本研究考慮在一個吸收、放射、散射且具空間可變折射係數介質中的輻射傳遞，推導由強度矩表示的積分方程式，並由Nyström法來求解。接著推導出輻射在具有可公式化的折射係數分佈的平板中行進的解析的路徑長。最後，基於這些精確的路徑長解析式，我們也發展出修正的球面諧和近似法。
我們首先利用積分表述式來研究在具有高度非等向散射及線性可變折射係數的冷平板中的輻射熱傳問題。此平板放置在不透明的基底上，平板與環境的介面考慮折射係數可為不連續，而平板與基底的介面則考慮折射係數連續，故假設為不反射。我們分別考慮具有外界入射的情形以及具有放射基底的問題作為應用此積分表述式的例子。我們也用蒙地卡羅法來求解這些問題。經由求解積分方程式所得到的半球反射率與半球穿透率與蒙地卡羅法所得到的結果相當一致。我們發現正的折射係數梯度會使正向的輻射增強。所以對於從環境入射的例子而言，無因次熱通量會隨著折射係數梯度的增加而增加。
其次，基於我們所推導的解析的路徑長，我們求解在輻射平衡平板或是在等溫平板中的輻射傳遞問題。對於輻射平衡的平板，邊界假設為黑邊界，而對於等溫平板，則其兩邊界皆考慮為折射係數可為不連續。我們也用離散方向法來求解輻射平衡平板的問題以作為比較之用。經由求解積分方程式所求得的無因次放射功率與無因次輻射熱通量與離散方向法所求得的結果相當吻合。對於具有正的折射係數梯度的輻射平衡平板而言，在折射係數變化不大的情形下，放射功率在下邊界的偏移量隨著光學厚度的增加而減少，但對於折射係數變化明顯時則未必如此。對於具有正折射係數梯度且邊界折射係數固定的不散射平板，其折射係數為線性分佈時所求得兩邊界的方向放射率小於其為斜率遞增的折射係數分佈時所求得的結果。
其三，考慮具有一維可變折射係數或二維可變折射係數的矩形冷介質，我們求解在該介質中輻射傳遞的積分方程式。邊界均假設為黑邊界。除了左邊界以外，其他的邊界溫度均為零。我們也使用離散方向法來求解相同的問題。其中考慮介質具有二維可變折射係數時所求得的結果也與使用蒙地卡羅曲線光追跡法所得的結果做比較。求解積分方程式所求得的結果與蒙地卡羅曲線光追跡法及離散方向法所求得的結果相當吻合。對於具有徑向遞減的折射係數分佈的介質，由於左邊界的放射隨著與原點的距離的增加而遞減並且部份的輻射被上邊界及下邊界吸收，而造成左邊界及右邊界的通量分佈不對稱。此外，當介質具有一維可變折射係數時，正的折射係數梯度會增強朝向下游的輻射。所以在右邊界的輻射通量隨著折射係數梯度的增加而增加。
最後，我們運用普通球面諧和近似法(PN-approximation)解由散射入射所造成相當散漫的部分，並嚴謹地處理衰減的入射強度來發展修正球面諧和近似法。而此修正的三階球面諧和近似法被用來分析一個暴露在散漫入射的折射平板中的輻射傳遞。對於不同的光學厚度、散射比、及變折射係數的組合，由修正三階球面諧和近似法所求得的結果與求解積分方程式與蒙地卡羅法所求得的結果頗為一致。相較於修正三階球面諧和近似法，普通三階球面諧和近似法在光學厚度不大或光學薄及散射較弱時無法獲得良好的結果。

In this work, we derive the integral equations of radiative transfer in terms of intensity moments for radiative transfer in an absorbing, emitting and scattering medium with a spatially varying refractive index (VRI). The integral equations are solved by the Nyström method. Further, the exact analytical path length of radiation traveling in a slab with formulated variable refractive index is derived. Finally, based on the exact analytical path length, a modified spherical harmonics approximation (PN-approximation) is developed.
We first apply the presented integral equations to study radiative heat transfer in a cold slab with higher-degree anisotropic scattering and linearly VRI. The slab lays on an opaque substratum. The refractive index may have a jump at the interface between the surroundings and the slab, while the interface between the slab and the substratum is assumed to be non-reflecting. To exemplify the application of the integral formulation, we consider the case with irradiation from external source in the surroundings and the case with an emitting substratum. We also solve the problems by the Monte Carlo method (MCM). The hemispherical reflectance and transmittance of the slabs obtained by solving integral equations are in excellent agreement with those obtained by the MCM. A positive gradient of refractive index ( ) enhances forward radiative transfer, and so the dimensionless radiative heat flux increases with the increase of for the cases with irradiation from the surroundings.
Secondly, based on the analytical path lengths, we solve the integral equations for radiative transfer in a slab at radiative equilibrium or for radiative transfer in an isothermal slab. The boundaries are assumed to be black for the slab at radiative equilibrium and the index jumps at both boundaries for the isothermal slab are considered. For comparison purpose, we also solve the radiative equilibrium problems by the discrete ordinates method (DOM). The dimensionless emissive power and dimensionless radiative heat flux obtained by solving integral equations show an excellent agreement with those obtained by the DOM. For the slab at radiative equilibrium and with positive gradient of refractive index, the jump of the emissive power at bottom boundary decreases with the increase of optical thickness for the cases with slightly varying refractive index, but the trend may not hold for the cases with significantly varying refractive index. For the non-scattering slab with positive gradient of refractive index and fixed refractive indices at the boundaries, the directional emittances at both boundaries for the case with linear refractive index are smaller than those for the case with a refractive index of slope-increasing profile.
Thirdly, we solve the integral equations for radiative transfer in a rectangular and cold medium with one-dimensional (1-D) VRI or two-dimensional (2-D) VRI. The boundaries are assumed to be black. Except the left boundary, the temperatures of other boundaries are zero. We also solve the problems by the DOM. The results for the case with 2-D VRI are compared with those obtained by the Monte Carlo curved ray tracing method (MCCRT). The results by solving integral equations are in excellent agreement with those obtained by the MCCRT and DOM. For the case with a radially decreasing refractive index, the fluxes at the left and at the right boundaries are asymmetric, because the emission from the left boundary decreasing with the distance from the original point and a part of radiation is absorbed by the top and the bottom boundaries. Besides, positive gradient of refractive index for the case with 1-D VRI enhances the downstream radiation, and so the radiative flux at the right boundary increases with the increase of the gradient of the refractive index.
Finally, a modified spherical harmonics approximation which applies the ordinary PN-approximation to solve the fairly diffuse part of radiation due to in-scattering and treats the attenuated incident intensity rigorously is developed. The modified third-order PN-approximation (MP3-approximation) is applied to analyze radiative heat transfer in a refractive slab exposed to diffuse irradiation. The results obtained by the MP3-approximation are in good agreement with the solutions of the integral equations and those obtained by the MCM for the cases with various combinations of optical thicknesses, scattering albedos and variable refractive indices. The ordinary P3-approximation, by contrast, does not perform well for the optically moderate and thin cases and the weak scattering cases.

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