Numerical Modelling of Turbulent Heat Transfer From Discrete Heat Soucers in Liquid-cooled Channel

8/13/2019 Numerical Modelling of Turbulent Heat Transfer From Discrete Heat Soucers in Liquid-cooled Channel

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~ Pergamon l n t J He a t M a s s T r a n s f er Vol. 41, No. 10, pp. 1157-1166, 1998 1998 Elsevier Science Ltd. Al l rights reservedPrinted in Great Britain00174310/98 $19.00+0.00PI I : S0017-9310(97)00257-3

N u m e r i c a l m o d e l l i n g o f t u r b u l e n t h e a t t ra n s f e rf r o m d i s c r e t e h e a t s o u r c e s i n a l i q u i d - c o o l e dc h a n n e lG. P. XU, K. W. TOU and C. P. TSO

School of Mechanical and Production Engineering, Nanyang Technological University,Singapore 639798R e c e i v e d 14O c t o b e r 1996a n d i n f i n a l f o r m 13A u g u s t 1997)

Abstract--Two-dimensional forced convection heat transfer between two plates with flush-mounted dis-crete heat sources on one plate to simulate electronics cooling is studied numericallyusing a finite differencemethod. The two plates form part of a liquid-cooled rectangular channel. The high Reynolds number formof the k-e turbulence model is used for the computations, which are performed for the liquids water and FC-72 over a range of Reynolds numbers from 104 to 1.5 x 105.The numerical procedures and implementation ofthe k-e model are validated by comparing the predictions with published experimental data of Mudawarand Maddox [11] and Incropera et al. [7] for a single plain heat source as well as with reported multi-chipmodule data of Incropera et al. and Gersey and Mudawar [12]. The effects of the ratio of the channelheight to the length of heat source and orientation on the heat transfer characteristics inside the channelare investigated. 1997 Elsevier Science Ltd.

I N T R O D U C T I O NIn the choice of cool ing fluid in electronic systems, airis widely used and will always be favored whereverpossible because of its economy and its ease of beinghandled. However, as circuit densities on a single sili-con chip conti nue to increase and as chips are packedin closer proximity on multi-chip modules, power den-sities continue to rise at both the chip and modulelevels, dissipating power more than 10 W/cm2 andbeyond, while chip temperature should be maint ainedbelow 85C. As it is becoming increasingly difficult torely on air cooling to dissipate the heat, l iquid coolingis considered, and it may be the only practical methodfor main taini ng reasonable component temperaturesin high power chips.

Although numerical investigations of turbulentheat transfer for air cooling of electronic systems werewidely reported, including those of Kim and A nan d[1], Asako and Faghri [2, 3], Knight and Crawford[4], scant numer ical work h ad been extended to liquidcooling. Heat transfer in lamin ar flow with one andtwo heat sources flush-mounted to one wall of a par-allel plate channel was considered numerically byRamadhyani e t a l . [5]. Moffatt e t a l . [6] and Incroperae t a l . [7] numerically predicted turbulent flow in arectangular chan nel using a relatively simple model :zero equation turbulent model combined with wallfunction. Mahaney e t a l . [8] extended the work ofIncropera e t a l . to the low Reynolds number regimewhere mixed convection becomes important.

Experimental studies on forced convection heattransfer from a single plain heat source as well as from

a multi-chip module had been carried out for liquidcooling by many investigators. One of the earliestworks us ing silicone oil and R-113 flowing over dis-crete sources with surface areas ranging from 1 to200 mm 2 was performed by Baker [9, 10]. Forcedconvect ion heat tr ansfer dat a for a single 12.7 mmsquare heat source and for a 4 x 3 array of heat sourceswere obtained by Incropera e t a l . [7] using water andFC-77 as the working fluid, and it was found that theupstream thermal boundar y affected that o f down-stream. Tests were conducted using a single heatsource hav ing the dimensi ons 12.7 12.7 mm, flush-mounted to one wall of a vertical rectangular channelwith 38.1 mm width an d 12.7 mm height at atmo-spheric pressure by Mudaw ar an d Madd ox [11], withFC-72 as the coolant. Although the slopes of the cor-relations obtained by Mudawar and Maddox a nd byIncropera e t a l . are almost identical, the data of theMudawar and Maddox was approximately 36 per centhigher than the latter. Experiments were performedby Gersey and Mudawar [12] and FC-72 on a seriesof nine in-line simulated microelectronic chips in aflow channel to ascertain the effect of or ient ation angleon the forced convection. The simulated chips,measuring 10 x 10 mm, were flush-mounted to onewall of a 20 x 5 mm flow channel. However, theyfound that the upstream thermal boundary had noeffect on that of downstream, and the data from thenine flush-mounted chips were correlated by a singleequation.

The objective of the present work is to studynumerically turbulent heat transfer with single andfour in-li ne flush-mounted heat sources for liquid

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11 58 G . P . X U e t al

C l ,c~c~DhfhHkLLiNULe~e~P rqReLSSLTw

c~N O M E N C L A T U R E

t u r b u l e n t m o d e l l i n g c o n s t a n t s vps p e c i fi c h e a t a t c o n s t a n t p r e s s u r e w[ J /kg K ] x 7t u r b u l e n t m o d e l l i n g c o n s t a n th y d r a u l i c d i a m e t e r o f c h a n n e l [ m] YLf r i c t i o n f a c t o rh e a t t r a n s f e r c o e f f ic i e n t [ W / m 2 K ]h e i g h t o f c h a n n e l [ m]t u r b u l e n t k i n e t i c e n e rg y [ m 2 / s 2 ] Fh e a t s o u r c e l e n g t h i n f l o w d i r e c t i o n [ m ]d i s t a n c e f r o m i n l e t to f i r s t h e a t s o u r c e[m]N u s s e l t n u m b e r b a s e d o n l e n g t h o fh e a t s o u r c e , = hL/2p r o d u c t i o n o f k i n e t i c e n e rg y [ P a lp a r a m e t e r d e f i n e d in e q u a t i o n ( l 3 )P r a n d t l n u m b e rh e a t f lu x [ W / m ER e y n o l d s n u m b e r b a s e d o n l e n g th o fh e a t s o u r c e , = vinL/vd i s t a n c e b e t w e e n h e a t e r s [ m ]s o u r c e t e r m e f ti n l e t t e m p e r a t u r e [ C ] fh e a t s o u r c e w a l l t e m p e r a t u r e [ C ] i nn o n d i m e n s i o n a l t e m p e r a t u r e n e a r t h e pw al l tv e l o c i t y i n x - d i r e c t i o n [ m / s] xv e l o c i t y i n y - d i r e c t i o n [ m / s] y

