On the condition of stability of the plane detonation wave

586

THIRD SYMPOSIU.~I ON COMBUSTION, FLAMF AND EXPLOSION PI[ENOMI';N\

ported can probably be relied upon only to within 2-3 atm. Improvements in the instruments and technique made recently are expected to yield more accurate data. Acknowledgment. This work was done as a dissertation under Profess~r E. Bright Wilson, Jr., of Harvard University, to whom the author is in debt for suggestion of the problem and for constant help in the course of the work. Dr. Win. D. Kennedy, Dr. D. F. Hornig and Mr. G. K. Fraenkel were part of a group working on the development and use of piezoelectric gauges, and nmch of the instrumentation was obtained from them. They also contributed many valuable suggestions.

REFERENCES

1. CAMPBELL~ LITTLER, AND WHITWORTH: Proc. Roy. Soc. (London), 137, 380 (1932). 2. RIMARSK[ AND KONSCHAK: Autogene Metallbearbeitung, 27, 209 (1934) 3. HENOERSON:Proc. of the Pacific Coast Gas Assoc., 32 (1941). 4. Combustion, Flames and Explosion of Gases, Cambridge (1938). 5. VON NEUMAN, JoaN: Theory of Detonation Waves, OSRD Report No. 549 (!942). 6. DORING, W.: Ann. Physik., 43, 421 (1943). 7. For an account of this see Becket: Zeit. Physik, 8, 321 (1922). 8. WENDLANDX:Z. Physik. Chem., 110, 637 (1924); 116, 227 (1925).

75

ON T H E C O N D I T I O N OF S T A B I L I T Y OF T H E P L A N E D E T O N A T I O N WAVE 1 By STUART R. BRINKLEY, JR.fl ANn JOHN G. KIRKWOOD 3 INTRODUCTION

The hydrodynamic-thermodynamic theory of the detonation velocity of plane detonation waves has been based upon the Rankine-Hugoniot (1, 2) equations for the continuity of mass, momentum, and energy across the detonation front, an equation of state and thermal data for the products of the detonation reaction, and the Chapman-Jouguet (3, 4) hypothesis which predicts the stable detonation velocity. Although calculations based upon this theory for gaseous explosives, for which the applicable equation of state is known, are in excellent agreement with experimental results (5), no completely satisfactory demonstration of the validity of the Chapman-Jouguet hypothesis has been presented. In the present communication, a purely dynamical consideration of the conditions for the existence of a stable solution to the equations of hydrodynamics leads to a verification of the Chapman-Jouguet hypothesis for the case of 'Work on manuscript completed August, 1948. Thts research is part of work being done at the Bureau of Mines on Project No. NA onr 29-48, supported by the Office of Naval Research. Physical Chemist, Explosives Research Section, Explosives Branch, Bureau of Mines, Pittsburgh, Pa. 3 Professor of Physical Chemistry, California Institute of Technology, Pasadena, Cal.

the plane detonation wave for which it is assumed that the effects of viscosity and heat conduction may be neglected, that the reaction rate of the detonation reaction is infinite, and that the Lagrange energy-time curve of the detonation products is initially monotone decreasing with finite initial slope. The first assumptions are in common ~ith the classical model employed in the theory of the detonation velocity, whereas the assumption as to the nature of the energy-time profile appears to be plausible on physical grounds. The Rankine-Hugoniot equations for the continuity of mass, momentum, and energy across the detonation front may be written in the form P -

p(D

-

Po = pouD, u)

= p~,

H - H0 --- 89 -- p0)(I/p0 -t- I/p),

(I)

where u is the particle velocity of the detonation products, D the detonation velocity, p the pressure, p the density, and H the specific enthalpy. The subscript zero refers to the intact explosive, supposed to bc at rest; the undesignated symbols refer to the products of the detonation reaction. We assume the existence of an equation of state and thermal data for the detonation products so

BURNING AND DETONATION OF EXPLOSIVES that the specific enthalpy can be expressed as a known function of the other two state variables for any particular explosive, H = H(p, o; H0).

(2)

Equations 1 and 2 can be employed to express all of the properties of the detonation state in terms of any one of them, say the pressure, for given initial conditions of state and material, u = u(p; oo, po, Ho), H = H(p;oo, Po, Ho),

D = D(p; oo, Po, Ho), o = o(P;oo, Po, Ho).

