Numerical and Experimental Investigation of Liquid−Liquid Two

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Numerical and Experimental Investigation of Liquid-Liquid Two-Phase Flow in Stirred Tanks Feng Wang and Zai-Sha Mao* Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100080, China

The experimental data on the holdup of the dispersed phase in a Rushton impeller agitated stirred tank are presented. Experimental measurement is performed utilizing the sample withdrawal method to obtain the local dispersed-phase holdup in a laboratory-scale stirred tank under a variety of operating conditions. Three-dimensional turbulent two-phase liquid-liquid flow in the stirred tank is also numerically simulated by solving the Reynolds-averaged NavierStokes equations of two phases formulated by the two-fluid model. The turbulence effect is formulated using a simple two-phase extension of the well-known k- turbulence model by adding an extra source term generated from the presence of the dispersed phase in the turbulent kinetic energy transport equation of the continuous phase. A modified “inner-outer” iterative procedure is employed to model the interaction of the rotating impeller with the wall baffles. The modelpredicted mean velocity, turbulence characteristics of the continuous phase, and holdup profiles of the dispersed phase are compared against the published experimental data and the present measurements to validate the computational procedure, and good agreement is found up to a rather high overall dispersed-phase holdup case (30 vol %). 1. Introduction Dispersion of two immiscible liquids in mechanically stirred tanks is commonly encountered in chemical and biochemical processes, for instance, in liquid-liquid extraction, suspension polymerization, etc. The purpose of such an operation is to mix two phases and increase the interfacial area by intensifying dispersion of one liquid into another and to enhance consequently the interphase heat/mass transfer and the chemical reaction. Investigations previously conducted on liquid-liquid dispersion in stirred tanks were mainly focused on measurements of the drop size distribution,1,2 which is a crucial factor for the heat and mass transfer. However, the rate of transfer heavily relies on the breakage and coalescence of droplets3,4 and is controlled by the phase dispersion and distribution in the macroscopic twophase flow field and turbulence structure. In a recent study, a nonintrusive LDA technique was used by Svensson and Rasmuson5 to measure the velocity field of the continuous phase in a liquid-liquid stirred tank, showing that the turbulence in the system was damped by the presence of the dispersed phase. On the other hand, the holdup profiles of the dispersed phase were measured in several attempts. Armenante and Huang6 presented experimental data on the dispersed-phase holdup using the sample withdrawal technique. The same method was used by Okufi et al.7 and Skelland and Moon.8 Other methods, such as the conductivity probe or ultrasonic method, are also available for measurement of the holdup of liquid-liquid dispersion systems. In general, the detailed experimental measurements of hydrodynamics of liquid-liquid dispersion in stirred tanks are difficult, and the report of the holdup distribution in the whole stirred tank is scarce. In recent years, the computational fluid dynamics (CFD) method has developed rapidly and is applied to * To whom correspondence should be addressed. Tel.: (+86) 10 6255 4558. Fax: (+86) 10 6256 1822. E-mail: zsmao@ home.ipe.ac.cn.

predict the hydrodynamic characteristics of singlephase9-11 and two-phase flows, such as solid-liquid12-14 and gas-liquid15,16 systems, in stirred tanks. It is reasonable to expect that the CFD method is helpful in understanding the flow and dispersion behavior of liquid-liquid two-phase systems in stirred tanks. However, the progress so far on liquid-liquid systems is “little”, as reviewed by Kuipers and Swaaij.17 This situation may mainly be attributed to the additional complexity originated from the drop deformation, breakage, coalescence, and occurrence of flow inside the drops. In the numerical simulation performed by Vikhansky and Kraft,18 the Eulerian-Lagrangian approach was employed to model the liquid-liquid two-phase flow in a rotating-disk contractor, while an algebraic slip mixture model was adopted to simulate the banded liquid-liquid Taylor-Couette-Poiseuille flow by Zhu and Vigil.19 In regards to the stirred tank, only a few studies20 were on the numerical simulation of complex, three-dimensional, recirculating, and turbulent liquidliquid two-phase flow in stirred tanks. The challenge in simulating the multiphase flow in stirred tanks arises from the accurate representation of the action of the impeller and appropriate modeling of the interphase interaction between continuous and dispersed phases. A benchmark work in CFD simulation of the multiphase flow in stirred tanks was presented by Gosman et al.,12 who numerically simulated gasliquid and solid-liquid two-phase flow in stirred tanks driven by a standard Rushton disk turbine impeller. In the development of the mathematical formulation, the two-fluid model was adopted and the drag force and added mass force were taken into account for describing the momentum exchange between phases. The impeller region was treated as a “black box”, and the experimental data were imposed on the boundary of the volume swept by the impeller blades as the necessary boundary conditions. However, the lack of experimental data at the impeller region prevented the black-box approach from becoming a general predictive tool for chemical

10.1021/ie049001g CCC: $30.25 © 2005 American Chemical Society Published on Web 06/03/2005

