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Geometric optimization of thermoelectric coolers in a confined volume using genetic algorithms
Applied Thermal Engineering 25 (2005) 2983–2997 www.elsevier.com/locate/apthermeng
Geometric optimization of thermoelectric coolers in a confined volume using genetic algorithms Yi-Hsiang Cheng *, Wei-Keng Lin Department of Engineering and System Science, National Tsing Hua University, 101, Sec. 2, Kuang Fu Road, Hsinchu 300, Taiwan, ROC Received 15 November 2004; accepted 12 March 2005 Available online 25 May 2005
Abstract The demand for thermoelectric coolers (TEC) has grown significantly because of the need for a steady, low-temperature operating environment for various electronic devices such as laser diodes, semiconductor equipment, infrared detectors and others. The cooling capacity and its coefficient of performance (COP) are both extremely important in considering applications. Optimizing the dimensions of the TEC legs provides the advantage of increasing the cooling capacity, while simultaneously considering its minimum COP. This study proposed a method of optimizing the dimensions of the TEC legs using genetic algorithms (GAs), to maximize the cooling capacity. A confined volume in which the TEC can be placed and the technological limitation in manufacturing a TEC leg were considered, and three parameters––leg length, leg area and the number of legs––were taken as the variables to be optimized. The constraints of minimum COP and maximum cost of the material were set, and a genetic search was performed to determine the optimal dimensions of the TEC legs. This work reveals that optimizing the dimensions of the TEC can increase its cooling capacity. The results also show that GAs can determine the optimal dimensions according to various input currents and various cold-side operating temperatures. 2005 Elsevier Ltd. All rights reserved. Keywords: Thermoelectric cooler; Genetic algorithms; Confined volume; Optimization
*
Corresponding author. Tel.: +886 3 5713583; fax: +886 3 5720724. E-mail address: [email protected] (Y.-H. Cheng).
1359-4311/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2005.03.007
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Nomenclature A H I L Lmin N Nmax P QH QL S TH TL Tmid Greek a q qr rc k
cross-sectional area of the TEC leg, m2 height of the confined volume, m input current, A length of the TEC leg, m minimum length of the TEC legs, m number of the TEC legs maximum number of the TEC legs power input to the TEC, W total dissipated heat from hot junction, W cooling capacity at cold junction, W cross-sectional area of the confined volume, m2 temperature of hot junction, K temperature of cold junction, K average of cold and hot side temperatures, K symbols Seebeck coefficient, V/K density of thermoelectric material, kg/m3 electrical resistivity, X m electrical contact resistance, X m2 thermal conductivity, W/m K
1. Introduction In recent years, the electronic industry has developed rapidly and the miniaturization of electronic components has been ongoing. As components have shrunk, the chip-level power density has continued to rise greatly. Therefore, thermal management is becoming a critical issue in system performance. Moreover, the electrical stability of many pieces of electronic equipment, such as laser diodes, semiconductor optical devices, infrared detectors, and others, should be improved, to ensure electrical stability and rapid electrical–optical transmission. Traditional thermal management devices, such as a fan and heat sink, cannot be used in such electronic equipment, in which the thermoelectric cooler (TEC) is therefore employed to control the thermal and electrical stabilities. When a TEC is used in a piece of electronic equipment, the TEC must be designed and constructed to have a specified cooling capacity and to operate with a proper coefficient of performance (COP). Therefore, a priori thermal analysis of the performance of the TEC is important in designing a TEC structure. Yamanashi et al. [1,2] and Chen [3] studied the optimum design of a multistage TEC system with a maximum COP. By applying finite-time thermodynamic method, Wu and coworkers [4] considered the effects of thermal resistances between the TEC and the two external heat exchangers, as well as of the number of the TEC legs, on the perfor-
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mance of a TEC system. Their results indicated that the thermal performance of the hot side heat exchanger strongly affected the COP and the cooling capacity. Under poor design conditions, the hot side heat exchanger may reduce the range of the operating current. Several studies have examined the optimization of the segmented TECs, and they have proposed algorithms that have optimized the internal structure of systems to maximize COP [5–7]. The scaling law [8] incorporates the effects of the size of the TEC legs on performance, and reveals that the maximum cooling capacity increases with the decrease of the leg length. Otherwise, the electrical contact resistance [9] and the thermal contact resistance [10] strongly affect the performance of the TEC. To operate a TEC at a higher cooling capacity generally reduces the COP for a fixed leg length of the TEC and a confined volume in which the TEC is placed. The performance of the TEC depends on many operating parameters. A traditional single-stage TEC of size 55 mm · 55 mm and a miniature TEC of size 9.6 mm · 9.6 mm can transfer heat at the rates of about 125 W and 20 W, respectively [11]. Optimizing the dimensions of the TEC legs can provides the advantage of higher cooling capacity than that of the traditional TEC. In recent years, genetic algorithms (GAs) have been widely applied to solve various complex problems. GAs are used as a computerized search and optimization technique that is based on the evolutionary principles of natural genetics and natural selection [12]. They start from a population of possible solutions and then select the most fitted ones. The fundamental genetic operations, which include selection, crossover and mutation, are applied in order to perform optimization. The GAs enable a global search to prevent being trapped at locally optimal solutions, and perform global optimization. Wei and Joshi [13] used GAs to optimize stacked micro-channel heat sinks, and the optimization reduced the overall thermal resistance. Wu and coworkers [14] presented a method for designing the flow path of an axial flow steam turbine stage, using an approach based on GAs. The objective of this work is to optimize a single-stage TEC in a confined volume to maximize its cooling capacity, under the requirement of minimum COP and the restriction on the maximum cost of the material. GAs are employed to optimize three variables––leg length, leg area and the number of legs. The optimization method presented in the following section establishes the search procedure for finding the optimal sizes of the TEC legs.
