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250 FIZIKA GORENIYA I VZRYVA
DETONATION AND SHOCK ADIABATS OF THE PRODUCTS OF RDX
N. M. Kuznetsov and K. K. Shvedov
Fizika Goreniya i Vzryva, Vol, 5, No. 3, pp. 362-369, 1969
UDC 534.222.2+532.598
A knowledge of the detonation and shock adiabats of the reac t ion products is impor tant for solving theore t - ical and prac t ica l problems associated with the use of explosives. The detonation adiabats can be used in ana- lyzing possible detonation reg imes , while the shock adiabats of the reac t ion products are n e c e s s a r y to es tabl ish a quanti tat ive re la t ion between the detonation p a r a m e t e r s of a given explosive and the p a r a m e t e r s of the shock waves created at the interface in different media.
We have calculated the detonation and shock adia- bats of the detonation products (DP) of RDX charges of vary ing densi ty (Pl). In these ca lcula t ions ,we employed the previously obtained [ 1] equation of state of the DP of RDX. The problem reduces to the joint solut ions of the equation of the detonation
! e - e , = T p ( v ' - . v ) + O(v,) O)
or shock
1 e - e , = y Ip + p , ) ( v . - v) (2)
adiabat and the equation of state
1 L P = Po 4- -~- "f (E -- Eo - ~),
Po = 15,4p 3 - 12, 69s 1000 arm
Eo = 0,77 p2 _ 1,26 p k J / g 2 ,_ o.~4ro [ Qe,pO ]
(exp 0 " ~ l) I . eal/mole exp 0 - - I
R~
e~ + 0.3 R (~ - 1)
0 = 3200 T = 103 �9 7"* T
2.7 '~4 q- 15 't:' ,~ := 0.72 p (T*) -~ ? = I 4- 1 + 2 . 6 x , '
T* = 0,3 p - Poe_, P~
�9 1.76 R 02 exp 0 c~, = ~ 2 , 6 2 4 R .
(exp 0 - - 1) ~
(3)
Here, v t, E l a n d v . , E , are the specific volume and specific in terna l energy for the s ta r t ing state of the explosive and for the state at the Chapman-Jouguet (C-J )po in t s , respect ive ly ; p . is the p r e s s u r e at the C-J points; p, v, p, E and T are the p r e s s u r e , spe- cific volume, densi ty, specific in terna l energy, and t empera tu re of the DP respect ive ly ; Q(vl) is the heat of explosion; P0 and E 0 are the e las t ic components
of p r e s s u r e and energy which depend only on density; T is the Gri ineisen constant of the DP;
T
a -= ~ (c,,/~) d 7 - c,, r/~; U
q~ - ~0 (p, T) is a function en te r ing into the express ion for the thermal component of the p r e s s u r e
p = po + v (v, T) ~ R----Z-r, (4)
for an ideal gas with p = 0 and q~ = 1; Cv* is thespe- cific heat of the ideal gas mixture ; and~ is the molec- u l a r weight of the DP. The heat of explos ion is assumed to be a known function of the charge densi ty Q = 1204 + + 172 Pi c a l / m o l e [2]. As in cons t ruc t ing the equation of s tate, it is assumed that the composi t ion of the Dp r ema ins constant as p va r i e s . For given values of v 1 and E 1 or v . , E . , p . , the sys tem of equations (1), (3) or (2), (3) makes i t possible , by ass igning a r b i t r a r y values of p, to find the cor responding values of p, E, and T.
Table 1 p re sen t s the resu l t s of a ca lcula t ion of these pa rame te r s for different values of Pl together with values of the mass veloci ty U. The pa rame te r s on the detonation (g) and shock (y) adiabats are p re - sented for each densi ty g rea te r than p . . Moreover , the rows corresponding to the symbol (c) give values of p and U calculated f rom the power law
p = A ~, (5)
where A and n are constants . The ranges of var ia t ion of the densi t ies and other
pa r ame te r s cor respond to all possible cases of re f l ec - tion of the detonation waves f rom dense media with different compress ib i l i t i e s up to an absolutely incom- p res s ib l e medium.