n o n d i m e n s i o n a l v e l o c i t y n e a r t h e w a l lw i d t h o f c h a n n e l [ m ]n o n d i m e n s i o n a l d i s t a n c e m e a s u r e df r o m t h e w a l lt o t a l c h a n n e l l e n g t h [ m] .

G r e e k s y m b o l sc o e f f ic i e n t i n e q u a t i o n ( 1 7 h )t r a n s p o r t c o e f f i c ie n t i n g e n e r a le q u a t i o n

e t u r b u l e n c e d i s s i p a t i o n r a t e [ N / s m 2]2 t h e r m a l c o n d u c t i v i t y [ W / m K ]# d y n a m i c v i s c o s i t y [ N s / m 2]v k in em at ic v i sco s i ty [ m2/s ]p den s i ty [ kg /m 3]a k , t ry , a t t u r b u l e n t m o d e l l i n g c o n s t a n t sz shea r s t r es s [ N /m 2]q~ gen er a l va r ia b le .

S u b s c r i p t sef fect ivef lu idin le tr e f e r r i n g t o p o i n t P n e a r t h e w a l lt u r b u l e n tr e f e r r i n g t o t h e x - d i r e c t i o nr e f e r r i n g t o t h e y - d i r e c t i o n .

c o o l a n t ( w a t e r w i t h P r = 5 .4 2 a n d F C - 7 2 w i t hP r = 9 .0 ) . T h e s t a n d a r d h i g h - R e y n o l d s n u m b e r k - em o d e l a n d w a l l f u n c t i o n a r e e m p l o y e d f o r t h e c o m -p u t a t i o n s . T h e k - e m o d e l i s a s e m i - e m p i r i c a l m o d e lt h a t h a s b e e n p r o v e n t o p r o v i d e e n g i n e e r in g a c c u r a c yi n a w i d e s p e c t r u m o f t u r b u l e n t f lo w s , in c l u d i n g s h e a rf l o w a n d w a l l - b o u n d e d f l o w s . T h e w a l l f u n c t i o nm e t h o d i s c o s t - e f fe c t i v e a s i t s u b s t a n t i a l l y r e d u c e s t h ec o m p u t e r s t o r a g e a n d C P U t i m e . T h i s m e t h o d i s j u s -t i f i e d s i n c e t h e h e a t s o u r c e s i n t h e p r e s e n t s t u d y a r ef l u s h - m o u n t e d a n d t h e r e is n o r a p i d c h a n g e i n c h a n n e lg e o m e t r y c a u s in g s t r o n g f l o w a c c e l e r a t i o n / r e t a r d a t i o no r f l o w s e p a r a t i o n / r e a t t a c h m e n t . C o n s e q u e n t l y , n os i g n if i ca n t d e p a r t u r e f r o m l o c a l o n e - d i m e n s i o n a l i t y i nt h e n e a r - w a l l r e g i o n i s a n t i c i p a t e d . T h e p r e s e n t r e s u l t sa r e c o m p a r e d w i t h r e l e v a n t e x p e r i m e n t a l d a t a . T h ee f f e c t s o f t h e o r i e n t a t i o n o f t h e f l o w c h a n n e l a n d t h er a t i o o f c h a n n e l h e i g h t t o t h e h e a t s o u r c e l e n g t h o nh e a t t r a n s f e r a r e i n v e s t i g a t e d .

M O D E L F O R M U L A T I O NDescription of the problem

T h e n u m e r i c a l m o d e l i s f o r m u l a t e d b a s e d o n s t e a d y ,t w o - d i m e n s i o n a l t u r b u l e n t h e a t t r a n s f e r i n a c h a n n e l

f o r m e d b e t w e e n t w o p l a t e s a s s h o w n i n F i g . 1 . T w om o d u l e s a r e c o n s i d e r e d ; in t h e f i rs t c a se , a s i n g l e h e a ts o u r c e f l u s h - m o u n t e d o n t h e l e f t w a l l a s s h o w n i n F i g .1 a ) , a n d i n t h e s e c o n d c a s e , f o u r i n - l i n e h e a t s o u r c e sm o u n t e d o n t h e l e f t w a l l a s s h o w n i n F i g . 1 b ) .

T h e r i g h t s u r f a c e i s a s s u m e d t o b e a d i a b a t i c , w h i l et h e l e ft i s a s s u m e d t o b e h e a t e d w i t h c o n s t a n t h e a tf lu x q~ g e n e r a t e d f r o m o n e o r m o r e h e a t e d e l e m e n t se a c h o f l e n g t h L a n d r e g u l a r l y s p a c e d a t d i s t a n c e sa l o n g t h e s u r f a c e . I t i s a s s u m e d t h a t t h e s u r f a c e o u t -s i d e t h e e l e m e n t s i s a d i a b a t i c . T h e d i s t a n c e f r o m t h ec h a n n e l e n t r a n c e t o t h e f i r s t s o u r c e , L i , a n d t h e d i s -t a n c e f r o m t h e l a s t s o u r c e t o t h e c h a n n e l e x i t , a r ee a c h e q u a l t o 5 L . F o r s i m p l i c i t y , t h e h e a t s o u r c e s a r ea s s u m e d t o b e s t r i p s o f i n f i n i t e l e n g t h i n t h e d i r e c t i o np e r p e n d i c u l a r t o t h e p a p e r , t h u s r e n d e r i n g t h e s i t u -a t i o n t w o - d i m e n s i o n a l . T h e w o r k i n g f l u i d is a s s u m e di n c o m p r e s s i b l e , a n d t h e f l u i d p r o p e r t i e s a r e c o n s t a n t .A C a r t e s i a n c o o r d i n a t e s y s t e m i s s e t u p a s s h o w n i nFig. 1.

