From One-step to Chain-branching Premixed Flame Asymptotics

The one-step model for chemistry has a successful history in providing a paradigm for describing many kinds of flame structure. If the activation energy is considered to be a large parameter, then an asymptotic picture emerges that is often simple enough to provide explicit solutions which describe such phenomena as structure, stability, ignition, and extinction. Qualitatively correct descriptions of many different kinds of burning have been achieved as a result. Nevertheless, the model must have its limitations. It has been found, for example, that the Zeldovich number required to describe stable flame balls, with heat loss in the burned gas, needs to be about three times the typical value found in hydrocarbon reactions. We examine the Zeldovich kinetic model, which has one autocatalytic reaction and one completion reaction, still using an asymptotic approach based on a large activation energy of the autocatalytic reaction; the exothermic completion reaction is taken to have a constant rate constant. Solutions describing planar flames and flame balls, found using this model, have the same overall qualitative features as those found using one-step chemistry, but these features also persist down to realistic Zeldovich numbers. In cases where the flame temperature in these solutions is arbitrarily close to an inhomogeneous crossover temperature, below which any chain-branching growth of radicals is suppressed, there can be an arbitrarily strong sensitivity to changes in temperature. In a broad sense, this is equivalent to an arbitrarily large Zeldovich number in the one-step model for chemistry. This offers some justification for continuing to use the one-step model, even in extreme cases where the actual Zeldovich number would seem to be unphysically large.