Journal of Advanced Computing (2012) 1

Transcription

Columbia International Publishing
American Journal of Heat and Mass Transfer
(2015) Vol. 2 No. 1 pp. 42-58
doi:10.7726/ajhmt.2015.1004
Research Article
The Use of Quadrature Method of Moments (QMOM)
in Studying the Mechanisms of
Aerosol Agglomeration and its Practical Use
Dong Liao1, Cheng Wu1, Ye Yao1*, Huiming Hu2, Fang Zhao 2, and Daolai Chen2
Received 18 January 2015; Published online 28 March 2015
© The author(s) 2015. Published with open access at www.uscip.us
Abstract
The efficiency of dust collector is relatively low when dealing with PM2.5. Agglomerating the aerosols under
acoustic field is a preliminary method to make the efficiency improve. Acoustic agglomeration of aerosol
mainly includes four mechanisms: orthokinetic interaction, hydrodynamic interaction, acoustic wake effect
and Brown agglomeration. All mechanisms have their own characteristics with the change of acoustic field.
Among witch orthokinetic interaction plays the leading role, while Brown agglomeration is the weakest.
Besides, Brown agglomeration is only related to the change of temperature and other air characteristics, thus
it changes little under the effect of acoustic field. Compared with experimental methods, simulating the
agglomerate process on computer is much faster and more economical. To overcome the flaws of traditional
ways in simulation of long-time, hard to make exact hypotheses and difficult to make the formulas close,
quadrature method of moments (QMOM) is employed when doing simulations, and taking it into practical
use. The results of QMOM show a good match with group method, while it takes less time and does not need
to make unnecessary hypotheses. Besides studying the characteristics of acoustic agglomeration, QMOM will
also be easy to be used in other fields of studies with similar dynamic equations.
Keywords: Aerosol; Agglomeration; Mechanisms; QMOM
1. Introduction
Acoustic agglomeration can promote the relative motion of fine particles under the hi-density
energy caused by high-strength acoustic field, thus improving the agglomeration rate of the fine
particles, which will largely improve the effect of dust collectors.
Researches on the mechanisms of acoustic agglomeration mainly include the mechanisms of
orthokinetic interaction, hydrodynamic interaction, acoustic wake effect and Brown agglomeration.
______________________________________________________________________________________________________________________________
*Corresponding e-mail: [email protected]
1 School of Mechanical Engineering, Shanghai Jiaotong University, China
2 School of Urban Construction and Safety Engineering, Shanghai Institute of Technology, China
42
Dong Liao, Cheng Wu, Ye Yao, Huiming Hu, Fang Zhao, and Daolai Chen /
American Journal of Heat and Mass Transfer (2015) Vol. 2 No. 1 pp. 42-58
Among these mechanisms, orthokinetic interaction is always to be regard as the main mechanism of
acoustic agglomeration (Chiu and Edwards, 1996; Rajendran et al., 1979), and plays the decisive
role. Hydrodynamic interaction is based on Bernoulli law, considering the motion of particles under
the effect of the flow field, which includes mutual radiation pressure interaction and acoustic wake
effect. This mechanism can be used to explain the force between long-distant particles and the
agglomeration among monodisperse particles (Hoffmann and Kooplmann, 1996). Besides, acoustic
wake effect can be used to explain the refill mechanism of acoustic agglomeration further. Brown
agglomeration is the agglomeration under the effect of Brownian motion (Otto and Fissan, 1999),
which is quite different from other mechanisms, for it is mainly defined by the temperature and the
particle size distribution while the intensity and the frequency seem to have little relationship to
this effect.
Based on the mechanisms above, related aerosol kinetic equations can be formed, and the kernel
functions related to numerical simulation of the particle agglomeration process can also be derived.
Then the law of the particle agglomeration process influenced by the changes of several
environment variations will also be concluded. By now, common numerical simulation methods
include group method, moment method and Monte Carlo method (Park and Lee, 2000). Among
these methods, moment method is quite a fast method compared with others. It will not be able to
get the particle size distribution, while many other important outcomes, like particle quantity
concentration and particle mass concentration etc., are easily to get. However, it is not easy to solve
the moment functions.
In order to overcome these flaws, a new method called quadrature method of moments (QMOM)
(Su et al. 2007) was proposed. With the help of QMOM, particle distribution moment functions can
be closed under any conditions (McGraw, 1997), for there is no need to make hypotheses of the
particle size distribution. The characteristic of this method is to integrate the dynamic equations
firstly, so it focuses on the overall change of the aerosols, and does not need to make hypothesis of
the aerosol dispersion at each time of calculation. Besides, since it considers the overall dispersion
of the aerosols, QMOM also takes less time to get the outcomes compared with other methods like
group method. It was introduced firstly in studying the growing mechanism of sulfuric acid droplet
(McGraw, 1997), then QMOM has been used to study the law of particle breaking, sedimentation
and growing etc. (Attarakih et al. 2009). While in particle agglomeration, some researchers have
put it in the study of orthokinetic interaction (Zhang, 2010), further studies are still needed. In this
paper, the QMOM is employed to study the mechanisms of hydrodynamic interaction, acoustic wake
effect and Brown agglomeration.
Table 1 Nomenclature
Symbols
Meanings
d
The diameters of particle
v
The volumes of the particles
c
The concentration of particle
u0
The vibration velocity of gas medium
43
s
The distance between particles i and j
f
The frequency
Lij
The length of related agglomeration volume
Tf
The time period of sound wave
l
Slip coefficient under Oseen flow field conditions
n
Slip coefficient under Stokes flow field conditions
h
An intermediate variable
I
Acoustic intensity
I0
Standard reference value of SPL
t
Time
k
The Boltzmann constant,k=1.38*10-23 J/K
r
Aerosol particles floating in the radius
gj1
The first element in the first line in the characteristic vector
D
Diffusion coefficient of particles i and j
m0
The number of aerosol particles concentration
N0
The number of aerosols
l,h,q
Intermediate variables for the purpose
L, w
The corresponding abscissa particle size and weighting factor
e, g
The characteristic value and characteristic vector for Jacobi matrix
Dm,
The average diameter of aerosols
SD
The standard deviation of particles
VSV
Velocity of sound velocity.
SPL
Sound pressure level
DC
The drag coefficient of particles in flow field
AT
Absolute temperature
n(r)
Distribution function for the number density of particles
44
mk(t)
The moment
Kor
The kernel function of orthokinetic agglomeration
KHy
The kernel function of Hydrodynamic Interaction
KAW
The kernel function of Acoustic Wake Effect
KBr
The kernel function of Brownian Agglomeration
Table 2 Greek Symbols
Symbols
Meanings
ω
The angle frequency
υ
The kinematical viscosity coefficient
μg
The slip coefficient
μp
The carrying coefficient
μ
Fluid viscosity coefficient
τ
The relaxation time
ρg
The gas medium
ρp
Density of particle
2. Agglomeration Models and Kernel Functions
2.1 Brief Introduction of Kernel Function
The agglomeration process leads to a reduction in the total number of particles and an increase in
the average size. The net rate of generation of particles of size k can be described as below
(Friedlander, 1977):

