Solids Settling Velocity and Distribution in Slurry Reactors with Dilute

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Ind. Eng. Chem. Res. 2001, 40, 4456-4462

GENERAL RESEARCH Solids Settling Velocity and Distribution in Slurry Reactors with Dilute Pseudoplastic Suspensions Davide Pinelli and Franco Magelli* DICMA-Department of Chemical, Mining and Environmental Engineering, University of Bologna, viale Risorgimento 2, 40136 Bologna, Italy

The solids distribution was investigated in a tank stirred with multiple Rushton turbines. The analysis of the behavior of suspensions made of solid particles and non-Newtonian liquids allowed us to better single out the role of the most significant parameters that define the solids profiles: particle settling velocity is confirmed to be a key parameter for this process. It is shown that the same concept of terminal velocity in these systems is to be considered with care. The settling velocity in the stirred liquid is usually intermediate between the terminal settling velocity in the still liquid and the terminal settling velocity calculated with an effective liquid viscosity based on Metzner and Otto’s concept. A correlation between the first and last settling velocities with the ratio of Kolmogoroff length scale and particle diameter is in very good agreement with that determined previously for Newtonian liquids. 1. Introduction Mixing in stirred tanks is a common practice in many areas of the chemical and process industry. An understanding of the complex fluid dynamic behavior prevailing in slurry agitated reactors and other equipment for solid-liquid treatment is crucial for rational design and reliable process operation. Three different topics are usually considered in the fluid dynamic context of solidliquid systems, namely, off-bottom particle suspension, spatial solids distribution inside the equipment, and solids separation at the withdrawal tube.1 This paper addresses the second of the abovementioned items. The main results obtained in the analysis of particle-liquid interaction, which is a key phenomenon for solids distribution, are reviewed in the following section. Suffice it to mention here that most of these studies arrived at the conclusion that the particle settling velocity in a stirred medium, considered either as a real suspension property or as a model parameter, is usually less than the settling velocity in a still liquid (the so-called terminal velocity). The aim of this paper is to discuss this last aspect further. The investigation was conducted with dilute suspensions of spherical particles in pseudoplastic liquids in order that the liquid environment “seen” by the single particles and the nature of particle-liquid interaction are affected by nonlinear rheological behavior, while particle-particle interactions are minimized. The analysis was performed in a tank of high aspect ratio stirred with multiple impellers, which allows for the solids concentration gradients along the vertical axis * Corresponding author: Dr. Franco Magelli, DICMADepartment of Chemical, Mining and Environmental Engineering, University of Bologna, viale Risorgimento 2, 40136 Bologna, Italy. Fax: +39-051-581200. Telephone: +39-0512093147. E-mail: [email protected].

to be maximized while those in the radial direction are neglected, thus making the study more straightforward. The experimental results are discussed in terms of the simple one-dimensional sedimentation-dispersion model with the particle velocity as a parameter. A limited number of ancillary experiments aimed at determining the terminal particle settling velocity and characterizing liquid macromixing in pseudoplastic fluids were also conducted and were instrumental in the above-mentioned analysis. 2. Background on Particle Settling Velocity in Stirred Media The first experimental analyses of particle-liquid interactions in stirred tanks date back to about 30 years ago. Different research groups2-5 have concluded that the average particle settling velocity in a stirred liquid is 30-60% of the settling velocity in the quiescent liquid at rest, Ut, thus implying a relevant drag coefficient increase. Although the techniques on which these results were based exhibit some limitations and the data themselves were relatively limited, the essence of these findings is consistent with turbulent statistical theory and with turbulence-related phenomena such as particulate mass transfer. More recently, detailed LDV measurements have revealed that the particles actually lag or lead the liquid in the zones characterized by upward or downward flow, respectively.6 On average, however, these last results do not seem to contradict the previous ones. Similarly, the drag coefficients of spheres falling in liquids have been reported to be affected by “external” turbulence (sometimes referred to as “free stream” turbulence), that is, turbulence other than that produced by particle motion itself.7 Although the form of this influence has been questioned recently,8 the general effect remains undisputed.

