Effects of thermal expansion on Taylor dispersion-controlled diffusion flames

Abstract

A theoretical analysis is developed to investigate the effects of gas expansion due to heat release on unsteady diffusion flames evolving in a pipe flow in which the mixing of reactants is controlled by Taylor's dispersion processes thereby extending a previously developed theory based on the thermo-diffusive model. It is first shown that at times larger than radial diffusion times, the pressure gradient induced by the gas expansion is, in the first approximation, small in comparison with the prevailing pressure gradient driving the flow, indicating that corrections to the background velocity profile are small. The corrections to the velocity components along with the leading-order mixing variables such as the concentrations, temperature and density are solved for a Burke–Schumann flame. Due to the dependence of the effective Taylor diffusion coefficients on the gas density, quantitative and sometimes qualitative departures in predictions from the thermo-diffusive model are observed.

Keywords:

• Taylor dispersion
• Burke–Schumann flame
• non-unity Lewis number
• Poiseuille flow
• thermal expansion

Acknowledgments

The authors are grateful to Forman A. Williams and Antonio L. Sánchez for helpful discussion and suggestions. The authors also express thanks to Amable Liñán for initiating this problem.

Disclosure statement

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

Notes

1 Combining (56(56) $\begin{array}{rl}& \frac{\mathrm{\partial }{\rho }_0}{\mathrm{\partial }\tau }+\frac{\mathrm{\partial }}{\mathrm{\partial }\xi }\left({\rho }_0{u}_e\right)=0,\end{array}$(56) ) and (57(57) $\begin{array}{rl}& {\rho }_0\frac{\mathrm{\partial }{T}_0}{\mathrm{\partial }\tau }+{\rho }_0{u}_e\frac{\mathrm{\partial }{T}_0}{\mathrm{\partial }\xi }=\frac{\mathrm{\partial }}{\mathrm{\partial }\xi }\left({\rho }_0\mathcal{D}\frac{\mathrm{\partial }{T}_0}{\mathrm{\partial }\xi }\right),\end{array}$(57) ) and using the relation ${\rho }_0{T}_0=1$ and boundary conditions, we can show that $u_e={\rho }_0\mathcal{D}\mathrm{\partial }{T}_0/\mathrm{\partial }\xi$.