Structure and extinction of reverse smolder waves in the presence of heat losses: A premixed-flame perspective

Abstract

A reverse smolder wave is structurally analogous to a premixed flame in the sense that the solid fuel and the gaseous oxidizer enter the reaction zone in a co-current manner. Such structural similarity is explored in this work to analytically examine the structural characteristics and extinction properties of planar reverse smolder waves that are driven by a forced oxidizer flow, based on the activation-energy asymptotic method. The analysis covers both the adiabatic condition and the non-adiabatic condition resulting from an external heat loss. Within the framework of the adopted diffusive-thermal model, two distinct reaction regimes are identified, namely the fuel-rich regime and the fuel-lean regime, separated in between by a stoichiometric state, which corresponds to a critical gas mass flux (for the adiabatic condition) or a critical heat loss coefficient (for the non-adiabatic condition). It is found that, under both adiabatic and non-adiabatic conditions, extinction occurs exclusively in the near-stoichiometric fuel-lean regime, in the sense that the values of characteristic quantities at the extinction limit are at most O(ε) away from those at the corresponding stoichiometric state, where ε is a small parameter inversely proportional to the activation energy. It is then demonstrated how this near-stoichiometric character can be exploited to seek a series-form representation of all the quantities characterizing the extinction limit, where the normalized initial oxygen mass fraction and the oxygen Lewis number are identified as two primary control parameters. A close examination of the differences between reverse smolder waves and premixed flames identifies a mechanism that may account for why reverse smolder waves are generally more robust than premixed flames when subjected to heat losses. Further, it is justified that limiting kinetics as a unified bottom-level mechanism can provide a uniform characterization of the extinction conditions of non-adiabatic reverse smolder waves.