G D. Roy et al. Progress in Energy and Combustion Science 30(2004)545-672 stage kinetics of hydrogen-oxygen reaction). Thus, one should anticipate the effect of additives on the induction period alone. Therefore, the effect of additives on the parameters of marginal detonations is much less pronounced than their effect on ignition delays With respect to the effect of additives on the ignition delays, various kinetic mechanisms of chain-thermal> reactions taking place under adiabatic or isothermal conditions have been analyzed in detail [133]. This study has demonstrated that the promoter effects depend on the D type of the ignition reaction and the nature of the additive. The production of active species from the promoter must be adjusted to the oxidation reaction of the basic fuel: it must be fast but not too fast, otherwise, the active species would recombine faster rather than they enter in the chain 0.60.8 propagation reactions. The promoter effect levels off as the promoter concentration increases, therefore, as follows 103/T/K1 from various estimates. there is no reason to add more than 15 or 20% of promoters to fuels. The reduction of ignition C,Hs +16.4% O,+ Ar mixture (1) and the same mixture with delays is less when the temperature is higher. The larger the 0.036% iso-C,,ONO,(2), 0.18% iso-C3H-ONO,(3),0.54% hydrocarbon molecule, the lower the promoting effect is-C3H7ONO2(4),0.036%N2F4(5),0.036%CH3ONO2(6,and Table 2 illustrates the effect of various additives (introduced 0. 18% N2 F4 (7)[133]. in amount of 1% with respect to fuel)on the ignition delays of 6CH4 1202+82Ar mixture at 1000 K and 1 atm. both the rich and lean detonation limits are very close to The most efficient promoters for hydrocarbons are those their flammability counterparts, whereas hydrocarbon-air that serve as homogeneous catalysts. Among them are mixtures with inhibitor additives (tetrafluoro-dibromo- organic nitrates and fluoronitrates. In the case of nitrates, ethane) detonate in a much wider range of the inhibitor pseudo-radicals NO react with the fuel or oxygen to produce concentration than they burn [134]. This is not surprisin radicals or atoms, and then recover to their initial state. In because the reaction mechanism governing propagation of the case of tetrafluorohydrazine, react with HO2 flames and detonations is quite different. Reactions in radicals to produce fluorin IcaIs I yl, and FNO. Then detonation waves are essentially of the self-ignition type, reactions F+ H,0= HF+OH, OH +CO= CO,+H hereas in flames they start in the preheat zone at relatively and FNO= F+ NO follow that introduce the No pseudo- low temperatures due to radical(mostly H atoms)diffusion radical in the system. The effect of most efficient additives to this zone Inhibitor additives suppress these reactions by on self-ignition of a 3. 6%0 C3H8 164% O2+ Ar mixture scavenging the radicals, and that is why the flame is is shown in Fig. 19 quenched. In as much as in detonation waves the reaction As far as the influence of additives in promoting the starts at a high temperature in the shocked gas, at which the detonation parameters is concerned, experiment shows that inhibitor molecules decompose very fast, it is not affected both the minimum energies of direct initiation of detonation by the additives(or sometimes can even be enhanced by the and limiting diameters of detonation can be reduced by a radicals formed in the course of inhibitor decomposition). factor usually not exceeding two This conclusion is supported by the data on self-ignition of ompare the detonation and flamm ydrocarbon-air mixtures with additives behind reflected ability limits under the same conditions. All the early shock waves in shock tubes [1351 experimental data furnished evidence that not all of It is worth to mention one experimental fact indicating the flammable mixtures could detonate, and only quite that the maximum velocity deficit(as compared to the CJ recently it has been discovered that this is not always true. detonation velocity) for the essentially 3D spinning mode For instance, for ethane-air and propylene-air mixtures does not exceed ten percent, which is in good agreement with the available theories of detonation limits (e.g. Zel'dovich theory [55,56,981). It is not unexpected, since of various additives(introduced in amount of 1% with respect this model is applicable to real detonations, but with a slight fuel)on the ignition delays of 6CH4+ 1202+ 82Ar mixture at nodification that the longest reaction zone influencing the 1000 K and I atm [133] limit is located near the walls just in front of the transverse Additive Cl2 CH3I H, (CH3)3N2 N2F? CHio CH3CHO detonation wave. This zone is most sensitive to the flow fluctuations due to the largest Arrhenius exponent E/RT. 121.21.22.5-3.03.01.5-22 The fact that the local decoupling may dramatically affect the detonation wave is clearly shown by experimental runs
the popular two-stage kinetics of hydrogen–oxygen reaction). Thus, one should anticipate the effect of additives on the induction period alone. Therefore, the effect of additives on the parameters of marginal detonations is much less pronounced than their effect on ignition delays. With respect to the effect of additives on the ignition delays, various kinetic mechanisms of chain-thermal reactions taking place under adiabatic or isothermal conditions have been analyzed in detail [133]. This study has demonstrated that the promoter effects depend on the type of the ignition reaction and the nature of the additive. The production of active species from the promoter must be adjusted to the oxidation reaction of the basic fuel: it must be fast but not too fast, otherwise, the active species would recombine faster rather than they enter in the chain propagation reactions. The promoter effect levels off as the promoter concentration increases, therefore, as follows from various estimates, there is no reason to add more than 15 or 20% of promoters to fuels. The reduction of ignition delays is less when the temperature is higher. The larger the hydrocarbon molecule, the lower the promoting effect. Table 2 illustrates the effect of various additives (introduced in amount of 1% with respect to fuel) on the ignition delays of 6CH4 þ 12O2 þ 82Ar mixture at 1000 K and 1 atm. The most efficient promoters for hydrocarbons are those that serve as homogeneous catalysts. Among them are organic nitrates and fluoronitrates. In the case of