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Journal of Heat Transfer | Volume 127 | Issue 11 | TECHNICAL PAPERS
TECHNICAL PAPERS: Radiative Heat Transfer

A Mixed-Mesh and New Angular Space Discretization Scheme of Discontinuous Finite Element Method for Three-Dimensional Radiative Transfer in Participating Media

[+] Author and Article Information
X. Cui

School of Mechanical and Materials Engineering,  Washington State University, Pullman, WA 99164

B. Q. Li1

School of Mechanical and Materials Engineering,  Washington State University, Pullman, WA 99164li@mme.wsu.edu

1

Corresponding author.

J. Heat Transfer 127(11), 1236-1244 (Jun 08, 2005) (9 pages) doi:10.1115/1.2039107 History: Received September 03, 2004; Revised June 08, 2005
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This paper presents a discontinuous finite element method for the numerical solution of internal thermal radiation problems in three-dimensional (3D) geometries using an unstructured mesh of mixed elements. Mathematical formulation, numerical implementation, and computational details are given. The different domain discretization methods are presented, and a new angular space discretization is also given. Numerical examples are presented for 3D radiative transfer in emitting, absorbing, and scattering media. Computed results compare well with analytical solutions whenever available. The localized formulation intrinsic in discontinuous finite elements is considered particularly useful for computational radiation heat transfer in participating media.
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Copyright © 2005 by American Society of Mechanical Engineers
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References

1
Modest, M. F., 1993, "Radiative Heat Transfer". McGraw-Hill, New York.
2
Reed, W. H., and Hill, T. R., 1973, “Triangular Mesh Methods for the Neutron Transport Equation,” Tech. Report LA-UR-73-479, Los Alamos Scientific Laboratory.
3
Li, B. Q., 2005, "Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer", Springer-Verlag, London.
4
Cui, X., and Li, B. Q., 2004, “A Discontinuous Finite Element Formulation for Internal Radiation Problems,” Numer. Heat Transfer, Part B, 46 , pp. 223–242.
5
Ai, X., and Li, B. Q., 2004, “A Discontinuous Finite Element Method for Hyperbolic Thermal Wave Problems,” Eng. Comput., 21 , pp. 577–597.
6
Chai, J. C., and Patankar, S. V., 2000, “Finite Volume Method For Radiation Heat Transfer,” "Advances in Numerical Heat Transfer", W.J.Minkowycz, and E.M.Sparrow, eds., Taylor & Francis, New York, Vol. 2 , Chap. 4.
7
Raithby, G. D., and Chui, E. H., 1990, “A Finite-Volume Method for Predicting a Radiant Heat Transfer in Enclosure with Participating Media,” ASME J. Heat Transfer, 112 , pp. 415–423.
8
Moder, J. P., Kumar, G. N., and Chai, J. C., 2000, “An Unstructured-Grid Radiative Heat Transfer Module for the National Combustion Code,” 38th Aerospace Sciences Meeting & Exhibit , Reno, NV.
9
Oden, J. T., Babuka, I., and Baumann, C., 1998, “A Discontinuous hp Finite Element Method for Diffusion Problems,” J. Comput. Phys.  [CrossRef], 146 , pp. 491–519.
10
Murthy, J. Y., and Mathur, S. R., 1998, “Finite Volume Method for Radiative Heat Transfer Using Unstructured Meshes,” J. Thermophys. Heat Transfer, 12 , pp. 313–321.
11
Kim, T. K., and Lee, H., 1998, “Effect of Anisotropic Scattering on Radiative Heat Transfer in Two-Dimensional Rectangular Enclosures,” Int. J. Heat Mass Transfer  [CrossRef], 31 , pp. 1711–1721.
12
Wiscombe, W. J., 1980, “Improved Mie Scattering Algorithms,” Appl. Opt., 19 , pp. 1505–1509.
13
Özisik, M. N., 1973, "Radiative Transfer", Wiley-Interscience, New York.
14
Siegel, R., and Howell, J. R., 1992, "Thermal Radiation Heat Transfer", 3rd Edition, Hemisphere, Washington, DC.
15
Kim, S. H., and Huh, K. Y., 2000, “A New Angular Discretization Scheme of the Finite Volume Method for 3-D Radiative Heat Transfer in Absorbing, Emitting and Anisotropically Scattering Medium,” Int. J. Heat Mass Transfer  [CrossRef], 43 , pp. 1233–1242.

Figures

Grahic Jump Location
+Schematic representation of radiative transfer in a participating medium (a) and definition of the direction of radiation intensity and symmetry boundary condition. (b) angular space discretization.
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Schematic representation of radiative transfer in a participating medium (a) and definition of the direction of radiation intensity and symmetry boundary condition. (b) angular space discretization.
Figure 1

Schematic representation of radiative transfer in a participating medium (a) and definition of the direction of radiation intensity and symmetry boundary condition. (b) angular space discretization.

