Download Agglomeration Kernel of Bipolar Charged Particles in the Presence

Aerosol and Air Quality Research, 17: 857–866, 2017
Copyright © Taiwan Association for Aerosol Research
ISSN: 1680-8584 print / 2071-1409 online
doi: 10.4209/aaqr.2016.09.0378
Agglomeration Kernel of Bipolar Charged Particles in the Presence of External
Acoustic and Electric Fields
Hao Chen, Tao Wang*, Zhongyang Luo**, Dong Zhou, Mengshi Lu, Mingchun He,
Mengxiang Fang, Kefa Cen
State Key Laboratory of Clean Energy Utilization, Zhejiang University, Hangzhou 310027, China
ABSTRACT
A theoretical study on the agglomeration kernel of bipolar charged particles in external fields is presented in order to
investigate particle agglomeration under the combined effects of a direct-current (DC) electric field and an acoustic field,
while taking into consideration particle collision efficiency. Three agglomeration kernels are considered in our model,
namely Coulomb agglomeration, electric agglomeration, and acoustic agglomeration. The model performance is validated
by comparing its predictions with the available exact solutions of peculiar cases. Our results indicate that the collision
efficiency increases when an electric field is applied simultaneously with an acoustic field. The size of the total
agglomeration kernel formed by the three processes listed above increases with the external electric field intensity and the
sound pressure level (SPL). Based on this model which includes collision efficiency, the electric and acoustic agglomeration
kernels both increase rapidly as the agglomeration nucleus particle size increases, eventually approaching saturation for
large diameters. On the other hand, the total agglomeration kernel decreases as the sound frequency increases within the
investigated range. The agglomeration kernels of particles with different relative dielectric constants, such as fly ash
particles, solid carbon particles and water droplets, are investigated as well.
Keywords: Agglomeration kernel; Collision efficiency; Electric field; Acoustic field.
INTRODUCTION
A number of different agglomeration technologies have
been proven to provide an efficient pretreatment method for
the removal of fine particles from flue gases with traditional
filtration devices. These techniques include electrical
agglomeration (Hautanen et al., 1995), acoustic agglomeration
(Gallego-Juarez et al., 1999), chemical agglomeration (Wei
et al., 2005), turbulent agglomeration (Derevich, 2007) and
magnetic agglomeration (Li et al., 2007).
Laitinen et al. (1996) have conducted an experimental study
of bipolar charged aerosol agglomeration with alternating
electric field in laminar gas flow, finding that the number
concentration of 0.1–1.0 µm sized particles decreases by 17%
–19%. Experiments of particle charging and agglomeration in
direct-current (DC) and alternating-current (AC) electric
fields were carried out by Ji et al. (2004) obtaining a 25–
*
Corresponding author.
Tel.: 0571-87952205; Fax: 0571-87951616
E-mail address: [email protected]
**
Corresponding author.
Tel.: 0571-87952440; Fax: 0571-87951616
E-mail address: [email protected]
29% reduction ratio of submicron particles in an AC field.
Tan et al. (2007) tested the effect of an external DC electric
field on bipolar charged aerosol agglomeration and found
that the relative decrease in number of sub-micron sized
particles was about 10.7%.
Based on the mechanism of Brownian motion, the
agglomeration kernel of electrically charged (including both
unipolar and bipolar charged) particles caused by the
Coulomb force in the absence of external fields has been
derived by Zebel (1958) and Fuchs (1964). Approximate
expressions for the agglomeration kernel of bipolar charged
particles in an external DC (Wang et al., 2005) and AC
(Tan et al., 2008) electric field, respectively, have been
proposed in agreement with previous numerical results
calculated by Koizumi et al. (2000). An analytic-numerical
model has been developed to study the kinematic coagulation
of charged particles in an AC electric field (Lehtinen et al.,
1995), demonstrating that the rate of kinematic coagulation
in the case of unipolar charging was negligible, while being
significant when fine particles and large particles had
opposite polarity.
Acoustic agglomeration has been extensively studied via
experiments (Hoffmann et al., 1993; Liu et al., 2009; Liu et
al., 2011), leading to the development of several acoustic
agglomeration theories (Mednikov, 1965; Hoffmann, 1997).
Orthokinetic collision, caused by different entrainment among
Chen et al., Aerosol and Air Quality Research, 17: 857–866, 2017
858
different-sized particles in an acoustic field, is widely
recognized as the dominant agglomeration mechanism in
polydisperse aerosols (Cheng et al., 1983). The Mednikov
model is generally employed to explain orthokinetic
agglomeration based on different particle entrainment in a
vibrating gas. In conventional calculations, the collision
efficiency (i.e., the ratio of the actual collision cross section
to the ideal, purely geometric one) is assumed to be unity
(Temkin, 1994; Hoffmann, 1997). However, this assumption
has been proven to overestimate the collision rate, especially
in the case of collisions between submicron and large
particles (Nakajima and Sato, 2003). Ezekoye and Wibowo
(1999) set the collision efficiency to a value as low as 5%
in order to obtain agreement between simulation and
experimental results. Dong et al. (2006) derived an empirical
formula to calculate the collision efficiency as a function of
the Stokes number. A numerical calculation of collision
efficiency was also performed by taking into account the
electrostatic interaction force (Nakajima and Sato, 2003).
Although agglomeration processes have been extensively
investigated, few studies concerning the combined effects of
two or three agglomeration technologies have been reported.
Sun et al. (2013) introduced a turbulent gas jet into an
acoustic agglomeration chamber to study the coupling of gas
jet and acoustic waves on inhalable particle agglomeration.
An experimental system was developed to measure how
the removal of fine particles emitted from coal combustion
is affected by an electric and an acoustic field, both
separately and simultaneously (Chen et al., 2015). However,
to the best of our knowledge, a model which combines
different agglomeration kernels in the simultaneous presence
of electric and acoustic fields has not yet been reported.
In this paper, a theoretical study on the agglomeration
kernel of bipolar charged particles in the presence of external
fields is presented, in order to study particle agglomeration
under the combined effects of a DC electric field and an
acoustic field, taking into account the particle collision
efficiency. The influence of particle size, electric field
intensity, SPL and sound frequency are investigated and
discussed, along with the effects of different relative dielectric
constants.
represents the viscous Stokes’ force, which is also called
acoustic driving force. The second and third terms are
electric
field force and Coulomb force, respectively. Where

