Completely Spectral Collocation Method For Solving Integral-Differential Form Of Radiative Transfer Equation

Radiation combined with other modes of heat transfer in participating media plays an important role in engineering applications, such as boilers, glass thermal forming, industrial furnaces, combustion chambers, optical and textile fiber processing, etc. It is generally required to solve a multidimensional radiation field using computational techniques. Determinations of heat transfer and temperature distributions in media, where multiple heat transfer modes are present (conduction, convection and/or radiation), are computationally difficult to predict. The descriptive equations are highly nonlinear. They involve radiative terms with integrals over the boundaries and volume of the medium that include temperature to the fourth power, plus convective and conductive terms that depend on derivatives of the temperature to the first and second powers.In practice, radiative heat transfer calculations are complex and consequently many approximate solutions are proposed, including, Monte Carlo method, ray tracing/nodal analyzing method, zonal method, finite volume method, discrete ordinates method, finite element method, discrete transfer method, and Lattice Boltzmann method. Different from the above methods, the spectral collocation method (SCM), as one kind of global methods, where the computation at any given point depends not only on information at neighboring points, but on information from the entire domain, can effectively overcome the shortages of low order accuracy and provide exponential convergence (in other words, spectral accuracy).The full spectrum K distribution (FSK) model for treating the spectral properties in absorbing-emitting medium represents an alternative to line-by-line (LBL) calculations which reduces the number of evaluations of the radiative transfer equation (RTE) from the order of a million to the order of ten without any significant loss of accuracy. For problems where an appropriate reference temperature can be defined, the FSK model is formally exact and consists only of a change of variables in the spectral domain.In this dissertation, completely spectral collocation method (CSCM), in which the angular domain and the spatial domain are all discritized for the interal-differential RTE, is extended to solve RTE in one-dimensional nonlinear scattering medium with space-dependent scattering coefficient, and anisotropic scattering medium with graded index. Furthermore, combined with discrete ordinates method (DOM), SCM-DOM also has a good flexibility to solve three dimensional steady and transient coupled radiative and conductive heat transfer problems in an absorbing, emitting, and scattering medium. Moreover, combined with full spectrum k-distribution model (FSK), SCM-FSK is extended to solve one-and two-dimensional non-gray gas radiative heat transfer problems.The main content of this dissertation is as follows:(1) SCM-DOM is presented to solve steady and transient coupled radiative and conductive heat transfer in three-dimensional absorbing, emitting, and scattering medium. The RTE, steady and transient energy conservation equation are both solved by the SCM on the same node system to predict coupled radiative and conductive heat transfer in an absorbing, emitting, and isotropically scattering participating medium. Four various test cases are taken as examples to verify the performance of the presented method for different parameters like, the scattering albedo, the conduction-radiation ratio, the wall emissivity, and the optical thickness. The predicted dimensionless radiative heat flux, dimensionless total heat flux and dimensionless temperature distributions agree well with those solutions in references. SCM-DOM is very efficient to solve transient three-dimensional coupled radiative and conductive heat transfer in semitransparent medium even using a small number of grids.(2) CSCM is developed to solve radiative heat transfer in the nonlinear anisotropic scattering. plane-parallel medium with space-dependent scattering coefficient. Both spatial and angular domains of RTE are discretized by CSCM. The exponential convergence characteristic of CSCM is investigated. The accuracy of CSCM is systematically compared with analytical solutions, lease square finite element method (LSFEM) results, and SCM-DOM results. These comparisons indicate that CSCM has a good accuracy. With the increase of collocation points, the relative mean square error of CSCM decreases exponentially. CSCM can obtain continuous solutions in spatial and angular domains, and results of CSCM are smoother and more accurate than those of SCM-DOM. Under the same collocation points in spatial and angular domains, CSCM is much more efficient than SCM-DOM.(3) CSCM is developed to solve radiative transfer in anisotropic scattering medium with graded index. CSCM is used to discretize both the angular domain and the spatial domain of RTE. The CSCM leads to more accurate expressions of the angular derivative term and the integral term of the RTE with graded index, and the finite difference approximation is avoided. Compared with the numerical results of other methods, and also the results of SCM-DOM, the CSCM can be the prior choice in accuracy and efficiency for various parameters, such as, the extinction coefficient, the scattering albedo, the scattering phase function, the refractive index gradient and the boundary emissivity. Using the same number collocation points in both spatial and angular domains, the CSCM is found much more efficient than the SCM-DOM. For space-dependent anisotropic scattering medium, compared with the SCM-DOM,the CSCM can successfully eliminate the ray effect and obtain more accurate results.(4) SCM-FSK is successfully applied to solve one- and two- dimensional non-gray gas (CO2/N2、H2O/N2 and CO2/H2O/N2) radiative heat transfer problems. For non-gray gas radiative properties, the FSK models using HITRAN2004 and HITEMP2010 database are adopted; for RTE, the spatial domain is expressed by Chebyshev polynomials and solved by the SCM, the angular domain is discretized by SRAP5 angular quadrature scheme. Compared with LBL solutions, the SCM-FSK can provide good accuracy for both one- and two- dimensional non-gray gas thermal radiation.