Numerical Simulation of LiquidSolid Flow in an Unbaffled Stirred Tank

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Ind. Eng. Chem. Res. 2008, 47, 2926-2940

Numerical Simulation of Liquid-Solid Flow in an Unbaffled Stirred Tank with a Pitched-Blade Turbine Downflow Xiangen Shan,† Gengzhi Yu,†,‡ Chao Yang,*,†,‡ Zai-Sha Mao,†,‡ and Weigang Zhang† Key Laboratory of Green Process and Engineering, Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China, and Jiangsu Institute of Marine Resource Exploitation, Lianyungang 222005, People’s Republic of China

Computational fluid dynamics (CFD) provides an efficient method for investigating highly complex fluid flows in mechanically stirred tanks, but less attention was given to unbaffled stirred tanks generated with axial impeller, which are frequently used in process industries. The present study is intended to evaluate the CFD predictions of key properties related to the mixing against measurements and to provide a detailed insight into the process. A three-blade 70° PBTD (Pitched Blade Turbine Downflow) was used to generate axial flows in a stirred tank, of which the local solid concentration profiles were numerically simulated. The liquid-solid system consisted of spherical silica beads in water. A method using “vector distance” in a Eulerian reference frame was proposed to resolve the impeller and the bottom of the tank of complex geometry. The predicted results of two-phase flows at various impeller speeds and solid holdups are presented. In addition, various Schmidt numbers and laminar viscosity coefficients of dispersion phase were tested to improve the predictions. In the experiments, an optical fiber probe was used to measure solid particle concentration, and the effects of impeller speed on solid holdup were examined. The solid particle distribution was affected with the impeller speed in the range of 113-173 rpm in the stirred tank. The radical concentration of solids particle presents different profiles at different axial cross sections. Good agreement between the experiments and simulations was observed. 1. Introduction In many industrial processes such as polymerization, fermentation, dissolution, crystallization, and catalysis reactions, many factors critically affect the suspension of particles in agitated vessels, including impeller speed, dispersion phase holdup, impeller off-bottom clearance, and so on. The chemical processes are frequently performed in a batchwise manner in baffled vessels operating under turbulent flow conditions, and they are heavily governed by the hydrodynamic and mixing characteristics, which, in turn, are dependent on both the vessel and impeller configurations. Many experimental investigations on hydrodynamics and mixing in vessels of different geometrical configurations agitated by various types of impeller have been performed in the past.1,2 Although baffled tanks are widely used for better mixing of solid particles and liquid, there are cases in which the use of unbaffled tanks may be desirable. Baffles are usually omitted in the case of very viscous fluids (those with a Reynolds number (Re) of 0; if point A is inside the impeller (located between two surfaces of a blade), as shown in the right sketch in Figure 1, and its distances to the two surfaces of the impeller are b a3 and b a4, the their dot product is b a3‚a b4 < 0; if A is just on the surface, their dot product is equal to zero. With this simple geometric rule, all the nodes on which the velocity components and pressure must be resolved can be identified, given that all surfaces of the impeller are already specified. Several three-dimensional integer arrays are defined to store the indices to flag if the velocity or pressure is going to be solved in the numerical program. Thus, the smooth blade surface is now approximated by a rough one. This would introduce some numerical errors to the simulation results; however, as the grid independence tests later indicated, the errors are expected to decrease as the grid is refined further. 2.2. Governing Equations. For the unsteady incompressible fluid flow under consideration, the Reynolds-averaged NavierStokes equations, combined with the standard k- turbulence model, were solved using the finite-volume discretization. In two-phase flow, if the mass and energy transports are not taken into account, the transient local equations based on mass and momentum conservation laws can be written as follows:14

Continuity equation: Navier-Stokes equation: -Rk

∂FkRk ∂ (F R u ) ) 0 + ∂t ∂xj k k kj

(1)

∂ ∂ (F R u ) + (F R u u ) ) ∂t k k ki ∂xj k k ki kj

∂Pk ∂ + (R τ ) + FkRkgi + Fki (2) ∂xj ∂xj k kij

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Ind. Eng. Chem. Res., Vol. 47, No. 9, 2008

Table 1. Diffusion Coefficients and Source Terms in the Governing Equations

a

Terms given in curly brackets ({‚‚‚}) are only used in a non-inertia reference frame.

