26 Propellants, Explosives, Pyrotechnics 26, 26Ä…32 (2001)
Reactive Flow and the Size Effect
P. Clark Souers*, Steve Anderson, Estella McGuire, Michael J. Murphy, and Peter Vitello
Lawrence Livermore National Laboratory, Livermore, CA 94550 (USA)
Summary the same whether P ‡ Q or P is used, although Q makes the
pulse start sooner. The edge of convergence occurs at roughly
The detonation reaction rate in ms 1 is derived from Size Effect data
4=hxei, where hxei is the average reaction zone length, i.e. we
using the relation Ä… DUs@Us=@yÄ… 1, where y ˆ 1=Ro, where Us is the
need a minimum of 4 zones in the explosive's reaction zone. It
detonation velocity for a ratestick of radius Ro and D is the in®nite-
will run with coarser zoning, but the coef®cients will greatly
radius detonation velocity. These rates are generally not constant with
radius and have pressure exponents ranging from < 5 to > 5. change. In almost all this work, we shall run slightly beyond
JWL‡‡, a simple Reactive Flow code, is run with one rate constant
the edge of convergence, e.g. if 4 zones=mm is the calculated
on many samples to compare its rates. JWL‡‡'s pressure exponents
edge, we shall run at 6 zones=mm. At this point, the detona-
vary from about 0.5 to 2.5, and failure occurs outside this range. There
tion velocity is only a mild function of the zoning and the
are three classes of explosives: (1) those for which the pressure
exponent is between 1 and 2 and the rate is nearly constant (e.g. porous number of zones is usually acceptable to users. We are
urea nitrate); (2) higher pressure explosives with a concave-down
running in a 2-D ALE (Arbitrary Lagrangian-Eulerian) code.
shape and large positive pressure exponents (dense TNT); and (3)
Figure 1 shows JWL‡‡ results for 1.62 g=cm3 TNT at a
explosives with negative pressure exponents and concave-up shapes
point on the axis. The cylinder radius is 5 mm and the square
(porous PETN). JWL‡‡ ®ts only the ®rst class well and has the most
trouble with class 3. The pressure exponent in JWL‡‡ is shown to be
zoning is 10 zones=mm at the edge of convergence. The sum
set by the shape of the Size Effect curve Ä… a condition that arises in
P ‡ Q rises slightly higher than P and it occurs sooner. The
order to keep a constant reaction rate for all radii. Some explosives
vertical line shows the value of 1 F for the peak P ‡ Q. In
have too much bend to be modeled with one rate constant, e.g. Comp.
ZND theory, we would expect a maximum pressure with
B near failure. A study with creamed TNT shows that the rate constant
need not be changed to account for containment. These results may
1 F ˆ 1. The 1 F peak value here is 0.35, and it will
well be pertinent to a larger consideration of the behavior of Reactive
move toward 1 as the zoning moves deeper into convergence.
Flow models.
We previously created a model of the Size Effect, which is
schematically illustrated in Figure 2(2Ä…4). The center region of
the cylinder, with radius Ro Re, detonates with an energy
1. Introduction Eo. For simplicity, it is shown as a cylindrical volume with no
curvature. The skin layer on the edge of width Re provides no
We previously described JWL‡‡, a simple Reactive Flow energy that inÅ»uences the detonation front. The center region
model for detonation(1). Like all Reactive Flow models, is divided into slices each Re thick (4 of them in Figure 2).
