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.     

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.     

On the derivation of vector radiative transfer equation for polarized radiative transport in graded index media

Light transport in graded index media follows a curved trajectory determined by Fermat's principle. Besides the effect of variation of the refractive index on the transport of radiative intensity, the curved ray trajectory will induce geometrical effects on the transport of polarization ellipse. This paper presents a complete derivation of vector radiative transfer equation for polarized radiation transport in absorption, emission and scattering graded index media. The derivation is based on the analysis of the conserved quantities for polarized light transport along curved trajectory and a novel approach. The obtained transfer equation can be considered as a generalization of the classic vector radiative transfer equation that is only valid for uniform refractive index media. Several variant forms of the transport equation are also presented, which include the form for Stokes parameters defined with a fixed reference and the Eulerian forms in the ray coordinate and in several common orthogonal coordinate systems.     

Finite element approach for radiative transfer in multi-layer graded index cylindrical medium with Fresnel surfaces

Because the optical plane defined by the incidence and reflection direction at a cylindrical surface has a complicated relation with the local azimuthal angle and zenith angle in the traditional cylindrical coordinate system, it is difficult to deal with the specular reflective boundary condition in the solution of the traditional radiative transfer equation for cylindrical system. In this paper, a new radiative transfer equation for graded index medium in cylindrical system (RTEGCN) is derived based on a newly defined cylindrical coordinate system. In this new cylindrical coordinate system, the optical plane defined by the incidence and reflection direction is just the isometric plane of the local azimuthal angle, which facilitates the RTEGCN in dealing with cylindrical specular reflective boundaries. A least squares finite element method (LSFEM) is developed for solving radiative transfer in single and multi-layer cylindrical medium based on the discrete ordinates form of the RTEGCN. For multi-layer cylindrical medium, a radial basis function interpolation method is proposed to couple the radiative intensity at the interface between two adjacent layers. Various radiative transfer problems in both single and multi-layer cylindrical medium are tested. The results show that the present finite element approach has good accuracy to predict the radiative heat transfer in multi-layer cylindrical medium with Fresnel surfaces.     

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.     

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.     

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.     

Discrete curved ray-tracing method for radiative transfer in an absorbing-emitting semitransparent slab with variable spatial refractive index

A discrete curved ray-tracing method is developed to analyze the radiative transfer in one-dimensional absorbing-emitting semitransparent slab with variable spatial refractive index. The curved ray trajectory is locally treated as straight line and the complicated and time-consuming computation of ray trajectory is cut down. A problem of radiative equilibrium with linear variable spatial refractive index is taken as an example to examine the accuracy of the proposed method. The temperature distributions are determined by the proposed method and compared with the data in references, which are obtained by other different methods. The results show that the discrete curved ray-tracing method has a good accuracy in solving the radiative transfer in one-dimensional semitransparent slab with variable spatial refractive index.     

Hybrid finite volume/ finite element method for radiative heat transfer in graded index media

The rays propagate along curved path determined by the Fermat principle in the graded index medium. The radiative transfer equation in graded index medium (GRTE) contains two specific redistribution terms (with partial derivatives to the angular coordinates) accounting for the effect of the curved ray path. In this paper, the hybrid finite volume with finite element method (hybrid FVM/FEM) (P.J. Coelho, J. Quant. Spectrosc. Radiat. Transf., vol. 93, pp. 89–101, 2005) is extended to solve the radiative heat transfer in two-dimensional absorbing-emitting-scattering graded index media, in which the spatial discretization is carried out using a FVM, while the angular discretization is by a FEM. The FEM angular discretization is demonstrated to be preferable in dealing with the redistribution terms in the GRTE. Two stiff matrix assembly schemes of the angular FEM discretization, namely, the traditional assembly approach and a new spherical assembly approach (assembly on the unit sphere of the solid angular space), are discussed. The spherical assembly scheme is demonstrated to give better results than the traditional assembly approach. The predicted heat flux distributions and temperature distributions in radiative equilibrium are determined by the proposed method and compared with the results available in other references. The proposed hybrid FVM/FEM method can predict the radiative heat transfer in absorbing-emitting-scattering graded index medium with good accuracy.     

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.     

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.     

Comparison of ray-tracing methods for semitransparent slab with variable spatial refractive index

The ray-tracing technique has the main difficulty in solving radiative transfer in the medium with variable spatial refractive index. Recently, three methods have been developed for the application of the ray-tracing technique in those medium. To compare and discuss the numerical characteristics of those methods, a semitransparent slab with variable spatial refractive index is taken as an example, and the reflectivity and the transmissivity of the slab are computed by the curved ray-tracing method, the multi-layer approach, and the discrete curved ray-tracing method, respectively. As the result, it is shown that, the discrete curved ray-tracing method gives the result with good accuracy and convergence characteristics than the multi-layer approach. Due to accounting physically inexistent reflection on the interface between sublayers, the multi-layer approach converges slowly.     

