Handbook of Shock Waves, Three Volume Set

CHAPTER

16.1

Chemical and Combustion Kinetics 16.1 Mass Spectrometric Methods for Chemical Kinetics in Shock Tubes RALPH D. KERN, H. J. SINGH AND Q. ZHANG Department of Chemistry, University of New Orleans, New Orleans, Louisiana, 70148, USA

16.1.1 Introduction 16.1.2 Coupling of a Time-of-Flight Mass Spectrometer to a Shock Tube 16.1.3 Chemical Kinetics Results from the TOF-Shock Tube Technique 16.1.4 Summary References

16.1.1

INTRODUCTION

The investigation of the rates and mechanisms by which chemical reactions occur presents a formidable array of obstacles to the experimentalist. The task is rooted in a diversity of experimental techniques and the skills of the workers performing them, although quantum mechanical calculations have established a more complete understanding of the complexities attendant to high-temperature chemical kinetics. The essential observations necessary to formulate a satisfactory mechanism for a particular reaction system involve the identification and measurement of the rates of formation and decay of the various reactants, intermediates, and products as a function of temperature and pressure in a well-defined environment. In the study of gas-phase reactions, it is advantageous to eliminate contributions from the surface of the reactor vessel. In addition, reliable thermodynamic data for the reaction species must be available either from tabulations, from the experiments, or calculated from theory. The proposed mechanism, consisting of forward and/or reverse rate Handbook of Shock Waves, Volume 3 Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISBN: 0-12-086433-9/$35.00

R. D. Kern, H. J.

Singh and Q. Zhang

constants along with the thermodynamic values for the enthalpies and entropies of pertinent species, is tested by modeling the observed reaction profiles as a function of temperature and pressure. The modeling should extend to data taken outside the range of temperature and pressure and reported by other workers employing different methods. The process is often iterative, and the critical values for the rate constants and thermodynamic quantities are uncovered by sensitivity analysis. The shock tube offers several advantages for the study of chemical kinetic systems: reactions may be investigated over a wide range of known pressures, temperatures, and reaction times in the gas phase in a uniform reaction zone; dynamic reactant, intermediate, and product analysis is possible; only small amounts of reactants dilute in an inert gas environment are required since the experiments demand single-shot analysis; elementary reactions can be observed in some cases (but it is more usual to encounter complex reaction systems); and the variance of the relative amounts of reactants is virtually unlimited. A discussion of each of these advantages follows. Applied to chemical problems the shock tube technique is unmatched for its ability to cover a wide range of temperatures (500-5000 K) and pressures (10-1000atm). These ranges encompass the practical interests of chemical kineticists. Relatively short observation periods (<5 ~ts), low temperatures (500-1300 K), and low pressures (<100 T) are most applicable to pre-reaction phenomena such as vibrational relaxation and incubation times. Higher temperature regimes (>3000 K) pertain to dissociations of molecules and radicals characterized by strong chemical bonds, e.g., N 2, 02, CO, and CN. High-pressure experiments (> 10 atm) are used to study combustion reactions related to engine performance and to establish high-pressure limits for dissociation reactions. One of the great advantages offered by the shock tube technique is the avoidance of reactions occurring on the tube walls. The role of surface reactions in conventional static reactors has always been a problem, but the relatively short observation times in shock tube work permit only a small fraction of the heated gas to travel to the walls (which are at room temperature), thereby eliminating surface reactions. The reaction zone is also devoid of sharp temperature gradients as are encountered in flame studies. Analysis of the reaction species by dynamic analytical techniques permits continuous real-time records, which are extremely difficult to achieve by absorption and/or emission spectroscopy. Usually only one or two species may be followed spectroscopically, and information about their respective spectral properties is also necessary. Time-of-flight (TOF) mass spectrometry is capable of identifying numerous species present during the observation period and recording their individual time histories. The TOF technique will be described in sufficient detail later on in this section.

16.1 Mass Spectrometric Methods for Chemical Kinetics in Shock Tubes The advantage of single-shot experiments employing the shock tube technique is that only a small percentage of reactant dilute in inert gas is used as compared to the relatively large amounts of reactants used in steadystate experiments such as flames and flow reactors. This opens the door to investigations of reactants that are either very expensive, difficult to synthesize and/or separate or hazardous and/or toxic and thus not available commercially. Often isotopically labeled compounds can be used to sort out some of the complexities of a reaction system. Some reactants, such as cyclopentadiene and methylcyclopentadiene, are available only in dimeric form from vendors and must undergo separation by distillation. The disadvantage of single-shot analysis is the requirement for fast detectors capable of recording vast amounts of data in often less than i ms observation time. Steady-state experiments enjoy the advantage of signal averaging, thereby reducing the scatter in the experimental data. The high dilution of the inert gas acts as a traditional heat bath, reducing temperature and pressure changes in the reaction zone due to enthalpy and mole changes in the reaction under study. Most chemical mechanisms consist of a sequence of elementary reactions. It is sometimes possible to isolate a particular chemical reaction and measure its rate in a shock tube experiment; several examples are found in the section on atomic resonance absorption spectroscopy. Practical systems of interest--such as flames, internal combustion engines, explosives, incineration, and atmospheric chemistrymrequire hundreds of elementary reactions to describe their behavior. The job of the chemical kineticist is simplify these systems and compile rate constants for as many elementary reactions as possible. For instance, to study the combustion of a complex fuel or fuel mixture, one might start with the studies of the pyrolyses of simpler fuels. Fuels of high molecular complexity often decompose into lower-molecular-weight compounds and/or radicals, and these subspecies then react with molecular or atomic oxygen and oxygen-containing radicals. The study of simpler pyrolyses and oxidations provide the key to building up the library of elementary reactions that may be applied to the modeling of more complex reaction systems. The study of flames to deduce the rates of elementary reactions first requires the flame to be ignited and maintained in some steady state for observation. This state is achieved within a rather narrow range of fuel/oxidizer ratio. Shock tube experiments are totally unaffected by this ratio. In fact, the ratio may be varied from 0 to oo, thus allowing the decomposition kinetics of the oxidizer to be investigated in the absence of the fuel and vice versa. Ignition delay times may also be recorded; these measurements are described in detail in another section. Nearly ideal environments for investigating problems in chemical kinetics can be achieved with the shock tube. The remaining part of the apparatus is the

R. D. Kern, H. J. Singh and Q. Zhang

analytical aspect. Various ingenious techniques have been adopted to monitor the individual rates of species present during the observations. However, the experimental objective is indeed daunting: to record simultaneously concentrations as a function of reaction time of all species from H atoms to highmolecular-weight (several hundred amu) products and intermediates that appear during the observation period. There is simply no one analytical technique that fulfills this objective. Nevertheless, the data from such techniques as TOE laser schlieren densitometry (LS), atomic resonance absorption spectroscopy (ARAS), single-pulse shock tube (SPST) end product analysis, and absorption and emission spectroscopy are complementary in the sense that each is capable of supplying a unique piece(s) of data. When viewed as a whole, these data enable chemical kineticists to assemble a coherent set of reactions that may be used to model some of the complicated features of practical systems. Each of these techniques is discussed in the following sections. The remainder of this section is devoted to the coupling of a TOF to a shock tube to analyze dynamically the chemical species in the reflected shock zone. Four reviews of chemical reactions behind incident and reflected shock waves have appeared in the Annual Review of Physical Chemistry: Bauer 1965; Belford and Strehlow 1969; Tsang and Lifshitz 1990; and Michael and Lim 1993. Another review of interest pertaining to dissociations of diatomic molecules studied by shock tube workers was published in 1976 (Kem 1976). The reader is referred to this and the following sections herein for an update of work in this area.

16.1.2 COUPLING OF A TIME-OF-FLIGHT MASS SPECTROMETER TO A SHOCK TUBE The first description of experimental results obtained from a TOF-shock tube apparatus appeared in 1961 (Bradley and Kistiakowsky 1961). The original version evolved as other workers and laboratories joined in the effort: Dove and Moulton 1965; Moulton and Michael 1965; Glass, Kistiakowsky, Michael, and Niki 1965; Gay, Kern, Kistiakowsky, and Niki 1966; Garnett, Kistiakowsky, and O'Grady 1969; Diesen and Felmlee 1963; Modica 1965; Ryason 1967; Barton and Dove 1969; and Kern and Nika 1971c. The only remaining TOF-shock tube apparatus operating is in the author's laboratory at the University of New Orleans. This apparatus has undergone many modifications since the initial description in 1970; a diagram of the latest version appears in Fig. 16.1.1. The description of the necessary features for successful experiments starts with the driver section, which is short in length relative to the driven section (13 in. vs 10.8 ft) and has an outside diameter of 11.7 in. This allows for the

16.1 MassSpectrometric Methods for Chemical Kinetics in Shock Tubes

FIGURE 16.1.1 Schematicof the TOF-shock tube apparatus.

placement of the spring-driven knife, which cuts the aluminum diaphragm into quadrants and thus preserves the diaphragm material and prevents diaphragm fragments from clogging the entrance to the TOF ion source (Dove and Moulton 1965). If by chance a small fragment from the diaphragm does find its way into the passage leading to the ion source, a lengthy disassembly process must be undertaken to clear the passage. Another essential feature is the placement of a ball valve 6.5 in. downstream of the diaphragm. The internal inside diameter of the ball valve matches that of the shock tube, I in. This allows a high vacuum to be maintained on the TOF side of the shock tube while the other side is raised to atmospheric pressure to change the diaphragm. The one-inch inside diameter of the shock tube is fixed by the TOF manufacturer's design of the entrance sleeve to the ion source. One of the most crucial and controversial aspects of the experiment is the sampling of the reflected shock zone. It was apparent early on that the boundary layer in the reflected shock zone starts to grow shortly after arrival of the shock wave at the end wall. If the hole in the center of the end wall were part of a fiat plate, a significant portion of the gas flowing from the reflected shock zone into the TOF ion source region would consist of the gas in the boundary layer. To minimize sampling from the boundary layer, a reentrant nozzle plate serves as the end wall (Dove and Moulton 1965) with the apex of the conical nozzle facing the reflected shock zone. The hole in the apex is ~0.1 mm in diameter to ensure hydrodynamic flow into the ion source; diffusive flow would favor masses of low amu and would therefore distort the sampling process with regard to chemical kinetics. Since the boundary

R. D. Kern, H. J. Singh and Q. Zhang

layer growth from the end wall to the apex is typically > 1 ms and is greater than the observation time of the experiment, <1 ms, the effects are truly minimized. The validity of the sampling has been tested repeatedly by comparing observed TOF equilibrium concentrations with the values calculated by thermodynamics and by comparing rates of reaction obtained by independent experiments. Two examples of many are the reactions of HC14- D 2 ~- DC14- HD (Kern and Nika 1971) and C2N 2 -}- H 2 ~- 2HCN (Brupbacher and Kern 1973). Both reactions were investigated using infrared emissions from HC1 and DC1 in the former and infrared emission from HCN in the latter. The experiments were conducted in separate shock tubes, one outfitted for time-resolved infrared emission traces and the other TOE Satisfactory agreements were obtained. This completes the shock tube side of the apparatus; a description of the mass spectrometer follows. A critical distance ~,3 mm, exists from the end plate to the electron beam in the ion source. Distances greater than 3 mm produce erratic results (Kern 1965). The operation of the TOF involves voltages as high as 3 kV. Typical vacuums are 2 x 10 -6 T. A problem arises when the shock wave reflects from the end wall, which increases the pressure in the shock tube to ~,0.5 atm. Subsequent flow from the reflected shock zone into the TOF will cause arcing of the high voltage due to the sudden pressure jump in the ion source region. Control of the pressure is attained by the open geometry of the ion source, the large volume of the TOF ('-'75 L), additional ballast volumes of "-,5 L each attached to ion source region, and two high-speed diffusion pumps. Alert operators turn off the TOF voltage when they hear the noise accompanying the diaphragm rupture and simultaneously open valves to high-capacity mechanical pumps connected to the shock tube. The TOF ion source usually runs at typical ionization intervals of 30 ~ts; during "-,1 ms obervation time, "-'30 mass spectra from 1-400amu are generated. Each mass spectra reveals the m/e of the various species that are present. Measurement of the peak heights as a function of reaction time are tabulated. Time zero for the reaction is determined by extrapolation of the shock velocity to the apex of the reentrant nozzle. The preshock pressures are 5 T of mixture in the shock tube and 2 x 10-6T in the TOE Static analysis of the test gas is performed routinely to check for impurities and/or leaks. Another necessity is the use of neon as the diluent. Neon has a relatively high ionization potential compared to argon and krypton. An electron energy of 30 eV will produce a peak height for neon, present at 96% of the mixture, comparable in height to the peak heights of the other reactants. If argon were employed as the major diluent, the peak height would be much higher and would result in saturation of the electron multiplier. The problem of pressure buildup in the ion source from shock arrival until a steady flow is achieved, ~-,60-80 ~ts, is solved by measuring peak height ratios.

16.1 Mass Spectrometric Methods for Chemical Kinetics in Shock Tubes For example in a strictly inert gas mixture of 99% Ne-1% Ar, the peak height ratio of Ar/Ne is ~0.4. Taking ratios also smooths the pressure fluctuations in the gas flow from the reflected shock zone to the ion source during the observation time. Although the individual peak heights for nonreacting species vary during a particular experiment, the ratio is reasonably constant, as shown in Fig. 16.1.2. For a reacting mixture, the observed peak heights for reactants, products, and intermediates are divided by the peak height of either neon or argon recorded in the same mass spectrum. To convert peak height ratio data to concentration-time profiles for each species, the individual mass spectral sensitivity must be measured or determined for each species. This task is accomplished by preparing a known amount of each species dilute in separate mixtures of neon-argon diluent. Each calibration mixture is shocked at nonreacting temperatures, and the measured peak height ratio is related to the known concentration thereby establishing the mass spectral sensitivity for acetylene, ethylene, or methane, as the case may be. Sometimes species are formed during a reaction that are difficult to synthesize and handle; e.g., triacetylene C6H 2. The procedure then depends on the carbon atom balance. The total carbon atom concentration is known at time zero. If the mass spectral sensitivities for the other species are known (e.g., C2H 2 and C4H2) , then the mass spectral sensitivity for C6H 2 may be deduced from the carbon atom balance. For radical intermediates, one must resort to reasonable estimates. The next major problem is data collection. In the first TOF-shock tube experiment, a rotating drum camera was mounted to an oscilloscope. With the 1.0

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FIGURE 16.1.2 Ratioof the peak heights of Ar/Ne as a function of observation time for a 99% Ne-1%Ar mixture at 1800K.

R. D. Kern, H. J. Singh and Q. Zhang

advent of 10,000 speed Polaroid film, a staircase generation circuit could be used to display 5 to 8 mass spectra on an oscilloscope screen with a stationary open lens camera to record the data. Four oscilloscopes triggered in series then captured the ~30 mass spectra recorded for a particular run. The various peak heights were usually read manually with the aid of magnifying equipment. Affordable transient digitizers have automated the once-tedious data reduction process. A digitizer has the capacity of storing one million points at a sampling rate of i ns. Since a typical TOF mass peak half-width is 40ns, 40 points are available to define the peak shape. Software has been written to determine the various amu values. Interesting parts of the recorded spectrum can be displayed and then peak heights of particular m/e values can be selected for automated measurements. Lastly, files can be created for input into the modeling programs (Kee, Rupley, and Miller 1989). The reaction profiles for the thermal dissociation of propyne serve as an example of a single TOF experiment. Displayed in Fig. 16.1.3 are the reactant decay profile and the growth profiles for three of the observed products: acetylene, diacetylene, and benzene. The solid lines represent the results of the modeling calculation for the proposed mechanism for propyne decomposition. The merits of the TOF technique are evident in Fig. 16.1.3; the reactant decay profile is recorded at 30 ps intervals, and the major products are identified and their temporal behavior are described along with their respective equilibrium plateaus. It is instructive to take a closer look at some of the details of Fig. 16.1.3. Time zero is established by the shock wave velocity and the known distance from the last velocity gauge to the end wall. The heart of the TOF circuitry is a free-running oscillator. If the first concentration of the reacting gas in the reflected shock is determined to take place 23 ~s after time zero and the ionizing intervals are selected to be 30 ~s, the reaction times recorded are at 23, 53, 83 . . . . . ~s. The concentration at time zero is calculated from the properties of the reflected shock zone under no-reaction conditions; for Fig. 16.1.3 the reflected zone total gas density is 2.1 x 10 -6 mol/cm 3. The initial concentration of propyne is 4.2 x 10 -8 mol/cm3; the initial carbon atom concentration is 7.59 x 1016 atom/cm 3 and is the basis for the carbon atom balance calculation. Since the mass spectral sensitivities of the products have been determined previously, the amount of "missing" carbon atoms may be determined. Their presence (not detected by TOF) is attributed to amounts of different minor carbon-containing products such as the polycyclic aromatic hydrocarbon compounds. The identities and trace amounts of these undetected (by TOF) species are revealed by other techniques, most likely SPST endproduct analysis. However, the latter technique does not reveal the respective time history profiles of such species. There are at least four other shortcomings of the TOF technique revealed in Fig. 16.1.3. First the technique cannot distinguish the respective amounts of

16.1

Mass Spectrometric Methods for Chemical Kinetics in Shock Tubes

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FIGURE 16.1.3 Reaction profiles recorded by the TOF technique for propyne decomposition at 1884 K and a total pressure of 0.34 atm. The solid lines are the modeling results. The dashed lines represent the model calculations for propyne and allene. Thus the solid line is the sum of the two dashed lines, which is the TOF observable m/e 40.

structural isomers. The m/e value at 40 is all due to propyne initially, but during the course of decay, allene is also formed by the propyne: allene isomerization reaction. This rate is known from other separate experiments. The solid line from the modeling calculation is for propyne; the TOF data points represent propyne + allene. Second, the total reflected shock zone

10

R. D. Kern, H.J. Singh and Q. Zhang

pressure for this run is 0.34 atm and cannot be varied appreciably due to the gas-flow conditions described previously for the TOE LS experiments provide the pressure-dependent rate data necessary to perform a Rice-RamspergerKassel-Marcus (RRKM) falloff calculation for propyne dissociation. Third, the LS data is capable of supplying precise data for the initial rate of propyne decomposition. Finally, ARAS experiments are required to record the H-atom profiles for this important reaction system. The complemetarity of the TOE LS, ARAS, and SPST techniques are illustrated amply for this moderately complex reaction system. Additional details and references to propyne pyrolysis will be presented later.

16.1.3 CHEMICAL KINETICS RESULTS FROM THE TOF-SHOCK TUBE TECHNIQUE There are two major areas of research pertinent to TOF results: isotopic exchange reactions and pyrolyses of hydrocarbons related to combustion science. The former is early work and will be dealt with briefly; the latter includes work that has been or will be published this year. Isotopic exchange systems are close to ideal examples for shock tube work. Since the reactions are thermoneutral, there is no temperature change in the reflected shock zone during the exchange process. The absence of a mole change ensures that the pressure is also constant. Exchange reactions are characterized by easily calculable equilibrium plateaus that allow a rigorous test of the TOF analytical results. Other workers and laboratories have employed the single-pulse technique, which allows the TOF results to be verified by independent sources. Whenever possible, internal kinetic and thermodynamic checks were performed independently at the University of New Orleans using separate shock tubes outfitted with different analytical techniques; namely, infrared emission spectroscopy and TOF mass spectrometry. Complementary data were obtained for the following reactions: HC1 + D2 ~ DC1 + HD (Kern and Nika 1971); HCN+D 2~DCN+HD (Kern and Brupbacher 1972); H B R + D 2~DBr + HD (Kern and Nika 1974). The results from each of these studies were compatible with regard to their respective Arrhenius expressions, quadratic time dependence for product formation, and reaction orders. Exchange reactions investigated solely by the TOF include the following: H 2+ D2 ~ 2HD (Kern and Nika 1971c); 2 H D ~ H 2 + D2 (Kern and Nika 1971b); CD4-iHC1 ~ CD3H + DCI (Kern and Nika 1972); 13C160 -~- 12C180 ~13C180 -~- 12C160 (Bopp, Kern, and O'Grady 1975); C2D2 + HC1 ~- C2HD +

16.1 Mass Spectrometric Methods for Chemical Kinetics in Shock Tubes

11

DC1 (Bopp and Kern 1975); 28N2 +30 N2 ~_ 214N15N (Bopp, Kern, and Niki 1977) 320 2 +36 02 ~ 216018 0 (Bopp, Kern, Niki, and Stack 1979). The exchange of H 2 and D 2 typify the results and problems associated with exchange reactions. The first shock tube studies of the forward and reverse reactions were performed with the SPST technique (Bauer and Ossa 1966; Burcat and Lifshitz 1967; Lewis and Bauer 1968). The forward reaction was characterized by fractional reaction orders on H 2 and D 2 whose sum is "-~1. There is also an inert gas dependence of ,-,1. Both these observations ruled out an elementary four-center reaction for the exchange of H 2 and D 2. TOF experiments (Kern and Nika 1971c) confirmed the complex nature of the exchange by reporting a nonlinear (quadratic) time dependence for the HD product growth. The observed equilibrium plateaus agreed with the statistical mechanical calculations of the equilibrium constants. The SPST data, corrected for quadratic growth, was in agreement with the TOF-derived Arrhenius parameters. The activation energy, ~44 kcal/mol, was too low for an atomic mechanism, which requires 110 kcal/mol, to be operative. A quadratic time dependence for product formation and first-order inert gas dependence are predicted by the atomic mechanism in agreement with the shock tube data. But first-order dependence is predicted for both H 2 and D2, which is not in agreement with the shock tube results. A mechanism involving a vibrational energy chain reaction results in a linear time dependence for product formation clue to the relatively short vibrational times for H 2 and D 2 (Kiefer and Lutz 1966 a & b). The development of both these mechanisms involved the steadystate approximation. A more detailed treatment is demanded by the vibrational excitation mechanism (VEX), (Bauer and Ossa 1966). The exchange is proposed to occur only by collisions in a critical vibrational energy manifold with molecules containing lesser amounts of vibrational energy. VEX causes a depletion of the population of vibrational levels and prevents the system from attaining a Boltzmann distribution of vibrational energy for a time greater than the vibrational relaxation times. Solution of this problem for the H 2 + D 2 exchange without invoking the steady-state approximation reveals a nonlinear time dependence for product formation. However, application of VEX to more complicated molecular exchange reactions is a formidable task. The presence of trace impurities in the reactant gas mixture in amounts sufficient to accelerate the exchange rate is a definite concern (Lifshitz and Frenklach 1977). However, the TOF experiments were conducted under conditions in which the presence of 02 was ~,20 ppm, an amount insufficient to affect the rate. In summary, SPST and TOF investigations of many simple exchange reactions presented convincing experimental evidence that these reactions occur by a series of steps and proceed via overall activation energies far below the values required by an atomic mechanism. There are some exceptions to this statement:

12

R. D. Kern, H. J. Singh and Q.

Zhang

15NO + C180and 15NO 4- N 2 (Bopp, Kem, Niki, and Stack 1980); 3202 4- 3602 at high temperatures (Bopp, Kern, Niki, and Stack 1979), and HBr 4- D2 (Kern and Nika 1974) may be explained by atomic mechanisms. However, the nonexistence of simple four-center transition states due to bimolecular collisions between reactants leading to products has been amply demonstrated. Although the validity of dynamic sampling of the reflected shock zone gas by the TOF technique has been supported by the agreement of the experimental equilibrium plateaus recorded with the values obtained from statistical mechanical calculations, the plateaus are temperature independent at the elevated temperature levels of the experiments, 1700-5400 K. A more stringent test resulted in the study of metathetical reactions in which K = f(T): C2N24H 2 ~ 2HCN (Brupbacher and Kern 1973), H 2 4- CO 2 -+ CO 4- H20 (Brupbacher, Kern, and O'Grady 1976); CNC1 4- H 2 -+ HCN 4- HC1 (Brupbacher, Esneault, Kern, Niki, and Wilbanks 1977), and H 2 4- Br2 --+ 2HBr (Bopp, Johnson, Kern, and Niki 1982). The various equilibrium plateaus as a function of temperature observed in these TOF experiments agreed with the respective thermodynamic calculations. Thus the internal reaction "thermometer" was indeed governed by the temperature of the reflected shock zone and verified by TOF measurements. The second major area of chemical kinetics investigated by the TOF technique is the thermal decompositions of hydrocarbon fuels pertinent to pre-particle soot formation chemistry. The focus of this work is to elucidate the nature of the ring-rupture and ring-formation mechanisms associated with the production of aromatic compounds. Early shock tube work on this problem (Graham, Homer, and Rosenfeld 1975a and b) employed the laser beam extinction (LEX) technique. The attenuation of the laser beam aligned perpendicular to the shock tube was related to the absorption and scattering of the beam by soot particles and polycyclic aromatic hydrocarbons (PCAHs). The data was ultimately related to soot yield as a function of temperature. The majority of the fuels tested revealed a bell-shaped dependence for the soot yield as depicted in Fig. 16.1.4. Different fuels exhibited different amplitudes for the soot yield maxima, but in general the temperature observed for the various maxima were reasonably close. As examples, toluene and benzene exhibit soot yields of ~95% whereas the yield for an equal amount of acetylene is 10%. The explanation for the bell-shaped behavior was that, at low temperatures, the thermal energy available to the fuel was insufficient to create the concentrations required for the radical pool, whereas at high temperatures the aromatic compounds such as the PCAHs were thermally unstable and decomposed into acyclic fuels and radicals. The optimum temperature, ~1800 K, maximized the radical concentrations necessary to form the aromatic rings required to ultimately create the soot particles but was not so high as to cause ring-rupture reactions to be dominant. This explanation provided a reasonable

16.1

13

Mass Spectrometric Methods for Chemical Kinetics in Shock Tubes

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and useful model for soot formation. However, the measurements were of a bulk property and were totally lacking in the detailed information required for molecular chemical kinetics. The TOF technique was applied to further the understanding of soot formation chemistry. Two questions were posed: what are the mechanisms by which selected acyclic fuels form aromatic rings? And what are the mechanisms by which aromatic and substituted aromatic fuels decompose? The compounds selected for detailed study are shown in the following list. Acyclic Compounds acetylene diacetylene ethylene vinylacetylene allene propyne propaargyl chloride propargyl bromide 1,2-bu tadiene 1,3-butadiene

Aromatic Compounds benzene toluene ethylbenzene chlorobenzene pyridine pyrazine pyrimidine furan cyclop entadiene

The results obtained for many of the pyrolyses listed were described in a review published in 1991 (Kern and Xie 1991). These reactions are indeed complex and the role of various analytical techniques such as LS, ARAS, and visible absorption, SPST, and TOF was discussed. The reader is referred to the following reviews for TOF work on: acetylene (Wu, Singh, and Kern 1987;

14

R. D. Kern, H. J. Singh and Q. Zhang

Kern, Xie, Chen, and Kiefer 1990), ethylene (Gay, Kem, Kistiakowsky, and Niki 1966), vinylacetylene (Kiefer, Mitchell, Kern, and Yong 1988), allene (Wu and Kem 1987), 1,2-butadiene (Singh, Wu, and Kern 1988), 1,3-butadiene (Kiefer, Wei, Kern, and Wu 1985), acetaldehyde and ethylene oxide ( Kern, Singh, and Xie 1990), benzene (Singh and Kern 1983; Kern, Wu, Skinner, Rao, Kiefer, Towers, and Mizerka 1985), toluene (Pamidimukkala, Kern, Patel, Wei, and Kiefer 1987), ethylbenzene (Pamidimukkala, and Kem 1986), and pyridine (Kern, Yong, Kiefer, and Shah 1988). One important observation that arose from the TOF pyrolytic studies is the correlation of the maxima of the bell-shaped benzene profiles with the LEX soot yield maxima (Kern, Wu, Yong, Pamidimukkala, and Singh 1988). It was pointed out that those fuels that exhibited the greatest amount of benzene formation were ranked in the same order for producing the greatest amount of soot: allene > 1,3-butadiene > vinylacetylene > C2H2. The comparisons were made for mixtures that contained the same carbon atom concentration, 2 x 1017 atoms/cm 3. This observation established the relationship between the behavior of a particular molecular compound, benzene, and the macroscopic measurement of soot yield by the LEX method. The role of chemi-ions in soot formation is an important consideration. The TOF-shock tube technique has been employed to identify and measure the reaction profiles of chemi-ions appearing in the oxidation of acetylene (Glass, Kistiakowsky, Michael, and Niki 1965) and ethylene (Gay, Glass, Kern, and Kistiakowsky 1967) and to determine their absence in pyrolytic (oxygen-free) mixtures of acetylene, ethylene, benzene, and acetaldehyde. Deuterated forms of the latter three fuels were also tested (Kem, Singh, and Xie 1990b). It was concluded that chemi-ions play only a minor role in the chemistry of soot formation. The establishment of a comprehensive mechanism for acetylene pyrolysis is essential; it is an integral part of the formulation of mechanisms to describe the decompositions of aromatic fuels, the later stages of hydrocarbon fuel combustion in rich mixtures, and a portion of the soot formation reaction. The progress made in understanding acetylene pyrolysis since the review by Kern and Xie (1991) follows, along with descriptions of reaction systems investigated by the TOF technique from 1991 to 2000. Many workers using a variety of experimental techniques have established that acetylene decomposition is second order with respect to acetylene and that the activation energy associated with the decomposition is "-,45 kcal/mol. It is generally accepted that the initiation step involves the formation of an activated vinylacetylene intermediate C4H4":

2 C2H 2 ~- C4H4"

16.1 MassSpectrometric Methods for Chemical Kinetics in Shock Tubes

15

The details of this step involve the isomerization of acetylene to singlet vinylidene followed by addition of vinylidene to acetylene (Kiefer and Von Drasek 1990; Kiefer, Sidhu, Kern, Xie, Chen, and Harding 1992). C2H2 ~ CH2--C: CH2-C:-}-C2H 2 ~ C4H~ The energized C4H4" has several reaction pathways available, depending on the temperature and pressure. At low temperatures, stabilized C4H4 (vinylacetylene) is formed; at high temperatures, C4H 2 is formed via the molecular dissociation of C4H4" and from the radical chain mechanism. The radical chain mechanism is initiated by the formation of i-C4H 3 isomer, H - C = C - C = C H 2. C4H~ --+ i-C4H 3 q-H The rate of a key chain propagation step is highly dependent on the thermochemistry of C2H, the ethynyl radical. H + C2H2 --~ C2H + H 2 The heat of formation of the ethynyl radical is derived from the C - H bond energy in acetylene. C2H 2 if- M ~ C2H -}- H + M

The reaction occurs at very high temperatures and has been studied in the lowpressure region (Frank and Just 1980). The value employed for the C - H bond energy in acetylene to fit the kinetic data using the standard theory was lower than the consensus of several recent measurements and theoretical calculations (AH~ -- 133 kcal/mol). This discrepancy in the falloff calculations has been remedied by including inharmonic effects in the acetylene ~ vinylidene isomerization (Kiefer, Mudipalli, Wagner, and Harding 1996). The currently accepted value for the heat of formation of C2H is 135.5 kcal/mol (Kiefer, Sidhu, Kern, Xie, Chen, and Harding 1992); included also in this reference are thermodynamic data for the other key species in the acetylene decomposition mechanism, the polyacetylenes and polyacetylenic radicals, and a complete mechanism to model the TOF and LS reaction profiles recorded from shock tube experiments employing acetylene and diacetylene mixtures over a wide range of temperatures and pressures. The acetylene mechanism was further tested by predicting the inhibiting effects on the decomposition rate by the addition of excess H2 to the initial C2H2 mixtures investigated by the TOF technique (Kern, Yao, and Zhang 1998). The reactions describing the decomposition of diacetylene are a subset of the mechanism for acetylene pyrolysis (Kiefer, Sidhu, Kern, Xie, Chen, and Harding 1992). Experimental work on C4H 2 is difficult because of the propensity of diacetylene to form soot (Slysh and Kinney 1961). The experi-

16

R. D. Kern, H.J. Singh and Q. Zhang

ments were performed with mixtures of C4H 2 and excess amounts of H 2 to inhibit soot formation (Kern, Xie, Chen, Sidhu, and Kiefer 1992). In a mixture containing 1% C4H2-30% H2-69% Ne, the formation of the polyacetylenes, C6H2 and CsH2, are suppressed and the following molecular reaction is dominant at early times (< 100 ~s): C4H 2 + H2 ~

2C2H 2

The modeling of the reaction profiles is clearly sensitive to the value employed for AfH~ (C4H2); the data strongly support 111 kcal/mol for the heat of formation for diacetylene. Shock tube experiments on ethylene dissociation led to the formulation of the prime dissociation step; 1,1 elimination of molecular hydrogen from the parent and formation of singlet vinylidene (Kiefer, Sidhu, Kumaran, and Irdam 1989). C2H4 --~ H2C=C:-}-H 2 Vinylidene isomerizes to acetylene to yield the products of the dominant molecular channel, C2H 2 and H 2. A TOF study of C2H4 dissociation was designed to monitor all of the major species simultaneously; i.e., the decay C2H 4 and the growth profiles of C2H 2 and H 2. It was found that a suitable adjustment of the TOF ion source and drift tube magnets could not be attained for m/e 2, 26, and 28 (Kern, Chen, and Yao 1996). Improved results were obtained using C2D 4. The mass spectral sensitivity of D 2 was sufficient to obtain meaningful D 2 profiles. However, it became evident that optimum magnet settings for D 2 resulted in increased scatter of the m/e 26 and 28 data and vice versa. The most reasonable compromise was to perform a separate set of experiments for the low-mass product profiles. A TOF investigation on allene dissociation reported benzene formation as one of the products and provided evidence that the propargyl radical is the most likely precursor (Wu and Kern 1987). Allene was also shown to be a fuel with strong propensity to form benzene (and subsequently soot) relative to other fuels (Kern, Wu, Yong, Pamidimukkala, and Singh 1988). The mutual isomerization of allene to propyne involves several intermediates, including cyclopentadiene (Kiefer, Kumaran, and Mudipalli 1994). However, the isomerization rate is somewhat slower than either the initial dissociation rate of allene or propyne, the rates of which have been measured at short times by LS (Kiefer, Mudipalli, Sidhu, Kern, Jursic, Xie, and Chen 1997). The dominant pathway for dissociation for either allene or propyne is C - H bond fission. A secondary reaction mechanism accounts for the longer reaction-time profiles recorded by the LS and TOF techniques; the major products recorded from allene and propyne by TOF are C2H2, CH4, C4H2, and benzene. The TOF carbon atom

16.1 Mass Spectrometric Methods for Chemical Kinetics in Shock Tubes

17

balance is not closed using only these products, which indicates the formation of soot. The propargyl radical, C3H3, is the key precursor to aromatic formation and thereafter soot formation; this proposal is believed to apply also to the primary routes to aromatics and soot formation in aliphatic fuels. The highly exothermic dimerization of propargyl to benzene may proceed via 1,5-hexadiyne and 1,2,4,5-hexatriene and via 1,2-hexadiene-5-yne (Melius, Miller, and Evleth 1993). The dimerization rate has been measured experimentally (Alkemade and Homann 1989; Morter, Farhat, Adamson, Glass, and Curl 1994; Atkinson and Hudgens 1999). Other TOF work on C3H 3 reactions include studies of C3H3C1 (Kern, Chen, Qin, and Xie 1995) and C3H3Br (Kern, Chen, Kiefer, and Mudipalli 1995). This concludes the dissociation of acyclic fuels and their tendencies to form benzene studied by the TOF technique. The next section is devoted to work on the decompositions of benzene, chlorobenzene, and toluene performed after the 1991 review (Kern and Xie 1991). The principal steps for benzene decomposition are initiated by C - H bond fission (Kiefer, Mizerka, Patel, and Wei 1985). C6H 6 -+ C6H 5 4- H C6H 6 4- H -+ C6H 5 4. H C6H 5 --~ CzH 2 4. C4H3 C4H 3 --~ C4H 2 4. H The model calculations of Kiefer et al. fit the TOF data on benzene dissociation (Kern, Wu, Skinner, Rao, Kiefer, Towers, and Mizerka 1985). It is interesting to note that the major products in benzene pyrolysis contain even numbers of carbon atoms, C2H 2 and C4H2, while the principal pathways to benzene formation involves the reaction of odd number carbon atoms, 2C3H 3 -+ C6H 6. Formation of C2H 2 and C4H 2 emanates from the decomposition of phenyl radical, C6H 5. A TOF study of chlorobenzene pyrolysis was performed to investigate C6H 5 dissociation (Kern, Xie, and Chen 1992). Phenyl is the only radical detected, HC1 is the only chlorine-containing product, and C2H 2 and C4H 2 are the major products. The total carbon atom balance for the experiments was only 25-40%; the deficit is attributed to soot formation. The mass balance is determined by subtracting the amount of carbon atoms contained in the observed products at the end of the sampling period ('~700 ~ts) from the carbon atoms in the initial concentration of chlorobenzene at the arrival of the reflected shock wave. The reaction profiles were modeled with a 60-step mechanism that utilized the rate

18

R. D. Kern, H.J. Singh and Q. Zhang

constants derived for phenyl decomposition from earlier work (Kiefer, Mizerka, Patel, and Wei 1985). Toluene dissociation is of particular interest since it serves as an important fuel additive and is known to produce large quantities of soot in fuel-rich combustion and in pyrolysis. There are two initiation channels: one forms benzyl radical, the other phenyl. C7H 8 --~ C7H 7 d- H C7H 8 --~ C6H 5 9 CH 3

The first channel (AHr = 88.0 kcal/mol, AS~ = 27.5 e.u.) is favored at lower temperatures, but at higher temperatures the second channel forming phenyl radical (AH r -102.8kcal/mol, ASr = 38.9 e.u.) is favored. The net overall reaction at high temperatures is essentially irreversible. C7H8 --~ C4H2 4- C2H2 4- CH3 4- H This proposal is based on the proposition that phenyl is more unstable than benzyl at elevated temperatures. The rate constant for the second channel dominates the first above 1300 K. A 28-step mechanism~which includes pressure-dependent rate constant expressions for the two initiation steps (Kern, Chen, Singh, Xie, Kiefer, and Sidhu 1993)~models successfully the LS and TOF toluene data (Pamidimukkala, Kern, Patel, Wei, and Kiefer 1987) and ARAS data (Braun-Unkhoff, Frank, and Just 1988). It does not fit the uv absorption spectroscopy data (Brouwer, Muller-Markgraf, and Troe 1988) nor the ARAS data of Rao and Skinner (1989). The pyrolyses of aromatic nitrogen-containing heterocycles are of considerable interest in coal and heavy-oil chemistry. The release of nitrogen oxides from the combustion of these fuels contribute to current air pollution problems. The decomposition kinetics of compounds such as pyrazine, pyrimidine, and pyridine relate to formation of NOx and also of soot. Hc4N~cH I

II

HC~N.-CH pyrazine

Hc~'N\cH I

II

HC-~c. N

Hc~-N\cH I

II

HC-~c/CH

H

H

primidine

pyridine

The LS and TOF techniques were applied to the thermolyses of these three aromatic azines (Kern, Yong, Kiefer, and Shah 1988; Kiefer, Zhang, Kern, Chen, Yao, and Jursic 1996; Kiefer, Zhang, Kern, Jursic, Chen, and Yao 1997). LS

16.1 MassSpectrometric Methods for Chemical Kinetics in Shock Tubes

19

experiments provided excellent rate data for the initiation reaction, C - H fission, for all three compounds as well as the early rates of chain acceleration. Time-resolved profiles of the major (HCN and C2H2C2H2) and minor (C4H 2 and cyanoacetylene products) were obtained by the TOF technique. Of particular interest are the temperature-dependent maxima and subsequent decay rates exhibited by the cyanoacetylene (HC3N) profiles and the extremely low concentrations attributed to cyanogen. No other species of higher mass were recorded. The process of constructing the TOF reaction profiles was tedious due to cracking pattern contributions from all three parent molecules that affected the product profiles for C2H 2 and HCBN. Appropriate corrections to the raw data were applied that were dependent on the various reaction temperatures. A 2% CsDsN mixture was employed to demonstrate that m/e 51 from the CsHsN mixture consists of primarily HCBN with little or no contribution from C4H 3. Upon full deuteration, C4D 3 would appear at m/e 54. However, there was an absence of m/e 54 observed in the deuterated pyridine experiments, which is consistent with the concentrations of C4D3 predicted by the model; i.e., concentrations levels below the detectability limit of the TOE A new chain mechanism for pyrazine decomposition consisting of 35 reactions, many of them pressure dependent, was formulated. The LS and TOF profiles were modeled independently over the temperature and total pressure ranges, 1000-2300 K and 150-350 T. The existence of a chain reaction with CN as the major chain carrier was confirmed. Heats of formation for several stable radical and molecular species are proposed; e.g., a value of 96 kcal/mol for HCBN was selected to match the late-time, high-temperature TOF profiles that represent partial equilibrium concentrations. The modeling of a particular HC3N profile is extremely sensitive to the value of AfH~ chosen with respect to the profile maximum, its subsequent decay rate, and the level of late-time plateau. To estimate heats of formation for radicals derived from imine N - H bond fission, high-level ab initio calculations using the GAUSSIAN 94 implementation of Density Functional Theory with 6-311 4- G(2d,2p) and 6-311 4-4- G(3df,3pd) basis sets were performed. A value of 89 kcal/mol for the imine N - H bond energy DO was obtained. Another significant test of the mechanism involved the recognition of the sensitivity of the HC3N profile to H 2. Indeed, the addition of 5% H 2 to the reaction mixture resulted in drastically lower TOF profiles for HC3N than those recorded in the presence of H 2. This observation supports a major departure in the mechanism as previously reported by a SPST shock tube study (Doughty, Mackie, and Palmer 1994), namely, that HC3N is produced via exothermic CN addition to parent pyrazine followed by ring opening and subsequent decomposition of the radicals N C - C H = C H - N = C H = N . and N C - C H = C H - N = C H . to yield 2HCN 4- HCBN overall instead of reactions

20

R. D. Kern, H.J. Singh and Q. Zhang

involving the pyrazyl radical. The addition of H 2 reduces the concentration of CN radicals via the reaction CN + H 2 ~ HCN + H, thereby reducing the HC3N concentrations. In contrast to the mechanism of HC3N formation that depends on reactions with the parent, C4H2 is produced via reaction with productsmnamely, C2H if- C2H2 ~ C4H 2 -}- H. As a consequence, the C4H 2 profiles exhibit monotonic growth. However, a reduction of C4H2 concentration is also predicted due to removal of the ethynyl radical via C2H if- H 2 ~ C2H2 if- H, which competes effectively with the C4H2 production channel. Other departures from the SPST work include a higher value for C - H fission to pyrazine to form the pyrazyl radical +H (105 vs 96.6 kcal/mol) and a kinetic explanation for low levels of cyanogen produced in both the SPST experiments and the TOF work. Using reactions from the core pyrazine mechanism, steps pertinent to pyrimidine and pyridine decomposition were added and the LS and TOF data were modeled. The initial step in the dissociations is C - H bond fission from the respective parents producing azyl radicals and H atoms. The respective barriers are 103 kcal/mol for pyrazine, 98 kcal/mol for pyrimidine, and 105 kcal/mol for pyridine. All of these barriers are lower than C - H fission for benzene, 112kcal/mol. The lower barriers are due to the additional contributions of resonance structures for the various azyl radicals. The azyl radicals are stabilized by neighboring N-C interactions. In contrast to the chain mechanisms characteristic of azine decomposition, the dissociation of a five-member ring heterocycle, furan, proceeds primarily via two competing molecular channels (Fulle, Dib, Kiefer, Zhang, Yao, and Kern 1998). (26.1)Iv C3H 4 + CO (51.4) *" C2H2 + C H ~ C = O

The numbers shown in parentheses represent the respective heats of reaction in kcal/mol for the molecular channels. Evidence for the molecular mechanism is derived from the observation of concave LS profiles, the experimented signature for a non-chain reaction, and the TOF product distribution, which consisted solely of C3H4, CO, C2H2, and CH2CO. The branching ratio, [C2H2]/[CO], was measured by the TOF as a function of temperature in the range of 1300-1700 K and fit to the following empirical expression. [C2H2]/[CO ] = (5.5 x 10 -9) T 2"~

The expression also fit the SPST data for furan decomposition at lower temperatures (1060-1260 K) (Lifshitz, Bidani, and Bidani 1986). This informa-

16.1 MassSpectrometric Methods for Chemical Kinetics in Shock Tubes

21

tion aided the modeling of the LS profiles. In addition to unimolecular dissociation measurements, the LS technique measured the vibrational relaxation times and incubation times for furan. Thus, a "cradle-to-grave" description of furan was provided by the complementarity of the LS and TOF techniques. Cyclopentadiene (CP) is an important species in coal and hydrocarbon combustion process (Brezinsky 1986; Emdee, Brezinsky, and Glassman 1992). TOF measurements identified the major product as acetylene with lesser amounts of allene/propyne, benzene, and diacetylene. LS experiments displayed profile shapes that were indicative of a weak chain reaction consistent with C - H fission as the initial step and formation of the relatively inactive cyclopentadienyl radical, c - C s H 5 (Kern, Zhang, Yao, Jusric, Tranter, Greybill, and Kiefer 1998). The initiation step can be represented by the following reaction.

(a)

(b)

(c)

A high-level density functional theory (DFT) calculation at the B3LYP/631G(d,p) level revealed that the double bonds of structure (a) are the shortest (1.371A). The longest single bond (1.483 A) is opposite the free radical site and the remaining two single bonds are each 1.437 A. These results support structure (a) as the most likely for c - C s H 5. The production of the major products, C2H 2 and H2, occurs via two essentially irreversible reactions involving c-C5H5, c - C s H 5 + M --~ C2H 2 q- C3H 3 -}- M C5H 6 if- H ~

c-CsH 5 + H 2

The c - C s H 5 radicals and H atoms are generated from the initial decomposition of CP, which is pressure dependent. The global reaction for c - C s H 5 decomposition is c - C s H 5 --~ C2H 2 + C3H 3. The initial and rate-determining step in the multistep process is a 1,2 H-atom shift.

I

22

R. D. Kern, H. J. Singh and Q. Zhang

An important input parameter in calculating the pressure dependence of c-C~H 5 decomposition is the reaction path degeneracy (RPD). DFT calculations show that the H atom being shifted is not in the plane of the ring. Furthermore, the barrier to pseudorotation is low in c-C~H 5 (Wang and Brezinsky 1998), thus allowing five equivalent structures above the ring and five below the ring, each with an RPD of two, for an RPD total of 20. The energy difference, AE0, between structures (d) and (a) was calculated by DFT to be 61.9 kcal/mol. Structure (e) then undergoes facile ring opening, leading eventually to the products C2H2 4-C3H 3. To test the irreversibility of the two main product-producing reactions, TOF experiments were performed with mixtures of C~H6 containing excess amounts of C2H2 and of H 2. The reaction profile rates were not inhibited by the presence of these excess reagents, thus supporting the irreversible nature of these two reactions. The modeling is sensitive to the thermochemical properties selected for c-C~Hs; evidence is presented for values of 65.3 kcal/mol for AfH~ and 66.0cal K-l/tool -1 for SO at 298 K (Kern, Singh, Zhang, Jursic, Kiefer, Tranter, Ikeda, and Wagner 1999). Furthermore, the mechanism was able to model H-atom profiles recorded by the ARAS technique (Roy, Frank, and Just 1996; Roy and Frank 1997; Roy, Horn, Frank, Slutsky, and Just 1998) at lower temperatures and higher pressures than those investigated by the LS and TOF techniques. The study of cyclopentadiene decomposition highlights the necessary features of a modern chemical kinetics investigation: the treatment of experimental data from different techniques such as TOE LS, and ARAS over a wide temperature and pressure range; high-level quantum mechanical calculations of transition state and transient species structures; application of RiceRamsperger-Kassel-Marcus (RRKM) theory to develop rate constant expressions as a function of temperature and pressure; determination of the thermochemical properties of key species by theoretical calculations and/or experiments; modeling of reactant and product profiles obtained from several sources; and use of sensitivity analysis to identify the relative importance of the rate constants.

16.1.4

SUMMARY

This chapter was devoted to a description of the technique for coupling a timeof-flight mass spectrometer to a shock tube, validation of the dynamic mass analysis, presentation of the high-speed data collection process, and application of the TOF technique to a wide variety of chemical kinetics problems ranging from relatively simple reaction systems involving isotopic exchange to complex thermal decomposition studies. Although the emphasis is obviously

16.1 Mass Spectrometric Methods for Chemical Kinetics in Shock Tubes

23

on the TOF technique, other complementary experimental methods are required for a complete understanding. Many such methods are described in other chapters of this handbook. The ultimate goal is to furnish rate constant expressions for elementary reaction steps as a function of temperature and pressure over a specified range along with the relevant thermochemical properties and a plausible reaction mechanism.

ACKNOWLEDGMENTS The TOF work on exchange reactions was supported by grants from the National Science Foundation. The work on hydrocarbon fuel pyrolyses was supported by the U.S. Department of Energy, Office of Basic Energy Services, Division of Chemical Sciences, Under Contract No. DEFG05-85ER 13400.

REFERENCES Alkemade, U. and Homann, K.H. (1989). Formation of C6H6 Isomers by Recombination of Propynyl in the System Sodium Vapor/Propynyl Halide. Z. Phys. Chem. N.E, 161: 19-34. Atkinson, D.B., and Hudgens, J.W. (1999). Rate Coefficients for the Propargyl Radical Self-Reaction and Oxygen Addition Reaction Measured Using Ultraviolet Cavity Ring-Down Spectroscopy. J. Phys. Chem., A103: 4292-4252. Barton, S.C., and Dove, J.E. (1969). Mass Spectrometric Studies of Chemical Reactions in Shock Waves: The Thermal Decomposition of Nitrous Oxide. Can. J. Chem., 47: 521-538. Bauer, S.H. (1965). Shock Waves. Annu. Rev. Phys. Chem., 16: 245-296. Bauer, S.H., and Ossa, E. (1966). Isotope Exchange Rates. III. The Homogeneous Four-Center Reaction H 2 + D 2. J. Phys. Chem., 45: 434-443. Belford, R.L., and Strehlow, R.A. (1969). Shock Tube Techniques in Chemical Kinetics. Annu. Rev. Phys. Chem., 20: 247-272. Bopp, A.E, Johnson, A.C., Kern, R.D., and Niki, T. (1982). Reaction of Hydrogen and Bromine Behind Reflected Shock Waves. J. Phys. Chem., 86: 805-807. Bopp, A.E, and Kern, R.D. (1975). The Exchange Reaction of Acetylene-d2 with Hydrogen Chloride. J. Phys. Chem., 79: 2579-2583. Bopp, A.E, Kern, R.D., and Niki, T. (1977). The Self-Exchange of Dinitrogen Behind Reflected Shock Waves. J. Phys. Chem., 81: 1795-1798. Bopp, A.F., Kern, R.D., Niki, T., and Stack, G.M. (1979). Self-Exchange of Oxygen Behind Reflected Shock Waves. J. Phys. Chem., 83: 2933-2935. Bopp, A.E, Kern, R.D., Niki, T., and Stack, G.M. (1980). Comparative Rates of Exchange Behind Reflected Shock Waves. II. 15NO -k C180 vs. 15NO q- N 2. J. Phys. Chem., 84: 2680-2682. Bopp, A.E, Kern, R.D., and O'Grady, B.V. (1975). The Self-Exchange of Carbon Monoxide Behind Reflected Shock Waves. J. Phys. Chem., 79: 1483-1487. Bradley, J.N., and Kistiakowsky, G.B. (1961). Shock Wave Studies by Mass Spectrometry. I. Thermal Decomposition of Nitrous Oxide. J. Chem Phys., 35: 256-263. Braun-Unkoff, M., Frank, P., and Just, T. (1988). A Shock Tube Study of the Thermal Decomposition of Toluene and of the Phenyl Radical at High Temperatures. In Proc. of the 22nd Symp. (Int.) on Combustion, 1053-1061. The Combustion Institute.

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Brezinsky, K. (1986). The High Temperature Oxidation of Aromatic Hydrocarbons. Prog. Energy Combust. Sci., 12: 1-24. Brouwer, L.D., Muller-Markgaf, W, and Troe, J. (1988). Thermal Decomposition of Toluene: A Comparison of Thermal and Laser-Photochemical Activation Experiments. J. Phys. Chem., 92: 4905-4914. Brupbacher, J.M., Esneault, C.P., Kern, R.D., Niki, T. and Wilbanks, D.E. (1977). The Reaction of Cyanogen Chloride with Hydrogen Behind Reflected Shock Waves. J. Phys. Chem., 81: 11281134. Brupbacher, J.M., and Kern, R.D. (1973). Reaction of Cyanogen and Hydrogen behind Reflected Waves. J. Phys. Chem., 77: 1329-1335. Brupbacher, J.M., Kern, R.D., and O'Grady, B.V. (1976). The Reaction of Hydrogen and Carbon Dioxide Behind Reflected Shock Waves. J. Phys. Chem., 80: 1031-1035. Burcat, A., and Lifshitz, A. (1967). Further Studies on the Homogeneous Exchange Reaction H 2 + D 2. J. Chem. Phys., 47: 3079. Diesen, R.W. and Felmlee, W.J. (1963). Mass Spectral Studies of Kinetics Behind Shock Waves. I. Thermal Dissociation of Chlorine. J. Chem. Phys., 39: 2115-2120. Doughty, A., Mackie, J.C., and Palmer, J.M. (1994). Kinetics of the Thermal Decomposition and Isomerization of Pyrazine (1,4-Diazine). In Proc. of the 25th Symp. (Int.) on Combustion, 893900, The Combustion Institute. Dove, J.E., and Mouhon, D. McL. (1965). Shock Wave Studies by Mass Spectrometry. III. Description of Apparatus; Data on the Oxidation of Acetylene and of Methane. Proc. Roy. Soc. (London), A283: 216-228. Emdee, J.L., Brezinsky, K., and Glassman, I. (1992). A Kinetic Model for the Oxidation of Toluene Near 1200 K. J. Phys. Chem., 96: 2151-2161. Frank, P., and Just, T. (1980). High-Temperature Thermal Decomposition of Acetylene and Diacetylene at Low Relative Concentrations. Combustion and Flame, 38: 231- 248. Fulle, D., Dib, A., Kiefer, J.H., Zhang, Q., Yao, J., and Kern, R.D. (1998). Pyrolysis of Furan at Low Pressures: Vibrational Relaxation, Unimolecular Dissociation and Incubation Times. J. Phys. Chem., 102: 7480-7486. Garnett, S.H., Kistakowsky, G.K., and O'Grady, B.V. (1969). Isotopic Exchange between Oxygen and Carbon Monoxide in Shock Waves. J. Chem. Phys., 51: 84-91. Gay, I.D., Glass, G.P., Kern, R.D., and Kistiakowsky, G.B. (1967). Ethylene-Oxygen Reaction in Shock Waves. J. Chem. Phys., 47: 313-320. Gay, I.D., Kern, R.D., Kistiakowsky, G.B., and Niki, H. (1966). Pyrolysis of Ethylene in Shock Waves. J. Chem. Phys., 45: 2371-2377. Glass, G.P., Kistiakowsky, G.B., Michael, J.V., and Niki, H. (1965). Mechanism of Acetylene-Oxygen Reaction in Shock Waves. J. Chem Phys. 42: 608-621. Graham, S.C., Homer, J.B., and Rosenfeld, J.L.J. (1975). Formation and Coagulation of Soot Aerosols. In Proc. of the lOth Int. Symp. on Shock Tubes (ed. Bershader), 621-631. Stanford University Press. Graham, S.C., Homer, J.B., and Rosenfeld, J.L.J. (1975b). Formation and Coagulation of Soot Aerosols Generated by the Pyrolysis of Aromatic Hydrocarbons. Proc. Royal Soc. (London), A344: 259-285. Kee, R.J., Rupley, EM., and Miller, J.A. (1989). Chemkin-II: A Fortran Chemical Kinetics Package for the Analysis of Gas-Phase Chemical Kinetics. Report SAND89-8009. Sandia National Laboratories, Livermore, CA. Kern, R.D. (1965). The Construction of a Time-of-Flight Mass Spectrometer to Study HighTemperature Gas Reactions Occurring in a Shock Tube. Ph.D. dissertation, University of Texas (Austin).

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Kern, R.D. (1976). Dissociation of Diatomic Molecules. In Comprehensive Chemical Kinetics, Selected Elementary Reactions, Vol. 18, pp. 1-37. Elsevier. Kern, R.D., and Brupbacher, J.M. (1972). The Reaction of Hydrogen Cyanide and Deuterium Behind Reflected Shock Waves. J. Phys. Chem., 76: 285-291. Kern, R.D., Chen, H., Kiefer, J.H., and Mudipalli, P.S. (1995). Thermal Decomposition of Propargyl Bromide and the Subsequent Formation of Benzene. In Proc. of the 25th Symp. (Int.) on Combustion, Combustion and Flame, 100: 177-184. Kern, R.D., Chen, H., Qin, Z., and Xie, K. (1995). Reactions of C3H3C1 with H2, C3H4, C2H2, and C2H4 Behind Reflected Shock Waves. In Proc. of 19th Int. Syrup. on Shock Waves (eds. Brun, Raymond, Cumitrescu, and Lucien), 113-119. Springer-Verlag. Kern, R.D., Chen, H., Singh, H.J., Xie, K., Kiefer, J.H., and Sidhu, S.S. (1993). Thermal Dissociation Studies of Toluene at High Temperatures. In Proc. of the 6th Toyota Conference on Turbulence and Molecular Processes in Combustion (ed. Takeno), 117-132. Elsevier. Kern, R.D., Chen, H., and Yao,J. (1996). High Temperature Thermolysis of Ethylene-d4. In Proc. of the 20th Int. Syrup. on Shock Waves (eds. Sturtevant, Shepherd, and Hornung), 833-838. World Scientific. Kern, R.D., and Nika, G.G. (1971a). A Complementary Shock Tube Technique Study of the Exchange of Hydrogen Chloride and Deuterium. J. Phys. Chem., 75: 171-180. Kern, R.D., and Nika, G.G. (1971b). Dynamic Sampling of the Deuterium Hydride Self-Exchange Behind Reflected Shock Waves. J. Phys. Chem., 75: 2441-2446. Kern, R.D., and Nika, G.G. (1971). The Rate of Exchange of Hydrogen and Deuterium Behind Reflected Shock Waves. Dynamic Analysis by Time-of- Flight Mass Spectrometry. J. Phys. Chem., 75: 1615-1621. Kern, R.D., and Nika, G.G. (1972). The Exchange Reaction of Methane-d 4 with Hydrogen Chloride. J. Phys. Chem., 78: 2809-2817. Kern, R.D., and Nika, G.G. (1974). The Exchange of Hydrogen Bromide and Deuterium Behind Reflected Shock Waves. J. Phys. Chem., 78: 2549-2554. Kern, R.D., Singh, H.J., and Xie, K. (1990a). A Shock Tube Study of the Thermal Decompositions of Acetaldehyde and Ethylene Oxide. In Proc. of the 17th Int Syrup. on Shock Waves and Shock Tubes (ed. Kim), 487-492. American Institute of Physics. Kern, R.D., Singh, H.J., and Xie, K. 1990 Identification of Chemi-ions Formed by Reactions of Deuterated Fuels in the Reflected Shock Zone. J. Phys. Chem., 94: 3333-3335. Kern, R.D., Singh, H.J., Zhang, Q., Jursic, B.S., Kiefer, J.H., Tranter, R.S., Ikeda, E. & Wagner, A.E (1999). Thermolysis of Cyclopentadiene in the Presence of Excess Acetylene or Hydrogen. In Proc. of the 322nd Int. Symp. on Shock Waves, in press. Kern, R.D., Wu, C.H., Skinner, G.B., Rao, V.S., Kiefer, J.H., Towers, J.A., and Mizerka, L.J. (1985). Collaborative Shock Tube Studies of Benzene Pyrolysis. In Proc. of the 20th Symp. (Int.) on Combustion, 789-797. The Combustion Institute. Kern, R.D., Wu, C.H., Yong, J.N., Pamidimukkala,, K.M., and Singh, H.J. (1988). Correlation of Benzene Production with Soot Yield Measurements as Determined from Fuel Pyrolyses. Energy and Fuels, 2: 454-457. Kern, R.D., and Xie, K. (1991). Shock Tube Studies of Gas-Phase Reactions Preceding the PreParticle Soot Formation Process. Prog. in Energy and Combust. Sci., 17: 191-210. Kern, R.D., Xie, K., and Chen, H. (1992). A Shock Tube Study of Chlorobenzene Pyrolysis. Comb. Sci. and Tech., 85: 77-86. Kern, R.D., Xie, K., Chen, H., and Kiefer, J.H. (1990). High Temperature Pyrolyses of Acetylene and Diacetylene Behind Reflected Shock Waves. In Proc. of the 20th Symp. (Int.) on Combustion, 789-797. The Combustion Institute.

26

R. D. Kern, H. J. Singh and Q. Zhang

Kern, R.D., Xie, K., Chen, H., Sidhu, S.S., and Kiefer, J.H. (1992). The Reaction of C4H2 and H 2 Behind Reflected Shock Waves. In Proc. of the 18th Int. Symp. on Shock Waves (ed. Takayama), 729-734, Springer-Verlag. Kern, R.D., Yao, J., and Zhang, Z. (1998). Inhibition of C2H2 Pyrolysis at High Temperatures by H 2 and HC1. In Proc. of the 21st Int. Symp. on Shock Waves (ed. Houwing), 269-272. Panther Publishing. Kern, R.D., Yong, J.N., Kiefer, J.H., and Shah, J.N. (1988). Shock Tube Studies of Pyridine Pyrolysis and Their Relation to Soot Formation. In Proc. of the 16th Int. Symp. on Shock Tubes and Waves (ed. Gronig), 437-442. VCH. Kern, R.D., Zhang, Q., Yao, J., Jursic, B.S., Tranter, R.S., Greybill, M.A., and Kiefer, J.H. (1998). Pyrolysis of Cyclopentadiene: Rates for Initial C - H Bond Fission and the Decomposition of c-CsH 5 Radical. In Proc. of the 27th Symp. (Int.) on Combustion, 143-150. The Combustion Institute. Kiefer, J.H., Kumaran, S.S., and Mudipalli, P.S. (1994). The Thermal Isomerization of Allene and Propyne. Chem. Phys. Lett., 224: 51-55. Kiefer, J.H., and Lutz, R.W (1966a). Vibrational Relaxation of Deuterium by a Quantitative Schlieren Method. J. Chem. Phys., 44: 658-667. Kiefer, J.H. and Lutz, R.W. (1966b) Vibrational Relaxation of Hydrogen. J. Chem. Phys., 44: 668672. Kiefer, J.H., Mitchell, K.I., Kern, R.D., and Yong, J.N. (1988). The Unimolecular Dissociation of Vinylacetylene: A Molecular Reaction. J. Phys. Chem., 92: 677-685. Kiefer, J.H., Mizerka, L.J. Patel, M.R., and Wei, H.C. (1985). A Shock Tube Investigation of Major Pathways in the High-Temperature Pyrolysis of Benzene. J. Phys. Chem., 89: 2013-2019. Kiefer, J.H., Mudipalli, PS., Sidhu, S.S., Kern, R.D., Jursic, B.S., Xie, K., and Chen, H. (1997). Unimolecular Dissociation in Allene and Propyne: The Effects of Isomerization on the LowPressure Rate. J. Phys. Chem., 101: 4057-4071. Kiefer, J.H., Mudipalli, PS., Wagner, A.E, and Harding, L. (1996). Importance of Hindered Rotations in the Thermal Dissociation of Small Unsaturated Molecules: Classical Formulation and Application to HCN and HCCH. J. Chem. Phys., 105: 8075-8096. Kiefer, J.H., Sidhu, S.S., Kern, R.D., Xie, K., Chen, H., and Harding, L.B. (1992). The Homogeneous Pyrolysis of Acetylene II: The High-Temperature Radical Chain Mechanism. Combust. Sci. and Tech., 82: 101-130. Kiefer, J.H., Sidhu, S.S., Kumaran, S.S., and Irdam, E.A. (1989). RRKM Model of C2H4 Dissociation: Heat of Formation of Vinylidene. Chem. Phys. Lett., 159: 32-34, and references therein. Kiefer, J.H., and Von Drasek, W.A. (1990). The Mechanism of Homogeneous Pyrolysis of Acetylene. Int. J. Chem. Kinetics, 22: 747-786. Kiefer, J.H., Wei, C.H., Kern, R.D., and Wu, C.H. (1985). The High-Temperature Pyrolysis of 1,3 Butadiene: Heat of Formation and Rate of Dissociation of Vinyl Radical. Int. J. Chem. Kinetics, 17: 225-253. Kiefer, J.H., Zhang, Q., Kem, R.D., Chen, H., Yao,J., and Jursic, B.S. (1996). Dissociation and Chain Reaction in the Pyrolysis of Pyrazine. In Proc. of 26th Symp. (Int.) on Combustion, 651-658. The Combustion Institute. Kiefer, J.H., Zhang, Q., Kern, R.D., Jursic, B.S., Chen, H., and Yao, J. (1997). Pyrolyses of Aromatic Azines: Pyrimidine, Pyrazine, and Pyridine. J. Phys. Chem., 101: 7061-7073. Lewis, D., and Bauer, S.H. (1968). Isotope Exchange Rates. VI. Homogeneous Self-Exchange in Hydrogen Deuteride. J. Amer. Chem. Soc., 90: 390-5396. Lifshitz, A., Bidani, M., and Bidani, S. (1986). Thermal Reactions of Cyclic Ethers at High Temperatures. 3. Pyrolysis of Furan Behind Reflected Shocks. J. Phys. Chem., 90: 5373-5377.

16.1

Mass Spectrometric Methods for Chemical Kinetics in Shock Tubes

27

Lifshitz, A., and Frenklach, M. (1977). The Reaction Between H 2 and D2 in a Shock Tube: Study of the Atomic vs Molecular Mechanism by Atomic Resonance Absorption Spectroscopy. J. Chem. Phys., 67: 2803-2810. Michael, J.V., and Lim, K.P. (1993). Shock Tube Techniques in Chemical Kinetics. Annu. Rev. Phys. Chem., 44: 429-458. Melius, C.E, Miller, J.A., and Evleth, E.M. (1993). Unimolecular Reaction Mechanisms Involving CH4, C4H4 and C6H6 Hydrocarbon Species. In Proc. of the 24th Symp. (Int.) on Combustion, 621-628, The Combustion Institute. Modica, A.P. (1965). Kinetics of the Nitrous Oxide Decomposition by Mass Spectrometry. A Study to Evaluate Gas-Sampling Methods Behind Reflected Shock Waves. J. Phys. Chem., 69: 21112116. Morter, C.L., Farhat, S.K., Adamson, J.D., Glass, G.P., and Curl, R.E (1994). Rate Constant Measurement of Recombination Reaction CBH3 + CBH3. J. Phys. Chem., 98: 7029-7035. Mouhon, D. McL., and Michael, J.V. (1965). Stationary Film Recording of Time-Resolved Mass Spectra. Rev. Sci. Instr., 36: 226-228. Pamidimukkala, K.M., Kern, R.D., Patel, M.R., Wei, C.H., and Kiefer, J.H. (1987). The HighTemperature Pyrolysis of Toluene. J. Phys. Chem., 91: 2148-2154. Pamidimukkala, K.M., and Kern, R.D. (1986). The High-Temperature Pyrolysis of Ethylbenzene. Int. J. Chem. Kinetics, 18: 1341-1353. Rao, V.S., and Skinner, G.B. (1989). Formation of H and D atoms in the Pyrolysis of Toluene-~, ~, ~-d3 Behind Shock Waves. J. Phys. Chem., 93: 1864-1869. Roy, K., and Frank, P. (1997). High Temperature Pyrolysis and Oxidation of Cyclopentadiene. In Proc. of the 21st Symp. (Int.) on Shock Waves, 403-407. Roy, K., Frank, P., and Just, T. (1996). Shock Tube Study of High-Temperature Reactions of Cyclopentadiene. Isr. J. Chem., 36: 275-278. Roy, K., Horn, C., Frank, P., Slutsky, V.G., and Just, T. (1998). High-Temperature Investigations on the Pyrolysis of Cyclopentadiene. In Proc. of the 26th Symp. (Int.) on Combustion, 326-329. The Combustion Institute. Ryason, P.R. (1967). Shock Tube-TOF Mass Spectrometer Apparatus with Cryosorption Pumping. Rev, Sci. Instr., 38: 609-611. Singh, H.J., and Kern, R.D. (1983). Pyrolysis of Benzene Behind Reflected Shock Waves. Combustion and Flame, 545: 49-59. Singh, H.J., Wu, C.H., and Kern, R.D. (1988). Thermal Decomposition of 1,2 Butadiene. Int. J. Chem. Kinetics, 20: 1731-1747. Slysh, R.S., and Kinney, C.R. (1961). Some Kinetics of the Carbonization of Benzene, Acetylene, and Diacetylene at 1200 ~ J. Phys. Chem., 65: 1044-1045. Tsang, W, and Lifshitz, A. (1990). Shock Tube Techniques in Chemical Kinetics. Annu. Rev. Phys. Chem., 41: 559-599. Wang, H., and Brezinsky, K. (1998). Computational Study on the Thermochemistry of Cyclopentadiene Derivatives and Kinetics of Cyclopentadiene Thermal Decomposition. J. Phys. Chem., A102: 1530-1541. Wu, C.H., and Kern, R.D. (1987). A Shock Tube Study of Allene Pyrolysis. J. Phys. Chem., 91: 6291-6296. Wu, C.H., Singh, H.J., and Kern, R.D. (1987). Pyrolysis of Acetylene Behind Reflected Shock Waves. Int. J. Chem. Kinetics, 19: 975-996.

CHAPTER

16.2

Chemical and Combustion Kinetics 16.2 The Application of Densitometric Methods to the Measurement of Rate Processes in Shock Tubes JOHN H. KIEFER Department of Chemical Engineering, University of Illinois at Chicago, Chicago, Illinois, 60607, USA

16.2.1 Introduction 16.2.2 Methods for the Observation of Gas Density 16.2.2.1 Atomic Absorption Methods 16.2.2.2 Rayleigh Scattering 16.2.2.3 Refractive Index Methods References

16.2.1 I N T R O D U C T I O N Densitometry for the measurement of rates in the shock tube involves the determination of total gas density as a continuous function of time by some nonintrusive method. The various techniques for accomplishing this were principal players in the earliest shock tube observations of rate processes. In fact, the very first observation of a rate process in a shock tube used optical interferometry to resolve the density variations caused by vibrational relaxation in shock-heated CO 2 and C12 (Smiley and Winkler 1954). This and other early work had a huge impact, demonstrating to chemists and physicists the potential of the shock tube for the observation of fundamental gas-kinetic processes at high temperatures. Since this initial effort, other densitometric Handbook of Shock Waves, Volume 3 Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISBN: 0-12-086433-9/$35.00

29

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J.H. Kiefer

methods have been devised and used, and hundreds of rate determinations have been reported. The virtues of densitometric methods m generality, accurate and unambiguous calibration, superior temporal/spatial resolution, precision of measurement, as well as the aforementioned non-intrusiveness have often made them the method of choice. For example, in recent years the exceptional resolution and sensitivity of laser schlieren (LS) densitometry has made it the dominant technique for the observation of very fast processes resulting from vibrational relaxation and/or chemical reaction at extreme temperatures. This chapter is limited to a description and comparison of the available densitometric techniques that have been applied to the measurement of rate processes in shock waves, including a complete listing, but with a somewhat cursory discussion of the results obtained through their use. As is often the case with such a large effort, a clearer specification of its scope can be had by listing some of the things it does not include. 1. Spectroscopic measurements of specific species concentration. Of course, when a given molecule is the entire gas, or a fixed fraction of it, total density can be inferred from its concentration. However, because reaction can easily distort this connection, and the calibration of absorption or emission may not be unambiguous, these popular methods are left to another review. 2. Detonations and ignition delays. Neither of these important problems involves the resolution of time dependence, so they are not considered here. 3. Gas-dynamic measurements. If shock-front structure determinations do not resolve rotational or other relaxation, they are not considered. Conventional schlieren, shadow, or interferometric observations of shock fronts, supersonic flow fields, and boundary layers are also excluded. 4. Formation and reactions of particles. No studies of condensation, evaporation, and burning of small particles are included. Finally, the author apologizes to the reader and to the authors of all the work that has been inadvertently left out of this survey. Unfortunately, some of the early work in densitometric rate determination is difficult to obtain at this late date. For one thing, a number of the earliest efforts are to be found only in institutional reports. Most of these are referenced in the papers cited, but were not always uncovered.

16.2

Densitometric Methods to the Measurement of Rate Processes in Shock Tubes

16.2.2 METHODS GAS DENSITY 16.2.2.1

31

F O R T H E O B S E R V A T I O N OF

ATOMIC ABSORPTION METHODS

16.2.2.1.1 Electron Beam Densitometry Electron beam densitometry, which has great sensitivity but a limited dynamic range that makes it most suitable for work at extremely low pressures, was first described by Venable and Kaplan (1955). Here an electron beam traverses the tube through very small diameter holes [0.1 mm or 0.05 mm (Duff 1959)] to a phosphor and photomultiplier, which measures the electron's transmission. Time resolution is as good as 0.3 Its (Duff 1959) and Venable (1955) has reported detecting density changes as small as 0.07 T in He. These methods have not been used for the determination of rates, evidently in large part because the density gradients that arise in any relaxing or reactive shock tend to deflect the beam from the exit hole (Ballard and Venable 1958). However, a measure of density can also be obtained by recording the fluorescence from atoms excited by passage of the beam (Schumacher and Gadamer 1958; Camac 1964), and this avoids the deflection problem. 16.2.2.1.2 X-Ray Absorption Here the transmission of a beam of soft X rays is recorded with a combination phosphor and photomultiplier (Kistiakowsky and Kydd 1956; Knight and Venable 1958). This transmission is reduced through absorption by heavy atoms in the shocked gas, usually dominated by those in a rare-gas diluent. Absorption is confined to the inner-shell electrons, is independent of temperature and any chemical bonding of the atoms, and thus measures the heavyatom density. This technique's principal advantage is an unambiguous calibration, completely insensitive to temperature and chemical bonding (Knight and Venable 1958; Kiefer and Lutz 1965a). Disadvantages are poor sensitivity and somewhat limited resolution. In many cases the sensitivity problem has led to the (expensive) use of krypton and xenon as diluents (see below). The apparatus in the most recent reported application, the O-atom recombination study by Kiefer and Lutz (1965a), was described as follows: "A pulse of 25 kV chromium X-rays passes through two collimating slits covered with 0.010 in. beryllium sheet and the 3 in. tube containing the gas sample to a phosphorphotomultiplier combination. The X-ray tube current was typically 250 mA. The signal from the multiplier was displayed on two Techtronix 315 oscilloscopes and photographed," A particularly fine example of an oscillogram taken

32

J . H . Kiefer

FIGURE 16.2.1 A typical oscilloscope record of the change in X-ray transmission across a shock in 95% Xe-5% 02 at an initial pressure of 19.9 T. The no-reaction temperature was 4510 K. The three sweep traces from top to bottom are the X-ray signal, a displaced baseline, and 100-~ts timing marks. Taken from Rink, Knight and Duff (1961).

from an earlier study of oxygen dissociation (Rink, Knight and Duff 1961) is reproduced in Fig. 16.2.1. The X-ray technique has been successfully applied to the determination of several diatomic dissociation rates and one recombination. In comparison with other reliable methods (some of which are listed below), these results appear quite accurate. Applications include a study of 0 2 dissociation (Rink, Knight, and Duff 1961) extended later to various rare gases as collision partners (Rink 1962a); H 2 (Rink 1962b; Gardiner and Kistiakowsky 1961); and D 2 (Rink 1962c). The study of O-atom recombination, mentioned above, with the atoms formed from dissociation of ozone, was evidently the last experiment to use this technique on shock waves in gases (Kiefer and Lutz 1965a). The method, although evidently quite reliable, has been supplanted by the equally reliable refractive index methods discussed below, methods having much superior sensitivity as well as improved resolution.

1 6 . 2 . 2 . 2 RAYLEIGH SCATTERING In this approach an intense, parallel light source is focused at a point in the free-stream flow and perpendicular scattering is observed. The scattered intensity in a clean gas is given by I = IoNVo~Z/24r 2 with I0 the incident intensity, N the number density of the scattering atoms, V the focal volume, 0~ the gas polarizability, and r the distance from scattering volume to detector. Successful use of this method requires a light source of extremely high intensity, and the only extant application of this approach employed a Q-

16.2

Densitometric Methods to the Measurement of Rate Processes in Shock Tubes

33

switched ruby laser (Reese 1978). The measurement is local, avoids the boundary layer, and is unaffected by shock tilt and nonplanarity, but it is restricted to one or two instances in time by the pulsed nature of the source. It is difficult to avoid interference from scattering from particles entrained in the shock or on the windows, as well as window reflections. For such reasons, this technique has never actually been used in a rate determination.

16.2.2.3 R E F R A C T I V E I N D E X M E T H O D S Light is slowed when it traverses any material medium, with the velocity c obeying the relation c / c o = n, where co is its velocity in vacuum and the ratio n is the refractive index, a property of the medium. For any fairly dilute gas (the medium of interest here), n = 1 4- Kp, where p is the density of the medium and K is the specific refractivity or Gladstone-Dale constant. This simple relation may be exploited in a number of useful ways, and these are considered in detail in what follows. All of these methods measure n, or its derivatives, and require a specification of K to extract density. Such specification is the topic of this section. Refractivity of gases has been reviewed and discussed on numerous occasions, and the issues relevant to shock wave experiments are well covered in the following citations. A general and very detailed m i f rather old m r e v i e w that fully presents the theory is to be found in Partington's treatise (1953). The application to shock wave densitometry has been quite completely presented in Kiefer (1981) and Gardiner, Hidaka, and Tanzawa (1981). The Gladstone-Dale relation is a low-density approximation to the LorenzLorentz formula, (n 2 -- 1 ) / ( n 2 4- 2) - - A p

That is, for small p, Ap ~ 2 ( n - 1)/3, so the specific refractivity is K = 3A/2. Individual molecule K i are commonly obtained from molar refractivities Ri, with K i ~ 3R~/2Mi, where M i is the molecular weight of the ith species. The molar and specific refractivities of a mixture are a d d i t i v e ~ e . g . , R = ~,R~,y.i with Zi the mole fraction of species i, so again K = 3 R / 2 M , with M = E z i M i . Molar refractivities vary with species and wavelength but appear to be independent of temperature and pressure (Partington 1953; Kiefer and Manson 1981). Gardiner, Hidaka, and Tanzawa (1981) present a very large number of molar refractivities for species commonly encountered in shock tube and combustion applications, taken from well-known reviews (Partington 1953; Landolt and BOrnstein 1962), and offer these for a range of wavelengths. The wavelength relation they recommend from theory and available data can be

34

J.H. Kiefer

written as R (cm3/mol) = 1 4 . 8 a / ( b - 2-2). Values of a and b are offered for all the 136 species tabulated. A widely applicable method for the estimation of molar refractivities of stable species is through the often-available refractive index of the liquid, nD, at a specified temperature and the NaD-line wavelength, 514 nm. Given a value

for/ID, R - M(n2D -- 1)p(n2D + 2) where M is again the molecular weight and p the liquid density in gm/cm 3 at the specified T. These R can now be approximately corrected to other wavelengths using the formula given above, with a and b values estimated from similar species in the Gardiner e t a | . paper. Radicals and other species for which llD is unavailable will need to be treated by the approximation of additive bond and atom refractivities (Partington 1953; Gardiner, Hidaka, and Tanzawa 1981). If the refractivity of a species or mixture can be constructed by summation of atom refractivities, it is constant even during chemical reaction (Kiefer 1981). Unfortunately, this can be a crude approximation, often poor for radicals and molecules with multiple bonds. Thus it may be necessary to take into account the variation in refractivity in reaction-kinetic experiments (Alpher and White 1959a; Alpher and White 1959b; Gardiner, Hidaka, and Tanzawa 1981). However, this is really only a serious matter in pure or concentrated reactant situations. Dilution with a heavy rare g a s ~ a good idea in general and feasible with the most sensitive methods~can usually reduce any variation of the refractivity to insignificance (Kiefer 1981). 16.2.2.3.1 Shock-Front Reflectivity The technique was devised by Hornig and coworkers (Cowan and Hornig 1950; Greene, Cowan, and Homig 1951; Greene and Hornig 1953; Andersen and Hornig 1956) and used to obtain some unique experimental measurements of shock-front structure. The method depends on the relation between density and reflectivity; for grazing incidence this is R - (1 + tan 40)K2p2(p2/pl

-

1)/4

with 0 the angle to the tube axis. The result is very sensitive to angle, but some deviation from 90 ~ is necessary to resolve variations through the shock. Typical angles are 70o-80 ~ and reflectivities are then small, around 10 - 5 - 1 0 -7. In addition to good estimates of shock thickness in rare gases (4-15%) with results in essential accord with theory (Mott-Smith 1951), a measure of shock thickness in some polyatomic species was possible, occasionally allowing a rough estimate of rotational relaxation times. Shock thicknesses were obtained

16.2

Densitometric Methods to the Measurement of Rate Processes in Shock Tubes

35

in H 2 (Greene and Hornig 1953; Andersen and Hornig 1959), 0 2 (Greene and Hornig 1953; Andersen and Hornig 1959; Levitt and Hornig 1962; Linzer and Hornig 1963); N 2 (Cowan and Hornig 1950; Greene, Cowan, and Homig 1951; Greene and Hornig 1953; Andersen and Hornig 1959; Linzer and Homig 1963); CO, HC1, C12, NH 3, and C H 4 (Andersen and Hornig 1959); CO2 and N20 (Andersen and Hornig 1956; 1959); and a H 2 4- 3 0 2 mixture (Levitt and Hornig 1962). The problem with these experiments was the low reflectivity whose observation was made difficult by interference from scattering. Nonetheless, these were seminal results, of considerable importance to the early development and understanding of shock phenomena. Thus it is somewhat surprising they have not been followed up using more modern apparatus. In particular, the collimation, monochromaticity, and intensity required in such experiments--so difficult to achieve with conventional light sources m should be enormously improved by the application of cw lasers. 16.2.2.3.2 Interferometry This is the classic technique for densitometry in shock waves. It has been widely used for the observation of shock structure and flow fields, for the measurement of vibrational relaxation times, and even for a number of dissociation rates. Almost all rate determinations have used spark interferometry with the Mach-Zender interferometer (MZI), although the considerably more powerful differential laser interferometer (Smeets 1971, 1977) has been used to observe vibrational relaxation and, in at least one instance (Oertel 1975), to determine dissociation rates in 02. The standard setup of an MZI is shown in Figure 16.2.2. Here a parallel beam is split by the partially reflecting mirror B1. After passage through the shock tube, the two beams are recombined at mirror B2 and their interference is recorded on the photographic plate. Fringes appear if the two mirrors are slightly misaligned, and movement (shifts) of these fringes measures density change Ap -- S2/KW, with S the fringe shift, 2 the wavelength, and KW the product of specific refractivity and distance across the tube. Examples with constant density and some showing vibrational relaxation are shown in Figure 16.2.3. Because the shift is large and rapid at the shock front, one usually cannot follow the movement of fringes across the front and is confined to measuring the much smaller postshock changes in density. Methods have been devised to deal with this, however, either using white light (Figure 16.2.3, upper left) or offset fringes (Ladenberg and Bershader 1954). In the interferometers used for shock wave determinations of rate, the light source has been a "spark" of short duration, for example, 0.3 ].ts (Blackman 1956); detectability is about 0.02 fringe, corresponding to a Ap/p -- 2 x 10 -3 for a fairly typical shock in Ar with 2 - 5 0 0 n m , p - - - l x 10 -5 , and

36

J. H. Kiefer \1,1

- , . - Light source ~r~

~

FilP~r Filte~ 1

_

_

Compensating section

Shock tube ,, _

:~

_

_

_

/! =__

Photographic plate

FIGURE 16.2.2 Typical Mach-Zehnder interferometer arrangement for shock tube measurements (Bradley 1962).

W = 10 cm. Thus sensitivity and resolution are more than adequate for many applications, although later techniques (discussed below) have improved both significantly. A much more detailed description of the application of interferometry to shock waves is available (and recommended) in Ladenberg and Bershader (1954). This review also has many lovely examples of flow-field interferograms. Other types of interferometers have been devised, each with some advantages over the MZI. Most do not seem to have had much application in rate determination. However, the laser-differential interferometer originally described by Smeets (1971, 1977) is well worth noting. This apparatus is claimed to be 2 to 3 orders of magnitude more sensitive than the conventional MZI, comparable to electron beam absorption. Here the parallel laser beam is split and subsequently combined with Wollaston prisms, and after passage

16.2 Densitometric Methods to the Measurement of Rate Processes in Shock Tubes

37

FIGURE 16.2.3 Interferograms showing the structure of shock waves in several gases. The picture of Ar is taken with white light to show the magnitude of the discontinuous jump across the shock front. The spark duration was about 0.5 ~ts (Griffith 1961).

through the tube the two beams are brought together onto a photomultiplier. As density varies between the closely spaced beams, fringes march across the detector, and this can be recorded. Because the beams are very close, the m e t h o d responds to differential change, close to the gradients determined by the laser schlieren technique described below. The increased sensitivity appears because, unlike photographic methods, the discernable fringe shift n o w

38

j.H. Kiefer

depends on the laser intensity and thus can be made very small. Resolution is effectively determined by the laser beam diameter and is typically well below 1 Its. The second paper of Smeets (1977) is a tutorial in the use of sophisticated optical methods in shock waves and should be read by anyone planning to use densitometry. Such a differential interferometer has been used by Oertel (1975, 1976, 1978) to measure both relaxation and dissociation of 02 in angled shock reflection and in direct reflection. This work appears to be unique in one respect m i t is evidently the only successful use of densitometry for rate measurement in a reflected shock wave. The reasons the reflected wave has been so avoided can be found in the difficulties encountered by Byron (1966) in his study of N 2 dissociation. He reported large, interfering perturbations arising from the bifurcation caused by interaction of the reflection with the incident boundary layer. However, it should be noted that this effect is absent for dilute mixtures in monatomic gases. Possibly for fear of such complications, all other experiments described in this review have used the incident shock. Nonetheless, the results reported by Oertel and coworkers appear to be quite satisfactory, close to more recent experiments using incident shock measurements. Although the laser differential interferometer should be fully competitive with any other densitometric technique, including the much more popular laser schlieren method, it has not been used much in rate measurement. However, as noted above, it was used to measure vibrational relaxation and dissociation in 02, and it has also been employed on the relaxation in CO 2 (Offenhauser and Frohn 1986), in pure H20 (Synofzik, Garen, Wortberg, and Frohn 1978), and to even observe rotational relaxation in H2 (Lensch and Groenig 1978). In the work on CO 2, relaxation times as a function of time through the incident shock show an order-of-magnitude increase in rate during the full relaxation. However, there seems to be some serious problem in both the CO 2 and the H20 work, inasmuch as well-established methods like conventional interferometry and laser schlieren are not even close to agreeing with either experiment. When this device is used at high sensitivity on relaxation or reaction, it seems possible that distortions could occur from differing schlieren deflection of the two beams, causing imperfect superposit i o n m a difficulty similar to that encountered with the equally sensitive electron beam apparatus described above. 16.2.2.3.2.1 Interferometric Measurements of Vibrational Relaxation

A major and pioneering application of MZI shock interferometry has been to the study of molecular vibrational relaxation. Some good examples of MZI relaxation interferograms are shown in Fig. 16.2.3. Beginning with its first

16.2

Densitometric Methods to the Measurement of Rate Processes in Shock Tubes

39

application to C12 and CO2 (Smiley and Winkler 1954), a very large number of studies has now appeared involving many small molecules. Most of these studies are dated before 1965 and are cited together with their results in early source-books on shock tubes, such as Greene and Toennies (1964), Bradley (1962), and Gaydon and Hurle (1963). The Greene and Toennies book is especially valuable for its detailed tabulation, which includes a summary of actual results. The field is also covered in several issues of the Annual Review of Physical Chemistry (Bauer 1965; Gordon, Klemperer, and Steinfeld 1968; Belford and Strehlow 1969), and the early work has been very thoroughly collected and neatly displayed in the correlation paper of Millikan and White (1963b). The proper treatment of shock wave relaxation data, where the temperature varies over the relaxation, has occupied a number of researchers [a listing of early work on this is given by Blythe (1961)]. The issue is how to apply the Bethe-Teller linear relaxation equation (Cottrell and McCoubrey 1961), dE/dt = (E~ - E)/z where E is the total vibrational energy at time t, Eoo is its equilibrium value, and z is the relaxation time, when both z and Eoo are varying with time. Blythe (1961) has given an accurate version of the Bethe approximation, dEoo/dE = - p , a constant, which should be more than adequate for most applications. Here P=Cvib[1--(~--1)M2/(1-M2)]/Cp~. The quantities appearing here are the vibrational heat capacity Cvib, the specific heat ratio for translation-rotation 7~, the Mach number M of the postshock flow relative to a sound speed using this ratio, and the translation-rotation heat capacity Cpa. Blythe compared this approximation to exact calculations given by Johannesen (1961) with excellent results. This convenient approximation was used in recent studies with the LS method, for example in Kiefer, Kumaran, and Sundaram (1993). The following is a list of all studies of relaxation that have used some form of interferometric densitometry. Diatomics

02 (Blackman 1956; Byron 1959; White and Millikan 1963a, 1963b, 1963c; White 1965a, 1965b, 1968; Oertel 1976), in 02, Ar, He, N2,, D2, CH4, C2H4, and H 2. Air (White and Millikan 1964), in air. N 2 (Blackman 1956; Byron 1959; White 1966b, 1967, 1968; Millikan and White 1963a; White 1968), in N2, H2, He, CO, CH 4 and C2H2. CO (Griffith, Brickl, and Blackman 1956; White 1966a; Matthews 1961). C12 (Smiley and Winkler 1954), in C12 and He.

