The document discusses using a discrete element method (DEM) simulation to study the mixing behavior of granular materials in a vibrated bed with the effect of electrostatic forces. The simulation finds that granular temperatures and mixing rates increase with higher electrostatic numbers. Specifically, granular temperatures increase linearly with electrostatic number, while mixing rate constants calculated from mixing degree increase as a power law function of the electrostatic number. The role of granular temperature in granular mixing is also discussed.

1. Mixing in vibrated granular beds with the effect of electrostatic force
Li-Shin Lu, Shu-San Hsiau *
Department of Mechanical Engineering, National Central University, Chung-Li, Taiwan 32054, ROC
Received 9 June 2004; received in revised form 29 June 2005; accepted 17 August 2005
Available online 14 October 2005
Abstract
In this study, the DEM (Discrete Elementary Method) is used to simulate the behavior of granular mixing in vibrated beds. First, the velocity
fields are simulated by the DEM model to examine the convective currents in a three-dimensional vibrated granular bed. Then, in order to
characterize the effect of electrostatic force on the granular flow, the electrostatic number Es is defined as the ratio of the electrostatic force to the
particle weight. Also, to quantify the quality of mixing, the mixing degree M is used by the well-known Lacey index. The top–bottom and the
side–side initial loading patterns of two groups of glass bead with different colors are employed to investigate the influence of the convective
currents on the granular mixing. To simplify the electrostatic effect, these two groups of glass beads are given opposite charges and the charged
strength is assumed to be constant. The simulation results demonstrate that the granular temperatures increase linearly with the increasing Es
number. Meanwhile, the mixing rate constants, calculated from the time evolutions of mixing degree, increase with the increasing Es number in
power law relations. The role of granular temperature in the granular mixing is also discussed in this paper.
D 2005 Elsevier B.V. All rights reserved.
Keywords: DEM; Vibrated bed; Mixing; Electrostatic force; Granular temperature
1. Introduction
The mixing of granules is economically important in many
different industries, including foodstuffs, pharmaceutical pro-
ducts, ceramics, chemicals, plastics, etc. The mixing process
can be regarded as a key process to add significant value to the
product, which has led to extensive research on the subject in
both numerical and experimental works. McCarthy et al. [1–3]
employed soft-sphere particle dynamics simulation to study
granular mixing in a rotating cylinder. To quantify the
characterization of granular mixer, Muzzio et al. [4–6]
investigated powder mixing in many kinds of tumbling
blenders. Meanwhile, numerous other researchers have focused
on the mixing of vibrated granular system in recent years.
Henrique et al. [7] used an event-driven molecular dynamics
(MD) algorithm to examine the efficiency of diffusion as the
mixing mechanism in a granular system with external random
acceleration. Using flow visualization and digital image
processing, Akiyama et al. [8] have experimentally shown that
convective motion and mixing of particles within vertical two-
dimensional vibrating beds were amenable to characterization
by fractal properties. Molina-Boisseau and Bolay [9] demon-
strated that the acceleration of vibration was the most important
parameter to influence the mixing of polymeric powders and
the glass beads in a shaker. With the discrete element model
simulation, Asmar et al. [10] estimated a mixing index, GMMI,
to quantify mixing in the three-dimensional vibrated particulate
bed.
In most discussions of granular mixing, the mechanisms
contributing to the creation of a mixture are usually described
as convective mixing, shearing mixing and diffusive mixing
[11,12]. Convection causes a movement of larger groups of
particles relative to each other (macromixing). Shearing mixing
is introduced by the momentum exchange of particles having
different velocities. Diffusive mixing is caused by the random
motion of the individual particles. The rate of mixing by
random motion is low compared with convective mixing
mechanism, but diffusion mixing is essential for microscopic
homogenization [11].
Achieving good mixing of particulate solids is the aim of
many industrial applications, and it is possible to produce
better mixtures by taking advantage of the attractive forces
0032-5910/$ - see front matter D 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.powtec.2005.08.028
* Corresponding author. Tel.: +886 2 2809 4617; fax: +886 2 2893 5295.
E-mail address: sshsiau@cc.ncu.edu.tw (S.-S. Hsiau).
Powder Technology 160 (2005) 170 – 179
www.elsevier.com/locate/powtec

2. between particles, such as liquid bridge force and electrostatic
force. Generally, the electrostatic effect of solid materials is a
serious problem in industries, especially for the works dealing
with powders and granular materials. However, various useful
applications of powder charging have been employed advan-
tageously in a wide range, e.g., xerography [13,14], separation
[15] and dry powder coating [16]. There was even a special
issue for electrostatic phenomena of particulate processes in
Powder Technology 135–136 (2003) 1. To get a satisfactory
granular mixture, using the electrostatic effect between
particles is a possible technique. For example, by giving
particles opposite charges through triboelectrification, the
ordered mixtures in pharmaceutical powder system may be
stabilized [17].
