Hydrodynamic characteristics of gas-irregular particle two-phase flow in a bubbling fluidized bed: An experimental and numerical study

    Hydrodynamic characteristics of gas-irregular particle two-phase flow in a bubbling fluidized bed: An experimental and numerical study Yurong He, Shengnan Yan, Tianyu Wang, Baocheng Jiang, Yimin Huang PII: DOI: Reference:

S0032-5910(15)30101-7 doi: 10.1016/j.powtec.2015.10.012 PTEC 11278

To appear in:

Powder Technology

Received date: Revised date: Accepted date:

31 March 2015 20 September 2015 5 October 2015

Please cite this article as: Yurong He, Shengnan Yan, Tianyu Wang, Baocheng Jiang, Yimin Huang, Hydrodynamic characteristics of gas-irregular particle two-phase flow in a bubbling fluidized bed: An experimental and numerical study, Powder Technology (2015), doi: 10.1016/j.powtec.2015.10.012

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ACCEPTED MANUSCRIPT Hydrodynamic characteristics of gas-irregular particle two-phase flow in a bubbling fluidized bed: an experimental and numerical study

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Yurong He*, Shengnan Yan, Tianyu Wang, Baocheng Jiang, Yimin Huang

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Tel.: +86 451 86413233

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School of Energy Science & Engineering, Harbin Institute of Technology, Harbin 150001, China

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*Corresponding author, Email address: [email protected]

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Highlights

Particle sphericities were taken into account in the Gidaspow drag model.

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A two-fluid model with the modified Gidaspow drag model was established.

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Flow behavior of irregular particles in a bubbling bed was researched.

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Particle Image Velocimetry was employed in the experiment.

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Results with a modified Gidaspow drag model were compared with the original one.

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Abstract:

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In practical applications, the particles in a bubbling fluidized bed are usually irregularly shaped. The gas-solid flow behavior of black millet particles with ellipsoidal shape was numerically and experimentally investigated. In the present work, a two-fluid model with a modified Gidaspow drag model has been built in a gas-solid bubbling bed to investigate the flow behavior of particles with irregular shape. Particle sphericities are taken into account in the Gidaspow drag model, which is a significant factor affecting the flow behavior of irregularly shaped particles. Particle concentration and velocity distribution were studied at various inlet superficial gas velocities and initial bed heights. Numerical results were compared with experimental results obtained by PIV (Particle Image Velocimetry). It is shown that the simulation results using the modified Gidaspow drag model

ACCEPTED MANUSCRIPT are in better agreement with the experimental results than using the original one, which indicates that the modified Gidaspow model is better suited for irregularly shaped particles.

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Keywords: Modified Gidaspow drag model, Irregularly shaped particles, Particle sphericity, Two-fluid model,

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Bubbling bed

1 Introduction

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Fluidized beds have a wide application in the fields of petroleum, chemical engineering, energy,

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environmental protection, pharmacy, food processing and so on. Whether in practical applications or theoretical research, the bubbling bed has been one of the most important research points for those involved in the field. In

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practical applications, the particles in a bubbling fluidized bed usually are irregularly shaped. Compared with

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regularly shaped particles, irregularly shaped particles can affect the gas-solid flow behavior of a bubbling

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fluidized bed. Therefore, research on the hydrodynamics of irregularly shaped particles in a bubbling fluidized bed is meaningful.

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Computational fluid dynamics (CFD), which is a powerful and efficient tool in investigating the gas-solid flow characteristics in a fluidized bed, has been adopted, studied and improved by a number of researchers[1-7]. The Two-fluid model (TFM) and the discrete element method (DEM) are the two major approaches. Simulations applying DEM can provide dynamic information that is difficult to obtain by conventional experimental techniques, such as the trajectories of and transient forces acting on individual particles[8,9]. DEM, therefore, is especially prevalent among investigators[10,11]. With the development of theory and application, more and more attention is paid to the hydrodynamics of irregularly shaped particles in particulate systems with the adoption of DEM. By comparison with a fluidized bed of spherical particles, Hilton et al.[12] presented the effect of grain shape on the dynamics of a fluidized bed, including increased pressure gradients within the bed and lower

ACCEPTED MANUSCRIPT fluidization velocities. Applying three dimensional CFD and DEM, Ren et al.[13] investigated the flow of corn-shaped particles in a cylindrical spouted bed with a conical base. Gas-solid flow patterns, pressure drop,

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particle velocity and particle concentration at various spouting gas velocity were studied. Zhou et al.[14]

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extended the CFD–DEM approach to consider the fluidization of ellipsoidal particles. Particles with aspect

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ratios ranging from 0.25 to 3.5, which represented cylinder-type and disk-type shape particles, were used in the simulation. The effect of the aspect ratio on the flow pattern, the relationship between pressure drop and gas

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superficial velocity, and microscopic parameters such as coordination number, particle orientation and force

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structure were investigated. Favier et al.[ 15 ] put forward a new method of representing non-spherical, smooth-surfaced, axi-symmetrical particles in DEM using model particles comprising overlapping spheres of

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arbitrary size whose centers are fixed in position relative to each other along the major axis of symmetry of the

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particle. Chung et al.[16] investigated the convection behavior of non-spherical particles in a vibrating bed using

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DEM. The proposed DEM adopted the Hertz-Mindlin no slip contact force model to model particle collisions and used the multi-sphere method to exactly represent the shape of paired particles. Moreover, the simulation

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results were validated with experimental results. Oschmann et al.[17] carried out a numerical investigation of mixing in a model type fluidized bed, which was based on three-dimensional DEM coupled with CFD. Various elongated particles shapes were researched including cylinders, plates and cuboids. What is more, comparisons to spherical particles were conducted. DEM is capable of providing micro-scale information of complex interactions and reproducing the macro-scale dynamic flow behavior and is extensively applied in investigations on gas flows. High computational demands are often needed considering the complexity of particle collision models[18-28]. In contrast, TFM is not limited by the particle number and requires less computational demands, and so becomes a more natural choice for hydrodynamic modeling of engineering scale systems [29-31]. To our knowledge, very few investigations on non-spherical particles of gas-solid fluidized beds using TFM have been presented.