10.1021/ie0010518 CCC: $20.00 © 2001 American Chemical Society Published on Web 09/05/2001

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Solids concentration profiles in tall tanks stirred with multiple Rushton turbines have been described by means of the simple one-dimensional sedimentationdispersion model9 that includes two parameters: the effective solids dispersion coefficient and the particle settling velocity (DeS and US, respectively). Although a model parameter, US can be regarded as representing the average behavior of the particle ensemble in the stirred medium, and it was deduced to be 30-100% of the settling velocity in the quiescent liquid.9 As an explanation for this reduction, particle interactions with turbulent eddies of a specific size range (measured in terms of the Kolmogoroff length scale λ) were invoked, and an empirical correlation between US/Ut and the ratio λ/dP was established. The microscale λ was calculated on the basis of average power consumption per unit mass. In view of the two order of magnitude variation of the local dissipation rate in a stirred tank,10-17 this represents a remarkable simplification that calls for improvement and refinement. Despite this weakness, the use of a settling velocity reduced with respect to the terminal onescalculated with the mentioned correlation or its variantssallowed the experimental solids concentration profiles obtained under different operating conditions in equipment of various geometries to be matched with those calculated with simple fluid dynamic models9,18-21 and CFD.22-24 On the contrary, use of the uncorrected terminal settling velocity produced anomalous results.22,25 The validity of the above-mentioned approach was supported also by Brucato et al.8 in an analysis of the solids distribution in Couette-Taylor flow. In view of the ordered vortex structures that characterize this flow situation, less scattered US/Ut data were obtained. These authors proposed a correlation of the drag coefficients (for the stirred and the quiescent liquids) as a function of λ/dP. Although different systems should be compared with caution, the results obtained in three adjacent research areas are also worth mentioning. A reduction in the particle settling velocity with increasing frequency was reported for oscillatory flows.26,27 Calculations of the energy dissipation for this type of motion reveal that a relationship between the velocity drop and the λ/dP ratio similar to that discussed above exists. Also, in gasliquid stirred tanks, the drag on gas bubbles seems to be modified by turbulence to a significant extent.28 Finally, Gore and Crowe29 described the modulation of turbulence intensity caused by the addition of particles to gas flows with a physical model centered on the use of the ratio of the particle diameter to a turbulent length scale. Apparently, evidence exists for the reduction of particle velocity in agitated turbulent media. Nevertheless, this evidence is to be regarded as circumstantial, and the question remains open as to the more suitable way to describe it. Indeed, this issue has been fully disregarded in some recent studies where reasonable fits of calculated vs experimental solids distributions could be achieved by using other parameters.30,31 This situation, therefore, stimulated us to search for additional, more convincing physical evidence that the mentioned phenomenon is not an artifact. As anticipated in the Introduction, the developed investigation is based on the simple idea that flow perturbation of shear-thinning fluids in the region near the particles caused by agitation should amplify these effects through

Figure 1. Geometrical characteristics of the tank. Table 1. Characteristics of the Solid Spherical Particles material

glass

glass

glass

plastic

plastic

dP (mm) Fs (g/cm3)

0.33 2.45

0.79 2.45

1.13 2.8

2.95 1.47

5.94 1.17

nonlinear rheology and affect the particle settling velocity through a reduction of the apparent local viscosity. 3. Experimental Equipment and Measurements 3.1. Experimental Stirred Tank and Conditions. The main body of the investigation was carried out in a vertical, cylindrical, flat-bottomed tank (T ) 23.2 cm, H/T ) 4), similar to that used in previous work.9 The vessel was made of acrylic glass with a fully closed design and was equipped with four vertical T/10 baffles. Agitation was provided by four identical, evenly spaced Rushton turbines (D ) 7.87 cm) mounted on the same shaft. The geometrical configuration of the tank is shown in Figure 1. The liquids used were two dilute solutions of highgrade Carbopol in water (0.05 and 0.07 w/w%, called S1 and S2, respectively, in the following, that exhibited non-Newtonian inelastic behavior) for the study of the solids distribution and settling velocity and aqueous solutions of poly(vinylpyrrolidone) (Newtonian behavior, viscosity up to 22 mPas) for preliminary and ancillary experiments. With the S1 and S2 solutions, pH was kept constant at 7.5 to ensure stable rheological properties. As the solids, monosized spherical particles of different diameter and materials were used (Table 1). The mean solids concentration was in the range of 1-5 g/L for the glass particles, 5-11 g/L for the plastic ones. The experiments were carried out at room temperature under batch conditions. The rotational speed was always higher than the “just suspended” condition. 3.2. Solids Concentration Measurements. The solids concentration profiles in the tank were determined by means of the nonintrusive optical technique and the related equipment described by Fajner et al.32 A laser diode and a silicon photodiode were used as the light source and the receiver, respectively. At each

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transparent tube (150 cm height, 15 cm in diameter) was used for this purpose. The time necessary for single particles to travel a given distance was measured with a stopwatch and the velocity was then calculated. Each measurement was usually repeated 10 times. No correction was introduced for wall effects because of the very low dP/T ratio nor for hindered settling. 4. The Fluid Dynamic Models

