Local Hydrodynamics Modeling of a Gas−Liquid−Solid Three-Phase

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Local Hydrodynamics Modeling of a Gas-Liquid-Solid Three-Phase Airlift Loop Reactor Xiaoqiang Jia, Jianping Wen,* Wei Feng, and Qing Yuan Department of Biochemical Engineering, School of Chemical Engineering and Technology, Tianjin UniVersity, Tianjin 300072, China

A three-dimensional (3D) transient model was developed to simulate the local hydrodynamics of a gasliquid-solid (GLS) three-phase airlift loop reactor (ALR) using the computational fluid dynamic (CFD) method, where the multiple size group (MUSIG) model was adopted to determine the bubble size distribution. Model simulation results such as the local time-averaged gas holdups and axial liquid velocities were validated by experimental measurements such as conductivity probes and laser doppler anemometer (LDA), under varied operating conditions, e.g., superficial gas velocities or initial solid loadings and at different locations of the ALR. Furthermore, the local transient hydrodynamics such as gas holdups, solid holdups, liquid velocities, as well as bubble size distribution were predicted reasonably by the developed model. Introduction Gas-liquid-solid (GLS) three-phase airlift loop reactors (ALRs) have been extensively used in recent years in fermentations, immobilized cell systems, and wastewater treatment processes because of their advantages in simple construction and operation, low investment and operational costs, very fine gas dispersion, definitely directed circulation flow, high mixing and mass transfer performance, and relatively low power requirements.1 The dispersion and interfacial heat- and masstransfer fluxes and the biochemical reaction rates are closely related to the fluid dynamics of the system through the liquidgas as well as liquid-solid contact area and the turbulence properties of the flow. The understanding of the complex multiphase flow phenomena will definitely facilitate the optimal design and scaleup of the ALR. The computational fluid dynamic (CFD) method has been used as a useful tool for understanding ALR flow behaviors in the past decade. Still, most of the literature is limited to twophase flows.2-6 Mudde and Van Den Akker presented 2D and 3D simulations of an ALR under steady-state conditions at low gas flow rates based on two-fluid model. Their investigations indicated that 3D simulation was more realistic compared with the 2D simulation.7 Van Baten and Krishna compared the hydrodynamics and mass transfer characteristics in an ALR and a bubble column (BC) using CFD techniques. They found that the ALR had higher circulation velocities at the same superficial gas velocity, which might be advantageous in many practical applications. They also concluded that CFD simulations can be powerful tools for the modeling and design of ALRs and BCs, especially in the context of describing the complex flow of the gas and liquid phases for different geometrical configurations.8 Blazˇej et al. carried out the simulation of two-phase flow in an ALR. The results showed that appropriate trends of the gas phase holdup and liquid phase velocities in the riser were followed, but the downcomer flow characterization was poor. The authors suggested the inclusion of more than one gas phase in future study, which led to the requirement of modeling bubble coalescence and breakup accompanied by a study of the bubble population in the ALR.9 * To whom correspondence should be addressed. Tel./Fax: 86-2227890492. E-mail address: [email protected].

However, reports of CFD modeling on GLS three-phase flows in ALRs are rather limited. Oey et al. treated the GLS three-phase flow as a GL pseudo-two-phase flow, in which the liquid phase consisted of the real liquid phase and the dispersed solid phase. The simplified model predicted the flow behavior well under low gas flow rates but deviated as soon as the gas fraction in the downcomer started to rise.10 The objective of this study is to develop a three-dimensional (3D) transient CFD model for simulating the local hydrodynamics of a GLS three-phase ALR, using the MUSIG model to determine the bubble size distribution. The model simulation results of the local time-averaged gas holdups and axial liquid velocities are to be validated by the experimental measurements under varied operating conditions and at different locations in the ALR. Furthermore, the dynamic behaviors of the local transient gas holdups, solid holdups, liquid velocities, and bubble size distributions are to be predicted by the developed model. Theory Governing Equations. In this work, a 3D transient CFD model is developed to simulate the local hydrodynamics of a GLS three-phase ALR. An Eulerian approach is adopted to describe the flow behavior of each phase. Liquid is considered to be the continuous phase while gas bubbles and solid particles are considered to be the dispersed phases. The continuity balance equation for each phase is

∂(FgRg) + ∇‚(FgRgug) ) 0 ∂t

(1)

∂(FlRl) + ∇‚(FlRlul) ) 0 ∂t

(2)

∂(FsRs) + ∇‚(FsRsus) ) 0 ∂t

(3)

where F is the density, R is the volume fraction, and u is the velocity vector of each phase.