40

j.H. Kiefer

Polyatomics CO 2 (Smiley and Winkler 1954; Griffith, Brickl, and Blackman 1956; Griffith and Kenny 1957; Greenspan and Blackman 1957; Johannesen, Zienkiewicz, Blythe, and Gerrard 1962; Simpson, Bridgman, and Chandler 1968a; Rees and Bhangu 1969; Offenh~iuser and Frohn 1986), in CO 2, H 2, D2, He, and H20. H20 (Synofzik, Garen, Wortberg, and Frohn 1978), in H20. N20 (Griffith, Brickl, and Blackman 1956; Simpson, Bridgman, and Chandler 1968b; Bhangu 1966), in N20. CH 4 (Griffith, Brickl, and Blackman 1956), in CH4. This body of work has established some important and rather general conclusions: (i) Relaxation in many diatomics was much slower than had been thought from ultrasonic data, which were evidently distorted by impurities and other problems. An extreme example is CO, where the ultrasonic results are highly scattered (Cottrell and McCoubrey 1961) and mostly about an order of magnitude too fast (Matthews 1961). (ii) The large deviations from equilibrium in shock wave experiments established that relaxation in diatomics and other small species is essentially linear; it accurately obeys the Bethe-Teller relaxation equation presented above. Yet it would seem this linearity does not persist in polyatomics. For example, most of the listed CO 2 studies concluded that the various modes must relax at differing rates. However, this is not consistent with more recent work using LS and other more sensitive methods, which convincingly demonstrate that CO 2 actually exhibits fully linear relaxation. The work on N20 (Simpson, Bridgman, and Chandler 1968b) also indicated a linear relaxation of all the modes. The matter is discussed at some length below, following presentation of the LS relaxation experiments. (iii) When the relaxation time is reduced to a standard pressure of one atm, the product P~: always closely follows the Landau-Teller temperature dependence, Pc = A exp BT -1/3, and does so quite accurately over a remarkably large range of temperatures (Millikan and White 1963b). Here the A and B are parameters characteristic of the relaxing molecule and its collision partner. 16.2.2.3.2.2 Interferometric Measurements of Reaction Rate

Only a rather small number of reactions have been examined with interferometry, but the results appear to be quite satisfactory, i n g o o d agreement with X-ray and LS data, as well as with that provided by other techniques. A list is given here. H 2 (Sutton 1962), in H 2 and Ar. D 2 (Sutton 1962), in D2 and Ar.

16.2

Densitometric Methods to the Measurement of Rate Processes in Shock Tubes

41

0 2 (Resler and Cary 1957; Byron 1959; Matthews 1959; Oertel 1975, 1976, 1978), in air, 02, Ar, and N 2. N 2 (Byron 1966; Cary 1965 ; Hornung 1972; Kewley and Hornung 1974a, 1974b), in N2, and Ar. CO2 (Ebrahim and Sandeman 1976), in CO 2. A particularly interesting instance is the detection of an incubation time for N 2 dissociation by Hornung (1972). This is a rather general phenomenon in which the dissociation is delayed while the vibration relaxes until internal energies are large enough to cause reaction. Such an incubation delay is a potentially important addition to the possible observables in a shock-induced dissociation (Tsang and Kiefer 1995). This phenomenon is discussed again at some length following the presentation of laser schlieren applications to dissociation.

16.2.2.3.3 Shadowgraph Shadow methods use a particularly simple optical arrangement to record strong variations in the second derivative of the density. A spark source is collimated by a lens, and the resulting beam then passes through the tube directly onto a photographic plate. Sharp gradients in density deflect the light toward increasing density, causing alternating dark and light zones. For a lovely illustration, see Fig. 16.2.4. The method is useful for the display of

FIGURE 16.2.4

Shadowgram of a shock diffracting over a knife edge (Schardin 1958).

42

j.H. Kiefer

shock fronts and other strong variations in density, but is difficult to make quantitative and has not been used for rate measurement.

16.2.2.3.4 Schlieren Techniques Schliere are refractive index or density gradients that deflect light as it passes through their medium. The phenomena are thus the same as in the shadow methods, but here the optics is arranged to record the actual deflections, i.e., the first derivative of the density. Both photographic and photodetector methods have been devised to observe and measure such schIieren deflections in shock waves. The former has been used to produce beautiful and detailed pictures of gross shock phenomena, but again has not been much used on rate processes. Photodetector methods include the integrated schlieren of Resler and Scheibe (1955), discussed below, and the very popular laser schlieren or narrow laser beam deflection method of Kiefer and Lutz (1965b, 1966a) and Kiefer (1981). A notable advantage common to both methods is that, by measuring the schlieren deflections, they are obviously not disturbed by the presence of those deflections.

16.2.2.3.4.1

Integrated Schlieren

The integrated schlieren technique was introduced by Resler and Scheibe (1955) (see also Witteman 1961; de Boer 1965). In this a broad, parallel beam of visible light is confined to a narrow width of constant intensity by a slit set wide enough to encompass the desired rate process in the tube. After passage, the beam is focused on a knife edge followed with a photomultiplier as shown in Fig. 16.2.5. As long as the leading edge of the beam remains in front of the shock, the signal is given by I(t) = ~K[p(t) - p(0)], where ~K is a product of instrumental constant and specific refractivity, p(t) is the postshock density, and p(0) its preshock value. The method works quite well even with conventional light sources, having a resolution near 0.1 mm and usable sensitivity. It has been successfully applied to relaxation in CO 2 (Witteman 1961, 1962; Daen and de Boer 1962) and D 2 (Moreno 1966). Note that it cannot provide density change across the shock front because the shock front produces large deflections that may even go off the detector. In any case, it would seem that a cw laser might again provide a superior light source for such measurements, but the diffraction arising from the edges of the confining slit could be broad and severe with a coherent source. It might be difficult to avoid detecting diffracted rays and distortion of the simple relation between intensity and density with such a system.

16.2 DensitometricMethods to the Measurement of Rate Processes in Shock Tubes

43

LIGHT SOURCE

LENS SHOCK. TUBE WALL

I

:

CONDITIONS

KNIFE EDGE PHOTOMULT|PLI[ R TUBE II ~

TO

I$111,,LG$r

FIGURE 16.2.5 Optics of the integrated schlieren system (Resler and Scheibe 1955). 16.2.2.3.4.2 Laser Schlieren

In recent years the preceding methods have largely fallen into disuse, and the LS method has come to dominate the field of rate measurement using shock densitometry. A typical current setup for this is shown in Fig. 16.2.6. In this configuration, it has several significant advantages. For one thing, the apparatus is quite simple, requiring neither sophisticated optics nor complex

He-Ne Laser ~,=633nm

l

ShockTube ~

Fixed 4.1. ~ . _ ~ Mirror ~_~ "-" Detector ~-~ ~

Rotating Mirror

Differential Amplifier

Lecroy Nullmeter Digitizer PC

FIGURE 16.2.6 Arrangementof a current laser schlieren (LS) experiment.

44

j.H. Kiefer

alignment procedures. Care must be taken to ensure that the tube is smooth near the observation station so the shock is free of following flow disturbances and that the shock is planar and accurately normal to the tube axis. But this requirement is easily satisfied with well-developed incident shock waves in a constant-area tube. To maintain maximum resolution, the beam must be accurately aligned normal to the tube axis. Again, this is easily accomplished by placing the (normal) window reflections in the same vertical plane as the incident beam. A further important advantage of the method, especially useful in reaction rate determinations, lies in the fact that the schlieren signal is directly proportional to local density gradient, already a derivative and directly proportional to rate (see below). Thus rates can be taken directly from the signal magnitude without extracting slopes. This feature makes for very precise rate determinations and maximizes the resolution by allowing a determination of rate from the earliest meaningful signal. The technique is also very sensitive, resolving angular deflections as small as 10-7 rad. These correspond to density gradients of just 5 x 10 -8 gm/cm 4 in Ar and are smaller than the expected boundary layer perturbations (de Boer and Miller 1976; Kiefer and Hajduk 1980; Kiefer 1981). The one real difficulty is that side-on viewing limits the resolution because of shock curvature and beam refraction/diffraction by the front. Because of this, a problem common to most densitometric methods, resolution at the front is roughly the time for shock passage through the beam, typically 0.5-1 ~ts or even more, depending on shock velocity and pressure. In any case, the easily achieved sensitivity and resolution are effectively unimproveable, being limited neither by optical configuration or detection electronics, but rather by shock curvature and gradients arising from growth of the boundary layer at the foot of the shock (Kiefer and Hajduk 1980; Kiefer 1981). The theory of rate measurement with the laser schlieren technique has been described in considerable detail by Kiefer and Lutz (1966a), Kiefer and Hajduk (1980), Kiefer, A1-Alami, and Hajduk (1981), and Kiefer (1981). The reader should especially note the detailed earlier review (Kiefer 1981) and the paper on schlieren optics (Kiefer, A1-Alami, and Hajduk 1981). The apparatus of Fig. 16.2.6 can be easily calibrated for angular sensitivity by spinning the rotating mirror at a known rotation rate and recording the signal resulting from the sweep across the split detector. The resulting signal is linear near its center, for small deflections, and experiments are easily confined to this linear range. Then, for a ray traversing a one-D shock flow, 0 = K W d p / d x , with W and K again the path length and specific refractivity, respectively. Although there is some averaging over the finite width of the beam (typically ~0.4 mm) as it crosses the shock tube, the effect of this is insignificant. In the absence of extremely rapid variations in density gradient, comparable to those at the shock front, the average over a Gaussian laser beam

16.2

Densitometric

Methods

to t h e M e a s u r e m e n t

of Rate Processes

in Shock Tubes

45

accurately records the density gradient. In fact, it does this exactly in the case of a truly exponential gradient (Kiefer and Lutz 1966a). This feature has obvious advantages in the study of relaxation, where an exponential density variation implies the exact same behavior in the gradient. As mentioned earlier, the signal generated by shock-front passage through the beam is a significant restriction on the resolution; in effect, the first 0.51 ~ts of postshock gradient are obscured, a serious matter with a technique otherwise having close to a 0.1 ~ts resolution. A related problem, and one also presenting some difficulty, concerns the location of the time-origin, the instant of shock heating and initiation of relaxation or reaction. For modeling of the density gradient in a temperature-varying flow, an accurate location of this starting point is essential. The matter is complex, and is discussed at some length in conjunction with the presentation of example experiments. Examples of raw LS signals recorded from shock waves in a rare gas are shown in Fig. 16.2.7. There being no relaxation or reaction here, the signal is entirely from passage of the front through the beam. This results in a schlieren "spike" with an initial negative and following positive excursion and is an unavoidable characteristic of all LS experiments using a split detector. (With a knife edge detector, only the positive signal is usually seen (Kiefer and Lutz

ii .... i i

ilt.

...................

i

i;t0, .............. iiill ii!iii iiii

1823K, 69torr

i

i

i~ ..........................................................................................................

:

................

: ...............

: ...............

: ...............

: ...............

: ...............

: ...............

: ................

. . . . . . . .

'. . . . . . . .

: . . . . . . .

: . . . . . . .

: . . . . . . .

', . . . . . . .

: . . . . . . .

'. . . . . . . .

9 ,.. ........ : ........

i

Krypton 1 4 0 0 K, 114 t o r r

'.

:

:

:

.

:

' . . . . . . .

: . . . . . . .

!

................................................................................................................................

. . . . . . . .

: . . . . . . . . . . . . . . .

~ . . . . . . . ~ .... i ....... i.lt ....

FIGURE

16.2.7

'. . . . . . . .

Krypton 1 2 2 0 K, 120 t o r r

: . . . . . . .

: . . . . . . .

i............. i ....... i ....... i

LS r e c o r d s i n p u r e Kr. T h e s m a l l c a r e t at t h e b o t t o m

time-origin location.

identifies the calculated

46

J.H. Kiefer

1966a) but it is of about the same width.) One example of a LS raw signal from a shock wave showing vibrational relaxation is displayed in Fig. 16.2.8. Here a near-exponential gradient is now evident following the schlieren spike. In Fig. 16.2.9, the linear relaxation of norbornene and methane is demonstrated in linear semilog plots of the density gradient. In the interpretation of experiments like these, and even more so in reactive examples like those shown below, an accurate location of the instant of process initiation is of considerable importance. The question of optimum time-origin location in LS experiments has been discussed several times (Kiefer and Lutz 1966a; Kiefer and Hajduk 1980; Kiefer 1981; Dove, Jones, and Teitelbaum 1973; Dove and Teitelbaum 1978; Kiefer, A1-Alami, and Hajduk 1981) and was even briefly a matter of some controversy

..........

.

..........

i

i. . . . . . . . . . . . i 9 ...................

.

!T2

i ....... :

. . ~ ..........................

................................

.

~ ...............

r~orDornene-Krypton (relaxed)= 1117 K

. . . . . . .

:. . . . . .. . .

. : ...............

.

~ ...............

:

.: . . . . .. . . . . . . . . . . . . . . . . . . .

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: ...............

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: ................

~ ...............

~ ...............

.. ...............

~ ...............

:

iliZilllll illilliiiii.ii.iiiiiillli..iii.ii.llii.illli!ii.ii..i i i2% Norbornene-Krypton i i i.......:' ii.... i.......::T2 (relaxed)- 1040K .....::.......!

.

. . . . . .

:.',

: ._:_~_

i ........ i . . . . . . .

_ ..J.

.

~

(

i........ i r ....... i . . . . . .

i ,.

'

.........

~ .

.

'.

.

.

~

~

.

~

i'i""i.......IT2 (relaxed)=631 K

...::.~

' ....

i ....................... : ............... :....... :....... :

. . . . . . . .

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....::"

!

::i!1::iii:::i: !:'....' . ...................

.

.

.

.

.

.

.

.

.

.

. ....

~ ...........................................................................................................

16.2.8 Unprocessed LS signals from a 2~ norbornene-krypton mixture. Reading up from the bottom, these show simple vibrational relaxation, vibrational relaxation followed by dissociation, and dissociation alone (Kiefer, Kumaran, and Sundaram 1993).

FIGURE

16.2

Densitometric Methods to the Measurement of Rate Processes in Shock Tubes

10 -4

"2% NB

\ ~

2% NB on-

dP

47

981 K, 60 Torr

om

'

10 "5

1

2

3 t (ILs)

4

5

xX x

i

|

1

3 t (~s)

.

4

5

1 0 0 % CH4 708 K, 29 Tort

10-5

d,O gm

10 -6 1

1

1

IKX

;

,

2

i=

3

l

....

4

t (~) FIGURE 16.2.9 Semilog plots of LS signals in a norbornene-krypton mixture and in pure methane, all showing linear vibrational relaxation.

48

J.H. Kiefer

(Dove, Jones, and Teitelbaum 1973; Kiefer and Hajduk 1980). This question was brought to the fore by experiments that indicated a short delay (incubation time) in the onset of dissociation in N20 (Dove, Nip, and Teitelbaum 1975) and by attempts to calculate a net Ap for relaxation in D 2 and H 2 (Kiefer and Lutz 1966a, 1966b, 1966c; Dove and Teitelbaum 1978), applying the obvious relation Ap -- u

dt o

Both incubation time and Ap are extremely sensitive to time-origin location; the incubation times are small and the fast exponential character of the relaxation puts most of the density change in the brief initial period blocked by the schlieren spike. Thus the accuracy of the incubation times was in doubt, and the D2/H 2 relaxation was found to have too small a net density change, comparing thermodynamic calculations (Kiefer and Lutz 1966c; Dove and Teitelbaum 1978). The problem with Ap is much reduced when the time-origin is relocated by full physical optics simulations of the schlieren spike (Kiefer, A1-Alami, and Hajduk 1981), but it does not go away (Kiefer 1981). These calculations demonstrated that the spike is a consequence of both diffraction and refraction, and that purely geometrical (refractive) treatments of the optics are misleading. The need for a physical optics approach becomes evident when one recognizes that a large signal will actually result even on passage of a perfectly plane and parallel shock. The phase of that portion of the beam traversing the highdensity region behind the shock is retarded relative to that portion in front, and the two portions then interfere when superposed at the detector. The theory and calculations presented by Kiefer, A1-Alami, and Hajduk (1981) are overly complex for this review and a presentation of applicable results will have to suffice. They ultimately present a linear correlation of signal intensity and the time between the actual calculated time origin and the zerocrossing point, defined as that point in time where the signal just crosses from negative to positive. They then plot a reduced time for this point, zc = -ut/ao, against -AS/PoSo, a normalized intensity of the deepest negative signal (AS). In this, t is the time from zero crossing to the time origin, u is the shock velocity, and a 0 is the (average) radius of the Gaussian laser beam at the tube. The product SoPo is the sum signal, which is generated by the full laser beam impinging on the detector. Their plot is reproduced here as Fig. 16.2.10. Obviously the correlation is rough, but the zero-crossing time is small and this procedure is thus fairly accurate. It has since been confirmed by many applications in relaxation and dissociation, applications to be discussed below. In Figs. 16.2.7 and 16.2.8 the t = 0 location found in this manner is identified by the/x symbol at the bottom of each graph.

16.2

Densitometric Methods to the Measurement of Rate Processes in Shock Tubes

0.7

I

49

I

0.5

0.3

0.1

-0.1

-0.3

-

9

/0

-0.5

-0.7 / 0.00

I 0.10

! 0.20 "

~

S/SoPo

I 0.30

J 0.40

(min.)

FIGURE 16.2.10 Plot of dimensionless time from zero crossing to the time origin in LS experiments against reduced intensity, after the calculations presented in Kiefer, A1-Alami, and Hajduk (1981).

16.2.2.3.4.2.1 Laser Schlieren Studies of Vibrational Relaxation. Our survey of the LS application literature quite properly begins with vibrational relaxation. This was the very first use of the technique, applied to the study of relaxation in 02, D2, and H 2 (Kiefer and Lutz 1965b, 1966a, 1966b), and these were actually the very first use of a laser in the measurement of any rate process. Examples showing the raw oscilloscope signals produced by relaxation in D 2 and H 2 are shown in Fig. 16.2.11. The work on H 2 relaxation is especially notable because this is in many ways the most difficult relaxation to investigate. With a characteristic temperature of 6325 K, the vibrational heat capacity of H 2 is miniscule even at rather high temperatures m in some of these 1100-2700 K experiments the net temperature change was only a few degrees m yet the relaxation is quite rapid. The low molecular weight and refractivity of H 2 compound the problem, because one cannot avoid using large fractions of H 2 in the necessary mix with a rare gas. As was noted earlier, these experiments and several subsequent efforts on this (Dove, Jones, and

50

J.H. Kiefer

r,

FIGURE 16.2.11 Top:LS oscillogramof ~ashock showing relaxation in 70% D2-30% Ar for an initial T = 1821 K (Kieferand Lutz (1966a). Bottom: LS oscillogramof relaxation in 30% H2-70% Ar (Kiefer and Lutz 1966b). In both, the bottom sweep contains 1-~tstiming marks. Teitelbaum, 1973; Dove and Teitelbaum 1974; 1979) and on D 2 (White 1965a, 1966a; Moreno 1966) all found the net Ap to be smaller than required by simple theory (Kiefer and Lutz 1966c). The estimation of net Ap for these fast relaxations is difficult, and some discrepancy remains, but much of it is removed with the more correct scheme for time-origin location discussed above. Numerous studies of relaxation in other species using the LS technique have now been completed and this problem has not reappeared [see Kiefer, Buzyna, Dib, and Sundaram (2000) for some convincing examples]. Perhaps there is some interference by incomplete rotational relaxation, as was originally suggested by Dove, Jones, and Teitelbaum (1973) and Dove and Teitelbaum (1974). The following is an attempt to present a complete list of LS investigations of vibrational relaxation, essentially covering the 20th century. If nothing more, its length should serve to convince the reader of the broad utility of the method. Diatomics

H 2 (Kiefer and Lutz 1966b; Dove, Jones, and Teitelbaum 1973; Dove and Teitelbaum 1974, 1979), in H2, He, Ne, and Kr. D 2 (Kiefer and Lutz 1966a, White 1965a, 1966a; Moreno 1966; Bird and Breshears 1972), in D2, Ar, He, and Kr.

16.2 Densitometric Methods to the Measurement of Rate Processes in Shock Tubes

51

HD (Simpson, Price, and Crowther 1975), in HD and Ar. HC1 and DC1 (Breshears and Bird 1969a), in HC1 and DC1. HBr and DBr (Kiefer, Breshears, and Bird 1969; Breshears and Bird 1970), in the pure gases. HI and DI (Kiefer, Breshears, and Bird 1969; Breshears and Bird 1970), in the pure gases. C1F (Santoro and Diebold 1978), in C1F and Ar. 0 2 (Kiefer and Lutz 1965a; Lutz and Kiefer 1966; Breen, Quy, and Glass 1973; Hanson and Flower 1973), in 02, Ar, and O atoms. N2 (Breshears and Bird 1968; Miyama and Endoh 1967; Kurian and Sreekanth 1988), in N 2, 0 2, H20, NH 3 and O atoms. NO (Breshears and Bird 1969c), pure. F 2 (Diebold, Santoro, and Goldsmith 1974; 1975), in F2, Ar, and He. C12 (Breshears and Bird 1969b), in C12, CO, HC1, and DC1. Polyatomics CO 2 (Simpson, Chandler, and Strawson 1969; Simpson and Chandler 1970; Simpson, and Simmie 1971; Simpson and Gait 1973; 1977; Andrews, Crowther, Price, Simpson and Gait 1975; Simpson, Gait and Simmie 1976a; Buchwald and Bauer 1972; Walsh and Bauer 1973), in CO 2, Ar, He4, N2, n-H2, p-H2, He3, Ne, H2, n-D2, o-D2, 02, H20, D20, CH4, CD4, and CH3F. COS (Simpson and Gait 1973; Simpson, Gait, and Simmie 1976b), in COS, He 3, He4, n-D2, o-D2, n-H2, and p-H 2. H20 (Kurian and Sreekanth 1988), in Ar and H20. N20 (Dove, Nip, and Teitelbaum 1975; Simpson, Gait, Price, and Foster 1982; Baalbaki, Teitelbaum, Dove and Nip 1986; Baalbaki and Teitelbaum 1986a), in N20, Ar, He4, He3, D2, HD, n-H 2, and p-H 2. C102 (Paillard, Duprr Youssefi, Lisbet, and Charpentier 1986), in Ar. C H 4 (Yasuhara, Yoneda, and Sato 1974; Jackson, Lewis, Skirrow, and Simpson 1979; Teitelbaum 1982; Baalbaki, Teitelbaum, Dove, and Nip 1986; Babji, Ahuja, and Rao 1983), in CH4, 0 2 , CO, Ar. CF 4 (Jackson, Lewis, Skirrow, and Simpson 1979), in CF 4 and CO. C2H2 (Baalbaki and Teitelbaum 1986b; 1986c), in C2H2 and Ar. C2H 4 (Teitelbaum 1982), in C2H 4 and Ar. SO2 (Kishore, Babu, and Rao 1980), in SO2, He, and Ar. SF 6 (Breshears and Blair 1973), in SF6 and Ar. CF3CI , CF3Br , CF3I (Kiefer and Sathyanarayana 1997), in parent molecules and Kr. c - C 4 H 4 0 (furan) (Fulle, Dib, Kiefer, Zhang, Yao, and Kern 1998), in C 4 H 4 0 and Kr.

52

J. H. Kiefer

(norbornene) (Kiefer, Kumaran, and Sundaram 1993), in Kr. C6H 6 (benzene), CsH 8 (cubane), C3H 6 (cyclopropane), C 4 H 4 0 , C 7 H 8 (norbornadiene), C2H40 (oxirane), C7H8 (quadricyclane) (Kiefer, Buzyna, Dib, and Sundaram 2000), in pure cyclopropane, furan, and oxirane, and all in Kr. C7H10

The above experiments both generally confirm and greatly extend the earlier ultrasonic results. They show the LS method is considerably more sensitive than MZI, and accordingly more accurate. One has only to see the beautiful results obtained by Teitelbaum o n C H 4 (Teitelbaum 1982) to appreciate the remarkable precision possible. The final Landau-Teller plot from this paper is shown in Fig. 16.2.12. Here the scatter is only about 2%, and extremely small deviations from linear mixing of the rates due to separate collisions with Ar and C H 4 c a n be discerned. As seen in the presentation of interferometric studies of relaxation, there are a number of important and general insights into the nature of relaxation in these LS experiments, many of which serve to confirm the earlier conclusions from interferometry. First, the reader will undoubtedly notice the large number of studies of relaxation in CO 2 by both LS and MZI. The molecule is certainly convenient, in that it has a large vibrational heat capacity, does not decompose

0.8

0.4 =L

1/2

(/1

13 O_ o

--0.4

xx /

~'~/~ ~ ..[

/x"x"'~X" !

0.10

I

I .....

0.12

(T/K )-'/3 I

I

0.14

FIGURE 16.2.12 Vibrational relaxation times (atm-~ts) in various mixtures of methane (the cited %) in argon. Taken from Teitelbaum (1982).

16.2 DensitometricMethods to the Measurement of Rate Processes in Shock Tubes

53

until very high temperatures, and is easily handled. For this reason it quickly became a standard for comparison and checking of various techniques and experimental setups. However, this does not fully explain the interest in CO2 inasmuch as oxygen, another popular and equally convenient molecule, can also suffice for this purpose. In fact, much of the interest in CO2 arises because it is polyatomic and has some widely spaced vibration frequencies. The important issue that motivated much of this work was whether its four vibrations relax together, as they appear to do in ultrasonic experiments on this and many other polyatomics (Cottrell and McCoubrey 1961). The absence of discernable deviation from a single relaxation in nearly all ultrasonic experiments is usually explained by the "series model" (Cottrell and McCoubrey 1961), in which fast inter- and intramolecular vibration-to-vibration (VV) transfer processes couple all the modes to that of the lowest frequency, the one most easily excited by translation/rotation-to-vibration (T/R-V) energy transfer. Then the entire vibrational heat capacity relaxes at the rate at which energy enters this lowest mode. This does seem to explain the ultrasonic observations, but here relaxation is only seen close to equilibrium. So the interesting question arises: Do the high temperature and far-fromequilibrium shock tube measurements also conform to the series description, or can one now discern different rates for different frequencies? As noted earlier, the interferometric relaxation experiments described above seem to indicate separate relaxation (Griffith, Brickl, and Blackman 1956; Greenspan and Blackman 1957; Offenh~iuser and Frohn 1986). The issue was finally laid to rest by the shock tube IR measurements of the relaxation time of the asymmetric stretch alone by Weaner, Roach, and Smith (1967), which showed full agreement with the rate of the bending modes. This was decisively confirmed by the extensive efforts of Simpson and coworkers (Simpson, Chandler and Strawson 1969; Simpson and Chandler, 1970; Simpson and Simmie 1971; Simpson and Gait 1973; Andrews, Crowther, Price, Simpson, and Gait 1975; Simpson, Gait, and Simmie 1976a). They demonstrated that relaxation was solidly exponential (linear) with a single relaxation time, and that relaxation times taken far from equilibrium were in excellent accord with ultrasonic, near-equilibrium values. Thus there was simply no credible evidence for complex relaxation in CO2, but this does not mean there is not such an effect in other species. Baalbaki and Teitelbaum (1986a, 1986b, 1986c) found indications of nonlinear relaxation in N20 and a severe effect in C2H2. Most theory would certainly suggest nonlinearity in polyatomic relaxation, but the experimental situation remains mixed. In the diatomics, relaxation seems linear; for example, Lutz and Kiefer (1966) found relaxation in O2-Ar mixtures to be very accurately so. Here the variation of relaxation time during an experiment was fully consistent with the temperature variation indicated by experiments

54

J.H. Kiefer

covering a wide range of temperature. An impressive counterexample is to be found in the very recent work of the author and his coworkers on m u c h larger molecules (Kiefer and Sathyanarayana 1997; Fulle, Dib, Kiefer, Zhang, Yao, and Kern 1998; Kiefer, Buzyna, Dib, and Sundaram 2000). Here the density gradients often exhibit a well-resolved local m a x i m u m a short time into the r e l a x a t i o n m a severe, even blatant nonlinearity. Figure 16.2.13 shows a few examples of this extreme nonlinear effect. Here the initial acceleration is probably caused by VV transfer to the ground and other low-lying states from the dense manifold of upper states where such transfer will always be near resonant. Another issue of considerable theoretical interest concerned the contribution of rotation to the rate of vibrational excitation on collision, the rotation residing in either the relaxing species or its collision partner (R-V transfer). The experiments of Breshears and his coworkers on the hydrogen and deuterium halides and the work of the Simpson group on the effects of H 2 and D 2 additions in CO2 and N 2 0 relaxation were the first to clearly show such rotational effects. The first solid indication was in the relaxation in HC1 and

F::' i'"""~ i ....... .':':':".".'Ii'i'/:"'k" ':"J,. IJt:''i' ~:....•I:..:.:.Pure 3'78'K' ...:..•i.:..Cyclopropane ..:.:11.4 .:.:.•....:.torr .:..:.i..:.:.:.......... i.... .....:i.:........ .:.•..:.:.•.:i.:........ :.i !i

, ..... i ....... i....-:,i. ..... ' ! ~ - - ~ - - - . ~ L................... ....... '. ....... ....... :: ....... ....... ....... ....... ....... !! x..........:::............... :...............i:............... :i ............... :::............... :::............... i............... .......

~ii

i ...........

,

............... ...... ::............... i ............... ::............... i ............... ....... ::............... .......

i::.::i !

...... ! ::::i::..::: i .....

.......

.......

.

...........

............

......

.

'

,

"i

........ :....... i ....... i....... ::....... : ....... : ....... :....... i ! ....... : ....... i ....... i ....... ?....... ! ....... i ....... : .......

:

.

.

.

.

.

.

i

................................................................................................................................. . , j

.

.

.

9

i ....... i' [1" .... 4% Norbomadiene:: ....... ::....... ::....... [ i ....... ! ' I : ~ 3 1 K ' . 6 4

t~

! i ...... i ....... i .... """[

FIGURE 16.2.13 Unprocessed LS signals showing nonlinear relaxation in various large molecule-Kr mixtures. Following the schlieren spike, there is a clear maximum in the relaxation gradient (Kiefer, Buzyna, Dib, and Sundaram 2000).

16.2 DensitometricMethods to the Measurement of Rate Processes in Shock Tubes

55

DC1 (Breshears and Bird 1969a), which showed pure HC1 relaxation to be much faster than pure DC1. This observation was not consistent with T-V theory, but fit well with R-V transfer theory (Moore 1965). R-V theory was also consistent with relaxation in the other hydrogen halides (Breshears and Bird 1969b; Kiefer, Breshears, and Bird 1969; Breshears and Bird 1970). The work of the Simpson group involved the study of the effect of D 2, H 2, p-H 2, o-H 2, and the D2's on CO2 (Simpson and Simmie 1971; Simpson and Gait 1973, Andrews, Crowther, Price, Simpson, and Gait 1975; Simpson, Gait, and Simmie 1976a), COS (Simpson and Gait 1973; Simpson, Gait, and Simmie 1976b), and N20 (Simpson, Bridgman and Chandler 1968b; Simpson, Gait, Price, and Foster 1982). In many of these experiments there is enhanced transfer by those partners (the effect was weak in COS and N20) with high rotational velocities; in particular, D 2 is significantly more efficient than He 4 in relaxing CO 2 despite having the same mass. In some of the experiments with H 2 additive, there were indications that enhancement of the relaxation rate may have been retarded by slow rotational relaxation of the H 2 (Simpson, Gait, and Simmie 1976a, 1976b; Simpson, Gait, Price, and Foster 1982). Another interesting finding, first seen in LS shock tube experiments, is the large acceleration (the "chemical effect") in relaxation rate for some diatomics with the O-atom as the collision partner when there is potential bonding (Kiefer and Lutz 1967; Breen, Quy, and Glass 1973; Breshears and Bird 1968). The resolution and accuracy of LS measurements are both nicely demonstrated by the reported experiments on relaxation in pure CF3C1, and CF3Br (Kiefer and Sathyanarayana 1997). Some examples are shown in Fig. 16.2.14. Here the relaxation is very fast and the shock tube data overlap ultrasonic measurements. As demonstrated in Table 16.2.1, the LS relaxation times TABLE 16.2.1 VibrationalRelaxation Times in Pure CF3CI and CF3Br Laser schlierena

Ultrasonicsb

T2 (K)

P2 (T)

P~ (ns-atm)

T (K)

Po (ns-atm)

CF3C1

492 452 484

36.6 26.9 21.8

213.4 210.5 221.5

CF3Br

534 516 497 489 481

12.6 13.2 14.0 15.2 16.4

180.6 179.6 181.1 178.1 165.4

300 373 474 574 300 372 472 574

250 280 260 240 180 250 230 205

Species

avibrationally relaxed post-incident-shock conditions bUltrasonic dispersioin measurements (Cottrell and McCaubrey 1961)

56

o

lIIl'

L L

q;r;~

I

I

i

I

~

1

I

l

n

l

~

l

l'l'l'~llll

ll,;11

~

,

i

l

;

~

'

(~mo/u~3)

o

IPlPll

Illlll

i;r~l

r',"~

IIFIil,

l

I

l

l

llll;l x

Jllltl

l~

~

I

[ I

ilqlpl'"'q

~

p

IIll~ J I I 0

]

I 1 -~

xPldP

I

I

I

~

o

J.H. Kiefer

16.2 DensitometricMethods to the Measurement of Rate Processes in Shock Tubes

57

actually have a superior self-consistency, but are in excellent agreement with the average magnitude of the ultrasonic results. One last observation relating to precision is worth a mention. In a few cases, the precision of LS data has allowed the resolution of small deviations from linear mixing, i.e., from additivity of the rates due to individual components of a mixture. In C H 4 mixtures with Ar (Teitelbaum 1982; Baalbaki, Teitelbaum, Dove, and Nip 1986), C2H 2 (Baalbaki and Teitelbaum 1986b, 1986c), and N20 (Baalbaki, Teitelbaum, Dove, and Nip 1986) such additivity simply did not hold. A reasonable explanation for this nonlinearity may be found in a significant contribution from VV transfer to the relaxation in these molecules. Each of the above studies has its own unique features and significance. It is therefore unfortunate, if unavoidable, that only broad generalized conclusions can be described in this review. The reader is encouraged to explore the original sources. 16.2.2.3.4.2.2 Laser Schlieren Studies of Reaction Rate. In recent years most of the applications of the LS technique have been to the determination of reaction rates. Because of its outstanding resolution, it has been able to produce reliable rate constants to higher temperatures (rates) than any other shock tube technique [see Kiefer and Kumaran (1993) for a striking example]. This is obviously a particularly valuable feature; it makes full use of the shock tube's high-temperature capability and often opens an entirely new temperature regime to kinetic investigation. It is worth mentioning that the determination of reaction rates is inevitably a more difficult application than relaxation, because the reactions one studies in shock waves have large activation energies and thus a strong variation of rate with temperature. The density gradient is a consequence of a temperature gradient resulting from endo- or exothermic processes in the post shock flow, so any such process cannot usually be isothermal [For an interesting exceptioin see Irdam, Kiefer, Harding and Wagner (1993)]. The problems associated with nonisothermal processes were discussed briefly in the section on relaxation; there the relatively weak dependence of relaxation rates on temperature make such problems a much less serious matter. Because of the sensitivity to temperature, it is usual to treat the gradient in reactive shocks as a modeling problem in which one tries to fit the experimental profile with a sufficiently complete mechanism using full thermodynamic data, detailed balance, and temperature-dependent rate constants. Of course, such modeling is no problem at all if there is just one reaction. The accuracy of the measured gradients then actually comes close to allowing the derivation of both magnitude and activation energy from a single experiment (Kiefer and Shah 1987; Kiefer, Kumaran, and Sundaram 1993). But the situation can become very difficult when there is complex secondary chemistry

58

J.H. Kiefer

as in chain decomposition. However, the proportionality of gradient and rate (see below), as well as the extremely high resolution of the LS method, usually permits a close estimate of 'initial' rate, the putative rate at t - 0 where the conditions are those of complete vibrational equilibrium but no reaction and are usually unambiguous. Clearly, a good estimate of this rate for fast reactions requires both an accurate time origin and some kind of model-assisted extrapolation of the gradient through the schlieren spike. This problem has been treated on several occasions and is discussed below in reference to specific examples. The method has proven to be a very accurate approach to measuring high-temperature reaction rates. In fact, the author knows of no examples where LS dissociation rates have proven to be inaccurate, whereas there are many examples where other methods have confirmed LS data in detail. Of course, the gradient is highly nonspecific, offering only an indirect indication of the responsible mechanism. In complex situations it cannot normally stand alone; other results from theory or experiment are needed to define the chemistry. This problem is especially vexing when an initial dissociation has more than one product channel. However, despite the patent inadequacy of a single measurement, it is often possible to derive considerable evidence on mechanism and even secondary rates from the full gradient variation. As noted above, in reactive shocks the gradient is almost proportional to endothermic rate. The relation, d p / & = p ~ ( ~ , A / ~ , - Cp~AN~)/tPoU(Cp~/M -- u~p~C~/p~I~)] i

was derived in an earlier review (Kiefer 1981), where it was shown to be exact for the usual ideal, one-D incident shock. Here r i is the reaction rate of reaction i, AH i is its heat of reaction, and AN i is the increase in mole number. The remaining quantities are local postshock properties, with the exception of P0, the density before the shock. Thus each reaction makes its own contribution to the total gradient with the factor p / [ P o u ( C p T / M - u2p~Cv/p2R)], in common. This summability of the individual reaction gradients is a considerable help in the interpretation of complex mechanisms and in their modeling. By simply calculating a gradient for each reaction throughout the integration using the above relation, it is easy to identify the principal sources of this gradient at all times. These individual gradients then offer a crude sensitivity analysis of the mechanism, one which at least responds to all components of the scheme that are involved in the heat balance. The following is again an attempt to provide a complete list of LS studies of reaction kinetics. Much of this material is also discussed in the reviews of Tsang and Lifshitz (1990) and Michael and Lim (1993). This list is then followed with a consideration of accomplishments and unresolved problems, emphasizing those that serve to illuminate the features of the LS technique.

16.2 DensitometricMethods to the Measurement of Rate Processes in Shock Tubes

59

Diatomics

H 2 (Breshears and Bird 1973a), in H2, H, Ar, and Xe. 02 (Breshears, Bird, and Kiefer 1971), in 02, He, Ar, Kr, and Xe. F 2 (Breshears and Bird 1973b), in F2 and Ar. C12 (Santoro, Diebold, and Goldsmith 1977), in C12, He, and Ar. Br 2 (Boyd, Burns, and Macdonald 1975; Macdonald, Burns, and Boyd 1977), in Br 2 and At. HC1 (Breshears and Bird 1972), in HC1 and At. This work on diatomic dissociation has again largely served to confirm and extend earlier studies using X-ray absorption, MZI, and other techniques like IR emission and absorption, as well as the powerful atomic resonance absorption (ARAS) methods. These results are unique to the shock tube, and are regarded here as one of the most significant accomplishments of the device. As before, the activation energies are all l o w - - w e l l below bond energies m and get lower with increasing temperature. This seems to be a solid and quite general result. Clearly, beyond some high energy on the ladder of bound states, collisions must be more likely to cause transitions to the continuum (dissociate) than to deactivate (Johnston and Birks 1972; Kiefer, Joosten, and Breshears 1975. Polyatomics CO 2 ~ CO + O (Kiefer 1974), in CO2 and Ar. N20 ~ N 2 + O (Dove, Nip, and Teitelbaum 1975), in N20 and Ar. SO 2 ~ SO + O (Kiefer 1975; Tyagaraju, Babu, and Rao 1980; Kiefer and Ramaprabhu 1982), in SO2 and Ar. NF 3 --+ NF 2 + F (Breshears and Bird 1978), in NF 3 and Ar. CH 4 -+ CH 3 + H (Gardiner, Owen, Clark, Dove, Bauer, Miller, and McLean 1974; Tabayashi and Bauer 1979; Kiefer and Kumaran 1993), in CH4, Ar and Kr. C2H2 ~ products (Tanzawa and Gardiner 1979, 1980; Kolln, Hwang, Shin, and Gardiner 1991; Kiefer, Sidhu, Kern, Xie, Chen, and Harding 1992), in Ar and Kr. c-C4H 8 -+ 2C2H 4 (Lewis, Feinstein, and Jeffers 1977), in Ar. CH3OH --~ CH 3 + OH (Crib, Dove, and Yamazaki 1984), in Ar. C3H 6 (propene) ~ CH 3 + C2H 3 (Kiefer, A1-Alami, and Budach 1982), in Kr. C3H 8 -+ C2H5 + CH 3 (A1-Alami and Kiefer 1983), in Kr. C2H 4 --+C2H2 + H2 (Tanzawa and Gardiner 1980; Kiefer, Kapsalis, A1Alami, and Budach 1983), in Ar and Kr. C2H4 ~ C2H 2 + H2 (Tanzawa and Gardiner 1980; Kiefer, Kapsalis, A1Alami, and Budach 1983), in Ar and Kr. C2H 6 -+ products (Olson, Tanzawa, and Gardiner 1979; Kiefer and Budach 1984), in Ar and Kr.

60

j. H. Kiefer 2CH 3 -~ C2H4 4- H 2 and/or C2H 6 (Su and Teilelbaum 1995), in Ar. 1,3-C4H 6 -~ products (Kiefer, Wei, Kern, and Wu 1985; Kiefer, Mitchell, and Wei 1988), in Kr. C6H 6 -~ C6H5 4- H (Kern, Wu, Skinner, Rao, Kiefer, Towers, and Mizerka 1985; Kiefer, Mizerka, Patel, and Wei 1985), in Kr. C3H4 -~ C3H3 4. H (Kiefer, Mudipalli, Sidhu, Kern, Jursic, Xie, and Chen 1997), in Ar and Kr. C8H10 (ethylbenzene) -~ CH 3 4. C7H 7 Mizerka and Kiefer 1986), in Kr. c-C6H10 --~ 1,3-C4H 6 4. C2H4 (Kiefer and Shah 1987), in Kr. C4H 4 --~ 2C2H 2 (Kiefer, Mitchell, Kern and Yong 1988), in Kr. C7H 8 (toluene) --~ CH 3 4- CsH 6 (Pamidimukkala, Kern, Patel, Wei, and Kiefer 1987), in Kr. C7H 8 (toluene) -~ C7H 7 (benzyl) 4- H (Pamidimukkala, Kern, Patel, Wei, and Kiefer 1987), in Kr. CsHsN -~ CsH4N 4- H (Kern, Yong, Kiefer, and Shah 1988; Kiefer, Zhang, Kern, Yao, and Jursic 1997), in Kr. C7H10 (norbornene) --~ c-CsH 6 4- C2H 4 (Kiefer, Kumaran, and Sundaram 1993), in Kr. C4H4N 2 (pyrazine) --~ C4H3N 2 4- H (Kiefer, Zhang, Kern, Chen, Yao, and Jursic 1996; Kiefer, Zhang, Kern, Yao, and Jursic 1997), in Kr. C4H4N 2 (pyrimidine) --~ C4H3N 2 4- H (Kiefer, Zhang, Kern, Yao, and Jursic 1997), in Kr. C3H60 3 (1,3,5-trioxane) -~ 3CH20 (Irdam and Kiefer 1990), in Kr. CH20 --~ H 4- HCO (Irdam, Kiefer, Harding, and Wagner 1993), in Kr. CsH9N (1,2,3,6-tetrahydropyrine) --~ 1,3-C4H6 4. CH3N (Sidhu, Kiefer, Lifshitz, Tamburu, Walker, and Tsang 1991), in Ar and Kr. C5H80 (3,4-dihydro-2H-pyran) -~ C2H4 4- C3H40 (Bessaris, Kiefer, Zhang, Walker, and Tsang 1995), in Ar and Kr. C4H40 (furan) -~ products (Fulle, Dib, Kiefer, Zhang, and Kern 1998), in Ar and Kr. c-CsH 6 -~ CsH 5 4- H (Kern, Zhang, Yao, Jursic, Tranter, Greybill, and Kiefer 1998), in Kr. CC14 -~ CC13 4- C1 (Michael, Lim, Kumaran, and Kiefer 1993), in Ar and Kr. CF3C1 -~ CF 3 4- C1 (Kiefer, Sathyanarayana, Lim, and Michael 1994), in Kr.. CF2HC1 -~ CF 2 4- HC1 (Su, Kumaran, Lim, Michael, Wagner, Dixon, Kiefer, and DiFelice 1996), in Ar and Kr. CF3Br -~ CF 3 4- Br (Kiefer and Sathyanarayana 1997), in Kr. CF3I --~ CF 3 4- I (Kiefer and Sathyanaryana 1997), in Kr. CHC13 -~ HC1 4- CC12 (Kumaran, Su, Lim, Michael, Klippenstein, DiFelice, Mudipalli, Kiefer, Dixon and Peterson 1997), in Ar and Kr.

16.2 DensitometricMethods to the Measurement of Rate Processes in Shock Tubes

61

The work on polyatomics listed here is so diverse, and the chemistry involved often so complex, that no generalizations are either possible or appropriate. Rather, just a few interesting and useful examples will be selected from this list for discussion, examples that serve to further emphasize the unique features of the LS method. As noted earlier, the measurement of density gradient, a quantity almost in direct proportion to rate, can simplify the extraction of rate and produce rate constants of superior precision and accuracy. This aspect of the technique makes itself evident in most all of the listed reaction experiments, but a few are especially notable. In the study of the simple one-channel molecular dissociations of the large species cyclohexene, norbornene, trioxane, dihydropyran, and tetrahydropyridine, Kiefer and coworkers were able to routinely obtain rate constants of exceptional precision. For example, Fig. 16.2.15 shows an Arrhenius plot (Kiefer and Shah 1987) of the derived rate constants for the retro-Diels-Alder dissociation of cyclohexene, where the data show mean deviations of less than 10%. With this precision small variations in rate are detectable, and this series of experiments was the first to clearly delineate unimolecular falloff in such a large molecule. Both the pressure variation and curvature of the Arrhenius plot in Fig. 16.2.15 are consequences of this falloff. A detailed analysis of all this large-molecule data has demonstrated that these features are fully consistent with those expected from standard RRKM theory (Kiefer 1998); the lines in Fig. 16.2.15 are the result of this "standard" theory. Thus the standard RRKM theory gives a good description of the falloff of large molecules at extreme temperatures. This is an important and encouraging finding for the modeling of combustion and especially soot formation chemistry at high temperatures. Precision is necessary but not sufficient; it must become accuracy to be of real value. Over the years, the LS rate measurements listed here have been extensively compared with those from other reliable shocktube diagnostic techniques such as atomic resonance absorption (ARAS). In particular, the several papers on dissociation in the small halogenated methanes were collaborations in which some of the LS data was overlapped by independent ARAS measurements in another shock tube. Without exception, these two methods gave the same results to the expected precision of the ARAS measurements. A particularly interesting example is provided by the early investigation of SO 2 dissociation. Here the LS results were immediately controversial, being a factor near 50 faster than rates derived from the seemingly solid uv absorption measurements of Olschewski, Troe, and Wagner (1965). However, subsequent experiments using ARAS to follow the O-atom product (Just and Rimpel 1977) lay within 3% of the LS data in their common region of temperature. Evidently the problem was the rapidity of the reaction, which was not resolved by Olschewski et al.; they actually saw the

62

J. H. Kiefer

.,oo

1 oo

13too

I

k~

k(s-1) (0) 2% in Kr, 369-538 ton" a ) 4% 367-520 torr (O) 2% 120-172 ton" 021)4% 110-152 torr

\\

\

C6Hlo ~ C4H6 + C284 \

o\ k I \ \

s:o

6:o

(t/r)xlo 4

7.'o

s~o

-

FIGURE 16.2.15 LS rate constants for the molecular dissociation of cyclohexene (Kiefer and Shah 1987). Note the self-consistency of the results as well as their curvature and evident pressure dependence; both of the latter features are indicative of unimolecular falloff.

subsequent dissociation of the SO product. There seems to be little doubt of the inherent accuracy of the LS method; the refractivities are typically unambiguous, the calibration is equally reliable, and there are no examples where the results disagree with other solid shock tube measurements. The direct measure of gradient with a small-diameter laser beam also provides exceptional resolution. This, perhaps the most notable virtue of the LS method, was most evident in the relaxation experiments and is also apparent throughout the above listing. However, gradient measurement also has its negative aspects; for one thing, rates must be fairly large for sensitive LS detection. In most cases a dissociation rate constant well above 1000 s -1 is

16.2 Densitometric Methods to the Measurement of Rate Processes in Shock Tubes

63

necessary. As remarked earlier, a consequence is that almost all the LS results lie at higher temperatures than any other m e t h o d when applied to the same problem. This can present a difficulty when comparing results with other techniques because there is little overlap of usable temperatures. This difficulty is apparent in the cited work on cyclohexene (Kiefer and Shah 1987), where there is a gap between LS and single-pulse data, and in CH 4 dissociation (Fig. 16.2.16), where extensive previous rate data stops at the 2800 K temperature of the lowest-temperature LS rate constants (Kiefer and Kumaran 1993). Again, this high-temperature capability has its uses; LS measurements on even the most exhaustively studied reaction will almost always provide something new.

10

10 10

3000

2000

I

I

12

11 Warnatz

10

"~ 10 o

E

~" 10 E

Davidson, et al.

0

0

10

Klemm, et al.

10

10 j

2

l

!

l

1

3

]

I

I.

I

I

I

I

I

I

4

I

l

5

I

]

j

j

I

I

I'

6

IO000/T

FIGURE 16.2.16 Comparisonof various recent results and recommendationsfor the dissociation of methane, CH4M ~ CH3 4- H 4- M, taken from Kiefer and Kumaran (1993). The LS data are the symbols; the lines are from Wamatz (1984), Klemm, Sutherland, Rabinowitz, Patterson, Quartemont, and Tao (1992), and Davidson, Dirosa, Chang, Hanson, and Bowman (1992).

64

J.H. Kiefer

One unique accomplishment of LS examinations of high-temperature reactions in polyatomics has been the resolution of unimolecular incubation (see Tsang and Kiefer 1995). This phenomenon was first noted in diatomics such as H2, D2, and 0 2 (Wray 1965; Watt and Myerson 1969; Breshears, Bird, and Kiefer 1971). Using LS, Dove, Nip, and Teitelbaum (1975), were the first to see the effects of incubation in a polyatomic, N20. Since then just two further examples have appeared: Incubation was resolved in LS studies of norbornene (C7H10) (Kiefer and Kumaran 1993) and furan (C4H40) (Fulle, Dib, Kiefer, Zhang, Yao, and Kern 1998). The resolution of this very short duration phenomenon is evidently difficult; it requires a slow vibrational relaxation at temperatures where dissociation is rapid, and this is not easily discovered. This phenomenon and these observations have been of considerable theoretical interest; see Tsang and Kiefer (1995) for a detailed discussion. An important aspect of LS studies of polyatomic decomposition that needs further consideration is the shape of the LS gradient profile and its implications for accurate rate determination and the extraction of secondary chemistry. If a single endothermic unimolecular dissociation somehow had a rate independent of temperature (an impossibility, to be sure), the LS gradient from this constant-rate first-order reaction would describe a perfect exponential in time and a linear semilog profile. The simplest deviation from this ideal behavior is a concave semilog profile with a rate falling as the temperature falls, as is seen in the molecular dissociations of vinylacetylene, cyclohexene, norbornene, trioxane, dihydropyran, and tetrahydropyridine. Examples of this behavior are found in Fig. 16.2.17. This kind of curvature is thus an inevitable and diagnostic feature of such molecular dissociation. Because the process is here a single reaction (or two reactions with known branching), it is then possible to model the entire gradient variation in time from the temperature dependence established by a set of such experiments. This is most beautifully illustrated in the decomposition of cyclohexene (Kiefer and Shah 1987) and norbornene (Kiefer, Kumaran and Sundaram 1993) as well as in those tetrhydropyridine examples illustrated in Fig. 16.2.17, where it was nearly possible to derive a good local activation energy from the curvature of semilog plots of the gradient in a single experiment. This simplicity, when combined with the precision of the data, then provides powerful evidence for the decomposition mechanism and its thermochemistry. The application of all the measured gradients to setting the rate also leads to higher precision in the rate constants. In the systems studied thus far, the above simplicity is the exception. Many examples actually exhibit convex profiles, and negative gradients may even appear at some time in the process. Examples showing such complex behavior are in Fig. 16.2.18. Of course, a convex profile implies acceleration and is therefore a strong indicator of chain reaction. The most extreme examples of

16.2

Densitometric Methods to the Measurement of Rate Processes in Shock Tubes

-

1267K, 478 torr 2% inKr

1256K, 467

65

torr

o

"~~ 10-4N

-

"U _

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FIGURE 16.2.17 LS semilog gradient profiles from the simple molecular dissociations in tetrahydropyridine (THP) (Sidhu, Kiefer, kifshitz, Tamburu, Walker, and Tsang 1991) and trioxane (Irdam and Kiefer (1990). The upper pair show the THP experiments and the lower those in trioxane. The curvature of these profiles is a consequence of the falling temperature as these endothermic reactions proceed and the consequent drop in rate of the first-order dissociation.

this convexity are found in the decomposition of benzene and especially in acetylene. In these, the chain acceleration dominates the profile; the initiating C-H bond fission is so difficult that the earliest gradient, arising from this alone, is very small, and when the chain takes over, the rise in rate and gradient is profound. It is comparatively difficult to extract an initial rate for the C-H fission in these experiments, but the subsequent acceleration is eloquent about the rate of the following chain process. This behavior permits the derivation of rates for several important chain reactions (Tanzawa and Gardiner 1979; 1980; Kolln, Hwang, Shin, and Gardiner 1991; Kiefer, Sidhu, Kern, Xie, Chen, and Harding 1992; Kiefer, Mizerka, Patel, and Wei 1985). In many chain decompositions the gradient starts positive, tracking the initiating unimolecular decomposition, and then becomes negative. Further on it may return to positive as the products of the exothermic reactions responsible for the negative values begin to decompose. This effect routinely appears in examples where initiation forms copious methyl radical, as in ethane, propane, or ethylbenzene dissociation, the subsequent strong exothermicity provided by recombination to ethane. Two examples of the gradient changing sign in propane are presented in Fig. 16.2.18. Such complex behavior is not always

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16.2 Densitometric Methods to the Measurement of Rate Processes in Shock Tubes

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easily interpreted, b u t w h e n this is accomplished the ups and d o w n s do translate into m o r e details of the m e c h a n i s m . In summary, the LS technique has b e c o m e the preferred m e t h o d for d e n s i t o m e t r y in shock waves, joining those other m o s t reliable and p r o v e n shock tube m e t h o d s like ARAS and single-pulse p r o d u c t analysis with a chemical thermometer. Of course, densitometric m e t h o d s only provide one parameter, but time variation of the density can be a strong indicator of m e c h a n i s m . In coordination with other m e t h o d s , the LS or other densitometric m e t h o d s offer a u n i q u e glimpse of the t h e r m o c h e m i s t r y of the reaction, and do so with superior resolution, high sensitivity, u n a m b i g u o u s calibration, and p r o v e n accuracy. If a m o d e l does not p r o d u c e the gradient profile in detail, it is clearly wrong. Some form of the presented densitometric techniques s h o u l d be a part of any shock tube examination of relaxation or reaction kinetics.

A C KN O W L E D G EMENT S The author is greatly indebted to Carolyn Moore, Weidong Zhang, and Balaji Ramamurthi for their invaluable assistance in the collection of the references. The support of the U.S. Department of Energy, Division of Chemical Sciences, under Grant No. DE-FG02-85ER13384, is also gratefully acknowledged.

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Gaydon, A.G. and Hurle, I.R. (1963). The Shock Tube in High-Temperature Chemical Physics. Reinhold, New York. Gordon, R.G., Klemperer, W and Steinfeld, J.I. (1968). Vibrational and rotational relaxation. Ann. Rev. Phys. Chem. 19: 215-250. Greene, E.F., Cowan, G.R. and Hornig, D.F. (1951). The thickness of shock fronts in argon and nitrogen and rotational heat capacity lags. J. Chem. Phys. 19: 427-434. Greene, E.E and Homig, D.E (1953). The shape and thickness of shock fronts in argon, hydrogen, nitrogen and oxygen. J. Chem. Phys. 21: 617-624. Greene, E.E and Toennies, J.P. (1964). Chemical Reactions in Shock Waves. Academic Press, New York. Greenspan, WD. and Blackman, V.H. (1957). Approach to thermal equilibrium behind strong shock waves in CO 2 and CO. Bull. Am. Phys. Soc. 2: 217-217. Griffith, W.C. (1961). Vibrational relaxation times. In Fundamental Data Obtained from Shock-Tube Experiments, ed., A. Ferri, pp. 242-260. Pergamon Press, New York. Griffith, W., Brickl, D. and Blackman, V. (1956). Structure of shock waves in polyatomic gases. Phys. Rev. 102: 1209-1216. Griffith, W.C. and Kenny, A. (1957). On fully-dispersed shock waves in carbon dioxide. J. Fluid Mech. 3: 286-288. Hanson, R.K. and Flower, W. (1973). Verification of a simple relationship for shock wave reflection in a relaxing gas. AIAA J. 11: 1777-1778. Hornung, H.G. (1972). Induction time for nitrogen dissociation. J. Chem. Phys. 56: 3172-3173. Irdam, E.A. and Kiefer, J.H. (1990). The decomposition of 1,3,5-trioxane at very high temperatures. Chem. Phys. Lett. 166: 491-494. Irdam, E.A., Kiefer, J.H., Harding, L.B. and Wagner, A.E (1993). The formaldehyde decomposition chain mechanism. Int. J. Chem. Kinet. 25: 285-303. Jackson, J.M., Lewis, RA., Skirrow, M.P. and Simpson, C.J.S.M. (1979). Vibrational relaxation of CF 4 and CH 4 and energy transfer between these gases and CO. J. Chem. Soc. Faraday Trans. II 75: 1341-1350. Johannesen, N.H. (1961). Analysis of vibrational relaxation regions by means of the Rayleigh-line method. J. Fluid Mech. 10: 25-32. Johnston, H. and Birks, J. (1972). Activation energies for the dissociation of diatomic molecules are less than the bond dissociation energies. Accs. of Chem. Res. 5: 327-335. Just, T. and Rimpel, G. (1977). The thermal decomposition of SO2 between 2500 and 3400 K. Proc. 11th Int. Symp. on Shock Tubes and Waves, B. Ahlbom, A. Hertzberg and D. Russell eds., pp. 226-231. Univ. of Washington Press. Kern, R.D., Wu, C.H., Skinner, G.B., Rao, V.S., Kiefer, J.H., Towers, J.A. and Mizerka, L.J. (1985). Collaborative shock tube studies of benzene pyrolysis. Proc. 20th Symp. (Int.) on Combustion, pp. 789-797. The Combustion Institute. Kern, R.D., Yong, J.N., Kiefer, J.H. and Shah, J.N. (1988). Shock tube studies of pyridine pyrolysis and their effect on soot formation. Proc. 16th Int. Symp. on Shock Tubes and Waves, Weinheim, ed., pp. 437-444. VCH Publishing. Kern, R.D., Zhang, Q., Yao, J., Jursic, B.S., Tranter, R.S., Greybill, M.A. and Kiefer, J.H. (1998). Pyrolysis of cyclopentadiene: Rates for initial C-H bond fission and the decomposition of the cyclopentadienyl radical. Proc. 27th Symp. (Int.) on Combustion, pp. 143-150. The Combustion Institute. Kewley, D.J. and Hornung, H.G. (1974a). Free-piston shock-tube study of nitrogen dissociation. Chem. Phys. Lett. 25: 531-536. Kewley, D.J. and Homung, H.G. (1974b). Non-equilibrium dissociating nitrogen flow over a wedge. J. Fluid Mech. 64: 725-736.

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Kiefer, J.H. (1974). Densitometric measurements of the rate of carbon dioxide dissociation in shock waves. J. Chem. Phys. 61: 244-248. Kiefer, J.H. (1975). Densitometric measurements of the rate of sulfur dioxide dissociation in shock waves. J. Chem. Phys. 62: 1354-1357. Kiefer, J.H. (1981). The laser schlieren technique in shock-tube kinetics. In Shock Waves in Chemistry, A. Lifshitz, ed., pp. 219-277. Marcel Dekker, New York. Kiefer, J.H. (1998). Some unusual aspects of unimolecular falloff of importance in combustion modeling. Proc. 27th Symp. (Int.) on Combustion, pp. 113-124. The Combustion Institute. Kiefer, J.H., A1-Alami, M.Z. and Budach, K.A. (1982). A shock-tube, laser-schlieren study of propene pyrolysis at high temperatures. J. Phys. Chem. 86: 808-813. Kiefer, J.H., A1-Alami, M.Z. and Hajduk, J.C. (1981). Physical optics of the laser-schlieren shock tube technique. Applied Optics 20: 221-230. Kiefer, J.H., Breshears, W.D. and Bird, P.E (1969). Vibrational relaxation of HBr and HI in shock waves. J. Chem. Phys. 50: 3641-3642. Kiefer, J.H. and Bucach, K.A. (1984). The very high temperature pyrolysis of ethane: Evidence against high rates for dissociative recombination reactions of methyl radicals. Int. J. Chem. Kinet. 16: 674-695. Kiefer, J.H,. Buzyna, L.L., Dib, A. and Sundaram, S. (2000). Observation and analysis of nonlinear vibrational relaxation of large molecules in shock waves. J. Chem. Phys. 113: 48-58. Kiefer, J.H. and Hajduk, J.C. (1980). Rate measurement. In Shock waves with the laser-schlieren technique. Proc. 12th Symp. on Shock Tubes and Waves, A. Lifshitz and J. Rom, eds., pp. 97-110. The Magnes Press, Jerusalem. Kiefer, J.H., Joosten, H.P.G. and Breshears, W.D. (1975). On the preference for vibrational energy in diatomic dissociation. Chem. Phys. Lett. 30: 424-428. Kiefer, J.H., Kapsalis, S.A., A1-Alami, M.Z. and Bucach, K.A. (1983). The very high temperature pyrolysis of ethylene and the subsequent reactions of product acetylene. Combustion and Flame 51: 79-93. Kiefer, J.H. and Kumaran, S.S. (1993). Rate of CH4 dissociation over 2800-4300K: The lowpressure limit rate constant. J. Phys. Chem. 97: 414-420. Kiefer, J.H., Kumaran, S.S. and Sundaram, J. (1993). Vibrational relaxation, dissociation and dissociation incubation times in norbornene. J. Chem. Phys. 99: 3531-3541. Kiefer, J.H. and Lutz, R.W. (1965a). Recombination of oxygen atoms at high temperatures as measured by shock-tube densitometry. J. Chem. Phys. 42: 1709-1714. Kiefer, J.H. and Lutz, R.W. (1965b). Simple quantitative schlieren technique of high sensitivity for shock tube densitometry. Phys. Fluids 8: 1393-1394. Kiefer, J.H. and Lutz, R.W. (1966a). Vibrational relaxation of deuterium by a quantitative schlieren method. J. Chem. Phys. 44: 658-667. Kiefer, J.H. and Lutz, R.W. (1966b). Vibrational relaxation of hydrogen.J. Chem. Phys. 44: 668-672. Kiefer, J.H. and Lutz, R.W. (1966c). Density change on vibrational relaxation in shock-heated hydrogen and deuterium. J. Chem. Phys. 45: 3888-3890. Kiefer, J.H. and Lutz, R.W. (1967). The effect of oxygen atoms on the vibrational relaxation of oxygen. Proc. 11th Syrup. (Int.) on Combustion, pp. 67-76. The Combustion Institute. Kiefer, J.H. and Manson, A.C. (1981). Refractive index change and curvature in shock waves by angled beam refraction. Rev. Sci. Inst. 52: 1392-1396. Kiefer, J.H., Mitchell, K.I., Kern, R.D. and Yong, J.N. (1988). Unimolecular dissociation of vinylacetylene: A molecular reaction. J. Phys. Chem. 92: 677-685. Kiefer, J.H., Mitchell, K.I. and Wei, H.C. (1988). The high temperature pyrolysis of 1,3 butadiene. II. Pulsed laser flash adsorptioin rate constants and consideration of possible molecular dissociation pathways. Int. J. Chem. Kinet. 20: 787-809.

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Kiefer, J.H., Mizerka, M.R., Pate1, M.R. and Wei, H.C. (1985). A shock tube investigation of major pathways in the high temperature pyrolysis of benzene. J. Chem. Phys. 89: 2013-2019. Kiefer, J.H., Mudipalli, P.S., Sidhu, S.S., Kern, R.D., Jursic, B.S., Xie, K. and Chen, H. (1997). Unimolecular dissociation in allene and propyne: The effect of isomerization on the lowpressure rate. J. Phys. Chem. 101" 4057-4071. Kiefer, J.H. and Ramaoprabhu, R. (1982). Detailed analysis of laser-schlieren measurements on the unimolecular dissociation of SO2. Chem. Phys. Lett. 86: 499-502. Kiefer, J.H. and Sathyanarayana, R. (1997). Vibrational relaxation and dissociation in the trifluoromethyl halides: CF3C1, CF3Br and CF3I. Int. J. Chem. Kinet. 29: 705-716. Kiefer, J.H., Sathyanarayana, R., Lim, K.P. and Michael, J.V. (1994). The thermal decomposition of CF3C1. J. Phys. Chem. 98: 12,278-12,283. Kiefer, J.H. and Shah, J.N. (1987). Unimolecular dissociation of cyclohexane at extremely high temperatures: Behavior of the energy transfer collision efficiency. J. Phys. Chem. 91: 3024-3030. Kiefer, J.H., Sidhu, S.S., Kern, R.D., Xie, K., Chen, H. and Harding, L.B. (1992). The homogeneous pyrolysis of acetylene. II. The high temperature radical chain mechanism. Comb. Sci. and Tech. 82: 101-130. Kiefer, J.H., Wei, H.C., Kern, R.D. and Wu, C.H. (1985). The high temperature pyrolysis of 1,3 butadiene: Heat of formation and rate of dissociation of vinyl radical. Int. J. Chem. Kinet. 17: 225-253. Kiefer, J.H., Zhang, Q., Kern, R.D., Chen, H., Yao, J. and Jursic, B.S. (1996). Dissociation and chain reaction in the pyrolysis of pyrazine. Proc. 26th Symp. (Int.) on Combustion, pp. 651-658. The Combustion Institute. Kiefer, J.H., Zhang, Q., Kern, R.D., Yao, J. and Jursic, B.S. (1997). Pyrolysis of aromatic azines: Pyrazine, pyrimidine and pyridine. J. Phys. Chem. 101: 7061-7073. Kishore, V.V.N., Babu, S.V. and Rao, V.S. (1980). A shock tube study of vibrational relaxation in SO2, SO2-Ar and SO2-He. J. Chem. Phys. 46: 297-305. Kistiakowsky, G.B. and Kydd, P.H. (1956). Gaseous detonation. IX. A study of the reaction zone by gas density measurements. J. Chem. Phys. 25" 824-835. Klemm, R.B., Sutherland, J.W., Rabinowitz, M.J., Patterson, P.M., Quartemont, J.M. and Tao, W. (1992). Shock tube kinetic study of CH 4 dissociation: 1726 K < T _< 2134 K.J. Phys. Chem. 96: 1786-1793. Knight, H.T. and Venable, D. (1958). Apparatus for precision flash radiography of shock and detonation waves in gases. Rev. Sci. Instr. 29: 92-98. Kolln, W.S., Hwang, S.M., Shin, K.S. and Gardiner, Jr., W.C. (1991). Re-evaluation of laser schlieren data for acetylene pyrolysis. AIAA J. 132: 372-385. Kumaran, S.S., Su, M.-C., Lim, K.P., Michael, J.V., Klippenstein, S.J., DiFelice, J., Mudipalli, P.S., Kiefer, J.H., Dixon, D.A. and Peterson, K.A. (1997). Experiments and theory on the thermal decomposition of CHC13 and the reactions of CC12. J. Phys. Chem. A101: 8653-8661. Kurian, J. and Sreekanth, A.K. (1988). Density gradient measurements of vibrational relaxation in (N 2 + H20 ) and (Ar 4- H20) mixtures. Proc. 16th Int. Symp. on Shock Tubes and Waves: Shock Tubes and Waves. H. Gr6nig, ed., pp. 411-419. VCH Verlagsgesellschaft mbH, D-6940 Weinheim. Ladenburg, R. and Bershader, D. (1954). Interferometry. In Physical Measurements in Gas Dynamics and Combustion, vol. IX, W. Ladenburg, ed., pp. 47-78. Princeton Univ. Press. Landolt, H.H. and B6rnstein, R. (1962). Physikalisch-Chemische Tabellen, 5th ed., Part 8. SpringerVerlag, Berlin. Lensch, G. and Grbnig, H. (1978). Experimental determination of rotational relaxation in molecular hydrogen and deuterium. Proc. 11 th Int. Symp. of Shock Tubes and Waves, B. Ahlborn, A. Hertzberg and D. Russell, eds., pp. 132-139. Univ. of Washington Press. Levitt, B. and Hornig, D.E (1962). Structure of detonation waves. J. Chem. Phys. 36: 219-227.

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Lewis, D.K., Feinstein, S.A. and Jeffers, P.M. (1977). Cyclobutane thermal decomposition rates at 1300-1500 K. J. Phys. Chem. 81: 1887-1888. Linzer, M. and Hornig, D.E (1963). Structure of shock fronts in argon and nitrogen. Phys. Fluids 6: 1661-1668. Lutz, R.W and Kiefer, J.H. (1966). Structure of the vibrational relaxation zone of shock waves in oxygen. Phys. Fluids 9: 1638-1642. Macdonald, R.G., Burns, G. and Boyd, R.K. (1977). Dissociation of Br2 in shock waves. J. Chem. Phys. 66: 3598-3608. Matthews, D.L. (1959). Interferometric measurement in the shock tube of the dissociation rate of oxygen. Phys. Fluids 2: 170-178. Matthews, D.L. (1961). Vibrational relaxation of carbon monoxide in the shock tube. J. Chem. Phys. 34: 639-642. Michael, J.V. and Lim, K.P. (1993). Shock tube techniques in chemical kinetics. Ann. Rev. Phys. Chem. 44: 429-458. Michael, J.V., Lim, K.P., Kumaran, S.S. and Kiefer, J.H. (1993). The thermal decomposition of carbon tetrachloride. J. Phys. Chem. 97: 1914-1919. Millikan, R.C. and White, D.R. (1963a). Vibrational energy exchange between N 2 and CO. The vibrational relaxation of nitrogen. J. Chem. Phys. 39: 98-101. Millikan, R.C. and White, D.R. (1963b). Systematics of vibrational relaxation. J. Chem. Phys. 39: 3209-3213. Miyama, H. and Endoh, R. (1967). Vibrational relaxation of nitrogen in shock-heated NH3-O2-N2 mixtures. J. Chem. Phys. 46: 2011-2012. Mizerka, L.J. and Kiefer, J.H. (1986). The high temperature pyrolysis of ethyl benzene: Evidence for dissociation to benzyl and methyl radicals. Int. J. Chem. Kinet. 18: 363-378. Moore, C.B. (1965). Vibrationnrotation energy transfer. J. Chem. Phys. 43: 2979-2986. Moreno, J.B. (1966). Shock-tube measurements of vibrational relaxation times in deuterium. Phys. Fluids 9:431-435. Mott-Smith, H.M. (1951). The solution of the Boltzmann equation for a shock wave. Phys. Rev. 82: 885-892. Oertel, H., Jr., (1975). Regular shock reflection used for studying oxygen dissociation. Proc. lOth Int. Shock Tube Symp.: Modern Developments in Shock Tube Research, G. Kamimoti, ed., pp. 605612. Shock Tube Research Society, Japan. Oertel, H., Jr., (1976). Oxygen vibrational and dissociation relaxation behind regular reflected shocks. J. Fluid Mech. 74: 477-495. Oertel, H., Jr., (1978). Oxygen dissociation behind oblique reflected and stationary oblique shocks. Arch. Mech. 30: 123-133. Offenhauser, E and Frohn, A. (1986). Theoretical and experimental study of vibrational nonequilibrium. Proc. 15th Int. Symp. on Shock Waves and Shock Tubes: Shock Waves and Shock Tubes, D. Bershader and R. Hanson, ed., pp. 327-333. Stanford Univ. Press. Olschewski, H.A., Troe, J. and Wagner, H.G. (1965). Der unimolekular zerfall von SO2. Zeitschrift fur Physikalische Chemie Neue Folge 44: 173-183. Olson, D.B., Tanzawa, T. and Gardner, Jr., WC. (1979). Thermal decomposition of ethane. Int. J. Chem. Kinet. 11: 23-44. Paillard, C., Dupr~, G., Yousseffi, S., Lisbet, R. and Charpentier, N. (1986). A shock tube study of the decomposition of chlorine dioxide. Proc. 15th Int. Symp. on Shock Waves and Shock Tubes: Shock Waves and Shock Tubes, D. Bershader and R. Hanson, ed., pp. 380-387. Stanford Univ. Press. Pamidimukkala, K.M., Kern, R.D., Patel, M.R., Wei, H.C. and Kiefer, J.H. (1987). High temperature pyrolysis of toluene. J. Phys. Chem. 91: 2148-2154.

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Partington, J.R. (1953). An Advanced Treatise on Physical Chemistry. Physico-Chemical Optics vol. IV. Longmans, London. Rabinowitz, M.J., Sutherland, J.W, Patterson, P.M. and Klemm, R.B. (1991). Direct rate constant measurement for H + CH4 --~ CH3 -4-H2, 897-1729 K using flash photolysis shock tube test. J. Phys. Chem. 95: 674-681. Reese, T. (1978). Local density measurement in argon shock waves by nonresonant 90~ scattering. Proc.. 11th Int. Symp. on Shock Tubes and Waves: Shock Tubes and Shock Wave Research, B. Ahlborn, A. Hertzberg, D. Russell eds., p-p. 439-445. Univ. of Washington Press. Rees, T. and Bhangu, J.K. (1969). Effects of small quantities of hydrogen, deuterium and helium on vibrational relaxation in carbon dioxide. J. Fluid Mech. 39: 601-610. Resler, E.L., Jr. and Cary, B.B. (1957). Dissociation of air behind strong shock waves. Proc. Conf. Chem. Aeron.: Threshold of Space, 320-327. Resler, E.L. Jr. and Scheibe, M. (1955). Instrument to study relaxation rates behind shock waves. J. Acous. Soc. Am. 27: 932-938. Rink, J.P. (1962a). Relative efficiencies of inert third bodies in dissociating oxygen. J. Chem. Phys. 36: 572-573. Rink, J.P. (1962b). Shock tube determination of dissociation rates of hydrogen. J. Chem. Phys. 36: 262-265. Rink, J.P. (1962c). Dissociation rates of deuterium. J. Chem. Phys. 36: 1398-1400. Rink, J.P., Knight, H.T. and Duff, R.E. (1961). Shock tube determination rates of oxygen. J. Chem. Phys. 34: 1942-1947. Santoro, R.J. and Diebold, G.J. (1978). Density gradient measurements of vibrational relaxation in Ar-C1F mixtures behind shock waves. J. Chem. Phys. 69: 1787-1788. Santoro, R.J., Diebold, G.J. and Goldsmith, G.J. (1977). Density gradient measurement of C12 dissociation in shock waves. J. Chem. Phys. 67: 881-886. Schardin, H. (1958). Ein beispiel zur verwendung des stosswellenrohres fuer probleme der instationaren gasdynamik. Z. Angew. Math. Physik 9: 606-621. Schumacher, B.W and Gadamere, E.O. (1958). Electron beam fluorescence probe for measuring by the local gas density in a wide field of observation. Can. J. Phys. 36: 659-671. Sidu, S.S., Kiefer, J.H,. Lifshitz, A., Tamburu, C., Walker, J.A. and Tsang, W. (1991). Rate of the retro-Diels-Alder dissociation of 1,2,3,6-tetrahydropyridine over a wide temperature range. Int. J. Chem. Kinet. 23: 215-227. Simpson, C.J.S.M., Bridgman, K.P. and Chandler, T.R. (1968a). A shock tube study of vibrational relaxation in carbon dioxide. J. Chem. Phys. 49: 513-522. Simpson, C.J.S.M., Bridgman, K.P. and Chandler, T.R. (1968b). A shock tube study of vibrational relaxation in nitrous oxide. J. Chem. Phys. 49: 509-513. Simpson, C.J.S.M. and Chandler, T.R.D. (1970). A shock tube study of vibrational relaxation in pure CO 2 and mixtures of CO 2 with the inert gases, nitrogen, deuterium and hydrogen. Proc. Roy. Soc. Lond. A. 317: 265-277. Simpston, C.J.S.M., Chandler, T.R.D. and Strawson, A.C. (1969). Vibrational relaxation in CO 2 and CO2-Ar mixtures studied using a shock tube and a laser-schlieren technique. J. Chem. Phys. 51: 2214-2219. Simpson, C.J.S.M. and Gait, P.D. (1973). Studies of vibration-rotation energy exchange using hydrogen and its isotopes. Proc. 9th Shock Tube Symp.: Recent Developments in Shock Tube Research, D. Bershader and W. Griffith eds., pp. 352-364. Stanford Univ. Press. Simpson, C.J.S.M. and Gait, P.D. (1977). The vibrational deactivation of the bending mode of CO 2 by 0 2 and by N 2. Chem. Phys. Lett. 47: 133-136. Simpson, C.J.S.M., Gait, P.D., Price, T.J. and Foster, M.G. (1982). A shock tube study of vibrational relaxation in pure N20 and mixtures of N20 with argon, helium-4, helium-3, deuterium, hydrogen deuteride, normal and para-hydrogen. Chem. Phys. 68: 293-302.

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Simpson, C.J.S.M., Gait, P.D. and Simmie, J.M. (1976a). Vibration-rotation energy exchange in carbon dioxide-hydrogen mixtures. Proc. Roy. Soc. Lond. A 348: 57-72. Simpson, C.J.S.M., Gait, P.D. and Simmie, J.M. (1976b). Vibrational relaxation in OCS mixtures. J. Chem. Soc. Fara. Trans. II 72: 417-422. Simpson, C.J.S.M., Price, T.J. and Crowther, M.E. (1975). High temperature vibrational relaxation in HD. Chem. Phys. Lett. 34: 181-183. Simpson, C.J.S.M. and Simmie, J.M. (1971). A study of vibrational-rotational energy exchange in a shock tube. Proc. Roy. Soc. Lond. A. 325: 197-206. Smeets, G. (1971). A high sensitivity laser interferometer for transient phase objects. Proc. 8th Int. Shock Tube Symp.: Shock Tube Research, J.L. Stollery, A.G. Gaydon and P.R. Owen, eds., paper No. 45. Barnes and Noble. Smeets, G. (1977). Flow diagnostics by laser interferometry. IEEE Trans. Aerosp. And Elect. Syst. AES-13: 82-90. Smiley, E.E and Winkler, E.H. (1954). Shock-tube measurements of vibrational relaxation. J. Chem. Phys. 22: 2018-2022. Su, M.-C., Kumaran, S.S., Lim, K.P., Michael, J.V., Wagner, A.E, Dixon, D.A., Kiefer, J.H. and DiFelice, J. (1996). Thermal decomposition of CF2HC1. J. Phys. Chem. 100: 15,827-15,833. Su, J.Z. and Teitelbaum, H. (1995). Kinetics of the decay of CH3 radicals in shock waves. Proc. 19th Int. Symp. on Shock Waves: Physico-Chemical Processes and Nonequilibrium Flow, R. Brun and L.Z. Dumitrescu eds., pp. 101-106. Springer. Sutton, E.A. (1962). Measurement of the dissociation rates of hydrogen and deuterium. J. Chem. Phys. 36: 2923-2931. Synofzik, R., Garen, W, Wortberg, G. and Frohn, A. (1978). Rotational and vibrational relaxation behind weak shock waves in water vapor. Proc. 11 th Int. Symp. on Shock Tubes and Waves: Shock Tube and Shock Wave Research, B. Ahlborn, A. Hertzberg and D. Russell, eds., pp. 127-131. Univ. of Washington Press. Tabayashi, K. and Bauer, S.H. (1979). The early stages of pyrolysis and oxidation of methane. Combustion and Flame 34: 63-83. Tanzawa, T., Hidaka, Y. and Gardiner, WC., Jr. (1979). Laser schlieren deflection in incident shock flow. Proc. of the 12th Int. Symp. on Shock Tubes and Waves, (eds. Lifshitz and Rom), pp. 555561. Magnes Press, The Hebrew University, Jerusalem. Tanzawa, T. and Gardiner, WC., Jr. (1979). Thermal decomposition of acetylene. Proc. 17th Symp. (Int.) on Combustion, pp. 563-573. The Combustion Institute. Tanzawa, T. and Gardiner, WC., Jr. (1980). Thermal decomposition of ethylene. Combustion and Flame 39: 241-253. Teitelbaum, H. (1982). Vibrational relaxation of polyatomic molecules in gas mixtures. Proc. 13th Int. Symp. on Shock Tubes and Waves, C.E. Treanor and J.G. Hall, eds., pp. 560-575. State Univ. of New York Press. Tsang, W and Kiefer, J.H. (1995). Unimolecular reactions of large polyatomic molecules over wide ranges of temperature. In Advanced Series in Physical Chemistry, Liu and Wagner, eds., pp. 58119. World Scientific Publishing, Singapore. Tsang, W and Lifshitz, A. (1990). Shock tube techniques in chemical kinetics. Ann. Rev. Phys. Chem. 41: 559-599. Tyagaraju, M., Babu, S.V. and Rao, V.S. (1980). Dissociation rate measurements in SO2 + Ar mixtures behind incident shock waves. Chem. Phys. 48: 411-416. Venable, D. (1955). Positive ion oscilloscope trigger for shocks in low density gases. Rev. Sci. Instr. 26: 729. Venable, D. and Kaplan, D.E. (1955). Electron beam method of determining density profiles across shock waves in gases at low densities. J. Appl. Phys. 26: 639-640.

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Walsh, P.M. and Bauer, S.H. (1973). Vibrational relaxation in CO2 with selected collision partners. II. Methane, tetradeuteriomethane and fluoromethane. J. Phys. Chem. 77: 1078-1082. Wang, H. and Bershader, D. (1966). Thermal equilibrium behind an ionising shock. J. Fluid Mech. 26: 459-479. Wamatz, J. (1984). Rate coefficients in the C/H/O system. In Combustion Chemistry, W.C. Gardiner, Jr., ed., pp. 243-360. Springer-Verlag, New York. Watt, W.S. and Myerson, A.L. (1969). Atom formation rates behind shock waves in oxygen. J. Chem. Phys. 51: 1638-1643. Weaner, D., Roach, J.F. and Smith, WR. (1967). Vibrational relaxation times in carbon dioxide. J. Chem. Phys. 47: 3096-3097. White, D.R. (1961). Optical refractivity of high temperature gases. III. The hydroxyl radical. Phys. Fluids 4: 40-65. White, D.R. (1965a). Vibrational relaxation of 0 2 in O2-D 2 mixtures of H 2 and D 2. J. Chem. Phys. 42: 447-449. White, D.R. (1965b). Vibrational relaxation of oxygen by methane, acetylene and ethylene. J. Chem. Phys. 42: 2028-2032. White, D.R. (1966a). Vibrational relaxation in C O - D 2 mixtures. J. Chem. Phys. 45: 1257-1258. White, D.R. (1966b). Vibrational relaxation of N 2 in N 2 - H 2 mixtures. J. Chem. Phys. 46: 19671968. White, D.R. (1967). Vibratioinal relaxation of N 2 in N 2 - H 2 mixtures. J. Chem. Phys. 46: 20162017. White, D.R. (1968a). Shock-tube study of vibrational exchange in N 2 - O 2 mixtures. J. Chem. Phys. 49: 5472-5476. White, D.R. (1968b). Vibrational relaxation of shocked N2-He, N2-CH 4 and N2-C2H 2 mixures. J. Chem. Phys. 48: 525-526. White, D.R. and Millikan, R.C. (1963a). Vibrational relaxation of oxygen.J. Chem. Phys. 39: 18031806. White, D.R. and Millikan, R.C. (1963b). Oxygen vibrational relaxation in O2-He and O2-Ar mixtures. J. Chem. Phys. 39: 1807-1808. White, D.R. and Millikan, R.C. (1963c). Oxygen vibrational relaxation in O 2 - H 2 mixtures. J. Chem. Phys. 39: 2107-2109. White, D.R. and Millikan, R.C. (1964). Vibrational relaxation in air. A/AAJ. 2: 1844-1846. Witteman, W.J. (1961). Instrument to measure density profiles behind shock waves. Rev. Sci. Instr. 32: 292-296. Witteman, W.J. (1962). Vibrational relaxation in carbon dioxide. II. J. Chem. Phys. 37: 655-661. Wray, K.L. (1965). Kinetics of 0 2 dissociation and recombination. Proc. lOth Symp. (Int.) on Combustion, pp. 523-537. The Combustion Institute. Yasuhara, M., Yoneda, K. and Sato, S. (1974). Vibrational relaxation measurements of methane by a laser schlieren. J. Phys. Soc. Japan 36: 555-557.

CHAPTER

16.3

Chemical and Combustion Kinetics 16.3 Atomic Resonance Absorption Spectroscopy with Flash or Laser Photolysis in Shock Wave Experiments JOE V. MICHAEL* and ASSA LIFSHITZ~ * Chemistry Division, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, Illinois, 60439, USA "~Department of Physical Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel

16.3.1 Introduction 16.3.2 Atomic Resonance Absorption Spectrometric Detection in Shock Tubes 16.3.2.1 Light Sources 16.3.2.2 Line Absorption Theory 16.3.2.3 Conclusions 16.3.2.4 Calibration Procedures 16.3.3 Flash and/or Laser Photolysis in Shock Tubes 16.3.3.1 Bimolecular Atom-Molecule Reactions 16.3.3.2 Bimolecular Radical-Molecule Reactions 16.3.3.3 Flash and/or Laser-Shock Tube Results 16.3.4 Summary References

The submitted manuscript has been created by the University of Chicago as Operator of Argonne National Laboratory ("Argonne") under Contract No. W-31-109-ENG-38 with the U.S. Department of Energy. The U.S. Government retains for itself, and others acting on its behalf, a paid-up, nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Government. Handbook of Shock Waves, Volume 3 Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISBN: 0-12-086433-9/$35.00

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78 16.3.1

j. v. Michael and A. Lifshitz

INTRODUCTION

With the development of computer codes for simulating chemical reaction mechanisms at high temperatures (Kee, Rupley, and Miller 1989; Radhakrishnan and Hindmarsh 1993; Radhakrishnan 1994), the uncertain issue in the use of these codes is the accuracy and availability of both the temperature and pressure dependences of rate constants. Most of the data used for simulating high-temperature (high-T) chemical kinetics--for example, in combustionm have been obtained with shock tube techniques, and these have been evaluated and used in building chemical mechanisms (Bowman et al. 1995; Pilling, Turanyi, Hughes, and Clague 1996). Shock tube studies on unimolecular dissociations of diatomic, triatomic, and polyatomic molecules have been reported. In addition, there are a large number of reported studies on bimolecular atom-molecule, radical-molecule, and radical-radical reactions. These high-T data then constitute a major part of the kinetics database used today for simulating combustion and other high temperature processes. Many detection methods, spectrometric and others, have been developed to follow the concentration profiles of species generated in the heated zone of shock tubes. Time-resolved concentrations of stable molecules, free radicals, and atoms have been successfully measured. However, this chapter will deal only with reactions involving atomic species and, specifically, with methods for their generation and detection. There are two methods for generation of atoms in shock tubes. The method used most is high-temperature thermal dissociation of molecules into atoms or into an atom and a radical species. Many studies have concentrated on the thermal dissociation rate alone; however, bimolecular rates can also be studied by observing the initially formed atom as it depletes due to atom-with-molecule reaction or the atomic formation from the concurrently formed radical with an added molecule. These types of experiments are reported in many of the references included in this review (particularly the work of Just, 1981b; Herzler and Frank, 1992; Mick, Matsui, and Roth, 1993; Lifshitz, Bidani, and Carroll, 1983; Rao and Skinner, 1989). A second and less used technique is photochemical generation of an atomic species by flash or laser photolysis. This method has been called the flash or laser photolysis-shock tube (FP- or LP-ST) method. A molecule, which on photolysis gives an atomic product, is included in the experimental mixture along with added reactant. Bimolecular rate constants can then be measured by observing atomic decay caused by reaction of the atom with the reactant. The only detection method for atomic species considered in the review is atomic resonance absorption spectrometry (ARAS). Most shock tube researchers are aware that shock tube chemists are and have been devising methods to measure rates of chemical reactions at high temperatures, which is a practical motivation for understanding high-T

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reaction systems. There are other motivations. These same researchers (usually those interested only in the physics of shock wave phenomena) are mostly unaware that methods have been used for decades to measure the same reaction rates at lower temperatures. Much of this work has supported modeling efforts in the planetary atmospheric chemistry field, including Earth's troposphere (both polluted and clean), stratosphere, and mesosphere. Experiments in static or slowly flowing reactors (where reaction is initiated by flash or laser photolysis) and in discharge flow reactors, have utilized sensitive spectrometric methods. Both types of low-T experiments have yielded a large volume of data, but these more traditional chemical kinetics methods are mostly limited to temperatures below ~ 800 K. It is also true that many shock tube chemists are not interested in the lowtemperature work simply because it can have little implication to hightemperature kinetics. Hence, evaluations (e.g., Mallard, Westley, Herron, and Hampson, 1994) of rate behavior for several reactions have been based only on low- and high-T determinations; i.e., there is little data for many reactions in the intermediate-T range between ~ 800 and 1300 K. Of course, there should be a continuous and smooth change in rate behavior over the entire temperature range. Shock tube physicists, fluid dynamicists, and aeronautical engineers are also mostly unaware that there are theoretical reasons why this continuous behavior should exist. However, one of the strong motivations for physical chemists is to supply accurate data over very large temperature ranges for reactions so that the results can be used to test modern theories of chemical kinetics. The FP- or LP-ST method is particularly suited for supplying results in the intermediate-T range for use by both combustion and theoretical chemists in practical applications and theoretical calculations, respectively.

16.3.2 ATOMIC RESONANCE ABSORPTION SPECTROSCOPIC DETECTION IN SHOCK TUBES The theory for emission and absorption of resonance radiation was developed in the early part of the 20th century and is thoroughly discussed in the classic book by Mitchell and Zemansky (1934). The use of the ARAS method in a shock tube dates back to the late 1960s in the pioneering work of Myerson, Thompson, and Joseph (1965) and Myerson and Watt (1968), who determined the dissociation rate of H 2 by following the thermal evolution of H atoms behind shock waves. These workers were the first to use the ARAS acronym. They then applied O-atom ARAS to the thermal decompositions of 0 2 and NO (Watt and Myerson 1969; Myerson 1973). Subsequently, Appel and Appleton

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(1974) measured the thermal dissociation and oxidation of D 2 using D-atom ARAS. Since this early work, the technique has been refined and detection has been extended to include other atomic species including N, C, C1, S, Si, Br, I, Cd, and Se atoms. Several review articles have been published during the past 20 years that consider ARAS results (Just 1981a, 1981b; Tsang and Lifshitz 1990; Michael and Lim 1993). In this section we will review some general principles of the technique pertinent to shock tube studies, and we will discuss such issues as the effect of light source line shape on the photometer sensitivity. Generally, the level of concentration observable with the ARAS technique is -~ 5 • 1012 atoms/cm -3 or less with a time resolution of ~ 10 Its. The ability to detect such small concentrations permits the designer of a kinetics experiment to use ppm concentrations of molecules from which atoms are produced. This has valuable consequences since small initial concentrations of molecules (a) ensure isothermal conditions close to those in a shock of pure monatomic gas and (b) substantially decrease the possibility of secondary reactions between atoms and between atoms and subsequently formed radicals. Hence, one can design experiments that can chemically isolate a specific reaction for chemical kinetics study. ARAS is a narrow-line spectroscopy and is therefore inherently different from commonly used broad-line spectrometry where a bandwidth is selected from a continuous light source with a monochromator. Most polyatomic absorber molecules have absorption coefficients that vary slowly with wavelength and, therefore, absorption always follows Beer's law. With a broad absorber and a narrow-line light emitter, Beer's law also holds provided the transition is not saturated. Absorption coefficients can change with temperature however, and this increases the complexity of the analysis. ARAS is similar in many respects to laser spectrometry since the line width of the absorber is always close to that of the emitter in the light source. However, depending on the ratio of line shapes between absorber and emitter, the system can be either in or out of resonance and, therefore, the photometer may or may not follow Beer's law. Since the bandwidth of the light source is defined by the line profile, curves of growth (i.e., absorbance versus concentration graphs where absorbance --ln(I/Io) and I and I0 are transmitted and initial intensities, respectively) can either follow a linear Beer's law or show negative deviations from linearity. The latter occurs if the absorber profile is substantially narrower than the emitter. An incomplete understanding of these relationships may cause serious ambiguities unless necessary precautions are taken. As a minimum, a proper calibration for each light source must be performed. Alternatively, if the line profile of the light source is known, absorption coefficients and the curve of growth can be calculated from spectroscopic quantities prior to the design of an experiment. In a section to follow, this will be specifically illustrated for Hatom ARAS.

