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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
22 Microelectronics Heat Transfer LaboratoryMicroelectronics Heat Transfer LaboratoryUniversity of WaterlooUniversity of Waterloo
• 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
33 Microelectronics Heat Transfer LaboratoryMicroelectronics Heat Transfer LaboratoryUniversity of WaterlooUniversity of Waterloo
• Numerical dataHassani1 - concentric circular cylinders
- concentric, base-attached double cones
Warrington et al.2 - concentric cubes- sphere in cubical enclosure- cube 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
44 Microelectronics Heat Transfer LaboratoryMicroelectronics Heat Transfer LaboratoryUniversity of WaterlooUniversity of Waterloo
• 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
55 Microelectronics Heat Transfer LaboratoryMicroelectronics Heat Transfer LaboratoryUniversity of WaterlooUniversity of Waterloo
• 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
66 Microelectronics Heat Transfer LaboratoryMicroelectronics Heat Transfer LaboratoryUniversity of WaterlooUniversity of Waterloo
• 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
77 Microelectronics Heat Transfer LaboratoryMicroelectronics Heat Transfer LaboratoryUniversity of WaterlooUniversity of Waterloo
• 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
88 Microelectronics Heat Transfer LaboratoryMicroelectronics Heat Transfer LaboratoryUniversity of WaterlooUniversity of Waterloo
• 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
99 Microelectronics Heat Transfer LaboratoryMicroelectronics Heat Transfer LaboratoryUniversity of WaterlooUniversity of Waterloo
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
1010 Microelectronics Heat Transfer LaboratoryMicroelectronics Heat Transfer LaboratoryUniversity of WaterlooUniversity of Waterloo
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
1111 Microelectronics Heat Transfer LaboratoryMicroelectronics Heat Transfer LaboratoryUniversity of WaterlooUniversity of Waterloo
• 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
1212 Microelectronics Heat Transfer LaboratoryMicroelectronics Heat Transfer LaboratoryUniversity of WaterlooUniversity of Waterloo
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
1313 Microelectronics Heat Transfer LaboratoryMicroelectronics Heat Transfer LaboratoryUniversity of WaterlooUniversity of Waterloo
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
1414 Microelectronics Heat Transfer LaboratoryMicroelectronics Heat Transfer LaboratoryUniversity of WaterlooUniversity of Waterloo
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
1515 Microelectronics Heat Transfer LaboratoryMicroelectronics Heat Transfer LaboratoryUniversity of WaterlooUniversity of Waterloo
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
1616 Microelectronics Heat Transfer LaboratoryMicroelectronics Heat Transfer LaboratoryUniversity of WaterlooUniversity of Waterloo
• 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