acceleration and heating of metal particles in condensed...

Proc. R. Soc. A (2012) 468, 1564–1590doi:10.1098/rspa.2011.0595

Published online 15 February 2012

Acceleration and heating of metal particles incondensed matter detonation

BY ROBERT C. RIPLEY1,2,*, FAN ZHANG1,3 AND FUE-SANG LIEN1

1Department of Mechanical Engineering, University of Waterloo, Waterloo,Ontario, Canada N2L 3G1

2Martec Limited, 1888 Brunswick Street, Suite 400, Halifax, Nova Scotia,Canada B3J 3J8

3Defence R&D Canada-Suffield, PO Box 4000, Station Main, Medicine Hat,Alberta, Canada T1A 8K6

For condensed explosives, containing metal particle additives, interaction of thedetonation shock and reaction zone with solid inclusions leads to high rates of momentumand heat transfer that consequently introduce non-ideal detonation phenomena. Duringthe time scale of the leading detonation shock crossing a particle, the acceleration andheating of metal particles are shown to depend on the volume fraction of particles, densepacking configuration, material density ratio of explosive to solid particles and ratio ofparticle diameter to detonation reaction-zone length. Dimensional analysis and physicalparameter evaluation are used to formalize the factors affecting particle accelerationand heating. Three-dimensional mesoscale calculations are conducted for matrices ofspherical metal particles immersed in a liquid explosive for various particle diameterand solid loading conditions, to determine the velocity and temperature transmissionfactors resulting from shock compression. Results are incorporated as interphase exchangesource terms for macroscopic continuum models that can be applied to practicaldetonation problems involving multi-phase explosives or shock propagation in denseparticle-fluid systems.

Keywords: condensed detonation; heterogeneous matter; mesoscale simulation

1. Introduction

Detonation in condensed explosives typically features wave propagation speedsfrom 6 to 9 mm ms−1 and peak pressures from 10 to 50 GPa. The detonation frontis often modelled by the von Neumann (VN) shock followed by a reaction zonein 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. Theaddition of heterogeneities such as metal particles into condensed explosivesintroduces microscopic interaction of the detonation shock and reaction zonewith the particles, which produces localized hot spots, transverse waves and high-pressure fluctuations that are manifested in macroscale detonation phenomena.

*Author for correspondence ([email protected]).

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Metal particles in condensed detonation 1565

Particles saturated in liquid explosives form a fundamental system termedwetted powder, or ‘slurry’ (Frost & Zhang 2009). The effect of a large volumefraction of metallic particle additives on the detonation properties of slurryexplosives 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 relatedto the particle properties, morphology and concentration. In general, addingparticles results in a velocity deficit below that of the neat explosive, because someof the chemical energy released goes into heating, and particularly, acceleratingthe particles.

Detonation failure depends on the competing effects of explosive sensitizationowing to the formation of hot spots near the particles, and the increasingdetonation reaction-zone length with momentum and energy absorbed intothe particles and detonation front curvature caused by lateral expansion.Engelke (1979) and Lee et al. (1995) extensively studied the critical diameterof 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 theexplosive detonation reaction time scale, and the resulting metal particlecombustion heat release occurs predominantly behind the sonic point in thedetonation. Baudin et al. (1998) suggested that for spherical aluminium particlesas small as 100 nm, there is insufficient time for particles to react withinthe detonation reaction zone. Kato et al. (2006) showed that aluminiumreaction contributes to the NM detonation only for particles smaller than2 mm. Furthermore, the results of Yoshinaka et al. (2007) for 30 mm aluminiumparticles shock-compressed to 13–30 GPa in condensed matter showed thatthe majority of particles did not melt. Thus, micrometric metal particles canusually be considered inert and intact within the condensed detonation reactionzone, and the interaction between the particles and explosive is dominated byshock compression.

Shock interaction and related momentum and heat transfer from the explosiveto the particles within the detonation zone are important mechanisms associatedwith macroscopic detonation initiation, wave propagation, stability and failurephenomena. On the other hand, the detonation transmits a strong shock intothe solid particles that rapidly accelerates and heats the metal as the wavepasses. Internal wave reflections and external interactions with neighbouringparticles dominantly affect the particle velocity and temperature owing toshock compression before viscous flow interaction takes over. While experimentalmethods can provide information on the bulk detonation response, at presentthe diagnostics do not have the resolution required to record the individualparticle behaviour. Mesoscale numerical simulation is a practical alternativeto gain insight into detonation at the grain scale (0.1 mm–1 mm) in condensedheterogeneous matter, where very high resolutions are required to capture thesmall geometrical features and shock interaction.

Ordered matrices of packed circles, spheres and simple polygons have typicallybeen used to represent layers of heterogeneous condensed matter in mesoscalesimulation. Shock interaction with cylindrical ‘particles’ has been simulated intwo-dimensions by Mader (1979), Milne (2000) and Zhang et al. (2003). Earlythree-dimensional models of Mader & Kershner (1982) have evolved to modern

Proc. R. Soc. A (2012)



1566 R. C. Ripley et al.

calculations of Baer (2002) and Cooper et al. (2006) that typically employ atleast 107 computational cells and contain O(100) particles. In general, onlya very small material sample (up to millimetre size) can be simulated usingmesoscale simulation.

