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Research Papers: Radiative Heat Transfer

Solution of the Radiative Transfer Equation in Three-Dimensional Participating Media Using a Hybrid Discrete Ordinates: Spherical Harmonics Method

[+] Author and Article Information
Maathangi Sankar

 Department of Mechanical and Aerospace Engineering, The Ohio State University Columbus, OH 43210

Sandip Mazumder1

 Department of Mechanical and Aerospace Engineering, The Ohio State University Columbus, OH 43210mazumder.2@osu.edu

1

Corresponding author.

J. Heat Transfer 134(11), 112702 (Sep 28, 2012) (12 pages) doi:10.1115/1.4007129 History: Received May 16, 2011; Revised May 21, 2012; Published September 28, 2012; Online September 28, 2012

In this article, a new hybrid solution to the radiative transfer equation (RTE) is proposed. Following the modified differential approximation (MDA), the radiation intensity is first split into two components: a “wall” component, and a “medium” component. Traditionally, the wall component is determined using a viewfactor-based surface-to-surface exchange formulation, while the medium component is determined by invoking the first-order spherical harmonics (P1 ) approximation. Recent studies have shown that although the MDA approach is accurate over a large range of optical thicknesses, it is prohibitive for complex three-dimensional geometry with obstructions, both from a computational efficiency as well as memory standpoint. The inefficiency stems from the use of the viewfactor-based approach for determination of the wall-emitted component. In this work, instead, the wall component is determined directly using the control angle discrete ordinates method (CADOM). The new hybrid method was validated for both two-dimensional (2D) and three-dimensional (3D) geometries against benchmark Monte Carlo results for gray media in which the optical thickness was varied over a large range. In all cases, the accuracy of the hybrid method was found to be within a few percent of Monte Carlo results, and comparable to the solutions of the RTE obtained directly using CADOM. Finally, the new hybrid method was explored for 3D nongray media in the presence of reflecting walls and various scattering albedos. As a noteworthy advantage, irrespective of the conditions used, it was always found to be computationally more efficient than standalone CADOM and up to 15 times more efficient than standalone CADOM for optically thick media with strong scattering.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Unstructured stencil showing all relevant geometric entities used for finite-volume integration of the governing equations

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Figure 2

Problem description for first test case: (a) geometry and boundary conditions, and (b) computational mesh used

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Figure 3

Comparison of nondimensional heat fluxes predicted using various methods with benchmark MC results for τh  = 0.1: (a) bottom wall, (b) side wall, and (c) top wall. The heat fluxes are normalized by σTh4.

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Figure 4

Comparison of nondimensional heat fluxes predicted using various methods with benchmark MC results for τh  = 1: (a) bottom wall, (b) side wall, and (c) top wall. The heat fluxes are normalized by σTh4.

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Figure 5

Comparison of nondimensional heat fluxes predicted using various methods with benchmark MC results for τh  = 10: (a) bottom wall, (b) side wall, and (c) top wall. The heat fluxes are normalized by σTh4.

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Figure 6

Geometry for the second test problem

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Figure 7

Nondimensional radiative heat fluxes on bottom wall for the second test case for τh  = 0.1: (a) computed using Monte Carlo method, (b) computed using standard CADOM, (c) computed using the hybrid CADOM-P1 method, (d) percentage error between Monte Carlo and CADOM, and (e) percentage error between Monte Carlo and CADOM-P1

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Figure 8

Nondimensional radiative heat fluxes on bottom wall for the second test case for τh  = 1: (a) computed using Monte Carlo method, (b) computed using standard CADOM, (c) computed using the hybrid CADOM-P1 method, (d) percentage error between Monte Carlo and CADOM, and (e) percentage error between Monte Carlo and CADOM-P1

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Figure 9

Geometry and boundary conditions for the third (nongray) test problem

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Figure 10

Comparison of nondimensional heat flux on the back wall: (a) standard CADOM, and (b) hybrid CADOM-P1

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