Research Papers: Radiative Heat Transfer

Differential Approximations to the Radiation Transfer Equation by Chapman–Enskog Expansion

[+] Author and Article Information
A. V. Gusarov

 ENISE, 58 rue Jean Parot, 42023 Saint-Etienne, Francegusarov@enise.fr

J. Heat Transfer 133(8), 082701 (May 02, 2011) (7 pages) doi:10.1115/1.4003724 History: Received October 07, 2010; Revised February 24, 2011; Published May 02, 2011; Online May 02, 2011

Direct numerical solution of the radiation transfer equation is often easier than implementation of its differential approximations with their cumbersome boundary conditions. Nevertheless, these approximations are still used, for example, in theoretical analysis. The existing approach to obtain a differential approximation based on expansion in series of the spherical harmonics is revised and expansion in series of the eigenfunctions of the scattering integral is proposed. A system of eigenfunctions is obtained for an arbitrary phase function, and explicit differential approximations are built up to the third Chapman–Enskog order. The results are tested by its application to the problem of a layer. The third-order Chapman–Enskog approximation is found to match the boundary conditions better than the first-order one and gives considerably more accurate value for the heat flow. The accuracy of the both first- and third-order heat flows generally increases with the optical thickness. In addition, the third-order heat flow tends to the rigorous limit value when the optical thickness tends to zero.

Copyright © 2011 by American Society of Mechanical Engineers
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Figure 4

Transforms of the coordinate system: (a) initial state (XYZ) with unit vectors of incident Ω′(θ′,φ′) and scattered Ω(θ,φ) radiation, (b) intermediate state (X1Y1Z) after rotation about axis (OZ) by angle φ, and (c) final state (X2Y1Z2) after rotation about axis (OY1) by angle θ, where the scattering angle ψ becomes the polar angle of vector Ω′

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Figure 3

Comparison of the first and third Chapman–Enskog approximations with numerical solutions for the normalized heat flow q between parallel absolutely black plates versus optical thickness Λ: lines, isotropic scattering; points, scattering by dispersed convex particles with diffusely reflecting surface

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Figure 2

Radiative transfer in gray absorbing scattering emitting medium between parallel absolutely black plates at the optical distance Λ=2. Comparison of the Chapman–Enskog method in the first (dashed lines) and third (solid lines) approximations and the numerical solution (dotted lines) for isotropic scattering (left) and scattering by dispersed convex particles with diffusely reflecting surface (right): spatial distribution of the normalized emissive power e (top) and angular distribution of the normalized radiation intensity I at a wall (bottom).

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Figure 1

Normalized Planck distribution W0, Rosseland weight function W1, and higher-order spectral functions W2 and W3



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