| Summary: | 博士 === 國立成功大學 === 機械工程學系碩博士班 === 100 === 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.
|
|---|