ui is the velocity of the particle, η is the viscosity of air,
u0 is the vibration
velocity amplitude of air, ω is the
angular frequency, Ea is the magnitude of the external
DC electric field, which is also called agglomeration
electric field in order to distinguish it from the charging
electric field, ε0 is the vacuum permittivity, r is the
center-to-center separation between the two particles, Cci is
the Cunningham correction factor, which can in turn be
expressed as (Fan et al., 2013):
Cci  1 


 
qi q j r
dui 3 di 
mi
u0 sin t  ui  Ea qi 

2
dt
Cci
4 0 r r


(1)
The life-hand side of Eq. (1) is the resultant force acting
on the particle. The first term on the right-hand side
(2)
where λ is the mean free path of air.
The charge acquired by the particle under corona discharge
is calculated by the following expression, combining field
charging and diffusion charging, which was derived by
Cochet (1961):


 2 2  2
qi   0  1 
 
di   1  2



di




    1 
2
 r
  Ec di
   r  2 




(3)
Here εr is the relative dielectric constant of the particle,
Ec is the magnitude of the electric field for particle charging,
which in the following is assumed to be 2 kV cm–1 if not
otherwise specified. To avoid ambiguity, we shall assume
that the fine particle is positively charged, while the nucleus
particle is negatively charged.
Agglomeration Kernel of Particles
The agglomeration kernel, employed to describe the
agglomeration rate, is defined as the number of
agglomerations between two group particles per unit time,
unit volume and unit particle number concentration, and it
is expressed by the following equation:
THEORETICAL ANALYSIS
Equations of Particles Motion
The agglomeration between two bipolar charged particles
in the presence of external acoustic and electric fields is
considered (Fig. 1). All nonlinear effects are neglected, as
well as image forces, particle interactions and gravitational
settling. The general kinetic equation for the motion of a
charged particle with diameter di, mass mi and charge qi
can be written as:
2 
 0.55di  
1.257  0.4 exp  
di 
  

