C A L C U L A T I O N OF C O M P O S I T I O N

A N D T H E R M O D Y N A M I C F U N C T I O N S

OF E X P L O S I O N P R O D U C T S

OF C O N D E N S E D E X P L O S I V E S

N. M. K u z n e t s o v , V. E . O k u n e v , a n d V. M. P o p o v

UDC 534.222.2

The calculat ion of chemica l compos i t ion is the mos t difficult pa r t in the comple te the rmodynamic cha rac t e r i za t ion of the products of explosion of condensed explos ives . Unlike the p r e s s u r e and density, the composi t ion of a gas is exper imenta l ly de te rmined under c a l o r i m e t r i c conditions at low p r e s s u r e s and t e m - p e r a t u r e s and not at the Jouguet point. The composi t ion, thus de termined, actual ly co r responds to the t ime of chemica l t emper ing that depends on the mass coordinates of the expanding products of explosion. The composi t ions of gases in the neighborhood of the Jouguet point and at the moment of quenching can cer ta in ly differ.

Information regard ing the equi l ibr ium composi t ion of a gas is n e c e s s a r y to calculate the heat of ex- plosion as a function of the the rmodynamic s ta te of the products . Besides , the calculat ion of gas compos i - tion is a n e c e s s a r y step in the subsequent the rmodynamic method of construct ing the equations of s ta te of the explos ionproduets which is not based on the use of exper imenta l gasdynamic data of detonating waves .

For calculat ing the equi l ibr ium gas composit ion, in pr inciple , it is sufficient to know the t he rmody- namic functions of individual components , energ ies of breaking the chemica l bonds, and var ia t ion of t h e r m o - dynamic fuctions in mixing the components . The the rmodynamic functions of individual components a r e de- t e rmined e i ther on the bas i s of impact c o m p r e s s i b i l i t y and calculat ions in approximat ions of the theory of f r ee volume [1, 2], or with the help of Vir ia l s e r i e s for p r e s s u r e . The f i r s t method is applied when the den- s i t ies a re high (p > P0, where P0 is the density- of the condensed phase under no rma l conditions), and the s e c - ond method of Vi r ia l s e r i e s is applied when the gas is suff iciently r a r e f i e d (p <<P0)-

In calculat ing a composi t ion it is normal ly a s sumed that the energy of breaking the chemica l bonds does not depend on densi ty and that the var ia t ion of the rmodynamic functions in mixing the components is de te rmined with the help of some approx imate ru le , for example , p re suming the addit iveness of vo lumes [1]. The higher the densi ty of the explosion products , the m o r e difficult i t is to evaluate the e r r o r s that or ig inate due to such assumpt ions in de te rmin ing the i r composi t ion and the rmodynamic functions, Never the les s , even when p > P0 the r e su l t s of calculat ion of the the rmodynamic functions show sa t i s f ac to ry ag reemen t with ex- p e r i m e n t a l gasdynamic data of detonating waves [1].

At the p re sen t moment , in view of the success fu l applications of the s ta t i s t i ca l theory" of s i n g l e - c o m - ponent dense gases , the re is a poss ibi l i ty of calculat ing the composi t ion and the rmodynamic functions of explosion products , when p~ P0. With such values of densi ty one can accept the energies of b reak ing the bond of an ideal gas as the actual values (with substant ia l ly higher re l iab i l i ty than in the case of l a rge com- p res s ions ) . When p _< P0, the effect ive radius of s h o r t - r a n g e r e c i p r o c a l fo rces that a r e respons ib le for chemica l bonds of a toms in molecules is an o rder less than the mean dis tance between the molecules , and, consequently, the energ ies of b reak ing the Chemical bonds di f fer negligibly f rom thei r " ideal" values .

It is known that with such dens i t ies the energy of mo lecu la r in terac t ion of van t ier Waals type is also a lmos t an o rde r tess than the energ ies of b reak ing the chemica l bonds. There fo re , it is sufficient to con- s ide r the fo rces of mo lecu la r in terac t ion approximate ly , using any of the known potential models . The ru l e s

Moscow. Trans la ted f rom Fizika Goreniya i Vzryva , Vol. 10, No. 6, pp. 791-797, N o v e m b e r - D e c e m - ber , 1974. Original a r t i c l e submit ted F e b r u a r y 8, 1974.