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Some additional precautions should be taken when using a sensitive detection method in a shock tube experiment. Since atom concentrations of "~ 5 x 1012 atoms/cm -3 are routinely measured, the gas-handling system must be very clean and the purity of monatomic gases used as diluents should be very high. Research-grade Ar and/or Kr for the driven and ultrahigh-purity He for the driver can all be obtained from commercial suppliers. Both clean gas handling and high gas purities are necessary to prevent spurious absorption due to impurity-generated atoms. This is particularly true when studies in the reflected shock regime are anticipated. Impurities absorbed on the shock tube walls can be stripped off into the gas phase by the incident wave and subsequently heated by the reflected wave, creating atoms (Lifshitz, Bidani, and Carroll 1981a). Most problems of this type can be overcome with careful designs for the shock tube equipment. Such designs include a highly polished shock tube inner surface (Michael 1989) that is bakeable to high temperatures (Just 1981b). In addition to increasing the quality of the shock waves, this procedure passivates the surface and diminishes surface adsorption. Also, allmetal vacuum flanges and special LiF and MgF 2 bakeable window adapters (Lifshitz, Bidani, and Carroll 1981b) can greatly minimize impurity problems. An efficient pumping system on the shock tube is also essential. Pump-down pressures of 10 - 7 to 10 -8 torr can be achieved with cryopumped diffusion or turbomolecular pumps. The crucial test for cleanliness is the observation of the level of absorption of H and/or O atoms in "pure" Ar and/or Kr bath gases at temperatures between 2000 and 3000 K.

16.3.2.1 LIGHT SOURCES The most commonly used light source in atomic absorption or fluorescence spectrometry is light emission from a plasma-generated microwave discharge, normally at 2.45GHz (Davis and Braun 1968). Diatomic or polyatomic molecules containing the atom to be studied are mixed with a flowing stream of He at 1-3 torr pressure and are injected into a glass tube containing crystal window apertures. The resultant mixture in the tube, at ppm up to a few percent levels, is discharged at 7-70 watts of microwave power using a tuned cavity that is cooled by flowing compressed air. The exact lamp conditions really depend on other experimental factors that are specific to a given apparatus. Hence, it is always true that some calibration procedure must be implemented to experimentally obtain the curve of growth for a given atomic species. This specifically means that photometer sensitivity can vary from laboratory to laboratory depending on the lamp conditions. The plasma emission will always contain a high intensity of radiation connecting the first excited atomic state to the ground atomic state; i.e., the

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highest energy resonance lines. These are almost always in the vacuum ultraviolet region of the spectrum and, therefore, the light must be detected with a solar blind photomuhiplier. As examples, for H atoms the resonance line is Lyman-a, the overlapped doublet, centered at 121.567 nm (Lifshitz, Skinner, and Wood 1979), for O atoms, the separated triplet at 130.217, 130.486, and 130.603 nm (Pamidimukkala, Lifshitz, Skinner, and Wood 1981), and for N atoms, the separated lines at 119.955, 120.022, and 120.071 nm where the splitting is due to nondegeneracy in the excited state (Wood, Skinner, and Lifshitz 1987). If the plasma opacity (the product of the atom concentration times the path length from one side of the plasma to the other, [A]sls) is small, the emission will approach a Doppler or Gaussian profile that is characterized by the plasma temperature and the mass of the emitting atom. One can ensure small opacity values either by limiting the distance ls or by limiting atomic concentration [A]s. This treatment disregards natural and/or pressure broadenings because both effects are small compared with Doppler broadening. However, if one wants to include these effects, then the analysis would require using a Voigt profile (Mitchell and Zemansky 1934; Lynch, Schwab, and Michael 1976).

16.3.2.2

LINE ABSORPTION THEORY

Many of the above points can be best illustrated by considering atomic line absorption theory for the Lyman-a transition in H atoms. This theory has been given in detail (Michael and Weston 1966; Barker and Michael 1968; Braun, Bass, and Davis 1970; Lynch, Schwab, and Michael 1976; Chiang, Lifshitz, Skinner, and Wood 1979; Maki, Michael, and Sutherland 1985) and is repeated here. In this case the muhiplet lines are overlapped--in contrast, for example, to O atoms where the line separation is so large that absorption from the triplets can be considered to be independent from one another (Pamidimukkala, Lifshitz, Skinner, and Wood 1981). For H atoms, the resonance lamp emission function E~ is E~ -- {1 - exp[-@ro1[H]111 exp(-(7/0q) 2) + 2~rol[H]l11 exp(-((7 + Av)/0q)2))]} x {exp[-(O-o2[H]2/2 exp(-72) + 2O'o2[H]2/2exp(-(? + Ay)2))]} (16.3.1) This equation then represents the product of absorption in layer 1 (over a distance ll from one side of the plasma to the other) and transmission in layer 2 (over a distance l2 from the front edge of the plasma to the lamp window). Layer 2 is the reversal layer and is generally taken to be at room temperature, 298K. kadenburg and Reiche (1913) were the first to derive this equation;

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16.3 Atomic Resonance Absorption Spectroscopy with Flash or Laser Photolysis

Mitchell and Zemansky (1934) and Braun and Carrington (1969) present a complete description. In the limit of low [H] in the plasma, the emission function becomes E~, -- ~ol[H]lll[exp(-(T/Otl) 2) + 2 exp(-((y + AT)/0fi)2)]

(16.3.2)

because both [H]I and [H]2 approach zero. Equation (16.3.2) then represents the low lamp opacity limit and is the Gaussian profile for the overlapped doublet. Transmission through a region that is external (region 3) to the light source containing an atomic concentration [H]3 (over a distance l3) is denoted by Ay, where A~ -- exp[-(o-03[H3]l 3 exp(-(7/0~3) 2) + 2cr03[H3]/3 exp(-((7 + A~)/0~3)2))]

(16.3.3) Both emission and transmission in regions 1 and 3, respectively, are referenced to the reversal layer, region 2, by the equations 0c,,- (T,/298 K) 1/2 and 7 = ( v - V o ) / 6 2 , where 6 n is (Vo/C)(2RTn/M) 1/2 and v0 is the center line frequency of the weak transition (2p1/2 ~- 2 $1/2). The absorption cross section at the line center is ~r0, = (f/~n)g-1/2(ge2/mc), where f is the oscillator strength, 0.1387, for the weak component (Bethe and Salpeter 1977), e and m are the charge and mass of the electron, c is the speed of light, R is the gas constant, and M is the atomic weight of the absorber (H atoms). ~r0,, therefore varies with T~ 1/2 and is equal to 1.1368 x 10 -13 cm 2 at 298 K. Since the lines overlap, the displacement between multiplets (1.094 x 101~ s -1) has to be included as A7 = 1.094 x 1010 S-1/~2 = 0.599. There are two good tests of the two-layer model for the emission function (Braun, Bass, and Davis 1970; Lifshitz, Skinner, and Wood 1979) given by Equation (16.3.1). In both cases, the Lyman-a H (or D) transition was measured spectroscopically in the vacuum ultraviolet at high order. Figure 16.3.1 shows an example of an experiment obtained with 1% H 2 and 1% D 2 in i torr He at 50watts microwave power (Tsang and Lifshitz 1990). It is clear that the two isotopes are well separated from one another (0.033 nm), allowing for D analysis in the presence of H or vice versa. Since the line center frequencies are strongly depressed, it is also clear that the two lines show substantial reversal. Lifshitz, Skinner, and Wood (1979) tested three different lamp configurations. Their designs progressively decreased the effects of the reversal layer by first decreasing l2 and second by decreasing the H 2 concentration in the plasma; i.e., both [H]I and [H]2. In agreement with theory, they found that Gaussian behavior was obtained when both the reversal layer thickness and the H-atom concentration in the plasma were small (low H 2 and low microwave power). On the other hand, when a 3.6-torr He source was

84

J. V. Michael and A. Lifshitz

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FIGURE 16.3.1 Hydrogenand deuterium emissions from a resonance lamp showing highly selfreversed line profiles. (Figure 1 from Tsang, W. and Lifshitz, A. (1990). Shock tube techniques in chemical kinetics. Ann. Rev.Phys. Chem. 41:559-599. Reproduced with the permission of Annual Reviews, Inc.) operated at a power of 70 watts (or 800 K plasma temperature) with 1% H2, the measured profile was highly reversed, as shown in Figure 16.3.2. The dotted line in the figure can be nearly reproduced using Equation (16.3.1) if [ H ] 2 - 1 x 1015 atoms/cm -3 (corresponding to 43% dissociation), [H]I = [H]2(298/800 ) for ll - 6 . 6 cm and l2 -- 1 cm. The theory agrees quite well with experiment; however, the experiment shows somewhat more broadening in the wings. The reason for the above consideration is that most workers use a monochromator to isolate atomic transitions in shock tube studies. Even with fast monochromators, the light intensity entering the slit must be high for photodetection, and this then requires the addition of large amounts of atom source molecules into the plasma. Hence, for H atoms, the condition used in most shock tube work is more similar to Figure 16.3.2 than to the Gaussian limit, Equation (16.3.2). It is then reasonable to ask what absorption sensitivity should be expected with a highly reversed lamp like that shown in the figure. Transmittance through an external absorber (region 3) is calculated from

T-

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(16.3.4)

--OO

Absorbance, ( A B S ) = - ln(I/Io), can be evaluated from Equation (16.3.4) as a function of concentration in the external absorber region for given source

16.3 AtomicResonance Absorption Spectroscopy with Flash or Laser Photolysis

85

i

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FIGURE 16.3.2 Lyman-~line profile for the highly reversed source described in the text........ Calculated from line-absorption theory. (Figure 2 from Lifshitz, A., Skinner, G. B., and Wood, D. R. (1979). Resonance absorption measurements of atom concentrations in reacting gas mixtures. I. Shapes of H and D Lyman-~ lines from microwave sources. J. Chem. Phys. 70:5607-5613. Reproduced with the permission of the American Institute of Physics.) emission profiles and absorber temperatures. If the profile shown in Figure 16.3.2 is used for the source and the absorber temperature is T 3 = 298 K, the effective cross section, O'eff = (ABS)/[H]3I 3, is 1.31 x 10 -16 c m 2 / a t o m -1. This indicates that almost none of the absorber atoms are in resonance with the emitter since the 298 K wavelength spread is located between the wings shown in Figure 16.3.2, where the intensity of emission is small. Similar calculations at T3 = 1600, 1800, 2000, and 2200 K are shown in the top panel of Figure 16.3.3. At all temperatures, Beer's law holds to within +0.5% up to [H]313 = 2 x 1013 a t o m s / c m - 2 ; however, r f increases from 1.70 to 2.45 • 10 -14 c m 2 / a t o m -1 over the T-range, reiterating the important need for devising calibration techniques for determining the curves of growth for a given lamp condition. In contrast to the profile shown in Figure 16.3.2, calculations can be made when the Gaussian profile limit [Equation (16.3.2)] is nearly obtained. The use of a m o n o c h r o m a t o r to isolate the transitions is then impossible since the lamp intensity is so low. In addition to the resonance radiation, spurious emission from other sources might be detectable over the spectral sensitivity range of the

86

J. v. Michael and A. Lifshitz

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FIGURE 16.3.3 Calculations of curves of growth for H atoms using Equation (16.3.4) with reversed and near-Doppler lamp profiles. The upper panel shows calculations with the highly reversed lamp profile of Figure 16.3.2 as a function of absorber temperature. The lower panel shows similar T-dependent calculations but with a near-Doppler lamp profile. ~ 1600 K , - - 1800 K, 2000 K, and . . . . . . 2200 K.

photomultiplier. In this case, the level of resonance radiation must be measured by some other method. Lee, Michael, Payne, and Stief (1978), Miller and Gordon (1983) and Michael, Sutherland, and Klemm (1985) advocate using an external room temperature absorber in front of the resonance lamp to assess the fraction of resonance light in the lamp. Hence, a fast discharge flow reactor that acts as a specific atomic filter is incorporated into the system. The actual plasma temperature in the lamp varies from one end to the other; however, an effective lamp temperature can be determined that gives a profile that can be fitted to a well-defined Doppler width (Maki, Michael, and Sutherland 1985). The effective temperature generally used is 490 K (50 W with substantial air cooling). For H atoms, the concentration of H 2 in the lamp is dictated by the impurity level in ultrahigh-purity grade He (~ 1-2 ppm H2). With a lamp pressure of 2 torr, [/-/]2 ~ 2 x 1011 atoms/cm -3

16.3

Atomic Resonance Absorption Spectroscopy with Flash or Laser Photolysis

87

for lI - - 6 cm and l2 - 4 cm. Equation (16.3.4) can then be evaluated at T3 - - 2 9 8 K for an atomic filter concentration and path length of [ H ] 3 - 2.5 • 1014 atoms/cm -3 and l3 - 4 cm, respectively, and the results show that > 99.6% of the Lyman-0~ radiation is removed. The atomic filter method then allows one to determine the fraction of detected light that is resonance radiation prior to a shock tube experiment. It should also to be noted that for Lyman-~ (and other resonance lines as well), crystal windows generally are used as optical components since these have sharp cutoffs and high transmission in the vacuum ultraviolet (Carver and Mitchell 1964). For Lyman-0~, either LiF (2 > 105 nm) or MgF 2 (2 > 115 nm) windows have been used. Also, additional spectral isolation can sometimes be achieved by passing the light before detection through a gas filter that selectively absorbs wavelengths other than those in resonance. For Lyman-0~, an air filter has been used because there is a deep transmission window in the 02 absorption spectrum that eliminates almost all wavelengths other than 121.6 nm (Lee 1955; Weissler 1956; Barker and Michael 1968). Line absorption calculations can then be carried out with the low opacity source described above for the same absorber temperatures as with reversed source (Figure 16.3.2), namely, 1600, 1800, 2000, and 2200 K. These calculations reveal that the photometer is only ~ 2% less sensitive than the Gaussian limit, Equation (16.3.2). The results are shown in the bottom panel of Figure 16.3.3. At all temperatures, Beer's law holds to within "~ 3% up to [H]313 - - 2 x 1013 atoms/cm-2; however, in contrast to the reversed profile case, the cr~ff values decrease only from 1.17 to 1.07 x 10 -13 cm2/atom -1 as temperature increases from 1600 to 2200 K. The results of Figure 16.3.3 show that the effective cross sections invert at some lamp condition. In the above illustration with the 490-K emitter, the inversion occurs at 50 ppm H atoms in 2 torr He, giving a nearly T-independent cross section of 8.2 x 10 -14 cm2/ atom. The most important point from the demonstration is that the near-Gaussian source is ~ 5 times more sensitive than the reversed source. In kinetics experiments, this gain in sensitivity can be an important advantage in cases where secondary reaction perturbations are possible, and, therefore, the calculations show that the lamp conditions are quite important in experimental design. This is also one reason why comparisons of ARAS photometers from different laboratories are difficult.

16.3.2.3 CONCLUSIONS Generalizations about ARAS photometry can be made by considering the above H-atom demonstration. For any atom, the sensitivity for detection is deter-

88

j. v. Michael and A. Lifshitz

mined by the extent of self-absorption and reversal in the lamp. Deviations from Beer's law are less pronounced with a narrower source. If the precise line profile and oscillator strength of a specific transition are known, the curve of growth can be calculated and absolute atom concentrations can be inferred directly from absorbance measurements. This strategy has been adopted for H(or D-) atom photometry by two research groups (Chiang, Lifshitz, Skinner, and Wood 1979; Maki, Michael, and Sutherland 1985; Lim and Michael 1994). For O atoms, a similar parameterized line-absorption calculational strategy has been used by the same two groups (Pamidimukkala, Lifshitz, Skinner, and Wood 1981; Michael and Lim 1992; Michael, Kumaran, and Su 1999). However, in most applications for other atomic species, the conditions used in a given laboratory depend on equipment configuration and, therefore, it is usually necessary to devise methods for measuring curves of growth.

1 6 . 3 . 2 . 4 CALIBRATION PROCEDURES In cases where a reversed lamp source is used, photometer sensitivity must be determined. For H atoms, at least four methods have been used, all of which rely on unimolecular decomposition processes. In the earliest work (Myerson, Thompson, and Joseph 1965; Myerson and Watt 1968), equilibrium constants for H 2 ~ 2H were used to calculate [H] in thermally equilibrated shock tube experiments at high temperatures. The same method was used by Roth and Just (1975). Appel and Appleton (1974) used the thermal decomposition of N20 under high concentration and high temperature conditions to completely dissociate a small [D2] through the sequence N20 + Ar--~ N 2 + O + Ar, O + D 2 - + OD + D, and O + OD --~ 02 + D. Hence, two D atoms were formed from every one initial D2 molecule. At lower temperatures, the thermal decomposition of hexamethylethane has been used (Chiang, Lifshitz, Skinner, and Wood 1979). Two molecules of t-butyl radicals are formed, and these rapidly dissociate to propene + H, giving two atoms for every molecule decomposed. In more recent times, the thermal decomposition of C2H5I (Herzler and Frank 1992; Kumaran, Su, Lim, and Michael 1996) has also been used at low temperatures since the initially formed C2H5 radicals rapidly decompose to ethylene + H. For O atoms, a quite satisfactory calibration method is the complete thermal dissociation of N20 to give N 2 + O. In this case, calibration is absolutely necessary because in all investigations the lamp sources are partially to strongly reversed. The experiments are unambiguous because quite low levels of N20 can be used and therefore, secondary reactions are too slow to affect the measured O-atom profile. This method has been used by a number of investigators (Roth and Just 1977; Pamidimukkala, Lifshitz, Skinner, and

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Atomic Resonance Absorption Spectroscopy with Flash or Laser Photolysis

89

Wood 1981; Thielen and Roth 1985; Fujii, Sagawai, Sato, Nosaka, and Miyama 1989; Davidson and Hanson 1990a; Michael and Lim 1992; Ross, Sutherland, Kuo, and Klemm 1997). For complete N20 dissociation, the temperature has to be greater than --, 1800 K. However, extrapolations to lower temperatures can be accomplished by fitting the results to parameterized line-absorption calculations (Pamidimukkala, Lifshitz, Skinner, and Wood 1981; Michael and Lim 1992). In agreement with experiment, the calculations suggest that the curve of growth is not strongly dependent on temperature if reversed sources are used. Recently, a procedure based on the H 4 - 0 2 --+ O4-OH reaction has extended the measurements to lower temperatures (Michael, Kumaran, and Su 1999). The lower-temperature lineabsorption calculations are in good agreement with those at higher temperatures, confirming a relatively T-independent curve of growth. For N atoms, lamp profiles are known for a variety of conditions, and lineabsorption calculations have been carried out (Wood, Skinner, and Lifshitz 1987). N-atom calibration has been accomplished by thermally decomposing either NO or N 2 at very high temperatures, allowing the system to equilibrate (Theilen and Roth 1985), and the line-absorption calculations agree well with experiment. Hanson and coworkers (Davidson, Snell, and Hanson 1990; Davidson and Hanson 1990a) have also used the same calibration method. At slightly lower temperatures, these latter authors devised another method: N20 is dissociated in the presence of N2, and the major reactions producing N atoms are then N 2 0 4- M --+ N 2 4- O 4- M, O 4- N 2 - + N O 4- N. C-atom ARAS has been used in kinetics experiments by Mozzhukin, Burmeister, and Roth (1989), Dean, Davidson, and Hanson (1990), and Dean, Davidson, and Hanson (1991a). Shock heating is convenient for releasing free atoms into the gas phase. Almost any atom can be produced and subsequently detected with high sensitivity using an ARAS technique. With thermal decomposition methods, a quite large chemical kinetics database already exists, and new investigations are continuing. Procedures have been and are being devised to measure atomic formation and/or depletion rates in both unimolecular and bimolecular reactive systems. Rate constants are derived from such data, if necessary, using chemical simulation methods (Bowman et al. 1995; Pilling, Turanyi, Hughes, and Clague 1996). Investigations have been carried out on C1 (Kruger and Wagner 1983; Lim and Michael 1993), S (Woiki and Roth 1992; Shiina et al. 1996), Si (Mick, Matsui, and Roth 1993), Br (Takahashi, Inoue, and Inomata 1996; Hranisavljevic, Carroll, Su, and Michael 1998), I (BraunUnkhoff, Frank, and Just 1990; Kumaran, Su, Lim, and Michael 1995; Takahashi, Inoue, and Inomata 1996), Cd (Smirnov, Votintsev, Zaslonko, and Moiseev 1990), and Se atoms (Votintsev, Moiseev, and Smirnov 1989) using ARAS methods. Extension to other atomic systems will undoubtedly occur in

90

j. v. Michael and A. Lifshitz

the future because new data are continually needed for a variety of applications. Many of the more recent studies based on thermal decomposition were referenced in preceding section. Photochemical generation can also be used to release atoms into the gas phase at high-T. As mentioned earlier, the combination of flash or laser photolysis with ARAS detection in a shock tube has been called the FP- or LP-ST technique. This technique and the results obtained with it are reviewed in detail next.

16.3.3 FLASH AND/OR IN SHOCK TUBES

LASER PHOTOLYSIS

The computer methods and codes for high-temperature chemical kinetics modeling in combustion, including flame modeling, are well developed (Kee, Rupley, and Miller 1989; Radhakrishnan and Hindmarsh 1993; Radhakrishnan 1994; Bowman et al. 1995; Pilling, Turanyi, Hughes, and Clague 1996). Sensitivity analysis and optimization techniques have been incorporated into this effort. However, numerical analytical manipulation of a given chemical mechanism (with given rate constants) cannot unambiguously fix a rate constant value for any reaction unless the mechanism is quite small and most of the rate constants are experimentally well determined. In the opinion of the authors, this fact has established a need for the FP- (LP-ST) technique, which allows for unambiguous determination of rate constants for a given isolated chemical reaction. In general, the data are obtained at temperatures that are usually less than those found in flames. Nonetheless, the fact that rate constants are chemically isolated and can be combined with lower and higher T-values then allows for confident extrapolation to flame temperatures. This process may involve the use of modern theories of chemical kinetics. Hence, there are two valuable advances from the method: (a) for use in practical hightemperature chemical modeling and (b) for testing and subsequently further developing modern chemical kinetics theory. Chemical kinetics research with combined flash photolysis and shock tube techniques was first reported by Burns and Hornig (1960). They photodissociated Br2 prior to shock heating and then observed Br2 formation rates as a function of temperature and bath gas from the recombination of Br atoms. The flash photolysis-shock tube (FP-ST) method was first used to measure OH radical kinetics. H20 was photodissociated at 2 > 165 nm in the reflected shock wave regime, and [OH]t was detected by optical absorption (Ernst, Wagner, and Zellner 1978; Niemitz, Wagner, and Zellner 1981). Recognizing that the high detection sensitivity of the ARAS method would offer ideal

16.3 Atomic Resonance Absorption Spectroscopy with Flash or Laser Photolysis

91

conditions for chemical isolation in kinetics studies, Michael, Stttherland, and Klemm (1985) were the first to combine shock tube, flash photolysis, and ARAS techniques. An excimer laser photolysis application in a shock tube (LPST) quickly followed (Davidson, Chang, and Hanson 1988). A schematic diagram of a typical apparatus is s h o w n in Figure 16.3.4. FP-ST experiments are always carried out in reflected shock waves. The reason is clear: The hot gas in the reflected regime is nearly at rest (Michael and Sutherland 1986). The shock tube then serves as a fast furnace to prepare hot gas in a well-defined thermodynamic state that is determined from the initial thermodynamic state of the static gas and the shock velocity. Hence, the FP-ST technique is simply an extension of the well-known kinetic spectroscopic method, where, in this case, the observed species is an atom that is produced as a result of photodissociation.

He Inlet

T DRIVER

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7m FIGURE 16.3.4 Schematic diagram of an LP (FP)-ST apparatus. Pmrotary pump. Dmoil diffusion pump. CT~liquid nitrogen baffle. GV~gate valve. G--bourdon gauge. B--breaker. DP--diaphragm. T--pressure transducers. M--microwave power supply. F--atomic filter. RL-resonance lamp. A--gas and crystal window filter. PM--photomuhiplier. DS--digital oscilloscope. MP~master pulse generator. TR--trigger pulse. DF--differentiator. AD--delayed pulse generator. LT laser trigger. XL~excimer laser. (Figure 4 from Michael, J. V. (1992b). Isotope effects at high temperatures. Isotope Effects in Gas-Phase Chemistry, ACS Syrup. Series 502, J. A. Kaye, ed. pp. 8093, American Chemical Society. Reproduced with the permission of the American Chemical 5ociety.)

92

J. v. Michael and A. Lifshitz

1 6 . 3 . 3 . 1 BIMOLECULAR ATOM-MOLECULE REACTIONS The mostly monatomic gas mixtures used in FP-ST experiments generally contain a small quantity of a source molecule AB, which yields an atomic species A on photodecomposition. AB + hv ~ A + B (16.3.5) A reactant molecule R, which reacts with the atom and removes it from the system, is also present in the mixture. A 4- R ~ Products (16.3.6) The temporal behavior of atomic concentration can then be calculated as d[A]/ dt - - k[R][A] (16.3.7) where k is the bimolecular rate constant for reaction (16.3.6). If [A] ~( [R], then [R] is virtually constant and Equation (16.3.7) can be integrated to give a pseudo-first-order rate law, ln[A]t -- -k[R]t 4- C (16.3.8) The first-order decay constant is the negative slope of a plot of ln[A]t against time, and the bimolecular rate constant is simply the decay constant divided by [R] for the conditions of the experiment. If prior calibration experiments confirm Beer's law behavior (i.e., a linear curve of growth), then (ABS)t = ~reff[A]tl is directly proportional to [A]t and the decay constant can be determined directly from plots of ln(ABS)t against time. In the above analysis, there are three possible complications that can affect the value determined for k. First, Beer's law must hold over the range of absorbances used in the analysis. Second, it is important to predetermine whether both the photolyte source and reactant molecules are thermally stable for the hightemperature conditions of a given experiment. If thermal decomposition of either or both occurs, then atom depletion may not be due to Reaction (16.3.6) but rather to species A, with a product of thermal decomposition. Thermal decomposition then determines the upper temperature limit in a given investigation. Third, the ARAS sensitivity must be roughly known so that secondary reaction complications can be assessed. If the sensitivity is low, then the derived value for k from the simple analysis might be too high. For example, if the pseudo-first-order decay constant for Reaction (16.3.6) is 2500 s -1 and rate constants for A + B and A + Product are both 1 x 10 -10 cm3/molecule-1/s -1, then the analysis based on Equation (16.3.8) will give values that are 3%, 6%, and 28% higher than the input value, 2500s -1, for [A]0 = [B]0 values of 5 • 1011, 1 x 1012, and 5 • 1012 atoms or molecules/cm -3, respectively. The point to this illustration is that high photometer sensitivity is very important in designing unambiguous FP-ST experiments.

16.3 Atomic Resonance Absorption Spectroscopy with Flash or Laser Photolysis

93

The reaction b e t w e e n C1 atoms and D 2 is a typical example of FP- or LP-ST results (Kumaran, Lim, and Michael 1994). The experiments in Ar diluent were straightforward a n d followed the procedures described above. The photolyte was COC12, and its relatively low absorption cross section at 1 9 3 n m required that relatively high concentrations be used even with an excimer laser energy of ~ 250 mJ. At this high [COC12]0, the onset of thermal d e c o m p o s i t i o n giving C1 atoms then d e t e r m i n e d the higher t e m p e r a t u r e limit (1224 K) for these experiments. The top panel of Figure 16.3.5 shows a typical

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Time/ms FIGURE 16.3.5 T5 = 992 K, P5 -----231 torr, P5 "- 2.248 • 10TM cm-3, XCOCI 2 = 63.26 ppm, and XD2 = 634.6 ppm. The top panel shows the Cl-atom transmittance following its formation from photolysis of COC12. The bottom panel shows the corresponding (ABS)cI profile. For (ABS)cl < 0.08, (ABS)c1 is proportional to [CI], and the linear semilog absorption profile then directly yields the pseudo-first-order time constant, k l s t = 2748 s -1. (Figure 5 from Kumaran, S. S., Lim, K. P., and Michael, J. V. (1994). Thermal rate constants for C14-H 2 and C1 4- D2 reactions between 298 and 3000 K. J. Chem. Phys. 101:9487-9498. Reproduced with the permission of the American Institute of Physics.)

94

j. v. Michael and A. Lifshitz

raw data record. In all such experiments, initial (ABS)r 1 was < 0.08. Therefore, the photometer behavior was in the linear portion of the curve of growth (Lim and Michael 1993). Hence, the data were analyzed according to Equation (16.3.8). The result in the top panel of Figure 16.3.5 then yields the semilog decay plot in the lower panel. The observed first-order decay constant from the negative slope is 2750s -1. [D2] in this experiment is 1.43 • 1013 molecules/cm -3, giving a bimolecular rate constant at 992 K of 1.92 x 10 -12 cm3/molecule-1/s -1. Additional D 2 experiments were then carried out as a function of temperature, ultimately resulting in a description of the T-dependence for the C1 + D 2 reaction.

1 6 . 3 . 3 . 2 BIMOLECULAR RADICAL-MOLECULE REACTIONS FP- or LP-ST experiments can also be designed to probe atomic products from a chemical reaction. If species B in Process (16.3.5) is a radical and one desires to measure the rate of B + R where one of the products from Reaction (16.3.6) is an atom X, then the formation of X can be observed using ARAS. [R] must be sufficiently high so that B is 100% converted to X. If Beer's law holds for IX] over the absorbance range used, then the kinetics follow the rate law, ln{[(ABS])o~- (ABS)t]/(ABS)oo} = - k [ R ] t , and the pseudo-first-order buildup constant is k[R]. Division by [R] gives the bimolecular rate constant. The C2H + C2H2--> C4H 2 + H reaction is a specific example of this type of experiment (Shin and Michael 1991a).

16.3.3.3 F L A S H A N D / O R LASER P H O T O L Y S I S S H O C K T U B E RESULTS Even though the technique is relatively recent, there have been a large number of FP- or LP-ST investigations to date. In the paragraphs that follow, the results from several laboratories are reviewed. The Stanford University group--Bowman, Hanson, and Davidsonmhave carried out several LP-ST studies following the first LP-ST study by Davidson, Chang, and Hanson (1988). They measured reaction rates for N + NO -+ N 2 + O and N + H 2 --~ NH + H (Davidson, Snell, and Hanson 1990; Davidson and Hanson 1990a). Using a novel pyrolysis-photolysis method, Davidson and Hanson (1990b) studied the reaction of N + CH 3 -+ H2CN-+-H and found rate constants between 1600 and 2000 K. Two FP-ST studies were performed using C-atom ARAS. In the first, Dean,

16.3 AtomicResonance Absorption Spectroscopy with Flash or Laser Photolysis

95

Davidson, and Hanson (1991a) measured rate constants for the reactions C 4- H2 --+ CH 4- H and C 4- 02 --~ CO 4- O. Dean, Davidson, and Hanson (1991b) then studied the reactions C 4- NO --+ CN 4- O and C 4- NO CO 4- N. Also, Davidson and Hanson (1990c) have determined rate constants for the O 4- H 2 --+ OH 4- H reaction. After the first FP-ST study using the ARAS technique (Michael, Sutherland, and Klemm 1985), several additional studies have been carried out at Brookhaven National Laboratory. Rate constants for the reaction NH 2 4- H 2 ~ H 4- NH 3 have been reported (Michael, Sutherland, and Klemm 1986) in a study where boundary layer corrections have been applied (Michael and Sutherland 1986). A study followed on the O 4- C H 4 --+ C H 3 4 - O H reaction between 760 and 1760 K (Sutherland, Michael, and Klemm 1986). By observing product H atoms, Sutherland and Michael (1988) measured the rate and equilibrium constants for the NH 3 4- H ~ NH 2 4- H 2 reaction. Rate constants for the O 4- H 2 ~ OH 4- H reaction have also been measured using O-atom ARAS (Sutherland et al. 1986). An H-atom FP-ST investigation on the H 4- H20 --+ H 2 4- OH reaction was done by Michael and Sutherland (1988), and this was followed by a study on the H 4- 02 ~ OH 4- O reaction by Pirraglia, Michael, Sutherland, and Klemm 1989). O-atom ARAS studies on the 0 4 - NH 3 --+ NH 2 4-OH (Sutherland, Patterson, and Klemm 1990a), O 4- C 2 H 4 (Klemm, Sutherland, Wickramaaratchi, and Yarwood 1990), and O 4- H20 ~ OH 4- OH (Sutherland, Patterson, and Klemm 1990b) reactions have been completed using the FP-ST technique. The H 4- C H 4 --+ C H 3 4- H 2 reaction rate constant was measured between 897 and 1728 K using H-atom ARAS (Rabinowitz, Sutherland, Patterson, and Klemm 1991). A direct measurement of rate constants for the O 4- NO 4- Ar ~ NO 2 4- Ar reaction was completed over the temperature range of 300 to 1341 K (Yarwood, Sutherland, Wickramaaratchi, and Klemm 1991). Matsui and coworkers have also carried out several investigations using the LP-ST technique. Koshi et al. (1990) studied the reactions N 4- NO ~ N 2 4- O and N 4- H 2 ~ NH 4- H, using the N-atom ARAS m e t h o d . N(4S) atoms were produced by the photolysis of NO. The reactions C2H 4- C2H 2 --+ C4H 2 4- H and C2H 4- H2(D2) --+ C2H2(HD) + H(D) were then studied by observing product H(D) atoms by Fukada, Koshi, and Matsui (1991) and Koshi, Fukada, Kamiya, and Matsui (1992). These workers then determined rate constants for H 4- H2S--+ H 2 4- SH and attempted to explain the non-Arrhenius behavior with CTST (conventional transition state theory) calculations (Yoshimura et al. 1992). Using both H- and O-atom ARAS, the technique was applied to CH4-O 2 mixtures (Ohmori, Yoshimura, Koshi, and Matsui 1992). LP-ST experiments were then carried out on O(3p) with a series of straightchain hydrocarbons and fluoromethanes using O-atom ARAS (Miyoshi, Ohmori, Tsuchiya, and Matsui 1993). Additional O-atom studies were reported

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on the reactions of O(3p) with a series of alkanes (Miyoshi, Ohmori, and Matsui 1993), on the reaction of O 4- H2S (Tsuchiya et al. 1994; Tsuchiya, Matsui, and Dupre 1995), and on the reactions of O 4-selected alkanes (Miyoshi, Tsuchiya, Yamauchi, and Matsui 1994; Miyoshi, Tsuchiya, Tezaki, and Matsui 1995; Miyoshi, Yamauchi, and Matsui 1996). Miyoshi, Yamauchi, and Matsui (1996) attempted to understand the relative reactivities of primary and secondary hydrogen atoms in alkanes. Rate constants for the reaction of O(3p) with Sill 4 were determined by Iida, Koshi, and Matsui (1996). These workers then turned their attention to the reaction of S(3p) atoms with several molecules in a series of three papers (Tsuchiya, Yamashita, Miyoshi, and Matsui 1996; Miyoshi, Shiina, Tsuchiya, and Matsui 1996; Shiina, Miyoshi, and Matsui 1998). To the credit of this group, many of the results from their numerous studies have been combined with lower-temperature work, and CTST theory, based on ab initio determinations of potential energy surfaces, has been applied. Rate constants have been theoretically calculated from room temperature or below to the high temperatures obtainable with shock tubes, and these constants have been compared to experiment. Michael and coworkers have continued using the FP- or LP-ST techniques for measuring bimolecular rate constants. Rate constants for the O + D 2 (Michael 1989) and O 4-C2H2(C2D2) (Michael and Wagner 1990) reactions were measured. Fisher and Michael (1990) studied the reaction D + D20 ~ D 2 4-OD between 1285 and 2261K. They also studied the fundamental reactions D + H 2 (Michael and Fisher 1990) and H + D 2 (Michael 1990; Michael, Fisher, Bowman, and Sun 1990). Lifshitz and Michael (1990) measured rate constants for the O + H 2 0 ~ O H + O H reaction between 1500 and 2400 K. Using both H- and D-atom ARAS, Shin and Michael (1991b) studied the reactions H + 0 2 and D 4-02 and, within experimental error, an isotope effect was not indicated. Similarly, an isotope effect was not found for the reactions C2H 4- C2H 2 and C2D 4- C2D2, nor was there an appreciable T-dependence (Shin and Michael 1991a). O-atom LP-ST studies on the reactions O 4- CH3C1, O 4- CH2C12 and O 4- CHC13 were completed by Ko, Fontijn, Lim, and Michael (1992) and Su et al. (1994). Michael and Lim (1992) studied the reaction N 4- NO ~ N 2 4- O using the LP-ST technique using N-atom ARAS. Lim and Michael (1993) used the pyrolysis-photolysis method to study the reaction O 4- CH 3 at high temperatures. [Much of this work has already been reviewed and compared to other studies (Michael 1991, 1992a, 1992b, 1992c)]. Kumaran, Lim, and Michael (1994) used Cl-atom ARAS to measure rate constants for C1 4- H 2 and D 2. The LP-ST results were combined with higher-temperature measurements, giving values between 699 and 3000 K. H-atom LP-ST results have been obtained for the H 4- CH 2CO and H 4- NO2 reactions (Hranisavljevic, Kumaran, and Michael 1998; Michael 1996).

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16.3.4 SUMMARY Almost all of the FP- or LP-ST work has been carried out in the four laboratories mentioned above. Most of the work has already been considered in previous reviews, and these can be consulted for details along with the original articles Oust 1981a, 1981b; Tsang and Lifshitz 1990; Michael and Lim 1993). Other methods involving ARAS analysis for measuring rate constants of bimolecular reactions are also continuing unabated, with much of the past work referenced above. Since these methods rely on thermal decompositions to produce the transient atomic species to be observed, it is generally true that unimolecular thermal rate behavior must first be studied before these methods can be used for subsequent monitoring of secondary atomic reactions. Hence, the systems are immediately complex; i.e., they involve two or more concurrent chemical reactions. If only one process depletes the atom, the results can be quite accurate. However, it is sometimes difficult to isolate one reaction, and then it is necessary to chemically model a multistep mechanism. If all rate constants in such a mechanism are known except one, then the fitted value can reflect a relatively accurate determination of the unknown rate constant. In most cases, the additional rate constants are not known with the necessary accuracy. Also, since thermal decomposition is the atomic source, the temperature-range over which experiments are performed can be substantially higher than with the FP- or LP-ST method. In the best work, experiments can be carried out systematically from simple to more complex mechanisms, and relatively good rate constants can be obtained hierarchically. In the case of radical detection (Davidson 2000), this modus operandi is about the only method available since the sensitivity for radical detection is substantially lower than with ARAS. This means that the elimination of secondary reactions, even when photolytic generation is available and used, is difficult if not impossible. Because FP- or LP-ST data are obtained under chemical isolation conditions, it is the view of the authors that these data should almost always be given first priority in mechanism-building efforts. Of course, researchers have been modeling high-temperature kinetics systems for decades using well-documented numerical methods for solving coupled first-order differential equations. Attempts at understanding complex reacting systems has become a separate field with its own history of successes. Therefore, researchers in this field sometimes resist attempts to change rate constants even when better measurements are available. There may be at least two causes for this reluctance. First, a set of particularly important macroscopic observations are adopted as being worthy of explanation. Then mechanisms with rate constants are used in simulations to try to explain the chosen observations. Optimization techniques

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may be used to secure the best possible explanation for the entire set of target observations. The optimized quantities are then taken to be the most correct even if certain of the rate processes and/or constants used in the simulation tum out to be wrong and/or inaccurate. This is the strategy used in GRI-Mech 2.11 (Bowman et al. 1995). The second cause is the continuing move toward commercialization of codes, including chemical mechanisms. If computer routines are purchased by industrial users, commercial owners may be quite reluctant to correct their codes even if additional, more accurate information becomes available. In our view, the possibility for effective chemical isolation of a given reaction is then the most important feature of the FP- or LP-ST method. As mentioned previously, the results can be used by theoreticians to develop and sharpen chemical kinetics theory. In general, the strategy involves first considering the potential energy for interaction of the reacting species using ab initio molecular structure calculations. Then some dynamics theory can be applied to calculate the thermal rate behavior. One such method would be CTST. Theory and experiment can then be compared. Success sometimes requires potential energy scaling of both energies and vibration frequencies. However, once the theory correlates with experiment, theory can then be used with substantial confidence to extrapolate to higher temperatures. Sometimes higher- and lower-temperature data do exist, and these can serve to further secure agreement between theory and experiment. The end result of this process is (a) to establish rate behavior at all temperatures (including high-T) for use in chemical modeling of complex mechanisms, and (b) to give advice to theoreticians on the accuracy and viability of their theoretical formalisms and/or theoretically derived quantities. Even though the FP- or LP-ST results are valuable for two important reasons, the implications to theoretical chemical physics have not been fully appreciated by some shock tube chemists. The same lack of appreciation for theory also exists in the tropospheric and stratospheric chemistry field, and the reasons are clear. Kineticists in these fields are only concemed with the practical implications of their results in chemical modeling. There also seems to be a lack of appreciation for chemistry by shock tube fluid dynamicists and aeronautical engineers. If the past is any measure of the future, these physicists will only become interested if and when they start choking on car exhaust and/or smoke stack emissions.

ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, under Contract No. W-31-109-Eng-38.

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REFERENCES Appel, D. and Appleton, J. P. (1974). Shock tube studies of deuterium dissociation and oxidation by atomic resonance absorption spectrophotometry. Proc. 15th Symp. (Int.) on Combustion, pp. 701-714. The Combustion Institute. Barker, J. R. and Michael, J. V. (1968). Experimental estimate of the oscillator strength of the 2P3/2,1/2 +__2S1/2 transition of the hydrogen atom. J. Opt. Soc. Am. 58:1615-1620. Bethe, H. and Salpeter, E. E. (1977). Quantum mechanics of one- and two-electron atoms, Plenum, New York. Bowman, C. T., Hanson, R. K., Davidson, D. E, Gardiner, Jr., W. C., Lissianski, V., Smith, G. P., Golden, D. M., Frenklach, M., and Goldenberg, M. (1995). GR/MechmAn optimized detailed chemical reaction mechanism for methane combustion. GRI-Mech 2.11. http://www.me.berkeley. edu/gri_mech/. Braun, W., Bass, A. M., and Davis, D. D. (1970). Experimental test of a two-layer model characterizing emission line profiles. J. Opt. Soc. Am. 60:166-170. Braun, W. and Carrington, T. (1969). Line emission sources for concentration measurements and photochemistry. J. Quant. Spectrosc. Radiat. Transfer 9:1133-1143. Braun-Unkhoff, M., Frank, P., and Just, T. (1990). High-temperature reactions of benzyl radicals. Ber. Bunsen-Ges. Phys. Chem. 94:1417-1425. Burns, G. and Homig, D. E (1960). A combined flash photolysis and shock wave method for the study of bromine atom recombination over a wide temperature range. Can. J. Chem. 38:17021713. Carver, J. H. and Mitchell, P. (1964). Ionization chambers for the vacuum ultra-violet. J. Sci. Inst. 41:555-557. Chiang, C.-C., Lifshitz, A., Skinner, G. B., and Wood, D. R. (1979). Resonance absorption measurements of atom concentrations in reacting gas mixtures. II. Calibration of microwave sources over a wide temperature range. J. Chem. Phys. 70:5614-5622. Davidson, D. E (2000). Spectroscopy. In Handbook of Shock Waves, G. Ben-Dor, O. Igra, T. Elperin, eds. Sec. 5.2, Part I. Academic Press, Burlington, MA. Davidson, D. E, Chang, A. Y., and Hanson, R. K. (1988). Laser photolysis shock tube for combustion studies. Proc. 22nd Symp. (Int.) on Combustion, pp. 1877-1885. The Combustion Institute. Davidson, D. E and Hanson, R. K. (1990a). High temperature reaction rate coefficients derived from N-atom ARAS measurements and excimer photolysis of NO. Int. J. Chem. Kinet. 22:843861. Davidson, D. E and Hanson, R. K. (1990b). Shock tube measurements of the rate coefficient for N + CH 3 --+ H 2CN 4- H using N-atom ARAS and excimer photolysis of NO. Proc. 23rd Symp. (Int.) on Combustion, pp. 267-273. The Combustion Institute. Davidson, D. E and Hanson, R. K. (1990c). A direct comparison of shock tube photolysis and pyrolysis methods in the determination of the rate coefficient for 0 4- H 2 --~ OH 4- H. Combust. and Flame 82:445-447. Davidson, D. E and Hanson, R. K. (1991). A shock tube study of reactions of C atoms with H 2 and 02 using excimer photolysis of C30 2 and C atom atomic resonance absorption spectroscopy. J. Phys. Chem. 95:183-191. Davidson, D. E, Snell, D. C., and Hanson, R. K. (1990). Shock-tube excimer photolysis and the measurement of N atom kinetic rates. Proc. 17th Int. Symp. on Shock Waves and Shock Tubes: Current Topics in Shock Waves, AlP Conf. Proc. 208, Y. W. Kim, ed., pp. 525-530. Am. Inst. of Physics. Davis, D. D. and Braun, W. (1968). Intense vacuum ultraviolet atomic line sources. Applied Optics 7:2071-2074.

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Dean, A. J., Davidson, D. E, and Hanson, R. K. (1990). C-atom ARAS diagnostic for shock tube kinetics studies. Proc. 17th Int. Symp. on Shock Waves and Shock Tubes: Current Topics in Shock Waves, AlP Conf. Proc. 208, Y. W Kim, ed., pp. 537-542. Am. Inst. of Physics. Dean, A. J., Davidson, D. E, and Hanson, R. K. (1991a). A shock tube study of reactions of C atoms with H 2 and 02 using excimer photolysis of C302 and C atom atomic resonance absorption spectroscopy. J. Phys. Chem. 95:183-191. Dean, A. J., Davidson, D. E, and Hanson, R. K. (1991b). A shock tube study of reactions of C atoms and CH with NO including product channel measurements. J. Phys. Chem. 95:3180-3189. Ernst, J., Wagner, H. G., and Zellner, R. (1978). A combined flash photolysis/shock wave study of the absolute rate constants for reactions of the hydroxyl radical with methane and trifluoromethane around 1350 K. Ber. Bunsen-Ges. Phys. Chem. 82:409-414. Fisher, J. R. and Michael, J. V. (1990). Rate constants for the reaction, D 4- D20 -+ D2 4- OD, by the flash photolysis-shock tube technique over the temperature range 1285-2261 K: Results for the back-reaction and a comparison to the protonated case. J. Phys. Chem. 94:2465-2471. Fujii, N., Sagawai, S., Sato, T., Nosaka, Y., and Miyama, H. (1989). Study of the thermal dissociation of N20 and CO2 using O(3P) atomic resonance absorption spectroscopy. J. Phys. Chem. 93:5474-5478. Fukuda, K., Koshi, M., and Matsui, H. (1991). Studies on the reactions: C2H 4- C2H2 ~ C4H2 4- H and C2H 4- H 2 ~ C2H2 4- H. Preprints, 202nd ACS Natl. Meet., Symp. Combust. Chem., Div. Fuel Chem. 36:1392-1399. Herzler, J. and Frank, P. (1992). High temperature reactions of phenylacetylene. Ber. Bunsen-Ges. Phys. Chem. 69:1333-1338. Hranisavljevic, J., Carroll, J. J., Su, M.-C., and Michael, J. V. (1998). Thermal decomposition of CF3Br using Br-atom absorption. Int. J. Chem. Kinet. 30:859-867. Hranisavljevic, J., Kumaran, S. S., and Michael, J. V. (1998). H 4- CH2CO --~ CH3 4- CO: A high pressure chemical activation reaction with positive barrier. Proc. 27th Symp. (Int.) on Combustion, pp. 159-166. The Combustion Institute. Iida, D., Koshi, M., and Matsui, H. (1996). Reaction of silane with atomic oxygen at high temperatures. Isr. J. Chem. 36:285-291. Just, T. (1981a). Chemical kinetic studies by vacuum UV spectroscopy in shock tubes. Shock Tubes and Shock Waves, Proc. 13th Int. Symp. on Shock Tubes and Waves: C. E. Treanor and J. G. Hall, eds., pp. 54-68. SUNY Press. Just, T. (1981b). Atomic resonance absorption spectrometry in shock tubes. In Shock Waves in Chemistry, A. Lifshitz, ed., pp. 279-318, Marcel Dekker, New York. Kee, R. J., Rupley, E M., and Miller, J. A. (1989). "Chemkin-II: A fortran chemical kinetics package for the analysis of gas-phase chemical kinetics." Report SAND89-8009; Sandia National Laboratories, Livermore, CA. Klemm, R. B., Sutherland, J. W., Wickramaaratchi, M. S., and Yarwood, G. (1990). Flash photolysisshock tube kinetic study of the reaction of atomic O(3p) with ethylene: 1052 K< T < 2284 K. J. Phys. Chem. 94:3354-3357. Ko, T., Fontijn, A., Lira, K. P., and Michael, J. V. (1992). A kinetics study of the O(3P) 4- CH3C1 reaction over the 556-1485 K range by the HTP and LP-ST techniques. Proc. 24th Symp. (Int.) on Combustion, pp. 735-742. Koshi, M., Fukada, K., Kamiya, K., and Matsui, H. (1992). Temperature dependence of the rate constants for the reactions of C2H with C2H2, H 2, and D2. J. Phys. Chem. 96:9839-9843. Koshi, M., Yoshimura, M., Fukuda, K., Matsui, H., Saito, K., Watanabe, M., Imamura, A., and Chen, C. (1990). Reactions of N(4S) atoms with NO and H 2. J. Chem. Phys. 93:8703-8708. KrUger, B. C. and Wagner, H. G. (1983). Measurement of absolute chlorine atom concentrations behind reflected shock waves. Proc. 14th Int. Symp. on Shock Tubes and Waves, R. E. Archer and B. E. Milton, eds., pp. 738-743. Sydney Shock Tube Symp. Publishers.

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Kumaran, S. S., Lim, K. P., and Michael, J. v. (1994). Thermal rate constants for the C14- H 2 and C1 4- D2 reactions between 296 and 3000 K. J. Chem. Phys. 101: 9487-9498. Kumaran, S. S., Su, M.-C., Lim, K. P., and Michael, J. V. (1995). Thermal decomposition of CF3I using I-atom absorption. Chem. Phys. Lett. 243:59-63. Kumaran, S. S., Su, M.-C., Lim, K. P., and Michael, J. V. (1996). The thermal decomposition of C2HsI. Proc. 26th Syrup. (Int.) on Combustion, pp. 605-611. The Combustion Institute. Ladenburg, R. and Reiche, F (1913). c~ber selektive absorption. Ann. d. Phys. 42:181-209. Lee, J. H., Michael, J. v., Payne, W. A., and Stief, L. J. (1978). Absolute rate of the reaction of N(4S) with NO from 196-400K with the DF-RF and FP-RF techniques. J. Chem. Phys. 69:3069-3076. Lee, P. (1955). Photodissociation and photoionization of oxygen (02) as inferred from measured absorption coefficients. J. Opt. Soc. Am. 45:703-709. Lifshitz, A., Bidani, M., and Carroll, H. E (1981a). The effect of minute quantities of impurities on shock tube kinetics. The reaction H 2 4- D 2 --~ 2HD. Proc. 13th Int. Syrup. on Shock Tubes and Waves: Shock Tubes and Shock Waves, C. E. Treanor and J. Hall, eds., pp. 602-609, SUNY Press. Lifshitz, A., Bidani, M., and Carroll, H. F (1981b). Vacuum uv window system for shock tubes bakeable to high temperature. Rev. Sci. Inst. 52:622-624. Lifshitz, A., Bidani, M., and Carroll, H. F (1983). The reaction H 2 4- D2 ~ 2HD. A long history of erroneous interpretation of shock tube results. J. Chem. Phys. 79:2742-2747. Lifshitz, A. and Michael, J. V. (1990). Rate constants for the reaction, O 4- H2O -~ OH 4- OH, over the temperature range, 1500-2400 K, by the flash photolysis-shock tube technique: Further consideration of the back reaction. Proc. 23rd Syrup. (Int.) on Combustion, pp. 59-67. The Combustion Institute. Lifshitz, A., Skinner, G. B., and Wood, D. R. (1979). Resonance absorption measurements of atom concentrations in reacting gas mixtures. I. Shapes of H and D Lyman-0~ lines from microwave sources J. Chem. Phys. 70:5607-5613. Lim, K. P. and Michael, J. v. (1993). The thermal decomposition of CH3C1 using the Cl-atom absorption method and the bimolecular rate constant for O 4- CH 3 (1609-2002 K) with a pyrolysis photolysis-shock tube technique. J. Chem. Phys. 98:3919-3928. Lim, K. P. and Michael, J. V. (1994). The thermal reactions of CH 3. Proc. 25th Syrup. (Int.) on Combustion, pp. 713-719. The Combustion Institute. Lynch, K. P., Schwab, T. C., and Michael, J. V. (1976). Lyman-~ absorption photometry at high pressure and atom density. Kinetic results for H recombination. Int. J. Chem. Kinet. 8:651-671. Maki, R. G., Michael, J. V., and Sutherland, J. W. (1985). Lyman-~ photometry: Curve of growth determination, comparison to theoretical oscillator strength, and line absorption calculations at high temperature. J. Phys. Chem. 89:4815-4821. Mallard, W. G., Westley, E, Herron, J. T., and Hampson, R. F (1994). "NIST chemical kinetics database--Ver. 6.0." NIST Standard Reference Data, Gaithersburg, MD. Michael, J. V. (1989). Rate constants for the reaction O 4- D 2 --~ OD 4- D by the flash photolysisshock tube technique over the temperature range 825-2487 K: The H 2 to D 2 isotope effect. J. Chem. Phys. 90:189-198. Michael, J. V. (1990). Rate constants for the reaction, H 4- D 2 -~ HD4- D, over the temperature range, 724-2061 K, by the flash photolysis-shock tube technique. J. Chem. Phys. 92:3394-3402. Michael, J. V. (1991). Thermal rate constant measurements by the flash or laser photolysis-shock tube method: Results for the oxidations of H 2 and D2, Preprints, 202nd ACS Natl. Meet., Syrup. Combust. Chem., Div. Fuel Chem. 36:1563-1570. Michael, J. V. (1992a). The measurement of thermal bimolecular rate constants by the flash photolysis-shock tube (FP-ST) technique: Comparison of experiment to theory. In Advances in Chemical Kinetics and Dynamics, vol 1. pp. 47-112. JAI.

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Ross, S. K., Sutherland, J. W., Kuo, S. C., and Klemm, R. B. (1997). Rate constants for the thermal dissociation of N20 and the O(3p) + N20 reaction. J. Phys. Chem. A101:1104-1116. Roth, P. and Just, T. (1975). Atom-Resonanzabsorptionsmessungen beim Thermischen Zerfall von Methan hinter Stosswellen. Ber. Bunsen-Ges. Phys. Chem. 79:682-686. Roth, P. and Just, T. (1977). Atomic absorption measurements on the kinetics of the reaction methane + atomic oxygen---~methyl+hydroxyl in the temperature range 1500 < T < 2250 K. Ber. Bunsen-Ges. Phys. Chem. 81:572-577. Shiina, H., Miyoshi, A., and Matsui, H. (1998). Investigation on the insertion channel in the S(3p) + H 2 reaction. J. Phys. Chem. 102:3556-3559. Shiina, H., Oya, M., Yamashita, K., Miyoshi, A., and Matsui, H. (1996). Kinetic studies on the pyrolysis of H2S. J. Phys. Chem. 100:2136-2140. Shin, K. S. and Michael, J. V. (1991a). Rate constants (296-1700K) for the reactions C2H 4- C2H 2 ~ C4H 2 4- H and C2D + C2D 2 ~ C4D 2 4- D. J. Phys. Chem. 95:5864-5869. Shin, K. S. and Michael, J. V. (1991b). Rate constants for the reactions, H + 0 2 ~ OH + O and D + 0 2 ~ OD 4-O, over the temperature range 1085-2277K by the flash photolysis-shock tube technique. J. Chem. Phys. 95:262-273. Smirnov, V. N., Votintsev, V. N., Zaslonko, I. S., and Moiseev, A. N. (1990). Kinetics of the thermal decomposition of dimethylcadmium. Kinet. Katal. 31:1041-1045. Su, M.-C., Lim, K. P., Michael, J. V., Hranisavljevic, J., Xun, Y. M., and Fontijn, A. (1994). Kinetics studies of the O(3p) 4- CHzC12 and CHC13 reactions over the 468-1355 and 499-1090 K ranges using two techniques. J. Phys. Chem. 98:8411-8418. Sutherland, J. w. and Michael, J. V. (1988). The kinetics and thermodynamics of the reaction H 4- NH 3 ~- NH 2 4- H 2 by the flash photolysis-shock tube technique: Determination of the equilibrium constant, the rate constant for the back reaction, and the enthalpy of formation of the amidogen radical. J. Chem. Phys. 88:830-834. Sutherland, J. W., Michael, J. V., and Klemm, R. B. (1986). Rate constant for the O(3p) 4- CH 4 --~ CH 3 4- OH reaction obtained by the flash photolysis-shock tube technique over the temperature range 760 < T < 1755 K. J. Phys. Chem. 90:5941-5945. Sutherland, J. W., Michael, J. V., Pirraglia, A. N., Nesbitt, E L., and Klemm, R. B. (1986). Rate constant for the reaction of O(3p) with H 2 by the flash photolysis-shock tube and flash photolysis-resonance fluorescence techniques: 504 < T < 2495 K. Proc. 21st Symp. (Int.) on Combustion, pp. 929-941. The Combustion Institute. Sutherland, J. W., Patterson, P. M., and Klemm, R. B. (1990a). Flash photolysis-shock tube kinetic investigation of the reaction of O(3p) atoms with ammonia. J. Phys. Chem. 94:2471-2475. Sutherland, J. W., Patterson, P. M., and Klemm, R. B. (1990b). Rate constants for the reaction system O(3p)4- H20 ~ OH 4- OH over the temperature range 1053 K to 2033 K using two direct techniques. Proc. 23rd Symp. (Int.) on Combustion, pp. 51-57. The Combustion Institute. Takahashi, K., Inoue, A., and Inomata, T. (1996). Direct measurements of rate coefficients for thermal decomposition of methyl halides using shock-tube ARAS technique. Proc. 20th Int. Symp. on Shock Waves, B. Sturtevant, J. E. Shepard, and H. G. Hornung, eds., pp. 959-964. World Scientific. Thielen, K. and Roth, P. (1985). Resonance absorption measurements of N and O atoms in high temperature NO dissociation and formation kinetics. Proc. 20th Symp. (Int.) on Combustion, pp. 685-693. The Combustion Institute. Tsang, W. and Lifshitz, A. (1990). Shock tube techniques in chemical kinetics. Ann. Rev. Phys. Chem. 41:559-599. Tsuchiya, K., Matsui, H., and Dupre, G. (1995). High temperature reaction of O(3p) 4- H2S. Proc. 19th Int. Symp. on Shock Tubes and Waves, Brun, Raymond, Cumitrescu, and Lucien, eds., pp. 71-76. Springer.

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Atomic Resonance Absorption Spectroscopy with Flash or Laser Photolysis

105

Tsuchiya, K., Yamashita, K., Miyoshi, A., and Matsui, H. (1996). Studies on the reactions of atomic sulfur (3p) with H 2, D 2, CH4, C2H 6, C3H8, n-C4H10, and i-C4H10. J. Phys. Chem. 100:17202-17206. Tsuchiya, K., Yokoyama, K., Matsui, H., Oya, M., and Dupre, G. (1994). Reaction mechanism of atomic oxygen with hydrogen sulfide at high temperature. J. Phys. Chem. 98:8419-8423. Votintsev, V. N., Moiseev, A. N., and Smirnov, V. N. (1989). Decomposition of hydrogen selenide in shock waves. Kinet. Katal. 30:225-226. Watt, W. S. and Myerson, A. L. (1969). Atom formation rates behind shock waves in oxygen. J. Chem. Phys. 51:1638-1643. Weissler, G. L. (1956). Photoionization in gases and photoelectric emission from solids. In Handbuch der Physik, vol. 21, S. Flugge, ed., pp. 304-382. Springer. Woiki, D. and Roth, P. (1992). Shock tube measurements on the thermal decomposition of COS. Ber. Bunsen-Ges. Phys. Chem. 96:1347-1352. Wood, D. R., Skinner, G. B., and Lifshitz, A. (1987). Measurement and modeling of the nitrogen resonance line profiles from an electrodeless discharge lamp. J. Chem. Phys. 87:5092-5096. Yarwood, G., Sutherland, J. W., Wickramaaratchi, M. A., and Klemm, R. B. (1991). Direct rate constant measurements for the reaction O 4- NO 4- Ar --~ NO 2 + Ar at 300-1341 K. J. Phys. Chem. 95:8771-8775. Yoshimura, M., Koshi, M., Matsui, H., Kamiya, K., and Umeyama, H. (1992). Non-Arrhenius temperature dependence of the rate constant for the H + H2S reaction. Chem. Phys. Lett. 189:199-204.

CHAPTER

16.4

Chemical and Combustion Kinetics 16.4

Single-Pulse Shock Tube

WING TSANG National Institute of Standards and Technology, Gaithersburg, MD 20899 ASSA LIFSHITZ Department of Physical Chemistry, Hebrew University of Jerusalem, Jerusalem, 91904, Israel

16.4.1 Introduction 16.4.2 The Single-Pulse Shock Tube 16.4.2.1 Configuration 16.4.2.2 Requirements 16.4.2.3 Limitations 16.4.2.4 Validation 16.4.3 Chemical Kinetics 16.4.3.1 General Considerations 16.4.3.2 Analytical Methods 16.4.3.3 Treatment of Data 16.4.3.4 Experimental Approaches 16.4.4 Complex Reaction Systems 16.4.4.1 Introduction 16.4.4.2 Determination of Reaction Mechanisms 16.4.4.3 Computer Simulation 16.4.5 Single-Reaction Studies 16.4.5.1 Justification 16.4.5.2 Experimental Configurations 16.4.5.3 Internal Standards and the Comparative Rate Technique 16.4.6 Specific Systems and Generalizations 16.4.6.1 Complex Reactions 16.4.6.2 Single-Step Kinetics 16.4.7 Summary and Future Directions References Appendix: Summary of Experimental Results Handbook of Shock Waves, Volume 3 Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISBN: 0-12-086433-9/$35.00

107

108 16.4.1

w. Tsang and A. Lifshitz INTRODUCTION

The basic characteristic of shock tubes is the pulse nature of the heating phenomenon. Studies on the chemistry of a reacting system obviously require some means of determining the concentration of reactants, products, and intermediates during the course of the reaction. In cases where complicated molecules have many reaction pathways, the analytical capabilities set the constraints on the value of the data obtained. In shock tube research, there have been two approaches to the analysis of the constituents of reacting systems. The first and most obvious is to make concentration measurements as a function of time immediately upon the passage of the shock wave. Since the time scale of shock tube studies are in the microsecond to low millisecond range, this generally means the use of optical techniques with spectroscopic detection of the molecules of interest. This sets limitations on the number of components that can be detected simultaneously (usually one or two) and the concentrations necessary for detection. This opens issues regarding the appropriate reaction pathways. For simple systems such as the decomposition of hydrogen molecules, this is not a problem: Obviously, the only channel for the decomposition of the hydrogen molecule is the formation of atoms. However, even here the assumption is that excited electronic states are not important. As the molecules of interest become more complex, uncertainties increase regarding the true significance of the real-time measurements on the one or two species that can be detected. It is here that single-pulse shock tube studies, with their capability of detecting the whole range of end products, fill a particular niche. Much of this chapter will deal with the use of the single-pulse shock tube to obtain information on the kinetics of decomposition of polyatomic molecules. One of the consequences of generating high temperatures is that polyatomic molecules can be decomposed. In the course of the process new molecules are created and, in the case of oxidation, the latent chemical energy in molecules is released through combustion. An understanding of such processes is an important direction in chemical research. The results are of vital importance for a host of technologies and natural phenomena that are, in general, centered around combustion and related processes. The detailed understanding of the chemistry in combination with modern computational techniques give to technologists a unique tool that can expand and indeed take the place of expensive and frequently uncertain physical testing. The key role of singlepulse shock tube studies is to provide the experimental database for use in simulations and in the testing and validation of predictive theories. A large amount of data have now been accumulated (Tsang 1981; Tsang and Lifshitz 1990) and reviewed. It is clear that for gathering information on the high-

16.4 Single-PulseShock Tube

109

temperature gas-phase behavior of complicated polyatomic molecules, singlepulse shock tube studies are one of the preferred tools. Nevertheless, for the unraveling of the complexities of such phenomena, the ideal situation is to simultaneously detect unstable intermediates and all final products. It is probably only by using all available tools that quantitative chemical kinetic results regarding rates and mechanisms can be unambiguously obtained. For the determination of reaction mechanisms, the important factors are effects on the yields of intermediates and products as a result of variations in the reactants and the physical and chemical environments. Traditionally, this has meant studying reactions in static or flow systems, quenching the reacting mixture and then subjecting it to detailed analysis. Unstable intermediates are not directly detected. Nevertheless, through changes in the reaction conditions much can be inferred. This is a particularly fruitful approach in the context of the tremendous advances in modern analytical capabilities since reaction conditions can be varied over enormous ranges and products at extremely low concentrations can be detected. Single-pulse shock tube experiments represent extensions of the classical static studies with the added feature of short heating time and no possible contributions from surface reactions. This chapter is divided into six main sections. The first deals with operational details and the basic physical phenomena. Evidence is provided to validate the general procedure. The second section contains a detailed discussion of how experiments are carried out and the type of kinetic problems that are accessible through single-pulse shock tube studies. The third section discusses kinetics studies, where the emphasis is on the understanding of the mechanisms and rate constants for the decomposition reactions of a particular molecule. The fourth section covers experiments concentrated on determining the rate expressions for a single reaction. Specific experiments and an outline of what has been learned is covered in the fifth section, followed by a brief summary and a discussion of possible future directions.

1 6 . 4 . 2 THE S I N G L E - P U L S E S H O C K TUBE

16.4.2.1 CONFIGURATION Figure 16.4.1 contains a schematic of the typical configuration of the singlepulse shock tube that is generally used at the present time (Klepeis 1961). Also included is a wave diagram that illustrates the physical phenomena. The various regions of interest are (1) the test gas at its initial configuration, (2) the gas behind the incident shock, (3) the driver gas immediately behind the diaphragm, (4) the initial driver gas, and (5) the test gas behind the reflected

110

w. Tsang and A. Lifshitz

shock. Also included are the particle paths (dotted lines), which are parallel to the driver and driven gas interface. The basic steps are (i)

the breaking of a diaphragm, leading to the formation of a shock wave; its passage through the test gas (region 1), resulting in a temperature and pressure step; (ii) the reflection of the shock wave from the end wall, leading to another temperature and pressure step; and (iii) the interaction of the shock wave with the interface, leading to the formation of an expansion wave cooling the shocked gas. During steps (ii) and (iii) the driver gas is being sucked into the dump tank, which initially was at the same pressure as the test gas. Ultimately the reflected shock is swallowed. The consequence is that the test gas feels a well-defined heating pulse. Due to the exponential dependence of rate constants on temperature, the temperature pulse behind the incident shock wave has no effect. A reproduction of a typical pressure trace is given in Fig. 16.4.2. Note the close correspondence with the idealized sketch in Fig. 16.4.1.

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4

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~2 L r r

| L

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3

4

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FIGURE 16.4.2 Photographof typical pressure trace.

The operation of the single-pulse shock tube is straightforward. It involves preparing the required mixture in a standard vacuum system, filling the dump tank and test section to the desired pressure, pressurizing the driver section, and breaking the diaphragm. Immediately thereafter a sample is removed for analysis. Since the volume of the dump tank is at least 1 order of magnitude larger than the test section, it is usually filled with the pure driven gas (argon, without the test gas). Thus a valve between the two sections is necessary. In recent years, to extend the range of molecules to those with lower vapor pressures, single-pulse shock tubes have been heated. It has been found that a temperature in the 100 to 200~ range is adequate for most purposes. Temperature uniformity is maintained within 1 or 2~ Although this does increase the complexity of the shock tube, no special problems have been encountered. The single-pulse shock tube described here is a variant of the original design described by Glick et al. (1955). The original design involved the dump tank being placed at the high-pressure end of the shock tube and maintained at vacuum. The composition of the driver gas was set so that the shock passed smoothly through the interface. Cooling was effected by the expansion wave generated from the dump tank. Thus it was necessary to properly time the breaking of the two diaphragms and to set a particular composition of the driver gas, a "tailored interface," so that the reflected gas passed smoothly across the interface. These operations are not necessary in the variant used today. Nevertheless, in the original form, the gas dynamics are particularly well described in terms of simple one-dimensional theory. In contrast, the actual behavior of the dump tank at the side of the test section has never been completely characterized. Confidence that this does not introduce significant errors into the data analysis is based on the consistency of the results that has

112

w. Tsang and A. Lifshitz

been obtained from such studies. However, this is not completely satisfactory and must be carefully considered, particularly if one is to add real-time detection to single-pulse shock tube studies.

16.4.2.2 R E Q U I R E M E N T S Probably the key physical effect for single-pulse shock tube work is the interaction of the shock wave with the driver-driven gas interface. For the generation of a rarefaction wave, the condition is a3 73- 1

a2 72- 1

where a is the speed of sound and 7 is the specific heat ratio. The subscripts are as given in Fig. 16.4.1. Since the temperature behind the incident shock must be higher than that behind the interface, the condition for quenching can only be achieved with a light driver gas such as hydrogen or helium and heavier driven gases such as argon or krypton. This expansion wave will then cool the heated gas, leading to the single-pulse feature. For hydrogen drivers and argon test mixtures, cooling rates as large as 1 million degrees per second can be achieved. The validity of the general procedure can be readily established by examination of the pressure history from such experiments (see Fig. 16.4.2). The dump tank prevents multiple reflections of the shock. Here again examination of the pressure traces guarantees that the process is operating in the desired manner. Actually, due to the high activation energy of most of the reactions that have been studied, the attenuation of the pulses ensures that none of the secondary pulses are anywhere near the temperature of the initial pulse. The real importance of the dump tank is probably to ensure that mixing between the driver and shocked gas is minimized. This is achieved by moving the driver and test gas interface far away from the sampling port. The authors have found that if the gas analyzer is directly connected to the shock tube and samples are taken within a few seconds after running the shock, losses of samples through mixing do not occur since material balance of better than 95% is easily achieved. However, this must be checked. As will be seen shortly, in cases where this is not achieved corrections must be made. The physical conditions that can be generated in a single-pulse shock tube are determined by the length, pressure, and temperature ratings (if it is heated) of the shock tube, subject only to the constraints of the relation given above. Usually this means a heating time of several hundred microseconds to milliseconds, pressures of a few to hundreds of bars and temperatures as high as 2000 K. This fills a very important niche between real-time shock tube

16.4

Single-Pulse Shock Tube

113

measurements and the turbulent flow reactors that are now widely used in combustion research (Linteris et al. 1991).

16.4.2.3 LIMITATIONS Due to the necessity of working in the reflected shock region, nonidealities will lead to temperatures considerably different than those determined from shock velocity measurements that assume ideal gas dynamic relations. The general situation has been discussed in detail by Michael and Sutherland (1986) and other authors Belford and Strehlow (1969). The physical situation is the buildup of the boundary layer behind the incident shock and its subsequent interaction with the reflected shock. It is known that these effects are minimized when heavy rare gases are used as the principal component of the test gas. Although it would seem possible to correct for such effects, an important uncertainty is whether the boundary layer is laminar or turbulent. A serious problem is the introduction of temperature gradients into the system, which would make the single-pulse shock tube no longer an isothermal reactor. However, this is probably more serious for real-time measurements than for single-pulse shock tube studies. Indeed, for single-pulse shock tube work this problem is largely eliminated through the use of an "average temperature". All the reactions in the mixture are occurring at this temperature. (We will discuss how this property can be utilized in a subsequent section). Ironically, these problems coupled with the wide range of unexpected phenomena first observed in shock tube experiments Tsang (1981) were in fact detrimental to the acceptance of the shock robe as a kinetics tool. In retrospect, it is apparent that the extension of the temperature range by shock tube experiments has led to a great increase in our understanding of the quantitative details of chemical kinetics. It is now clear that the chemistry can be used to infer properties of the shocked gas and hence to calibrate the reaction conditions. Thus, while it is important to be aware of the problems, there are a variety of essentially chemical means of amelioration. The increasing acceptance of single-pulse shock tube and indeed of all shock tube results has been largely due to the steady accumulation of self-consistent data and greater understanding of the problems intrinsic to high-temperature chemistry. Although the uncertainty in temperature should be borne in mind when considering single-pulse shock tube data, this problem should be kept in perspective. Generally speaking, it is difficult to carry out experiments at high temperatures. The alternative experimental approaches are studies in flow or static reactors. In both cases, heating is by heat transfer from hot walls.

114

w. Tsang and A. Lifshitz

Considerable care must be taken to make sure that the actual reaction time is as determined by flow and thermocouple measurements. Past experience has indicated that investigators using classical techniques have usually treated such issues without necessary care and this has led to considerable errors in the literature. This is in contrast to the situation with shock tube studies, where, ironically, awareness of the physical problems has led to an underestimation of the validity of results. In any kinetics experiment, the time required to achieve the desired temperature must be considerably shorter than that of the time-dependent phenomenon of interest. This sets a definite limit on the type of reactions that can be studied using any method. The temperatures that can be interrogated classically are considerably lower than those in shock tube experiments. Thus certain reaction channels are only accessible through the latter. Furthermore, the dependence on heating from hot walls obviously means that the reacting gas must at some time or other make contact with the wall. Thus the possibility of surface reactions must always be considered in classical studies. In contrast, in shock tube studies the heating is the result of the passage of the shock wave and no hot walls are required. Indeed, the walls of the shock tube are cold and hence much less reactive. Furthermore, the short reaction time (in comparison to classical methods) sets definite limits on how many reactive species can move into the main gas stream. Thus surface effects can be eliminated from consideration. All reactions must therefore originate from one or a series of gas phase processes. The determination of such pathways in a quantitative manner is one of the main contributions of single-pulse shock tube studies. The reaction time in single-pulse shock tube experiments is a function of the entire length of the tube and the relative length of the low- and highpressure sections. Since this is not readily changed, measurements in most experiments are all made at one particular time. This can be restrictive since for a given temperature one cannot carry out reactions to any desired extent of reaction by varying the time. As a result, at very low temperatures it is necessary to rely on product formation. The results may therefore be affected by trace impurities. The technique is obviously restricted to the determination of the concentration of all the stable products, and thus all possibilities of determining mechanisms and rate constants on the basis of the temporal behavior of the reactants and products are lost. On the other hand, the prospect of determining a large number of products at a particular time represents a powerful capability. The short residence time means that the operation must be at much higher temperatures. Thus chain lengths are generally shorter than in classical situations. This is not only due to the short reaction time; the higher radical concentrations naturally make the termination reactions more important. Finally, the temperatures used in single-pulse shock tube are very close to those actually found in combustion and pyrolytic processes. Thus extensive extrapolations are no longer needed for practical applications.

16.4

115

Single-Pulse Shock Tube

16.4.2.4 VALIDATION The ultimate validation of any method is through comparison of the results with those from other techniques. Present-day confidence in the results of the single-pulse shock tube method basically comes from such comparisons as well as from the internal consistency of the data. Generally speaking, it appears that the predictions from fluid dynamics considerations are much more pessimistic than would be warranted by the current results, particularly where some means of internally calibrating the behavior of system is used. Figures 16.4.3a-c contain results from single-pulse shock tube, static, and flow experiments. The reactions selected involve direct formation of stable products and are thus particularly straightforward in the context of single-pulse shock tube studies. The results bear particularly on the issues dealing with nonideal behavior described earlier. Even the most cursory examination of the data will reveal that the single-pulse shock tube studies produce results that are at least

A = diethoxymethane t

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al = 1.58x1013e~(-23350/T) (Bigley and Wren, 1972) a2 = 4.05x1013exp(-23752/T) (Bigley and Wren, 1972) a3 = 7.94x1013exp(-23148/T)(Gordon and Norris, 1965) a4 = 1.15x1013exp(-23454/T) (Cross et al, 1976) a5 = 1.07x1013exp(-23300/T) (Herzler et al, 1997) FIGURE 16.4.3a Rate constants for decomposition of diethoxymethane to form ethylene + C2HsOCH2OH. The initial listing refers to static and flow experiments. Those on the bottom are from shock tube studies. The expressions in brackets are shock tube determinations with temperature from shock velocities. Others are from comparative rate studies. The dotted lines are extrapolations from comparative rate studies.

W. Tsang and A. Lifshitz

116

B = t-butylchloride

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1000/T bl = 7.gx1012exp(-20933/T) (Asahina and Onozuka, 1964) b2 = 5x1013exp(-22594/T) (Maccoll and Wong, 1968) b3 = 5.9x1013exp(-22665/T) (Failes and Stimson,1962) b4 = 8.5x1013exp(-22997/T) (Heydtmann et a1,1975) b5 = 1.9x1014exp(-23148/T) (Brearly et al, 1936) b6 = 2.5x1012exp(-20833/T) (Barton and Onyon, 1949) b7 = 7xl 013exp(_22488/T) (Tsang, 1964b,a) [b8 = 7.9x1013exp(-23248/T) (Tsang, 1964c)] FIGURE 16.4.3b Rateconstants for decomposition of t-butyl chloride to form isobutene 4- HCI. The initial listing to refers to static and flow experiments. Those on the bottom are from shock tube studies. The expressions in brackets are shock tube determinations with temperature from shock velocities. Others are from comparative rate studies. The dotted lines are extrapolations from comparative rate studies. as good as those from more standard techniques and that the classical studies have the problems described earlier. Hence it can be concluded that uncertainties in rate constants arising from fluid dynamic effects are rarely more than a factor of 1.5 and should not be reflected in the activation energies by more than a few percent. Finally, it will be noted that the single-pulse shock tube and classical studies span a total rate constant range of close to 10 orders of magnitude. In the case of the flow experiments, the differences in temperatures are between 100 to 200 ~. Since single-pulse shock tube studies are particularly suitable for studies on the decomposition of larger molecules, an important problem is frequently their low vapor pressure at room temperature. To increase the vapor pressure of the compounds in question and to prevent their adsorption on cold surfaces, the shock tube, the gas-handling manifold, the GC injection unit, and all the

16.4

117

Single-Pulse Shock Tube

C = cyclohexene

/ QII

0

v

C2 ~

-

r

,6

0.8

i

!

0.9

1.0

. . . . . . . . . . . . . . . . . . . . . . . . i

i

i

1.1

1.2

1.3

i

~1 .

1.4

1.5

IO00/T cl = 1.4x1017exp(-36584/T) (Smith and Gordon, 1961) c2 = 1.2x1012exp(-27727/T) (Kraus et al, 1957) c3 = 8.9 xl012 exp(-28935/T) (Kuchler,1939) c4 = 1.45x1015exp(-33313/T) (Uchiyama et al, 1964) [c5 = 1.45 x1015exp(-32955/T) (Bamard and Parrott,1976)] c6 = 1.8 x 1015exp(-33002/T) (Newman et al, 1980) c7 = 1.4 x1015exp(-33500/T) (Tsang,1973) [c8 = 1.5 xl015 exp(-33666/T) (Hidaka et al, 1984)] [c9 = 2.5x1015exp(-33716/T) (Skinner et al, 1981)] FIGURE 16.4.3c Rate constants for decomposition of cyclohexene to form 1,3-butadiene + ethylene. The initial listing refers to static and flow experiments. Those on the bottom are from shock tube studies. The expressions in brackets are shock tube determinations with temperature from shock velocities. Others are from comparative rate studies. The dotted lines are extrapolations from comparative rate studies.

other parts of the system must be heated. Heating to temperatures of 100 to 200~ depending on the compound to be studied, is in most cases enough. The heating is normally computer controlled and the temperature is kept constant to within +2~

16.4.3 CHEMICAL KINETICS 16.4.3.1 GENERAL CONSIDERATIONS Kinetics studies can be divided into two parts" the determination of a reaction mechanism and the assignment of individual rate constants. If the mechanistic

118

w. Tsang and A. Lifshitz

results are ambiguous, one literally does not know what time-dependent phenomena is being measured. For mechanistic determinations, highly accurate time-dependent measurements are not really needed; the key aim is the identification of reaction products under a wide variety of physical and chemical conditions. Due to the wealth of new instrumentation that facilitates quantitative measurements of a few species, this aspect of kinetics is often forgotten. Nevertheless, when one is faced with the issue of the decomposition of a complex molecule, this problem cannot be neglected. In this context, the capability of single-pulse shock tube studies to detect all products formed in the course of particular reaction with well-established analytical methodology represents a powerful mechanistic tool, especially when certain products are not detected. Obviously, if a molecule is a characteristic product of a certain pathway, its absence is the strongest evidence for the unimportance of such a reaction channel. However, such a finding is not completely unambiguous; the molecule could have been destroyed in the course of the work. From a mechanistic point of view, not having to worry about surfacegenerated processes is a tremendous advantage. Surface processes are much less understood than those occurring in the gas phase. If such processes make contributions, it will usually be necessary to consider transport phenomena. Obviously, products from surface reaction must diffuse to the main part of the reacting gas mixture. Single-pulse shock tube studies are completely analogous to static reactor investigations but are mechanistically less complicated. The added advantage, as will be developed below, is the extremely short heating time. The main complication from shock tube experiments is that of deciding whether products are formed through a series of radical-induced decomposition, or directly as a consequence of a single unimolecular or a series of unimolecular processes, or a combination of all the possibilities. Particularly if one wishes to obtain quantitative results, this remains a formidable task. Nevertheless, there are a variety of means m t h r o u g h the addition of inhibitors or initiators or varying reaction conditions over large ranges--for resolving such ambiguities. These techniques are especially pertinent for single-pulse shock tube applications, where concentrations can be easily varied over wide ranges. For example, the addition of sufficiently large quantities of radical inhibitor can often alter the nature of the reaction products or drastically lower rate constants. The invariance of the yields of a particular product in the presence of radical inhibitors gives very satisfactory evidence for the involvement of a direct unimolecular reaction channel. The addition of a source of radicals can indicate what compounds arise from radical-induced decomposition. In making decisions regarding mechanisms, increasing knowledge on the nature of chemical reactivity in the gas phase also plays an important role. It

16.4

Single-Pulse Shock Tube

119

should be noted that with a complex molecule it is not really feasible to experimentally eliminate all possible reaction channels. Instead, recourse is made to the body of past kinetic and thermodynamic properties that indicate which are the most likely or unlikely pathways. The role of the experiments is then to differentiate between the various possibilities. All of these issues are strongly influenced by the analytical methodology and the treatment of the data.

1 6 . 4 . 3 . 2 ANALYTICAL METHODS 16.4.3.2.1 Gas Chromatography: Determination of Product Concentrations The most commonly used analytical tool in the operation of the single-pulse shock tube is the gas chromatograph, using a variety of detectors such as FID (flame ionization detector), NPD (nitrogen/phosphorus detector), MSD (mass selective detector), TCD (thermal conductivity detector) and others, depending on the compounds to be analyzed. Whereas peaks are normally identified by their retention times on various gas chromatographic columns, the help of an MSD is very often needed for ultimate identification. Gas chromatograms of postshock mixtures contain peaks corresponding to the various products obtained as a result of shock heating and what is left of the reactant. However, to determine extent of reaction, the peak area of the reactant behind the reflected wave prior to shock heating must be known. There are in principle two methods by which this information can be obtained. In one method, the peak area is determined separately in a chromatogram of the unshocked sample and the peak height is then normalized to the pressure at which the postshock sample is introduced into the gas chromatograph. Since, however, there are changes in the sensitivity of the GC detectors from one chromatogram to another and gas originating from the driver section can be mixed with the postshock gas, such a procedure can introduce considerable error and scatter in the data. In the second method, the determination of the product concentrations and the reactant in each experiment is based on a single chromatogram. The peak area of the reactant prior to shock heating is calculated from the sum of the normalized peak areas of the reactant and products. This method is based on the assumption of a complete mass balance. This assumption, which is not always correct, must be verified. In cases where it does not exist, an error may be introduced.

120

w. Tsang and A. Lifshitz

The concentrations of the reaction products C5(pri) using the single chromatogram method are calculated from their GC peak areas from the following relations (Lifshitz et al. 1987a): Cs(Pri) = A(Pri)t/S(Pri) x (Cs(reactant)o/A(reactant)o) Cs(reactant)0 = Pl • %(reactant) • (ps/Pl)/lOORT1 A(reactant)0 -- A(reactant)t + 1/n c ~

N(pri) x A(Pri)t/S(pri)

In these relations n c is the number of carbon atoms in the reactant molecule, C5(reactant)0 is the concentration of the reactant behind the reflected shock prior to decomposition, and A(reactant)0 is the calculated GC peak area of the reactant prior to decomposition, where A(pr/) t is the peak area of a product i in the shocked sample, S(pr/) is its sensitivity relative to the reactant, and N(pr/) is the number of its carbon atoms. Ps/PI is the compression behind the reflected shock and T 1 is the temperature of the shock tube. The typical gas chromatograms shown in Fig. 16.4.4 were taken from a study on the decomposition of 4-methyl pyrimidine. The GC analysis was performed on two detectors, an FID (upper trace) for all carbon-containing products and an NPD (lower trace) for compounds containing C - N bonds, for which the latter is much more sensitive (see the caption for peak identification). The chromatograms taken on the two detectors are combined to one chromatogram using the reactant peak or a peak of a high-concentration product as a standard. 16.4.3.2.2 Hidden Peaks: GC-MS There are cases where reaction products cannot be separated on the GC columns and assistance of a gas chromatography-mass spectrometry (GC-MS) becomes necessary. A problem of this nature and its solution can be seen in the study of the thermal decomposition of 2-methylfuran (kifshitz et al. 1997b). In this study the reaction products 1-butyne and 1,2-butadiene could not be separated from the large peak of vinylacetylene (C4H4) and were hidden behind it. These two C4H 6 isomers were discovered in a series of experiments using GC-MS. In Fig. 16.4.5, such chromatogramsmwith m/z 54, 53, 39, and 27, which are characteristic to C4H6 isomers, and m/z 52, characteristic to C4H4m are shown. As can be seen, there are two peaks, slightly separated, of C4H6 hidden behind a large peak of vinylacetylene. These two peaks were identified as 1,2-butadiene and 1-butyne. The identification was based on the relative heights of m/z 39 and 54, which differ considerably in these two isomers (Royal Society of Chemistry 1991; Stein 1998). This method, which uses the SIM (selected-ion monitoring) mode of the GC-MS, has been used successfully in other studies as well (kifshitz et al. 1998, unpublished d).

16.4

121

Single-Pulse Shock Tube /J

140

(s)

-(5)

i

>

E

105

(1)

)

1

~

(201

-

| c: ~

70

i

1 (2)

FID 1

Ii

(2)

35 (!) 13.

1 I

23a4 I

t

I

I

I

b I

1

I

0

I

56

I

t

!

[

I

!

I

1

10

1

|

1

m

I

I

1

I

I

I

1 /J

20

30

/I

I

1 a I

40

I

I

50

I

I

J

60

Retention. time (min) .

250

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

, ;,//it"

(2)

200

(41

.c:

(i)

NPD

1

(4)

100

:40)

jL_ 0

2,,

! 3 4 5,

6 ,

7, ,

10 t

15

20

25

11 !

30

I

,/j,,,,

35

45

12 55

Retention time (min) FIGURE 16.4.4 Gas chromatograms of a postshock mixture of 4-methylpyrimidine taken on FID md NPD detectors. The numbers on the peaks indicate multiplication factors. The reactant was ;ubjected to shock temperature of 1296 K. The list of products is as follows: 1. FID: 1, methane; 2, ethylene; 3, ethane; 4, acetylene; 5, allene; 6, propyne; 7, hydrogen cyanide; 8, cyanoacetylene; 9, acetonitrile; 10, acrylonitrile; 11, propylnitrile; 12, (cis, trans) crotonitrile, vinylacetonitrile; 13, pyrimidine; 14, pyrazine; 15, methylpyrimidine; (a) and (b) are the peaks of the internal standard: (a) 1,1-difluoroethylene, (b) 1,1,1-trifluoroethane 2. NPD: 1, cyanogen; 2, hydrogen cyanide; 3, cyanoacetylene; 4, acetonitrile; 5, acrylonitrile; 6, propylnitrile; 7, cis crotonitrile; 8, vinylacetonitrile; 9, trans-crotonitrile; 10, pyrimidine; 11, pyrazine; 12, methylpyrimidine.

W. Tsang and A. Lifshitz

122

120

~100

m r

~

C4H4 (m/z)=52

!!i 'i

'1 ~3IBUtadlene

/A\ /

12 2- Butyne/~

10

80

m

-

i

|

/AI

c m

] t ~ It 1 , 2 - B u t a d i e n e

o 60

[

. . . .

27

i i

,',\1

~ 4O |

line m/z 54 . . . . . 53

/"

/ ~'' ,"" \)!!,:,1 ,Z

/c~,

~

/

~ 1-Butyne

ij}'

?0

'

O. !

17.0

17.5

18.0

18.5

19.0

19.5

Retention time, min FIGURE 16.4.5 GC-MSchromatogram using the selected-ion monitoring mode of the mass selective detector. Peaks of 1,2-butadiene and 1-butyneare hidden behind a large peak of C4H4and can be analyzed only with the assistance of the GC-MS. (See Color Plate 1).

16.4.3.3

T R E A T M E N T OF DATA

16.4.3.3.1 Product Distribution The experimental product distribution in the postshock mixtures is the most important piece of information obtained in the study of any reaction system. This information is essentially the concentration (mole percent) of the products at different temperatures. It is obtained from the experimental gas chromatograms, an example of which was shown in Fig. 16.4.4 for the decomposition of 4-methyl pyrimidine. The method by which the concentration of each product is evaluated from the areas under its corresponding peak in the gas chromatogram was discussed in the preceding section. The product distribution obtained in the study on the decomposition of tetrahydrofuran is shown, as an example, in Fig. 16.4.6 in the form of mole percent vs temperature (Lifshitz et al. 1986a). As will be discussed later, plots of mole percent vs temperature form the basis for computer simulation of the overall decomposition process.

