Geometric optimization of micro-thermoacoustic cooler for heat management in electronics

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1 Geometric otimization of micro-thermoacoutic cooler for heat management in electronic K.artibu, B.Sun and M.A.E. Kaunda Deartment of Mechanical Engineering Cae Peninula Univerity of echnology Box 65 Cae own 8000, South Africa Abtract A a reult of miniaturiation, electronic roduct are hrining in ize and weight but with greater reure for cot reduction. eat fluxe have increaed coniderably and hence thermal management become crucial from the reliability oint of view. hermoacoutic heat engine rovide a ractical olution to the roblem of heat management in microcircuit where they can be ued to um heat or roduce ot cooling of ecific circuit element. owever, the mot inhibiting characteritic of thermoacoutic cooling i it current lac of efficiency. A multiobjective otimization aroach i reented to model and otimize a mall-cale thermoacoutic regenerator. Otimization of multile objective comonent i conidered and global otimal olution have been identified uing the eilon contraint method. refrigerator. he reence of a tac rovide heat exchange with the ound field and the generation or abortion of acoutic ower. With a uitable geometry ubtantial amount of heat can be moved a demontrated, for examle, by Garrett and ofler [7]. An intereting and imortant feature of uch engine i that the erformance deend on geometric factor and ga arameter [8]. I. INRODUCION hermal management ha alway been a concern for comuter ytem and other electronic. Comutational eed will alway be limited by the amount of noie roduced by comuter chi. Since mot noie i generated by wated heat, comuter comonent and other emiconductor device oerate fater and more efficiently at lower temerature [1]. he need to manage heat fluxe of order W/cm and higher in microcircuit ha emhaized the imortance of develoing device which can coe with uch heat level. Many intereting device have been rooed for uch alication. hey range from forced convection cooling device to thermoelectric device, heat ie, liquid coolant, and evaorative ray cooling device []. ere thermoacoutic device are rooed (Fig. 1); uch engine can convert heat to ound or ue ound to um heat. hey are a new alication in the area of thermal management but baed on their erformance and adatability to microcircuit they how much romie [3]. If thermoacoutic cooling device could be caled for comuter alication, the electronic indutry would realize longer lifetime for microchi, increaed eed and caacity for telecommunication, a well a reduced energy cot [4]. he baic mechanic behind thermoacoutic are already well undertood. A detailed exlanation of the way thermoacoutic cooler wor i given by Swift [5] and Wheatly et al. [6]. Reearch i focuing on otimizing the method o that thermoacoutic cooler can comete with commercial Fig. 1. (a) Acoutic ot-cooler interfaced with circuit (b) Prime mover interfaced with circuit [3] Otimization technique a a deign ulement are everely under-utilized, and reviou effort in the otimization of thermoacoutic device are rare. Minner et al. [9], Wetzel [10], Benoin [11] and ijani et al. [1] utilized a linear aroach while trying to otimize the device. Additionally, mot tudie (the excetion being the Minner et al. tudy) vary only a ingle arameter, holding all ele fixed and ignored thermal loe to the urrounding. hee Parametric tudie are unable to cature the nonlinear interaction inherent in thermoacoutic model with multile variable, and can only guarantee locally otimal olution. Zin et al [13] and ra et al. [14] illutrate the otimization of thermoacoutic ytem, while taing into account thermal loe to the urrounding that are tyically diregarded. hey ue mathematical analyi and otimization and illutrate the conflicting nature of objective comonent conidered in their modeling aroach. In ite of their introductory nature, the reented wor are imortant contribution to thermoacoutic a it merge the theoretical otimization aroach with thermal invetigation in thermoacoutic. herefore ince everal conflicting objective have been identified, an effort to effectively imlement the eilon contraint method for roducing the Pareto otimal olution in a multiobjective otimization mathematical rogramming method i carried out in our aroach. hi ha been imlemented in the widely ued modeling language /13/$ IEEE 57

