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
HYDRO-REACTIVE COMPUTATIONS WITH A TEMPERATURE
DEPENDENT REACTION RATE
Y. Partom
Rafael, P.O. Box 2250, Haifa 31021 Israel
Hydro-reactive computations are usually performed with a reaction model containing a pressure
dependent reaction rate (PDRR). A well-known example is the "Ignition & Growth" (I&G) reaction
model introduced by Lee and Tarver some twenty years ago. Performing such computations it has
become evident that in many cases the results obtained seem unreliable. For these cases using a
temperature dependent reaction rate (TDRR) may produce better results. We're using a surface burn
reaction model that we've developed some twenty years ago. Originally we used it with a TDRR.
Some years ago we introduced it into the PISCES code and used it with a PDRR. In this work we
reintroduce the TDRR into the model with the purpose of comparing the performance of the two
reaction rates. We first calibrate the rates to reproduce the pop-plot of PBX-9502. We then run the
code with the two rates for several ID and 2D situations. The resulting differences are qualitatively
as expected, but previously we could not have estimated them quantitatively.
It is generally accepted that reaction rate should
be mainly dependent on temperature. Nevertheless,
most reaction models use PDRR. This may be
justified for single shock initiation, but not for
complex initiation and detonation processes.
Typical examples for which using PDRR is
expected to be in error are short shock, ramp and
double shock initiation, detonation in a rod, and
corner turning.
In what follows we reintroduce TDRR into our
SB reaction model in PISCES. We then show
results of computations with PDRR and TDRR,
calibrated to the same pop-plot (of PBX-9502), for
the cases mentioned above.
INTRODUCTION
Hydro-reactive computations are usually done with
a reaction rate that is mainly pressure dependent
(PDRR). The I&G [1] rate equation is given
generally by:
W = W(W,p/ Po ,P)
(1)
where
W=reaction
progress
parameter,
W =reaction rate, p=mass density and P=pressure.
Some twenty years ago we developed a reaction
model based on ignition at hot spots and
propagation of burn surfaces [2-4]. We used there a
temperature dependent reaction rate (TDRR):
W = W(W,TS)
(2)
where Ts=temperature of the unreacted explosive.
A few years ago we introduced our surface burn
(SB) reaction model into the PISCES code [5]. On
that occasion we changed the reaction rate to
PDRR, to be able to compare with I&G predictions.
SURFACE BURN (SB) REACTION MODEL
The physical picture behind our SB model is
described in [2-4]. We report here only the main
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equations. We use separate equations of state
(EOS) for the unreacted explosive (s for solid) and
for the products (g for gas).
Es=Es(Ps,Vs);Eg=Eg(pg,Vg)
(3)
where E=specific internal energy, V=specific
volume. Conservation of energy in the solid and in
the gas are given by:
P = (V-E V W)/D V ; f s = D T s P
Vg=DVgP +EVgW;V.=Dv.P
where:
D Ts =-(q/c Vs +r s T s /v s )D Vs
nVs
-q)+3Es/dVs
dEg/dP
Es+(Ps+q)Vs=0
E g +(p g +q)v g -
(4)
"(P + q)+aj g /SV g
-(p g+ q)(v s -V g )£-(Q + E s -E g )f = 0
where q=artificial viscosity and Q=heat of reaction.
To complete the set of equations we need four mix
rules. Three of them are:
ps =p
r
r
r
g =p
V=WVg+(l-W)Vs
Vg
~ W
(9)
(P + q)+dE g /dV g
Dv=WDVg+(l-W)DV8
Ev=WEVg+Vg-V.
where Cv=heat capacity
coefficient.
(5)
E = WE g +(l-W)E s
The fourth mix rule is hidden in Eqs. (4). We
assume there that solid swept by a burning surface
joins the gas, and that the solid remains adiabatic.
The solid and gas temperatures are therefore
different. For PDRR we use:
W = R P y(w)(P/P ref ) a ;P>P l g
(6)
w =o
(8)
and
r=Gruneisen
CALIBRATION
We calibrate the reaction rate parameters to
reproduce the experimental pop-plot in a ID
calculation. The EOS parameters that we use for
PBX-9502 are:
p 0 = 1.895g/cm3 ; Q = 3.73kJ/g
C 0 =3.32 ; S = 2.22
r s =l;C V s =.0012kJ/g
(10)
A = 473.1 GPa ; B = 9.544GPa
• p<p
, JT ^ ITlg
where y(W) is a (nonnalized) burn topology
function (BTF) described in [3], RP and a are
parameters to be calibrated from a pop-plot, Pref is
an arbitrary reference pressure (we use 5 GPa), and
Pig is the threshold pressure for hot spot ignition
(we use 1 GPa). For TDRR we use:
W = R T y(w)exp(-T*/TS); Ts > Tlg
R j =4 ; R 2 =1.7 ; w = .48
where Co and S are parameters of the solid
Hugoniot and A,B,Ri,R2,w are the JWL EOS
parameters for the gas. The calibrated parameters
for PDRR and TDRR are:
R p = 0.0235 jis"1 ; a =4.1
(11)
R T =4741^ ; T * = 2 7 7 0 K
In the calibration process we use the same cell size
as later in the applications (10 cells/mm). For
TDRR there is a significant dependence on cell
size.
