本文以微分近似法推導一維變折射係數介質輻射熱傳之近似解，並和蒙地卡羅法模擬的結果作比較。本文首先考慮Fresnel邊界之純散射介質的輻射熱傳問題，其折射係數分佈函數為線性或指數，探討不同光學厚度、散射相函數係數以及折射係數分佈對兩種方法結果的影響，結果顯示光學厚度的增加、折射係數變化率的減少以及Fresnel邊界的加入，皆可提升微分近似結果的準確性；而散射相函數係數的不同，對於結果的準確性沒有明顯的改變。接著考慮散漫反射邊界，介質為輻射平衡且內部有均勻熱生成之輻射熱傳問題，在不同光學厚度、散射比、散射相函數係數以及折射係數分佈等情況，對兩種方法結果的影響，結果顯示光學厚度的增加、折射係數變化的減少以及反射邊界的加入，仍會提高微分近似結果的準確性；但散射比和散射相函數係數的不同，對結果的準確性卻無明顯的改變。Fresnel邊界受角度變化影響相對於散漫反射邊界較大，導致兩種方法結果的差距也相對增加，而隨著介質內折射係數變化的增加，兩種邊界的結果會越來越接近。

The analytical solutions of differential approximation (DA) for radiative transfer in a planar refractive medium is derived, and the results are compared with the accurate numerical solutions by the Monte Carlo Method (MCM), in this thesis. The first case considered is radiative transfer in a purely scattering medium with Fresnel boundaries and either linearly or exponentially spatial variations of refractive index. Solutions obtained by the DA and the MCM are conducted for different optical thicknesses, scattering phase function coefficients and gradients of the refractive index. The results show that the accuracy of the DA solution increases as the optical thickness increases or the gradient of the refractive index decreases. Besides, the boundary reflection also increases the accuracy of the DA solution. The influence of the scattering phase function coefficient on the accuracy of the DA solution is relatively less. The second case considered is radiative equilibrium in a medium with diffuse boundaries and heat generation. The result shows that the accuracy of the DA solution increases as the optical thickness increases or the gradient of the refractive index decreases with either black or diffuse boundaries. The influence of the scattering phase function coefficient and scattering albedo on the accuracy of the DA solutions is also less noticeable. The angular dependence in the case with Fresnel boundary is greater than that in the case with diffuse boundary；the increase of the dependence increases the discrepancy between the results of the MCM and the DA. However, the results of Fresnel and diffuse boundaries get close as the gradient of the refractive index increases.

[1] R. Siegel and C. M. Spuckler, "Variable refractive index effects on radiation in semitransparent scattering multilayered regions," Journal of Thermophysics and Heat Transfer, vol. 7, pp. 624-630, 1993.
[2] P. Ben Abdallah and V. Le Dez, "Thermal emission of a semi-transparent slab with variable spatial refractive index," Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 67, pp. 185-198, 2000.
[3] P. Ben Abdallah and 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, vol. 65, pp. 595-608, 2000.
[4] K.-Y. Zhu, Y. Huang, and J. Wang, "Curved ray tracing method for one-dimensional radiative transfer in the linear-anisotropic scattering medium with graded index," Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 112, pp. 377-383, 2011.
[5] L. H. Liu, "Discrete curved ray-tracing method for radiative transfer in an absorbing-emitting semitransparent slab with variable spatial refractive index," Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 83, pp. 223-228, 2004.
[6] L. H. Liu, K. Kudo, and B. X. Li, "Comparison of ray-tracing methods for semitransparent slab with variable spatial refractive index," Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 85, pp. 125-133, 2004.
[7] Y. Huang, X.-L. Xia, and 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, vol. 74, pp. 217-233, 2002.
[8] Y. Huang, X.-L. Xia, and H.-P. Tan, "Temperature field of radiative equilibrium in a semitransparent slab with a linear refractive index and gray walls," Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 74, pp. 249-261, 2002.
[9] D. Lemonnier and V. Le Dez, "Discrete ordinates solution of radiative transfer across a slab with variable refractive index," Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 73, pp. 195-204, 2002.
[10] C. C. Chang and C.-Y. Wu, "Azimuthally dependent radiative transfer in a slab with variable refractive index," International Journal of Heat and Mass Transfer, vol. 51, pp. 2701-2710, 2008.
[11] L. H. Liu, "Finite volume method for radiation heat transfer in graded index medium," Journal of Thermophysics and Heat Transfer, vol. 20, pp. 59-66, 2006.
[12] L. H. Liu, L. Zhang, and H. P. Tan, "Finite element method for radiation heat transfer in multi-dimensional graded index medium," Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 97, pp. 436-445, 2006.