ACCEPTED MANUSCRIPT Marques et al.[32] investigated experimentally the air-carton mixtures flow dynamics in conical spouted beds (CSBs) and presented comparisons with characteristic fluid dynamics obtained by using TFM. The flow

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behavior of air-carton disks is experimentally investigated by analyzing data of bed pressure drop, air velocity

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and fountain height. Sau et al.[33] carried out experimental and numerical studies for the hydrodynamics in a

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gas-solid tapered fluidized bed. Glass beads (spherical) of 2.0 mm and dolomites (non-spherical particles) of 2.215 mm in diameter were used in the work. Experimental results were compared with simulation results

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applying TFM. Bed pressure drop, bed expansion ratio and solid volume fraction were mainly discussed. Hence,

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it is of great significance to investigate the influence of particle shape on the hydrodynamics of gas-solid fluidized beds with the adoption of TFM.

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In this work, we studied the flow behavior characteristics of black millet particles in a bubbling bed, such

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as velocity, granular temperature, concentration distribution and so on. The bubble properties including bubble

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diameter and velocity were also investigated. In the end, we studied the frequency and power spectral magnitude of bed voidage and made a wavelet multi-scale analysis of particle concentration pulsation. After

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this introduction, the paper is structured as follows: second part, experimental method; third part, numerical method; fourth part, results and discussion; fifth part, conclusions; sixth part, acknowledgements.

2. Experimental Method 2.1 Experimental Setup In a pseudo two-dimensional bubbling fluidized bed constructed of plexiglass (W × D × H = 0.16 m × 0.03 m × 0.6 m), irregularly shaped black millet particles with a density of 1343.5 kg/m3, an average diameter of 1 mm, and particle sphericity of 0.96 were fluidized with air at a superficial gas velocity of 0.6 m/s, 0.7 m/s, 0.8 m/s, 0.9 m/s and 1.0 m/s. Considering the electrostatic interaction between particles and walls of the fluidized bed, the air entering into the bed needed to be humidified. The air was humidified up to a relative humidity of

ACCEPTED MANUSCRIPT 50-60%. Moreover, we conducted a series of experiments to measure the particle-particle and particle-wall restitution coefficients. The experiments are performed with and without humidified air. The surface of the

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particles were covered with water by humidifying the air to a relative humidity 50-60%. It was found that the

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restitution coefficients under the two conditions are nearly the same. So we assume that the effect of water is

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small in the simulation. Fig. 1a and 1b give a diagram and a schematic illustration of the experimental setup used in this work. To ensure the quality of the obtained images, a light with a direct-current power supply was

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details of the experimental setup are listed in Table 1.

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utilized to illuminate the bed. A high-speed digital camera was used to record the images of the bed. Further

2.2. Particle Velocity Measurements.

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A nonintrusive technique, particle image velocimetry (PIV), is applied for the measurement of an

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instantaneous velocity field in one plane of a flow in a bubbling fluidized bed. No additional tracer particles are

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needed in gas-particle flows since the discrete particles can readily be distinguished to visualize the particle movement. The flow on the front of the bed is illuminated using a light source. A high-speed digital camera was

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used to record images of the particles for the illuminated front plane. Two subsequent images of the flow, separated by a short time delay, were divided into small interrogation areas. Cross-correlation analysis was adopted to determine the volume-averaged displacement of the particle images between the interrogation areas in the first and second image. A cross-correlation analysis yields a dominant correlation peak corresponding to the local average particle displacement. The average particle velocity was then calculated from the displacement vector and the time delay between the two images. Further detailed information about the application of PIV can be found in reference[34,35]. (a) Diagram

(b) Schematic illustration

Figure 1 Photograph and schematic illustration of the bubbling fluidized bed Table 1 Details of the PIV Experimental Setup

ACCEPTED MANUSCRIPT 3. Numerical Method 3.1 Governing Equations of the Gas and Solid Phase

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3.1.1 Continuity Equation

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For the gas phase:

(1)

  s  s      s  s us   0 t

(2)

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  g g      g g ug   0 t

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For the solid phase:

density, us is the solid velocity.

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3.1.2 Conservation of Momentum Equation

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where, εg is voidage, ρg is the gas density, ug is the gas velocity, εs is the solid concentration, ρs is the solid

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For the gas phase:

(3)

  s s us      s s us us    sPg  Ps    τ s   s s g  gs  ug  us  t

(4)

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For the solid phase:

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  g g ug      g g ug ug    gPg    τg   g g g  gs  us  ug  t

where, Pg is the gas pressure, τg is the gas stress tensor, g is the gravity acceleration, βgs is the drag coefficient in a control volume, Ps is the solid pressure, τs is the solid stress tensor. 3.1.3 Stress Equation For the gas phase:



τ g   g g ug   ug 

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  23     u I g

g

g

(5)

For the solid phase: 2  T  τs   s s us   us     s  s  s    us I   3  

(6)

ACCEPTED MANUSCRIPT where, μg is the gas viscosity, I is the identity matrix, μs is the solid viscosity, λs is the solid bulk density. 3.1.4 Solid Phase Pressure

Ps,k   s s 1  2 g0 s (1  e)

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(7)

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where, Ps,k is the kinetic part of the solid pressure, θ is the granular temperature, g0 is the radial distribution

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function. 3.1.5 Radial Distribution Function

3.1.6 Solid Phase Shear Viscosity

1

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4  10 s d p    s2 s d p g0 1  e   5  96 1  e   s g0

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s,k

   

(8)

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where, εs,max is the maximum solid volume fraction.

1/3

  

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   g 0  1   s    s,max 

 4  1  5 g0 s 1  e  

2

(9)

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where, μs,k is the kinetic part of the shear viscosity of particles μs, d is the particle diameter, e is the restitution coefficient.

