TITLE: BQUATION OF STATE FUR DETONATION PRODUCTS

AUTHOR(S): William Chester Davis, H-3

LA-uR--85-1129

DE85 009608

SUBMITTED TO: 8th Symposium on Detonation~lbuquerquc, NM, .JuIY 15-19, lg~s.

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La3A[MASTER

NationalLaboratory

Hsrnluumla.wIHISmclJMIHlIs Umtlmo

Detls (s-018)

EQuATION OF STATE FOR DE TOltATION PR(JWCTS

u. c. DRVISLos Alamos Matlona! Laboratory

Los A!tmos. IS.M. 87545

The concepts of hydrodymanlcs and thermodynamics as th~y applyto equations of state for ●xplosive products ●re collectgd andalscussed. The physics Behind the behavior of dmtsm gases Izconsidered. Some Ideas about ●pplications are presented.This paper IS Intended as an fntrnduction to the subjec’ ofequation of state for detonation groducts.

1. lNTROOUCTION

The concepts ana formulas thatare pertinent to the development ●nduse of an equation of state for ●xplo-sive Products gases. taken from lvydro-oynamlcs. thermodynamics. and the Phy-sics of gases are collected and d~scus-Sed in this paper. perhaps havingthem collected fn on@ Place will helpclarify the confusing subject IISU&llycalled ‘equation of sthte= by thozewno work with explosives.

The sec~nd and thlrrj \eCtlOnS ●redevOtCd to the equations of hydro-dynamics and their iolut!ons. Thefourth section prtsents thermodynamicsfor use with nydrovlynam~cs, and thefifth a discussion of Incomplete equa-tion- of state as they are used forexplostvas. Tne sixth sectfon DreSentSthe simple Dhysical prlnclplel thatdttermfne the general form tor ●neouatlon of state. Sections seven,eight. ●nd nine discuss ●ngineeringapplications, $the choice of a fltt ngform for an ●quation of state. and tnecalibration of th~ fittfng form.

Tnig ptper 1s Intended to be an{ntroductton to the mysteries of thesuDJect, and is certain’y not the finalaescrfptlon of all the lntrlcaciet.

11. FQuATIONs OF tvYORODTiIAM1llS

The ●quations for the conserva-tion of ■ass, momentum, and energy.for flo. in one dimension, can aewritten as

v - vu -ox [2-I)

Cl”vp X+A-O (2-2)

p-j -2~lJ + V(pu)x . B

(2-3)

where v Is the speclf’c volume. u isthe particle velocity. p is the pres-sure. and E IS the specific internalener~y. The dot dtnote$ the totaltime derivative such that v _ 8v/at +u avlax. and u.. - aulsx is ● parti41derl ative.

!in Eq. (2-3) the term E +

1/2u f$ the SUm of the internal ●ndkinetic ●nerglest ●nd Is the totaiSPeCifiC ●nergy Of the fluld !J1.mQnt.The term A in Eq. (2-Z) representsnonequilibrium processes thet tran$fermomentum3 usually viscous effects.The term E in Ea. (2-3j representsnonequtlibr!um proc?sse$ that transfer●nergy, USti&lly ViSCOUS and thermaldlffuslon Drocesses.

In addition to these eauatlons.there {S an ●quilibrium ●quation ofstate for the material

E - E(D.v) (2-4)

that describes the ●auflibrium materialDropertles. The equllibr!um equationof state can be used to ●xpand theterm in ~ in Eq. (2-3) as

.E - Epb “ Ev@ . (2-5

where

Ep . (tE/ap) v’ Ev- (sElav)p . (2-6

1

-Of llUSMCUMEHI1SWHO

@

oSViS {S-018)

.

Equation (2-3) can be written. ●fterdoing the Indicated alfferentlatlon●nd substltutln terms from EVJS. (2-1),

!(2-2), and (2-5 , as

V(EV + g)P- (P/v)9 = (v/Ep)(u4 * 6)/V .

~EP

(2-7)

The coefficients that describe thematerial properties hhve their ownspecial names. The coefficient(v/E ) IS called the 6runeisen gammaand ~S represented as

vlE=r.P

(2-8)

The coefficient V(EV +P)/PEp Iscalled the aatabatlc gamma ana Is writ-ten

V(EV + P)/PED = Y . (2-9)

dltn these defin!tlons, Ea. (2-7)becomes

.Dlp - YVIV =r(uA + RJIPV . (2-lo)

The conservation eau.st!ons, Eas.(2-l). (2-2), ama(2-1), can now bereplacea by

v - vu x .0 (2-11)

U+vp *A.Ox (2-12)

DID - Tvlv - r(uA * B)/Dv . (2-13)

A1l the oe$crlption of the material ISgiven in the two der{vat+ves, T and I_.[f ths two derivatives are given ●sf:nctfon$ of p #nd v, the ●gua:ionsare a complete set of three eauationswith tnree d9pendent vtrialVleS.

111. SOLUTIONS OF THE EQUATIONS

The equations af hydrodynamicshave simpla Xolutionl fOr soacialCd&es, hnd tnese solutions allow someinsights fnto the physical meaning ofthe vhrious terms in the ●auatlons.

