Analytical Model for the Downward Flame Spread over a Thermally Thin Fuel into an Opposed Flow

Abstract

We derive a new analytical expression for the flame front speed in the downward combustion of thin solid fuels based on a model that consists of a preheating region plus a pyrolyzing one. The solid phase is modeled by means of a simple but physically realistic behavior of the main variables in these two regions, whereas the gas phase follows classical reaction–diffusion–convection equations for the fuel mass fraction, oxygen mass fraction, and gas temperature. Both fuel and oxygen mass fractions are expressed in terms of the gas temperature by using a simplified vertical flame model. This approach allows us to reduce the combustion model into a single reaction–diffusion–convection equation of a single variable (gas temperature) for both regions. Matching conditions at the flame leading edge gives us an analytical expression for the burning rate that depends on the kinetics parameters of the combustion reaction and the opposed flow. In contrast with de Ris's classical formula, our analytical expression predicts (1) extinction limits at low values of oxygen concentration and at large values of opposed flow (blowoff), (2) a decrease of the burning rate as gravity increases, and (3) a finite value of the burning rate as the solid thickness tends to zero. A comparison with experimental data is also performed with reasonable results near the extinction limits.