Analysis and Optimization of the Thermal Performance of Microchannel Heat Sinks

Transcription

1 Purdue Univerit Purdue e-pub R Reearh Publiation ooling ehnologie Reearh enter 5 Anali and Otimization o the hermal Perormane o Mirohannel Heat Sin Dong Liu Univerit o Houton S V. Garimella Purdue Univerit, urehg@urdue.edu Follo thi and additional or at: htt://do.lib.urdue.edu/oolingub Liu, Dong and Garimella, S V., "Anali and Otimization o the hermal Perormane o Mirohannel Heat Sin" (5). R Reearh Publiation. Paer 59. htt://d.doi.org/.8/ hi doument ha been made available through Purdue e-pub, a ervie o the Purdue Univerit Librarie. Pleae ontat eub@urdue.edu or additional inormation.

2 Anali and Otimization o the hermal Perormane o Mirohannel Heat Sin Dong Liu and Sureh V. Garimella ooling ehnologie Reearh enter Shool o Mehanial Engineering Purdue Univerit, Wet Laaette, IN USA Abtrat A number o modeling aroahe o inreaing level o omleit or the anali o onvetive heat traner in mirohannel are reented and omared. A detailed omutational luid dnami (FD) model i ued to obtain baeline reult againt hih the dierent aroimate aroahe are omared. hee inlude a -D reitane model, a in aroah, to in-liquid ouled model, and a orou medium aroah, all o hih are amenable to loed-orm olution or the temerature ield. he good agreement beteen the eat and aroimate method indiate that ith areull hoen aumtion, thee analtial reult an lead to adequate derition o the thermal erormane, hile alloing eaier maniulation o mirohannel geometrie or the uroe o otimization. Pratial otimization roedure are develoed to minimize the overall thermal reitane o mirohannel heat in, uing eah o the ive aroahe. Keord: mirohannel; otimization; heat in; eletroni ooling Submitted or ubliation in International Journal o Numerial Method or Heat and Fluid Flo, Februar 3, and in revied orm, Ma 3. orreonding author: (765)494-56, urehg@en.urdue.edu - -

3 Nomenlature A mirohannel ro-etional area in temerature A in ro-etional area b temerature at the bae o the in A area o heat in eii heat u m luid temerature mean lo veloit D h hdrauli diameter W idth o heat in Re rition ontant mirohannel idth h heat traner oeiient in thine H height o heat in Gree Smbol H mirohannel deth aet ratio o mirohannel thermal ondutivit in eiien L length o heat in dnami vioit m ma lo rate n number o mirohannel Nu Nuelt number P reure q heat lu q heat removal rate Q volume lo rate R thermal reitane Re Renold number t ubtrate thine thermal reitane denit o luid reure dro temerature dierene Subrit and Suerrit hannel luid i inlet olid in all - -

4 Introdution he otential or handling ultra-high heat lue ha urred intenive reearh into mirohannel heat in (uerman and Peae, 98; Weiberg and Bau, 99; Sobhan and Garimella, ). For imlementation in ratial deign, the onvetive heat traner in mirohannel mut be analzed in onjuntion ith the hoie and otimization o the heat in dimenion to enure the required thermal erormane. Deign roedure are alo needed to minimize the overall thermal reitane. he ou o thi aer i to reent a omrehenive diuion and omarion o ive dierent (aroimate) analtial model o inreaing ohitiation, hih oer loed-orm olution or ingle hae onvetive heat traner in mirohannel. A general FD model i irt et u to obtain an eat olution. Detail o the aroimate model and the aumtion involved are then reented along ith a omarion o the thermal reitane redition rom thee model. Otimization o the thermal erormane o mirohannel heat in i then diued. Derition o the Problem he mirohannel heat in under onideration i deited in Fig.. Material or abriation ma inlude ondutive material uh a oer and aluminum or modular heat in, or ilion i the mirohannel are to be integrated into the hi. For onervative etimate o thermal erormane, the lid (to late) ma be to be inulated. he idth o individual mirohannel and intervening in ( + ) i tiall mall omared to the overall heat in dimenion W, and numerou hannel are aommodated in arallel lo ath

