A reactive flow model for heavily aluminized cyclotrimethylene-trinitramineBohoon Kim, Jungsu Park, Kyung-Cheol Lee, and Jack J. Yoh

Citation: Journal of Applied Physics 116, 023512 (2014); doi: 10.1063/1.4887811 View online: http://dx.doi.org/10.1063/1.4887811 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/116/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Hot-spot contributions in shocked high explosives from mesoscale ignition models J. Appl. Phys. 113, 233513 (2013); 10.1063/1.4811233 One-dimensional plate impact experiments on the cyclotetramethylene tetranitramine (HMX) based explosiveEDC32 J. Appl. Phys. 112, 064910 (2012); 10.1063/1.4752865 Sparse Partial Equilibrium Tables in Chemically Resolved Reactive Flow AIP Conf. Proc. 706, 1061 (2004); 10.1063/1.1780421 A Reaction Zone Enthalpy Balance Model to Simulate ShocktoDetonation Transition and Unsteady DetonationWave Propagation AIP Conf. Proc. 706, 943 (2004); 10.1063/1.1780392 Shock loading and reactive flow modeling studies of void induced AP/AL/HTPB propellant AIP Conf. Proc. 429, 373 (1998); 10.1063/1.55692

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A reactive flow model for heavily aluminized cyclotrimethylene-trinitramine

Bohoon Kim,1 Jungsu Park,2 Kyung-Cheol Lee,1 and Jack J. Yoh1,a)

1Department of Mechanical and Aerospace Engineering, Seoul National University, Seoul 151-742,South Korea2Agency for Defense Development, Daejeon 305-600, South Korea

(Received 26 April 2014; accepted 27 June 2014; published online 11 July 2014)

An accurate and reliable prediction of reactive flow is a challenging task when characterizing an

energetic material subjected to an external shock impact as the detonation transition time is on the

order of a micro second. The present study aims at investigating the size effect behavior of a

heavily aluminized cyclotrimethylene-trinitramine (RDX) which contains 35% of aluminum by

using a detonation rate model that includes ignition and growth mechanisms for shock initiation

and subsequent detonation. A series of unconfined rate stick tests and two-dimensional

hydrodynamic simulations are conducted to construct the size effect curve which represents the

relationship between detonation velocity and inverse radius of the charge. A pressure chamber test

is conducted to further validate the reactive flow model for predicting the response of a heavily

aluminized high explosive subjected to an external impact. VC 2014 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4887811]

I. INTRODUCTION

We characterize a multi-purpose high explosive (HE)

which is comprised of 50% RDX (cyclotrimethylene-trinitr-

amine, C3H6N6O6) and 35% aluminum powder with 15%

HTPB (hydroxyl-terminated polybutadiene) binder, and its

initial density after pressing is 1.78 g/cc. To simulate the

shock-to-detonation transition (SDT) that is dominantly

driven by pressure, Lee-Tarver1 and JWLþþ (Jones-

Wilkins-Leeþþ)2 models are used in general. Lee-Tarver

considers generation of hotspots and their effect during ini-

tiation. The model requires predetermination of parameters

both empirically and ad hoc, making it difficult to be univer-

sal for generalized energetic material. The JWLþþ model

offers simplification to Ref. 1 by ignoring the initiation step

and using only the single growth step for a detonating explo-

sive. The methodology to determine model parameters are

somewhat qualitative, and the reactive flow model would

need to be recalibrated each time a new explosive is formu-

lated. Once calibrated for one explosive, it does not manifest

as to how to change the unknown parameters for other

explosives.

Souers and Vitello3 provided a useful approach to deter-

mine the unknown parameters of JWLþþ model. They pro-

vided an analytical approach that allows prediction of the

rate constants in terms of the size effect measurements. The

method is restricted to a growth step only, and an empirical

fit to a diameter effect is required to constrain unknown

parameters, indicating that there are still additional

unknowns to be determined.

The present study aims at developing a modified reac-

tive flow model that includes both ignition and growth steps

while attempting to overcome limitations of aforementioned

JWLþþ model. In addition, the unknown parameters

are handled theoretically to minimize ambiguity of the

numerical iterations. In order to validate the reactive flow

response of a heavily aluminized RDX, two distinct experi-

ments, namely, the unconfined rate stick test and a pressure

chamber test have been carried out.