JWL‡‡ must describe the Size Effect (change of detonation The average detonation energy is set by the volumes. We
velocity with cylinder radius), the detonation front curvature also assume that
and the decreased detonation velocity for explosive corner
Eo P U2 2Ä…
s
turning. JWL‡‡ consists of an unreacted Murnahan equa-
tion-of-state, a reacted C-Form JWL EOS, a ``partial pres-
where we need the energy-detonation velocity relationship
sure'' analytic pressure=sound speed mixer, and a rate term
here. This gives:
of the form
EodilutedÄ… pRo ReÄ…2hxei Re 2 Us 2
dF
ˆ ˆ 1 ˆ
ˆ GP ‡ QÄ…b1 FÄ…1Ä…
EofullÄ… pR2hxei Ro D
o
dt
3Ä…
Here, F is the burn fraction, G the rate constant, P the
pressure, Q the monotonic arti®cial viscosity, and b the
which may be solved to give the Eyring equation
pressure exponent. A single exponential rate constant is
Us Re hxei
used in each case, which presumably describes the inherent
ˆ 1 ˆ 1 4Ä…
D Ro sRo
rate of decomposition. The exponent b ranges from about 0.5
to 2.5 in the code. If the zoning is converged, the results are
In Eyring's original equation(5) Re was the reaction zone
length, but here it is the skin layer thickness. The reaction
zone thickness lies at right angles to the skin layer and is
given by hxei. The ratio of the two is s. If we assume an
* Corresponding author; e-mail: [e-mail] exponential reaction rate with one distance constant being
# WILEY-VCH Verlag GmbH, D-69469 Weinheim, 2001 0721-3115/01/0302Ä…0026 $17.50‡:50=0
Propellants, Explosives, Pyrotechnics 26, 26Ä…32 (2001) Reactive Flow and the Size Effect 27
@F DUs
ms 1 ˆ 5Ä…
@t @Us=@yÄ…
We substitute from Eq. (3) to get
@F Us 1 sUs
ms 1 ˆ ˆ
@t Re 1 ‡y=ReÄ…@Re=@yÄ… hxei
1
6Ä…
1 ‡y=hxeiÄ…@hxei=@yÄ… y=sÄ…@s=@yÄ…
The relation sUs=hxei is what we might have guessed
would be the rate, because it is just velocity divided by
distance. We would get this if the last two derivatives at the
right in Eq. (6) are canceled. We could have gotten Us=hxei
from the Eyring equation, but we would not have known s.
Using these equations, we can calculate the rate from the Size
Effect data, then obtain the number of distance constants s.
Figure 1. Pressure (P), arti®cial viscosity (Q) and the unreacted burn
We next need to relate the rate to pressures. The ®rst
fraction (1 F) at steady state for 1.62 g=cm3 TNT on the axis.
possibility is given by
DUs Us 2c
GP ‡ QÄ…b1 FÄ…jaxis GPc
cj
@Us=@yÄ… D
7Ä…
This equates the rate with the average pressure in the
interior region with a radius less than Ro Re. In the code,
this is a broad pressure plateau that runs nearly constant
almost to the cylinder edge. In ZND theory, the front of the
reaction zone is at the spike pressure and the back is at C-J, so
that the average pressure is close to the C-J pressure. The ®rst
term on the right is how the code would see it; the second
term is an analytic representation. The two representations
have different pressure exponents b and c. The third term uses
a C-J pressure, Pcj, from the CHEETAH thermochemical
code(6), and lowers it accordingly to the radius, as given by
P U2. The ZND value of 1 F ˆ 1 is assumed. A plot of
s
Eq. (7) gives the c values listed in Table 1(7Ä…18). Many of the
high pressure explosives have c > 5; a few explosives have
Figure 2. Schematic of the detonating cylinder where the reacting
c < 5, and only a small number are in range of 1Ä…2 that
volume has Żat ends.
matches the b values used in the code.
The second pressure possibility is at the inner edge of the
skin layer, where the cylinder ®rst experiences the outside
world. Here we would have
Re, then the reaction zone is s distances long. The model
DUs
went on to apply a detonation front curvature with a
GP ‡ QÄ…b1 FÄ…jskin 8Ä…
geometric algorithm for equating the skin layer with the
@Us=@yÄ…
missing energy. The result gave hxei without ever having to
This is more dif®cult to work with because we cannot
evaluate s along the way.
guess the skin layer pressure ahead of time.
A plot of Us data versus y ˆ 1=Ro constitutes the Size
Effect curve. In®nite-radius is at the left at y ˆ 0; failure lies
to the far right. The data commonly have a concave-down
shape; concave-up exists but is rare.
3. Fitting the Size Effect Curves
It is necessary to calibrate a Reactive Flow model to the
experimental detonation velocities. The reacted JWL is
2. Determination of the Reaction Rate obtained from CHEETAH plus any cylinder test data. The
JWL is usually set so that it generates the in®nite-radius
We need a way of getting the approximate reaction rate detonation velocity, D, which is slightly different from a
from the available data, and we try the analytic rate program ``Burn'' JWL, where any measured value is used.
