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Applied Thermal Engineering 28 (2008) 895–906 www.elsevier.com/locate/apthermeng

Optimization of compact heat exchangers by a genetic algorithm G.N. Xie a

a,1

, B. Sunden

b,2

, Q.W. Wang

a,*

State Key Laboratory of Multiphase Flow in Power Engineering, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China b Division of Heat Transfer, Lund University, P.O. Box 118, SE-221 00 Lund, Sweden Received 26 October 2006; accepted 4 July 2007 Available online 18 July 2007

Abstract In this study a plate–fin type Compact Heat Exchanger (CHE) is considered for optimization. The optimization method uses a Genetic Algorithm (GA) to search, combine and optimize structure sizes of the CHE. The minimum total volume or/and total annual cost of the CHE are taken as objective functions in the GA, respectively. The geometries of the fins are fixed while three shape parameters are varied for the optimization objectives with or without pressure drop constraints, respectively. Performance of the CHE is evaluated according to the conditions of the structure sizes that the GA generated, and the corresponding volume and cost are calculated. It is shown that with pressure drop constraints the optimized CHE provides about 30% lower volume or about 15% lower annual cost, while without pressure drop constraints the optimized CHE provides about 49% lower volume or about 16% lower annual cost than those presented in the literature.  2007 Elsevier Ltd. All rights reserved. Keywords: Compact heat exchanger CHE; Genetic algorithm; Optimization; Volume; Annual cost; Pressure drop

1. Introduction Compact Heat Exchangers (CHEs), including two types of heat exchangers such as plate–fin types and fin-and-tube (tube–fin) types, are widely used for gas–gas or gas–liquid applications. CHEs are employed in many industrial processes in chemical and petroleum engineering, refrigeration and cryogenics, Heating and Ventilation, Air-Conditioning (HVAC), aeronautics and astronautics, automotive, electric and electronic equipments, etc. CHEs own merits of compactness, small volume, low weight, high effectiveness and low cost. Plate–Fin Compact Heat Exchangers (PFCHEs) are widely used in gas–gas applications such as microturbine regenerators and recuperators. PFCHEs are extensively used in automobile, naval and aeronautical *

Corresponding author. Tel./fax: +86 29 82663502. E-mail addresses: [email protected] (G.N. Xie), [email protected] (B. Sunden), [email protected] (Q.W. Wang). 1 Tel./fax: +86 29 82663502. 2 Tel.: +46 46 2228605; fax: +46 46 2224717. 1359-4311/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2007.07.008

applications. PFCHE (as shown in Fig. 1) has high thermal effectiveness, because fins are employed on both sides to interrupt boundary layer growth due to the disturbance of the flow, and has high thermal conductivity due to the thin thickness of plate. In addition, PFCHE has high compactness (up to 1000–2500 m2/m3) and has low weight because of the use of aluminum. Many high-performance fin structures have been developed and applied to industrial engineering applications, e.g., wavy fin, offset strip fin, perforated fin, and louvered fin. Heat exchanger optimization is an important field and full of challenges. The task of optimization may be considered as a design process, in which any possible candidates will be evaluated based on requirements. Savings of materials or energy, as well as capital cost and operating cost, are common objectives for industrial applications of heat exchangers. On the other hand, heat exchanger design involves complex processes, including selection of geometrical parameters and operating (dynamic) parameters for the design, cost estimation and optimization. Generally the design task is a complex trial-and-error process, because

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G.N. Xie et al. / Applied Thermal Engineering 28 (2008) 895–906

Nomenclature A Ac C CV CA Cin Cop f Gc h j kel L1 L2 L3 n N Nu pf

heat transfer area (m2) minimum free flow area (m2) annual cost ($) moderate cost and volume price per unit area ($ m2) annual cost of investment ($) annual cost of operation ($) friction factor volumetric velocity (m3 s1) heat transfer coefficient (W m2 K1) Colburn factor price of electrical energy ($ MW h1) hot stream flow length (m) cold stream flow length (m) no-flow length (m) exponent of nonlinear increase with area increase number of flow arrangements Nusselt number penalty factor

Fig. 1. Typical multi-stream plate–fin heat exchanger.

all possible values of the geometrical parameters are combined in turn the designed heat exchanger is evaluated with respect to the conditions of specified requirements, i.e., outlet temperature, heat duty and pressure drop. In this sense, there is always the possibility that the designed results are not the optimum. Therefore researchers try to optimize thermal devices by means of optimization techniques, and many interesting and well-organized works have been presented. In recent years, much additional literature has appeared on optimization of heat exchangers, e.g., [1–13].

