CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
CALCULATION OF CHEMICAL DETONATION WAVES WITH
HYDRODYNAMICS AND A THERMOCHEMICAL EQUATION OF
STATE
W. M. Howard, L. E. Fried, P. C. Souers and P. A. Vitello
Lawrence Livermore National Laboratory, 7000 East Avenue L-282, Livermore, CA 94550
Abstract. We model detonation waves for solid explosives, using 2-D Arbitrary Lagrange Eulerian
(ALE) hydrodynamics, with an equation of state (EOS) based on thermochemical equilibrium, coupled
with simple kinetic rate laws for a few reactants. The EOS for the product species is based on an
exponential-6 potential model, whose parameters are fitted to a wide range of shock Hugoniot and static
compression data. We show some results for the non-ideal explosive, urea nitrate. Such a model is a
powerful tool for studying such processes as initiation, and detonation wave propagation as a function of
cylindrical radius.
CHEETAH [2], which allows for products to be in
thermochemical equilibrium with an arbitrary
number of reactants coupled via arbitrary rate laws.
CHEETAH calculates the chemical composition of
the reacting high explosive gases, and then predicts
the EOS of the gases using a high-pressure fluid
EOS.
The transformation of the solid high
explosives into a reacting fluid of small product
molecules is based on simplified chemical kinetic
schemes, which are determined from measured
detonation velocities as a function of size.
The EOS for products can be based on an
exponential-6 potential [3] for supercritical fluids,
with a Murnaghan form [4] for solids and liquids.
For the exponential-6-based EOS, the parameters are
chosen to reproduce the shock Hugoniot and static
compression data for a wide range of supercritical
fluids [2.3,5]. This is an accurate and efficient
complete equation of state of the exponential-6 fluid
based on HMSA integral equation theory and Monte
Carlo calculations. Unlike previous detonation
product sets, this model is not fit to detonation data.
For the purposes of this study, we are interested
in so-called "non ideal" high explosives, because
they reveal interesting hydrodynamic and kinetic
effects. We define non-ideal explosives as those
with a finite (more than 1 mm) reaction zone. Non-
INTRODUCTION
We plan to study the effect of equation of state
(EOS), hydrodynamics and kinetics rate laws on the
behavior of non ideal (and ideal) explosives. In
particular, we use an Arbitrary-Lagrange-Eulerian
(ALE) hydrodynamics code, an equation of state for
gas and solid products that is only fit to shock
hugoniot and static compression data, and simple
simple pressure-dependent kinetic rate laws. A
single kinetic rate law parameter is our only
adjustable parameter. We find that for Urea Nitrate,
with a single parameter, we can reproduce the
detonation velocities for confined and unconfined
detonations for a wide range of sizes, as well as
radius of curvature data. The use of ALE
hydrodynamics allows us to predict detonation wave
curvature, in contrast to the Wood-Kirkwood model,
where the curvature is an input parameter. We also
find that when we use our predicted radius of
curvature-size relationship in a Wood Kirkwood
model we can replicate the hydrodynamics results
reasonably well.
An earlier version of this work has been reported
in ref [1]. They discuss linking an in-line
thermochemical code with a 2-D hydrodynamics
code. The EOS is based on the thermochemical code
161
ideal
ideal explosives
explosives are
are often
often poorly
poorly modeled
modeled by
by
Chapman-Jouget
Chapman-Jouget (CJ)
(CJ) or
or Zeldovich-von
Zeldovich-von NeumannNeumannDoring
Doring (ZND)
(ZND) theory,
theory, where
where itit isis assumed
assumed that
that
thermodynamic
equilibrium
of
the
thermodynamic equilibrium of the detonation
detonation
products
productsare
arereached
reached instantaneously.
instantaneously. For
For example,
example,
itit isis found
found experimentally
experimentally that
that the
the detonation
detonation
velocity
velocity ofof non-ideal
non-ideal explosives
explosives can
can vary
vary sharply
sharply
from
fromthe
theCJ
CJvalue
valueand
anddepend
depend on
on the
the charge
charge radius.
radius.
InIn addition,
addition, the
the detonation
detonation wave
wave fronts
fronts exhibit
exhibit
significant
significant curvature,
curvature, because
because the
the reactants
reactants are
consumed
consumedon
onaatime
time scale
scale comparable
comparable with
with that
that of
the
thepassage
passageofofthe
thedetonation
detonationwave.
wave.
