A thick reaction zone model for premixed flames in two-dimensional channels

Abstract

Direct interactions between the flow field and the chemical reaction in premixed flames occur when the reaction zone thickness is comparable to, or greater than flow length scales. To study such interactions, a laminar model is considered that has direct bearings to steadily propagating deflagrations in a Hele-Shaw channel with a background plane Poiseuille flow. The study employs asymptotic analyses, pertaining to large activation energy and lubrication theories and considers a distinguished limit where the channel width is comparable to the reaction zone thickness, with account being taken of thermal-expansion and heat-loss effects. The reaction zone structure and burning rates depend on three parameters, namely, the Peclet number, $\mathcal{P}$, the Lewis number, $Le$ and the ratio of channel half-width to reaction zone thickness, ${\lambda }_{\ast }$. In particular, when the parameter ${\lambda }_{\ast }$ is small wherein the reaction zone is thick, transport processes are found to be controlled by Taylor's dispersion mechanism and an explicit formula for the effective burning speed $S_T$ is obtained. The formula indicates that $S_T/S_L\propto 1/Le$ for $\mathcal{P}\gg 1$, which interestingly coincides with a recent experimental prediction of the turbulent flame speed in a highly turbulent jet flame. The results suggest that the role played by differential diffusion effects is significant both in the laminar and turbulent cases. The reason for the peculiar $1/Le$ dependence can be attributed, at least in our laminar model, to Taylor dispersion. Presumably, this dependence may be attributed to a similar but more general mechanism in the turbulent distributed reaction zone regime, rather than to diffusive-thermal curvature effects. The latter effects play however an important role in determining the effective propagation speed for thinner reaction zones, in particular, when ${\lambda }_{\ast }$ is large in our model. It is found that the magnitude of heat losses at extinction, which directly affects the mixture flammability limits, is multiplied by a factor $1/{Le}^2$ in comparison with those corresponding to the no-flow case in narrow channels.

Keywords:

• Taylor dispersion
• flow-reaction interaction
• preferential diffusion effects
• Poiseuille flow
• thick reaction zone

Disclosure statement

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

Notes

1 We let the modified Peclet number $\mathcal{P}$ and the other parameter ${\lambda }_{\ast }$ introduced in the abstract to emerge naturally through the analysis provided in §3 and §5.

2 It will turn out that key parameter characterising the flame is not the cold temperature Peclet number $Pe$, but the Peclet number $\mathcal{P}$ at the adiabatic flame temperature. It is indeed appropriate to formulate the original problem from the outset for the limits $ϵ\to 0$ and $\mathcal{P}\sim 1$ since $\alpha \sim 1$.

3 We note parenthetically that if V is plotted as a function of A by keeping ϵ fixed, then one obtains (not shown herein) curves without exhibiting the bending effect, similar to those presented in figure 3 of [4], figure 8 of [1] and (as the green lines of) figure 3 of [3]; note that in [1], Peclet number is defined with respect to $S_L$ so that keeping their Peclet number fixed is equivalent keeping our ϵ constant.

Additional information

Funding

This research was supported by the UK EPSRC (Engineering and Physical Sciences Research Council) through [grant number EP/V004840/1].