A Variationally Formulated Problem of the Stationary Heat Conduction in a Plate with Radiation Reduction Factor Increased under Temperature

Abstract

The equipment uses heat-shielding and structural materials that, when exposed, absorb radiation both on the surface and in the volume. In a variety of technical devices, absorption processes of penetrating radiation of materials and structural elements are typical for a number of process steps and operating conditions. Absorption of radiation penetrating into material volume may significantly affect the temperature state and runability of construction made of such material. The process of material-absobed penetrating radiation is associated with transition of the electromagnetic wave energy into the excitation energy of this material microparticles that, after all, leads to increasing internal energy and temperature growth. With radiation passing through the layer of material its flow density and hence energy of penetrating radiation decreases exponentially with increasing distance from the exposed layer surface. This law was experimentally established by the French physicist P. Bouguer and bears his name. In general, a certain fraction of this energy is radiated and dissipated in the material volume, and the rest is absorbed. A mathematical model describing these processes is an equation of the radiative energy transfer. In mathematical modeling of thermomechanical processes there is a need to consider the effect of penetrating radiation on the temperature state of materials and construction elements. The P. Bouguer law is used also when the volume radiation and scattering of penetrating radiation in the material can be neglected, but it is necessary to take into account its absorption. In this case, a negative indicator of the exponential function is represented by the product of the distance from the irradiated surface and integral or some average absorption factor that is constant for a given material and spectral distribution of penetrating radiation. However, with increasing power of radiation passing through the material layer there is a dependence of the absorption factor on the local intensity of this radiation. Furthermore, it can be a significant dependence of this factor on the local value of the material temperature, reflecting the above-mentioned relationship between the absorption of electromagnetic wave energy and the excitation of material microparticles. This process can be described by Boltzmann distribution function that comprises the energy to activate microparticles and the local value of temperature. This paper presents a variational formulation of the nonlinear problem of stationary heat conduction in a plate for the case when the radiation reduction factor in relation to the Bouguer law depends on the local temperature. This formulation includes a functional that can have several fixed points corresponding to different steady states of the plate temperature. Analysis of the properties of this functional enabled us to identify the stationary points, which correspond to the realized temperature distribution in the plate.