570 G.D. Roy et al. Progress in Energy and Combustion Science 30(2004)545-672 using the Arrhenius equation), o= p/po is the density ratio, the heat evolution stage), from various critical diameters. a, (v= 1, 2, 3)are the constants, and index s labels proper- and from energies of direct detonation initiation in tubes. All ties at the lead shock wave these semi-empirical methods suffer significant error In Ref. [151 the critical energy of spherical detonation associated with uncertainty of the measured cell sizes tiation is defined as: (because in most of practically important mixtures the cell structure is quite irregular)and with rather too approximate E3 equations that do not take into account all the gasdynamic and chemical factors and nonuniqueness of the relation where Mc is the mach number of the cj de between the detonation front thickness. cell size. and J=9/(vD1) reaction time. The method based on measurements of the In Ref [152], the critical energy of spherical detonation initiation energy of plane detonation (as illustrated by initiation is considered to be proportional to the induction Fig. 21)has some advantages since it does not require ne length, Lind, which is calculated on the basis of measurements of poorly reproducible parameters and admits detailed kinetic mechanism measuring initiation energies of mixtures with very low ES=BL reactivity in the laboratory-scale equipment. The limiting diameter of detonation propagation in tubes is the lowest of where the coefficient b is determined from a measured value all the types of critical diameters usually measured of E, for a fixed mixture composition and then considered detonation studies, and a length of the tube which limits Ter, constant for other mixtures of the given fuel. an be taken as large as required to make measurements The formulas of other available models show a much with mixtures possessing very low detonability greater discrepancy when compared with experimental data The measured minimum initiation energies for some and therefore are not discussed here fuel-oxygen mixtures are listed in Table 4[155]. For the Fig. 22 [153] shows the comparison of predicted and FAMs, the values of E3(in kg of Tetryl)were measured in measured critical explosive charge mass, me, required for Ref [156](see Table 5) of spherical detonations of ethylene-air an Another serious problem, which arises in detonation pending on fuel concentration. initiation experiments and may cause misleading inferences, The correlation between me and Es is given by is the rate of heat deposition by the initiation source. In this Epl respect, all the sources can be divided into two groups: the telkg TNT] 4.520×10° rst one represents sources where the blast wave with the maximum amplitude at the front is formed already within In general, the agreement between the predicted and the source and the second comprises sources with energy measured results can be treated as satisfactory deposition distributed in time. High explosives and Thus, the energy of direct detonation detonating gases are typical representatives of the first estimated from measured detonation cell sizes, ignition group, while electrical devices can be related to the second delays(or more precisely reaction times that include also e. For the first group, the governing parameter is △ 口◇ 1234 ▲5 6 10 盒 △ 中 Fig 22. Critical explosive charge mass, me, for initiation of spherical detonations vs. molar fraction of fuel, yr(a)C2H4-air mixture Symbols- xperiments [143]: 1-5--detonations, 6--deflagration. Curve 7-models [143 144], curve 8-models [150, 154]:(b )H2-air mixture ymbols-experiments[143 ] 1-5-detonations Curve 6--models [143, 144], curve 7-models [ 150, 154
using the Arrhenius equation), s ¼ r=r0 is the density ratio, anðn ¼ 1; 2; 3Þ are the constants, and index s labels properties at the lead shock wave. In Ref. [151], the critical energy of spherical detonation initiation is defined as: E3 ¼ 2197pg0JM2 CJ 16 p0a3 where MCJ is the Mach number of the CJ detonation and J < q=ðvD2 CJÞ: In Ref. [152], the critical energy of spherical detonation initiation is considered to be proportional to the induction zone length, Lind; which is calculated on the basis of a detailed kinetic mechanism: E3 ¼ BL3 ind where the coefficient B is determined from a measured value of E3 for a fixed mixture composition and then considered constant for other mixtures of the given fuel. The formulas of other available models show a much greater discrepancy when compared with experimental data and therefore are not discussed here. Fig. 22 [153] shows the comparison of predicted and measured critical explosive charge mass, mc; required for initiation of spherical detonations of ethylene–air and hydrogen–air mixtures depending on fuel concentration. The correlation between mc and E3 is given by: mc½kg TNT ¼ E3½J 4:520 £ 106 In general, the agreement between the predicted and measured results can be treated as