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Proc. R. Soc. A (2012) 468, 1564–1590
doi:10.1098/rspa.2011.0595
Published online 15 February 2012
Acceleration and heating of metal particles in
condensed matter detonation
BY ROBERT C. RIPLEY1,2, *, FAN ZHANG1,3
AND
FUE-SANG LIEN1
1 Department
of Mechanical Engineering, University of Waterloo, Waterloo,
Ontario, Canada N2L 3G1
2 Martec Limited, 1888 Brunswick Street, Suite 400, Halifax, Nova Scotia,
Canada B3J 3J8
3 Defence R&D Canada-Suffield, PO Box 4000, Station Main, Medicine Hat,
Alberta, Canada T1A 8K6
For condensed explosives, containing metal particle additives, interaction of the
detonation shock and reaction zone with solid inclusions leads to high rates of momentum
and heat transfer that consequently introduce non-ideal detonation phenomena. During
the time scale of the leading detonation shock crossing a particle, the acceleration and
heating of metal particles are shown to depend on the volume fraction of particles, dense
packing configuration, material density ratio of explosive to solid particles and ratio of
particle diameter to detonation reaction-zone length. Dimensional analysis and physical
parameter evaluation are used to formalize the factors affecting particle acceleration
and heating. Three-dimensional mesoscale calculations are conducted for matrices of
spherical metal particles immersed in a liquid explosive for various particle diameter
and solid loading conditions, to determine the velocity and temperature transmission
factors resulting from shock compression. Results are incorporated as interphase exchange
source terms for macroscopic continuum models that can be applied to practical
detonation problems involving multi-phase explosives or shock propagation in dense
particle-fluid systems.
Keywords: condensed detonation; heterogeneous matter; mesoscale simulation
1. Introduction
Detonation in condensed explosives typically features wave propagation speeds
from 6 to 9 mm ms−1 and peak pressures from 10 to 50 GPa. The detonation front
is often modelled by the von Neumann (VN) shock followed by a reaction zone
in the idealized one-dimensional Zel’dovich–von Neumann–Döring (ZND) wave
(Fickett & Davis 1979), which travels at a minimum speed for the Chapman–
Jouguet (CJ) condition where a sonic point terminates the reaction zone. The
addition of heterogeneities such as metal particles into condensed explosives
introduces microscopic interaction of the detonation shock and reaction zone
with the particles, which produces localized hot spots, transverse waves and highpressure fluctuations that are manifested in macroscale detonation phenomena.
*Author for correspondence ([email protected]).
Received 2 October 2011
Accepted 13 January 2012
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Metal particles in condensed detonation
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Particles saturated in liquid explosives form a fundamental system termed
wetted powder, or ‘slurry’ (Frost & Zhang 2009). The effect of a large volume
fraction of metallic particle additives on the detonation properties of slurry
explosives has been investigated by a number of researchers, e.g. Engelke (1979),
Baudin et al. (1998), Gogulya et al. (1998), Haskins et al. (2002), Zhang et al.
(2002) and Kato et al. (2006), where the bulk propagation speed has been related
to the particle properties, morphology and concentration. In general, adding
particles results in a velocity deficit below that of the neat explosive, because some
of the chemical energy released goes into heating, and particularly, accelerating
the particles.
Detonation failure depends on the competing effects of explosive sensitization
owing to the formation of hot spots near the particles, and the increasing
detonation reaction-zone length with momentum and energy absorbed into
the particles and detonation front curvature caused by lateral expansion.
Engelke (1979) and Lee et al. (1995) extensively studied the critical diameter
of nitromethane (NM) containing silica glass beads; Kurangalina (1969)
and Frost et al. (2005) studied NM detonation failure with aluminium powders.
Particle ignition delay and reaction times are typically greater than the
explosive detonation reaction time scale, and the resulting metal particle
combustion heat release occurs predominantly behind the sonic point in the
detonation. Baudin et al. (1998) suggested that for spherical aluminium particles
as small as 100 nm, there is insufficient time for particles to react within
the detonation reaction zone. Kato et al. (2006) showed that aluminium
reaction contributes to the NM detonation only for particles smaller than
2 mm. Furthermore, the results of Yoshinaka et al. (2007) for 30 mm aluminium
particles shock-compressed to 13–30 GPa in condensed matter showed that
the majority of particles did not melt. Thus, micrometric metal particles can
usually be considered inert and intact within the condensed detonation reaction
zone, and the interaction between the particles and explosive is dominated by
shock compression.
Shock interaction and related momentum and heat transfer from the explosive
to the particles within the detonation zone are important mechanisms associated
with macroscopic detonation initiation, wave propagation, stability and failure
phenomena. On the other hand, the detonation transmits a strong shock into
the solid particles that rapidly accelerates and heats the metal as the wave
passes. Internal wave reflections and external interactions with neighbouring
particles dominantly affect the particle velocity and temperature owing to
shock compression before viscous flow interaction takes over. While experimental
methods can provide information on the bulk detonation response, at present
the diagnostics do not have the resolution required to record the individual
particle behaviour. Mesoscale numerical simulation is a practical alternative
to gain insight into detonation at the grain scale (0.1 mm–1 mm) in condensed
heterogeneous matter, where very high resolutions are required to capture the
small geometrical features and shock interaction.
Ordered matrices of packed circles, spheres and simple polygons have typically
been used to represent layers of heterogeneous condensed matter in mesoscale
simulation. Shock interaction with cylindrical ‘particles’ has been simulated in
two-dimensions by Mader (1979), Milne (2000) and Zhang et al. (2003). Early
three-dimensional models of Mader & Kershner (1982) have evolved to modern
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R. C. Ripley et al.
calculations of Baer (2002) and Cooper et al. (2006) that typically employ at
least 107 computational cells and contain O(100) particles. In general, only
a very small material sample (up to millimetre size) can be simulated using
mesoscale simulation.
While much of the foregoing literature investigates the influence of
heterogeneities on the detonation wave structure and propagation velocity,
limited work has addressed the acceleration and heating imparted on metal
particles in condensed matter during explosive detonation. Zhang et al. (2003)
showed that light metal particles achieve 60–100% of the shocked fluid velocity,
and demonstrated that the material density ratio of the explosive to solid
particle is an important factor affecting particle acceleration. The influence of
other parameters affecting particle acceleration and heating have been studied,
including particle matrix properties and explosive reactivity (Ripley et al. 2005,
2006, 2007).
In the present work, a quantitative description of the resultant momentum and
heat transfer is sought in conjunction with determination of the principal shock
interaction mechanisms. This is achieved using a theoretical dimensional analysis
to identify key parameters, applying three-dimensional mesoscale continuum
modelling of packed particle matrices saturated in liquid explosive to understand
the mechanisms and behaviour of the key parameters, and by compiling results
into transmission factors that quantify the momentum and heat transfer from
the explosive to the particles. Finally, the transmission factors are incorporated
into momentum and heat exchange source terms developed for a macroscopic
fluid dynamics framework, which is suitable for modelling detonation shock
compression in metal particle-condensed explosive systems.