II. UNCONFINED RATE STICK TESTS

A series of unconfined rate stick tests for heavily alumi-

nized RDX is performed, where the detonation velocities

were measured as a function of cylindrical charge diameter.

The steady detonation wave is observed by utilizing a charge

length sufficiently long enough to damp out any booster

overdrive or initiation transients which can persist for up to

five times the charge diameter. A charge length-to-diameter

ratio of 10 is chosen to adequately capture the fully devel-

oped detonation velocities in all tests.

The samples are prepared by compacting the heavily

aluminized RDX into a precisely machined mono-cast

Nylon. Electronic pins were utilized for measurement of the

propagation velocity of detonation wave. If the installed pin

is exposed to a high temperature-pressure shock wave, the

pin breaks and signal generator transmits a signal. Then a

high speed counter, a Time-to-Digital Converter (TDC)

measures the arrival time of the shock wave, sensing an

ascending (or descending) waveform of input pulses. We

used pin CA-1041 (DYNASEN, Inc.) and a high speed coun-

ter Lec-4208 (LeCroy, Inc.) as the wide range real-time

TDC. The TDC is designed to cover the time measurements

performed in real time, and requiring a wide dynamic range

with high resolution.

A rate stick packed in a cylindrical mono-cast Nylon of

90 mm diameter is shown in Fig. 1. 16 different diameters

used were 13.5 mm, 14.0 mm, 15.0 mm, 16.0 mm, 17.0 mm,

18.0 mm, 20.0 mm, 25.0 mm, 30.0 mm, 35.0 mm, 40.0 mm,

45.0 mm, 50.0 mm, 70.0 mm, 90.0 mm, and 110.0 mm. We

considered a wide range of diameters to secure enough dataa)Email: jjyoh@snu.ac.kr. Tel.: 82-2-880-9334.

0021-8979/2014/116(2)/023512/9/$30.00 VC 2014 AIP Publishing LLC116, 023512-1

JOURNAL OF APPLIED PHYSICS 116, 023512 (2014)

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to establish the size effect behavior of heavily aluminized

RDX.

The propagation of the detonation front along the length

of each charge was measured using five equally spaced

sensor probes. The TDC with a bandwidth of 1 GHz or fre-

quency resolution of 1 ns (10�9 s) was used during each test.

The measured probe data for 30 mm diameter test are shown

on a plot of probe position versus trigger time in Fig. 2. The

time-distance diagram proves that all initiation transients

have in fact dampened and that the self-sustained propaga-

tion velocity is constant by the end of a charge with

L/D¼ 10 because the time dependent slope of the pin posi-

tion is steadily maintained. The probe data minimizing the

influence of initiation transients were used to get a fully

developed detonation velocity.

The chemical kinetic information comes from the size

effect data where the change in steady state or fully devel-

oped detonation velocity, Us, is given in terms of radius, R0.

The detonation velocity is plotted as a function of the inverse

radius, 1/R0, and the curve is extrapolated back to zero to

obtain the infinite radius detonation velocity, D.

III. A REACTIVE FLOW MODEL

The rate of product mass fraction consists of ignition

and growth terms4 such that

dkdt¼ I 1�kð Þb g�að Þx

þG1 1�kð ÞckdPyþG2 1�kð ÞekgPz; g¼ qq0

�1:(1)

Here, k is the burned mass fraction, and constants I, b, a, x,G1, c, d, y, G2, e, g, and z are the unknown parameters. k is

reaction progress variable (k¼ 0 unreacted state and k¼ 1

reacted state). P is the pressure, t is the time, and q0 and q are

the initial and current densities, respectively. The Mie-

Gruneisen equation of state (EOS)5 in Eq. (2) is for the

unreacted solid HE, and the isentropic JWL EOS6 in Eq. (3)

is used for the reacted gaseous product

Punreacted ¼ PH þ Cqðe� eHÞ; (2)

Preacted ¼ Ae�R1ðq0=qÞ þ Be�R2ðq0=qÞ þ Cðq0=qÞ�ð1þxÞ: (3)

Here, PH and eH are pressure and internal energy of a

reference state that follows the Hugoniot curve, and C is

the Gruneisen gamma. PH and eH can be expressed as

follows:

PH ¼ C20

1

q0

� 1

q

� ��1

q0

� S1

q0

� 1

q

� �� �2

; (4)

eH ¼ C20

1

q0

� 1

q

� �2�2

1

q0

� S1

q0

� 1

q

� �� �2

: (5)

C0, S are the bulk sound speed and linear Hugoniot slope

coefficient, respectively. Also the shock speed relations are

S ¼ dUshock=dUparticle; (6)

C0 ¼ ð@P=@qÞ1=2; (7)

Ushock ¼ C0 þ SUparticle; (8)

where Ushock is the shock wave velocity and Uparticle is the

material particle velocity.