5 ontinuum equation or onervation o ma, momentum and energ, reetivel, or the onvetive heat traner in mirohannel heat in an be ritten a (Fedorov and Vianta, ; oh, et al., ): V () V () V V P V or the luid (3) or the in (4) hi et o equation aume tead-tate ondition or inomreible, laminar lo, ith radiation heat traner negleted. With an aroriate et o boundar ondition, thee equation rovide a omlete derition o the onjugate heat traner roblem in mirohannel. ae in Figure FD Model A numerial model a ormulated to olve or the three-dimenional heat traner in mirohannel uing the ommerial FD otare aage, FLUEN (Fluent, 998). he imulation a erormed or three dierent et o dimenion a lited in able I. hee three ae are hoen to imulate eeriment in the literature (uerman and Peae, 98) that have oten been ued or validating numerial tudie (Weiberg and Bau, 99; oh, et al., ; Ru, et al., ). ae in Figure he omutational domain, hoen rom mmetr onideration, i hon in Fig.. he to urae i adiabati and the let and right ide are deignated mmetri boundar ondition. A uniorm heat lu i alied at the bottom urae. In the reent or, ater i ued a the - 4 -

6 oring luid ( = 997 g/m 3, = 479 J/gK, =.855 g/m, and =.63 W/mK evaluated at 7), and ilion i ued a the heat in ubtrate material ith = 48 W/mK. ae in able I In the numerial olution, the onvetive term ere diretized uing a irt-order uind heme or all equation. he entire omutational domain a diretized uing a 564 (--z) grid. o veri the grid indeendene o the onvetive heat traner reult, three dierent mehe ere ued in the luid art o the domain: 5, 37, and 55. he thermal reitane hanged b 3.4% rom the irt to the eond meh, and onl b.3% uon urther reinement to the third grid. Hene 37 grid ere ued in the luid domain or the reult in thi or. he agreement beteen the eerimental and redited value o thermal reitane in able I validate the ue o the numerial redition a a baeline againt hih to omare the aroimate aroahe onidered in thi or. he numerial reult ma alo be ued to hed light on the aroriate boundar ondition or the roblem under onideration. For intane, it i oten aumed in mirohannel heat in anale that the aial ondution in both the olid in and the luid ma be negleted. Uing the numerial reult or ae a an eamle, the aial ondution through the in and luid ere ound to aount or.3% and.% o the total heat inut at the bae o the heat in, reetivel. hu the aumtion o negligible aial ondution aear valid or heat traner in the ilion mirohannel onidered. ae in Figure 3 o alternative boundar ondition have been ommonl ued at the bae o the in in mirohannel anale (Zhao and Lu, ; Samalam, 989; Sabr, ): - 5 -

7 q'' (5) or d q'' (6) d in hih Eq. (5) imlie that the imoed heat lo evenl into the luid via the bottom o the mirohannel and into the in via the bae o the in, hile Eq. (6) imlie that all the heat rom the bae travel u the bae o the in. learl, neither o thee to etreme ae rereent the atual ituation orretl. he omuted heat lu in the ubtrate in the immediate viinit o the in bae i hon in Fig. 3 or ae. he heat lue into the luid and the in are 55.5 W/m and 333 W/m, reetivel. Hene, the error aoiated ith emloing Eq. (5) and (6) a the boundar ondition at the bae o the in ould be 5% and 4%, reetivel. A reaonabl aurate alternative or the boundar ondition ould be develoed a ollo: q h L hh L b b (7) Hene, the ratio o the heat diiated through the vertial ide o the in to that loing through the bottom urae o the mirohannel into the luid i H /, or. hi lead to a more reaonable boundar ondition at the bae o the in: d d q '' (8) hi ondition reult in a heat lu o 366 W/m through the bae o the in, hih i ithin % o the omuted eat value o 333 W/m. In light o thi diuion, Eq. (8) i imoed a the thermal boundar ondition at the bae o the in or all the ive aroimate model develoed in thi or

8 Aroimate Analtial Model In vie o the omleit and omutational eene o a ull FD aroah or rediting onvetive heat traner in mirohannel heat in, eeiall in earhing or otimal oniguration under ratial deign ontraint, imliied modeling aroahe are ought. he goal i to aount or the imortant hi, even i ome o the detail ma need to be ariied. Five aroimate analtial model (Zhao and Lu, ; Samalam, 989; Sabr, ; Kim and Kim, 999) are diued, along ith the aoiated otimization roedure needed to minimize the thermal reitane. he ou in thi diuion i on the develoment o a et o thermal reitane ormulae that an be ued or omarion beteen model, a ell a or otimization o mirohannel heat in. A hon in Fig., or the roblem under onidertaion, the luid lo arallel to the -ai. he bottom urae o the heat in i eoed to a ontant heat lu. he to urae remain adiabati. he overall thermal reitane i deined a R o = q A ma " (9) in hih ma = (,o -,i ) i the maimum temerature rie in the heat in, i.e. the temerature dierene beteen the ea temerature in the heat in at the outlet (,o ) and the luid inlet temerature (,i ). Sine the thermal reitane due to ubtrate ondution i iml R ond t LW () the thermal reitane R alulated in olloing model ill not inlude thi term: R R R () o ond - 7 -

