Behaviour of combustion waves in one-step and two-step models

Premixed combustion waves in models with various reaction mechanisms have drawn the attention of researchers for a long period of time. Such models are typically described by parabolic reaction-diffusion systems of partial differential equations relating to energy conservation and the conservation of the chemical species involved in the various steps of the combustion reaction. Reaction-diffusion systems corresponding to combustion are distinguished by the strong nonlinear dependence of the reaction rates on temperature, which are often modelled by the Arrhenius law or some large-exponent power law. Advances in computational power have allowed detailed numerical study of reaction mechanisms involving a variety of different steps. While these investigations have been useful in providing some quantitative predictions for observed phenomena, there is still uncertainty about the reliability of these complex models when applied to the prediction of the generic behaviour of flames. To avoid this uncertainty considerable effort has been put into the study of reduced reaction mechanisms that involve only one or two steps. One-step models have the advantage of being relatively simple, allowing analytical investigations into phenomena such as ignition and extinction of flames, which have been proved useful and qualitatively correct. However, in many reactions models assuming one-step reaction mechanisms can lead to misleading results. The logical next step is to consider two-step reaction mechanisms that have shown promise in capturing the essential behaviour of more complicated reaction schemes. In this paper we discuss the generic properties of travelling combustion wave solutions to models involving one-and two-step reaction mechanisms assuming Arrhenius kinetics. In particular, for one-step models we discuss the dependence of the speed of the propagating flame front on the activation energy, Lewis number and the effects of heat loss, all of which can be controlled in a laboratory setting. In addition we present a summary discussion on the stability properties of one-step combustion waves. Stability of combustion waves is an important issue in applications such as self-propagating high-temperature synthesis (SHS) of advanced materials, where pulsating instabilities in combustion waves can lead to undesirable laminar irregularities in the product material. Understanding the dependence of flame speed and stability on the various prescribable parameters is therefore a subject of considerable practical interest. The generic properties of solutions of the one-step adiabatic and nonadiabatic models are compared to those of an adiabatic model assuming a two-step chain branching and recombination mechanism. For the two-step reaction scheme it is found that the fuel Lewis number has a considerable effect on the qualitative dependence of the wave speed on the activation energy, while the Lewis number for radicals does not greatly alter the generic behaviour. In particular, when the fuel Lewis number is less than unity, travelling combustion wave solutions are stable and propagate with a speed uniquely defined by a monotonically decreasing function of the activation energy. Solutions exist for all values of the activation energy up to a finite value corresponding to a wave speed of zero (extinction). On the other hand, when the fuel Lewis number is greater than unity the wave speed is double-valued, with unstable 'slow' solutions and 'fast' solutions that are either stable or that become unstable due to the onset of pulsations.