## Abstract

The problem of the onset of thermal explosion in a slab subject to time-periodic surface temperature variation has been investigated where the slab is symmetrically heated by an exothermic zero-order chemical reaction. The main purpose of the investigation was to obtain the critical Frank-Kamenetskii parameter $\delta_c$($\epsilon$, $\omega$) as a function of amplitude $\epsilon$ and frequency $\omega$/2$\pi$ of the surface temperature oscillation. We have shown that with values of this parameter, $\delta$, such that 0 < $\delta \leqslant \delta_c$($\epsilon$, $\omega$) steady temperature oscillations can be maintained within the slab, but that $\delta$ > $\delta_c$($\epsilon$, $\omega$) must lead to thermal explosion. For a period of oscillation of 24 h and suitable values for the thermal diffusivity and half-width of the slab, $\omega \thickapprox$ 2$\pi$, and the parameter range 0 $\leqslant \epsilon \leqslant$ 4 covers ambient temperature fluctuations likely to be encountered by hazardous materials, such as in the hold of a ship in tropical seas. The problem has been examined in three different ways. (i) A comparison theorem for partial differential equations has been used to determine an analytical bound on $\delta$, the result shows that stable oscillations exist for all $\omega$, provided $\delta \leqslant \delta_c$ e$^{-\epsilon}$ ($\delta_c$ = 0.878), this represents a lower bound to the stability surface in ($\delta, \epsilon, \omega$) space. (ii) Perturbation theory, for small amplitude, has been used to determine the critical parameter in the form $\delta_c$($\epsilon, \omega$) = $\delta_c$ + $\epsilon^2\delta_1(\omega$) + ... with $\delta_1(\omega$) a function of frequency. Comparison with the exact numerical solution shows that this gives values which differ by less than about 0.2% for $\omega \geqslant$ 2$\pi$ and 0 $\leqslant \epsilon \leqslant$ 1. (iii) The energy conservation equation has been solved numerically over a rectangular mesh representing the half-width of the slab and one period of steady oscillation. For given $\epsilon, \omega$ such solution could be found for $\delta$ sufficiently small; it is shown that the breakdown of the numerical process is associated with criticality, allowing the limiting parameter to be determined. This method has been used to obtain curves of $\delta_c$($\epsilon, \omega$) versus $\epsilon$ for $\omega$ = 2$\pi$, 4$\pi$, 8$\pi$ and the range 0 $\leqslant \epsilon \leqslant$ 4.

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