Particle velocity distribution function around a single bubble in gas-solid fluidized beds

PTEC-14889; No of Pages 12 Powder Technology xxx (2019) xxx

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Particle velocity distribution function around a single bubble in gas-solid fluidized beds Runjia Liu a,b, Zongyan Zhou b, Rui Xiao a,⁎, Mao Ye c, Aibing Yu b a b c

Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, College of Energy and Environmental Engineering, Southeast University, Nanjing, 210096, China Laboratory for Simulation and Modelling of Particulate Systems, Department of Chemical Engineering, Monash University, Clayton, VIC 3800, Australia National Engineering Laboratory for MTO, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, China

a r t i c l e

i n f o

Article history: Received 21 June 2019 Received in revised form 27 October 2019 Accepted 2 November 2019 Available online xxxx Keywords: Bubbling fluidized bed PIV DEM Tri-peak model Particle velocity distribution

a b s t r a c t Particle image velocimetry (PIV) is employed in this work to measure particle flow field in a two-dimensional fluidized bed to obtain particle velocity distribution function around a single bubble. Discrete element method (DEM) is also used to investigate particle velocity distribution at the individual particle scale. Both experimental and simulation results show that the velocity distribution of particles surrounding a single bubble can be described by tri-peak model which is a linear superposition of three Maxwellian distributions. A tri-peak distribution model based on the fluid and particle control mechanisms is theoretically derived. Three kinds of models such as tri-peak model, bi-peak model and single-peak model are proposed and compared. The error analysis shows that compared with other models, the tri-peak model can profile particle velocity distribution more accurately. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Bubbling fluidized beds have been widely used in industries such as biomass energy production, pharmaceuticals, chemical engineering and pollution control. Bubbles play an important role in particle mixing [1] and heat and mass transfer [2] in gas-solid fluidized beds, and bubble dynamics have been carried out extensively in the past decades [3,4]. For example, Davidson and Harrison [5] developed a model to describe the bubble motion behaviour in bubbling fluidized beds. Numerical and experimental studies show that the characteristics of bubbles such as size and shape can be affected by many factors including bed distributor [6], gas inlet velocity, bed dimensions, particle size [7,8] and particle shape [9]. There are some studies focusing on the pressure wave attenuation [10], particle mixing [1,11], and cloud formation around bubbles [12,13]. Tremendous efforts also have been made to investigate and understand the particle speed distribution surrounding bubbles, for example, [14–25]. For the research on particle velocity for the particles around single bubble, most of the studies focused on the motion of a single bubble as it rose and then broke through the top surface [14,15]. By using PIV application, the initial particle velocity distributions of the bubble eruption at the bed surface were observed and discussed by Santana et al. [16]. Their results show that avoiding coalescence of bubbles at the bed surface can lead to less particle entrainment out of the bed and ⁎ Corresponding author. E-mail address: [email protected] (R. Xiao).

consequently to shorter fluidized beds. The PIV experiments by Muller et al. [17] suggest that particle velocity around bubble, particularly its roof as the bubble broke through the surface of the bed can be predicted by potential flow analysis. By using PIV combined with Digital Image Analysis (DIA), particle velocity field for single bubble inside the bed was observed by Laverman et al. [18]. However, none of these authors presents the discussion of concrete velocity distribution curve. Many studies [19–21] suggests that the particle velocity distribution function in a gas-solidtwo-phase system should satisfy the Maxwellian distribution function. Some researchers [22] employed the Maxwellian distribution to describe the particle motion in particlefluid two-phase systems. The Maxwellian distribution function in such two-phase systems was originally verified by Carlos and Richardson [23,24] via experimentation. These authors investigated the velocity distribution of 570 particles in a fluidized bed, and found that the distribution was in a good agreement with the Maxwellian distribution function. Gidaspow et al. [20] applied the findings of Carlos and Richardson to derive the kinetic theory of granular flow (KTGF). However, some researchers suggested that the particle velocity distribution in a two-phase system does not completely follow the Maxwellian distribution. For example, Goldschmidt et al. [25] reported that in a dense fluidized bed, elastic particles follow a homogeneous Maxwellian distribution while inelastic and coarse particles follow a heterogeneous Maxwellian distribution. Lu et al. [26] reached a similar conclusion via hard-sphere model simulation and proved that higher inelasticity of particles results in more conspicuous heterogeneity. Kumaran [27] simulated the collisions of 500 particles in a cube and

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Please cite this article as: R. Liu, Z. Zhou, R. Xiao, et al., Particle velocity distribution function around a single bubble in gas-solid fluidized beds, Powder Technol., https://doi.org/10.1016/j.powtec.2019.11.007