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3.2 Frictional Stress Model

3.2.1 Particulate Stress Tensor

τs  τ k  τf

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where, τk is the kinetic stress tensor, τf is the frictional stress tensor. 3.2.2 Solids Pressure

 Ps,k  Ps,f  s   s,min Ps    s   s,min  Ps,k

(11)

where, Ps,f is the frictional part of the solid pressure, εs,min is the solid concentration at the transition point when frictional stresses become important.

ACCEPTED MANUSCRIPT 3.2.3 Solids Viscosity

 s,k  s,f  s   s,min  s   s,min  s,k

s  

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(12)

3.2.4 Frictional Pressure of Particles s

  s,min 

n

s,max   s 

(13)

p

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Ps,f

 F 

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where, μs,f is the frictional part of the shear viscosity of particles μs.

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where, F, n and p are the empirical material constants. The values of the empirical parameters εs,min, F, n and p are taken to be 0.5, 0.05, 2.0 and 5.0 according to the expression originally developed by Johnson et al.[36]. Note

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that the expression was developed in a fully developed flow down an inclined chute. However, since then, the

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expression has been used in a range of particle flow applications, more or less without changing the empirical

great care.

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3.2.5 Frictional Viscosity

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constants and without proving the accuracy of the model. Hence, the outcome of the model must be treated with

s,f 

Ps,f sin  

(14)

2 I 2D

where, Ψ is the angle of internal friction, I2D is the second invariant of the strain rate tensor. The value of Ψ is taken to be 30º for black millet particles[37]. 3.3 Gas-Solid Drag Model 3.3.1 Original Gidaspow Model[38,39]

 g ( ug  us )(1   g )  g (1   g )2  gs  150  1.75  g d p2 dp  g 1   g   g 1.65 3  gs  CD ug  us 4 dp

εg > 0.8

εg ≤ 0.8

(15)

ACCEPTED MANUSCRIPT  24 1  0.15 Re0.687    CD   Re  0.44

(16)

Re > 1000

 g d p  ug  us  g

(17)

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s

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Re 

Re  1000

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where，βgs is the inter-phase momentum exchange coefficient, dp is the particle diameter, CD is the standard drag coefficient for a particle, Re is the Reynolds number. 3.3.2 Modified Gidaspow Model[38,40]

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 g ( ug  us )(1   g )  g (1   g )2  gs  150  1.75  g (d p )2 d p

εg ≤ 0.8

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 g 1   g   g 1.65 3  gs  CD ug  us 4 d p

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b3 Re  24 b2  1  b1 Re   b4  Re CD   Re 0.44 

Re  1000

(18)

(19)

Re > 1000

b1  exp  2.3288  6.4581  2.4486 2 

(20)

b2  0.0964  0.5565

(21)

b3  exp  4.905  13.8944  18.4222 2  10.2599 3 

(22)

b4  exp 1.4681  12.2584  20.7322 2  15.8855 3 

(23)

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εg > 0.8



s S

(24)

where,  is the shape factor, s is the surface area of a sphere having the same volume as the particle, S is the actual surface area of the particle. Since the surface area of a particle is hard to measure, we adopt the following method. A practical formula for computing the shape of black millet particles must be such that the obtained value approaches as closely as possible the degree of true sphericity. Hence, to improve the accuracy and reliability, we adopted the equation suggested by Wadell[41], which is shown below.



dc Dc

(25)

ACCEPTED MANUSCRIPT In this formula, dc is the diameter of a circle equal in area to the area obtained in the standard size when the black millet particle rests on one of its larger faces, more or less parallel to the plane of the longest and

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intermediate diameters, and Dc is the diameter of the smallest circle circumscribing the black millet particle

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reproduction of the standard size. It is assumed that the value obtained from this formula generally approaches

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the value of the degree of true sphericity. To ensure the accuracy of our work, we performed an experiment to determine the particle sphericity. We randomly selected 100 particles and measured the sphericities of particles

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one by one. Then we made an averaged calculation of the obtained values. Finally we got the statistically

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calculated sphericity. According to the calculation, the approximate sphericity value of a black millet particle is 0.96.

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3.4 Initial and Boundary Conditions

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A sketch of the bubbling fluidized bed constructed in our work is presented in Fig. 1. Black millet particles

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are introduced into the bed with a constant solid flux through the bottom of the bed. Air enters the fluidized bed uniformly to fluidize the particles. The redistribution of all particles is completed in the course of the simulation

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under the forces acting on them. For the gas phase, a no-slip condition is employed at the walls and a pressure boundary condition is used at the top of the bed. Most of the physical parameters used in the simulation correspond with the experimental settings. To ensure the accuracy of our work, we studied the particle size distribution. We randomly selected 100 particles and used a high-pixel camera to record the images of particles. Then we employed in-house developed code in Matlab to analyze the distribution of particle sizes. The mathematical expectation is 1 mm. To simplify the calculation work, we take a uniform particle diameter of 1 mm in the simulation. In our future work, more efforts will be put on the particle size distribution in the simulation. The simulation was run for 30 s and time averaged calculations were performed during the last 20 s. With respect to the restitution coefficient, we adopt the method proposed by LoCurto et al.[42], which regards the e value as the ratio of the square root of the initial height of drop and the height of rebound. Drop and

ACCEPTED MANUSCRIPT rebound heights were measured only from the black millet particles that fell with minimal rotation and whose rebound trajectories were almost vertical. Then the obtained e values were averaged and the mean value was

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Table 2 Summary of parameters employed in the simulation

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around 0.7. The simulation parameters are summarized in Table 2.

t=26.16 s

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4. Results and Discussion t=26.24 s

t=26.32 s

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t=26.48 s

t=26.40 s

superficial gas velocity of 0.6 m/s.

t=7.68 s

t=7.76 s

t=7.84 s

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t=7.60 s

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Figure 2 Snapshots of the instantaneous particle positions and corresponding particle velocity fields at a

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t=7.92 s

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Figure 3 Snapshots of the instantaneous particle positions and corresponding particle velocity fields at a superficial gas velocity of 0.8 m/s.