Cohesive granular mixing is of considerable industrial
importance and has intrigued researchers in recent years.
Although electrostatic force is one of the major cohesive
forces, especially for the small and dry particles, the
electrostatic effect has received less attention than the liquid
bridge effect on the subject of granular mixing [3,18,19]. The
purpose of this study is to examine the mixing characteristics of
a vibrated granular bed with the effect of electrostatic force in a
simplified situation.
The current study uses the three-dimensional soft particle
discrete-element method (DEM) to simulate the behavior of
particles in vibrated granular beds. The DEM is able to give
intimate detailed events and conditions within the bed,
including contact forces, velocities and granular temperatures.
In order to examine the influence of the convective mixing
mechanism on the vibrated granular bed, two cases are
analyzed with different initial loading patterns. To characterize
the effect of electrostatic force on the granular motion, the
electrostatic number, Es, is defined as the ratio of electrostatic
force to the weight of particle. Furthermore, to quantify the
performance of mixing, the mixing degree, M, is used by the
so-called Lacey index [20]. The mixing rate constant, k, is
calculated from a least-square fit using the results of M. In this
simulation work, the correlation between granular temperatures
and Es number is initially found in order to explore the role of
electrostatic force in the granular system. In addition, the
relationships between mixing rate constants and Es number are
also obtained to investigate the influence of electrostatic effect
on the mixing behavior.
2. Simulation method
Use of the DEM to simulate granular flow was first
developed by Cundall and Strack [21] and has been applied
in various simulation models in several fields, including
granular mixing [10,22–24]. This method simulates the
individual dynamics of every particle in an assembly by
numerically integrating their accelerations resulting from all
forces, including normal and tangential contact forces, gravity
forces and/or electrostatic forces. At the beginning of every
time step, the particle positions are recorded, and then the
particle interactions can be evaluated. All the forces acting on
each particle in the system are first calculated. Subsequently,
Newton’s second law is used to determine the accelerations,
which are then integrated with time to find the velocities and
positions of each particle at the new state. This process is
repeated until the end of the simulation.
2.1. Contact forces
When two particles or a particle and a solid boundary come
into contact, the deformation may be approximated as an
overlap area of the particle pressed against each other (Fig. 1).
Therefore, the contact forces between them depend on the
relative velocities of the particles in the overlap area and the
material properties of the particles. Based on the positions of
invading points, Ci and Cj, the vector of invasion in the normal
direction is given by
dn
ij ¼
Y
CjCj
Y
ð1Þ
Then, let Vo
i
Y
and Vo
j
Y
be the velocities of the mass centers of
particle i and particle j, respectively. In addition, xi
Y
and xj
Y
are
the angular velocities of particles i and j. Therefore, the
velocities of invading points Ci and Cj are determined as
follows:
VC
i
Y
¼ Vo
i
Y
þ xi
Y
‘ OiCi
Y
ð2Þ
VC
j
Y
¼ Vo
j
Y
þ xj
Y
‘ OjCj
Y
ð3Þ
Then the relative velocity of the two invading points is found
from
VC
ij
Y
¼ VC
i
Y
VC
j
Y
ð4Þ
The normal and tangential components of the relative velocity
are calculated as follows:
VCn
ij
Y
¼
VC
ij
Y
I OiOj
Y
j OiOj
Y
j2
OiOj
Y
ð5Þ
VCs
ij
Y
¼ VC
ij
Y
VCn
ij
Y
ð6Þ
The normal contact force is modeled by a linear spring and
dashpot, so that
Fn
cij
Y
¼ Kn dn
ij
Y
þ c0n
VCn
ij
Y
ð7Þ
where Kn and c0n are the normal spring constant and damping
coefficient, respectively. The spring and the frictional slider are
j
C
Oi
O
Ci
j
Fig. 1. Inelastic contact of two particles as overlap.
