The method of lines solution of discrete ordinates method for radiative heat transfer in cylindrical enclosures

A radiation code based on method of lines solution of discrete ordinates method for radiative heat transfer in axisymmetric cylindrical enclosures containing absorbing-emitting medium was developed and tested for predictive accuracy by applying it to (i) test problems with black and grey walls (ii) a gas turbine combustor simulator enclosing a non-homogeneous absorbing-emitting medium and benchmarking its steady-state predictions against exact solutions and measurements. Comparisons show that it provides accurate solutions for radiative heat fluxes and can be used with confidence in conjunction with CFD codes based on the same approach.     

On the use of a meshless method for solving radiative transfer with the discrete ordinates formulations

A meshless method is presented for solving the radiative transfer equation in the discrete ordinates approach. It is shown that the primitive variables formulation is unstable for low values of the absorption coefficient while the even parity formulation is always stable and accurate.     

Finite element formulation of the first- and second-order discrete ordinates equations for radiative heat transfer calculation in three-dimensional participating media

Two finite element methods (FEMs), FEDOM1 and FEDOM2 (standing for the first and the second finite element discrete ordinates methods, respectively), are formulated and numerically tested. The reference second-order discrete equation is modified in its scattering terms and is applied to the problems of absorbing/emitting and anisotropically scattering media by using the FEM. Numerical features of the developed FEMs are compared with one of the discrete ordinates interpolation method (DOIM), which uses a finite difference scheme. Prediction results of radiative heat transfer by these two FEMs are compared with reference solutions and verified in three-dimensional enclosures containing participating media. The results of FEDOM1 and FEDOM2 agree well with exact solutions for the problem of absorbing/emitting medium with various range of optical thickness. Generally, the two FEMs show more accurate results than DOIM. And FEDOM1 shows more accurate results than FEDOM2 in most of the test problems. Both of the developed FEMs show reasonable results compared with published Monte Carlo solutions for the tested absorbing/emitting and anisotropically scattering media. Although the FEDOM2 is not as accurate as the FEDOM1, it shows its own advantages that it reduces CPU time and memory space of dependent variable to half.     

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.     

Computation of radiative image formation in isolated source and collimated irradiation problems

Due to the ray effect, it is not suitable to employ the discrete ordinates method to calculate the radiation field and the image-formation process in radiative problems with isolated radiative sources (such as point and line sources, isolated medium or boundary source). In this paper, a hybrid method, named Monte Carlo-discrete ordinates method (MCDOM) is developed. Firstly, the Monte Carlo method is used to calculate the emission process. Secondly, the discrete ordinates method is employed to calculate the scattering process, correspondingly, an alternative energy partitioning method is proposed to combine the above two conventional methods. Thirdly, the DOS+ISW algorithm (JQSRT, 2003, 78: 437-453) is used to calculate the image-formation process. Finally, the MCDOM is applied to computing the image formation of an endoscope, which was used to study the hydrodynamics of circulating fluidized beds (Powder Technology, 2001;114:71-83).     

Discrete ordinates interpolation method for radiative heat transfer problems in three-dimensional enclosures filled with non-gray or scattering medium

The discrete ordinates interpolation method (DOIM) is applied to three groups of problems of radiative heat transfer in three-dimensional rectangular enclosures containing non-gray or scattering medium. The original DOIM is first extended to a gray gas model using a new geometric interpolation scheme. It is applied to participating media for different scattering phase functions and optical thicknesses. For the non-gray gas model, the DOIM coupled with the narrow band-based weighted-sum-of-gray-gases (WSGG) model is developed. A few test problems with real gases such as pure H₂O and a mixture of CO₂, H₂O and N₂ are taken. The wall heat flux is calculated and compared with the exact solutions or reference values. All results of test problems are found to be reliable in this study. The DOIM closely reproduces the Monte Carlo reference solutions for different scattering phase functions and optical thicknesses. The non-gray gas results are compared with reference calculations based on the statistical narrow band model and they also show good agreements. The DOIM shows a remarkable merit in the computation time and the grid compatibility, to prove its usefulness for engineering applications.     

