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

Abstract

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 (ID) parabolic partial differential equation, TMOL engen ders 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 pa rameter. Essentially, the adjoint second-order ordinary differential equations that result are of quasi-stationary nature, and depending on the coordinate system may have con stant or variable coefficients. In this work, TMOL is applied to the unsteady ID heat equation in simple bodies (large plate, long cylinder, and sphere) with temperatureinvariant 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 o f time. As a consequence, usage of the more involved second adjoint quasistationary heat equation accounting for two consecutive time jumps come to be unnecessary.

Prediction of the temperature–time history in ordinary bodies induced by surface heat flux utilizing the enhanced method of discretization in time and the finite difference method

“…As mentioned in the Introduction, the MDT or the transversal method of lines (TMOL) was applied by Campo and Garza (2014) to analyze unsteady heat conduction in a subset of simple ordinary bodies (large plane wall, long cylinder and sphere) possessing uniform initial temperature and constant thermo-physical properties and receiving uniform surface heat flux.…”

“…The transformability of a linear partial differential equation of parabolic type with continuous time into a linear ordinary differential of second order with discrete time embedded as a time parameter can be carried out with the method of discretization in time (MDT) (Rektorys, 1982), also called the transversal method of lines (Rothe, 1930). Campo and Garza (2014) implemented the MDT to analyze unsteady heat conduction in a subset of bodies in one dimension (large plane wall, long cylinder and sphere) influenced by uniform surface heat flux and constant thermo-physical properties. Because MDT has intrinsic truncation errors of first order, it is expected that the results using one time jump are valid for “small time,” the “early regime.” Thereby, the computed MDT results produce reasonable approximate temperature histories in the three bodies during incipient heating.…”

Hybrid Analytical/Numerical Solution of the Unsteady Heat Conduction Equation Subject to Unequal Robin Boundary Conditions

“…As additional information, the Method of Lines (MOL) and its variants the Numerical Method of Lines (NMOL), the Transversal Method of Lines (MOL) and the Improved Transversal Method of Lines (MOL) have been reported by Campo and collaborators (see References [16][17][18][19][20][21].…”

“…Consequently, the coupled system of three ordinary differential equations of first-order consists in eqs. ( 15), ( 10) and (18). Further, the initial conditions in eq.…”