10.1021/ie061697l CCC: $37.00 © 2007 American Chemical Society Published on Web 06/08/2007

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The momentum balance equation for each phase is

∂(FgRgug) + ∇‚(FgRgugug) ) -Rg∇p + ∂t ∇‚(Rg µeff,g(∇ug + (∇ug)T)) + FgRgg - MI,lg (4) ∂(FlRlul) + ∇‚(FlRlulul) ) -Rl∇p + ∂t ∇‚(Rl µeff,l(∇ul + (∇ul)T)) + FlRlg + ΜI,lg + MI,ls (5) ∂(FsRsus) + ∇‚(FsRsusus) ) -Rs∇p + ∂t ∇‚(Rs µeff,s(∇us + (∇us)T)) + FsRsg - MI,ls (6) where p is the pressure, µeff is the effective viscosity, g is the gravitational acceleration, and MI is the interphase momentum transfer force. The phasic volume fractions satisfy the compatibility condition:

Rg + R l + R s ) 1

(7)

Interphase Momentum Transfer. In this study, only drag force and lift force between the continuous phase and the dispersed phases are considered. The drag force exerted by the dispersed phase on the continuous phase is calculated as

MD,lg )

3 CD,lg F R |u - ul|(ug - ul) 4 dg l g g

(8)

MD,ls )

3 CD,ls F R |u - ul|(us - ul) 4 ds l s s

(9)

where CD is the drag coefficient and d is the diameter. The drag coefficient exerted by the gas phase on the liquid phase is obtained by the Ishii-Zuber drag model11

CD,lg ) 8 24 2 (1 + 0.15Rem0.687), min Eo0.5E(Rg), (1 - Rg)2 max Rem 3 3 (10)

(

(

))

where Rem is the mixture Reynolds number: Eo is the Eotvos

Rem )

Fl dg|ug - ul| µm

µm ) µl(1 - Rg)-2.5µ* µg + 0.4 µl µ* ) µg + µ l

(11) (12) (13)

number:

g(Fl - Fg)dg2 Eo ) σ

(14)

E(Rg) is the correction term:

E(Rg) )

1 + 17.67f(Rg)6/7 18.67f(Rg)

(15)

f(Rg) )

µl (1 - Rg)0.5 µm

(16)

The drag coefficient exerted by the solid phase on the liquid phase is obtained by the Wen Yu drag model:12

CD,ls ) Rs-1.65 max

24 (1 + 0.15(Re′) (Re′

0.687

),0.44

)

Re′ ) RsRe Re )

(17) (18)

Fl ds|us - ul| µl

(19)

The lift force acting perpendicular to the direction of relative motion of the two phases is given by

ML,lg ) CLFlRg (ug - ul) × (∇ × ul)

(20)

ML,ls ) CLFlRs (us - ul) × (∇ × ul)

(21)

where CL is the lift coefficient, the value of which is 0.5. The interphase momentum transfer between the two dispersed phases, virtual mass force, and turbulent dispersion force between the continuous phase and the dispersed phases are all neglected in this study, as adding of these forces cannot bring any obvious refinement to the current simulation results but only convergence difficulties. Turbulence Closure. The eddy viscosity hypothesis is assumed to hold for each turbulent phase. Diffusion of momentum in each phase is governed by an effective viscosity

µeff,l ) µl + µ T,l + µg,l + µs,l

(22)

µeff,g ) µg + µ T,g

(23)

