Propagation and Quenching of Detonation Waves in Particle
Laden Mixtures
YIGUANG JU and CHUNG K. LAW*
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Propagation and quenching of curved detonation waves in particle laden mixtures were investigated analytically
and numerically using the square wave model and quasi-steady state assumption, respectively. An analytical
expression describing the combined effects of heat and momentum losses because of the interaction between
the particle and gas phases, as well as the effect of wave curvature on detonation velocity, was obtained.
Detonation quenching and multiple detonation regimes were examined. Numerical simulation of the
detonation wave structure was also made. The results showed that particle heat loss, momentum loss and wave
curvature considerably reduce the detonation speed and cause detonation quenching. It is also shown that, for
fixed particle volume fractions, smaller particles cause a larger heat loss from the gas phase and result in a lower
detonation velocity and narrow detonation limit. � 2002 by The Combustion Institute
NOMENCLATURE Ta reduced activation energy, see Eq. 5

vp particle volume normalized by (t0c0)3

a1 constant in Eq. 11
w velocity normalized by c0


B0 reaction frequency

x streamwise coordinate normalized by t0c0

b1 constant in Eq. 13

xig ignition length normalized by t0c0

c0 acoustic wave speed of unshocked gas,

YF mass fraction of fuel
RgT0

Zeldovich number, see Eq. 9
c acoustic wave speed normalized by c0

variance of detonation velocity, see Eq. 22
Cf particle drag coefficient
specific heat ratio
CQ particle heat loss coefficient
i loss terms defined in Eq. 19
cs density of solid phase normalized by 0

heat conductivity
D wave velocity normalized by c0


dp diameter of particles normalized by t0 c0 viscosity

density

E activation energy
0 density of unshocked gas
FD normalized drag in Eq. 4
p particle density in fluid
fp particle volume fraction, np vp
s density of solid material
G0 effect of drag, see Eq. 23

time normalized by t0
H0 effect of heat loss, see Eq. 23
r characteristic chemical reaction time
j geometry factor
temperature perturbation defined in Eq. 9
M Mach number

chemical reaction rate normalized by 0/t0
Ł
np number density of particles normalized by

