Int J Avance Design an Manufacturing Technology, Vol. 8/ o. 3/ September - 2015 67 A Hybri Approach Base on the Genetic Algorithm an Monte Carlo Metho to Optimize the 3-D Raiant Furnaces B. Kamkari* Young Researchers an Elite Club, Yaegar-e-Imam Khomein (RAH) Shahre-Rey Branch, Islamic Aza University, Tehran, Iran E-mail: kamkari@ut.ac.ir *Corresponing author Sh. Mahjoub Department of Engineering, Majlesi branch, Islamic Aza University, Isfahan, Iran E-mail: sh_mahjoub@yahoo.com L. Darvishvan Department of Mechanical Engineering, University of Tehran, Iran E-mail: l.arvishvan@ut.ac.ir Receive: 6 January 2015, Revise: 12 March 2015, Accepte: 15 March 2015 Abstract: This stuy presents an optimization methoology to obtain the uniform thermal conitions over the 3-D esign boy (DB) in 3-D raiant furnaces. For uniform thermal conitions on the DB surfaces, optimal temperature of the heater an the best location of the DB insie the furnace are obtaine by minimizing an objective function. The raiative heat transfer problem is solve on the basis of the Monte Carlo metho (MCM) to calculate the heat fluxes on the DB surfaces. The genetic algorithm (GA) is use to minimize the objective function efine base on the calculate an esire heat fluxes. The results inicate that thermal conitions on the DB surfaces are greatly influence by the location of the DB an heater temperature. It is conclue that the introuce metho is well capable to achieve the uniform thermal conitions on the DB surfaces by fining the optimal values for temperature of the heater an the best location for the DB insie the raiant furnace. Keywors: Genetic algorithm, Inverse Bounary Design Problems, Monte Carlo Metho, Raiation Heat Transfer Reference: Kamkari, B., Mahjoub, Sh., an Darvishvan, L., A Hybri Approach Base on the Genetic Algorithm an Monte Carlo Metho to Optimize the 3-D Raiant Furnaces, Int J of Avance Design an Manufacturing Technology, Vol. 8/o. 2, 2015, pp. 67-75. Biographical notes: B. Kamkari is currently Assistant Professor at the Department of Mechanical Engineering of Yaegar-e-Imam Khomeini (RAH) Branch, Islamic Aza University, Tehran, Iran. He receive his MSc in Mechanical Engineering from Isfahan University of Technology, an his PhD from Tehran University. S. Mahjoub is a PhD caniate in the School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran. He receive his MSc in mechanical engineering in 2003 from Sharif University of Technology, Tehran, Iran. He is currently a lecturer of thermo-flui in the Department of Engineering, Majlesi Branch, Islamic Aza University, Isfahan, Iran. L. Darvishvan receive her MSc in mechanical engineering in 2011, from the School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran. She is currently working on the raiative heat transfer an esign of raiant enclosures.
68 Int J Avance Design an Manufacturing Technology, Vol. 8/ o. 3/ September 2015 1 ITRODUCTIO Raiant furnaces are use in various engineering applications such as semiconuctor prouction, thermal operation, surface coating an rying processes. Among various heating evices, raiant furnaces are able to prouce esire uniform thermal conitions, heat flux an temperature on the esign surfaces. Proviing uniform thermal conitions on the esign boy (DB) surfaces requires consiering some system parameters incluing arrangement an powers of the heaters, geometry an material properties of the furnace an location of the DB insie the furnace. Design of the raiant furnaces with the objective of satisfying the uniform thermal conitions over the DB surfaces is categorize as inverse bounary esign problems which are known to be ill-pose [1]. A goo number of stuies have been given in literature to esign the raiant furnaces using special mathematical tools such as regularization an optimization tools [1-3], [4-9]. Daun presente a comprehensive review of applying regularization methos to the inverse esign of thermal systems [2]. Recently, optimization methos have been significantly applie in inverse raiative heat transfer problems [8-14]. In optimization methos, an objective function which epens on the esign parameters is efine so that its minimum point correspons to the ieal esign outcome. This objective function which epens on the esign parameters is minimize by employing the specialize numerical algorithm. Therefore, optimization methos change the esign parameters systematically to satisfy the esign goals. Among the optimization methos, GA has receive much attention for its