Modeling of Bubble-Structure-Dependent Drag for Bubbling Fluidized

Sep 5, 2014 - The predictions by the present model with consideration of bubble effects ... research means to investigate the behavior of full-size sy...
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Modeling of Bubble-Structure-Dependent Drag for Bubbling Fluidized Beds Shuai Wang, Huilin Lu,* Qinghong Zhang, Guodong Liu, Feixiang Zhao, and Liyan Sun School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, China S Supporting Information *

ABSTRACT: Considering the effect of bubble-emulsion structures in bubbling fluidized beds, a bubble-structure-dependent drag coefficient model is developed. Accelerations in the bubble and emulsion phases are incorporated into the solution of the drag coefficient. Meanwhile, the influence of solid pressure and bubble-induced added mass force is also taken into account. In combination with the two-fluid model, flow behaviors in two-dimensional and three-dimensional bubbling fluidized beds are simulated. The predictions by the present model with consideration of bubble effects are in more reasonable agreement with the experimental results compared to the Gidaspow drag model. It is shown that the present model obtains a zonal distribution of the drag coefficient with solid concentration, which reveals that the drag coefficient not only depends on the local solid concentration but also is greatly influenced by the local velocities. In recent years, filtered TFMs have been developed to account for the effects of unresolved structures for coarse-grid simulations of gas-particle flows.10−12 Constitutive models for filtered TFMs are deduced through filtered results obtained from highly resolved simulations. Parmentier and Simonin13 modified the effective relative velocity appearing in the filtered drag through a functional subgrid drift velocity model. The bed expansion was well predicted using coarse grids with the model. Sarkar et al.14 constructed a subgrid drag model for cylinder suspension. It was found that the presence of cylinders resulted in a reduction in the filtered interphase drag force. Schneiderbauer and Pirker15 deduced two kinds of closures for the unresolved terms of the solid stress and drag force in filtered TFMs: one derived from the filtered data and the other on the basis of the assumption that the cluster formation led to heterogeneity inside the fluidized beds. The simulated results revealed that the prediction with subgrid modifications using coarse grids was in fairly good agreement with the highly resolved simulation. The energy minimization multiscale (EMMS) approach was widely used to deal with the impact of heterogeneous structures on the drag force.16,17 Chalermsinsuwan et al.18 calculated the drag coefficient by employing a simplified EMMS model and performed a simulation in a thin bubbling fluidized bed with Geldart A particles. Wang and Liu19 revised the EMMS model using an implicit cluster diameter expression and applied it to the simulation of bubbling fluidized beds. These models derive from the calculation of cluster-based drag and are very questionable when applied to dense bubbling fluidized beds. Shi et al.20 proposed a bubble-based EMMS model and treated the bubble phase as the mesoscale structure. In this model, the accelerations for the emulsion and bubble phases were

1. INTRODUCTION Bubbling fluidized beds have been widely applied to various industrial processes, such as coal gasification, fluid-catalyticcracking regeneration, and polypropylene production.1,2 Fundamental knowledge of the hydrodynamic characteristics is essential for the design and scale-up of such reactors. Computational fluid dynamics (CFD) is a promising research means to investigate the behavior of full-size systems and complex phenomena in multiphase flow, which is difficult for current measurement techniques. In the last 2 decades, substantial progress has been made to develop mathematical models to describe the behavior of bubbling-fluidized-bed systems with acceptable accuracy.3,4 A two-fluid model (TFM) has proven to be successful in the prediction of the hydrodynamics of a bubbling fluidized bed.5,6 Reuge et al.7 pointed out that the solid stress model greatly influenced the simulation of bubbling fluidized beds. The prediction using the Princeton model was in better agreement with measured data for bed characteristics compared to the Schaeffer model. On the basis of multiscale analysis, Wang and Ge8 developed a mathematical model to characterize the kinetic feature of coarse particles. The results indicated that the unresolved subgrid structures affected the interphase drag coefficient and particle-phase stresses, which should be considered explicitly. For bubbling fluidization, bubbles have significant effects on physical mechanisms, which play roles similar to those of clusters in fast fluidized beds. The gas tends to form dispersed bubbles, which results in the reduction of gas−solid interaction compared to the standard drag on the basis of a uniform particle distribution assumption. It was reported that very fine grids can capture local nonuniform patterns and the grid size should be on the order of 2−4 particle diameters in bubbling fluidized beds.9 Owing to the limitations of the computational resource, coarse-grid simulation with appropriate subgrid models is required to reflect the temporal−spatial multiscale structure for a large-scale industrial fluidized bed. © 2014 American Chemical Society

