Efficient and accurate method for radiation transfer problems

An efficient method of analysis, which utilizes trial functions based on Case's 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 Chandrasekhar's characteristic H-functions and their moments.     

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.     

双向蒙特卡罗法模拟参与性介质中的辐射换热

双向蒙特卡罗法继承了正向蒙特卡罗法和反向蒙特卡罗法两种方法的优势,不仅可以解决各种各向异性散射问题,还能简易直接地解决辐射能量的方向性和定位置问题。为了验证双向蒙特卡罗法模拟热辐射传递,与文献中其它方法计算结果进行了比较。最后,利用双向蒙特卡罗法模拟计算圆柱形参与性介质中的辐射换热。     

Three-dimensional vector radiative transfer in a semi-infinite, Rayleigh scattering medium exposed to spatially varying polarized radiation: generalized reflection matrix

Three-dimensional vector radiative transfer in a semi-infinite medium exposed to spatially varying, polarized radiation is studied. The problem is to determine the generalized reflection matrix for a multiple scattering medium characterized by a 4×4 scattering matrix. A double integral transform is used to convert the three-dimensional vector radiative transfer equation to a one-dimensional form, and a modified Ambarzumian's method is then applied to derive a nonlinear integral equation for the generalized reflection matrix. The spatially varying backscattered radiation for an arbitrarily polarized incident beam can be found from the generalized reflection matrix. For Rayleigh scattering and normal incidence and emergence, the generalized reflection matrix is shown to have five non-zero elements. Benchmark results for these five elements are presented and compared to asymptotic results. When the incident radiation is polarized, the vector approach used in this study correctly predicts three-dimensional behavior, while the scalar approach does not. When the incident radiation is unpolarized, both the vector and scalar approaches predict a two-dimensional distribution of the intensity, but the error in the scalar prediction can be as high as 20%.     

Approximate solution for describing polarization characteristics of radiation in a Rayleigh scattering medium

An approximate analytical solution for the 4 × 4 Green’s matrix of the problem of polarized radiation transfer in a plane-parallel layer of an absorptive Rayleigh scattering medium is proposed. It permits one to perform fast estimates of angular distributions of the Stokes parameters that are created by an incident beam with an arbitrary polarization state at different levels in a layer when the layer thickness, absorption magnitude, and albedo of the underlying surface are varied. The developed solution is compared with data obtained by the numerical doubling method. The value of the scattering coefficient for a circularly polarized radiation is shown to be somewhat smaller than that for linearly polarized radiation.     

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．     

Two-dimensional radiative transfer in a finite scattering planar medium

The source function, radiative flux, and intensity at the boundaries are calculated for a two-dimensional, scattering, finite medium subjected to collimated radiation. The scattering phase function is composed of a spike in the forward direction super-imposed on an isotropic background. Exact radiative transfer theory is used to formulate the problem and Ambarzumian's method is used to obtain results. Using the principle of superposition, the results for any step variation in incident radiation are expressed in terms of universal functions for the semi-infinite step case. Two-dimensional effects are most pronounced at large optical thicknesses and albedos.     

Two-dimensional linearly anisotropic scattering in a semi-infinite cylindrical medium exposed to a laser beam

A modification of Ambarzumian's method is used to develop the integro-differential equations for the source function, flux, and intensity at the boundary of a two-dimensional, semi-infinite cylindrical medium which scatters linearly. The incident radiation is collimated, normal to the top surface of the medium, and is dependent only on the radial coordinate. The radial variation is assumed to be a Bessel function or a Gaussian distribution. The Gaussian boundary condition is used to simulate a laser beam. Numerical results are presented in graphical and tabular forms for both boundary conditions. Results for forward and backward scattering phase functions are compared with those for isotropic scattering. A method is presented for extending these results to the problem of a strongly anisotropic phase function which is made up of a spike in the forward direction superimposed on a linear phase function.     