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Figure 1. Configurations of the stirred tank and Rushton impeller. T ) 0.154 m, H ) T, B ) 0.1T, D ) 0.062 m, d ) 0.75D, b ) 0.25D, and w ) 0.2D.

engineering. In other publications13,14 with the two-fluid approach applied to solid-liquid stirred tanks, the inner-outer iterative procedure21 or the “sliding-grid” method22 was employed to represent the action of the impeller. However, only the drag force was considered, and the turbulence modification by the presence of particles was neglected in those two works. Wang and Mao23 improved the inner-outer iterative procedure by combining it with the “snapshot” approach proposed by Ranade and Van den Akker24 so as to keep the unsteady turbulence properties, which has been applied to simulate the flow in single-phase and two-phase stirred tanks.25-27 The major objective of the present paper is to present the experimental measurements on the local dispersedphase holdup in a laboratory-scale liquid-liquid stirred tank for the purpose of validating the numerical procedure. The three-dimensional mean flow field and turbulence characteristics of liquid-liquid two-phase flow in a baffled stirred tank with a six-bladed disk turbine impeller are numerically simulated using the Eulerian-Eulerian two-fluid model. The holdup profiles of the dispersed phase are also obtained at the same time. Simulation results are compared with experimental data obtained in this work and reported by Svensson and Rasmuson.5 2. Experimental Procedure Experiments are carried out in a vertical, cylindrical, flat-bottomed tank made of Perspex equipped with four equally spaced baffles with width B ) T/10. The impeller is a flat six-bladed disk (Rushton) turbine with diameter D ) 0.062 m. The configurations of the stirred tank and the impeller are shown in Figure 1. It is noted that a relatively larger impeller (D ) 0.4T) than the one traditionally used (D ) T/3) is chosen in this study in order to provide better dispersion. The clearance between the center plane of the impeller and the tank bottom varied from C ) 0.2T to 0.5T. Nine copper sampling tubes with 4.5 mm i.d. are mounted midway between two baffles vertically along the tank wall from near the bottom (z ) 0.1H) to near the free liquid surface (z ) 0.9H). Sample tubes can reach the tank interior from r/R ) 0.1 to 0.9 (Figure 2) if not prohibited by the sweeping impeller blade. The experimental setup is illustrated in Figure 2. The stirred tank is charged with tap water as the continuous phase and n-hexane as the dispersed phase. The total height of the two liquids is always equal to the diameter of the tank (T ) H). The overall dispersed-phase holdup

Figure 2. Experimental setup: 1, stirred tank; 2, impeller; 3, motor; 4, sample tube; 5, graduated cylinder; 6, peristaltic pump; 7, valve.

varies from 5% to 30% by volume. The motor runs at an impeller speed equal or higher than the critical one at which the dispersed drops start to appear in the bulk continuous-phase flow. The impeller speed is measured using a digital laser tachometer and is accurate within (5 rpm. In all runs of the experiment, 1 h is needed to achieve a steady state of dispersion in the present investigation, though Armenante and Huang6 claimed that the equilibration period was only 10-15 min. The peristaltic pump is activated after the steady-state dispersion is achieved, and the dispersion is partly circulated through the sampling tubes back to the tank to ensure the sampling system delivers the representative samples on holdup. Then the flow is directed to the graduated cylinder to collect about 15 mL of the liquid mixture. Because there is a significant difference in volatility between the two liquids, the sample is vaporized at 80 °C after centrifuging and weighting, so as to determine the local holdup of the dispersed light phase (n-hexane), which boils at 68 °C under atmospheric pressure (n-hexane is always the dispersed phase in the present experiment). The total quantity of the samples in each run is about 3% of the total system, and the bulk flow is deemed not changed significantly by the sample withdrawal. Adequate amounts of liquids are added into the tank to keep the liquid inventory of the system at the original values. The reproducibility of the holdup measurements is about 3.3%. The same procedure is repeated for various axial and radial positions, impeller speeds, impeller locations, and average concentrations of the dispersed phase. The impeller speed is varied in order to cover the range from the incipient dispersion to the nearly uniform dispersion state as judged by visual inspection. 3. Mathematical Model 3.1. Governing Equations. In the present work, a “two-fluid” model based on the Eulerian-Eulerian approach is adopted to model the liquid-liquid two-phase flow in a stirred tank because this method shows many computational advantages over the Eulerian-Lagrangian approach in the case of high dispersed-phase contents. The Eulerian-Eulerian “two-fluid” model assumes that the continuous and dispersed phases are all continua interpenetrating into and interacting with each other in the whole domain under consideration.28 The pressure field of the system is shared by two phases. As far as the interphase friction is concerned, the droplets are treated as rigid spheres without deformation of the

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interface, and the fluid flow inside the droplet as well as the processes of breakage and coalescence is neglected. For treatment of the turbulent flow of a liquid-liquid dispersion in stirred tanks, it is necessary to carry out the Reynolds decomposition and then take the time average of the governing equations. In the present work, the Reynolds time-averaging procedure results in the following time-averaged mass and momentum conservative equations:

∂(RkFk) ∂ (R F u + FkR′ku′ki) ) 0 + ∂t ∂xi k k ki

(1)

In the time-averaged eqs 1 and 2, the pressure fluctuation related terms and triple correlation terms are neglected. The viscous shear terms are dropped out in comparison with the turbulent shear terms. The interphase force term is calculated with the timeaveraged parameters and its fluctuation also omitted, as Ranade and Van den Akker24 suggested. Several turbulent fluctuation correlation terms appear and have to be related to known or calculable quantities via either algebraic relations or differential equations to close the momentum balance equations. In this work, the velocity correlations, namely, the Reynolds stresses, are modeled following the Boussinesq hypothesis as

u′kiu′kj ) -νkt

(

)

νkt ∂Rk σt ∂xi

where σt is the turbulent Schmidt number for the phase dispersion and was set to unity in this work after some numerical trials. The governing equations with effective viscosities are mathematically closed further with the aid of a simple extension of the widely tested standard k- twoequation turbulent model for single-phase flow to the multiphase system.12,24,26 The dispersed phase is assumed to affect the turbulence through interphase momentum exchange, and the contribution of the dispersed phase to the turbulence in the continuous phase is expressed by extra terms in the source terms Sφ in the transport equations for the turbulent kinetic energy k and the turbulent kinetic energy dissipation rate :

(6)

and the turbulent viscosity of the dispersed phase, µdt, is given by29

µdt ) Kµct

(7)

with

K)

Fdu′diu′di

(8)

Fcu′ciu′ci

A correlation of u′di to u′ci was derived from a Lagrangian analysis of the particle response to much larger eddies as12

u′di ) u′ci[1 - exp(-t1/tp)]

(9)

where t1 ) 0.41k/ is the mean eddy lifetime and tp is the particle response time obtained by the Lagrangian integration of the motion equation of a swarm of droplets moving through a fluid eddy of given velocity distribution with the expression

(3)

(4)

(5)

where φ can be either k or  and σφ is the model parameter describing turbulent dispersion of k. For details of the model formulation, refer to the previous paper by Wang and Mao.23 The turbulent viscosity of the continuous phase, µct, is expressed as

tp )

where k is the turbulent kinetic energy, δij is the Kronecker delta, and νkt is the kinematic turbulent viscosity of phase k. The correlation of the velocity and holdup fluctuations, R′ku′ki, which appears in both continuity and momentum equations, represents the transport of both mass and momentum by dispersion, respectively. The simplest way of modeling such correlations is to assume gradient transport as well, which gives

R′ku′ki ) -

)

µct ) CµFck2/

∂(RkFkuki) ∂P ∂ + (R F u u ) ) -Rk + RkFkgi + ∂t ∂xi k k ki kj ∂xi ∂ (R u′ u′ + ukiR′ku′kj + ukjR′ku′ki) (2) Fki - Fk ∂xi k ki kj

∂uki ∂ukj 2 + + kδij ∂xj ∂xi 3

(

µct ∂φ ∂ ∂ ∂ (RcFcuciφ) ) Rc + Sφ (RcFcφ) + ∂t ∂xi ∂xi σφ ∂xi

4Fddd 2FdCDRd|ud - uc|

(10)

The turbulent flow is unsteady because of the interaction of the rotating impeller with the stationary wall baffles, and the flow pattern becomes axisymmetrical once it is fully developed. Ranade and Van den Akker24 suggested that the time derivative terms in governing equations can be ignored without much error in most regions of the tank except for the impeller-swept volume. A snapshot of the flow can capture the main features of flow in the impeller region at that particular instant. In the present work, the flow field in the impeller-swept volume is simulated in a noninertial reference frame rotating with the impeller, and the time derivative terms are absent. Therefore, the resulting formulation of the mass and momentum conservation equations for phase k in the general form in a cylindrical coordinate system reads

1 ∂ ∂ 1 ∂ (rFkRkukrφ) + (F R u φ) + (FkRkukzφ) ) r ∂r r ∂θ k k kθ ∂z 1 ∂ RkΓφ,eff ∂φ ∂φ 1 ∂ rRkΓφ,eff + + r ∂r ∂r r ∂θ r ∂θ ∂ ∂φ RΓ + Sφ (11) ∂z k φ,eff ∂z

(

)

( (

) )

where Γeff ) Γlam + Γt. The body force term Fki is included as part of Sφ. 3.2. Interphase Interaction Term. The interphase interaction term, Fki, in eq 2 represents the momentum exchange between the continuous and dispersed phases,

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Figure 3. Effects of the interphase interaction terms on the predicted velocity components (Rav ) 0.05; N ) 540 rpm). Left: below the impeller (z ) 10 mm). Right: above the impeller (z ) 103 mm).

