Pomraning-Eddington approximation for radiative transfer in a spherical turbid medium

The Pomraning-Eddington approximation is used to solve the radiative transfer problem for anisotropic scattering in a spherical homogeneous turbid medium with diffuse and specular reflecting boundaries. This approximation replaces the radiative transfer integro-differential equation by a second-order differential equation which has an analytical solution in terms of the modified Bessel function. Here, we calculate the partial heat flux at the boundary of anisotropic scattering on a homogeneous solid sphere. The calculations are carried out for spherical media of radii 0.1, 1.0 and 10 mfp and for scattering albedos between 0.1 and 1.0. In addition, the calculations are given for media with transparent, diffuse reflecting and diffuse and specular reflecting boundaries. Two different weight functions are used to verify the boundary conditions. Our results are compared with those given by the Galerkin technique and show greater accuracy for thick and highly scattering media.     

Pomraning-Eddington approximation for radiative transfer in a spherical turbid medium

Abstract

The Pomraning-Eddington approximation is used to solve the radiative transfer problem for anisotropic scattering in a spherical homogeneous turbid medium with diffuse and specular reflecting boundaries. This approximation replaces the radiative transfer integro-differential equation by a second-order differential equation which has an analytical solution in terms of the modified Bessel function. Here, we calculate the partial heat flux at the boundary of anisotropic scattering on a homogeneous solid sphere. The calculations are carried out for spherical media of radii 0.1, 1.0 and 10 mfp and for scattering albedos between 0.1 and 1.0. In addition, the calculations are given for media with transparent, diffuse reflecting and diffuse and specular reflecting boundaries. Two different weight functions are used to verify the boundary conditions. Our results are compared with those given by the Galerkin technique and show greater accuracy for thick and highly scattering media.     

Heat conduction in a random medium

The method of time-ordered cumulants is used to investigate the behavior of heat pulses in a one-dimensional medium in which the thermal conductivity is random. A partial differential equation is obtained for the average temperature profile; it is the heat equation modified by the addition of a fourth-order spatial derivative. A solution is obtained by asymptotic series. The first two spatial moments of the average temperature profile are evaluated and are shown to tend to those of a Gaussian whent is large. Finally, an equation is obtained for the covariance function.Alfred P. Sloan Fellow.     

Heat transfer by radiation and conduction in fibrous media without axial symmetry

Coupled radiative and conductive heat transfer in a fibrous medium formed by silica fibres is investigated in this paper by not taking account of the axial symmetry for the distribution of fibres or the boundary conditions. Radiative properties of the medium are calculated by using the Mie theory. The model obtained depends only on optical parameters (indices of silica) and on morphological parameters (diameter and orientation of the fibres, density of the medium). Simulations make it possible to study the strongly anisotropic behaviour of the scattering of the radiation by a fibre and to study the influence of various parameters on the radiative properties of the medium. The results of the Mie theory make possible the simulation of the heat transfer coupled by radiation and conduction. To do this, we introduce a new numerical scheme able to simulate heat transfer in the lack of axial symmetry. With this model, we can show the effects of distribution of fibres and temperature on the thermal behaviour of the medium as well as showing the importance of the phenomenon of scattering in fibrous media.     

Equivalence of MTF of a turbid medium and radiative transfer field

The equivalence of the modulation transfer function (MTF) of a turbid medium and the transmitted radiance from the medium under isotropic diffuse illumination is demonstrated. MTF of a turbid medium can be fully evaluated by numerically solving a radiative transfer problem in a plane parallel medium. MTF for a homogenous single layer turbid medium is investigated as illustration. General features of the MTF in the low and high spatial frequency domains are provided through their dependence on optical thickness, single scattering albedo, asymmetrical factor, and phase function type.     