satisfactory. Thus, the energy of direct detonation initiation can be estimated from measured detonation cell sizes, ignition delays (or more precisely reaction times that include also the heat evolution stage), from various critical diameters, and from energies of direct detonation initiation in tubes. All these semi-empirical methods suffer significant errors associated with uncertainty of the measured cell sizes (because in most of practically important mixtures the cell structure is quite irregular) and with rather too approximate equations that do not take into account all the gasdynamic and chemical factors and nonuniqueness of the relation between the detonation front thickness, cell size, and reaction time. The method based on measurements of the initiation energy of plane detonation (as illustrated by Fig. 21) has some advantages since it does not require measurements of poorly reproducible parameters and admits measuring initiation energies of mixtures with very low reactivity in the laboratory-scale equipment. The limiting diameter of detonation propagation in tubes is the lowest of all the types of critical diameters usually measured in detonation studies, and a length of the tube which limits rcr; can be taken as large as required to make measurements with mixtures possessing very low detonability. The measured minimum initiation energies for some fuel–oxygen mixtures are listed in Table 4 [155]. For the FAMs, the values of E3 (in kg of Tetryl) were measured in Ref. [156] (see Table 5). Another serious problem, which arises in detonation initiation experiments and may cause misleading inferences, is the rate of heat deposition by the initiation source. In this respect, all the sources can be divided into two groups: the first one represents sources where the blast wave with the maximum amplitude at the front is formed already within the source and the second comprises sources with energy deposition distributed in time. High explosives and detonating gases are typical representatives of the first group, while electrical devices can be related to the second one. For the first group, the governing parameter is Fig. 22. Critical explosive charge mass, mc; for initiation of spherical detonations vs. molar fraction of fuel, cf (a) C2H4 –air mixture. Symbols— experiments [143]: 1–5—detonations, 6—deflagration. Curve 7—models [143,144], curve 8—models [150,154]; (b) H2 –air mixture. Symbols—experiments [143]: 1–5—detonations. Curve 6—models [143,144], curve 7—models [150,154]. 570 G.D. Roy et al. / Progress in Energy and Combustion Science 30 (2004) 545–672
G D. Roy et al. Progress in Energy and Combustion Science 30(2004)545-672 571 Table 4 This can be done by varying the geometry Measured minimum initiation energies for some fuel-oxygen confinement, for example, by initiating the detonation in a ixtures [155] tube and then letting it enter an unconfined cloud, or placing Fuel Fuel %o in the most E3 o) obstacles on the way of the blast wave(with a low blockage detonable mixture ratio), or else by he charges with shells(dense shells allow the blast wave generated by the primary explosion to decay 0.11 more slowly ). Even in semi-confined areas, one can reduce ubstantially the minimum charge capable of initiating Allene 3333 ometry, for example, by spreading the same amount of HE over a solid surface. Two conditions are to be met here Vinylmethyl ether 0.62 in order to get reliable initiation with the same amount of 25.0 HE. First, the layer should not be thinner than that providing Ethylene the critical energy for plane detonation initiation, and the second, the lateral rarefaction wave should not merge at the Diethyl ether charge axis until the blast wave travels beyond the critical Detonation can also be initiated by sources, which do not te strong shock romising technique for detonation initiation with relatively weak sources has been the energy released due to detonation of a charge, provided suggested and validated experimentally by Frolov et al that the blast wave entering the mixture to be initiated has [158, 159]. Here, distributed external energy sources are the parameters higher than those of the lead shock wave of used to artificially induce exothermic reactions behind a C detonation. It is shown [157] that detonating gases give relatively weak shock wave in order to stimulate strong the same energy of direct detonation initiation as high coupling between the shock wave and energy deposition In explosives do solely when the shock amplitude produced by the experiments, a weak shock wave was accelerated in the the initiating mixture ('donor')is not lower than the shoc reactive mixture by means of in-phase triggering of seven pressure in the wave leading the C detonation in the test electrical discharges in the course of shock wave propa- mixture (acceptor) gation along the tube. Detonation-like regimes have been r the sources of the second obtained at a distance of 0.6-0.7 m in the stoichiometric (e.g. electrical discharges), both experiments and calculations show that gaseous propane-air mixture under normal conditions in a there is one parameter on which the critical energy depends. smooth-walled 2 in.