01976 Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. No part o f th& publication may be reproduced, stored in a retrieval system, or transmi.tted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission o f the publisher. A copy of this article is available from the publisher for $15.00.

713

i { °//,

0,7~

~,5 2,s 45 4,5 T'10 -~ off

Fig. 1

log xi

0,9

0,7

o,5

o,J

/ / I ,2

N / 6

J

,I 2~E ~,5 4~E

Fig. 2

Fig. 1. Dependence of concentra t ions x i on t e m p e r a t u r e at the r e l a - t ive densi ty N/VL=100 . 1) 100XN2; 2) 100xcoo; 3) 100XH20; 4) 104xoH; 5)104XH; 6) 104xNO; 7) 100XH2; 8) 100xCO2; 9)z104xo2 .

Fig. 2. Dependence of concentra t ions x i on t e m p e r a t u r e at the r e l a - t ive densi ty N / V L =400. (The notation is the same as in Fig. 1).

TABLE 1

Com- ponent ai" ~" ~/k, °K Source

N~ H~O CQ CO H~ NO OH H 02

3,656 2,9 3,8l 3,7 2,9 3,4 3,8 2,1 3,5

91,5 500 205 100 33

120 175 750 If3

i91 [101 [lt1 [121 [121 [121 [10I [10] [121

of de te rmin ing the the rmodynamic functions of mixing have been studied be t t e r under the condition p_~ P0, pa r t i cu l a r ly i~ supe rc r i t i c a l t e m p e r a - tu re f ields [3, 4], than in the field of l a rge compres s ions .

In the p re sen t a r t i c l e the method of calculat ing the composi t ion and the rmodynamic functions of reac t ing gases , when p <~P0 is being r e - ported. The actual calculat ions have been ca r r i ed out fo r the products of detonation of Hexogen. The f r ee energy of individual components has been calculated on the bas is of the Rowlinson method [5, 6] of h i g h - t e m p e r a - tu re approximat ion with Lennard-Jones potential of molecu la r in te rac - tion.

The l i t e ra ture shows var ious p a r a m e t e r s of I ~ n n a r d - J o n e s poten- t ial obtained by dif ferent methods (on the basis of Vi r ia l coefficients of gases , t he rmodynamic p r o p e r t i e s of c r y s t a l s , t r a n s f e r coeff icients , method

of mo lecu la r beams , etc.), which do not ag ree well with one another . The mos t na tura l way to calculate the the rmodynamic functions of gases is to use the p a r a m e t e r s obtained by the f i r s t of the above-ment ioned methods. The in terac t ion of dipole molecules of wa te r should not be modelled by some d i p a r a m e t e r poten- t ia l in the context of such a wide range of t he rmodynamic var iab le s and with the s a m e degree of accu racy that can be obtained for the s imp le s t molecules having no constant dipole moment . Since the method that will be subsequently applied to the calculat ions is essen t ia l ly connected with the applicat ion of Lennard- Jones potential , the s a m e potent ial was used to ca lcula te the the rmodynamic functions of water . To de- s c r i b e the the rmodynamic p r o p e r t i e s of wa t e r in a be t te r way, the cold and only dens i ty-dependent compo-

v nents of energy E o = - - f PodV and p r e s s u r e P0 were taken into account,

oa

Po~----- --Ap ~.

The coeff icient A and power m were de te rmined with an accu racy of Ap0/p 0 ~ 10% by means of in t e r - polation of data on the equation of s ta te of wa te r [7] and detonation products of Hexogen [8] containing about 30~ water . After c a r ry ing out the numer i ca l ca lcula t ion of the composi t ion and the rmodynamic functions of explosion products of Hexogen with va r i ab l e p a r a m e t e r s A and m , a c o m p a r i s o n was made with the r e - sul ts given in [8]. The values of m and A found by that method were equal to 1.5 and 18.103 a tm, r e s p e c - t ively.

714

T A B L E 2*

,+ .K I :;m }.:[;,1 =++' I +".° [=co. I.+o ] =,: I ="°l .o,, I =" I .o.