L.-S. Lu, S.-S. Hsiau / Powder Technology 160 (2005) 170–179 171

3. both considered in tangential contact. The tangential force is
first calculated from the spring-displacement law,
Fks
cij
Y
¼ Ks ds
ij
Y
ð8Þ
where Ks is the tangential spring constant, and ds
ij
Y
is the
tangential vector of invasion cumulating from the initial contact
point,
ds
ij
Y
¼ ds
ij
YV
þ VCs
ij
Y
IDt ð9Þ
Then the friction force is calculated using a friction coefficient
l:
Fsr
cij
Y
¼ l Fn
cij
Y
ð10Þ
The tangential force used in this simulation is the smaller force
of Fks
cij
Y
and Fsr
cij
Y
:
Fs
cij
Y
¼ min Fks
cij
Y
; Fsr
cij
Y
ð11Þ
Finally, the total contact force can be expressed as follows:
Fcij
Y
¼ Fn
cij
Y
þ Fs
cij
Y
ð12Þ
2.2. Electrostatic forces
Higashiyama et al. [25] triboelectrified polymer particles
using a conveyer with different plastic films, which could
electrify the same particles to charge with opposite sign. For a
glass material, one particle will become charged positively by
contact with PerChloro Erhy Lene, since some negative
charges move from the glass to the PerChloro Erhy Lene and
thus create net positive charges on the initially neutral glass
particle. Oppositely, rubbing a neutral glass particle with fur,
the glass particle will become a charged particle with net-
negative charges due to receiving the negative charges moving
from the fur. In the current simulation, two groups of glass
beads with the identical diameter, d, are oppositely charged
before being loaded into the vibrated bed. In order to simplify
the model, the surface charge density of each spherical particle,
r (C/m2
), is assumed to be constant throughout the vibration
process, which means that all of the particles have the same
charge strength:
q ¼ pd2
r ð13Þ
The Coulombic force between each pair of charged
particles, i and j, is calculated as follows [11]:
Feij ¼
qiqj
4pere0l2
ð14Þ
where relative permitivity (r =1 (air), permitivity of free space
(0 =8.8510 12
C2
N 1
m 2
, and l is the distance between
the centers of these two particles.
In this simulation, the boundaries, floor and side-walls of
the bed are assumed to be uncharged. However, the vicinal
charged particle i are adhered to the surfaces by the image-
charge force, which is given by
Feij ¼
q2
i
4pere0 2lV
ð Þ2
ð15Þ
where lVis the distance between the particle’s center and its
vicinal boundary.
2.3. Equations of motion
After the forces acting on particle i have been determined,
Newton’s second law is used to evaluate the particle motion.
For the translational motion, the relationship can be written as
follows:
mi
d Vi
Y
dt
¼ Fgi
Y
þ
X
Fcij
Y
þ Feij
Y
ð16Þ
where mi is the mass of particle i, Fgi
Y
is the gravitational force
on the particle. The sum of the right-hand side in Eq. (16) is the
total force acting on the particle i, Fi
Y
. On the other hand, the
rotational motion of the particle is governed by the following
equation:
Ii
d xi
Y
dt
¼
X
OiCi
Y
‘ Fi
Y
ð17Þ
where I is the moment of inertia.
The velocities Vo
i
Y n
, xi
Y n
and position xi
Y n
at the end of the
time step are then determined using an explicit numerical
integration
Vo
i
Y n
¼ Vo
i
Y n1
þ Vo
i
Y
n
Dt ð18Þ
xi
Y n
¼ xi
Y n1
þ Vo
i
Y n
Dt ð19Þ
xi
Y n
¼ xi
Y n1
þ xi
Y
n
Dt ð20Þ
where n denotes the present time step.
3. Simulated system
In the present study, all the granular materials are frictional
and inelastic spherical glass beads with mass density q =2500
kg/m3
and diameter d =1 mm. The granular beds are energized
by vertical sinusoidal oscillations at vibrated acceleration of 3g
and amplitude of 2d, where g is the gravity acceleration. Table
1 shows the general model data used in the simulations. For the
comparison with our previous experiment [26], the one-layer
granular bed is first considered to verify the validity of our
DEM program. For this, 600 (3020) glass beads are initially
loaded into the one-layer granular bed (see Fig. 2), which
consists of one floor and two sidewalls with friction. On the
other hand, the three-dimensional vibrated granular beds, with
L.-S. Lu, S.-S. Hsiau / Powder Technology 160 (2005) 170–179
172

4. one frictional floor and four frictional sidewalls of glass plane,
are used to investigate the behavior of granular mixing. In
order to examine the influence of convective mixing mecha-
nism, two groups of glass beads, identical but with different
colors, are loaded into the three-dimensional vibrated bed in
two different initial patterns, top–bottom loading (Case 1) and
side–side loading (Case 2). Fig. 3(a) and (b) displays the
settings of the initial condition for the two loading patterns with
equal amounts of each component. The total number of glass
beads in the three-dimensional bed is 1728 (121212). To
investigate the influence of the electrostatic force on the
granular mixing, these two groups of glass beads with different
colors are charged oppositely before being put into the beds by
means of the description in Section 2.2. In addition, the average
charge density of glass bead is around 10 AC/m2
after contact
with PerChloro Erhy Lene film from our preliminary measure-
ment. Therefore, the simulations are carried out with a constant
charge density, r, ranging from 0 to 20 AC/m2
with an interval
of 2 AC/m2
, where r =0 means there is no effect of electrostatic
force. Note that the charge density is assumed to be constant in
each simulation since all of the materials are the same, resulting
in less charge transfer.
