Discontinuous finite element method for radiative heat transfer in semitransparent graded index medium

To avoid the complicated and time-consuming computation of curved ray trajectories, a discontinuous finite element method based on discrete ordinate equation is extended to solve the radiative transfer problem in a multi-dimensional semitransparent graded index medium. Two cases of radiative heat transfer in two-dimensional rectangular gray semitransparent graded index medium enclosed by opaque boundary are examined to verify this discontinuous finite element method. Special layered and radial graded index distributions are considered. The predicted dimensionless net radiative heat fluxes and dimensionless temperature distributions are determined by the discontinuous finite element method and compared with the results obtained by the curved Monte Carlo method in references. The results show that the discontinuous finite element method has a good accuracy in solving the multi-dimensional radiative transfer problem in a semitransparent graded index medium.     

Finite element method for modeling radiative transfer in semitransparent graded index cylindrical medium

Both Galerkin finite element method (GFEM) and least squares finite element method (LSFEM) are developed and their performances are compared for solving the radiative transfer equation of graded index medium in cylindrical coordinate system (RTEGC). The angular redistribution term of the RTEGC is discretized by finite difference approach and after angular discretization the RTEGC is formulated into a discrete-ordinates form, which is then discretized based on Galerkin or least squares finite element approach. To overcome the RTEGC-led numerical singularity at the origin of cylindrical coordinate system, a pole condition is proposed as a special mathematical boundary condition. Compared with the GFEM, the LSFEM has very good numerical properties and can effectively mitigate the nonphysical oscillation appeared in the GFEM solutions. Various problems of both axisymmetry and nonaxisymmetry, and with medium of uniform refractive index distribution or graded refractive index distribution are tested. The results show that both the finite element approaches have good accuracy to predict the radiative heat transfer in semitransparent graded index cylindrical medium, while the LSFEM has better numerical stability.     

Transient coupled radiative and conductive heat transfer in a semitransparent layer of ceramic

This work considers transient radiative and conductive heat transfer in a semitransparent layer of ceramic, submitted to several thermal and radiative boundary conditions. Each side of the layer is exposed to hot or cold radiative surroundings, while each boundary is heated or cooled by convection. The solution procedure must provide accurate temperature distribution in the layer, so a nodal analysis based on Hottel's zonal method extended by ray tracing method is carried out. A finite difference method with non-uniform space and time increments is used to solve the transient energy equation, including a radiative heat source, coupled to a equation of radiative transfer. Variable spacing was used to concentrate grid points in regions with large temperature gradients. The influence of refractive index, optical thicknesses and conduction-radiation parameters is investigated.     

Discrete transfer method applied to radiative transfer in a variable refractive index semitransparent medium

Application of the discrete transfer method (DTM) has been extended to the analysis of radiative heat transfer in a variable refractive index participating medium. To validate the DTM formulation, radiative heat transfer in an absorbing, emitting and isotropically scattering planar medium was considered. The participating medium was assumed to be in radiative equilibrium. For both constant and variable refractive indices of the medium, the DTM results were compared with those available in the literature. The DTM was found to provide accurate results.     

An inverse analysis to determine conductive and radiative properties of a fibrous medium

In present paper, a modified factor of extinction coefficient and an equivalent albedo of scattering were defined taking into account anisotropic scattering in fibrous insulation. An inverse conduction-radiation analysis in an absorbing, emitting and scattering medium was conducted for the simultaneous estimation of the conductive and radiative properties using the experimentally measured temperature responses for external temperatures up to 980 K. The estimated properties were validated by comparing the predicted and measured results under transient and steady-state condition. It was found that the calculated results corresponded well with the experimental data within an average of 3.1% under transient condition and 9.8% under steady-state condition. This confirms the good behavior of the model and the validity of results.     