V Vt

volume (m3) volumetric flow rate (m3 s1)

Greek symbols a1, a2 weighting factors q fluid density (kg m3) g efficiency of the pump or fan s hours of operation per year e thermal effectiveness DP pressure drop (kPa) DPmax allowable pressure drop (kPa) Subscripts 1,2 inlet, outlet c cold side f fin h hot side m mean max maximum min minimum

For optimization of plate–fin heat exchangers, Wang et al. [4,5] developed a new methodology to design multistream plate–fin heat exchangers in heat exchanger networks, and they validated the suggested method by an industrial case study. Jia et al. [6,7] used traditional optimization methods to search and optimize a plate–fin heat exchanger, and developed a new and efficient software, in which optimization, database and process drawing are included. Reneaume et al. [10–12] proposed mathematical formulations for optimization of plate–fin heat exchangers and developed a tool for the computer-aided design, they also discussed the solution strategies under numerous design and operating constraints, and Successive Quadratic Programming (SQP) or Simulated Annealing (SA) or Branch and Bound (BB) algorithms were used to solve the corresponding optimization problems. In recent years, applications of Genetic Algorithms (GAs) in thermal engineering have received much attention for solving real-world problems [14]. Applications of GAs into heat exchangers optimizations have suggested that GAs have a strong ability of search and combined optimization and can successfully optimize and predict thermal problems. Thus applications of GAs in the field of thermal engineering are new challenges. At this point, the GA technique may be used in the geometrical optimization of heat exchangers in order to obtain optimal results under specified design objectives within the allowable pressure drops. For example, the fin-and-tube heat exchanger performance was predicted using a GA [15]. Shell-and-tube heat exchangers are optimized under four strategies by using SA and GA, and the corresponding performances are

G.N. Xie et al. / Applied Thermal Engineering 28 (2008) 895–906

compared [16]. A new design method was proposed to optimize a shell-and-tube heat exchanger from an economic point of view by a GA [17]. Optimizations of the geometries of cross-wavy and cross-corrugated primary surface recuperators were studied via GAs [18,19]. Heat transfer correlations for compact heat exchangers were obtained using GAs, and in turn these correlations were used to estimate their performances [20–22]. In addition, plate–fin heat exchangers were optimized by means of GAs [23–25]. Mishra et al. [23] used a GA to optimize a multilayer plate–fin heat exchanger, in which the specified heat duty and flow restrictions were alternately considered, and the effects of additional constraints on optimum design were investigated. Ozkol et al. [24] determined the optimum geometry of a heat exchanger body by a GA, in which the ratio of the number of heat transfer units (NTU) to the airside pressure drop was taken as the objective function, and two examples demonstrated GA method. Xie et al. [25] optimized a plate–fin heat exchanger through a GA, the fin geometries were scaled to be searched and optimized for minimum total weight or maximum effectiveness. The aforementioned research works draw much attention to geometrical optimization of plate–fin heat exchangers by GAs. However, the pressure drop constraints are clearly considered in [23–25], but the objective of minimum total annual cost is not included in [24,25]. Generally, the optimal design of a heat exchanger is conducted under the constraint of an allowable pressure drop, while the contrary case, without pressure drop constraints, is not considered often. In some applications the pressure drop is not the most critical aspect. It might be that the gain/benefit of the optimized performance is worthwhile even if a pressure drop penalty is needed. Therefore, the motivation of this study is optimal design of compact heat exchangers with or without constraints on the allowable pressure drop, based on GA auto-search, combination and optimization techniques. On the other hand, for heat exchanger optimization, the trade-off between heat transfer and pressure drop may be considered (as shown in Fig. 2). In general, a higher flow velocity means a higher heat transfer coefficient and hence a smaller heat transfer area and correspondingly a lower

Total cost

Power cost

Cost Area cost

Velocity Fig. 2. Economic optimization of heat exchangers.

