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Modification of the agglomeration kernel and simulation of the flow pattern in acoustic field with fine particles
Powder Technology 356 (2019) 930e940
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Powder Technology journal homepage: www.elsevier.com/locate/powtec
Modification of the agglomeration kernel and simulation of the flow pattern in acoustic field with fine particles Jianxiang Zheng*, Yukai Li, Zongqun Wan, Wenpeng Hong, Long Wang School of Energy and Power Engineering, Northeast Electric Power University, Jilin, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 30 September 2018 Received in revised form 10 August 2019 Accepted 7 September 2019 Available online 11 September 2019
This paper studies the acoustic agglomeration of fine particles of the coal fly ash in the acoustic field agglomeration device, based on the coupling population balance model (PBM) for acoustic agglomeration of fine particles and the modified kinetic theory of the fine particle flow. The new root mean square of the agglomeration kernel function operates according to the acoustic agglomeration mechanism. We present the theoretical investigation of the problem and then compare it with experimental results. The results show that the agglomeration process of particles in different size ranges is dominated by different agglomeration mechanisms. Moreover, we show the numerical simulation of the acoustic agglomeration process of the particles using the new agglomeration kernel function, which is consistent with the experimental results. In addition, the particle flow in the acoustic field agglomeration device presents a typical turbulent structure, and a large particle agglomeration occurs near the wall and outlet. © 2019 Elsevier B.V. All rights reserved.
Keywords: Modified kinetic theory Population balance model Collision efficiency Agglomeration kernel
1. Introduction The emission of dust pollutants from the coal-fired power plants is one of the main sources of the particulate matter of the air in China [1]. Traditional particle filtration devices, i.e., the electrostatic precipitator, cyclone separator and fabric filter, can capture >99% of the particles whose diameters are >2.5 mm[2]. However, the collection efficiency of the fine particulate matter (PM2.5) with the aerodynamic diameter of <2.5 mmis extremely low. The PM2.5 easily absorbs harmful substances, and causes a substantial harm to the human health [3e4]. Therefore, it is necessary to use physical or chemical means to pretreatment fine particles before entering into the traditional filter device. In general, five types of particle peragglomeration technologies exist, including electrical [5], magnetic [6], turbulent [7], chemical [8] and acoustic [9] agglomerations. The acoustic agglomeration uses the high intensity sound field for forcing the particles to collide. Owing to the irregular surface meshing of the particles and strong surface attraction of the molecules (van der Waals force, liquid bridge force, etc.), the collision of particles leads to the formation of bigger particles in the acoustic field. The acoustic agglomeration represents the most potential
* Corresponding author at: School of Energy and Power Engineering, Northeast Electric Power University, No. 169, Changchun Road, Chuanying District, Jilin 132012, China. E-mail address: [email protected] (J. Zheng). https://doi.org/10.1016/j.powtec.2019.09.022 0032-5910/© 2019 Elsevier B.V. All rights reserved.
particle pretreatment technologies, which exhibits high efficiency and is not affected by high temperature flue gas. Such a technology has been extensively studied in the past few years. The first experimental study on the agglomeration of suspended particles inside the acoustic field is reported in Ref. [10], where the agglomeration of the suspended particles at the standing wave nodes as acoustic agglomeration was shown. The experimental investigation of the acoustic agglomeration of the Zn particles inside the high-temperature gas was shown in Ref. [11], where the optimal agglomeration effect was achieved at 210 Hz and 991 Hz. Compared with the silent wave action, the average particle size was increased by 50% and the particle number concentration was decreased by 60%. Afterwards, the experimental demonstration of the agglomeration of the coal fly ash was reported in Ref. [12], using high frequency (20KHz) and low frequency (1000-1800 Hz) sound sources. The results in Ref. [12] revealed that, a low frequency sound source is more suitable for the particle agglomeration, where the optimal agglomeration frequency of the coal fly ash was shown to be 1400 Hz. Vladas in Ref. [13] conducted an acoustic agglomeration experiment on the particles in the cyclone separator. The results showed that the agglomeration of 1e5 mm particles occurs for the sound pressure level of between 130 dB and 170 dB. Hence, the capture efficiency was improved by 10.3% compared with the conventional cyclone separator. In general, the experiment quantitatively shows the validity of the acoustic agglomeration technology, however, the complexity of the acoustic agglomeration process and limitation of the testing methods, it is difficult to study
J. Zheng et al. / Powder Technology 356 (2019) 930e940
the acoustic agglomeration process experimentally. With the development of the computing technology, the numerical simulation provides an effective tool for the study of the acoustic agglomeration. At present, the acoustic agglomeration mechanism of the particles is composed of the orthokinetic agglomeration mechanism, hydrodynamic mechanism and the Brownian agglomeration mechanism. The particle agglomeration in the acoustic filed was numerically simulated using the Monte Carlo method in Ref. [14]. By a simple linear addition of all agglomeration mechanisms, it is found that the predicted results are not adapted to the extensive experimental conditions. Next, the numerical simulation of the hydrodynamic effect of the aerosol particles under the Oseen flow conditions was reported [15]. Through the study of the particle size and acoustic frequency, it is demonstrated that the hydrodynamic effect represents the main agglomeration mechanism of the mono-disperse aerosol particles, or the main complementary mechanism of the orthokinetic mechanism of the multi-dispersed aerosol particles. The acoustic agglomeration effect of the PM2.5 in standing waves is then investigated in Ref. [16] by liner addition of different sound agglomeration kernels. The results revealed that the orthokinetic agglomeration plays a leading role in the acoustic wave agglomeration, while the brown diffusion effect has minor effect on the particle agglomeration. The current scientific researches on the mechanism of the acoustic agglomeration include the linear combination of all agglomeration mechanisms and the coupling effect between them. This article investigates the acoustic agglomeration process of the cohesive fine particles in the acoustic field agglomeration device. First, we establish the modified kinetic theory of fine particle flow to describe the flow of fine particles. This theory is based on the Gidaspow’s classical kinetic theory of granule flow (KTGF)[17], which consists of the viscous collision of particles. Next, a population balance model (PBM) will be introduced to describe the change of the particle size distribution in agglomeration process [18]. Then, we employ the collision efficiency to modify the orthokinetic agglomeration kernel function, and propose the root mean square of the agglomeration kernel for the particles of the acoustic agglomeration. Finally, the root mean square of the agglomeration kernel function is compiled into the PBM using the User defined function (UDF) to simulate the acoustic field agglomeration.
v ! ! ! εg rg u g þ V, εg rg u g u g ¼ εg Vp þ εg V,tg vt ! ! þ εg rg g b u g u p (3) In the viscous fluid, the stress tensor of the gas phase may be expressed as.
"
T
v ! ! ! εp rp u p þ V, εp rp u p u p ¼ V,ðTk þ Tc Þ þ εp rp g vt ! ! þb ug up
The particle phase is expressed as.
v rp εp þ V, rp εp ! up ¼ 0 vt
(2)
(4)
(5)
Tk ¼ εs rs qgran I
(6)
5 ! V, u s I Tc ¼ ps I 2εs ms0 S þ 6
(7)
Next, the contact bonding energy of the particles is denoted by Ec, so that by changing the particle diameter, the contact energy will change [19]. Instantaneous velocity distribution function of the fine particles in the agglomeration process is reconstructed by the contact energy. The non-consequential coefficient of the fluctuation energy reads.
Rc ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6Ec
(8)
prs e2 D3po qgran
The correction coefficient for the dynamic pressure of the particle phase is given by.
xp ðRc Þ ¼
e Rc 2 e 1 1þe
(9)
and the correction coefficient for the shear viscosity of the particle phase may be represented by.
The gas-phase is expressed as.
(1)
2 3
The momentum conservation equation of the particles phase reads.
2.1. Mass conservation equation
v rg εg þ V, rg εg ! ug ¼ 0 vt
#
! ! tg ¼ εg mg V! u g þ V u g εg mg V, u g
xv ðRc Þ ¼
2. Mathematical model
931
" ! ! 2 e R2 R4 R4c R6c eRc 1 þ c c þ lnRc 1þe 2 32 4 16 # R8 g 1 6 g 4 Rc þ Rc 1 þ þ c 32 32 8 8
(10)
Then, we achieve the dynamic pressure Ps and the shear viscosity ms of the viscos-ity particle phase on the basis of the particle dynamics.
ps ¼ ps0 1 þ xp ðRc Þ
(11)
ms ¼ ms0 ð1 þ xv ðRc Þ Þ
(12)
where
ps0 ¼ 2ð1 þ eÞε2s rs g0 qgran 2.2. Momentum conservation equation The momentum conservation equation of the gas phase may be represented by.
ms0
4 ¼ εs rs g0 Dð1 þ eÞ 5
(13)
rffiffiffiffiffiffiffiffiffiffi
qgran p
(14)
The deformation rate of the viscous particles may be expressed
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by.
Sij ¼
1 ! 1 Wij þ Wji V, u s dij 2 3
(15)
For a non-viscous particle system, the viscosity is zero, and the collision stress is represented by a non-viscous calculation form. It has the same result as Ding and Gidaspow [17]. 2.3. Conservation equation for the fluctuation energy of the particle phase The conservation equation may be expressed as
3qgran v ! 3qgran εs rs þ V, εs rs u s ¼ V,qc vt 2 2
! ðTk þ Tc Þ : V u s þ g (16)
The flux of the fluctuation energy is given by.
qc ¼ ks Vqgran
(17)
and the conductivity of the fluctuation energy reads.
qffiffiffiffiffiffiffiffiffiffi 4 ks ¼ pffiffiffiε2s rs g0 D qgran
(18)
p
here, we can apply the contact bonding energy Ec to the reconstructed instantaneous velocity distribution function [19]. Then, we achieve the collision energy dissipation of the cohesive particles, which may be simplified to.
(19)
(20)
xr2 ðRc Þ ¼ erfcðRc Þ 1 R2c þ exp R2c
2 pffiffiffi Rc þ
p
8 pffiffiffi R3c 15 p (21)
The particle dynamic viscosity may be then expressed as.
4 5
mp ¼ ε2p rp Dg0 ð1 þ eÞ
rffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffi
qgran 10rp D pqgran 4 1 þ g0 εp ð1 þ eÞ þ 96ð1 þ eÞεp g0 5 p
2
(22) Considering an inviscid particle system, the particle collision energy dissipation rate may be used to deduce the classical particle kinetic theory model. Then, the fluctuation energy exchange between the gas phase and the particle phase reads.
