Thermal Design and Optimization of Heat Sinks J. Richard Culham
Thermal Design andOptimization of Heat Sinks
J. Richard Culham
Outline
Background
Modelling Approach
Validation
Optimization
Future Work
Summary
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
75 100 125 150100
101
102
103
Component Failure Rate
Junction Temperature ( C)
Nor
mal
ized
Fai
lure
Rat
e
o
FC: 58.2 C
NC: 96.1 C
(2 m/s)
o
o
o
Pentium 233 MHz@ 7.9 W
no heat sinkFC: 99.5 C
NC: 136.4 Co
Pentium 233 MHz@ 7.9 W
with heat sink
(2 m/s)
Intel Design Specification:
T = 75 Cjo
GaN - SiC
Si - SiO2
Moore’s Law (1965)
1975 1980 1985 1990 1995 2000
10M
1M
100K
10K
(transistors)
500
25
1.0
0.1
0.01
(mips)
40048080
8086
80286
80386
80486
Pentium
Pentium III
Micro 2000
➣ each new chip contains roughly twice as muchcapacity as its predecessor
➣ a new generation of chips is released every18 - 24 months
From: www.intel.com➥ in 26 years, the
population of transistors per
chip has increasedby 3,200 times
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
Why Use Natural Convection?
simplicity: ➥ low maintenance ➥ lower power consumption ➥ less space (notebook computers)
less noise
fail safe heat transfer condition
Thermal Resistance
Heat source (junction)
Heat sink (air)
contactresistance
materialresistance
spreadingresistance
filmresistance
Rh Afilm ≡
•1 • increased heat transfer coefficient
immersion cooling (boiling) impingement cooling forced air natural convection
• increased surface area spreaders heat sinks
Plate Fin H.S. Pin Fin H.S.
Radial Fin H.S. Specialty H.S.
Plate Fin Pin Fin
Turned Fin Spiral Fin
Plate Fin Pin Fin
Turned Fin Spiral Fin
Plate Fin Pin Fin
Turned Fin Spiral Fin
Plate Fin Pin Fin
Turned Fin Spiral Fin
Heat Sink Model
Plate fin heat sink
Natural convection
Isothermal
Steady state
Working fluid is air i.e. Pr = 0.71
Modelling Procedure
Exterior surfaces
Interior surfaces
● fins : top, bottom, ends & tip● base plate: top, bottom, ends and back
● fins : side walls● channel base
g
LL
H t
b
bpt
fN
Given:
Find:
dimensions & temperature
Nu vs. Rab b
= •g T b b
L
βα ν∆ 3
= h b
k f
Exterior Surfaces
Boundary layer
Diffusion
● lower Rayleigh numbers● thick boundary layers
● higher Rayleigh numbers● thin boundary layers
Diffusion Model
Nu S SL D
L DA A plate
GM
GM0
30 76
3
1 0 8688
1 2= = [ ] + ( )
+
* *
..
SL L
L LA plate
*[ ] =+( )2 1 1 2
2
1 2
π
SL L
L LA plate
*
ln[ ] = ( )2 2
41 2
1 2
π
1 0 5 01 2. .≤ ≤L L
5 0 1 2. < < ∞L L
L
L L
D
12
3
GM
Exterior Boundary Layer Model
Nu G F RaA A A
= • •(Pr) /1 4
F(Pr).
( . / Pr) / /=+[ ]
0 670
1 0 5 9 16 4 9
(in terms of the surface area)
Where:
GH W L H W
L W L H H WA= + +
• + • + •
2
0 6251 84 3 4 3
7 6
3 4/
/ /
/
/. ( )
)
Rag T A
A=
( )β
αν
∆3
W
L
H
g
Interior Surfaces
Control surfaces
Channel flow
● Elenbaas model with adjustment for end wall● combined flow : developing + fully developed
● open surfaces with energy migration
Parallel Plates Model
Nu Ra Rab b b= − −( ){ }124
1 353 4
exp //
Nu Nu Nub fdm
devm m
= +{ }− − −1/
Elenbaas, 1941
Churchill, 1977
fd - fully developed
dev - developing flowNu Rafd b= 124
Nu G F Radev A b= • •(Pr) /1 4
b
L
g
- body gravity functionGA
F(Pr)- Prandtl number function
Comprehensive Model
Nu Nu
Nu Nu Nu
Nu= ++
+
+0
2 3 4
1
11 1{
1 24 44 34 44
{
diffusion channel flow
externalboundarylayer flow
Model Validation
cuboids ➊ plate - Karagiozis (1991), Saunders (1936) ➋ cube - Chamberlain (1983), Stretton (1988) ➌ rectangular prism - Clemes (1990)
parallel plates ➊ Elenbaas (1942), Aihara (1973), Kennard (1941)
Karagiozis (1991)
Van de Pol & Tierny (1978)
Limiting Cases
Heat Sinks
Modelling Domain
10-4 10-2 100 102 104 106 108 101010-3
10-2
10-1
100
101
102
103
Plate spacing
Aspect ratio
fully-developedlimit
Boundar y layer limit
b
Rab
Nu
b
CUBE PRISM FLAT PLATE ‰ ‰ PLATES HEAT SINK
+Nu = Nu 0 +Nu12Nu + 43Nu Nu+-2 -2 -1/2
10-2 10-1 100 101 102 103 104 10510-1