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Particle image velocimetry measurements were performed in the fluidized bed for a period of 40 s. Fig. 2 and Fig. 3 show the snapshots of the instantaneous particle positions and corresponding particle velocity fields at the superficial gas velocities of 0.6 m/s and 0.8 m/s, respectively. The process of bubble growth is distinctly presented in the figure, which is as follows: firstly a small bubble appears at the bottom of the bed. Then the bubble rises under the gas force, gradually becomes bigger and eventually breaks up at the top. The top boundary of the bubble is stretched downward due to the forces acting on the bubble in the bed. Then a tip appears and rapidly develops into a finger-type or knife-like structure. It starts at the critical point of the top of the bubble and ends around the wake of the lower part. When the downward development speed of the finger-type structure exceeds the horizontal one and the bubble will split perpendicularly. When the bubble reaches the top boundary of the bed, the acting force is different from that in other places. Thus, the bubble will

ACCEPTED MANUSCRIPT break up. To our delight, we observed this process and presented in these two figures. In other words, it manifests that our experiment is rigorous and relatively precise, which provides a reference for the following

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simulation. As reported before, bubbles in a fluidized bed are not stable. Instead, they constantly split and

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coalesce with others in their rising process. Under certain conditions, the process, namely the splitting and

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growth of bubbles, reaches a dynamic equilibrium. At this stage, the mix of gas and particles is enough. More details of the flow field in the fluidized bed is shown in Fig. 4.

(b) Particles’ velocity.

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(a) Snapshot of particles’ position.

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Figure 4 Instantaneous volume fraction and particle velocity in a bubbling fluidized bed at a superficial gas velocity of 0.8 m/s

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Fig. 4 shows recorded pictures and corresponding instant velocity field at the superficial gas velocity of

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0.8 m/s. Black millet particles move upward in the center of the bed and downward near the walls. As has been

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argued, the bubble first appeared at the bottom of the bed, then rose, became larger, which was affected by the gas, and eventually broke up at the top of the bed. It can be found that the mixing of gas and solid phases is

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more violent with the increase of superficial gas velocity. (a) Experimental result

(b) Simulation result

Figure 5 Comparison of instantaneous void fraction (usp = 0.8 m/s) (a) Experimental result

(b) Simulation result

Figure 6 Comparison of instantaneous void fraction (usp=0.9 m/s) Fig. 5 and Fig. 6 show the comparison of the instantaneous void fractions at the superficial gas velocities of 0.8 m/s and 0.9 m/s. As is clearly depicted in the figure, the bubble shape and position are nearly the same. Simulation results show good agreement with the experimental ones. There are several small and inconspicuous bubbles in the bed. As discussed before, they are in the splitting and coalescing process of the bubbles. When the bubble rises, it will supplant the others around it. Especially, smaller ones will be pushed to the side. Bigger

ACCEPTED MANUSCRIPT bubbles move faster than the smaller ones, thus the smaller ones gradually move into the trailing vortex of the bigger bubbles. Then the smaller bubbles are accelerated, quickly catch up with bigger ones and are stretched in

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the vertical direction. This process lasts until the smaller bubbles enter into the bottom of the bigger ones.

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Bubbles of different sizes merge and coalesce via this process. Generally, bubbles in fluidized beds always

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appear with the absorption of smaller bubbles by the trailing vortex of bigger ones. Even in the case of two adjacent bubbles, whose sizes are nearly the same, the coalescence is completed via a similar process.

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Fig. 7 and Fig. 8 show the comparison of the solid-phase time-averaged velocity at the superficial gas

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velocities of 0.8 m/s and 0.9 m/s. We can see that the trend of the particle motion in the simulation results is the same as that in the experimental results. The particle moves upward in the center and downward at the walls of

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the bed. Moreover, the violence of particle motion increases with the increase of the superficial gas velocity.

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The solid-phase time-averaged velocity is a little higher for the simulation results, especially at the side walls.

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We assume that this may be due to a little difference between the experimental and the simulation specularity coefficient of the side walls. According to Loha et al.[43], model predictions are sensitive to the specularity

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coefficient. In other words, the specularity can influence the hydrodynamic behavior of a bubbling fluidized bed. Different specularity coefficients result in different particle velocities, granular temperature and particle volume fraction not only close to the wall but also in the central region. Selecting an absolutely exact value of specularity coefficients is hard. In addition, more experimental validations are needed, which will be a time-consuming work. To sum up, the deviation between the experiment and simulation is acceptable. (a) Experimental result

(b) Simulation result

Figure 7 Comparison of the solid-phase time-averaged velocity (usp=0.8 m/s) (a) Experimental result

(b) Simulation result

Figure 8 Comparison of the solid-phase time-averaged velocity (usp=0.9 m/s) (a)

(b)

ACCEPTED MANUSCRIPT (c)

(d) (e)

(f)

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Figure 9 Comparison of the solid-phase time-averaged horizontal and vertical velocity

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We conducted a comparison of the simulation with the original Gidaspow model, simulation with a

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modified Gidaspow model and the experiments. The solid phase velocity in the horizontal and vertical directions at the superficial gas velocities of 0.7 m/s, 0.8 m/s and 0.9 m/s, as shown in Fig. 9. With respect to

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the velocity in the horizontal direction, it has two peak values near the center of the horizontal direction. The

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velocity in the vertical direction has a sole peak value, approximately in the center of the horizontal direction. In other words, the vertical velocity is larger in the center of the bed and decreases towards the wall. The

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horizontal velocity is also larger near the center and smaller at the walls of the bed. Overall, a better agreement

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between simulations and experiments was obtained in the comparison of the velocity in the vertical direction.

