Multiplicity in combustion wave behaviour for a model with competing exothermic reactions

Combustion, while being part of our everyday experience, is a complex phenomenon involving a multitude of reactions and dynamics. There is great value in understanding combustion through mathematical modelling, not only for its own sake, but also because of the applications of the developed model to combustion-based industrial processes or to that perennial environmental threat to populace, property and wildlife — the bushfire. In this work, we present a mathematical model of a combustive process where the combustion wave propagates through a fuel shaped like a rod, as it may be in self-propagating high temperature synthesis of metals. Said fuel is assumed to have chemical properties such that all reactions occurring during combustion may be lumped together as two different heat producing (exothermic) reactions. The model presented considers the idealised case of there being no heat lost in the reactions (i.e. the adiabatic limit). The two reactions, say A and B, compete for the available fuel. In one limit of the model parameters A will dominate and consume the majority of the fuel. At another extreme reaction B will dominate. In both cases our model reduces to the well-known “one-step” model. The one-step model allows a single wave front solution per system parameter. At intermediate parameter values no reaction dominates to the complete exclusion of the other. We argue there must exist some parameter value at which reactions A and B are consuming the fuel equally. Either side of this “cross-over” point one reaction begins to dominate. The “cross-over” concept enables the determination of the region in parameter space that allows a multiplicity of solutions. That is, for a given activation energy, there are multiple allowable combustion waves which typically have large differences in propagation speed. Lastly, the boundary for the transition from unique solutions to multiple solutions is calculated numerically and it is demonstrated that within this multiplicity region the flame front speed is an ‘S’–shaped function of the model parameters.