Analytical solution to the problem of heat transfer using boundary conditions of the third kind

Abstract

Non-stationary heat transmission within solid bodies is described using parabolic and hyperbolic equations. Currently, numerical methods for studying the processes of heat and mass transfer in the flows of liquids and gases have disseminated. Modern programs allow the automatic construction of computational grids, solutions to the systems of equations and offer a wide range of tools for analysis. Approximate analytical solutions have significant advantages compared to numerical ones. In particular, the solutions obtained in an analytical form allow performing parametric analysis of the system under study, configuration and programming of measurement devices, etc. Based on the joint use of additional desired function and additional boundary conditions in the integral method of heat balance, a method of mathematical modeling for the heat transfer process in a plate under symmetric boundary conditions of the third kind has been developed. Using the heat flux density as a new desired function, the method for solving heat conduction problems with boundary conditions of the third kind has been proposed. Finding a solution to the partial differential equation with respect to the temperature function presents integrating an ordinary differential equation with respect to the heat flux density on the surface of the studied zone. It has been shown that isotherms appear on the surface of the plate with a certain initial velocity which depends on the heat transfer intensity. The calculation results have been compared to the exact solution. The presented method can be used in determining the heat flux density of buildings and heating devices, finding heat losses during convective heat transfer and designing heat transfer equipment. The results can be applied to increase the validity and reliability of the calculation of actual losses and balance of thermal energy. The method reliability, validity and a high degree of approximation with about 3% inaccuracy have been demonstrated. The accuracy of the solution depends on the number of approximations performed and is determined by the degree of the approximating polynomial.