Thermal Design and Optimization of Heat Sinks

• increased heat transfer coefficient
immersion cooling (boiling)
impingement cooling
forced air
natural convection
• increased surface area
spreaders
heat sinks

1. Thermal Design and
Optimization of Heat Sinks
J. Richard Culham

2. Outline
Background
Modelling Approach
Validation
Optimization
Future Work
Summary

3. 40 Watts! What’s the big deal?
Light Bulb

➢ Power: 40 W
➢ Area: 120 cm
➢ Flux: 0.33 W/cm
Pentium III
Silicon Package
➢ Power: 40 W
➢ Area: 1.5 cm
➢ Flux: 26.7 W/cm
➢ Rj-c: 0.94 C/W
➢ Rj-a: 6.8 C/W
(no heat sink)
➢ Rj-a: 2.5 C/W
(heat sink)
2 2
❋ 0.25 micron CMOS technology
❋ 9.5 million transistors
❋ 450 - 550 MHz
2 2
80 x

4. 75
100
125
150
10
0
10
1
10
2
10
3
Component Failure Rate
Junction Temperature ( C)
Normalized Failure Rate
o
FC: 58.2 C
NC: 96.1 C
(2 m/s)
o
o
o
Pentium 233 MHz
@ 7.9 W
no heat sink
FC: 99.5 C
NC: 136.4 C
o
Pentium 233 MHz
@ 7.9 W
with heat sink
(2 m/s)
Intel Design
Specification:
T = 75 C
j
o
GaN - SiC
Si - SiO
2

5. Moore’s Law (1965)
1975 1980 1985 1990 1995 2000
10M
1M
100K
10K
(transistors)
500
25
1.0
0.1
0.01
(mips)
4004
8080
8086
80286
80386
80486
Pentium
Pentium III
Micro 2000
➣ each new chip contains roughly twice as much
capacity as its predecessor


➣ a new generation of chips is released every
18 - 24 months

From: www.intel.com
➥
in 26 years, the
population of
transistors per
chip has increased
by 3,200 times

6. IC Trends: Past, Present & Future
1980
1999

2003

2006
2012
Comp. Per Chip
0.2 M
6.2 M
18 M
39 M
100 M
Frequency (MHz)
5
1250
1500
3500
10000
Chip Area (sq. cm)
0.4
4.45
5.60
7.90
15.80
Max. Power (W)
5
90
130
160
175
Junction Temp. (C)
125
125
125
125
125
From: David L. Blackburn, NIST

7. Why Use Natural Convection?
simplicity:
➥ low maintenance
➥ lower power consumption
➥ less space (notebook computers)
less noise
fail safe heat transfer condition

8. Thermal Resistance
Heat source

(junction)
Heat sink

(air)
contact
resistance
material
resistance
spreading
resistance
film
resistance
R
hA
film
≡
•
1
•
increased heat transfer coefficient

immersion cooling (boiling)
impingement cooling
forced air
natural convection
•
increased surface area

spreaders
heat sinks

9. Plate Fin H.S.
Pin Fin H.S.
Radial Fin H.S.
Specialty H.S.

10. Plate Fin
Pin Fin
Turned Fin
Spiral Fin

11. Plate Fin
Pin Fin
Turned Fin
Spiral Fin

12. Plate Fin
Pin Fin
Turned Fin
Spiral Fin

13. Plate Fin
Pin Fin
Turned Fin
Spiral Fin

14. Heat Sink Model
Plate fin heat sink
Natural convection
Isothermal
Steady state
Working fluid is air i.e. Pr = 0.71

15. Modelling Procedure
Exterior surfaces Interior surfaces
t fins : top, bottom,
ends & tip
t base plate: top,
bottom, ends
and back
t fins : side walls
t channel base
g
LL
H
t
b
bp
t
f
N
Given:
Find:
dimensions & temperatureNu vs. Ra
b b
=•
gTbb
L
β
αν
∆
3
=
hb
k
f

16. Exterior Surfaces
Boundary layer
Diffusion
t lower Rayleigh
numbers
t thick boundary
layers
t higher Rayleigh
numbers
t thin boundary
layers

17. Diffusion ModelNu S S
LD
LD
AA
plate
GM
GM
0
3
076
3
1 08688
12
==
[]
+
()
+






**
.
.
S
LL
LL
A
plate
*
[]
=
+
()
21
12
2
12
π
S
LL
LL
A
plate
*
ln
[]
=
()
22
4
12
12
π
10 50
12
..≤≤LL
50
12
.<<∞LL
L
L L
D
12
3
GM

18. Exterior Boundary Layer Model
Nu G F Ra
AA A
=• •(Pr)
/14
F(Pr)
.
( . /Pr)
/
/
=
+
[]
0670
105
916
49
(in terms of the surface area)
Where:
G
HWLHW
LW LHHW
A
=
++
•+•+•





