Premixed flames modelled with thermally sensitive intermediate branching kinetics

Abstract

The foundations of a relatively simple two-step kinetic scheme for flame chemistry are outlined, involving a model chain branching process that should adopt the activation temperature of a rate-limiting branching reaction in order to offer a broad approximation for hydrocarbon flames. A model energetic intermediate reactant then acts as a buffer between fuel consumption and the release of heat, as the intermediate is converted into products through a completion reaction step. By taking the rate of the latter reaction to be linear in the concentration of the intermediate, which is consistent with the final state being an equilibrium in a broader chemical system, a form of the model is arrived at which admits asymptotic solutions in a thermodiffusive context with constant coefficients. These are developed to second order for large values of the activation energy of the branching reaction and are found to involve the same trends that are seen for lean methane and hydrogen flames calculated using detailed chemical and transport models. Linear stability analysis identifies the ranges of Lewis numbers in which cellular or oscillatory instability can arise, with the latter form of instability disappearing above a threshold heat of reaction. These and the underlying flame solutions themselves depend on the heat of reaction and the degree of heat loss but not on the activation temperature of the branching reaction, to leading order. Near the limit of flammability a direct parallel arises with one-step kinetic models for premixed flames.

Keywords:

• Premixed flames
• Two-step kinetic model
• Intermediate chain-branching chemistry
• Flame stability

Acknowledgements

Conversations with a number of people have helped in the development of the ideas presented in this article. In particular the author would like to thank Charlie Westbrook, Forman Williams, Barry Greenberg, Grisha Sivashinsky, Kal Seshadri, Amable Liñán, Rodney Weber, Joel Daou and Norbert Peters for useful discussions and suggestions (that were not always followed). Funding from the EPSRC is gratefully acknowledged as is the much valued assistance of Anna Zinoviev.

Notes

¹One electron volt is about 1.602177×10⁻¹⁹ joules so that one electron volt per molecule represents 1 ev× A ₀≈96 485.3 J mole⁻¹, where Avogadro's number is A ₀≈6.022137×10²³ molecules per mole.