100
101
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
Rab
Nu
b
CUBE PRISM FLAT PLATE ‰ ‰ PLATES HEAT SINK
+Nu = Nu 0 +Nu12Nu + 43Nu Nu+-2 -2 -1/2
10-2 10-1 100 101 102 103 104 10510-1
100
101
Clemes (1990) Model
g
units inmm
50.43 x 50.43 X 510.6
Rab
Nu
b
CUBE PRISM FLAT PLATE ‰ ‰ PLATES HEAT SINK
+Nu = Nu 0 +Nu12Nu + 43Nu Nu+-2 -2 -1/2
10-3 10-2 10-1 100 101 102 103 104 105 10610-1
100
101
102
43.2
W
L
H
g
Rab
Nu
b
L x W x H (mm) Karagiozis (1991) 150x170x9.54 Model Saunders (1936) 76x230x.00254 Model
CUBE PRISM FLAT PLATE ‰ ‰ PLATES HEAT SINK
+Nu = Nu 0 +Nu12Nu + 43Nu Nu+-2 -2 -1/2
10-1 100 101 102
103 104 10510-2
10-1
100
101
Ra b
Nu
b
g
b
Elenbaas (1942) Aihara (1973) Kennard (1941) Model
CUBE PRISM FLAT PLATE ‰ ‰ PLATES HEAT SINK
+Nu = Nu 0 +Nu12Nu + 43Nu Nu+-2 -2 -1/2
10-3 10-2 10-1 100 101 102 103 104 105 10610-2
10-1
100
101
102
H/b Karagiozis Model Van de Pol & Tierney0.75 1.19 2.98
Nu
b
Rab
H
b
L
t tbp
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 constraintsi.e. given: • heat input • max. temp. • max. outside dimensions find: the most efficient design
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?
Entropy Balance (local)
′′′ = ∇ • ′′ − ′′ • ∇ +ST
QT
Q TDs
Dtgen1 1
0 02 ρ
′′′ = ∇( ) +ST
k TTgen
1 1
02
2
0
µφ
1st law ofthermodynamics
Gibb’sEquation
heat transfer viscous dissipation
′′′ = +
− +
+• •
S dV m sQ
Tm s
Q
T
dS
dtgenout in
cv
0 0
conservation of mass + +
Q s mx x x, ,
Q s mz z z, ,
Q s my y y, ,
•
•
•
•
•
•
Entropy Balance (external & internal)
SQ
Td
Q
TgenwA
B
B
= ′′ −∫∫ σ•
Extended surface
Passage geometr y
dx
m`
A
T w
•′ = ′ + −
S
Q T
T
m
T
dP
dxgenw∆
02
0ρ
irreversibilities due to: wall-fluid ˘T fluid friction
•
AQ B
T B T w
Q
irreversibilities due to base-wall ˘T
Total Entropy Generation
S Sgen gen= ∑ ∑ ∑• •
differential level
elemental level
component level
dxdy
dz
differentialcontrolvolume
Extended surface• fin • channel flow
System• fins • base plate
= +Q R
T
F U
TB total d2
02
0
= +Q
T
F U
TB B dϑ
02
0
where:
QB −ϑ B −T0 −FD −
Rtotal −
base heat flow rate
base - stream temp. difference
ambient temperature
drag force
total fin resistance
U - specified
- fan curve
- buoyancy induced
Example: Heat Sink Optimization
Board spacing ” H
L
W
Step 1: Determine problem constraintsi) power input, Q
ii) maximum chip temperature, Tmax iii) geometry , H, L, W
Example: Heat Sink Optimization
Board 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
Example: Heat Sink Optimization
Board spacing ” H
L W
H
Step 3: Optimize heat sink i) number of fins
ii) fin thickness iii) fin spacing iv) base plate thickness
Single Parameter EGM
0 10 200
0.05
0.1
0.15
0.2
0.25
Number of Fins
En
tro
py
Gen
erat
ion
(W
/K)
14
H = L = W = 50 mm Fin thickness = 1 mm Base plate thickness = 1.25 mmQ = 50 W Find: number of fins
Multi-Parameter Minimization Procedure
S f x x x xgen N= ( )1 2 3, , , ,K•
∂∂S
xg i Ngen
ii= = =0 1 2 3() , , , , ,K
•
∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂
δδδ
g x g x g x
g x g x g x
g x g x g x
x
x
x
g
g
g
1 1 1 2 1 3
2 1 2 2 2 3
3 1 3 2 3 3
1
2
3
1
2
3
=
where: g guess g actual g guess xi i i i( ) ( )≈ + ′( ) • δ➥
iterate until
∂xi → 0
Newton-Raphson Method with Multiple Equations and Unknowns
Future Work
Goal: Develop a comprehensive model to find the best heat sink design givena 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
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
The End
Karagiozis Heat Sink Model
Nu Nu A Nu A C RaC
Racub f chfd
f
n
l bb
mn n
cub ch m= • + •( ) + +
−
− −
11
11
1
1 41
**
//
where:
CH
blm= +
0 509 0 0135 0 6. ( . ) , .min
CH b
H b H b
H bH b
=( )
( ) ≥
=( )
<
12 51
1
1411
1
3 17
3 17
./
/ , /
/, /
.
.
mH
b1 1 2 0 64 0 56= +
. , . .min
n at t mm
at t mm
at t mm
1 1 20 1 95 4 96
1 57 3 0 9 67
1 44 2 23 14 96
= → == → == → =
. . .
. . .
. . .
Modified flat plate model ➛ correction term at low Ra