本研究的目的主要在探討連續變化可變折射係數介質內的方位角不對稱輻射熱傳遞現象。邊界條件為入射處邊界不反射，另ㄧ邊界為完全吸收的黑體。原本非方位角對稱的輻射熱傳遞方程式，將輻射強度中的方位角經由傅立葉級數的展開轉成獨立於方位角的方程式，經由離散方向法求得數値解。文中平行光入射於折射係數梯度較大的介質所得到的方向-半球穿透率會較大，隨著入射極角的增加，方向-半球穿透率會減小，受到散射相函數的影響，線性前向散射時，得到較大的方向-半球穿透率，而方向-方向反射率在與原入射方向方位角相同時大於與入射方向方位角相差π，後向散射的情形則相反。散射相函數並不會對吸收率造成影響。Rayleigh散射的方向-方向反射率在與入射方向方位角相相差π時較大，而與入射方向方位角同時較小。光學厚度越厚，方向-半球穿透率越小，方向-半球反射率越大。散射比越大，方向-半球穿透率與方向-半球反射率越大，吸收率越小。

The purpose of this work is to investigate azimuthally dependent radiative heat transfer in a medium with a continuous variation of refractive index. The boundary exposed to collimated iradiation is non-reflective, and the other is black. Then, azimuthally unsymmetric radiative transfer equation can be transformed to a set of azimuthally independent equations by expanding the intensity in a Fourier series in terms of azimuthal angle. Numerical solutions of these equations are calculated by discrete ordinates method. A collimated light illuminating into a medium with greater gradient of refractive index yields greater direction-hemispherical transmittance than that of a medium with a small gradient of refractive index. With increasing incident polar angle, the direction-hemispherical transmittance decreases. The effect of scattering phase function is that forward linear scattering gets greater direction-hemispherical transmittance. Bidirectional reflectance which is in the same azimuthal angle of incident ray is greater than that differs π-radian azimuthal angle from incident ray. Effect of backward scattering is reversal. Phase function does not affect the absorptance. Rayleigh scattering bidirectional reflectance in the same azimuthal angle of incident ray is lower than that differing π-radian from incident ray azimuthally. When the optical thickness increases, the direction-hemispherical transmittance decreases, and the direction-hemispherical reflectance increases. When the albedo is greater, direction-hemispherical transmittance and direction-hemispherical reflectance is larger and the absorptance is smaller.

1. S. Kumar, J. D. Felske, “Radiative transport in a planar medium exposed to azimuthally unsymmetric incident radiation,” Journal of Quantitative Spectroscopy and Radiative Transfer, 35, pp. 187-212, 1986.
2. M. F. Modest, “Oblique collimated irradiation of an absorbing, scattering, plane-parallel layer,” Journal of Quantitative Spectroscopy and Radiative Transfer, 45, pp. 309-312, 1991.
3. C. Proulx, D. R. Rousse, R. Vaillon, J.-F. Sacadura, “Handling of collimated irradiation on a plane-parallel participating medium,” ASME HTD, 366-1, pp. 85-92, 2000.
4. R. Siegel, C. M. Spuckler, “Variable refractive index effects on radiation in semitransparent scattering multilayered regions,” Journal of Thermophysics and Heat Transfer, 7, pp. 624-630, 1993.
5. P. Ben Abdallah, V. Le Dez, “Thermal emission of a semi-transparent slab with variable spatial refractive index,” Journal of Quantitative Spectroscopy and Radiative Transfer, 67, pp. 185-198, 2000.
6. P. Ben Abdallah, V. Le Dez, “Temperature field inside an absorbing-emitting semi-transparent slab at radiative equilibrium with variable spatial refractive index,” Journal of Quantitative Spectroscopy and Radiative Transfer, 65, pp. 595-608, 2000.
7. P. Ben Abdallah, V. Le Dez, “Thermal emission of a two-Dimensional rectangular cavity with spatial affine refractive index,” Journal of Quantitative Spectroscopy and Radiative Transfer, 66, pp. 555-569, 2000.
8. P. Ben Abdallah, A. Charette, V. Le Dez, “Influence of a spatial variation of the thermo-optical constants on the radiative transfer inside an absorbing-emitting semi-transparent sphere,” Journal of Quantitative Spectroscopy and Radiative Transfer, 70, pp. 341-365, 2001.
9. Y. Huang, X. L. Xia, H. P. Tan, “Radiative intensity solution and thermal emission analysis of a semitransparent medium layer with a sinusoidal refractive index,” Journal of Quantitative Spectroscopy and Radiative Transfer, 74, pp. 217-233, 2002.
10. D. Lemonnier, V. Le Dez, “Discrete ordinates solution of radiative transfer across a slab with variable refractive index,” Journal of Quantitative Spectroscopy and Radiative Transfer, 73, pp. 195-204, 2002.
11. L. H. Liu, H. P. Tan, Q. Z. Yu, “Temperature distributions in an absorbing-emitting-scattering semitransparent slab with variable spatial refractive index,” International Journal of Heat and Mass Transfer, 46, pp. 2917-2920, 2003.
12. L. H. Liu, “Finite element solution of radiative transfer across a slab with variable spatial refractive index,” International Journal of Heat and Mass Transfer, 48, pp. 2917-2920, 2005.
13. M. F. Modest, Radiative Heat Transfer, 2th ed, Academic, Boston, 2003.
14. S. Chandrasekhar, Radiative transfer, Dover Publications, New York, 1960.
15. M. N. Özisik, Radiative Transfer, Wiley-Interscience, New York, 1973.
16. L. H. Liu, L. M. Ruan, “Numerical approach for reflections and transmittance of finite plane-parallel absorbing and scattering medium subjected to normal and diffuse incidence,” Journal of Quantitative Spectroscopy and Radiative Transfer, 75, pp. 637-646, 2002.
17. I. W. Busbridge, S. E. Orchard, “Reflection and transmission of light by a thick atmosphere according to a phase function: 1+ ,” Astrophysical Journal, 149, pp. 655-644, 1967.