897

capital cost. However, higher velocity will generally lead to higher pressure drop and hence a higher power consumption and correspondingly a higher power cost [1–7]. The heat exchanger area and the pressure drop are mainly associated with capital cost (investment cost) and power cost (operating cost), respectively. Thus, the important objective, minimum cost, should be considered ahead of the optimum design. In addition, in some practical applications, where the CHE size/volume is critical for compactness and the total cost should be low, moderate volume and cost are needed to consider for the specified requirements, and usually the importances between them are not equal for considerations. In this paper, a genetic algorithm based optimization technique has been demonstrated for a plate–fin compact heat exchanger. The objectives of the minimum total volume or/and total annual cost of the heat exchangers are considered for optimization. The fin geometries are taken from an available database, and three shape parameters are varied for the optimization objectives with or without pressure drop constraints, respectively. The used method, GA, is not new, however, the application of GA into plate–fin heat exchanger optimizations for different objectives with/without different constraints is attractive, and the results are optimistically interesting, which are useful for further research or references. 2. Physical model and design parameters According to the established knowledge, high effectiveness, small volume (hence low weight) and low cost are the common objectives in heat exchanger design. In engineering practice, there are two design requirements. One is to decrease volume and annual cost of the heat exchanger as much as possible under a specified effectiveness and an allowable pressure drop. The other is to increase the effectiveness as much as possible. Thus, before the optimal design is carried out, the optimization objective should be considered based on different requirements. Without loss of generality, in the present work the minimum total volume or/and total annual cost are considered. Besides, the minimum pressure drops of the two sides now can be taken as an objective in this study, in turn to discuss the difference between the optimal results. The appearance of GAs and many successful applications of them in scientific fields, have shown that GAs own the ability of strong search and combined optimization. Thus in the present study GA is used for the optimization process instead of the traditional methods, and for optimization of structure sizes under specified design objectives. 2.1. Plate–fin compact heat exchanger, PFCHE In the present study a microturbine recuperator is considered [26]. A typical core of a PFCHE is shown in Fig. 3. For such a PFCHE, two streams are in cross-flow

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G.N. Xie et al. / Applied Thermal Engineering 28 (2008) 895–906 0.1

Gas-side surface characteristics Kays&London, Plain-fin 19.86

Gas

f

0.01

j

Air

Curve-fitting

(a) The core of a recuperator

Re/103 1

10

(a) Gas-side surface Air-side surface characteristics Kays&London, Plain-fin 1/8-15.61

(b) Plain triangular fin

0.1

(c) Offset-strip fin Fig. 3. Core of two-stream plate–fin heat exchanger.

with different mass flow rates under specified performances, e.g., given duty, allowable drops. There are many geometrical parameters which may be taken as optimization variables such as shape length L1, L2, L3, fin pitch (fin density) Fp, fin height (plate height) Fs, fin thickness df and number of flow arrangements N. In this work, the thickness of the plate and fins are assumed to be constant, and thus are not to be optimized as they satisfy initial requirements, and almost do not affect the performance of the heat exchanger. The materials of plate and fin are aluminum with a thermal conductivity of 190W/(m K), density of 2790 kg/m3. Plain triangular fins (Fig. 3b) and offset strip fins (Fig. 3c) are employed on each side, and the fin parameters are taken from Kays and London [27]. The corresponding surface characteristics are shown in Fig. 4.

f

j

Curve-fitting

0.01

Re/103 1

10

(b) Air-side surface Fig. 4. Surface basic characteristics for two side.

bined in a genetic algorithm routine. A brief description of the genetic algorithm and the selection of variables and parameters can be found in the next section. 3. Genetic algorithm 3.1. Simple descriptions

2.2. Design parameters Since the fin structures (surface characteristics) are completely taken from a database, the fin geometries of the two sides are fixed. That is, in this study the fin density and fin height of both sides are not modified, so that other parameters and correlations can be picked up safely from references. Scaling fin geometry may be dangerous to design and optimization, and additional discussion is addressed in Section 6. Therefore, three shape parameters L1, L2, L3 are selected for optimization, as these parameters are sufficient for the design of a PFCHE when the fin geometry is fixed. The three design parameters are searched and com-

The genetic algorithm is maintained by a population of parent individuals that represent the latent solutions of a real-world problem. For example, the designer might encode the design parameters into corresponding binary strings, and then all the binary strings are connected into a binary string, which is represented as an individual. A certain number of sets of design parameters accordingly become a population of parent individuals. Fig. 5 shows the flow chart of the Genetic Algorithm (GA). Each individual is assigned a fitness based on how well each individual fits a given environment and then is evaluated by survival of the fitness. Fit individuals go through the

G.N. Xie et al. / Applied Thermal Engineering 28 (2008) 895–906 gen=0 Initialize population

Yes

Satisfy conditions (gen>1000) ?

No

Fitness (rating performance of heat exchanger)

Evaluation

output results of evolution

Selection

END

899

choices for the hot stream flow length L1, 200 choices for the hot stream flow length L2, 250 choices for the stack height. Thus, here the optimization problem is considered as a large scale, discrete and combined optimization problem. Based on these choices of optimization parameters, the total number of possible combinations is 200 * 200 * 250 = 107. This number means that it will require enormous time and work until the global optimization solution is reached. So the novel strategy or technique of auto-search and combination will be necessary to present, and the GA might be just the one, which may do the above job. The detailed principles of the coding and decoding processes can be found in the above-mentioned classical books.