4. Electrostatic number and mixing index
In order to quantify the electrostatic effect on the flow
of granular material, the electrostatic number, Es, is
X
Y
Z
Vibrated Floor
Fig. 2. Schematic representation of the one-layer granular bed.
X
Y
Z
Vibrated Floor
CASE1
(a)
X
Y
Z
Vibrated Floor
CASE 2
(b)
Fig. 3. Initial loading patterns in the three-dimensional granular bed. (a) Case 1:
top–bottom loading pattern; (b) case 2: side–side loading pattern.
Table 1
Data for simulations of vibrated granular bed
Parameter Value Unit
Dimensionless vibrated acceleration C (=ax2
/g) 3
Vibrated amplitude a 0.002 m
Particle diameter d 0.001 m
Particle density q 2500 kg/m3
Frictional coefficient l
Particle/particle 0.5
Particle/wall 0.8
Normal damping coefficient c0n
Particle/particle 4.17102
N/(m/s)
Particle/wall 8.33102
N/(m/s)
Normal spring coefficient Kn
Particle/particle 5900 N/m
Particle/wall 11800 N/m
Tangential spring coefficient Ks
Particle/particle 5900 N/m
Particle/wall 11800 N/m
Time step Dt 5.0106
s
L.-S. Lu, S.-S. Hsiau / Powder Technology 160 (2005) 170–179 173

5. defined as the ratio of the electrostatic force to the particle
weight
Es ¼
Fe
Fg
¼
p2
d4
r2
4pe0l
2
4
3
p
d
2
3
qg
¼
3dr2
2qge0l
2
ð21Þ
where l̄ is the average distance between particles in the bed.
Note that the Fe in Eq. (21) only considers the electrostatic
force of one pair of particles, while the electrostatic force of
one particle is actually affected by all of its neighbor
particles. However, the definition of Es in Eq. (21) is still a
good physical and simple dimensionless number to charac-
terize the electrostatic effect in the granular system.
To quantify the quality of mixing in the binary mixture, we
use the well-known Lacey index based on statistical analysis
[20]. For this purpose, the three-dimensional bed is divided into
several cubic cells, each with volume of 8 mm3
. The variance
S2
for the concentration of a reference component in each cell
is defined in the following way:
S2
¼
X
N
i¼1
/i /m
ð Þ2
N 1
ð22Þ
where N is the number of cells in the bed, /i is the
concentration of the reference component in cell i, /m is the
overall concentration of the reference component. In a two-
component system, the theoretical maximum and minimum
values of mixture variance, S0
2
and Sr
2
, are the variance in a
completely segregated mixture and that in a perfectly mixture,
respectively. These are given by
S2
0 ¼ /m 1 /m
ð Þ ð23Þ
S2
r ¼
/m 1 /m
ð Þ
n
ð24Þ
where n is the number of particles contained in a cell. Then, a
dimensionless and normalized index, M, is used to characterize
the degree of mixing:
M ¼
S2
0 S2
S2
0 S2
r
ð25Þ
which is the ratio of mixing achieved to mixing possible.
5. Results and discussion
5.1. Sidewall convection
In this study, the long-term average velocity method [27] is
used to calculate the velocity field in the vibrated granular bed.
The displacements and velocities are determined by averaging
over 200 vibration cycles. Fig. 4(a) shows the simulation result
of the velocity field in the one-layer vibrated granular bed
without any cohesion force. Two symmetric convection rolls
appear clearly in the bed. The vertical frictional sidewalls
promote the convection rolls, where particles fall in a thin layer
next to the walls. Therefore, the particles have to rise at the
central part due to the conservation of mass. This simulation
result is in agreement with the previous experimental result for
the same vibrated condition [26].
The velocity field in the three-dimensional vibrated granular
bed without any cohesion force is presented in Fig. 4(b). Due to
the influence of the four frictional sidewalls, the symmetric
convective flows are upward in the central zone and downward
near the side surfaces, like a fountain jetting from the center
and spraying around the sidewalls. Furthermore, the velocity
field of the central section and the diagonal section are shown
in Fig. 4(c) and (d), respectively, to observe the internal
convection currents, which are similar to the results of Lan and
Rosato [28].