(t0 c0) 3

p pressure normalized by 0 c02

Subscript
Q chemical heat release of unit mass of fuel
normalized by c02

0 initial state of unshocked gas
Qc normalized drag
CJ values of CJ detonation

R0 universal gas constant
g gas

Rc radius of curvature normalized by t0c0, see

p particle phase
Eq. 2
ig ignition

T temperature normalized by T0
N, CJ Neumann state of CJ detonation

t0 reference time scale defined in Eq. 7

T0 temperature of unshocked gas
Superscript
*Corresponding author. E-mail: [e-mail] 0 parameters of adiabatic CJ detonation
COMBUSTION AND FLAME 129:356 364 (2002)
0010-2180/02/$ see front matter � 2002 by The Combustion Institute
PII S0010-2180(02)00342-5 Published by Elsevier Science Inc.
PROPAGATION AND QUENCHING OF DETONATION WAVES 357
INTRODUCTION The dynamics of detonation waves in uncon-
fined flow fields was theoretically analyzed by
Recent interest in the development of high Klein and Stewart [9], and He and Clavin [10].
speed propulsion systems such as the pulse Their results showed that wave curvature can
detonation engine and the ram-accelerator has also dramatically reduce the wave speed and
led to renewed interest in the study of the even quench the detonation wave.
fundamental mechanisms of detonation initia- For various practical reasons, liquid fuels are
tion, propagation, and quenching. However, ex- preferred in pulse detonation engines. The em-
perimental studies of detonation propagation ployment of liquid fuel dramatically enriches
are complicated by turbulent transition, near the phenomena because of momentum and
wall inhomogeneity, tube resistance, wall heat energy exchange between the two phases. Par-
loss, and phase interactions. Earlier experimen- ticularly, conduction heat loss from the gas
tal and theoretical investigations on the effects mixture to the droplets may have a profound
of tube resistance and the viscous boundary effect in affecting the detonation velocity. In
layer were studied by Shchelkin [1], Fay [2] and addition, most of the detonation waves in prac-
Dabora et al. [3]. The results showed that wall tical detonation engine are curved, particularly
resistance results in a lower wave speed than the during the initiation process. Therefore, the
Chapman-Jouguet (CJ) detonation velocity and coupling between the curvature effect and loss
that there is a detonation quenching limit. mechanisms in two-phase media may accelerate
Based on Shchelkin s analysis, an empirical det- the reduction of detonation speed. Earlier nu-
onation quenching limit for mixture concentra- merical studies of detonation propagation in gas
tions was defined by Dabora et al. Systematic mixtures in the presence of evaporating and/or
experimental investigations on tube friction and inert particles were conducted by Fan and
wall inhomogeneity were later conducted by Sichel [11], Gelfand et al. [12], Uphoff, H�nel,
Lee, Knystautas, and Freiman [4], and Lyamin and Roth [13], and Eidelman and Yang [14].
and Pinaev [5]. The results showed that there The results demonstrated significant effects of
exist multiple propagation regimes, the quasi- particle drag on the detonation speed and initi-
detonation regime, and the low velocity detona- ation energy. Recently, numerical simulations
tion regime. of the multi-dimensional diesel-air and kero-
The first attempt to interpret the reduction of sene-liquid oxygen detonation were conducted
the wave speed and the existence of multiple by Smirnov et al. [15] and Navaz and Berg [16].
detonation regimes was conducted by Zeldovich However, a theoretical description of the com-
et al. [6] by introducing the momentum and heat bined effects of particle drag, heat loss and
losses due to wall friction and cooling with a curvature on detonation propagation has not
quasi-one-dimensional formulation. The nu- been made.
merical results using one-step chemistry suc- Therefore, the purpose of the present work is
cessfully demonstrated the reduction of detona- to analytically and numerically study the com-
tion velocity and the existence of the detonation bined effects of particle drag, heat loss, and
limit. In addition, it was also shown that com- curvature on the detonation velocity and
pared to wall friction, wall heat loss exerts only quenching limit in particle laden mixtures, using
a weak influence on the detonation velocity the quasi-one-dimensional formulation. Here,
near the detonation limit. By extending this inert particles are employed to simplify the
model to detailed chemistry, Kusharin et al. [7] coupling of fuel concentration and ignition in
numerically calculated the detonation velocity the analysis, but still retain the merits in exam-
of H2/CO/CO2/air mixtures in the detonation ining the effect of heat loss and drag effects of
tube and showed a good agreement with the the spray. In the next section the governing
experimental data. Recently, by employing the equations are formulated. This is followed by an
Zeldovich model, Brailovsky and Sivashinsky [8] asymptotic analysis for the detonation velocity
numerically identified multiple propagation re- as well as numerical results of the effects of heat
gimes: the high velocity sub-CJ quasi-detona- loss, drag, and curvature on the detonation
tion regime and the supersonic choking regime. velocity.
358 Y. JU AND C. K. LAW
GOVERNING EQUATIONS
1
h YFQ T w2 (3)
g
1 2
In the coordinate of the moving wave front with
where Q is the normalized chemical heat release
curvature Rc and speed D, the normalized one-
per unit mass of fuel and the specific heat
dimensional governing equations for mass, mo-
ratio. The particle drag force FD and the con-
mentum, energy, and species conservations in a
duction heat loss from gas to particles Qc are,
particle laden mixture can be written as
respectively, given as
g gwg j
g D wg 0 FD 3Cf dp wp wg np
x Rc x
(4)
Qc 2CQ dp Tg Tp np
gwg D gw2 p
g
g
x
where Cf and CQ are the coefficients of particle
drag and conduction heat loss from gas phase to
j
g D wg wg FD
particle phase, the heat conduction coefficient
Rc x
of gas phase, and dp the diameter of particle.
The rate for the one-step chemical reaction is
h p D h
g gwg gwg
x
Ta

