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