9 he olloing aumtion are made or the mot imliied anali:. Stead-tate lo and heat traner. Inomreible, laminar lo 3. Negligible radiation heat traner 4. ontant luid roertie 5. Full develoed ondition (hdrodnami and thermal) 6. Negligible aial heat ondution in the ubtrate and the luid 7. Averaged onvetive heat traner oeiient h or the ro etion. In the aroimate anale onidered, thi et o aumtion i rogreivel relaed. Model - D Reitane Anali ae in Figure 4 In addition to maing aumtion to 7 above, the temerature i aumed uniorm over an ro etion in the imlet o the model. For ull develoed lo under a ontant heat lu, the temerature roile ithin the mirohannel in the aial diretion i hon in Fig. 4. he three omonent o the heat traner roe are: q, o b, o ond A () t q ha ( ) (3) onv b al, o, i q Q (4) he overall thermal reitane an thu be divided into three omonent: - 8 -

10 =, o, i q'' LW R o = ma q '' LW = Rond Ronv Ral (5) in hih the three reitane ma be determined a ollo:. ondutive thermal reitane R ond t LW (6). onvetive thermal reitane R onv nhl( H ) (7) tanh ith in eiien mh. mh 3. alori thermal reitane: R al Q (8) Model - Fin Anali In thi model, aumtion to 7 above are adoted, and the luid temerature roile i onidered one-dimenional (averaged over -z ro etion), = (). he temerature ditribution in the olid in i then: ith boundar ondition: d h d (9) d d q '' () - 9 -

11 - - H d d () It ollo that mh m H q m inh oh ' ' ) ( ), ( () here m = (h/ ) /. he luid temerature () an be obtained rom an energ balane: q d d m ' ' (3) ith (=) =. he bul luid temerature i then: H u q m '' (4) and Eq. () an be reritten a: mh m H q m inh oh ' ', H u q m '' (5) he thermal reitane i thu: LW q L LW q R ' ', ' ' W H u LW mh mh m m inh oh (6) Model 3 - Fin-Fluid ouled Aroah I Folloing the ame line o reaoning a in the in anali (Model ) and adoting aumtion to 7 above, but averaging the luid temerature onl in the z diretion (Samalam, 989), the energ equation in the in an be ritten a:, h (7)

12 ith q '' J (8) H (9) he energ balane in the luid i rereented b: um h, (3) and it i aumed that (=, ) =. Subtituting h = Nu /D h into Eq. (3) ield umdh Nu Nu (3) Deining X = /a and Y = /a here umdh a Nu, the olution to Eq. (3) an be ritten a X X X ' X Y X ', Y e Hene, Eq. (7) an then be tranormed to in hih X, Y Y, dx ' (3) X X X ' X, Y X ', Y e dx (33) ' Solving Eq. (33) b Lalae tranorm, a (34) Dh (35) Nu ( Y, ) Y ( ) (36) he boundar ondition in Eq. (8) and (9) beome - -

13 Y Y J a (37) in hih H H a ~ (38) Y H Y, and J i deined in Eq. (8). he olution to thi tem o equation i he invere Lalae tranorm ield the temerature: X, Y L Y, Ja oh Y H (39) inh H Ja H H 6 { X Y H n n n X o ~ e } n n ~ n Y H H (4) in hih n ~ n / H ~. n / H hi i a raidl onverging ininite erie or hih the irt three term adequatel rereent the thermal reitane: L, Ja L / a '' '' H R q LW q LW H 3 LW (4) Model 4 - Fin-Fluid ouled Aroah II In thi model, aumtion to 7 above are again adoted, eet that aial ondution in the in i not negleted (Sabr, ). he governing equation in the olid in and liquid, reetivel, are thereore: (,, z) (4) - -

14 - 3 - ),, ( )),, ( ( z z V (43) At the in-luid interae, the ondition i i h z z (44) in hih the averaged loal luid temerature i / / ) ( / m u dz v (45) ith / / / m vdz u. Along ith Eq. (8), the olloing boundar ondition al: z z L H z z (46) Integrating Eq. (4) over z rom - / to, the in temerature varie a z (47) in hih / / dz. ombining Eq. (44) and (46), i h (48) Auming i, ine Bi = h( /)/ <<, Eq. (48) beome h (49) I the aial ondution term ere negleted, Eq. (49) ould redue to Eq. (7). Sine ull develoed ondition are aumed and aial ondution in the luid i negleted, Eq. (43) ma be integrated over z rom to / to ield

