In case of slow subsonic flames the pressure wave typically has gradually growing front with the typical peak overpressure less than 1 bar and duration of several seconds. In case of fast supersonic flames which are usually fast turbulent flames the pressure wave transforms into shock which is characterized by abrupt step-wise changes in media characteristics with typical overpressures of order of magnitude of 10 bar and its duration of - ms.
In the limiting case of detonation a shock wave is also formed but with higher peak values and shorter duration. For example, in case of detona-tion of stoichiometric hydrogen-air mixture at normal conditions an amplitude of detonation wave reaches 17 bar with duration of less than ms. Pressure waves generated by combustion process decay while they propagate from their source. In far zone the amplitude of such pressure wave is defined only by the chemical energy released during com-bustion.
This means that for the same amount of hydrogen independently on the intensity and charac-teristics of the combustion process, the overpressure value will be the same. On the other hand see e. Dorofeev S. Journal de Physique de France IV, 12 7 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.
As the flame propagates closer to the walls the flame front velocity slows down due to compression of the unburnt fresh gas. 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. 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.
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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, :. 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. BradleyD , Ya. Zeldovich, , MolkovVV , A. Dahoe, 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 ; burning velocity of unstretched hydrogen-air flame was obtained in A. For example, a stoichiometric hydrogen-air deflagration in a closed vessel was modelled in MolkovVVf. 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. The mathematical theory of combustion and explosions. Consultants Bureau, New-York, Bradley D. BibTeX Molkov V. BibTeX Dahoe A. Journal of Loss Prevention in the Process Industries to be published , pp. Molkov V. Dobashi R. Journal of Loss Prevention in the Process Industries, Combustion Science and Technology, The temperature gradient exists in the gas burnt in a closed vessel, rising from the gas burnt last to the gas burnt first B.
Lewis, Zeldovich, and was theoretically estimated in a paper L. Flamm, In a closed vessel combustion the gas temperature changes due to the heat release and due to adiabatic compression. Generally, an intermediate portion of unburnt gas in a closed vessel may be pre-compressed due to combustion of initial portions of gas, then it burns, producing burnt mixture with a higher temperature, and then these combustion products are compressed again due to combustion of following portions of gas.
At initial stage of process close to ignition point combustion occurs at initial pressure and temperature and then the combustion products undergo adiabatic compression with corresponding temperature rise. Thus, these gas portions will have different initial conditions, different combustion products temperature and different composition. The Mache effect is known to reduce the explosion pressure below the one for uniform temperature distribution. Zeldovich, Lewis B.
Combustion, Flames and Explosion of Gases.
Dynamics of Explosions (Progress in Astronautics and Aeronautics)
Academic Press, London, Flamm L. Combustion of an explosive gas mixture within a closed vessel. Wien: Ber. Vented deflagration pressure dynamics has multi-peak structure. The number of peaks and their magnitude depend on various explosion conditions. Detailed analysis of this peak structure was performed in paper .
Four distinct pressure peaks were registered for the case of vented deflagration in low-strength enclosures Figure 1.
The following deflagration phenomena and pressure peak are identified for vented deflagration with initially closed vent cover. After ignition deflagration develops in closed vessel until moment a when vent cover is released. So, pressure peak P 1 is associated with vent release. At moment b , when time derivative of pressure is equal to zero, the volume of gases produced by combustion inside the vessel is equal to volume of gases flowing out of the vessel, i. It is known that venting of combustion products is more efficient for reducing explosion pressure compared to flammable mixture as for the same pressure drop at the vent a velocity of flow is proportional to square root of temperature.
This explains that moment c on pressure transient corresponds to the beginning of burnt gas outflow. Helmholz oscillations could be generated after external explosion with characteristic wave length longer than resonator vessel size. The Helmholz oscillations can induce Rayleigh-Taylor instability .
Increase of mass burning rate due to increase of flame front area implies growth of pressure with time. At moment e the contact of flame front with walls of the vessel is sufficient for the heat losses to compensate the heat release by combustion and pressure peak P 3 is formed.
At final stage of the explosion when combustion proceed mainly close to walls and in corners high-frequency oscillations may be generated. Rayleigh criterion is applied in this case: "If heat be given to the air at the moment of greatest condensation, or taken from it at the moment of greatest rarefaction, the vibration is encouraged" .
Oscillations of peak P 4 can be eliminated by taking adequate measures, such as lining the interior of the vessel to be protected with damping material. It is important to note that when the vent cover failure pressure is increased to 7. For relatively high vent release pressures, which are characteristic for process equipment, only two pressure peaks one - vent release, another - mixture burnout can be observed.
Cooper M. Taylor, G. I, Proc. London , A Rayleigh, L.
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Rayleigh, J. For gaseous explosions the venting process itself is known to cause the flame to accelerate . Gov-erning equations for turbulent vented gaseous deflagrations were derived from the first principles in paper . The inverse problem method for vented gaseous deflagrations has been developed  and efficiently used over the years of research allowing to gather unique data on venting generated turbu-lence.
For example, an analogue to the Le Chatelier-Brown principle for vented gaseous deflagrations  was revealed by this method. The universal correlation for vented deflagrations was developed for the first time in  followed by the closure of this fundamentally new vent sizing approach with the correlation for venting generated turbulence, presented for the first time two years later in . Two of our previous articles were devoted to the problem of inertial vent covers in explosion protection . Recently our original correlations for vent sizing were developed further to include experimental data on fast burning mixtures, such as near stoichiometric and rich hydrogen-air mixtures, and test data on elevated initial pressures .
This fact of discharge coefficient dependence on con-ditions was recognised already about 20 years ago by various authors, e. The following correlation for venting generated turbulence has been obtained by processing a wide range of experimental data on vented gaseous deflagration [11, 13]. The empirical correlation for the DOI number gives the dependence of turbulence level as enclosure scale in power 0. This is in agreement with conclusions of fractal theory with corresponding fractal dimension 2. The turbulent combustion intensifies with increase of the Bradley number as follows from the correla-tion.
It means that an increase of venting area F will be accompanied by an increase of turbulence factor. The increase of burning velocity has opposite effect, i. The DOI number is increasing with increase of initial pressure in enclosure. The turbulence level for vented deflagration in conditions of experiments  increased from at initial atmospheric pres-sure to at initial pressure 7 atmospheres.
Hence the increase of initial pressure from 1 to 7 at-mospheres leads to about four-fold increase of the turbulence level. This result demonstrates explicitly that it is venting that is responsible for a substantial increase in the turbulence level, but not just an elevated initial pressure itself.