Therefore,
Therefore, one
one requires
requires the
the interaction
interaction of
chemicalkinetics
kineticswith
withthe
thedetonation
detonation wave
wave in
in order
order
chemical
havean
anacceptable
acceptablerepresentation
representation of
of detonation
detonation in
totohave
non-ideal explosives.
explosives. Wood
Wood and
and Kirkwood
Kirkwood [6]
non-ideal
(WK) proposed
proposed aa two
two dimensional
dimensional steady
steady state
state
(WK)
kinetic detonation
detonation theory
theory that
that solves
solves many
many of
of the
kinetic
limitationsofofCJ
CJ(or
(orZND)
ZND)theory.
theory. WK
WK considered
considered a
limitations
cylindricalcharge
chargeofofinfinite
infinite length.
length. They
They solved
solved the
cylindrical
hydrodynamic Euler
Euler equations
equations in
in the
the steady
steady state
hydrodynamic
limitalong
along the
the central
central streamline
streamline of
of the
the cylinder.
cylinder.
limit
Radial
expansion
was
treated
as
a
source
term
in the
the
Radial expansion was treated as a source term in
1-D
flow
along
the
streamline.
1-D flow along the streamline.
Earlier,we
wereported
reported [7]
[7] on
on implementing
implementing WK
WK
Earlier,
theory into
into the
the thermochemical
thermochemical CHEETAH
CHEETAH code.
code.
theory
Oneproblem
problem with
with WK
WK theory
theory isis that
that the
the rate
rate of
of
One
radial
expansion
along
the
center
streamline,
which
radial expansion along the center streamline, which
dependson
onthe
the charge
charge size,
size, must
must be
be specified.
specified. In
In
depends
addition,
one
must
also
specify
the
kinetic
rate
laws
addition, one must also specify the kinetic rate laws
forthe
the decomposition
decomposition ofof the
the high
high explosive,
explosive, for
for
for
which
there
is
little
experimental
or
theoretical
which there is little experimental or theoretical
knowledge.
Using the
the ALE
ALE hydrodynamics,
hydrodynamics,
knowledge.
Using
coupled with the in-line CHEETAH code, can help
coupled with the in-line CHEETAH code, can help
us gain insight into these parameters. For example,
us gain insight into these parameters. For example,
the effective kinetic rates can be inferred from the
the effective kinetic rates can be inferred from the
detonation velocity size effect.
detonation velocity size effect.
We can then take the kinetic rate laws and rate
We can then take the kinetic rate laws and rate
of radial expansion from the hydrodynamic
ofcalculation
radial expansion
from the hydrodynamic
and use them as input into WKcalculation
and
use
them
as input
into thatWKCHEETAH calculations. For
the case
we
CHEETAH
calculations.
For
the
case
that we
describe in this study, we find that this procedure
describe
in this
find that
this the
procedure
works very
well.study,
That we
is, when
we take
rate of
works
very
well.
That
is,
when
we
take
the
of
radial expansion along the streamline and the rate
kinetic
radial
expansion
along
the
streamline
and
the
kinetic
rate laws from the ALE calculation as input into the
rate
laws from themodel,
ALE calculation
as input into
the
WK-CHEETAH
the WK-CHEETAH
model
WK-CHEETAH
model,
the
WK-CHEETAH
model
reproduces the ALE calculations reasonably well.
reproduces
ALE calculations
reasonably
well.
That is, thethedetonation
velocity, and
the pressures
That
is, the detonation
and are
thereasonably
pressures
and densities
behind thevelocity,
shock front
and
densities
behind
front are reasonably
matched
between
the the
twoshock
calculations.
matched between the two calculations.
An additional
additional advantage
advantage ofof the
the using
using the
theALE
ALE
An
hydrodynamics
is that
that we
we can
can also
also make
make use
use ofof
hydrodynamics is
strength and
and melt
melt models
models for
for metals
metals[9].
[9]. This
Thisallows
allows
strength
for
the
study
of
the
interaction
of
detonation
waves
for the study of the interaction of detonation waves
with
metals.