5.2. Granular temperature
The random motions of particles are quantified by granular
temperature, which is defined as the specific fluctuation kinetic
energy of particles and serves as a key property of granular
material flows. To calculate the local granular temperature, the
granular bed is divided vertically into several layers with the
height of Dh =0.99d. The local granular temperature in vertical
layer y is calculated from
T y
ð Þ ¼
buV2
þ bmV2
þ bwV2
3
ð26Þ
where u, v and w denote the velocity components of a particle
in x, y and z directions, respectively. The fluctuating velocities,
uV
, vVand wV
, are defined as the difference between the particle
velocity and the average local velocity in layer y. The brackets,
b, denote an ensemble average for the layer y. The
dimensionless average granular temperature of the whole bed
is defined as
Td ¼
X
y
nyT y
ð Þ
gd
X
y
ny
ð27Þ
where ny is the number of particles within the layer y.
To examine the effect of electrostatic force between particles
on the granular temperature, Td is calculated for different Es
numbers. Computationally, the Es is most simply varied
through changes in the surface charge density r. The
magnitudes of r in the simulations of this study are carried
out at 0–20 AC/m2
, resulting in the corresponding Es at 0–
1.882, as presented in the abscissa of Fig. 5. The dimensionless
granular temperatures are plotted against the electrostatic
number in Fig. 5. In both Case 1 and Case 2, Td increases
linearly with increasing Es. The larger granular temperature
indicates that the fluctuated energy is greater and the granular
bed is more excited. That is, the hotter agitation in the granular
bed results from the stronger electrostatic force of particles,
since the electrostatic effect promotes the activity of particles.
In addition, the two groups of particles with opposite charges
L.-S. Lu, S.-S. Hsiau / Powder Technology 160 (2005) 170–179
174

6. are completely segregated at the initial loading conditions in
both cases (see Fig. 3(a) and (b)). Naturally, every particle
tends to be attracted by the oppositely charged particles and to
be repulsed from the identically charged particles. This effect is
the extra mechanism to excite the fluctuated motions of
particles and, therefore, enlarge the granular temperature in
the bed.
In Fig. 5, it is noteworthy that the values of Td for different
Es numbers and the slope of the straight line by linear fitting of
Case 1 are almost the same as those of Case 2, although the
initial loading patterns of oppositely charged particles are
different in these two cases. This suggests that the dimension-
less granular temperatures are correlated very well with the
electrostatic number.
5.3. The time evolution of mixing degree
In this study, the initially completely segregated systems
(Fig. 3(a) and (b)) are used to observe the histories of mixing
degree and the times to reach stationary mixing state in the
mixing processes. Firstly, to simplify the influence of loading
pattern on the mixing degree, two groups of particles with
different colors are all neutralized, so no electrostatic force acts
between particles in both Case 1 and Case 2. The variations of
mixing degree with the dimensionless time t (vibrated cycles)
are presented in Fig. 6. This shows that for both cases, the
mixing degrees gradually increase until the stationary mixing
states are reached to a certain value, M =0.82. Fig. 6 also shows
the mixing times, smix1 for Case 1 and smix2 for Case 2, which
-0.015
-0.01
-0.005
0
0.005
0.01
X (m)
Z
0
0.01
0.02
0.03
Y
(m)
(a)
0
0.005
0.01
0.015
0.02
Y
(m)
-0.005
0
0.005
X (m)
-0.005
0
0.005
Z (m)
(b)
0
0.005
0.01
0.015
0.02
Y
(m)
-0.005
0
0.005
X (m)
-0.005
0
0.005
Z (m)
(c)
0
0.005
0.01
0.015
0.02
Y
(m)
-0.005
0
0.005
X (m)
-0.005
0
0.005
Z (m)
(d)
Fig. 4. The velocity field in the vibrated granular bed under the vibrated condition of G=3.0 and a =2d. (a) The one-layer vibrated granular bed, (b) the three-
dimensional vibrated granular bed, (c) the central section in the three-dimensional vibrated granular bed.
L.-S. Lu, S.-S. Hsiau / Powder Technology 160 (2005) 170–179 175

7. denote the times required for the mixing degrees increase from
zero to an equilibrium value (M =0.82). The duration from start
to smix can be defined as the initial mixing stage, which is the
main mixing stage since the mixing state is stable after smix. It
is noted that in the initial mixing stage, the mixing degree of
Case 1 is greater than that of Case 2, and smix1 is shorter than
smix2. Thus, the mixing speed in Case 1 is much faster
comparing with Case 2. It is mainly due to the convective
motion of particles in the vibrated granular bed. As mentioned
above in Section 5.1, the convective currents in the three-
dimensional bed are symmetrically toward the four side
surfaces, and the interflows of particles are vertically in both
central zone and sidewalls. Thus, the initial mixing of the top–
bottom loading case (Case 1) can be improved significantly by
the vertical interflows. However, the side-symmetric convec-
tion flow and the vertical interflows of particles have very little
effect on promoting the side–side mixing of particles (Case 2).
