Exothermic systems with diminishing reaction rates: Mathematical analysis

Abstract

There are many practical instances of materials in which the rate of an exothermic reaction decays with time t like ( t + t pr) − α, t pr ≥ 0, 0 ≤ α < 1. The origin of such systems is described in Boddington et al. in Proc. Royal. Soc. (1980). For such systems critical behavior is characterized easily by the conditions separating the two types of behavior: supercritical ones which reach infinite temperatures in a finite time and subcritical ones which pass through a maximum and then (asymptotically with time) fall to zero. The problem of determining the critical conditions does not, for all values of the exponent α, admit to analytical solution even for the uniform temperature case (Semenov model). The results of this paper give an asymptotic estimate for the critical values of a dimensionless parameter δ 1 in the expression for the rate of heat evolution δ 1 cr ∼ δ 0 cr (1 − 1 2 αln α) for small α, where δ 0 cr is the critical value of the usual Frank-Kamenetskii parameter (for α = 0) for the same geometry and Biot number and includes the uniform temperature case. Here τ pr will be the dimensionless parameter corresponding to the time τ pr in the rate equation. Although the true form of δ 1 cr versus α cannot be determined analytically, it can be shown that for τ pr = 0, δ 1 cr = 0 at α = 1; and for 0 < τ pr, δ 1 cr as a function of α has only one maximum and δ 1 cr → 0 + as α → ∞. Further, for the uniform temperature case it is shown that when τ pr = 0, δ 1 cr ∼ 1 − α near α = 1. The analytical results are supported by the computed results reported for the sphere with infinite Biot number for the case τ pr = 0. This shows that the maximum of δ 1 cr occurs for α ⋍ 0.04 and is close to the classical value for α = 0.