Thermal-runaway approximation for ignition times of branched-chain explosions

An asymptotic analysis for high activation energy of the branching step is developed for predicting ignition times of branched-chain explosions on the basis of a criterion of thermal runaway in homogeneous, isobaric, adiabatic systems. The chemistry includes an initiation step, a branching step, and a recombination step and leads to a nonlinear second-order ordinary differential equation for the temperature as a function of time under adiabatic conditions. One or both of the branching and recombination steps must be exothermic, whereas the initiation step can be endothermic or exothermic. A two-term expansion is derived in a small parameter representing the ratio of the initiation rate to the branching rate, yielding explicit expressions for the ignition time. At leading order, the ignition time is found to be inversely proportional to the net branching rate, the proportionality constant being the logarithm of the ratio of the branching rate to the initiation rate multiplied by energetic and rate parameters associated with branching and recombination. Resulting ignition times are shown to correspond closely with those calculated by numerical integration of the full equations on the basis of a temperature-inflection criterion, except near crossover, where the branching rate equals the recombination rate. Application of the theory to fuel-rich hydrogen-oxygen systems demonstrates good agreement.