The theory of thermal explosions with simultaneous parallel reactions. III. Disappearance of critical behaviour with one exothermic and one endothermic reaction

Abstract

Many real reaction systems involve more than one simultaneous reaction. In parts I and II the situation of two parallel reactions with an average activation energy was formulated and analysed in three class A geo­metries: an infinite slab, an infinite circular cylinder and a sphere, when both reactions were exothermic. This paper extends the treatment to the case when one reaction is exothermic and the other endothermic, such as occurs when a wet reactive substance oxidizes. The treatment in this, as in the earlier parts, makes the realistic simplifying assumptions that reactant consumption can be neglected, the two reactions are chemically independent, and that the stationary steady-state theory can be applied. Here, as in parts I and II, the simple geometries of the infinite slab, infinite circular cylinder, and sphere are all that are considered, with spatially distributed temperature and with infinite Biot number presumed on the boundary (Frank-Kamenetskii boundary conditions). (The behaviour described here will also hold for arbitrary shapes and finite Biot number.) The combination of one exothermic and one endothermic reaction, although not excluded in the previous parts (parts I and II), gives rise to the phenomenon of the ‘disappearance of criticality’ or ‘transition’ which also arises with a single reaction in the case of very low ( E < 4 RT a ) activation energy, where the Frank-Kamenetskii approximation is no longer valid. This transition, in both cases, is manifested by a change in the number of steady-state solutions, where solution is unique after critical behaviour has disappeared. By using a variational method it is possible to determine numerically the values of the parameters that characterize transition in the parameter space. If the activation energy of the endothermic reaction ( E 2 ) is greater than that of the exothermic reaction ( E 1 ) then critical behaviour can occur and the extent of the régime that exhibits critical behaviour decreases as a function of E 2 / E 1 . Monotonic behaviour is also observed in the occurrence of the other parameters used at transition. Analytical proof of all this behaviour has not yet been achieved but some general results are given to this end.