While much of the foregoing literature investigates the influence ofheterogeneities on the detonation wave structure and propagation velocity,limited work has addressed the acceleration and heating imparted on metalparticles 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 solidparticle is an important factor affecting particle acceleration. The influence ofother 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 andheat transfer is sought in conjunction with determination of the principal shockinteraction mechanisms. This is achieved using a theoretical dimensional analysisto identify key parameters, applying three-dimensional mesoscale continuummodelling of packed particle matrices saturated in liquid explosive to understandthe mechanisms and behaviour of the key parameters, and by compiling resultsinto transmission factors that quantify the momentum and heat transfer fromthe explosive to the particles. Finally, the transmission factors are incorporatedinto momentum and heat exchange source terms developed for a macroscopicfluid dynamics framework, which is suitable for modelling detonation shockcompression 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 ofa liquid explosive containing a high volume fraction of metal particles. Liquidexplosives are considered homogeneous in the absence of additives, and detonationof liquid explosives has a reaction length (LR) ranging from a few microns to afew millimetres. Sheffield et al. (2002) estimated the reaction zone in NM to beabout 300 mm, whereas Engelke & Bdzil (1983) predicted a NM reaction length of36 mm. The detonation reaction length can be reduced to a few micrometres whenNM is sensitized by liquid diethylenetriamine or triethylamine. Nitroglycerin is aliquid above 13◦C, with a reaction-zone length of 210 mm (Dobratz & Crawford1985). Isopropyl nitrate is another liquid explosive with an estimated reaction-zone length on the order of 1 mm (Zhang et al. 2002). Trinitrotoluene (TNT) isa 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-zonelength of 10−6 < LR < 10−3 m is assumed.

Various metal particles have been added to liquid explosives to form a denseheterogeneous 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 dpdenotes a spherical particle diameter. A parameter relating the particle diameter





VN

0 0 0

(a) (b) (c)

dp/LR<<1 dp/LR~1 dp/LR>>1

VN

CJCJ

CJ

VN

Figure 1. Schematic of a Zel’dovich–von Neumann–Döring-type detonation superimposed onparticle matrices to illustrate the relative length scales. Nomenclature: 0, initial state; VN, vonNeumann; CJ, Chapman–Jouguet.

and reaction length scale can therefore be defined as d = dp/LR. A potentialrange of 10−5 < d < 103 can be derived from the ranges of dp and LR asjustified earlier. Three regimes for detonation interaction with particles can beidentified according to the value of d, as illustrated schematically in figure 1. Inreality, the leading shock structure is three-dimensional with transverse wavesand transmission into particles, followed by a multitude of shocks reflectedfrom 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 berepresented by a Heaviside step function. Let us define a shock interactiontime 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 responseis represented by a single length scale of the particle diameter. For an inertplanar shock crossing a particle, the dynamic response of the particle and thesurrounding flow field at any given time can be scaled by the particle diameterusing geometrical similarity when employing inviscid governing equations andrate-independent material models (Zhang et al. 2003). Thus, the computationalresults for a system of liquid explosive containing particles of a given size can bescaled to systems of the same liquid explosive with any diameter particles withinthe small particle limit.

(b) Large particle limit (dp/LR � 1)

For d = dp/LR → ∞, the reaction length becomes negligibly small and thedetonation wave can therefore be considered as a discontinuity of the CJ front,separating the fresh explosive from its detonation products. The interaction





consists of diffraction of a thin CJ detonation front dictated by the curvedboundary of the particle, followed by unsteady expanding products flow controlledby the rear boundary. The particle acceleration and heating are then characterizedby the single length scale of the particle diameter, similar to the small particlelimit (dp/LR � 1), whereas the Heaviside VN shock is replaced with the CJ shockand 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 thereforemost complex because both the particle diameter and reaction-zone length scalesplay a role in the particle acceleration and heating. In this regime, the particleinteracts with both the VN shock front and the expanding reacting flow in thedetonation reaction zone that terminates at a sonic point. Locally, the reactionzone is affected by the particle presence, resulting in a decreased reaction-zonelength at hot spots and an increased reaction-zone length in the expansion flowaround the particle.

(d) Definition of transmission factors

In order to describe the effect of the acceleration of solid particles in shock anddetonation of a condensed explosive, a velocity transmission factor, a, is definedas 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 perfectreflection off a rigid body and 1 for perfect transmission into a particlewith the same material properties as those of the host fluid. Similarly, atemperature transmission factor, b, is defined as the ratio of the particle mass-averaged temperature, Tp, after an interaction time, over the shocked fluidtemperature, 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 particlein condensed matter. For a dense solid particles-fluid system, transmissionfactors are measured after t = 2tS such that the immediate effect of wavereflections both within particles and in the voids between neighbouring particlesis included. This comprises the majority of the acceleration and heating owing toprimary shock transmission, while subsequent internal waves further influencethe final velocity and temperature achieved during shock compression. Thistime frame also accounts for the influence of transverse wave reflections andupstream/downstream particle reflections in a densely packed matrix.





3. Factors affecting particle acceleration and heating

(a) Dimensional analysis

The force acting on a particle during detonation of a liquid explosive containingdense solid particles is assumed to be a function of relevant dimensionalparameters, e.g. Fd = f (dp, D0, rf0, rs0, fs0, p0, mf0, LR), where dp is the particlediameter, D0 is the detonation velocity, rf0 is the initial density of the explosivefluid, rs0 is the initial particle material density, fs0 is the initial solid particlevolume fraction, p0 is the ambient pressure, mf0 is the molecular viscosity ofthe explosive and LR is the detonation reaction-zone length. The Buckingham ptheorem (Buckingham 1914) considers primary dimensions (length, mass, timeand temperature) to reduce the number of parameters describing a physicalbehaviour into a minimum set of dimensionless groups. Assuming dp, D0and rf0 are independent variables among the nine parameters in the forceexpression, p theorem yields that there are six dimensionless groups, which areas follows:

P1 = Fd

rf0D20d2

p= Cd, P2 = rf0D0dp

mf0= Re, P3 = rf0D2

0

p0= gM 2

0 ,

P4 = rf0

rs0, P5 = dp

LR= d and P6 = fs0.

