Heat Transfer and Fluid Flow through Microchannels a theoretical approach

Transcription

1 Proeeding o the 5th IASME/WSEAS Int. Conerene on eat raner, hermal Engineering and Environment, Athen, Greee, Agt 5-7, eat raner and Flid Flow throgh Mirohannel a theoretial approah N.AMANIFARD, M.BORJI, A.K. AGI * Falty o Engineering he Univerity o Gilan P.O. Box 3756, Raht IRAN * aghi@gilan.a.ir, Abtrat: hree dimenional heat traner and water low harateriti in a et o retanglar mirohannel heat in or advaned eletroni ytem invetigated in thi paper. he ll Navier-Stoe approah i employed or thi ind o narrow hannel or the water low aement. he omplete orm o the energy eqation aompanying the diipation term i alo lined to the momentm eqation. he allated thermal ditribtion how good agreement with the orreponding experimental predition. hen thee relt are ompared with poro medim approah in mall dimenion. In thi approah mirohannel i modeled a a lid-atrated poro medim, and i nmerially olved baed on the Forhheimer-Brinmanextended Dary eqation or the lid low and two eqation model or heat traner between the olid and lid phae. Key-Word: Poro medim- Mirohannel heat in- eat traner- Veloity proile. Introdtion he inreaing inorporation o eletroni ytem reqire innovative, mall ale and highly eetive ooling tehniqe or the removal o a large amont o heat rom a mall area in order to avoid it temperatre rom riing igniiantly and operate eletroni devie at an optimm temperatre. Many tehniqe have been developed or ontrolling and removing the heat generated in h a ae. One h method i to e mirohannel heat in. Mirohannel heat in i a trtre with many miro ale hannel o large apet ratio bilt on the ba o the mirohip, and a liqid i ored throgh thee paage to arry ot the energy. he mirohannel heat in at irt propoed by ermann and Peae [], they demontrated that the mirohannel heat in, oniting o miroretanglar low paage, have a higher heat traner oeiient in laminar low regime than that in trblent low throgh onventionally-ized devie. Sine pioneering wor o erman and Peae [], many experimental, analytial and nmerial invetigation are reported. Peng and Wang [] and Peng and Peteron [3] ytematially examined the ored low and heat traner harateriti o water and binary mixtre lowing throgh retanglar mirohannel. It wa oberved that laminar low tranition orred at Reynold nmber between 00 and 700. And the ritial tranition Reynold nmber diminihed with the dereae in the ize o the mirohannel. Frthermore, the eet o apet ratio and the hydrali diameter on the low and heat traner in mirohannel were invetigated. hee relt provided notieable experimental data and oniderable manietation that the low and heat traner in mirohannel are trongly dependent pon the type and propertie o the woring lid a well a geometri parameter o mirohannel [4], and thereore may be dierent rom what typially or in the marohannel. Wilding et al [5] analyzed low o water and vario biologial lid in gla-apped ilion mirohannel. he data illtrated an approximately 50% inreae in the Dary rition ator rom the theoretial relt. Similar relt were oberved by Jiang et al [6] who tdied low o water throgh retanglar and trapezoid ro-etion hannel. he mirohannel ed in thi tdy were ormed by ething a ilion btrate and apping it with a gla waer. Alternatively, in view o the mall dimenion o the mirotrtre, Koh and Colony [7] modeled the mirohannel a a poro medim by ing Dary law to deribe the low. Later, Kim and Kim [8] and Kim et al. [9] have preented analytial oltion or veloity and temperatre ditribtion or ored onvetion in mirohannel heat in by ing the Brinman-extended Dary eqation or the