Governin9 equationsT h e g i v e n f l o w f i e l d m u s t s a t i s f y t h e c o n t i n u i t y

e q u a t i o n


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Num erical mode l l ing of tu rbu le n t heat t ransfer 1159

Z

. T ra i l i n g ed g e

,y edge[ i

Y L

Tx( a ) S i n g l e h e a t s o u r c e ( b ) F o u r i n - l in e h e a t s o u r c e s

Fig. 1. Physical mode l for coolin g cha nne l in electronic equip me nt.

O p l ~ ) a p v )0-- ~ + ~ = O. (1 )B a s e d o n t h e f o r e g o i n g a s s u m p t i o n s , t h e e q u a t i o nf o r a g e n e r a l v a r i a b l e t r a n s p o r t e d b y c o n v e c t i o n a n dd i f f u s io n i n t u r b u l e n t f l ow i s g iv e n b y

a pu ) a pv~p) r + F + S .O x + O y a y(2)

I n t h i s e q u a t i o n , t h e g e n e r a l v a r i a b l e q~ r e p r e s e n t s u ,v , T , k , o r e , w h e r e F a n d S r e p r e s e n t t h e a p p r o p r i a t et r a n s p o r t c o e f fi c ie n t a n d s o u r c e t e r m r e s p e ct i ve l y , a ss h o w n i n T a b l e 1.I n t h e t a b l e ,

/~err = ~t + #, (3)I t = C . p k 2 / e 4 )

o yP k = I z [ L\ax kay j ~ + ~ ) j . 5 )

F o r h o r i z o n t a l f l o w , 0 = 0 ; f o r v e r t i c a l u p f l o w ,0 = 90 .

W a l l f u n c t i o nF o r c o m p l e t i o n o f t h e p r o b l e m f o r m u l a t i o n , t h e

w a l l f u n c ti o n m e t h o d i s e m p l o y e d n e a r t h e c h a n n e lw a l l . S i n c e a f i n i te d if f e r e n c e m e t h o d i s u s e d , t h eb o u n d a r y c o n d i t i o n s h o u l d b e s p e c if ie d in t e r m s o fw a l l f l u x e s b e t w e e n t h e w a l l a n d t h e a d j a c e n t g r i dp o i n t ( d e n o t e d b y P ) a s s h o w n i n F i g . 2 . T h e r e g i o nc l o se t o a s o l i d w a l l c a n b e d i v i d e d i n t o t w o s u b l a y e r s ,( a ) a l a m i n a r o r v i s c o u s s u b l a y e r w h e r e v i s c o u s ef f e ct sa r e d o m i n a n t a n d ( b ) a t u r b u l e n t s u b la y e r . T h e w a l lv e lo c it y c a n b e o b t a i n e d f r o m L a u n d e r a n d S p a l d in g[14] as :

Tab le 1 . General var iab les and correspond ing d i f fusion coefficien ts an d sourcesr s

u + 1 ~ tv + ,T / P r + a rk + I~,/ake + ,/a~

aptOx + Ob~o~ aulOx)]tOx + a~ O vt O x) ]l O y- pg cos 0- aplOy + ob~o~ oulay)]lax + aluo~ avlay)ltOy- pg sin 00P k - p e( C iP ke - C2pe:) / k

C~ = 1.44, C2 = 1.99, C,, = 0.09, ak = 1.0, cr = 1.3, at = 0.9 [13].


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1160 G P. XU et al .Viscous Turbulentsublayer sublayer

Fig. 2. Location of points adjacent to wall.

u+ = 2.5Inx,C+5.5, forxl > 11.5P (6)c+ CC +P p, forx,f d 11.5 (7)

where

x; = xp(c;4k;i2)Y (8)v+ _ vp(C; 4k;2)P - L4lP (9)

(10)The wall turbulent viscosity at point P can be cal-culated from

p,+,%

The wall temperature can be obtained from [l] asT+ = (Tw T,JpC,,C, 4k;2

P ,I4% = (r,[2.5 ln(9x,+) + P,J

where(12)

p =9(:-1)(:,. (13)

The relation between the heat flux and temperature atthe wall surface is given by Fouriers law as

q;;. = ,Y (14)

where /I, is the thermal conductivity in the fluidbetween the wall and the position P. On rearranging,

1, = ~ 5 wpa,[2.5 ln(9x,+) + PC] ifxc > 11.5. (15)

The dissipation rate of kinetic energy is calculatedfrom

C3/4k3/25 B Y0.4x, (16)

It should be noted that the dissipation rate E in the Eequation is also employed to prescribe the value nearthe wall control volume.Boundary condi t ions

The boundary conditions are specified as followsAtx=O,x=H, u=v=O. (17a)

Aty=O, u=O. (17b)Aty=O, v=v,,. (17c)Aty=O, T= T,,. (174

At x = 0, x = H (except at the heat sources),dTyg 0. (17e)

At the heat sources, q:, = - I F:. (170

At the outlet, 8Ty = Y,, - = 0.ay (1%)Equations (17~) and (17d) prescribe a uniform vel-ocity and a uniform temperature at the channelentrance, and equation (17g) assumes the streamwiseenergy diffusion flux to be negligible at the outlet.