which consists of a linear combination of different momentum exchange mechanisms. Usually, three different forces are taken into account, namely, the drag force Fdrag, the added mass force Fam, and the lift force Flift. However, no convergent opinion on the contribution of these forces to the interaction of two phases has been reached so far. It is recommended by Gosman et al.12 that the drag force and the added mass force had much influence on the gas-liquid and solid-liquid two-phase flow in stirred tanks. Ljungqvist and Rasmuson30 reported that the added mass and lift forces had no remarkable influence on the slip velocities of the solidliquid flow in stirred tanks except in the impeller region. The analysis presented by Joshi31 suggested that the added mass and lift forces were important in numerical simulation of the multiphase flow in a bubble-column reactor. In the present study, the effects of these three forces are examined. The drag force between the continuous and dispersed phases can be expressed by24

3FcRcRdCD|ud - uc|(udi - uci) Fci,drag ) -Fdi,drag ) 4dd (12) where CD is the nondimensional drag coefficient depending on the droplet Reynolds number

Fcdd|ud - uc| Red ) µc,lam

(13)

and the local value of the droplet diameter is calculated as Nagata32 suggested:

dd ) 10-2.316+0.672Rdνc,lam0.0722-0.914(σg/Fc)0.196

tion of droplets was taken into account:

CD )

24 (1 + 0.1Rem0.75) Rem

(15)

with

Fcdd|ud - uc| , µm µd,lam + 0.4µc,lam Rd µm ) µc,lam 1 exp -2.5Rm Rm µd,lam + µc,lam

Rem )

(

) (

)

The added mass force arises from the acceleration of one phase relative to another and can be calculated from34

Fci,am ) -Fdi,am ) ∂uci ∂uci ∂udi ∂udi + ucj + udj -CamRdFc ∂t ∂xj ∂t ∂xj

[(

) (

)]

(16)

where Cam is expressed as35

Cam ) 0.5(1 + 2.78Rd)

(17)

For a rigid spherical particle moving in a nonuniform flow field, a lift force perpendicular to the average flow direction occurs. In the present study, the lift force is given by36

Fci,lift ) -Fdi,lift ) CliftRdFcijkklm(udj - ucj)

∂ucm (18) ∂xl

(14)

In the present work, the correlation used by Ishii and Zuber33 is adopted for the advantage that the deforma-

where Clift is the lift coefficient with a value of 0.5. In addition, the centrifugal force Fk,cen ) RkFk(ω × r) × ω and the Coriolis force Fk,cor ) 2RkFkω × uk should

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The governing equations presented above are solved by using the control volume approach and the SIMPLE algorithm.37 The pressure field is obtained using a pressure-correlation formula derived by combining the two continuity equations after being normalized with the respective phase density as Carver38 suggested. The discretized equations for physical variables are solved iteratively through an ADI algorithm. The nonlinearities in the phase momentum and turbulence equations are tackled with a standard under-relaxation technique. The convergence of the solution is judged by relative residuals R(φ), as practiced previously.23 For momentum and turbulence transport equations, the convergence criterion has to be decreased below 10-6, and for the continuity equation, it is 10-4.

Figure 4. Velocity vectors of continuous and dispersed phases (Rav ) 0.05; N ) 540 rpm). Left: continuous phase. Right: dispersed phase.

also be included in Fk when a noninertial reference frame is used. 4. Numerical Procedure In the present paper, the three-dimensional turbulent flow field and the holdup profiles of the dispersed phase in the liquid-liquid dispersion in a stirred tank are numerically simulated using an in-house CFD code developed on the basis of that originally used for gasliquid flow.23 Half of the tank is chosen as the computational domain because the time-averaged flow field is periodical in the azimuthal direction. The computational grid adopted to run all simulations consists of 36 × 36 × 90 meshes in radial, azimuthal, and axial directions, respectively. The action of the impeller is modeled using a modified inner-outer iterative procedure, which was detailed by Wang and Mao23 and is not addressed here for the reason of brevity.

5. Results and Discussion 5.1. Mean Flow Field. In this study, the computational procedure presented above is employed to simulate the flow field of the liquid-liquid dispersion in a stirred tank for which the experimental data are available.5 The stirred tank is a flat-bottom cylindrical tank with diameter T ) 0.14 m and is equipped with four vertical wall baffles with width B ) T/12. The liquid depth in the tank is H ) T. Agitation is provided by a six-bladed disk turbine impeller with diameter D ) T/3. The liquid-liquid model system is composed of an aqueous NaI solution (in 0.03% Na2S2O3) (Fc ) 1340 kg‚m-3; µc,lam ) 1.4 × 10-3 Pa‚s) as the continuous phase and silicon oil (Fd ) 940 kg‚m-3; µd,lam ) 11.0 × 10-3 Pa‚s) as the dispersed phase. The model predictions of the mean axial and radial velocity components of the continuous phase are illustrated in Figure 3, and the agreement with the data is quite reasonable. From the trial computation, it is observed that the effects of added mass and lift forces on the model predictions are of less significance, suggesting that the drag force shows the dominating effects

Figure 5. Comparison of the predicted mean velocities with experimental data for Rav ) 0.01 and N ) 540 rpm. Left: below the impeller (z ) 10 mm). Right: above the impeller (z ) 103 mm).