dck 1 k
  K  i, j  di d j  dvi  ck  K  i, k  di  dvi
0
dt 2 0
(1)
where the first item in the right of the formula is the collision frequency function; the second
represents the rate of loss of particles of size k by collision with all other particles; di and dj is the
diameters of particle i and j; vi and vj are the volumes of the particles, and vk=vi+vj; particle k is
agglomerated from particles i and j; vk is the volume of particle k; ck is the concentration of particle
k; K(i,j) is the kernel function of particles i and j, which represents the number of collisions
occurring per unit time per unit volume between the two classes of particles.
To get the numerical solution with computer, Eq. (1) is converted into discrete form called
Smoluchowski function as follows (Sarabia, 2003):
45
m
dck 1
  K  i, j  ci c j  ck  K  i, k  ci
dt 2 i  j k
i 1
(2)
The Smoluchowski function is based on the idea that the particles are grouped in terms of the
diameter from small to large.
2.2 Orthokinetic Agglomeration Mechanism
Under the effect of acoustic field, the motions of the particles will be quite different due to the
differences in mass, volume, inertial and etc., which will cause relative motions among the particles
and finally lead to agglomeration. Assuming that only two types of particles, big and small, exist in
the model and all of them are spherical. Big particles play the role of collecting kernel; the small
particles around them are waiting to be agglomerated. The space that a big particle can move under
the effect of acoustic field is called agglomeration volume (Mednikov, 1965), only in which the big
and small particles are possibly to agglomerate. Then assuming that the concentrations of small
particles in and out of the agglomeration volume are the same, and new small particles would
immediately fill in after the particles in the agglomeration volume are agglomerated.
In numerical simulation, the kernel function of orthokinetic agglomeration can be written as below
(Medikov, 1965):
  i  j
2
1
KijOr  u0  di  d j 
2
1   2 i2 1   2 2j
(3)
where di and dj are the diameters of particles i and j. And the relaxation time is:
 pd 2
=
18 g
(4)
where ρp and ρg represent the density of particle and gas medium, respectively; ν is the kinematical
viscosity coefficient; u0 is the vibration velocity of gas medium, which can be conducted from
acoustic field intensity I (W/m2):
u0 
2I
 g  VSV
(5)
where VSV is velocity of sound velocity; I is acoustic intensity, which can be expressed by Eq. (6).
In practical, sound pressure level (SPL (dB)) usually represents the intensity of acoustic field, which
can be expressed by:
SPL
I  10 10  I 0
(6)
where SPL is sound pressure level, dB; I0 is the standard reference value.
2.3 Hydrodynamic Interaction
According to orthokinetic agglomeration, the forces among particles are limited to the space of
agglomeration volume, and only exist between the big and small particles when ignoring the
electrostatic force. However, experiment manifested that the forces exist among monodisperse
particles, and the distance is far beyond the space of agglomeration volume. Considering the
46
movement of flow medium under the effect of acoustic field, the hydrodynamic interaction is
introduced, which mainly includes the effect of mutual radiation pressure interaction and acoustic
wake effect.
For the effect of mutual radiation pressure interaction (Danilov and Mironov, 1984), when the
directions of the sound wave and the line of connecting the two particles are vertical, the two
particles will be closer since the space for the medium decreases and speed of the medium
increases, which leads to a drop of pressure between the two particles. When the directions of the
sound wave and the line of connecting the two particles are parallel, the resistance to the medium
will reduce the medium speed, which leads to an increase in pressure between the two particles.
Thus the two particles move apart.
Because of the effect of acoustic wake (Hoffmann, 1997), if relative motion between particles and
flow medium exists, the wake will formulate behind a particle for the asymmetry of flow field under
Oseen flow conditions. The pressure in the wake is lower, so the particles within this space will
move closer. Under the effect of acoustic field, the movement of flow medium is reciprocating, so
the direction of the wake will also change. Thus, the particles in the wake will gradually move closer
until agglomeration.
Of the two effects, the theoretical effect of mutual radiation pressure interaction is much weaker
than that of acoustic wake (González et al., 2003). Analyzing the forces among particles in the flow
field based on Bernoulli law, the kernel function can be conducted as follows (Wang, 2012):
2  3u 
u
1

