G D. Roy et al. Progress in Energy and Combustion Science 30(2004)545-672 not account for finite-rate chemical kinetics and assumes that all of the fuel is in a vaporized state Of importance is the effect of relatively small fuel additives(up to 20%)on the detonation parameters I is+3200F cla cipated that small additives can hardly influence the aracteristics of steady C detonation waves. Indeed thermodynamic calculations performed in Ref. [95] for the iso-octane-air and n-heptane-air mixtures with admixed hydrogen peroxide(HP) vapor (see Fig. 4) reveal a weak dependence of detonation velocity(Fig 4a), as well as temperature(Fig. 4b), pressure(Fig. 4c). and molecular mass(Fig. 4d) of detonation products on the molar fraction of the additive, a A great body of the experimental data on detonation parameters reveals that, for ove of mixtures and shapes of charges, the measured wave velocities and pressures are fairly well consistent with the ideal thermodynamic calculations. It should be emphasized that recorded pressure profiles exhibit intense oscillations and, therefore, a comparison of calculations with exper iment in this case is often uncertain to a large extent. Usually, in compliance with the thermal theory of detonation limits [98, deviations of the measured detonation wave velocities in mixtures with arrhenius [100] and computed 2 [101] gas temperatures bout 10% even for marginal detonations. However, there (a) and donatio stoichiometric C2H2-O2 re exceptions for special types of detonations heavily onditions temperature and pressul affected by energy and momentum losses in which heat d to c conditions: Tc= 3937K and release kinetics senses only little variations of the gas Pc:33.3 atm, respectively. rameters. These waves require special consideration One of the most important features of detonation waves (see Section 2.2.5). n homogeneous mixtures is the instability that results in Detonation parameters that are measured their essentially 3D and unsteady nature. A major feature of (intrinsically) integration over the duct cross-sect as density measured by absorption of X-rays) detonation wave propagation is shown in Fig. 6a and b Fig 6a shows a typical footprint of detonation on the sooted profiles subject to less pronounced oscillations [99]. The foil mounted on the tube wall [102 Fig 6b[103] in terms veraged density at the end of the zone where the reaction of a series of pressure maps at evenly spaced intervals. The keeps going and transverse waves are still intense is indeed consistent with thermodynamic calculations. Recent connection of the paths of triple points produces the cellular detailed measurements of temperature and pressure histories structure that has become a characteristic feature of gaseous behind a detonation front [1001, using advanced laser detonations. The dimensions of the cellular structure diagnostics, revealed that trends in measurements agree with longitudinal size b and transverse size a-are related to the simulations [101] although certain discrepancies in the properties of the material and the chemical reaction profiles are apparent(Fig. 5) nechanism. Long chemical reaction times or induction This certainly makes the ID ZNd theory very useful times correlate with large detonation cells even though it does not adequately describe peculiarities of The structure of most propagating detonations is usually the detonation wave structure. There are several reasons for much more complex than that shown in Fig. 6, sometimes the detonation parameters to deviate from ideal thermo there are substructures within a detonation cell. and dynamic calculations: wave instability, incomplete reaction sometimes the structures are very irregular. Moreover at the sonic(C)) plane (if it exists in multidimensional ome detonations exhibit essentially 3D structure [1041 waves), and momentum and energy losses. Therefore, a Ref. [104], detailed measurements of the detonation reasonably good agreement between the calculated and structure in a square-section tube were made. The measured detonation parameters is observed only in long schematics of transverse motion of front shocks in cases ducts of a diameter exceeding the limiting value. At short of 2D and 3D detonation structures are shown, respectively, distances from the initiator(of about 1 or 2 m)and in narrow in Fig. 7a and b ducts, the deviations can be quite significant as obvious even Modeling of detonation waves initiated and propagating from id calculations with finite reaction kinetics in real combustion chambers is an efficient method for
not account for finite-rate chemical kinetics and assumes that all of the fuel is in a vaporized state. Of importance is the effect of relatively small fuel additives (up to 20%) on the detonation parameters. It is anticipated that small additives can hardly influence the characteristics of steady CJ detonation waves. Indeed, thermodynamic calculations performed in Ref. [95] for the iso-octane–air and n-heptane–air mixtures with admixed hydrogen peroxide (HP) vapor (see Fig. 4) reveal a weak dependence of detonation velocity (Fig. 4a), as well as temperature (Fig. 4b), pressure (Fig. 4c), and molecular mass (Fig. 4d) of detonation products on the molar fraction of the additive, cA: A great body of the experimental data on detonation parameters reveals that, for overwhelming majority of mixtures and shapes of charges, the measured wave velocities and pressures are fairly well consistent with the ideal thermodynamic calculations. It should be emphasized that recorded pressure profiles exhibit intense oscillations and, therefore, a comparison of calculations with experiment in this case is often uncertain to a large extent. Usually, in compliance with the thermal theory of detonation limits [98], deviations of the measured detonation wave velocities in mixtures with Arrhenius reaction kinetics from those calculated do not exceed about 