The Heat Transfer Problem in a Rigid and Nonconvex Gray Body with Temperature Dependent Thermal Conductivity

This work studies coupled steady-state conduction-radiation heat transfer in a non-convex gray body when the thermal conductivity temperature-dependent. The gray-body assumption is an improvement with respect to the black body model, because in this model a portion of the incident radiant energy can be reflected from the body boundary. The problem is mathematically described by a nonlinear partial differential equation subjected to a nonlinear boundary condition involving a Fredholm operator which arises from the non-convexity of the body. In this problem the absolute temperature distribution is the unknown, as in the case of a black body. The Kirchhoff transformation is employed to linearize the partial differential equation, giving rise to new boundary conditions. The solution of the problem is constructed by a proposed iterative procedure, employing  sequences that involve the temperature and the radiosity. The convergence is explicitly demonstrated.  Besides, an error estimate, for each element, is presented. It is remarkable that the results obtained for black bodies are a particular case of this work. In other words, the results presented in reference  [1]  consists of a particular case of this paper, obtained when the emissivity is equal to one.