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By calculation, it is found that the deviation between experiment and simulation with the modified Gidaspow drag model is about 30%. Also, the results of horizontal velocity by using the modified model are closer to the

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real values. Though it may not be that perfect, it will be our motivation of making an improvement for a more accurate drag model. It is assumed that the difference of the velocity in the horizontal direction between simulation and experiment is acceptable. Hence, we acquired a relatively satisfactory result. It is noteworthy that the simulation results with the modified Gidaspow model are closer to the experimental ones. This validates the accuracy of the modified model in this work, which provides a reference for our future related studies. Granular temperature, also called fluctuating kinetic energy, is a quantity that is proportional to the square of the particle velocity fluctuations about the mean. The particle stresses () per unit bulk density over a given frame are calculated from the definition for kinetic stresses given by Gidaspow[44]. Ci C j

1

n

 r , t    cik  r , t   vi  r , t  c jk  r , t   v j  r , t  n k 1

(26)

ACCEPTED MANUSCRIPT where, Ci is the particular particle velocity in the ith direction; ci is the instantaneous particle velocity in the ith direction; vi is the hydrodynamic velocity for particles in the ith direction; r is the position vector; t is time. The

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granular temperature (θ) over a single frame is defined as the mean of the particle normal stresses per unit bulk

1 CiCi ( r, t ) 3

(27)

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 (r, t ) 

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density.

Granular temperature of particles for various superficial gas velocities at different bed heights are shown in

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Fig. 10 and Fig. 11. As presented in the figure, particle granular temperature at the top of the bed is relatively

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higher than that in the middle and at the bottom of the bed. This denotes that particle motion at the top of the bed is relatively more violent than in other places. Additionally, it is obvious that the bed at a higher superficial

(a)

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behavior of gas and black millet particles.

D

gas velocity has a higher granular temperature value. This implies that there exists a more violent mixing

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(b)

(c)

Figure 10 Granular temperature at different bed heights usp=0.7 m/s

usp=0.8 m/s

usp=0.9 m/s

Figure 11 Time-averaged particle granular temperature Bubble size, which is one of the most important parameters for the design and simulation of a fluidized-bed reactor, has been studied by many researchers. As reported in prior literatures, various equations for estimating bubble diameters in fluidized beds have been proposed and tested. In this paper, the simulated bubble diameter was compared with that predicted by different investigators who presented different correlations of the bubble diameter. The average area-equivalent bubble diameter was calculated by the following equation:

ACCEPTED MANUSCRIPT db 

4 Ab

(28)



where, db is the average area-equivalent bubble diameter, Ab is the bubble area. As shown in Fig. 12, bubble

(b)

(c)

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(a)

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diameter increases with the increase of bed height.

Figure 12 Bubble diameter distribution along the bed height

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Mori et al.[45] developed a relation of the bubble size and growth in fluidized beds of various diameters. A maximum bubble diameter determined from the bubble coalescence was incorporated in the relation to relate

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the effect of the bed diameter on the bubble size. Choi et al.[46] proposed a generalized bubble-growth model that takes into account the volumetric bubble flow rate and bubble coalescence-splitting frequencies. Moreover,

TE

D

the model can be successfully applied to beds with Geldart’s group A, B and D particles. It is clearly shown that the simulation results of bubble diameter are closer to those predicted by Choi’s formulation. Generally

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speaking, Mori’s relation is especially suitable for particles whose minimum fluidized velocity ranges from 0.5 to 20 cm/s and with a diameter ranging from 0.006 to 0.045 cm. By contrast, Choi’s relation is better with beds

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of Geldart’s group A, B and D particles. On the other hand, the black millet employed in this investigation, with a diameter of 1 mm, is in the range of Geldart’s group D particles. Furthermore, the minimum fluidized velocity of the black millet is 0.375 m/s, which agrees with the relation proposed by Wen & Yu[47]. The relations proposed by Mori et al. and Choi et al. are shown below. Mori: dbm  db  exp(0.3h / Dt ) dbm  db 0

(29)

db 0  0.00376(usp  umf )2

(30)

dbm  0.652  At (usp  umf ) 

2/5

(31)

where Dt is the bed diameter, dbm is maximum bubble diameter, db0 is the initial bubble diameter formed at the

ACCEPTED MANUSCRIPT surface of the perforated plate, At is the cross-sectional area of the bed, and umf is the minimum fluidization velocity.

g 1/3

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1.787(usp  umf )2/3 h2/3

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db 

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Choi:

(a)

(b)

(32)

(c)

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Figure 13 Bubble velocity distribution along the bed height The bubble rising velocity along the bed height is shown in Fig. 13. As expected, the bubble rising velocity

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increases with bed height due to the increase of bubbles’ sizes and their coalescence. Besides, it can be found that at the same bed height, the bubble rising velocity is higher with higher superficial gas velocity. This can be

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D

attributed to the more violent mixing behavior of the gas and solid phases. It should be noted that we compared the simulation results with the formulation proposed by Davidson and Harrison [48], and a good agreement was

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obtained. The expected results demonstrate that the modified drag model is applicable to black millet particles

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with an irregular shape. The following is the relation of Davidson and Harrison.

vb  0.711 gdbm  usp  umf

(33)

where vb is the velocity of a single train of bubbles rising along the center line. Figure 14 Simulated instantaneous concentration of particles Fig. 14 shows the instantaneous concentration of particles at a superficial gas velocity of 0.9 m/s. As shown here, the concentration of particles fluctuates fiercely with time. The fluctuation of the local volume fraction of the gas phase is mainly caused by the motion of bubbles. The small bubbles first form near the bottom of the bed, then ascend and become larger as they pass through the bed. With the mutual collisions of bubbles, the volume of the small ones become larger via coalescence. Ultimately, the bubbles get to the top of the bed and break. The breakup of bubbles causes the fluctuation of the bed surface, which strengthens the mixing of particles and makes the pressure drop of the bed fluctuate dramatically. The bubble movements result

ACCEPTED MANUSCRIPT in the motion of particles that forms the repeated cycles of particle movements in the bed. Consequently, the fluctuation of the concentration of particles mirrors the spatio-temporal pattern of the gas-solid flow in the bed.

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Figure 15 Bubble frequency at different superficial gas velocities

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Bubble frequency, which reflects the number of bubbles at a certain height of the bed per unit time, is one

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of the most crucial parameters for investigating the bubble behavior in a bubbling fluidized bed. Fig. 15 shows the bubble frequency at different bed heights and superficial gas velocities. In the case of 0.7 m/s, it is obvious

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that there is a drop of bubble frequency at the bed height of 0.2 m, which is due to the fact that a bubble is less

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likely to appear at the top of the bed. Meanwhile, we find that the bubble frequency mostly increases with the increase of superficial gas velocity in the cases of 0.8 m/s and 0.9 m/s. Clearly, the expansion of the bed in the

D

bubbling fluidization becomes larger as the superficial gas velocities increase, which can be seen from Fig. 17.