2. Mathematical modelling Operation of a TEC is based on the Peltier effect. When a DC current passes through a pair of p- and n-type semiconductor materials, one side of the junctions is cooled and the other side is heated. Therefore, TEC acts as a solid sate device that can pump heat from one junction to the other junction when a DC current is applied. Fig. 1 refers to a TEC with multiple thermoelectric legs sandwiched between two substrates, and depicts the directions of heat flows associated with the Peltier effect, the Joule heating effect and the Fourier effect. The heat balance equations at the hot junction and the cold junction for a single-stage TEC are Eqs. (1) and (2). The heat flows aITL and aITH caused by the Peltier effect are absorbed at the cold junction and released from the hot junction, respectively. Joule heating I2(qrL/A + 2rc/A) due to the flow of electrical current through the material is generated both inside the TEC legs and at the contact surfaces between the TEC legs and the two substrates. TEC is operated between temperatures TH and TL, so heat
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TH QFourier
QJoule
QPeltier TL Fig. 1. Schematic diagram of heat flow in a TEC.
conduction kA(TH TL)/L occurs through the TEC legs. The COP is defined as the ratio of the heat absorbed at the cold junction to the total power input to TEC, as described in the following 1 L 2rc kAðT H T L Þ ; ð1Þ QH ¼ N aIT H þ I 2 qr þ 2 A L A 1 2 L 2rc kAðT H T L Þ ; ð2Þ QL ¼ N aIT L I qr þ 2 A L A COP ¼
QL QL ¼ P QH QL
ð3Þ
where QL is the cooling capacity at the cold junction and QH is the heat dissipated rate from the hot junction; N is the number of the TEC legs; a, qr, rc and k are the Seebeck coefficient, the electrical resistivity, the electrical contact resistance and the thermal conductivity, respectively; I is the input current; A and L are the cross-sectional area and the length of the TEC leg, and TH and TL represent the temperatures of the cold and hot junctions of TEC, respectively. Evidently, the total cost of the thermoelectric material is as specified in Eq. (4), which determines that the total cost can be determined by multiplying the cost per kg by the total weight of the material. The material parameters associated with the general thermoelectric material, Bismuth Telluride, are involved in the calculation procedures, so the density of 7.7 g cm3 is used. Consequently, the thermoelectric material cost almost $2200 per kg. Cost ¼ qðN A LÞ ðcost per kgÞ
ð4Þ
where q is the density of the thermoelectric material. In the calculation, the thermoelectric material properties, including the Seebeck coefficient (a), the electrical resistivity (qr) and the thermal conductivity (k), are also considered to depend on temperature, and can be generalized as follows [15] a ¼ ½263.38 þ 2.78 T mid 0.00406 T 2mid 106 ;
ð5Þ
qr ¼ ½22.39 0.13 T mid þ 0.00030625 T 2mid 106 ;
ð6Þ
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k ¼ 3.95 0.014 T mid þ 0.00001875 T 2mid
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ð7Þ
where Tmid = (TH + TL)/2 (in K).
3. Optimization 3.1. Optimization algorithms and process When a TEC is placed in a confined volume, the dimensions of the TEC legs are bounded. Given this restriction, the requirements of cooling capacity and COP cannot always be met simultaneously. As the COP increases with the leg area, the cooling capacity may decrease because the total available volume is limited. As the leg area is reduced, the cooling capacity generally increases. A smaller leg area and a greater number of legs yield greater cooling capacity. Otherwise, the contact resistance between the TEC legs and the two substrates imposes a lower bound on the leg length. When the leg length is below than this lower bound, the cooling capacity declines enormously. As the leg lengths in the range of 0.3–1 mm, the leg areas in the range of 0.09–6 mm2 and a single-stage TEC was placed in a volume of 1 mm · 100 mm2, the maximum cooling capacity and the corresponding COP calculated from Eqs. (1)–(3) were as shown in Fig. 2, in which the applied current is 1 A, the hot side and cold side temperatures were both set to 323 K, and the maximum cost of the thermoelectric material was limited to $385 dollars. Therefore, the objective of the optimization calculation is to determine the optimal leg length, the leg area and the number of legs in this confined volume, to maximize the cooling capacity and yield the required COP. Additionally, the total cost of the material is limited and the technological limitation in manufacturing TEC legs is adopted. Given these considerations, leg length, leg area and the number of legs are regarded as variable parameters to be optimized. The main parameters must be classified into four groups to optimize the design of TEC in a confined volume. 8
15 L=0.3mm L=0.4mm L=0.6mm L=0.8mm L=1.0mm
L=0.3mm
6 L=0.4mm
10
4
COP
QL (W)
L=0.6mm L=0.8mm L=1.0mm
5 2
0
0 0
2
4
6
A (mm2) Fig. 2. Maximum cooling capacity and the corresponding COP as functions of the leg area and the leg length. (The applied current is 1 A, the hot side and cold side temperatures were both set to 323 K, and the maximum cost of the thermoelectric material was limited to $385.)
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Group A. Fixed parameters: (1) Total volume in which TEC can be placed (determined by height (H) and cross-sectional area (S)); (2) Temperature at the hot junction of TEC (TH); (3) Temperature at the cold junction of TEC (TL); (4) Input current (I); (5) Lower-bound on leg length (Lmin), adopted from the technological limitation in manufacturing TEC legs. Group B. Variables: (1) Length of legs (L); (2) Cross-sectional area of legs (A); (3) Number of legs (N). Group C. Constraints: (1) COP > required minimum COP (COPmin); (2) Total cost < defined maximum cost (Costmax). Group D. Objective functions: Maximum cooling capacity. The optimization calculation begins with a population P of n randomly generated individuals, which represent n potential solutions. The individuals of a population are P = [L, A, N]. Herein, the leg length has upper and lower bounds when the technological limitation in manufacturing TEC legs (Lmin) and the height of the available volume (H) are given. Therefore, the variation in leg length can be formalized as L 2 [Lmin, H]. The range of leg areas is also constrained by the technological limitation in manufacturing legs (L2min ) and the total cross-sectional area in the available volume (S), so A 2 ½L2min ; S. The number of legs is related to the area of a single leg, because the total area in the limiting volume is restricted. Hence, the maximum number of legs will be Nmax = S/A, and N satisfies N 2 [1, Nmax]. The Genetic coding was implemented as follows to maximize the cooling capacity of TEC. *
(1) Real number encoding: a population of feasible starting points, F ¼ ½L; A; N , is randomly generated to be within all geometrical bounds, but may violate the constraints described in Group C. (2) Arithmetic crossover: the convex combination is selected for use in this GAs, and it is imple* * mented by combining two vectors (chromosomes) F1 and F2 , as *
*
*
*
*
*
F10 ¼ kF10 þ ð1 kÞF20 F20
¼ kF 2 þ ð1 kÞF 1 where k 2 [0, 1].