4s ¼ 3bqgran
(23)
! ! 3C εp εg rg u g u p 2:65 εg b¼ d 4D
(24)
where
8 < 24 1 þ 0:15Re0:687 Cd ¼ Re : 0:44
vnðV; tÞ þ V,½ f ðV; tÞ ¼ SðVÞ vt
Re < 1000 Re 1000
(27)
where n(V; t) is the number density function of the particles phase，describing the distribution of the number of particle phase entities in the volume space V and the time space t. S(V) is the source term describing the change of the number density of the particles phase caused by the microscopic behavior of the particles phase, e.g., the agglomeration, breakage and growth. The Van der Waals force dominates the agglomeration process between the fine particles, and the agglomerates are not easy to break. Therefore, here we only consider the agglomeration behavior between the particles. The microscopic description of the agglomeration behavior may be expressed as.
SðVÞ ¼
ðV 1 b V V 0 ; V 0 n V V 0 ; t n V 0 ; t dV 0 2 nðV; tÞ
(28) b V; V 0 n V 0 ; t dV 0
0
xr1 ðRc Þ ¼ exp R2c 1 þ R2c 2 5
The population balance equation (PBE) is a transport equation of the number density that used to determine the both spatial and temporal evolutions of the number density. The population balance equation with particle volume as the internal coordinates can be expressed as.
∞ ð
i q h g ¼ ε2s ks rs g0 qgran f24D gran 1 e2 xr1 ðRc Þ h ipo 5ðV,us Þ 1 e2 xr2 ðRc Þ
(26)
2.4. Population balance model (PBM)
0
rffiffiffiffiffiffiffiffiffiffi
! ! Re ¼ rg εg D u g u p mg
(25)
where b(V, V0 ) is the agglomeration kernel function of particles with volume of V and volume of V0 . The first term on the right side of Eq. (28) represents the increase in the number of particles V formed by the agglomeration of particles of the volume V0 with particles of volume V - V0 . The second term on the right of Eq. (28) represents the agglomeration of particles of volume V with particles of volumeV0 , resulting in the decrease in the number of particles of volume V. The solution domain of Eq. (27) is usually semi-infinite, and there is no general analytical solution. To solve this equation, one may use the discrete method, moment method or Monte Carlo method. Thanks to the strong computational ability of the discrete method, it can directly solve the particle size distribution. It is especially effective for numerical calculation with a particle size range span of 2e3 orders of magnitude. Then, the discrete method is used to discretize the continuous particle size distribution into a series of different size groups. Finally, the discrete form of the population balance equation will be solved numerically. 2.5. Agglomeration mechanisms of particles inside the acoustic field This section analyzed the kernel function of particle agglomeration and the agglomeration mechanisms. The kernel function is a formula that considers both the particle size and the force in the different flow regime [20]. Therefore, in the wide screening particle size distribution region, there are various physical effects that cause the particles to agglomerate. In the sound field turbulence concentrator, the physical effects of sound field force, hydrodynamic force, Brownian heat motion, and gravity make the particles agglomerate [21]. Therefore, the formula of a single particle
J. Zheng et al. / Powder Technology 356 (2019) 930e940
agglomerate cannot accurately represent the true particle agglomeration mechanism. Therefore, this paper considers the coupling of multiple forces in the concentrator and the coupling of various agglomeration mechanisms. Here, we investigate the effects of the orthokinetic agglomeration mechanism(bor), hydrodynamic mechanism (bhy) and Brownian agglomeration mechanism(bBr)on the particle agglomeration process. Because the particle size of our research is <10 mm, differential settlement by the gravity can be ignored [7]. In the acoustic agglomeration process of the particles, the agglomeration mechanisms are coordinated and partly overlapped, and the impact scale of the agglomeration is different. Geng in Ref. [22] pointed out that the simple linear addition of the agglomeration kernel function in the process of the agglomeration would lead to overestimation of the true agglomeration kernel function by 30%. For improving the description accuracy of the agglomeration kernel function, we propose a new roots mean square of the agglomeration kernel function on the basis of the full consideration for three kinds of the agglomeration mechanisms. It can expressed as:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
b0 Or þ b2Hy þ b2Br
b Di ; Dj ¼
(29)
In this way, we get the formula of the sound field agglomeration. The following sections are the detail analysis of three agglomeration mechanisms. 2.5.1. Orthokinetic mechanism The orthokinetic agglomeration mechanism is first proposed by Mednikov [23] in 1965, which represents the most important acoustic agglomeration mechanism. It is mainly based on the theory of acoustic entrainment. The collision of particles with different sizes occurs due to the difference between their relative velocities. We assume that there are two types of spherical particles with different sizes in the agglomeration area [23]. Large particles act as the collecting particle, the smaller as the collected particle. In the process of agglomeration, the concentration of the particles is invariable, that is, with the continuous agglomeration of particles, small particles can instantaneously fill into the agglomeration volume. Through the continuous optimization of the researchers, the following reasonable co-agglomerated kernel functions are achieved.
bor Di ; Dj
2 u tti ttj 1 ¼ Di þ Dj Ug rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ u2 t2ti 1 þ u2 t2tj
r D2 tt ¼ p i ; Ug ¼ 18mg SPL
I ¼ 10 10 I0
sffiffiffiffiffiffiffiffiffiffi 2I rg ug
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reaches 150 dB and the particle diameter is 10 mm, Re is only 0.01. Therefore, it is reasonable to assume that the gas flow within the agglomerated volume is in the Stokes flow region [25]. The Stokes number represents the ratio of particle inertia to diffusion. The value of St is smaller, the inertia of the particles is smaller. On the contrary, the larger the value, the larger the particle inertia. The follow-up of particle motion is less obvious. The collision efficiency associated with the Stokes number can fully express the collision of solid particles in the gas-solid flow field. In this paper, the particle size is in the micron and submicron range, and even at very high sound source levels, the Reynolds number is considered to be very small, and the entire flow is in the Stokes region. Therefore, it is reasonable to use the calculation formula for the collision efficiency related to the Stokes number in calculating the collision efficiency between particles of this case.
a¼
B St St þ A
(33)
where A and B are constants，i.e., A ¼ 0.65 and B ¼ 3.7, St is the Stokes number, as.
St ¼
rp mg Ug D22 18D1 h
(34)
In Eq. (34), D1is the large particle, D2 is the small particle, and h is the relative carrying coefficient between large and small particles, so that.
uðtt1 tt2 Þ h ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 þ u2 t2t1 1 þ u2 t2t2
(35)
Through the above analysis, the formula (34) is applied to the stoke flow regime. When the particle size is below 10 mm, Calculated by the formula (34), the value of St can be between 0.0012 and 10.95. Then, we achieve the modified orthokinetic agglomeration kernel function.
b’or Di ; Dj ¼ abor Di ; Dj
(36)
(30)
(31)
(32)
with I0 being the standard reference value which is equal to 10-12w/ m2 . In the particle agglomeration process, due to the external flow field and the interference between particles, some small particles cannot collide with large particles, especially for the submicron particles, where the collision efficiency is very low [24]. Therefore, it is necessary to modify the agglomeration kernel function by introducing the collision efficiency, representing the fraction of the total number of small particles that can collide with large particles. In acoustic agglomeration, the particles are small, and the relative velocity between the particles and the gas medium caused by the sound waves is not high, thus the Re number is very low. It can be obtained by calculation that even when the sound pressure level
2.5.2. Hydrodynamic interaction According to the orthokinetic agglomeration kernel function, the collision agglomeration cannot occur between the particles of the same size since there is no relative movement between them. In fact, the agglomeration of particles exists in a mono-disperse particle system [26], The hydrodynamics is the main mechanism of this type of agglomeration [27]. The mechanism of hydrodynamics can also explain the collision and agglomeration of particles when the distance between the particles is larger than the amplitude of the acoustic wave. The hydrodynamic mechanism fully considers the influence of the motion of the fluid medium in the acoustic field on the particles, which becomes much more important with the reduction of the particle spacing. The kernel function of the fluid dynamics may be expressed as.
1 4
bHy Di ; Dj ¼ p Di þ Dj 2 Uij
(37)
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Uij ¼
3Ug Ug 2 2 2Di li þ 2Dj lj þ Di li þ D2j l2j 8pd pm
d
þ
3mg 9Ug 2 2 Di li þ D2j l2i Di þ Dj d 2 2 2 p d 64pd 3Ug2 16d2 u
(38)
li lj li qi dlj qj Di dDj
d ¼ kB TB ¼
where li, hi, and qi are intermediate variables, given by.
li ¼
mgi
; hi ¼ 2
1 þ hi mgi
mp þ hi m2gi 9rg Ug ; qi ¼ prp uDi 1 þ hi m2gi
(39)
With mgi and mp being the slip and entrainment coefficients in the Stokes flow, respectively.
uti 1 ; mp ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mgi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ ðutti Þ2 1 þ ðutti Þ
(40)
2.5.3. Brownian agglomeration mechanism In addition to the influence of the acoustic waves, the particles in the fluid medium carry on the Brownian motion simultaneously. The phenomenon of the particle agglomeration caused by the diffusion of the particles due to the Brownian motion is called the Brownian agglomeration [28]. Here, the particle size range is 0.02110 mm. For Brown agglomeration, there is a vital dimensionless number Kn (Knudsen number). This is the ratio of gas molecule’s free path to particle diameter. Otto et al. [29]considered that for a free molecular region with Kn > 50, the collision coefficient is usually obtained based on the molecular motion theory. For a continuous-near continuum region with Kn < 1,the coefficient is obtained based on Fick’s diffusion theory. However, for the transition region of 1 < Kn < 50, there is no such mature theory as molecular motion theory or diffusion theory, and the Fuchs’semiempirical absorption sphere theory is used [30]. In the free molecular region, Friedlander further derived a collision coefficient of particles from the molecular motion theory [31]. Einstein deduced the expression of collision coefficient of particles in the near-continuum region [32].(Because Kn ¼ l/(3v/4p)1/3: where n is the particle volume. pffiffiffi l is the average free-range of the gas molecule, l ¼ kB T= 2pd2 p,where kB is Boltzmann constant ¼ 1.38 1023 J/K, T is absolute temperature ¼ 403 K, d is particle size ¼ 0.021e6.256 mm, p is the pressure inside the device ¼ 105pa), after calculation, the maximum value of Kn ¼ 2.6 103 < <1, so it belongs to continuous-near Media area. The Brownian agglomeration kernel function first derived by Smoluchowski [33] is then obtained by substitution in (41),
b di þ dj ¼ 4p di þ dj
di dj þ 2 2
(41)
For ultrafine particles that follow Stokes-Cunning’s law, a Brownian force or a diffusion forces acts on the particles to move the particles toward a low concentration region. It is equal to the force that the gas prevents the particles moving. This force is Stokes resistance, ie [34].