K (i, j ) 
dN ij
ni n j dt
(4)
where Nji is the number of collision agglomerations, ni, nj
indicate the number of particle i, j per unit volume,
respectively. In this paper, particle i is assumed to be a small
particle called a fine particle, while the label j indicates a
large particle called a nucleus particle. It should be pointed
out that the theory also works for particles of similar sizes.
Based on the fact that the Coulomb force between two
particles acts in a short distance range while the applied
external fields are effective over a long-distance range, the
two effects can be considered independent of each other
(Wang et al., 2005). Thus, the agglomeration kernel of bipolar
charged particles in external acoustic and electric fields is
composed of two parts. The first part is the agglomeration
Chen et al., Aerosol and Air Quality Research, 17: 857–866, 2017
859
Fig. 1. The motions of two bipolar charged particles in external acoustic and electric fields. The directions of the acoustic
and electric field are parallel to the x-axis and y-axis, respectively.
kernel resulting from the Coulomb force without the effects
of external fields, while the second is caused by external
fields without the effect of Coulomb force.
r  s    ui  u j  dt
Coulomb Agglomeration Kernel
The velocity of fine particles can be obtained by solving
Eq. (1), neglecting the effects of external fields and ignoring
the inertial term since the aerodynamic relaxation time is
shorter than a few microseconds for a fine particle in air:
The total number of agglomerations between particles up
to time t becomes:
t
(6)
0
r
Nij  ni n j  4 r 2 dr  ni n j
s

ui 
Cci qi q j
2
12 2 0 di r
(5)
Fig. 2 shows the velocity of the fine particle as a function
of center-to-center separation, charging electric field intensity
and nucleus particle diameter. It can be seen that the velocity
becomes very small when the center-to-center separation is
greater than 20 µm since the Coulomb force is negligible in
the long-distance region. Conversely, the velocity significantly
increases with decreasing the center-to-center separation,
being inversely proportional to square of the separation.
Large charging electric field intensity and large nucleus
particle diameter are both beneficial to the increase of fine
particle velocity, which is strongly influenced by these two
factors according to Eq. (3) and Eq. (5).
If we assume that the nucleus particle is located at its
initial position and does not move, in terms of relative
movement, it takes a time t for the fine particle to reach the
nucleus particle at a velocity of ui + uj from the surface of
starting position
r to the collision surface s (Since
the


velocity ui is in the same direction as the vector r , scalars
are used for simplification in the derivation):
4
 r 3  s3
3


(7)
Solving Eqs. (4), Eq. (5) and Eq. (7), the Coulomb
agglomeration kernel can be obtained by:
K k (i, j )  4 r 2  ui (r )  u j (r )  
qi q j  Cci Ccj 



d j 
3 0  di
(8)
External Fields Agglomeration Kernel of Particles
In the presence of external fields, by using the method
proposed by Williams and Loyalka (1991), the agglomeration
kernel can be expressed as:
K F (i, j )  S  V 

d
4
i
 
2 
 d j  ui  u j
(9)
where s is the ideal collision cross section and V is the
relative velocity. It should be noted that this equation cannot
be directly applied to real situations in this form, due to the
discrepancy between the real and ideal cross sections, as
discussed in the following sections.
Following the motions of particles in the presence of
acoustic and electric fields, the external fields agglomeration
Chen et al., Aerosol and Air Quality Research, 17: 857–866, 2017
860
Fig. 2. Velocity of a positively charged 1 µm fine particle in the vicinity of a nucleus particle with negative charge.
kernel consists of two parts in the x and y directions. One
is caused by the electric field in the y direction, with the
two particles moving in opposite directions. The other is
caused by the acoustic field in x direction, with the two
particles moving in the same direction. By solving the
equations of motion, Eq. (1), in both directions, the relative
velocity can be obtained:
uijy  Ea
uijx  u0

2

 i qi

mi
 j qj
mj
 Ea
 i  j
1   2 i2 1   2 2j
Cci qi
3 di

Ccj q j
3 d j
cos t  i   j 
(11)
  i  j
u0
(10)
1   2 i2 1   2 2j
 
Here τi and τj indicate the relaxation time of the i and j
 d j 2 Ccj
 di 2 Cci
particles,  i 
, j 
, ρ is the particle
18
18
density, and the quantity |cos(ωt – φi – φj)| is replaced by its
average value in half cycle as 2/π (Temkin, 1994).
Substituting Eqs. (10) and (11) into Eq. (9), the following
expression for the agglomeration kernel can be obtained:
K
F
 i, j   Ea

