NUMERICAL SIMULATION OF ACOUSTIC AGGLOMERATION OF DUST
PARTICLES IN HIGH TEMPERATURE EXHAUST GAS
Sergey V. Komarov and Masahiro Hirasawa
Institute of Multidisciplinary Research for Advanced Materials, Tohoku University,
2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan
Phone +81-22-217-5177, email : komarov@tagen.tohoku.ac.jp
A numerical simulation is made on the agglomeration of dust particles (Zn, size 0.2 to 32
m) which are exposed to a powerful standing sound wave of a certain frequency (35 to
1000 Hz). The particles are carried by a high temperature carrier gas (Ar, 100 to 900ºC)
inside a vertical sonoprocessing chamber (I.D. 0.11 m, height 1.7 m) from the bottom to
the top. A computer code, in which the particles agglomeration due to standing sound
wave motion is modeled, is developed and implemented in the commercially available
CFD software PHOENICS. The prediction the agglomeration rate is made for various
particle sizes, sound frequencies and sound pressures. The computational results are
compared with the experimental results obtained under the same conditions. Both the
computational and experimental results show that the agglomeration rate becomes
larger as the sound pressure increases. The finer are the particles in the exhaust gas, the
higher must be the frequency of sound in order to remove the particles from the gas.
Introduction
Many metallurgical processes are accompanied by formation of large amount of dust
in exhaust gas. Because the dust consists of very fine particles, treatment of such a
dust-loaded gas has required costly and technically complicated equipment. At present,
increasingly stringent regulation of air pollutant emission calls for further investigation
in improving the efficiency of exhaust gas purification methods.
The present study deals with a method of cleaning exhaust gas based on application of
powerful sound waves. The method has received the name “acoustic agglomeration”.
The acoustic agglomeration of dust particles has been long recognized as a method of
the dust removal from exhaust gas (1,2).
The first investigations on application of power sound waves to high temperature
metallurgical processes were undertaken in the former Soviet Union. The obtained
results revealed that sound wave is an attractive tool for the control of dust content in
exhaust gas carried off from metallurgical units (3,4). However, the efficiency of
acoustic agglomeration under high temperature is strongly dependent on such
parameters as sound frequency, intensity, temperature, and, physical properties of the
gas and dust particles. The effects of these parameters have been poorly understood yet.
Several mathematical models of the acoustic agglomeration are available today, but a
number of disadvantages inherent to these models prevent us to apply the models to the
metallurgical processes. The followings are the main disadvantages: (1) the models
consider particles to be stationary relative to the sound source: (2) the models ignore the
mutual motion among the particles: (3) the models ignore temperature variations within
the gas phase.
The goal of the present study is to develop a mathematical model that is applicable to
the prediction of the rate of agglomeration among various size of particles which are
exposed to sound waves of various frequencies and intensities under conditions of
variable temperature inside the sonoprocessing chamber. A part of the computed results
12 -
are to be verified experimentally.
Theory
Particle entrainment coefficient

For a better understanding of the model content, the acoustic agglomeration mechanism
is briefly reviewed here. One of the main origins of acoustic agglomeration is a relative
motion of two particles of different size affected by sound wave. Mednikov (5) showed
that when a particle of diameter dp is exposed to a sound wave of frequency f, amplitude
of the particle oscillation, Ap is differ from that of sound wave, A. The ratio of Ap to A is
called “entrainment coefficient”. If the particle is small enough to obey Stokes’ law (Re
< 0.1), the entrainment coefficient,  can be given by the following equations.
Ap
 p d p2
1


(1)