Let us first consider the impo~tantcese of a steady shock wave propagat-ing in the ●aterial. Steaay mearsIndependent of time, afvd thu$ the Dar-tial derivatives witn reimect to time

In Eas. (2-l). (2-2), and (2-3) areall zero. The ●~uations become

Uvx - vu - 0x (3-1)

Uu + Vvx -k.ox :3-2]

UEX ● U2UX + Vup + VPU = B .x x (3-3)

The fir$t eauation can be immediatelyintegrated to give VIW = constant. Ifwe require that the shock wave belocalized near x - 0 with the materialflowing in the po$itlve direction fromnegative values af x at veloclty Uo,anti set the specific volume In theundisturbed material at Vo, then thesolution iS

U/v . Uolv0. (3-4)

For the solution of the next two ●aua-tlons more Information about A and Bis needea. In the Navier-Stokeseauatfons,

(3-5]A.v . - +[

(4/3judx1

[ 1B/u ■ (vo/uo) ~ (4/3)uuux - k;x (3-5)

Eauation L3-2) can be written. usingEa. (3-4) -na (3-5), as

[ 1(IIofVo)ux * D, - + (4i3)IIUx - CI{3-71

and immealately integrated to give.with the boundary conditions ~m9Gsed.

D - Do - OOUO(UO - u] - (4i3)uU .(3-s)x

whrre CIo - lIVO. After alviain?bY u. and using Ea. (3-1) to el mlnate

ux/u, one can write Ea. (3-3) as

Ex - “.= ● VP= + PVX

[ 1- (Vo/Uo) % (4/3)II”Ux - klx . ( 3-9 )

●nd tni~ Cen b~ integrated to give,with the i)ovndary cond!tlons impG$ed

(E ● Dv) - (E. - fJovo) - +(U: - u 2,

[ 1- (Vo/Uo) (4/3)~uu= + klx . (3- 0)

Far from the shock in the re ion oflar e potltlve x, the terms ?n Eas.(3-1) and (3-10) containing derivatives

2

~vfS (S-018)

nave decreased to zero, atid the Eqs.(3-4), (3-8), and {3-10) cm be writtenas the usual jump conditions for asteady shock wave. These are

Vlvo - Ulu o (3-11)

or

Vlv - 1 - (U. - u)/uoo

(3-12)

for ttve conservation of ●ass, from Eq.(3-4). Eauat on (3-12) is written tocorrespond toin laboratorymass ti:loclty(3-6) tkecomes

D - Do = noUo

the ❑ore familiar formcoordinates, whe”e theis (U. - u). Eq~ation

‘o - u) , (3-131

~lreaay In its familiar form. It canalso be written. using Eq. (3-!2) toeliminate (Uo - u), in its Rayleighline form ~S

(3-14)

Equation (3-10) can be written in itsseveral fmlllar forms a:

IE+P) - (EO-DOVO) - ; u: - &J2(3-15)

E - E. - + (UO-.J)2 + Do(vO-V).

E -Eo-i ~ (l)*Do)(#o-v) .

Eauation !3-16) IS obtained f~omby substltutifia u = Dfi *

3-16)

3-;7)

3-15)

