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The behavior of detonation waves at concentration gradients
312
COMBUSTION A N D F L A M E 84:312-322 (1991)
The Behavior of Detonation Waves at Concentration Gradients G. O. THOMAS,* P. SUTTON, and D. H. EDWARDS Department of Physics, University College of Wales, Aberystwyth, Dyfed, UK SY23 3BZ.
Two detonation tubes were used to Study detonation propagation in concentration gradients. Each tube, 22 x 10 m m and 50 nun diameter in cross section, respectively, was fitted with slide valves that with the tubes mounted vertically allowed known concentration gradients to form by diffusion. The peak-transmitted shock strength when a detonation is incident on an abrupt planar interface with an inert gas was found to agree reasonably well with the simple Paterson-Glass model. A significant improvement was found in the predicted values when allowance was made for the Taylor expansion and energy and momentum losses to the tube walls. Under certain conditions a secondary pressure pulse was observed behind the transmitted shock; the origin of this pulse is the explosion of the unreacted gas between the transmitted shock and the unburned gas. When incident on gradients of fuel concentration, the transmitted detonation wave velocity and cell size adjusted rapidly, to values appropriate to the local gas composition and wall boundary layer losses. Finally, it was shown that the mechanism of detonation initiation in a weaker mixture by an incident detonation propagated across a concentration gradient differs in one important respect from that in a homogeneous mixture in that a secondary shock is formed as an intermediate stage.
INTRODUCTION Most studies on the ignition and propagation of detonation waves are with systems of uniform composition. However, fuel-air clouds, irrespective of the manner they are produced, will exhibit varying degrees of inhomogeneity, and the way in which detonation waves propagate in such clouds has received scant attention. Because of the lack of information on the influence of concentration gradients on wave behavior it is often tacitly assumed in hazard analyses that the presence of such gradients will hamper rather than enhance the propagation of a detonation wave, but this assumption is by no means proven. The present study therefore aims at elucidating some of the factors that control the behavior of detonations in the presence of compositional variations. A preliminary study of the problem has been *To whom all correspondence should be addressed.
0010-2180/91/$3.50
reported by Donato et al. [1]. However, because of the uncertainties that existed in the production of the gradients this is of qualitative interest only. Similarly, the fundamental studies of Strehlow et al. [2] on the transmission of a detonation wave into an inert gas across a compositional interface were primarily concerned with the decay rates of the transverse wave systems rather than the strengths of the frontal shocks. However, a work that has a direct bearing on the present investigation is that described by Bull et al. [3] on sympathetic detonations. These authors determined the maximum gap lengths of air that spherical detonation fronts, of different radii, could travel and subsequently cause reignition in a similar unconfined system. More recently, Bjerketvedt et al. [4] have reported an extensive experimental study of the transmission of a detonation across an inert region of air. They were able to establish, inter alia, some important conclusions regarding the Copyright © 1991 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc. 655 Avenue of the Americas, New York, NY 10010
DETONATION WAVES AT CONCENTRATION GRADIENTS enhancement of the transmission efficiency as the steepness of the concentration gradient at the inert region decreased. The significance of these conclusions is discussed in the present article in light of the new results that are described.
EXPERIMENTAL Before meaningful results could be obtained it was essential to be able to create well-defined concentration gradients in the system under study. Preliminary experiments showed that, in a horizontal detonation tube, the vertical, free interface, created by the rapid removal of a thin diaphragm remained reasonably planar for less than 1 s before distorting due to buoyancy effects. Consequently, horizontal tubes could only be used for the study of abrupt gradients if the tube is fired within a fraction of a second following the removal of the diaphragm. In all our work therefore vertical tubes were used. The first was described by Edwards et al. [5] with an internal cross section of 22 x 10 mm and comprising three sections: a driver section that was used to generate a stable detonation in the donor gas, and a third, acceptor section. A stable free interface was created between donor and acceptor gases by removing a purpose-built slide. The second tube comprised the same three sections, but was constructed from 50-mm-diameter stainless-steel tube. Concentration gradients of any desired steepness could be easily generated by allowing the gases initially separated by the slide to diffuse for a given time. In order to check the quality of the diffusional gradients set up by the system a separate section was built to study the density profiles using a Mach-Zehnder interferometer. In this case the only effect that the removal of the diaphragm appeared to have was to produce a highly damped oscillation of the interface, with a period of about 0.5 s, generating a disturbed region of gas with a thickness of about 5 m m on either side of the interface. At later times, the concentration gradients that developed as a result of diffusion were in good agreement with those predicted by the solution of Boyd et al. [6] for Fick's law. Piezoelectric pressure transducers mounted in
313
the tube wall recorded the detonation and shock motion and the related pressure profiles. The positions of the reaction zones were determined using 3-cm wavelength microwaves launched by an aerial placed in the end wall of the acceptor section. When these are reflected by the ionized burned gases the reflected signal is detected by the launching aerial and the recorded interference pattern enables the velocity of the reflecting surface to be found. Ionization probes provided reference points and thus gave an absolute determination of the reaction zone position. This method has proved extremely useful in monitoring reaction zones over long lengths of tube, e.g, Ref. 7. A window section was available for taking both spark and streak schlieren photographs. Unfortunately the closest edge of the window was 7 cm from the interface so that it was not possible to observe the interactions at the interface itself. In addition to the window section a smoked-foil section was also available for observation of the detonation wave structure both upstream and downstream of the slide valve. In tubes of small internal dimensions, boundary layer losses can greatly influence the nonsteady Taylor expansion wave, which, in turn, will affect the dynamics of the wave interaction process. It was therefore decided to construct a second detonation tube, of the same design as the first, but of circular cross section and 50 nun diameter. A number of experiments were repeated in this tube, thus enabling the assessment to be made of the magnitude of the boundary layer effect. I N T E R A C T I O N OF A D E T O N A T I O N WAVE WITH A PLANAR I N E R T INTERFACE When a planar detonation front interacts with normal incidence on a plane interface separating the reactive gas from an inert medium the transmitted wave is always a shock, whereas the refleeted wave may be either a shock or a rarefaction, depending on the relative acoustic impedances of the reactive and inert gas. The problem is to determine the strength of the transmitted shock together with the flow field behind
314 it. If allowance were to be made for the three-dimensional character of the detonation front then the problem would be intractable. Some idealization is therefore necessary to remove the transverse wave interactions. One way, termed the square-wave model, is to assume a plane shock front followed by uniform flow, the duration of which is determined by the kinetics of the thermoneutral induction reactions. These reactions are terminated by the onset of exothermic reactions which accelerate the flow to constant equilibrium or Chapman-Jouquet (CJ) values. More recently, calculations have been made using the Random Choice Method (RCM) code and a onestep model of detonation. Assuming infinitely fast kinetics, CJ values of particle velocity, pressure, sound speed, and so on are assigned to the wave front which then propagates as determined by the RCM algorithm. This approach has the distinct advantage that the appropriate Taylor expansion wave is modeled correctly. In the present article, calculated values for several interactions using both of the above models are described and compared with the observed results in the two detonation tubes. The Transmitted Shock Wave The problem of shock wave refraction, at both normal and oblique incidence, at a planar gaseous interface was first investigated by Poachek and Seegar [8]. Paterson [9] and later Glass [10] extended these analyses to determine the behavior of a detonation front at a boundary with an inert gas. They assumed that the effect of the induction and reaction zones of the interaction process could be disregarded because of the extreme brevity of their duration. The detonation is therefore assumed to be a wave across which the gas parameters are raised instantaneously to those values obtaining at the CJ plane. This idealized interaction is depicted in Fig. 1. A shock S i, which is incident on the interface, I, gives rise to a transmitted shock, S t under all circumstances, whereas the reflected wave, R, can be either a shock or a rarefaction depending upon the ratio of the acoustic impedances across L The nature of this wave and the strength of S t is determined by the
G . O . THOMAS ET AL.
