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Research Papers: Radiative Heat Transfer

Completely Spectral Collocation Solution of Radiative Heat Transfer in an Anisotropic Scattering Slab With a Graded Index Medium

[+] Author and Article Information
Jing Ma

Key Laboratory of National Education Ministry
for Electromagnetic Processing of Materials,
Northeastern University,
Shenyang 110819, China

Ya-Song Sun

The Beijing Key Laboratory
of New and Renewable Energy,
North China Electric Power University,
Beijing 102206, China

Ben-Wen Li

Key Laboratory of National Education Ministry
for Electromagnetic Processing of Materials,
Northeastern University,
Shenyang 110819, China
e-mail: heatli@hotmail.com;
heatli@epm.neu.edu.cn

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the Journal of Heat Transfer. Manuscript received March 19, 2013; final manuscript received July 4, 2013; published online October 25, 2013. Assoc. Editor: Zhixiong Guo.

J. Heat Transfer 136(1), 012701 (Oct 25, 2013) (10 pages) Paper No: HT-13-1150; doi: 10.1115/1.4024990 History: Received March 19, 2013; Revised July 04, 2013

A completely spectral collocation method (CSCM) is developed to solve radiative transfer equation in anisotropic scattering medium with graded index. Different from the Chebyshev collocation spectral method based on the discrete ordinates method (SP-DOM), the CSCM is used to discretize both the angular domain and the spatial domain of radiative transfer equation. In this approach, the angular derivative term and the integral term are approximated by the high order spectral collocation scheme instead of the low order finite difference approximations. Compared with those available data in literature, the CSCM has a good accuracy for a wide range of the extinction coefficient, the scattering albedo, the scattering phase function, the gradient of refractive index and the boundary emissivity. The CSCM can provide exponential convergence for the present problem. Meanwhile, the CSCM is much more economical than the SP-DOM. Moreover, for nonlinear anisotropic scattering and graded index medium with space-dependent albedo, the CSCM can provide smoother results and mitigate the ray effect.

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Figures

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Fig. 5

Temperature distributions in the case of n = 1.2+0.6sin(πx/L), ɛ0 = ɛL = 1, β = 1 and ω = 0.9.

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Fig. 1

The flow chart of the implementation steps

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Fig. 3

Effects of the number of collocation points in both spatial and angular domains on temperature distributions. (a) Spatial direction and (b) angular direction.

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Fig. 9

The exit distribution of radiation intensity at x = 0 in the case of anisotropic incidence

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Fig. 10

The forward radiative heat flux distribution in the case of anisotropic incidence

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Fig. 4

Temperature distributions in the case of ɛ0 = ɛL = 1, β = 1, ω = 0.9 and B3 anisotropic scattering function

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Fig. 6

Temperature distributions in the case of n = 1.2+0.6sin (πx/L), β = 1, ω = 0.9 and F1 anisotropic scattering function

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Fig. 2

Comparisons of temperature distributions got by the CSCM, the SP-DOM, and the Monte Carlo curved ray-tracing method in the case of n(x) = 1.2+0.6x/L, ɛ0 = ɛL = 1.0, β = 1.0 and ω = 0.8, respectively

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Fig. 7

Contour plots of the radiation intensity in the case of anisotropic incidence, anisotropic scattering and n(x) = 1.0+0.1x/L

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Fig. 8

Contour plots of the radiation intensity in the case of anisotropic incidence, anisotropic scattering and n(x)=1.0+0.3x/L

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