1YF exp , Ta E/R0T0
Ł

r
DFD D wp FD Q
T
(5)
YF YF
wg
Ł
where E is the activation energy, R0 the univer-
x
sal gas constant and r the characteristic reac-
np npwp j
tion time
np D wg 0
x Rc x
1 1/ (6)
t0B0
r
wp wp D
vp s vp swp p FD

x
Here B0 is the frequency factor. Variables with
overbars are dimensional while those without
Tp Tp
are quantities respectively normalized by using
vp scs vp scswp Q
x

the density ( 0), pressure (p0), temperature (T0)


and gas constant (Rg) of the preshock mixture,
p gTg (1)
and
where is the time, x the coordinate in flow
T0 2 exp Ta/T0
N,CJ N,CJ
direction, the density, p the pressure, w the

t0 (7)

B0 1 QTa
velocity, T the temperature, YF the mass fraction
of fuel, and vp the particle volume. Respectively, s Here T0 denotes the temperature of the
N,CJ
and cs are the particle density and specific heat for
Neumann state of the adiabatic planar CJ det-
the solid phase. Variables with subscripts g and p
onation.
denote the gas and particle phases, respectively.
The boundary conditions for Eq. (1) are given
Rc is the curvature of shock front
by
x ; Tg 1, g 1, wg wp DCJ,
t
Rc D s ds (2)

Tp 1, YF 1, np np0, p 1
0

x xCJ; wg Tg (8)
and the enthalpy h is given by
PROPAGATION AND QUENCHING OF DETONATION WAVES 359
ANALYTICAL SOLUTION energy and species conservation equations, and
the boundary conditions are given as
Because the characteristic flow time is much
d
longer than the chemical reaction time, the
x 1YF exp a1
1
dx
problem can be assumed quasi-steady. In the
limit of large activation energy ( Ą ), the
dYF
temperature and fuel concentration in the
b1x 1YF exp
1
dx
post-shock induction region can be expanded
using the Neumann state temperature in the
x Ą 0, 0, YF 1 (10)
form of
Here a1 represents the effects of curvature,
Tg Ta
particle drag and heat loss of gas phase on the
1 / , (9)
TN,CJ TN,CJ
evolution of the gas temperature and is defined
as
By substituting Eq. 9 into Eq. 1 using the
quasi-steady state assumption, the gas phase
Ta 1
a1 gw2c2j D wg /R c2 w2 Q w2 wp wg wpc2 FD (11)
g g g
T2 gwg w2 c2
NCJ g
where c is the acoustic wave speed. It is noted By employing the Hugoniot relation [17, 10],
here that both the positive wave curvature and the temperature variation in the Neumann state
can be given as a function of the variation of the
the particle drag result in an increase of the gas
detonation speed
temperature. The constant x1 is the length
scale of the ignition length of Neumann state
TN,CJ 2 1 D0 2 DCJ 2 DCJ
CJ
as a1 Ą 0

0
T0 1 D0 2 2 D0 DCJ
N,CJ CJ CJ
(16)
w2 c2 T0 2 1 1
g NCJ
x 1 exp Ta

1
w2 c2 wgT2 TNCJ T0
g NCJ NCJ
where D0 represents the CJ velocity of the
CJ
adiabatic gas detonation. Using Eq. 14, the
(12)
relation between ignition length and the varia-
and
tion of detonation velocity is given as
2 DCJ
T2 w2 c2
NCJ g
xig x0 exp (17)

ig,CJ
b1 (13)
D0
CJ
1 QTa w2 c2
g
By integrating Eq. 1 using the boundary con-
By further assuming that a1x1 is O(1/ ) and
dition given in Eq. 8, the mass, momentum and
that x1 and a1 are constants in the induction
energy equations can be written as
region, the ignition length can be given as
gwg D 1
TN,CJ T0
N,CJ gw2 p D2 1 2 3
g
xig x0 exp (14)