Crossover

3.3. Genetic operators and parameters Mutation gen=gen+1

Fig. 5. Flow chart of Genetic Algorithm (GA).

process of survival selection, crossover and mutation, resulting in creating next generation, called child individuals. A new population is therefore formed by selection of good individuals from parent and child individuals. After some generations, the algorithm has converged to a best individual, which probably represents the best solution of the given problem. More details about description of genetic algorithms can be found in many books [28–30]. 3.2. Ranges of the variables Binary string is adopted for encoding the variables of a given model. The search ranges and binary string lengths of the three design parameters are listed in Table 1. The selection of upper bound and lower bound for shape parameters seems somewhat arbitrary, however their bound values are around the corresponding points from references, and one/ user can freely change the bound values based on practical space constraints, e.g., height or footprint. Considering the ability of computer handling bit operation and engineering application, the computational precision is set to three decimals (0.001). An individual of a model needs 24-character binary. From Table 1 it can be seen that, there are 200 Table 1 Search ranges and string lengths of design parameters Variable

Unit

Search range

Precision

Choices

String length

Hot-stream flow length, L1 Hot-stream flow length, L2 No-flow length, L3

m

0.2–0.4

0.001

200

8

m

0.5-0.7

0.001

200

8

m

0.75–1.0

0.001

250

8

In the present study, tournament selection, uniform crossover and one-point mutation were selected. Niching (sharing) and elitism were adopted [31–33]. Tournament selection: random pairs of individuals are selected from the population under a given probability, and the better (with larger fitness) of each pair is allowed to mate, thus one child is created, which becomes a mix of the two parent individuals. Uniform crossover: it is possible to obtain any combination of two parents individuals, e.g., there exist two individuals, 011100 and 101011. Thus the children could be 111010 and 001001. One-point mutation: there is a small probability that one or more of the children will be mutated, e.g., 111010 could be 101010. Niching (Sharing): the fitness of each individual is adjusted according to similar degree, which is evaluated by a specified sharing function. The selection is conducted based on the new fitness. This technique ensures the variety of individuals (solutions) so that the global optimization solution will be obtained rather than the local optimization solution. Elitism: The best parent is reproduced (copied) into the new population. After the new population is generated, GA checks if the best parent has been replicated. The size of population and maximum evolution generation are set to 50 and 1000, respectively. Probability of crossover and mutation are set to 0.5 and 0.005, respectively. The selection of genetic parameters is a trial-anderror process, and with the variation of these parameters, results are not exactly identical. However, they are very close to one another. On the other hand, in this study the selection of genetic parameters and operators is based on the Carroll’s recommendations [31,32]. In addition, De Jone [34] tested a great deal of numerical cases and addressed that when the probability of mutation is set to the same order of 0.001, the local optimization might be avoided. The task of optimization process is conducted in FORTRAN language, the main program is for running GA,

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and the subroutine program is for carrying out the rating performance process under the combinational parameters. The idea for optimization of heat exchangers and definition of design objectives can be found in the next section. 4. Optimization In the present study, four objective functions are considered in the design process. The first is the minimum total cost and the second is the minimum total volume, the third is moderate cost and volume and the last is the minimum total pressure drop. The traditional rating method, the eNTU method, is used in the rating performance of the PFCHE. The detailed expressions and procedures are as follows. 4.1. Rating performance Once the geometrical (structure) sizes have been obtained by the GA-based individuals decoded, rating performance can be carried out under the specified requirements. The e-NTU method is used. Details of the design process can be found in Shah [35]. For the rating performance of a heat exchanger, the data of heat transfer, j-factors, and friction factors, f-factors, are required for various Reynolds numbers. This is because in the optimization process the shape parameters vary as GA searches and combines, in return leading to changing Reynolds numbers (flow rate is fixed while cross-section of passage is changed). Therefore it is difficult and nonconvenient to look for j and f from the figure of the reference since there are many possible combinations in the search process of GA. Fortunately, ahead of GA search relations among j and f vs. Re are in-house obtained by curve-fitting experimental data from Kays and London [27]. In this study, a five-order polynomial relation is used to correlate the data (see Fig. 4, curve line). Thus once the relations have been built, the computer-aided search and acquisition can be conducted. The deviation between the original results from [26] and the correlated results by the computer-aided program are listed in Table 2. It should be noted that the pressure drop is determined by Table 2 Comparisons by validating the rating performance Variables jh fh jc fc hh hc e Q DPha DPca a

Unit

Ref. [26]

This work

Error (%)

W/(m2 K) W/(m2 K)

0.013 0.055 0.017 0.065 70.44 154.82 0.807 160.1 0.312 2.105

0.01251 0.05298 0.0169 0.06518 68.55 153.9 0.8 158.6 0.309 2.108

3.77 3.67 0.59 0.28 2.68 0.59 0.87 0.94 0.96 0.14

kW kPa kPa

Calculated by Eq. (1).