A, B, C, R1, and R2 are the material dependent JWL

parameters with x being the Gruneisen coefficient of

Eq. (3). Equations (2) and (3) are combined into a single

expression Eq. (9) using the product mass fraction (k) and

FIG. 1. A rate stick packed in a cylindrical mono-cast Nylon tube.

FIG. 2. Captured position versus trigger time plotted for a 30 mm diameter

case.

023512-2 Kim et al. J. Appl. Phys. 116, 023512 (2014)

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reactant depletion (1� k). The standard mixture rule applies

to the internal energy and volume ratio such as

Ptotal ¼ ð1� kÞPunreacted þ kPreacted; (9)

etotal ¼ ð1� kÞeunreacted þ kereacted; (10)

�total ¼ ð1� kÞ�unreacted þ k�reacted: (11)

The Lee-Tarver model is comprised of (i) ignition term

that represents formation of the hotspots by the rapid compres-

sion, and (ii) first growth term that describes the effect of the

propagation of the reacting waves in the substance and second

growth term that represents completion due to a detonation

transition. For determination of 12 unknowns here, a curve fit-

ting method with optimization technique is usually needed.

If the ignition term involving the initial density ratio is

omitted, a JWLþþ model2 is generally recovered, and its

simple form is

dkdt¼ G 1� kð Þ Pþ Qð Þb: (12)

Here, G is the growth constant, b is the pressure sensitivity,

and Q is the artificial viscosity. The determination of the

unknowns is relatively easier, and thus, the usability of a

simplified form of the rate law is greater if the process is

mostly growth dependent.

An alternative to these two limiting forms of the reactive

flow equation can be casted into a following form such that:

dkdt¼ I 1� kð Þga þ G 1� kð ÞPb; g ¼ q

q0

� 1: (13)

Here, the model resembles a full Lee-Tarver, but it now

includes an initiation step which is missing in JWLþþ. Four

unknown parameters having the major significance in view

of detonation are kept, namely, I, a, G, and b.

The initiation step requires defining the ignition constant,

I, while the reactant depletion (1� k) is time-resolved in the

governing equation with a compression g¼q/q0� 1. The

ignition of high explosive occurs by compression because of

the shock wave propagation. Hotspots are formed in a shocked

high explosive, leading to a void collapse. Any void or gas

bubble that exists in high explosives may provide a potential

site for local adiabatic compression that leads to the localized

heating well beyond the activation energy for detonation.

Wackerle et al.7,8 reported that hotspot formation due to

void collapse depends on the shock pressure, Ps, and the

reaction rate is proportional to P2 which is known experi-

mentally. To investigate the shock-pressure-squared depend-

ence of ignition, compression sensitivity, a, must be set to 4

because the relationship between pressure and density is

parabolic based on the Rankine-Hugoniot argument. Once

again, the pressure squared term is dominant in the shock

induced ignition, and the pressure is proportional to com-

pression squared. As such the compression sensitivity being

4 is the best choice for most of the SDT problems.

Souers and Vitello3 provided a useful approach to deter-

mine the unknown parameters of JWLþþ model. They ana-

lytically linked the growth term with the unconfined rate

stick data. The rate stick test provides detonation velocity

versus the inverse radius of a cylindrical charge. With such

size effect information, the unknowns of the growth term are

determined. That is, the relationship between non-

dimensional detonation velocity and a non-dimensional

inverse radius is derived using a detonation velocity equation

of unconfined rate stick following Eyring et al.9 as:

Us

D¼ 1� ð0:4=½1� Us=Dð Þ2�0:8Þ�1 teUs

R0

; (14)

where Us is the detonation velocity for radius R0, and D is

the detonation velocity at infinite radius. te is the time to

cross the reaction zone during which Eq. (14) is valid.

