Unifying approaches to the stationary-state theory of thermal explosion. Part 1

Abstract

In two linked papers alternative approaches to classical conductive thermal explosion theory are developed. New results are presented (together with known features) in tabular form for convenient reference. Interrelationships between different geometries are displayed for the first time in a simple way. Among the quantities that can be readily extracted are: (i) expressions for temperature–position profiles θ(ρ), for central temperature excesses (θ₀) and for temperature gradients at the surface (Γ) in stable stationary states, (ii) the same information at criticality and, with less precision, in unstable states, (iii) the same information for endothermic systems, (iv) limiting forms appropriate to small departures from isothermal behaviour, (v) effectiveness factors η(a measure of the increase in reaction rate caused by non-isothermal effects) and expressions for errors in activation energies ΔE/E, (vi) algebraic relationships between the dimensionless rate of heat production (Frank-Kamenetskii's δ) and θ₀.

Part 1 shows how a single parameter l can be used to yield very simple expressions for all the major quantities of interest for the infinite slab (j= 0), the infinite cylinder (j= 1) and for the sphere (j= 2). For exothermic reactions, the range 1 > l > ½ corresponds to stable states, the value l=½ to criticality and the range ½ > l > 0 to unstable solutions. Thermoneutral reactions have l= 1, and endothermic reactions have l > 1.

Examples of simple general formulae that emerge are: [graphic omitted].

Wake's problem (an exponential form for the temperature-dependent thermal conductivity) can be slotted into an identical framework.