No abstract

No abstract

This paper presents a formulation of the spherical harmonics method for analyzing radiative heat transfer through participating planar media. The proposed approach expand only the radiative intensity into a finite series of Legendre polynomials, while the scattering phase function is directly averaged over incident and scattered radiations without its expansion in series of Legendre polynomials as classical spherical harmonics method assumes. A matrix algorithm formulation is then implemented, which enable computer of spherical harmonics solution for higher order without difficulty. Radiative heat transfer through anisotropic scattering media under diffuse incidence is examined. The Mie theory for spherical particles is used to account scattering phase function of participating isotropic media. Different scattering phase functions, scattering albedo and optical thicknesses are used in the numerical analysis. Excellent agreements on radiative heat fluxes, hemispherical transmittance and reflectance are obtained between spherical harmonics methods with and without expansion of scattering phase function in series of Legendre polynomials. For high angular discretization, computational time requirements of each of the two spherical harmonics methods are comparable, while the present spherical harmonics formulation is time consuming for low order angular discretization. However, for low and high angular discretizations, spherical harmonics with and without scattering phase function expansion in series of Legendre polynomials computational times are extremely small. The results also indicate that for the same angular discretization order, spherical harmonics methods are more accurate than the discrete ordinates method.

The radiative transfer problems in a participating inhomogeneous scalar planar atmosphere, subjected to diffuse or collimated incidence, are investigated using the discrete spherical harmonics method. In developing the method, the radiative intensity is expanded in a finite series of Legendre polynomials and the resulting first-order coupled differential equations of radiance moments are expressed in a set of discrete polar directions. The method is applied to homogeneous/inhomogeneous atmospheres of various anisotropic scattering degrees and thicknesses, and reflective boundary conditions. The discrete spherical harmonics method albedo, transmittance, and radiative intensity predictions agree well with benchmark literature results. Additionally, numerical predictions show that the discrete spherical harmonics method using Mark boundary conditions are more efficient than using Marshak boundary conditions.

A discrete spherical harmonics method is developed for the radiative transfer problem in inhomogeneous polarized planar atmosphere illuminated at the top by a collimated sunlight while the bottom reflects the radiation. The method expands both the Stokes vector and the phase matrix in a finite series of generalized spherical functions and the resulting vector radiative transfer equation is expressed in a set of polar directions. Hence, the polarized characteristics of the radiance within the atmosphere at any polar direction and azimuthal angle can be determined without linearization and/or interpolations. The spatial dependent of the problem is solved using the spectral Chebyshev method. The emergent and transmitted radiative intensity and the degree of polarization are predicted for both Rayleigh and Mie scattering. The discrete spherical harmonics method predictions using 36 streams are found in good agreement with benchmark literature results. The maximum deviation between the proposed method and literature results and for polar directions | | ≥ 0.1 is less than 0.5% and 0.9% for the Rayleigh and Mie scattering, respectively. These deviations for directions close to zero are about 3% and 10 % for Rayleigh and Mie scattering, respectively.