15 / u dz z (5) Uing the boundar ondition in Eq. (44), thi redue to u m h (5) he olloing dimenionle variable are introdued: X / L, Y / H and (5) in hih hh / q' ' (53) he tem o equation above an be at in dimenionle term: X Y A mh (54) X S (55) Y mh Y (56) X Y Y X Y Y (57) X (58) here A H / L and the modiied Stanton number (Sabr, ) i given b S hl um. Emloing imilar tehnique a adoted or Model 3, the in temerature i obtained a - 4 -

16 oh mh Y X, Y mh o n Y n X (59) inh mh n in hih the irt term in the ininite erie rovide reult o aetable aura (< 5% deviation rom the omlete erie): i i X S X e X 3 i (6) i ith S S mh A S S mh e e S e e e S e A S A 3 mh he thermal reitane i thu obtained a, R q'' LW q'' LW mh mh i S e 3 mh LW oh i hh inh i i (6) In mot ratial ae, (mh /A) >> S/, and Eq. (6) redue to mh oh SA R mh S hh inh mh mh LW (6) - 5 -

17 Model 5 - Porou Medium Aroah he onvetive heat traner roe in mirohannel an alo be treated a being imilar to that in a luid-aturated orou medium, ith the etended r equation ued or luid lo and a volume-averaged to-equation model ued or heat traner, a demontrated in Vaai and ien (98). Folloing the anali o Kim and Kim (999), a to-equation model an be emloed to obtain the volume-averaged roertie over a rereentative elementar volume or the olid region and the luid region earatel. he momentum equation and boundar ondition are d d d u u (63) d K u at, H (64) here u i the volume-averaged veloit, = /( + ) i the oroit, and K = / i the ermeabilit. Equation (63) and (64) ma be ritten a U d U dy P (65) uing the dimenionle arameter U = at Y =, (66) u U, u m K, H Y, H K d P. um d he olution to the momentum equation i then oh U Poh Y inh Y inh (67) he volume-averaged energ equation or the in and luid, reetivel, are: - 6 -

18 e ha (68) ith boundar ondition e u ha (69) at = (7) at = H (7) in hih a i the etted area er unit volume, h the loal heat traner oeiient deined a the ratio o the interaial heat lu to the olid-luid temerature dierene, and e and e the eetive ondutivitie o the olid and luid, deined a e For ull develoed lo under ontant heat lu, it i non that, e d d = ontant (7) and q' ' umh (73) he energ equation (68) and (69) and boundar ondition an thu be ritten in dimenionle orm a d D dy (74) d U D (75) dy ith at Y = (76) - 7 -

19 - 8 - dy d dy d at Y = (77) in hih, hah D, H q ' ', and H q ' '. Subtituting the olution obtained or veloit, Eq. (74) and (75) an be olved to give 3 [ Y Y P 5 4 inh oh Y D Y D ] inh inh oh oh Y Y (78) oh [ oh inh ] ] inh P Y Y Y Y (79) in hih D N oh D D N inh inh inh oh

20 D 3 D D 5 4 D N N D D D oh Y inh D Finall, the thermal reitane an be obtained a R K H, b (8) 3 W um LW in hih, b i the bul mean luid temerature, deined a, b U dy UdY Ke eature o the ive aroimate model diued above, inluding the aumtion, governing equation and reitane ormulae develoed, are ummarized in able II. Aement o the Aroimate Model For the mirohannel arameter lited in able I, thermal reitane ere omuted ith Fluent a ell a rom the ive aroimate model. he reult are hon in able III. It an be een that all the aroimate model ould rovide aetable redition or the thermal reitane o the mirohannel heat in, ith the maimum deviation being 7.8%. Model through 5 are more omle to al than Model, and involve the olution o the dierential governing equation. In ite o it imliit, Model aear to adequatel rereent the hi o the heat traner roblem, and i reommended or ue in the deign and otimization - 9 -

