COMBUSTION, EXPLOSION, AND SHOCK WAVES 125

ISENTROPIC EXPANSION OF THE DETONATION PRODUCTS OF RDX

N. M. Kuznetsov and K. K. Shvedov

Fizika Goreniya i Vzryva, Vol. 3, No. 2, pp. 203-210, 1967

UDC 534.222.2

In [i] the authors obtained an equation of state for

the detonation products of RDX that is suitable for

calculating the parameters of state and motion of the

detonation products (DP) over the entire density range from the Jouguet points (for any possible charge den-

sity) to zero density. The equation of state [i] expresses, in transcen-

dental form, the relation between internal energy E,

pressure p, density p and temperature T and makes

it possible to obtain any of these functions from two

others. It does not contain the entropy S in explicit

form and in order to use it to obtain isentropic rela-

tions it is necessary to employ a differential equation

expressing the condition of constant S, for example,

d E = p a ~ (1) p~

Equat ion (1), or soroe o ther equiva lent r e l a t i on , cannot be i n t eg ra t ed ana ly t i ca l ly but only by n u m e r i - cal methods .

This paper p r e s e n t s the r e s u l t s of a n u m e r i c a l in - t e g r a t i o n of Eq. (1) with the equat ion of s ta te [1]. The density dependence of p, the thermal part of the

pressure PT, the work of adiabatic expansion of the

DP R = - - p - - ~ , the speed of sound c, and the

i s e n t r o p i e exponent n have been obtained by m e a ns of c o m p u t e r ca lcu la t ions for a fami ly of ad iabat ic e u r v e s . Two v a r i a n t s of the ca lcu la t ions were made for the two m o s t typ ica l p r o b l e m s in which the data ob ta ined can be used.

The f i r s t v a r i a n t (Table 1) d e s e r i b e s the v a r i a t i o n of the p a r a m e t e r s for i s en t rop i c expans ion of the DP f rom the Jouguet points . Each individual par t of Table 1 is e h a r a e t e r i z e d by i ts own value of the charge den- s i ty Pl, which e o r r e s p o n d s to c e r t a i n va lues of the

dens i ty Pz of the DP and the o ther p a r a m e t e r s at the ,Jouguet point. (The r e l a t i o n between Pl and p~ used at the head of each table was taken f rom [11.) The l ines of cons t an t en t ropy of Table 1 a re r e q u i r e d for s tudying the gas d y n a m i c s and k ine t i cs of expans ion

of the DP. The second v a r i a n t (Table 2) r e p r e s e n t s i s en t rop i c

expans ion f rom in i t i a l s ta tes in which the dens i ty of the products is equal to the density of the charge, and

the internal energy to the heat of explosion (~instan-

taneous explosion"). Such states are obtained, for

example, after equalization of the pressure upon ex-

plosion of a charge in a rigid case. Table 2 describes

the variation of the thermodynamic parameters of the "gas bubble" formed under these conditions when its expansion is sufficiently slow (so that the state of the

explosion products at any moment of time does not

depend on the coordinates). In particular, if the vol-

ume and the density of the charge are known, then,

from the data of Table 2, it is possible to find the

pressure of the explosion products as a function of the volume of the expanded "gas bubble." Each indi- vidual part of Table 2 gives one adiabatic curve of the explosion products with initial density equal to a cer-

tain charge density.

{ !

:~ 3,5 1,9 "L5 2.0 g/era 3

Fig. 1

The error of the tabulated data as a whole is de-

termined by the error of the equation of state of [11. The related error in the value of n means that the

pressure at points p < p2 on the isentropic curves is

determined with a certain error which for given/)2 increases with decrease in po Estimates show that

for expansion to p = 0 the maximum error s is

I0-15%. For illustration Fig. 1 shows the curves of varia-

tion of pressure divided by density for expansion from

the Jouguet points. Each curve corresponds to a cer-

tain value of the density and other parameters at the Jouguet point. Similar curves are also obtained for

expansion from initial states corresponding to "in-

stantaneous explos ion ."