Therefore, the convective mixing mechanism affects the top–
bottom loading much more significantly than the side–side
loading in the vertical vibrated bed. It thus results in the top–
bottom loading with more rapid mixing, since the convective
mixing is the fastest mixing mechanism compared with the
shearing mixing and diffusive mixing, especially in the initial
mixing stage [6,11].
To observe the influence of electrostatic force on the mixing
degree, various constant charge densities are employed in the
simulations. Fig. 7(a) and (b) show the time evolutions of
mixing degree in different Es number. In both Case 1 and Case
2, it is not surprised that the mixing degree increases faster for
larger Es, since the stronger electrostatic force results in the
larger granular temperature and excites the mobility of
particles. The fluctuating motions of particles, of course, can
promote the granular mixing. Also, it is noted that the
equilibrium value of mixing degree is higher and closer to
M =1 for the case with larger Es. Thus, with the increase of
electrostatic force, the binary mixture of particles with opposite
charges can achieve better mixing status due to the character-
t (cycles)
M
0 100 200 300
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Case 1
Case 2
τmix1 τmix2
Fig. 6. The time evolutions of mixing degree of Case 1 and Case 2 without the
effect of electrostatic force, in which smix is the time required for the mixing
degree increases from zero to equilibrium value of the stable mixture.
t (cycles)
M
0 100 200 300
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Es=0
Es=0.077
Es=0.305
Es=0.926
Es=1.882
Case 1
(a)
t (cycles)
M
0 100 200 300
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Es=0
Es=0.077
Es=0.305
Es=0.926
Es=1.882
Case 2
(b)
Fig. 7. The time evolution of mixing degree at different Es number. (a) Case 1
and (b) Case 2.
Es
T
d
0 0.5 1 1.5 2
0
0.001
0.002
0.003
Case1
Case2
Fig. 5. The dimensionless average granular temperature Td plotted against the
electrostatic number Es.
L.-S. Lu, S.-S. Hsiau / Powder Technology 160 (2005) 170–179
176

8. istic of electrostatic effect, i.e., attractive force between
oppositely charged particles and repulsive force between
identically charged particles.
5.4. Mixing rate constant
From the time evolutions of mixing degree in the previous
section, we predict that the mixing degrees increase exponen-
tially with time as they approach the equilibrium values [1,3,5].
The exponential curves in the initial mixing stage are fitted to
an equation of the form
M ¼ 1 exp kt
ð Þ ð28Þ
where k is the mixing rate constant, which can be used to
quantify the mixing speed in the initial stage. For each case, k
is obtained from a least-squares fit of the data in Fig. 6 using
Eq. (28). Fig. 8 shows (1M) plotted against t for Case 1 and
Case 2 in logarithmic-normal scale. Before (1M) reaches the
equilibrium value, the data are fitted to a straight line by the
least-squares method; and the mixing rate constant, k, can be
determined from the slope of the straight line. Using the same
approach, the mixing rate constants in various Es number in
both Case 1 and Case 2 can be calculated from Fig. 7 and Eq.
(28). The mixing rate constants are plotted against Es number
in Fig. 9. Corresponding to the results of the time evolutions of
mixing degree, it is expected that the mixing rate constants
increase with increasing Es. Furthermore, Fig. 9 demonstrates
that the mixing rate constants depend on the Es number by
some power law relations:
k ¼ k0 þ 0:132Es0:535
for Case 1
ð Þ ð29Þ
and
k ¼ k0 þ 0:128Es0:721
for Case 2
ð Þ ð30Þ
where k0 is the mixing rate constant of the simulation without
the electrostatic effect (Es=0). The values of k0 are 0.025 and
0.013 in Case 1 and Case 2, respectively. The k0 in Case 1 is
larger than that in Case 2 due to the effect of the initial loading
pattern, as described in Fig. 6. Because of the difference of k0,
the coefficient and the power in Eq. (29) are different from
those in Eq. (30). However, the values of k in Case 1 and Case
2 are eventually very close when Es is large enough (Es1.5).
In fact, the relation between Td and Es in Case 1 is almost the
same as that in Case 2 (see Fig. 5), indicating that the influence
of electrostatic force on particles’ fluctuation motions in Case 1
is almost the same as that in Case 2. Also, the larger fluctuation
motion can promote the granular mixing as mentioned in
Section 5.3. Consequently, with the increase of Es, the effect of
the fluctuation motion on the granular mixing is greater in both
cases and it even becomes the dominant effect at large Es
value. It should be noted that the power law relations between k
and Es shown in Fig. 9 are valid at Es2.0. In fact, while
Es2.0, the relations between k and Es of these two cases are
almost the same and therefore different from Eq. (29) or Eq.