1500 0,220 1270 2000 0,2201 1730 2500 0,2201 2190 3000 0,220 t 2640 3500 0,220 t 3090 4000 0,218 3540 4500 0,216 3990

N/VL=200

57901 0,3331 0,263170 0,263 70 --6 5 0,3331 0,265168 C~ - - 8 - -43 --6 8 ~5 1 0,265 68 --3 6

8145 0,3331 0,265,67 0,265157 --3 1 --4 4 --4 7 - -42--6 --4 2 --3 2 --5 --3 8 99851 0,332] 0,263 67 0,266167 --3 4

11480 t 0,329[ 0,255105 0,266169 --33 --3 7 --3 6 --33 --4 --3 9 --3 16 1 0,267}73 O, 239159 0,3221 130301 --3

N]VL=600

1500 2000 2500 3000 3500 4000 4500

1500 2000 2500 3000 3500 4000 4500

N/VL=800

o,+I +, ++, o,+, o+ il o,+o,+_,_,, 0,881] 16400 t 55981 0,333} 0,285148 0,285[48--3 0,8811 202401 69161 0,333 0,281 53 --3 0,281153--3 0,8811 238601 8280[ 0,333 0,277156 0,277156--3 0,8801 27300[ 96831 0,3331 0,274[58 0,275158--3 0.880 30600 11120 / 0,332 / o,27116o o,27216l--3 0,878 33780 12600[ 0,33l/ 0,267151 0,271163--3

1500 2000 2500 3000 3500 4000 4~30

0 + 60+0 31 + +3 + +3" 0,66t I 8980 t 5673)0,333)0,281152--3 0,28t 5 2 - - 3 2 - - 8 9 - - 8 2 0,661 i It2001 7006[ 0,3331 0,278155 0,278155 --32 --6 4 --6 4 ~54 0,66I I 13300j 8383} 0,3331 0,276157 --3 0,276157 --3--3 23 Z~ --4--5 82 ~___4 0~6601 154001 97991 0,333} 0,273159 0,274159

--3 i --3 2 0.6591 174001 I12501 0,3321 0,270160 0,272161 --3 1 13 0,657[ 194001 12750i 0,330 / 0,264[60 0,271/64 --34

-1214-1112 =~ - - 8 l

- 6 1 - 6 2 - 5 --5 2 - s 1 - 4 --4 +4 1 +4 5 --4 4--4 1 --3 1 --3 3 - - 3

* The

N/VL= I000

0,223140 -31 0,293140 -+ -127_7-12_7 :~_5. t} °,285148 -3[ °,285,48 -+[+ -9 15 -i [ ' 0,280[53 -31 0,280153 - 3 [ 7 13 1 -5 + 0,270157--31 0,276157 --3/I .--5 15 --6 7 0,273160 --31 0,273160 --311 --4 13 _--4 0, 270162 --31 0,270163 --315 --4 /l ---- 0,266164 --31 0,269165 --312 --3 t4 --4 | 1 --3

table uses the nota t ion A -- B-=A • 10 -B .

1,i011 2107o1 42021 0,3331 1,1011 286601 55061 0,333 I 1.1011 349301 68061 0,333 I I;1011 40680] 81551 0,3331 1,1011 46100 9543 0,3331 I,I00 51200 10970 0,333[ 1,099 56000 2430 0,3321

--17 --12 --9 --7 --6 --5 --4

--17 --13 --10 --8 --6 --5 --5

--18 --13 --10 - - 8 --7 --6 --5

T A B L E 3

N/VL=IO0 p, IF, T'°K! /cmstatI n -F, p, calTg g/cm*

1500 2000 250O 3000 3500 4000 4500

0,110 575 0,110] 777 0.1101 980 0,1101 1180 0,110[ 1390 0,108 t590 0,100 1800

N VL=400

- F , cal/g

Ip) at, m

3~80 l 4;801 4444 5736 56401 7081 67701 8470 78801 9898 8970 11370

101O0 12890

4506 0,440 58141 0,440 I 71731 0,440] 85781 0,440/ 1oo3o L 0,44o I

11540 0)439 13120 0,436

The in t e rac t ion of r a d i c a l s H and OH a l so cannot be mod- el led by L e n n a r d - J o n e s potent ia l , but s ince the concen t r a t ions of these componen ts in the explos ion p roduc t a r e smal l , the de - s c r i p t i on of the i r i n t e rac t ion wi th the help of L e n n a r d - J o n e s p o - ten t ia l does not in t roduce a l a rge e r r o r in the r e su l t s . F o r be t - t e r a g r e e m e n t with the equat ion of s ta te of the explos ion p r o d - ucts of Hexogen [8] the p a r a m e t e r n and n/2 of the potent ia l was a l so va r i ed . With the high dens i t i es and t e m p e r a t u r e s under cons ide ra t ion , the value n e a r e s t to the op t imum was n=10, which c o r r e s p o n d s to a l e s s " r ig id" potent ia l in c o m p a r i s o n with n = 12. In the case of s t i l l h igher dens i t i es , in s t r ong shock waves