2 Proeeding o the 5th IASME/WSEAS Int. Conerene on eat raner, hermal Engineering and Environment, Athen, Greee, Agt 5-7, lid low. More reently Amaniard et al. [0] nmerially invetigated the eet o eletrial doble layer (EDL) near the olid/liqid interae, on three dimenional heat traner harateriti and prere drop o water low throgh a retanglar mirohannel. he relt illtrate that, the liqid low in retanglar mirohannel i inlened igniiantly by the EDL, partilarly in the high eletri potential, and hene deviate rom low harateriti deribed by laial lid mehani. In thi paper the ompatibility and eetivene o Navier-Stoe and energy orrelation in miroale hannel i nmerially veriied baed on the experimental relt o erman and Peae []. Figre () Shemati o mirohannel Phyial model and omptational domain A hemati view, phyial, and omptational domain o h mirohannel i depited in Fig (), Fig (), and Fig (3), repetively. eat i removed primarily by ondtion throgh the olid and then diipated away by onvetion o the ooling lid in the mirohannel. he mirohannel ha been tdied i made o ilion with thermal ondtivity () o 48 W / m. K. At the bottom, a niorm heat lx o q" arie rom an eletri hip that i onneted to the mirohannel. At the top o the hannel, there i a Pyrex plate whih mae an adiabati ondition. he width o mirohannel and the wall thine are repreented by W and W, repetively. he thine o the ilion btrate throgh whih the heat lx i tranormed to the ooling lid lowing in hannel an be imply reognized a t -, aording to Fig (). he total length and width o mirohannel are L h and W t whoe vale o ive dierent ae are in able (). Moreover, teady inompreible and laminar lid low and teady heat traner, with negligible radiative heat traner and ontant olid and lid property have been amed in omptation. he inlet temperatre o ooling water throgh the hannel i 0ºC. A mirohannel in the enter part o the plate will be onidered in rrent wor. A a relt o the ymmetry o the retanglar hannel, we will enter the omptational domain in a hal hannel a hown in Fig (3). Figre () phyial model Figre (3) Comptational domain

3 Proeeding o the 5th IASME/WSEAS Int. Conerene on eat raner, hermal Engineering and Environment, Athen, Greee, Agt 5-7, able () Five dierent ae o mirohannel Deription W ( µ m) ( µ m) (m) L t q (W / m ) Inlet Veloity ( m/ ) Apet ratio Nelt nmber Cae /6 0/ Cae /4 8 / Cae / Cae /4 790 / Governing Eqation Aming a laminar lly developed low in retanglar hannel in poitive x-diretion, the omponent o veloity atiy ( y, z) and v w 0 in term o Carteian oordinate ytem. he eqation o motion i written a ollow: dp + () y z µ dx A preented in Fig (), a ilion waer plate with a large nmber o mirohannel i onneted to the hip. A liqid i ored to low throgh thee mirohannel to remove the heat. All mirohannel are amed to have a niorm retanglar ro-etion with geometri parameter a hown in able. For a teady-tate, lly developed, laminar low in a mirohannel, the energy eqation (with onideration o the axial thermal ondtion in low diretion and the vio diipation) or the ooling liqid tae the peii orm: α x x y z () ρ µ C p + y z Where and α are the temperatre and the thermal diivity o the ooling liqid, repetively, C i the peiied heat apaity o the p ooling liqid. Baed on preented omptational domain, the adiabati ondition an be ed along the hannel ymmetri enter line: z 0 0 (3) z At the bottom o hannel, a niorm heat lx o q" i impoed over the heat in, and an be expreed a: y 0 q" (4) y ear i the thermal ondtivity o the liqid oolant. Sine the thermal ondtivity o the gla i abot two-order o magnitde lower than that the top bondary i inlted. hi i a onervative amption whih will lead to light nderetimation o the overall heat traner oeiient. hi amption yield: y 0 (5) y 4 Nmerial oltion method In rrent wor inite volme method o Patanar [3] i ed to olve the ontinity, momentm, and energy eqation nmerially. Sine a detailed diion o the FVM i available in Patanar [3], only a very brie deription o the main eatre o thi method i given here. In the FVM, the domain i divided into a nmber o ontrol volme h that there i one ontrol volme rronding eah grid point. he grid point i loated in the enter o a ontrol volme. he governing eqation i integrated over eah ontrol volme to derive an algebrai eqation ontaining the grid point vale o the dependent variable. he diretization eqation then expree the onervation priniple or a inite ontrol volme jt a the partial dierential eqation expree it or an ininiteimal ontrol volme. he relting oltion implied that the integral onervation o qantitie h a ma, momentm, and energy i exatly atiied or any ontrol volme and o ore, or the whole domain. he power-low heme i ed to model the ombined onvetion- 3