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Therefore, for the case of 0.7 m/s, the bed expansion at the bed height of 0.2 m may be the largest, while it is

appears.

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not the largest in the case of higher superficial gas velocities. That is why the previously discussed issue

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Figure 16 Frequency and power spectral magnitude of bed voidage Figure 17 Bed expansion at different superficial gas velocities

Fig. 16 shows the power spectrum density of the bed void at the bed height of 0.105 m. The dominant frequency of the bed voidage at the superficial gas velocity of 0.9 m/s is about 2.1 Hz. The dominant frequency ( f ) of porosity oscillations in the bubbling bed can be estimated by an analytical solution, as follows (Gidaspow et al., 2001[49]; Jung et al., 2005[50]):

1 f  2

1/2

 g     H0 

  3 s /  g  2   s     s0  

1/2

(34)

where εs0 and H0 are some initial solid volume fraction and initial bed height. The time averaged solid volume fraction at the center of the bed is approximately 0.38. The initial bed height is 0.15 m. The calculated main frequency for porosity oscillations obtained from the above equation is 1.9 Hz. This shows a reasonably good

ACCEPTED MANUSCRIPT agreement with the main frequency in Figure 15. It reveals that the equation mentioned before is suitable for irregularly shaped particles, which is of great significance. (b)

T

(a)

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Figure 18 The wavelet multi-scale analysis of particle concentration pulsation

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Fig. 18 presents the original signal, different scale signals, detail signals, and wavelet coefficients. Based on the wavelet transformation, the original signal can be decomposed into signals and detail signals of different

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scales. The reconstruction of the original signal can be done with the last scale signal. First, the original signal

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is decomposed into scale 1 signal (a1) and scale 1 detail signal (d1). Scale 1 detail signal captures information at high frequencies. Applying wavelet filtering, we can get a series of detail signals which are of different scales.

D

The Scale 8 signal is the last signal to be reserved in the process of the wavelet transformation. Hence, the

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reconstruction of the original signal can be done from the scale 8 signal and other detail signals. As shown in

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Fig. 18 (b), alternating peak values appear between 1.5625Hz and 3.125Hz, which is in accord with the result

5 Conclusions

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that the dominant frequency of bed voidage at the superficial gas velocity of 0.9 m/s is about 2.1 Hz.

The hydrodynamics of a gas-solid two-phase flow in a bubbling fluidized bed has been experimentally studied using a nonintrusive optical measurement technique, particle image velocimetry (PIV). The gas-solid two-phase flow in the bubbling fluidized bed has also been simulated with a TFM-CFD model. Taking particle sphericity into account, the Gidaspow drag model was tested for the sake of investigating flow behaviors of black millet particles with irregular shape in a bubbling fluidized bed. The velocity distribution and particle granular temperature distribution under different operating conditions were studied. Good agreements were obtained when comparing the TFM simulation and experiment with black millet particles. Moreover, the bubble frequency, the frequency and power spectral magnitude of the bed voidage were researched in detail in this

ACCEPTED MANUSCRIPT work and compared with theoretical models. Prior expectations were reached, which augments our confidence and experience in the related work of gas-solid two-phase flow.

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In general, the original Gidaspow model that does not take into consideration particle sphericity fits

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spherical particles well, but may not be suitable when researching the flow behaviors of irregularly shaped

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particles. With integration of sphericity into the drag model, acceptable results were achieved. The Gidaspow drag model with the consideration of particle sphericity is better with irregularly shaped particles. Hence, it is

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essential that the influence of particle shape in flow behaviors of gas-solid fluidized bed is taken into account.

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Experimental results in this work indicate that PIV can be an efficient tool for quantitative flow visualization in

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D

a bubbling fluidized bed, even at the relatively high particle volume fractions encountered in these systems.

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Nomenclature bubble area (m2)

At

cross-sectional area of the bed (m2)

CD

drag coefficient

D

depth of the fluidized bed (m)

Dc

diameter of the smallest circle (m)

Dt

bed diameter (m)

db

average area-equivalent bubble diameter (m)

dbm

maximum bubble diameter (m)

db0

initial bubble diameter (m)

dc

diameter of a circle (m)

dp

particle diameter (m)

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Ab

ACCEPTED MANUSCRIPT restitution coefficient

f

dominant frequency (Hz)

fb

bubble frequency (Hz)

g

gravity acceleration (m·s-2)

g0

radial distribution function

h

bed height (m)

H

total height of the fluidized bed (m)

I

identity matrix

I2D

second invariant of the strain rate tensor

Pg

gas pressure (Pa)

Ps

solid pressure (Pa)

Ps,f

frictional part of the solid pressure (Pa)

Ps,k

kinetic part of the solid pressure (Pa)

Re

Reynolds number

s

surface area of a sphere having the same volume as the particle (m2)

S

actual surface area of the particle (m2)

ug

gas velocity (m·s-1)

us

solid horizontal velocity (m·s-1)

umf

the minimum fluidization velocity (m·s-1)

usp

Superficial gas velocity (m·s-1)

vs

solid vertical velocity (m·s-1)

W

width of the fluidized bed (m)

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D

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e

ACCEPTED MANUSCRIPT Greek letters inter-phase momentum exchange coefficient (kg·m-3·s-1)

βgs

drag coefficient in a control volume (kg/(m2·s))

εg

bed voidage

εs

solid concentration

εs,max

maximum solid volume fraction

εs,min

solid concentration at the transition point when frictional stresses become important

θ

granular temperature (m2/s2)

λs

solid bulk density (Pa·s)

μg

gas viscosity (Pa·s)

μs

solid viscosity (Pa·s)

μs,k

kinetic part of the shear viscosity of particles (Pa·s)

μs,f

frictional part of the shear viscosity of particles (Pa·s)

ρg

gas density (kg·m-3)

ρs

solid density (kg·m-3)

τg

gas stress tensor (Pa)

τk

kinetic stress tensor (Pa)

τf

frictional stress tensor (Pa)

τs

solid stress tensor (Pa)



shape factor

Ψ

angle of internal friction (º)

Subscripts

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CE P

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D

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T

β

ACCEPTED MANUSCRIPT bubble

g

gas phase

mf

minimum fluidization

p

particle

s

solid phase

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T

b

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6 Acknowledgements

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This work is financially supported by the National Natural Science Foundation of China (Grant No. 51322601), the National Natural Science Foundation for Creative Research Groups of China (Grant No. and

the

Fundamental

Research

D

51421063),

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HIT.BRETIV.201315).