10% even for marginal detonations. However, there are exceptions for special types of detonations heavily affected by energy and momentum losses in which heat release kinetics senses only little variations of the gas parameters. These waves require special consideration (see Section 2.2.5). Detonation parameters that are measured involving (intrinsically) integration over the duct cross-section (such as density measured by absorption of X-rays) exhibit profiles subject to less pronounced oscillations [99]. The averaged density at the end of the zone where the reaction keeps going and transverse waves are still intense is indeed consistent with thermodynamic calculations. Recent detailed measurements of temperature and pressure histories behind a detonation front [100], using advanced laser diagnostics, revealed that trends in measurements agree with simulations [101] although certain discrepancies in the profiles are apparent (Fig. 5). This certainly makes the 1D ZND theory very useful even though it does not adequately describe peculiarities of the detonation wave structure. There are several reasons for the detonation parameters to deviate from ideal thermodynamic calculations: wave instability, incomplete reaction at the sonic (CJ) plane (if it exists in multidimensional waves), and momentum and energy losses. Therefore, a reasonably good agreement between the calculated and measured detonation parameters is observed only in long ducts of a diameter exceeding the limiting value. At short distances from the initiator (of about 1 or 2 m) and in narrow ducts, the deviations can be quite significant as obvious even from 1D calculations with finite reaction kinetics. One of the most important features of detonation waves in homogeneous mixtures is the instability that results in their essentially 3D and unsteady nature. A major feature of detonation wave propagation is shown in Fig. 6a and b. Fig. 6a shows a typical footprint of detonation on the sooted foil mounted on the tube wall [102], Fig. 6b [103] in terms of a series of pressure maps at evenly spaced intervals. The connection of the paths of triple points produces the cellular structure that has become a characteristic feature of gaseous detonations. The dimensions of the cellular structure— longitudinal size b and transverse size a—are related to the properties of the material and the chemical reaction mechanism. Long chemical reaction times or induction times correlate with large detonation cells. The structure of most propagating detonations is usually much more complex than that shown in Fig. 6, sometimes there are substructures within a detonation cell, and sometimes the structures are very irregular. Moreover, some detonations exhibit essentially 3D structure [104]. In Ref. [104], detailed measurements of the detonation structure in a square-section tube were made. The schematics of transverse motion of front shocks in cases of 2D and 3D detonation structures are shown, respectively, in Fig. 7a and b. Modeling of detonation waves initiated and propagating in real combustion chambers is an efficient method for Fig. 5. Measured 1 [100] and computed 2 [101] gas temperatures (a) and pressures (b) for detonation of stoichiometric C2H4 –O2 mixture at normal conditions. Peaks on temperature and pressure curves correspond to CJ conditions: TCJ ¼ 3937 K and pCJ ¼33.3 atm, respectively. G.D. Roy et al. / Progress in Energy and Combustion Science 30 (2004) 545–672 555
G.D. Roy et al. Progress in Energy and Combustion Science 30(2004)545-672 Below. one of the feasible models is considered that is general enough to be capable of explaining the physical meaning of averaged measured detonation parameters and their behavior in time behind the detonation front it is a good supplement to any numerical solution of 3D or 2D detonation problems since unlike numerical results, this approach provides the means of easy access to the general properties of detonation waves Following Voitsekhovsky et al. [105], assume that the nt can by turbulent' motion including pressure, density, entropy, and velocity fluctuations, which is specified by some averaged fluctuation parameters and by the averaged unidirectional flow. Since this formulation of the problem remains ID(no lateral flux of mass, momentum and energy)one can write the conservation equations in the following form Fig. 6.(a) Soot footprint of 2C0+O2 detonation [102]. M (b) Computed sequence of pressure contours from a computatio of a detonation cell for a mixture of H: 0,: Ar/ 2: 1: 7. The cell lengt bis about 77 mm and the ratio of the width a to the length of the cell alb =0.61. The computed detonation velocity is 1623 m/s (D/(ho+D/)=Es (CJ velocity is 1619 m/s). The black lines superimposed on the where v is the specific volume, h is the enthalpy, Ms,Is, and contours trace the paths of the triple points [103] Es are, respectively, the mass, momentum, and energy fluxes optimizing the performance of a propulsion device. Taking averaged over the tube cross-section. The detonation wave into account the realistic detonation (or reactive shock) assumed to propagate along the and integration is structure in modeling codes would necessitate prohibitively performed over the flow cross-section. As a steady fine 3D computational grids, normal simulations are not detonation wave is considered, the integrands attempted to resolve the fin parameters averaged over the spatial computational cells. some time which is much longer than the characteristic are, consider the wave parameters averaged over the To make the equations more convenient for further coordinate normal to the wave propagation direction(analog ualitative analysis they are presented in the form: Most of the measurements with detonation waves, except those intended to study their structure, are performed Po+ D-lo=Pa +(ucav/avx(1+K) treating them as a ID phenomenon. It is necessary to [y(y-1)lPawvav+D-2 know what correspondence these measurements have in relation to the averaged parameters in detonation waves =[yy-1)vo+a2a/2-1-B)- BD回国 四c回国面 DAc国 囚cD国画 日量区函 Fig. 7. Schematic diagram of the front shocks at different locations of the cycle of the detonation cell [104](a) The case represented here is'in (rectangular type). The arrows show the motion of the four triple point lines generating the ctahedron faces. I-development of lach stem inside the central octahedron. ll-development of incident wave inside the central exhibits no slapping waves(diagonal type)and the shocks are canted at an angle of 45" to the tube out of eight triple point lines generating the central octahedron faces. For clarity, the motion of the other four triple point lines is not shown. I-development of Mach stem inside the central octahedron, Il-development of incident wave inside the central octahedron