4 Proeeding o the 5th IASME/WSEAS Int. Conerene on eat raner, hermal Engineering and Environment, Athen, Greee, Agt 5-7, diion eet in the tranport eqation. he SIMPLER algebrai o Patanar i ed to reolve the prere-veloity opling. he relting algebrai eqation are olved ing a line-by-line ri-diagonal matrix Algorithm. 5 Mirohannel a a poro medim In view o the mall dimenion o the mirotrtre, the mirohannel i modeled a a lid-atrated poro medim. By ollowing a proedre deribed by Chien-in Chen [] the low i amed to be laminar and both hydrodynamially and thermally lly developed. Alo, the thermal phyial propertie are amed to be ontant. o relax the ontant-lid temperatre amption along the height o the hannel wall, the mirohannel heat in i modeled a a lidatrated poro medim, a hown in Fig (4). he Forhheimer-Brinman-extended Dary eqation propoed by Vaai and ien [] or the lid low and the volme-averaged two-eqation model or heat traner are ed. he mmary o eqation and bondary ondition by ing ollowing nondimenional grop an be expreed in dimenionle orm a ollow, and omplete eqation an be oberved in Chen wor []; Y y, U Where the braet D w w, θ, θ qw qw ( ε ) ( ε ) repreent a volmeaveraged vale, and ε,, and (6) D K d p εµ dx are poroity, thermal ondtivity o olid and Daian onvetive veloity, repetively. Da d U ΓU U + 0 ε dy (7) d θ ~ Bi( θ ) 0 θ dy (8) d θ ~ r Bi( θ ) SU 0 θ dy (9) U 0 at Y 0, (0) θ 0 at Y 0 () θ θ θ Y Y Where K Da, 0 at y ρ DCK Γ εµ () (3) Are the Dary nmber and the inertial ore parameter, repetively. And S i given by S (4) 0 UdY he parameter K r and B ~ i are the eetive thermal ondtivity ratio and the eqivalent Biot nmber in poro medim model, repetively, and an be written a e ε ~ ha r, Bi 0 rα (5) ( ε ) e Where K, C,, h, a, and are the permeability, inertial ore oeiient, eetive thermal ondtivity o olid, intertitial heat oeiient, wetted area per volme, and eetive thermal ondtivity o the lid, repetively. he poroity, the wetted area per volme, and eetive thermal ondtivitie an be expreed a w ε, a (6) w + t w + t e ( ε ), ε (7) And i the thermal ondtivity o lid. Permeability and the intertitial heat traner oeiient an be written a εµ εw K (8) ( dp ) dx q" 5 h (9) w w he energy arried by the lid a it move pat a given ro etion i m ρ da (0), m A ere ṁ i the ma low rate and e e, i the bl m mean lid temperatre ing volme-averaged vale. For ontant-lid propertie,, m A h m A da () N () h w (3) w q, m 4

5 Proeeding o the 5th IASME/WSEAS Int. Conerene on eat raner, hermal Engineering and Environment, Athen, Greee, Agt 5-7, Where, i the onventional bl mean m temperatre and an be written a, m da A (4) m A veloity bondary layer i wrong. Fig. 5 how that the veloity proile throgh whole hannel i not developed and the entry region eet are notieable. he main reaon o prere drop in thee ae i the developing veloity proile whih inreae the wall hear tre. hee relt are not onitent with erman experiment. So there mt be ome modiiation in the governing eqation to reah the arate relt. able () hermal reitane omparion Figre (4) the eqivalent poro medim 6 Relt and diion For olving the eqation everal grid trtre were ed. he grid denity o in z, y, and x diretion i onidered to be appropriate. hermal reitane an be imply ompted ing max ( z) in R( z) (5) q In eqation (5), ot and in repreent the meared otlet and inlet temperatre o ooling water, repetively, and q i the heat lx. he ompted thermal reitane in or dierent ae are ompared with thoe o experimental relt ondted by erman and Peae [] in table (). It i hown that iiently reaonable agreement exit in h omparion and, thereore, ll Navier-Stoe approah an be deployed or h mirohannel low and heat traner omptation with a hydrali diameter o abot 00 µm. he Nelt nmber (N) are ompted a hd h N (6) K And hydrali diameter D h i deined a ollow W Dh (7) + W We an ee in ae 4 with 790 w heat lx, only m C inreaing in ooling water temperatre. Fig (5) illtrate the veloity proile in vario y-z etion in Cae. A it an be een, the veloity gradient near the hannel wall i very large. hi large veloity gradient how that the wall hear tre i oniderably large; o the prere drop along the hannel will be igniiant. Althogh the thermal bondary layer o low throgh hannel i lly developed, the amption o lly developed Cae R( m K / W ) W Error q( ) (%) m Experimental Nmerial Figre (5) Veloity proile in y-z etion or Cae In Fig (6), we an ee the dimenionle veloity proile or vario vale o apet ratio and inertial ore parameter in poro medim approah. A illtrated in thi Figre, the veloity proile how a reaonable agreement in two method at apet ratio abot 3 to 6. Bt the deviation o two method appear with inreaing the inertial ore parameter (Γ) and dereaing the apet ratio. he eet o vario problem parameter on the Nelt nmber i illtrated in Fig (7) and (8). It i lear rom thi igre and table that the Nelt nmber inreae with inreaing the apet ratio in two method. A depited in Fig (7), or a given 5

