Approximate Analytical Solution of the Problem of Thermal Conductivity for a Tube Wall with Variable Heat-Transfer Coefficients along the Perimeter

Abstract

Based on the definition of an additional function and additional boundary conditions in the integral heat balance method, an approximate analytical solution is obtained for a nonstationary two-dimensional heat conduction problem for an infinite hollow cylinder under boundary conditions of the third kind with variable heat-transfer coefficients in the circumferential direction. An additional function describes the change in temperature over time at one particular point in the spatial variable. Using it, one can reduce the solution of the original partial differential equation to the integration of an ordinary differential equation, from which the eigenvalues of the boundary value problem are found. That is, a different concept of determining eigenvalues based on the time equation with respect to an additional function is presented, in contrast to classical methods, where eigenvalues are found when solving the Sturm–Liouville boundary value problem for a region of a spatial variable. The assignment of additional boundary conditions is the solution of the original equation at the boundary points. It is shown that solving the equation at the boundary points leads to its solution inside the considered region as well. Additional boundary conditions are derived using the original differential equation and basic boundary conditions. Repeatedly differentiating the equation with respect to the spatial variable, and the boundary conditions with respect to time, by comparing the obtained relations, one can find any number of additional boundary conditions. The accuracy of solving the equation inside the considered region depends on the number of approximations, and, consequently, on the number of additional boundary conditions used. The approximate analytical solution obtained in this way is characterized by a simple design that is convenient for use in engineering applications.