Monte Carlo discrete curved ray-tracing method for radiative transfer in an absorbing-emitting semitransparent slab with variable spatial refractive index

A Monte Carlo discrete curved ray-tracing method is developed to analyze the radiative transfer in one-dimensional absorbing-emitting semitransparent slab with variable spatial refractive index, in which the Monte Carlo method is combined with the discrete curved ray-tracing method. A problem of radiative equilibrium with linear variable spatial refractive index is taken as an example to examine the accuracy of the proposed method. The temperature distributions and the dimensionless radiative heat flux are determined by the proposed method and compared with the data in references, which are obtained by other different methods. The results show that the Monte Carlo discrete curved ray-tracing method has a good accuracy in solving the radiative transfer in one-dimensional semitransparent slab with variable spatial refractive index.     

多维梯度折射率介质内辐射传递的扩散近似

推导多维梯度折射率介质内稳态辐射传递的扩散近似方程.使用有限元法对扩散近似进行离散和求解,利用两个二维半透明介质的稳态辐射传递问题验证该扩散近似的精度及适用性.算例考虑介质为均匀折射率及梯度折射率两种情况.利用扩散近似分别求解辐射平衡时的边界热流、介质内温度场分布,并与辐射传递方程的求解结果进行对比分析.结果表明:介质折射率变化、散射特性、光学厚度及散射反照率均直接影响扩散近似的精度;在光学厚及强散射条件下,该扩散近似可以作为一种快速算法应用于梯度折射率介质稳态辐射传递的求解.     

Theoretical study of combined radiative conductive and convective heat transfer in semi-transparent porous medium in a spherical enclosure

A new method for the solution of the radiative transfer equation in spherical media based on a modified discrete ordinates method is extended to study radiative, conductive and convective heat transfer in a semi-transparent scattering porous medium. The set of differential equations is solved using the fourth-order Runge-Kutta method. Various results are obtained for the case of combined radiative and conductive heat transfer, as well as for the interaction of those modes with convection. The effects of some radiative properties of the medium on the heat transfer rate are examined.     

配置点谱方法求解一维圆柱梯度折射率介质中的辐射传热

发展一种配置点谱方法，计算包含半透明各向异性散射介质的圆柱系统中的辐射传热.介质具有梯度折射率，并且散射反照率随空间位置变化.通过与精确解或其他方法的结果对比，验证配置点谱方法的准确性.结果表明：配置点谱方法仅采用少量的节点数就可以得到准确的辐射热流量.     

Generalization of the spherical harmonic method to radiative transfer in multi-dimensional space

The basic radiative transfer equation in three-dimensional space is expressed in terms of three commonly used coordinate systems, namely, Cartesian, cylindrical and spherical coordinates. The concept of a transformation matrix is applied to the transformation processes between the Cartesian system and two other systems. The spherical harmonic method is then applied to decompose the radiative transfer equation into a set of coupled partial differential equations for all three systems in terms of partial differential operators. By truncating the number of partial differential equations into four along with further mathematical analyses, we obtain a modified Helmholtz equation. For each coordinate system, analytical solutions in terms of infinite series are obtained whenever the equation is solvable by the technique of separation of variables with proper boundary conditions. Numerical computations are carried out for one dimensional radiative transfer to illustrate the applicability of the technique developed in the present study.     

Temperature field of radiative equilibrium in a two-dimensional graded index medium with gray boundaries

In this paper, a numerical method is presented for the study of the radiative transfer in a two-dimensional graded index semitransparent medium with diffuse gray boundaries. The numerical method is a combination of the linear refractive index bar model, the discrete curved ray-tracing technique and the pseudo source adding method (LRIB-CRTP). In the traditional ray-tracing technique, it is difficult to deal with the diffuse gray boundary while solving the radiative transfer. Using the pseudo source adding method, the diffuse gray boundary of the medium can be treated as a black boundary. We have also studied the radiative equilibrium temperature field of the medium and analyzed the influence of some parameters involved. The results show that the directional discrete number is important for the medium having a large absorption coefficient. The results also show that the refractive index distribution greatly influences the temperature field, whereas the linear absorption coefficient distribution has little influence on the temperature field.     

半透明梯度折射率介质内辐射熵传递方程及其数值模拟

在非相干辐射条件下,基于Planck光谱辐射熵强度定义,导出半透明梯度折射率介质内光谱辐射熵传递方程,以及局部辐射熵产率理论表达式.基于离散坐标法对辐射熵传递方程进行数值求解.以一维半透明梯度介质平板为例,对辐射熵方程及其算法进行验证.平板整体无因次辐射熵产的计算结果与宏观热力学定律的结果一致.     

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.