KijHy    di  d j   0  2di bi  2d j b j  0 di2bi2  d 2j b 2j 
4


 8 s 
(7)
2

9u0
3u0
2 2
2 2

di bi  d j b j 
bi b j  bi qi  b j q j  di  d j  
64 s 2
16s 2





 pi  H i  gi2
 pj  H j  gj2
qi 
; qj 
1  H j  gi2
1  H j  gj2
 gi
 gj
; bj 
bi 
2
1  H j  gj2
1  H i  gi
9  g u0
9  g u0
; Hj 
Hi 
 p di
 p d j
 j
 i
;  gj 
 gi 
2
2
1   i 
1   j 
 pi 
1
1   i 
2
;  pj 
1
1   j 
2
(8)
(9)
(10)
(11)
(12)
where H, b, and q are intermediate variables for the purpose to make the functions seem clearer; s
is the distance between particles i and j; μp and μg are the carrying coefficient and slip coefficient.
47
2.4 Acoustic Wake Effect
After one agglomeration period, small particles in agglomeration volume will be supplemented
immediately. Considering the long-distant effect of acoustic wake, the mechanism of refill can be
well explained by acoustic wake effect (Dong et al., 2006). Then combining mechanisms of
orthokinetic agglomeration and acoustic wake effect, the expanding kernel function in numerical
simulation is provided as below (Wang, 2012):
 fu0  di  d j   i   j


6u0  di li  d j l j  T f

  1


1
2
2 
2


L
2
ij
1   i  1   j  

nj
ni
; lj 
li 
1  2h j n 2j  h 2j n 4j
1  2hi ni2  hi2 ni4
2
KijAW 
u0  i   j
Lij 
ni 
hi 
1   i 
2
; nj 
(14)
(15)
1   2 i2 1   2 2j
 i
(13)
 j
1   j 
2
4u0  g
4u0  g
 ; hj 

