AD-A2 3 8 948

hAZ. •AD-E402 217

Technical Report ARAED-TR-91013

AN EXPLOSIVES PRODUCTS THERMODYNAMIC EQUATION OF STATEAPPROPRIATE FOR MATERIAL ACCELERATION AND OVERDRIVENDETONATION: THEORETICAL BACKGROUND AND FORMULATION

Ernest L. Baker

July 1991

U.S. ARMY ARMAMENT RESEARCH, DEVELOPMENT ANDENGINEERING CENTER

Armament Engineering Qireciorate

Picatinrny Arsenai, New JerseyUS ARMY

ARMAMENT MUNmONS& CHEMICAL COMMAND

ARMAMENT RO3E CENTER

Approved for public release; distribution is unlimited.

91-06254! 1 . ll i i Ii ! J' l i IS g'•"• . 0' jl -8 ,•I,,,:, !I II, Il! ll

S.. . , ! , , ,

The views, opinlo.-s, and/or findings contained in tihisreport are those of the author(s) and should not bhconstrued as an official Department of the Army position,policy, or decision, unless so designated by otlierdocumentation.

The citation An this report of the names of commx.rcitlfimis of commercially available products or sysiems Ll.oe3nct constitute official endo;sement by oi" approval of theU.S. Government.

Destroy this report when no longer needed by anymethod that will prevent disclosure of contents orreconstruction of the document. Do not rcturn to theoriginator.

REPORT DOCUMENTATION PAGE Fom ppovedPrbfc tipotrtg b- fnrdr for IN, c.lfanbon of informatinon I. estimated 1. -. 9.g I F-,n par respon., incfrufirg the tio m for r-vi-ing io-Iriuiiorns, n,a rcfhing *,I song dst a sorn,, grifhorirng andf maintainin o the d&Ianiieded, arid c Von p anoard -anning 0r. -1l4dioo ofInforom ton. Send oonrme no regarding fhis burden oalmnstl. or any offror aefrinrt ofIh tir Collection of iotorronfoir, jocknit ng sger ingnnfir for red-nkr g this fnnnidno

to~ ~ ~ ~ ~,= I~e~igo Honrrfoatar %1`,n~ . 0 r nr'ora forfoiotion Operations aod 11"drs. 1215 Jef~ffaon flee. f"ghn.y SoUt. 1204, Arington. VAr 2M22 4302, and fo the Of fice of MAsnageornf and B-nd~f,

1. AGENCY USE ONLY (Leos" blank)I 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

I July 19914. TITLE AND SUBTITLE 5. FUNDING NUMBERS

AN EXPLOSIVES PRODUCTS EQUATION OF STATE APPROPRIATEFOR MATERIAL ACCELERATION AND OVERDRIVEN DETONATION:rHEORETICAL. BACKGROUJND AND FORMULATION

6. AUTHOR(S)

Ernest L. Baker

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATIONARDEC, AED REPORT NUMBERATTN: Energetic Mvaterials and Warheads Div. Technical Report ARAED-TR-91013ATTN: SMCAR-AEE-WWPicatinny Arsenal, NJ 07806-5000

9. SPONSORING/MONITORII4G AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOIRING/MONITORING

ARDEC, IMVD AGENCY REPORT NUMBER

STINFO BrATTN: SMICAR-IMI-IPicatinny Arsenal, NJ 07806-5000 as

11. SUPPLEMENTARY NOTES

12s. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTIOI' CODE

Approved for public release, distribution is unlimited~.

13. ABSTRACT (MaximuLm 200 words)

Many modern explosive applications require modeling of both overdriven detonation and lower pressuredetonations products expansion. Common thermodynamic equations of state used for warheads designhave been demonst rated to provide poor representations of overdriven detonation. The Jones-Wilkens-Lee-Baker (JWLB) has been formulated to provide a more accurate representation. This report provides adetailed theoretical background and equation of state formu~ation for JWLI3.