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requirement that both the pressure and particle velocity be equal across the contact surface C, This requirement may be expressed graphically as shown in Fig. 2. The (velocity, pressure) curves, labeled a, b, c, d, and e, are Rankine-Hugoniot curves that represent the possible states obtaining behind shock waves over a range of Mach numbers in various gases. Similarly, using CJ values and the equation of state for the reaction products, curve A is obtained, and is centered on the CJ point. Movement to the left of this point corresponds to a rarefaction propagating through the detonation products, whereas movement to the right corresponds to a shock. The intersections of a, b, c, d, and e into A give the solutions for the interaction of the detonation-inert gas interface. It can be seen from this that all the gases give rise to a reflected rarefaction when a detonation wave in oxyacetylene is incident on them, except for SF6, in which case the reflected wave is a shock. Calculated values of the Math numbers of the transmitted shock waves from the interaction of a
DETONATION WAVES AT CONCENTRATION GRADIENTS
TABLE 1 Calculated Values of Mach Numbers of Transmitted Shock Waves from the Interaction of a Detonation in C2H 2 + 2.502 with Several Inert Gases
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Calculated Mach No. (Paterson Theory)
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Experimental Mach Number 22 x 10 nun 4, - 50 m m 2.4 4.4 4.3 4.3
2.9 4.7 4.8
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40 PRESSURE RATIO
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Fig. 2. Hugoniot relations for shocks in (a) He, 0a) Air, (c) CO z, (d) Ax, (e) SF6, and (A) for shock and rarefactions in detonation products of C2H 2 + 2.502.
detonation in C2H 2 + 2.5 0 2 with several inert gases are given in Table 1. Experimental values from both detonation tubes are also shown for comparison. In both tubes the observation point was 7 cm away from the position of the interface. Good agreement between theory and experiment is obtained in the 50-mm-diameter tube, whereas in the smaller rectangular tube there is a slight discrepancy between the two values. It will be shown that this arises from the effect of the enhanced wall boundary layer, which occurs in the smaller tube, on the Taylor expansion wave.
and application of the analysis of Shapiro [12] to calculate the modified pressures and temperatures. Using values of Cy determined by the previous investigations of Edwards et al. [13], pressure profiles for a detonation wave in C2H 2 + 2.502 at an initial pressure of 100 torr are given in Fig. 4a for the 5-cm-diameter tube with Cy = 0.015 and Fig. 4b for the smaller, 22 x 10 mm tube and a value of Cy = 0.035. The agreement with experimental profiles in both cases is very good. The next step was to use the pressure profile data provided in Fig. 3, together with the related density and particle velocity profiles, to examine the interaction with a planar air interface. The results obtained for the transmitted shock for both detonation tubes are displayed in Figs. 4 and 5 as an (x, t) trajectory and ( M s, x) plot, respectively. It will be noted that the agreement beo
An Improved Model Using an RCM Code A serious weakness of the Paterson-Glass model is the fact that it ignores the effect of the nonsteady expansion that follows the detonation wave. At best, therefore, it can only claim to give the true instantaneous values of the wave parameter immediately after interaction. One numerical code that allows the profile of the incident detonation to be taken into account is the RCM. The program used for the present calculations was based on that of Saito and Glass [11]. This allows the Taylor expansion wave to be included, while momentum and heat transfer losses to the tube wall can be accounted for by a friction factor (7/
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Fig. 3. Measured and calculated detonation pressure-time histories with boundary layer losses, l , rectangular wave guide; O, 50 m m diameter pipe. Solid lines are calculated profiles: (a) C / = 0.015, (b) Cf = 0.035.
316
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value. The magnitude of this plateau value is somewhat less than that predicted by the idealized Paterson-Glass model.
Fig. 4. Comparisonof experimental distance time plots with corresponding RCM calculations for C2H 2 + 2.502 into air. Initial pressure is 100 torr, (a) rectangular tube, (b) 50 mm diameter tube. Experimental; l , shock, I , contact surface. RCM; [], shock, (D, contact surface.
The Secondary Pressure Peak
tween the theoretical and experimental data is much better in the case of the 50-mm-diameter tube than the 22 × 10 m m tube: this is due to the more pronounced influence of the wall boundary layer in the smaller tube. A further point to be observed in the velocity plots is that the strength of the transmitted shock takes a time of the order 100 ps, corresponding to a distance o f approximately 40 mm, to attain a reasonably steady
From four pressure transducers placed upstream of the interface the pressure-time histories behind the transmitted shock in the acceptor gas are obtained. An example of these records is shown in Fig. 6 for C2H 2 + 2.502 and CO 2 at a Po of 100 torr as donor and aceeptor systems, respectively. Also shown in the diagram are the trajectories of the shock and flame fronts. A notable feature of the shock profiles, apart from the oscillations deriving from the transverse
DETONATION WAVES AT CONCENTRATION GRADIENTS
317
contrast to this is provided by air and CO 2, which give rise to a weaker R, and SF6 where R is a weak shock; conditions that are more conducive to an explosion.