(Lllv)(uo - ~11.‘wlI’C~ ~s Gbtblngafrom Fas. (3-13) and (3-11). Equatlol!3-17) is obtalnea from (3-16) Dy sl,b-stitucinq for one of tht berms(IIO - U) from EL. (3-13), and forthe ottver cne In the squartvl term from[q.(3-12). These @quationS descrfbcthe condlt~ons far lrom the shock wave,

~~~s~’~~ ‘he Properties on tl.e twotn~ s Ock.

Thm ilota!ls of the shock it~olfcan BQ o~tainca by ~l,teyratfng Eus.[3-8] ●nd (3-10), considering them asthe diffcrwntial Qauaticns that des-cribe the shock Itself. Some addi-tional assumptions about the ●qu4tionof state ar.d the values Of the shear

vlscnsity “ and thermal COnfJuCtivity kart rtau!red. The problem 1s celltreated by Hayes (l).

The jump In ●ntropy is ●lso lnter-estlmg. Tha ●quations show that theentropy Is Increased by the dlsslpatlveproctsses In the shock. From the firstlaw of thcrsodynemlcs

TdS = dE + pdv , (3-18)

on~ can write

15X _ Ex + DVX . (3-19)

Using Eqs. (3-7) ●nd (3-9) to substi-tute for terms on the )hs. one T;nds

s= -[

- (v:T) :x (4/3)uux 1+(VO,’UL)(lIT) +

[ 1(4/3)uuux+kTx (3-20)

‘itke equation can be Slm91iffef to

Ocucsx - (4/3)u(ux)2/T * (k/T) ~(TJ .

(3-21)

Integration then gives

I

xOouo(s - so) “ + (:)

‘o

( 3-22)

Iihile the fir$t term on the rh$ i$zero tar from the shock, th~ two inte-gral terms are positive contributionsto the entropy.

At thlz point, although it ha%n-tn!ng to do wfth finding s~ecialsolutions to the ●auctions, let uslook briefly ●t the viscous terms inthe equations,

‘:~::!:;t::dbf2 :3;n:rp~-t of 8 in Eas(3-2) and (3-3). Often writars usethe term -viscous pressure,” ususllydenoted by q, and It is fdentltiedwith tne terms {n EVJS. (3-5) ●nd (3-6}●s

q.- (4/3)uux . (3-?3)

For numer!cal snlut!on of the ●Qua-tions, artificial viscusity ~S used tn

•~ke the solutlons of the equations$tablc to perturbations by numericalnoise. Many different forms heve beenused for q. Me see that a has thedimensions of pressure, ●nd that thecoefficient of u In Ea. (3-23) mustbe a product of ~enslty, velocity, ●ndaistance. The Important dl$tance for

numerical stability Is the ❑esh spac-ing AX for the calculation! and thedensity o must be the locaI density,but the valoclty term can be chosen to●ake the numerical oscillations damIn Optimal fashion. Some populerchoices are the Landsnoff form

q- - OCAXUX (3-24)

using the sound velocity, theRlcntmyer-von Neum&n or quadratic form

a- - o(Axux)&tux (3-25)

uS$n9 Axux aS a \@lGCity, and theHarlow or PIC form

a- - Suixu x (3-26!

using the local particle velocity. In● numerical calculation all tnree forms●ay be used in linear ccmblnatfon,with dimensionless multfpllOrS chosenfor optimum damping.

NOW let US tu-n away from the

$tron~ shock wave, and look for Solu-tions correspondlnof an inflnlteslma? l;s~~~b~~~~~g~t’Onsound have. Me wish to Consider auniform medium with no strong gradi-ents, so the viscou~ and heat conduc-tion terms are ne ligible.

!we use

Eqs. (2-11). (2-1 ), ar., (2-13). re-written here as

.v - vu -ox (3-27)

.u + Vp -0a (3-281

. .DID ● Tvlv - 0 . (3-29)

Ue look for sclutlons for lnflnlte$iaalwaves ●oving at constant velocity cwithout chenge of shaDe, described by

v-vo

* #lf(x-et) (3-30)

IJ - 0 - Ulf(x-et) (3-31\

D-DO + Plf(x-et) . (3-32)

owls (s-018)

where VI, u . and II\Differcntla ing, an~ ;;~<;;~~;ll;s

higher than first order in the smallperturbations. we find

v - - Cv f’1 ( 3-33

u= - Ulf’ ( 3-34

u - - culf’ ( 3-35

PI = Dlf’ (3-36)

.D-- cD1f’ . (3-37)

Substituting these values into theoriginal differential eauations gives

Cv 1 - ‘Ovl.0 (3-38)

Cul - ‘OD1-0 [3-39)

P1 + (Ynolvo)vl - 0 - (3-40)

From Eas. (3-3b) and 13-39) we find

~2 2- - V_#l/V1 , (3-;1)

Corresllonaing to the usual deflnltiofi

c? - - V%D18VIS (;..42)

‘f 01 a’”:~:;:::::”;;::;; a:;..ln,Using ECIS

21 - c Ipv 00” (3-.”3)

corresponding to the usual deflnlt{on,after we substitute from Ea. (3-42).

T- - (vlrll(apisv)~ . (3-44)

TrILIS me ha~e shown t~at our eauationsdescribe a medium that trans.,its soundwaves ●ncl the ~, dtfined by Ea. (2-9).lC $lmply the square of thQ dimefision-less ,nund $veed.

In Eqs. (3-27’ throu h (3-23”- thedissiIaati.e termCA a~d i -ire . .glocted. Inclusion of these termsallows for dissipation of energy, ●ndthere?ore attenuation of scund. Formost cases of Dhy$ical interest, thedamping is small. The sound velocityremains ttiat for the r,ond~ssipativecase. Discussions of the damDing era

&

Oevls (S-olo)

given by Bond, Watson, ●nd Welch (2),and by Lighthlll (3).

Iv. THERMOOYMAMICS FOR HYORODYMAMICS

Compressible flows usually contelnlarge raglons where the flow Is ap-proximately isentroplc separ%ted fromother Isentropic regions Py smallregions rn.?re the flow is strol~glynonequilibrium ●nd nonisentropic. Thenatural thernodynamlc potential todescribe such flows is tha specificinternal ●nergy E, written as

E = E(S,V) (4-1)

-here S is tne specific entropy and v1S the specific volume. For theregions of isentropic flow the poten-tial E i> a function only of volume.