(2)
A
18 g
1   2 2
Here,  (=2f) is the angular frequency,  is the particle relaxation time (Eq.2) which is
a function of the particle density p, diameter dp and the gas dynamic viscosity g.
Figure 1 presents  as a function of frequency for three particle diameters and two
temperatures corresponding to those at the lower (800 ºC) and upper (20 ºC) parts of the
experimental apparatus used in the present study. The three particle sizes represent a
typical set of small (dp = 1 m), middle (dp =
0
10
1
8 m) and large (dp = 22 m) particles
observed in the experiments. Oscillation
amplitude of the smaller particle is
-1
practically not different from that of sound
10
2
wave even at higher frequencies. On the
other hand,  drastically decreases as
dp (m)
T
frequency increases for middle and larger
-2
3
1
10 1
0
particles. Since real dust is always a mixture
20 C
2
8
0
of particles of various sizes, sound wave of a
800 C
3 22
certain frequency should force smaller
-3
10
particles in oscillation, while larger particles
1
2
3
10
10
10
can remain stationary. This promotes a
f (Hz)
collision between the particles and their
Fig.1 Frequency dependence of
agglomeration. This effect is often called
entrainment coefficient
“orthokinetic mechanism”. In the present
mathematical
model,
the
particle
agglomeration was assumed to proceed only due to the orthokinetic mechanism.
Agglomeration kernel
The core of the proposed mathematical model is the agglomeration kernel. An
expression for the agglomeration kernel is based on the behavior of two groups of
particles of different size located in sound wave. Firstly, suppose that the particles of
one group are enough big for the effect of sound motion to be neglected (= 0) while
the particles of the other group are enough small for their inertial motion to be neglected
(= 1). In this case, all small particles, contained within a volume Va, should be collided
with the big particles during one acoustic oscillation cycle. Here, Va is often referred to
as “agglomeration volume”. As shown in Fig.2, the agglomeration volume is a cylinder,
22 -
the radius of which is the sum of radiuses of the big and small particles (R1 and R2), and
the height of which is equal to twice the amplitude, A, of the displacement of the small
particle. However, in the practical cases of particle agglomeration, both the big and
small particles are vibrated, i.e. 0<  < 1. In this case, the collision probability is
expressed as Va multiplied by 1 - 2 . The agglomeration kernel, 12, is hence given
by the following equation (3), where f is the sound wave frequency.
12  2 ( R1  R2 ) 2 Af 1  2
(3)
Considering the angular frequency  =
2f, and also considering the amplitude
of acoustic velocity in sound wave that is
given by the product of A and , one can
obtain another expression for the
agglomeration kernel as following.
12  ( R1  R2 ) 2 U a 1  2
(4)
The physical meaning of the
agglomeration kernel is the frequency of
collision between two particles of
different sizes when the volume concentrations of the particles of the both sizes are
unity.
Mathematical formulation
The motion of the dust containing gas within the vertically arranged sonoprocessing
chamber was simulated as a non-isothermal two-phase turbulent flow. For
simplification of computational procedure, the simulation was carried out in two steps.
At first, the gas phase flow was simulated on the assumption that the volume fraction of
particles in the gas phase was small enough so that any influence of particle presence on
the gas flow and gas properties was ignored. Then, mass transfer equations of particles
were solved on the assumption that the vertical component of particle velocity, Wp, was
smaller than that of gas velocity by the particle terminal velocity, WT, while the
velocities of gas and particles in the horizontal direction are the same. In the case of
small Reynolds number (Re < 0.1), WT of a spherical particle can be determined from
Eq.5.
 p d p2 g
WT 
(5)
18
The following assumptions were also made: (1) all particles are spherical; (2)
physical properties of gas are dependent of temperature; (3) particles are involved in
the eddy motion inside the sonoprocessing chamber in the same way as the fluid
particles of the gas; (4) two particles agglomerate at every collision; (5) once a particle
contact to the surface of the wall or bottom of the sonoprocessing chamber, it is
removed from the gas phase regardless with the temperature at the surface.
The fluid equations were formulated on the basis of the standard k- model for a
non-isothermal gas flow in axially symmetrical coordinates. The k- model includes the
following governing equations: gas-phase mass conservation equation, gas-phase
momentum equations in the r and z direction, conservation equations for the turbulent
kinetic energy and its dissipation, and gas-phase energy equation. The details of the
model can be found in textbooks on hydrodynamics (6).
Based on the polydispersivity of dust system, the ensemble of the particle system was
32 -
assumed to consist of M groups of particles of different sizes. The mass transfer
equation of the particles in k-th group can be written as the following.
 Ck   (VCk )  (WCk  WT Ck )   1  
 Ck   
 Ck  