28 P. C. Souers, S. Anderson, E. McGuire, M. J. Murphy, P. Vitello Propellants, Explosives, Pyrotechnics 26, 26Ä…32 (2001)
Table 1. Summary Data for Explosives Studied with JWL‡‡
Explosive Zones Entire size effect curve Large radius
only, all b ˆ 1
Density Ref. to mm at Pcj power concav. Ave. Best JWL ‡‡ ®ts
(g=cm3) fail last pt. (GPa) c b Rate bG Rate G
Comp. B, full curve 1.71 7 F 8 27 > 5 0.32 73 > 2.75 130 210
TNT, creamed, uncon. 1.62 8 F 5 20 > 5 0.32 22 1.75 255 40 85
ANFO 50% HMX 1.60 9 F 3 22 1.45 0.27 3.8 1.5* 94 6.1 45
ANFO 30% HMX 1.51 9 F 2 17 1.51 0.22 2.8 1.5* 82 3.9 30
TNT, cast 1.57 10 4 19 > 5 0.21 53 2.75 2030 90 140
TNT, press <385 mm 1.46 11 F 6 15 > 5 0.19 64 2.0 1190 70 150
Comp. B, less last 4 pts 1.71 7 8 27 > 5 0.16 88 2.5 1250 130 210
Amatol 50Ä…50, cast 1.53 12 F 1 18 2.17 0.15 1.2 1.0 to 2.0 7.
Ä…100 1.5 10
TNT, pressed 1.62 11 F 12 20 > 5 0.13 90 2 to 2.5 1700. 100 360
Ä…3900
TNT, creamed, conf. 1.62 8 3 20 > 5 0.12 50 1.75 255 70 130
TNT, press <385 mm 0.80 11 F 3 5 1.08 0.12 1.4 1.30** 149 2 50
ANFO Kl emulsion 1.16 13,14 F 2 9 1.08 0.07 2.1 1.26** 57 2.6 43
TNT, press <385 mm 1.55 11 10 17 4 0.06 56 1.0 250 65 250
TNT, < press 385 mm 1.00 11 F 3 7 1.54 0.00 3.0 1.0 58 4 58
TNT, press 200 mm 1.45 15 4 14 < 5 0.0 48 0.88** 36 45 125
Urea Nitrate, granular 0.69 16 F 1.5 3 0.1 0.0 1.3 1.0 50 1.2 50
EDNA 1.00 17 3.0 9 1.5 0.0 4.9 1.0 60 5.1 60
PETN, 150Ä…210 mm 1.00 17 F 10 9 < 5 0.10 33 0.0 20 25 220
HANFO 1.07 18 F 2.5 6 1.0 0.20 0.22 1.0 2.5 0.16 2.5
*Failing at small radii, **obtained using the GLO optimizer
Sometimes, the scatter in the data is such that neither the
shape of the curve is obvious nor the extrapolated value of D.
The calibration is done by setting b and trying different G's
until the smallest-radius detonation velocity, possibly near
failure, is obtained. If G is too small or b too large, the code
run will fail. A set of b, G values are found that match the
smallest radius. Then we run a larger radius about halfway up
the Size Curve, looking for the b, G combination that gives
the best ®t. The in®nite-radius detonation velocity should be
®xed by the code, so that large radii runs will approach D.
The length of the cylinder is at least 10 times the radius. The
starting end of the cylinder is usually slightly overdriven by
starting the nodes above the C-J velocity.
There are three behavior groups.
Group 1. Low pressure explosives with a clearly unique b,
G solution, a concave-down Size Effect curve shape and
c b. This is shown in Figure 3 for 0.69 g=cm3 urea nitrate,
which has a unique solution near b ˆ 1; G ˆ 50, showing a
straight line in the Size Effect curve(16). JWL‡‡ was run
at the edge of convergence at 2 zones=mm and also at
Figure 3. Group 1: a unique solution for b ˆ 1, G ˆ 50 is found for
6 zones=mm and the results are close. The runs at b ˆ 0:5
0.69 g=cm3 urea nitrate. The b ˆ 0:5 and 2 curves are ®xed at the
and 2 ®t the smallest-radius point near failure, but the rest of small radius but bow out at the intermediate radii. The diamonds have
5 times ®ner zoning.
the curves bows outward. This is the only group where,
because c b, the JWL‡‡ results will closely model the
actual data.