the op t imum value of the power of the repu l s ion potent ia l fo r the s a m e componen t s is, in a c c o r d a n c e with [1], equal to nine.

The avai lable data in the l i t e r a tu r e about the p a r a m e t e r s ai and e i of L e n n a r d - J o n e s potent ia l a r e r e - lated, as a ru le , to the c a s e n = 1 2 and have been used in the ca lcu la t ions . It can be shown that at high t e m - p e r a t u r e s the e r r o r a s s o c i a t e d with the subs t i tu t ion of such p a r a m e t e r s in I ~ n n a r d - J o n e s potent ia l (10-5) is not l a rge , and it is c o m p a r a b l e with the a c c u r a c y of the potent ia l mode l i tself .

The c o m p l e t e compi la t ion of va lues of cr i and ~ fo r al l the componen t s under c o n s i d e r a t i o n is g iven in Table 1.

The p a r a m e t e r s ¢ij and ~ j of the potent ia l of i n t e rac t ion of d i f fe ren t molecu les and the e f fec t ive pa - r a m e t e r s a and e of the potent ia l o f the m ix tu r e a r e d e t e r m i n e d by the r e s p e c t i v e r u l e s of combina t ion

and a p p r o x i m a t e r e l a t i o n s [3]

715

o. 3 : ~ 3 . ~ 3 = oq X~Xs, ~ = ~ ~. ~so~ix~x r (1) i i l 1

Here x ~ N~ , t--~i, ], and N l is the number of pa r t i c l e s of the / - th component.

At the p r e sen t moment the re a r e sufficiently detailed tables of the rmodynamic functions of un icom- ponent gases with Lennard - Jones in teract ion potent ial calculated by the P e r c u s - Y e v i c k method [13]. At high t e m p e r a t u r e s (T ' - - - -kT /e> 12) the data avai lable in the tab les can be p resen ted with good accu racy by the s i m p l e r Rowlinson approximat ion [3, 4, 13]. According to [5, 6], at high t e m p e r a t u r e s the molecules with Lennard - Jones potent ia l can be considered as rigid sphe res with d i a m e t e r s that depend on t e m p e r a - t a re .

The equation of s ta te of the r igid sphe res [14, 15] has the following fo rm:

pV 1 q= y, + y~ -- y~ . (2) N~ hT (l - - yt)3 ,

where in the Rowlinson approximat ion

* 3 Ni(t ~ ~V'5 CV:~-~n [1 + * ( T ~ ) / n ] • V Y ~ = - ' ~ - ~ ~! ......

Here n is the power of the potent ia l and 4~(T*) is the function whose table of values is given in [16].

The f ree energy of the unicomponent gas calculated af ter taking Eq. (2) into account has the fo rm

Fi W eVi { m ikT '? /2 ] F~i .

= - I,-, l t _, + + + . . . . . . . . . . . . . . . . . . . 3 i 2 y ~ _ _ 3 ,

where m i is the m a s s of the molecules of type i, and Fbi is the f r ee energy of rotat ional , t rans la t ional , and e lec t ron ic degree of f r eedom of the ideal gas having molecules of type i.

S imi lar ly , for the f r ee energy of a mix tu re of a given composi t ion we have

N#~r S , ~ k 2 - -~ ) J XsF~' + ~ (~' o); (3)

N = Z N ~ ; x s ~ l V J N .

The p a r a m e t e r s of potent ia l that f o r m a pa r t of (3) depend on the sought concentra t ions of the compo- nents and are determined by the relations (1).