Received: Revised: Accepted: Published: 15776

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Figure 1. Grid resolution of the gas−solid bubbling fluidized bed.

T1-7, where μg represents the gas viscosity and is assumed to be constant in the current work. At a high solid concentration, particles are closely packed and influenced by sustained contact with multiple neighbors. The frictional stress of sliding contacts is dominant. Here, the friction stress model proposed by Srivastava and Sundaresan25 is adopted to consider the frictional contribution, which modifies the Savage26 model to take strain rate fluctuations into account. The kinetic theory is still used to describe the kinetic and collisional contribution. To provide a smooth transition between the shear flow and frictional regime, a transitioning function φ is introduced.27 The kinetic and frictional components of solid pressure are expressed by eqs T1-9 and T1-10. The shear and frictional viscosities of the particle phase are given by eqs T1-14 and T1-15. 2.2. BSD Drag Coefficient Model. In the previous study, the CSD drag model was proposed to describe the effect of mesoscale structures in fast fluidized beds.23 The local heterogeneous flow is resolved into three phases: the dense phase in the form of clusters, the gas-rich dilute phase, and the interface. Bubbles, as typical mesoscale structures, lead to a nonuniform drag solution in bubbling fluidized beds. Analogously, the local flow within the grid is separated into three subsystems: the emulsion phase, the bubble phase, and the interphase, as displayed in Figure 1. Here, it is assumed that the bubble phase only includes the gas phase and the distribution of particles in the emulsion phase is uniform. Hence, the BSD drag force consists of two contributions: one from the drag component in the emulsion and the other from the bubble-induced drag component, which is expressed as follows:

introduced. Compared with empirical relations and experimental results, the model agreed well with the available data. Wang et al.21 and Lv et al.22 established drag models on the basis of local structures, where local structural parameters are obtained by solving the mass and momentum conservation equations and empirical correlations of the bubble velocity and bubble diameter. The above models used the global hydrodynamic conditions to deduce the correlation between the drag coefficient and voidage, which neglected the dependence of the drag coefficient on the local information in the control volume. In this paper, a bubble-structure-dependent (BSD) drag model is developed on the basis of the previous clusterstructure-dependent (CSD) drag model.23 The heterogeneity within a computational cell caused by bubbles is described by resolving the overall system into subsystems. The pressure gradient due to particle−particle collision interaction and the bubble-induced added mass force are incorporated into calculation of the drag coefficient. The accelerations in the emulsion and bubble phases are also taken into account. By means of two-dimensional (2D) and three-dimensional (3D) simulations of bubbling fluidized beds, the bubble-based drag model can give better predictions of the experimental data.

2. MATHEMATICAL MODEL An Eulerian−Eulerian multiphase model is adopted in the present study. The main assumptions of the model are as follows: (1) mass transfer between the phases is neglected because of no reaction; (2) the diameter and density of the particles are uniform. The governing equations consist of the conservation equations of mass and momentum. The kinetic theory of granular flow is used to close the model, as reviewed by Gidaspow.24 Detailed information can be found in Table S1 in the Supporting Information (SI). 2.l. Hydrodynamic Model. The continuity equations for the gas and solid phases are given by eqs T1-1 and T1-2 (see the SI). The momentum conservation equations of the gas and solid phases are expressed by eqs T1-3 and T1-4, where τ and β denote the stress tensor and the gas−solid drag coefficient, respectively. To describe the fluctuating energy of the particles, granular temperature θ is introduced and defined as θ = ⟨C2⟩/3, where C represents the fluctuating velocity of the particles. The conservation equation of granular temperature is described by eq T1-5. Constitutive relations are used to close governing equations. The gas and solid stress tensors are expressed as eqs T1-6 and

