LITERATURE CITED

I. M. E. Gill and E. D. Lingley, US Patent No. 3249047 (1966). 2. J. L. Austing, A. J. Julis, et al., Propellants, Explosives, Pyrotechnics, 9, 193 (1984). 3. K. Yukhanson and P. Person, Detonation of Explosives [Russian translation], Mir, Moscow

(1973). 4. F. P. Bowden and A. D. Yoffe, Initiation and Growth of Explosions in Liquids and Solids,

Cambridge University Press, New York (1985). 5. F. P. Bowden and A. D. Yoffe, Fast Reactions in Solids [Russian translation], IL, Mos-

cow (1962). 6. A. F. Belyaev, V. K. Bobolev, et al., Transition from Combustion to Explosion in Con-

densed Systems [in Russian], Nauka, Moscow (1973). 7. A. G. Merzhanov, V. V. Barzykin, and V. T. Gontovskaya, Dokl. Akad. Nauk SSSR, 148,

No. 6 (1963). 8. U. I. Gol'dshleger, K. V. Pribytkov, and V. V. Barzykin, Fiz. Goreniya Vzryva, 9, No.

1 (1973). 9. V. S. Berman and Yu. S. Ryazantsev, Prikl. Mat. Mekh., 40, No. 6 (1976). i0. V. I. Krylov, V. V. Bobkov, and P. I. Monastyrskii, Computational Methods. Part 2 [in

Russian], Nauka, Moscow (1979). ii. G. R. Otei and Kh. A. Duaier, Raket. Tekh. Kosmonavt., 17, No. 6 (1979). 12. V. J. Calderbank, Course in Programming in FORTRAN-IV, Methuen, New York. 13. M. H. Friedman, Combust. Flame, ii, No. 3 (1967). 14. M. H. Friedman, Combust. Flame, 13, No. 6 (1969). 15. G. Dauge, J. P. Giraudov, and R. Ficat, 15th Symposium (International) on Combustion,

Tokyo (1974). 16. I. S. Klochkov and N. D. Manachinskii, Fiz. Goreniya Vzryva, 9, No. 4 (1973).

MODEL OF THE DEVELOPMENT OF REACTION IN AN EXPLOSIVE

WITH SHOCK-WAVE LOADING

A. N. Rabotinskii, S. P. Smirnov, V. S. Solov'ev, and E. V. Kolganov

UDC 662.215.5+534.222.2

At present, in investigating the decomposition of explosives under the action of shock waves (SW), two approaches may be distinguished. The first is based on experimental data on the parameters of a flow of reacting medium [1-4]. The method used in [I, 2] does not require the use of any model assumptions and deals completely with experimental results for one of the mechanical quantities of the flow and the propagation rate in a weak-explosion medium. On the basis of these data, a quantity qualitatively characterizing the relaxation process may be calculated [5]

where t is the time; p, pressure; v, specific volume; c, velocity of sound. Of course, this method leaves aside the question of the activation mechanisms of the material and is further decomposition.

In [3, 4], the decomposition of the explosive was judged on the basis of s mass frac- tion Of final reaction products x. In this case, consideration was limited to experimental data for one of the mechanical quantities. However, model assumptions regarding the equa- tion of state of the initial explosive and the reaction products must be used here, as well as assumptions regarding the relation between the initial explosive and the reaction prod- ucts in the mixture. As shown in [6]~ these assumptions may lead to distorted concepts re- garding the desired kinetics.

Moscow. Translated from Fizika Goreniya i Vzryva, Vol. 23, No. i, pp. 60-64, January- February, 1987. Original article submitted March 27, 1986.

54 0010-5082/87/2301-0054512.50 O 1987 Plenum Publishing Corporation

The second approach is based on phenomenological description of shock-wave initiation. Common to the models introduced here is the assumption that the reaction develops from sources [8-11]. The number of sources is determined by the thermal component of the specific internal energy, and the decomposition of the material is a process of burnup of the explosive sur- rounding the source. On the basis of these ideas, the kinetic equations of the decomposition of the explosive are written. The parameters of the equation are determined in comparing calculation and experiment.

The difficulty arising in this description of the decomposition process [I0] indicates that existing concepts regarding the shock-wave initiation are still far from perfect.

In the present work, a model of the development of reaction in an explosive based on the concept of a branched chain mechanism of increase in the number of reactive particles is considered. This model extends the theory of decomposition of crystalline materials [12] to the case of shock-wave initiation.

