Semi-analytical solution of unsteady heat conduction in plain walls with equal surface temperatures: The Transversal Method Of Lines (TMOL) delimited to the “small time” sub-domain

The present paper addresses uni-directional, unsteady, heat conduction in a plane wall with uniform initial temperature and equal temperature prescribed at the surfaces. The thermal diffusivity of the solid is assumed to be nearly invariant with temperature. A hybrid analytical /numerical procedure named the Transversal Method Of Lines (TMOL) transforms the one-dimensional, unsteady heat conduction equation along with the initial temperature in rectangular coordinates into an equivalent one-dimensional, “quasi-steady” heat conduction equation applied at a pre set time. In other words, a partial differential equation of parabolic type is converted into an ordinary differential equation of second order with the space coordinate as the independent variable and the time absorbed as an embedded parameter. In this work, the analytical/numerical mid-plane and mean temperature profiles, as well as the analytical-numerical total heat transfer are produced by various time levels within TMOL framework, namely 1−TMOL, 2−TMOL and 3−TMOL. Overall, from a qualitative standpoint, the analytical/numerical mid-plane and mean temperature profiles and total heat transfer with the the 3−TMOL approach exhibit excellent quality over the crucial “small time” sub-domain evidencing relative errors that stay within a 5% margin.