Figure 2. Example of rheological behavior of the non-Newtonian solutions: O b, S1 solution (n ) 0.73); 0 9, S2 solution (n ) 0.58); open symbols, 18 °C; solid symbols, 30 °C.

vertical position, the light beam passed through the vessel horizontally along a chord about 1 cm off the axis, approximately midway between pairs of two opposed vertical baffles, and the intensity attenuation caused by the presence of the solid particles was detected. Because the radial solids concentration gradients are fairly limited in baffled tanks,33-36 each measurement was representative of the solid concentration on the whole horizontal plane. The system was calibrated for each particle fraction. For each experimental condition, the solids concentration was measured at 32 elevations. 3.3. Liquid Phase Characterization. The macromixing state of the liquid was characterized with the flow model described in section 4.2. Its parameter was determined by matching the experimental mixing time curves, which were obtained by means of the classic pulse-response technique.37 A rapid injection of KCl solution (tracer) was made at the base of the tank, and the resulting concentration was detected at the top as a function of time. The transient tracer concentration was measured with a conductivity cell. Occasionally, the injection and detection points were exchanged to verify the absence of any tracer-driven buoyancy phenomena. 3.4. Rheological Measurements. The rheological properties of the non-Newtonian solutions used in the experiments were determined with a rotational viscometer (Rotovisko, Haake, Germany). The pseudoplastic behavior of the liquids was interpreted with the twoparameter power-law equation

τ ) ksγn

4.1. Solids Concentration Profiles. To interpret the solids concentration profiles in the tank, the onedimensional sedimentation-dispersion model was adopted. With the z axis directed upward (so that z ) 0 and z ) H correspond to the bottom and the top of the vessel, respectively), the mass balance for a nonreacting solid phase in a stirred suspension is

∂CS ∂CS ∂2CS - (UL + US) ) DeS 2 ∂t ∂z ∂z

(2)

According to the system of coordinates adopted, US is negative for FS > FL. The dimensionless solution of eq 2 for batch systems and steady-state conditions is9,20

CS*(ζ) )

PeS 1 - exp(-PeS)

exp(-PeS ζ)

(3)

where PeS ) -USH/DeS. The model has been used for calculating PeS from the experimental profiles, for which purpose a best-fit technique was employed. 4.2. Macromixing State of the Liquid. The classic axial dispersion model was adopted to interpret the liquid behavior

∂CL ∂2CL ∂CL ) DeL - UL 2 ∂t ∂z ∂z

(4)

For batch conditions, the dependence of theoretical tracer concentration CL (at the generic position z) on the time after a pulse disturbance (at t ) 0 and z ) 0) has been derived.38 In dimensionless form, it can be given at ζ ) 1 as ∞

CL* ) 1 + 2

cos(kπ) exp(-k2π2ϑ) ∑ k)1

(1)

with ϑ ) tDeL/H2 (5)

Because the values of ks and n are affected by a number of factors (including solution aging, duration of agitation, and possible addition of salts), such measurements were performed in correspondence with each specific experiment. As an example, at 30 °C, ks was equal to about 63 and 203 mPa sn-1 and n to 0.68 and 0.59 for the 0.05 and 0.07 w/w% solutions, respectively. An example of rheological curves on log-log coordinates is shown in Figure 2. The extrapolation to γ f 0 of the lines correlating the experimental data on a linear plot did not reveal the presence of any yield stress (although this evaluation is questionable, as the lower measured shear rates were not small enough for a valid estimate to be made). 3.5. Terminal Settling Velocity. The particle settling velocity in a still liquid, Ut, was also determined for single particles in all of the liquids. A vertical

The value of parameter DeL was determined by fitting eq 5 to the experimental mixing time curves. Results The results will be presented in three separate steps: particle settling in non-Newtonian still liquids, axial dispersion coefficients for the liquid in pseudoplastic media, and solids distribution and particle settling velocity in the pseudoplastic solutions. As anticipated, the first two pieces of information are necessary for developing the last point. 5.1. Particle Settling Velocity in Quiescent Liquids. Only a limited number of experiments of this type was performed. The Ut values measured with the Newtonian liquids were in very good agreement with literature data39,40 and are not discussed here; they were

Ind. Eng. Chem. Res., Vol. 40, No. 20, 2001 4459 Table 2. CD Expressions from the Literature for Pseudoplastic Liquids Acharya et al.43 CD )

24X1(n) F1 + ReP Re F2

ReP < 1000

(8A)

P

X1(n) ) 3(3n-3)/2

-22n2 + 29n + 2 , n(n + 2)(2n + 1) F1 ) 10.5n - 3.5, F2 ) 0.32n + 0.13 (8B)

Gu Dazhi and Tanner41 CD )

Figure 3. Drag coefficient CD for spheres. s, standard curve for Newtonian liquids;39,40 O, S1 solution (n ) 0.73); 9, S2 solution (n ) 0.59).