 Deff r

 Deff
   S (6)


t  r
z
r  z
 z  k
 r  r 
Equation (6) represents time and spatial variations of particle number density, Ck, of
k-th group due to convection, diffusion and acoustic agglomeration (Sk). The particle
effective diffusion coefficient, Deff, is defined as

Deff  t (7)
Sct
where t is the turbulent viscosity of gas, and Sct is the turbulent Schmidt number which
is assumed to be unity in the present study.
The source/sink term Sk represents the rate of change in the particle number density
of the k-th group. Following T.L.Hoffmann(7), Sk can be given as
M
dCk 1 k
Sk 
   ij Ci C j  Ck   ik Ci (8)
dt
2 i jk
i 1
Here, the first term on the right side represents the “birth” of particles in k-th group due
to acoustically driven collisions between particles whose size is smaller then that of
particles of k-th group. The second term accounts for the “death” of particles in k-th
group due to agglomeration. In the case of polydispersed system, the general
agglomeration kernel,  ij, is represented by the following equation for the collisions
between the particles of i-th group and those of j-th group.
ij  ( Ri  R j ) 2 U a i   j
( 9)
In the present study, the acoustic agglomeration is considered under the condition of
standing sound wave. In the case of standing wave, the amplitude of acoustic velocity,
Ua, is a function of distance, z, from the sound source and the wave number k (=/c),
and is represented by
U a  U a ,0 cos( kz) (10)
Here c is the sonic velocity, and Ua,0 is the acoustic velocity amplitude at z = 0.
The commercial CFD code PHOENICS was used for the present numerical
simulation. Incorporating the equations (1), (2), (5) to (10) into the code allowed
predictions of the agglomeration rate for various particle sizes, sound frequencies and
sound pressures.
Experimental
The computational domain, boundary and initial conditions were set according to the
experimental conditions.
Figure 3 shows main parts of the experimental setup. Since the system was axially
symmetric, only the right half of the setup is presented in the figure. The working space
of the experimental apparatus was comprised of three graphite tubes of 108 mm in
I.D.(1) mounted inside an outer stainless steel shell (2), and a stainless steel adapter (3).
The lower part of the shell was installed inside a resistance furnace (4) heating a
graphite crucible (5) up to 9000C under an atmosphere of Ar. Sound waves were
generated by a powerful loudspeaker (6) fixed at the top of the adapter. A perforated
graphite disk (7) was placed between the lower and middle graphite tubes. The disk was
served as a sound wave reflecting plate providing a formation of a standing wave in the
42 -
space between the perforated disk and loudspeaker. The conditions of resonance
standing wave were determined prior to the experiments.
About 10 g of pure Zn was loaded into the crucible to
produce Zn vapor. The vapor was carried by Ar gas
(flow rate 2.6 Nl/min) and transported through the
perforated disk to the lower part of the sonoprocessing
chamber. Inside the sonoprocessing chamber,
temperature decreases gradually from about 8000C at the
disk to 400C at the loudspeaker. The temperature
gradient caused condensation of Zn vapor into liquid
drops and/or solid particles that were exposed to the
sound wave during 5 minutes as they rose through the
sonoprocessing chamber.
The chamber was equipped with four side tubes A, B,
C and D for dust sampling. The holes served also as a
gas outlet. Two methods were used for taking the dust
particle sample. The first one is the injection of the
particle containing gas into a pure water bath (200 ml).
Then, the particle size distribution in the suspension was
analyzed with a laser particle sizer. The second sampling
method is trapping the dust particles on a quartz plate
(11 cm) inserted into the sonoprocessing chamber
through one of the four side tubes A to D in Fig.3. The
plate was placed horizontally at the center of the cross
sectional area of the graphite tube. The particles trapped
on the plate were analyzed with SEM for size and shape.
Computational domain and boundary conditions
The numerical simulation was conducted in a computational domain shown in Fig.3.
The domain size was the same as the inner space of the sonoprocessing chamber. The
domain was divided into non-uniform grids of 708 computational cells in the axial and
radial directions, respectively. The cell size became smaller near the domain solid
boundaries and gas outlet as shown in Fig.3 for the case where the gas leaves the
sonoprocessing chamber through the outlet A.
A complete list of the boundary conditions is presented in Table 1. A non-slip
boundary condition is imposed for velocities V and W at the domain walls. The vertical
Table 1. Boundary conditions
P
V
Side wall
N.s.
0
W
0
k
0