By way of example, Fig. 1 shows the detonation adiabats , which are cha rac te r i zed by an intense in- c rease in p r e s s u r e as the volume dec reases . Fo r Pl = 1.0 g / c m ~, an inc rease in the veloci ty of the front by 10% on the lower branch cor responds to an approx- imate ly 50% decrease in p r e s s u r e . This sens i t iv i ty of the p r e s s u r e to changes in the veloci ty of the front can be used in an exper imenta l invest igat ion of the poss ib i l i ty of rea l i za t ion of unde rcompres sed detona- tion reg imes (we note that above-normal f ront veloc- i t ies were observed in ce r t a in cases of detonation of low-densi ty condensed explosives in [3]).
COMBUSTION, EXPLOSION, AND SHOCK WAVES 251
h, g / c m 3
1,8
1,6
1,4
Table 1
p, g / e r a 3
3.2
3.0
2,8
2.6
2.39 t,8
3.0
2,8
2,7
2.5
2.12" 2.0 1.6
2.6
2.4
2.2
P a r a m e t e r s
g Y C
g Y C
C
C
C
C
C
C
C
C
C
P, 1 0 0 0 a t m
78?.3 737.2 836.7
629.7 611.9 678,5
507.1 5{}3.3 562,5
409,2 409.0 437.6
343,2* 164,5
742.2 667.5 729,5
582.9 548. I 582.3
515.8 495,7 529.6
407.8 402.0 420.8
264.8* 224.6 125.3
545,5 496.6 543,0
419.6 398.5 426.3
335.7 318.0 328.6
U. Km/sec
4.36 4,21 4.46
3.74 3.68 3,86
3.11 3.15 3, 30
2.64 2,63 2.73
2.17" o o
4,65 4.40 4.64
3.95 3,82 4.0
3.65 3,55 3,75
3,03 3.0 3,15
2 , 0 3 * 5,30
4,23 4.0 4.13
3.53 3.43 3.54
2.95 2.88 2,93
E , k J ]g
15.78 14.38
13,31 12.72
11.34 11.23
9.81 9.86
8.4" 6.32
17.0 14,54
14,0 12.79
12,73 12,0
10.76 10.54
8.21" 7.58 6.18
14,98 13.10
12.25 11.4
10.4 9.98
T.IO ~, ~
5.8 5.2
5,0 4.7
4.4 4.3
4.0 4.0
3 . 9 " 3.6
6.9 5,6
5.8 5,2
5.3 4,95
4.8 4,6
4 , 1 " 4,0 3.8
6.8 5.8
5.8 5.3
5.5 5.0
252 F I Z I K A G O R E N I Y A I V Z R Y V A
T a b l e 1 ( c o n t ' d )
Pl, g/cm 3 p, g]cm 3 Parameters p, 1000 atm U, kin/see E, kJ/g T.10 a, ~
251.2 2.32 8,73 4 .8 2 ,0 249.3 2 .36 8 ,65 4.73
1 , 4 248.4 2.21 - - - -
,89" - - 2 0 3 . 6 " 1.91" 7,85* 4,2* .4 92 . l oo 6 ,04 4 .0
O 533.6 4.71 17,0 8 .9 2 .4 y 44~}.7 4 .20 13,13 6 .4
b 417,6 4 , 1 0 - - - -
0 392.1 3 . 8 5 13.3 7 .2 2 ,2 y 3-18.7 3.59 t i . 3 4 6 .0
356.4 3,53 - - - -
1 . 2 0 292,4 3.12 10.74 6 ,5 2 , 0 y 271.8 2.98 9.76 5 ,5
C 257.5 2.91 - - - -
0 20%6 2,43 8 .79 5 .8 1 , 8 y 186.8 2.28 7.52 4 .5
C 190.7 2.20 - - - -
1.628* - - 144,5" 1 , 7 8 ' 7 , 4 " 4.5* 1.20 63.8 oo 5.,~9 4 .0
0 3?2.1 3 .95 13,38 9 .4 1.9 y 268.8 3.47: 10,47 6 .7
C " 223.9 3,23 - - - -
0 2 5 1 . 1 3 , 6 5 11.32 7 ,5 1 . 8 y 2 z l , l 3 1 9,55 6 .0
1 .0 c 192.5 2.91 - - - - 0 1;,6,8 2 .42 8 .86 5 .4
1 ,6 y 1-19.5 2 , , ,6 8.13 5.1 C 143.0 2.32 -- - -
l , : 3 9 * - - lO0.O* 1,68" 6.(~5 * 4 .6" 1.20 66,2 3 .32 6,30 4 .2 1,0 42.7 co 5;75 4 ,3
0 20~.4' 3.61 12, 1 9.1 1 ,6 y 176,9 3,22 '9.82 6 .8
C 142,6 2.95 I - - - - I
0 163,0 3. I 10.35 7. 8 1 . 5 y 154.8 2,85 8.98 6 .5
c 122;8 2,65 - - - -
0 . 8 0 122,5 2,56 8.88 6 .3
8 .24 5 .7 1 ,4 y 115.7 2.45 c ] 104.3 2 ,35
l , 14" - - 64 .0" 1.54" 0 ,8 24.8
*These Values correspond to C h a p m a n - - ~ o u g u e t points .