µeff,s ) µs + µ T,s

(24)

where µ is the molecular viscosity, µT is the turbulence induced viscosity, and µg,l and µs,l are the particle (gas bubble and solid particle) induced viscosities of the liquid phase. The turbulence viscosity of the continuous phase is obtained by the k- model:

()

µT,l ) CµFl

k2 

(25)

The description of the standard k- model is

((

))

µ T,l ∂(Flk) + ∇‚(Flulk) ) ∇‚ µl + ∇k + Pk - Fl ∂t σk

((

(26)

))

µ T,l ∂(Fl)  ∇ + (C1Pk - C2Fl) + ∇‚(Flul) ) ∇‚ µl + ∂t σ k (27) where k is the turbulence kinetic energy, defined as the variance of the fluctuations in velocity (m2/s2), and  is the turbulence eddy dissipation, defined as the rate at which the velocity fluctuations dissipate (m2/s3). The turbulence production due to viscous forces Pk is the following:

2 Pk ) µ T,l∇ul‚(∇ul + ∇ulT) - ∇‚ul(3µ T,l∇‚ul + Fl k) 3 (28)

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The additional particle induced eddy viscosities of the continuous phase are the following:13

µg,l ) CµpFlRgdg|ug - ul|

fBV )

(29)

The turbulence viscosity of the dispersed phase is calculated

µs,l ) CµpFlRs ds|us - ul|

where fBV is the breakup fraction, with

(30)

π mi ) Fgdi3 6

µT,g )

Fg µ Fl T,l

(31)

µT,s )

Fs µ Fl T,l

(32)

The values of the constants are the standard ones: C1 ) 1.44, C2 ) 1.92, Cµ ) 0.09, Cµp ) 0.6, σk ) 1.0, σ ) 1.3. The value of the molecular viscosity of the solid phase is set to be the same as that of the water, since its variations do not bring obvious changes to the simulation results.14 Bubble Size Distribution. The bubble size distribution is determined by the MUSIG model, e.g., population balance model with coalescence and breakup effects.15 This MUSIG model considers that several bubble groups with different diameters di can be represented by an equivalent phase with the Sauter mean diameter dg. In this study, 10 bubble groups with the diameters ranging from 1 to 15 mm are considered based on the equal diameter discretization. The population balance is written as

∂(FgRg fi) + ∇‚(FgRg fiug) ) Si ∂t

(33)

where Si is the source term due to bubble coalescence and breakup; fi is defined as the ith group fraction, which has a relationship with the gas holdup as follows:

niVi ) Rg fi

(34)

where ni is the ith group number density and Vi is the ith group volume. The Sauter mean diameter can thus be written as follows:

fi

∑di

(35)

The source term Si for the ith group is given by the following:

Si ) Bbreakup - Dbreakup + Bcoalescence - Dcoalescence (36) The right-hand of the above equation is the birth and death source due to bubble breakup and coalescence, respectively. The net source due to breakup to the ith group is

Bi ) FgRg(

Bji fj - fi ∑Bij) ∑ j>i j 0.977). With the addition of the solid particles, the local time-averaged gas holdup decreases. It is clear that the effect of the changes in the initial solid loadings on the local time-averaged gas holdups shown in Figure 5 is smaller compared with the effect of the changes in the superficial gas velocities shown in Figure 3. Figure 6 exhibits the comparison between the model simulated results and the LDA measured data of the local time-averaged axial liquid velocities under different superficial gas velocities in the riser and the downcomer of the ALR. The model predictions agree well with the LDA measurements (R2 > 0.990). From this figure, it is obvious to see that the local timeaveraged axial liquid velocity increases with the increase in the superficial gas velocities. Larger value of the local time-averaged axial liquid velocity occurs at the central line of the ALR compared with the periphery. While in the downcomer, the

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Figure 7. Validation of model simulated local time-averaged axial liquid velocities at different axial positions (Y ) 0 m, Ug,sup ) 0.02 m/s, s,0 ) 2%).