The phase holdups comply with the normalization condition:

∑ R˜ k ) 1

(3)

To get momentum variation in the average sense, the instantaneous governing equations should be averaged using a time-averaged method. The Reynolds time-averaged method is applied to two-phase turbulent flow, which is similar to that in the single flow (i.e., the method called Reynolds-averaged Navier-Stokes equations (RANS) simulation). Transient terms are retained only when the time-dependent sliding-grid approach is adopted and disappear in steady-state simulations. Fr, Fθ, and Fz are body forces that include both gravity and the centrifugal Coriolis terms, which occur only

when a rotating reference frame is used:

Fr ) F(ω2r + 2ωuθ)

(4)

Fθ ) F(-2ωur)

(5)

Fz ) -Fg

(6)

In eq 2, τr, τθ, and τz are the row vectors that comprise the traceless part of the effective stress tensor τij. For a Newtonian fluid and under the effective (laminar-eddy) viscosity assumption, the six components of the effective stress tensor are given in explicit forms by

Ind. Eng. Chem. Res., Vol. 47, No. 9, 2008 2929

τrr ) 2µeff τθθ ) 2µeff

(

(

)

∂ur 1 ∂ur ∂ ∂uθ , τrθ ) µeff +r ∂r r ∂θ ∂r r

)

(

(7a,b)

)

∂ur ∂uz 1 ∂ur ur + , τrz ) µeff + (7c,d) r ∂θ r ∂z ∂r

τzz ) 2µeff

(

)

∂uz 1 ∂uz ∂uθ , τθz ) 2µeff + ∂z r ∂θ ∂z

(7e,f) Figure 2. Configuration of the impeller in this work.

with µeff ) µ + µt, where the eddy viscosity is expressed using the k- model:15

µt ) CµF

() k2

(8)

The transport equations for k and are16

∂ ∂ (F R u k) ) (F R k) + ∂t c c ∂xi c c ci µct ∂Rc µct ∂k ∂ ∂ Rc + k + Sk (9) ∂xi σk ∂xi ∂xi σk ∂xi

(

) (

)

∂ ∂ (F R u ) ) (F R ) + ∂t c c ∂xi c c ci µct ∂Rc µct ∂ ∂ ∂ Rc + + S (10) ∂xi σ ∂xi ∂xi σ ∂xi

(

) (

)

in which the values of the Schmidt numbers are given as σk ) 1.3 and σ ) 1.0. The source terms in the k and transport equations are

Sk ) Rc[(G + Ge) - Fc] S ) Rc

(11)

(k)[C (G + G ) - C F ] 1

e

2 c

(12)

the continuous phase. According to the analysis by Hjertager and Morud,18 the value of Cb was set at 0.02 in this simulation. The reference values of the model constants are the consensus ones: Cµ ) 0.09, C1 ) 1.44, C2 ) 1.92, σk ) 1.0, and σ ) 1.3. Some simulations were done using the modified values Ck ) 0.125 and C2 ) 1.60 that were suggested by Abujelala and Lilley19 and also adopted by Ranade and Joshi.20 In consideration of the strong vortex in the discharge zone, C1 was modified as follows:21

C1 ) 1.44 + Rf )

[

( )]

Ge )

∑d

Cb|F|(

∑

Rf ) 0

(for z < C - 1.5w, z > C + 1.5w)

Fci,drag )

4dd

CD ) (14)

24 (1 + 0.15Red0.687) Red

CD ) 0.44

(17) (18)