Diffusion force ¼ 3pmnd=Cc
molecule; the other is that the kinetic energy of particle motion is equivalent to the kinetic energy of gas molecules, that is KE ¼ 3kBT/ 2, the third is that the Brownian force acting on the particles is osmotic pressure, osmotic pressure is given by van’t Hoff’s law. Based on the above three points, Einstein gives the diffusion coefficient of the particles of the Cunningham correction zone as,
(42)
where m is the dynamic viscosity and Cc is Cunningham slip factor Einstein [33] proposed three viewpoints: one is that the Brownian motion of ultrafine particles is the same as the motion of a giant gas
kB T k TC ¼ B f 3pmd
(43)
where B is the particle mobility; f is the coefficient of friction of the resistance to the particle. The derivation process of formula (43) can be found in the literature [32]. It can be seen from the above diffusion coefficient that the higher the temperature of the particles, the larger the diffusion coefficient of the particles, and the more obvious the Brownian agglomeration. Since the diffusion force of the left term of the formula (42) is obtained by the Osmotic pressure. The Osmotic pressure is a macroscopic quantity, but (42) the right side are for a specific single particle, and we can see Eq. (43) embodies the macroscopic and microscopic unity. Substituting Eq. (43) into Eq. (42).
2k T Ci Cj b di ; dj ¼ B di þ dj þ 3mg di dj
! (44)
where di and dj respectively represent the diameters of particles i and j; mg represents the dynamic viscosity; kB represents the Boltzmann constant; T represents the absolute temperature; Ci, Cj represents the Cunningham slip factor [34].
C ¼ 1þ
lm ½2:514 þ 0:8expð 0:55D=lm Þ D
(45)
The movement of the small-sized particles is highly susceptible to the influence of the Brownian force and the frictional force of the particles, which causes collision agglomeration. The smaller the particle, the higher the temperature, then the effect of the Brownian force will be greater, and the effect of friction is smaller. The more likely the particles are to collide, the Brownian motion must be considered when the small-sized particles collision agglomeration occurred. 3. Theoretical analysis 3.1. Collision efficiency Fig. 2 shows the collision efficiency between the collecting particles (i.e. Dj1¼10 mm, Dj2¼5 mm, Dj3¼1 mm) and the collected particles. It may be seen from this figure that, when the collected particle size is <2 mm, the collision efficiency rapidly decreases as the particle size decreases. For instance, for the collected particle size of 0.5 mm, the collision efficiency with the collecting particle size of 5 mm and 10 mmare only 0.044 and 0.012, respectively, which largely weakens the influence of orthokinetic agglomeration mechanism. When the collected particle is >2 mm, the collision efficiency is basically maintained at about 0.7. Moreover, when the collected particles are sufficiently close to the collecting particles, the collision efficiency exhibits a downward trend. This occurs mainly because of the low relative velocity between two types of particles, where the collected particles are more likely to bypass the collecting particles, and the orthokinetic agglomeration effect is weakened. Hence, the conventional orthokinetic agglomeration kernel function significantly overestimated, especially in the submicron region. Therefore, in order to be more realistic, we must use
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the collision efficiency to modify the conventional orthokinetic agglomeration kernel function. (See Fig. 1.) 3.2. The comparison of the agglomeration kernel Fig. 3 shows the influence of different collecting particle size (Dj1 ¼ 1 mm，Dj2 ¼ 5 mm，Dj3 ¼ 10 mm) on the agglomerate kernel function. The acoustic frequency and sound pressure level is 1400 Hz and 147 dB, respectively [35]. When the collected particle size Di belongs to submicron regime, due to the low collision efficiency, the contribution of the orthokinetic mechanism to the particle agglomeration is very small compared with other agglomeration mechanism. When the collected particle size Di is smaller than the collecting particle size Dj, an optimum collected particle size exists. At this an optimum collected particle size, the orthokinetic kernel function reaches to peak value. When the two kinds of particles size (Di and Dj) is close enough, the relative velocity between the particles is very small. Hence, according to the orthokinetic agglomeration mechanism, the orthokinetic kernel function decreases sharply. As shown in Fig. 3 (a), the Brownian agglomeration plays a dominant role in the small particle size range. By increasing the
Fig. 1. Range of values of St number.
Fig. 3. Influence of particle size on the agglomerate kernel function.
Fig. 2. Collision efficiency between the collecting particles and collected particles.
particle size, the Brownian agglomeration kernel function will decrease. When the particle size increases to a certain critical particle size, the Brownian agglomeration kernel function value begins to rise slowly. The critical particle size increases with the increase of the collecting particle size. However, when the critical
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Fig. 6. Ratio N/N0 of the particle number density as a function of time.
particle size is >10 mm, it has no practical significance for the Brownian agglomeration, which is not shown in Fig. 3 (c). When the collected particle size is >1.2 mm, the order of the magnitude of the Brownian agglomeration kernel function is much lower than that of the other two agglomeration kernel functions. In this regime the effect of the Brownian agglomeration on the particle agglomeration is can be neglected. For the collecting particle sizes of 5 mmand 10 mm, shown in Fig. 3 (b), (c), the hydrodynamic mechanism plays the dominant role in the submicron regime, and the hydrodynamic kernel function increases with the increase of the collected particle size. By comparing the results of Fig. 3, we may see that the order of the magnitude of the agglomeration kernel function of three types increases obviously with the increase of the collecting particle size. As shown in the Fig. 3. The optimization of the agglomeration kernel function by the root mean square of the agglomeration kernel function is mainly reflected in two aspects compared with the linear agglomeration kernel function. i.e., when the collected particle size is in the range of 0.2e0.8 mm, the agglomeration kernel function is optimized by 20e30%; on the other hand, when the distance between the particle size Di and Dj is close enough, the agglomeration kernel function is optimized by 13e25%. With the increase of the particle size, the root mean square of agglomeration kernel function is close to the orthokinetic agglomeration kernel function. This is also consistent with the conclusion that the main agglomeration mechanism is the orthokinetic agglomeration mechanism. Fig. 4. Structure of the acoustic field agglomeration device.
4. Simulation and discussions 4.1. Model validation
Fig. 5. Distribution of the particles number density for different particle sizes.
The verification of the mathematical model in this paper is based on the experimental results of Liu [35]. In Liu’s experiment, he compared the conditions of different sound pressure levels and different acoustic frequencies. The results show that the acoustic wave frequency is 1400 Hz and the sound pressure level is 147 dB, which is the recommended case. The focus on this paper is on the agglomeration mechanism and the correction of agglomeration kernel functions. Therefore, this study selects aforementioned working condition and the physical model for simulation. The simulation results are representative. Fig. 4 shows the structure of the acoustic field agglomeration device. The acoustic generator is located above the acoustic field agglomeration device. The inlet boundary condition is set to the velocity inlet, the outlet boundary condition is the pressure outlet, and the wall surface is the non-slip boundary condition. In addition, the fly ash particles enter from the upper part of the agglomeration
J. Zheng et al. / Powder Technology 356 (2019) 930e940
chamber. Specific simulation parameters are shown in Table 1. The population balance model (PBM) is numerically solved by the discrete method. The particle sizes of 0.021e6.256 mm measured in the flue before the dust collector are divided into 12 ranges, as shown in Table 2 below. It may be seen from Fig. 5 that, while the acoustic wave is not acting, the initial particle size distribution presents a three-mode distribution with peak values at 0.08 mm, 0.63 mm, and 1.75 mm. The number density in the whole particle size range of the coal-fired fly ash decreases significantly under the application of acoustic compared with the experiment data [35], especially in the submicron region. This indicates that a large number of fine particles are agglomerated under the action of sound waves. When the particle size is 2.2e6.256 mm, the simulation results are slightly larger than the experimental values, mainly due to the simulation does not consider the phenomenon of differential sedimentation of larger particles in the experimental process. The simulation process is basically consistent with the experimental results. Fig. 6 shows the N/N0 ratio of the particle number concentration evolved with the time in the outlet. The ratio is the concentration with the acoustic wave action to that without acoustic wave action. The number of concentration decreases with the extension of the
937
action time of the acoustic wave, and the decreasing speed gradually slows down. The particle number concentration decreased by 45.06% in 3e5 s and 27.18% in 5e7 s. This is mainly because in the early stage of particle agglomeration, the particle number concentration is higher, the particle distance is smaller, and the agglomeration process between particles is easier. With the process of the agglomeration occur, the particle number concentration decreases, and the agglomeration effect between particles is weakened. The results demonstrate the effectiveness of the acoustic agglomeration. The simulation results are in good agreement with the experimental results. 4.2. Simulation result discussions Fig. 7 shows the instantaneous concentration distribution of the particles phase for the simulation time of 0.1 s, 2 s,4 s, 6 s. It also may be seen from Fig. 7 that fine particles easily agglomerate in the high intensity turbulence regime and near the wall field. At 0.1 s, the particles are in a freely dispersed state because the sound field has not yet funcationed. By acting the acoustic wave, small eddies begin to appear in the upper part of the acoustic agglomeration device at 2 s, and the agglomeration is promoted at the edge of the
Fig. 7. The instantaneous concentration distribution of the particles phase for the simulation time of 0.1 s, 2 s,4 s, 6 s.
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J. Zheng et al. / Powder Technology 356 (2019) 930e940
Fig. 8. Distribution of the particle phase fluctuation temperature for different heights. (Starting point (height ¼ 0 m) at the bottom of the device).
Fig. 9. Distribution of the particle phase diameter in different height. (Starting point (height ¼ 0 m) at the bottom of the device).