d
d
i
 d j   Cci qi Ccj q j


dj
12  di

2
i
dj 
2
2
straight line toward the nucleus particle. This assumption
is, however, not completely accurate, since the trajectory of
the fine particle will be deflected from the straight line due
to entrainment of the air flow. In order to obtain an effective
collision cross section value, the trajectory of the fine
particle should be calculated within the airflow. Theoretically,
the trajectory should be three dimensional, including also
the z-component of the flow. However, due to the axial
symmetry of the air flow, a factor φ, which indicates the
angle between the y and z axes measured in the clockwise
direction (Fig. 1), is introduced in order to perform a 2-d
trajectory calculation.
The air flow is in the Stokes region due to the small
Reynolds number. Therefore, the stream function for the
well-known Stokes flow (Turcotte and Schubert, 2014),
which satisfies the no-lip velocity boundary condition at
the surface of the nucleus particle, can be expressed as:




u0  i   j
(12)
1   2 i 2 1   2 j 2
Collision Efficiency of Particles
In the above derivation of the external fields agglomeration
kernel, we assumed that the fine particle moves along a
U
2
 2 d j 3 3d j r  2

r 
 sin 

16r
4 

(13)
where U denotes the approaching velocity of the airflow.
The angular and radial components of the velocity in
terms of the stream function can then be obtained (Fig. 1):
Ur 

d j 3 3d j 
1 



U
1

 cos 
 16r 3 4r 
r 2 sin  


U  

d j 3 3d j 
1 
 U  1 

 sin 
 32r 3 8r 
r sin  r


(14)
(15)
The x and y components of the air flow can be expressed
by:
Ux = Ur cosθ – Uθ sinθ
(16)
Uy = Ur sinθ – Uθ cosθ
(17)
Chen et al., Aerosol and Air Quality Research, 17: 857–866, 2017
Thus the equations of motion for the fine particle can be
written as:
mi
d 2 x 3 di

Cci
dt 2
dx 

U x    f x
dt 

(18)
mi
d 2 y 3 di

Cci
dt 2
dy 

U y    f y
dt 

(19)
Here fx, fy respectively indicate other forces in the x and
y direction, except for the viscous force. In the case of
acoustic agglomeration, such forces need not be taken into
consideration, while in this paper, the electric force is
considered as a factor enhancing collision probability.
When an electric field is applied perpendicularly to the
acoustic field, these forces can be expressed as:
fx = 0
fy = Eaqi cosφ
(20)
 2
  y c d  , yc  0   
0
Sr  
y
 c 0 2
  yc d , yc  0  
 0
K  i , j 
C

d
qi q j
3 0
 E
d
a

d
12
i
 dj 
2
dj
i
 dj 
i
Ccj

ci
2



C q

 d
ci

i
i
2
Ccj q j
dj



(24)
u 0  i   j
1  i
2
2
1  j
2
2
Eq. (24) shows three distinct contributions to the total
agglomeration kernel. The first term on the right-hand side
describes the Coulomb agglomeration kernel caused by the
sole Coulomb force. The second and third terms account for
the electric and acoustic agglomeration kernels, respectively.
Here, the effect of hydrodynamic interactions on particles
agglomeration is neglected in order to consider the acoustic
agglomeration kernel to be only caused by the orthokinetic
collision mechanism.
RESULTS AND DISCUSSIONS
(22)
In order to eliminate the unit, a dimensionless parameter
called collision efficiency is introduced, which is defined as:
  2
 4  yc d
 0
, yc  0  
2


