Entropy Generation Minimization of Pin Fin Heat Sinks by Means of Metaheuristic Methods

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1 Indan Journal of Scence and Technology Entropy Generaton Mnmzaton of Pn Fn Heat Snks by Means of Metaheurstc Methods Amr Jafary Moghaddam * and Syfollah Saedodn Department of Mechancal Engneerng, Semnan Branch, Islamc Azad Unversty, Semnan, Iran; amrjafary45@yahoo.com; S_Sadodn@ust.ac.r Abstract In ths paper utlzaton of metaheurstc methods such as genetc algorthm and partcle swarm optmzaton for determnng desgn parameters of pn fn type heat snks through mnmzng generated entropy s dscussed. Ths paper follows recent studes on entropy generaton mnmzaton of plate type heat snks by evolutonary optmzaton methods. Keywords: Pn Fn Heat Snk, Entropy Generaton Mnmzaton, Metaheurstc Algorthms.. Introducton Heat snks as coolng devces are used for heat transfer especally n electronc crcuts. There are dfferent structures and geometres of heat snks whch have dfferent advantages of frmness, compactness and heat transfer qualty. Most studed and desgned heat snks for electronc devces have been of plate fn geometry [4]. A plate fn heat snk s shown n Fgure. To acheve good performance of plate fn heat snks, entropy generaton mnmzaton s used n the paper of Culham and Muzychka [4], whch ntroduces smultaneous optmzaton of geometrc parameters, heat dsspaton, used materal and flud flow condtons. Utlzaton of computatonal flud dynamcs for plate fn heat snk desgn s presented n []. A smple practcal model for desgnng the plate fn heat snk as also proposed [6]. In recent years metaheurstc methods have been ncreasngly used for the problem of heat snk optmum desgn. Utlzaton of genetc algorthms [3], partcle swarm optmzaton [7], mperalst compettve algorthm [8], and harmony search [9] for example have been proposed to fnd optmum values of parameters for plate fn heat snks. Another geometry for heat snks s pn fn geometry, for whch pns of dfferent cross secton could be assumed (Fgure 2). In Yu et al. [2] expermental study for comparson of fully plate fn heat snk and hybrd platepn fn heat snk s presented. Selecton of pns cross secton by CFD smulatons and optmzaton has been proposed [2]. Entropy generaton mnmzaton for pn fn heat snks s proposed n [8], and genetc algorthm based optmzaton of them s also presented n Hajabdollah et al. [6] by means of thermal modelng. In ths paper dfferent methods of metaheurstc optmzaton are used for mnmzaton of generated entropy n heat transfer through pn fn heat snk by determnng optmum parameters of ts geometry. Prevous studes have been done for utlzaton of some of these methods for plate fn heat snks. The am of ths paper s to use two methods for pn fn heat snks also, and then comparng the results. Usng metaheurstc methods lke GA * Correspondng author: Amr Jafary Moghaddam (amrjafary45@yahoo.com)

2 Amr Jafary Moghaddam and Syfollah Sadodn 4887 and PSO s a general way to overcome hard optmzaton problems. In contrast to other case specfc methods, these methods are based on smple prncples and could be adapted to many problems wth hghest generalzaton. In secton 2 of ths paper, the concept of entropy generaton mnmzaton s ntroduced and modelng of pn fn heat snk entropy generaton rate wll be presented. In secton 3 genetc algorthms are dscussed. PSO and ts modfcatons s the subject of secton 4 and mperalst compettve algorthm s presented n secton 5. In secton 6 Fgure. Heat snk wth plate fn geometry. the mentoned metaheurstc methods are used for determnng optmum parameters of pn fn heat snk through entropy generaton mnmzaton. Fnally the results are dscussed and compared. 2. Entropy Generaton Mnmzaton Method of entropy generaton mnmzaton s ntroduced and developed by Bejan [2], to combne concepts of flud dynamcs and heat transfer for systems and models n whch both aspects are present. Ths method s useful for fnte-sze and fnte-tme systems whch are not consdered as deal nfnte thermodynamc systems. The method s applcable n dfferent domans of engneerng and scence such as low-temperature devces, heat transfer n electronc devces, and power plants. The concept s based on assumpton of a systemenvronment confguraton depcted n Fgure 3. In ths confguraton, the system has mass of M, energy of E and entropy of S at the moment, has an nput mass transfer rate m n and an output mass transfer rate m out, and has heat transfer wth n+ temperature reservors wth heat transfer rates ( QQ,,..., Q n ). Also the net work done on system s W. The frst and second laws of thermodynamcs for ths system are n the forms of equaton and 2 respectvely de dt n Q W + mh mh n out () S gen ds n Q ms ms dt T n out (2) Fgure 2. Heat snk wth pn fn geometry. In whch h s the sum of enthalpy and energy of boundary stream. The method of entropy generaton mnmzaton wll be the process of makng the equaton 2 to be ts mnmum possble value. Elmnatng Q from equatons and 2, and lettng S g, one obtans the exergy analyss relaton d W dt E T S n T T Q m h T S rev ( ) + + n mh ( TS out ) We have also the relaton W W TS (4) rev gen (3) Vol 6 (7) July 23

