Detonation Waves in Gases

CHAPTER VIII

Detonation Waves in Gases

1. Introductory Remarks The earliest observations of detonation waves were made by Berthelot and Vieille and Mallard and Le Chatelier. While studying the propagation of flames in tubes they found that under certain conditions combustible gas mixtures propagate flames in tubes with speeds far greater than had been measured previously. The propagation reaches enormous velocities, from 1000 to 3500 m/sec, depending on the gas mixture, or many times the velocity of sound at ordinary temperatures and pressures. Detonation waves are shock waves which are sustained by the energy of the chemical reaction that is initiated by the shock compression. They develop from flames in tubes by coalescence of flame-generated pressure pulses into shock waves, and they propagate spherically in suitably strong mixtures when initiated by a small charge of high explosive. Their rate of propagation is limited by the rate at which a shock wave can travel, and hence it has been possible to develop the theory of propagation on the basis of hydrodynamics alone to such extent that detonation velocities may be computed from the physical properties of the explosive medium. There are, however, other aspects of detonation phenomena which are only partially related to hydrodynamic processes and thus are outside the scope of the classical theory. These comprise transition from flame to detonation, limits of detonability, and pulsation and spin of detonation waves. In addition, various problems arise from observations on the ignition of explosive gases by weak shocks, and from other incidental observations. We shall review these topics, beginning with an outline of the classical theory of shock waves and steady-state detonation waves. 1

2

3

2. Theory of Shock and Detonation Waves The theory of shock and detonation waves is largely founded on the work of Chapman, Jouguet, and Becker. The theory comprises two phases, namely, the formation of shock waves in a compressible medium, specifically a gas, and main­ tenance of the shock wave by the energy of chemical reaction. These will be considered separately. 4

5

6

532

533

2. THEORY OF SHOCK AND DETONATION WAVES A.

SHOCK WAVE IN A NEUTRAL GAS IN A TUBE

Becker has shown how the formation of a shock wave can be visualized in a simple manner. Consider a long tube closed at the left (Fig. 286) by a piston. A small velocity dw is imparted to the piston. This movement produces in the gas a weak compression wave that travels from left to right with the velocity of sound. At a given instant (b) the gas to the right of the wave front is unchanged and at rest, while the gas between the wave front and the piston is compressed adiabatically by an amount dp and has the velocity dw. The velocity of the piston is now increased by another increment dw, whereby a second compression wave is produced in the gas which follows the first (c). By frequent repetition of this procedure the velocity of the piston is brought to the final velocity w. Thus, there is produced within the mass of gas a step-like set of waves, the particles in the upper step having the velocity w (d). Concerning the further history of this set of waves it is noted that the upper steps of the terrace have a greater velocity than the lower steps. This is so since the temperature and, therefore, the velocity of sound, is larger in the gas in the upper steps and also since the gas itself has a higher flow velocity. Consequently, the individual steps will draw together in the manner of a telescope, and the wave front will become increasingly steeper (e andf). As soon as they are completely merged, a shock wave with an extremely steep pressure gradient is formed. After the piston has attained its constant velocity w, the further course of events is that a column of gas of ever increasing length is pushed ahead of the piston at the same velocity w; work must constantly be performed by the piston in order to compress and set in motion this lengthening gas column. The shock wave forms the head of this column and travels at a constant but greater velocity than the column of gas behind it. To an observer moving with the wave the gas enters it with a velocity u\ in the plane 1, and leaves it compressed at the smaller velocity u in the plane 2; the planes are normal to the flow in the undistributed and fully compressed medium, 2

TÎTTTTTH ΓΠΤΠ

EI FIG. 286. Becker's model of shock-wave formation.

534

VIII.

DETONATION W A V E S IN GASES

respectively, w is given by W

=

U\ —

(D

u . 2

If the piston were suddenly stopped, a zone of rarefaction would form at the face of the piston and gradually spread into the compressed gas ahead. The zone of rarefaction itself would have a definite front and would travel as a rarefaction wave pursuing the shock wave. It would finally overtake the latter, and both would travel forward together and ultimately degenerate to a sound wave. The hydrodynamic treatment of the foregoing problem of the piston moving with an initial acceleration to a final velocity allows a determination of the time required for the shock wave to build up. However, for the present purpose we are only interested in the stationary shock wave itself, that is, in the determination of the velocities ui and w and the pressure-temperature conditions within the wave. In the shock wave there appear entirely different relationships of temperature and pressure than those governing the usual adiabatic (isentropic) compression. Consider a unit mass of gas in front of the wave having the volume vi and pressure p before, and v and p after compression. The work done by the piston on the unit of mass is obviously P2(vi — v ), since after the establishment of the wave the pressure on the piston is always p . This work goes to increase the internal energy of the unit of mass by an amount ΔΕ and to impart to it the kinetic energy w /2, so that 2

x

2

2

2

2

2

ΔΕ =p (v 2

x

- v) - —· 2

(2)

If the unit of mass were compressed in the ordinary isentropic manner, for example by enclosing it in a cylinder and moving the piston against it sufficiently slowly so that the pressure throughout the gas is at each moment equalized and smaller than the pressure on the face of the piston by only an infinitesimal amount, the increase in the internal energy would be J*^ ρ dv. This is by no means identical with Eq. (2), the value of the integral always being smaller as will be seen by the following calculation. Consider a coordinate system moving with the wave front. Since the process is stationary, the masses, momenta, and energies of the gas passing through a plane in front of and behind the wave front must be equal. Considering the passage of a unit mass of gas through a unit of area, one obtains from fundamental hydrodynamics 1

(3)

(4)

E\ + — + p i v i

+ P2V2.

(5)

2.

535

THEORY OF SHOCK AND DETONATION WAVES

The equations represent the conservation of mass, momentum, and energy, respec­ tively, for the steady state. E and E are the internal energies, i.e., thermal energy content, of the unit mass before and after passage through the wave. Eq. (5) is derived from the energy theorem for flow where resistance occurs. In the present case the pressure rise in the shock wave offers resistance to flow. Under these conditions a part of the mechanical energy is converted irreversibly to heat. Eq. (5) is not applicable to flow in which the pressure and volume changes are isentropic (reversible). The symbols used here have been rather consistently used in the literature on this subject, so that their retention is advisable. By substituting values of «îand wifrom Eqs. (3) and (4) into Eq. (5), one obtains x

2

7

E - E = AE = i(Pi + Pi)(vi ~ v ). 2

x

2

(6)

This is the Hugoniot equation which for this type of compression replaces the integral / ' ρ dv of isentropic compression. Taking a perfect gas as an example, it can easily be shown that for the same volume change Eq. (6) will lead to a larger Δ £ and therefore a higher temperature. The physical interpretation of the mechanism by which the gas entering the wave front is compressed according to Eq. (6) and not according to the ordinary adiabatic relationship may be proposed as follows. It should be remembered that in isentropic compression the assumption is made that the compression takes place so slowly that the external force on the piston is only infinitesimally larger than the opposing force exerted by the pressure of the gas on the piston face. This will be the case as long as the piston velocity is small compared to the average molecular velocity and will therefore, in fact, be true for rather high piston velocities because molecular velocities are high. When the piston velocity becomes of the order of magnitude of molecular velocities, the degradation of the kinetic energy of the piston into random molecular motion, i.e. thermal energy, makes a significant additional contribution to the increase of internal energy of the compressed gas. The term Δ £ of Eq. (6) comprises both the isentropic compression energy and the energy gained by degradation of piston energy. For very small volume changes the Hugoniot equation reduces to the differ­ ential form of the isentropic equation, dE = —p dv. The latter equation is valid for piston velocities up to the velocity of sound. As the velocity is increased beyond sound velocity the difference between the piston force and the opposing force becomes large. In an actual shock wave the piston is represented by the column of compressed gas that moves in the direction of the shock wave. ν

ν 2

From Eqs. (1), (3), and (4) the velocity of propagation D of the shock wave into the gas at rest is found to be (7) and the velocity, w, of the gas behind the wave, often referred to as particle velocity,

536

VIII. DETONATION WAVES IN GASES

is

7

Pi — P\ Vi -

v

(8)

2

The temperature and pressure in the wave for any flow velocity w can now be determined for a perfect gas by introducing the gas law pv = nRT,

(9)

where η is the number of moles per unit mass and R is the molar gas constant, and the energy equation AE = c (T v

~ Γι),

2

(10)

where c is the average spécifie heat at constant volume between the temperatures Τ and T before and after passage through the wave front. From the above system of equations Table 1 has been calculated by Becker. The values of c used by Becker are not accurate at high temperatures, but the essential result is the same. In the next to the last column is given the temperature that should result from adiabatic compression alone. The last column is of interest in estimating the effect of a shock wave as it strikes an obstacle. This force (total impulse i) comprises the static pressure difference p — p and the force of the flow of the mass of gas behind the wave front, p w . With p = l/v and the value of w from Eq. (8), one obtains v

λ

2

v

2

x

2

2

2

2

i = (Pi - p 0 ( v i / v ) .