14. SUBJECITERMS 15, NUMBER OF PAGES

1716. PRICE CODE

7 SECURITY CLASSIFICAIION 18. SECURITY CLASSIFICATION '-19. SECURITY CLASSIFICATION 20. LIMImIAiON OF ABSTRACT

I.INCLA35iF!1_D UNCLASSIFIED UNCLASSIFIED SAR

NSN 54001 .8~-500Stanrlitrd Form 29LI (Rny 2 89)NSN 740-0-2805500Proscribed by ANSI Sid. Z39-1 298-102

CONTENTS

Page

Background 1

Equation of S!ý,te Formulation 1

Speed of Sound 3

Adiabatic Gamma 4

Principal Isentrope PropertPs 4

Reacted Products Hugoniot 4

Conclusion 5

Symbols 5

References 6

Distribution List 9

SI

1R I

BACKGROUND

Many modem explosive applications require finit- element and finite differencemodeling of both overdriven detonation and lower pressure detonation products expansion.Large detonation wave shaping and multiple point initiation are typical overdriven detonationapplications. Lower pressure detonation products expansion is required for materialacceleration applications, such as explosively formed penetrators and shaped charges. Currentthermodynamic equation of states used in dynamic finite element and finite differenceprograms are either parameterized to give agreement with thermochemical calculations [1] orexperimental copper cylinder explosive expansion experiments [2,3]. Thermochemicalcalculations have proven to be very useful for the prediction of explosive products properties,particularly near and above the Chapman-Jouguet state [4,5]. Unfortunately, they do notreprodu,_e the products expansion behavior accurately enough for typical warheads design.Currently, thermodynamic equations of state (JWL, Wilkens) [2,3] used for warheads designare normally calibrated to give agreement with copper cylinder explosive expansionexperiments. These equatious of state have not been calibrated for high pressures above theChapman-Jouguet state. Experimentation [6] and comparison with thermochemicalcalculations (Figures 1,2, and 3) have demonstrated that a poor description of the high pressureregion exists. In order to achieve a suitable equation of state for both overdriven and lowerpressure products expansion, an appropriate equation of state form has been derived. Aprevious preliminary report [7] briefly describes the equation of state and parameterizationmethodology, but does not include details. This report provides a detailed theoreticalbackground and equation of state formulation. A forthcoming report will provide a detaileddescription of the parameterization methodology.

EQUATION OF STATE FORMULATION

The equation of state form was chosen so as to adequately describe the high pressureregime produced by overdriven uetonatttn, and, yet t h., ..... r.... re.....n....beh..i.r

required for standard material acceleration modeling. To this end, the derived form is basedon the Jones-Wilkens-Lee (JWL) equation of state [1] due to its computational robustness, andasymptotic approach to an ideal gas at high expansions. Additional exponential terms and avariable Gruneisen parameter have been added to adequately describe the high pressure regionabove the Chapman-Jouguet state. The resulting equation of state form, namedJones-Wilkens-Lee-Baker (JWLB), is

P. A,[-+ e-R-iV -t-V-4- C(1 V(wl 1

where,

,=L(AAi +BAi)d - + (0 (2)

For consistency with the 3WL equation of state, V is defined as a specific volume ratio, V

1

po/p and E is defined as E a poe where e is the specific internal energy. The JWLB equationof state form is based on a first order expansion around the principal isentrope:

A, .'ie'R iV + CV"Q°o'I). (3)

Using the Gruneisen Parameter,

X alp1V' (4)

the isentropic identity,

P dE! (5)

o . E A ._ e-Ri V +C v- ~ -RVj C-O

Ef- i i + - -- -e V -CV.j, (6)i 1 i

and the Chapman-Jouguet condition,

Eoj = E. + +i(Pcj + Po)(Vo - V•)

ie-,iVJ+ ,C -W C , (7)

the final form may be derived

P = V7E - E•) + PA

= 4(E-- 'i'-"V ) +2 Aiie-vi+ CV" (8)' R

* Final Form.

