Least-squares finite element method for radiation heat transfer in graded index medium

To avoid the complicated and time-consuming computation of curved ray trajectories, a least-squares finite element method based on discrete ordinate equation is extended to solve the radiative transfer problem in a multi-dimensional semitransparent graded index medium. Four cases of radiative heat transfer are examined to verify this least-squares finite element method. Linear and nonlinear graded index are considered. The predicted dimensionless net radiative heat fluxes are determined by the least-squares finite element method and compared with the results obtained by other methods. The results show that the least-squares finite element method is stable and has a good accuracy in solving the multi-dimensional radiative transfer problem in a semitransparent graded index medium, while the Galerkin finite element method sometimes suffers from nonphysical oscillations.     

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.     

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.     

Benchmark numerical solutions for radiative heat transfer in two-dimensional medium with graded index distribution

In graded index media, the ray goes along a curved path determined by Fermat principle. Generally, the curved ray trajectory in graded index media is a complex implicit function, and the curved ray tracing is very difficult and complex. Only for some special refractive index distributions, the curved ray trajectory can be expressed as a simple explicit function. Two important examples are the layered and the radial graded index distributions. In this paper, the radiative heat transfer problems in two-dimensional square semitransparent with layered and radial graded index distributions are analyzed. After deduction of the ray trajectory, the radiative heat transfer problems are solved by using the Monte Carlo curved ray-tracing method. Some numerical solutions of dimensionless net radiative heat flux and medium temperature are tabulated as the benchmark solutions for the future development of approximation techniques for multi-dimensional radiative heat transfer in graded index media.     

Least-squares finite element formulations for one-dimensional radiative transfer

We present least-squares-based finite element formulations for the numerical solution of the radiative transfer equation in its first-order primitive variable form. The use of least-squares principles leads to a variational unconstrained minimization problem in a setting of residual minimization. In addition, the resulting linear algebraic problem will always have a symmetric positive definite coefficient matrix, allowing the use of robust and fast iterative methods for its solution. We consider space-angle coupled and decoupled formulations. In the coupled formulation, the space-angle dependency is represented by two-dimensional finite element expansions and the least-squares functional minimized in the continuous space-angle domain. In the decoupled formulation the angular domain is represented by discrete ordinates, the spatial dependence represented by one-dimensional finite element expansions, and the least-squares functional minimized continuously in space domain and at discrete locations in the angle domain. Numerical examples are presented to demonstrate the merits of the formulations in slab geometry, for absorbing, emitting, anisotropically scattering mediums, allowing for spatially varying absorption and scattering coefficients. For smooth solutions in space-angle domain, exponentially fast decay of error measures is demonstrated as the p-level of the finite element expansions is increased. The formulations represent attractive alternatives to weak form Galerkin finite element formulations, typically applied to the more complicated second-order even- and odd-parity forms of the radiative transfer equation.     

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 method for radiation heat transfer in multi-dimensional graded index medium

In graded index medium, ray goes along a curved path determined by Fermat principle, and curved ray-tracing is very difficult and complex. To avoid the complicated and time-consuming computation of curved ray trajectories, a finite element method based on discrete ordinate equation is developed to solve the radiative transfer problem in a multi-dimensional semitransparent graded index medium. Two particular test problems of radiative transfer are taken as examples to verify this finite element method. The predicted dimensionless net radiative heat fluxes are determined by the proposed method and compared with the results obtained by finite volume method. The results show that the finite element method presented in this paper has a good accuracy in solving the multi-dimensional radiative transfer problem in semitransparent graded index medium.     

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.     

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.     

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.     

Monte Carlo ray-tracing simulation for radiative heat transfer in turbulent fluctuating media under the optically thin fluctuation approximation

The Monte Carlo ray-tracing method (MCRT) based on the concept of radiation distribution factor is extended to solve radiative heat transfer problem in turbulent fluctuating media under the optically thin fluctuation approximation. A one-dimensional non-scattering turbulent fluctuating media is considered, in which the mean temperature and absorption coefficient distribution are assumed and the shape of probability density function is given. The distribution of the time-averaged volume radiation heat source is solved by MCRT and direct integration method. It is shown that the results of MCRT based on the concept of radiation distribution factor agree with these of integration solution very well, but results of MCRT based on the concept of radiative transfer coefficient do not agree with these of integration solution. The solution of time-averaged radiative transfer equation by the concept of radiative transfer coefficient should be treated with caution.     

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.     

