In this paper, thermal resistance of a 2D flux channel with nonuniform convection coefficient in the heat sink plane is studied using the method of separation of variables and the least squares technique. For this purpose, a two-dimensional flux channel with discretely specified heat flux is assumed. The heat transfer coefficient at the sink boundary is defined symmetrically using a hyperellipse function which can model a wide variety of different distributions of heat transfer coefficient from uniform cooling to the most intense cooling in the central region. The boundary condition along the edges is defined with convective cooling. As a special case, the heat transfer coefficient along the edges can be made negligible to simulate a flux channel with adiabatic edges. To obtain the temperature profile and the thermal resistance, the Laplace equation is solved by the method of separation of variables considering the applied boundary conditions. The temperature along the flux channel is presented in the form of a series solution. Due to the complexity of the sink plane boundary condition, there is a need to calculate the Fourier coefficients using the least squares method. Finally, the dimensionless thermal resistance for a number of different systems is presented. Results are validated using the data obtained from the finite element method (FEM). It is shown that the thick flux channels with variable heat transfer coefficient can be simplified to a flux channel with the same uniform heat transfer coefficient.

References

References
1.
Lee
,
S.
,
Song
,
S.
,
Au
,
V.
, and
Moran
,
K. P.
,
1995
, “
Constriction/Spreading Resistance Model for Electronics Packaging
,”
4th ASME/JSME Thermal Engineering Joint Conference
,
Maui, HI
, pp.
199
206
.
2.
Song
,
S.
,
Lee
,
S.
, and
Au
,
V.
,
1994
, “
Closed-Form Equation for Thermal Constriction/Spreading Resistances With Variable Resistance Boundary Condition
,”
Intelligent Energy and Power Systems Conference
,
Atlanta, GA
, pp.
111
121
.
3.
Das
,
A. K.
, and
Sadhal
,
S. S.
,
1999
, “
Thermal Constriction Resistance Between Two Solids for Random Distribution of Contacts
,”
Heat Mass Transfer
,
35
(
2
), pp.
101
111
.
4.
Lam
,
T. T.
, and
Fischer
,
W. D.
,
1999
, “
Thermal Resistance in Rectangular Orthotropic Heat Spreaders
,”
ASME Advances in Electronic Packaging
, Vol.
26-3
,
American Society of Mechanical Engineers
,
New York
, pp.
891
896
.
5.
Ellison
,
G.
,
1991
, “
Extensions of a Closed Form Method for Substrate Thermal Analyzers to Include Thermal Resistances From Source-to-Substrate and Source-to-Ambient
,”
7th IEEE Semi-Therm Symposium
, pp.
140
148
.
6.
Ellison
,
G.
,
1995
, “
Thermal Analysis of Microelectric Packages and Printed Circuit Boards Using an Analytic Solution to the Heat Conduction Equation
,”
Adv. Eng. Software
,
22
(
2
), pp.
99
111
.
7.
Ellison
,
G.
,
1996
, “
Thermal Analysis of Circuit Boards and Microelectronic Components Using an Analytical Solution to the Heat Conduction Equation
,”
12th IEEE Semi-Therm Symposium
, pp.
144
150
.
8.
Muzychka
,
Y. S.
,
Yovanovich
,
M. M.
, and
Culham
,
J. R.
,
2004
, “
Thermal Spreading Resistance in Compound and Orthotropic Systems
,”
J. Thermophys. Heat Transfer
,
18
(
1
), pp.
45
51
.
9.
Muzychka
,
Y. S.
, and
Yovanovich
,
M. M.
,
2001
, “
Thermal Resistance Models for Non-Circular Moving Heat Sources on a Half Space
,”
ASME J. Heat Transfer
,
123
(
4
), pp.
624
632
.
10.
Muzychka
,
Y. S.
,
Culham
,
J. R.
, and
Yovanovich
,
M. M.
,
2003
, “
Thermal Spreading Resistance of Eccentric Heat Sources on Rectangular Flux Channels
,”
ASME J. Electron. Packag.
,
125
(
2
), pp.
178
185
.
11.
Muzychka
,
Y. S.
,
Stevanovic
,
M.
, and
Yovanovich
,
M. M.
,
2001
, “
Thermal Spreading Resistances in Compound Annular Sectors
,”
J. Thermophys. Heat Transfer
,
15
(
3
), pp.
354
359
.
12.
Muzychka
,
Y. S.
,
2006
, “
Influence Coefficient Method for Calculating Discrete Heat Source Temperature on Finite Convectively Cooled Substrates
,”
IEEE Trans. Compon. Packag. Technol.
,
29
(
3
), pp.
636
643
.
13.
Muzychka
,
Y. S.
,
Yovanovich
,
M. M.
, and
Culham
,
J. R.
,
2006
, “
Influence of Geometry and Edge Cooling on Thermal Spreading Resistance
,”
J. Thermophys. Heat Transfer
,
20
(
2
), pp.
247
255
.
14.
Muzychka
,
Y. S.
,
Bagnall
,
K.
, and
Wang
,
E.
,
2013
, “
Thermal Spreading Resistance and Heat Source Temperature in Compound Orthotropic Systems With Interfacial Resistance
,”
IEEE Trans. Compon., Packag., Manuf. Technol.
,
3
(
11
), pp.
1826
1841
.
15.
Bagnall
,
K.
,
Muzychka
,
Y. S.
, and
Wang
,
E.
,
2013
, “
Application of the Kirchhoff Transform to Thermal Spreading Problems With Convection Boundary Conditions
,”
IEEE Trans. Compon., Packag., Manuf. Technol.
,
4
(
3
), pp.
408
420
.
16.
Bagnall
,
K.
,
Muzychka
,
Y. S.
, and
Wang
,
E.
,
2014
, “
Analytical Solution for Temperature Rise in Complex, Multi-Layer Structures With Discrete Heat Sources
,”
IEEE Trans. Compon., Packag., Manuf. Technol.
,
4
(
5
), pp.
817
830
.
17.
Muzychka
,
Y. S.
,
2014
, “
Spreading Resistance in Compound Orthotropic Flux Tubes and Channels With Interfacial Resistance
,”
J. Thermophys. Heat Transfer
,
28
(
2
), pp.
313
319
.
18.
Yovanovich
,
M. M.
, and
Marotta
,
E. E.
,
2003
, “
Thermal Spreading and Contact Resistances
,”
Heat Transfer Handbook
,
A.
Bejan
, and
A. D.
Kraus
, eds.,
Wiley
,
New York
, pp.
261
393
.
19.
Yovanovich
,
M. M.
,
2005
, “
Four Decades of Research on Thermal Contact, Gap and Joint Resistance in Microelectronics
,”
IEEE Trans. Compon. Packag. Technol.
,
28
(
2
), pp.
182
206
.
20.
“ASUS Releases NVIDIA GeForce GT 520 Silent Low Profile Graphics Card,” http://www.legitreviews.com
21.
“Asus' Formula Rampage Motherboard X48 Goes Old-school With DDR2,” http://www.techreport.com
22.
Kelman
,
R. B.
,
1979
, “
Least Squares Fourier Series Solutions to Boundary Value Problems
,”
Soc. Ind. Appl. Math.
,
21
(
3
), pp.
329
338
.
23.
Maple 10, Waterloo Maple Software
, Waterloo, ON, Canada.
24.
COMSOL Multiphysics
® Version 4.2a.
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