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Research Papers: Conduction

A Multipoint Flux Approximation of the Steady-State Heat Conduction Equation in Anisotropic Media

[+] Author and Article Information
Amgad Salama

Department of Mathematics and
Computer Science,
King Abdullah University of
Science and Technology (KAUST),
Thuwal 23955-6900, KSA;
Nuclear Research Center,
Atomic Energy Authority,
13759 Cairo, Egypt
e-mail: amgad.salama@kaust.edu.sa

Shuyu Sun

Department of Mathematics and
Computer Science,
King Abdullah University of
Science and Technology (KAUST),
Thuwal 23955-6900, KSA

M. F. El Amin

Department of Mathematics and
Computer Science,
King Abdullah University of
Science and Technology (KAUST),
Thuwal 23955-6900, KSA;
Mathematics Department,
Faculty of Science,
Aswan University,
81718 Aswan, Egypt

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the Journal of Heat Transfer. Manuscript received May 10, 2012; final manuscript received October 25, 2012; published online March 20, 2013. Assoc. Editor: Bruce L. Drolen.

J. Heat Transfer 135(4), 041302 (Mar 20, 2013) (6 pages) Paper No: HT-12-1214; doi: 10.1115/1.4023228 History: Received May 10, 2012; Revised October 25, 2012

In this work, we introduce multipoint flux (MF) approximation method to the problem of conduction heat transfer in anisotropic media. In such media, the heat flux vector is no longer coincident with the temperature gradient vector. In this case, thermal conductivity is described as a second order tensor that usually requires, at least, six quantities to be fully defined in general three-dimensional problems. The two-point flux finite differences approximation may not handle such anisotropy and essentially more points need to be involved to describe the heat flux vector. In the framework of mixed finite element method (MFE), the MFMFE methods are locally conservative with continuous normal fluxes. We consider the lowest order Brezzi–Douglas–Marini (BDM) mixed finite element method with a special quadrature rule that allows for nodal velocity elimination resulting in a cell-centered system for the temperature. We show comparisons with some analytical solution of the problem of conduction heat transfer in anisotropic long strip. We also consider the problem of heat conduction in a bounded, rectangular domain with different anisotropy scenarios. It is noticed that the temperature field is significantly affected by such anisotropy scenarios. Also, the technique used in this work has shown that it is possible to use the finite difference settings to handle heat transfer in anisotropic media. In this case, heat flux vectors, for the case of rectangular mesh, generally require six points to be described.

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References

Figures

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Fig. 5

Temperature distribution in longitudinal section (comparison with Ref. [23])

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Fig. 4

Temperature contours when the medium is anisotropic

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Fig. 3

Temperature contours when the medium is isotropic

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Fig. 2

Computational domain based on the work of Zhang [23]

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Fig. 1

Schematic diagram of the contributing cells of the fluxes at a generic node

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Fig. 6

Temperature contours and heat flux for isotropic medium (case 1)

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Fig. 7

Temperature contours and heat flux for anisotropic medium (case 1)

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Fig. 8

Temperature contours and heat flux for isotropic medium (case 2)

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Fig. 9

Temperature contours and heat flux for anisotropic medium (case 2)

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