6 . 7 8 * L 5 . 6
5 ,0" 4,1
COMBUSTION, EXPLOSION, AND SHOCK WAVES 253
In Fig. 2, the shock adiabats are shown in p-U co- ordinates . The f igures on the curves indicate the charge density. The curves for different Pl are s i m i l a r and can be descr ibed by an in terpola t ion fo rmula of the form
p=kU +4U2 + U a. (6)
The coefficient k is a function O f Pi- In the range pi = = ( 1 . 2 - 1 . 8 ) g / c m 3, k = l l 7 P l - 71 a n d k = 9 6 p l - 45 a t p l = ( 0 . 7 - 1 . 2 ) g / c m a.
According to the exper imenta l data of [l] , the depen- dence of the p r e s s u r e at the C-J point on charge densi ty is expressed by the in terpola t ion formula
p = 22 Pl + (7) 1 + 0.2 p~
Express ions (6) and (7) make it possible to calculate , without making any m e a s u r e m e n t s , the kinetic p a r a m - e te r s of the shock wave created in var ious media by the detonation of RDX charges of any densi ty, ff the shock adiabats of the media are known. Such ca lcu la - t ions are most eas i ly made by a graphic cons t ruc t ion on the p-U diagram. F igure 2 p resen t s the shock adia- bats of copper, a luminum, and paraff in (p) taken f rom [4-6]. The in te r sec t ion of the dece lera t ion or expan- sion curves drawn through the c - J point and the shock adiabats of the ma t e r i a l s gives the cor responding values of U and p (Table 2).
The exper imenta l values of the mass velocity in the above-ment ioned media associa ted with RDX charges
2 t
400 . . . . . . .
I e
200 ] --
O, Z5 0.50 0.7,5 o,crn3/~g
Fig. 1. Detonation Hugoniots of RDX and Michelson (Rayleigh) l ines : 1)01 =
= 1.6 g/cm3; 2) Pl = 1.0 g / c m 3.
of var ious densi t ies are as follows: UA1 = 1.84 k m / s e c atp l= 1.767 g/cm3; UCu = 1.0 k m / s e c a tp l = 1.72 g /
/ c m 3 [8]; Up is equal to 3.16 and 1.9 k m / s e c atpl = 1.7 and 1.0 g / cm 3 , respect ive ly . In meta l s , these values of the mass veloci t ies were obtained by the plate veloc-
6O0
\ \ .
400
" , . . . ,
200 . ...
o I ] f 2 J U,.km/sec
Fig. 2. Shock adiabats of the detonation p ro - ducts of RDX and cer ta in solids in p-U coordi - nates : 1) calculated f rom the formula p = Apn;
2) Chapman-Jouget points .
ity method [7], and in paraffin, by the e lec t romagnet ic method [9] f rom the point of discont inui ty on the U = = U(t) profile at the explos ive-paraf f in in terface . It should be noted that, in the exper imenta l de te rmina t ion of mass veloci t ies in meta ls by the plate veloci ty method, ser ious d is tor t ion of the r e su l t s is poss ib le owing to the effect of the p r e s s u r e spike and the ref lected shock.
Clear ly , the most re l iable resu l t s will be those ob- tained for quite thick plates, when the elevated p res - sures are completely damped, and for charges with ra the r large d iamete rs and lengths, in which case the drop in m a s s velocity in the expansion wave beyond the C-J point of the detonation wave and its analog in the metal will be smal l , and a Substantial difference be- tween the thickness of the plate on which the m e a s u r e - ments are made and the th ickness cor responding to the posit ion of the C-J point will not have much effect on the resul t .
