Thermal explosion and times-to-ignition in systems with distributed temperatures Il. The influence of reactant consumption

Abstract

In part I, we studied thermal explosions in systems with distributed temperatures, and more particularly the evolution of reactant temperatures in time. Reactant consumption, however, was ignored. Here we consider the influence of reactant consumption for the whole range of Biot number from the Semenov extreme (β = 0) to the Frank-Kamenetskii extreme (β → ∞). We concentrate on numerical results for the three simplest geometries (infinite slab, infinite cylinder and sphere), but our route is valid for arbitrary geometry. Earlier treatments in this field are extended by considering not only generalized temperature-dependences of reaction rate, f (θ), but also generalized concentration-dependences of reaction rate, g ( w ). An important influence of reactant consumption is to modify the critical value for the Frank-Kamenetskii parameter δ, δ cr /δ 0 = 1 + ϕ( g w /B) ⅔ , where the coefficient ϕ depends on the geometry, g w represents an effective order of reaction (to be evaluated at initial temperature and concentration), B is a dimensionless adiabatic temperature rise and δ 0 is the critical value of δ when reactant consumption is ignored ( B → ∞). The value of the constant ϕ is readily calculated from certain simple integrals. Its dependence on geometry and its variation with Biot number β are presented and discussed. It is shown that for β → ∞, reactant consumption has a smaller effect on the critical behaviour than in the case where β = 0, and the same trend is found for intermediate values of β. The role of diffusion of reactant is examined. It is found that the behaviour, as gauged by the leading-order terms of an asymptotic analysis, falls into one of two classes, so that diffusion either may be entirely ignored or is so rapid that concentrations are uniform. Results for the two classes show a remarkably close correspondence and permit a uniform treatment. Results are given for the time taken by a system either to ignite or, otherwise, to reach a maximum temperature. For important limiting cases simple asymptotic formulae are given for these times; they constitute good approximations over wide ranges. For moderately supercritical systems the time to ignition, t ign , differs by little from the ignition time t ign (B → ∞) for the case of zero reactant consumption: t ign / t ign ( B → ∞) = 1+ G ( g w / B ) (δ-δ 0 /δ) -3/2 + O ( g w / B ) 2 . For the Arrhenius temperature dependence with large activation energy the constant G is about unity for all cases .