3 4888 Entropy Generaton Mnmzaton of Pn Fn Heat Snks by Means of Metaheurstc Methods Fgure 3. The system-envronment confguraton assumed for defnton of entropy generaton mnmzaton method [2]. two-dmensonal steady lamnar flow wth unform velocty normal to pns axs and wthout bypass, and adabatc fn tps. For the heat snk depcted schematcally n Fgure 4, one obtans the entropy generaton rate as S gen Q m P hs T + r T amb 2 amb (5) In equaton 5, the resstance of heat snk s n whch the bulk materal resstance s hs m + fns (6) m tbp (7) ka Fgure 4. The schematc structure of pn fn heat snk. whch tells us the relaton of destroyed power and entropy generaton rate. So t s necessary to mnmze entropy generaton rate to gan best workng performance of systems. To desgn pn fn heat snk, n Khan et al. [8] the modelng of system for the am of entropy generaton mnmzaton s developed. In ths paper ths modelng procedure s followed. The assumptons of ths model are: Isotropc materal, ncompressble and unform flud, and resstance of fns s fns N + + for whch we have c fn bp c fn bp ha C C h A h h A fn fn fn bp bp (8) (9) Vol 6 (7) July 23

4 Amr Jafary Moghaddam and Syfollah Sadodn 4889 defned usng h fn m tanh( mh) mh 4h fn kd () For an nlne array of cylndrcal fns a dmensonless heat transfer coeffcent s defned as Nu Dfn hfnd C e 2 3 DPr () v and C s determned by geometrc parameters as C ST ST SL. + exp. 5. (2) S T For base plate of heat snk, heat transfer coeffcent s also shown that to be Nu L hbpl 75 L 2. e Pr 3 (3) v Mass flow rate and pressure dfference are calculated by m ru N S HD a T T P f ru 2 / 2N max L (4) The pressure drop s also related to structural parameters as n whch f K K S T S. 9 S The maxmum flow velocty s T L e e D D S S T T Umax max Ua, Ua ST SD (5) (6) (7) wth defnton of S D ST SL (8) The resultng optmzaton problem s a nonlnear one, n whch the optmum values of heat snk parameters are to be determned subject to constrants on assumed values for some of system temperatures and szes. So t s useful to utlze metaheurstc methods to solve ths nonlnear problem. In the next sectons some of such methods are dscussed. 3. Genetc Algorthm (GA) One of the most well-known metaheurstc optmzaton methods s genetc algorthm whch s based on a model of evoluton of chromosomes [5]. GA s used n many applcatons of engneerng n recent years []. Ths algorthm selects the most fttng solutons of a problem due to a predefned ftness functon. As n evolutonary theory the best ndvduals n a generaton are the most probable to survve and reproduce. In GA, next generaton of solutons (chromosomes) are determned by means of some basc operatons on chromosomes of current generaton. The crossover operaton makes new ndvduals by combnng some parts of a soluton wth some parts from another soluton, and t may be consdered as a model for reproducton. Another operaton s the mutaton n whch values of some elements of some chromosomes wll change randomly to another values; as a model for mutatons of genes n nature. It s also possble for some of the most fttng chromosomes to survve and be present n next generaton drectly. The man procedure of GA s as follows:. An ntal set of chromosomes (possble solutons of the problem) are set as ntal populaton. Each chromosome s a vector of numbers (genes), and each of ths numbers defnes one of optmzaton varables. 2. For all ndvduals n populaton, the ftness s calculated by ftness functon of problem. Then ndvduals are sorted due to ther ftness values. 3. Probabltes of crossover, mutaton and drect transfer are determned from ftness values for each ndvdual. 4. In crossover process, some pars of ndvduals are selected (wth a selecton strategy) and make new ndvduals (offsprngs) as shown n Fgure 5. Vol 6 (7) July 23