9 di  p
9 d j  p
(16)
(17)
where Tf is the time period of sound wave; l and n are slip coefficient under Oseen and Stokes flow
field conditions, respectively; Lij is the length of related agglomeration volume; h is an intermediate
variable, to make the functions seems clearer.
2.5 Brownian Agglomeration
Besides the effect of acoustic field, the diffusion motion of particles caused by Brownian motion is
also one of the agglomeration mechanisms. Unlike other mechanisms, Brown agglomeration is not
related to the intensity and frequency of acoustic field, and its kernel function is as follows:
(18)
KijBr  2 Di  D j di  d j



where Di and Dj are diffusion coefficient of particles i and j, and their functions are below:
Di =
k  AT
k  AT
; Dj =
DC j
DCi
DCi =3 di ; DC j =3 d j
(19)
(20)
where DCi and DCj are the drag coefficients of particles in flow field; μ is fluid viscosity coefficient; k
is the Boltzmann constant, and AT is absolute temperature.
3. The Use of Quadrature Method of Moments and Results
3.1 Theoretical Model and Algorithm
48
QMOM algorithm of aerosol particle size distribution of k order of moments can be defined as
(Marchisio et al., 2003a):

mk   n(r )r k dr
0
k  0,1, 2
(21)
where r stands for the radius of aerosol particles; n(r) is the distribution function for the number
density of particles.
Different moments have their own physical meanings (McGraw et al., 1998), and it is helpful to
analyze some properties of aerosol particles. Moments as m0, m1 and m2 can be got when k equals
to 0, 1 and 2. In this paper, only m0 is used, which means the number of aerosol particles
concentration. For other moments like m1 and m2 stand for the total of particle diameters and
superficial areas, respectively, which are not closely related to this study. Based on aerosol dynamic
formulas, the density formulas can be approximated gotten as below (Marchisio et al., 2003b):
k /3

