Particle Dispersion and Circulation Patterns in a 3D Spouted Bed with

Article pubs.acs.org/IECR

Particle Dispersion and Circulation Patterns in a 3D Spouted Bed with or without Draft Tube Kun Luo, Shiliang Yang, Ke Zhang, Mingming Fang, and Jianren Fan* State Key Laboratory of Clean Energy Utilization, Zhejiang University, Hangzhou 310027, P. R. China ABSTRACT: Solid dispersion and circulation properties in a 3D spouted bed with or without a draft tube are reported based on the CFD-DEM coupling approach. The distribution characteristics of the local and global solid dispersion coefficients are studied. Meanwhile, the effects of tube configuration and bed diameter on these two aspects are investigated. The results show that the large lateral dispersion coefficient concentrates in the spout−annulus interface and the lower part of the fountain region, while the vertical one lies in the central region of the spout. Insertion of a draft tube results in not only large systematic vertical and lateral dispersion coefficients but also more regular particle circulation patterns in the lateral and vertical directions. Furthermore, the larger the entrainment distance of the draft tube, the higher the local and systematic dispersion coefficients are and also an increasing particle circulation rate appears. Extension of the draft tube above the bed surface leads to a larger particle circulation rate and more vigorous dispersion system. With the scale-up of apparatus, both the lateral and the vertical dispersion coefficients are enhanced and solid circulation becomes more energetic.

1. INTRODUCTION Due to its high coefficient on the fluid−particle interaction and the ability to process solid phase with large diameter, a spouted bed has been widely applied in several physical and chemical operations such as coating of tablets, gasification and combustion of coal or waste material, drying of granular materials, and solid mixing.1−4 Three typical regions existing in a spouted bed, namely, the spout region, the annulus region, and the fountain region, exhibit significantly distinctive gas−solid flow behaviors. Therefore, a steady circulation of solid phase is established in a spouted bed. However, the falling of particles from the annulus into the spout at all bed levels makes controlling the particle circulation pattern difficult. Thus, a draft tube is inserted into the system to reduce the random falling of solid phase from the spout−annulus interface and make operation of the system more stable. Introduction of the draft tube remarkably changes the dynamic behavior of gas−solid flow in the system. Extensive experiments have been conducted by many researchers focusing on investigation of the influence of the draft tube on the flow behavior of gas and solid phases in spouted beds. Clafin et al.5 measured the particle motion and gas distribution in a 0.3 m spouted beds with porous and solid wall draft tubes experimentally. The results showed that the porous tube has a higher annulus air flow than the solid tube at low separation distances and a lower pressure drop for a given annulus flow. Khoe et al.6 discussed the drying characteristics in a draft tube spouted bed based on the flow patterns of gas and solid. They reported that the drying rate is determined by the heat transfer in the recirculation zone and in the draft tube and concluded that the draft tube spouted bed is a promising dryer design for heat-sensitive particles with slow intraparticle mass transfer coefficients. San José et al.7 conducted experimental research on the bed stability of a conical spouted bed with a nonporous draft tube, within which an original correlation © 2013 American Chemical Society

calculating the minimum spouting velocity in conical spouted beds with draft tube has been proposed. Besides the experimental research on the hydrodynamics of a spouted bed with a draft tube, numerical simulation has proved to be an effective method for investigation of latent gas−solid movement mechanisms in the system. Azizi et al.8 presented numerical research on the gas−solid flow in a a spouted bed with a nonporous draft tube based on the two-fluid model (TFM).9 The results showed that the gas velocity in the annulus region increases longitudinally because of the presence of the porous draft tube. Cezar et al.10 numerically studied the fluid dynamics of a draft tube continuous spouted bed using TFM. Zhao et al.11 took a numerical simulation in a spouted bed with a draft tube by the discrete element method (DEM) to investigate the gas−solid hydrodynamics. The design of a spouted bed with a draft tube depends on the solid movement property in the system. An important characteristic of the solid movement can be described by the particle dispersion and circulation properties for the whole system. The dispersion coefficient is usually adopted for understanding the solid diffusivity activity. Lots of research has been conducted in the investigation of the solid dispersion coefficient in a bubbling fluidized bed by many researchers.12−15 Few reports have been found on this topic in a spouted bed, especially for a system with a draft tube. For the particle circulation behavior of spouted bed, much research has been carried out. Chatterjee et al.16 conducted an experiment to explore the effects of particle diameter and apparent particle density on the internal solid circulation rate in an air-spouted bed. Berruti et al.17 experimentally studied the effects of operating and geometry parameters on the solids Received: Revised: Accepted: Published: 9620

