The radiative transport equation(RTE) is a multi-dimensional integro-differential equation, which is very difficult to solve even in one-dimensional cases due to its directional nature. As a result approximate methods need to be used. The two most popular methods are the discrete ordinates (or SN) method and its variants, and the spherical harmonic (or PN) method. Unfortunately neither of those methods are accurate over a large range of optical thickness. For example, the SN method is inaccurate for optically thick situation, whereas the PN approximation is inaccurate for optically thin situations, especially for cases when a cold medium is bounded by hot walls. This thesis focusses on the spherical harmonics method, and different approaches for improving accuracy were investigated.
In the first approach, a higher order approximation to the spherical harmonics solution of the RTE, the P3 approximation, was explored. The P3 approximation results in 4 strongly coupled linear second-order elliptic PDEs with Robin-type boundary conditions. A Finite-Volume formulation was developed to discretize the equations. The resulting algebraic equations were solved using a coupled solver, which employs the Binary Spatial Partitioning Algorithm to partition the domain into sub-domains followed by solution of the equation in each sub-domain using a Pre-conditioned General Minimum REsidual Solver(GMRES). The mesh used to discretize the domain was unstructured in nature. The Finite-volume formulation and the solution of the P3 equation on unstructured meshes is a contribution of the present work. On comparing the results with benchmark Monte Carlo results, and with results obtained using the P1 approximation, it was found that accurate results are obtained only for optically thick cases, and the improvement of P3 over P1 was only marginal. The P3 model was also coupled with a flow solver for a 2D laminar flame computation. It was found that the P3 equations required an additional 14:5% CPU time as compared to 9:5% for the P1 approximation. In conclusion, the P3 approximation was not found to be substantially superior to P1 approximation, and thus other avenues to improve the accuracy of the P1 method were investigated.
The second approach was to use the Modified Differential approximation, which was initially proposed to remove the shortcomings of the P1 in optically thin situations. In this approach, the radiation intensity is split into a wall-emitted ballistic component, and a medium-emitted diffusive component. The ballistic component is determined using a combination of surface-to-surface exchange relations and ray-tracing algorithms. The diffusive component is determined by invoking the P1 approximation. Despite being in existence for two decades, the MDA approach has found limited use in practical engineering application. One of the contributions of this thesis was to extend it to arbitrary, inhomogeneous 3D media with obstructions. Ray-tracing was used to handle geometries having obstructions and inhomogeneous media. On comparing with Monte-Carlo results, MDA results were found to agree well for wide range of optical thickness, and showed considerable improvement when compared to P1 approximation. The calculation and storage of view-factors presented the most significant challenge in the 3D case and methods to handle this challenge are discussed.