The Asymptotic Structure of Strained Chain-Branching Premixed Flames

ABSTRACT

The asymptotic structure of stretched chain-branching premixed flames with unity Lewis numbers is analyzed with the Zel’dovich-Liñán two-step mechanism, including (I) temperature-sensitive autocatalytic chain-branching step and (II) first-order chain-recombination step with combustion heat release. Depending on the order-of-magnitude of the Damköhler number ratio between the branching and recombination reactions, three distinct asymptotic limits, namely the fast, intermediate and slow recombination regimes, emerge with their own distinct multi-layer asymptotic structures. Our attention is focused on the asymptotic chain-branching flame-structure analysis within the framework of the intermediate recombination regime, in which the recombination layer is asymptotically thicker than the branching layer, but thinner than the outer convective-diffusive layer. The multi-layer asymptotics, involving the Damköhler number asymptotics for the recombination layer and the activation-energy asymptotics for the branching layer, yields the chain-branching flame-structure solution. The calculation results reveal the unique characteristics of strained chain-branching flames. First, the chain-carrier concentration and temperature at the branching reaction sheet are found to be constant irrespective of the strain rate. The chain-carrier concentration increases as the recombination reaction becomes slower. Moreover, the chain-carrier concentration at the branching reaction sheet is found to be proportional to the laminar flame speed. However, no quasisteady extinction was observed in any calculation results because the branching-reaction rate manages to maintain its strength thanks to the invariant branching-layer temperature. It is worthwhile to note that the present two-step model for chain-branching flames is perhaps the simplest asymptotic model, involving the minimum number of kinetic parameters to properly describe the asymptotic structure without losing any physical essence.

KEYWORDS:

• Chain-branching prexmied flame
• Zel’dovich Liñán two-step mechanism
• asymptotic structure
• intermediate recombination regime
• Damköhler number ratio

Acknowledgments

As some readers may already know, J.S. Kim, the second author of this paper, studied the science of combustion in University of California at San Diego, where Paul Libby was an inaugural member of the school of engineering and had taught fluid mechanics for over three decades. Kim also learned the key analytic techniques of flame analysis, such as the boundary layer theory and shooting method, from Paul Libby, and was inspired by the simple and effortless analytic approaches that Paul Libby demonstrated time after time. Paul Libby was a role model to be followed by the aspiring combustion theoreticians like Kim. When S.R. Lee and J.S. Kim, the authors of this paper, were told of the CST special issue to commemorate the Paul Libby’s ${100}^{\mathbf{t}\mathbf{h}}$ birth year, we decided to write a paper on strained premixed flames with two-step Zel’dovich-Liñán model. Of course, the paper has to be written á la Paul Libby, that is, simple and clear, to be the fitting tribute to the Paul Libby’s old-school mastery of analytic simplicity. We feel only sorry that Paul Libby passed away before the CST special issue is published, and hope that this paper has the “simple” quality meeting the Paul Libby’s simplicity standard. Paul Libby’s coolness and wisdom will be missed by many who ever became to know him. SRL was supported by the Research Program funded by Seoul National University of Science and Technology.

Disclosure statement

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

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Notes

¹ Liñán rediscovered usefulness of the two-step mechanism (Liñán 1971). In order to distinguish so many different reaction mechanisms proposed by Zel’dovich, the two-step mechanism is usually called the Zel’dovich-Liñán mechanism.

² Hereafter, the three distinguished limits sequentially emerging with increasing recombination-zone thickness will be called the fast recombination regime, intermediate recombination regime and slow recombination regime as Liñán (1974) used similar terms, such as the diffusion-flame regime, premixed-flame regime, partial burning regime, and ignition regime, depending on the distinguished limits associated with the order-of-magnitude of the reaction rate. Of course, in this two-step mechanism, each regime will have its own distinct multi-layer structure as well as order-of-magnitude for its reaction rate.

³ It should be kept in mind that the above ordering for the reduced Damköhler number $\mathrm{\Lambda }$ is unqiue for the intermediate regime. If a different distinguished limit, among the fast recombination regime and slow recombination regime, from the intermediate recombination regime, is employed, the corresponding reduced Damköhler number will take a different ordering with the kinetic parameters.

⁴ This problem is first solved by Liñán (1974) for the premixed-flame regime under extremely strong super-adiabaticity. The relevant solution can be found in the subsection (c) of the Appendix C of the Liñán’s paper.

Additional information

Funding

This paper is to be considered for the Special Issue in Honor of Professor Paul A. Libby on the Occasion of His 100th Birthday with Guest Editors Antonio Sánchez and Kal Seshadri