Some important characteristics of the equation of state are that the Gruneisen Parameter A, isrepresented as an analytic function of specific volume, V. X + 1 approaches a constant

sadiabatic gamma, p = w + 1 for large li, so that ideal gas behavior is asymptotically

approached. The principal isentrope desciiption is cssentially identical to JWL, with theexception of an increased number of exponential terms. It has been found that for mostexplosives, three exponential terms (instead of two in JWL) are adequate to describe theprincipal isentrope over both the high pressure region above the Chapman-Jouguet state andthe lower pressure expansion region. It is important to note thait the internal energyreferencing is defined by (7). The value of E. must be consistent so that,

2

E0 eWRiVCJ +Cv i1~E R.(zw . 0 Voj (P~j + Po)(Vo Vcj).(9i 1

Normally, the equation of state parameters are chosen so that E. has the value E. = p0AH,where AH is the heat of detonation. This is consistent with the initial internal energy of theurireacted material having a value of zero.

SPEED OF SOUND

The generalized speed of sound is often required for implementation into a finite elementor finite difference program. The speed of sound may be easily derived as follows,

d),

(3p r~ dV a I e-R iV - r~ [1- R2 -R.VTV = i - e --Ai V- Rie.

+C -( 5 (1 1W+l)- C(0 + 1)(1- _-XV-(co+2)

EX + A 0E E 0A (10)

V v VTOV

ax(5), (10) *- i -.V- -iv Ri + e-RiV

- c((O+1)(1-b + ý V-,,2

EX _ EaX (11)c V=O' 2 -V-dV

C 2 : 3poC = V2P (12)

a xA•2 Ai [ (9 dVR i V 22V] e-R.iV

+ C (w+(Il>+)( + I V-" + EX + VP - EV- (13)Co av

A - Xi. -_(AXi V + Bki)RxieR )iV (14)

3

ADIABATIC GAMMA

Another useful quantity is the generalized adiabatic gamma. The adiabatic gamma may alsobe easily derived,

,jnPI =Y vp (15)7- '9lnV s -as

+)V P aV

(14)* 2 =. VA e'RiV - ,AV + 13 xiav L X21 )RRki

PRINCIPAL ISENTROPE PROPERTIES

Often, properties along the isentrope that passes through the Chapman-Jouguet state are ofparticular interest. The Gruneisen parameter is given by (2). The adiabatic g-rn1r1 is givenby,

(3) * • P•= A•AiRie' RiV - C(w+I)V(,°+2 ) (17)

VNAiR R i V + C(o+1)V('o-)

(3), (15) • = I' V(18)A e 'Rit + CV-((O+1)

The l jXedpU o1 NURLI llalog thIC eIM pricia l1t ,ttUrjj1 givcit t)y,

(12) Z 7 = YAiRiV2e-RiV + C(w+l)V"° (19)

REACUED PRODUCIS LIUGONlOT

Another often used state space locus is the leactcd prodtIcts IlJugoniot. Assuming the initialpressure to be -ero, conservation gives,

Mass: pjD = p(D-u) (20)

Momentum: P = p,Du (21)

Energy: +2 E 1'D-.+ +4 t (22)

4

(20) 4 u = L-P-D = (1-V)D (23)

P

(23),(20) # P = p.(1-V)D2 * D2 P (24)

( 2 2,( 2 4) P Eo P + P E (25)

4 E)E 0 1- V 2P = - 1 /(22).V (24 :P 2i--R+, p 2 I--T-) po

*E=.+P -V P2 -PV2(1 -V) 2TM-D

= E. + 2- (1-V2-2V+2V2) +E +2_(V2"2V+1)

= E. + P(1--V (26)2

(1), (26) (2 E6) 2V )

= E1 + (!Ai [1iŽ] e'RiV + C(1 X ý"('°+1))(2. (27)

Fo + A.1- -- e-RiV + C(1-

SE = ~(28)2V

CONCLUSION

An advance,! tr!.rmodynamic equation of state applicable to problems involvingoverdriven detonation ,ml1 material acceleration has been developed and calibrated for severalexplosives. This m1nuwc.ipt has presented a theoretical background and formulation for thenew equation of ,.tt. A, -,,•O,-- -, reorrt xill decribe the parameterizatiofl methodoloyv.The new equation of state nmii'tains required low pressure expansion behavior while providinga better high pres:;ure descripticA 1pplicable to overdriven detonation. Typical applications ofthe new equationi of state in,71uve wave shaped and peripherally initiated munitions. Finally,the equation of state asymptotically approaches the constant ganmma equation of state at highexpansions. It is therefore believed that the equation of state will be applicable to extremelylarge volume expansion applicatiois, such as blast, without a loss of accuracy in the higherpressure regions. In order to use the equation of state for very large cxpansions, propercalibration to very large volume expansion experimentation will be required.

