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.     

Coupled conductive-radiative heat transfer problem for two-layer slab

The coupled conductive radiative transfer problem in two homogeneous layers slab of anisotropic scattering with specularly reflecting boundaries has been considered. A Galerkin-iterative technique is used to solve the coupled conductive radiative heat equations in integral forms for the two layers. Numerical results are obtained for the temperature, the conductive, radiative and the total heat fluxes for the two homogeneous layers with isotropic and anisotropic scattering. The calculations are also carried out for homogeneous plane parallel medium with anisotropic scattering which show good agreement with the published calculations.     

Effect of reflecting modes on combined heat transfer within an anisotropic scattering slab

Under various interface reflecting modes, different transient thermal responses will occur in the media. Combined radiative-conductive heat transfer is investigated within a participating, anisotropic scattering gray planar slab. The two interfaces of the slab are considered to be diffuse and semitransparent. Using the ray tracing method, an anisotropic scattering radiative transfer model for diffuse reflection at boundaries is set up, and with the help of direct radiative transfer coefficients, corresponding radiative transfer coefficients (RTCs) are deduced. RTCs are used to calculate the radiative source term in energy equation. Transient energy equation is solved by the full implicit control-volume method under the external radiative-convective boundary conditions. The influences of two reflecting modes including both specular reflection and diffuse reflection on transient temperature fields and steady heat flux are examined. According to numerical results obtained in this paper, it is found that there exits great difference in thermal behavior between slabs with diffuse interfaces and that with specular interfaces for slabs with big refractive index.     

Radiative heat transfer in a planar medium with anisotropic scattering and directional boundaries

Radiation heat transfer in an absorbing, emitting and scattering medium has been the subject of many previous investigations. Most solutions are numerically complex and the existing analytical solutions are restricted in application by the simplifying assumptions involved. A plane-parallel medium is considered which scatters anisotropically. The boundaries are considered to be specular reflectors, as predicted by Fresnel's relations, while the diffusely incident radiation is refracted according to Snell's law. The emission is restricted to a medium with a uniform temperature distribution. Approximate closed-form solutions for the radiative heat flux and incident intensity are presented for dielectric layers and linear anisotropic scattering. Numerical results are also presented and show that the effects of directional boundaries, anisotropic scattering, scattering albedo and optical depth are accurately predicted by the approximate solution.     

The use of the itFN method for radiative transfer problems with reflective boundary conditions

The F_N method is used to compute the net radiative heat flux relevant to radiative transfer in an anisotropically scattering, plane-parallel medium with specularly and diffusely reflecting boundaries.     

The PN method for radiative transfer problems with reflective boundary conditions

The P_N method is used to compute the partial heat fluxes relevant to radiative transfer in an anisotropically scattering, plane-parallel medium with specularly and diffusely reflecting boundaries.     

Radiative transfer in an isotropically scattering two-region slab with reflecting boundaries

The problem of radiative heat transfer in an absorbing, emitting, isotropically scattering two-layer slab with diffusely and specularly reflecting boundaries is solved by the F_N method and results are presented for the transmissivity and reflectivity of the slab.     

Stochastic radiative transfer in a finite plane-parallel medium with general boundary conditions

The time-independent radiative transfer problem in a scattering and absorbing planar random medium with general boundary conditions and internal energy source is considered. The medium is assumed to consist of two randomly mixed immiscible fluids, with the mixing statistics described as a two-state homogeneous Markov process. The problem is solved in terms of the solution of the corresponding free-source problem with simple boundary conditions which is solved using Pomraning-Eddington approximation in the deterministic case. A formalism, developed to treat radiative transfer in statistical mixtures, is used to obtain the ensemble-averaged solution. The average partial heat fluxes are calculated in terms of the albedoes of the source-free problem. Results are obtained for isotropic and anisotropic scattering for specular and diffused reflecting boundaries.     