Group 2. High pressure explosives with a non-unique b,
G solution, a concave-down Size Effect curve shape as given only at b ˆ 0:25 and 4.5. The reason for the non-uniqueness
by 1.62 g=cm3 TNT in Figure 4 and c b(11). It is not appears to be the 1 F term, which is varying at the edge
possible to decide on the best ®t by visually comparing the of convergence. While P ‡ Q may be slightly different
runs with the data. All curves show a concave-down shape, between b values, the variability of 1 F makes the rates
including b ˆ 1. All curves from 1:5 < b < 2:5 appear by similar. 1 F is closer to 1 in urea nitrate in group 1 than in
eye to ®t, although b ˆ 2 may be the closest. Points at TNT here. If the zoning is made extremely ®ne, then we
b ˆ 0:5 and 3.5 are also close; deviation is evident for sure expect the TNT 1 to approach 1 F and a unique rate to
Propellants, Explosives, Pyrotechnics 26, 26Ä…32 (2001) Reactive Flow and the Size Effect 29
Figure 4. Group 2: Size Effect curve for 1.62 g=cm3 TNT, a high-
density explosive with a concave-down Size Effect curve. The values
Figure 6. Plotting the bend of the Size Effect curve using dimen-
of b, G are listed. The curves from b ˆ 1:0 to 2.5 all ®t fairly well, so
sionless variables. The amount of bend, b, is shown for the full Comp.
that no unique solution is evident.
B curve (squares). TNT 1.55 g=cm3 has almost no bend (circles) and
porous PETN is concave-up and has a negative b (triangles).
to 1.0 mm. It is possible that ®xing b for the best bend in the
Size Effect curve may not optimize another property.
The degree of the bend in the Size Effect curve is
important. The curve runs from y ˆ 0 (in®nite radius) and
D to the smallest radius yx with detonation velocity Usx. We
try the dimensionless units
y Us Usx
Y ˆ ; U ˆ 9Ä…
yx D Usx
The amount of bend, b, is the deviation of this curve from
the straight line, as seen in Figure 6. Equation (9) is de®ned so
that concave-up is positive and concave-down is negative. A
straight line would have a zero value. The b values are listed
in Table 1(7Ä…18). Because of the scatter in the data, b is rarely
an accurate number. Some explosives have so much scatter
that b is set to 1. If large radius shots were not done, some
explosives have dubious extrapolations to y ˆ 0 and the D
value. This, in turn, can change the estimate of b. It is not
Figure 5. Group 3: 1.0 g=cm3 PETN is slightly concave-up and
necessary to ®t every point down to failure. Equation (9) works
cannot be accurately ®t. The b ˆ 0, G ˆ 20 curve is the closest.
for any segment of the data. If a problem covers only a small
part of the Size Effect curve, the bend and b will be reduced.
We are suggesting that, as long as G must be a constant,
appear. However, with c > b, we will always force the
that b and G are selected to create a Size Effect curve with a
results into a form particular to JWL‡‡.
given shape. Low power explosives (Pcj 10 GPa) gener-
Group 3. Explosives with a non-unique b, G solution, a
ally show a straight line for b ˆ 1 with concave-down for
concave-up Size Effect curve shape as given by 1.0 g=cm3 b > 1 and concave-up for b < 1. High power explosive
PETN in Figure 5, and c (negative) < b(17). JWL‡‡ wants to
curves (Pcj 20 GPa) are all concave-down, but the b ˆ 1
give a concave-down shape, so that no perfect ®t can be made.
curve usually is straight once it has moved away from the
Maximum concavity-up was obtained using b ˆ 0, G ˆ 20
large-radius region.
but the smallest-radius point never could be ®t. For the 10 mm
Table 1 summarizes the data for all explosives run in this
radius, the axis pressure was the same for 0 < b < 2, but the
report. In each case, the most likely value of b is selected, and
edge lag in the detonation front curvature decreased from 1.6
it varies accordingly to the extent of the Size Effect data
30 P. C. Souers, S. Anderson, E. McGuire, M. J. Murphy, P. Vitello Propellants, Explosives, Pyrotechnics 26, 26Ä…32 (2001)
Although we can estimate b, we cannot estimate G ahead
of time except for group 1. Hand-®tting appears necessary for
group 2 explosives over the entire curve.