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confirmed that elastic particles follow a Maxwellian distribution, coarse particles approximately follow a Maxwellian distribution, and the inelastic particle velocity distribution was a superposition of a Maxwellian distribution and an exponential distribution. Leszczynski et al. [28] analysed particle velocity distribution functions in different areas of a circulated fluidized bed and concluded that the result is a linear combination of two Maxwellian distributions. Based on energy minimization multi-scale (EMMS) theory and heterogeneity, Wang et al. [29] theoretically determined that the dual-peak distribution in a gas-solid two-phase system is a superposition of two Maxwellian distributions. Liu et al. [30] analysed particle velocity fluctuations via large-scale direct numerical simulation (DNS), showing that particle fluctuation velocity (PFV) distribution matches the typical dualdiscrete mode. Some studies [31–35]found that the particle velocity distribution significantly deviates from Maxwell distribution law due to the effects of dissipation due to inelasticity of interparticle collisions. In particular, particle speed distribution will follow an stretched exponential law: P(v)~ exp [−|v/v0|ζ], where the exponent ζ = 3/2, P is the velocity distribution function. v and v0 is the particle velocity and “thermal velocity”. These studies demonstrate that the particle velocity distribution in a gas-solid system has a certain level of heterogeneity. Various researchers have proposed different velocity distribution functions under different conditions. However, detailed information such as particle velocity distribution around bubbles is required to describe particle behavior for a better understanding of bubble motion. In this work, based on a typical mesoscopic-scale heterogeneous structure surrounding a bubble, the velocity distribution function of particles is investigated. Furthermore, a tripeak expression of the velocity distribution function is proposed based on the fluid and particle control mechanisms. Three kinds of models of tri-peak, bi-peak and single-peak models are compared and discussed in detail. 2. Method description 2.1. Experimental setup The technique of particle image velocimetry (PIV) provides the possibility for the study of particle behaviour. For example, Duursma et al. [36] used PIV to produce vector maps of the gas-phase flow in the

freeboard region of fluidized beds. Bokkers et al. [37] used PIV to obtain the particle velocity profile in the vicinity of a bubble in a dense fluidized bed. Link et al. [38] investigated particle movement in a spout-fluid bed by the PIV technique and hard-sphere based DPM simulation, and showed a similar influence of the background fluidization velocity on the spout behaviour. Dijkhuizen et al. [39] extended the PIV method to simultaneously measure the local instantaneous granular temperature, and found that direct lighting gives better results than indirect lighting in the experimental setup. Muller et al. [40] analysed the motion and eruption of a bubble at the surface of a two-dimensional (2D) fluidized bed using PIV. In this study, the experiments were performed in a pseudo-2D fluidized bed, as shown in Fig. 1. The fluidized gas is air, and particles are 1 mm glass balls in diameter with a narrow grain size distribution. The bed layer thickness is 1.5 mm. This ensures that only one layer of observable particles exists along the thickness direction, which ensures the accuracy of images captured and velocity field processing and analysis. The minimum fluidization velocity Umf is 0.887 m/s which was calculated by the relationship of pressure drop with gas superficial velocity. Image capture was performed using Olympus i-VELOCITY automatic high-velocity camera, and the recording rate was 1000 fps. The captured images were processed to identify particles via grey scale differences and calculate the particle spatial distribution. As particles were present in large quantities and at high density, the particles in two consecutive frames were matched via cross-correlation theory, and the crosscorrelation of particle images was calculated to obtain the velocity field and velocity distribution. In this work, cross-correlation analysis was performed via fast Fourier transform (FFT) to obtain the particle velocity [41]. The analytical process is shown in Fig. 2. Briefly, in the same position, two sampled regions, f(m,n) and g(m,n), with identical dimensions were selected. Images of f and g were subjected to FFT to obtain expressions F(m,n) and G(m,n) in the frequency domain. Then, the spatial frequency domain operation was performed to calculate the cross-correlation function Φ in the frequency domain. Φ was subjected to reverse Fourier transform to obtain the cross-correlation function ϕ. The cross-correlation peak was identified to determine the identical individual particle in two images. Then the displacement of particle will be calculated and then divided by the time interval between the two images to obtain the particle velocity.

Fig. 1. (a) Schematic of the PIV experiment device, and (b) Pseudo-2D cold fluidized bed.

Please cite this article as: R. Liu, Z. Zhou, R. Xiao, et al., Particle velocity distribution function around a single bubble in gas-solid fluidized beds, Powder Technol., https://doi.org/10.1016/j.powtec.2019.11.007

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Fig. 2. Cross-correlation calculation procedure for the analysis of PIV experiment data.

2.2. Discrete element method (DEM)

and

DEM model tracks each particle using Newton's second law of motion, and the fluid phase is described using the averaged equations of motion. DEM method has been used and documented somewhere [42–45]. For convenience, it is briefly given below.