nitrates, pseudo-radicals NO react with the fuel or oxygen to produce radicals or atoms, and then recover to their initial state. In the case of tetrafluorohydrazine, radicals react with HO2 radicals to produce fluorine atom, hydroxyl, and FNO. Then reactions F þ H2O ¼ HF þ OH, OH þ CO ¼ CO2 þ H, and FNO ¼ F þ NO follow that introduce the NO pseudoradical in the system. The effect of most efficient additives on self-ignition of a 3.6% C3H8 þ 16.4% O2 þ Ar mixture is shown in Fig. 19. As far as the influence of additives in promoting the detonation parameters is concerned, experiment shows that both the minimum energies of direct initiation of detonation and limiting diameters of detonation can be reduced by a factor usually not exceeding two. It is of interest to compare the detonation and flammability limits under the same conditions. All the early experimental data furnished evidence that not all of the flammable mixtures could detonate, and only quite recently it has been discovered that this is not always true. For instance, for ethane–air and propylene–air mixtures both the rich and lean detonation limits are very close to their flammability counterparts, whereas hydrocarbon–air mixtures with inhibitor additives (tetrafluoro-dibromoethane) detonate in a much wider range of the inhibitor concentration than they burn [134]. This is not surprising, because the reaction mechanism governing propagation of flames and detonations is quite different. Reactions in detonation waves are essentially of the self-ignition type, whereas in flames they start in the preheat zone at relatively low temperatures due to radical (mostly H atoms) diffusion to this zone. Inhibitor additives suppress these reactions by scavenging the radicals, and that is why the flame is quenched. In as much as in detonation waves the reaction starts at a high temperature in the shocked gas, at which the inhibitor molecules decompose very fast, it is not affected by the additives (or sometimes can even be enhanced by the radicals formed in the course of inhibitor decomposition). This conclusion is supported by the data on self-ignition of hydrocarbon–air mixtures with additives behind reflected shock waves in shock tubes [135]. It is worth to mention one experimental fact indicating that the maximum velocity deficit (as compared to the CJ detonation velocity) for the essentially 3D spinning mode does not exceed ten percent, which is in good agreement with the available theories of detonation limits (e.g. Zel’dovich theory [55,56,98]). It is not unexpected, since this model is applicable to real detonations, but with a slight modification that the longest reaction zone influencing the limit is located near the walls just in front of the transverse detonation wave. This zone is most sensitive to the flow fluctuations due to the largest Arrhenius exponent E=RT: The fact that the local decoupling may dramatically affect the detonation wave is clearly shown by experimental runs, Fig. 19. Arrhenius plots of measured ignition delays for 3.6% C3H8 þ 16.4% O2 þ Ar mixture (1) and the same mixture with 0.036% iso-C3H7ONO2 (2), 0.18% iso-C3H7ONO2 (3), 0.54% iso-C3H7ONO2 (4), 0.036% N2F4 (5), 0.036% CH3ONO2 (6), and 0.18% N2F4 (7) [133]. Table 2 Effect of various additives (introduced in amount of 1% with respect to fuel) on the ignition delays of 6CH4 þ 12O2 þ 82Ar mixture at 1000 K and 1 atm [133] Additive Cl2 CH3I H2 (CH3)2N2 N2F2 C4H10 CH3CHO ti;na=ti 1.2 1.2 1.2 2.5–3.0 3.0 1.5–2 2 G.D. Roy et al. / Progress in Energy and Combustion Science 30 (2004) 545–672 565
566 G.D. Roy et al. Progress in Energy and Combustion Science 30(2004)545-672 suppressed by liquid films [136]. The shadowgraphs enough. The Zel'dovich theory is essentially based on this demonstrate that the local reaction zone destruction is assumption. A comparison of the two components of the followed by a very quick disappearance of the reaction front reaction time (inductio and energy release time fer) all over the tube cross-section shows that for many hydrocarbon-air mixtures they become Thus, the models proposed for planar detonation waves equal to each other in the direct vicinity of the limits. This is describe at least qualitatively the marginal behavior of 3d just confirmation of the above statement. Indeed, the etonation waves. The exact solution of the 3D unsteady temperature sensitivity of find is very high whereas that of problem for the marginal detonation is extremely time fer is very low, therefore, when fer dominates, gasdynamic onsuming. Therefore, for practical purposes a very simple fluctuations do not strongly change the overall reaction zone relation may be suggested. The spinning wave can be quasi length. As calculations of initiation of id detonation by a pin pitch is larger than th point explosion demonstrate, the minimum initiation energ aximum length of the reaction zone. Otherwise, the lso corresponds to the situation at which find becomes less amount of the energy released behind the detonation front than fer after the first dip of the initiating -wave velocity will fluctuate, leading to periodical (or aperiodical) decay of during this dip their relation reverses he lead shock wave and, hence to instantaneous lengthen- In some practical situations the reactive-gas charge can ing of the reaction zone beyond the limit where the reaction be stratified, i.e. a nonreactive gas would serve as its outer completely decouples with the shock front. Thus, one may boundary, instead of solid walls. The critical diameter for write for the limiting detonation diameter Td uf(u is the detonation propagation in this case should be much greater particle velocity in front-fixed frame of reference and I, is Experiments with unconfined cylindrical mixture charges the characteristic reaction time), where in accordance with support this assertion. Rarefaction waves spreading inward many measurements, the angle of the helical spin trajectory the charge result in a peculiar gasdynamic pattern of the is assumed to be close to 45. Analysis of many spin tracks How with various types of transverse and longitudinal shows that because of the fluctuation of the reaction zone perturbations. The detonation cell size and velocity change length the above inequality should be changed to d,= uf, at periodically in the radial and axial directions, the average velocity is 20-30%o lower than its CJ value. Interestingly, This simple formal model of detonation limits allows the critical diameter of unconfined charges for hydro- explaining the virtual independence of the lean detonation carbon -air mixtures is nearly identical [135] with the limits in lean hydrocarbon-air mixtures and the two limits in critical diameter for detonation transition from a narrow hydrogen-air mixtures from the tube diameter when it tube into a wider tube( see Section 2.2. 