βBSD =

εgFgs Uslip

=

εg Uslip

(neFde + nbFdb)

(1)

where nb and ne are the number densities of bubbles and particles in the emulsion phase. Fde and Fdb denote drag components in the emulsion phase and generated by bubbles, respectively. Detailed correlations are listed in Table S2 in the SI.28−31 To solve the mutiscale drag coefficient, six independent parameters are required, that is, the volume fraction of bubbles (δb), the voidage in the emulsion phase (εe), the superficial gas velocity in the emulsion phase (Uge), the superficial solid velocity in the emulsion phase (Use), the velocity of the bubble (Ub), and the diameter of the bubble (db), which can be obtained by solving a set of nonlinear equations. 15777

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2.2.1. Solid Momentum Equation of the Emulsion Phase. For simplification, it is assumed that the solid stress of the emulsion phase is neglected. The solid momentum equation in the emulsion phase along the flow direction is expressed as follows:

εg = (1 − δ b)εe + δ b

2.2.6. Stability Criterion. In this model, there are six independent variables with five hydrodynamic equations (eqs 2−6). Hence, the stability criterion is required to close the equations. In the bubbling-fluidized-bed system, particles tend to array themselves with minimal energy loss by drag and gas tends to form bubbles and select an upward path with least resistance. The extremum condition of energy dissipation consumed by the drag force is given as

neFde + nbFdb = (1 − δ b)(1 − εe)∇pg + (1 − δ b)(1 − εe) (ρs − ρg )(g + as,e) + ∇ps

⎡ (1 − δ b)Us,e (1 − δ b)Us,e ⎤ ⎥ /∂z as,e = ∂⎢ 1 − εe ⎦ ⎣ 1 − εe

(2)

Ndf =

(2a)

For each grid cell, the local information (ug, us, εg, and Δp/ Δz) is obtained from the TFM and used to solve six independent variables by employing five hydrodynamic equations (eqs 2−6) and a stability condition (eq 7). In this way, the solution of the BSD drag coefficient can be achieved. Here, the BSD drag coefficient is solvable in the range (εmf, εd). εmf denotes the minimum fluidizing gas volume fraction, and εd is the voidage where the ratio of the BSD and Gidaspow drag coefficients is 1. Once this interval is exceeded, the Gidaspow drag coefficient model is still adopted. The bubblebased drag model is used to describe the effect of unresolved bubbles in the grid. When the bubble size exceeds the grid size, the bubble can be directly solved by the grid. 2.3. Boundary Conditions and Numerical Solution Procedure. Initially, the particles are loaded in the static bed. The uniform gas velocity is set at the bottom inlet. The atmospheric pressure is prescribed at the top outlet. For the wall, there is no slip boundary condition for the gas phase and a partial slip boundary condition for the solid phase is selected with a specularity coefficient of 0.5.34 The simulations are conducted with a modified K-FIX code, which is a general purpose CFD code for modeling the hydrodynamics in fluid−solid systems and has been successfully used for the simulations of circulating fluidized beds and spouted beds.35,36 The time step varies from 10−4 to 10−6 s, and the maximum convergence residual of 10−3 is set. Simulations last for 40 s, and flow variables are time-averaged from 10 to 40 s when the quasi-steady state has been reached. It costs about 1 week using the BSD drag model.