Fab;t ¼



−kt δt −ηt υab;t ; for jFab;t j ≤μ f jFab;n j −μ f jFab;n j t ab ; for jFab;t j Nμ f jFab;n j

ð8Þ

2.2.1. Governing equations The fluid flow is solved by the averaged continuity and momentum equations of the continuum which are given as

Here k is the spring stiffness, η the damping coefficient, nab the normal unit vector, δn the overlap, δt the tangential displacement, μf the friction coefficient and υab the relative velocity between two particles. The drag force Fd,i is a combination of Ergun equation [47] for dense regime and Wen-Yu's drag model [48] for dilute regime:


 ∂ ερg

Fd;i ¼ 3πμ g ε2 dp ðu−vi Þ f ðεÞ

∂t


 þ ∇  ερg u ¼ 0


 ∂ ερg u ∂t

ð1Þ


 þ ∇  ερg uu ¼ −ε∇p−Sp −∇  ðετ Þ þ ερg g

ð2Þ

where ε presents the porosity, g the gravity acceleration, ρg the gas density, u the gas velocity, τ the viscous stress tensor, and p the pressure of the gas phase. The source term Sp is: 1 Sp ¼ V

Z X N

Fd;i δðr−ri ÞdV

ð3Þ

2

d ri dt

2

¼ Fcont;i þ Fd;i −V i ∇p þ mi g

ð4Þ

¼ Ti

ð5Þ

2

Ii

d Θi dt

2

8 150ð1−εÞ 1:75 Rep > < þ ; εb0:8 18 ε3 18ε3 f ðεÞ ¼ Cd > : Rep ε−4:65 ; ε ≥0:8 24

ð10Þ

In this study, particle Reynolds number Rep is less than five. Therefore, for low Reynolds numbers cases, the drag coefficient C d ¼

24 3 Rep Þ follows Oseen [49]. ð1 þ Rep 16

i¼0

where V is the volume of fluid cell and N the number of particles in the fluid cell. Fd,i is the drag force acting on particle i. The δ-function ensures that the drag force acts as a point force at the particle centre [46]. The Newtonian equation is used to track the motion of particles. The equations for the translational and rotational motion of particle i are: mi

ð9Þ

where mi is the mass of particle, the first and second terms are the contact force and drag force, respectively. The third term represents the pressure drag induced by the pressure gradient around the particle. Ii is the moment of inertia, Θithe angular displacement, and Ti the torque of particle. The contact force Fcont,i includes both normal and tangential components which are given by Fcont;i ¼

X 

Fab;n þ Fab;t



ð6Þ

contactlist

Soft-sphere model is used to calculate the contact force. The normal and tangential components are respectively given by: Fab;n ¼ −kn δn nab −ηn υab;n

ð7Þ

2.2.2. Simulation conditions The simulation parameters are listed in Table 1. A pseudo-2D fluidized bed model with 100 mm in width, 7 mm in depth and 1000 mm in height is used to simulate experimental fluidized bed. The depth of the fluidized bed in CFD-DEM simulations is larger than that in the experiments. Large depth is more close to real cases, and provide more meaningful results than experiment. The parameters such as gas temperature, gas viscosity and molar mass, particle size and density, are

Table 1 Simulation parameters. Parameter

Value

Unit

Gas temperature, T Shear viscosity of gas, μg Molar mass of gas, M Number of particles, Npart Diameter of particle, D Density of particle, ρs Bed width, w Bed depth, d Bed height, h Normal restitution coefficient, en Normal restitution coefficient wall, en,w Tangential restitution coefficient, et Tangential restitution coefficient, et,w Friction coefficient, μ Time step, Δt Time step for data save, Δts

293 1.8 × 10−5 2.9 × 10−2 80,000 1.0 × 10−3 2400 1.0 × 10−1 7.0 × 10−3 1.0 0.97 0.97 0.33 0.33 0.1 1.0 × 10−5 1.0 × 10−3

(K) (Pas) (kg/mol) (−) (m) (kg/m3) (m) (m) (m) (−) (−) (−) (−) (−) (s) (s)

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determined based on air properties used in the experiment. For restitution coefficient and friction coefficient, the values were chosen from our previous works [50]. The sensitivity of model prediction based on particle spring stiffness k was examined. The results showed that the overlapδn would less than 5% of particle diameter and the gas-solid flow characteristics were almost independent of the value of spring stiffness after the value exceeds 104N/m. This is consistent with the work by Peng et al. [43]. Therefore, the values of 7 × 104 N/m and 2 × 104 N/m for the normal and tangential spring stiffness were chosen in this work. 3. Results and analysis 3.1. Comparison of experimental and simulation results Fig. 3 shows the bubble snapshots and flow patterns obtained from experiment (top row) and simulation (bottom row), demonstrating consistent results. Also, the shape of bubbles agree with typical characteristic as described by Kunii et al. [2]. Bubbles are not spherical but flattish and even concave at the bottom. This is because the rising bubble drags a wake of particles up the bed behind it and those particles change the bubble shape. After the first leading bubble was formed near the distributor, the air through the central jet was not able to form the next bubble immediately. Instead, an elongated void or multiple small bubbles are formed. This result is consistent with the previous DEM simulation of Pandit et al. [8] Those void/multiple small bubbles exhibit another phenomenon called bubble coalescence. Experimental [9] and simulation [6] studies have shown that the leading bubble often deaccelerates before the trailing bubble accelerates and catches up, and then the merged bubble moves up the bed as a single bubble. A similar trend can be observed in this study. Fig. 4 presents the corresponding particle velocity vector field around the bubble from experiment (left) and simulation (right). Note that in experiment, limited particles velocity vector are detected by PIV technology because only one layer of observable particles exists along the thickness direction. More