4). Experiments also exceeds a certain value. The above limiting condition show that even light confinement, like a wire spiral, suggests a logarithmic extension of the limits with tube significantly reduces the critical diameter [136 diameter (indeed, as tube diameter, d, increases, the In previous discussions, the issues dealing with limits of temperature behind the lead shock wave can drop as log d, detonability in single-shot studies were considered. One ce t r exp(ErT), this in turn means that the energy loss more issue extremely important for pulse detonation from the mixture increases also logarithmically). However, propulsic propulsion is the limits of detonability in a pulse mode, in reality this dependence is significantly weaker. In real which is recently investigated by Baklanov et al. [137]. For waves, instability generates not only transverse waves but detonation experiments with gaseous mixtures, a 3 m long longitudinal waves as well. These oscillations cause tube was used. The tube was water-cooled, and the puls periodical fluctuations of the reaction zone length. In tubes frequency was varied from 0.5 to 10 Hz. Mixtures of methane of large diameter. these oscillations affect the reaction zone ith oxygen-enriched air at normal pressure were studied more substantially. This is because, first, the wavelength of Predetonation distances and detonation velocities were the dominant longitudinal oscillations becomes comparable measured as functions of oxidizer- to-fuel ratio, a. The effect ith the reaction zone length, and, second, when the mixture of different vortex generators on shortening the predetona- approaches the limit inherent in large-diameter tubes, the tion distance was also studied. It has been shown that the temperature behind the lead shock wave drops and the e/rT predetonation distance is very sensitive and exhibits a factor gets so high that the reaction zone cannot tolerate even well-known U-shaped behavior. An example of measured very small perturbations(because it will tend to infinity at the dependence of the predetonation length on a for the elongation stage). This implies that the actual reaction zone methane-oxygen-enriched air mixture is given in Fig. 20 length should be much shorter than that permitted by the Presented in the same figure is the measured dependence Zel'dovich theory. One-dimensional numerical calculations of the predetonation distance on a for the case when a vortex of detonation initiation with heat losses taken into account generator is inserted in the detonation tube. The vortex show that the widening of the detonation limits with diameter generator is the inverted Schelkin spiral: on a part of the occurs much slower than logarithmically inner surface of the tube the thread was machined. It follows It should be emphasized that the sharp transition fromfrom the figure, that for fuel-rich mixtures (a 1)the detonation go to condition must occur only when the dependence of Lopt on a is not affected by the vortex temperature sensitivity of the reaction rate is high generator while for fuel-lean mixtures(a> 1) the vortex
suppressed by liquid films [136]. The shadowgraphs demonstrate that the local reaction zone destruction is followed by a very quick disappearance of the reaction front all over the tube cross-section. Thus, the models proposed for planar detonation waves describe at least qualitatively the marginal behavior of 3D detonation waves. The exact solution of the 3D unsteady problem for the marginal detonation is extremely time consuming. Therefore, for practical purposes a very simple relation may be suggested. The spinning wave can be quasisteady solely when the spin pitch is larger than the maximum length of the reaction zone. Otherwise, the amount of the energy released behind the detonation front will fluctuate, leading to periodical (or aperiodical) decay of the lead shock wave and, hence, to instantaneous lengthening of the reaction zone beyond the limit where the reaction completely decouples with the shock front. Thus, one may write for the limiting detonation diameter pdl . utr (u is the particle velocity in front-fixed frame of reference and tr is the characteristic reaction time), where in accordance with many measurements, the angle of the helical spin trajectory is assumed to be close to 458. Analysis of many spin tracks shows that because of the fluctuation of the reaction zone length the above inequality should be changed to dl ¼ utr at the limit. This simple formal model of detonation limits allows explaining the virtual independence of the lean detonation limits in lean hydrocarbon–air mixtures and the two limits in hydrogen–air mixtures from the tube diameter when it exceeds a certain value. The above limiting condition suggests a logarithmic extension of the limits with tube diameter (indeed, as tube diameter, d; increases, the temperature behind the lead shock wave can drop as log d; since tr , expðE=RTÞ; this in turn means that the energy loss from the mixture increases also logarithmically). However, in reality this dependence is significantly weaker. In real waves, instability generates not only transverse waves but longitudinal waves as well. These oscillations cause periodical fluctuations of the reaction zone length. In tubes of large diameter, these oscillations affect the reaction zone more substantially. This is because, first, the wavelength of the dominant longitudinal oscillations becomes comparable with the reaction zone length, and, second, when the mixture approaches the limit inherent in large-diameter tubes, the temperature behind the lead shock wave drops and the E=RT factor gets so high that the reaction zone cannot tolerate even very small perturbations (because it will tend to infinity at the elongation stage). This implies that the actual reaction zone length should be much shorter than that permitted by the Zel’dovich theory. One-dimensional numerical calculations of detonation initiation with heat losses taken into account show that the widening of the detonation limits with diameter occurs