and
ð8Þ
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The leg area and the number of legs are related to each other, so the crossover operation in this study must also include an assurance that the crossover operation is reasonable. Appendix A lists the particular determinations in the crossover procedure. (3) Non-uniform mutation: each potential individual solution xk is replaced by a real value x0k , which is randomly selected from the values between the two-sided bounds. In the non-uniform mutation, the alterations are larger in the earlier generations and are smaller in the later ones. or x0k ¼ xk þ Dðt; xU k xk Þ
x0k ¼ xk þ Dðt; xk xLk Þ
ð9Þ
and t b Dðt; yÞ ¼ y r 1 T where r is a random number in [0, 1]; t is the generation number, T is the total number of generations, and b is the degree of non-uniformity. Like the crossover operation, the mutation operation involves determining the leg area and the number of legs. Appendix B presents the considerations. The crossover and mutation operations are herein designed to keep all variables within their reasonable domains. (4) Top pop-size selection: this part includes the evaluation of constraints and penalty functions. When individuals violate one or more constraints, penalty functions in Eqs. (10) and (11) are used to deal with the constrained problems, and the fitness value can thus be determined from b * COPmin COP ; ð10Þ gcop ðF Þ ¼ COPmin
Cost Costmax gcost ðF Þ ¼ 1 þ Costmax *
3 * 1 gcop ðF Þ 5 fitness ¼ QL ðF Þ4 * gcost ðF Þ
d ;
ð11Þ
2
*
ð12Þ
where gcop and gcost are the penalty functions used for violations of minimum COP and maximum cost, respectively. The penalty functions gcop and gcost are equal to zero and one, respectively, when the constraints COPmin and Costmax are satisfied. This scheme ensures the individuals that fulfill all of the constraints have a fitness of one. The fitness of those individuals that violate any constraints will be [0, 1). The next generation is formed by selecting the best n individuals in the previous generation according to their fitness values. 3.2. Test for genetic search The GAs used in this work were tested to determine their convergence rate and prevent illogical divergence. A TEC placed in a confined volume with a height of 1 mm and cross-sectional area of 100 mm2 was first considered. The GAs coded in this study for a manufacturing technique in which the available leg length was 0.3 mm, randomly generated leg lengths from 0.3 mm to
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1 mm, and leg areas randomly from 0.3 mm · 0.3 mm to 100 mm2. The maximum cost of the material was $385. The temperatures of the hot and cold junctions were both set to 323 K initially to demonstrate the search ability, so the properties of the thermoelectric material were calculated at an average temperature of 323 K. The calculation also considered the effect of the electrical contact resistance (rc), which was 108 X m2. When the requirement for a minimum COP is ignored, the optimal search in GAs should be able to determine the global maximum cooling capacity, which is circled in Fig. 3. Figs. 4–9 display the convergence during the genetic search. The search starts with a random population of feasible solutions, and the population almost converges after 20 runs. The optimization of such a single-stage TEC in a confined volume produces a maximum cooling capacity of 6.3 W and a corresponding COP of 0.55. The optimal dimensions are a leg length of 0.3 mm, a leg area of 0.36 mm2, and the number of legs is 208. The genetic search converges so quickly that the entire search process is completed in several seconds. A comparison of the maximum cooling 15
8 L=0.3mm
6
COP
QL (W)
10 4
5 2
0
0 0
2
4
6
A (mm2) Fig. 3. Illustration of the genetic search.
1.0 0.9 0.8
L (mm)
0.7 0.6 0.5 0.4 0.3 0.2 0
20
40
Generation Fig. 4. Convergence curve of the leg length (L).
60
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N
300 200 100 0 0
20
40
60
Generation Fig. 5. Convergence curve of the number of legs (N).
3.0 2.5
A (mm2)
2.0 1.5 1.0 0.5 0.0 0
20
40
60
Generation Fig. 6. Convergence curve of the leg area (A). 8 7 6
QL (W)
5 4 3 2 1 0 0
20
40
60
Generation Fig. 7. Convergence curve of the cooling capacity (QL).
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COP
3 2 1 0 0
20
40
60
Generation Fig. 8. Convergence curve of the COP.
600 500
Cost ($)
400 300 200 100 0 0
20
40
60
Generation Fig. 9. Convergence curve of the cost of the material.
capacity obtained from the genetic search with the analytical result reveals that the genetic result is exactly the same as the analytical result. This procedure demonstrated and verified the search ability of the GAs in this investigation, and indicated the efficiency of the convergence. Figs. 4–9 demonstrate that the optimal results obtained by applying the GAs also satisfied all geometric and implicit constraints.
4. Results When a TEC is placed in a confined volume, its cooling capacity may be reduced if the COP increases because the leg area increases. However, the value of COP may drop after the cooling
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capacity is increased, because the leg area is then smaller. The contact resistance between the TEC legs and the two substrates also imposes a lower bound on the leg length. The volume in which TEC can be placed and the technique used to machine the TEC legs are considered, and a new optimization approach is implemented based on GAs to optimize the dimensions of TEC. This approach can compute the optimal dimensions of the TEC to maximize its cooling capacity, while fulfilling the COP requirements. In this study, the hot side temperature of TEC is 323 K and the maximum cost of the material is $385. The input current and the cold side temperature are varied. Table 1 presents the main results. The data include the optimal dimensions of the TEC in a confined volume of 1 mm · 100 mm2. The COP requirement is initially neglected, and this case sufficiently demonstrates the method of GAs for optimizing TEC for various input currents. Table 1 demonstrates that the TEC can reach its maximum cooling capacity even under various input currents. This optimization technique based on GAs is advantageous for designing electronic systems that comprise a TEC, because the electrical design of any electronic system must be seriously considered. Table 1 The optimal dimensions of the TEC legs for various input currents I (A) 0.1 0.2 0.5 1 2 4 6 8
Maximum cost = $385, TL = 323 K, TH = 323 K
L (mm) 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30
A (mm2)
N
QL (W)
COP
0.09 0.10 0.18 0.36 0.75 1.45 2.36 3.19
840 778 426 208 96 52 32 24
4.18 6.0 6.3 6.3 6.3 6.3 6.3 6.3
1.89 0.79 0.44 0.47 0.54 0.47 0.54 0.56
The hot side temperature of TEC is 323 K and the maximum cost of the material is $385. The technological limitation in manufacturing legs is 0.3 mm. The requirement of COP is ignored.
8 Dimensions are fixed (A=3.19 mm2, L=0.3 mm, N=24) Dimensions are optimized
7
QL (W)
6 5 4 3 2 1 0 0
2
4
6
8
I (A)
Fig. 10. Effects of the optimized dimensions on the cooling capacities under various input currents.