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I N T E R A C T I O N OF A D E T O N A T I O N W I T H VARYING C O N C E N T R A T I O N GRADIENTS I
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Fig. 6. Typical distance-time plots for C2H 2 + 2.502 into CO2, showing evidenceof a secondarypressure peak. Solid line, shock; dashed line, flamelocus.
waves, is the evidence of a secondary peak that forms some time after the decoupling of the shock and reaction zone of the incident wave. A fairly extensive study of the secondary peak revealed three things. Firstly, the magnitude of the peak depends on the acoustic impedance of the acceptor; the higher the value of the latter the stronger is the peak developed. Thus the effect was found to be strongest in SF6, whereas in He it was nonexistent. Secondly, the sharper the concentration gradient between the donor and acceptor (i.e., the shorter the diffusion time), the larger the peak. In gradual gradients, as we shall see in the following section, the secondary peak is much less pronounced. Finally, no evidence of such a peak was found with the corresponding RCM solutions. From this evidence, it would appear that the secondary peak originates from an explosion of the unreacted gas between the transmitted shock and the burned gas boundary. If the interface were perfectly plane, then no such pocket could exist; however, despite the utmost care, a mixing region of the order of 40 mm wide was always generated, as can be seen from the microwave velocity data discussed below. The reactive gas in this region is therefore diluted and will exhibit a longer induction time than the orginal mixture. It is the delayed reaction of this pocket that, it is postulated, gives rise to the observed secondary pressure rise. The strength of the reflected expansion, R, in He is large and will therefore tend to inhibit such an explosion. A
The next stage of the investigation involved the creation of concentration gradients of varying steepness between the detonable donor and the inert acceptor. Although some previous studies, referred to in the Introduction, have been reported in which finite gradients have been used, none of these has been able to specify the magnitude of the gradient with any precision. Indeed, in most of them a large uncertainty exists in both the nature and size of the gradient. In the present work carefully designed slide valves enabled very weil-defined gradients to be set up by diffusion in both rectangular and circular detonation tube. The progress of the detonation front through a concentration gradient was determined from velocity measurements, and at the same time the wave front structure was recorded by means of smoked foils.
Velocity Measurements A convenient way of measuring the velocity of a detonation wave in a tube is the microwave doppler technique, which has the advantage of providing a continuous monitor of the velocity with distance; see, for example, Ref. 13. In the present study the rectangular tube was employed because it ideally accommodates 3 cm wavelength microwaves. The results obtained for detonations in C2H 2 + 2.502 propagating into argon are presented in Fig. 7. Gradients of concentration were obtained by allowing diffusion to take place between the oxyacetylene and argon, for varying times after the removal of the slide. The solid curves represent the calculated CJ values appropriate to the particular composition. Ideally, of course, these curves should be coincident, but the
318
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Fig. 7. Velocity distance plots for a range of diffusion times for C2H 2 + 2.502 into argon, Po = 100 torr. (a) 0 s, (b) 120 s, (c) 240 s, and (d) 600 s. Solid lines are calculated CJ velocities correspondingto local gas composition. discrepancy observed, which increases with the degree of dilution, can be accounted for by the effects of the wall-boundary layer• From a knowledge of the concentration gradients and the velocity-distance record obtained by the microwave records, velocity-dilution values can be obtained; these are plotted in Fig. 8. It is seen that the detonation velocities in the gradients agree well with the corresponding measured steady-state CJ value. Hence we conclude from this agreement that the velocity at any plane in a concentration gradient adjusts very rapidly to the value appropriate to the dilution and boundary layer effects at that plane. Further evidence o f this rapid equilibration to the degree of concentration is provided by the smoked-foil records described below. Smoked-foil
taken in the region of concentration gradient provide a better test than velocity to the wave adjustments to local concentration conditions. Several inert gases were examined in this way over a range of diffusional gradients. The result with
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Fig. 8. Measured velocities as a function of local gas composition. Solid line is calculated CJ; O, measured steady state velocities. Diffusiontimes are (I-7) 120 s, (11) 240 s, (×) 360 s, and (+) 600 s.
DETONATION WAVES AT CONCENTRATION GRADIENTS
319
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Fig. 9. Variation of measured cell size with distance after 6 rain diffusion time at a C2H 2 4- 2.502/CO 2 interface. Open points are from measurements of steady state detonations at corresponding compositions.
carbon dioxide are shown in Fig. 9. A 6-rain diffusion time was used in this test. The full line represents the cell size for a steady-state detonation as a function of CO 2 dilution. Bearing in mind the difficulty in measuring the cell size in systems with some degree of irregularity, the agreement between the steady-state date and those obtained in the gradients is very good. Thus, taken together, the velocity and cell-size data indicate that the cell structure adjusts rapidly to the local concentration conditions. This supports the conclusion previously established by Strehlow et al. [2]. TRANSMISSION OF A D E T O N A T I O N THROUGH A CONCENTRATION GRADIENT Having established the behavior of the transmitted shock caused by the normal incidence of a
detonation wave both at an abrupt and gradual interface with an inert gas, we can proceed to examine the effectiveness of this wave in reestablishing a detonation in an acceptor mixture. This problem is obviously a complex one, and for this preliminary study we have restricted our attention to one combination of donor-inert-acceptor system, viz. C2H 2 + 2.502, Air and C2H 2 + 2.502 + 6N2, all at an initial pressure of 100 torr. Various concentration profiles were produced by allowing diffusion to occur at the first interface for known intervals of time, whereas the second interface was in each experiment an abrupt one and produced by removing the slide just prior to firing the tube. Tube sections 160, 260, and 330 mm long, respectively, could be inserted between the two slide sections. This enabled a range of concentration gradients to be set up between the first and second interfaces.
320
G . O . THOMAS ET AL. ° O
Experimental Results Superimposed pressure and distance-time plots of shock and flame front motion are shown in Figs. 10a and 10b for 0 and 30 s diffusion times, respectively, and an initial interface separation of 162 mm in the 22 x 10-mm tube. Both exhibit flame acceleration in the acceptor section and transit to detonation, the transition occurring earlier with an initial 30 s diffusion time. With an initial interface separation of 260 mm, transition to detonation again occurred in all cases, although the time and distance over which transition occurred was not strongly influenced by the initial diffusion times, which ranged from 0 to 120 s. A typical distance-time plot is shown in Fig. 11. No transitions were observed with an interface separation of 330 mm. For the 50-mmdiameter tube, an initial interface separation of 260 mm was used, but in this case transition to detonation was observed very close to the second interface for all diffusion times.