(3)

Henceforth, our notation will not show the implicit dependence of the several quantities upon the given initial conditions of state. The third of equations 1 and equation 2 consist of a relation, for fixed initial conditions, between the pressure and density that is most conveniently considered on the p, v-plane, v = 1/o. This p,vcurve is the well-known "dynamic adiabatic" of Hugoniot, and it is the locus of detonation states that are compatible with the conservation conditions. Since, from equation 1, D 2 = -Vo'(p - po)/(v - Vo),

(4)

the square of the detonation velocity corresponding to each such state is given by the negative slope of the chord from the initial state (P0, v0) to the final state (p,v). I t has been shown that the Hugoniot curve is tangent to a member of the family of adiabatic p,v-curves at the points of contact of the tangents from (P0, vo) and it is more strongly curved than the adiabatics in the portion of the curve corresponding to detonation, for which Vo >V. The thermodynamic-hydrodynamic theory of detonation waves is completed by the hypothesis of Chapman and Jouguet that the velocity of the stable detonation wave is the minimum one compatible with the conservation conditions. The detonation velocity is thus assumed to be given by the tangent to the Hugoniot curve from (P0, v0). The Chapman-Jouguet condition has been discussed by many writers (6, 7, 8, 9). It has been shown that the hypothesis is equivalent to the assumption that the detonation velocity equals the sum of particle velocity u and the local Euler sound velocity c in the detonation products, D ~ u + c.

(5)

It has also been shown that D > u + c when p < p* and that D < u + c when p > p*, where

587

p* denotes the pressure of the Chapman-Jouguet state. In support of the hypothesis, detonation states with p > p*, D < u + c, have been shown to be unstable for an arbitrary period of time because the wave corresponding to such a state would be penetrated and retarded by a following rarefaction wave, which would have a velocity of u + c relative to a stationary origin. When p = p*, D = u + c, such rarefaction waves could no longer overtake the detonation wave. The argument employed to exclude detonation states for which p < p* has been thermodynamic in character. Every chord from the point (po, Vo) to the Hugoniot curve except the tangent intersects the curve at two points, say p' and p", for which p' > p* and p" < p*. It has been shown that S(p') > S(p"), where S is the entropy. It is stated as a consequence that the detonation state for which p = p' is a more probable state than that for which p = p" and that detonation states for which p < p* are therefore excluded. But the states for which p > p* have already been shown to be unstable. Therefore, the Chapman-Jouguet state is the stable detonation state. Scorah (8) has shown that the work content for the Chapman-Jouguet state is a minimum and inferred that this state corresponds to a maximum degradation of energy. In the subsequent discussion, we shall employ without proof several well-known properties of the Hugoniot curve. These relations are derived in detail in the references cited. We shall denote the pressure for which D = u + c by p*, and we shall employ the inequalities D >u +cwhenp D < u +cwhenp

p*.

Since D is a minimum at p = p*, d D / d p = 0 when p = p*, d D / d p < 0 when p > p*, p < p*. Tm~ STABILITY CO~mIXlON The methods of a previously published (10) theory of the propagation of shock waves may be employed for a purely dynamical discussion of the condition of stability for a plane detonation wave in a semi-infinite slab of explosive. We assume, as in the classical model of the detonation wave, that the reaction rate of the detonation reaction is infinite and that viscosity and heat conduclion may be neglected; the detonation wave is thus represented by a mathematical discontinuity, across which the Rankine-Hugoniot equations may

588

T H I R D SYMPOSIUM ON C O M B U S T I O N , F L A M E AND E X P L O S I O N P H E N O M E N A

be applied, whether or not the velocity of propagation of the discontinuity is constant. It is convenient to write the equations of hydrodynamics in the form

p Ou ----4po OXo

10p --0, pc ~ Ot

Ou 1 0 p _ O, O-t + po Oxo u = (Ox/Ot)~o,

equations, equations 1, and the equation of state of the detonation products. If equations 8 can be supplemented by a fourth relation berween the partial derivatives, it will be possible to solve for each of the four derivatives as a function of p and, with the aid of equation 7, to formulate an ordinary differential equation,

dp _ Op Op = F(p), dt Ot -4- D ~oxo (6)

where t is the time and x the Euler coordinate at time t of an element of fluid with Lagrange coordinate x0. The Euler sound velocity c is equal to [6p/~p)s] ~2. Equations 6 are supplemented by the equations of state of the fluid composed of the products of the detonation reaction and the entropy transport equation bS/~t = 0. We shall not explicitly use the latter equation. Equations 6 are of a hybrid form, in that we employ the Lagrange coordinates x0 and t as independent variables but retain the Euler equation of contirs Equations 6 are subject to initial conditions specified on a curve in the xo, t-plane and to the Rankine-Hugoniot equations at the detonation front. The latter constitute supernumerary boundary conditions which are compatible with the differential equations and specified initial conditions only if the detonation front follows an implicitly prescribed curve xo(t) in the xo,t-plane. We denote a derivative in which the detonation front is stationary by