The initial equation describing the change in concentration of the reactive particles takes the form

dn/dt = (= - ~). n/6t, ( 1 )

where n is the dimensionless concentration; t, time; 6t, mean lifetime of a reactive particle; ~, probability of branching; 7, probability of chain breaking. In Eq. (i), the number of active particles formed on account of the Maxwell-Boltzmann energy distribution is neglected. Assuming, as in [12], that 7 = =x/xi, where x and xi are the current degree of decomposition of material and that at the point of inflection on the kinetic curve x = x(t), and using the expression for the reaction rate dx/dt = n/6t, it is found that

dn/dz = ~( t - - x/xO., ( 2 )

Integrating Eq. (2) with the initial conditions n = nf, x = 0 when t = 0 leads to the expres- sion

n = nf + ~(x -- ~/2x,~. ( 3 )

Suppose that the shock wave with a pressure amplitude pf induces reactive molecules with di- mensionless concentration nf in a Lagrangian particle of the explosive; a plane one-dimen- sional problem is considered.

Using Eqs. (i)-(3), it is found that

dxldt = t / ~ n f + ~ ( z ' - ~ / 2 ~ ]. ( 4 ) Integrating Eq. (4) leads to the kinetic equation

.~.~.at.- [ "~1~ + a - ~ ~ + t ]

where A = [i + p61~) z12; R(pf/p0)8;(Pg/<)(2 + ] < = ~/2R. Suppose that nf is related to pf as follows:

nf = P0 =I GPa, P = Pf/P0.

The unknown parameters in Eq. (5) are determined by comparing theoretical and experi- mental p-s dependences (p is the amplitude of the perturbing SW; s is the transition length from the initiating SW to a detonation SW). The theoretical dependence p(s is found by the model of [13], where the possibility of using a simplified model of the overtaking of the rear unloading wave by the initiating SW is shown. The relation obtained between the time for the appearance of the rear wave ~ and the distance s at which it is overtaken by the initiating SW is

s TD(a.+ 2b=)/b~. \ " (6)

Here D is the velocity of the initial SW; a and b are parameters in the relation D = a + bu for the initial explosive; u is the mass velocity of the medium.

In the present work, it is assumed that the appearance of detonation corresponds to the overtaking of the reaction wave by the initiating SW. A reaction wave is understood to mean propagation of a definite level of decomposition of the material x... in space. This description agrees with the data of [14], where the correctness of the given approach was verified on the basis of consideration of the decomposition process of the explosive.

55

Denoting the time of motion of the level x~ by t.~, and using Eqo (6), a relation is ob- tained between the length of the transitional section s and tr

l , = t , O (a + 2bu)/bu. ( 7 )

After substituting Eq. (5) into Eq. (7) when x ffi x~, an expression for the theoretical p-s dependence is obtained

8t [ z,/.~+A--t A+l]Da+2bu z,= ~ln T~S:~7./~ a - t j b. " (S)

Comparison of the experimental and theoretical p-s dependences shows that xr has no in- fluence on the accuracy of the approximation; therefore, this method is only used to find A = 6t/~; K = ~/2R; 8. These parameters are used to estimate the rate of energy liberation directly behind the SW front.

The rate of decomposition of the explosive at the front, according to Eq. (4), is found from the relation

(dz /d t ) f =. n f /S t = R (pf/po)"/St.

Expressing R/6t in terms of the parameters of the model, it is found that

(dz/dt) f = (pf/p,) ~ 2 x A . ( 9 )

In [15, 16] , e x p e r i m e n t a l v a l u e s o f t h e k i n e t i c c h a r a c t e r i s t i c ~ [5] a t t h e f r o n t o f t he initiating shock wave were given for cast TNT. The relation between ~ and the rate of energy liberation is [15]

~=rpQ.., (i0)

where F is the Gruneisen coefficient; p, density; 0p,V, rate of energy liberation at constant pressure and volume. Using the expression

Q~. , = Q . dz /d t (11)

as an estimate, where Q is the specific heat of explosion of TNT, and taking account also of Eqs. (9) and (I0), it is found that

= FpQ (Pf/Po) P/2xA. (12)

Using Eq. (12), x, may be found on the basis of the condition of best agreement of the theoretical - Eq. (12) - and experimental [15, 16] dependences of ~ on pf at the SW front.

The parameters of the model are calculated and tested for cast TNT of density 1.62.103 kg/m 3. Data on the p-s dependence are taken from [17], on the D-u relation from [18], and on the dependence F(v) from [19]; Q ffi 4.19"103 kJ/kg.

