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For exothermic reactions obeying the Arrhenius equation in circumstances in which heat flow is purely conductive, critical conditions for thermal explosion are satisfied when a dimensionless group $\delta \equiv a^{2}QEAc\_{0}^{\text{m}}$e$^{-E/RT\_{\text{a}}}/\kappa RT\_{\text{a}}^{2}$ attains a critical value $\delta \_{\text{cr}}=ca$ 0.88, 2 or $ca$. 3.32 for the infinite slab, infinite cylinder or sphere. This result (Frank-Kamenetskii 1938) of stationary state treatment is appropriate so long as activation energies are not too low or ambient temperatures are not too high: $E\gg RT\_{\text{a}}$ (or $\epsilon \equiv RT\_{\text{a}}/E\ll $1). Criticality persists for $E$ decreasing or $T$ increasing so long as $\epsilon $ is smaller than a transitional value $\epsilon \_{\text{tr}}$. At this transitional value $\epsilon \_{\text{tr}}$, only continuous behaviour is possible: ignition phenomena disappear. Accurate transitional values for the reduced ambient temperature $\epsilon $, for the critical value of $\delta $, and for the reduced central temperature excess $\theta \_{\text{m}}$, have been calculated by quadrature for the infinite slab. The following results are obtained under Frank-Kamenetskii boundary conditions $(\alpha \rightarrow \infty)$ for two common temperature dependences. [Note: Table omitted. See the image of page 441 for this table.] The most striking features of these results are threefold. The first is the size of the central temperature excess $T\_{\text{m}}-T\_{\text{a}}$ achieved in the steady state: not only is $\theta \_{\text{m, tr}}$ nearly four times as great as the classical critical value (1.187 for $RT\_{\text{a}}/E\rightarrow $ 0), but the absolute magnitude of heating at the centre $T\_{\text{m}}-T\_{\text{a}}$ is comparable with $T\_{\text{a}}$, say 400 K or more. The second is the suddenness with which criticality is lost: temperature jumps that would be recognized experimentally as explosive persist to within 0.025 of $\epsilon \_{\text{tr}}$. The third is the utility of the Semenov approximation: no problem in thermal explosion theory should be investigated until its main features have been sought for under Semenov assumptions. The shape of the critical locus (the bifurcation set), a curve in three dimensional space $(\delta,\epsilon,\theta \_{\text{m}})$, is illustrated for two rate laws and for Frank-Kamenetskii and Semenov extremes. Both extremes generate the same general dependence on $\epsilon $: both correspond to a cusp catastrophe with the cusp at the criticality-continuity transition point. The well known Semenov transitional value for Arrhenius kinetics, $\epsilon =\frac{1}{4}$ exactly, is a rigorous upper bound on $\epsilon _{\text{tr}}$ for arbitrary Biot number, including the Frank-Kamenetskii extreme, and similarly for other rate laws. [Note: Table omitted. See the image of page 442 for this table.] |
