Detonation ignition from a temperature gradient for a two-step chain-branching kinetics model

The evolution from a linear temperature gradient to a detonation is investigated for combustible materials whose chemistry is governed by chain-branching kinetics, using a combination of high-activation-temperature asymptotics and numerical simulations. A two-step chemical model is used, which captures the main properties of detonations in chain-branching fuels. The first step is a thermally neutral induction time, representing chain initiation and branching, which has a temperature-sensitive Arrhenius form of the reaction rate. At the end of the induction time is a transition point where the fuel is instantaneously converted into chain-radicals. The second step is the main exothermic reaction, representing chain termination, assumed to be temperature insensitive. Emphasis is on comparing and contrasting the results with previous studies that used simple one-step kinetics. It is shown that the largest temperature gradient for which a detonation can be successfully ignited depends on the heat release rate of the main reaction. The slower the heat release compared to the initial induction time, the shallower the gradient has to be for successful ignition. For example, when the rate of heat release is moderate or slow on the initial induction time scale, it was found that the path of the transition point marking the end of the induction stage should move supersonically, in which case its speed is determined only by the initial temperature gradient. For steeper gradients such that the transition point propagates subsonically from the outset, the rate of heat release must be very high for a detonation to be ignited. Detonation ignition for the two-step case apparently does not involve the formation of secondary shocks, unlike some cases when one-step kinetics is used.