optimizing the performance of a propulsion device. Taking into account the realistic detonation (or reactive shock) structure in modeling codes would necessitate prohibitively fine 3D computational grids, normal simulations are not attempted to resolve the fine wave structure, yielding parameters averaged over the spatial computational cells. In order to understand how adequate the simulation results are, consider the wave parameters averaged over the coordinate normal to the wave propagation direction (analog of the 1D ZND model). Most of the measurements with detonation waves, except those intended to study their structure, are performed treating them as a 1D phenomenon. It is necessary to know what correspondence these measurements have in relation to the averaged parameters in detonation waves. Below, one of the feasible models is considered that is general enough to be capable of explaining the physical meaning of averaged measured detonation parameters and their behavior in time behind the detonation front. It is a good supplement to any numerical solution of 3D or 2D detonation problems since unlike numerical results, this approach provides the means of easy access to the general properties of detonation waves. Following Voitsekhovsky et al. [105], assume that the flow behind the detonation front can be represented by ‘turbulent’ motion including pressure, density, entropy, and velocity fluctuations, which is specified by some averaged fluctuation parameters and by the averaged unidirectional flow. Since this formulation of the problem remains 1D (no lateral flux of mass, momentum and energy) one can write the conservation equations in the following form: D=v0 ¼ Ms p0 þ D2 =v0 ¼ Is ðD=v0Þðh0 þ D2 =2Þ ¼ Es where v is the specific volume, h is the enthalpy, Ms; Is; and Es are, respectively, the mass, momentum, and energy fluxes averaged over the tube cross-section. The detonation wave is assumed to propagate along the z-axis and integration is performed over the flow cross-section. As a steady detonation wave is considered, the integrands are time independent (which implies that they are averaged over some time which is much longer than the characteristic pulsation time equal to b=D). To make the equations more convenient for further qualitative analysis they are presented in the form: D=v0 ¼ uz;av=vav p0 þ D2 =v0 ¼ pav þ ðu2 z;av=vavÞð1 þ K0 Þ ½g=ðg 2 1Þpavvav þ D2 =2 ¼ ½g=ðg 2 1Þp0v0 þ u2 z;av=2 2 ½qð1 2 bÞ 2 H0 Fig. 7. Schematic diagram of the front shocks at different locations of the cycle of the detonation cell [104]. (a) The case represented here is ‘in phase’ (rectangular type). The arrows show the motion of the four triple point lines generating the central octahedron faces. I—development of Mach stem inside the central octahedron, II—development of incident wave inside the central octahedron. (b) The case represented here exhibits no slapping waves (diagonal type) and the shocks are canted at an angle of 458 to the tube wall. The arrows show the motion of the four out of eight triple point lines generating the central octahedron faces. For clarity, the motion of the other four triple point lines is not shown. I—development of Mach stem inside the central octahedron, II—development of incident wave inside the central octahedron. Fig. 6. (a) Soot footprint of 2CO þ O2 detonation [102]. (b) Computed sequence of pressure contours from a computation of a detonation cell for a mixture of H2:O2:Ar/2:1:7. The cell length b is about 77 mm and the ratio of the width a to the length of the cell, a=b <0.61. The computed detonation velocity is 1623 m/s (CJ velocity is 1619 m/s). The black lines superimposed on the contours trace the paths of the triple points [103]. 556 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 K detonations may result in both lower and higher detonation energy and enthalpy, respectively, Id is the velocity velocities as compared to the plane Cj state depending on fluctuation, B is the reaction progress variable, and y is the ratio of the equilibration rates for all the three major the ratio of specific heats, subscriptav' signifies averaging. components governing interconversions of the momentum The Rayleigh -Michelson line in this case is no longer a and energy within the zone between the lead front and the straight line because it depends on K, and the Hugoniot h effective Cj plane, where mean flow velocity is sonic. in general a usual shape, although it depends on both Kand The above qualitative model allows one to explain H. Thus, the governing equations differ from the conven- experimental observations [99] that seemed to be somewhat tional ZND ones by the presence of K and H, which surprising(Fig. 8). The first observation(see Fig Sa)is the characterize the momentum and energy fractions contained averaged-density variation behind the detonation front. in the fluctuations. Both the Hugoniot and the Rayleigh- Numerous measurements of X-ray absorption by heavy Michelson lines depict not only the final state but noble gases(Xe or