16.4.3.3.2

Arrhenius

Parameters

Another way of presenting the experimental results is an Arrhenius plot of the production rates of the various decomposition products. Figures 16.4.7 and

16.4

Single-Pulse Shock Tube

123

100 ! . . . . . |

t0

.~\~

~

:

,

,

9

0.25% THF in argon ,0, = ! 00 Torr CO

"~

i

<" "

L, , ~

.............."~.~: . ........

. .........

.............

/",,

"'

. .:..., .~............

L.;H

":" S "

"a--~-"~"~='~'~.

"

,."/< ,5'/4>' "a',<...-----f ././* C4H2 C4H8

O, 1 v / ; ' / '17'/ !....<./,/, .

1:100

~

I

1200

~

......7,,; 7. _

1300

1400

1500

T (K) FIGURE 16.4.6 Color Plate 2).

Product distribution in the decomposition of shock-heated tetrahydrofuran. (See

C2H s

102

_I

i

4., 101 0 13 0 =,_

ii

10 o

l

/og [,4, s-1] = 13.94 E = 61.1 k c a l / m o l

10-1 0.8

0.9

1.0

1.1

1 0 0 0 I T (K -1) FIGURE 16.4.7 An Arrhenius plot of log(klst) vs 1/T for the production rate of ethane in the decomposition of propylene oxide. The slope of the line and the preexponential factor do not correspond to a true first-order rate constant.

124

W. Tsang and A. Lifshitz

10 2

9

10~ -7"

Acetone

lO 0 ","t,

U~ 10.1

~ m S

i9

o

= 1 0 .2

"o 0

o.

0. 3

10 .4

Flowers (1977) ; ~ ~ ,

I 0 -s

k= 1 10 -e 0.8

|

|

7xl |

|

i

1.0

|

|

i,,

|

1.2

,

|

|

|

|

1.4

I

1.6

1000/T (K -1) FIGURE 16.4.8 A plot of log(klst) vs lIT for the isomerization rate of propylene oxide--~ acetone. Results obtained at lower temperatures are shown for comparison.

16.4.8 show Arrhenius plots for the formation rate constants (kproduct) of two products, ethane and acetone, taken from the study of the decomposition of propylene oxide (Lifshitz and Tamburu 1994). The products were obtained by shock heating a mixture of 0.3% propylene oxide in argon. For acetone, the rate constant obtained at lower temperatures using a different experimental technique is also shown for comparison (Flowers 1977). Values of Arrhenius temperature dependencies Ea for product formation, as obtained from the slopes of these plots and their corresponding preexponential factors, are given as an example for the decomposition of propylene oxide in Table 16.4.1. The rate constants are expressed as first-order rate constants and are calculated for each product from the relation kproduc t - -

[product]t [reactant] ~ - [reactant]t • ktotal

where ktota 1 is calculated using the relation ktota1 - - ln{[reactant]t/[reactant]o}/t In this specific system, there are four isomerization products macetone, propanal, methyl-vinyl ether, and allyl alcohol m and several decomposition products as a result of shock heating (Lifshitz and Tamburu 1994). The Arrhenius parameters representing true rate constants are only those for

16.4

125

Single-Pulse Shock Tube

TABLE 16.4.1 First-Order Arrhenius Parameters for Product Formation in the Decomposition of Propylene Oxide

Product

logA (s -1)

E (kJ/mol)

k (S -1 ) at (950 K)

T (K)

Total decomposition Propanal Acetone Allyl alcohol Methyl-vinyl-ether Carbon monoxide Methane Ethane Ethylene Acetylene Propylene 4- propane Allene Propyne Acetaldehyde

14.00 14.26 14.00 12.90 13.21 16.57 14.21 13.94 16.50 15.44 13.20 14.94 14.27 12.72

236 245 251 239 246 306 272 256 301 313 249 328 305 254

10.1 6.35 1.66 0.581 0.483 0.567 0.181 0.562 0.865 1.62 • 10 -2 0.326 7.62 x 10 -4 3.33 x 10 -3 5.71 • 10 -2

835-1165 835-1165 835-1165 925-1110 855-1110 950-1175 915-1200 915-1170 875-1150 915-1170 905-1175 1020-1215 1035-1215 925-1170

products that are formed by unimolecular reactions, that is, directly from the reactant. However, this holds only for the range where the decomposition is relatively small. As can be seen in Fig. 16.4.8, the Arrhenius curve bends at high temperatures due to massive decomposition of the product resulting from free radical attack. The parameters for the formation of the other products, ethane for example, do not represent elementary unimolecular reactions because of the involvement of free radical reactions in their production. Also, the rate parameters for the total decomposition, although they are presented as parameters of a first-order rate constant, do not imply that the total decomposition obeys a first-order dependence. This presentation simply provides a convenient way to summarize general rates for product formation and reactant disappearance and can also form another basis for computer modeling.

16.4.3.3.3 Mass Balance Since the method by which species' concentrations are determined from their GC peak areas assumes a mass balance, and this does not always exist, it is important to examine the validity of such an assumption. The mass balance problem arises from possible loss of products caused by adsorption of material on the shock tube walls or on the injection system on the way to the gas chromatograph. To minimize adsorption, the shock tube and the gas injection

126

w. Tsang and A. Lifshitz

system are heated, as has been mentioned earlier, but the problem still exists. The adsorption on the walls is particularly severe where high-boiling-point products or products containing active functional groups are concerned. The possible loss of material was examined, for example, in the study of the thermal reactions of quinoline and isoquinoline (Laskin and Lifshitz 1998). In several tests, 0.3% xenon was added to the reaction mixture as a marker and ratios of [quinoline]/[xenon] were measured by GC-MS in the unshocked mixture and in several shocked samples. By comparing these ratios in the shocked and in the unshocked samples, it was possible to determine the relative loss of the reactant in each test. This relative loss was compared to the corresponding values calculated by the sum ( 1 / 9 ) ~ N(Pri) x A(Pri)t/S(Pri) , which represents the concentrations of all the products normalized by the number of their carbon atoms (1/9 is the ratio of nitrogen to carbon in quinoline) (see Section 16.4.3.2.1). These comparisons are shown graphically in Fig. 16.4.9. Except for two tests, the ratios are practically the same, showing scatter in both directions. In this particular study, there was no significant loss of products in the analyses. Another type of a mass balance check involves the determination of the concentration of carbon atoms relative to another atom in the postshock mixtures. For example, in the study of the decomposition of 2,3-dimethylox-

100

r

~

~

from Xenon m of p r o d u c t s

0

t_ 80

I

cl. e'"-- 60

I

_o e-" 40

20

ft. /.-----;

1515 1560 1570 1600 1640 1725 1750 1750 2000

Temperature, K FIGURE 16.4.9 Depletionof quinoline as a result of shock heating, calculated from the sum of the products (1/9) ~ N(pri) x A(Pri)t/S(Pri) and from the ratio [quinoline]t/[xenon] in comparison to the ratio [quinoline]o/[xenon].

16.4

127

Single-Pulse Shock Tube 100

10

..r 0 pq

0.1

0.01 0.01

0.1

1

10

100

0.25xZ(no,~[C~]) FIGURE 16.4.10

Oxygen-carbon balance among the products in the decomposition of 1,2-

dimethyl-oxiran.

iran (Lifshitz and Tamburu 1995) the question was to what extent the ratio [O]/[C] among the decomposition products in the postshock mixture retains the original ratio of the reactant molecule. The mass balance of oxygen vs carbon is shown in Fig. 16.4.10. The concentrations of the oxygen-containing species are plotted against one-fourth of the sum of the concentrations of all the decomposition products, each multiplied by the number of its carbon atoms. The 45 ~ line in the figure represents a complete oxygen-carbon balance. Within the limit of the experimental scatter, there is no deviation in this study from a complete mass balance.

1 6 . 4 . 3 . 4 EXPERIMENTAL APPROACHES Single-pulse shock tube studies can be divided into two broad classifications. We will call them multiple- and single-reaction studies. The first consists of studies of the decomposition of a compound by itself or in a reactive media at a convenient concentration, and then attempting to unravel the detailed chemistry through chemical kinetics modeling with the aim of fitting all of the :oncentration profiles. The second focuses on the determination of the rate

128

w. Tsang and A. Lifshitz

expression of a particular reaction. Here the aim is to carry out studies under conditions where one particular reaction is predominant and then to determine highly accurate rate constants and expressions for the particular process. The two types of experiments are in a sense complementary. The first type, where a variety of reactions occur, enables one to identify the key reactions in a process. This is closer to real-world situations and through modeling gives considerable information about the reaction process. Particularly useful are the results where the reaction conditions are varied since these frequently can confirm the postulated mechanism. With such results, one can then carry out studies under conditions where a particular process is the predominant reaction and determine rate expressions without mechanistic ambiguity. In the best situation, this can be visualized as an iterative process since one can now feed the highly accurate data into the results and make further adjustments to refine the data. In a number of cases the mechanism may be so straightforward so that such studies merge into the second category. It should be emphasized that with surface reactions removed from consideration, all single-pulse shock tube observations must be accounted for by gas-phase reactions. Thus it is necessary to account for all species on the basis of increasingly well-established principles. In the following sections we will discuss each of these two approaches in detail.

16.4.4

COMPLEX

16.4.4.1

REACTION

SYSTEMS

INTRODUCTION

By complex reaction systems we refer, in the context of this section, to thermal reactions of organic molecules whose decomposition mechanisms involve many elementary reactions such as unimolecular initiations, chain propagations, unimolecular decomposition of unstable intermediates, terminations, and others. Among the free radical reactions we can find abstractions, dissociative and nondissociative recombinations, radical attachments, and so on. Due to the complexity of these systems, we cannot always determine rate parameters of a single elementary reaction. More often, the only information that can be obtained is simply the product distribution in the postshock mixtures as a function of temperature and concentration. In most cases, we find many products and thus have to devise quite complicated mechanisms to understand the thermal reactions of these molecules. To obtain a quantitative understanding and verification of a mechanism, we must compose a kinetics scheme containing a large number of species and elementary reactions and run computer experiments under the same initial conditions as in the laboratory.

16.4

Single-Pulse Shock Tube

129

The kinetics scheme is numerically integrated, and the computed product distribution is then compared to the one obtained in the laboratory. An agreement between the two will support the overall reaction mechanism that led to the construction of the scheme. When a reaction mechanism has been suggested, a considerable amount of information is still needed before a computer simulation can be performed. This includes Arrhenius rate parameters for the elementary reactions in the scheme and the thermochemical properties of the species involved. These properties can be obtained from existing databases or can be estimated if such databases are unavailable. We will review in this section the type of experimental data that are obtained in the study of complex reaction systems using the single-pulse shock tube technique, as well as the mode of data reduction. We will also discuss what experiments should be performed to determine whether a given reaction product is obtained by a unimolecular process or by consecutive free radical reactions. In addition, we will discuss the issue of how to compose complex reaction schemes for computer modeling, what databases can be used for kinetic and thermochemical information, and how to perform the modeling and the sensitivity analysis. In the course of the presentation of these issues, we will show examples from a variety of studies and discuss briefly a number of specific systems.

16.4.4.2 D E T E R M I N A T I O N OF R E A C T I O N MECHANISMS 16.4.4.2.1 Isotope Labeling Among the molecules of interest, some are available commercially or can be synthesized in isotopic form and can thus be used as labeled reactants. The determination of the isotope distribution in the products when labeled reactants are being used can shed light on the detailed mechanisms in a complex reaction system. For example, the distinction between unimolecular and multimolecular chain reactions can be made by mixing fully hydrogenated with fully deuterated reactants, generating a shock wave, and looking for scrambling in the products. If the reaction is carried out at temperatures where isotope exchange reactions do not play an important role, fully hydrogenated or fully deuterated products will indicate a single skeleton origin, or unimolecular process. Isotope scrambling among the products clearly indicates a multimolecular reaction. Reactants labeled in specific locations in the molecule can provide clues on specific bond cleavage in the reactant molecules. This type of analysis can be seen, as an example, in the study on the decomposition

130

w. Tsang and A. Lifshitz

of tetrahydrofuran (Lifshitz et al. 1986a). In this study ethylene was found to be a major reaction product in the pyrolysis and as such it is probably a primary product. This assumption, however, was verified by shocking a mixture containing 1% tetrahydrofuran and 1% tetrahydrofuran-d8 in argon, and examining the isotope distribution in the ethylene in the postshock mixture. If the only ethylenes formed are C2H4 and C2D 4 and no scrambled products are found, then the ethylene retains the composition of the original skeleton and is it thus being formed directly via ring cleavage. At high temperatures some isotope exchange may occur, but at the low-temperature end of the range this should be minimal. Figure 16.4.11 shows a mass spectrum of the ethylenes in the postshock mixture, taken with a high-resolution mass spectrometer. In Fig. 16.4.11a, the raw spectrum as obtained in the GC-MS analysis is shown. It contains parent ions, as well as daughter ions arising from the cracking pattern of the parents. Several ions in these two groups overlap since an extremely high resolution power of the mass spectrometer is required to separate them. Common and even high-resolution mass spectrometers cannot separate, for example, the C2D2 + ions from C2H4 + at m/z 28. The first is a daughter ion of C2D4 +, whereas C2H4 + is a parent. Since, however, the mass spectra of both C2D4 +

lOO -

(a)

28

80-

>, t~ e-

6O

27 32

40 30

t-

20

tO

o

.>

26

29

25 "11

I,

i

,

I, I,

|

31 |,

[C2HJI[C2DJ=1.65 80-

|

(b)

28

.,..,. t...,=

60-

C2D

C2H4

E

40

C2DH 3

20 0

24

29 |

27 I

26

4

32

1

28

, 30

C2DaH 31 I 32

m/z FIGURE 16.4.11 Isotopic distribution of ethylene in postshock mixture of 50% THF and 50% THF-d8 diluted in argon. The upper diagram (a) shows the original GC-MSspectrum. In the lower diagram (b) only the parent ions are left. The intense peaks of C2H4 and C2D4 verify the assumption that the ethylene molecule retains the skeleton of the original tetrahydrofuran molecule.

131

16.4 Single-Pulse Shock Tube

and C2H4 + are known, one can remove the daughter ions from the spectrum and leave only the parents. In Fig. 16.4.11b, the spectrum contains only parent ions after removing all the peaks that belong to the daughter ions and peaks at m/z 28 and 32 coming from the air background. Separating N2 and 02 from the spectrum also requires a high-resolution mass spectrometer; however, the mass difference between C2H 4 and N2 and between C2D 4 and O2 is much larger than that of C2D2+ and C2H4. As can be seen, the peak at m/z 30 that is present in Fig. 16.4.11a disappears completely in Fig. 16.4.11b since it corresponds to the species C2D3+ (62% of the parent C2D4+). Some residues of m/z 29 and 31 are still present due to either small isotope exchange or some deviation from the published cracking pattern, which is somewhat instrument dependent. These results clearly show that both ethylenes preserve the original skeleton of the tetrahydrofuran and are thus formed by unimolecular ring cleavage and not by free radical reactions. After establishing the fact that ethylene is formed directly by ring cleavage, it still must be determined from what locations in the ring it is eliminated: It can be formed via elimination from tetrahydrofuran at the 2-3 (or 5-4) positions or from the 3-4 positions.

OH2

)~

OH2 > OHm- CH~ + (CH2-CH2-O)

C . ~ ~

CH=

CH2

> C H2= CH2 + (CH2-O-CH2)

To clarify which channel is operative, a mixture containing 3,3,4,4-tetrahydrofuran-d4 in argon was shock heated. Elimination of ethylene from the 3-4 positions leads to the production of CD2--CD2

OH=

OH=

.~ C D2= CD2 * (CH~,-O-CH2)

whereas elimination from the 2-3 (or 5-4) positions leads to the production of a scrambled ethylene, CD2--CH2.

Cu/~4~-~3~ D2 CH~/,5~ I ' ~ C H ~

> CD2- CH2+ (CD2-CH2-O)

132

W. Tsang and A. Lifshitz

loo F

30

(a)

sof E

28

60.

i

e-

40

.E

2(1

e,o (I) >

0

.w,

27

29

26 l,

31

,L

I,

'

[C2D2H2]I[C2D2] ~ 2.2

30

8(1

ID L_

32

-

,

/,

(b)

C2D2H2

617

32

40

20

C2DsH C2D4 27 ,

0 24

'

26

29

|

,

31

II

28

30

32

34

m/z FIGURE 16.4.12 Isotopic distribution of ethylene in postshock mixture of 3,3,4,4-tetrahydrofuran-d4 in argon. The upper diagram (a) shows the original GC-MS spectrum. The lower diagram (b) shows that only the parent ions are left. The intense peaks of C2D4 and C2H2D2 show that ethylene is formed by elimination from both 2-3 (4-5) and 3-4 positions.

The results of these experiments are shown in Fig. 16.4.12. In Fig. 16.4.12a, the original GC-MS spectrum is shown. In Fig. 16.4.12b, the spectrum after removing the air background and the peaks resulting from the cracking pattern of C2D4 + and C2D2H2+ is shown. As can be seen, both molecules are present, which means that both channels take place. The ratio of CD2=CH2 to CD2=CD2 is roughly 2:1, indicating that their rates are practically identical except for a statistical factor of two as there are two ways to produce CH2=CD2 (2-3 and 4-5) and only one way to produce CD2=CD2 (3-4). 16.4.4.2.2 Free Radical Scavengers An additional method to assess the extent of free radical involvement in the production of a specific product is the use of free radical scavengers. The most commonly used scavenger in the single-pulse shock tube research is toluene.

+R" ....

---->

+RH

133

16.4 Single-PulseShock Tube

Its function as a free radical scavenger is based on the very high stability of the benzyl radical that is formed by abstraction of a hydrogen atom from its methyl group by an active radical such as H ~ OH ~ CH3 ~ etc. The benzyl radical, which is a very stable and unreactive species, does not react at all in the time frame of the single-pulse shock tube regime. At high temperatures, however, the decomposition of toluene shown here might add free radicals to the system so that there is a temperature limit in the use of this particular scavenger.

OH3

[~~"

Toluene has been used as a scavenger in several single-pulse shock tube studies. Typical examples are the studies of the thermal decomposition of isoxazole (Lifshitz and Wohlfeiler 1992a) and 5-methylisoxazole (Lifshitz and Wohlfeiler 1992b). These molecules are five-membered ring heterocyclic compounds with nitrogen and oxygen as heteroatoms. c.

Their thermal decomposition was studied in the temperature range of 8501100 K, where the decomposition of toluene in the time scale of 1-2 ms is negligible. In both cases the main products in the decomposition are nitriles, acetonitrile (CH3CN) in isoxazole and ethylnitrile (C2HsCN) in 5-methylisoxazole. As major products of the decomposition, it can be assumed that they are formed directly by ring cleavage (either by one or two steps).

HC~c

CH3CN+ CO

C2HsCN+ CO However, to examine the extent of free radical involvement in the formation rate of these two products, mixtures containing a large excess of toluene {[toluene]/[isoxazole] ~ 10 and {[toluene]/[5- methylisoxazole] ,~ 10} were prepared and the results were compared to runs without toluene. As can be

134

W. Tsang and A. Lifshitz

I 101

_ 9 - without

toluene w-I

o - with toluene =.--,

x o t~ p..-,

Z

10 ~

0 "i" 0 1 0 -1

900

950

1000

1050

T (K) FIGURE 16.4.13 Ratios of [CH3CN]t/[isoxazole]t at different temperatures with toluene at a ratio of [toluene]0/[isoxazole] 0 = 10 and without toluene. The production rate of CH3CN is independent of the presence of toluene, indicating that it does not involve free radical reactions.

seen in Figs. 16.4.13 and 16.4.14, the data points with and without toluene coincide, showing no effect of the latter on the production rate of acetonitrile from isoxazole and ethylnitrile from 5-methylisoxazole. This is a clear indication that they are formed by unimolecular reactions directly from the reactant molecules.

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16.4 Single-PulseShock Tube

135

Reaching a conclusion regarding the mechanism is more complicated for the production of acetylene and hydrogen cyanide from isoxazole. Both can in principle be produced by a direct elimination from the ring, as shown here,

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136

w. Tsang and A. Lifshitz

toluene. The figure shows a large effect of toluene on the production rate of acetylene; it is reduced by more than a factor of 3, indicating that free radical reactions are an important factor in its formation. The concentration of HCN (not shown) was found to be affected to a much lesser extent by the presence of toluene. This means that both unimolecular elimination and (probably) the reaction CH 3CN + H ~ -~ HCN + CH3 ~ play a role in its production.

1 6 . 4 . 4 . 3 COMPUTER SIMULATION 16.4.4.3.1 Reaction Schemes and Modeling To model an observed product distribution, one must construct a reaction scheme, which in most cases contains a large number of species and elementary reactions. After the scheme has been constructed, the computer experiments are performed in the following way. The kinetics scheme is numerically integrated under the same initial conditions as those used in the laboratory experiments, to yield concentration time history. The concentration of each product at a time equal to the experimental dwell time is obtained from the calculations, and a product distribution for a given temperature is constructed. The calculations are carried out at different temperatures so that the product distribution as a function of temperature can be obtained. Calculations at different compositions can also be done. There are a number of "experimental conditions" under which a set of coupled differential equations representing an overall kinetics scheme can be numerically integrated: under constant density, under constant pressure, or more precisely coupled to the shock equations. The latter is very cumbersome and requires considerable, and sometimes inaccessible, computer time. However, if the temperature change during the dwell time is small, which is true when small reactant concentrations are being used, the integration of the reaction scheme coupled to the shock equations is unnecessary. The second best choice is the use of a constant density assumption. Here the shock equations are solved without the chemistry to establish the initial conditions behind the reflected shock, and the kinetics scheme is then numerically integrated starting at these conditions. There are several codes available for numerically solving a set of coupled differential equations. All use the Gear integration algorithm. The most commonly used code is Chemkin-II, "A Fortran Chemical Kinetic Package for Analysis of Gas Phase Chemical Kinetics," (Kee et al. 1992), but there are many other codes as well. The ability to perform simulation experiments depends very much on the ability to compose a suitable reaction scheme and on the availability of correct

16.4 Single-PulseShock Tube

137

rate parameters. Too many guessed rate constants and the omission of "important" elementary reactions make the simulation a hard task to perform. An example of a scheme describing the thermal decomposition of 2,5dimethylfuran is shown in Table 16.4.2 (Lifshitz et al. 1998). This particular scheme contains 50 species and 181 elementary reactions. The rate constants listed in the table are given as k = A exp(-O/T) in units of cm 3, mo1-1, and s -1. Column 1 lists all the elementary reactions in the scheme, column 2 gives the preexponential factors, and column 3 gives the activation energy (E/R) of each reaction. In some cases, where curvature in the Arrhenius plots exists (normally when the rate constants are given for a wide temperature range), the preexponential factors are expressed as A' x T n. Columns 4 and 5 give the rate constants of the forward and the back reactions calculated at 1250 K, and columns 6 and 7 give the standard enthalpy AH~ and entropy AS~ of the particular elementary reaction at the same temperature. The rate constants of the back reactions are calculated from the rate constants of the forward reactions using the relation kr = kf/Keq, where Keq is the equilibrium constant of the particular elementary reaction. This constant is calculated from the thermodynamic properties of the species that participate in the reaction using the relation Keq- e x p ( A S ~ 1 7 6 (RT) -av, where Av is the change of the number of moles in the reaction. The Arrhenius parameters for unimolecular reactions, such as isomerizations and others, are determined directly from the rate of production of these products. The other reactions in the scheme are either estimated or taken from various literature sources such as the NIST-Kinetic Standard Reference Database 17 (Westley et al. 1998) and other compilations (Warnatz 1984; Atkinson et al. 1999; Baulch et al. 1994; Wang and Frenklach 1994, 1997; Miller and Bowman 1989; Tsang and Hampson 1986). The parameters for the reactions that are taken from the various compilations and from the NIST-Kinetic Database are, in many cases, a best fit to a number of entries. The thermodynamic properties of the species are taken from various literature sources (Stull et al. 1969; Pedley et al. 1986; Melius 1999; Burcat and McBride 1997; Burcat 1999; Stein et al. 1994, 1998; Tsang and Hampson, 1986) or estimated by various methods including the NIST Standard Reference Database 25 (Stein et al. 1994) (Structure and Properties program, SP). In most of the decomposition studies, where very low reactant concentrations are being used, the system is not very sensitive to the precise values of the equilibrium constants. This behavior has been demonstrated in sensitivity studies, as will be discussed later. Figure 16.4.16 shows the overall decomposition of 2,5-dimethylfuran (Lifshitz et al. 1998) expressed as mole percent vs temperature. The data points are the experimental results and the line is the best fit to the calculated points (+) taken at 25-K intervals. Figures 16.4.17-16.4.21 show comparisons

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148

w. Tsang and A. Lifshitz

between the experimentally measured yields and the calculated yields using the reaction scheme shown in Table 16.4.2. The agreement for most of the reaction products is satisfactory considering the large number of products obtained in this decomposition. 16.4.4.3.2 Sensitivity Analysis To establish a better agreement between the computed and the experimental product distribution by varying estimated rate parameters, a sensitivity analysis of the specific reaction scheme is run. Rate parameters of all the elementary reactions in the scheme are systematically varied by an arbitrary factor, and the effects of such variations on the concentration of the products are examined. The sensitivity factor in this case, Sij, is defined as

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I r-I

I

152

w. Tsang and A. Lifshitz

Table 16.4.3 shows that only a relatively small number of elementary steps affect the product distribution in the sense that their elimination from the scheme affects the yield of at least one product. The majority of the elementary reactions that compose the scheme do not affect the distribution at all. They are normally left in the kinetics scheme for completeness and applicability beyond the temperature range of a given series of experiments. It should be mentioned, however, that the sensitivity analysis is performed by removing a single reaction at a time. When a group of reactions is eliminated from the scheme, there can be a strong effect on particular products even though the elimination of only one step, as shown in the table, does not have an effect. Most of the sensitivities that appear in Table 16.4.3 are self-explanatory. They enable one to follow the sequence of steps that lead to a particular product. Cyclopentadiene, for example, is formed by reaction 126 and is turned off completely when this reaction is removed. However, if either the main channel for the formation of CH3(C4H20)CH 2" (reaction 17) or its decomposition to C H 2 = C H - C H - - C H - C H 2 ~ and CO (reaction 40) are removed from the scheme, the yield of cyclopentadiene drops almost to zero. This indicates that the formation sequence of cyclopentadiene is R17--~ R40---~ R126. Many similar examples can be found by examining the sensitivity data shown in the table. In conclusion, the sensitivity analysis of a complex kinetics scheme where there are many parallel routes for product formation and many consecutive steps is a necessary procedure. Without it, it is almost impossible to understand the overall reaction mechanism.

16.4.5

SINGLE-REACTION

STUDIES

16.4.5.1 J U S T I F I C A T I O N We will now describe the determination of rate constants and expressions of single-step thermal reactions with high accuracy. As a preliminary, it is necessary to consider the nature and need for such properties. Single-step thermal reactions are processes involving the interaction of molecules in Boltzmann distributions. The reason for the interest in thermal reactions is that they represent the lowest-level kinetic property related to theory. Thus accurate expressions can form the basis for predictions, extrapolations, and interpolations. Indeed, accurate kinetic data of this type also form the basis for determination of the thermodynamic properties of unstable reaction intermediates. Under certain high-temperature conditions, the distribution functions are perturbed. However, the means to treat such situations are increasingly well established. Practically, this means that it may be necessary

16.4

Single-Pulse Shock Tube

153

to adjust the thermal rate constants from shock tube studies to reflect the physical environment. The need for such capabilities originate from the well-known fact that the high-temperature behavior of polyatomic molecules under combustion or pyrolytic conditions can be extremely complex in terms of the multiplicity of single-step reactions. The modern tendency is to try to describe the overall reaction or global process using these single-step reactions as the input data for computer programs that increasingly can reproduce the phenomenon of direct interest. The multiple-reaction studies described above are one aspect of such work. The rationale for such an approach, in contrast to the more traditional empirical approach of developing correlations of the observed temporal behavior of reactants and products for the system of interest, arises from the much wider applicability of the more fundamental approach. One cannot yet faithfully describe a complex process purely through computer simulation. Instead computer simulation represents a technique that can aid in the conceptualization or optimization of a process and it is in fact the first step of an iterative procedure where simulations and experiments go hand in hand. The emphasis on theory originates from the fact that for the simulation of any system scores of chemical reactions must be considered. It is patently impossible to determine the mechanisms and rates of reaction of all the processes under the desired conditions. Hence theory is needed to provide large portions of the database. Obviously, for any given overall process all reactions are not equally important. However, as a first cut it is important not to neglect any possible process. Then through the use of sensitivity analysis subroutines in the simulation packages one can describe the overall process in terms of the key reactions. As the physical and chemical environment is changed, the relative importance of particular reactions may also be changed. The thermal rate expressions that are determined can be used directly to extract the properties of the transition state (Benson 1976). Thus from the standard rate expression one finds that

k = kT/h exp(AS+/R) exp(-AE+/RT) where AS+ and AE + are the entropies and energies of activation, respectively. It should be emphasized that high-pressure rate expressions represent the only means of experimentally determining such properties of the transition state. Such expressions are of special importance for developing correlations and as a check of the validity of results of ab initio calculations from programs such as Gaussian. In addition, if the rate expression for the reverse process is known, then it will be possible to determine the thermodynamics of the overall process. Alternatively, if the thermodynamics are known, then kinetics data in one direction leads to the rate expression for the reverse. The breaking of carbon-carbon bonds is the initiating reaction in many combustion and pyrolysis processes. There is also considerable data on the rate

154

w. Tsang and A. Lifshitz

expressions for the reverse radical combination process. Combination of the two sets of rate expression leads to the thermodynamic properties of the radicals that are formed during decomposition. Of particular importance are their heats of formatj0n, which are fundamental to any understanding of their reactivity. For malJy years the heats otr formation of organic radicals were thought to be well established (McMillen and Golden 1982). As will be seen below, the results from single-pulse shock tube studies demonstrate that there were in fact serious systematic errors in the generally used heats of formation. This finding led to a variety of new experiments that completely validated the shock tube results and resolved many long-standing problems dealing with the internal consistency of the databases. In the following section, we describe the type of reaction kinetics that the single-pulse shock tube method is uniquely qualified to study. At the fundamental level practically all chemical kinetic processes can be divided into unimolecular and bimolecular processes. In the former category, molecules are excited by collisions until the first or several decomposition channels are accessed. If energy transfer by collisions is sufficiently rapid, then the Boltzmann distribution will be maintained at all times. The decomposition will then be truly unimolecular with the rate constants being pressure independent. If energy transfer effects are present, unimolecular rate constants will be pressure dependent. Since the fundamental quantities are the highpressure rate constants and quantities defining energy transfer, it will be necessary to convert the measured rate constants to these quantities. However, if this is overlaid by other decomposition processes--for example, possible contributions from radical mechanisms--it becomes virtually impossible to make such conversions. The resulting data becomes extremely difficult to extrapolate to different pressures. Thus only if the individual unimolecular reaction can be isolated can one confidently carry out such extrapolations.

1 6 . 4 . 5 . 2 EXPERIMENTAL CONFIGURATIONS The conditions necessa~ for the determination of high-quality kinetics data of the type described above are fairly obvious. One would like a situation where the only process that can occur is the reaction of interest. If this is not achievable, then one should at least understand or have an estimate of the contributions from possible competitive processes. We mentioned earlier the intrinsic advantage of shock tube studies being almost completely independent of surface processes, the lessened importance of radical-induced decomposition, and in any case the possibility of suppressing the latter. We illustrate the latter by the following rough calculations. Consider a typical single-pulse shock tube experiment operating at a temperature of 1100 K, 2 bar pressure of

155

16.4 Single-PulseShock Tube

argon and with a heating time of 500 gs. Under such conditions it is easy to calculate that any molecule will suffer approximately 3 million collisions. The consequence is that a compound at the 0.3 ppm level can suffer only 1 collision with a product from its decomposition. This demonstrates the value of the short heating time in the sense of reducing the possibility of long chains. Furthermore, very few chemical reactions proceed on every collision, thus the same effect can be realized at considerably higher concentrations. This inhibiting effect can be further amplified through the addition of a radical scavenger, which was discussed in an earlier section. Here, a more quantitative approach is used. The present aim will be to indicate the concentrations necessary to ensure all radical contributions are suppressed. For many of the studies, methylated benzenes have been used. The reactions they undergo are R ~ + C6HsCH 3 =:} RH + C6HsCH2" =~ CH3 ~ + C6HsR CH3 ~ + C6HsCH 3 =~ CH 4 + C6HsCH2 ~ where C6H5 is a phenyl group, C6HsCH2 ~ is a benzyl radical, and R~ is any reactive radical. For our purposes, the primary concern is the hydrogen atom and methyl radicals. The overall effect is to convert reactive radicals into less reactive benzyl radicals. The lessened reactivity of the benzyl type radical is brought about by its resonance stabilization, which lowers the b e n z y l - H bond energy by nearly 50 kJ/mol. In the short time of a single-pulse shock tube experiment and working with appropriately small concentrations of the target molecule, there is simply no time for resonance-stabilized radicals such as benzyl to attack the molecule being tested. Thus the only reaction a target molecule can undergo is unimolecular decomposition. The benzyl radicals that are formed will largely recombine with themselves or with other radicals that may be present. It should be noted that radical combination reactions are known to be fast and largely unaffected by resonance stabilization. With such a scenario, the benzyl radical is itself a radical scavenger. The amount of scavenger needed to bring about substantial scavenging of the radicals depends on the relative rate constants of radical attack on the benzylated and test molecules. We have found that ratios of 50:1 to 100:1 are sufficient. The actual amount needed can be readily established by carrying out experiments with different ratios and observing the situation when the unimolecular decomposition rate constant becomes invariant. Frequently, the products from a chain reaction may be different than that from the unimolecular decomposition. This makes the appropriate ratio particularly easy to determine.

156

w. Tsang and A. Lifshitz

Figures 16.4.22 and 16.4.23 show modeling results on the decomposition of neopentane that illustrate some of the issues discussed here. The rate constants used in the simulation are listed in Table 16.4.4. The results in Fig. 16.4.22 give an indication of the concentrations of various species of concern. Note the relatively larger amounts of the benzyl-type radicals in comparison to the methyl and especially hydrogen atoms. From Fig. 16.4.23 it can be seen, as expected, that progressively larger amounts of the inhibitor leads to decreasing rate constants of neopentane disappearance or isobutene appearance until a

le+1 Mesitylene

le+O le-1

Neopentane

le-2

Isobutene

Methane

Dimesitylene

le-3

m-Xylene

le-4 le-5 r

Mesityl

o '1~

le-6

o :E

le-7

__

t-Butyl Ethyl-dimethylbenzene

le-8 le-9 Methyl

le-10 le-ll

Ethane

le-12 le-13 le-14

.... 100

,

,

200

300

'

,

' ..........

400

, 500

.....

, 600

700

time (l~sec)

FIGURE 16.4.22 Temporalconcentration of reactants, intermediates, and products during the decomposition of 100ppm neopentane in 1% mesitylene at 1250K and 2bar pressure as determined from simulations using data from Table 16.4.4.

16.4

157

Single-Pulse Shock Tube 0.5 w ~ 9 ,,,,=,,

0.4 2

"~ 0.3 c 0

:= 0.2 0.1 ~"

)0

=

i'"

|

|

|

200

300

400

500

600

700

time (psec) FIGURE 16.4.23 Fractional yields (in terms of isobutene formed) during neopentane decomposition as a function of relative quantities of mesitylene/neopentane ratio as determined from simulations using data from Table 16.4.4.

minimum value is reached. Although there may be considerable uncertainties in the rate constants used in these simulations, the results in Figs. 16.4.22 and 16.4.23 are useful in setting guidelines for the relative quanitities of the scavenger and target molecule necessary to achieve accurate results. The general procedure for using radical scavengers to isolate unimolecular reactions was originated by Szwarc (1950). Unfortunately, the rate expressions that he obtained are now known to be in error. This is largely due to the fact that in the flow system used by Szwarc the actual temperatures and heating times were improperly estimated. This led to rate expressions whose rate parameters were much too small. In addition, the ratios of scavengers to test molecules were frequently not large enough, which led to additional errors. An obvious requirement is that the stability of the scavenger be substantially larger than that of the test molecule. Thus there are molecules for which the methylated benzenes are not suitable as scavengers. For such cases we have even adopted methane as the scavenger. Here we make use of the fact that methyl radicals are not particularly reactive in the context of the short heating time of single-pulse shock tube experiments. Thus methane is highly stable and at sufficiently low concentrations of the test molecule it will not be able to abstract a hydrogen (Tsang and Cui 1990). In the case where radicals are formed, it is very important to establish the ultimate fate of the radical since these are not detected from the final product analysis. This can lead to complications in the case where the radical lifetimes are as long as the heating times or longer. A particularly simple situation occurs if t h e lifetimes are much shorter than the heating time where the decomposition products include at least one stable molecule. This is the case for many non-resonance-stabilized organic radicals. For such compounds lifetimes are on the order of a few microseconds under single-pulse shock tube conditions. Thus quantitative conversions can be expected. More thermally stable radicals such methyl or resonance-stabilized species can be expected to appear in the final product analysis as compounds formed from

158

i,,-4

!

,,-i

-.....

o~ ~

09

~

+

a

t

.

~.~

~

+ +

++

~

+

~

~+++

, ~

,'-~

,::,

W. Tsang and A. Lifshitz

>

~.~

e~

e~

~ ' ~

+

~

N

~

~

+4-++

16.4 Single-PulseShock Tube

159

combination reactions. It should be possible to trap the more reactive radicals with hydrogen donors and use the products as measures of reactions. In fact, this has been done for perfluoromethyl and phenyl radicals with cyclopentane as the hydrogen donor (Tsang 1986; Robaugh and Tsang 1986a). 16.4.5.2.1 Unimolecular Reactions

The comparative rate technique was developed for use in the determination of rate constants for unimolecular decomposition. Most of the experiments have been carried out in this area. In the following, we give a typical example. Figures 16.4.24 to 16.4.26 contain data on the decomposition of 1,7-octadiene (Tsang and Walker 1992). The concentration determinations are summarized in Fig. 16.4.24. The important observation is, except for in allene, the invariance of yields as a function of the inhibitor-to-reactant concentration ratio. The reason for the variation of the allene concentration is that the allyl decomposition reaction is in the kinetic region for the timescale of these experiments. Indeed, the study of the stability of the allyl radical was the main thrust of the work. Note that although there is indeed a small effect on the 1,5hexadiene concentration; most of the allyl radical recombines. Thus the allyl decomposition process is manifested to a much lesser degree in the concentration of this combination product. The general mechanism is summarized in Fig. 16.4.25 and is completely consistent with the nature of the product distribution given in Fig. 16.4.24. For the present, we will concentrate on the initial decomposition process, the results of which are shown in Fig. 16.4.26. It can be seen that the data show a minimal amount of scatter. This is to be expected since all the results are traceable to gas chromatographic measurements. The rate expressions are as follows" k(1,7-octadiene - allyl + 4-pentenyl) - 1.2 4- 0.8 • 1016 exp(-35,700 4- 400/T) s -1 k(1,7-octadiene - propene 4- 1,4-pentadiene) - 3 4- 1.5 x 1012 e x p ( - 27, 900 4- 270) s -1 The uncertainties are the statistical variations and are in the range of factors of 1.5 in the A-factor and 2-4 kJ/mol in the activation energy. Also included in Fig. 16.4.26 are the rate constants for the decomposition of hexene-1 (Tsang 1978c). It can be seen that, except for the reaction pathway degeneracy, the rate constants are very close to each other. This is one of the great advantages of the comparative rate technique. The physical uncertainties cancel out and it is now possible to make intercomparisons between rate constants for different molecules and thus derive correlations that can lead to accurate estimates.

160

W. Tsang and A. Lifshitz

E ~

"o

1,7octadiene

A

AA6

0 l9 9 ethylene

2

9 9

Q. X

9

0 0 y--

I

0 ._J

-2 1040

9

~

I

[]

9 O

[]

D" 1,4pentadiene ,-~ o~ O i

o

"o

0

O 9

O

O O

O 9

cyclopentene

e

r

i

1060

1080

.... ~

[

1100

[. . . . .

1120

1140

i

,

1160

1180

1200

Temp(K) e-

2

"a m

1

1,5 hexadiene

V

~:7

V

0 r~.

9 0

O 9

O

9 Q.

9

allene

21

X

0 0

-2

0 _J

-3

y.-

1040

|

1060

~

v

J

i

i

1080 1100 1120 1140 1160 1180 1200 Temp(K)

FIGURE 16.4.24 Fractionalyields of products from 1,7-octadiene decomposition as a function of temperature. Hollow symbolsml00ppm 1,7-octadiene in 1% mesitylene. Solid symbolsm 100ppm 1,7-octadiene in 1% mesitylene. Pressure- 2.5 bar. The comparative rate m e t h o d is of course not a panacea for all m e a s u r e m e n t problems in single-pulse shock tube kinetics. The requirements for isolation of reactions from each other was mentioned earlier. All the requirements for "good practice" must still be observed. For example, the cool b o u n d a r y layer and unreacted gases in the sampling tubes leads to a "dead space." Thus it is probably good practice to carry conversions to no more than 50% to 60%. As noted earlier, since the residence time is not easily changed, there is the temptation to carry out reactions to very high extents of reaction. Failure to

16.4

161

Single-Pulse Shock Tube allene + H

T

allyl + ethylene

allyl + 4-pentenyl

7

\

.i* cyclopentene + H

1,7 octadiene

propene + 1,4 pentadiene allyl + allyl = 1,5 hexadiene FIGURE 16.4.25

Mechanism for the thermal decomposition of 1,7-octadiene.

take this into account can lead to results that show drastic curvatures in the standard Arrhenius plots. This is not to say that Arrhenius plots cannot be curved. Indeed, transition state theory requires a temperature-dependent Afactor and hence curved Arrhenius plots. However, over the limited range of a single set of shock tube experiments curvature is inevitably either an experimental artifact or evidence of multiple-reaction channels.

25 .........,

2.0

I,,,Q

O 1.5-

-....

,

_..J

1.0

-

0.5

i

0.85

i

i

-,...

I

I

0.90

0.95

1.00

1000/T FIGURE 16.4.26 Arrhenius plots for the decomposition of 1,7-octadiene via bond breaking (1,7octadiene ~ allyl 4- pentenyl-4) (square), and retroene reaction (1,7-octadiene ~ propene 4- 1,4pentadiene) (diamond).

162

w. Tsang and A. Lifshitz

16.4.5.2.2 H y d r o g e n Atom Attack After determining the mechanism and rate constants for a set of bond-breaking reactions, it is possible to use this as a means of generating radicals to attack other compounds. A key constraint is that the target compound must be much more stable than the radical source. For example, hexamethylethane decomposition has proven to be the most convenient as a source of H atoms. The processes are (t-C4H9) 2 ~ 2t-C4H9 ~ t-C4H9 ~ --+ i-C4H 8 + H ~

(fast)

Thus the isobutene yields provide a means of counting the number of hydrogen atoms released into the system. If the reaction is now carried out in large amounts of another molecule and hydrogen atom reaction leads to a specific product, then the ratio of the concentration of that product to the isobutene yield is a direct measure of the fraction of hydrogen atoms that have reacted with the target compound. Recall that the unimolecular reactions discussed earlier have all been studied in vast amounts of methylated benzenes, thus it is straightforward to study this aspect of the reaction system. A specific example is the reaction of hydrogen atoms with toluene. The reactions of interest are H ~ 4- C6HsCH3 ~ C6H 6 4- CH3 ~ --~ C 6H~ CH2 ~ 4- H2 Thus the yields of benzene are a measure of the importance of the displacement reaction. The difference in the concentration of isobutene and benzene represents the contribution from the abstraction reaction. These results thus establish the branching ratio for hydrogen atom attack on toluene. The results can now be placed on an absolute basis by adding large amounts of methane so as to produce a substantial reduction in the benzene yield. This is due to the removal of hydrogens from the reaction H" + CH 4 -+ CH3 ~ q-H 2 On the basis of the well-established rate expression for this reaction (Tsang and Hampson 1986),

k(H' + CH 4 --+ CH3 ~ +

H 2 ) - 1014 e x p ( - 7 0 0 0 / T ) c m 3 / m o l - 1 / s -1

and making use of the differences of concentrations of benzene formed without methane, the rate constants for the two channels in hydrogen atom attack on toluene can be established. The determined rate expressions for H-atom attack on the methylated benzenes can in turn be used as an intemal standard. This is a particularly useful approach since the demethylated compound is a direct product.

16.4

163

Single-Pulse Shock Tube

Furthermore, the benzybtype radicals created by abstraction are unreactive and therefore cannot attack the target compound. Some results on the hydrogenatom-induced decomposition on trichloroethylene (Tsang and Walker 1995) can be found in Figs. 16.4.27 to 16.4.29. Note the large number of products in Fig. 16.4.27. Each product represents a particular channel for decomposition. Figure 16.4.28 gives the mechanism for decomposition. The large amounts of hydrogen were added to convert reactive radicals such as C1 to H atoms, thus amplifying the concentrations. The invariance of the results despite the changes in reaction conditions confirms the postulated mechanism and the derived rate constants. Figure 16.4.29 gives rate constant data in terms of the ratio of rate constants for the possible reaction channels with trichloroethylene and that for methyl displacement from mesitylene. The rate expressions are k(H ~ + C2CI3H -- C H 2 C C I 2 + C1 ~

k(H ~ + mesitylene -- m-xylene + H')

= 0.89 -t- 0.2 exp(816 + 122/T)

k(H" + C2C13H - CHC1CHC1 + C1) = 0.55 + 0.1 exp(-691 + 220/T) k(H ~ + mesitylene -- m-xylene + H ~ k(H ~ + C2CI3H -- HCI + ~

= 5.6 4- 1 exp(-3431 + 122/T)

k(H" + mesitylene -- m-xylene + H')

Note again the small statistical uncertainties. As in the case with unimolecular decomposition, also included in Fig. 16.4.29 are comparisons from similar studies. As before, this lays the basis for accurate estimation.

sum

of products

2.0 =,..,-= t-

1,1-dichlorethylene

O3

"o 1.5

2,4 dimethylbenzyl (est)

_

r 0 --t "0

e

meta-xylene

1.0

chloroacetylene

X

trans-dichloroethylene oc~ 0.5

cis-dichlorethylene

._1

dichloroacetylene

0.0

,

I

990

,

,

,

I

1020

,

,

,

,

t

,

,

1050

,

,

I

1080

,

,

,

,

I

......

1110

Temp (K) FIGURE 16.4.27 Product yields normalized to hydrogen atoms, released from the hydrogenatom-induced decomposition of trichloroethylene.

164

w. Tsang and A. Lifshitz CI2C=CH2+ CI displacement, ,

fHCIC=CCIH

+ CI

CI2C=CCIH + H HCI + CI2C=CH* ~ *CIC=CCIH

"~

H2 + CI2C=CCI FIGURE 16.4.28

CICCH abstraction

~- CCICCI

Mechanism for the hydrogen-atom-induced decomposition of trichloroethylene.

The present result represents a graphic demonstration of the capability of single-pulse shock tube studies for uniquely determining the contributions from the various channels of a complex chemical process. Furthermore, through the use of an internal standard, it can be seen that as before it is possible to make quantitative comparisons between different processes. The most serious problem in this type of study is the accounting of all the hydrogen atoms. Some possible reaction channels that may lead to confusion

r

0.4

E d.

_ m:~ o - ~ o - ~

~-z~,-

o ~aAx-- o o ~ - 8x- - - ~

o~---

~

"~

0.0

"IrO c~ - 0 . 4 rO

oao_oo

+

I

_

v

-0.8 o ._1 '"

0.88

'

'

'

'

9

'

i

0.92

.

.

.

.

'

'

!

. . . . .

0.96

,

"

9

''

~

'

'

i

,

,

1.00

1000/T FIGURE 16.4.29 Rate constants for addition and abstraction reactions of hydrogen atoms with trichloroethylene. Circles m mixture containing 200 ppm hexamethylethane, 2% trichloroethylene, and 1% mesitylene. Triangles m mixture containing 100 ppm hexamethylethane, 2% trichloroethylene, 0.5% mesitylene, and 20% hydrogen. Dashed lines--best fits through points. Filled symbols--hydrogen abstraction of chlorine atom. Hollow symbols mdisplacement processes. The higher values are for displacement of the least-substituted chlorine. The lower values are for the more highly chlorine compound and is the sum of cis- and trans-species. Lines A and B are the comparable values times 0.5 for H attack on tetrachlorethylene leading to displacement and times 0.75 for H attack on tetrachlorethylene leading to abstraction.

16.4 Single-PulseShock Tube

165

include reaction with the hydrogen atom generator itself (hexamethylethane) or reactions with the radicals that are present in the system. These effects can be minimized by making ratios, of radical source and radical sink as large as possible. By carrying out reactions at a variety of ratios, any such contributions should be discernible. Due to the multichannel nature of these reactions, it has not been possible to make extensive comparisons of results from other type of experiments. However, recently the Stuttgart group (Horn et al. 1998) has made direct studies of hydrogen atom attack on phenol and obtained results that completely reproduce the earlier data from the single-pulse shock tube studies. Besides hydrogen atoms, labile organics can also be used generate radicals. One can then study their stability in the unimolecular sense. For example, the use of 1,7-octadiene as a precusor for allyl radicals was discussed earlier. Similarly, 2,4-dimethypentene-1 (Tsang 1973b) has been used to generate isobutenyl radicals and to determine the rate expression for its decomposition. More recently, n-pentyl iodide has been used as a precursor for n-pentyl radicals (Tsang et al. 1998). In this case, rate constants for decomposition are extremely fast and thus not accessible to single-pulse shock tube experiments. It is however quite straightforward to determine branching ratios for decomposition to form ethylene and propene and the corresponding methyl radicals and hydrogen atoms. The experiments described here, particularly those involving the resonance-stabilized radicals, are essentially inaccessible to classic techniques. On the other hand, the more stable radicals such as benzyl and propargyl need even higher temperatures.

16.4.5.3 INTERNAL STANDARDS AND THE COMPARATIVE RATE T E C H N I Q U E The isolation of unimolecular reactions by the use of scavengers gives one the capability to study several unimolecular reactions simultaneously in the same single-pulse shock tube experiment (Tsang 1964a, 1964b). An obvious prerequisite is that the molecular components of these reactions do not interact with each other. At present this is validated only for systems that produce reactive intermediates of the type that are found in hydrocarbons systems. Other systems, where different types of highly reactive radicals are created, must be treated on an individual basis. A great danger in carrying out this type of experiment is that one always obtains highly precise data that are only directly meaningful if the reactions do not "talk" to each other. The advantage of studying several reactions at once is that one of the reactions can be used as an internal standard. Specifically, if the rate expression of a unimolecular decomposition has been determined by some other means,

166

w. Tsang and A. Lifshitz

then that expression can be used as a basis for estimating the reaction temperature. This then removes all the problems originating from the nonideal behavior of real shock tubes. Thus for example, suppose that for a unimolecular reaction with the rate expression k(A = B + C) = A s e x p ( - E ~ / R T ) s -1

one obtains from an particular experiment k(A = B + C) -- -(I/t)ln(Af/Ai) -- - ( l / t ) l n ( [ A i - B(or C)]/Ai)

where t is the heating time and the subscripts f and i refers to the final and initial concentrations, respectively. One can then obtain an average reaction temperature T on the basis of the relation 1 / T = lnA~ - lnk(A = B + C) --- lnA~ - l n ( - 1 / t ) l n ( [ A i - B(or C)]/Ai)

Thus on the basis of a measured heating time and concentrations it is possible to obtain an average reaction temperature. Since the other decomposing species is in the same bath, it must be suffering the exact reaction temperature and the same heating time. This is the basis of the comparative rate measurements that have been used for many years. It is probably the most accurate means of obtaining unimolecular rate constants and expressions. A detailed uncertainty analysis has been published (Tsang 1964b). The increase in precision arises from the sole use of concentration measurements to determined rate expressions. Obviously, if the activation energy for the two processes is the same, there will be no errors in the relative rate constants. Since rate constants are exponentially dependent on the temperature, it is in fact difficult to carry out studies with tremendously different activation energies. Thus the key requirement is easily satisfied. An important input in this procedure is obviously the rate expression for the standard reaction. For this purpose we have chosen decomposition processes that lead directly to the production of stable molecules and for which surface effects have been accounted for. At the present time, the standard to which all the published rate expressions derived from internal standards are related is the reverse-Diels-Alder decyclization of cyclohexene (Tsang 1981), or k(c-C6Hlo -- C2H 4 q- 1,3C4H6) -- 1015"15e x p ( - 3 3 , 5 0 0 / T )

S-1

The estimated uncertainty is probably no more than a factor of 1.5 in the Afactor and 1 - 2 k J/tool in the activation energy. The uncertainty in the rate constants is on the order of a factor of 1.2. Verification of this rate expression can be made through comparison with other well-established rate expressions in the literature. In most applications for which chemical kinetics are important, it is in fact the relative rate constants that are of prime concern. The present type of study represents direct determinations of such quantities.

16.4 Single-PulseShock Tube

167

Since all properties are determined from Concentration measurements, it is estimated that the relative rate constant should not exceed factors of 1.05. The type of reaction epitomized by cyclohexene decomposition, where stable products are directly formed, represent a category of processes that is particularly easy to study using the single-pulse shock tube technique. This is because no active radicals are formed and one can indeed carry out studies in the absence of a scavenger. On the other hand, it is always wise to study the effect of adding scavengers to ensure that radical-induced decompositions are not making contributions. It should be noted that many such reactions m for example, the dehydrohalogenation of alkyl halidesm are particularly sensitive to surface effects, thus special "seasoning" processes must be undertaken to eliminate such processes in static reactors..These problems do not arise in shock tube experiments. This category of reactions, involving multiple bond breaking and bond formation, is particularly important at the present time because predictive capabilites for the rate constants or expressions for such processes are still uncertain. Many such r e a c t i o n s ~ for example, the decyclization of small ring compounds ~ represent an active research field in physical organic chemistry. The shock tube experiments extend the temperature and pressure ranges where such measurements can be made. In this application, chemical kinetics used to define the properties behind the reflected shock wave. This removes all the uncertainties brought about by the nonideal behavior. This general approach has not been applied by investigators with interest in high-temperature fluid dynamics. Nevertheless, as our understanding of gas-phase chemistry increases, it is clear that there are many interesting possibilities.

16.4.6 SPECIFIC SYSTEMS AND GENERALIZATIONS We will now summarize results from specific systems that have been studied. The data can be found in the appendix. The first section deals with data summarized in Appendix 16.4.9.1.1, which contains information on studies where the process involves a number of elementary reactions. Appendix 16.4.9.1.2 contains information on a number of isomerization processes that occur in parallel with the decomposition reactions listed in Appendix 16.4.9.1.1. The information testifies to the complex rearrangements involving bond breaking and bond formation that even intermediate sized molecules can undergo. The prediction of these pathways is one of the great challenges of theoretical chemistry. The second section deals rwith the rate expressions that have been determined from single-pulse shock wave studies. A number of the

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generalizations from such results will be discussed. The actual values can be found in Appendix 16.4.9.2.

16.4.6.1

COMPLEX REACTIONS

Appendix 16.4.9.1 provides a list of single-pulse shock tube studies of complex reaction systems. It lists the compounds that were studied, the temperature range, the products obtained under reflected shock heating, and references to the original manuscripts. It also states in what studies computer modeling has been performed. The compounds that will be reviewed are divided into three categories: a. Saturated and unsaturated aliphatic hydrocarbons, with and without functional groups. b. Aromatic hydrocarbons containing one or two rings, with and without functional groups. c. Heterocyclic compounds containing oxygen, nitrogen, and both, as the heteroatoms. We will briefly discuss the main features of the thermal reactions of these compounds. It should be mentioned that many of these reactions have been studied using shock tubes with a variety of diagnostics in addition to the single-pulse shock tube. In particular, one should note the many publications of Hidaka et al., who used diagnostic techniques such as IR emission, UV, and laser absorption for time-resolved information in addition to product identification using the single-pulse shock tube mode. For further details and references to studies using shock tubes and other experimental methods, the reader is referred to the original articles. 16.4.6.1.1 Aliphatic Hydrocarbons A large number of aliphatic hydrocarbons m including those having heteroatoms such as oxygen, nitrogen, and sulfur m h a v e been investigated using sing!e-pulse shock tubes. There are also some data on the decomposition of halocarbons, which involve complex mechanisms. Studies on the decompositions of methane, ethane, acetylene, acetylene in the presence of SO2, propane, propylene, allene and propyne, 2-butene, 1-butyne, 2-butyne, 1,2-butadiene, vinylacetylene, cyclopentadiene, and octane in the presence of hydrogen were reported in the literature. The decomposition of methane was studied by many groups and the mechanism is fairly well understood. The reactions of the methyl radicals produced in the initiation step were studied in detail (Hidaka et al. 1990) to explain the production of ethylene and propyne. Ethane

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decomposes by cleavage of the C - C bond, and C2H4, C2H2, and CH 4 are identified as the decomposition products (Burcat et al. 1973; Hidaka et al. 1985b). Acetylene decomposition was studied up to a temperature of 2400 K (Colket 1986). Upon decomposition it yields both aliphatic and aromatic products. C4H4, C4H2, and C6H 2 are the aliphatic products, and C6H6 and C8H6 (phenyl acetylene) are the major aromatic products. A mixture of acetylene and sulfur dioxide when elevated to high temperatures yields CO, CS2, C2H4, and CO2 (Fifer et al. 1971). Propane yields upon decomposition CH4, C2H4, C2H6, and C3H 6 (Lifshitz et al. 1973; Lifshitz and Frenklach 1975; Hidaka et al. 1989c). The initiation step is C3H8 --~ C2H5 + CH3, followed by unimolecular split of H atoms from ethyl radicals and free radical attack on propane. Propylene has stronger C - C bonds than propane. To account for the distribution of the reaction products which are C2H4, C2H2, CH4, and allene and propyne (Burcat 1975; Hidaka et al. 1992b), three initiation steps had to be assumed (Hidaka et al. 1992b): C3H 6 -~ CH 3 -b C2H 3, C3H6 -~ H + C3H5, and C3H 6 --~ CH 4 -F C2H 2. The structural isomers allene and propyne have different decomposition channels (Lifshitz et al. 1976; Hidaka et al. 1989b). Propyne gives high yields of methane and acetylene, whereas allene gives a high yield of ethylene (and cumulene), which was shown to be formed by a bimolecular reaction between two allene molecules: CH2--C=CH 2 + CH2---C--CH 2 --+ C2H 4 nt- C H 2 - - C = C - - C H 2. Their major reaction, however, is allene ~ propyne interisomerization, and they equilibrate at a much higher rate then they decompose. Their decomposition channels had to be determined before isomerization takes place, that is, at extremely low extents of reaction. 2-butene, both cis and trans, decomposes to yield methane, propylene, and butadiene, while the cis ~ trans isomerization does not reach an equilibrium (.Jeffers and Bauer 1974). Cyclopentadiene decomposes by H-atom ejection from the CH2 group followed by successive fl-scissions (Burcat and Dvinyaninov 1996). Its main decomposition products are C2H 2 and C2H 4. Both 1butyne (Hidaka et al. 1995b) and 2-butyne (Hidaka et al. 1993a) isomerize to 1,3- butadiene and 1,2-butadiene prior to decomposition. The main decomposition products are CH4, C2H2, C2H 4, C6H6, allene, and propyne. 1,2butadiene (Hidaka et al. 1995a) also isomerizes to other C4H 6 isomers at a faster rate than it decomposes and yields the same products as the other C4H 6 isomers. Vinylacetylene (Colket 1986; Hidaka et al. 1992a) has been assumed to have three initiation reactions: C4H 4 ~ C4H 3 if- H, C4H 4 ~ C2H 2 -+- C2H2, and C4H 4 --+ C4H 2 q - H 2. The major reaction products were C2H2, C4H2, C6H2, and C6H6. The decomposition of octane diluted in argon gives mainly ethylene, methane, propylene, and small yields of ethane. When diluted in a mixture of 50% hydrogen and 50% argon, the yields of methane and ethane increase and the overall decomposition rate of octane increases by 1 order of magnitude (Doolan and Mackie 1983).

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Among the oxygen-containing species, the decomposition of propanal (Lifshitz et al. 1990), crotonaldehyde (CH3CH--CHCHO) (Lifshitz et al. 1989b), propanoic acid (Doolan et al. 1986), methanol (Hidaka et al. 1989d), and nitromethane (Zhang and Bauer 1997) were investigated using the single-pulse shock tube technique. The oxygen in the aldehydes ends up as carbon monoxide, whereas other oxygen-containing species appear in very low yields or do not appear at all. Propanoic acid yields both carbon monoxide and carbon dioxide. The reaction products in methanol are CO, CH20, CH 4, C2H6, and C2H4. The initiation reaction in nitromethane is cleavage of the relatively weak C - N bond in the molecule, followed by methyl group attack on the reactant. CH4, C2H6, C2H4, and C2H2 appeared as the main carbon-containing reaction products. Oxidative decomposition of methane in the presence of N20 and the decomposition of ethane in the presence of NO have been reported. Methane in the presence of N20 yields mainly C2H4 and C2H6, but at high temperatures carbon monoxide and acetylene perdominate (Mackie and Hart 1990). Ethane in the presence of NO yields a considerable amount of HCN resulting from the attack of CH3 radicals on NO: CH 3 4-NO--~ CH3NO ~ HCN 4- H20 (Lifshitz et al. 1993b). The decomposition of CH3CN (Lifshitz et al. 1987b; Ikeda and Mackie 1996; Lifshitz and Tamburu 1998), CH2=CHCN (Lifshitz et al. 1989a) and butene nitriles (Doughty and Mackie 1992a) were investigated and the product distribution was reported. The major products in the decomposition of CH3CN are CH4, HCN, and C2H2, together with numerous other products. The initiation reaction in this decomposition is an H-atom ejection from the C - H bond in the molecule, followed by dissociative attachment of H atom to CH3CN to produce HCN and methyl radical: CH3CN 4- H'--~ CH3" 4- HCN. The production of HCN and C2H2 from CH2=CHCN is believed to proceed via a four-center transition state. There is not much reported work on single-pulse shock tube decompositions of halocarbons where complex reaction systems are concerned. The overall process is dominated by the direct dehydrohalogenation process. Propargyl chloride (CH2C1C--CH) isomerizes to chloroallene (CHCI=C--CH2) as the main reaction (Kumaran et al. 1996) and equilibrates before considerable decomposition begins. The major decomposition products in addition to isomerization to chloroallene are C2H2, C6H6, C4H2, C6H~C1, and others (Lifshitz and Suslensky unpublished e). 16.4.6.1.2 Aromatic Hydrocarbons Among the aromatic compounds studied with the single-pulse shock tube are benzene, benzonitrile, o-dichlorobenzene, toluene, indene, and phenol. Benzene decomposes by H-atom ejection and opening of the ring in the fl-

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position to the radical site (Laskin and Lifshitz 1997a). Its main decomposition products are C2H2 and C4H2. However, at the lower-temperature end biphenyl is the major product due to an attack of phenyl radicals on benzene. The thermal reactions of benzonitrile (Lifshitz et al. 1997a) are similar to those of benzene, yielding as major products C2H2, C4H2, C6H6, C H = C - C N , and HCN. O-dichlorobenzene decomposes to yield C2H2, C4H2, C6HsC1, and C6H6 as the major decomposition products (Lifshitz et al. unpublished b). The decomposition of toluene was studied up to a temperature of 1800 K and a large number of products, including conjugated aromatic rings, were obtained (Colket and Seery 1994). The initiation is H-atom ejection from the methyl group at low temperatures and methyl group elimination at the higher end of the temperature range. Indene is cyclopentadiene fused to benzene. The decomposition begins by ejection of H atoms from the CH2 group and further decomposition of the indenyl radical (Laskin and Lifshitz 1999). However, indanyl, which is obtained by H-atom attachment to the cyclopentadiene ring in indene, plays a very important role in the decomposition. This channel is not important in the decomposition of cyclopentadiene itself. Phenol decomposes in a unimolecular reaction to carbon monoxide and cyclopentadiene (Burcat and Olchanski 1999). This is the most important step in the decomposition, although ejection of H atoms from the hydroxyl group takes place as well. The fragmentation of the formed cyclopentadiene is responsible for the production of lower-molecular-weight species. Carbon monoxide, cyclopentadiene, ethylene, acetylene benzene, and methane were identified as the major decomposition products. 16.4.6.1.3 Heterocyclic Compounds With the introduction of a heteroatom to a cyclic hydrocarbon, the symmetry of the molecule decreases and the number of distinguishable reaction channels increases. Three-membered rings, the epoxy family of molecules, where a CH2 group has been replaced by an O atom is a good example. Three molecules in this group have been thoroughly investigated using the single-pulse shock tube (Lifshitz 1995). Oxiran (ethylene oxide) can isomerize only to acetaldehyde (Lifshitz and Ben-Hamou 1983). Propylene oxide (Lifshitz and Tamburu 1994), and 2,3-dimethyloxiran (Lifshitz and Tamburu 1995), yield four isomerization products: aldehyde, alcohol, ether, and ketone. All the three compounds yield decomposition products at relatively low temperatures due to the formation of thermally excited isomers before being de-excited to the ground Boltzmann distribution (Benson 1964). Considerable effort has been devoted to the study of 5-membered heterocyclics containing oxygen (Lifshitz 1988, 1989). The thermal reactions of hydrogenated furans with or without substituents have been investigated.

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and A. Lifshitz

Tetrahydrofuran (Lifshitz et al. 1986a) is a kinetically stable heterocyclic compound. It undergoes several unimolecular ring cleavage processes, but no isomerization products were identified in the postshock mixtures. With the introduction of one double bond, the thermal stability decreases. Thus the decomposition of 2,5-dihydrofuran (Lifshitz et al. 1986c) and 2,3-dihydrofuran (Lifshitz and Bidani 1989) occurs at considerably lower temperatures. The main reaction in 2,5-dihydrofuran is H2 elimination from the 2-5 positions forming furan. The extent of fragmentation is by orders of magnitude below the main reaction. There is no H2 elimination from 2,3-dihydrofuran. The major reaction here is isomerization to cyclopropanecarboxaldehyde (C3HsCHO). However, fragmentationmparticularly to CO + C3H6 and the formation of other products mdoes occur at rates comparable to the isomerization. 4,5- dihydro-2-methyl-furan (Lifshitz and Laskin 1994) isomerizes to cyclopropane-methylketone and 3-pentene-2-one with a very low extent of decomposition. With the introduction of an additional double bond, f u r a n ~ t h e most stable p r o d u c t ~ is formed. Furan does not isomerize (Lifshitz et al. 1986b); its main decomposition channel is the formation of carbon monoxide and propyne initiated by H-atom migration from position 5 to position 4 in the ring. Other decomposition channels forming acetylene and other products do exist as well. The thermal reactions of 2-methylfuran (Lifshitz et al. 1997b), 2,5-dimethylfuran (Lifshitz et al. 1998), and 2-furonitrile (Laskin and Lifshitz 1996a) have also been investigated using the single-pulse shock tube technique. Whereas the - C N group in 2-furonitrile does not migrate, migration of a hydrogen atom or a methyl group with the elimination of carbon monoxide in 2-methylfuran and 2,5-dimethylfuran is again the major reaction channel. Four isomers of C4H6 m 1,3-butadiene, 2-butyne, 1-butyne, and 1,2-butadiene~ are formed in the decomposition of 2-methylfuran and the same four isomers of C4H6 and several isomers of C~H8 are formed in the decomposition of 2,5dimethylfuran. Unpublished single-pulse shock tube data are also available on the thermal reactions of 2,3-dihydrobenzofuran (Lifshitz et al. unpublished b) and isodihydrobenzofuran (phtalan) (Lifshitz et al. unpublished a). 2,3-dihydrobenzofuran is 2,3-dihydrofuran fused to benzene in the 4-5 positions of the furan ring. Its thermal stability is higher, and the temperature at which products begin to appear following a dwell time of ~2 ms is approximately 200 K higher. Bond cleavage occurs in the furan ring, whereas the benzene ring stays intact. This results in the formation of several derivatives of benzene, including isomerization products. Dihydrobenzofuran and isodihydrobenzofuran undergo similar isomerization. By cleavage of the 1 - 3 bond in the furan ring and H-atom migration from position 3 to position 1-,2,3-dihydrobenzofuran gives o-hydroxy styrene and phtalan gives o-tolualdehyde.

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The addition of a nitrogen atom to the furan ring in position 2, adjacent to the oxygen, decreases the symmetry of the molecule. The stability of the molecule is also decreased due to the weak N - O bond (--~315kJ/mol). In view of the weak N - O bond, this compound m isoxazole (Lifshitz and Wohlfeiler 1992a)--and two of its derivativesm 5-methylisoxazole (Lifshitz and Wohlfeller 1992b) and 3,5-dimethylisoxazole (Lifshitz et al. 1995)~have been studied behind reflected shocks at much lower temperatures than used with furan and its derivatives. Migration from position 5 to position 4 in both isoxazole and 5-methylisoxazole and elimination of carbon monoxide is still the major channel of decomposition. H-atom migration in isoxazole yields carbon monoxide and acetonitrile, whereas methyl group migration in 5methylisoxazole yields carbon monoxide and propylnitrile. In 3,5-dimethylisoxazole, the main channel is isomerization to 2-methyl-3-oxo-butyronitrile. Moving the oxygen atom to position 3 in the ring (oxazole) increases the stability of the ring close to that of furan (Lifshitz and Wohlfeiler unpublished), and the decomposition occurs at much higher temperatures than in the isoxazoles. Another group of five-membered heterocyClics is the pyrrole group of molecules. The decompositions of pyrrolidine, pyrrole, N-methylpyrrole, and 2,4-dimethylpyrrole have been studied in the single-pulse shock tube. Pyrrolidine (tetrahydropyrrole), like its isoelectronic tetrahydrofuran, is a kinetically stable molecule and decomposes at temperatures somewhat below those of tetrahydrofuran (Lifshitz et al. 1987a). The main products are ethylene and hydrogen cyanide; only traces of pyrrole are found. No isomerization products were identified in the postshock mixtures. There are no single-pulse shock tube data on the decomposition of the two isomers of dihydropyrrole. Pyrrole is a resonance-stabilized molecule and is kinetically very stable. The temperature range over which it decomposes is similar to that of furan. In contrast to furan, its major reactions are isomerizations to the various isomers of butenenitriles. Its main decomposition products are hydrogen cyanide and acetonitrile (Mackie et al. 1991; Lifshitz et al. 1989c). Pyrrole ring fused to benzene (indole) has also been studied (Laskin and Lifshitz 1997b). It isomerizes to benzyl cyanide, o- and m-tolunitrile and yields also decomposition products resulting from cleavage of the fivemembered ring. N-methylpyrrole fragmentizes to small molecules such as methane and ethane, but also undergoes methyl group migration to other locations in the ring (Lifshitz et al. 1993a; Doughty et al. 1994a). Ejection of a hydrogen atom from the methyl group produces the methylene pyrrole radical, which undergoes ring expansion to yield hydropyridile and finally pyridine (Dubrikova and Lifshitz 2000). 2,4-dimethylpyrrole undergoes ring cleavage to small fragments as well as ring expansion processes to yield 2-picoline, 4picoline, and pyridine (Lifshitz et al. unpublished c).

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Among the six-membered heterocycles, pyridine (Mackie et al. 1990), 2picoline (methylpyridine) (Terentis et al. 1992; Doughty and Mackie 1992b), 3-picoline (Jones et al. 1996), and pyrimidine (Doughty and Mackie 1994) have been studied, as have quinoline and isoquinoline, which consist of pyridine fused to benzene in two different locations. The initiation of pyridine decomposition is an H-atom ejection from the a-position to the N atom. The pyridile radical, which is resonance stabilized by some 6 kcal/mol, decomposes by successive ]~-scissions. The major decomposition products are C2H2, HCN, CH_=C-CN, and C4H2 . The different isomers of picoline decompose to yield practically the same products, with C2H2, HCN, and CH 4 being the major ones. In pyrimidine, the major products are HCN, CH2=CHCN, and CH_=C-CN. Quinoline and isoquinoline (Laskin and Lifshitz 1998) decompose by breaking the pyridine ring in the a-position to the nitrogen. The two isomers give identical decomposition products and yield the same major products: C2H2, C6H6, C6HsCN, and CH=C-CN. The identical products and yields is explained by assuming that the decomposition routes of the two isomers are coupled by 1-indene-imine radical.

1 6 . 4 . 6 . 2 SINGLE-STEP KINETICS 16.4.6.2.1

Introduction

The results in Appendix 16.4.9.2 deal mostly with unimolecular decompositions and include practically all the known mechanisms for unimolecular decomposition. The total number of items in these tables is close to 200; the review of Benson and O'Neal (1969) covers about 500 items. This is thus a significant contribution to the base of experimental results on unimolecular reactions. Also included are results on hydrogen atom attack on organics, which demonstrate the possibilities derived from the work on unimolecular decompositions. All of these results are indicative of the capabilities of the single-pulse shock tube studies for studying multichannel processes. For all but the simplest molecules, this is probably the rule rather than the exception. The reactions that have been covered are divided into seven different categories. The first, summarized in Appendix 16.4.9.2.1, contains bondbreaking processes. These are conceptually the most straightforward: A chemical bond is broken and two radicals are formed, and the reverse reaction has no barrier. Thus the activation energy can be related to the bond dissociation energy. Since radicals are formed, all the studies have been carried out with inhibitors. Internal standards have also been used in all the studies. The reactions are frequently the initiating processes in chain decompositions.

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Rate expressions are also important as inputs for the calculation of thermodynamic properties of radicals. More generally, they serve as the experimental basis for testing of theories on the kinetics of bond breaking. Appendix 16.4.9,2.2 contains data on retroene reactions. For unsaturated hydrocarbons, these are processes that often accompany the bond-breaking reactions. When heteroatoms are present, the activation energy is lowered so that they become the predominating process. The reactions proceed through a six-membered transition state. Appendix 16.4.9.2.3 summarizes reactions on molecular elimination. A prerequisite for such a process to be important is the presence of a highly polar group. Thus the general trends upon methyl substitution for the iodides chlorides and bromides track the situation in solution and much lower temperatures. Indeed, it has been found that the activation energy is 28% of the ion dissociation energy (Tsang 1964b). It is unfortunate that the earlier work on the fluorinated compounds were carried out without internal standards; the accuracy of the rate expressions and constants are larger than those using intemal standards so extrapolations to related reactions may be more uncertain. Appendix 16.4.9.2.4 contains results involving decyclization processes. Many of these reactions involve biradical intermediates. These reactions have long been a favorite subject for physical organic chemists. Single-pulse shock tube studies extend the available temperature range and permit the investigation of more stable compounds such as cyclopentane and cyclohexane. These reactions and those involving larger tinged compounds are related to the bondbreaking processes summarized in Appendix 16.4.9.2.1 except that the radicals cannot escape. In these cases, the main products are in fact those arising from internal disproportionation. The results in Appendix 16.4.9.2.5 on cis-trans isomerization and other isomerizations can be considered a subclass of the decyclization processes discussed previously if one considers a double-bonded molecule a two-membered ring. The studies on both class of reactions benefit particularly from the ability of the researchers to synthesize the particular molecules that they wish to study. Appendix 16.4.9.2.6 contains results on a number of organometallic compounds. Much further work is necessary. Intermediates, final products, and patterns of reactivity have very little relationship with the organics discussed earlier. A particular problem is that the unsaturated compounds, unlike the situation for the hydrocarbons, are highly reactive. Thus without appropriate trapping agents they cannot serve as markers for the extent of reactions. Furthermore, the lessened stability puts most of these reactions into the energy transfer region. There remains a need for making appropriate corrections.

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Appendix 16.4.9.2.7 contains data on hydrogen attack on unsaturates. This represents a truly unique set of results in the sense of the absence of comparable results even at lower temperatures. It builds on the data from hydrocarbon decomposition. This general approach has also been used to study the stability of resonance-stabilized radicals (Tsang 1973b; Tsang and Walker 1990) as well as the branching ratio for alkyl radical decomposition (Tsang et al. 1995). A number of the results summarized in Appendix 16.4.9.2.2 are given as a function of pressure. These represent results in the falloff region. An important problem in existing single-pulse shock tube studies is the difficulty in carrying out studies across large pressure ranges. Thus there is always the question as to whether reported results are at the high-pressure limit. In practically all the comparative rate studies, the pressure was varied by a factor of 3 and except for the smallest compounds pressure effects could not be observed. The implication is that the pressure dependence is less than p0.O3. There is an obvious need for more work.

16.4.6.2.2 A-Factors A particularly interesting observation from the results is the relative invariance of the A-factors with respect to reaction type. It would appear that the only important contribution to variations are in the reaction path degeneracy, which is suggestive of transition states that are very localized with respect to structure. This can provide a very simple empirical basis for prediction: It means that a measurement at one temperature can result in accurate rate expressions. Unfortunately, it has not been possible to verify such observations from determinations using other experimental methods. Some of these considerations are illustrated in Fig. 16.4.30, which contains the A-factors for a variety of different reaction types as given in Appendix 16.4.9.2 and restricted to comparative rate studies. This similarity would not be particularly surprising for molecules that decompose through normal or perhaps tight complexes. However, for bondbreaking reactions, there are no reaction barriers for combination. Hence the exact position of the transition state is unclear. It is known that for such reactions the A-factors are large, thus signifying loose transition states. The comparative rate work set definite limits on permissible values. Even more interesting is that such studies can distinguish between cases of reactions where one of the radicals contains structures with resonance energy. From Fig. 16.4.30 it can be seen that these reactions lead to A-factors about half an order of magnitude lower those where simple alkyl radicals are the sole products. This is in accord with the picture of radical combination according

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FIGURE 16.4.30 A-factors for various types of thermal decomposition processes. (A) retroene reactions, (B) dehydrohalogenation (I, Br, C1), (C) R-X ~ R + X, X ~ Br, (D) ResAlkyl ~ Res + Alkyl, (E) Alkyll-Alkyl2~- Alkyll + Alkyl2. to the Gorin model (Benson 1976). However, note that the A-factors are all much smaller than that derived on the basis of orbiting radicals.

16.4.6.2.3 Bond Energies Probably the most consequential results from single-pulse shock tube work have been the revision of the bond energies of a large number of hydrocarbon compounds or, equivalently, the heats of formation of hydrocarbon radicals (Tsang 1985). Until the advent of the single-pulse shock tube results, these values have been considered to be well established. Since the breaking and formation of chemical bonds is the basis of chemical change, these are the essential factors with regard to chemical reactivity. The single-pulse shock tube studies showed that these established values were too small by 16 to 37 kJ/mol. These studies also imply that the stability or unimolecular lifetimes for branched alkanes at the shock tube temperatures were in error by as much as 3 orders of magnitude. These are enormous errors and their resolution has solved many long-standing problems, including issues dealing with stability of alkyl radicals and the need to postulate barriers for the cyclization of biradicals. With these new values experimental results on radical decomposition and addition to olefins obey detailed balance and cyclization of small biradicals have small or no barriers. A summary of bond energy results derived from single-pulse shock experiments can be found in Table 16.4.5. Also included are the generally accepted results as summarized by McMillen and Golden (1982) and newer results given in a subsequent review by Berkowitz et al. (1995). The

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TABLE 16.4.5 Summary of Heats of Formation Determined by Shock Tube Studies and Recommendations of McMillen and Golden (1982) and Berkowitz et al. (1995)

Radicals C2H5 n-CBH7 i-CBH7 s-C4H9 i-C4H9 t-C4H9 t- C5Hll C3H5 (allyl) C3H5 (propenyl) C4H7 (isobutenyl) C4H7 (methylallyl) C3H3 (propargyl) C4H5 (methylpropargyl) C6H5 (phenyl) C6HsCH2 (benzyl) C6H50 (phenoxy) CH3CO (acetyl) CH3COCH2 CF3 2-Hydroxyethyl 2-Hydroxypropyl 2-Aminopropyl NH2

Single-pulse shock tube results (ref) (kJ/mol) 117.1 98.4 85.3 63.2 60.2 48 32.2 174 267 138 158 351.4 312.5 341 205 55.3 -13.8 -12.6 -460 -56.9 -91.6 96.3 185.3

Tsang, 1981 Tsang, 1981 Tsang, 1981 Tsang, 1981 Tsang, 1981 Tsang, 1981 Tsang, 1981 Tsang and Walker, 1992 Cui et al., 1988 Tsang, 1981 Tsang, 1981 Tsang, 1981 Tsang, 1981 Robaugh and Tsang, 1996b Walker and Tsang, 1990 Walker and Tsang, 1990 Tsang, 1984a Tsang, 1984a Tsang, 1986 Tsang, 1976 Tsang, 1976 Tsang, 1978e Tsang, 1978e

McMillen and Golden Berkowitz et al. (kJ/mol) (kJ/mol) 108 87.8 76.1 57.7 34.7 163

121 89.9 67 51.4 171

140 338 330 200 -24 -24

203 -10

-63.6 --101 185

degree of agreement between the latter and the earlier shock tube results are extraordinarily good. It should be noted that a great deal of the subsequent work was instigated by the shock tube results.

16.4.6.2.4 Rate Expressions for Bond Cleavage The accurate rate expressions for bond-breaking reactions can also be used to make reliable estimates for the parameters for other bond-breaking processes. Specifically, one can make use of the fact that for many radicals the geometric mean rule is well established. Then for any system where the rate expressions for the decomposition of AA to 2A and BB to 2B has been measured, the rate expression for AB to A and B can be determined using only the thermodynamic properties of the stable compounds. Thus as the database for the decomposition of organic compounds into radicals is built up through experiments, rate

16.4

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expressions for many others involving the radicals that have been formed can be rigorously calculated. Actually, this has proven to be less useful than once thought due to the empirical observations regarding the similarities in the Afactors for similar types of bond-breaking reactions and the relationship between the experimental activation energy and the bond dissociation energy. Of course, this is applicable only at or near the single-pulse shock tube temperatures. At lower temperatures, as will be discussed below, the Afactors become strongly dependent on molecular structure. Experimental studies of the unimolecular region usually cover a small temperature range. Thus the best fit of experimental results can only be expressed in terms of the standard Arrhenius form. Of course, transition state theory is inconsistent with such a form; hence the frequent use of a temperature-dependent A-factor in rate expressions. There is, however, very little experimental data bearing on this issue. Combination of shock tube results for bond breaking together with rate constants for combination at or near room temperature and the thermodynamic properties of the radicals that are formed permit derivation of rate expressions for decomposition from room temperature to close to 1200 K. These are summarized in Table 16.4.6 for the formation of the three simple alkyl radicals. It can be seen that for these bondbreaking reactions there is a negative curvature of the Arrhenius plot and that the extent of this curvature increases with methyl substitution. The constancy of the A-factors for decomposition of the three prototypical alkanes at the higher temperatures is lost. Indeed, it appears that at room temperature the Afactors are strongly dependent on the size of the alkyl radicals and range from n e a r 1017 s - 1 for ethane to values as large as 1019 to 10 20 s - 1 for hexamethylethane. An interesting issue is whether this decrease in A-factor will continue as one goes to even higher temperatures. In any case, these observations should be a crucial test for any general theory of radical combination. The consequence of the single-pulse shock tube work on unimolecular reactions is that there is now sufficient data for empirical estimation. This is particularly important for bond-breaking processes from a practical point of

TABLE 16.4.6 Rate Expressions for Combination and Decomposition (at the High Pressure Limit) of Some Simple Alkyl Radicals over the Temperature Range 300-1200 K (Tsang and Kiefer, 1995, Tsang, 1978d) Reactions

C2H5-C2H 5 ~ 2-C2H 5 i-C3H7-i-C3H 7 ~ 2-i-C3H 7 t-C4H9-t-C4H 9 ~ 2-t-C4H 9

Rate constants for combination (cm3/mo1-1/s -1)

Rate constants for dissociation (s -1)

1 x 1013 6 x 1012(300/T) 0"7 1.4 x 1012(300/T) 1"5

4.4 x 1025(1/T) 27 exp(-4441/T) 1.6 x 1031(1/T) 4"2 exp(-43,987/T) 5.5 x 1038(1/T) 6"45exp(-41,065/T)

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view since these are the initiation processes in combustion and pyrolytic applications. As noted earlier, with the correct bond energies discrepancies in the measured rate constant for radical decomposition and that derived from the reverse addition and the thermodynamics have also been resolved. Thus the databases including such reactions are consistent with regard to the thermochemistry. 16.4.6.2.5 Hydrogen Atom Attack Appendix 16.4.9.2.7 summarizes the data on the rate constants for hydrogen atom attack on a number of organic compounds. Many of these compounds are complicated unsaturated compounds with abstraction as well as displacement pathways. The latter is actually a composite reaction involving addition, followed by radical decomposition. In all the cases studied, the assumed decomposition process is more favored than the reverse hydrogen emission reaction. Radical lifetimes are extremely short. Hence, physically, the process has the appearance of a displacement process. It can be seen that at the reaction conditions abstraction is slightly favored over displacement. The rate expressions for the former are characterized by larger parameters or looser transition states. Thus, as the temperature is increased abstraction will be increasingly favored. Chlorine for hydrogen substitution in all cases reduces the rate constants. In general, it appears that electronegative group leads to slower rate constants for displacement. Also of interest are the results on the abstraction and displacement of flourine atoms: The rate constants are the smallest of all that have been measured. Here as before the general trends make the results valuable as a basis for estimation.

16.4.7

SUMMARY AND FUTURE

DIRECTIONS

In this review we have summarized the experimental work that has been carried out using the single-pulse shock tube. The large body of information that has now been accumulated can be extremely useful for the detailed understanding of a variety of important practical processes. The accumulated data also represent the basic ingredients for the calibration and validation of theory. There is very little doubt that the single-pulse shock tube represents an extremely powerful tool for the study and quantitative understanding of hightemperature gas-phase processes. This is especially the case for organic processes since it is possible to build on a broad base of earlier knowledge. Of great importance is the fact that shock tube studies cover a range of temperatures that are inaccessible to standard flow and static experiments but are important in many technical processes such as combustion. When coupled

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with some means of determining the actual conditions in the shock tube, such as the internal standard technique or isolation of the reaction to be studied, the results that are produced are probably the most reliable of all methods for determining high-temperature thermal rate constants and expressions. The logical extension of this work is in the studies of highly unsaturated organics and organometallics, more extreme conditions (particularly in terms of pressures), and the use of the single-pulse feature in combination with realtime detection capabilities. It is increasingly clear that the decomposition of the highly unsaturated organics proceeds through a multitude of possible reaction channels. Many of these involve complex rearrangements. There is at present very little predictive capability on these issues. This is fundamentally a very important scientific question as well as an extremely important practical problem since the issue is essentially the quantitative understanding of polynuclear aromatic hydrocarbon and soot formation and all their related problems. The high-quality data for the purely organic compounds described here depended on isolation of the individual reactions for study. There is at present insufficient understanding of the decomposition processes associated with many inorganic compounds, so there are uncertainties regarding how various reactive intermediates can be trapped. The knowledge that has been attained for the organics is applicable to radicals such as H, O, OH, etc. The question regarding the trapping of reactive inorganics associated with the decomposition of the organometalic compounds is still open. In the case of silicon compounds, an important issue is the proper trapping of the silyene radical. Here again interesting science is mixed with important applications, such as the gas-phase contribution to chemical vapor deposition. A particularly interesting extension of the work summarized here will be the carrying out of studies at pressures much higher than those used here (up to approximately 10 bar). As mentioned earlier, as the temperature increases the energy transfer effect becomes more important. The tendency will then be to distort the distribution function, leading to lower rate parameters. We have mentioned our uneasiness with respect to the systems studied here, although the data does not suggest any serious problems. But for three or four carbon systems, distortions arising from this source can be very important. It would also be very important to have, for a number of systems, the unimolecular rate constants over extended pressure ranges. With such data it may finally be possible to test procedures for extrapolation to the true high-pressure limits. Finally, it is clear that much valuable information can be obtained if real-time information can be combined with single-shock tube experiments. There is no question that temporal information combined with final product analysis of all species will permit much more definitive conclusions regarding mechanisms and rate constants.

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FIGURE 16.4.5 GC-MS chromatogram using the selected-ion monitoring mode of the mass selective detector. Peaks of 1,2-butadiene and 1-butyne are hidden behind a large peak of C4H 4 and can be analyzed only with the assistance of the GC-MS.

FIGURE 16.4.6

Product distribution in the decomposition of shock-heated tetrahydrofuran.

FIGURE 16.4.17 Experimental and calculated mole percents of the four isomers of C4H6. The points are the experimental values and the lines are the best fit through the calculated points shown here as crosses.

FIGURE 16.4.18 Experimental and calculated mole percents of methane, ethane, and propylene. The symbols are the experimental points and the lines are the best fit through the calculated points shown here as crosses.

FIGURE 16.4.20 Experimental and calculated mole percents of allene, propyne, and acetylene. "[he symbols are the experimental points and the lines are the best fit through the calculated points shown here as crosses.

FIGURE 16.5.1

Wheat grains arranged on a chessboard in a geometrical series.

CHAPTER

16.5

Chemical and Combustion Kinetics 16.5

Ignition Delay Times

A s s a Lifshitz Department of Physical Chemistry, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel

16.5.1 Introduction 16.5.2 Basic Concepts 16.5.3 Methodology 16.5.3.1 Experimental Methods 16.5.3.2 Design of an Experiment and Data Processing 16.5.3.3 Modeling Procedures 16.5.4 Kinetic Systems 16.5.4.1 Introductory Remarks 16.5.4.2 Ignition of Small Molecules, The Loop Concept 16.5.4.3 Thermal Ignition without Chain Branching, N20 + COS, N20 q- CO 16.5.4.4 The Concept of Energy Branching, H2 + C12, H2 + F2

16.5.4.5 Correlation of Ignition Delay Times with Bond Dissociation Energies. The Role of Initiation vs. Chain Branching 16.5.4.6 The Dependence of the Ignition Delay Time on the Fuel Concentration 16.5.4.7 Inhibiting Effects of the Diluent 16.5.4.8 Effect of Additives 16.5.5 Computer Modeling 16.5.5.1 Reaction Scheme 16.5.5.2 Sensitivity Spectrum 16.5.6 Summary References

Handbook of Shock Waves, Volume 3

Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISBN:0-12-086433-9/$35.00

211

212 16.5.1

A. Lifshitz INTRODUCTION

A useful method for studying the kinetics of combustion reactions is the measurement and modeling of the induction period that precedes the ignition of a fuel when the fuel is suddenly elevated to a high temperature in the presence of an oxidant. The great potential of this method has been recognized by many combustion kineticists, and a very large volume of experimental results and kinetics modeling has been published. In this chapter, we will discuss the basic aspects of this phenomenon known as induction time or ignition delay time: its measurements behind shock waves and its use in determining mechanisms of combustion reactions. In view of the huge volume of available information on this topic, we will limit our discussion to data on ignition delay that were obtained behind shock waves in shock tubes and to modeling studies that are based on results obtained in shock tube experiments. We will not cover, at least not in detail, ignition delay studies that utilized heating methods other than shock waves. Information on ignition delay and on rate parameters for computer modeling appears in individual papers and in several databases. The GRI-Mech. 3.0 (Smith et al. 1999) contains (among other information) a large number of studies on ignition delay of small hydrocarbons, particularly methane. It also contains graphical presentations of ignition delays, reaction schemes, results of computer modeling, and sensitivity spectra. The reader is encouraged to look for information in this report and the NIST Chemical Kinetics Database No. 17 (Westley et al. 1998), which is a large database for reaction rate parameters. In view of the large number of entries in these and other compilations, we shall not refer to each individual study in most cases, but rather to the databases where references to the studies can be found.

