PHASE-PLANE ANALYSIS OF MULTIPLICITY AND THERMAL RUNAWAY IN A HOLLOW SPHERE SUBJECT TO A CONSTANT INTERNAL RADIATIVE POWER SOURCE

In this paper, we present an analysis of the steady-state solutions of a classical equation describing thermal runaway in spherical symmetry , modified to study the hazard presented by a fractured optical fibre in a flammable atmosphere. Exothermic reaction is coupled with a radia tion source that imposes a temperature gradient at the edge of an iner t spherical core within the reaction zone. Reactant consumption is neg lected. The governing equation is integrated directly and the steady-s tate profiles are. examined with the Frank-Kamenetskii parameter, delt a, as the bifurcation parameter. Gross multiplicity of steady states i s seen to be a feature for certain parameter values. Subsequently, a p hase-plane approach is used which renders some features of the solutio ns more accessible. In particular, the point at which the system cease s to exhibit criticality, which is important from the hazard assessmen t point of view, is easily identified.