Kr)revealed the following characteristic intermediate ones as well, for each state there exist its features of the density profiles:(i)the density everywhere own values of B, K, and H'. These three quantities var behind the lead shock front(peak marked 1)is significantly more or less independently since they characterize different lower than that predicted for the von Neumann spike, p processes(chemical reaction, mechanical equilibration, and ii)the minimum density(between peaks marked I and 2) mixing of the gases). Although K and H seem to be is always slightly below (10-15%)the ideal CJ value, interdependent, their variation with departure from the lead Pc, and (iii) after the minimum, the density grows front(or leading control surface)may differ. Unfortunatel fy their dependence on the distance from the lead front, because th would necessitate the exact solution of the 3D unsteady problem. The only thing which is definite about these quantities is that all of them vanish as the system approaches equilibrium(but certainly at different rates) The only curves, which one can plot in the plpo vs. v/vo diagram, are the planar-shock and final-detonation Hugo. (a) nots. The shock Hugoniot does not represent any real state since it is never reached, because within the framework of the averaged approach the curved lead fronts(as in Fig. 6). first, spread the discontinuity in space, and, second, start locally the reaction virtually at the instant when the first portions of the lead front touch the control surface, i.e. K 1-B, and Hare finite from the very beginning. An analysis of the equations shows that the fluid keeps gaining some nergy(either from the reaction or, which is much more likely, due to conversion of the excess kinetic and thermal energy of fluctuations into average internal energy and wo-electrode ionization gauge kinetic energy of the unidirectional flow)downstream of the averaged C plane. Since this heat deposition occurs in the supersonic region of the steady flow it chokes partial the flow and should generate a compression wave in it. A It follows from the above considerations that real ane ones but one has to take nt the OH-emission peculiarities of the energy evolution profiles which now should also include various types of energy and momentum conversions in the flow. Both experiment and numerical calculations [106, 107 show that there is some unburnt mixture that escapes the zone attended by strong pressure ( c) waves, thus burning of these mixture pockets(although small in the volume fraction) downstream of the Cj plane Fig. 8. Records of X-ray absorption(a), pressure and ionization must also be taken into account Characteristic of multifont current(b), and OH-emission(c)behind a detonation wave in 6%o propane 30%o O2+ 29%o N2+ 35%o Kr mixture; D= 1569 m/s detonations, treated as ID waves, are low parameters at the the longitudinal cell size b=20-25 mm [99]. Time interval lead shock front(von Neumann spike). The effect of between peaks I and 2(14-18 us) characterizes the maximum ransverse waves and spatial inhomogeneity of multifont duration of chemical energy deposition behind the detonation wave
Here, K0 ¼ u02 av=u2 z;av and H0 are the fluctuations of kinetic energy and enthalpy, respectively, u0 is the velocity fluctuation, b is the reaction progress variable, and g is the ratio of specific heats, subscript ‘av’ signifies averaging. The Rayleigh–Michelson line in this case is no longer a straight line because it depends on K0 ; and the Hugoniot has in general a usual shape, although it depends on both K0 and H0 : Thus, the governing equations differ from the conventional ZND ones by the presence of K0 and H0 ; which characterize the momentum and energy fractions contained in the fluctuations. Both the Hugoniot and the Rayleigh– Michelson lines depict not only the final state but intermediate ones as well, for each state there exist its own values of b; K0 ; and H0 : These three quantities vary more or less independently since they characterize different processes (chemical reaction, mechanical equilibration, and mixing of the gases). Although K0 and H0 seem to be interdependent, their variation with departure from the lead front (or leading control surface) may differ. Unfortunately, at present it is inexpedient even to try to specify their dependence on the distance from the lead front, because this would necessitate the exact solution of the 3D unsteady problem. The only thing which is definite about these quantities is that all of them vanish as the system approaches equilibrium (but certainly at different rates). The only curves, which one can plot in the p=p0 vs. v=v0 diagram, are the planar-shock and final-detonation Hugoniots. The shock Hugoniot does not represent any real state since it is never reached, because within the framework of the averaged approach the curved lead fronts (as in Fig. 6), first, spread the discontinuity in space, and, second, start locally the reaction virtually at the instant when the first portions of the lead front touch the control surface, i.e. K0 ; 1-b, and H0 are finite from the very beginning. An analysis of the equations shows that the fluid keeps gaining some energy (either from the reaction or, which is much more likely, due to conversion of the excess kinetic and thermal energy of fluctuations into average internal energy and kinetic energy of the unidirectional flow) downstream of the averaged CJ plane. Since this heat deposition occurs in the supersonic region of the steady flow it chokes partially the flow and should generate a compression wave in it. It follows from the above considerations that real multifront detonations propagating in tubes may be treated on average as plane ones but one has to take into account the peculiarities of the energy evolution profiles which now should also include various types of energy and momentum conversions in the flow. Both experiment and