The equations of chemical equilibrium follow from the conditions of minimum free energy with re- spect to the change of compos i t ion in each reac t ion at a given t e m p e r a t u r e and densi ty:

Hx s = gp (T) / V kzvs [ , " e>:p - - + E - -

/

• V . ~ x ? ' i - -

2e Oiler f • k---~ ~ x { ~ i - -

where Kp(T) a r e the equi l ibr ium constants of cor responding (p' _3# V and ¢P'UkT a r e the de r iva t ives of ~pfor Na3/V and

N o / " i r i c coeff lc ents of reac t ion .

v J ] - - +'~ " (4)

!

reac t ions for an ideal gas as given in [17], £/kT, r e spec t ive ly , and vj a r e the s to ich iomet-

At T < 5000°K and with sufficiently high dens i t ies the main products of detonation of Hexogen C3H6N606 could be N2, H20, CO2, CO, H 2, NO, OH, H, and 02. The fo rmat ion of condensed oxygen is a l so not excluded. However, on the bas i s of avai lable expe r imen ta l data [18] i ts propor t ion in the explosion product of Hexogen is v e r y smal l . The compos i t ion of explosion products a l so include polya tomic components CH4, N20 , NO2, etc. The sma l l concentra t ions of al l such undetermined components can be calculated by the following ap- proximat ion , using the values of bas ic components that have a l r eady been de te rmined . The evaluat ions show that the m a x i m u m values of XCH 4, XNO2, and XN20 a re as fol lows: XCH4=2.35 - 10 -2 (N/WL=1000, T=1500°IQ,

716

XCH¢ = 6.68 • 10 -3 (N/VL = 1000, T=2000OK), XNO ° = 2 . 4 7 . 1 0 -5 (N/VL = 100, T = 4500°K), XN20=2.05.10 -5 (N/ VL=100, T=4500°K). Here L denotes Loschmi~t ' s number .

The concentra t ion of undetermined a tomic oxygen is max imum, where N / V L = 100 and T = 4500°K, and is equal to 5 • 10 -3. It is evident f r o m Table 2 that in the case of sufficiently good approximat ion xo2 may not be considered.

The comple te s y s t e m of equations de te rmin ing the equi l ibr ium values of re la t ive concentra t ions x i of the above-ment ioned nine components at a given t e m p e r a t u r e and p a r t i c l e - n u m b e r densi ty N/VL includes the equation of normal iz ing , th ree equations of maintaining the re la t ive composi t ion of the e lements N, O, C, and H, and the equation [Eq. (4)] for f ive independent reac t ions . This s y s t e m of equations was solved numer ica l ly by the method of Newtonian i te ra t ion . The f r ee energy, p r e s s u r e , and composi t ion of the gas were calculated at the t e m p e r a t u r e s 1500 °, 2000 °, 2500 °, 3000 °, 4000 °, and 4500°K and "dens i t ies" N/VL= 100,200, 400, 600, 800, and 1000. The r e s u l t s of calcula t ions a re given in Tables 2 and 3 and Figs. 1 and 2.

The e r r o r s in the ca lcula t ions of the concentra t ions of components a r e caused by the e r r o r s of using the reac t ion potent ia l and i ts p a r a m e t e r s and a lso by the e r r o r s of approximat ion (1). The evaluat ions show that the indefini teness of the p a r a m e t e r s of the reac t ion potential r e su l t s in an e r r o r of about 5% for all the components and of about 10% for H20 in the concentra t ion calculat ion. The e r r o r s assoc ia ted with the inac- curacy of the model of the potent ial i tself , as well as with the averaging (1), a re difficult to evaluate. Thei r magni tude, however , d e c r e a s e s with i nc rea se in t e m p e r a t u r e , and apparent ly at t e m p e r a t u r e s of the o rde r of 1000°K the total e r r o r in the de te rmina t ion of composi t ion is c lose to what has been given above.

Simi lar calculat ions were a lso ca r r i ed out for an ideal gas. Qualitatively, the dependence of compos i - tion on t e m p e r a t u r e and density is the s ame as in the case of an ideal gas. The concentra t ions of molecules with la rge bond energy (CO 2 and N2) i nc rea se with d e c r e a s e in t e m p e r a t u r e . Inc rease in densi ty at i so- t h e r m s re su l t s in the inc rease of re la t ive concentra t ions of polyatomic components . However, the quanti- ta t ive d i f fe rence between the r e a l and ideal gases va r i e s f rom a few pe rcen t at N / V L = 100 to tens of p e r - cent at N /VL=400 . Such a d i f ference leads to a significant var ia t ion of in ternal energy and other t h e r m o - dynamic functions of a gas.

