Finite element formulation of the discrete-ordinates method for multidimensional geometries

Radiative heat transfer in a multidimensional participating medium was predicted using the discrete-ordinates (DO) method. The even parity radiative transfer equation (RTE) is formulated for an absorbing, isotropically scattering, and re-emitting medium enclosed by gray walls. The even parity RTE is spatially discretized with the finite element method. The solution accuracy and convergence are discussed for several element types. Several test enclosures are modeled. Results are compared with the standard DO control volume formulation, exact solutions, and the P3 differential approximation. Solutions are found for enclosures with either absorbing or isotropically scattering media. Results compare well with published results. Also, the use of the method for complex geometries and unstructured grids is discussed. Nomenclature Eh = emissive power, 6T4, W/m2 G = incident energy, J47r / dft, W/m2 G* = nondimensional incident energy, G/Eb I = intensity, 7(r, ft), W/m2-sr L = characteristic length, m M = total number of discrete ordinate directions N = total number of global nodes NJ = basis function n = surface normal Q* = nondimensional net wall heat flux, qlEb q = heat flux, J2ir |rt-ft|7 dft, W/m2