d
d



i
j

  
y
 c 0 2
4
  yc d
 0
, y  0  
 d d 2 c


i
j

an electric field, Ea = 2 kV cm–1, and an acoustic field, SPL
= 142 dB, ƒ = 1400 Hz. If the acoustic field is applied
solely, the critical radius is 1.88 µm and the real collision
cross section is 11.09 µm2 (Fig. 4). When an electric field
is added as well, the critical radius becomes 3.61 µm for φ
= 0 and 0.19 µm for φ = π, with a real collision cross
section of 14.4 µm2. The collision efficiency increases from
0.116 to 0.152 upon adding the electric field. It is clear that
the simultaneous presence of acoustic and electric fields
can increase the collision efficiency.
Combining Eq. (8), Eq. (12) and taking the collision
efficiency into consideration, the total agglomeration kernel
of bipolar charged particles in external acoustic and electric
fields can be expressed as:
(21)
Fig. 3 shows the real collision cross section of the fine
particle. The starting x position is determined from the
relative displacement between two particles (Dong et al.,
2006). By solving the above equations numerically, the
critical trajectory can be obtained. If the starting y position
of the fine particle is less than the critical radius yc, it can
definitely collide with the nucleus particle, otherwise it will
escape from collision. The critical radius yc is a function of
the angle φ, since the force fy changes with the angle. If the
critical radius yc ≤ 0 when φ = π, the collision cross section
S (as shown in Fig. 3(b)) will be calculated twice, thus it
should be eliminated in the calculation. As a result, different
expressions for the real collision cross section are used:
861
(23)
Two fly ash particles, with which the density is 2500
kg m–3 and the relative dielectric constant is 1.7, suspend in
Accuracy of the Model
In order to validate the calculated Coulomb agglomeration
kernel of bipolar charged particles in the absence of
external fields, the expression of Eq. (8), is compared with
the classical solution (Zebel, 1958; Fuchs, 1964) derived
from the conventional Brownian motion of two charged
particles, and with numerical results (Koizumi et al., 2000)
taken from the available literature. Here, the particles are
solid carbon particles with a relative dielectric constant of
3, and the charging electric field intensity is 2.5 kV cm–1.
Our results show good agreement with the theoretical and
numerical values, despite being obtained from different
concepts and derivation procedures (Fig. 5).
The present calculation of particle collision efficiency
displays good agreement with the results found by Zhang
et al. (2009) in the presence of the acoustic field only. The
particles in this case are fly ash particles with diameters of
1 and 10 µm. Fig. 6 shows that, in the presence of an acoustic
field, the collision efficiency is relatively low and stable for
low SPL values, increasing dramatically when the SPL
becomes high. If an electric field is applied, the collision
862
Chen et al., Aerosol and Air Quality Research, 17: 857–866, 2017
Fig. 3. The real cross section: (a) yc > 0 when φ = π; (b) yc ≤ 0 when φ = π.
Fig. 4. Critical trajectory of 1 a µm particle around a nucleus particle of 10 µm in external fields. Here, Ea = 2 kV cm–1;
SPL = 142 dB; ƒ = 1400 Hz.
Fig. 5. Agglomeration kernel of bipolar charged particles by the Coulomb force without the effect of external fields.
efficiency increases proportionally to the agglomeration
electric field intensity. The collision efficiency shows a
non-monotonic change with the SPL in the simultaneous
presence of acoustic and electric fields. There exists a
critical SPL value, corresponding to the lowest collision
efficiency. When the SPL changes, the vibrating amplitude
and velocity change accordingly. The action time of the
electric field, which is also the time required for a fine
particle to travel from the starting position to the collision
surface, will be influenced consequently.
Influence of Particle Size, Electric Field Intensity, SPL
and Frequency
Particle Size
The effects of nucleus particle size on the agglomeration
kernel are shown in Fig. 7 (both with and without considering
Chen et al., Aerosol and Air Quality Research, 17: 857–866, 2017
863
Fig. 6. SPL effects on the collision efficiency in external fields at a frequency of 1000 Hz for a pair of particles of size 1
and 10 µm.
