A general approach to solve heat conduction problems with internal heat sources using resistance and quadrupole concepts

Abstract

In the heat transfer literature, resistance and quadrupole concepts are used to solve heat conduction problems with or without internal heat generation, see for example Maillet et al. (2000). However, in most cases, the heat transfer is unidirectional. In this paper, we present a general approach to solve conduction problems with internal sources. This is done by simply introducing a partition coefficient β. The pertinence of this approach is mathematically demonstrated for both steady state and transient problems. Under steady state conditions, β represents a resistance partition coefficient while under transient conditions β is a quadrupole partition coefficient. In each case, the resulting equivalent thermal circuits are presented. Finally, the proposed approach is applied to electrothermal problems with heat generation due to the joule effect, the result is remarkable, that is, the partition coefficient β is a constant equal to 1/2.