numerical calculations [106,107] show that there is some unburnt mixture that escapes the zone attended by strong pressure waves, thus burning of these mixture pockets (although small in the volume fraction) downstream of the CJ plane must also be taken into account. Characteristic of multifront detonations, treated as 1D waves, are low parameters at the lead shock front (von Neumann spike). The effect of transverse waves and spatial inhomogeneity of multifront detonations may result in both lower and higher detonation velocities as compared to the plane CJ state depending on the ratio of the equilibration rates for all the three major components governing interconversions of the momentum and energy within the zone between the lead front and the effective CJ plane, where mean flow velocity is sonic. The above qualitative model allows one to explain experimental observations [99] that seemed to be somewhat surprising (Fig. 8). The first observation (see Fig. 8a) is the averaged-density variation behind the detonation front. Numerous measurements of X-ray absorption by heavy noble gases (Xe or Kr) revealed the following characteristic features of the density profiles: (i) the density everywhere behind the lead shock front (peak marked 1) is significantly lower than that predicted for the von Neumann spike, rs; (ii) the minimum density (between peaks marked 1 and 2) is always slightly below (10–15%) the ideal CJ value, rCJ; and (iii) after the minimum, the density grows Fig. 8. Records of X-ray absorption (a), pressure and ionization current (b), and OH-emission (c) behind a detonation wave in 6% propane þ 30% O2 þ 29% N2 þ 35% Kr mixture; D ¼ 1569 m/s, the longitudinal cell size b ¼ 20–25 mm [99]. Time interval between peaks 1 and 2 (14–18 ms) characterizes the maximum duration of chemical energy deposition behind the detonation wave. G.D. Roy et al. / Progress in Energy and Combustion Science 30 (2004) 545–672 557
G.D. Roy et al. Progress in Energy and Combustion Science 30(2004)545-672 )and then decays in the Taylor haracterizes approximately the maximum duration of rarefaction wave. Th away is the detonation wave hemical energy deposition behind the detonation front. In from the spinning less pronounced is the vo terms of distance, this is the time representative of the ngitudinal detonation cell size b. The second observation concerns the averaged pressure Purely gasdynamic studies of the effective sonic plane (see Fig. 8b, upper beam). It should be noted that the 1641, indicate that the Cj plane is positioned at a distance of pressure fluctuations are more pronounced therefore the several cell sizes from the lead shock front. The effective CJ ccuracy with which the average profile is drawn is les plane is defined as a site where either the attached oblique satisfactory, but nevertheless, the pressure recorded within shock wave at a sharp wedge, over which a detonation wave the von Neumann spike(marked 1)is definitely lower than propagated started to depart from the tip, or as a cross- that predicted by the ZND model and the minimum pressure section in a tube covered with a thin film( destroyed by the (marked 2)is 10-15% lower than the ideal CJ pressure. N detonation wave) at which the lateral rarefaction wave Thus, the two types of measurements ascribe unequivoca ceased to affect the detonation wave velocity the real detonation wave to the weak(underdriven) branch It is remarkable that the third rise in the density signal in of the equilibrium Hugoniot. This behavior finds a simple Fig 8a(marked 3)also coincides with analogous rises(also explanation within the framework of the above model. marked 3) in all the signals, which means that this rise is Furthermore. measurements of the ionization currer accompanied by an increase in the average internal energy the oh emission(Fig &b, lower beam, and Fig &c), as well of the reaction products, i.e. it signifies an effective energy as recent PLIF measurements [108(Fig. 9)are also ain. This energy gain was discussed in the above model consistent with the above model The possibility of detonation wave velocities higher or Indeed, the model states that the major chemical reaction lower than Dc predicted by the model is also confirmed by is completed within a zone which is smaller than the cell experiment size, and in conformity with this, both the OH emission and Thus, in spite of a very qualitative treatment of the the ionization current peak at the very beginning of the cell problem in the model, it provides quite reasonable and logical size thus indicating that the reaction rate passed its explanations of many experimental findings. The weakest maximum. It is worth noting that OH emission is most point of the model is the relative rates of change o ngitudinal, b, or transverse, a, cell size and D, since it likely due to the recombination of O and H atoms and the just states that they may vary, but not specifying how. Since ionization, although close to the thermal one, still can be this variation is very important for the structure of detonation influenced by chemi-ionization within the zone with intense waves, the model calls for further development. Probably the chemical reaction. That is why it is believed that the first measurements of the average parameters behind the detona- peaks(marked 1)in Fig. 8b(low beam) and c reflect the behavior of the chemical reaction. The next peaks(marked tion waves will help one to specify these rates and ascribe 2)that are present in all the signals are presumably due to them to one or another hydrodynamic or chemical processes The marginal spinning detonations are known to show the collision of the tails of the transverse waves behind th good consistency with the acoustic modes in the detonation reaction zone. This collision raises the temperature of products [109, 110]. It is expedient to note that not only the reaction products to a very high value, which