Inverse solutions to radiative-transfer problems with partially transparent boundaries and diffuse reflection

Analytical techniques are used to solve a class of inverse radiative-transfer problems relevant to finite and semi-infinite plane-parallel media. While the assumption of isotropic scattering is made, diffuse reflection is allowed at the surface, for the semi-infinite case, and at both surfaces for the case of a finite layer. For the general case based on a semi-infinite medium, a cubic algebraic equation is used to define the basic result, but for the specific case of a semi-infinite medium illuminated by a constant incident distribution of radiation, very simple exact expressions are developed for the albedo for single scattering ? and the coefficient for diffuse reflection ρ. Analytical results are also developed (again in terms of a cubic algebraic equation) for the case of a finite layer with equal reflection coefficients relevant to the two surfaces. For the general case of a finite layer with unequal reflection coefficients, two specific formulations are given. The first algorithm is based on a system of three quadratic algebraic equations for the two reflection coefficients ρ₁ and ρ₂ and the single-scattering albedo ?. Secondly, an elimination between these three algebraic equations is carried out to yield two coupled algebraic equations for ρ₁ and ρ₂ plus an explicit expression for ? in terms of ρ₁ and ρ₂. In addition, an exact expression for τ₀, the optical thickness of the finite layer, is developed in terms of ?, ρ₁ and ρ₂. As is typical with the considered class of inverse problems in radiative transfer, all surface quantities are either specified or considered available from experimental measurements. All basic results are tested numerically.     

A solution of radiative transfer in isotropic plane-parallel atmosphere by integrating Milne equation

In this paper we solve the inversion problem of the radiative transfer process in the isotropic plane-parallel atmosphere by iterative integrations of the Milne integral equation. As a result, we obtain the scattering function in the form of a cubic polynomial in optical thickness. The author has already solved the same problem by iterative integrations of Chandrasekhar's integral equation. In the Milne integral equation, both the cosines of the viewing angles and the optical thickness are integral variables, while in Chandrasekhar's integral equation the cosines of the viewing angles are variables but the optical thickness is not. We derive several series of exponential-like functions as intermediate derivations. Their convergences are evaluated by the author's previous work in the solution of Chandrasekhar's integral equation. The truncated scattering function up to the third order in optical thickness thus obtained is identical to that obtained from Chandrasekhar's integral equation, though their apparent forms are different. Chandrasekhar pointed out that the solution of Chandrasekhar's integral equation does not have a uniqueness of solution. The Milne equation, in contrast, has been proven to have a unique solution. We discuss the uniqueness of the solution by these two methods.     

A new 1D approximation for the solution of 2D radiative transfer problems

The simplified M-1D algorithm (M stands for modified) for the calculation of horizontal distribution of reflected brightness coefficient in 2D regions with large homogeneous pixels is presented. This algorithm is based upon modified 3^?N−1 1D-transport equations (where N is the number of large pixels) instead of one 2D-transport equation, usually used in such problems. The method does not rely on empiric assumptions on both optical properties of atmosphere or diffuse radiation intensity. Numerical results demonstrating the accuracy of the presented algorithm for simulating brightening and shadowing effects in a vicinity of the jump of optical properties given. The accuracy of the M-1D approximation strongly depends on the geometry and illumination conditions. However, it remains below 15% and can reach 1% for all cases studied and is strongly higher than the accuracy of usually employed independent pixel approximation. Time reduction via replacing 2D problem via modified 1D problem is about 17 times for all cases considered in this paper.     

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.     

Modified finite-volume method based on a cell vertex scheme for the solution of radiative transfer problems in complex 3D geometries

A modified finite-volume method based on a cell vertex scheme was applied to solve radiative transfer problems within a participating medium of complex three-dimensional shaped domain. The computational spatial domain of interest was divided into four-node tetrahedron elements with unstructured meshes while the adopted formulation was combined with a closure relation based on an exponential scheme. The studied medium was assumed to be grey, non-scattering and was bounded by black surfaces. Our results were then compared with those found in other articles on the subject. The approach shows a very good level of performance for wall heat transfer evaluation. Accurate results were obtained on coarse computational meshes and solution errors were found to decrease with grid refinement.     

Comparison of the radiative transfer equations for constant and spatially varying refraction

The radiative transfer equation for scattering media with constant refraction index (RTE) and the radiative transfer equation for scattering media with spatially varying refraction index (RTEvri) are compared by using the principle of conservation of energy. It is shown that the RTEvri, not only accounts for the spatial variations of refraction index, but also contains a term that accounts for the divergence of the rays. The latter term is missing in the RTE. A corrected RTE is proposed.     

A simplified implementation of the discrete-ordinates method for a class of problems in radiative transfer with polarization

A simplified implementation of the analytical discrete ordinates (ADO) method in radiative transfer with polarization is presented in this work. The class of problems that can be solved with the simplified ADO approach consists of problems defined in plane-parallel geometry and driven by external illumination in the form of obliquely incident parallel rays. Numerical results of benchmark quality are tabulated for the albedo problem with polarization and Lambert reflection. The new results improve on a tabulation made available in a previous work by the authors that was based on the (less accurate) spherical harmonics method.     

