The Asymptotic Structure of Strained Chain-Branching Premixed Flames Under Nonadiabatic Conditions

ABSTRACT

The present paper, which investigates the chain-branching premixed-flame dynamics under nonadiabatic conditions, is the third and final contribution to a series of studies on premixed flames with the modified Zel’dovich-Liñán two-step mechanism under the three most prevailing physical influences, namely (i) flame stretch, (ii) differential diffusion, and (iii) heat loss to or gain from the flame downstream. Asymptotic analysis for the chain-branching premixed-flame structure is carried out within the framework of the intermediate recombination regime, in which the chain-recombination layer with a characteristic thickness of $\mathrm{O}({\beta }^{-1/2})$ is asymptotically thicker than the inner chain-branching layer but thinner than the outer convective-diffusive layer. The combustion characteristics are presented by the $\mathcal{M}$–$\mathrm{\Delta }$ plots, where the specific reaction intensity $\mathcal{M}$ is a measure of the reaction intensity and the reduced recombination Damköhler number $\mathrm{\Delta }$ is a measure of the inverse of the flame stretch. The $\mathcal{M}$–$\mathrm{\Delta }$ plots are found to exhibit a striking resemblance to the AEA counterparts of Libby and Williams (1983), in that the S-shaped $\mathcal{M}$–$\mathrm{\Delta }$ curves emerge for sufficiently large downstream heat losses. The Damköhler number ratio $D_I/D_{\mathit{II}}$ is found to be another key parameter controlling the chain-branching flame dynamics. The greater $D_I/D_{\mathit{II}}$, the slower the recombination reaction and the thicker the recombination layer tends to be, which results in a reduced overall nonlinearity for the Zel’dovich-Liñán two-step mechanism. Consequently, abrupt extinction is less likely observable in the much slower recombination regime. Combining with the results of the previous two papers by the authors, the overall combustion characteristics of strained chain-branching premixed flames with the Zel’dovich-Liñán two-step kinetics is found to be in qualitatively good agreement with that of one-step AEA, unless the recombination step is too slow or too fast to be reasonably described by the intermediate recombination regime.

KEYWORDS:

• Zel’dovich-liñán two-step mechanism
• chain-branching premixed flame
• intermediate recombination regime
• downstream heat transfer
• activation-energy asymptotics

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

¹ Libby and Williams published a fourth paper (Libby and Williams 1984) on interacting duel premixed flames. The problem involves unnecessarily too many parameters and its physical relevance is only secondary to the above-mentioned three papers dealing with flame stretch, nonunity Lewis number, and heat loss or gain. Therefore, the authors do not intend to pursue the case of interacting duel chain-branching premixed flames.

² The Zel’dovich number $\mathit{\beta }$ measures the ratio of the activation energy for the thermal energy released by combustion. AEA employs the large Zel’dovich number as the primary expansion parameter, and the exothermic reaction is confined within a thin reactive layer with a characteristic thickness of $\mathrm{O}({\beta }^{-1})$.

³ This complete S-cure flame-response behavior is also cited in the “Turbulent Combustion” chapter (page 419) of Combustion Theory by Forman Forman Williams (1985) as a typical behavior of strained laminar premixed flamelet.

⁴ Up to date, there does not yet exist an exact asymptotic solution for the laminar flame speed of one-dimensional chain-branching premixed flames, so that we do not have the precise parametric dependence of the laminar flame speed which could serve as a reference characteristics to verify the future theoretical and numerical investigation on a variety of chain-branching premixed flame configurations. Asymptotic analysis for one-dimensional chain-branching flames could be a worthy subject of combustion analysis.

⁵ The conventional crossover condition is obtained by a point or bulk balance without considering the $Y_Z$ and $Y_F$ distributions. In contrast to the conventional method to extract the crossover condition, this newly proposed crossover condition can take into account the distributed fuel and chain-carrier profiles, which we expect provides a much more accurate and universal crossover condition, and perhaps flammability condition too.

Additional information

Funding

SRL was supported by the Research Program funded by Seoul National University of Science and Technology.