Fluid 1000 kg/m3 1.004 × 10-3 Pa s Particle 1970 kg/m3 80 µm

(19a) (19b)

in which Red is the Reynolds number of the particles:

value 0.3 m 0.42 m 0.08 m 0.16 m 0.06 m 0.08 m 0.03 m

(for Red < 1000)

(for Red g 1000)

Red )

Tank

density, Fd diameter, dp

3RcRdCD|ud - uc|(udi - uci) Fdi,drag ) -Fci,drag

Table 2. System Dimensions and Properties

density, Fl viscosity, µ (20 °C)

(16b)

According to their simulation, Ljungqvist and Rasmuson22 found that the drag force has a critical role in the liquid-solid flow in a stirred tank, whereas the influence of virtual mass and lift force can be neglected. Therefore, the drag force only was taken into consideration in this paper. The expressions of drag force used in this paper are given as follows:

(13)

where Cb is an empirical coefficient. When Cb ) 0, it means that the energy induced by the dispersed phase dissipates in the interface and has no influence on the turbulent energy of

tank diameter, T fluid depth, H impeller diameter, D impeller height, C blade height, Hb blade width, Wb shaft diameter, dl

(15)

where CD is the drag force coefficient:16

∂uci ∂xj

(udi - uci)2)1/2

parameter

(G + Ge)

1 ∂ ucθ u′cru′cθr ∂r r (for C - 1.5w < z < C + 1.5w) (16a)

where G is the dissipation function and Ge is the extra dissipation function that is due to the dispersion phase, which is mainly dependent on the drag force F between the continuous phase and the dispersion phase:17

G ) -FcRcu′ciu′cj

0.8RfFc

dd|ud - uc|Fc µc,lam

(20)

The governing equations can be summarized in Table 1, where

[( ) ( [ ( )

) ( )] ] [ ] [ ]

∂ucr 2 ∂ucz 2 1 ∂ucθ ucr 2 + + + + ∂r r ∂θ r ∂z 2 1 ∂ucz ∂ucθ 2 ∂ ucθ 1 ∂ucr µct r + + + µct + ∂r r r ∂θ r ∂θ ∂z ∂ucr ∂ucz 2 (21) µct + ∂z ∂r

G ) 2µct

2.3. Initial and Boundary Conditions. For a symmetric structure, only one-third of the tank was simulated and periodical

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2.3.1. Wall Function. The wall function was used on all solid walls, including the impeller, the wall of the tank, and the bottom of the tank. The no-slip condition (u ) V ) w ) 0) is appropriate for velocity components at solid walls. The normal component of velocity can simply be set to zero at the boundary. The implementation of wall boundary conditions in turbulent flows was according to Gosman and Ideriah’s proposal:23 when y+ > 11.63, it can be assumed that the flow was turbulent and the wall function was used, where

y+ )

Figure 3. Shape and characteristic dimensions of an unbaffled stirred tank.

1 F C 1/4k1/2yP µk 1 µ

(22)

and a near-wall flow was taken to be laminar if y+ < 11.63. 2.3.2. Top Free Surface. The free surface was treated as a no-shear surface that is characterized by the following conditions:

ucz ) udz ) 0,

∂φ )0 ∂z

(φ * ucz, udz)

(23)

Zero-normal derivative conditions, similar to eq 23, were also imposed on k and . When the differential stress model was used, the shear components of the Reynolds stress tensor vanished on the free surface, while zero-normal derivative conditions were imposed on all remaining components. 2.3.3. The Axis of Symmetry.

ucr ) ucθ ) udr ) udθ ) 0,

∂φ )0 ∂r (φ * ucr, ucθ, udr, udθ) (24)

3. Simulation Methods Figure 4. Mesh of an unbaffled stirred tank.

conditions were imposed on the planes θ ) 0 and θ ) 2π/3. The tangential velocity, -ωr, was imposed on the peripheral and bottom walls (which are still in the laboratory reference frame). The conventional linear logarithmic “wall functions” were used on all solid walls.