半透明平板边界入射辐射热流密度的反问题

对一维半透明平板内辐射、导热及边界对流耦合换热过程进行了研究。提出了一种由一侧边界出射辐射强度反演另一侧边界入射辐射热流密度的方法。通过对各向异性散射、吸收系数、散射系数、边界外侧来流温度、对流换热系数、半透明平板的导热系数和平板厚度等参数对反演精度影响的分析表明,方法是可行的。     

多维梯度折射率介质内辐射传递的扩散近似

推导多维梯度折射率介质内稳态辐射传递的扩散近似方程.使用有限元法对扩散近似进行离散和求解,利用两个二维半透明介质的稳态辐射传递问题验证该扩散近似的精度及适用性.算例考虑介质为均匀折射率及梯度折射率两种情况.利用扩散近似分别求解辐射平衡时的边界热流、介质内温度场分布,并与辐射传递方程的求解结果进行对比分析.结果表明:介质折射率变化、散射特性、光学厚度及散射反照率均直接影响扩散近似的精度;在光学厚及强散射条件下,该扩散近似可以作为一种快速算法应用于梯度折射率介质稳态辐射传递的求解.     

A theoretical study of radiation between two concentric spheres using a modified discrete ordinates method associated with Legendre transform

A modified discrete ordinates method (DOM) is used in spherical participating media. The radiative intensity is broken up into two components. One component is traced back to the enclosure's source. It is called direct intensity. The other component is rather traced back to the contribution of the medium itself. It is called diffuse intensity. Thus, the radiative transfer equation (RTE) is transformed into two simultaneous equations: a direct RTE and a diffuse RTE. The direct RTE is solved analytically. The diffuse RTE is solved numerically using the DOM. The streaming angular derivative term appearing in spherical geometry is modeled by making use of the Finite Legendre Transform. We study a pure radiation transfer problem between two concentric spheres. The medium is assumed to be gray and isotropically scattering. The limiting spheres are considered to be opaque, gray, diffusely emitting and diffusely reflecting with uniform emissivity over each surface. The obtained results are compared with available cases reported in the literature. In particular, relative importance of the direct radiation in optically thin media is studied.     

A parametric study of radiative transfer with anisotropic scattering in a one-dimensional system

Radiative heat transfer in an absorbing, emitting, anistropically-scattering, one-dimensional medium is analyzed. Unlike many of the existing works, the present analysis does not require a known temperature distribution within the medium. Assuming a model of linear anistropic scattering, the transfer equation and the energy equation are solved simultaneously by utilizing a recently developed successive approximation technique. Closed-form approximate solutions and accurate higher-order results are both presented. Calculations show that the relative importance of the anistropic scattering effect generally decreases with decreasing wall emissivity and decreasing optical thickness. For radiative equilibrium without internal heat generation, it is demonstrated that the anistropic-scattering heat-transfer results can be approximated quite adequately by the isotropic-scattering result with the introduction of the concept of an effective optical thickness. For media with internal heat generation, an interesting effect of the scattering albedo is observed. It is established that, in the limit of a large scattering albedo, the temperature of the medium approaches a constant value that is independent of anistropic-scattering effects and wall emissivity. The exact limiting expressions for the temperature and apparent emissivity of an isothermal slab are found.     

Estimation of emissivities in a two-dimensional irregular geometry by inverse radiation analysis using hybrid genetic algorithm

An inverse radiation analysis is presented for estimating the wall emissivities for an absorbing, emitting, scattering media in a two-dimensional irregular geometry with diffusely emitting and reflecting opaque boundaries from the measured temperatures. The finite-volume method was employed to solve the radiative transfer equation for 2D irregular geometry. The hybrid genetic algorithm which contains local optimization algorithm was adopted to estimate wall emissivities by minimizing an objective function, while reducing computation time. It was found that an increase in the standard deviation in measurements significantly deteriorated the estimation of wall emissivities. Thus, a very accurate measurement was required in inverse radiation for better estimation of wall emissivities, especially, in a high temperature environment.     