flexibility, globality, robustness, parallelism an simplicity. Gosselin et al., [15] reviewe the utilization of genetic algorithms in heat transfer problems. They mentione that inverse heat transfer problems are one of the three main families of heat transfer problems which are solve using GA. Safavineja et al. [6] implemente micro-genetic algorithm to optimize the number an location of the heaters in 2-D raiant enclosures compose of the specular an iffuse surfaces. They also stuie the effects of the refractory surface characteristics (i.e., iffuse an/or specular) on the optimal solution. In another stuy, Safavineja et al. use this metho to solve the inverse bounary esign problem in 2-D raiant enclosures with absorbing-emitting meia [7]. For the uniform thermal conitions, Chopae et al. [13] investigate the constraints on the size of a 3-D esign object that can be place insie a 3-D raiant furnace. They foun the constraints on the size of the esign objects that can have uniform thermal conitions. Monte Carlo metho (MCM) is one of the most efficient commonly use numerical solutions of the raiation heat transfer problems. In a goo stuy, Howell [16] presente a comprehensive review of applying the MCM in raiative heat transfer. The MCM simulates the raiation heat transfer by tracing the history of a ranom sample of photons from their points of emission until absorbing all of them. To the best of the authors knowlege, none of the previous works in esign of raiant furnaces, has employe MCM combine with the GA. Also, the effect of location of the 3-D DB on uniformity of the heat fluxes over the DB surfaces has not been consiere before. Hence, in this stuy, effects of the location of the DB as well as the temperature of the heater have been consiere to prouce the esire uniform thermal conitions on the surfaces of the 3-D DB. The DB is locate insie a rectangular raiant furnace with transparent meium. To achieve the uniform thermal conition, the optimal temperature of the heater an the best location of the DB have been obtaine. The MCM has been use to solve the irect raiative heat transfer formulate base on the total iffuse-specular raiation istribution factor (RDF). The GA has been employe to fin the best location of the DB an temperatures of the heaters by minimizing the objective function efine base on the sum of the square error between calculate an esire heat fluxes on the DB. 2 PROBLEM FORMULATIO 2.1. Optimization Strategy Fig. 1 shows the geometry of the problem uner consieration. The rectangular furnace is equippe with a heater on its top wall. The other surfaces of the furnace are aiabatic which are assume to be iffusespecular reflectors with a specularity ratio of 0.8 an a reflectivity of one. That is, they reflect the entire irraiations from other surfaces. The heater an DB surfaces are iffuse emitters with emissivity values of 0.8 an 0.4, respectively. They are also consiere to be iffuse-specular reflectors with a specularity ratio of 0.2. The surfaces of the DB an furnace are ientifie with numbers 1 to 6 for the DB an 6 to 12 for the furnace. The aims of this optimization problem are to fin the temperature of the heater an the best location of the DB to prouce a uniform heat flux of 100 kw/m 2 over all surfaces of the DB with specifie temperature of 500 o K. This is accomplishe first by specifying the esire temperature over the DB surfaces, then, obtaining the heat fluxes on the surfaces of the DB,
Int J Avance Design an Manufacturing Technology, Vol. 8/ o. 3/ September - 2015 69 afterwars, efining the objective function using these calculate heat fluxes as follows, F ( q ( i) q ) i1 c 2 (1) Where q an c q are calculate an esire heat fluxes over the DB surfaces, respectively. is the number of the DB surfaces. Fig. 1 Schematic of the 3-D raiant furnace an esign boy The objective function is subject to the following constraints, 0.1 xc 0.5 0.1 yc 0.5 0.1 zc 0.5 1000 TH 4000 ( m) ( m) ( m) ( K) (2) These constraints are accoring to the limitations of the furnace imensions an power of the heater. Since the bottom of the raiant furnace is symmetrical about the X an Y axis, to avoi repetitions of locations, the problem is solve in a quarant. To stuy the effect of the location of the DB, some possible locations are chosen in a quarant an their coorinates are given in Table 1. By minimizing the objective function with respect to the unknown parameters using the GA metho, the esire thermal conitions on the DB surfaces are achieve. The raiative heat