detailed information for particle velocity can be obtained in the simulation results. The colours represented different value of particle velocity. As shown in Fig. 4, particles below the bubble have large velocity because they are stirred up by the bubble trailing vortex. Particles around bubble have negative velocity. Some studies [12,51] use “cloud pattern” to describe the region of those particles. Particles far from bubble have low velocity. This is because that according to Eqs. (9) and (10), lower porosity ε results in lower drag force and decreases particle velocity. On the other hand, the results from both experiment and simulation show that particles at the top drop along both sides of bubble and particles at the bottom constitute the bulk of trailing wake, and rise along the bubble. The similar trend can also be found in the early theoretical studies [12]. Fig. 5 shows particle velocity distribution determined via experiment and simulation under various gas velocities. The velocity distribution is the average of 30 snapshots for each gas velocity. Note that the particle number in each snapshot is different. Therefore, we define f ðcÞdc F(c) as FðcÞdc ¼ , where F(c)dc indicates the particle number n ratio in the velocity range [c, c + dc]. Compared with the experiment, particle velocity distribution curves from simulation is more smooth because of more data points obtained from simulation. Although the bubbles have different shapes, the velocity distribution curves are basically consistent and follow similar patterns. For example, all the figures show the feature of “multi-peaks” for velocity distribution. This means that the Maxwellian single-peak velocity distribution curve may not accurately reflect the particle velocity distribution surrounding a bubble. When the gas velocity changes, the peak also changes. The first peak on the left decreases with the increase of gas velocity. The second peak shifts to the right and other peaks become more obvious compared with the low gas velocity cases. The same trend can also be found in the study by Alexander et al. [52]. Their results show that increasing gas velocity lead to a broader particle velocity distribution with a longer tail of particles with high speeds. They believed that this trend are attributed to the increased gas flow rate which leads to an increase in the number

Fig. 3. Snapshots of flow patterns (top - experiments; bottom – DEM simulation) under various gas velocities: (a) Ug = 1.22 m/s, (b) Ug = 1.33 m/s, (c) Ug = 1.44 m/s, (d) Ug = 1.66 m/s (size of image: 80×100 mm).

Please cite this article as: R. Liu, Z. Zhou, R. Xiao, et al., Particle velocity distribution function around a single bubble in gas-solid fluidized beds, Powder Technol., https://doi.org/10.1016/j.powtec.2019.11.007

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Fig. 4. Comparison of particle velocity vector field: experiment (the left figure) vs DEM simulation (the right figure) in a fixed research domain(80×80 mm).

0.25

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Ug=1.22m/s

Ug=1.33m/s 0.20

Experiment Simulation

Particle number ratio [%]

Particle number ratio [%]

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Ug=1.44m/s 0.20

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particle velocity [m/s]

(c)

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particle velocity [m/s]

particle velocity [m/s]

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particle velocity [m/s]

(d)

Fig. 5. Particle speed distribution determined via experiment and simulation under various gas velocities in a fixed research domain(80×80 mm): (a) Ug =1.22 m/s, (b) Ug =1.33 m/s, (c) Ug =1.44 m/s and (d) Ug =1.66 m/s.

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of large, fast moving bubbles, causing particles around them to move at faster speeds. In this study, another possible reason is that bubble size will affect the particle number ratio in a size-fixed image and change the height for each peak shown in Fig. 5. 3.2. Particle velocity distribution in different domain The selected domain size may impact the characteristics of particle velocity distribution. Therefore, the effect of domain size on particle velocity distribution is examined. Fig. 6 shows a particle velocity vector field for a typical bubble. Three different sized domains are chosen for analysis, and the corresponding particle velocity distribution curves are shown in Fig. 7. Based on the method of Li et al. [53], the particles are divided into three categories according to particle-fluid interaction mechanism. The particles inside the bubble correspond to the rightmost peak of the distribution curve in domain 1. The particles in this region are in dilute phase and are stirred up by the bubble trailing vortex. They have a small quantity but high velocity, belonging to the fluid dominating (FD) category. The particles around the bubble correspond to the second peak in domain 1 or the peak in domain 2 shown in Fig. 7. The particles in this region are in relatively dense phase and fall into the particle-fluid compromising (PFC) category. The remaining particles are less susceptible to the gas velocity. Therefore, the velocity is close to zero, and they belong to the particle dominating (PD) category. This corresponds to the first peak near the zero point in domain 3. Domain 2 could be better to describe the velocity feature in Fig. 6 (Ug = 1.22 m/s), because particles far from bubble is not the interest of study. However, for large gas velocity cases, bubbles may be large and the information near the bubble may be missing. Therefore, in this study, to fully and comprehensively describe the characteristics of particle movement around a single bubble, particle velocity distribution for domain 3 (80 × 80 mm) will be used and discussed in the following analysis. 3.3. Tri-peak model 3.3.1. Velocity distribution function The velocity distribution function is the foundation of particle dynamics theory. Normally, the particle velocity distribution follows a Maxwellian single peak distribution: !  32 1 ðc−uÞ2 f M ðc=u; n; θÞdxdc ¼ n exp − dxdc 2πθ 2θ

ð11Þ

where θ is the granular temperature, u is the particle average velocity, c is the particle velocity and n is the particle quantity density.