increases spinning detonation exhibits such a consistency, but the real the thermal luminosity and the ionization. Hence, the time multifont detonations show also a resonance behavior with interval between the peaks(14-18 us in Fig. 8b and c) the higher acoustic modes. This is not surprising because as in spinning modes, finite-amplitude waves in the products generated by the transverse waves near the lead front degenerate very rapidly into acoustic or quasi-acoustic ones because of the high(as compared to the transverse wave velocity of sound in the products. Moreover, the shock waves in the products dissipate their energy at a high rate just by transferring it to dissociation of the highly heated gas and thus approach rapidly the acoustic velocity. Exper- mental photographs show quite clearly that the waves in the products can propagate at a velocity slightly higher than the acoustic one only near the detonation front, further down- stream they rapidly are converted into the conventional acoustic modes. Thus the close coupling between the two types of waves supporting each other is inherent in al Fig 9. OH PLIF (a)and Schlieren(b)images in H2-N2 0-3N2 detonation waves mixture at initial pressure 20 kPa [108]. Chemical activity nearly From the aforesaid, it may be inferred that although the ceases at a distance of about one detonation cell size average chemical reaction time certainly affects directly
slightly again (peak marked 2) and then decays in the Taylor rarefaction wave. The farther away is the detonation wave from the spinning mode the less pronounced is the von Neumann spike. The second observation concerns the averaged pressure (see Fig. 8b, upper beam). It should be noted that the pressure fluctuations are more pronounced therefore the accuracy with which the average profile is drawn is less satisfactory, but nevertheless, the pressure recorded within the von Neumann spike (marked 1) is definitely lower than that predicted by the ZND model and the minimum pressure (marked 2) is 10–15% lower than the ideal CJ pressure, pCJ: Thus, the two types of measurements ascribe unequivocally the real detonation wave to the weak (underdriven) branch of the equilibrium Hugoniot. This behavior finds a simple explanation within the framework of the above model. Furthermore, measurements of the ionization current and the OH emission (Fig. 8b, lower beam, and Fig. 8c), as well as recent PLIF measurements [108] (Fig. 9) are also consistent with the above model. Indeed, the model states that the major chemical reaction is completed within a zone which is smaller than the cell size, and in conformity with this, both the OH emission and the ionization current peak at the very beginning of the cell size thus indicating that the reaction rate passed its maximum. It is worth noting that OH emission is most likely due to the recombination of O and H atoms and the ionization, although close to the thermal one, still can be influenced by chemi-ionization within the zone with intense chemical reaction. That is why it is believed that the first peaks (marked 1) in Fig. 8b (low beam) and c reflect the behavior of the chemical reaction. The next peaks (marked 2) that are present in all the signals are presumably due to the collision of the tails of the transverse waves behind the reaction zone. This collision raises the temperature of the reaction products to a very high value, which increases the thermal luminosity and the ionization. Hence, the time interval between the peaks (14–18 ms in Fig. 8b and c) characterizes approximately the maximum duration of chemical energy deposition behind the detonation front. In terms of distance, this is the time representative of the longitudinal detonation cell size b: Purely gasdynamic studies of the effective sonic plane [64], indicate that the CJ plane is positioned at a distance of several cell sizes from the lead shock front. The effective CJ plane is defined as a site where either the attached oblique shock wave at a sharp wedge, over which a detonation wave propagated started to depart from the tip, or as a crosssection in a tube covered with a thin film (destroyed by the detonation wave) at which the lateral rarefaction wave ceased to affect the detonation wave velocity. It is remarkable that the third rise in the density signal in Fig. 8a (marked 3) also coincides with analogous rises (also marked 3) in all the signals, which means that this rise is accompanied by an increase in the average internal energy of the reaction products, i.e. it signifies an effective energy gain. This energy gain was discussed in the above model. The possibility of detonation wave velocities higher or lower than DCJ predicted by the model is also confirmed by experiment. Thus, in spite of a very qualitative treatment of the problem in the model, it provides quite reasonable and logical explanations of many experimental findings. The weakest point of the model is the relative rates of change of longitudinal, b; or transverse, a; cell size and D; since it just states that they may vary, but not specifying how. Since this variation is very important for the structure of detonation waves, the model calls for further development. Probably the measurements of the average parameters behind the detonation waves will help one to specify these rates and ascribe them to one or another hydrodynamic or chemical processes. The marginal spinning detonations are known to show good consistency with the acoustic modes in the detonation products [109,110]. It is expedient to note that not only spinning detonation exhibits such a consistency, but the real multifront detonations show also a resonance behavior with the higher acoustic modes. This is not surprising because as in spinning modes, finite-amplitude waves in the products generated by the transverse waves near the lead front degenerate very rapidly into acoustic or quasi-acoustic ones because