(11)

2

Therefore, the effect of the shock wave exceeds the static pressure difference by the factor Vi/v . 2

TABLE 1 SHOCK WAVES IN AIR FOR DIFFERENT FLOW VELOCITIES

A

T, adiabatic compression °K

P\

330 426 515 794 950 1,710 2,070

1.63 11.4 34.9 296 699 14,300 37,600

2

Pi

Vl

w,

P\

v

m/sec

D m/sec

T, shock wave, °K

2 5 10 50 100 1,000 2,000

1.63 2.84 3.88 6.04 7.06 14.3 18.8

175 452 725 1,795 2,590 8,560 12,210

452 698 978 2,150 3,020 9,210 12,900

336 482 705 2,260 3,860 19,100 29,000

2

T\ = 273°K.

a

2

i

537

2. THEORY OF SHOCK AND DETONATION WAVES

Becker investigated further the microscopic structure of the wave front itself, assuming that the processes of heat conduction and friction occur in the ordinary way. Obviously, the pressure and temperature gradients within the wave front cannot become infinitely large. After they have reached a certain steepness, the above processes will prevent the attainment of perfect discontinuity between the gases on either side of the wave front. The latter attains a certain width. Becker's calculation leads to numerical values of the width that already for small values of the velocity of the shock wave is of the order of the mean free path, and for higher though actually observable* velocities becomes even smaller than the average distance between two molecules. The equations of hydrodynamics based on continuum physics would not seem to be applicable to the problem, and the numerical results obtained by Becker, particularly for the higher velocities, would not appear to be real. However, recon­ sideration of the problem, taking account of the increase of viscosity and thermal conductivity with temperature, leads to the result that even the strongest shock waves in air are a few free paths thick. For numerical calculations of wave parameters, such as have been performed by Becker, the shock wave equations are reformulated in terms of the Mach number M = Die and the ratio of specific heats 7 = c /c . Here c is the velocity of sound, equal to Λ/yRT/m, where m is the molecular weight. 8

9

p

B.

v

DETONATION WAVE

We now shall consider that a shock wave is traveling in a medium of combustible gas. Chemical reaction is initiated in the shock front as described in Section 2, Chapter IV, for mixtures of hydrocarbons and oxygen, and if the shock temperature is sufficiently high, the reaction is completed within a distance from the shock front that is comparable to the diameter of the confining tube. This means that plane 1 lies in the unburned gas and plane 2 in the burned gas. In the following we shall not use the subscripts u and b as in Chapter V on combustion waves, but shall continue to use 1 and 2 as in the preceding section on shock waves. Obviously, Eqs. (3) to (5) remain unchanged in describing the flow of mass, momentum, and energy in the steady state. Therefore Eqs. (6)-(8) are directly applicable. However, it is noted that the energy difference given by the Hugoniot equation represents only the change in internal energy caused by compression, whereas now the temperature of the gas passing through plane 2 has been increased due to the energy AE released in the chemical reaction. Therefore, instead of Eq. (10), and still assuming a perfect gas, we shall write C

kE = c (T -T )-&E , v

2

l

c

* By setting off a charge of high explosive at the end of a tube filled with air.

(12)

538

VIII. DETONATION WAVES IN GASES

where c is now the mean specific heat of the burned gas at constant volume between T\ and T . If n is the number of moles per unit mass of the burned gas, v

2

2

pv 2

2

=

n RT . 2

(13)

2

For calculating the wave parameters, five equations are now available, namely, (6)-(8), (12), and (13); they contain six unknowns, namely, D, w, p , v , Δ £ , and Γ . Since it has been found experimentally that the detonation velocity is a constant for a given mixture, an additional relationship is required. The sixth relationship will be introduced after a consideration of the behavior of the system in a p-v diagram. From the Hugoniot Eq. (6) and after substituting Eqs. (12) and (13), the curve shown in Fig. 287 can be constructed, on which lie all pairs of values of p , v for a given pair of values p , v i , the latter being represented by point A. This curve is known as the Hugoniot or Η curve. Point G corresponds to a pressure p , obtained by burning the gas adiabatically in its own volume (vi = v ), that is, ΔΕ = 0. Point F corresponds to combustion at constant pressure with an increase in volume cor­ responding to the work done, — ΔΕ = —p(v — v ). In a neutral gas, that is, for ΔΕ = 0, and n = n , points G and F would merge with A. The values D and w corresponding to any pair of values p , v are obtained by the following proce­ dure. If α is the angle BAp , then according to Eqs. (7) and (8) and since tan α = 2

2

2

2

2

x

2

2

x

€

2

2

x

2

2

x

(Pi

~ P\)l(v

x

~

v ), 2

D = vi V t a n α

(14)

p

Pi

ν FIG. 287. Hugoniot curve.

2.

THEORY OF SHOCK AND DETONATION WAVES

539

and w = (vi — v ) V t a n a .

(15)

2

Hence, if a straight line* is drawn from some point /? , v on the H curve to the point A, the velocities D and w are found from the slope tan α = D lv\. It is seen that any such straight line, other than the tangent, intersects the Η curve at two points. With reference to the part of the curve above G it is noted that on decreasing the slope tan α from line ACB to the tangent AJ, the lower points of intersection with the Η curve move to higher pressures and smaller specific volumes whereas the particle velocity, w, increases in accordance with the larger momentum change on passage of the gas through the wave front. It is noted, however, that D decreases and attains a minimum value at the point of tangency. Conversely, the upper points of intersection move to lower pressures and larger specific volumes and the particle velocity decreases. On this part of the curve D likewise decreases toward the point of tangency. The lower part of the Η curve, FK, representing a decrease in pressure and increase in volume, corresponds to rarefaction. Here the flow velocity w is negative, that is, the burned gas no longer moves in the same direction as the wave, as is the case in detonation, but in the opposite direction. Therefore, this part of the curve should correspond to the propagation of an ordinary plane combustion wave. Indeed Eq. (14) may be written in the form 2

2

2

(16) Using the symbols of Chapter V, subscripts 1 and 2 become u and b, respectively, and 1/vi becomes p , l / v becomes p^, and D becomes S . Equation (16) then becomes M

2

u

which is Eq. (2) in Chapter V. Between G and F on the H curve, V t a n a is imaginary. Therefore, this region does not correspond to any actual process. The curve from G upward represents an increase in pressure and a decrease in volume corresponding to compression. Here, the flow velocity is positive, i.e., the burned gas moves in the same direction as the flame front. It is on this part of the curve that the point corresponding to actual detonation must lie. As an hypothesis, Chapman proposed that this is the point of contact, 7, of the tangent drawn from A. At this point J the adiabatic-compression relation dE = ρ dv obtains, as is readily verified by differentiating Eq. (6) and

* These lines are commonly termed Rayleigh lines.

540

VIII. DETONATION WAVES IN GASES

introducing the equation of tangency P2-P\ Vi -

v

= — dpldv.

(18)

2

It follows that at point J the ratio (p — P\)l(v\ - v ) corresponds to the ratio —dpldv for adiabatic pressure and volume changes in medium 2, such as occur in a sound wave traveling in medium 2. Therefore, at point J Eqs. (7), (8) and (17) yield 2

2

D = w + v V — dpldv — w + c , 2

(19)

where the term c is the velocity of sound in medium 2, that is, the burned gas. Eq. (19) furnishes the sixth relationship required for calculating the wave parameters. It shows that the detonation velocity D equals the sum of the particle velocity w and the sound velocity c in the burned gas. At points above 7, toward B, D would be smaller and at points below, larger than this sum. The state defined by point J is known as the Chapman-Jouguet state. A partial justification of Chapman's hypothesis, showing that a detonation wave in a state above point J would automatically change to a state corresponding to point J or lower, has been given by Jouguet. He pointed out that a rarefaction wave behind the wave would pursue the latter with a velocity equal to the sum of the sound velocity c and the particle velocity w. Therefore, at any point Β above J the rarefaction wave would overtake and weaken (slow down) the detonation wave. A rarefaction wave must form in a tube since there is no piston moving behind the wave to contain the gas expansion. Concerning the part of the Η curve below 7, the following consideration has been advanced by Becker. To a given value of tan α > —dpldv there correspond two values of the detonation velocity, one at C and the other at Β (Fig. 287). It can be shown that at Β the entropy of the gas is always larger than at C. Considering now that the burned gas at the moment of its formation will tend toward the state of greatest probability in the statistical mechanical sense, i.e., greatest entropy, one should conclude that the gas, in choosing between the two alternatives, will decide on the point B, so that the part of the curve below J will not correspond to an actual physical process. This leads to the total result that the detonation wave, being mechanically unstable above J and thermodynamically improbable below J, will only find it possible to travel with a speed corresponding to the point J. Becker's thermodynamic argument for the exclusion of points below 7, as well as alternative theories designed to justify Chapman's hypothesis, are obviated if one abandons the concept that chemical reaction is complete with the attainment of peak pressure, and that therefore a single Hugoniot curve suffices for describing the pressure and density changes in the detonation wave. Actually the reaction rate is finite and hence there must be a reaction zone of finite width with attendant temper­ ature and pressure gradients. This problem has been investigated independently by Zeldovich, von Neumann, and Doling. The three authors arrive at the common viewpoint that the front of the detonation 10

11

12

541

2. THEORY OF SHOCK AND DETONATION WAVES

wave is formed by steep pressure and temperature gradients, so that the fresh incoming gas is rapidly raised to a high temperature and pressure without appreciable chemical change. At the front of the detonation wave, therefore, the conditions resemble an ordinary shock wave traveling in a neutral gas. Chemical reaction is initiated by the high temperature and pressure, and the enthalpy release continues behind the shock front to some level of completion of chemical reaction. The gas flow in the reaction zone is of the type described by Eqs. (3)-(5); therefore, to each layer of the detonation wave there corresponds a Hugoniot curve; and the conditions inside the wave must be determined from consideration of the family of Hugoniot curves. An example of a family of Hugoniot curves calculated by Gordon is shown in Fig. 288 for a mixture of 20% H in air. Each curve corresponds to a different wave layer characterized by the fraction ξ of completion of reaction, ξ increases from 0 in the plane of unreacted gas to 1 in the plane of complete reaction. The curves are calculated from Eqs. (6) and (12) with values of Δ Ε corresponding to the fractions ξ. If the wave is stable, the various planes move with equal velocity D , and it follows from the linear Eq. (16) that the pressures p and volume ratios v /vi are determined by the intersections of the curves with a straight line drawn from the zero point (corresponding to initial conditions p , v/vi = 1) at the slope D /v . If, as assumed in Fig. 288, the line is tangent to the curve ξ = 1, there are two points of intersection 13