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(3.1)

In (3.1), Cd is called an ‘effective’ drag coefficient, Re is the particle Reynoldsnumber and M0 is the Mach number of the detonation shock (M0 = D0/af0, whereaf0 is the sound speed). Note that the flow compressibility is represented by theshock Mach number, instead of the flow Mach number commonly used. Thus,the force expression can be rewritten in a non-dimensional form:

Cd = f(

Re, M0,rf0

rs0, fs0, d

). (3.2)

Similarly, heat transfer to a particle during detonation in a solid particle-explosive 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 theexplosive and cp is the explosive fluid heat capacity. Using dp, D0, rf0 and T0as the independent variables, the p theorem gives seven dimensionless groups:

P1 = rf0D0dp

mf0= Re, P2 = D2

0

cpT0= (g − 1)M 2

0 ,

P3 = rf0D30dp

kf0T0= PrReM 2

0 (g − 1), P4 = Qc

rf0D30d2

p= Nu

PrReM 20 (g − 1)

,

P5 = rf0

rs0, P6 = dp

LR= d and P7 = fs0.

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(3.3)

In (3.3), Nu is an ‘effective’ Nusselt number (Nu = hdp/kf , where h is theconvective heat transfer coefficient) and Pr = mfcp/kf is the Prandtl number.





Finally, the non-dimensional form of the heat transfer equation can beexpressed by

Nu = f(

Re, Pr , M0,rf0

rs0, fs0, d

). (3.4)

The same non-dimensional parameters can also be obtained from the mass,momentum and energy conservation equations that govern the flow. Definingdimensionless variables x∗

i = xi/dp, t∗ = t/tS = tD0/dp, u∗i = 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

vt∗ + vr∗f u

∗f,i

vx∗i

− [fs0](

vr∗f f∗

s

vt∗ + vr∗f f∗

su∗f,i

vx∗i

)+

[fs0

rs0

rf0

] (vr∗

s f∗s

vt∗ + vr∗s f∗

su∗s,i

vx∗i

)= 0,

(3.5)which shows the non-dimensional parameters of fs0 and rf0/rs0.

Let additional dimensionless variables be defined as p∗ = p/rf0D20 , s∗ =

sdp/mf0D0, m∗ = m/mf0, F ∗p = Fpd3

p/F0, E∗ = E/D20 , Q∗

R = QR/D20 , q∗ = qdp/kf0T0,

where F0 is a reference force, E is the total specific energy and QR representsa chemical reaction energy source. Substituting all the dimensionless variablesinto the three-dimensional Navier–Stokes equations for the fluid phase of theparticles-explosive system, one derives the linear momentum equations and energyequation of the fluid phase in dimensionless form. For example, the energyequation becomes

vr∗f E

∗f

vt∗ + vr∗f u

∗f,iE

∗f

vx∗i

+[

1gM 2

0

] (vp∗

f u∗f,i

vx∗i

)− [fs0]

(vr∗

f f∗sE

∗f

vt∗ + vr∗f f∗

su∗f,iE

∗f

vx∗i

)

−[

fs0

gM 20

] (vp∗

f f∗su

∗f,i

vx∗i

)−

[1

Re

] (vu∗

f,jsij∗

vx∗i

)−

[1

PrReM 20 (g − 1)

] (vq∗

i

vx∗i

)

= [Cd](F ∗p,iu

∗p,i) +

[Nu

PrReM 20 (g − 1)

]Q∗

c +[

dp

LR

]r∗

f u∗Q∗R. (3.6)

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 momentumequation, which has been omitted for brevity.

For dilute particles-gas flow (fs0 � 1; rf0/rs0 � 1) with a small particle-limitingscale (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 fluidsystems shows the additional parameters, including the density ratio of fluid tosolid particle, solid volume fraction and ratio of particle diameter to detonationreaction-zone length.

(b) Discussion of the parameters

This dimensional analysis is aimed at studying the particle accelerationand heating during detonation shock compression in a dense solid particle-condensed explosive system. There are two major forces acting on a solidparticle during the detonation process: shock compression and viscous drag, with





corresponding time scales characterized by the shock interaction time, tS, andvelocity relaxation time, tV (given in Rudinger (1980)), respectively. The ratio ofshock interaction time to viscous relaxation time is

tS

tV= 3mfCdRe

4dprsD0= 3

4

(rf

rs

) (uf

D0

)Cd, (3.7)

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 shockinteraction 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 ofspheres in compressible flow). Thus, shock compression is the dominant forcefor the particle acceleration during the detonation process and the viscous dragcan be assumed to be negligible. Consequently, the Reynolds number can beconsidered less important in the effective drag equation (3.2) and the flow can beassumed to be inviscid.

Similarly for heat transfer, the ratio of shock interaction time to thermalrelaxation (convection) time, tT (given in Rudinger (1980)), is

tS

tT= 6kfNu

dprscsD0= 6Nu

Re

(cp

cs

) (rf

rs

) (uf

D0

). (3.8)

For detonation of the inert aluminium particles-liquid NM system, the thermalrelaxation time is two orders of magnitude greater than the shock interactionfor dp = 0.1 mm and at least three orders of magnitude greater than the shockinteraction time for dp > 1 mm. Therefore, shock compression heating controlsthe particle temperature during the detonation process and convective heattransfer can be assumed negligible. Consequently, the Reynolds number andPrandtl number can therefore be considered less important in the Nusselt numberequation (3.4), and the flow can be assumed to be completely inviscid andnon-heat-conducting.