For
example,
we
can
compare
the
with metals. For example, we can compare the
cylinder wall
wall expansion
expansion data
datawith
withour
ourcalculations
calculationstoto
cylinder
validate material
material strength
strength models
models and
and detonation
detonation
release models.
models. In
In our
our calculations,
calculations,we
weare
are
energy release
able to
to resolve
resolve the
the von
von Neumann
Neumann pressure
pressure
also able
reaction zone
zone and
andstudy
studytheir
theirdependence
dependence
spike and reaction
kinetic rate
rate laws
laws and
andEOS.
EOS.
on the kinetic
3.6
Cheetah
Cheetahwith
withW-K
W-Kradial
radialexpansion
expansion
Detonation velocity (km/s)
X"V
42, 3.4
3.4
ALE/Cheetah
ALE/Cheetahcalculations
calculations
3.2
^3.2
|
3.0
|,0
0> 2.8
2.8
O 2.6
2.6
2.4
I24
2.2
O 2.2
0.02
0.02
Data from Sandstrom et al.
I
0.04
0.04
I
0.06
0.06-1
RR
0.08
0.10
0.10
0.08
(mm
-1
0.12
0.12
)
(mm "
Figure 1. Detonation velocity as a function of inverse cylinder
Figure 1. Detonation velocity as a function of inverse cylinder
size for urea nitrate. The red squares are data for unconfined urea
size for urea nitrate. The red squares are data for unconfined urea
nitrate[10]. The blue squares are results from the
nitrate [10]. The blue squares are results from the
ALE/CHEETAH simulations, while the black circles are results
ALE/CHEETAH simulations, while the black circles are results
from WK/CHEETAH calculations. For this case, the ALE and
from
WK/CHEETAH
calculations.
case, the ALE and
WK calculations
agree quite
well with For
one this
another.
WK calculations agree quite well with one another.
RESULTS
RESULTS
We have chosen to study the detonation of urea
We(CH
haveNchosen
to study the detonation of urea
nitrate
4 2O-HNO3), because it exhibits nonnitrate
(CH
N
O-HNO
it exhibitshave
non4
2
3), becauseproperties
ideal behavior and its detonation
ideal
behavior
and
its
detonation
properties
have
been well studied. Urea nitrate’s detonation velocity
been
wellstrongly
studied.onUrea
nitrate's
depends
charge
radius,detonation
and the velocity
depends strongly
chargeThere
radius,
the
detonation
front ison
curved.
is and
experimental
detonation
front
is
curved.
There
is
experimental
detonation velocity data for both confined and
detonation velocity
data
bothrange
confined
and
unconfined
cylinders,
forfor
a wide
of cylinder
unconfined
cylinders,
for
a
wide
range
of
cylinder
sizes [10]. In addition, there is also a single
sizes [10]. wave
In addition,
there
is also a single
detonation
curvature
measurement.
detonation
curvature
For ureawave
nitrate
we take measurement.
the heat of formation to
be For urea nitrate we take the heat of formation to
be
162
since the rate law was fit to the detonation velocity
since the rate law was fit to the detonation velocity
size effect only. On the outer edge of the cylinder,
size effect only. On the outer edge of the cylinder,
the
calculation predicts somewhat more curvature
the calculation predicts somewhat more curvature
than
is seen in the experiment.
than is seen in the experiment.
The
calculations predict
predict that
thatthe
the
The WK-CHEETAH
WK-CHEETAH calculations
reaction
zone
length
is
about
5
mm
and
that
most
of
reaction zone length is about 5 mm and that most of
the
urea
nitrate
is
consumed
within
the
reaction
the urea nitrate is consumed within the reaction
zone.
calculated von
von Neumann
Neumann spike
spike isis about
about
zone. The
The calculated
60
kbars
with
a
spatial
width
of
about
1
mm.
The
60 kbars with a spatial width of about 1 mm. The
calculated
and density
density spatial
spatial profiles
profiles agree
agree
calculated pressure
pressure and
reasonably
with the
the ALE
ALE calculations,
calculations, with
with the
the
reasonably well
well with
WK
calculations
showing
a
somewhat
more
rapid
WK calculations showing a somewhat more rapid
fall
the detonation
detonation wave.
wave.
fall off
off with
with distance
distance behind
behind the
H of =-563kJ/mole,
Hof = -563 kJ/mole,
and
for the Murnaghan parameters for the unreacted
and for the Murnaghan parameters for the unreacted
urea
ureanitrate,
nitrate,we
wetake
take
BB
0 = =8.17GPaandB'
8.17 GPa and B0’ ==7.0.
7.0.
o
o
The
correspond
Theunconfined
unconfinedurea
ureanitrate
nitrateexperiments
experiments
correspond
3
3
totoananinitial
density,
p
=
0.69
g/cm
.