21 o ratial mirohannel heat in. ae in able III It ma be noted that in Model, the luid temerature i onidered to be onl a untion o the -oordinate, and the in temerature i olved in a trul -D manner. he thermal reitane ereion rom Model i thereore idential to that rom Model. Alo, in Model 3 and 4, ine -D temerature ield are onidered in both the in and the luid, the ne term H 3n Land Q n H L aear, in addition to the other term in the imler Model and. he dierene beteen Model 3 and 4 i that the aial ondution term aear eliitl in the in equation o Model 4, hile it i negleted in Model 3. In the alulation above, ereion or Nuelt number Nu and the rition ontant Re are needed or omuting the onvetive heat traner oeiient h and the average veloit u m in the mirohannel. In all ive aroimate model diued in the reent or, the lo i aumed to be thermall and hdrodnamiall ull develoed. Hene the olloing relation are ued in term o mirohannel aet ratio (Inroera and DeWitt, 996; Shah and London, 978): Nu d 8.35( ) (8) 3 Re 96(.3553 /.9467 /.7 / d /.537 / ) (8) Hoever, the ull develoed aumtion i not ala valid, eeiall or mirohannel ith the larger hdrauli diameter and hort length. With hdrodnami and thermal length deined a Re and * Re Pr, the olloing relation (Samalam, 989; Harm, et al., L L D h L L D h 999) ould be emloed intead o Eq. (8) and (8): * Nu 3.35( L ) Pr,.3 *. L (83) - -

22 * Nu.87( L ) Pr,.5 *. 3 L (84) / 3. a Re ( Re).57, L. 5 (85) d ( ) L In the reent alulation, L/D h ith moderate Renold number, o that the hdrodnamiall ull develoed ondition i atiied. For the oring luid in thi tud (ater, Prandtl number ~ 5.8), Nuelt number alulated rom Eq. (8) and (83) are lited in able IV. he deviation beteen the to et o reult i ithin 6%, and thereore the aumtion o thermall ull develoed ondition i aetable. In general, develoing thermal eet hould be areull onidered beore ull develoed ondition are aumed. ae in able IV Otimization he otimization o mirohannel heat in deign an be motivated uing the thermal reitane aroah in Model. A indiated in Eq. (8), R al i inverel roortional to the ma lo rate. When the reure head along the mirohannel length i reribed a the ontraint, R al ill dereae a inreae hen H reahe the maimum alloable value. Hoever, the onvetive heat traner oeiient h ill inreae hen D h dereae, leading to a redution in R onv, a hon b Eq. (7). he heat traner rom the ubtrate through the in ill alo be enhaned i the in eiien inreae, hih require a larger in thine. Hoever, the inreae in ill redue the number o mirohannel/in air in a heat in or a reribed heat in ize. Due to thee ometing ator, there eit an otimal mirohannel dimenion that minimize the overall thermal reitane. - -

23 In order to otimize the thermal erormane o a mirohannel heat in, the olloing variable mut be eiied rom imlementation ontraint:. hermal ondutivit o the bul material ued to ontrut the heat in ( S );. Overall dimenion o the heat in (L and W rom the ize o the hi, H and t rom abriation and trutural onideration); 3. Proertie o the oolant (,,, ); and 4. Alloable reure head (P). o illutrate the roedure, the eamle onidered ue ater a the oring luid to ool a hi ith L = W = m and a given reure head o P = 6 Pa. he heat load i W/m. he mirohannel heat in i to be made o ilion ith t = m and H = 4 m. he luid roertie are evaluated at 7. he otimization roe involve inding the otimal mirohannel geometr (hannel idth, in thine and aet ratio = H / ) that ill minimize thermal reitane. Solution to the olloing equation ould ield the otimum: R (86) R (87) In thi or, the otimization omutation ere erormed uing the ommerial olver MALAB (MathWor, ). he otimized reult derived rom the ive aroimate model are lited in able V. he otimal thermal reitane value reorted rom the ive model agree to ithin %. It ma alo be noted that the minimum thermal reitane i ala attained at the larget alloable aet ratio. In ratial deign, the aet ratio ould be determined b the limit on the mirohannel deth and the ubtrate thine. - -

