Thermal explosions and the disappearance of criticality at small activation energies: exact results for the slab

Abstract

For exothermic reactions obeying the Arrhenius equation in circumstances in which heat flow is purely conductive, critical conditions for thermal explosion are satisfied when a dimensionless group = QE »/ kR T attains a critical value £Cr = ca. 0.88, 2 or ca. 3.32 for the infinite slab, infinite cylinder or sphere. This result (Frank-Kamenetskii 1938) of stationary state treatm ent is appropriate so long as activation energies are not too low or ambient temperatures are not too high: E > RTa (or e = R Ta/E 1). Criticality persists for E decreasing or T increasing so long as e is smaller than a transitional value etr. At this transitional value etr, only continuous behaviour is possible: ignition phenomena disappear. Accurate transitional values for the reduced ambient temperature e, for the critical value of 8, and for the reduced central temperature excess 0m, have been calculated by quadrature for the infinite slab. The following results are obtained under Frank-Kamenetskii boundary conditions (α ->∞) for two common temperature dependences.