2

Ε

2

2

2

x

40 ι

1

x

1

1

542

VIII. DETONATION WAVES IN GASES

on each curve ξ < 1. One notes that there are apparently two alternative paths by which a mass element passing through the wave from ξ = 0 to ξ = 1 may satisfy the conservation laws embodied in Eqs. (6), (12), and (14) and at the same time change its pressure and volume monotonically not discontinuously, with distance of travel. The element may either enter the wave in the /?, ν state corresponding to the upper point of intersection on the curve ξ = 0, and then move through the sequence of states downward along the upper part of the straight line; or it may enter in the ρ, ν state corresponding to the lower point of intersection on the curve ξ = 0, and move upward through the sequence of states that end at the point of tangency at ξ = 1. However, the lower sequence of states starts with the zero point itself, corresponding to initial conditions of pressure, density and temperature where no chemical reaction occurs, whereas the upper sequence starts with a shock front of high temperature and pressure capable of initiating chemical reaction. This leads to the conclusion that the upper sequence represents the real physical changes occurring in the wave, and that the lower sequence is impossible. Thus, on entering the shock front the state of a mass element changes abruptly from zero to the upper point of intersection and then passes gradually through the upper sequence of states to the point of tangency at ξ = 1. It follows that between planes ξ = 0 and ξ = 1 the pressure decreases and the specific volume increases. At the same time, the tempera­ ture also increases. In the example of Fig. 288, the pressure in the wave front is about 23 atmospheres, the volume ratio v / v i is 0.20, and the temperature is calculated to be 1350°K. In the plane of complete reaction, ξ = 1, the pressure has decreased to 13 atmospheres while the volume ratio and the temperature have increased to 0.56 and 2425°K, respectively. The plane corresponding to the point of tangency of the straight line in the Hugoniot diagram is called the Chapman-Jouguet or C-J plane. Between this plane and the shock front lies the reaction zone in which the shock energy is generated by thermal expansion. The momentum equilibrium between any two planes 1 and 2 in this zone, where plane 1 is closer to the shock front than plane 2, is described by Eq. (16); it is seen to be exactly the same as in the ordinary combustion wave, where the pressure also rises in the direction from the burned to the unburned gas. However, a combustion wave has no stationary pressure front because it travels with subsonic velocity. On the other hand, if the rate of flame propagation has strong positive characteristics with respect to pressure, temperature, and turbulence, it is easily seen that in the confinement of a tube, where pressure pulses condense to a shock front, a combustion wave may generate a detonation wave. Behind the C-J plane the expansion process continues isentropically and a rare­ faction wave is formed which pursues the detonation wave. Jouguet's argument, previously given for the case of a single Hugoniot curve, now applies to states above the point of tangency on the curve for complete reaction, ξ = 1. Lower points along the curve are unattainable because, as explained above, the path to such points leads through the impossible lower sequence of states. Thus, the more realistic concept of 2

543

3. THE CALCULATION OF DETONATION VELOCITY

the detonation wave as a reaction zone of finite width provides a ready proof for the validity of Chapman's hypothesis. Brinkley and Richardson have extended the hydrodynamic theory by examining the more general case that the rarefaction wave enters the reaction zone itself, instead of being confined to the region of complete combustion behind the surface ξ = 1. They find that for any specified position of the rarefaction wave within the reaction zone a stable regime is preserved between the shock front and the front of the rarefaction wave; that is, the detonation wave continues to propagate, albeit at lower velocity. Of particular interest are their further conclusions that, if the chemical reaction is thus still proceeding in the rarefaction wave, a pressure pulse must eventually overtake its front and enter into the detonation zone where it increases the strength and velocity of the shock front. This may lead to an instability comprising alternate periods of acceleration and periods during which the rarefaction wave moves faster than the shock front, so that the wave slows down while more chemical reaction occurs and another pressure pulse is building up behind the wave; or it may happen that the alternate phases merge to the extent that transition to stable detonation results, with the Chapman-Jouguet state at ξ = 1. In the case of free three-dimensional expansion behind the wave, as in unconfined cylindrical charges of solid explosives, stable detonation is possible even with the Chapman-Jouguet state at ξ < 1, because the reacting rarefaction wave is stabilized by sidewise expansion; in such system the detonation velocity decreases with decreasing charge diameter, and below some critical diameter detonation does not occur. In the case of fully developed stable detonation, where the Chapman-Jouguet state is at ξ = 1, one can calculate the detonation velocity from the equations derived previously for a single Hugoniot curve representing the /?, ν states of the reaction products. Such calculations are presented below. 14

3. The Calculation of Detonation Velocity and Comparison with Experiment Numerical calculations, now generally programmed on computers, are usually made on the basis of a single Hugoniot curve for complete reaction, ξ = 1. Writing the differential equation for isentropic compression* in the burned gas,

dpi dv

Ί2Ρ2

(20)

2

*This follows from the basic equation ρ dv — — nC dT and the equation of state for a perfect gas, which in differential form becomes ρ dw + ν dp = nR dT, and further, 7 = C IC = ( C + R)IC , where C is the molar heat capacity and η the number of moles. v

P

v

V

V

V

544

VIII. DETONATION WAVES IN GASES

where 7 i s th e rati o o f specifi c heat s a t constan t pressur e an d volume , th e equation s given i n th e precedin g sectio n reduc e t o

4vV? -

/

2

c (T v

2

1\ Vi

ΠχΤχ

1 + - + - ^ = 0 , V 72/ v nTy 2

- r , ) - A Z ?

c

- | ^ - l ) ( n Pi _

νχη Τ

P\

ν η Τχ

2

2

D = - Vy n RT v

2

2

2

2

2

T

2

(21)

2

+ ηχΤχ ^

= 0,

(23)

2

λ

= - Vy p v .

2

(22)

v

2

2

2

(24)

2

2

w is given by Eq. (8). Eq. (24) states that the velocity of the detonation wave is v i / v times the velocity of sound in the burned gas. The known quantities in the equations a r e ρ χ , ν χ , Τ χ , η χ , and AE . The quantities c and 72 are found from tables of thermodynamic data. In case of thermal disso­ ciation the last quantities depend on pressure as well as temperature. The unknown quantities are p v , T , and D for which there are four equations. The quantity n is different from ηχ only in case of dissociation and is solved for by means of the usual thermodynamic equations together with values of c and 72. Eqs. (21)-(24) are solved for the four unknowns by trial and error. In case thermal dissociation can be neglected, a value of T is assumed and 72 for this temperature is detennined. These values for T and 72 together with the known values of n , n , and Τ χ are introduced into Eq. (21), which is solved for v i / v . The latter is introduced into Eq. (22), which is then solved to give an improved value of T . The calculations are repeated until values of v i / v and T are found that satisfy both (21) and (22). These are then used in Eqs. (23) and (24) to determine p and D, w can now be calculated from Eq. (8). The calculation requires inclusion of the various dissociation equilibria. To each equilibrium there corresponds an equilibrium constant. The dissociation, being a heat absorbing process, increases the specific heat, and since it leads to an increase in the number of moles, increases the specific volume. The first critical experimental test of the applicability of the Chapman-Jouguet theory was carried out by Lewis and Friauf on mixtures of hydrogen and oxygen diluted with various gases. The dissociation equilibria considered were 2

C

v

2y

2

2

2

v

2

2

{

2

2

2

2

2

2

15

H o ^ H + è0 2

2

2

H 0 ^ |H + O H 2

2

H ^ ± 2H . 2

Oxygen equilibri a wer e no t included , bu t thi s doe s no t significantl y alte r th e results . This als o applie s t o nitroge n equilibria . I n Tabl e 2 th e first colum n contain s th e

3.

THE CALCULATION OF DETONATION

545

VELOCITY

TABLE 2 COMPARISON OF CALCULATED AND EXPERIMENTAL DETONATION VELOCITIES OF MIXTURES OF HYDROGEN, OXYGEN, AND NITROGEN

Explosive mixture (2H (2H (2H (2H (2H (2H (2H (2H (2H (2H

2

2

2

2

2

2

2

2

2

2

+ + + + + + + + + +

0 ) 0 ) 0 ) 0 ) 0 ) 0 ) 0 ) 0 ) 0 ) 0 ) 2

2

2

2

2

2

2

2

+ + + + + -f + + +

Concentrations, percent burned gas

atm

T, °K

Calculated

Experimental*

OH

18.05 17.4 15.3 14.13 17.37 15.63 14.39 17.25 15.97 14.18

3583 3390 2970 2620 3367 3003 2685 3314 2976 2650

2806 2302 1925 1732 2378 2033 1850 3354 3627 3749

2819 2314 1922 1700 2407 2055 1822 3273 3527 3532

25.3 28.5 13.5 6.3 14.7 5.5 2.1 5.9 1.2 0.3

2

2

2

Detonation velocity, m/sec

0

10 30 50 1N 3N 5N 2H 4H 6H

2

2

2

2

2

2

2

2

2

H 6.9 1.8 0.2 0.07 3.3 0.9 0.2 6.5 3.0 1.1

"From Lewis and Friauf. p = 1 atm; T = 29ΓΚ. ''Data from Dixon, and Payman and Walls. 15

{

x

16

16

explosive mixtures obtained by adding the indicated number of moles of gas to a stoichiometric mixture of hydrogen and oxygen. The second and third columns contain the pressure and temperature in the wave. The sixth and seventh columns contain the calculated percentages of the radicals OH and atoms H in the mixture corresponding to equilibrium at the temperature T and the pressure p . Except for large excess of hydrogen the agreement between the calculated and experimental velocities is good. The influence of dissociation on the velocity is quite marked. If dissociation is neglected, the calculated velocities are considerably larger—of the order of several hundred meters per second in the high-temperature detonations. The equilibrium concentrations of radicals shown in the last two columns suggest that the chain-carrier concentration in the reaction zone must be very large indeed. This, together with the high temperature, provides an explanation for the extreme rapidity of the reaction. The chain-carrier concentration grows to values approxi­ mating those of the reactants themselves. Since at these temperatures virtually each collision is successful, one can understand that the reaction is completed within a few molecular free paths. In this connection it is interesting to note that according to earlier calculations of Lewis on a number of different mixtures, the molecular velocities of chain carriers attain values of the order of the detonation velocity. It is seen from Table 2 that dilution of the stoichiometric mixture with hydrogen results in an increase in the detonation velocity despite the reduced temperature T . This reflects the influence of the much reduced density of the mixture, Eq. (24) for 2

2

15

17

2

546

VIII.

