Phase-plane analysis of multiplicity and thermal runaway in a hollow sphere subject to a constant internal radiative power source

Abstract

In this paperwe present an analysis of the steady-state solutions of a classical equation describing thermal runaway in spherical symmetrymodified to study the hazard presented by a fractured optical fibre in a flammable atmosphere. Exothermic reaction is coupled with a radiation source that imposes a temperature gradient at the edge of an inert spherical core within the reaction zone. Reactant consumption is neglected. The governing equation is integrated directly and the steady-state profiles are examined with the Frank-Kamenetskii parameterbas the bifurcation parameter. Gross multiplicity of steady states is seen to be a feature for certain parameter values. Subsequentlya phase-plane approach is used which renders some features of the solutions more accessible. In particularthe point at which the system ceases to exhibit criticalitywhich is important from the hazard assessment point of viewis easily identified