The computation of the analytical solution of the steady temperature distribution in multilayered media can become numerically unstable if there are different longitudinal (i.e., the directions parallel to the layers) boundary conditions for each layer. In this study, we develop a method to resolve these computational difficulties by approximating the temperatures at the junctions step-by-step and solving for the thermal field separately in only the single layers. First, we solve a two-layer medium problem and then show that multilayered media can be represented as a hierarchy of two-layered media; thus, the developed method is generalized to an arbitrary number of layers. To improve the computational efficiency and speed, we use varying weighting coefficients during the iterations, and we present a method to decompose the multilayered media into two-layered media. The developed method involves the steady-state solution of the diffusion equation, which is illustrated for 2D slabs using separation of variables (SOV). A numerical example of four layers is also included, and the results are compared to a numerical solution.
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September 2014
This article was originally published in
Journal of Heat Transfer
Research-Article
A Method for Computing the Analytical Solution of the Steady-State Heat Equation in Multilayered Media
Ivor Dülk,
Ivor Dülk
1
Department of Measurement and
Information Systems,
e-mail: divor@mit.bme.hu
Information Systems,
Budapest University of Technology and Economics
,Magyar tudósok körútja 2. I ép.
,Budapest 1117
, Hungary
e-mail: divor@mit.bme.hu
1Corresponding author.
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Tamás Kovácsházy
Tamás Kovácsházy
Department of Measurement and
Information Systems,
e-mail: khazy@mit.bme.hu
Information Systems,
Budapest University of Technology and Economics
,Magyar tudósok körútja 2. I ép.
,Budapest 1117
, Hungary
e-mail: khazy@mit.bme.hu
Search for other works by this author on:
Ivor Dülk
Department of Measurement and
Information Systems,
e-mail: divor@mit.bme.hu
Information Systems,
Budapest University of Technology and Economics
,Magyar tudósok körútja 2. I ép.
,Budapest 1117
, Hungary
e-mail: divor@mit.bme.hu
Tamás Kovácsházy
Department of Measurement and
Information Systems,
e-mail: khazy@mit.bme.hu
Information Systems,
Budapest University of Technology and Economics
,Magyar tudósok körútja 2. I ép.
,Budapest 1117
, Hungary
e-mail: khazy@mit.bme.hu
1Corresponding author.
Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received August 14, 2013; final manuscript received June 2, 2014; published online June 27, 2014. Assoc. Editor: Danesh / D. K. Tafti.
J. Heat Transfer. Sep 2014, 136(9): 091303 (11 pages)
Published Online: June 27, 2014
Article history
Received:
August 14, 2013
Revision Received:
June 2, 2014
Citation
Dülk, I., and Kovácsházy, T. (June 27, 2014). "A Method for Computing the Analytical Solution of the Steady-State Heat Equation in Multilayered Media." ASME. J. Heat Transfer. September 2014; 136(9): 091303. https://doi.org/10.1115/1.4027838
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