Fig. 7. Effects of nucleus particle size on the agglomeration kernel of a 1 µm particle in external fields with collision
efficiency. Here, Ec = 2 kV cm–1; SPL = 142 dB; ƒ = 1400 Hz.
the collision efficiency). The fine particle diameter is fixed
at 1 µm, while the nucleus particle diameter varies from 2 to
10 µm. Both particles are fly ash particles. The acoustic field
has a SPL of 142 dB and a frequency of 1400 Hz, while the
intensity of the agglomeration electric field is 2 kV cm–1.
When particle collision efficiency is not taken into account,
both the acoustic and the electric agglomeration kernels
rapidly increase with the agglomeration nucleus particle
diameter. However, if the collision efficiency is considered,
the increasing trend becomes slower at large nucleus particle
diameters. This is due to the collision efficiency increasing
at first and then decreasing when the nucleus particle
diameter changes from 2 to 10 µm. When the particle sizes
approach a similar value, the relative velocity between two
particles is low. Thus the fine particle carried by the airflow
can easily get through the nucleus particle leading to a low
collision efficiency. The particle relative displacement,
however, exhibits little changes for large particle size ratios.
Therefore, in these conditions the collision efficiency becomes
small even though the nucleus particle diameter is large.
Electric Field Intensity
The effects of the electric field intensity on the
agglomeration kernel are presented in Fig. 8. The fine and
nucleus particle diameters are 1 and 10 µm, respectively.
The sound frequency is 1400 Hz and the SPL is 142 dB.
The figure shows that, if the collision efficiency is considered,
the electric and acoustic agglomeration kernels increase when
the electric field intensity changes from 1 to 4 kV cm–1. An
intense agglomeration electric field can both enhance the
collision efficiency and electric agglomeration. Increasing
the collision efficiency enlarges the both the electric and
the acoustic agglomeration kernels, which means that the
agglomeration electric field intensity has dramatic influence
864
Chen et al., Aerosol and Air Quality Research, 17: 857–866, 2017
Fig. 8. Effects of the electric field intensity on the agglomeration kernel at a frequency of 1400 Hz and a SPL of 142 dB
for a pair of particles of size 1 and 10 µm.
on electric agglomeration. Considering the collision efficiency
significantly decreases the size of the total agglomeration
kernel. That is because the collision efficiency is as low as
0.12 when the agglomeration electric field intensity is 1
kV cm–1, and it only increases to 0.28 when the electric
field intensity increases to 4 kV cm–1.
The agglomeration kernels of different type of particles
are shown in Table 1, comparing fly ash particles, solid
carbon particles and water droplets with relative dielectric
constant of 1.7, 3.0, 80, respectively. The agglomeration
process occurs between two particles with diameters of 1
and 10 µm, in a varying charging and agglomeration electric
field combined with a fixed acoustic field, with a SPL of
142 dB and a sound frequency is 1400 Hz. It is clear that
water droplets are more effective for agglomeration than fly
ash particles and solid carbon particles in the same condition.
The larger the relative dielectric constant, the larger the
charge capacitance, which means more charge can be held
under the same corona charging condition. Increasing the
charge level can both enhance the agglomeration kernel
and the collision efficiency. It can also be found that the
agglomeration kernel of water droplets increases up to 8
times from 7.3 × 10-12 m3 s–1 to 58 × 10–12 m3 s–1 as the
charging electric field intensity increases from 1 kV cm–1
to 3 kV cm–1, while a much smaller increase (by a factor
1.6) is found for the same variation of the agglomeration
electric field intensity.
SPL and Frequency
The effects of SPL and frequency on the agglomeration
kernel with and without collision efficiency are illustrated
in Fig. 9. The agglomeration electric field intensity is fixed
at 2 kV cm–1. The SPL of the acoustic field takes the values
136 dB, 142 dB and 148 dB, while its frequency changes
from 800 to 2400 Hz, which is commonly used in acoustic
agglomeration experiments. Both particles are fly ash
particles with diameters of 1 and 10 µm.
When the collision efficiency is not taken into account,