Funds

for

the

Central

Universities

(Grant

No.

ACCEPTED MANUSCRIPT 1

Black millet ρp = 1343.5kg/m3 dp = 1mm

T

2

1 Humidity meter 2 Fluidized bed 3 Light source 4 Camera 5 Computer 6 Flux meter 7 Fan

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4

0.6m

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3

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0.15m

7

6

5

0.16m

1.0 m/s

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D

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(a) Diagram (b) Schematic illustration Figure 1 Photograph and schematic illustration of the bubbling fluidized bed

1.0 m/s

1.0 m/s

1.0 m/s

1.0 m/s

150

150

150

100

100

100

100

100

0

50

100

x (mm)

0

150

0

50

50

100

0

150

y (mm)

y (mm)

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y (mm)

50

50

0

y (mm)

150

y (mm)

150

50

0

50

100

0

150

50

0

50

x (mm)

x (mm)

100

0

150

x (mm)

0

50

100

150

x (mm)

t=26.16 s t=26.24 s t=26.32 s t=26.40 s t=26.48 s Figure 2 Snapshots of the instantaneous particle positions and corresponding particle velocity fields at a superficial gas velocity of 0.6 m/s.

1.0 m/s

1.0 m/s

1.0 m/s

1.0 m/s

1.0 m/s

150

150

100

100

100

100

100

0

50

100

x (mm)

150

0

0

50

100

x (mm)

150

0

50

50

50

50

0

y (mm)

y (mm) 50

y (mm)

150

y (mm)

150

y (mm)

150

0

50

100

x (mm)

150

0

0

50

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x (mm)

150

0

0

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x (mm)

150

ACCEPTED MANUSCRIPT t=7.60 s

t=7.68 s

t=7.76 s

t=7.84 s

t=7.92 s Figure 3 Snapshots of the instantaneous particle positions and corresponding particle velocity fields at a superficial gas velocity of 0.8 m/s. 1.0 m/s

g

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T

0.15

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y (m)

0.1

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0.05

0

0

0.05

0.1

0.15

x (m)

g 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5

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D

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(a) Snapshot of particles’ position. (b) Particles’ velocity. Figure 4 Instantaneous volume fraction and particle velocity in a bubbling fluidized bed at a superficial gas velocity of 0.8 m/s.

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(a) Experimental result (b) Simulation result Figure 5 Comparison of instantaneous void fraction (usp = 0.8 m/s)

g 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5

(a) Experimental result (b) Simulation result Figure 6 Comparison of instantaneous void fraction (usp=0.9 m/s)



0 0 0 0 0 0 g0 0 00 00 0 0 0 0 0 0 0 0

ACCEPTED MANUSCRIPT 1.0 m/s

1.0 m/s

0.1

0.1

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0.05

0.05

0

0.05

0.1

0.15

x (m)

0

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0

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y (m)

T

0.15

y (m)

0.15

0

0.05

0.1

0.15

x (m)

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(a) Experimental result (b) Simulation result Figure 7 Comparison of the solid-phase time-averaged velocity (usp=0.8 m/s)

1.0 m/s

D

1.0 m/s 0.15

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0.15

0.05

0

y (m) 0.05

0.05

0.1

0

0.15

x (m)

0

0.05

0.1

x (m)

(a) Experimental result (b) Simulation result Figure 8 Comparison of the solid-phase time-averaged velocity (usp=0.9 m/s) 0.12 Sim. with original Gidaspow model h=0.1m Sim. with modified Gidaspow model h=0.1m 0.10 exp. h=0.1m u =0.7m/s 0.08 0.06 0.04 0.02 0.00 -0.02 -0.04 -0.06 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

0.3 0.2

x (m)

vs (m/s)

sp

us (m/s)

0

0.1

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y (m)

0.1

0.1 0.0 -0.1 -0.2

Sim. with original Gidaspow model h=0.1m Sim. with modified Gidaspow model h=0.1m exp. h=0.1m usp=0.7m/s

-0.3 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

x (m)

0.15

ACCEPTED MANUSCRIPT (a) Sim. with original Gidaspow model h=0.1m Sim. with modified Gidaspow model h=0.1m exp. h=0.1m usp=0.8m/s

0.08

0.3 0.2

vs (m/s)

0.04 0.00

-0.04

0.1 0.0 -0.1 -0.2

T

0.12

us (m/s)

(b) 0.4

Sim. with original Gidaspow model h=0.1m Sim. with modified Gidaspow model h=0.1m exp. h=0.1m usp=0.8m/s

-0.3

-0.12 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

-0.4 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

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x (m)

(c)

x (m)

(d)

0.3

0.16

Sim. with original Gidaspow model h=0.1m Sim. with modified Gidaspow model h=0.1m exp. h=0.1m usp=0.9m/s

0.12 0.08

0.2

vs (m/s)

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us (m/s)

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-0.08

0.04 0.00

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-0.04 -0.08

-0.12 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

0.0

-0.1 -0.2

Sim. with original Gidaspow model h=0.1m Sim. with modified Gidaspow model h=0.1m exp. h=0.1m usp=0.9m/s

-0.3 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

x (m)

D

x (m)

0.1

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(e) (f) Figure 9 Comparison of the solid-phase time-averaged horizontal and vertical velocity