2. Regimes for detonation interaction with particles
A natural starting point for condensed explosives is to consider a system of
a liquid explosive containing a high volume fraction of metal particles. Liquid
explosives are considered homogeneous in the absence of additives, and detonation
of liquid explosives has a reaction length (LR ) ranging from a few microns to a
few millimetres. Sheffield et al. (2002) estimated the reaction zone in NM to be
about 300 mm, whereas Engelke & Bdzil (1983) predicted a NM reaction length of
36 mm. The detonation reaction length can be reduced to a few micrometres when
NM is sensitized by liquid diethylenetriamine or triethylamine. Nitroglycerin is a
liquid above 13◦ C, with a reaction-zone length of 210 mm (Dobratz & Crawford
1985). Isopropyl nitrate is another liquid explosive with an estimated reactionzone length on the order of 1 mm (Zhang et al. 2002). Trinitrotoluene (TNT) is
a liquid above 81◦ C, where the reaction-zone length is 0.9–1.1 mm (Dobratz &
Crawford 1985). Thus for liquid explosives, a range for detonation reaction-zone
length of 10−6 < LR < 10−3 m is assumed.
Various metal particles have been added to liquid explosives to form a dense
heterogeneous system. This includes magnesium, aluminium, titanium, steel,
copper and tungsten in a size range of 10−8 < dp < 10−3 m (Zhang et al. 2001;
Frost et al. 2002, 2005; Haskins et al. 2002; Kato et al. 2006), where dp
denotes a spherical particle diameter. A parameter relating the particle diameter
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Metal particles in condensed detonation
(a)
(b)
VN
(c)
VN
VN
CJ
CJ
0
dp/LR<<1
CJ
0
dp/LR~1
0
dp/LR>>1
Figure 1. Schematic of a Zel’dovich–von Neumann–Döring-type detonation superimposed on
particle matrices to illustrate the relative length scales. Nomenclature: 0, initial state; VN, von
Neumann; CJ, Chapman–Jouguet.
and reaction length scale can therefore be defined as d = dp /LR . A potential
range of 10−5 < d < 103 can be derived from the ranges of dp and LR as
justified earlier. Three regimes for detonation interaction with particles can be
identified according to the value of d, as illustrated schematically in figure 1. In
reality, the leading shock structure is three-dimensional with transverse waves
and transmission into particles, followed by a multitude of shocks reflected
from particles.
(a) Small particle limit (dp /LR 1)
At the limit of d = dp /LR → 0, the detonation shock front is considered inert
(i.e. the von Neumann shock) during the early interaction, which can then be
represented by a Heaviside step function. Let us define a shock interaction
time scale as the time required for the detonation front to cross a particle:
tS = dp /D0 , where D0 is the shock velocity. Within the shock interaction time,
the detonation reaction-zone length is no longer a parameter and the response
is represented by a single length scale of the particle diameter. For an inert
planar shock crossing a particle, the dynamic response of the particle and the
surrounding flow field at any given time can be scaled by the particle diameter
using geometrical similarity when employing inviscid governing equations and
rate-independent material models (Zhang et al. 2003). Thus, the computational
results for a system of liquid explosive containing particles of a given size can be
scaled to systems of the same liquid explosive with any diameter particles within
the small particle limit.
(b) Large particle limit (dp /LR 1)
For d = dp /LR → ∞, the reaction length becomes negligibly small and the
detonation wave can therefore be considered as a discontinuity of the CJ front,
separating the fresh explosive from its detonation products. The interaction
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R. C. Ripley et al.
consists of diffraction of a thin CJ detonation front dictated by the curved
boundary of the particle, followed by unsteady expanding products flow controlled
by the rear boundary. The particle acceleration and heating are then characterized
by the single length scale of the particle diameter, similar to the small particle
limit (dp /LR 1), whereas the Heaviside VN shock is replaced with the CJ shock
and expanding products rear flow.
(c) Intermediate regime (dp /LR ∼ 1)
The case of d = dp /LR ∼ 1 lies between the above two limits and is therefore
most complex because both the particle diameter and reaction-zone length scales
play a role in the particle acceleration and heating. In this regime, the particle
interacts with both the VN shock front and the expanding reacting flow in the
detonation reaction zone that terminates at a sonic point. Locally, the reaction
zone is affected by the particle presence, resulting in a decreased reaction-zone
length at hot spots and an increased reaction-zone length in the expansion flow
around the particle.
(d) Definition of transmission factors
In order to describe the effect of the acceleration of solid particles in shock and
detonation of a condensed explosive, a velocity transmission factor, a, is defined
as the ratio of the particle mass-averaged velocity, up , after an interaction time,
t, over the shocked fluid velocity, uf1 :
a=
up (t)
.
uf1
(2.1)
In general, the velocity transmission factor varies between 0 for perfect
reflection off a rigid body and 1 for perfect transmission into a particle
with the same material properties as those of the host fluid. Similarly, a
temperature transmission factor, b, is defined as the ratio of the particle massaveraged temperature, Tp , after an interaction time, over the shocked fluid
temperature, Tf1 :
b=
Tp (t)
,
Tf1
(2.2)
where 0 ≤ b ≤ 1 for most metals. In (2.1) and (2.2), t = O(tS ) with tS = dp /D0 .
The characteristic shock interaction time, tS , is used for a single particle
in condensed matter. For a dense solid particles-fluid system, transmission
factors are measured after t = 2tS such that the immediate effect of wave
reflections both within particles and in the voids between neighbouring particles
is included. This comprises the majority of the acceleration and heating owing to
primary shock transmission, while subsequent internal waves further influence
the final velocity and temperature achieved during shock compression. This
time frame also accounts for the influence of transverse wave reflections and
upstream/downstream particle reflections in a densely packed matrix.
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Metal particles in condensed detonation
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3. Factors affecting particle acceleration and heating
(a) Dimensional analysis
The force acting on a particle during detonation of a liquid explosive containing
dense solid particles is assumed to be a function of relevant dimensional
parameters, e.g. Fd = f (dp , D0 , rf0 , rs0 , fs0 , p0 , mf0 , LR ), where dp is the particle
diameter, D0 is the detonation velocity, rf0 is the initial density of the explosive
fluid, rs0 is the initial particle material density, fs0 is the initial solid particle
volume fraction, p0 is the ambient pressure, mf0 is the molecular viscosity of
the explosive and LR is the detonation reaction-zone length. The Buckingham p
theorem (Buckingham 1914) considers primary dimensions (length, mass, time
and temperature) to reduce the number of parameters describing a physical
behaviour into a minimum set of dimensionless groups. Assuming dp , D0
and rf0 are independent variables among the nine parameters in the force
expression, p theorem yields that there are six dimensionless groups, which are
as follows:
⎫
rf0 D02
Fd
rf0 D0 dp
2 ⎪
= Cd , P2 =
= Re, P3 =
= gM0 ,⎪
P1 =
⎪
⎬
mf0
p0
rf0 D02 dp2
(3.1)
⎪
rf0
dp
⎪
⎪
⎭
P4 =
, P5 =
= d and P6 = fs0 .
rs0
LR
In (3.1), Cd is called an ‘effective’ drag coefficient, Re is the particle Reynolds
number and M0 is the Mach number of the detonation shock (M0 = D0 /af0 , where
af0 is the sound speed). Note that the flow compressibility is represented by the
shock Mach number, instead of the flow Mach number commonly used. Thus,
the force expression can be rewritten in a non-dimensional form:
rf0
, fs0 , d .
