Phase-plane analysis of criticality for thermal explosions with reactant consumption

Abstract

The effect of reactant consumption on the critical conditions for thermal ignition can only be determined approximately, since the differential equations are intractable. For the uniform tem perature case and with first order reactions, the dependence of the critical value of the parameter a from the criterion of a point of inflexion in the phase plane (see tex t for explanation of symbols), on the dimensionless adiabatic tem perature rise B is shown rigorously to be a continuous, monotonic decreasing function of B -1 . This is in line with the approximate methods of this paper and previous authors. Bounds of the form a cr [B -1 ) = a cr (0) (1-0(B -q )), 1/2 ⩽ q ⩽2/3 are rigorously established (see (v) of §4). Also it is shown th at this definition of criticality only exists for RT a / E ⩽ 1/4 - B -1 By using backwards integration in two steps from the point at which the trajectory first touches the locus of points of inflexion, it is first shown that a very good linear approximation (with the Frank-Kamenetskii approximations for the reaction rate) is a cr [B -1 )= 1/3 e 2 (1-4 B -1 ), and then this is improved to give the better estimate (see figure 5). This last estimate gives the best agreement with the computer solution, and appears to be superior in overall fit to earlier estimates.