16.5.2

BASIC CONCEPTS

Before discussing the great potential of ignition delay studies for determining chemical mechanisms of combustion reactions, we first must clarify what an ignition process is, under what conditions it occurs, and why is it preceded by an induction period. To understand this phenomenon, consider the following legend. The legend tells us that the inventor of the chess game was asked to state what reward he felt he should receive for his very sophisticated invention (there is still controversy over when and where this happened). Modest as he was, his reply was very simple. He asked for the wheat grains that were to be placed on the chessboard in the manner shown in Fig. 16.5.1. One grain was to be placed on the first square, two on the second, four on the third,

16.5

Ignition Delay Times

FIGURE 16.5.1 Plate 6).

213

Wheat grains arranged on a chessboard in a geometrical series. (See Color

eight on the fourth, sixteen on the fifth, and so on until the 64th square was filled. At first glance, the number of grains required to fulfil this request did not seem to be of great concern. It becomes obvious, however, that the number passes the limits of imagination as the last several squares are filled. This is illustrated in Fig. 16.5.2, where the percent of grains out of the total is plotted against the square number on the board. We can see here the typical behavior of a process that advances according to a geometrical series or, in other words, advances exponentially. Very little happens during a considerable part of the process, and almost everything happens during a very short section toward the end.

The legend says that it took some time to realize that so many wheat grains could not be obtained in the entire country, and the great inventor was executed for his exaggerated request. If now, in a figure similar to Fig. 16.5.2 one replaces the abscissa by a time scale and the ordinate by the percent energy released in an exothermal reaction, a phenomenon known as an ignition will be seen. It is characterized by a behavior where very little energy is released during a given period of time (an induction time) and almost the entire heat of reaction is released during a very short time at the end. Such an energy release is the ignition, and it is preceded by the induction time, also known as the ignition delay time. The ignition in a mixture of propylnitrile and oxygen diluted in argon is shown as an example in Fig. 16.5.3. The pressure, which is an indication of the energy release in the combustion process, is almost constant during an induction period of about 2 ms, then it suddenly rises from its plateau value

214

A. Lifshitz

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at the onset of ignition. This induction period and the ignition phenomenon, which resemble the wheat grain legend, are the direct outcome of the exponential nature of the overall reaction rate, which results from the presence of chain-branching reactions and an adiabatic temperature rise during the

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16.5 IgnitionDelay Times

215

course of the exothermal reaction. As can be seen in Fig. 16.5.3 the ignition delay time is a readily measurable quantity. Its magnitude depends on the initial pressure, the composition of the ignited mixture, and the temperature, and it is determined by the elementary chemical reactions that govern the process. The measurement of its duration and the determination of its dependence on the experimental variables of the system provide a very powerful tool for elucidating the reaction mechanism and the combustion kinetics. It is useful to summarize the dependence of the ignition delay times on the composition of the system and on the temperature in a simple parametric relation that can later serve as a basis for computer modeling. It has been shown in numerous ignition studies behind shock waves (see Table 16.5.2, p. 220) that the general relation between the induction times and the concentrations is very similar to the relation between a rate of a chemical reaction and the concentrations (Lifshitz et al. 1971); that is, tignition--AI-Ic~ii where tignition is the ignition delay time, Ci is a concentration of a component i, and ]~i is a parameter somewhat similar to an empirical reaction order. It has also been shown that the concentration independent parameter A can be presented as A = 10~ exp[E/RT] an expression very similar to a rate constant (except that A decreases with temperature). The parameters E and ~i are determined by a complex kinetics scheme. They are experimentally determined quantities and provide a very useful means to summarize the experimental results in a quantitative manner. After establishing an empirical relation as above and determining the parameters, one can perform computer simulations under conditions similar to laboratory experiments and try, for a given reaction scheme, to reproduce these parameters. One can then arbitrarily vary the magnitude of the various rate constants in the kinetics scheme and examine the influence of such variations on the magnitude of the ignition delay time and its dependence on the concentrations and the temperature. From the results of such simulations, the role of each reaction in the overall mechanism can be elucidated. By employing such methods, many interesting combustion schemes have been analyzed, and details of the kinetics and an understanding of the oxidation mechanisms have been obtained. The following sections discuss several such systems.

216

A. Lifshitz

16.5.3 METHODOLOGY 16.5.3.1

EXPERIMENTAL MEASUREMENTS

To measure the induction time that precedes the ignition of a fuel, a premixed, gas-phase, reaction mixture of a fuel and an oxidant is suddenly raised to a high temperature to initiate a combustion reaction. The shock tube is an ideal tool for performing such an experiment. The gas in the tube is very rapidly heated to a high temperature, and it is maintained and reacts at that temperature for a measurable period of time, usually in the millisecond range. Induction times shorter than this dwell time can thus be measured. There are a number of experimental properties that can signify the ignition phenomenon: pressure (most of the studies shown in Table 16.5.2); heat flux (Burcat et al. 1971b; Lifshitz and Schechner 1975); total emission and emission of specific species in the hot gas (Baug~ et al. 1997; Borisov et al. 1988a); UV, IR, and laser absorption (Eubank et al. 1981a, 1981b; Gardiner et al. 1987, Petersen et al. 1996); density gradients (Gardiner et al. 1966); and combinations of these properties (Borisov et al. 1983) (and there are many additional studies describing experiments using these properties). The most commonly used method, both in laboratory and computer modeling, is pressure observations (Fig. 16.5.3) A miniature high-frequency pressure transducer is placed at the end plate of the tube's driven section, and the pressure is followed under the reflected shock conditions. If the experiment is carried out in a single-pulse shock tube, where the reaction mixture is very rapidly cooled at the end of the dwell time, a sample can be withdrawn from the tube to be analyzed for stable products at pre- or post-ignition conditions (Burcat and Radhakrishnan 1985, Burcat et al. 1996a). A comparison between the experimental and the calculated product distribution can add to our ability to elucidate the overall mechanism. Since ignition delay times can be measured with reasonable accuracy and low scatter, the parameters E and ]~i can be determined by least squares analysis with relatively high confidence.

16.5.3.2

D E S I G N OF AN E X P E R I M E N T AND

DATA P R O C E S S I N G To determine the parameters E and ]~i, a number of reaction mixtures are prepared and the induction times in each mixture are measured over a wide temperature range. The compositions of the mixtures and the initial pressures are chosen in such a manner that trial values for the ]~i can be independently reduced by comparisons between induction times in a pair of experiments.

16.5

217

Ignition Delay Times

TABLE 16.5.1 Mixture Compositions and Initial Pressures of Six Groups of Experiments in Acetonitrile Ignition Delay Studies Group A B C D E F

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Final v a l u e s of t h e s e p a r a m e t e r s , as well as the v a l u e of E, are later o b t a i n e d by a least s q u a r e s analysis of all the data points. A n e x a m p l e of s u c h a p r o c e d u r e is d e m o n s t r a t e d in Figs. 1 6 . 5 . 4 - 1 6 . 5 . 7 , o b t a i n e d f r o m the i g n i t i o n s t u d y of a c e t o n i t r i l e ( C H 3 C N ) (Lifshitz et al. 1997). F o r this study, t h r e e r e a c t i o n m i x t u r e s w e r e p r e p a r e d a n d r u n at different initial p r e s s u r e s , as s h o w n in Table 16.5.1. A c o m p a r i s o n b e t w e e n i n d u c t i o n times r u n w i t h m i x t u r e s D a n d F (Fig. 16.5.4) gives the effect of a t w o f o l d difference in the o x y g e n c o n c e n t r a t i o n in m i x t u r e s w h e r e the fuel c o n c e n t r a t i o n s a n d the initial p r e s s u r e s are the same. :, ...........

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(The deviation from an exact factor of 2 comes to compensate for the different shock compression in the two mixtures.) The figure shows a strong enhancing effect of the oxygen, and ]~oxygen can be calculated from the distance between the points in the two groups of experiments. The solid lines on the figure are the results of computer modeling, which will be discussed later. A similar plot is shown in Fig. 16.5.5 where mixtures B and D are compared. The figure shows the effect of the fuel concentration on the induction times. The oxygen concentrations and the initial pressures in the two mixtures are the same, but there is a twofold difference in the fuel concentrations. Figure 16.5.5 shows a small inhibiting effect of the fuel; in the mixture with the high fuel concentration, the induction times are somewhat longer. The effect of the inert gas concentration on the induction times can be seen in Fig. 16.5.6, where mixtures A and F are compared. In these two mixtures, the concentrations of both the fuel and the oxygen are the same. The mixture with the low percentage of the components is run at Pl -" 120 Torr, whereas the mixture with the high percentage is run at an initial pressure of P] = 60 Torr. The only difference between the two mixtures is thus the third-body density. A small inhibiting effect of the argon can be seen.

16.5

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16.5

m

I~ition Delay Times

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A. Lifshitz 4.5

E=41.7 kcal/mol

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4.0 ..< O

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1 000IT (K -~) FIGURE 16.5.7 A plot of log2 vs 1/T, where 2 --tignition/{[CH3CN]~176 0"34}for data points from six series of experiments. With the above correlation, all the points scatter along one line with a slope of 41.7kcal/mol. (See Color Plate 10).

temperature, and pressure--provides the experimental parameters in the empirical relation tignition -- 10 ~ exp(E/RT) [-[ C~i' so that a great number of tests can be expressed in terms of the values of 0~, E and ]~i- One agreement between the computer simulations and the laboratory experiments is the comparable induction times, but more significant are the similar (at least qualitatively) values of E and ]~iThe computer simulations are carried out in the following way. A kinetics scheme believed to describe the overall combustion is numerically integrated using the same initial conditions as used in the laboratory experiments. This yields concentration, pressure, and temperature time history during the experimental dwell time. The ignition can be detected by a sudden rise of the pressure from its plateau value. An example of calculated pressure profiles at 1500 and 1600 K are shown in Fig. 16.5.8 for a mixture of CH3CN + 0 2 diluted in argon, using a reaction scheme that will be discussed later. There are a number of experimental conditions under which a set of coupled differential equations representing an overall kinetics scheme in a shock tube can be numerically integrated: under constant density, under constant pressure, or more precisely coupled to the shock equations. The

225

16.5 Ignition Delay Times

r

5.5

E 9~,

5.0

1600 K

t~ tl) '--

K

4.5

oo

o~ I1)

1 500

~,, ~,i 4.0

ignition

'un"'~

3.5 ~ 0

,

, 200

, 400

600

800

t(ps)

FIGURE 16.5.8 Calculated pressure profiles for the ignition of a mixture of CH3CN + 02 in argon at 1500 and 1600 K, using the reaction scheme shown in Table 16.5.3. The ignition points can be easily determined.

latter is very cumbersome and requires considerable and sometimes inaccessible computer time, depending on the number of species and elementary reactions involved. The second best is the use of constant density conditions, especially when the pressure rise upon ignition is measured at the end plate of the driven section. With this method, the shock equations are solved without chemistry to establish the initial conditions behind the shock wave, and the kinetics scheme is then numerically integrated starting at these conditions. Since there are temperature changes during the induction time, the specific assumption of constant density or constant pressure might affect the "progress of the reaction". The temperature increase during the induction time is a very important factor. It will be evaluated differently, whether constant density (ATp = AH/Cv) or constant pressure (ATp = AH/Cp) is assumed. It has been found that whereas the actual value of tignition might be somewhat affected by the method of calculation (shorter tignition for constant p), its effect on E and flimwhich are always evaluated by ratios of induction timesmis small. Most of the calculations that will be referred to in this chapter were carried out under constant density assumption. One purpose of the computer simulations is to perform sensitivity analysis on the reaction scheme. Rate constants of all the elementary steps, both forward and reverse, are varied in turn by an arbitrary factor or eliminated completely from the scheme, and the effect of such an operation on the induction times is examined. In addition, the influence of these changes on the experimental parameters E and fli can also be obtained. Such a sensitivity

226

A. Lifshitz

analysis provides a very important tool for elucidating the role of each one of the elementary steps in the overall mechanism.

16.5.4 KINETICS SYSTEMS 16.5.4.1

INTRODUCTORY REMARKS

The kinetics scheme that gives a full account on a mechanism of a combustion reaction is composed of many elementary reactions of different types. In the past, these were divided into three categories: initiation, propagation (straight and branched chains), and recombinations as chain-breaking reactions. Many discussions were thus centered around these types of elementary reactions. From numerous modeling studies performed in recent years, it has become clear that the initiation, for example, although a necessary step that determines the behavior of the system at the beginning of the process, becomes an unimportant step when chain-branching reactions exist (Seery and Bowman 1970 in: Smith et al. 1999, GRI-Mech. 3.0, ig.lb). In addition, recombinations have very often negligible effect on the progress of the reaction since free radical loss due to recombinations are overcompensated for by chain-branching reactions. As will be shown later, other types of reactionsmsuch as competition reactions, for examplemdo exist and are very important in determining the character of the overall process.

16.5.4.2 I G N I T I O N OF SMALL M O L E C U L E S : THE LOOP CONCEPT Let us examine the ignition characteristics of three fuels: COS (Lifshitz et al. 1975), C2N 2 (Lifshitz et al. 1974; Lifshitz and Frenklach 1980), and CH 4 (Lifshitz et al. 1971; Tsuboi and Wagner 1974). All react with oxygen to produce either CO or CO 2, depending on the fuel/oxygen ratio, and ignite at high temperatures with the release of a considerable amount of energy: COS + 0 2 --~ CO + SO2

AH~ = - 6 3 kcal/mol

C2N 2 q- 0 2 --)- 2CO 4- N 2

AH0 = - 1 2 6 kcal/mol

1/2CH 4 + 0 2 --~ 1/2CO 2 + H20

AH~ = - 9 6 kcal/mol

At first glance, the first two reactions, except for the extent of exothermicity, seem to be very much alike. However, an examination of the power depen-

16.5

227

Ignition Delay Times

dencies of the ignition delay times on the concentrations reveals an entirely different behavior of the ignition parameters ~i in the parametric relation tignition - A I-[ C~i i In the COS 4- 02 system, a strong dependence of tignition on the oxygen concentration ( ~ ( o 2 ) = - 1 . 1 2 ) (an enhancing effect) and a weak inhibiting effect of the fuel (]~(cos)= 4-0.30) is observed. On the other hand, the C2N 2 4- 0 2 system shows a very strong dependence on the fuel concentration (]~(C2N2) = --1.01) and a rather weak dependence on the oxygen concentration (]~(o2) = -0.21). The ignition of methane gives a picture similar to that of COS. Its ignition delay is very strongly shortened by the increase in the oxygen concentration ( / / ( o 2 ) = - 1 . 0 3 ) and is somewhat higher when the methane concentration is increased (fl(cH4)= 4-0.33). The question is whether the completely different behavior of COS and C2N 2 on one hand and the similarity between the COS and C H 4 o n the other, can be explained on the basis of the same line of reasoning or whether this behavior is the result of independent phenomena. An examination of the kinetics schemes of the three systems shows that the following pairs of reactions, SO + 0 2 ~ SO 2 + O

(1)

o + c o s -~ c o + s o

(2)

CN + 0 2 --~ N CO + O O 4- C2N 2 --+ NCO + CN

(i) (2)

CH 3 4- 02 -~ CH 3 4- O

(1)

O 4- CH 4 --~ OH 4- CH 3

(2)

have the highest rates in the schemes. It shows also that the rates of the two reactions in each system are exactly the same. This behavior results from the fact that each pair of reactions forms a closed loop. (One of the reactions in each loop [reaction 2] is also a chain-branching reaction.) The question that arises now is which one of the two reactions in each loop determines the rate of the loop and what are the conclusions that can be drawn from the answer to this question. In the COS 4-02 system, the C - S bond is the weakest bond in the fuel (D O -'~ 74 cal/mol). It is much weaker than the bond in the oxygen molecule (D o = 120 kcal/mol). Therefore, the activation energy of reaction 1 is much higher than the activation energy of reaction 2 and thus k 2 is much higher than k I . For roughly equal concentrations of the fuel and the oxygen, the rate of the loop will be determined by the rate of reaction 1, rate = k1[O2][SO~ Increas-

228

A. Lifshitz

ing the oxygen concentration will increase the rate of the loop, and one can thus expect that the overall rate of the ignition process will increase. This will be reflected by shortening the ignition delay time. Increasing the fuel concentration, on the other hand, will not affect the ignition delay at all. In fact, the computer modeling of the kinetics scheme (Fig. 16.5.9) shows that the increase in the fuel concentration simply decreases the oxygen atom concentration so as to keep the rate of reaction 2 unchanged, equal to the rate of reaction 1. This analysis is compatible with the experimental observations [Lifshitz et al. 1975). In the C2N 2 4- 02 system, the dissociation energy of the C - C bond in C2N 2 (D O ~ 135 kcal/mol) (Stein et al. 1994) is considerably higher than the energy of O - O bond in oxygen. Therefore, k 1 )>)>k 2 and thus for roughly equal concentrations of the fuel and the oxygen, the rate of the loop will be determined by the rate of reaction 2, rate = k2[O'][C2N2]. Increasing the fuel concentration will increase the rate of the loop, and one can thus expect that the overall rate will increase. Increasing the oxygen concentration in this system will result in a decrease in the CN" concentration (Lifshitz et al. 1974), and the rate of the loop will not be affected. Indeed, the ignition delay times are only slightly dependent on the oxygen concentration but highly dependent on the cyanogen concentration. Methane, although very different from carbonyl sulfide in many respects, behaves in a similar manner. Its induction time is enhanced by the oxygen and

[o] 8

4% C O S , 12% 0 2

o ~"

"4 10

\

0 ~15

8% COS, 12% 0 2

12

0 I

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t (ps) FIGURE 16.5.9 Profilesof oxygen atoms in two different concentrations of COS increasing the COS concentration results in decreasing the O-atom concentration so as to keep the rate of the reaction COS 4- O --> CO 4- SO unchanged.

16.5 IgnitionDelay Times

229

is slightly inhibited by the methane (Lifshitz et al. 1971; Tsuboi and Wagner 1974). The reaction rate constant of the reaction O ~ + C H 4 is several orders of magnitude higher than the rate constant of the reaction CH] + 02 (Westley et al. 1998). Therefore, the latter determines the rate of the loop and the increase in its rate will shorten the induction period. The reaction O ~ 4-CH 4 has no effect on the induction time (Seery and Bowman 1970 in: Smith et al. 1999, GRI-Mech. 3.0, ig.lb). The strong dependence of the induction time on the oxygen concentration and its very small dependence on the methane concentration is the direct result of the nature of the loop. In the three systems just described, the character of the thermal loop determines to a large extent the behavior of the overall oxidation. Such a behavior is typical to systems where the thermal decomposition prior to oxidation is not significant. In higher hydrocarbonsmpropane, for examplem where there is an early decomposition, the loop loses some of its significance.

16.5.4.3

THERMAL IGNITION WITHOUT

BRANCHING: N20

-+- C O S ,

N20

CHAIN

+ CO

As previously mentioned, chain-branching reactions are the dominant factor responsible for the ignition mode in combustion. However, there are cases where branching reactions cannot take place and the ignition results from heat release during the course of the reaction. If heat is released in a system of chemical reactions under adiabatic conditions, the temperature goes up and the rate increases exponentially, which is a condition for an ignition mode. This behavior can be found in systems where the oxidant is a molecule containing oxygen but it is not a molecular oxygen. Typical examples are the oxidation of carbonyl sulfide and carbon monoxide with nitrous oxide (N20) as an oxidant (Lifshitz and Kahana 1978, Borisov et al. 1978b). A mixture of N20 and COS is a very reactive mixture that readily ignites when raised to high temperatures, although chain branching does not take place (COS + O ~ ~ CO + SO~ The exponential behavior of the rate, which brings about the ignition phenomenon, results from an adiabatic release of energy and temperature rise during the course of the reaction. Since the supply of oxygen atoms comes from the selfdissociation of N20, the induction period is highly dependent on the initiation reaction, also fl(N20) = --1.09. Another test, which may indicate to what extent the temperature increase during the induction time is a dominant factor in ignition process, is a variation in the specific heat of the system. Indeed, when the heat capacity of the COS + N20 + Ar mixture is artificially increased by a factor of 2 in the modelling calculations, the induction times increase by close to 90% (Lifshitz and Kahana 1978). When chain-branching reactions are the

230

A. Lifshitz

dominant factor, the variation in the specific heat has only a minor effect on the ignition delay times.

16.5.4.4

T H E C O N C E P T OF ENERGY

BRANCHING- H 2 q- C12, H 2 q- F 2 The concept of energy branching was raised by Semenov (1962), who suggested the possibility of energy branching in the hydrogen chlorine reaction, where vibrationally excited HC1 formed in the H + C12 --~ HCI* + C1 reaction can transfer its vibrational energy directly to molecular chlorine and enhance its dissociation. To rationalize the fact that multiquantum transitions must occur very efficiently, he suggested a formation of a [HC13] complex, the lifetime of which is sufficiently long to allow a complete equilibration of the vibrational energy among the various degrees of freedom of the complex. This activated complex then dissociates, obeying the theory of unimolecular reactions, with one of the channels being [HC13]*---~ HC1 + C1~ + Cl'. This idea was verified by modeling the results of shock tube measurements of the ignition delay times in mixtures containing C12 and H 2 diluted in argon (Lifshitz and Schechner 1975). A reaction scheme based on a simple exothermal chain propagation, H ~ + C12 --+ HC1 + C1~ C1~ + H 2 @ HC1 + H" could not reproduce the experimental parametric relation. Also, the calculated induction times were longer than the measured ones for the same temperature, pressure and composition by about a factor of 2 and, the calculated Arrhenius temperature dependence was ~40% higher than the measured value. When the energy-branching reactions HCl*(v)i + C12 ~ [HC13]* --+ HCI(v -- 0) + C1 + C1 (i - 0 - 4) were added to the reaction scheme with the right vibrational distribution in HC1 molecules, an excellent agreement with the experimental results was obtained. The preexponential factors in this process were assumed to be independent of v and the activation energies were calculated from the relation Eactivation,v - - EC12 --E v

+

Eo

where Ec12 is the bond dissociation energy of C12, E v is the vibrational energy of the HC1, and E 0 is its ground state energy. Translation --~ vibration deactivation was also taken into account (Lifshitz and Schechner 1975).

16.5 IgnitionDelay Times

231

Energy branching is believed to be a dominant factor in the thermal reactions of C12 and F 2 with hydrogen, where the HX is obtained in vibrationally excited states.

1 6 . 5 . 4 . 5 CORRELATION OF IGNITION DELAY TIMES WITH BOND DISSOCIATION ENERGIES: THE ROLE OF INITIATION VS CHAIN BRANCHING Ignition delay times vary by orders of magnitude from one fuel molecule to another. The temperature must be changed by hundreds of degrees to obtain identical ignition delays in different systems. Ignition delays (300ps, for example) in CH 4 and CH3NO 2 are measured at temperatures that differ by some 700 K (Lifshitz et al. 1990). This is demonstrated in Fig. 16.5.10, which shows plots of log tignition vs 1/T for a series of fuel molecules. As can be seen, there is a large spread in the reactivity of these molecules. The question is what the molecular properties are with which the ignition delays correlate. To answer this question, let us examine Fig. 16.5.11, which correlates the ignition delay times of six different molecules with their lowest bond dissociation energy. The figure shows a plot of 1/T vs Dc_ x, where the temperatures correspond to the same ignition delay time of 316 l~s (log tignition = 2.5) for all the six molecules. The temperatures at which the ignition delays are 316 ~s were calculated from the parametric relations of each system (Table 16.5.2) at

1600 I

3.5

1400 I-

1200 I

I

CH3-CN 0 (t)

1000 I

I

900 I

CH3-Br C H 3-

3.0

7-- 2.5 .9 ._ C O) O

2.0

.p-

CH3-CH 3 NO

o 1.5

2

I 1 I * J I I I I l | I I I * I I I i I ] i i i 1 i i i 0.6

0.8

1.0

|

1.2

1000/T (K -~) FIGURE

energies.

16.5.10 A plot of

log(tignition , ~S)

VS 1/T for fuels with different bond dissociation

232

A. Lifshitz

~ignition = 318 IJs 1.1 1.0

1000

0.9

1100 C H3- N H 2 " ~ I I CH3"C H3

0 X

I---

o.8

1300

C2HsCN9 ~ o.z

1500

CHa"oNto

0.6

CHa-H 0.5

1700

,,,,,,~,,IL,,,i,,J,l~,,,,,,,,

50

7O

9O

110

Dc. x (kcal/mol) FIGURE 16.5.11 A plot of 1/T vs De_ x, which is the weakest bond in the fuel. The temperature on the ordinate at each point corresponds to an ignition delay time of 318 ~s (1ogtignition "- 2.5).

similar fuel, oxygen, and argon concentrations. The bond dissociation energy is a good measure of the initiation rate, so the correlation shown in the figure is essentially the relation between the ignition delay times and the rate of initiation. The stronger the bond dissociation energy (lower initiation rate), the higher the temperature that is required to obtain the same ignition delay time. The question that now arises is, to what extent does this strong dependence of the ignition delay on the rate of initiation express itself in the sensitivity analysis of the kinetics scheme? In many cases it does not. In the system C H 4 + 0 2 , for example, the rate of initiation, C H 4 + M ~ CH~ + H ~ M, does not appear at all in the sensitivity list (Seery and Bowman 1970 in: Smith et al. 1999, GRI-Mech. 3.0, ig.lb). Doubling the initiation rate COS + M ~ CO + S ~ M in the system COS + 02 decreases the ignition delay by only 8% (Lifshitz et al. 1975). At first glance, these two observations might look contradictory. Why is there such a strong dependence of the induction times on the rate of initiation on one hand, and no dependence on this rate in the sensitivity analysis of the kinetics scheme on the other? In shock tube studies, ignition can occur only if the delay time does not exceed a period of 1 to 2 ms, the available reaction times in the shock tube. Moreover, at longer times, heat transfer and diffusion of hydrogen atoms to the walls as well as surface recombinations slow down the process and prevent the

16.5 IgnitionDelay Times

233

ignition from taking place. When the dissociations are too slow, the initial concentrations of free radicals are insufficient for the chain branching to bring them up to a critical value in a matter of a few milliseconds. Ignition will therefore not occur. For ignition to occur, the initial concentrations of free radicals must reach a minimum value, so that branching can increase their number exponentially until the required critical value is achieved. One can therefore state that the temperature range over which ignition will take place is determined almost solely by the initiation rate of one of the reacting partners, as is demonstrated in Fig. 16.5.10. However, once sufficient free radicals are initially formed, the role of initiation in increasing the concentrations of the free radicals becomes negligible compared to the role of the chain branching. Therefore, over the temperature range where ignition can occur, the sensitivity of the system to the initiation reactions is normally small. Both the experimental findings and the explanation hold only when chainbranching reactions are the major factor in the kinetics scheme. In systems where branching does not OccurmCOS + 2N20 ~ SO 2 + 2N 2 + CO (Lifshitz and Kahana 1978) and N20 + CO ~ N 2 + CO2 (Borisov et al. 1978b), for example--and the ignition is the result solely of the temperature increase during the course of the reaction, the findings are completely different. The sensitivity of the ignition delay to the dissociation rate of N20 is high. In the system C O S + N 2 0 , it decreases by 45% for a twofold increase in the dissociation rate constant (Lifshitz and Kahana 1978).

16.5.4.6 T H E D E P E N D E N C E OF THE I G N I T I O N DELAY T I M E ON THE F U E L C O N C E N T R A T I O N 16.5.4.6.1 Inhibiting Effects due to Competition Reactions: Hydrocarbons In addition to the importance of initiation, termination, and chain branching, competition reactions play a very important role in the progress of the ignition process. In fact, as we shall see later, the dominant factor that determines the dependence of the induction times on the fuel concentration is the competition on active free radicals between the fuel and the oxygen. In the hydrocarbon system, methane for example, hydrogen atoms are produced by the initiation reaction CH 4

q- M ~ CH~ + H ~ + M

234

A. Lifshitz

After hydrogen atoms have been produced, a competition between the fuel and the oxygen on the hydrogen atoms takes place: H"/7 +CH4 --+ CH~ + H 2 "N + 0 2 _ + O H ' + O ~ The net rate of these two reactions are roughly the same for the same oxygen and fuel concentration. However, whereas the reaction with oxygen is a chainbranching reaction that is extremely important to the ignition process, the reaction with methane, which competes on the H atoms, is a much less important linear chain and has therefore an inhibiting effect. It not only prevents the chain branching from taking place, it also replaces a reactive hydrogen atom by a considerably less active methyl radical. The positive power dependence, fl(CHO -- +0.33 (Lifshitz et al. 1971; Tsuboi and Wagner 1974), is the direct result of the effect of the reaction of H ~ with C H 4. Indeed, if this step is artificially removed from the reaction scheme, the calculated positive power dependence disappears, and the induction times become independent of the methane concentration. This behavior changes at very low equivalence ratios, 1/10, for example, where flmethane is negative. Under these conditions, the ability of the fuel to compete with the oxygen diminishes drastically and then no longer inhibits the progress of the reaction. Similar behavior is found in higher hydrocarbons, R1-R2, both aliphatic and aromatic, where H atoms are not the main product of the dissociation. Here the competition between the hydrocarbon and the oxygen on R~ is in the following form (Baker and Skinner 1972; Burcat et al. 1972; Hidaka et al, 1981; Burcat et al. 1986b; Hidaka et al. 1983; Burcat and Dvinyaninov 1995; Burcat et al. 1996b; Burcat et al. 1979; Thyagarajan 1990):

RI/7 +R1-R2 '+ R~ + R1H "~ + 0 2 ---> R1O~ + O ~ In the first reaction, a small radical is producing a larger and less-reactive species whereas the second reaction is a chain branching. As can be seen in Table 16.5.2, a positive power dependence on the fuel concentration is common to almost all hydrocarbons and is dictated by this competition reaction. At high temperatures, however, when the dissociation of the higher hydrocarbons becomes quite significant, the dissociation begins to appear among the reactions that shorten the ignition delay. The system COS + 02 is another example of a system with a series of competition reactions of different nature that determine the inhibiting effect (positive 13) of COS (Lifshitz et al. 1975). The process begins with the formation of sulfur atoms by the self-dissociation of COS: COS + M --~ CO + S~ + M

16.5

235

Ignition Delay Times

The oxygen molecule and carbonyl sulfide compete on the S atoms in the same manner as in the methane oxygen system: S" 7 + 0 2 ~ S O ' + O" + C O S --+ CO + S2

The reaction with the oxygen is a chain-branching reaction, whereas the reaction with COS is a termination. The latter is in fact the reason for the inhibiting effect of the fuel. There are other competition reactions in this system of a quite different nature, such as the two parallel reactions of oxygen atoms with COS:

c o s + o" "~ c o + so" CO2 + S~ The first reaction is simply a propagation reaction, which important as it may be does not lead to branching. The second reaction produces sulfur atoms that do enter into a branching reaction: S" -~- 0 2 --> SO" -~- O"

The first reaction has an inhibiting effect and the second has an accelerating effect, so they practically cancel one another. The inhibiting effect of the COS fuel comes solely from the reaction S~ + COS --~ CO + $2 which is practically a termination reaction. It prevents the S" atoms from entering into a chain-branching reaction with the oxygen molecule. 16.5.4.6.2 Enhancing Effects due to Thermal Excitation: The Epoxy Group of Molecules In contrast to the ignition characteristics of hydrocarbons where the fuels have positive ]~fuel values, there are fuels in which other factors play a role in the oxidation process and inverse the picture. A very interesting example of such a behavior is the epoxy group of molecules. As can be seen in Table 16.5.3, the common feature in the ignition of this group is the strong enhancing effect of the fuel (a large negative value for/~fuel) and the large inhibiting effect of the diluent. Although the competition reactions discussed earlier play a role in any C - H system, including the epoxy group of molecules, their inhibiting effect is overcompensated in the present line of fuels by two other factors that do not exist in simple hydrocarbons. The three-membered epoxy ring, being an unstable structure, tends to quickly open and isomerize to more stable molecules such as ketones, aldehydes, alcohols, and ethers (Lifshitz and Tamburu 1994, 1995). In view

A. Lifshitz

236 TABLE 16.5.3 Experimental Ignition Parameters for Various Epoxy Fuels: z = 10~exp(E/RT)[Fuel]~[O2]&[M]&

o~

E (kcal/mol)

//1

//2

Ethylene oxide, 02, Ar

- 12.62

29.6

-0.44

-0.72

Ethylene oxide, 02, N2 Propylene oxide, 02, Ar

-12.81 -14.28

30.8 33.6

-0.82 -1.07

-0.91 -0.52

1,2-Epoxybutane, 02, Ar - 14.56

31.5

-0.86

-0.72

2,3-Epoxybutane, 02, Ar -15.73

34.3

-0.92

-0.76

Components

]~3

Reference

0.78 Lifshitzand Suslensky [1995] 1.48 Burcat [1980] 0.78 Lifshitzand Suslensky [1995] 0.92 Lifshitzand Suslensky [1995] 0.89 Lifshitzand Suslensky [1995]

of the large differences in the heats of formation of the epoxy molecules and some of their stable isomers, the more stable molecules are produced with excess thermal energy. This excess energy is equal to Eact + AHr, where Eact is the activation energy of the isomerization process and AH r is its exothermicity. An energy diagram for the isomerization of propylene oxide is shown as an example in Fig. 16.5.12. The thermally excited acetone, and to a lesser extent propanal, can now either lose their energy by collision with the diluent to produce stable isomers, or, as shown in Fig. 16.5.12, can decompose to produce free radicals that support the chain reactions in the process. This feature, which is common to all the epoxy molecules, is responsible for the high production rate of free radicals. Hydrocarbon fuels dissociate from a state of a Boltzmann distribution corresponding to the temperature of the bulk. Since in most cases this is a slow process, the contribution of simple fuel dissociation to free radical production is negligible. Chain branching takes over at the very early stages of the reaction. The competition reaction H ~ 4- RH -~ H 2 4- R ~ which is a fast reaction, is not compensated by any enhancing reactions in which the fuel is involved. In view of the dissociation from a thermally excited state, the dissociation in the epoxy group of molecules is very fast compared to a normal hydrocarbon dissociation. Its contribution to the overall production of free radicals is therefore important during most of the induction time and can compete with the contribution from the chain-branching processes. This contribution overrides the inhibiting effect of the fuel and leaves an overall enhancing effect. It expresses itself experimentally by the high negative power dependence of the induction times on the fuel concentration (Lifshitz and Suslensky 1995).

16.5

237

Ignition Delay Times

energy level of the thermally excited isomers C H +CHO

CH2CH=CH2+OH o!

......

/ _ E-59+2 kcal/mol [

....

i

....

" kcal/mol AH=62 kcal/mol [

......

.... I .........

] [

AI4--~ 1 k~al/mol

kcal/mol

[

CH3CH--CH 2 ........ - " " '" """ . . . . . . . . . . . . . I . . . . . . . . . . . . 4 , - 9 . 4 \ / CH2=CH-O-CH ~ [ -'- ~--O -22.7 [ CH2=CHCH2OH I

,-3o.8

CH3COCH 3 FIGURE 16.5.12 Energy diagram for the isomerization of propylene oxide. Acetone and, to some extent, propanal are formed in a thermally excited state with enough energy to dissociate from this state before being de-excited.

When the dissociation from the thermally excited state was introduced into the kinetics scheme, a very good agreement between the experimental and calculated induction times and power dependencies was obtained. When this step was removed from the scheme, the agreement was very poor. An additional factor in the enhancing effect of the fuel is the large amount of heat released in the process of isomerizations. There are several nitrogen- and oxygen-containing fuels that show a negative power dependence on the fuel concentration (see Table 16.5.2). Not all are clear and easily explained unless there are no competition reactions that can cause inhibition. One very important group of fuels is the unstable hydrazines (Abid et al. 1991; Catoire et al. 1994, 1997, 1999). Their enhancing effect results probably from their strong exothermic decomposition during the induction period. Frenklach and coworkers discussed a general relation between exothermicity and induction times (Rabinowitz and Frenklach 1987). Another group of molecules is the organic amines ( - N H 2) (Lifshitz et al. 1991; Lifshitz and Suslensky 1999). It has already been suggested in the study of the ignition of monomethylamine that the normal inhibiting effect of the fuel is compensated by additional reactions that are not effective in the ignition of simple hydrocarbons. These reactions are involved in the produc-

238

A. Lifshitz

tion of H202, which then decomposes quite rapidly to two OH radicals. The sensitivity analysis of the monomethylamine ignition (Lifshitz et al. 1991) shows that the ignition delay times in this system are highly sensitive to the reaction CH3NH 2 + HO~ --~ CH~NH 2 + H202 The ignition schemes do not show sensitivity to the equivalent reactions in simple hydrocarbons.

16.5.4.7

I N H I B I T I N G E F F E C T S OF THE D I L U E N T

As can be seen in Table 16.5.2, in almost all cases where the power dependence of the ignition delay on the diluent concentration//diluent was determined the dependence was positive. Increasing the diluent concentration inhibits the process. In hydrocarbon and many other fuels the diluent (nitrogen if air is being used, or simply argon) plays two different roles during the induction period. One is the role of a third body in dissociation-recombination reactions. Due to chain branching, the concentrations of free radicals in the system overshoot their equilibrium concentrations at the early stages of the reaction. The role of argon or nitrogen as third bodies is thus to enhance recombinations rather than dissociations and to decrease the overall concentration of free radicals. This would manifest itself in a positive, although small, power dependence. The second effect of the diluent is a cooling effect. Since in many systems heat is released as the reaction proceeds, the temperature and the overall rate increase. An increase in heat capacity for the same amount of heat release--by increasing the diluent concentrationmdepresses the temperature elevation and thus inhibits the progress of the reaction. In the epoxy group of molecules, for example, a considerable amount of heat is generated in the process of isomerization so that this effect becomes quite significant. As can be seen in Table 16.5.3,//argon in the epoxy group is high compared to the values that we find for most of the other fuels (Table 16.5.2). There is an additional and more significant inhibiting effect of the diluent in the epoxy group: the quenching of the thermally excited species that are formed in the isomerization (Fig. 16.5.12) (Lifshitz and Tamburu 1994). Since these excited species can lose their excess thermal energy by collisions with argon (or nitrogen), an increase in the diluent concentration will increase the quenching rate and prevent the thermally excited molecules from dissociating to free radicals.

16.5

239

Ignition Delay Times

16.5.4.8

E F F E C T OF ADDITIVES

The addition of small amounts of fuels or oxidizers with lower bond dissociation energies shortens the induction times and varies their temperature and concentration dependencies. Borisov et al. (1988) and Zamansky and Borisov (1992) have studied the effects of a large number of additives on hydrocarbon ignition. They examined the effects of additives such as (CH3)2N2, CH3ONO , CH3ONO 2, N2F 2, n-C3H7NO 2, iso-C3H7NO2, and several others. The promoting effects of these additives was attributed to their fast decomposition, which produced a high concentration of free radicals. Propane and/or ethane added to methane (Smith et al. 1999, GRI-Mech. 3.0; Lifshitz et al. 1971; Frenklach and Bornside 1984; Yang et al. 1996), hydrogen to cyanogen (Lifshitz and Bidani 1980, 1986), and nitrogen dioxide to methane (Dorko et al. 1975; Dabora 1975; Burcat 1977) were also examined. In some cases the power dependencies of the ignition delay on the additive concentrations were determined (Spadaccini and Colket 1994; Lifshitz and Bidani 1980, 1986). These dependencies always had, as expected, negative 13 values. Interesting studies on the ignition of methane in the presence CH3-X (where X = C1, Br, or I) were reported by Baug6 et al. (1997) and Burcat et al. (1996c). The presence of these additives, which are known as inhibitors of combustion processes in general as well as flame retardants, cause a decrease in the induction times compared to those in pure methane. These observations imply that the inhibitions in halocarbons caused by the recombinations of Br or C1 with H and by other inhibiting reactions may be very significant, but are not strong enough to override the high decomposition rate due to the weak C-Br and C-C1 bonds.

16.5.5 COMPUTER MODELING

16.5.5.1 REACTION SCHEME The purpose of the computer modeling is to examine the validity of a suggested mechanism in light of the experimental observations. The main issue is to reproduce the induction periods and, perhaps more significantly, the parameters E and ]~i in the parametric relation tignition - 10~ exp[E/RT] I~ C3i i which is obtained by a least squares analysis of all the experimental data. The modeling procedures were discussed in section 16.5.3.3. In Table 16.5.4, the kinetics scheme of the ignition in a mixture of CH3CN and 02

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diluted in argon is shown as an example, and the mechanism is discussed here. The scheme contains 35 species and 111 elementary reactions. The rate constants listed in the table are given as k = A e x p ( - - E / R T ) in units of cm 3, s -I, and mo1-1, and kcal/mol. Column 1 lists all the elementary reactions in the scheme, column 2 gives the preexponential factors, and column 3 gives the activation energy of each reaction. In some cases, where curvature in the Arrhenius plots exists (normally when the rate constants are given for a wide temperature range), the preexponential factors are expressed as X • T n. Columns 4 and 5 give the rate constants of the forward and back reactions calculated at 1600 K, and columns 6 and 7 give, respectively, the standard entropy AS~ and standard enthalpy AH~ of the elementary reactions at the same temperature. The rate constants of the back reactions are calculated from the rate constants of the forward reactions using the relation k r = kf/Keq , where Keq is the equilibrium constant of each elementary reaction, calculated from the thermodynamic properties of the species that participate in the reaction, using the relation

Keq- e x p ( A S ~ 1 7 6

• (RT) A~

where Av is the change in the number of moles in the reaction and R = 82.01 cm 3 atm/(K.mol). The Arrhenius parameters for reactions in the scheme are either estimated or taken from various literature sources such as the NIST Chemical Kinetics Database No. 17 (Westly et al. 1998), GRI-Mech. 3.0 (Smith et al. 1999), and other compilations (Warnatz 1984; Atkinson et al. 1999; Baulch et al. 1994; Miller and Bowman 1989; Tsang and Hampson 1986). The parameters for the reactions that are taken from the various compilations and from the NIST Chemical Kinetics Database are, in most cases, a best fit to a number of entries. The thermodynamic properties of the species are taken from various literature sources (Smith et al. 1999, GRI-Mech. 3.0; Stull et al. 1969; Pedley et al. 1986; Melius 1999; Burcat and McBride 1997; Burcat 1999; Stein et al. 1994, 1998; Tsang and Hampson 1986) or are estimated by various methods including the NIST Standard Reference Database 25 (Stein et al. 1994). Calculated pressure profiles for this system were shown in Fig. 16.5.8. The agreement between the experimental and the calculated delay times of the CH3CN + 02 + Ar system can be seen in Figs. 16.5.4-16.5.6. The points on the figures are the experimental delay times, and the lines pass through three calculated induction times at 1500, 1600, and 1700 K, using the reaction scheme shown in table 16.5.4 and marked as crosses on the figures. Figure 16.5.13 shows a plot of log2 vs 1 / T for 60 calculated induction times, 10 calculations at the composition, and initial pressure of each one of the six

16.5

245

Ignition Delay Times -0.2

E=46.2 kcal/mol

-0.4

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-1,0 = tign/[OH3 O N] 0.43[02] -1lS[Ar] ~ te] -1.2 ' 0.55

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FIGURE 16.5.13 A plot of log2 vs 1/T where 2 - tignition/{[CH3CN ] " [02]- " [Ar] " } for 60 calculated points corresponding to the six series used in the experiments. With the above correlation, all the calculated points scatter along one line with a slope of 46.2 kcal/mol. (See Color Plate 11).

series of experiments. The line that passes through the calculated points on the figure was evaluated by a least squares analysis with the parameters ,~ - - tignition / {[CH3 C N ] ~ 1 7 6

}

compared to the experimental parameters 0 12

2--tignition/{[CH3CN]"

0 76

[O2]-"

0 34

[Ar]" }

(Lifshitz et al. 1997). Also, the calculated Arrhenius temperature dependence E in the relation 2 - 10 a exp[E/RT] which was obtained from the slope of the line in Fig. 16.5.13, was 46 kcal/mole as compared to 41.7 kcal/mol in the experiment. The calculated induction times are close to the measured values. The calculated parameters, overestimate somewhat the concentration effect as compared to the experimental parameters, but the general behavior is very much alike. There is a small inhibiting effect of the fuel and a strong enhancing effect of the oxygen. Such an agreement can be considered as quite satisfactory. It can be improved, however, by further refinement of the kinetics scheme. In any case, the main features are clearly seen in the present scheme.

246

A. Lifshitz

1 6 . 5 . 5 . 2 SENSITIVITY SPECTRUM To e s t a b l i s h a b e t t e r a g r e e m e n t b e t w e e n the c o m p u t e d a n d the e x p e r i m e n t a l i n d u c t i o n t i m e s a n d their p a r a m e t e r s , b y v a r y i n g e s t i m a t e d rate c o n s t a n t s , a s e n s i t i v i t y analysis of the specific r e a c t i o n s c h e m e is r u n . Rate p a r a m e t e r s of all t h e e l e m e n t a r y r e a c t i o n s in the s c h e m e are s y s t e m a t i c a l l y v a r i e d b y a n a r b i t r a r y factor, S, a n d the effect of s u c h v a r i a t i o n s o n the i n d u c t i o n times are e x a m i n e d . Alternatively, the r e a c t i o n s , o n e at a time, are r e m o v e d c o m p l e t e l y f r o m t h e s c h e m e ( t h e i r rate c o n s t a n t s are m u l t i p l i e d b y a v e r y s m a l l n u m b e r , 1 x 10 -10 for e x a m p l e ) . By e x a m i n i n g t h e effect of s u c h a n o p e r a t i o n , t h e role of e a c h e l e m e n t a r y step in the overall k i n e t i c s s c h e m e can be elucidated. O n e c a n also p e r f o r m sensitivity analysis w i t h r e s p e c t to v a r i a t i o n s ( o r rather, u n c e r t a i n t i e s ) in t h e v a l u e s of AH~ of species w h o s e t h e r m o d y n a m i c p r o p e r t i e s w e r e e s t i m a t e d or are n o t k n o w n a c c u r a t e l y e n o u g h ( S m i t h et al. 1999, G R I - M e c h . 3.0). I n c o r r e c t v a l u e s of t h e t h e r m o d y n a m i c f u n c t i o n s r e s u l t in e r r o n e o u s v a l u e s for t h e rate c o n s t a n t s of t h e b a c k reactions. Table 16.5.5 s h o w s t h e sensitivity s p e c t r u m of the kinetics s c h e m e of t h e i g n i t i o n of a c e t o n i t r i l e C H 3 C N (Lifshitz et al. 1997). This is the sensitivity of t h e i n d u c t i o n times to a n i n c r e a s e b y a factor of 3 of the rate c o n s t a n t s a n d to

TABLE 16.5.5

Sensitivity Spectrum of CH3CN Ignition (Numbers are given in percent) S = 1.0 x 10 -10

No.

Reaction

1 2 4 5 6 7 8 17 38 66 67 68 85 94 100 104 105

CH2CN~ 4-H ~ CH2CN~ 4- H2 CH3CN 4- CH3~ ~ CH4 4- CH2CN~ CH3CN 4- O ~ --~ NCO ~ 4- CH3~ C H 3 C N4- O ~ ~ CH2CN~ 4- OH ~ C H 3 C N4- OH~ ~ CH3~ 4- HOCN C H 3 C N4- OH~ ~ CH2CN~ 4- H20 0 2 4- H~ ~ OH ~ 4- O ~ CH20 4- Ar ~ CO 4- H 2 4- Ar CH3. 4- 02 ~ CH30 ~ 4- O ~ CH3" 4- 02 ~ CH20 4- OH ~ CH3" 4- CH3~ --~ C2H6 CN" 4- 02 ~ NCO" 4- O" NCO" 4- CO ~ CN~ 4- CO2 NCO" 4- H20 ~ HNCO 4- OH" NCO" 4- 02 --~ NO 4- CO 4- O" NCO ~ 4- HCN ~ HNCO 4- CN ~ CH3CN ~

CH3CN 4- H~ ~

S = 3.0

1500 K

1700 K

1500 K

1700 K

297 -8 -25 318 -15 505 -26 22 49 165

667 -12 -14 125 -7 109 -26 44 8 79 11 -12 36

-29 11 31 -10 37 -10 134 -18

-33 20 26

-52 -15

-49 -25

-25 28 -28 -24

-7 22 -24 -21

-11 102 22 -24 45 30

-22 33 16

13 79 -23

16.5 IgnitionDelay Times

247

elimination of reactions from the kinetics scheme, at 1500 and 1700K, respectively. The table gives the percent change in the induction times as a result of such operations. Reactions that show an effect of less than 5% both at 1500 K and 1700 K were not included in the table. A common feature to many reactions is the decreased sensitivity as the temperature increases. The number of channels that contribute to the progress of the reaction increases as the temperature increases so that the effect of eliminating a single reaction decreases. The table also shows that only a relatively small number of elementary steps affect the ignition delay times directly. The majority of the elementary reactions in the scheme have no effect at all. The question that arises is why should we leave so many elementary reactions in the scheme if their elimination has no effect on the ignition delay. There are several reasons why this is done. First, the sensitivity analysis is done by removing only a single reaction at a time. When a group of reactions is eliminated from the scheme there can be a strong effect, even though the elimination of only one step, as was shown in Table 16.5.4, does not affect the induction times. Moreover, if many "unimportant" steps are removed, the pressure profiles do not show a clear ignition. An example is shown in Fig. 16.5.14 for the system CH3CN 4-02, where a scheme of only 26 elementary reactions that show an effect on the ignition delay was used in the calculations. As can be seen in that figure it is impossible to determine at what point ignition occurred. Figure 16.5.14 should be compared to Fig. 16.5.8, where a more complete scheme was used in the calculation. The ignition points in Fig. 16.5.8 can be very

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t(us) FIGURE 16.5.14 Calculated pressure profiles of the CH3CN4-02 reaction using a concise reaction scheme of only 26 reactions. An ignition point cannot be detected.

248

A. Lifshitz

accurately determined. Another good reason for using a comprehensive scheme is for the sake of completeness and applicability beyond the temperature range of a given series of experiments. It is informative to examine briefly some of the details in the mechanism of the ignition process in acetonitrile in light of the findings given in Table 16.5.5. As can be seen, the initiation reaction is an H-atom ejection from acetonitrile. This reaction is 93-94 kcal/mol endothermic, whereas the dissociation channel CH3CN--+ CH 3 + CN is about 120-121 kcal/mol endothermic (Stein et al. 1994) and thus is negligible even though its preexponential factor is more than one order of magnitude higher. The dissociation of molecular oxygen, 0 2 + M --+ O + O + M, is also several orders of magnitude slower and does not contribute to increasing the concentrations of the free radicals in the system. The reason behind leaving this reaction in the scheme is not because of its contribution to the production of free radicals; on the contrary, it is left because of the effect that the reverse reaction might have on the O-atom concentration. Since the O-atom concentration overshoots its equilibrium value, the reverse reactionmwhich is the recombination of O atomsmmight affect the kinetics. Also, the reaction CH 3CN + 0 2 ~ CH 2CN ~ + HO~, which can, in principle, play the role of an initiation reaction, under the initial conditions used in the calculations is two orders of magnitude slower than the process of H-atom ejection from acetonitrile. When the dissociation of acetonitrile is removed from the scheme, the reaction still proceeds due to the other two initiation reactions mentioned above, but the ignition delay times are much longer. Table 16.5.5 shows that this behavior becomes more significant at high temperatures due to the high activation energy of the initiation and its ability to supply a considerable amount of free radicals at high temperatures. The reactive free radicals~H ~ O ~ and OH~ with the reactant CH3CN via two reaction channels: an abstraction of an H atom to form CH2CN ~ and a dissociative attachment to the CN group, followed by a displacement of HCN (when H atoms are concerned) and the formation of CH~. Dissociative attachment is much faster than abstraction. As can be seen in Table 16.5.5, the channels of the first type (reactions 2, 4, 6, and 8) have an inhibiting effect. Their removal from the scheme shortens the ignition delay times. The channels of the second type (reactions 5 and 7) have an enhancing effect. The reactive radicals produce by an abstraction a very inert radical CH2CN ~ As a relatively big molecule, abstraction by CH2CN ~ is rather slow. Also, its self-dissociation is extremely slow because of the strong C H z - C N ~ (138-139kcal/mol) and H - C H C N ~ (110-112kcal/mol) bonds (Stein et al. 1994). The net result is that the abstraction reactions that produce CH 2CN ~ to some extent play the role of free radical scavengers and inhibit the progress of the overall reaction.

16.5 Ignition Delay Times

249

Another significant feature is the relative role of the two major branching reactions H" + 02 ~ OH" + O ~ (reaction 17) and CH~ + 02 --~ CH30 ~ + O ~ (reaction 66). As can be seen in Table 16.5.5, the removal of reaction 66 from the scheme is more inhibiting than the removal of reaction 17. This behavior results from the much higher concentration level of methyl radicals as compared to hydrogen atoms. As can be seen, the sensitivity analysis of a complex kinetics scheme where there are many parallel routes and many consecutive steps is a necessary procedure for understanding the overall reaction mechanism.

1 6 . 5 . 6 SUMMARY We have demonstrated that ignition delay measurements, computer modeling, and sensitivity analysis can shed light on the kinetics mechanisms of fuel combustion. We have clarified what an ignition phenomenon is and discussed the conditions under which ignition in mixtures of a fuel and an oxidant can occur. We have also shown that the very sharp temperature rise behind shock waves and the conditions that prevail in a shock tube are perfectly suitable for studying combustion reactions in an ignition mode. There are a number of methods by which the progress of a combustion reaction can be followed to determine an ignition point. We discussed the several methods used in shock tube research. We also covered in detail the mode of data processing and the reduction of parametric equations that relate induction times with the temperature, pressure, and composition. We used the modeling procedures and detailed modeling of the kinetics scheme of one specific system, CH3CN nt- 0 2 in argon, and its sensitivity spectrum as an example. We reviewed the dependence of the induction times on the concentrations of the fuel, the oxidant and the diluent, and its relation to the fuel and to the oxidant properties for several systems. We also examined the dependence in connection with the detailed reaction mechanisms. We included a long list of ignition delay measurements and the parameters relating the ignition delays to the concentrations and temperature. An ignition delay time is a global phenomenon. Each experiment yields just one point. Although determining ignition delay time is a very useful tool, additional information such as distribution of reaction products prior to ignition and concentration time history of unstable intermediates will enhance our understanding of combustion reaction mechanisms.

250

A. Lifshitz

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FIGURE 16.4.20 Experimental and calculated mole percents of allene, propyne, and acetylene. "[he symbols are the experimental points and the lines are the best fit through the calculated points shown here as crosses.

FIGURE 16.5.1

Wheat grains arranged on a chessboard in a geometrical series.

FIGURE 16.5.4 Experimental (points) and calculated (lines) ignition delays in two different mixtures of CH 3CN + 02 in argon. From the distance between the two series of experiments, the power dependence with respect to oxygen,/3(o2), can be determined. A strong enhancing effect of oxygen can be seen.

FIGURE 16.5.5 Experimental (points) and calculated (lines) ignition delays in two different mixtures of CH 3CN + 02 in argon. From the distance between the two series of experiments, the power dependence with respect to the fuel,/3(ot3r ), can be determined. A small inhibiting effect of acetonitrile can be seen.

FIGURE 16.5.6 Experimental (points) and calculated (lines) of ignition delays in two different mixtures of CH 3CN + 0 2 in argon. From the distance between the two series of experiments, the power dependence with respect to the diluent argon, fl(Ar), can be determined. A very small inhibiting effect of argon can be seen.

FIGURE 16.5.7 A plot of log2 vs 1/T, where 2 = tignition/{[CH3CN ] 0 "12 [ 0 2 ] - "0 76 [ m r ]0 34 } for data points from six series of experiments. With the above correlation, all the points scatter along one line with a slope of 41.7 kcal/mol.

FIGURE 16.5.13 A plot of log2 vs 1/T where ~ = tignition/[[CH3CN]~176 } for 60 calculated points corresponding to the six series used in the experiments. With the above correlation, all the calculated points scatter along one line with a slope of 46.2 kcal/mol.

CHAPTER

16.6

Chemical and Combustion Kinetics 16.6

Particulate Formation and Analysis

HAI WANG Department of Mechanical Engineering, University of Delaware, Newark, Delaware, 197'16, USA

16.6.1 Introduction 16.6.2 Particle Size Distribution Function 16.6.3 Particle Analysis Techniques 16.6.3.1 Laser Light Extinction and Scattering 16.6.3.2 Complex Refractive Index 16.6.3.3 Light Emission 16.6.3.4 Other Detection Techniques 16.6.4 Soot Formation 16.6.4.1 Induction Time 16.6.4.2 Soot Yield 16.6.4.3 Soot Growth Rate 16.6.4.4 REM and TEM Studies 16.6.5 Summary References

16.6.1 INTRODUCTION The reaction of gaseous mixtures following shock heating often results in the production of condensed-phase materials. In most cases, these materials are produced in the form of particulate matter which is typically found during the pyrolysis of hydrocarbons. Although these compounds tend to decompose to small fragments following shock heating, under almost all conditions the fragmentation process is accompanied with the production of carbon or soot particulate. Depending on the shock condition and the chemical nature of the reactant, nearly the entire carbon mass in the initial reactant can be converted to soot within the time period observable by the shock tube technique (Graham et al. 1975a, 1975b). Handbook of Shock Waves, Volume 3 Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISBN: 0-12-086433-9/$35.00

257

258

H. Wang

In studies of gas-phase reaction kinetics, the formation of particulate is undesirable because it causes serious complications, ranging from increases in noise-to-signal ratios in optical measurements to difficulties in GC and/or MS sampling and analysis. A "clean" shock tube experiment is often accomplished by using highly diluted reactant mixtures or by monitoring only the initial stages of reaction. On the other hand, the shock tube technique provides a very useful tool for the study of the chemical mechanism of soot formation during the pyrolysis and oxidation of hydrocarbons at high temperatures. Indeed, a variety of soot studies have been carried out using shock tubes to generate well-defined reaction conditions. Table 16.6.1 provides a summary of these investigations, along with a summary of the reactants and the principal measurement technique employed. Findings from some of the studies listed in the table were reviewed by Haynes and Wagner (1981) and by Wagner (1994). In addition to soot formation, the shock tube technique has also been applied to a variety of particulate formation problems. It was used to understand the chemical processes relevant to the synthesis of nano-powder from gases (Steinwandel et al. 1981a; Steinwandel and Hoeschele 1985; Carmer and Frenklach 1989; Frenklach et al. 1996; Herzler et al. 1998). It was also employed to generate supersaturated metal vapors so that the nucleation of metal aerosols and the subsequent coagulation of the aerosol particles could be examined under well-defined conditions (Graham and Homer 1973a, 1973b; Freund and Bauer 1977; Frurip and Bauer 1977a, 1977b, 1977c; Stephens and Bauer 1981; Steinwandel et al. 1981a; Steinwandel and Hoeschele 1985). The shock tube technique has been widely used to study heterogeneous reaction kinetics. In this case, the initial reactant mixture contains small particles suspended in an inert diluent or a reactive gas-diluent mixture. There have been a number of studies in which the ignition of metal particles was examined following shock heating of metal particles suspended in a reactive gas (Fox et al. 1977; Fursov et al. 1979; Boiko et al. 1989; Roberts et al. 1993). Studies on the pyrolysis and oxidation of pulverized coal have been extensively reported (Nettleton 1977; Nettleton and Stirling 1974; Lowenstein and von Rosenberg 1977; Seeker et al. 1978; Frieske et al. 1981; Szydlowski et al. 1981; Ural et al. 1981; Banin et al. 1997; Commissaris et al. 1998). The reaction rate of carbon particles with molecular oxygen has been measured in shock tubes over a wide range of conditions (Park and Appleton 1973a, 1973b; Brandt and Roth 1989; Roth and Brandt 1990; Roth et al. 1990); these studies are, however, outside of the field covered here. A quantitative measurement of the concentration of the particles formed behind shock waves is not trivial. The difficulties may arise from extensive variations in the chemical and/or the physical properties of the particle matter. Particles produced from a shock tube may range from a cluster of a few

16.6

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Particulate Formation and Analysis

0

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~

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o o

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259

260

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H. Wang

E .r-

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16.6

261

Particulate Formation and Analysis

molecules to a macroscopic entity with properties identical to a bulk material. The size of the particle may vary correspondingly, from a few nanometers to tens or hundreds of micrometers. Particles may also vary in physical appearance, ranging from an amorphous material to a single crystal. The morphology of the particles may also vary from a single particle to an aggregate containing tens to thousands of single particles. The elementary particle processes are intrinsically more difficult to describe than gas-phase reaction processes. There are a number of elementary mechanisms of particle formation and growth, one or more of which may contribute at a particular stage of particle formation processes. These mechanisms include particle inception from gaseous species (the formation of small clusters consisting of a limited amount of monomers), particle-particle coagulation, gas-particle surface reaction, and gas condensation onto the particle surface. All of these processes have been observed in various stages of soot formation in flames and in shock tubes (Haynes and Wagner 1981; Bockhom et al. 1982; Graham et al. 1975a; Frenklach and Wang 1991). Shock-generated particulate matter is invariantly multi-dispersed. Therefore before a more detailed discussion can be made, it is necessary to gain a familiarity with the particle size distribution function (PSDF), which is to be introduced in Section 16.6.2. We shall then discuss the particle analysis techniques in Section 16.6.3. In Section 16.6.4 shock tube studies of soot formation are overviewed, with a large set of experimental data compiled and presented. The data compilation is a consequence of a wide range of data available for soot formation in shock tubes, and is motivated by the need of a reference data set for future work in this area. In the last two sections (16.6.5 and 16.6.6), we discuss the applications of shock tube techniques to studies relevant to nano-particle synthesis and of the homogeneous nucleation of particles from supersaturated vapors.

16.6.2 PARTICLE SIZE DISTRIBUTION FUNCTION A particle ensemble can be viewed as a population of particles each of which consist of an integral number of molecules or atoms. The population can be characterized by the number density of particles of size i, which is denoted by N i (Seinfeld 1986). The total particle number density N is given by o~

N -- ~ N i i--1

(16.6.1)

262

H. Wang

The particle size can be characterized by a number of parameters, including particle mass, number of molecules or atoms in a particle, diameter, surface areas, and volume. If particles are spherical or near spherical, it is convenient to define the size by the particle diameter. When the mass density is known, the diameter-based size distribution can be readily converted to the mass-based distribution. Because of infinite summation, Equation (16.6.1) is not very usable. In practice, the particle size distribution is usually divided into K number of size bins. The infinite summation is then replaced by a summation of K terms of N i, each representing the number density of the ith size bin with diameters ranging from D i to Di+ 1. Letting AD i --Di+ 1 - D i ~ O, we arrive at a continuous distribution function. In doing so, we replace the discrete size index i by the size distribution function n(D), where n(D)dD is defined as the number of particles per unit volume having diameters in the range from D to D + dD. The total number density of particles is then N -

Jo

n(D)~

(16.6.2)

In many cases only the mean properties of the particle size distribution function are measurable. The mean particle diameter is defined by

12

(D} = ~

Dn(D)dD

(16.6.3)

The degree of spread of particle sizes is characterized by the variance of the distribution, defined as cr2 _ 1 N

( D - (D})n(D)dD

(16.6.4)

In soot studies, the extent of particle formation is conveniently defined by the volume fraction of particles (Haynes and Wagner 1981), fv - IO -~ 1 rcD3n(D) dD

(16.6.5)

It will be shown that this definition is quite useful in laser light extinction measurement because the volume fraction is proportional to an optical property directly measurable in a laser light extinction experiment. In addition to soot volume fraction, soot yield is also commonly used to characterize the efficiency of carbon conversion to soot. The yield is related to the volume fraction by the expression Yield - Pfv/([C]Mc)

(16.6.6)

16.6

263

Particulate Formation and Analysis

where p is the mass density of soot, M c is the atomic mass of carbon, and [C] is the C-atom concentration in the mixture. Because the diameter may span several orders of magnitude in a particle ensemble, it is often convenient to use a logarithmic size distribution, n(logD)dlog D, defined as the number density of particles in the size range log D to log D 4- d log D. The total number density is N --

n(logD)dlogD

(16.6.7)

Note that n(logD) is not the same distribution function as n(D). It can be shown that the two size-distribution functions are related by the expression n(log D) = (In IO)Dn(D)

(16.6.8)

A particle size distribution function can also be completely characterized by the moments of distribution function (Hogg and Craig 1970; Johnson and Leone 1977). The rth moment is defined by M r --

;o

Drn(D)dD

(16.6.9)

Obviously, the zeroth moment M 0 is equal to the total number density N. The reduced rth moment is defined by Mr

1

#~= N =-N

Io

D"n(D)dD

(16.6.10)

Thus the first reduced moment Pl is the mean diameter (D), and the second and third reduced moments are proportional to the mean surface area and particle volume, respectively.

16.6.3 PARTICLE ANALYSIS TECHNIQUES 1 6 . 6 . 3 . 1 LASER LIGHT EXTINCTION AND SCATTERING A widely used particle measurement technique is that of light extinction. Figure 16.6.1 presents a schematic of the apparatus commonly employed in shock tube studies of soot and other particulate matter. In this technique, a laser operating in the visible or infrared wavelength region is used to generate the incident laser beam. The argon-ion, helium-neon, and helium-cadmium lasers have been frequently used in previous studies. The incident light beam passes through the shock tube near the end wall. When particles are present, the incident light is attenuated due to light absorption and scattering by the

264

FIGURE 16.6.1 shock waves

H. Wang

Schematic of a typical optical system for detection of particles formed behind

particles. The attenuation of the laser beam is monitored by a photomultiplier. A narrowband interference filter can be inserted in the path of laser beam to eliminate the interference of light emission from the particles (Yoshizawa et al. 1979; Frenklach et al. 1983a, 1983b). According to the Beer-Lambert law (Kerker 1969; Van de Hulst 1981), the extinction of light by small particles across an optical length l obeys a relation given by It -- = exp(-kextl ) Io

(16.6.11)

where It/I o is the transmittance or the ratio of the transmitted to incident light intensities, kext is the extinction coefficient, and l is the optical length or the internal diameter of the shock tube. For spherical absorbing particles, the extinction coefficient is related to the total cross-sectional area of the particles multiplied by the extinction efficiency Qext,

j

o~ ~ D 2

(16.6.12)

o

In the above equation, the extinction efficiency is a function of the wavelength, the complex refractive index of the absorbing material, and the particle

16.6 Particulate Formation and Analysis

265

diameter. In the small-particle-size limit, i.e., the Rayleigh region with o~ = reD~2 <_ 0.3, the extinction coefficient is given by (Hottel and Sarofim

1967; D'Alessio 1982) o

kext = - -/~- Im 1'~2l q--

~3n(D)dD

(16.6.13)

where 2 is the wavelength, IM{} refers to the imaginary part of a complex number, and r h - - n - ik is the complex refractive index of the particle material. A comparison of equation (16.6.13) with Equation (16.6.5) yields the proportionality between hext and the particle volume fraction, 6~ 2 {ff12-- 1} k~xt - - - ~ Im rh2 + 2 f~

(16.6.14)

Thus the measurement of transmittance gives the particle volume fraction, if the refractive index value is known. In principle, light scattering measurements can also be made in a shock tube experiment. The interpretation of the scattered light, however, is not as straightforward as that of the light extinction measurement (Graham et al. 1975a). The basic theory of light scattering is contained in the classic books by Van de Hulst (1981) and Kerker (1969). The subject has been reviewed by D'Alessio (1982) in the context of laser light scattering diagnostics of fuel-rich flames. The technique is equally applicable to shock tube experiments. To relate the intensity of scattered light and particle properties, a scattering coefficient Q is introduced, which is proportional to the intensity of scattered light. In the Rayleigh region, the scattering coefficient for the vertical component of the scattered light due to a vertically polarized incident laser beam is given by 22 ff~2

1

Q~ - ~-~2 r ~ 2 + 2

j;

o~6N(D)dD_

~-~ r ~ 2 + 2

I

N(D 6)

(16.6.15)

The polarization of the laser beam is achieved by an optical polarizer (D'Alessio 1982). In the Rayleigh region, Q~ is independent of the angle between the incident and scattered light beams. For convenience, the scattered light intensity is measured in a 90 ~ angle, as shown in Figure 16.6.1. To enhance the signal-to-noise ratio, a mechanical chopper is used on the path of the incident light. The scattered light signal I s can be converted to the scattering coefficient Qvv by considering the scattering volume, the solid angle aperture of the detection optics, and the intensity of the laser beam within the scattering volume (D'Alessio 1982). These quantities are, however, difficult to define in practice. Instead, the conversion of the scattered light signal to the scattering coefficient

266

H. Wang

is usually accomplished by a calibration procedure (Graham et al. 1975a). In the simplest form, this calibration may be performed by measuring the scattered light intensity of argon and assuming that the scattering volume and the solid angle in the calibration are equal to those of the particle experiments. Using an identical incident laser intensity and optical setup, it is possible to determine a calibration factor, given by ktcalib -- NAr~Ar/Is,Arand Qvv - ktcalibIs, where NAr is the number density of argon during the calibration procedure, O'Ar is the scattering cross section of an argon atom, and Is,Ar is the intensity of the scattered light from argon. For an incident laser beam operating at the usual wavelength of, say, 632.8nm, the Rayleigh approximation is valid for particles with diameter smaller than 60 nm. For earlier stages of particle formation, this approximation is expected to be adequate. However, the detection of particles of large sizes at later stages of reaction may require the use of Mie theory (Van de Hulst 1981). For ~ _< 1 and n~ _< 1, the Mie coefficients can be expanded in a series (Penndorf 1962), allowing the extinction and scattering coefficients to be expressed respectively by (Cadwell et al. 1994) kext ~ ~

[KICX3 + K30~5 + K4cx6 -+- K50ff]N(D)dD

"~ 922 [K6~6 + K8e8 + K90c9 + Kloel~ Qvv- 16re2 0

(16.6.16)

(16.6.17)

where K i are given by K 1 = 6Ala3 K3 = 6Alas + 6Big5 + 10A~5 K4 = 6AIR6

K5 -- 6A~7 4- 6B~7 4- I OAza74- 10Bza74- 14A~7 K6 _ (AR3)2+ (A/3)2 I

I

I

I

K8 -- 2(A~3A~5 4-A13A15) K9 -- 2(Ala3A~6+ A13A16) R q_ 2A/3 (A/7 _53 B27I _ 7 A~7) K10 _ 2AR3 (AR7 _ 5 BR7 _ 67 A37)

In the above equations, A and B are functions of the complex refractive index of the particle material, originally derived by Penndorf (1962) but reproduced in Tables 16.6.2 and 16.6.3 for convenience. It can be readily shown that the Rayleigh approximation is derived by neglecting the terms higher than 0c3 in Equation (16.6.16) and c~6 in Equation (16.6.17).

16.6

267

Particulate Formation and Analysis TABLE 16.6.2

Elements of Mie Coefficient Expansion (after Penndorf 1962)

3

5

6

7

~P2

~(PIQ2 q- P2Q1) 1 v~ ~-~

4(p12 - e~)

B~,j A~,,

1 $2

1 (PiR 2 q_ P2R1 ) -945 -1 W2 11 T2

Af,j B~,j

-AZl,j -B/,j -A~j

~ v~

2P1

g2(P1 Q1 -

P2Q2)

145 V1 ~ S1

8p1P2

Y~l (PiR 1 _ P2R2) 915 Wl 11T 1

-B~j

~ Vl

I - A 3,j

1175 U1

a The entities P, Q, R, S, T, U, V, and W in the table are listed in Table 16.6.3.

Figure 16.6.2 presents typical transmission and scattering profiles observed during the pyrolysis of ethylbenzene behind an incident shock wave at 1750 K (Graham et al. 1975a; Haynes and Wagner 1981). It is seen that the absorptions at 488 nm and 663 nm begin immediately after the passage of the shock front. There is, however, no detectable absorption at 3.39 ~tm at the early stage of the reaction. This feature is generic to the pyrolysis of all hydrocarbon compounds, as will be discussed in subsequent sections. The absorption at visible wavelengths is most certainly caused by pre-soot species. This effect is consistent with the observation of Cundall et al. (1978), who detected absorption of these species in acetylene pyrolysis. In that study, the absorption intensity was found to be larger at smaller wavelengths. However, Rawlins et al. (1983a, 1983b) concluded that light extinction at 632.8 nm is primarily caused by soot particles. Figure 16.6.3 shows the soot volume fraction profile and the pressure record for benzene pyrolysis behind a reflected shock wave with T5 = 1890 K, P 5 - - 5 0 bar, and the total C-atom concentration equal to 8.3 • 10 -7 mol/cm 3 (Bauerle et al. 1994). The dispersion quotient (DQ) or two-color method has been used to provide additional information of particle size and number density (Lester and Wittig 1975; Wittig et al. 1990; MCfller and Wittig 1994; Frenklach et al. 1996). This method is based on the extinction of two laser beams along the same optical path but at different wavelengths. Frenklach et al. (1996) used a He-Ne laser with the wavelength at 632.8 nm and a He-Cd laser at 441.6 nm,

268

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H, Wang

E 8

I:h

L'-

269

16.6 Particulate Formation and Analysis

whereas Mfiller and Wittig (1994) used the laser beams at 488 and 632.8nm. The dispersion quotient DQ is defined by

~D2Qext(~l,17tl,D)n(D)dD ~~

|n (I~o)21

DQ-~=

ln(/~)

(16.6.18)

22

For a given complex refractive index and an assumed particle size distribution function n(D), the dispersion quotient is only a function of mean particle

0

l

............

,

,

Shock arrival

0.7

~

9

~

0.8

~

0.9

_x_i 'i ,-~

>

,

,

,

(a)

t

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1.5 ~I,,.~ ,.~

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~

0.5

r,~

0.0

__~_~~r~j

(b)

I . . . .

I000

200O

30OO

Time (Its) FIGURE 16.6.2 Transmissionand scattering of light during shock tube pyrolysis of ethylbenzene in argon at 1750K (Graham et al. 1975a)

270

H. Wang 2.0

1.5

- 4

,o ~

•

- 2 1.0

r~

-0

0.5-

0.0I

i

0

500

,,I

1000

Time

I

1

1500

2000

(las)

FIGURE 16.6.3 Soot volume fraction profile from light extinction and pressure record for benzene pyrolysis behind a reflected shock wave. T5 = 1890 K, P5- 50bar, and [C]= 8.3 • 10-e mol/cm3 (Bauerle et al. 1994). The fv data were obtained from light extinction at 632.8 nm, initially derived with the complex refractive index value of Lee and Tien (1981), and were rescaled here using the ffa value of Dalzell and Sarofim (1969). The dashed line is fit to data using the apparent first-order rate equation df~/dt=kf(fv,oo-f~ ) with kf = 2 2 0 0 s -1 and [v,oo = 1.26 x 10 - 6 . The induction time of soot appearance is denoted by z (= 0.124 ms).

diameter (D) (MOller and Wittig 1994), which can be solved with the Mie theory using, for example, the approach of Aden (1951). A log-normal distribution is commonly assumed for the calculation of the right-hand side of Equation (16.6.18), N

n(D) -- x / ~ D In ag

[ ( l n D - l n l D g ) 2] exp 2 In 2 ag

(16.6.19)

where ag is the standard deviation and/Dg is the m e d i u m diameter for which exactly one-half of the particles are smaller and one-half are larger (Seinfeld 1986). In principle, the standard deviation varies as a function of time from the onset of particle nucleation. For a particle growth process dominated by coagulation in the free-molecule regime, however, the PSDF tends to a selfserving distribution (Lai et al. 1972; Graham and Homer 1973a, 1973b; Graham et al. 1975a). In flames, In ag tends to a constant value of 0.34 after a short reaction time. A wide range of In ag values have been reported or used for soot produced from shock robes. Miiller and Wittig (1994) used a In ag value of 0.5 in the evaluation of the dispersion quotient [Equation (16.6.18)], whereas Bauerle

16.6

271

Particulate Formation and Analysis

et al. (1994) and Knorre et al. (1996) reported a geometric standard deviation ln~rg ~ 0.2 for soot produced from shock tube pyrolysis of a variety of hydrocarbon compounds. The dispersion quotient technique may severely overestimate the particle size. Frenklach et al. (1996) showed that irrespective of the assignment of the rh values, the particle sizes determined by the dispersion quotient technique is substantially larger than those determined from transmission electron microscopy. Assuming that the particles are monodispersed results in even larger discrepancies. More sophisticated light extinction and scattering techniques have been developed for studies of particles formation in shock tubes. For example, retrieval of soot aggregate morphology is possible by detection of the scattered light intensities at different scattering angles (di Stasio et al. 1996; di Stasio and Massoli 1997).

1 6 . 6 . 3 . 2 COMPLEX REFRACTIVE INDEX Perhaps the largest uncertainty in the optical measurement of soot volume fraction or yield is the complex refractive index. In earlier studies (Graham et al. 1975a, 1975b; Wang et al. 1981; Frenklach et al. 1983a), soot yields exceeding 100% were observed from light extinction experiments. It is most likely that this overshoot is caused by uncertainties in the complex refractive index. Indeed, a wide range of complex refractive indexes have been reported for carbon particles. The most frequently used values of refractive index are summarized in Table 16.6.4 for soot at 632.8 nm. It is seen that the discrepancy is significant among the published values of refraction index. In particular, the imaginary part of the complex number, (rh 2 - 1)/(~ 2 + 2), varies by as much as a factor of 1.7 when comparing the results of Lee and Tien (1981) and Menna and D'Alessio (1981). In Rayleigh region, the soot yield or soot volume

TABLE 16.6.4 Most Frequently used Values of Complex Refractive Index (rh = n - ik) of Soot Material n 1.57 1.85 1.70

k

Im{(rn2 - 1)/(rn2 + 2)}

I(rn2 - 1)/(rn2 4- 2)1

Reference

0.56 0.48 0.75

0.26 0.18 0.30

0.47 0.51 0.57

Dalzell and Sarofim 1969a Lee and Tien 1981 Menna and D'Alessio 1981

Used in the present compilation of soot yield data

272

H. Wang

fraction evaluated using these index values should different by the same amount. The complex refractive index varies markedly with the wavelength for flame soot. Figure 16.6.4 shows the variation of the index of refraction as a function of the wavelength, from several representative sources. It is known that the refractive index also varies with temperature. However, Lee and Tien (1981) showed that this variation is not significant: changes in n and k did not exceed 30% from 1000 to 1600 K, which are well within the absolute uncertainty of the refractive index itself. For this reason, the temperature dependence of the refractive index is usually ignored above 1000 K. The data shown in Figure 16.6.4 represent an average of the refractive index at these temperatures. Even larger uncertainties exist for silicon particles produced in shock heating of silane and disilane. Frenklach et al. (1996) observed that under the same silicon loading and at a reaction time of I ms, the transmittance at 632.8 and 441.6nm initially increases and reaches a peak at around 11001200 K. The transmittance then decreases up to a temperature of "~ 1300 K, above which it remains unchanged until 1900K. On the basis of kinetic considerations, they concluded that the decrease in transmittance above 11001200 K can only be explained by a change in the complex refractive index of

5i ~176176~176176176176 ~176176176176

~.~

~

-"

"o

tp,,q

1 0.8

.... ..

~...-'" .~"~

0.6

0.4

0.2

......... Dalzell & Sarofim (1969) ~ Lee & Tien (1981) - - -Chang & Charalampopoulos (1990) I

I

I

I,

I

I

I

I

I

,,,

I

1000

I

I

I

I

I,,I

[

100120

Wavelength (nm) FIGURE 16.6.4 Real and imaginary parts of the complex refractive index, ffa = n - ik, of soot material

16.6

Particulate Formation and Analysis

273

the silicon particles because of the transition of particles from a solid-phase material to liquid droplets. Using a detailed kinetics model, Frenklach et al. (1996) fitted the transmittance profiles by a unique contour of n versus k for experiments conducted at temperatures above 1286 K. This contour encompasses the complex refractive index of liquid silicon (Jellison and Lowndes 1987), r h - 3 . 5 - 5.2i at 2 = 632.8nm and r h - 2 . 6 - 4 . 8 i at 441.5nm. A further note from that study is that although the melting temperature of bulk silicon is 1683 K, submicron particles can melt at much lower temperatures than that of the bulk material (Buffat and Borel 1976). 16.6.3.3

LIGHT EMISSION

The onset of light emission due to radiation of particles has been used to characterize the induction time of soot formation in shock tubes (Mar'yasin and Nabutovskii 1973; Gosling et al. 1973; Graham et al. 1975a; Graham 1981; Fussey et al. 1978; Tanzawa and Gardiner 1979). The detection of light emission is accomplished simply by a photomultiplier. The peak sensitivity of the photomultiplier should be around the wavelength of maximum radiation intensity of a blackbody, e.g., around 800 nm at 1600 K. A filter must be used to reject light having wavelength below 730 nm, so that the major emission bands from the C 2, CH, and CN radicals can be excluded (Fussey et al. 1978). A typical emission record is shown in Figure 16.6.5, where the onset of emission or soot formation is clearly identifiable. Monochromatic, infrared emission diagnostics was reported by Parker et al. (1990) for soot formation behind reflected shock waves. The emission measurements were made using calibrated, bandpass-filtered radiometers. The emission data are converted to emissivity by taking the ratio of the observed radiance to that of a blackbody. The emissivity e is related to the absorption coefficient KabS by Kabs =

- ln[1 - e(2)] l = O'abs(~)[C]soo t

(16.6.20)

where Crabs(/~) is the effective absorption cross section and [C]soo t is the number density of carbon in soot. The value of the absorption cross section can be evaluated from the index of refraction of the particle material (Parker et al. 1990; Van de Hulst 1981).

1 6 . 6 . 3 . 4 OTHER DETECTION T E C H N I Q U E S Laser Doppler anemometry and particle sizing using laser interferometry (Farmer 1972) has been introduced for particle measurements in shock

274

H. Wang

Induction time

o

al

0

..

~

/

I

I

~

200

~

~

I

400

.

,

,

,

I

600

9

~

,

.

.!.

800

.i..

1000

Time (laS)

~ Onset of emission Shock arrival

t

Contract surface arrival

FIGURE 16.6.5 A typical light emission trace of hydrocarbon pyrolysisin shock tube (Fussey et al. 1978) tubes (Frenklach et al. 1983c). Particle detection methods other than the lightbased measurements have been reported. For example, the material balance of carbon and hydrogen in the products of acetylene pyrolysis in a shock tube was used to quantify the production of soot (Mar'yasin and Nabutovskii 1969, 1970). Vaughn et al. (1981) reported a gravimetric analysis of the solid residue formed from benzene pyrolysis using a removable liner in the end section of the shock tube. Particle samples can be collected on substrates affixed on or near the end wall of a shock tube (Frenklach et al. 1996). The particle sample on the substrate can then be analyzed by transmission electron microscopy (Williams and Carter 1996) to examine the particle size distribution and morphology or by electron diffraction (Cowley 1992) to examine the crystal structure.

16.6.4 SOOT FORMATION 16.6.4.1

INDUCTION TIME

The induction time z can be determined from the intersection of the tangent at the inflection point of a light transmittance or soot volume fraction curve with

16.6

275

Particulate Formation and Analysis

the time axis (see, for example, Figure 16.6.3), or by the onset of light emission (see, for example, Figure 16.6.5). Unlike the ignition delay time, the induction time of soot formation is somewhat ambiguous. In the case of ignition delay measurements, the reaction or heat release rate beyond the induction time is so rapid that different methods yield essentially similar quantitative results. This is not the case for soot formation. Fussey et al. (1978) showed that light emission tends to give shorter induction time than light extinction at 632.8nm. Frenklach et al. (1983a) showed that the induction period determined by light extinction in the infrared (3.39 ~tm) are longer than those in the visible (632.8nm). Indeed, an examination of Graham's transmittance traces (Graham et al. 1975a) in Figure 16.6.2 readily points to the difference in the induction time determined at different wavelengths. For this reason and because the induction time correlation does not seem to provide any indication of the nature of a rate-determining step, the real significance of the induction period has been questioned by Haynes and Wagner (1981). Nonetheless, we shall summarize the past experimental results below because these data provide a semiquantitative guidance of sooting tendency. Although ignition delay and induction time of soot formation are completely different physical properties, they share similarity in the methods of correlation. Following the idea of ignition delay correlation (Burcat et al. 1970; Lifshitz et al. 1971), Gosling et al. (1973) correlated the induction time of soot formation with temperature and the initial hydrocarbon concentration in the form of

(E)

zPi_ic -- A exp ~-~

(16.6.21)

where z is the induction time and PHC is the molar density of hydrocarbon. Figure 16.6.6 presents the induction time data of Gosling et al. (1973) and of Fussey et al. (1978) for acetylene, ethylene, and ethane pyrolysis in argon behind incident shock waves. The induction times are plotted in the form of the product of induction time and the C-atom concentration, z[C], which is equivalent to the left-hand side of Equation (16.6.21). Figure 16.6.6 shows that under the same carbon loading, acetylene tends to produce soot earlier than ethylene and ethane. Fussey et al. (1978) found that Equation (16.6.21) does not yield a unique correlation for experiments conducted at different pressures. To account for the pressure effect, they correlated the induction time of soot formation in the form of

(E)

z[C]n -- A exp ~-~

(16.6.22)

H. Wang

276

'~,

A

I0-3

c~i~ C~H~ ,~.~o ~~ 2 ~ 0 c~

~" 10-4 9

A ~"

AA A

A

I

4.5

5.0

,

I,

I

I

I

5.5

6.0

6.5

7.0

104 K / T FIGURE 16.6.6 Inductiontimes of soot formation multiplied by the C-atom concentration as a function of temperature for acetylene, ethylene, and ethane pyrolysis in argon behind incident shock waves. The induction times were determined by the onset of light emission. Solid symbols: data for P2 = 1-2bar from the appendix of Fussey et al. (1978) with initial concentrations [C2H2]0 -" 1.33-2.89 x 10-7 mol/cm3, [C2H4]0 -- 1.79-2.90 x 10-7 mol/cm3, [C2H6]0 = 1.672.63 x 10-7mol/cm3. Open symbols: data for pressures of 0.02-0.04bar from Gosling et al. (1973).

for the pyrolysis of ethane, ethylene, and acetylene in the pressure range of 1-12bar. Figure 16.6.7 presents the selected data and their fits with n - 0.41, E -- 31 kcal/mol for acetylene; n - 0.23, E -- 28 kcal/mol for ethylene; and n - 0 . 4 2 , E -- 36 kcal/mol for ethane (Fussey et al. 1978). Using light extinction at 632.8nm, Yoshizawa et al. (1978) measured the induction time of soot formation from acetylene pyrolysis behind reflected shock waves for pressures between 2.8 and 3.5 atm. They found that the induction time exhibits a first-order dependence on the initial acetylene concentration. Using a similar technique,however, Frenklach et al. (1983b) found that at 632.8 n m and 3.39 ~m the induction time of soot formation from acetylene pyrolysis is not affected notably by the C-atom concentration. Figure 16.6.8 presents the data of Yoshizawa et al. (1978) and Frenklach et al. (1983b). The data at T < 2200 K can be correlated by 1: = 2.9 x 10-3[C]-~

-~

exp(13 , 000/ T)

(~s, m o l / c m 3)

The m a r k e d deviation of the data points from the regression line, however, indicates a certain inadequacy of the empirical correlation. The correlation exhibits only a weak dependence on both the total carbon loading and the argon concentration. The high-pressure data of B6hm et al. (1998) and Knorre

16.6

277

Particulate Formation and Analysis

102 g []

C2H4

D

101

C2~

O []

0

9

lOo • t~

lO-I

4.4

)

t

I

I

4.8

5.2

5.6

6.0

.

t

6.4

. 6.8

104 K / T FIGURE 16.6.7 Induction time of soot formation from acetylene, ethylene, and ethane pyrolysis in argon behind incident shock waves (Fussey et al. 1978). The induction times were determined by the onset of light emission. Filled symbols: data for pressures of 1-2 bar and with initial concentrations [C2H2] 0 -- 1 . 3 3 - 2 . 8 9 x 10 -7 mol/cm 3, [C2H6] 0 = 1 . 6 7 - 2 . 6 3 x 10 -7 mol/cm 3. Open symbols: data for pressures of 10-12bar with initial concentrations [C2H2] 0 = 1 . 6 8 - 2 . 1 4 x 10 -6 mol/cm 3, [C2H4] 0 = 1 . 2 9 - 1 . 7 0 x 10-6mol/cm 3, [C2H6] 0 = 1 . 3 1 - 1 . 6 9 x 10 -6 m o l / c m 3. The lines are fits to data: z x [C] n = A e x p ( E / R T ) with n = 0.41, E = 31 kcal/mol ol for acetylene, n = 0.23, E - - 2 8 k c a l / m o l for ethylene, and n = 0.42, E = 36kcal/mol for ethane.

et al. (1996) are also shown in Figure 16.6.8. Knorre et al. (1996) correlated data collected at elevated pressures up to 60 bar and reported the expression z = A e x p ( 2 7 , 4 0 0 / T ) [ C ] -~ The dependence of z on C-atom concentration was found to be similar to that of benzene as well as mixtures of benzene and acetylene under similar pressures. The systematic deviation of the correlations obtained at low and high pressures is obvious. Finally, it is interesting to see the change in the induction time dependence on temperature at around 2300 K (Yoshizawa et al. 1978), which cannot be described by the simple correlation (16.6.22). Such a behavior has not been observed from the pyrolysis of other hydrocarbon compounds. Despite the uncertainty that is likely to exist in the general correlation equation of soot induction time, it is certain that a correlation exists between the induction time and temperature if the C-atom concentration is held fixed. The total pressure affects very little the induction time (Bauerle et al. 1994; Knorre et al. 1996), as evidenced by the weak dependence of z on the argon concentration in induction time correlation reported by Wang et al. (1981) and by Frenklach et al. (1983b). In general, an increase in the C-atom concentra-

278

H. W a n g

10a o 0

~.

101

x

t

xx

+ X X r n O +4~ X .4 ^ , , . ~ +4.+. + X~D 0 ~ n ~ "~ v O4- +.+, 9 ++ + O0 ~ +

10o

< r~ •

~ ~'O+ " +

+

OtJ

J

-I

%

~#+

v , - + ~++Jrr

lO-1

I

I

3.5

4.0

.....

