Some features of temperature field definition in axisymmetric problems

Abstract

The solution of a plane problem of temperature field distribution in axisymmetric formulation is under discussion. All phases essential for problem solving, from the problem definition and obtaining governing equations till getting the results and analyzing the received data, are adduced. Two methods of problem solving are presented: the finite difference method and the finite element method. Mistakes and discrepancies made by some authors in their problem solutions are pointed out. The Fourier's Law of Heat Conduction is used to determine the temperature field. If solved with the use of the finite difference method, the Fourier's heat conduction equation looks as the second-order nonhomogeneous differential equation; if solved with the use of the finite element method, the corresponding functionality for elliptic equation is used. A non-stationary temperature field is analyzed, since water and gas supply pipes are the examples of bodies in terms of the axisymmetric problem. In terms of numerical solution different authors have several approaches: approximation of the speed of temperature rise before and after the integration of elliptic equations. Equations solution takes the form of a tridiagonal matrix which can be solved with the use of the Gauss method in the MatLab software system. In contrast to the majority of authors who calculate the stiffness matrix by coordinates of the center of the finite element, we got the exact analytical solution in terms of the given function form.