Now, all k’s are collected on the left of the growth term

in Eq. (13), and the equation is integrated such thatð1

1� kð Þ dk ¼ð

GPbdt ! �ln 1� kð Þ ¼ GPbte: (15)

Finally, Eq. (16) specifies the relationship between

dimensionless detonation velocity and the charge inverse

radius

RD

R0

¼ �0:4 1� Us=Dð Þ2h i�0:8

ln 1� Us=Dð Þ2� � Us

D

� �2b�1

1� Us

D

� �: (16)

The non-dimensional and generalized radius (RD) and

growth constant (G) are given by

RD ¼ DðGPbÞ�1;G ¼ DðRDPbÞ�1: (17)

This means that the Eyring equation can be expressed in

terms of the detonation velocity and the charge size, defined

through a pressure term b such that the size effect can be

derived by changing the value b in Eq. (16).

The curve fitting procedure for a heavily aluminized

RDX is performed to determine the pressure sensitivity term,

b. The optimal size effect curve of the analytical solution is

drawn based on the pressure sensitivity set to b¼ 0.7 with

maximum detonation velocity at zero inverse radius being

7.387 mm/ls, and the generalized radius equals 0.75 mm.

The analytical line which is an optimal fit is characteristi-

cally high for RD/R0< 0.05 and low for 0.05<RD/R0< 0.12.

Now, the Ignition constant is defined by the ratio of igni-

tion to growth, namely,

dkdtI

Ignition

¼ I 1� kð Þga;dkdtG

Growth

¼ G 1� kð ÞPb: (18)

Here, tI and tG are the total time for completion. The dimen-

sionless ignition and growth (IG) takes the form

IG ¼ Iga

GPb� 0:1 or I ffi 0:1GPbg�a; (19)

where the four parameters (I, a, G, and b) decide the intrinsic

characteristics of initiation and detonation. By assigning 0.1

to this number, ignition constant I can be calculated from

Eq. (19), suggesting that 10% of the full reaction time is

023512-3 Kim et al. J. Appl. Phys. 116, 023512 (2014)

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spent for growth. Thus, if IG is zero, the model reduces to a

JWLþþ rate law where the rapid growth step is of only rele-

vance in the reactive flow process.

The constants of ignition I and growth G are set to

3.177� 108 s�1 and 0.7045 s�1 Pa�b, respectively. The pres-

sure sensitivity of the explosive is 0.7, and the compression

sensitivity is 4. Table I shows comparison of the pressure sen-

sitivity values and the literature values. The sensitivity of each

energetic material was derived by the theoretical argument

without any speculations based on numerical trials, which is an

improvement from the earlier reactive flow models.

IV. RATE STICK SIMULATION

The multi-material numerical simulation of shock-

to-detonation transition is conducted with a hydrocode.13 A

level-set based reactive Ghost Fluid Method (GFM) that

imposes exact boundary conditions at material interfaces is

developed to reproduce the experimental size effect data of

the unconfined rate stick tests.

The conservative laws of mass, momentum, and energy

equations in two-dimensional axisymmetric coordinate are

defined by

@ ~U

@tþ @

~E

@zþ @

~F

@r¼ ~S ~Uð Þ; (20)

where the vectors represent the conservative variables U, spa-

tial fluxes in axial and radial directions E and F, and the source

term S that represents a variety of multi-material loading con-

ditions. In the Eulerian form, each vector is given as follows:

~U ¼

q

quz

qur

qE

266664

377775 ~E ¼

quz

qu2z þ p

quruz

uz qEþ pð Þ

266664

377775 ~F

¼

qur

quzur

qu2r þ p

ur qEþ pð Þ

266664

377775~S ¼ �u

1

r

qur

quruz

qu2r

ur qEþ pð Þ

266664

377775; (21)

where u¼ 0, 1 for rectangular and cylindrical coordinates,

respectively. Here, q is the density, and uz and ur are velocity

components in axial and radial coordinates. E ¼ eþðu2

z þ u2r Þ=2 is the total energy per unit mass with the spe-

cific internal energy e, and p is the hydrostatic pressure,

respectively. The governing equations are solved by a third-

order Runge-Kutta in time, and convex ENO (essentially

non-oscillatory) method discretizes the spatial fluxes.14

The rate of burnt mass production is governed by

@ qkið Þ@tþ@ qkiuj

�@xj

¼ wi; (22)

where wi is the reaction rate and ki is the reaction progress

variable or burned mass fraction. The model is closed by

specifying the equation of state, and the form for the rate

law. Both Mie-Gruneisen EOS and JWL EOS are used to

provide closure to the formulation. The parameters related to

these constitutive relations are given in Table II.