Figure 8. Validation of model simulated local time-averaged axial liquid velocities under different initial solid loadings (Y ) 0 m, Z ) 0.4 m, Ug,sup ) 0.015 m/s).

direction of the local time-averaged axial liquid velocities is opposite to that in the riser because of the liquid recirculation as the result of the pressure difference between the upper and the lower end of the reactor. Figure 7 exhibits the comparison between the model simulated results and the LDA measured data of the local time-averaged axial liquid velocities at different axial positions in the riser and the downcomer of the ALR. The model predictions agree well with the LDA measurements (R2 > 0.994). The distribution profile of the local time-averaged axial liquid velocities along the X-axis in the riser is basically axial symmetry, which becomes flatter as the axial position in the riser increases. However, the downward local time-averaged axial liquid velocities in the downcomer of the ALR change little with the axial position, which also proves that the ALR has a better flow pattern compared with the traditional BC. The comparison between the model simulated results and the LDA measured data of the local time-averaged axial liquid velocities under different initial solid loadings in the riser and the downcomer of the ALR is illustrated in Figure 8. Good agreements between the model simulations and the experimental measurements are assured (R2 > 0.993). The increase in the initial solid loadings brings a negative effect on the local timeaveraged axial liquid velocities as shown in Figure 8; a possible reason for this might be the reduced formation of fast-rising large bubbles in the presence of solid particles.

From Figures 3-8, it is clear to see that the experimental measurements are reliable since the relative standard deviations of the experimental data are all within (3% as shown in the corresponding error bars, and the agreements between the model simulated results and corresponding experimental measured data of local time-averaged gas holdup and axial liquid velocity profiles in the riser and the downcomer of the ALR are very good. It is indicated that this developed CFD model can be adopted for simulating the time-averaged dynamic behaviors of the local hydrodynamic characteristics of the GLS threephase ALR. As the understanding of the local transient dynamic behaviors of the whole reactor is very important and the measuring of that is rather difficult, model predictions of the local transient hydrodynamics as well as the gas bubble size distribution are necessary for understanding and analysis of the factors which may affect the multiphase reaction processes such as liquid flow and mass transfer characteristics. Model Prediction of Local Transient Hydrodynamics. Figure 9 is the model prediction of the dynamic behaviors of the local transient hydrodynamics of the GLS three-phase ALR under the fixed superficial gas velocity and initial solid loading at the vertical section. The nine charts in Figure 9 visually exhibit the model predicted instantaneous dynamic changes of the local transient gas holdups, liquid velocities, and solid holdups from up to down and at different times of t ) 20, 110, and 200 s from left to right, respectively. Gas bubbles are carried downward by the liquid circumfluence in the downcomer and near the inner wall of the tube as well. A gas bubble plume is formed and moves back and forth in the riser resulting from the interphase momentum transfer forces. Large vortices alongside the bubble plume are engendered, which further results in the nonuniform distribution of the solid particles within the reactor. It can be concluded from the figure that it is the pseudoperiodical wiggle of the gas bubble plume that originates the dynamic changes in the liquid velocities as well as the solid holdup distribution. Furthermore, the flow pattern in the ALR is more regular compared with the bubble column. Moreover, model calculation found that the volume-averaged gas holdup in the riser is higher than that in the downcomer, while the volume-averaged solid holdup in the riser is lower than that in the downcomer, which may result in better performance for the multiphase reaction in the riser. Model Prediction of Bubble Size Distribution. In multiphase reaction processes, the dispersion and interfacial heatand mass-transfer fluxes, as well as the biochemical reaction rates, are closely related to the fluid dynamics of the system through the liquid-gas as well as liquid-solid contact area. Since the diameter of the solid particles is normally fixed, the dynamic changes of the gas bubble diameters will play an important role in the study of the multiphase reaction mechanisms. Then, this developed CFD model is used to predict the transient dynamic behaviors of the gas bubble size distribution, which further can be used as a way to evaluate the momentum, heat, and mass transfer performances of the GLS three-phase ALR. Figure 10 is the model predicted bubble size distribution of the GLS three-phase ALR under the fixed superficial gas velocity and initial solid loading at the vertical section. The 30 charts within Figure 10 respectively illustrate the model predictions of the size fractions of the 10 bubble groups 1-10 at different times of t ) 20, 110, and 200 s. Bubbles with the diameters ranging from 1 to 15 mm are divided into 10 bubble groups, where the average diameter of each group is 1.7, 3.1, 4.5, 5.9, 7.3, 8.7, 10.1, 11.5, 12.9, and 14.3 mm, respectively.