-diameter tube. Moreover, it has been found that the total critical detonation initiation energy was this is the source power or characteristic time of energy significantly less than that required for direct detonation evolution. Fast energy evolution means that all the energy of initiation with a single electric discharge the electrical discharge has been released before the onset of In Ref [1601. this technique has been applied to spray detonation so that this energy deposition can be approxi- mately treated as an instantaneous explosion on the time detonation initiation(see Section 2.3.3). Here, spontaneous scale relevant to detonation initiation. At longer times, or or stimulated (e.g. by electrical discharge) ignition of lower source power, a part of the energy deposited does not reactive mixture is used to amplify the shock wave. Frolov contribute to the blast wave production and therefore is los et al. explain the approach by means of simple ID so that more energy should be introduced during the first calculations shown in Fig. 23a-d. Case (a) presents the stage of the electrical discharge, which is most important for rimary(attenuating) shock wave produced by initiator detonation initiation cated at the closed end- wall of the tube Case (b)) show the situation when the external ignition source mounted at a In practice, initiation of detonation can be achieved even certain distance from the end-wall(shown as a horizontal with energy sources somewhat weaker than the critical one. bar with an arrow) is triggered somewhat prior to the Table 5 rimary shock arrival. The external ignition source Measured minimum initiation energies E, (in kg of Tetryl)for some facilitates ignition of the mixture producing a local pressure fuel-air mixtures [156] peak, and the primary shock wave is slightly amplified. Case (c)shows nearly resonantconditions, when the external Methane Ethane Propane n-Butane i-Butane Ethylene ignition source is triggered nearly in phase with primary shock arrival. Finally, case (d) corresponds to resonant 0.08 0.1 0015 (003)30.05)(005)(008)(0010) conditions, when the external ignition source is triggered just in phase with primary shock arrival. Clearly, in case(d) external stimulation of reaction results in detonation b Insufficient to cause gas detonation. nitiation. With increasing the time delay of triggering
the energy released due to detonation of a charge, provided that the blast wave entering the mixture to be initiated has the parameters higher than those of the lead shock wave of CJ detonation. It is shown [157] that detonating gases give the same energy of direct detonation initiation as high explosives do solely when the shock amplitude produced by the initiating mixture (‘donor’) is not lower than the shock pressure in the wave leading the CJ detonation in the test mixture (‘acceptor’). For the sources of the second group (e.g. electrical discharges), both experiments and calculations show that there is one parameter on which the critical energy depends, this is the source power or characteristic time of energy evolution. Fast energy evolution means that all the energy of the electrical discharge has been released before the onset of detonation so that this energy deposition can be approximately treated as an instantaneous explosion on the time scale relevant to detonation initiation. At longer times, or lower source power, a part of the energy deposited does not contribute to the blast wave production and therefore is lost, so that more energy should be introduced during the first stage of the electrical discharge, which is most important for detonation initiation. In practice, initiation of detonation can be achieved even with energy sources somewhat weaker than the critical one. This can be done by varying the geometry of the confinement, for example, by initiating the detonation in a tube and then letting it enter an unconfined cloud, or placing obstacles on the way of the blast wave (with a low blockage ratio), or else by HE charges with shells (dense shells allow the blast wave generated by the primary explosion to decay more slowly). Even in semi-confined areas, one can reduce substantially the minimum charge capable of initiating semi-spherical detonation just by varying the charge geometry, for example, by spreading the same amount of HE over a solid surface. Two conditions are to be met here in order to get reliable initiation with the same amount of HE. First, the layer should not be thinner than that providing the critical energy for plane detonation initiation, and the second, the lateral rarefaction wave should not merge at the charge axis until the blast wave travels beyond the critical radius. Detonation