Grahic Jump Location
+Schematic illustration of arrangement of mixed elements
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Schematic illustration of arrangement of mixed elements
Figure 2

Schematic illustration of arrangement of mixed elements

Grahic Jump Location
+3D discretization and calculated radiative heat flux q* along the middle line of top surface y∕L=0.5, and z∕L=1.0. (a) The cube is discretized with 3072 tetrahedral elements or 1024 wedge elements or 512 hexahedral elements or 736 elements of mixed hexahedra and pentahedra. (b) Comparison of radiative heat fluxes calculated using different meshes shown in (a). The absorption coefficient varies from κ=0.1ndκ=1.0. (c) Comparison of the DFE and the finite volume methods for the solution of RTE for κ=10.
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3D discretization and calculated radiative heat flux q* along the middle line of top surface y∕L=0.5, and z∕L=1.0. (a) The cube is discretized with 3072 tetrahedral elements or 1024 wedge elements or 512 hexahedral elements or 736 elements of mixed hexahedra and pentahedra. (b) Comparison of radiative heat fluxes calculated using different meshes shown in (a). The absorption coefficient varies from κ=0.1ndκ=1.0. (c) Comparison of the DFE and the finite volume methods for the solution of RTE for κ=10.
Figure 3

3D discretization and calculated radiative heat flux q* along the middle line of top surface y∕L=0.5, and z∕L=1.0. (a) The cube is discretized with 3072 tetrahedral elements or 1024 wedge elements or 512 hexahedral elements or 736 elements of mixed hexahedra and pentahedra. (b) Comparison of radiative heat fluxes calculated using different meshes shown in (a). The absorption coefficient varies from κ=0.1ndκ=1.0. (c) Comparison of the DFE and the finite volume methods for the solution of RTE for κ=10.

Grahic Jump Location
+Unstructured tetrahedral mesh and calculated q* in a tetrahedral cavity filled with an absorbing and emitting medium. (a) Unstructured mesh consisting of 1062 tetrahedral elements. (b) Comparison of radiative heat fluxes calculated by the DFE and ray-tracing methods.
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Unstructured tetrahedral mesh and calculated q* in a tetrahedral cavity filled with an absorbing and emitting medium. (a) Unstructured mesh consisting of 1062 tetrahedral elements. (b) Comparison of radiative heat fluxes calculated by the DFE and ray-tracing methods.
Figure 4

Unstructured tetrahedral mesh and calculated q* in a tetrahedral cavity filled with an absorbing and emitting medium. (a) Unstructured mesh consisting of 1062 tetrahedral elements. (b) Comparison of radiative heat fluxes calculated by the DFE and ray-tracing methods.

Grahic Jump Location
+(a) Solid angle distributions of two different angular schemes: the equivalent scheme and the uniform scheme. (b) The boundary heat flux along X∕L at Z∕L=0.5, Y∕L=1.0. The absorption coefficient of this case is κ=1.0 and without scattering.
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(a) Solid angle distributions of two different angular schemes: the equivalent scheme and the uniform scheme. (b) The boundary heat flux along X∕L at Z∕L=0.5, Y∕L=1.0. The absorption coefficient of this case is κ=1.0 and without scattering.
Figure 5

(a) Solid angle distributions of two different angular schemes: the equivalent scheme and the uniform scheme. (b) The boundary heat flux along X∕L at Z∕L=0.5, Y∕L=1.0. The absorption coefficient of this case is κ=1.0 and without scattering.

Grahic Jump Location
+Radiative transfer in a cubic cavity filled with an absorbing, emitting, and scattering medium. (a) Different scattering functions. (b, c) Radiative heat flux q* distributions for different anisotropic scattering phase functions calculated by the DFE method; here the Monte Carlo solution for isotropic scattering is plotted as a reference. The q* distribution is along the middle line (y∕L=0.5 and z∕L=1.0) at the top surface of the cube. The extinction coefficients are equal to 1.0 (b) and 2.0 (c).
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Radiative transfer in a cubic cavity filled with an absorbing, emitting, and scattering medium. (a) Different scattering functions. (b, c) Radiative heat flux q* distributions for different anisotropic scattering phase functions calculated by the DFE method; here the Monte Carlo solution for isotropic scattering is plotted as a reference. The q* distribution is along the middle line (y∕L=0.5 and z∕L=1.0) at the top surface of the cube. The extinction coefficients are equal to 1.0 (b) and 2.0 (c).
Figure 6

Radiative transfer in a cubic cavity filled with an absorbing, emitting, and scattering medium. (a) Different scattering functions. (b, c) Radiative heat flux q* distributions for different anisotropic scattering phase functions calculated by the DFE method; here the Monte Carlo solution for isotropic scattering is plotted as a reference. The q* distribution is along the middle line (y∕L=0.5 and z∕L=1.0) at the top surface of the cube. The extinction coefficients are equal to 1.0 (b) and 2.0 (c).

Grahic Jump Location
+Comparison of heat flux distributions calculated by the DFE, finite volume, and Monte Carlo methods for an isotropic medium. The radiative heat flux distribution is along the middle line (y∕L=0.5 and z∕L=1.0) at the top surface of the cube. The scattering coefficient is 0.5.
View Large  |  Save Figure  |  Download Slide (.ppt)
Comparison of heat flux distributions calculated by the DFE, finite volume, and Monte Carlo methods for an isotropic medium. The radiative heat flux distribution is along the middle line (y∕L=0.5 and z∕L=1.0) at the top surface of the cube. The scattering coefficient is 0.5.
Figure 7

Comparison of heat flux distributions calculated by the DFE, finite volume, and Monte Carlo methods for an isotropic medium. The radiative heat flux distribution is along the middle line (y∕L=0.5 and z∕L=1.0) at the top surface of the cube. The scattering coefficient is 0.5.

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