Analytical Modeling of Forced Convection. Microelectronics Heat Transfer Laboratory. University ofwaterloo. Waterloo, Ontario, Canada

Transcription

1 Analyical Modeling of Forced Convecion in Sloed lae Fin Hea Sinks. Teersra, J.R. Culham y and M.M. Yovanovich z Microelecronics Hea Transfer Laoraory Deparmen of Mechanical Engineering Universiy ofwaerloo Waerloo, Onario, Canada ABSTRACT Analyical models are developed for he average hea ransfer rae in forced convecion-cooled, sloed n hea sinks. These models for he upper and lower ounds can e used o invesigae he eecs of slo size and placemen on hea sink performance. Experimenal measuremens are performed for a variey of slo conguraions over a range of Reynolds numers, and hese daa are compared wih he proposed analyical models. An approximae model is proposed ha predics he experimenal resuls for he average hea ransfer rae o wihin a 1% RMS dierence. NOMENCLATURE A channel surface area, m A a approach ow cross secion area, m A o hea sink ow cross secion area, m channel spacing, m g graviaional acceleraion (ms ) h average hea ransfer coecien, Wm K H n heigh, m k n hermal conduciviy, WmK k f uid hermal conduciviy, WmK L aseplae lengh, m _m mass ow rae, kgs n composie model cominaion parameer Graduae Research Assisan y Research Associae rofessor and Memer, ASME z rofessor and Fellow, ASME N numer of ns N s numer of slos Nussel numer, (Q)(k f AT ) Nu i ideal Nussel numer, 1 slo pich, m r randl numer, Q oal hea ransfer rae, W R hermal resisance, 1h A, o CW Reynolds numer, U Re channel Reynolds numer, (L) S slo widh, m T a inle air emperaure, o C T f lm emperaure, (T s + T a ), o C T s aseplae emperaure, o C T emperaure dierence, T s T a, o C n hickness, m U average velociy inchannel, ms U a approach velociy, ms W aseplae widh, m Greek Symols hermal diusiviy (m s) hermal expansion coecien (1K) n eciency, Nu i kinemaic viscosiy (m s) Suscrips dev developing fd fully developed LB lower ound UB upper ound

2 INTRODUCTION Finned hea sinks are commonly used for enhancing hea ransfer in air-cooled microelecronics and power elecronics applicaions. The use of nned hea sinks reduces he hermal resisance and operaing emperaures of componens and assemlies y increasing he availale surface area for convecive hea ransfer. The plae n hea shown in Fig. 1a) is one of he mos common hea sink conguraions used in curren applicaions, consising of a parallel array of hin a plaes aached o a single, hermally conducive aseplae. Forced convecion hea ransfer occurs in he channels formed eween adjacen ns y fan or lower-driven airow in a direcion parallel o he aseplae. Figure 1: Schemaic of Hea Sink Geomery a) lae Fin Hea Sink ) Sloed lae Fin Hea Sink An enhancemen of he hermal performance of plae n hea sinks achieved hrough he inroducion of sloed secions in he n surfaces has een proposed (R-Thea, 1999), as shown in Fig. 1). lacing slos in he ns a regular inervals causes new hermal oundary layers o e iniiaed a each n secion, resuling in a reducion of he average oundary layer hickness and an increase in he average hea ransfer coecien. However, hese slos also conriue o a decrease in he availale surface area for convecion, whichmay reduce he oal hea ransfer from he hea sink. An opimal sloed plae n hea sink design is one ha alances he enhancemen in he hea ransfer coecien wih he susequen reducion in surface area. In order o opimize all he hea sink parameers, including he relaive size and placemen of slos, analyical models are required ha predic hea sink hermal performance as a funcion of slo geomery and overall n area. These analyical design ools can e used for parameric and rade-o sudies early in he design process, prior o more cosly and ime consuming numerical simulaions or prooype esing. There are no models or correlaions currenly availale in he lieraure for sloed n hea sinks, and no experimenal or numerical resuls have een presened ha address his prolem. The ojecive of he curren sudy is o develop analyical models for he average hea ransfer rae of sloed n hea sinks for he full range of Re from fully developed o laminar developing ow. The proposed models will include n eecs o accoun for he variaion in n emperaure, an imporan facor in he high aspec raio hea sinks examined in his sudy. Experimenal measuremens will examine he performance of various sloed hea sink conguraions as a funcion of slo size and spacing, and hese daa will e used o validae he proposed models. MODEL DEVELOMENT rolem Descripion The prolem of ineres in his research sudy involves forced convecion hea ransfer from air-cooled plae n hea sinks wih sloed ns, as shown in Fig. 1). The hea sink consiss of a parallel array of N ns of hermal conduciviy k aached o a recangular aseplae, wih dimensions as shown in Fig. 1. The aseplae is assumed o e relaively hick wih a large value of hermal conduciviy, such ha spreading resisance can e negleced and he aseplae can e reaed as isohermal. The lower surface and edges of he aseplae are assumed o e adiaaic, such ha all hea ransferred y he hea sink is dissipaed from he ns y convecion. The sloed n hea sinks consis of uniformly sized and spaced slos cu ino he ns from he n ip o he aseplae, such ha he remaining n secions are separaed from each oher and conneced only y he aseplae. The size and spacing of he slos can e fully descried y wo geomeric parameers as shown in Fig., he slo pich, and he slo widh S. For he susequen analysis, i is convenien o nondimensionalize he slo pich y he aseplae lengh,