Transient radiation and conduction heat transfer inside a plane-parallel participating gray medium with boundaries having different reflecting characteristics

A hybrid ray-tracing method is developed for the solution to the radiative transfer in a plane-parallel participating medium having one specular surface and another diffuse surface. By this method, radiative transfer coefficients (RTCs) for specular–diffuse (S–D) surfaces are deduced. The medium surfaces are considered to be semitransparent. The effects of convection–radiation parameter, conduction–radiation parameter and refractive index on transient coupled heat transfer are investigated. Results show that the temperature curves of the medium having S–D surfaces is higher than those of the medium having S–S surfaces (two specular surfaces); the total heat flux at steady state for the S–D surfaces is lower than that for the S–S surfaces.     

Heat conduction from a spherical nano-particle: status of modeling heat conduction in laser-induced incandescence

Laser-induced incandescence (LII) of nano-second pulsed laser heated nano-particles has been developed into a popular technique for characterizing concentration and size of particles suspended in a gas and continues to draw increased research attention. Heat conduction is in general the dominant particle cooling mechanism after the laser pulse. Accurate calculation of the particle cooling rate is essential for accurate analysis of LII experimental data. Modelling of particle conduction heat loss has often been flawed. This paper attempts to provide a comprehensive review of the heat conduction modelling practice in the LII literature and an overview of the physics of heat conduction loss from a single spherical particle in the entire range of Knudsen number with emphasis on the transition regime. Various transition regime models developed in the literature are discussed with their accuracy evaluated against direct simulation Monte Carlo results under different particle-to-gas temperature ratios. The importance of accounting for the variation of the thermal properties of the surrounding gas between the gas temperature and the particle temperature is demonstrated. Effects of using these heat conduction models on the inferred particle diameter or the thermal accommodation coefficient are also evaluated. The popular McCoy and Cha model is extensively discussed and evaluated. Based on its superior accuracy in the entire transition regime and even under large particle-to-gas temperature ratios, the Fuchs boundary-sphere model is recommended for modeling particle heat conduction cooling in LII applications. PACS 44.05.+e; 44.10.+i; 47.45.-n; 61.46.Df; 78.70.-g     

Time-dependent radiation transfer in a semi-infinite stochastic medium with Rayleigh scattering

The time-dependent radiation transfer in a semi-infinite stochastic medium of binary Markovian mixture with Rayleigh scattering is presented. A formalism, developed to treat radiation transfer in statistical mixtures, is used to obtain the ensemble-averaged solution. The average reflectivity, radiant energy and net flux are computed for specular-reflecting boundary. For the sake of comparison, we use two different weight functions in our calculations.     

Radiative transfer in a spherical, emitting, absorbing and anisotropically scattering medium

The atmospheres of planets (including Earth) and the outer layers of stars have often been treated in radiative transfer as plane-parallel media, instead of spherical shells, which can lead to inaccuracy, e.g. limb darkening. We give an exact solution of the radiative transfer specific intensity at all points and directions in a finite spherical medium having arbitrary radial spectral distribution of: source (temperature), absorption, emission and anisotropic scattering. The power and efficiency of the method stems from the spherical numerical gridding used to discretize the transfer equations prior to matrix solution: the wanted ray and the rays which scatter into it both have the same physico-geometric structure. Very good agreement is found with an isotropic astrophysical benchmark [Avrett EH, Loeser R. Methods in radiative transfer. In: Kalkofen W, editor. Cambridge: Cambridge University Press; 1984. pp. 341-79]. We introduce a specimen arbitrary forward- side-back phase scattering function for future comparisons. Our method directly and exactly addresses spherical symmetry with anisotropic scattering, and could be used to study the Earth's climate, nuclear power (neutron diffusion) and the astrophysics of stars and planets.     

On the solution of a vectorial radiative transfer equation in an arbitrary three-dimensional turbid medium with anisotropic scattering

The authors developed a numerical method of the boundary-value problem solution in the vectorial radiative transfer theory applicable to the turbid media with an arbitrary three-dimensional geometry. The method is based on the solution representation as the sum of an anisotropic part that contains all the singularities of the exact solution and a smooth regular part. The regular part of the solution could be found numerically by the finite element method that enables to extend the approach to the arbitrary medium geometry. The anisotropic part of the solution is determined analytically by the special form of the small-angle approximation. The method development is performed by the examples of the boundary-value problems for the plane unidirectional and point isotropic sources in a turbid medium slab.     