I

I

4.5

5.0 10 4

......IJ

5.5

I

6.0

6.5

K/T

FIGURE 16.6.8 Induction time of soot formation from acetylene pyrolysis in argon behind reflected shock waves. The induction times were determined by laser light extinction at 632.8 nm. Triangles (Frenklach et al. 1983b): 1.09%C2H2 in argon, [C] = 3.34-3.42 x 10 -7 mol/cm 3. Open diamonds (Frenklach et al. 1983b): 4.65% C2H2 in argon, [C] = 3.22-3.44 x 10 -7 mol/cm 3. Circles (Frenklach et al. 1983b): 4.65% C2H2 in argon, [C] = 8.08-8.56 x 10 -7 mol/cm 3. Squares (Frenklach et al. 1983b): 20% C2H2 in argon, [ C ] - 7.78-9.12 x 10 -7 mol/cm 3. x (Yoshizawa et al. 1978): 2.5% C2H2 in argon, P5 = 2.8-3.5atm. Crosses (Yoshizawa et al. 1978): 5% C2H2 in argon, Ps =2.8-3.5atm. Filled diamonds (BOhm et al. 1998): [C]=3.8 x 10-6 mol/cm 3, p5 = 56bar. Solid line: fit to the low-pressure data of Yoshizawa et al. (1978) and Frenklach et al. (1983b) z - 2.9 x 10-3[C]-~ -2"95 exp(13, O00/T) (its, m o l / c m 3) for T < 2200 K. Dashed line (Knorre et al. 1996): from the fit of data at 60bar in the form of z x [C]~ - A exp(O/T).

tion r e d u c e s the i n d u c t i o n time. T h e e x t e n t of this i n f l u e n c e varies a m o n g different h y d r o c a r b o n s . F i g u r e 16.6.9 p r e s e n t s the i n d u c t i o n t i m e data of e t h y l e n e pyrolysis in reflected s h o c k w a v e s at 5 0 b a r a n d the C - a t o m c o n c e n t r a t i o n e q u a l to 4.26 x 10 -6 m o l / c m 3 (Bauerle et al. 1994). T h e c o r r e l a t i o n yields z(].ts) = 2.6 • 10 -4 e x p ( 2 6 , 0 0 0 / T ) . Bauerle et al. (1994) n o t e d t h a t o n l y a s m a l l c h a n g e in the i n d u c t i o n t i m e was o b s e r v e d w h e n the C - a t o m c o n c e n t r a t i o n was varied f r o m 3 x 10 -6 to 1.6 • 1 0 - S m o l / c m 3. F i g u r e 16.6.10 s h o w s the i n d u c t i o n times of allene a n d 1 , 3 - b u t a d i e n e pyrolysis at a C - a t o m c o n c e n t r a t i o n of ~ 3 . 3 x 10 -7 m o l / c m 3 ( F r e n k l a c h et al. 1983b). It is s e e n t h a t i n d u c t i o n t i m e of allene is s h o r t e r t h a n t h a t of 1 , 3 - b u t a d i e n e by a b o u t a factor of 3. U n d e r the c o m p a r a b l e c o n d i t i o n , the i n d u c t i o n time for allene is j u s t slightly l o n g e r t h a n that for t o l u e n e . T h e i n d u c t i o n t i m e of 1 , 3 - b u t a d i e n e , o n the o t h e r h a n d , is close to that of a c e t y l e n e u n d e r c o m p a r a b l e c o n d i t i o n s .

16.6

279

Particulate Formation and Analysis 104

103

3J f

102

101

Z 4.4

46

7

jo

4.8

50

5.2

5.4

5.6

5.8

6.0

10 4 K / T FIGURE 16.6.9 Induction time of soot formation from ethylene pyrolysis in argon behind reflected shock waves (Bauerle et al. 1994). The induction times were determined by laser light extinction at 632.8nm. Ps = 50bar; [C]= 4.25 x 10 -6 mol/cm 3.

10"

103

102

10 ~ z,.0

i

i

i

i

i

4.5

5.0

5.5

6.0

6.5

7.0

10 4 K / T FIGURE 16.6.10 Induction time of soot formation from allene (0.726% in argon) and 1,3butadiene (0.54% in argon) pyrolysis behind reflected shock waves (Frenklach et al. 1983b). In both experiments, [C]= "~3.3 • 10 -7 mol/cm 3. The induction times were determined by laser light extinction at 632.8 nm. The data can be correlated by z(~ts)= 1.33 x 10 -2 exp(18, 200/T) for allene and z(~s)= 7.54 x 10 -2 exp(17, 100/T) for 1,3-butadiene.

280

H. Wang

Figure 16.6.11 presents the induction time data of soot formation from benzene pyrolysis in argon behind reflected shock waves (Bauerle et al. 1994; Knorre et al. 1996; B6hm et al. 1998). All of the data were collected at the elevated pressures of 50 to 60bar. Bauerle et al. (1994) noted a strong dependence of ~ on the C-atom concentration and reported an empirical correlation equation, z = 3.1 x 10 -10 exp(31,300/T)[C] -~ (Its, mol/cm3). This correlation was obtained by varying the C-atom concentration from 3.3 x 10 -7 to 4.2 x 10 -5 mol/cm 3. Under comparable conditions, the onset of sooting in benzene pyrolysis is 1 order of magnitude faster than that of ethylene and n-hexane. Figure 16.6.12 shows the induction time data for n-hexane pyrolysis behind reflected shock waves (Hwang et al. 1991; Bauerle et al. 1994). Again all data reported in the literature are obtained at elevated pressures, ranging from 20 to 100 bar. Very little pressure dependence was observed for hexane, and the data show an activation energy of about 51 kcal/mol (Bauerle et al. 1994). For a variation of the C-atom concentration between 5 x 10 -6 to 4 x 10 -5 mol/cm 3, hardly any dependence of ~ on [C] was found. Figure 16.6.13 presents the induction time data for toluene pyrolysis behind reflected shock waves (Wang et al. 1981; Frenklach et al. 1983a; Parker et al. 1990). Although Wang et al. (1981) did not report actual data, they

lO'

lo3 o 102

101

J 1

o

04.0

.

I

i

i

4.5

5.0

5.5

,,.

i

i

6.0

6.5

7.0

104 K / T FIGURE 16.6.11 Induction time of soot formation from benzene pyrolysis in argon behind reflected shock waves. The induction times were determined by laser light extinction at 632.8 nm. Circles (Bauerle et al. 1994): P5 = 50bar, [C]= 4.2 • 10 -6 mol/cm 3. Diamonds (B6hm et al. 1998): P5 -- 60bar, [C]-- 4.2 x 10 -6 mol/cm 3. Line (Knorre et al. 1996): from the fit at 60bar in the form of ~ x [C]~ - - A e x p ( O / T ) .

281

16.6 Particulate Formation and Analysis 10 4

oo

10 3

*9 ,~,,,'%~ a

oo % a ~ * O o

;

102

#4 ~ 9

T

l

lOa

4.5

5.0

9

. . . . . .

1.

.

5.5

.

.

.

.

6.0

104 K / T FIGURE 16.6.12 Induction time of soot formation from n-hexane pyrolysis in argon behind reflected shock waves. The induction times were determined by laser light extinction at 632.8 nm. Circles (Hwang et al. 1991): P5 -- 20bar, [C]= 4.3-5.7 x 10-6 mol/cm 3. Squares (Hwang et al. 1991): p~ = 63bar, [C]-- 6.1-6.8 x 10-6 mol/cm 3. Triangles (Hwang et al. 1991): P5 = 100bar, [C]= 4.6-6.2 x 10-6 mol/cm 3. Diamonds (Bauerle et al. 1994): P5 = 20-100bar, [C]= 5.3 x 10-6 mol/cm 3. Line: from Geck (1975).

p r e s e n t e d an empirical correlation in the form of z = 5.4 x 10 -9 exp(17,900/T)[C]-~ -~ (ITS, m o l / c m 3 ) , w h i c h fits very well the data collected at later times. The e x p o n e n t for the C-atom c o n c e n t r a t i o n is closed to - 1 w h i c h points to a strong d e p e n d e n c e of i n d u c t i o n time on the C-atom concentration. The total pressure does n o t seem to affect the i n d u c t i o n time, as indicated by the small e x p o n e n t of the argon concentration. This observation is consistent with the data of Parker et al. (1990), w h o s h o w e d hardly any d e p e n d e n c e of z w h e n the pressure is varied from 10 to 3 0 a t m . The i n d u c t i o n time of soot formation has also been studied in fuel-rich h y d r o c a r b o n oxidation (Wang et al. 1981; Wittig et al. 1990; MMler and Wittig 1994; Kellerer et al. 1996). Similar to the correlation of ignition delay times (Burcat et al. 1970; Lifshitz et al. 1971), an empirical equation in the form of z -- A exp(E/RT)[C]~[O2]fl[Ar] ~

(16.6.23)

has been used to correlate the i n d u c t i o n time of soot formation. Figure 16.6.14 shows such a correlation for toluene oxidation b e h i n d reflected s h o c k waves, as reported by W a n g et al. (1981). The correlation yields Z-

5.8 • 10 -15 exp(32,200/T)[C]-191[O2] 0 "78 [Ar]- 0 36

(its, m o l / c m 3)

282

H. W a n g

1 0 -2

O

t~

~.

10 "3

A

l0 4

rj 10-s :~ 3.5

I

I

!

1

4.0

4.5

5.0

5.5

,

_1,

I

I

6.0

6.5

7.0

7.5

10 4 K / T FIGURE 16.6.13 Induction time of soot formation from toluene pyrolysis in argon behind reflected shock waves. The induction times were determined by laser light extinction at 632.8 nm. Circles (Frenklach et al. 1983a): 0.311% toluene in argon, [C]= 3.32 x 10 - 7 mol/cm 3. Squares (Frenklach et al. 1983a): 1.75% toluene in argon, [C]- 2.69-3.32 x 10 - 7 mol/cm3. Triangles (Parker et al. 1990): p = 10-30 bar, 0.021-0.064% toluene in argon (specific conditions under which the induction times were measured were not given. Here it is assumed that the mixture contains 0.03% toluene and P5 = 20bar); line: empirical correlation reported by Wang et al. (1981), ~ = 5.4 x 10-9 exp(17, 900/T)[C]-~ -~ (~ts, mol/cm3).

C o m p a r e d to toluene pyrolysis, the d e p e n d e n c e on the C-atom c o n c e n t r a t i o n increases w h e n o x y g e n is present. An increase in C-atom c o n c e n t r a t i o n decreases the i n d u c t i o n time, whereas an increase in the oxygen c o n c e n t r a t i o n increases the i n d u c t i o n time. W i t h the presence of oxygen, the influence of pressure is s o m e w h a t larger than that of toluene pyrolysis. Mflller and Wittig (1994) s h o w e d that the i n d u c t i o n time of soot formation from m e t h a n e oxidation can be well correlated with the partial pressure of the fuel.

16.6.4.2 SOOT YIELD Soot yield is defined as the fraction of carbon atoms a c c u m u l a t e d in soot, and it characterizes the conversion efficiency of a h y d r o c a r b o n to soot (Frenklach et al. 1988). In almost all studies, the soot yield is d e t e r m i n e d by the laser light extinction technique. Figure 16.6.15 shows typical soot yield traces as a function of time, observed d u r i n g the pyrolysis of a t o l u e n e - a r g o n m i x t u r e b e h i n d reflected shock waves (Frenklach et al. 1983a). Similar traces have been

16.6

283

Particulate Formation and Analysis

0

10"4

0 E

:::t.

10 ~

,~o

~D el.

~o ooodP 0 o

u.____l

A

~~

1 0 "~

,(.).

10 -8

9

I

5.0

0

,.

I

I

5.5

6.0

. . . . . . .

I

6.5

,

I

7.0

7.5

104 K / T FIGURE 16.6.14 Induction time of soot formation from toluene-oxygen-argon mixtures behind reflected shock waves (Wang et al. 1981). The induction times were determined by laser light extinction at 632.8nm. Circles: 0.15% C7H8-0.0875% O2-argon, [ C ] - - 7 . 5 • 10 -7 mol/cm 3. Squares: 0.6% C7H8-0.35% O2-argon, [C]= 7.5 • 10 -7 mol/cm 3. Open triangles: 0.2% CFH 80.35% O2-argon, [C]= 2.5 • 10 -7 mol/cm 3. Diamonds: 0.3% C7H8-0.875% O2-argon, [C]-- 7.5 x 10 -7 mol/cm 3. Filled triangles: 0.3%C7Hs-0.21%O2-argon [C]-- 7.5 x 10 -7 mol/cm 3. Line: correlation given by z - 5.8 x 10 -15 exp(32, 200/T)[C]-l'91[O2]~ -0"36 (Its, mol/cm3).

reported by Knorre et al. (1996) for a benzene-acetylene mixture. It is seen that the dependence of these time traces on temperature is quite complex. At a temperature of 1495 K, little soot is observed. As temperature increases, carbon is converted to soot in larger rates and greater amounts. A further increase in temperature beyond 1729 K, however, brings the soot yield down at long reaction times, even though the rate of soot conversion may still be larger at small reaction times. Graham et al. (1975b) noted that at a fixed reaction time, the soot yield from the pyrolysis of various aromatics and related compounds exhibits a pronounced bell-shaped dependence on temperature. This dependence is illustrated in Figure 16.6.16 for data collected at a reaction time of 2.5 ms. The maximum yield is attained at a temperature around 1750 K with nearly 100% conversion of carbon to soot. Graham and coworkers suggested that the observed behavior is caused by the competition between the fragmentation rate of the parent hydrocarbon and the rate of molecular growth process, both of which increase as the temperature is elevated. More detailed shock tube analyses of soot yield followed. However, before the results of these analyses can be presented and systematically compared, we

284

H. W a n g

100 ~1729 K 1799 K

80 1672 K

"c3

60

e~

o o r,r

40 2059 K .. 1587 K

20

0

I

0

I, ~,

,

I

1

223 Ii

2

1495

1"

3

Time (ms) F I G U R E 16.6.15

T h e time trace of soot yield from the pyrolysis of 0.311% t o l u e n e in argon

([C]= 3.3 • 10-7 mol/cm3) behind reflected shock waves (Frenklach et al. 1983a). The traces are smoothed from the experimental profiles obtained from light extinction at 632.8nm, using the complex refractive index fit = 1.56-0.56i (Dalzell and Sarofim 1969).

shall discuss several issues related to soot yield measurements. It is important to note that the term soot is rather ambiguous because the lower bound of the particle sizes cannot be unambiguously determined. In practice, the term soot means an ensemble of particles that attenuate a laser beam. Because of different sensitivities of laser attenuation by particles at different wavelengths (Figure 16.6.2), soot yields at lower conversions may differ significantly when different wavelengths are used. Frenklach et al. (1983a) showed that at 0.5ms the temperatures of maximum soot yield differ quite significantly during the pyrolysis of toluene, as seen in Figure 16.6.17. In particular, for temperatures between 1600 and 1800 K, the result at 632.8 nm shows about 7% of conversion, whereas at 3.32~tm no soot particles are observed. The observed difference is consistent with the possibility of marked light attenuation by pre-particles at 632.8nm, whereas these particles do not attenuate light at 3.32 ~tm, as discussed previously. For this reason, it is important to report soot yield data along with the wavelength used in the measurement. A second ambiguity in the soot yield measurement arises from the uncertainty in the complex refractive index (Table 16.6.4). Figure 16.6.17

16.6

285

Particulate Formation and Analysis

80 r9

[] 4+ n

0 []

0

benzene

9

ethylbenzene

•

toluene

[]

indene

9 9 + 9 A

cyclohexatriene 1,4-cyclohexadiene cyclopentadiene acetylene pyridine

.+

[]

20

0

I

,

+

"

A

i...... I 1600

,

, 1800

2000

2200

I .... 9 2400

Temperature (K) FIGURE 16.6.16 Yields of soot at a reaction time of 2.5 ms from the pyrolysis of various hydrocarbons in argon in incident shock waves, aU with [C] = 3.3 • 10 -7 mol/cm 3 (Graham et al. 1975b). The data were collected from light extinction measurements at 4 4 8 n m with Im{(fla2 - 1)/(fla2 + 2)} -- 0.254 (Graham et al. 1975a). Lines are fits to data.

120 t

o 9 O

9 O/~'-"~

100

0.5 ms, 632.9 nm 0.5 ms, 3.32 larn 2 ms, 632.9 nm .

"d "~.,

0 1400

1600

1800

2000

2200

2400

Temperature (K) FIGURE 16.6.17 Effect of laser wavelength on the measured yields of soot at two reaction times, for the pyrolysis of 0.311% toluene in argon ([C] - 3.3 x 10 -7 mol/cm 3) behind reflected shock waves (Frenklach et al. 1983a). The complex refractive indexes used for the calculation of soot yields are fla - 1.56-0.56i at 632.8 nm and fla - 2.28-1.39i at 3.39 ~tm from Dalzell and Sarofim (1969). Yields over 100% are obtained at 3.39 ~tm, indicating the uncertainty in fla. Lines are fits to data.

286

H. Wang

shows that at the reaction time of 2 ms, the maximum soot yields at 632.8 nm and 3.32 ~tm differ markedly. Moreover, the yield at 3.32 ~tm exceeds 100%, indicating the uncertainty of the complex refractive index. This problem was noted by Graham (1975a) and further discussed in the work of Frenklach et al. (1983a, 1983b). The data in Figure 16.6.17 were converted from transmittance using r h - 1 . 5 6 - 0.56i at 632.8nm and r h - 2 . 2 8 - 1.39i at 3.39 ~tm from Dalzell and Sarofim (1969). The use of refractive index reported by Lee and Tien (1981) does not resolve the problem. To emphasize this ambiguity, Frenklach et al. (1983a, 1983b) reported their results in the form of soot yield multiplied by Im{(ffi 2 - 1)/(ffl 2 + 2)}. Here we shall uniformly use the data collected at 632nm and the refractive index of Dalzell and Sarofim (1969) to facilitate data comparison, although the absolute soot yield is questionable. Figure 16.6.18 shows the variation of soot yield as a function of temperature at the reaction times of 0.5, 1.0, 1.5, and 2.0 ms for acetylene pyrolysis behind reflected shock waves (Frenklach et al. 1983b). It is seen that soot yield exhibits a pronounced bell-shaped dependence on temperature, as noted by Graham et al. (1975b). The position of maximum soot yield is not universal under a given shock condition, but it is dependent on reaction time. The Catom concentration was also found to affect the shape of the soot yield curve, as seen in Figure 16.6.19. A larger C-atom concentration generally leads to a higher conversion to soot.

15 +j~,

--e-

/+ \ y

~,

10

Q O ~

5

/

T~-r

0.5 ms

o \

1.0m

t ~12:50m:

-

+

\ 0

1600

, ~

~ ' ~ ' r ~ - ' - - ' L ' ~) n ' T ~ " , - -

1800

2000

I

2200

J

I

2400

,

I

2600

)

I

2800

~

I

3000

J

3200

Temperature (K) FIGURE 16.6.18 Yields of soot at the reaction times of 0.5, 1.0, 1.5, and 2.0ms from the pyrolysis of a 1.09% acetylene-argon mixture ([C] = 8 . 1 - 8 . 6 x 10 -7 mol/cm 3, P5 - 1.27-2.33 bar) behind reflected shock waves (Frenklach et al. 1983b). Lines are fits to data.

16.6

287

Particulate Formation and Analysis

10

%C2H2 [C]•

_

r x oo of O~

_

"O -g

o

4.65

?

o o

0 1600

.~-.,-...~

.~...er~

1800

iZXl

2000

Ps

(raot/ern3) (bar) 1.09 3.3 2.14-3.87 zx 4.65 3.3 0.52-0.91 8.3

1.27-2.33

Oo/j/oO ,

I

2200

,

I

2400

~

I

~

2600

I

2800

,

I

3000

,

I

3200

Temperature (K) FIGURE 16.6.19 Yields of soot at a reaction time of 1.5 ms from the pyrolysis of acetylene-argon mixtures behind reflected shock waves (Frenklach et al. 1983b). The data were obtained from light extinction at 632.8 nm. Lines are fits to data.

Under the same C-atom concentration, the pyrolysis of benzene leads to the largest conversion to soot, as seen in Figure 16.6.20, when compared to acetylene, allene, vinylacetylene, and 1,3-butadiene (Frenklach et al. 1988, 1990). At temperatures below 2100 K, acetylene leads to the lowest amount of conversion to soot. Below the same temperature, vinylacetylene and 1,3butadiene have approximately the same sooting tendency. Soot formation from allene is much faster and in much larger quantities than from vinylacetylene and butadiene. Frenklach et al. (1983b) noted that the characteristics of soot formation from allene are similar to those of aromatics. In addition, the maximum soot yields for benzene and allene are achieved at lower temperatures than those for acetylene and 1,3-butadiene. At elevated pressures, the dependence of soot yield on temperature is still a bell-shaped curve, as seen in Figure 16.6.21 (Bauerle et al. 1994). Comparing between aromatic and nonaromatic hydrocarbons, the absolute difference in the maximum soot yield at elevated pressures is not as large as that at low pressures (cf. Figure 16.6.20). Benzene still has a higher propensity to soot than nonaromatic hydrocarbons (ethylene and n-hexane). Figure 16.6.22 presents the soot yields obtained in argon-diluted hydrocarbon mixtures: (a) allene-acetylene, (b) 1,3-butadiene-acetylene, and (c) benzene-acetylene (Frenklach et al. 1986a, 1988). The yields from each single component are also shown in the same figure. It is seen that the interaction of

H. Wang

288

9 * +

2f,

~

_

o

1.09%C2H 2 + At' 0.726% C3H4 + Ar 0.54% C4H4 + Ar 0.54% C4H6 + Ar

1-

~

o or ~

:lZ/ 7 o o ,I

!

~

1600

,I

~

1800

I

,

2000

I

~,

2200

Temperature

I

2400

~

,,,t

2600

(K)

FIGURE 16.6.20 Yields of soot at a reaction time of i ms from the pyrolysis of acetylene-, allene-, vinylacetylene-, butadiene-, and benzene-argon mixtures ([C] = 3.3 x 10 -7 mol/cm 3) behind reflected shock waves (Frenklach et al. 1988, 1990). The data were obtained from light extinction at 632.8nm, initially reported for the complex refractive index value of Menna and D'Alessio (1981). The data shown here are recalculated using the complex refractive index value of Dalzell and Sarofim (1969).

~'

80 / / /

--6mol/cm3, P5 = 50 bar

O

C6H6-Ar, [C] = 4.2•

9

C2H4-AI',[C] = 4.2x 10-~ mol/cm3, P5 = 50 bar

+

n-C~Hl4-Ar, [C] = 5.3•

-~ mol/cm3, P5 = 20-100 bar

6~I 40 8

200

1500

2000

2500

Temperature (K) FIGURE 16.6.21 Maximum yields of soot from the pyrolysis of benzene-, ethylene-, and nhexane-argon mixtures behind reflected shock waves (Bauerle et al. 1994). The data were obtained from light extinction at 632.8 nm, initially using the value of the complex refractive index of Lee and "lien (1981). The data presented in the plot are rescaled by using the ffa value of Dalzell and Sarofim (1969). Lines are fits to data.

16.6

Particulate Formation and Analysis 25

+ o a

,-, 20 ~.

289

(a)

0.726% C3H4 + 1.09% C2H2 + Ar 0.726% C3H4 + At" 1.09% C2H 2 + AT

"1::1 15-

+'

~

o o

105n_

0

.,4-

-~h

+'

~

.....

)

8 ~

+ o a

76x:l

5-

9,..4

4

o o

(b)

0.54% C4H6 + 1.09% C2H2 + Ar 0.54% C4I-I~ + Ar 1.09% C2H2 + Ar + +

.%

3

+

~+

+

2

+

1

0 70

(c)

+ 0.311% C6I-I~ + 1.09% C2I-I2 + Ar o 0.311% C6I-I~ + Ar &" 50 .......... 1.09% C2H2 + Ar 6O

_

,--, 40 0 ,., o o

30 20 10 0

.m. O+

1600

-r'-,.

1800

2000 2200 Temperature (K)

....

t ......

2400

~,

I

2600

FIGURE 16.6.22 Yields of soot at a reaction time of i ms from the pyrolysis of single-component and binary mixtures behind reflected shock waves (Frenklach et al. 1988). The data were obtained from light extinction at 632.9nm, initially reported for the complex refractive index value of Menna and D'Alessio (1981). The data shown here are recalculated using the complex refractive index value of Dalzell and Sarofim (1969). For single-component mixtures, [C]-3.3 x 10 -7 mol/cm3; for binary mixtures, [C] = ~ 6 . 7 x 10 -7 mol/cm 3. Lines are fits to data.

the binary components

is r a t h e r c o m p l e x .

like a l l e n e a n d b e n z e n e ,

t h e a d d i t i o n o f a c e t y l e n e l e a d s to l i t t l e c h a n g e i n t h e

soot yield, while

for a weakly

sooting

In strongly sooting hydrocarbons

hydrocarbon

like 1,3-butadiene,

the

a d d i t i o n o f a c e t y l e n e s e e m s to s h o w s o m e s y n e r g i s t i c e f f e c t w i t h r e s p e c t to s o o t formation.

290

H. Wang

Addition of hydrogen strongly suppresses soot formation from acetylene (Wang et al. 1981; Frenklach et al. 1988), as seen in Figure 16.6.23. This effect is evident at both near-atmospheric pressure (Figure 16.6.23a) and at elevated pressures (Figure 16.6.23b). A similar effect of soot suppression was observed in the shock tube pyrolysis of toluene (Wang et al. 1981). This effect was

5

-

+

4.65% C.2H 2 + Ar, Ps = 1.27-2.33 bar

o

4.65% C z H 2 + 4.65% H 2 + At', P5 - 1.3-1.8 bar

44-

++

@ O

/+

+

~

(a) 0

'

t

.

t

v

,,

v

I

,

I

,

I

,

8070-

60" 0 9 t,...t

0 o

oO.,

.... ..C2H2 "- o. o~

:

50-

/

40-

O

C2H2/I'I2-1/1, [C] -- 2x10 -6 m o l / c m 3 Ps = 60 bar

,.

6"..

...

C2H2/H2=l/1, [C] = 4x10 -6 mol/cm 3

"'"% . . "

:

9

3020-

/

10 0 1400

I

"

,

I

1800

,

I

,'"'"

I

2200

,

{

~

I

2600

~,,

i

,

I

,

3000

Temperature (K) FIGURE 16.6.23 Effect of hydrogen addition on the yield of soot from the pyrolysis of acetylene behind reflected shock waves. (a) Data at a reaction time of i ms for pressures at 1.27-2.33 bar from Frenklach et al. (1988); (b) maximum soot yields at the pressure at 60bar (Knorre et al. 1996). Both data sets were obtained from light extinction at 632.8 nm. The data in panel (a) were initially reported for the complex refractive index value of Menna and D'Alessio (1981). The data shown here are recalculated using the complex refractive index value of Dalzell and Sarofim (1969). In panel (b), the dotted lines are for the pyrolysis of pure acetylene or ethylene with total C atom concentration [C] - 4 x 10 -6 mol/cm 3. Lines are fits to data.

16.6

291

Particulate Formation and Analysis

attributed to a reduction of aromatic radical concentrations by H 2 via the back reaction of the H abstraction of the growing aromatic molecules (Frenklach 1984b, 1988). The complex bahavior exhibited for the dependence of soot yield on temperature extends to chlorinated hydrocarbons. Frenklach et al. (1986b) examined soot formation from the pyrolysis of chlorinated hydrocarbons behind reflected shock waves. Figure 16.6.24 shows the soot yields at 0.5 ms of pyrolysis time, comparing chloromethane-, dichloromethane-, trichloromethane-, tetrachloromethane-, methane-, and acetylene, all with the same Catom concentration of ~8.3 • 10 -7 mol/cm 3 and under similar pressure. Marked differences in the sooting characteristics were observed for these compounds. Upon shock heating, chlorinated compounds generally produce soot much faster and in larger quantities than do methane and acetylene. There are notable differences in the dependence of the soot yield on temperature. Dichloromethane exhibits a bell-shaped dependence at temperatures below 2400 K and appears to have the largest sooting propensity of all the chloromethane compounds. The maximum soot yield from the pyrolysis of trichloromethane is larger than that of dichloromethane, but this maximum is obtained at some 700 to 800 K higher than that for dichloromethane. For tetrachloromethane, the soot yield profile shifts to even higher temperatures. Chloromethane, on the other hand, has sooting propensity not so different from that of acetylene. 6O

~

40

~ 9

30

9.3% CHCI 3

"

9.3% CC14

9

rm

20 H2

10

1500

2000

2500

3000

Temperature (K) FIGURE 16.6.24 Yields of soot at a reaction time of 0.5 ms from the pyrolysis of chloromethane-, dichloromethane-, trichloromethane-, tetrachloromethane-, methane-, and acetylene-argon mixtures ([C] = ~ 8 . 3 • 10 -7 mol/cm 3) behind reflected shock waves (Frenklach et al. 1986b). The data were obtained from light extinction at 632.9 nm.

292

H. W a n g

50 4.65% CH2CI 2 + Ar [C] = 1.7x10 -7 mol/cm 3

40

Reoflected shock -~ o ~

3O

20 r~

lO d]~ 0

'

9~1 1600

Incident shock ,

i ....

2000

,

i

1

2400

2800

I 3200

Temperature (K) FIGURE 16.6.25 Yields of soot at a reaction time of i ms from the pyrolysis of dichloromethane behind incident and reflected shock waves (Frenklach et al. 1986b). The data were obtained from light extinction at 632.8 nm. Lines are fits to data.

An interesting yet unsettling observation by Frenklach et al. (1986b) was that soot yields determined behind incident and reflected shock waves may be qualitatively similar but quantitatively different. Figure 16.6.25 shows soot yields at a reaction time of i ms from the pyrolysis of dichloromethane, comparing the results obtained behind incident and reflected shock waves. It is seen that the soot yield determined behind reflected shock waves are larger than those from incident shock waves, by as much as a factor of 2. This difference was attributed to the possibility of additional hydrogen atoms formed from impurities behind reflected shock waves (Lifshitz and Frenklach 1977; Lifshitz et al. 1983), which promote soot formation by providing additional H abstraction from aromatic molecules and thereby increase the rate of aromatic growth reactions (Frenklach et al. 1984b). In general, an increase in pressure promotes soot formation from the pyrolysis of hydrocarbon, although the actual response of soot yield varies among different hydrocarbons. In some cases, an increase in pressure increases the soot yield over an entire temperature range. Figure 16.6.26 shows that for ethylene pyrolysis at a constant C-atom concentration behind reflected shock waves, a higher pressure leads to a larger soot yield at a given temperature (Bauerle et al. 1994). In other cases, an increase in pressure first shifts the soot bell curve to lower temperatures (Frenklach et al. 1983a, 1983b), as seen in Figure 16.6.27. A further increase in pressure does not shift the soot bell to even lower temperatures, nor does it increase the soot yield. This can be seen

16.6

293

Particulate Formation and Analysis

100

Ethylene-argonmixture [C] = 4.2x10 -6 mol/cm 3

80

i ......

[] +

I00 bar 50 bar

ix

25 bar

20 0

1700

1800

1900

2000

21oo

2200

Temperature (K) FIGURE 16.6.26 Maximum yields of soot from the pyrolysis of ethylene in argon behind reflected shock waves at three different pressures (Ps) for a constant C-atom concentration (Bauerle et al. 1994). The data were initially obtained from light extinction at 632.8nm. The reported data were derived from the complex refractive index value of Lee and Tien (1981) and were rescaled here using the rh value of Dalzell and Sarofim (1969). Lines are fits to data.

in Figure 16.6.27 by comparing the data of Frenklach et al. (1983a), determined at 1.83-3.06bar, to the data of Parker et al. (1990), determined at a much higher pressure (30 bar). Note that the C-atom concentrations in the experiments shown in Figure 16.6.27 are about equal, thus any difference observed for soot yield is presumably due to the influence of pressure. Hwang et al. (1991) and Bauerle et al. (1994) showed that for pressures between 20 and 100 bar, the soot yield remains the same for n-hexane pyrolysis behind reflected shock waves. This pressure insensitivity is similar to the behavior just discussed of toluene at elevated pressures. In general, the addition of oxygen causes the soot bell curve to shift to lower temperatures (Frenklach et al. 1984a, 1990). Figure 16.6.28 demonstrates the influence of oxygen when it is added to acetylene, allene, vinylacetylene, 1,3butadiene, and toluene. The most drastic effect is seen in the case of acetylene, where the addition of oxygen shifts the bell curve to a temperature as much as 500 K lower than in the case of acetylene pyrolysis, without significantly affecting the maximum soot yield. The enhanced soot production at low temperatures was attributed to the fact that the addition of oxygen results in the formation of reactive species, which promote the pyrolysis reactions (Frenklach et al. 1984a). The addition of oxygen to toluene at subatmospheric pressure of no. 4 bar (Figure 16.6.28e), on the other hand, has little effect on

294

H. Wang

o 80

1.75%C.TH8 + Ar, [C] = 3.3• -7 mol/cm3 (P5 - 0.31--0.53 bar)

+

0.311%CTH8 + At, [C] = 3.3• -7 tool/era3 (P5 = 1.83-3.06 bar)

ix

CTH s + At, [C] = 3•

60

-7 mol/cm 3, P5 = 30 bar []

~

9.,

O o r~

40

20-

0 + 1500

i

D

I

2000

2500

,,

i

3000

Temperature (K) FIGURE 16.6.27 Yields of soot at a reaction time of i ms from the pyrolysis of toluene-argon mixtures behind reflected shock waves. The low-pressure data (Frenklach 1983a, 1984a) were measured with light extinction at 632.8 nm and the data at 3 0 a t m (Parker et al. 1990) from light extinction at 389 and 633 nm or from emission measurements. Lines are fits to data.

soot formation at temperatures below 2150 K, but it decreases soot production at higher temperatures. Under a higher-pressure condition (1.87-3.08 bar), the addition of oxygen to toluene simply reduces the soot yield over the entire temperature range without significantly affecting the shape of the soot bell curve, as seen in Figure 16.6.29. The effect of oxygen on soot formation from allene, vinylacetylene, and 1,3-butadiene is somewhat intermediate between the effects observed for acetylene and toluene. The promotion effect of oxygen at low temperatures is obvious for vinylacetylene and 1,3-butadiene, yet such an effect is hardly seen for allene. Wang et al. (1981) showed that an increase in the amount of oxygen addition reduces the yield of soot from the pyrolysis of toluene, but such an effect is more drastic at the higher-temperature end of the bell curve. Figure 16.6.30 shows that an increase in the oxygen addition suppresses soot formation, notably toward high temperatures. Lowering both the initial toluene and oxygen concentrations does not affect the soot yield at the lowtemperature side of the bell curve, but it increases soot yield at higher temperatures. These results again illustrate the competition between the soot-suppression effect of oxygen by oxidizing the precursor to soot and the soot-promotion effect by increasing the concentrations of reactive species.

16.6

295

Particulate F o r m a t i o n and A n a l y s i s 5 "

" A +

4.65% C.2H2 + Ar 4.65% C 2 H 2 + 1.5%

(a) O=+

ha"

[C]= 8.3x+I0-7mol/cmS ~

2O A +

(b)

0.726% CsH, + Ar 0.726% C 3 H 4 + 0.726% 02 + Ar [C] = 3.3x 10-I tool/all3

..i o o

I. zx +

4

!

(c)

0.54% C i H 4 + Ar 0.54% C4H, + 0.54% Oa + At" [C] = 3.3x 10-v mol/cm 3

3 4-

2-

A

1

0 4

o ....4

(d)

Ar

0.54% C,Hs + 3

+

0.54% C4H s + 0.54% 02 + Ar [C] = 3.3x 10 -7 moI/cm3 ,~

A

1.75% C.~Hs + Ar 1.75% C~H s + 1.75% 02 + Ar

2

1

0 7O

60

+

50 ....,

[(3] = 3.3x10-~ mol/arn3

./ +

4O

(e) ......_& a

a~

3O a +~

2O

~

AaA

l0 1

200

1

!

1400

1600

.

.

.

1800

.

2000

.

2200

2400

I

2600

i

2800

I

Temperature (K) FIGURE 16.6.28 Yields of soot at a reaction time of i ms from the pyrolysis of (a) acetylene, (b) allene, (c) vinylacetylene, (d) 1.3-butadiene, and (e) toluene in argon behind reflected shock waves, with or without oxygen addition (Frenklach et al. 1984a, 1990). All data were obtained from light extinction at 632 nm, initially reported for the complex refractive index value of Menna and D'Alessio (1981). The data shown here are recalculated using the complex refractive index value of Dalzell and Sarofim (1969). Lines are fits to data.

296

H. Wang

60-

zx o

0.311% C_,TH8 + Ar 0.311% C,TH8 + 0.311% 02 + Ar

50-

40

-

"d 30 o r~

20

-

10 -

0

1400

~

m~ _

1600

1800

2000

2200

o

q

,

2400

Temperature (K) FIGURE 16.6.29 Yields of soot at 1ms from the pyrolysis of toluene in argon ([C] = 3.3 x 10 -7 mol/cm 3, P5 = 1.87-3.08 bar) behind reflected shock waves, with or without oxygen addition (Frenklach et al. 1984a). The data were obtained from light extinction at 632.9nm, initially reported for the complex refractive index value of Menna and D'Alessio (1981). The data shown here are recalculated using the complex refractive index value of Dalzell and Sarofim (1969). Lines are fits to data.

120 100 -

o o ~ ~ t ~ 6

o

0.3% C7H$ + 0.21% 02 +Ar

o zx

0.3% C7H$ + 0.875% 02 + Ar 0.15% CTH,+ 0.0875% 02 +Ar

80 9,-, o

r~

60

40

20 0 1400

1600

1800

2000

2200

2400

2600

Temperature (K) FIGURE 16.6.30 Yields of soot at a reaction time of 2.5 ms for toluene-oxygen-argon mixtures ([C]-- 7.5 • 10 -7 mol/cm 3) behind reflected shock waves (Wang et al. 1981). The data were obtained from light extinction at 632.8nm and initially reported for I m { ( r h 2 - 1 ) / (rh 2 + 2)} --0.292. Here the data are shown for the complex refractive index value of Dalzell and Sarofim (1969). Lines are fits to data.

297

16.6 Particulate Formation and Analysis

8070~',

60-

,~

50-

0.35%C6H 6 + 0.875% 02 + Ar o 0.30%C7Hs + 0.875% 02 + At ~x 0.157%C7H8 + 0.5% C2H2 + 0.875% 02 + Ar

[]

~ .p,,

;~

~_.-~

o~

40-

O c

3020100

1300

1400

1500

1600

1700

1800

1900

T e m p e r a t u r e (K) FIGURE 16.6.31 Yields of soot at a reaction time of 2.5 ms for benzene-, toluene-, toluene + acetylene-oxygen-argon mixtures ([C] = 7.5 x 10 - 7 mol/cm3) behind reflected shock waves (Wang et al. 1981). The data were obtained from light extinction at 632.8 nm and initially reported for I m { ( t n 2 - 1)/(tn 2 q- 2)} = 0.292. Here the data are shown for the complex refractive index value of Dalzell and Sarofim (1969). Lines are fits to data.

For the same C-atom concentration and with the same a m o u n t of oxygen addition, the soot yield from the pyrolysis of b e n z e n e is higher than that of toluene. Figure 16.6.31 shows that the n u m b e r of aromatic rings in the reactant influences the a m o u n t of soot p r o d u c t i o n (Wang et al. 1981). This observation is further s u p p o r t e d by the evidence that for the same C-atom concentration, the toluene-acetylene m i x t u r e produces less soot than toluene alone. In mixtures of h y d r o c a r b o n - o x y g e n - i n e r t with similar total C-atom concentrations, the soot yields at elevated pressure, say, at 40 bar, are similar a m o n g various alkane c o m p o u n d s , including m e t h a n e , propane, and nheptane at a given temperature (Kellerer et al. 1996), as d e m o n s t r a t e d in Figure 16.6.32. The d e p e n d e n c e of soot yield on temperature remains to be that of a bell-shaped curve with m a x i m u m a r o u n d 1800 K. Pressure is seen to have a drastic effect on soot yield in an oxidizing e n v i r o n m e n t (Kellerer et al. 1996). Figure 16.6.33 shows that for the reaction of an n - h e p t a n e - o x y g e n argon m i x t u r e at the equivalence ratio of 5, the soot yield increases by about a factor of 2 w h e n the pressure is increased from 30 to 50 bar. The notable effect of pressure in an oxygen-containing m i x t u r e is s o m e w h a t different from that in the pyrolysis of n-hexane, where it is observed that pressure has little effect on soot yield (Hwang et al. 1991; Bauerle et al. 1994).

298

H. Wang

30-

a

C3Hs.Oz_Ar' [C]_-.6.0x10..6 moi/r 3

P5 = 40 bar

x

n_CTHl6_O2_Ar' [C]=5.9x10-6 mol/cm 3

0=5

zx

CH4-O2-Ar,[C]=7.6x10-~ mol/crn3 •

ID ~

20-

g r~ 10-

0 1600

I

i

t

1700

I

J

1800

I

I

,,

1900

,

,

2000

,

,,

2100

2200

Temperature (K) FIGURE 16.6.32 Yields of soot for the reaction of propane-, n-heptane, and methane-oxygenargon mixtures with similar C-atom concentrations, behind reflected shock waves (Kellerer et al. 1996). The data were obtained from light extinction at 632.8nm and initially reported for the complex refractive index value of Lee and Tien (1981). Here the data are shown for the complex refractive index value of Dalzell and Sarofim (1969). Line is fit to data.

n-CTHl6 + 02 + 99%Ar, (r = 5) 30-

"m

[]

30 bar

0

40 bar

zx

50 bar

20 z )

o o r~

zx

10

0

1600

,

I

1700

,

I

1800

,

I

1900

,

i

2000

,

2100

Temperature (K) FIGURE 16.6.33 Yields of soot for the reaction of n-heptane-oxygen-argon mixtures at three different pressures, behind reflected shock waves (Kellerer et al. 1996). The data were obtained from light extinction at 632.8nm and initially reported for the complex refractive index value of Lee and Tien (1981). Here the data are shown for the complex refractive index value of Dalzell and Sarofim (1969). Lines are fits to data.

299

16.6 Particulate Formation and Analysis

16.6.4.3

SOOT GROWTH RATE

It has been shown (Bauerle et al. 1994) that the time trace of soot volume fraction profile can be empirically fitted by the first-order rate equation (Haynes and Wagner 1981),

df. dt = - h f ( f v

- fv, oo)

(16.6.24)

where the rate constant kf is used as a measure for the rate of soot formation, and fv, oo is the final soot volume fraction. Indeed, the soot volume fraction profile beyond the induction time can be well fitted into Equation (16.6.24), as demonstrated in Figure 16.6.3. Obviously, to determine kf the reaction time must be long enough so that the soot volume fraction approaches its final value, fv,~. This requirement could pose some problems as the soot volume fraction may continue to rise before the contact surface arrives. For shock tube experiments at sufficiently high pressures and high temperatures, this is usually not a problem because the rate of soot formation is large enough so that the soot volume fraction should reach a plateau within, say, 2ms of reaction time. Bauerle et al. (1994) and Knorre et al. (1996) showed that the rate constants kf can be well represented in an Arrhenius plot when they are normalized by the total C-atom concentration. No influence of pressure was observed for kf/[C] for acetylene, ethylene, benzene, n-hexane, and mixtures of acetylene and benzene over a wide range of pressure (20-100 bar) and temperature (1600-2500 K). For acetylene, ethylene, n-hexane, and mixtures of acetylene and benzene, kf/[C] approaches a maximum at a temperature of about 2000 K, whereas no maximum was observed for benzene. For a given C-atom concentration, the rate constant for benzene is at least 1 order of magnitude larger than those for ethylene and n-hexane. The apparent activation energy for ethylene and n-hexane in the temperature range from 1600 to 2000 K, and benzene and mixtures of benzene and acetylene in the temperature range from 1600 to 2500 K is about 48 kcal/mol. For benzene, the pre-exponential factor was reported to be 1.2 x 1015/s-1 (Knorre et al. 1996). Tanke et al. (1998) showed that the apparent rate constant can be significantly affected by additives, such as iron pentacarbonyl.

16.6.4.4

REM AND TEM STUDIES

Soot particles can be collected on carbon-film-coated copper grids mounted near the end wall of a shock tube. These particles have been analyzed by transmission electron microscopy (TEM) and raster electron microscopy

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(REM). Under a variety of shock tube conditions, the morphology of soot was found to be similar to that in hydrocarbon flames. These particles are aggregates of many individually spherical particles (Bauerle et al. 1994; Knorre et al. 1996), known as the primary particles. The average diameter of the primary particles is usually around 10nm. The particle size can be described by a log-normal distribution (Equation 16.6.19) with a geometric standard deviation In crg-~ ~0.2. The particle size does not seem to change markedly over a wide range of temperature, pressure, and C-atom concentration (Bauerle et al. 1994).

16.6.5

NANO-PARTICLE

SYNTHESIS

Despite the fact that the shock tube technique is ideal for kinetics studies of particulate formation at high temperatures, as demonstrated by the work reported for soot formation, the use of the shock tube for particle or nanopowder synthesis has been limited. Steinwandel and coworkers (1981a, 1985) generated silicon clusters or particles from the pyrolysis of silane behind incident and reflected shock waves. The light extinction technique at two wavelengths [e.g., 248 and 366nm, Steinwandel and Hoeschele (1985)] was used to detect and quantify the particles. A comprehensive shock tube study of silicon particle formation from the pyrolysis of silane and disilane diluted in argon and hydrogen was reported by Frenklach and coworkers (Frenklach et al. 1996). The experiments were conducted behind incident shock waves, at temperatures from 900 to 2000 K and pressures from 0.2 to 0.7 atm. The formation of particles was monitored by light extinction at two wavelengths. The fractional yield, particle size, and number density were simultaneously determined. Within the available reaction time, particles begin to form at temperatures around 1000 K for silane and around 900 K for disilane. The transmittance reaches a maximum at around 1100 to 1200 K, although it is most likely that the observation is caused by a change in the index of refraction, as discussed previously. Silicon particles were collected on a substrate mounted on the end wall of the shock tube. The particles were analyzed with electron diffraction, TEM, and secondary ion mass spectrometry. It was found that loosely aggregated particles are formed, consisting of nearly spherical primary particles that ranged from 10 to 40 nm in diameter and contained ~ 15% hydrogen on an atomic basis. The formation of binary-component particles has also been examined in shock tubes. Carmer and Frenklach (1989) studied the formation of silicon carbide (SIC) particles in a silane-methane-argon mixture behind incident and reflected shock waves at temperatures between 800 and 3650 K and pressures of 0.46 to 4.16 atm. The progress of particle formation was monitored by light

16.6

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Particulate Formation and Analysis

extinction at 632.8 nm, and the formed particles were analyzed by TEM and electron diffraction. It was found that (a) no particles were formed at temperatures below 900K; (b) particles formed between 900 and 1400K contained silicon only, having diameters ranging from 10 to 50nm; and (c) above 1400 K, particles contained both silicon and carbon. The ratio of t-SiC to Si increases steadily from 1400 to 1700 K, and remains constant above 1700 K. At very high temperatures, T~ > 2800 K, particles having diameters of 500 nm were observed. These particles consisted of//-SIC and were in the form of thin single-crystal platelets. A growth rate on the order of 106 lam/h was measured, which is unseen for silicon carbide formed by any other technique. The experimental observations led the authors to postulate a two-stage process of SiC particle formation, involving the homogeneous nucleation of SiC particles, along with Si particles, which are etched by hydrocarbon intermediates. The resulting products can add to the growing SiC particles by coelescence and reaction with the particles. Herzler et al. (1998) studied the formation of titanium nitride (TIN) particles from mixtures of TiC14-NH3-H 2 behind reflected shock waves at temperatures between 1400 and 2500 K and pressures of 1 to 2.3 bar. The formation of particles was monitored by light extinction, and the TiN molecules detected by laser absorption spectroscopy (A21-I ~-- X2X]). In most experiments, an induction time of particle formation was observed, which was found to be dependent only on temperature. The dependence of induction time on temperature was found to be similar to that in soot formation from hydrocarbon pyrolysis. The TiN molecule profile also shows an induction time, after which the TiN concentration increases steadily, reaches a maximum, and then decreases, presumably due to consumption by particle nucleation and growth.

16.6.6 HOMOGENEOUS METAL PARTICLES

NUCLEATION

OF

In a series of papers, Bauer and coworkers examined the homogeneous nucleation of metal particles by shock heating volatile metal-bearing compounds diluted in argon (Freund and Bauer 1977; Frurip and Bauer 1977a, 1977b, 1977c; Stephens and Bauer 1981). In these experiments, supersaturated metal vapor at controlled densities and temperatures were generated following the fast decomposition of the organometallics within the shock front. Thus, unlike soot formation from shock heating hydrocarbon compounds, the production of particles is not influenced by the pyrolysis kinetics of the reactant. For this reason, the shock tube technique is ideally

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suited for the examination of homogeneous nucleation theory, as was done in the work of Bauer and coworkers. The homogeneous nucleation of metal particles was followed by light extinction, scattering, and/or density gradients behind the shock front. The heat of condensation was measured for iron clusters as a function of particle size (Freund and Bauer 1977). The critical supersaturation ratio for the onset of rapid condensation from the vapors was determined for iron, lead, and bismuth as a function of temperature (Frurip and Bauer, 1977a, 1977b). The cluster growth rates were measured with light scattering (Frurip and Bauer 1977c). In all studies, Bauer and coworkers illustrated the difficulties in predicting the nucleation phenomenon by thermodynamic equilibrium or by the classical nucleation theory. Based on these studies, a self-consistent model had emerged (Bauer and Frurip 1977), featuring the kinetic phenomenon in the nucleation process and a kinetic criterion for the onset of condensation. Bauer's work was further extended to metal oxides by shock heating mixtures of Fe(CO)5 with N20 and Sill 4 with N20 (Stevens and Bauer 1981). Similar metal-vapor condensation experiments were reported by Steinwandel and coworkers (Steinwandel et al. 1981a, 1981b; Steinwandel and Hoeschele 1985, 1986). In these experiments, the nucleation of particles was detected by time-resolved integral atomic absorption spectroscopy for metal vapors and by light extinction for the metal particles. Again, large discrepancies were found to exist between the experimental data and the classical nucleation theory. The necessity of considering the nonequilibrium behavior is again emphasized (Steinwandel and Hoeschele 1986).

16.6.7

SUMMARY

In this chapter we discussed the methods and results of particulate formation in shock tubes. We have seen that the shock tube technique is ideally suited for the study of soot formation from gases at high temperatures. Induction time and soot conversion data have provided the much-needed information regarding the mechanism of soot formation and the influence of various parameters and fuel structures on soot formation (Haynes and Wagner 1981; Frenklach et al. 1984b, 1986a; Frenklach 1988; Wagner 1994). The use of shock tubes for the study of particulate formation relevant to material synthesis is limited. However, past studies in this area have opened a venue for approaching a basic understanding of the intricate kinetics of particle formation in chemical vapor deposition. In addition to the obvious advantage of well-defined conditions in which spatial transport of gas and particles is entirely eliminated, the relative ease of laser diagnostic techniques make the shock tube technique an ideal tool for this purpose. It is reasonable to

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303

a n t i c i p a t e t h a t this t e c h n i q u e will c o n t i n u e to b e w i d e l y u s e d in r e s e a r c h areas i n v o l v i n g t h e f o r m a t i o n of c o n d e n s e d - p h a s e m a t e r i a l s f r o m gases a n d in n a n o science and technology.

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Cundall, R. B., Fussey, D. E., Harrison, A. J., and Lampard, D. (1978). Shock tube studies of the high temperature pyrolysis of acetylene and ethylene. J. Chem. Soc. Faraday Trans. I 74:14031409. Cundall, R. B., Fussey, D. E., Harrison, A. J., and Lampard, D. (1979). High temperature pyrolysis of ethane and propylene. J. Chem. Soc., Faraday Trans. I 75:1390-1394. D'Alessio, A. (1982). Laser light scattering and fluorescence diagnostics of rich flames produced by gaseous and liquid fuels. In Particulate carbonmFormation during combustion, D. C. Siegla and G. W. Smith, eds., pp. 207-259. Plenum, New York. Dalzell, W. H. and Sarofim, A. E (1969). Optical constants of soot and their application to heat-flux calculations. J. Heat Transfer 91:100-104. di Stasio, S. and Massoli, P. (1997). Morphology, monomer size and concentration of agglomerates constituted by Rayleigh particles as retrieved from scattering/extinction measurements. Comb. Sci. Tech. 124:219-247. di Stasio, S., Massoli, P., and Lazzaro, M. (1996). Retrieval of soot aggregate morphology from light scattering/extinction measurements in a high-pressure high-temperature environment. J. Aerosol Sci. 27:897-913. Farmer, W. M. (1972). Measurement of particle size, number density, and velocity using a laser interferometer. Appl. Opt. 11:2603-2612. Fox, T. W., Rackett, C. W., and Nicholls, J. A. (1977). Shock wave ignition of magnesium powders. Proc. 11th Int. Symp. on Shock Tubes and Waves: Shock Tube and Shock Wave Research, pp. 262268. University of Washington Press, Seattle. Frenklach, M. (1988). On the driving force of PAH production. Proc. 22nd Symp. (Int.) on Combustion, pp. 1075-1082. The Combustion Institute. Frenklach, M., Clary, D. W., Gardiner, Jr., W. C., and Stein, S. E. (1984b). Detailed kinetic modeling of soot formation in shock4ube pyrolysis of acetylene. Proc. 20th Symp. (Int.) on Combustion, pp. 887-901. The Combustion Institute. Frenklach, M., Clary, D. W., Gardiner, Jr., W. C., and Stein, S. E. (1986a). Effect of fuel structure on pathways to soot. Proc. 21st Symp. (Int.) on Combustion, pp. 1067-1076. The Combustion Institute. Frenklach, M., Hsu, J. P., Miller, D. L., and Matula, R. A. (1986b). Shock-tube pyrolysis of chlorinated hydrocarbons: Formation of soot. Combustion and Flame 64:141-155. Frenklach, M., Ramachandra, M. K., and Matula, R. A. (1984a). Soot formation in shock-tube oxidation of hydrocarbons. Proc. 20th Symp. (Int.) on Combustion, pp. 871-878. The Combustion Institute. Frenklach M., Taki, S., Durgaprasad, M. B., and Matula, R. A. (1983b). Soot formation in shocktube pyrolysis of acetylene, allene, and 1,3-butadiene. Combustion and Flame 54:81-101. Frenklach M., Taki, S., Li Kwok Cheong, C. K., and Matula, R. A. (1983c). Soot particle size and soot yield in shock tube studies. Combustion and Flame 51:37-43. Frenklach M., Taki, S., and Matula, R. A. (1983a). A conceptual model for soot formation in pyrolysis of aromatic hydrocarbons. Combustion and Flame 49:275-282. Frenklach M., Ting, L., Wang, H., and Rabinowitz, M. J. (1996). Silicon particle formation in pyrolysis of silane and disilane. Isr. J. Chem. 36:293-303. Frenklach M. and Wang, H. (1991). Detailed modeling of soot particle nucleation and growth. Proc. 21st Symp. (Int.) on Combustion, pp. 1559-1566. The Combustion Institute. Frenklach M., Yuan, T., and Ramachandra, M. K. (1988). Soot formation in binary hydrocarbon mixtures. Energy and Fuels 2:462-480. Frenklach M., Yuan, T., and Ramachandra, M. K. (1990). Soot formation in shock-tube pyrolysis and oxidation of vinylacetylene. Proc. 17th Int. Symp. on Shock Waves and Shock Tubes: Current Topics in Shock Waves, pp. 475-480. American Institute of Physics, New York.

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Freund, H.J. and Bauer, S. H. (1977). Homogeneous nucleation of metal vapors. 2. Dependence of the heat of condensation on cluster size. J. Phys. Chem. 81:994-1000. Frieske, H. J., Seelbach, E., and Adomeit, G. (1981). Fuel pyrolysis of pulverized lignite in a singlepulse shock-tube. Proc. 13th Int. Syrup. on Shock Tubes and Waves: Shock Tubes and Waves, pp. 790-799. State of University of New York Press, Albany. Frurip, D. J. and Bauer, S. H. (1977a). Cluster growth rates in supersaturated lead vapor. Proc. 1 lth Int. Symp. on Shock Tubes and Waves: Shock Tube and Shock Wave Research, pp. 451-458. University of Washington Press, Seattle. Frurip, D. J. and Bauer, S. H. (1977b). Homogeneous nucleation of metal vapors. 3. Temperature dependence of the critical supersaturation ratio for iron, lead, and bismuth. J. Phys. Chem. 81:1001-1006. Frurip, D. J. and Bauer, S. H. (1977c). Homogeneous nucleation of metal vapors. 4. Cluster growth rates from light scattering. J. Phys. Chem. 81:1007-1015. Fursov, V. P., Shevtsov, V. I., Gusachenko, E. I., and Stesik, L. N. (1979). Role of the process of evaporation of volatile metals in the mechanism of their high-temperature oxidation and flaming. Combust. Explos. Shock Waves 16:247-254. Fussey, D. E., Gosling, A. J., and Lampard, D. (1978). A shock-tube study of induction times in the formation of carbon particles by pyrolysis of the C2 hydrocarbons. Combustion and Flame 32:181-192. Geck, C. C. (1975). "Untersuchung der bildungsgeschwindigkeit von rutg bei der pyrolyse yon /~thylen hinter reflectierten stotgweller." Ph.D. thesis, Universit~it GOttingen, G6ttingen. Gosling, A. J., Lampard, D., and Fussey, D. E. (1973). A shock tube study of the formation of carbon particles during the pyrolysis of hydrocarbons. In Combustion Institute European Symposium, E J. Weinberg, ed., pp. 388-393. Academic Press, London. Graham, S. C. (1976). The collisional growth of soot particles at high temperatures. Proc. 16th Syrup. (Int.) on Combustion, pp. 663-669. The Combustion Institute. Graham, S. C. (1981). The modeling of the growth of soot particles during the pyrolysis and partial oxidation of aromatic hydrocarbons. Proc. Roy. Soc. Lond. A 377:119-145. Graham, S. C. and Homer, J. B. (1973a). Coagulation of molten lead aerosol. Fogs and Smokes, Symposia of the Faraday Society, pp. 85-96. Chemical Society, London. Graham, S. C. and Homer, J. B. (1973b). Light-scattering measurements on aerosols in a shock tube. Proc. 9th Shock Tube Symp.: Recent Developments in Shock Tube Research, pp. 712-719. Stanford University Press, Stanford. Graham, S. C., Homer, J. B., and Rosenfeld, J. L.J. (1975a). The formation and coagulation of soot aerosols generated by the pyrolysis of aromatic hydrocarbons. Proc. Roy. Soc. Lond. A 344:259285. Graham, S. C., Homer, J. B., and Rosenfeld, J. L.J. (1975b). The formation and coagulation of soot aerosols. Proc. l Oth Int. Shock Tube Symp.: Modern Developments in Shock Tube Research, pp. 621-631. Shock Tube Research Society, Kyoto. Haynes, B. S. and Wagner, H. Gg. (1981). Soot formation. Prog. Energy Combust. Sci. 7:229273. Herzler, J. Leiberich, R., Mick, H. J., and Roth, P. (1998). Shock tube study of the formation of TiN molecules and particles. Nanostructured Mater. 10:1161-1171. Hogg, R. V. and Craig, A. T. (1970). Introduction to mathematical statistics. Macmillan, London. Hooker, W. J. (1959). Shock tube studies of acetylene decomposition. Proc. 7th Syrup. (Int.) on Combustion, pp. 949-952. Butterworth. Hottel, H. C. and Sarofim, A. E (1967). Radiative heat transfer. McGraw-Hill, New York. Hwang, S. M., Vlasov, P., Wagner, H. Gg., and Wolff, T. (1991). A shock tube study of soot formation following n-heptane pyrolysis. Z. Phys. Chem. Neue. Folge 173:129-139.

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Jellison, Jr. G. E. and Lowndes, D. H. (1987). Measurements of the optical properties of liquid silicon and germanium using nanosecond time-resolved ellipsometry. Appl. Phys. Lett. 51:352354. Johnson, N. L. and Leone, E C. (1977). Statistics and experimental design in engineering and physical sciences. Wiley, New York. Kellerer, H., M~ller, A., Bauer, H.-J., and Wittig, S. (1996). Soot formation in shock tube under elevated pressure conditions. Comb. Sci. Tech. 113-114:67-80. Kerker, M. (1969). The scattering of light and other electro-magnetic radiation. Academic Press, New York. Knorre, V. G., Tanke, D., Thienel, T., and Wagner, H. Gg. (1996). Soot formation in the pyrolysis of benzene/acetylene and acetylene/hydrogen mixtures at high carbon concentrations. Proc. 26th Symp. (Int.) on Combustion, pp. 2303-2310. The Combustion Institute. Lai, E S., Friedlander, S. K., Pich, J., and Hidy, G. M. (1972). The self-preserving particle size distribution for Brownian coagulation in the free-molecule regime. J. Colloid Interface Sci. 40:395-405. Lee, S. C. and Tien, C. L. (1981). Optical constants of soot in hydrocarbon flames. Proc. 18th Symp. (Int.) on Combustion, pp. 1159-1166. The Combustion Institute. Lester, T. W and Wittig, S. L. K. (1975). Particle growth and concentration measurements in sooting homogeneous hydrocarbon combustion systems. Proc. l Oth Int. Shock Tube Symposium: Modern Developments in Shock Tube Research, pp. 632-639. Lifshitz, A., Bidani, M., and Carroll, H. E (1983).The reaction of H 2 4- D2 ~ 2HD. A long history of erroneous interpretation of shock-tube results. J. Chem. Phys. 79:2742-2747. Lifshitz, A. and Frenklach, M. (1977). The reaction between H 2 and D2 in a shock tube: study of the atomic vs. molecular mechanism by atomic resonance absorption spectrometry. J. Chem. Phys. 67:2803-2810. Lifshitz, A., Scheller, K., Burcat, A., and Skinner, G. B. (1971). Shock tube investigation of ignition in methane-oxygen-argon mixtures. Combustion and Flame 16:311-321. Lowenstein, A. I. and von Rosenberg, Jr. C. W. (1977). Shock tube studies of coal devolatilization. Proc. 1lth Int. Symp. on Shock Tubes and Waves: Shock Tube and Shock Wave Research, pp. 366374. University of Washington Press, Seattle. Mar'yasin, I. L. and Nabutovskii, Z. A. (1969). An investigation of the kinetics of the pyrolysis of benzene in shock waves. I. Kinetics and Catalysis 10:800-806. Mar'yasin, I. L. and Nabutovskii, Z. A. (1970). Investigation of the kinetics of the pyrolysis of acetylene in shock waves. II. Kinetics and Catalysis 11:706-711. Mar'yasin, I. L. and Nabutovskii, Z. A. (1973). Investigation of the kinetics of carbon black formation during the thermal pyrolysis of benzene and acetylene in a shock wave. III. Kinetics and Catalysis 14:139-144. Menna, P. and D'Alessio, A. (1981). Light scattering and extinction coefficients for soot forming flames in the wavelength range from 200nm to 600 nm. Proc. 19th Symp. (Int.) on Combustion, pp. 1421-1428. The Combustion Institute. M~ller, A. and Wittig, S. (1994). Experimental study on the influence of pressure on soot formation in a shock tube. In Soot Formation in Combustion: Mechanisms and Models, H. Bockhorn, Ed., pp. 350-368. Springer-Verlag, Berlin. Nettleton, M. A. (1977). Shock-wave chemistry in dusty gases and fogs--Review. Combustion and Flame 28:3-16. Nettleton, M. A. and Stirling, R. (1974). Influence of additives on burning clouds of coal particles in shocked gases. Combustion and Flame 22:407-414. Park, C. and Appleton, J. P. (1973a). Shock tube measurements of soot oxidation rates. Combustion and Flame 20:369-379.

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Park, C. and Appleton, J. P. (1973b). Shock tube measurements of soot oxidation rates at combustion temperatures and pressures. Proc. 9th Shock Tube Symp.: Recent Developments in Shock Tube Research, pp. 793-803. Stanford University Press, Stanford. Parker, T. E., Foutter, R. R., and Rawlins, W. T. (1990). Soot initiation and particle growth in the pyrolysis of toluene at high inert gas pressures. Proc. 17th Int. Symp. on Shock Waves and Shock Tubes: Current Topics in Shock Waves, pp. 481-486. American Institute of Physics, New York. Penndorf, R. B. (1962). Scattering and extinction coefficients for small absorbing and nonabsorbing aerosols. J. Opt. Soc. Am. 52:896-904. Rawlins, W. T., Cowles, L. M., and Krech, R. H. (1983b). Optical signatures of soot formation in the pyrolysis of toluene near 2000 K. Paper presented at the Fall Technical Meeting of the Eastern States Section of the Combustion Institute, Providence, RI. Rawlins, W. T., Tanzawa, T., Schertzer, S. P., and Krech, R. H. (1983a). "Synthetic fuel combustion: Pollutant formation. Soot initiation mechanisms in burning aromatics." Physical Sciences Report TR-361. Roberts, T. A., Burton, R. L., and Krier, H. (1993). Ignition and combustion of aluminum/ magnesium alloy particles in 0 2 at high pressures. Combustion and Flame 92:125-143. Roth, P. and Brandt, O. (1990). Shock tube measurements of soot oxidation rates by using a rapid tuning IR-laser. Proc. 17th Int. Symp. on Shock Waves and Shock Tubes: Current Topics in Shock Waves, pp. 506-511. American Institute of Physics, New York. Roth, P., Brandt, O., and yon Gersum, S. (1990). High temperature oxidation of suspended soot particles verified by CO and CO 2 measurements. Proc. 23rd Symp. (Int.) on Combustion, pp. 1485-1491. The Combustion Institute. Seeker, W. R., Wegener, D. C., Lester, T. W., and Merklin, J. E (1978). Single pulse shock tube studies of pulverized coal ignition. Proc. 17th Symp. (Int.) on Combustion, pp. 155-166. The Combustion Institute. Seinfeld, J. H. (1986). Air pollution. Wiley, New York. Steinwandel, J., Dietz, T., Joos, V., and Hauser, M. (1981a). Condensation kinetics of iron and silicon in the vapor phase. Proc. 13th Int. Symp. on Shock Tubes and Waves: Shock Tubes and Waves, pp. 700-706. State of University of New York Press, Albany. Steinwandel, J., Dietz, T., Joos, V., and Hauser, M. (1981b). Homogene kondensation von tibers/~ttigtem eisendampf. Kinetische untersuchungen mit einem stogwellenrohr. Ber. Bunsenges. Phys. Chem. 85:683-686. Steinwandel, J. and Hoeschele, J. (1985). Spectroscopic detection of particles from shock-waveinduced decomposition of Sill4. Chem. Phys. Lett. 116:25-29. Steinwandel, J. and Hoeschele, J. (1986). Spectroscopic investigation of the homogeneous nucleation of nickel induced by shock pyrolysis of Ni(CO) 4. J. Chem. Phys. 85:6765-6772. Stephens, J. R. and Bauer, S. H. (1981). Investigation of homogeneous nucleation of Fe, Si, Fe/Si, FeO x, and SiOx vapors and their subsequent condensation. Proc. 13th Int. Symp. on Shock Tubes and Waves: Shock Tubes and Waves, pp. 691-699. State of University of New York Press, Albany. Szydlowski, S. L., Wegener, D. C., Merklin, J. E, and Lester, T. W. (1981). Short residence-time pyrolysis and oxidative pyrolysis of bituminous coals. Proc. 13th Int. Symp. on Shock Tubes and Waves: Shock Tubes and Waves, pp. 800-808. State of University of New York Press, Albany. Tanke, D. (1995). "Rugbildung in der kohlenwasserstoffpyrolyse hinter stogwellen." Ph.D. Dissertation, Universit/it G6ttingen, G6ttingen. Tanke, D., Wagner, H. Gg. and Zaslonko, I. S. (1998). Mechanism of the action of iron-bearing additives on soot formation behind shock waves. Proc. 27th Symp. (Int.) on Combustion, pp. 1597-1604. The Combustion Institute. Tanzawa, T. and Gardiner, Jr. W. C. (1979). Thermal decomposition of acetylene. Proc. 17th Symp. (Int.) on Combustion, pp. 563-573. The Combustion Institute.

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Ural, E. A., Sichel, M., and Kauffman, C. W. (1981). Shock wave ignition of pulverized coal. Proc. 13th Int. Symp. on Shock Tubes and Waves: Shock Tubes and Waves, pp. 809-817. State of University of New York Press, Albany. Van de Hulst, H. C. (1981). Light scattering by small particles. Dover, New York. Vaughn, S. N., Lester, T. W., and Merklin, J. E (1981). A single pulse shock tube study of soot formation from benzene pyrolysis. Proc. 13th Int. Symp. on Shock Tubes and Waves: Shock Tubes and Waves, pp. 860-868. State of University of New York Press, Albany. Wagner, H. Gg. (1994). The influence of operating conditions on the formation of soot and hydrocarbons in flames. Hazardous Waste Hazardous Mater. 11:5-29. Wang, T. S., Matula, R. A., and Farmer, R. C. (1981). Combustion kinetics of soot formation from toluene. Proc. 18th Symp. (Int.) on Combustion, pp. 1149-1158. The Combustion Institute. Williams, D. B. and Carter, C. B. (1996). Transmission electron microscopy: A textbook for materials science. Plenum, New York. Wittig, S., M~ller, A., and Lester, T. W. (1990). Time-resolved soot particle growth in shock induced high pressure methane combustion. Proc. 17th Int. Symp. on Shock Waves and Shock Tubes: Current Topics in Shock Waves, pp. 468-474. American Institute of Physics, New York. Yoshizawa, Y., Kawada, H., and Kurokawa, M. (1978). A shock-tube study on the process of soot formation from acetylene pyrolysis. Proc. 17th Symp. (Int.) on Combustion, pp. 1375-1381. The Combustion Institute.

CHAPTER

17

Detonation Waves in

Gaseous Explosives JOHN H. S. LEE Professor of Mechanical Engineering, McGill University, Montreal, Quebec, Canada

17.1 17.2 17.3 17.4 17.5 17.6

17.1

Introduction The Structure of Nonideal Detonations Initiation of Detonation Waves Detonation Limits Theory of Nonideal Detonations Concluding Remarks

INTRODUCTION

A gaseous explosive mixture can sustain two modes of combustion distinguished by their propagation mechanism. A deflagration wave propagates via molecular (or turbulent) transport of heat and chemical species from the reaction zone to the unburned mixture ahead of it to effect ignition. According to classical theory, a detonation wave propagates via autoignition induced by the adiabatic compression of the gas by the leading shock front ahead of the reaction zone. A deflagration wave is essentially a diffusion front and propagates at low subsonic speeds with a pressure drop across the wave. A detonation wave is a compression shock wave and must necessarily propagate at supersonic speeds. In smooth tubes, experiments indicate that a unique detonation velocity is obtained for a given explosive mixture at a given initial state. This unique detonation velocity is only weakly dependent on boundary conditions (i.e., tube diameter) in general. Typical propagation velocity of detonations in stoichiometric fuel-air mixtures at standard initial conditions is on the order of 1800 m/s. The corresponding detonation pressure is typically about 20 bar. Shortly after the detonation phenomenon had been identified by Mallard and LeChatelier 1 and Berthelot and Vielle 2 in the late 1800s, Chapman 3 proposed a theory whereby the detonation velocity could be computed.

Handbook of Shock Waves, Volume 3 Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISBN: 0-12-086433-9/$35.00

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Chapman noted that a unique solution to the one-dimensional conservation equations across the detonation wave corresponds to the minimum velocity solution where the Rayleigh line is tangent to the equilibrium Hugoniot curve. Using this as a criterion, the detonation velocity can be determined from the conservation laws and thermodynamic data for the product species (i.e., heat of formation, heat capacity, equilibrium constant). The computed detonation velocity from Chapman's theory agrees quite well with experimental observations in general. Jouguet 4 in the early 1900s independently demonstrated that the minimum velocity solution also corresponds to the condition of sonic flow (relative to the shock) at the end of the reaction zone. Thus, the sonic condition can also serve as an independent criterion for determining a unique solution from the conservation laws. Jouguet's criterion (i.e., sonic flow) and Chapman's criterion (i.e., minimum velocity or tangency solution) are in fact identical and differ only in the iteration method used in obtaining the solution from the conservation equations. Chapman's criterion is simply a postulate, but Jouguet's criterion can provide some physical explanation as to why the tangency solution should be chosen. If the flow is sonic behind the detonation, expansion waves associated with the relaxation of the highpressure detonation products cannot travel upstream to the reaction zone and quench the reactions. Thus, the steady detonation can be isolated from the nonsteady flow downstream. The same physical argument also applies when the flow is supersonic at the end of the reaction zone, corresponding to the lower intersection point of the Rayleigh line and the equilibrium Hugoniot. However, the arguments used to eliminate this supersonic or weak detonation solution become more complicated. Chapman and Jouguet's criteria provide the necessary condition to close the set of conservation equations and permit a unique detonation solution to be obtained. In general, the Chapman-Jouguet (C-J) solution is found to agree quite well with experimental measurements, especially for conditions well within the detonation limits in large-diameter, smooth tubes. Thus, very early in the study of the detonation phenomenon, a successful theory had been formulated for the prediction of the detonation state in a given explosive. It is interesting to note that the very reason for the success of the C-J theory (i.e., determination of the detonation solution without the need to consider the nonequilibrium structure of the wave and the propagation mechanisms) is also the cause of its limitations. For example, the C-J theory cannot provide any information on the detonation initiation requirements, the effect of boundaries and confinement that leads to failure (i.e., detonability limits), or the critical conditions that permit the transition from deflagration to detonation. The preceding questions can only be resolved by considering the detailed physical and chemical processes inside the nonequilibrium reaction zone itself and the influence of external conditions on these processes. To determine the non-

Detonation Waves in Gaseous Explosives

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equilibrium, rate-dependent parameters (i.e., initiation energy, limits, etc.), a model for the detonation structure is required. The next major development in detonation theory occurred in the early 1940s when Zeldovich, 5 von Neumann, 6 and DOring 7 independently proposed a model for the structure of detonation waves. The ZND model for detonation structure assumed a leading shock front, which compresses the explosive mixture to a sufficiently high temperature to initiate rapid chemical reactions in its wake. The subsequent expansion of the high-pressure reacting gases provides the momentum change to sustain the propagation of the leading shock front. Thus, the detonation is sustained by the chemical energy release via the work done in the expansion behind the shock front. The ChapmanJouguet theory involves only the conservation laws across the entire detonation complex (shock-reaction zone) and hence does not contain any information on the propagation mechanism. However, the ZND model provides the propagation mechanism, i.e., autoiginition via adiabatic shock heating and expansion work from the high-pressure reacting gases to maintain the shock. If the reaction rates are known, then the ZND model permits the details of the detonation structure to be computed. The effect of initial and boundary conditions on the propagation of the detonation wave can also be determined based on the ZND model for the structure. Therefore, in principle, velocity deficit, initiation energy, detonation limits, etc., can all be predicted based on the ZND model. However, early attempts to develop quantitative theories to predict these so-called dynamic parameters (i.e., critical energy, velocity deficit, detonation limits, critical diameter) using the ZND model failed to produce results in accord with experiments. For example, critical energies for direct initiation of spherical detonations are found to be three orders of magnitude less than experimental values, s The reason for this discrepancy is that the ZND structure does not correspond to reality. The structure of real detonations is three-dimensional and transient even though the average overall velocity is constant and close to the theoretical C-J value. The next major advances in detonation research occurred in the late 1950s and early 1960s when the three-dimensional, unstable structure that characterizes all real detonation fronts was conclusively demonstrated both theoretically and experimentally. Using high-speed photographic observations (schlieren and interferometry), fast-response pressure and temperature gauges, and recordings on smoked foils inscribed by the passing detonation front, it was shown that all self-sustained detonations have a three-dimensional cellular structure formed by an ensemble of interacting transverse shock waves sweeping laterally across the leading shock front of the detonation wave. The boundaries of the shock intersections define the observed cellular pattern when the detonation is observed "head-on." The trajectories of the triple shock intersections of the front forms the characteristic "fish scale" pattern on a

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carbon-soot coated foil placed in the path of the cellular detonation. Theoretical stability analyses (e.g., Erpenbeck 9 and Zaidel lO) also confirmed that the ZND structure is unstable to small perturbations. In spite of all the conclusive evidence that real detonations are unstable, the ability of the C-J theory to predict remarkably well the detonation velocity caused a great deal of reluctance to relinquish the C-J theory and seek alternate models. Detonation research appears to focus on different aspects at various times over the four decades. However, the main objective has always been to u n d e r s t a n d the complex, three-dimensional structure and to reconcile this complexity within the simple, one-dimensional C h a p m a n - J o u g u e t theory that has always been the "Rock of Gibraltar" on the subject. The central t h e m e of detonation research had been repeatedly stated by the p r o m i n e n t workers in the field from the early 1960s to the present. At the 4th AGARD C o l l o q u i u m in Milan in 1960, O p p e n h e i m 11 ended his review on the d e v e l o p m e n t and structure of plane detonation waves with the following remarks: Although according to classical observations the detonation appears as an extremely steady process, on closer inspection, it seems that it is neither uniform in time nor in space. Consequently the detonation may form an essentially nonsteady, non-uniform regime so that in order to explain its precise nature, multidimensional effects in space as well as its irregular behavior in time have to be taken into account. In fact, in view of this evidence one should express amazement that the one dimensional steady flow theory was so successful in rationalizing so many experimental observations. O p p e n h e i m thus recognized the importance of integrating the three-dimensional, n o n s t e a d y effects of the structure into the one-dimensional steady C-J theory. In a later review article by Fay 12 in 1962 on the structure of gaseous d e t o n a t i o n waves, he remarked: The peculiar disadvantage of detonation research is that it was too successful at too early a date. The quantitative explanation of the velocity of such waves given over fifty years ago by Chapman and Jouguet has not been improved upon and has perhaps intimated further inquiry. Thus, Fay also emphasized the need to improve u p o n the C-J theory to account for the effects due to the three-dimensional transient structure. The same views on the future direction of detonation research were again echoed by Davis 13 in an article in Scientific American on the detonation of explosives: It had been thought that even if turbulence or other similar small scale perturbations arose from the mechanical flow of material in the reaction zone, the effects of such perturbations on the chemical reactions would be small enough not to warrant any qualitative change in the detonation models. Most workers now believe that the effects are not negligible and that detonation theory must be extended to take into account the effects of turbulence and of curved detonation waves, both of which are influenced by the type of inert material that surrounds the explosive.

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The preceding remarks made by these three prominent detonation researchers spanning the past four decades form the basis of this article, the aim of which is to elaborate on what they said and to demonstrate that these threedimensional fluctuations in the reaction zone are indeed essential for the propagation of the detonation. Furthermore, this article also points the way to the modification of the classical Chapman-Jouguet criterion and the ZND theory to take these "turbulence" effects into consideration for the description of real detonations. We shall define all real detonations as nonideal detonations, in contrast to the classical theories of Chapman-Jouget, Zeldovich, von Neumann, and D6ring that address ideal and one-dimensional laminar detonations. Classical detonation theories of ideal detonations have been discussed in most textbooks and are not covered in this article, even though the importance of their thorough understanding cannot be overemphasized. A good summary of recent development in detonations (experimental, numerical, and various analytical studies of detonation stability) can also be found in numerous review articles in the past two decades. Of particular interest are the books by Strehlow, 14 Fickett and Davis, is and Glassman 16 that provide a more complete discussion of classical theory as well as some of the essential results for real detonations. For completeness, the present article summarizes key results obtained in the past four decades and puts them into proper perspective for interpretation. An attempt is made to provide a general framework for the development of a theoretical description of real detonations where all the nonideal effects are accounted for as source terms in the quasi-one-dimensional conservation equations for the detonation structure. The future challenge lies in the formulation of appropriate models to describe the nonideal effects (i.e., curvature, turbulence, etc.). The subject of detonation encompasses many specialized areas in physics, chemistry, applied mathematics, and computational physics, as well as engineering. As such, the researchers in the field come from a variety of disciplines. Thus, any general discussion of detonation phenomena is bound to be biased toward the authors' personal interpretation. This article is no exception. It is more of a view than a review article.

17.2

THE STRUCTURE

OF NONIDEAL

DETONATIONS The ideal one-dimensional ZND model for the detonation structure is not realized experimentally. Toward the late 1950s and early 1960s, experimental evidence of the universal three-dimensional "turbulent" structure of real detonations began to appear. The random density distribution within the

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reaction zone of gaseous detonations is best illustrated by the interferograms of D. R. White. 1 Figure 17.1 shows a typical example of interferograms of detonations in hydrogen-oxygen mixtures at different initial pressures. The random, turbulence-like density variations within the reaction zone are quite evident. At lower initial pressure, the fluctuations take on a more regular periodic pattern of a larger scale. In the late 1950s, an alternative method was used by Soviet researchers for observing the detonation structure. Denisov and Troshin 2 and Schelkhin and Troshin 3 applied the smoked-foil technique (first

FIGURE 17.1 Interferograms of gaseous detonations in hydrogen-oxygen mixture diluted by xenon at various initial pressures. Reprinted with permission from White, D.R., Turbulent structure of gaseous detonations, Phys. Fluids 4, 465-480, 9 American Institute of Physics (1961).

Detonation Waves in Gaseous Explosives

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used by Mach to record the triple point trajectory of a three-shock "Mach interaction") to investigate the structure of real detonations. A characteristic "fish scale" pattern is left on the s m o k e d foil u p o n the passage of a d e t o n a t i o n wave. A typical laser schlieren cinematography of a propagating d e t o n a t i o n in low pressure H2-O2 mixtures with the "fish-scale" pattern i m p r i n t e d on the soot-coated w i n d o w of the detonation channel is illustrated in Fig. 17.2. This

FIGURE 17.2 Stroboscopic laser schlieren photographs of a detonation wave in H 2 - O 2 mixture propagating in a two-dimensional channel with one of the windows coated with soot. Reprinted from Lee, J.H.S., Soloukhin, R.I. and Oppenheim, A., Current views on gaseous detonation, Astronautica Acta 14, 565-584 (1969), with permission of Elsevier Science.

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conclusively demonstrated that the "writing on the wall" is due to the shock interactions occurring at the detonation front itself. The "end-on" normal reflection of a detonation wave on a smoked foil produces a characteristic cellular pattern, quite similar to the morphology of a turbulent flame front (Fig. 17.3). The cell boundaries in Fig. 17.3 are formed by the intersections of transverse shock waves with the leading normal shock front of the detonation. Thus, a real or nonideal detonation front consists of an ensemble of transverse shock waves sweeping across the leading normal shock front. The cell boundaries correspond to triple shock Mach intersections where the temperature, and hence the chemical reaction rate, is most intense. Self-luminous

FIGURE 17.3 Smoked-foilrecord of the end of reflection of a detonation wave in C2H 2 Jr-0 2 mixture propagating in a 25-mm diameter tube. Reprinted with permission, from the Annual Review of Fluid Mechanics, Volume 16, 9 1984, by Annual Reviews www.AnnualReviews.org.

Detonation Waves in Gaseous Explosives

317

"head-on" photography of a detonation also revealed the similar cellular pattern shown in the smoked record of Fig. 17.3. From experiments and numerical simulations, we can reconstruct an idealized picture of the cellular detonation front showing the leading shock, transverse waves, and reaction zones at different times. Figure 17.4 is a schematic diagram illustrating the details of the structure of a cellular detonation front at different times as it propagates from left to right. The trajectories of the triple shock intersections result in the characteristic "fish-scale" pattern are illustrated in Fig. 17.2. Although the entire cellular detonation front propagates at a constant average velocity quite close to the Chapman-Jouget value, large velocity fluctuations occur locally. The local velocity can fluctuate between the limits of about 1.5 to 0.5 times the average C-J velocity depending on the mixture and its composition. Thus, the detonation locally propagates in a pulsating cyclic manner. Starting at the beginning of the cycle at point A when a pair of transverse waves has just collided, the detonation is highly overdriven (~1.5 Vc_J) locally. The overdriven detonation then decays with progressive decoupling of the reaction front from the leading shock. Toward the end of a cycle at point D, the shock velocity can be as low as 50% of the C-J velocity before a pair of transverse

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waves collide again and start the next pulsating cycle. The decay of the overdriven detonation in a pulsating cycle is not continuous in general. An initial rapid decay to about 0.5 Vc_J first occurs when the detonation has propagated about half the cycle length (i.e., "BC"). A "quasi-steady" regime where the shock velocity remains practically constant then occupies the second half of the pulsating cycle. For the first half of the cycle from "•' to "BC," the strong leading shock front serves as the Mach stem to the incident shocks of the adjacent weaker part of the leading shock. For the second half of the cycle, i.e., "BC" to "D," the weak leading shock then becomes the incident shock to the adjacent Mach stems. Thus, the leading shock locally alternates between incident shock and Mach stems of a triple shock Mach interaction as the transverse shocks collide and reflect in a cyclic manner. At the beginning of a pulsating cycle, the leading shock (Mach stem) is sufficiently strong that the reaction zone is intimately coupled to it. For the second half of the cycle where the leading front has decayed and becomes the incident shock, the reaction front is decoupled from it and the mixture is now burned behind the transverse waves. Experimental studies of the detailed structure of a cellular detonation front were carried out in the late 1950s and early 1960s by various researchers; most notable among them were Voitsekhovskii and co-workers, 4 Schott, s Edwards, 6 Strehlow, Z Takai et al., s and Van Tiggelen and co-workers. 9 Because of the three-dimensional transient nature of the cellular detonation front, it is extremely difficult to obtain detailed information on the detonation structure experimentally. However, numerical simulation of the cellular detonation structure has been proven to be extremely useful in providing detailed information on the complex wave interaction processes and the corresponding transient flow field. Numerical simulation of one-dimensional, unstable, pulsating detonation using the method of characteristics was first carried out by Fickett and Wood 1~ as early as 1966. Since then, more thorough studies on one-dimensional pulsating detonations have been carried out by Abouseif and Toong 11 and Moen et al. 12 In a one-dimensional simulation, the detonation instability is manifested by the periodic longitudinal pulsation of the detonation front with the velocity fluctuating typically between the limits of 1.5 to 0.5 Vc_J. This velocity fluctuation is similar to the local velocity fluctuation in a three-dimensional cellular detonation front. The stability limit is governed by the activation energy of the one-step Arrhenius rate law usually assumed in theoretical and numerical studies. The cyclic pulsations change from harmonic oscillations to nonlinear and eventually to chaotic as the value of the activation energy is increased from its value at the stability limit. Similar behavior is also obtained when the degree of overdrive is reduced (for a given unstable value of the activation energy). Figure 17.5 illustrates the behavior of a one-dimensional pulsating detonation as the activation energy is increased beyond the stable value (Fig. 17.5a to 17.5c), and when the degree of overdrive is reduced

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(for a fixed unstable value of the activation energy (Fig. 17.5d to 17.5f). The numerical result is in accord with the stability analyses of Erpenbeck ~3 or Lee and Stewart. ~4 The numerical simulations provide details of the nonsteady flow field behind the unstable detonation and elucidate the important cyclic amplification process leading to the overdriven phase at the beginning of each pulsating cycle. Simulation of two-dimensional cellular detonations began in the late 1970s with the pioneering works of Taki and Fujiwara, ~50ran and co-workers at NRL, 16 and Markov. 17 Since then, rapid advances in computational power and numerical algorithms have brought these simulations to a very high degree of accuracy. The majority of these simulations are for two-dimensional detonations using a single-step Arrhenius reaction rate model. However, simulations of three-dimensional detonations or two-dimensional detonations using more complex reaction models have also been carried out. Most of the early twodimensional simulations essentially demonstrate the ability to reproduce qualitatively the experimentally observed features of the cellular structure. Simulations of three-dimensional cellular detonation have proven to be of rather limited value thus far because of the difficulty in displaying the complex three-dimensional structure and in the reduction of the vast amount of numerical information obtained in a comprehensible manner. Most of the simulations also suffer from a dependency on the resolution of the numerical computation. However, the recent investigations by Gamezo et al. 18'19 were carried out much more carefully and thus can elucidate a number of physical issues and enhanced our current understanding of unstable detonations. In Gamezo's simulations, great attention was exercised to ensure that the results obtained are independent on the resolution of the numerical computations. It is of interest to review Gamezo's results in order to present a more complete picture of our current understanding of the cellular detonation structure. Since the transient development of the cellular structure depends on the initial conditions, it is worthwhile to first briefly describe the computational procedures in Gamezo's numerical investigations. The two-dimensional reactive Euler equations are solved using the FCT technique 2~ for a two-dimensional domain bounded by free slip solid walls at the top and bottom. The reactants enter at the right boundary at specified initial conditions and exit at the left boundary of the computational domain. The domain is defined by a uniform Eulerian grid with a resolution chosen so that the detonation cell size and regularity of the structure are independent of the grid size used. The height of the computational domain is also chosen to be sufficiently large to accommodate at least two to three detonation cells. Thus, the structure is not influenced by the dimension of the channel. A single-step Arrhenius rate law of the form

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with 0~ being the reaction progress variable, and A, E a and R represent the preexponential factor, activation energy, and the gas constant, respectively. The parameters and initial conditions were chosen to represent approximately stoichiometric 2H 2 + 0 2 mixtures at P0 = i bar and TO = 293 K. The numerical computation is initiated by imposing a planar shock with a square wave profile and a pressure of 2pcJ near the left boundary. This strong initial shock wave induces chemical reactions that produce an overdriven detonation advancing toward the right of the domain and decays asymptotically to a steady C-J detonation. With a proper choice of the frame of reference, the detonation can be made to eventually stabilize near the right boundary of the domain. The preexponential scale factor A is also chosen to ensure that the length of the computation domain always contains several detonation cells. Thus, even though this initial condition does not correspond to a real experimental initiation process, the transient development of the instability leading to the eventual steady cellular C-J detonation from an initial overdriven wave is described in the numerical simulation. The time-integrated maximum pressure contour from the numerical simulation is found to correspond to the trajectories of the triple point of Mach interactions, which have been demonstrated to be similar to the writing on the smoked-foil records. The numerical results therefore indicate that the peak pressure in a cellular detonation is also localized at the triple shock intersections. A typical "numerical smoked foil" corresponding to a value of the activation energy Ea/RT* = 2.1 (where T* -- 1709 is the shocked temperature) is shown in Fig. 17.6. We note that in the initial overdriven state, the detonation is stable and the pressure is uniform across the planar front. Instability develops and cell formation appears only later on as the overdriven detonation decays to near its final C-J velocity. In the final frame, one can see that the cell pattem is also quite regular for this particular value of the activation energy. For a regular cell pattern, the transverse waves are weak and correspond to acoustic perturbations sweeping across the leading shock front. Weak transverse waves are analogous to Mach waves in a supersonic flow and they play a minor role in the propagation mechanism of the detonation wave. For higher values of the activation energy, i.e. E ~ / R T * = 4.9 and 7.4, the corresponding development of the cellular pattern from the initial overdriven state is illustrated in Figs. 17.7 and 17.8. For higher activation energies, the strength of the transverse waves are much stronger and the cell pattem also becomes more irregular with increasing activation energy. These numerical results are in qualitative agreement with experimental observations and stability theory. The growth of initially weak transverse perturbations to form the final cellular detonation as the overdriven wave decays to its final C-J value is similar to the experimental observation of Strehlow et al. 21 for reflected shock initiation of detonation. The dependence of the regularity of

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the cellular pattern on the activation energy is also in accord with the observations of Ulyanitskii, zz Manzhalei, z3 and Moen e t a | . z4 and stability analysis (e.g., Lee and Stewart14). The value of the stability parameter Ea/RT* essentially provides a measure of the temperature sensitivity of the reaction, i.e., the steepness of the exponential dependence of reaction rate on temperature. Thus, for large values of the activation energy, large fluctuations of the reaction rate result from small temperature variations and render the system highly unstable. Note that for qualitative purposes, numerically generated smoked records of the cellular pattern are superior to real ones because of the difficulty in depositing a uniform coating of soot on a surface in practice. For the purpose of demonstrating the qualitative features of the cellular structure, numerical experiments have proven to be more informative. Gamezo et al. 18 also carried out an interesting numerical experiment by subjecting the detonation to large random fluctuations in the chemical energy release and observed its response. For large activation energies where the cellular pattern is highly irregular, he found that the external perturbations have little influence on the propagation of the detonation, i.e., little change in the irregular cell pattern is observed. The insensitivity of unstable detonations to the external perturbations is due to the fact that the detonation is already highly unstable and there is a continuous decay and growth of transverse waves in the unstable front (as manifested by the irregular cell pattern). Thus, only when the external perturbations dominate the removal and formation of new cells at a rate comparable to the natural frequency of the intrinsic instability itself can an external influence be observed. For highly unstable detonations (i.e., high activation energy), therefore, the detonation is more "robust" and less vulnerable to perturbations from boundary conditions and confinement. This is in accord with the experimental observations of the detonation failure and reinitiation in the critical tube diameter phenomenon and also in the near limit, unstable propagation of the detonation wave for high activation energies. For more stable detonations (i.e., lower values of the activation energy and a more regular cell pattern) the transverse waves are weaker and play only a minor role in the propagation mechanism. Growth rates of perturbations are slower and thus the detonation cannot recover from external disturbances sufficiently fast. Therefore, the detonation can fail more easily. The variation of the shock strength in a pulsating cycle (along the cell axis) is found to depend also on the activation energy. Figure 17.9 shows the variation of the local shock velocity as a function of the scaled distance z / L c, where Lc is the length of a cell cycle in the direction of propagation. For highly unstable detonation (i.e., curve c for Ea/RT* = 7.4), the velocity fluctuation varies between 1.7 and 0.6 Vc_J, whereas for a more stable detonation with an activation energy (curve a where E a / R T * = 2.1), the velocity fluctuation is smaller and varies only between 1.2 and 0.9 Vc_J. This result is also in accord

Detonation Waves in Gaseous Explosives

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with the experimental observation of Lee et al. 25 of the velocity fluctuation of near limit detonations. It was found that the velocity fluctuations are larger for more unstable detonations with higher values of the activation energy. Perhaps the most important result obtained from numerical simulation is the demonstration that unreacted "pockets" of the explosive mixtures are left behind the leading shock front. These unreacted pockets were first numerically obtained by Oran et al. 26 as early as 1982, and they persistently showed up even in the more refined computations carried out over the past 20 years and this warrants their acceptance as a real phenomenon. Schlieren photographs taken earlier by Edwards et al. 27 have already indicated the existence of these unburned pockets. However, it is difficult to be conclusive about the nature of these nonuniform density regions observed in schlieren photographs without additional, independent diagnostics. Thus, the combination of real and numerical experiments had identified the possibility of unreacted pockets left behind by the cellular detonation front. That these "unreacted pockets" are not a consequence of the simplified chemistry model used had also been proven recently by Matsuo and co-workers 28 where the detailed kinetics of H 2 + 0 2 are used in the numerical computations. Figure 17.10 shows the temperature field behind a cellular detonation for the case of E a / R T * = 7.4. Apart from the dark regions corresponding to unburned gas pockets, Fig. 17.10

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0.6 FIGURE 17.9 Variation of the local shock front velocity at the cell axis as a function of the scaled distance z/L c, where Lc is the length of the detonation cell (curve a, E / R T * = 2.1; curve b E/RT* --4.9; curve c, E/RT* -- 7.4). Reprinted from Gamezo, V. Desbordes, D. and Oran, E.S. Formation and evolution of two-dimensional cellular detonations, Combustion and Flame 116, 154165 (1999), with permission of Elsevier Science.

