M. H. Haggag

Radiative transfer in an inhomogeneous sphere

Abstract We connect the problem of radiation transfer in a diffusely-reflecting sphere containing an energy source and source-free radiation transfer with isotropic boundary conditions. An equation for the radiation heat flux is obtained for a polynomial source. In the special case of a uniform source, the radiation heat flux and the angular flux are obtained in terms of the albedo of the second problem. Numerical results are presented for inhomogeneous and homogeneous spheres, and the results for the homogeneous sphere are compared with those of Mordant.

EFFICIENT AND ACCURATE METHOD FOR RADIATION TRANSFER PROBLEMS

Abstract An efficient method of analysis, which utilizes trial functions based on Cases eigenvalues, is developed for solving radiation transfer in an absorbing and scattering homogeneous semi-infinite plane-parallel medium subjected to externally incident radiation. Expressions for the forward and backward intensities, reflectivity and total radiation intensity are included. Numerical results are given and compared involving different forms of the externally incident radiation on the boundary surface. It is shown that the solution converges rapidly to the exact results and that lower-order solutions predict values of the physical parameters that are accurate to five figures in all values of the single-scattering albedos in the range 0.1 ≤ ω ≤ 1. The method has been also used to get approximate formulae for calculating Chandrasekhars characteristic H-functions and their moments.

Albedo problem for finite plane-parallel medium

Abstract The problem of radiation transfer through a scattering and absorbing finite plane-parallel medium is solved using an efficient and accurate method of analysis which utilizes trial functions based on Cases eigenvalues plus a linear combination of exponential integral functions. The proposed trial functions are used on the integral equation reducing it to a system of algebraic equations to be solved for the expansion coefficients which are used to calculate some interesting physical quantities such as the angular radiation intensity and the reflection and the transmission coefficients. Numerical results are obtained for two different external incidence on the left boundary, x=0. The results are compared with the exact results and with those calculated by the Pomraning–Eddington variational method.

Bivariational calculations for radiation transfer in an inhomogeneous participating medium

Equations for radiation transfer are obtained for dispersive media with space-dependent albedo. Bivariational bound principle is used to calculate the reflection and transmission coefficients for such media. Numerical results are given and compared.

Padé approximant in radiative transfer

Abstract A functional relation is obtained between radiative transfer in an inhomogeneous medium with internal sources and diffuse reflection. The intensity of the emerging radiation for a linear source is obtained by using the Pade approximation. The single scattering albedo is assumed to decrease exponentially with optical depth. Numerical results are given.

A Fourier transform solution of radiation transfer in plane-parallel medium with space-dependent albedo

A method is presented for solving radiation transfer problems involving space-dependent single-scattering albedoc(x) for a grey plane non-emitting medium with isotropic scattering. Expressions for the exit distributions and the reflection and transmission coefficients, relevant to a medium having a slab geometry, are given. The solution of the linear transport equation is performed on the basis of integral Fourier transforms. Numerical results are obtained in the case of exponentially varying single-scattering albedo and compared with previous results.

Diffuse reflection and transmission of radiation for Rayleigh phase function

Diffuse reflection and transmission coefficients in a plane parallel medium are calculated for a Rayleigh phase-function averaged over polarization and Rayleigh polarized phase-function. This is calculated by imbedding the finite medium into a semi-infinite scattering and absorbing medium. Numerical calculations for semi-infinite albedo are compared with Pomraning results. The albedos for finite medium are calculated via the imbedding equations which converge for large τ to the value of semi-infinite medium.

The solution of radiation transfer problem for a slab with space-dependent albedo and anisotropic scattering

The transport of thermal radiation has been considered within a finite slab which absorb and scatter anisotropically. The problem involves the space-dependent single-scattering albedow(x). Two approximations are taken forw(x). In the first it is represented in exponential form asw(x)=w0 exp(−x/s), wherew0 ands are given constants andx is the optical variable. The second approximation assumes the formw(x) = ∑r=0Rdr*pr(x/a), wheredr* are known expansion coefficients anda is the half optical thickness of the slab. Analytic expressions for the forward, backward radiation intensities and fluxes are given in each approximation. The solution of the linear transport equation is performed on the basis of integral Fourier transforms.

Radiation transfer in dispersive media

Equations for the reflection coefficients associated with radiation scattered by dispersive media are obtained. The Pade approximant technique is used to calculate the reflection coefficients. The results for the (0/1) Pade approximant lead to computationally useful results that compare well with the exact results.

Transport calculations in finite slab geometry

The problem of particle reflection and transmission through slabs, which scatters isotropically, is analytically investigated. The solution is based on a projectional procedure, equivalent to a variational approach. Expressions for the reflection and transmission functions are given. Numerical results concern the above mentioned physical quantities are given and compared with the data available in literature.