Pomraning–Eddington approximation for radiative transfer in a homogeneous solid cylinder

Abstract

The radiative transfer in a solid cylinder containing a homogeneous turbid medium with anisotropic scattering is considered. The medium has a diffuse and specular reflecting boundary illuminated by an external incidence and contains an internal energy source. This general problem can be solved in terms of the solution of the corresponding source-free problem with a specular reflecting boundary and isotropic external incidence. The Pomraning–Eddington approximation is used to solve the source-free problem. Three different weight functions are used to verify the boundary condition to find the constants of the solution. The partial flux, the irradiance and the net flux at the boundary for the general problem are calculated.     

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.     

Non-gray radiative transfer in plane-parallel media with reflecting boundaries

A generalized equation of radiative transfer in the two-group picket-fence model is analyzed for a plane parallel, emitting, absorbing and isotropically scattering medium containing uniform heat sources and having boundary surfaces which are diffuse emitters and diffuse reflectors and are maintained at uniform but arbitrary temperatures. The solution of the general problem is expressed by the superposition of simpler problems which are solved by the application of the normal-mode-expansion technique. Highly accurate numerical results are presented for the temperature distribution and the radiative heat flux in the medium.     

An inverse source problem in radiative transfer

The spherical-harmonics method is used to develop a solution to an inverse source problem in radiative transfer. It is assumed that, with the exception of the inhomogeneous source term, all aspects of the radiation-transport problem are known, and we seek to determine the inhomogeneous source term from specified angular distributions of radiation exiting the two surfaces of a homogeneous plane-parallel medium. Anisotropic scattering is included in the monochromatic radiative-transfer model and general reflecting boundary conditions are considered.     

Radiative transfer with anisotropic scattering in an isothermal slab

Two types of anisotropic scattering, linear anisotropic scattering and Rayleigh anisotropic scattering, are considered in the analysis of radiative transfer for an isothermal, plane-parallel medium confined between gray, diffuse walls. The problem is formulated in terms of a coupled pair of integral equations containing the intensity-moments as the unknown variables. These intensity-moments are shown to be the components of the source function. The set of equations is then solved both numerically and in closed form. The results reveal clearly the effects of anisotropic scattering on important characteristics such as heat flux directional emittance and incident radiant energy per unit area. These effects are well predicted by the approximate closed-form solution.     

Radiative transfer in spatially heterogeneous,two-dimensional,anisotropically scattering media

A method is presented for solving the radiative transfer equation for a general anisotropically scattering and emitting medium exposed to arbitrary boundary radiation conditions. The method allows, in principle, for quite arbitrary spatial variability in the scattering and extinction properties of the medium. We formulate the method in the context of 2-dimensional radiative transfer and describe general solution procedures, based on the principles of invariant imbedding, which are applied in the form of doubling algorithms to obtain solutions for optically thick media. Some selected results are shown to demonstrate the versatility of the approach.     

Theoretical study of combined radiative conductive and convective heat transfer in semi-transparent porous medium in a spherical enclosure

A new method for the solution of the radiative transfer equation in spherical media based on a modified discrete ordinates method is extended to study radiative, conductive and convective heat transfer in a semi-transparent scattering porous medium. The set of differential equations is solved using the fourth-order Runge-Kutta method. Various results are obtained for the case of combined radiative and conductive heat transfer, as well as for the interaction of those modes with convection. The effects of some radiative properties of the medium on the heat transfer rate are examined.     

Scattering of electromagnetic waves by a randomly rough surface as a boundary problem of laser radiation interaction with light-scattering materials and media

The light scattering by a rough surface with random Gaussian fluctuations of roughness is studied in the case of coarse roughness, whose parameters—mean deviation and correlation length—are much greater than the radiation wavelength. Closed analytical solutions of the problem are presented in terms of radiophysics for the boundary conditions of an ideal conductor and the impedance boundary conditions. These solutions are formulated in terms of a photometric scattering indicatrix. The possibility of their application to the problems of photometry and theory of radiative transfer and scattering in turbid media, in particular, in simulation of the process of boundary scattering of laser radiation by rough surfaces of biological tissues and media, is discussed.     

Statistical characteristics of photon distributions in a semi-infinite turbid medium: Analytical expressions and their application to optical tomography

The paper focuses on the determination of statistical characteristics of photon distributions in a semi-infinite turbid medium, specifically the photon average trajectory and the root-mean-square deviation of photons from the average trajectory, with an approach based on the diffusion approximation to the radiative transfer equation. We show that the Dirichlet and Robin boundary conditions used for this purpose give close results. We derive exact analytical expressions for the case of the Dirichlet boundary condition. To demonstrate the practical value of our results we consider approximate solution of the inverse problem of time-domain diffuse optical tomography with the flat layer transmission geometry. The problem is solved with the method of photon average trajectories which are constructed with analytical expressions derived for a semi-infinite medium.     

Radiative properties of scattering and absorbing dense media: theory and experimental study

We investigate the validity of the radiative transfer equation to model transmission of light through an absorbing and scattering medium. Assuming that radiative transfer equation is valid, the inverse scattering problem for non-polarized radiative transfer in one-dimensional absorbing and scattering media is solved using a parameter identification method. We discuss how to identify the albedo, phase function and extinction coefficient of the medium. We present experimental data that confirm that this approach is robust and can be used to make reliable predictions of the behavior of scattering absorbing systems.     