much slower than logarithmically. It should be emphasized that the sharp transition from detonation go to no-go condition must occur only when the temperature sensitivity of the reaction rate is high enough. The Zel’dovich theory is essentially based on this assumption. A comparison of the two components of the reaction time (induction time tind and energy release time ter) shows that for many hydrocarbon–air mixtures they become equal to each other in the direct vicinity of the limits. This is just confirmation of the above statement. Indeed, the temperature sensitivity of tind is very high whereas that of ter is very low, therefore, when ter dominates, gasdynamic fluctuations do not strongly change the overall reaction zone length. As calculations of initiation of 1D detonation by a point explosion demonstrate, the minimum initiation energy also corresponds to the situation at which tind becomes less than ter after the first dip of the initiating-wave velocity (during this dip their relation reverses). In some practical situations the reactive-gas charge can be stratified, i.e. a nonreactive gas would serve as its outer boundary, instead of solid walls. The critical diameter for detonation propagation in this case should be much greater. Experiments with unconfined cylindrical mixture charges support this assertion. Rarefaction waves spreading inward the charge result in a peculiar gasdynamic pattern of the flow with various types of transverse and longitudinal perturbations. The detonation cell size and velocity change periodically in the radial and axial directions, the average velocity is 20–30% lower than its CJ value. Interestingly, the critical diameter of unconfined charges for hydrocarbon–air mixtures is nearly identical [135] with the critical diameter for detonation transition from a narrow tube into a wider tube (see Section 2.2.4). Experiments also show that even light confinement, like a wire spiral, significantly reduces the critical diameter [136]. In previous discussions, the issues dealing with limits of detonability in single-shot studies were considered. One more issue extremely important for pulse detonation propulsion is the limits of detonability in a pulse mode, which is recently investigated by Baklanov et al. [137]. For detonation experiments with gaseous mixtures, a 3 m long tube was used. The tube was water-cooled, and the pulse frequency was varied from 0.5 to 10 Hz. Mixtures of methane with oxygen-enriched air at normal pressure were studied. Predetonation distances and detonation velocities were measured as functions of oxidizer-to-fuel ratio, a: The effect of different vortex generators on shortening the predetonation distance was also studied. It has been shown that the predetonation distance is very sensitive to a and exhibits a well-known U-shaped behavior. An example of measured dependence of the predetonation length on a for the methane–oxygen-enriched air mixture is given in Fig. 20. Presented in the same figure is the measured dependence of the predetonation distance on a for the case when a vortex generator is inserted in the detonation tube. The vortex generator is the inverted Schelkin spiral: on a part of the inner surface of the tube the thread was machined. It follows from the figure, that for fuel-rich mixtures ða , 1Þ the dependence of LDDT on a is not affected by the vortex generator while for fuel-lean mixtures ða . 1Þ the vortex 566 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 propagation of the detonation wave and e is the internal energy of the gas within this zone. For convenience, the gas parameters can be related to the steady Cj detonation wave with the finite reaction zone because. as it will be shown below, usually rer is markedly larger than the detonation ce size. This is accounted for by the fact that the distance between the lead shock front and the effective Cj plane, Lcj in multifont detonations is greater than the longitudinal cell size b, and since the rarefaction wave that follows the blast wave produced by the initiator is ves pe coenergy en for a longer distance in order to preclude its decay. Thus, physical considerations suggest that rer should be significantly longer than Lc. If the total energy averaged over the [0, Ter l interval is removed from the integral sign in Eqs. (1)-(3)one arrives at the following simple relations: Fig. 20. Measured predetonation length ys. oxidizer-to-fuel ratio a E1kI'rer E3=kgre for methane-oxygen-enriched air mixture without (1)and with (2) where indices 1, 2, and 3 denote planar, cylindrical, and vortex generator [137]. pherical cases, respectively, and k (v=l, 2, and 3)are the length or provides a noticeable effect on the predetonation corresponding constants. If one goes further assuming that th. a number of experiments have been performed to an approximate proportionality between the reveal the dependencies of detonability limits and pre induction zone length, Lind final, and rer, the above detonation length on the Reynolds number of the inflow of relationships can be rewritten as follows burned combustible mixture in a detonation tube. E= kInd Reynolds number was varied by means of changing the pulse frequency and overall mass flow rate. It has been where k is constant. This latter equation was derived first by shown that an increase in the flow velocity inside the elovich et al. 154. Of course, the above relationships are quite far from the exact ones and are only capable of predicting the general 2.2.3. Direct initiation trends, because they still are based on the concept of a The energy required to initiate detonation directly should mooth id wave and do not take into account the real certainly be evolved at a high rate and in the amount capable structure of detonation waves. Moreover. the flow co of generating a blast wave with an amplitude at least close to ditions behind detonation waves of various geometry that of the shock wave propagating at the Cj velocity and are different, therefore the averaged total energies also will slightly depend on rer. For this reason a direct than that of the reaction induction time. Since the heat experimental study was undertaken to verify the validity behind the detonation front is evolved within a finite time of these relations [138 the critical energy for detonation initiation should exceed Fig. 21 presents the results of measurements of critical ome finite value determined as nergies of detonation initiation