Finite element formulation of first- and second-Order discrete ordinates equations for radiative heat transfer calculation in 3-D participating media

Abstract

Two finite element methods, FEDOM1 and FEDOM2 (standing for the first and the second finite element discrete ordinates methods, respectively), are formulated and numerically tested in three-dimensional participating media. 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 finite element method (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. With a few exceptions, the errors are generally reduced as the number of ordinates and grids increases. Improvement of accuracy by increasing the grids is not as notable as that by increasing the ordinates in this problem. Both of the developed FEMs show reasonable results compared with published Monte Carlo solutions for the tested absorbing / emitting and anisotropically scattering media. Isotropically and different two anisotropically scattering phase functions, F2 and B2, are examined. Generally, the accuracy is increased as the number of ordinates and grid increases. The FEDOM1 shows closer results to the Monte Carlo computations than the FEDOM2. The FEDOM1 and FEDOM2 are applied to the nonorthogonal grid to further check its validity. The accuracy of the FEDOM1 is shown to be higher than that of the FEDOM2 for all range of optical thickness. Generally, the two FEMs show more accurate results than DOIM but the FEMs require more calculation time and memory consumption because their matrix band width is larger than that of the DOIM. And FEDOM1 sh...