December 20, 2012 March 22, 2013 June 12, 2013 June 12, 2013 dx.doi.org/10.1021/ie303555g | Ind. Eng. Chem. Res. 2013, 52, 9620−9631

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circulation rates at ambient temperature. Muir et al.18 carried out an experiment in a 20 cm semicylindrical spout-fluid bed with a draft tube to investigate the effect of various physical parameters on the solid circulation rate. Mann et al.19 modeled solid circulations in spouted beds by measuring the concentration of tracer particles. Ren et al.20 investigated the particle circulation by tracking two tracer particles with different densities in a spouted bed based on the CFD-DEM coupling approach. The results showed that the fraction of time spent in the annulus of a cycle is much longer than that spent in a fountain and spout together. Takeuchi et al.21 simulated a spouted bed with the discrete element method to observe the particle circulation of the system. However, an investigation of the influence of the draft tube configuration on the circulation property has not been reported. As mentioned above, investigations of local and systemic dispersion behaviors and particle circulation in a spouted bed with a draft tube have become a prerequisite for design and operation of the system, while few reports appear on the comparison of these important aspects for a spouted bed. To overcome this gap, the present work carries out a numerical simulation based on the CFD-DEM coupling approach in a 3D conical-base spouted bed with or without a draft tube to explore the dispersion characteristic and particle circulation behavior of the solid phase, together with the influence of the presence of a draft tube on these two aspects. Furthermore, the effects of the tube configuration on the particle circulation and the dispersion property of the solid phase in the spouted bed are discussed. Finally, variation tendencies of these two aspects are studied with the scale-up of the geometry.

∂(εgρg k u j) ∂ (εgρg k) + ∂t ∂xj =

− εgρg εt

∂t

+

∂(εgρg u i) ∂xi

=0

=

εt2 k

(4)

where σε = 1.3 and σk = 1.0 are turbulent Prandtl numbers for εt and k, respectively. c1 and c2 are constants: c1 = 1.44, c2 = 1.92. Sp is the momentum source term exerted by the particles on the current computational cell, which can be obtained by summing up all the fluid forces impacting on the particles locating in the current cell as n

Sp =

∑ (fd ,i + f p,i)/ΔV i=1

(5)

where fd,i and fp,i are the fluid−particle interaction force on particle i and n is the total number of particles in the current computational grid. 2.2. Governing Equations for Solid Phase. The motion of the solid phase is described using the discrete element method, in which the movement of a specific particle is governed by Newton’s second law. The governing equations for solid motion can be expressed as mp

(1)

Ip

⎡ ∂p ∂ ⎢ − Sp + ρg εg g + εg(μ + μt ) ∂xi ∂xj ⎢⎣

⎤ ⎛ ∂u j ∂u i ⎞⎥ ⎜⎜ ⎟⎟ + ∂xj ⎠⎥⎦ ⎝ ∂xi

εgc1εt ∂u i ⎛ ∂u j μ ⎞ ∂ε ⎤ ∂u i ⎞ ∂ ⎡⎢ ⎛ ⎜⎜ ⎟ εg ⎜μ + t ⎟ t ⎥ + μt + ∂xj ⎢⎣ ⎝ σk ⎠ ∂xj ⎥⎦ ∂xj ⎝ ∂xi ∂xj ⎟⎠ k − εgc 2ρg

∂(εgρg u iu j) ∂ (εgρg u i) + ∂t ∂xj = −εg

(3)

∂(εgρg εt u j) ∂ (εgρg εt) + ∂t ∂xj

2. COMPUTATIONAL METHOD 2.1. Governing Equations for Fluid Phase. The continuity and momentum equations for the fluid phase are described with the Navier−Stokes equations based on the local average variables, which are formulated as ∂(εgρg )

μ ⎞ ∂k ⎤ ∂u ⎛ ∂u j ∂u i ⎞ ∂ ⎡⎢ ⎛ ⎟ εg ⎜μ + t ⎟ ⎥ + εgμt i ⎜⎜ + ∂xj ⎢⎣ ⎝ σk ⎠ ∂xj ⎥⎦ ∂xj ⎝ ∂xi ∂xj ⎟⎠

dvp dt

dωp dt

= mp g + fd + fc + f p

(6)

