Solution of radiative heat transfer in nonhomogeneous participating media has been an important research subject for many engineering and scientific applications. Most of the applications encountered in actual systems contain participating media with nonuniform radiative property distributions in the multi-dimensional geometries. Computational limitations are presently a major factor in dictating the present state-of-the-art since modeling real properties and geometries is very computationally expensive, usually much more expensive than the flow simulations in the same system. Due to the integro-differential nature of the radiation transport, many algorithms, many more than those for the Navier-Stokes equations, have been developed in the past to solve the radiation transfer equation (RTE). Although many methods, such as the discrete ordinates method, the Monte Carlo method, the finite element method, the finite volume method, and the YIX method, can be applied to arbitrarily complex geometry and spectral properties in principle, little is known on their efficiency and accuracy relative to each other.

Because of the complexity of the RTE and the wide variety of methods available, error estimation of the computations is usually unavailable, and it is not uncommon that large differences can be found in the results for the same problem using different methods. This became apparent at the above mentioned Symposium. In that Symposium, participants were asked to solve a three-dimensional problem consisting of a nongray mixture of spherical carbon particles and CO2 gas contained in a rectangular enclosure. The problem was intended to model a coal-fired furnace. Despite major property and geometry simplifications made in defining the problem, larger than expected variations in the predictions were found. Similar situations can be found in journal publications where large differences among results are not unusual.

Therefore, it is critical to provide some benchmark solutions to the radiation heat transfer community. The need for benchmarks has been also reflected in a recent workshop on the use of high-performance computing to solve participating media radiative heat transfer problems (held in March 1993 at the Sandia National Laboratories), in which participants were asked to identify 5 classes of highly challenging, nationally important problems relating to the use of high-performance computing in the participating media radiative heat transfer. The first problem identified is the development of benchmark solutions for RTE.

Following the first Symposium in 1992, the YIX method has been used in a separate benchmarking effort with the Monte Carlo and the finite element methods. They are used to solve the radiative heat transfer within a unit cubical enclosure with nonhomogeneous participating media. With the first order accurate distance quadrature and piecewise constant integrand, the YIX solutions show 1 to 3% difference of surface heat fluxes in cases E1 and E2 as compared with the finite element solutions on the same grid.¹

Cases E1 and E2 have uniform temperature distribution inside the cold and black enclosure. Using the same piecewise constant radiative property distribution as the YIX solutions, The Monte Carlo solutions have the same order of error. It is also found that the YIX solutions have bigger difference at the core region where the optical thickness is larger.

In this paper, three higher order interpolation schemes, i.e., piecewise linear, trilinear, and parabolic, are presented to improve solution accuracy in generating benchmarks with YIX method. Detail and systematic error analysis indicate that superconvergence exists for these interpolations. Especially notable is the piecewise trilinear interpolation. It has excellent convergent rate as compared with the order of its interpolation error. The result shows that one order of magnitude reduction in error can be achieved without resorting to finer grid. Significant computational time and memory can be saved with high order interpolation. This has important implications when coupling the RTE calculation with the flow code.

Other than the interpolation error, the integration errors of the YIX method, which include distance and angular quadrature, are also examined. The use of discrete ordinates set in the angular quadrature is studied rigorously and compared with the use of Simpson rule. The convergence rate of the discrete ordinates sets is superior due to its spherical symmetry.

A related issue with angular quadrature is the ray effect. It is shown that for the problem that could cause ray effect in the solution, the effect can be eliminated by using large number of angular quadrature points. It is further determined that an adaptive angular quadrature scheme has the potential of removing ray effect without significant increase of the computational time. On the other hand, the use of high order distance quadrature is found that, without the corresponding higher order interpolation, little benefit can be obtained from it. The main intent of these results is to provide a verified set of solutions which can be useful as benchmarks when developing other methods.

For the calculation of the multidimensional spectral radiative heat transfer quite a lot of models have been suggested to describe the radiative properties of combustion gases and soot as well as the multidimensional radiative heat exchange have been evaluated seperately at the Institut für Thermische Strömungsmaschinen (ITS) in special test cases. It has been found that the Discrete Ordinates method should be best suited to combustion systems with respect to accuracy and computation time. However the Discrete Ordinates method has not been evaluated in real combustion systems.

Therefore, the main objective of this paper is the application of the Discrete Ordinates method to calculate the multidimensional radiative heat transfer in a "“real"” propane fired three-dimensional cylindrical model combustor featuring high gradients of temperature and concentrations of radiative active species. The information about the reacting flow-field inside this model combustor has been provided by both experiments and numerical analysis.

The radiation calculations are performed on a spectral basis using narrow-band models for an accurate representation of the radiative properties of combustion gases. To avoid "“sweeping"” in angular space the Discrete Ordinates Equations describing radiative exchange in a three-dimensional cylindrical coordinate system have been discretized by a “purely” spatially applied Finite Volume technique for the first time. This has been achieved by formulating the discrete direction vector in a cylindrical coordinates, too.

The different applied S_n-approximations (S₄, S₆, S₈ ) are evaluated by comparing the calculated spectra to measured spectra and calculated spectra obtained from 1D-calculations which serve as reference data. It has been proven, that multidimensional spectral radiative heat transfer in a real cylindrical combustion system featuring steep gradients of temperature and concentration of radiative active species can be predicted accurately using the models presented. As expected, the calculations of radiative heat transfer through the reaction zone is most critical, due to the steep gradients of scalar variables. However, using a fine grid size of 18998 grid points, highly accurate results are already obtained by application of the S₄-approximation. Therefore considering CPU-time consumption and accuracy, the S₄-approximation is best suited to radiation calculations in combustors.

By using above methods, the steady-state cooling environment of a meeting room with a cooled ceiling panel is analyzed. The local thermal influence of a heated window on the cooling environment is examined. In the analysis, a three-dimensional model of a human body is also used. It is divided into three parts: head, body and legs. The skin temperature and clothing combination can be assigned to each part of the body independently. The thermal sensation i of the human body in the room is evaluated quantitatively by calculating the heat loss of the human body and the predicted mean vote (PMV).

Our analysis method contributes to the systematic design of the radiant cooled space. It can determine the arrangement of furniture and the temperature of the cooling panel to obtain satisfactory thermal sensation in the space. It also enables us to improve the environment where thermal radiation sources such as heated windows cause thermal discomfort.

DEA’s of the system are calculated. The discrete-ordinates weights are expressed in terms of the product of cross-sectional weights and axial directional weights. For the cross-sectional weights and directions, Sanchezand Smith’s method or Chevyshef method is used. The abscissas and wights of Gaussian quadrature are utilized for axial direction integration to incorporate absorbing and emitting effects by the infinite layer.

The effects of optical thickness as well as the number of spatial and angular divisions on the accuracy of the DEA results are studied. The results are presented at the optical thickness values of 0, 0.1, 1, and 10. When optical thickness is greater than 5, if the relative errors of DEA values should be less than about 2%, then it is needed more spatial and angular divisions than current study. As optical thickness increases at a given spatial and angular divisions, the errors increase. For a given optical thickness, the errors are reduced as the number of spatial and angular divisions increase. If there is shading due to protrusion in the system, the accuracy of the results depends on both the number of spatial and angular divisions.

The results are compared with the DEA prediction by the S-N discrete-ordinates methods of order upto S-10. The results indicate that the higher order S-N method than S-10 is necessary to increase the accuracy of computation. The results are also compared with the numerical integration values of the DEA’s expression. In conclusion, whether there is shading in the system or not, direct exchange areas can be accurately obtained by using the discrete-ordinates method.