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For comparison, a leg length of 0.3 mm, a leg area of 3.19 mm2, and 24 legs are used to calculate the maximum cooling capacity under various currents. Additionally, the results of the optimization search indicate the effects of the optimization. Fig. 10 reveals that optimizing the cooling capacities of TEC improves performance. After optimization, the cooling capacities can be promoted even if the TEC is operated at a lower current. Table 2 presents the optimal design variables for various cold side temperatures from 293 K to 333 K. As the cold side temperature is increased, the optimal leg area decreases and the number of legs increases. The cooling capacity increases rapidly with the cold side temperature; this improvement becomes obvious as the cold side temperature exceeds the hot side temperature by a larger margin. This result is consistent with the statements of Huang [16], who stated that the cooling capacity was large when a TEC was operated in an enforced regime. Fig. 11 plots the effects of the optimized dimensions on the cooling capacities, for various cold side temperatures. The cooling capacity increases with the cold side temperature, although the dimensions of TEC are fixed. Fortunately, the cooling capacity has a large margin for improvement if the dimensions of TEC, corresponding to the cold side temperature, are optimized.
Table 2 The optimal dimensions of the TEC legs for various cold side temperatures TL (K) 283 293 303 313 323 333
L (mm)
Maximum cost = $385, I = 1 A, TH = 323 K
– 1.00 0.75 0.33 0.30 0.30
A (mm2)
N
QL (W)
COP
– 0.51 0.49 0.40 0.36 0.35
– 44 60 170 210 216
– 0.17 0.67 2.40 6.31 10.2
– 0.06 0.18 0.23 0.46 0.70
The hot side temperature of TEC is 323 K and the maximum cost of the material is $385. The technological limitation in manufacturing legs is 0.3 mm. The requirement of COP is ignored.
12 Dimensions are fixed (A=0.57 mm2 , L=1mm, N=48) Dimensions are optimized
10
QL (W)
8 6 4 2 0 20
30
40
50
60
TL (˚C)
Fig. 11. Effects of the optimized dimensions on the cooling capacities under various cold-side temperatures.
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Table 3 The optimal dimensions of the TEC legs for various input currents I (A) 0.1 0.2 0.5 1 2 4 6 8
Required COP = 1, maximum cost = $385, TL = 273 K, TH = 273 K L (mm)
A (mm2)
N
QL (W)
0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3
0.09 0.12 0.29 0.56 1.13 2.26 3.42 4.54
840 640 262 134 66 34 22 16
4.18 5.49 5.57 5.62 5.62 5.62 5.62 5.62
The hot side temperature of TEC is 323 K and the maximum cost of the material is $385. The technological limitation in manufacturing legs is 0.3 mm. The required COP is 1.
Table 3 presents the optimal design variables for various currents, with the requirement that COP should be at least one. The leg area must be increased to tolerate a higher input current. After optimization, the maximum cooling capacities can be forced to reach their maximum values.
5. Conclusion A new method was developed for optimizing the dimensions of a single-stage TEC in a confined volume. Based on GAs, the leg length, the leg area, and the number of legs can be optimized to obtain the maximum cooling capacity, while satisfying the constraints. The total cost of the material is regarded as one constraint and the technological limitation in manufacturing TEC legs is considered. Under these constraints, the optimization can improve the cooling capacity. The lower bound on the leg length set by the electrical contact resistance between the legs and the two substrates can also be determined. The optimal search in GAs converges so rapidly that the optimization process is time and cost effective. The approach based on GAs can be used effectively to optimize the dimensions of a TEC system. Further work to optimize TEC should involve several areas, such as the heat exchange capabilities of the two heat reservoirs on the cold and hot substrates and the heat leakage through the air enclosed in the cold and hot substrates. Moreover, the use of two- or multi-stage TECs is more practicable when a larger range of operating temperatures is required. Further work should optimize both the internal structures, including the size of each leg, the ratio of the numbers of legs in different stages, and the external heat transfer conditions, in such a case.
Acknowledgement The authors would like to thank Prof. T.S. Zhao and the anonymous reviewers for their helpful comments.
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Appendix A. Considerations of crossover operation (leg area and number of legs) Leg area and the number of legs are as follows: A01 ¼ kA1 þ ð1 kÞA2 N 01 ¼ kN 1 þ ð1 kÞN 2 N check ¼
S A01
Acheck ¼
S N 01
The following if-conditions are randomly set. If N 01 > N check then N 01 ¼ N check or If A01 > Acheck then A01 ¼ Acheck : Appendix B. Considerations of mutation operation (leg area and number of legs) Leg area and the number of legs are as follows: If the leg area is selected to be mutated then t b A01 ¼ A1 þ ðS A1 Þ r 1 or T t b A01 ¼ A1 ðA1 Amin Þ r 1 T N check ¼
S A01
If N 01 > N check then N 0max 6 N 01 ¼ N check If the number of legs is selected to be mutated, then t b N 01 ¼ N 1 þ ðN max N 1 Þ r 1 or T t b N 01 ¼ N 1 ðN 1 1Þ r 1 T Acheck ¼
S N 01
If A01 > Acheck then Amax 6 A01 ¼ Acheck :
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References [1] M. Yamanashi, Y. Kibayashi, F. Toyada, M. Azechi, Optimization design in thermoelectric cooling systems, in: 15th International Conference on Thermoelectrics, 1996, pp. 220–222. [2] M. Yamanashi, A new approach to optimum design in thermoelectric cooling system, Journal of Applied Physics 80 (9) (1996) 5494–5502. [3] J. Chen, Comment on ‘‘A new approach to optimum design in thermoelectric cooling system’’, Journal of Applied Physics 82 (12) (1997) 6368–6369. [4] Y. Bi, L. Chen, C. Wu, S. Wang, Effect of heat transfer on the performance of thermoelectric heat pumps, Journal of Non-Equilibrium Thermodynamics 26 (2001) 41–51. [5] T. Wartanowicz, A. Czarnecki, Optimization of a multistage thermoelectric cooling system concerning maximum COP, in: 15th International Conference on Thermoelectrics, 1996, pp. 307–310. [6] X.C. Xuan, K.C. Ng, C. Yap, H.T. Chua, Optimization of two-stage thermoelectric coolers with two design configurations, Energy Conversion and Management 43 (2002) 2041–2052. [7] H. Lai, Y. Pan, J. Chen, Optimum design on the performance parameters of a two-stage combined semiconductor thermoelectric heat pump, Semiconductor Science and Technology 19 (2004) 17–22. [8] L. Rushing, A. Shakouri, P. Abraham, J.E. Bowers, Micro thermoelectric coolers for integrated applications, in: 16th International Conference on Thermoelectrics, 1997, pp. 646–649. [9] U. Ghoshal, Y.S. Ju, A. Miner, M.B. Ketchen, Advanced electronic microcooler, in: 18th International Conference on Thermoelectrics, 1999, pp. 113–116. [10] X.C. Xuan, Investigation of thermal contact effect on thermoelectric cooler, Energy Conservation and Management 44 (2003) 399–410. [11] Melcor, Thermoelectric Handbook, Melcor Thermoelectric, Trenton, NJ, 1995. [12] M. Gen, R.W. Cheng, Genetic Algorithms and Engineering Design, Wiley, New York, 1997. [13] X. Wei, Y. Joshi, Optimization study of stacked micro-channel heat sinks for micro-electronic cooling, IEEE Transactions on Components and Packaging Technologies 26 (1) (2003) 55–61. [14] X. Qin, L. Chen, F. Sun, C. Wu, Optimization for a steam turbine stage efficiency using a genetic algorithm, Applied Thermal Engineering 23 (2003) 2307–2316. [15] Y.-H. Cheng, W.-K. Lin, S.-D. Xu, An algorithmic method for predicting the thermal performance of thermoelectric cooling modules, in: 6th International Symposium on Heat Transfer, Beijing, China, June 2004, pp. 159–164. [16] R. Chein, G. Huang, Thermoelectric cooler application in electronic cooling, Applied Thermal Engineering 24 (2004) 2207–2217.