(3.2)
Cd = f Re, M0 ,
rs0
Similarly, heat transfer to a particle during detonation in a solid particleexplosive system is assumed to be Qc = f (dp , D0 , rf0 , rs0 , fs0 , T0 , mf0 , kf0 , cp , LR ),
where T0 is the ambient temperature, kf0 is the thermal conductivity of the
explosive and cp is the explosive fluid heat capacity. Using dp , D0 , rf0 and T0
as the independent variables, the p theorem gives seven dimensionless groups:
⎫
D02
rf0 D0 dp
⎪
2
⎪
= Re, P2 =
= (g − 1)M0 ,
P1 =
⎪
⎪
⎪
mf0
cp T 0
⎪
⎪
⎪
⎪
⎬
3
rf0 D0 dp
Q
Nu
c
2
(3.3)
P3 =
= PrReM0 (g − 1), P4 =
=
,
kf0 T0
rf0 D03 dp2 PrReM02 (g − 1) ⎪
⎪
⎪
⎪
⎪
⎪
⎪
rf0
dp
⎪
⎪
⎭
, P6 =
= d and P7 = fs0 .
P5 =
rs0
LR
In (3.3), Nu is an ‘effective’ Nusselt number (Nu = hdp /kf , where h is the
convective heat transfer coefficient) and Pr = mf cp /kf is the Prandtl number.
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R. C. Ripley et al.
Finally, the non-dimensional form of the heat transfer equation can be
expressed by
rf0
Nu = f Re, Pr, M0 ,
, fs0 , d .
(3.4)
rs0
The same non-dimensional parameters can also be obtained from the mass,
momentum and energy conservation equations that govern the flow. Defining
dimensionless variables xi∗ = xi /dp , t ∗ = t/tS = tD0 /dp , ui∗ = ui /D0 , r∗f0 = rf /rf0 ,
r∗s = rs /rs0 and f∗s = fs /fs0 , and substituting them into the continuity equation
for the solid particles-explosive system, one obtains
∗ ∗
∗ ∗
∗ ∗
∗ vr∗f uf,i
vr∗f f∗s uf,i
vr∗s f∗s us,i
vr∗f
vrf fs
vrs fs
rs0
+
f
= 0,
+
−
[f
]
+
+
s0
s0
vt ∗
vxi∗
vt ∗
vxi∗
rf0
vt ∗
vxi∗
(3.5)
which shows the non-dimensional parameters of fs0 and rf0 /rs0 .
Let additional dimensionless variables be defined as p∗ = p/rf0 D02 , s∗ =
sdp /mf0 D0 , m∗ = m/mf0 , Fp∗ = Fp dp3 /F0 , E ∗ = E/D02 , QR∗ = QR /D02 , q ∗ = qdp /kf0 T0 ,
where F0 is a reference force, E is the total specific energy and QR represents
a chemical reaction energy source. Substituting all the dimensionless variables
into the three-dimensional Navier–Stokes equations for the fluid phase of the
particles-explosive system, one derives the linear momentum equations and energy
equation of the fluid phase in dimensionless form. For example, the energy
equation becomes
∗ ∗ ∗ ∗ ∗
∗ ∗
∗
∗
Ef∗
vpf uf,i
vr∗f f∗s uf,i
vrf fs Ef
vr∗f Ef∗ vrf uf,i Ef
1
− [fs0 ]
+
+
+
vt ∗
vxi∗
vxi∗
vt ∗
vxi∗
gM02
∗ ∗ ∗ ∗ ∗ ∗
vuf,j sij
vpf fs uf,i
fs0
1
1
vqi
−
−
−
∗
∗
2
2
vxi
Re
vxi
vxi∗
gM0
PrReM0 (g − 1)
Nu
dp
∗ ∗
∗
= [Cd ](Fp,i up,i ) +
(3.6)
r∗f u∗ QR∗ .
Qc +
2
LR
PrReM0 (g − 1)
Thus, (3.5) and (3.6) provide the non-dimensional parameters of Re, Pr, M0 ,
fs0 , d, Cd and Nu. The same Re, M0 , fs0 and Cd are also derived for the momentum
equation, which has been omitted for brevity.
For dilute particles-gas flow (fs0 1; rf0 /rs0 1) with a small particle-limiting
scale (dp /LR 1), the drag coefficient function (3.2) and Nusselt number function
(3.4) approach the well-established classic forms of Cd = f (Re, M0 ) and Nu =
f (Re, Pr, M0 ). The dimensional analysis for dense solid particles-reactive fluid
systems shows the additional parameters, including the density ratio of fluid to
solid particle, solid volume fraction and ratio of particle diameter to detonation
reaction-zone length.
(b) Discussion of the parameters
This dimensional analysis is aimed at studying the particle acceleration
and heating during detonation shock compression in a dense solid particlecondensed explosive system. There are two major forces acting on a solid
particle during the detonation process: shock compression and viscous drag, with
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Metal particles in condensed detonation
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corresponding time scales characterized by the shock interaction time, tS , and
velocity relaxation time, tV (given in Rudinger (1980)), respectively. The ratio of
shock interaction time to viscous relaxation time is
uf
3mf Cd Re 3 rf
tS
(3.7)
Cd ,
=
=
4dp rs D0
4 rs
D0
tV
where Cd denotes a classic drag coefficient. For particles larger than 0.1 mm,
the viscous relaxation time is an order of magnitude greater than the shock
interaction time for a typical example of aluminium particles-liquid NM system
(rs = 2785 kg m−3 for inert aluminium, rf,CJ = 1539 kg m−3 , uf,CJ = 1.764 mm ms−1 ,
DCJ = 6.612 mm ms−1 for NM detonation, and assuming Cd ∼ 1 for an array of
spheres in compressible flow). Thus, shock compression is the dominant force
for the particle acceleration during the detonation process and the viscous drag
can be assumed to be negligible. Consequently, the Reynolds number can be
considered less important in the effective drag equation (3.2) and the flow can be
assumed to be inviscid.
Similarly for heat transfer, the ratio of shock interaction time to thermal
relaxation (convection) time, tT (given in Rudinger (1980)), is
6kf Nu
6Nu cp
rf
uf
tS
=
=
.
(3.8)
Re cs
rs
D0
t T d p rs cs D 0
For detonation of the inert aluminium particles-liquid NM system, the thermal
relaxation time is two orders of magnitude greater than the shock interaction
for dp = 0.1 mm and at least three orders of magnitude greater than the shock
interaction time for dp > 1 mm. Therefore, shock compression heating controls
the particle temperature during the detonation process and convective heat
transfer can be assumed negligible. Consequently, the Reynolds number and
Prandtl number can therefore be considered less important in the Nusselt number
equation (3.4), and the flow can be assumed to be completely inviscid and
non-heat-conducting.
The detonation shock Mach number of solid and liquid explosives at their
theoretical maximum density has a narrow range of 2.5 < M0 < 4 in general.
Shock velocity and pressure effects were investigated in a previous work (Zhang
et al. 2003) by varying the inert shock pressure from 5 to 20 GPa. While the
resulting momentum transfer to particles remained proportional to the shocked
fluid velocity, the variation in velocity transmission after the shock interaction
was less than 10 per cent for a shock velocity range of 4–9 mm ms−1 for metal
particles in cyclotrimethylenetrinitramine (RDX) explosives.
For a step shock wave passing a spherical metal particle in condensed matter,
the velocity transmission factor, a, was studied for magnesium, beryllium,
aluminium, nickel, uranium and tungsten in liquid NM and various solid
RDX densities subjected to a shock pressure range of 5–20 GPa (Zhang et al.
