Existence, uniqueness and construction of the solution of the energy transfer problem in a rigid and non-convex blackbody with temperature-dependent thermal conductivity

Abstract

In this paper, we study the steady-state (coupled) conduction-radiation heat transfer phenomenon in a non-convex opaque blackbody with temperature-dependent thermal conductivity. The mathematical description consists of a nonlinear partial differential equation subjected to a nonlinear boundary condition involving an integral operator that is inherently associated with the non-convexity of the body. The unknown is the absolute temperature distribution. The problem is rewritten with the aid of a Kirchhoff transformation, giving rise to linear partial differential equation and a new unknown. An iterative procedure is proposed for constructing the solution of the problem by means of a sequence of problems, each of them with an equivalent minimum principle. Proofs of convergence as well as existence and uniqueness of the solution are presented. An error estimate, for each element of the sequence, is presented too.Mathematics Subject Classification. 35J60 · 35J65.

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

“…Although many times the black body assumption may be a good approximation for engineering purposes, thermal radiant energy is emitted, absorbed and reflected by any real body. This article may be considered as a generalization of [1], aiming at a more realistic description, since real bodies boundaries actually reflect part of the incident radiant energy. When the body is surrounded by a rarefied atmosphere or when the temperature levels are high, this thermal radiation exchange must be taken into account.…”

“…Also existence and uniqueness of the solution of this solution are demonstrated. Reference [1] may be considered as a particular case of the present work, which arises when the emissivity is equal to one (that means black body). The mathematical description for a gray body is much more complex than the description for a black body, due to the boundary conditions.…”

“…As a first approximation, the body could be treated as a blackbody. Nevertheless, a blackbody [1] does not reflect thermal radiant energy and, many times, this fact does not allow a good mathematical description for bodies in which the reflection is not negligible. The simplest physical model for taking into account the reflection of thermal radiant energy is based on the gray body assumption.…”

“…Φ 0 ≡ 0 and Θ 0 ≡ 0. The existence of each element of the sequences is ensured from the coerciveness of the associated operator[19],[1]. The solution of the problem (3.3), denoted by T and such that ω = ℵ[T ], is the limit of the sequence [Φ 1 , Φ 2 , Φ 3 , .…”