The data obtained make it poss ib le to t r ace the de- pendence of n on dens i ty on the l ines of cons tan t en - t ropy. In o r de r to s impl i fy the gasdynamic ca l cu la - t ions it is often a s s u m e d that n = cons t , or even m o r e spee i f i ea l ly n = 3, s ince in this case the computa t ions are simpler. The justification for this approximation is usually based on the relation between p and p for the state in which the temperature is so low that the

thermal pressure is small as compared with the

pressure of elastic repulsion of the compressed mo- lecules. In [i] it was shown that in the DP of RDX the

thermal pressure at the Jouguet points is not less

than the elastic pressure even at maximum charge

density. To a considerable extent this also applies to

the exp~%nding detonation products and lower charge

densities. As may be seen from Tables 1 and 2, the ratio of the thermal part of the pressure to the total pressure during expansion of the DP from maximum

densities rapidly increases and reaches unity at p -~

126 FIZIKA GORENIYA I VZRYVA

Table 1

p, g / c m 3 T.I~,'~K PT, 1 0 0 0 a t m p, 1 0 0 0 a t m R, k J / g

3 4 5

c, km/sec

2.4 2.2 2.0 1,8 1.6 1,4 1.2 1.0 0.8 0.6 0.4 0.2 0.05

2.2 2.0 1.8 1.6 1,4 1,2 1.0 0.8 0.6

0.2 0.05

2.0 1,8 1,6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.05

1.8 1.6 t,4 1.2 1.0 0.8 0.6 0.4 0.2 0.05

1.6 1.4 1.2 1.0 0.8

4.16 3.83 3.51 3.20 2.91 2.64 2.40 2.18 1.96 1,74 1.49 1.16 0.71

4.26 3.91 3.58 3.27 2,98 2.72 2,48 2.24 2.00 1.72 1.36 0,86

4.35 4.00 3~67 3.36 3,08 2.82 2.56 2.29 1.98 1.57 1.02

4.46 4.10 3,78 3.48 3.19 2,91 2.60 2.26 1.81 1.19

4,58 4,23 3.91 3.75 3.44

P l = 1.8 g / c m 3, P2 = 2 .4 g / c m 3

208,1 171.0 137.5 107.5 81.10 58,43 39.67 25,04 14.54 7.654 3.463 1.068 0.134

348.4 0.000 274.0 1.18 210,3 2.28 156.5 3.29 111.9 4.21 75.99 5.03 48.14 5.75 27.84 6.36 14.36 6.87 6.445 7,27 2.433 7.59 0.687 7.90 O. I O4 8.27

P l = 1 .66 g / c m 3, P2 = 2 .2 g / c m 3

187.2 150.3 117,3 88.30 63.47 43.05 27.25 15.96 8.518 3,917 1.232 0.160

290.2 0,00 223.1 1.17 166,3 2.24 119.1 3,22 81.04 4.10 51,53 4,87 30.O6 5.23 15.78 6.07 7.308 6.52 2.887 6.9O 0.851 7.27 0.130 7.73

P l = 1.51 g / c m 3, P2 = 2 .0 g / c m 3

163.9 127.6 95.89 68.79 46.65 29.64 17.50 9.462 4.415 1.412 0.189

236.7 0,00 176.6 1.15 126.7 2,19 86.36 3.13 55.12 3,95 32.43 4.65 17.32 5-25 8.252 5.74 3.385 6,18 1.032 6,63 0,159 7,18

P 1 = 1 .34 g / c m 3, P 2 = 1.8 g / c m 3

138.5 103.8 74.35 50.44 32.18 19.16 10.49 4.955 1.608 0.220

187.5 0.00 134,6 1.12 91.91 2.11 58.91 2.99 34.97 3.75 18.98 4.39 9.276 4.94 3.925 5.44 1.227 5.97 0,190 6.63

P l = 1 .18 g / c m 3, P2 = 1.6 g / c m 3

111,9 142.7 0.00 80.04 97,60 1.07 54.35 62.83 2-00 43.84 49.09 242 27.23 28,25 3.17