The influence of anisotropic scattering on the radiative intensity in a gray, plane-parallel medium calculated by the DRESOR method

Even though there have been many ways to treat complex anisotropic scattering problems, in most of the cases only the radiation flux or its dimensionless data were provided, and radiative intensity with high directional resolution could merely be seen. In this paper, a comprehensive formulation for the DRESOR method was proposed to deal with the anisotropic scattering, emitting, absorbing, plane-parallel media with different boundary conditions. The method was validated by the data from literature and the integral formulation of RTE. The DRESOR value plays an important role in the DRESOR method, and how it is determined by the anisotropic scattering was demonstrated by some typical results. The intensities with high directional resolution at any point can be given by the present method. It was found that the scattering phase function has little effect on the intensity for thin optical thickness, for example, 0.1. And there is the largest boundary intensity for the medium with the largest forward scattering capability, and the smallest one with the largest backward scattering capability. An attractive phenomenon was observed that the scattering of the medium makes the intensity at boundary can not reach the blackbody emission capability with the same temperature, even if the optical thickness tends to very large. It was also revealed that the scattering of the medium does not mean it cannot alter the magnitude of the energy; actually, stronger scattering causes the energy to have more chance to be absorbed by the medium, and indirectly changes the energy magnitude in the medium. Finally, it is easy to deduce all the associated quantities such as the radiation flux, the incident radiation and the heat source from the intensity, just as done in literature.     

Heat transfer in a spherical turbid medium with conduction and radiation

The heat transfer through a spherical media with conduction and radiation is considered. The medium is considered to be turbid and anisotropically scattering with diffusely reflecting boundaries of constant temperatures. The radiative transfer problem is solved using the Galerkin method. An iterative method is used to solve the nonlinear relation between the radiative transfer equation and the conductive energy equation. Calculations are carried out and compared for a homogeneous, isotropically scattering medium with isothermal, transparent boundaries. The results show good agreement with previous work. Calculations are carried out for inhomogeneous media with isotropic, and forward and backward anisotropic scattering. The boundaries of the media are considered to be isothermal and may be transparent or diffusely reflecting boundaries. The calculations are used to study the effects of the single scattering albedo, the anisotropic scattering parameter, the conduction-radiation parameter and the heat source.     

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.     

Source function expansion method for radiative transfer in a two-layer slab

Radiative heat transfer in an isotropically scattering, absorbing and emitting two-layer slab with specularly reflecting boundaries is solved by expanding the source function by Legendre polynomials in the space variable in the integral form of the equation of radiative transfer. The reflectivity and the transmissivity of the slab for an externally incident isotropic radiation are determined. The S-1 solution yields results which are sufficiently accurate for most engineering applications.     

Influence of the divergence of a light beam on the characteristics of collinear diffraction

Two types of anisotropic collinear diffractions of light on a diffraction grating are theoretically investigated, including that where the light is scattered forward in the direction of the incident radiation and that where the light is scattered backward in the direction toward the incident radiation. For both types, two-dimensional transfer functions are calculated, and the character of their transformation upon variation of the light wavelength and the period of the diffraction grating is analyzed. The dependence of the integrated diffraction efficiency and transmission band of diffraction filters on the divergence angle of the light beam is studied.     

Optical properties of pigmented coatings taking into account particle interactions

A T-matrix approach is used to obtain the orientation-averaged scattering and absorption cross sections of randomly oriented particle clusters, and the average angular distribution of the radiation scattered by them. The coefficients involved in the expansion of the phase function are obtained from this T-matrix approach, and used in a multiple scattering formalism to characterize the angular distribution of the diffuse radiation propagating through a particulate coating perpendicularly illuminated with collimated visible radiation. Asymmetry between forward and backward propagating diffuse radiation intensities is taken into account by means of this multiple scattering approach, which is based on solving the radiative transfer equation for successive scattering order contributions. A four-flux model is applied to compute the reflectance in terms of wavelength of the incident radiation and particle concentration. An application of the formalism is carried out to predict the optical properties of titanium dioxide pigmented polymer coatings, in terms of the pigment volume fraction and the degree of aggregation.     