= Tp

(7)

where mp, Ip, vp, and ωp are the mass, momentum inertia, translational velocity, and angular velocity of the particle, respectively. The forces involved are the gravity force mpg, the drag force exerted by the fluid phase fd, the contact force fc exerted between the colliding particle−particle or the particle− wall pair, and the pressure gradient force fp(= −Vp∇p).22 Tp stands for the torque exerted on the particle by other colliding particles or walls. The drag force fd on a specific particle in the dense flow is modeled as

(2)

where εg (= 1 − (∑in= 1Vpi)/(ΔV) is gas voidage in the current computational cell. ρg, u, and p are the density, velocity vector, and pressure of the fluid phase, respectively. g is the gravity acceleration. μ represents the dynamic viscosity of the fluid phase, and μt is the turbulent viscosity calculated from the k − ε turbulence model as μt = cμρgk2/εt, in which cμ (=0.09) is a constant. k and εt are the turbulent kinetic energy and its dissipation rate, respectively. The transport equations for k and εt are formulated as

fd = βpf (u − vp)Vp/(1 − εg)

(8)

where βpf is the fluid−particle interaction coefficient. The correlation proposed by Gidaspow9 is applied to calculate the coefficient, which is expressed as 9621

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Figure 1. Geometry and grid representation of the 3D spouted bed. (a) Geometry of the spouted bed; (b) front and top views of grid distribution.

⎧ ρ ε (1 − εg )|u − vp| ⎪3C g g εg−2.65 εg ≥ 0.8 D ⎪4 dp ⎪ ⎪ 150(1 − ε )2 μ g βpf = ⎨ εg < 0.8 2 ⎪ εgd p ⎪ 1.75ρg (1 − εg )|u − vp| ⎪ + ⎪ dp ⎩

⎧ 24 (1 + 0.15Re 0.687 ) Re p < 1000 ⎪ p C D = ⎨ Re p ⎪ 0.44 Re p ≥ 1000 ⎩

contact force when the sliding occurs between the colliding pair. 2.3. Computational Method for Dispersion Coefficient. As Liu et al.12 pointed out, two methods exist for estimation of the solid dispersion coefficient, namely, the macro method based on solving the transient particle concentration profile by fitting the Ficked-type equation and the micro method based on statistics on the position displacement of particles in the system. With the advantage of the CFD-DEM coupling approach, the position of each particle can be obtained in the tracking process. Thus, the microcalculation method is chosen in the present work. The dispersion coefficient of a particle can be calculated from Einstein’s expression in the system. For a specific tracked particle i, the local dispersion coefficient is calculated as

(9)

(10)

where Rep is the particle Reynolds number, which is estimated as Re p =

εgρg |u − vp|d p μ

Di =

(11)

For calculation of the contacting forces between the colliding pair of the particle−particle or particle−wall, the soft-sphere contact model originally proposed by Cundall and Strack23 is adopted in the current work to predict the interaction force as

fcij = fcnij + fctij

(12)

D fcnij = f Scnij + f cn ij = ( − k nδnij − ηn vtij · nij)nij

(13)

(Δri)2 2Δt

(15)

where Δri is the local displacement of current particle in time interval Δt. Moreover, the local dispersion coefficient of the solid phase in a specific computational cell is estimated by averaging the dispersion coefficients of all particles located in the current cell. Then, an instantaneous distribution field of dispersion coefficient in the whole system can be obtained. However, the instantaneous value of the dispersion coefficient in a specific computational cell is strongly dependent on the number of particles at the considered moment, leading to a large fluctuation if the change of particles number is obvious. In order to investigate the local distribution property of the dispersion coefficient, the time-averaged dispersion coefficient is obtained statistically over all time intervals as

fctij = f Sctij + f ctDij = min(−k tδtijtij − ηt (vtij ·tij)tij , μp |fcnij|tij) (14)

where δ, k, and η are the particle displacements and the spring and damping coefficients between the colliding pair, respectively. The subscripts n and t stand for the normal and tangential directions, respectively. μp is the friction coefficient in the colliding pair, which is used for calculation of the tangential 9622


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Dl =

∑i = 1 Di ,local N

=

1 N

N

Article

⎛ (Δr )2 ⎞ 1 i ⎟= ⎝ 2Δt ⎠ 2 × Δt × N

Table 2. Physical and Numerical Parameters for Simulation

∑⎜ i=1

gas phase (air) temperature, K spouting velocity, m/s pressure, atm solid phase (bean)

N

∑ ((Δri) ) 2

(16)

i=1

where l is the computational cell label, N is the total number of particles in the current cell, and Di,local is the local dispersion coefficient of particle i that locates in the cell l. For the instantaneous dispersion coefficient of the whole system at a specific time instant, the dispersion coefficient can be estimated by averaging the dispersion coefficient over all particles in the system. The coefficient can be calculated as D=

1 NP

NP

∑ i=1

(Δri)2 1 = 2Δt NP

NP

∑ i=1

no. diameter, m interparticle restitution coefficient interparticle friction coefficient geometry

(17)

where NP is the total number of particles in the system and ri is the position of particle i.

