Thermal explosions with extensive reactant consumption: a new criterion for criticality

In classical treatments of thermal runaway, reactant consumption is commonly ignored. Criticality is readily identified as the disappearance of steady states. The distinction between subcritical and supercritical systems is sharp. For explosive behaviour, infinite excess temperatures $(\theta \rightarrow \infty)$ are reached in a finite time; non-explosive reaction is characterized by a low stationary state of self-heating $(\theta \leqslant 1)$. These are divided discontinuously by a critical value for the Semenov number $\psi$: $\psi = \frac{QVAc^m_0 e^{-E/RT}_\mathrm{a}}{\chi S(RT^2_\mathrm{a}/E)}; \psi_{\mathrm{cr}} = \mathrm{e}^{-1}; \theta_{\mathrm{cr}} = 1.$ When reactant consumption occurs, this sharpness disappears. Each temperature-time history evolves to a maximum temperature excess $\Delta T*$ or $\theta*$ before decaying back to ambient. A new criterion is presented here for thermal ignition in systems with extensive reactant consumption. It is based on the dependence of $\theta*$ on the initial reaction rate. We identify criticality with the maximum sensitivity of $\theta*$ to $\psi$, and look for the point of inflexion in the function $\theta*(\psi)$. Such a definition is closely related to that usedimplicitly by an experimentalist. The critical value of the Semenov number is derived as an integral equation. This is solved numerically to yield $\psi_{\mathrm{cr}}$ as a function of the reaction exothermicity $B$. The calculations lead to values for $\psi_{\mathrm{cr}}$ up to 2 or 3 times greater than the classical value found when reactant consumption is ignored. There is a smooth transition between new and old values close to the classical limit (at very large $B$). Our results coincide with the predictions from asymptotic analysis for $B$ greater than ca. 50. They peel away significantly as $B$ diminishes through the range $50 > B > 4$, a range that can be met in practice for dilute gases or solid masses of low reactivity.