The detonation shock Mach number of solid and liquid explosives at theirtheoretical maximum density has a narrow range of 2.5 < M0 < 4 in general.Shock velocity and pressure effects were investigated in a previous work (Zhanget al. 2003) by varying the inert shock pressure from 5 to 20 GPa. While theresulting momentum transfer to particles remained proportional to the shockedfluid velocity, the variation in velocity transmission after the shock interactionwas less than 10 per cent for a shock velocity range of 4–9 mm ms−1 for metalparticles 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 solidRDX 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 tostrongly depend on the initial density ratio of explosive to metal and can beexpressed by

a = 1a + b

(a + b

rf0

rs0

)rf0

rs0, (3.9)

where a and b are constants independent of the particle and explosive matter.





The volume fraction of solid particles is defined by fs = npmp/rs, where npis 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 powderstypically have fs0 ≈ 0.5; random packing in particle beds produces higher volumefractions, for instance, fs0 = 0.58–0.62 in slurries containing Al particles saturatedin liquid NM (Haskins et al. 2002; Frost et al. 2005). For ordered packing ofmono-sized spheres, fs0 = 0.74 is a theoretical geometrical maximum for a close-packed matrix. The particle acceleration and heating depends on the solid volumefraction 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 reaction-zone length, LR, which is an additional length scale that influences the particleacceleration 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 detonationinteraction (figure 1). The effect of the volume fraction of solid particles andthe ratio of particle diameter to detonation reaction-zone length is investigatedin this paper using a mesoscale modelling approach.

4. Mesoscale computational approach

A system of spherical Al particles in NM forms the prototype heterogeneousmixture for the study of detonation interaction with particles. Nitromethane(CH3NO2) is a uniform, low-viscosity liquid explosive and is therefore assumed tofollow ZND detonation theory. The incident shock pressures encountered in theNM 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 thatonly volumetric strain occurs in the particles. Hence, neglecting viscosity andthermal conductivity as justified earlier, the hydrodynamic response of both liquidNM and solid aluminium particles can be computed using the three-dimensionalinviscid Euler equations.

The governing equations are solved using the Harten, Lax, van Leer withcontact correction (HLLC) approximate Riemann solver scheme (Batten et al.1997). The three materials are tracked using mass fraction continuity equationsfor the liquid NM explosive, NM gaseous detonation products and solid Alparticles treated as inert, with equations of state for each described below. Themixture of unreacted explosive and detonation products within the detonationzone and the material boundaries at the metal particle surface are treated usingcontinuum mixture methods following Benson (1992).

(a) Equations of state

The NM and the aluminium particles are modelled using the Mie-Grüneisenequation of state (EOS):

p = GS

n(e − eH) + pH, T = (e − eH)

cv+ TH (4.1)





Table 1. Material and shock Hugoniot parameters from Mader (1998).

parameter nitromethane aluminium

r0 (g cm−1) 1.128 2.785C (mm ms−1) 1.647 5.350S 1.637 1.350GS 0.6805 1.7cv (kJ kg−1 K−1) 1.7334 0.9205FS 5.41 −14.24GS −2.73 −95.75HS −3.22 −155.2IS −3.91 −102.9JS 2.39 −23.53

which is an expansion approximation around the shock Hugoniot. In (4.1), n is thespecific volume, e is the specific internal energy and GS is the so-called Grüneisengamma, which is the negative of the log slope along an isentrope. The Hugoniotstates (subscript H) are determined using the pressure relation:

pH =(

Cn0 − S(n0 − n)

)2

(n0 − n), (4.2)

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 ofWalsh & 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 Hugoniotparameters used for NM and aluminium are summarized in table 1, which wereselected 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):

p = A exp(−R1n

n0

)+ B exp

(−R2n

n0

)+ C

(n

n0

)−u−1

. (4.4)

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 thermochemicalcode and Becker–Kistiakowsky–Wilson–Sandia library (Fried et al. 1998).

(b) Nitromethane reaction model

The NM detonation model follows Mader (1998), who simulated reaction-zonelengths ranging from 0.24 to 70.5 mm. For this work, the length of the reactionzone has arbitrarily been fixed and the particle diameter is varied to study a





x/dp

pr

essu

re (

GP

a)

50 100 150 2000

5

10

15

20

25

Figure 2. Detonation wave pressure profiles over a running distance of 200dp, where dp = 1 mm, onan 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 bedconsisting of 10 layers of metal particles.

range of d = dp/LR. A single-step Arrhenius reaction model was employed for theNM detonation,

uf = rfYF Z exp(−Ea

RTf

), (4.5)

with Z = 8.0 × 1012 s−1 and Ea = 1.672 × 105 J mol−1 K−1. A heat of detonationof DHdet = 5.725 kJ cm−3 was determined using the Cheetah chemical equilibriumcode (Fried et al. 1998). The resulting reaction-zone length is 2 mm when measuredfrom the VN spike to the sonic point (location where D = uf − af ). In thismodel, the reaction is 99.9 per cent complete at the CJ point. A comparisonof the numerical CJ point with the Cheetah equilibrium results shows reasonableagreement. Figure 2 shows the resulting detonation wave pressure profiles for aclosed boundary condition at x/dp = 0. Other more sophisticated kinetics schemesand modern equations of state are established in the literature, e.g. Lysne &Hardesty (1973), Hardesty (1976) and Winey et al. (2000); however, the presentapproach is still adequate to study the mechanical and thermal interaction ofparticles with shock and detonation flow.

Prior to the detonation wave entering a packed particle matrix, the detonationis run out to a distance of 1 mm (five times the running distance depicted infigure 2). This was performed to reduce the gas expansion rate in the Taylorwave during the interaction with the metal particles. Numerical simulation of theNM 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 forLR = 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-dimensionalmeshes of the same resolution containing the particle bed. For dp = 1 mm, theparticles are resolved with 100 cells per dp, which is sufficient to provide agrid-independent solution.

