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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
3Weak h-dependency of spreading Rth in convectively cooled substrates
Introduction
► calculation of thermal resistance(maximum temperature used)
Problem formulation
PTR source
th =
4Weak h-dependency of spreading Rth in convectively cooled substrates
IntroductionEarlier works in literature (1)
Fisher et al. (Trans ASME 1996) Ellison (IEEE Trans CPT 2003)
5Weak h-dependency of spreading Rth in convectively cooled substrates
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.
6Weak h-dependency of spreading Rth in convectively cooled substrates
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
7Weak h-dependency of spreading Rth in convectively cooled substrates
Outline
► Introduction► Exact calculations► Approximate model► Discussion► Conclusions
8Weak h-dependency of spreading Rth in convectively cooled substrates
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=
9Weak h-dependency of spreading Rth in convectively cooled substrates
Exact calculationsMethod comparison
Al2O3 substrate (k = 22 W/mK)
22 Ra π=
10Weak h-dependency of spreading Rth in convectively cooled substrates
Exact calculationsLTCC [4 W/mK]
heat source10mm x 10mm
11Weak h-dependency of spreading Rth in convectively cooled substrates
Exact calculationsAl2O3 [22 W/mK]
heat source10mm x 10mm
12Weak h-dependency of spreading Rth in convectively cooled substrates
Exact calculationsSi [160 W/mK]
heat source10mm x 10mm
13Weak h-dependency of spreading Rth in convectively cooled substrates
Exact calculationsCu [380 W/mK]
heat source10mm x 10mm
14Weak h-dependency of spreading Rth in convectively cooled substrates
Exact calculationsGeneral tendency
Rth = A ln(h) + B
15Weak h-dependency of spreading Rth in convectively cooled substrates
Outline
► Introduction► Exact calculations► Approximate model► Discussion► Conclusions
16Weak h-dependency of spreading Rth in convectively cooled substrates
Approximate modelLayout
17Weak h-dependency of spreading Rth in convectively cooled substrates
( ) ( ) 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
18Weak h-dependency of spreading Rth in convectively cooled substrates
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
19Weak h-dependency of spreading Rth in convectively cooled substrates
Approximate modelResults (1)
20Weak h-dependency of spreading Rth in convectively cooled substrates
Approximate modelResults (2)
21Weak h-dependency of spreading Rth in convectively cooled substrates
Approximate modelResults (3)
22Weak h-dependency of spreading Rth in convectively cooled substrates
Outline
► Introduction► Exact calculations► Approximate model► Discussion► Conclusions
23Weak h-dependency of spreading Rth in convectively cooled substrates
DiscussionModel discrepancy
► correct slope but underestimates Rth
► reason: heat source is on top surface, not in substrate
24Weak h-dependency of spreading Rth in convectively cooled substrates
DiscussionRth vs. h relation
► :weak dependency
( )( )hlnfRth =
25Weak h-dependency of spreading Rth in convectively cooled substrates
DiscussionSlope
► independent of source dimensionkd4
1Aπ
−=
26Weak h-dependency of spreading Rth in convectively cooled substrates
Outline
► Introduction► Exact calculations► Approximate model► Discussion► Conclusions
27Weak h-dependency of spreading Rth in convectively cooled substrates
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 +⋅≈
28Weak h-dependency of spreading Rth in convectively cooled substrates
Acknowledgements
The authors wish to thank
for supporting the presented work.