24 ae in able V onluion Five aroimate analtial model or rediting the onvetive heat traner in mirohannel heat in are reented and omared. loed-orm olution rom thee model are omared to ull FD imulation and eerimental reult, and the eia o the dierent model aeed. Otimization roedure are diued or minimizing the thermal reitane o the heat in. he reult obtained demontrate that the model develoed oer uiientl aurate redition or ratial deign, hile at the ame time being quite traightorard to ue. Anoledgement he author anoledge the inanial uort rom member o the ooling ehnologie Reearh enter (htt://idget.en.urdue.edu/~r), a National Siene Foundation Indutr/Univerit ooerative Reearh enter at Purdue Univerit. Reerene Fedorov, A. G. and Vianta, R. (), hree-dimenional onjugate heat traner in the mirohannel heat in or eletroni aaging, International Journal o Heat Ma raner, Vol. 43, no. 3, Fluent Uer Guide, (998), Fluent In. Lebanon, Ne Hamhire. Harm,. M., Kazmierza, M. J. and Gerner, F. M. (999), Develoing onvetive heat traner in dee retangular mirohannel, International Journal o Heat Fluid Flo, Vol., no., Inroera, F. P. and DeWitt, D. P. (996), Fundamental o Heat and Ma raner, John Wile & Son, N. Y

25 Kim, S. J. and Kim, D. (999), Fored onvetion in mirotruture or eletroni equiment ooling, Journal o Heat raner, Vol., no.3, Matlab, Verion 6., (), he MathWor, In., Nati, MA. Ru, J. H., hoi, D. H. and Kim, S. J. (), Numerial otimization o the thermal erormane o a mirohannel heat in, International Journal o Heat Ma raner, Vol. 45, no. 3, Sabr, M.-N. (), ranvere temerature gradient eet on in eiien or mirohannel deign, Journal o Eletroni Paaging, Vol. 3, no. 4, Samalam, V. K. (989), onvetive heat traner in mirohannel, Journal o Eletroni Material, Vol. 8, no. 5, Shah, R. K. and London, A. L. (978), Laminar lo ored onvetion in dut, Advane in Heat raner, Sulement, Aademi Pre. Sobhan,. B. and Garimella, S. V. (), A omarative anali o tudie on heat traner and luid lo in mirohannel, Miroale hermohial Engineering, Vol. 5, no. 4, ien,. L. and Kuo, S. M. (987), Anali o ored onvetion in mirotruture or eletroni tem ooling, Pro. Int. Sm. ooling eh. or Eletroni Equiment, Honolulu, HI, oh, K.., hen, X. Y. and hai, J.. (), Numerial omutation o luid lo and heat traner in mirohannel, International Journal o Heat Ma raner, Vol. 45, no., uerman, D. B. and Peae, R. F. (98), High-erormane heat ining or VLSI, IEEE Eletroni Devie Letter EDL-, Vaai, K. and ien,. L. (98), Boundar and inertia eet on lo and heat traner in orou media, International Journal o Heat Ma raner, Vol. 4, no., Weiberg, A. and Bau, H. H. (99), Anali o mirohannel or integrated ooling, International Journal o Heat Ma raner, Vol. 35, no., Zhao,. Y. and Lu,. J. (), Anali o mirohannel heat in or eletroni ooling, International Journal o Heat Ma raner, Vol. 45, no. 4,

26 FIGURE APIONS Figure. Shemati o a mirohannel heat in. Figure. omutational domain. Figure 3. Heat lu ditribution at the bae o the in. Figure 4. emerature roile in a mirohannel heat in

27 able I. omarion o thermal reitane. ae 3 (m) (m) H (m) H (m) P (Pa) q (W/m ) R e (/ W) *..3.9 R num (/ W) * uerman and Peae (98); L = W = m - 6 -

28 - 7 - able II. Summar o aroimate analtial model. Model emerature Aial ondution Governing equation hermal reitane (R) Fin Fluid Fin Fluid -D -D X X Not ued ( ) nhl H Q (88) -D -D X X A hp d d q d d m Q H nhl ) ( (89) 3 -D -D X X h, h u m, / 3 Ja L a H H LW (9) 4 -D -D X ),, ( z ),, ( )),, ( ( z z V L H n Q Q L nhh (9) 5 -D -D X X u K u d d d d e ha ha u e, 3 b m H K W u LW (9) (X not onidered, onidered)

29 able III. Overall thermal reitane. hermal ae reitane (/ W) 3 R o,num R o,model...9 R o,model...9 R o,model R o,model R o,model

30 able IV. Nuelt number. ae 3 Nu d Nu

31 able V. Otimal dimenion. Model R o (m) (m) (/ W)

32 L W H t H z Fig. Shemati o a mirohannel heat in. / / H H z q Fig. omutational domain

33 Heat lu (W/m ) X =.3 m X =.5 m X =.7 m X =.9 m 5 5 Fin Fluid Z ( m) Fig. 3 Heat lu ditribution at the bae o the in.,o,i bae,o,o bae,i,i Fig. 4 emerature roile in a mirohannel heat in