The control of the time step increment is determined

automatically, where the step size is given by

DtCFL ¼ CFLmin Dz;Drð Þ

max uþ c; u; u; u� c½ �: (23)

Here, the sound speeds for Mie-Gruneisen EOS (unreacted)

and JWL EOS (reacted) are given by

c2unreacted ¼

1

qq0

1� gCg

PH þ Cq e� eHð Þ�

þ 2S

1þ g 1� Sð Þ PH � CqeHð Þ�; (24)

TABLE I. Comparison of the pressure sensitivity of selective energetic

materials.

Energetic materials

Pressure sensitivity

b - Present b - Others

LX-17 1.7 0.67 (Ref. 10)

ANFO-K1 1.38 1.3 (Ref. 11)

ANFO emulsion with 30% HMX 1.7 1.5 (Ref. 12)

Creamed TNT 1.0 1.75 (Ref. 12)

Urea nitrate (0.69 g/cm3) 1.0 1.0 (Ref. 12)

HANFO 0.95 0.8 (Ref. 3)

TNT 1.0 …

HMX emulsion 1.7 …

Non-ideal AN emulsion k1a 1.45 1.5 (Ref. 3)

Extremely non-ideal AN 0.8 0.8 (Ref. 3)

ANFO prill 1.65 1.5 (Ref. 3)

Potassium chlorate/sugar 1.05 0.8 (Ref. 3)

TABLE II. Equation of state coefficients and rate law parameters for the

heavily aluminized RDX.

Model parameter Value

Reactant C0 (mm/ls) 2.60

S 1.86

C 0.99

Product A (GPa) 2633

B (GPa) 8.59

C (GPa) 1.09

R1 6.68

R2 1.11

w (J/g K) 0.09

Chemical kinetics I (s�1) 3.18� 108

A 4.0

G (s�1 Mbar�b) 3.53� 107

B 0.7

TABLE III. Chapman-Jouguet values of the heavily aluminized RDX.

C-J condition Pcj (GPa) 16.14

qcj (g/cc) 2.2

ecj (kJ/cc) 1.54

Tcj (K) 4410

c 4.24

023512-4 Kim et al. J. Appl. Phys. 116, 023512 (2014)

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c2reacted ¼

q0

q2AR1 exp �R1

q0

q

� ��

þBR2 exp �R2

q0

q

� �þ C

1þ x

q0=qð Þ2þx

#: (25)

The combined sound speed is then calculated by using

c2total ¼ ð1� kÞc2

unreacted þ kc2reacted: (26)

To evaluate the predictability of a modeled SDT, the

unconfined rate stick experiment is conducted. A series of

cylindrical charges with 0.1 m in length and inverse radii

being 0 mm�1 to 0.125 mm�1 is used and shock-initiated by

an impact on one side with 300 m/s. The prescribed impact

was sufficient for pressure rise above the threshold value of

the initiation.

The C-J conditions for heavily aluminized RDX are

given in Table III obtained by using a thermo-chemical

code.15 Most of the aluminum particles do not participate in

the reaction during the detonation phase16 as these particles

in micron size requires a significantly long time to be oxi-

dized. Since the reaction time is ultra-short, these metal par-

ticles do not have enough time to participate in the reaction.

The RDX would still follow its sole reaction path without

the particle effect. For a different particle size distribution,

however, smaller particles are ignited in a relatively shorter

time, thus providing extra heat release of the reaction.

Figure 3 shows computational domain. Two distinct

materials which are void and explosive were initially brought

to a contact by a zero level set represented by a black line in

the figure. Six different radiuses are considered in the simu-

lation, which were carried out on a uniform mesh of size

1/120 mm.

Figure 4 is a plot of the detonation wave structures illus-

trated according to grid resolution. The lines are represented

by dotted (h¼ 0.05 mm), dashed (h¼ 0.0167 mm), and solid

(h¼ 0.0083 mm). As seen from the figure, optimal mesh size

of h¼ 0.0083 mm seems to support convergence of a detona-

tion wave.

The length of the reaction zone of heavily aluminized

RDX is obtained from the detonation pressure profile and the

burned mass fraction in Fig. 5, and it is approximately

4.0 mm. At that time, pZND¼ 35 GPa, pCJ¼ 16 GPa, and

pZND=pCJ¼ 2.1875. For reference, the length of the reaction

zone of a pure RDX is 0.4 mm–0.5 mm.17,18 This suggests

FIG. 3. Computational domain of a 2-D cylindrical duct for unconfined rate

stick simulation.

