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Research Papers: Conduction

Analysis of Heat Conduction in a Heterogeneous Material by a Multiple-Scale Averaging Method

[+] Author and Article Information
James White

6017 Glenmary Road,
Knoxville, TN 37919
e-mail: JWhiteTechnology@gmail.com

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received August 24, 2014; final manuscript received February 4, 2015; published online March 24, 2015. Assoc. Editor: K. Tafti.

J. Heat Transfer 137(7), 071301 (Jul 01, 2015) (11 pages) Paper No: HT-14-1562; doi: 10.1115/1.4029774 History: Received August 24, 2014; Revised February 04, 2015; Online March 24, 2015

In order to better manage computational requirements in the study of thermal conduction with short-scale heterogeneous materials, one is motivated to arrange the thermal energy equation into an accurate and efficient form with averaged properties. This should then allow an averaged temperature solution to be determined with a moderate computational effort. That is the topic of this paper as it describes the development using multiple-scale analysis of an averaged thermal energy equation based on Fourier heat conduction for a heterogeneous material with isotropic properties. The averaged energy equation to be reported is appropriate for a stationary or moving solid and three-dimensional heat flow. Restrictions are that the solid must display its heterogeneous properties over short spatial and time scales that allow averages of its properties to be determined. One distinction of the approach taken is that all short-scale effects, both moving and stationary, are combined into a single function during the analytical development. The result is a self-contained form of the averaged energy equation. By eliminating the need for coupling the averaged energy equation with external local problem solutions, numerical solutions are simplified and made more efficient. Also, as a result of the approach taken, nine effective averaged thermal conductivity terms are identified for three-dimensional conduction (and four effective terms for two-dimensional conduction). These conductivity terms are defined with two types of averaging for the component material conductivities over the short-scales and in terms of the relative proportions of the short-scales. Numerical results are included and discussed.

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Figures

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Fig. 1

Profiles of effective thermal conductivity as functions of short length scale ratio and area ratio (ka = 1,kb = 5)

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Fig. 2

Steady-state temperature profiles over a square cross section for three heterogeneous cases and one homogeneous material configuration (fixed surface temperatures and no heat generation)

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Fig. 3

Steady-state temperature profiles for three heterogeneous cases with uniform heat generation (fixed surface temperatures)

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Fig. 4

Comparison of time-dependent temperature solutions for a heterogeneous and homogeneous material, with each having the same arithmetically averaged conductivity (A = 1,ka = 1,kb = 5,Δt¯ = 2(10-5), black line) for heterogeneous case, (k = 3, red line) for homogeneous case (a) solution after 100 dimensionless time steps of heterogeneous solution, (b) 250 time steps, (c) 500 time steps, and (d) 1000 time steps

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Fig. 5

Steady-state temperature profile for a heterogeneous material with fixed surface temperatures and uniform heat generation (A = 1, ka = 1,kb = 5)

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Fig. 6

Steady-state temperature profile for a homogeneous material with fixed surface temperatures and uniform heat generation (k = 3)

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Fig. 7

Steady-state temperature profile for one insulated surface and three fixed temperature surfaces, with uniform heat generation (A = 1/5, ka = 1,kb = 5)

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Fig. 8

Steady-state temperature profile for one insulated surface and three fixed temperature surfaces, with uniform heat generation (A = 1, ka = 1,kb = 5)

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Fig. 9

Steady-state temperature profile for one insulated surface and three fixed temperature surfaces, with uniform heat generation (A = 5, ka = 1,kb = 5)

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Fig. 10

Steady-state temperature profile of a homogeneous material for one insulated surface and three fixed temperature surfaces, with uniform heat generation (k = 3)

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Fig. 11

Steady-state temperature profile for different heterogeneous materials in left and right halves of square cross section (fixed surface temperatures and uniform heat generation)

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Fig. 12

Steady-state temperature profile for different homogeneous materials in left and right halves of square cross section (fixed surface temperatures and uniform heat generation)

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Fig. 13

Convergence characteristics of the steady-state solutions

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