Figure 5. Fiber-optical probe system for solids holdup measurement.

The solid-liquid flows in an unbaffled stirred tank with an elliptical bottom and without wall baffles were numerically simulated. The diameter of the tank (T) is 0.3 m, and the shaft diameter (d1) is equal to T/10 throughout this study. The height of the tank is 0.45 m, and the height of the elliptical bottom is 0.12 m. The impeller is a three-bladed PBT (70°) with a diameter (D) of T/2. The water height (H) is equal to 0.42 m.


4. Experimental Section

Figure 6. Relationship between voltage and concentration.

The system dimensions and properties are listed in Table 2. The configuration of the impeller is shown in Figure 2. The shape and grid of the tank are shown in Figures 3 and 4. To study the solid concentration distribution in the stirred tank, the average diameter of solid particles was assumed to be 80 µm, the impeller off-bottom clearance was 2H/5, the solid holdup was given as Rd,av ) 0.005 (volume fraction), and the impeller speeds were varied as 113, 142, and 173 rpm.

Figure 7. Simulation results with different grids (N ) 173 rpm, C ) 0.16 m).

Some preliminary experimental determinations of uniformity of solid suspension in an unbaffled stirred tank were performed previously by the present authors.24 The experimental setup and procedure are briefly described in this section. Experiments were conducted in a stirred tank with an elliptical bottom, as described in section 3. The material of the tank and impeller is transparent Plexiglas with a thickness of 8 mm. To investigate the solid dispersion in axial and radial directions in the unbaffled tank, the experiments were performed as follows. First, the impeller off-bottom clearance and agitation speed were set and the solid holdup was set at Rd,av ) 0.005. The measurement of the local solids concentration, using a PC-6A fiber optic probe (manufactured by Institute of Process Engineering, Chinese Academy of Sciences), was conducted in an obscured environment to prevent daylight from interfering with the optical measuring technique. The impeller then was fixed according to the impeller off-bottom clearance. Figure 5 schematically shows the fiber-optic system in this work. The optical fiber probe was composed of two bundles of quartz fibers encased in a 3-mm-inside-diameter stainless steel probe tip. One bundle of fibers acts as light projectors, carrying light from a source and projecting it onto the passing swarm of particles. The other interspersed bundle acts as light receivers, transmitting the light reflected by the particles to a phototransistor that converts the light to an electrical signal. An amplifier

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Figure 8. Influence of the Schmidt number (σt) on the predicted local solids concentration (N ) 173 rpm, C ) 0.16 m).

Figure 9. Influence of µd,lam on the predicted local solids concentration (N ) 173 rpm, C ) 0.16 m).

increases the resulting signal to a voltage range of 0-5 V. The signal is then input to a personal computer via an analog/digital (A/D) converter. The relative error of the probe measurement is 1/200 of the full measurement range, and an accuracy of 0.5% for the concentration measurements is observed. For details of this probe, see the other author reports.25,26 Considering that the diameter and density of the particle used in this study were relatively small, the calibration was conducted in a 1-L beaker and the particles were assumed to be uniformly separated. As shown in Figure 6, the system with the solid concentration ranges from 0.002 to 0.2 was calibrated. A secondorder polynomial was used to fit the experimental data. The correlation between voltage and solid concentration is

Φ ) 1.36003 × 10-4 + 0.01381V + 0.00683V2 (25) For a good suspension in the tank, the agitation speed should be greater than or at least equal to the critical speed (Njs) required

for complete suspension, which is also known as the complete off-bottom suspension speed, as defined by Zwietering,27 corresponding to the state that no solid particles remain stationary on the bottom longer than 1-2 s. The critical agitation speed was determined to be ∼113 rpm in this work, and all measurements were then conducted at a speed above it at room temperature. The solid particle suspension and flow reached the stabilized state normally after 10 min of stirring. The signal sampling frequency was set at 2000 Hz, and 32 s were normally spent for each measurement. Three measurements were retrieved at every measuring point, and the average was recorded as the local solids concentration. After a measuring point was finished, the location of the probe was adjusted to the next one. 5. Results and Discussion 5.1. Computational Grid. In this study, four sets of gridss 30 × 50 × 70 (r × θ × z), 30 × 50 × 90, 30 × 50 × 120, and