Geometric optimization of thermoelectric coolers in a confined volume using genetic algorithms Yi-Hsiang Cheng *, Wei-Keng Lin Department of Engineering and System Science, National Tsing Hua University, 101, Sec. 2, Kuang Fu Road, Hsinchu 300, Taiwan, ROC Received 15 November 2004; accepted 12 March 2005 Available online 25 May 2005
Abstract The demand for thermoelectric coolers (TEC) has grown significantly because of the need for a steady, low-temperature operating environment for various electronic devices such as laser diodes, semiconductor equipment, infrared detectors and others. The cooling capacity and its coefficient of performance (COP) are both extremely important in considering applications. Optimizing the dimensions of the TEC legs provides the advantage of increasing the cooling capacity, while simultaneously considering its minimum COP. This study proposed a method of optimizing the dimensions of the TEC legs using genetic algorithms (GAs), to maximize the cooling capacity. A confined volume in which the TEC can be placed and the technological limitation in manufacturing a TEC leg were considered, and three parameters––leg length, leg area and the number of legs––were taken as the variables to be optimized. The constraints of minimum COP and maximum cost of the material were set, and a genetic search was performed to determine the optimal dimensions of the TEC legs. This work reveals that optimizing the dimensions of the TEC can increase its cooling capacity. The results also show that GAs can determine the optimal dimensions according to various input currents and various cold-side operating temperatures. 2005 Elsevier Ltd. All rights reserved. Keywords: Thermoelectric cooler; Genetic algorithms; Confined volume; Optimization
*
Corresponding author. Tel.: +886 3 5713583; fax: +886 3 5720724. E-mail address: [email protected] (Y.-H. Cheng).
1359-4311/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2005.03.007
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Nomenclature A H I L Lmin N Nmax P QH QL S TH TL Tmid Greek a q qr rc k
cross-sectional area of the TEC leg, m2 height of the confined volume, m input current, A length of the TEC leg, m minimum length of the TEC legs, m number of the TEC legs maximum number of the TEC legs power input to the TEC, W total dissipated heat from hot junction, W cooling capacity at cold junction, W cross-sectional area of the confined volume, m2 temperature of hot junction, K temperature of cold junction, K average of cold and hot side temperatures, K symbols Seebeck coefficient, V/K density of thermoelectric material, kg/m3 electrical resistivity, X m electrical contact resistance, X m2 thermal conductivity, W/m K
1. Introduction In recent years, the electronic industry has developed rapidly and the miniaturization of electronic components has been ongoing. As components have shrunk, the chip-level power density has continued to rise greatly. Therefore, thermal management is becoming a critical issue in system performance. Moreover, the electrical stability of many pieces of electronic equipment, such as laser diodes, semiconductor optical devices, infrared detectors, and others, should be improved, to ensure electrical stability and rapid electrical–optical transmission. Traditional thermal management devices, such as a fan and heat sink, cannot be used in such electronic equipment, in which the thermoelectric cooler (TEC) is therefore employed to control the thermal and electrical stabilities. When a TEC is used in a piece of electronic equipment, the TEC must be designed and constructed to have a specified cooling capacity and to operate with a proper coefficient of performance (COP). Therefore, a priori thermal analysis of the performance of the TEC is important in designing a TEC structure. Yamanashi et al. [1,2] and Chen [3] studied the optimum design of a multistage TEC system with a maximum COP. By applying finite-time thermodynamic method, Wu and coworkers [4] considered the effects of thermal resistances between the TEC and the two external heat exchangers, as well as of the number of the TEC legs, on the perfor-
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mance of a TEC system. Their results indicated that the thermal performance of the hot side heat exchanger strongly affected the COP and the cooling capacity. Under poor design conditions, the hot side heat exchanger may reduce the range of the operating current. Several studies have examined the optimization of the segmented TECs, and they have proposed algorithms that have optimized the internal structure of systems to maximize COP [5–7]. The scaling law [8] incorporates the effects of the size of the TEC legs on performance, and reveals that the maximum cooling capacity increases with the decrease of the leg length. Otherwise, the electrical contact resistance [9] and the thermal contact resistance [10] strongly affect the performance of the TEC. To operate a TEC at a higher cooling capacity generally reduces the COP for a fixed leg length of the TEC and a confined volume in which the TEC is placed. The performance of the TEC depends on many operating parameters. A traditional single-stage TEC of size 55 mm · 55 mm and a miniature TEC of size 9.6 mm · 9.6 mm can transfer heat at the rates of about 125 W and 20 W, respectively [11]. Optimizing the dimensions of the TEC legs can provides the advantage of higher cooling capacity than that of the traditional TEC. In recent years, genetic algorithms (GAs) have been widely applied to solve various complex problems. GAs are used as a computerized search and optimization technique that is based on the evolutionary principles of natural genetics and natural selection [12]. They start from a population of possible solutions and then select the most fitted ones. The fundamental genetic operations, which include selection, crossover and mutation, are applied in order to perform optimization. The GAs enable a global search to prevent being trapped at locally optimal solutions, and perform global optimization. Wei and Joshi [13] used GAs to optimize stacked micro-channel heat sinks, and the optimization reduced the overall thermal resistance. Wu and coworkers [14] presented a method for designing the flow path of an axial flow steam turbine stage, using an approach based on GAs. The objective of this work is to optimize a single-stage TEC in a confined volume to maximize its cooling capacity, under the requirement of minimum COP and the restriction on the maximum cost of the material. GAs are employed to optimize three variables––leg length, leg area and the number of legs. The optimization method presented in the following section establishes the search procedure for finding the optimal sizes of the TEC legs.
