Thermal explosion theory for a slab with time-periodic surface temperature variation

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 #c(e, w) as a function of amplitude e and frequency w/2n of the surface temperature oscillation. We have shown that with values of this parameter, such that 0 < e, w) steady temperature oscillations can be maintained within the slab, but that 8 > 8c{e, w) 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, o) 27i, and the parameter range 0 < e ^ 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 8,the result shows th at stable oscillations exist for all oj, provided 8^ 8C e~e (8C = 0.878), this represents a lower bound to the stability surface in (8, e, oj) space, (ii) Perturbation theory, for small amplitude, has been used to determine the critical parameter in the form oj) — 8C + e1 281(oj) + ... with £x(w) a function of frequency. Comparison with the exact numerical solution shows that this gives values which differ by less than about 0-2 % for oj^ 2k and 0 ^ e^ 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 e, oj such solution could be found for 8 sufficiently small; it is shown th at 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 £c(e, oj) versus e for oj = 2 4n, 8tt and the range 0 ^ ^ 4.