3 where T s and T a are he source and he inle air emperaures, respecively, and Q is he oal hea ransfer rae for a single channel. The oal surface area in he channel, A, is deermined as a funcion of he dimensionless slo widh: A LH 1 S (3) Figure : Schemaic of Slo Geomery L, and he slo widh y he pich, S. The values of hese dimensionless parameers vary over he range from 0 o 1: 0 <L 1 0 S < 1 for he full range of possile slo conguraions. Two dieren slo arrangemens, inline and saggered, will e examined in his sudy as shown in Fig.. The analyical model will assume a uniform velociy of magniude U hrough he channels formed eween he ns, wih no \leakage" of air ou he op of he hea sink. This model assumpion is valid for shrouded hea sinks, where all he airow is conained wihin he channels or for use in conjuncion wih a ow ypass model for unshrouded hea sinks, such as Wirz e al. (1994) or Simons and Schmid (1997). By assuming ha he eecs of he aseplae and he shroud can e negleced, he hea sink is modeled as an array ofn 1 parallel plae channels. This approach is valid in cases where he n heigh is much larger han he n spacing, H >>, such as in he high aspec raio hea sinks used in power elecronics applicaions. The accuracy of his modeling approach will e reduced when H, where he eecs of he aseplae and shroud can no longer e negleced. The independen and dependen variales are nondimensionalized using he n spacing as he characerisic lengh. The Reynolds numer is dened as: U (1) The average hea ransfer rae is non-dimensionalized using he Nussel numer: Q ka (T s T a ) () All air properies are evaluaed a he lm emperaure: T f T s + T a (4) The analyical models for he sloed n hea sink will e ased on a previously developed modeling procedure for he plae n hea sink (Teersra e al., 1999), which will e summarized in he following secion. lae Fin Hea Sink Model An analyical model for forced convecion cooled plae n hea sinks has een developed y Teersra e al. (1999) using he same assumpions presened in he previous secion. The model uses a wo sep procedure ha rs deermines he funcional relaionship for he average hea ransfer coecien for an isohermal D parallel plae channel, and applies his value o a n analysis o predic he average hea ransfer rae for he hea sink. The rs sep in he modeling procedure involves he prolem of forced convecion hea ransfer eween isohermal parallel plaes. Asympoic soluions for he wo limiing cases, fully developed ow and simulaneously developing hermal and hydrodynamic ow, are comined using he inverse formulaion of he Churchill and Usagi (197) composie soluion echnique: Nu h(nu fd ) n +(Nu dev ) ni 1n () The asympoic soluions for he Nussel numer are presened y Teersra e al. (1999) as follows: Nu fd Re r Nu dev 0:664 p s Re r :6 p Re (6) (7) where he channel Reynolds numer, Re, analogous o he channel, or Elenaas Rayleigh numer in naural convecion, is dened as: Re L (8)