These r equ i r emen t s are completely sat isf ied by the data of [7] (plate th ickness i = 2o 5 mm; an inc rease or decrease i n / b y approximately 2.5 mm does not affect the r e su l t ) . The mass velocity in copper was obtained for a plate th ickness of 5 ram, which c lea r ly is also close to the Chapman-Jouguet state. A compar i son of the above calculated and exper imenta l values of the mass veloci t ies shows that they coincide within the
Table 2
g/cm~ P AI, PC;u, Up, kin/see PP' P'" UAl'km/sec 1000 atm UCu, krn/sec 1000 atm 1000 atm
I 1,75 ] 1.85 1,60 I 1.62 1.40 1,38 1,20 1.14 1.0 I 0,90 0,8 ] 0.68
393,0 324.0 272.0 208,0 160,0 116,0
1.08 0.92 0,79 0,62 0,48 0,34
528,0 440.0 355.0
266:0 200.0 140.0
3,00* / !90,0" 2,76 170,0 2,47 142,0 2,12 114,0 1.82 92.0
1,47 68,0
*pj = 1,7.
254 FIZIKA GORENIYA I VZRYVA
l imi t s of exper imenta l accuracy (3-5 %). Us ing the shock adiabats obtained for DPof RDX, we can calculate with the same accuracy the k inemat ic pa r ame te r s of the shock waves in other media.
g. 6
/ / / /
j t I
/ 200 400 600 p40~ atm
Fig. 3. P r e s s u r e - t e m p e r a t u r e dependence on the shock adiabats of the detonation pro- ducts (the f igures on the curves indicate the
charge densi ty.
It is of in te res t to es t imate the accuracy of the f r e - quently employed s impl i f ied calculat ion of the shock wave p a r a m e t e r s in p-U coordinates which uses the ptD l ines (D is the detonation velocity) instead of the shock adiabats of the DP. Obviously, the e r r o r of these calculat ions depends on the re la t ion between the densi ty and compress ib i l i t y of the detonation products and the medium. The dashed l ines in Fig. 2 r ep re sen t m i r r o r images of the shock adiabats of the DP and the p ~ l ines for RDX charges with pt = t . 6 and 1.0 g / cm 3 drawn through the co r r e spond ingC-J points. As may be seen f rom the f igure, in paraff in, these l ines give a lmost ident ical values of Q and p. In 1, the p 1D l ines give values of p that are too low by about 3% for Pl = 1.6 g / c m 3 and by about 8%for pi = 1.0 g / c m 3. In copper, these d i sc repanc ies are even g rea te r and consti tute about 7 and 12%for Pi equal to 1.6 and 1.0 g / c m 3, respect ive ly . These compar i sons indicate that the use of p iD l ines instead of the shock adiabats of the DP fo rca lcuIa t ing the shock p a r a m e t e r s in var ious media may lead to substant ia l ea ro r s .
We now analyze another method of approximately cons t ruc t ing the shock ad iaba t s of the Dp, in which a power- law re la t ion between p r e s s u r e and density is employed ( Eq . ( 5 ) ).The constants A and n in (5) are de termined f rom the p a r a m e t e r s at the C-J point. As may be seen f rom Table 1, at large values of p (150- 300 thousand a tm) both p and U on l ines (5) differ f rom the values calculated f rom Eqs. (2) and (3) for the same p. However, in p-U coordinates , in the p re s - sure range in which the calculat ions were made, curves (5) and the shock adiabats of theDP coincide c o r r e c t to about 3%. This is very impor tant , s ince the above- ment ioned graphic method of calculat ing the k inemat ic pa r ame te r s of the medium is usual ly ca r r i ed out in p-U coordinates . At lower p r e s s u r e s (50 < p , < 100 thousand atm), the difference in the behavior of the cor responding curves becomes more marked even in
p-U coordinates . This is quite apparent f rom Fig. 2. At p, approximately equal to 100 000 atm or less , n is a l ready considerably reduced [1]. If on l ine (5)we as - sume that the p a r a m e t e r n depends on density and purely formal ly subs t i tu te in (5) the va r iab le values of n i P ) d e t e r m i n e d exper imenta l ly f rom C-J po in t s , the curves in quest ion again a lmost coincide.
Thus, in t h e p r e s s u r e range where n = const or the re la t ion n = n(p) is known, polytrope (5) can be successful ly employed to calculate the p a r a m e t e r s of the ref lected shock waves with sufficient accuracy.
