Heat-Conduction Problems with Time-Variable Heat-Transfer Coefficients

Based on the definition of additional boundary conditions and an additional sought function (ASF), an approximate analytical solution of the heat-conduction problem for an infinite plate subject to Newton’s symmetrical boundary conditions with a time variable heat-transfer coefficient is obtained. In accordance with the integral thermal balance method, the solution is subdivided into two stages in time. The first and the second stage include the time intervals corresponding to an irregular and a regular heat-transfer mode, respectively. At the first stage, a function characterizing the displacement with time of the temperature disturbance front along the ξ coordinate is adopted as the ASF. At the second stage, a function characterizing the change with time of the temperature at the symmetry center, in which the boundary condition of no heat transfer is specified, is considered as the ASF. Owing to the use of ASFs at both stages, it becomes possible to boil down the solution of the initial differential equation with partial derivatives to integration of an ordinary differential equation with respect to the ASF. By solving this equation at the second stage, the eigenvalues are found, which are determined in the classical methods from the Sturm–Liouville boundary value problem, in which a transcendental trigonomertic equation is solved for each particular Biot number using a numerical method. Hence, in this case, another technique for determining eigenvalues is considered, which makes it possible to obtain a formula from which eigenvalues for each particular Biot number can be found. The form in which the additional boundary conditions are given is such that their satisfaction in finding the sought solution would be equivalent for the case of solving the initial differential equation at the boundary points. It is shown that the solution of the equation at the boundary points leads to its solution also inside the domain considered; in this case, direct integration of the equation along the spatial variable is excluded and is only limited to fulfilling the thermal balance integral, i.e., the averaged initial differential equation.