Spectral collocation method with a flexible angular discretization scheme for radiative transfer in multi-layer graded index medium

The spectral collocation method (SCM) is employed to solve the radiative transfer in multi-layer semitransparent medium with graded index. A new flexible angular discretization scheme is employed to discretize the solid angle domain freely to overcome the limit of the number of discrete radiative direction when adopting traditional S_N discrete ordinate scheme. Three radial basis function interpolation approaches, named as multi-quadric (MQ), inverse multi-quadric (IMQ) and inverse quadratic (IQ) interpolation, are employed to couple the radiative intensity at the interface between two adjacent layers and numerical experiments show that MQ interpolation has the highest accuracy and best stability. Variable radiative transfer problems in double-layer semitransparent media with different thermophysical properties are investigated and the influence of these thermophysical properties on the radiative transfer procedure in double-layer semitransparent media is also analyzed. All the simulated results show that the present SCM with the new angular discretization scheme can predict the radiative transfer in multi-layer semitransparent medium with graded index efficiently and accurately.     

The DRESOR method for radiative heat transfer in a one-dimensional medium with variable refractive index

This paper extends the DRESOR (Distribution of Ratios of Energy Scattered by the medium Or Reflected by the boundary surface) method to radiative transfer in a variable refractive index medium. In this method, the intensity is obtained from the source term along the curved integration paths determined only by the variable refractive index, and the DRESOR values are calculated by the Monte Carlo method in which the propagation of the energy bundles are affected by Snell's law. With given temperatures on the black boundaries of a one-dimensional medium, the temperature distribution inside the medium with a variable scattering property is calculated under the condition of radiative equilibrium. It is shown that the DRESOR method has a good accuracy in the cases studied. For an isotropic-scattering medium with the same optical thickness, the scattering albedo has no effect on the temperature distribution, which can be obtained from the general equations and can be seen as an extension of what exists for a constant refractive index; however, the different refractive index causes obvious changes in the temperatures inside the medium. The effect of anisotropic scattering on the temperature distribution cannot be ignored, although it is still weaker than the effect caused by variation in the refractive index.     

Nonaxisymmetric radiative transfer in inhomogeneous cylindrical media with anisotropic scattering

In this paper, the control volume finite element method (CVFEM) is applied for the first time to solve nonaxisymmetric radiative transfer in inhomogeneous, emitting, absorbing and anisotropic scattering cylindrical media. Mathematical formulations as well as numerical implementation are given and the final discretized equations are based on similar meshes used for convective and conductive heat transfer in computational fluid dynamic analysis. In order to test the efficiency of the developed method, four nonaxisymmetric problems have been examined. Also, the grid dependence and the false scattering of the CVFEM are investigated and compared with the finite volume method and the discrete ordinates interpolation method.     

Application of the spatial averaging theorem to radiative heat transfer in two-phase media

The spatial averaging theorem is applied to rigorously derive continuum-scale equations of radiative transfer in two-phase media consisting of arbitrary-type phases in the limit of geometrical optics. The derivations are based on the equations of radiative transfer and the corresponding boundary conditions applied at the discrete-scale to each phase, and on the discrete-scale radiative properties of each phase and the interface between the phases. The derivations confirm that radiative transfer in two-phase media consisting of arbitrary-type phases in the range of geometrical optics can be modeled by a set of two continuum-scale equations of radiative transfer describing the variation of the average intensities associated with each phase. Finally, a Monte Carlo based methodology for the determination of average radiative properties is discussed in the light of previous pertinent studies.     

Curved ray tracing method for one-dimensional radiative transfer in the linear-anisotropic scattering medium with graded index

The curved ray tracing method (CRT) is extended to radiative transfer in the linear-anisotropic scattering medium with graded index from non-scattering medium. In this paper, the CRT is presented to solve one-dimensional radiative transfer in the linear-anisotropic scattering gray medium with a linear refractive index and two black boundaries. The predicted temperature distributions and radiative heat flux at radiative equilibrium are determined by the proposed method, and numerical results are compared with the data in references. The results show that the CRT has a good accuracy for radiative transfer in the linear-anisotropic scattering medium with graded index and the dimensionless emissive power and dimensionless radiative heat flux depend on the dimensionless refractive index gradient. It can also be seen that the dimensionless refractive index gradient has important effects on the temperature discontinuity at the boundaries.     

Benchmark solutions of radiative transfer equation for three-dimensional rectangular homogeneous media

Three-dimensional steady-state radiative integral transfer equations (RITEs) for a cubic absorbing and isotropically scattering homogeneous medium are solved using the method of “subtraction of singularity”. Surface integrals and volume integrals are carried out analytically to eliminate singularities, to assure highly accurate solutions, and to reduce the computational time. The resulting system of linear equations for the incident energy is solved iteratively. Six benchmark problems for cold participating media subjected to various combinations of externally uniform/non-uniform diffuse radiation loads are considered. The solutions for the incident energy and the net heat flux components are given in tabular form for scattering albedos of ω=0, 0.5 and 1.     

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.