An integral equation for radiation transport in an infinite cylinder with Fresnel reflection

We formulate an integral equation for radiation transport in an infinitely long cylinder. Scattering is included and sources arise from incidence on the surface and from an internal volume source. Internal reflection at the surface follows the Fresnel law. For the special case of no scattering and no axial variation of the source, we obtain an exact solution which we have compared numerically with some results of Tian and Chiu [Radiative absorption in an infinitely long hollow cylinder with Fresnel surfaces. JQSRT 2006;98:249]. We have also obtained closed form analytical solutions for no scattering with a line source along the cylinder axis and also that for the case of a spatially constant volume source. The general case when there is variation in the axial direction is also presented and the special case of a radially uniform plane source at the origin is explored in some detail. The axial solution is compared with an approximate areal average method introduced by Larsen [A one-dimensional model for three dimensional transport in a pipe. Transp Theory Stat Phys 1984;13:599; Larsen EW, Malvagi F, Pomraning GC. One dimensional models for neutral particle transport in ducts. Nucl Sci Eng 1986;93:13] in another context