Solution of nonaxisymmetric stationary problem of heat conductivity for polar-orthotropic ring plate of variable thickness with account of heat transfer with external environment

Abstract

The solution of the nonaxisymmetric stationary problem of the heat conductivity for profiled polar-orthotropic annular plates considering the heat exchange with external environment through the bases is presented. Thermophysical characteristics of the material of the plate are assumed to be temperature-independent. A constant temperature T1∗ is maintained on the inner contour of the ring plate and on the outer contour N equidistant point sources of heat with the same temperature T2∗ each are applied. Plate temperature is higher than ambient temperature T0 (T0 < T1∗ < T2∗). It is assumed that the temperature does not vary in thickness of a thin ring plate. The temperature values on the contours of the annular plate are given. There are no internal heat sources in the plate. The temperature distribution in such plates will be nonaxisymmetric. Analytical solutions of the stationary heat conductivity problem for the following anisotropic annular plates are presented: the plate of constant thickness, the back conical and the conical plate. The Volterra integral equation of the second kind corresponding to the given differential equation of the stationary heat conductivity for profiled anisotropic annular plates is written to obtain the solution in the general case. The kernels of the integral equation for anisotropic annular plates of power and exponential profiles are given explicitly. The solution of the integral equation is written by using the resolvent. It is indicated that due to the presence of irrational functions in the kernels of the integral equation it is necessary to apply numerical methods in the calculation of iterated kernels or numerically solve the Volterra integral equation of the second kind. A formula for the calculation of temperatures in anisotropic annular plates of an arbitrary profile is given.

Keywords

• polar-orthotropic annular plate
• temperature
• stationary equation of heat conductivity
• differential equation
• volterra integral equation of the second kind
• plate of a constant thickness
• вack conical plate
• сonical plate
• plate of a power profile
• plate of an exponential profile