DETONATION W A V E S IN GASES

the velocity showing that the latter is inversely proportional to the square root of the density of the burned gas. This suggests the performance of experiments with helium and argon as diluents. The addition of helium to a stoichiometric mixture should increase the velocity above that of the stoichiometric mixture because of the decreased density. On the other hand, the addition of argon should effect a decrease in the velocity because of the increased density. Therefore, one should find the rather remarkable result that two inert monatomic gases, differing in atomic weight but identical in their thermal effect, affect the detonation velocity in opposite directions. Since, for the same amount of inert gas added, Γ , y , and vi/v are the same, the velocity should be inversely proportional to the square root of the density of the burned gas. How well this is fulfilled can be seen from the results in Table 3 obtained by Lewis and Friauf. The experimental fact that dilution with the inert gas, helium, increases the detonation velocity provides a particularly clear confirmation of the theory. The thermodynamic data used by Lewis and Friauf were not the present-day bandspectroscopic data. Both the experiments and calculations have been repeated by Berets, Greene, and Kistiakowsky The agreement between the newer and older experimental velocities is, on the whole, rather satisfactory; and the calculations with the accurate thermochemical data have not introduced a discrepancy of more than 1 %. Figure 289 shows the percentage deviation of the experimental detonation velocities from the theoretical for the data given in Ref. 18. As with Lewis and Friauf's results, the experimental data are preponderantly below the theoretical. The differences are not large except for weak mixtures in the neighborhood of limits of detonability They appear to be a consequence of lateral loss not allowed for in the one-dimensional theory. This point will be discussed further. 2

2

2

15

18

TABLE 3 DETONATION VELOCITIES IN MIXTURES OF HYDROGEN, OXYGEN, HELIUM, AND A R G O N

E x p l o s i v e mixture (2H (2H (2H (2H (2H (2H (2H (2H (2H

2

2

2

2

2

2

2

2

2

+ -1+ + + + + + +

0 ) 0 ) 0,) 0 ) 0 ) 0 ) 0 ) 0 ) 0 ) 2

2

2

2

2

2

2

2

+ + + + + + + +

1.5He 3He 5He ( 2 . 8 2 H e + 1.18A) ( 1 . 5 H e + 1.5A) 1.5 A 3A 5A

P2»

D e t o n a t i o n velocity, m/sec

atm

Tt, °K

Calculated

Experimental

18.05 17.60 17.11 16.32 16.68 17.11 17.60 17.11 16.32

3583 3412 3265 3097 3175 3265 3412 3265 3097

2806 3200 3432 3613 2620 2356 2117 1907 1762

2819 3010 3130 3160 2390 2330 1950 1800 1700

From Lewis and Friauf. ρ, = 1 atm; 7, = 291°K. 15

A

4. +

547

MEASUREMENT OF DETONATION VELOCITY

2

-121

1

1

0

1

20

I

I

40

I

I

I

60

I

I

80

100

HYDROGEN,PERCENT

FIG. 289. Deviations of experimental detonation velocities from theory (Berets, Greene, and Kistiakowsky ). Triangles: H + 0 ; squares: H + 0 + He; circles: H + 0 + A . Solid symbols 10-cm pipe. Open symbols 1.2-cm pipe. 18

2

2

2

2

2

2

4. Measurements of Detonation Velocity; Limits of Detonability; Pulsating and Spinning Detonations For the measurement of the velocity of detonation Berthelot and Vielle developed a chronoelectric method whose principle is used even today for determining velocities in high explosive cartridges with refinements afforded by modern electronic tech­ niques. Probes are mounted in the tube at a measured distance apart which either make or break an electric circuit on arrival of the wave. The make system is made possible by the conductivity of the flame gas due to its ion content. The break system demands that the shock rupture a wire or metal foil. In the latter case it is necessary to construct identical probes so that the rupture process occurs in equal time intervals at both stations. In gases where the shocks are not excessively destructive piezoelectric gages set into the tube wall provide a signal of the pressure in the passing shock wave. A tourmaline crystal has been found to be most suitable for such work. With sufficient refinements very exact and reproducible velocities may be obtrained by such electric signal methods. The finer details of the process are explored by photographic methods. The pioneering work was done by Mallard and Le Chatelier, who obtained records of the wave on moving photographic plates. The insensitive emulsions of that period made it necessary for these investigators to work with very actinic mixtures such as C S + 6NO and C S + 3 0 . Later investigators used drum cameras in which a film moves at right angles to the direction of motion of the wave. An image of the wave is formed on the film. Figure 290 illustrates the principle of the method for 19

1318

20

2

2

2

548

VIII. DETONATION WAVES IN GASES

FIG. 2 9 0 . Diagram illustrating schlieren photographic record obtained from a pressure wave. C , drum; S, slot; L, lens; D, diaphragm; W, wave; M, concave mirror.

schlieren photography. In records of this type the velocity of the wave is determined from the slope of the track. In the case of a stable detonation wave the slope is constant. An equivalent photographic procedure is furnished by spreading the image of the disturbance on a stationary film by means of a rotating mirror. In general, mirror cameras have higher resolution than drum cameras because a small mirror can be rotated at a higher speed than a large drum. Schlieren photography also makes it possible to obtain still pictures of detonation waves in gases, by means of powerful light flashes of microsecond duration. Figure 291 is a typical photographic record of a stable detonation wave; it represents detonation in a 5 H + 0 mixture. Many records of this type for various mixtures 2

2

15

FIG. 291. Direct photograph of detonation wave (Lewis and Fraiuf ). Mixture composition, 5 H + 0 ; visible tube length, 4 7 cm; detonation velocity, 3 4 3 0 m/sec; particle velocity, 8 2 7 m/sec. 15

2

2

4.

549

MEASUREMENT OF DETONATION VELOCITY

have been published by Dixon and others. One observes that the detonation wave approaching from the top right moves at constant speed to the very end of the closed glass tube. Here a reflected wave is thrown back into the burned gas and travels along a brilliantly luminous path. The dark vertical lines are marked strips of paper on the tube placed at distances of 20 cm between the inner edges. Of interest are the striae, which are caused by luminous lead particles carried from the lead pipe that was attached to the glass tube at the right. The glass tube was found to be coated with a fine deposit of lead from previous explosions. It can be seen that as the flame enters the glass tube these particles move with great velocity. At the point of reflection of the wave their velocity drops to zero and increases again in the opposite direction in the path of the reflected wave. The movement of these particles as they emerge from the wave front should correspond approximately to the velocity w. Measurements by Lewis and Friauf show reasonably good agreement with the value of w calculated from the equations given above. The velocity of the particles falls off with time behind the wave front. Extensive measurements of detonation velocities have been made by numerous investigators. Tables 4 - 9 contain representative data. The change of detonation velocity with mixture composition is further illustrated in Figs. 292-297 for H - 0 , H -air, N H - 0 , C H - 0 , C H - 0 , C H -air. It is noted that hydrocarbon 2

2

21

2

3

2

3

8

2

2

2

2

2

2

TABLE 4 DETONATION VELOCITIES OF 2 H

Effect of temperature and pressure Temperature = 10°C Pressure, mm Hg Detonation velocity, m/sec Temperature = 100°C Pressure, mm Hg Detonation velocity, m/sec

+ 0

2

2

MIXTURES

1

200 2627

300 2705

500 2775

760 2821

1100 2856

1500 2872

390 2697

500 2738

760 2790

1000 2828

1450 2842

— —

Effect of added

gases

Moles of added gas

Rate of detonation of 2 H + 0 = 2821 m/sec Detonation Detonation Moles of velocity Moles of velocity added gas m/sec added gas m/sec

22

2

2H 4H 6H

3268 3527 3532

2

2

2

Effect of tube diameter Tube diameter, mm 2H + 0 2H + 4 0 2H + 0 + 3N

io 30 50

2

1N 3N 5N

2328 1927 1707

2

2

2

2

2

2

Detonation velocity m/sec 2426 2055 1815

23

2

2

2

2

2

2

2

9 2821 1927 2055

12.7

—

15 2828

1921

—

—

2089

550

VIII. DETONATION WAVES IN GASES TABLE 5 EFFECT OF TEMPERATURE ON DETONATION VELOCITY

OF C H + 2 0 MIXTURE 2

4

0

2

Temperature, °C Rate of detonation, m/sec

10 2581

100 2538

"From Dixon.

TABLE 6 EFFECT OF WATER VAPOR ON DETONATION VELOCITY OF 2CO

2

D r i e d with H S 0 and P 0 Dried with H S 0 Saturated with H 0 a t 10° a t 20° at 20° at 20° a t 28° a t 35° a t 35° a t 35° a t 45° at 45° a t 45° at 55° a t 65° a t 75° 2

2

4

2

5

4

0

a 2

Detonation velocity, m/sec

Pressure, mm Hg

P e r cent H 0

Condition

+

— —

760 760

1264 1305

1.2 2.3 2.3 2.3 3.7 5.6 5.6 5.6 9.5 9.5 9.5 15.6 24.9 38.4

760 400 760 1100 760 400 760 1100 400 760 1100 760 760 760

1676 1576 1703 1737 1713 1616 1738 1782 1570 1693 1742 1666 1526 1266

2

°From Dixon.

TABLE 7 DETONATION VELOCITIES IN ACETYLENE ALONE (98%

Pressure, kg/cm Detonation velocity, m/sec 2

°From Berthelot and le Chatelier.

5 1050

10 1100

12 1280

PURE)

15 1320

0

20 1500

30 1600

551

4. MEASUREMENT OF DETONATION VELOCITY TABLE 8 DETONATION VELOCITIES IN METHANE,

ACETYLENE,

AND CYANOGEN"

Hg

Velocities in m / s e c

+ |0, 2470 C H + 20 2581 C H + 10 2716 C N + 20 2321

CH + 2 0 2322(2146) C H + 30 2368 C H + |0 2391 C N + 30 2110

10°C and 760 m m CH + 2528 C H + 2507 C H + 2961 <ΛΝ + 2728 4

0

CH

2

2

4

0

2

2

2

0

2

2

0

2

4

4

2

4

2

2

2

2

2

2

2

2

2

4

2

2

2

2

2

2

2

"From Bone and Townend.