in fuel-oxygen (filled symbols) and fuel-air (open symbols) mixtures. The E1≥ measured energies in the graph are grouped near a straight for plane geometry (3) line with a slope equal to 3.0, which is in line with the E2=2 Zel'dovich relation E,=kInd. According to this relation, the critical energy for spherical detonation initiation, E3, is proportional to the reaction time to the third power and the critical energy for plane detonation initiation, El, is r cylindrical geometry proportional to the reaction time to the first power. Hence, in logarithmic coordinates log E3- log En the slope of the E3 E3(E1)-dependence should be 3.0. Although this relation- progress behind the lead shock e) in both the temperature where rer is a certain critical radius which specifies the rear gradient in the reaction zone eometries are boundary of the zone behind the lead front of the blast different. Anyway, this empirical rting the wave possessing an energy sufficient to support further general theoretical model is very helpful in assessing
generator provides a noticeable effect on the predetonation length. A number of experiments have been performed to reveal the dependencies of detonability limits and predetonation length on the Reynolds number of the inflow of unburned combustible mixture in a detonation tube. Reynolds number was varied by means of changing the pulse frequency and overall mass flow rate. It has been shown that an increase in the flow velocity inside the chamber results in widening detonability limits. 2.2.3. Direct initiation The energy required to initiate detonation directly should certainly be evolved at a high rate and in the amount capable of generating a blast wave with an amplitude at least close to that of the shock wave propagating at the CJ velocity and with duration of the pressure pulse comparable or longer than that of the reaction induction time. Since the heat behind the detonation front is evolved within a finite time, the critical energy for detonation initiation should exceed some finite value determined as E1 $ ðrcr 0 re þ r u2 2 !dr for plane geometry ð3Þ E2 $ ðrcr 0 2pr re þ r u2 2 !dr for cylindrical geometry ð4Þ E3 $ ðrcr 0 4pr 2 re þ r u2 2 !dr for spherical geometry ð5Þ where rcr is a certain critical radius which specifies the rear boundary of the zone behind the lead front of the blast wave possessing an energy sufficient to support further propagation of the detonation wave and e is the internal energy of the gas within this zone. For convenience, the gas parameters can be related to the steady CJ detonation wave with the finite reaction zone because, as it will be shown below, usually rcr is markedly larger than the detonation cell size. This is accounted for by the fact that the distance between the lead shock front and the effective CJ plane, LCJ; in multifront detonations is greater than the longitudinal cell size b; and, since the rarefaction wave that follows the blast wave produced by the initiator is very steep, the energy deposited to the mixture must support the reactive wave even for a longer distance in order to preclude its decay. Thus, physical considerations suggest that rcr should be significantly longer than LCJ: If the total energy averaged over the ½0;rcr interval is removed from the integral sign in Eqs. (1)–(3) one arrives at the following simple relations: E1 ¼ k1rcr;1; E2 ¼ k2r 2 cr;2; E3 ¼ k3r 3 cr;3 where indices 1, 2, and 3 denote planar, cylindrical, and spherical cases, respectively, and kn (n ¼1, 2, and 3) are the corresponding constants. If one goes further assuming that there exists an approximate proportionality between the induction zone length, Lind ¼ tindu; and rcr; the above relationships can be rewritten as follows: En ¼ kLn ind where k is constant. This latter equation was derived first by Zel’dovich et al. [54]. Of course, the above relationships are quite far from the exact ones and are only capable of predicting the general trends, because they still are based on the concept of a smooth 1D wave and do not take into account the real structure of detonation waves. Moreover, the flow conditions behind detonation waves of various geometry are different, therefore the averaged total energies also will slightly depend on rcr: For this reason a direct experimental study was undertaken to verify the validity of these relations [138]. Fig. 21 presents the results of measurements of critical energies of detonation initiation in fuel–oxygen (filled symbols) and fuel–air (open symbols) mixtures. The measured energies in the graph are grouped near a straight line with a slope equal to 3.0, which is in line with the Zel’dovich relation En ¼ kLn ind: According to this relation, the critical energy for spherical detonation initiation, E3; is proportional to the reaction time to the third power and the critical energy for plane detonation initiation, E1; is proportional to the reaction time to the first power. Hence, in logarithmic coordinates log E3 2 log E1 the slope of the E3ðE1Þ-dependence should be 3.0. Although this relationship follows from the dimensional analysis, this consistency is somewhat surprising, because the conditions for reaction progress behind the lead shock front (e.g. the temperature gradient in the reaction zone) in both geometries are different. Anyway, this empirical correlation supporting the general theoretical model is very helpful in assessing Fig. 20. Measured predetonation length vs. oxidizer-to-fuel ratio a for methane–oxygen-enriched air mixture without (1) and with (2) vortex generator [137]. G.D. Roy et al. / Progress in Energy and Combustion Science 30 (2004) 545–672 567