DP ¼

G2c A qm f 2q1 Ac q1

ð1Þ

In this study for simplicity the pressure losses of entrance effect/expansion, flow acceleration and exit effect/constriction are neglected, since the fraction of them to the total pressure drop may be a small value, say 2% or smaller. From Table 2, it can be seen that the errors are small, and acceptable from an engineering optimization point of view. Therefore the computer-aided rating performance is valid. The designs of distributors and headers are not considered in this study. 4.2. Objective functions • OB1: When the minimum total annual cost is considered in optimization process, the objective function in GA is now set [6], as follows:Total annual cost = annualized cost of heat transfer area + operating cost of pump/ compressor TACðxi Þ ¼ C in þ C op C in ¼ C A  An     DPV t DPV t þ k el s C op ¼ k el s g g h c

ð2Þ ð2aÞ ð2bÞ

Here CA and kel are the price per unit area and electrical energy, respectively, n and s are the exponent of nonlinear increase with area increase and the hours of operation per year, respectively. DP, Vt and g are pressure drop, volumetric flow rate and pump/compressor efficiency, respectively. Since in practical applications the heat exchangers are operated under specified requirements, and consumption of pumping power is necessary to transfer the fluid flow through the passages in the heat exchangers, pressure loss/drop will be inevitable. In addition, in industrial applications, some constraints of the pumping power are critical and must be considered. The pressure drop must be below a specified maximum value. Thus, the heat exchanger optimization is a constrained optimization process with the following inequality conditions:  DP h < DP h;max ; DP c < DP c;max Constraints: ð3Þ xmin < xi < xmax Here, the DPh,max, DPc,max are the maximum allowable pressure drop on the hot and cold sides, respectively. In order to compare with previous works, two such values are set, namely, 0.3 kPa and 2 kPa. x refers to the optimization design variable to be optimized. xmax and xmin refer to the upper and lower bounds of the design variables, respectively. On the other hand, for cross-flow heat exchangers, it is economic that the effectiveness e should not be less than a certain value emin, generally say 0.75. For constrained optimization, when the geometrical sizes, which generated by GA, can not satisfy the specified performance (through rating routine), a penalty function

G.N. Xie et al. / Applied Thermal Engineering 28 (2008) 895–906

901

has to be imposed/conducted on the objective function. In this paper, the step-wise penalty factor is defined as follows:  1 DP < DP max and e P emin pf ¼ ð4Þ 0 otherwise

ical units, and a direct summation will introduce numerical errors, in which the low value (V, order of 0.1) will be ‘‘eated’’ by the high value (C, order of 103). • OB4: When the minimum pressure drop is considered in optimization process, the volume and cost are not considered in this way.

Thus the fitness of the individuals should be adjusted into following equation:

For OB4 : fitness ¼ pf  ðDP h;max =DP h þ DP c;max =DP c Þ

ð5Þ

4

Cmax is a constant assigned to 10 . The case of invalid design, that is, minimum fitness (is zero), will be removed from the possible combinations, not being updated in the GA evolution. On the contrary, in some applications the pressure drop of the heat exchanger is not the most critical part of the total pressure drop of the thermal plant. It might well be that the gain by the optimized performance is evident over the degree of penalty by the scaled pressure drop. Therefore, it is interesting that the optimized results are different if the pressure drop constraint is removed. Here the GA optimization with pressure drop constraints is called GA1 while GA optimization without pressure drop constraints is called GA2. • OB2: When the minimum volume is considered in the optimization process, the objective function in GA is now set, as follows: Volume = hot stream flow length * cold stream flow length * no-flow length V ¼ L1  L2  L3 For OB2 : fitness ¼ pf  V max =V

ð6Þ ð7Þ

The constraints are similarly handled, see Eq. (4). • OB3: When the moderate cost and volume are considered in the optimization process, a multi-objective problem may occur. With respect to multi-objective optimizations, there might exist many advanced methods to deal with, and it is possible that there exists one novel algorithm or methodology matching one kind of real-world problem. However, considering the sum of weighted objectives as the total objective function has gained a remarkable popularity in optimization problem, since it is relatively simple to be implemented in engineering applications, and one can adjust weighting factors to represent the degree of importance between all objectives. Thus in this case the objective function in GA is now set, as follows: C max V max CV ¼ a1 þ a2 C V For OB3 : fitness ¼ pf  CV