2
0625
18
43 43
76
34
/
//
/
/
.()
)
Ra
gTA
A
=
()
β
αν
∆
3
W
L
H
g

19. Interior Surfaces
Control surfaces
Channel flow
t Elenbaas model
with adjustment
for end wall
t combined flow :
developing +
fully developed
t open surfaces
with energy
migration

20. Parallel Plates Model
Nu Ra Ra
bb b
=−−
()
{}
1
24
135
34
exp /
/
Nu Nu Nu
b fd
m
dev
m
m
=+
{}
−−
−1/
Elenbaas, 1941Churchill, 1977
fd
- fully developed
dev
- developing flow
Nu Ra
fd b
=
1
24
Nu G F Ra
dev A b
=• •(Pr)
/14
b
L
g
- body gravity function
G
A
F(Pr)
- Prandtl number function

21. Comprehensive Model
Nu Nu
Nu Nu Nu
Nu=+
+
+
+
0
234
1
1
11
{
124443444
{
diffusion channel
flow
external
boundary
layer flow

22. Model Validation
cuboids
plate - Karagiozis (1991), Saunders (1936)
cube - Chamberlain (1983), Stretton (1988)
rectangular prism - Clemes (1990)

parallel plates

Elenbaas (1942), Aihara (1973), Kennard (1941)

Karagiozis (1991)

Van de Pol & Tierny (1978)
Limiting Cases
Heat Sinks

23. Modelling Domain
10
-4
10
-2
10
0
10
2
10
4
10
6
10
8
10
10
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
Plate spacing
Aspect ratio
fully-developed
limit
Boundar y layer
limit
b
Ra
b
Nu
b

24. CUBE
PRISM
FLAT PLATE
‰ ‰ PLATES
HEAT SINK
+
Nu
=
Nu
0
+
Nu
1
2
Nu
+
43
Nu Nu
+
-2
-2
-1/2
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
-1
10
0
10
1
L x W x H (mm)
Chamberlain (1983) 43.2 x 43.2 x 43.2
Stretton (1984) 38.1 x 38.1 38.1
Model

43.2
W
L
H
g
Ra
b
Nu
b

25. CUBE
PRISM
FLAT PLATE
‰ ‰ PLATES
HEAT SINK
+
Nu
=
Nu
0
+
Nu
1
2
Nu
+
43
Nu Nu
+
-2
-2
-1/2
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
-1
10
0
10
1
Clemes (1990)
Model

g
units in
mm
50.43 x 50.43
X 510.6
Ra
b
Nu
b

26. CUBE
PRISM
FLAT PLATE
‰ ‰ PLATES
HEAT SINK
+
Nu
=
Nu
0
+
Nu
1
2
Nu
+
43
Nu Nu
+
-2
-2
-1/2
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
-1
10
0
10
1
10
2
43.2
W
L
H
g
Ra
b
Nu
b
L x W x H (mm)
Karagiozis (1991) 150x170x9.54
Model
Saunders (1936) 76x230x.00254
Model

27. CUBE
PRISM
FLAT PLATE
‰ ‰ PLATES
HEAT SINK
+
Nu
=
Nu
0
+
Nu
1
2
Nu
+
43
Nu Nu
+
-2
-2
-1/2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
-2
10
-1
10
0
10
1
Ra
b
Nu
b

g
b
Elenbaas (1942)
Aihara (1973)
Kennard (1941)
Model

28. CUBE
PRISM
FLAT PLATE
‰ ‰ PLATES
HEAT SINK
+
Nu
=
Nu
0
+
Nu
1
2
Nu
+
43
Nu Nu
+
-2
-2
-1/2
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
-2
10
-1
10
0
10
1
10
2
H/b Karagiozis Model Van de Pol
& Tierney
0.75
1.19
2.98
Nu
b
Ra
b
H
b
L
t t
bp

29. Which is the Right Tool?
➥ design is known a priori
➥ used to calculate the
performance of a given
design, i.e. Nu vs. Ra
➥ cannot guarantee an
optimized design
Analysis Tool
Design Tool
vs.
➥ used to obtain an
optimized design for a
set of known constraints
i.e. given: • heat input
• max. temp.
• max. outside
dimensions
find: the most efficient
design

30. Optimization Using EGM
➥
entropy production amount of energy degraded
to a form unavailable for work
➥
lost work is an additional amount of heat that could
have been extracted
➥
degradation process is a function of thermodynamic
irreversibilities e.g. friction, heat transfer etc.
➥
minimizing the production of entropy, provides a
concurrent optimization of all design variables
Why use Entropy Generation Minimization?

