Novel Bubble–Emulsion Hydrodynamic Model for Gas–Solid Bubbling

Article pubs.acs.org/IECR

Novel Bubble−Emulsion Hydrodynamic Model for Gas−Solid Bubbling Fluidized Beds Xi Gao, Li-Jun Wang,* Cheng Wu, You-Wei Cheng, and Xi Li Department of Chemical and Biological Engineering, Zhejiang University, Hangzhou 310027, China ABSTRACT: A bubble−emulsion two-fluid model is proposed for the simulation of gas−solid flow in bubbling fluidized beds. This work proposes an analogous research program, according to the analogies in the flow regime and the mechanism between a gas−solid bubbling fluidized bed and a gas−liquid bubble column. The research includes a complete description of the constitutive relationship for interphase forces around the bubbles, a comprehensive investigation of correlations for bubble and emulsion phase properties, and an in-depth analysis of the mechanism of radial nonuniformity in a fluidized bed. The model is verified through comparison of simulated results with experimental data of both Geldart A and B particles. The model is conducted on a coarse grid which shows more potential for the simulation of commercial fluidized equipment.

1. INTRODUCTION Bubbling fluidized beds and risers have been widely used in industrial applications. In recent years, with the megaton-scale fluidized beds of fluid catalytic cracking (FCC), methanol to olefins (MTO), and indirect coal liquefaction (CTL) devices having been commissioned into production, the enlargement and enhancement of fluidized bed reactors have become the tendency of current technology development. Quantitative understanding of the hydrodynamics of fluidization is required for the design and scale-up of commercial reactors. In the 1960s, Davidson and Harrison1 proposed a slow bubble model, which laid the foundation for subsequent researchers to establish a common two- and three-phase reactor model.2 Since the 1990s, the two-fluid model (TFM) based on the kinetic theory of granular flow (KTGF) has now become the main tool for fluidization studies.3,4 KTGF-based TFM established on a solid physical basis has been widely accepted and applied in various fields after development by many researchers.4 However, the standard TFM has difficulty in reflecting the temporal−spatial multiscale structures in fluidized beds, which is not appropriate for large-scale industrial scale simulation directly. The nonlinear nature of the interaction between gas and solid makes the flow intrinsically unstable, and the distribution patterns of two existing phases are nonuniform and time-varying complex multiscale structures.5−7 A typical phenomenon is that, at low solid concentration (such as riser flow), the dispersed phase particles will spontaneously form a variety of amorphous aggregates (cluster), while at high solid concentration (such as bubbling fluidization), gas forms a large number of dispersed bubbles or voids, which makes the drag force between two phases significantly lower than the standard drag force calculated based on a uniform particle distribution assumption. Thus the solid holdup and velocity distribution is seriously distorted.8−10 Some authors proved that only a very small space grid and time step is required to capture a local nonuniform pattern, while the larger grid and step size will deny such a partial structure.11,12 Explicitly capturing the mesoscale structures demands that the grid size should be of the order of 2−4 particle diameters in bubbling fluidized beds (BFBs)9,13,14 or of the order of 10 particle diameters in circulating fluidized beds.12,15−19 Due to © 2013 American Chemical Society

the limitation of computational efficiency, it is difficult to simulate a industrial fluidized bed with a diameter greater than 1 m. Thus coarse grid simulation with suitable subgrid scale (SGS) models for the unresolved SGS structural effect is needed. In order to reflect the small-scale nonuniform structure on the constitutive relation, a variety of methods were proposed in the literature, such as the empirical equivalent drag model,8,20−25 the energy minimization multiscale (EMMS) method,6,26,27 and the filtered model;12,15 for a review, see Wang et al.13 From the literature review for bubbling fluidization simulation, some authors make some modifications to the original KTGF to make it more applicable, such as to account for the solid frictional stress;28 some others focus on the problem of the aggregation behavior of particles. The basic idea of coarse grid simulation is to develop a mesoscale model to reflect the local nonuniform structure on the mechanical constitutive relations.29 However, simulation of commercial bubbling fluidized beds still has a long way to go, because industrial bubbling fluidized beds are usually 5−12 m in diameter; even with coarse grid simulation using SGS models, the required number of grid cells and the corresponding times would definitely become prohibitive. Therefore, models that are developed to describe the hydrodynamics of very large systems will have to rely on closures for the bubble behavior, which has not received sufficient attention.30 The flow type in a BFB can be viewed as follows: the emulsion phase is considered as the continuous phase; the air bubble phase is considered as the dispersed phase. Kuipers and coauthors31,32 developed a Euler−Lagrange type model, the discrete bubble model (DBM): the bubbles are modeled as discrete elements and are tracked individually during their rise through the emulsion phase, which is considered as a continuum. Shi et al.33 extended the EMMS method to the bubbling fluidized bed (BFB), considering the presence of the bubble phase in the subgrid optimization to calculate the particle-based Received: Revised: Accepted: Published: 10835

