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
MOMENTUM TRANSFER DURING SHOCK INTERACTION WITH
METAL PARTICLES IN CONDENSED EXPLOSIVES
Fan Zhang1, Paul A. Thibault2, Rick Link2, and Alexander L. Conor3
1
Defence Research Establishment Suffield, PO Box 4000, Stn Main, Medicine Hat, AB T1A 8K6 Canada
2
Combustion Dynamics Ltd., Halifax, NS B3J 3J8 Canada
3
University of Toronto, Toronto, ONM5S3G8 Canada
Abstract. Detonation propagation in a condensed explosive with compressible metal particles can
result in significant momentum transfer between the explosive and the particles during the shockparticle interaction. Consequently, the classic assumption of a "non-momentum-transfer shock" used in
multiphase continuum detonation initiation and propagation models may not be valid. This paper
addresses this issue by performing numerical and theoretical calculations in liquid explosives and RDX
with various compressible metal particles under conditions of detonation pressure. The results showed
that immediately behind the shock front the velocity of particles such as Al and Mg can achieve 60 94 % the value of the shocked velocity of the explosive.
Therefore, the frozen shock assumption may fail
and momentum transfer during the shock-particle
interaction together with that behind the shock front
could influence the detonation initiation and
structure. Detonation velocity deficit was observed
experimentally in condensed explosives with 0.1
jiim aluminum particles [5-6]. The objective of the
present paper is to calculate the momentum transfer
during the shock interaction with compressible
metal particles in condensed matter under
conditions of detonation pressures.
INTRODUCTION
Theoretical models for detonation in a fluidsolid particle system have mostly been based on
multiphase fluid dynamics models taking mass,
momentum and heat transfer between the phases
into consideration. [1-2]. In these models, a frozen
shock-particle interaction is often assumed in
which the solid particles are not accelerated during
crossing of the shock front. Behind the shock front,
a viscous drag force is assumed to determine the
momentum transfer between the fluid phase and the
particles. For detonation in two-phase mixtures of
gas and solid particles, the shock-particle
interaction time is several orders of magnitude
smaller than the velocity relaxation time related to
viscous drag. Thus, the particle crosses the shock
front with negligible changes in its velocity.
Momentum loss behind the shock front plays an
important role in the detonation velocity deficit for
gas-particle systems [3-4]. However, for detonation
in a condensed explosive containing metal particles
of 0.1 to 1 |Lim, the shock-particle interaction time
is about the same order as or one order of
magnitude less than the velocity relaxation time.
SHOCK OVER A SINGLE PARTICLE
It was assumed that detonation initiation and
decomposition of the condensed explosive would
not start within the shock front. It was also assumed
that the reaction time scale of solid particles is
larger than the shock-particle interaction time so
that the particles can be considered chemically
frozen within the shock front. Thus, the explosive
material was described using the Murnagham
equation of state excluding temperature effect. The
solid particle material was described using the
HOM equation of state [7]. The Euler equations for
934
mass and momentum were used to model the flow
of the non-reacted explosive, and the governing
equations for inviscid plastic flow were used for
modeling of dynamic response in metal particles.
The interaction between the explosive and the
particles was solved by matching their boundary
conditions of pressure and particle velocity.
Numerical solution of these closure equations was
obtained using the IPS AS code [8]. The resolution
for the calculations corresponded to 20 cells or
elements for a particle radius using a cylindrically
axi-symmetrical mesh to represent a spherical
particle immersed in the fluid. Calculations were
conducted using a 10 |im diameter particle. Since
the flow of the explosive and the metal particle is
assumed inviscid and the equations of state and the
constitutive model contain no rate-dependent terms,
the results can be scaled to any other particle
diameters using simple geometric similarity
arguments.
Ins
2 ns
3 us
compressibility as water, and RDX with a bulk
initial density of 1.4 g/cm3 and 1.8 g/cm3. It can be
seen in Fig. 1 that an aluminum particle is severely
deformed during the shock interaction process due
to the fact that the incident shock pressure far
exceeds the yield stress of the material. The
deformation is directly related to the relative
velocity between the leading and trailing edges of
the particle. The early-phase of the particle
acceleration is mainly controlled in the direction of
motion by the transmitted shock and the rarefaction
reflected off the downstream end of the particle.
