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Dynamics of flame propagation and pressure buildup in closed space

Flame and pressure dynamics in a closed vessel has been studied best in the simplest case of a spherical vessel with central ignition and spherical flame propagation. At the initial stage of combustion the spherical flame front expands similar to that one in unbounded space – the pressure rise is negligible, vessel walls are relatively far from the flame front to affect the combustion process and the flame front velocity is equal to V_{ff} = s_u E_i , where s_u - burning velocity of the mixture, E_i = \rho _i / \rho _b - expansion coefficient of the combustion products. As the flame propagates closer to the walls the flame front velocity slows down due to compression of the unburnt (fresh) gas. In vicinity of walls the unburnt gas doesn’t expand any more and the flame front velocity equals simply to the burning velocity V_{ff} = s_u .

In spite of decelerating flame velocity, the speed of pressure built up is rising owing to the increasing with radius flame front area and, thus, mass burning rate. The most of pressure rise occurs at the final stage of combustion when the flame front is close to the vessel wall. According to the model developed in (DB, Ya. Zeldovich, 1985), for a mixture with expansion coefficient , only 20% of pressure rise occurs during flame propagation over 80% of vessel radius. The pressure during combustion in a closed vessel may be assumed constant across a vessel as the sound speed is much faster than the flame front velocity and pressure equilibration time is much shorter than the combustion time.

Combustion process in a closed vessel implies there is no overall gas expansion and the process is accompanied by a pressure rise in a vessel. As the heat of reaction is not spent on work of expansion, but goes solely into raising the internal energy of the gas, the combustion products in a closed vessel have larger temperature compare to combustion of the same fuel-oxidiser composition at a constant pressure.

Pressure dynamics with time may be found starting from the balance equation for the burnt mixture mass fraction (Ya.Zeldovich, 1985):

m{{d\eta } / {dt}} = \rho \left( {p,\,T_u } \right)\,s_u \left( {p,\,T_u } \right)\,A\left( t \right),

where m - mass of gas in the vessel, \eta - fraction of burnt mixture, t - time, \rho \left( {p,\,T_u } \right) - density of unburnt mixture as a function of pressure P in vessel and temperature of unburnt mixture T_u , s_u \left( {p,\,T_u } \right) - burning velocity, A\left( t \right) - flame area. The flame and pressure dynamics with time may be obtained provided the dependence of the burning velocity with temperature and pressure for a particular mixture and a flame shape are known. Different integral balance models are available from literature, e.g. (BradleyD:1976), (Ya.Zeldovich, 1985), (MolkovVV:1981), (A.Dahoe, 2005).

One may use an integral balance model together with inverse problem method to obtain burning velocity of mixture from deflagration pressure dynamics in a closed vessel. Burning velocity and baric index for stoichiometric hydrogen-air deflagration in a large-scale vessel were obtained in (MolkovVV:2000); burning velocity of unstretched hydrogen-air flame was obtained in (A.Dahoe, 2005) in a range of equivalence ratios \Phi = 0.5 - 3.0 from a small-scale closed vessel experiments.

More versatile method of modelling flame and pressure dynamics in closed vessels may be computational fluid dynamics (CFD) methods, which don’t require assumption about the shape of a flame front, but, instead, model its development from first principles. For example, a stoichiometric hydrogen-air deflagration in a closed vessel was modelled in (MolkovVV:2004f). There the application of large-eddy simulation (LES) method, which is an advanced method of modelling turbulent and reacting flows, allowed to resolve development of the flame front including the effect of hydrodynamic instabilities.

Zeldovich Ya.B., Barenblatt G.I., Librovich V.B., Makhviladze G.M. The mathematical theory of combustion and explosions. Consultants Bureau, New-York, 1985.
Bradley D. and Mitcheson A. (1976) Mathematical Solutions for Explosions in Spherical Vessels. Combustion and Flame, 26:201-217.(BibTeX)
Molkov V.V. and Nekrasov V.P. (1981) Dynamics of Gas Combustion in a Constant Volume in the Presence of Exhaust. Fizika Goreniya i Vzryva, 17:4.(BibTeX)
Dahoe A.E. Laminar burning velocities of hydrogen–air mixtures from closed vessel gas explosions. Journal of Loss Prevention in the Process Industries (to be published), pp.1–15, 2005.
Molkov V.V. Dobashi R., Suzuki M. and Hirano T. (2000) Venting of Deflagrations: Hydrocarbon-Air and Hydrogen-Air Systems. Journal of Loss Prevention in the Process Industries, 13:397-409.(BibTeX)
Molkov V.V., Makarov D. and Grigorash A. (2004) Cellular structure of explosion flames: modeling and large-eddy simulation. Combustion Science and Technology, 176:851-865.(BibTeX)

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