Mixed Solids Distribution in Stirred Vessels: Experiments and

Mar 30, 2007 - The distribution of dilute, mixed solids in Newtonian fluids in a baffled vessel stirred with multiple impellers has been studied exper...
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Ind. Eng. Chem. Res. 2007, 46, 2885-2891

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Mixed Solids Distribution in Stirred Vessels: Experiments and Computational Fluid Dynamics Simulations Giuseppina Montante* and Franco Magelli Department of Chemical, Mining and EnVironmental Engineering, UniVersity of Bologna, Viale Risorgimento 2, I-40136 Bologna, Italy

The distribution of dilute, mixed solids in Newtonian fluids in a baffled vessel stirred with multiple impellers has been studied experimentally and simulated by means of computational fluid dynamics. For the sake of experimental simplicity, fractions of two solids of different densities and equal size have been used. The experimental part, which was primarily intended to provide data for the validation of the simulation procedure, showed that the two phases act independently and that simple distribution superposition holds true. The computational approach is an extension of that already used for the calculation of monodisperse particle distribution in stirred vessels and is based on the solution of sets of RANS equations, one for each of the phases involved in the process coupled with the “homogeneous” k- model for closure. The sliding grid method has been selected for the simulation of the baffled vessel. As in the case of a single solid fraction, the momentum transfer between each solid phase and the liquid has been modeled only through the drag forces with the influence of turbulence being accounted for. As shown by comparison with experimental data, the simulation results are very realistic predictions of the solids distribution along the vessel vertical coordinate. 1. Introduction Treatment of solid-liquid mixtures in agitated vessels is a widespread operation in the chemical and allied industries. Among the aspects of interest for equipment design and process control, the knowledge of solids distribution inside the stirred suspension and the attainment of sufficient suspension homogeneity are important issues. While the solid phase is usually characterized by a certain size distribution and possibly different physical properties, this topic has generally been tackled by considering monodisperse particles only, sometimes of a suitable average sizesthe exceptions to this simplified approach being rather limited.1-4 Computational fluid dynamics (CFD) methods have started being used to predict the behavior of solid-liquid stirred vessels. Either Eulerian-Lagrangian or Eulerian-Eulerian approaches have been applied, with the use of black box techniques in the first instances, and entirely predictive procedures have been developed in the past few years.5-15 So far, Reynolds averaged Navier-Stokes (RANS) based multifluid model approaches have led to acceptable predictions of solids distribution in stirred vessels in most cases, while, more recently, complex simulations based on large eddy simulations (LES)16,17 or direct numerical simulations (DNS)18 of the liquid flow and the solution of particle motion equations (Lagrangian tracking) have provided advances in solid-liquid and particle-particle interactions modeling. However, these last approaches seem to be more suited to study the interphase interactions than to be applied to industrial cases, mainly due to their great computational requirements. The case of suspensions with particle size distribution has been addressed by Sha et al.,9 who used a multifluid model frame. However, they did not provide a direct comparison of the simulation predictions with experimental data, while this * To whom correspondence should be addressed. Tel.: +39 051 2090406. Fax: +39 051 6347788. E-mail: giusi.montante@ mail.ing.unibo.it.

aspect is crucial to validate the adopted models and procedures. Most recently, Ochieng and Lewis15 performed fully predictive Eulerian simulations of mono- and polydisperse nickel particles in stirred vessels. The effect of particle size and solids loading on the solids distribution was evaluated from the simulations results, but the modeling approach was validated only for dilute conditions and monodisperse solid fractions. This work is aimed at investigating the distribution of mixed solids in dilute stirred suspensions. To make the experiments easier, pairs of solid particle fractions having the same size but rather different densities have been considered, anyhow exhibiting significantly different settling behaviors. The multiphase modeling approach adopted in the present work is based on the use of three different sets of Reynolds averaged Navier-Stokes (RANS) equations: one for the liquid phase (continuous) and one for each of the two dispersed solid fractions. The suitability of the modeling approach will be assessed by comparing the predicted vertical profiles of solids concentration with the corresponding experimental data, thus also evaluating the model assumption of negligible interactions between the solid types and the influence of system turbulence on the coupling among the solid and liquid phases. 2. Experimental Section The experimental part of the investigation was carried out in a cylindrical, flat-bottomed Perspex vessel (T ) 23.2 cm diameter, H ) 4T height, 39.6 L volume), similar to that used in a previous investigation.19 The vessel had a lid and was equipped with four vertical T/10 baffles. Agitation was provided with four, evenly spaced Rushton turbines (D ) 7.87 cm, D/T ) 0.34) mounted on the same shaft; the lower impeller was at a distance T/2 above the base. Though this impeller configuration may not be the optimal one for process requirements, the main objective of this work (i.e., the study of equipment behavior and simulation with mixed solids) is unaffected by this choice. The liquids used were water and an aqueous solution of poly(vinylpyrrolidone) (Newtonian behavior, viscosity 0.9 and 5.3