(c) Initial particle packing configurations

Realistic heterogeneous explosive mixtures feature random packing and non-spherical particles. To fundamentally address the acceleration and heating





(a) (b) (c)

x

y

z x

y

z x

y

z

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 sphericalparticles are considered for various inter-particle spacings, covering a range ofvolume fractions from 0.093 ≤ fs0 ≤ 0.740. The regular spacing and monodispersesize assumptions avoid dense particle–particle collisions because all particlesare subjected to the same acceleration processes. Three idealized packingconfigurations were considered as illustrated in figure 3; however, the face-centredcubic lattice arrangement was selected for most of the calculations because theclosest packing represents the saturation limit, whereas the solid volume fractioncan be varied by adjusting the spacing between particles. Candidate particlesare studied six to eight layers into the matrix, where the interaction has becomequasi-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-symmetricmodel of a single spherical particle, whereas the dense limit (fs0 → 1) is simulatedusing a one-dimensional model of a semi-infinite solid slab. In three-dimensions,plane-symmetric domains containing 20–40 particles were resolved using up to32 million mesh points and 100 central processing units in parallel. Reflectiveboundary conditions were used to represent a periodic solution in a semi-infinitearray 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 particlematrix (dp = 10 mm, dp/LR → 0) resulting from a 10.1 GPa inert Heaviside stepshock travelling from left to right. The deformed particles resemble a saddle shapeand, owing to the inviscid hydrodynamics, are strongly influenced by the complexshock reflections from neighbouring particles. The deformation of the first layeris different because the reflected shock wave is not subsequently re-reflected fromupstream particles. The severe deformation in the rear flow indicates that thealuminium particles will likely be damaged or fragmented if material shear stressand failure are considered.





1400T(K)

T(K)

view A-A(a)

(b)view B-B

1300120011001000900800700600500400

14001300120011001000900800700600500400

Figure 4. Inert shock interaction in a close-packed matrix of aluminium particles: resultingdeformation 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 eightlayers of packed particles. The pressure reaches a peak at the perfect reflectionvalue of 45.4 GPa and rapidly expands to a sustained quasi-steady pressureplateau centred below the one-dimensional wave transmission value of 20.9 GPa.The reverberating pressure oscillates with a period of 0.5tS corresponding tothe wave transit time within the interstitial pores contained between packedparticles. These resonant fluctuations would be dampened with random packingand non-spherical particles as shown by Baer (2002).

Figure 5b shows the mass-averaged particle velocity for three packingconfigurations. Within 1tS, the velocity in all three matrices exceeds the one-dimensional wave transmission value of 1.11 mm ms−1 owing to the internalrarefaction as the wave exits the trailing edge of the particle. Subsequent shockreflection and interaction with neighbouring particles causes velocity fluctuationsproportional to the particle spacing that vary with packing configuration. Atime scale of 2tS is sufficient to capture one full interaction cycle that includesthe internal wave reverberation and successive expansions and compressionsfrom upstream, downstream and neighbouring particles. Because severe particledeformation and resonant fluctuations occur after the shock interaction timescale in the rear flow, they do not influence the evaluation of the velocity andtemperature transmission factors.

Figure 6 provides a summary of the velocity and temperature transmissionresults for the small particle limit, using a 10.1 GPa inert shock. Comparisonof the packing configurations shows that the transmission factors for the simple





pres

sure

(G

Pa)

0 1 2 3 40

10

20

30

40

50

60(a) (b)

t/tSt/tS

part

icle

vel

ocity

(m

m μ

s–1)

–2 0 1 2 3 4 5

0

0.2

0.4

0.8

1.0

1.2

1.4

0.6

–1

Figure 5. Results in the small particle limit: (a) particle leading edge pressure histories for a close-packed matrix (fs0 = 0.74); (b) mass-averaged particle velocity for various matrices of packedspherical particles (face centred: dashed-dotted line f = 0.74; solid line, f = 0.68; dashed line,f = 0.52).

volume fraction, fs0 volume fraction, fs0

a

0 0.2 0.4 0.6 0.8

(a) (b)

1.0 0 0.2 0.4 0.6 0.8 1.00.5

0.6

0.7

0.8

0.9

1.0 close packed spheresbody-centred spheressimple cubic spheressolid slabsingle spherecubic fittwo cylinders11 cylinders

close packed spheresbody-centred spheressimple cubic spheressolid slabsingle spherecubic fit

b

0.4

0.3

0.5

0.6

0.7

0.8

0.9

1.0

Figure 6. Velocity and temperature transmission factors for a 10.1 GPa inert shock interaction withpacked particle matrices. Cylinder results (solid symbols) are from Zhang et al. (2003).

cubic packing were considerably different. This is primarily due to this particularpacking matrix that provides two propagation channels: one through the lineararray of stacked particles, and the other uninhibited though the column of liquidin the void space. The simple cubic packing is an unrealistic configuration inpractice and is not given further consideration. The present three-dimensionalresults are compared with two-dimensional cylindrical results of Zhang et al.(2003). The agreement is improved for a higher number of two-dimensionalcylinders, as this better approximates an infinite array of particles as in thethree-dimensional configuration.





Both the velocity and temperature transmission factors for the close-packedand body-centred matrices follow an inverted U-shaped function. The maximumtransmission factors occurred for 0.25 ≤ fs0 ≤ 0.40, whereas the minimumtransmission factors resulted at the dilute limit (fs0 = 0) and high volumefraction limit (fs0 = 1). In between the volume fraction limits, the spacing of theparticles affects the arrival time and magnitude of reflected shocks from upstreamand downstream particles, in addition to affecting the local flow diffraction.The maximum transmission factors are a result of superposition of transmittedshock 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 thedetonation wave over a single aluminium particle indicated that the particlevelocity is first increased as the detonation front crosses the particle and is thenreduced, subject to the Taylor expansion flow conditions. The Taylor expansionafter the thin detonation front reduces the detonation flow velocity, thus changingthe direction of the drag and reversing the momentum transfer direction fromthe particles to the detonation expansion flow. The large particle limit wasstudied preliminarily using dp = 30 mm (Ripley et al. 2006), which correspondsto dp/LR = 15. The unsteady Taylor expansion effect was minimized by runningthe detonation sufficiently far from the initiation location.