Figure 10. Vector plots of flow field in a stirred tank (N ) 173 rpm, Rd,av ) 0.005, C ) 0.16 m: (A) continuous phase, r-z plane; (B) dispersion phase, r-z plane; (C) continuous phase, r-θ plane; and (D) dispersion phase, r-θ plane).

30 × 50 × 140swere chosen to determine the reasonable computational grid. The test simulations were conducted at N ) 173 rpm, Rd,av ) 0.005, and C ) 0.16 m, which are also typical experimental conditions. The predicted results using different grids are listed in Figure 7, from which the difference between 30 × 50 × 120 and 30 × 50 × 140 is very small. Thus, the 30 × 50 × 140 grid is sufficient for spatial computational accuracy at lower agitation speeds and is chosen for subsequent simulations.

5.2. Influence of Schmidt on Solid Particle Distribution. The Schmidt number σt is affected by the diameter of particle and the scale of turbulent flow, but few reports are available regarding the systematical experiments and theoretical models on σt. It is found that the simulation result was sensitive to σt; therefore, to eliminate this influence, a reasonable value of σt must be proposed. As shown in Figure 8, the influence of σt on the predicted local solids concentrations is very slight when the

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Figure 11. Contour plots of solids concentrations at different impeller speeds (Rd,av ) 0.005, C ) 0.16 m): (A) N ) 113 rpm, (B) N ) 142 rpm, and (C) N ) 173 rpm.

Schmidt numbers are 1.0 and 2.0. Therefore, the Schmidt number used in this paper is given as σt ) 1.0. 5.3. Influence of µd,lam on solid particle distribution. The determination of the laminar viscosity coefficient (µd,lam) of the dispersion phase is a hot topic, in regard to studying two-phase flow, especially in gas-solid fluidized beds. In this simulation,

µd,lam was set as a dynamic parameter, varying between 10-4 Pa s and 10-2 Pa s. The influence of µd,lam on the predicted local solids concentration is very small, as shown in Figure 9. Therefore, µd,lam was set to be that of the continuous-phase laminar viscosity coefficient, which was µd,lam ) µc,lam ) 10-3 Pa s.


Figure 12. Solid particle concentration distribution at different impeller speeds (Rd,av ) 0.1, C ) 0.16 m): (A) N ) 173 rpm, (B) N ) 500 rpm, and (C) N ) 800 rpm.

5.4. Computational Velocity Field. Figure 10 shows the twodimensional vector plots, representing the continuous phase and

dispersion phase, respectively, under the conditions of Rd,av ) 0.005, C ) 0.16 m. The figures depict one-half of a vertical

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Figure 13. Predicted and experimental solids holdup (N ) 113 rpm, Rd,av ) 0.005, C ) 0.16 m): (A) h ) 0.35 m, (B) h ) 0.28 m, (C) h ) 0.24 m, (D) h ) 0.20 m, and (E) h ) 0.14 m.