6 Proeeding o the 5th IASME/WSEAS Int. Conerene on eat raner, hermal Engineering and Environment, Athen, Greee, Agt 5-7, apet ratio, a larger poroity will prode a larger Nelt nmber. Fig (7) nveil that the variation o inertial ore parameter an only relt a light imilar variation in Nelt nmber. It evident that or a peiied apet ratio, onorm o two method i dirpted with inreaing the poroity. Bt in mall poroitie two oltion how a good aommodation. Figre (6) Dimenionle veloity proile or vario vale o α and Γ 7 Conlion In thi wor the heat traner in or geometri type o mirohannel heat in ha been invetigated. he nmerial relt are obtained or thermal reitane and the Nelt nmber and how a good agreement with experimental data. he Nelt nmber i ond to inreae with inreaing the apet ratio. he relt alo gave the reqired arane o ing the ll Navier-Stoe approah or the mirohannel with hydrali diameter abot 00 µm. he veloity proile how an entry length area in the hannel and thereore, a high gradient temperatre and veloity proile were ahieved. In addition, baed on the Chen wor et. al. [], the eet o vario parameter a; hannel apet ratio, inertial ore parameter, and the poroity on the Nelt nmber and veloity proile in poro medim approah are invetigated. he omparion between two method howed a good agreement at apet ratio abot 3 to 6 or veloity proile and in mall poroitie or Nelt nmber. Figre (7) Overall Nelt nmber v. α at eleted vale o ε and Γ Figre (8) Overall Nelt nmber v. r at eleted vale o ε and Γ Reerene: [] D.B. ermann, and R.F.W. Peae, ighperormane eat Sining or VLSI, IEEE Eletron. Dev.Lett.EDL-, pp. 6-9 (98). [] X.F. Peng, G.P. Peteron, and Wang, B.X., eat traner harateriti o water lowing throgh mirohannel. Experimental eat traner 7,pp (994). [3] Peng, X.F., and Peteron, G.P., Fored onvetion eat raner o ingle-phae binary mixtre throgh mirohannel. Experimental hermal and lid iene pp 98-04(996). [4] B.X. Wang, X.F. Peng, Experimental Invetigation on liqid ored onvetion heat traner throgh mirohannel, International Jornal o eat and Ma raner 37 (pple.), pp 73-8(994). [5] P. Wilding M. A. Shoner, and L. J. Kira., Maniplation and Flow o Biologial Flid in Straight hannel miromahined in Silion, Clin. Chem., vol. 40, pp , (994). [6] X. N. Jiang, Z.Y. Zho, J. Yao, Y. Li, and X. Y. Ye, Miro-lid Flow in Mirohannel, In pro. randeer 95, Stoholm, Sweden, Jne 5-9, pp , (995). [7] J.C.Y. Koh, R. Colony, eat traner o mirotrtre or integrated irit, Int. Commn. eat Ma raner 3 (986) [8] S.J. Kim, D. Kim, Fored onvetion in 6

7 Proeeding o the 5th IASME/WSEAS Int. Conerene on eat raner, hermal Engineering and Environment, Athen, Greee, Agt 5-7, mirotrtre or eletroni eqipment ooling, ASME J. eat raner (999) [9] S.J. Kim, D. Kim, D. Y. Lee, on the loal thermal eqilibrim in mirohannel heat in, Int. J. eat Ma raner 43 (000) [0] N. Amaniard, M. Borji, A. K. aghi, eat raner in poro medim, Brazilian J. Chemial engineering 4 (007)-in pre. [] Chien-in Chen, ored onvetion heat traner in mirohannel heat in, Int. J. eat and Ma raner 50 (007)8-89. [] K. Vaai, C.L ien, Bondary and inertia eet on low and heat traner in poro media, Int, J, eat Ma raner 4(98) [3] S.V.Patanar, Nmerial eat raner and Flid Flow, emiphere, New Yor,