On the electromagnetic energy density and energy transfer rate in a medium with dispersion due to conduction

The simplest dispersion relation determined by dissipation due to conduction is considered; the electromagnetic energy density in a plane monochromatic wave and its (phase and group) velocity are determined, as well as the energy and momentum transfer rates. It is shown that the energy density at low frequencies in this case has the form of the electrostatic density, in which the permittivity is replaced by its real part, and the energy transfer rate in a plane electromagnetic wave is equal to the phase velocity. The group velocity may exceed the speed of light.     

Analysis of combined conduction and radiation heat transfer in presence of participating medium by the development of hybrid method

The current study addresses the mathematical modeling aspects of coupled conductive and radiative heat transfer in the presence of absorbing, emitting and isotropic scattering gray medium within two-dimensional square enclosure. A blended method where the concepts of modified differential approximation employed by combining discrete ordinate method and spherical harmonics method, has been developed for modeling the radiative transport equation. The gray participating medium is bounded by isothermal walls of two-dimensional enclosure which are considered to be opaque, diffuse and gray. The effect of various influencing parameters i.e., radiation-conduction parameter, surface emissivity, single scattering albedo and optical thickness has been illustrated. The adaptability of the present method has also been addressed.     

Radiative transfer in a two-dimensional rectangular medium exposed to diffuse radiation

The integral equation for radiative transfer in a two-dimensional rectangular scattering medium exposed to diffuse radiation is solved numerically by removing the singularity. This method yielded accurate results except at very large optical thicknesses. Graphical and tabular results for the source function, flux, and intensity are presented. The source function is also calculated using the first term of a Taylor series expansion. The Taylor series is fairly accurate for small optical thicknesses and columnar geometries. A method is presented for extending these results to the problem of a strongly anisotropic scattering phase function which is made up of a spike in the forward direction superimposed on an isotropic phase function.     

Non-Fourier heat conduction with radiation in an absorbing, emitting, and isotropically scattering medium

This article numerically analyses the combined conductive and radiative heat transfer in an absorbing, emitting, and isotropically scattering medium. The non-Fourier heat conduction equation, which includes the time lag between heat flux and the temperature gradient, is used to model the conductive heat transfer in the medium. It predicts that a temperature disturbance will propagate as a wave at finite speed. The radiative heat transfer is solved using the P₃ approximation method. In addition, the MacCormack's explicit predictor-corrector scheme is used to solve the non-Fourier problem. The effects of radiation including single scattering albedo, conduction-to-radiation parameter, and optical thickness of the medium on the transient and steady state temperature distributions are investigated in detail. Analysis results indicate that the internal radiation in the medium significantly influences the wave nature. The thermal wave nature in the combined non-Fourier heat conduction with radiation is more obvious for large values of conduction-to-radiation parameter, small values of optical thickness and higher scattering medium. The results from non-Fourier-effect equation are also compared to those obtained from the Fourier equation. Non-Fourier effect becomes insignificant as either time increases or the effect of radiation increases.     

Attenuation of a narrow light beam in a turbid medium

In model scattering media, in which the extinction coefficient and the probability of photon lifetime vary, the relation between the illumination near a parallel light beam and the optical depth is studied. It is shown that in media of low turbidity, Bouguer's Law holds at small penetration depths, because the scattered radiation is insignificant in comparison to the total energy balance. For beams of finite width, with constant , the illumination increases with increasing values. The importance of in relations describing the attenuation of a parallel light beam in turbid media is clarified.     

Multiple scattering of polarized light in a turbid medium

It is shown that multiple scattering of polarized light in a turbid medium can be represented as independent propagation of three basic modes: intensity and linearly and circularly polarized modes. Weak interaction between the basic modes can be described by perturbation theory and gives rise to “overtones” (additional polarization modes). Transport equations for the basic and additional modes are derived from a vector radiative transfer equation. Analytical solutions to these equations are found in the practically important cases of diffusive light propagation and small-angle multiple scattering. The results obtained are in good agreement with experimental and numerical results and provide an explanation for the experimentally observed difference in depolarization between linearly and circularly polarized waves.