Gelfand type problem for two phase porous media

TL;DR

This paper extends the classical Gelfand problem to two-phase porous media, analyzing how inter-phase heat exchange influences thermal explosion and connecting it to the classical problem in the limit of infinite heat exchange.

Contribution

It introduces a two-phase generalization of the Gelfand problem, demonstrating the effects of inter-phase heat exchange on thermal explosion behavior.

Findings

Thermal explosion occurs only when stationary temperature distributions are absent.

Inter-phase heat exchange delays the onset of thermal explosion.

In the limit of infinite heat exchange, the problem reduces to the classical Gelfand problem.

Abstract

We consider a generalization of the Gelfand problem arising in Frank-Kamenetskii theory of thermal explosion. This generalization is a natural extension of the Gelfand problem to two phase materials, where, in contrast to the classical Gelfand problem which utilizes single temperature approach, the state of the system is described by two different temperatures. We show that similar to the classical Gelfand problem the thermal explosion occurs exclusively due to the absence of stationary temperature distribution. We also show that the presence of inter-phase heat exchange delays a thermal explosion. Moreover, we prove that in the limit of infinite heat exchange between phases the problem of thermal explosion in two phase porous media reduces to the classical Gelfand problem with renormalized constants.