Tne aifferent~al expansiun of Eq.(4-1) Is

aE = T dS - pev (4-2)

where

T . (tE/~S)v and p - - (aE/av)S (4-3)

Tne independent variatvles S and v, andtne ~arlalsles obtained from the firstpartial derlvatlves, are the variablesof thermodynamics.

The derivatives of these variablescan be axpressaa as second darivatlvesaf the potential= There are thrgeIndependent second ~erlvatfve3. SO allthe derivatlvss can he ●xpressed interms of tsvree independent secondaerlvatives. In what fallows, we US?the subscript notation far differen-tiatlcn. sa that, for examgle.

E - (s2E/ai2)SVv(4-4)

ana tne independent variable held con-stant fs obvious from the context.The aef{nttions used hCre are

T.VE Vvll) - - (V/D)(#D/dV)S (4-5)

r - - VE~W/T - - (v/T)(aT/*v)j

- - (V/T)(IJ/tSlv (4-6)

g _ DvE@T2 - Dv/CvT . (4-7)

Tne$@ three partial derivatives formtne standard set for Ivydroaynamics;

●ll othor thcrwsdyneaic first dwiv~-tives can be written in terms ef t-.

The ■oessing of these secoad tiri-vetlves tkset form th~ standard set forhydrodynamics ●sy be ●adm C1OSFOF byconsidering tho follouiag ●xtrgssl.as:

1- - (> in pla in V)S ( 4-8)

r. - (s lnT/t In V)S . (4-9 )

now suppose that v and r are const8nts.Then one can integrate to find, on ahisentrope, thet

pvy- constant (4-10)

TVr = constant . (4-11)

Slmllarly, one can write

r(TSlpv) - (t in pla in S)v (4-12)

g(TS/pv) = (B in 7/s in S)v (4-13)

and integrate these tc get expressionson the curves of constant valume.(The factor TS/pv ●nters because wedid not use S when we made the secondderivatives dltiensionless.) Since wedo nat measure S, DerhaPS the ratfo ofthe twa.

Tr\p9 - canstant (4-14)

on a curve of constant volume, is ■areuseful.

In the real physical case T, r,and g are not constants, yet the ●x-pre$slons obtdined this way ●re tan-gents to the real curves at pdlntswhere the exponents tseve the chosenvalues.

Thermadynamlcs books usually useanother standavd set af derivatives,abtainea from the 61bbs potential, 6 =6(T,D), defined as

CD = -TGTT (4-15)

B - GTPIV IS-16)

CT - - GpD/v . (4-17)

The reason for this chaice is, afcourse. that many experiments are donewith either T ar p held constant, ●nd6(T,p) is the natural Dotentlal. Thisusual standard set can be expressed interms of the hydrodynamic standard setas follows:

5

Cp - (pvlgT

s - (rlgT)

owls (s-018)

- ~2/gj11(T (4-18)

iT - rzlp) (4-19)

(4-23)

so the sound velocity Increases withpressure only if 6 $s greater than 1.

‘1 - PI(Y - f_2/g) . (4-20)

The denamlnator In each expression canbe shown to be - (v/D)(tCl/8v)T. andIt ●ust be positive to ensure mechani-cal staolllty (A). For an Ideal gasthe denominator IS one.

The choice of symbols 1s hopelesslyconfusing. Various authors use varioussymbols; worse still, they use the samesymbols with different ❑eanings. Untilsome standardization takes Place.readers will just have to resign them-selves to being very careful to checkthe deflnltlons Perhaps the mostbothersome Is our definition of theediabatic gamma. The symbol Y hasbeen widely used for many years toaenote the rat~o cf speciftc heats;aur deflnltion, common In nyarodynam-ics, is given by Eq. (4-5). The ad~a-batlc gamma &nri the ratio of sPeciflcheats are ldenttcal for an ideal gas,but not for real 3ases. as can be seenby Combining Eqs. (4-7) and !4-18).

One higher derivative Is Imvortantin the context of empirical equaaionsof state. lt IS called the ‘fundamen-tal derivative of gas dynamics- ojThompson (5!, ana IS defined es

6 .+ [y + 1 - (v/y)(aT/w)~] - (~-:1)

For ordinary materials, 6 is posltfv~.Its importance 1s that when G ~s 9osi-tive, compression snock waves f~rm.If 6 is neqatlwe, rarefaction snocksform. For ttre purposes of thfs 9aDJ1 .

one must be careful not to choose formsfor gamma that leaa to 6 less than zerounless ra=efaction shocks are desired.6 can also tie written, usin the nata-t19n Of Eqs- (4-5) through !4-7) as

G - - VEVV*12EVV , (4-22)

It is often glibly sate that the ~on-dition for compression shocks to formis that thw sound speed must increasefilth kr$ssura. Really tne conditionis th4t hi her DrassurQ waves frombehind mut~ 0vart3t0 tha front, andthey travel ●t velocity u * c ratherthan c, and u also inc~aases withpressJre. The difference can bo made●specially clear by relating G to thesederivatives. It can DC shown thct

However,

w - ““con the characteristics b~hlna theshock. so compression shocks willa% lnng as 6 Is greater than zero

u JKCOHPLETE EQUATION OF STATE

4-24)

form

E = E(S,V) is a comPlete ●quation~f state. All thermodynamic deriva-tives can be obtained from It. E _E{D,v) IS not a comDlete eauat~on ~fstate. but IS very useful for hydrody-namics.

The relationship between E(S,V)and E(D,v) is ●asy to see. If one hasE(S,V), then - Ev - P(S,V). Inprinciple, at least, this ●xpressionfor p can be inverted to give S\P,v).ane then S can be eliminate in E(S,V)to give E(D.v). However. there is ncway to go backward; that is, one CannOtget back from E!D,v) to E(S,V).fherefore, EiP,-i Is incomirlete.