cross section of the stirred tank with the shaft and impeller in Figures 10a and 10b. The flow patterns for the two phases seem to be quite similar, naturally because the solid particles are small, with a terminal velocity of ∼4.85 × 10-2 m/s, and they follow the liquid flow very closely. To better reveal the underlying structure of the flow, the axial, tangential, and radial velocity components are all displayed in Figure 10. Secondary circulation loops in the flow field are revealed both above and below the impeller, in addition to the main circulation near the blade tips. These circulation regions

are the primary cause of the segregation of solid particles in the system, and their positions are dependent on the impeller speed. In the r-θ plane, a vortex is formed behind the impeller in both continuous and dispersion phases, because of relatively low pressure in this area. 5.5. Contour Profile of the Solids Concentration. Figure 11 shows the influence of impeller speeds on the solids concentration distributions. The same mass fraction of silica particles at Rd,av ) 0.005 was stirred at different impeller speeds. From the contour profiles of the solids concentration, a relatively



high concentration area exists below the impeller, which corresponds to the zone of low pressure in the flow field. The high concentration near the wall in the upper tank region in Figure 11 can be attributed to the circumferential flow and centrifugal force. A similar observation was made by Ochieng and Lewis10 and Derksen28 with a Lagrangian simulation approach. The solids collided with the wall, and, after losing their momentum, the liquid flow cannot carry them through. It is expected that the particles would lose momentum at this point and, instead of continuing with their initial trajectory, they had a tendency to settle. The concentration near the shaft is almost

zero, caused by the central vortex. With the increase of the impeller speed, the concentration below the impeller increased, and the zones of low concentration near the free surface and shaft shrank. Because the impeller speed varied in a narrow range, there is little difference among the contour profiles of the solids concentration in this simulation. For the system with high solids loading, the same mass fraction of silica particles at Rd,av )0.1 was stirred under different impeller speed ranges, as shown in Figure 12. The axial solids concentration profiles were predicted separately at speeds of 173, 500, and 800 rpm, to give some trend on

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uniformity of the solids suspension at higher solids loading. At 173 rpm, the solid distribution was such that a substantial proportion of the particles remained moving and disturbed at the bottom of the tank, although this velocity was far greater than Njs ) 113 rpm. A reasonable suspension was achieved with an impeller speed of 500 rpm. As shown in Figure 12, as the impeller speed increased, more and more particles were transformed to the upper tank region. However, the increase in the impeller speed from 500 rpm to 800 rpm did not result in higher homogeneity in the tank, which means that after the

impeller speed reaches a certain level, the efficiency that is involved in achieving higher homogeneity solely by increasing the impeller speed is quite low. This is rather contrary to the intuition of the engineers. A vortex existed in the lower impeller zone near the bottom of the tank, which can be attributed to the high shear stress of the continuous phase. 5.6. Comparison between Predicted and Experimental Results. The entire stirred tank can be divided into three zones, based on the position of the impeller: the upper zone, the impeller discharge zone, and the lower zone. The solids