2 GAMS [15] (General Algebraic Modeling anguage, A a reult, Gam code are written to define, to analyze, and olve otimization roblem to generate et of Pareto otimal olution unlie reviou tudie. he remainder of thi aer i organized in the following fahion: the fundamental comonent of our mathematical model characterizing the tanding wave thermoacoutic heat engine are reented in Section. Section 3 illutrate our otimization rocedure. Section 4 conider multiobjective otimization uing eilon contraint method. In Section 4, we conclude by uggeting oible future extenion of thi wor. weighting factor emhai. w i to rovide aroriate uer-defined II. MODEING APPROAC In thi ection, our modelling aroach for the hyical tanding wave engine deicted in Fig. i dicued; the develoment of our mathematical model and it correonding otimization i included. he roblem i reduced to a two dimenional domain, becaue of the ymmetry reent in the tac. wo contant temerature boundarie are conidered namely one convective boundary and one adiabatic boundary, a hown in Fig.. For our model, only the regenerator geometry i conidered; the model conider variation in oerating condition and the interdeendency of tac location and geometry. Five different arameter are conidered to characterize the regenerator: : Stac length, : tac height, Za: tac lacement (with Za=0 correonding to the cloed end of the reonator tube), d: channel diameter, and N: number of channel. hoe arameter have been allowed to vary imultaneouly. Five different objective a decribed by ra et al. [14] namely two acoutic objective (Acoutic wor W of the thermoacoutic engine and vicou reitance R V through the regenerator [16]) and three thermal objective (convective heat flow Q conv, radiative heat flow Q rad, and conductive heat flow Q cond ) are conidered to meaure the quality of a given et of variable value that atifie all of the contraint. Becaue wor i the only objective to be maximized, we intead minimize it negative magnitude along with all of the other comonent. Ultimately, otimizing the reulting roblem generate otimal objective function value [ ] = W, R V, Q conv, Q rad, Q cond olution [ ] =,, d, Za, N G and otimal x. Since the five objective are conflicting in nature [14], a multiobjective otimization aroach ha been ued. hoe objective are conflicting in the ene that, if otimized individually, they do not hare the ame otimal olution. Since we otimize multile objective comonent imultaneouly, each objective comonent ha been given a Fig.. Comutational domain According to wang and Maud [17], the method for olving multiobjective mathematical rogramming roblem can be claified into three categorie, baed on the hae in which the deciion maer involve in the deciion maing roce exreing hi/her reference: the a riori method, the interactive method and the a oteriori or generation method. he a oteriori (or generation) method give the whole icture (i.e. the Pareto et) to the deciion maer, before hi/her final choice, reinforcing thu, hi/her confidence to the final deciion. In general, the mot widely ued generation method are the weighting method and the ε -contraint method. hee method can rovide a rereentative ubet of the Pareto et which in mot cae i adequate. he baic te toward further enetration of the generation method in our multiobjective mathematical roblem i to rovide aroriate code in a Gam environment and roduce efficient olution. III. IUSRAION OF E OPIMIZAION PROCEDURE OF E REGENERAOR he five variable,, d, Za, N may only tae value within the certain lower and uer bound. he feaible domain for a thermoacoutic regenerator i defined a follow: min max min max d min d d max (1) Za min Za Za max N min N N max,, d, Za R + and N Ζ + Additionally, the total number of channel N of a given diameter d i limited by the cro-ectional radiu of the reonance tube. herefore the following contraint relation can be determined: N( d + t w ) () where t w rereent the wall thicne around a ingle 58

