Thermal Explosion and Times to Ignition in Systems with Distributed Temperatures. IV. Behaviour at Points of Criticality and Transition

The conductive theory of thermal explosion in systems with distributed temperatures expresses conditions for the breakdown of stability as the critical value of a single dimensionless group δ, Frank-Kamenetskii's parameter. For values of δ above critical, there is thermal runaway; for values of δ less than critical, the system is quiescent. Analytic solutions for excess temperatures as a function of position and time are not generally available for these systems, but can be found when δ is close to the critical value, by using the techniques of asymptotic analysis. In this paper we consider two related problems. First, the approach of temperature to the critical, steady-state profile is found when δ takes precisely its critical value. The second concerns systems where criticality is just being lost through the coalescence of two critical points. It establishes how the new `transitional' steady-state profile is then approached. In both circumstances we neglect reactant consumption. We allow for an arbitrary dependence of reaction rate on temperature, and an arbitrary Biot number in the boundary conditions. The solutions are found in terms of leading-order descriptions of the behaviour of the reduced temperature-excess as it moves towards the steady state profile.