(9)

for the peak pressure p as a function of the time l. We define an energy function, 5(xD) as the sum of the adiabatic work w0 per unit area done by the initiating shock and the total energy released per unit area by the explosive contained in the section bounded by the planes at the origin and at x0. Then

A(zo) = wo +

off podp(xo)]'

dxo,

(10)

where e(p) is the specific-energy increment from the intact explosive to the detonation products corresponding to peak pressure p after the detonation products have expanded to P0 along the adiabatic S(p). In the case of explosives capable of sustaining a detonation w~.ve, the quantity zX is clearly positive and it is finite for finite x0. According to the laws of thermodynamics, the energy function equals the total work done per unit area at the plane x0. Therefore,

A(xo) = [

p'(t')u'(t') dt',

(11)

d t o(Zo)

d_ dt

O + D__.O Ot OXo

(7)

If the operator d/dt is applied to the first of the Rankine-Hugoniot equations, equations 1, and if equations 6 are specialized for the detonation front, x = x0, three relations are obtained between the four partial derivatives of pressure and particle velocity with respect to time and distance.

p Ou ---+ po OXo

10p -0, pc 20t

Ou 10p v 0 O - t - + -Po - - Oxo - _ , Ou

~/-+ D

Ou Oxo

g Op po Oxo

9 Op - O, po D O t

(8)

where g = 1 - oou (riD~alp). All of the coefficients in equations 8 can be expressed as functions of the pressure alone by means of the Rankine-Hugoniot

where p' and u' are the pressure and particle velocity, respectively, behind the detonation front (unprimed symbols being reserved henceforth for quantities at the detonation front), to(xo) is the time of arrival of the detonation front at x0, and where the integral is taken along a path of constant Lagrange coordinate x0. The assumptions underlying equations I0 and 11 break down if the detonation wave can be overtaken by shock waves built up in its rear. This will not be the case if the pressure-time curve is initially monotone decreasing with asymptotic value p0. If the excess pressure p' - P0 has a negative phase, a shock wave wilt develop in the negative part of the pressure-time curve but cannot overtake the detonation wave. In this case, our formulation of the energy equation will apply to the positive phase if the time integral of equation 11 is extended not to infinity but to the time

589

B U R N I N G A N D D E T O N A T I O N OF E X P L O S I V E S

at which the excess pressure of the positive phase vanishes. The integral of equation 11 may be expressed in reduced form,

Employing the second of the Rankine-Hugoniot equations, equations 1, to eliminate po/m the quantity G m a y be expressed in the alternative form

a(Xo) = pu~u,

G = 1 - ((D - u)/c) 2. It follows that

1 / u = -- (O l o g _ p ' u ' ~ Ol ] t = to(~o)

10p p Ot vfxo) = fo

f ( x o , r) dr,

10u u Ot '

r -- t - to(xo) #

f(xo, r) = p'u'/pu.

(12)

vp~ u A

I

~

9 l+g+(p/pouD)(l+g-G)

1

'

(13)

where G = I (poD~pc) ~', g = 1 - pou(dD/dp). I t is evident that G < 1. Since d D / d p <~ O, it follows that g >/ 1. Therefore, the denominator of the term in the brackets is always positive. Since the coefficient of the term in the brackets is positive, it follows that -

dp/dt > 0 when G < O, dp/dt = 0 when G = 0, dp/dt < 0 when G > 0.

D > u +cwhenG

<0,

D = u +cwhenG

= O,

D < u +cwhenG

>0.

(B)

Combining relations (A) and (B),

The function f(xo, r) is the energy-time integrand, normalized by its peak value at the detonation front, expressed as a function of x0 and a reduced time 9 which normalizes its initial slope to - 1 if u does not vanish. We assume that u does not vanish and that f is a monotone decreasing function of r. Equations 12 yield the desired fourth relation, supplementing equations 8 between the four partial derivatives. This set is exact, involv]ng integrals of equations 6 for a knowledge of f(xo, r). I f f is a monotone decreasing function of and if u does not vanish, then u is positive and is finite and positive. Elimination of u between the first two of equations 12 and combination with equations 8 yields a set of four equations that may be solved for the four partial derivatives, and an ordinary differential equation for the peak pressure can be formulated with the aid of equation 9. The resulting expression can be written in the form dp dt

(14)

(A)

dp/dt > 0 w h e n D

> u + c

dp/dt = 0 w h e n D = u + c , dp/dt < 0 when D < u + c.