4 Comining he asympoic soluions using Eq. () resuls in a model for he isohermal parallel plae channel ha is valid for he full range of Rayleigh numer, as shown in Fig. 3. Using a cominaion parameer value n 3hechannel model was validaed wih resuls from a series of numerical simulaions for a wide range of Re, wih a maximum dierence of % and an RMS dierence of less han 1% Fully-Developed Developing Flow Composie Soluion Figure 3: Channel Model Behavior In mos hea sink designs, he goal of maximizing he n surface area is ofen achieved a he expense of n eciency. Tall, hin ns can lead o increased emperaure dierences eween he ns and aseplae due o increased conducive resisance, and he eeciveness of he n may e reduced. This eec is mos pronounced in forced convecion applicaions, where srong convecion from he ns ends o remove hea more quickly han i can e replaced y conducion. In order o apply he parallel plae model o he prolem of he plae n hea sink, he model presened y Teersra e al. (1999) includes a n analysis o accoun for hese eecs. Treaing he isohermal parallel plae channel model as an ideal value, Nu i, he n eciency is dened y: Nu i (9) where is he average hea ransfer rae for a channel including n eecs. Assuming an adiaaic n ip condiion, he n eciency can e deermined as a funcion of he dimensionless parameers: anh s s Nu i k f k Nu i k f k H H H H L +1 (10) L +1 Therefore, for a plae n hea sink wih N ns, he oal hea ow rae Q can e deermined y: Q N k f A (T s T a ) Nu i (11) " Re Nu i r 3 + 0:664 p s 3!3 Re r13 1+ p 13 3:6 (1) Re Experimenal measuremens were performed y Teersra e al. (1999) for an air cooled plae n hea sink, and he model was found o e in good agreemen wih he daa, wihin.1% RMS dierence. Sloed Hea Sink Models Unlike he plae n hea sink, where clearly dened limiing cases wih asympoic soluions exis, he sloed n hea sink is a complex prolem for which exac soluions are no possile. However, ased on he model for he plae n hea sink i is possile o derive models for he upper and lower ounds on he soluion. These ounds are dened as he soluions o geomerically-similar prolems ha provide upper and lower limis for he resuls, and all daa for a paricular hea sink conguraion will fall eween hese ounds. The original premise for he use of sloed ns was ha new hermal oundary layers would e iniiaed a each n secion, herey reducing he average oundary layer hickness and increasing he average hea ransfer coecien. Lower and upper ounds for he hermal performance of he hea sink are dened ased on wo limiing cases for he oundary layer growh. The lower ound corresponds o he siuaion where no new hermal oundary layers are formed, resuling in hicker oundary layers and lower average hea ransfer coecien values. The upper ound is achieved when new hermal and hydrodynamic oundary layers are iniiaed a each n secion, independen of any upsream eecs. Models for he lower and upper ounds, ased on he plae n hea sink model, will e developed in he following secions. Lower Bound The lower ound on he average hea ransfer for he sloed n hea sink has een descried as occurring when no new hermal oundary layers are formed in he channels eween he ns. This lower ound

5 is modeled y an equivalen channel wih lengh L LB corresponding o he comined lengh of all he n secions, as shown in Fig. 4. This lower ound n lengh can e deermined y: L LB L N S S (13) where N S is he numer of slos in he ns. In he case of saggered slo arrangemen, L LB can e deermined y averaging he oal n secion lenghs on each side of he channel, as shown in Fig. 4). For long hea sinks where << L,henumer of slos can e approximaed y N S L, which leads o he following expression for he lower ound lengh: L LB L 1 S (14) Based on his new, lower ound lengh, wo parameers in he plae n hea sink model will e modied: Re LB Re 1 S (1) L LB L 1 S (16) Figure 4: Schemaic of Lower and Upper Bound Models Susiuing hese lower ound expressions ino he plae n hea sink model gives he following: Nu LB LB Nu i (17) v 0 1 anh u Nu k f H H B L C k 1 S +1A LB v 0 1 (18) u Nu k f H H B L C k 1 S +1A " Re LB Nu i r 3 + 0:664 p s!3 3 Re LB r13 1+ p 13 3:6 (19) Re LB Upper Bound Unlike he lower ound, which assumes ha no new oundary layers were formed as a resul of he slos, he upper ound model assumes he es possile case, ha a new oundary layer forms a each n secion independen ofany upsream eecs. This ound is modeled y focusing on he channel formed eween wo n secions, as shown in Fig. 4. The lengh of his equivalen channel, L UB, corresponds o he lengh of a single n secion: L UB S (0) When expressed in erms of he previously dened dimensionless slo pich and widh, his upper ound lengh ecomes: L UB L 1 S (1) L For inline slos he n spacing is used in he upper ound model, while for saggered slos he widh of he equivalen channel is equal o. As in he lower ound model, he wo parameers in he hea sink model ha depend on he n lengh are modied as follows: Re UB L UB Re L(1 S ) L L(1 S ) () (3)

