Exothermic systems with diminishing reaction rates: temperature evolution, criticality and spontaneous ignition in the sphere

The behaviour of an exothermic, spherical mass, in which reaction rate diminishes with time t according to the law $\mathrm{rate} \infty (t+t{_pr})^{-\alpha}, t_{pr} \geqslant 0, 0 \leqslant \alpha < 1,$ is studied by numerical computation. (Analytical solutions are not attainable.) This law is a useful empirical representation of various complex systems of great practical importance: coal, sawdust, wool, polypropylene and fish meal. Central-temperature evolutions fall into two families: supercritical ones reach infinite temperatures in a finite time; subcritical ones pass through a maximum and then fall to zero. These diverge from a common stem (corresponding to criticality) which tends to infinite temperatures in an infinite time. Critical conditions and ignition times are reported: the important parameters are the decay exponent $\alpha$, the activation energy E and a dimensionless rate of heat-evolution $\delta_1$, analogous to the constant $\delta$ of Frank-Kamenetskii. Attainment of the value $\theta^*_0$ = 1.607, which is the classical critical value of the reduced central temperature excess in the stationary state, does not have special significance; larger values of $\theta_0$ are attained in subcritical systems. The evolving temperature-position profiles (except in two special epochs) closely match those appropriate to stationary states having the same value of $\theta_0$. Both stable and unstable stationary-state solutions are generated, even in the course of subcritical, time-dependent behaviour. This behaviour is not in general quasi-stationary, although there are two major regions in which the matched steady-state profiles (both stable and unstable) correspond to values of $\delta$ that are close to the instantaneous values of $\delta$ for the evolving system. It is shown that there are circumstances in which a region away from the centre may reach a maximum temperature and subsequently cool for a long time, but in which ignition ultimately occurs. A simple, approximate, analytical model is advanced to explain the main features of the observed behaviour.