Table 1 Parameters used in simulation. Parameter
Value
Unit
Height of device Width of device Initial diameter of particles Density of particles Acoustic frequency Sound pressure level Inlet velocity Total mesh number
1500 108 0.021e6.256 2400 1400 147 0.5 15,300
mm mm mm kg/m3 Hz dB m/s e
turbulent eddy. This indicates that particles form agglomeration under the action of the acoustic wave and the turbulent flow. However, the agglomeration range is not large. With the continuous action of the acoustic waves, the vortex size increases and the turbulence intensity has further enhanced. The flow of the particles phase in the acoustic agglomeration device present a kind of ringannular flow construction. Fig. 8 shows the distribution of the particle phase fluctuation temperature in different heights. It may be seen from Fig. 8 that the fluctuation temperature of the particles phase can reflect the extent to the particles agglomeration. Since the particle fluctuation temperature is a measure of the particle pulsation intensity, in the center of the acoustic field agglomeration device, the concentration of particle phase is dilute and the probability of the agglomeration is lower. Hence, the fluctuation temperature of particles is smaller. In the near wall area, the particle concentration is higher and the fluctuation is more intense. Thus, the particle fluctuation temperature is larger. As the height increases, the lowest value of the particle pulsation temperature moves to the right along the radial position, while its lowest value gradually increases. Since the particle temperature indicates the severity of particle collision agglomeration, the higher the particle concentration, the more easily the particles collide with the agglomerates, so the higher the particle temperature. Because the inlet is above the left side, the particles will accumulate near the right wall due to inertia. Therefore, it can be seen from Fig. 7 that the instantaneous concentration distribution of the particle phase increases with height, the high concentration region of the particle gradually shifts to the right side and the concentration value increases, and the temperature of the particles on both sides also increases. Due to the higher particle concentration, the particle collision is more intense and enhances the particle pulsation temperature. At the same time, with the flow direction, after the particles collide and agglomerate, the number of particles decreases. Therefore, at h ¼ 450 mm, the average collision pulsation intensity of the particles is the lowest, and as can be seen from Fig. 7, since the center of the vortex is close to the right side, as the height increases, the lowest value of the pulsation temperature of the particles moves to the right along the radial position. Fig. 9 shows the distribution of the particle phase diameter for different heights. It can be seen from Fig. 9 that the agglomeration process is affected by both the turbulent and the acoustic wave. Due to the effect of the high concentration of particles near the inlet and the acoustic generator located above the acoustic agglomeration device near the inlet position, the particles easily agglomerates near the inlet. It may be seen from Figs. 7 and 8 that under continuous action of sound waves, the size of the vortex in the acoustic agglomeration device has continuously increased, resulting in a higher particle concentration and a high fluctuation temperature in the near wall region at any height, and the particles are more likely to agglomerate. As the height decreases, the agglomeration between the particles continues to occur, and the average diameter of the particles increases continuously. However, due to the decrease of the particle concentration, the collision probability between the particles decreases, and the change trend of the
Table 2 The number density distribution of the particle size. Bin
The average particle size /mm
Number density percentage(n)
Bin
The average particle size /mm
Number density percentage(n)
Bin-0 Bin-1 Bin-2 Bin-3 Bin-4 Bin-5
6.256 3.074 1.945 1.224 0.758 0.481
0.0010 0.0038 0.0077 0.0115 0.0102 0.0082
Bin-6 Bin-7 Bin-8 Bin-9 Bin-10 Bin-11
0.314 0.200 0.120 0.073 0.041 0.021
0.0088 0.0826 0.3902 0.1857 0.1685 0.1218
J. Zheng et al. / Powder Technology 356 (2019) 930e940
particle size along the axial direction gradually slows down. In addition, the unevenness distribution of the particle diameter has gradually decreased. According to the calculation data analysis, the average diameter of the outlet particles is stable at around 5.16 mm, and the average diameter of the particles has increased by about 106% compared with the average diameter 2.5 mm of the particles at the inlet.
bhy bbr Ug I C lm T
a 5. Conclusion
St
h We have studied the mechanism of the acoustic agglomeration of the coal fly ash particles in the acoustic field agglomeration device. Next, the root mean square of the agglomeration kernel function has proposed for numerical simulation of the acoustic agglomeration process. The contributions and observations of this paper may be summarized as follows. (1) The coupling of the modified kinetic theory of fine particle flow and the population balance model for the coal fly ash agglomeration in the acoustic field is established. The viscous particle dynamics model considering the cohesive collision in the particle aggregation process is used to simulate the agglomeration flow process. At the same time, the root mean square of the agglomeration kernel function is used to closed the population balance model. (2) Compared with the linear additive agglomeration kernel function, the agglomeration kernel function is optimized by 20%e30%. When the particle size range is 0.2e0.8 mm; the agglomeration kernel function is optimized by 13%e25% when the particle size is fully close. (3) The fluctuation temperature of the particles can reflect the extent to the particles agglomeration. The particle fluctuations temperature in the near wall region is high and easy to agglomeration. (4) The continuous effect of the acoustic wave enhances the agglomeration effect of particles. The average particle size of the outlet position has increased by 106% with respect to the inlet position.
Notation
r
ε ! u g p
b tg tt mg ms Tk Tc
qgran
Ec S Rc e Dpo V t
bor
density (kg/m3) concentration (m3) speed vector (m/s) gravity acceleration (kg/(m$s2)) pressure (pa) the momentum exchange coefficient (kg$s/m3) stress tensor of gas phase (pa) the relaxation time (s) the dynamic viscosity (pa$s) the shear viscosity (pa$s) dynamic stress (pa) viscous collision stress (pa) the particle fluctuation temperature (K) the contact bonding energy (J) shear stress tensor (kg/ms2) ration of contacting bonding energy to fluctuation energy (–) normal restitution coefficient (–) initial particle diameter before agglomeration (m) volume of particle (m3) time (s) the orthokinetic agglomeration kernel function (m3/s)
Uij d
mp kB Sij qc kp D g0 ks Re Cd
Fs u g xp(Rc) xv(Rc) xr(Rc)
939
the hydrodynamics agglomeration kernel function (m3/ s) the Brownian agglomeration kernel function (m3/s) the gas velocity amplitude (m/s) the sound field strength (w/m2) the cunningham slip factor (–) the molecule mean free path (m) the absolute temperature (k) the collision efficiency (–) the stokes number (–) the relative carrying coefficient (–) the relative velocity between two particles (m/s) the particle distance (m) the entrainment coefficient (–) the Boltzmann constant (J/k) deformation rate (–) flux of fluctuation energy (kg/s3) the thermal conductivity (w/m$k) the agglomeration diameter (m) radial distribution function (–) the conductivity of the fluctuation energy (kg/(m$s)) the Reynolds number (–) the drag coefficient (–) the fluctuation energy exchange coefficient (–) angular velocity (rad/s) collision dissipation of energy (m2/s2) correction dynamic pressure coefficient (–) correction shear viscosity coefficient (–) collision dissipation coefficient (–)
Acknowledgments The financial support provided by Jilin Province Science and Technology Department 13th Five Year Key Technology R&D Project (20180201001SF) and Jilin Provincial Department of Education “13th Five Year” Scientific Research Project (JJKH20190703KJ) is gratefully acknowledged. References [1] C.L. Senior, J.J. Helble, A.F. Sarofim, Emissions of mercury, trace elements, and fine particles from stationary combustion sources, Fuel Process. Technol. 65e66 (2000) 2000263e2000288. ~talo, Jukka Hautanen, Electrostatic precipitator penetration function for [2] S. Yla pulverized coal combustion, Aerosol Sci. Technol. 129 (1998) 17e30. [3] J. Lewtas, Air pollution combustion emissions: characterization of causative agents and mechanisms associated with cancer, reproductive, and cardiovascular effects, Mutat. Res. Rev. Mutat. Res. 636 (2007) 95e133. [4] U. Franck, A.M. Leitte, P. Suppan, Multiple exposures to airborne pollutants and hospital admissions due to diseases of the circulatory system in Santiago de Chile, Sci. Total Environ. 468e469 (2014) 746e756. [5] J. Hautanen, M. Kilpelainen, E.I. Kauppinen, K. Lehtinenl, Electrical agglomeration of aerosol particles in an alternating electric field, Aerosol Sci. Technol. 22 (1995) 181e189. [6] Y.W. Li, C.S. Zhao, X. Wu, D.F. Lu, S. Han, Aggregation mechanism of fine fly ash particles in uniform magnetic field, Korean J. Chem. Eng. 24 (2007) 319e327. [7] I.V. Derevich, Coagulation kernel of particles in a turbulent gas flow, Int. J. Heat Mass Transf. 50 (2007) 1368e1387. [8] W. Zhao, Y. Liu, J.J. Bao, J.F. Geng, L.J. Yang, Experimental research on fine particles removal from flue gas by chemical agglomeration, Proc. CSEE. 33 (2018) 52e58. [9] D. Liao, C. Wu, Y. Yao, H. Hu, The use of quadrature method of moments (QMOM) in studying the mechanisms of aerosol agglomeration and its practical use, Am. J. Heat Mass Tran. 2 (2015) 42e58. [10] H.S. Patterson, W. Cawood, Phenomena in a sounding tube, Nature. 127 (1931) 667. [11] S.V. Komarov, T. Yamamoto, T. Uda, M. Hirasawa, Acoustically controlled behavior of dust particles in high temperature gas atmosphere, Iron Steel Inst. Jpn. 44 (2004) 275e284. [12] J. Liu, J. Wang, G. Zhang, J. Zhou, K. Chen, Frequency comparative study of coal-fired fly ash acoustic agglomeration, J. Environ. Sci. 23 (2011) 1845e1851. [13] V. Vekteris, V. Strishka, D. Ozarovskis, V. Mokshin, Experimental investigation of processes in acoustic cyclone separator, Adv. Powder Technol. 25 (2014)
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1118e1123. [14] C. Sheng, X. Shen, Modelling of acoustic agglomeration processes using the direct simulation Monte Carlo method, J. Aerosol Sci. 37 (2006) 16e36. [15] G.X. Zhang, J.Z. Liu, J. Wang, J.H. Zhou, K.F. Cen, Numerical simulation of acoustic wake effect in acoustic agglomeration under Oseen flow condition, Sci. Bull. 57 (2012) 2404e2412. [16] F. Fan, Z. Peng, E. Doroodchi, B. Moghtaderi, Direct simulation Monte Carlo method for acoustic agglomeration under standing wave condition, Aerosol Air Qual. Res. 17 (2017). [17] D. Gidaspow, Multiphase flow and fluidization, Conti. Kinet. Theory Description. 95 (1994) 1e29. [18] D. Ramkrishna, Population Balances: Theory and Applications to Particulate Systems in Engineering, Academic Press, San Diego, CA, 2000. [19] J.X. Zheng, Kinetic Theory of Cohesive Particles Flow and Numerical Simulation of Gas-Solid Two-Phase Flows. Doctor Dissertation of Harbin Institute of Technology, 2008 (March). [20] K.M. Kellogg, Continuum theory for rapid cohesive-particle flows: general balance equations and discrete-element-method-based closure of cohesionspecific quantities[J], J. Fluid Mech. 832 (2017) 345e382. [21] N. Harris, Modelling of Particle Paths Passing Through an Ultrasonic Standing Wave[C]//Ultrasonics International 30 June e 3 July 2003 Granada, Elsevier B.V., Spain, 2003. [22] J. Geng, Simulation of aerosol coagulation and deposition under multiple flow regimes with arbitrary computational precision, Aerosol Sci. Technol. 47 (2013) 530e542. [23] E.P. Mednikov, Chas V. Larrick, Acoustic coagulation and precipitation of aerosols, Appl. Mech. Mater. 34 (1965) 1087e1103.