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FIGURE 17.10 The temperature field for a cellular detonation wave illustrating the presence of unburned gas regions behind the leading shock front (E/RT* = 7.4). Reprinted from Gamezo, V. Desbordes, D. and Oran, E.S. Formation and evolution of two-dimensional cellular detonations, Combustion and Flame 116, 154-165 (1999), with permission of Elsevier Science.

also illustrates the random nature of the temperature distribution, quite similar to that of a turbulent flame structure. Gamezo 18 also reported that the unreacted pockets become larger and penetrate more deeply into the wake of the leading shock front as the activation energy increases (i.e., for a more unstable detonation). The existence of these unreacted gas regions has profound implications on the mechanism of detonative combustion. According to the classical ZND model, the ignition is effected by the leading shock front. In a real detonation where the leading shock is no longer planar, but consists of an ensemble of interacting shocks, the shock ignition mechanism can still be preserved provided that autoignition can still be achieved by this collection of interacting shocks (i.e., transverse and

Detonation Waves in Gaseous Explosives

329

incident shocks and Mach stems). However, the discovery of "unreacted gas pockets" that were swept downstream showed that not all the mixtures can be burned via autoignition by shock heating in a real detonation front. Large regions of gases are therefore quenched by the nonsteady expansion associated with the transient interacting shock waves during their induction periods. In reality, these unburned gas pockets have to undergo combustion eventually since they are embedded in a sea of hot combustion products. However, the burning will occur via a diffusion mechanism at the boundaries of these pockets as in a flame. Hence, in an unstable cellular detonation, the combustion mechanism is no longer solely due to shock ignition, but diffusional transport also plays a role in the conversion of all the unburned mixture that crosses the leading shock front to final products. With a combination of the two combustion mechanisms, the sharp distinction between deflagration and detonation can no longer be made. A consequence of these unreacted gas pockets that are swept downstream and burned eventually via diffusion will be a prolonged reaction zone length. Figure 17.11 compares the "effective" space and time averaged reaction zone of a cellular detonation with the corresponding one-dimensional ZND structure. For low activation energy where the induction period is small, the "pockets" are small and react fairly rapidly. Hence, the difference between the "averaged" reaction zone length and the one-dimensional ZND value is small. For large activation energy, the "averaged" reaction zone length becomes larger as the "unreacted pockets" can now penetrate further behind the leading front. If the pockets penetrate past the rarefaction waves, the energy released can no longer be fed back to sustain the shock, resulting in a decrease in the detonation velocity. This mechanism results in a velocity deficit and contributes to a departure from ideal behavior that arises from the intrinsic instability of the detonation front itself and not from any external influences such as boundary conditions and confinement. In spite of the "turbulence"-like structure of a real detonation front, a characteristic length scale can be obtained from the cellular pattern recorded on a smoked foil placed on the wall of the detonation tube. This length scale represents the averaged transverse wave spacing and is equivalent to the averaged dimension of the detonation cells as recorded on a smoked foil of an "end-on" reflection of the detonation shown in Fig. 17.3. Since a cellular detonation front consists of an ensemble of pulsating detonations with a pulsating cycle length of about 1.5 times, the transverse wave spacing, the cell size 2 can represent a direct measure of the reaction zone thickness of a real detonation. A true reaction zone thickness would require the hydrodynamic fluctuations (i.e., turbulence and transverse shock waves) to decay and a complete equilibrium condition to be reached. Nevertheless, the cell size 2 (or equivalently the length of the local pulsations L~ ~ 1.52) can serve as a

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quantitative measure of the reaction zone length of a real detonation. The cell size is typically between one and two orders of magnitude greater than the ideal one-dimensional ZND reaction length, depending on the definition used to define the reaction length. 29 An attempt to incorporate the hydrodynamic effects on the chemical reaction processes by defining a so-called "hydrodynamic thickness" for a real detonation front was first made by Soloukhin et al. 3~ Later attempts to determine the hydrodynamic thickness by Vasiliev et al. 31 by measuring the location of the sonic surface and by Edwards et al., 32 who measured the attenuation of the transverse pressure fluctuations, gave an estimate of about 2 to 4 cell lengths for the hydrodynamic thickness. To determine the hydrodynamic thickness numerically would require the shock

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and turbulence dissipation rates to be computed. For realistic values of the Reynolds number, direct Navier-Stokes simulation of cellular detonations is not currently possible. The cell size has proven to be an extremely useful length scale to characterize the sensitivity of an explosive mixture. Knowledge of the cell size permits the dynamic detonation parameters (i.e., critical initiation energy, detonation limits, critical tube diameter) to be estimated. 33 Although a smoked foil record can easily be obtained experimentally, the interpretation of the foil to deduce a "representative" size for the detonation cells requires a certain degree of judgment on the part of the observer. This is due to the irregular cell pattern as well as the presence of substructure (representing higher harmonics) superimposed on the dominant cell pattern observed in most explosive mixtures. Even in the numerically generated foils, the deter-

332

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mination of a representative cell size from an irregular cell pattern such as that shown in Fig. 17.8 is not easy. Only in special cases (e.g., C2H2-O 2 highly diluted with argon) when the cell pattern is relatively regular and devoid of substructure can we identify the cell size easily. However, with experience and using a long sample of smoked foil (which records the detonation structure over a long distance of propagation), a fairly objective value of the cell size can usually be obtained. Attempts have been made by Shepherd and Tieszen 34 and Lee et al. 35 to use digital image processing techniques to quantify the cell size spectrum and obtain a most probable value. However, the nonuniformity of the soot deposit as well as the erratic erosion of the soot itself by the detonation renders the digitization of an actual experimental smoked foil extremely difficult. In spite of the difficulties in the analysis of experimental smoked-foil records, the cell sizes for a wide array of gaseous explosive mixtures have been determined. A compendium of detonation cell size data has been made by Shepherd and co-workers 36 and represents a valuable source of information on the detonation sensitivity of various explosive mixtures. The relationship between the cell size 2 and the ZND reaction length g had been proposed by Schelkhin and Troshin, 3 i.e., 2 = Ag, where A is a proportionality constant. In the past few decades, the detailed chemical kinetic schemes and their rate constants for the oxidation of most of the common fuels have been fairly well established. This permits the ZND reaction zone length to be computed. The constant A can be determined by matching with one experimental data point, and hence the cell size can be predicted over any desired range of composition and initial conditions. Depending on the definition of the theoretical ZND reaction zone length, the constant A is found to be in the range 30-60. Figure 17.12 shows the correlation of the experimental cell size data with a linear dependence law 2 = Ag for various fuel-air mixtures. 3r The constant A is obtained by matching to the experimental value for the cell size )o at stoichiometric composition, i.e., q5 = 1. The agreement appears to be fairly reasonable; however, this can be misleading since the U-shaped curves are fairly steep and do not show the differences well especially for lean mixtures (i.e., q5 < 1) It was recognized at the outset that a simple linear dependence cannot possibly describe the highly nonlinear gas-dynamic effects involved in the chemical reaction zone of the cellular detonation front. However, attempts to seek a general functional dependence of A on the various properties of the mixture have not been successful. In the most recent attempt by Gavrikov et al., 38 the ratio A = 2/g is assumed to be dependent on two stability parameters, the activation energy and a parameter that describes the relationship between the chemical energy and the critical internal energy of the mixture. Furthermore, instead of computing the ZND reaction zone length based on the

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von Neumann state at Vc_J, a higher value of the shock strength (averaged between the maximum of 1.6 Vc_J and Vc_J) is used. The new correlation appears to provide a better fit of the experimental data for hydrogen-air mixtures. However, these correlations are empirical and their general validity has yet to be determined. It appears that, in spite of all its drawbacks, the estimate of cell size from experimental smoked foils still remains as the best alternative to assess the sensitivity of an explosive mixture for some time to come. The unstable nature of real detonations has now been conclusively proven by theoretical stability analysis and confirmed by experimental observations. It

334

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would be of value to end our discussions on the structure of real detonations by speculating on the reasons as to why Nature requires the detonation to have such a complex three-dimensional cellular structure. That the three-dimensional cellular structure is absolutely necessary for a self-sustained detonation to propagate at its C-J velocity was conclusively demonstrated by Duprr et al. 39 and Teodorczyk and Lee. 4~ In their respective experiments, a fully established cellular detonation propagating at its constant C-J velocity in a smooth tube is made to enter a section of the tube (of the same diameter) in which the wall is lined with an acoustic absorbing material. The transverse waves, upon reflection from the wall, are now significantly attenuated in this damping section of the tube. Thus, the transverse waves are progressively being eliminated as the detonation propagates along this damping section, and as a result, the cells enlarge and disappear, and the detonation front eventually becomes planar as in the ZND model. As cells disappear and the detonation front becomes planar, the reaction zone is observed to decouple from the shock and the velocity of this decoupled wave drops to about one half of its initial C-J value. Figure 17.13 shows a sequence of high-speed framing schlieren photographs of the decay of a cellular C-J detonation wave. The velocity eventually drops to half its initial velocity. This experiment implies that without the transverse shock interactions with the leading shock front (hence no cells), the chemical reactions cannot proceed at a sufficiently fast rate for the reaction zone to be coupled to the shock front. The decoupling then leads to a reduction of the detonation velocity. Hence, the cellular structure provides the condition for a more rapid chemical reaction rate than the planar ZND structure. It should be noted that in order for the detonation to fail, the tube diameter (or height of the channel) should not be large compared to the cell size so that the rate at which new transverse waves are formed near the tube axis is less than the rate at which they are attenuated at the tube wall. For example, if the tube diameter (or channel height) is greater than the critical value (i.e., d > 132 or h > 102), then reinitiation would be able to occur near the tube axis even though all the transverse waves are eliminated at the boundary defined by the tube wall. The necessity for the shock wave interactions in the reaction zone is to attain sufficiently high reaction rates to sustain the propagation of the detonation wave. This is associated with the shock wave enhancement of the turbulent mixing rate in the reaction zone. In D. R. White's original description of the unstable detonation structure as being "turbulent," he may not have considered that shock wave interactions can be considered as part of the turbulence mechanisms. Conventional notions of turbulence are derived essentially from studies of low-speed, incompressible flows where shear and vorticity interactions constitute the principal energy dissipation mechanism. In high-speed, compressible turbulent flows, the intense pressure fluctuations generate an ensemble of shock waves. The presence of shock waves brings in

Detonation Waves in Gaseous Explosives

335

FIGURE 17.13 Decay of a cellular detonation to a deflagration wave due to the damping of the transverse shock waves by an acoustic attenuating wall.

336

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FIGURE 17.13

(continued)

Detonation Waves in Gaseous Explosives

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an array of additional mechanisms of vorticity generation and dissipation. For example, the baroclinic mechanism of vorticity production due to the interaction between pressure and density gradient fields becomes important for high-speed compressible flows. Nonlinear shock-shock interactions (e.g., Mach interactions) produce vorticity even in the absence of boundaries to create the velocity gradient fields. Shock-vortex interactions also lead to breakdown of the large-scale mean flow to increase kinetic energy dissipation rates. With the presence of strong density interfaces associated with the chemical reactions, increased turbulence production also arises from the various interface instability mechanisms (e.g., Richtmyer-Meshkov, TaylorMarkstein) due to shock wave-density interface interactions. The effects of shock waves cannot be excluded from the consideration of high-speed compressible turbulence. As early as 1955, Lighthil141 had already recognized the inherent role of shock waves in compressible turbulence. Because of its clarity, it is of interest to quote directly from Lighthill's eloquent statement of compressible turbulence: Extending the picture to three dimensions, one may imagine the turbulence to consist not only of the usual vortex motions, but also of a three-dimensional statistical assemblage of N-Waves, that is, of shock waves of all shapes rushing about in all directions with regions of more gradual expansion between them and with continual interactions taking place between pairs of shock waves (including unions, regular intersections and Mach intersections) and, to a lesser extent, between them and the longitudinal expansion waves and shear turbulence. The interactions between shock waves actually create additional vorticity; also a single shock wave along which entropy increase is non-uniform creates vorticity in proportion to the gradient of that increase. Thus, to some extent, the shock wave system can generate new turbulence. Whether as a result of all this, any kind of equipartition between the energy of longitudinal and shearing motion is likely to be set up can only be a matter of opinion. The Author feels rather that the system has become one in which the division of the motion of turbulence on the one hand and sound (or shock waves) on the other is ahnost without significance.

Thus, if we broaden the conventional definition of turbulence to include the role played by shock waves, then detonations can simply be regarded as an extension of turbulent deflagrations to the high-speed compressible regime. Returning to the original question as to why Nature requires real detonations to have such a complex cellular structure, the response is simply that the additional, powerful dissipative mechanisms associated with shock wave interactions have to be recruited to produce the required reaction rates to cope with the high propagation speed of the detonation wave. In a largediameter, smooth tube, or in a spherical wave, the ensemble of interacting shock waves have to be produced via hydrodynamic instability, that is, growth of small perturbations. Thus, the onset of detonations is abrupt when the propagation mechanism changes from the diffusional transport of the deflagra-

338

J . H . S . Lee

FIGURE 17.14 Flame acceleration due to turbulent mixing enhancement by transverse shock waves generated by wall obstacles.

339

Detonation Waves in Gaseous Explosives

tion regime to autoignition via shock heating and turbulent mixing of the detonation regime. In a very rough tube, experiments have indicated that the acceleration to detonation is relatively smooth, without the usual abrupt jump to an overdriven detonation that subsequently decays to a C-J wave. 42 This is due to the dominant role now played by the rough wall in generating the vorticity and pressure waves necessary for the detonation regime. There is no longer the need to rely on instability to form the transverse waves. Figure 17.14 illustrates a sequence of framing schlieren photographs of the flame acceleration process in a very rough-walled tube. The train of transverse waves (shock waves generated by the roughness on the walls of the channel) is clearly demonstrated. The mixing enhancement due to the interaction of these transverse waves with the turbulent reaction zone leads to a continuous increase of the burning rate and acceleration of the deflagration until it reaches a final steady-state velocity. With the rough wall controlling the production of transverse waves, the final steady-state velocity can span the entire spectrum of supersonic speeds (up to the C-J velocity) depending on the tube diameter, the dimension of the wall roughness, and the natural cell size of the mixture. The existence of a continuous spectrum of high-speed deflagrations (or quasidetonations) implies that the combustion mechanism is also continuous, with transverse shock waves entering and enhancing the turbulent burning rate as the deflagration accelerates. Without the rough wall, the transverse shocks can only be formed from the growth of small perturbations from instability. Thus, the flame has to accelerate to a sufficiently high speed to generate the critical conditions for this to occur. With a rough-walled tube, the transverse pressure waves can now be generated continuously by the wall protrusions. Without a distinct difference between either the propagation mechanism or the propagation speed, a unique separation between the two phenomena of high-speed, turbulent deflagration and detonation can no longer be made. Only in situations where the influence of boundary conditions is small and the detonation has to rely on self-generation of transverse waves from instability can detonation phenomena be uniquely defined. However, in terms of the mechanism of combustion, detonations and high-speed, turbulent deflagrations can be considered to be similar if shock wave interaction is considered as an inherent part of high-speed compressible turbulence. Hence, it is the nature of how the transverse waves are being generated that separates the detonation from deflagration phenomena.

17.3

INITIATION

OF DETONATION

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In general, a detonation wave can be obtained via the process of a transition from a deflagration wave (DDT, i.e., deflagration to detonation transition) or

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J.H.S. Lee

from the decay of a strong blast wave generated by a powerful ignition source (i.e., direct initiation). Direct initiation is a relatively well-defined phenomenon characterized by a critical energy required for the initiation of a detonation wave. DDT, however, is an ill-defined phenomenon that includes a spectrum of different processes leading to the formation of the detonation wave. In the classical experiments of Mallard and LeChatelier 1 and Berthelot and Vielle 2 where a long, smooth combustion tube is used, the DDT process is characterized by a continuous acceleration of the deflagration followed by an abrupt transition to a detonation. Figure 17.15 shows a typical streak photograph of DDT in a smooth tube. Ignition is effected by a hot jet of combustion products from a small orifice from the reflection of a detonation wave on the left side of the orifice plate. Subsequent to ignition, the deflagration wave accelerates continuously until an abrupt transition to detonation occurs. The flame acceleration is mainly due to the effect of turbulence and hence includes all processes that can lead to an increase in the flame area and transport rates across the flame surface. Thus, the intrinsic instability of the flame, the various interface instability mechanisms associated with the acceleration, and acoustic and shock wave interactions with the flame, as well as the velocity gradient and turbulence induced in the displacement flow of the unburned gases ahead of the propagating flame, are all responsible for increasing the burning rates. A detailed discussion of all the various flame acceleration mechanisms can be found in the review articles by Lee and Moen 3 and Shepherd and Lee. 4 The different flame acceleration mechanisms are very sensitive to initial and boundary conditions (e.g., type and strength of the igniter, its location, tube geometry and size, wall roughness, closed or opened ends of the tube). Hence, the relative roles played by each of the different mechanisms in the DDT process differ under different conditions. As a result, the transition distance (i.e., the distance the flame travels from ignition to the location of the onset of detonation) can be orders of magnitude different for the same explosive mixture under different conditions. The transition distance is not a unique parameter that can characterize the DDT phenomenon. Note that the transition distance is also referred to as the "run-up distance" and in some of the older literature, it is referred to as the "induction distance. ''5 The transition distance is typically 50-100 tube diameters in smooth circular tubes with a weak spark ignition at a closed end. Attempts have been made in the early studies of DDT to correlate the "induction distance" to the properties of the mixture (at least for one geometry of smooth circular tubes); however, these correlations lack generality and are of limited value in terms of promoting fundamental understanding of the DDT phenomenon. The termination of the flame acceleration phase of DDT is the abrupt onset of detonation. Early views of the DDT process were that the flame must accelerate to a sufficiently high velocity so that the precursor shock in front of

Detonation Waves in Gaseous Explosives

341

FIGURE 17.15 Streak photograph of DDT in C2H 2 - O 2 mixtures in a smooth tube. Ignition is by a hot gas jet from the reflection of a detonation wave off an orifice plate. Reprinted, with permission, from the Annual Review of Physical Chemistr3; Volume 28, i{~, 1977, by Annual Reviews www.AnnualReviews, org.

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J . H . S . Lee

it can result in autoignition of the mixture. 6'7 However, later investigations indicated that the onset of detonation can correspond to a variety of different conditions. Perhaps the best illustration of the different modes of DDT is given by Urtiew and Oppenheim. s From their excellent stroboscopic laser schlieren photographs, they demonstrated that the onset of detonation can originate from a "hot spot" in the turbulent flame zone, in the precursor shock-boundary layer interaction region and at the interface formed from the coalescence of two shock waves. The "hot spot" is a rather arbitrary definition of the location of the origin of the incipient detonation kernel and may not actually represent a local high-temperature region. The origin of the hot spot has not been clearly established. In the article by Meyer, Urtiew and Oppenheim, 9 they demonstrated that the origin of the hot spot cannot be due to the adiabatic compression of the mixture from the precursor shock waves. They carefully followed the thermodynamic history of the particle due to the various wave compression processes and found that only about 4% of the induction process occurred when the onset of detonation happened. Hence, they concluded: Gasdynamic processes of compression ahead of the accelerating front are entirely insufficient to bring about the transition to detonation. The occurrence of this event must be due, therefore, to other phenomena of which the most influential should be those associated with heat and mass transfer from the flame.

Thus, Oppenheim and co-workers implicitly credited turbulent mixing as the dominant mechanism that brings about autoignition for the onset of detonation. The typical process of the onset of detonation from a hot spot in the turbulent mixing zone is illustrated in Fig. 17.16. The incipient detonation kernel grows to catch up with the precursor shock front to form an overdriven detonation. The detonation kernel becomes a shock wave when it propagates back into the combustion products where there is no unbumed mixture. This shock is known as the retonation wave. When the spherical detonation kemel grows and reflects from the tube walls, a transverse shock wave is formed that reverberates between the walls and attenuates slowly as the detonation and the retonation waves move apart. All of these features are also illustrated in the streak photograph of the transition processes shown in Fig. 17.15. The autoignition process required for the onset of detonation can be brought about by free radicals obtained via thermal dissociation at high temperatures and also by other means. For example, direct initiation of detonation in hydrogen-chlorine mixtures at room temperature by photodissociation of chlorine had been demonstrated by Lee et al. 1~ Furthermore, Knystautas et al. 11 have also showed that by rapid, turbulent mixing of combustion products with the unbumed mixture in a turbulent jet, direct initiation of detonation can also be achieved. In this case, the free radicals to

Detonation Waves in Gaseous Explosives

343

FIGURE 17.16 Stroboscopic laser schlieren photograph of the onset of detonation in H202 mixtures. Detonation kernel originates at bottom wall in the turbulent flame brush. (Courtesy of A. K. Oppenheim.)

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j.H.S. Lee

induce autoignition are present in the hot combustion products originally. These experiments showed conclusively that autoignition is a necessary condition for the onset of detonation and can be achieved by means other than by shock heating. Thus, the DDT phenomenon encompasses all the different processes that can bring about the onset of detonation from an initial deflagration wave. However, it is now generally acknowledged that rapid turbulent mixing leading to autoignition constitutes the key mechanism of DDT. Considerable attention has been devoted to the onset of detonation resulting from shock-flame interaction. 12-17 Enhancement of turbulent mixing due to shock-flame interaction has proven to be a very efficient means of generating the necessary conditions for the onset of detonation. Shock-flame interaction also provides a well-defined initial condition for numerical simulation as well as in actual experiments. In addition to the crucial role played by shock waves in the propagation mechanism of self-sustained detonation waves as discussed in the previous section, we now see that shock wave enhancement of turbulent mixing also provides an effective mechanism for generating the critical conditions for the onset of detonation in the DDT process. Although autoignition at a local region (hot spot) of the turbulent mixing zone represents the start of the detonation formation process, the shock or compression wave from the energy released at the hot spot must also be capable of amplifying rapidly to form the detonation wave. Even for an instantaneous constant volume explosion of the hot spot, the shock generated is only of the order of M ~ 2.5. Experiments indicate that the incipient detonation kernel is formed within a very short distance of the shock propagation from a local hot spot. Thus, a further requirement for the onset of detonation is the condition necessary for the rapid amplification of the shock wave formed from the explosion of the hot spot. From a study of the photochemical initiation of detonation, Lee et al. 1~ had proposed the shock wave amplification by coherent energy release (SWACER) mechanism as being responsible for the onset of detonation. In the photochemical initiation process, a gradient of chlorine atoms is generated in the direction of the irradiation. Thus, a gradient in the induction time is also generated, resulting in the sequential autoignition of the mixture along the induction time gradient. A progressive reaction front is thus obtained, although the propagation speed of this reaction wave is a phase velocity predetermined by the initial induction time gradient and not by any physical or transport process. The termination of the induction period is followed by the recombination or a rapid energy release process that then generates a shock or compression wave. If the gradient field is such that the reaction wave path is coincident with the shock wave trajectory, then the chemical energy released is in phase with the shock propagation, resulting in a very rapid amplification of the shock wave. This amplification process is similar to the laser concept and led Lee to adopt a

Detonation Waves in Gaseous Explosives

345

similar acronym (i.e., SWACER). In an earlier investigation, Zeldovich et al. i s had also considered an induction time gradient resulting from a temperature gradient. It was shown that shock amplification also requires the coincidence of the path of the shock with that of the reaction wave. Figure 17.17 illustrates results of the numerical simulation of the SWACER mechanism in an induction time gradient field in an H2-C12 mixture from the photodissociation of C122~ Figure 17.17a illustrates the supercritical regime where the irradiation intensity I0 = 2 k W / c m 2 is above the critical level to produce an optimum induction time gradient for the onset of detonation due to the SWACER mechanism. The critical regime shown in Fig. 17.17b (I0 = 1 k W / c m 2) indicates that the initial gradient field did not lead to the formation of a detonation wave immediately, but created the conditions corresponding to the quasi-steady regime similar to blast initiation (at around profile 5). A second longitudinal acceleration process occurs between profiles 5 and 7, resulting in the onset of an overdriven detonation. In Fig. 17.17c, the intensity is much higher (I0 = 15 k W / c m 2) than the critical and thus gives a much smaller induction time gradient. The SWACER mechanism is suppressed and a progressing volumetric explosion front results. In the limit of even higher irradiation, the entire mixture undergoes a constant volume explosion. It is interesting to note that the three regimes of direct initiation do not correspond to the irradiation energy but on achieving of an appropriate induction time gradient. Thus, we see that successful initiation of detonation requires a combination of induction gradient and heat release profiles as well as a critical length of the gradient field itself. Because of the nonlinear effect of the shock wave on the initial induction time gradient profile as the shock amplifies, it is difficult to derive an analytical criterion for the SWACER mechanism. Numerical simulations had first been carried out by Yoshikawa 19'2~ on the H2-C12 system, because the detailed kinetics are well established. Since then numerous studies on the SWACER mechanism have been carried OUt 19-26 with the gradient field formed via turbulent mixing. A detailed, recent review of this mechanism is given by Bartenev and Gelfand 27 in which a complete bibliography on the subject can be found. It is also of great practical importance to know if DDT can occur in a given explosive mixture under given initial and boundary conditions. The criterion for the possibility of DDT is closely related to that for the detonability limits. It is clear that if the conditions are outside the detonability limits, then DDT cannot occur. However, the reverse is not true, that is, DDT may not be possible even though the conditions are well within the detonability limits. A typical example is the case for spherical detonations. In purely unconfined geometry, almost all the flame acceleration mechanisms are ineffective, and it is extremely difficult for a spherical deflagration to accelerate and transit to a spherical detonation. However, if a powerful ignition source is used for direct

346

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L

.

.

.

.

.

.

.

.

.

.

.

I

0

10

d'~tance (cm)

20

Po

40-

b

30

20

10

10

dismr'~e (cm)

20

P/Po

I 10

C

0

10

20

30

diMatt~:(cm).

FIGURE 17.17 Onset of detonation due to SWACER mechanism in an induction time-gradient field generated by photodissociation of C12 in an H2-C12 mixture. The supercritical (a), critical (b); and subcritical (c) regimes are illustrated. 9 1992 AIAA, reprinted with permission.

Detonation Waves in Gaseous Explosives

347

initiation, the spherical detonation can readily sustain itself once initiated. Since DDT depends on the effectiveness of the various flame acceleration mechanisms, it is no longer possible to establish a general criterion for the occurrence of DDT. The onset of detonation, that is, growth of the incipient detonation kernel from a local hotspot, appears to be sufficiently general in all DDT processes. Hence, it is possible to establish a criterion for the successful growth of the detonation kernel subsequent to its formation from a hotspot. In an extensive experimental study by Peraldi et al. 2s on the transition in circular tubes of three different tube diameters (i.e., 5, 15, and 30 cm) in a variety of fuel-air mixtures (hydrogen, acetylene, ethylene, propane, and methane) at standard conditions, it was initially found that DDT requires that the tube diameter D be at least equal to a detonation cell size 2, i.e. D / 2 > 1. In a rough tube with periodically spaced orifice plates of orifice diameter d, it was found that DDT also requires that the minimum transverse dimension to be of the order of a cell size, i.e., d / 2 > 1. Although this is a necessary condition, it is not sufficient by itself, since DDT also requires sufficiently rapid turbulent mixing to achieve autoignition and the subsequent amplification of the shock wave from the explosion of the "hotspots" (i.e., the SWACER mechanism). Thus, we see that DDT is a more demanding phenomenon requiring a set of critical conditions that have to be met simultaneously before it can occur. Direct initiation is a relatively simpler phenomenon than DDT, requiring only a powerful ignition source to generate a strong blast wave of sufficient duration. The strong blast wave then decays asymptotically to a C-J detonation as it propagates away from the ignition source. For direct initiation, the initial strength of the shock wave required is typically above the C-J detonation velocity, and the duration (i.e., the period of time in which the shock is above some critical value) should be long enough to permit a sufficient amount of energy release by the chemical reactions to sustain the shock and prevent its further decay. The strength and duration of the initiating shock wave depend on the energy-time characteristics of the ignition source. However, in the limit of an instantaneous point, line, or planar energy source, an ideal blast wave is obtained and the direct initiation process can then be characterized by a single parameter, i.e., the ignition source energy. It has been well established that condensed explosives approach the ideal point source since the energy density is extremely high (i.e., dimension of the explosive charge is small compared to the explosion length R o -- (Eo/Po)l/3). Extensive studies have been carried out in the 1960s and 1970s on direct initiation using electric sparks in an attempt to correlate the critical energy with the energy time profile and the power density of the source. These studies have been summarized in the review article by Lee. 29 With the interest in less sensitive fuel-air explosives, the critical energies required for direct initiation are met using condensed explosive charges in practice. The characteristics of the initiating blast wave are therefore

348

J.H.S. Lee

closely approximated by the ideal point blast in the direct initiation of fuel-air mixtures with condensed explosive charges. The direct initiation phenomenon by an ideal point blast wave is determined by a single parameter of the source energy only. For a given explosive mixture at a given initial thermodynamic state, it is found that there exists a critical minimum value of the blast energy below which a detonation is not obtained. The reaction zone is found to separate from the decaying blast wave as it propagates away from the source and the shock eventually decays to an acoustic wave while the decoupled reaction front propagates subsequently as a spherical deflagration wave. For ignition energies below the minimum critical value, the phenomenon is generally referred to as the subcritical regime (Fig. 17.18a). For initiation energy above the critical value, the blast wave decays asymptotically to a C-J detonation wave. The reaction zone is always coupled to the shock, and the onset of cellular instability is observed as the blast decays near the C-J velocity of the mixture. This is referred to as the supercritical regime (Fig. 17.18b). There exists a narrow range of initiation energy around the critical value in which the phenomenon is more complex. It is found that the blast at first decays below the C-J velocity with a distinct separation of the reaction zone from the shock front. This is followed by a quasi-steady period in which the shock propagates at a relatively constant velocity with the reaction front trailing behind at approximately the same velocity (i.e., the separation distance is constant). At the termination of the quasi-steady period, hot spots began to appear near the reaction front surface from which detonation wavelets or kernels develop. These detonation wavelets then grow rapidly to engulf the entire shock surface, forming an asymmetrical detonation that eventually becomes more spherically symmetric as it expands. At around the minimum initiation energy, the velocity of the shock wave in the quasi-steady regime is typically of the order of half the C-J velocity. The initiation processes near the minimum initiation energy is referred to as the critical regime (Fig. 17.18c). The formation of hot spots and the subsequent growth of the detonation kernels to eventually form the cellular C-J detonation wave in the critical regime are similar to the onset of detonation in DDT. The quasi-steady regime is also analogous to the final stage of DDT at the onset of detonation. Thus, DDT and direct initiation differ only in the initial processes involved in the creation of the critical condition required for the onset of detonation. In the critical regime of direct initiation, the formation of hot spots is due to the instability and hydrodynamic fluctuations in the reaction front during the quasi-steady period. The critical energy for direct initiation can readily be measured experimentally and can serve as a quantitative measure of the sensitivity of a gaseous explosive. Matsui and Lee3~ had suggested the use of the critical energy for the assessment of the detonation hazard of explosive gases. Extensive measure-

Detonation Waves in Gaseous Explosives

FIGURE 17.18

349

(a) Subcritical regime of blast initiation of C2H2-O 2 mixture by laser spark.

ment of the critical initiation energy for fuel-air were carried out by Bull and co-workers at Shell's Thornton Research Centre in the 1970s. 31-33 The dependence of the critical energy on the equivalence ratio for various fuelair mixtures is illustrated in Fig. 17.19. These typical U-shaped curves are qualitatively similar to those for the dependence of the detonation cell size 2 on the equivalence ratio. In the pioneering study of direct initiation of spherical detonations by Zeldovich et al., 34 a dependence of the critical energy on the cube of the induction zone length of the detonation was proposed. This cubic dependence can be shown from dimensional considerations since the blast wave is characterized by the explosion length R o --(E/po) 1/3, while the detonation wave is represented by its chemical length scale s (the induction distance); thus, the critical energy E "~ s Since the cell size 2 is proportional to the ZND induction length s the critical energy is E ~ 23 also. Following Zeldovich's pioneering work on direct initiation, numerous attempts have been made to develop a quantitative theory from Zeldovich's criterion. All these attempts evolved around the choice for an appropriate critical shock strength and its duration. Zeldovich's original criterion states that direct initiation requires that the distance traveled by the decaying blast must

350

J.H.S. Lee

FIGURE 17.18 (b) Supercritical regime of blast initiation in H2-C12 mixture by an electrical spark discharge. be at least equal to the induction zone length by the time the shock strength has decayed to the C-J value of the mixture. Using the similarity solution of ideal point blast of Taylor,35 Lee et al. 36 showed that Zeldovich's theory underestimated the critical energy by about three orders of magnitude. If a more representative reaction zone length such as the "hydrodynamic thickness" were used, a more reasonable agreement with the experiment can be achieved. In the detonation kemel theory of Lee and Ramamurthi, 37 the shock strength corresponding to the quasi-steady regime was used instead of the C-J velocity. This value of the shock strength is quite close to the autoignition limit for the mixture. The critical distance (or the detonation kernel size) was determined from a balance between the blast decay rate and the chemical energy release rate. It was argued that a balance between these two competing rates led to the quasi-steady condition of the critical regime. Reasonably good agreement between experimental results for acetylene-oxygen and hydrogenoxygen mixtures and the prediction using the detonation kernel theory was obtained. In a later study, Lee et al. 38 proposed an alternate kernel size based on the equivalence of the surface area of the kernel (i.e., 4~R .2) and the critical tube diameter of the mixture (i.e., d c = 132). The kernel size is thus

Detonation Waves in Gaseous Explosives

351

FIGURE 17.18 (c) Critical regime of blast initiation in C2H4-O 2 mixture with a laser spark ignition. Reprinted, with permission, from the Annual Review of Physical Chemistry, Volume 28, 9 1977, by Annual Reviews www.Annual Reviews.org.

R* = 3.252, and using a value for the quasi-steady shock strength of Mcj/2 in the strong blast theory, the following expression for the critical energy was obtained: E = 14.5rtTp0M~j23.

(17.1)

From independent measurements of the detonation cell size 2, the above expression predicts reasonably well the critical energy for fuel-air mixtures when compared with experiments (see Fig. 17.19). Note that the cubic dependence on 2 renders the critical energy extremely sensitive to the accuracy in the measurement of the cell size itself. Since the uncertainty in measurement of the cell size from smoked foils can be quite large, the semi-empirical expression obtained is probably as good as can be expected if it is based on the cell size 2. Since the similarity solution for strong blast decay can readily be applied to geometries other than spherical, similar expressions for the critical initiation

352

J. H. S. Lee

.I

I C2H6

i

, |

....

FUEL ELSWORTH

CsH8

H2

C4HIo

O

c~4 ~ c2%

-

-

U

C4HIo

0

0 ,ea)

H2

C2H4

II,LI r.O "r" 0

C~H2 ,I .05 .02

I 0

I I

,

I 2 EQUIVALENCE

,

I 5 RATIO ~Z~

,

4

FIGURE 17.19 Critical energy for direct initiation in fuel-air mixtures initially at atmospheric pressure and room temperature. Reprinted, with permission, from the Annual Review of Fluid Mechanics, Volume 16, 9 1984, by Annual Reviews www.AnnualReviews.org.

energies for cylindrical and planar detonations can readily be derived. The experimental realization of an instantaneous planar energy source is very difficult. However, using the hypersonic blast wave analogy, 39 the critical energy for the direct initiation of cylindrical detonation can be obtained indirectly via the direct initiation by a hypervelocity projectile 4~ Similar to a hypervelocity projectile, a condensed phase detonation in a detonating cord 41 can also be used to obtain cylindrical initiation. Figure 17.20 illustrates

Detonation Waves in Gaseous Explosives

353

different regimes of direct initiation using a PETN detonating cord in ethyleneair mixtures. In the supercritical regime shown in Fig. 17.20a, a conical detonation is established with the energy source traveling along the cord at a velocity VDc of about 6.3 km/sec. The cone angle of the detonation is - - sin -1 V c j / V D c , where Vc_J is the detonation velocity of the ethylene-air mixture at 1.8 km/sec. The cone angle obtained is approximately 16 ~ in good agreement with experimental observation. Note that according to the hypersonic blast analogy, the conical detonation is equivalent to a cylindrical detonation under the transformation z = Vc)ct where z is the distance along the axis from the apex of the cone, and Vt)c is the velocity of the condensedphase detonation along the cord. Thus, the instantaneous photograph of the conical detonation can be interpreted as a streak photograph of an expanding cylindrical detonation. For the critical regime shown in Figs. 17.20b and 17.20c, we see that the conical detonation is formed from the growth of discrete detonation kernels originating from hot spots distributed on the conical blast wave surface. For the subcritical case in Fig. 17.20d, the weak conical deflagration can be seen just ahead of the condensed-phase detonation products. The decoupled blast wave in air cannot be seen in the self-luminous photographs. The initiation phenomena illustrated in Fig. 17.20 for cylindrical detonations are identical to those for spherical detonations shown in Fig. 17.18.

FIGURE 17.20 Regimesof cylindrical blast initiation in C2H4-air mixture by a PETN detonation zord: (a) supercritical, (b, c) critical, (d) subcritical. (CourtesyofM. Radulescu). (See Color Plate 12).

354

j.H.S. Lee

The analogous expression for the critical energy (per unit length) for the direct initiation of cylindrical detonation can readily be obtained as E-

10.12p0M2122.

(17.2)

According to the hypersonic blast wave analogy, the work done (per unit length) by the drag force on a hypervelocity projectile is equivalent to the cylindrical blast wave energy. Equating the work done to the blast energy results in the following expression relating to the critical diameter of the hypervelocity projectile d and its velocity Mo~ to the C-J velocity Mc_J and the cell size of the mixture:

M~

2

~ = 5.3Mcj d" A comparision of the critical diameter of the projectile, predicted by the preceding expression, to the experimental results obtained by Higgins 4~ is shown in Fig. 17.21. The agreement is reasonably good in spite of the fact that the velocity of the projectile in Higgins' experiment is only slightly higher than the C-J velocity of the detonating gas. Hence, we would not expect the hypersonic blast analogy to be valid under this circumstance. In the more recent study using a detonating cord, 41 the velocity of the condensed phase denotation along the PETN cord is of the order 6-8 km/sec. This is 3 to 4 times the C-J velocity of the mixture, and thus the hypersonic blast analogy should be more applicable in this case. The experimental results obtained using the PETN cord are also found to be in good agreement with the prediction of Eq. (17.2). Since the behavior of an ideal blast wave is characterized by a single length scale, that is, the explosion length R0 (E/po)l/j+l; where j = 2, 1, 0 for the spherical, cylindrical and planar geometries, respectively, and the detonation sensitivity is represented also by a chemical length scale (e.g., the cell size 2), dimensional considerations indicate that the ratio R0/2 should be invariant. In other words, for the same explosive mixture, hence 2, the critical explosion length R0 should be the same. This explosion length invariance was first suggested by Lee 29 to permit the critical energy for any geometry to be estimated if it is known for one particular geometry. This explosion length invariance was recently verified experimentally by Radulescu et al. 42 where the critical energy of ethylene-air mixtures was determined for both spherical and cylindrical geometries. The explosion length determined from the critical energy obtained was found to be identical for the two geometries. The ratio of the critical explosion length to the detonation cell size was also found to be invariant (i.e., R~/2 ~, 32 for the case of ethylene-air mixtures) in accordance -

-

355

Detonation Waves in Gaseous Explosives 2.00

9 1.75

E

Experiment with Spheres Cylindrical Initiation (Theory)

1.50

v

1.25

Q.

1.00

(3

.m=

0

0.75 VSPHERE = V C j

0.25 . . . . . . . . . . . 0.4 0.6

I, 0.8

9

,, I,,, 1.0

1 , , .... I . . . . I , , ,., I . . . . I . . . . I . . . . I , , 1.2 1.4 1.6 1.8 2.0 2.2 2.4

i 2.6

Sphere Diameter (cm) FIGURE 17.21 Critical initial pressure of mixture for direct initiation by a hypervelocity projectile as a function of the sphere diameter. (Courtesy of A. Higgins)

with ideal blast wave theory. Thus, ideal blast initiation is a fairly wellestablished phenomenon. The ideal blast initiation problem is also amenable to numerical simulation. The one-dimensional reactive Euler equations with simplified kinetic rate laws can be integrated with a high degree of accuracy with current numerical algorithms. Thus, the detailed flow field of the transient development of the detonation wave can be determined numerically much more readily than experimentally. Although the structure of self-sustained detonations is invariably three-dimensional, the growth of cellular instability appears to take place only during the final phase of the initiation process as the overdriven detonation decays to its C-J state. Thus, it appears that the entire blast initiation process can perhaps be described by just one-dimensional simulations. Although the development of local hot spots at the termination of the quasi-steady regime appears to give the impression that the onset of detonation is a three-dimensional phenomenon, the formation of hot spots is due to small fluctuations that greatly amplify because of the strong exponential temperature dependence of the induction time. This results in a temporal and spatial variation of hot spots on the surface of the reaction front. However, the essential physics of the onset of detonation, that is, the SWACER process, is

356

j.H.S. Lee

not influenced by the spatial distribution of hot spots. The important threedimensional event of cellular instability comes later when the overdriven detonation decays to a self-sustained C-J detonation. Using a single-step Arrhenius reaction rate law, the three regimes of direct initiation (i.e., supercritical, critical, and subcritical) for the planar geometry are shown in Fig. 17.22, where the shock pressure is plotted against the distance (normalized with respect to the half ZND reaction length) from the ignition source. 43 The numerical results are in good agreement with experiments and reproduce the same qualitative behavior of the three regimes. At the critical regime, we note that the blast pressure decays to about one-half of the C-J value, as observed experimentally, before reaccelerating to an overdriven detonation at the termination of the quasi-steady period. The advantage of numerical simulation is that the detailed transient flow field of the initiation process can be obtained readily. Figure 17.23 illustrates the temperature profiles behind the blast wave for the subcritical and the 120

100

8O

t/

S u p e r c n t "" ical

t-

~. 60

.

Critical initiation

4O Pv.n. 2O

0 0

50

100

150

200

250

300

Xsh

FIGURE 17.22 Numerical simulation of planar blast initiation showing the three regimes. Reprinted from Higgins, A. and Lee,J.H., Comments on criteria for direct initiation of detonation, Phil. Trans. R. Soc., Lond. A 357 3503-3521 (1999).

Detonation Waves in Gaseous Explosives

357

supercritical regimes. In Fig. 17.23a, for the subcritical regime, the progressive decoupling of the reaction front from the shock front as it decays can be observed. In the supercritical regime (Fig. 17.23b), decoupling does not occur as the blast decays asymptotically to a C-J detonation. Note that for a sufficiently high value of the activation energy, the C-J detonation is unstable. However, in a one-dimensional simulation, the instability is manifested by a longitudinal pulsating detonation. Thus, for a high (unstable) value of this activation energy, the initiating blast decays to a pulsating detonation oscillating about the C-J state of the mixture. 12 a) 10

E

,

0

.

,

~

~

~

,

|

,

9

~

I

,

,

,

20

40

60

80

00 . . . . . .20 ..

40

60

80

100

120

140

12 (b) 10 86421

,

100

'

120

,

,

,

1

140

x . . . . . . . . . .

FIGURE 17.23 Temperatureprofiles behind blast wave for (a) subcritical and (b) supercritical regimes. Reprinted from Higgins, A. and Lee, J.H., Comments on criteria for direct initiation of detonation, Phil. Trans. R. Soc., Lond. A 357 3503-3521 (1999).

358

J.H.S. Lee

In the critical regime shown in Fig. 17.24, we can see that the reaction zone decouples from the decaying blast initially. During the quasi-steady regime, the shock amplitude as well as the temperature profile behind the shock appears to be stationary (e.g., 81 _< x <_ 110). At the end of the quasi-steady period, the reaction zone speeds up rapidly to coalesce with the shock front to form an overdriven detonation. The corresponding pressure profiles of the critical regime are shown in Fig. 17.25. Since the decay rate of a shock wave is proportional to the expansion gradient behind the shock, we note that the pressure gradient diminishes as the blast decays towards the quasi-steady regime. During the quasi-steady period when the shock front propagates at a relatively constant velocity, the expansion gradient is very small and the pressure profile is almost fiat. During the acceleration phase, the pressure gradient becomes positive (i.e., increasing pressure with distance away from the shock). The shock pressure amplitude increases as the gradient behind it steepens until the shock becomes an overdriven detonation. The reaction zone (indicated by the arrows) progressively accelerates toward the shock front during the final acceleration phase. Comparing the pressure profiles of the quasi-steady regime with those shown in Fig. 17.17 for photochemical initiation where the gradient of chemical activity is obtained by photodissociation we may conclude that direct initiation involves a similar mechanism (i.e., SWACER). In blast initiation, the gradient of chemical activity is a result of the decaying blast, giving each particle crossing the shock a different initial condition and thermodynamic history. Thus, at any instant of time, there is a gradient of chemical activity behind the shock front with the most intense reaction occuring at the reaction zone and decreasing toward the shock front. With the proper gradient of chemical activity, the pressure waves originating at the reaction front will trigger and couple with the energy release time sequence and amplify as it propagates toward the shock. This SWACER event is quite evident from the pressure profiles shown in both Figs. 17.17 and 17.25. Hence, we see that even from a one-dimensional simulation, the key mechanism for the onset of detonation can be captured. Three-dimensional cellular instability of the front appears to play a role in the self-sustained propagation of the detonation itself. Numerical simulations (as well as experimental observations) indicated that the formation of detonation involves an acceleration process. Thus, a criterion for direct initiation should address the requirements for the amplification of compression waves in the reacting flow behind the shock. Almost all existing theories for direct initiation are essentially based on a failure criterion (i.e., the condition in which a detonation quenches) instead of a criterion for shock wave amplification from the detonation. In the original criterion of Zeldovich et al.,34 he stated that the blast must decay to the C-J velocity at a radius at least equal to the ZND reaction zone length. Subsequent refinement of Zeldovich's

Detonation Waves in Gaseous Explosives

359

10

(a) 8

i e E

2

10

(b) 8

II E

2

~

~'5 ..... ;8'

81

"

~

',;~,-

" ~)

" "g3'

'

'

L

12

(c) 10

je ~-4 2

DO

105

110

115 x

120

125

130

FIGURE 17.24 Temperature profiles for the critical regime of planar blast initiation. Reprinted from Higgins, A. and Lee, J.H., Comments on criteria for direct initiation of detonation, Phil. Trans. R. Soc., Lond. A 357 3503-3521 (1999).

360

J. H. S. Lee 35

(a) 30 25

i

o. 20 15

1

1 5

_ ,

I

20

......

l

25

,

,

,

30

35

, ,_~

40 x

i

. . . .

45

~ ,

50

55

65

60

24

(b)

22 20 is 18

=

~ 16 14

12

lo,2

~

75

78

81

84

i~

|,

87 x

. . . . .

90

!

93

. . . . .

1

96

102

99

I

(c)

| 20 9

~q~

! .......

los

11o

11s

12o

~2s

x

FIGURE 17.25 Pressure profiles for the critical regime of planar blast initiation. Arrow indicates location of location zone. Reprinted from Higgins, A. and Lee, J.H., Comments on criteria for direct initiation of detonation, Phil. Trans. R. Soc., Lond. A 357 3503-3521 (1999).

Detonation Waves in Gaseous Explosives

361

criterion essentially involves more appropriate choices for the minimum shock strength (instead of the C-J value) and the minimum radius (other than the ZND induction length). The choice of a minimum shock radius is essentially the same as the requirement for a maximum allowable curvature of the shock when it reaches a certain critical strength (e.g., C-J velocity or half C-J velocity corresponding to the quasi-steady regime value) as in the initiation theory of He and Clavin. 44 A similar failure criterion based on curvature was also suggested by Edwards et al. 45 earlier for the analogous critical tube diameter problem. For a planar blast wave where there is no curvature of the shock front, the decay rate of the shock is governed by the pressure gradient behind the shock. This is generally referred to as an unsteady effect, in contrast to the curvature effect that also results in the decay of the shock. Both effects are essentially the same, since the decay rate of a shock is governed by the expansion gradient behind it. Whether the gradient is due to longitudinal expansion as in a planar blast or resulting from the lateral expansion due to flow divergence behind a curved shock is of secondary significance. In a spherical or cylindrical blast, both unsteady effect and curvature contribute simultaneously toward the local shock decay rate. Thus, it would be more appropriate to specify just a critical shock decay rate without the need to consider its cause as due to curvature of unsteadiness. In the initiation theory of Eckett et al., 46 the non-steady effect was shown to be more important than curvature. However, from the foregoing discussions, it may be concluded that the direct initiation criterion should address the amplification process rather than the initial decoupling of the reaction front from the decaying blast wave. It appears that the formation of a detonation wave is governed by the final processes for the onset of detonation, that is, rapid amplification of a compression pulse or shock in a gradient field of chemical activity. Whether it is DDT or direct initiation, this final stage of the formation process is common. In DDT, it is the flame acceleration mechanisms that bring about the necessary conditions for the onset of detonation. In direct initiation, the same critical conditions are obtained from decaying reactive blast waves. Thus, the fundamental process of detonation formation is essentially one of rapid, coherent amplification of compression waves in an appropriate gradient field of chemical activity.

17.4 DETONATION

LIMITS

Detonation limits generally refer to the critical conditions (e.g., chemical composition, initial pressure) beyond which self-sustained propagation of a detonation wave is not possible. The Chapman-Jouguet theory cannot predict the limits since the rate processes in the reaction zone are not considered. The ZND theory for the structure, however, can, in principle, be used to determine

362

J . H . S . Lee

detonation limits since the nonequilibrium rate processes in the reaction zone are considered. Detonation limits result from the competition between rate processes, and detonation failure occurs when the energy release rate has been drastically reduced and fails to maintain the shock at its C-J velocity. From a purely chemical kinetic consideration, a significant reduction in the reaction rate can occur when the rapid chain branching mechanism is overidden by a chain termination reaction. This is, in essence, the basis of the theory proposed by Belles 1 for the detonation limits for hydrogen-oxygen mixtures. The onset of the detonation limits is assumed to be based on the competition between the chain branching reaction H 4 - 0 2 --~ OH 4-O and the chain termination reaction of H + 0 2 + M - ~ HO 2 + M. The critical shock temperatures are thus those that satisfy the second explosion limit of the hydrogen-oxygen reaction, that is, twice the chain branching reaction rate must be greater than (or equal to) the chain termination rate. The von Neumann shock temperature can be computed from the Chapman-Jouguet detonation velocity, which can readily be determined for a given mixture composition. Thus, the limiting mixture compositions can be found from consideration of the competing reaction rates. Reasonable agreement with experimental observation was obtained. However, from the discussions given in the previous section on the cellular structure of real detonations, we note that gas-dynamic processes have to be considered apart from chemical kinetics alone in the prediction of the limits. In the pioneering work on the detonation structure, Zeldovich 2 had already attempted to address the problem of the velocity dependence on tube diameter and detonation limits by considering heat and momentum losses. These transport processes are described by "source terms" in the momentum and energy equations for the detonation structure. With the presence of source terms, the solution can no longer be determined by the Chapman-Jouguet criterion, and the conservation equations have to be integrated using the shock-jump conditions as initial values. The criterion to seek the correct solution requires that flow gradients be finite when the sonic singularity is encountered. For a given friction factor and heat transfer coefficient, it was found that there can exist more than one possible solution that satisfies this criterion. However, only one solution is a stable solution, and the choice for the correct solution requires further stability analysis. It is found that there also exists a maximum friction factor when no solution can be obtained. This may be used to define the detonation limits due to frictional losses. We discuss the theory of nonideal detonations in more detail in a later section. Zeldovich's theory of detonation limits could, in principle, give quantitative predictions if the appropriate source terms are used to model the heat and momentum losses. In Zeldovich's analysis, Schlichting's friction formula for turbulent boundary layer is used and heat transfer is handled via Reynolds'

Detonation Waves in Gaseous Explosives

363

analogy. From the discussions of the turbulent, cellular structure given in the previous section, it is clear that a more sophisticated model must be used to describe the turbulent losses associated with the cellular detonation. If such models are developed to account for the transient, three-dimensional fluctuations in the reaction zone, then the one-dimensional ZND equations could be integrated to yield quantitative predictions of the limits (as well as the dependence of detonation velocity on pipe diameter and wall roughness). Such models have yet to be developed, and at present there are no quantitative theories for the detonation limits. Detonation limits have to be determined experimentally in general. However, the detonation limits are not universal and are specific for given boundary conditions (i.e., tube dimension, geometry of cross section, wall roughness, etc.). In many publications and books, detonation limits are still given in terms of the chemical composition of the mixture only. GuCnoche and Manson 3 have explicitly pointed out as early as 1954 that "detonation limits are no longer characteristic of the combustion mixture if the tube diameter changes . . . . " In addition to the tube dimension, the geometry of the tube cross sectional area, and the nature of the confinement (e.g., rigid wall, acoustic absorbing wall, yielding confinement, etc.), as well as wall roughness, all contribute to the determination of the limiting mixture compositions. Experiments indicate that there exists a continuous spectrum of steady-state detonation velocities ranging from the C-J detonation velocity to the sound speed of the mixture when the boundary conditions are varied (e.g. rough walled tubes and porous media) hence it is difficult to define uniquely the detonation limits, since what constitutes a steady detonation wave can no longer be precisely defined. Before we deal with this class of quasi-detonation phenomena, let us first discuss the classical case of the rigid wall "smooth" circular tube to illustrate the complexity of the near limit phenomenon. Consider a long, smooth, rigid, circular tube of a given diameter where the composition limits are to be determined. To determine the limits, the method of initiating the detonation must first be considered. If a weak ignition source is used, then a flame will first be generated and the detonation will result from the transition from the deflagration to a detonation. The length of the tube must be sufficiently long for the transition to take place and, furthermore, since the detonation formed is initially overdriven, a further length of the detonation tube is required to ensure that steady-state Chapman-Jouguet velocity is eventually attained. Near the limits when the mixture is rather insensitive, the "run-up" distance to detonation may be quite substantial in general. To reduce the "run-up" distance, a "Schelkhin spiral" or repeated obstacles can be inserted near the ignition end of the tube to facilitate the flame-acceleration process. Since the detonation is formed from the acceleration of an initially slow deflagration, and the flame-acceleration process

364

J.H.S. Lee

involves physical mechanisms other than those responsible for the propagation of the detonation wave itself, it is not obvious that the transition limits should also correspond to the detonation limits. Experiments by Peraldi et al. 4 have demonstrated that in smooth, circular tubes, the transition limits are given by D / 2 > 1, that is, the tube diameter D must be at least of the order of the cell size 2. In rough tubes with repeated circular orifice plates as obstacles, a similar criterion was also observed, that is, d / 2 > 1, where d is now the diameter of the orifice plate. For more complex tube cross-sectional geometries or other obstacle configurations, the transition limits have not been determined but would be specific for the boundary conditions used. If the transition limits are given by D / 2 >_ 1, then it is clear that the propagation limits of the detonation should be at least equal to, or perhaps wider than, the transition limits (if other means of initiation of the detonation is used). Indeed, if a very powerful ignition source were used to initiate the detonation wave instantaneously (i.e., direct initiation), the combustion wave formed would be found to be able to propagate at a velocity near the Chapman-Jouguet velocity for very long distances as a "metastable" wave. This had been demonstrated by Wolanski et al. s where a powerful H 2 4 - 0 2 driver tube is used to initiate a detonation in CH4-air mixtures in a 6.35cm-square tube. The detonation cell size for stoichiometric CH4-air mixtures is estimated to be of the order of 30 cm6; thus D / 2 ~- 0.2, significantly below the transition limit of D/~. ~ 1 found by Peraldi et al. 4 Using a powerful igniter, Wolanski et al. s in fact initiated a "metastable," single-headed spin detonation over a range of CH4-air compositions. This metastable detonation can fail abruptly. Saint-Cloud 7 and Donato 33 had showed that these metastable waves outside the detonation limits would fail when subjected to a finite perturbation. Thus, it is important to avoid using too strong an ignition source in determining the detonation limits. Since the cell size 2 is a characteristic length scale of the detonation, it is reasonable to expect that the limits criterion should also be defined by the ratio of the tube diameter to the detonation cell size. The transition limits found by Peraldi et al. 4 of D / 2 ----- 1 had, in fact, been proposed earlier by Schelkhin s to be the detonation limits also. Since the single-headed spinning detonation is the lowest stable detonation mode according to the acoustic theory of Manson 9 and Fay 1~ this led Dove and Wagner 11 to propose that the onset of singleheaded spin should be used to define the detonation limits. From Manson and Fay's acoustic theory of spinning detonation, the spin-pitch to diameter ratio is given by p / D = ~ U / k n c , where c is the local sound speed, U is wave velocity, and kn is the first root of the derivative of the Bessel function of order n (i.e., k 1 - - 1.841, k 2 = 3.045, etc.). In the study of near limit detonation propagation in circular tubes, Moen et al. 12 found good agreement between theory and experiment for the onset of single-headed spin in C2H4-O2-N 2 mixtures in three tubes of diameters 28mm, 48 and 145, respectively. Moen et al. 12

Detonation Waves in Gaseous Explosives

365

assumed that if the pitch of a single-headed spin detonation can be considered as the length of the cell (i.e., ~ 1.62), then the D / 2 criterion for the limits based on the onset of single-headed spin is D / 2 TM 1.7. Moen et al. 12 found good correlation with experimental data for ethylene-air mixtures. In a more extensive study of limits in circular tubes, Dupre et al. 13 found that the criterion of D / 2 -- 1/re (first proposed by Kogarko and Zeldovich 14 and Lee 15) appears to correlate better with experimental data. A physical explanation for the criterion was given by Lee 15 as follows" Since the tube circumference reDrepresents the largest characteristic length scale of the cross-sectional area of the tube, the largest characteristic time or period of the circumferential mode of the acoustic vibration should be of the order rcD/c where c is sound speed of the detonation products. For the periodic chemical processes in a cellular detonation front, the characteristic chemical time scale can be represented by 2/c Thus, for a resonant coupling between the acoustic vibration and the periodic chemical processes in the cellular detonation, we arrive at the criterion 2 = reD.... Measurements of the detonation velocity near the lean limit for H2-air mixtures in the 49-mm diameter tube and a 38-mm diameter tube by Dupre et al. 13 indicate that near the limits defined by D / 2 < 1/re (or 2 / D > rc -- 3.14), the detonation is very unstable and exhibits large velocity fluctuations (Figs. 17.26a and 26b). For rigid circular tubes, the velocity deficit near the limits seldom exceed 10% in general. In fact, Manson et al. 16 have used the velocity fluctuation to define the wave stability and limits. Fig. 17.27 shows the extensive experimental results of Dupre et al. 13 for the velocity deficit of H2-air mixtures in tube diameters ranging from 38 to 152 mm. Most of the data fall in the region bounded by D / 2 - - 1 / r e and A V / V c j - 0 . 1 . This indicates that the onset of single-headed spin as given by D / 2 - 1/re defines quite well the detonation limits and that the velocity deficit seldom exceeds 10% at the limits for smooth tubes. As the detonation limits are approached, the detonation velocity exhibits very large fluctuations. Using a microwave doppler interferometer to obtain a continuous monitoring of the detonation velocity, Lee et al. lz have studied the unstable, near-limit propagation of the detonation wave. Six different types of unstable, near-limit behavior were identified and shown in Table 17.1. The velocity-time histories as well as the velocity histograms for the six modes of propagation are illustrated in Fig. 17.28. In the stable mode (Fig. 17.28a), the detonation propagates at a constant velocity very close to its theoretical C-J value. The stability of this mode is indicated by the very narrow band in the velocity histogram. The mean value is only slightly below the theoretical C-J value. For the rapid fluctuation mode shown in Fig. 17.28b, the velocity spectrum is wider and the deficit of the mean velocity is also larger and of the order of A V / V c j "" 0.2 or V ~ 0.8 Vc3. For the stuttering mode (Fig. 17.28c), the velocity alternates between Vc_J and 0.6 Vc_J for long periods of

366

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TABLE 17.1 Mode 1 2 3 4 5 6

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Brief description

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the wave propagation (e.g., about a period of 1 ms or about 26 tube diameters of wave propagation). The velocity histogram also clearly demonstrates the two dominant velocities of this "stuttering mode." Further away from the limits, we have the so-called "galloping detonation" mode shown in Fig. 17.28d where the velocity fluctuates between 0.5 Vc_J and 1.5 Vc_J. For most of the duration of a galloping cycle, the detonation remains at the low velocity phase of about 0.5 Vc_J. Rapid acceleration to an overdriven detonation of about 1.5 Vr occurs at the end of the galloping cycle. The overdriven detonation then decays to the C-J where it will remain for a short duration before the wave fails and decays to a low velocity of ~0.5 Vc_J again. The period of the galloping cycle is of the order of 100 tube diameters and the phenomenon is reminiscent of the transition from deflagration to detonation. For the so-called "low velocity stable" mode (Fig. 17.28e), the detonation decays to about 0.5V c_J and remains at this velocity for over 65 tube diameters with little variation. The detonation of this metastable low-velocity regime is dependent on the tube diameter, and small perturbations can cause it to accelerate as in the galloping or fail. Still further away from the limits, the initial C-J detonation fails and the velocity drops again to a metastable state at about 0.5 Vc_J and remains at this low velocity for long distances of travel. However, the low velocity regime is much less stable and velocity perturbations can lead to failure, that is, the detonation fails. Since the detonation fails (Fig. 17.28f) easily under perturbations and does not reinitiate again. This mode is referred to as the failure mode. Note that all these unstable regimes are dependent on the tube diameters. If a larger tube is used, some of these unstable regimes may not be observed. For the rapidly fluctuating mode (Fig. 17.28b), the value of D / 2 ~ 0.3 corresponds to the onset of single-headed spin detonation. The other unstable modes are all "outside" the limits as defined by the D / 2 ~- rc criterion. Thus, even in smooth tubes, what constitutes the detonation limits is difficult to define since there exists a spectrum of unstable phenomena beyond the stable single-headed spin detonation that is the lowest detonation mode according to the acoustic theory of Manson 9 and Fay. 1~ Hence, the specification of the detonation limit depends on what is considered as a "bona fide" detonation wave. Stability (i.e., narrow velocity fluctuation about the C-J value) and a maximum deviation of the mean velocity from the C-J velocity (e.g., velocity deficit A V / V c j < 0.1) could be used as a limit criterion. From Fig. 17.28b, we see that the criterion defined by D / 2 ~ rc also provides a fairly reasonable criterion for the detonation limits in circular, smooth tubes in terms of the stability of the detonation wave. Of particular interest is the mechanism in which the detonation re-accelerates from about 0.5 Vc_J to an overdriven detonation in the galloping mode. Moen et al. 12 have measured the pressure-time history of a galloping detonation that illustrates the reacceleration mechanism (Fig. 17.29). In the top trace

372

J . H . S . Lee

FIGURE 17.29 Pressure records of onset of detonation resulting from the amplification of transverse pressure waves in the reaction zone. (a) Amplification of transverse pressure waves. (b) Formation of overdriven detonation and subsequent decay to a C-J wave. Moen, I.O., Donato, M., Knystautas, R., and Lee, J.H., Proc. Combust. Inst., 18:1615-1622 (1981).

of Fig. 17.29a, one can observe the normal shock preceding the reaction zone where intense transverse pressure fluctuations occur. The transverse pressure fluctuations amplified and also longitudinal compression waves are being sent forward toward the shock front resulting in a pressure rise ahead of the reaction zone. In the first pressure trace of Fig. 17.29b, we can see that the longitudinal compression waves steepen to form a shock, and this shock also catches up with the leading shock front. When it catches up with the leading shock, an overdriven detonation is formed (third trace of Fig. 17.29b). This overdriven detonation then decays subsequently to a C-J detonation (third and fourth traces of Fig. 17.29b). These pressure histories illustrate that the

Detonation Waves in Gaseous Explosives

373

mechanism for the reacceleration of the decoupled shock reaction zone complex is due to the amplification of the transverse pressure fluctuation in the turbulent reaction zone behind the leading shock front. This mode of transition is different from the one due to hot spots in the turbulent flame brush as illustrated in the studies by Urtiew and Oppenheim. 18 For the propagation of detonations in rough or obstacle-filled tubes, the limits are difficult to define because of the existence of multiple quasi-steady propagation regimes in which the combustion wave is supersonic. For a given tube and obstacle geometry (i.e., blockage ratio and obstacle spacing), it was found that as the sensitivity of the mixture is varied (e.g., equivalence ratio ~b or the initial pressure), there corresponds a so-called quasi-detonation regime (1000m/s_< V < 2 0 0 0 m / s for fuel-air mixtures) and the choking regime ( 8 0 0 m / s < V <1100m/s) where the combustion wave is supersonic. For a given tube diameter and obstacle configuration, abrupt transitions in the wave velocity occur at well-defined mixture compositions from one regime to the other. In the so-called choking regime, the wave speed is of the order of the sound speed of the hot combustion products (i.e., V-~ 900m/s). This is referred to as the choking regime since the propagation mechanism is credited to the near sonic flow of the combustion gas in the reaction zone in the obstacle field of the tube (i.e., frictional choking). Further reduction in the mixture sensitivity results in another abrupt transition from the high-speed "choking regime" to a slow, subsonic deflagration of velocities about 200m/s and less. For the case of hydrogen-air mixtures in both circular and orificeplate obstacles and in a two-dimensional, square, cross-sectional tube with a cylindrical rod array as obstacles, the various propagation regimes are shown in Fig. 17.30. At the abrupt transition point from quasi-detonation to the choking regime, it is found that the critical compositions are defined by d / 2 (or w / 2 ~ 1) where d and w denote the orifice diameter (for circular orifice plate) and spacing between the cylindrical rod obstacles (for square tube) respectively. Thus, this is in accord with the transition limits of d / 2 "~ 1 observed by Peraldi et al. 4 discussed earlier. High-speed framing schlieren photographs 19 indicate that in the quasi-detonation regime, continuous initiation of detonation occurs upon reflections from obstacles and the wall followed by failure of the detonation upon diffraction over the obstacles. In the choking regime, the rapid combustion is due to the enhanced turbulent transport due to the interactions of the transverse pressure waves in the reaction zone. 2~ Thus, in very rough or obstacle-filled tubes, the criterion of the detonation limits really depends on the type of supersonic combustion wave phenomenon considered since there is no longer a unique C-J detonation phenomenon. In tubes with acoustic absorbing walls, the transverse waves are attenuated upon reflection. It is obvious that the detonation limits would now be dependent on the nature of the absorbing material of the tube wall. In fact,

374

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Detonation Waves in Gaseous Explosives

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in a tube with a very thin plastic wall (i.e., yielding confinement), Murray and Lee 22'23 found that the limit criterion varies in the range 19 > D/2 > 7.5 when the thickness of the tube wall varies between 0 and 0.22 kg/m 2. The yielding confinement case is similar to the critical tube diameter problem where the inertia of the yielding wall now controls the strength of the inward propagating expansion waves as the yielding wall expands. Limits are governed by the possibility of quenching of the detonation due to adiabatic cooling as the expansion waves propagate toward the axis of the tube. The strength of the expansion fan depends on the inertia, and hence the wall thickness of the confinement. The preceding discussions show that the detonation limits depend strongly on boundary conditions, i.e., confinement, as well as the nature of the confinement itself (i.e., rough wails, acoustic absorbing or yielding walls). We note that the onset of limits is associated with the failure of the detonation as a result of the perturbations generated by the boundaries. In other words, we can say that failure is due to the inability of the detonation to maintain self-propagation under the conditions imposed by the boundaries. To illustrate the limiting case of the influence of boundary conditions on the limits, it is best to consider the failure and reinitiation mechanisms of a C-J detonation wave emerging from a rigid tube or channel into a sudden area enlargement. Under the confinement by the rigid wall of the tube, the detonation can propagate steadily at its C-J velocity. However, upon emerging from the tube, the confinement is suddenly removed. Expansion of the highpressure detonation products generates rarefaction waves that converge toward the tube axis. The adiabatic cooling associated with the inward propagating expansion waves quenches the detonation and the reaction front is decoupled from the leading shock. Thus, a failure wave (i.e., the boundary between the coupled and decoupled part of the detonation front) originating at the outer edge of the tube exit also progressively propagates toward the axis at the same time as rarefaction waves. However, the trajectory of the failure wave does not necessarily coincide with the head of the rarefaction fan. For mixtures with very high activation energies (i.e., reaction rate is extremely temperature sensitive), a small perturbation may cause the detonation to fail. In this case, the trajectory of the failure wave corresponds closely to the head of the rarefaction fan. On the other hand, for mixtures with low activation energies, the failure wave is less sharply defined and the decoupling occurs more gradually. Hence, there is a more progressive attenuation of the detonation beginning at the head of the expansion wave until the decoupling reaches a certain critical value and the detonation then fails completely. The attenuated portion of the detonation front gives rise to a curvature and eventual failure occurs when the overall curvature of the detonation wave reaches a certain critical value. In contrast to the high activation energy failure mode, which is

376

J . H . S . Lee

abrupt and is due to the adiabatic cooling by the expansion waves, the failure mode for the case of low activation energy is more gradual and is a result of excessive curvature of the attenuated detonation front. The two failure modes are illustrated by the sketches shown in Figs. 17.31 and 17.32. In Fig. 17.31, the failure wave corresponds closely to the head of the rarefaction fan, whereas in Fig. 17.32 the failure wave lags behind, and the region bounded by the head of the rarefaction and the failure waves is an attenuated, curved detonation front. Failure eventually occurs when the global curvature of the attenuated, curved detonation exceeds a certain critical value. The two failure modes can be demonstrated experimentally from smokedfoil records or from open shutter photographs in a thin, two-dimensional channel. Figure 17.33 illustrates the two failure modes for C2H2-O2 mixtures where the luminosity of the intense chemical reactions at the triple points are sufficiently localized to give good-quality, open shutter records of the transverse wave trajectories equivalent to a smoked-foil record. In Fig. 17.33, we can see the head of the rarefaction fan originating at the corner of the tube exit and propagating at the sound speed of the products toward the axis and thus coinciding with a transverse wave trajectory. The failure wave, denoted by the destruction of the cellular structure, is similar to the head of the rarefaction fan (or a transverse wave trajectory originating from the edge of the tube axis for high activation energy mixtures). The failure wave does not correspond to the head of the expansion fan (as indicated by the transverse wave trajectory), but

FIGURE 17.31

Failure and local reinitiation for mixtures with high activation energies.

Detonation Waves in Gaseous Explosives

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FIGURE 17.32 Failure due to excessive global curvature of the detonation front for mixtures with low activation energies. Bull, D.C., 1979, Concentration limits to the initiation of unconfined detonation in fuel/air mixtures, Trans IChemE, 57: 219-227. Reproduced by permission of IChemE.

lies more closely to the original tube boundary. The attenuation of the curved detonation front by the expansion fan is manifested by an increase in the detonation cell size, leading to eventual failure of the detonation when the curvature becomes excessive. In Fig. 17.34, we note that reinitiation for the high activation energy case occurs locally at hot spots near the tube axis. For the low activation energy case, the attenuated curved detonation continues to propagate following the original tube boundary. It failed to grow to fill the unconfined space because of the excessive curvature at the boundary. From the preceding discussions, we see that detonation can fail in two different ways depending on the temperature sensitivity (i.e., the activation energy) of the mixture. Thus, the stability of the detonation front is also closely linked to the failure mechanism and hence influences the detonation limits as well. Continued propagation of the detonation front depends on its ability to negate the perturbations generated at the boundary. Experiments indicate that there exists a critical diameter of the tube above which the detonation can reinitiate itself and continue to propagate as a spherical detonation in the unconfined space beyond the tube exit. For unstable mixtures (i.e., high activation energy) it was found that the critical tube diameter is about 13 times

378

J . H . S . Lee

FIGURE 17.33 Open shutter photographs illustrating the failure of detonation in mixtures with high activation energy (a) and low activation energy (b). (Courtesy A. Vasiliev.)

Detonation Waves in Gaseous Explosives

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FIGURE 17.34 Successful local reinitiation for mixtures with high activation energy (a) and continued propagation of detonation in unconfined space for low activation energy mixtures (b). 9 1995 AIAA.

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J.H.S. Lee

the cell size. This was first observed by Soloukhin and Mitrofanov,24 and the universality of this correlation was pointed out by Edwards et al. 25 Later, Knystautas et al. 26'27 carried out extensive experimental verification of this correlation for both fuel-oxygen and fuel-air mixtures. That there should exist a correlation between the cell size and the critical tube diameter can be explained as follows: Referring to Fig. 17.31, we see that the divergence of the stream tubes after the passage of the head of the expansion fan diminishes as the fan spreads out when it converges toward the tube axis. The adiabatic cooling of the shocked mixture depends on the area divergence. If we evoke some stability criterion (e.g., Schelkhin's criterion of A2/2 > 1 for failure) for the local explosion phenomenon in each stream tube, we see that there must exist a certain critical distance from the outer edge of the tube when the stream tube divergence becomes sufficiently small (i.e., A2/2 < 1) so that the reaction is not quenched by adiabatic cooling, and detonation reinitiates at this location. The maximum distance is the radius of the tube itself, because if the reinitiation does not occur when the expansion has penetrated to the tube axis, then the detonation would fail. For a universal correlation to exist, this critical distance (when scaled to the tube diameter) must be independent of the kinetics and energetics of the mixture. This requires that the reaction zone decouple abruptly from the diffracted shock so that the chemical reactions do not influence the diffraction processes (i.e., the diffraction is essentially the same as in a nonreactive mixture). In this case, the divergence of the stream lines is only a function of the distance from the axis and independent of the mixture. For high activation energy (i.e., unstable) mixtures, decoupling occurs abruptly at the head of the expansion fan and the local thermodynamic state there is essentially governed by the scaled distance from the axis as in nonreactive mixtures. Thus, we would expect a general critical diameter-cell size correlation to exist. The numerical value of dc - 1 3 2 depends on the criterion used for the initiation at the local stream tube itself. For stable mixtures (low activation energy) such as C2H2-O 2 with high concentration of argon dilution, the failure mode is different and hence the dc ~ 132 correlation cannot be expected to hold in this case. Indeed, Moen et al. 28 found that for C2H2-O 2 diluted with 75% argon, d c ~- 242 instead of dc ~ 132 for undiluted mixtures. Similarly, Shepherd et al. 29 reported that dc ~ 20 - 302 for the same C2H2-O 2 mixtures but diluted with 80% argon. It is known that with heavy argon dilution, the shock temperature is high, thus rendering the mixture stable with a highly regular cell pattern. For stable mixtures, the failure mode corresponds to an excessive curvature of the curved detonation front. It is more of a global phenomenon involving the entire curved detonation front than a local explosion event at a particular stream tube. The failure criterion for this case can be based on a velocity deficit of the attenuated curved front. Exceeding a certain critical value of the velocity deficit

Detonation Waves in Gaseous Explosives

381

will result in too low a shock temperature to permit coupling of the reaction front to the shock. Cellular instability (transverse waves) plays a minor role (if any) in the propagation of stable detonation. From this discussion of the critical tube diameter phenomenon, we can see that detonation limits are determined by the ability of the detonation to survive the perturbations created by boundary conditions and continue to reinitiate itself to sustain steady propagation. Thus, limits are specific for boundary conditions as stated as early as 1957 by Guenoche and Manson. 3~ It is unfortunate that this fact is still not recognized by all, and detonation limits data based solely on chemical composition are still being reported. Perhaps the true detonation limit is that of a spherical detonation where there is neither confinement nor boundary. The limits for spherical detonation are now governed by the chemical and gas-dynamic processes within the detonation structure itself. The experimental determination of the detonation limits of unconfined spherical detonation results in problems due to the difficulty involved in the initiation of the spherical detonation near the limits. In an unconfined geometry, almost all of the effective flame acceleration mechanisms are no longer operational. Thus, transition from spherical deflagration to detonation is practically never observed except for extremely sensitive mixtures such as acetylene with pure oxygen. 31 In general, spherical detonations are generated via direct blast initiation using a large, solid explosive charge or using planar detonation energy from a tube of sufficient diameter into an unconfined volume. Since the critical energy for direct blast initiation is proportional to the cube of the detonation cell size and the cell size increases rapidly as the limits are approached, the amount of explosive required for direct initiation becomes extremely large as the limits are approached. Thus, the detonation formed remains overdriven for a long distance from the ignition source; hence requiring that the scale of the experiments be very large. The variation of the critical initiation energy with chemical composition is typically a U-curve with the initiation energy increasing steeply as the lean and rich limits are approached. As an operational definition, one could fix an upper limit for the initiation energy and define the concentration limits for spherical detonation on the basis of the upper bound of the initiation energy chosen. This approach has been proposed by Bull, 32 and Fig. 17.35 illustrates the critical initial energy as a function of equivalence ratio for various fuel-air mixtures at atmospheric conditions. For an initiation energy of i g of tetryl (line AA), the concentration limits for hydrogen-air and acetylene-air mixtures are found to be 0.9 < ~ _< 1.23 and 1.10 < ~ _< 2.85, respectively. All the other fuels cannot be detonated with a charge of i g of tetryl. However, with 10 g of tetryl (line BB), the concentration limits for H2-air , C2H2-air , and C2H4-air are found to be 0.64 _ ~ _< 2.10, 0.65 _< ~, and 1.05 _< ~ _ 1.65, respectively. The alkane group of hydrocarbon fuels is outside the limits with

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X.

FIGURE 17.35 Detonability limits for spherical detonation in fuel-air mixtures as defined by the value of critical energy for direct initiation. (Courtesy D. Bull.)

10 g of tetryl. However, if the initiation energy is increased to 100 g of tetryl (line CC), the following are the limits for the fuels: H2-air, 0.48 < ~b < 2.88, C2H2-air 0.46 < q~ < oo, C2H4-air 0.75 < ~b < 2.35, C3Hs-air 0.96 < ~b < 1.35, and C2H4-air 0.86 < ~b < 1.45. Methane-air even at stoichiometric concentrations cannot be detonated with 100g of tetryl. Specifying the maximum initiation energy provides an operational definition of the limits of unconfined spherical detonations. This perhaps would be sufficient for practical purposes for hazard assessment. However, the fundamental physics of the limits of spherical detonation is not elucidated. From our previous discussions of the detonation structure, note that there exists a given repre-

383

Detonation Waves in Gaseous Explosives

sentative cell size for a given mixture. Thus, for a spherical detonation where the surface area increases as the square of the radius, the rate of growth of new detonation ceils must be able to cope with the rate of increase of the surface area for the averaged cell size to remain constant. The detonation will attenuate and fail if the cell size increases to about twice its normal dimension according to Schelkhin's criterion. For a small radius (i.e., small when compared to the cell size) where the curvature is large, the lateral expansion would quench the wave by lowering the reaction rate and slowing the growth of new cells. Hence, the spherical wave needs to be overdriven and the role of the initiation blast energy is to ensure that the decaying blast wave is sufficiently overdriven prior to the ability of the detonation to develop cells sufficiently rapid to cope with the surface area increase. At larger radii when the curvature becomes smaller and the spherical wave now approaches a planar wave, the normal transverse wave generation by instability would be able to maintain steady propagation. Thus, spherical detonation captures the essential mechanism of cell generation via instability. From a fundamental point of view, detonation limits, in essence, define the condition in which perturbations can grow fast enough to maintain the cellular structure necessary for rapid detonative combustion. Any initial and boundary conditions that influence the development and growth of the instability processes would alter the condition for self-sustained propagation of the detonation. Thus, detonation limits are not unique properties of the mixture only and are very specific of geometry, boundary, and initial conditions. The study of limits, however, can provide the understanding of the mechanism required for detonative combustion.

17.5

THEORY OF NONIDEAL

DETONATIONS

All real detonations are nonideal, in the sense that they deviate from the Chapman-Jouguet and one-dimensional ZND models. There are numerous definitions for nonideal detonations, and it is difficult to define unambiguously the extent to which a detonation has to depart from ideal behavior before it is referred to as nonideal. Hence, we shall adopt a general definition based on the criterion in which the steady detonation solution is obtained from the conservation laws rather than on the physical properties of the detonation wave. For ideal detonations, the steady-state solution is determined from the conservation equations of mass, momentum, and energy (for a control volume bounded between two equilibrium planes) together with an equation of state giving the internal energy of the explosive and products as a function of thermodynamic states. Since there are five unknowns and four equations, an additional equation is required to close the set of equations. This fifth equation is the Chapman-Jouguet criterion for ideal C-J detonations. Note that Chap-

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J . H . S . Lee

man and Jouguet's criteria are different even though they both refer to the same solution itself. In Chapman's theory,1 he recognized that the conservation laws permit only a unique solution when the Rayleigh line is tangent to the equilibrium Hugoniot. The tangency condition also corresponds to the minimum velocity solution. For velocities greater than the minimum, the Rayleigh line intersects the equilibrium Hugoniot at two points, thus giving two possible solutions: a strong or overdriven detonation (upper intersection point) and a weak detonation solution (lower intersection point). In accordance with experimental observations that a unique detonation velocity is usually observed for a given explosive mixture, Chapman simply adopted the minimum velocity or the tangency condition as a criterion to solve for the steady detonation solution from the conservation equations. Jouguet, 2 in his independent investigations, found that the tangency or minimum velocity solution also corresponds to a sonic condition at the equilibrium plane (relative to the shock front) at the rear boundary of the detonation wave. Thus, the sonic condition can also be used as an alternate criterion to iterate for the same minimum velocity solution. Above and below the tangency point on the Hugoniot gives subsonic and supersonic flows for the strong and weak detonation solutions, respectively. Only at the tangency (or minimum velocity) point is the flow sonic. Although the two conditions, i.e., tangency (or minimum velocity) and sonic condition at the equilibrium plane, both refer to the same solution, nevertheless, they are independent criteria when we proceed to iterate for the detonation solution from the conservation equations. Both criteria are referred to as the Chapman-Jouguet criterion, and the solution obtained based on it is known as the ChapmanJouguet solution. The use of the sonic condition for determining the detonation solution leads to a problem regarding the proper sound speed that should be used to define the sonic condition. In a reacting mixture, there are two possible sound speeds, depending on whether the reaction rates are so slow that the composition remains unchanged across the sound perturbation (frozen sound speed) or whether the reaction rates are so fast that equilibrium conditions can adjust immediately to the change of state across the sound wave (equilibrium sound speed). The use of the frozen sound speed in Jouguet's criterion leads to a weak detonation solution on the equilibrium Hugoniot (detonation velocity greater than the minimum but pressure less than the tangency solution). However, it is not possible to proceed from the strong to the weak detonation solution if the integral curve starts from the von Neumann state. Once it reaches the strong detonation solution on the equilibrium Hugoniot, continuation to the weak detonation solution along the Rayleigh line would violate the second law of thermodynamics. However, if the equilibrium sound speed is used in Jouguet's criterion, the minimum velocity or tangency solution is obtained. The appropriate choice for the sound

Detonation Waves in Gaseous Explosives

385

speed was extensively debated in the late 1 9 5 0 s 3-7 and eventually the conclusion was that the equilibrium sound speed should be the correct choice since that recovers the tangency solution on the equilibrium Hugoniot. The important thing to note is that the Chapman-Jouguet theory is based on the conservation laws for a control volume across the detonation relating conditions ahead of the wave (unreacted) to the conditions downstream (products) where the product gases are at equilibrium. The nonequilibrium reaction zone need not be considered. Thus, the C-J theory is based purely on equilibrium thermodynamics. Note that the C-J theory does not require the assumption of an infinitely thin reaction zone as is often erroneously stated in textbooks and publications. Since only the steady, one-dimensional conservation laws are used, initial and boundary conditions do not enter into the solution, and thus Chapman-Jouguet detonations are independent of the initial and boundary conditions, depending only on the properties of the explosive mixture. We shall define this unique solution (i.e., ChapmanJouguet) as the ideal detonation solution. Chapman and Jouguet's investigations predated, by almost half a century, the theory for the structure of the detonation wave by Zeldovich, 8 von Neumann, 9 and DOring 1~ of the early 1940s. The ZND model for the detonation structure consists of a leading shock front followed by a chemical reaction zone whose thickness depends on the kinetic rates of the chemical reactions. Whereas the C-J theory does not provide any information on the propagation mechanism for the detonation wave, the ZND theory implies that ignition of the explosive mixture is achieved via the adiabatic compression by the leading shock front. The temperature behind the shock increases further in the reaction zone as chemical energy is being released. However, the pressure decreases from a peak value behind the leading shock (i.e., the "von Neumann spike") as the products expand in the reaction zone. It is the expansion of the products that provides the work to sustain the propagation of the shock front, preventing its decay and thus maintaining a constant velocity of propagation. To determine the distribution of the gas-dynamic and thermodynamic states inside the reaction zone, we can integrate the differential form of the steady, one-dimensional conservation equations (referenced to the shock coordinates) together with the kinetic rate equations for the chemical reactions. This can be achieved by first determining the Chapman-Jouguet velocity from the global conservation laws across the entire detonation structure using the C-J criterion. Then, with the C-J velocity known, the so-called von Neumann state across the leading shock can be obtained from the Rankine-Hugoniot equation for a normal shock. With the von Neumann state as the initial condition, the differential form of the conservation laws can then be integrated until the reactions are completed and the equilibrium conditions are approached. Note that the final equilibrium state obtained asymptotically in the integration

386

J.H.S. Lee

should recover the same values determined from the C-J theory, since the correct von Neumann state is used in the integration. Thus, the structure is determined after the C-J detonation velocity has been found. Alternately, we could also determine the solution for the ZND structure without first using the C-J solution to find the von Neumann state to start the integration. Without assuming the C-J velocity, the detonation solution is not known a priori and we have to iterate for it via the the integration of the ZND equations for the structure. We first assume an arbitrary shock velocity, and hence the state behind the shock can be determined from the RankineHugoniot equations for a normal shock. Using this state as the initial condition, the ZND equations can then be integrated (together with the kinetic rate equations). Eventually, the integral will encounter the sonic condition and become singular (i.e., infinite gradients). By choosing different initial shock strengths, we can iterate for the particular solution that is regular when the sonic singularity is encountered. The regularity of the solution at the sonic singularity then serves as the criterion for the determination of the correct solution. Although the iteration for a singularity-free solution for the structure recovers the C-J solution, the criterion used for the iteration (i.e., regularity at the sonic singularity) is based on a mathematical rather than a physical requirement. Note that the three criteria that can be used to determine the steady detonation solution are all independent (i.e., Chapman's tangency or minimum velocity solution, Jouguet's sonic condition at the equilibrium plane, and the regularity requirement at the sonic singularity within the detonation structure).