transfer problem is solve using the MCM. A brief formulation for the MCM is provie in the following section. Further etails of the MCM can be foun in [17], [18]. Table 1 Some possible locations of the DB insie the furnace Case o. Center coorinates of esign boy 1 (0.1, 0.1, 0.1) 2 (0.1, 0.3, 0.1) 3 (0.1, 0.5, 0.1) 4 (0.3,0.3,0.1) 5 (0.3,0.5,0.1) 6 (0.5,0.5,0.1) 7 (0.1,0.1,0.3) 8 (0.1,0.3,0.3) 9 (0.1,0.5,0.3) 10 (0.3,0.3,0.3) 11 (0.3,0.5,0.3) 12 (0.5,0.5,0.3) 13 (0.1,0.1,0.5) 14 (0.1,0.3,0.5) 15 (0.1,0.5,0.5) 16 (0.3,0.3,0.5) 17 (0.3,0.5,0.5) 18 (0.5,0.5,0.5) 2.2. Direct raiative heat transfer problem In this stuy, the MCM base on the total raiation istribution factor (RDF), is applie to moel the raiative heat transfer. The RDF is the fraction of total raiation emitte iffusely from surface i an absorbe by surface j, ue to irect raiation an to all possible iffuse an specular reflections within the enclosure [18]. Base on the RDF efinition, the net heat flux from surface i in a enclosure compose of the e surfaces can be written as, e 4 q ( ) ( W i i T j D 2 ) (3) m j1 Where i, Ai, an T i are emissivity, surface area an temperature of the surface i, respectively. is Stefan- Boltzman constant an D is the RDF. In this equation, is Kronecker elta efine as follows: 1 i j (4) 0 i j In this stuy, the MCM is employe to calculate the RDFs. A logic block iagram that appears in Fig. 2 shows the proceure of the MCM to obtain the RDF. In this metho, first, a large number of energy parcels, calle photons, are emitte from ranomly selecte points on the surface i an in ranomly uniform irections. Then, these photons are trace through a series of reflections until they are finally absorbe by one of the enclosure surfaces calle j. Subsequently, the RDF ( D ) is numerically equal to the ratio of the number of photons absorbe by surfaces j ( n ) to the
q (kw/m 2 ) 70 Int J Avance Design an Manufacturing Technology, Vol. 8/ o. 3/ September 2015 total number of photons emitte from surface i ( n i ). It can be well estimate using a large amount of emitte photons from surface i. 0.4, respectively. They are also iffuse-specular reflectors with a specularity ratio of 0.2. With the objective of estimating the heat flux istributions, temperatures of the furnace an DB surfaces are set to 2000 o K an 500 o K, respectively, an the DB is locate at (0.5, 0.5, 0.5). Calculate heat fluxes base on the RDF an RSF formulations have been compare in Fig. 3. An excellent agreement between calculate heat fluxes using the RDF an RSF methos is observe. 100 50 Shape Factor Metho Distribution Factor Metho 0-50 -100-150 -200-250 -300-350 -400-450 -500 7 8 9 10 11 12 Surface umber Fig. 3 Comparison of the heat fluxes obtaine using the raiation istribution factor metho an raiation shape factor metho Fig. 2 A logic block iagram for the MCRT metho The accuracy of the RDF results is compare against the results obtaine using the raiation shape factor (RSF) approach. The net heat flux of surface i in an enclosure compose of e surfaces base on the RSF is given by [19] : e 1 ( ) j 1 j j F q j j e 4 ( F ) T j, i 1,2,..., e j 1 (5) Where is Kronecker elta efine in Eq. (4) an F is RSF between surfaces i an j. For this valiation stuy, all surfaces of the furnace an DB are assume to be iffuse emitters with emissivity values of 0.8 an 2.3. Optimization base on the genetic algorithm The GA is a search technique that is base on the principles of the natural selection an survival of the fittest iniviuals in the nature. Here, a real-value GA [20] is use as an optimization tool shown in Fig. 4. Fining a global optimum an spee of convergence are greatly influence by the choices of GA parameters incluing population size, crossover an mutation ratio [15]. In this optimization problem, to achieve the optimum values, several settings of the GA parameters are trie. Initially, the process of the optimization starts with generating ranomly an initial population in appropriate intervals of the optimization variables. Here, 100 iniviuals are consiere in the population an each iniviual has 4 variables. In GA metho, the iniviuals of the population are reprouce by means of crossover an mutation operators to obtain a new generation. In this stuy, the probability of crossover has been fixe at 0.8. In orer to increase the iversity in the population an prevent the GA from early convergence to a local optimum solution, the mutation operator is use in the GA metho. In this stuy, 20% of the population is chosen to mutate.