0
initial density, ρo = 0.69 g/cm .
The
Theresults
resultsofofour
ourcalculations
calculationsare
aresummarized
summarized in
in
Figure
Figure1.1.The
Therate
ratelaw
lawfor
forthe
thedecomposition
decompositionof
ofurea
urea
nitrate
nitratethat
thatbest
bestrepresents
representsthe
theexperimental
experimental data
data isis
given
by:
given by:
12
dλ
dtdt
Time= 50 µs
l
—— ==l.l-P(GPa)\isec
1.1 ⋅ P(GPa) µsec,-1,
P=0.05 5 R (cm)
where
whereA,λrepresents
representsthe
theamount
amountofofunburned
unburnedreactant
reactant
normalized
normalizedtotovary
varybetween
between00(all
(allunburned)
unburned) and
and 11
(all
(all burned).
burned). Since
Since all
all reaction
reaction products
products are
are
assumed
assumedtotobebeininthermochemical
thermochemical equilibrium,
equilibrium, the
the
exact
exactdecomposition
decomposition pathway
pathway isis unimportant.
unimportant. We
We
have
have chosen
chosen toto use
use aa very
very simple
simple P-dependent
P-dependent
kineticrate
ratelaw
lawfor
forthe
thedecomposition
decomposition ofof the
the urea
urea
kinetic
nitrate. We
Wefind
findthat
thatthis
thischoice,
choice,while
whilesimpler
simpler than
than
nitrate.
mostreactive
reactive flow
flow rate
rate laws
laws for
for high
high explosive
explosive
most
initiation, isis adequate
adequate toto model
model steady-state
steady-state
initiation,
detonation over
over the
the range
range ofof diameters
diameters provided
provided
detonation
here.
here.
The WK-CHEETAH
WK-CHEETAH calculations
calculations require
require the
the
The
spatialderivative
derivative ofof the
the radial
radial velocity
velocity along
along the
the
spatial
centerlineofofthe
thedetonation
detonationflow.
flow. InInWK
WKtheory
theory (see
(see
centerline
Eq.(1)
(1)ininref
ref [8]),
[8]),this
this depends
depends on
on the
the radius
radius of
of
Eq.
curvature
of
the
detonation
wave,
which
we
take
curvature of the detonation wave, which we take
fromthe
theALE
ALEcalculations.
calculations.
from
Figure2 2shows
showsthe
thecalculated
calculated detonation
detonation wave
wave
Figure
unconfinedurea-nitrate.
urea-nitrate. For
For the
the hydrodynamic
hydrodynamic
forforunconfined
calculationswe
wehave
haveused
usedaazonal
zonalresolution
resolution of
of 20
20
calculations
zones/cm.
Notice
that
the
curvature
front
has
some
zones/cm. Notice that the curvature front has some
irregularities. For
For the
the unconfined
unconfined case,
case, the
the
irregularities.
detonation
wave
front
exhibits
a
significant
detonation wave front exhibits a significant
curvature. However,
However, curvature
curvature measurements
measurements are
are
curvature.
availableonly
onlyatat25.6
25.6mm
mmradius.
radius.Figure
Figure33shows
showsthe
the
available
measureddetonation
detonationwave
wavefront
frontcurvature
curvature (red
(red line
line
measured
and open triangles) for the case of an unconfined
and open triangles) for the case of an unconfined
26.5 mm radius urea nitrate. The blue dots represent
26.5
mm radius urea nitrate. The blue dots represent
the results from our ALE calculation. The detonation
the results from our ALE calculation. The detonation
front curvature is a prediction of the current model,
front curvature is a prediction of the current model,
4
P=0.0
3
2
1
0
13
3
14
14
15
15
16
16
17
17
18
Z
(cm)
Z(cm)
Figure 2.