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As noted earlier, one factor that complicates the interpretation of the data obtained in this investigation, particularly in the small rectangular tube, is the strong influence of the wall boundary layers on both the shock and flame fronts. Because of this tube dependence a strict quantitative evaluation of the data cannot be attempted. Nevertheless the body of data obtained is useful in elucidating the broad features of the transmission mechanism. Thus in the 22 x 10-mm tube, with air as the inert gas, the strength M t of the transmitted shock is 3.8, rapidly decaying to a plateau value of 3.3, whereas for the 50-mm tube it is 4.5. In both these cases the postshock temperatures (795 K and 1214 K, respectively) are such that ignition of the gas occurs, but with a significantly longer induction time to ignition for the lower temperature. This means that, in the smaller tube, direct ignition of the acceptor does not occur during the experimental observation time, whereas in the 50-ram-diameter tube it does. Moreover, the transition process observed in the 50-mm-diameter tube is very rapid whereas in the smaller tube it
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Fig. 10. Distance-time and superimposed pressure histories for detonation transmission across an inert gap, length 162 mm, from CzH 2 + 2.502 into air into CzH z + 2.502 + 6N z at an initial pressure of 100 torr. Diffusion time (a) 0 s and (b) 30 s. Solid line, shock front locus, dashed line, flame front motion determined from microwave measurements.
occurs at a distance of the order of 1 m. Both transitions, however, are similar in their mechanism in many aspects and only differ essentially in their time scale: the details of these transition mechanisms have been described by Urtiew and Oppenheim [14] and Lee and Moen [15]. It is also interesting to compare the present results with those of Edwards et al. [5], who investigated the initiation of detonation in the same acceptor mixture by steady planar incident
DETONATION WAVES AT CONCENTRATION GRADIENTS
In the present work, the incident shock speeds are less in the rectangular tube than those observed previously by Edwards et al. [5], and the acceleration mechanism is different. As can be seen from Fig. 11, detonation occurs as a result of flame acceleration only, and the transmitted shock plays no role other than to preheat the mixture. The accelerating reaction front gives rise to a second shock front, and transition to detonation then occurs behind this accelerating shock, in a manner reminiscent of the mechanism discussed above. Thus, although the mechanisms whereby the conditions for transition develop are different in the two studies, both appear to exhibit the same conditions at the time of detonation inception. This should not be surprising, because for transition to detonation to be successful the conditions must be such that self-sustaining transverse wave structure must develop and propagate away from the point of initiation.
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Fig. 11. Distance time and superimposed pressure histories for detonation transmission across an inert gap, length 263 ram, from C2H2 + 2.502 into air into C2H2 + 2.502 + 6N2 at an initial pressure of 100 tort. Diffusion time 30 s. Solid line, shock front locus; dashed line, flame front motion determined from microwavemeasurements. waves using the same 22 x 10-mm tube. In their results, the incident wave Mach number is close to 4.5 and the reaction, which occurs after a short induction time, soon couples with and strengthens the incident shock. The result is the formation of a localized explosion due to a coherent energy release by particles exhibiting a gradient in induction time, caused by the continual increase in temperature behind the accelerating shock front. It is this localized explosion that forms into a detonation wave.
1. The initial strength of the shock wave transmired following the normal interaction of a detonation wave with an abrupt planar interface with an inert gas agrees reasonably well with the values calculated by the PatersonGlass model, which ignores Taylor expansion wave and wall losses. At later times, the influences of all losses on the pressure profile, particularly in the smaller tube, is well predicted by an RCM model using empirical values of wall friction coefficients. 2. Inevitably, some imperfections exist in the interface separating the detonation mixture and the inert gas. These give rise to trapped pockets of unreacted but shock-heated gas between the frontal shock and the flame boundary. When this pocket explodes, it causes a secondary pressure pulse to propagate behind the transmitted shock. The strength of this pulse is determined by the relative acoustic impedances of the inert and burned gases. 3. When a detonation is incident upon a region with varying fuel concentration, the velocity
322 and frontal structure of the transmitted detonation at a particular plane rapidly adjust to the values appropriate to that plane. 4. In order that a shock should transform to a detonation in a homogeneous system in a smooth-walled tube its strength must be of the order 0.6 M o . For weaker shocks the induction times to exothermic reaction are too long to allow the reaction wave and shock front to couple. However, if a fuel concentration gradient is present, flame acceleration can occur, which produces a second shock wave in the unburned gas initially preheated by the transmitted shock. Transition to detonation occurs when the reaction wave and second shock couple. Smooth concentration gradients can, therefore, facilitate the transmission process in confined mixtures. On the other hand, abrupt gradients may dissociate the front shock and reaction zone and cause the detonation to fail.
G . O . THOMAS ET AL. 3.
4. 5. 6. 7. 8. 9.
10. 11. 12.
13. 14.
REFERENCES 1. Donato, M., Donato, L., and Lee, J. H., First Specialist Meeting (International) of the Combustion Institute, Bordeaux, 1982. 2. Strehlow, R. A., Adamcykz, A. A., and Stiles, R. J., Astronaut. Acta 17:509 (1972).
15.
Bull, D. C., Elsworth, J. E., McLoed, M. A., and Hughes, D. J., Prog. Astronaut. Aeronaut. 75:61 (1981). Bjerketvedt, D., Sonju, O. K., and Moen, I. O., Prog. Astronaut. Aeronaut. 106:109, 1986. Edwards, D. H., Thomas, G. O., and Williams, T. L., Combust. Flame 43:187 (1981). Boyd, C. A., Stein, N., Steingrimsson, V., and Rumpel, N. F., J. Chem. Phys. 19:458 (1951). Edwards, D. H., Hooper, G., and Morgan, J. M., J. Phys. D. Appl. Phys. 7:242 (1974). Poachek, H., and Seegar, R. J., Phys. Rev. 84:922 (1951). Paterson, S., Fourth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1953, p. 468. Glass, I. I., UTIAS Report No. 12, pt. 1 (1958). Saito T., and Glass, I. I., UTIAS report No. 240 (1979). Shapiro, A. H., The Dynamics and Thermodynamics of Compressible Fluid Flow, Ronald Press, 1953, Vol. 1, p. 159. Edwards, D. H., Brown, D. R., Hopper, G., and Jones, A. T., J. Phys. D. AppL Phys. 3:365 (1970). Urtiew, P. A., and Oppenheim, A. K., Proc. R. Soc. A 295:13 (1966). Lee, J. H., and Moen, I. O., Prog. Ener. Combust. Sci. 6:359 (1980).
Received 26 June 1990; revised 27 October 1990
COMBUSTION A N D F L A M E 84:312-322 (1991)
The Behavior of Detonation Waves at Concentration Gradients G. O. THOMAS,* P. SUTTON, and D. H. EDWARDS Department of Physics, University College of Wales, Aberystwyth, Dyfed, UK SY23 3BZ.