In view of the situation that the experimental valuesof k and E are not known at the inlet, some reasonableassumptions can be made. The inlet kinetic energy ofturbulence is estimated according to a certain per-centage of the square of the average inlet velocity

k,, = civ,, (17h)where v,, is the average inlet velocity and c( is a per-centage between 0.5% and 1.5%, following Patankaret al . [151 for fully developed turbulent flow.

The inlet dissipation is calculated according to theequationE,, = O.lk; (17i)

Prediction results are insensitive to inlet k and E values[151 since the viscous effects are small compared tothe fluid inertia at high Reynolds number. This hasbeen confirmed after a few numerical trials.

The outlet conditions for k and E are not specified,but are given at the inlet as per equations (17h) and(17i). The problem is then approximated by a one-way space coordinate of parabolic nature under theaction of fluid flow and when convection mode oftransport dominates the diffusion mode. In this situ-ation, the solution is largely controlled by theupstream condition and very little by the downstreamone.


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Numerical modelling of turbulent heat transfer 1161R e y n o l d s n u m b e r , N u s s e l t n u m b e r

For calcula tion of the Reynolds number , L is selec-ted as the characteristic dimension, that is,

v i n LR e L = (18) ~V ~7

m ; _ q ; L 1 9 ) ~2 (T ,. ,- T~,)2

The local Nusselt number is defined asN b l L i - -

while the average Nussel t num ber is calculated fromNUL,i

i = IN u L - - -m (20)where m is the total nu mber of grid points in the heatsources.

N U M E R I C A L S O L U T I O NThe numer ical procedure Semi-Implicit Method for

Pressure-Linked Equation (SIMPLE) is used to solvethe basic conservation equations. The finite volumetechnique has been described in detail by Patanker[16]. This algor ithm provides a remarkably successfulimplicit method for simulating incompressible flowincluding those involved in the cooling of electronicequipment [1-3, 17]. The set of discretization equa-tions for each variable is solved by the line by lineprocedure, which is the combi nation of the Tri-Diagonal Matrix Algorithm (TDMA) and the Gauss-Seidal iteration technique. The SIMPLE algorithmsolves the pressure equation to obtain the pressurefield and solves the pressure-correction equation tocorrect the velocities.

The convergence criterion used in this computati onis that the value of the mass flux residuals (mass flow)Rsum in each con trol volume takes a value less than10 -9, and relative values of velocities u and v andtemperature cease to vary by more than 10 -5 betweentwo successive iterations. The under- relaxa tion factorvalues for velocity, pressure, turbulence kineticenergy, energy dissipation rate and turbulen t viscosityare set to 0.5, 0.8, 0.1-0.4, 0.1-0.4, and 0.5, respec-tively. A sufficient num ber of iterations, typically 2000to 10 000, are performed to obta in a converged solu-tion, with the number of interactions depending onthe Reynolds number, the geometry and the workingfluid.G r id - in d e p e n d e n c e

Grid independence is established by examining thewall tempera ture dist ribu tion at the surface of the heatsource. A non- uni for m mesh with a large con-centration of nodes in heat sources is set up. Thecomputational region consists of 50 grid lines in thex-direction and 93 grid lines in the y-direction, thelatter i nclud ing 30 grid lines in the single heat source.

6 0

5 0

4 0

3 0

2 0

N o t e : M e s h e sResu l ts fo r 60x93 mesh es i sc l os e t o t h a t f o r 5 0 x 9 3 m e s h e s. 3 0 X 4 8

- - - 4 0 X 6 3- - 5 0 X 9 3. . . . 6 0 9 3

4 5 6 7 8y L

Fig. 3. Effect of grid size on temperature distribution(L = 12.7 mm, H/L = 1 , ReL = 1.5 x 105).

For the purpose of the grid independence study, the R eis chosen to be 1.5 105. Fou r different non-u niforml yspaced grid sizes are used 30 x 48 (with 15 grid pointsin the y-direct ion of the heat source), 40 x 63 (with 20grid points in the y-direction of the heat source),50 93 and 60 x 93 (with 30 grid points in the y-direction of the heat source). The wall temperaturedistr ibuti on for the various grid sizes and the Nusseltnumbe rs are shown in Fig. 3 and Table 2, respectively.Grid independence is declared when maximumchanges in temperature distribution, Nusselt numb erand friction factor are less than 3%. The results fromthe last three grids are acceptable.

R E S U L T S A N D D I S C U S S I O N SS i n 9 p l a i n h e a t s o u r c e

Figure 4 shows the comparison of the numericallypredicted with the experimen tally determined Nusseltnumber obtained by Mudawa r and Maddox [11] forthe single plain heat sources using FC-72 as the work-ing fluid. The numerical results agree well with theexperimental data, with differences of about 10%.

Figure 5 shows the wall temperature distributionsat Reynolds numb er 7 l04 an d 1.5 105, respec-

Table 2. Effect of grid size on Nusselt number and frictionfactor (ReL = 1.5 l0s, P r = 9.0, FC-72)Grid NuL f20 x 48 980.9 0.011730 x 48 1113.0 0.013240 63 1142.0 0.013650 x 93 1141.0 0.013660 93 1139.0 0.0135


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1 16 2 G . P . X U e t a l .1 0 3

a 102

- - P r e s e n t m o d e l. . . . M u d a ~

1 0 1 , , , , , , , , I , , , , , , , , I1 0 s 1 0 4 1 0 s

R eLFig. 4 . Com par ison of p resen t model wi th the exper imentald a t a o f M u d a w a r a n d M a d d o x [ 1 1] H / L = 1, L = 12.7ram).