6 Susiuing hese upper ound parameers ino he hea sink model gives he following expressions for he upper ound for he inline slo arrangemen: Nu UB UB Nu i (4) v 0 1 anhu Nu k f H H B L i 1 S +1C A L UB v 0 1 () u Nu k f H H B L i 1 S +1C A L Nu i " Re UB r 3 + 0:664 p s Re UB r13 1+ p 3:6! (6) Re UB The upper ound model for he saggered n arrangemen is deermined in a similar fashion y susiuing for he channel widh in Eqs. (4-6). EXERIMENTAL MEASUREMENTS Experimenal ess were performed o validae he proposed ounds models using commercially-availale hea sink prooypes wih a variey of slo piches and widhs, as summarized in Tale 1. Maching pairs of hea sinks were fasened ogeher in a ack-o-ack arrangemen, and wo 300 W pencil heaers were press- ino holes drilled a he inerface eween he aseplaes. Given he symmery of his conguraion, i is assumed ha all he hea dissipaed y he heaers was equally disriued eween he wo hea sinks. Temperaure measuremens were performed using mil T-ype copper-consanan hermocouples conneced o a Fluke Helios daalogger. Baseplae emperaures were measured using 4 hermocouples aached o he aseplae of one of he hea sinks a he locaions indicaed y T 1 - T 4 indicaed in Fig.. These hermocouples were locaed close o and far away from he heaers, such ha an arihmeic average of heir measured values would provide a represenaive value for he average aseplae emperaure T s. The maximum dierence noed eween he hermocouple readings during he ess was less han 1% of he average emperaure, validaing he assumpion of a uniform aseplae emperaure. Figure : Schemaic of Experimenal Apparaus Tale 1: Summary of Experimenal Tess Tes L S Slo Arrangemen Saggered Saggered To minimize ow ypass eecs he hea sink assemly was placed inside a lexiglas shroud, as shown in Fig.. The inner dimensions of he shroud were wihin one channel of each of he sides of he hea sink, and he aseplae edges were narrowed o reduce conducion losses o he shroud. The shroud and hea sink assemly were suspended in he cener of a 300 mm 300 mm es secion of a verical, open-circui wind unnel, and he spaces eween he shroud and he es secion were lled o preven ypass. Velociies were measured approximaely 00 mm upsream of he hea sink using a Danec ho wire anemomeer, as shown in Fig.. Tess were performed a one power level, Q 00W, for a range of approach velociy values, U a ms. Each eswas allowed o reach seady-sae over a 3-4 hour period, and he resuls were recorded when he measured values remained unchanged for a 1 minue period.

7 0 1 / L S / / L S / R ( o C / W) 4 3 Saggered Saggered a) ) Figure 6: Experimenal Measuremens for Sloed Hea Sinks: a) vs. Re )R vs. Re Radiaion ess were performed for each of he hea sink prooypes in a vacuum environmen and he resuls were used o reduce radiaion eecs from he convecive ess. In all cases, he relaive magniude of he radiaive componen was less han 1% of he oal hea ransfer, as expeced in forced convecion applicaions. In order o compare he measured values wih he predicions of he ounds models, hey mus e convered in erms of he previously dened dimensionless parameers, Re and. By a simple coninuiy relaionship he average velociy in he channels formed eween he hea sink ns is relaed o he approach velociy: U A a U a (7) A o where A a and A o are he cross secional areas of he inle and hea sink secions, respecively. The resuling channel velociy can e expressed in erms of he Reynolds numer dened in Eq. (1), where he properies are evaluaed a he lm emperaure. The oal hea ransfer rae, Q, (less radiaion) is divided y he oal numerofchannels per side, N, which includes he wo half channels formed eween he ouer ns and he shroud, o give he per-channel hea ransfer rae. This value is non-dimensionalized y he previously dened Nussel numer: Q N (8) kat In order o examine he alance eween he convecion enhancemen of he slos and he reducion in surface area an addiional dependen variale, he hermal resisance R, is dened: R 1 ha (9) where h is he average hea ransfer coecien, which can e relaed o he Nussel numer y is deniion: h k The experimenal daa are ploed in Fig. 6 for all slo conguraions and Reynolds numers examined in he curren sudy. Figure 6 a) presens he Nussel numer vs. Re for he 4 inline and saggered slo conguraions for he range of Reynolds numer 40 Re 180. The daa in Fig. 6 a) indicaes ha a decrease in he slo pich and an increase in he slo size conriues o a susanial increase in he Nussel numer. For he slo con- guraion L 0:09 S 0:4, he is approximaely 30% higher han for any of he remaining cases, L 0:11 S0:, for he full range of Re. For he n spacings and slo piches esed, saggering he slos had only a small eec on he average resuls. MODEL VALIDATION The proposed models for he lower and upper ounds on he average Nussel numer for he sloed n hea sink are compared wih he experimenal daa in Fig. 7 a) - f). In all of he cases repored in Fig. 7 he daa are conained wihin he upper and