Analysis of the radiative transfer equation with highly assymetric phase function

This paper considers a scalar radiative transfer problem with high scattering anisotropy. Two computational methods are presented based on decomposition of the diffuse light field into a regular and anisotropic part. The first algorithm (DOMAS) singles out the anisotropic radiance in the forward scattering peak using the Small-Angle Modification of RTE. The second algorithm (DOM2+) separates the single scattering radiance as an anisotropic part, which largely defines the fine detail of the total radiance in the backscattering directions. In both cases, the anisotropic part is represented analytically. With anisotropy subtraction, the regular part of the signal, which requires a numerical solution, is essentially smoothed as a function of angles. Further, the transport equation is obtained for the regular part that contains an additional source function from the anisotropic part of the signal. This equation is solved with the discrete ordinates method. A conducted numerical analysis of this work showed that algorithm DOMAS has a strong advantage as compared to the standard discrete ordinates method for simulation of the radiance transmission, and DOM2+ is the best of the three for the reflection computations. Both algorithms offer at least a factor of three acceleration of convergence of the azimuthal series for highly anisotropic phase functions.     

Assessment of the re-ordered wide band model for non-grey radiative transfer calculations in 3D enclosures

The objective of the present study is to evaluate variations of the re-ordered wide band model for non-grey radiative transfer calculations in 3D enclosures using the discrete ordinates method. First, the performance of various angular and spatial discretisation schemes of the discrete ordinates method is investigated. Then, several formulations, averaging procedures, and scaling methods of the re-ordered wide band model are tested, and the results are validated against those of a statistical narrow band model. The grey gases formulation using three optimised absorption coefficient is found to be the most efficient method.     

Numerical simulation of radiative heat transfer from non-gray gases in three-dimensional enclosures

The discrete ordinates and the discrete transfer methods are applied to the numerical simulation of radiative heat transfer from non-gray gases in three-dimensional enclosures. Several gas radiative property models are used, namely the correlated k-distribution (CK), the spectral line-based weighted-sum-of-gray-gases (SLW) and the weighted-sum-of-gray-gases (WSGG) methods. The results are compared with recently published accurate calculations based on the statistical narrow band model. The WSGG model is computationally efficient, but often yields relatively large errors. It should be used only if moderate accuracy is sufficient. The SLW model is the best alternative regarding the compromise between accuracy and numerical efficiency. However, an optimization of the coefficients of the model is essential to reduce the computational requirements, especially in the case of gas mixtures. The CK model is the most accurate of the methods evaluated here, but too time consuming for engineering applications, although recent developments may partly overcome this shortcoming.     

Simultaneous derivation of intensities and weighting functions in a general pseudo-spherical discrete ordinate radiative transfer treatment

The retrieval of atmospheric constituents from measurements of backscattered light requires a radiative transfer forward model that can simulate both intensities and weighting functions (partial derivatives of intensity with respect to atmospheric parameters being retrieved). The radiative transfer equation is solved in a multi-layer multiply-scattering atmosphere using the discrete ordinate method. In an earlier paper dealing with the upwelling top-of-the-atmosphere radiation field, it was shown that a full internal perturbation analysis of the plane-parallel discrete ordinate solution leads in a natural way to the simultaneous generation of analytically-derived weighting functions with respect to a wide range of atmospheric variables. In the present paper, a more direct approach is used to evaluate explicitly all partial derivatives of the intensity field. A generalization of the post-processing function is developed for the derivation of weighting functions at arbitrary optical depth and stream angles for both upwelling and downwelling directions. Further, a complete treatment is given for the pseudo-spherical approximation of the direct beam attenuation; this is an important extension to the range of viewing geometries encountered in practical radiative transfer applications. The numerical model LIDORT developed for this work is able to generate intensities and weighting functions for a wide range of retrieval scenarios, in addition to the passive remote sensing application from space. We present a number of examples in an atmosphere with O₃ absorption in the UV, for satellite (upwelling radiation) and ground-based (downwelling radiation) applications. In particular, we examine the effect of various pseudo-spherical parameterizations on backscatter intensities and weighting functions with respect to O₃ volume mixing ratio. In addition, the use of layer-integrated multiple scatter output from the model is shown to be important for satellite instruments with wide-angle off-nadir viewing geometries.     