Int J Avance Design an Manufacturing Technology, Vol. 8/ o. 3/ September - 2015 71 3 RESULTS AD DISCUSSIO Fig. 4 A flowchart for the genetic algorithm metho In each iteration, objective function values (costs of the iniviuals) are calculate for all iniviuals. Then, these values are ranke an the weakest iniviuals are iscare in orer that the average of the cost values ecreases in new generation until an acceptable solution is foun. The number of generations, that is one of the most commonly use stop criteria, shoul be efine at the start of the optimization process. The best of the iniviuals in each generation is chosen until no improvement in the cost values is observe. This stuy aims at optimizing the raiant furnace using the GA an MCM. During the optimization process, the temperature of the heater an the best location of the DB are foun to provie a uniform heat flux of 100 kw/m 2 over all surfaces of the DB with specifie temperature of 500 o K. Prior to presenting the results of the optimization problem, first the effects of the esign parameters on the heat fluxes over the DB surfaces are iscusse. For this purpose, the heat fluxes over the DB surfaces at 18 locations emonstrate in Table 1 are calculate an the results are illustrate in Fig. 5. To stuy the effect of the DB location on the calculate heat flux, the temperature of heater is set to 4000 o K. It can be observe from Fig. 5 that while the temperature of the heater is constant, heat fluxes on the DB surfaces vary with changing the location of the DB. It can also be unerstoo from Fig. 5 that the uniform thermal conitions on the DB surfaces nee careful selection of the esign parameters incluing the temperature of the heater an location of the DB insie the furnace. To have a better iea of how the location of the DB influence the heat fluxes over the DB surfaces, the average heat fluxes on all surfaces of the DB in each location are illustrate in Fig. 6. In this investigation, heater temperature is set to 4000 o K. As it can be observe from Fig. 6, with changing the location of the DB, the average heat fluxes over the DB surfaces changes. Fig. 7 shows the effect of the heater temperature on the objective function values for the DB locate at (0.5, 0.5, 0.5). It is clearly evient from Fig. 7 that as the temperature of the heater increases, the objective function value ecreases until it reaches its minimum value. Then the objective function value begins to rise with any further increase in temperature. In orer to evaluate how the location of the DB affects the objective function values, heat fluxes on the DB surfaces an accoringly the objective function values are obtaine at 18 locations an the results are shown in Fig. 8 in one quarant of the furnace. In this Figure, heater temperature is set to 4000 o K. As it can be observe from Fig. 8, the objective function values vary with changing the location of the DB. Base on the above mentione stuies, we come to the conclusion that for proviing the uniform thermal conitions over all surfaces of the 3-D DB in the 3-D raiant furnace, temperature of the heater an location of the DB must be chosen appropriately through an optimization process.
q (kw/m 2 ) q (kw/m 2 ) Objective function (kw/m 2 ) q (kw/m 2 ) q ave (kw/m 2 ) 72 Int J Avance Design an Manufacturing Technology, Vol. 8/ o. 3/ September 2015 6400 6200 6000 Center of esign boy is locate at: (0.1, 0.1, 0.1) (0.1, 0.3, 0.1) (0.1, 0.5, 0.1) (0.3, 0.3, 0.1) (0.3, 0.5, 0.1) (0.5, 0.5, 0.1) 5800 5750 5700 T= 4000 K 5800 5650 5600 5600 5400 5550 5200 Design boy surface number 6400 6200 6000 5800 5600 5400 (a) Z=0.1 m Center of esign boy is locate at: (0.1, 0.1, 0.3) (0.1, 0.3, 0.3) (0.3, 0.3, 0.3) (0.1, 0.5, 0.3) (0.3, 0.5, 0.3) (0.5, 0.5, 0.3) 5200 Design boy surface number 6400 6200 6000 5800 5600 5400 (b) Z=0.3 m Center of esign boy is locate at: (0.1, 0.1, 0.5) (0.1, 0.3, 0.5) (0.3, 0.3, 0.5) (0.1, 0.5, 0.5) (0.3, 