2. Mathematical modelling Operation of a TEC is based on the Peltier effect. When a DC current passes through a pair of p- and n-type semiconductor materials, one side of the junctions is cooled and the other side is heated. Therefore, TEC acts as a solid sate device that can pump heat from one junction to the other junction when a DC current is applied. Fig. 1 refers to a TEC with multiple thermoelectric legs sandwiched between two substrates, and depicts the directions of heat flows associated with the Peltier effect, the Joule heating effect and the Fourier effect. The heat balance equations at the hot junction and the cold junction for a single-stage TEC are Eqs. (1) and (2). The heat flows aITL and aITH caused by the Peltier effect are absorbed at the cold junction and released from the hot junction, respectively. Joule heating I2(qrL/A + 2rc/A) due to the flow of electrical current through the material is generated both inside the TEC legs and at the contact surfaces between the TEC legs and the two substrates. TEC is operated between temperatures TH and TL, so heat
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TH QFourier
QJoule
QPeltier TL Fig. 1. Schematic diagram of heat flow in a TEC.
conduction kA(TH TL)/L occurs through the TEC legs. The COP is defined as the ratio of the heat absorbed at the cold junction to the total power input to TEC, as described in the following 1 L 2rc kAðT H T L Þ ; ð1Þ QH ¼ N aIT H þ I 2 qr þ 2 A L A 1 2 L 2rc kAðT H T L Þ ; ð2Þ QL ¼ N aIT L I qr þ 2 A L A COP ¼
QL QL ¼ P QH QL
ð3Þ
where QL is the cooling capacity at the cold junction and QH is the heat dissipated rate from the hot junction; N is the number of the TEC legs; a, qr, rc and k are the Seebeck coefficient, the electrical resistivity, the electrical contact resistance and the thermal conductivity, respectively; I is the input current; A and L are the cross-sectional area and the length of the TEC leg, and TH and TL represent the temperatures of the cold and hot junctions of TEC, respectively. Evidently, the total cost of the thermoelectric material is as specified in Eq. (4), which determines that the total cost can be determined by multiplying the cost per kg by the total weight of the material. The material parameters associated with the general thermoelectric material, Bismuth Telluride, are involved in the calculation procedures, so the density of 7.7 g cm3 is used. Consequently, the thermoelectric material cost almost $2200 per kg. Cost ¼ qðN A LÞ ðcost per kgÞ
ð4Þ
where q is the density of the thermoelectric material. In the calculation, the thermoelectric material properties, including the Seebeck coefficient (a), the electrical resistivity (qr) and the thermal conductivity (k), are also considered to depend on temperature, and can be generalized as follows [15] a ¼ ½263.38 þ 2.78 T mid 0.00406 T 2mid 106 ;
ð5Þ
qr ¼ ½22.39 0.13 T mid þ 0.00030625 T 2mid 106 ;
ð6Þ
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k ¼ 3.95 0.014 T mid þ 0.00001875 T 2mid
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ð7Þ
where Tmid = (TH + TL)/2 (in K).
3. Optimization 3.1. Optimization algorithms and process When a TEC is placed in a confined volume, the dimensions of the TEC legs are bounded. Given this restriction, the requirements of cooling capacity and COP cannot always be met simultaneously. As the COP increases with the leg area, the cooling capacity may decrease because the total available volume is limited. As the leg area is reduced, the cooling capacity generally increases. A smaller leg area and a greater number of legs yield greater cooling capacity. Otherwise, the contact resistance between the TEC legs and the two substrates imposes a lower bound on the leg length. When the leg length is below than this lower bound, the cooling capacity declines enormously. As the leg lengths in the range of 0.3–1 mm, the leg areas in the range of 0.09–6 mm2 and a single-stage TEC was placed in a volume of 1 mm · 100 mm2, the maximum cooling capacity and the corresponding COP calculated from Eqs. (1)–(3) were as shown in Fig. 2, in which the applied current is 1 A, the hot side and cold side temperatures were both set to 323 K, and the maximum cost of the thermoelectric material was limited to $385 dollars. Therefore, the objective of the optimization calculation is to determine the optimal leg length, the leg area and the number of legs in this confined volume, to maximize the cooling capacity and yield the required COP. Additionally, the total cost of the material is limited and the technological limitation in manufacturing TEC legs is adopted. Given these considerations, leg length, leg area and the number of legs are regarded as variable parameters to be optimized. The main parameters must be classified into four groups to optimize the design of TEC in a confined volume. 8
15 L=0.3mm L=0.4mm L=0.6mm L=0.8mm L=1.0mm
L=0.3mm
6 L=0.4mm
10
4
COP
QL (W)
L=0.6mm L=0.8mm L=1.0mm
5 2
0
0 0
2
4
6
A (mm2) Fig. 2. Maximum cooling capacity and the corresponding COP as functions of the leg area and the leg length. (The applied current is 1 A, the hot side and cold side temperatures were both set to 323 K, and the maximum cost of the thermoelectric material was limited to $385.)