Inverse radiation problem of temperature distribution in one-dimensional isotropically scattering participating slab with variable refractive index

In this paper, an inverse analysis is performed for estimation of source term distribution from the measured exit radiation intensities at the boundary surfaces in a one-dimensional absorbing, emitting and isotropically scattering medium between two parallel plates with variable refractive index. The variation of refractive index is assumed to be linear. The radiative transfer equation is solved by the constant quadrature discrete ordinate method. The inverse problem is formulated as an optimization problem for minimizing an objective function which is expressed as the sum of square deviations between measured and estimated exit radiation intensities at boundary surfaces. The conjugate gradient method is used to solve the inverse problem through an iterative procedure. The effects of various variables on source estimation are investigated such as type of source function, errors in the measured data and system parameters, gradient of refractive index across the medium, optical thickness, single scattering albedo and boundary emissivities. The results show that in the case of noisy input data, variation of system parameters may affect the inverse solution, especially at high error values in the measured data. The error in measured data plays more important role than the error in radiative system parameters except the refractive index distribution; however the accuracy of source estimation is very sensitive toward error in refractive index distribution. Therefore, refractive index distribution and measured exit intensities should be measured accurately with a limited error bound, in order to have an accurate estimation of source term in a graded index medium.     

Time-dependent polarized radiation transfer in semi-infinite binary Markovian media

The problem of time-dependent radiation transfer in a semi-infinite plane-parallel random medium with Rayleigh scattering phase function including polarization is considered. The random medium is assumed to consist of two immiscible mixed materials with specular reflecting boundary. The mixing statistics of the two components of the medium is described by the two-state homogeneous Markovian statistics. A formalism, developed to treat radiative transfer in statistical mixtures, is used to obtain the ensemble-averaged solution. Two different weight functions are used to obtain the numerical results for the ensemble-average for reflectivity, radiant energy, and net flux of the medium at any time.     

DRESOR法对瞬态辐射传递问题的研究

用基于蒙特卡洛法(Monte Carlo Method,MCM)的DRESOR法(Distributions of Ratios of Energy Scattered by the medium Or Reflected by the boundary surface)求解入射辐射经过介质散射、壁面反射传递后辐射强度随时间变化的瞬态辐射传递方程(Transient RadiativeTransfer Equation,TRTE)问题。通过在系统内计算一单位瞬态入射辐射对介质的DRESOR数分布,就能计算任意时间内入射辐射在系统内时间响应特性,这样有效提高数值方法处理瞬态辐射问题的通用性。并且能够获得高方向分辨率的辐射强度随时间变化的结果,这是目前大多数数值处理方法比较难做到的,显示出了DRESOR法处理瞬态入射辐射问题的能力．     

Analysis of collimated radiation in participating media using the discrete transfer method

A general formulation of the discrete transfer method is provided to analyze radiative heat transfer problems in a participating medium subjected to collimated radiation. The formulation is validated by considering 1-D planar absorbing, emitting and anisotropically scattering gray medium in radiative equilibrium. Anisotropy of the medium is approximated by linear anisotropic phase function. For the purpose of comparison, the problem is also solved analytically. Results are obtained for different angles of incidence of the collimated radiation. At a given angle of incidence, results are obtained for forward, isotropic and backward scattering situations. Heat flux results are compared over a wide range of values of the extinction coefficient. Emissive power distributions in the medium are also obtained for some cases. The discrete transfer method results are found to compare very well with the analytic results.     

漫反射各向异性散射介质内的温度热流场

本文采用射线踪迹结合节点分析法和谱带模型,研究了漫反射不透明边界下吸收、发射、各向异性散射介质内的热辐射传递过程。考虑介质辐射能的入射和散射方向,导出漫反射、不透明边界、各向异性散射介质的辐射传递系数。在辐射平衡的情况下,考察了表面发射率和散射反照率对介质内辐射热流和温度场的影响。研究表明,介质不透明边界处存在温度跃迁现象,而且,内界面发射率越大,相应界面温度跃迁越小。     

Complete matrix solution of radiative transfer equation for PILE of horizontally homogeneous slabs

This article covers the analytical solution of the discretized radiative transfer equation in the matrix form. The equation is discretized according to the discrete ordinates method. The solution is based on the representation of the light field in a scattering medium as a superposition of an anisotropic and a smooth regular parts. The first of them is calculated analytically using the smoothness of the solution angular spectrum. The regular part is obtained from a radiative transfer equation boundary problem with the anisotropic part as a source function by discrete ordinates method with a scaling transformation and a matrix-operator method applied. There is no limitation of the scattering law in a medium.     

Two analytic methods applied to radiative transfer in the linear-anisotropic scattering medium with graded index and diffuse gray boundaries

The curved ray-tracing method is extended to radiative transfer in the graded index medium with diffuse gray boundary conditions instead of black boundary conditions and the pseudo-source adding method is extended to the case of the linear-anisotropic scattering medium with graded index from non-scattering medium. Furthermore, the equivalence of the two methods is verified by formulation derivation. As exact analytical solutions, both the methods have high accuracy and fast computational speed. The predicted temperature distributions and dimensionless radiative heat flux at radiative equilibrium are determined by the proposed methods, and the numerical results are compared with the data in references. The results show that the present methods have a good accuracy. Influences of various combinations of refractive index and boundary emissivities on the temperature distributions and dimensionless radiative heat flux are also investigated.     

Tau method approximation for radiative transfer problems in a slab medium

An approximate method for solving the radiative transfer equation in a slab medium with an isotropic scattering is proposed. The method is based upon constructing the double Legendre series to approximate the required solution using Legendre tau method. The differential and integral expressions which arise in the radiative transfer equation are converted into a system of linear algebraic equations which can be solved for the unknown coefficients. Numerical examples are included to demonstrate the validity and applicability of the method and a comparison is made with existing results.