A semi-analytical numerical method for time-dependent radiative transfer problems in slab geometry with coherent isotropic scattering

We describe a semi-analytical numerical method for coherent isotropic scattering time-dependent radiative transfer problems in slab geometry. This numerical method is based on a combination of two classes of numerical methods: the spectral methods and the Laplace transform (LTS_N) methods applied to the radiative transfer equation in the discrete ordinates (S_N) formulation. The basic idea is to use the essence of the spectral methods and expand the intensity of radiation in a truncated series of Laguerre polynomials in the time variable and then solve recursively the resulting set of “time-independent” S_N problems by using the LTS_N method. We show some numerical experiments for a typical model problem.     

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.     

Simultaneous solution of Kompaneets equation and radiative transfer equation in the photon energy range 1-125 keV

Radiative transfer equation in plane parallel geometry and Kompaneets equation is solved simultaneously to obtain theoretical spectrum of 1-125 keV photon energy range. Diffuse radiation field are calculated using time-independent radiative transfer equation in plane parallel geometry, which is developed using discrete space theory (DST) of radiative transfer in a homogeneous medium for different optical depths. We assumed free-free emission and absorption and emission due to electron gas to be operating in the medium. The three terms n, n² and (∂n/∂x_k) where n is photon phase density and x_k=(hν/kT_e), in Kompaneets equation and those due to free-free emission are utilized to calculate the change in the photon phase density in a hot electron gas. Two types of incident radiation are considered: (1) isotropic radiation with the modified black body radiation I^MB[1] and (2) anisotropic radiation which is angle dependent. The emergent radiation at τ=0 and reflected radiation τ=τ_max are calculated by using the diffuse radiation from the medium. The emergent and reflected radiation contain the free-free emission and emission from the hot electron gas. Kompaneets equation gives the changes in photon phase densities in different types of media. Although the initial spectrum is angle dependent, the Kompaneets equation gives a spectrum which is angle independent after several Compton scattering times.     

A comparison of spatial discretization schemes for differential solution methods of the radiative transfer equation

A comparison of discretization schemes required to evaluate the radiation intensity at the cell faces of a control volume in differential solution methods of the radiative transfer equation is presented. Several schemes developed using the normalized variable diagram and the total variation diminishing formalisms are compared along with essentially non-oscillatory schemes and genuinely multidimensional schemes. The calculations were carried out using the discrete ordinates method, but the analysis is equally valid for the finite-volume method. It is shown that the S schemes of the genuinely multidimensional family perform quite well, particularly in problems with discontinuous radiation intensity fields. However, they are time consuming, and so they do not always become more attractive regarding the trade-off between accuracy and computational requirements, in comparison with other high-order schemes that, although being less accurate, are also more economical.     

Discrete ordinates solution of coupled conductive radiative heat transfer in a two-layer slab with Fresnel interfaces subject to diffuse and obliquely collimated irradiation

The coupled conductive radiative heat transfer in a two-layer slab with Fresnel interfaces subject to diffuse and obliquely collimated irradiation is solved. The collimated and diffuse components problems are treated separately. The solution for diffuse radiation is obtained by using a composite discrete ordinates method and includes the development of adaptive directional quadratures to overcome the difficulties usually encountered at the interfaces. The complete radiation numerical model is validated against the predictions obtained by using the Monte Carlo method.     

Difference-equation methods for the solution of radiative transfer problems in media with discontinuities

Difference-equation methods are developed for solving the equation of transfer in media with discontinuities in their physical properties. These should prove useful in calculating the radiation field in dynamical atmospheres having shocks. Two examples with thermal or scattering source functions, for which exact solutions can be obtained, are used to evaluate the accuracy of the techniques, which prove quite satisfactory.     

A quasi-analytic solution of the radiative transfer equation for three-dimensional atmospheres

In this study, we present a new solution of the three-dimensional (3-D) radiation transfer equation (RTE). The solution employs a discretization technique to separate the independent variables involved in the 3-D RTE, and the doubling-adding method to solve the RTE explicitly and quasi-analytically. The remarkable feature of the present solution is the application of scaling-function expansion to those terms that are dependent on horizontal coordinates. Scaling-function expansion is suitable for representing irregular horizontal inhomogeneities with small-scale variations. By applying scaling-function expansion, the 3-D RTE can be formulated in the form of a vector-matrix differential equation; matrices involved in the equation are generally sparse and dominantly diagonal matrices, and this considerably reduces the labor involved in matrix calculations. We tested the performance of the present solution via radiative transfer calculations of solar radiation in horizontally inhomogeneous two-dimensional cloud models. The calculated results indicate that even if the resolution of the scaling-function expansion is too coarse in regions around small-scale variations, the influence does not spread problematically to other regions far from the variations; this illustrates the advantage of the scaling-function expansion. The present solution can be used to investigate quantitatively and to estimate the effects of cloud spatial inhomogeneity on the corresponding radiation field.     