ig,CJ
(18)
T0
N,CJ
T w2 D2
g
Q 4
1 2 1 2
where x0 is the ignition length for the Neu-
ig,CJ
mann state of the adiabatic planar gas CJ det-
where 1, 2, 3 and 4, respectively, represent
onation
the mass loss because of curvature, momentum
loss because curvature and particle drag, and
1 M0 2 wgN,CJ
NCJ
heat loss because of cooling of the gas phase by
x0 (15)
ig,CJ
1 M0 2
N,CJ
the particles,
360 Y. JU AND C. K. LAW
2jD0 xig 2jD0 2xig
CJ CJ
1 , 2
1 Rc 1 Rc
6 dpnpD0 xig
CJ
3 (19)
1
D0 wp 2 dpnp TN,CJ 1 xig
CJ
4 1 3

D0 D0
CJ CJ
By using the CJ state condition in Eq. 8, the through kinetic improvement is very important
relation of the detonation velocity as a function for the reduction of the detonation velocity
deficit. Moreover, it is also noted that Eq. 21
of the losses ( i) can be obtained as
reduces to the results of Klein and Stewart [9]
2 D2 1 DCJ 2 DCJ 3 2
CJ
and He and Clavin [10] in the limit of G0 Ą 0,
2 2 1 D2 1 1 2
H0 Ą 0.
CJ
(20)
The relation between the detonation veloc-
2 1 D2 2Q 2 4
CJ
ity deficit and the particle volume fraction fp
0
2 1
is shown in Fig. 1. The dimensional and
non-dimensional parameters employed here
By further assuming that the effects of curva-

are: t0 1.68 10 7s, 0 1.12 kg/m3, p0

ture, particle drag and particle cooling are

1.0 105 Pa, T0 300 K, Rg 297 J/kg-K,
O(1/ ) and using the Hugoniot relation for the
1.4, Q 13.2, D0 5.2, 7.16,
CJ
adiabatic detonation wave, the perturbation re-
0.12, 0.025, cs 2.29, s 2805, and the
lation of the normalized velocity deficit, (DCJ
particle diameter is 10 m. It is seen that by
D0 )/ D0 , with O(1/ ) can be obtained directly
CJ CJ
allowing for heat loss from the gas phase to
from Eq. 20 as
the particles and neglecting the curvature and
drag effects (Cf 0, Rc ), the detonation
4j 2
exp 2 G0 H0 np x0

ig,CJ speed decreases rapidly as the particle con-
2 1 Rc
centration increases. There is a critical value
(21)
for the particle volume fraction beyond which
the quasi-steady detonation wave does not
where
exist. This critical point is the quenching limit
of the detonation wave. When the particle
DCJ D0
CJ
2 (22)
drag is taken into account, the detonation
D0
CJ
denotes the deficit of the detonation velocity,
and G0 and H0 are given as
0
G0 6 dp 1 DCJ 1
(23)
H0 2 2 1 dp TN,CJ 1 D0 3
CJ
Equation 21 is the propagation equation of
the detonation wave. The first term on the r.h.s.
of Eq. 21 denotes the curvature effect, the
second and third terms (G0 and H0), respec-
tively, represent the effects of particle drag and
heat loss from the gas phase to the particles.
Equation 21 clearly indicates that the total
deficit of the detonation velocity through above
three mechanisms is proportional to the ignition
length. Therefore, a decrease of ignition time Fig. 1. Relation between and fp.
PROPAGATION AND QUENCHING OF DETONATION WAVES 361
velocity is rapidly reduced. The quenching ture, particle drag and heat loss from gas
limit is dramatically narrowed from fp 0.022 phase to particles, the detonation wave is
to 0.0076. This result implies that in a particle quenched at fp 0.0057, which is much
laden mixture, unlike the rough tube detona- smaller than that with heat loss alone. There-
tion [6], both the heat loss from the gas phase fore, in the study of detonation propagation in
to the particles and the particle drag play two phase flow, the combined effects of drag,
important roles in reducing the detonation conduction heat loss to the solid phase and
velocity. In addition, when the detonation the curvature effect need to be considered
wave is curved (Rc 3000), the detonation simultaneously.
velocity deficit becomes more dramatic. Fig-
ure 1 further shows that curvature causes a
significant detonation deficit even at zero NUMERICAL RESULTS
particle volume fraction. As can be seen in
Eq. 21, curvature plays an increasingly impor- Equation (1) has been solved numerically in the
tant role in affecting the detonation wave phase plane [17]. The velocity equation in the
speed when the wave curvature becomes phase coordinate of fuel mass fraction is given
smaller. With the combined effects of curva- as
dwg wg jc2 g
D wg 1 Q wg wp wp)FD g( 1)Q (24)
Ł
dYF gc2 M2 1 Rc
Ł
where M is the Mach number. The equations for solution downstream. However, when the wave
the gas temperature, density and particle tem- velocity is equal to the CJ detonation velocity,
perature and velocity can be derived similarly the CJ plane, at which the flow Mach number
and reaches unity, occurs exactly at the point of
complete fuel consumption. For adiabatic pla-
dTg 1 dwg wpFD Q
nar detonation waves, since the first three terms
Q wg