The lateral unloading produces a secondary effect
of deceleration of the particle.
The particle velocity, based on a mass average
over the particle elements, is compared in Fig. 2 for
different metals for a 101.3 kbar liquid shock. The
earlier time histories within about 2 ns display a
shock interaction process where the particle
acceleration decreases with the particle density. A
velocity transmission factor a is defined as the
ratio of the particle mass-averaged velocity us after
the shock interaction time T = d/D$ over the
shocked fluid velocity ui: a = us/ul9 where d is
the particle diameter and D0 is the shock velocity.
The values of a calculated for various metal
particles are summarized in Table 1. Among the
material properties and shock strength investigated,
a is most dependent on the initial density ratio of
the explosive to metal Po/ps() (see Fig. 4a). In
spite of large difference of the sound speed
between magnesium and beryllium and between
tungsten and uranium, the velocity transmission
factor remains almost the same for the same shock
pressure. A curve fit of the numerical data suggests
4 us
FIGURE 1. Pressure contours for an aluminum particle
subjected to a 101.3 kbar shock in water.
2 ns
3 ns
4 ns
(b) 7ns
FIGURE 2. Particle velocity histories in magnesium, aluminum,
nickel and tungsten subjected to a 101.3 kbar water shock.
Calculations were performed for metal
particles subjected to a shock in explosive under
conditions of various particle material density,
particle acoustic impedance and shock strength.
Explosives considered in the analysis included a
liquid explosive with the same initial density and
FIGURE 3. Pressure contours for cylindrical aluminum
particles subjected to a 101.3 kbar water shock, a) Two particles
with rigid sidewall; b) a cluster of particles.
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TABLE 1. Velocity transmission factors
Material
PsO
g/cm3
Magnesium
Beryllium
Aluminum
Nickel
Uranium
Tungsten
Magnesium
Aluminum
Magnesium
Aluminum
CsO
Pi
m/s
kbar
Water, p0 = 1.0g/cm3:
4700
50.7
1.770
101.3
202.6
7975
101.3
1.870
2.785
5350
101.3
4646
101.3
8.860
18.98
2540
101.3
4060
101.3
19.30
202.6
RDXp 0 =1.4g/cm 3 :
4700
101.3
1.770
2.785
5350
101.3
RDXp 0 =1.8g/cm 3 :
4700
101.3
1.770
5350
101.3
2.785
wave reflected off the downstream particle in the
direction of motion.
Figure 3b displays a matrix of cylindrical
aluminum particles compacted by a 101.3 kbar
liquid shock into a tight cluster of particles. The
particles are largely deformed and coalesce due to
the time lag between the acceleration of the leading
(bottom) and trailing (top) particles. However, the
initially non-uniform velocity in the particle
quickly reaches and oscillates around an average
value behind the shock front due to the multiple
interactions between particles. The transmitted
velocity for multiple particles, computed at a time
equal to twice the shock-particle interaction time
1, is summarized in Fig. 4b. The results indicate
that the transmitted particle velocity decreases with
increase in particle volume fraction ^.
a
0.776
0.790
0.803.
0.781
0.600
0.227
0.108
0.108
0.114
0.940
0.754
1.008
0.802
a
where a = 3.947 and b = -1.951 with respect to the
grid resolution used.