Radiative intensity solution and thermal emission analysis of a semitransparent medium layer with a sinusoidal refractive index

The radiative intensity in a sinusoidal refractive index semitransparent medium layer is solved by the curved ray-tracing method in combination with the pseudo-source adding method. One boundary of the medium layer is an opaque diffuse substrate wall. The other boundary is a semitransparent specular or diffuse surface, from which the medium thermal emission emerges. With considering a linear temperature distribution, the radiative intensity formulae are, respectively, deduced under the two boundary conditions. On the basis of the radiative intensity solutions, the directional and hemispherical emission of the medium layer with a specular surface as well as the hemispherical emission of that with a diffuse surface are calculated. The influences of the optical thickness, sinusoidal refractive index distribution and linear temperature distribution on the thermal emission are investigated. The results show that the effects of refractive index and temperature distribution are significant and are different under the two reflecting modes of the surface.     

Verification of numerical simulation method for entropy generation of radiation heat transfer in semitransparent medium

The numerical simulation method of radiative entropy generation in participating media presented by Caldas and Semiao [Entropy generation through radiative transfer in participating media: analysis and numerical computation. JQSRT 2005;96:423-37] is extended to analyze the radiative entropy generation in the enclosures filled with semitransparent media. A discrete ordinates method is used to solve radiative transfer equation and radiative entropy generation. Two different examples are employed to verify the numerical simulation method of radiative entropy generation in the enclosure. Numerical results of dimensionless radiative entropy generation of enclosure are identical to that of entire thermodynamics analysis for the enclosure system. This numerical simulation method can be used in the entropy generation analysis of high-temperature systems such as boilers and furnaces, in which radiation is the dominant mode of heat transfer.     

Discontinuous spectral element method for solving radiative heat transfer in multidimensional semitransparent media

A discontinuous spectral element method (DSEM) is presented to solve radiative heat transfer in multidimensional semitransparent media. This method is based on the general discontinuous Galerkin formulation. Chebyshev polynomial is used to build basis function on each element and both structured and unstructured elements are considered. The DSEM has properties such as hp-convergence, local conservation and its solutions are allowed to be discontinuous across interelement boundaries. The influences of different schemes for treatment of the interelement numerical flux on the performance of the DSEM are compared. The p-convergence characteristics of the DSEM are studied. Four various test problems are taken as examples to verify the performance of the DSEM, especially the performance to solve the problems with discontinuity in the angular distribution of radiative intensity. The predicted results by the DSEM agree well with the benchmark solutions. Numerical results show that the p-convergence rate of the DSEM follows exponential law, and the DSEM is stable, accurate and effective to solve multidimensional radiative transfer in semitransparent media.     

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.     

A sensitivity function-based conjugate gradient method for optical tomography with the frequency-domain equation of radiative transfer

The Sensitivity Function-based Conjugate Gradient Method (SFCGM) is described. This method is used to solve the inverse problems of function estimation, such as the local maps of absorption and scattering coefficients, as applied to optical tomography for biomedical imaging. A highly scattering, absorbing, non-reflecting, non-emitting medium is considered here and simultaneous reconstructions of absorption and scattering coefficients inside the test medium are achieved with the proposed optimization technique, by using the exit intensity measured at boundary surfaces. The forward problem is solved with a discrete-ordinates finite-difference method on the framework of the frequency-domain full equation of radiative transfer. The modulation frequency is set to 600 MHz and the frequency data, obtained with the source modulation, is used as the input data. The inversion results demonstrate that the SFCGM can retrieve simultaneously the spatial distributions of optical properties inside the medium within a reasonable accuracy, by significantly reducing a cross-talk between inter-parameters. It is also observed that the closer-to-detector objects are better retrieved.     

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.     

Equalization of acoustic source using multi-pole sources and source strength estimation using inverse method

Sound measured at various points around the environment can be evaluated by a series of multi-pole sources and their acoustic strength can be acquired. In this numerical study, a method, called the inverse method, was examined to achieve this goal. A variety of arrangements of different sources were considered and the acoustic strength of these sources was acquired. Through the application of the mismatch criterion, good conformity was observed between these sound models and the original sound. Furthermore, with regard to results, sound was generated via different source arrangements which showed acceptable agreement with the original sound. Finally, an arrangement named ‘sources vertical arrangement’ was selected as the best approach.     

