CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Hone
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
A COMPLETE EQUATION OF STATE
FOR DETONATION PRODUCTS IN HYDROCODES
O. Heuze
CEA DAM/ lie de France, B.P. 12 , 91680 Bruyeres-le-Chatel, FRANCE
Recent works have shown new requirements for the equation of state of detonation products in
hydrocodes. Experimental results about sound speed in detonation products implied that a new model has
to be built. Moreover, the advent of teraflop computers will allow the use of new reactive flow models
based on Arrhenius kinetics which need realistic temperatures given by the equation of state. In this paper,
we describe a new model that proposes solutions to these problems. It is a complete equation of state for
both detonation products and initial explosive that provides access to all thermodynamic properties
including temperature, entropy and free energy. Several procedures are proposed to determine the
parameters of the model, from thermochemical calculations or experimental data.
INTRODUCTION
temperatures such that it may be used with Arrhenius
kinetics for modeling the transition to detonation.
For ten years, several proposals have been done
[1,2,3,4] to provide improved models compared to
the well known JWL equation of state [5]. Recently,
the improvement of both experimental [6] and
numerical [7] tools allows to obtain more
information to predict the behavior of explosives.
However, equations of state used in hydrocodes are
not able to provide such information.
The aim of the present paper is to propound a
new model called CW2 corresponding to these
requirements.
In a former paper [8], we have shown the validity
of the Mie Griineisen assumption for detonation
products of high explosives. This leads to a linear
P(E) (pressure versus internal energy) relationship,
very useful for hydrocode calculations.
We introduced a potential EK(V) for the cold
contribution, and a temperature/volume relationship
which gives a good agreement with thermochemical
calculations. The parameters of the whole model are
adjusted from thermochemical calculations or
detonation experiments.
Moreover, it reproduces the linear shock
velocity versus particle velocity relationship, and
then is suited to the initial explosive. It gives realistic
THE CW2 MODEL
We define the whole model from the free
energy, as function of volume and temperature, and 9
parameters (F0 ,a, k , JLL, Cv 8, pr,0r,Sr):
A(V,T) = E K (V) + C v .T[ln(x)-6.x]-S r T
(1)
with :
EK(V)=y[?i(p/Pr)+Hln(p/pr)]
(2)
x = 0ry/T
(3)
= ( P /p r )'°exp[a((p/ Pr ) 10 -l)/2]
(4)
The first derivatives of A(V,T) give pressure
and entropy:
m/\
5-x)
S(V,T)=C v [l-ln(x)]+S r
(5)
(6)
with:
(7)
450
and
r=r0+a(P/Pr)2
Let us compute P(1/V,E) and E(1/V,T) from
thermochemical calculations. Choose arbitrary
parameters pr,0r,Sr.
P(1/V,E) gives the slope of P versus E at constant
volume from which we deduce the Griineisen
coefficient F. A linear regression of F versus p2 gives
F0 and a. Then it is possible to determine y.
Then we calculate A(V) = P(1/V,E=0). A second
linear regression of V.A(V)/y versus density yields A
and ji . At that point, it is possible to make hydrocalculations without temperature.
From
E(1/V,T), we calculate (E-Ek(V))/T
versus x. A third linear regression gives Cv and 8.
The nine parameters of our model are now
determined: three arbitrary parameters, and six
parameters calculated from three linear regressions.
(8)
then:
: v .T(l-5.x)
(9)
F0 is the Griineisen coefficient at null density.
Cv is the specific heat capacity at constant
volume
pr) 0r, Sr are respectively reference density,
temperature and entropy. They are arbitrary
parameters.
0r is similar to a Debye temperature.
MAIN PROPERTIES
From (5) and (7), we obtain this equation of state
in the Mie-Griineisen form:
PARAMETERS FROM CHAPMAN-JOUGUET
STATE DATA
(10)
Notice that:
From given p0, E0 and Chapman-Jouguet data
DCJ, PCJ? TCJ and the pressure of combustion at
constant volume Pv (Pv -0.46 P C j ), we can
calculate the main parameters.
The detonation products properties are not very
sensitive to Fo, a and Cv. For instance, for high
explosive, we can take the following average values:
Fo-0.13, a~0.28 and Cv~2200 J/kg.K.
We deduce:
dy/dV = r(V)/V
Let us assume that:
dE K (V) T(V).