can also be initiated by sources, which do not produce strong shock waves. A promising technique for detonation initiation with relatively weak sources has been suggested and validated experimentally by Frolov et al. [158,159]. Here, distributed external energy sources are used to artificially induce exothermic reactions behind a relatively weak shock wave in order to stimulate strong coupling between the shock wave and energy deposition. In the experiments, a weak shock wave was accelerated in the reactive mixture by means of in-phase triggering of seven electrical discharges in the course of shock wave propagation along the tube. Detonation-like regimes have been obtained at a distance of 0.6–0.7 m in the stoichiometric gaseous propane–air mixture under normal conditions in a smooth-walled 2 in.-diameter tube. Moreover, it has been found that the total critical detonation initiation energy was significantly less than that required for direct detonation initiation with a single electric discharge. In Ref. [160], this technique has been applied to spray detonation initiation (see Section 2.3.3). Here, spontaneous or stimulated (e.g. by electrical discharge) ignition of reactive mixture is used to amplify the shock wave. Frolov et al. explain the approach by means of simple 1D calculations shown in Fig. 23a–d. Case ðaÞ presents the primary (attenuating) shock wave produced by initiator located at the closed end-wall of the tube. Case ðbÞ) shows the situation when the external ignition source mounted at a certain distance from the end-wall (shown as a horizontal bar with an arrow) is triggered somewhat prior to the primary shock arrival. The external ignition source facilitates ignition of the mixture producing a local pressure peak, and the primary shock wave is slightly amplified. Case ðcÞ shows nearly ‘resonant’ conditions, when the external ignition source is triggered nearly in phase with primary shock arrival. Finally, case ðdÞ corresponds to resonant conditions, when the external ignition source is triggered just in phase with primary shock arrival. Clearly, in case ðdÞ external stimulation of reaction results in detonation initiation. With increasing the time delay of triggering Table 5 Measured minimum initiation energies E3 (in kg of Tetryl) for some fuel–air mixtures [156] Methane Ethane Propane n-Butane i-Butane Ethylene 22a 0.04 (0.03)b 0.08 (0.05) 0.08 (0.05) 0.1 (0.08) 0.015 (0.010) a Extrapolated. b Insufficient to cause gas detonation. Table 4 Measured minimum initiation energies for some fuel–oxygen mixtures [155] Fuel Fuel % in the most detonable mixture E3 (J) Acetylene 40.0 ,0.11 Ethylnitrite 31.0 0.31 Allene 28.6 0.31 Methyacetylene 28.6 0.31 Ethylene oxide 40.0 0.31 Vinylmethyl ether 28.6 0.62 Cyclopropane 25.0 0.62 Ethylene 33.0 0.62 Vinylfluoride 40.0 0.88 Propylene 25.0 1.25 Diethyl ether 40.0 2.50 Propane 22.2 2.50 Ethane 28.6 8.75 Acetaldehyde 40.0 12.50 G.D. Roy et al. / Progress in Energy and Combustion Science 30 (2004) 545–672 571
572 G.D. Roy et al. Progress in Energy and Combustion Science 30(2004)545-672 Normalized Distance (c) Normalized Distance 千(1 000 Normalized distan Normalized Distance Fig. 23. Calculated temporal evolution of pressure waves generated by a hot spot and external energy deposition in a reacting gas:(a)hot spot mixture, (b) hot spot ignitio yal:(c) hot spot ignition followe (d)resonant triggering of energy source resulting ved by ge by triggering of external energy source(shown with a bar and arrow) far prior to gering of external energy source nearly resonant with shock wave arrival; and the external ignition source, the situation becomes again volume at the initial stage of the process. Since the pressure very similar to that shown in Fig. 23b and a. The important rise near the travelling compression wave front shortens the feature of the phenomenon is that the dynamics of the ignition delays in this region. this wave initially driven by system is very sensitive to the triggering time of the external the self-ignition front propagating due to natural termination igniter, other parameters kept unchanged. Note that, in fact, of induction periods in subsequent mixture layers converts the idea of using a sequence of external igniters to initiate gradually into a self-supporting wave and no longer needs detonation goes back to Zel'dovich and Kompaneetz [98 .A the ignition delay gradient. This type of detonation initiatio energy deposition from external sources in the inert medium applications than the direct initiation of detonation ctical ID computational study of shock-wave amplification by ay turn out much more was reported by Thibault et al. 1941 Chemical additives may also reduce the energy require There are examples available in the literature of direct to initiate detonation by blast waves. This effect may be detonation initiation by injecting hot turbulent jets [161] or readily estimated from the Zel'dovich