Thermal explosions, criticality and the disappearance of criticality in systems with distributed temperatures. II. An asymptotic analysis of critically at the extremes of Biot number (( Bi ) -> 0, ( Bi ) -> ∞ for generalized reaction rate-laws

Numerical attacks on the problem of criticality in thermally igniting systems with generalized resistance to heat-transfer are expensive in computer time and particular to the cases studied. We show here how very general circumstances may be treated analytically by straightforward perturbation methods (asymptotic expansions). Asymptotic expressions of considerable precision can be made (i) starting from the Semenov extreme ((Bi) = 0) in terms of (Bi), and (ii) starting from the Frank-Kamenetskii extreme ((Bi) $\rightarrow \infty$) in terms of (Bi)$^{-1}$. They apply to any geometry and they are presented here for the infinite slab, the infinite cylinder and the sphere as expressions for critical values of the Frank-Kamenetskii or Semenov parameters and for critical centre-temperature $\theta$ in terms of Biot number. The importance of the method is that it can cope with any temperature-dependence of rate coefficient f($\theta$), and although the asymptotic expansions are strictly valid only at the extremes, for $\delta$ they together cover the whole range of Biot numbers. Numerical comparisons are given for the case f($\theta$) = e$^\theta$, for which the results are well known and for the case $f(\theta) = \exp[\theta/(1 + \epsilon\theta)]$, corresponding to Arrhenius kinetics.