Unifying approaches to the stationary-state theory of thermal explosion. Part 2

Abstract

A new route to classical stationary-state thermal-explosion theory for the infinite slab, infinite cylinder and sphere is presented. The results arise in terms of a single parameter x, and are tabulated in such a way that inter-relationships are highlighted. The key quantity δ(the dimensionless rate of heat production at ambient temperature) is related to x by a quadratic equation with coefficients appropriate to each shape. Criticality emerges systematically for the three geometries because the expressions have natural maxima (at x=x_cr). Exothermic systems have two solutions for x for any small value of δ(<δ_cr). One root lies in the range 0 < x < x_cr, and corresponds to stable stationary states; the second x > x_cr, yields unstable profiles. Endothermic systems are characterized by unique, negative values for x, in the range –1 < x < 0.

The formulae are neat and easily manipulated. Explicit relationships between δ and the central temperature excess θ₀, for example, can be generalized for all three shapes (slab, cylinder, sphere with j= 0, 1, 2, respectively) as [graphic omitted]. Critical values and limiting forms under near-isothermal conditions for the various quantities may be set out as follows: [graphic omitted]

The physical significance of the parameter x is that it measures the excess over unity of the effectiveness factor η: this relationship is exact for the cylinder and close for the slab and sphere.