0
T
T(z)
Top wall
N.s.
0
0
0
0
20C
Inlet
-gW02
0
W0
k0
0
800C
Ck(k)
Outlet
0
N.s.
N.s.
k
0
r

0
r
T
0
r
N.s.
52 -
Ck
C k
0
r
C k
0
z
F (%)
N.s. denotes non-setting condition.
component of gas velocity at the domain inlet, W0, was set to 0.0167 m/s according to
the experimental condition.
To determine the turbulent kinetic energy, k0, and its dissipation rate, 0, at the domain
inlet, the initial level of turbulence was set at a level of 5%. The velocities at the domain
outlet were computed from the pressure difference at the domain input and output
(-gW02 and 0, respectively). Logarithmic wall functions were used in the calculation of
the shear stress, turbulent kinetic energy, its dissipation, and the velocity components
parallel to the domain boundaries. Temperatures at the domain walls and the inlet were
those measured in preliminary experiments.
Particular attention was given to the analysis of initial particle size distribution, Ck(k),
at the domain inlet. According to the commonly accepted standpoint, two phenomena
are responsible for the formation of dust particles from vapor under a temperature
gradient. The first one is homogeneous nucleation of particles, and the second one is a
condensation of vapor on the surface of the formed nuclei. Unfortunately, a lack of
knowledge about the mechanism of these phenomena gives no way of describing the
rates of particle nucleation and growth mathematically. In the present study, it was
assumed that the formation of particles was finished before they enter into the
sonoprocessing chamber and that the further changes in the particle size occur only due
to their gravitational settling or acoustic agglomeration. Direct observation of the
particles at the chamber inlet was impossible due to high temperature. However, some
information on the initial particle size distribution could be retrieved from the following
experimental findings. The size of the particles trapped at the levels A to D by the
bubbling technique were found to obey to
the lognormal distribution of Eq.(11).
100
 (s  s ) 2 
dF
1
Without sound