SYMBOLS

P- Pressure

-) Density

V Ratio of Specific Volume to Initial Specific

5

Volume

E a Specific Internal Energy divided by InitialSpecific Volume

a Gruneisen Parameter

7 Adiabatic Gamnma

C Sound Speed

D) Detonation Velocity

U mMass Velocity

A 1l Heat of Detonation

B ;., R xi, wo Equation of State Constants

8 Isentropic

ýj Chapman-Jouguet State

I Iniial Cniin

REFERENCES

1. C.L. Mader, Numcr-ical Modelinig of Detonations, University of California Press, 1979.

2. M.L. Wilkens , "The 'Equation of State of PBX 9404 and LXO4-O1", Lawerence RadiationLabhoratory, Livermore, Rept. UCRL-7797, 1964.

3. E.L. Lee, H.C. Horning, and J.W. Kury, "Adiabatic Expansion of High ExplosiveDetonation Produc;ts", University of California Report No. UCRL-50422, 1968.

4. C.L. Mader, "FORTRAN BKW - A Code for Computing the Detonation Properties of'Explosives', University of California Technical Report LA-3704, 1967.

5. M. Cowperthiwaite and W.1 . Zwisler, "TIGER Computer Program D)ocumentation",Stanford Rcsearchi Institute Pu~blication No. Z106.

6. L.G. Green, C.M. Tarver, and D.J. Eirskine, "Reaction Zone Strcturle III SupracoinpOIc~)IssedDetonating Explosives", University of California Rleport No. UCRL-1O 186-2.

7. E.L. Baker, and J. Orosz, "Advanced Warhead Concepts: An Advanced ThermodynamicEquation of State for Overdriven Detonation", Pic'xtinny Arsenal T'echnical ReportARAED-TR-91007, 1991.

6

x0o 0 EOS Calculations

S_.-JWL CJ State- -- BKWR CJ State

.o .300 - -JCZ3 CJ Statc

S.200-

Co

S .100-BKWR Isentrope•- JCZ3 lscntrope

.0 - ,-J-W L Isentrope

.000 1.000 2.000 3,000Specific Volun 1- (Cely) x1o o =

Figure 1. Pressure versus specific volume for the principal isentrope of octol 75/25 below the

Chapman-Jouguet state. The thermochemical calculations (t3KWR and JCZI) agree lairly well

with the standard JWL.

x10 o EOS Calculations

• .900 ,,-JCZ3 Ilugoniot

JCZ3 Isentrope

,70o0 1-o .500 -

k- .L0goi --- I1\YV~i.Il.u." i.400 JWL Isentrope

.300L-ugoniot - . ..

.lo00 .325 .350 .375 .400

Specific Voi0u,',.,c (cc/g) X10

Figure 2. Pressure versus specific volume for the principal isentrope and reactive thlgoliot ofoctol 75/25 above the Chapnian-Jouguet state. The standard JWL underpredicts the highpressure region.

X10 0EOS Calculations

.900 - ~ JCZ3

S .800 -

C .600 B(W

ýý .500

S.400 __

S .300.900 JWL

.100

.000 1.000 2.000 3.000S~w'LfC 'olu~mc cc g) xI io

Fivure 3. Tile Gruneiserl parameter versu.s specific, voltime for Owe principali SentrLIOpt :1[11d

reactive Hjugonliot.