10.1021/ie060616i CCC: $37.00 © 2007 American Chemical Society Published on Web 03/30/2007

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Table 1. Properties of the Glass and Bronze Particles Suspension: Case 1

liquid glass bronze

F (kg/m3)

µ (mPa s)

997.5 2450 8410

5.33

dp (µm)

Cav (g/L)

327 327

1.50 1.50

N (rpm)

Ut (cm/s)

Us (cm/s)

1.40 5.94

1.34 5.70

Ut (cm/s)

Us (cm/s)

4.40 0.71

2.40 0.38

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Table 2. Properties of the Glass and PMMA Particles Suspension: Case 2

liquid glass PMMA

F (kg/m3)

µ (mPa s)

997.5 2450 1150

0.99

dp (µm)

Cav (g/L)

327 327

1.90 1.70

N (rpm) 1146

mPa s, respectively). As the solids, spherical particles of poly(methyl methacrylate) (PMMA), glass, and bronze were used: they had the same size (dp ) 0.33 mm) but different densities (FS ) 1.15, 2.45, and 8.41 kg/L, respectively), thus resulting in rather different settling velocities in the same liquid. The mean solids concentration in each run was in the range 1.5-3.6 g/L. The experiments were carried out at room temperature in batch conditions. The rotational speed was always higher than the “just suspended” condition, and fully turbulent flow regime was attained in all cases. The solids concentration was determined by means of the nonintrusive optical technique used previously,19 which consists of traversing the stirred suspension with a light beam along a chord about 1 cm off the axis, approximately midway between adjacent vertical baffles. A laser diode and a silicon photodiode were used as the light source and the receiver, respectively; the solids concentration was determined from the measured light attenuation by means of a calibration curve. The same calibration could be used for either the monodisperse or mixed solids as the attenuationsapart from solids concentrationsdepends on the particle size, which was unchanged in all conditions. Each measurement was considered representative of the solids concentration on the whole horizontal plane, as already assessed in several studies.19-23 Measurements at 32 elevations were effected so that a vertical concentration profile was obtained for each condition. The experiments regarded two different cases, namely the suspension of a glass and bronze mixture (case 1) and of glass and PMMA (case 2); the physical properties of the mixtures and the experimental conditions are summarized in Tables 1 and 2.

following RANS continuity and momentum equations were solved:

(

The multiple impeller stirred vessel was simulated by adopting the sliding grid method implemented in the CFD code CFX-4.3. The computational domain was described by a structured grid consisting of about 140 000 cells (196 × 30 × 24 cells along the axial, radial, and angular directions, respectively) over an azimuthal extension of π, equivalent to 280 000 cells in the whole vessel. For selected conditions, the effect of grid density on the results was checked; namely the grid was refined by a factor of 4 in the tangential direction, i.e., 96 cells instead of 24 (further refinements were not allowed due to computer memory limitations). The calculated mean variables were almost unchanged relative to the case of the coarser grid, in agreement with previous results,25 while the total computational time for obtaining the converged solution increased from 29 to 120 h on a 3.6 GHz processor. The continuous phase and the two dispersed solid fractions were modeled with the “multifluid model”.26 The phases have been regarded as interpenetrating continua, and the

(1)

∂ (φ F U ) + ∇‚φkFkUkUk + ∇‚φkτk ) ∂t k k k -φk∇pk + φk∆Fg + FIK (2) with k ) L (liquid), k ) S1 (solids fraction 1), and k ) S2 (solids fraction 2). This is essentially the same approach used by Sha et al.9 Only the effect of the liquid fluid dynamics on the particles was considered (one-way coupling) because of the very low solids concentration of the suspension (about 10-3 by volume). For modeling the turbulent dispersion of the solids fraction the eddy diffusivity hypothesis was introduced. The turbulent Schmidt number, σt, has been assumed to be equal to 0.8. It is worth recalling that, contrary to what happens with dynamic phenomena, the sensitivity of the results to this parameter is negligible in steady-state solid-liquid systems.10 The momentum transfer between the liquid and each solid phase was taken into account only by the drag force, FIK. Additional forces, such as the Magnus lift, Saffman lift, Basset force, and added mass force, were not included, as their contribution to the velocity field and particle concentration distribution was found to be unimportant with respect to the drag force in dilute solid-liquid suspensions.5,11,14 The following form for the drag force was selected:27

