Thermal explosions, critically and transition in systems with variable thermal conductivity. Distributed temperatures

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Abstract
The conductive theory of thermal explosion introduced by Frank-Kamenetskii neglects any temperature dependence of thermal conductivity κ. In certain circumstances this is too simple a view, and the present paper evaluates the consequences of assuming a realistic dependence on temperature. Two important forms are considered linear: κ/κ0= 1 +α(T–Ta)= 1 + aθ=h1(θ) square root: κ/κ0=(T/Ta)1/2=(1 +εθ)=h2(θ) In these equations θ=(T–Ta)/(RT2 a/E) denotes the dimensionless excess temperature and ε=RTa/E reflects the ambient temperature Ta. The heat-balance equation in the stationary state becomes div [h(θ) grad θ]+δ exp [θ/(1 +εθ)]= 0, where δ is the conventional dimensionless measure of the reaction rate. This equation has been solved numerically for the three ‘class A’ geometries (sphere, infinite cylinder and infinite slab) for ε≠ 0 (Arrhenius form, general case) and for ε→ 0 (exponential approximation) subject to the condition θ= 0 at the boundary. The following features have been established: (a) temperature-position profiles for subcritical and critical circumstances for different a and ε, (b) critical values of the parameters δ and θ0 and (c) the disappearance of criticality at high temperatures or low activation energies (transition). The results are compared with their simpler prototypes for the Semenov uniform-temperature case when the heat-transfer coefficient χ is not constant but depends upon temperature.

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