Accounting for reactant consumption in the thermal explosion problem. Part I: Mathematical foundation

Abstract

New definitions for the criticality conditions of the thermal explosion problem are founded on the mathematical behavior of the governing equations. The paper deals with uniform temperature and concentration (Semenov problem). It is well known that the results can be applied to the distributed temperature and concentration case by the use of correction factors. It is shown that criticality can be defined in the temperature-time plane as accepted by most authors. However, using our definitions of criticality in the temperature-concentration plane confirms the previous findings of Adler and Enig. They showed that the classically defined critical state in the temperature-time plane is always a subcritical state in the temperature-concentration plane. At the same time, the critical state in the temperature-concentration plane is always a supercritical state in the temperature-time plane. However, the critical state in the temperature-concentration plane is in agreement with that in the Semenov number (Ψ)-temperature plane. It is shown that the critical states in all planes coincide only when n = 0 or B = ∞ and agree with the well known results neglecting reactant consumption. The difference between the critical and ignition temperatures is discussed. It is shown that as B approaches infinity, the solution for Φ as a function of τ for any value of n approaches the solution for n = 0. Hence for B = ∞, substituting for n = 0 in Alder and Enig results produces the classical Semenov result. This resolves the objections expressed against these results before. At the same time the locus of the critical states for n = 0, with finite B is determined. The effect of the degree of reaction on the induction time for adiabatic systems is demonstrated. The conditions required for ignition of subcritical systems is demonstrated. The conditions required for ignition of subcritical systems are discussed as well as the effect of initial conditions on criticality.