2003). The particle velocity after the shock interaction time, tS , was found to
strongly depend on the initial density ratio of explosive to metal and can be
expressed by
rf0 rf0
1
a+b
,
(3.9)
a=
a+b
rs0 rs0
where a and b are constants independent of the particle and explosive matter.
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The volume fraction of solid particles is defined by fs = np mp /rs , where np
is the number density of particles and mp is the mass of an individual particle,
and has a range of 0 ≤ fs ≤ 1. For example, in Tritonal (80 wt% TNT, 20 wt%
Al), the solid volume fraction is about fs0 ≈ 0.13. Loose polydisperse powders
typically have fs0 ≈ 0.5; random packing in particle beds produces higher volume
fractions, for instance, fs0 = 0.58–0.62 in slurries containing Al particles saturated
in liquid NM (Haskins et al. 2002; Frost et al. 2005). For ordered packing of
mono-sized spheres, fs0 = 0.74 is a theoretical geometrical maximum for a closepacked matrix. The particle acceleration and heating depends on the solid volume
fraction and packing configuration.
For inert shock interaction, there is only a single characteristic length scale,
i.e. the particle diameter, dp . The detonation process takes place in a reactionzone length, LR , which is an additional length scale that influences the particle
acceleration and heating. Therefore, the ratio d = dp /LR appears in equations (3.2)
and (3.4) and has been used to characterize three regimes of the detonation
interaction (figure 1). The effect of the volume fraction of solid particles and
the ratio of particle diameter to detonation reaction-zone length is investigated
in this paper using a mesoscale modelling approach.
4. Mesoscale computational approach
A system of spherical Al particles in NM forms the prototype heterogeneous
mixture for the study of detonation interaction with particles. Nitromethane
(CH3 NO2 ) is a uniform, low-viscosity liquid explosive and is therefore assumed to
follow ZND detonation theory. The incident shock pressures encountered in the
NM detonation are 13–23 GPa, far exceeding the yield strength of aluminium,
which is 0.035 GPa for 99 per cent commercially pure aluminium (Callister 1997).
Therefore, the aluminium material strength has been neglected by assuming that
only volumetric strain occurs in the particles. Hence, neglecting viscosity and
thermal conductivity as justified earlier, the hydrodynamic response of both liquid
NM and solid aluminium particles can be computed using the three-dimensional
inviscid Euler equations.
The governing equations are solved using the Harten, Lax, van Leer with
contact correction (HLLC) approximate Riemann solver scheme (Batten et al.
1997). The three materials are tracked using mass fraction continuity equations
for the liquid NM explosive, NM gaseous detonation products and solid Al
particles treated as inert, with equations of state for each described below. The
mixture of unreacted explosive and detonation products within the detonation
zone and the material boundaries at the metal particle surface are treated using
continuum mixture methods following Benson (1992).
(a) Equations of state
The NM and the aluminium particles are modelled using the Mie-Grüneisen
equation of state (EOS):
p=
Proc. R. Soc. A (2012)
GS
(e − eH ) + pH ,
n
T=
(e − eH )
+ TH
cv
(4.1)
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Metal particles in condensed detonation
1573
Table 1. Material and shock Hugoniot parameters from Mader (1998).
parameter
nitromethane
aluminium
r0 (g cm−1 )
C (mm ms−1 )
S
GS
cv (kJ kg−1 K−1 )
FS
GS
HS
IS
JS
1.128
1.647
1.637
0.6805
1.7334
5.41
−2.73
−3.22
−3.91
2.39
2.785
5.350
1.350
1.7
0.9205
−14.24
−95.75
−155.2
−102.9
−23.53
which is an expansion approximation around the shock Hugoniot. In (4.1), n is the
specific volume, e is the specific internal energy and GS is the so-called Grüneisen
gamma, which is the negative of the log slope along an isentrope. The Hugoniot
states (subscript H) are determined using the pressure relation:
2
C
pH =
(n0 − n),
(4.2)
n0 − S (n0 − n)
in which the coefficients C and S are determined from experiments (Marsh 1980).
The shock Hugoniot parameters are fit to the linear approximation D0 = C + Suf1 ,
where D0 is the shock velocity and uf1 is the fluid velocity.
The temperature-density behaviour is modelled using fitting to the data of
Walsh & Christian (1955):
ln TH = FS + GS (ln n) + HS (ln n)2 + IS (ln n)3 + JS (ln n)4 .
(4.3)
The temperature fitting coefficients (FS , GS , H S , I S and J S ) and Hugoniot
parameters used for NM and aluminium are summarized in table 1, which were
selected from those available for several common materials (Mader 1998).
The expansion of the gaseous detonation products is represented by the Jones–
Wilkins–Lee EOS (Lee et al. 1968):
−u−1
−R2 n
n
−R1 n
+ B exp
+C
.
(4.4)
p = A exp
n0
n0
n0
The fitting coefficients A = 277.2 GPa, B = 4.934 GPa, C = 1.223 GPa, R1 =
4.617, R2 = 1.073 and u = 0.379 are calculated using the Cheetah thermochemical
code and Becker–Kistiakowsky–Wilson–Sandia library (Fried et al. 1998).
(b) Nitromethane reaction model
The NM detonation model follows Mader (1998), who simulated reaction-zone
lengths ranging from 0.24 to 70.5 mm. For this work, the length of the reaction
zone has arbitrarily been fixed and the particle diameter is varied to study a
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R. C. Ripley et al.
pressure (GPa)
25
20
15
10
5
0
50
100
x/dp
150
200
Figure 2. Detonation wave pressure profiles over a running distance of 200dp , where dp = 1 mm, on
an one-dimensional mesh with a resolution of 100 cells per dp corresponding to a total of 20 000 cells.
The shaded region identifies the extent of the detonation wave that interacts with a packed bed
consisting of 10 layers of metal particles.
range of d = dp /LR . A single-step Arrhenius reaction model was employed for the
NM detonation,
−Ea
uf = rf YF Z exp
,
(4.5)
RTf
with Z = 8.0 × 1012 s−1 and Ea = 1.672 × 105 J mol−1 K−1 . A heat of detonation
of DHdet = 5.725 kJ cm−3 was determined using the Cheetah chemical equilibrium
code (Fried et al. 1998). The resulting reaction-zone length is 2 mm when measured
from the VN spike to the sonic point (location where D = uf − af ). In this
model, the reaction is 99.9 per cent complete at the CJ point. A comparison
of the numerical CJ point with the Cheetah equilibrium results shows reasonable
agreement. Figure 2 shows the resulting detonation wave pressure profiles for a
closed boundary condition at x/dp = 0. Other more sophisticated kinetics schemes
and modern equations of state are established in the literature, e.g. Lysne &
Hardesty (1973), Hardesty (1976) and Winey et al. (2000); however, the present
approach is still adequate to study the mechanical and thermal interaction of
particles with shock and detonation flow.
Prior to the detonation wave entering a packed particle matrix, the detonation
is run out to a distance of 1 mm (five times the running distance depicted in
figure 2). This was performed to reduce the gas expansion rate in the Taylor
wave during the interaction with the metal particles. Numerical simulation of the
NM detonation was performed on a high-resolution mesh with a 10−8 m cell size,
giving three to five cells across the shock and 200 cells in the reaction zone for
LR = 2 mm. A steady detonation was obtained after a running distance of 0.1 mm.
