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 timetaccording to rate∝t─αor, more generally, rate∝(t+tpr)─α. Heretpris a ‘prior reaction’ time and the exponentαlies 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 dθ/ dτ=ψ1eθ/τα─θ. For any given value ofαthe behaviour of the system is solely determined by the value ofψ1, the role of which is analogous to that played by the Semenov numberψunder zero-order conditions (α= 0). In terms of real variables, the Newtonian cooling timetNemerges as the natural yardstick for time, andτ=t/tN. The parameterψ1represents a dimensionless rate of heat release of the system after one Newtonian time-scale has elapsed (i. e. atτ= 1), andθhas its usual meaning as a dimensionless temperature excess. The dependences of critical values ofψ1onαand 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 inψ1(disappearance of criticality) for non-zero values ofRTa/E.

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