3 channel. he relation between the tac erimeter and the cro ectional area A a determined by Swift [16] i given by: A = (3) d + tw he following boundary condition mut alo be enforced: 1. Contant hot ide temerature ( h ),. Contant cold ide temerature ( C ), 3. Adiabatic boundary, modeling the central axi of the cylindrical tac: = 0; (4) r r= 0 4. Free convection and radiation to urrounding (at ) with temerature deendent heat tranfer coefficient ( h ), emiivity ε, and thermal conductivity ( ): r r= = h 4 4 ( ) + ε ( ) S b S he acoutic ower er channel ha been derived by Swift [15]. he following equation can be derived for N channel: (5) ( γ 1) ( ) ( ) crit. 1 δ vρu c 1+ ε π W = ωn δ ( ) d + t w ρ (6) he thermal enetration deth δ, the vicou enetration deth δ v and the critical temerature are given by the following equation: K δ = (7) ρ c ω δ V = m μ ρ ω m ω1 mc u1 crit. = (9) ρ With K being the thermal conductivity, ρ m the mean denity, c the contant reure ecific heat, μ i the diffuivity of the woring fluid and ω the oerating frequency. he amlitude of the dynamic reure and ga velocity u due to the tanding wave in the tube are given by: πza = max co (10) λ πza u = u max in λ (11) max with u max = ρ.c (1) he heat caacity ratio can be exreed by [16]: ( ρc δ ) g tanh( ( i + 1) y 0 / δ ) ε = (13) ρc δ tanh i + 1 l / δ ( ) (( ) ) (8) hi exreion can be imlified to value of ε = y 0 / δ if y0 / δ < 1 and ε = 1if y0 / δ > 1 [13], where y 0 half of the channel height i, l i half of the wall thicne and δ i the olid thermal enetration deth. Jut a the total acoutic ower of the tac wa deendent on the total number of channel, the vicou reitance alo deend on hi value. he following equation can be derived: μ μ R V = = (14) A Cδ V N δ V ( d + t w ) π N he heat tranfer coefficient and the heat flux to the urrounding were etimated uing a linear temerature rofile. In thi model, the actual temerature ditribution throughout the tac i taen into account by utilizing MAAB finite element toolbox [18], which cature the temerature deendence of the heat tranfer coefficient. Only the temerature ditribution at the hell urface and the temerature gradient at the cold ide are of interet. ra et al. [13] have derived the final urface temerature ditribution a a function of axial direction Za. It i given by: S C Za ln h he = (15) he convective heat tranfer coefficient and the radiative heat flux to the urrounding are aumed to be deendent on the temerature. he total convective heat tranfer acro the cylindrical hell in it integral form can be decribed by: π ( ( x) )( ( x) ) Q = h dxdρ (16) conv 0 0 It i hown for the cae of a horizontal tube ubject to free convection [18], the heat tranfer coefficient h i derived from the Nuelt number, which i a non-dimenional heat tranfer coefficient a follow: h = (17) g ( ) Nu 1 4 D Nu = Ra Pr (18) hi exreion deend on the Prandtl number, which can be exreed by: ν Pr = α (19) 3 gβ( S ) 8 Ra = να (0) where Pr i the Prandtl number, S i the urface temerature, i the (contant) temerature of the urrounding, ν i the vicoity of the urrounding ga, and α i the thermal diffuivity of the urrounding ga (air). he temerature ditribution tated in Eq.15 i then ued to determine the convective heat tranfer to the urrounding. 59

4 After integrating we derive the following heat flow exreion: C Q conv = πh (1) C ln he radiation heat flux become increaingly imortant a increae, a hown in the following equation: π 4 4 ε( ( x) ) Q = dxdρ () rad Where B 0 0 B i the Stefan Boltzmann contant, and ε i the urface emiivity, which deend on the emitted wavelength, and in turn i a function of temerature. After integrating we derive the following heat flow exreion: 4 4 C 4 Q rad = π Bε (3) C 4 ln he conductive heat flux i rereentative of the heat lo acro the cold end of the domain. A the temerature gradient there i non-zero, a heat flux mut be reent. It i aumed that thermal energy i removed via the cooling water flow. Similar to the cylindrical hell, thi heat flux ha to be integrated over the whole urface rereenting the cold ide: π Q cond = zz drdρ (4) r 0 0 Where the value of the axial thermal conductivity zz i determined by the following equation: St w + gd zz = (5) t w + d he temerature ditribution i ued to determine the temerature gradient at the to urface Za, r =.he general tatement of Fourier law of thermal conduction ΔQ Δ exreed a = A i ued to determine the heat Δt Δx C C ln flow. herefore: = (6) z z= zz And Q cond = π C ln (7) C IV. EMPASIZING A OBJECIVE COMPONENS All the exreion involved in our mathematical model (MPF) have been reented in the reviou ection. ogether with the following exreion, they rereent a non-linear mixed integer rogram: ( MPF) min ξ = w1( W) + wr V + w3qconv + w4qrad + w5qcond,,z,d,n a (8) here i no ingle otimal olution that imultaneouly otimize all the two objective function. In thee cae, the deciion maer are looing for the mot referred olution. o find the mot referred olution of thi multiobjective model, we aly the augmented ε -contraint method (AUGMECON) a rooed by Mavrota [0]. he ε - contraint method ha everal imortant advantage over traditional weighted method. hee advantage are lited in [0]. In the conventional ε -contraint method, there i no guarantee that the obtained olution from the individual otimization of the objective function are Pareto otima or efficient olution. In other to overcome thi deficiency, the lexicograhic otimization for each objective function to contruct the ayoff table for the multiobjective mathematical rogramming (MMP) i rooed here in other to yield jut Pareto otimal olution. he mathematical detail of comuting ayoff table for MMP roblem can be found in [1]. he augmented ε -contraint method for olving model (Eq.35) can be hown a follow: ( ) 3 max F x e... r r3 r F = x e.t. ( ) ( x) 3 3 F 3 = e max F x = e x S and ( ) + i R (9) o illutrate our aroach, we conider the thermoacoutic coule (AC) a decribed in []. It conit of a arallellate tac laced in helium-filled reonator. All relevant arameter are given in able I and able II. ABE I: Secification for hermoacoutic coule Parameter Symbol Value Unit Ientroic coefficient ν 1.67 Ga denity ρ g/m 3 Secific heat caacity c J/g.K Dynamic vicoity μ g/m. Maximum velocity u max 670 m/ Maximum reure max Pa Seed of ound c 100 m/ hicne late t w M Frequency f 696 z hermal conductivity elium g 0.16 W/(m.K) hermal conductivity tainle teel Iobaric ecific heat caacity S 11.8 W/(m.K) c J/(g.K) 530

5 ABE II: Additional arameter ued for rogramming Parameter Symbol Value Unit emerature of the urrounding 98 K C 300 K 700 K Contant cold ide temerature Contant hot ide temerature Wavelength λ m hermal exanion β 1 / 1/K hermal diffuivity α.1117e-5 m-1 he following contraint (uer and lower bound) have been enforced on variable in other for the olver to carry out the earch of the otimal olution in thoe range:.lo = 0.005; Za.lo = 0.005;.lo = 0.005; d.lo >. δ.u = 0.05; ; d.u < 4. δ (30) We ue lexicograhic otimization for the ayoff table, the alication of model (Eq.9) will rovide only the Pareto otimal olution (lie in the reviou cae), avoiding the wealy Pareto otimal olution. Efficient olution of the rooed model have been found uing AUGMENCON method and the INDOGOBA olver. o ave comutational time, the early exit from the loo a rooed by Mavrota [0] ha been alied. he integer variable N ha been given value of 0 to 50. hi roce generate otimal olution correonding to each integer variable. he following ection reort only et of Pareto olution obtained: Fig.3. Otimal tructural variable hee otimal olution are then ued to contruct Figure 4 and Figure 5 rereenting reectively acoutic wor, vicou reitance, conductive, convective and radiative heat fluxe obtained for different value of N. For maximum erformance of the device, the acoutic ower ha to be maximized and the vicou reitance and thermal loe minimized. he conflicting nature of the five objective can be oberved in thoe rofile. For intance, while the acoutic ower, the vicou reitance, the convective and the radiative heat fluxe decreae between 6 and 31 channel, the conductive heat flux increae (with *, d*, * and Za* given reectively in able III). ABE III: Non-dominated olution found by AUGMENCON N d Za CPU time () Figure 3 rereent the Pareto olution grahically; it how that for a ecific number of channel correond a ecific tac height, a ecific tac oition and a ecific acing between late. here i not only a ingle otimal olution that otimize the geometry of the regenerator and highlight the fact that the geometrical arameter are interdeendent, which uort the ue of a multiobjective aroach for otimization. Fig.4. Acoutic ower, vicou reitance lotted a a function of N 531