A selected spatial wave distribution was used to initialize three-dimensional
meshes of the same resolution containing the particle bed. For dp = 1 mm, the
particles are resolved with 100 cells per dp , which is sufficient to provide a
grid-independent solution.
(c) Initial particle packing configurations
Realistic heterogeneous explosive mixtures feature random packing and nonspherical particles. To fundamentally address the acceleration and heating
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Metal particles in condensed detonation
(a)
y
z
(b)
x
y
z
y
(c)
x
z
x
Figure 3. Geometrical arrangements of packed spherical particles: (a) simple cubic, fpacked = 0.52;
(b) body-centred, fpacked = 0.68; (c) face-centred (close packed), fpacked = 0.74.
of metal particles during detonation, ordered matrices of uniform spherical
particles are considered for various inter-particle spacings, covering a range of
volume fractions from 0.093 ≤ fs0 ≤ 0.740. The regular spacing and monodisperse
size assumptions avoid dense particle–particle collisions because all particles
are subjected to the same acceleration processes. Three idealized packing
configurations were considered as illustrated in figure 3; however, the face-centred
cubic lattice arrangement was selected for most of the calculations because the
closest packing represents the saturation limit, whereas the solid volume fraction
can be varied by adjusting the spacing between particles. Candidate particles
are studied six to eight layers into the matrix, where the interaction has become
quasi-steady in the absence of starting and end effects (Ripley et al. 2006, 2007).
The dilute limit (fs0 → 0) is simulated using a two-dimensional axi-symmetric
model of a single spherical particle, whereas the dense limit (fs0 → 1) is simulated
using a one-dimensional model of a semi-infinite solid slab. In three-dimensions,
plane-symmetric domains containing 20–40 particles were resolved using up to
32 million mesh points and 100 central processing units in parallel. Reflective
boundary conditions were used to represent a periodic solution in a semi-infinite
array of ordered particles.
5. Mesoscale results
(a) Results for the small particle limit (dp /LR 1)
Figure 4 illustrates the particle deformation in a three-dimensional particle
matrix (dp = 10 mm, dp /LR → 0) resulting from a 10.1 GPa inert Heaviside step
shock travelling from left to right. The deformed particles resemble a saddle shape
and, owing to the inviscid hydrodynamics, are strongly influenced by the complex
shock reflections from neighbouring particles. The deformation of the first layer
is different because the reflected shock wave is not subsequently re-reflected from
upstream particles. The severe deformation in the rear flow indicates that the
aluminium particles will likely be damaged or fragmented if material shear stress
and failure are considered.
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(a)
view A-A
(b)
view B-B
R. C. Ripley et al.
T(K)
1400
1300
1200
1100
1000
900
800
700
600
500
400
T(K)
1400
1300
1200
1100
1000
900
800
700
600
500
400
Figure 4. Inert shock interaction in a close-packed matrix of aluminium particles: resulting
deformation and temperature distribution with fs0 = 0.428: (a) matrix viewed on the [100] plane;
(b) matrix viewed on the [110] plane.
Figure 5a shows the pressure histories at the leading edges of the first eight
layers of packed particles. The pressure reaches a peak at the perfect reflection
value of 45.4 GPa and rapidly expands to a sustained quasi-steady pressure
plateau centred below the one-dimensional wave transmission value of 20.9 GPa.
The reverberating pressure oscillates with a period of 0.5tS corresponding to
the wave transit time within the interstitial pores contained between packed
particles. These resonant fluctuations would be dampened with random packing
and non-spherical particles as shown by Baer (2002).
Figure 5b shows the mass-averaged particle velocity for three packing
configurations. Within 1tS , the velocity in all three matrices exceeds the onedimensional wave transmission value of 1.11 mm ms−1 owing to the internal
rarefaction as the wave exits the trailing edge of the particle. Subsequent shock
reflection and interaction with neighbouring particles causes velocity fluctuations
proportional to the particle spacing that vary with packing configuration. A
time scale of 2tS is sufficient to capture one full interaction cycle that includes
the internal wave reverberation and successive expansions and compressions
from upstream, downstream and neighbouring particles. Because severe particle
deformation and resonant fluctuations occur after the shock interaction time
scale in the rear flow, they do not influence the evaluation of the velocity and
temperature transmission factors.
Figure 6 provides a summary of the velocity and temperature transmission
results for the small particle limit, using a 10.1 GPa inert shock. Comparison
of the packing configurations shows that the transmission factors for the simple
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Metal particles in condensed detonation
(a) 60
(b)
particle velocity (mm μs–1)
1.4
pressure (GPa)
50
40
30
20
10
1.2
1.0
0.8
0.6
0.4
0.2
0
0
0
1
2
t/tS
3
4
–2
–1
0
1
2
3
4
5
t/tS
Figure 5. Results in the small particle limit: (a) particle leading edge pressure histories for a closepacked matrix (fs0 = 0.74); (b) mass-averaged particle velocity for various matrices of packed
spherical particles (face centred: dashed-dotted line f = 0.74; solid line, f = 0.68; dashed line,
f = 0.52).
(a)
(b)
1.0
close packed spheres
body-centred spheres
simple cubic spheres
solid slab
single sphere
cubic fit
two cylinders
11 cylinders
0.9
0.8
a
1.0
close packed spheres
body-centred spheres
simple cubic spheres
solid slab
single sphere
cubic fit
0.9
0.8
0.7
b
0.6
0.7
0.5
0.6
0.4
0.3
0.5
0
0.2
0.4
0.6
volume fraction, fs0
0.8
1.0
0
0.2
0.4
0.6
0.8
volume fraction, fs0
1.0
Figure 6. Velocity and temperature transmission factors for a 10.1 GPa inert shock interaction with
packed particle matrices. Cylinder results (solid symbols) are from Zhang et al. (2003).
cubic packing were considerably different. This is primarily due to this particular
packing matrix that provides two propagation channels: one through the linear
array of stacked particles, and the other uninhibited though the column of liquid
in the void space. The simple cubic packing is an unrealistic configuration in
practice and is not given further consideration. The present three-dimensional
results are compared with two-dimensional cylindrical results of Zhang et al.
(2003). The agreement is improved for a higher number of two-dimensional
cylinders, as this better approximates an infinite array of particles as in the
three-dimensional configuration.
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R. C. Ripley et al.
Both the velocity and temperature transmission factors for the close-packed
and body-centred matrices follow an inverted U-shaped function. The maximum
transmission factors occurred for 0.25 ≤ fs0 ≤ 0.40, whereas the minimum
transmission factors resulted at the dilute limit (fs0 = 0) and high volume
fraction limit (fs0 = 1). In between the volume fraction limits, the spacing of the
particles affects the arrival time and magnitude of reflected shocks from upstream
and downstream particles, in addition to affecting the local flow diffraction.
The maximum transmission factors are a result of superposition of transmitted
shock waves that act in a coherent manner.