the total agglomeration kernel keeps almost unchanged in
the range of calculated sound frequency, while it decreases
with increasing sound frequency if collision efficiency is
included. Even though the sound frequency does not affect
the Coulomb agglomeration kernel, it can reduce the external
fields agglomeration kernel by decreasing the collision
efficiency. This happens because the action time of the electric
force decreases with increasing sound frequency, leading to
a low critical radius, and to an according decrease of the
collision efficiency. Fig. 9 shows that the total agglomeration
kernel becomes larger as the SPL increases. In fact, a large
SPL means that the approaching velocity of the airflow is
large, causing a low deflection of the trajectory and favoring
the collision between the particles. Furthermore, Eq. (24)
Table 1. Comparison of the agglomeration kernel between a pair of particles of size 1 and 10 µm for different materials.
Here, SPL = 142 dB; ƒ = 1400 Hz.
Density
(kg m–3)
Fly Ash
Solid Carbon Particle
Water Droplet
2500
2200
1000
Relative
dielectric
constant
1.7
3.0
80
Agglomeration kernel (10–12 m3 s–1)
Charging Electric field Ec (kV cm–1)
1
1
1
2
Agglomeration Electric field Ea (kV cm–1)
1
2
3
1
6.4
6.7
9.4
13
7.0
8.0
9.6
15
7.3
9.3
12
26
3
1
21
28
58
Chen et al., Aerosol and Air Quality Research, 17: 857–866, 2017
865
Fig. 9. Frequency and SPL effects on the agglomeration kernel with an external electric field of 2 kV cm–1 for a pair of
particles of size 1 and 10 µm.
shows that a large approaching velocity also increases the
acoustic agglomeration kernel even though the collision
efficiency is unaffected. While a high-intensity acoustic field
consumes a lot of energy, a suitable SPL should be chosen
for conducting experiments.
CONCLUSIONS
Three agglomeration processes of polydisperse bipolar
charged particles in the presence of external acoustic and
electric fields are studied, namely Coulomb, electric and
acoustic agglomeration. The effects of particle collision
efficiency are included in the model. Parametric studies are
conducted to investigate the effects of particle size, electric
field intensity, SPL and sound frequency on the agglomeration
kernel. The main results obtained from this study can be
summarized as follows:
(1) The particle collision efficiency becomes large when
an external electric field is applied in the presence of an
acoustic field. However, variations of the intensity of
such electric field do not significantly affect the
collision efficiency.
(2) The total agglomeration kernel increases with the external
electric field intensity and sound pressure level (SPL).
Water droplets are more effective for agglomeration than
fly ash and solid carbon particles in the same conditions,
due to their large relative dielectric constant.
(3) If collision efficiency is considered, the electric and
acoustic agglomeration kernels both increase rapidly
with the nucleus particle diameter, but the increasing
rate becomes slower at large diameters.
(4) The total agglomeration kernel is reduced as the sound
frequency increases within the investigated range, due
to the decrease of collision efficiency.
ACKNOWLEDGEMENTS
The authors gratefully acknowledge the kind support of
the National Basic Research Program (973) of China (No.
2013CB228504).
REFERENCES
Chen, H., Luo, Z., Jiang, J., Zhou, D., Lu, M., Fang, M.
and Cen, K. (2015). Effects of simultaneous acoustic and
electric fields on removal of fine particles emitted from
coal combustion. Powder Technol. 281: 12–19.
Cheng, M.T., Lee, P.S., Berner, A. and Shaw, D.T. (1983).
Orthokinetic agglomeration in an intense acoustic field.
J. Colloid Interface Sci. 91: 176–187.
Cochet, R. (1961). Lois charge des fines particules
(submicroniques) etudes theoriques–controles recents
spectre de particules. Coll. Int. la Physique des Forces
Electrostatiques et Leurs Application, Centre National
de la Recherche Scientifique, Paris 102: 331–338.
Derevich, I.V. (2007). Coagulation kernel of particles in a
turbulent gas flow. Int. J. Heat Mass Transfer 50: 1368–
1387.
Dong, S., Lipkens, B. and Cameron, T. (2006). The effects
of orthokinetic collision, acoustic wake, and gravity on
acoustic agglomeration of polydisperse aerosols. J.
Aerosol Sci. 37: 540–553.
Ezekoye, O. and Wibowo, Y. (1999). Simulation of acoustic
agglomeration processes using a sectional algorithm. J.