It should be noted that using the steady conservation equations implicitly assumed the existence of a steady-state detonation solution. In practice, certain values of the properties of the explosive mixture (e.g., activation energy) may result in that no steady-state solution is possible. For example, stability analyses indicated that if the activation energy (for the case of a single-step Arrhenius reaction rate law) exceeds a certain critical value, the detonation is unstable and hence no steady-state solutions could be obtained. On the other hand, we could always determine a steady C-J detonation solution from the steady conservation laws or the corresponding ZND solutions for the detonation structure irrespective of the activation energy. However, if we perform a stability analysis on the ZND solution, we can determine the stability limits within which steady detonations could be realized. In other words, an additional stability analysis is required to establish the existence of stable detonations. Under certain initial and boundary conditions, the determination of steady detonation solution may even require the complete solution of the transient (i.e., reactive Euler) equations for the region bounded by the detonation and the near boundary. If steady solutions exist, then they will be achieved asymptotically from the time-dependent solution. G. I. Taylor 11 was

Detonation Waves in Gaseous Explosives

387

the first to point out that the existence of a steady-state detonation solution depends on the ability to match the nonsteady expansion flow of the detonation products to the steady boundary conditions at the detonation front and the near boundary. For a C-J wave, the sonic condition at the equilibrium C-J plane corresponds to a C + characteristic where its trajectory d x / d t - u 1 + c 1 in the x - t plane (the subscript "1" denotes the values at the equilibrium C-J plane). The C-J criterion requires D - u 1 + c 1, where D is the C-J detonation velocity. Thus the detonation trajectory coincides with the C + characteristic at the head of the rarefaction fan, and it is possible to match the nonsteady expansion flow of the products to the steady C-J detonation state for planar waves. However, for diverging, cylindrical, and spherical detonations, matching the centered rarefaction fan to a C-J wave resulted in a singularity in the form of an infinite expansion gradient at the CoJ plane. That created some doubts as to the existence of steady C-J diverging detonations 12-16 from the singular nature of the mathematical solution. Similarly, for the case of converging cylindrical and spherical detonations, no steady detonation solution compatible to the nonsteady flow behind the detonation could be obtained. Converging detonations are nonsteady and amplify with decreasing radius. Thus, the compatibility between the nonsteady flow of the products to the steady boundary conditions at the detonation front can also serve as a criterion for determining if a steady detonation solution is possible. For an ideal C-J detonation, the structure is described by the steady, onedimensional conservation equations (i.e., inviscid Euler equations without body forces). Departure from ideal behavior can be handled by additional source terms in the conservation equations. For example, for a curved detonation front, the fluid particles behind the shock front undergo an expansion due to flow divergence. Detonation waves in condensed explosives are practically impossible to confine, and the yielding confinement results in a flow divergence and hence a curved detonation front. Gaseous detonations propagating in a rigid tube are also curved because of flow divergence from the negative displacement thickness of the wall boundary layer. 17 For a small curvature, the two-dimensional flow divergence can be considered as quasione-dimensional by introducing a mass loss term in the continuity equation. Effect of friction, heat transfer, and body forces can also be analyzed by introducing appropriate momentum, heat loss, and dissipation terms in the conservation of momentum and energy equations. With the introduction of mass, momentum, and energy source terms into the ZND equations for the detonation structure to describe nonideal detonations, the steady-state solution can no longer be determined using the C-J criterion of ideal detonations. The remaining alternative is to evoke the mathematical requirement of a regular solution when the sonic singularity is encountered while solving for the detonation structure. Hence, it seems appropriate to define nonideal detona-

388

j.H.S. Lee

tions as those whose solutions must be determined from the equations for the structure where the nonideal effects can be accounted for by additional source terms in the conservation equations. It is of interest to discuss some salient features of nonideal detonation solutions arising from different source terms. Since almost all real detonations, in practice, are nonideal, their theoretical descriptions will most likely be achieved via the appropriate models for the source terms used to describe the nonideal behavior. It is neither feasible nor desirable to solve exactly the transient, three-dimensional Navier-Stoke equations in order to achieve a complete description of real, turbulent cellular detonations with various nonideal effects arising from boundary conditions. The quasi-steady, one-dimensional conservation equations for the ZND detonation structure in a reference frame moving with the shock can be written as d -~ (pu) = m d dx d

(17.3)

(pu 2 + p) - Dm - f 1

(~(p~ + p)) - ~ Dm 2 - Df + q

i u2 P ~ D2Q + e -- P(7 - 1) 2 '

(17.4) (17.5)

(17.6)

where we have assumed a perfect gas equation of state. In these equations, m, f, and q are the source terms in the mass, momentum, and energy equations, respectively, to account for departure from ideality due to curvature, friction, and heat losses. D is the detonation velocity, and the other variables have their usual meaning. The parameter 2 denotes the degree of reaction, that is, (2 -- 0 denotes an unreacted mixture while 2 = 1 represents fully reacted products. Note that the distance x is measured from the shock front and the particle velocity, u, is relative to the shock coordinates moving at velocity D. The set of ordinary differential equations can be combined to yield a single equation for the particle velocity,

du --

(7 - 1)[p2Q + q] + uL-Tu(D - u) + c2] + f[Tu - D(7 - 1)] p(c2 _ u2) ,

(17.7)

where c2 - 7PIP is the sound speed and 2 = d2/dt is the reaction rate. For an ideal detonation, all the source terms vanish (i.e., m - - f = q = 0), and Eq. (17.7) reduces to --

c2_u2

.

(17.8)

389

Detonation Waves in Gaseous Explosives

To proceed further, a reaction rate law has to be specified prior to the integration of Eq. (17.8). For a single-step Arrhenius reaction, we can write i =

A(1

-

-

,

(17.9)

where A and E (the preexponential factor and activation energy) are constants to be specified. For a given value of the chemical energy release Q, the C-J velocity D can first be obtained, and the von Neumann states, u 1, c 1 can then be determined from the normal shock equations. Equations (17.8) and (17.9) can then be integrated simultaneously from the shock front x = 0, where 2 = 0, u = u 1, c = c 1 until the equilibrium plane 2 - 1. Note that at the equilibrium plane for a C-J detonation, c 2 = u 2 and the denominator or Eq. (17.8) vanishes. However, the numerator also vanishes because ,~ = 0 when equilibrium is reached. Note that the C-J solution need not be determined first. We can alternately start with an arbitrary value for the shock velocity and then integrate Eq. (17.8) from the shock front backwards. When the sonic singularity c2 - u 2 is encountered, the numerator is now finite and thus d u / d x ~ oo. We can start again and choose another value for the shock velocity and iterate until we get the correct solution in which the numerator vanishes simultaneously as the denominator vanishes. This regular solution found via iteration should correspond to the C-J or minimum velocity solution when Eq. (17.8) for ideal detonation structure is used. Thus, we see that the use of the mathematical requirement of a regular solution at the sonic singularity can lead to a unique detonation solution that also corresponds to the classical C-J solution determined from the global conservation laws using the Chapman-Jouguet criterion. It is of interest to note that there exists a class of nonideal detonations where the detonation velocity is actually greater than the equilibrium C-J velocity. This nonideal effect can arise from the nature of the chemical reaction scheme itself. For example, if we consider the reaction to proceed from reactant A to product C via an intermediate species, B, and A --~ B is exothermic whereas B ~ C is endothermic, then the energy release profile will have a peak value in the nonequilibrium reaction zone prior to relaxing to a lower final equilibrium value. This energy "overshot" occurs if the exothermic reaction step A --~ B is much more rapid than the endothermic step B--~ C. For such a chemical system, the C-J criterion could always be used to determine an equilibrium detonation solution where only the equilibrium value of the energy release is used. However, if the ZND equation for the detonation structure is used instead, the steady detonation velocity obtained is found to be greater than the equilibrium C-J velocity since the solution could be based on a higher nonequilibrium value of the energy release. If the velocity is greater than the CJ value, then the solution now lies on the weak branch of the equilibrium

390

J.H.S. Lee

Hugoniot curve. However, it is now possible to reach this weak detonation solution from the von Neumann state without violating the second law of thermodynamics because the upper strong detonation solution is never encountered while proceeding from the Rayleigh line down to the final weak detonation solution on the equilibrium Hugoniot. Such detonations are referred to as pathological detonations and were first postulated by von Neumann. It was Zeldovich and Ratner 18 who first pointed out that the Hz-C12 system met the requirement for such pathological detonations. In the hydrogen-chlorine system, the exothermic reaction of H 2 + C12 ~ 2HC1 via the Nernst chain is competing against a much slower endothermic dissociation reaction of chlorine (i.e., C12 + M -+ 2C14-M). Pathological detonations have been studied both theoretically and verified experimentally by Dionne. z5 A good qualitative demonstration of the existence of pathological detonations can be obtained using the simple two-step reaction model suggested by Fickett and Davis19, that is, A -+ B exothermic and B --~ C endothermic with an Arrhenius law for both reactions: d21dt = A ( 1 - 2 1 ) e x p ( )~-~ E1 (17.10) dt = A1 (21 - 22) exp ~-~

The reaction progress variable, 21 and 22 can be related to the mass fraction of reactant A and product C, that is, 21 - 1 - XA, 22 -- Xc (where 0 < XA < 1 and 0 < X c < 1). The net chemical energy release is given by where Q -- 21Q1 4- ~,2Q2, where Q1 and Q2 denote the heat release in the exothermic and endothermic reactions. The differential equation for the ZND structure can now be written as du

dx =

(7 - 1)[2Q1 + 22Q2] c2 _ u2 .

(17.11)

This equation can be integrated for any chosen value of the shock velocity. The so-called "eigenvalue" or the singularity-free solution can be obtained by evoking the criterion that the desired integral curve is regular at the sonic singularity u - c (i.e., ~1~1 4- ~2~2 -- 0). The equilibrium C-J velocity will be based on a value of Qe = Q1 + Q2 since at equilibrium 21 = 1 and 22 = 1 (with Q2 being negative for the endothermic reaction, i.e., Qe = Q1 - Q2). However, when we integrate Eq. (17.11), we will reach the sonic singularity u = c prior to chemical equilibrium and the value of the heat release there, that is, Q* ~ Qe- The value Q* depends on the relative rates of the exothermic and endothermic reactions (i.e., their activation energies E 1 and E 2 in Eq. 17.10). For a very slow endothermic reaction, that is, E 2 >> E 1, Q*---> Q1 when u--+ c, the detonation velocity will then be

Detonation Waves in Gaseous Explosives

391

governed by QI" For a very fast endothermic reaction, there will be no "overshot," that is, E 2 ~ El, and complete equilibrium will be reached when u --~ c at the C-J plane. The detonation velocity is essentially the equilibrium C-J value. Thus, depending on the relative rates of the exothermic and endothermic reactions, the detonation velocity will be in between the equilibrium and the eigenvalue velocity based on just the exothermic value of QI" Hence, we see that the detonation velocity is not always dependent on the equilibrium value of the heat release as in ideal C-J theory. The correct detonation velocity must be determined using the mathematical requirement of a regular solution when the sonic singularity is reached in the reaction zone. The sonic plane is thus embedded within the reaction zone prior to chemical equilibrium for pathological detonations. Since the pathological detonation velocity is greater than the equilibrium CJ value, the state where the Rayleigh line finally intersects the equilibrium Hugoniot is now at the weak detonation branch. This is permissible since the upper intersection point is at a nonequilibrium state and does not correspond to the equilibrium, strong detonation solution. Using the steady ZND equation for the structure, that is, Eq. (17.8) or (17.11), we could always obtain a steady detonation solution. However, whether the solution obtained is stable or not has to be determined from separate considerations. The standard way to do this would be to carry out a further stability analysis of the steady ZND solution obtained. This kind of stability analysis was pioneered by Erpenbeck, 2~ and perhaps the most accessible treatment of the subject was given by Lee and Stewart. 21 For a single-step Arrhenius reaction rate model, stability analysis indicates that there exists a critical value of the activation energy about which the detonation is unstable, that is, the longitudinal, pulsating mode in one dimension or the transverse cellular mode in two (or three) dimensions. Alternately, the existence of a stable, steady solution could also be obtained by integrating the transient, reactive Euler equations for given initial conditions. If the detonation is stable, then the transient solution will asymptotically approach the steady-state solution as determined from the integration of the ZND equations for the structure (i.e., Eq. 17.8 or 17.11). Figure 17.36 shows the case of stable and unstable pathological detonations initiated directly by a strong blast wave. Figure 17.36a shows the asymptotic decay of the strong blast to a steady solution that corresponds identically to the solution from integration of Eq. (17.11). With a blast energy near the critical value, the initiating blast decays below the steady value and reaccelerates back to an overdriven detonation that subsequently relaxes to the same steady-state solution obtained from the ZND equation (i.e., Eq. 17.11). However, for a higher value of the activation energy (for the exothermic reaction, i.e., E 1), a nonsteady pulsating detonation solution is obtained. Stability analysis of pathological detonations has been carried out by Sharpe, 22'23 and it was also found that the stability limit

392

J. H. S. Lee 100

j

I

I

I

I

I

I

I

I

~ 80 r~

60 40

0 0

100

200 300 400 500 Normalized distance

0

600

,,,

7O 60 50 4o .N

30

z 10 I

0 0

I

I

I

.I

I

I

50 100 150 200 250 300 350 400 Normalized distance

F I G U R E 17.36 Asymptotic decay of initiating blast to steady pathological d et o n at i o n (a) and to an unstable, pulsating detonation (b). (Courtesy of J. P. Dionne)

is determined by the value of the exothermic activation energy in accord with the transient analysis. Pathological detonations provide a good demonstration that the classical equilibrium C-J solution does not always correspond to the correct steady-state detonation solution of a given explosive mixture. The detailed kinetic scheme of the reactions can even influence the steady detonation velocity. The C-J theory involves the conservation laws across a control volume bounded by the

393

Detonation Waves in Gaseous Explosives

shock and the equilibrium plane, and consideration of the processes within the control volume is not required. The C-J theory therefore failed to predict the correct steady-state solution for pathological detonation, since the structure has to be considered. On the other hand, the analysis of the ZND structure (using the mathematical requirement of regularity at the sonic singularity) can always permit a steady detonation solution to be determined. For the case of nonideal detonations where source terms are present, the steady detonation solution can only be determined from the integration of the ZND structure equations (e.g., Eq. 17.7). Nonideal detonations cannot be determined from the global conservation laws for a control volume bounded by two equilibrium planes and the C-J detonation. Nonideal detonations arising from friction and heat transfer were first considered by Zeldovich 8 in his pioneering study of detonation structure. We briefly discuss this case again to demonstrate that the source terms themselves can also influence the stability of the detonation other than just the activation energy. Furthermore, the presence of source terms can also lead to a multiplicity of possible steady-state solutions, again requiring additional stability analyses to arrive at the appropriate choice for a steady solution. The source term for friction suggested by Zeldovich is given by

f

=

kfPUabs]Uabs[

(17.12)

where uo~s is the particle velocity in the fixed reference frame and kf is the friction factor. An example of kf for a rough tube is given by Schlichting as + 1.74

(17.13)

where ks is the equivalent sand roughness and R is the tube radius. The absolute value of the velocity used in Eq. (17.12) ensures that friction is always opposing the flow. For the purpose of investigating the qualitative effect of friction, we shall assume kf to be a constant for simplicity, and we also ignore heat transfer. The ZND equation with friction alone, that is, m = q - - 0 , reduces to

where the source term is given by

f

=

-kfp(M-

u)[M-

ul.

Since the pressure now is no longer an algebraic function of the particle velocity when a source term is present in the momentum equations, it is

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J . H . S . Lee

necessary to integrate simultaneously the differential equation for the conservation of momentum together with Eq. (17.14), that is,

i dp

+

du

= -I

or

dp

du - -?M~-

7f.

(17.15)

Consider a simple, single-step Arrhenius reaction rate law. We write 2 as

i - a(1 - X) xp -)--d-

(17.16)

where the perfect gas equation of state has been used to express the exponent in the form shown above. Equations (17.14) to (17.16) form a coupled set of differential equations that can be integrated for a given shock velocity. The parameters to be specified are the values for ),, the friction factor kI, the heat release Q, the preexponential factor A, and the activation energy E. The eigenvalue solution is again determined by the mathematical requirement of regularity at the sonic singularity. Possible steady-state solutions as a function of the friction factor, kI (for fixed values of ?,, Q, A, E) can have an interesting multivalued behavior depending on the activation energy. For very low values of the activation energy, the steady detonation velocity, M, is found to be a single-valued function of kf, decreasing from the ideal C-J solution when kI = O, and approaching sonic velocity as kI increases (Fig. 17.37a). However, for larger values of the activation energy, the M versus kf curve can have a Z-shaped dependence (gig. 17.37b), giving rise to multiple values of M for a single value of kI. Referring to Fig. 17.37b, we first see that M decreases steadily from its C-J value when kI = 0 until it reaches some critical value M1~ for increasing values of kf (upper branch). No solution exists beyond this maximum value of the friction factor. However, for values of kI less than this maximum value, solutions are again possible (lower branch) until a second critical value corresponding to a certain minimum value of kI is reached. No solutions for smaller kI exist beyond this minimum value of kI. However, the solution can continue when kI is now increased again, thus giving the M versus kI curve a characteristic multi-value Z-shape. Such a Z-shaped curve was also found by Stewart and Yao24 in their study of the nonideal effect of curvature on the detonation velocity (in the study of Stewart and Yao24 on nonideal detonations, m -r 0, but f = q = 0). Thus, multivalue behavior appears to be possible for certain cases of nonideal detonations where source terms are present in the conservation equations for the detonation structure. With a multivalued

395

Detonation Waves in Gaseous Explosives

7

I

6i

I

"•5 ~4

2 3

zl i

[

0

0

,,

[

0.5 1 Normalized friction factor k. !

I

I

1.5

I

6 "~ 5

~4 o

b 3

.N o

z

1 0

I

0

I

I

I

0.01 0.02 0.03 0.04 Normalized friction factor kf

0.05

FIGURE 17.37 Variation of detonation velocity with friction factor (a) for stable, low activation energy, E J R T - - 22; and (b) for unstable, high activation energy, E a / R T - - 32. (Coutesy of J. P. Dionne)

solution, there exists the task of selecting the correct physical solution from the various possibilities. The existence of a maximum friction (or curvature) beyond which no solution can be found appears to be in accordance with experimental observations where detonation fails when the velocity deficit exceeds a certain critical value. However, only one steady detonation solution (or none) is generally observed in an experiment.

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With friction, it is interesting to note that there exists a critical value of the shock velocity, M c a , at the lower branch when the shock velocity is equal to the sound speed of the products. Since UreZ --C~ at the sonic plane, M c a ~ - c b implies that the absolute velocity (with respect to the fixed laboratory coordinates) vanishes, that is, uabs ~ M - ure l ---- M -- c b ~- O. In other words, the products are now at rest at the sonic plane. When the flow is at rest, the friction term f - - 0 and the numerator of Eq. (17.14) can only go to zero if ----0, and this can only occur when 2 - 1. Thus, the M - M c a solution corresponds to complete chemical reaction at the sonic plane as in an ideal C-J detonation. Since M c a - c b - ure l and 2 -- 1, an analytical solution can readily be obtained for Mca, that is, Mca -- v / 7 ( 7 - 1)Q 4- 1. It is interesting to note that this critical velocity is independent of the activation energy. Experiments in very rough and obstacle-filled tubes indicate that there exists a steady-state detonation velocity close to the sound speed of the products. However, it is not clear that the mathematical solution obtained for M c a does indeed describe the experimental observations, because the friction source term used grossly oversimplifies the complex physical processes of a real detonation propagating in a rough tube. It is also of interest to note that the M - McR solution represents a propagating constant-volume explosion, that is, the sonic plane of the detonation front corresponds to a moving boundary of an expanding constant-volume explosion. This can be seen from the fact that since the flow is at rest behind the detonation, the normalized specific volume is unity from the conservation of mass. As the mixture is stationary in front and behind the detonation front, the detonation then becomes a propagating front where the explosive mixture undergoes a constant-volume explosion as the detonation propagates. For M > Mca, the generalized C-J criterion can be used to determine the steady detonation solution, since a sonic plane exists within the reaction zone. However, if M < M c a , the shock velocity is now below the sound speed of the products. Thus, the solution no longer encounters a sonic singularity because the flow is subsonic throughout. Therefore, for M < Mca, the generalized C-J criterion can no longer be used determine an eigenvalue velocity for the detonation. To continue the solution beyond M < M c a , an alternate criterion was proposed by Dionne. 2~ Noting that for the case where M -- M c a the flow is at rest (i.e., M - Urel) when chemical equilibrium is reached (2 - 1), Dionne suggested that this condition could be used as a criterion to seek an eigenvalue detonation solution. The continuation of the Z-shaped curve below M ~ M c a in Fig. 17.37b is based on this criterion. Note that all these criteria used to

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determine the steady solution for nonideal detonations are based on mathematical considerations, and their validity has to be established via comparison with experiments. In a transient analysis, that is, numerical solution of the time-dependent reactive Euler equations, no criterion has to be imposed. If a steady-state solution exists, then it would be approached asymptotically from the transient solution. Furthermore, the selection of an appropriate solution in the case of a multivalue possibility as well as the stability of the solution can all be determined in a transient analysis. For the case of nonideal detonation due to friction discussed earlier, Dionne 25 has carried out a numerical solution of the time-dependent conservation equations. For the initial conditions in the transient analysis, Dionne considered that the detonation is initiated by an ideal blast wave. As in the ideal case without friction, he found that the stability of the detonation is also governed by the activation energy. However, friction tends to increase the effective activation energy and renders the detonation more unstable for the same value of the activation energy. With friction, a significantly lower activation energy has to be used before the initiating blast wave can decay to a steady, stable state solution asymptotically. If it is stable, then the steady solution obtained from the transient analysis is found to be identical to the one determined from the integration of the equations for the ZND structure. For M -- MCR,the asymptotic steady solution is also found to agree with the result from the steady ZND equations even though there is no longer a sonic singularity, and a different criterion has to be used to obtain the solution from the steady ZND equations. The transient analysis also indicates that the particle velocity (with respect to the fixed laboratory coordinates) behind the detonation vanishes and the velocity of the detonation front is equal to the sound speed of the products as predicted by the steady ZND solution. For larger values of the friction factor where M < Mca, the asymptotic steadystate solution from the transient analysis also agrees with the corresponding one from the steady ZND equation. Thus, for sufficiently low values of the activation energy where the detonation is stable (even with friction), numerical solution of the nonsteady equations gives an asymptotic steady solution identical to the one from the steady ZND equation, even though no criterion has been used to determine the solution. For higher values of the activation energy where the steady ZND equations led to multivalue solutions for the same friction factor, the selection of the appropriate solution now requires additional considerations (e.g., stability analysis). The transient analysis should naturally recover the correct steadystate solution asymptotically for large times. Since friction tends to increase the activation energy, it is found that the blast decays to a highly unstable, pulsating detonation. Whereas the ZND equations always yield a steady solution that requires further stability analysis to determine if the solution

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obtained is stable, the transient analysis automatically gives the unstable solution if conditions are outside the stability limits. Figure 17.38 shows the unstable pulsating detonation obtained as the initiating blast decays asymptotically. The three possible steady-state solutions determined from the ZND equation are also shown only for comparison. It is clear that if we solve the transient problem from given initial conditions, the true solution can be obtained, whether it is stable or not. However, the analysis of the ZND structure does not always lead to a unique solution or indicate its stability. To account for other nonideal effects such as curvature and heat transfer, additional source terms have to be added in the ZND equations. In general, the various source terms compete to drive the reacting flow behind the leading shock toward sonic conditions. The criterion used to seek the "eigenvalue" solution is simply a mathematical requirement that the solution be regular at the sonic singularity. Using the classical C-J criterion (Chapman's tangency solution or Jouguet's sonic condition at the equilibrium plane) for ideal detonations, the mathematical requirement of a regular solution at the singular sonic point is satisfied automatically since at chemical equilibrium the numerator of Eq. (17.8) vanishes when the denominator becomes zero as the flow becomes sonic. For nonideal detonations, the sonic condition no longer corresponds to chemical equilibrium; thus, the classical C-J criterion is no longer valid. However, if we retain the mathematical requirement of a regular solution at sonic singularity, then an eigenvalue solution can always be

80

....... I

I

I

I

I

I

7O 60

~ 5o ~ 40 ~ 30

F

~

10 0

.,i 0

200

I

I

400

600

I

I

I

800 I 0 0 0 1 2 0 0 1 4 0 0

Normalized distance FIGURE 17.38 Asymptoticdecay to pulsating chaotic detonation from the numerical solution of the transient reactive Euler equations. The three dotted lines denote the steady-state eigenvalue solutions from the ZND equation for the structure.

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obtained even with the presence of various source terms in the ZND equation for the structure (e.g. Eq. 17.11 or 17.14). The requirement of a singularity free solution is, in essence, a generalization of the classical C-J criterion and was thus formally referred to as the "generalized Chapman-Jouguet" criterion by Bdzi126 in his study of curvature effects on the propagation of detonation. Although the generalized C-J criterion can be used when a sonic singularity is present, there may be special situations where the flow is entirely subsonic. Such a special case was found by Dionne in his study of the effect of momentum losses on the propagation of the detonation. In a detailed examination of the integral modes of the steady-state solutions corresponding to various nonideal effects, Dionne 2~ found that they all possess a similar feature. This led him to propose a more general criterion for obtaining the eigenvalue solution from the steady ZND equations. Dionne proposed the criterion, "The desired detonation solution should be one that corresponds to the minimum detonation velocity that permits a regular solution." This criterion also encompasses the original C-J criterion since it also seeks a minimum velocity solution that is regular at the sonic point (which happens to be the equilibrium plane for ideal detonations). Analysis of the steady ZND structure does not address the question of stability. A transient analysis (i.e., solution of the reactive Euler equation for given initial conditions) will lead to the proper steady-state asymptotic solution. The need to consider the nonsteady flow behind the detonation front and seeking a detonation solution that can be matched to the transient flow behind the detonation was first recognized by G. I. Taylor 11 in 1950. However, almost all theoretical analyses since then have been focused solely on the detonation front. Numerical solution of the one-dimensional reactive Euler equations has become a fairly straightforward task. Inclusion of source terms to account for the various nonideal effects should not impose too many additional problems. Although real detonations are inherently three-dimensional, a onedimensional (in the direction of propagation) analysis with all the threedimensional effects described by appropriate source terms can provide an adequate description of the nonideal (real) detonation. This is in the same spirit as large eddy simulations for turbulent flows. Thus, the challenge in the theoretical description of nonideal detonations lies in the development of appropriate models (source terms) for the nonideal effects.

17.6 CONCLUDING

REMARKS

The instability of almost all self-sustained detonation waves has by now been demonstrated conclusively by the investigations of the past five decades. The one-dimensional, laminar ZND structure cannot describe the three-dimen-

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sional turbulent structure of real detonations. It is appropriate to conclude with a brief discussion of the influence of the unstable detonation structure on the further development of detonation theory. Regarding the prediction of the steady detonation state, the equilibrium C-J theory can always be used to provide a good approximation. Standard equilibrium codes (e.g., STANJAN 1 and CHEETAH 2) are available to facilitate the computation of the detonation states of most of the common explosives. In principle, the unstable structure should not have a large influence on the steady detonation state itself. This is because the C-J theory does not involve a consideration of the detonation structure. In fact, the C-J theory was formulated prior to the ZND model. The C-J theory is based on the steady, onedimensional conservation laws for a control volume bounded by two equilibrium planes normal to the direction of propagation. Thus, if the threedimensional fluctuations are small, or if they are damped out sufficiently fast to result in a steady one-dimensional flow at the downstream equilibrium plane, then the one-dimensional conservation laws for the control volume are still valid. If there is no heat loss in the reaction zone, then the downstream equilibrium state still will lie on the adiabatic equilibrium Hugoniot. A Rayleigh line can still be defined on the basis of the steady, one-dimensional conservation equations (i.e., continuity and momentum) if external friction effects are not significant. The state at the upstream equilibrium plane is always the initial state of the unbumed explosive. Also, the upstream boundary of the control volume does not necessarily have to represent a planar normal shock as in the ZND model. We may consider the three-dimensional cellular front (from the intersections of an ensemble of transverse shocks with the leading incident shocks and Mach stems) to be downstream of the arbitrary planar upstream boundary of the control volume. In that case, the Rayleigh line (whose slope is based on the steady, averaged velocity) no longer terminates at the normal shock Hugoniot that defines the von Neumann states of the ZND structure. However, the von Neumann state is not required in the equilibrium C-J theory. The conservation laws involve only the upstream and downstream equilibrium states. Hence, the solution will still follow the Rayleigh line from the initial state to a final state on the equilibrium Hugoniot for the products. Also, the C-J criterion can still be used to determine a unique solution. In applying the C-J theory to real detonations, we note that the assumptions involved are only that of a steady, averaged one-dimensional flow across a control volume bounded by two equilibrium planes for the upstream unbumed mixture and the downstream product gases. A minimum velocity or the tangency solution can still be obtained as in the original C-J theory for ideal detonations. The validity of this solution depends, as always, on how well it compares with experiments. Thus, even for real detonations, a unique tangency or minimum velocity solution can always be obtained on the basis of an averaged one-dimensional steady flow if

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the energy losses are negligible, so that the adiabatic equilibrium Hugoniot can adequately represent the locus of states for the products at the downstream boundary of the control volume. The three-dimensional transient condition at the cellular leading shock front only requires us to forsake the assumption of a planar normal shock for the upstream boundary. However, the von Neumann state is not used in the C-J theory in the first place. As long as we can assume a steady, averaged one-dimensional flow with negligible energy losses so that the downstream state is represented on the equilibrium Hugoniot for the products, a unique tangency or minimum velocity solution can be obtained. The merit of this solution will depend on its agreement with experimental observations, and in practice, it is found that the theoretical C-J velocity agrees well with experiments even for highly threedimensional, spinning detonations near the limits. The equilibrium C-J theory will continue to play a fundamental role in detonation theory to provide a first estimate of the detonation state of an explosive. To account for nonideal effects arising from interaction with the boundaries (e.g. friction, curvature), the structure of the reaction zone has to be considered. If we are seeking an averaged steady detonation state, the onedimensional steady conservation equations for the structure have to be used. In a quasi-one-dimensional model, the transient, three-dimensional, nonideal effects are represented by the use of source terms in the mass, momentum, and energy equations. However, the fluctuations in the nonequilibrium reaction zone of a real detonation may be very large. Hence, the quasi-steady, onedimensional flow approximation may not be valid inside the reaction zone. To assume quasi-steady flow in the reaction zone, the three-dimensional fluctuations have to be small to permit them to be treated as perturbations on a mean flow. In the previous discussion on the C-J theory, the downstream equilibrium plane can always be taken sufficiently far downstream so that the perturbations can decay. However, within the reaction zone itself, the steady, one-dimensional assumption now requires the three-dimensional fluctuations to be small throughout as compared to the mean flow properties. This then permits us to still write the steady one-dimensional equations for the structure to describe the mean flow field. The three-dimensional effects appear as perturbation on the mean, steady, one-dimensional flow. Since the integration of the differential equations for the structure requires the appropriate starting conditions, the von Neumann state has to be assumed at the upstream boundary of the control volume. The selection of the correct integral curve can now be based on the generalized C-J criterion, requiring that the solution be regular when the sonic singularity is encountered. For nonideal detonations, the sonic plane is also longer and corresponds to the equilibrium plane. Thus, in order to determine an eigenvalue detonation velocity, the assumption that the three-dimensional fluctuations are small has to be imposed. This assumption permits a mean-

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ingful averaged steady one-dimensional flow within the nonequilibrium reaction zone to be defined. Whereas the C-J theory requires only a meaningful averaged steady flow at the upstream and downstream boundaries of the control volume, the use of the ZND theory requires that the steady, onedimensional flow assumption be also valid locally throughout the nonequilibrium reaction zone (between the boundaries of the control volume). Thus, the prediction of the velocity from a consideration of the detonation structure is less accurate than just using the C-J theory, since more assumptions have to be met. It should be noted that since real detonations are also unstable, the local velocity of the front can have a large temporal fluctuation (e.g., 0.5 Vcj _< V < 1.6Vcj). The mean detonation velocity (averaged over local space and time variations) can have large, longitudinal variations in the direction of propagation near the detonation limits. However, the description of this very unstable, near limit behavior requires more than just a consideration of the detonation structure. The entire flow field behind the detonation must now be considered as well. In fact, the proper treatment of detonation should then involve the solution of the transient reactive Euler equations under the given initial conditions. The final steady detonation is then obtained asymptotically as time approaches infinity. No criterion will be necessary to arrive at the final steady state. In fact, the C-J criterion (or the generalized C-J criterion) is essentially a statement for smooth matching of the steady flow at the detonation front to the nonsteady flow in the rear of the sonic plane. Such a statement becomes superfluous if the entire nonsteady flow field is also computed. The future challenge in detonation theory, as with turbulence, lies in the development of subgrid models to handle the small-scale fluctuations. Wellestablished algorithms now exist for the accurate computation of the onedimensional (in the direction of propagation) reactive Euler equations for large-scale mean flow. Detailed chemistry does not pose any additional problems. The difficulty lies in the exact Navier-Stokes simulation of the compressible turbulence flow structure of the local cell dynamics of shockshock, shock-vorticity, and shock-density interface interactions. The DNS (direct Navier-Stokes simulation) of the microscopic processes at the subcellular level then permits the mesoscale modeling of intercellular dynamics and the interactions with boundaries to be described. The mesoscale models will then lead further to the development of models for the appropriate source terms to take into account all the nonideal effects in one-dimensional Euler equations for the macroscopic propagation of nonideal detonation fronts. A similar approach is now being adopted in condensed-phase detonations. Molecular dynamics are used for the description at the intra- and intermolecular level. This is followed by the mesoscale modeling at the crystal level, where the effects of deflects, dislocations, grain boundaries, and voids are being

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described. The mesoscale level serves as the intermediate step linking processes at the atomic scale to the detonation phenomenon at the macroscopic scale of continuum hydrodynamics. The current status of mesoscale modeling of heterogeneous reactive materials has been demonstrated by Baer. 3 The initiation of detonation is a relatively simpler problem to resolve than the turbulent detonation structure. In essence, it involves achieving a quantitative theory for the shock wave amplification in a gradient field (i.e., the SWACER mechanism). Whether it is DDT or direct initiation, the SWACER is the final event for the onset of detonation. In the predetonation stage of DDT, all the turbulent flame acceleration mechanisms are involved. This flame acceleration phase of DDT serves to bring about a sufficiently rapid turbulent mixing rate to result in the autoignition condition. The successful formation of a detonation subsequent to the local autoexplosion still requires the presence of a favorable gradient field of induction time for SWACER to occur. In direct initiation, the initial strong blast decays to the quasi-steady regime where the shock strength is close to the autoignition limit. An induction time gradient field is already generated by the transient shock decay process. Following a local fluctuation of reaction rate and the formation of an explosion center at the reaction zone, the onset of detonation is achieved when the shock from the local explosion event can be rapidly amplified via the SWACER mechanism. The SWACER process is essentially one-dimensional with the propagating shock in phase with the energy release wave. Numerical simulation of the direct initiation of one-dimensional planar detonation has also demonstrated the SWACER process to be responsible for the onset of detonation. Experiments and numerical simulations have also showed that cellular instability only appears during the decay of the overdriven detonation subsequent to its formation via the SWACER process. Thus, it appears that cellular instability only arises to provide the enhanced, turbulent mixing for a rapid reaction rate during the self-sustained phase of propagation of the detonation. Cellular instability does not appear to play a role in the initiation process. It is appropriate to comment on the definition of a detonation wave in view of the existence of an almost continuous spectrum of possible steady propagation velocity from the C-J value to the sonic velocity of the mixture under different conditions. It should be noted that this continuous spectrum of propagation speed is the consequence of the dominant role played by the boundary conditions in controlling the turbulent flow structure in the reaction zone. Significant momentum losses result in a large deviation from the C-J velocity, but steady propagation is still maintained by the strong turbulence generated by the rough boundaries. In the absence of strong boundary effects, the turbulence must arise from the intrinsic instability of the reaction front. In the absence of strong boundary effects in the structure, the detonation limits are now governed by the ability of the detonation to undergo cellular instability

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and develop the required turbulent burning rate to maintain the supersonic propagation of the combustion wave. In an unconfined, spherical detonation the limits are completely governed by the instability of the front. Thus, the selfgeneration of turbulence via cellular instability marks the distinction between deflagration and detonation. In smooth tubes (small wall effects) the detonation limits can be scaled according to the cell size. Also in spherical detonation, only the properties of the mixture define the limits. It should also be noted that when boundary effects do not control the propagation (e.g., smooth tubes) and natural cellular instability is responsible for generating the turbulence in the reaction zone, the velocity of the detonation seldom deviates significantly from its C-J value (e.g., <10%). Thus, one may define "true detonations" as those that could maintain their high turbulent burning rate via instability, and the class of supersonic wave phenomena controlled by boundary conditions (very rough walls and obstacle-filled tubes) should be referred to as "quasi detonations." Although this article is devoted entirely to a discussion of gaseous detonation, it is of interest to point out its differences from condensed-phase detonation. Because of the extreme detonation pressures in condensed explosives (e.g., 20-40 GPa), material strength becomes negligible and the detonation processes can also be described by the hydrodynamic equations as in gaseous detonations. Thus, the basic theory for condensed-phase detonations is essentially the same as that for gaseous detonations, differing only in the equation of state used to describe the products. The Chapman-Jouguet theory, which involves only the hydrodynamic conservation laws and equilibrium thermodynamics, is thus equally applicable to condensed explosives. Regarding the theory for the detonation structure itself, the ZND model based on the hydrodynamic conservation equations is still valid. However, for the ignition mechanism we must examine closely the chemical reaction processes, that is, initiation and propagation. Gas-phase reactions are well known and are based on collisions between the reacting molecules. Initiation begins with active species that are generated by thermal decomposition, and the reactions then propagate via chain branching processes. Since gaseous detonations are essentially supersonic turbulent deflagrations, the compressible turbulent mechanisms of shock-shock, shock-vortex, and shock-interface, as well as direct shock heating, all contribute toward the enhancement of transport and mixing to bring about the rapid reaction rate required for the supersonic propagation of the detonation. Thus, hydrodynamic transport processes play a dominant role in the reaction zone of gaseous detonations. In condensed-phase detonation, the reaction mechanism is relatively poorly understood. Collisional theory does not play an important role in the decomposition of the energetic molecules of most high explosives. The use of the standard exponential temperature-dependent Arrhenius rate law failed to

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describe adequately the reaction processes in condensed-phase detonations. This necessitates the addition of more and more empirical terms to the reaction rate laws in order to fit experimental data. That the reaction rate in condensed explosives is not dominated by temperature alone was demonstrated in a simple experiment by Dremin et al. 4 who found that, with identical detonation pressure, the time required for reaction in porous explosive charges filled with a liquid filler is equal to the reaction time in charges without the liquid filler. For the same pressure, the shock temperature is much lower for the case with the liquid filler. This suggests that temperature does not play the controlling role in the reaction, and that the mechanical process within the shock front itself is responsible for the decomposition of the explosive molecule. It has also been well established that the mechanical properties of the material, such as heterogeneity, voids, impurities, and grain boundaries, have a strong influence on the sensitivity of the explosive, that is, the relative ease in which the initiation of the chemical reactions can be effected. There is strong experimental evidence to demonstrate that mechanical processes can play an important, if not dominant, role in the initiation of chemical reactions in condensed-phase detonations. As early as 1940, Bridgman 5 had investigated the initiation of a number of solid explosives under high mechanical stresses of compression and shear. Although the results were inconclusive, a few of the explosives detonated under the influence of high mechanical stresses alone. In recent years, a number of Russian researchers have carried out similar studies on the initiation of explosive reactions in organic and inorganic compounds under high compression and shear. 6-11 They also concluded that detonations can be initiated at low temperatures by the action of high mechanical stresses alone. It is interesting to note that Edward Teller, 12 in the early 1960s, had attempted to provide a plausible explanation for the mechanical initiation of explosive decomposition of materials. He postulated that mechanical stresses can result in a lowering of the activation energy. The mechanical work associated with the compression and shear is assumed to contribute toward the production of an appropriate activated state that forms the lowest barrier between the reactant and product molecules, thus facilitating the decomposition of the reactant molecule. Recognizing that ignition can result from both thermal and mechanical deformation, a phenomenological model had been proposed by Benderskii et al. 1~ where the overall activation energy is the difference between the thermal and the mechanical energy of deformation. Thus, an increase in the mechanical deformation energy will lower the overall activation energy in accordance with the experimental observation of the existence of a critical pressure for ignition at normal temperatures. In the numerical study of Baer 3 on heterogeneous explosives, the chemical reaction is also assumed to be triggered by a critical pressure threshold as opposed to the normal temperature criteria. The possible role that mechanical processes play

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in the initiation of chemical reactions has a profound influence on the propagation mechanism of condensed-phase detonations, making them distinctively different from the mechanism of gas-phase detonations. Thus, even though the dynamic detonation processes are governed by the hydrodynamic equations and thermodynamics, the ignition processes are now controlled by the detailed structure and the mechanical properties of the material. Gaseous explosives are structureless and homogeneous. Hydrodynamic instability brought about the heterogeneous turbulent structure in reaction zone enhances transport, and hence reaction rate. Liquid explosives are also structureless and relatively homogeneous (even though heterogeneous molecular clusters can be present). Detonations in liquid explosives have been shown to possess a similar cellular structure to gaseous detonations. In solids, however, structure and heterogeneity are inherent in the material. Thus, the hydrodynamic instability mechanism may not be required for detonations in solid explosives. If material structure can dictate the mechanism in the reaction zone, then a universal theory for detonations in solid media may not be possible. The reaction's mechanism will be specific of the material properties itself and can be engineered into the explosives via its mechanical structure.

REFERENCES 17.1 INTRODUCTION 1. Mallard, E., and LeChatelier, H. "Sur la vitesse de propagation de rinflammation dans les melanges explosifs," C.R. Acad. Sci., Paris, Vol. 93, pp. 145-148 (1881). 2. Berthelot, M., and Vielle, P. "Sur la vitesse de propagation des phenomenes explosifs dans les gaz," C.R. Acad. Sci., Paris, Vol. 94, seance du 16 Janvier, pp. 101-108 (1882), seance du 27 Mars, pp. 822-823 (1882); Vol. 95, seance de 24 Juillet, pp. 151-157 (1882). 3. Chapman, D. L. "On the Rate of Explosion in Gases," Fluid. Mag., Vol. 47, 5th series, No. 284, pp. 90-104, January (1889). 4. Jouguet, E. "Sur la propagation des reactions chimiques dans le gaz,"J. Math Pures Appl., 6 ieme S~rie, tome 1, V.60 fasc. 4, pp. 347-425 (1905); tome 2, V.61 fasc. 1, pp. 1-86 (1906). 5. Zeldovich,Y. B. "On the Theory of the Propagation of Detonation in Gaseous Systems,"JETP, Vol. 10, pp. 542-568 (1940) (in Russian). Translated in National Committee for Aeronautics Technical Memorandum No. 1261, November (1950). 6. von Neumann, J. "Progress Report on Theory of Detonation Waves," Office of Scientific Research and Development, Report No. 549, May 4 (1942). 7. D6ring, W "The Detonation Process in Gases," Ann. Phys. (Folge 5, Bd. 43, No. 6-7, $421-436 (1943) (in German); "The Velocity and Structure of Very Strong Shock Waves in Gases," Ann. Phys. (Folge 6), Bd. 5, No. 3-5, $113-150 (1949) (in German). 8. Lee, J. H. S., Lee, B. H. K., and Knystautas, R. "Direct Initiation of Cylindrical Gaseous Detonation," Phys. Fluids, Vol. 9, pp. 221-222 (1966).

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9. Erpenbeck, J.J. "Stability of Idealized One-Reaction Detonations," Phys. Fluids, Vol. 7, p. 684 (1964). 10. Zaidel, R.M. "The Stability of Detonation Waves in Gaseous Mixtures," Dokl. Akad. Nauh. SSSR, Vol. 136, p. 1142 (1961). 11. Oppenheim, A.K., "Development and Structure of Plane Detonation Waves," 4th Colloquium on Combustion and Propulsion, Pergamon Press, pp. 186-258 (1961). 12. Fay, J.A. "The Structure of Gaseous Detonation Waves," 8th Symposium (International) on Combustion, The Williams and Wilkins Co., Baltimore, pp. 30-39 (1962). 13. Davis, W "The Detonation of Explosives," Scientific American, Vol. 256, pp. 106-112 (1987). 14. Strehlow, R., Combustion Fundamentals, McGraw Hill Book Co., New York (1984). 15. Fickett, W., and Davis, W, Detonation, University of California Press (1979). 16. Glassman, I., Combustion, 3rd Edition, Academic Press (1996).

1 7 . 2 THE STRUCTURE OF NONIDEAL DETONATIONS 1. White, D.R. "Turbulent Structure of Gaseous Detonations," Phys. Fluids, 4:465-480 (1961). 2. Denisov, Yu. N. and Troshin, Ya. K. "Pulsating and Spinning Detonation of Gaseous Mixtures in Tubes," Dokl. Akad. Nauk. SSSR (Phys. Chem.), Vol. 125, pp. 110-113 (1959). 3. Schelkhin, K. I., and Troshin, Ya. K. "Gasdynamics of Combustion," English translation: NASATT-F-23 (1964). 4. Voitsekhovskii, B. V., Mitrofanov, V. V., and Topchian, M. E. "Structure of a Detonation Front in Gases," Novosibirsk: 1XD-V0 Sibirsk, Otdel, Akad. Nauk. SSST (1963). English translation: Wright Patterson Air Force Base Rept. FTD-MT-64-527 (HW-633, 821) (1966) see also Fiz. Goreniya Vzryva 5, pp. 385-395 (1969). 5. Schott, G.L. "Observation of the Structure of Spinning Detonation," Phys. Fluids, Vol. 8, pp. 850-865 (1965). 6. Edwards, D. H. "A Survey of Recent Work on the Structure of Detonation Waves," 12th Symposium (International) on Combustion, Combustion Institute., Pittsburg, PA, pp. 819-828 (1969). 7. Strehlow, R. "Gas Phase Detonations: Recent Development," Combustion and Flame, Vol. 12, pp. 81-101 (1968). 8. Takai, R., Yoneda, K., and Hikita, T. "Studies of Detonation Wave Structure," 15th Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, pp. 69-78 (1974). 9. Libouton, J.C., Dormall, M., and Van Tiggelen, P.J. "Reinitiation Process at the end of a Detonation Cell," Progress in Astronautics and Aeronautics, Vol. 75, pp. 358-369 (1981); Libouton, J. C., and Van Tiggelen, P.J. "Influence of the Composition of the Gaseous Mixture on the Structure of Detonation Waves," Acta Astronautica, Vol. 3, 759-769 (1976). 10. Fickett, W., and Wood, WW "Flow Calculations for Pulsating One-Dimensional Detonations," Phys. Fluids, Vol. 9, pp. 903-916 (1966). 11. Abouseif, G. E. and Toong, T. Y. "Theory of Unstable Detonations," Combustion and Flame, Vol. 45, pp. 67-94 (1982). 12. Moen, I.O., Funk, J.W, Ward, S., Rude, G., and Thibault, P "Detonation Length Scales for Fuel-Air Mixtures," Progress in Astronautics and Aeronautics, Vol. 94, pp. 55-77 (1985). 13. Erpenbeck, J. J. "Stability of Idealized One Reaction Detonations," Phys. Fluids, Vol. 7, p. 684 (1964). 14. 14. Lee, H. I. and Stewart, D. S. "Calculation of Linear Stability: One Dimensional Instability of Plane Detonations," Journal of Fluid Mechanics, Vol. 216, p. 103-132 (1990).

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15. Taki, S., and Fujiwara, T. "Numerical Analysis of Two-Dimensional Unsteady Detonations," AIAA Journal, Vol. 16, pp. 73-77 (1978). 16. Oran, E.S., Boris, J.P., Young, T., Flanagan, M., Burks, T., and Picone, M. "Numerical Simulations of Detonations in Hydrogen-Air and Methane-Air Mixtures," 18th Symposium (International) on Combustion, Combustion Institute, Pittsburgh, PA., pp. 1641-1649 (1981). 17. Markov, V.V. "Numerical Simulations of the Formation of Muhifront Structure of Detonation Waves," Dokl. Akad. Nauk. SSSR, Vol. 258, pp. 314-317 (1981). 18. Gamezo, V., Desbordes, D. and Oran, E. S. "Formation and Evolution of Two-Dimensional Cellular Detonations," Combustion and Flame, Vol. 116, pp. 154-165 (1999). 19. Gamezo, V., Desbordes, D., and Oran, E.S. "Two Dimensional Reactive Flow Dynamics," Shock Waves, Vol. 9, pp. 11-17 (1999). 20. Oran, E. S., and Boris, J. P., Numerical Simulation of Reactive Flows, Elsevier, New York (1987). 21. Strehlow, R., Liangminas, R., Watson, R. H., and Eyman, J. R. "Transverse Wave Structure in Detonations," 11th Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, pp. 683-691 (1967). 22. U'lyanikskii, V. Yu. "Role of Flashing and Transverse-Wave Collisions in the Evolution of a Muhifrontal Detonation Wave Structure in Gases," Fizika Goreniya Vzryva, Vol. 17, p. 127 (1981). 23. Manzhalei, V.I. "Fine Structure of the Leading Front of a Gas Detonation," Fizika Goreniya Vzryva, Vol. 13, p. 3 (1977). 24. Moen, I. O., Sulmistras, A., Thomas, G. O., Bjerketvedt, D. and Thibault, R "Detonation Length Scales for Fuel-Air Explosives," Progress in Astronautics and Aeronautcs, Vol. 106, pp. 220-243 (1986). 25. Lee, J. J., Dupre, G., Knystautas, R., and Lee, J. H. "Doppler Interferometry Study of Unstable Detonations," Shock Waves, Vol. 5, pp. 175-181 (1995). 26. Oran, E. S., Young, T. R., Boris, J. P., Picone, J. M., and Edwards, D. H. "A Study of Detonation Structure: The Formation of Unreacted Gas Pockets," 19th Symposium (International) on Combustion, pp. 573-582, The Comb. Inst., Pittsburgh, PA (1982). 27. Oran, E. S., Young, T. R., Boris, J. R, Picone, J. M., and Edwards, D. H. "A Study of Detonation Structure: The Formation of Unreacted Gas Pockets," NRL Memo Rept. 4866 (1982). 28. Matsuo, A., and Inaba, K. "Cellular Structure of H2-air planar Detonation with Detailed Chemical Reaction Model," Symposium on Shock Waves, Japan 2000, March 16--18, Tokyo University, Bunkyo, Tokyo, pp. 385--388 (2000). 29. Shepherd, J. E. "Chemical Kinetics of Hydrogen-Air-Diluent Detonations," Progress in Astronautics and Aeronautics, Vol. 106, p. 263 (1998). 30. Lee, J. H. S., Soloukhin, R.I., and Oppenheim, A. "Currrent Views on Gaseous Detonations," Astronautica Acta, Vol. 14, pp. 565-584 (1969). 31. Vasiliev, A. A., Gavrilenko, T. P. and Topchian, M. E. "On the Chapman-Jouguet Surface in Multi-Headed Detonations," Astronautica Acta, Vol. 17, pp. 499-502 (1972). 32. Edwards, D.H., Jones, A.T. and Phillips, D.E. "The Location of the Chapman-Jouguet Surface in a Muhiheaded Detonation Wave," Journal Phys. D. Applied Physics, Vol. 9, pp. 1331-1342 (1976). 33. Lee, J. H. S. "Dynamic Parameters of Detonations," Ann. Rev. Fluid Mechanics, Vol. 16, pp. 311336 (1984). 34. Shepherd, J. E., and Tieszen, S.R. "Detonation Structure and Image Processing," Sandia National Laboratories Report SAND86-0033 (1986). 35. Lee, J. J., Frost, D. L., Lee, J. H. S., and Knystautas, R. "Digital Signal Processing Analysis of Soot Foils," Progress in Astronautics and Aeronautics, Vol. 153, pp. 182-202 (1992); Lee, J. J., Garinis, D., Frost, D. L., Lee, J. H. S., Knystautas, R. "Two-Dimensional Autocorrelation Function Analysis of Smoked Foil Patterns," Shock Waves, Vol. 5, No. 3, pp. 169-174 (1995).

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36. Kaneshige, M., and Shepherd, J.E. "Detonation Database," Technical Report, FM97-8 GALCIT, July 1997. Revised September 3 (1999). 37. Knystautas, R., Guirao, C. Lee, J. H. S., and Sulmistras, A. "Measurements of Cell Size in Hydrocarbon-Air Mixtures and Predictions of Critical Tube Diameter, Critical Initiation Energy, and Detonability Limits," Progress in Astronautics and Aeronautics, Vol. 94, pp. 23-37 (1984). 38. Gavrikov, A. I., Efimenko, A. A., and Dorofeev, S. B. "A Model for the Detonation Cell Size Prediction from Chemical Kinetics," Combustion and Flame, Vol. 120, pp. 19-33 (2000). 39. Dupre, G., Peraldi, O., Lee, J. H. S., Knystautas, R. "Propagation of Detonation Waves in an Acoustic Absorbing Walled Tube," Progress in Astronautics and Aeronautics, Vol. 114, pp. 248263 (1988). 40. Teodorcyzk, A. and Lee, J. H. S. "Detonation Attenuation by Foams and Wire Meshes Lining the Walls," Shock Waves, Vol. 4, pp. 225-236 (1995). 41. Lighthill, M.J. "The Effects of Compressibility on Turbulence" in Gasdynamics of Cosmic Clouds, IAV Sym. No. 2, Ed. by H. C. van de Hulst and J. M. Burgers, pp. 121-129 (1955). 42. Lee, J. H. S., Knystautas, R. and Frieman, A. "High-Speed Turbulent Deflagration and Transition to Detonation in H2-Air Mixtures," Combustion and Flame, Vol. 56, pp. 227-239 (1984).

17.3 INITIATION OF DETONATION WAVES 1. Mallard, E., and Le Chatelier, H. "Sur la vitesse de propagation de l'inflammation dans les m~langes explosifs," C.R. Acad. Sci., Paris, Vol. 93, pp. 145-148 (1881). 2. Berthelot, M., and Vielle, P. "Sur la vitesse de propagation des phCnomenes exposifs dans les gaz," C.R. Acad. Sci., Paris, Vol. 94, pp. 101-108 s~ance du 16 Janvier (1882); pp. 822-823 s~ance du 27 Mars (1882); Vol. 95, pp. 151-157 seance de 24 Juillet (1882). 3. Lee, J. H. S. and Moen, I. O. "The Mechanisms of Transition from Deflagration to Detonation in Vapour Cloud Explosion," Progress Energ. Comb. Sci., Vol. 6, pp. 359-389 (1980). 4. Shepherd, J. E. and Lee, J. H. S. "On the Transition from Deflagration to Detonation," in Major Research Topics in Combustion, eds. M.Y. Hussaini, A. Kumar, R.G. Voight, Springer Verlag, pp. 439-471 (1992). 5. Bollinger, L. E., Fong, M. G., and Edse, R. "Theoretical Analysis and Experimental Measurements of Detonation Induction Distances at Atmospheric and Elevated Initial Pressures," presented at the American Rocket Society 14th Annual Meeting, Washington, D.C., November 16-20 (1959). 6. Bone, W. A., Fraser, R. P., and Wheeler, W. H. "A Photographic Investigation of Flame Movements in Gaseous Explosions, Part VII," Phil. Trans. Roy. Soc. A, Vol. 235, pp. 29-68 (1936). 7. Brinkley, S. R. and Lewis, B. "On the Transition from Deflagration to Detonation," 7th Symposium (International) on Combustion, Butterworth Scientific Pub., London, pp. 807-811 (1959). 8. Urtiew, P., and Oppenheim, A. K. "Experimental Observations of the Transition to Detonation in an Explosive Gas," Proc. Roy. Soc. A, Vol. 295, pp. 13-28 (1966). 9. Meyer, J. W, Urtiew, P., and Oppenheim, A. K. "On the Inadequacy of Gasdynamic Processes for Triggering the Transition to Detonation," Combustion and Flame, Vol. 14, pp. 13-20 (1970). 10. Lee, J. H. S., Knystautas, R., and Yoshikawa, N. "Photochemical Initiation of Gaseous Detonations," Acta Astronautica, Vol. 5, pp. 971-982 (1978). 11. Knystautas, R., Lee, J. H., Moen, I., and Wagner, H. G. "Direct Initiation of Spherical Detonation by a Hot Turbulent Gas Jet," Colloquium on Fire and Explosion, pp. 1235-1245 (1978).

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12. Scarinci, T., Lee, J. H., Thomas, G. O., Bambrey, R., and Edwards, D. H. "Amplification of a Pressure Wave by Its Passage through a Flame Front," Progress in Astronautics and Aeronautics, Vol. 152, pp. 3-24 (1992). 13. Thomas, G. O., Sands, C. J., Bambrey, R. J. and, Jones, S. A. "Experimental Observations of the Onset of Turbulent Combustion Following a Shock-Flame Interaction," Proceedings of the 16th International Colloquium on the Dynamics of Explosions and Reactive Systems, Cracow, Poland, August 3-8, pp. 2-5 (1997). 14. Khokhlov, A. M., Oran, E. S., and Thomas, G. O. "Numerical Simulation of Deflagration to Detonation Transition: The Role of Shock-Flame Interactions in Turbulent flames," Combustion and Flame, Vol. 117, pp. 323-329 (1999). 15. Oran, E. S., and Khokhlov, A. M. "Deflagrations, Hot Spots and the Transition to Detonation," Phil. Trans. Roy. Soc. Lond. A, Vol. 357, pp. 3539-3551 (1999). 16. Khokhlov, A. M., and Oran, E. S. "Numerical Simulation of Detonation Initiation in a Flame Brush: The Role of Hot Spots," Combustion and Flame, Vol. 119, pp. 400-416 (1999). 17. Khokhlov, A. M., Oran, E. S., Chtchelkanova, A. Yu., and Wheeler, J. C. "Interaction of a Shock with a Sinusoidally Perturbed Flame," Combustion and Flame, Vol. 117, pp. 99-116 (1999). 18. Zeldovich, Ya. B., Librovich, V. B., Makhviladze, G. M., and Sivashinsky, G. L. "On the Development of Detonation in a Non-Uniformly Preheated Gas," Astronautica Acta, Vol. 15, pp. 313-321 (1970). 19. Yoshikawa, N. "Coherent Shock Wave Amplification in Photochemical Initiation of Gaseous Detonation," Ph.D. Thesis, Dept. of Mech. Eng., McGill University (1980). 20. Yoshikawa, N., and Lee, J. H. "Formation and Propagation of Photochemical Detonations in Hydrogen-Chlorine Mixtures," Progress in Astronautics and Aeronautics, Vol. 153, pp. 95-104 (1992). 21. Zeldovich, Ya. B., Gelfand, B. E., Tsyganov, S. A., Frolov, S. M., and Polenov, A. N. "Concentration and Temperature Non-Uniformities of Combustible Mixtures as Reason for Pressure Waves Generation," Progress in Astronautics and Aeronautics, Vol. 114, pp. 99-123 (1988). 22. Weber, H. J., Mack, A., and Roth, P. "Combustion and Pressure Wave Interaction in Enclosed Mixtures Initiated by Temperature Non-Uniformities," Combustion and Flame, Vol. 97, pp. 281295 (1994). 23. He, L., and Clavin, P. "Theoretical and Numerical Analysis of the Photochemical Initiation of Detonations in Hydrogen-Oxygen Mixtures," 25th Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, p. 34 (1994). 24. Dorofeev, S. R., Kochurko, A. S., and Chaivanov, B. R. "Detonation Onset Condition in Spatially Non-Uniform Combustible Mixtures," Proceedings of 6th International. Symposium on Loss Prevention, Vol. 4, pp. 22.1-22.19 (1989). 25. Khokhlov, A. M., Oran, E. S., and Wheeler, J. C. "A Theory of Deflagration to Detonation Transition in Unconfined Flames," Combustion and Flame, Vol. 108, pp. 503-517 (1997). 26. Montgomery, C. J., Khokhlov, A .M., and Oran, E. S. "The Effect of Mixing Irregularities on Mixed Region Critical Length for Deflagration to Detonation Transition," Combustion and Flame, Vol. 115, pp. 38-50 (1998). 27. Bartenev, A. M., and Gelfand, B. E. "Spontaneous Initiation of Detonations," Progress in Energy and Combustion Science, Vol. 26, pp. 29-55 (2000). 28. Peraldi, O., Knystautas, R., and Lee, J. H. "Criteria for Transition to Detonation in Tubes," 21st Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, pp. 16291637 (1986). 29. Lee, J. H. S. "Initiation of Gaseous Detonation," Ann. Rev. Phys. Chem., Vol. 28, pp. 75-104 (1977).

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30. Matsui, H., and Lee, J. H. "On the Measure of the Relative Detonation Hazards of Gaseous Fuel-Oxygen and Air Mixtures," Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, pp. 1269-1280 (1978). 31. Bull, D. C., Elsworth, J. E., and Hooper, G. "Initiation of Spherical Detonation in Hydrocarbon/Air Mixtures," Acta Astronautica, Vol. 5, pp. 997-1006 (1978). 32. Bull, D. C. "Concentration Limits to the Initiation of Unconfined Detonation in Fuel/Air Mixtures," Trans. IChem E, Vol. 57, pp. 219-227 (1979). 33. Atkinson, R., Bull, D. C., and Shuff, P. J. "Initiation of Spherical Detonation in Hydrogen/Air," Combustion and Flame, Vol. 39, pp. 287-300 (1980). 34. Zeldovich, Ya. B., Kogarko, S .M., and Semenov, N. N. "An Experimental Investigation of Spherical Detonation of Gases," Soviet Phys. Tech. Phys., Vol. 1, pp. 1689-1713 (1956). 35. Taylor, G. "The Formation of a Blast Wave by a Very Intense Explosion," Proc. Roy. Soc. of London A, Vol. 201, pp. 175-186 (1949). 36. Lee, J. H. S., Lee, B. H. K., and Knystautas, R. "Direct Initiation of Cylindrical Gaseous Detonations," Physics of Fluids, Vol. 9, pp. 221-222 (1966). 37. Lee, J. H. S. and Ramamurthi, K. "On the Concept of the Critical Size of a Detonation Kernel," Combustion and Flame, Vol. 27, pp. 331-340(1976). 38. Lee, J. H. S., Knystautas, R., and Guirao, C. M. "The Link between Cell Size, Critical Tube Diameter, Initiation Energy and Detonability Limits," Proc. of Specialist Meeting in Fuel-Air Explosions, ed. J.H. Lee and C.M. Guirao, University of Waterloo Press, pp. 157-187, (1982). See also Knystautas, R., Guirao, C. M., Lee, J. H. S., and Sulmistras, A. "Measurements of Cell Size in Hydrocarbon-Air Mixtures and Predictions of Critical Tube Diameter, Critical Initiation Energy and Detonability Limits," Progress in Astronautics and Aeronautics, Vol. 94, pp. 23-37 (1985). 39. Hayes, W. D. "On Hypersonic Similitude," Q. Appl. Math, Vol. 5, pp. 105-106 (1947). See also Anderson, J.D, Hypersonic and High Temperature Gas Dynamics, McGraw Hill, New York, pp. 117-137 (1989). 40. Higgins, A. J. "Investigation of Detonation Initiation by Supersonic Blunt Bodies," Ph.D. dissertation, University of Washington, Seattle (1996). See also Lee, J.H.S. "Initiation of Detonation by a Hypervelocity Projectile," A/AA, Vol. 173, pp. 293-310 (1997). 41. Higgins, A. J., Radulescu, M. I., and Lee, J. H. S. "Initiation of Cylindrical Detonation by Rapid Energy Deposition Along a Line," 27th Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, pp. 2215-2223 (1998). 42. Rodulescu, M. I., Higgins, A. J., Lee, J. H. S., and Murray, S. B. "On the Explosion Length Invariance in Direct Initiation of Detonation," 28th Symposium (International) on Combustion, The Combustion Instutute, Pittsburgh, PA (2000). 43. Mazaheri-Body, K. "Mechanism of the Onset of Detonation in Blast Initiation," Ph.D. Thesis, McGill University, Dept. of Mechanical Engineering (1997). 44. He, L. and Clavin, P. "On the Direct Initiation of Gaseous Detonation by an Energy Source," Journal of Fluid Mechanics, Vol. 277, pp. 227-249 (1994). 45. Edward, D. H., Thomas, G. O., and Nettleton, G. O. "The Diffraction of a Planar Detonation Wave at an Abrupt Area Change," Journal of Fluid Mechanics, Vol. 95, pp. 79-96 (1979). 46. Eckett, C. A., Quirk, J., and Shepherd, J. "An Analytical Model for Direct Initiation of Detonation," Proceedings of 21st International Symposium on Shock Waves, Springer-Verlag, Vol. 1, pp. 383-388 (1998). Also to be published in Journal of Fluid Mechanics.

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1 7 . 4 DETONATION LIMITS 1. Belles, E E., and Ehlers, J. G. "Shock Wave Ignition of Hydrogen-Oxygen-Diluent Mixtures near Detonation Limits," ARS Journal, pp. 215-220, February (1962). 2. Zeldovich, Ya. B. "On the Theory of the Propagation of Detonation in Gaseous System," Zh. Eksp. Teor. Fiz., Vol. 10, pp. 542-568 (1940). English translation: NACA TM.1261 (1960). 3. Guenoche, H., and Manson, N. "etude de l'influence du diam~tre des tubes sur la celerite des ondes explosives," Rev. Inst. Franc. Petrole, tome 9, pp. 214-220 (1954). 4. Peraldi, O., Knystauas, R., and Lee, J. H. "Criteria for Transition to Detonation in Tubes," 21st Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, pp. 16291637 (1986). 5. Wolanski, P., Kauffman, C. W, Sichel, M., and Nicholls, J.A. "Detonation of Methane-Air Mixtures," 18th Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA (1981). 6. Moen, I. O., Funk, J. W, Ward, S. A., Rude, G. M., and Thibault, P. A. "Detonation Length Scales for Fuel-Air Explosives," Progress in Astronautics and Aeronautics, Vol. 94, pp. 55-79 (1985). 7. Saint-Cloud, J. P., Guerrand, C., Brochet, C. and Manson, N. "Quelques Particularites des Detonations Tres Instables dans les Melanges Gazeux," Astronautica Acta, Vol. 17, pp. 487-498 (1972). 8. Schelkhin, K. "Instability of Combustion and Detonation of Gases, Soy. Phys. Usp., Vol. 8(5), p. 780 (1966). 9. Manson, N. "Sur la structure des ondes explosifs helicoidales," Compt. Rend. Acad. Sci., Vol. 222, p. 46 (1946). 10. Fay, J. A. "A Mechanical Theory of Spinning Detonations," J. Chem. Phys., Vol. 20, p. 942 (1952). 11. Dove, J. E., and Wagner, H. G. G. "A Photographic Investigation of the Mechanisms of Spinning Detonation," 8th Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, pp. 589-600 (1960). 12. Moen, I. O., Donato, M., Knystautas, R., and Lee, J. H. "The Influence of Confinement on the Propagation of Detonations near the Detonability Limits," 18th Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, pp. 1615-1622 (1981). 13. Dupre, G., Knystautas, R., and Lee, J. H. "Near Limit Propagation of Detonation in Tubes," Progress in Astronautics and Aeronautics, Vol. 106, pp. 244-259 (1986). 14. Kogarko, S. M., and Zeldovich, Ya. B. "On the Detonation of Gaseous Mixtures," Dokl. Akad. Nauk. SSSR, Vol. 63, p. 553 (1948). 15. Lee, J. H. "Dynamic Parameters of Gaseous Detonations," Ann. Rev. Fluid. Mech., Vol. 16, pp. 311-336 (1984). 16. Manson, N. Brochet, C., Brossard, J., and Pujol, Y. "Vibratory Phenomena and Instability of Self-Sustained Detonations in Gases," 9th Symposium (International) on Combustion, Academic Press, London, pp. 461-469 (1963). 17. Lee, J. J., Dupre, G., Knystautas, R., and Lee, J. H. "Doppler Interferometry Study of Unstable Detonations," Shock Waves, Vol. 5, No. 3, pp. 175-181 (1995). 18. Urtiew, P., and Oppenheim, A.K. "Experimental Observations of the Transition to Detonation in an Explosive Gas," Proc. Royal Soc. A, Vol. 295, pp. 13-28 (1966). 19. Teodorczyk, A., Lee, J.H., and Knystautas, R. "Propagation Mechanism of Quasi-Detonation," 22nd Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, pp. 1723-1731 (1988).

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20. Teodorczyk, A., Lee, J. H. S. and Knystautas, R. "The Structure of Fast Turbulent Flames in Very Rough Obstacle-Filled Channels," 23rd Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, pp. 735-741 (1990). 21. Dupr~, G., Peraldi, O., Lee, J.H., and Knystautas, R. "Propagation of Detonation Waves in an Acoustic Absorbing Walled Tube," Progress in Aeronautics and Astronautics, Vol. 114, pp. 248263 (1988). 22. Murray, S. B., and Lee, J. H. S. "The Influence of Yielding Confinement of Large-Scale Ethylene Air Detonations," Progress in Astronautics and Aeronautics, Vol. 94, pp. 80-103 (1985). 23. Murray, S. B., and Lee, J. H. S. "The Influence of Physical Boundaries on Gaseous Detonation Waves," Progress in Astronautics and Aeronautics, Vol. 106, pp. 329-366 (1986). 24. Soloukhin, R. I., and Mitrafanov, V. V. "The Diffraction of Multifront Detonation Waves," Soviet Physics Dokl., Vol. 9, No. 12, pp. 1055-1058 (1965). 25. Edwards, D. H., Thomas, G. O., and Nettleton, M. A. "The Diffraction of a Planar Detonation Wave at an Abrupt Area Change," Journal of Fluid Mechanics, Vol. 95, No. 1, pp. 79-96 (1979). 26. Knystautas, R., Lee, J. H., and Guirao, C.M. "The Critical Tube Diameter for Detonation Failure in Hydrocarbon-Air Mixtures," Combustion and Flame, Vol. 48, No. 1, pp. 63-83 (1982). 27. Knystautas, R., Guirao, C., Lee, J. H., and Sulmistras, A. "Measurements of Cell Size in Hydrocarbon-Air Mixtures and Predictions of Critical Tube Diameter, Critical Initiation Energy and Detonation Limits," Progress in Astronautics and Aeronautics, Vol. 94, pp. 23-37 (1985). 28. Moen, I. O., Sulmistras, A., Thomas, G. O., Bjerketvedt, D., and Thibault, P. A. "Influence of Cellular Regularity on the Behavior of Gaseous Detonations," Progress in Astronautics and Aeronautics, Vol. 106, pp. 220-243 (1986). 29. Shepherd, J. E., Moen, I. O., Murray, S., and Thibault, P.A. "Analysis of the Cellular Structure of Detonations," 21st Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, pp. 1649-1658 (1986). 30. Gu~noche, H., and Manson, N. "Effect of the Charge Diameter on the Velocity of Detonation Waves in Gas Mixtures," 6th Symposium (International) on Combustion, pp. 631-635 (1957). 31. Schelkhin, K. I., and Troshin, Ya. K. "Gasdynamics of Combustion," NASA-TT-F-23 (1964). 32. Bull, D. C. "Concentration Limits to the Initiation of Unconfined Detonation in Fuel-Air Mixtures," Trans I Chem E, Vol. 57, pp. 219-225 (1979). 33. Donato, M. "The Influence of Confinement on the Propagation of Near Limit Detonation Waves," Ph.D.Thesis, McGill University Dept. of Mechanical Engineering, Montreal, Quebec (1982).

17.5 THEORY OF NONIDEAL DETONATIONS 1. Chapman, D. L. "On the Rate of Explosion in Gases," Philos. Mag., Vol. 47, pp. 90-104 (1899). 2. Jouguet, E. "On the Propagation of Chemical Reactions in Gases," J. de Mathematiques Pures et Appliqu~es, Vol. 1, pp. 347-425 (1905). Continued in Vol. 2, pp. 5-85 (1906). 3. Wood, W, and Kirkwood, J. G. "Structure of a Steady Plane Detonation Wave with Finite Reaction Rate,"J. Chem. Phys., Vol. 22, pp. 1915-1919 (1954). 4. Wood, W, and Kirkwood, J. G. "On the Existence of Steady State Detonations Supported by a Single Chemical Reaction," J. Chem. Phys., Vol. 25, pp. 1276-1277 (1956). 5. Duff, R. E. "Calculation of Reaction Profiles behind Steady State Shockwaves I. Application to Detonation Waves," J. Chem. Phys., Vol. 28, pp. 1193-1197 (1958). 6. Wood, W, and Parker, F R. "Structure of a Centered Rarefaction Wave in a Relaxing Gas," Phys. Fluids, Vol. 1, pp. 230-241 (1958). 7. Wood, W, and Salzburg, Z. W "Analysis of Steady State Supported One-Dimensional Detonations and Shocks," Phys. Fluids, Vol. 3, pp. 549-566 (1960).

414

J . H . S . Lee

8. Zeldovich, Ya. B. "On the Theory of the Propagation of Detonation in Gaseous System," Zh. Eksp. Teor. Fiz, Vol 10, pp. 542-568 (1940). Also in NACA TM 1261 (1960). 9. von Neumann, J. "Theory of Detonation Waves," John von Neumann, Collected Works, Ed. A.J. Tanb, Macmillan, Vol. 6 (1942). 10. D6ring, W. "On Detonation Processes in Gases," Ann. Phys., Vol. 43, pp. 421-436 (1943). 11. Taylor, G. I. "The Dynamics of the Combustion Products behind Plane and Spherical Detonation Fronts in Explosives," Proc. Roy. Soc. A, Vol. 200, pp. 235-247 (1950). 12. Courant, R., and Frederiche, K., Supersonic Flow and Shock Waves, Interscience Pub., New York, p. 429 (1948). 13. Zeldovich, Ya. B., and Kompaneets, A.S., Theory of Detonations, Academic Press, New York (1960). 14. Manson, W and Ferrie, E "Contribution to the Study of Spherical Detonation Waves," 4th Symposium (International) on Combustion, pp. 486-494 (1952). 15. Lee, J. H. S., Lee, B. H., and Shanfield, I. "Two-Dimensional Unconfined Gaseous Detonation Waves," lOth Symposium (International) on Combustion, pp. 541-551 (1965). 16. Lee, J. H. S. "Gasdynamics of Detonations," Acta Astronautica, Vol. 17, pp. 455-466 (1972). 17. Fay, J. G. "Two Dimensional Gaseous Detonations: Velocity Deficit," Phys. Fluids, Vol. 2, p. 283 (1959). 18. Zeldovich, Ya. B., and Ratner, S. B. "Calculation of the Detonation Velocity in Gases," Acta Physicochimica USSR, Vol. XIV, No. 5, pp. 587-612 (1941). 19. Fickett, W., and Davis, W., Detonation, Univ. of Calif. Press, Chapter 5 (1979). 20. Erpenbeck, J. "Stability of Steady State Equilibrium Detonations," Phys. Fluids, Vol. 5, pp. 604614 (1962). 21. Lee, H. I., and Stewart, D. S. "Calculations of Linear Instability: One Dimensional Unstability of Plane Detonations," J. Fluid Mech., Vol. 216, p. 103-132 (1990). 22. Sharpe, G. J. "Linear Stability of Pathological Detonations," J. Fluid Mech., Vol. 401, pp. 311338 (1999). 23. Sharpe, G. J., and Falle, S. A. "One-Dimensional Non-Linear Stability of Pathological Detonation," submitted. 24. Stewart, D. S., and Yao, J. "The Normal Detonation Shock Velocity-Curvature Relationship for Materials with Nonideal Equation of State and Multiple Turning Points," Combustion and Flame, Vol. 113, pp. 224-235 (1998). 25. Dionne, J. P. "Theoretical Study of the Propagation of Nonideal Detonations," Ph.D. Thesis, Dept. of Mech. Eng., McGill University, March (2000). 26. Bdzil, J. B. "Steady State Two-Dimensional Detonation," J. Fluid Mech., Vol. 108, pp. 195-226

(1981).

17.6 CONCLUDING REMARKS 1. Reynolds, W. C. "The Element Potential Method for Chemical Equilibrium Analysis: Implementation in the Interactive Program STANJAN," 3rd Edition, Mechanical Engineering Dept., Stanford University (1986). 2. Fried, L. E. "CHEETAH 1.0 User's Manual," Energetic Materials Center, Lawrence Livermore National Lab., UCRL-MA-117541 (1994). 3. Baer, M. R. "Computational Modeling of Heterogeneous Reactive Materials at the Meso-Scale," in Shock Compression of Condensed Matter, ed. M.D. Furnish, L.C. Chabildas, R.S. Hixson, The American Institute of Physics, pp. 27-33 (1999).

Detonation Waves in Gaseous Explosives

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4. Dremin, A. N., Savrov, S. D., Trofimov, V. S., and Schevedov, K. K., Detonation Waves in Condensed Media, Nauka, Moscow, p. 171 (1970). 5. Bridgman, P.D. "The Effect of High Mechanical Stress on Certain Solid Explosives," J. Chem. Phys., Vol. 15, No. 5, pp. 311-313 (1947). 6. Enikolopyan, N. S., Mkhitaryan, A. A., Karagezyan, A. S., and Khzardzhyan, A. A. "Critical Events in the Explosion of Solids under High Pressure," Dokl. Akad. Nauk. SSSR, Vol. 292, pp. 887-890 (1987). 7. Enikolopyan, N. S., Mkhitaryan, A. A. and Karagezyan, A. S. "Explosive Chemical Reactions of Metals and Oxides of Salts in Solids," Dokl. Akad. Nauk. SSSR, Vol. 294, pp. 912-915 (1987). 8. Enikolopyan, N. S., Khzardzhyan, A. A., Gasparyan, E. E., and Voreva, V. B. "Kinetics of Explosive Chemical Reactions in Solids," Dokl. Akad Nauk. SSSR, Vol. 294, pp. 1151-1154 (1987). 9. Enikolopyan, N. S., Voreva, V. B., Khzardzhyan, A. A., and Ershov, V. V. "Explosive Chemical Reactions in Solids," Dokl. Akad. Nauk. SSSR, Vol. 292, pp. 1165-1169 (1987). 10. Benderskii, V. A., Filipov, P. Q., and Ovchinmikov, M. A. "Ratio of Thermal and Deformation Ignition in Low Temperature Solid Phase Reactions," Dokl. Akad. Nauk. SSSR, Vol. 308, pp. 401-405 (1989). 11. Benderskii, V. A., Misochko, E. Ya., Ovchinmikov, A. A., and Filipov, P. G. "A Phenomenological Model of Explosion in Low Temperature Chemical Reactions," Sov. J. Chem. Phys., pp. 1171182 (1982). 12. Teller, E. "On the Speed of Reactions at High Pressures," J. Chem. Phys., Vol. 36, pp. 901-903 (1962).

INDEX

A Acetylene pyrolysis, 14-22 A-factors, 176-177 Aliphatic hydrocarbons, 168-170 Allene dissociation, 16 Aromatic hydrocarbons, 168, 170-171 Arrhenius parameters, 122-125 Atomic absorption, 31-32 Atomic resonance absorption spectroscopy (ARAS), 4, 10, 22, 59, 61, 67 background information, 78-79 calibration procedures, 88-90 conclusions, 87-88 detection in shock tubes, 79-90 development of, 79-80 flash and/or laser photolysis in shock tubes, 90-98 light sources, 81-82 line absorption theory, 82-87

B Beer-Lambert law, 264 Beer's law, 80, 87, 88, 92, 94 Benzene dissociation, 17 Bethe-Teller linear relaxation equation, 39, 40 Bimolecular atom-molecule reactions, 92-94 Bimolecular radical-molecule reactions, 94 Bond cleavage, rate expressions for, 178-180 Bond energies, 177-178

C Chain branching, 226-229 ignition versus, 231-233 without, 229-230 Chapman-Jouguet (C-J) theory, 310-311,312, 383-399 Chemi-ions, 14 Chemkin-II, 136 Chlorobenzene dissociation, 17-18 Cis-trans isomerization, 175, 206-207 Comparative rate technique, 165-167 Complex refractive index, 271-273 Cooling effect, 238 Critical regime, 348 CTST theory, 96 Cyclopentadiene (CP), 21

D Decyclization, 175, 204-205 Deflagration to detonation transition (DDT), 339, 340-348 Deflagration wave, 309 Densitometric methods, shock tubes and atomic absorption, 31-32 background information, 29-30 direct initiation, 347-361 electron beam, 31 interferometry, 35-41 Rayleigh scattering, 32-33 refractive index, 33-67 schlieren techniques, 42-67

Handbook of Shock Waves, Volume 3 Copyright 9 2001 by Academic Press. All rights of reproduction in any form reserved. ISBN: 0-12-086433-9/$35.00

417

418 Densitometric methods, shock tubes and (continued) shadowgraph, 41-42 shock-front reflectivity, 34-35 Density functional theory, 19, 21, 22 Detonation waves in gaseous explosives acoustic absorbing walls, 373-375 autoignition process, 342, 344 cell size and tube diameter, relationship between, 377, 380-383 Chapman-Jouguet (C-J) theory, 310-311, 312, 383-399 computing detonation velocity, 309-310 conclusions, 399-406 development/evolution of, 309-313 failure modes, 375-377 from friction and heat transfer, 393-396 hot spot, 342 initiation of, 339-361 interferograms, 314 length scale, 329-332 limits, 361-383 nonideal detonations, structure of, 313-339 nonideal detonations, theories of, 383-399 numerical simulations, 322, 326-328, 356-357, 358 pathological, 390-393 reacceleration, 371-373 rigid, smooth, circular tube example, 363-364 rough/obstacle-filled tubes, 373 shock wave amplification by coherent energy release (SWACER), 344-345, 355-356, 358 smoked-foil/fish-scale technique, 314-317 transition/induction/run-up distance, 340, 363 two-dimensional, 321-322 types of unstable, near-limit behavior, 365, 367 unreacted pockets, 328-329 velocity time histories, 365-371 ZND model for detonation, 311,312, 361-363, 383, 385-386, 391-399 Diacetylene, decomposition of, 15-16 Diatomics interferometric measurements of vibrational relaxation and, 39 laser schlieren and reaction rates and, 59

Index laser schlieren and vibrational relaxation and, 50-51 Differential laser interferometer, 35 Diluents, effects of, 238 2,5-dimethylfuran, decomposition of, 137-148 sensitivity analysis, 148-152 Dispersion quotient (DQ), 267, 269-271

E Electron beam densitometry, 31 Endothermic rate, 58 Energy branching, 230-231 Epoxy group of molecules, ignition delay times, 236-238 Ethylene dissociation, 16 Euler equation one-dimensional reactive, 355 two-dimensional reactive, 321

F First-order decay constant, 92, 94 Flame ionization detector (FID), 119, 120 Flame modeling, 90 Flash and/or laser photolysis in shock tubes background information, 78-79, 90-91 bimolecular atom-molecule reactions, 92-94 bimolecular radical-molecule reactions, 94 results, 94-96 summary, 97-98 Free radical scavengers, 132-136 Furan decomposition, 20-21, 172-173

G Gas chromatography, 119-120 Gas chromatography-mass spectrometry (GC-MS), 120-122 Gas density, observation methods for atomic absorption, 31-32 Rayleigh scattering, 32-33 refractive index, 33-67 Gaseous explosives, detonation waves in. See Detonation waves in gaseous explosives Gear integration algorithm, 136 Gladstone-Dale constant, 33

419

Index

H Heterocyclic compounds, 168, 171-174 Hidden peaks, 120-122 Homogeneous nucleation of metal particles, 301-302 Hot spot, 342 Hydrocarbons, ignition delay time, 233-235 Hydrogen atom attack, 162-165, 176, 180, 209-210 Hypersonic blast wave analogy, 352-354

I Ignition delay times basic concepts, 212-215 computer modeling, 239-249 experimental measurements, 216 experiment design and data processing, 216-219 modeling procedures, 219-226 reaction scheme, 239-245 sensitivity analysis, 225-226, 246-249 summary, 249 Ignition delay times, kinetics systems additives, effects of, 239 categories of, 226 diluents, effects of, 238 energy branching, 230-231 epoxy group of molecules, example, 236-238 fuel concentration, role of, 233-238 hydrocarbons, example, 233-235 ignition versus chain branching, 231-233 loop concept, 226-229 without chain branching, 229-230 Induction time, 212, 213 soot formation and, 274-282 Integrated schlieren, 42 Interferometry, 35-41 reaction rate measurements, 40-41 vibrational relaxation measurements, 38-40 Isoquinoline, 174 Isotope labeling, 129-132 Isotopic exchange reactions, 10-12

L Landau-Teller temperature dependence, 40 Laser beam extinction (LEX), 12, 14 Laser-differential interferometer, 36-38 Laser Doppler anemometry, 273-274

Laser interferometry, 273-274 Laser light extinction and scattering, 263-271 Laser schlieren densitometry (LS), 30 advantages and description of, 43-49 convex profiles, 64-65 mass spectrometry and, 4, 10, 18-19, 20-21 reaction rates and, 57-67 vibrational relaxation and, 49-57 Light emission, 273 Line absorption theory, 82-87 Loop concept, 226-229 Lorenz-Lorentz formula, 33

M Mach-Zender interferometer (MZI), 35, 38-39, 59 Mass balance, 125-127 Mass selective detector (MSD), 119 Mass spectrometric methods, shock tubes and background information, 1-4 time-of-flight (TOF), 2, 4 time-of-flight, chemical kinetics, 10-22 time-of-flight, coupling of, 4-10 Mie theory, 266, 270 Molar refractivities, 33-34 Molecular elimination, 175, 202-203

N Nano-particle synthesis, 300-301 NIST-Kinetic Standard Reference Database, 137, 212, 244 Nitrogen/phosphorus detector (NPD), 119, 120

O O-atom recombination study, 31, 32 Organometallic compounds, 175, 208

P Particle size distribution function (PSDF), 261-263 Particulate formation and analysis See also Soot formation background information, 257-261 complex refractive index, 271-273 homogeneous nucleation of metal particles, 301-302

420 Particulate formation and analysis (continued) laser light extinction and scattering, 263-271 light emission, 273 nano-particle synthesis, 300-301 other detection techniques, 273-274 summary of, 302-303 PETN, 353, 354 Polyatomics interferometric measurements of vibrational relaxation and, 40 laser schlieren and reaction rates and, 59-67 laser schlieren and vibrational relaxation and, 51-57 Polycyclic aromatic hydrocarbons (PCAHs), 12 Pyrazine decomposition, 18-20 Pyridine decomposition, 18, 20, 174 Pyrimidine decomposition, 18, 20, 174 Pyrrole decomposition, 173-174

Q Quinoline, 174

R Rankine-Hugoniot equation, 385, 386 Raster electron microscopy (REM), 299-300 Rayleigh approximation, 266 Rayleigh scattering, 32-33, 265 Reaction path degeneracy (RPD), 22 Reaction rates interferometric measurements of, 40-41 laser schlieren and, 57-67 Refractive index methods, 33 interferometry, 35-41 schlieren techniques, 42-67 shadowgraph, 41-42 shock-front reflectivity, 34-35 Retroene reactions, 175, 201 Reynolds' analogy, 362-363 Rice-Ramsperger-Kassel-Marcus (RRKM), 10, 22, 61 Rotation-vibration (R-V) transfer, 54-55

S Schlichting's friction formula, 362 Schlieren techniques integrated, 42 laser, 4, 10, 18-19, 20-21, 30, 43-67

Index Sensitivity analysis, 148-152, 225-226, 246-249 Shadowgraph, 41-42 Shock-front reflectivity, 34-35 Shock tubes See also Densitometric methods, shock tubes and; Mass spectrometric methods, shock tubes and atomic resonance absorption spectroscopic detection in, 79-90 flash and/or laser photolysis in, 90-98 particulate formation problems and, 258 soot formation and, 258 Shock wave amplification by coherent energy release (SWACER), 344-345, 355-356, 358 Silicon carbide particle formation, 300-301 Silicon particle formation, 300 Single-pulse shock tubes (SPSTs), 4, 10-11, 20 background information, 108-109 configuration of, 109-112 limitations, 113-114 requirements, 112-113 summary and future directions, 180-181 validation, 115-117 Single-pulse shock tubes, chemical kinetics analytical methods, 119-121 Arrhenius parameters, 122-125 bond cleavage, rate expressions for, 178-180 bond energies, 177-178 cis-trans isomerization, 175, 206-207 complex (main processes), 193-197 complex (parallel isomerization), 198-200 decyclization, 175, 204-205 experimental approaches/configurations, 127-128, 154-165 gas chromatography, 119-120 general issues, 117-119 hidden peaks, 120-122 hydrogen atom attack, 162-165, 176, 180, 209-210 mass balance, 125-127 molecular elimination, 175, 202-203 organometallic compounds, 175, 208 product distribution, 122 retroene reactions, 175, 201 single-step, 174-180 treatment of data, 122-127 Single-pulse shock tubes, reaction systems A-factors, 176-177

421

Index aliphatic hydrocarbons, 168-170 aromatic hydrocarbons, 168, 170-171 background information, 128-129 complex, 168-174 computer simulation, 136-152 determination of reaction mechanisms, 129-136 experimental configurations, 154-165 free radical scavengers, 132-136 heterocyclic compounds, 168, 171-174 hydrogen atom attack, 162-165, 176, 180, 209-210 internal standards and comparative rate technique, 165-167 isotope labeling, 129-132 justification for thermal rate constants, 152-154 reaction schemes and modeling, 136-148 sensitivity analysis, 148-152 single-reaction studies, 152-167 unimolecular reactions, 159-161 Soot formation See also Particulate formation and analysis chemistry, 12-22 chlorinated hydrocarbons, effects of, 291 hydrogen, effects of, 290-291 induction time, 274-282 primary particles, 300 raster and transmission electron microscopy studies, 299-300 soot growth rate, 299 soot yield, 282-298 studies on, 259-260 Steady-state approximation, 11 Steady-state detonation solution, 386-387 Subcritical regime, 348 Supercritical regime, 348

T Thermal conductivity detector (TCD), 119 Thermal decompositions of hydrocarbon fuels and soot formation chemistry, 12-22

Time-of-flight (TOF), 2, 4 chemical kinetics, 10-22 chemi-ions, 14 coupling of, to a shock tube, 4-10 isotopic exchange reactions, 10-12 peak height ratios, 6-7 pyrolyses of hydrocarbons, 10 thermal decompositions of hydrocarbon fuels and soot formation chemistry, 12-22 Titanium nitride particle formation, 301 Toluene dissociation, 18 as a free radical scavenger, 132-136 Transmission electron microscopy (TEM), 299-300 Two-color method, 267, 269-270

U Unimolecular incubation, 64 Unimolecular reactions, 159-161

V Vibrational excitation mechanism (VEX), 11 Vibrational relaxation interferometric measurements of, 38-40 laser schlieren and, 49-57 Vibration-to-vibration (VV) transfer, 53-54, 57 von Neumann state, 385-386

X X-ray absorption, 31-32, 59

Z ZND (Zeldovich, von Neumann, and D/3ring), model for detonation, 311, 312, 361-363, 383, 385-386, 391-399