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Figure 9. Prediction of local transient hydrodynamic characteristics (Ug,sup ) 0.015 m/s, s,0 ) 2%).

It can be seen that small bubbles, viz. groups 1-5, prevail in almost the whole reactor with size fractions mostly larger than 0.2, while large bubbles, viz. groups 6-10, have size fractions mostly smaller than 0.2. It should be noticed that there are

locations with bubble group size fractions larger than 0.2 for group 10 in the downcomer, which might be the result of sustaining bubble coalescence as the obvious gas recirculation there.

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Figure 11. Prediction of volume-averaged transient gas bubble distribution (Ug,sup ) 0.015 m/s, s,0 ) 2%).

in the riser while the big bubbles mainly gather in the downcomer. This results in a more uniform bubble size distribution in the downcomer than that in the riser. The bubble size distribution is fluctuating drastically with time, which may be attributed to the presence of solid particles in the multiphase flow. The presence of solid particles may bring good effects on the gas-liquid mass transfer as the specific area of the gas bubbles increases. Further work on the CFD modeling of the real multiphase reaction processes coupling liquid flow, mass transfer, and intrinsic bioreaction kinetics in the GLS threephase ALR is underway. Concluding Remarks

Figure 10. Prediction of local transient gas bubble distribution (Ug,sup ) 0.015 m/s, s,0 ) 2%).

For better demonstration of the gas bubble size distribution in the ALR, the volume-averaged bubble size fractions of group 1-10 at t ) 20, 110, and 200 s in the riser and the downcomer are calculated and shown in Figure 11. As obviously can be seen in this plot, the small bubbles of groups 1-5 take a big part of all the gas bubbles (fraction sum > 0.87 in the riser and >0.57 in the downcomer). And, the small bubbles mainly gather

A 3D transient CFD model for simulating the local hydrodynamics of a GLS three-phase ALR reactor, with the MUSIG model adopted to determine the bubble size distribution, was validated to be reliable by time-averaged data of the experimental measurements. The local time-averaged gas holdups and axial liquid velocities increase with the increase in the superficial gas velocity and decrease with the increase in the solid loading. The central line of the riser has higher local time-averaged gas holdups and axial liquid velocities compared with the periphery. With the rise in the axial position, the distribution profiles of the local time-averaged gas holdups and axial liquid velocities along the X-axis in the riser become flatter. There is obvious liquid and gas recirculation in the downcomer. Local transient hydrodynamics of the three-phase ALR such as gas holdups, solid holdups, liquid velocities, and the bubble size distribution were also predicted reasonably by the developed CFD model. Gas bubbles are carried downward by liquid circumfluence in the downcomer and near the inner wall of the tube. The pseudoperiodical wiggle of the bubble plume originates the dynamic changes in the liquid velocities as well as the solid holdup distribution. Small bubbles prevail in the whole reactor while the bubble size distribution is more uniform in the downcomer. The study on the dynamic changes of the bubble size may be helpful for understanding the mass transfer characteristics of the three-phase reaction in the ALR. This model is under improvement for simulating three-phase reactions using immobilized microorganisms, where three-phase liquid flow, interphase mass transfer, and intrinsic bioreaction kinetics should be coupled together. Moreover, this developed model can be further applied for optimizing the design and construction of the ALR and the operation of the fermentation process, as well as facilitating the reactor scaleup strategies.

