature from the inlet air temperature for a ventilation rate 280m-3 h-1 ................................................................................................ 80 Fig. 4-3 Schematic representation of the well-mixed zone concept in ventilated room filled with obstacles ................................................................................................................... 83 Fig. 4-4 The third order block diagram of system .................................................................... 87 Fig. 4-5 Third order fitted model and residual plot for 280 m3 h-1 inlet air flow rate (63.6 volume changes per hour): (a) position 27; (b) position 30 ............................................. 90 Fig. 4-6 Two dimensional contours of parameter β1 in the obstacle chamber at the ventilation rate 280 m3 h-1 at positions of vertical planes: (a) rear plane; (b) front plane .................. 91 Fig. 4-7 Comparison of parameter β1 at the different ventilation rates (160-280 m3 h-1) ........ 92 Fig. 4-8 Parameter α2 at several positions at ventilation rates 160 m3 h-1 and 280 m3 h-1 ...... 94 Fig. 4-9 Parameters β1 and α2 value at the difference of ventilation rates (position 33) ........ 95 Fig. 5-1 Boxes of potatoes and inserted sensor in potato Experiments .................................. 101 Fig. 5-2 Step up air inlet (a) and step down air inlet (b) of dynamic experiment ;(c) The measured dynamic response of the airspace and potato temperatures from the air temperature inlet at ventilation rate 280m3 h-1 ............................................................... 102 Fig. 5-3 Scheme of heat transfer in a selected well mixed zone containing several potatoes 105 Fig. 5-4 Third order block diagram of system ........................................................................ 109 Fig. 5-5 Temperature distribution in airspace with ventilation rate 200 m3 h-1 ...................... 112 Fig. 5-6 Temperature distribution in bulk of potatoes with ventilation rate 200 m3 h-1 ......... 112 Fig. 5-7 Dynamic temperature data at position 6 (a) and 30 (b) ............................................ 114 Fig. 5-8 Partial contour of parameter β1 (s-1) at front plane with ventilation rate 200 m3 h-1 116 Fig. 5-9 Comparison β1 and α2 at position (1) at ventilation rate from 160 to 320 m3 h-1 ... 117
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List of symbols and abbreviations Cp,0 heat capacity of air supply, J kg-1 oC-1 Cp,buff heat capacity air in the WMZ2, J kg-1 oC-1 Cp,i heat capacity air in the WMZ1, J kg-1 oC-1 Cp,m heat capacity of air in the ball, J kg-1 oC-1 heat capacity of material, J kg-1 oC-1 K1 model parameter K2 model parameter K3 model parameter Km model parameter k1 heat transfer coefficient between WMZ1 and WMZ2, J s-1 m-2 oC-1 k2 heat transfer coefficient between buffer zone and environment, J s-1 m-2 oC-1 km heat transfer coefficient between WMZ1and inside ball, J s-1 m-2 oC-1 heat transfer coefficient between WMZ1and material, J s-1 m-2 oC-1 m mass of material, kg qc part of internal heat production entering WMZ1, J s-1 S1 surface area between WMZ1 and WMZ2, m2 S2 surface area between WMZ2 and environment, m2 Sm surface of one ball, m2 surface of one material, m2 T0(t) inlet temperature, oC t0(t) inlet temperature at small temperature perturbations about steady state, oC o T0(t-τ) supply air temperature, C o t0(t-τ) supply air temperature at small temperature perturbations about steady state, C o Tbuff(t) temperature in the buffer zone (WMZ2), C tbuff(t) buffer temperature at small temperature perturbations about steady state, oC Ti(t) temperature in the well mixed zone i (WMZ1), oC Tlab temperature of environment, oC Tm(t) temperature inside ball, oC temperature of material, oC tm(t) temperature inside ball at small temperature perturbations about steady state, oC temperature of material at small temperature perturbations about steady state, oC − temperature in the WMZ1 at the steady state condition, oC Ti −
temperature inside material at the steady state condition, oC
−
input temperature at the steady state condition, oC
Tm
T0 −
T buff V Vin V1 VOL vol buff
voli
buffer temperature at the steady state condition, oC part of the ventilation rate entering the WMZ1, m3 s-1 ventilation rate, m3 s-1 volume of well mixed zone, m3 considerable volume, m3 volume of the buffer zone WMZ2, m3 volume of air in the WMZ1, m3
3 volm volume of potato, m -1 ζ ratio of volume and surface of material, m ε porosity of bulk material
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α1 model parameter -1 α2 parameter related to overall heat transfer coefficient, s β1 γ0
γbuff γi γm
cooling rate, s-1 local volumetric concentration of fresh air rate, s-1 density of air supply, kg m-3 density of air in the buffer zone (WMZ2), kg m-3 density of air in the WMZ1, kg m-3 density of air in the ball or density of material kg m-3 density of material kg m-3
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1 Chapter 1 Introduction and Objectives
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Chapter 1 Introduction and Objectives 1.1 Introduction Heating and cooling are common thermal processes in the post harvest and food industry. These thermal processing techniques are widely used to maintain quality and safety of food products and to extend shelf life of products (Wang & Sun, 2003). During heating/cooling applications, temperature, particularly the fluctuating temperature during the process, was found to be a major factor contributing to product losses during processing, storage and distribution of food (Vigneault et al., 2006). Postharvest losses of horticultural crops are estimated to be as high as 25% to 50% of the production due to poor postharvest handling techniques, mainly poor temperature management (Nunes & Emond, 2003).
In post harvest, temperature is the variable of the postharvest environment that has the greatest impact on the storage life of vegetables (Nunes & Emond, 2003). Keeping fruits and vegetables within their optimal ranges of temperature and relative humidity is the most important factor in maintaining their quality and minimizing postharvest losses (Michael Knee, 2002). Lower temperatures may cause chilling injury (the lowest safe temperature depend on the type of fruit or vegetable, e.g. lower than 11.5-13oC for bananas, 2-3oC for apples, 3oC for oranges, etc. (Wang, 2004)), and higher temperatures can reduce the storage life of the product. Kader (1991) reported that every 10°C increase in temperature accelerates deterioration and the rate of loss in nutritional quality by two to threefold. In storage of meat, Olafsdottir et al. (2006) found that reducing the storage temperature of cold fillets from 1.5oC to 0.5oC increased the minimum sensory life from 12.5 to 15 days.
The variation of temperature as a function of time or fluctuating temperature in processes is also a problem that needs to be controlled during heating and cooling. The temperature variability caused by poorly designed or controlled heating, ventilating, and air conditioning 3
(HVAC) equipment, is often taken into account by the evaluation of safety factors. In the storage of grain, temperature fluctuations may result in increased water loss, and condensation may develop on the product from the surrounding air, leading to the growth of microorganisms (Wills et al., 1998). Temperature fluctuations cause different other problems in relation to food storage and processes. They cause heat shock proteins in living cells in the post harvest of fruits and vegetables resulting in variation of the rate of moisture losses (Morimoto et al., 2003). The shelf life of frozen food is a complex concept that depends on the characteristics of the food product and the environmental conditions (Giannakourou et al., 2006). When temperature fluctuations occur during frozen storage, the crystal ice in a system will be varied (increasing the size or amount of ice crystal) (Reid, 1997) and influence the quality loss during frozen storage (Giannakourou et al., 2006) (e.g. moisture absorption and redistribution in foods, drip loss during thawing etc.). Additionally, the variation of temperature is also a cause of the growth of different pathogen particles as micro-bacteria and other deteriorations in the cold chains (Bogataj et al., 2005). In thermal processing, the variation of temperature with 1oC will lead to the change about 25% in the process lethality (Lewis, 2006). The retort temperature changes occasionally during the production. This results in an incomplete sterilization of the cans, which could lead to growth of nonpathogenic spoilage organisms or to toxic production of Clostridium botulinum. Therefore, temperature deviations represent a possible threat to the health of the consumer as well as to the economic survival of the food processor (Houlzer and Hill, 1977).
The spatial variation of temperature or the three dimensional distribution of product temperature in a process room during the heat treatment, is strongly dependent on the uniformity of the environment because of heating and cooling actions. In many cases, environmental temperature and fluid flow velocity around the products are not
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homogeneously distributed throughout the volume of a process room. This results in an imperfect mixed fluid, and a non-uniformity of the product temperature occurs during the heat treatment (Sun, 2007).
Uniformity of product temperature is one of the most important considerations in food processing. Non-uniform temperature distributions during heating, cooling, and storage may reduce the quality of food, safety level and thus cause economical losses. For example, in thermal processing of food, uniform heating is desirable to achieve a predetermined level of sterility with minimum destruction of the colour, texture, and nutrients of food products (Sun, 2007). Thermal equipments always contain regions in which the temperature of the heating medium is lower than the control temperature due to poor circulation of steam or steam/air mixtures (Tucker et al., 2001), Smout et al. (2000) reported that the coefficient of variation ranging from 14.9% to 63.1% for the thermal death time (F values) at different positions within the retort. DHSS (1994) suggested that the medium temperature should be the same for all the containers within the process vessel to within 0.5oC (no two points should be more than 1oC apart). In grain storage, the temperature difference within the grain bulk causes natural convection within the grain bulk, which results in moisture movement from a high to a low temperature area, and increases the chances for the distribution of insects, mites and fungi and thus brings about the deterioration of grain quality (Brooker et al., 1992, Canchun et al., 2000).
Consequently, control of 3D uniformity of product temperature during heating/cooling applications need to be realised to: - Reduce the product temperature gradient to control exact reactions occurring in food during the processing (inactivate microorganism, biological and chemical reactions, etc).
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- Control the hazard and food loss due to the non-uniformity of the temperature like coolest or hottest points because of inadequate heating and cooling (prevent a local heat respiration, chilling injury, air movement, etc.). - Optimize the process to save energy and to prevent weight loss during the heat treatment (Simpson, 2004, Canchun et al., 2000). - Create products coming to market with uniform characteristics.
Several mathematical modelling approaches have been applied to model temperature distribution in food processes and storage (i.e. Computation fluid dynamic (CFD), zoned models, multi zoned model, data based modelling, etc.). In these methods, CFD (based on the continuity equation and the set of the Navier-Stokes equations) offers a powerful design and analysis tool for the agri-food industry (Sun, 2002). Zoned modelling (based on the conservation equations of mass and energy to each cell where the air is treated as a perfect gas (Daoud and Galanis, 2007)) is used for predicting indoor temperature in a displacement ventilated room (Song et al., 2008). Data based modelling (based on quantitative data analysis) is a tool for predicting and control. Table 1.1 shows that most studies focused on modelling the temperature distribution to predict 3D airspace temperature and airflow pattern for optimal design under steady state conditions (i.e. Hoang et al. (2000), Nahor et al. (2005), Mirade and Picgirard, (2006), etc.) and unsteady state conditions (i.e. Ozcan et al. (2003), Therdthai et al. (2004)). In these studies, uniformity of medium temperature was used as a condition to realise the uniformity of product temperature. In reality however, there is no unique relationship between the uniformity of product temperature and medium temperature because of the local differences in the heat transfer coefficient.
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Dynamic modelling of the temperature of a process room filled with biological products was introduced in several studies. Mirade (2003) used a two-dimensional computational fluid dynamics model with time-dependent boundary conditions (i.e. an unsteady model). The CFD code Fluent v.5.4.8 was used to build a numerical model in this research. The homogeneity of the distribution of the air velocity in an industrial meat dryer was discussed and this research provided a rational approach to help in the operation and design of modern dryers. A three-dimensional computational fluid dynamics (CFD) model with moving grids was developed by Therdthai et al. (2004) to simulate the dynamic response during an industrial continuous baking process. This model can be used to optimise temperature profile in design to reduce the variations in product quality. Verdijck (1999), Verdijck and Straten (2002), Verdijck et al. (2005) developed a control methodology for model-based product quality control applied to climate controlled processing of agro-material. Model Predict Control (MPC) algorithm was used based on the non-linear control algorithm. This is the local controller to optimise energy efficiency without harming the product, and can be applied in storage, transportation, and drying processes etc. This model is limited to control directly the product quality (e.g. sugar in potato storage) and its quality variation in local climate controlled operations (e.g. optimisation of temperature), there is no information about the control input. Van Brecht et al. (2003) developed a control algorithm to increase the 3D uniformity in temperature during the entire incubation process. The uniformity of air temperature, airflow pattern and air quality are factors that need to be controlled in this process. The effect of heat flow direction to eggs was also considered in this research, a heat transfer coefficient was obtained from the air velocity, physical characteristics of air and egg, and temperature difference between air and egg. In practise, it is quite impossible to place a sensor exactly perpendicular to the airflow direction in a porous medium. The air circulation is continuously changing direction and the measuring device insertion generally disturbs the
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air pathway (Alvarez and Flick, 1999). Additionally, it is not easy to measure the egg temperature during the incubator process.
From the state of the art in literature (see Table 1.1), we can see that 19 of 32 existing models for this problem are focussing on temperature in steady state conditions. Out of 19 models in literature, only seven models handle the dynamic behaviour of temperature but only six models were used for control purposes. No model was developed to control 3D temperature distribution in the bulk of products in heating/cooling operations.
From this literature survey, we can conclude that no dynamical models have been used that focus on controlling the dynamic behaviour of 3D temperature distribution in the bulk of products during heating/cooling processes and that contain physically meaningful parameters.
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Table 1-1 Researches related to the improvement of product temperature during heat treatments No.
Authors
1
Alvarez & Flick, 1999
2
Alvarez & Trystram, 1996
3
Canchun et al., 2000
4
Gill et al., 1996
5
Feustel & Dieris, 1991 Hoang et al., 2000
6 7
Inard et al., 1996
8
Fukuyo et al., 2003 Keesman et al., 2003
9
Studies
Objectives
Design
Predict
Control
Analysis of heterogeneous cooling of agricultural products inside bins Part I: aerodynamic study Design of a new strategy for the control of the refrigeration process: fruit and vegetables conditioned in a pallet
Analysis air flow pattern to improve the uniformity of airspace temperature
Yes
No
Using numerical model to describe the refrigeration process between the air and agricultural products. Finding the hottest and coldest points to control over treatment Finite element method to calculate the temperature distribution in airspace during storage Using temperature history in the microbiology model to control product quality during storage
Yes
Finite element prediction of transient temperature distribution in a grain storage bin Control of product temperatures during the storage and transport of bulk containers of manufacturing beef. A survey of air flow models for multizone structures Analysis of the air flow in a cold store by means of computational fluid dynamics Prediction of air temperature distribution in buildings with a zonal model Thermal uniformity and rapid cooling inside refrigerators Optimal climate control of a storage facility using local weather forecasts
Dimension
Dynamic
No
Air space or products Airspace
3D
No
Yes
Yes
Airspace
-
No
Yes
Yes
No
Airspace
2D
No
No
Yes
Yes
Products
Random points
No
Zonal model
Yes
Yes
No
Airspace
3D
No
CFD to calculate air flow pattern in cool store
Yes
Yes
No
Airspace
3D
No
Calculation of mass air flows between two zones based on zonal method Using CFD to calculate the temperature distribution Optimal climate control of a potato storage facility by controlling air inlet
Yes
Yes
No
Airspace
3D
No
Yes
Yes
No
Airspace
3D
No
No
Yes
Yes
Point
No
9
Table 1.1 Researches related to the improvement of product temperature during heat treatments (Continuous) No.
Authors
Studies
Objectives
Design
Predict
Control
Find a spatial homogeneity of the air velocities above the product based on the pressure profiles and the air velocities in the drying chamber by CFD Predict air velocity and optimize design by using CFD method
Yes
Yes
Yes
Predict 3D temperature and velocity profiles in baking process
10
Mathioulakis et al., 1998
Simulation of air movement in a dryer by computational fluid dynamics: Application for the drying of fruits
11
Mirade and Picgirard, 2006
12
Mirade et al., 2004
13
Mirade, 2003
14
Moureh et al., 2002
15
Moureha & Flick, 2004
16
Nahor et al., 2005
Improvement of ventilation homogeneity in an industrial batch-type carcass chiller by CFD investigation Characterization and CFD modelling of air temperature and velocity profiles in an industrial biscuit baking tunnel oven Prediction of the air velocity field in modern meat dryers using unsteady computational fluid dynamics (CFD) models Numerical and experimental study of airflow in a typical refrigerated truck configuration loaded with pallets Airflow pattern and temperature distribution in a typical refrigerated truck configuration loaded with pallets CFD model of the airflow, heat and mass transfer in cool stores
Dimension
Dynamic
No
Air space or products Airspace
3D
No
Yes
No
Airspace
3D
No
Yes
Yes
No
Airspace
3D
No
the homogeneity of the distribution of the air velocity in an industrial meat dryer by using CFD
Yes
Yes
No
Airspace
2D
Yes
Using CFD to improve and to optimise air-distribution systems in refrigerated vehicles
Yes
Yes
No
Airspace
3D
No
Using CFD to improve and to optimise air-distribution systems in refrigerated vehicles
Yes
Yes
No
Airspace
3D
No
Using CFD to calculate velocity, temperature distribution in cool stores. Using model to predict airspace and product temperature
Yes
Yes
No
Airspace
3D
No
10
Table 1.1 Researches related to the improvement of product temperature during heat treatments (Continuous) No.
Authors
Studies
17
Ozcan et al., 2005
18
Smout et al., 2000
19
Therdthai et al., 2004
20
Várszegi, 2001 Van Brecht et al., 2003
Predicting 3D spatial temperature uniformity in food storage systems from inlet temperature distribution Non-uniformity of lethality in retort processes based on heat distribution and heat penetration data. Three-dimensional CFD modelling and simulation of the temperature profiles and airflow patterns during a continuous industrial baking process Modelling cooling methods for horticultural produce. Quantification and control of the spatiotemporal gradients of air speed and air temperature in an incubator. A modelling and control structure for product quality control in climate-controlled processing of agro-material Optimisation of product quality and minimisation of its variation in climate controlled operations Direct product quality control for energy efficient climate controlled transport of agromaterial
21
22
Verdijck & Van Straten, 2002
23
Verdijck et al., 2005
24
Verdijck et al., 2005
Objectives
Design
Predict
Control
Dimension
Dynamic
No
Air space or products Airspace
Control air space
No
Predict
3D
Yes
Relation of non-uniformity of environment and non-uniformity of product temperature.
Yes
No
No
Products
3D
No
Using CFD to predict temperature distribution in baking process
Yes
Yes
No
Airspace
3D
Yes
Calculate cooling rate
Yes
Yes
No
Products
1D
No
DBM to control airspace temperature
Yes
Yes
Yes
Airspace
3D
Yes
Control airspace temperature in storage room in relation to product quality parameters
No
Yes
Yes
Products
Point
Yes
Direct control of product quality of agro-material and minimisation of its quality variation Control product quality by climate control
Yes
Yes
No
Airspace
1D
No
No
Yes
Yes
Airspace
Point
Yes
11
Table 1.1 Researches related to the improvement of product temperature during heat treatments (Continuous) No.
Authors
Studies
Objectives
Design
Predict
Control
25
Verdijck, 1999
Yes
Verboven et al., 2000
Optimal storage conditions based on a model of product quality under an effect of the storage conditions. CFD to predict the temperature distribution in oven
Yes
26
Yes
27
Vigneault C., 2005
Calculate air velocity based on the cooling rate value
28
Delgado and Sun, 2001
29
Janssens ,1999
30
De Moor, 1996
Model-based product quality control for a potato storage facility Computational fluid dynamics modelling and validation of the temperature distribution in a forced convection oven A new approach to measure air distribution through horticultural crop packages. Heat and mass transfer models for predicting freezing processes – a review Dynamic modelling of heat and mass transport in imperfectly mixed fluid: A comparison of two approaches Modelling and control of energy and mass transfer in imperfectly mixed fluid
31
Berckmans et al., 1992
Methods to predict moisture and temperature in product during freezing process Using Data based mechanistic modelling approach to modelling the spatio-temporal temperature distribution in an empty room. Dynamic modelling approach was used to evaluate the dynamic behaviour of the microenvironmental factors (temperature and humidity) within an imperfect mixed space. Data based mechanistic model to find a physically meaning parameter to control 3D imperfectly mixed spaces
32
Patrick et al., 1995
New model concept to control the energy and mass transfer in a three-dimensional imperfectly mixed ventilated space Climatic Control of a Storage Chamber Using Fuzzy Logic
Using fuzzy logic for the control of MIMO in storage chamber (temperature and humidity of airspace)
12
Dimension
Dynamic
Yes
Air space or products Product
1D
Yes
Yes
No
Airspace
3D
No
Yes
Yes
No
Products
3D
No
No
Yes
No
Products
Point
-
No
Yes
Yes
Airspace
3D
Yes
No
Yes
Yes
Airspace
3D
Yes
No
Yes
Yes
Airspace
3D
Yes
No
No
Yes
Airspace
-
-
There are many methods to control a process. The Model predictive control (MPC) is an advanced method of process control (usually understood as techniques more advanced than a standard PID control (Tatjewsky et al., 2007)) that has been used in the process industries since the 1980s, such as in chemical plants and oil refineries. Models used in MPC include white-box models, black-box models, and grey-box models (some physical insight is available, but several parameters remain to be determined from observed data).
Several reasons to use MPC were listed by Tatjewski (2007). -
Firstly, the MPC algorithms can directly take into account constraints on both process inputs and outputs, which often decide on the quality, effectiveness, and safety of the production.
-
Secondly, they generate process inputs taking into account internal interactions within the process, due to the direct use of the model. Thus, they can be applied to processes with difficult dynamics and to multivariable control, even when numbers of manipulated and controlled variables are uneven.
-
Thirdly, the principle of operation of these algorithms is comprehensible and relatively easy to explain to engineering and operator staff, which is a very important aspect when introducing new techniques into industrial practice.
Using the model predictive control concept to control the uniformity of 3D product temperature in process room is shown in Fig. 1.1.
13
Control output 3D temperature of the bulk of products Control inputs -Ventilation rate -Inlet temperature
Measuring
Measuring Dynamic Modelling
Physically meaningful parameters for control
Model Predictive Control (MPC)
Fig. 1-1 Scheme of model predictive control of 3D temperature distribution in a room filled with biological product In Fig. 1.1, to control the 3D temperature of the bulk of products we need: (1) To use real time feedback of 3D product temperature by using multiple sensors to obtain product temperature during processing, (2) To predict the 3D product temperature at several positions in the process room by changing inputs, (3) To use a model predictive controller to control 3D product temperature distribution.
To control the product temperature at a specific position in a process room, we can use a single temperature sensor with a feedback loop in a temperature or ventilation rate controller with a black-box model. To control the uniformity of 3D product temperature in a process room, all product temperatures in 3D should be measured by the controller. This is not realistic in practice, especially in case of biological products (damage of product due to the high respiration rate and internal heat production when temperature sensors are inserted inside products). To avoid damaging of products, product simulators (Vigneault et al., 2002) can be 14
applied but without physically meaningful parameters in a model, it is impossible to predict and control the product temperature around simulators.
The Data Based Mechanistic (DBM) model approach has been shown to allow accurate modelling (e.g. De Moor and Berckmans, 1993; Janssens et al., 2000) which was applied to the problem of modelling imperfect mixing in the forced ventilated buildings (Janssens et al., 2004), cars (Quanten et al., 2004), incubators (Van Brecht et al., 2002). This approach is unique for controlling imperfectly mixed fluids since it is a hybrid between the extremes of mechanistic and data based modelling, and provides a physically meaningful description of the dominant internal dynamics of heat and mass transfer in the imperfectly mixed fluid. Physically meaningful parameters obtained from DBM could be used to predict and control 3D temperature distribution in ventilated room. The dynamics models from Berckmans et al. (1992a,b), De Moor (1996), Janssens (1999), and Van Brecht (2004) were developed from the heat and mass transfer in an empty room (heat transfer from airspace to products is neglected). Therefore, the ability of models was limited in an empty room and in a room filled with biological products where energy transfer from airspace to material is not important (i.e. in egg incubator). So far, it is impossible to predict product temperature and control dynamic behaviour of the 3D product temperature during cooling/heating processes.
15
1.2 Objectives The general objective of this PhD is to study the possibility of real time monitoring and control of the three dimensional (3D) temperature distribution in a room filled with biological products to improve the uniformity of product quality during heating/cooling applications.
The first objective consists of investigating the 3D temperature distribution in a ventilated room filled with obstacles and analyse the effect of the varying inlet temperature and ventilation rate on the dynamic behaviour of the 3D temperature distribution the airspace (airspace was defined as air surrounding the products in Figure 1.2) and the product temperature. Energy exchange between airspace and products Air inlet Product Product Product
Airspace Product Product
Air outlet
Volume of well-mixed zone voli
Product Temperature Tp(t)
Airspace Temperature Ti(t)
Figure 1.2 Fig. 1-2 Energy exchanges in the Well Mixed Zone (WMZ)
16
The second objective focuses on developing a model for controlling the 3D temperature distributed in airspace and product temperature by air inlet temperature.
The third objective is to find physical meaningful parameters in the model for the dynamic behaviour of the 3D temperature distribution in the airspace and product temperature that could be used for control purposes.
17
1.3
Brief outline of next chapters
In previous studies the dynamic response 3D distribution of the air temperature was studied in empty rooms and in incubators. Second order transfer function models were made with physical meaning in the parameters. These models were made for control purposes and this mainly to control the uniformity of airspace temperature in a building, car, incubator, etc. In these models however the heat transfer from the airspace to the products is not considered. Consequently so far it is not possible to control the 3D temperature distribution of the product by using control inputs.
The main objective of this research is the real time modelling of the dynamic response of 3D temperature distribution of the product in conditioned rooms filled with biological products. In order to archive to this objective, this PhD works is organized as follows
In chapter 1 a literature review is given of existing models in the field of ventilation and control of temperature distribution in conditioned process rooms and ventilated buildings. This will lead to a more detailed formulation of the objectives of this thesis.
Chapter 2 describes the test installations and the modelling methods that were used in this research. - First we did measurements in an empty ventilated room equipped with all sensors and requirements to measure and visualise dynamic responses of 3D temperature distribution to variations of the air inlet temperature. - Than the room was filled with obstacles with regular shape and well known characteristics: plastic balls. In a next step these balls were filled with water to increase the 18
complexity of the heat transfer in the system. After this we did experiments with real biological products (potatoes, apples) to gain mechanistic understanding of the process. In a last step we used different size and type of balls for validation purposes. - A simplified refined instrumental variable method (SRIV) will be used to estimate the parameters in a fitted model that related the control inputs to the temperature distribution in the room. - A Data based mechanistic model was used to explain the physical meaning of the parameters in the model.
In chapter 3, the three-dimensional temperature distributions in a ventilated empty room and a room filled with obstacles are compared experimentally to evaluate the presence of stored products, on the temperature uniformity. During experiments, step inputs in inlet air temperature are applied and temperature responses at 36 sensor locations in the room are recorded to develop a data-based mechanistic (DBM) model of the temperature response at different positions in the room. To study the influence of stored product and obstacles on 3D temperature distribution we had to start with the modelling of an empty room for comparison reasons.