DISTRIBUTION LIST

CommanderU. S. Army Armament Research, Development and Engineering CenterATWN: SMCAR-.IMI-1 (5)

SMCAR-COSMCAR-TDSMCAR-TDCSMCAR-AE (3)SMCAR-AEE (3)

Picatinny Arsenal, NJ 07806-5000

CormmanderU. S. Army Armament Research, Development and Engineering CenterATT[N: AMSMC-GCL(D)

AMSMC-DSM-B(D)Picatinny Arsenal, NJ 07806-5000

CommanderIU. S. Army Materiel CommandATI'N: AMCCN-C

AMCLD,D. Vitali5001 Eisenhower Avenue

lexandria, V A. 22333-0001

CommanderU. S. Army Armament, Munitions and Chemical CommandAT[N: AMSMC-DCG

AMSMC-DSM-AAberdeen Proving Ground,MD 21010-5423

CommanderU. S. Army Depot Systems CommandAYITFN: AMSDS-OV

AMSDS-TAMSDS-Sý I-SPAAMSDS-CG

Chambersburg,PA i7201-4170

AdministratorDefense Technical Information CenterATTN: Accessions Division (2)Cameron StationAlexandria, VA 22304-6145DirectorU. S_ Army Materiel Systems Analysis ActivityANfTN: AMXSY-MP

AMXSY-RWAberdeen Proving Ground,MD 21005..5066

CommanderChemical Res,ýarch,Development and Engineering CerterU. S. Army Armament, Munitions and Chemical CommandA'TN: SMCCR-SPS-IL

SMCCR-COSMCCR-MUSMCCR-SPS-MSMCAR-RSP-A

Aberdeen Proving Ground, MD 21010-5423

DirectorBallistic Research LaboratoryATrN: AMXBR-OD-ST

SLCBR-TB-EEAberdeen Proving Ground,MD 21005-5066

ChiefBenet Weapons Laboratory, CCACArmament Research,Development and Engineering CeaterU. S. Army Armament, Munitions and Chemical CommandATTN: SMCAR-CCB-TLWatervliet, NY 12189-5000

CommanderU. S. Army Armament,Munitions and Chemical CommandAl7 N: SMCAF -FSP-D

SMCAR-DSAMSMC-IMF-L

Rock Island, IL 61299-6000

DirectorU. S. Army TRADOC Systems Analysis ActivityATFN: ATAA-SLWhite Sands Missile Range, NM 88002

CommanderAnava We apns CenterATFN: L. Smith, Code 320

A. Amster, Code 385R. Reed, Jr., Code 388K J. Graham, Code 3835

China Lake, CA 93555

DirectorLawrence Livermore National LaboratoryUniversity of CaliforniaP.O. Box 808ATFN: M. Finger

C. TarverL. Green

Livermore, CA 94550

10

DirectorLos Alamos National LaboratoryAT'TN: L. Hull

T. BennionJ. ChapyakJ. Ritchie

Los Alamos, NM 87545

DirectorSandia National LaboratoryAlbuquerque, NM 87185

CommanderNaval Surface Weapons CenterATTN: Code G13Dahlgreen, VA 22448

CommanderNaval Surface Weapons CenterATTN: L. Roslund, R122

M. Stosz, R 121Code X211, LibraryE. Zimet, R13

J.W. Forbes, R13S.J. Jacobs, R10C.DickinsonJ. Short, R12

Silver Spring,MD 20910

Air Force Armament LaboratoryATTN: J. Forster, AFATL-MNW

G. Parson, AFATL-MNEEglin AFB,FL 32542-5434

Dyna East CorporationATTN: William Flis3201 Arch StreetPhiladelphia, PA 19104

Aerojet Ordnance CompanyATTN: Dan Donati2521 Michelle DrTustin, CA 92680

Applied Ordnance Technology, Inc.ATTN: NIMIC, Eddie Norton1000 Century Plaza, Suite 212, BX2410630 Little Patuxent ParkwayColumbia, MD 21044

11

CommanderChemical Research, Development and Engineering CenterU.S. Amiy Armament, Munitions and Chemical CommandATTN: SMCCR-MSIAberdeen Proving Ground, MD 21010-5423

Royal OrdnanceATTN: Dr. Peter R. LeeWestcott, AylesburyBuckinghamshire, Iip180NZ England

12