In Fig. i, theoretical(1) and experimental (2, 3) dependences of ~ on pf are shown. In plotting curve 3, account is taken of the dependence of the sound velocity on the specific volume in accordance with the data of [19]. Curve 2 corresponds to the estimate of ~ for the maximum and minimum possible sound velocities. The calculation employs the model param- eters A = 139,665 ~sec, 8 ffi 3.12, < = 0.02, x~ ffi 0.3.

Supposing, as in [20], that the action on the explosive of a SW of pressure amplitude pj corresponding to the Chapman-Jouguet point is analogous to the action of a detonation wave, the chemical-peak time (tch) should follow from Eq. (5) at pj. Estimation using the parameter values obtained leads to the value tch = 0.2 ~sec. The time of chemical reaction in the detonation wave determined in [5] is 0.26 ~sec, i.e., the theoretical time is in good agreement with the experimental result~

In constructing the model, it is assumed that detonatiDn is ensured by decomposition of the material to some degree xe. Hence, energy liberation Q, corresponding to this de- gree of decomposition must be sufficient to maintain steady detonation-wave parameters. Since x, ~ 0.3 is found, it follows that in the approximation in Eq~ (Ii)

Q, = Q.O,3 = 1,26.tO'm'2/S Pa

where Q = 4.19"103 kJ/kg is the specific calorimetric heat of explosion of TNT.

56

GPa

r

4O

5

5 10 f5 .of pGPa

Fig. i

It is clear from gasdynamic considerations that the specific energy required to main- tain the SW with steady-detonation parameters is uj=/2, where uj is the mass velocity corre- sponding to the Chapman-Jouguet point. Using uj = 1.62"103 m/sec from [5], the estimate obtained for the required energy is -1.31"106 m2/sec 2.

The value obtained is close to Q,, which supports the value x, = 0.3 found using the model here proposed.

Thus, a model of the decomposition of explosive based on the concept of a b;anched chain mechanism of increase in the number of reactive particles is proposed in the present work. The model has been tested for data on cast TNT. Good agreement of the theoretical and experi- mental chemical-peak time is found. The prediction of the model regarding the degree of decomposition of the explosive ensuring the maintenance of steady-detonation parameters is in agreement with the gasdynamic estimate.

LITERATURE CITED

I. A. A. Vorob'ev, V. S. Trofimov, K. M. Mikhailyuk, et al., Fiz. Goreniya Vzryva, 21, No. 2, 106 (1985).

2. A. A. Vorob'ev, K. M. Mikhailyuk, and V. S. Trofimov, in: Ist All-Union Symposium on Macroscopic Kinetics and Chemical Gasdynamics [in Russian], Vol. 2, Chernogolovka (1984) Part i.

3. A. N. Dremin and G. I. Kanel', Fiz. Goreniya Vzryva, 13, No. i, 85 (1977). 4. J. Uokerli, G. Johnson, and P. Khallek, in: Detonation and Explosives [Russian trans-

lation], Mir, Moscow (1981). 5. A. N. Dremin, S. D. Savrov, V. S. Trofimov, et al., Detonation Waves in Condensed Media

[in Russian], Nauka, Moscow (1970). 6. V. S. Trofimov, Fiz. Goreniya Vzryva, 17, No. 5, 93 (1981). 7. V. S. Solov'ev and V. N. Postnov, in: Chemical Physics of Combustion and Explosion

Processes. Detonation [in Russian], Chernogolovka (1977). 8. M. V. Batalova, M. S. Bakhrakh, and V. N. Zubarev, Fiz. Goreniya Vzryva, 16, No. 2,

105 (1980). 9. V. F. Lobanov, S. M. Karakhanov, and S. A. Bordzilovskii, Fiz. Goreniya Vzryva, 18,

No. 3, 90 (1982). 10. L. Grin, E. Nidik, E. Li, and K. Tarver, in: Detonation and Explosives [Russian trans-

lation], Mir, Moscow (1981). ii. R. Chaiken and G. Edwards, in: Detonation ~nd Explosives [Russian translation], Mir,

Moscow (1981). 12. S. Bon, in: Solid-State Chemistry [Russian translation], V. Garner (ed.), IL, Moscow

(1961). 13. J. B. Ramsay, Acta Astron., 6, 771 (1979). 14. W. H. Andersen, Prop. Explos. Pyrot . , 9, No. 2, 39 (1984).

57