Introduction: Plate Fin Heat Sinks Heat transfer enhancement for air cooled applications: { increase eective surface area { decrease thermal resistanc

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1 Heat Transfer Laoratory Microelectronics ofwaterloo University Analytical Forced Convection Modeling of Plate Fin Heat Sinks P. Teertstra, M.M. Yovanovich and J.R. Culham Department of Mechanical Engineering Waterloo, Ontario, Canada T.F. Lemczyk R-Theta Inc. Mississauga, Ontario, Canada

2 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 common conguration Heat convected y ow through channels etween ns

3 Introduction: Heat Sink Selection Numerous plate n heat sinks currently availale Heat sink selection depends on many factors: { performance { availale airow { dimensional constraints { cost Quick and accurate design tools are required: { predict performance early in design { perform trade o or \what if?" studies { alternative to numerical simulation, experiments

4 Ojectives Develop analytical forced convection model for average heat 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 heat sink and compare results with model

5 Uniform velocity through channels no \leakage" out edges: with XV Prolem Denition - Plate Fin Heat Sink Array of N plates in perfect contact with aseplate Baseplate assumptions: { isothermal { adiaatic lower surface, edges { shrouded heat sink { with ow ypass model for un-shrouded heat sinks Heat sink modeled as N ; 1 parallel plate channels

6 = U Re Q w ; T a ) ka(t Prolem Denition - Parallel Plate Channel Assume H: { 2D channel ow { neglect aseplate, shroud eects Isothermal oundary conditions Reynolds numer: Nusselt numer: Nu = A =2LH

7 Nu = h(nu fd ) ;n +(Nu dev ) ;n i;1=n Model Development - Parallel Plate Channel Forced convection solutions availale for 2 limiting cases: { fully developed ow { developing ow Churchill and Usagi (1972) composite solution: Developing Flow log( Nu ) Fully-Developed Composite Solution log( Re )

8 Enthalpy alance assuming uid exit temperature = T w XV Q = _mc p (T w ; T a ) Re? = Re L Model Development: Fully Developed Flow Asymptote Fully developed ow asymptote: Nu = 1 2 Re? Pr where channel Reynolds numer dened as:

9 Laminar forced convection solution in entrance of at, duct (Sparrow, 1955): rectangular XV dev =0:664 p Re? Pr1=3 1+ 3:65 p Nu? Re Nu L =0:664 p ReL Pr 1=3! 1=2 Model Development - Developing Flow Asymptote valid for Pr 1. Approaches the at plate solution for large Re? :

10 Nu = Re? XV ;n p? Re 0:664 Pr1=3 1+ 3:65 Re! 1=2 1 n =3 Model Summary - Parallel Plate Channel ;1=n 3 ;n 2 0 Pr A Comination parameter from 10 0 FLOTHERM solutions Nu % RMS dierence etween Numerical Data Model Re * model and numerical data

11 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 i Nu where Nu i \ideal" value from channel model

12 Model Development - Heat Sink with Non-Isothermal Fins Assume adiaatic condition at n tip: s = tanh (mh) m = hp h = Nu i k f c ka mh tanh s 2 Nu i k f H H t +1 L k t = s 2 Nu i k f H H t +1 L k t

13 Nu Nu i = XV Re? ;3 H H H t H t p? Re 0:664 Pr1=3 1+ 3:65 Re 1;3 3! ;1=3 1=2 7 5 Model Summary N = Nu ka (T s ; T a ) Q t L +1 s tanh 2 Nu i k f k = Nu i s t L +1 2 Nu i k f k 0 2 Pr A 2

14 High aspect ratio heat sink H= 20 Approach velocity measured with wire anemometer hot Temperatures measured at 4 on aseplate locations Radiation losses measured in experiment separate XV Experimental Apparatus Back-to-ack arrangement Heat sinks in Plexiglas shroud

15 U a Q tot Nu = Ao 2 Experimental Procedure Measurements performed for the following cases: m=s W Re? related to approach velocity U a y continuity:? U2 = Re = L A U a L Nusselt numer: tot (Q =2) ; T s ; T a LH) kn(2

16 Model Validation Nu Re * Experimental Data Channel Model Heat Sink Model

17 Analytical forced convection model developed for heat transfer rate for plate n heat sinks average XV Summary and Conclusions { 2D channel model for shrouded high aspect ratio heat sinks, H { Temperature variation etween ns and aseplate Model in excellent agreement with measured values: { 2.1 %RMS dierence { 6 % maximum dierence

18 Acknowledgments The authors gratefully acknowledge the support of: R-Theta Materials and Manufacturing Ontario