(2b)

2.2.2. Momentum Equations of Gas in the Emulsion Phase and Bubbles. For the flow of gas in the emulsion phase and bubbles, we assume that the stress of gas is neglected. From the momentum equations of gas in the emulsion phase and bubbles at the steady state, we derive a pressure drop balance equation between gas in the emulsion phase and bubbles: δb neFde = nbFdb − δ bρg (ag,e − ag,b) + ∇pb (1 − δ b)εe

(3)

⎡ (1 − δ b)Ug,e (1 − δ b)Ug,e ⎤ ⎥ /∂z , ag,e = ∂⎢ ⎢⎣ ⎥⎦ εe εe ag,b = ∂(δ bUb × δ bUb)/∂z

(3a)

where ag,e and ag,b are the accelerations of gas in the emulsion phase and bubbles. The added mass force can be written as ∇pb = C b(1 − εe)δ b[εeρg + (1 − εe)ρs ] {ag,b − [εeag,e + (1 − εe)as,e]}

(3b)

33

where the coefficient Cb is calculated by C b = 0.5

1 + 2δ b 1 − δb

(3c)

3. RESULTS AND DISCUSSION 3.1. Comparison with the Experiment of Laverman et al. A 2D simulation is carried out on the basis of the experimental setup of Laverman et al.,37 as displayed in Figure 2. In their experiments, a pseudo-2D fluidization system with a height of 0.7 m, a width of 0.30 m, and a depth of 0.015 m was tested. Glass beads with a narrow particle size distribution of 400−600 μm were selected as fluidized particles. Digital image analysis and particle image velocimetry were used to obtain the bubble information and instantaneous emulsion-phase velocity profiles. Detailed operating conditions and physical parameters can be found in Table 1. From the study of Li et al.,38 the influence of the front and back walls was significant. Here, the frictional model proposed by Li and Zhang39 is applied to consider the wall effect, where the friction force and solid fluctuation energy supplied by the front and back walls are accounted for. A mesh-independent investigation is previously conducted to evaluate the impact of the grid size on the predictions. Uniform

2.2.3. Mass Conservation Equation of Gas. The gas flow within the grid cell can be characterized by two-phase structures including gas in the emulsion phase and bubbles. Considering the mass balance of gas, the velocity of the gas phase in the grid cell requires ug =

[(1 − δ b)Ug,e + δ bUb] εg

(4)

2.2.4. Mass Conservation Equation of Particles. The flow of the particles in the grid is characterized by dispersed particles in the emulsion phase. Hence, the solid velocity in the grid cell requires us =

(1 − δ b) Us,e (1 − εg)

1 (neFdeUg,e + nbFdbUbδ b) → minimum (1 − εg)ρs (7)

where as,e represents the accelerations of particles in the emulsion phase. The solid pressure gradient takes the empirical particulate stress model and can be calculated by32 ∇ps = 10−8.76[(1 − δ b)εe] + 5.43∇[(1 − δ b)(1 − εe)]

(6)

(5)

2.2.5. Overall Gas Volume Fraction. According to the definition of voidage in a grid cell, we have 15778

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grids with different sizes are adopted. Figure 3 displays the time-averaged axial profile of a solid volume fraction under different grid resolutions with two drag models. We can observe that, although there is a similar axial distribution of the solid concentration for the two drag models, the discrepancy between them is apparent. The prediction by the Gidaspow drag model with coarser grids differs from the others. When finer grids are employed, a similar curve can be captured for the two drag models, whereas the BSD drag model still obtains a better prediction using a coarser grid resolution. This implies that the BSD drag model can allow a coarser grid scheme without losing accuracy. Figure 4 shows a time sequence of the solid concentration using two different drag models. Both models can predict the movements of bubbles. The bubbles form near the inlet, pass through the bed with coalescence, and erupt at the bed surface. However, the difference between them can be clearly distinguished. In general, the BSD drag model captures a larger bubble diameter than the Gidapsow drag model, which is attributed to the fact that the interaction between the emulsion phase and bubbles becomes weak with consideration of the bubble effects and the coalescence of bubbles occurs easily. The gas tends to form a larger bubble to pass through the bed. An instantaneous local solid concentration fluctuation is shown in Figure 5. A low solid concentration means a high gas volume fraction. When the solid concentration is lower than 0.2, the gas is thought to pass through the bed in the form of bubbles. The peak of the concentration reflects a bubble frequency. It can be found that there is an intense oscillation at the center region. This indicates that the passing of bubbles mainly appears at the center of the bed with the bursting of bubbles. Figure 6 displays the measured and simulated axial solid velocity at two different heights. The prediction of the Gidaspow drag model is also shown. Roughly, both models can describe the nonuniform distribution of the solid velocity along the radial direction. The negative velocity is captured close to the wall. By comparison, the simulated result by the present model is in better agreement with the measured result, while the Gidaspow drag model overestimates the solid velocity. With the application of the BSD drag model to consider the bubble effects, the experimental data can be more accurately matched. To investigate the motion of bubbles, contour plots of voidage are converted to binary images. The software package Image-Pro Plus40 is applied to identify the bubbles and their coordinates. The gas volume fraction with a threshold value of 0.8 is used as a cutoff point to demarcate bubble boundaries. The bubble size is represented by the area of the bubble (Ab) in the 2D bubbling fluidized bed. Equivalent bubble diameters are calculated from the measured void areas:

Figure 2. Schematic diagram of a simulated 2D bubbling fluidized bed.

Table 1. Parameters for Numerical Simulation parameter particle density particle diameter reactor height reactor diameter inlet gas velocity gas viscosity gas density initial static bed height initial concentration of the particles grid size (Dx × Dz) restitution coefficient of particle−particle restitution coefficient of particle−wall specularity coefficient

experiment (Laverman et al., 2008) 2500 485 0.7 0.3 0.45 1.8 × 10−5 1.2 0.3

simulation

unit

2500 485 0.7 0.3 0.45 1.8 × 10−5 1.2 0.3 0.6

kg/m3 μm m m m/s Pa·s kg/m3 m

1.0 × 1.0

cm × cm

0.95 0.9 0.5

db,eq =

4Ab /π

(8)

Figure 7 presents the distribution of equivalent bubble diameters with the bed height together with the experimental results of Laverman et al.37 We can see that the BSD drag model predicts a profile similar to that of the measured data. Small bubbles are found at the bottom of bed, and with increasing bed height, the equivalent diameters of bubbles have a rising trend. Compared to the Gidaspow drag model, the BSD drag model gives larger bubble diameters, which are in better agreement with the experimental data.

Figure 3. Axial profile of the solid concentration with different grid sizes.

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Figure 4. Instantaneous concentrations of particles in bubbling fluidized beds: (a) Gidaspow drag model; (b) BSD drag model.

Figure 5. Instantaneous concentrations of particles as a function of time.

Usually, the bubble rising velocity mainly depends on the diameter of the bubbles. Figure 8 shows variation of the bubble rising velocity with the diameter of the bubbles. The profiles of the bubble rising velocity obtained from the correlation proposed by Hilligardt and Werther41 and the experiments of Laverman et al.37 are also plotted. It can be observed that, as the bubble diameter increases, the bubble rising velocity is promoted. Roughly, the trends are consistent. However, the difference is obvious. In this work, the bubble rising velocity is obtained by means of tracking the movement of the individual bubble center. The wall effects, coalescence, and breakup of bubbles have a strong influence on the bubble rising velocity, even if the bubble sizes are similar. More detailed insight into the bubble diameter and bubble rising velocity is still needed for a deep understanding of the dynamics behavior in bubbling fluidized beds. To evaluate the effects of solid pressure due to particle− particle collisional interaction and bubble-induced added mass force, the variations of solid pressure and added mass force with solid concentrations are displayed in Figure 9. It can be observed that there is a clear difference between them. With the solid volume fraction increased, the contribution of solid pressure becomes significant. This is attributed to the fact that a higher solid concentration promotes the probability of collision

Figure 6. Lateral profiles of the time-averaged axial solid velocity: (a) z = 0.105 m; (b) z = 0.245 m.

between particles. While the bubble-induced added mass force is weakened at a high solid concentration, it dominates at a low solid volume fraction, where the movement of bubbles is intense. The accelerations in the dense and dilute phases have a great influence on the predicted results in the simulation of risers.42 15780

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Figure 10. Profile of accelerations with the solid concentration. Figure 7. Distribution of the bubble diameter with the bed height.