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Acknowledgment The authors wish to acknowledge the financial support provided by the Key National Natural Science Foundation of China (No. 20336030), Program for Science and Technology Development of Tianjin (No. 04318511120), Natural Science Foundation of Tianjin (No. 07JCZDJC01500), the Research Fund for the Doctoral Program of Higher Education (No. 20060056010), Program for New Century Excellent Talents in University, Program for Changjiang Scholars and Innovative Research Team in University, and Program of Introducing Talents of Discipline to Universities (No. B06006). Nomenclature B ) birth source (kg/m3‚s) Bij ) specific breakup rate (1/s) CB ) constant (dimensionless) CD ) drag coefficient (dimensionless) Cij ) specific coalescence rate (m3/s) CL ) lift coefficient (dimensionless) C1 ) constant (dimensionless) C2 ) constant (dimensionless) Cµ ) constant (dimensionless) Cµp ) constant (dimensionless) D ) death source (kg/m3‚s) d ) diameter (m) E(Rg) ) correction term (dimensionless) Eo ) Eotvos number (dimensionless) f ) size fraction (dimensionless) fBV ) breakup fraction (dimensionless) g ) gravitational acceleration (m/s2) h ) film thickness (m) k ) turbulence kinetic energy (m2/s2) MD ) drag force (N/m3) MI ) interphase momentum transfer force (N/m3) ML ) lift force (N/m3) m ) mass (kg) n ) number density (1/m3) Pk ) turbulence production (kg/m‚s3) p ) pressure (Pa) Re ) particle Reynolds number (dimensionless) Rem ) mixture Reynolds number (dimensionless) rij ) equivalent radius (m) S ) source term (kg/m3‚s) Sij ) cross-sectional area of the colliding (m2) t ) time (s) tij ) time required for coalescence (s) Ugo ) superficial gas velocity at the distribution plate (m/s) u ) velocity vector (m/s) ut ) turbulent velocity (m/s) V ) volume of bubbles (m3) Xjki ) mass fraction (dimensionless) Greek Letters R ) volume fraction (dimensionless) β ) constant (dimensionless)  ) turbulence eddy dissipation (m2/s3) ηij ) collision efficiency (dimensionless) µ ) molecular viscosity (Pa‚s) µeff ) effective viscosity (Pa‚s) µg,l ) gas particle induced viscosity of the liquid phase (Pa‚s) µs,l ) solid particle induced viscosity of the liquid phase (Pa‚s) µT ) turbulence induced viscosity (Pa‚s) F ) density (kg/m3) σ ) surface tension (N/m)