3. SIMULATION SETUP The simulation is carried out in the framework of CFD-DEM coupling, for which a detailed coupling description has been well documented in the literature.24−27 For solving the governing equations for the fluid phase, the PISO28 is adopted for decoupling of the pressure and velocity in a collocated grid. In order to solve algebra equations obtained from discretization of governing equations, specific boundary conditions should be assigned to the variables of the system boundaries. The velocity−inlet boundary condition is applied for the inlet of bed, while the nonslip boundary condition is used for the wall and the draft tube of the spouted bed. For the outlet of the system, pressure−outlet boundary condition is chosen for the pressure. The simulation is carried out in a 3D conical-base spouted bed with or without a draft tube, which has the same geometry parameters as the experiment conducted by Vieira Neto et al.29 The sketch of the geometry is shown in Figure 1, in which three tube configurations are considered with the parameters listed in Table 1. The bed has an internal diameter of 0.21 m and height

0.07 0.12

0.04 0.15

0.27

density, kg/m3 spring constant, N/s particle−wall restitution coefficient particle−wall friction coefficient

1.8 × 10−5 1.2 28.8 1170 420 0.7 0.32

0.21 0.035 1.0 0.19 1.0 × 10−7 4848 (without tube) 5568 (Ld = 0.12 m, Hd = 0.07 m) 5972 (Ld = 0.15 m, Hd = 0.04 m) 5936 (Ld = 0.15 m, Hd = 0.07 m)

numbers of particles used in each case are 26 719, 31 354, 35 072, 38 480, and 41 026, respectively. The calculation time step is 1.0 × 10−7 s, and a real time of 90 s for each case is simulated, while data from 10 to 90 s are utilized for the time statistic of the local dispersion coefficient in a specific computational cell.

4. RESULTS AND DISCUSSION 4.1. Model Validation. The total pressure drop under different air flow rates reflects the latent dynamic behavior of the solid phase in the system. Figure 2 presents the experimental and simulated pressure drop curves by approach of increasing air flow rate in spouted beds with different tube configurations. The variation tendency of the pressure drop obtained from the simulation shows a similar behavior with experimental results in all three systems. The pressure drop of the system increases sharply with enlarging air flow rate until a peak is reached. This maximum value of the pressure drop corresponds to the energy needed to break the packed state of the bed. Subsequently, the pressure drop decreases sharply until a steady spouting regime is constructed in the system. For all three profiles, it can be concluded that the derivation between the experimental data and the simulation result is small under a small flow rate. In the process of the flow regimes changing from packed bed to spouting state, the calculation results are larger than the experimental data, which can be attributed to the fact that the transition region occupied by the pressure drop is large. Under the situation of a large flow air rate, the simulated results agree well with the experimental profiles. Hence, the proposed model can be used to investigate the solid dispersion and circulation behaviors of the system. 4.2. Qualitative Description of Dispersion Coefficient. The distribution property of the lateral dispersion coefficient reflects the vigorous mixing property of the solid phase. Figure 3a shows the contour plot of the time statistical lateral dispersion coefficient of the solid phase in slice Z = 0.105 m of a spouted bed under Ug = 37 m/s, from which excellent symmetrical distribution can be found. The largest lateral dispersion coefficient mainly exists in the region between the

Table 1. Parameters of the Entrainment Distance and Length of Draft Tube Hd (m) Ld (m)

viscosity, Pa·s density, kg/m3 molecular weight, kg/mol

22 000 6 × 10−3 0.71

diameter of the vessel, m diameter of spouting inlet, m vessel height, m initial bed height, m time step of calculation, s total number of grid

(ri − ri0)2 2Δt

(i = 1, 2, ..., NP)