exp 
 (11)
ds  s 2
2 s2 
80 Sampling point A

Here F is the cumulative distribution of
60
particles; s = log di, where di is the particle
40
diameter; s is the arithmetic mean of log
d; and s is the standard deviation of log d.
20
Second finding was that the order of
0
magnitude of the size of smallest particles
leaving the chamber was roughly
1
10
d (m)
estimated as 10-7 m, while the order of
Fig.4 Particle size cummulative distribution
magnitude of the size of biggest particles
in the chamber was found to be 10-5 m.
Based on these findings, the minimum and maximum diameters of particles in the
initial distribution were taken as 0.1 and 32 m, respectively, in the present simulation.
The standard deviation was found from best-fitting curve to the experimental
distribution for the sampling point A shown in Fig.4. The lognormal particle size
distribution was in good agreement with the experimental observations at s = 0.2.
Results and discussion
Analysis of the distributions of gas velocity and temperature over the computational
domain reveals that the temperature gradient along the height of the sonoprocessing
chamber gives rise to buoyancy forces that drive a vigorous convection inside the
chamber. Figure 5 presents velocity vector (a) and temperature (b) fields over the
62 -
computational domain. For the sake of convenience, all figures showing spatial
distributions of computational variables are stretched by 5 times in the radial direction
relative to the vertical direction. As seen in Fig.5 (a), there is a big circulatory loop
inside the chamber with the upward flow close to the chamber axis and downward flow
close to the chamber walls. In the loop, the vertical component of gas velocity is
increased from its initial value at the sonoprocessing chamber inlet (0.0167 m/s) by
more than ten times at a distance of about 10 cm above the inlet. This results in fast
change in the particle number density during about 30 seconds after the particles start
entering the chamber, and then the density remains almost unchanged as the time
passes. Therefore, in the followings, the
discussion is based on the results of
calculation at 30 s.
Figures 6 to 8 are the contour plots of
particle number density calculated for three
typical sizes of particles; small (1 m),
middle (8 m) and large (26 m),
respectively. The letters a), b) and c) under
the plots correspond to different
computational conditions: a) without sound
application, b) and c) with sound
application at SPL= 158 dB and frequencies
210 (b) and 990 (c) Hz, respectively. As
seen in the figures, in the absence of sound
a
b
application, small and middle particles fill
the chamber volume completely and their
number densities reach very high values. At
the same time, the number density of large
particles is close to 0 over the volume
Fig.5 Velocity vector (a) and temperature
(Fig.8(a)) because the gas velocity at the
(b) fields
sonoprocessing chamber inlet is not high
enough for the large particles to be picked
up by the upward gas flow.
The sound application results in a significant decrease in number density of the small
and middle particles, on the one hand, and in an increase in number density of the large
particles, on the other hand. However, the effect of sound on the particle number
density distribution is dependent on the frequency of the sound. For relatively higher
frequency ( 990 Hz ), the decrease in the number density of the small particle is
especially significant (Fig.6(c)). On the other hand, relatively lower frequency (210 Hz)
is more effective in reducing the number density of middle particles as it follows from a
comparison of Fig7(a), (b) and (c). The results can be explained in the terms of the
frequency dependence of entrainment coefficient (Fig.1) and agglomeration kernel
(Eq.9). Oscillation magnitude of small particles is practically independent of frequency.
However, the middle particles are oscillated with a much smaller magnitude at f = 990
Hz as compared to that at f = 210 Hz. Since the rate of collision between particles is
proportional to the difference in entrainment coefficients of the particles, the rates of
collision and agglomeration between smaller and bigger particles should be enhanced
as the frequency increases. This explains the difference in number density distributions
of small particles between at frequencies 210 and at 990 Hz.
In the case of middle particles, their number density is governed by the rates of two
contrary processes. The first one is the birth of the middle particles from smaller ones.
72 -
The second one is the particle death due to their agglomeration with other particles and
Fig.7 Number density contour of
middle particles (8 m)
going to another particle group as described
by Eq.(8). As it follows from the above
consideration, the higher frequency should
enhance the particle birth rate. On the other
hand, because the entrainment coefficient of
middle particle at lower frequency is larger
than that at higher one, the wave of 210 Hz
should result in a faster rate of the middle
particle death than that at 990 Hz. This is the
reason why the number density of middle
particles is higher at f = 990 than that at f =
210 Hz.
The same explanation is applied to the
behavior of the large particles. Exposing the
gas that contains particle to sound wave
a)
b)
c) resulting
a)
b)
c)
causes the particle
agglomeration,
Fig.6 Number density contour ofin the large particle formation in the
small particles (1 m)
sonoprocessing chamber. The formation rate
is especially high at the lower part of the
a)
b)
c)
sonoprocessing chamber where the number
Fig.8 Number density contour of
density of smaller particles is large. Because
large particles (26 m)
the agglomeration between small and middle
particles is enhanced at higher frequencies,
sonoprocessing of gas by the high frequency
wave (990 Hz) should result in a formation of big particles in a larger quantity than that
achieved by sonoprocessing with the low frequency wave (210 Hz). Higher frequency
is beneficial for the increase in the number density of the large particle also from the
82 -
p (%)
viewpoint of particle death rate because of the large difference in entrainment
coefficients at frequencies of 210 and 990 Hz. The rate of agglomeration of a big
particle with another particles of similar size should be much smaller at f =990 than that
at f = 210 Hz, which assures that the death rate of the large particle is smaller at higher
frequency.
Thus, the numerically predicted results of the number density of differently sized
particles reveal that the acoustically driven agglomeration is responsible for a change in
distribution of the particle number density in such a way that the number of small
particles reduces while the number of big particles increases. These results can be
verified by the experiments. Figure 9 shows distribution of particle in size for two
cases: a) without sonoprocessing and b) with sonoprocessing with frequency of 990 Hz.
The particle samples were taken at the point A (Fig.3) with the bubbling technique. At a
quick look at, it is obvious that the sonoprocessing causes a shift of the particle size
distribution toward larger sizes. Here, it is noted that we could find neither small (less
than or equal to1 m) nor big (larger than or equal to16 m) particles in the samples due
to the inherent limitation of the particle sampling method. In the case of bubbling
technique, very fine particles are carried away from the bubbler together with carrier
gas flow, while very large
20 Without sound With sound, f =992 Hz
particles fall down and stick
SPL = 152 dB to the bottom part of the
16
sampling tube during they
fly through sampling tube.
12
It is also noted that the
presence of very fine and
8
large particles in the
sonoprocessing
chamber
4
has been confirmed by the
other sampling methods.
0
d
(