Two detonation tubes were used to Study detonation propagation in concentration gradients. Each tube, 22 x 10 m m and 50 nun diameter in cross section, respectively, was fitted with slide valves that with the tubes mounted vertically allowed known concentration gradients to form by diffusion. The peak-transmitted shock strength when a detonation is incident on an abrupt planar interface with an inert gas was found to agree reasonably well with the simple Paterson-Glass model. A significant improvement was found in the predicted values when allowance was made for the Taylor expansion and energy and momentum losses to the tube walls. Under certain conditions a secondary pressure pulse was observed behind the transmitted shock; the origin of this pulse is the explosion of the unreacted gas between the transmitted shock and the unburned gas. When incident on gradients of fuel concentration, the transmitted detonation wave velocity and cell size adjusted rapidly, to values appropriate to the local gas composition and wall boundary layer losses. Finally, it was shown that the mechanism of detonation initiation in a weaker mixture by an incident detonation propagated across a concentration gradient differs in one important respect from that in a homogeneous mixture in that a secondary shock is formed as an intermediate stage.
INTRODUCTION Most studies on the ignition and propagation of detonation waves are with systems of uniform composition. However, fuel-air clouds, irrespective of the manner they are produced, will exhibit varying degrees of inhomogeneity, and the way in which detonation waves propagate in such clouds has received scant attention. Because of the lack of information on the influence of concentration gradients on wave behavior it is often tacitly assumed in hazard analyses that the presence of such gradients will hamper rather than enhance the propagation of a detonation wave, but this assumption is by no means proven. The present study therefore aims at elucidating some of the factors that control the behavior of detonations in the presence of compositional variations. A preliminary study of the problem has been *To whom all correspondence should be addressed.
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reported by Donato et al. [1]. However, because of the uncertainties that existed in the production of the gradients this is of qualitative interest only. Similarly, the fundamental studies of Strehlow et al. [2] on the transmission of a detonation wave into an inert gas across a compositional interface were primarily concerned with the decay rates of the transverse wave systems rather than the strengths of the frontal shocks. However, a work that has a direct bearing on the present investigation is that described by Bull et al. [3] on sympathetic detonations. These authors determined the maximum gap lengths of air that spherical detonation fronts, of different radii, could travel and subsequently cause reignition in a similar unconfined system. More recently, Bjerketvedt et al. [4] have reported an extensive experimental study of the transmission of a detonation across an inert region of air. They were able to establish, inter alia, some important conclusions regarding the Copyright © 1991 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc. 655 Avenue of the Americas, New York, NY 10010
DETONATION WAVES AT CONCENTRATION GRADIENTS enhancement of the transmission efficiency as the steepness of the concentration gradient at the inert region decreased. The significance of these conclusions is discussed in the present article in light of the new results that are described.
EXPERIMENTAL Before meaningful results could be obtained it was essential to be able to create well-defined concentration gradients in the system under study. Preliminary experiments showed that, in a horizontal detonation tube, the vertical, free interface, created by the rapid removal of a thin diaphragm remained reasonably planar for less than 1 s before distorting due to buoyancy effects. Consequently, horizontal tubes could only be used for the study of abrupt gradients if the tube is fired within a fraction of a second following the removal of the diaphragm. In all our work therefore vertical tubes were used. The first was described by Edwards et al. [5] with an internal cross section of 22 x 10 mm and comprising three sections: a driver section that was used to generate a stable detonation in the donor gas, and a third, acceptor section. A stable free interface was created between donor and acceptor gases by removing a purpose-built slide. The second tube comprised the same three sections, but was constructed from 50-mm-diameter stainless-steel tube. Concentration gradients of any desired steepness could be easily generated by allowing the gases initially separated by the slide to diffuse for a given time. In order to check the quality of the diffusional gradients set up by the system a separate section was built to study the density profiles using a Mach-Zehnder interferometer. In this case the only effect that the removal of the diaphragm appeared to have was to produce a highly damped oscillation of the interface, with a period of about 0.5 s, generating a disturbed region of gas with a thickness of about 5 m m on either side of the interface. At later times, the concentration gradients that developed as a result of diffusion were in good agreement with those predicted by the solution of Boyd et al. [6] for Fick's law. Piezoelectric pressure transducers mounted in
313
the tube wall recorded the detonation and shock motion and the related pressure profiles. The positions of the reaction zones were determined using 3-cm wavelength microwaves launched by an aerial placed in the end wall of the acceptor section. When these are reflected by the ionized burned gases the reflected signal is detected by the launching aerial and the recorded interference pattern enables the velocity of the reflecting surface to be found. Ionization probes provided reference points and thus gave an absolute determination of the reaction zone position. This method has proved extremely useful in monitoring reaction zones over long lengths of tube, e.g, Ref. 7. A window section was available for taking both spark and streak schlieren photographs. Unfortunately the closest edge of the window was 7 cm from the interface so that it was not possible to observe the interactions at the interface itself. In addition to the window section a smoked-foil section was also available for observation of the detonation wave structure both upstream and downstream of the slide valve. In tubes of small internal dimensions, boundary layer losses can greatly influence the nonsteady Taylor expansion wave, which, in turn, will affect the dynamics of the wave interaction process. It was therefore decided to construct a second detonation tube, of the same design as the first, but of circular cross section and 50 nun diameter. A number of experiments were repeated in this tube, thus enabling the assessment to be made of the magnitude of the boundary layer effect. I N T E R A C T I O N OF A D E T O N A T I O N WAVE WITH A PLANAR I N E R T INTERFACE When a planar detonation front interacts with normal incidence on a plane interface separating the reactive gas from an inert medium the transmitted wave is always a shock, whereas the refleeted wave may be either a shock or a rarefaction, depending on the relative acoustic impedances of the reactive and inert gas. The problem is to determine the strength of the transmitted shock together with the flow field behind
314 it. If allowance were to be made for the three-dimensional character of the detonation front then the problem would be intractable. Some idealization is therefore necessary to remove the transverse wave interactions. One way, termed the square-wave model, is to assume a plane shock front followed by uniform flow, the duration of which is determined by the kinetics of the thermoneutral induction reactions. These reactions are terminated by the onset of exothermic reactions which accelerate the flow to constant equilibrium or Chapman-Jouquet (CJ) values. More recently, calculations have been made using the Random Choice Method (RCM) code and a onestep model of detonation. Assuming infinitely fast kinetics, CJ values of particle velocity, pressure, sound speed, and so on are assigned to the wave front which then propagates as determined by the RCM algorithm. This approach has the distinct advantage that the appropriate Taylor expansion wave is modeled correctly. In the present article, calculated values for several interactions using both of the above models are described and compared with the observed results in the two detonation tubes. The Transmitted Shock Wave The problem of shock wave refraction, at both normal and oblique incidence, at a planar gaseous interface was first investigated by Poachek and Seegar [8]. Paterson [9] and later Glass [10] extended these analyses to determine the behavior of a detonation front at a boundary with an inert gas. They assumed that the effect of the induction and reaction zones of the interaction process could be disregarded because of the extreme brevity of their duration. The detonation is therefore assumed to be a wave across which the gas parameters are raised instantaneously to those values obtaining at the CJ plane. This idealized interaction is depicted in Fig. 1. A shock S i, which is incident on the interface, I, gives rise to a transmitted shock, S t under all circumstances, whereas the reflected wave, R, can be either a shock or a rarefaction depending upon the ratio of the acoustic impedances across L The nature of this wave and the strength of S t is determined by the
G . O . THOMAS ET AL.