沙尘气溶胶粒子群的散射和偏振特性

根据Mie散射理论,以对数正态分布函数描述沙尘气溶胶粒子群的粒径尺度分布,计算了沙尘气溶胶粒子群在0.2～40μm波段间对太阳短波辐射和地球大气长波辐射的单次散射反照率、散射相矩阵函数,揭示了不同相对湿度时,沙尘粒子群对入射辐射的散射和偏振的特征。结果表明,沙尘粒子群的单次散射反照率随着入射波长的增加有较大起伏,不同相对湿度条件下,变化趋势基本一致;在可见光、近红外波段单次散射反照率随湿度增加而变大,湿度95%时非常接近于1;大于10μm的热红外波段单次散射反照率随相对湿度增加而减小,具有较强的吸收辐射能力。散射辐射强度受湿度影响较小,随散射角的增加呈现先减小后增大的趋势,且增大的趋势随着波长的增加而减弱;不同波段上,线偏振和圆偏振随散射角和相对湿度变化存在差异;在前向和后向仅对入射辐射为圆偏振辐射产生圆偏振散射;散射光的偏振特性及其湿度差异主要表现在后向散射区,多以拱形形式体现。拱顶峰值散射角位置存在差异,且峰值散射角随相对湿度的降低向后向漂移。     

A two-step strategy for numerical simulation of radiative transfer with anisotropic scattering and reflection

This article presents a two-step procedure for the computation of radiative heat transfer with anisotropic scattering and reflection. It is based on a concept that the coincident processes of absorption and scattering/reflection can be separated factitiously. All medium elements and wall surfaces are supposed to be pure-absorbing when receiving incident radiation. Afterwards they emit the scattered/reflected radiations. The absorption of both the initial and the secondary radiations can be assessed by the direct exchange area. It is needed to repeat the processes for a few times until the radiations are substantially absorbed. For anisotropic scattering/reflection, a vector summation obtains the directional distribution of emissive power. The method is validated by several benchmark computations in terms of emissive power and heat transfer coefficients. It is shown that the method gives more accurate solution than the isotropic scaling for the heat transfer in anisotropically scattering media.     

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.     

Three-dimensional radiative transfer in an isotropically scattering, plane-parallel medium: generalized X- and Y-functions

The topic of this work is the generalized X- and Y-functions of multidimensional radiative transfer. The physical problem considered is spatially varying, collimated radiation incident on the upper boundary of an isotropically scattering, plane-parallel medium. An integral transform is used to reduce the three-dimensional transport equation to a one-dimensional form, and a modified Ambarzumian's method is used to derive coupled, integro-differential equations for the source functions at the boundaries of the medium. The resulting equations are said to be in double-integral form because the integration is over both angular variables. Numerical results are presented to illustrate the computational characteristics of the formulation.     

Benchmark solutions of radiative integral transfer equations for three-dimensional emitting–absorbing and scattering media bounded by gray walls

Three-dimensional dimensionless radiative integral transfer equations (RITEs) for a cubic emitting–absorbing and isotropically scattering homogeneous medium of constant properties bounded by gray walls are solved using the method of “product integration”. The resultant system of linear equations for the incident energy is solved iteratively. Evaluation of the accuracy of the numerical solution is achieved by using the computer codes to make predictions for an idealized enclosure in which the exact analytical solution is possible. Comparison of the analytical and numerical values of medium temperatures and heat fluxes has shown that our method is logically correct and has good accuracy. Four benchmark problems for participating medium subjected to various combinations of internally uniform/non-uniform source terms are solved. To make the benchmark problems more complex and more general, different scattering albedo (ω=0.1, 0.5, 0.9), different wall emissivity (ε=0.1, 0.5, 0.9) and different optical thickness (τ=0.1, 1.0, 5.0) are considered. The solutions for the temperature of the medium and the heat flux components are given in tabular form and they may serve as standard values to assess the accuracy of other methods more conveniently.