m)
Both the predicted and
1
10
1
10 p
experimental results show a
Fig.9 Particle size distribution
significant reduction in the
total number density of particle due to the acoustic agglomeration. Figure 10 is a plot of
the relative total number density of particles, N/N0, versus sound pressure at a
frequency of 210 Hz. Here, N0 denotes the total number density in the absence of sound
1.0
Experiment
Simulation
N/N0
0.8
f = 210 Hz
0.6
0.4
0.2
0
1
2
3
Ps (kPa)
Fig.10 Dependence of total relative number density of
on particles sound pressure
92 -
wave. As seen in the figure, the higher the sound pressure the smaller is the total
number density of particles. The mechanism of particle reduction is explained as
follows. The particles agglomeration induces the increase in the average size of particle
in the gas phase. This results in an increase in average particle terminal velocity (Eq.5),
which leads, in turn, to an enhanced precipitation of enlarged particles onto the
sonoprocessing chamber bottom and thus to the decrease in total number density of
particles. However, in Fig.10, the computed results predict more steep reduction in
N/N0 with sound pressure than the experimental result. This result is supposedly
attributed to an overestimating of agglomeration probability originated from the first
and fourth assumptions in the proposed mathematical model mentioned in the previous
section. Actually, non-spherical particle formed during agglomeration must have a
larger drag coefficient relative to gas flow than the spherical one. If some particles do
not agglomerate but rebound each other after collision, then the efficiency of acoustic
agglomeration must be smaller than the present model. In these cases, the rate of
agglomeration can become smaller than the calculation in the present study. The effect
of various particle shapes and a coefficient of particle rebounding should be included in
an improved model. Such an improvement is to be made in the next step of our
investigation.
Conclusions
A mathematical model of the agglomeration for differently sized particles under
different conditions of sonoprocessing of high temperature particle laden exhaust gas
was developed and implemented in the CFD software PHOENICS. The sound
frequency and sound pressure were found to be of prime importance in controlling the
distribution of particles in size and their number density in exhaust gas. Generally,
acoustically driven agglomeration results in the decrease in number density of fine
particles and increase in number density of large particles. According to the results of
numerical simulation, the finer the particles in the exhaust, the higher must be the
frequency of sound in order to proceed the agglomeration between the finer and larger
particles. The total number density of particles in exhaust gas is drastically decreased as
the sound pressure increases. The results of model calculation agree with experimental
results qualitatively well.
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