1 0 3

n 102

- - P r e s e n t m o d e l- - - - E x p e r i m e n t a l r e s u l t [ 7 ]- - - N u m e r i c a l r e s u l t [ 7 ] ~ / -

J

1 0 1 ~ ~ I I1 0 3 1 0 4 1 0 5

R e LFig. 6 . Com par ison of p resen t model wi th the exper imentald a t an d n u mer i ca l re su lt s o f In c ro p era et al. [7] (H = 11.9mm , L = 12 .7 ram).

70160

G 5 0o ._ .~ 40.=.E 3O

2010

. . . . . R e L = 7 X 10 4I tl - - R e L = 1 . 5 X 1 0 sI q = 1 0 W / c m2

T i n = 3 0 o Ci FC-72 (Pr=9.0)

0 I I I i I I I I I I0 1 2 3 4 5 6 7 8 9 10 11y/L

Fig . 5 . Dis t r ibu t ion s o f wal l temperatures .

t i v e l y . T h e w a l l t e m p e r a t u r e s a r e s h a r p l y p e a k e d a tt h e l e a d i n g e d ge o f t h e h e a t s o u r c e a n d i n c r e a s e w i t hi n c r e a s i n g d i s t a n c e t o t h e t r a i l i n g ed g e . T h e t r e n d o ft h e r e s u l t i s si m i l a r t o t h e r e s u l t s o b t a i n e d b y R a m -a d h y a n i e t a l . [5 ] f o r l a m i n a r f l ow , a n d b y M o f f a t t e tal. [9 ] f o r t u r b u l e n t f lo w . H o w e v e r , t h e N u s s e l t n u m -b e r p r e d i c t e d b y t h e p r e s e n t m o d e l i s a b o u t 3 0 %h i gh e r t h a n t h a t b y t h e z e r o e q u a t i o n t u r b u l e n t m o d e lu s e d b y M o f f a t t et al. [ 6 ] . C o m p a r e d t o t h e e x p e r -i m e n t a l r es u l ts a n d n u m e r i c a l r e s u lt s r e p o r t e d b yI n c r o p e r a et al. [ 7 ] f o r t h e s i n g l e p l a i n h e a t s o u r c eu s i n g w a t e r a n d F C - 7 7 a s t h e w o r k i n g f l u id , it is f o u n dt h a t t h e r e s u l t p r e d i c t e d b y t h e p r e s e n t m o d e l i s a ls oa b o u t 3 0 % h i g h e r t h a n t h e i r r e s u l ts a s s h o w n i nFig . 6 .

F l o w a n d t h e r m a l f i e l d a n a l y s isT h e r e p r e s e n t a t iv e v e l o c i t y v e c t o r s a r e p r e s e n t e d i n

F i g . 7 , s h o w i n g t h a t t h e v e l o c i ty is q u it e u n i f o r m i nt h e c h a n n e l . T h i s m a y b e d u e t o t h e w a l l f u n c t i o nm e t h o d b e i n g u se d i n t h e p r e s e n t f o r m u l a t i o n s o t h a tt h e v e l o c i t y n e a r t h e w a l l c a n n o t b e d e t a i l e d b y t h en u m e r i c a l m e t h o d . A n o t h e r r e a s o n m a y b e t h a t t h e rei s n o o b s t a c l e i n t h e f l o w c h a n n e l , a s t h e s i m u l a t e de l e c t r o n ic c h i p i s f l u s h - m o u n t e d o n t h e w a l l.

T h e v e l o c i ty p r o fi l es i n t h e y - d i r e c t i o n a t t h r e e c h a n -n e l l o c a t i o n s y / L = 4 . 9 2 , 5 . 5 5 a n d 6 . 1 2 ) a r e s h o w n

10 0

7 . 5

. J

5 0

2 5

0 . 0

T TTT~ T

TTT ~T TTT

TTTI~TTI~IT

~I~TTTITTTT

~ TTTTTT TT

TTT~r TT~t

T TTT TT

[TT TTT

T T T T T I T ~ I

I I T T T T T I T I [

T T I ~ T T T T r I

~ T T t T I I F ? I T0.1 .0x / L

Fig. 7. Predicted velocity vec tor plot ReL = 7 104, FC-72).


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Numerical modelling of turbulent heat transfer 1163

1 .5 ,---g|

1

0 . 5

lBV(~.'r'4.92) eV(~. 5.56) v(y/L=6.12)D T (y/L=4.92) O T (y/t.=5.55) A T (y/L=:6.12)

q~, = 10 W / c m 2T = = 3 0 0 C

0 O f l 0 f l O 0 p

70

6 0

E5 0 1a4 o l

3 0

0 . . . . . . . . . . . . . . . . . . . 2 00 0.0~ 0.1 0.15 0.2x / L

Fig. 8. Velocity profiles and temperature profiles at the mid-dle of and near the heat source ReL = 7 x 10 4, FC-72).

Table 3. The effect of H/L on average Nusselt numberReL = 7 10 4, FC-72)H/L NUL1 0 661.20.5 691.70.25 781.30.125 882.5

in Fig. 8. It is found that the velocity profiles for theturb ulent flow is flat with velocity increasing sharplyat the first point near the wall; the velocity near thewall is defined by the wall function.