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liquid phase. Its composition and rheological properties change with particle type and operating conditions. Second, there are no physical interfaces between the bubble and emulsion phases in the fluidized bed and gas can freely cross two phases, which causes the bubble size, bubble rising velocity, and even constitutive relations to be different from those in the liquid phase. Third, the boundary conditions of the emulsion phase at the wall are different from those of the liquid phase. These differences endow the two-fluid model of a BFB with some new contents that are quite different from those of the bubble column. The following work will carry out these new concepts and develop a novel bubble−emulsion model for bubbling fluidization.

local average drag force and input the EMMS drag to TFM. The work of Krishna and coauthors34−36 in the period from 1993 to 2001 is more enlightening; the authors first noted the flow similarity between fluidized beds and bubble column and conducted systematic comparative studies. They proposed a bubble−emulsion two-fluid model for the BFB similar to the bubble column; the emulsion phase is considered as a liquidlike continuum. The turbulence of the emulsion phase is modeled using a k−ε model; the bubble phase is assumed to be laminar. The model can simulate large-scale fluidized beds with diameters ranging from 0.1 to 6 m, and shows a greater potential for simulation of industrial fluidization devices than the KTGF-based TFM. However, their model only considered the drag force, and they found that the assumption of cylindrical axisymmetry prevents lateral motion of the dispersed bubble phases and leads to an unrealistic gas bubble holdup distribution wherein a maximum holdup is experienced away from the central axis. We believe that the problem is caused by the lack of a radial distribution mechanism in their model. This work focuses on the hydrodynamic modeling, and a time-averaged model has been developed for the simulation of a BFB. The authors believe that, for the simulation of a BFB, employing a two-fluid model similar to that of bubble columns is more rational than the standard TFM. First of all, the bubble and emulsion phases in fluidized beds are real rather than virtual; many experiments have proved that the distribution of particles in the emulsion phase or dense phase is generally uniform, although there are some different views. Thus the emulsion phase can be assumed to be a continuum to avoid the problem of handling particle agglomeration, and KTGF theory is no longer needed instead of the general continuum model, such as the k−ε turbulence model. The advantage of this approach is that it is easy to consider the effect of a long-term and large-scale average process, which is suitable for industrial scale applications. Second, fluidized bed and bubble column reactors have similar flow patterns and flow mechanisms, both belonging to bubble-driven multiphase flow. The research focus is on the interphase forces around the bubbles, such as drag, lift, turbulent dispersion, and wall forces. The general TFM for BFB focuses on the description of particles, and lacks the descriptions of bubbles, thereby making it difficult to reflect the main flow mechanism and laws. Additionally, in recent years, some new progress and new results obtained in bubble column simulations can be alternatively brought to the simulation of fluidized beds. Previous work37,38 reveals that work on the drag force has received enough attention while other forces have been overlooked, which makes it difficult to model the radial nonuniformity. We believe the problem is caused by the insufficient consideration of the mechanical mechanism. Attention must be paid to the lateral forces, because in the lateral direction the balance of these forces determines the distribution of bubbles, though they are much smaller compared with the drag force or the gravity force in the axial direction. The objective of the present work is to transplant some effective new method and new results of bubble column simulation to the study of fluidized beds, to try new ideas to solve the problem of fluidized bed simulation. The flow pattern and mechanism similarity between a fluidized bed and a bubble column provides the possibility for this analogous research. However, it must be borne in mind that the two reactors have obvious differences. First, the emulsion phase is a gas−solid mixture or inflatable flow of particles, which is different from a

2. MODEL FORMULATION 2.1. Resolution of Bubbling Fluidized Beds. When a gaseous phase is introduced uniformly through the bottom of a packed bed of particles, bubbles begin to appear for gas velocities exceeding the minimum bubbling velocity. In the bubbling fluidization regime, a small portion of the entering gas is used to keep the solids in suspension forming a emulsion phase while the major portion of the gas flows in the reactor in the form of bubbles. The emulsion phase gas voidage (δe) remains practically constant in the heterogeneous flow regime.36 This implies that a constant amount of gas equal to Umb is needed to keep the particles in suspension. The excess gas (Ug − Umb) rises up the column in the form of a bubble phase or a dilute phase.