2. Pressure
Pressure profiles
profiles for
Figure
for the
the detonation
detonation of
of 40
40 mm
mm radius
radius
unconfined urea
urea nitrate.
nitrate. Notice
unconfined
Notice the
the structure
structure in
in the
the detonation
detonation
wave.
Figure 44 is
is the
the calculated
Figure
calculated radius
radius of
of curvature
curvatureatatRR
=
0
(center
line)
for
unconfined
urea
= 0 (center line) for unconfined urea nitrate,
nitrate, as
as aa
function of
of charge
charge radius.
function
radius. The
The curvature
curvature was
was
assumed to
to be
be quadratic.
quadratic.
assumed
163
we take
take the
the radius
radius of
of curvature
curvature as
as aa function
function of
of
we
radius from
from the
the ALE/Cheetah
ALE/Cheetah calculations
calculations
charge radius
and use them in
in the
the WK
WK model
model to
to calculate
calculate the
the rate
rate
of radial
radial divergence,
divergence, then
then the
the WK
WK model
model can
can
replicate the
the ALE
ALE results.
results. This
This demonstrates
demonstrates that
that
replicate
model provides
provides an
an adequate
adequate
the simple WK model
description of
of the
the hydrodynamic
hydrodynamic flow
flow inin non-ideal
non-ideal
description
detonation.
detonation.
4
ALE
ALEhydro
hydrocalculation
calculation
Z (mm)
3
2
Experiment
ACKNOWLEDGEMENTS
ACKNOWLEDGEMENTS
1
This work was
was performed under
under the
the auspices
auspices ofof
the United States
States Department
Department of
of Energy
Energy by
by the
the
Lawrence Livermore
Livermore National
National Laboratory
Laboratory under
under
Contract No. W-7405-ENG-48
W-7405-ENG-48
0
-20
-10
-10
0
10
10
20
(mm)
RR (mm)
REFERENCES
REFERENCES
Figure 3.3. The
The detonation
detonation wave-front
wave-front curvature for unconfined
Figure
26.5 mm
mm radius
radius urea
urea nitrate.
nitrate. The
The red
red line
line is
is the measurements
26.5
[9],while
whilethe
theblue
bluedots
dots are
are from
from our
our ALE/Cheetah
ALE/Cheetah calculation.
calculation.
[9],
1.
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9. Sandstrom, F. W., Abernathy, R. L., Leone, M. G. and
9. Sandstrom, F. W., Abernathy, R. L., Leone, M. G. and
Banks, M. L., “Diameter Effect and Detonation Front
Banks, M. L., "Diameter Effect and Detonation Front
Curvature of Ideal and Non Ideal Explosives”, Shock
Curvature of Ideal and Non Ideal Explosives", Shock
Compression of Condensed Matter – 1999, AIP, 825,
Compression of Condensed Matter - 1999, AIP, 825,
1999
160
160
140
140
£
0)
120
120
Radius of curvature (mm)
^
100
100
|
o
80
80
•R
60
60
3
40
40
E
20
20
10
10
15
20
25
30
35
40
15
20
25
30
35
40
Cylinder
radius
(mm)
Cylinder radius (mm)
Figure4.4.The
Theradius
radius of
of curvature
curvature for
for unconfined
unconfined urea-nitrate
urea-nitrate as
Figure
as aa
function of
of charge
charge radius,
radius, as
as calculated
calculated with
with ALE/Cheetah.
ALE/Cheetah. This
function
This
result can
can be
be used
used by
by WK
WK Cheetah
Cheetah to
to calculate
calculate the
the detonation
detonation
result
velocity as a function of charge radius.
velocity
as a function of charge radius.
CONCLUSIONS
CONCLUSIONS
Weapplied
appliedboth
both our
our WK
WK Cheetah
Cheetah model
model and
and our
our
We
ALE/Cheetah
model
to
the
calculation
of
the
ALE/Cheetah model to the calculation of the
detonation of urea nitrate. We find that both models
detonation of urea nitrate. We find that both models
reproduce the experimental detonation velocity size
reproduce the experimental detonation velocity size
effect reasonably well with a simple kinetic rate law.
effect reasonably well with a simple kinetic rate law.
The detonation front curvature as a function of
The detonation front curvature as a function of
charge radius is a prediction of the current model. If
charge radius is a prediction of the current model. If
1999
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