ð8Þ

ð10Þ The constraints are similarly handled, see Eq. (4). A case study and the corresponding results can be found in next section. 5. A case study and optimized results To demonstrate the described procedure, a case study is considered as follows. Optimization a PFCHE heat exchanger with the following duty (as listed in Table 1): 0.6 m3/s air at 4 C is cooling 1.2 m3/s gas from 240 C to 51 C. The inlet pressure is 110 kPa. The plate thickness is 0.4 mm, and both fins and plates are made from aluminum. Plain triangular fins and offset strip fins are employed on each side, and the fin parameters are taken from Kays and London [27]. For the economic calculations, the cost per unit area, CA, is assigned 100$/m2, the exponent for the nonlinear increase with area, n, is assigned to 0.6, the hours of operation is assumed to 6500 h/year, the price of electricity is assumed to 30$/MWh, the pump efficiency is assumed to 0.5. Note that the cost parameters, e.g., CA, n , kel, depend on current practice experiences or recommended values [1,6]. The real results will depend on these parameters but the proposed procedure is general and is applicable for any selected values. The selected values thus represent a case study. The evolution process for minimum cost is shown in Fig. 6. At the beginning of the evolution process (For GA1 less than 300 generations while For GA2 less than 50 generations), the individuals with higher fitness are saved, and the individuals with small fitness are removed. 3400

3300

3200

Cost, $

For OB1 : fitness ¼ pf  ðC max  CÞ

3100

GA1

3000

GA2

ð9Þ 2900

The constraints are similarly handled, see Eq. (4). Note that the sum of the weighting factors is equal to unity, that is a1 + a2 = 1. Cmax and Vmax are the scales for cost and volume, respectively, by which C and V are made compatible. This is because C and V have different phys-

2800 0

200

400

600

800

Generation Fig. 6. Evolution process for minimum cost, $.

1000

902

G.N. Xie et al. / Applied Thermal Engineering 28 (2008) 895–906

Table 3 Optimized results for minimum cost Geometrical parameters

Performance 3

L1, m L2, m L3, m V, m Reference 0.300 GA1 0.235 GA2 0.200

Cost, $ DPh, kPa DPc, kPa

0.600 0.898 0.16164 3572.84 0.3093 0.500 1.000 0.11750 3047.67 0.2979 0.500 0.994 0.09942 2973.29 0.2387

2.1084 1.9995 2.5122

0.20 0.18

Volume, m

3

0.16 0.14

GA1 0.12 0.10 0.08

GA2

0.06 0

200

400

600

800

1000

Generation Fig. 7. Evolution process for minimum volume, m3.

After certain generation (larger than 700,100 generations, respectively), the differences between every individual are relatively large, in turn the variation of fitness for minimum cost is small, finally a level off value is found. Compared with results from the references, the optimized results for minimum cost are listed in Table 3. From the table, it is seen that under pressure drop constraints the total cost decreases by about 15%, and the total volume also decreases by about 27%, while without pressure drop constraints the total cost decreases by about 16.5%, and the total volume also decreases by about 38%. For GA1 optimization, the allowable pressure drops of the two sides are almost fully utilized. For GA2 optimization, due to the without pressure drop constraints, the airside pressure drop is beyond/over the allowable pressure drop, increasing by about 25.5%, while the gas-side pressure drop decreases by 20%. Therefore, if the total pressure drop is allowed to slightly exceed the permitted limit in some practical applications, the benefits from the optimized results might be considerable.

Similar trend of the evolution process for minimum total volume is shown in Fig. 7. The optimized results for minimum total volume are listed in Table 4. It is seen that with pressure drop constraints the total volume decreases by about 30%, and the total cost also decreases by about 13.8%, while without pressure drop constraints the total volume decreases by about 49%, and the total cost also decreases by about 13.7%. For GA1 optimization, the allowable pressure drop of air-side is almost fully utilized. For GA2 optimization, the pressure drops of both sides are over the allowable pressure drops, increasing by about 22% and 63%. Thus, even if total pressure drop is allowed to slightly exceed the limit in some practical applications, the benefits from the optimized results might not be so considerable. The evolution process for objective functions, cost, volume and pressure drops are shown in Fig. 8, the two weighting factors for cost and volume are equal to 0.5, which is CV = 0.5 * Cmax/C + 0.5 * Vmax/V. The optimized results are listed in Table 5. As expected, the results of CV are between C and V. With pressure drop constraints the total cost decreases by about 14.5%, and the total volume also decreases by about 28%, while without pressure drop constraints the total cost decreases by about 15.2%, and the total volume also decreases by about 46.2%. For GA1 optimization, the allowable pressure drop of the airside is almost fully utilized. For GA2 optimization, the pressure drops of both sides are over the allowable pressure drops, increasing by about 19.5% and 53.2%, respectively. The results of this case can be helpful to designer in order to complete the optimization of heat exchangers, in which the designer can change the weighting factors according to the degree of importance between cost and volume. Obviously the results may be included between the results of two extreme cases: minimum cost and minimum volume. Based on practical requirements and constraints, one can change the upper and lower bounds of the geometrical parameters, and change the weight factors, to achieve the corresponding optimizations. Compared with results from the references, the optimized results for minimum pressure drop are listed in Table 6. In this case, the optimized geometrical parameters will be arrived within the bounding values. Since minimum pressure drops are required, the sizes/volume may be increased as much possible. It can clearly be interpreted that the higher flow length, hence the higher heat transfer area and minimum free flow area, in turn leading to the smaller velocity. The smaller velocity, hence the smaller pressure