In chapter 4, a data-based mechanistic model for temperature responses at different positions in both airspace and obstacles was developed. Plastic balls with known characteristics are used as obstacles. Step increases in air inlet temperature are applied while the airspace temperature and the air inside the balls are recorded. Using a compact model several physically meaningful parameters are found to represent the temperature distribution in the airspace and in the obstacles as well. This is different from all work that has been done in the past with the well mixed zone principle. 19
In chapter 5, a data-based mechanistic modelling approach is developed for room filled with real biological products. Boxes of potatoes are used as biological material. Step inputs on an air inlet temperature were applied while airspace and the dynamic responses of potato temperatures were recorded. Several physically meaningful parameters are found to present the temperature distribution in airspace and in the bulk of products.
In chapter 6, the evaluation of the model and an example of using a data based mechanistic model in case of the cooling of apples. Two methods to calculate the cooling rate are conducted and the comparisons of the estimated cooling rate and heat transfer coefficient are made.
Finally, the main conclusions and suggestions for further studies will be formulated in chapter 7.
20
2 Chapter 2 Materials and Methods
21
22
Chapter 2 Materials and Methods 2.1 Introduction First the 3D temperature distribution was modelled in an empty room based on temperature measurements in 36 positions in dynamic experiments. Then, the room was filled up with obstacles. To have obstacles with well known characteristics and similar shape plastic balls filled with air where used. To study the increase of the model order the ball in next step where filled with water this implies a new thermal buffering capacity For validation experiments a different size of ball with water was used to test the model performances in a new situations. 2.2 Technical descriptions Two installations (Figs. 2.1 and 2.9) were used in this research. They were installed at the Laboratory of Measure, Model & Manage Bioresponses (M3-BIORES), Katholieke Universiteit Leuven, Belgium. For the model development, the test chamber in Fig. 2.1 was used. This chamber is based on the test installation which was described in several publications (e.g. De Moor & Berckmans, 1996; Price et al., 1999; Van Brecht et al., 2000, 2004; Young et al., 2000; Zerihun Desta et al., 2004, 2005; Özcan et al., 2005; Van Buggenhout et al., 2005, 2006), with some adaptations, as described below.
The laboratory test chamber (Fig. 2.1) was a mechanically ventilated room with a length of 3 m, a height of 2 m, a width of 1.5 m and a volume of 9 m3. It has a slot inlet (Fig. 2.2) in the left sidewall just beneath the ceiling. The slotted air inlet, which has a width of 0.6 m and a height of 0.036 m, was positioned 1.55 m above the floor. The initial direction of the incoming air jet is not horizontal, but downward. The inlet angle with the horizontal is -10°. On the floor of the test installation, 12 large tubes with a diameter of 90 mm were installed (Fig. 2.3) and were connected to the air outlet (Fig. 2.4). These tubes were perforated with 24 holes with a diameter of 0.01 m and the holes positions were as in Fig. 2.5. A series of five 23
aluminium-heating elements (300 W totals) was installed at the bottom to physically simulate the internal heat production
An envelope chamber of length 4 m, width 2.5 m and height 3 m was constructed around the test chamber to minimise disturbing effects of varying laboratory conditions (fluctuating temperature, opening doors, etc.). The enveloping chamber was not ventilated. The volume of the buffer zone was 21 m3. The main test room and the envelope chamber were both constructed of transparent plexiglas in order to be able to visualise the airflow pattern. The thickness of the plexiglas walls is 0.075 m. The heat conduction coefficient is 0.16 W m-1 oC-1.
.
24
2.5m
3m
1.5m
Thermocouples 3m
Air inlet 0.036m
0.8m
0.6m
Rear sensor plane
Front sensor plane
2m
1.6m 1.55m
0.4m
Heat generation elements
Air outlet
Holes Φ 0.01m
Ducts Φ 0.09m
Fig. 2-1 First of the laboratory test installation
25
Envelope chamber
Fig. 2-2 Slot of air inlet
Fig. 2-3 Air outlet systems
Fig. 2-4 Air outlet tube
0.04m
Φ 0.09m 0.1m
Φ 0.01m
Fig. 2-5 Holes on the air outlet system
26
The laboratory test chamber was provided with a mechanical ventilation system. The air was extracted from the laboratory by a centrifugal fan. To achieve a desired ventilation rate, a movable cone was used (Fig 2.6). The position of the cone was regulated by a computercontrolled step motor.
Fig. 2-6 Control of the ventilation rate through the test room The mechanical ventilation system enables an adequate control of the ventilation rate in the range 70 - 420 m3 h-1 and this with an accuracy of 10 m3 h-1. These ventilation rates result in rather high inlet velocities (0.44 - 2.61 m s-1), air refreshment rates (7.6 - 46.6 refreshments per hour), and Reynolds numbers (1837 - 11021 at the inlet section of the test chamber). The high Reynolds numbers indicated that turbulent airflow prevails in the ventilation chamber (Welty et al., 1984).
The ventilation rate was measured in the supply duct from the pressure drop over an orifice and feedback to the ventilation PID-control system. The measurement error is 10 m3 h-1. To avoid that the thin plexiglas walls of the test chamber are subject to overpressure during the experiments (up to 100 Pa at a ventilation rate of 300 m³ h-1), an exhaust fan was provided in the outlet terminal.
A heat exchanger (Fig. 2.7 (b)) was provided in the supply air duct to regulate the temperature of the inflowing air. The supply air temperature is measured by a thermocouple in the inlet 27
section of the test chamber (in range 10 - 30°C), which was fed back to the on/off control system.
(a)
(b)
Fig. 2-7 Cooling unit (a) and heat exchanger (b) To measure the spatio-temporal temperature distribution in the test chamber, 36 temperature sensors were positioned in a 3-D measuring grid covering a large part of the room. The configuration of the 3-D sensor grid consisting of temperature in the test chamber was shown more detailed in Fig. 2.8. Air inlet
34
28
22
10
16
4 0.8m
35 31
29 25
23 19
11
17 7
13
5 1 0.8m
36 Height
30
24
12
18
6
32
26
20
14
8
2
33
27
21
15
9
3
Width 0.5m
0.8m Air outlet
0.375m
0.4m Length Front plane
Rear plane
Fig. 2-8 Numbered sensors in the test installation 28
The sensors are located in two vertical xy-planes: a front sensor plane (0.375 m from the front wall) and a rear sensor plane (0.375 m from the back wall) (Fig. 2.1). The temperature sensors have an average accuracy of respectively 0.1°C. The time constant of the sensors is less than 3 seconds. In addition to the 3-D distribution of temperature, the ventilation rate and the temperature of the supply air, the buffer zone temperature and the laboratory air temperature were measured during the experiments. A measurement and data collection unit (DATASCAN 7000, Measurement System Ltd.) with 192 channels was used for the data acquisition. The measurement frequency of the data logger can be programmed up to 400 readings per second.
This thesis is focusing on the 3D temperature distribution (bulk products and its surrounding air) in a ventilated process room filled up with obstacles. The hypothesis is that the 3D temperature distribution in airspace and products (defined in chapter 1) results mainly in an air mass flow through the obstacles and the interaction with the product temperature.
In the first step, the objective was to analyse whether the data based mechanistic model (DBM) method as developed for an empty room, could be applied in a ventilated room filled with obstacles. To have similar identical and well known obstacles, we used balls as obstacles in the first series of experiments. The second series of experiments potatoes in boxes were used.
29
Inlet
Ball Temperature sensor Outlet Heat generation element
Duct
Fig. 2-9 Room filled with balls The empty room was first filled with 480 plastic balls (Fig. 2.9), each with a diameter of 0.19 m and a weight of 0.12 kg to physically simulate uniform obstacles. These obstacles represented the physical presence of material in heating/cooling processes. The porosity, defined as airspace divides to total bulk volume in this experiment, was approximately 0.27. Later 144 boxes of potatoes (Fig. 2.10), each box (0.4 x 0.6 x 0.3 m) containing approximately 4.17 kg of potatoes (total 480 kg of potatoes), were used as biological products in heating/cooling applications. Product temperatures were monitored by an inserted thermocouple at the centre of a product (Figs. 2.11 and 2.12) with a Keithley 2700 data logger system. These measured potatoes were close to thermocouples that measured the temperature in the airspace.
30
Fig. 2-10 Room filled with boxes of potatoes
(a)
(a) Randomised balls in room
(b)
(b) Inserted thermocouple in the ball
Fig. 2-11 Position of balls in the test room and measuring air temperature inside the ball
(b)
(a)
(a) Potatoes in boxes
(b) Inserted thermocouple in the potato
Fig. 2-12 Product temperature measurements
31
To be able to modelling the heat exchange between airspace and products (in a large room was not enough energy to transfer to the products, therefore, the dynamic response is too low (See Fig. 2.17)), a uniform zone was selected in a small test installation (Fig. 2.13). This is a transparent mechanically ventilated room (located in the large test installation as described above) with 0.85 m (length) x 0.6 m (width) x 0.85 m height. It has a slot inlet (width 0.6 m and height 0.036 m). Ten T type thermocouples were installed in this test installation to measure inlet (one sensor), airspace (4 sensors), product (4 sensors) and buffer zone (one sensor) temperature. A Keithley 2700 data logger system was used for data recording. In this experiment, the air supply system used was the same as in the big room. The ventilation rate was adjusted by an air-extracted system (open and close system) to maintain the same ranges of ventilation rate through material in a big chamber (Fig. 2.14).
32
0.85m Air inlet
0.036m
0.4m 0.6m
Extracted air
0.85m
0.6m Sensor positions
Air extracted system
Air outlet
Fig. 2-13 Test installation for validating the obtained model
Fig. 2-14 Air extracted system To compare the obtained physically meaningful parameters (which related to heat transfer coefficient) by different methods and calculate the values of heat transfer coefficient based on the obtained physically meaningful parameters and the known physical characteristic of material. The room was filled with the table tennis balls (with pure water inside as known physical characteristic) was used and apples as an example for real biological products.
33
2.3
Materials and methods
2.3.1. Material In this research, two groups of material were used to develop and validate the models. Firstly, a group of non-biological products with known thermal characteristics were used, 480 plastic balls with diameter 0.19 m replaced the biological products to find a fitted model with physically meaningful parameters in a process (air inside the balls as a product temperature). Secondly, a room filled with 144 boxes of potatoes was used as a biological product. Finally, balls with diameter 0.04 m were filled with pure water as a well-known physical characteristic to validate the best-fitted model. Finally, biological products (apples) were used as an example for room filled with biological products.
The physical characteristics of materials are shown in Table 2.1 Table 2-1 Physical characteristics of materials used in this research Material
Number
Porosity
Experiments
0.2 per ball
480
0.27
(1), (2)
0.04
0.035
160
-
(4)
Potatoes
-
480 total
-
0.29
(3)
Apples
-
50 total
-
-
(4)
Plastic balls
Colour
Red,
Thickness
Diameter
Weight
(mm)
(m)
(kg)
1
0.19
0.45
yellow, blue, black (Fig. 2.9) Water filled
White
plastic balls
34
2.3.2. Methods 2.3.2.1 Research methods Data based mechanistic (DBM) modelling approach was applied to estimate the 3D temperature distributions in a room filled with biological products from the inlet air temperature.
Four steps of this study were conducted in this thesis (see Fig. 2.15.) (1) Adapting a data based mechanistic model approach from an empty room to a room filled with obstacles. In this test, air inside the ball was assumed as airspace in a test chamber and DBM was applied to model the dynamic response of airspace from inlet air temperature. If, it is able to model this process, the fitted model should be suggested. (2) Developing a data based mechanistic model for a room filled with plastic balls as obstacles, analysing the heat exchange from the environment to air inside the balls to find physically meaningful model parameters. (3) Developing a data base mechanistic model for a room filled with biological products, analysing the heat exchange from the environment to biological products (solid material) to find a physically meaningful parameter for online predicting the product temperature and real time control of 3D product bulk temperature in heating/cooling applications. (4) Evaluating the ability of the model can be used for cooling of biological products.
35
1
Testing a well mixed zone approach in ventilated room filled with non biological products
2
Data based mechanistic model development for room filled with non biological products
Order transfer function Physically meaningful parameters
3
Extending Data based mechanistic model for room filled with biological products
Physically meaningful parameters
4
Evaluating the ability of the model
Suggestion the order of the transfer function can be applied
Fig. 2-15 Researching methods to control 3D product bulk temperature in heating/cooling applications 2.3.2.2 Data collection To obtain dynamic data sets for dynamic modelling, dynamic step up (a) and step down (b) of air inlet temperature (Fig. 2.16) were conducted with different step changes depending on each experiment while maintaining the airflow rate constant. The inlet, airspace and product temperature profiles at a specific ventilation rate and certain position in test installation is
Temperature, oC
Temperature, oC
illustrated as an example in Fig. 2.17.
17
11
17
11
(a) 2000
Time, s
(b) 2000
10000
Time, s
Fig. 2-16 Step up (a) and step down (b) experiments: Inlet temperature setting; measured response
36
10000
o
Temperature ( C)
22 21 20 19
Inlet air temperature 0
100
200
300
400
500
600
700
500
600
700
500
600
700
Time (min)
o
Temperature ( C)
25 20 15
Airspace temperature 10
0
100
200
300
400
25
o
Temperature ( C)
Time (min)
20 15
Potato temperature 10
0
100
200
300
400
Time (min)
Fig. 2-17 Step up inlet air temperature and dynamic temperature changes in airspace and potatoes at ventilation rate 280 m3 h-1
2.3.2.3 Data Based Mechanistic (DBM) modelling approach In biological research, most of the time researchers use so-called mechanistic model to modelling processes. When scientists are observing the reality and developing a mechanistic mathematical model, they look though the glasses of their knowledge and education to decide what different model pair should be used to describe the complex reality. The disadvantage, however, is that in most cases, due to the complexity of the studied systems, this mostly results in very complex model structures with many parameters. For design purposes, this might be interesting since sensitivity analysis can be done. The disadvantage of this approach is that too many parameters are difficult to be given a correct value in real time modelling for control purposes. Another limitation is that the researcher is limited by his knowledge to discover new insight since he is projecting his existing knowledge in reality.
37
Another method for modelling process is “Data based mechanistic modelling”. This term was first used in Young & Lees (1993) but the basic concepts of this approach to model dynamic systems were developed over many years. This method was developed originally to analyse measured time-series data particularly in relation to signal processing, forecasting, and automatic control systems (Young & Chotai, 2001).
To determine the physical nature of an imperfectly mixed fluid based on experimental data. The Data Based Mechanistic (DBM) model approach has been shown to allow accurate modelling (e.g. De Moor and Berckmans, 1993; Janssens et al., 2000) which was applied to the problem of modelling imperfect mixing in the ventilated buildings (Janssens et al., 2004), cars (Quanten et al., 2004), etc. This DBM approach is unique for controlling imperfectly mixed fluids since it is a hybrid between the extremes of mechanistic and data based modelling and provides a physically meaningful description of the dominant internal dynamics of heat and mass transfer in the imperfectly mixed fluid (Berckmans and Goedseels, 1986; Berckmans, 1986; Berckmans et al., 1992).
Van Brecht (2004) listed the advantages of data based mechanistic modelling in Table 2.2
38
Table 2-2 Comparison between mechanistic, data based and data based mechanistic modelling Mechanistic approach (M) + Physical insight Suited for Design
High computational power Validation problem Not suited for control
Data based approach (DB)
+ Low No physical computational insight power Suited for Not suited for control design No validation problem
Many assumptions: turbulence model, boundary Conditions
Data based mechanistic approach (DBM) + Physical Not suited for insight design Suited for control Low computational power No validation problem
The main difference from data based ‘black box’ models and data based mechanistic models require a mechanistic interpretation. Data based mechanistic modelling is a model identification technique consisting of two phases (Fig. 2.18).
39
Dynamic Dynamic data data
Identification Identification experiments experiments
Identification Identification of of minimally minimally parameterised parameterised model model
Physical Physical knowledge knowledge
Physical Physical interpretation interpretation of of the the identified identified model model
No
1 Data Data based based phase phase
2 Mechanistic Mechanistic phase phase
Model Model physical physical meaningful? meaningful?
End End Fig. 2-18 Data-based mechanistic (DBM) modelling technique (Young et al., 2002) The first phase in DBM approach involves the objective identification and estimation of a minimally parameterised transfer function (TF) model from the experimental dynamic data (data based phase). The DBM principle only accepts this TF model, if the model gives not only a good identification of the dynamic data, but in the second phase also provides a physically meaningful description of the system under study (mechanistic phase).
In order to identify and model the temperature dynamics in a ventilated room, it would be preferable to perform experiments in which the inlet temperature is changed sharply in a ‘sufficiently exciting’ manner (Young, 1984). In other words, the input should be chosen to induce changes in the outlet temperature that are sufficiently informative to allow for the unambiguous estimation of the dominant dynamic characteristics. 40
The continuous-time SRIV algorithm (Young & Price, 1999) is used to identify the linear TF (ordinary differential equation) model between dynamic input and airspace temperature, with the coefficient of determination RT2 and the YIC identification criterion employed as model structure identification criteria (Young, 1989). A schematic overview of the general concept shows in Fig. 2.14. Input – output data
Use discrete time SRIV and convert parameters to their continuous time Equivalent for pre-filters
Generate auxiliary model
Update the pre-filters and auxiliary model
Generate filtered derivatives for input, output and auxiliary model
Instrumental variables
Algorithm converges ?
Return the model parameters
Fig. 2-19 Diagrammatic representation of the continuous time SRIVC algorithm (Young, 1989)
It is a logical extension of the more heuristically defined state variable filter (SVF) and follows from the optimal Refined Instrumental Variable (RIV) and SRIVC algorithms for discrete time identification (Garnier & Young, 2004), was selected. This method has been used for many years in a wide range of practical applications (e.g. De Moor & Berckmans, 1996; Price et al., 1999; Young et al., 2000; Janssens et al., 2004; Quanten et al., 2003, 2004; Desta et al., 2004, 2005; Van Buggenhout et al., 2006). A continuous-time transfer function model for a single-input single output (SISO) system has the following general form:
41
x (t ) =
B(s) u (t − τ ) and y(t ) = x(t ) + e(t ) A( s )
y (t ) =
B(s) u (t − τ ) + e (t ) A( s )
or
where A(s) and B(s) are the following polynomials in the derivative operator s =
d dt
A( s ) = s n + a1 s n −1 + ... + a n −1 s + a n B ( s ) = b0 s m + b1 s m −1 + ... + bm −1 s + a m
and τ is any pure time delay in time units. This transfer function model structure is denoted by the triad [n, m, τ]. u(t) is the input signal, x(t) is the ‘noise free’ output signal and y(t) is the noisy output signal. Initially, the noise e(t) is considered as zero mean, white noise with Gaussian amplitude distribution, it can also be written in the following differential equation form, which is often more familiar to physical scientists. d n y (t ) d n −1 y (t ) d m u (t ) + a + ... + a y ( t ) = b + ... + bm u (t − τ ) + μ (t ) 1 n 0 dt n dt n −1 dt m
where μ (t ) is defined as μ (t ) = A( s)e(t )
Note that s is used as the derivative operator in the transfer function model because of the very close relationship to the Laplace transform operator, which is often used for the incorporation of initial conditions of the variables and their derivatives. Initially, it is assumed that there are no major effects of any initial conditions remaining on the observed time series y(t) and u(t) that would complicate the estimation of the model parameters.
42
The SRIVC algorithm uses a recursive technique to analyze the data and to provide the best applicable model. The best-selected model is based on the coefficient of determination ( RT2 ) and on the Young Information Criterion (YIC). Particularly, the coefficient of determination ( RT2 ) is a widely applicable statistical index that represents the model fit by using the equation: ⎛δ 2 ⎞ RT2 = 1 − ⎜ 2 ⎟ ⎜δ ⎟ ⎝ y⎠
where δ 2 is the variance of the model residuals and δ y2 is the variance of the data. The Young Information Criterion (YIC) is a more complex index since it estimates the model fit and the efficiency of parameters by applying the following formula: ⎛δ 2 ⎞ YIC = ln⎜ 2 ⎟ + ln{NEVN } ⎜δ ⎟ ⎝ y⎠
where NEVN is the normalised error variance form (Young et al., 1980, Young et al., 1989, Young & Beven, 1994).
NEVN =
p 1 δ 2 ii ∑ p k αi
where p = m + n + 1 is the total number of parameters estimated, α i is the estimate of the ith parameter in the vector α for all the model parameters.
α= [a1 a2 ... an b0 b1 ... bm]T The first term of YIC is simply a relative logarithmic measure of how well the model explains the data: the smaller the model residuals the more negative the term becomes. The second logarithmic term, on the other hand, tends to become large (less negative) when the model is over-parameterized and the parameter estimates are poorly defined. Consequently, the criterion attempts to identify a model that explains the data well but with the minimum of statistically well-defined (low-variance) parameters. 43
44
3 Chapter 3 Modelling of Three-dimensional Air Temperature Distribution in Porous Media
45
46
Chapter 3 Modelling of Three-dimensional Air Temperature Distribution in Porous Media 3.1 Introduction The uniform quality of a product in a storage room has been greatly related to the air and
product temperature distribution (Ville & Smith, 1996, Chua et al., 2002; Verboven et al., 2004; Chao & Wan, 2004; Somkiat et al., 2005). It is now widely recognised that the air in a ventilated process is never perfectly mixed due to the existence of multiple airflow regions, the presence of stagnant zones and the occurrence of short-circuiting of air to the exhaust outlet. The spatially heterogeneous heat and mass transport phenomena in imperfectly mixed fluids are complex dynamic processes with considerable uncertainty about their nature (Janssens et al., 2004). In non-isothermal conditions, incomplete air mixing gives rise to three-dimensional (3D) temperature gradients (D’Alfonso et al., 1994; De Moor & Berckmans, 1993) that have a major impact on process quality, energy usage and process efficiency.
Although computational fluid dynamics (CFD) can be applied successfully to modelling velocity, temperature distributions to a detailed level (Campen & Bot, 2003); it must be used with care. One must be aware of the fact that a CFD model constitutes the culmination of a large number of assumptions and approximations resulting in models that lack the necessary accuracy to be appropriate for control purposes (Oltjen & Owens, 1987). The use of an insufficiently dense grid, the selection of an improper turbulence model and carelessly specified boundary conditions can lead to erroneous results.
When applied to the problem of active process control of the spatio-temporal heat and mass distribution in a ventilated air space or agricultural and industrial process rooms, such numerical CFD models are restrictive owing to their exceptional complexity. To overcome 47
this difficulty, a modelling approach was developed, which as a hybrid between the extremes of data based modelling, provides a physically meaningful description of the dominant internal dynamics of heat and mass transfer in the imperfectly mixed fluid (Berckmans & Goedseels, 1986; Berckmans et al., 1992a; Janssens et al., 2004). Data based mechanistic modelling (DBM) for 3D temperature distribution approach was developed for two reasons: (1) A DBM model is relatively compact, characterises the dominant modal behaviour of the dynamic system and is, therefore, an ideal basis for model based control system design (Camacho & Bordons, 1999; Maciejowski, 2002). (2) The model structure of a DBM model provides a physically meaningful description of the process dynamics and can, therefore, easily be applied to a wide range of imperfect mixing processes: in different fluids (air, water, etc.), on different scales (micro and macro scale) and in numerous application areas.
The objectives of this chapter is the quantification of the 3D spatial uniformity of air temperature in a ventilated room filled up with obstacles and tries to resolve the causes by using a DBM approach. 3.2
Materials and methods
3.2.1. Laboratory test room A mechanically ventilated laboratory test room described in Part 2.1, Chapter 2 (Fig. 2.1) was
used in this study. 3.2.2. Experiments Two configurations were used for the experiments: empty chamber (Figs. 2.1) and chamber
filled with obstacles (plastic balls) (Figs. 2.2). (Plastic balls were used as the well known obstacles which can be extended to heating/cooling processes)
48
For the room filled with balls as obstacles, the test installations were filled with 480 plastic balls (Fig 2.2), each with a diameter of 0.19 m and a weight of 0.12 kg to physically simulate uniform obstacles. These obstacles represented the physical presence of material in storage processes. The porosity, defined as airspace divides to total bulk volume, in this experiment was approximately 0.27.
To obtain dynamic data for modelling, experiments were carried out with step changes in inlet temperature from 11 to 17°C and 17 to 11°C (Fig 3.2) while maintaining a constant ventilation rate.
49
Temperature, oC
Temperature, oC
17
11
17
11
(a) 2000
Time, s
(b) 2000
10000
Time, s
10000
(a)
(c) Fig. 3-1 Step up (a) and step down (b) experiments (c) The measured dynamic response of airspace and buffer temperature from the inlet air temperature for a ventilation rate 280m-3 h1
, inlet temperature setting;
, measured response
The ventilation rate was varied between 120 and 280 m3 h−1 (13.3–31.1 volume changes per hour for empty room) in different experiments, while the internal heat production was maintained at 300 W in each experiment. The experimental design was in Table 3.1
50
Table 3-1 Overview of the dynamic experiments No. Experiments
Ventilation rates (m3 h-1)
1
120
2
120
3
140
4
140
5
160
6
160
7
180
8
180
9
240
10
240
11
250
12
250
13
260
14
260
15
270
16
270
17
280
18
280
Dynamic experiments Temperature (oC) Total time of experiment (s) 11 2000 17 10000 17 2000 11 10000 11 2000 17 10000 17 2000 11 10000 11 2000 17 10000 17 2000 11 10000 11 2000 17 10000 17 2000 11 10000 11 2000 17 10000 17 2000 11 10000 11 2000 17 10000 17 2000 11 10000 11 2000 17 10000 17 2000 11 10000 11 2000 17 10000 17 2000 11 10000 11 2000 17 10000 17 2000 11 10000
To be able to visualise the airflow pattern, the test installation was equipped with a smoke injection system, white smoke (3-ethylglycol (30% by weight), propylene glycol (30% by weight) in water) was injected at the air inlet. Four halogen lamps of 500 W each illuminated 51
the injected smoke, which was recorded by a black and white charge coupled device (CCD) camera (Hitachi KP-M1E/K) (Fig. 3.2). The camera was positioned in front of the test chamber. The back wall of the chamber was painted in black to ensure a good contrast with the white smoke pattern. Sixteen consecutive images of the recorded smoke pattern were digitised with a digitalisation card (Matrox Frame Grabber, PIP 1024 B) controlled by a personal computer. The time interval between the consecutive images was 0.7 seconds. The digitised images are 256 by 256 pixel matrices, resulting in a corresponding pixel size of 1.2 by 0.8 cm. The pixels have a grey value varying from 0 (black) to 255 (white). An algorithm was used to segment the smoke pattern out of the background of the test installation (Van Brecht et al., 2000). 4
1 3 2
y z
5 x
Fig. 3-2 Three-dimensional view of the test installation. (1) Air inlet, (2) Air outlet, (3)(4) Halogen lamp illumination, (5) Charge coupled device (CCD) camera. 3.2.3. Data-based mechanistic modelling approach Data based mechanistic modelling approach was applied to estimate the temperature
distributions in the obstacle room from inlet temperature. This approach is illustrated in Fig. 2.12. The most parametrically efficient model structure was first defined mathematically from the available dynamic data in an inductive manner, based on a generic class of black-box models (normally linear or non-linear differential equations or their difference equation equivalents). After this initial black-box modelling stage, the model is interpreted in a 52
physically meaningful, mechanistic manner based on the nature of the system under study and the physical laws that are most likely to control its behaviour (Young, 2002). 3.2.3.1 Mechanistic phase in the empty room
The DBM approach represents the imperfectly mixed fluid in a process room by a number of well-mixed zones (WMZs) in the room (Well mixed zone was defined as a zone around a temperature sensor in which there exists a good mixing and acceptably low temperature difference (Berckmans, 1986; Berckmans and Goedseeds, 1988; Berckmans et al., 1992a). The dynamic temperature response in the well mixed zone to the variation of air inlet temperature was considered for control purposes. Since the temperature within each well mixed zone has to be controlled by the control in puts (temperature of the in coming air, ventilation rate, etc.). The response of each well mixed zone temperature to the control inputs matter. The interaction between well mixed zones is considered as a disturbing factor taking into account in the real time model of each well mixed zone. These WMZs exist in every imperfectly mixed fluid. A schematic representation of a WMZ in a process room with ventilation rate Vin in m3 s−1) and with internal sensible heat production qc in J s−1 is given in Fig. 3.3.