The shock adiabats calculated by the two methods in p-U var iab les coincide, despite the fact that the equation of state (3) has l i t t le in common with a power- law dependence of p on p. Even at high p r e s s u r e s , the the rmal par t of the p r e s s u r e is not less than the e l a s - tic component , for which, s t r i c t l y speaking, the power- law dependence of p o n p i s assumed. However, as noted in [10],the product Tq~(p, T) in the express ion for the thermal par t of the p r e s s u r e PT ~ pT q~(p, T) on the cons tan t -en t ropy l ines is also a lmost a power function of p, and, at the ampli tudes in question, the change of entropy in the ref lected waves is smal l .
Assuming that a s i m i l a r p ic ture is also observed for other explos ives , we can calcula te their shock adiabats on the bas is of data on the dependence of p, p, and n at the C-J points on pt. For TNT, at p r e s s u r e s cor responding to a charge densi ty of 1 . 3 - 1 . 6 g / c m 3, and at higher p r e s s u r e s , the coefficient of polytropy can be assumed constant and equal to 3.3 [9]. The dependence of the detonation velocity on p i in this range of densi t ies has the fo rm: D = ]872 + 3187 Pl m / s e c [11]. These data are sufficient to de te rmine the detonation p a r a m e t e r s and cons t ruc t the shock adia- bats in accordance with a re la t ion of type (5). In p-U coordinates , the shock adiabats of the DP of TNT charges with Pl = 1 . 3 - 1 . 6 g / c m 3 have the form
p=kU+4 Us+ 1,2 U 3 1000 atm
k= 110,7 pi--74.
A graphic solution using these adiabats gives the values of the mass velocity in a luminum and copper p resen ted in Table 3. The calculated and exper imenta l values of U coincide co r r ec t to 3.5 %.
The re la t ion between t empera tu re and the p r e s s u r e on the shock adiabats at varous charge densi t ies is
Table 3
r kin/see
p,, g/cm 3 :calen
copper
I expt.[8]
I 1 ;59 0,70 1 0.71 1,45 0.5~ 1 0.60 1.35 0.50 0.50
Aluminum
colon, expt. [ 12]
1.22 1.2 1.05 0.93
presented in Fig. 3. The shape of the curves depends on Pl- The sharp var ia t ion of t empera tu re on the shock adiabat at re la t ive ly smal l p (50-150 thousand aim Pt =
COMBUSTION, EXPLOSION, AND SHOCK WAVES 255
= 0 . 8 - 1 . 2 g /cm ~ is at tr ibutable to the fact that at these p res su res the tempera ture is higher than at higher p res su res and the Grilneisen constant is va r i - able.
Qualitatively the same picture should be observed for the heating of the explosive compressed in the det- onation front, since the equations of state employed are also suitable for descr ibing a solid. This should be taken into account in determining the pa ramete r s , and especia l ly the react ion t ime, in a detonation wave by the plate velocity method. The shock waves ref lec t - ed from the metal b a r r i e r will increase the p re s su re and temperature of both the explosive i tself and the products in the react ion zone, which may cons ider - ably accelera te the decomposition of the explosive and disturb the s t ruc ture of the detonation wave [12].
REFERENCES
1. N. M. Kuznetsov and K. K. Shvedov, FGV [Combustion, Explosion, and Shock Waves], 2, 4, ] 966.
2. A. Ya. Apin and Yu. A. Lebedev, Dokl. AN SSSR, 114, 4, 819, 1957.
3. K. K. Andreev and L. F. Belyaev, Theory of Explosives [in Russian], Oborongiz, Moscow, 1960.
4. R. G. McQueen and S. P. Marsh, J. Appl. Phys . , 31, ]253, ]960.
5. A. V. Al ' t shuler , A. A. Bakanova, and R. F. Trunin, Doki. AN SSSR, 121, 1, 1958.
6. A. N. Dremin and I. A. Karpukhin, PMTF, 3, 184, 1960.
7. W. E. Deal, J. Chem. Phys . , 17, 3, 1957. 8 . V . S . Ilyukhin, P. F. Pokhil, et a l . , Dokl. AN
SSSR, 131, 4, 1960. - 9. A. N. Dremin and K. K. Shvedov, PMTF, 2,
1964.
10. N. M. Kuznetsov and K. K. Shvedov, FGV [Combustion, Explosion, and Shock Waves], 3, 2, 1967.
11. M. I. Urizar , E. James , et a l . , Phys. of Fluids, 4, 2, 1961.
12. V. A. Veretennikov, A. N. Dremin, and K. Ko Shvedov, FGV [Combustion, Explosion, and Shock Waves], 3, 1965.
18 October 1968 Moscow
