An implicit differential equation governing lumped capacitance, radiation dominated, unsteady, heat transfer

Abstract

Purpose

Although most physical problems in fluid mechanics and heat transfer are governed by nonlinear differential equations, it is less common to be confronted with a “so – called” implicit differential equation, i.e. a differential equation where the highest order derivative cannot be isolated. The purpose of this paper is to derive and analyze an implicit differential equation that arises from a simple model for radiation dominated heat transfer based upon an unsteady lumped capacitance approach.

Design/methodology/approach

Here we discuss an implicit differential equation that arises from a simple model for radiation dominated heat transfer based upon an unsteady lumped capacitance approach. Due to the implicit nature of this problem, standard integration schemes, e.g. Runge‐Kutta, are not conveniently applied to this problem. Moreover, numerical solutions do not provide the insight afforded by an analytical solution.

Findings

A predictor predictor‐corrector scheme with secant iteration is presented which readily integrates differential equations where the derivative cannot be explicitly obtained. These solutions are compared to numerical integration of the equations and show good agreement.

Originality/value

The paper emphasizes that although large‐scale, multi‐dimensional time‐dependent heat transfer simulation tools are routinely available, there are instances where unsteady, engineering models such as the one discussed here are both adequate and appropriate.

Citation

De Chant, L.J. (2012), "An implicit differential equation governing lumped capacitance, radiation dominated, unsteady, heat transfer", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 22 No. 7, pp. 896-906. https://doi.org/10.1108/09615531211255770