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Figure 6. Comparison of the predicted mean velocities with experimental data for Rav ) 0.10 and N ) 540 rpm. Left: below the impeller (z ) 10 mm). Right: above the impeller (z ) 103 mm).

Figure 7. Effect of the average holdup of the dispersed phase on simulated radial rms values.

on numerical simulation of the mean flow field of liquid-liquid dispersion systems in stirred tanks. The model predictions of the mean velocity field of the continuous and dispersed phases are illustrated in Figure 4 in the form of vector plots in the r-z plane. It is readily seen that the well-documented flow pattern generated by a disk turbine impeller is present; that is, two ring vortices exist above and below the impeller plane, and a high-velocity radial impeller stream with a slight tilt due to the lighter dispersed phase is also predicted. The velocity fields of the two phases are very similar to each other in most of the domains. However, the dispersed phase tends to float upward at the top of the tank close to the impeller shaft, as can be seen from the corresponding velocity vector plot, because the motion of the dispersed phase is more influenced by buoyancy in the top section of the tank. A comparison of numerically predicted radial variations of axial and radial mean velocities of the continuous phase versus the experimental data5 is shown in

Figures 5 and 6 corresponding to liquid-liquid dispersion systems with different average dispersed-phase holdup. Meanwhile, the predicted mean velocities of the dispersed phase are also presented. It should be noted that a “space-averaging” procedure proposed by Harvey et al.39 is adopted to the model predictions because the experimental data are obtained from averaging of the measurements in many revolutions over the time instants when the phase is detected, and the possibility of detection is directly proportional to the phase concentration. Therefore, the model-predicted velocity components at a radial location are averaged along the azimuthal direction with the local holdup as the weight:

uk )

|

∑Rkuk ∑Rk r,z

(19)

In Figures 5 and 6, it is clearly seen that, in general, the profiles of predicted radial variations of axial and

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Figure 8. Effect of the average holdup of the dispersed phase on simulated tangential rms values.

Figure 9. Effect of the average holdup of the dispersed phase on simulated axial rms values.

Figure 10. Measured axial variations of dispersed-phase holdup at C ) T/3 and Rav ) 0.10.

radial mean velocities are similar to those in the experimental data both below and above the impeller. The agreement between the two is satisfactory even for the case in which the average dispersed-phase holdup is up to 10%, though neither the breakage/coalescence mechanisms nor the interaction between droplets is considered. It can also be observed that, because the density difference between the continuous and dispersed phases is less than that in gas-liquid and solid-liquid flows, the droplets move by following closely the continuous phase. However, the axial velocity profiles of the dispersed phase are always above those of the

continuous phase, especially at the location above the impeller, because of the fact that the droplets are lighter in density. There are some discrepancies between the model predictions and measurements, though the general trend and order of magnitude of the axial and radial velocity profiles of the continuous phase are well captured by the computational procedure used in the present study. It could be attributed to the defects of the isotropic k- turbulence model adopted in this study when applied to complex anisotropic turbulent flow in stirred tanks. An improved multiphase turbulence

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model is required for more accurate simulation of the liquid-liquid two-phase flow in stirred tanks. 5.2. Turbulence Features. The influence of the dispersed phase on the turbulence structure has also been experimentally investigated and reported in terms of the root-mean-square (rms) of the fluctuating velocities by Svensson and Rasmuson.5 In the present study, the rms of the continuous phase in the system with various average dispersed-phase holdup is calculated through eq 3 given i ) j. The predicted radial, tangential, and axial rms values are depicted in Figures 7-9, respectively. It is clearly observed that the rms values of the continuous phase decrease with an increase of the average holdup of the dispersed phase, especially in the region close to the impeller, which indicates that the presence of a larger amount of the dispersed phase creates a stronger suppression on the turbulence of the continuous phase. This is in agreement with the conclusion of Svensson and Rasmuson.5 The predicted maximum rms values are 0.43 and 0.32 m/s corresponding to Rav ) 0.0 and Rav ) 0.25, respectively, in close comparison to 0.45 and 0.30 m/s obtained in their experiment. 5.3. Holdup Profiles of the Dispersed Phase. A typical holdup profile determined in the present experiment is shown in Figure 10. The axial profiles of dispersed-phase holdup at different radial locations are obviously different when the impeller stirring speed is 300 rpm. The holdup is largest at the top of the tank near the impeller shaft, suggesting the incomplete dispersion of the dispersed phase at lower stirring speed. The dispersed phase is stratified above the bulk dispersion as a clear liquid “pool” around the shaft where the dispersion is most difficult. For an impeller stirring speed of 400 rpm, which is higher than the critical impeller speed for complete dispersion, the homogeneity of the dispersed phase is significantly improved, as shown in Figure 10b. The local holdup contour maps of the dispersed phase in the vertical cross section are presented in Figure 11 corresponding to various operating conditions. As can be seen from the figure, more dispersed phase is drawn into the liquid bulk with the off-bottom clearance of the impeller increased from C ) 0.2T to 0.5T, suggesting that the impeller clearance plays an important role for the liquid-liquid dispersion. Despite the different average dispersed-phase holdup, a similar trend is observed. In the present study, the local holdup profiles of the dispersed phase are predicted numerically for the tank described in section 2. The influence of the different interphase force terms on the predicted holdup profiles of the dispersed phase is examined and shown in Figure 12. It can be observed that the interphase drag dominates over the added mass and lift forces, partly because of the similar physical properties of the continuous and dispersed phases. The calculated contour plots of the normalized dispersed-phase holdup are shown in Figure 13, and the contours are labeled with the value of the relative holdup, R/Rav. Overall, the predictions are able to capture roughly the experimental trends, as demonstrated by the measurements shown in Figure 11. For a lower impeller stirring speed (N ) 300 rpm), it can be seen from Figure 13a that the dispersed phase seems to accumulate around the impeller shaft at the top of the tank, corresponding with the experimental findings. With an increase of the impeller speed, as shown in