Most of the experiments in hyrJro-aynamics are mechanical ●xperiments.Their variables are p and v. Tempera-ture anC entropy are not measuredquantities and they cannot be i,,ferredfrom E(p.v\. Ontheotherhand, thevariables that cannot be ‘easured mustnot be really ne~ded, or they could bemeasured. Fur many purposes the in-complete ●ouation of state E - E(D.v)Is ~deq.ate.

Tne differentialDlete cauation of s’

dE - EDaD + Evdv .

If we use Ea. (5-1)ativd with respectwe get

of this incom-

ate is

(5-1)

t~ find the deriv-ova? constant S

(sE/av)S = Ep(tplav)S + Ev ; (5-2)

but Wa know that

(tEltv)S - - P (5-31

and us~ng this we can rearrange Ea.(5-2) to ql.e

- (JD/av)S = (EW * D)/ED . !5-4)

6

Davis (s-018)

Comparing this result with ‘Eq. (4-5)we find

Y = V(EV + p)/PEP“

(5-5)

Sliailarly, tak$ng the derivativewith respect to S at constant v onecan show that

(aplaS)v - T/Ep . (5-6)

Using this result with Eq. (4-6) onefinds that

r= v/Ep . (5-7)

The incomplete equation of stateE(p,v) thus gives the adlisbatlc gamma.snd the (iruneisen gamma. As shown InSec. II, it is adequate for simplehydrodynamics.

From the incomplete equation ofstate, g cafinot be determined; however,a differential equation for g can beobtained, and g is thus deternlned,except for a constant, along an isen-trope where Y and r are known. Thedifferential equation Is obtained byrequirin

!that the partl&l derivatives

of E(S,V do not depend on the orderof differentiation,Essv. E

so that E~~s =and E s are given

Eqs. (4-~f and (~-7). Some manlpula-tton gives the differanttal equation

(v/9)(09/av)~

- r+ 1- ~ - (rp/g)(9r/8p)v . (5-8)

A dtfferenttal equation for the tem-perature is also available, so temper-ature can be determined, within a con-stant multiplierOne form of Eq. IMn:san ‘Sentrope”

(v/T)(iT/tV)* - -r . (5-9)

If r is known, this can be integratedimmediately, except for the constant.

Expressions for T and g are especiallyuseful for detecting flaws In thechoice of a form for an empiricalequatton of state. One may not knowexactly what to expect for T or g, butona can axpact them to be smooth, pos-itive, and monotone on the isentrope.

VI. PHYSIC4L PRINCIPLES

It was shown In Section II that the~diabatic ?;:ma and the Gruneisengamma are important features of

the ●quation of stat. for hydrodyn--ics. If. Section 111 it was shows thatthe adiabatic gaama is the square ofthe dimensionl~ss sound speed, or

Here we ask what we know @bout thebehavior of the adiabatic gamma ●ndthe 6runeisen gamma ●s functions ofspecific volume ●long an isentrope.

Molecules interact with each otherwhen the distance between their centersis ● few tenths of ● nanometer. Ifi●

gas at room temperature ●nd pressurethe avera e distance between moleculesis about ! ~m, so nest of the time amolecule drifts at thermal velocity,unaffected by any other Molecule. Adisturbance, such as a sound wave, istransmitted through the gas by mole-cules traveling at thermal velocities,and the velocity of ● sound wave isabout two-thirds of the average thtrmalvelocity. Collisions are rare events.The details of the molecular interac-tion have a trivial effect on the vel-ocity of sound.

If the gas, originally at roomtemperature and pressure, Is compresseda thousandfold, so the number of mole-~:;:s2~nxa1:~~i: oc:~t;m:;~~, i;:;eases

average intermolecular spacing de-creases from 3.3 nm to 0.33 nm. Theeffect of a disturbance, a sound wave,is transmitted by molecules that drifta short distance and then collfde withanother molecule. The motion Is thentransmitted through the m~lecule bythe electrical forces at nearly thevelocity of light. Then there isanoth~r thermal drift, but only for ashort distance. Thus when the inter-molecular s~aclng is of the same orderas the molecular stzo, the speed ofsound increases above the low densityvalue. The change takes place, forordinary explosive products, at a den-sity near onemeter. The ad?;;;t?~rg;~;!~;;;ti-square of the dimensionless soundvelocity, changes markadly withspeclflc volume In this region.

As compression is continued up theisentrope, the sound velocity con-tinues to increase. Its value dependsin detail on the exact form of themolecular interaction. The adiabaticgamma, however, levels off at nearly aconstant value. This happens becausewe have dafined gamma by normalizingwith respect to PV, as shown in Eq.(6-l). Because of the energy in the

7

oSViS (S-018). ——.

mlecular Interactions, pv/RT increasesto lar ● values. For most reasonable

!forms or the molecular repulsion, theadiaba%lc gamma seems t~ be nearly con-stant ●t small specific volume. It iSeasy to show that If the repulslve po-tential for the interaction ●nergy oftwo molecules varies Inversely as thenth power of the separation, there isan upper Lound for the adlabatlc gmaa,

Y < 1 + n13 . (6-2)