8 4 3 1 / L 0.09 S / / L 0.11 S / 0. Daa Daa a) ) / L 0. S / / L 0.44 S / 0. Daa Daa c) d) / L 0. S / 0. Saggered / L 0.44 S / 0. Saggered Daa Daa e) f) Figure 7: Comparison of Bounds wih Experimenal Daa lower limis for he full range of Re.For each of he slo conguraions esed he daa moves smoohly eween he limis, from he lower ound for small Re o he upper ound for large Re.For small L values and larger values of S, such as in Fig. 7 a) he daa ends owards he upper ound wih each n secion acing independenly, and here is a signican dierence eween he ounds. For large L and small values of S,suchasshown in Fig. 7 d) and f), he dierence eween he lower and upper ounds, and he daa approaches oh limis for he range of Re esed. Figure 7 also shows ha here is lile dierence eween he inline and saggered slo arrangemens for he range of conguraions esed. The curren ounds models can e used o formulae a design ool o predic sloed hea sink perfor-

9 mance for he range of slo conguraions examined in his sudy, 0:11 L 0:44 and S 0:. For his range of values, he average Nussel numer can e approximaed y he arihmeic mean of he lower and upper ounds: Nu LB + Nu UB (30) which resuls in an RMS percen dierence of 1% for he range of channel Reynolds numer: 40 Re 180. For Re > 180 or L < 0:1 he upper ound model should e used o predic he average Nussel numer, Nu UB. Because of he limied range of S values in he curren daa i is dicul o formulae a reliale procedure o predic opimal slo pich and widh as a funcion of Re and he surface area A. As shown in Figs. 6 a) and ), he hermal performance of a sloed hea sink depends on oh he enhancemen of he hea ransfer and he reducion in surface area, and he predicion of he \es" values of L and S may change as a funcion of Re. Addiional sudy and experimenal daa for oher values of S are required o develop a general procedure for opimizing slo size and pich ha is applicale o he full range of he independen parameers. REFERENCES R-Thea Inc., 1999, 60 Kesrel Rd., Mississauga, ON, LT 1Y9, Canada, hrough personal communicaion. Teersra,., Yovanovich, M.M., Culham, J.R and Lemczyk, T.F., 1999, \Analyical Forced Convecion Modeling of lae Fin Hea Sinks," 1h Annual IEEE Semiconducor Thermal Measuremen and Managemen Symposium, San Diego, CA, March 9-11, pp SUMMARY AND CONCLUSIONS Models for he lower and upper ounds on he average dimensionless hea ransfer coecien for sloed n hea sinks have een developed. Experimenal measuremens have een performed for a range of slo conguraions, S 0: 0:09 L 0:44, for he range of Reynolds numer, 40 Re 180. The daa has een shown o lie wihin he proposed ounds for he full range of es condiions, wih he daa moving smoohly from he lower o upper limis. An approximae model for predicing he performance of sloed n hea sinks ased on an arihmeic mean of he ounds has een proposed, wih an RMS percen dierence from he model of 1%. The need for addiional daa o develop a general mehod suiale for he opimizaion of he slo size and pich asafunc- ion of he independen parameers has een idenied. ACKNOWLEDGMENTS The auhors graefully acknowledge he coninued nancial suppor of R-Thea Inc. and Maerials and Manufacuring Onario.