The temperature distributions in the flow channelnear the wall have also be studied and given in Fig. 8at the same locations. The results are similar to thoseof velocity, but the temperature distributions are moreunif orm than those o f velocity. This temperatu re pro-file suggests that there is practically no heat transferin the fluid across the channel as the heat transferprocess has been dominated and completed near thewall. The direct liquid cooling of electronic chips isdifferent from the air cooling in that the temperaturedifference of the liqu id between the entrance and theexit of the channel is small, less than 0.5C as cal-culated from the energy balance equation. However,because the heat capacity pCp) of liquid is generallymuch higher than that of air, the heat flux dissipatedby the electronic chips in liquid cooling is higher. Thetemperature in the first point near the wall is veryclose to the inlet temperature. The temperature dis-tributions between the first point and the wall alsohave been specified by the wall function. The tem-perature distribution before the heat sourcey / L = 4.92) is a little lower than that after the heatsource y / L = 6.12), showing that the effect of theupstream thermal boun dary layer on the downstreamis weak. Similar experimental results were reported byGersey and Mudawar [12].Effec t o f channel heiyht on heat transfer performanceThe effect of the geometric parameter, expressed bythe ratio of channel height to the heat source lengthH / L , on the average Nusselt numbe r at R e z = 7 x 104is shown in Table 3. It is found that the ratio H / L haslittle effect on the Nusselt number in the range of 1 to0.125. The average Nusselt nu mber decreases slightlywith increases in the ratio. For example, the differencein average Nusselt num ber between the ratio H / L = 1and H / L = 0.5 is only 5%. Similar results wereobtained by Gersey and Mudawar in experimentalmulti-chip cooling using FC-72 as the working fluid.

They found that the experimental data of H / L = 0.5and H / L = 0.2 can be correlated by the same cor-relation; the channel height has little effect on theaverage Nusselt numbe r. A weak dependence on H / Lwas also founded by Incropera et al. by using theirnumerical computation. The effect of ratio of channelheight to the length of heat source H / L on the walltemperature distr ibuti on is shown in Fig. 9.However, for the air cooling of protruded discreteheat sources, the effects are noticeable. Kim andAna nd [1] numerically showed that the average Nus-selt numbe r is proportional to H/L)-814 in turbulentflow. McEntir e and Webb [19] experimentally showedthat the average Nusselt number is proportional toH/L ) -4 in the range of l03 < ReL < 104. Olivos andMaju mdar [20] numerically showed that the averageNusselt number decreases with increases in the ratioof channel height to length of heat source H / L inlaminar flow. It is anticipated that their temperatureprofile is not as uniform as in the present case in aliquid-cooled channel.Effect of orientation on the single-phase forced con-vection heat transfer

The effect of orientation on the forced convectionheat transfer is studied as shown in Table 4. It is found

1 0 09080

G 70o .I j= 60=9 = 5 o

o302010

- - H / L = 1- - - H / L = 1 / 2- - - - H /L=1/4H/L= 1/8~ ~ . . _~ q =10 W/cm z

TI,=30CFC-72 (Pr=9,0)

0 I I I I [ I I I I I0 1 2 3 4 5 6 7 8 9 10 11y/L

Fig. 9. Effect of H/L on the distribution of wall temperatureReL = 7 x 104).


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1164 G. P. XU et al .Table 4. Effect of orientation on the average Nusselt number (FC-72)

R e L = 2 1 0 4 R e z = T x 1 0 4 R e L = 1.5105Horizontal flow (0 = 0) N u L = 336.0 N u L = 661.2 N u L = 1141Vertical upflow (0 = 90) Nuz . = 336.2 N u L = 661.0 NUL = 1141

that orientation has no or little effect on the forcedconvection. The result is consistent with the con-clusion that body force terms in the governing equa-tions can be neglected for forced convection [21]. Theresult can also be supported from the order of mag-nitude analysis on dimensionless groups in the gov-erning equations. The effect of orientation had alsobeen investigated experimentally for nine in- line dis-crete heat sources by Gersey and Muda war [3]. Similarresults were obtained from their study; whose datafor different angles of orie ntati on were correlated wellby a single correlation, and there were no dimensionalor dimens ionless item reflecting the angle an d bodyforce in the correlation. From both numerical andexperimental results, it is concluded that orientationhas little effect on forced convection heat transfer indirect liquid cooling. But the effects of ori entat ion onheat transfer must be considered in free convectionand phase change circumstances.F o u r i n li n e h e a t s o u r c e sFigure 10 shows the comparison between thenumerically predicted results for the four in-line heatsources (as in Fig. 1 b)) and the experimentally deter-mined Nusselt number obtained by Incropera e t a l .using water as the working fluid. Although all thechips have similar heat transfer coefficients, it is fou ndthat the Nusselt number for the first chip is slightly

higher than the values of other chips as predicted bypresent model. However, not only the experimentaldata of Incropera e t a l . are about 30% lower than thevalues predicted by present model, but the effect ofthe numb er of rows o f heat sources on heat transfercoefficient is also inconsi stent with the present model.They reported that, for an array of the heat sourcesconsisting of four rows of three sources per row,upstream thermal bound ary layer development causesthe average Nusselt num ber to decrease with increas-ing row numb er, unti l a fully-developed cond itio n isreached at approximately the fourth row. Values forthe first row were considerably larger than those forthe last three rows, with the percentage differencebetween the rows decreasing with increasing Reynoldsnumber. But the present study finds that the nu mberof chips have little effect on the thermal boundarylayer.Compared to the experimental data obtained byGersey and Mudawar with FC-72 on a series of ninein-line simulated chips in a flow channel, the numericaldata agree well with their results as shown in Fig. 11.Because Gersey and Mudaw ar fo und that the nine in-line simulated chips had same heat transfercoefficients, all the data from the nine flush-mountedchips were correlated by a single correlation.

Figure 12 shows the wall temperature distributionsof four in-line heat sources for FC-72 at Reynolds

103

. 102

] - - Present model Chip 1[ ] Chip 2A Chip 3 6 ~Chip 4 fExpe rimental correlations [7]0 First rowo Second row ~A 1-hird rowFourth row ~ . . ~

/ /C f

1 0 1 i , k , i i i ~ L I : J , r h i l l103 104 105R eL

Fig. 10. Comparison of predictions and the experimentaldata of Incropera et al . [7] for four rows of heat sources.