53
Tlab Buffer zone
volbuff
Tbuff(t)
Air inlet T0(t-τ), V, γ0, Cp,0
Vin T0(t)
Ti(t)
WMZ
qc
Air outlet
voli V, Ti(t), γi, Cp,i
Fig. 3-3 Schematic representation of the well-mixed zone concept: Vin, air flow rate; V, part of the total ventilation rate entering the well-mixed zone i; voli , volume of well-mixed zone; vol buff , volume of the buffer zone; T0(t), supply air temperature at time t; Tbuff(t), temperature of air in the buffer zone at time t; Tlab, laboratory temperature; qc, part of the total heat production in the room entering the well-mixed zone i; t, time;τ, advective time delay; Cp,0, Cp,i, heat capacity of the air in the well-mixed zone i and the supply air; γi , γ0, densities of the air in the well-mixed zone i and supply air To describe the dynamic behaviour of temperature in each of the considered n WMZs, standard heat transfer theory was applied. In the case of a constant ventilation rate, this resulted in first-order differential equation of the form (Janssens et al., 2004) dTi ( t ).vol i .γ i .C p,i dt
= V .T0 ( t − τ )γ 0 .C p,0 − V .Ti (t )γ i .C p,i + qc + k1 .S1 (Tbuff ( t ) − Ti ( t ))
where: Ti(t) is the temperature in the well-mixed zone, °C. T0(t−τ) is the input temperature, oC. V is the part of the ventilation rate that enters the well-mixed zone, m3 h−1. voli is the volume of well mixed zone, m3. Tbuff(t) is the temperature in buffer zone, °C. k1 is the heat transfer coefficient between WMZ1 and WMZ2, J s−1 m−2 °C−1. S1 is the surface area of heat exchange between inside room and buffer zone, m2. 54
(3.1)
qc is a part of the internal heat entering the well-mixed zone in J s−1. Cp,0, Cp,i are the specific heats of air inlet and air in the well-mixed zone i, J kg−1 °C−1.
γ0, γi are the densities of the air supply and air in the well-mixed zone i in kg m−3.
The buffer zone consists of a volume of air between the central chamber walls and the envelope chamber walls. There is a heat exchange between the air in the buffer zone and the air in the central chamber. In contrast to the zonal and nodal models in the literature (Dalicieux et al., 1992; Li et al., 1992), the different WMZs here are considered as decoupled or non-interactive zones. Each WMZ is considered to be coupled with the control inputs Vin and T0. The reason for considering non-interactive 3D zones is that: (1) The resulting n models (the response of temperature in each WMZ to changes in the conditions of air temperature inlet is described by a single model, resulting in n first order models for n WMZs) that can be used for controlling the conditions in the n well-mixed zones each time only using the air inlet conditions. (2) Since the product in all positions within the ventilated process needs to be dried, the focus on the WMZ model concept is to model the movement of supplied air to a particular zone. Finally, all zones must be reached by using the process control inputs. This provides two major advantages: (1) it is not required to determine the interactions (air and heat flows) between the different zones that generate lots of inaccuracies in zoned and nodal modelling; (2) when considering non-interactive WMZs, the spatio-temporal model is also less complex and more appropriate for control purposes.
Under the assumption that γ0≈γi≈γ and Cp,0≈Cp,i≈Cp, differential Eqn (3.1) was rewritten by Janssens et al. (2004) as
55
⎛ V qc dTi (t ) k 1 . S1 k1 .S1 ⎞⎟ V = + T0 (t − τ ) + Tbuff (t ) − ⎜ Ti (t ) + ⎜ vol vol .γ .C ⎟ vol i .γ .C p dt voli voli .γ .C p i p ⎠ ⎝ i
(3.2) The associated steady state equations are as follows: ⎛ V qc k1 .S1 k1 .S1 ⎞⎟ V + =0 T0 + T buff − ⎜ Ti + ⎜ ⎟ voli .γ .C p voli voli .γ .C p ⎝ voli voli .γ .C p ⎠
(3.3)
with, T 0 , T i , T buff , input, room and buffer zone temperature at steady-state condition in °C. If only small temperature perturbations t0(t−τ), tbuff(t) and ti(t) are considered around steady state, then (Janssens et al., 2004) ⎛ V dt i ( t ) k1 .S1 k1 .S1 ⎞⎟ V t0 (t − τ ) + tbuff (t ) − ⎜ t (t ) = + ⎜ vol vol .γ .C ⎟ i dt voli voli .γ .C p i p ⎠ ⎝ i
(3.4)
dti (t ) = β1t0 (t − τ ) + K1tbuff (t ) − α1ti (t ) dt where the coefficient β1, K1 and α1 are given by ⎛ V
k .S
(3.5)
β1 =
V ; voli
K1 =
k1 .S1 ; voli .γ .C p
⎞
1 1 ⎟ = β 1 + K1 + α1 = ⎜⎜ ⎟ vol vol i .γ .C p ⎠ ⎝ i
Equation (3.5) can be expressed in a continuous-time transfer function form as
ti (t ) =
K1 β1 t0 (t − τ ) + tbuff (t ) s + α1 s + α1
(3.6)
It has already been shown that the spatio dynamic temperature response of 36 sensors in an imperfectly mixed ventilated empty room can be successfully modelled with the DBM concept [Eqn (3.6)] for a wide range of ventilation rates to variations of the supply air temperature. It has been demonstrated that the modelling approach is applicable to different fluids (air, water), to process rooms of different scale (macro-scale, micro-scale) and to
56
processes with different flow conditions (turbulent flow, laminar flow), (De Moor, 1996, Janssens et al., 2004).
It has been shown that the model parameters β1, K1 and α1 in Eqn (3.6) have a physical meaning (Berckmans et al., 1992a; Janssens et al., 2004). Parameter α1 in s−1 is the sum of β1 in s−1 and K1 in s−1. It is also the reciprocal of the time constant τ (s) of the first-order model. Parameter K1 is the local transmission coefficient of heat exchange between the WMZ and the buffer zone. Parameter β1 is the local volumetric concentration of fresh airflow rate (Berckmans et al., 1992a; Janssens et al., 2004) or the local outside air change rate (Li et al., 1994) in the WMZ. It is the amount in m3 of fresh supply air flowing into the considered WMZ per unit of time in second divided by the volume of the WMZ in m3.
A first order heat balance can be written for the buffer zone: dTbuff ( t ) vol buff .γ buff .C p,buff dt
= k1 .S1 (Ti ( t ) − Tbuff ( t )) + k 2 .S 2 (Tlab − Tbuff ( t ))
(3.7)
where: Tbuff(t) is the temperature in the buffer zone, °C. volbuff is the volume of buffer zone, m3. k2 is the heat transfer coefficient between buffer zone and environment, W m−2 °C−1; S2 is the surface of heat exchange between buffer zone and environment, m2. Tlab is the environment temperature, °C. γbuff is the density of air in buffer zone, kg m−3. Cp,buff is the specific heat of air in buffer zone, J kg−1 °C−1. with Cp,buff=Cp and γbuff=γ, Eqn (3.7) can be simplified analogous to dtbuff ( t ) dt
= K 3 .ti (t ) − ( K 3 + K 2 )tbuff (t )
57
(3.8)
where the coefficients K1, K2 and K3 are: K1 =
k1 .S1 k 2 .S 2 k1.S1 ; K2 = ; K3 = voli .γ .C p voli .γ .C p volbuff .γ .C p
These first orders of differential Eqns (3.5) and (3.8) can be converted to a transfer function (TF) form by using the Laplace operator. These yields
ti (t ) =
K1 β1 t 0 (t − τ ) + tbuff (t ) s + α1 s + α1 tbuff (t ) =
(3.9)
K3 ti ( t ) s + ( K2 + K3 )
(3.10)
The block diagram of Eqns (3.9) and (3.10) is given in Fig. 3.4. t0(t-τ)
β1
1 s + α1
+
ti(t)
K1 K3 s + (K 2 + K3 )
tbuff(t)
Fig. 3-4 Block diagram of the feedback connected the second order transfer function: s, Laplace operator; t0(t−τ), supply air temperature at time; tbuff(t), temperature of air in the buffer zone at time; ti(t), temperature of air in the well mixed zone; t, time; τ, advective time delay; β1, local volumetric concentration of fresh air rate; K1, K2, K3, α1, model parameters Combining transfer functions in Eqns (3.9) and (3.10), the second order continuous-time TF model for the central chamber-buffer zone system becomes (s is Laplace operator)
ti (t ) =
b0 s + b1 t0 (t − τ ) s + a1s + a2 2
where: b0=β1 ; b1 = β1 ( K 2 + K 3 )
a1 = α1 + K 2 + K 3 a2 = (K 2 + K 3 )α1 − K1 K 3 = (K 2 + K 3 )α1 − (α1 − β1 )K 3
58
(3.11)
3.2.3.2 Data-based phase in the empty room
The dynamic data obtained from dynamic experiments were used. In ‘data-based phase’, the modelling technique was used to fit data to several equations and evaluate each equation by statistical meaning. Although other techniques are available, the simplified refined instrumental variable (SRIV) approach was employed as a method for model identification, since it not only yields consistent estimates of the parameters but also exhibits close to optimum performance in the model order reduction context (Young, 1984). By combining the mechanistic and data based phase, a second-order fitting equation could be selected to estimate the heat exchange process in the empty room. 3.3 Results and discussions The uniformity of the air temperature distribution was calculated by the well mixed zone
approach with the acceptable temperature differences and the cause of the non uniformity was captured by applying the DBM approach both for the empty room and for the room filled with obstacles. 3.3.1. Uniformity of air temperature in the empty test installation A lot of applications require perfect mixing in certain regions (eg. storage of fruits and
vegetables, thermal processing, cooling, etc.). One of the control strategies in these applications can be the positioning of a specific well mixed zone. The well mixed zones were defined as the maximum 3D volume in which an acceptable temperature difference is measured. The well mixed zones were identified by visualizing the measured 3D temperature distribution (Fig. 3.5a).Then the average temperature is calculated at the considerate time and the position and volumes of well mixed zones were calculated with an acceptable temperature difference.
The temperature uniformity index Itemp is a useful index to quantify the spatial homogeneity of temperature in a ventilated airspace, and is defined as the volumetric part in % of airspace 59
with the temperature between the limiting value Tavg–ΔT and Tavg+ΔT, where Tavg is the average of air temperature in the chamber, and ΔT is the acceptable temperature difference .
ΔT is the range of temperature which hasn’t an effect on the quality and efficiency of the process, ΔT is defined by processes (Janssen, 1999). For example, in an oven for microchips the acceptable temperature difference is 0.02oC. In this thesis we used 0.2oC. n
I temp =
∑ vol i =1
i
(3.12)
VOL
where: voli is the volume of the well-mixed zone i in m3 ( voli =0 if temperature is not in Tavg±ΔT); VOL is a considerable volume in m3; ΔT is the acceptable temperature difference in °C; and Tavg is the average of air temperature in the chamber in °C.
As a representative example, Figs 3.5a and 3.5b visualise the measured 3D temperature distribution in the empty room under the following steady state condition (duration for steady state is 2 h). This figure illustrates that the empty room has an imperfectly mixed air volume with a non-uniform temperature distribution, and the volume part of airspace with acceptable temperature difference 0.2°C is 66.7%.
Air inlet
Air outlet
60
Outlet
Inlet
Width, m
0.8
2.0 1.6
0 1.2
1.6 0.8 0.8 Height, m 0 21.9
22.2
Length, m
0.4
22.4
0 22.6
22.8
23.1
23.3
Temperature scale, oC
23.5
23.7
23.9
24.2
(a)
Width, m
0.8
2.0 1.6
0 1.2
1.6 0.8 0.8
0.4
Height, m 0
Length, m
0
Fig. 3-5 (a) Visualisation of measured three-dimensional temperature distribution in the empty room under the following steady-state conditions: ventilation rate 120 m3 h−1, air supply temperature 17°C and internal heat production 300W; (b) visualisation of the 3D zone in the empty room with acceptable temperature difference 0.2°C and average temperature in a well mixed zone 23.1±0.2 oC 61
The measured temperature uniformity index is shown in Fig. 3.6 as a function of the acceptable temperature difference ΔT and the ventilation rate. The higher the acceptable temperature difference, the more volume of air in the test installation fulfils the condition of uniformity. The higher the ventilation rate, the better the air inside the room is mixed, resulting in a higher uniformity index. The temperature distribution in airspace is depend on airflow pattern (Berckmans and De Moor, 1993) and the relation between uniformity index to ventilation rates were non-linear (Özcan et al., 2005), but the high velocity in airspace is
Temperature uniformity index, %
resulted in the turbulent flow (high in the Reynol number) occurred in this experiment.
100 90 80 70 60 50 160 150 140 Ventilation rate, m3 h-1
0.6
130 120
0.2
0.8
1
0.4 o Acceptable C oC Acceptabletemperature temperaturegradient, difference,
Temperature uniformity index, %
(a)
100 90 80 70 60 50 280 270 260 3 -1
Ventilation rate, m h
250 240
0.2
0.4
0.6
0.8
1
Acceptable temperature gradient, oC Acceptable temperature difference, oC
(b)
Fig. 3-6 Temperature uniformity index (Itemp) of the empty room: (a) low ventilation rates (120–160 m3 h−1); (b) high ventilation rates (240–280 m3 h−1) with an acceptable temperature difference from 0.2 to 1°C
62
3.3.2. Data-based phase in the empty room Applying continuous time SRIV algorithm (Young, 1981) to estimate the parameters in the
first and second order transfer function is based on coefficient of determination RT2 and minimisation of the Young identification criterion (YIC) value. Positions (1) and (32) in the empty room were used as examples for calculating.
From Table 3.2, it is obvious that the first and the second-order transfer function can be applied in the empty room, but the second order has a higher value for RT2 than the first order. This result is in good agreement with previous research (Janssens et al., 2004). Table 3-2 The model parameter estimates with for sensor position (1) and (32) in the empty room with ventilation rate is 280 m3 h−1 Sensors
Order
[m ,n,τ]
of TF First
[0,1,10]
order 1
Second
[1,2,10]
order
First 32
[0,1,10]
order Second order
[1,2,10]
Parameter
Standard error
RSE,
estimates
of parameters
%
a1= 0.0297
0.00069
2.32
b0= 0.0155
0.00035
2.26
a1= 0.1135
0.00680
5.99
a2= 0.0004
0.00000
0.00
b0= 0.0427
0.00230
5.39
b1= 0.0003
0.00000
0.00
a1= 0.0723
0.00170
2.35
b0= 0.0481
0.00110
2.29
a1= 0.2458
0.01120
4.56
a2=0.0010
0.00010
10.00
b0= 0.1350
0.00590
4.37
b1= 0.0007
0.00010
14.29
RT2
YIC
0.0189
0.9759
-10.58
0.0076
0.9904
-8.23
0.0141
0.9886
-11.34
0.0035
0.9972
-10.06
SE
values
TF, transfer function; RSE, relative standard error of parameters (in %, it is the estimate's standard error (SE) divided by the estimate itself. The larger the RSE, the less precise the estimate); SE, standard error of equations; RT2 , coefficient of determination; YIC, Young identification criterion; m, n and τ, denominator, numerator and time delay; a1, a2, a3, b0, b1, parameters in the first and second order of transfer function. By modelling the temperature responses at each sensor location in the experimental chamber, physical meaningful model parameters were derived. The most important parameter in 63
relation to uniformity of air temperature is the local volumetric concentration of fresh airflow rate β1. In Fig. 3.7, the local volumetric distribution of the concentration of fresh airflow rate is shown on a vertical plane in the test installation as a function of the ventilation rate in the empty test installation. (At the low ventilation rate (120 m3 h−1) β1 changes in the range of 0.0013–0.0975 compared to 0.0042–0.135 at the high ventilation rate (280 m3 h−1)).
Fig. 3-7 Partial contours of parameter β1 in the empty chamber: (a) front plane at the ventilation rate is 120 m3 h−1; (b) front plane at the ventilation rate is 280 m3 h−1 Further, the contour plots calculated from the measured dynamic temperature distribution relate well to the airflow pattern (Fig. 3.7). The incoming air rapidly moves from the top to the bottom through holes on the duct system and coming out. Therefore, the air circulates 64
much more slowly at the top positions. The performance of β1 value (Fig. 3.7) is also in good agreement with the visualised airflow pattern in the smoke injection experiment (Fig. 3.8).
(a)
(b) Fig. 3-8 Visualisation of the airflow pattern with smoke: (a) 120 m3 h−1; (b) 280 m3 h−
65
3.3.3. Uniformity of air temperature in the test installation filled with obstacles Figures 3.9(a) and (b) visualise the measured 3D temperature distribution in the obstacle
room under the following steady-state condition. The volume part of airspace with acceptable temperature difference 0.2°C is 13.89 % of bulk volume.
Inlet
Outlet
(a)
66
Width, m
0.8
2.0 1.6
0 1.2
1.6 0.8 0.8 Height, m
0.4 0
Length, m
0
Fig. 3-9 (a) Visualization of measured three-dimensional airspace temperature distribution in obstacle room under the following steady state conditions: ventilation rate 120 m3 h−1, air supply temperature 17 °C and internal heat production 300 W; (b) Visualisation of the 3D zone in the obstacle room with acceptable temperature difference 0.2°C, a well mixed zone 23.6±0.2 oC The performance of these plots has the same form as the empty room (Fig. 3.10). At low ventilation rates and low acceptable temperature differences, the uniformity index is less. The lower the acceptable temperature difference, the less the volume of air in the test installation fulfils the condition of uniformity. The lower the ventilation rate, the less the air inside the room is mixed, resulting in a lower uniformity index. More detail of the comparison of the uniformity index in both cases is shown in Fig. 3.14.
67
Temperature uniformity index, %
100 80 60 40 20 160 150 140 3 -1
Ventilation rate, m h
130 120
0.4
0.2
0.6
0.8
1 o
Acceptable temperature gradient, C Acceptable temperature
difference, oC
Temperature uniformity index, %
(a)
100 80 60 40 20 280 270 260 3 -1
Ventilation rate, m h
250 240
0.2
0.4
0.6
0.8
1
o
Acceptable temperature gradient, C
Acceptable temperature difference, oC
(b) Fig. 3-10 Temperature uniformity index of airspace (Itemp) of the obstacle chamber at (a) low ventilation rates (120–160 m3 h−1); (b) high ventilation rates (240–280 m3 h−1)
3.3.4. Data-based mechanistic modelling approach in the obstacles room The continuous-time SRIV algorithm was used to identify and estimate the parameters in the
first, second and third order transfer function at 36 positions in the room. Positions 1 and 32 were used as examples in this chapter (Fig. 3.11).
68
21.5
20.5
o
Tem perature, C
21
20
19.5
19
18.5 0
1000
2000
3000
4000
5000 Time, s
6000
7000
8000
9000
10000
5000 Time, s
6000
7000
8000
9000
10000
(a) 20.5 20
o
Tem perature, C
19.5 19 18.5 18 17.5 17 16.5 0
1000
2000
3000
4000
(b)
Fig. 3-11 (a) The output of the first-order, second-order transfer function model compared with the measured airspace temperature response at the sensor position (1); (b) at the sensor position (32) in the obstacle chamber at ventilation rate 280 m3 s−1; , first order; , second order; noise line, experiment data
69
Table 3-3 The model parameter estimates with for sensor position (1) and (32) (in the obstacle room with ventilation rate is 280 m3 h−1) Sensors
Order of TF
[m, n,τ]
1
First order Second order
[0,1,10]
32
[1,2,10]
Third order
[2,3,10]
First order Second order
[0,1,10]
Third order
[2,3,10]
[1,2,10]
Parameter estimates a1= 0.0167 b0= 0.0085 a1= 0.0913 a2= 0.0003 b0= 0.0284 b1= 0.0002 a1= 0.1520 a2= 0.0022 a3= 0.0000 b0= 0.0373 b1= 0.0008 b2= 0.0000 a1= 0.1156 b0= 0.0936 a1= 0.3746 a2= 0.0024 b0= 0.2574 b1= 0.0002 a1= 0.9171 a2= 0.0224 a3= 0.0001 b0= 0.5666 b1= 0.017 b2= 0.000
Standard error of parameters 0.0032 0.0015 0.0028 0.0000 0.0011 0.0000 0.0176 0.0007 0.0000 0.0027 0.0002 0.0000 0.0026 0.0021 0.0024 0.0001 0.0093 0.0001 0.0317 0.0013 0.0000 0.0193 0.0009 0.0000
RSE, %
SE
RT 2
YIC values
1.92 1.18 4.49 0.00 3.87 0.00 11.5 31.8 0.0 7.42 2.5 0.00 2.25 2.24 3.71 4.17 3.61 5.00 3.46 5.80 0.00 3.41 5.29 0.00
0.0154
0.97
-10.84
0.0028
0.99
-9.60
0.0027
0.99
-5.61
0.008
0.99
-12.07
0.0015
0.99
-11.58
0.0013
0.99
-10.35
TF, transfer function; RSE, relative standard error of parameters; SE, standard error of equations; RT2 , coefficient of determination; YIC, Young identification criterion; m, n and τ, denominator, numerator and time delay; a1, a2, a3, b0, b1, b2, parameters in the first and second order of transfer function. Table 3.3 shows that the best identified transfer function model in terms of YIC identification criterion is a first order model. The best-fitting equation is the third-order model associated with the smallest standard error and highest value for the coefficient of determination ( RT2 ). The residuals plot of the first, second and third order models are shown in Fig. 3.13. From this figure, there is a significant difference between the first and second order model, but not between the second and third order model. Although, the third order model gives the best fitting equation, it is more complicated than the second order model (high in YIC values). So,
70
the second order of transfer function can be selected to estimate the air temperature distribution in the obstacle room with accuracy comparable to the third order model.
Fig. 3-12 Residual plots of the first, second and third order model at the sensor position (32) in the obstacle room with the ventilation rate 280 m3 h−1 The partial contour of β1 is equal to b0 parameter in a second-order transfer function of rear and front planes as presented in Fig. 3.13. β1 value at the position (32) is higher than the position (1). A high velocity in the obstacle room with the same ventilation rate is the cause of a high β1 in the position (32), but β1 in the obstacle room is not proportional to the air velocity. There is a significant relation with the heat transfer to objects and the airspace velocity distribution in the obstacle room.
The partial value of β1 for the front plane is presented in Fig. 3.13.
71
Fig. 3-13 Partial contours of parameter β1 in the obstacle chamber at the front plane with the ventilation rate 280 m3 h−1 3.3.5. Comparison between the empty test installation and the test installation filled with obstacles The same ventilation rate and acceptable temperature difference resulted in a different
temperature uniformity index of airspace with the empty room and room filled with obstacles (Fig. 3.14). With an acceptable temperature difference below 0.6°C there is a large difference between the two experiments. At an acceptable temperature difference above 0.8°C, however, there was not a significant difference. At higher ventilation rate, a higher temperature uniformity index was measured in the chamber due to the transport of internal heat production.
72
Temperature uniformity index, % Temperature uniformity index, %
100 80 60 40 20 0
120
140
150
160 240 250 Ventilation rate, m3 h-1 (a)
260
270
280
120
140
150
160 240 250 3 -1 Ventilation rate, m h (b)
260
270
280
100 80 60 40 20 0
Fig. 3-14 Comparison between the temperature uniformity index of airspace (Itemp) for two values of the acceptable temperature difference 0.2°C (a) and 1°C (b); , empty room; , obstacle room
The airspace (was defined as air surrounding the balls in Fig. 1.2) in the obstacle chamber was smaller than in the empty chamber (approximately 27% in comparison with the empty room), so the air velocity in the obstacle chamber was relatively higher in comparison with the empty chamber at the same ventilation rate of inlet air. The high velocity in the obstacle chamber means that there is more fresh air coming to each place in the chamber. It is clear that the value of β1 in the chamber with obstacle is high compared to that for β1 in the empty chamber but the value for β1 is not proportional to the air velocity in the obstacle room. A part of energy has been transferred to the air inside the obstacles so the β1 should be smaller than for the same air velocity in the empty room.
Acceptable temperature difference and local volumetric concentration of fresh air rate β1 have great relationship with uniformity index in both cases. Those are parameters to be considered to control the uniformity of temperature final products during thermal processing and storage processes. 73
3.4 Conclusions To control the uniformity of product quality in thermal processing and storage processes, in
this study, an earlier developed data-based mechanistic (DBM) approach was extended from an imperfect mixing process in an empty mechanically ventilated laboratory test chamber to a test installation filled with obstacles. The DBM model, with physically meaningful model parameters, could model the spatio-dynamic temperature response at 36 positions in the room to a variation of the supply air temperature with an accuracy of 0.1°C.
The local fresh air concentration in the empty test installation showed good qualitative agreement with the smoke visualisation experiments. This means that dynamic temperature experiment could be used to quantify the air temperature distribution inside the room.
Local fresh air rate in porous media was higher than in the empty room. For example, 0.028 and 0.247 compared to 0.042 and 0.135 at the positions (1) and (32), respectively, at the same rate of ventilation input (280 m3 h−1). This was explained by the fact that the local volumetric concentration of fresh air rate in the porous media is not only determined by air distribution but it also by the heat transfer to the objects.
A model from the second order and a coefficient of determination RT2 >0.99 was found to predict temperature distribution in the obstacle room with a very small error (<0.1oC). This model is an important first step to control the temperature distribution.