0.5, 0.5) (0.5, 0.5, 0.5) 5200 Design boy surface number (c) Z=0.5 m Fig. 5 Calculate heat fluxes over the esign boy surfaces at 18 locations (Temperature of the heater is set to 4000 o K) 5500 7 8 9 10 11 12 13 14 15 16 17 18 Location number Fig. 6 The average heat fluxes on the esign boy surfaces at 18 locations (Heater temperature is set to 4000 o K) 2500 2000 1500 1000 500 Center of esign boy is locate at: 150 100 50 0 1000 1250 1500 1750 2000 (0.5, 0.5, 0.5) 0 1000 1500 2000 2500 3000 3500 4000 Heater temperature (K) Fig. 7 Effect of the heater temperature on the objective function values when the esign boy is locate at (0.5, 0.5, 0.5) Optimal results are liste in Table 2. The heat flux istribution on the DB surfaces at the optimal point is presente in Fig. 9. It can be seen that the calculate heat fluxes at the optimal point are well close to the esire one (100 kw/m 2 ) with an acceptable uniformity an a variance of 0.09 kw/m 2. Design parameters Optimal value Table 2 Optimal values of the esign parameters x c(m) 0.1 y c (m) 0.1 z c (m) 0.1 T H (K) 1464
q (kw/m 2 ) Objective function value (kw/m 2 ) q (kw/m 2 ) Int J Avance Design an Manufacturing Technology, Vol. 8/ o. 3/ September - 2015 73 110 108 106 Desire heat flux Heat flux at optimal point 104 102 100 98 96 94 92 (a) Z=0.1 m 90 Design boy surface number Fig. 9 Heat flux istribution on the DB surfaces at the optimal point (b) Z=0.3 m In orer to clarify the obtaine results, Fig. 10 shows the heat fluxes over the top an bottom surfaces of the DB at the first 6 locations presente in Table 1 (Z=0.1 m) when temperature of the heater is set to optimum temperature (T H =1464 o K). As it can be observe, when the DB moves from center to the corner of the furnace, the heat flux on bottom surface increases an it ecreases on the top surface of the DB. Consequently, the objective function value ecreases until it reaches to the minimum point. So, the ifference between the heat fluxes over bottom an top surfaces an objective function value are minimum at (0.1, 0.1, 0.1). 101 100.8 100.6 100.4 100.2 100 99.8 99.6 99.4 99.2 Bottom surface of DB Top surface of DB Objective function 0.15 0.14 0.13 0.12 0.11 0.1 (c) Z=0.5 m Fig. 8 The objective function values in one quarant of the furnace (Temperature of the heater is set to 4000 o K). 99 0.09 Design boy location number Fig. 10 Calculate heat fluxes over the top an bottom surfaces of DB at 6 locations (Z= 0.1m) when the heater temperature is set to optimum temperature (T H =1464 o K)
74 Int J Avance Design an Manufacturing Technology, Vol. 8/ o. 3/ September 2015 4 COCLUSIO This stuy emonstrates the successful application of combination of the MCM an GA for esign an optimization of the 3-D raiant furnaces to satisfy the uniform thermal conitions on the surfaces of the 3-D DBs. Effects of ifferent parameters incluing the heater temperature an location of the DB on achieving the uniform thermal conitions on the DB surfaces were investigate. It was shown that while the temperature of the heater is constant, heat fluxes over the DB surfaces vary with changing the location of the DB. Hence, the GA was use in conjunction with the MCM to fin the optimal location for the DB as well as the temperature of the heater to satisfy the uniform heat flux an temperature istributions on the DB surfaces. Results showe that the calculate heat fluxes on the DB surfaces are in goo agreement with the esire one. It was conclue that the GA-MCM is an efficient combine metho to optimize the 3-D raiant furnaces which can be use by esigners to fin a prior knowlege about the suitable temperature of the heater an the optimal location of a prouct to provie the uniform thermal conitions on its surfaces. 