TABLE 9 DETONATION VELOCITIES OF VARIOUS MIXTURES AT ROOM TEMPERATURE AND ATMOSPHERIC

Mixture

Detonation velocity, m /sec

2H + 0 2CO + 0 CS + 3 0 CH + 2 0 CH + 1.50 + 2.5N C H + 3.50 C H + 30 C H + 20 + 8N C H + 1.50 C H + 1.50 + Nj

2821 1264 1800 2146 1880 2363 2209 1734 2716 2414

2

2

2

2

2

4

2

4

2

2

6

2

4

2

4

2

2

2

2

a

2

2

2

2

2

2

2

PRESSURE

0

Detonation velocity, m/sec

Mixture C H + 30 C H + 60 1-C4H10 +

40

3

8

2

3

8

2

2

i-C Hi + 8 0 C H + 80 C H + 8 0 + 24N C H + 7.50 C H + 22.50, C H OH + 3 0 C H OH + 3 0 + 12N 4

0

5

1 2

6

1 2

6

6

6

6

2

6

2

6

2

2

2

2

2

2

2

2

2600 2280 2613 2270 2371 1680 2206 1658 2356 1690

From Laffitte.

mixtures and C2N2 mixtures show maxima that are widely displaced from the stoichiometric composition for combustion to C 0 and H 0 ; instead they are fairly close to compositions corresponding to combustion to CO and H 0 . As shown in Figs. 292-297, there exist limits of detonability, analogous to limits of flammability, beyond which stable detonation is not observed. A number of such limits are listed in Table 10. In common with the steady-state theory of the combustion wave, the steady-state 2

2

2

552

VIII. DETONATION WAVES IN GASES 3,900

theory of the detonation wave does not predict the existence of limits. Information on the phenomenon is obtained from experimental work; and a theoretical under­ standing is offered by consideration of losses incurred as the reaction behind the shock front becomes slower with mixture dilution and the width of the reaction zone correspondingly increases. The degeneration of detonation in the neighborhood of such limits has been studied in some detail by Wendlandt. For the criterion of the establishment of a stable detonation wave he takes the constancy of the velocity as determined in the first and last portions of a long tube, using a stoichiometric mixture of hydrogen and oxygen as an initiating charge contained in an extension of the tube at a sufficient distance from the first point of measurement. With progressive dilution of a mixture, a point is reached at which there is an abrupt decrease in the velocity. For example, in hydrogen-air mixtures at atmospheric pressure the velocities are 1620, 1480, and 27

2,500

1,000 18.2 20 40 HYDROGEN,PERCENT FIG. 293. Detonation velocities of hydrogen-air mixtures (Breton ). 21

3,500

ο

I I

T

£ 3,000 or UJ CL co or

•

f f 2,500 ΰ ο

—I UJ

>

Ο

+

h-

< § 2,000 ι— LU

Ω

1,500 0

Y///W/À 20 25.4

40 A M M O N I A , PERCENT

70 75.4 8 0

60

FIG. 294. Detonation velocities of ammonia-oxygen mixtures (Breton ). Detonations are very minous outside the mixture ranges indicated by shading. 21

554

VIII.

k

j

0

D E T O N A T I O N W A V E S IN

GASES

[

r

C\J ι O 1 iD ι

+

3.1

10

+

l

00 1 χ 1 co 1 ο 1 16.6

ο ι C\J ι

oo ,

X

co.

o 20 22.2

PROPANE,

30

37

40

PERCENT

FIG. 295. Detonation velocities of propane-oxygen mixture (Breton ). 21

3,000

0 3.6

20

40 60 ACETYLENE, PERCENT

80

92

FIG. 296. Detonation velocities of acetylene-oxygen mixtures (Breton ). 21

100

4.

555

MEASUREMENT OF DETONATION VELOCITY

2,500

10

4.2

12.8

20

30

30.5

ACETYLENE, PERCENT

FIG. 297. Detonation velocities of acetylene-air mixtures (Breton ). 21

TABLE 10 LIMITS OF DETONABILITY*

Mixture H -0 H -air CO-0 , CO-0 , (CO + (CO + NH -0 CaH -0 2

2

2

moist well dried H )-0 H )-air

2

2

3

2

2

2

8

2

Î-C4H10—0

C C C C

2

2

4

4

2

2

H —0 H -air Hi O(ether)-O H O-air 2

2

2

0

10

"From Laffitte.

2

Lower limit, per c e n t fuel

Upper limit, per cent fuel

15 18.3 38

90 59 90 83 91 59 75 37 31 92 50 >40 4.5

— 17.2 19 25.4 3.2 2.8 3.5 4.2 2.6 2.8

556

VIII. DETONATION WAVES IN GASES

1050 m/sec for 19.6, 18.8, and 17.6% H , respectively. Below a certain percentage of hydrogen, the detonation wave is no longer stable, its velocity decreasing as it progresses along the tube. Below some other percentage of hydrogen, the flame, after traveling a considerable distance, which is dependent upon the composition of the mixture and the source of ignition, is extinguished, after which a shock wave travels with decreasing velocity through the explosive mixture without the occurrence of chemical transformation. In hydrogen-air mixtures the detonation limit is about 18.5% H , and the velocity of the stable detonation wave is 1250 m/sec; the limit at which the flame is extinguished after some distance is about 10% H . Figure 298 illustrates these results. The dashed lines correspond to shock waves generated by the same initiating charge as they would travel in the first and last parts of the tube if no reaction occurred. These curves were obtained by measuring the velocities of these waves in air in the same apparatus and correcting for the density of the rnixtures. For any mixture below the limit the steady-state theory predicts a value of the detonation velocity. These values are shown in Fig. 298. It is seen that a mixture of 2

2

2

700

500

3QOl 5

ι

.

•

ι

I

ι

ι

ι

ι

I

ι

ι

ι

ι

10 15 Hydrogen, percent

I 20

FIG. 298. Limit of detonability in hydrogen-air mixture (Wendlandt ). x Observations in first part of tube. · Observations in last part of tube. — · — Theoretical detonation velocity for complete reaction in the wave front. Shock waves in first and last part of tube without chemical reaction (explanation in text). 27

4.

557

MEASUREMENT OF DETONATION VELOCITY

even 5% H in air, that is, well below the conventional limit of flammability for downward propagation, should, according to the steady-state theory, propagate a detonation wave at a velocity of 900 m/sec, which illustrates the limitations of the theory Further studies have been carried out by Gordon, Mooradian, and Harper. These experimenters used a detonation tube 12 m long and 20 mm in diameter. The front 2-m section contained the driver gas, usually 2H + 0 . This initiating mixture was separated from the experimental section by a cellophane diaphragm. The pressure of the driver gas was usually made higher than that of the experimental mixture in order to overdrive the detonation. Both pressure limits and composition limits were found. As in Wendlandt's experiments, the overdriven detonation in an above-limit mixture settled down to a final steady-state velocity, but in a below-limit mixture it suddenly failed and degenerated into a simple shock wave without combustion. The composition of the limit mixture was not essentially altered by increasing the strength of the initiating shock. The mixtures used in this study comprised hydrogen with air, hydrogen with oxygen, and stoichiometric hydrogen-oxygen with argon and helium. The hydrogen percentages (or oxygen percentages) at the limit are plotted in Fig. 299 versus initial pressure. It is seen that, as the pressure is increased, the limit mixtures become weaker, i.e., more dilute, until finally a maximum dilution is attained which does not change with further increase of pressure. 2

28

2

2

The detonations were found to be mostly of the spinning type, a phenomenon that is always observed near limits of detonability and is described below. In some

Percentage hydrogen

Percentage hydrogen

FIG. 299. Limits of detonability in various hydrogen mixtures (Gordon, Mooradian, and Harper ). 28

558

VIII. DETONATION WAVES IN GASES

hydrogen-air rnixtures the detonation was found to be pulsating in the manner predicted by Brinkley and Richardson's theory. An effect decidedly indicative of the role played by chemical kinetics was found when it was discovered that traces of water vapor strongly inhibit the limits of these hydrogen-oxygen mixtures. Thus, a 15% H -air mixture was found to support stable detonation when its moisture content was reduced to 0.005% H 0 , but did not detonate stably when the moisture content was raised to 0.05% H 0 . The effect is suggestive of the reaction H + 0 + H 0 = H 0 + H 0 , which, as is known from the kinetics on the hydrogen-oxygen system, efficiently removes hydrogen atoms from the reacting system. Another fact was discovered when detonation theory was applied to compute temperatures in the reaction zone. It was found that for limit compositions in the region where the effect of pressure vanishes, the computed temperatures at the shock front are substantially the same, namely 1100°K for all hydrogen-oxygen-diluent systems investigated. This temperature evidently represents the initiating temperature which the rnixture must attain in order to react at sufficiently high rate. Although the limit mixtures differ widely in their contents of chemical enthalpy, they are similar with respect to the temperature characteristics of their reaction rates. It appears, therefore, that a shock front temperature of 1100°K is the minimum for establishment of a detonation wave in such hydrogen-oxygen-diluent mixtures. The observed composition limits are thus definitely linked to chemical-kinetic factors. The pressure limits have been interpreted by Zeldovich as being due to heat losses to the tube. The author estimates the width of the reaction zone behind the shock front from chemical-kinetic considerations and the rate of heat loss from heat transfer considerations. The heat loss results in a drop of the detonation velocity and thus lowers the temperature at the shock front by a given amount. The decrease of the shock-front temperature in turn lowers the reaction rate according to Arrhenius' law; this in turn increases the width of the reaction zone and hence increases the heat loss. At some critical rate of heat loss detonation becomes impossible. The author estimates that this condition obtains when the detonation velocity has dropped by some 10 to 15% below the ideal Chapman-Jouguet value. Data obtained by Gordon and associates show that in many cases the deviation from theoretical velocity in the neighborhood of the low-pressure limit does indeed lie in the 10 to 15% range. However, these authors point out that this result must be partly fortuitous because near-limit mixtures give spinning detonations which scarcely conform to the idealized model of Zeldovich. In any case the model is inadequate because it fails to take into account the fact that the width of the reaction zone is governed by the length of the induction period which principally determines the distance between the shock front and the ChapmanJouguet plane in which the reaction is completed. The induction period is an Arrhenius-type function of the shock temperature as described in Section 2 of Chapter IV for mixtures of hydrocarbons and oxygen. The shock temperature is diminished 2

2

2

2

2

2

2

10

28

4.

