THE TEMPERATURE DEPENDENCE OF UNSTEADY HEAT CONDUCTION IN SOLIDS

Abstract

A mathematical model of the process of unsteady thermal conductivity of solids is proposed in the case where the dependence of the thermal characteristics of the medium (heat capacity, density and thermal conductivity coefficient) on temperature cannot be neglected in the heat conduction equation. Based on the experimental data equations of thermal conductivity are obtained for the cases of high (Т» θ) and low (Т«θ) temperatures (θ is the Debye temperature). Both in the case of high and low temperatures, the temperature dependences of the heat capacity and the thermal conductivity coefficient are power-law, which allows us to bring the original heat conduction equation to a form that allows the use of the classical method of variable separation in solving the corresponding boundary value problems for the heat conduction equation. The solution of the thermal conductivity equation is considered in the approximation, in which the free path of phonons is limited and does not depend on temperature, so that the temperature behavior of the thermal conductivity coefficient is determined only by the temperature dependence of the heat capacity. Exact analytical solutions for boundary value problems modeling thermal conductivity in dielectrics and metals in the polycrystalline state are obtained. The solutions relating to both areas with fixed and moving boundaries are considered. In order to solve boundary value problems with moving boundaries, in the framework of the proposed model of thermal conductivity, the functional transformation of a special kind is used. This allows reducing the original problem to the problem with fixed boundaries, but with the transformed heat conduction equation. The obtained results can be used in engineering studies of the kinetics of some physical and chemical processes in solids and liquids - diffusion, sedimentation, viscous flow, neutron deceleration, fluid flow through a porous medium, electrical oscillations, sorption, drying, combustion, etc.

Keywords

• equation of unsteady thermal conductivity
• debye temperature
• high and low temperatures
• fixed and moving boundaries