103

n 10==

Present modelChip 1Chip 2A C h ip 3 ~ /

~' Chip 4. . . . Experimental c ~

/ / / / / / ~

101 i i i J f r l k l ~ i i i J i l l i1 0 ~ 1 0 4 1 0 5R~Fig. 11. Comparison of prediction and the experimental dataof Gersey and Mudawar [12] for multi-chip module.


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Num erica l mode l l ing o f tu rbu len t hea t t rans fe r I 1657

6 0

~ 3 o

~ 20 q =10 W/cmT i n = 3 0 C

1 0 F C - 72 ( P r=9 .0 )

0 I I I r I I I I I I L I I I 1 I0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7

y LFig . 12 . The wa l l tempera tu re d is t r ibu t ions fo r four in - l inehea t sources with FC-72 (ReL = 1 105).

7 0

O

REe

6 0

5 O

4 0

3 0

2 0

1 0

0 0

q = 1 0 W / c m 2 . . . . R e L = I X 1 0 4T jn= 30C R eL = 5X 1 04W a t e r ( P r = 5 . 4 2 ) R e L = I X 1 0s

/ I I I II I I t I it I I I I II I I I I II I I I I II I I II I I I I I i

I I I I I I I I I I I I I I I I1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7

y LFig . 13 . The w a l l temp era tu re d is t r ibu t ions fo r fou r in - l ineheat sources with water.

n u m b e r 1 1 05 . T h e r e i s l i t tl e c h a n g e i n t e m p e r a t u r ed i s t r i b u t i o n a m o n g c h i p s . T h e w a l l t e m p e r a t u r e d i s-t r i b u t i o n s o f w a t e r a r e s h o w n i n F i g. 1 3 w h i c h a l s oi n d i c a t es t h a t t h e n u m b e r o f c h i p s h a v e l i t tl e e f f ec t o nt h e w a l l t e m p e r a t u r e .

C O N C L U S I O N SD i r e c t l i q u i d c o o l i n g o f s i m u l a t e d c h i p s i n a r ec -

t a n g u l a r c h a n n e l h a s b e e n s t u d i e d n u m e r i c a l l y b ys o l v i n g t h e t w o - d i m e n s i o n a l g o v e r n i n g e q u a t i o n su s i n g t h e k - e m o d e l f o r t u r b u l e n t c l o s u r e . T h e k e yf i n d i n g s a r e a s f o l l o w s :1. T h e s o l u t i o n t e c h n i q u e i s v a l i d a t e d b y c o m p a r i n g

t h e e x p e r i m e n t a l d a t a f o r a s i n gl e p l a i n h e a t s o u r c ei n a r e c t a n g u l a r c h a n n e l [ 1 1 ] .

2 . T h e w a l l t e m p e r a t u r e s a r e s h a r p l y p e a k e d a t t h el e a d i n g e d g e o f t h e h e a t s o u r c e a n d i n c r e a s e w i t hi n c r e a s i n g d i s t a n c e t o t h e t r a i l i n g e d g e .

3 . T h e n u m e r i c a l s t u d i e s s u g g e s t t h a t t h e h e a t t r a n s f e rp r o c e s s i s d o m i n a t e d n e a r t h e w a l l o f t h e h e a t s o u r -c e s a n d a s a r es u l t a n a r r o w c h a n n e l w i t h s m a l lH / L c a n b e u s e d f o r l i q u i d c o o l i n g .

4 . T h e a v e r a g e N u s s e l t n u m b e r p r e d i c t e d b y p r e s e n tm o d e l i s a b o u t 3 0 % h i g h e r t h an t h e e x p er i m e n t a lr e s u lt s a n d n u m e r i c a l r e s u lt s r e p o r t e d b y I n c r o p e r aet al. [7].

5 . T h e a v e r a g e N u s s e l t n u m b e r d e c r e a s e s s l ig h t l y w i t hi n c r e as e s i n th e r a t i o o f c h a n n e l h e i g h t t o t h e l e n g t ho f h e a t s o u r c e. I t i s c o n c l u d e d t h a t t h e t h e r m a lc o n d i t i o n s a r e m o r e r e p r e s e n t a t i v e o f e x t e r n a l fl o w .

6 . T h e o r i e n t a t i o n o f f l o w c h a n n e l h a s l i tt l e e f f ec t o nf o r c e d c o n v e c t i v e h e a t t r a n s f e r i n l i q u i d c o o l in g .

7 . F o r f o u r p l a i n h e a t s o u r c e s , a l l h e a t s o u r c e s h a v es i m i l a r s i n g l e - p h a s e h e a t t r a n s f e r c o e f f i c i e n t s .

8 . C o m p a r e d t o t h e e x p e r i m e n t a l d a t a o b t a i n e d b yG e r s e y a n d M u d a w a r [ 12 ] f o r m u l t i - c h i p m o d u l e ,t h e n u m e r i c a l d a t a a g r e e w e ll w i t h t h e i r d at a . H o w -e v e r , p r e s e n t r e s u l t s a r e i n c o n s i s t e n t w i t h t h e d a t ao f I n c r o p e r a e t a l . [7].