In Fig. 1 we show computed pop-plots for both
PDRR and TDRR. The experimental pop-plot is
represented by two crosses at 10 and 20 GPa input
pressure.
W =0
'.T^T^
where RT and T* are parameters to be calibrated
from a pop-plot, and Tig is a threshold temperature
for hot spot ignition (we use 400 K). The BTF
starts from a finite small value, increases to one,
and goes to zero at W=l. We do not change y(W)
in the calibration process.
To solve the model equations across a time step
we express them in rate form by time
differentiation. We integrate the system of rate
equations across a time step using a standard ODE
solver. The system of rate equations is:
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the delay of the second shock. As for ramp loading
here too TDRR run distance is higher and for the
same reason.
Input pressure (GPa)
FIGURE 1. Calibration check. Computed pop-plots with PDRR
and TDRR. Data represented by two crosses at 10 and 20 GPa.
0
0.2
0.4
0.6
0.8
1
1.2
Ramp duration (microsec)
ID EXAMPLES
FIGURE 3. Ramp loading. Run to detonation versus ramp
duration for a ramp loading to 15 GPa.
The ID examples include short shock, ramp and
double shock initiation. In Fig. 2 we compare the
results for short shock initiation.
0.4
0.6
0.8
1
1.2
Delay time (microsec)
FIGURE 4. Double shock initiation (5 to 15 GPa). Run to
detonation versus the delay of the second shock.
0.1
0.2
0.3
0.4
0.5
Pulse duration (microsec)
FIGURE 2. Short shock initiation. Run to detonation versus pulse
duration for 15 GPa input pressure.
2D EXAMPLES
We see from Fig. 2 that while for PDRR the
dependence on pulse duration is sharp, for TDRR it
is more gradual. This is because even upon
complete pressure release the residual temperature
is still substantial.
In Fig. 3 we compare results of ramp loading to
15 GPa. We see that TDRR produces a larger run
distance to detonation. This is expected as shock
heating from a ramp is smaller than from a single
shock to the same pressure.
In Fig. 4 we compare results of initiation from a
double shock, a pre-shock of 5 GPa and a second
shock to 15 GPa. We show run to detonation versus
The 2D examples include detonation in a rod
and corner turning.
In Fig. 5 we show W contours at 2 jo,s for
detonation in a rod with PDRR and TDRR. A
PMMA impactor starts the detonation wave. We
see from Fig. 5 that for PDRR there is an unreacted
layer 1 mm thick near the boundary, while for
TDRR the unreacted layer near the boundary is
almost non-existent. In Table 1 we give our results
for detonation in a rod in terms of
propagation/extinction of the wave. We see from
Table 1 that for PDRR the failure diameter is
between 8 and 9 mm (lines 1 and 3), while for
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differences between predictions with PDRR and
TDRR. Predictions with TDRR seem more reliable,
but more tests are needed to establish that.
TDRR it is between 7 and 8 mm (lines 7 and 8).
The difference is within the scatter of experimental
data.
Euaa E* LML MVBM oo: 01- acco a.ME RH
PDRR
TDRR
FIGURE 5. Detonation in a rod. Comparison of PDRR and
TDRR. W contours at 2 |1S. Rod radius is 4 mm and PMMA
impactor velocity is 4 km/s.
TABLE 1. Summary of rod detonation computations.
Rod
P/E
Reaction No.
Impact
Radius
rate
Velocity
(mm)
(km/s)
1
4
4
E
2
4.5
4
E
PDRR
3
4
4.5
P
4
3.8
4.5
E
3
4
5
E
P
3.5
4
TDRR
6
P
7
4
4
8
4
3.5
E
P=Propagation, E=Extinction
FIGURE 6. Comer turning. W contours for PDRR (at 5.5 jos) and
TDRR (at 6 ^is).
In Fig. 6 we show corner turning results from
computations with PDRR and TDRR. We see W
contours at 5.5 jis (PDRR) and 6 j^s (TDRR). We
see from Fig. 6 that for PDRR corner turning is
very slow while for TDRR it is much faster, with a
corner turning distance of about 12 mm (close to
experimental data for PBX-9502).
REFERENCES
1.
Lee, E. L. and Tarver, C, M., Phys. Fluids 23,
2362-2372(1980).
2. Partom, Y., "A Void Collapse Model for Shock
Initiation", lihsymp. OnDet, 506-511 (1981).
3. Partom, Y., "Characteristics Code for Shock
Initiation", LANL Report, LA-10773 (1986).
4. Partom, Y., "Surface Burn Model for Shock
Initiation", JJDP Symp. IV, 161-170 (1995).
5. Partom, Y., "Failure Diameter of Confined
Explosive Rods", SCCMSymp., 829-832 (1999).
CONCLUSIONS
To check the importance of using TDRR we
reintroduce it into our SB reaction model. We
calibrate Both PDRR and TDRR to reproduce the
pop-plot of PBX-9502, and then compare
predictions with the two rates. We find significant
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