Theoretical analysis of steady nonadiabatic premixed laminar flames

A one-dimensional, steady, non-adiabatic, premixed laminar flame is assumed semi-infinite with the burner at the origin, and the investigation centers on the asymptotic behavior of the temperature and species mass fractions in the burned region near infinity. After a consideration of the general $N$-species problem, specific results are obtained for the global two-step reaction ${v_r}R \to {\mu _i}I$, ${v_r}I \to {\mu _p}P$, where $R$, $I$, $P$ denote reactant, intermediate, and product species, respectively, and ${v_r}$, ${v_i}$, ${\mu _i}$, ${\mu _p}$ are stoichiometric coefficients. Assuming Arrhenius kinetics, it is shown that the classical linearized asymptotic theory is not applicable unless ${v_r} = {v_i} = 1$, in which case the approach to burned equilibrium is an exponential decay. Consequently, a nonlinear theory applicable to arbitrary ${v_r}$ and ${v_i}$ is presented which shows that in general the asymptotic decay is algebraic. It is further shown that boundedness of the solution at infinity permits the arbitrary specification of only three boundary conditions on the original sixth-order differential system. This result is illustrated by a comprehensive analytical example and the computational implications for the general $N$-species problem are discussed.