A(V) = -E K (V)
(11)
dV
V
We notice that A(V) corresponds to P(V,E=0), and
equation (10) can be written:
(12)
Then V.A(V)/y is a linear function of density:
V.A(V)/y = A, / V + ILL
Yc/
(13)
PO(YC/ +I )
Notice the linear relationship of F versus square
of density.
We remark that x is constant on isentropic
curves. This property is very interesting to determine
analytically the isentropic expansion.
It is possible to invert equations and express
internal energy from volume and entropy:
Py =
We introduce Kj and K2:
K
fy__\
Po
_
E =EK(V)+y.Cv.0r exp[(S-So)/Cv-l)]
(14)
DETERMINATION OF PARAMETERS FROM
THERMOCHEMICAL CALCULATIONS
_r
!
Yc/L
YC/+ 1
Then we obtain the parameters:
From former considerations, we are now able to
determine the parameters of our model.
451
To
\l=C
8=-
PARAMETERS FOR A LINEAR D-u
RELATIONSHIP
MIXING LAW
The Mie-Griineisen form is generally used for
solids. Then the idea to use the CW2 equation of
state for the initial explosive becomes obvious.
Let us assume the linear relationship between
shock velocity and particle velocity:
D=c+su
and introduce:
T! = P / Po
This gives the Hugoniot pressure PH:
With one set of parameters for the initial
explosive and another set for its detonation products,
we define the equation of state for any mixture
corresponding to a burn fraction a.
We take a constant value for the three reference
parameters, and the six other parameters (pi =(T0 ,a,
X , ji, Cv, 5) follow the law:
With the linearity of the parameters versus a , we
do not need to make any other assumption about the
mixture: isotherm, isobaric . . .
Then such a model is particularly useful for
homogeneous liquid explosives with an Arrhenius
kinetics.
The CW2 model gives:
At the initial state, we obtain:
P = Po
11=1
PH=0
From which we deduce:
A, + M. + (r0+a) = E0
We introduce:
RESULTS
We compare our model with recent experimental
results obtained by TANG et al. The sound speed was
measured versus pressure on the upper part of the
reactive Hugoniot of PBX9501.
Our
parameters
were
obtained
from
thermochemical calculations and the three linear
regressions.
A significant discrepancy could be seen with the
original JWL equation of state, but a better
agreement was obtained with a modified form.
Figure I shows that the results obtained with our
model is in good agreement with the experimental
points. This implies its good qualitative behavior.
W2 (r0 ,a) = 2(F0 + a}3 + (F0 + d)2 - 6a
CONCLUSION
This study is a step of a general method to provide a
predictive and efficient equation of state of
Then, we obtain:
452
10000
9500
9000
8500
8000
7500
7000
6500
300
400
500
600
700
800
900
1000
P(kb)
Figure I. Sound speed versus pressure for PBX9501
detonation products for hydrocodes. We propose a
single equation of state valid for both the explosive
and its detonation products, including temperature
and entropy, expressed both in P(V,T) and P(V,E).
Parameters can be deduced from thermochemical
calculations or experimental data.
Then it can also be used with chemical reactions.
It provides temperature then it allows to be used with
Arrhenius kinetics, particularly useful
for
homogeneous liquid explosives.
We have shown that, without particular
adjustment, it is in good agreement with recent
measurements of sound speed versus pressure.
[3] BAKER E.L. , Technical report ARAERD-TR91013(1991)
[4] HEUZE O, BAUER P., "A simple method for the
calculation of the detonation properties of CHNO
explosives" Symposium HDP, La Grande Motte,
pp. 225-232 (1989)
[5] KURY J.W. "Metal acceleration by Chemical
explosives", 4th Symposium on Detonation
(1965)
[6] TANG P., HIXSON R.S., FRITZ J.N. "Modeling
PBX9501 overdriven release experiments" APS
Shock Compression in Condensed Matter,
Amherst(1997)
[7] TARVER C., APS Shock Compression in
Condensed Matter, Amherst (1999)
[8] HEUZE O. , "An equation of state from
detonation products in hydrocodes", 27th
International Pyrotechnics Seminar, Grand
Junction CO (2000).
REFERENCES
[1] DAVIS W.C., "Equations of state for Detonation
products" 10th Int. Symposium on Detonation,
Boston (1993)
[2] BYERS BROWN W., BRAITHWAITE M.,
"Development of the Williamsburg Equation of
State to model non-ideal Detonation" " 10th Int.
Symposium on Detonation, Boston (1993)
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