Solution of adiabatic and nonadiabatic combustion problems using step-function reaction models

We consider the use of step-functions to model Arrhenius reaction terms for traveling wave solutions to combustion problems. We develop a methodology by which the Arrhenius reaction rate term is replaced by a suitably normalized step-function. The resulting model introduces interior interfaces and allows the conservation equations for energy and species to be solved explicitly within the subdomains bounded by the interfaces. The problem can then be reduced to a small number of nonlinear algebraic equations governing appropriate interface conditions. We apply this methodology to a variety of single-reaction problems and show that the resulting solutions agree with those obtained by the well-known front δ-function) approximations for large Zeldovich numbers. We then consider multiple reaction problems, specifically problems involving two independent reactions and problems involving sequential reactions. For these problems we compare the results with simpler front models as well as with Arrhenius kinetics. We show that the step-function models are generally superior to the front models where available and agree, both qualitatively and with reasonable quantitative accuracy, with solutions obtained via full Arrhenius kinetics.