[13] N. A. Krishna and S. C. Mishra, "Discrete transfer method applied to radiative transfer in a variable refractive index semitransparent medium," Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 102, pp. 432-440, 2006.
[14] J. M. Zhao and L. H. Liu, "Solution of radiative heat transfer in graded index media by least square spectral element method," International Journal of Heat and Mass Transfer, vol. 50, pp. 2634-2642, 2007.
[15] C.-Y. Wu and M.-F. Hou, "Solution of integral equations of intensity moments for radiative transfer in an anisotropically scattering medium with a linear refractive index," International Journal of Heat and Mass Transfer, vol. 55, pp. 1863-1872, 2012.
[16] C.-Y. Wu and M.-F. Hou, "Integral equation solutions based on exact ray paths for radiative transfer in a participating medium with formulated refractive index," International Journal of Heat and Mass Transfer, vol. 55, pp. 6600-6608, 2012.
[17] Y. Huang, X.-G. Liang, and X.-L. Xia, "Monte Carlo simulation of radiative transfer in scattering emitting,absorbing slab with gradient index," Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 92, pp. 111-120, 2005.
[18] L. H. Liu, H. C. Zhang, and H. P. Tan, "Monte Carlo discrete curved ray-tracing method for radiative transfer in an absorbing-emitting semitransparent slab with variable spatial refractive index," Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 84, pp. 357-362, 2004.
[19] L. H. Liu, "Benchmark numerical solutions for radiative heat transfer in two-dimensional medium with graded index distribution," Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 102, pp. 293-303, 2006.
[20] C.-Y. Wu, "Monte Carlo simulation of transient radiative transfer in a medium with a variable refractive index," International Journal of Heat and Mass Transfer, vol. 52, pp. 4151-4159, 2009.
[21] C.-Y. Wu, "Monte Carlo simulation of radiative transfer in a refractive layered medium," Heat and Mass Transfer, vol. 46, pp. 607-613, 2010.
[22] Y. Zhang, H.-L. Yi, and H.-P. Tan, "The lattice Boltzmann method for one-dimensional transient radiative transfer in graded index gray medium," Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 137, pp. 1-12, 2014.
[23] T. Khan and A. Thomas, "Comparison of PN or spherical harmonics approximation for scattering media with spatially varying and spatially constant refractive indices," Optics Communications, vol. 255, pp. 130-166, 2005.
[24] M. Bhuvaneswari and C.-Y. Wu, "Differential approximations for transient radiative transfer in refractive planar media with pulse irradiation," Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 110, pp. 389-401, 2009.
[25] K.-Y. Zhu, Y. Huang, and J. Wang, "Combination of spherical harmonics and spectral method for radiative heat transfer in one-dimensional anisotropic scattering medium with graded index," International Journal of Heat and Mass Transfer, vol. 62, pp. 200-204, 2013.
[26] A. R. Degheidy, M. Sallah and M. T. Attia, "Radiative heat transfer in variable refractive index slab media using Pomraning–Eddington approximation," International Journal of Heat and Mass Transfer, vol. 70, pp. 688-692, 2014.
[27] M.-F. Hou, C.-Y. Wu, and Y.-B. Hong, "A closed-form solution of differential approximation for radiative transfer in a planar refractive medium," International Journal of Heat and Mass Transfer, vol. 83, pp. 229-234, 2015.
[28] A. R. Degheidy, M. Sallah, M. T. Attia, and M. R. Atalla, "On Galerkin method for solving radiative heat transfer in finite slabs with spatially-variable refractive index," International Journal of Heat and Mass Transfer, vol. 100, pp. 416-422, 2016.
[29] A. Sharma, D. V. Kumar, and A. K. Ghatak, "Tracing rays through graded-index media: a new method," Applied Optics, vol. 21, pp. 984-987, 1982.
[30] A. Sharma, "Computing optical path length in gradient-index media: a fast and accurate method," Applied Optics, vol. 24, pp. 4367-4370, 1985.
[31] A. Sharma and A. K. Ghatak, "Ray tracing in gradient-index lenses: computation of ray-surface intersection," Applied Optics, vol. 25, pp. 3409-3412, 1986.
[32] C.-Y. Wu and H.-Y. Tseng, "折射介質輻射熱傳的數值光線覓跡蒙地卡羅模擬,"碩士論文,國立成功大學,台南市,2012.
[33] M. F. Modest, Radiative heat transfer, 3rd ed., Academic Press, New York, 2013.
[34] R. Siegel, M. Pinar Mengüç, and J. R. Howell, Thermal Radiation heat transfer, 5th ed., CRC Press, Taylor & Francis , 2011.