Figure 11. Profiles of the solved bubble diameter and number density of the bubbles with the solid concentration. Figure 8. Variation of the bubble rising velocity with the bubble diameter.

Figure 12. Profile of the drag component with the solid concentration. Figure 9. Profile of the solid pressure gradient and added mass force with the solid concentration.

concentration increases, the accelerations in the emulsion and bubble phases decrease and tend to zero. The acceleration of the bubble phase is higher than that of the emulsion phase. This implies that the rate of momentum change in the bubble phase is more obvious. Figure 11 shows the distributions of the solved bubble size and number density of the bubbles with the solid

Here, the accelerations in the emulsion and bubble phases with the solid concentration are shown in Figure 10. The accelerations in the emulsion and bubble phases are directly calculated from local independent variables. We can find that both accelerations display a similar trend. As the solid 15781

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Figure 16. Size distribution of quartz sand particles.

Figure 13. Profile of the drag coefficient with the solid concentration.

Table 2. Parameters for 3D Experiment and Numerical Simulation parameter particle density mean particle diameter reactor height reactor diameter inlet gas velocity gas viscosity gas density initial static bed height initial concentration of the particles grid size (Dr × Dθ × Dz)

Figure 14. Experimental system of a bubbling fluidized bed mounted with an ECT sensor.

restitution coefficient of particle−particle restitution coefficient of particle−wall specularity coefficient

concentration. Here, the bubble density takes on a definition similar to that of the cluster density in the previous CSD drag model. The bubble size is directly solved by the momentum equation in the cell. It can be observed that, with increasing solid volume fraction, the bubble diameter is decreased and the bubble density is improved. This implies that, at a high solid volume fraction, more small bubbles appear in the emulsion phase. The effect of the bubbles is gradually weakened. There are two parts of contributions from the emulsion and bubble phases to the multiscale drag force in bubbling fluidized beds. Figure 12 shows the distribution of the two contributions with solid concentrations. It can be seen that, with increasing solid concentration, the drag component in the emulsion phase

experiment 2650 530 1.2 0.06 0.55/0.75 1.8 × 10−5 1.2 0.2

simulation 2650 530 1.2 0.06 0.55/0.75 1.8 × 10−5 1.2 0.2 0.6

unit kg/m3 μm m m m/s Pa·s kg/m3 m

0.25 × 0.05π × cm × rad × 1.0 cm 0.95 0.8 0.5

increases. On the contrary, the contribution from the bubbles decreases. This is due to the fact that there are small number densities of bubbles at a high solid concentration. However, when the solid volume fraction is lower, the bubble-induced drag component is comparable to that in the emulsion phase, which indicates that the bubble-induced drag component plays an important role in determination of the total drag force. The distribution of the drag coefficient with the solid concentration is shown in Figure 13. We can find that both drag coefficients have a rising trend with increasing solid

Figure 15. Schematic diagram of an ECT system. 15782

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Figure 17. Comparisons of simulated results (a and c) and measured data (b and d) at different operating velocities (z = 0.2 m).