σk ) constant (dimensionless) σ ) constant (dimensionless) τ ) actual time during the collision (s) υ ) kinematic viscosity (m2/s) ξ ) dimensionless size of eddies in the inertial subrange of isotropic turbulence (dimensionless) Subscripts 0 ) initial condition g ) gas phase i ) ith bubble group j ) jth bubble group k ) kth bubble group l ) liquid phase o ) orifice s ) solid phase sup ) superficial Literature Cited (1) Wen, J. P.; Jia, X. Q.; Cheng, X. R.; Yang, P. Characteristics of three-phase internal loop airlift bioreactors with complete gas recirculation for non-Newtonian fluids. Bioprocess Biosyst. Eng. 2005, 27 (3), 193205. (2) Bagatin, R.; Andrigo, P.; Protto, L.; Meastri, F.; Di Muzio, F.; Masi, M.; Carraa, P. Fluid dynamics of gas-liquid airlift reactors through CFD packages. PVP computational technologies for fluid/thermal/structural/ chemical systems with industrial applications. ASME 1999, 397-1, 293303. (3) Cartland Glover, G. M.; Blazˇej, M.; Generalis, S. C.; Markosˇ, J. Three-dimensional gas-liquid simulation of an airlift bubble column reactor. Chem. Pap. 2003, 57 (6), 387-392. (4) Oey, R. S.; Mudde, R. F.; Van Den Akker, H. E. A. Numerical simulations of an oscillating internal-loop airlift reactor. Can. J. Chem. Eng. 2003, 81 (3-4), 684-691. (5) Van Baten, J. M.; Ellenberger, J.; Krishna, R. Hydrodynamics of internal air-lift reactors: Experiments versus CFD simulations. Chem. Eng. Process. 2003, 42 (10), 733-742. (6) Van Baten, J. M.; Ellenberger, J.; Krishna, R. Using CFD to describe the hydrodynamics of internal air-lift reactors. Can. J. Chem. Eng. 2003, 81 (3-4), 660-668. (7) Mudde, R. F.; Van Den Akker, H. E. A. 2D and 3D simulations of an internal airlift loop reactor on the basis of a two-fluid model. Chem. Eng. Sci. 2001, 56 (21-22), 6351-6358. (8) Van Baten, J. M.; Krishna, R. Comparison of hydrodynamics and mass transfer in airlift and bubble column reactors using CFD. Chem. Eng. Technol. 2003, 26 (10), 1074-1079. (9) Blazˇej, M.; Cartland Glover, G. M.; Generalis, S. C.; Markosˇ, J. Gas-liquid simulation of an airlift bubble column reactor. Chem. Eng. Process. 2004, 43 (2), 137-144. (10) Oey, R. S.; Mudde, R. F.; Portela, L. M.; Van Den Akker, H. E. A. Simulation of a slurry airlift using a two-fluid model. Chem. Eng. Sci. 2001, 56 (2), 673-681. (11) Ishii, M.; Zuber, N. Drag coefficient and relative velocity in bubbly, droplet or particulate flows. AIChE J. 1979, 25 (5), 843-855. (12) Wen, C. Y.; Yu, Y. H. Mechanics of fluidization. Chem. Eng. Prog. 1966, 62, 100-111. (13) Sato, Y.; Sekoguchi, K. Liquid velocity distribution in two-phase bubbly flow. Int. J. Multiphase Flow 1975, 2 (1), 79-95. (14) Schallenberg, J.; Enβ, J. H.; Hempel, D. C. The important role of local dispersed phase hold-ups for the calculation of three-phase bubble columns. Chem. Eng. Sci. 2005, 60 (22), 6027-6033. (15) Olmos, E.; Gentric, C.; Midoux, N. Numerical description of flow regime transitions in bubble column reactors by a multiple gas phase model. Chem. Eng. Sci. 2003, 58 (10), 2113-2121. (16) Luo, S. M.; Svendsen, H. Theoretical model for drop and bubble breakup in turbulent dispersions. AIChE J. 1996, 42 (5), 1225-1233. (17) Prince, M.; Blanch, H. Bubble coalescence and break-up in airsparged bubble columns. AIChE J. 1990, 36 (10), 1485-1499. (18) Chen, P.; Dudukovic, M. P.; Sanyal, J. Three-dimensional simulation of bubble column flows with bubble coalescence and breakup. AIChE J. 2005, 51 (3), 696-712. (19) Padial, N. T.; Vander Heyden, W. B.; Rauenzahn, R. M.; Yarbro, S. L. Three-dimensional simulation of a three-phase draft-tube bubble column. Chem. Eng. Sci. 2000, 55 (16), 3261-3273.

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(20) Krishna, R.; Van Baten, J. M.; Urseanu, M. I.; Ellenberger, J. Design and scale up of a bubble column slurry reactor for Fisch-Tropsch synthesis. Chem. Eng. Sci. 2001, 56 (2), 537-545. (21) Cui, Z.; Fan, L. S. Turbulence energy distributions in bubbling gasliquid and gas-liquid-solid flow systems. Chem. Eng. Sci. 2004, 59 (8-9), 1755-1766. (22) Feng, W.; Wen, J. P.; Fan, J. H.; Yuan, Q.; Jia, X. Q.; Sun, Y. Local hydrodynamics of gas-liquid-nanoparticles three-phase fluidization. Chem. Eng. Sci. 2005, 60 (24), 6887-6898.

(23) Liu, T. J.; Bankoff, S. G. Structure of air-water bubbly flow in a vertical pipe-II. Void fraction, bubble velocity and bubble size distribution. Int. J. Heat Mass Transfer 1993, 36 (4), 1061-1072.

ReceiVed for reView December 31, 2006 ReVised manuscript receiVed May 1, 2007 Accepted May 10, 2007 IE061697L