concentrations at h ) 0.35, 0.28, 0.24, 0.20, and 0.14 m were selected to compare the predictions with the experiments at an impeller off-bottom clearance of 0.16 m. The contrast of solids concentrations in the stirred tank at impeller speeds of 113, 142, and 173 rpm is presented in Figures 13-15. From the bottom of the tank to the zone near the surface, five sets of radial solid particle concentrations were compared between the predicted and experimental results. In the upper zone with h ) 0.35 and 0.28 m, the solids concentration would increase with the radial coordinate. In the impeller discharge zone (h ) 0.24 and 0.20 m), the solid concentration increased with the radial coordinate at a slower rate. In the lower zone (h ) 0.14 m), the solids concentration decreased in the radial direction, because solid particles here had a tendency to gather more easily at the bottom of the tank. It is found that the trends of change in the simulation and experiments are similar; however, some discrepancies still exist, especially in the region close to the shaft and near the free surface. This is probably because the free surface condition was adopted in the simulation, but, in fact, a vortex was formed at the surface when the system was agitated. We even modeled the free surface using a concave curve, based on experimental observation, but no obvious improvement in the predicted flow field and the solid particle distribution was observed. Significant nonuniformity was also detected in the direct proximity to the wall, as well as at the shaft above the impeller. 6. Conclusions The water-silica two-phase system in an unbaffled elliptical bottom tank that was stirred with a pitched-blade impeller was numerically simulated. A method to address the complex boundary of the impeller by a regular fixed Eulerian computational grid was proposed, and the solid particle distributions in the stirred tank were predicted. The simulations were contrasted against the experiments on the solids concentration uniformity, using a PC-6A fiber-optic probe. The following conclusions can be drawn: (1) According to the contrast between the simulations and the experiments, the present numerical method to resolve an impeller of complex geometry was cost-effective, and good agreement between the predicted and experimental solid holdups was attained. The method of using “vector distance” to examine the complex impeller can be extended to predict the solid particle suspension, flotation, precipitation, and dispersion in stirred tanks. The approximation of a smooth surface with a zig-zag rough surface seems to work for two-phase flow with reasonable accuracy. The method, combined with the k- model, was tested to simulate the solid-liquid two-phase flow in an unbaffled tank with a solids loading of Rd,av ) 0.1, while the experimental part, for further validation, will be conducted in the future. (2) The solids concentrations varied largely in the tank, even when the agitation rate was well above the critical stirring speed. In the upper zone, the solids concentrations increased with the radial coordinate and the discrepancy of concentrations at the region close to the shaft and the wall seemed more obvious. In the impeller discharge zone, the solids concentrations increased moderately with the radial coordinate. In the lower zone, the solid concentrations decreased in the radial direction. (3) Increasing the impeller speed would improve the solid particle suspension. The local solids concentrations in the upper zone increased with the impeller speed, but the local solid concentrations in the impeller discharge zone decreased with the impeller speed.

(4) In the bulk tank, the trend of change in solids concentration was very consistent between the simulations and experiments. Because of the simplification of the numerical model in the simulations, the discrepancy in the zone near the free surface and close to the shaft still exists. While in the zone close to the tank wall, the predicted and experimental concentration profiles are quite similar. Nomenclature a ) length of macro axis (m) b ) length of minor axis (m) C ) impeller off-bottom clearance (m) C1, C2, Cu ) coefficients of the turbulent model Cb ) empirical constant of the Ge equation D ) impeller diameter (m) d ) particle diameter (µm) f ) mass diffusion term (kg/s) F ) force (kg m-2 s-2) Fc ) Coriolis force (kg m-2 s-2) Fr ) centrifugal force (kg m-2 s-2) G ) dissipation function (kg m-1 s-3) Ge ) extra turbulent kinetic energy (kg m-1 s-3) g ) acceleration due to gravity (m/s2) h ) axial height (m) H ) liquid height (m) k ) turbulent kinetic energy (m2/s2) N ) impeller speed (s-1) P ) pressure (Pa) R ) impeller radius (m) Red ) Reynolds number S ) source term t ) time (s) T ) diameter of stirred tank (m) u ) velocity (m/s) V ) volume (m3) w ) impeller width (m) yp ) distance (m) z ) axial position (m) Greek Symbols R ) volume fraction Γ ) diffuse coefficient (Pa s) ) turbulent energy dissipation rate (m2/s3) θ ) tangential (rad) µ ) viscosity (Pa s) ν ) kinematic viscosity (m2/s) F ) density (kg/m3) τ ) stress tensor (Pa) ω ) angular velocity (rad/s) Φ ) local concentration Subscripts av ) average c ) continuous phase d ) dispersion phase eff ) efficient i ) i-direction j ) j direction z ) z-direction k ) k-phase lam ) laminar flow RMS ) time-average fluctuation velocity r ) radial θ ) tangential

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z ) axial t ) turbulent flow Acknowledgment This work was financially supported by the National Natural Science Foundation of China (Nos. 20236050, 20306028, 50574081, 20676134) and the National Basic Research Program of China (Nos. 2004CB217604, 2007CB613507).

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ReceiVed for reView September 11, 2007 ReVised manuscript receiVed January 10, 2008 Accepted January 26, 2008 IE071225M

Numerical Simulation of LiquidSolid Flow in an Unbaffled Stirred Tank

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