FIG. 4. Convergence of detonation wave front shown with grid resolution.

FIG. 5. Structure of detonation wave representing the reaction zone of

heavily aluminized RDX.

FIG. 6. Density contours for uncon-

fined rate stick simulation of 0 (top),

0.075 mm�1 (middle), and 0.125 mm�1

(bottom) inverse radius cases with den-

sity range 0.1–2.4 g/cc.

023512-5 Kim et al. J. Appl. Phys. 116, 023512 (2014)

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that the reaction zone length of a heavily aluminized RDX is

10 times that of a pure RDX.

Figure 6 depicts density contours that account for the

detonation front propagation and expansion of exhaust gas

outwards in the cases of infinite (0 mm�1), 13.33 mm

(0.075 mm�1), and 8 mm (0.125 mm�1) radius. Behind the det-

onation front, most of reactants are consumed and high pres-

sure product gas gives rise to an extended reaction zone length.

The reaction starts at the shock front and ends along the sonic

point (C-J state). The compression region occurs just ahead of

the reaction zone in both cases. Results show that the detona-

tion velocity is strongly dependent on the size of the charge.

Figure 7 describes propagation of detonation wave for

0 and 0.125 mm�1 unconfined rate sticks. A series of pres-

sure profiles shows the transition and stabilization of a von

Neumann spike. The figure indicates that there are two dis-

tinct regions: (i) run up region that represents fluctuation of

detonation pressure, and (ii) steady region that describes the

convergence of detonation flow.

There is a difference in the detonation velocity of 0 and

0.125 mm�1 rate sticks. The mean Zeldovich-von Neumann-

FIG. 7. Propagation of detonation front in 0 and 0.125 mm�1 unconfined

rate sticks.

FIG. 8. Size effect curve for the unconfined rate stick tests for heavily

aluminized RDX.

FIG. 9. (a) Schematic of a charge detonation simulation and (b) calculated

pressure history at each gauge.

TABLE IV. Comparison of the fully developed detonation velocity for dif-

ferent charge radiuses.

Inverse radius (mm�1)

Detonation velocity (m/s)

Numerical Experimental

0 7403.3 7387 (est.)

0.025 7348.9 7322 (est.)

0.050 7207.1 7197

0.075 7018.3 7044 (est.)

0.100 6829.9 6848

0.125 6544.2 6565

023512-6 Kim et al. J. Appl. Phys. 116, 023512 (2014)

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Doering (ZND) pressures in the steady region for these

rate sticks are calculated as being pZNDjr¼0 mm�1 ¼ 35 GPa,

pZNDjr¼0:125 mm�1 ¼ 27 GPa, respectively. This suggests that

pressure gradient determines the detonation velocity of a rate

stick as such that the chemical energy pumping in the

shocked region is decreased as the stick diameter is reduced.

The size effect curves showing detonation velocity ver-

sus inverse radius are shown in Fig. 8. The hydrocode simu-

lation is shown to reproduce the detonation velocities of six

sample radii of unconfined rate sticks (0 mm�1, 0.025 mm�1,

0.050 mm�1, 0.075 mm�1, 0.100 mm�1, and 0.125 mm�1).

Table IV shows comparison of the fully developed

FIG. 10. Detonation of RDX followed by the aluminum afterburning at different times: (a) Pressure, (b) RDX-product mass fraction, and (c) Al-product mass fraction.

023512-7 Kim et al. J. Appl. Phys. 116, 023512 (2014)

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detonation velocity. The size effect data using the present

rate law and parametric study are in good agreement with

the experimental data, suggesting that the model effectively

reduced steps in the parameter determination and solved the

over-simplification issue related to JWLþþ model.

V. PRESSURE CHAMBER TEST OF A HEAVILYALUMINIZED RDX AND ITS SIMULATION

Another validation of the model is pursued using the

pressure chamber test. A spherical charge of 3.5 cm radius

with 1.78 g/cc density is placed in a chamber maintained at

1 bar and detonated. Two pressure gauges are placed at two

radial distances at 1.6 m and 1.9 m from the initial position

of a charge. The purpose of the test was to obtain a primary

detonation signal (pressure) followed by the sequent pressure

peak representative of the aluminum afterburning. In order

to simulate both chemical processes of detonation and aero-

bic reaction of aluminum, we use a multi-ignition reaction

scheme for combined RDX and aluminum.