568 G.D. Roy et al. Progress in Energy and Combustion Science 30(2004)545-672 the blast wave being only insignificantly supported by the energy evolved in the reaction zone of a very small radius at CHg 7%E the maximum reaction zone length in the end of the H pulsation (in the case of plane initiation, the heat transfer 3% ay cause detonation failure during the pulsations). The 935% presence of the transverse waves generating hot spots that do not allow large reaction zone pulsations makes the blast s&5% wave-reaction zone complex less vulnerable to longitudinal fluctuations and thereby facilitates the initiation process (despite the larger overall thickness of the detonation front) It should be also noted that the discrepancy between the CH incorrectness of the global reaction rate equation used in 5.010.0 many studies. More detailed reaction schemes improved the E/MJ/m agreement, although it still remained insufficient to consider such calculations as a quantitative method for evaluating the Fig. 21. Critical energy of direct initiation of spherical detonation critical energy of direct detonation initiation. (E3)vs. the critical energy of plane detonation initiation(E,)for Calculations using the znd model with inclusion of the arious fuel-oxygen(filled symbols)and fuel-air(open symbols detailed kinetics reveal [ 134] that there are three character- istic ranges of the blast wave mach number within which the the detonability of low-reactivity combustible mixtures For nature of the process is different. In the vicinity of an example, the available data on the critical energy of initiator, with almost instantaneous energy release, the blast detonation initiation in unconfined methane-air mixtures wave initiates a reaction with extremely short ignition k consensus, ranging from I kg TNT to more than 100 kg delays but the overall energy release is negative due to TNT. Based on the presented correlation, a value of about dissociation of the reaction products. In the second stage, the 10 kg TNT is most reasonable reaction, which is already exothermic on the whole The minimum energy of direct detonation initiation is a becomes weakly coupled with the lead blast wave, i.e very attractive criterion for calculating it numerically using these two fronts depart from each other in time. This the Znd model. however the first calculations showed a departure may continue until wave velocity is dramatic discrepancy between the calculated and measured reached which, depending on the energy of the source and energies(sometimes up to 10 for spherical detonations) on the thermodynamic and kinetic parameters of the The calculated energies were always higher than the mixture, can drop even to 0.6D]J, where Dc is the measured ones. This was ascribed to the three-dimension- ermodynamic detonation velocity. The cell size during ality of the real process of detonation onset. This this stage of the initiation process grows and sometimes explanation is quite plausible. First, there is a direct disappears (or to be more precise, of the triple experimental evidence of formation of strong gasdynamic points become illegible) for a short period when the energy perturbations on the incipient detonation front. A single is close to the critical one, thus indicating that the transverse strong transverse wave arises when the blast wave front area waves may attenuate(but not vanish). The third stage is reinitiation of the detonation (if it has been converted for a small, and then the number of perturbations grows qu while into a decoupled nonsmooth shock wave and reaction as the wave front departs from the initiation site. Second, the front) or its acceleration. The CJ state is usually attained transverse waves shorten appreciably the overall reaction after one or several oscillations of the detonation velocity zone attaching it to the lead shock wave and thereby stabilizing the detonation wave. Thus, the transverse waves, The deeper the dip of the detonation velocity, i.e. the closer the source energy to the critical value, the larger is its on the one hand make the initiation of the reaction in th overshoot that follows the minimum. The nature of the detonation wave easier but, on the other, they extend the overshoot is quite clear, after decoupling a large mass of the overall reaction time and, what is particularly importan the distance from the lead shock front to the effective C generates a compression wave within this zone which then surface. As the calculations show, the initiation process is overtakes the lead shock wave and amplifies it. characterized by detonation velocity pulsations of a very It is natural ct the position of the minimum on high-amplitude due to longitudinal instability caused by the the D vs. distance curve with the critical radius introduced rarefaction waves traveling between the initiation centre and arlier, because anyway there is no strict definition of the detonation front. These pulsations naturally increase The experimental data indicate that final transition to the CJ periodically the length of the reaction zone(if the detonation tate occurs approximately at r= 2re wave is treated as a ZND one)and at the beginning may be a Numerical calculation cannot provide quite reliable data reason of detonation failure due to the too rapid decay of on the critical energy of detonation initiation because
the detonability of low-reactivity combustible mixtures. For example, the available data on the critical energy of detonation initiation in unconfined methane–air mixtures lack consensus, ranging from 1 kg TNT to more than 100 kg TNT. Based on the presented correlation, a value of about 10 kg TNT is most reasonable. The minimum energy of direct detonation initiation is a very attractive criterion for calculating it numerically using the ZND model. However, the first calculations showed a dramatic discrepancy between the calculated and measured energies (sometimes up to 104 for spherical detonations). The calculated energies were always higher than the measured ones. This was ascribed to the three-dimensionality of the real process of detonation onset. This explanation is quite plausible. First, there is a direct experimental evidence of formation of strong gasdynamic perturbations on the incipient detonation front. A single strong transverse wave arises when the blast wave front area is small, and then the number of perturbations grows quickly as the wave front departs from the initiation site. Second, the transverse waves shorten appreciably the overall reaction zone attaching it to the lead shock wave and thereby stabilizing the detonation wave. Thus, the transverse waves, on the one hand, make the initiation of the reaction in the detonation wave easier but, on the other, they extend the overall reaction time and, what is particularly important, the distance from the lead shock front to the effective CJ surface. As the calculations show, the initiation process is characterized by detonation velocity pulsations of a very high-amplitude due to longitudinal instability caused by the rarefaction waves traveling between the initiation centre and the detonation front. These pulsations naturally increase periodically the length of the reaction zone (if the detonation wave is treated as a ZND one) and