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Numerical Study of Thermal Spreading Resistance in Body-Fitted Curvilinear Coordinates

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Abstract

One of the most important problems encountered in the thermal management of microelectronics is thermal spreading resistance. This occurs either due to the heat transfer by the conduction mechanism from one solid to another with different cross-sectional areas, or as a result of the heat flow through a conductive solid with a variable cross-sectional area. In this study, both geometric conditions are considered simultaneously. A C++ program code is developed to calculate the thermal spreading resistance in arbitrary curved-edge heat spreaders. A method for automatic numerical generation of a body-fitted curvilinear coordinate system is applied to solve the heat conduction equation on the orthogonal curvilinear grids. A set of Poisson equations is then used to generate two-dimensional grids with grid control along all of the boundaries. The finite difference method is employed to discretize the partial differential equations of the problem. In addition, the Maxwell coordinate system is also used as a special case to demonstrate the similarity of grids generated by numerical and analytical methods. The numerical results of the study are compared with the exact solution, thus illustrating the performance of the approach. Finally, the temperature distribution and thermal spreading resistance are determined for the different shapes of curved edges.

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Abbreviations

a:

Length of the spreader top surface (m)

A:

Area (\(\hbox {m}^{2}\))

A, B:

Relations defined in Eq. 20

b:

Length of the spreader bottom surface (m)

c:

Length of the heat source (m)

J:

Jacobian value

M:

Control function

n:

Unit normal vector, positive in the outward direction

P:

Control function

W, W\(^{*}\) :

Weighting function scheme

x, y, z:

Cartesian coordinates (m)

\(\upalpha ,\upbeta ,\upgamma \) :

Functions defined in Eq. 19

\(\upvarepsilon , \tau \) :

Constant value defined in Eq. 16

\(\updelta \) :

Height of heat spreader (m)

\(\uptheta \) :

Angel

\(\upxi ,\upeta \) :

Curvilinear coordinates (m)

a:

Spreader top surface length

avg:

Length-averaged value

c:

Heat source

E:

East node neighbor

F:

Function

i:

Nodes in x direction

j:

Nodes in y direction

min:

Minimum value

max:

Maximum value

NW:

North-West node neighbor

NE:

North-East node neighbor

P:

Main node

S:

Heat sink, spreading

SW:

South-West node neighbor

S:

South node neighbor

SE:

South-East node

t:

Total value

W:

West node neighbor

\(\infty \) :

Fluid temperature, convective sink case

a, b, c, d:

Arbitrary values used in Eq. 21

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Authors and Affiliations

  1. Babol University of Technology, P.O. Box 484, Babol, Iran

    Yousef Rahmani, Mofid Gorji Bandpy & Davood Domiri Ganji

Authors
  1. Yousef Rahmani
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  2. Mofid Gorji Bandpy
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  3. Davood Domiri Ganji
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Correspondence to Yousef Rahmani.

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Rahmani, Y., Bandpy, M.G. & Ganji, D.D. Numerical Study of Thermal Spreading Resistance in Body-Fitted Curvilinear Coordinates. Int. J. Appl. Comput. Math 3, 2873–2888 (2017). https://doi.org/10.1007/s40819-016-0271-7

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