Thermal explosion, times to ignition and near-critical behaviour in uniform-temperature systems. Part 2.—Generalized dependences of rates on temperature and concentration

Abstract

This paper extends the range of earlier but narrower treatments of the time evolution of reactant temperature in exothermic, homogeneous systems near to criticality. It considers (i) generalized temperature dependences of the reaction rate coefficient, ƒ(θ), and (ii) generalized concentration dependences of the reaction rate, g(w). Here θ is the dimensionless temperature excess and w=(c/c₀) is the fractional concentration.

When reactant consumption can be neglected, the temperature dependence of the rate of reaction is represented by k∝ exp (–E/RT_a)ƒ(θ). The Arrhenius form implies ƒ(θ)= exp [θ/(1 +εθ)], and the case of ε→ 0 corresponds to the exponential approximation. Time to ignition has the form [graphic omitted], where M′={2π²/[ƒ(θ₀)ƒ_θθ(θ₀)]}^1/2. This correctly generates the value M′= [graphic omitted]2π/e for the exponential approximation.

When transition is reached, our results show that M′→∞; therefore t_ign lengthens without limit even for a fixed degree of supercriticality.

A similar systematic approach allows the influence of reactant consumption on the critical value of the Semenov number to be assessed for a very wide variety of reaction rate expressions (and not merely first or nth order). We find the general form: ψ_cr=ψ₀[1 + 2.946(ƒ₀/ƒ_θθ)^1/3(g_w/B)^2/3]. (For an nth-order deceleratory reaction g_w=n.)