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RESEARCH PAPERS: Solution Methods

A Finite Element Treatment of the Angular Dependency of the Even-Parity Equation of Radiative Transfer

[+] Author and Article Information
R. Becker1

Institut für Thermische Strömungsmaschinen, Universität Karlsruhe, Kaiserstraße 12, 76128 Karlsruhe, Germanybecker@its.uni-karlsruhe.de

R. Koch, H.-J. Bauer

Institut für Thermische Strömungsmaschinen, Universität Karlsruhe, Kaiserstraße 12, 76128 Karlsruhe, Germany

M. F. Modest

Department of Mechanical and Nuclear Engineering, Pennsylvania State University, University Park, PA 16802

1

Corresponding author.

J. Heat Transfer 132(2), 023404 (Dec 04, 2009) (13 pages) doi:10.1115/1.4000233 History: Received November 13, 2008; Revised March 31, 2009; Published December 04, 2009; Online December 04, 2009

The present article introduces a new method to solve the radiative transfer equation (RTE). First, a finite element discretization of the solid angle dependence is derived, wherein the coefficients of the finite element approximation are functions of the spatial coordinates. The angular basis functions are defined according to finite element principles on subdivisions of the octahedron. In a second step, these spatially dependent coefficients are discretized by spatial finite elements. This approach is very attractive, since it provides a concise derivation for approximations of the angular dependence with an arbitrary number of angular nodes. In addition, the usage of high-order angular basis functions is straightforward. In the current paper, the governing equations are first derived independently of the actual angular approximation. Then, the design principles for the angular mesh are discussed and the parameterization of the piecewise angular basis functions is derived. In the following, the method is applied to one-dimensional and two-dimensional test cases, which are commonly used for the validation of approximation methods of the RTE. The results reveal that the proposed method is a promising alternative to the well-established practices like the discrete ordinates method (DOM) and provides highly accurate approximations. A test case, which is known to exhibit the ray effect in the DOM, verifies the ability of the new method to avoid ray effects.

Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

Structure of the spatial and angular linear system

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Figure 2

Triangulation of the order N=2 and the derived octahedral angular finite element mesh

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Figure 3

Test case A: Radiative heat flux q in an emitting-absorbing plane-parallel medium

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Figure 4

Test case A: Incident radiation G in an emitting and absorbing parallel medium

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Figure 5

Test case A: rms values of net heat flux predictions

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Figure 6

Test case A: rms values of incident radiation predictions

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Figure 7

Contour plots of the exact solution of the even-parity equation

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Figure 8

Test case A: Comparison of the rms(q) calculated with the finite element method, the DOM, and the P1-approximation

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Figure 9

Test case B: Incident radiation G

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Figure 10

Test case B: rms values of the net heat flux

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Figure 11

Test case B: rms values of the incident radiation

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Figure 12

Test case C: Radiative heat flux profiles at the walls of an emitting and absorbing two-dimensional medium

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Figure 13

Test case C: Radiative heat flux profiles at the walls of an emitting and absorbing two-dimensional medium

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Figure 14

Test case C: Incident radiative intensity in the centerline of an emitting and absorbing two-dimensional medium

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Figure 15

Test case C: Incident radiative intensity in the centerline of an emitting and absorbing two-dimensional medium

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Figure 16

Test case D: Radiative heat flux at the top of a purely isotropically scattering two-dimensional medium

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Figure 17

Test case D: Incident radiative intensity in the center of a purely isotropically scattering two-dimensional medium

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