dYF dYF g
Ł
of the numerator of Eq. 24 are zero, both the
denominator and numerator of the r.h.s. of Eq.
d g j g g dwg
24 vanish at the CJ point. As such, fuel must be
D wg
dYF Rc dYF
Ł Ł
completely consumed at the CJ plane. However,
for curved non-adiabatic detonation waves, be-
(25)
dTp wg Q

dYF cswp p
Ł
dwp j p
D wp
dYF Rc
Ł
Equations 24 and 25 are solved using the
ODE solver [18]. The dependence of the Mach
number on the fuel mass fraction for the adia-
batic gas detonation is shown in Fig. 2. The
upper and lower branches, respectively, denote
supersonic combustion (without a leading
shock) and detonation. It is seen that for a wave
velocity larger than the CJ detonation velocity,
the Mach number behind the shock wave is
subsonic. On the other hand, when the wave
velocity is less than the detonation velocity, the
flow reaches the sonic point before the fuel is
Fig. 2. Phase diagram of Mach number and YF for adiabatic
fully burned and there is the no steady-state gas detonation.
362 Y. JU AND C. K. LAW
Fig. 4. Effect of particle heat loss on DCJ. The dashed line
Fig. 3. Phase diagram of Mach number and YF for detona-
is the analytical result for dp 10 m.
tion in particle laden mixture.
certain range of particle loading, there are two
cause of the existence of losses the fuel is not
detonation regimes.
fully consumed at the CJ plane and thus yields a
The dependence of the detonation velocity on
decrease of the detonation speed.
the volume fraction of particles with heat loss
We first consider the effect of heat loss from
alone is plotted in Fig. 4. It can be seen that the
the gas phase to the particles on detonation
detonation velocity decreases rapidly with in-
propagation. The relation between the Mach
creasing particle volume fraction. For dp 20
number and fuel concentration for typical shock
m, detonation is quenched by the heat loss
wave speeds (D) is plotted in Fig. 3. It is seen
from the gas phase to the particles at fp 0.086.
that for wave speeds larger than the CJ velocity
In addition, with decreasing particle diameter,
(D 1.05DCJ), the flow Mach number in- the effect of heat loss from the gas phase to the
creases initially as the reaction proceeds and
particles becomes much more significant. In
peaks at Yp 0.02. Further completion of
addition, the analytical result for dp 10 mis
chemical reaction results in a reduction of the
also given in Fig. 4. It is seen that the analytical
Mach number. Thus, flow in the entire region is
result gives a reasonable result for detonation
subsonic. This is the case of the overdriven
quenching induced by heat loss, but yields a
detonation. For D DCJ the Mach number
higher detonation velocity. This is because the
increases monotonically with the completion of
nonlinear coupling between the wall cooling
reaction and reaches the sonic point (CJ plane)
effect and the temperature is not included in the
at Yp 0.03. This is the so-called sub-CJ
analysis.
quasi-detonation regime and its CJ velocity is
The corresponding distance between the
lower than the adiabatic CJ velocity because shock and the CJ plane and the fuel concentra-
fuel is not yet fully consumed at the CJ plane tion on the CJ plane are plotted in Fig. 5 for dp
[4]. For D 0.84 DCJ, the Mach number 10 m. It is seen that the distance between
reaches unity with an infinite gradient. This the CJ plane and shock decreases sharply at
implies that quasi-steady solutions do not exist. small particle volume fractions. This is under-