0.30

domain 1 domain 2 domain 3

0.25

Particle number ratio [%]

6

0.20 0.15 0.10 0.05 0.00 0.0

0.1

0.2

0.3

0.4

0.5

0.6

Particle velocity [m/s] Fig. 7. Particle speed distribution at different selected domains.

Wang et al. [29] suggested that the single-peak distribution based on gas molecular dynamics theory was not an accurate representation of the heterogeneity in a fluidized bed. Therefore, a dual-peak distribution based on EMMS theory was proposed: ! 32 1 ðc−udi Þ2 f ðcÞ ¼ ð1− f Þndi exp − 2πθdi 2θdi !  32 ðc−ude Þ2 1 exp − þ fnde 2πθde 2θde 

ð12Þ

where f represents the proportion of dense phase, the first term represents the velocity distribution of dilute-phase particles, the second term represents the velocity distribution of dense-phase particles, and the overall velocity distribution is a superposition of the two. It is worth noting that particle velocity c in Eqs. (11) and (12) is a vector with three components. If velocity components in different directions were investigated, velocity distributions in the vertical direction would be wider than those in the horizontal direction. As this work only involves the value of the particle velocity, scalar functions and velocity distribution functions are investigated. Eqs. (11) and (12) cannot be used directly. On the other hand, because the velocity determined via the experiment only has components in two directions, the velocity distribution functions should be two-dimensional. In this work, the Maxwellian distribution is used as an example to derive the two-dimensional velocity distribution function and velocity

Fig. 6. Particle velocity field for a typical bubble (Ug = 1.22 m/s): (a) domain 1: 40×40 mm, (b) domain 2: 60×60 mm, and (c) domain 3: 80×80 mm).

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0.30

For the particle velocity distribution in a 2D experiment, Eq. (16) can be rewritten as:

exp fitting curve

0.25

 2 !  2 !   c−ufd c−upfc 1 0 c þ f fnpfc c exp − θpfc 2θfd 2θpfc  2 !    c−upd 1 0 c þ 1−f fnpd exp − θpd 2θpd 

Particle number ratio [%]

f ðcÞ ¼ ð1−f Þnfd

0.20

0.15

1 θfd



exp −

ð17Þ 0.10

0.05

0.00 0.0

0.1

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Particle velocity [m/s] Fig. 8. Single-peak distribution curves compared with experimental data (Ug = 1.33 m/s).

distribution function. For more details about the derivation process, see Appendix A. The Maxwellian velocity distributions for two-dimensional particle motion are as follows: !

  1 ðc−uÞ2 f M ðc=u; n; θÞ ¼ n exp − c θ 2θ

ð13Þ

Based on the above, in this work, the original dual-peak distribution curve was modified to match the tri-peak particle velocity distribution of particles around single bubble. Using a derivation method similar to Wang et al. [29], the dense phase of the second term in the dual-peak distribution in Eq. (2) is expanded: ! 3 1 ðc−ude Þ2 2 exp − nde 2πθde 2θde  2 !  3 c−upfc 1 0 2 exp −  f npfc 2πθpfc 2θpfc  2 !  3  c−upd 1 0 2 exp − þ 1− f npd 2πθpd 2θpd 

ð14Þ

Eq. (17) describes the motion of particles around a bubble. The first term represents the Maxwellian velocity distribution of dilute-phase particles; the second term represents the Maxwellian velocity distribution of dense-phase particles, which are significantly affected by the fluid; and the third term represents the Maxwellian velocity distribution of dense-phase particles that are less susceptible to the fluid. These three terms correspond to the three peaks in the analysis described above. The parameters in Eq. (17) (such as ufd, θfd) are difficult to obtain in the experiment. For convenience, Eq. (17) may be presented as follows: f ðcÞ ¼ Af M1 ðcÞ þ Bf M2 ðcÞ þ Cf M3 ðcÞ

8 !   > 1 ðc−a2 uÞ2 > > > ð Þ exp − f c ¼ n c M1 > > a1 θ 2a1 θ > > > > > ! >   2 < 1 ðc−b2 uÞ exp − c f M2 ðcÞ ¼ n > b1 θ 2b1 θ > > > > ! >   > > > 1 ðc−c2 uÞ2 > > exp − c f ð c Þ ¼ n > : M2 c1 θ 2c1 θ

1 2πθdi

3

2

ðc−udi Þ exp − 2θdi

2

!

  nfd

1 2πθfd

3 2

 exp −

c−ufd 2θfd

2 !

Then, the final tri-peak distribution curve is as follows: 

1 2πθfd

f ðcÞ ¼ ð1−f Þnfd

0

þ f fnpfc



1 2πθpfc

 0 þ 1− f fnpd



3 2

3 2

2

ðc−ufd Þ exp − 2θfd

exp −

1 2πθpd

3 2

 c−upfc 2θpfc 

exp −

0.25

exp fitting curve

0.20

ð15Þ

!

ð19Þ

Here n, uand θ are particle number, the value of average particle velocity and granular temperature for all particles in the selected domain. The correction factors are empirical which may be related to gas flow rate, bubble diameter, etc.

