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Abstract
A model is presented for a one-dimensional laminar premixed flame, propagating into a rich, off-stoichiometric, fresh homogenous mixture of water-in-fuel emulsion spray, air and inert gas. Due to its relatively large latent heat of vaporisation, the water vapour acts to cool the flame that is sustained by the prior release of fuel vapour. To simplify the inherent complexity that characterises the analytic solution of multi-phase combustion processes, the analysis is restricted to fuel-rich laminar premixed water-in-fuel flames, and assumes a single-step global chemical reaction mechanism. The main purpose is to investigate the steady-state burning velocity and burnt temperature as functions of parameters such as initial water content in the emulsified droplet and total liquid droplet loading. In particular, the influence of micro-explosion of the spray’s droplets on the flame’s characteristics is highlighted for the first time. Steady-state analytical solutions are obtained and the sensitivity of the flame temperature and the flame propagating velocity to the initial water content of the micro-exploding emulsion droplets is established. A linear stability analysis is also performed and reveals the manner in which the micro-explosions influence the neutral stability boundaries of both cellular and pulsating instabilities.
Acknowledgements
The partial support of the Lady Davis Chair is gratefully acknowledged by JBG.
Nomenclature
| = | the radius of the inner water content | |
| = | pre-exponential constant | |
| = | the original external radius of the WIFE droplet | |
| = | evaporation coefficients of fuel and water, respectively | |
| = | specific heats of liquid and gas phases, respectively | |
| = | droplet radius | |
| = | terms in Damköhler number expansion | |
| = | chemical Damköhler number | |
| = | mass diffusion coefficient | |
| = | thermal diffusion coefficient | |
| = | activation energy | |
| = | evaporation coefficient | |
| f | = | micro-explosion factor |
| h | = | thickness of the fuel outer shell before micro-explosion |
| = | Heaviside function | |
| = | wave number | |
| = |
| |
| = | average inter-droplet distance | |
| = | non-dimensional latent heat of vaporisation of fuel and water, respectively | |
| = | characteristic length | |
| = | Lewis number | |
| = | mass fraction and normalised mass fraction, respectively | |
| = | heat of reaction | |
| = | universal gas constant | |
| = | the mole fraction of oxygen in the fresh mixture | |
| = | evaporation rate | |
| = | time and normalised time coordinate, respectively | |
| = | terms in temperature expansion | |
| = | temperature | |
| = | burning velocity | |
| = | (dimensional) chemical source term | |
| = | spatial and normalised spatial coordinates, respectively | |
| = | initial water content in each WIFE droplet | |
| = | stability analysis parameters, Equation (85) | |
| = | stoichiometric coefficient | |
| = | stretched spatial coordinate | |
| = | delta function | |
| = | total initial liquid load (fuel + water) | |
| = | Laplacian (Equation (29)) | |
| = | small parameter | |
| = | perturbed flame front surface | |
| = | equivalence ratio | |
| = | thermal conductivity | |
| = | thermal diffusivity | |
| = | thermal expansion ratio | |
| = | density | |
| = | viscosity | |
| = | perturbation frequency | |
| = | normalised spatial coordinates | |
| = | normalised time coordinate | |
| = | normalised large activation energy |
Suffixes
| = | adiabatic | |
| = | burned value | |
| = | relating to water in droplets | |
| = | relating to fuel in droplets | |
| = | fuel | |
| = | relating to gas phase | |
| = | relating to liquid phase | |
| = | unburned value | |
| = | water | |
| = | oxygen | |
| = | value at the onset of vaporisation | |
| – | = | relating to steady-state solution |
| ˜ | = | relating to perturbation |
| ´ | = | modified |
| * | = | dimensional dependent and independent variables |
Disclosure statement
No potential conflict of interest was reported by the authors.