298 37 1

0.07 0.15

of 1.0 m. The internal diameter of the draft tube is 0.035 m and aligned at different heights of the bed to investigate the influence of the bed geometry on the dynamic behavior of the system. Information on related physical and numerical parameters is given in Table 2. The initial packed bed height is 0.19 m. In order to calculate the solid dispersion coefficient, every particle is assigned a unique label to track the trajectory for the postprocessing. To investigate the impact of bed diameter on the dispersion and circulation behavior of the solid phase in a spouted bed with a draft tube, the system with a tube configuration of Ld = 0.12 m and Hd = 0.07 m is amplified with five scaling up ratios of 1.2, 1.4, 1.6, 1.8, and 2.0. The calculation grid cell numbers are 6532, 7546, 8793, 9624, and 10 878, respectively. Moreover, the 9623



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Figure 3. Lateral dispersion coefficient in slice Z = 0.105 m of spouted bed with and without tube, Ug = 37 m/s: (a) without draft tube; (b) with draft tube, Ld = 0.12 m, Hd = 0.07 m.

Figure 2. Experimental and simulated pressure drop curves with increasing air inflow rate in spouted beds. (a) Spouted bed without draft tube. (b) Spouted bed with Hd = 0.04 m, Ld = 0.15 m. (c) Spouted bed with Hd = 0.07 m, Ld = 0.12 m. Figure 4. Contour plots of vertical solid dispersion coefficient in slice Z = 0.105 m of spouted bed with or without draft tube under Ug = 37 m/s: (a) without draft tube; (b) with draft tube, Ld = 0.12 m, Hd = 0.07 m.

central part of the fountain and the wall of bed, resulting from the vigorous lateral movement behavior of the solid phase after being ejected into the free domain. Moreover, a relatively large lateral dispersion coefficient appears in the spout of the system, which is due to the random falling event of particles into the spout region from the spout−annulus interface. The effect of a draft tube on the lateral dispersion of the solid phase in a spouted bed is illustrated in Figure 3b, within which the time statistical lateral dispersion coefficient in slice Z = 0.105 m of a system with a tube of Ld = 0.12 m and Hd = 0.07 m under Ug = 37 m/s is obtained. It can be observed that the presence of the draft tube remarkably alters the distribution characteristic of the lateral dispersion coefficient. Not only is the maximum value much larger than that of the system without the draft tube, but also the domain with a large lateral dispersion coefficient is enlarged especially in the axial direction. The result indicates that the lateral dispersion behavior of the ejected particles is enhanced with the presence

of the draft tube. Moreover, no explicit large value of the lateral dispersion coefficient can be observed in the spout region with a draft tube due to the presence of a draft tube limiting solid exchange between the spout and the annulus. Vertical solid transportation is stronger than the lateral one since gas is introduced from the bottom of the bed. Figure 4a shows the contour plot of the time statistical vertical dispersion coefficient in slice Z = 0.105 m of a spouted bed under Ug = 37 m/s. It can be concluded that the vertical dispersion coefficient distributes symmetrically and a strong dispersion behavior exists in the central part of the spout, resulting from vigorous vertical movement of the solid phase in this region. Moreover, the maximum value of the vertical dispersion coefficient 9624



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Figure 5. Quantitative comparison of the time statistical lateral dispersion coefficient of solid phase at different heights in slice Z = 0.105 m of spouted bed without draft tube, Ug = 37 m/s: (a) conical region; (b) cylindrical region with spout; (c) fountain region.

Figure 6. Quantitative comparison of the vertical solid dispersion coefficient in slice Z = 0.105 m of spouted bed without draft tube, Ug = 37 m/s: (a) conical region; (b) cylindrical region with spout; (c) fountain region.

locating in the outlet region of the spout is nearly 0.0154 m2/s, which is 8 times the lateral maximum value. The influence of the draft tube on vertical solid dispersion can be obtained from comparison of the contour plots in Figure 4. The region with a large vertical dispersion coefficient is uplifted to the bed surface in the presence of a draft tube. On the other hand, the existence of a draft tube enhances the vertical dispersion behavior in the near central part of the fountain region. 4.3. Quantitative Description of Dispersion Coefficient. The variation tendency of the lateral dispersion coefficient in slice Z = 0.105 m of a spouted bed without a draft tube is shown in Figure 5. The lateral dispersion coefficient has a large value in the near central part of the spout, as illustrated in Figure 5a and 5b, and increases with height elevation in the conical region, while no significant

changes appear in the spout region in the cylindrical part of the system. However, obviously a different distribution behavior appears in the fountain, as can be seen in Figure 5c, in which the lateral dispersion coefficient is observed to increase initially until a maximum is reached in the halfway point from the column center and then decrease until the near wall region. Moreover, the peak value shows an increasing tendency and then decreases with bed elevation, reflecting that the strongest lateral motion of the solid phase in the fountain region is not in the lowest part but the middle place of this region. The distribution profiles of the time-averaged vertical solid dispersion coefficient are presented in Figure 6. It is shown that the solid vertical dispersion coefficient decreases along the 9625



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Figure 7. Lateral dispersion coefficient in systems with different tube configurations under Ug = 37 m/s: (a) Y = 0.03 m; (b) Y = 0.16 m; (c) Y = 0.26 m.