formula. Indeed, me chemical compounds [ 162], as well as by irradiating since the energy of direct initiation depends on the reaction the photosensitive gas [1631, leading to mixture self- time reduction of either find or fer will reduce Er. There are ignition. The mechanism of detonation initiation in these many chemical additives capable of reducing find at high cases is essentially based on the idea first put forward by temperatures within an order of magnitude, but fer is almost Zel'dovich et al. [93] and then developed in many insensitive to additives studied, therefore Er is reduced by theoretical studies [164, 165]. This is self-ignition or flame additives to a much lesser extent than the induction time (at some stage of the process) front acceleration and shock (usually within a factor of less or only slightly higher than wave amplification in mixtures with temperature or 10, instead of several orders as would be expected from the concentration gradients. These produce a gradient of Zel'dovich formula). But nevertheless, as experiment ignition delays, which affects energy release behind a hows, small amounts of organic nitrates, nitrites, or weak compression wave, formed due either to the initial compounds containing NF2 groups, as well as of unsaturated pressure disturbance or to very intense reaction in a certain or higher hydrocarbons being added(in concentrations not
the external ignition source, the situation becomes again very similar to that shown in Fig. 23b and a. The important feature of the phenomenon is that the dynamics of the system is very sensitive to the triggering time of the external igniter, other parameters kept unchanged. Note that, in fact, the idea of using a sequence of external igniters to initiate detonation goes back to Zel’dovich and Kompaneetz [98]. A 1D computational study of shock-wave amplification by energy deposition from external sources in the inert medium was reported by Thibault et al. [94]. There are examples available in the literature of direct detonation initiation by injecting hot turbulent jets [161] or some chemical compounds [162], as well as by irradiating the photosensitive gas [163], leading to mixture selfignition. The mechanism of detonation initiation in these cases is essentially based on the idea first put forward by Zel’dovich et al. [93] and then developed in many theoretical studies [164,165]. This is self-ignition or flame (at some stage of the process) front acceleration and shock wave amplification in mixtures with temperature or concentration gradients. These produce a gradient of ignition delays, which affects energy release behind a weak compression wave, formed due either to the initial pressure disturbance or to very intense reaction in a certain volume at the initial stage of the process. Since the pressure rise near the travelling compression wave front shortens the ignition delays in this region, this wave initially driven by the self-ignition front propagating due to natural termination of induction periods in subsequent mixture layers converts gradually into a self-supporting wave and no longer needs the ignition delay gradient. This type of detonation initiation may turn out much more convenient in many practical applications than the direct initiation of detonation. Chemical additives may also reduce the energy required to initiate detonation by blast waves. This effect may be readily estimated from the Zel’dovich formula. Indeed, since the energy of direct initiation depends on the reaction time reduction of either tind or ter will reduce En: There are many chemical additives capable of reducing tind at high temperatures within an order of magnitude, but ter is almost insensitive to additives studied, therefore En is reduced by additives to a much lesser extent than the induction time (usually within a factor of less or only slightly higher than 10, instead of several orders as would be expected from the Zel’dovich formula). But nevertheless, as experiment shows, small amounts of organic nitrates, nitrites, or compounds containing NF2 groups, as well as of unsaturated or higher hydrocarbons being added (in concentrations not Fig. 23. Calculated temporal evolution of pressure waves generated by a hot spot and external energy deposition in a reacting gas: (a) hot spot ignition of reactive mixture, (b) hot spot ignition followed by triggering of external energy source (shown with a bar and arrow) far prior to shock wave arrival; (c) hot spot ignition followed by triggering of external energy source nearly resonant with shock wave arrival; and (d) resonant triggering of energy source resulting in detonation initiation [159,160]. 572 G.D. Roy et al. / Progress in Energy and Combustion Science 30 (2004) 545–672