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requirement that both the pressure and particle velocity be equal across the contact surface C, This requirement may be expressed graphically as shown in Fig. 2. The (velocity, pressure) curves, labeled a, b, c, d, and e, are Rankine-Hugoniot curves that represent the possible states obtaining behind shock waves over a range of Mach numbers in various gases. Similarly, using CJ values and the equation of state for the reaction products, curve A is obtained, and is centered on the CJ point. Movement to the left of this point corresponds to a rarefaction propagating through the detonation products, whereas movement to the right corresponds to a shock. The intersections of a, b, c, d, and e into A give the solutions for the interaction of the detonation-inert gas interface. It can be seen from this that all the gases give rise to a reflected rarefaction when a detonation wave in oxyacetylene is incident on them, except for SF6, in which case the reflected wave is a shock. Calculated values of the Math numbers of the transmitted shock waves from the interaction of a
DETONATION WAVES AT CONCENTRATION GRADIENTS
TABLE 1 Calculated Values of Mach Numbers of Transmitted Shock Waves from the Interaction of a Detonation in C2H 2 + 2.502 with Several Inert Gases
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315
Inert Gas
Calculated Mach No. (Paterson Theory)
He Air Ar CO 2 SF6
2.8 4.6 4.9 5.1 6.3
(b)
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Experimental Mach Number 22 x 10 nun 4, - 50 m m 2.4 4.4 4.3 4.3
2.9 4.7 4.8
(e) (A)
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20
a
I
I
40 PRESSURE RATIO
I
60
Fig. 2. Hugoniot relations for shocks in (a) He, 0a) Air, (c) CO z, (d) Ax, (e) SF6, and (A) for shock and rarefactions in detonation products of C2H 2 + 2.502.
detonation in C2H 2 + 2.5 0 2 with several inert gases are given in Table 1. Experimental values from both detonation tubes are also shown for comparison. In both tubes the observation point was 7 cm away from the position of the interface. Good agreement between theory and experiment is obtained in the 50-mm-diameter tube, whereas in the smaller rectangular tube there is a slight discrepancy between the two values. It will be shown that this arises from the effect of the enhanced wall boundary layer, which occurs in the smaller tube, on the Taylor expansion wave.
and application of the analysis of Shapiro [12] to calculate the modified pressures and temperatures. Using values of Cy determined by the previous investigations of Edwards et al. [13], pressure profiles for a detonation wave in C2H 2 + 2.502 at an initial pressure of 100 torr are given in Fig. 4a for the 5-cm-diameter tube with Cy = 0.015 and Fig. 4b for the smaller, 22 x 10 mm tube and a value of Cy = 0.035. The agreement with experimental profiles in both cases is very good. The next step was to use the pressure profile data provided in Fig. 3, together with the related density and particle velocity profiles, to examine the interaction with a planar air interface. The results obtained for the transmitted shock for both detonation tubes are displayed in Figs. 4 and 5 as an (x, t) trajectory and ( M s, x) plot, respectively. It will be noted that the agreement beo
An Improved Model Using an RCM Code A serious weakness of the Paterson-Glass model is the fact that it ignores the effect of the nonsteady expansion that follows the detonation wave. At best, therefore, it can only claim to give the true instantaneous values of the wave parameter immediately after interaction. One numerical code that allows the profile of the incident detonation to be taken into account is the RCM. The program used for the present calculations was based on that of Saito and Glass [11]. This allows the Taylor expansion wave to be included, while momentum and heat transfer losses to the tube wall can be accounted for by a friction factor (7/
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Fig. 3. Measured and calculated detonation pressure-time histories with boundary layer losses, l , rectangular wave guide; O, 50 m m diameter pipe. Solid lines are calculated profiles: (a) C / = 0.015, (b) Cf = 0.035.
316
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Fig. 5. Transmitted shock Mach number M t versus distance for C2H2 + 2.502 into (a) air, (b) CO2, and (c) helium. Solid line in each case is ideal Paterson solution. Experimental values for the 50-ram-diameter tube for (ll) 4.42 m and ([]) 1.5 m driver length; O, rectangular tube.
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value. The magnitude of this plateau value is somewhat less than that predicted by the idealized Paterson-Glass model.
Fig. 4. Comparisonof experimental distance time plots with corresponding RCM calculations for C2H 2 + 2.502 into air. Initial pressure is 100 torr, (a) rectangular tube, (b) 50 mm diameter tube. Experimental; l , shock, I , contact surface. RCM; [], shock, (D, contact surface.
The Secondary Pressure Peak
tween the theoretical and experimental data is much better in the case of the 50-mm-diameter tube than the 22 × 10 m m tube: this is due to the more pronounced influence of the wall boundary layer in the smaller tube. A further point to be observed in the velocity plots is that the strength of the transmitted shock takes a time of the order 100 ps, corresponding to a distance o f approximately 40 mm, to attain a reasonably steady
From four pressure transducers placed upstream of the interface the pressure-time histories behind the transmitted shock in the acceptor gas are obtained. An example of these records is shown in Fig. 6 for C2H 2 + 2.502 and CO 2 at a Po of 100 torr as donor and aceeptor systems, respectively. Also shown in the diagram are the trajectories of the shock and flame fronts. A notable feature of the shock profiles, apart from the oscillations deriving from the transverse
DETONATION WAVES AT CONCENTRATION GRADIENTS
317
contrast to this is provided by air and CO 2, which give rise to a weaker R, and SF6 where R is a weak shock; conditions that are more conducive to an explosion.