at the beginning may be a reason of detonation failure due to the too rapid decay of the blast wave being only insignificantly supported by the energy evolved in the reaction zone of a very small radius at the maximum reaction zone length in the end of the pulsation (in the case of plane initiation, the heat transfer may cause detonation failure during the pulsations). The presence of the transverse waves generating hot spots that do not allow large reaction zone pulsations makes the blast wave–reaction zone complex less vulnerable to longitudinal fluctuations and thereby facilitates the initiation process (despite the larger overall thickness of the detonation front). It should be also noted that the discrepancy between the calculated and measured E3 is ascribed partially to the incorrectness of the global reaction rate equation used in many studies. More detailed reaction schemes improved the agreement, although it still remained insufficient to consider such calculations as a quantitative method for evaluating the critical energy of direct detonation initiation. Calculations using the ZND model with inclusion of the detailed kinetics reveal [134] that there are three characteristic ranges of the blast wave Mach number within which the nature of the process is different. In the vicinity of an initiator, with almost instantaneous energy release, the blast wave initiates a reaction with extremely short ignition delays but the overall energy release is negative due to dissociation of the reaction products. In the second stage, the reaction, which is already exothermic on the whole, becomes weakly coupled with the lead blast wave, i.e. these two fronts depart from each other in time. This departure may continue until a minimum wave velocity is reached which, depending on the energy of the source and on the thermodynamic and kinetic parameters of the mixture, can drop even to 0:6DCJ; where DCJ is the thermodynamic detonation velocity. The cell size during this stage of the initiation process grows and sometimes disappears (or to be more precise, the traces of the triple points become illegible) for a short period when the energy is close to the critical one, thus indicating that the transverse waves may attenuate (but not vanish). The third stage is reinitiation of the detonation (if it has been converted for a while into a decoupled nonsmooth shock wave and reaction front) or its acceleration. The CJ state is usually attained after one or several oscillations of the detonation velocity. The deeper the dip of the detonation velocity, i.e. the closer the source energy to the critical value, the larger is its overshoot that follows the minimum. The nature of the overshoot is quite clear, after decoupling a large mass of the gas between the two fronts, self-ignites in hot spots and this generates a compression wave within this zone which then overtakes the lead shock wave and amplifies it. It is natural to connect the position of the minimum on the D vs. distance curve with the critical radius introduced earlier, because anyway there is no strict definition of rcr: The experimental data indicate that final transition to the CJ state occurs approximately at r ¼ 2rcr: Numerical calculation cannot provide quite reliable data on the critical energy of detonation initiation because of Fig. 21. Critical energy of direct initiation of spherical detonation ðE3Þ vs. the critical energy of plane detonation initiation ðE1Þ for various fuel–oxygen (filled symbols) and fuel–air (open symbols) mixtures at normal initial pressure [138]. 568 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 the uncertainty of the heat release kinetics, therefore sem The numerical calculations for gaseous mixtures within empirical approaches seem to be most attractive. That is the framework of a lD detonation model [140, 141] describe why a large number of studies are devoted to derivation of in detail a qualitative pattern of ID initiation: attenuation of such semi-empirical relations. an initiating wave at initiator energy E<E,(v= 1, 2, or 3) One of the first approaches was suggested by Troshin and formation of a detonation wave at E2 Ep. In Ref. [139]. He defined rcr as a radius at which two conditions are [142 a quantitative approach for calculating E3 with a satisfied simultaneously, namely, the velocity of the blast arameter taken from experiments is suggested and wave generated by a strong point explosion with the energy implemented for stoichiometric hydrogen-air mixture E3 equals Dc, and the chemical energy released within this within the framework of detailed kinetics model. Other region equals that deposited by the source. The following mixtures needed new calculations. expression was then derived for E3 About 20 approximate models for a ID detonation 0.59pcJ initiation in gaseous mixtures are known so far. All were E3 analyzed previously in Refs. [143, 144. Such models allow the estimation of a value of Ep with some accuracy. where q is the heat effect of chemical reactions. For a In a multifont detonation wave, at any instant of time, stoichiometric hydrogen-oxygen mixture the critical radius the induction zone differs significantly(up to two orders of agnitude) for various elements of the detonation wave was expressed through the length of the induction zone, front. In this case, the use of a uniform ignition delay for the entire front(as in ID models)can strongly misrepresent the 3=402a initiation conditions. The reason for this is that ignition event is governed by a local temperature in the hot spots Thus it has been shown that the critical radius must be much rather than by the average temperature. Such spots in a real larger than the induction zone length Lind and longer that the detonation wave are the sites of collisions of transverse ngitudinal detonation cell size b waves. The account of nonone-dimensional collisions of The above expression is not conducive to be used for shock-wave configurations in a realistic detonation front quick estimation of E,. Therefore, it has been suggested to allows the level of the critical initiation energy to be assume that the above relation between Lind and r derived for oxygen-hydrogen mixtures holds for other a model of multipoint initiation was suggested in Ret. [145 mixtures as well. The coefficient of proportionality nd then modified in Refs. [146, 1471 relating Es and find varies from mixture to mixture within According to the latest version of