As before, we assume that none of the aluminum

particles reacted during the detonation of RDX. The after-

burning of the aluminum is modeled following the work of

Ref. 19, which was originally applied for aluminum combus-

tion in a gas phase. The aluminum ignition mechanism at

high pressure condition is utilized to predict the afterburning

of a heavily aluminized RDX.

A two-step Arrhenius-type mechanism for modeling the

subsequent burning of aluminum that follows a detonation of

RDX is:

T > 2000 K; kAl > 0:01;

dkAl

dt¼ A q0 1� kAlð Þ

�d2

p exp � Ea

RT

� �;

Ea ¼ 22:8 kcal=mol; A ¼ 1� 10�2 �

; (27)

T < 2000 K; kAl � 0:01;

dkAl

dt¼ A q0 1� kAlð Þ

�exp � Ea

RT

� �;

Ea ¼ 60 Kcal=mol; A ¼ 6:25� 1010 �

; (28)

qAl0 ¼ aq0; qAl ¼ ð1� kAlÞqAl0 ; (29)

kT ¼ akAl þ bkRDX; (30)

where dp or the particle diameter is 5 lm, while all other con-

stants are referred from Ref. 19. kAl is the final product mass

fraction of aluminum. A is the pre-exponential factor, and Ea

is the activation energy. qAl0 is the initial density of alumi-

num in the mixture, and qAl is the calculated density of

unreacted aluminum. a is 0.35 for aluminum and b is 0.65

for combined RDX/binder, both of which represent the initial

mass percent of the mixture. Then kT represents total final

product mass fraction. Also, the heat release due to alumi-

num reaction is _wAlqAl, which is added to the energy source

S of Eq. (21), where _wAl is the aluminum reaction rate and

qAl is the enthalpy of reaction of aluminum (85 000 kJ/kg).20

As for the EOS of gas phase aluminum, we adapted the

Noble-Abel equation of the form

PAl ¼akAlqRT

1� nAn; nAn :� 0:75ð Þ; (31)

where R denotes gas constant, n is the number of moles per

unit volume, and An is an empirical constant. Also, total

pressure is defined as

PTotal ¼ ð1� kRDXÞPunreacted þ kRDXPreacted þ kAlPAl: (32)

Figure 9 shows calculated pressures taken from gauges

placed at 0.04 m (gauge 1), 1 m (gauge 2), 1.6 m (gauge 3),

and 1.9 m (gauge 4) along the radial distance from a charge

center at zero. The radially detonating RDX in the chamber

shows a rapid decrease in the pressure.

Figure 10 shows a series of timed images of the chamber

test. Aluminum is ignited at 650 ls behind the detonated

RDX gases. The burnt aluminum mass propagates toward

the center as well as radially outward, because the fresh alu-

minum fuel is distributed within the radius of the hot product

gas of RDX. Both primary peak of RDX and secondary peak

of the afterburning are shown in the simulations.

Figure 11 shows a comparison of pressures calculated

and measured at gauge locations 1.6 and 1.9 m. Calculated

RDX peak slightly over-predicts the measurement, while the

FIG. 11. Comparison of pressure calculated and measured (a) at 1.6 m and

(b) at 1.9 m gauge locations.

023512-8 Kim et al. J. Appl. Phys. 116, 023512 (2014)

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subsequent aluminum peak agrees well with the measure-

ment. From these figures, the primary wave average velocity

is 1730 m/s and the subsequent afterburning velocity is 410

m/s as estimated at gauge 1.9 m. One fourth of the RDX det-

onation wave speed approximates the average velocity of the

afterburning of aluminum.

VI. CONCLUSION

A reactive flow model aimed at quantifying the detona-

tion response of energetic materials subjected to an external

shock impact is presented. A detailed explanation for deter-

mining the free parameters of the model is provided. Two

validation experiments are performed using a heavily alumi-

nized RDX. The unconfined rate stick experiment offered

detonation velocity as a function of the charge radius. A

pressure chamber test offered the time delay between first

and subsequent reaction peak. The hydrocode simulation that

utilized the present chemical mechanism confirmed that the

proposed reactive flow model reproduced the experimental

measurements, and it can overcome the limitations set by the

earlier reactive flow models.

ACKNOWLEDGMENTS

This work was supported by ADD-HH-5 through the

Institute of Advanced Aerospace Technology at Seoul

National University.

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