Numerical solution of radiative and conductive heat transfer in concentric spherical and cylindrical media

A new technique is presented to improve the performance of the discrete ordinates method when solving the coupled conduction-radiation problems in spherical and cylindrical media. In this approach the angular derivative term of the discretized one-dimensional radiative transfer equation is derived from an expansion of the radiative intensity on the basis of Chebyshev polynomials. The set of resulting differential equations, obtained by the application of the S_N method, is numerically solved using the boundary value problem with the finite difference algorithm. Results are presented for the different independent parameters. Numerical results obtained using the Chebyshev transform method compare well with the benchmark approximate solutions. Moreover, the new technique can easily be applied to higher-order S_N calculations.     

A least square finite element formulation of the collimated irradiation in frequency domain for optical tomography applications

This paper presents an extension of the least square finite element formulation associated to the discrete ordinates method to solve collimated irradiation problems in frequency domain. The features of the method are shown with a separation of the intensity into its collimated and scattered parts for a better handling of discontinuities due to the boundary conditions of Dirichlet type used in optical tomography applications. Numerical tests are used to gauge the accuracy of the model in both isotropic and anisotropic scattering media, with and without frequency modulation. The results show that the method is accurate compared to some reference solutions.     

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.     

Evaluation of solution methods for radiative heat transfer in gaseous oxy-fuel combustion environments

The exact solution to radiative heat transfer in combusting flows is not possible analytically due to the complex nature of the integro-differential radiative transfer equation (RTE). Many different approximate solution methods for the solution of the RTE in multi-dimensional problems are available. In this paper, two of the principal methods, the spherical harmonics (P₁) and the discrete ordinates method (DOM) are used to calculate radiation. The radiative properties of the gases are calculated using a non-gray gas full spectrum k-distribution method and a gray method. Analysis of the effects of numerical quadrature in the DOM and its effect on computation time is performed. Results of different radiative property methods are compared with benchmark statistical narrow band (SNB) data for both cases that simulate air combustion and oxy-fuel combustion. For both cases, results of the non-gray full spectrum k-distribution method are in good agreement with the SNB data. In the case of oxy-fuel simulations with high partial pressures of carbon dioxide, use of gray method for the radiative properties may cause errors and should be avoided.     

Discrete ordinates interpolation method incorporated into a flow and energy solver for solution of combined heat transfer problems

Combined conduction-radiation and natural convection-radiation in two-dimensional enclosures containing gray absorbing/emitting medium are numerically investigated. The discrete ordinates interpolation method (DOIM) is used to solve the radiative transfer equation (RTE). It is incorporated into a commercial software (FLUENT^®) by using user-defined function (UDF) to be used in a finite volume-based code for fluid flow computation. Two issues are critically examined: accuracy and versatility. Cases of combined conduction-radiation are considered first and the results are compared with other benchmark solutions to validate the accuracy. Additional problems are also tested to verify the capability of handling unstructured grid system and irregular geometry. Combined natural convection-radiation problem is then examined varying the optical thickness. The radiation effect is investigated through the profiles of velocity, temperature distributions and streamlines. The results are compared with discrete ordinates (DO) solutions, Rosseland solutions and P1 solutions which are offered by FLUENT^® package. The accuracy and other numerical characteristics of DOIM are scrutinized. The DOIM shows very successful results from the viewpoint of accuracy and grid compatibility. It is proved to be a reliable future numerical tool for combined heat transfer problems in engineering applications.     