(30). However, the surface charge density of glass bead is
difficult larger than 20 AC/m2
(corresponding Es=1.882)
through simple contact electrification; thus, the results of k
are not shown in Fig. 9. Nevertheless, Fig. 9 demonstrates that
the mixing rate constant is strongly influenced by the Es
number. It can be very useful to increase the mixing rate
constant by taking advantage of the attractive and repulsive
force of the electrostatic effect.
6. Conclusion
This paper illustrates the usefulness of DEM simulations in
studying the behavior of granular mixing in three-dimensional
vibrated beds. In order to investigate the granular mixing with
the effect of electrostatic force between particles, this study
defines four dimensionless parameters: Electrostatic number,
Es; dimensionless average granular temperature, Td; mixing
degree, M; and mixing rate constant k. Two cases of different
initial loading pattern are used to examine the effect of
convective motion on the binary mixing system. Without the
t (cycles)
1-M
0 100 200 300
10
-2
10
-1
100
Case 1
Case 2
CurveFit 1
CurveFit 2
Fig. 8. (1M) plotted against the dimensionless time t in logarithmic scale, in
which the straight lines are fitted curves by the least-square method.
Es
k
0 0.5 1 1.5 2
0
0.1
0.2
0.3
0.4
0.5
Case 1
Case 2
CurveFit 1
CurveFit 2
k=0.025+0.132Es
0.535
k=0.013+0.128Es
0.721
Fig. 9. The mixing rate constant k plotted against the Es number of Case 1 and
Case 2, in which the lines are fitted curves by least-square method.
L.-S. Lu, S.-S. Hsiau / Powder Technology 160 (2005) 170–179 177

9. electrostatic effect, the mixing degree, M, of the top–bottom
loading case increases faster than the case of side–side loading
due to the symmetric convection flow in the vertical vibrated
bed. However, with the electrostatic effect, the influences of
electrostatic force on the granular temperature are almost the
same in these two cases, indicating that the granular
temperature is in good correlation with the Es number. In
addition, the mixing rate constants increase with the increasing
Es number in power law relations. The power law relations of
these two cases are different due to the different initial loading
patterns. Nevertheless, the mixing rate constants of these two
cases are eventually very close, since the effect of the
fluctuation motion is greater for larger Es numbers. Generally,
the quality and speed of granular mixing can be enhanced
through taking advantages of the electrostatic effect.
List of symbols
a vibrated amplitude (m)
c0 damping coefficient (N s/m)
d particle diameter (m)
Es electrostatic force
F total force acting on the particle (N)
Fc contact force (N)
Fe electrostatic force (N)
Fg gravitational force (N)
g gravity acceleration (m/s2
)
I the moment of inertia (m4
)
Kn normal spring constant (N/m)
Ks shear spring constant (N/m)
k mixing rate constant
k0 mixing rate constant at Es=0
l the distance between the centers of two particles (m)
lV the distance between the particle’s center and its
vicinal boundary (m)
l̄ the average distance between particles in the bed (m)
M mixing degree
m the mass of particle (kg)
N the number of cells in the bed
n the number of particle
q charge strength (C)
S2
The variance for the concentration of a reference
component in each cell
Td dimensionless granular temperature
t dimensionless time (vibrated cycle)
u the velocity component of particle in x direction (m/s)
Vo
the velocity of the mass center of particle (m/s)
VC
the velocity of the invading point of particle (m/s)
v the velocity component of particle in y direction (m/s)
w the velocity component of particle in z direction (m/s)
Greek letters
Dh the height of vertical layer (m)
Dt time step (s)
d the vector of invasion
(0 permitivity of free space (C2
N1
m2
)
(r relative permitivity
/ the concentration of the reference component in a cell
l frictional coefficient
q particle density (kg/m3
)
r surface charge density (C/m2
)
x angular velocity of particle (rad/s)
Acknowledgements
The authors gratefully acknowledge the financial support
from the National Science Council of the Republic of China
(grants NSC 92-2212-E-008-007 and NSC 93-2212-E-008-
002) and from Kuang Wu Institute of Technology (grant KW
92-ME-C03) in Taiwan.
References
[1] D.V. Khakhar, J.J. McCarthy, T. Shinbrot, J.M. Ottino, Transverse flow
and missing of granular materials in a rotating cylinder, Phys. Fluids 9
(1997) 31–43.
[2] J.J. McCarthy, J.M. Ottino, Particle dynamics simulation: a hybrid
technique applied to granular mixing, Powder Technol. 97 (1998)
91–99.