There are two exceptions, shown by the circles in Figure 8.
These are ANFO emulsions with 30% and 50% HMX. The
30% emulsion is shown in Figure 8, and it extends from
7mm=s to under 3 mm=s. This is an extremely wide range of
detonation velocities. Near failure, the detonation velocity of
the unreacted and reacted species is about the same. The
model is failing below 4 mm=ms, so that the effective b is
smaller than the calculated one. In effect, the code detonation
is tailing off at small radii in this special case.
Using Eqs. (7) and (8), we next obtain the three rates of Eq.
(7), which are summarized in Table 2. The ®rst is the one
derived from the derivative of the detonation velocities used
in the JWL‡‡ ®t. The second is the rate on the axis using
P ‡ Q and 1 F. The third is the same but at the inner edge
of the skin layer. The three rates are in agreement considering
the roughness of the code-extracted data. 1 F is closer to 1
Figure 7. Variation of the pressure exponent b as a function of the
for urea nitrate than for TNT. It is the variability of 1 F that
bend of the Size Effect curve, b. The triangles are ANFO emulsion with
HMX and are exceptions because of creeping failure at small radii. creates the multiple solutions for TNT.
From above, we see that a high-pressure explosive will
probably have a smaller derivative than a low-pressure
explosive. The smaller derivative translates to a higher rate,
which raises c and b. This is the reason why b roughly rises
with the C-J pressure. The use of the bend, b, is another way
to look at it. The best way would be to use the de®nition of the
rate itself, but unfortunately, 1 F is generally unknown at
the edge of convergence. The urea nitrate, with 1 F 1
and b c, is better behaved in JWL‡‡ than the high-
pressure TNT.
Most people do not design explosives near failure. We
can consider the upper part of the Size Effect curve from
in®nite-radius to twice the failure radius where almost all
design work will be done. We ®rst use Eq. (5) averaged
over the smaller set of large radii to get a new rate, which
will be larger. We ®nd from Table 1 that all of the
explosives can be ®t using b ˆ 1, which is a considerable
simpli®cation.
Finally, we consider creamed TNT, one of the few
explosives which Size Effect curve has been measured bare
and with 3 mm steel con®nement(8). The parameters
b ˆ 1:75, G ˆ 255 ®t both curves as shown in Figure 9.
Thus, containment is obtained for this example using
JWL‡‡
Figure 8. ANFO emulsion with 30% HMX. This explosive has an
exceptionally long range of detonation velocities before experimental
failure occurs. The code is slowly failing for all low velocities.
