The dependence of criticality on activation energy when reactant consumption is neglected

Abstract

In the study of thermal ignition, when reactant consumption is neglected, criticality is usually taken as the point at which there is a large jump in the maximum of the stationary-state temperature attained, for a small change in the external parameters. If the stationary-state equations are taken in their exact form, without the Frank-Kamenetskii approximationε ≡ RT a/ E≈ 0, criticality vanishes for ε ⩾ 0.25, that is, for low activation energies E ⩽ 4 RT a, as in the uniform temperature case. However, unlike the latter case, in the distributed temperature case, as considered here, it is not yet been determined that criticality exists for all activation energies E ⩾ 4 RT a. In this paper a model is introduced for which the value of ε, say ε 0, at which criticality vanishes can be determined precisely for the geometries of an infinite slab, infinite circular cylinder, and a sphere. This model is not exactly that given for the Arrhenius form, but very close to it. That is, exp ( θ/(1 + εθ) is replaced by exp ( θ/(1 + εθ m)), but it does not make the crucial approximation ε ≈ 0 as many previous analytic treatments have done. For these three shapes it is shown that, at infinite Biot number [Display omitted] The second and third columns list the corresponding value of the critical value of the Frank-Kamenetskii parameter δ and the maximum dimensionless temperature rise, respectively. For finite values of the Biot number Bi, ε 0 is a decreasing function of Bi, from ε 0 = 0.25 for Bi small and approaching the value given here from above as Bi becomes large. This is calculated explicitly in the simplest of these three shapes, that is, the infinite circular cylinder, in this paper.