For dp/LR � 1, the VN shock can be neglected and the detonation interactionwith particle matrices can be better represented by a CJ shock. The jump in thehost liquid does not follow the Hugoniot because the shock is reactive, where thechemical reaction rate is essentially infinite and the post-shock flow contains hotexpansion 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 aNM–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 atsuccessive particle leading edges; this leads to a flattening of the particles duringdeformation.

The particle velocity and temperature histories are shown in figure 8. Within2tS, both the incident shock and internal rarefaction accelerate the particle, whilean increase in particle temperature owing to shock compression is followed by adecrease from the rarefaction expansion before lateral compression continues theparticle heating. Figure 8a illustrates a decrease in particle velocity over 3–4 tSas a result of a reflected wave returning from the downstream particles. Furtherparticle acceleration must be influenced by viscous drag, which is beyond thescope of the present paper. Similarly, the particle heating is only affected by shockcompression in the non-heat-conducting assumption valid within the time frameconsidered, 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 three-dimensional mesh showed that the shock is both transmitted into the metal atthe particle forward stagnation point, and also reflected back into the reactionzone, thereby increasing the reaction rate. A Mach stem forms as the shock in theexplosive diffracts over the particle. Depending on the impedance ratio, the shock





3200

r (kg m–3)

30002800

x

y

z

260024002200200018001600140012001000

(a)

r (kg m–3)

x

y

z3200

(b)

30002800260024002200200018001600140012001000

Figure 7. Fluid and particle density distribution and particle deformation for Chapman–Jouguetshock propagation through a particle matrix: (a) fs0 = 0.520 and (b) fs0 = 0.093.

t/tS t/tS

part

icle

vel

ocity

(m

m μ

s–1)

–2 –1 0 1 2 3 4 5 6 –2 –1 0 1 2 3 4 5 6–0.5

0

0.5

1.0

1.5

2.0(a) (b)

part

icle

tem

pera

ture

(K

)

0

800

1000

600

400

200

Figure 8. (a) Mass-averaged particle velocity and (b) temperature for detonation interaction at thelarge particle limit (fs0 = 0.093).





30pressure (GPa)

2826242220181614121086420

x

y

z

Figure 9. Pressure distribution and particle deformation for detonation (dp/LR = 0.5) through apacked particle matrix (fs0 = 0.428).

inside the metal particle may travel ahead of the incident shock, transmittingan oblique shock into fresh explosive ahead of the detonation front. Behind thediffracting detonation shock, expansion of the flow on the backside of the particlecauses the local reaction-zone length to increase. At the particle trailing edge,either the diffracted shock arrives first, or the converging shock inside the particleis retransmitted back into the fresh explosive. This can initiate the explosivelocally ahead of the detonation shock. Subsequently, a strong rarefaction formswithin the metal particle, contributing to both a large acceleration and a decreasein temperature.

For packed particle matrices, the situation is more complex with reverberatingwaves and lateral interactions that further influence the particle and surroundingflow. Figure 9 illustrates the irregular propagation pattern of the detonationfront in an Al–NM matrix (dp = 1 mm, dp/LR = 0.5), where the detonation travelswithin the explosive contained in the voids and narrow channels between particlesin addition to shock propagation within the metal. In close-packed matrices,transmission of the shock through the particle can subsequently pre-compressand initiate detonation in the fresh explosive on the far side of the particle priorto the diffracted shock arrival (hot-spot mechanism). For the conditions depictedin figure 9, the NM behind the particle trailing edge reaches 800 K owing to shocktransmission and prior to reaction.

Figure 10 shows numerical pressure gauge results for detonation in a close-packed matrix, with the peak reflected VN shock pressure followed by oscillationsat a frequency proportional to particle diameter and later by the Taylor waveexpansion. Figure 10a shows a precursor shock with magnitude of about 10 GPa(temperature approx. 1300 K) that resulted from the reflection of the shocktransmitted through the particle trailing edge into the void prior to arrival ofthe diffracted VN shock. Figure 10b shows a smaller precursor shock exiting thetrailing edge and a larger peak pressure owing to collision of the diffracted shocksbehind the particle trailing edge.

Figure 11 shows the particle velocity and temperature histories in a packedparticle matrix for dp/LR = 0.2, where the detonation reaction zone is muchlarger than the particle size. This is evident in the particle velocity history result,





t/tS

pres

sure

(G

Pa)

0 5 10 15 20t/tS

0 5 15

0

20

40

60(a) (b)

10 20

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 behindtrailing edges. Each curve presents successive results from the first, fifth, ninth, 13th and17th layers.

0 5 10 15 0 5 10 15t/tSt/tS

0

part

icle

vel

ocity

(m

m μ

s–1)

–0.5

0

0.5

1.0

1.5

2.0(a) (b)

part

icle

tem

pera

ture

(K

)

1500

1000

500

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 thendecreases mainly over 5tS during the expansion inside the reaction zone. The 2tSassumption is especially justified in the temperature history where the first peakis reached between 1 and 2tS.