A schematic plot of the variation ofgamma with speclf!c volume is shown inFig. 1. It always has this generalshape, but details of the potential,and effectsfrom ohase changes and phaseseparation can cause small perturba-tions In local regions.

It should be mentioned that thisdiscussion is to be applied to the tem-peratures and volumes of interest forexplosives. That 1s, regions where thethermal energy of the molecule is muchless than Its ionization energy. I/henthere Is appreciable ionization anddissociation, new effects are impor-tant.

The Grunelsen gamma has behaviorvery similar to that of the adiabaticgamma. At very large specific volume,

r- Y- 1 , (6-3)

so it has a value near 0.3. It in-creases as the volume decreases, andlevels off at O.fI or 0.7’at small vol-ume. Figure 1 shows a schematic dia-gram of the usual behavlcr of theGruneisen gamma.

VII. ENGINEERING APPLICATIONS

The preceding sections hava beendevotad to the properties of the ther-modynamic equation of state of detona-tion product gases. The words “equa-tion of state” are often used to denotesomething very different.

Exploslv@ systems are usually de-signed with the help of computer pro-grams that solve the hydrodynamic equa-tions. Often the computed system dif-fers markedly from the actual physic:lsystem. For exansple, the shape may beideallzed by neglecting glue joints,small vofds, or plastic potting com-pounds. The intttatfon is usually lde-allzed In important ways and not com-puted In detail. The chamical reactionzone fs not modeled properly. The pro-perties of the material bein ~ driven,perhaps metal or rock, ●re s mpllfted.And fn the tnterest of getting things

3

2

I

J===0.1 I 10 100

v

Fig. 1. The dependence of the adfa-batic and Grune\sen gammas onspecific volume.

done, sometimes the mesh s{ze is madelarge and the computer does not givean accurate solutiun of the equations.All these defects are accommodated byadjusting the ‘equation of state.”

A system designer, then, cannot usea real, thermodynamic equation ofstate. Hhat he needs Is an approximateform that will allow him to desigu afirst approximation to the requl~edsystem using his computer, with all itsdefects. Then he must test hls firstdesign, and use the results to changethe “equat{on of state.” Then he musttry again. If the requirements havetight tolerances, this iteration can bevery expensive and tfme consuming.

One of the problems with the f~t-ting forms In common use 1s that theyhave so many adjustable constants.There are too many ways to adjust the“equation of state” for a good fit tothe experiment. An “equation of state”with just one adjustable parameter pro-vides all the necessary adjustment, butstill allows the user to make system-atic changes: so (much positive forth!s defect, so much the other sfgn foranother, etc.

The one adjustable parameter for afittfng form wfll probably not be oneof the constants In the form but somereal physical paralneter of importancefor the system bel~g designed. Forexample, for reproducing the results

8

.- .?hVf$ (S48)

of ● cylinder test, the iaportant par-ameter is the cylinder wall energy atlarge expansion, and this Is the onethat aust be adjusted. The constantsIn the fitting form aust be varied sothat the other calibration par~etersare held fixed, and only this ●nergyis changed. For a shock wave In air,the low pressure expansion fs impor-tant, and for an overdrlven or conver-gent detonation the high pressureregion is the one to adjust. The vitalthing Is to adjust only that faportantregion, and to use only one parameter.In this nay, a systematic understand-ing o? the adjustments that correctfor various defects can be obtained.

VIII. DESIGNING A FITTING FORM

The usual approach, and the onediscussed hete, is to find a fittingform for the incomplete equation ofstate discussed in Section V,

E = E(P,v) . (8-1)

It is convenient and customary tochoose a particular isentrope, usuallythe principal isentrope that passesthrough the Chapman-Jouguet point, andfind ? fit for it. Because it is aparticular isentrope, any function onthe isentrope can be expressed as afunction of volume only. Thus on theisentrope the specific internal energyis

ES = E$(v) , (8-2)

where the subscript S is used to indi-cate that the subscripted variable isto be taken on the particular isen-trope. The definition of the Gruneisengamma, given in Eq. (2-8), is

r = vIEP’

(8-3)

so in the immediate neighborhood ofthe isentrope the energy may be ex-panded as

E(P,v) - E$(v)+(v/r)[p-pS(v)] . (8-4)

To make a useful equation of state, itis assumed that Eq. (8-4) appliesthroughout the region of interest, andthat the Gruneisun gamma ts a functionof volume only,

r -r(v) . (8-5)

These two assumptions are not as badas they might seem, because the entropyproduced in shock processes inexplosive-driven systems is never

large. ●nd the rqion of”iatarost is anarrow strip always close to theprincipal isentrope.

Perhaps physi:al intultlom Is bestfor the ferm of the ●diabatic gane enthe principal isentrope, ●s was dis-cussed in Section VI. If its form ischosen, one has

