conduction shape factor models for 3-d · pdf fileaiaa-2003-0159 conduction shape factor...

AIAAAIAA--20032003--01590159

Conduction Shape Factor Models Conduction Shape Factor Models for 3for 3--D EnclosuresD Enclosures

P. Teertstra, M. M. Yovanovich and J. R. Culham

Microelectronics Heat Transfer LaboratoryDepartment of Mechanical Engineering

University of WaterlooWaterloo, Ontario, Canada

http://www.mhtlab.uwaterloo.ca

AIAA 41st Aerospace Sciences Meeting and Exhibit ConferenceAIAA 41st Aerospace Sciences Meeting and Exhibit ConferenceReno, NV January 6, 2003Reno, NV January 6, 2003

OutlineOutline

11 Microelectronics Heat Transfer LaboratoryMicroelectronics Heat Transfer LaboratoryUniversity of WaterlooUniversity of Waterloo

• Introduction• Literature review• Model development• Application and validation of models• Summary and Conclusions

IntroductionIntroduction


• Analytical models for conduction shape factors in enclosures

heated inner body

cooled surrounding enclosure

arbitrarily-shaped, concentricboundaries

isothermal boundary conditions

• Limiting case for natural convection models for enclosures

Ti

To

Ti To>

Literature ReviewLiterature Review


• Numerical dataHassani1 - concentric circular cylinders

- concentric, base-attached double cones

Warrington et al.2 - concentric cubes- sphere in cubical enclosurecube in spherical enclosure

• Analytical model – Hassani and Hollands3

, concentric spheres – linear superposition

, other boundary shapes, dependent on geometry, inner area, aspect ratio

limited to geometrically similar boundary shapes

( ) iimmm ASASSSS 51.3,, 0

10 ==+= ∞∞ δ

1=m1>m

Problem DefinitionProblem Definition


• Conduction shape factor

• Dimensionless shape factor

( )oi

o

iA

A

TTTrT

dAn

Si

i

−−

=

∂∂

−= ∫∫v

ψ

ψ

( )oiiiA TTAk

QASS

i −==∗

s

ss

d d

h

d

h

b

a

c

Inner and Outer Boundary Shapes

Model DevelopmentModel Development


• Thermal resistance for spherical shells

• Recast based on gap thickness

• General model based on concentric sphere solution

= inner body conduction shape factor in full-space region

= effective gap spacing

( )oiA

oi ddS

ddkR

i −=⇒⎟⎟

⎠

⎞⎜⎜⎝

⎛−= ∗

1211

21 ππ

πδπ

δπδδ 2

)2(2+=⇒

+=⇒

−= ∗ i

Aii

io dSddk

Rddi

∗

iASR ,

∗∞

∗ += SA

Se

iAi δ

∗∞S

eδ

Effective Gap Spacing Effective Gap Spacing –– Integral ModelIntegral Model


• Local gap thickness can be calculated in spherical coordinates for certain geometries

• Example: cube in spherical enclosure

• Effective gap spacing from area average

( ) ( ) ( )

2sectan

40

cossin2,,,

2,

1 πφθπθ

θφθφρθφρθφδ

≤≤≤≤

=−=

−

io sd

( ) θφφθφδπ

δπ π

θ

dddd

o

oe sin

4,24 24

0

2

sectan2

1∫ ∫

−

=

x=s /2i

x, =0θ

y

z, =0φ

φ

θ

( )θφδ ,

Effective Gap Spacing Effective Gap Spacing –– TwoTwo--rule Modelrule Model


• Equivalent spherical enclosure preservesinner body surface area,

enclosed volume,

• Conduction shape factor with two-rule model

( )[ ]26

623

31 ioe

iiioii

ddAVVVdAd −==+== δ

πππ

∗∞

∗ +

−⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+

= S

AV

S

i

Ai

161

2313

31

π

π

iAV

Model ValidationModel Validation


• Models validated for seven enclosure configurations

geometrically similar boundary shapes• cubes, cylinders, base-attached double cones

different boundary shapes• cube in spherical enclosure

• sphere in cubical enclosure

• cuboid in cubical enclosure

• circular cylinder in cubical enclosure

• Validation performed using numerical dataexisting data from literature

FLOTHERM7 CFD simulations

Geometrically Similar Boundary ShapesGeometrically Similar Boundary Shapes


391.3

116

1

2313

+

−⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛+

=∗

i

o

A

ss

Si

π

π• Concentric cubes

• Cylinders

• Base-attached double cones

443.3

116

21

2313

+

−⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛+

=∗

i

o

A

dd

Si

π

1=dh

1=dh

471.3

1121

2313

+

−⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛+

=∗

i

o

A

dd

Si

π

Geometrically Similar Boundary ShapesGeometrically Similar Boundary Shapes


V 1/3 / √ Ai

S* √Ai

10-1 100 101100

101

102

⎯

Spheres

CubesCylindersDouble Cones

CylindersDouble ConesCubes

Two-term Model

Hassani1

Warrington et al.2

⎧⎨⎩

⎯

⎧⎨⎩

Exact Solution

Enclosures with Different Boundary ShapesEnclosures with Different Boundary Shapes


• Cube in spherical enclosure

integral method

two-rule method

• Sphere in cubical enclosure

integral method

two rule method

• Cuboid in cubical enclosure

two-rule method

• Circular cylinder in cubical enclosure

two-rule method

⎟⎟⎠

⎞⎜⎜⎝

⎛−= 6107.0

21

i

oie s

dsδ

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

216107.0

i

oie d

sdδ

( )acaba 175.2,785.3, ==

( )5.0=dh

Cube in Spherical EnclosureCube in Spherical Enclosure


V 1/3/ √Ai

S* √Ai

d o/s

i

0.5 1 1.5 22

4

6

8

10

12

14

0

2

4

6

8

10Warrington et al.2Integral ModelTwo-rule Model

⎢

⎢

do / si

⎧⎨⎩

Sphere in Cubical EnclosureSphere in Cubical Enclosure


V 1/3/ √Ai

S* √Ai

s o/d

i

0.5 1 1.5 22

4

6

8

10

12

14

0

2

4

6

8

10Warrington et al.2Integral ModelTwo-rule Model

⎢

⎢

so / di

⎧⎨⎩

Cuboid in Cubical EnclosureCuboid in Cubical Enclosure


V 1/3/ √Ai

S* √Ai

s o/b

1 2 3 4 52

4

6

8

10

12

14

0

2

4

6

8

10Two-rule ModelFLOTHERM7 Data

⎢

⎢

so / b

⎧⎨⎩

Circular Cylinder in Cubical EnclosureCircular Cylinder in Cubical Enclosure


V 1/3/ √Ai

S* √ Ai

s o/d

i

0.5 1 1.5 22

4

6

8

10

12

14

0

1

2

3

4

5Two-rule ModelFLOTHERM7 Data

⎢

⎢

so / di

⎧⎨⎩

Summary and ConclusionsSummary and Conclusions


• General model for conduction shape factors for isothermal 3-D enclosures

• Two models for effective gap spacingintegral method – limited to specific geometries

two-rule method – applicable to all enclosures

• Excellent agreement (<3% RMS) with numerical data for geometrically similar boundary shapes

• Good agreement for enclosures with different boundary shapes

3 – 5 % RMS when 131 >iAV

AcknowledgementsAcknowledgements


• MMO - Materials and Manufacturing Ontario

• CMAP - Center for Microelectronics Assembly and Packaging

conduction shape factor models for 3-d · pdf fileaiaa-2003-0159 conduction shape factor...

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