G D. Roy et al. Progress in Energy and Combustion Science 30(2004)545-672 higher than 15-20% with respect to the fuel)to simple sufficient to ignite the mixture and give rise to detonation hydrocarbon gases (like propane, methane, ethane) do indicated in Fig. 24c and d. reduce Es for initiation of spherical detonation by factors a device capable of creating a collapsing toroid quite suitable for practical purposes. It should be empha- detonation wave front has been designed and manufactured ized that the effect of promoters is strongly dependent on in Ref. [1301(Fig. 25). The goal is to generate pressures and the nature of both the fuel and the additive, and therefore the temperatures at the focal point of the collapsing detonation optimum concentration and the type of promoter should b wave that will be sufficient to initiate detonations in sought for individually for each fuel insensitive FAMs inside a detonation tube without blocking Thus, the critical initiation energy is heavily affected by the flow path. This toroidal initiator uses a single spark and the 3D structure of detonation waves, which implies that its an array of small-diameter channels to generate and merge calculation should be based not only on reliable chemical any detonation waves to create a single detonation wave kinetic data, but on the 3D unsteady computer codes. with a toroidal front. Testing was performed with stoichio- Therefore, at present, there is not much hope that numerical metric propane-oxygen mixtures at po= I bar. Images of modeling will furnish quite reliable and easily accessible the detonation front show a nearly circular wave front information on Ey. The semi-empirical relations based on (Fig. 26). To determine the pressure increase achieved by measured parameters relevant to the heat evolution kinetics toroidal focusing, pressure transducers were mounted on a are almost the only source for estimating Er(although the radial line with the central transducer located on the central results obtained using these relations exhibit uncertainty axis of the initiator tube. A typical set of pressure traces is within an order of magnitude for spherical detonations) shown in Fig. 27. The outermost three pressure transducers The overwhelming majority of these semi-empirical show a gradually decreasing pressure wave as the radius of procedures use the only reliably measured parameter the imploding torus decreases. The relevant to the kinetics of heat evolution in detonation transducer. however. recorded a value above its maximum waves, namely, the detonation cell size, which unfortunately reliable operating range. This value was four times larger is not uniquely related to the real reaction zone length than the c pressure for the mixture. detonation waves The most reliable direct measurements of Er is a time 2. 2. 4. Detonation transition Since the energy required to detonation FAMs, therefore it is relatively seldom used. It should be particularly in FAMS, is so large that it ely difficult emphasized that E, can be varied within a limited range by from a ypical energy source. Detonation is known to arise both physical and chemical means fixture preconditioning can substantially reduce the most readily in long narrow ducts. Hence, in practice. it is initiation energy. The most illustrative examples of this important to estimate the probability of transition of preconditioning are initiation of detonation after reflection detonation to unconfined or semi-confined large mixture volumes from where it can be excited by weak sources. For of weak shock waves from concave end-plates, after this reason, numerous investigations were conducted to imploding shock waves, and in an expanding flow Incident tudy critical conditions for detonation transition from a shock waves preheat the mixture and generate after tube to an unconfined mixture cloud or to a tube of a much reflection a hot spot (a region of finite size with a temperature gradient and high temperature at the center larger diameter. Here a parameter controlling the transition is the critical diameter of the narrow tube. d. The values of capable of self-igniting the mixture). The temperature this critical diameter were estimated for many fuel-oxygen gradient favors fast coupling between the compression and FAMs. They range from millimeters to more than one wave generated by mixture self-ignition at the hot spot meter. It is natural to expect that der should depend on the enter and heat release in the adjacent mixture layers which distance between the lead shock front and the effective cj ends up in detonation onset. Depending on the fuel type and plane, Lc, which can be expressed through the transverse end-plate geometry, the shock Mach number needed to cell size, a, of the detonation front. Experiments show that initiate detonation in a FAM initially at room temperature usually the ratio of der to the cell size, dera, is close to13 for can be reduced to about 2, which means a significant round tubes and 7 for slots [84]. Although there are some eduction of the energy to be deposited for generating mixtures where the ratio reaches even 46 detonation. As an example, Fig. 24 shows the results of Fig. 28a-c present three series of Schlieren photographs computer simulation [166] of detonation initiation behind shock wave reflected from the lateral wall of cylindrical detonation reignition(near-critical case) and detonation cavity filled with stoichiometric hydrogen-oxygen mixture decay (subcritical case) in hydrogen-oxygen-argon at To=300 K, Po=0. 