## Abstract

Thermal explosions occur when reactions evolve heat too rapidly for a stable balance between heat production and heat loss to be preserved. Even when reactions are kinetically simple, and obey the Arrhenius equation, the differential equations for heat balance and reactant consumption cannot be solved explicitly to express temperatures and concentrations as functions of time unless strong simplifications are made. This difficulty exists for the spatially uniform (Semenov) as well as for the distributed temperature (Frank-Kamenetskii) case. Solutions become possible if strong simplifications are made (no reactant consumption; approximations to the Arrhenius term). Ignition is then represented by the threshold at which stationary states disappear. A single parameter (see appendix for definitions and symbols) summarizes the criteria for ignition. In the spatially uniform case, the Semenov parameter $\psi $ = $VQEAc_{0}^{m}$ e$^{-E/RT_{\text{a}}}/\chi SRT_{\text{a}}^{2}$ has the critical value e$^{-1}$. In the distributed temperature case, the Frank-Kamenetskii parameter $\delta $ = $d^{2}QEAc_{0}^{m}$ e$^{-E/RT_{\text{a}}}/\kappa RT_{\text{a}}^{2}$ has critical values that depend on the geometry. The present paper is concerned especially (1) with the relation between the spatially uniform and distributed temperature cases, (2) with the effect of approximations to the Arrhenius term, and (3) with Frank-Kamenetskii's treatment of ignition under conditions where reactant consumption is not completely ignored. Quadratic approximations to the Arrhenius term, i.e. e$^{-E/RT}$ $\approx $ e$^{-E/RT_{\text{a}}}$($p$ + $q\theta $ + $r\theta ^{2}$), are exploited. They offer explicit analytical solutions where the traditional exponential approximation e$^{-E/RT}$ = e$^{-E/RT_{\text{a}}}$ e$^{\theta /(1+\epsilon \theta)}$ $\approx $ e$^{-E/RT_{\text{a}}}$ e$^{\theta}$ leads to far less tractable equations. In the uniform temperature case, when reactant consumption is neglected, both approximations to the Arrhenius term yield a natural definition of the induction time before explosion, and Frank-Kamenetskii's derivation of the limiting form for marginally supercritical zero-order conditions is illuminated by the comparison. If the fact that quotient $\epsilon $ = $RT_{\text{a}}$/$E$ though usually small is not zero ($\epsilon $ = 0 is effectively implied by the exponential and the quadratic approximations), is taken into account, important qualitative differences are made in principle. One of these is that temperatures no longer rise without limit (except in adiabatic conditions, when they eventually rise linearly with time). This means there is no possibility of following Frank-Kamenetskii's procedure of defining induction periods as the location of a vertical asymptote in a temperature-time history. In practice, however, most observable effects are small until $\epsilon $ approaches the value $\frac{1}{4}$. (At this point, criticality disappears.) The critical value of the Semenov criterion $\psi $ for explosion, and the critical value of the stationary temperature excess now satisfy the relations: e$\psi $$_{\text{cr}}$ = 1 + $\epsilon $ + $O(\epsilon ^{2})$, $\theta $$_{\text{cr}}$ = 1 + 2$\epsilon $ + $O(\epsilon ^{2})$. At these values there is a large jump in the stationary thermal regime. For the distributed temperature case, with a temperature step at the boundary, but for which reactant consumption is again ignored, numerical investigation by variational techniques of the critical value for the Frank-Kamenetskii parameter $\delta $ shows a similar dependence on $\epsilon $ to that above and this is so for all Biot numbers. It cannot be as concisely expressed, since the equations are not soluble in closed form, but the shifts in criticality and in centre temperature will show similar dependences on $\epsilon $: an illustrative example is given for a sphere with infinite Biot number. The parameter $\delta $$_{\text{cr}}$ can still be readily calculated accurately under conditions of low Biot number, and there the identity $\delta /Bi$ = $\psi Sd/V$ both indicates that $\delta $$_{\text{cr}}$/$Bi$ rather than $\delta $$_{\text{cr}}$ is the useful parameter to tabulate and suggests an easy route to approximate values for $\delta $$_{\text{cr}}$ under conditions of low heat transfer at the boundary ($Bi$ $\rightarrow $ 0; Semenov case). When reactant consumption is not ignored, the problem is again qualitatively different, and the quantitative differences are more likely to be significant in practice than those arising from finite $E$. Temperatures are now bounded whether or not the Arrhenius term is approximated and remain bounded even in the adiabatic case. Once again, induction periods may not be logically defined on the existing bases. This problem can be evaded in the uniform temperature case by a procedure (Frank-Kamenetskii 1945) that identifies an 'equivalent zero-order' system and so enables a one-step transformation to be made from systems for which induction periods and critical parameters may be defined in an obvious manner to systems for which the definition is elusive. The argument is again easier to follow when a quadratic approximation is employed, and the particular quadratic having the same value, slope and curvature as exp $\theta $ at $\theta $ = 1 leads to the expressions $\frac{t_{\text{i}}}{t_{\text{ad}}}$ = $\pi $ $\surd \frac{2}{e\omega}$ - $\frac{2}{\text{e}}$ + $O(\omega ^{\frac{1}{2}}$), e$\psi _{\text{cr}}^{-1}$ $\approx $ 1 - 2.70($m/B)^{\frac{2}{3}}$ + 1.33($m/B)$. The treatments of Thomas (1961) and of Kassoy & Linan (1977) inevitably lead to the same type of result. All these approaches are unsatisfactory and the errors resulting from the various approximations in them are hard to assess. In the distributed temperature case, even these approaches fail, because induction periods can no longer be calculated explicitly for marginally super-critical conditions.

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