The Temperature Condition of the Plate with Temperature-Dependent Thermal Conductivity and Energy Release

Abstract

The temperature state of a solid body, in addition to the conditions of its heat exchange with the environment, can greatly depend on the heat release (or heat absorption) processes within the body volume. Among the possible causes of these processes should be noted such as a power release in the fuel elements of nuclear reactors, exothermic or endothermic chemical reactions in the solid body material, which respectively involve heat release or absorbtion, heat transfer of a part of the electric power in the current-carrying conductors (so-called Joule’s heat) or the energy radiation penetrating into the body of a semitransparent material, etc. The volume power release characterizes an intensity of these processes. The extensive list of references to the theory of heat conductivity of solids offers solutions to problems to determine a stationary (steady over time) and non-stationary temperature state of the solids (as a rule, of the canonical form), which act as the sources of volume power release. Thus, in general case, a possibility for changing power release according to the body volume and in solving the nonstationary problems also a possible dependence of this value on the time are taken into consideration. However, in real conditions the volume power release often also depends on the local temperature, and such dependence can be nonlinear. For example, with chemical reactions the intensity of heat release or absorption is in proportion to their rate, which, in turn, is sensitive to the temperature value, and a dependence on the temperature is exponential. A further factor that in such cases makes the analysis of the solid temperature state complicated, is dependence on the temperature and the thermal conductivity of this body material, especially when temperature distribution therein is significantly non-uniform. Taking into account the influence of these factors requires the mathematical modeling methods, which allow us to build an adequate nonlinear mathematical model of the heat conductivity process in the volume power release body. Quantitative analysis of these models requires using the numerical methods, as a rule. At the same time, such a simple body, which is an unlimited plate of the constant thickness allows us, under certain assumptions, to solve analatically a nonlinear heat conductivity problem taking into account the thermal conductivity of the plate material and the power release intensity versus temperature. This solution enables us to reveal a number of significant effects that have impact on the thermal state of the plate, including those related to conditions of available steady temperature distribution, and it can be used to test the results obtained by numerical methods.