Particle number ratio [%]



ð18Þ

where A,B and C represent the proportion of each peak, respectively. f M1 (c), fM2 (c) and f M3 (c) are the correction of the overall Maxwellian distribution which represent FD, PFC and PD particles, respectively.

where f' represents the proportion of PFC particles in the total volume. The first term which represents the dilute phase can be regarded as particles by fluid dominating (FD) category:

ndi

7

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2 !

c−upd 2θpd

0.00 0.0

2 ! ð16Þ

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Particle velocity [m/s] Fig. 9. Bi-peak distribution curves compared with experimental data (Ug = 1.33 m/s).

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0.40

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exp fitting curve

Ug=1.22m/s, R=0.9721 Ug=1.33m/s, R=0.9401 Ug=1.44m/s, R=0.9075 Ug=1.66m/s, R=0.8166 Ug=1.88m/s, R=0.6434

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F(c)dc for fitting curve

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line: y=x

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Fig. 10. Tri-peak distribution curves compared with experimental data (Ug = 1.33 m/s).

3.4. Fitting curves with different model In this section, three kinds of Maxwellian models are proposed and compared. The single-peak model distribution is used first to fit the average experimental data: f ðcÞ ¼ Bf M2 ðcÞ

ð20Þ

Fig. 8 shows single-peak distribution curves compared with experimental data. It is difficult to profile the real particle velocity distribution. This is because a Maxwellian distribution is to describe the motion of random gas molecule collisions or fine particles in a fluidized bed. However, for a small regional single bubble, especially for Geldart D particles, the heterogeneity become obvious and the effect of gas on particles becomes complex and diversified. Therefore, Maxwellian single-peak distribution cannot reflect particle movement accurately. For the bi-peak model, particle velocity distribution is a linear combination of two Maxwellian forms: f ðcÞ ¼ Bf M2 ðcÞ þ Cf M3 ðcÞ

ð21Þ

0.30

0.20 0.15 0.10 0.05

0.10

0.15

0.20

0.35

0.40

Ug=1.22m/s, R2=0.9723 Ug=1.33m/s, R2=0.9486 Ug=1.44m/s, R2=0.9481 Ug=1.66m/s, R2=0.8962 Ug=1.88m/s, R2=0.8505

0.35

line: y=x

0.05

0.30

0.40

0.25

0.00 0.00

0.25

Fig. 9shows the bi-peak distribution curves compared with experimental data. The curve using Eq. (12) basically matches the particle velocity distribution curve obtained via measurement. This is because compared with single-peak model, bi-peak model can reflect two kinds of particles which are influenced by two kinds of particle-fluid interaction mechanisms, respectively. In some studies of single bubbles, particles in the bubble phase were neglected [5]. This is because nfd in Eq. (17) is very small compared with npfc and npd in the case of a single bubble. In other words, there are two primary particle-fluid interaction mechanisms occurring: PFC and PD. However, it cannot reflect particles which belongs to the fluid dominating (FD) category. According to the experimental data and analysis in Section 3.3, the particle velocity distribution is a linear combination of three Maxwellian forms. The particle velocity distribution bases on the tri-peak model is used to fit the experimental data by using Eq. (18). To highlight the advantage of Eq. (13), the estimated distributions for the tri-peak model are presented against the experimental data in Fig. 10. For the bubbles in different gas velocity, the tri-peak model distribution reflects the real changes in the experimental data correctly.

F(c)dc for fitting curve

F(c)dc for fitting curve

0.30

0.20

Fig. 12. Correlation coefficient and F(c)dc for the experimental data and bi-peak model.

Ug=1.22m/s, R=0.3686 Ug=1.33m/s, R=0.6161 Ug=1.44m/s, R=0.7571 Ug=1.66m/s, R=0.6817 Ug=1.88m/s, R=0.4244

0.35

0.15

F(c)dc for experimental data

Particle velocity [m/s]

0.40

0.10

0.25

0.30

0.35

0.40

0.25

line: y=x

0.20 0.15 0.10 0.05 0.00 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

F(c)dc for experimental data

F(c)dc for experimental data Fig. 11. Correlation coefficient and F(c)dc for the experimental data and single-peak model.

Fig. 13. Correlation coefficient and F(c)dc for the experimental data and tri-peak model.

Please cite this article as: R. Liu, Z. Zhou, R. Xiao, et al., Particle velocity distribution function around a single bubble in gas-solid fluidized beds, Powder Technol., https://doi.org/10.1016/j.powtec.2019.11.007

R. Liu et al. / Powder Technology xxx (2019) xxx Table 2 Value of correction factors. Factors a1 b1 c1 a2 b2 c2 Value 0.035–0.047 0.40–0.67 0.0013–0.0086 1.57–3.44 0.12–0.42 0.0001