[

]

35.2 20.9 +n 1(ReP*)1.03 (ReP*)1.11

0.2 < ReP* < 16

[

16 < ReP* < 100 (9B)

]

37 + 0.25 + 0.36n (ReP*)1.1 with ReP* ) 2-ndPnUt2-nFL/ks CD )

(9A)

Tripathi et al.42 CD )

24X2 (n) ReP

0.1 < ReP < 10 (10A)

X2(n) ) values from numerical computation; see table in the original paper CD )

[

24 0.42 (1 + 0.15ReP0.687) + ReP 1 + 42 500Re

-1.16 P

]

F3

1 < ReP < 200 (10B)

F3 ) 1.00276 + 0.001714 ReP +

0.0924 ReP

(10C)

Darby 44 Figure 4. Dimensionless axial dispersion coefficient as a function of Reynolds number. O, S2 solution (n ) 0.59); 0, demineralized water; 9, PVP solution (Newtonian); s, correlation for Newtonian liquids.37

just considered as a suitable means for system calibration. From the experimental data, the drag coefficient

CD ) 4gdP(FS - FL)/FLUt2

(6)

was calculated. The results obtained with solutions S1 and S2 are plotted in Figure 3. The two data sets, which are characterized by n ) 0.59 and 0.73, respectively, are slightly higher than the standard Newtonian curve for ReP < 0.2 and slightly lower for ReP > 1. This is in general agreement with previous findings;41,42 the “crossover” particle Reynolds number is almost the same as that given by Tripathi et al.42 The experimental settling velocities for the nonNewtonian liquids were also compared with the values calculated with the standard equation

Ut ) (4dP|FS - FL|g/3FLCD)0.5

(7)

For these calculations, the drag coefficient was evaluated with expressions available in the literature for pseudoplastic liquids (Table 2) based on either theoretical treatments or experiments. The agreement between the theoretical and experimental values obtained in this work was fair for ReP < 0.2 (especially with eqs 11A-C and 12A,B) and just acceptable in the intermediate regime. In any event, the level of approximation pro-

CD ) (F4 + 4.8F50.5/xxReP)2

ReP < 100

(11A)

F4 ) [(1.82/n)8 + 34]-1/8

(11B)

1.33 + 0.37n 1 + 0.7n3.7

(11C)

F5 )

Matijasic and Glasnovic45 CD )

24 X (n) + 0.653 ReP 3

ReP < 1000

X3(n) ) -1.26n + 2.3

(12A) (12B)

vided by all of the equations shown in Table 2 is sufficient for the subsequent discussion. 5.2. Liquid Phase Behavior. The modeling of the macromixing behavior of the liquid phase in tanks of the same geometry as that of the present study has been discussed in previous work.9 Various flow models were used, and, for the simple axial dispersion model, an empirical correlation of the parameter DeL as a function of Re was given for Newtonian systems.37 Mixing time experiments were then performed with the two solutions S1 and S2 to search for possible effects of the viscous anomaly on the axial dispersion coefficient. The experimental normalized curves were fitted with eq 5 and the DeL parameter was evaluated for each condition. The Reynolds number was calculated with the classical method of Metzner and Otto46 by replacing the dynamic viscosity µ with an apparent viscosity, which for a power-law liquid, can be given by

ηav ) τ/γav ) ks(KN)n-1 with K ) 11

(13)

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Figure 5. Example of solids concentration profiles in the stirred tank. N ) 21.7s-1; plastic particles (dP ) 2.95 mm, FS ) 1.47 g cm-3); 9, S1 solution; O, S2 solution.