Figure 11. Contour plots of measured normalized holdup of the dispersed phase and Rav ) 0.10. Left: N ) 300 rpm. Right: N ) 400 rpm.

Figure 13b, the distribution of the dispersed phase inside the stirred tank becomes more homogeneous. The simulated axial and radial profiles of dispersedphase holdup are compared in Figures 14 and 15 with measurements obtained in the present experiment. The simulation results, in general, are in agreement with the experimental data especially for a higher impeller stirring speed. The comparison indicates that the computational approach adopted here is suitable and predicts semiquantitatively the distribution of the dispersed phase inside stirred tanks. However, the predicted results are noticeably above the experimental data, especially at the low impeller speed, which could be

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Figure 12. Influence of the different interphase force terms on the predicted axial variations of dispersed-phase holdup (Rav ) 0.10; N ) 300 rpm).

Figure 13. Numerically predicted contour plots of normalized holdup of the dispersed phase at C ) T/3 and Rav ) 0.10: (a) N ) 300 rpm; (b) N ) 400 rpm.

attributed to ignoring the breakage and coalescence of droplets. With an increase of the impeller speed, the agreement is improved significantly. The effects of the clearance of the impeller plane off the bottom of the tank on the holdup profiles of the dispersed phase are also numerically simulated. The simulated results are shown in Figure 16 for C ) 0.2T and 0.5T. As can be seen from the figure, the model predictions are in fair agreement with the experimental data. In Figure 17, the predicted axial variations of dispersed-phase holdup are compared with the measurements for the system with a high average holdup of the dispersed phase, Rav ) 0.30. Both calculation and measurements show a sudden increase at the top of the tank up to a maximum of 3.3, indicating the presence of a separated layer of the dispersed phase when the impeller stirring speed is 300 rpm. The axial and radial variations of dispersed-phase holdup become moderate when the impeller stirring speed is increased up to 400 rpm. In general, the simulation results and the measurements are in close agreement for the high average

Figure 14. Comparison of the axial and radial variations of dispersed-phase holdup between simulation and experimental data at C ) T/3, Rav ) 0.10, and N ) 300 rpm.

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Figure 15. Comparison of the axial and radial variations of dispersed-phase holdup between simulation and experimental data at C ) T/3, Rav ) 0.10, and N ) 400 rpm.

Figure 16. Comparison of the axial variations of dispersed-phase holdup between simulation and experimental results at Rav ) 0.10 and N ) 400 rpm. Left: C ) 0.2T. Right: C ) 0.5T.

holdup case, although the multidroplet interaction has not been incorporated into the mathematical formulation. 6. Conclusions and Remarks In the present study, experimental investigation and numerical simulation are carried out on the hydrodynamics characteristics of liquid-liquid two-phase flow in a stirred tank. The measurements of the dispersed-

phase holdup distribution demonstrate the effects of the stirring speed, impeller location, and average dispersedphase holdup. The three-dimensional flow field is simulated using the two-fluid approach with incorporation of the phase holdup fluctuation correlations appearing in the Reynolds time-averaged governing equations. An extension of the k- turbulence model is employed to describe the turbulence in the system, and the effect of the dispersed phase is accounted for by introducing source terms into

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Figure 17. Comparison of the axial variations of dispersed-phase holdup between simulation and experimental results with C ) T/3 and Rav ) 0.30.

the k and  transport equations. The model predictions reflect the correct trends demonstrated by the experimental data on the mean flow field, turbulence characteristics, and local holdup profiles, even in the cases with a large average holdup of the dispersed phase up to 0.30. Although predicted results are not in perfect agreement with experimental data, for example, the predicted holdup profiles of the dispersed phase at a low impeller stirring speed deviate from the experimental measurements, the computational procedure described in the present work performs rather well in predicting the flow field of the liquid-liquid dispersion and is considered reliable. The effects of different interphase forces on the simulation results of liquid-liquid flow are also examined, and it is found that the drag force plays the dominant role over the added mass and lift forces. Nevertheless, further improvement of modeling of the liquid-liquid two-phase flow in stirred tanks is desired to achieve quantitative agreement with experimental data. The drop size distribution model accounting for breakage and coalescence is required to be implemented to improve the prediction. The turbulence model also needs further development. Anisotropic turbulence models such as the algebraic stress model, etc., would be expected to be more accurate for describing the complex multiphase turbulent flow in stirred tanks.