Vs - Ys(v) . ( 8-6)

The definition of the ●diabatic gemma,Eq. (3-44), can then be written as

dp$ips = - Ysdvlv B (8-7)

and then integration gives

Ps = Ps(v) . (8-8)

The internal energy on the isentropecan be obtained from the first law ofthermodynamics with the entropy heldconstant,

dES - -psdv . (8-9)

Each integration introduces a constantof integration; the one from the pres-sure eq”jation, Eq. (8-7), allows oneto choose Jhe particular isentrope,making it pass through a chosen p,vpoint, and the one from the energyequation, Eq. (8-9), sets the zero ofenergy, usually taken so the energy iszero at infinite volume.

The program outlined here seemsvery simple, but when one attempts tocarry it through, it quickly becomesapparant that the integrals cannot beexpressed in closed form if Ys(v) ischosen with efiough complexity to givea reasonable representation of itsreal form. One response to this dif-ficulty is to let the integrals beexpressed as interpolations in tablesobtained f?om numerical integration,or as series expansions. An~ther pos-sible response is to divide the volumeInto small intervals with simple fitsin each interval. And a third responseis to start with the energy represent~dby a sum of functions, so that

E$(v) ● Zaidi(v) . (R-1o)

Then by differentiation one fi}lds

Ps(v) - - zaid~(v) , (8-11)

and

9

ONIS (S-018)

YS(V) - v z ai6; (v)/ ~a#;(v) . (8-12)

This form for YS can then be fit t~the chosen form for Y (v).

?The

widely used JUL equat on of state isof this type.

A form for the 6runeisen gammamust also be chosen. Many workershave chosen it to be constant. Itseem that a better choice is to giveit the sane form as the adiabaticgamma, hut with values near thosediscussed at the end of Section VI.

An expression for the adiabat:cgamma off the principal isentrope isobtained by using the definition ofgamma, Eq. (2-9), and substituting inthe partial derivatives obtained fromEq. (8-4). After simplifying theresult by substituting from Eqs. (8-7)and (8-9), the result is

Y(P*V) - (PSIP)YS

+ (1-pS/p)(r+l-d in r/d In v). (8-13)

This expression makes it clear that adiscontinuity in the slope of the6runeiseo gamma will lead to a discon-tinuity in the adiabatic gamma itself.Similarly, Eq. (4-21) shows that adiscontinuity in the slope of the adi-abatic gamma will lead to a discontin-uity in G. Such discontinuities arenonphysical, but it isn’t clear whatspurious effects might appear tn iicalculation where an equation of statewith discontinuities in the slopes ofeither of the gamm..ss on the isentropewas used.

Expressions for new isentropes,above or below the principal isentrope,are obtained by integrating Eq. (8-13).Hugoniot curves are obtained by usingthe equation of state, Eq. (8-4), andthe !iugoniot relation

(8-14)

and eliminating E. Particle velocitieson the isentrope are obtained by inte-grating

dp/du - ● civ = ● (YP/V)l’2 . (8-15)

Even for simple choices of functionsfor the adiabatic gamma, the integra-tion almost always has to be donenumerically.

IX. CALIBRATION

The calibration of a fitting formfor an equation of state opens oppor-tunity for prejudice and personal pre-ference. There are no absolute rules.The importance of various measurementsto the calibration depends on theapplication.

Calibration of an eauation of statebegins with tie Chapman:Jouguet state.-A subscript j denotes that state inwha: follows. First there is the re-quirement that the principal ise~:::pepass through the point p , v .there is the addition.sl #equfremelitthat the Rayleigh line and the Hugon-iot curve be tangent at that point(the Chapman-Jouguet condition); thisrequirement is met by requiring

Pj = Po$(Yj + 1)

and

‘jlvo= Yj/(Yj + 1) .

(9-1

(9-2

One might proceed, for axample, bymeasuring the detonation velocity andthe CJ pressure. Then Yj can beobtained from Eq. (9-l), and vj fromEq. (9-2).

The problem with this approach isthat apparently ISOone knows how tomeasure CJ pressure. One need onlythumb through the seven DetonationSymposium volumes to see that therewas no more agreement in 1981 thanthere was in 1951, and thdt thediscussions qet more and more complexwith time. For calibrating an equationof state, one need only realize thatif it made a lot of difference, itwould have been measured by now. Formany purposes the exact value is notvery important. This fact has led tothe development of “rules for gammkWVthat give the value for the adiabaticgamma at the CJ point simply in termsof the initial density of the explo-sive. A simple rule that workssatisfactorily is

‘d. 1.6+ (),8 PO . (9-3)

The initial density and the measureddetonation velocity can then be issedwith Eqs. (9-1) and (9-2) to find pjand vj.

The second thing to get right inthe calibration of an equation of state

10

owls (s-018)