Development of a finite element model for coupled radiative and conductive heat transfer in participating media

Considering the geometrical applicability, a finite element model (FEM) for coupled radiative-conductive heat transfer has been developed which is applicable to enclosures of arbitrary geometry in present research. The present work provides a solution of coupled heat transfer in a rectangular, cylindrical or annulus enclosure with black or gray walls containing an absorbing-emitting-scattering medium. It is also applied to study the influence of conductive/radiation coefficient, albedo and wall emissivity on the temperature distribution in the medium. Compared with the results available in other references, the present FEM has no limitation with respect to geometry and can predict the coupled radiative-conductive heat transfer in participating media accurately.     

Computation of direct intensity and medium intensity by using the discrete ordinates method

On the basis of the discrete ordinate scheme with (an) infinitely small weight(s), an easy-to-use and comprehensive method, named multi-rays method, is developed to calculate total, direct and medium intensities in arbitrarily specified directions. In doing this, for each of the specified directions, three identical discrete directions with infinitely small weights are employed to represent the three intensities. The new method is verified with two standard test problems, and is used to compute the intensities in two anisotropically scattering problems.     

Matrix formulations of radiative transfer including the polarization effect in a coupled atmosphere-ocean system

A vector radiative transfer model has been developed for a coupled atmosphere-ocean system. The radiative transfer scheme is based on the discrete ordinate and matrix operator methods. The reflection/transmission matrices and source vectors are obtained for each atmospheric or oceanic layer through the discrete ordinate solution. The vertically inhomogeneous system is constructed using the matrix operator method, which combines the radiative interaction between the layers. This radiative transfer scheme is flexible for a vertically inhomogeneous system including the oceanic layers as well as the ocean surface. Compared with the benchmark results, the computational error attributable to the radiative transfer scheme has been less than 0.1% in the case of eight discrete ordinate directions. Furthermore, increasing the number of discrete ordinate directions has produced computations with higher accuracy. Based on our radiative transfer scheme, simulations of sun glint radiation have been presented for wavelengths of 670 nm and 1.6 μm. Results of simulations have shown reasonable characteristics of the sun glint radiation such as the strongly peaked, but slightly smoothed radiation by the rough ocean surface and depolarization through multiple scattering by the aerosol-loaded atmosphere. The radiative transfer scheme of this paper has been implemented to the numerical model named Pstar as one of the OpenCLASTR/STAR radiative transfer code systems, which are widely applied to many radiative transfer problems, including the polarization effect.     

The method of lines solution of discrete ordinates method for non-grey media

A radiation code based on method of lines (MOL) solution of discrete ordinates method (DOM) for radiative heat transfer in non-grey absorbing-emitting media was developed by incorporation of a gas spectral radiative property model, namely wide band correlated-k (WBCK) model, which is compatible with MOL solution of DOM. Predictive accuracy of the code was evaluated by applying it to 1-D parallel plate and 2-D axisymmetric cylindrical enclosure problems containing absorbing-emitting medium and benchmarking its predictions against line-by-line solutions available in the literature. Comparisons reveal that MOL solution of DOM with WBCK model produces accurate results for radiative heat fluxes and source terms and can be used with confidence in conjunction with computational fluid dynamics codes based on the same approach.     

Meshless approach for coupled radiative and conductive heat transfer in one-dimensional graded index medium

A meshless local Petrov-Galerkin (MLPG) approach is employed for solving the coupled radiative and conductive heat transfer in a one-dimensional slab with graded index media. The angular distribution term in discrete ordinate equation of radiative transfer within a one-dimensional graded index slab is discretized by a step scheme, and the meshless approach for radiative transfer is based on the discrete ordinate equation. A moving least-squares approximation is used to construct the shape function. Two particular test cases for coupled radiative and conductive heat transfer within a one-dimensional graded index slab are examined to verify this new approximate method. The temperatures and the radiative heat fluxes are obtained. The results are compared with the other benchmark approximate solutions. By comparison, the results show that the MLPG approach has a good accuracy in solving the coupled radiative and conductive heat transfer in one-dimensional graded index media.