volume fraction. The BSD drag coefficient is relatively low compared to the Gidaspow drag coefficient, which means that the drag coefficient becomes weak when the bubble effect is considered. Meanwhile, the BSD drag model obtains a zonal distribution of the drag coefficient, which is different from that by the Gidaspow drag model. This indicates that the drag coefficient not only depends on the local solid concentration but also is greatly influenced by other parameters such as local velocities. Schneiderbauer et al.43 conducted a comparative analysis of subgrid drag modifications in bubbling fluidized beds. The results revealed that, although different dependencies on the void fraction and slip velocity led to the discrepancy in drag modifications, the main features of bubbling fluidized beds can be well predicted by these subgrid modifications. Meanwhile, Milioli et al.44 pointed out that it was necessary to consider the dependency of the drag coefficient on the slip velocity. 3.2. Comparison with the 3D Experimental Results. 3.2.1. Experiment Using an Electrical Capacitance Tomography (ECT) Sensor. Experiments are conducted in a 3D bubbling fluidized bed, as illustrated in Figure 14. The column is 0.06 m in inner diameter and 1.2 m in height with a static bed height of 0.2 m. Uniform fluidizing air is injected into the column through the distributor at the bottom. The flow rate is measured using a flowmeter. Particle concentrations are measured by means of an ECT system. Figure 15 displays a schematic diagram of an ECT system. It consists of an electrical capacitance sensor with 12 measurement electrodes, a capacitance data acquisition system, and a data postprocesser. The electrical capacitance sensor is placed around a measurement section of the flow field. The capacitance signals acquired from the sensors are transmitted to a capacitance data acquisition system having a capturing rate of 140 frames/s. The linear back-projection image reconstruction method45 is employed to derive the image from the capacitance data.

Figure 18. Time-averaged profiles of the solid concentrations along the lateral direction: (a) ug = 0.55 m/s; (b) ug = 0.75 m/s.

Quartz sand particles are selected as the bed material in this experiment. The size distribution of the particles is displayed in Figure 16. The mean diameter is 0.53 mm, which belongs to Geldart B particles. The flow behaviors in bubbling fluidized beds are tested under two superficial gas velocities. The corresponding 3D simulations are also performed according to experimental conditions. Detailed parameters and conditions in the experiment and simulation are listed in Table 2. 3.2.2. Experimental and Simulated Results in a 3D Fluidized Bed. Figure 17 displays the contour plot of an instantaneous solid volume fraction obtained by simulation and experiment. It can be observed that the model prediction can capture a distribution similar to that of the measured result. There is an accumulation of particles toward the walls owing to the wall friction. The motion of bubbles in the center leads to a lower solid volume fraction. In comparison to the solid distribution at low velocity, a high inlet velocity enhances the lateral discrepancy of the solid concentration. The lateral profiles of time-averaged solid holdup under different drag models are shown in Figure 18. We can find that the solid concentration is low in the center regime and increases toward the wall. The BSD drag model can obtain a fair prediction with experimental data. The traditional Gidaspow drag model 15783

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underpredicts the solid concentration, which is related to the overlook of the mesoscale structure effect in the grid. The prediction by the BSD drag model can agree reasonably with the experimental data using coarse-grid resolution. From the profiles of solid holdup using two different inlet velocities, a similar trend can be observed. A higher operating velocity results in a significant lateral discrepancy in the solid distribution.

4. CONCLUSION A BSD drag coefficient model is developed to account for the effect of the bubbles and incorporated into the TFM to simulate the hydrodynamic characteristics in bubbling fluidized beds. The effects of solid pressure due to particle−particle collisional interaction and bubble-induced added mass force are evaluated. At a high solid volume fraction, the contribution of solid pressure is significant, while the bubble-induced added mass force dominates at a low solid volume fraction. A zonal distribution of the drag coefficient with the solid volume fraction is obtained by the BSD drag model, which implies that the drag coefficient depends not only on the local solid concentration but also on other parameters such as local velocities. To further verify the model, 3D simulations of bubbling fluidized beds under different operating velocities are carried out. An ECT sensor system is employed to measure the lateral distribution of the solid volume fraction in the bed. By comparisons with measured data, the BSD drag model gives a better prediction than the conventional drag model. The drag force plays a vital role in the simulation of bubbling fluidized beds with Geldart A particles. In future work, the BSD drag model will be extended to the prediction of fluidization with Geldart A particles in bubbling fluidized beds.

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ASSOCIATED CONTENT

* Supporting Information S

Details of the equations used in the CFD modeling listed in Tables S1 and S2 and a listing of the nomenclature used in this paper. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +0451 8641 2258. Fax: +0451 8622 1048. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was financially supported by the National Natural Science Foundation of China (Grants 51390494 and 21276056).

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