Acknowledgment The authors gratefully acknowledge the financial support by the National Natural Science Foundation of China (Grant 20236050).

Nomenclature B ) width of the baffle (m) C ) clearance of the impeller plane off the bottom of the stirred tank (m) Cam ) added mass force efficient CD ) drag coefficient Clift ) lift force coefficient d ) diameter of the droplet (m) d ) diameter of the disk of the impeller (m) D ) diameter of the impeller (m) F ) interphase force (N) Fam ) added mass force (N) Fcen ) centrifugal force (N) Fcor ) Coriolis force (N) Fdrag ) drag force (N) Flift ) lift force (N) g ) gravity acceleration (m‚s-2) H ) height of the liquid in the stirred tank (m) k ) turbulent kinetic energy (m-2‚s-2) K ) proportional factor N ) impeller stirring speed (rpm) P ) pressure (Pa) R ) radius of the stirred tank (m) Red ) particle Reynolds number R(φ) ) relative residue r ) radial coordinate (m) r ) radial vector (m) S ) source term t ) time (s) T ) diameter of the stirred tank (m) t1 ) mean eddy lifetime (s) tp ) particle response time (s) u ) mean velocity component (m‚s-1) u′ ) fluctuation of the velocity component (m‚s-1) u ) velocity vector (m‚s-1) w ) height of the impeller blade (m) z ) axial coordinate starting from the tank bottom (m)

Ind. Eng. Chem. Res., Vol. 44, No. 15, 2005 5787 Greek Symbols R ) holdup Rm ) maximum packing holdup of the dispersed phase δij ) Kronecker delta  ) turbulent kinetic energy dissipation rate (m2‚s-3) ijk ) permutation tensor Γ ) diffusion coefficient (Pa‚s) µ ) viscosity (Pa‚s) λ ) Kolmogorov length (m) F ) density (kg‚m-3) σ ) interfacial tension (N‚m-1) σt ) Schmidt number θ ) azimuthal coordinate (rad) τ ) shear stress (Pa) φ ) general variable ω ) angular speed (rad‚s-1) Subscripts av ) averaged am ) added mass force c ) continuous phase d ) dispersed phase drag ) drag force eff ) effective i, j ) i and j directions k ) kth phase lam ) laminar lift ) lift force r, θ, z ) radial, azimuthal, and axial directions rms ) root mean square t ) turbulence

Literature Cited (1) Calabrese, R. V.; Chang, T. P. K.; Dang, P. T. Drop breakup in turbulent stirred-tank contactors. Part I. Effect of dispersedphase viscosity. AIChE J. 1986, 32, 657-666. (2) Zhou, G.; Kresta, S. M. Correlation of mean drop size and minimum drop size with the turbulence energy dissipation and the flow in an agitated tank. Chem. Eng. Sci. 1998, 53, 20632079. (3) Hinze, J. O. Fundamentals of the hydrodynamics mechanism of splitting in dispersion process. AIChE J. 1955, 1, 289295. (4) Wichterle, K. Drop breakup by impeller. Chem. Eng. Sci. 1995, 50, 3581-3586. (5) Svensson, F. J. E.; Rasmuson, A. LDA measurements in a stirred tank with liquid-liquid system at high volume percentage dispersed phase. Chem. Eng. Technol. 2004, 27, 335-339. (6) Armenante, P. M.; Huang, Y.-T. Experimental determination of the minimum agitation speed for complete liquid-liquid dispersion in mechanically agitated vessels. Ind. Eng. Chem. Res. 1992, 31, 1398-1406. (7) Okufi, S.; de Ortiz Perez; E. S.; Sawistowski, H. Scale-up of liquid-liquid dispersion in stirred tanks. Can. J. Chem. Eng. 1990, 68, 400-406. (8) Skelland, A. H. P.; Moon, L. J. Agitator speeds in baffled vessels for uniform liquid-liquid dispersions. Ind. Eng. Chem. Process Des. Dev. 1978, 17, 473-478. (9) Harvey, P. S.; Greaves, M. Turbulent in an agitated vessel, Part II. Numerical solution and model predictions. Trans. Inst. Chem. Eng. 1982, 60, 201-210. (10) Kresta, S. M.; Wood, P. E. Prediction of the threedimensional turbulent flow in stirred tanks. AIChE J. 1991, 37, 448-460. (11) Derksen, J.; Van der Akker, H. E. A. Large eddy simulation of the flow driven by a Rushton turbine. AIChE J. 1999, 45, 209221. (12) Gosman, A. D.; Lekakou, C.; Politis, S.; Issa, R. I.; Looney, M. K. Multidimensional modeling of turbulent two-phase flow in stirred vessels. AIChE J. 1992, 38, 1946-1956. (13) Micale, G.; Montante, G.; Grisafi, F.; Brucato, A.; Godfrey, J. CFD simulation of particle distribution in stirred vessels. Trans. Inst. Chem. Eng. 2000, 78A, 435-444.