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Group A. Fixed parameters: (1) Total volume in which TEC can be placed (determined by height (H) and cross-sectional area (S)); (2) Temperature at the hot junction of TEC (TH); (3) Temperature at the cold junction of TEC (TL); (4) Input current (I); (5) Lower-bound on leg length (Lmin), adopted from the technological limitation in manufacturing TEC legs. Group B. Variables: (1) Length of legs (L); (2) Cross-sectional area of legs (A); (3) Number of legs (N). Group C. Constraints: (1) COP > required minimum COP (COPmin); (2) Total cost < defined maximum cost (Costmax). Group D. Objective functions: Maximum cooling capacity. The optimization calculation begins with a population P of n randomly generated individuals, which represent n potential solutions. The individuals of a population are P = [L, A, N]. Herein, the leg length has upper and lower bounds when the technological limitation in manufacturing TEC legs (Lmin) and the height of the available volume (H) are given. Therefore, the variation in leg length can be formalized as L 2 [Lmin, H]. The range of leg areas is also constrained by the technological limitation in manufacturing legs (L2min ) and the total cross-sectional area in the available volume (S), so A 2 ½L2min ; S. The number of legs is related to the area of a single leg, because the total area in the limiting volume is restricted. Hence, the maximum number of legs will be Nmax = S/A, and N satisfies N 2 [1, Nmax]. The Genetic coding was implemented as follows to maximize the cooling capacity of TEC. *
(1) Real number encoding: a population of feasible starting points, F ¼ ½L; A; N , is randomly generated to be within all geometrical bounds, but may violate the constraints described in Group C. (2) Arithmetic crossover: the convex combination is selected for use in this GAs, and it is imple* * mented by combining two vectors (chromosomes) F1 and F2 , as *
*
*
*
*
*
F10 ¼ kF10 þ ð1 kÞF20 F20
¼ kF 2 þ ð1 kÞF 1 where k 2 [0, 1].
and
ð8Þ
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The leg area and the number of legs are related to each other, so the crossover operation in this study must also include an assurance that the crossover operation is reasonable. Appendix A lists the particular determinations in the crossover procedure. (3) Non-uniform mutation: each potential individual solution xk is replaced by a real value x0k , which is randomly selected from the values between the two-sided bounds. In the non-uniform mutation, the alterations are larger in the earlier generations and are smaller in the later ones. or x0k ¼ xk þ Dðt; xU k xk Þ
x0k ¼ xk þ Dðt; xk xLk Þ
ð9Þ
and t b Dðt; yÞ ¼ y r 1 T where r is a random number in [0, 1]; t is the generation number, T is the total number of generations, and b is the degree of non-uniformity. Like the crossover operation, the mutation operation involves determining the leg area and the number of legs. Appendix B presents the considerations. The crossover and mutation operations are herein designed to keep all variables within their reasonable domains. (4) Top pop-size selection: this part includes the evaluation of constraints and penalty functions. When individuals violate one or more constraints, penalty functions in Eqs. (10) and (11) are used to deal with the constrained problems, and the fitness value can thus be determined from b * COPmin COP ; ð10Þ gcop ðF Þ ¼ COPmin
Cost Costmax gcost ðF Þ ¼ 1 þ Costmax *
3 * 1 gcop ðF Þ 5 fitness ¼ QL ðF Þ4 * gcost ðF Þ
d ;
ð11Þ
2
*
ð12Þ
where gcop and gcost are the penalty functions used for violations of minimum COP and maximum cost, respectively. The penalty functions gcop and gcost are equal to zero and one, respectively, when the constraints COPmin and Costmax are satisfied. This scheme ensures the individuals that fulfill all of the constraints have a fitness of one. The fitness of those individuals that violate any constraints will be [0, 1). The next generation is formed by selecting the best n individuals in the previous generation according to their fitness values. 3.2. Test for genetic search The GAs used in this work were tested to determine their convergence rate and prevent illogical divergence. A TEC placed in a confined volume with a height of 1 mm and cross-sectional area of 100 mm2 was first considered. The GAs coded in this study for a manufacturing technique in which the available leg length was 0.3 mm, randomly generated leg lengths from 0.3 mm to
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1 mm, and leg areas randomly from 0.3 mm · 0.3 mm to 100 mm2. The maximum cost of the material was $385. The temperatures of the hot and cold junctions were both set to 323 K initially to demonstrate the search ability, so the properties of the thermoelectric material were calculated at an average temperature of 323 K. The calculation also considered the effect of the electrical contact resistance (rc), which was 108 X m2. When the requirement for a minimum COP is ignored, the optimal search in GAs should be able to determine the global maximum cooling capacity, which is circled in Fig. 3. Figs. 4–9 display the convergence during the genetic search. The search starts with a random population of feasible solutions, and the population almost converges after 20 runs. The optimization of such a single-stage TEC in a confined volume produces a maximum cooling capacity of 6.3 W and a corresponding COP of 0.55. The optimal dimensions are a leg length of 0.3 mm, a leg area of 0.36 mm2, and the number of legs is 208. The genetic search converges so quickly that the entire search process is completed in several seconds. A comparison of the maximum cooling 15
8 L=0.3mm
6
COP
QL (W)
10 4
5 2
0
0 0
2
4
6
A (mm2) Fig. 3. Illustration of the genetic search.
1.0 0.9 0.8
L (mm)
0.7 0.6 0.5 0.4 0.3 0.2 0
20
40
Generation Fig. 4. Convergence curve of the leg length (L).
60
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N
300 200 100 0 0
20
40
60
Generation Fig. 5. Convergence curve of the number of legs (N).
3.0 2.5
A (mm2)
2.0 1.5 1.0 0.5 0.0 0
20
40
60
Generation Fig. 6. Convergence curve of the leg area (A). 8 7 6
QL (W)
5 4 3 2 1 0 0
20
40
60
Generation Fig. 7. Convergence curve of the cooling capacity (QL).
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COP
3 2 1 0 0
20
40
60
Generation Fig. 8. Convergence curve of the COP.
600 500
Cost ($)
400 300 200 100 0 0
20
40
60
Generation Fig. 9. Convergence curve of the cost of the material.
capacity obtained from the genetic search with the analytical result reveals that the genetic result is exactly the same as the analytical result. This procedure demonstrated and verified the search ability of the GAs in this investigation, and indicated the efficiency of the convergence. Figs. 4–9 demonstrate that the optimal results obtained by applying the GAs also satisfied all geometric and implicit constraints.