6 Powered by CPDF ( Fig.5. Conductive, convective and radiative heat fluxe lotted a a function of N V. CONCUSION In order for a thermoacoutic engine to be cometitive on the current maret, they have to be otimized in order to imrove it overall erformance. Previou tudie have relied heavily uon arametric tudie. hi wor target the geometry of the thermoacoutic regenerator and ue multiobjective otimization aroach to find the otimal et of geometrical arameter that otimize the device. Five different objective have been identified; a weight ha been given to each of them to allow the deigner to lace deired emhai. Mixed-integer nonlinear rogramming for thermoacoutic regenerator ha been imlemented in Gam. For the cae of multiobjective otimization, a nonlinear rogramming for thermoacoutic regenerator ha been imlemented in Gam. We have alied an imroved verion of a well-nown multiobjective olution method, i.e., the eilon contraint method called augmented eilon contraint method (AUGMENCON). he reult found how the interdeendence between the geometrical arameter of the regenerator which uort the ue of our multiobjective aroach to otimize the geometry of thermoacoutic engine. Although the efficient olution of rooed model could be found uing AUGMENCON method by uing commercial otimization olver INDOGOBA, it hould be noted that the correonding comutational time grow exonentially with roblem ize. herefore, develoing heuritic or metaheuritic olution method could be of great interet. ACKNOWEDGMEN hi reearch wa uorted by the Deartment of mechanical engineering at the Cae Peninula Univerity of echnology, Cae own, South Africa. electronic", hermomechanical Phenomena in Electronic Sytem - Proceeding of the Interociety Conference,. 41. [4] Garrett, S.., ofler,.j., Perin, D.K. hermoacoutic refrigeration. Refrigeration and air conditioning technology worho. June 3-5, [5] Swift G.W hermoacoutic engine. J Acout Soc Am, vol. 4, [6] Wheatley J.C, ofler, Swift G.W, Migliori A Undertanding ome imle henomena in thermoacoutic with alication to acoutical heat engine. Am J Phy, vol. 53, [7] Garrett, S. and ofler,. hermoacoutic Refrigeration, echnology 001. NASA Conference Publication 3136, vol., 1991, [8] Swift, G.W. 1995, "hermo acoutic engine and refrigerator", Phyic oday, vol. 48, no. 7,. -8. [9] Minner, B.., Braun, J.E. & Mongeau,.G. 1997, "heoretical evaluation of the otimal erformance of a thermoacoutic refrigerator", ASRAE ranaction, [10] Wetzel, M. & erman, C. 1997, "Deign otimization of thermoacoutic refrigerator", International Journal of Refrigeration, vol. 0, no. 1, [11] Benoin, E. Numerical tudy of thermoacoutic heat exchanger, Ph.D. thei, John oin Univerity (001). [1] ijani, M.E.., Zeeger, J.C.. & De Waele, A..A.M. 00, "he otimal tac acing for thermoacoutic refrigeration", Journal of the Acoutical Society of America, vol. 11, no. 1, [13] Zin, F., Waterer,., Archer, R. & Schaefer,. 009, "Geometric otimization of a thermoacoutic regenerator", International Journal of hermal Science, vol. 48, no. 1, [14] ra, A.C., Zin, F., Prooyev, O.A., Schaefer, hermoacoutic heat engine modeling and deign otimization. Journal of Alied hermal Engineering, vol. 31, [15] Generalized Algebraic modelling Sytem, (GAMS), [online] available: htt:// [16] Swift, G.W, hermoacoutic: a unifying erective for ome engine and refrigerator. Acoutical ociety of America, Melville NY, 00. [17] wang, C.., Maud, A Multile Objective Deciion Maing. Method and Alication: A tate of the art urvey. ecture Note in Economic and Mathematical Sytem Vol Sringer-Verlag, Berlin. [18] he MathWor, Inc., MAAB Uer Guide. he Math Wor, Inc., 007. [19] Baehr,.D., Stehan, K. W a rme - und Stoff u bertragung. (tranl: eat and Ma ranfer), fourth ed. Sringer, eidelberg, 004. [0] Atchley, A.A. ofler,., Muzzerall, M.., Kite, Chianing A.O Acoutically generated temerature gradient in hort late. Journal of Acoutical Society of America, vol. 88,. 51. [1] Mavrota, G., 009. Effective imlementation of the ε-contraint method in multiobjective mathematical rogramming roblem. A. Math. Com. 13, [] Aghaei, J., Amjady, N., Shayanfar,.A Multi-objective electricity maret clearing conidering dynamic ecurity by lexicograhic otimization and augmented eilon contraint method. Al. Soft. Com., vol. 11, no. 4, REFERENCES [1] Yuan, S.W.K. & Jung, , "hermal management of comuter ytem uing active cooling of Pule ube refrigerator", American Society of Mechanical Engineer, EEP,.. [] Johi, Y.K. & Garimella, S.V. 003, "hermal challenge in next generation electronic ytem", Microelectronic Journal, vol. 34, no. 3, [3] Abdel-Rahman, E., Azenui, N.C., Korovyano, I. & Symo, O.G. 00, "Size conideration in interfacing thermoacoutic cooler with 53