(b) Results for the large particle limit (dp /LR 1)
In the dp /LR 1 case, early simulation results (Ripley et al. 2005) for the
detonation wave over a single aluminium particle indicated that the particle
velocity is first increased as the detonation front crosses the particle and is then
reduced, subject to the Taylor expansion flow conditions. The Taylor expansion
after the thin detonation front reduces the detonation flow velocity, thus changing
the direction of the drag and reversing the momentum transfer direction from
the particles to the detonation expansion flow. The large particle limit was
studied preliminarily using dp = 30 mm (Ripley et al. 2006), which corresponds
to dp /LR = 15. The unsteady Taylor expansion effect was minimized by running
the detonation sufficiently far from the initiation location.
For dp /LR 1, the VN shock can be neglected and the detonation interaction
with particle matrices can be better represented by a CJ shock. The jump in the
host liquid does not follow the Hugoniot because the shock is reactive, where the
chemical reaction rate is essentially infinite and the post-shock flow contains hot
expansion products. Figure 7 illustrates a CJ shock (D0 = 6.69 mm ms−1 , PCJ =
13.8 GPa, uf1,CJ = 1.827 mm ms−1 and rf1,CJ = 1.551 g cm−3 ) travelling through a
NM–Al matrix. For a particle spacing of 1dp , the volume fraction is fs0 = 0.093,
and the CJ shock front profile approaches that of a planar wave prior to arrival at
successive particle leading edges; this leads to a flattening of the particles during
deformation.
The particle velocity and temperature histories are shown in figure 8. Within
2tS , both the incident shock and internal rarefaction accelerate the particle, while
an increase in particle temperature owing to shock compression is followed by a
decrease from the rarefaction expansion before lateral compression continues the
particle heating. Figure 8a illustrates a decrease in particle velocity over 3–4 tS
as a result of a reflected wave returning from the downstream particles. Further
particle acceleration must be influenced by viscous drag, which is beyond the
scope of the present paper. Similarly, the particle heating is only affected by shock
compression in the non-heat-conducting assumption valid within the time frame
considered, although the downstream NM products are hot (Tf1,CJ = 3657 K).
(c) Results for the intermediate regime (dp /LR ∼ 1)
Computation of diffraction of a reactive shock over a single particle in a threedimensional mesh showed that the shock is both transmitted into the metal at
the particle forward stagnation point, and also reflected back into the reaction
zone, thereby increasing the reaction rate. A Mach stem forms as the shock in the
explosive diffracts over the particle. Depending on the impedance ratio, the shock
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Metal particles in condensed detonation
(a)
y
r (kg m–3)
3200
3000
2800
2600
2400
2200
2000
1800
1600
1400
1200
1000
z
x
(b) y
r (kg m–3)
3200
3000
2800
2600
2400
2200
2000
1800
1600
1400
1200
1000
z
x
Figure 7. Fluid and particle density distribution and particle deformation for Chapman–Jouguet
shock propagation through a particle matrix: (a) fs0 = 0.520 and (b) fs0 = 0.093.
1.5
800
particle temperature (K)
(b) 1000
particle velocity (mm μs–1)
(a) 2.0
1.0
0.5
0
–0.5
–2
–1
0
1
2
t/tS
3
4
5
6
600
400
200
0
–2 –1
0
1
2
t/tS
3
4
5
6
Figure 8. (a) Mass-averaged particle velocity and (b) temperature for detonation interaction at the
large particle limit (fs0 = 0.093).
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R. C. Ripley et al.
y
z
x
pressure (GPa)
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
Figure 9. Pressure distribution and particle deformation for detonation (dp /LR = 0.5) through a
packed particle matrix (fs0 = 0.428).
inside the metal particle may travel ahead of the incident shock, transmitting
an oblique shock into fresh explosive ahead of the detonation front. Behind the
diffracting detonation shock, expansion of the flow on the backside of the particle
causes the local reaction-zone length to increase. At the particle trailing edge,
either the diffracted shock arrives first, or the converging shock inside the particle
is retransmitted back into the fresh explosive. This can initiate the explosive
locally ahead of the detonation shock. Subsequently, a strong rarefaction forms
within the metal particle, contributing to both a large acceleration and a decrease
in temperature.
For packed particle matrices, the situation is more complex with reverberating
waves and lateral interactions that further influence the particle and surrounding
flow. Figure 9 illustrates the irregular propagation pattern of the detonation
front in an Al–NM matrix (dp = 1 mm, dp /LR = 0.5), where the detonation travels
within the explosive contained in the voids and narrow channels between particles
in addition to shock propagation within the metal. In close-packed matrices,
transmission of the shock through the particle can subsequently pre-compress
and initiate detonation in the fresh explosive on the far side of the particle prior
to the diffracted shock arrival (hot-spot mechanism). For the conditions depicted
in figure 9, the NM behind the particle trailing edge reaches 800 K owing to shock
transmission and prior to reaction.
Figure 10 shows numerical pressure gauge results for detonation in a closepacked matrix, with the peak reflected VN shock pressure followed by oscillations
at a frequency proportional to particle diameter and later by the Taylor wave
expansion. Figure 10a shows a precursor shock with magnitude of about 10 GPa
(temperature approx. 1300 K) that resulted from the reflection of the shock
transmitted through the particle trailing edge into the void prior to arrival of
the diffracted VN shock. Figure 10b shows a smaller precursor shock exiting the
trailing edge and a larger peak pressure owing to collision of the diffracted shocks
behind the particle trailing edge.
Figure 11 shows the particle velocity and temperature histories in a packed
particle matrix for dp /LR = 0.2, where the detonation reaction zone is much
larger than the particle size. This is evident in the particle velocity history result,
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Metal particles in condensed detonation
pressure (GPa)
(a) 60
(b)
40
20
0
0
5
10
t/tS
15
20
0
5
10
t/tS
15
20
(a) 2.0
(b) 1500
particle velocity (mm μs–1)
particle temperature (K)
Figure 10. Gauge histories from detonation (dp /LR = 0.5) in a close-packed matrix (fs0 = 0.74)
with 20 layers of particles: (a) pressure at particle leading edge; (b) pressure in voids behind
trailing edges. Each curve presents successive results from the first, fifth, ninth, 13th and
17th layers.
1.5
1.0
0.5
0
–0.5
0
5
t/tS
10
15
1000
500
0
0
5
t/tS
10
15
Figure 11. (a) Particle velocity and (b) temperature histories for dp /LR = 0.2 and fs0 = 0.74.
where the velocity first increases during interaction with the VN shock, and then
decreases mainly over 5tS during the expansion inside the reaction zone. The 2tS
assumption is especially justified in the temperature history where the first peak
is reached between 1 and 2tS .
Figure 12a shows the detonation shock velocity through the particle matrices
immersed in liquid explosive. The bulk propagation velocity was measured using
wave time of arrival between consecutive numerical gauge stations located in an
array perpendicular to the averaged detonation wavefront. Under the detonation
conditions studied here, i.e. aluminium particles in NM, shock waves reflected
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R. C. Ripley et al.
