Modelling thermal runaway and criticality in systems with diminishing reaction rates: the uniform temperature (Semenov) approximation

Abstract

Various complex exothermic oxidations of considerable technical importance can be represented by an empirical rate-law in which the isothermal reaction rate diminishes with the elapsed time t according to rate ∝t ─ α or, more generally, rate ∝ ( t + t pr ) ─ α . Here t pr is a ‘prior reaction’ time and the exponent α lies between 0 and 1. We have computed the generalized behaviour of such a system with a near-Arrhenius dependence of reaction rate on temperature under Semenov conditions, i. e. uniform internal temperature. Temperature-time histories fall into three categories. In subcritical behaviour, temperatures pass through a finite maximum and then decay asymptotically to zero. In supercritical behaviour, temperatures rise steeply to infinite values. Critical behaviour is the frontier between these: a common temperature-time stem from which the other temperature histories diverge and which itself tends to infinite values at infinite times. The rate equation can be written in a general dimensionless form d θ / d τ = ψ 1 e θ / τ α ─ θ . For any given value of α the behaviour of the system is solely determined by the value of ψ 1 , the role of which is analogous to that played by the Semenov number ψ under zero-order conditions ( α = 0). In terms of real variables, the Newtonian cooling time t N emerges as the natural yardstick for time, and τ = t / t N . The parameter ψ 1 represents a dimensionless rate of heat release of the system after one Newtonian time-scale has elapsed (i. e. at τ = 1), and θ has its usual meaning as a dimensionless temperature excess. The dependences of critical values of ψ 1 on α and times to ignition are reported. The model reproduces many features of the distributed temperature case. It also allows the investigation of transition from discontinuous to continuous responses to slow changes in ψ 1 (disappearance of criticality) for non-zero values of RT a / E .