Transversal Method of Lines for Unsteady Heat Conduction With Uniform Surface Heat Flux

The transversal method of lines (TMOL) is a general hybrid technique for determining approximate, semi-analytic solutions of parabolic partial differential equations. When applied to a one-dimensional (1D) parabolic partial differential equation, TMOL engenders a sequence of adjoint second-order ordinary differential equations, where in the space coordinate is the independent variable and the time appears as an embedded parameter. Essentially, the adjoint second-order ordinary differential equations that result are of quasi-stationary nature, and depending on the coordinate system may have constant or variable coefficients. In this work, TMOL is applied to the unsteady 1D heat equation in simple bodies (large plate, long cylinder, and sphere) with temperature-invariant thermophysical properties, constant initial temperature and uniform heat flux at the surface. In engineering applications, the surface heat flux is customarily provided by electrical heating or radiative heating. Using the first adjoint quasi-stationary heat equation for each simple body with one time jump, it is demonstrated that approximate, semi-analytic TMOL temperature solutions with good quality are easily obtainable, regardless of time. As a consequence, usage of the more involved second adjoint quasi-stationary heat equation accounting for two consecutive time jumps come to be unnecessary.