weak heat transfer coefficient depen- dency of thermal ... fileweak h-dependency of spreading r th...

2nd European ATW on Micropackaging and Thermal ManagementSession 3A – February 1st, La Rochelle, FRANCE

Weak heat transfer coefficient depen-dency of thermal spreading resistance

in convectively cooled substrates

B. Vermeersch and G. De MeyGhent University, Belgium

2Weak h-dependency of spreading Rth in convectively cooled substrates

Outline

► Introduction► Exact calculations► Approximate model► Discussion► Conclusions


Introduction

► calculation of thermal resistance(maximum temperature used)

Problem formulation

PTR source

th =


IntroductionEarlier works in literature (1)

Fisher et al. (Trans ASME 1996) Ellison (IEEE Trans CPT 2003)


IntroductionEarlier works in literature (2)

T.S. Fisher, F.A. Zell, K.K. Sikka & K.E. Torrance:Efficient heat transfer approximation for the chip-on-substrate problemTransactions of the ASME – Journal of Electronic Packaging 118 pp. 271-279, 1996.


Introduction

► Rth vs. normalizedsubstrate thickness withBiot number (Bi = hd/k) as a parameter

► approximate solutionaccurate but still verycomplicated, with lot of variables

New approach

► thickness not a real design parameter but determined bytechnology (e.g. Si: 300µm)

Rth vs. heat transfer coefficient

► simple model to provideinsight


Outline



Exact calculations

► Lee, Song & Moran (ASME conference 1995)

► Carslaw & Jaeger (Heat in solids, Oxford Press)

Infinite series solution

( )( )

( )( )

( )

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

τλλ+

λ+ζλ⋅

τλζλ⋅

λλελ+⎟

⎠⎞

⎜⎝⎛ ζ+ε= ∑

∞

=1nn

n

nn

n

n

n20

2n

n1

tanhBi

1

Bitanh

coshcosh

JJ2

Bi1

kpaz)T(r,

khbBi,b/r,b/z,b/t,b/a ==γ=ζ=τ=ε

( )2

2

kh2

n

kh

nn

2d2tan

−αα=α

( )( ) ( )[ ]( ) ( ) ( )[ ] ( ) ( )[ ]

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛ α−++α

α+αα⋅α+ααπ

−= ∑

∞

=

−+−

v

n

1n kh

kh2

n

nkh

nnnkh

nnkt4'yy'xxC

Cktexp

2d2dsindcoszsinzcos

kt4exp

t;'rrG2

2

22vrr

( ) ( )∫∞

=

=0t

DC dtt;'rrG'rrG

( )∑∞

=⎟⎠⎞⎜

⎝⎛ −+−α=

1n

22n0n )'yy('xxKC ( ) ( )∫∫=

sourceDC 'dy'dx'rrGz,y,xT rr

( ) 0J n1 =λ

steadystate:

EASY

HARD

a10b 5=


Exact calculationsMethod comparison

Al2O3 substrate (k = 22 W/mK)

22 Ra π=


Exact calculationsLTCC [4 W/mK]

heat source10mm x 10mm


Exact calculationsAl2O3 [22 W/mK]



Exact calculationsSi [160 W/mK]



Exact calculationsCu [380 W/mK]



Exact calculationsGeneral tendency

Rth = A ln(h) + B


Outline



Approximate modelLayout


( ) ( ) R0TkdhT2 ≥ρ=ρ−ρ∇

( ) ( ) finiteisT,PddTkRd2

R

∞→ρ=ρ

⋅π−=ρ

( ) ( )( )L/K

L/KdRk2

LPT1

0

ρρ⋅

⋅⋅π⋅=ρ

( ) ( )( )LR

1LR

LR

0th K

Kkd2

1PRTR ⋅

π==

Approximate modelCalculations (1)

► Heat equation

► Boundary conditions

► Solution

2L1


Approximate modelCalculations (2)

1LR <<

hkdL =

( ) ( ) ( ) ( )20 xOxln2lnxK +γ−−≈

For small arguments:

( ) ( )21 xO1xKx +≈⋅

( ) ( ) ( )kd2

ln2lnhln

kd41R kd

R

th πγ−−

+π

−≈

( ) BhlnARth +≈

k larged largeh small

R small


Approximate modelResults (1)


Outline



DiscussionModel discrepancy

► correct slope but underestimates Rth

► reason: heat source is on top surface, not in substrate


DiscussionRth vs. h relation

► :weak dependency

( )( )hlnfRth =


DiscussionSlope

► independent of source dimensionkd4

1Aπ

−=


Outline



Conclusions

► surface heat source on substrate with bottom-sideconvective cooling

► Rth vs. h► for wide variety of k and d:

in range of at least 3 decades (h = 1 – 1000 W/m²K)► explanation with simple model► weak ln(h) dependency due to compensation

(less cooling = more spreading)

( ) BhlnARth +⋅≈


Acknowledgements

The authors wish to thank

for supporting the presented work.

weak heat transfer coefficient depen- dency of thermal ... fileweak h-dependency of spreading r th...

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