5 489 Entropy Generaton Mnmzaton of Pn Fn Heat Snks by Means of Metaheurstc Methods Fgure 5. Crossover operaton n GA. 5. In mutaton process, some ndvduals are selected and some of ther elements are changed randomly. 6. Some of most fttng ndvduals are selected to go to the next generaton drectly and wth no change. 7. New generaton s made from crossover, mutaton and drect transfer. Then the procedure from stage 2 s terated for new generaton. By the teratve procedure mentoned above, the new generatons wll get better gradually. After some number of teratons the populaton converges to optmum values. The flowchart for man operatons n GA s shown n Fgure Partcle Swarm Optmzaton (PSO) Partcle Swarm Optmzaton s another well-known metaheurstc optmzaton method whch s ntroduced n Kennedy and Eberhart [7] by modelng the movements of a swarm n a search space. The possble solutons of the problem are set to be the possble postons of partcles n the search space, and the ftness functon determnes the ftness of each poston. The partcles have nformaton sharng about ther best experenced poston. There s a memory for swarm to mantan the best experenced poston from all partcles n the swarm. Also, each partcle has ts own memory for ts best experenced poston. The velocty of a partcle s determned from the sum of vectors from partcle poston ponted to best postons (Fgure 7). The new poston of partcle wll be determned from ths velocty and the procedure of updatng the partcle s Fgure 6. Flowchart of GA. velocty and poston s terated. The man procedure of PSO s as follows:. Intal postons and veloctes of partcles n search space are set randomly. For each partcle the ftness of ts poston s determned by ftness functon. The best fttng poston among all swarm s set as global best poston (gbest), and for -th partcle, ts poston s set for personal best poston (pbest ). Vol 6 (7) July 23

6 Amr Jafary Moghaddam and Syfollah Sadodn 489 Fgure 7. Determnng the velocty of partcle n PSO. 2. Velocty of -th partcle s updated as equaton (9) V ( t+ ) IV ( t) + rc gbest x ( t) + rc 2 2 pbest x( t), (9) where I s nerta factor, r and r 2 are random numbers, and C and C 2 are constant coeffcents. 3. Poston of -th partcle s updated also, as equaton (2) x( t+ ) x( t) + d V ( t+ ), (2) n whch the δ determnes the tme step. 4. Ftness values for the new postons are calculated by ftness functon, gbest and pbest values are updated, and the procedure s terated from stage 2. After enough numbers of teratons, the swarm converges to the poston wth optmum ftness. Ths best poston represents the optmum values for optmzaton varables. The flowchart for conventonal PSO algorthm s depcted n Fgure 8. Some modfcatons of PSO are proposed n lterature. In Mult-Populaton PSO (MPSO), the whole swarm s dvded to some subpopulatons, whch are evolved separately or have some sorts of nformaton sharng and mgratons of partcles from subpopulatons to another ones []. In Fuzzy PSO (FPSO), man equatons of PSO are consdered to have fuzzy varables [4]. Chaotc and Quantum PSO (CPSO and QPSO) are modfcatons of PSO wth local search propertes [5, 9]. PSO s used n many applcatons of engneerng n recent years [3]. 5. esults and Dscusson In ths secton the results of entropy generaton mnmzaton of pn fn heat snk by GA and PSO algorthms are presented. The man task s to fnd desgn parameters D, H, Fgure 8. Flowchart of PSO. U a, N T and N L so the entropy generaton rate of equaton (5) be n ts mnmum value. For GA the chromosome s set to be a vector of numbers each presentng one of the mentoned desgn varables. For PSO the desgn varables are coordnates values of partcle postons. For sake of comparson both populaton szes n GA and PSO are set wth same value and equal to 25, and maxmum teratons of algorthms are set to Vol 6 (7) July 23