0.0006 Sim. h=80mm Sim. h=100mm Sim. h=120mm usp=0.7m/s

 (m/s)

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2

0.0005 0.0004 0.0003 0.0002 0.0001

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

x (m)

(a) 0.0010

0.0010 Sim. h=80mm Sim. h=100mm Sim. h=120mm usp=0.8m/s

 (m/s)

2

 (m/s)

0.0008 2

0.0008

Sim. h=80mm Sim. h=100mm Sim. h=120mm usp=0.9m/s

0.0006

0.0006

0.0004

0.0004

0.0002 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

0.0002 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 x (m)

x (m)

(b) (c) Figure 10 Granular temperature at different bed heights

ACCEPTED MANUSCRIPT

 (m/s) 2  (m/s) 0.11

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0.11 0.099 0.088 0.077 0.066 0.055 0.044 0.033 0.022 0.011 0

2

0.11 0.099 0.088 0.077 0.066 0.055 0.044 0.033 0.022 0.011 0

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 (m/s)

2

0.099 0.088 0.077 0.066 0.055 0.044 0.033 0.022 0.011 0

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0.11 0.099 0.088 0.077 0.066 0.055 0.044 0.033 0.022 0.011 0

2

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 (m/s)

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usp=0.7 m/s usp=0.8 m/s usp=0.9 m/s Figure 11 Time-averaged particle granular temperature 0.20

0.12

0.04

0.05

0.10

0.15

0.08

0.20

0.00 0.00

0.25

0.04

0.05

0.20

0.25

0.00 0.00

0.7

1.1

(a)

0.20

0.25

1.5

Dav. & Har. Sim. usp = 0.8m/s

1.4

1.0 0.9

0.7 0.00

0.15

0.20

0.25

(c)

1.6

0.8 0.10 0.15 h (m)

0.10

h(m)

vb (m/s)

1.2

0.8

0.05

0.05

h(m)

1.3

Dav. & Har. Sim. usp = 0.7m/s

vb (m/s)

vb (m/s)

0.15

1.4

0.9

0.6 0.00

0.10

(b) Figure 12 Bubble diameter distribution along the bed height

1.3

1.0

Mori Choi Sim. usp = 0.9 m/s

0.08

AC (a)

1.1

0.12

0.04

h (m)

1.2

0.16

dB(m)

0.08

0.00 0.00

0.20

Mori Choi Sim. usp = 0.8 m/s

TE dB(m)

0.12

0.16

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dB (m)

0.16

0.20 Mori Choi Sim. usp = 0.7 m/s

1.3

Dav. & Har. Sim. usp = 0.9m/s

1.2 1.1 1.0

0.05

0.10 0.15 h (m)

0.20

0.25

0.9 0.00

0.05

(b) Figure 13 Bubble velocity distribution along the bed height

0.10 0.15 h (m)

(c)

0.20

0.25

1.0

1.0

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.5

0.4

0.4

usp = 0.9m/s, h = 0.05m

12

14

16

18

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0.3 10

0.3 10

20

t (s)

14

16

18

20

t (s)

1.0

0.9

0.9

0.8

0.8

g

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0.7 0.6

D

0.5 0.4 14

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12

16

18

0.7 0.6 0.5 0.4

usp = 0.9m/s, h = 0.15m

20

usp = 0.9m/s, h = 0.2m

0.3 10

12

14

16

t (s)

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t (s)

Figure 14 Simulated instantaneous concentration of particles

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4.4 4.0

usp = 0.7m/s usp = 0.8m/s usp = 0.9m/s

3.6

fb (Hz)

g

12

usp = 0.9m/s, h = 0.1m

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1.0

0.3 10

IP

0.5

0.6

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g

g

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3.2 2.8 2.4 0.00

0.05

0.10

0.15

0.20

0.25

h (m) Figure 15 Bubble frequency at different superficial gas velocities

18

20

ACCEPTED MANUSCRIPT 0.5

T

0.3

IP

h (m)

0.4

0.7

0.8 usp (m/s)

0.9

1.0

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0.1 0.6

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0.2

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60

24

TE

36

h = 0.105 m

D

48

usp = 0.9 m/s

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Power Spectral Magnitude

Figure 16 Bed expansion at different superficial gas velocities

12 0

2

4 6 8 Frequency (Hz)

10

Figure 17 Frequency and power spectral magnitude of bed voidage

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(a) (b) Figure 18 The wavelet multi-scale analysis of particle concentration pulsation

ACCEPTED MANUSCRIPT Table 1 Details of the PIV Experimental Setup camera

FlowSense 4M Camera

camera frame rate

15 Hz

pixel pitch

7.40 μm

camera resolution

2048×2048 pixels2

restitution

T

particle-particle collision particle-wall collision restitution

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0.70±0.01

IP

0.70±0.01

Table 2 Summary of parameters employed in the simulation Parameter

Value (units)

1 (mm)

Particle density, ρp

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Particle diameter, dp

2560 (kg/m³) 0.7 (-)

Particle-wall restitution coefficient, ew

0.7 (-)

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Particle-particle restitution coefficient, e

Time interval, △t Bed width, W

TE

0.16 (m) 0.03 (m) 0.60 (m) 15 (-)

CFD grid number, Ny

60 (-)

Gas temperature, T

25 (℃)

Shear viscosity of air, μg

1.82×10-5（kg/m∙sec）

Density of air, ρg

1.206 (kg/m³)

Pressure, P

1.15 ×10-5 (Pa)

Superficial gas velocity, usp

0.7, 0.8, 0.9 (m/s)

Initial solid height, H0

0.15 (m)

Initial solid volume fraction, εs0

0.36 (-)

Specularity coefficient, φ

0.5 (-)

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CFD grid number, Nx

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Bed height, H

D

Bed depth, D

10-5

ACCEPTED MANUSCRIPT 1

2

3

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1 Humidity meter 2 Fluidized bed 3 Light source 4 Camera 5 Computer 6 Flux meter 7 Fan

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5

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Graphical abstract

References

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