74
4 Chapter 4 Data Based Mechanistic Modelling for Control of Three Dimensional Temperature distribution in a Ventilated Room Filled with Obstacles
75
76
Chapter 4 Data Based Mechanistic Modelling for Control of Three Dimensional Temperature distribution in a Ventilated Room Filled with Obstacles 4.1 Introduction In chapter 3, the Data Based Mechanistic modelling (DBM) approach from an empty room
was adapted for application in a room filled with obstacles. The physically meaning of the parameters was not yet defined for a room filled with obstacles.
In this chapter, a mechanistic phase will be developed to find physically meaningful parameters based on the heat transfer from airspace to air inside balls in the test installation.
Uniformity of product quality in several processes such as thermal processing, cool storage, egg incubation so far are not realised by the used control systems. Many researchers and methods have been suggested to design and control the three-dimensional temperature distribution in process room. (e.g. Van Brecht et al., 2003; Verboven et al., 2004; Qian Zou et al., 2006a; Qian Zou et al., 2006b).
In computational fluid dynamics (CFD), the conservation equations for mass, momentum and thermal energy are solved for all nodal points of a two or three-dimensional grid or around the object under investigation. A well-known example of a CFD model is PHOENICS (Spalding, 1981). Velocity and air temperature distribution in storage rooms was modelled using CFD to optimise cold storage designs (Mirade, 2003; Frederic et al., 2004; Verboven et al., 2004; Nahor et al., 2005). Despite the advantages to optimise and for design, A CFD models suffer from their complex nature and cannot be used in applications where reduced order model is mandatory, e.g. in controller algorithms (Zerihun Desta et al., 2004).
77
In this PhD, Data based mechanistic model (DBM) approach was developed for two reasons: (1) A data based mechanistic model is relatively compact, characterises the dominant modal behaviour of the dynamic system and is, therefore, an ideal basis for model-based control system design (Camacho & Bordons, 1999; Maciejowski, 2002). (2) the model structure of a data based mechanistic model provides a physically meaningful description of the process dynamics and can, therefore, be easily applied to a wide range of imperfect mixing processes: in different fluids (air, water), on different scales (micro and macro scale) and in numerous application areas (Janssens et al., 2004).
The objectives of this chapter were to extend an existing data based mechanistic modelling approach to rooms filled with obstacles and to model the heat transfer to the obstacles in ventilated porous media. Data based mechanistic modelling approach was applied to identify the model parameters for online adaptive control of three dimensional temperature distribution in a room filled with obstacles. 4.2
Materials and methods
4.2.1. Laboratory test room In this experiment, the room described in Chapter 2 (Fig. 2.1) was used to develop a model.
As explained in chapter 2, the test installation was filled up with 480 plastic balls (Fig. 2.2) with identical and well-known characteristics, each with a diameter of 0.19 m, weight 0.12 kg. These obstacles represented the physical presence of material in heating and cooling applications. Ten calibrated thermocouples were inserted inside ten balls (Fig. 2.4b) to measure the air temperature inside balls positioned as shown in Fig. 4.1. The porosity based on airspace in the room and obstacle volume in this experiment was approximately 0.27.
78
Inlet
Outlet
Air inlet
34
28
22
10
16
4 0.8m
35 31
29 25
23 19
11
17 7
13
5 1 0.8m
36 Height
30
24
12
18
6
32
26
20
14
8
2
33
27
21
15
9
3
Width 0.5m
0.8m Air outlet
0.375m
0.4m Length
Thermocouple positions in airspace Thermocouple positions in airspace and inside balls (3,4,6,15,18,27,30,33,34,36)
Fig. 4-1 Numbered sensors in the test chamber 4.2.2. Experiments Experiments were conducted in two configurations of this room: empty room as a reference;
and room filled with 480 plastic balls.
In both configurations, the ventilation rates were varied between 120 to 280 m3 h-1 (13.3 to 31.1 volume changes per hour for empty room, and 27.2 to 63.2 volume changes per hour for 79
obstacle room). In each experiment at a specific ventilation rate, after 2000 seconds of steady state, the inlet temperature was adjusted from 11 to 17oC and from 17 to 11oC. After each step, the temperatures in the room and in the obstacles were monitored for 10,000 seconds
Temperature, oC
Temperature, oC
(Fig. 4.2). The overview of the dynamic experiments was in Table 4.1.
17
11
17
11
(a) 2000
Time, s
(b) 2000
10000
Time, s
10000
Fig. 4-2 Step up (a), step down (b) of dynamic experiments,(c) The measured dynamic response of airspace and air inside ball temperature from the inlet air temperature for a ventilation rate 280m-3 h-1 ; , inlet temperature setting; , measured response
80
Table 4-1 Overview of the dynamic experiments No. Experiments
Ventilation rates (m3 h-1)
1
120
2
120
3
140
4
140
5
160
6
160
7
180
8
180
9
240
10
240
11
260
12
260
13
280
14
280
Dynamic experiments Temperature (oC) Total time of experiment (s) 11 2000 17 10000 17 2000 11 10000 11 2000 17 10000 17 2000 11 10000 11 2000 17 10000 17 2000 11 10000 11 2000 17 10000 17 2000 11 10000 11 2000 17 10000 17 2000 11 10000 11 2000 17 10000 17 2000 11 10000 11 2000 17 10000 17 2000 11 10000
4.2.3. Data based mechanistic modelling approach This approach was described in Chapter 2 (Part 2.2.2.2) 4.2.4. Data based identification and estimation High frequent data (10 seconds interval) were obtained from dynamic experiments. In the
‘Data based phase’, a dynamic transfer function model was fitted through data and evaluated on its accuracy. Although other techniques are available, in this study the simplified refined instrumental variable (SRIV) approach was used as a method for model identification, since it 81
not only yields consistent estimates of the parameters but also exhibits close to optimum performance in the model order reduction context (Young, 1984).
The method to select the best-fitted model based on the coefficient of determination RT2 , Young Information Criterion YIC and standard error SE was in Chapter 2 (Part 2.2.2.2). 4.2.5. Mechanistic phase in the ventilated porous media Research on ventilated rooms (De Moor & Berckmans, 1993) has demonstrated that the
empty test chamber that is used in this study is an imperfectly mixed airspace with considerable spatio-temporal gradients of temperature. Within such an imperfectly mixed airspace it is always possible to define a number of well mixed zones (WMZs) around temperature sensors in which there exists a good mixing at an acceptably low temperature difference (Berckmans & Goedseeds, 1986; Berckmans et al., 1992a,b).
Assuming such a well mixed zone (WMZ1) (Fig. 4.3) with a volume V1 in which there are several sphere materials (diameter d in m) located. The volume of V1 has porosity ε, the surface of heat exchange between WMZ1 and the buffer zone is S1, and the airspace in the well mixed zone V1 is voli . Assuming sufficient mixing air in the buffer zone (WMZ2) to guarantee equal wall temperature and have the volume vol buff and surface of heat exchange between buffer zone and environment S2.
Assuming that the air inside each ball is well mixed and that heat absorbed by plastics material is negligible to simplify here the order of transfer function (mass of material will be considered in the next experiment), heat transfer in the system can be presented as in Fig. 4.3.
82
kk22SS22(T -T buff(t)) (Tlab (t)) lab-Tbuff
Tlab
WMZ2 Tbuff(t), volbuff, γbuff, Cp,buff, S2 Air enter the WMZ1 T0(t-τ), V, γ0, Cp,0 VT VT00(t(t-ττ))γγ00CCp,0 p,0
kk1SS1(T (t)-T (t)) 1 1(Tbuff buff(t)-Tii(t)) WMZ1 (V (V1) Ti(t), ε, S1
VT VTii(t) (t)γγiiCCp,i p,i
voli Air exits the WMZ V, Ti(t), V1, γ1, Cp,i
Ball kkmmSSmm(T (Tii(t)-T (t)-Tmm(t)) (t)) qc
Air inside ball Tm(t), γm, Cp,m, volm,,Sm, km
Fig. 4-3 Schematic representation of the well-mixed zone concept in ventilated room filled with obstacles The original heat balance differential equation (3.1) was extended with an addition tern (n.km.sm(Tm(t)-(Ti(t)) accounting for the heat transfer from airspace to the air inside ball as Eqn (4.1).
dTi (t ).voli .γ i .C p ,i
= V .T0 (t − τ )γ 0 .C p,0 − V .Ti (t )γ i .C p ,i + qc dt + n.k m .S m (Tm (t ) − Ti (t )) + k1 .S1 (Tbuff (t ) − Ti (t )) where: Ti(t) is the temperature in the well mixed zone, oC. T0(t-τ) is the input temperature, oC . V is part of the ventilation rate enter the well-mixed zone, m3 h-1.
voli is the volume of air in well mixed zone V1, m3. n is the number of material in well mixed zone. 83
(4.1)
Tm(t) is the air temperature inside ball, oC. Tbuff(t) is the temperature in buffer zone, oC. k1 is the heat transfer coefficient between WMZ1 and WMZ2, J s-1 m-2 oC-1. km is the heat transfer coefficient between WMZ1 and inside ball; J s-1 m-2 oC-1. Sm is the surface of the ball, m2. S1 is the surface heat transfer between WMZ1 and WMZ2, m2 qc is a part of the internal heat production entering the WMZ1, J s-1. Cp,0, Cp,i are the specific heat of air inlet and air in the well mixed zone i, J kg-1 oC-1.
γ0, γi are the density of air supply and air in the well mixed zone i, kg m-3.
For the energy exchange between the ventilated room and the buffer zone in Eqn 3.2 was used dTbuff ( t ).vol buff .γ buff .C p ,buff dt
= k1 .S1 (Ti ( t ) − Tbuff ( t )) + k 2 .S 2 (Tlab − Tbuff ( t ))
(3.2)
where, vol buff is the volume of buffer zone in m3; k2 is the heat transfer coefficient between buffer zone and environment in J s-1 m-2 oC-1; Tlab is the environment temperature in oC; γbuff is the density of air in buffer zone in kg m-3; Cp,buff is the specific heat of air in buffer zone in J kg-1 oC-1.
For the heat transfer between a WMZ1 and volume of the obstacle: m.C p ,m
dTm (t ) = k m .S m (Ti (t ) − Tm (t )) dt
(4.2)
where, m is the mass of air inside one ball in kg; Cp,m is the specific heat of air in the ball in J kg-1 oC-1; km is the heat transfer coefficient between WMZ1 and inside ball in J s-1 m-2 oC-1
84
Within the volume of WMZ1 (V1), n sphere materials are located (d diameter) with porosity
ε, the volume of the obstacles in WMZ1 is
in WMZ1 is n
(1 − ε )
ε
voli and the volume of n sphere materials
π.d 3 (1 − ε ) π.d 3 ; So vol i = n 6 ε 6
Let Sm be the surface of one sphere material, then total surface of all obstacles in the WMZ1 will be nSm. For a sphere material S m = πd 2 , and consequently: (1 − ε )
ε
vol i = n
π.d 3 d d or = n.π.d 2 = n.S m 6 6 6
n.S m =
6(1 − ε )voli ε .d
(4.3)
In Eqns (4.1), (3.2) and (4.3), if it is assumed that density and specific heat capacity remains constant, which named γ and Cp; therefore, the heat balance equations can be rewritten as: dTi (t ) qc 6(1 − ε )k m V V (T (t ) − Ti (t )) = + T0 (t − τ ) − Ti (t ) + dt voli .γ .C p ε .γ .C p .d m voli voli k1 .S1 + (Tbuff (t ) − Ti (t )) voli .γ .C p dTbuff ( t ) dt
=
k1 . S1 k 2 .S 2 (Ti (t ) − Tbuff ( t )) + (Tlab − Tbuff (t )) vol buff .γ.C p vol buff .γ.C p
dTm (t ) S m .k m = (Ti (t ) − Tm (t )) dt m.C p
(4.1)
(3.2)
(4.4)
Under steady state conditions Eqns (4.1), (3.2) and (4.4) will become _ _ − qc 6(1 − ε )k m − k1 .S1 V − V − T0− Ti+ + (T m − Ti ) + (T buff − Ti ) = 0 ε .γ.C p .d voli .γ.C p voli .γ.C p voli voli
− − − k1 .S1 k 2 .S 2 (T i − T buff ) + (Tlab − T buff ) = 0 volbuff .γ.C p volbuff .γ.C p
85
(4.5)
(4.6)
S m .k m − − (T i − T m ) = 0 m.C p −
−
−
(4.7)
−
where, T0 , Ti , Tbuff , Tm are the temperatures in the steady state condition in oC If we only consider small temperature perturbations (t0(t-τ), tbuff(t), ti(t), tm(t)) around steady state condition, subtract Eqn (4.1) from Eqn (4.5), Eqn (3.2) from Eqn (4.6), and Eqn (4.4) from Eqn (4.7) results in: dti (t ) 6(1 − ε )k m V V t0 (t − τ ) − ti ( t ) + (t (t ) − ti (t )) = dt voli voli ε .γ.C p .d m k1 .S1 + (tbuff (t ) − ti (t )) voli .γ.C p dt buff (t ) dt
=
k 1 . S1 k 2 .S 2 ( ti (t ) − t buff ( t )) + tbuff (t ) vol buff .γ.C p vol buff .γ.C p
dtm (t ) Sm .k m = (ti (t ) − tm (t )) dt m.C p
(4.8)
(4.9)
(4.10)
with
β1 =
k1.S1 V k 2 .S 2 k1.S1 ; K1 = ; K2 = ; K3 = ; voli voli .γ.C p volbuff .γ.C p volbuff .γ.C p Km =
6(1 − ε )k m 6.k m ; α2 = ε .γ.C p .d γ.d .C p
Then, the set of Eqns (4.8), (4.9), (4.10) can be rewritten as: dti (t ) = β1t0 (t − τ ) − β1ti (t ) + K m (tm (t ) − ti (t )) + K1 (tbuff (t ) − ti (t )) dt dt buff ( t ) dt
= K 3 ( ti ( t ) − t buff ( t )) + K 2 .t buff ( t )
dtm (t ) = α2 (ti (t ) − tm (t )) dt
(4.11)
(4.12)
(4.13)
After converting Eqns (4.11), (4.12) and (4.13) with the Laplace operator, with
α1 = β1 + K1 + K m , the transfer functions results in: 86
ti (t ) =
β1 K1 K t0 (t − τ ) + tbuff (t ) + m tm (t ) s + α1 s + α1 s + α1
tbuff (t ) =
(4.14)
K3 ti (t ) s + (K 2 + K3 )
tm (t ) =
(4.15)
α2 ti (t ) s + α2
(4.16)
The block diagram of these three first order transfer functions Eqns (4.14), (4.15) and (4.16) is given in Fig. 4.4.
This figure shows that the temperature of the air between the obstacles ti (t ) can be controlled by adjusting the supply air temperature t0 (t − τ ) when the estimated parameters β1, α1, α2, K1, K2, K3 are known.
tm(t)
α2 s + α2
Km t0(t-τ)
β1
1 s + α1
+
ti(t)
K1 tbuff(t)
K3 s + (K 2 + K3 )
Fig. 4-4 The third order block diagram of system Replace Eqns (4.14) and (4.15) in Eqn (4.16), results in Eqn (4.17) t i (t ) =
⎞ ⎛ K ⎞⎛ α ⎞ ⎛ K1 ⎞⎛ K3 β1 ⎟⎟ti (t ) + ⎜⎜ m ⎟⎟⎜⎜ 2 ⎟⎟ti (t ) ⎟⎟⎜⎜ t 0 (t − τ ) + ⎜⎜ s + α1 ⎝ s + α1 ⎠⎝ s + α 2 ⎠ ⎝ s + α1 ⎠⎝ s + ( K 2 + K 3 ) ⎠
87
(4.17)
Minimised Eqn (4.17). We obtained the following third order continuous- time transfer function model for the inlet temperature and the temperature in the well mixed zone in room filled with obstacles was obtained.
ti (t ) =
b0 .s 2 + b1 .s + b2 t0 (t − τ ) s 3 + a1 .s 2 + a 2 .s + a3
(4.18)
where, b0 = β1 ;
b1 = β1 (α2 + K 2 + K3 ) ; b2 = β1 .α2 ( K 2 + K 3 )
with
a1 = α1 + α2 + K 2 + K3
a2 = K 2 .α2 + K 2 .α2 + α2 .α1 + α1.K 2 + α1 .K 3 − K1 .K 3 − K m .α2 a3 = α1.K 2 .α2 + α1.K 3 .α2 + K1 .K 3 .α2 − K m .α2 .K 2 − K m .α2 .K 3
The comparison of the mechanistic model [Eqn. 4.18] with a third order transfer function shows that the model parameter β1 =
V equals b0 , which is the same formula of β1 in case voli
of an empty ventilated room [β1 has been defined as the local volumetric concentration of fresh air flow rate (Berckmans et al., 1992a)]. Therefore, β1 obtained from the third order transfer function in the ventilated porous media has the same physical meaning as β 1 in the empty room (from the second order transfer function).
Estimated parameters (b0, b1, b2, a1, a2, a3) from the data based phase are used to calculate α1,
α2, Km, K1 values which are related to the heat transfer parameters from the mechanistic phase.
88
4.3
Results and discussions
4.3.1. The third order transfer function in ventilated porous media ‘Data based phase’ applied to ventilated porous media resulted in a third order transfer function [Eqn (4.18)] presenting the heat transfer in a control volume filled with obstacles.
Table 4-2 Third order model parameter associated statistical measures for step up experimental data at 280 m3 h-1 inlet flow rate Positions 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 34 35 36
a0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
a1 0.058 0.057 0.003 0.068 0.053 0.024 0.067 0.048 0.005 0.076 0.071 0.009 0.093 0.072 0.032 0.093 0.076 0.027 0.139 0.162 -0.005 0.104 0.158 0.033 0.105 0.136 0.029 0.113 0.026 -0.001 0.111 0.225 0.085 0.010 0.348 0.018
a2 -0.00025 -0.00026 -0.00008 -0.00031 -0.00025 -0.00015 -0.00030 -0.00024 -0.00001 -0.00035 -0.00030 -0.00010 -0.00029 -0.00027 -0.00019 -0.00028 -0.00027 -0.00015 -0.00034 -0.00025 -0.00008 -0.00027 -0.00040 -0.00012 -0.00035 -0.00040 -0.00012 -0.00035 -0.00016 -0.00011 -0.00030 -0.00048 -0.00013 -0.00036 -0.00069 -0.00014
a3 -0.0000027 -0.0000026 -0.0000002 -0.0000029 -0.0000024 -0.0000011 -0.0000030 -0.0000021 -0.0000002 -0.0000032 -0.0000032 -0.0000004 -0.0000049 -0.0000035 -0.0000013 -0.0000050 -0.0000038 -0.0000013 -0.0000078 -0.0000102 0.0000004 -0.0000058 -0.0000088 -0.0000020 -0.0000052 -0.0000070 -0.0000018 -0.0000058 -0.0000012 0.0000004 -0.0000061 -0.0000128 -0.0000058 -0.0000048 -0.0000200 -0.0000008
b0 0.0184 0.0200 0.0034 0.0186 0.0149 0.0066 0.0226 0.0144 0.0036 0.0189 0.0218 0.0045 0.0355 0.0227 0.0001 0.0252 0.0216 0.0046 0.0534 0.0647 0.0053 0.0321 0.0382 0.0071 0.0396 0.0668 0.0136 0.0420 0.0125 0.0073 0.0407 0.1130 0.0383 0.0346 0.1878 0.0119
89
b1 -0.000031 -0.000040 -0.000034 -0.000026 -0.000015 -0.000016 -0.000039 -0.000027 -0.000033 -0.000008 -0.000017 -0.000033 -0.000002 0.000001 0.000065 0.000036 0.000011 0.000022 0.000057 0.000150 -0.000095 0.000065 0.000193 0.000074 -0.000028 -0.000075 -0.000019 -0.000011 -0.000052 -0.000118 0.000031 0.000002 0.000077 -0.000023 -0.000013 -0.000075
b2 -0.0000018 -0.0000019 -0.0000002 -0.0000019 -0.0000016 -0.0000007 -0.0000021 -0.0000015 -0.0000002 -0.0000021 -0.0000023 -0.0000003 -0.000004 -0.0000026 -0.0000010 -0.0000033 -0.0000027 -0.0000010 -0.0000061 -0.0000081 0.0000003 -0.0000043 -0.0000064 -0.0000018 -0.0000039 -0.0000058 -0.0000016 -0.0000043 -0.0000011 0.0000003 -0.0000046 -0.0000105 -0.0000049 -0.0000035 -0.0000168 -0.0000007
RT 2 0.990 0.997 0.934 0.996 0.997 0.998 0.995 0.926 0.998 0.987 0.997 0.998 0.998 0.997 0.999 0.964 0.977 0.994 0.998 0.997 0.998 0.949 0.908 0.998 0.997 0.996 0.998 0.998 0.998 0.996 0.998 0.998 0.994 0.994 0.990 0.999
YIC -6.29 -8.87 0.58 -7.92 -7.63 -8.42 -7.89 -2.58 -7.42 -3.27 -7.41 -8.35 -2.52 -0.78 -5.62 -3.31 -2.01 -5.91 -8.58 -8.87 -5.75 -3.30 -3.23 -13.36 -7.71 -7.91 -7.28 -6.17 -9.94 -3.48 -7.97 -1.11 -9.03 -5.99 -0.81 -9.65
Table 4.2 shows typical examples of parameter estimates for 36 sensor positions within the obstacles (airspace) with corresponding coefficient of determination RT2 and YIC values. In this illustration, a step rise in inlet temperature at an airflow rate of 280 m3 h-1 was used to generate the data. The third order fitted model is showed in Fig. 4.5. In Table 4.2, for each sensor position, the coefficient of determination RT2 is very high (>0.99) and YIC value is low in the most estimations. The predicted values were shown in Fig. 4.5a, fit very well with the calculation of the third transfer functions, resulting in a mean prediction error of 0.1oC (Fig. 4.5b).
(b)
(a)
Fig. 4-5 Third order fitted model and residual plot for 280 m3 h-1 inlet air flow rate (63.6 volume changes per hour): (a) position 27; (b) position 30
4.3.2. Physical meanings of estimated parameters V Value β1 = b0 is equal to and is a local volumetric concentration of fresh airflow rate for voli a well mixed zone at a specific sensor position. The distribution of β 1 in a vertical plane 90
within the test installation can be shown as a partial contour plots for front and rear plane as in Fig. 4.6. From this figure, it is obvious that close to the air inlet position β 1 has a higher value than at other positions. The resistance of air movement in a room filled with obstacles resulted in the high of β1 at the air inlet positions.
Fig. 4-6 Two dimensional contours of parameter β1 in the obstacle chamber at the ventilation rate 280 m3 h-1 at positions of vertical planes: (a) rear plane; (b) front plane Also within the ventilated room, the airflow pattern of fresh air corresponds to zones with higher β 1 values. The difference between the rear and front plane in Fig. 4.6 can probably be explained by a non-symmetrical distribution of obstacles in the room.
At the same position (sensor 33), with a higher ventilation rate, a higher β 1 value was found (Fig. 4.7). This can be explained by the fact that β 1 is related to the air refreshment in the room. At other positions however (sensor 3), an increase in ventilation rate does not have a significant effect on β 1 . This means that β 1 not only depends on air replacement but also on the airflow pattern. 91
0.04 0.035
0.025
1
Parameter β , s
-1
0.03
0.02 0.015 0.01 0.005 0
160
180
200
220
240
260
280
Ventilation rate, m 3 h -1
Fig. 4-7 Comparison of parameter β1 at the different ventilation rates (160-280 m3 h-1) , position 3; , position 33
In ‘the data based phase’, the simplified refined instrumental variable (SRIV) technique was also used to identify parameters in Eqn (4.16) for modelling the temperature in the obstacles. Results are presented in Table 4.3. Table 4-3 First order model parameters in a relationship between temperature in WMZ1 ( ti (t ) ) and material temperature ( tm (t ) ) with associates statistics measure for step up experiment at 280 m3 h-1 Sensor positions
a0
a1
b0
RT 2
YIC values
3 6 15 18 27 30 33 36
1 1 1 1 1 1 1 1
0.0225 0.0121 0.0228 0.0112 0.0179 0.0226 0.0170 0.0264
0.0195 0.0115 0.0228 0.0108 0.0174 0.0212 0.0168 0.0263
0.9995 0.9994 0.9995 0.9987 0.9995 0.9998 0.9990 0.9997
-17.6490 -17.9800 -17.2940 -16.3210 -18.2790 -19.8300 -16.9460 -19.0890
92
Differences between a1 and b0 values, % 13.30 5.25 0.11 3.86 2.92 6.28 1.33 0.56
Table 4.3 shows a typical example of parameter estimates for eight sensor positions to model the temperature inside the obstacles with corresponding coefficient of determination RT2 and YIC values.
From Eqn (4.16), that has the same structure as a first order transfer function
y (t ) =
b0 u(t ) with y(t ) = tm (t ) ; u (t ) = ti (t ) , α2 values can be estimated from a1 or b0. a0 .s + a1
The estimated parameters a1 and b0 are not very different (maximum 13.3% at position (2) and minimum 0.11% at position (15)) in Table 4.3 and based on Eqn (4.18), these parameters are an estimate for α 2 .
At the same time, α 2 has a relationship with the heat transfer coefficient (km) and consequently, the heat transfer coefficient km can be calculated with known parameters α 2 , γ, d and Cp from equation α2 =
6.k m γ .d .C p
Otherwise, α 2 is the same structure with “cooling rate” in Newton’s law for cooling processes. The “cooling rate” value presents the change in product temperature for unit change of cooling time for each degree of temperature difference between the product and its surrounding. It depends on many factors including the rate of heat transfer, the difference in temperature between the product and the cooling medium, the nature of the cooling medium, the physical properties of the product, etc. (Brosnan & Sun, 2001).
The results of the α2 calculations for two-ventilation rates 160 and 280 m3 h-1 are presented in Fig. 4.8. The α 2 values are usually high in case of high ventilation rates due to increased 93
convection, but α 2 does not only depend on ventilation rate but also on the airflow pattern expressed by the β 1 value.
0.045 0.04
Parameter Cooling rate,αs2-1, s-1
0.035 0.03 0.025 0.02 0.015 0.01 0.005 0
3
6
15
18
27
Sensor positions
30
33
36
Fig. 4-8 Parameter α2 at several positions at ventilation rates 160 m3 h-1 and 280 m3 h-1 , ventilation rate 160 m3 h-1; , 280 m3 h-1
4.3.3. Calculating parameters from the third fitted model in the ventilated porous media It can be concluded that, the values in the set of transfer functions [Eqns (4.14), (4.15) and (4.16)] in this model for ventilated porous media, are based on estimated values from a third order transfer function in relationship between inlet temperature (t0(t)) and the temperature in WMZ1 (ti(t)) [Eqn (4.18)] combined with a the first order transfer function in a relationship between temperature inside the obstacles (tm(t)) and the temperature in WMZ1 (airspace) (ti(t)) [Eqn (4.16)].