1 bar. The initial intensity of the mixture [1671 shock wave in the channel attached to the cavity is as low as To understand the reason why the ratio dera is nearly Ms=2.2. The local pressure and temperature peaks in the constant and why it is close to the above numbers, it is gasdynamic focus formed after shock reflection ar necessary to analyse the flow pattern near a step-wise
higher than 15–20% with respect to the fuel) to simple hydrocarbon gases (like propane, methane, ethane) do reduce E3 for initiation of spherical detonation by factors quite suitable for practical purposes. It should be emphasized that the effect of promoters is strongly dependent on the nature of both the fuel and the additive, and therefore the optimum concentration and the type of promoter should be sought for individually for each fuel. Thus, the critical initiation energy is heavily affected by the 3D structure of detonation waves, which implies that its calculation should be based not only on reliable chemical kinetic data, but on the 3D unsteady computer codes. Therefore, at present, there is not much hope that numerical modeling will furnish quite reliable and easily accessible information on En: The semi-empirical relations based on measured parameters relevant to the heat evolution kinetics are almost the only source for estimating En (although the results obtained using these relations exhibit uncertainty within an order of magnitude for spherical detonations). The overwhelming majority of these semi-empirical procedures use the only reliably measured parameter relevant to the kinetics of heat evolution in detonation waves, namely, the detonation cell size, which unfortunately is not uniquely related to the real reaction zone length in detonation waves. The most reliable direct measurements of En is a time consuming and very expensive procedure, particularly for FAMs, therefore it is relatively seldom used. It should be emphasized that En can be varied within a limited range by both physical and chemical means. Mixture preconditioning can substantially reduce the initiation energy. The most illustrative examples of this preconditioning are initiation of detonation after reflection of weak shock waves from concave end-plates, after imploding shock waves, and in an expanding flow. Incident shock waves preheat the mixture and generate after reflection a hot spot (a region of finite size with a temperature gradient and high temperature at the center capable of self-igniting the mixture). The temperature gradient favors fast coupling between the compression wave generated by mixture self-ignition at the hot spot center and heat release in the adjacent mixture layers which ends up in detonation onset. Depending on the fuel type and end-plate geometry, the shock Mach number needed to initiate detonation in a FAM initially at room temperature can be reduced to about 2, which means a significant reduction of the energy to be deposited for generating detonation. As an example, Fig. 24 shows the results of computer simulation [166] of detonation initiation behind a shock wave reflected from the lateral wall of cylindrical cavity filled with stoichiometric hydrogen–oxygen mixture at T0 ¼ 300 K, p0 ¼ 0:1 bar. The initial intensity of the shock wave in the channel attached to the cavity is as low as Ms ¼ 2:2: The local pressure and temperature peaks in the gasdynamic focus formed after shock reflection are sufficient to ignite the mixture and give rise to detonation as indicated in Fig. 24c and d. A device capable of creating a collapsing toroidal detonation wave front has been designed and manufactured in Ref. [130] (Fig. 25). The goal is to generate pressures and temperatures at the focal point of the collapsing detonation wave that will be sufficient to initiate detonations in insensitive FAMs inside a detonation tube without blocking the flow path. This toroidal initiator uses a single spark and an array of small-diameter channels to generate and merge many detonation waves to create a single detonation wave with a toroidal front. Testing was performed with stoichiometric propane–oxygen mixtures at p0 ¼ 1 bar. Images of the detonation front show a nearly circular wave front (Fig. 26). To determine the pressure increase achieved by toroidal focusing, pressure transducers were mounted on a radial line with the central transducer located on the central axis of the initiator tube. A typical set of pressure traces is shown in Fig. 27. The outermost three pressure transducers show a gradually decreasing pressure wave as the radius of the imploding torus decreases. The central pressure transducer, however, recorded a value above its maximum reliable operating range. This value was four times larger than the CJ pressure for the mixture. 