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Research Papers: Conduction

A Method for Computing the Analytical Solution of the Steady-State Heat Equation in Multilayered Media

[+] Author and Article Information
Ivor Dülk

Department of Measurement and
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,
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 136(9), 091303 (Jun 27, 2014) (11 pages) Paper No: HT-13-1416; doi: 10.1115/1.4027838 History: Received August 14, 2013; Revised June 02, 2014

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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References

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Figures

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Fig. 1

Model of the multilayered medium

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Fig. 2

The developed calculation method

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Fig. 3

The straightforward two−layer partitioning

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Fig. 4

Proposed method to decompose the multilayered medium into two-layered media

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Fig. 5

Lumped model definitions and symbols

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Fig. 6

Lumped network representation of the layers

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Fig. 7

The four-layered structure used for simulation

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Fig. 8

Temperature at the left-hand side of layer #1

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Fig. 9

Temperature at the interface of layers #1 and #2

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Fig. 10

Temperature at the interface of layers #2 and #3

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Fig. 11

Temperature at the interface of layers #3 and #4

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Fig. 12

Temperature at the right-hand side of layer #4

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