R E F E R E N C E S1 . Kim, S . H. and Anand , N. K. , Turbu len t hea t t rans fe rbe tween a se rie s o f pa ra l le l p lates with surface -m ounted

discrete heat sources. A SM E Journal o f Heat Trans fer ,1994, 116, 577-587.2 . Asko , Y. and Faghri , M . , Three d imens iona l hea t t rans -fe r ana lys is o f a rrays o f hea ted square b locks . A SM EHT D- Vo l. 171, 1991, pp. 135-141.3 . Asko , Y. and Faghri , M . , Pred ic t ion o f tu rbu len t hea tt rans fe r in the en trance o f an a rray o f hea ted b locksus ing low-Reynolds -number k e mode l . Numer. HeatTransfer, 1995, Par t A, 28, 263 277.4 . K n ig h t , R . W. a n d C ra w fo rd , M. E . , N u m e r ic a l p re -d ic t ion o f tu rbu len t f low and hea t t rans fe r in channe lswith pe r iod ica l ly va ry ing c ross sec t iona l a rea . Pro-ceedings o f the 1988 National Heat Transfer Co ~[erenee,AS M E, Vo l. 1, 1988, pp. 669-676.5 . Ramadhyani , S . , Moffa t t , D. F . and lnc ropera , F . P . ,Conjuga te hea t t rans fe r f rom smal l i so the rmal hea tsources embedded in a la rge subs tra te . InternationalJournal of Heat and M ass Transfer , 1985, 28, 194 5-1952.6 . Moffa t t , D. F . , Ramadhyani , S . and Inc ropera , F . P . ,Conjuga te hea t t rans fe r f rom wal l embedded sources intu rbu len t channe l f low. In Heat TransJer in ElectronicEquipment, e d . A . B a r -C o h e n , A S ME H T D -V o l . 5 7 ,1986, pp. 177 182.7 . Inc ropera , F . P . , Kerby , J. S ., Moffa t t , D. F . and Ram -adhyani , S . , Convec t io n hea t t rans fe r f rom d isc re te hea tsources in a rectangular channel. International Journalo f Heat and Mass Tram fer , 1986, 29, 1051 1058.8 . Mahaney , H. V. , Inc ropera , F . P . and Ramadhyani , S . ,C o m p a r i s o n o f p re d ic t ed a n d m e a s u re d m ix e d c o n -vec t ion hea t t rans fe r f rom an a rray o f d isc re te sourcesin a hor izon ta l rec tangula r channe l . International Journalo f Heat and M ass Trans fer , 1990, 33, 1233 1245.9 . Baker , E . , L iqu id coo l ing o f mic roe lec tron ic dev ices byfree and fo rced convec t ion . Microelectronics andReliability, 1972, 11 , 213 532 .10. Baker, E . , L iqu id imm ers ion coo l ing o f smal l e lec tron ic


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1 16 6 G P X U e t a l

devices. Microelectronics and Reliability, 1973 , 12 , 16 3-173.1 1. M u d a w a r , I . a n d M a d d o x , D . E . , E n h a n c e m e n t o f c r i ti -c a l h e a t f l u x f r o m h i g h p o w e r m i c r o e l e c t r o n i c h e a ts o u r c e s i n a f l ow c h a n n e l . I n Heat Transfer in Electronics,e d . R . K . S h a h , A S M E H T D - V o l . 1 1 1 , 1 9 89 , p p . 5 1 - 5 8 .1 2. G e r s e y , C . O . a n d M u d a w a r , I . , E f f ec t s o f o r i e n t a t i o no n c r i t i c a l h e a t f lu x f r o m c h i p a r r a y s d u r i n g f l o w b o i l i n g .I n Advances in Electronic Packagin , Vol. 1 , ed. T. C.W i l l i a m a n d A . H i r o y u k i , A S M E , N e w Y o r k , 1 9 92 , p p .123-134.1 3. M a r k a t o s , N . C . , T h e m a t h e m a t i c a l m o d e l l i n g o f t u r -bu lenc e f lows . Appl. Ma th. Modelling, 1986, 10, 190-220.1 4. L a u n d e r , B . E . a n d S p a l d i n g , D . B . , T h e n u m e r i c a l c o m -p u t a t i o n o f t u r b u l e n t f lo w s. Comp. Methods Appl . Mech .Eng., 1974, 3, 269-289.1 5. P a t a n k a r , S . V . , S p a r r o w , E . M . a n d I v a n o v i c , M . , T h e r -m a l i n t e r a c t i o n a m o n g t h e c o n f i n i n g w a ll s o f a t u r b u l e n trec i rcu la t ing f low. International Journal of Heat andMass Transfer , 1978 , 24 , 269-274 .

16 . P a tan kar , S . V . , Nu me ricalH eat Transfer and Fluid Flow.H e m i s p h e r e P u b l i s h i n g C o m p a n y , N e w Y o r k , 1 9 80 .1 7. K i m , S . H . a n d A n a n d , N . K . , L a m i n a r d e v e l o p i n g f lo wa n d h e a t t r a n s f e r b e t w e e n a s e r i e s o f p a r a l l e l p l a te s w i t hs u r f a c e - m o u n t e d d i s c r e t e h e a t s o u r c e s . InternationalJournal of He at and M ass Transfer , 1994 , 37 , 2231-2244 .1 8. G e r s e y , C . O . a n d M u d a w a r , I . , N u c l e a t e b o i l i n g a n dc r i ti c a l h e a t f lu x f r o m p r o t r u d e d c h i p a r r a y s d u r i n g f l o wboi l ing . A S M E Journa l o f E lectron ic Packagin , 1993,115 , 78-88 .1 9 . M c E n t i r e , A . B . a n d W e b b , B . W . , L o c a l f o r c e d c o n -v e c ti v e h e a t t r a n s f e r fr o m p r o t r u d i n g a n d f l u s h -m o u n t e dt w o - d i m e n s i o n a l d i s c r e t e h e a t s o u r c e s . InternationalJournal of Heat and Mass Transfer , 1990, 33, 1233-1245.2 0 . O l i v o s , T . a n d M a j u m d a r , P . , A c o m p u t a t i o n a l m o d e lf o r f o r c e d c o n v e c t i o n c o o l i n g i n e l e c tr o n i c c o m p o n e n t s .Journal o f Electronics M anufacturing, 1995, 5, 183-192.21 . Tao , W. Q . , Numerical Heat Transfer . X i ' a n J i a o t o n gUniv ers i ty P ress , PRC ( in Chinese ) , 1988 .

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Numerical Modelling of Turbulent Heat Transfer From Discrete Heat Soucers in Liquid-cooled Channel

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