6.32 5.87 5.41 4.95 4.48 3,99 3-46 2.89 2-28 1,68 1:14 0.73 05I

6.02 5.56 5.09 4.61 4.10 3,56 2.97 2,35 1.75 1.22 0.80 0,57

5.71 5.24 4,74 4.22 3-66 3.05 2.43 1.82 1,29 0.88 0-63

5.39 4.88 4.34 3.76 3.14 2.50 1 90 1,37 0.95 0-69

5.02 4.46 3.86 354 2.90

2.75 2.77 2.79 2.82 2,87 2.97 2.99 3.00 2.90 2,64 2,16 1.54 1.26

2.75 2,77 2.81 2,85 2.91 2.95 2.94 2,81 2 52 2,06 1 52 1,26

2.76 2,79 2.84 2.89 2.92 2.88 2.72 2,42 1-98 1-49 1.26

2.79 2.83 2.87 2.88 2.82 2.64 2,33 1.92 1.48 1.26

2.82 2.86 2.85 2.82 2.67

0 II

II II

�9

0

C~

�9

�9

CfJ

�9

�9

P~

<~

128 FIZIKA GORENIYA I VZRYVA

-~ 0.8 g / c m 3. The r e l a t i o n be tween the t o t a l and t h e r - m a l p r e s s u r e s and the t e m p e r a t u r e of the expanding DP for the c a s e P2 = 2.2 g / cm 3 (P l= 1.66 g/era ~) i s i l - l u s t r a t e d in Fig . 2, which shows the r a t i o s pip , pT/p , and T a s funct ions of Oz. C l e a r l y , o t h e r e x p l o s i v e s wi th high c a l o r i e i t y have a s i m i l a r r e l a t i o n be tween the e l a s t i c and t h e r m a l p a r t s of the p r e s s u r e .

~o/,0~ 1 0 0 0 a r m /r.~ g/era 3

100 P/2/~ i" 40

50 .i . ~/ 2 . 0

0 0 0.5 "I,0 1.5 2.0 g]cm 3

Fig . 2

In sp i t e of the fac t that the equa t ion of s t a t e of [1] has l i t t l e in c o m m o n with the p o w e r dependence of p on p, as fo l lows f r o m T a b l e s 1 and 2, the va lue of n

v a r i e s l i t t l e up to p = 0.8 g /e ra ~ even at ! a r g o in i t i a l d e n s i t i e s c lo se to 2 . 7 - 3 .

H e r e the quant i ty n i s m a i n l y of t h e r m a l o r ig in . In the e x p r e s s i o n [1] for the t h e r m a l p a r t of the p r e s s u r e ~T ~ p T ~ ( p , T ) the t e m p e r a t u r e and the funct ion q~ d e - c r e a s e wi th i s e n t r o p i c expans ion to p ~ 0.5 g /cm a, so tha t the v a r i a t i o n of PT i s p r o p o r t i o n a l to p to the p o w e r 2 . 5 - 3 .

I t i s c l e a r f r o m the t abu l a t ed d a t a tha t fo r i s e n - t r o p i c e x p a n s i o n f r o m s t a t e s wi th l a r g e d e n s i t i e s (p -~ - 2 . 4 - 1 . 4 ) the va lde of n p a s s e s t h rough a b r o a d m a x - i m u m . The v a r i a t m n of n fo r expans ion of the DP at P2 = 2.2 g / c m 3 (Pl ~ 1.66 g / cm 3) is i l l u s t r a t e d in F ig . 3.

r , / i / - ?