Development and comparison of different spatial numerical schemes for the radiative transfer equation resolution using three-dimensional unstructured meshes

In the present work four different spatial numerical schemes have been developed with the aim of reducing the false-scattering of the numerical solutions obtained with the discrete ordinates (DOM) and the finite volume (FVM) methods. These schemes have been designed specifically for unstructured meshes by means of the extrapolation of nodal values of intensity on the studied radiative direction. The schemes have been tested and compared in several 3D benchmark test cases using both structured orthogonal and unstructured grids.     

Effect of radiative transfer of heat released from combustion reaction on temperature distribution: A numerical study for a 2-D system

Both light and heat are produced during a chemical reaction in a combustion process, but traditionally all the energy released is taken as to be transformed into the internal energy of the combustion medium. So the temperature of the medium increases, and then the thermal radiation emitted from it increases too. Chemiluminescence is generated during a chemical reaction and independent of the temperature, and has been used widely for combustion diagnostics. It was assumed in this paper that the total energy released in a combustion reaction is divided into two parts, one part is a self-absorbed heat, and the other is a directly emitted heat. The former is absorbed immediately by the products, becomes the internal energy and then increases the temperature of the products as treated in the traditional way. The latter is emitted directly as radiation into the combustion domain and should be included in the radiation transfer equation (RTE) as a part of radiation source. For a simple, 2-D, gray, emitting-absorbing, rectangular system, the numerical study showed that the temperatures in reaction zones depended on the fraction of the directly emitted energy, and the smaller the gas absorption coefficient was, the more strong the dependence appeared. Because the effect of the fraction of the directly emitted heat on the temperature distribution in the reacting zones for gas combustion is significant, it is required to conduct experimental measurements to determine the fraction of self-absorbed heat for different combustion processes.     

On the use of Fresnel boundary and interface conditions in radiative-transfer calculations for multilayered media

The ADO (analytical discrete ordinates) method is used to establish a concise and accurate solution for a multi-layer radiative-transfer problem with Fresnel boundary and interface conditions. A finite plane-parallel medium composed of a number (K) of sub-strata with different material properties is considered to be illuminated by isotropically incident radiation. While a general result is obtained, emphasis in the numerical work is given to computing accurately the currents and the intensities that exit each of the two exterior surfaces. Monochromatic forms (with anisotropic scattering) of the radiative-transfer equation are used, and numerical results are given for several specific cases. The complications introduced by the Fresnel boundary and interface conditions are well resolved, so that the numerical results obtained are thought to define a very high standard.     

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.     

Fast approximate technique for the cumulative wavenumber model to modeling radiative transfer in a mixture of real gas media

In the cumulative wavenumber (CW) model, the total range of the absorption cross-section Cη is subdivided into the supplementary absorption cross-section of gray gases C_j, j=1,…,n, where n is the number of gray gases; and the wavenumber region is subdivided into intervals Δ_i=[η_i−1, η_i], i=1, 2,…,p, where p is the number of intervals. The intersection of the two spectral subdivisions is used to define the modeling of the fractional gray gas D_ij. In the CW model, we solve the radiative transfer equation (RTE) in every subinterval D_ij; then it is necessary to solve n x p times the spectral form of the RTE for complete spectral integration. In this work, the CW model is used with a numerical approximation technique based on additive properties of radiative intensity to reduce the solution of RTE to n new fractional gray gas D_j for complete spectral integration. The CW model was first coupled with the discrete ordinates method and the accuracy of the simplified technique and the algorithm was first examined for one-dimensional homogeneous media; results are compared with line-by-line calculations and it is found that the CW model with the simplified technique is exact for the homogeneous media examined. Also, the fast approach is tested in the diffuse reflecting boundaries case. The CW model is implemented in a bi-dimensional enclosure containing real gases in isothermal cases. Afterwards, this approximate technique is extended to non-isothermal and non-homogeneous cases; the results are compared with line-by-line calculations taken from literature and good agreement was found. The results obtained using the acceleration technique for the CW model agree with the results of original CW model. With this acceleration technique the CPU time decreases p times. Spectral database HITRAN and HITEMP are used to obtain the molecular absorption spectrum of the gases.     

VLIDORT: A linearized pseudo-spherical vector discrete ordinate radiative transfer code for forward model and retrieval studies in multilayer multiple scattering media

We describe a new vector discrete ordinate radiative transfer model with a full linearization facility. The VLIDORT model is designed to generate simultaneous output of Stokes vector light fields and their derivatives with respect to any atmospheric or surface property. We develop new implementations for the linearization of the vector radiative transfer solutions, and go on to show that the complete vector discrete ordinate solution is analytically differentiable for a stratified multilayer multiply scattering atmospheric medium. VLIDORT will generate all output at arbitrary viewing geometry and optical depth. The model has the ability to deal with attenuation of solar and line-of-sight paths in a curved atmosphere, and includes an exact treatment of the single scatter computation. VLIDORT also contains a linearized treatment for non-Lambertian surfaces. A number of performance enhancements have been implemented, including a facility for multiple solar zenith angle output. The model has been benchmarked against established results in the literature.