The result of all estimated parameters at sensor position (33) at different ventilation rates is shown in Table 4.4. 94
Table 4-4 Calculated parameters at position (33) with ventilation rate from 220 to 280 m3 h-1 Ventilation rates, m3 h-1 220 240 260 280
β1
α1
α2
0.0279 0.0254 0.0363 0.0383
0.0814 0.0541 0.0891 0.0767
0.0143 0.0153 0.0164 0.0169
K1 0.0496 0.0247 0.0484 0.0338
Km 0.0039 0.0041 0.0044 0.0046
Comparison of local volumetric concentration of fresh air rate β1 and cooling rate α2 at the different ventilation rates is shown in Fig. 4.9. In general, higher values of β1 result in higher
α2 values. This can be explained by the fact that the heat transfer coefficient from the obstacles (expressed by α2) has a good relationship with air flow pattern and resulting air velocity in the media (expressed by β1), so estimated values from data based mechanistic model are in agreements with the heat transfer theory. The air flow pattern in a process room resulted from the convection and buoyancy effects. If buoyancy changes, the airflow pattern will change and heat transfer from airspace to products is also changed. But β1 in this research is the combination of convection and buoyancy effects which were obtained by online modelling. This technique could be used to control the 3D temperature distribution. 0.09
β
0.08
1
α
2
β1 and α2 (s -1)
0.07 0.06 0.05 0.04 0.03 0.02 0.01 0
220
240 260 3 -1 Ventilation rate (m h )
280
Fig. 4-9 Parameters β1 and α2 value at the difference of ventilation rates (position 33) 95
4.4 Conclusions A data-based mechanistic model was applied to model temperature distribution in the room filled with obstacles. A third-order transfer function was generated from heat transfer theories, to represent the dynamic response of heat exchange processes in the air within the ventilated porous media. By fitting the data from a dynamic experiments to a third-order transfer function, close agreement with local temperature measurements was obtained ( RT2 > 0.99), corresponding to an error of <0.1oC.
By introducing a mechanistic phase in the model, several parameters in the fitted model could be related to physical meaningful parameters. Two important parameters have been extracted: local volumetric concentration of fresh air β1 and “cooling rate” α2. β1 values were related to the air temperature distributions in the porous media and α2 values were used to present the heat transfer rate from airspace to products.
Estimated values of β1 and α2 were found at the difference of ventilation rates and positions had good agreements with the general heat transfer theories in the air ventilated in the porous media.
Further more, a new method to calculate the cooling rate was developed in this chapter. It was obtained from the dynamic temperature data of product and its surrounding medium temperature.
96
5 Chapter 5 Data Based Mechanistic Modelling of Three Dimensional Temperature Distribution in Ventilated Rooms Filled with Biological Material
97
98
Chapter 5 Data Based Mechanistic Modelling of Three Dimensional Temperature Distribution in Ventilated Rooms Filled with Biological Material 5.1 Introduction As described in chapter 4, Data based mechanistic modelling approach was successfully applied in a room filled with balls. A third order transfer function from the dynamic response of the temperature airspace between the balls to variation of from inlet air temperature was found as the best-fitted model. This is in correspondence with the general theory of heat transfer from airspace to the air inside balls. In practise, there is no application with the heat transfer from airspace to the air inside the ball. Therefore, data based mechanistic model for a room filled with biological products should be developed to control 3D temperature distribution in the bulk of products during heating/cooling processes.
In heating and cooling applications, conductive heat transfer through solid food products is normally modelled by Fourier’s equation of heat conduction under the boundary conditions of the governing equations. Most heat transfer models can only be solved analytically for simple cases. Numerical methods are useful for estimating the thermal behaviour of foods under complex but realistic conditions such as variation in initial temperature, non-linear and nonisotropic thermal properties, irregular-shaped bodies and time dependent boundary conditions (Puri & Anantheswaran, 1993). In solving the models, the finite difference and finite element methods are widely used (Ahmad, et al., 2001; Erdogdu et al., 1998a,b, 1999; Jia et al., 2000a,b,c,d, 2001; Wang & Sun, 2002a,b,c,d,e; Zhang & Fryer, 1995; Zhou et al., 1995). In recent years, the finite volume method was the main computational scheme used in commercial computational fluid dynamics (CFD) software packages. Computational fluid dynamics is a simulation tool, which uses powerful computer and applied mathematics to model of fluid flow situations for the prediction of heat, mass and momentum transfer and optimal design in industrial processes (Bin Xia & Da-Wen Sun, 2006). For analysing complex 99
flow behaviour (Scott & Richardson, 1997; Sun, 2002), temperature distribution in storage room (Ville et al., 2002, Chua et al., 2002, Verboven et al., 2004; Chao et al., 2004), in the food industry (Lijun & Sun, 2003), etc.
Despite that CFD has already showed to be a valuable tool to design and optimise in many areas, CFD models suffer from their complex nature and can not be used in applications where reduced order model is mandatory, e.g. in controller algorithms (Zerihun et al., 2004). The term “data based mechanistic modelling” (DBM) was first used by Young and Lees in 1993. This approach to modelling is applied to the problem of modelling imperfect mixing in the forced ventilation of buildings, based on previous research concerned with the modelling and control of glasshouse and environmental systems. In DBM models, the model structure is first identified using objective methods of time series analysis based on a given, general class of time series model (here linear, continuous-time transfer functions or the equivalent ordinary differential equations). But the resulting model is only considered fully acceptable if, in addition to explaining the data well, it also provides a description that has relevance to the physical reality of the system under study. Note that, in the present study, such mechanistic interpretation is aided by the use of differential equation models since the equations of heat and mass transfer are normally formulated in these terms (Young et al., 2000).
Objectives of this chapter focused on: (1) Modelling to predict airspace temperature in a room was filled with biological products. (2) Developing a data based mechanistic model for controlling of 3D temperature distribution in air-ventilated rooms filled with biological products. (3) Finding an algorithm for real time monitor and online adaptive to control for 3D temperature in both an individual biological product and a local moving air. 100
5.2
Material and method
5.2.1. Laboratory test room The test installation was used in this study is a room described in Chapter 2 (Part 2.1, Fig. 2.1). This room was filled up with 144 boxes, each box (0.4 x 0.6 x 0.3 m) contained approximate 4.17 kg of potatoes (total 480 kg of potatoes) representing the physical presence of material in thermal processing and storage processes. The 36 thermocouples (type T) were inserted inside 36 potatoes (central of potato) to measure the potato temperature (Fig. 5.1). We know that in really, there is a temperature difference in the potato. In this work, however we assumed a uniform temperature in a potato within the acceptable temperature difference in well mixed zone. It is possible to test this assumption by using the Biot number (if an acceptable temperature difference gives positive effects on the quality of products). The position of these 36 potatoes was close to thermocouples that measure temperature in the airspace, the numbered sensor in Fig. 2.8.
Fig. 5-1 Boxes of potatoes and inserted sensor in potato Experiments
101
Dynamic experiments were conducted in an air-ventilated room filled with the boxes of potatoes. The ventilation rates were varied between 160 to 320 m3 h-1 (16.6 to 37.6 volumes change per hour). In each experiment at a specific ventilation rate, after 2 hour of steady state, the inlet temperature was adjusted from 6 to 20oC and from 20 to 6oC. After each step, the temperatures in air space and potatoes were monitored for 10 hours with interval of 30
Temperature, oC
Temperature, oC
seconds per sample (Fig. 5.2). The overview of the dynamic experiments was in Table 5.1.
20
6
2
Time, h
20
6
2
12
(b)
22
o
Temperature ( C)
o
Temperature ( C)
(a)
12
Time, h
21 20
Inlet air temperature
19 0
100
200
300
400 Time (min)
500
600
700
400
500
600
700
500
600
700
22 20 18
Airspace temperatures
16 14
0
100
200
300
o
Temperature ( C)
Time (min) 22 20 18
Potato temperatures
16 14
0
100
200
300
400
Time (min)
(c) Fig. 5-2 Step up air inlet (a) and step down air inlet (b) of dynamic experiment ;(c) The measured dynamic response of the airspace and potato temperatures from the air temperature inlet at ventilation rate 280m3 h-1 , inlet temperature setting; , measured response 102
Table 5-1 Overview of the dynamic experiments No. Experiments
Ventilation rates (m3 h-1)
1
160
2
160
3
200
4
200
5
240
6
240
7
280
8
280
9
300
10
300
11
320
12
320
Dynamic experiments Temperature (oC) Total time of experiment (h) 6 2 20 10 20 2 6 10 6 2 20 10 20 2 6 10 6 2 20 10 20 2 6 10 6 2 20 10 20 2 6 10 6 2 20 10 20 2 6 10 6 2 20 10 20 2 6 10
5.2.2. Data based mechanistic modelling approach Data based mechanistic (DBM) modelling approach was applied to estimate the temperature distributions in a room filled with biological products from the inlet temperature. The principle of this method was presented in Chapter 2 (Part 2.2.2.2) and Fig. 2.12.
5.2.2.1 Data based phase The time-series data obtained from dynamic experiments were used in the ‘Data based phase’ by means of mathematical identification techniques. Although other techniques are available, the simplified refined instrumental variable (SRIV) approach to identifying continuous-time transfer function models was employed as a method for model identification, since it not only 103
yields consistent estimates of the parameters but also exhibits close to optimum performance in the model order reduction context (Young, 1984). Comparison of the fitted model based on coefficient of determination ( RT2 ), Young Critical Identification (YIC) and standard error was used to select the best-fitted model. In the present example, the continuous-time transfer function model obtained from this SRIV identification phase of the DBM analysis has a third order structure. This forms the basis for subsequent mechanistic modelling, which involves the definition of a mechanistic model that has the same third order structure, as discussed in the next Section 5.3.3
5.2.2.2 Data based mechanistic model in room filled with biological products Research on ventilated rooms (De Moor & Berckmans, 1993) has demonstrated that the empty test chamber that is used in this study is an imperfectly mixed airspace with considerable spatio-temporal gradients of temperature. Within such an imperfectly mixed airspace, it is always possible to define a number of well mixed zones (WMZs) around temperature sensors in which there exists a good mixing at acceptably low temperature difference (Berckmans & Goedseeds, 1986; Berckmans, De Moor, M., & De Moor, B, 1992).
Assuming such a well-mixed zone (WMZ1) with a volume V1 in which n potatoes are located and voli is a volume of airspace. The volume of V1 has porosity ε, and the surface of heat exchange between WMZ1 and the buffer zone (WMZ2) is S1. It is assumed that there is perfectly mixed air in WMZ2 and the volume is vol buff and surface of heat exchange between WMZ2 and environment is S2.
A uniform temperature in each potato was assumed (uniformity within an acceptable temperature difference of 0.2oC), the respiration heat from potatoes was considered to be 104
constant during the experiments and the humidity changes in a well-mixed zone are neglected. The heat transfers in this system are shown in Fig. 5.3
kk2SS2(T -T (t)) 2 2(Tlab lab-Tbuff buff(t))
Tlab
WMZ2 Tbuff(t), volbuff, γbuff, Cp,buff, S2 kk1SS1(T (t)-T (t)) 1 1(Tbuff buff(t)-Tii(t))
Air enter the WMZ1 T0(t-τ), V, γ0, Cp,0 VT VT00(t(t-ττ))γγ00CCp,0 p,0
(V1) WMZ1 (V voli, Ti(t), ε, S1, γi, Cp,i
Potato
qc
kkmSSm(T (t)-T (t)) m m(Tii(t)-Tmm(t))
VT VTii(t) (t)γγiiCCp,i p,i
Air exits the WMZ1 V, Ti(t), γi, Cp,i
Potato Tm(t), γm, Cp,m, volm, Sm, km, ζ
Fig. 5-3 Scheme of heat transfer in a selected well mixed zone containing several potatoes The partial differential equation Eq. 4.1 for heat transfer from the airspace to products in WMZ1 was applied.
dTi (t )voli .γi .C p,i
= V .T0 (t − τ )γ 0 .C p ,0 − V .Ti (t )γ i .C p,i + qc dt + n.k m .S m (Tm (t ) − Ti (t )) + k1 .S1 (Tbuff (t ) − Ti (t ))
(4.1)
Applying Eq. 3.2 for the heat balance in WM2 (inside buffer zone): dTbuff ( t ).vol buff .γ buff .C p,buff dt
= k1 .S1 (Ti ( t ) − Tbuff ( t )) + k 2 .S 2 (Tlab − Tbuff ( t ))
105
(3.2)
Assuming density and specific heat of air inlet, air in the well mixed zone (WMZ1) and in the buffer zone are equal (γ0 = γ1 = γbuff = γ ; Cp,0 = Cp,i = Cp,buff = Cp). In addition, the ratio of volume of potato and surface denotes as ξ, ξ =
volm , where volm , Sm are volume and surface Sm
of potato.
There are n potatoes located in the consideration well mixed zone V1 (WMZ1) with a porosity
ε, air space ( voli ) in relation to volume of one material volm in WMZ1 can be expressed as: V1 − Vi =
(1 − ε )
ε
voli = n.vol m = n.ξ .S m
or
n. S m =
(1 − ε ) vol i ε .ξ
(5.1)
Equation (4.1) can be written
dTi (t ) qc n.k m .S m k1 .S1 V V = + (Tbuff (t ) − Ti (t )) (Tm (t ) − Ti (t )) + T0 (t − τ ) − Ti (t ) + voli .γ .Cp dt voli .γ .C p voli .γ .C p voli voli (5.2) Replacing
nS m (1 − ε ) = in Eqn (4.2), results in Eqn (5.3) voli εξ
dTi ( t ) qc (1 − ε ) k m k1 . S1 V V T0 ( t − τ ) − Ti ( t ) + (Tm ( t ) − Ti ( t )) + = + (Tbuff ( t ) − Ti ( t )) vol i vol i dt vol i .γ .C p ε .γ .C p .ξ vol i .γ .C p
(5.3) Applying the Eq. 4.2 for the heat transfer from the airspace in the well-mixed zone (WMZ1) to the potato
m.C p,m
dTm (t ) = S m .k m (Ti (t ) − Tm (t )) dt
or
dTm (t ) Sm .km = (Ti (t ) − Tm (t )) dt m.C p,m With
m = volm .γ m and, ξ =
volm Eqn (4.2) can be minimized to Eqn (5.4) Sm
106
(4.2)
dTm (t ) km = (T (t ) − Tm (t )) γ m .C p,m .ξ i dt
(5.4)
The set of three equations (4.1), (3.2), (5.4) present the heat transfer in an air ventilated potato room. These three equations had the same structure of Eqns 4.8, 4.9 and 4.10 with modification for shape and density of material in the α2 values. dTi (t ) qc = β1T0 (t − τ ) − β1Ti (t ) + + K m (Tm (t ) − Ti (t )) + K1 (Tbuff (t ) − Ti (t )) (4.8) dt voli .γ .C p dTbuff ( t ) dt
= K 3 (Ti ( t ) − Tbuff (t )) + K 2 (Tlab − Tbuff ( t ))
(4.9)
dTm (t ) = α 2 (Ti (t ) − Tm (t )) dt
(4.10)
With
β1 =
V ; voli
K1 =
k1 .S1 ; voli .γ .C p
α1 = β1 + K1 + K m ; α 2 =
K2 =
k 2 .S 2 ; volbuff .γ .C p
K3 =
k1.S1 ; volbuff .γ .C p
Km =
(1 − ε )k m ; ε .γ .C p .ξ
km γ m .C p,m .ξ
Under the steady state condition Eqns (4.8), (4.9) and (4.10) will be
β1 .T0 − Ti (β1 + K m + K1 ) +
qc + Tm (K m ) + Tbuff .K1 = 0 voli .γ .C p
Ti (K 3 ) − Tbuff (K 3 + K 2 ) + Tlab . K 2 = 0
Ti .α 2 − α 2 .Tm = 0
where, Ti air space temperature under steady state condition, oC T0 inlet temperature under steady state condition, oC
107
(5.5) (5.6) (5.7)
Tm product temperature under steady state condition, oC o Tbuff buffer zone temperature under steady state condition, C
With a very small temperature perturbation around steady state (t0(t-τ), tbuff(t), ti(t) and tm(t)) , and after subtracting Eqns (4.8), (4.9) and (4.10) from Eqns (5.5), (5.6) and (5.7) results in (the same structure in case of balls with the modification of the shape factor (ς) and density (ρm) of potato in the α2 value) dt i ( t ) = β1 .t0 (t − τ ) − ( β1 + K m + K1 )ti (t ) + K m .t m (t ) + K1 .tbuff (t ) dt dt buff ( t ) dt
= K 3 ( ti ( t ) − tbuff ( t )) − K 2 .t buff ( t )
dt m (t ) = α 2 (ti (t ) − t m (t )) dt
(4.11)
(4.12)
(4.13)
Laplace transformation on Eqns (4.11), (4.12) and (4.13) result in Eqns (4.14), (4.15) and (4.16).
ti (t ) =
Km β1 K1 tbuff (t ) + t m (t ) t0 (t − τ ) + s + α1 s + α1 s + α1 tbuff (t ) =
K3 ti (t ) s + (K 2 + K3 )
t m (t ) =
α2 ti (t ) s +α2
(4.14)
(4.15)
(4.16)
The set of three equations (4.14), (4.15), (4.16) is presented in Fig.5.4 (the same structure with Fig. 4.4)
108
tm(t)
α2
s + α2 Km t0(t-τ)
β1
1 s + α1
+
ti(t)
K1
K3 s + (K2 + K3 )
tbuff(t)
Fig. 5-4 Third order block diagram of system Moreover, the set of Eqns (4.14), (4.15), (4.16) can be minimized to the same form of Eqn (4.18). ti (t ) =
b0 .s 2 + b1 .s + b2 t0 (t − τ ) s 3 + a1 .s 2 + a 2 .s + a3
(4.18)
Equation (4.18) is a third order of transfer function with
b0 = β1
(5.8)
b1 = β1 (α 2 + K 2 + K 3 )
(5.9)
b2 = β1 .α 2 ( K 2 + K 3 )
(5.10)
a0 = 1
a1 = α1 + α 2 + K 2 + K 3 a2 = K 2 .α 2 + K 2 .α 2 + α 2 .α1 + α1 .K 2 + α1 .K 3 − K1.K 3 − K m .α 2 a3 = α1 .K 2 .α 2 + α1.K 3 .α 2 + K1.K 3 .α 2 − K m .α 2 .K 2 − K m .α 2 .K 3
Equations (5.8), (5.9) and (5.10) can be minimized to Eqn (5.11)
α 22 .β1 − b1.α 2 + b2 = 0 or α 22 .b0 − b1 .α 2 + b2 = 0 109
(5.11)
With α2 value obtained from a root of Eqn (5.11). From α 2 =
km , with γm, Cp,m and ξ γ m .C p,m .ξ
are constant, a heat transfer coefficient (km) is proportional to α2. So α2 is a value presenting the ability of heat transfer from well mixed zone to biological products (α2 value (s-1) is the same structure with ‘Cooling rate’ term in Newton’s law of cooling process. In cooling storage process, cooling rate is an important parameter to predict product temperature, cooling time and air velocity distribution). In the other way, Km parameter is related to heat transfer coefficient km and the relationship between α2 and Km is: Km =
(1 − ε ) ⎡ C p,m .γ m ⎤ ⎥α ⎢ ε ⎢⎣ C p .γ ⎥⎦ 2
(5.12)
Consequently, the parameters from the fitted model (third order transfer function) are basic to identify two important values, which presented for temperature distribution in airspace (β1) and products (α2).
5.3
Results and discussion
5.3.1. Temperature distribution in airspace and potatoes during experiment Temperature distribution in airspace and potatoes are presented in Figs. 5.5 and 5.6, after 5 hours in the steady state condition with constant ventilation rate 200 m3 h-1 (23.5 volumes change per hour). These figures have shown the non-uniformity of temperature in each case.
110
Air inlet
Air outlet
111
Air outlet
Air inlet
Fig. 5-5 Temperature distribution in airspace with ventilation rate 200 m3 h-1
Air outlet
Air inlet
Fig. 5-6 Temperature distribution in bulk of potatoes with ventilation rate 200 m3 h-1
112
5.3.2. Uniformity of air and potato temperature in experiment The temperature uniformity index Itemp is a useful index to quantify the spatial homogeneity of temperature in airspace and bulk of potatoes. The equation is more description in Chapter 3 (Part 3.3.1).
The temperature uniformity index in airspace is related to ventilation rate and acceptable temperature difference in Table 5.2 and for bulk of potatoes in Table 5.3. Table 5-2 Temperature uniformity index as a function of acceptable temperature difference and ventilation rates in the airspace Acceptable temperature differences, oC 0.20 0.40 0.60 0.80 1.00
Ventilation rates, m3 h-1 160 2.78 16.67 33.33 44.44 66.67
200 2.78 8.33 22.22 36.11 52.78
240 5.56 19.44 27.78 38.89 63.89
280 11.11 19.44 25.00 33.33 50.00
300 13.89 13.89 27.78 36.11 58.33
320 5.56 19.44 22.22 36.11 61.11
Table 5-3 Temperature uniformity index as a function of acceptable temperature difference and ventilation rates in bulk of potatoes Acceptable temperature differences, oC 0.20 0.40 0.60 0.80 1.00
Ventilation rates, m3 h-1 160 16.67 27.78 36.11 41.67 55.56
200 8.33 25.00 36.11 38.89 66.67
240 5.56 22.22 36.11 47.22 66.67
280 5.56 16.67 27.78 38.89 55.56
300 8.33 19.44 25.00 33.33 41.67
320 8.33 19.44 30.56 36.11 52.78
Higher acceptable temperature difference and higher ventilation rate resulted in a higher uniformity index in both cases. Maximum Itemp reached 67.67% for both airspace and potatoes. The uniformity of potato temperature depends on the uniformity index of air space and airflow pattern in the room. 113
5.3.3. Data based phase In data based phase, the set of time-series data from experiment were used. Fig. 5.7 shows the inlet, airspace and potato temperature at position (6) and (30) at ventilation rate 200 m3 h-1 as illustration for calculation. In this figure, the airspace temperature reached the inlet temperature at the end, while potato temperature changed slowly in the experiment.
Fig. 5-7 Dynamic temperature data at position 6 (a) and 30 (b) with ventilation rate 200 m3 h-1:
… , inlet temperature, oC; ⎯, airspace temperature, oC; , potatoes temperature, oC
The continuous-time SRIV algorithm in the CAPTAIN Toolbox for Matlab (Taylor et al., 2007) was used to identify and estimate the parameters in the third order transfer function at 36 positions in the room at the ventilation rates changes from 160 to 320 m3 h-1. Position 6 and 30 at ventilation rate 200 m3 h-1 were selected as examples in this paper. A fully functional
version
of
the
CAPTAIN
Toolbox
can
be
downloaded
from
http://www.es.lancs.ac.uk/cres/captain/). The full results of the SRIV identification analysis are given in Table 5.4. 114
Table 5-4 Estimated parameters from the third order of transfer function at position (6) and (30) with ventilation rate of 280 m3 h-1 as an example of calculating method Sensor positions
[m, n,τ]
Parameter estimates
RT2
YIC values
SE
[2,3,30]
a1= 0.7568 a2= -0.0067 a3= -0.0003 b0= 0.0740 b1= -0.0003 b2= -0.00001 a1= 0.2586 a2= -0.0022 a3= -0.00001 b0=0.04501 b1=0.00024 b2=-0.0001
0.997
-9.714
0.002
0.996
-9.7322
0.005
6
[2,3,30] 30
with RT2 , coefficient of determination; YIC, young identification criterion; SE, standard error of determination; m, n and τ , denominator, numerator and time delay; a1, a2, a3, b0, b1, b2, parameters in the first and second order of transfer function.
From Table 5.4, the third order transfer function gave the best fit based on the coefficient of determination (>99%), very low of YIC values (<-9.7) and standard error.
5.3.4. Mechanistic phase The second step in data based mechanistic modelling approach is interpreted in a physically meaningful way based on the nature of the system (Young & Garnier, 2006). In this research, the third order of fitted model was selected to seek the physically meaning of this process. From Eqns (4.18) and (5.11), two importance values were found. β1 value is equal to b0, defined as volumetric of fresh air concentration in many publications presented for air temperature distribution in airspace, and α2 value as a root of a second order equation with coefficients related to numerator in a third order of transfer function Eqn (4.18). α2 is also a ‘cooling rate’ term related to heat transfer coefficient from well-mixed zone to material presented for product temperature in an air-ventilated room filled biological products. In Fig.
115
5.8 the partial contour of β1 values at front plane (matrix of six columns and three rows in the test installation) with the ventilation rate 200 m3 h-1.
Fig. 5-8 Partial contour of parameter β1 (s-1) at front plane with ventilation rate 200 m3 h-1 In this figure, β1 values are very high compared to β1 in an empty room in previous study (Janssen, 2004). By solving Eqn (5.11), α2 values were obtained in Table 5.5 in comparison with β1. Table 5-5 α2 values from numerator of the third order transfer function at ventilation rate 200 m3 h-1 b0 b1 b2 Positions α2 1 0.0452 -0.00015 -0.000009 0.0130 2 0.0075 -0.00008 -0.000001 0.0098 3 0.0045 -0.00004 -0.000001 0.0099 4 0.0573 -0.00015 -0.000012 0.0134 5 0.1341 0.00004 -0.000037 0.0167 6 0.0730 -0.00032 -0.000014 0.0119 7 0.0299 -0.00015 -0.000006 0.0118 8 0.0307 -0.00005 -0.000008 0.0159 9 0.0020 -0.00004 0.000000 0.0033 30 0.04508 -0.00020 -0.000009 0.0121
116
Equation (5.24) showed relationship between b0 (or β1) and α2, the results in this table are in good agreement with the heat transfer theory (high velocity is resulting in high heat transfer coefficient). As shown in Fig. 5.9.
Fig. 5-9 Comparison β1 and α2 at position (1) at ventilation rate from 160 to 320 m3 h-1 , β1 value, s-1; , α2 value, s-1
Therefore, it is possible to use α2 values as the parameters presented for the product temperature in a-ventilated room filled with biological products.
5.3.5. Analysis of the order of transfer functions Table 5.6 shows that the order of the transfer functions depends on the numbers of equations that we considered to analyse the transport processes. In heating and cooling applications for example, the heat transfer from the airspace to the products should be considered. Moreover the product temperature should be integrated in the model for control purposes since it is the product that is the main target rather than the air.
117
Table 5-6 Analysis and suggestion of the order of transfer function in several processes
Analysis from The research
Prediction for the processes
Processes
Heat transfer to well mixed zone
Heat loss
Heat transfer to products
Mass transfer
Empty room
1
1
0
Room filled with balls
1
1
0
Room filled with biological products Drying process
1
1
1
? Not considered ? Not considered ? Not considered
1
1
1
1
1
1
Freezing process
1 (for diffusion process) 1 For latent heat of crystal
The suggestion for orders of transfer function Second order Second order or third order Third order
Fourth order
Fourth order
In the empty room the model order is 2 explained by the process of heat transfer to the air in the well mixed zone and the heat losses from the room to the surroundings. When the room is filled with obstacles and we do not consider the plastic of the balls than the model order remains 2. If we however consider the plastic than the model order is increasing to 3 since in the mechanistic model there is an equation more. If we fill the balls with water there is another buffering capacity in the process and the model order will increase to third order. Regarding drying processes, the Fick’s law equation can be added in the model to explain the moisture loss during drying process and the model order would increase to fourth order.