/ $

.... 2

0 0.5 1.0 1.5 g/era 3

Fig , 3

Q u a l i t a t i v e l y , the p r e s e n c e of a m a x i m u m of n i s in a g r e e m e n t wi th the da t a ob ta ined by Z u b a r e v [2] in s tudy ing the g a s d y n a m i c p i c t u r e of expans ion of the DP behind a s e l f - s i m i l a r de tona t ion f ron t . The m a x i - m u m of n i s v e r y b r o a d and w e a k and t h e r e f o r e i t s t r u e va lue and p o s i t i o n on each a d i a b a t i c c u r v e m a y be qui te d i f f e r en t . The e r r o r s in d i f f e r e n t i a t i n g the a p p r o x i m a t e e x p r e s s i o n s ev iden t l y accoun t fo r a c e r - t a i n d i s c r e p a n c y b e t w e e n the v a l u e s of n(P2) and the p o s i t i o n s of the m a x i m a of the funct ion n = n(p2) ob - t a i n e d by c a l c u l a t i o n and f r o m the e x p e r i m e n t a l r e - l a t i o n n = n(p~) a t the Jougue t po in t s . C a l c u l a t i o n g i v e s a m a x i m u m of n a t P2 - 1.5 g / c m 3, but e x p e r i - m e n t a l l y i t i s o b s e r v e d a t P2 - 1.8 g /e ra 3.

In c o n s t r u c t i n g the equa t ion of s t a t e [1] the s p e e d of sound d e t e r m i n e d f r o m the Jougue t cond i t i on was u s e d on ly fo r d e n s i t i e s g r e a t e r than 1.7 g / c m ~, and

only fo r d e t e r m i n i n g the Gr t ine i sen cons tan t . T h e r e - f o r e the e x p e r i m e n t a l da t a on the s p e e d of sound can be u sed fo r checking the a c c u r a c y of the equat ion of s t a t e .

The s p e e d of sound c a l c u l a t e d by d i f f e r en t i a t i ng the equa t ion of s t a t e of i l l ( see Tab le 1) a g r e e s wi th in 20% wi th the e x p e r i m e n t a l da ta of [1] o v e r the e n t i r e r ange of d e n s i t i e s p; a t the Jougue t po in ts f r o m 0.7 to 2.4 g / cm 3. If we c o n s i d e r tha t in d i f f e r en t i a t i ng the a p p r o x i m a t e e x p r e s s i o n s the a c c u r a c y is u s ua l l y much r e d u c e d , the r e s u l t ob ta ined fo r the s p e e d of sound m u s t be c o n s i d e r e d (at l e a s t as a check on the a c c u r a c y on the equat ion of s t a t e ) p e r f e c t l y s a t i s f a c - t o r y . The variation of the speed oi sound for faun- tropic expansion of the DP from the initial state is shown in Fig. 3.

Figure 4 shows graphs of the ratio of the detona- tion pressure at the Jouguet point to the pressure in the p r o d u c t s fo r an in s t an taneous exp los ion p / p ' and the ana logous t e m p e r a t u r e r a t i o s T / T ' as a funct ion

p l~ ' - - T/ 7"

2.4, ~ t . 4 "...<,.,! 2,2 ' ' 1,2

2.0 ~ 1,0

t~ :o8 0,5 %0 ~,5 g/cm3

Fig . 4

of the initial charge density Pl- The values of p, T, p', and T' were calculated for the same charge den- sities. For perfect gases with a constant heat capac- ity p/p' is the same for all densities and equal to two, while the ratio T/T' = k + (i/2)k and for k = 9/7 (di- atomic gas with excited vibrational degree of free- dom) is approximately equal to 0.9 [3]. As may be seen from Fig. 4, the ratio p/p' is close to two at large densities (Pl -~ 1.5-I.8 g/cm ~) and increases

with d e c r e a s e in dens i ty , r e a c h i n g 2.4 at Pl < ] .0 g / c m $ . The r a t i o T / T ' is a l m o s t cons tan t at a l l d e n s i - t i e s and is a p p r o x i m a t e l y equal to 1.1. T h e s e c h a r - a c t e r i s t i c s of the b e h a v i o r of the r a t i o s p / p ' and T / T ' a r e d e t e r m i n e d by the spec i f i c p r o p e r t i e s of the equa - t ion of s t a t e of the DP ( v a r i a b i l i t y of n, hea t c a p a c i t y , e tc . ) and they should be t aken into account when, a s s o m e t i m e s h a p p e n s , the ac tua l d e t o n a t i o n p a r a m e t e r s a r e r e p l a c e d by the in s t an taneous exp los ion p a r a m - e t e r s [4].

