Theory of Thermal Explosions with Simultaneous Parallel Reactions. I. Foundations and the One-Dimensional Case

Many real, exothermic systems involve more than one simultaneous reaction. Even when they are chemically independent, interactions must arise through their several responses to the collective generation of heat. A simple and unifying approach is possible to the behaviour of such systems below and up to criticality. It introduces a communal activation energy E as the basis for dimensionless quantities (θ, δ, ɛ and so on) but otherwise involves only familiar ideas from basic thermal explosion theory. The definition of E is E = RT^2d(ln Z)/dT,where Z = Σ Z_i. Here, Z is the rate of energy release per unit volume (the power density) by the whole system and Z_i is the contribution of the constituent i. This enables us to define and use the conventional dimensionless parameter δ for the whole system and for its constituent reactions. We illustrate affairs by considering a pair of concurrent, exothermic reactions; heat is transferred solely by conduction towards the faces (temperature T_a) of an infinite slab of thickness 2a and conductivity kappa. For a constituent reaction (i = 1, 2 here) δ_i = (Ea^2/kappa RT^2_a)Z_i(T_a) and for the whole system δ = δ_1 + δ_2 (+...) for two (or more) reactions. We find that the condition δ > δ_cr guarantees instability, where δ_cr is always less than 0.878. The bounds 0.65 < δ_cr < 0.878 are good enough for a substantial range of relative sizes of activation energy 0.2 < E_1/E_2 < 5. We also pursue the problem numerically and present solutions for critical δ and critical central temperature excess over the whole composition range for a pair of simultaneous exothermic reactions.