Figure 8. Comparison of vertical dispersion coefficient in slice Z = 0.105 m of systems with different tube configurations under Ug = 37 m/s: (a) Y = 0.03 m; (b) Y = 0.16 m; (c) Y = 0.26 m.

radial direction in the region below the bed surface. A large dispersion coefficient in the central part first increases and then decreases with bed elevation, while in the fountain region, as shown in Figure 6c, the vertical dispersion coefficient across the bed width is large in the center of the bed and decreases until a valley is reached halfway from the center and then increases again, followed by a decreasing tendency to the near wall region. Moreover, the dispersion in the central part of the system decreases along the axial direction due to the deceleration of the solid phase under the mutual effect of a reduced drag force and gravity. The presence of a draft tube changes the flow behavior of gas and solid phases in a spouted bed, resulting in a different solid dispersion behavior in the system. Figure 7 illustrates the comparison of the time statistical lateral solid dispersion

coefficient at three heights of spouted beds with different tube configurations under Ug = 37 m/s. As expected, the local distribution characteristic changes a lot under different tube configurations. The lateral dispersion behavior is reduced with the presence of a draft tube in the region below the bed surface, reflecting the restriction effect of a draft tube on solid changing from the spout−annulus interface. However, a different impact of the tube on the lateral dispersion coefficient appears in the fountain region of systems with a different tube configuration. It is explicitly shown that the lateral dispersion behavior in this region is enhanced in the presence of the draft tube for all systems investigated. On the other hand, the effect of tube geometry on the lateral dispersion property can be inferred from profile comparison at all three heights. The higher the 9626



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Figure 10. Time evolution profiles of solid dispersion coefficients under different scale-up ratios: (a) lateral dispersion coefficient; (b) vertical dispersion coefficient.

Figure 9. Time evolution of the dispersion coefficient in three directions of spouted beds with different tube configurations, Ug = 37 m/s: (a) in the X direction; (b) in the Z direction; (c) in the Y direction.

about the influence of the tube geometry on the dispersion behavior of the whole system. Hence, the systematic dispersion coefficient is adopted to represent the overall dispersion property of the system and adopted to be a criterion to investigate the influence of the tube on system performance. Figure 9 shows the time evolution profiles of the systematic dispersion coefficients in spouted beds with different tube configurations. Profiles of the systematic dispersion coefficients in each direction are captured to fluctuate around a value in steady operation condition. On the other hand, in all systems considered, a spouted bed without a draft tube shows the smallest dispersion value in all three directions and also the smallest fluctuation compared with that in the system with a draft tube. Thus, it can be concluded that insertion of a draft tube results in an increase of the systematic dispersion coefficient and larger system fluctuation behavior. For the dispersion property of a spouted bed with a draft tube, the largest dispersion coefficient exists in a system with tube of Ld = 0.15 m and Hd = 0.07 m while the lowest profile is corresponding to a system with Ld = 0.15 m and Hd = 0.04 m. The systematic dispersion coefficient of a spouted bed with Ld = 0.12 m and Hd = 0.07 m lies in the middle of them. Hence, it can be concluded that a small distance between the bottom of tube with the inlet of the system leads to a small systematic dispersion coefficient in all directions, reflecting a less vigorous movement system. Extension of the draft tube above the bed surface has the strongest dispersion behavior in all directions.

entrainment distance is, the larger the dispersion coefficient can be observed. Moreover, extension of the draft tube above the bed surface results in a larger dispersion system. 4.4. Effect of Tube Configuration on Vertical Dispersion. The effect of the tube configuration on the local vertical dispersion coefficient is shown in Figure 8. The vertical dispersion coefficient of the solid phase decreases in the central part of the conical region with the presence of the draft tube, and the lowest value appears in a spouted bed with a smaller entrainment distance. Moreover, a decreasing tendency of the vertical dispersion coefficient can be observed at heights in the draft tube region, as can be seen in Figure 8b, due to the relatively small changes of solid velocity in this region. Figure 8c illustrates the distribution profiles of the vertical solid dispersion coefficient at height Y = 0.26 m of systems with different tube configurations. A larger dispersion coefficient is observed on the existence of a draft tube as compared with that in the spouted bed without a draft tube. Hence, the presence of a draft tube enhances the vertical dispersion behavior in the fountain region of a spouted bed. 4.5. Systematic Dispersion Coefficient. As illustrated above, the impact of a draft tube on the local dispersion property shows a remarkable difference in the different regions of the system. Thus, a macroscopic view should be investigated 9627