118
5.4 Conclusions The main objective of this chapter was the introduction of a data based mechanistic modelling approach to control 3D product temperature in an air-ventilated room filled with potatoes as an example of biological products. Dynamic temperature data at 72 sensor positions in a room (36 for airspace and 36 for products), and input temperature at different ventilation rates were used to find transfer functions for each sensor position.
For modelling the airspace temperature in the room, a third order transfer function gave the best results in agreement with observed data. The coefficient of determination reached 99%.
This third order model with high accuracy could be interpreted with a physical meaning of this system. β1 value, defined as the volumetric fresh air concentration presented for airspace temperature distribution and α2 value, ‘Cooling rate’ was a measure for the product temperature, and a predication of the time required to reach a predefined product temperature during the storage process.
Doing so, a technique was developed to calculate the cooling rate of the product from dynamic measurements of air inlet temperature and the temperature in the airspace between the products.
From ‘fresh air rate concentration’ and ‘cooling rate’ which were located in the model and the information of biological material physical characteristics, the algorithm for real time monitoring and online adaptive control of 3D temperature distribution in products was developed.
119
120
6 Chapter 6 Validation of a Data Based Mechanistic Model for Online Calculation of the Cooling Rate during Cooling Processes
121
122
Chapter 6 Validation of a Data Based Mechanistic Model for Online Calculation of the Cooling Rate during Cooling Processes 6.1 Introduction Data based mechanistic model for a room filled with biological material was described in more detail in chapter 5. Two important parameters were extracted: -
Local volumetric of fresh air concentration (β1) was used to identify and control the airflow pattern in a room filled with biological products.
-
“Cooling rate” α 2 was used to estimate and control the rate of heat transfer from medium to products. Further more, it is also used to online predict product temperature (non-invasive method) in heating/cooling applications.
In storage of fruits and vegetables, the rate of heat transfer between product and environment or the cooling rate, is a critical value for efficient removal of field heat in order to achieve food cooling. As a form of energy, heat always seeks equilibrium. In the case of cooling, the sensible heat (or field heat) from the product is transferred to the cooling medium.
The cooling rate studied so far comes from Newton’s cooling law assuming a uniformity of the product temperature during heat treatments and its value results from the calculation of a heat transfer process from fluid flow to product under the steady state condition. The cooling rate value presents the change in product temperature for unit change of cooling time for each degree of temperature difference between the product and its surrounding. It depends on many factors including the rate of heat transfer, the difference in temperature between the product and the cooling medium, the nature of the cooling medium, the physical properties of the product, etc. (Brosnan & Sun, 2001). The cooling rate is a common parameter to estimate the product temperature, the cooling time during cold storage, and the cooling rate is an important
123
parameter to estimate air velocity and air flow pattern in boxes during the storage of horticultural products (Vigneault et al. 2005). The objective of this chapter is to study the possibility of a data based mechanistic modelling approach and cooling rate (chapter 5) to calculate the heat transfer coefficient for real time monitoring of the 3D product temperature distribution in a ventilated room filled with products.
6.2
Material and method
6.2.1. Laboratory test room The laboratory test room used in this study was presented in Chapter 2 (Part 2.2) (Fig. 2.9). One well-mixed zone (1 m3) with several products was selected in this experiment. Normally, the scale of a room was affected by the estimated parameter values. With the known model structure we can estimate online airflow pattern and use this information for online control of 3D temperature distribution in a process room of any dimension scale.
Air inlet, airspace and product temperature were monitored with type T thermocouples (accuracy 0.1oC), and recorded by a Keithley-2700 data logger system. The product temperature was measured by the inserted thermocouples at the centre of materials. (See Chapter 2)
Big balls with air inside were used for model development (more details in chapter 3 and 4 where the mass of balls is neglected). In this chapter, small balls with water inside were used with known physical characteristics of material (density, heat capacity and diameter) to check the ability of the model for calculating the heat transfer coefficient based on the “cooling rates”. In this research, the uniformity of product temperature was assumed; therefore the size of the products is not affected by the heat transfer coefficient from airspace to product in a selected well mixed zone. 124
The test installation was filled up with plastic sphere materials with pure water inside (as reference with known physical properties); the diameter of the plastic sphere material was 0.04 m (table tennis ball). In a next step apples, with a diameter of approximately 0.1 m were used to demonstrate the calculation procedure.
6.2.2. Experiment Dynamic step experiments (steps on inlet temperature) were conducted in the room filled with sphere materials filled with water. To simulate an air force cooling of fruit and vegetable, the ventilation rates were adjusted 20, 100, and 120 m3 h-1 for each experiment (20, 100, 120 air change per hour for each experiment).
Steps on inlet air temperature were set from 20 to 5oC. Inlet, airspace and product temperature were recorded with 10 seconds interval and the total experimental time was 50,000 seconds. For the next step, a room filled with apples was used as an example of the application of data based mechanistic modelling to demonstrate the real time estimation of the cooling rate.
6.3 Data based mechanistic model approach This approach was divided in two phases: data based phase and mechanistic phase as described more detailed in Chapter 2
6.3.1. Data based phase The dynamic data obtained from dynamic experiments were used. The simplified refined instrumental variable (SRIV) approach was employed as a method for model identification (more details in Chapter 2).
6.3.2. Mechanistic phase A mechanistic phase represents the imperfectly mixed fluid in a room filled with obstacles as demonstrated in Chapter 4. Equations are as follows to describe the dynamic behaviour from temperature at position i in the room filled with biological products 125
ti (t ) =
β1 K1 K t0 (t − τ ) + tbuff (t ) + m tm,i (t ) s + α1 s + α1 s + α1 tbuff (t ) =
K3 ti (t ) s + ( K 2 + K3 )
(4.15)
α2 ti (t ) s + α2
(4.16)
tm,i (t ) =
where, β1 =
(4.14)
k1 .S1 k 2 .S 2 V k1.S1 ; K1 = ; K2 = ; K3 = ; voli voli .γ .C p volbuff .γ .C p volbuff .γ .C p
Km =
(1 − ε )k m ,i k m ,i ; α1 = β1 + K1 + K m ; α 2,i = ε .γ .C p .ξ γ m .C p,m .ξ
with: tm,i the product temperature in the well mixed zone i km,i the heat transfer coefficient in the well mixed zone i
α2,i the cooling rate in the well mixed zone i
and the set of Eqns (4.14), (4.15), (4.16) can be minimized to the third order of transfer function.
ti (t ) =
b0 .s 2 + b1 .s + b2 t0 (t − τ ) s 3 + a1 .s 2 + a 2 .s + a3
(4.18)
with,
b0 = β1
(5.18)
b1 = β1 (α 2, i + K 2 + K 3 )
(5.19)
b2 = β1 .α 2 ,i ( K 2 + K 3 )
a0 = 1 a1 = α1 + α 2 , i + K 2 + K 3 a 2 = K 2 .α 2 ,i + K 2 .α 2 ,i + α 2 ,i .α 1 + α 1 . K 2 + α 1 . K 3 − K 1 . K 3 − K m .α 2 ,i
a 3 = α 1 .K 2 .α 2 ,i + α 1 .K 3 .α 2,i + K 1 .K 3 .α 2 ,i − K m .α 2,i .K 2 − K m .α 2 ,i .K 3
126
(5.20)
From Eqns. (5.18), (5.19) and (5.20), two important values were found:
b0 = β1 =
V is the local volumetric concentration of the fresh air flow rate to a well mixed voli
zone i. This value presents the amount of fresh air to a well mixed zone i, and it is linked to the airflow pattern in the room.
A new parameter, representing the temperature distribution between the products, was found from this study (as discussed in Chapter 5 (Part 5.2.2)). This is the ‘cooling rate’ term in a well-mixed zone i α 2,i =
k m ,i (see Chaper 5). It can be used not only for real time γ m .C p,m .ξ
estimating of 3D cooling rate, but also for predicting a cooling time (the speed of the cooling process, and how long it takes for the product to reach a desirable temperature) or the product temperature (how much is the change of temperature in a considerable time).
As a result, in Chapter 5, two methods were proposed to estimate the cooling rate: In the first method, the cooling rate was obtained from a data based phase by the first order transfer function [Equation (4.16)] tm,i (t ) =
α2 ti (t ) . This equation has the same structure s + α2
as the first order transfer function that relates the product temperature to the fluid temperature within a well-mixed zone. y (t ) =
b0 u(t ) where tm,i(t)= y(t) is the product temperature a0 .s + a1
in a well mixed zone i with time, ti(t)=u(t) is the airspace temperature in a well-mixed zone i with time, and b0, a0, a1 are estimated parameters. The cooling rate in a well-mixed zone i is
α 2, i = b0 = a1 . In the second method, the cooling rate is obtained from the root of the equation (see Chapter 5) 127
α 22,i .b0 − b1 .α 2 ,i + b2 = 0
(5.11)
where b0, b1, b2 are estimated parameters from the dynamic response of the airspace temperature in a well mixed zone i to steps inlet temperature. 6.4
Results and discussions
6.4.1. The third order transfer function for the heat transfer in a room filled with products A third order transfer function [Eq. 4.18] was applied to calculate the dynamic response of
airspace temperature in a well mixed zone i and steps inlet temperature at different ventilation rates. The results are in Table 6.1. From this table, very high coefficient determination ( RT2 ) and very low in YIC were obtained. Table 6-1 Estimated parameters from the third order transfer function at the different ventilation rates Ventilation rates (m3 h-1) Parameters
20
100
120
a0
1
1
1
a1
0.17
0.13
0.17
a2
0.0007
0.0004
0.0012
a3
0.0000005
0.00000013
0.0000015
b0
0.034
0.043
0.052
b1
0.00036
0.0003
0.00062
b2
0.0000004
0.0000001
0.000001
RT2
0.9998
0.9999
0.9999
YIC
-12.5
-13.9
-13.7
0.0012
0.0007
0.0007
o
SE ( C)
In addition, with the air volumes change per hour from 20 to 120 increases the β1 by 0.53% at a considered zone. These results are similar to the previous studies on effects of ventilation rates on the β1 in a room filled with balls in Chapter 4 (from 49 to 63.3 increases the β1 by 0.4%). Estimated parameters b0, b1, and b2 were used to calculate the cooling rate values in the next steps. 128
6.4.2. Cooling rate from the first order transfer function Estimated parameters from the dynamic response of the product and airspace temperature are
in Table 6.2., b0 and a1 values must have the same value. In the results in Table 6.2, no significant differences of b0 and a1 values were found to correspond with a high coefficient of determination ( RT2 >99.9) and very low YIC value (YIC<-16.6). This result agrees with previous experiments (Chapter 3, part 3.2). Cooling rate values as a function of ventilation rate are presented in Table 6.3. Table 6-2 Estimated parameters from the first order transfer function at the different ventilation rates Parameters a0 a1 b0 RT2 YIC SE
Ventilation rates (m3 h-1) 20 100 1 1 0.006 0.008 0.005 0.008 0.998 -16.6 0.016
0.998 -16.7 0.012
120 1 0.008 0.007 0.998 -16.7 0.013
Table 6-3 Cooling rate values at the different ventilation rates (1st order TF) Ventilation rates (m3 h-1) Cooling rate (s-1) 20 100 120 α2 0.0055 0.0083 0.0077
Table 6.3 shows the cooling rate at the different ventilation rates from 20 to 120 m3 h-1. The cooling rate value increased with ventilation rates from 20 to 100 m3 h-1 and decreased at ventilation rates from 100 to 120 m3 h-1. The cooling rate covers a heat transfer coefficient which is high at high ventilation rates due to a high velocity obtained in the airspace, explained in case of increasing of the cooling rate (from 0.055 to 0.0083 corresponding to a changing ventilation rate from 20 to 100 m3 h-1). The high ventilation rate resulted in an air flow pattern change explained the cooling rate decreasing from 0.0083 to 0.0077 when the air flow rates change from 100 to 120 m3 h-1. 129
6.4.3. Cooling rate as found from the estimated parameters of the third order transfer function
In the equation α 22,i .b0 − b1 .α 2 ,i + b2 = 0 , α2,i is a root of the second order equation where b0, b1 and b2 are parameters from the third transfer function of the dynamic response of airspace to input temperature. By solving the second order equation based on estimated parameters in Table 6.1, two roots were obtained in the Table 6.4 Table 6-4 Roots of second order equation Roots of 2nd equation Root 1 Root 2
20 0.0092 0.0013
Ventilation rates (m3 h-1) 100 0.0059 0.0004
120 0.0094 0.0025
Two roots are received from the second order equation [Eqn.5.11]. It is hard to select a good root, but comparing with the results in Table 6.3, the values of root 1 in Table 6.4 are the nearest values in the Table 6.3 at the same ventilation rate, and these are selected as a cooling rate value. 6.4.4. Comparison of ‘Cooling rate’ values obtained by two methods Cooling rate values obtained from two methods are shown in Table 6.5. From this table the
differences between values are in the range of 22 to 66%. Several assumptions (well-mixed zone, uniform product temperature, density of product and air, specific heat of product and air at constant temperature, etc.) were causes of the differences. Table 6-5 Comparison cooling rate values from two methods at different ventilation rates Cooling rates (s-1)
α2,i from 1 TF (Product –
20
Ventilation rates (m3 h-1) 100
120
st
airspace temperature)
α2,i from 3rd TF (airspace –
inlet temperature) Different of two methods (%)
0.0055
0.0083
0.0077
0.0092 66
0.0059 29
0.0094 22
130
6.4.5. Calculation of a heat transfer coefficient from ‘Cooling rate’ with known physical properties of material A major problem in heat transfer is the estimation of heat transfer coefficients to be used for
design purposes. Data based mechanistic model approach was used to estimate a heat transfer coefficient under unsteady state condition. The heat transfer coefficient (km,i) is a function of the cooling rate ( k m ,i = α 2 ,i .γ m .C p,m .ξ ) , the material physical properties and the geometry of products, in those experiments. The known physical characteristics of material are as follows: the diameter is 0.04 m; the density of water is 1000 kg m-3; and the specific heat of water is 4186 J kg-1 oC-1. The result of the estimated heat transfer coefficient at different ventilation rates is in Table 6.6. Table 6-6 Estimated bulk heat transfer coefficient from known physical characteristics of material and ‘Cooling rate’ term from model parameters at different ventilation rates Heat transfer coefficients (W m-2 oC-1) km from 1st TF km from 3rd TF Different of two methods (%)
Ventilation rates (m3 h-1) 20 100 120 153.9 232.4 214.2 256.7 164.6 262.3 40 41 18
The heat transfer coefficient from the first order transfer function was in range of some literature (the heat transfer coefficient in free convection is very low, about 60 W m-2 K-1. (Romeo T. Toledo, 2007)) has proved the ‘data based mechanistic’ model approach is a powerful tool to estimate the heat transfer coefficient under unsteady state condition, and it is able to online monitor this value to control the uniformity of energy transfer in a process room.
The difference between heat transfer coefficients from two methods is not so high, from 1840% (see Table 6.6). It is possible to calculate a heat transfer coefficient value in a well mixed zone i from the dynamic response of airspace temperature i and the steps inlet temperature, 131
and able to get a real time monitor 3D heat transfer coefficient value during heat treatment processes. 6.4.6. An example for biological products Apples were selected as a demonstration of the application of data based mechanistic model
for forced air-cooling of biological products, with a selected ventilation rate of 20 m3 h-1. The results for a selected well-mixed zone i are in the Table 6.7. Table 6-7 Estimated parameters from Data based mechanistic model for forced air-cooling of apples at ventilation rate 20 m3 h-1 Methods First order transfer function (Product ~ airspace temperature)
Estimated parameters a0 = 1 a1 = 0.0049 b0 = 0.0046
Cooling rate (s-1) 0.0048
RT2 = 0.9997 Third order transfer function (Airspace ~ inlet temperature)
YIC = -21 SE = 0.0019 a1 = 0.103 a2 = 0.0003 a3 = 0.00000016 b0 = 0.016 b1 = 0.0002 b2 = 0.00000013
0.0126
RT2 = 0.9998 YIC = -10.4 SE = 0.00076
The results from Table 6.7 showed the cooling rate obtained from two methods: from the first order transfer function for a dynamic response of product and airspace temperature and from the third order transfer function for a dynamic response of airspace and inlet air temperature in a room filled with apples as an example for biological products. These results proved that the data based mechanistic model worked under the experimental conditions. The assumptions on the uniformity of temperature in a considered zone resulted in the different cooling rate values between two methods. Although there is a difference, but cooling rates obtained in both methods are always presented for heat transfer from fluid flow to product and
132
it is possible to use this value to monitor or control the uniformity of product temperature in a storage room. 6.5 Conclusions The cooling rate is an important value during the cooling process and can be obtained by two
methods; from the first order transfer function of a dynamic response of product and an airspace temperature, and from the solution of parameters from the third order transfer function of a dynamic response of airspace and a step inlet temperature. The differences between values from the two methods were estimated in the range of 22-66% at ventilation rates from 20-120 m3 h-1.
It is possible to calculate a bulk heat transfer coefficient by applying a data based mechanistic modelling approach and known product physical properties. The results from the first order transfer function of the dynamic response of product and airspace temperature were in range with some literature. A heat transfer coefficient was obtained by the two methods with different values from 18 to 40%. The results from the third order transfer function of the dynamic response of airspace and a step inlet temperature create a new method for calculating and monitoring bulk heat transfer coefficient values at different positions in a process room without measuring the product temperature in different heat treatment processes. An example with biological products in case of forced air cooling proved the possibility of a data based mechanistic modelling approach for real time monitoring, predicting and adaptive control of 3D temperature distribution in bulk products during food processing and storage.
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7 Chapter 7 General conclusions and Suggestions for future research
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Chapter 7 General conclusions and Suggestions for future research 7.1 General conclusions This study was about the improvement of the uniformity of the product temperature during
heating/cooling applications with conditioned air. In this study, a Data Based Mechanistic model (DBM) was developed for real time control of the 3D temperature distribution in a room filled with biological products.
This research was carried out in four steps: (1) Adapting a data based mechanistic model approach from an empty room to a room filled with obstacles. (2) Developing a data based mechanistic model for a room filled with plastic balls as obstacles, analysing the heat exchange from the environment to the air inside balls to find physically meaningful model parameters. (3) Developing a data base mechanistic model for a room filled with biological products, analysing heat exchange from environment to the biological products to find physically meaningful parameters for real time control of 3D product bulk temperature in heating/cooling applications. (4) Evaluating the ability of the model to be used for cooling of biological products.
In a first step, it was analysed wherever DMB method, as developed for an empty room, could be applied in a ventilated room filled with obstacles. An empty room (3x2x1.5 m) and a room filled with 480 plastic balls as simulative products were used and 36 thermocouples were installed in two vertical planes. Experiments were carried out with step changes in inlet temperature from 11 to 17oC and 17 to 11oC while maintaining a constant ventilation rate. Next, the ventilation rate was varied between 120 and 280 m3 h-1 (13.3 to 31.1 volumes changed per hour for the empty room and 27.2 to 63.2 volume changes per hour for a room 137
with the balls) in different experiments. The measured temperature uniformity index under steady-state conditions (duration for steady state was 2 h) in both rooms was calculated at different acceptable temperature differences. The results showed that the temperature uniformity index in the empty and filled ballrooms was dependent on the ventilation rates. High ventilation rates in combination with high acceptable temperature differences resulted in a high temperature uniformity index. The temperature uniformity index in both cases reached 100% with a ventilation rate of 240 m3 h-1 and acceptable temperature difference of 1oC, due to adequate mixing.
By adopting the mechanistic phase from an empty room to a room filled with obstacles to model the dynamic response of airspace from inlet air temperature, it was shown that DBM approach works in a room filled with obstacles.
In the second step, data based mechanistic modelling was further developed for a ventilated room filled with 480 balls. The dynamic response of airspace temperature, air inside the ball after steps on air inlet temperature was used to find a best-fitted model based on the coefficient of determination ( RT2 ), Young Information Criterion (YIC) and standard error (SE). A third order transfer function from the dynamic response of airspace from inlet air temperature was found as the best fitted model in correspondence with the general theory of
heat transfer from airspace to the balls.
Physically meaningful parameters were found from the fitted model: there are the local volumetric concentration of fresh air flow rate (β1), and cooling rate α2 which was related to an overall heat transfer coefficient from the airspace to the air inside the balls.
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In this step, a new method to calculate the cooling rate under unsteady state condition was found from Data based mechanistic modelling. This method can be used for real time controlling and predicting product temperature (Non-invasive method).
The distribution of local volumetric of fresh air concentration values in a room filled with obstacles represented an airflow pattern as in an empty room, and α2 values contains a heat capacity, density, diameter of air inside the ball and a heat transfer coefficient from airspace to air inside the ball. So cooling rate α2 explained the amount of energy which transfer from airspace to air inside the ball, and it can be used to calculate the temperature of air inside the ball.
For the third step, the data based mechanic model was developed for a room filled with solid material with different shapes as biological products. 144 boxes of potatoes (480 kg) were filled in the process room, 72 thermocouples (36 for the airspace and 36 for potatoes) were installed in the room to record the dynamic response of airspace and product temperature from steps on inlet air temperature. The ventilation rate was varied between 160 and 320 m3 h1
(16.6 to 37.6 volumes change per hour) for each experiment. The temperature uniformity
index of airspace and potato bulk was calculated at different acceptable temperature differences (0.2 to 1oC) and ventilation rates (160 to 320 m3 h-1). The calculated temperature uniformity index for airspace varied from 2.7 to 66%, and during potato bulk from 8.0 to 66%. The cumulative energy in potatoes during the process is not only dependent on the temperature in airspace but also on the airspace velocity and the physical characteristic of material.
A third order transfer function was found as the best fitted model and was obtained from
the dynamic response of airspace temperature from step on inlet air temperature. The equation 139
explained a heat transfer process in the room filled with the boxes of potatoes. Two physically meaningful parameters were found from model parameters.
An important parameter found in this step is the local ‘cooling rate’. The local cooling rate in a model explained the speed of energy transfer to the product from the airspace. The local cooling rate value contains both different temperature (between airspace and product) and the airspace velocity. The local cooling rate is an important parameter in the model that can be used for real time control of the 3D temperature distribution in a room filled with biological products.
A new technique was developed to calculate the cooling rate of the product from dynamic
measurements of air inlet temperature and the temperature in the airspace between the products was found in this part. This method can be used to control the 3D uniformity of product temperature in heating/cooling applications.
The last step of this research was the evaluation of the model. Two methods to calculate the local cooling rate and to estimate the product temperature from DBM model were found
in the second and third steps. - The first method is a general method (direct method): The dynamic response of the product temperature from airspace temperature was used and estimated parameters from the first order transfer function were applied to calculate a local cooling rate of the product.
- The second method is a non measuring product temperature method (indirect method), the dynamic response of the airspace from inlet air temperature was used, and estimated parameters from the third order transfer function were applied to calculate a local cooling rate of the product.
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Results of the local cooling rate with balls filled with pure water with ventilation rates, from 20, 100 and 120 m3 h-1 (20, 100 and 120 volumes change per hour) were calculated. The differences of the local cooling rates between the two methods at the different ventilation rates were 22 to 26%. The used assumptions such as no gradient temperature in the product
and uniformity of temperature and velocity in a well-mixed zone can explain the different values of the two methods.
This model has been proven to calculate the local heat transfer coefficient from the local cooling rate with known product physical characteristics (density, size and specific heat).
The results from this estimate for airspace to water inside the balls at ventilation rates 20, 100 and 120 m3 h-1 are 153 to 214 W m-2 oC-1 from first order transfer function, and 256 to 262 W m-2 oC-1 from third order transfer function.
In this research, Data based mechanistic model approach was successfully applied to model the temperature in a room filled with biological products and it extends for the 3D temperature distribution by using a well-mixed zone approach. Physically meaningful parameters were found such as a local volumetric of fresh air concentration and a local ‘cooling rate’. Their existence in a model are a key factor to explain an energy transfer within a fluid flow, a speed of energy transfer from fluid flow to product and a fluid flow velocity around products during heating/cooling processes. The local cooling rate is a physically meaningful parameter in the model that can be used in a control algorithm to predict the product temperature. It can be applied in real time control of the 3D temperature uniformity index in heating and cooling applications.
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7.2 Suggestions for future research Data based mechanistic model approach in this research is an initial study to apply a transfer
function model in the food industry. The dynamic response of the temperature to varying temperature of the coming air was modelled by the third order of transfer function. The heat exchange in this research is focused on the sensible heat (neglecting the latent heat and moisture loss). This model can be extended for heat and mass transfer processes such as sterilization, pasteurization, freezing, thawing, baking oven, drying process etc. For the drying process, the fourth order transfer function is suggested. One order for the heat transfer in a well mixed zone, one for heat loss, one for heat transfer to product and one for the diffusion of water from product to air surrounding (Fick s’ law equation). For freezing and thawing processes, the fourth order transfer function is also suggested with one order for the heat transfer process by the latent heat.
Data based mechanistic model in this thesis provides a new idea for further study to calculate a heat transfer coefficient. In addition, it is possible to extend to calculating a mass transfer coefficient in several mass transfer processes such as extracting, soaking, osmotic dehydration etc. to online control the uniformity of products and to predict a chemical concentration in product and fluid flow.
The model structure and physically meaningful parameters were found in this research should be analysis for model predictive control to design a controller for controlling 3D temperature distribution in the bulk of product during heating/cooling applications.
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List of publications International peer previewed articles
1. V T Thanh, E Vranken, D Berckmans (2008) Data Based Mechanistic Modelling for Control of Three Dimensional Temperature Distribution in a Room Filled with Biological Products. Journal of Food Engineering. 86(3), 422-432.
2. V T Thanh, E Vranken, A Van Brecht, D Berckmans (2007) Data Based Mechanistic Modelling for Control of Three Dimensional Temperature Distribution in a Room Filled with Obstacles. Journal of Biosystems Engineering. 98(1), 54-65.
3. V T Thanh, A Van Brecht, E Vranken, D Berckmans (2007) Modelling of Three Dimensional Air Temperature Distribution in Porous Media. Journal of Biosystems Engineering. 96(3), 345-360. (Nominated for EurAgEng Outstanding Paper Award 2008)
Proceeding
V T Thanh, A Van Brecht, E Vranken, D Berchmans (2004). Modelling of Three Dimensional Air Temperature Distributions in Porous Media. Communications in Agricultural and Applied Biological Sciences. 69(2), 281-284.
Poster
V T Thanh, A Van Brecht, E Vranken, D Berchmans (2004). Modelling of Three Dimensional Air Temperature Distributions in Porous Media. Applied Biological Sciences in GENT, 29 September 2004.