[24] Y. Nakajima, T. Sato, Electrostatic collection of submicron particles with the aid of electrostatic agglomeration promoted by particle vibration, Powder Technol. 135 (2003) 266e284. [25] A.T. Chwang, T.Y.T. Wu, Hydromechanics of low-Reynolds-number flow. 2: singularity method for stokes flows, J. Fluid Mech. 67 (2006) 787e815. [26] D.T. Shaw, K.W. Tu, Acoustic particle agglomeration due to hydrodynamic interaction between monodisperse aerosols, J. Aerosol Sci. 10 (1979) 317e328. [27] S.D. Danilov, Average force acting on a small sphere in a traveling-wave field in a viscous fluid, Akusticheskii Zhurnal. 31 (1985) 45e49. [28] E. Otto, H. Fissan, Brownian coagulation of submicron particles, Adv. Powder Technol. 10 (1999) 1e20. [29] S.H. Park, The log-normal size distribution theory of brownian agglomeration for the entire particle size range, J. Aerosol Sci. 30 (1999) S23eS24. [30] N.A. Fuchs, The mechanics of aerosols, Phys. Today 18 (1965) 73. [31] K. Sheldon, Smoke, Dust, and Haze :Fundamentals of Aerosol Dynamics, Oxford University Press, 2000, 2nd ed[M]. [32] F.O. Einst, Self-preservation and gelation during turbulence-induced coagulation, J. Aerosol Sci. 37 (2006) 0142. [33] F. Filbet, Numerical Simulation of the Smoluchowski Coagulation Equation [M], Society for Industrial and Applied Mathematics, 2004. [34] H. Masuda, K. Higashitani, H. Yoshida, Powder Technology Handbook, Third edition, CRC Press, 2006, p. 3. [35] J.Z. Liu, G.X. Zhang, J.H. Zhou, J. Wang, W. Zhao, Experimental study of acoustic agglomeration of coal-fired fly ash particles at low frequencies, Powder Technol. 93 (2009) 20e25.
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Modification of the agglomeration kernel and simulation of the flow pattern in acoustic field with fine particles Jianxiang Zheng*, Yukai Li, Zongqun Wan, Wenpeng Hong, Long Wang School of Energy and Power Engineering, Northeast Electric Power University, Jilin, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 30 September 2018 Received in revised form 10 August 2019 Accepted 7 September 2019 Available online 11 September 2019
This paper studies the acoustic agglomeration of fine particles of the coal fly ash in the acoustic field agglomeration device, based on the coupling population balance model (PBM) for acoustic agglomeration of fine particles and the modified kinetic theory of the fine particle flow. The new root mean square of the agglomeration kernel function operates according to the acoustic agglomeration mechanism. We present the theoretical investigation of the problem and then compare it with experimental results. The results show that the agglomeration process of particles in different size ranges is dominated by different agglomeration mechanisms. Moreover, we show the numerical simulation of the acoustic agglomeration process of the particles using the new agglomeration kernel function, which is consistent with the experimental results. In addition, the particle flow in the acoustic field agglomeration device presents a typical turbulent structure, and a large particle agglomeration occurs near the wall and outlet. © 2019 Elsevier B.V. All rights reserved.
Keywords: Modified kinetic theory Population balance model Collision efficiency Agglomeration kernel
1. Introduction The emission of dust pollutants from the coal-fired power plants is one of the main sources of the particulate matter of the air in China [1]. Traditional particle filtration devices, i.e., the electrostatic precipitator, cyclone separator and fabric filter, can capture >99% of the particles whose diameters are >2.5 mm[2]. However, the collection efficiency of the fine particulate matter (PM2.5) with the aerodynamic diameter of <2.5 mmis extremely low. The PM2.5 easily absorbs harmful substances, and causes a substantial harm to the human health [3e4]. Therefore, it is necessary to use physical or chemical means to pretreatment fine particles before entering into the traditional filter device. In general, five types of particle peragglomeration technologies exist, including electrical [5], magnetic [6], turbulent [7], chemical [8] and acoustic [9] agglomerations. The acoustic agglomeration uses the high intensity sound field for forcing the particles to collide. Owing to the irregular surface meshing of the particles and strong surface attraction of the molecules (van der Waals force, liquid bridge force, etc.), the collision of particles leads to the formation of bigger particles in the acoustic field. The acoustic agglomeration represents the most potential
* Corresponding author at: School of Energy and Power Engineering, Northeast Electric Power University, No. 169, Changchun Road, Chuanying District, Jilin 132012, China. E-mail address: [email protected] (J. Zheng). https://doi.org/10.1016/j.powtec.2019.09.022 0032-5910/© 2019 Elsevier B.V. All rights reserved.
particle pretreatment technologies, which exhibits high efficiency and is not affected by high temperature flue gas. Such a technology has been extensively studied in the past few years. The first experimental study on the agglomeration of suspended particles inside the acoustic field is reported in Ref. [10], where the agglomeration of the suspended particles at the standing wave nodes as acoustic agglomeration was shown. The experimental investigation of the acoustic agglomeration of the Zn particles inside the high-temperature gas was shown in Ref. [11], where the optimal agglomeration effect was achieved at 210 Hz and 991 Hz. Compared with the silent wave action, the average particle size was increased by 50% and the particle number concentration was decreased by 60%. Afterwards, the experimental demonstration of the agglomeration of the coal fly ash was reported in Ref. [12], using high frequency (20KHz) and low frequency (1000-1800 Hz) sound sources. The results in Ref. [12] revealed that, a low frequency sound source is more suitable for the particle agglomeration, where the optimal agglomeration frequency of the coal fly ash was shown to be 1400 Hz. Vladas in Ref. [13] conducted an acoustic agglomeration experiment on the particles in the cyclone separator. The results showed that the agglomeration of 1e5 mm particles occurs for the sound pressure level of between 130 dB and 170 dB. Hence, the capture efficiency was improved by 10.3% compared with the conventional cyclone separator. In general, the experiment quantitatively shows the validity of the acoustic agglomeration technology, however, the complexity of the acoustic agglomeration process and limitation of the testing methods, it is difficult to study
J. Zheng et al. / Powder Technology 356 (2019) 930e940
the acoustic agglomeration process experimentally. With the development of the computing technology, the numerical simulation provides an effective tool for the study of the acoustic agglomeration. At present, the acoustic agglomeration mechanism of the particles is composed of the orthokinetic agglomeration mechanism, hydrodynamic mechanism and the Brownian agglomeration mechanism. The particle agglomeration in the acoustic filed was numerically simulated using the Monte Carlo method in Ref. [14]. By a simple linear addition of all agglomeration mechanisms, it is found that the predicted results are not adapted to the extensive experimental conditions. Next, the numerical simulation of the hydrodynamic effect of the aerosol particles under the Oseen flow conditions was reported [15]. Through the study of the particle size and acoustic frequency, it is demonstrated that the hydrodynamic effect represents the main agglomeration mechanism of the mono-disperse aerosol particles, or the main complementary mechanism of the orthokinetic mechanism of the multi-dispersed aerosol particles. The acoustic agglomeration effect of the PM2.5 in standing waves is then investigated in Ref. [16] by liner addition of different sound agglomeration kernels. The results revealed that the orthokinetic agglomeration plays a leading role in the acoustic wave agglomeration, while the brown diffusion effect has minor effect on the particle agglomeration. The current scientific researches on the mechanism of the acoustic agglomeration include the linear combination of all agglomeration mechanisms and the coupling effect between them. This article investigates the acoustic agglomeration process of the cohesive fine particles in the acoustic field agglomeration device. First, we establish the modified kinetic theory of fine particle flow to describe the flow of fine particles. This theory is based on the Gidaspow’s classical kinetic theory of granule flow (KTGF)[17], which consists of the viscous collision of particles. Next, a population balance model (PBM) will be introduced to describe the change of the particle size distribution in agglomeration process [18]. Then, we employ the collision efficiency to modify the orthokinetic agglomeration kernel function, and propose the root mean square of the agglomeration kernel for the particles of the acoustic agglomeration. Finally, the root mean square of the agglomeration kernel function is compiled into the PBM using the User defined function (UDF) to simulate the acoustic field agglomeration.
v ! ! ! εg rg u g þ V, εg rg u g u g ¼ εg Vp þ εg V,tg vt ! ! þ εg rg g b u g u p (3) In the viscous fluid, the stress tensor of the gas phase may be expressed as.
"
T
v ! ! ! εp rp u p þ V, εp rp u p u p ¼ V,ðTk þ Tc Þ þ εp rp g vt ! ! þb ug up
The particle phase is expressed as.
v rp εp þ V, rp εp ! up ¼ 0 vt
(2)
(4)
(5)
Tk ¼ εs rs qgran I
(6)
5 ! V, u s I Tc ¼ ps I 2εs ms0 S þ 6
(7)
Next, the contact bonding energy of the particles is denoted by Ec, so that by changing the particle diameter, the contact energy will change [19]. Instantaneous velocity distribution function of the fine particles in the agglomeration process is reconstructed by the contact energy. The non-consequential coefficient of the fluctuation energy reads.
Rc ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6Ec
(8)
prs e2 D3po qgran
The correction coefficient for the dynamic pressure of the particle phase is given by.
xp ðRc Þ ¼
e Rc 2 e 1 1þe
(9)
and the correction coefficient for the shear viscosity of the particle phase may be represented by.
The gas-phase is expressed as.
(1)
2 3
The momentum conservation equation of the particles phase reads.
2.1. Mass conservation equation
v rg εg þ V, rg εg ! ug ¼ 0 vt
#
! ! tg ¼ εg mg V! u g þ V u g εg mg V, u g
xv ðRc Þ ¼
2. Mathematical model
931
" ! ! 2 e R2 R4 R4c R6c eRc 1 þ c c þ lnRc 1þe 2 32 4 16 # R8 g 1 6 g 4 Rc þ Rc 1 þ þ c 32 32 8 8
(10)
Then, we achieve the dynamic pressure Ps and the shear viscosity ms of the viscos-ity particle phase on the basis of the particle dynamics.
ps ¼ ps0 1 þ xp ðRc Þ
(11)
ms ¼ ms0 ð1 þ xv ðRc Þ Þ
(12)
where
ps0 ¼ 2ð1 þ eÞε2s rs g0 qgran 2.2. Momentum conservation equation The momentum conservation equation of the gas phase may be represented by.
ms0
4 ¼ εs rs g0 Dð1 þ eÞ 5
(13)
rffiffiffiffiffiffiffiffiffiffi
qgran p
(14)
The deformation rate of the viscous particles may be expressed
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J. Zheng et al. / Powder Technology 356 (2019) 930e940
by.