7 4892 Entropy Generaton Mnmzaton of Pn Fn Heat Snks by Means of Metaheurstc Methods The convergence curves of GA and PSO, -whch show the mnmum entropy generaton rate n each teraton- are shown n Fgures 9 and respectvely. Both GA and PSO have found optmum solutons for desgn parameters fnally, but PSO have faster behavor, both n computatonal tme and n fndng the optmal soluton. The results are summarzed n Table. 6. Concluson In ths paper metaheurstc methods of GA and PSO are utlzed for desgnng pn fn heat snk wth objectve of mnmzaton of generated entropy. Both methods could fnd optmum solutons for desgn varables properly, but PSO was faster n fndng the soluton and n total elapsed computaton tme. esults from ths study showed that Fgure. Convergence curve for PSO. Table. esults of entropy generaton mnmzaton wth GA and PSO Optmum desgn (D, H, U a, N T, N L ) Mnmum Entropy Generaton rate Total Elapsed computaton tme Iteraton of fndng the optmum Fgure 9. GA (4mm, 2mm, 5m/s,, ) Convergence curve for GA. PSO (4mm, 2mm, 5m/s,, ).48 W/K.48 W/K 5.8 sec 5.73 sec 35 5 usng metaheurstc methods one can fnd more sutable desgns for pn fn geometry of heat snks whch leads to more effcency of electronc devces, and lower rsks of falures and damages due to generated heat. 7. eferences. Aref, A., Davoud, M., & Davoud, M. (22). Optmal placement and estmaton of DG capacty n dstrbuton network s usng Genetc Algorthm-based method. Indan Journal of Scence and Technology, 5(3), Bejan, A. (996). Entropy generaton mnmzaton: The new thermodynamcs of fnte sze devces and fnte tme processes. Journal of Appled Physcs, 79(3), Boroujen, S. M. S., Hemmat,., Delafkar, H., & Boroujen, A. S. (2). Optmal PID power system stablzer tunng based on partcle swarm optmzaton. Indan J. Sc. Technol, 4(4), Culham, J.., & Muzychka, Y. S. (2). Optmzaton of plate fn heat snks usng entropy generaton mnmzaton. Components and Packagng Technologes, IEEE Transactons on, 24(2), Goldberg, D. E. (989). Genetc algorthms n search, optmzaton, and machne learnng. 6. Hajabdollah, F., afsanjan, H. H., Hajabdollah, Z., & Hamd, Y. (22). Mult-objectve optmzaton of pn fn to determne the optmal fn geometry usng genetc algorthm. Appled Mathematcal Modellng, 36(), Kennedy, J., & Eberhart,. (995, November). Partcle swarm optmzaton. In Neural Networks, 995. Proceedngs., IEEE Internatonal Conference on (Vol. 4, pp ). IEEE. 8. Khan, W. A., Culham, J.., & Yovanovch, M. M. (25). Optmzaton of pn-fn heat snks usng entropy generaton Vol 6 (7) July 23

8 Amr Jafary Moghaddam and Syfollah Sadodn 4893 mnmzaton. Components and Packagng Technologes, IEEE Transactons on, 28(2), Lu, B., Wang, L., Jn, Y. H., Tang, F., & Huang, D. X. (25). Improved partcle swarm optmzaton combned wth chaos. Chaos, Soltons & Fractals, 25(5), Nu, B., Zhu, Y., & He, X. (25). Mult-populaton cooperatve partcle swarm optmzaton. Advances n Artfcal Lfe, Park, K., Oh, P. K., & Lm, H. J. (26). The applcaton of the CFD and Krgng method to an optmzaton of heat snk. Internatonal journal of heat and mass transfer, 49(9), Saht, N., Durst, F., & Gerema, P. (27). Selecton and optmzaton of pn cross-sectons for electroncs coolng. Appled thermal engneerng, 27(), Sanaye, S., & Hajabdollah, H. (2). Thermal-economc mult-objectve optmzaton of plate fn heat exchanger usng genetc algorthm. Appled Energy, 87(6), Sh, Y., & Eberhart,. C. (2). Fuzzy adaptve partcle swarm optmzaton. In Evolutonary Computaton, 2. Proceedngs of the 2 Congress on (Vol., pp. 6). IEEE. 5. Sun, J., Feng, B., & Xu, W. (24, June). Partcle swarm optmzaton wth partcles havng quantum behavor. In Evolutonary Computaton, 24. CEC24. Congress on (Vol., pp ). IEEE. 6. Wu, H. H., Hsao, Y. Y., Huang, H. S., Tang, P. H., & Chen, S. L. (2). A practcal plate-fn heat snk model. Appled Thermal Engneerng, 3(5), Yousef, M., Enayatfar,., & Darus, A. N. (2a). Optmal desgn of plate-fn heat exchangers by a hybrd evolutonary algorthm. Internatonal Communcatons n Heat and Mass Transfer. 8. Yousef, M., Darus, A. N., & Mohammad, H. (2b). Second law based optmzaton of a plate fn heat exchanger usng Imperalst Compettve Algorthm. 9. Yousef, M., Enayatfar,., Darus, A. N., & Abdullah, A. H. (22). Optmzaton of Plate-fn heat exchangers by an mproved harmony search algorthm. Appled Thermal Engneerng. 2. Yu, X., Feng, J., Feng, Q., & Wang, Q. (25). Development of a plate-pn fn heat snk and ts performance comparsons wth a plate fn heat snk. Appled thermal engneerng, 25(2), Vol 6 (7) July 23