Rate-Ratio Asymptotic Analysis of Strained Premixed Laminar Methane Flame Under Nonadiabatic Conditions

ABSTRACT

Motivated by the pioneering activation-energy asymptotic analysis of strained laminar premixed flames in counterflow by Libby and his coworkers, a rate-ratio asymptotic analysis is carried out to elucidate the structure and predict the critical conditions of extinction of strained premixed methane flames. Steady, axisymmetric, laminar flow of two counterflowing streams: a reactive mixture stream and a product stream toward a stagnation plane is considered. The temperature of the reactive mixture stream is $T_1$ and it is made up of methane (CH₄), oxygen (O₂) and nitrogen (N₂), while the temperature of the product stream is $T_2$, and it is made up of O₂, carbon dioxide, water vapor and N₂. The asymptotic flame structure is presumed to be made up of a thin reaction zone where all chemical reactions take place. On one side of the reaction-zone is an inert, preheat zone containing the reactants and on the other side a post-flame zone made up of products. Analysis of the preheat zone gives matching conditions that is required to analyze the structure of the reaction zone. A four-step, reduced mechanism is used to describe the chemical reactions. The reaction zone is presumed to be made up of an inner layer, where CH₄ is consumed. The hydrogen (H₂) and carbon monoxide that are formed in this layer are consumed in an oxidation layer that is made up of two layers: an H₂-oxidation layer and a CO-oxidation layer. The results of the analysis are used to predict the flame location, $y_r$, flame temperature, $T_r$, and the speed of the convective flow, $v_{\mathrm{b}}$, in the reaction zone as a function of the strain-rate, $a_{\mathrm{u}}$. Classical C-shaped curves were obtained when $y_r$, $T_r$ and $v_{\mathrm{b}}$ are plotted as a function of $a_{\mathrm{u}}$ and they were used to predict extinction. A key finding of this work is that $v_{\mathrm{b}}$ is proportional to $T_r-T_0$, where $T_0$ is the crossover temperature predicted by the rate-ratio asymptotic analysis. Whether abrupt extinction will take place or not was found to depend on the value of $T_2$ relative to $T_0$, which is different from the predictions of activation-energy asymptotic analysis where $T_2$ must be compared with the value of the adiabatic temperature. Similar to the analysis of Libby and his coworkers, the rate-ratio asymptotic analysis predicts the existence of “negative flame speeds,” where the convective flow and the diffusive flow of reactants in the reaction zone, are in opposite directions. The predictions of the rate-ratio asymptotic analysis were found to agree with the results of computations with detailed chemistry and previous experimental data.

KEYWORDS:

• Asymptotic analysis
• premixed flames
• extinction
• flame speed
• counterflow

Acknowledgments

We are honored to contribute this paper to this special issue of Combustion Science and Technology in honor of Professor Paul A Libby. The corresponding author was fortunate to have had numerous hours of technical discussions with Prof. Libby on various aspects of fluid mechanics of combustion, some of which motivated the present work. We regret, like many other contributors to this special issue, that we did not have a chance to discuss this work with Prof. Libby. We thank Prof. Williams for his contributions and helpful discussions.

Disclosure statement

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

Correction Statement

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

Notes

¹ In the computational study, the counterflowing streams were injected into the mixing layer from boundaries that are separated by a finite distance $L$. Since plug flow boundary conditions were applied, the inviscid flow outside the boundary layer around the stagnation plane in the computational study is inviscid but rotational (Seshadri and Williams 1978).