of the high (as compared to the transverse wave) velocity of sound in the products. Moreover, the shock waves in the products dissipate their energy at a high rate just by transferring it to dissociation of the highly heated gas and thus approach rapidly the acoustic velocity. Experimental photographs show quite clearly that the waves in the products can propagate at a velocity slightly higher than the acoustic one only near the detonation front, further downstream they rapidly are converted into the conventional acoustic modes. Thus the close coupling between the two types of waves supporting each other is inherent in all detonation waves. From the aforesaid, it may be inferred that although the average chemical reaction time certainly affects directly Fig. 9. OH PLIF (a) and Schlieren (b) images in H2 –N2O–3N2 mixture at initial pressure 20 kPa [108]. Chemical activity nearly ceases at a distance of about one detonation cell size. 558 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 both the detonation cell size and the distance between the principal features. That is why it is still used almost without lead shock front and the effective Cj plane, the relation any modifications by the specialists. Thus, it is expedient to between these characteristic parameters is not unique. The apply the term detonations to supersonic reactive waves in only definite statement that can be made at this stage of hich the exothermic reaction is coupled with the shock our knowledge of the process is the relative order of the wave supported by the heat released behind it and which, in magnitudes of these times: the average length of the the absence of external energy sources, approach in the fina chemical reaction zone is less than the detonation cell in a steady state characterized by a plane (or zone size, while the separation of the lead shock wave and the positioned downstream of the lead wave front nonpermeable average C plane is longer than the cell size. It should be for weak gasdynamic perturbations. This definition relies on however, noted that the maximum(local)energy conversion the physical model of the wave and therefore defies zone (meaningful for the ZND model) size can be misinterpretation. commensurate with the cell size Unfortunately, the complexity of the gasdynamic pattern Since the cell size a (or b)as the most readily in real detonation waves and the absence( for this reason)of measurable quantity is often used as the kinetic character- simple analytical relations capable of predicting the istic of both the reaction zone length(Ln kya)and the C behavior of detonations under various conditions that may position (La= ka), it should be be encountered in practice shifts the emphasis towards the proportionality coefficients k1 and k2 are functions of finding empirical or semi-empirical correlations. Especially ny parameters and therefore the relations which are this concerns marginal and unsteady detonation waves, derived for a restricted number of mixtures are applicable which are of the greatest practical importance since they only to them and not necessarily should hold for other define the conditions for onset and propagation of detona- mixtures. This can be illustrated, for example, by the tion. This is the wave type, which is expected to be normally dence of transverse cell size a on the observed in combustion chambers of PDEs. Detonation temperature at the lead shock front, Ts- A typical Arrhenius wave instability is inherent not only in gaseous mixtures but plot of a is shown in Fig. 10, from which it is obvious that heterogeneous systems as well. Therefore deriving reliable unlike the ignition delays the cell size a exhibits a expressions that relate the heat release kinetics in detonation definitely non Arrhenius behavior with a too high effective Ives to the wave structure and parameters easily recorded activation energy in multihead detonation waves and to in experiment and specify the averaged parameters appear low energy when only few heads are present in the front. ing in the equations of the above model is the goal of future Hence the exact relations between the cell size a, reaction experimental and theoretical investigati zone length Lrz, and the physical detonation front thickness The basic quantitative characteristics of any reactive Lc(that is the zone where a detonation wave is vulnerable tonation limits. limiting diam to any external perturbations) still await more sophisticated detonation propagation, the minimum energy of direct and detailed numerical and analytical studies detonation initiation. and the critical diameter of detonation After analyzing steady-state modes of propagation of transmission to unconfined charge. The knowledge of the f detonations in defining c reactio one may formulate the definition detonation wave structure allows one to understand the to distinguish them from other nature of the detonation limits and correctly predict detonation it is appropriate to behavior of detonation waves under various conditions proceed essentially from the ZND model since despite over- nd detonability of FAMs realization of the process it nevertheless reflects all its 2.2.2. Detonability limits One of the major practical problems is classification of combustible mixtures with respect to their detonability. The problem is not simple since the critical conditions for detonation depend on several initial conditions that the nondetonability criterion may be used only as applied to a articular situation. However, there are some general experimental approaches that provide sufficient information 0.0 to assess, at least relatively, the detonability of various mixtures. These are the approaches based on the concepts of 500.5 0.65 octane number (on) dating back to late 1920s [111-113]. concerning detonation run-up distance [401, critical initiation energy [54, 1141, and limiting tube diameter Fig. 10. Arrhenius plot se size a of the nation cell [54-56, 115]. Contrary to other approaches, the On concept In propane-oxygen diluted by various amounts of is usually applied to a test fuel in a piston engine to assess nitrogen and krypton initial pressure erature the detonability in terms of the percentage of iso-octane(by [99]. Ts is the post-shock temperature volume)in the n-heptane
