Bifurcation and Asymptotic Analyses of Hydrogen - Diffusion Flames

The objective of this study is to investigate the structure of the steady, counterflow, hydrogen-air diffusion flame by bifurcation and asymptotic methods in two different regimes of interest. First, ignition for oxidizer-stream temperatures T_infty larger than or of the order of the crossover temperature T_ {c} associated with the second explosion limit of hydrogen is considered. Two types of solutions are identified, a frozen solution that always exists because for high-temperature ignition all rates of relevant steps are proportional to concentrations of intermediate radicals, and an ignited solution, represented by a branch of the curve giving the maximum concentration of radicals in terms of the strain time. The form of this ignited branch is investigated for two-step reduced chemistry. It is found that for T_infty> T_{c } this branch bifurcates from the frozen solution if the strain time is increased to a critical value. For T_infty larger than a value T_{rm s}>T_{ rm c} the effects of chemical heat release are small and ignition is always gradual in the sense that the limiting ignited-branch slope is positive. For T _infty in the range T_ {rm c}<T_infty < T_ {rm s} the heat release associated with the radical-consumption step causes the limiting ignition -branch slope to become negative producing abrupt ignition which leads to an S curve. For values of T_ infty below crossover the ignited branch appears as a C-shape curve unconnected to the frozen solution. Although the qualitative behavior that emerges agrees well with numerical results, it is shown that at least three overall steps, with O and H atoms as the chain-branching species and a detailed model adopted for the flow field, are necessary to provide accuracy. The second regime considered is that of a vigorously burning flame corresponding to strain times larger than the characteristic chemical time of three-body recombination reactions. Under these conditions, it is shown that the reactants can coexist only within a thin reaction zone separating two radical-free equilibrium regions. Matching the solutions from the inner reaction region with those from the outer equilibrium regions, determined by introduction of appropriate conserved scalars, yields a first-order asymptotic solution to the problem, which compares favorably with results of numerical integrations. The results help to improve understanding of flame structure and provide information needed in design of high -speed air-breathing propulsion devices.