4. Results When a TEC is placed in a confined volume, its cooling capacity may be reduced if the COP increases because the leg area increases. However, the value of COP may drop after the cooling
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capacity is increased, because the leg area is then smaller. The contact resistance between the TEC legs and the two substrates also imposes a lower bound on the leg length. The volume in which TEC can be placed and the technique used to machine the TEC legs are considered, and a new optimization approach is implemented based on GAs to optimize the dimensions of TEC. This approach can compute the optimal dimensions of the TEC to maximize its cooling capacity, while fulfilling the COP requirements. In this study, the hot side temperature of TEC is 323 K and the maximum cost of the material is $385. The input current and the cold side temperature are varied. Table 1 presents the main results. The data include the optimal dimensions of the TEC in a confined volume of 1 mm · 100 mm2. The COP requirement is initially neglected, and this case sufficiently demonstrates the method of GAs for optimizing TEC for various input currents. Table 1 demonstrates that the TEC can reach its maximum cooling capacity even under various input currents. This optimization technique based on GAs is advantageous for designing electronic systems that comprise a TEC, because the electrical design of any electronic system must be seriously considered. Table 1 The optimal dimensions of the TEC legs for various input currents I (A) 0.1 0.2 0.5 1 2 4 6 8
Maximum cost = $385, TL = 323 K, TH = 323 K
L (mm) 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30
A (mm2)
N
QL (W)
COP
0.09 0.10 0.18 0.36 0.75 1.45 2.36 3.19
840 778 426 208 96 52 32 24
4.18 6.0 6.3 6.3 6.3 6.3 6.3 6.3
1.89 0.79 0.44 0.47 0.54 0.47 0.54 0.56
The hot side temperature of TEC is 323 K and the maximum cost of the material is $385. The technological limitation in manufacturing legs is 0.3 mm. The requirement of COP is ignored.
8 Dimensions are fixed (A=3.19 mm2, L=0.3 mm, N=24) Dimensions are optimized
7
QL (W)
6 5 4 3 2 1 0 0
2
4
6
8
I (A)
Fig. 10. Effects of the optimized dimensions on the cooling capacities under various input currents.
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For comparison, a leg length of 0.3 mm, a leg area of 3.19 mm2, and 24 legs are used to calculate the maximum cooling capacity under various currents. Additionally, the results of the optimization search indicate the effects of the optimization. Fig. 10 reveals that optimizing the cooling capacities of TEC improves performance. After optimization, the cooling capacities can be promoted even if the TEC is operated at a lower current. Table 2 presents the optimal design variables for various cold side temperatures from 293 K to 333 K. As the cold side temperature is increased, the optimal leg area decreases and the number of legs increases. The cooling capacity increases rapidly with the cold side temperature; this improvement becomes obvious as the cold side temperature exceeds the hot side temperature by a larger margin. This result is consistent with the statements of Huang [16], who stated that the cooling capacity was large when a TEC was operated in an enforced regime. Fig. 11 plots the effects of the optimized dimensions on the cooling capacities, for various cold side temperatures. The cooling capacity increases with the cold side temperature, although the dimensions of TEC are fixed. Fortunately, the cooling capacity has a large margin for improvement if the dimensions of TEC, corresponding to the cold side temperature, are optimized.
Table 2 The optimal dimensions of the TEC legs for various cold side temperatures TL (K) 283 293 303 313 323 333
L (mm)
Maximum cost = $385, I = 1 A, TH = 323 K
– 1.00 0.75 0.33 0.30 0.30
A (mm2)
N
QL (W)
COP
– 0.51 0.49 0.40 0.36 0.35
– 44 60 170 210 216
– 0.17 0.67 2.40 6.31 10.2
– 0.06 0.18 0.23 0.46 0.70
The hot side temperature of TEC is 323 K and the maximum cost of the material is $385. The technological limitation in manufacturing legs is 0.3 mm. The requirement of COP is ignored.
12 Dimensions are fixed (A=0.57 mm2 , L=1mm, N=48) Dimensions are optimized
10
QL (W)
8 6 4 2 0 20
30
40
50
60
TL (˚C)
Fig. 11. Effects of the optimized dimensions on the cooling capacities under various cold-side temperatures.
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Table 3 The optimal dimensions of the TEC legs for various input currents I (A) 0.1 0.2 0.5 1 2 4 6 8
Required COP = 1, maximum cost = $385, TL = 273 K, TH = 273 K L (mm)
A (mm2)
N
QL (W)
0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3
0.09 0.12 0.29 0.56 1.13 2.26 3.42 4.54
840 640 262 134 66 34 22 16
4.18 5.49 5.57 5.62 5.62 5.62 5.62 5.62
The hot side temperature of TEC is 323 K and the maximum cost of the material is $385. The technological limitation in manufacturing legs is 0.3 mm. The required COP is 1.
Table 3 presents the optimal design variables for various currents, with the requirement that COP should be at least one. The leg area must be increased to tolerate a higher input current. After optimization, the maximum cooling capacities can be forced to reach their maximum values.
5. Conclusion A new method was developed for optimizing the dimensions of a single-stage TEC in a confined volume. Based on GAs, the leg length, the leg area, and the number of legs can be optimized to obtain the maximum cooling capacity, while satisfying the constraints. The total cost of the material is regarded as one constraint and the technological limitation in manufacturing TEC legs is considered. Under these constraints, the optimization can improve the cooling capacity. The lower bound on the leg length set by the electrical contact resistance between the legs and the two substrates can also be determined. The optimal search in GAs converges so rapidly that the optimization process is time and cost effective. The approach based on GAs can be used effectively to optimize the dimensions of a TEC system. Further work to optimize TEC should involve several areas, such as the heat exchange capabilities of the two heat reservoirs on the cold and hot substrates and the heat leakage through the air enclosed in the cold and hot substrates. Moreover, the use of two- or multi-stage TECs is more practicable when a larger range of operating temperatures is required. Further work should optimize both the internal structures, including the size of each leg, the ratio of the numbers of legs in different stages, and the external heat transfer conditions, in such a case.
Acknowledgement The authors would like to thank Prof. T.S. Zhao and the anonymous reviewers for their helpful comments.
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Appendix A. Considerations of crossover operation (leg area and number of legs) Leg area and the number of legs are as follows: A01 ¼ kA1 þ ð1 kÞA2 N 01 ¼ kN 1 þ ð1 kÞN 2 N check ¼
S A01
Acheck ¼
S N 01
The following if-conditions are randomly set. If N 01 > N check then N 01 ¼ N check or If A01 > Acheck then A01 ¼ Acheck : Appendix B. Considerations of mutation operation (leg area and number of legs) Leg area and the number of legs are as follows: If the leg area is selected to be mutated then t b A01 ¼ A1 þ ðS A1 Þ r 1 or T t b A01 ¼ A1 ðA1 Amin Þ r 1 T N check ¼
S A01
If N 01 > N check then N 0max 6 N 01 ¼ N check If the number of legs is selected to be mutated, then t b N 01 ¼ N 1 þ ðN max N 1 Þ r 1 or T t b N 01 ¼ N 1 ðN 1 1Þ r 1 T Acheck ¼
S N 01
If A01 > Acheck then Amax 6 A01 ¼ Acheck :
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