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 ($\theta, \delta, \epsilon$ and so on) but otherwise involves only familiar ideas from basic thermal explosion theory. The definition of E is $E = RT^2d(ln Z)/dT,\text{where} Z = \Sigma Z_i$. 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 $\delta$ 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) $\delta_i = (Ea^2/\kappa RT^2_a)Z_i(T_a)$ and for the whole system $\delta = \delta_1 + \delta_2$ (+...) for two (or more) reactions. We find that the condition $\delta > \delta_{cr}$ guarantees instability, where $\delta_{cr}$ is always less than 0.878. The bounds 0.65 < $\delta_{cr}$ < 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 $\delta$ and critical central temperature excess over the whole composition range for a pair of simultaneous exothermic reactions.