the model, the energ a factor of 1.4, which is less than the spread of the of individual hot spots, Ehs, and the critical initiation experimental data on lind. Therefore an average value of energies(for v=l, 2, 3), are defined by the following this proportionality coefficient, 4.2 x 1020 J/s, was chosen for practical use [135]. The data presented in Table 3 Ehs=4220,PDa,b2 demonstrate quite a good agreement of the estimates with measured E3 E1 Ehs =AlpoDc b More sophisticated studies based on the analysis of the detonation wave structure lead also to relations that are m(dera) essentially one version of the Zel'dovich formula or another Some authors attempted to estimate Es for spherical E3= 2 tan o(dcr/a)-bEhs=A3 PoDab detonations from the data on the critical diameter of where tan p= alb, e is a parameter in the detor mixture (see Section 2.2.4) or from the limiting diameter model [1481, der is the critical diameter for reinitiation of of detonation propagation in a tube spherical detonations under diffraction(see Section 2.2. 4), a, is the parameter of the strong explosion model, and A, is Table 3 Comparison of calculated and measured values of E3 for various Other approximate models for estimating Er are worth stoichiometric fuel-air mixtures [1351 noting. In Refs. [149, 1501, the following relationships are Fuel in ai Ethane Propane Methane (stoich) rer =8v(0+01-2)Lin kTs 0018 0.007 kg TNT) E=a,pe 0.015 10-100 (estimates) where E is the effective activation energy of the induction period(within the framework of the average descriptie
the uncertainty of the heat release kinetics, therefore semiempirical approaches seem to be most attractive. That is why a large number of studies are devoted to derivation of such semi-empirical relations. One of the first approaches was suggested by Troshin [139]. He defined rcr as a radius at which two conditions are satisfied simultaneously, namely, the velocity of the blast wave generated by a strong point explosion with the energy E3 equals DCJ; and the chemical energy released within this region equals that deposited by the source. The following expression was then derived for E3 : E3 ¼ 4 3 pr 3 cr;3 0:31 rCJu2 CJ 2 þ 0:59pCJ gCJ 2 1 2 p0 gCJ 2 1 2 r0q " # where q is the heat effect of chemical reactions. For a stoichiometric hydrogen–oxygen mixture the critical radius was expressed through the length of the induction zone, Lind; as Rcr;3 ¼ 40 Lind gCJ 2 1 Thus, it has been shown that the critical radius must be much larger than the induction zone length Lind and longer that the longitudinal detonation cell size b: The above expression is not conducive to be used for quick estimation of E3: Therefore, it has been suggested to assume that the above relation between Lind and rcr;3 derived for oxygen–hydrogen mixtures holds for other mixtures as well. The coefficient of proportionality relating E3 and t 3 ind varies from mixture to mixture within a factor of 1.4, which is less than the spread of the experimental data on tind: Therefore an average value of this proportionality coefficient, 4.2 £ 1020 J/s3 , was chosen for practical use [135]. The data presented in Table 3 demonstrate quite a good agreement of the estimates with measured E3: More sophisticated studies based on the analysis of the detonation wave structure lead also to relations that are essentially one version of the Zel’dovich formula or another. Some authors attempted to estimate E3 for spherical detonations from the data on the critical diameter of detonation transition from a tube into the unconfined mixture (see Section 2.2.4) or from the limiting diameter of detonation propagation in a tube. The numerical calculations for gaseous mixtures within the framework of a 1D detonation model [140,141] describe in detail a qualitative pattern of 1D initiation: attenuation of an initiating wave at initiator energy E , En (n ¼ 1; 2, or 3) and formation of a detonation wave at E $ En: In Ref. [142], a quantitative approach for calculating E3 with a parameter taken from experiments is suggested and implemented for stoichiometric hydrogen–air mixture within the framework of detailed kinetics model. Other mixtures needed new calculations. About 20 approximate models for a 1D detonation initiation in gaseous mixtures are known so far. All were analyzed previously in Refs. [143,144]. Such models allow the estimation of a value of En with some accuracy. In a multifront detonation wave, at any instant of time, the induction zone differs significantly (up to two orders of magnitude) for various elements of the detonation wave front. In this case, the use of a uniform ignition delay for the entire front (as in 1D models) can strongly misrepresent the initiation conditions. The reason for this is that ignition event is governed by a local temperature in the hot spots rather than by the average temperature. Such spots in a real detonation wave are the sites of collisions of transverse waves. The account of nonone-dimensional collisions of shock-wave configurations in a realistic detonation front allows the level of the critical initiation energy to be significantly lowered (in comparison with 1D models). Such a model of multipoint initiation was suggested in Ref. [145] and then modified in Refs. [146,147]. According to the latest version of the model, the energy of individual hot spots, Ehs; and the critical initiation energies (for n ¼ 1; 2, 3), are defined by the following formulae: Ehs ¼ 412 anr0D2 CJb2 E1 ¼ pðdcr=aÞ 4b Ehs ¼ A1r0D2 CJb E2 ¼ pðdcr=aÞ 2 Ehs ¼ A2r0D2 CJb2 E3 ¼ 2p tan wðdcr=aÞ 2 bEhs ¼ A3r0D2 CJb3 where tan w ¼ a=b; 1 is a parameter in the detonation cell model [148], dcr is the critical diameter for reinitiation of spherical detonations under diffraction (see Section 2.2.4), an is the parameter of the strong explosion model, and An is the constant. Other approximate models for estimating En are worth noting. In Refs. [149,150], the following relationships are suggested: rcr < 8vg2 ðss þ s21 s 2 2Þ LindRTs 3E En ¼ anp0 8rcr v v where E is the effective activation energy of the induction period (within the framework of the average description Table 3 Comparison of calculated and measured values of E3 for various stoichiometric fuel–air mixtures [135] Fuel in air (stoich.) Ethane Ethylene Propane Methane E3calc (kg TNT) 0.018 0.007 0.07 120 E3exp (kg TNT) 0.035 0.015 0.08 10–100 (estimates) G.D. Roy et al. / Progress in Energy and Combustion Science 30 (2004) 545–672 569