To evaluate the strength of the tri-peak model, a statistics method is introduced. The correlation coefficient (R) between the experimental data and fitting curve of all three model was calculated. The correlation coefficient should be located in the range of 0–1, and R = 1 indicates the perfect fit. By this method, the particle velocity distributions and correlation coefficients for the single-peak, bi-peak and tri-peak models are given in Fig. 11, Fig. 12 and Fig. 13, respectively. Fig. 11 shows that the correlation coefficient is low for the singlepeak model. Gas velocity will affect the accuracy of single-peak model significantly. The effect of gas velocity on accuracy shows a trend that R increases from Ug = 1.22 m/s and reach a highest value at Ug = 1.44 m/s, then it decreases at Ug = 1.88 m/s. The reason can be found in Fig. 5. For the cases of low Ug (1.22 m/s and 1.33 m/s) and high Ug (1.66 m/s and 1.88 m/s), there are at least two peaks which represent PD and PFC particles or PD, PFC and FD particles, respectively. Therefore, although the parameter B in Eq. (11) can be adjusted to fit the experimental data, single-peak model is poor at profiling the characterization of real particle velocity distribution. At Ug = 1.44 m/s, for the same region, the proportion of PFC particles (the second peak in Fig. 7) increases and becomes the dominant factor. Therefore, this case has the highest R in single-peak model. Fig. 12 shows the accuracy of the bi-peak model. The correlation coefficients are more than 0.9 at Ug = 1.22 m/s, 1.33 m/s and 1.44 m/s which means that compared with the single-peak model, the bi-peak model can profile the particle velocity distribution accurately when particles fall into PD and PFC category. However, the accuracy decreases at high Ug. Fig. 13 shows the accuracy of the tri-peak model. The difference between the tri-peak model and bi-peak model is small for the low Ug case. For high Ug case, the trailing vortex effect becomes evident. For the same analysis region, the proportion of FD particles (the third peak in Fig. 9.) increases. Therefore, the tri-peak model has a better match with the particle velocity distribution curve obtained via experiment. To sum up, the single-peak model can only reflect one kind of particle motions. It is difficult to profile particle movement in a fluidized bed because there is at least two kinds of particle-fluid interaction mechanisms acting on particles. The bi-peak model can reflect two kinds of particle motions, so it can fit the case when the proportion of FD particles can be neglected. For some complex cases, when there are three kinds of mechanisms acting on particles, the tri-peak model can be used to profile the particle velocity distribution more accurately. 4. Further discussion of tri-peak model In this study, for tri-peak model, Eq. (18) is used to fit the experimental distribution curves and the correction factors are showed in Table 2. The results shows that for particles around a single bubble, the relationship of granular temperature and average particle velocity for each peak will be θpfc N θfd N θpd and ufd N upfc N upd. For PD particles, the granular temperature and average particle velocity are far lower than other two kinds of particles. This is because the movement of PD particles are affected by particle-particle collision and the impact of drag force is limited which results in particles in this domain have low velocity. For PFC particles, the largest value of granular temperature shows the violent velocity fluctuation. This is because the movement of PFC particles are

9

affected by both fluid and particles, and the complex mechanism such as the wake makes particle velocity diversified. For FD particles, the granular temperature is between the PFC and PD particles. Obviously the average particle velocity is larger than other two kinds of particles because of the effect of drag force. In gas-solid two-phase systems, heterogeneous structures due to the existence of bubbles are difficult to describe accurately and quantitatively. The conventional method describes heterogeneous structure indirectly via porosity, bubble equivalent diameter and other dynamics parameters. In this work, the velocity distribution function is used to describe the heterogeneous structure in terms of single-, bi- and tri-peak models. The results show that the more peaks there are in the distribution function, the more detailed characterization of particle velocity distribution can be obtained. Analysing the velocity distribution function may be a new method to describe heterogeneous structures in gas-solid two-phase systems. However, it should be noticed that the different regions identified (PD, PFC and FD) in this study is not strictly equal to prior understanding of emulsion phase, cloud phase and bubble phase (wake region). The former is based on particle velocity distribution, while the latter is based on porosity and bubble velocity. For example, the size U br þ 2U mf =εmf of cloud in Davidson [54] was calculated by: dc ¼ U br −U mf =εmf Therefore, particle velocity or other conclusions for the three regions in this study may only provide a crude approximation on those in prior theory. 5. Conclusions In this work, the particle velocity distribution for particles around a single bubble was measured, and the data were collected via experimentation and simulation. Three kinds of Maxwellian distribution models are proposed and discussed. The following conclusion can be drawn. (1) The simulation results by DEM compare relatively well with the experiment. Both results show that particle velocity distribution function does not follow the Maxwellian distribution but is instead a linear superposition of multiple Maxwellian velocity distributions. (2) Based on various particle-fluid interaction methods, a tri-peak distribution function of bubble-forming particles is derived. Three kinds of models such as tri-peak model, bi-peak model and single-peak model are proposed to fit the experimental data. The error analysis shows that compared with other models, the tri-peak model can profile particle velocity distribution more accurately. (3) The value of granular temperature and average particle velocity for each peak was calculated. Based on those value and their physics meaning, the characteristics of particles for each peak were analysed. Declaration of Competing Interest We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Acknowledgement The authors greatly acknowledge the funding from the projects supported by the National Science Foundation for Distinguished Young Scientists of China (Grant no. 51525601), and supports provided by SIMPAS and Monash University.