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Figure 11. Typical trajectory of a specific particle labeled 15000, Ug = 37m/s in time interval 10−45 s: (a−c) X, Y, Z in a system without a tube; (d− f) X, Y, Z in a system with a tube of Ld = 0.12 m, Hd = 0.07 m; (g−i) X, Y, Z in a system with a tube of Ld = 0.15 m, Hd = 0.04 m; (j−l) X, Y, Z in a system with a tube of Ld = 0.15 m, Hd = 0.07 m.

On the other hand, it can be found that the systematic dispersion coefficient in the lateral direction is at a scale of 10−4 m2/s, while a scale of 10−3 m2/s can be observed for the vertical dispersion coefficient, which has a more vigorous dispersion behavior of the solid phase in the vertical direction from a macroscopic view. On the other hand, scale-up is an interesting and important aspect of the spouted bed, which limits its capacity and wide usage on an industrial scale. Extensive research both experimentally and numerically has been conducted in recent years.30−32 In the current work, the influence of bed diameter on the solid dispersion property is explored based on calculation results in systems with different scale-up ratios of the geometry. The time evolution profiles of the solid

dispersion coefficient are illustrated in Figure 10. In general, both the lateral and the vertical dispersion coefficients fluctuate around a constant value and exhibit an increasing tendency with scale-up of the geometry. Meanwhile, the fluctuation quantity is larger in the system with a relatively larger bed diameter. Hence, it can be inferred that a more vigorous dispersion system can be achieved with scale-up of the geometry. 4.6. Particle Circulation. A typical trajectory of a specific particle in the system is adopted to investigate the latent movement mechanisms of the solid phase. Due to the advantage of the CFD-DEM coupling approach, the position of a particle initially locating in 0.14079, 0.03004, and 0.17063 m is tracked in the calculation process. The time evolution position profiles of the particle are presented in Figure 11a−c 9628



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Figure 12. Vertical trajectories of a specific particle in five systems with different scale-up ratios: (a) scale-up ratio of 1.2; (b) scale-up ratio of 1.4; (c) scale-up ratio of 1.6; (d) scale-up ratio of 1.8; (e) scale-up ratio of 2.0.

for a spouted bed without a draft tube under Ug = 37 m/s. A particle is observed to undergo chaotic movement in the lateral direction. Relatively regular movement characteristic can be obtained in the axial direction with two types of movement behaviors, namely, gross circulation and local circulation. After being entrained into the spout, the tracer moves upward vertically in the spout and then is injected into the fountain followed by a falling process with a sharp height decrease. Subsequently, the particle settles down on the bed surface and then moves downward slowly. In the falling process, the particle

is occasionally entrained into the spout from the spout− annulus interface in some situations, and then a small circular pattern is formed. While sometimes it falls down until the bed bottom, a gross circulation of the particle is formed. Three cases of small circular patterns and four gross circulations of the particle can be observed in Figure 11b. After entrainment into the spout region, another similar cycle appears in the following tracking event. The moving trajectories of the specific particle of systems with different tube configurations are shown in Figure 11d−l. 9629



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Generally speaking, more relatively regular behavior can be captured in the presence of a draft tube especially along the axial direction. In the vertical direction, the tracked particle undergoes upward movement from the lowest position with a maximum height reached in a small time interval. The number of small circulation is reduced since the presence of a draft tube prevents random falling of particles from the annulus to the spout. Moreover, the influence of the tube configuration on the movement behavior of the solid phase can be captured from comparison of the position profiles of tracer in Figure 11g−l. It can be observed that both the vertical and the lateral motions of the tracer are accelerated in a system with a tube of Ld = 0.15 m and Hd = 0.07 m compared with the other two configurations. The spouted bed with a draft tube of Ld = 0.15 m and Hd = 0.04 m exhibits the worst circulation performance in both the lateral and the vertical directions. Moreover, an important conclusion is that extension of the draft tube above the bed surface gives rise to the increase of the particle circulation rate and a more vigorous movement system, which is especially beneficial to solid mixing and heat exchanging between gas−solid phases. On the other hand, the impact of the bed diameter on particle circulation is explored with system scale-up. The vertical trajectories of a specific particle are tracked in five systems with different scale up ratios, which are presented in Figure 12. Obviously, the vertical movement of the particle is enhanced with amplification of the geometry. The larger the bed geometry, the higher the circulation rate. Moreover, the maximum height that the particle can reach rises with increasing bed diameter.

