On Approximate Solutions for Unsteady Conduction in Slabs with Uniform Heat Flux

Abstract

In this paper, the Transversal Method of Lines (TMOL) or Rothe's method is employed to obtain analytical expressions of simple form for the unsteady one-dimensional heat conduction in a slab. Initially, the slab is maintained at a uniform temperature, and then a uniform heat flux is applied to its surfaces. Implementation of TMOL generates a sequence of adjoint ordinary differential equations, where the spatial coordinate is the only independent variable and the time becomes a parameter. In spite of the anticipated expectations that the semi-discrete solutions produced by TMOL would yield accurate temperature responses for short times only, detailed calculations demonstrate the opposite trend. Surprisingly, the temperature results associated with two equal time steps are excellent not only for short times, but during the entire heating period.