(a) 7.4
(b) 7.0
fs0 = 0
fs0 = 0.093
fs0 = 0.740
shock velocity (mm μs–1)
7.2
7.0
6.6
6.8
6.4
6.6
6.2
6.4
6.0
6.2
5.8
6.0
0
5
x/dp
10
cheetah (inert Al)
mesoscale
6.8
15
5.6
0
20
40
60
80
mass fraction (%)
100
Figure 12. Detonation velocity through packed particle matrices (dp = 1 mm; LR = 2 mm): (a)
unsteady propagation velocity through several layers of aluminium particles with distance
measured from the first layer; (b) quasi-steady propagation velocity for various aluminium
mass fractions.
from the leading edge of the particles cause an increase in the local NM
density and pressure, thereby increasing the detonation velocity. Furthermore,
shock waves travelling within the metal particle are transmitted into the fresh
liquid explosive ahead of the detonation wave diffracting around the curved
particle surface, thereby pre-compressing the explosive and increasing the local
detonation velocity. These two local hot-spot factors both contribute to bulk
propagation speeds in excess of the CJ value in fresh NM. In the present numerical
calculations, the greatest bulk propagation velocities (up to 7.4 mm ms−1 ) were
observed for the highest metal mass fraction condition in combination with
the smallest particle diameter (or longest reaction zone) within the region of the
first particle layer. Afterwards, the momentum and energy transferred into the
particles within the detonation zone competes with the local hot-spot factors,
resulting in a quasi-steady propagation velocity with a deficit. The velocity
deficit increases with an increase in solid volume fraction and a decrease in
particle diameter. The maximum velocity deficit corresponded to a wave speed
of 5.3 mm ms−1 and occurred for the small particle limit.
Figure 12a shows increasing instability in the detonation front velocity for
higher solid volume fractions, which is an expected feature due to higher
momentum and heat loss. The initially transient shock velocity becomes quasisteady after travelling a distance of 6dp into the matrix. Figure 12b illustrates
the quasi-steady detonation shock velocity as a function of increasing metal mass
fraction. In comparison with Cheetah chemical equilibrium predictions using inert
aluminium, where velocity and temperature equilibrium is assumed for all phases
(Fried et al. 1998) and essentially d → 0, the mesoscale results for detonation
shock velocity are consistently higher because the relative velocity of the solid
phase remains below the flow velocity following the shock interaction. Similarly,
the Cheetah equilibrium temperature is also lower than the mesoscale detonation
results owing to the same phase-non-equilibrium nature.
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Metal particles in condensed detonation
particle velocity, uP (mm μs–1)
(a)
(b)
2.0
0.8
0.7
0.6
1.5
0.5
1.0
d = 0.1
d = 0.2
d = 0.5
d=1
d=2
d = 10
0.5
0
0
2
t/tS
4
6
a 0.4
0.3
0.2
fitting function
mesoscale
0.1
0
10–3 10–2
0.1
1
d
10
102
103
Figure 13. Shock compression acceleration of a single particle (fs0 = 0) in a detonation flow: (a)
mass-averaged velocity for various particle diameters; (b) velocity transmission factor versus d and
fitting function.
(d) Particle acceleration
The mass-averaged particle velocity history achieved during detonation
interaction with a single particle is illustrated in figure 13a. Because a fixed
numerical mesh resolution was maintained (100 cells mm−1 ), such that the number
of cells in the ideal detonation reaction zone (LR = 2 mm) was unchanged,
the number of cells across the particle diameter increased with d and the
corresponding computational effort increased exponentially. Thus for a large d,
limited duration particle velocity histories are available. For d < 0.5, the peak
particle velocity exceeds the CJ value, illustrating a significant influence of the VN
spike on the particle acceleration. For t/tS 1, the particle velocity equilibrates
below the CJ value (uCJ = 1.827 mm ms−1 ) in the absence of Taylor expansion.
Figure 13b shows the mesoscale results of the single particle velocity and
corresponding transmission factor as a function of d = dp /LR . Both the velocity
and the velocity transmission factor decrease from VN to CJ mainly over the
interval from 0.1 ≤ d ≤ 1. The single particle acceleration results are bound by the
small particle limit (d → 0) and the large particle limit (d → ∞). The resulting
velocity transmission factor was fit to the sigmoidal function:
a = aCJ +
aVN − aCJ
,
1 + exp[− log(d/d0 )/w]
(5.1)
where d0 = 0.38 and w = 0.25. In equation (5.1), aCJ = a(d → ∞) = 0.351 and
aVN = a(d → 0) = 0.669.
Figure 14a illustrates the particle velocity transmission in an Al–NM matrix
as a function of volume fraction. Close-packed spheres are used, and the volume
fraction is adjusted by changing the inter-particle spacing. For the various d
considered, there is a twofold range in particle velocity between the interaction
limits. There is weak similarity in the results for various d, which are bound
by limiting cases of d → 0 for VN shock interaction and d → ∞ for CJ shocked
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R. C. Ripley et al.
particle velocity, uP (mm μs–1)
(a) 2.5
(b)
0.8
2.0
0.6
1.5
a 0.4
1.0
d →0 (VN shock)
d = 0.2
d = 0.5
d = 15
d → ∞(CJ shock)
0.5
0
0.2
0.4
0.6
0.8
volume fraction, fs0
1.0
d →0 (VN shock)
d = 0.2
d = 0.5
d = 15
d → ∞(CJ shock)
fitting function
0.2
0
0.2
0.4
0.6
0.8
volume fraction, fs0
1.0
Figure 14. Shock compression acceleration results in a particle matrix: (a) range of
transmitted particle velocity between interaction limits; (b) multi-variable fitting for velocity
transmission factor.
flow conditions. The maximum a generally occurs at fs0 = 0.2, above which the
transmission decreases linearly for increasing fs0 . The regime of primary interest
for dense granular flow in condensed explosives is from 0.2 < fs0 < 0.6.
The velocity transmission factor results were fit to a function of volume fraction
and detonation reaction-zone length: a = a(fs0 , d). The volume fraction effect
was represented by a second-order polynomial function of fs0 . The reaction-zonelength influence is exhibited as a shift in the velocity transmission factor, which
was represented using an exponential function of d and a fixed offset. The resulting
function contains five fitting constants, as follows:
a = c1 f2s0 + c2 fs0 + c3 exp(−c4 d) + c5 ,
(5.2)
where c1 = 0.2, c2 = 0.1, c3 = 0.4, c4 = 3.3 and c5 = 0.36. A comparison of the
fitting function with the mesoscale results is given in figure 14b. The agreement
is reasonable for d < 1; the largest differences occurred for high volume fractions
in the large particle limit.
(e) Particle heating
Figure 15a shows the mass-averaged shock compression temperature for a
single particle for various d. Smaller particles achieve higher temperatures,
although the maximum temperature during shock interaction is much less
than the CJ flow temperature (TCJ = 3657 K). Oscillations in the temperature
histories are caused by successive compressions and expansions owing to wave
reverberation inside the particle.
Figure 15b shows the mesoscale results of the single particle temperature and
corresponding transmission factors, b, which are monotonic decreasing functions
for increasing d = dp /LR . Both the temperature and the temperature transmission
factor decrease from the VN to CJ limiting values mainly over the interval
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Metal particles in condensed detonation
particle temperature, TP (K)
(a)
(b)
1000
800
0.4
0.3
600
b 0.2
400
d = 0.1
d = 0.2
d = 0.5
d=1
d=2
d = 10
200
0
0
2
t/tS
4
6
0.1
fitting function
mesoscale
0
10–3 10–2
0.1
1
d
10
102
103
Figure 15. Shock compression heating of a single particle (fs0 = 0) in detonation flow: (a) massaveraged temperature of single particles of various diameters; (b) temperature transmission factors
with fitting function.
from 0.1 ≤ d ≤ 10. Similar to the velocity transmission factor, the temperature
transmission factor for a single particle is fit to the sigmoidal function:
b = bCJ +
bVN − bCJ
,
1 + exp[− log(d/d0 )/w]
(5.3)
where d0 = 0.70, w = 0.45, bCJ = 0.170 and bVN = 0.337.