Moreover, a more intense dispersion behavior can be obtained with amplification of the geometry. (4) Two circulation patterns can be captured for the movement behavior of the solid phase in a spouted bed. Regular particle circulation is formed with insertion of a draft tube both laterally and vertically with reduction of the local circulation pattern. With a higher entrainment distance, particle circulation is accelerated both vertically and laterally. Meanwhile, extension of the draft tube above the bed surface makes particle circulation more vigorous. The larger the bed diameter is, the more vigorous solid circulation appears.

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AUTHOR INFORMATION

Corresponding Author

*Fax: +86-0571-87991863; E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from the National Natural Science Foundations of China (Grant nos. 50976098 and 51176170) and the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (2007B4) is sincerely acknowledged.

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5. CONCLUSIONS The dispersion and particle circulation properties of the solid phase have been investigated in a 3D spouted bed with or without a draft tube from the results obtained in the framework of the CFD-DEM coupling approach. Furthermore, the effects of draft tube configuration and bed diameter on these two important aspects of the system are studied. On the basis of the simulation results, the following conclusions can be drawn. (1) For a spouted bed without a draft tube, the large lateral dispersion coefficient concentrates in the offset central part of the lower region in the fountain and the spout− annulus interface while the large vertical dispersion coefficient appears in the central part of the spout. For the spouted bed with a draft tube, the large lateral dispersion coefficient lies in the offset of the fountain while the large vertical dispersion coefficient exists in the outlet of the spout and the lower part of the fountain. (2) In the presence of a draft tube, the local lateral dispersion behavior is reduced in the region below the bed surface while it is enlarged in the fountain of the bed. Extension of the draft tube above the bed surface leads to the most vigorous motion in the central region of the bed along the axial direction. A larger entrainment distance leads to a higher vertical dispersion coefficient at all heights of the system. (3) The lateral and vertical systematic dispersion characteristics are enhanced with insertion of the draft tube. If the entrainment distance is small enough, the systematic dispersion characteristic is suppressed. On the other hand, extension of the draft tube above the bed surface strengthens the systematic dispersion characteristic.

NOMENCLATURE CD = drag force coefficient D = dispersion coefficient, m2/s dp = particle diameter, m fc = contact force, N fcn,ij = normal contact force, N fct,ij = tangential contact force, N fd = drag force, N fp = pressure gradient force, N g = gravitational acceleration, m/s2 Hd = entrainment distance, m Ip = particle moment of inertia, kg·m2 k = turbulent energy, m2/s2 kn = spring coefficient in normal direction, N/m kt = spring coefficient in tangential direction, N/m Ld = length of draft tube, m mp = particle mass, kg N = number of time interval for statistic NP = total number of particle in the system n = total number of particles located in a specific cell p = pressure, Pa r = position, m r = particle radius, m Rep = particle Reynolds number Sp = particle drag sink term, N/m3 Δt = time step, s t = time, s Tp = torque on particle, N·m Ug = spouting velocity, m/s u = fluid velocity in the current cell, m/s vp = particle velocity, m/s Vp = particle volume, m3 ΔV = volume of the current cell, m3

Greek Symbols

βpf = interphase momentum transfer coefficient, kg/(m3·s) 9630



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δnij = normal displacements between particle i and particle j, m δtij = tangential displacements between particle i and particle j, m εg = voidage εt = dissipation rate of turbulent kinetic, m2/s3 μ = gas dynamic viscosity, kg/(m·s) μp = friction coefficient μt = gas turbulence viscosity, kg/(m·s) ρg = gas density, kg/m3 ω = particle angular velocity, 1/s ηn = damping coefficients in normal direction, kg/s ηn = damping coefficients in tangential direction, kg/s

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Subscripts

c = contact force d = drag force g = fluid phase i = particle i j = particle j n = normal component p = particle phase t = tangential component

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REFERENCES

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Particle Dispersion and Circulation Patterns in a 3D Spouted Bed with

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