(C)

Furthermore, we have seen that p < p*whenD > u +c, p > p*whenD < u +c,

(D)

where p* denotes the peak pressure of the Chapman-Jouguet detonation state. In consequence of these relations, it is seen that if a detonation state is initiated with a peak pressure greater than that for the Chapman-Jouguet state, the pressure will decrease in the course of propagation of the wave to that for the ChapmanJouguet state and will thereafter remain constant. Correspondingly, if a detonation state is initiated with a peak pressure less than that for the Chapman-Jouguet state, the pressure will increase in the course of propagation of the wave to that for the Chapman-Jouguet state and thereafter remain constant. Since all of the properties of the detonation state are functions of the peak pressure only, these quantities will also remain constant after the pressure becomes constant. In particular, the stable value of the detonation velocity is the constant value predicted by the hypothesis of Chapman-Jouguet. This discussion of the validity of the ChapmanJouguet hypothesis for a plane detonation wave in an explosive for which the reaction rate is infinite has involved a purely dynamic argument which does not require identification of the probability of the existence of a state with its entropy. We have assumed that the Lagrange energy-time curve is initially monotone decreasing with a finite slope. Although these assumptions appear to be reasonable o n physical grounds, they prevent a complete

590

THIRD SYMPOSIUSI ON COMBUSTION~ FLAME AND EXPLOSION PHENOMENA

6. BECKER, R.: Z. Elektrochem., 23, 40, 304 (1917); Z. Phys., 8, 321 (1922) 7. JOST, W.: Z. Elektrochem., 41, 183 (1935); Z. Phys. Chem., 42, 136 (1939). 8. SCORAH,R. L.: J. Chem. Phys., 3, 425 (1935). 9. KISTIAKOWSKY,G. B., AND WILSON-, E. B., JR.: OSRD Rep. 114 (available from Off. Tech. Serv., U. S. Dept. Commerce, Report PBL 32715). See also the summary by H. L. Dryden, F. D. Murnaghan, and H. Bateman, Bull. Nat. Res. Council, No. 84, 1932, p. 551. 10. BRINKLEY,S. R. JR., AND KIRKWOOD,J. G.: Phys. Rev., 71, 606 (1947).

identification of the detonation wave in an actual explosive with the model that has been employed in the present discussion. REFERENCES 1. RANKINE, W. J. M.: Trans. Roy. Soc. London,

A 160, 277 (1870). 2. HUGONIOT, H.: J. de l'6cole polyt., 57, 3, 1887; 5,S', 1 (1888). 3. CHM'MAN,D. L.: Phil. Mag., (5), 47, 90 (1889). 4. JOUGtTET,E.: Comptes Rendus, 132, 573 (1901). 5. LEWIS, B., AND FRIAr:F, J. B.: J. Am. Chem. Soc., 52, 3905 (1930).

76

T H E P R O P E R T I E S OF G A S E S AT H I G H P R E S S U R E S W H I C H CAN BE D E D U C E D F R O M E X P L O S I O N E X P E R I M E N T S ~ By H. JONES INTRODUCTION

Following the successful theory of the detonation of gases ~ontained in tubes formulated by D. Chapman (1) and E. Jouguet (2), at the beginning of the century, attempts have been made, at first notably by Schmidt (3), to apply this theory to solid explosives. In order to determine the velocity of the detonation wave theoretically a knowledge of the equation of state of the product gases of the explosion is required in a range of temperature and pressure far beyond those realizable in ordinary laboratory work. The pressures involved are of the order l0 s atmospheres and temperatures of the order of 3000~ Since the detonation velocity is a function of the loading density of the explosive, and since these velocities can be measured with considerable accuracy, it is natural that attempts should have been made to reverse the argument and to deduce information about the equation of state from the observed detonation rates. I t is the purpose of this paper to discuss the extent of the information concerning the equation of state which can be obtained in this way from detonation velocities. Care has to be taken when applying the ChapThe author is indebted to the Director-General of Scientific Research (Defence), Ministry of Supply, for permission to publish thls paper.

man-Jouguet theory to rods of solid explosives because this theory applies only to a strictly one dimensional detonation wave. It has been shown (4) that the lateral expansion in the reaction zone of a detonation wave causes a diminution in the velocity, which is given by an expression of the type

(Uo/U)' = 1 + f(l/r)

(1)

where f is a function which can be calculated approximately and which diminishes rapidly as l/r diminishes. U is the actual detonation velocity and U0 that which would be given by a correct application of the Chapman-Jouguet theory, r is the radius of the charge and l the length of the reaction zone. Thus the only data which should be used in an attempt to obtain information about the equation of state are velocities determined for charges of diameter large enough to give the maximum detonation rate. For most high explosives, charges of diameter of about one inch usually give the maximum velocity, but for the low power explosives, e.g. those containing much ammonium nitrate, the observed velocity will not at all correspond to the limiting value, which alone can be used with the Chapman-Jouguet theory to yield information on the equation of state.