559

MEASUREMENT OF DETONATION VELOCITY

by friction loss at the tube wall which decreases the pressure pulses by which the reaction maintains the pressure and temperature in the shock front. The high activation energy of the Arrhenius relation between shock temperatures and induction period causes the induction period to be sensitive to changes of shock temperature, and this effect increases as the shock temperature is lowered by dilution of the mixture. Thus, there exists a critical dilution by which the shock temperature is progressively lowered and the width of the reaction zone progressively increased due to friction loss. In this way the detonation wave becomes extinct. The critical dilution evidently depends on various factors such as tube diameter and surface roughness. In the regime of stable detonation any chemical factor that causes the induction period to increase also causes the shock temperature to decrease due to increase of the width of the reaction zone and consequently increase of friction loss, thus causing the detonation velocity to decrease. This is illustrated by the decrease of the detonation velocity in C O - 0 mixtures with increasing dryness (see Table 6 and Chapter HI) and by the suppression of detonation in a 15% H -air mixture by a small addition of water vapor. The above considerations show in a general way that the detonation velocity decreases below the ideal value as the tube diameter is decreased. This effect has been investigated further by Fay who considers that the growth of the viscous boundary layer on the tube wall is principally responsible for generating a divergence of flow in the reaction zone of the detonation front, thus changing the propagation from a one-dimensional to a two-dimensional problem. The author has developed this concept into a quantitative theory which reasonably accounts for the difference between measured and ideal detonation velocities in the few mixtures investigated. On approaching the limits from a composition and pressure regime of stable detonation, the initial indications of degeneration of the ideal Chapman-Jouguet state are found in the phenomena of pulsating and spinning propagation. We present here several illustrative photographs obtained by Bone, Fraser, and Wheeler. Figure 300a shows details of a process in which a shock wave is formed ahead of the flame and in time becomes sufficiently strong to ignite the mixture and proceed as an over­ driven detonation wave. The example refers to a moist 2CO + 0 mixture in a tube 1.3 cm in diameter. The flame enters the picture at the upper right comer of Fig. 300b at a speed of 1275 m/sec. In front of it a shock wave is forming, and the distance between the flame front and the wave is rapidly diminishing. Figure 301 is a shadow photograph of the shock wave with the flame about 1 m behind (not shown in the photograph). In Fig. 302 the flame has approached to within 10 cm of the wave, the latter being considerably reinforced. The shadow of the wave consists of a dark and light zone owing to the sharp change of density gradient. One notes the remarkable planeness of this wave; the criss-cross pattern behind the wave is ascribed to strains created within the glass. When the flame reaches a point 6.37 cm behind the shock wave, autoignition is seen to occur in the latter (Fig. 300b). From this origin two new flame fronts start; both move forward, one at an initial speed of 2380 2

2

29

30

2

560

VIII.

DETONATION W A V E S IN

GASES

FIG. 300. (a) Detonation started ahead of combustion wave (Bone, Fraser, and Wfieeler ). Moist mixture of 2CO + 0 ; tube diameter 1.3 cm; visible tube length about 48 cm. 30

2

m/sec but quickly rising to 3260 m/sec, and the other at an average speed of 350 m/ sec. The latter flame front shortly meets the original flame front, giving rise to a wave traveling through the burned gas at a speed of 875 m/sec. Two other waves are set up, one at the point of origin of ignition and the other just ahead of it, both eventually traveling with a speed of about 1320 m/sec. The detonation wave slows down and passes out of the picture at a decreasing speed of 1980 m/sec and eventually reaches (25 cm farther on) a constant speed of 1760 m/sec. In Fig. 303 the described process is seen to be repetitive, thus giving rise to pulsating propagation.

FIG. 3 0 0 . (b) Analysis of part (a). Numbers in diagram denote velocity in meters per second.

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ FIG. 301. Shadow photograph of shock wave about 1 m ahead of combustion wave. (Bone, Fraser, and Wheeler ). Moist mixture of 2CO + 0 . Tube diameter 1.3 cm. 30

·'

J

^

2

^

k

.

,

- -

FIG. 302. Shadow photograph of shock wave about 10 cm ahead of combustion wave. (Bone, Fraser, and Wheeler ). Conditions as in preceding figure. 30

Whereas a basis for understanding the described phenomenon is provided by Brinkley and Richardson's theory, the phenomenon of spin is incongruous with the model of one-dimensional wave propagation that historically has formed the basis of detonation theory. The spin was discovered by Campbell and Woodhead and was studied further photographically by a number of investigators. Figure 304 shows a glass tube used in six successive detonations of a moist 2CO + 0 mixture. Since a lead tube preceded the glass tube, the inner surface of the latter had become coated 14

31

2

30

562

VIII. DETONATION WAVES IN GASES

FIG. 303. Successive auto-ignitions ahead of combustion wave (Bone and Fraser ). Moist mixture of 2 C 0 + 0 ; visible tube length, 1 m. Wave entering at right at 900 m/sec. 30

2

FIG. 304. Helical path traced by detonation wave (Bone and Fraser ). 30

with a thin gray film of lead. In this film a helical path had been traced by the detonation. Similar traces in dust deposits on the tube wall were observed earlier by Campbell and co-workers. Bone and Fraser report that spinning detonations crack glass tubes in spiral fashion, and Bone et al. observed that a detonation wave in a silvered glass tube cut a spiral track by volatilizing the deposited silver during its passage. Campbell and Finch, by photographing the detonation wave head-on, were able to show that a luminous zone, occupying only a fraction of the cross section, rotates around the axis of the tube along the surface as the detonation wave progresses. The pitch of the helix thus formed is of the order of magnitude of the tube diameter. A photographic time record of a spinning detonation is always un­ dulated as shown in the example of Fig. 305a. In the few cases where comparisons 30

30

32

4.

MEASUREMENT OF DETONATION VELOCITY

563

have been made, the rotational frequency calculated from the trace at the tube wall and the average detonation velocity was found to agree closely with the frequency of the undulations in the photographic track. ' Of added curiosity is the fact that more or less the same path at the tube wall of spiral track should have been followed by six successive detonations in the same tube. In the experiments of Campbell and Finch this regularity was not observed, the rotation being sometimes clockwise and at other times counterclockwise. The tracks are not always clearly cut. In the literature the helical path at the tube wall and the undulations in the photographic time record of the detonation have been generally accepted as being due to the same cause and the term spin is applied to the time track undulations. The frequency of the spin for a given mixture composition has been found to be inversely proportional to the tube diameter. The spin is not confined to circular tubes. Bone, Fraser, and Wheeler obtained spin in tubes of triangular, square, and rectangular cross sections; with comparable diameters the values of the frequency were of comparable magnitude. Bone and Fraser inserted a 1-mm ridge along the inner wall of a 12-mm diameter tube which was shown to be without effect on the spin. These authors conclude that the spin does not involve the rotation of the gas mass as a whole but only of the "head" of the detonation. Such rotation of the gas would indeed be highly improbable for reasons of fluid mechanics. Figure 305a is the continuation of the record shown in Fig. 300a some 25 cm to the left of the latter picture, in which the beginning of the spin is already observable. The photograph was taken with a mirror camera of high-speed and high-resolving power, developed by Fraser. It shows not only the sinuous track of the wave front (frequency 44,300 s e c ) but also two sets of striae criss-crossing and producing at their crossing points the impressions of horizontal bands or "tails" emanating from the troughs of the undulations. These appear on the photograph as pairs of divergent bright lines subtending an angle of 22° (see Fig. 305b). One of these lines appears to be due to luminous gas or particles traveling forward in the rear of the wave. Careful inspection shows that the line is not straight but curved as one would expect from the gradual decrease of particle velocity. At some distance from the flame track the velocity is estimated to be 780 m/sec. The other line which is very straight represents the luminous track of a compression wave traveling backward through the burned gas and generally but not always producing a strong increase of luminosity when passing through a mass of particles associated with one of the discrete particle tracks. The compression wave has a velocity of 320 m/sec. Since the particle velocity w is 780 m/sec, the shock wave travels with an absolute velocity of 1100 m/sec, which should be the sound velocity. A careful inspection of the figure reveals that the undulations and pairs of particle tracks and compression waves do not match. For seven pairs of troughs and crests of the undulations, one counts eleven pairs of particle tracks and shock waves. Remarkably, there are only seven rows of luminous intersections that form the nearly horizontal bands, and each of these rows appears to originate in a trough of the undulations. 30 32

33

30

-1

564

VIII. DETONATION WAVES IN GASES

FIG. 305. (a) Spinning detonation (Bone, Fraser, and Wheeler ). Moist mixture of 2CO + 0 . Tube diameter 1.3 cm. Average detonation velocity 1760 m/sec. (b) Next page, analysis of Fig. 305a (Bone, Fraser, and Wheeler ). 30

30

2

4.

565

MEASUREMENT OF DETONATION VELOCITY

Manson and later Fay have studied the problem of spin, considering it to be an acoustic vibration or traveling wave which is maintained by the chemical reaction in some unspecified manner. According to the model, a vibratory motion is main­ tained in the gas immediately behind the shock front; the motion has components in a plane normal to the tube axis, resulting in one or more pressure "heads" at the tube wall that rotate around the axis. The frequency of rotation is derived from acoustic theory and is found to be in very good agreement with experimentally observed spin frequencies. Such agreement was also found with a few exceptions, by Gordon et al. for their measured spin frequencies. Martin and White obtained 34

35

2S

36

566

VIII. DETONATION WAVES IN GASES

snapshot interferograms of spinning detonations in hydrogen-oxygen mixtures, at pressures and mixture compositions similar to those studied by Gordon et al. These snapshots show a somewhat nonplanar shock front with density gradients normal to the tube axis behind the front, lending support to the model of Manson and Fay. Vibrations of this sort are strongly damped by the mass flow through the detonation front; however, a driving force is provided by the increase of reaction rate in the pressure heads. At the shock front the gas is unreacted but attains a state of temperature and pressure which induces self-accelerating reaction, viz., reaction preceded by an induction period. If the phenomenon were ideally one-dimensional, the reaction zone would consist of a zone of constant depth behind the shock front within which the mixture is substantially unreacted, followed by a much narrower zone within which the reaction rapidly goes to completion. It is readily seen that this system cannot be stable because any slight pressure fluctuation inside the induction zone locally shortens the induction period and destroys the one-dimensional symmetry. The spin evidently represents such pressure fluctuations in the induction zone, governed by acoustic law. Within the pressure heads reaction is completed ahead of the time that it would normally take following the initial compression in the shock front. The shock front is, of course, disturbed and nonplanar because of such localized pressure generation. One may expect that complex phase relations exist between the main shock and these local pressure pulses, and that these phase relations govern the periodic bursts of reaction rate of which the striae in Fig. 305a are indicative. Spinning detonation would thus appear to be the normal mode of propagation whenever an induction zone exists of sufficient width so that the transverse vibrations responsible for the phenomenon are not damped out by the mass flow. The spin is thus associated with the weaker mixtures and disappears in strong mixtures where induction times are very short and induction zones are very narrow.