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.     

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 new inverse approach for the equivalent gray radiative property of a non-gray medium using a modified zonal method and the complex-variable-differentiation method

Non-gray radiative properties of an absorbing, emitting, non-gray participating medium significantly increase the difficulty of solving the radiative transfer equation. This paper presents a new inverse approach for the equivalent gray radiative property of a non-gray medium. In this approach, the unknown equivalent gray radiative properties are treated as the optimization variables, and the errors to be minimized are the differences between the calculated temperatures and the measured ones. The measured data are simulated by solving the direct problem, in which a modified zonal method together with the Edwards exponential wide-band model is employed. In the inverse problem, the sensitivity coefficients are first calculated by the complex-variable-differentiation method, and then the least-square method and the Newton-Raphson iterative method are employed to minimize the target function. The effectiveness and efficiency of the inverse problem are demonstrated in an example, and another case is given to show the accuracy and potential of the proposed algorithm. The effects of the measurement error and the number of measurement points on the accuracy of the inverse analysis are also investigated in detail.     

Solution of the equation of radiative transfer for interlocked doublets by double interval spherical harmonic method

The double-ordinate spherical harmonic method presented by Wilson and Sen (Publ. Astron. Soc. Jpn. 15 (1963) 351) has been used to solve the equation of radiative transfer in the Milne-Eddington model for interlocked doublets. Solutions have been obtained in the first and second approximation in a particular case η₁=1; .     

A vector radiative transfer model of coupled ocean-atmosphere system using matrix-operator method for rough sea-surface

A vector radiative transfer model termed PCOART has been developed for the coupled ocean-atmosphere system, using the matrix-operator (or adding-doubling) method, which considers the rough sea-surface. The theoretical formulations of the solution of the vector radiative transfer equation of the coupled ocean-atmosphere system, and the reflection-transmission matrices and internal radiation sources for rough sea surface are described. The model intercomparison is performed for several radiative transfer problems in the atmosphere and ocean, and the results show that PCOART can exactly predict the radiance fields for both flat and rough sea surface. Also, the polarizing remote sensing data from POLDER is used to test the capacity of PCOART to simulate the polarization radiance at the top-of-atmosphere, which shows that PCOART can perfectly reproduce the linear polarization reflectance measured by POLDER. PCOART can not only simulate the total radiance field in the coupled ocean-atmosphere system with wind-induced rough sea surface but also predict the polarization radiance field both in the atmosphere and in the ocean, which can serve as a good tool for the ocean optics and ocean color remote sensing communities.     

Analysis of radiative transport in a cylindrical enclosure—An application of the modified discrete ordinate method

Application of the modified discrete ordinate method (MDOM) proposed by Mishra et al. [Mishra SC, Roy HK, Misra N. Discrete ordinate method with a new and simple quadrature scheme. J Quant Spectrosc Radiat Transfer 2006;101:249-262.] has been extended for calculation of volumetric radiative information in a cylindrical enclosure. Radiatively, the medium inside a diffuse gray 1-D concentric cylinder is absorbing, emitting and scattering. Three types of problems, viz., an isothermal medium representing non-radiative equilibrium case, a non-isothermal medium representing radiative equilibrium situation and the case of a combined mode conduction and radiation heat transfer have been used to test the robustness of the MDOM. Temperature/emissive power and heat flux/energy flow rate distributions in the medium have been found for the effects of various parameters like the extinction coefficient, the scattering albedo, the boundary emissivity and the conduction-radiation parameter. To check the accuracy of the results of the MDOM, results have been compared with those available in the literature and also by obtaining the radiative information using the finite volume method. MDOM has been found to provide accurate results.     

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.     

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.