Table 4 Optimized results for minimum volume Geometrical parameters

Reference GA1 GA2

Performance 3

L1, m

L2, m

L3, m

V, m

Cost, $

DPh, kPa

DPc, kPa

0.300 0.239 0.224

0.600 0.500 0.511

0.898 0.984 0.750

0.16164 0.11316 0.08584

3572.84 3078.68 3084.75

0.3093 0.2888 0.3673

2.1084 1.9918 3.2610

G.N. Xie et al. / Applied Thermal Engineering 28 (2008) 895–906

drops. For GA2 optimization, without pressure drop constraint, the cold flow length arrives at upper bound value while the hot flow length arrives at the lower bounding value. This is understandable because a smaller hot flow length leads to higher cold flow pressure drop. In turn,

903

the gas-side pressure drop is about 2.3 times smaller while the air-side pressure drop is about 1.75 times higher than those in reference [26]. Thus the case of GA2 may be referred to applications where attention is drawn on decrease of gas-side pressure drop.

0.20 0.18

Volume, m3

0.16 0.14

GA1 0.12 0.10 0.08

GA2

0.06 0

200

400

600

800

1000

Generation (a) Evolution process 3200

0.16

Volume,m

Cost,$

3

0.14

3100

GA1

GA1

0.12

0.10

GA2 0.08

0.06

GA2 3000

0.04

0

200

400

600

800

1000

0

200

Generation

400

600

800

1000

Generation

(b) Cost and volume 4

0.50

0.45

ΔPc , kPa

ΔPh , kPa

GA2 0.40

GA2 0.35

3

GA1

2

0.30

GA1 0.25 1

0.20 0

200

400

600

800

1000

0

200

Generation

400

600

Generation

(c) Two sides pressure drop Fig. 8. Evolution process for moderate cost and volume.

800

1000

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G.N. Xie et al. / Applied Thermal Engineering 28 (2008) 895–906

Table 5 Optimized results for moderate cost and volume CV = 0.5 * Cmax/C + 0.5 * Vmax/V Geometrical parameters

Reference GA1 GA2

Performance 3

L1, m

L2, m

L3, m

V, m

Cost, $

DPh, kPa

DPc, kPa

0.300 0.238 0.223

0.600 0.500 0.501

0.898 0.979 0.778

0.16164 0.11650 0.08692

3572.84 3051.25 3029.69

0.3093 0.2885 0.3586

2.1084 1.9951 3.0650

Table 6 Optimized results for minimum pressure drop Geometrical parameters

Reference GA1 GA2

Performance

L1, m

L2, m

L3, m

V, m3

Cost, $

DPh, kPa

DPc, kPa

0.300 0.400 0.200

0.600 0.700 0.700

0.898 1.000 1.000

0.16164 0.2800 0.1400

3572.84 4451.46 3715.26

0.3093 0.2545 0.1390

2.1084 1.4312 3.5176

6. Discussions The optimized results in the present work are based on theoretical optimization design. With respect to the volume or/and annual cost as objectives in the optimization process, the results are optimum compared to those in Ref. [26]. The results show that it is effective to use a genetic algorithm technique to search and combine optimal parameters for heat exchangers under different requirements, e.g., space, cost and pressure drop. Making use of a genetic algorithm provides strong ability of search and combined optimization, which effectively might be better than the traditional method, since there is always the possibility that the results from the latter process are not optimal. As seen from the optimization procedure, the optimized results may depend on the selection and application of empirical correlations. Errors (deviations) between optimized results and applied results (later through manufacturing and tests based on the design results) will surely be generated. The correlations used in the present work, are based on computer-aided curve-fitting, in turn leading to fitting deviations between experimental data and correlated data. Another discussion about escalation of the fin geometry should be addressed here. In general it is reasonable that slightly scaling the fin geometry is suggested ahead of the heat exchanger design, and empirical correlations are usually applied to predict the corresponding performance. However, Shah [36] pointed out that if the accuracy of the prediction is not less than the degree of escalation, there exists additional material that would be wasted. Therefore it might be dangerous if no limit is set on the fin geometry in the design and prediction. Besides, no existing rules can refer to the degree of escalation. Sometimes one may overly scaling the fin geometry, which can not satisfy the geometry analogy so that the predicted performance, Nu or j and f, is beyond the practical requirements, not only the problem of wasting material. In this sense, it is best suggested that the