Aerosol Sci. 30: 1117–1138.
Fan, F.X., Yang, X.F. and Kim, C.N. (2013). Direct
simulation of inhalable particle motion and collision in a
standing wave field. J. Mech. Sci. Technol. 27: 1707–1712.
Fuchs, N. (1964). The Mechanics of Aerosols. Pagamon,
New York.
Gallego-Juarez, J.A., De Sarabia, E.R.F., Rodriguez-Corral,
G., Hoffmann, T.L., Galvez-Moraleda, J.C., RodriguezMaroto, J.J., Gomez-Moreno, F.J., Bahillo-Ruiz, A.,
Martin-Espigares, M. and Acha, M. (1999). Application
of acoustic agglomeration to reduce fine particle emissions
from coal combustion plants. Environ. Sci. Technol. 33:
3843–3849.
Hautanen, J., Kilpeläinen, M., Kauppinen, E.I., Lehtinen,
K. and Jokiniemi, J. (1995). Electrical agglomeration of
aerosol particles in an alternating electric field. Aerosol
Sci. Technol. 22: 181–189.
866
Chen et al., Aerosol and Air Quality Research, 17: 857–866, 2017
Hoffmann, T.L., Chen, W., Koopmann, G.H., Scaroni, A.W.
and Song, L. (1993). Experimental and numerical-analysis
of bimodal acoustic agglomeration. J. Vib. Acoust. 115:
232–240.
Hoffmann, T.L. (1997). An extended kernel for acoustic
agglomeration simulation based on the acoustic wake
effect. J. Aerosol Sci. 28: 919–936.
Ji, J.H., Hwang, J., Bae, G.N. and Kim, Y.G. (2004).
Particle charging and agglomeration in dc and ac electric
fields. J. Electrostat. 61: 57–68.
Koizumi, Y., Kawamura, M., Tochikubo, F. and Watanabe,
T. (2000). Estimation of the agglomeration coefficient of
bipolar-charged aerosol particles. J. Electrostat. 48: 93–
101.
Laitinen, A., Hautanen, J., Keskinen, J., Kauppinen, E.,
Jokiniemi, J. and Lehtinen, K. (1996). Bipolar charged
aerosol agglomeration with alternating electric field in
laminar gas flow. J. Electrostat. 38: 303–315.
Lehtinen, K.E., Jokiniemi, J.K., Kauppinen, E.I. and
Hautanen, J. (1995). Kinematic coagulation of charged
droplets in an alternating electric field. Aerosol Sci.
Technol. 23: 422–430.
Li, Y., Zhao, C., Wu, X., Lu, D. and Han, S. (2007).
Aggregation experiments on fine fly ash particles in
uniform magnetic field. Powder Technol. 174: 93–103.
Liu, J.Z., Zhang, G.X., Zhou, J.H., Wang, J., Zhao, W.D.
and Cen, K.F. (2009). Experimental study of acoustic
agglomeration of coal-fired fly ash particles at low
frequencies. Powder Technol. 193: 20–25.
Liu, J.Z., Wang, J., Zhang, G.X., Zhou, J.H. and Cen, K.F.
(2011). Frequency comparative study of coal-fired fly
ash acoustic agglomeration. J. Environ. Sci.-China 23:
1845–1851.
Mednikov, E.P. (1965). Acoustic coagulation and precipitation
of aerosols, In Acoustic Coagulation and Precipitation of
Aerosols, Consultants Bureau.
Nakajima, Y. and Sato, T. (2003). Electrostatic collection
of submicron particles with the aid of electrostatic
agglomeration promoted by particle vibration. Powder
Technol. 135: 266–284.
Sun, D., Zhang, X. and Fang, L. (2013). Coupling effect of
gas jet and acoustic wave on inhalable particle
agglomeration. J. Aerosol Sci. 66: 12–23.
Tan, B., Wang, L. and Zhang, X. (2007). The effect of an
external dc electric field on bipolar charged aerosol
agglomeration. J. Electrostat. 65: 82–86.
Tan, B.H., Wang, L.Z. and Wu, Z.N. (2008). An
approximate expression for the coagulation coefficient
of bipolarly charged particles in an alternating electric
field. J. Aerosol Sci. 39: 793–800.
Temkin, S. (1994). Gasdynamic agglomeration of aerosols.
I. Acoustic waves. Phys. Fluids 6: 2294–2303.
Turcotte, D.L. and Schubert, G. (2014). Geodynamics.
Cambridge University Press.
Wang, L.Z., Zhang, X.R. and Zhu, K.Q. (2005). An
analytical expression for the coagulation coefficient of
bipolarly charged particles by an external electric field
with the effect of coulomb force. J. Aerosol Sci. 36: 1050–
1055.
Wei, F., Zhang, J.Y. and Zheng, C.G. (2005). Agglomeration
rate and action forces between atomized particles of
agglomerator and inhaled-particles from coal combustion.
J. Environ. Sci.-China 17: 335–339.
Williams, M.M.R. and Loyalka, S.K. (1991). Aerosol
Science: Theory and Practice. Pergamon, New York.
Zebel, G. (1958). Zur theorie des verhaltens elektrisch
geladener aerosole. Colloid. Polym. Sci. 157: 37–50.
Zhang, G.X., Liu, J.Z., Zhou, J.H. and Cen, K.F. (2009).
Simulation and analysis of collision efficiency in
acoustic agglomeration. CIESC J. 60: 42–47.
Received for review, September 1, 2016
Revised, November 28, 2016
Accepted, December 6, 2016