153
10th PhD Symposium on
154
Appendix
155
156
Appendix Appendix A-1 Dynamic temperature response from variation of inlet air temperature in the empty room and room filled with obstacles at ventilation rate 280 m3 h-1
22 20 18 16
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
0
1000
2000
3000
4000
5000 Time (s)
6000
7000
8000
9000
10000
22.5 22 21.5 21 20
o
Inlet temperature (C)
o
Buffer temperature (C)
o
Inside temperature ( C)
Empty room
18 16 14
24 22 20 18 16
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
0
1000
2000
3000
4000
5000 Time (s)
6000
7000
8000
9000
10000
23
o
Buffer temperature (C)
o
Inside temperature ( C)
Room filled with obstacles
22.5 22
20
o
Inlet temperature (C)
21.5
18 16 14
157
Appendix A-2 b0 from the second order of transfer function in the empty room and room filled with balls Numbered sensors 3 Rear plane
2 1
Front plane
3 2 1
34 35 36
28 29 30
22 23 24
16 17 18
10 11 12
4 5 6
1
2
3
4
5
6
31 32 33
25 26 27
19 20 21
13 14 15
7 8 9
1 2 3
1
2
3
4
5
6
At ventilation rate is 240 m3 h-1 Empty room 3 0.4220 0.2182 0.5543 0.2303 0.0772 0.2727 Rear 2 0.9089 0.2311 0.1840 0.1556 0.1690 0.0791 plane 1 0.2488 0.3940 0.1737 0.1524 0.1690 0.2317
Front plane
1 2 3 4 5 6 3 0.2677 0.1625 0.1765 0.1873 0.2180 0.1926 2 0.0000 0.2026 0.1408 0.1179 0.1254 0.1190 1 0.0000 0.6332 0.0416 0.1299 0.1566 0.1939 1
2
3
4
5
6
Room filled with balls 3 0.0198 0.1766 1.9361 0.2724 0.0127 Rear 2 0.7344 0.1878 0.0940 0.0263 0.0917 plane 1 0.2365 0.0569 0.0624 0.0444 0.0917
1.5695 0.0319 -0.0082
1 2 3 4 5 3 0.5190 0.1247 1.4725 0.2032 0.2505 2 0.4699 0.2159 0.0250 0.0185 0.0460 1 0.2596 0.0950 0.0258 0.0248 0.0006
6 0.3272 0.0211 0.0176
Front plane
1
2
3
158
4
5
6
Numbered sensors 3 Rear plane
2 1
Front plane
3 2 1
34 35 36
28 29 30
22 23 24
16 17 18
10 11 12
4 5 6
1
2
3
4
5
6
31 32 33
25 26 27
19 20 21
13 14 15
7 8 9
1 2 3
1
2
3
4
5
6
At ventilation rate is 280 m3 h-1 Empty room 3 0.1414 0.2257 0.3141 0.2521 0.0635 0.2576 Rear 2 0.8071 0.2806 0.1763 0.2134 0.2342 0.0814 plane 1 0.2103 0.4624 0.2484 0.2031 0.2342 0.2310
Front plane
1 2 3 4 5 6 3 0.2047 0.1672 0.1653 0.1967 0.1853 0.2362 2 0.6193 0.1832 0.1719 0.1556 0.1729 0.1616 1 0.3111 0.3283 0.0459 0.1633 0.2268 0.2548
1 2 Room filled with balls 3 0.0442 0.3036 Rear 2 1.5285 0.2788 plane 1 0.3872 0.0787
3
4
5
6
0.1392 0.0991 0.1136
0.2374 0.0494 0.0892
0.0251 0.1337 0.1337
0.1708 0.0603 0.0200
1 2 3 0.3810 0.2549 2 0.8666 0.3108
3 0.2257 0.0373
4 0.3319 0.0347
6 0.1437 0.0428
1 0.4618 0.1465
0.0366
0.0611
5 0.2377 0.0915 0.0064
0.0490
3
4
5
6
Front plane
1
2
159
Appendix B.1 Modelling results for the room filled with balls at the ventilation rate is 160 m3 h-1 RT2 b1 b2 Sensor positions a0 a1 a2 a3 b0
YIC
1
1
0.056
-0.0004
-2E-06
0.005
-5E-07
-7E-07
0.986
0.924
2
1
0.046
-0.0003
-2E-06
0.007
-2E-05
-7E-07
0.991
-6.873
3
1
0.008
-0.0001
-5E-07
0.002
-5E-06
-2E-07
0.811
4.766
4
1
0.031
-0.0003
-9E-07
0.003
-1E-05
-3E-07
0.876
-1.826
5
1
0.055
-0.0003
-2E-06
0.007
-6E-06
-9E-07
0.992
-5.308
6
1
0.026
-0.0002
-9E-07
0.003
-1E-05
-4E-07
0.996
-8.350
7
1
0.014
-0.0002
-4E-07
0.002
-1E-05
-2E-07
0.910
-1.909
8
1
0.065
-0.0004
-3E-06
0.013
-3E-05
-1E-06
0.995
-8.545
9
1
-0.005
-5E-05
2.4E-07
0.002
-3E-05
7.9E-08
0.975
-3.242
10
1
0.037
-0.0003
-1E-06
0.004
-1E-05
-4E-07
0.952
-3.509
11
1
0.076
-0.0004
-3E-06
0.013
-3E-05
-1E-06
0.992
-7.586
12
1
0.004
-1E-04
-2E-07
0.002
-2E-05
-1E-07
0.997
-6.247
13
1
0.046
-0.0003
-1E-06
0.006
-2E-05
-6E-07
0.990
-7.032
14
1
0.090
-0.0005
-4E-06
0.020
-4E-05
-2E-06
0.996
-8.796
15
1
-0.022
0.00034
-2E-06
#####
1.9E-05
2.1E-08
16
1
0.024
-0.0002
-8E-07
0.004
-2E-05
-3E-07
-0.722
3.346
17
1
0.073
-0.0004
-3E-06
0.016
-3E-05
-2E-06
0.994
-7.977
18
1
0.034
-0.0002
-1E-06
0.004
-3E-07
-7E-07
0.997
-1.231
19
1
0.021
-0.0002
-8E-07
0.003
-1E-05
-3E-07
-0.543
4.103
20
1
0.146
-0.0006
-8E-06
0.038
-4E-05
-4E-06
0.997
-8.152
21
1
0.012
-0.0001
-5E-07
0.003
-1E-05
-3E-07
0.997
-7.960
22
1
0.045
-0.0003
-2E-06
0.005
2.7E-06
-8E-07
0.920
1.561
23
1
0.087
-0.0004
-4E-06
0.019
-1E-05
-2E-06
0.995
-6.159
24
1
0.026
-0.0002
-8E-07
0.003
-4E-06
-5E-07
0.719
2.819
25
1
0.019
-0.0001
-6E-07
0.002
-9E-06
-2E-07
0.974
-4.352
26
1
0.081
-0.0004
-4E-06
0.024
-7E-05
-2E-06
0.996
-9.167
27
1
0.004
-9E-05
-2E-07
0.004
-4E-05
-2E-07
0.993
-4.030
28
1
0.022
-0.0002
-1E-06
0.002
3.6E-06
-3E-07
0.907
0.239
29
1
0.003
-1E-04
-1E-07
0.003
-3E-05
-6E-08
0.996
-4.905
30
1
0.027
0.00021
2.5E-07
0.003
8.5E-05
1.6E-07
0.999
-6.059
31
1
0.032
-0.0002
-1E-06
0.004
-2E-05
-4E-07
0.600
0.371
32
1
0.160
-0.0005
-9E-06
0.053
-5E-05
-5E-06
0.994
33
1
0.032
-0.0004
-3E-06
0.003
-7E-05
-2E-06
34
1
0.053
-0.0004
-1E-06
0.007
-3E-05
-6E-07
0.989
-7.120
35
1
0.742
-0.0025
-4E-05
0.276
-0.0003
-2E-05
0.997
-10.079
36
1
0.000
-9E-05
6.6E-08
0.004
-5E-05
3.2E-08
0.998
2.584
160
-6.570 40.531
Table B.2 Modelling results for the room filled with balls at the ventilation rate is 180 m3 h-1 RT2 b1 b2 Sensor positions a0 a1 a2 a3 b0
YIC
1
1
0.036
-0.0002
-1E-06
0.006
-1E-05
-5E-07
0.993
-6.723
2
1
0.047
-0.0002
-2E-06
0.012
-3E-05
-1E-06
0.988
-6.414
3
1
0.002
-6E-05
-1E-07
0.002
-2E-05
-8E-08
0.997
-4.276
4
1
0.048
-0.0003
-1E-06
0.007
-2E-05
-6E-07
0.985
-6.204
5
1
0.059
-0.0002
-2E-06
0.010
1.2E-05
-1E-06
0.996
-7.113
6
1
0.033
-0.0002
-1E-06
0.006
-8E-06
-6E-07
0.997
-7.841
7
1
0.022
-0.0002
-5E-07
0.005
-3E-05
-3E-07
0.992
-7.388
8
1
0.078
-0.0003
-3E-06
0.021
-3E-05
-2E-06
0.994
-7.520
9
1
0.011
-0.0001
-3E-07
0.003
-2E-05
-2E-07
0.994
-6.810
10
1
0.031
-0.0002
-7E-07
0.006
-3E-05
-3E-07
0.968
-5.032
11
1
0.051
-0.0003
-2E-06
0.012
-3E-05
-9E-07
0.982
-6.052
12
1
0.009
-1E-04
-3E-07
0.003
-2E-05
-2E-07
0.986
-4.788
13
1
0.052
-0.0002
-2E-06
0.008
9E-07
-8E-07
0.990
-1.057
14
1
0.131
-0.0005
-5E-06
0.033
1.9E-05
-3E-06
0.990
-4.774
15
1
0.006
-7E-05
-4E-07
0.001
2E-06
-2E-07
0.994
-0.483
16
1
0.074
-0.0003
-2E-06
0.014
-2E-05
-1E-06
0.995
-7.681
17
1
0.103
-0.0003
-5E-06
0.026
1.7E-05
-3E-06
0.997
-7.149
18
1
0.044
-0.0002
-2E-06
0.005
3E-05
-1E-06
0.998
-9.040
19
1
0.050
-0.0002
-2E-06
0.010
-2E-05
-8E-07
0.989
-6.407
20
1
0.132
-0.0003
-6E-06
0.038
3.4E-05
-4E-06
0.998
-8.342
21
1
0.030
0.00017
-2E-07
0.001
9.6E-05
-6E-08
0.998
-2.716
22
1
0.074
-0.0003
-3E-06
0.017
-2E-05
-1E-06
0.996
-7.623
23
1
0.079
-0.0002
-4E-06
0.020
1.3E-05
-2E-06
0.997
-6.583
24
1
0.017
-0.0001
-6E-07
0.003
-6E-06
-3E-07
0.995
-5.117
25
1
0.045
-0.0003
-1E-06
0.007
-2E-05
-6E-07
0.993
-7.285
26
1
0.083
-0.0003
-4E-06
0.028
-3E-05
-2E-06
0.998
-8.967
27
1
0.000
-7E-05
9.9E-08
0.004
-6E-05
6.2E-08
0.987
3.861
28
1
0.036
-0.0002
-1E-06
0.008
-3E-05
-6E-07
0.992
-7.423
29
1
0.007
-9E-05
-2E-07
0.004
-4E-05
-1E-07
0.998
-7.068
30
1
0.045
0.00057
1.4E-06
0.002
0.00021
9.2E-07
0.999
-9.385
31
1
0.051
-0.0003
-2E-06
0.009
-2E-05
-8E-07
0.994
-7.423
32
1
0.179
-0.0004
-9E-06
0.066
2.1E-05
-6E-06
0.998
-6.750
33
1
0.084
-0.0001
-5E-06
0.024
0.00014
-4E-06
0.995
-8.047
34
1
0.041
-0.0002
-2E-06
0.007
-8E-06
-7E-07
0.994
-6.431
35
1
0.460
-0.0007
-2E-05
0.190
0.00015
-2E-05
0.998
-9.177
36
1
0.000
-9E-05
2.5E-07
0.005
-7E-05
1.7E-07
0.998
-1.394
161
Table B.3 Modelling results for the room filled with balls at the ventilation rate is 200 m3 h-1 RT2 b1 b2 Sensor positions a0 a1 a2 a3 b0
YIC
1
1
0.052
-0.0002
-2E-06
0.008
1.2E-06
-8E-07
0.980
-0.114
2
1
0.065
-0.0002
-3E-06
0.016
-8E-06
-1E-06
0.993
-4.683
3
1
0.001
-6E-05
-4E-08
0.002
-2E-05
-3E-08
0.997
-2.350
4
1
0.042
-0.0002
-2E-06
0.006
-2E-06
-6E-07
0.964
-0.623
5
1
0.081
-9E-05
-5E-06
0.014
0.0001
-2E-06
0.995
-8.250
6
1
0.041
-0.0001
-2E-06
0.007
1.8E-05
-1E-06
0.997
-7.854
7
1
0.040
-0.0002
-2E-06
0.007
-5E-07
-7E-07
0.965
3.338
8
1
0.103
-0.0002
-5E-06
0.027
6.7E-05
-3E-06
0.996
-8.055
9
1
0.030
-0.0001
-1E-06
0.005
1.3E-05
-7E-07
0.995
-6.577
10
1
0.058
-0.0002
-2E-06
0.008
-4E-06
-7E-07
0.936
-0.354
11
1
0.097
-0.0001
-5E-06
0.023
6.5E-05
-3E-06
0.992
-6.662
12
1
0.048
-9E-05
-3E-06
0.007
5.9E-05
-1E-06
0.997
-8.875
13
1
0.064
-0.0002
-3E-06
0.011
2.2E-05
-1E-06
0.985
-5.383
14
1
0.128
-0.0002
-6E-06
0.032
9.4E-05
-4E-06
0.996
-8.712
15
1
0.022
-0.0001
-1E-06
0.000
3.3E-05
-5E-07
0.997
-4.298
16
1
0.083
-0.0003
-3E-06
0.014
-2E-05
-1E-06
0.980
-5.038
17
1
0.199
-0.0002
-1E-05
0.053
0.00018
-6E-06
0.996
-8.388
18
1
0.060
-0.0002
-3E-06
0.007
7.1E-05
-2E-06
0.997
-9.359
19
1
0.073
-0.0002
-4E-06
0.015
1.4E-05
-2E-06
0.989
-4.503
20
1
0.150
-7E-05
-9E-06
0.044
0.00014
-5E-06
0.997
-7.369
21
1
0.018
-3E-05
-1E-06
0.002
5.6E-05
-1E-06
0.998
-4.996
22
1
0.115
-0.0002
-6E-06
0.030
3.1E-05
-3E-06
0.987
-4.897
23
1
0.081
-0.0002
-4E-06
0.026
1.6E-05
-2E-06
0.995
-5.620
24
1
0.018
-0.0002
-4E-07
0.004
-2E-05
-2E-07
0.979
-4.690
25
1
0.060
-0.0001
-3E-06
0.008
2.4E-05
-1E-06
0.988
-5.527
26
1
0.136
2.3E-06
-9E-06
0.046
0.00015
-5E-06
0.997
-0.549
27
1
0.036
3.2E-05
-3E-06
0.010
0.00013
-3E-06
0.996
-2.856
28
1
0.097
-0.0002
-5E-06
0.025
3.9E-05
-3E-06
0.996
-7.663
29
1
0.019
-0.0001
-1E-06
0.007
-3E-05
-6E-07
0.982
-4.581
30
1
0.057
0.00118
5.7E-06
0.002
0.00033
3.6E-06
0.999
-4.461
31
1
0.072
-0.0002
-3E-06
0.013
2.8E-06
-1E-06
0.989
-2.168
32
1
0.215
5.6E-05
-1E-05
0.084
0.00025
-9E-06
0.998
-6.375
33
1
0.084
8.6E-05
-6E-06
0.028
0.00019
-4E-06
0.998
-8.812
34
1
0.039
-0.0001
-2E-06
0.008
-2E-06
-7E-07
0.966
0.388
35
1
0.481
0.00047
-3E-05
0.206
0.00078
-2E-05
0.998
-9.321
36
1
0.007
-0.0001
-4E-08
0.007
-7E-05
-4E-08
0.998
-1.890
162
Appendix B-4 Modelling results for the room filled with balls at the ventilation rate is 220 m3 h-1 RT2 b1 b2 Sensor postions a0 a1 a2 a3 b0
YIC
1
1
0.026
-0.0002
-8E-07
0.008
-4E-05
-4E-07
0.934
-3.071
2
1
0.054
-0.0002
-2E-06
0.017
-2E-05
-1E-06
0.995
-6.704
3
1
-0.007
-5E-05
3.6E-07
0.003
-5E-05
2.1E-07
0.997
-7.582
4
1
0.040
-0.0001
-2E-06
0.008
4.6E-06
-8E-07
0.991
-3.683
5
1
0.055
-0.0001
-3E-06
0.013
2.5E-05
-2E-06
0.997
-8.146
6
1
0.029
-0.0001
-1E-06
0.006
-2E-06
-7E-07
0.997
-4.596
7
1
0.046
-0.0002
-2E-06
0.011
-1E-05
-1E-06
0.992
-5.876
8
1
0.075
-0.0002
-4E-06
0.019
3.7E-05
-2E-06
0.996
-7.783
9
1
0.026
-0.0001
-1E-06
0.005
8.5E-06
-6E-07
0.996
-5.757
10
1
0.052
-0.0002
-2E-06
0.011
-2E-05
-8E-07
0.975
-4.941
11
1
0.076
-0.0002
-4E-06
0.021
3.2E-05
-2E-06
0.996
-7.430
12
1
0.020
-1E-04
-9E-07
0.005
-1E-06
-6E-07
0.998
-3.467
13
1
0.078
-0.0002
-4E-06
0.021
4.1E-05
-2E-06
0.995
-7.371
14
1
0.102
-0.0002
-5E-06
0.027
6.1E-05
-3E-06
0.996
-8.482
15
1
0.015
0.00034
9.6E-07
0.001
1.9E-05
5.6E-07
0.998
-9.007
16
1
0.037
-0.0002
-1E-06
0.009
-4E-05
-5E-07
0.981
-6.150
17
1
0.095
-0.0002
-5E-06
0.026
6.9E-05
-3E-06
0.996
-8.715
18
1
0.027
-0.0001
-1E-06
0.005
1E-05
-7E-07
0.998
-7.327
19
1
0.087
-0.0001
-4E-06
0.027
4.2E-05
-3E-06
0.997
-8.243
20
1
0.122
-9E-05
-7E-06
0.041
0.00012
-4E-06
0.998
-8.727
21
1
0.038
0.0005
1.6E-06
0.005
0.00018
1.1E-06
0.998
-7.657
22
1
0.062
-0.0002
-3E-06
0.017
-4E-06
-2E-06
0.994
-3.322
23
1
0.094
-0.0002
-5E-06
0.027
6E-05
-3E-06
0.997
-8.610
24
1
0.008
0.00018
8.7E-07
0.003
1.4E-05
5.7E-07
0.995
-2.651
25
1
0.052
-0.0002
-2E-06
0.015
-3E-05
-1E-06
0.994
-7.443
26
1
0.103
-0.0002
-5E-06
0.043
-1E-05
-3E-06
0.998
-5.390
27
1
0.028
-8E-05
-2E-06
0.012
5.2E-06
-1E-06
0.996
-3.772
28
1
0.070
-0.0001
-3E-06
0.021
1.7E-05
-2E-06
0.997
-7.286
29
1
0.010
-0.0001
-1E-07
0.007
-6E-05
-1E-07
0.998
-6.841
30
1
0.042
0.00071
2.9E-06
0.006
0.00028
2.1E-06
0.858
0.185
31
1
0.062
-0.0002
-3E-06
0.017
-6E-06
-1E-06
0.996
-5.189
32
1
0.155
-0.0001
-8E-06
0.068
7.7E-05
-6E-06
0.998
-8.132
33
1
0.062
-6E-05
-4E-06
0.025
5E-05
-3E-06
0.998
-8.459
34
1
0.080
-0.0001
-4E-06
0.023
3.5E-05
-2E-06
0.997
-8.457
35
1
0.399
0.00043
-3E-05
0.187
0.00074
-2E-05
0.998
-9.984
36
1
0.007
-0.0001
1.2E-07
0.008
-1E-04
7.9E-08
0.999
-5.963
163
Appendix B-5 Modelling results for the room filled with balls at the ventilation rate is 240 m3 h-1 RT2 b1 b2 Sensor postions a0 a1 a2 a3 b0
YIC
1
1
0.026
-0.0002
-8E-07
0.008
-4E-05
-4E-07
0.934
-3.071
2
1
0.054
-0.0002
-2E-06
0.017
-2E-05
-1E-06
0.995
-6.704
3
1
-0.007
-5E-05
3.6E-07
0.003
-5E-05
2.1E-07
0.997
-7.582
4
1
0.040
-0.0001
-2E-06
0.008
4.6E-06
-8E-07
0.991
-3.683
5
1
0.055
-0.0001
-3E-06
0.013
2.5E-05
-2E-06
0.997
-8.146
6
1
0.029
-0.0001
-1E-06
0.006
-2E-06
-7E-07
0.997
-4.596
7
1
0.046
-0.0002
-2E-06
0.011
-1E-05
-1E-06
0.992
-5.876
8
1
0.075
-0.0002
-4E-06
0.019
3.7E-05
-2E-06
0.996
-7.783
9
1
0.026
-0.0001
-1E-06
0.005
8.5E-06
-6E-07
0.996
-5.757
10
1
0.052
-0.0002
-2E-06
0.011
-2E-05
-8E-07
0.975
-4.941
11
1
0.076
-0.0002
-4E-06
0.021
3.2E-05
-2E-06
0.996
-7.430
12
1
0.020
-1E-04
-9E-07
0.005
-1E-06
-6E-07
0.998
-3.467
13
1
0.078
-0.0002
-4E-06
0.021
4.1E-05
-2E-06
0.995
-7.371
14
1
0.102
-0.0002
-5E-06
0.027
6.1E-05
-3E-06
0.996
-8.482
15
1
0.015
0.00034
9.6E-07
0.001
1.9E-05
5.6E-07
0.998
-9.007
16
1
0.037
-0.0002
-1E-06
0.009
-4E-05
-5E-07
0.981
-6.150
17
1
0.095
-0.0002
-5E-06
0.026
6.9E-05
-3E-06
0.996
-8.715
18
1
0.027
-0.0001
-1E-06
0.005
1E-05
-7E-07
0.998
-7.327
19
1
0.087
-0.0001
-4E-06
0.027
4.2E-05
-3E-06
0.997
-8.243
20
1
0.122
-9E-05
-7E-06
0.041
0.00012
-4E-06
0.998
-8.727
21
1
0.038
0.0005
1.6E-06
0.005
0.00018
1.1E-06
0.998
-7.657
22
1
0.062
-0.0002
-3E-06
0.017
-4E-06
-2E-06
0.994
-3.322
23
1
0.094
-0.0002
-5E-06
0.027
6E-05
-3E-06
0.997
-8.610
24
1
0.008
0.00018
8.7E-07
0.003
1.4E-05
5.7E-07
0.995
-2.651
25
1
0.052
-0.0002
-2E-06
0.015
-3E-05
-1E-06
0.994
-7.443
26
1
0.103
-0.0002
-5E-06
0.043
-1E-05
-3E-06
0.998
-5.390
27
1
0.028
-8E-05
-2E-06
0.012
5.2E-06
-1E-06
0.996
-3.772
28
1
0.070
-0.0001
-3E-06
0.021
1.7E-05
-2E-06
0.997
-7.286
29
1
0.010
-0.0001
-1E-07
0.007
-6E-05
-1E-07
0.998
-6.841
30
1
0.042
0.00071
2.9E-06
0.006
0.00028
2.1E-06
0.858
0.185
31
1
0.062
-0.0002
-3E-06
0.017
-6E-06
-1E-06
0.996
-5.189
32
1
0.155
-0.0001
-8E-06
0.068
7.7E-05
-6E-06
0.998
-8.132
33
1
0.062
-6E-05
-4E-06
0.025
5E-05
-3E-06
0.998
-8.459
34
1
0.080
-0.0001
-4E-06
0.023
3.5E-05
-2E-06
0.997
-8.457
35
1
0.399
0.00043
-3E-05
0.187
0.00074
-2E-05
0.998
-9.984
36
1
0.007
-0.0001
1.2E-07
0.008
-1E-04
7.9E-08
0.999
-5.963
164
Appendix B-6 Modelling results for the room filled with balls at the ventilation rate is 260 m3 h-1 RT2 b1 b2 Sensor postions a0 a1 a2 a3 b0
YIC
1
1
0.041
-0.0001
-2E-06
0.013
-2E-05
-1E-06
0.993
-6.693
2
1
0.053
-0.0002
-2E-06
0.018
-2E-05
-1E-06
0.996
-7.211
3
1
0.003
-6E-05
-2E-07
0.003
-2E-05
-1E-07
0.997
-4.805
4
1
0.043
-0.0002
-2E-06
0.011
-2E-05
-8E-07
0.993
-7.485
5
1
0.058
-0.0001
-3E-06
0.014
3E-05
-2E-06
0.997
-8.297
6
1
0.029
-0.0001
-1E-06
0.007
-4E-06
-7E-07
0.997
-5.789
7
1
0.061
-0.0002
-2E-06
0.017
-1E-05
-1E-06
0.994
-5.562
8
1
0.064
-0.0002
-3E-06
0.017
9.8E-06
-2E-06
0.996
-5.697
9
1
0.016
-0.0001
-6E-07
0.004
-1E-05
-4E-07
0.996
-6.830
10
1
0.043
-0.0002
-1E-06
0.011
-5E-05
-6E-07
0.982
-6.466
11
1
0.073
-0.0002
-3E-06
0.022
9.6E-06
-2E-06
0.997
-5.772
12
1
0.020
-1E-04
-8E-07
0.005
-5E-06
-6E-07
0.998
-6.213
13
1
0.081
-0.0002
-4E-06
0.027
2.8E-05
-3E-06
0.997
-7.407
14
1
0.081
-0.0002
-4E-06
0.024
2.6E-05
-2E-06
0.997
-7.751
15
1
0.023
-0.0001
-9E-07
0.000
4.1E-05
-6E-07
0.998
-6.400
16
1
0.053
-0.0002
-2E-06
0.012
9E-06
-1E-06
0.991
-4.795
17
1
0.070
-0.0002
-3E-06
0.021
2.6E-05
-2E-06
0.997
-7.656
18
1
0.028
-0.0001
-1E-06
0.005
2.2E-05
-8E-07
0.997
-7.824
19
1
0.114
-3E-05
-7E-06
0.038
0.00016
-5E-06
0.997
-7.119
20
1
0.166
0.00017
-1E-05
0.056
0.00037
-8E-06
0.997
-9.431
21
1
0.143
0.00128
-8E-07
0.001
0.00091
-5E-07
0.998
-7.168
22
1
0.068
-0.0001
-3E-06
0.019
3.3E-05
-2E-06
0.994
-7.038
23
1
0.102
-0.0002
-5E-06
0.029
7.3E-05
-3E-06
0.997
-9.161
24
1
0.008
-7E-05
-4E-07
0.005
-2E-05
-3E-07
0.998
-6.631
25
1
0.083
-0.0001
-4E-06
0.026
4.4E-05
-3E-06
0.996
-7.904
26
1
0.124
-9E-05
-7E-06
0.053
9.8E-05
-5E-06
0.998
-8.180
27
1
0.031
-2E-05
-2E-06
0.012
5.4E-05
-2E-06
0.998
-6.665
28
1
0.091
-1E-04
-5E-06
0.029
7.4E-05
-3E-06
0.997
-8.915
29
1
0.013
-0.0001
-4E-07
0.008
-6E-05
-3E-07
0.998
-8.250
30
1
0.017
0.00013
7.6E-07
0.009
4E-05
5.5E-07
0.999
-8.159
31
1
0.079
-0.0002
-4E-06
0.026
7.8E-06
-2E-06
0.997
-5.312
32
1
0.247
0.00012
-2E-05
0.110
0.00038
-1E-05
0.998
-8.209
33
1
0.096
0.00012
-7E-06
0.036
0.00025
-6E-06
0.998
-9.432
34
1
0.098
-0.0001
-5E-06
0.030
7.3E-05
-3E-06
0.997
-8.980
35
1
0.466
0.00037
-3E-05
0.229
0.00084
-2E-05
0.998
-9.510
36
1
0.015
-0.0001
-5E-07
0.009
-6E-05
-4E-07
0.998
-8.257
165
Appendix B-7 Modelling results for the room filled with balls at the ventilation rate is 280 m3 h-1 RT2 YIC Sensor positions a0 a1 a2 a3 b0 b1 b2 1
1
0.058
-0.0002
-3E-06
0.018
-3E-05
-2E-06