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I N T E R A C T I O N OF A D E T O N A T I O N W I T H VARYING C O N C E N T R A T I O N GRADIENTS I
I 200
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Fig. 6. Typical distance-time plots for C2H 2 + 2.502 into CO2, showing evidenceof a secondarypressure peak. Solid line, shock; dashed line, flamelocus.
waves, is the evidence of a secondary peak that forms some time after the decoupling of the shock and reaction zone of the incident wave. A fairly extensive study of the secondary peak revealed three things. Firstly, the magnitude of the peak depends on the acoustic impedance of the acceptor; the higher the value of the latter the stronger is the peak developed. Thus the effect was found to be strongest in SF6, whereas in He it was nonexistent. Secondly, the sharper the concentration gradient between the donor and acceptor (i.e., the shorter the diffusion time), the larger the peak. In gradual gradients, as we shall see in the following section, the secondary peak is much less pronounced. Finally, no evidence of such a peak was found with the corresponding RCM solutions. From this evidence, it would appear that the secondary peak originates from an explosion of the unreacted gas between the transmitted shock and the burned gas boundary. If the interface were perfectly plane, then no such pocket could exist; however, despite the utmost care, a mixing region of the order of 40 mm wide was always generated, as can be seen from the microwave velocity data discussed below. The reactive gas in this region is therefore diluted and will exhibit a longer induction time than the orginal mixture. It is the delayed reaction of this pocket that, it is postulated, gives rise to the observed secondary pressure rise. The strength of the reflected expansion, R, in He is large and will therefore tend to inhibit such an explosion. A
The next stage of the investigation involved the creation of concentration gradients of varying steepness between the detonable donor and the inert acceptor. Although some previous studies, referred to in the Introduction, have been reported in which finite gradients have been used, none of these has been able to specify the magnitude of the gradient with any precision. Indeed, in most of them a large uncertainty exists in both the nature and size of the gradient. In the present work carefully designed slide valves enabled very weil-defined gradients to be set up by diffusion in both rectangular and circular detonation tube. The progress of the detonation front through a concentration gradient was determined from velocity measurements, and at the same time the wave front structure was recorded by means of smoked foils.
Velocity Measurements A convenient way of measuring the velocity of a detonation wave in a tube is the microwave doppler technique, which has the advantage of providing a continuous monitor of the velocity with distance; see, for example, Ref. 13. In the present study the rectangular tube was employed because it ideally accommodates 3 cm wavelength microwaves. The results obtained for detonations in C2H 2 + 2.502 propagating into argon are presented in Fig. 7. Gradients of concentration were obtained by allowing diffusion to take place between the oxyacetylene and argon, for varying times after the removal of the slide. The solid curves represent the calculated CJ values appropriate to the particular composition. Ideally, of course, these curves should be coincident, but the
318
G . O . T H O M A S ET AL.
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Fig. 7. Velocity distance plots for a range of diffusion times for C2H 2 + 2.502 into argon, Po = 100 torr. (a) 0 s, (b) 120 s, (c) 240 s, and (d) 600 s. Solid lines are calculated CJ velocities correspondingto local gas composition. discrepancy observed, which increases with the degree of dilution, can be accounted for by the effects of the wall-boundary layer• From a knowledge of the concentration gradients and the velocity-distance record obtained by the microwave records, velocity-dilution values can be obtained; these are plotted in Fig. 8. It is seen that the detonation velocities in the gradients agree well with the corresponding measured steady-state CJ value. Hence we conclude from this agreement that the velocity at any plane in a concentration gradient adjusts very rapidly to the value appropriate to the dilution and boundary layer effects at that plane. Further evidence o f this rapid equilibration to the degree of concentration is provided by the smoked-foil records described below. Smoked-foil
taken in the region of concentration gradient provide a better test than velocity to the wave adjustments to local concentration conditions. Several inert gases were examined in this way over a range of diffusional gradients. The result with
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The spacing of the transverse tion, and hence the cell size, the degree of departure o f the the CJ value. For this reason,
waves in a detonais very sensitive to wave velocity from smoked foil records
40'
60'
;0
10
'o
ARGON DILUTION (%)
Fig. 8. Measured velocities as a function of local gas composition. Solid line is calculated CJ; O, measured steady state velocities. Diffusiontimes are (I-7) 120 s, (11) 240 s, (×) 360 s, and (+) 600 s.
DETONATION WAVES AT CONCENTRATION GRADIENTS
319
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,
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t
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Fig. 9. Variation of measured cell size with distance after 6 rain diffusion time at a C2H 2 4- 2.502/CO 2 interface. Open points are from measurements of steady state detonations at corresponding compositions.
carbon dioxide are shown in Fig. 9. A 6-rain diffusion time was used in this test. The full line represents the cell size for a steady-state detonation as a function of CO 2 dilution. Bearing in mind the difficulty in measuring the cell size in systems with some degree of irregularity, the agreement between the steady-state date and those obtained in the gradients is very good. Thus, taken together, the velocity and cell-size data indicate that the cell structure adjusts rapidly to the local concentration conditions. This supports the conclusion previously established by Strehlow et al. [2]. TRANSMISSION OF A D E T O N A T I O N THROUGH A CONCENTRATION GRADIENT Having established the behavior of the transmitted shock caused by the normal incidence of a
detonation wave both at an abrupt and gradual interface with an inert gas, we can proceed to examine the effectiveness of this wave in reestablishing a detonation in an acceptor mixture. This problem is obviously a complex one, and for this preliminary study we have restricted our attention to one combination of donor-inert-acceptor system, viz. C2H 2 + 2.502, Air and C2H 2 + 2.502 + 6N2, all at an initial pressure of 100 torr. Various concentration profiles were produced by allowing diffusion to occur at the first interface for known intervals of time, whereas the second interface was in each experiment an abrupt one and produced by removing the slide just prior to firing the tube. Tube sections 160, 260, and 330 mm long, respectively, could be inserted between the two slide sections. This enabled a range of concentration gradients to be set up between the first and second interfaces.
320
G . O . THOMAS ET AL. ° O
Experimental Results Superimposed pressure and distance-time plots of shock and flame front motion are shown in Figs. 10a and 10b for 0 and 30 s diffusion times, respectively, and an initial interface separation of 162 mm in the 22 x 10-mm tube. Both exhibit flame acceleration in the acceptor section and transit to detonation, the transition occurring earlier with an initial 30 s diffusion time. With an initial interface separation of 260 mm, transition to detonation again occurred in all cases, although the time and distance over which transition occurred was not strongly influenced by the initial diffusion times, which ranged from 0 to 120 s. A typical distance-time plot is shown in Fig. 11. No transitions were observed with an interface separation of 330 mm. For the 50-mmdiameter tube, an initial interface separation of 260 mm was used, but in this case transition to detonation was observed very close to the second interface for all diffusion times.