iS tne am~unt of energy available forthe sy<:em under consideration. Foralmost any system there is a “cut-offpressure”, where once the exoloslveproducts have expanded to the volumewhere the pressure reaches this value,little additional work *S done on thesystem. Either the metal or rockbreaks, allowing the qases to ●scape,or the time is too long and the addi-tional acceleration too late, or someother external condltlon makes theenergy remaining in the prod~cts use-less. For metal systems driven byexplosive. the cut-off pressure isabout 0.1 Wa in many cases. It hasbecome customary to fix the energydelivered for an expansion oowr totnat pressure witn a calibration ex-periment. Perhaps the best known ex-periment is the cylinder test (6). Fornigh-density, nigh-energy explosivesthe cut-off pressure comes at an ex-pansion of six or seven times the i-i-tial volume, and the cylinder test Is

resignea to measare an appropriatevalue. For other explosives and Otner

uses, alternative tests nave been used.

Tne Jacobs engine and tne Fickett-Jacobs cvc!e described by Fickett andDavis (7) make it easy to understandtnis calibration. Figure 2 is a dia-gram Gf tne cycle; it Is cescribed In

the captiofi. Tne area between tne

Dase line. the Rayleign line, anfl theprincipal isentroDe 1s equal to themaximum useful work tnat coul0 be ob-

tained from tne explosive. Some oftnat energy Is not .sef.1, because thepress.re is too low for the aGDl4ca-tian. Tnerefore, the diagram must betruncated, as shown in Fig. 3, at theilmiting useful pressure. For tneuSEfUl energy Cal toration, the area tothe ieft of tne truncation line mustre made proportional to the energy

9@talned from the test. Several rulesfar an approx~mate ~al!bration have

been used. The total area Is Eo,ano the area to the right of the trUfi-cation line is E5, so tne rule is

E - E,, - Etest 5.

(9-4)

The eguatlon of state Daramet ‘5 areadjusted to satisfy ●nis rela onsnlp.

A thiro calibration golnt IS to

set the total energy. Eo. eatial totne calculated chemic~~ energy of theexplosive. Altnougn it is sstneticallySatisfying to nave the work availableequal tn the chemical energy, it isnot a!. lmDol cant calibration Dolnt.In the first place, there is no aDDll-catlon of exolosiwes wnere tne energy

Fig. 2.

Fig. 3.

v

The Flckett-Jacobs cycle.The initial state, unreacted●xplosive Is at golnt O. TheCJ state Is at point 1. TheDroduct gases expand agaln$ta Pfston from Point 1 topoint 2, doing useful work.The gases are cooled so theycontract from 2 to 3, andthis ●nergy IS lost to thesystem.reacted ~% ~a~~s~r~ackinto the t!riginal ●xplosive.TO Get all the gas uniformlyinto the CJ state. work mustbe aone on it, 6,.J this workis rsp~ezented by the arza0-1-4. The maximum usefulsutDut wnrk 4s the area O-1-2.

II

In most application. of ex-DIOS!VeS, the product gasasdo useful work only when theirDressure is above some cut-off pressure, shown here atpo{nt 5. The useful work ISthen represented by the nreaO-1-5-6-O.

11

hViS (S-018)

●t low pressure Is tmportant; If lowpressure could do the work, onewouldn’t use exploslve. And In thesecond place, heat energy Gf the ex-plosfve is not used (see the segment2-3 In Fig. 2), and ts rejected to thesurroundings. Therefore, E. Is notequal to the chei%lcal energy. It iSreally more Important to get th~ soundvelocity about right at low pressurethan to worry about adjusting for “hetotal energy.

These calibrations d~termine theprtncipal isentrope. The remainingcalibration Is for the Gruneisen gamma,

It influences the values for statesofr the pr!ocipal isentrape. 9ver-driven, collidlng, and convergentdetorlations provide the deta for de-termining 6runeisen gamma. So far,there have been no definitive callbr&-tions. If overdrive detonations areimportant in the application of theequation of stfite, Grun@ison gamma$hould be adjusted to fit the data.Otherwise, the choice is not important,and a simple form or evun a cc.nstafitvalue cai. be used tiith little effecton the calculations.

ACKNOWLEDGEMENTS

U. Fir.kett, M. Finger, b. D. Lam-bourn, E. L. Lee, C. L. Nader, M. S,Shaw, R. C. Heingart, and many othershave helped me to understand somethings about equations of state, and Iwish to thank them all for their help,

REFERENCES

1.

2.

3.

4.

5.

6.

7.

U. L!. Hayes,Gasdynamic OFundamentalsEmmons, Ed.,AerodynamicsPrinceton UnPrinceton, N,1958.

‘The ?asic Theory ofscontinuities,” inof 6asdynamics, H. U.Vol . III, High Speedand Jet Propulsion,versity Press,J .* pp. 416-481,

J. U. Eand, K. M. Matson, and J.A. Helch, Atomic Theory of Gas;~;:mics, pp. 2746 Add-b ison -

ey, Reading, Mass., 1965.

J. Lighthill, Haves in Fluids, pp.76-85, Cambr!d-ersity fress,Cambridge, Mass., 1979.

H. 8. Callen, Thermodynamics, p.Nib John Ulley and Sons, New York,

.

J. W. Kury et al., ‘Yetal Acceler-ation by Chemical Explosives,”Proceedings Fourth Syrposlum(International) on Detonation,ACR-126, Office of Naval I!esearch,pp. 3-12, 1965.

W. Fickett and W. C. Oavis,Detonation, Univ. of Californiapress, ~keley, California, pp.35-39, 1979.

12