However, for D 0.73 DCJ another quasi- standable for the adiabatic detonation, since the
steady detonation regime exists. This slow mode CJ plane is at Yp 0, chemical reaction takes
of detonation is similar to the slow mode burn- an infinite length to consume the fuel com-
ing of the laminar flame, and is unstable. For D pletely. The distance reaches its minimum point
0.73 DCJ Fig. 3 shows that there exists a at fp 0.01 and increases again with further
subsonic burning regime and this is related to increase of fp. In addition, the distance on the
the so-called choking regime [5]. Therefore, at a slow mode branch is much longer than that on
PROPAGATION AND QUENCHING OF DETONATION WAVES 363
Fig. 7. Detonation structure: dependences of pressure, gas
Fig. 5. Dependence of xCJ and YFCJ on fp.
temperature and velocity, particle temperature and velocity
on the fuel mass fraction.
the fast mode, yielding a larger loss from drag
and heat conduction. On the other hand, the
strates that curvature significantly influences the
unburned fuel concentration increases mono-
detonation speed.
tonically on the fast mode branch with increas-
The detonation structure of the particle laden
ing particle volume fraction. This is the reason
mixture subject to heat loss from the gas phase
why the detonation velocity decreases rapidly
to the particles and particle drag (Cf CQ 1
and monotonically with increasing fp.
and Rc is infinity) is shown in Fig. 7. It is seen
The combined effect of heat loss from the gas
that, with the progress of the chemical reaction,
phase to the particles, particle drag and wave
gas pressure decreases dramatically. However,
curvature on the detonation velocity is shown in
there is a maximum temperature point for the
Fig. 6. It is seen that particle drag also consid-
gas phase at YF 0.1. A further progress of
erably reduces the detonation velocity. This is in
combustion causes a decrease of gas tempera-
agreement with the analytical results in the last
ture, showing a significant cooling effect of the
section. When the detonation wave curvature is
particle phase to the gas phase. In addition,
500, which is two-orders of magnitude larger
than the ignition length of the planer CJ deto- compared to the rapid increase of the gas
nation, the numerical solution also demon- velocity behind the shock, the particle velocity
changes very little. Therefore, it is reasonable to
assume a constant particle velocity in the induc-
tion zone in the square wave model.
The dependence of the detonable limit for
particle volume fraction on the particle diame-
ter is shown in Fig. 8. It is seen that with the
decrease of the particle size, the detonable limit
for particle volume fraction decreases dramati-
cally, showing the strong cooling effect of small
particles. Therefore, the role of heat loss from
the gas phase to the particle phase in the
detonation of particle laden mixtures is much
more important than that of the wall heat loss in
the tube detonation [6]. Furthermore, it is seen
that both the curvature and the particle drag
narrow the detonable limit for a given particle
Fig. 6. Relation between detonation velocity and fp. volume fraction.
364 Y. JU AND C. K. LAW
of Dr. Gabriel D. Roy. YJ would like to thank Dr.
Longting He at Princeton University for his helpful
discussions.
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This research is supported by the Office of
Received 11 May 2001; revised 18 December 2001; accepted 2
Naval Research under the technical management January 2001