Thermal explosion in a hot gas mixture with fuel droplets: a two reactant model

Abstract

We extend previous analyses of thermal explosion in a gas-droplets mixture to permit a more complete description of the chemistry via a single-step two-reactant model of general order, rather than the prior deficient reactant model. A detailed mathematical analysis has been carried out of this new physical model that encompasses oxidizer effects (in both fuel rich and fuel lean situations) on the thermal explosion of a hot combustible mixture of gases and cool evaporating fuel droplets. The closed mathematical formulation involves a singularly perturbed system of four highly non-linear ordinary differential equations. The entire dynamical picture of the system is qualitatively exposed by exploiting the geometrical version of the powerful asymptotic approach known as the method of integral manifolds (MIM).

It was found that the system's behaviour can be classified according to the values of nine dimensionless parameters. All possible types of dynamical behaviour of the system were studied and the parametric regions of their existence were delineated, with emphasis on the underlying physico-chemical processes at play. Both conventional explosive and delayed regimes were found to occur, including the freeze delay regime. Whereas this latter important regime had been associated with physically unviable operating conditions in previous deficient reactant models, it was found that the current use of a single-step two-reactant chemical kinetic model renders the freeze delay regime physically plausible. Due to its practical importance the delayed regimes were analysed in detail and explicit analytical formulae for delay and evaporation times were extracted. The predictions were found to agree rather well with the results of direct numerical simulations.

It was also found that the stoichiometry of the initial mixture per se does not lead to a natural classification of different sorts of regimes. Rather, the ratio of two key parameters plays the dominant role in defining the relevant fast variables and their associated dynamical regimes, irrespective of the initial mixture stoichiometry.