The r e l a t i o n be tween the w o r k and hea t of e x p l o - s i o n m a y s e r v e a s an i n t e g r a l e s t i m a t e of the a c c u - r a c y of the equat ion of s t a t e , In the c a s e of an i n s t a n - t aneous exp los ion the w o r k of a d i a b a t i c expans ion to p = 0 ( R ) should be equa l to the hea t of exp lo s ion Q. The va lue of R ~ can be c a l c u l a t e d f r o m the R = R{P) da t a at the l o w e s t de ns i t y p = 0.05 g / cm 3 p r e s e n t e d in the t a b l e s , if to R(p) we add the w o r k of expans ion of a p e r f e c t gas p(p) /p(k - 1)[p=0.05, w h e r e f o r e s t i m a t - ing p u r p o s e s k m a y be t aken equal to the P o i s s o n a d i - aba t i c exponen t fo r the DP Cp/C v at low t e m p e r a t u r e s , i . e . , wi th a l l owance for the t r a n s l a t i o n a l and v i b r a - t i ona l d e g r e e s of f r e e d o m of the m o l e c u l e s only. In this case the value ~f k for the DP o~ RDX of the given

COMBUSTION, EXPLOSION, AND SHOCK WAVES 129

Table 2

| /

p, g/cm3 r.loL ~ PT, 1000 atrn p, 1000 atm R, kJ/g [ c, kin/see / n

�9 ] "t 1 2 3 4 5 6 7

1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0,2 0.05

1.6 1.4

1.2 1.0 0,8 0.6 0.4 0.2 0.05

1.4 1.2 1,0 0.8 0.6 0.4 0.2 0.05

1.2 1.0 0,8 0.6 0.4 0.2

0 . 0 5

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0,i 0.05

o.s 0.7

3.63 3.32 3.03 2.77 2.52 2.28 2.03 1.75 1.38 0,88

3.82 3.51 3.22 2.95 2.68 2.40 2.08 1.66 1.08

4.02 3.71 3.41 3.11 2.79 2:42 1.95 1.29

4.20 3.87 3.54 3.18 2.77 224 !.51

4.33 4.15 3.97 3.77 3.57 3.36 3.12 2.85 2.53 2.08 1.72

4.40 4.19

Pl = 1.8 g/cm 3

118.6 167.5 89.25 120.01 64.14 81.71 43.51 51.97 27.55 30.35 16.15 15.97 8.634 7.425 3,979 2.949 1,254 0.874 0.163 0.134

Pl = 1o5 g/cm 3

98.72 129.5 7O.78 88.35 48.01 56.47 30.53 33.33 18.09 17.91 9,824 8.615 4.606 3.576 1.481 1.100 0.200 0.171

Pl = 1.4 g/cm 3 77.38 94.94 52,52 60.98 33.57 36.37 20.09 19.91 11.06 9.850 5.259 4,229 1.718 1.337 0.238 0.209

P 1 = 1.2 g/cm 3

56.92 65.39 36.58 39,38 22,10 21.92 12,30 11.09 5.916 4.886 1.957 1.576 0 276 0.247

Pl = l,og/cm3

39.51 42.31 31.08 32.1 I 24,06 23.88 18,26 17,37 13.52 12.31 9,667 8.443 6,563 5.533 4.098 3.379 2.192 1.811 0.807 0.697 0.314 0.285

