Application of a non-asymptotic approach to prediction of the propagation of a flame through a fuel and/or oxidant droplet cloud

The mathematical analysis of laminar premixed spray propagation has generally been based on exploiting the inverse of the large chemical activation energy as an appropriate parameter for asymptotic analysis. In the current work we apply a modification of a recently suggested non-asymptotic approach for gaseous flames which makes use of a different approximation. In it, only the Arrhenius exponential term in the reaction rate expression is approximated using a step function chosen so that the two functions are in proximity in an integral sense. Application of this approach is more amenable and is shown to yield a simple formula for the burning velocity of a flame propagating through a cloud of fuel and/or oxidant droplets, for the fuel rich off-stoichiometric case in which the only reactant present in the chemical reaction term is the deficient oxidant which appears linearly. Results computed with the new analytical solutions are presented and a comparison is made with the predictions using the usual large activation energy approach. In addition, a double spray is considered for the first time in which both liquid oxidant and liquid fuel feature as sprays of droplets in the unburned pre-mixture. Such a situation arises in rocket engines in which two initially separate spray streams mix in a turbulent shear flow so that locally one dimensionally propagating double spray premixed flames are created. The analysis leads to an analytical expression for the laminar burning velocity dependent on the spray- and gas-related parameters. Typical thermal and velocity maps in parametric space are presented.