Figure 16 shows the particle temperature and corresponding b transmission
factors in the Al particle-liquid explosive matrix as a function of volume fraction.
The temperature transmission factors are scaled using the VN shocked fluid
temperature. Similar to particle acceleration, the particle heating is bound
between the small particle and large particle-limiting cases, and the effect of
solid volume fraction on b is reduced for larger particles (d → ∞). For a given
volume fraction, b increases with the reaction-zone length (decreasing d). For
volume fractions relevant to dense granular flow in condensed explosives, i.e.
0.2 < fs0 < 0.6, the range of b is limited to 0.32–0.40 across two orders of
magnitude for the ratio of dp /LR .
6. Application to macroscopic models
Having gained an understanding of the particle interactions with the explosive
fluid and neighbouring particles during shock and detonation in an idealized dense
or packed particle-condensed matter system, let us try to apply the mesoscale
results to formulate macroscopic momentum and energy transfer functions. These
functions can then be incorporated into interphase exchange source terms that
are used during the shock compression time scale. The resulting approach can
be applied to macroscopic continuum modelling of practical problems such
as detonation in a multi-phase explosive or shock propagation in a dense
particle-fluid system.
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R. C. Ripley et al.
particle temperature, TP (K)
(a)
(b)
1500
0.6
1000
0.4
b
500
0.2
d →0 (VN shock)
d = 0.2
d = 0.5
d = 15
d → ∞(CJ shock)
0
0
0.2
0.6
0.8
0.4
volume fraction, fs0
1.0
d → 0 (VN shock)
d = 0.2
d = 0.5
d = 15
d → ∞(CJ shock)
0
0
0.2
0.4
0.8
0.6
volume fraction, fs0
1.0
Figure 16. Shock compression heating results in a particle matrix: (a) range of transmitted particle
temperature between the interaction limits; (b) corresponding temperature transmission factors.
During the detonation shock interaction time, the momentum transfer rate, i.e.
the force, Fp , and the heat transfer rate, Qp , acting on a macroscale solid particlecondensed explosive control volume containing np particles each with mass, mp ,
can be approximated by
Fp = np mp
and
Qp = np mp cs
dup
auf1 − up0
≈ n p mp
dt
t
dTp
bTf1 − Tp0
≈ n p m p cs
.
dt
t
(6.1)
(6.2)
Because the mesoscale results showed that the increase in mass-averaged
particle velocity was mostly linear within the interaction time scale, a simple
acceleration is assumed: up (t) = up0 + (up1 − up0 )(t − t0 )/t for t0 < t < t0 + t,
where t0 is the shock arrival time at the particle leading edge. Combining the
standard definition for drag force on spherical particles,
Fp =
pdp2
8
np rf |uf − up |(uf − up )Cd ,
(6.3)
with (6.1) and assuming up0 = 0 and uf ≈ uf1 , an effective drag coefficient is
obtained for the shock compression interaction:
Cd (t) =
4rs D0
a
,
3rf uf1 (1 − ((t − t0 )/t)a)2
for t0 < t < t0 + t.
(6.4)
In (6.4), a = a(rf0 /rs0 , fs0 , d) can be obtained from the mesoscale results in §5,
keeping in mind the limitations of uniform spherical particles in ordered matrices.
Similarly, the particle heating rate in the mesoscale results can be assumed
constant within the interaction time, and the particle temperature can
therefore be approximated by a linear function: Tp (t) = Tp0 + (Tp1 − Tp0 )
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Metal particles in condensed detonation
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(t − t0 )/t. Combining the standard definition for convective heat transfer on
spherical particles,
(6.5)
Qp = pdp np kf (Tf − Tp )Nu,
with (6.2) and assuming Tf ≈ Tf1 , an effective Nusselt number is obtained for the
shock compression interaction:
Nu(t) =
bTf1 − Tp0
dp cs rs D0
,
6kf
(1 − ((t − t0 )/t)b)Tf1 − (1 − ((t − t0 )/t))Tp0
for t0 < t < t0 + t.
(6.6)
Note that the physical parameters Cd (equation (6.4)) and Nu (equation (6.6))
feature a time-dependence in order to facilitate their numerical implementation.
The effective drag force and heat transfer models are suitable for two-phase
macroscopic models of dense particles such as Baer & Nunziato (1986).
7. Conclusions
Dimensional analysis showed that the particle acceleration force and heat transfer
during detonation shock compression in a dense solid particle-condensed explosive
system are a function of the material density ratio of explosive to particle,
rf0 /rs0 , the volume fraction, fs0 and the ratio of particle diameter to detonation
reaction-zone length, d = dp /LR , apart from Reynolds number, Re, Prandtl
number, Pr and Mach number, M0 . While viscosity and heat conduction are
important in later time, they can be neglected when compared with the other
parameters during the shock compression time scale. Thus, the acceleration
force and heat transfer can be described by an effective drag coefficient, Cd =
f (rf0 /rs0 , fs0 , d, M0 ), and the heat transfer is represented by an effective Nusselt
number, Nu = f (rf0 /rs0 , fs0 , d, M0 ). Mesoscale simulations of spherical aluminium
particles immersed in liquid NM were conducted by varying fs0 and d at a
given M0 and rf0 /rs0 , which were known to be important parameters. The full
range of fs0 and d was studied, where d ranged between the small particle
limit, with essentially inert shock interaction, to the large particle limit, with
infinitely thin detonation front interaction followed by detonation products
expansion flow.
Features of heterogeneous detonation were explored including: detonation
instability and velocity deficit; pressure front fluctuations with peaks up to
four times the CJ detonation pressure and periods proportional to the particle
size; and, transverse waves and hot spot characteristics of locally enhanced
pressure and temperature fronts. These physics are consistent with macroscopic
phenomena observed in published experiments. Detonation failure was not
considered in this study because the mesoscale calculations were infinite diameter
(no charge edge effects).
From the mesoscale simulation, a shock compression velocity transmission
factor, a = f (rf0 /rs0 , fs0 , d, M0 ), and temperature transmission factor, b =
f (rf0 /rs0 , fs0 , d, M0 ) were obtained to summarize the acceleration and heating
behaviour within the detonation shock interaction time. The maximum particle
acceleration occurred at fs0 = 0.25, whereas the maximum shock compression
heating occurred over a wider range of solid volume fraction fs0 = 0.43–0.74.
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R. C. Ripley et al.
Scaling of the transmitted velocity and temperature using the von Neumann state
is convenient because it is easily obtained from analytical shock relationships;
however, this scaling showed a strong dependence on the ratio of particle diameter
to reaction-zone length.
Overall, velocity and temperature transmission factors obtained using
mesoscale calculations can be tabulated and fit to simple functions of density
ratio, volume fraction and the ratio of particle diameter to detonation reactionzone length, for a given Mach number. Fitting in this fashion allows practical
models to be developed for engineering calculations. As an example, the
results were applied to formulate functions for macroscopic momentum and
energy transfer between the two phases during detonation shock compression.
These functions can then be used as the interphase exchange source terms
applied to macroscopic continuum modelling of practical problems such as
detonation of a multi-phase explosive or shock propagation in a dense
particle-fluid system.
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