Figure 1. Bubble−emulsion two-fluid model for bubbling fluidized beds.

As shown in Figure 1, the bubbling fluidized system is resolved into two phases or subsystem, i.e., the emulsion phase and the bubble phase. At a first approximation, the bubble phase is assumed only consisting of gas, omitting the contribution of particles. Also, the net gas exchange between the bubble phase and the emulsion phase is not considered in the present work. 2.2. Emulsion Phase Properties. The emulsion phase is a gas−solid mixture or inflatable flow of particles, which may be viewed as a pseudofluid with mean density ρe and viscosity μe.39 ρe = ρp (1 − δe) + ρg δe

(1)

μe = μg [1 + 2.5αp + 10.05αp2 + 0.00273 exp(16.6αp)] (2) 40

Abrahamson and Geldart found that, with the increase of the superficial gas velocity, the superficial gas velocity in the 10836

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emulsion phase (Uge) and δe change with Ug, and these changes can be represented for Geldart A and B solids by ⎛ δe ⎞3⎛ 1 − αmf ⎞ ⎛ Uge ⎞0.7 ⎟=⎜ ⎟ ⎟⎜ ⎜ ⎝ αmf ⎠ ⎝ 1 − δe ⎠ ⎝ Umf ⎠

⎯u are the volume fraction and the phase timewhere αk and → k averaged velocity of the emulsion phase (k = e) and the bubble phase (k = b). τk and τkRe represent phase stress tensor and phase ⎯→ Reynolds stress tensor, respectively. FIk represents the interphase momentum exchange term between bubble and emulsion phases. 3.2. Interphase Forces. The formulation of interphase forces is a key point for ⎯→ the accurate simulations of fluidized beds. The force term FIk in eq 9 can be calculated by decomposing it into five independent physical terms: ⎯→ ⎯→ → ⎯ ⎯⎯⎯→ ⎯→ ⎯ ⎯→ FIk = FD + FL + FVM + FW + FT (10)

(3)

In this work, the superficial gas velocity in the emulsion phase Uge is denoted as Umb; thus δe can be calculated from eq 3. Due to the inherent instability of the system to produce bubbles, the minimum bubbling velocity is difficult to measure, especially for Geldart A particles. Generally, the well-known experimental correlation of Abrahamsen and Geldart40 is used to calculate the minimum bubbling velocity. 2300ρg 0.126 μg 0.523 exp(0.716F45) Umb = Umf d32 0.8g 0.954 (ρp − ρg )0.934

They represent drag force, lift force, added mass force, wall lubrication force, and turbulent dispersion force, respectively. According to Sokolichin et al.,48 the contribution of added mass force becomes negligible for the simulation results. Thus, all forces except the added mass force were considered in the present model. The momentum exchange due to drag force was written as ρ ⎯⎯⎯→ ⎯⎯→ 3 ⎯u − → ⎯u |(→ ⎯ → ⎯ FDb = − FDe = αbαe e C D|→ b e ue − ub) 4 dbe (11)

(4)

where F45 is the mass fraction of particles having a diameter less than 45 μm. For Geldart B particles, commonly considered Umb equals Umf. Namely, the voidage of the emulsion phase, δe, is equal to αmf. 2.3. Bubble Phase Properties. Many experimental relations have been derived for the bubble diameter and bubble rise velocity in a fluidized bed; for a good review, see ref 41. Liu et al.42 found that the model proposed by Horio and Nonaka43 is probably the most accurate of those, because it makes an allowance for jetting and bubble splitting, as well as growth by coalescence. This model was also used by Shi et al.33 to calculate the averaged diameter of bubble swarms (dbe). Thus it is chosen in the present work. It is given by dbe = [−γm + (γm 2 + 4dbm/Dt )0.5 ]2 Dt /4