Figure 12a shows the detonation shock velocity through the particle matricesimmersed in liquid explosive. The bulk propagation velocity was measured usingwave time of arrival between consecutive numerical gauge stations located in anarray perpendicular to the averaged detonation wavefront. Under the detonationconditions studied here, i.e. aluminium particles in NM, shock waves reflected





x/dp

shoc

k ve

loci

ty (

mm

μs–1

)

0 5 10 156.0

6.2

6.4

6.6

6.8

7.0

7.2

7.4(a) (b)

fs0=0fs0=0.093

fs0=0.740

mass fraction (%)0 20 40 60 80 100

5.6

5.8

6.0

6.2

6.4

6.6

6.8

7.0cheetah (inert Al)mesoscale

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 distancemeasured from the first layer; (b) quasi-steady propagation velocity for various aluminiummass fractions.

from the leading edge of the particles cause an increase in the local NMdensity and pressure, thereby increasing the detonation velocity. Furthermore,shock waves travelling within the metal particle are transmitted into the freshliquid explosive ahead of the detonation wave diffracting around the curvedparticle surface, thereby pre-compressing the explosive and increasing the localdetonation velocity. These two local hot-spot factors both contribute to bulkpropagation speeds in excess of the CJ value in fresh NM. In the present numericalcalculations, the greatest bulk propagation velocities (up to 7.4 mm ms−1) wereobserved for the highest metal mass fraction condition in combination withthe smallest particle diameter (or longest reaction zone) within the region of thefirst particle layer. Afterwards, the momentum and energy transferred into theparticles within the detonation zone competes with the local hot-spot factors,resulting in a quasi-steady propagation velocity with a deficit. The velocitydeficit increases with an increase in solid volume fraction and a decrease inparticle diameter. The maximum velocity deficit corresponded to a wave speedof 5.3 mm ms−1 and occurred for the small particle limit.

Figure 12a shows increasing instability in the detonation front velocity forhigher solid volume fractions, which is an expected feature due to highermomentum and heat loss. The initially transient shock velocity becomes quasi-steady after travelling a distance of 6dp into the matrix. Figure 12b illustratesthe quasi-steady detonation shock velocity as a function of increasing metal massfraction. In comparison with Cheetah chemical equilibrium predictions using inertaluminium, where velocity and temperature equilibrium is assumed for all phases(Fried et al. 1998) and essentially d → 0, the mesoscale results for detonationshock velocity are consistently higher because the relative velocity of the solidphase remains below the flow velocity following the shock interaction. Similarly,the Cheetah equilibrium temperature is also lower than the mesoscale detonationresults owing to the same phase-non-equilibrium nature.





t/tS

part

icle

vel

ocity

, uP

(mm

μs–1

)

0 2 4 6

0

0.5

1.0

1.5

2.0

(a) (b)

d = 0.1d = 0.2d = 0.5d = 1d = 2d = 10

d

a

10–3 10–2 0.1 1 10 102 103

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

fitting functionmesoscale

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 andfitting function.

(d) Particle acceleration

The mass-averaged particle velocity history achieved during detonationinteraction with a single particle is illustrated in figure 13a. Because a fixednumerical mesh resolution was maintained (100 cells mm−1), such that the numberof cells in the ideal detonation reaction zone (LR = 2 mm) was unchanged,the number of cells across the particle diameter increased with d and thecorresponding computational effort increased exponentially. Thus for a large d,limited duration particle velocity histories are available. For d < 0.5, the peakparticle velocity exceeds the CJ value, illustrating a significant influence of the VNspike on the particle acceleration. For t/tS � 1, the particle velocity equilibratesbelow 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 andcorresponding transmission factor as a function of d = dp/LR. Both the velocityand the velocity transmission factor decrease from VN to CJ mainly over theinterval from 0.1 ≤ d ≤ 1. The single particle acceleration results are bound by thesmall particle limit (d → 0) and the large particle limit (d → ∞). The resultingvelocity 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 andaVN = a(d → 0) = 0.669.

Figure 14a illustrates the particle velocity transmission in an Al–NM matrixas a function of volume fraction. Close-packed spheres are used, and the volumefraction is adjusted by changing the inter-particle spacing. For the various dconsidered, there is a twofold range in particle velocity between the interactionlimits. There is weak similarity in the results for various d, which are boundby limiting cases of d → 0 for VN shock interaction and d → ∞ for CJ shocked





volume fraction, fs0 volume fraction, fs0

part

icle

vel

ocity

, uP

(mm

μs–1

)

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.00

0.5

(a) (b)

1.0

1.5

2.0

2.5

d →0 (VN shock)d = 0.2d = 0.5d = 15d → ∞(CJ shock)


a

0

0.6

0.8

fitting function

0.4

0.2

Figure 14. Shock compression acceleration results in a particle matrix: (a) range oftransmitted particle velocity between interaction limits; (b) multi-variable fitting for velocitytransmission factor.

flow conditions. The maximum a generally occurs at fs0 = 0.2, above which thetransmission decreases linearly for increasing fs0. The regime of primary interestfor 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 fractionand detonation reaction-zone length: a = a(fs0, d). The volume fraction effectwas represented by a second-order polynomial function of fs0. The reaction-zone-length influence is exhibited as a shift in the velocity transmission factor, whichwas represented using an exponential function of d and a fixed offset. The resultingfunction contains five fitting constants, as follows:

a = c1f2s0 + c2fs0 + c3 exp(−c4d) + c5, (5.2)

where c1 = 0.2, c2 = 0.1, c3 = 0.4, c4 = 3.3 and c5 = 0.36. A comparison of thefitting function with the mesoscale results is given in figure 14b. The agreementis reasonable for d < 1; the largest differences occurred for high volume fractionsin the large particle limit.

(e) Particle heating

Figure 15a shows the mass-averaged shock compression temperature for asingle particle for various d. Smaller particles achieve higher temperatures,although the maximum temperature during shock interaction is much lessthan the CJ flow temperature (TCJ = 3657 K). Oscillations in the temperaturehistories are caused by successive compressions and expansions owing to wavereverberation inside the particle.