both the detonation cell size and the distance between the lead shock front and the effective CJ plane, the relation between these characteristic parameters is not unique. The only definite statement that can be made at this stage of our knowledge of the process is the relative order of the magnitudes of these times: the average length of the chemical reaction zone is less than the detonation cell size, while the separation of the lead shock wave and the average CJ plane is longer than the cell size. It should be, however, noted that the maximum (local) energy conversion zone (meaningful for the ZND model) size can be commensurate with the cell size. Since the cell size a (or b) as the most readily measurable quantity is often used as the kinetic characteristic of both the reaction zone length (Lrz ¼ k1aÞ and the CJ plane position ðLCJ ¼ k2aÞ; it should be emphasized that the proportionality coefficients k1 and k2 are functions of many parameters and therefore the relations which are derived for a restricted number of mixtures are applicable only to them and not necessarily should hold for other mixtures. This can be illustrated, for example, by the dependence of transverse cell size a on the reciprocal temperature at the lead shock front, Ts: A typical Arrhenius plot of a is shown in Fig. 10, from which it is obvious that unlike the ignition delays the cell size a exhibits a definitely nonArrhenius behavior with a too high effective activation energy in multihead detonation waves and too low energy when only few heads are present in the front. Hence the exact relations between the cell size a; reaction zone length Lrz; and the physical detonation front thickness LCJ (that is the zone where a detonation wave is vulnerable to any external perturbations) still await more sophisticated and detailed numerical and analytical studies. After analyzing steady-state modes of propagation of supersonic reaction waves one may formulate the definition of detonations in order to distinguish them from other regimes. When defining detonation it is appropriate to proceed essentially from the ZND model since despite overidealization of the process it nevertheless reflects all its principal features. That is why it is still used almost without any modifications by the specialists. Thus, it is expedient to apply the term detonations to supersonic reactive waves in which the exothermic reaction is coupled with the shock wave supported by the heat released behind it and which, in the absence of external energy sources, approach in the final run a steady state characterized by a plane (or zone) positioned downstream of the lead wave front nonpermeable for weak gasdynamic perturbations. This definition relies on the physical model of the wave and therefore defies misinterpretation. Unfortunately, the complexity of the gasdynamic pattern in real detonation waves and the absence (for this reason) of simple analytical relations capable of predicting the behavior of detonations under various conditions that may be encountered in practice shifts the emphasis towards finding empirical or semi-empirical correlations. Especially this concerns marginal and unsteady detonation waves, which are of the greatest practical importance since they define the conditions for onset and propagation of detonation. This is the wave type, which is expected to be normally observed in combustion chambers of PDEs. Detonation wave instability is inherent not only in gaseous mixtures but heterogeneous systems as well. Therefore deriving reliable expressions that relate the heat release kinetics in detonation waves to the wave structure and parameters easily recorded in experiment and specify the averaged parameters appearing in the equations of the above model is the goal of future experimental and theoretical investigations. The basic quantitative characteristics of any reactive mixture are the detonation limits, limiting diameters of detonation propagation, the minimum energy of direct detonation initiation, and the critical diameter of detonation transmission to unconfined charge. The knowledge of the detonation wave structure allows one to understand the nature of the detonation limits and correctly predict behavior of detonation waves under various conditions and detonability of FAMs. 2.2.2. Detonability limits One of the major practical problems is classification of combustible mixtures with respect to their detonability. The problem is not simple since the critical conditions for detonation depend on several initial conditions that the nondetonability criterion may be used only as applied to a particular situation. However, there are some general experimental approaches that provide sufficient information to assess, at least relatively, the detonability of various mixtures. These are the approaches based on the concepts of octane number (ON) dating back to late 1920s [111–113], concerning detonation run-up distance [40], critical initiation energy [54,114], and limiting tube diameter [54–56,115]. Contrary to other approaches, the ON concept is usually applied to a test fuel in a piston engine to assess the detonability in terms of the percentage of iso-octane (by volume) in the n-heptane–iso-octane blend that matches Fig. 10. Arrhenius plot of the transverse size a of the detonation cell in propane–oxygen mixtures diluted by various amounts of nitrogen and krypton at normal initial pressure and temperature [99]. Ts is the post-shock temperature. G.D. Roy et al. / Progress in Energy and Combustion Science 30 (2004) 545–672 559