[3] J.J. McCarthy, D.V. Khakhar, J.M. Ottino, Computational studies of
granular mixing, Powder Technol. 109 (2000) 72–82.
[4] D. Brone, A. Alexander, F.J. Muzzio, Quantitative characterization of
mixing of dry powders in V-blenders, AIChE J. 44 (1998) 271–278.
[5] M. Moakher, T. Shinbrot, F.J. Muzzio, Experimentally validated computa-
tions of flows, mixing and segregation of noncohesive grains in 3D
tumbling blenders, Powder Technol. 109 (2000) 58–71.
[6] O.S. Sudah, D. Coffin-Beach, F.J. Muzzio, Quantitative characterization
of mixing of free-flowing granular material in tote (bin)-blenders, Powder
Technol. 126 (2002) 191–200.
[7] C. Henrique, G. Batrouni, D. Bideau, Diffusion as a mixing mechanism in
granular materials, Phys. Rev. E 6301 (2000) 1304.
[8] T. Akiyama, T. Iguchi, K. Aoki, K. Nishimoto, A fractal analysis of solids
mixing in two-dimensional vibrating particle beds, Powder Technol. 97
(1998) 63–71.
[9] S. Molina-Boisseau, N.L. Bolay, The mixing of polymeric powder and the
grinding medium in a shaker bead mill, Powder Technol. 123 (2002)
212–220.
[10] B.N. Asmar, P.A. Langston, A.J. Matchett, A generalized mixing index in
distinct element method simulation of vibrated particulate beds, Granul.
Matter 4 (2002) 129–138.
[11] K. Gotoh, H. Masuda, K. Higashitani, Powder Technology Handbook,
Marcel Dekker, New York, 1997.
[12] B.H. Kaye, Powder Mixing, Chapman Hall, UK, 1997.
[13] E.J. Gutman, G.C. Hartmann, Triboelectric properties of two-component
developers for xerography, J. Imaging Sci. Technol. 36 (1992) 335–349.
[14] L.B. Schein, Electrophotography and Development Physics, 2nd edR,
Laplacian Press, Morgan Hill, CA, 1996.
[15] D.K. Yanar, B.A. Kwetkus, Electrostatic separation of polymer powders,
J. Electrost. 35 (1995) 79–87.
[16] W. Kleber, B. Makin, Triboelectric powder coating: a practical approach
for industrial use, Part. Sci. Technol. 16 (1998) 43–53.
[17] J.N. Staniforth, J.E. Rees, Electrostatic charge interactions in ordered
powder mixes, J. Pharm. Pharmacol. 34 (1982) 69–76.
[18] D. Geromichalos, M.M. Kohonen, F. Mugele, S. Herminghaus, Mixing
and condensation in a wet granular medium, Phys. Rev. Lett. 90 (2003)
168702.
[19] H.M. Li, J.J. McCarthy, Controlling cohesive particle mixing and
segregation, Phys. Rev. Lett. 90 (2003) 184301.
[20] P.M.C. Lacey, Developments in the theory of particulate mixing, J. Appl.
Chem. 4 (1954) 257.
[21] P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular
assemblies, Géotechnique 29 (1979) 47–65.
L.-S. Lu, S.-S. Hsiau / Powder Technology 160 (2005) 170–179
178

10. [22] P.W. Cleary, Guy Metcalfe, K. Liffman, How well do discrete element
granular flow models capture the essentials of mixing processes?, Appl.
Math. Model. 22 (1998) 995–1008.
[23] M.J. Rhodes, X.S. Wang, M. Nguyen, P. Stewart, K. Liffman, Study of
mixing in gas-fluidized beds using a DEM model, Chem. Eng. Sci. 56
(2001) 2859.
[24] B.N. Asmar, P.A. Langston, A.J. Matchett, J.K. Walters, Validation tests
on a distinct element model of vibrating cohesive particle systems,
Comput. Chem. Eng. 26 (2002) 785–802.
[25] Y. Higashiyama, Y. Ujiie, K. Asano, Triboelectrification of plastic
particles on a vibrating feeder laminated with a plastic film, J. Electrost.
4 (1997) 63–68.
[26] S.S. Hsiau, C.H. Chen, Granular convection cells in a vertical shaker,
Powder Technol. 111 (2000) 210–217.
[27] C.R. Wassgren, Vibration of granular materials. PhD thesis, California
Institute of Technology, CA, USA 1997.
[28] Y. Lan, A.D. Rosato, Convection related phenomena in granular dynamics
simulations of vibrated beds, Phys. Fluids 9 (1997) 3615–3624.
L.-S. Lu, S.-S. Hsiau / Powder Technology 160 (2005) 170–179 179