Figure 15b shows the mesoscale results of the single particle temperature andcorresponding transmission factors, b, which are monotonic decreasing functionsfor increasing d = dp/LR. Both the temperature and the temperature transmissionfactor decrease from the VN to CJ limiting values mainly over the interval





t/tS

part

icle

tem

pera

ture

, TP

(K)

0 2 4 60

200

400

600

800

1000(a) (b)

d = 0.1d = 0.2d = 0.5d = 1d = 2d = 10

b

0

0.2

0.1

0.3

0.4

fitting functionmesoscale

10–3 10–2 0.1 1d

10 102 103

Figure 15. Shock compression heating of a single particle (fs0 = 0) in detonation flow: (a) mass-averaged temperature of single particles of various diameters; (b) temperature transmission factorswith fitting function.

from 0.1 ≤ d ≤ 10. Similar to the velocity transmission factor, the temperaturetransmission 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 fluidtemperature. Similar to particle acceleration, the particle heating is boundbetween the small particle and large particle-limiting cases, and the effect ofsolid volume fraction on b is reduced for larger particles (d → ∞). For a givenvolume fraction, b increases with the reaction-zone length (decreasing d). Forvolume 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 ofmagnitude for the ratio of dp/LR.

6. Application to macroscopic models

Having gained an understanding of the particle interactions with the explosivefluid and neighbouring particles during shock and detonation in an idealized denseor packed particle-condensed matter system, let us try to apply the mesoscaleresults to formulate macroscopic momentum and energy transfer functions. Thesefunctions can then be incorporated into interphase exchange source terms thatare used during the shock compression time scale. The resulting approach canbe applied to macroscopic continuum modelling of practical problems suchas detonation in a multi-phase explosive or shock propagation in a denseparticle-fluid system.





volume fraction, fs0

part

icle

tem

pera

ture

, TP

(K)

0 0.2 0.4 1.0

0

500

1000

1500(a) (b)

b

0

0.4

0.6

0.6 0.8volume fraction, fs0

0 0.2 0.4 1.00.6 0.8

0.2



Figure 16. Shock compression heating results in a particle matrix: (a) range of transmitted particletemperature 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 particle-condensed explosive control volume containing np particles each with mass, mp,can be approximated by

Fp = npmpdup

dt≈ npmp

auf1 − up0

t(6.1)

and

Qp = npmpcsdTp

dt≈ npmpcs

bTf1 − Tp0

t. (6.2)

Because the mesoscale results showed that the increase in mass-averagedparticle velocity was mostly linear within the interaction time scale, a simpleacceleration 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 thestandard definition for drag force on spherical particles,

Fp = pd2p

8nprf |uf − up|(uf − up)Cd, (6.3)

with (6.1) and assuming up0 = 0 and uf ≈ uf1, an effective drag coefficient isobtained for the shock compression interaction:

Cd(t) = 4rsD0

3rfuf1

a

(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 assumedconstant within the interaction time, and the particle temperature cantherefore be approximated by a linear function: Tp(t) = Tp0 + (Tp1 − Tp0)





(t − t0)/t. Combining the standard definition for convective heat transfer onspherical particles,

Qp = pdpnpkf (Tf − Tp)Nu, (6.5)with (6.2) and assuming Tf ≈ Tf1, an effective Nusselt number is obtained for theshock compression interaction:

Nu(t) = dpcsrsD0

6kf

bTf1 − Tp0

(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-phasemacroscopic models of dense particles such as Baer & Nunziato (1986).

7. Conclusions

Dimensional analysis showed that the particle acceleration force and heat transferduring detonation shock compression in a dense solid particle-condensed explosivesystem 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 detonationreaction-zone length, d = dp/LR, apart from Reynolds number, Re, Prandtlnumber, Pr and Mach number, M0. While viscosity and heat conduction areimportant in later time, they can be neglected when compared with the otherparameters during the shock compression time scale. Thus, the accelerationforce 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 Nusseltnumber, Nu = f (rf0/rs0, fs0, d, M0). Mesoscale simulations of spherical aluminiumparticles immersed in liquid NM were conducted by varying fs0 and d at agiven M0 and rf0/rs0, which were known to be important parameters. The fullrange of fs0 and d was studied, where d ranged between the small particlelimit, with essentially inert shock interaction, to the large particle limit, withinfinitely thin detonation front interaction followed by detonation productsexpansion flow.

Features of heterogeneous detonation were explored including: detonationinstability and velocity deficit; pressure front fluctuations with peaks up tofour times the CJ detonation pressure and periods proportional to the particlesize; and, transverse waves and hot spot characteristics of locally enhancedpressure and temperature fronts. These physics are consistent with macroscopicphenomena observed in published experiments. Detonation failure was notconsidered in this study because the mesoscale calculations were infinite diameter(no charge edge effects).

From the mesoscale simulation, a shock compression velocity transmissionfactor, 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 heatingbehaviour within the detonation shock interaction time. The maximum particleacceleration occurred at fs0 = 0.25, whereas the maximum shock compressionheating occurred over a wider range of solid volume fraction fs0 = 0.43–0.74.





Scaling of the transmitted velocity and temperature using the von Neumann stateis convenient because it is easily obtained from analytical shock relationships;however, this scaling showed a strong dependence on the ratio of particle diameterto reaction-zone length.

Overall, velocity and temperature transmission factors obtained usingmesoscale calculations can be tabulated and fit to simple functions of densityratio, volume fraction and the ratio of particle diameter to detonation reaction-zone length, for a given Mach number. Fitting in this fashion allows practicalmodels to be developed for engineering calculations. As an example, theresults were applied to formulate functions for macroscopic momentum andenergy transfer between the two phases during detonation shock compression.These functions can then be used as the interphase exchange source termsapplied to macroscopic continuum modelling of practical problems such asdetonation of a multi-phase explosive or shock propagation in a denseparticle-fluid system.

References

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