Визначення двовимірних нестаціонарних температурних полів у плитах та панелях з плоскопаралельними межами за наявності джерел тепла

Abstract

Non-stationary two-dimensional problems of thermal conductivity for plates and panels with plane-parallel boundaries in the presence of volume-distributed non-stationary heat sources are formulated. A method of constructing a solution to the formulated heat conduction problems for the considered bodies is proposed. The technique uses the approximation of the temperature distribution in both elements by the thickness variable by a cubic polynomial. The coefficients of the approximation polynomial are given through the integral over the thickness variable temperature characteristics and the conditions for the boundary values of the temperature on the outer surfaces of the plate and panel. As a result, the initial two-dimensional initial boundary value problems for the temperature for the plate and panel are reduced to one-dimensional initial boundary value problems for the integral temperature characteristics. To construct the solution of the initial-boundary value problem for the integral characteristics of the temperature in the case of a plate infinite in longitudinal and transverse coordinates, the integral Laplace transform in time and the integral Fourier transform in the longitudinal coordinate were used. The solution of the problem on the integral temperature characteristics in the case of the panel is found using the integral Laplace transform in time and the finite integral transform in the transverse coordinate. Expressions of integral temperature characteristics for the plate and panel are obtained in the form of convolutions of functions corresponding to homogeneous solutions of initial-boundary value problems for integral temperature characteristics and functions describing the available non-stationary heat sources in these bodies and given surface temperature values. The general solutions of the two-dimensional initial boundary value problems of thermal conductivity for slabs and panels are recorded for the presence of arbitrarily variable spatial coordinates of non-stationary heat sources and the conditions of convective heat exchange with the external environment on the surfaces of the considered bodies.