Treatment of the Unsteady Heat Equation Subject to Heat Flux Boundary Conditions: The Method of Discretization in Time Complemented With Regression Analysis

Abstract

The Method of Discretization in Time (MDT) is a general hybrid technique intended to alleviate partial differential equations of parabolic type. The MDT engenders a sequence of adjoint second order ordinary differential equations, wherein the space coordinate is the independent variable and the time appears as an embedded parameter. Essentially, the resulting adjoint second order ordinary differential equations are considered of “quasi-stationary” nature. In this work, the MDT is applied to the unsteady heat equation in simple bodies (large plate, long cylinder and sphere) with temperature-invariant thermophysical properties, constant initial temperature and surface heat flux boundary conditions. In engineering applications, the surface heat flux is customarily provided by electrical heating, radiative heating and pool fire heating. Using a single time jump and the first adjoint “quasi-stationary” heat equation, it is demonstrated that the approximate, semi-analytical MDT temperature solutions expressed in terms of the space coordinate and excluding time are easily obtainable for each simple body. As a direct consequence, usage of the second adjoint “quasi-stationary” heat equation engaging two time jumps come to be unnecessary. As a sound replacement, regression analysis is applied to the deviations of the dimensionless surface temperature as a function of the dimensionless time. Thereafter, the outcomes are articulated with the approximate, semi-analytical MDT temperature solutions.