FIK ) 3. CFD Model and the Computational Approach

)

µtk ∂ ∇φk ) 0 (φkFk) + ∇‚ φkFkUk ∂t σt

φS1(FS1 - FL)g US12

|US1 - UL|(US1 - UL) +

φS2(FS2 - FL)g US22

|US2 - UL|(US2 - UL) (3)

Therefore, the momentum transfer between the liquid and each solid fraction is dependent on the particle settling velocity, Us. For each solid type Us was computed with the following correlation:28

(

)

Us 16λ ) 0.4 tanh - 1 + 0.6 Ut dp

(4)

In turn, the particle terminal velocity, Ut, was computed using the Schiller and Naumann correlation,29 while the Kolmogoroff length scale, λ, was evaluated from the average power dissipation, , based on the experimental power number. As a result, a constant particle settling velocity value was considered at all vessel locations.

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Figure 1. Case 2. Comparison of experimental solid concentration profiles. Open symbols, S1-S2-L system; filled symbols, normalized sum of solid profiles in S1-L and S2-L systems. Figure 3. Case 1. Vertical volumetric fraction distribution maps of bronze (a) and glass (b) particles.

Figure 2. Case 1. (a) Liquid velocity vector plot in a vertical plane midway between two baffles. (b) Radial profiles of the axial velocity at different elevations between the first and second impellers. Solid lines, liquid; dotted lines, bronze particles. The horizontal dotted lines represent the axial position z/T and the zero velocity value of the associated profiles.

For the RANS equations closure, the development of turbulence models for multiphase applications is still limited. The “homogeneous” k- model was employed; it coincides with the “standard” k- model adopted for single-phase flows, where the physical properties of the solids-liquid mixture are adopted. Other formulations of the multiphase k- model were found to be inadequate for application to solid-liquid suspensions.13 For each simulation, periodic boundary conditions were adopted in the azimuthal direction and the calculations were

Figure 4. Axial profiles of dimensionless mixed solids concentrations. Symbols, experimental data; line, CFD results. Conditions: see Table 1.

started from still fluid and uniformly distributed particles in the vessel. No-slip boundary conditions via conventional “wall functions” were adopted at the walls. About 20 s of impeller rotation was required to obtain the solution. As for the time steps, in all cases a value corresponding to an impeller rotation of 60° was adopted. Indeed, Brucato et al.12 have shown that practically the same steady-state results are obtained with large time step sliding grid simulations (that reproduce exactly the

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Figure 5. Case 1. Radial profiles of solids volume fraction at different elevations. Open symbols, glass particles; filled symbols, bronze particles. The horizontal dotted lines represent the axial position z/T and the zero concentration value of the associated profiles.

Figure 6. Case 2. Vertical volumetric fraction distribution maps of glass (a) and PMMA (b) particles.

same geometric configuration at the end of each time step) and time-averaged results (over smaller time steps, typically adopted for refining the solution). For each time step 30 internal iterations were required for obtaining settled residuals. The other details on the numerical solution procedure do not differ from those adopted in previous works10,13 and are, therefore, omitted here. The simulations regarded case 1 and case 2, whose details are given in Tables 1 and 2. 4. Results and Discussion The vertical concentration profiles obtained for the cases of single or mixed solids exhibit the same generic trend as discussed in previous papers.19,28 For case 2, measurements were performed for both the mixed solids system and the single solids fractions (with the same concentration as in the mixed solids suspension), at the same rotational speed. The experimental mixed solids concentration profile and the normalized sum of the two single profiles obtained with each solid fraction are almost coincident, as shown in Figure 1. This proves that the two phases act independently of each othersat least at this dilute concentrationsand provides an important indication for the modeling. Thus, for proper description of the system behavior, the particle-particle interaction phenomena may be ignored in the model equations. The solution of the model equations has provided the velocity flow field for each of the three phases and the distribution of the solids fractions in the whole vessel volume. For the present stirred vessel geometric configuration, impeller spacing was large enough to allow almost independent recirculation loops for each impeller,30 as can be observed in Figure 2a, where the liquid velocity vector plot relevant to case 1 is shown for a

