Stationary problem of complex heat transfer in a system of semitransparent bodies with boundary conditions of diffuse reflection and refraction of radiation

Abstract

A boundary value problem describing complex (radiation-conductive) heat transfer in a system of semitransparent bodies is considered. Complex heat transfer is described by a system consisting of a stationary heat equation and an equation of radiative transfer with the boundary conditions of diffuse reflection and diffuse refraction of radiation. The dependence of the radiation intensity and optical properties of bodies on the frequency of radiation is taken into account. The unique existence of the weak solution to this problem is established. The comparison theorem is proven. Estimates of the weak solution are derived, and its regularity is established.

Unique solvability of a stationary radiative‐conductive heat transfer problem in a system consisting of an absolutely black body and several semitransparent bodies

“…Proof We use some ideas of the method proposed in Krizek and Liu 107 for proving comparison theorems for quasilinear elliptic equations. Versions of this method were used in the literature 41,42,54‐57 for the radiative‐conductive heat transfer problems. Let 0 < δ < 1 be a parameter.…”

“…In the past 30 years, a large number of papers have been devoted to the solvability of complex heat transfer problems in radiation‐opaque or radiation‐semitransparent materials (cf. other works 16–47,48–64 ). Note that, in the literature, 46–51,53,59–64 the radiation transfer equation is changed by its diffusion P 1 approximation.…”

“…We do not take into account the dependence of the radiation intensity and the properties of semitransparent materials on the radiation frequency. To study the solvability of the problem under consideration, we use the technique developed in the articles 12,41,55,56 …”