(5)

dbm = 1.49g −2[(Ug − Umf )πDt 2]0.4

(6)

γm = (2.56 × 10−2)(Dt /g )0.5 /Umf

(7)

where CD is the drag coefficient. The drag coefficient for bubble swarms is complicated due to their deformation, breakage, and coalescence; most of the correlations for bubble swarms are some kinds of extension with the phase fraction incorporated into ones for single bubbles. The drag coefficient used in the present work was proposed by Karamanev and Nikolov,49 and is expressed as ⎧ 24 (1 + 0.15Reb 0.687) Re ≤ 1000 ⎪ Re b ⎪ 0.413 CD = ⎨ + ⎪ 1 + 16.3Reb−1.09 ⎪ ⎩ 0.44 Re > 1000

where the Reynolds number is defined as ⎯u − → ⎯u |ρ /μ Re = d |→ b

Here Dt is the column diameter.

b

e e

e

where CL is the lift force coefficient. The physical mechanism of the lift force is so complex that there is a dispute regarding the magnitude of the lift force coefficient and even the sign of it (see ref 51). Joshi44 recommended that the lift force coefficient should be regarded as an adjustable parameter to match the experimental results. In the present model, the lift force coefficient was set as 0.1. Lubrication wall force is expressed by ⎯⎯⎯→ ⎯⎯→ FWb = − FWe ⎯u − → ⎯u |2 ⎡ ⎛ αeαbρe |→ d ⎞⎤ ⎯ b e =− max⎢0, ⎜C1 + C2 b ⎟⎥→ nr ⎢⎣ ⎝ db y ⎠⎥⎦

(8)

(14)

∂ ⎯u ) + ∇·(α ρ → ⎯ → ⎯ (αkρk→ k k k uk uk ) ∂t ⎯→ = −αk∇p + ∇·( τk + τkRe) + αkρk g ⃗ + FIk

be

This expression is close to the Schiller and Naumann correlation50 for low bubble Reynolds numbers. The lift force was formulated as ⎯→ ⎯ ⎯→ ⎯u − → ⎯u ) × ∇ × → ⎯u FLb = − FLe = −C Lαbαeρe (→ (13) e b e

3. MATHEMATICAL MODELING 3.1. Governing Equations of Bubble−Emulsion Model. In this work, separate mean transport equations of mass and momentum are solved for each phase and coupled through interphase transfer terms. The computational fluid dynamics (CFD) model was based on the Favre averaging twofluid model with the Eulerian−Eulerian approach; thus no volume fraction fluctuations are introduced into the continuity equations. For bubble columns, Joshi,44 Jakobsen et al.,45 Rafique et al.,46 Ekambara et al.,47 and Sokolichin et al.48 have given details pertaining to the formulation of governing equations for the axisymmetric two-dimensional flows. Using a similar procedure, the equations of continuity and momentum for bubbling fluidized beds can be written as ∂ ⎯u ) = 0 (αkρk ) + ∇·(αkρk→ k ∂t

(12)

⎯n is the normal vector from the wall. The lubrication where → r force coefficients taken by Chen52 are C1 = −0.01 and C2 = 0.05, which means that the wall force only acts within the distance of

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⎛ ρ ⎞ ρ εm = ⎜⎜αe e + αb b C t 2⎟⎟εe ρm ⎠ ⎝ ρm

5 times the diameter of bubbles in a bubble column. However, it is impractical in fluidized beds, because the bubble diameter in fluidized beds is much larger than that in bubble columns. C2 was assumed to be 0.003 in this work, so the wall force is acting only near the wall. The turbulent dispersion force is also important for the determination of radial profiles of gas holdup. It can be expressed as53−55 ⎯→ ⎯ ⎯→ ⎯ FTb = FTe = −ρe ub′ iue′i∇αb (15)

The turbulence response coefficient Ct is set as 1 as the gas volume fraction is larger than 6% in the present system. For more detailed information about Ct, please see ref 56. The model parameters in the turbulence model are provided in Table 1. Table 1. Constants in Turbulence Model

In eq 15, the covariance ub′ iue′i is related to the turbulent kinetic energy of both phases, which can be expressed as56

ub′ iue′i = 2C tke

(25)