Figure 7. Glass (0, )) and PMMA (], s) axial profiles of dimensionless concentration. Symbols, experimental data; lines, CFD results using Us values.

vertical plane midway between two baffles. The independence of each impeller from the others31,32 has led to consideration of an overall power number, equal to that of a single turbine (value taken from the literature) multiplied by the turbine number, for the evaluation of λ. Unsurprisingly, very small differences were found between the velocity fields of the liquid and the two solid phasesssuch a feature having also been checked experimentally for the case of a suspension with a single solid fraction.33 In

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Figure 8. Glass and PMMA mixed solids axial profiles of dimensionless concentration. Symbols, experimental data; lines, CFD results. Ut values (s); Us values ()).

particular, negligible differences were found for the radial velocity components of the three phases; also, the tangential velocities of the liquid and of the two solid phases were almost coincident at all locations, while small differences could be observed only between the axial velocity of the liquid and that of the mixed solids. In Figure 2b the radial profiles of the axial velocities for the liquid and bronze particles are shown for case 1. Only profiles at selected elevations comprised between the first impeller and the second impeller (i.e., z/T ) 0.6, 0.8, 1, 1.2, 1.4 from the vessel bottom) are considered, as identical features were found in each vessel module. As can be observed, the bronze particles move faster than the liquid where the recirculation loop is directed downward (i.e., r/T < 0.4 for z/T ) 0.6 and 0.8 and r/T > 0.4 for z/T ) 1.2 and 1.4), while they exhibit a slightly smaller velocity values with respect to the liquid phase in the regions of upward flow. The glass particles

exhibit the same qualitative behavior (not shown), though the differences relative to the corresponding liquid velocity are less evident than those of the bronze particles, due to the smaller density difference and slip velocity. The axial velocity vanishes for all phases at z/T ) 1, giving quantitative evidence that each impeller gives rise to independent recirculation loops. The solids concentration distribution features important differences for the two solid phases for both cases 1 and 2, due to the dissimilar settling behavior of the solid fractions. The vertical volumetric fraction distributions for case 1 are shown in parts a and b of Figure 3 for the bronze and glass particles, respectively. As can be observed, the bronze particles are mainly concentrated in the lower part of the vessel, up to about z ) T, while the decrease in the glass-phase fraction from the vessel bottom to the top is much less marked. It is worth noting that for this condition little difference was found between the particle terminal velocity in still liquid, Ut, and Us as calculated by eq 4. Therefore, the simulation was performed by adopting the Ut values, which are only slightly higher than Us for both solids (see Table 1). At each elevation, the solid concentration of both particle fractions was averaged along the radius, and then the total solid concentration was calculated as the sum of the predicted values of the two solid fractions. Finally, a dimensionless axial concentration profile was obtained by dividing the total concentration at each axial location by the total concentration of the solids inside the whole vessel. The axial dimensionless profiles of total solids concentrations are compared with the corresponding experimental data in Figure 4. As can be observed, the calculated profile is in good agreement with the experimental data, thus confirming that the simple, selected computational approach is appropriate for the prediction of the mixed solids distribution. Having ascertained the accuracy of the simulation approach, the details of the predicted solids distributions can be analyzed further to obtain additional information as per the different attitudes of the two solid fractions to distribute throughout the vessel cross sections. As an example, the radial profiles of glass and bronze particles calculated at selected elevations between the first impeller and the second impeller are shown in Figure 5. Above the lower impeller, up to z/T ) 1.0, the bronze particle concentration experiences significant variations along the radius, while at the same elevations the glass particle profiles deviate

Figure 9. Case 2. Radial profiles of solids volume fraction at different elevations. Open symbols, PMMA; solid symbols, glass. (4, 2) z/H ) 0.18; (O, b) z/H ) 0.48; (0, 9) z/H ) 0.76.