5. Transition from Flame to Detonation When combustion waves are initiated in tubes, laminar steady-state flames develop only when the ignition source is placed at an open end of the tube and the other end is closed, and when the tube is sufficiently narrow so that the vortex flow in the unburned gas ahead of the flame is stabilized by viscous drag at the tube wall. This type of flame has been described in Chapter V on pages 305 to 309. Outside the laminar steady-state regime the propagation is always self-accelerating. The accel­ eration is caused by the increase in wave area as the flame elongates and the flow becomes turbulent; in addition, the burning velocity increases as the unburned gas ahead of the flame is preheated and precompressed by the compression waves that are generated by the mass acceleration in the combustion wave. Depending on various conditions the waves may telescope into a shock front and the shock front may develop into a detonation wave.

5.

TRANSITION FROM FLAME TO DETONATION

567

Implicit in this mechanism of transition from flame to detonation is the existence of a predetonation run of the flame. Table 11 shows distances traversed by a hydrogenoxygen flame in a 25 mm tube before onset of detonation. The distance decreases as the pressure is increased, which is consistent with increase of burning velocity of such a mixture with pressure. During the predetonation run, the gas in the tube is set in motion and turbulence develops by friction at the wall. A turbulent boundary layer is thus formed at the wall which grows in depth until it occupies the whole cross-section of the tube. Turbulence is augmented by the fluctuations of the flame surface, as described in Chapter VI on pages 439 to 441, and by the passage of compression waves through the combustion zone, as described below. In ordinary pipe flow, the distance from the pipe end at which the flow enters to the point where turbulence is fully developed increases proportionately with increas­ ing pipe diameter. If it is true that the development of turbulence by friction at the wall has an important influence on the predetonation run in an initially quiescent gas, it follows that, other conditions being equal, the length of the detonation run similarly increases with increasing tube diameter. This is indeed found to be the c a s e ' although, as one may expect from the complicating factors involved, the relation is not necessarily linear. Brinkley and Lewis, however, have quoted experi­ ments by Shuey on acetylene detonations in long closed tubes of diameters between in. and at several pressures, initiated by a squib, in which the detonation run did increase linearly with tube diameter, being about 60 tube diameters in all cases. This coincides with Nikuradse's observation that in pipe flow fully developed turbulence occurs within a distance of about 60 diameters. When obstructions were placed in the tube near the ignition source, the detonation run was sharply reduced, an observation that emphasizes the important role of turbulence generation in the predetonation run. A sudden large increase of flame surface occurs when a shock wave passes through 37

38

39

40

41

42

TABLE 11 DISTANCE TRAVERSED BY FLAME BEFORE ONSET OF DETONATION OF

2H + 0 2

a b 2

Initial pressure, atm

Distance, cm

1 2 3 4 5 6 6.5

70 60 52 44 35 30 27

Tube diameter, 25 mm. *Laffitte and Dumanois

a

37

568

VIII.

DETONATION WAVES IN

GASES

a combustion wave. Markstein made photographic studies of initially laminar flames in tubes that are being met head-on by shock waves. The effect that is produced by the passage of a shock front across such density discontinuity is similar to the effect of sudden release of pressure by rupture of a diaphragm. A rarefaction wave propa­ gates backward into the unburned gas, and a jet of unburned gas develops which penetrates deeply into the burned gas. The shear between burned and unburned gas in this flow configuration produces extreme turbulence, so that a sudden large increase of the burning rate occurs and trains of compression waves are formed. The details of events occurring in such shock wave-flame interactions are ex­ tremely complex since the initial state of the flame, the intensity and direction of the initial shock wave, turbulence generation and interactions of the disturbed zone with reflected and secondarily formed shock waves all contribute to the process. With respect to effects produced by the generation of extreme turbulence, Karlovitz has suggested that, when turbulent wrinkling of the combustion surface becomes so strong that the pockets of unburned gas in the folds of the combustion wave on the average attain dimensions of the order of the wave width, the preheat zones of juxtaposed wave elements merge with each other and the turbulent flame brush collapses in a strong burst of reaction. This might lead to the formation of a single strong shock conducive to a detonation wave. This mechanism of initiation of detonation has been confirmed experimentally by Oppenheim et al. During a detonation run, interaction of the flame with a counter-running shock wave may occur when a shock wave that runs ahead of the flame is reflected from the closed end of the tube and meets the flame before the latter has developed into detonation. An illustration of such process is furnished by Fig. 306 which shows simultaneous direct and schlieren photographs of an ethylene-oxygen flame that attained high velocity but did not develop detonation. The direct photograph on the left shows only luminous gas. Compression waves increase the luminosity of the burned gas so that their tracks become visible. As usual the schlieren photograph brings out details that are otherwise invisible. On the right is an explanatory diagram of the two records. The directions of travel of the film and the flame image, and the length of film travel in 1 msec, are indicated in the figure. The width of the record corresponds to the length of the closed detonation tube of 30 cm. The wave is initiated by the spark S at the upper right and is seen to accelerate gradually At some point a compression wave A emerges from the wave front. It travels ahead of the flame through the unburned gas to the end of the tube where it is reflected and returns along path B. As it meets the flame, it continues along path B' through the burned gas, while the flame is pushed back and slugs of unburned gas are driven through the flame front to a considerable distance, as is clearly shown by the schlieren tracks. A rarefaction wave propagates from the flame front into the unburned gas, forming a dark band roughly in line with the track of the flame before it had collided with the shock wave. The front of the turbulent flame brush moves irregularly along path F. Vigorous combustion takes place, as indicated by a series of plainly visible 43

44

45

46

5.

TRANSITION FROM FLAME TO DETONATION

569

compression waves which move into the unburned gas at velocities exceeding the rarefaction wave and are reflected at the tube end back toward the flame. The compression wave that had traveled along path Β ' is seen to undergo several more reflections as it travels through the now completely burned gas. The compression wave that travels along path A does not immediately start out as a strong shock; rather, it is initially a comparatively weak pressure wave which is overtaken and reinforced during its travel by numerous other pressure waves origi­ nating in the combustion zone. These pressure waves are quite visible in the original schlieren photograph but are somewhat obscured in the reproduction. Their coales­ cence into a shock front does not, in the present example, result in detonation, but observations on other systems have shown that a detonation wave may develop at this stage of the process. An example is provided by schlieren kinematographic records of flames in tubes containing detonable propane-oxygen-nitrogen mixtures,

570

VIII. DETONATION WAVES IN GASES

which were obtained by Schmidt, Steinicke, and Neubert. Each successive flame of these films clearly shows the contours of the turbulent flame brush and the emergence of compression waves from the combustion zone. Near the combustion zone the compression waves form irregular patterns of oblique surfaces, showing that in the turbulent flame brush the gas flow is accelerated irregularly over the tube cross section. As a consequence, parts of the flame surface move forward above average velocity, and other parts are retarded, so that the turbulent flame brush increases in length. Such stage of growth is followed by rapid burn-out, supporting Karlovitz' view of a sudden collapse of the combustion surface. This is accompanied by enhanced emission of compression waves, and soon afterwards centers of autoignition appear in the column of gas between the flame front and the shock front. The appearance of these centers marks the transition from the type of burning induced by heat transfer across the combustion wave to the type induced by heat generated by shock compression. The final stage consists in an extremely rapid burn-out of the remainder of the unburned gas behind the shock front, accompanied by much luminosity and followed by the establishment of the detonation wave. Similar kinematographic records showing many details of the flame and shock waves were obtained by Salamandra, Bazhenova, and Naboko, who studied the predetonation period in hydrogen-oxygen mixtures. A shock wave may run counter to the advancing flame brush, as illustrated by the record shown in Fig. 306, or it may run up from behind, as illustrated by the record shown in Fig. 307, which has been obtained by Dixon. In the latter case the ignition source was placed at some distance from the closed end of the tube and the flame can be seen to advance initially in both directions. On reaching the nearer end, the reflected pressure pulses combine to form a compression wave, which travels through the burned gas and overtakes the other flame front. Here it is partly reflected; at the same time, the flame brush is distinctly accelerated by the disturbance. This inter­ action between flame and compression wave does not produce in this case immediate detonation although a detonation wave does develop eventually at a point outside the range of observation in this record. However, other cases are known in which such compression waves, advancing from the rear, did produce instantaneous detonation. The records and the other evidence that have been quoted have clarified the mechanism of the detonation run, during which a flame, which travels initially at a velocity of the order of meters per second with respect to the unburned gas, is accelerated to a velocity of the order of thousands of meters per second. It is seen that initially the combustion wave elongates and becomes turbulent due to friction of the gas flow at the tube wall. In this state, accelerations of mass flow by the combustion process occur nonuniformly within the turbulent flame brush; thus, pressure pulses are generated nonuniformly over the tube cross section and propagate laterally as well as longitudinally. The lateral components are reflected at the tube wall, so that inside the turbulent flame brush trains of pressure waves move along the tube upstream as well as downstream. The waves moving upstream accelerate 47

48

49

40

5.