σk

σε

σΘ

Cμ

C1ε

C2ε

C3ε

1.0

1.3

1.0

0.09

1.44

1.92

1.2

(16)

where Ct is the turbulence response coefficient, which stands for the ratio of the dispersed phase velocity fluctuations to those of the continuous phase. 3.3. Turbulence Modeling. The turbulence modeling is based on the mixture k−ε model. Behzadi et al.56 reported that dispersed k−ε turbulence models are limited in application to dilute systems and the turbulent fluctuations can no longer be assumed to be dominated by the continuous phase as the dispersed phase fraction increases. For another reason, some experiments suggest that, in bubble-driven systems, as the gas phase fraction increases beyond 6%, both phases tend to fluctuate as one entity, which suggests that the use of one set of equations for k and ε is appropriate. Indeed, for bubbly flow where ρe ≫ ρb, the mixture turbulent transport equations tend to those for the continuous phase, unless αb is very close to 1. The turbulent kinetic energy and the energy dissipation rate were calculated from their governing equations:

4. SIMULATION CONDITIONS AND STRATEGY 4.1. Test Cases. Two fluidized beds with particles belonging to Geldart A (see ref 57) and Geldart B (see ref 58) were selected as numerical test cases. The selection of these cases was driven by plentiful experimental information at different operating conditions. Simulation conditions and system properties are reported in Tables 2 and 3.

⎛ μt,m ⎞ ∂ (ρm k m) + ∇·(ρm k m⎯→ um) = ∇·⎜ ∇k m⎟ + Gk ,m − ρm εm ∂t ⎝ σε ⎠

Table 3. Summary of Parameters Used in the Simulation of Taghipour et al.58

Table 2. Summary of Parameters Used in the Simulation of Zhu et al.57 dp ρp ρg μg H0

dp ρp ρg μg H0

(17)

∂ (ρ εm) + ∇·(ρm εm ⎯→ u m) ∂t m ⎛ μt,m ⎞ ε = ∇·⎜ ∇εm⎟ + m (C1εGk ,m − C2ερm εm) σ k ⎝ ε ⎠ m

(18)

(19)

εb = C t 2εe

(20)

have been involved. The mixture quantities and properties can be related to those of the continuous and dispersed phases as follows: ρm = αbρb + αeρe (21) ⎯u + α ρ→ ⎯ C t 2αbρb→ b e e ue ⎯→ um = C t 2αbρb + αeρe μt,m =

(22)

(αeμet + αbμ bt C t 2)ρm αeρe + αbρb C t 2

⎛ ρ ⎞ ρ k m = ⎜⎜αe e + αb b C t 2⎟⎟ke ρm ⎠ ⎝ ρm

2.75 × 10−4 m 2500 kg/m3 1.225 kg/m3 1.7894 × 10−5 Pa·s 0.4 m

Umf εmf Ug Δt

Umf εmf Ug Δt

0.003 m/s 0.44 0.2, 0.3, 0.4 m/s 1.0 × 10−4−1.0 × 10−2

0.065 m/s 0.39 0.38, 0.46 m/s 1.0 × 10−4−1.0 × 10−2

The axisymmetric coordinate system was adapted in the simulation. Symmetric boundary conditions were applied at the center. The grids are created in a CAD program called GAMBIT 2.2.30. The uniform structured mesh is implemented with 28 × 492 grids (radial × axial) for Geldart A particles and 28 × 200 grids (radial × axial) for Geldart B particles. Other meshes are also used for the grid dependency analysis, which will be described in sections 5.1.1 and 5.2.1. 4.2. Simulation Code and Numerical Algorithm. The previously given conservation equations were solved by a finite volume method by the commercial CFD code FLUENT 6.3.26 (Ansys Inc.) in double precision mode. The pressure−velocity coupling was resolved using the SIMPLE algorithm.59 To avoid numerical diffusion, a second-order scheme was used for all variables except for the volume fraction equations, for which the QUICK discretization scheme was used. The additional terms of interphase forces were imbedded in FLUENT code with the help of user-defined functions (UDFs). Generally, it took about 60−100 s of simulated physical time to reach the steady state when almost all physical parameters did not change with time.

