Taylor-diffusion-controlled combustion in ducts

Abstract

An analysis is presented for the Burke–Schumann flame established when a fuel tank discharges with mean velocity U along a circular duct of radius a filled initially with air. Attention is focused on effects of interactions of shear with transverse diffusion resulting in enhanced longitudinal dispersion. The analysis accounts for preferential-diffusion effects arising for non-unity values of the fuel Lewis number $L_{\mathrm{F}}$, with the Peclet number $\mathit{P}\mathit{e}=Ua/D_o$ based on the thermal diffusivity $D_o$ taken to be of order unity for generality. The solution to the associated Taylor-dispersion problem is described for times $t^{\prime }$ much larger than the characteristic diffusion time across the pipe $a^2/D_o$, when the flame is embedded in a mixing region of increasing longitudinal extent moving with the mean velocity. At leading order in the limit $t^{\prime }\gg a^2/D_o$, the longitudinal flame location, the burning rate, and the peak temperature are found to be a function of the effective Lewis number $L_{\mathrm{e}\mathrm{f}\mathrm{f}}=L_{\mathrm{F}}(1+{\mathit{P}\mathit{e}}^2/48)/(1+L_{\mathrm{F}}^2{\mathit{P}\mathit{e}}^2/48)$, whose value changes from $L_{\mathrm{e}\mathrm{f}\mathrm{f}}=L_{\mathrm{F}}$ for $\mathit{P}\mathit{e}\ll 1$ to $L_{\mathrm{e}\mathrm{f}\mathrm{f}}=1/L_{\mathrm{F}}$ for $\mathit{P}\mathit{e}\gg 1$. As a result of this variation, the flame exhibits preferential-diffusion effects that depend fundamentally on $\mathit{P}\mathit{e}$, with important implications in designs of microcombustion devices employing narrow channels and pipes.

Keywords:

• Taylor dispersion
• Burke–Schumann flame
• differential diffusion
• poiseuille flow

Disclosure statement

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