TRANSITION FROM FLAME TO

571

DETONATION

FIG. 307. Direct photograph of development of detonation in a 16.6% CS -0 mixture (Dixon ). 2

2

49

the unburned gas within the brush in the direction opposite to the flame travel, so that shear flow between burned and unburned gas develops similar to the flow observed by Markstein in his studies of penetration of combustion waves by shock waves. The acceleration of the overall burning rate by this process is enhanced by the precompression and preheating of the unburned gas that takes place as the pressure waves move downstream and coalesce into a shock front. At some stage of the process rapid bum-out of the flame brush occurs. This may lead to the final intensi­ fication of the shock front that produces detonation, or the process may repeat itself until such intensification occurs. In addition, it may happen that the flame brush is met by a reflected shock wave and is thus greatly accelerated. Brinkley and Lewis have considered the relation between detonation run and 40

572

VIII. DETONATION WAVES IN GASES

destructiveness of the detonation. During the run the pressure and temperature of the unburned gas are raised by the precompression wave. This precompression is the greater the longer the run of the flame brush. When the detonation wave is established, it propagates into a medium of pressure which is greater than the initial pressure prior to ignition, with the result that the amplitude of the detonation wave may be very significantly greater than would be the case if the detonation wave were estab­ lished immediately upon ignition. This point is bourne out by experimental obser­ vations quoted by the authors. The detonation wave attains a maximum amplitude when the preceeding flame run consumes nearly all of the explosive gas with which the container was initially filled prior to initiation of the detonation wave, thus resulting in maximum precompression of the unburned gas. These considerations can be employed to calculate in a simple manner an upper limit to the detonation pressure. Such a limiting pressure can be employed for conservative design calcu­ lations in practical situations. Because of the precompression, transient, nonsteady detonations may occur in mixtures that normally do not detonate, so that limit-of-detonability data cannot be relied upon in usual practical situations. The thermodynamic regimes corresponding to the various interactions that are observed in the schlieren records of Schmidt et al. have been evaluated and mapped out by Oppenheim. Similarly, Popov has treated the records of Salamandra and co-workers. An example of the role of chemistry in the predetonation period is shown by experiments of Shchelkin and Sokolik. These investigators determined the depend­ ence of the predetonation distance for fixed spark conditions in a C H i + 8 0 + 2N mixture as a function of pressure, with and without addition of 1.2% of Pb(Et) . Their data are shown in Fig. 308. Pb(Et) has a definite inhibiting effect. In addition there is a curious stepwise increase of the distance with decreasing pressure. As mentioned earlier, in the fully developed detonation wave, Pb(Et) is without effect. In another series of experiments a somewhat rich C H i - 0 mixture at a pressure of about 300 mm Hg was preheated to temperatures from 300 to 400°C for short periods of time preceding passage of the igniting spark. In this temperature range the mixture reacted in the manner described in Chapter IV, and after an induction period τι, cool flames appeared. The predetonation path was dependent on the length of the heating period preceding sparking in the manner shown in Fig. 309. The shaded area corresponds to the appearance of cool flame. When the mixture was ignited imme­ diately after the appearance of cool flame the predetonation run decreased consid­ erably; but it increased again when ignition was delayed. It is clear that sensitization of the mixture due to accumulation of aldehydes and peroxides markedly facilitates development of the detonation wave. The experimental conditions are seen to be a hybrid between the usual detonation of an unsensitized mixture and the spontaneous formation of ignition centers in the preheated end charge of the Otto engine. 50

51

48

52

5

2

2

2

4

4

4

5

2

2

1

9

0

Γ Τ

50

90

130

170

210

PRESSURE, mm Hg

FIG. 3 0 8 . Predetonation distance from igniting spark in pentane-oxygen mixtures with and without lead tetraethyl (Shchelkin and Sokolik ). Curve 1, C H , + 8 0 + 2 N . Curve 2 , C H + 8 0 + 2 N + 1.2% Pb(Et) . Tube diameter, 2 8 mm; condensed spark, 0 . 0 2 microfarads at 3 0 0 0 to 4 0 0 0 volts. 52

5

2

4

2

2

2

5

12

2

574

VIII. DETONATION WAVES IN GASES

References 1. M. Berthelot and P. Vielle, Compt. Rend. 93, 18 (1881). 2. E. Mallard and H. Le Chatelier, ibid. 93, 145 (1881). 3. P. Laffitte, Compt. Rend. Acad. Sci. 177,178 (1923); N. Manson and F. Ferrie, "Fourth Symposium on Combustion," p. 486, Williams & Wilkins, Baltimore, 1953. 4. D. L. Chapman, Phil. Mag. [5] 47, 90 (1899). 5. E. Jouguet, J. Mathématique, p. 347 (1905); p. 6 (1906); "Mécanique des Explosifs," Ο. Doin, Paris, 1917; see also, L. Crussard, Bull. Soc. Ind. Minérale St.-Etienne 6, 109 (1907). 6. R. Becker, Z. Physik 8, 321 (1922); Z. Elektrochem. 42, 457 (1936). 7. See, for example, P. P. Ewald, T. Pôschl, and L. Prandtl, "The Physics of Solids and Fluids" (translated by J. J. Dougall and W. M. Deans), 2nd Ed., p. 358, et seq, Blackie & Son Ltd., London, 1936. 8. L. H. Thomas, /. Chem. Phys. 12, 449 (1944); G. R. Cowan and D. F. Hornig, ibid. 18, 1008 (1950). 9. See, for example, A. G. Gaydon and I. R. Hurle, "The Shock Tube in High-Temperature Chemical Physics," Reinholt Publishing Corp., New York, 1963. 10. Ya. B. Zeldovich, /. Exptl. Theoret. Phys. (U.S.S.R.) 10, 542 (1940); "Teoriya Goreniya i Detonatsii i Gazov." U.S.S.R. Acad. Sci., Moscow, 1944; A. S. Kompaneets, "Teoriya i Detonatsii." U.S.S.R. Acad. Sci., Moscow, 1955. 11. J. von Neumann, O.S.R.D. Rept. No. 549 (1942); Ballistic Research Lab. File No. X-122. 12. W. Doring, Ann. Physik 43, 421 (1943). 13. W. E. Gordon, "Third Symposium on Combustion and Flame and Explosion Phenomena," p. 579. Williams & Wilkins, Baltimore, 1949. 14. S. R. Brinkley, Jr., and J. M. Richardson, "Fourth Symposium on Combustion." p. 450. Williams & Wilkins, Baltimore, 1953. 15. B. Lewis and J. B. Friauf, J. Am. Chem. Soc. 52, 3905 (1930). 16. H. B. Dixon, Phil. Trans. Roy. Soc. A184, 97 (1893); W. Payman and J. Walls, J. Chem. Soc. p. 420 (1923). 17. B. Lewis, J. Am. Chem. Soc. 52, 3120 (1930). 18. D. G. Berets, E. F. Greene, and G. B. Kistiakowsky, /. Am. Chem. Soc. 72, 1080 (1950). 19. M. Berthelot and P. Vielle, Compt, Rend. Acad. Sci. 94, 101, 149, 822 (1882); 95, 151, 199 (1882); Ann. Chim. et Phys. 28, 289 (1883). 20. E. Mallard and H. Le Chatelier, Ann. mines [8] 4, 274, 335 (1883); Compt. Rend. Acad. Sci. 130, 1755 (1900); 131, 30 (1900). 21. J. Breton, Ann. office natl. combustibles liquides 11, 487 (1936); Theses Faculté des Sciences, Univ. Nancy, 1936. 22. H. B. Dixon, Phil. Trans. Roy. Soc. A184, 97 (1893); A200, 315 (1903). 23. C. Campbell, J. Chem. Soc. p. 2483 (1922). 24. M. Berthelot and H. Le Chatelier, Compt. Rend. Acad. Sci. 129, 427 (1899). 25. W. A. Bone and D. T. A. Townend, "Flame and Combustion in Gases," pp. 177, 179. Longmans, Green, London, 1927. 26. Compiled by P. Laffitte, "Science of Petroleum," Vol. IV, p. 2995. Oxford Univ. Press, London and New York, 1938. 27. R. Wendlandt, Z. physik. Chem. 110, 637 (1924); 116, 227 (1925). 28. W. E. Gordon, A. J. Mooradian, and S. A. Harper, "Seventh Symposium on Combustion," p. 752. Butterworths, London, 1959. 29. James A. Fay, Physics of Fluids 2, 283 (1959). 30. W. A. Bone and R. P. Fraser, Phil. Trans. Roy. Soc. A228, 197 (1929); A230, 363 (1932); W. A. Bone, R. P. Fraser, and W. H. Wheeler, ibid. A235, 29 (1936).

REFERENCES

575

31. C. Campbell and D. W. Woodhead, J. Chem. Soc. p. 3010 (1926); p. 1572 (1927). 32. C. Campbell and A. C. Finch, J. Chem. Soc. p. 2094 (1928). 33. H. Guénoche, Rev. inst. franc, pétrole et Ann. combustibles liquides 4, 15 (1949). 34. Ν. Manson, "Propagation des détonations et des déflagrations dans les mélanges gazeux," Office natl. d'études et recherches aer. et Inst. franc, des pétroles, Paris, 1947; Compt. Rend. Acad. Sci. 222, 46 (1946). 35. J. A. Fay, J. Chem Phys. 20, 942 (1952). 36. F. J. Martin and D. R. White, "Seventh Symposium on Combustion," p. 856. Butterworths, London, 1959. 37. P. Laffitte and P. Dumanois, Comp. rend. 183, 284 (1926). 38. P. Laffitte, Ann. phys. [101 4, 623 (1925). 39. Κ. I. Shchelkin, J. Tech. Phys. (U.S.S.R.) 17, 613 (1947). 40. S. R. Brinkley, Jr., and B. Lewis, "Seventh Symposium on Combustion," p. 807. Butterworths, London, 1959. 41. H. M. Shuey, private communication (1958). 42. J. Nikuradse, Verein deut. Ing. Forschungsheft 356 (1932). 43. G. H. Markstein, "Sixth Symposium on Combustion," p. 387. Reinhold, New York, 1957. 44. B. Karlovitz, see ref. 40. 45. A. J. Laderman, R A. Urtiew, and A. K. Oppenheim, "Ninth Symposium on Combustion," p. 256, Academic Press, 1963. See also A. K. Oppenheim, Α. J. Laderman, and P. A. Urtiew, Combustion and Flame 6, 193(1962). 46. W. Payman and H. Titman, Proc. Roy. Soc. A152, 418 (1935). 47. E. Schmidt, H. Steinicke, and U. Neubert, "Fourth Symposium on Combustion," p. 658. Williams & Wilkins, Baltimore, 1953. 48. G. D. Salamandra, T. V. Bazhenova, and I. M. Naboko, "Seventh Symposium on Combustion," p. 581. Butterworths, London, 1959. 49. Η. B. Dixon, Phil. Trans. Roy. Soc. A200, 315 (1903). 50. A. K. Oppenheim, "Seventh Symposium on Combustion," p. 837 Butterworths, London, 1959. 51. V. A. Popov, "Seventh Symposium on Combustion," p. 799. Butterworths, London, 1959. 52. Κ. I. Shchelkin and A. Sokolik, Acta Physicochim. (U.R.S.S.) 7, 581, 589 (1937).