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from an almost constant value only close to the vessel wall. Smaller variations are exhibited by the bronze particles when approaching the second impeller, but the profile shape still highlights the particle tendency to concentrate close to the wall and the vessel axis up to z/T ) 1.4. When moving toward the vessel top (not shown), the concentration of the bronze particles significantly decreases and so do their radial gradients; the glass particles, which are more uniformly distributed along the vessel height, exhibit also much smaller radial variations. Overall, the radial distribution exhibits more significant deviations from homogeneity for the heavier particles, particularly where the solids concentration is higher. For case 2, a Us reduction of about 54% with respect to Ut was estimated by eq 4; therefore, two sets of simulations were performed, with either values of particle settling velocities, Us and Ut. The predicted flow fields show the same qualitative features already described for case 1. The concentration distributions of the glass and the PMMA particles obtained with Us values are shown in parts a and b, respectively, of Figure 6. In this case, the glass particles are the denser phase and, as can be observed, they exhibit greater vertical gradients with respect to PMMA particles. These last are almost homogeneously distributed in the whole vessel volume, as expected because of the little density difference from that of water. The results of the S1-S2-L simulation are compared with the experimental profiles obtained with the single solids fractions in Figure 7, and, as can be observed, the agreement is very good for both particle fractions. It is worth stressing that a good fit was obtained by using the relevant Us values, while overestimated settling behavior and marked profile discrepancy were found when using the Ut values (not shown in the figures). The lumped effect of Ut and Us for the mixed solids can be clearly appreciated in Figure 8, where the data obtained in the experiment with these solids are compared with the concentration profiles predicted with the two velocity values. Apparently, with the adoption of eq 4 for the estimation of particle settling velocity, much better agreement can be obtained, as discussed previously for monodisperse systems.13 The radial distributions are characterized by much smaller variations with respect to case 1, as can be observed in Figure 9. In particular, the PMMA particles are uniformly distributed at all vessel cross sections, while limited variations are exhibited by the glass particle profiles. However, in the lower part of the vessel the glass distribution exhibits the same qualitative trend as the bronze particles (i.e., higher concentration close to the vessel wall and the shaft) with a maximum volumetric fraction difference along the radius about 4 times smaller than with the bronze particles. Similar behavior had been detected with the same suspension in a bigger vessel stirred with triple PBTs.24

solid fraction was found to act independently of the other in the suspension. In view of these results, it is believed that the adopted approach can be easily extended to suspensions of particles of a wider spread of characteristics and/or sizes by simply considering additional independent solid phases. It is to be stressed that the mentioned approach can be adopted only up to suspension concentrations for which one-way coupling is an acceptable approximation. The effect of turbulence on particle settling velocity has to be considered for obtaining accurate solids concentration values in the vessel, thus avoiding overestimation of the solids settling behavior. Acknowledgment This work was financially supported by the Italian Ministry of University and Research (MIUR) and the University of Bologna under Project PRIN 2005. The assistance of Dr. D. Fajner and Mr. F. Orlandini in carrying out the experimental program is gratefully acknowledged. These results have been presented in an abridged form at the 12th European Conference on Mixing, Bologna, June 27-30, 2006. Nomenclature C ) mass concentration of solids [kg m-3] Cav ) average mass concentration of solids [kg m-3] D ) impeller diameter [m] dp ) particle diameter [m] g ) gravity acceleration [m s-2] H ) tank height [m] N ) agitation speed [s-1] p ) pressure [Pa] r ) radial coordinate [m] T ) tank diameter [m] U ) mean velocity vector [m s-1] Us ) particle settling velocity in turbulent fluids [m s-1] Ut ) particle terminal velocity in still fluids [m s-1] z ) axial coordinate [m] Greek Symbols  ) turbulent dissipation [W kg-1] φ ) volumetric fraction λ ) Kolmogoroff length scale [m] F ) liquid density [kg m-3] µ ) liquid viscosity [Pa s] µt ) turbulent viscosity [m2 s-1] σt ) turbulent Schmidt number τ ) stress tensor [Pa] Literature Cited

5. Conclusions The behavior of a stirred vessel where a suspension of mixed solids was agitated with multiple Rushton turbines has been investigated by experiments and simulations. In particular, the vertical distribution of solids concentration was measured by an optical technique and the flow field of all the intervening phases and their concentration distribution in the vessel was calculated with CFD tools. The computational approach was an extension of that already developed for single solids fractions to the case of a mixture of solids of different physical properties. For simplicity, only two solid fractions of different densities and equal size were considered. The simulations have provided good predictions of the solids distribution as compared with the experiments. For the investigated dilute conditions, each

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ReceiVed for reView May 18, 2006 ReVised manuscript receiVed February 15, 2007 Accepted February 26, 2007 IE060616I