In deriving the above equations the relations k b = C t 2ke

6.5 × 10−5 m 1780 kg/m3 1.225 kg/m3 1.7894 × 10−5Pa·s 1.2 m

(23)

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4.3. Boundary and Initial Conditions. Initially, the column was filled with emulsion phase, that is αb = 0, up to the level that matches the bed height at minimum bubbling fluidization conditions measured in the experiment. Above this level, the initial gas holdup was αb = 1. Boundary Conditions at the Inlet. The boundary conditions at the inlet are set by prescribing a fixed inlet velocity related to the superficial velocity with the bubble fraction set to 1.0. A parabolic profiles velocity inlet is used. ubin = 2(Ug − Uge)[1 − (y/R t)2 ]

where I and dh are the turbulent intensity (set to 0.05) and the hydraulic diameter of the inlet, respectively. Boundary Conditions at the Outlet. Pressure outlet conditions are employed for both phases. Boundary Conditions at the Wall. A no-slip boundary is used at the wall. A standard wall function, common in single phase turbulent flows, is used to model the near-wall region.

5. RESULTS AND DISCUSSION 5.1. Simulation Results of Geldart A Particles. 5.1.1. Grid Size Independency. To confirm that the CFD results are independent of the mesh size, the simulations were performed for 28 × 492, 23 × 411, 20 × 352, 18 × 308, and 15 × 247 grids (radial × axial), which equal grid sizes of 5 mm × 5 mm, 6 mm × 6 mm, 7 mm × 7 mm, 8 mm × 8 mm, and 10 mm × 10 mm, respectively. The computed cross-sectional averaged solid volume fractions in the axial direction for different grid sizes are plotted in Figure 2. Simulation conditions and system properties are reported in Table 2. According to the results it can be concluded that, for all mesh cases, the correct hydrodynamic behavior could be captured. The mesh size 28 × 492 is selected as a base case for providing reasonably mesh independent results. 5.1.2. Axial and Lateral Solid Concentration Profiles. Comparison for the numerical results using the bubble− emulsion model and experimental results57 is illustrated in Figure 3. Figure 3 shows the simulated axial profiles of solid concentrations at different superficial gas velocities. It can be seen that the bubble−emulsion model can well predict the axial profiles of the solid fraction. Figure 4 shows the computed emulsion phase volume fraction contour and radial profiles of solid concentrations (αp = αe(1 − δe)) between simulation and experiments. The radial profiles show a core−annulus structure at all heights, and the simulation gives satisfactory agreement with experimental data. 5.2. Simulation Results of Geldart B Particles. 5.2.1. Grid Size Independency. A grid size independency study was also conducted for Geldart B particles. The simulations were performed for 28 × 200, 23 × 167, 20 × 143, 18 × 125, and

(26)

The inlet conditions for the turbulent kinetic energy and dissipation rate are estimated by k in =

3 (ubinI )2 2

εin = Cμ3/4

k in 3/2 0.07dh

(27)

(28)

Figure 2. Simulated axial profiles of solid fraction due to the effect of grid size.

Figure 3. Simulated axial profiles of solid fraction for (a) Ug = 0.2 m/s and (b) Ug = 0.4 m/s. 10839

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Figure 4. Simulated emulsion phase volume fraction and radial profiles of solid fraction for Ug = 0.3 m/s.

Figure 5. Simulated axial profiles of solid fraction due to the effect of grid size.

15 × 100 grids (radial × axial), which equal grid sizes of 5 mm × 5 mm, 6 mm × 6 mm, 7 mm × 7 mm, 8 mm × 8 mm, and 10 mm × 10 mm, respectively. The computed solid volume fractions for different grid sizes are plotted in Figure 5. Simulation conditions and system properties are reported in Table 3. According to the results it can be concluded that, for all mesh cases, the correct hydrodynamic behavior could be captured. The mesh size 28 × 200 is selected as a base case for providing reasonably mesh independent results. 5.2.2. Lateral Solid Concentration Profiles. Figure 6 shows the computed emulsion phase volume fraction contour and

radial profiles of solid concentrations between simulation and experiments. The radial profiles show a core−annulus structure at all heights, and the simulation gives acceptable agreement with experimental data considering the scatter of the experimental results. 5.3. Effects of Lift Force Coefficient. As mentioned earlier, the lift force is the most controversial among the interfacial forces. Several authors regard the lift force coefficient as an adjustable parameter to match the experimental results. Figure 7 shows the parameter sensitivity of the model predictions with respect to CL. Clearly, the lift force has a 10840