dmk (t) 1 
  n( , t)  K ( ,  )(  3   3 ) n(  , t) d  d 
0
dt
2 0


(22)
  L n( L, t)  K ( L,  ) n( , t )d  dL
k
0
0
k=0, 1, 2…
where the first item in the right of the equation means the rate of birth of particles due to
aggregation of smaller particles; the second represents the rate of death of particles due to
agglomeration with other particles; n(λ, t) is the number density function in terms of the particle
volume; K(λ, β) is the volume-based kernel function; L stands for the abscissa particle size; λ and β
stand for the volume of the particles; t is time.
When taking k from 1 to N, N corresponding formulas are gotten. By using QMOM, the N
corresponding formulas are closed and will be solved easily (Marchisio et al., 2003c). And the
moments of the particle size distribution are:
Nq
mk (t)   wi (t)Lki (t)
(23)
i 1
where w and L stand for weights and abscissas determined through PD(product-difference)
algorithm.
In order to calculate the corresponding abscissa particle size Li and weighting factor Wi, PD
(product-difference) algorithm can be used (Gordon, 1968; Dette and Studden, 1997). Assuming the
characteristic value and characteristic vector for Jacobi matrix is e and g by PD, we have：
Li  ei
wi  m0 g j12
(24)
where gj1 is the first element in the first line in the characteristic vector.
The initial value of Mk can be calculated by the following formula:
mk (t  0)  N0  exp(k ln(Dm )  k 2 (ln SD)2 / 2)
(25)
where Dm is the average diameter of aerosols; SD stands for the standard deviation of particles.
49
Initial amount: input
frequency, sound pressure
level, temperature, grain
density, geometric standard
deviation, particle size,
time and etc
Through the
formula to
calculate Mk
initial value
when t = 0
Agglomeration
kernel function
Kor，Kaw，Khy，Kbr
Computing the
next moment the
eigenvalue Li(t)
and eigenvector
Wi (t)
Again through the
calculation of
eigenvalue and
eigenvector to get
Mk(t)
Y
t<tm
N
Draw and print
Mk(t)
Fig. 1. QMOM simulate aerosol reunification process flow chart
The calculation procedure for simulating the process of aerosol acoustic agglomeration is presented
in Fig.1 (Zhang, 2010).
50
3.2 Numerical Simulation of Acoustic Agglomeration
3.2.1 The agglomeration effect of four mechanisms
In numerical simulation, except for Brownian motion simulation, all parameters stay unchanged
except for sound frequency which ranges from 10 to 100000Hz. The basic conditions are given as
below: sound pressure level (SPL) is 150dB; temperature is 30 degrees Celsius; the initial number
of aerosols (N0) is 100000; particle density is 2400kg/m3; the standard deviation (SD) of particles is
1.6; the simulation time is 4 seconds.
(1) The Influence of Frequency on Orthokinetic Agglomeration
Dm=6m，SD=1.6
N0 (105 m-3)
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
10
100
1000
10000
100000
1000000
f(HZ)
Fig. 2. The influence of frequency on the acoustic agglomeration with Kor
Fig.2. presents the influence of frequency on the orthokinetic agglomeration, where N0 is the
remaining number of particles after agglomerating 4 seconds, and it is equal to the moment m 0.
Smaller N0 means better agglomeration effect. Here the initial number of N0 is set to 100000. As the
frequency changes, an optimal frequency for agglomeration exists. When the average diameter of
aerosols is 6μm, the optimal frequency is about 4000Hz. Since orthokinetic is the leading effect in
agglomeration process, this frequency is quite close to the optimal frequency in practice (Shuster et
al., 2002).
(2) The Influence of Frequency on Hydrodynamic Interaction Mechanism
51
N0(105m-3)
1.00
Dm=6m,SD=1.6
0.95
0.90
0.85
0.80
0.75
10
100
1000
10000
100000
1000000
f (HZ)
Fig. 3. The influence of frequency to acoustic agglomeration of Khy
Fig. 3 shows the number change of aerosols under the effect of hydrodynamic interaction, where
the average particle size is 6μm; standard deviation of the aerosols is 1.6. It is found that the effect
of hydrodynamic mechanism becomes more significant as frequency goes higher. However, when it
reaches over 12000Hz, the effect of kernel function grows slowly. Thus when considering the effect
of hydrodynamic, there is no such thing of an optimal frequency as it is in orthokinetic effect, but a
sudden-change point, above which the effect can only grow slowly. And even if the frequency comes
to 1000000Hz, over 75% of the particles still remain.
(3) The Frequency Influence on the Acoustic Wake Effect
1.1
Dm=6m,SD=1.6
1.0
N0(105m-3)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
10
100
1000
10000
100000
1000000
f(HZ)
Fig. 4. The influence of frequency to acoustic agglomeration of Kaw
52
The influence of frequency on the acoustic wake effect is shown in Fig. 4. The effect goes in the
same way of that in hydrodynamic mechanism. As frequency becomes higher, the effect is more
obvious. The principle of acoustic wake effect explains that, the higher the frequency, the
differences between the speed of particles and fluid go larger. And the effect of acoustic wake goes
more obvious as well. But before the frequency reaches a specific point, the effect of acoustic wake
is extremely low. But when it comes to high frequency zone, the effect becomes much stronger.
(4) The Influence of Temperature on Brownian Motion
0.89
Dm=5m,SD=1.6
0.88
N0(105m-3)
0.87
0.86
0.85
0.84
0.83
0.82
0.81
0
200
400
600
800
1000
T(℃ )
Fig. 5. The influence of temperature on acoustic agglomeration of Kbr
Fig. 5. shows the influence of temperature on acoustic agglomeration of Brownian motion, where
the following two parameters in Brownian motion kernel function are changed: one is the average