SHOCK OVER MULTIPLE PARTICLES
a
i.o
1.0
0.8
0.9
0.6
0.8
0.4
Two-dimensional calculations were conducted
for a cluster of cylindrical aluminum particles 10
|im in diameter subjected to a 101.3 kbar liquid
shock. Although the results of the cylindrical
particle calculations cannot be directly applied to
spherical particles, they do provide a qualitative
trend for the effect of the particle volume fraction
on the velocity transmitted to the particle. Figure 3a
displays pressure contours for two particles with a
5 |om gap in the direction of motion. The effect of
neighboring particles in the lateral direction is
simulated by introducing a rigid wall at the
symmetry plane between the particles and the
lateral particles with a 5 pn gap. It can be seen that
the early-phase of the particle acceleration is
controlled in the direction of motion by: 1) the
transmitted shock, 2) the rarefaction occurring at
the downstream end of the particle, 3) the reflected
rarefaction occurring at the upstream end of the
particle, 4) the wave reflected off the downstream
particle. Around the leading (bottom) particle, the
waves reflected off the lateral particles collide and
produce a secondary pressure pulse to the bottom
particle. This produces an acceleration which is less
significant than the deceleration effect from the
(a)
0.2
0.0
O water
+ RDX1.4
D RDX1.8
.......curve fit~
• Theory
i I
0.0 0.2 0.4 0.6 0.8 1.0
(b)
—9— Linear model
• 2 cylinders * 11 cylinders
0.7
0.6
0.5
o.O 0.2 0.4 0.6 0.8
1.0
Po/PsO
FIGURE 4. Velocity transmission factors for: a) a single
particle, and b) multiple particles.
A SIMPLE MODEL
To get more insight into the mechanism, a ID
planar analytical model was established. It was
assumed that the impedance matching of the
explosive and particle material results in a reflected
shock in the explosive and a transmitted shock in
the particle. This yields an equation for the
transmitted velocity in the particle us3. As the
surrounding explosive shock with the velocity D0
reaches the particle trailing edge, the transmitted
shock with a velocity Ds0 still propagates in the
particle if Ds0 < D0; or is already reflected off the
particle trailing edge and the reflected rarefaction
runs back into the particle if Ds0 > D0. Taking into
account the shock velocity difference, a momentum
balance rule is assumed after the shock-particle
interaction time, that is,
936
The results predicted using equation (5) for
multiple aluminum particles are displayed in Fig.
4b and compared with the numerical calculations.
(2)
if
where the subscript "s" denotes the mass-averaged
state after the shock-particle interaction time r ,
subscripts "3" and "4" represent the transmitted
shock and the reflected rarefaction state
respectively. Distances d3 and d4 are defined by the
shock-particle interaction time,
d4
£>,n
CONCLUSIONS
For a charge of a condensed explosive with
metal particles, the present study indicates that the
momentum transfer from the condensed explosive
to the metal particles is significant during the
particle crossing of the shock front. The particle
velocity after the shock-particle interaction strongly
depends on the initial density ratio of explosive to
metal, but is relatively insensitive to the other
parameters, such as the particle acoustic
impedance, shock strength and bulk explosive
shock Hugoniot. The transmitted particle velocity
decreases with increase in the particle volume
fraction. The significant momentum transfer during
the particle crossing of the shock front must be
taken into account when modeling the shock
initiation and detonation structure for two-phase
mixtures of condensed explosive and metal
particles.
(3)
£L
For simplicity, the flow velocity u^ behind the
rarefaction wave is assumed to be u^ ~ 2us3 and
also ps ~ Ps3 ~ Ps4 is assumed. Substitution of
equation (3) into equation (2) yields
for
(4)
1+ -
£>c,
--1
for
[)
This work was supported under the auspices of
the DND contract W7702-8-R696.
where as=us3/ul. The theoretical velocity
transmission factors calculated from equations (4)
are also displayed in Fig. 4a. Comparison of the
theoretical results with the numerical calculations
shows fairly good agreement, except for the
theoretical values are generally larger than the
numerical ones since the ID theory does not
consider the loss caused by the lateral deformation
and expansion.
For multiple particles, the particle velocity
calculated varies with the selection of the time for
the shock interaction with the multiple particles. It
may also vary with the geometric arrangement of
the particles for a given particle volume fraction.
For complicated cases involved in multiple
particles, it may simply assume a linear model for
the velocity transmission factor between the value
for a single particle, a, for the particle volume
fraction <|) = 0, and the value for the transmitted
shock in the solid, ocs, at <|) = 1, that is,
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937
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