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Figure 6. Simulated emulsion phase volume fraction and radial profiles of solid fraction for (a) Ug = 0.38 m/s and (b) Ug = 0.46 m/s.

significant impact on the solid radial distribution: with the increase of the lift coefficient, the radial profile of the solid volume fraction becomes steeper, because a larger lift force acts on bubbles and pulls them to the column center. It appears that the results with CL = 0.1 fit better with the experimental data. 5.4. Formation Mechanism of Radial Nonuniformity. Due to the complexity of fluidization behavior, many of the inherent laws including the formation mechanism of radial nonuniformity have not yet been revealed. The basic characteristics of fluidized beds are the coexistence of gas gathered bubbles and particle gathered emulsions. The complexity lies in the nonuniformity and polymorphism of bubbles and emulsions. Nonuniformity can be described by the porosity distribution, which has not yet been theoretically quantitatively described. The porosity distribution is closely related to the bubble movement, so the understanding of the nonuniformity depends on the investigation of the bubble motion.

Figure 7. Parametric sensitivity of model predictions with respect to lift force coefficient (CL). 10841

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Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by the Dalian Institute of Chemical Physics, Chinese Academy of Science, and partially supported by the Natural Science Foundation of China (Nos. U1162125, 20906077), National 863 Plan Project (No. 2011AA05A205), and Fundamental Research Funds for the Central Universities (2013QNA4035).

■ Figure 8. Radial profiles of forces in bubble phase momentum equation.

In the fully developed region of a BFB, the radial component of the bubble phase momentum balance can be simplified as 0 = FLr + FTr + FWr duez dα b − 2C tρe ke dr dr ⎡ ⎛ αeαbρe (ub − ue)2 d b ⎞⎤ − max⎢0, ⎜C1 + C2 ⎟⎥ ⎢⎣ ⎝ db y ⎠⎥⎦

= −C Lαbαeρe (uez − ubz)

(29)

Greek Symbols

αk = volume fraction δe = emulsion phase gas voidage εk = turbulent dissipation rate, m2·s−3 μk = viscosity, Pa·s μk,t = turbulent viscosity, Pa·s ρk = density, kg·m−3 τk = phase stress tensor, N·m−2

The radial distributions of these forces from the computation are displayed in Figure 8 for Geldart A particle at an inlet velocity of 0.3 m/s. The results show that the radial nonuniformity is mainly determined by the lift force, the turbulent dispersion force, and the lubrication wall force. All the forces acting on the bubbles determine the movement of the bubble in the radial direction, and this creates a radial profile for the solid fraction. The wall force is limited to the near-wall region. Beyond the wall proximity, the balance between the lift and the turbulent dispersion force determines the radial nonuniformity.

τkRe = phase Reynolds stress tensor, N·m−2 Subscripts

6. CONCLUSION A novel bubble−emulsion two-fluid model was developed to simulate the hydrodynamics of Geldart A and Geldart B particles in bubbling fluidized beds. The model was formulated based on analogies in the flow regime and the mechanism between gas−solid bubbling fluidized beds and gas−liquid bubble columns. The following conclusions can be drawn from this study: 1. The mesh size independency study shows that the bubblebased model can be conducted on a coarse grid, which is more reasonable for simulating practical engineering devices. 2. The simulated axial and radial solid volume fractions fit well with experimental data obtained from Zhu et al.57 and Taghipour et al.58 under different operation conditions. 3. Through detailed analysis of the different contributions of forces in the lateral direction, we reveal that radial nonuniformity is mainly determined by the balance among the lift force, the turbulent dispersion force, and the wall force.

■

NOTATION CD = drag coefficient, dimensionless dbe = averaged diameter of bubble swarms, m Dt = column diameter, m Fk = interphase forces g = gravitational acceleration, m·s−2 kk = turbulent kinetic energy, m2·s−2 p = pressure, Pa Reb = bubbles Reynolds number uk = velocity, m·s−1 Umb = minimum bubbling velocity, m·s−1 Umf = minimum fluidization velocity, m·s−1 Ug = superficial gas velocity, m·s−1 Uge = superficial gas velocity in emulsion phase, m·s−1

■

b = bubble phase e = emulsion phase g = gas m = mixture of bubble−emulsion phases p = particle

REFERENCES

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AUTHOR INFORMATION

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