4. Limitations of the Model
For high pressure explosives, b and G are set by the bend of
the Size Effect curve, which is a serious limitation. It is done
selected. Figure 7 shows a plot of these b's as a function of the
because this is the easy way. This leads us to b 2:5 for
bend, b. The important result is that the pressure coef®cient b
1.62 g=cm3 TNT. When this is substituted into the Eq. (5)
reÅ»ects in large part the bend of the Size Effect curve. The ®t
rate, it says that G is not a constant. To make G constant, a
of the squares in Figure 7 is given by
value of c 9 is needed, which cannot be used in the code
without failure. So b and c are different for both groups 2 and
3, and we cannot decide whether a constant G exists for the
b 0:95 ‡ 6:16b 10Ä…
real explosive. Comp. B has a too extreme bend to ®t with any
Propellants, Explosives, Pyrotechnics 26, 26Ä…32 (2001) Reactive Flow and the Size Effect 31
Table 2. Comparison of JWL‡‡ Rates from the Axis and the Inside Edge of the Skin Layer
Radius Detvel @Us=@y P ‡ Q (Mb) 1 7 F Rate (ms71)
(mm) (mm=ms) (mm2 ms)
on axis calcd skin axis skin left side right, axis right, skin
Urea Nitrate 1 3.60
b ˆ 1, G ˆ 50 40 3.40 10.0 0.042 0.027 0.94 0.90 1.2 2.0 1.2
20 3.16 9.1 0.035 0.026 0.92 0.92 1.3 1.6 1.2
12 2.85 8.2 0.027 0.024 0.90 0.92 1.2 1.2 1.1
10 2.69 7.5 0.0245 0.021 0.81 0.92 1.3 1.0 1.0
7.5 2.38 6.5 0.019 0.017 0.71 0.92 1.3 0.7 0.8
TNT 1.62 g=cm3 1 7.03
b ˆ 2, G ˆ 1700 1.6 6.653 1 0.232 0.20 0.49 0.60 24 45 41
b ˆ 2, G ˆ 1700 2.5 6.857 0.75 0.254 0.22 0.44 0.50 33 48 41
b ˆ 2, G ˆ 1700 6 6.995 0.48 0.265 0.22 0.68 0.50 52 81 41
b ˆ 2, G ˆ 1700 10 7.018 0.42 0.258 0.22 0.26 0.50 60 29 41
b ˆ 1.0, G ˆ 340 6 6.998 0.5 0.26 0.21 0.92 0.68 50 106 51
b ˆ 1.5, G ˆ 750 6 6.991 0.45 0.26 0.22 0.90 0.65 56 103 53
b ˆ 2, G ˆ 1700 6 6.995 0.4 0.265 0.22 0.68 0.50 63 81 41
b ˆ 2.5, G ˆ 3900 6 7.005 0.3 0.27 0.24 0.67 0.60 84 83 59
Urea nitrate: 7.5 mm near failure
b, G pair. It can be roughly ®t at small radii with b ˆ 2 by 5. References
using G ˆ 625 for radii larger than 2.3 mm, G ˆ 450 at
(1) P. C. Souers, S. Anderson, E. McGuire, and P. Vitello,
2.2 mm and G ˆ 415 at 2.1 mm.
JWL‡‡ is descended from ``Ignition & Growth''(19Ä…20) ``JWL‡‡: A Simple Reactive Flow Code Package for Detona-
tion'', Propellants, Explosives, Pyrotechnics 25, 54Ä…58 (2000).
and is a stripped-down version. ``Ignition & Growth'' has
(2) P. C. Souers and R. Garza, ``Kinetic Information from Detonation
more variables that could be adjusted to create the Size Effect Front Curvature'', 11th Int. Detonation Symposium, Snowmass
Village, CO, August 30Ä…September 4, 1998, pp. 447Ä…453.
curve bend, but it would appear generally that Reactive Flow
(3) P. C. Souers and R. Garza, ``Size Effect and Detonation Front
models will have severe problems modelling many explosive
Curvature'', American Physical Society, Topical Group on Shock
features, simply because using a starting and ending state is
Compression of Condensed Matter, Amherst, MA, July 28Ä…
too simple. Finally, we have used only one rate constant, August 1, 1997, pp. 325Ä…328.
(4) P. C. Souers, ``Size Effect and Detonation Front Curvature'',
which models the Size Effect. The extension of JWL‡‡ to
Propellants, Explosives, Pyrotechnics 22, 221Ä…225 (1997).
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the containment.
32 P. C. Souers, S. Anderson, E. McGuire, M. J. Murphy, P. Vitello Propellants, Explosives, Pyrotechnics 26, 26Ä…32 (2001)
neering Research, Stockholm, SveBeFo Report 17, 1995. (Finn (19) S. G. Cochran and J. Chan, ``Shock Initiation and Detonation
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Non-Ideal Explosives'', Shock Compression of Condensed
Acknowledgements
Matter Ä… 1999, American Institute of Physics, pp. 825Ä…828.
This project was done under the ASCI Program with Randy Simpson
(Additional data from Michael Leone, private communication,
the High Explosives Manager. This work was performed under the
1999.)
auspices of the U.S. Department of Energy by the Lawrence Livermore
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