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Research Papers: Electronic Cooling

# Thermal Resistance in a Rectangular Flux Channel With Nonuniform Heat Convection in the Sink Plane

[+] Author and Article Information
M. Razavi

Faculty of Engineering and Applied Science,
Memorial University of Newfoundland,
Street John's, NL A1B 3X5, Canada
e-mail: m.razavi@mun.ca

Y. S. Muzychka

Faculty of Engineering and Applied Science,
Memorial University of Newfoundland,
Street John's, NL A1B 3X5, Canada

S. Kocabiyik

Department of Mathematics and Statistics,
Memorial University of Newfoundland,
Street John's, NL A1C 5S7, Canada

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received December 7, 2014; final manuscript received June 15, 2015; published online July 21, 2015. Assoc. Editor: Amy Fleischer.

J. Heat Transfer 137(11), 111401 (Jul 21, 2015) (9 pages) Paper No: HT-14-1786; doi: 10.1115/1.4030885 History: Received December 07, 2014

## Abstract

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.

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## References

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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
“ASUS Releases NVIDIA GeForce GT 520 Silent Low Profile Graphics Card,”
“Asus' Formula Rampage Motherboard X48 Goes Old-school With DDR2,”
Kelman, R. B. , 1979, “Least Squares Fourier Series Solutions to Boundary Value Problems,” Soc. Ind. Appl. Math., 21(3), pp. 329–338.
Maple 10, Waterloo Maple Software, Waterloo, ON, Canada.
COMSOL Multiphysics® Version 4.2a.

## Figures

Fig. 1

A sample of heat sink with variable heat transfer coefficient

Fig. 2

Example of systems with nonuniform heat sink: (a) ASUS NVIDIA GeForce GT 520 silent low profile graphics card [20] and (b) ASUS R.O.G. Rampage formula motherboard [21]

Fig. 3

Two-dimensional flux channel with a central heat source and a variable heat transfer coefficient

Fig. 4

Variable heat transfer coefficient for half of the slab by considering h(x)/ho=1-(x/c)m

Fig. 5

Variable heat transfer coefficient for half of the slab by considering h(x)/h¯=(m+1)/m[1-(x/c)m]

Fig. 6

Dimensionless thermal resistance for Bio = 1 and τ = t/c = 0.1

Fig. 7

Dimensionless thermal resistance for Bio = 1 and τ = t/c = 0.5

Fig. 8

Dimensionless thermal resistance for Bio = 10 and τ = t/c = 0.1

Fig. 9

Dimensionless thermal resistance for Biavg = 0.1 and τ = t/c = 0.1

Fig. 10

Dimensionless thermal resistance for Biavg = 1 and τ = t/c = 0.1

Fig. 11

Dimensionless thermal resistance for Biavg = 1 and τ = t/c = 0.5

Fig. 12

Dimensionless thermal resistance for Biavg = 10 and τ = t/c = 0.1

Fig. 13

Dimensionless thermal resistance for Biavg = 100 and τ = t/c = 0.1

Fig. 14

Left: 2D flux channel with dimensionless thickness of τ = 0.1 and right: 2D flux channel with dimensionless thickness of τ = 0.5

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