Analytical Forced Convection Modeling of Plate Fin Heat Sinks

Analytical Forced Convection Modeling ofPlate Fin Heat SinksP. Teertstra, M.M. Yovanovich and J.R. CulhamMicroelectronics Heat Transfer LaboratoryDepartment of Mechanical EngineeringUniversity ofWaterlooWaterloo, Ontario, CanadaT.F. LemczykR-Theta Inc.Mississauga, Ontario, CanadaSEMI-THERM XVMarch 9, 1999Microelectronics Heat Transfer LaboratoryUniversity ofWaterloo

Introduction: Plate Fin Heat Sinks Heat transfer enhancement for air cooled applications:{ increase eective surface area{ decrease thermal resistance{ reduce operating temperatures Plate n heat sinks most commonconguration Heat convected by ow through channels between nsSEMI-THERM XVMarch 9, 1999Microelectronics Heat Transfer LaboratoryUniversity ofWaterloo

Introduction: Heat Sink Selection Numerous plate n heat sinks currently available Heat sink selection depends on many factors:{ performance{ dimensional constraints{ available airow{ cost Quick and accurate design tools are required:{ predict performance early in design{ perform trade o or \what if?" studies{ alternative to numerical simulation, experimentsSEMI-THERM XVMarch 9, 1999Microelectronics Heat Transfer LaboratoryUniversity ofWaterloo

Objectives Develop analytical forced convection model for averageheat transfer rate for plate n heat sinks:{ laminar ow{ full range of developing and fully developed ow{ non-isothermal ns Perform experimental measurements for commercial heatsink and compare results with modelSEMI-THERM XVMarch 9, 1999Microelectronics Heat Transfer LaboratoryUniversity ofWaterloo

Problem Denition - Plate Fin Heat Sink Array ofN plates in perfect contact with baseplate Baseplate assumptions:{ isothermal{ adiabatic lower surface, edges Uniform velocity through channelswith no \leakage" out edges:{ shrouded heat sink{ with ow bypass model for un-shrouded heat sinks Heat sink modeled as N ; 1 parallel plate channelsSEMI-THERM XVMarch 9, 1999Microelectronics Heat Transfer LaboratoryUniversity ofWaterloo

Problem Denition - Parallel Plate Channel Assume b H:{ 2D channel ow{ neglect baseplate, shroud eects Isothermal boundary conditions Reynolds number: Nusselt number:Nu b =Re b = UbQbkA(T w ; T a ) A =2LHSEMI-THERM XVMarch 9, 1999Microelectronics Heat Transfer LaboratoryUniversity ofWaterloo

Model Development - Parallel Plate Channel Forced convection solutions available for 2 limiting cases:{ fully developed ow { developing ow Churchill and Usagi (1972) composite solution:i;1=nNu b =h(Nu fd ) ;n +(Nu dev ) ;nDeveloping Flowlog( Nu b)Fully-DevelopedComposite Solutionlog( Re b)SEMI-THERM XVMarch 9, 1999Microelectronics Heat Transfer LaboratoryUniversity ofWaterloo

Model Development: Fully Developed Flow Asymptote Enthalpy balance assuming uid exit temperature = T wQ = _mc p (T w ; T a ) Fully developed ow asymptote:Nu b = 1 2 Re? b Prwhere channel Reynolds number dened as:Re ? b = Re b b LSEMI-THERM XVMarch 9, 1999Microelectronics Heat Transfer LaboratoryUniversity ofWaterloo

Model Development - Developing Flow Asymptote Laminar forced convection solution in entrance of at,rectangular duct (Sparrow, 1955):Nu dev =0:664 p Re ? b Pr1=3 1+ 3:65p Re?bvalid for Pr 1.! 1=2 Approaches the at plate solution for large Re ? b :Nu L =0:664pReL Pr 1=3SEMI-THERM XVMarch 9, 1999Microelectronics Heat Transfer LaboratoryUniversity ofWaterloo

Model Summary - Parallel Plate Channel26 Re?bNu b = 4Pr2;n+0! 11=2p@ 0:664 Re?b Pr1=3 1+ p 3:65 ARe?b;n375;1=nNu b10 110 0Numerical Data10 -1 Combination parameter fromFLOTHERM solutionsn =3 2.1 % RMS dierence betweenmodel and numerical dataModel10 -210 -1 10 0 10 1 10 2Re b*SEMI-THERM XVMarch 9, 1999Microelectronics Heat Transfer LaboratoryUniversity ofWaterloo

Model Development - Heat Sink with Non-Isothermal Fins High aspect ratio heat sinks:{ tall thin ns, small spacing{ increased surface area for convection{ eciency reduced Fin eciency : = Nu bNu iwhere Nu i \ideal" value from channel modelSEMI-THERM XVMarch 9, 1999Microelectronics Heat Transfer LaboratoryUniversity ofWaterloo

Model Development - Heat Sink with Non-Isothermal Fins Assume adiabatic condition at n tip:stanh (mH)hP = m = h = Nu i k fmHkA c b =tanhss2 Nu ik fk2 Nu ik fkHbHbHt H tL +1t tL +1SEMI-THERM XVMarch 9, 1999Microelectronics Heat Transfer LaboratoryUniversity ofWaterloo

Model SummaryNu bQ = Nu b kA (T s ; T a )b= Nu i tanh26 Re?bNu i = 4Pr2ss2 Nu ik fk2 Nu ik fk;3+ NHbHbHt H tt L +1 tL +10! 1 31=2;3;1=3p@ 0:664 Re?b Pr1=3 1+ p 3:65 A 7Re? 5bSEMI-THERM XVMarch 9, 1999Microelectronics Heat Transfer LaboratoryUniversity ofWaterloo

Experimental Apparatus High aspect ratio heat sink H=b 20 Back-to-back arrangement Heat sinks in Plexiglas shroud Approach velocity measured withhot wire anemometer Temperatures measured at 4locations on baseplate Radiation losses measured inseparate experimentSEMI-THERM XVMarch 9, 1999Microelectronics Heat Transfer LaboratoryUniversity ofWaterloo

Experimental Procedure Measurements performed for the following cases:U aQ tot2 3 4 5 6 7 8 m=s100 200 300 400 500 W Re ? related to approach velocity U b a by continuity: Re ? = Ub2 Ao bbL = A 2U aL Nusselt number:Nu b =(Q tot =2) bkN(2 LH) ; T s ; T aSEMI-THERM XVMarch 9, 1999Microelectronics Heat Transfer LaboratoryUniversity ofWaterloo

Model Validation543Nu b2Experimental DataChannel ModelHeat Sink Model110 20 30 40 50Re b*SEMI-THERM XVMarch 9, 1999Microelectronics Heat Transfer LaboratoryUniversity ofWaterloo

Summary and Conclusions Analytical forced convection model developed foraverage heat transfer rate for plate n heat sinks{ 2D channel model for shrouded high aspect ratioheat sinks, b H{ Temperature variation between ns and baseplate Model in excellent agreement with measured values:{ 2.1 %RMSdierence{ 6 % maximum dierenceSEMI-THERM XVMarch 9, 1999Microelectronics Heat Transfer LaboratoryUniversity ofWaterloo

AcknowledgmentsThe authors gratefully acknowledge the support of: R-Theta Materials and Manufacturing OntarioSEMI-THERM XVMarch 9, 1999Microelectronics Heat Transfer LaboratoryUniversity ofWaterloo