On Coupled Conductive-radiative Heat-transfer Problems in a Sphere

-A sphere-to-plane transformation technique and the spherical harmonics method are used, along with Hermite cubic splines, to define an iterative technique for solving a class of nonlinear radiative transfer problems in a sphere. Computational aspects of the technique are discussed, and the method is used to establish numerical results for several test problems. 1. I N T R O D U C T I O N In a paper published in 1975, Wu and Siewert t generalized a transformation technique reported by Mitsis 2 in order to reduce a class of radiation transport problems formulated in spherical geometry to more easily solved "pseudo" problems that have plane symmetry. Although the paper by Wu and Siewert t has been essentially overlooked by researchers in the radiative heat-transfer field, the transformation technique was used by Siewert and Grandjean 3 in 1979 to solve two problems, formulated in terms of neutron transport theory, that are of interest in the field of radiative transfer? It is clear that the technique of transforming from spherical problems to plane problems, as discussed by Wu and Siewert, 1 is not sufficiently general to solve all problems with spherical symmetry; however, problems in a solid sphere, with a diffusely reflecting surface, that are based on isotropic scattering can be solved in this manner for an arbitrary inhomogeneous source term and for an arbitrary distribution of radiation incident on the boundary. In order to demonstrate the merits of the sphere-to-plane transformation technique, 1 we use the technique along with the spherical harmonics method s-v and Hermite cubic splines to solve a spherical version of the steady-state problem in combined-mode (conduction and radiation) heat transfer that has been formulated by OZl~lk. 4 As (~Zl§lk 4 has reviewed carefully the numerous works that have contributed to this field of study, we do not repeat a review here; however, we do note that the present work draws heavily from our recent solution s for the plane geometry case. Also, we note that Thynell and OZl~lk, 9 Tsai and (~Zl~lk, 10 Jia et al tl and Thynel112 have reported numerical results for coupled problems with spherical symmetry. The work of Jia et al" addresses simultaneous radiation and conduction between concentric spheres, and so the problems considered in the work are clearly outside the class of problems solved in Ref. 1. The paper ~2 by Thynell includes an effect of anisotropic scattering in the equation of transfer, and so the heat transfer problems in a solid sphere that were solved in the work are also outside the class of problems solved in Ref. 1. We consider the equation of transfer written as ~ I(r,#)q 1-1~2 f-# w f ' w)~rn2 # ~r -r I(r, #) + I(r, #) = -~ -, I(r, #') dlt" + (1 _---~-T4(r), (1) for r E (0, R) and /z e [ 1, 1]. We seek a solution to Eq. (1) subject to the boundary condition ;o 1 I(R,--#) =E an2 T4 + 2 0 I(R, # ' )# ' dkt' (2) 7~