Pl = 0.8 g/cm 3

26.01 25.83 19.82 18.93

0.00 1.00 1.88 2.66 3.33 388 4.33 4.71 5,10 5.57

0.00 0.97 1.81 2.54 3.15 3.66 4.12 4.59 5.19

0.0O 0.92 1.70 2.38 2.96 3.49 4.06 4~78

0.00 0.86 1,59 2.24 2.84 3.51 4.36

0,00 0.42 0.80 1.17 1.51 1.85 2.19 2.55 2.96 3.50 3.94

0.00 0.40

5,11 4.63 4,12 3.57 2.98 2,36 1.76 1,23 0.81 0.58

4.79 4.27 3.70 3.09 2.45 1,85 1.32 0.91 0.66

4,41 3.82 3,18 2.54 1.94 1.41 099 0.72

3.92 3.27 2.63 2.02 1,50 1.07 0.79

3,36 3.03 2,70 '2,39 2,10 1.83 1.58 1.35 1.15 0.96 0.84

2.78 2,47

2,80 2,85 2.91 2.95 2.93 2.80 2.50 2.05 1.51 1.26

2.84 2.89 2.91 2.86 2.69 2,38 1.96 1 A9 1.26

2.86 2,86 2.79 2.60 2.29 1.89 1,47 1.26

2.82 2.72 2.52 2.21 1.84 1.46 1.25

2.65 2.57 2.45 2.31 2.15 1.98 1.80 1.62 1,45 1.31 1.25

I 2.39 2.25

0 0 L~

M

COMBUSTION, EXPLOSION, AND SHOCK WAVES 131

eompositon [i] is equal to 1.38. The corresponding values R~ = 1440 keal/kg at Pl = 1.55 g/era 3 and R~ = = 1290 kcal/kg at Pl = 1.05 g/cm 3 are in satisfactory agreement with the experimental values of the heat of explosion [5], 1480 and 1380 kcal/kg, respectively.

In c o n c l u s i o n , we wi l l c o m p a r e the c a l c u l a t e d va I - ues of the work of a d i a b a t i c expans ion R(p) for an in - s t a n t a n e o u s exp los ion with the ex i s t ing e x p e r i m e n t a l da ta . As a l r e a d y noted , th i s c o m p a r i s o n p r e s u p p o s s e s the s low expans ion of the "gas bubble" unde r the ex - p e r i m e n t a l cond i t i ons , which i s ev iden t ly r e a l i z e d in e x p e r i m e n t s with a T r a u z l b lock . The work of expan - s ion of the DP of RDX to a t m o s p h e r i c p r e s s u r e ob- t a i n e d by th i s me thod in [6] was 1190 k e a I / k g a t Pl = = 1.55 g / c m 3 and 1150 k c a l / k g at Pl = 1.05 g /era 3. The c a l c u l a t e d v a l u e s a r e equaI to 1380 and 1200 k c a l / k g , r e s p e c t i v e l y . The r a t i o s of the e x p e r i m e n t a l va lue s of R to the c a l c u l a t e d va lue s a r e a p p r o x i m a t e l y 0.86 and 0.96. T h e s e v a l u e s a r e c l o s e the the m a x i m u m e f f i c i ency (90-92%) d e t e r m i n e d in [6] a s the r a t i o R/Q, w h e r e R i s the work of expans ion to a t m o s p h e r - ic p r e s s u r e .

REFERENCES

i. N. M. Kuznetsov and Ko K. Shvedov, FGV [Combustion, Explosion, and Shock Waves], llI, I, 1967.

2. V. N. Zubarev, PMTF [Journal of Applied

M e c h a n i c s ] , 2, 1965. 3. Ya, B. Ze['dovich and A. S. Kompaneets, Det-

onation Theory [in Russian], Gostekhizdat, 1955. 4. F. A. Baum, K. P. Stanyukovieh, and B. I.

Shekhter, Physics of Explosion [in Russian], Fiz- matgiz, 1959.

5. A. Ya. Apin and Yu. A. Lebedev, DAN SSSR, 114, 4, 1957.

6. A. F . Be lya e v and R. Kh. K u r b a n g a l i n a , in: P h y s i c s of Exp los ion [in Russ i an ] , no. 5, I zd -vo AN SSSR, 1956.

24 N o v e m b e r 1966 Moscow