fin geometry is fixed so that the corresponding surface characteristics are forwarded/borrowed to apply for prediction. If the fin geometry has to be scaled for practical applications, much attention should be drawn on the errors, which are generated from the predicted performance, e.g., h and DP, since the original correlations are not necessarily valid for the new geometry analogy. It is possible that the experimentally tested results of the scaled geometry are far from the predicted performance, which is obtained from the empirical correlations of the existing geometry. In this work, the plate–fin compact heat exchanger is demonstrated, however, other types of heat exchangers such as fin-and-tube heat exchangers, primary surface recuperators, shell-and-tube heat exchangers may be optimized based on the GA optimization method. For example, when optimizing a shell-and-tube heat exchanger, the geometrical parameters, including tube diameter, tube pitch, baffle pitch, number of flow arrangement and baffles, thickness of tube and baffles and so on, can be searched and optimized via the genetic algorithm integrated with a traditional design method. Besides, although two kind of fins are employed on both sides, other types from databases [27], e.g., plain rectangular fins, plain wavy fins, strip fins, perforated fins, louvered fins, may be considered in the optimization task. Thus, the genetic algorithm may be applied in heat exchanger problems to search and optimize the complex thermal devices or networks. 7. Conclusions This paper demonstrates successful application of a genetic algorithm for optimization of a plate–fin compact heat exchanger. A generalized procedure has been developed to carry out the optimization to find the minimum volume and/or annual cost and pressure drop of the heat exchangers, respectively, based on the e-NTU and the genetic algorithm technique. A case study has been

G.N. Xie et al. / Applied Thermal Engineering 28 (2008) 895–906

presented to show the optimized results by the proposed method. It is concluded that the genetic algorithm can provide a strong ability of auto-search and combined optimization in the optimization design of heat exchangers compared to the traditional designs in which a trial-anderror process may be involved. By application of the genetic algorithm in the optimal design the heat exchanger configurations/structures can be optimized according to different design objectives such as minimum surface area and cost. The method can be transferred for use in optimization design of different types of heat exchangers, but also for the presented plate–fin heat exchangers but with different fins, e.g., perforated fins, slotted fins and louvered fins. Acknowledgements This work was supported by the National Natural Science Foundation of China and the Program for New Century Excellent Talents in University of China (Grant No. NCET-04-0938). References [1] H. Martin, Economic optimization of compact exchangers, in: R.K. Shah (Ed.), First International Conference on Compact Heat Exchangers and Enhancement Technology for the Process Industries, Banff, Canada, 1999, pp. 75–80. [2] K. Muralifrishna, U.V. Shenoy, Heat exchanger design targets for minimum area and cost, Transactions of the Institution of Chemical Engineers 78 (2000) 161–167. [3] M.R. Jafari Nasr, G.T. Polley, An algorithm for cost comparison of optimized shell-and-tube heat exchangers with tube inserts and plain tubes, Chemical Engineering Technology 23 (2000) 267–272. [4] L. Wang, Performance analysis and optimal design of heat exchangers and heat exchanger networks, PhD thesis, Division of Heat Transfer, Department of Heat and Power Engineering, Lund Institute of Technology, 2001. [5] L. Wang, B. Sunden, Design methodology for multistream plate–fin heat exchangers in heat exchanger networks, Heat Transfer Engineering 22 (2001) 3–11. [6] R. Jia, B. Sunden, Y. Xuan, Design and optimization of compact heat exchangers. in: R.K. Shah (Ed.), Third International Conference on Compact Heat Exchangers and Enhancement Technology for the Process Industries, Davos, Switzerland, 2001, pp. 135–142. [7] R. Jia, B. Sunden, Optimal design of compact heat exchangers by an artificial neural network method, in: Proceedings of HT2003, ASME Summer Heat Transfer Conference, Paper No. HT200347141, 2003. [8] A. Unuvar, S. Kargici, An approach for the optimum design of heat exchangers, International Journal of Energy Research 28 (2004) 1379–1392. [9] A. Traverso, A.F. Massardo, Optimal design of compact recuperators for micro-turbine application, Applied Thermal Engineering 25 (2005) 2054–2071. [10] J.M. Reneaume, N. Niclout, MINLP optimization of plate fin heat exchangers, Chemical and Biochemical Engineering Quarterly 17 (2003) 65–76. [11] J.M. Reneaume, N. Niclout, Optimal design of plate–fin heat exchangers using both heuristic based procedures and mathematical programming techniques, in: R.K. Shah (Ed.), Third International Conference on Compact Heat Exchangers and Enhancement Technology for the Process Industries, Davos, Switzerland, 2001, pp. 135– 142.

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