31. Entropy Balance (local)
′′′=∇• ′′− ′′•∇ +S
T
Q
T
QT
Ds
Dtgen
11
0 0
2
ρ
′′′=∇
()
+S
T
kT
T
gen
110
2
2
0
µφ
1st law of
thermodynamics
Gibb’s
Equation
heat transfer viscous dissipation
′′′=+






−+






+
••
SdV ms
Q
T
ms
Q
T
dS
dt
gen
out in
cv
00conservation
of mass
+ +
Qsm
xx x
,,
Qsm
zz z
,,
Qsm
yy y
,,
•
•
•
•
•
•

32. Entropy Balance (external & internal)
S
Q
T
d
Q
T
gen
wA
B
B
=
′′
−
∫∫
σ
•
Extended surface
Passage geometr y
dx

m`
A
T
w
•
′=
′
+−




S
QT
T
m
T
dP
dxgen
w
∆0
2
0
ρ
irreversibilities
due to: wall-fluid ˘T fluid friction
•
AQ

B
T

B
T
w
Q
irreversibilities due to
base-wall ˘T

33. Total Entropy Generation
SS
gen gen
=∑ ∑ ∑
•
•
differential
level
elemental
level
component
level
dx
dy
dz
differential
control
volume
Extended surface
• fin
• channel flow
System
• fins
• base plate
=+
QR
T
FU
T
Btotal d
2
0
2
0
=+
Q
T
FU
T
BB d
ϑ
0
2
0
where:
Q
B
−
ϑ
B
−
T
0
−
F
D
−
R
total
−
base heat flow rate
base - stream temp. difference
ambient temperature
drag force
total fin resistance
U
- specified
- fan curve
- buoyancy induced

34. Example: Heat Sink Optimization
Board spacing ” H
L
W
Step 1: Determine problem constraints
i) power input, Q
ii) maximum chip temperature, Tmax
iii) geometry , H, L, W

35. Example: Heat Sink OptimizationBoard spacing ” H
L
W
H
Step 2: Set maximum heat sink volume
i) package foot print = L x W
ii) maximum height - board spacing minus
package height

36. Example: Heat Sink OptimizationBoard spacing ” H
L
W
H
Step 3: Optimize heat sink
i) number of fins
ii) fin thickness
iii) fin spacing
iv) base plate thickness

37. Single Parameter EGM
0
10
20
0
0.05
0.1
0.15
0.2
0.25
Number of Fins
Entropy Generation (W/K)
14
H = L = W = 50 mm
Fin thickness = 1 mm
Base plate thickness = 1.25 mm
Q = 50 W
Find: number of fins

38. Multi-Parameter Minimization Procedure

Sfxxxx
gen N
=
()
123
,,,,K
•
∂
∂
S
x
gi N
gen
i
i
== =0123(), , , , ,K
•
∂∂ ∂∂ ∂∂
∂∂∂∂ ∂∂
∂∂ ∂∂ ∂∂
δ
δ
δ
gx gx gx
gx gx gx
gx gx gx
x
x
x
g
g
g
11 12 13
21 22 23
31 32 33
1
2
3
1
2
3




















=










where:
g guess g actual g guess x
ii i i
()()≈+ ′
()
•
δ
➥
iterate until ∂
x
i
→0
Newton-Raphson Method with Multiple Equations and Unknowns

39. Future Work
Goal: Develop a comprehensive model to
find the best heat sink design given
a limited set of design constraints
Physical Design Thermal Cost
Standards
¥ heat sink type
¥ material
¥ weight
¥ dimensions
¥ surface finish
¥ maximum volume
¥ boundary conditions
¥ max. allowable temp.
¥ orientation
¥ flow mechanism

¥ labour
¥ manufacturing
¥ material

¥ noise
¥ exposure to
touch

40. Summary
Heat sink design requires both a
selection tool & an analysis tool
Selection is based on:

➥ physical constraints - geometry, material, etc.
➥ thermal-fluid conditions - bc’s, properties, etc.
➥ miscellaneous conditions - cost, standards etc.

Analysis is based on simulating a
prescribed design

41. The End

42. Karagiozis Heat Sink Model
Nu Nu A Nu A C Ra
C
Ra
cub f ch
fd
f
n
l b
b
m
n
n
cub ch m
=•+•
()
++










































−
−
−
1
1
1
1
1
14
1
*
*
/
/
where:
C
H
b
l
m
=+






0509 00135 06.(.),.
min
C
Hb
Hb Hb
Hb
Hb
=
()
()
≥
=
()
<
125
1
1
141
1
1
317
317
.
/
/,/
/
,/
.
.
m
H
b
1
12 064 056=+






.,. .
min
nattmm
at t mm
at t mm
1
120 195 496
157 30 967
144 223 1496
=→ =
=→ =
=→ =
.. .
.. .
.. .
Modified flat plate model ➛ correction term at low Ra