Please cite this article as: R. Liu, Z. Zhou, R. Xiao, et al., Particle velocity distribution function around a single bubble in gas-solid fluidized beds, Powder Technol., https://doi.org/10.1016/j.powtec.2019.11.007

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R. Liu et al. / Powder Technology xxx (2019) xxx

Appendix A. Appendix For a detailed derivation of the three-dimensional Maxwellian distribution, please refer to the literature on gas molecular dynamics theory. In this paper, the derivation of the two-dimensional Maxwellian distribution is provided and results in the formula proposed by Carlos et al. [24]. A.1. Derivation of the two-dimensional Maxwellian-Boltzmann distribution via the Boltzmann H theorem When a system is in a balanced state, the distribution function lnfis the sum of invariants. Therefore, it is represented as a linear combination of basic collision invariants: c2 2 A0, A1, A2 are determined based on the definition of the distribution function, where A1 consists of two components, A1x, A1y: Z n¼ f dc Z 1 c f dc u¼ n Z 2 1 c E¼ f dc n 2 ! 1 A1 A1 Let A3 ¼ exp A0 þ ; then, 2 A2 ln f ¼ A0 þ A1  c−A2 

" #   1 A1 2 A2 f ¼ A3 exp − c− 2 A2 When solving the integral, because c is a two-dimensional velocity vector, the integral should be a double integral in the x and y directions. After substitution of f, Z n ¼ fdc " #   Z 1 A1 2 ¼ A3 exp − c− A2 dc A2 2 ( "    #) A1y 2 1 A1x 2 þ cy − ¼ A3 ∬ exp − A2 cx − dcx dcy A2 A2 2 ( "    #)     A1y 2 A1y 1 A1x 2 A1x d cy − ¼ A3 ∬ exp − A2 cx − þ cy − d cx − A2 A2 A2 A2 2 " "  2 # rffiffiffiffiffiffi Z   # rffiffiffiffiffiffi  Z A1y 2 A1y 2 1 A1x A2 A1x 1 A2 ¼ A3 exp − A2 cy − d d exp − A2 cx − cx − cy − A2 2 A2 A2 2 A2 A2 2 2 2π " # AZ2   1 1 A1 2 cA3 exp − c− u¼ A2 dc n 2 A2 " " # ! #      Z  Z 1 A1 1 A1 2 A1 1A1 1 A1 2 A1 c− A3 exp − c− A3 exp − c− A2 d c− A2 dc ¼ þ ¼ A2 A2 A2 A2 A2 n 2 n A2 2 ¼ A3

E¼

1 n

Z

" #   ðc−uÞ2 1 A1 2 A3 exp − c− A2 dc 2 2 A2

 ðc−uÞ2 1 A3 exp − ðc−uÞ2 A2 dc 2 2

 Z A3 A2 2 2 c0 exp − c0 dc0 ¼ 2n 2


  A3 A
¼ ∬ c2x þ c2y exp − c2x þ c2y dcx dcy 2 2n

Z          Z Z Z A3 A2 A2 A2 A2 c2x exp − c2x dcx ¼ exp − c2y dcy þ c2y exp − c2y dcy exp − c2x dcx 2n 2 2 2 2 sffiffiffiffiffiffipffiffiffisffiffiffiffiffiffi sffiffiffiffiffiffipffiffiffi sffiffiffiffiffiffi ! p ffiffiffi p ffiffiffi A3 2 2 π 2 2 2 π 2 πþ π ¼ 2 A2 A2 A2 2n A2 A2 2 A2 1 ¼ A2

¼

1 n

Z

Please cite this article as: R. Liu, Z. Zhou, R. Xiao, et al., Particle velocity distribution function around a single bubble in gas-solid fluidized beds, Powder Technol., https://doi.org/10.1016/j.powtec.2019.11.007

R. Liu et al. / Powder Technology xxx (2019) xxx

11

2 1 where E is energy with two degrees of freedom:E ¼ RT ¼ . 2 A 2 The velocity distribution function is as follows: " # n ðc−uÞ2 exp − dc f ðcÞdc ¼ 2πRT 2RT The speed distribution differential is as follows:   f cx ; cy dcx dcy ¼

"  2 # ðcx −ux Þ2 þ cy −uy n exp − dcx dcy 2πRT 2RT

cosφ − c sinφ ¼c The integral polar coordinate transform should be multiplied by the Jacobi determinant: Jðc; φÞ ¼ sinφc cosφ dcx dcy ¼ cdcdφ ðφ∈½0; 2π Þ Thus:   ∬ f cx ; cy dcx dcy ¼ ∬ f ðc; φÞcdcdφ ¼

Z 2πf ðcÞcdc

Then, the speed distribution differential is as follows: " # n ðc−uÞ2 exp − f ðcÞ ¼ cdc RT 2RT In a particle flow system, the granular temperature is θ = RT. Then, the speed distribution differential is as follows:

f ðcÞ ¼

" # n ðc−uÞ2 exp − cdc θ 2θ

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Please cite this article as: R. Liu, Z. Zhou, R. Xiao, et al., Particle velocity distribution function around a single bubble in gas-solid fluidized beds, Powder Technol., https://doi.org/10.1016/j.powtec.2019.11.007