particle size and standard deviation, another is the temperature. In this case, the standard deviation
and average particle size are given as 1.6 and 5μm, respectively; sound pressure level (SPL)
=150dB; the initial number of aerosols is 100000; particle density is 2400kg/m3; the temperature
range is 273-1273K. This effect is quite weak compared with other effects.
3.2.2 The Comparison between QMOM and Group Method in Kernel Functions
In this section, the results are compared to the researches previously (Wang, 2012). More changes
of parameters, like the aerosol’s average diameter and standard deviation, of the simulation process
are take into account, and the changes of kernel functions (Kor, KHy, KAW, KBr), which represent the
reduction number of N0 at each simulation time step, are shown in the 4 agglomeration
mechanisms. Here larger kernel function means better agglomeration effect.
(1) The Comparison of Orthokinetic Agglomeration
53
Fig. 6 shows the influence of sound frequency on kernel function Kor at different diameters and
SDs. The diameters of 2.5μm, 5μm and 10μm are set in QMOM. The standard deviation of the
aerosol is 1.8, a relatively large parameter, which represents the dispersion degree of the particles.
And in the paper (Wang, 2012) using group method, the large particle is as large as 5 times of the
small ones, a large number as well.
The optimal frequencies in agglomeration for the diameter of 2.5μm, 5μm and 10μm are 15859Hz,
39910Hz, 1010Hz respectively in QMOM, which accord well with the frequencies in group method
of 15850Hz, near to 4000Hz and 1000Hz. Besides, the overall changing tendencies of the two
methods accord with each other, too. While QMOM needs less time to get the conclusion.
1.35
di=2.5m,SD=1.8
di=5m ,SD=1.8
1.30
di=10m ,SD=1.8
Kor(103s-1)
1.25
1.20
1.15
1.10
1.05
1.00
10
100
1000
10000
100000
1000000
f(HZ)
Fig. 6. The influence of frequency on the kernel function of orthokinetic agglomeration
(2) The Comparison of Hydrodynamic Interaction Mechanism
As it is shown in Fig. 7, the change of kernel function KHy at different aerosol dispersion condition is
given. In hydrodynamic mechanism, the sudden-change point is the most important sign to show
the results under different particle size distribution conditions. Here the diameters of the particles
are 5μm, 10μm and 10μm, and the standard deviations are 0.6, 1.4 and 1.8. The standard deviations
have the same tendency to the particle size disperse in group method. And the sudden-change
frequencies of QMOM and group method (Wang, 2012) are near to 1000Hz, accord with each other
as well.
54
di=10m,SD=1.4
1.8
di=10m,SD=1.8
di=5m ,SD=0.6
1.7
Khy(108s-1)
1.6
1.5
1.4
1.3
1.2
1.1
1.0
10
100
1000
10000
100000
1000000
f(HZ)
Fig. 7. The influence of frequency to the kernel function of hydrodynamic interaction mechanism
(3) The Comparison of Acoustic Wake Effect
1.08
di=2.5m,SD=1.6
1.07
di=5m ,SD=1.6
Kaw(×10-6s-1)
1.06
1.05
1.04
1.03
1.02
1.01
1.00
0.99
10
100
f(HZ)
1000
10000
Fig. 8. The influence of frequency on the kernel function of acoustic wake effect
55
Fig. 8. Represents the effect of sound frequency on kernel function KAW at two different aerosol
dispersion conditions. The results also show a sudden-change point, while it is on the opposite
direction. Similar to that in group method, the diameters of the particles are 2.5μm and 5μm. Since
the diameters of large particles is as five times as that of the small particles, standard deviation in
QMOM is set to 1.6. Comparing with group method, the sudden-change frequencies are near to
1000Hz (Wang, 2012). The two methods are in accord with each other.
(4) The Comparison of Brownian Motion
1.90
1.88
di=2.5m,SD=0.1
1.86
di=1m ,SD=2.3
1.84
Kbr(1011s-1)
1.82
1.80
1.78
1.76
1.74
1.72
1.70
1.68
1.66
10
100
1000
T(Celsius)
Fig. 9. The influence of temperature to the kernel function of Brownian motion
As it is shown in Fig. 9. As temperature rises, the effect of Brownian agglomeration is better. This is
because of the rising temperature leads to higher possibility for particles to collision, which causes
the Agglomeration efficiency to rise (Oubella et al., 2014).
In the QMOM program, the sound source frequency is 1400Hz and sound pressure level is 150dB,
which are the same to group method (Wang, 2012). The temperature varies from 10 Celsius to
1000 Celsius when the grain size are di=2.5μm, SD=0.1 and di=1μm, SD= 2.3.
4. Conclusion
By analyzing the results of four mechanisms, the outcomes of QMOM have a good match with that of
group method, which has turned out to be reliable, and take less time. Besides, QMOM takes the
56
overall distribution of the aerosol into consideration by dealing with the average diameter and
standard deviation. By this way, the simulation process is closer to real conditions for it does not
need to make unnecessary hypotheses.
Among the four mechanisms, Brownian motion is the weakest while orthokinetic is the strongest.
And in general conditions, the influence of Brownian motion can be ignored. Hydrodynamic
interaction and acoustic wake effect are stronger as sound frequency becomes larger, but
orthokinetic agglomeration still plays the leading role. The optimal agglomeration frequency can
also be able to get by using QMOM, and this will be helpful to enhance the effect of dust collector in
practical use. Besides frequency and temperature, QMOM will also be able to study the influence of
other parameters like sound intensity, agglomeration time and etc. and other fields as well.
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