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As noted earlier, one factor that complicates the interpretation of the data obtained in this investigation, particularly in the small rectangular tube, is the strong influence of the wall boundary layers on both the shock and flame fronts. Because of this tube dependence a strict quantitative evaluation of the data cannot be attempted. Nevertheless the body of data obtained is useful in elucidating the broad features of the transmission mechanism. Thus in the 22 x 10-mm tube, with air as the inert gas, the strength M t of the transmitted shock is 3.8, rapidly decaying to a plateau value of 3.3, whereas for the 50-mm tube it is 4.5. In both these cases the postshock temperatures (795 K and 1214 K, respectively) are such that ignition of the gas occurs, but with a significantly longer induction time to ignition for the lower temperature. This means that, in the smaller tube, direct ignition of the acceptor does not occur during the experimental observation time, whereas in the 50-ram-diameter tube it does. Moreover, the transition process observed in the 50-mm-diameter tube is very rapid whereas in the smaller tube it
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400 600 TIME (microseconds)
(b)
~o O30 g SECOND INTERFACE
°
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200
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400
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600
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I
800
TIME (microseconds)
Fig. 10. Distance-time and superimposed pressure histories for detonation transmission across an inert gap, length 162 mm, from CzH 2 + 2.502 into air into CzH z + 2.502 + 6N z at an initial pressure of 100 torr. Diffusion time (a) 0 s and (b) 30 s. Solid line, shock front locus, dashed line, flame front motion determined from microwave measurements.
occurs at a distance of the order of 1 m. Both transitions, however, are similar in their mechanism in many aspects and only differ essentially in their time scale: the details of these transition mechanisms have been described by Urtiew and Oppenheim [14] and Lee and Moen [15]. It is also interesting to compare the present results with those of Edwards et al. [5], who investigated the initiation of detonation in the same acceptor mixture by steady planar incident
DETONATION WAVES AT CONCENTRATION GRADIENTS
In the present work, the incident shock speeds are less in the rectangular tube than those observed previously by Edwards et al. [5], and the acceleration mechanism is different. As can be seen from Fig. 11, detonation occurs as a result of flame acceleration only, and the transmitted shock plays no role other than to preheat the mixture. The accelerating reaction front gives rise to a second shock front, and transition to detonation then occurs behind this accelerating shock, in a manner reminiscent of the mechanism discussed above. Thus, although the mechanisms whereby the conditions for transition develop are different in the two studies, both appear to exhibit the same conditions at the time of detonation inception. This should not be surprising, because for transition to detonation to be successful the conditions must be such that self-sustaining transverse wave structure must develop and propagate away from the point of initiation.
Ta
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CONCLUSIONS /
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500
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1 0 0 0 1 5 0 0 2000 TIME (microseconds)
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2500
Fig. 11. Distance time and superimposed pressure histories for detonation transmission across an inert gap, length 263 ram, from C2H2 + 2.502 into air into C2H2 + 2.502 + 6N2 at an initial pressure of 100 tort. Diffusion time 30 s. Solid line, shock front locus; dashed line, flame front motion determined from microwavemeasurements. waves using the same 22 x 10-mm tube. In their results, the incident wave Mach number is close to 4.5 and the reaction, which occurs after a short induction time, soon couples with and strengthens the incident shock. The result is the formation of a localized explosion due to a coherent energy release by particles exhibiting a gradient in induction time, caused by the continual increase in temperature behind the accelerating shock front. It is this localized explosion that forms into a detonation wave.
1. The initial strength of the shock wave transmired following the normal interaction of a detonation wave with an abrupt planar interface with an inert gas agrees reasonably well with the values calculated by the PatersonGlass model, which ignores Taylor expansion wave and wall losses. At later times, the influences of all losses on the pressure profile, particularly in the smaller tube, is well predicted by an RCM model using empirical values of wall friction coefficients. 2. Inevitably, some imperfections exist in the interface separating the detonation mixture and the inert gas. These give rise to trapped pockets of unreacted but shock-heated gas between the frontal shock and the flame boundary. When this pocket explodes, it causes a secondary pressure pulse to propagate behind the transmitted shock. The strength of this pulse is determined by the relative acoustic impedances of the inert and burned gases. 3. When a detonation is incident upon a region with varying fuel concentration, the velocity
322 and frontal structure of the transmitted detonation at a particular plane rapidly adjust to the values appropriate to that plane. 4. In order that a shock should transform to a detonation in a homogeneous system in a smooth-walled tube its strength must be of the order 0.6 M o . For weaker shocks the induction times to exothermic reaction are too long to allow the reaction wave and shock front to couple. However, if a fuel concentration gradient is present, flame acceleration can occur, which produces a second shock wave in the unburned gas initially preheated by the transmitted shock. Transition to detonation occurs when the reaction wave and second shock couple. Smooth concentration gradients can, therefore, facilitate the transmission process in confined mixtures. On the other hand, abrupt gradients may dissociate the front shock and reaction zone and cause the detonation to fail.
G . O . THOMAS ET AL. 3.
4. 5. 6. 7. 8. 9.
10. 11. 12.
13. 14.
REFERENCES 1. Donato, M., Donato, L., and Lee, J. H., First Specialist Meeting (International) of the Combustion Institute, Bordeaux, 1982. 2. Strehlow, R. A., Adamcykz, A. A., and Stiles, R. J., Astronaut. Acta 17:509 (1972).
15.
Bull, D. C., Elsworth, J. E., McLoed, M. A., and Hughes, D. J., Prog. Astronaut. Aeronaut. 75:61 (1981). Bjerketvedt, D., Sonju, O. K., and Moen, I. O., Prog. Astronaut. Aeronaut. 106:109, 1986. Edwards, D. H., Thomas, G. O., and Williams, T. L., Combust. Flame 43:187 (1981). Boyd, C. A., Stein, N., Steingrimsson, V., and Rumpel, N. F., J. Chem. Phys. 19:458 (1951). Edwards, D. H., Hooper, G., and Morgan, J. M., J. Phys. D. Appl. Phys. 7:242 (1974). Poachek, H., and Seegar, R. J., Phys. Rev. 84:922 (1951). Paterson, S., Fourth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1953, p. 468. Glass, I. I., UTIAS Report No. 12, pt. 1 (1958). Saito T., and Glass, I. I., UTIAS report No. 240 (1979). Shapiro, A. H., The Dynamics and Thermodynamics of Compressible Fluid Flow, Ronald Press, 1953, Vol. 1, p. 159. Edwards, D. H., Brown, D. R., Hopper, G., and Jones, A. T., J. Phys. D. AppL Phys. 3:365 (1970). Urtiew, P. A., and Oppenheim, A. K., Proc. R. Soc. A 295:13 (1966). Lee, J. H., and Moen, I. O., Prog. Ener. Combust. Sci. 6:359 (1980).
Received 26 June 1990; revised 27 October 1990