2.2.4. Detonation transition Since the energy required to initiate detonation, particularly in FAMs, is so large that it is extremely difficult to generate the conditions where direct initiation can result from a typical energy source. Detonation is known to arise most readily in long narrow ducts. Hence, in practice, it is important to estimate the probability of transition of detonation to unconfined or semi-confined large mixture volumes from where it can be excited by weak sources. For this reason, numerous investigations were conducted to study critical conditions for detonation transition from a tube to an unconfined mixture cloud or to a tube of a much larger diameter. Here a parameter controlling the transition is the critical diameter of the narrow tube, dcr: The values of this critical diameter were estimated for many fuel–oxygen and FAMs. They range from millimeters to more than one meter. It is natural to expect that dcr should depend on the distance between the lead shock front and the effective CJ plane, LCJ; which can be expressed through the transverse cell size, a; of the detonation front. Experiments show that usually the ratio of dcr to the cell size, dcr=a; is close to 13 for round tubes and 7 for slots [84]. Although there are some mixtures where the ratio reaches even 46. Fig. 28a–c present three series of Schlieren photographs relevant to detonation transition (superctitical case), detonation reignition (near-critical case) and detonation decay (subcritical case) in hydrogen–oxygen–argon mixture [167]. To understand the reason why the ratio dcr=a is nearly constant and why it is close to the above numbers, it is necessary to analyse the flow pattern near a step-wise G.D. Roy et al. / Progress in Energy and Combustion Science 30 (2004) 545–672 573
G.D. Roy et al. Progress in Energy and Combustion Science 30(2004)545-672 Fig. 24. Numerical simulation of initiation of on in stoichiometric H,-0, mixture behind a shock wave reflected from wall of cylindrical cavity [166]. Initial shock Mach n the channel attached to the cavity is 2.2. Upper halves of figures show isobars. lower--isochors for different time instants re the time of spontaneous ignition:(a)-33μs,(b)-8μs,(c)+11μs,and(d)+39μs change of the tube cross-section(Fig. 29a [55, 1681).When detonation wave 1 exits from a channel it generates a diffracted shock front 2 at the periphery(Fig. 29b[169) The temperature drop in this wave portion is so large that ignition ceases behind it. Thus, transverse waves 3 travelling over the detonation front meet no partners to collide with at the periphery. The soot tracks show clearly(Fig. 29a)that a kind of a phase wave of cell disappearance originates at the tube rim and propagates toward the tube axis. This wave propagates at the velocity of transverse wave motion(which is approximately 0. 6Dcj). Lateral expansion of the gas at the tube rim produces a rarefaction wave fan, the head of which (curve 4 in Fig. 29b) spreads toward the tube axis. It spreads through the unburnt mixture compressed by the lead shock Fig. 25. Schematic of annular detonation wave initiator(coverin front of the detonation wave, since it is this wave which can shell omitted for clarity)[130]
change of the tube cross-section (Fig. 29a [55,168]). When detonation wave 1 exits from a channel it generates a diffracted shock front 2 at the periphery (Fig. 29b [169]). The temperature drop in this wave portion is so large that ignition ceases behind it. Thus, transverse waves 3 travelling over the detonation front meet no partners to collide with at the periphery. The soot tracks show clearly (Fig. 29a) that a kind of a phase wave of cell disappearance originates at the tube rim and propagates toward the tube axis. This wave propagates at the velocity of transverse wave motion (which is approximately 0:6DCJ). Lateral expansion of the gas at the tube rim produces a rarefaction wave fan, the head of which (curve 4 in Fig. 29b) spreads toward the tube axis. It spreads through the unburnt mixture compressed by the lead shock front of the detonation wave, since it is this wave which can Fig. 24. Numerical simulation of initiation of detonation in stoichiometric H2 –O2 mixture behind a shock wave reflected from the lateral wall of cylindrical cavity [166]. Initial shock Mach number in the channel attached to the cavity is 2.2. Upper halves of figures show predicted isobars, lower—isochors for different time instants relative to the time of spontaneous ignition: (a) 233 ms, (b) 28 ms, (c) þ11 ms, and (d) þ39 ms. Fig. 25. Schematic of annular detonation wave initiator (covering shell omitted for clarity) [130]. 574 G.D. Roy et al. / Progress in Energy and Combustion Science 30 (2004) 545–672