(14) Montante, G.; Micale, G.; Magelli, F.; Brucato, A. Experiments and CFD predictions of solid particle distribution in a vessel agitated with four pitched blade turbines. Trans. Inst. Chem. Eng. 2001, 79A, 1005-1100. (15) Morud, K. E.; Hiertager, B. H. LDA measurements and CFD modeling of gas-liquid flow in a stirred vessel. Chem. Eng. Sci. 1996, 51, 233-249. (16) Bakker, A.; Van den Akker, H. E. A. A computational model for the gas-liquid flow in stirred reactors. Trans. Inst. Chem. Eng. 1994, 72A, 594-606. (17) Kuipers, J. A. M.; Swaaij, W. P. M. Application of computational fluid dynamics to chemical reaction engineering. Rev. Chem. Eng. 1997, 13, 1-118. (18) Vikhansky, A.; Kraft, M. Modelling of RDC using a combined CFD-population balance approach. Chem. Eng. Sci. 2004, 59, 2597-2606. (19) Zhu, X. Y.; Vigil, R. D. Banded liquid-liquid TaylorCouette-Poiseuille flow. AIChE J. 2001, 47, 1932-1940. (20) Tsouris, C.; Tavlarides, L. L. Breakage and coalescence model for drops in turbulent dispersions. AIChE J. 1994, 40, 395406. (21) Brucato, A.; Ciofalo, M.; Grisafi, F.; Micale, G. Numerical prediction of flow fields in baffled stirred vessels: a comparison of alternative modeling approaches. Chem. Eng. Sci. 1998, 53, 3653-3684. (22) Luo, J. Y.; Gosman, A. D.; Issa, R. I.; Middleton, J. C.; Fitagerald, M. K. Full flow field computation of mixing in baffled stirred reactors. Trans. Inst. Chem. Eng. 1993, 71A, 342-344. (23) Wang, W. J.; Mao, Z. S. Numerical simulation of gas-liquid flow in a stirred tank with a Rushton impeller. Chin. J. Chem. Eng. 2002, 10, 285-295. (24) Ranade, V. V.; Van den Akker, H. E. A. A computational snapshot of gas-liquid flow in baffled stirred reactors. Chem. Eng. Sci. 1994, 49, 5175-5192. (25) Sun, H. Y.; Wang, W. J.; Mao, Z.-S. Numerical study of the whole three-dimensional flow in a stirred tank with anisotropic algebraic stress model. Chin. J. Chem. Eng. 2002, 10, 15-24. (26) Wang, F.; Wang, W. J.; Mao, Z.-S. Numerical study of solid-liquid two-phase flow in stirred tanks with Rushton impeller. Part I. Formulation and simulation of flow field. Chin. J. Chem. Eng. 2004, 12, 599-609. (27) Wang, F.; Mao, Z.-S.; Shen, X. Q. Numerical study of solidliquid two-phase flow in stirred tanks with Rushton impeller. Part II. Prediction of critical impeller speed. Chin. J. Chem. Eng. 2004, 12, 610-614. (28) Ishii, M. Thermo-fluid dynamic theory of two-phase flow; Eyrolles: Paris, 1975. (29) Grienberger, J.; Hofmann, H. Investigations and modeling of bubble columns. Chem. Eng. Sci. 1992, 47, 2215-2220. (30) Ljungqvist, M.; Rasmuson, A. Numerical simulation of the two-phase flow in an axially stirred vessel. Trans. Inst. Chem. Eng. 2001, 79A, 533-546. (31) Joshi, J. B. Computational flow modeling and design of bubble column reactors. Chem. Eng. Sci. 2001, 56, 5893-5933. (32) Nagata, S. Mixing: principles and applications; Halsted Press: New York, 1975. (33) Ishii, M.; Zuber, N. Drag coefficient and relative velocity in bubbly, droplet or particulate flows. AIChE J. 1979, 25, 842855. (34) Anderson, T. B.; Jachson, R. A fluid mechanical description of fluidized beds. Ind. Eng. Chem. Fundam. 1967, 6, 527-539. (35) Van Winjgaarden, L. Hydrodynamic interaction between gas bubbles in liquid. J. Fluid Mech. 1976, 77, 27-53. (36) Auton, T. R. The lift force on a spherical body in rotational flow. J. Fluid Mech. 1987, 183, 199-218. (37) Patankar, S. V. Numerical heat transfer and fluid flow; McGraw-Hill: New York, 1980. (38) Carver, M. B. Numerical computation of phases separation in two fluid flow. J. Fluids Eng. 1984, 106, 147-153. (39) Harvey, A. D.; Lee, C. K.; Rogers, S. E. Steady-state modeling and experimental measurement of a baffled impeller stirred tank. AIChE J. 1995, 41, 2177-2186.

Received for review October 14, 2004 Revised manuscript received April 21, 2005 Accepted May 3, 2005 IE049001G