Pure-triplet scattering for radiative transfer in binary discrete random finite media with reflective boundaries and internal source of radiation

The stochastic solution of the monoenergetic radiative transfer equation in a finite slab random medium with pure-triplet anisotropic scattering is considered. The random medium is assumed to consist of two randomly mixed immiscible fluids labelled by 1 and 2. The extinction function, the scattering kernel, and the internal source of radiation are treated as discrete random variables, which obey the same statistics. The theoretical model used here for stochastic media transport assumes Markovian processes and exponential chord length statistics. The boundaries of the medium under consideration are considered to have specular and diffuse reflectivities with an internal source of radiation inside the medium. The ensemble-average partial heat fluxes are obtained in terms of the average albedos of the corresponding source-free problem, whose solution is obtained by using the Pomraning-Eddington approximation. Numerical results are calculated for the average forward and backward partial heat fluxes for different values of the single scattering albedo with variation of the parameters that characterize the random medium. Compared to the results obtained by Adams et al. in the case of isotropic scattering based on the Monte Carlo technique, it can be demonstrated that we have good comparable data.     

Transient combined radiation and conduction in a one-dimensional non-gray participating medium with anisotropic optical properties subjected to radiative flux at the boundaries

Transient combined heat transfer by radiation and conduction is investigated in non-gray participating media with anisotropic optical properties. The medium is irradiated by a time-dependent source. Different kinds of boundaries are considered: black, opaque, transparent and semi-transparent. The heat transfer equations are solved numerically in a one-dimensional configuration. Comparisons are made with test cases taken from the literature, and the results obtained demonstrate the accuracy of the present numerical model. The influence of transparent, semi-transparent and opaque boundaries on an insulating fibrous medium is also studied.     

Application of multidimensional scheme and the discrete ordinate method to radiative heat transfer in a two-dimensional enclosure with diffusely emitting and reflecting boundary walls

Radiative heat transfer is the dominant mode of heat transfer in many engineering problems, including combustion chambers, space, greenhouses, rocket plume sensing, among others. The aim of this study is to develop an efficient method capable of eliminating ray effects in complex 2D situations and to use the developed code for other problems including combined conduction and convection in connection with CFD codes. A complete genuinely multidimensional discretization in two-dimensional discrete ordinates method is formulated to solve radiative heat transfer in a rectangular enclosure composed of diffusely emitting and reflecting boundaries and containing homogeneous media that absorbs, emits and scatters radiation. A new genuinely multidimensional differencing scheme is used to solve the radiative transfer equation with S₄, S₆, S₈, T₆, T₇, T₈ and T₉ angular quadrature schemes. Different cases are analyzed and the results are compared when possible with those obtained by others researchers.     

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.     

Time-dependent radiative transfer using Chapman-Enskog maximum entropy method

The time-dependent problems of radiative transfer involve a coupling between radiation and material energy fields and are nonlinear because of proposed temperature dependence of the medium characteristics in semi-infinite medium with Rayleigh anisotropic scattering. By means of the limited flux, Chapman-Enskog and maximum entropy technique the time-dependent radiative transfer equation has been solved explicitly. The maximum entropy method is used to solve the resulting differential equation for radiative energy density. The calculations are carried out for temperature (normalized dimensionless) Θ(x,τ), radiative energy density and net flux with Rayleigh and anisotropic scattering for different space at different times.     

Pure-triplet scattering for radiative transfer in binary discrete random finite media with reflective boundaries and internal source of radiation

The stochastic solution of the monoenergetic radiative transfer equation in a finite slab random medium with pure-triplet anisotropic scattering is considered. The random medium is assumed to consist of two randomly mixed immiscible fluids labelled by 1 and 2. The extinction function, the scattering kernel, and the internal source of radiation are treated as discrete random variables, which obey the same statistics. The theoretical model used here for stochastic media transport assumes Markovian processes and exponential chord length statistics. The boundaries of the medium under consideration are considered to have specular and diffuse reflectivities with an internal source of radiation inside the medium. The ensemble-average partial heat fluxes are obtained in terms of the average albedos of the corresponding source-free problem, whose solution is obtained by using the Pomraning–Eddington approximation. Numerical results are calculated for the average forward and backward partial heat fluxes for different values of the single scattering albedo with variation of the parameters that characterize the random medium. Compared to the results obtained by Adams et al. in the case of isotropic scattering based on the Monte Carlo technique, it can be demonstrated that we have good comparable data.