The r e su l t s of calculat ions were compared only with the calculat ions of equi l ibr ium composi t ion of explosion products of Hexogen ca r r i ed out by Cook [19] by the phenomenological method based on the ap- pl icat ion of veloci ty of detonation waves . The data in Table 2 ag ree with [19] with s a t i s f ac to ry accu racy only for XN2 (the d i f fe rence being about 1.5%). For the r emain ing components at c e r t a in t e m p e r a t u r e s and densi t ies the d i f ference in the values of XH20, X c o , and x H is 10%, and the d i f ference for xCO " is 30-40%. Such a d i f fe rence can be due to the fac t that the exper imenta l data used in the phenomenologica~ method [19] a re g ro s s ty insufficient for formulat ing the s y s t e m of equations that unequivocally de t e rmines the equil ib- r i u m composi t ion of a gas. A closed s y s t e m of equations has been obtained in [19] by using additional a s - sumptions that a r e not unequivocally connected with cer ta in exper imen ta l facts .

Thus, a method has been developed to calcula te the equi l ibr ium composi t ion and the rmodynamic func- t ions of reac t ing nonionized gas mix tures , when p_< P0. The method is based on the use of Lennard - Jones reac t ion potent ials and the Rowlinson h igh - t empe ra tu r e approximat ion for the equation of s ta te of individual components . The composi t ion, p r e s s u r e , and f r ee energy of detonation products of Hexogen at the t e m p e r a - tu res 1500°-4500°K and densi t ies 0.1-1.0 g / c m 3 have been calculated.

L I T E R A T U R E C I T E D

1. V . N . Zubarev and G. S. Telegin, Dokl. Akad. Nauk SSSR, 14_~7, 5 (1962). 2. V . N . Zubarev and G. S. Telegin, Dokl. Akad. Nauk SSSR, 15.~8, 2 (1964). 3. T . W . LeLand, J . S. Rowlinson, et at. , T rans . Far . Soc., 64, 546 (1968). 4. H . N . Temper l ey , (editor), Phys ics of Simple Liquids - Exper imenta l Studies, A m e r i c a n E l sev i e r

(1968). 5. J . S . Rowlinson, Mol. Phys. 7, 4 (1964). 6. J . S . Rowlinson, Mol. Phys. , 8, 2 (1964). 7. N . M . Kuznetsov, Zh. p r i M . Mekhan. Tekh. F iz . , 1 (1961)o 8. N . M . Kuznetsov and K. K. Shvedov, Fiz. Goreniya i Vzr)wa, No. 3, 2 (1967). 9. A . A . Antonovich, M. A. Plotnikov, and G. Ya. Savel ' ev , Zh. Pr ik l . Mekhan. Tekh. Fiz . , 3 (1969).

10. V . P . Glushko (editor), Thermodynamic and Thermophys ica l P r o p e r t i e s of Products of Burning [in Russian], Vol. 1, Izd. VINITI, Moscow (1971).

717

11. P .M. Kessel 'man, Yu. I. Blank, and V. I. Mogilevskii, in: Thermophysical Propert ies of Gases [in Russian], Nauka, Moscow (1970).

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13. A.M. Berezhkovskii, N. M. Kuznetsov, and L V. Fryazinov, Zh. Prikl. Mekhan. Tekh. Fiz., 2 (1972). 14. A. Khan. Phys.Rev., 134, 2A (1964). 15. N .F . Carnahan and K. E. Starling, J. Chem. Phys., 51, 2 (1969). 16. R. Chen. D. Henderson, and S. Davidson, Proc. Nat. Acad. Sci., USA, 54, 6 (1965). 17. V . P . Gtushko (editor), Thermodynamic Propert ies of Individual Substances, [in Russian], Izd. Akad.

Nauk SSSR, Moscow (1962). 18. A. Ya. Apin and Y. A. Lebedev, Dokl. Akad. Nauk SSSR, 114, 4 (1957). 19. M.A. Cook, The Science of High Explosives, Van Nostrand Reinhold, New York (1959).

718