5 OMECLATURE D F F e q c q TH Greek symbols i REFERECES Raiation istribution factor between surfaces i an j Raiation shape factor between surfaces i an j Objective function umber of esign boy surfaces umber of enclosure surfaces Heat Flux over esign surfaces Calculate heat fluxes Temperature of heater Stefan-Boltzman constant Emissivity of surface i Kronecker elta [1] Erturk, H., Ezekoye, O. A., an Howell, J. R., Comparison of three regularize solution techniques in a three-imensional inverse raiation problem, J. Quant. Spectrosc. Raiat. Transfer, Vol. 73, 2002, pp. 307-316. [2] Daun, K., França, F., Larsen, M., Leuc, G., an Howell, J. R., Comparison of methos for inverse esign of raiant enclosures, J. Heat Transfer, Vol. 128, 2005, pp. 269-282. [3] Liu, D., Wang, F., Yan, J. H., Huang, Q. X., Chi, Y., an Cen, K. F., Inverse raiation problem of temperature fiel in three-imensional rectangular Enclosure containing inhomogeneous, anisotropically scattering meia, Int. J. Heat Mass Transfer, Vol. 51, 2008, pp. 3434-3441. [4] Bayat,., Mehraban, S., an Sarvari, S. M., Inverse bounary esign of a raiant furnace with iffuse-spectral esign, Int. Commun. Heat Mass Transfer, Vol. 37, 2010, pp. 103-110. [5] Lee, K. H., Beak, S. W., an Kim, K. W., Inverse raiation analysis using repulsive particle swarm optimization algorithm, Int. J. Heat Mass Transfer, Vol. 51, 2008, pp. 2772-2783. [6] Safavineja, A., Mansouri, S. H., Sakurai, A., an Maruyama, S., Optimal number an location of heaters in 2-D raiant enclosures compose of specular an iffuse surfaces using micro-genetic algorithm, Appl. Therm. Eng., Vol. 29, 2009, pp. 1075-1085. [7] Safavineja, A., Mansouri, S. H., an Hosseini Sarvari, S. M., Inverse bounary esign of 2-D raiant enclosures with absorbing-emitting meia using micro-genetic algorithm, IASME Transaction, Vol. 2, 2005, pp. 1558-1567. [8] Chopae, R. P., Mishra, S. C., Mahanta, P., Maruyama, S., an Komiya, A., Uniform thermal conitions on 3-D objects: Optimal power estimation of panel heaters in a 3-D raiant enclosure, Int. J. Therm. Sci., Vol. 51, 2012, pp. 63-76. [9] Chopae, R. P., Mishra, S. C., Mahanta, P., an Maruyama, S., Estimation of power of heaters in a raiant furnace for uniform thermal conitions on 3-D irregular shape objects, Int. J. Heat Mass Transfer, Vol. 55, 2012, pp. 4340-4351. [10] Liu, F. B., An application of particle swarm optimization to ientify the temperature-epenent heat capacity, Heat an Mass Transfer, Vol. 48, 2012, pp. 99-107. [11] Sencan Sahin, A., Kılıc, B., an Kılıc, U., Optimization of heat pump using fuzzy logic an genetic algorithm, Heat an Mass Transfer, Vol. 47, 2011, pp. 1553-1560. [12] Moparthi, A., Das, R., Uppaluri, R., an Mishra, S. C., Optimization of Heat Fluxes on the Heater an the Design Surfaces of a Raiating-Conucting Meium, umerical Heat Transfer, Part A, Vol. 56, 2009, pp. 846-860. [13] Chopae, R. P., Mishra, S. C., Mahanta, P., Maruyama, S., umerical Analysis of an Inverse Bounary Design Problem of a 3-D Raiant Furnace with a 3-D Design Object, umerical Heat Transfer, A, Vol. 60, 2011, pp. 25-49. [14] Amiri, H., Mansouri, H., an Coelho, P. J., Inverse Optimal Design of Raiant Enclosures with Participating Meia: A Parametric Stuy, Heat Transfer Engineering, Vol. 34, o. 4, pp. 288-302, 2013. [15] Gosselin, L., Tye-Gingras, M., an Mathieu-Potvin, F., Review of utilization of genetic algorithms in
Int J Avance Design an Manufacturing Technology, Vol. 8/ o. 3/ September - 2015 75 heat transfer problems, International Journal of Heat an Mass Transfer, Vol. 52, 2009, pp. 2169-2188. [16] Howell, J. R., The Monte Carlo metho in raiative heat transfer, J. Heat Transfer, Vol. 120, 1998, pp. 547-560. [17] Moest, M. F., Raiative Heat Transfer, secon e., Acaemic Press, USA, 2003, pp. 644-676. [18] Mahan, J. R., Raiation Heat Transfer-A Statistical Approach, John Wiley & Sons, ew York, 2002, pp.305 333. [19] Siegel, R., Howell, J. G., Thermal Raiation Heat Transfer, fourth e., McGraw-Hill, ew York, 1972, pp. 207-248. [20] Haupt, S. E., Haupt, R. L., Practical genetic algorithm, secon e., pp. 51-65, John Wiley, ew Jersey, 2004.