0.990
-6.293
2
1
0.058
-0.0003
-3E-06
0.020
-4E-05
-2E-06
0.997
-8.869
3
1
0.003
-8E-05
-2E-07
0.003
-3E-05
-1E-07
0.933
0.581
4
1
0.068
-0.0003
-3E-06
0.019
-3E-05
-2E-06
0.996
-7.919
5
1
0.054
-0.0003
-2E-06
0.015
-1E-05
-2E-06
0.997
-7.627
6
1
0.024
-0.0001
-1E-06
0.007
-2E-05
-7E-07
0.998
-8.420
7
1
0.068
-0.0003
-3E-06
0.023
-4E-05
-2E-06
0.995
-7.888
8
1
0.048
-0.0002
-2E-06
0.014
-3E-05
-1E-06
0.926
-2.575
9
1
0.005
-1E-04
-2E-07
0.004
-3E-05
-2E-07
0.998
-7.423
10
1
0.076
-0.0003
-3E-06
0.019
-8E-06
-2E-06
0.987
-3.267
11
1
0.071
-0.0003
-3E-06
0.022
-2E-05
-2E-06
0.997
-7.407
12
1
0.009
-0.0001
-4E-07
0.005
-3E-05
-3E-07
0.998
-8.355
13
1
0.093
-0.0003
-5E-06
0.035
-2E-06
-4E-06
0.998
-2.521
14
1
0.072
-0.0003
-4E-06
0.023
6.2E-07
-3E-06
0.997
-0.783
15
1
0.032
-0.0002
-1E-06
0.000
6.5E-05
-1E-06
0.999
-5.624
16
1
0.093
-0.0003
-5E-06
0.025
3.6E-05
-3E-06
0.964
-3.314
17
1
0.076
-0.0003
-4E-06
0.022
1.1E-05
-3E-06
0.977
-2.014
18
1
0.027
-0.0002
-1E-06
0.005
2.2E-05
-1E-06
0.994
-5.907
19
1
0.139
-0.0003
-8E-06
0.053
5.7E-05
-6E-06
0.998
-8.582
20
1
0.162
-0.0003
-1E-05
0.065
0.00015
-8E-06
0.997
-8.865
21
1
-0.005
-8E-05
4E-07
0.005
-1E-04
3.6E-07
0.998
-5.752
22
1
0.105
-0.0003
-6E-06
0.032
6.4E-05
-4E-06
0.949
-3.297
23
1
0.159
-0.0004
-9E-06
0.038
0.00019
-6E-06
0.908
-3.231
24
1
0.033
-0.0001
-2E-06
0.007
7.4E-05
-2E-06
0.998
-13.361
25
1
0.105
-0.0004
-5E-06
0.040
-3E-05
-4E-06
0.997
-7.710
26
1
0.136
-0.0004
-7E-06
0.067
-8E-05
-6E-06
0.996
-7.910
27
1
0.030
-0.0001
-2E-06
0.014
-2E-05
-2E-06
0.997
-7.282
28
1
0.113
-0.0004
-6E-06
0.042
-1E-05
-4E-06
0.998
-6.166
29
1
0.027
-0.0002
-1E-06
0.013
-5E-05
-1E-06
0.998
-9.937
30
1
-0.001
-0.0001
4E-07
0.007
-0.0001
3.5E-07
0.996
-3.484
31
1
0.112
-0.0003
-6E-06
0.041
3.1E-05
-5E-06
0.998
-7.966
32
1
0.225
-0.0005
-1E-05
0.113
1.8E-06
-1E-05
0.998
-1.114
33
1
0.086
-0.0001
-6E-06
0.038
7.7E-05
-5E-06
0.998
-9.027
34
1
0.100
-0.0004
-5E-06
0.035
-2E-05
-3E-06
0.994
-5.986
35
1
0.349
-0.0007
-2E-05
0.188
-1E-05
-2E-05
0.990
-0.806
36
1
0.019
-0.0001
-8E-07
0.012
-7E-05
-7E-07
0.998
-9.646
166
Appendix B-8 Modelling results (1st TF) for the room filled with balls at the ventilation rate is 160 m3 h-1 (Dynamic response of ball temperature from airspace temperature) RT2 a1 b0 YIC Sensors a0 1 1 0.008 0.007 0.998 -15.741 2 1 0.009 0.008 0.998 -15.971 3 1 0.025 0.023 1.000 -17.677 4 1 0.042 0.042 1.000 -18.373 5 1 0.034 0.032 0.999 -15.163 6 1 0.017 0.016 0.999 -16.344 7 1 0.025 0.025 1.000 -17.940 8 1 0.023 0.025 0.998 -14.610 Appendix B-9 Modelling results (1st TF) for the room filled with balls at the ventilation rate is 180 m3 h-1 RT2 Sensors a0 a1 b0 YIC 1 1 0.009 0.009 1.000 -18.465 2 1 0.009 0.009 0.999 -18.089 3 1 0.027 0.024 0.999 -17.052 4 1 0.037 0.037 1.000 -18.729 5 1 0.031 0.028 0.999 -16.389 6 1 0.016 0.016 1.000 -18.136 7 1 0.025 0.023 0.999 -17.231 8 1 0.015 0.013 0.999 -16.760 Appendix B-10 Modelling results (1st TF) for the room filled with balls at the ventilation rate is 200 m3 h-1 RT2 Sensors a0 a1 b0 YIC 1 1 0.009 0.008 0.999 -16.901 2 1 0.010 0.010 0.999 -16.664 3 1 0.033 0.030 0.996 -13.404 4 1 0.037 0.036 1.000 -18.405 5 1 0.029 0.027 0.999 -16.417 6 1 0.017 0.016 1.000 -18.641 7 1 0.017 0.014 0.998 -15.588 8 1 0.013 0.012 0.999 -18.251
167
Appendix B-11 Modelling results (1st TF) for the room filled with balls at the ventilation rate is 220 m3 h-1 RT2 a1 b0 YIC Sensors a0 1 1 0.010 0.009 0.998 -16.027 2 1 0.010 0.009 0.998 -15.372 3 1 0.029 0.026 1.000 -17.884 4 1 0.036 0.035 1.000 -18.333 5 1 0.029 0.027 0.999 -15.547 6 1 0.017 0.016 0.999 -16.948 7 1 0.018 0.014 0.999 -17.071 8 1 0.014 0.015 0.999 -17.562 Appendix B-12 Modelling results (1st TF) for the room filled with balls at the ventilation rate is 240 m3 h-1 RT2 a1 b0 YIC Sensors a0 1 1 0.010 0.009 0.999 -16.795 2 1 0.010 0.010 0.998 -15.712 3 1 0.022 0.020 0.999 -16.077 4 1 0.031 0.030 1.000 -18.177 5 1 0.025 0.022 0.999 -16.618 6 1 0.022 0.021 0.999 -14.906 7 1 0.018 0.015 1.000 -18.269 8 1 0.015 0.016 0.999 -18.173 Appendix B-13 Modelling results (1st TF) for the room filled with balls at the ventilation rate is 260 m3 h-1 RT2 a1 b0 YIC Sensors a0 1 1 0.010 0.009 0.999 -16.462 2 1 0.011 0.010 0.998 -15.941 3 1 0.023 0.021 1.000 -19.254 4 1 0.030 0.030 1.000 -18.421 5 1 0.024 0.021 0.999 -16.974 6 1 0.019 0.018 0.999 -15.917 7 1 0.018 0.016 1.000 -18.345 8 1 0.016 0.017 0.998 -16.212
168
Appendix B-14 Modelling results (1st TF) for the room filled with balls at the ventilation rate is 280 m3 h-1 RT2 a1 b0 YIC Sensors a0 1 1 0.012 0.012 0.999 -17.980 2 1 0.011 0.011 0.999 -16.321 3 1 0.023 0.021 1.000 -19.830 4 1 0.026 0.026 1.000 -19.089 5 1 0.022 0.019 1.000 -17.649 6 1 0.023 0.023 1.000 -17.294 7 1 0.018 0.017 1.000 -18.279 8 1 0.017 0.017 0.999 -16.946
169
Appendix C-1 Modelling results for the room filled with potatoes at the ventilation rate is 160 m3 h-1 RT2 YIC Positions a0 a1 a2 a3 b0 b1 b2 α2 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
1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0.097 0.081 1.663 0.107 0.309 1.300 0.089 0.056 0.120 0.000 0.110 1.361 0.061 0.131 0.000 0.054 0.035 0.076 0.097 0.064 0.057 0.115 0.207 0.041 0.224 0.432 1.731 0.125 0.431 0.220 0.422 1.634
-0.001 -0.001 -0.014 -0.001 -0.002 -0.011 -0.001 0.000 -0.001 0.000 -0.001 -0.010 -0.001 -0.001 0.000 0.000 0.000 -0.001 -0.001 -0.001 -0.001 -0.001 -0.001 0.000 -0.001 -0.002 -0.016 -0.001 -0.002 -0.002 -0.001 -0.003
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
33 34 35 36
1 1 1 1
0.008 0.148 0.706 0.004
0.000 -0.001 -0.002 0.003
0.000 0.000 0.000 0.000
0.020 0.008 0.076 0.018 0.038 0.116 0.016 0.009 0.005 0.000 0.020 0.113 0.013 0.020 0.000 0.012 0.005 0.005 0.027 0.006 0.002 0.030 0.038 0.002 0.050 0.110 0.031 0.033 0.128 0.028 0.113 0.681 0.057 0.027 0.270 0.015
170
-0.00012 -0.00003 0.00001 -0.00006 -0.00004 -0.00043 -0.00009 -0.00006 -0.00003 0.00000 -0.00008 -0.00005 -0.00013 -0.00006 0.00000 -0.00012 -0.00006 -0.00001 -0.00024 -0.00003 -0.00001 -0.00018 -0.00004 -0.00001 -0.00008 -0.00007 0.00034 -0.00021 -0.00019 -0.00004 -0.00006 0.00026
0.00000 0.00000 -0.00002 0.00000 -0.00001 -0.00002 0.00000 0.00000 0.00000 0.00000 0.00000 -0.00003 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 -0.00001 0.00000 -0.00001 -0.00002 -0.00002 0.00000 -0.00002 -0.00001 -0.00002 -0.00013
0.990 0.995 0.996 0.994 0.995 0.996 0.993 0.995 0.992 0.000 0.991 0.995 0.988 0.993 0.000 0.993 0.996 0.992 0.987 0.997 0.995 0.988 0.994 0.995 0.987 0.993 0.993 0.987 0.994 0.996 0.989 0.996
-7.609 -7.901 -6.450 -7.945 -7.368 -9.750 -8.300 -8.936 -7.685 0.000 -7.661 -7.405 -7.102 -8.145 0.000 -8.051 -8.744 -6.046 -6.952 -9.101 -7.548 -7.266 -6.910 -7.209 -5.403 -6.243 -8.167 -6.977 -7.900 -8.294 -3.890 -6.541
0.010 0.016 0.017 0.014 0.015 0.012 0.011 0.011 0.012 0.000 0.013 0.015 0.008 0.014 0.000 0.008 0.007 0.017 0.007 0.014 0.016 0.010 0.015 0.017 0.014 0.015 0.030 0.009 0.013 0.015 0.014 0.014
0.00127 -0.00006 0.00001 -0.00057
0.00000 -0.00001 -0.00006 0.00002
0.986 0.986 0.996 0.996
-8.604 -6.134 -1.536 -6.610
-0.002 0.014 0.014 -
Appendix C-2 Modelling results for the room filled with potatoes at the ventilation rate is 200 m3 h-1 RT2 YIC Positions a0 a1 a2 a3 b0 b1 b2 α2 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
1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1
0.184 0.046 0.079 0.262 1.098 0.757 0.133 0.183 0.025 0.156 0.182 0.278 0.192 0.000 -0.012 0.375 0.046 0.037 0.393 0.070 0.002 0.416 0.148 0.134 0.220 0.158 0.000 0.234 0.403 0.259 0.261 0.814
-0.001 0.000 -0.001 -0.002 -0.008 -0.007 -0.001 -0.001 0.000 -0.001 -0.001 -0.002 -0.001 0.000 0.000 -0.002 -0.001 0.000 -0.002 -0.001 0.000 -0.002 -0.001 -0.001 -0.001 -0.001 0.000 -0.001 -0.002 -0.002 -0.001 -0.003
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
33 34 35 36
1 1 1 1
-0.001 0.081 0.445 0.017
0.000 -0.001 -0.002 0.003
0.000 0.000 0.000 0.000
0.045 0.008 0.005 0.057 0.134 0.074 0.030 0.031 0.002 0.035 0.038 0.027 0.049 0.000 0.000 0.096 0.006 0.004 0.153 0.009 0.001 0.123 0.038 0.016 0.078 0.054 0.000 0.077 0.148 0.045 0.100 0.393 0.051 0.025 0.198 0.014
171
-0.00015 -0.00008 -0.00004 -0.00015 0.00004 -0.00032 -0.00015 -0.00005 -0.00004 -0.00016 -0.00011 -0.00005 -0.00017 0.00000 -0.00001 -0.00018 -0.00006 -0.00005 -0.00032 -0.00008 -0.00002 -0.00018 -0.00019 -0.00010 -0.00037 -0.00026 0.00000 -0.00030 -0.00036 -0.00020 -0.00034 -0.00063
-0.00001 0.00000 0.00000 -0.00001 -0.00004 -0.00001 -0.00001 -0.00001 0.00000 -0.00001 -0.00001 -0.00001 -0.00001 0.00000 0.00000 -0.00002 0.00000 0.00000 -0.00003 0.00000 0.00000 -0.00003 -0.00001 0.00000 -0.00001 -0.00001 0.00000 -0.00001 -0.00003 -0.00001 -0.00002 -0.00007
0.993 0.998 0.998 0.995 0.993 0.997 0.995 0.996 0.995 0.985 0.992 0.997 0.992 0.000 0.995 0.995 0.997 0.993 0.995 0.998 0.997 0.996 0.991 0.989 0.994 0.991 0.000 0.994 0.994 0.997 0.995 0.996
-8.120 -10.072 -9.053 -8.645 -5.965 -9.742 -8.942 -7.926 -7.934 -6.546 -7.775 -8.730 -7.845 0.000 0.436 -8.473 -9.502 -7.585 -8.354 -9.958 -5.773 -8.297 -8.045 -7.170 -8.628 -7.930 0.000 -8.553 -8.779 -9.732 -8.633 -8.639
0.013 0.010 0.010 0.013 0.017 0.012 0.012 0.016 0.003 0.012 0.014 0.016 0.013 0.000 -0.002 0.014 0.010 0.010 0.013 0.011 0.002 0.014 0.012 0.011 0.011 0.012 0.000 0.011 0.013 0.012 0.012 0.013
0.00122 -0.00023 -0.00048 0.00037
0.00000 0.00000 -0.00004 0.00008
0.995 0.992 0.996 0.999
-7.729 -8.002 -9.155 -15.117
-0.001 0.008 0.012 -
Appendix C-3 Modelling results for the room filled with potatoes at the ventilation rate is 240 m3 h-1 RT2 YIC Positions a0 a1 a2 a3 b0 b1 b2 α2 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 34 35 36
1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0.270 0.046 0.064 0.191 0.156 0.662 0.122 0.184 0.039 0.369 0.190 0.455 0.200 0.000 -0.020 0.431 0.057 1.454 3.129 0.092 0.017 0.693 0.116 0.228 0.794 0.326 0.027 1.992 0.459 0.273 1.912 1.642 0.070 0.465 0.724 1.930
-0.001 0.000 -0.001 -0.001 -0.001 -0.005 -0.001 -0.001 -0.001 -0.001 -0.001 -0.003 -0.001 0.000 0.000 -0.001 -0.001 -0.014 -0.010 -0.001 0.000 -0.001 -0.001 -0.002 0.000 -0.001 0.000 -0.003 -0.001 -0.002 0.002 -0.001 -0.001 -0.001 -0.002 -0.012
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.061 0.008 0.005 0.041 0.018 0.073 0.028 0.032 0.003 0.087 0.039 0.046 0.043 0.000 0.000 0.103 0.009 0.094 0.012 0.011 0.002 0.190 0.029 0.033 0.312 0.115 0.002 0.771 0.190 0.060 0.826 0.842 0.013 0.163 0.346 0.528
172
0.00018 -0.00008 -0.00004 -0.00003 -0.00008 -0.00016 -0.00011 -0.00004 -0.00004 0.00020 -0.00004 -0.00007 0.00002 0.00000 -0.00001 0.00011 -0.00007 -0.00015 -0.00098 -0.00008 -0.00003 0.00045 -0.00013 -0.00027 0.00082 0.00000 -0.00003 0.00112 -0.00012 -0.00029 0.00375 0.00130 -0.00015 0.00022 -0.00024 -0.00200
-0.00002 0.00000 0.00000 -0.00001 0.00000 -0.00002 -0.00001 -0.00001 0.00000 -0.00002 -0.00001 -0.00001 -0.00001 0.00000 0.00000 -0.00002 0.00000 -0.00002 -0.00021 0.00000 0.00000 -0.00005 -0.00001 0.00000 -0.00007 -0.00003 0.00000 -0.00016 -0.00004 -0.00001 -0.00020 -0.00017 0.00000 -0.00004 -0.00006 -0.00008
0.994 0.996 0.996 0.994 0.982 0.997 0.994 0.996 0.994 0.975 0.995 0.996 0.992 0.000 0.625 0.995 0.995 0.994 0.988 0.997 0.995 0.996 0.993 0.995 0.995 0.995 0.987 0.995 0.995 0.996 0.995 0.996 0.996 0.995 0.996 0.992
-7.734 -9.283 -8.677 -6.065 -6.933 -9.213 -8.486 -7.442 -6.967 -5.719 -6.701 -8.745 -4.596 0.000 5.034 -7.333 -8.809 -8.120 -4.870 -9.454 -6.096 -9.225 -8.180 -8.730 -6.619 -0.476 -1.755 -8.284 -6.684 -9.639 -8.687 -7.834 -9.067 -7.075 -7.306 -10.015
0.018 0.009 0.010 0.015 0.012 0.014 0.012 0.015 0.006 0.018 0.015 0.015 0.016 0.000 0.002 0.016 0.011 0.015 0.013 0.011 0.004 0.017 0.012 0.007 0.016 0.015 0.007 0.015 0.014 0.011 0.018 0.015 0.006 0.016 0.013 0.010
Appendix C-4 Modelling results for the room filled with potatoes at the ventilation rate is 280 m3 h-1 RT2 YIC Positions a0 a1 a2 a3 b0 b1 b2 α2 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 34 35
1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0.156 0.067 0.056 0.140 0.182 0.992 0.136 0.291 0.050 0.250 0.252 0.271 0.116 2.821 0.000 0.301 0.097 0.120 0.484 0.055 0.018 0.231 0.189 0.135 2.134 0.420 0.010 4.980 1.234 0.864 6.811 3.265 0.074 1.088 1.347
-0.001 -0.001 -0.001 -0.001 -0.001 -0.007 -0.001 -0.002 -0.001 -0.001 -0.001 -0.002 -0.001 -0.021 0.000 -0.001 -0.001 -0.001 -0.002 -0.001 0.000 -0.001 -0.001 -0.001 -0.004 -0.002 0.000 -0.012 -0.005 -0.007 -0.004 -0.008 -0.001 -0.002 -0.005
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 -0.001 0.000 0.000 -0.001 0.000 0.000 0.000 0.000
0.045 0.013 0.006 0.036 0.025 0.140 0.037 0.055 0.006 0.067 0.058 0.032 0.034 0.398 0.000 0.079 0.016 0.013 0.161 0.007 0.003 0.069 0.056 0.021 0.980 0.164 0.002 1.996 0.555 0.198 3.497 1.836 0.016 0.433 0.696
173
-0.00008 -0.00009 -0.00007 -0.00008 -0.00005 -0.00029 -0.00009 -0.00001 -0.00007 -0.00007 -0.00002 -0.00008 -0.00012 -0.00007 0.00000 0.00000 -0.00007 -0.00008 0.00007 -0.00007 -0.00006 0.00000 -0.00014 -0.00017 0.00123 -0.00002 -0.00003 0.00208 -0.00023 -0.00085 0.00833 -0.00033 -0.00017 0.00049 -0.00043
-0.00001 0.00000 0.00000 -0.00001 -0.00001 -0.00003 -0.00001 -0.00001 0.00000 -0.00002 -0.00001 -0.00001 -0.00001 -0.00010 0.00000 -0.00002 0.00000 0.00000 -0.00004 0.00000 0.00000 -0.00002 -0.00001 0.00000 -0.00020 -0.00004 0.00000 -0.00041 -0.00011 -0.00003 -0.00075 -0.00033 0.00000 -0.00009 -0.00013
0.995 0.996 0.997 0.995 0.990 0.996 0.994 0.996 0.996 0.987 0.995 0.995 0.993 0.995 0.000 0.996 0.996 0.994 0.995 0.997 0.997 0.994 0.993 0.995 0.996 0.996 0.988 0.995 0.996 0.997 0.996 0.996 0.996 0.996 0.997
-7.363 -9.128 -9.098 -7.772 -6.612 -9.856 -7.786 -5.138 -8.392 -5.404 -4.943 -8.563 -7.731 -6.974 0.000 -1.581 -9.005 -7.731 -5.726 -9.629 -8.096 -1.113 -7.758 -8.555 -8.867 -3.754 -3.377 -9.866 -7.109 -10.204 -9.867 -5.181 -8.845 -8.557 -7.704
0.014 0.012 0.008 0.014 0.015 0.014 0.014 0.016 0.007 0.015 0.016 0.014 0.013 0.016 0.000 0.016 0.014 0.012 0.015 0.010 0.003 0.016 0.014 0.009 0.015 0.015 0.004 0.015 0.014 0.011 0.016 0.013 0.007 0.015 0.013
Appendix C-5 Modelling results for the room filled with potatoes at the ventilation rate is 300 m3 h-1 RT2 YIC Positions a0 a1 a2 a3 b0 b1 b2 α2 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 34 35 36
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0.194 0.046 0.040 0.203 0.468 0.180 0.208 0.219 0.212 0.601 0.630 0.307 0.128 -0.591 -1.661 0.185 0.062 0.129 0.502 0.108 0.035 0.298 0.194 0.085 5.769 0.560 0.012 1.459 0.686 5.118 6.059 0.034 2.399 4.656 0.815
-0.001 0.000 0.000 -0.001 -0.003 -0.002 -0.001 -0.001 -0.002 -0.001 -0.002 -0.002 -0.001 0.026 0.014 -0.001 -0.001 -0.001 -0.001 -0.001 0.000 -0.001 -0.001 -0.001 -0.007 -0.002 0.005 0.313 -0.006 -0.006 -0.001 -0.006 0.000 -0.002 -0.006 -0.006
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 -0.014 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 -0.001 0.000 0.002 0.017 0.000 0.000 -0.001 -0.001 0.000 0.000 -0.001 0.000
0.048 0.009 0.004 0.042 0.050 0.028 0.045 0.043 0.011 0.130 0.121 0.029 0.031 -0.005 -0.04 0.046 0.011 0.011 0.147 0.012 0.002 0.079 0.056 0.011 2.554 0.216 -0.012 0.654 0.168 2.562 3.348 -0.119 0.894 2.396 0.248
174
0.00000 -0.00009 -0.00003 0.00006 0.00010 -0.00020 0.00007 0.00000 0.00001 0.00077 0.00046 0.00001 -0.00003 -0.01447 -0.00085 -0.00008 -0.00008 -0.00003 0.00037 -0.00012 -0.00002 0.00013 -0.00009 -0.00008 0.00543 0.00025 0.00242 -0.10143 -0.00055 -0.00084 0.00778 0.00625 0.00205 0.00332 0.00654 -0.00138
-0.00001 0.00000 0.00000 -0.00001 -0.00002 0.00000 -0.00001 -0.00001 0.00000 -0.00004 -0.00004 -0.00001 -0.00001 -0.00137 0.00003 -0.00001 0.00000 0.00000 -0.00004 0.00000 0.00000 -0.00002 -0.00001 0.00000 -0.00056 -0.00005 0.00001 0.01245 -0.00012 -0.00003 -0.00058 -0.00069 -0.00001 -0.00023 -0.00056 -0.00003
0.995 0.996 0.997 0.995 0.986 0.986 0.996 0.996 0.997 0.990 0.995 0.995 0.993 0.995 0.996 0.994 0.991 0.721 0.997 0.994 0.994 0.993 0.997 0.996 0.997 0.997 0.997 0.997 0.979 0.997 0.997 0.989
0.240 -8.931 -8.829 -7.032 -6.567 -7.435 -7.551 7.124 -6.548 -7.788 -9.730 -6.155 -4.952 -7.627 -8.985 -7.639 -7.676 6.073 -8.228 -7.448 -7.237 -8.168 -10.765 -8.346 -8.901 -10.635 -7.380 -9.467 -7.401 -10.669 -10.743 -8.694
0.016 0.010 0.012 0.018 0.019 0.009 0.018 0.017 0.020 0.022 0.020 0.018 0.016 2.545 0.000 0.000 0.011 0.016 0.018 0.012 0.015 0.017 0.015 0.011 0.016 0.016 0.000 0.000 0.013 0.010 0.017 0.015 0.018 0.017 0.009