Sij ¼
1 ! 1 Wij þ Wji V, u s dij 2 3
(15)
For a non-viscous particle system, the viscosity is zero, and the collision stress is represented by a non-viscous calculation form. It has the same result as Ding and Gidaspow [17]. 2.3. Conservation equation for the fluctuation energy of the particle phase The conservation equation may be expressed as
3qgran v ! 3qgran εs rs þ V, εs rs u s ¼ V,qc vt 2 2
! ðTk þ Tc Þ : V u s þ g (16)
The flux of the fluctuation energy is given by.
qc ¼ ks Vqgran
(17)
and the conductivity of the fluctuation energy reads.
qffiffiffiffiffiffiffiffiffiffi 4 ks ¼ pffiffiffiε2s rs g0 D qgran
(18)
p
here, we can apply the contact bonding energy Ec to the reconstructed instantaneous velocity distribution function [19]. Then, we achieve the collision energy dissipation of the cohesive particles, which may be simplified to.
(19)
(20)
xr2 ðRc Þ ¼ erfcðRc Þ 1 R2c þ exp R2c
2 pffiffiffi Rc þ
p
8 pffiffiffi R3c 15 p (21)
The particle dynamic viscosity may be then expressed as.
4 5
mp ¼ ε2p rp Dg0 ð1 þ eÞ
rffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffi
qgran 10rp D pqgran 4 1 þ g0 εp ð1 þ eÞ þ 96ð1 þ eÞεp g0 5 p
2
(22) Considering an inviscid particle system, the particle collision energy dissipation rate may be used to deduce the classical particle kinetic theory model. Then, the fluctuation energy exchange between the gas phase and the particle phase reads.
4s ¼ 3bqgran
(23)
! ! 3C εp εg rg u g u p 2:65 εg b¼ d 4D
(24)
where
8 < 24 1 þ 0:15Re0:687 Cd ¼ Re : 0:44
vnðV; tÞ þ V,½ f ðV; tÞ ¼ SðVÞ vt
Re < 1000 Re 1000
(27)
where n(V; t) is the number density function of the particles phase，describing the distribution of the number of particle phase entities in the volume space V and the time space t. S(V) is the source term describing the change of the number density of the particles phase caused by the microscopic behavior of the particles phase, e.g., the agglomeration, breakage and growth. The Van der Waals force dominates the agglomeration process between the fine particles, and the agglomerates are not easy to break. Therefore, here we only consider the agglomeration behavior between the particles. The microscopic description of the agglomeration behavior may be expressed as.
SðVÞ ¼
ðV 1 b V V 0 ; V 0 n V V 0 ; t n V 0 ; t dV 0 2 nðV; tÞ
(28) b V; V 0 n V 0 ; t dV 0
0
xr1 ðRc Þ ¼ exp R2c 1 þ R2c 2 5
The population balance equation (PBE) is a transport equation of the number density that used to determine the both spatial and temporal evolutions of the number density. The population balance equation with particle volume as the internal coordinates can be expressed as.
∞ ð
i q h g ¼ ε2s ks rs g0 qgran f24D gran 1 e2 xr1 ðRc Þ h ipo 5ðV,us Þ 1 e2 xr2 ðRc Þ
(26)
2.4. Population balance model (PBM)
0
rffiffiffiffiffiffiffiffiffiffi
! ! Re ¼ rg εg D u g u p mg
(25)
where b(V, V0 ) is the agglomeration kernel function of particles with volume of V and volume of V0 . The first term on the right side of Eq. (28) represents the increase in the number of particles V formed by the agglomeration of particles of the volume V0 with particles of volume V - V0 . The second term on the right of Eq. (28) represents the agglomeration of particles of volume V with particles of volumeV0 , resulting in the decrease in the number of particles of volume V. The solution domain of Eq. (27) is usually semi-infinite, and there is no general analytical solution. To solve this equation, one may use the discrete method, moment method or Monte Carlo method. Thanks to the strong computational ability of the discrete method, it can directly solve the particle size distribution. It is especially effective for numerical calculation with a particle size range span of 2e3 orders of magnitude. Then, the discrete method is used to discretize the continuous particle size distribution into a series of different size groups. Finally, the discrete form of the population balance equation will be solved numerically. 2.5. Agglomeration mechanisms of particles inside the acoustic field This section analyzed the kernel function of particle agglomeration and the agglomeration mechanisms. The kernel function is a formula that considers both the particle size and the force in the different flow regime [20]. Therefore, in the wide screening particle size distribution region, there are various physical effects that cause the particles to agglomerate. In the sound field turbulence concentrator, the physical effects of sound field force, hydrodynamic force, Brownian heat motion, and gravity make the particles agglomerate [21]. Therefore, the formula of a single particle
J. Zheng et al. / Powder Technology 356 (2019) 930e940
agglomerate cannot accurately represent the true particle agglomeration mechanism. Therefore, this paper considers the coupling of multiple forces in the concentrator and the coupling of various agglomeration mechanisms. Here, we investigate the effects of the orthokinetic agglomeration mechanism(bor), hydrodynamic mechanism (bhy) and Brownian agglomeration mechanism(bBr)on the particle agglomeration process. Because the particle size of our research is <10 mm, differential settlement by the gravity can be ignored [7]. In the acoustic agglomeration process of the particles, the agglomeration mechanisms are coordinated and partly overlapped, and the impact scale of the agglomeration is different. Geng in Ref. [22] pointed out that the simple linear addition of the agglomeration kernel function in the process of the agglomeration would lead to overestimation of the true agglomeration kernel function by 30%. For improving the description accuracy of the agglomeration kernel function, we propose a new roots mean square of the agglomeration kernel function on the basis of the full consideration for three kinds of the agglomeration mechanisms. It can expressed as:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
b0 Or þ b2Hy þ b2Br
b Di ; Dj ¼
(29)
In this way, we get the formula of the sound field agglomeration. The following sections are the detail analysis of three agglomeration mechanisms. 2.5.1. Orthokinetic mechanism The orthokinetic agglomeration mechanism is first proposed by Mednikov [23] in 1965, which represents the most important acoustic agglomeration mechanism. It is mainly based on the theory of acoustic entrainment. The collision of particles with different sizes occurs due to the difference between their relative velocities. We assume that there are two types of spherical particles with different sizes in the agglomeration area [23]. Large particles act as the collecting particle, the smaller as the collected particle. In the process of agglomeration, the concentration of the particles is invariable, that is, with the continuous agglomeration of particles, small particles can instantaneously fill into the agglomeration volume. Through the continuous optimization of the researchers, the following reasonable co-agglomerated kernel functions are achieved.
bor Di ; Dj
2 u tti ttj 1 ¼ Di þ Dj Ug rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ u2 t2ti 1 þ u2 t2tj
r D2 tt ¼ p i ; Ug ¼ 18mg SPL
I ¼ 10 10 I0
sffiffiffiffiffiffiffiffiffiffi 2I rg ug
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reaches 150 dB and the particle diameter is 10 mm, Re is only 0.01. Therefore, it is reasonable to assume that the gas flow within the agglomerated volume is in the Stokes flow region [25]. The Stokes number represents the ratio of particle inertia to diffusion. The value of St is smaller, the inertia of the particles is smaller. On the contrary, the larger the value, the larger the particle inertia. The follow-up of particle motion is less obvious. The collision efficiency associated with the Stokes number can fully express the collision of solid particles in the gas-solid flow field. In this paper, the particle size is in the micron and submicron range, and even at very high sound source levels, the Reynolds number is considered to be very small, and the entire flow is in the Stokes region. Therefore, it is reasonable to use the calculation formula for the collision efficiency related to the Stokes number in calculating the collision efficiency between particles of this case.
a¼
B St St þ A
(33)
where A and B are constants，i.e., A ¼ 0.65 and B ¼ 3.7, St is the Stokes number, as.
St ¼
rp mg Ug D22 18D1 h
(34)
In Eq. (34), D1is the large particle, D2 is the small particle, and h is the relative carrying coefficient between large and small particles, so that.
uðtt1 tt2 Þ h ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 þ u2 t2t1 1 þ u2 t2t2
(35)
Through the above analysis, the formula (34) is applied to the stoke flow regime. When the particle size is below 10 mm, Calculated by the formula (34), the value of St can be between 0.0012 and 10.95. Then, we achieve the modified orthokinetic agglomeration kernel function.
b’or Di ; Dj ¼ abor Di ; Dj
(36)
(30)
(31)
(32)
with I0 being the standard reference value which is equal to 10-12w/ m2 . In the particle agglomeration process, due to the external flow field and the interference between particles, some small particles cannot collide with large particles, especially for the submicron particles, where the collision efficiency is very low [24]. Therefore, it is necessary to modify the agglomeration kernel function by introducing the collision efficiency, representing the fraction of the total number of small particles that can collide with large particles. In acoustic agglomeration, the particles are small, and the relative velocity between the particles and the gas medium caused by the sound waves is not high, thus the Re number is very low. It can be obtained by calculation that even when the sound pressure level
2.5.2. Hydrodynamic interaction According to the orthokinetic agglomeration kernel function, the collision agglomeration cannot occur between the particles of the same size since there is no relative movement between them. In fact, the agglomeration of particles exists in a mono-disperse particle system [26], The hydrodynamics is the main mechanism of this type of agglomeration [27]. The mechanism of hydrodynamics can also explain the collision and agglomeration of particles when the distance between the particles is larger than the amplitude of the acoustic wave. The hydrodynamic mechanism fully considers the influence of the motion of the fluid medium in the acoustic field on the particles, which becomes much more important with the reduction of the particle spacing. The kernel function of the fluid dynamics may be expressed as.
1 4
bHy Di ; Dj ¼ p Di þ Dj 2 Uij
(37)
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Uij ¼
3Ug Ug 2 2 2Di li þ 2Dj lj þ Di li þ D2j l2j 8pd pm
d
þ
3mg 9Ug 2 2 Di li þ D2j l2i Di þ Dj d 2 2 2 p d 64pd 3Ug2 16d2 u
(38)
li lj li qi dlj qj Di dDj
d ¼ kB TB ¼
where li, hi, and qi are intermediate variables, given by.
li ¼
mgi
; hi ¼ 2
1 þ hi mgi
mp þ hi m2gi 9rg Ug ; qi ¼ prp uDi 1 þ hi m2gi
(39)
With mgi and mp being the slip and entrainment coefficients in the Stokes flow, respectively.
uti 1 ; mp ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mgi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ ðutti Þ2 1 þ ðutti Þ
(40)
2.5.3. Brownian agglomeration mechanism In addition to the influence of the acoustic waves, the particles in the fluid medium carry on the Brownian motion simultaneously. The phenomenon of the particle agglomeration caused by the diffusion of the particles due to the Brownian motion is called the Brownian agglomeration [28]. Here, the particle size range is 0.02110 mm. For Brown agglomeration, there is a vital dimensionless number Kn (Knudsen number). This is the ratio of gas molecule’s free path to particle diameter. Otto et al. [29]considered that for a free molecular region with Kn > 50, the collision coefficient is usually obtained based on the molecular motion theory. For a continuous-near continuum region with Kn < 1,the coefficient is obtained based on Fick’s diffusion theory. However, for the transition region of 1 < Kn < 50, there is no such mature theory as molecular motion theory or diffusion theory, and the Fuchs’semiempirical absorption sphere theory is used [30]. In the free molecular region, Friedlander further derived a collision coefficient of particles from the molecular motion theory [31]. Einstein deduced the expression of collision coefficient of particles in the near-continuum region [32].(Because Kn ¼ l/(3v/4p)1/3: where n is the particle volume. pffiffiffi l is the average free-range of the gas molecule, l ¼ kB T= 2pd2 p,where kB is Boltzmann constant ¼ 1.38 1023 J/K, T is absolute temperature ¼ 403 K, d is particle size ¼ 0.021e6.256 mm, p is the pressure inside the device ¼ 105pa), after calculation, the maximum value of Kn ¼ 2.6 103 < <1, so it belongs to continuous-near Media area. The Brownian agglomeration kernel function first derived by Smoluchowski [33] is then obtained by substitution in (41),
b di þ dj ¼ 4p di þ dj
di dj þ 2 2
(41)
For ultrafine particles that follow Stokes-Cunning’s law, a Brownian force or a diffusion forces acts on the particles to move the particles toward a low concentration region. It is equal to the force that the gas prevents the particles moving. This force is Stokes resistance, ie [34].
Diffusion force ¼ 3pmnd=Cc
molecule; the other is that the kinetic energy of particle motion is equivalent to the kinetic energy of gas molecules, that is KE ¼ 3kBT/ 2, the third is that the Brownian force acting on the particles is osmotic pressure, osmotic pressure is given by van’t Hoff’s law. Based on the above three points, Einstein gives the diffusion coefficient of the particles of the Cunningham correction zone as,
(42)
where m is the dynamic viscosity and Cc is Cunningham slip factor Einstein [33] proposed three viewpoints: one is that the Brownian motion of ultrafine particles is the same as the motion of a giant gas
kB T k TC ¼ B f 3pmd
(43)
where B is the particle mobility; f is the coefficient of friction of the resistance to the particle. The derivation process of formula (43) can be found in the literature [32]. It can be seen from the above diffusion coefficient that the higher the temperature of the particles, the larger the diffusion coefficient of the particles, and the more obvious the Brownian agglomeration. Since the diffusion force of the left term of the formula (42) is obtained by the Osmotic pressure. The Osmotic pressure is a macroscopic quantity, but (42) the right side are for a specific single particle, and we can see Eq. (43) embodies the macroscopic and microscopic unity. Substituting Eq. (43) into Eq. (42).
2k T Ci Cj b di ; dj ¼ B di þ dj þ 3mg di dj
! (44)
where di and dj respectively represent the diameters of particles i and j; mg represents the dynamic viscosity; kB represents the Boltzmann constant; T represents the absolute temperature; Ci, Cj represents the Cunningham slip factor [34].
C ¼ 1þ
lm ½2:514 þ 0:8expð 0:55D=lm Þ D
(45)
The movement of the small-sized particles is highly susceptible to the influence of the Brownian force and the frictional force of the particles, which causes collision agglomeration. The smaller the particle, the higher the temperature, then the effect of the Brownian force will be greater, and the effect of friction is smaller. The more likely the particles are to collide, the Brownian motion must be considered when the small-sized particles collision agglomeration occurred. 3. Theoretical analysis 3.1. Collision efficiency Fig. 2 shows the collision efficiency between the collecting particles (i.e. Dj1¼10 mm, Dj2¼5 mm, Dj3¼1 mm) and the collected particles. It may be seen from this figure that, when the collected particle size is <2 mm, the collision efficiency rapidly decreases as the particle size decreases. For instance, for the collected particle size of 0.5 mm, the collision efficiency with the collecting particle size of 5 mm and 10 mmare only 0.044 and 0.012, respectively, which largely weakens the influence of orthokinetic agglomeration mechanism. When the collected particle is >2 mm, the collision efficiency is basically maintained at about 0.7. Moreover, when the collected particles are sufficiently close to the collecting particles, the collision efficiency exhibits a downward trend. This occurs mainly because of the low relative velocity between two types of particles, where the collected particles are more likely to bypass the collecting particles, and the orthokinetic agglomeration effect is weakened. Hence, the conventional orthokinetic agglomeration kernel function significantly overestimated, especially in the submicron region. Therefore, in order to be more realistic, we must use
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the collision efficiency to modify the conventional orthokinetic agglomeration kernel function. (See Fig. 1.) 3.2. The comparison of the agglomeration kernel Fig. 3 shows the influence of different collecting particle size (Dj1 ¼ 1 mm，Dj2 ¼ 5 mm，Dj3 ¼ 10 mm) on the agglomerate kernel function. The acoustic frequency and sound pressure level is 1400 Hz and 147 dB, respectively [35]. When the collected particle size Di belongs to submicron regime, due to the low collision efficiency, the contribution of the orthokinetic mechanism to the particle agglomeration is very small compared with other agglomeration mechanism. When the collected particle size Di is smaller than the collecting particle size Dj, an optimum collected particle size exists. At this an optimum collected particle size, the orthokinetic kernel function reaches to peak value. When the two kinds of particles size (Di and Dj) is close enough, the relative velocity between the particles is very small. Hence, according to the orthokinetic agglomeration mechanism, the orthokinetic kernel function decreases sharply. As shown in Fig. 3 (a), the Brownian agglomeration plays a dominant role in the small particle size range. By increasing the
Fig. 1. Range of values of St number.
Fig. 3. Influence of particle size on the agglomerate kernel function.
Fig. 2. Collision efficiency between the collecting particles and collected particles.
particle size, the Brownian agglomeration kernel function will decrease. When the particle size increases to a certain critical particle size, the Brownian agglomeration kernel function value begins to rise slowly. The critical particle size increases with the increase of the collecting particle size. However, when the critical
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Fig. 6. Ratio N/N0 of the particle number density as a function of time.
particle size is >10 mm, it has no practical significance for the Brownian agglomeration, which is not shown in Fig. 3 (c). When the collected particle size is >1.2 mm, the order of the magnitude of the Brownian agglomeration kernel function is much lower than that of the other two agglomeration kernel functions. In this regime the effect of the Brownian agglomeration on the particle agglomeration is can be neglected. For the collecting particle sizes of 5 mmand 10 mm, shown in Fig. 3 (b), (c), the hydrodynamic mechanism plays the dominant role in the submicron regime, and the hydrodynamic kernel function increases with the increase of the collected particle size. By comparing the results of Fig. 3, we may see that the order of the magnitude of the agglomeration kernel function of three types increases obviously with the increase of the collecting particle size. As shown in the Fig. 3. The optimization of the agglomeration kernel function by the root mean square of the agglomeration kernel function is mainly reflected in two aspects compared with the linear agglomeration kernel function. i.e., when the collected particle size is in the range of 0.2e0.8 mm, the agglomeration kernel function is optimized by 20e30%; on the other hand, when the distance between the particle size Di and Dj is close enough, the agglomeration kernel function is optimized by 13e25%. With the increase of the particle size, the root mean square of agglomeration kernel function is close to the orthokinetic agglomeration kernel function. This is also consistent with the conclusion that the main agglomeration mechanism is the orthokinetic agglomeration mechanism. Fig. 4. Structure of the acoustic field agglomeration device.
4. Simulation and discussions 4.1. Model validation
Fig. 5. Distribution of the particles number density for different particle sizes.
The verification of the mathematical model in this paper is based on the experimental results of Liu [35]. In Liu’s experiment, he compared the conditions of different sound pressure levels and different acoustic frequencies. The results show that the acoustic wave frequency is 1400 Hz and the sound pressure level is 147 dB, which is the recommended case. The focus on this paper is on the agglomeration mechanism and the correction of agglomeration kernel functions. Therefore, this study selects aforementioned working condition and the physical model for simulation. The simulation results are representative. Fig. 4 shows the structure of the acoustic field agglomeration device. The acoustic generator is located above the acoustic field agglomeration device. The inlet boundary condition is set to the velocity inlet, the outlet boundary condition is the pressure outlet, and the wall surface is the non-slip boundary condition. In addition, the fly ash particles enter from the upper part of the agglomeration
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chamber. Specific simulation parameters are shown in Table 1. The population balance model (PBM) is numerically solved by the discrete method. The particle sizes of 0.021e6.256 mm measured in the flue before the dust collector are divided into 12 ranges, as shown in Table 2 below. It may be seen from Fig. 5 that, while the acoustic wave is not acting, the initial particle size distribution presents a three-mode distribution with peak values at 0.08 mm, 0.63 mm, and 1.75 mm. The number density in the whole particle size range of the coal-fired fly ash decreases significantly under the application of acoustic compared with the experiment data [35], especially in the submicron region. This indicates that a large number of fine particles are agglomerated under the action of sound waves. When the particle size is 2.2e6.256 mm, the simulation results are slightly larger than the experimental values, mainly due to the simulation does not consider the phenomenon of differential sedimentation of larger particles in the experimental process. The simulation process is basically consistent with the experimental results. Fig. 6 shows the N/N0 ratio of the particle number concentration evolved with the time in the outlet. The ratio is the concentration with the acoustic wave action to that without acoustic wave action. The number of concentration decreases with the extension of the
937
action time of the acoustic wave, and the decreasing speed gradually slows down. The particle number concentration decreased by 45.06% in 3e5 s and 27.18% in 5e7 s. This is mainly because in the early stage of particle agglomeration, the particle number concentration is higher, the particle distance is smaller, and the agglomeration process between particles is easier. With the process of the agglomeration occur, the particle number concentration decreases, and the agglomeration effect between particles is weakened. The results demonstrate the effectiveness of the acoustic agglomeration. The simulation results are in good agreement with the experimental results. 4.2. Simulation result discussions Fig. 7 shows the instantaneous concentration distribution of the particles phase for the simulation time of 0.1 s, 2 s,4 s, 6 s. It also may be seen from Fig. 7 that fine particles easily agglomerate in the high intensity turbulence regime and near the wall field. At 0.1 s, the particles are in a freely dispersed state because the sound field has not yet funcationed. By acting the acoustic wave, small eddies begin to appear in the upper part of the acoustic agglomeration device at 2 s, and the agglomeration is promoted at the edge of the
Fig. 7. The instantaneous concentration distribution of the particles phase for the simulation time of 0.1 s, 2 s,4 s, 6 s.
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Fig. 8. Distribution of the particle phase fluctuation temperature for different heights. (Starting point (height ¼ 0 m) at the bottom of the device).
Fig. 9. Distribution of the particle phase diameter in different height. (Starting point (height ¼ 0 m) at the bottom of the device).
Table 1 Parameters used in simulation. Parameter
Value
Unit
Height of device Width of device Initial diameter of particles Density of particles Acoustic frequency Sound pressure level Inlet velocity Total mesh number
1500 108 0.021e6.256 2400 1400 147 0.5 15,300
mm mm mm kg/m3 Hz dB m/s e
turbulent eddy. This indicates that particles form agglomeration under the action of the acoustic wave and the turbulent flow. However, the agglomeration range is not large. With the continuous action of the acoustic waves, the vortex size increases and the turbulence intensity has further enhanced. The flow of the particles phase in the acoustic agglomeration device present a kind of ringannular flow construction. Fig. 8 shows the distribution of the particle phase fluctuation temperature in different heights. It may be seen from Fig. 8 that the fluctuation temperature of the particles phase can reflect the extent to the particles agglomeration. Since the particle fluctuation temperature is a measure of the particle pulsation intensity, in the center of the acoustic field agglomeration device, the concentration of particle phase is dilute and the probability of the agglomeration is lower. Hence, the fluctuation temperature of particles is smaller. In the near wall area, the particle concentration is higher and the fluctuation is more intense. Thus, the particle fluctuation temperature is larger. As the height increases, the lowest value of the particle pulsation temperature moves to the right along the radial position, while its lowest value gradually increases. Since the particle temperature indicates the severity of particle collision agglomeration, the higher the particle concentration, the more easily the particles collide with the agglomerates, so the higher the particle temperature. Because the inlet is above the left side, the particles will accumulate near the right wall due to inertia. Therefore, it can be seen from Fig. 7 that the instantaneous concentration distribution of the particle phase increases with height, the high concentration region of the particle gradually shifts to the right side and the concentration value increases, and the temperature of the particles on both sides also increases. Due to the higher particle concentration, the particle collision is more intense and enhances the particle pulsation temperature. At the same time, with the flow direction, after the particles collide and agglomerate, the number of particles decreases. Therefore, at h ¼ 450 mm, the average collision pulsation intensity of the particles is the lowest, and as can be seen from Fig. 7, since the center of the vortex is close to the right side, as the height increases, the lowest value of the pulsation temperature of the particles moves to the right along the radial position. Fig. 9 shows the distribution of the particle phase diameter for different heights. It can be seen from Fig. 9 that the agglomeration process is affected by both the turbulent and the acoustic wave. Due to the effect of the high concentration of particles near the inlet and the acoustic generator located above the acoustic agglomeration device near the inlet position, the particles easily agglomerates near the inlet. It may be seen from Figs. 7 and 8 that under continuous action of sound waves, the size of the vortex in the acoustic agglomeration device has continuously increased, resulting in a higher particle concentration and a high fluctuation temperature in the near wall region at any height, and the particles are more likely to agglomerate. As the height decreases, the agglomeration between the particles continues to occur, and the average diameter of the particles increases continuously. However, due to the decrease of the particle concentration, the collision probability between the particles decreases, and the change trend of the
Table 2 The number density distribution of the particle size. Bin
The average particle size /mm
Number density percentage(n)
Bin
The average particle size /mm
Number density percentage(n)
Bin-0 Bin-1 Bin-2 Bin-3 Bin-4 Bin-5
6.256 3.074 1.945 1.224 0.758 0.481
0.0010 0.0038 0.0077 0.0115 0.0102 0.0082
Bin-6 Bin-7 Bin-8 Bin-9 Bin-10 Bin-11
0.314 0.200 0.120 0.073 0.041 0.021
0.0088 0.0826 0.3902 0.1857 0.1685 0.1218
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particle size along the axial direction gradually slows down. In addition, the unevenness distribution of the particle diameter has gradually decreased. According to the calculation data analysis, the average diameter of the outlet particles is stable at around 5.16 mm, and the average diameter of the particles has increased by about 106% compared with the average diameter 2.5 mm of the particles at the inlet.
bhy bbr Ug I C lm T
a 5. Conclusion
St
h We have studied the mechanism of the acoustic agglomeration of the coal fly ash particles in the acoustic field agglomeration device. Next, the root mean square of the agglomeration kernel function has proposed for numerical simulation of the acoustic agglomeration process. The contributions and observations of this paper may be summarized as follows. (1) The coupling of the modified kinetic theory of fine particle flow and the population balance model for the coal fly ash agglomeration in the acoustic field is established. The viscous particle dynamics model considering the cohesive collision in the particle aggregation process is used to simulate the agglomeration flow process. At the same time, the root mean square of the agglomeration kernel function is used to closed the population balance model. (2) Compared with the linear additive agglomeration kernel function, the agglomeration kernel function is optimized by 20%e30%. When the particle size range is 0.2e0.8 mm; the agglomeration kernel function is optimized by 13%e25% when the particle size is fully close. (3) The fluctuation temperature of the particles can reflect the extent to the particles agglomeration. The particle fluctuations temperature in the near wall region is high and easy to agglomeration. (4) The continuous effect of the acoustic wave enhances the agglomeration effect of particles. The average particle size of the outlet position has increased by 106% with respect to the inlet position.
Notation
r
ε ! u g p
b tg tt mg ms Tk Tc
qgran
Ec S Rc e Dpo V t
bor
density (kg/m3) concentration (m3) speed vector (m/s) gravity acceleration (kg/(m$s2)) pressure (pa) the momentum exchange coefficient (kg$s/m3) stress tensor of gas phase (pa) the relaxation time (s) the dynamic viscosity (pa$s) the shear viscosity (pa$s) dynamic stress (pa) viscous collision stress (pa) the particle fluctuation temperature (K) the contact bonding energy (J) shear stress tensor (kg/ms2) ration of contacting bonding energy to fluctuation energy (–) normal restitution coefficient (–) initial particle diameter before agglomeration (m) volume of particle (m3) time (s) the orthokinetic agglomeration kernel function (m3/s)
Uij d
mp kB Sij qc kp D g0 ks Re Cd
Fs u g xp(Rc) xv(Rc) xr(Rc)
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the hydrodynamics agglomeration kernel function (m3/ s) the Brownian agglomeration kernel function (m3/s) the gas velocity amplitude (m/s) the sound field strength (w/m2) the cunningham slip factor (–) the molecule mean free path (m) the absolute temperature (k) the collision efficiency (–) the stokes number (–) the relative carrying coefficient (–) the relative velocity between two particles (m/s) the particle distance (m) the entrainment coefficient (–) the Boltzmann constant (J/k) deformation rate (–) flux of fluctuation energy (kg/s3) the thermal conductivity (w/m$k) the agglomeration diameter (m) radial distribution function (–) the conductivity of the fluctuation energy (kg/(m$s)) the Reynolds number (–) the drag coefficient (–) the fluctuation energy exchange coefficient (–) angular velocity (rad/s) collision dissipation of energy (m2/s2) correction dynamic pressure coefficient (–) correction shear viscosity coefficient (–) collision dissipation coefficient (–)
Acknowledgments The financial support provided by Jilin Province Science and Technology Department 13th Five Year Key Technology R&D Project (20180201001SF) and Jilin Provincial Department of Education “13th Five Year” Scientific Research Project (JJKH20190703KJ) is gratefully acknowledged. References [1] C.L. Senior, J.J. Helble, A.F. Sarofim, Emissions of mercury, trace elements, and fine particles from stationary combustion sources, Fuel Process. Technol. 65e66 (2000) 2000263e2000288. ~talo, Jukka Hautanen, Electrostatic precipitator penetration function for [2] S. Yla pulverized coal combustion, Aerosol Sci. Technol. 129 (1998) 17e30. [3] J. Lewtas, Air pollution combustion emissions: characterization of causative agents and mechanisms associated with cancer, reproductive, and cardiovascular effects, Mutat. Res. Rev. Mutat. Res. 636 (2007) 95e133. [4] U. Franck, A.M. Leitte, P. Suppan, Multiple exposures to airborne pollutants and hospital admissions due to diseases of the circulatory system in Santiago de Chile, Sci. Total Environ. 468e469 (2014) 746e756. [5] J. Hautanen, M. Kilpelainen, E.I. Kauppinen, K. Lehtinenl, Electrical agglomeration of aerosol particles in an alternating electric field, Aerosol Sci. Technol. 22 (1995) 181e189. [6] Y.W. Li, C.S. Zhao, X. Wu, D.F. Lu, S. Han, Aggregation mechanism of fine fly ash particles in uniform magnetic field, Korean J. Chem. Eng. 24 (2007) 319e327. [7] I.V. Derevich, Coagulation kernel of particles in a turbulent gas flow, Int. J. Heat Mass Transf. 50 (2007) 1368e1387. [8] W. Zhao, Y. Liu, J.J. Bao, J.F. Geng, L.J. Yang, Experimental research on fine particles removal from flue gas by chemical agglomeration, Proc. CSEE. 33 (2018) 52e58. [9] D. Liao, C. Wu, Y. Yao, H. Hu, The use of quadrature method of moments (QMOM) in studying the mechanisms of aerosol agglomeration and its practical use, Am. J. Heat Mass Tran. 2 (2015) 42e58. [10] H.S. Patterson, W. Cawood, Phenomena in a sounding tube, Nature. 127 (1931) 667. [11] S.V. Komarov, T. Yamamoto, T. Uda, M. Hirasawa, Acoustically controlled behavior of dust particles in high temperature gas atmosphere, Iron Steel Inst. Jpn. 44 (2004) 275e284. [12] J. Liu, J. Wang, G. Zhang, J. Zhou, K. Chen, Frequency comparative study of coal-fired fly ash acoustic agglomeration, J. Environ. Sci. 23 (2011) 1845e1851. [13] V. Vekteris, V. Strishka, D. Ozarovskis, V. Mokshin, Experimental investigation of processes in acoustic cyclone separator, Adv. Powder Technol. 25 (2014)
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