The dimensionless DeL/ND2 parameter was then plotted against Re. As is apparent in Figure 4, the agreement between non-Newtonian data and the available correlation for Newtonian liquids is excellent. 5.3. Solid Particle Distribution in Stirred NonNewtonian Liquids. The solids concentration profiles obtained with the pseudoplastic solutions S1 and S2 (Figure 5) are qualitatively similar to those reported with Newtonian liquids.9 The “more viscous” the liquid, the lower the departure from suspension homogeneity. From each experimental profile, the PeS values were obtained by means of eq 3 through a best-fit technique. The settling velocity US was then calculated from the PeS definition. For this calculation, DeS was taken as equal to the dispersion coefficient for the liquid, DeL, in view of the results of previous direct measurements of these effective parameters47,48 as well as liquid and particle turbulent diffusivities.49 For the S2 solution, the values of US were always one order of magnitude higher than the terminal velocity calculated with the equations given in Table 2, whereas US and Ut were comparable (with US e Ut) for the S1 solution. Similarly, the theoretical profiles calculated with eq 3 based on the model parameter -UtH/DeL compared rather poorly to the experimental ones for the more concentrated solution. Bearing in mind that the liquid neighboring the particles can hardly be considered at rest and, instead, it is affected by the turbulence produced by the impellers, the use of an average apparent liquid viscosity was devised as a means of calculating a reference particle settling velocity Ut* that better reflects this situation. The effective terminal velocity Ut* was then calculated with the relationships valid for the Newtonian liquids39,40 by replacing the dynamic viscosity with the effective one ηav (see eq 13). For the S2 solution, the values of Ut dropped by one order of magnitude from those calculated with the correlations for quiescent liquids and became comparable to US; in contrast, Ut* values were similar to Ut for the S1 solution, thus having no meaningful effect. Therefore, under all conditions, Ut e US e Ut*. The ratio US/Ut* was finally plotted against the ratio λ/dP, which, according to a previous study,9 gives a generic account of particle-eddy interaction. The reference microscale λ was calculated with ηav/F as an estimate of kinematic viscosity. These data are shown in Figure 6, where the correlation line obtained with many particles and Newtonian liquids50 is also plotted for comparison. Despite the rather empirical and sim-

Figure 6. US/Ut* vs λ/dP: s, best-fit line for Newtonian liquids;50 data points, suspensions of spherical particles in the non-Newtonian liquids (S1, Carbopol 0.05%; S2, Carbopol 0.07%).

plistic approach and the several approximations, the agreement between the non-Newtonian and the Newtonian data is good. Not only is the trend for the former data described nicely by the latter, but also the two sets of values exhibit reasonable agreement. This gives, therefore, additional support to the existing evidence discussed in Section 2 that particle settling velocity in a stirred medium is affected by stirring and external turbulence. Conclusions Solids concentration profiles of monosized particles in pseudoplastic liquids were measured in a high-aspect ratio tank stirred with four Rushton turbines. The profiles were interpreted with the axial dispersionsedimentation model, thus allowing PeS values to be obtained as a single model parameter. From the PeS definition, the particle settling velocity in the stirred liquid, US, was then calculated and compared with the particle settling velocity in the unstirred liquid, Ut. This last is strongly dependent, all the other parameters being equal, on the liquid viscosity. When Ut is calculated with the standard power-law relationships (established for quiescent liquids), its value is one order of magnitude lower than US, at least for the more viscous solutions and under the tested conditions. On the contrary, the assumption that the effective viscosity of the power-law liquid is reduced by stirring (according, in particular, to the simple Metzner and Otto model) allows for an effective terminal velocity, Ut* to be calculated, that is quantitatively comparable to US. The ratio US/Ut* exhibits essentially the same dependence on λ/dP as for the Newtonian liquids, i.e. a reduction of the former as the latter is decreased. It is interesting to note that US/Ut* ≈ 1 for ReP ) 0.05-1, a condition where the particles follow the liquid quite closely. In contrast US becomes less than Ut* for ReP > 1, in correspondence with the development of velocity fluctuations and dynamic interaction between eddies and particles that result in drag increasesas happens with the Newtonian liquids. Obviously, more careful descriptions of the flow field in these systems with means more sophisticated than the simple model used here would provide better insight into this topic. Notation In the following, nole that L indicates units of length, M indicates units of mass, and T indicates usints of time.

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Nomenclature CL ) tracer liquid concentration (M L3) CL* ) dimensionless tracer concentration CD ) drag coefficient CS ) local volumetric solids concentration (M L3) CS* ) CS(ζ)/CS,av ) dimensionless local solids concentration CS,av ) average volumetric solids concentration (M L3) D ) turbine diameter (L) DeL ) dispersion coefficient for the liquid (L2 T-1) DeS ) dispersion coefficient for the solid (L2 T-1) dP ) solid particle diameter (L) Fi(n) ) functions of n, see eqs 8, 10, and 11 H ) tank height (L) K ) constant, eq 13 ks ) consistency index (M L-1 Tn-2) n ) flow behavior index N ) rotational speed (T-1) P ) power consumption (M L2 T-3) PeS ) -USH/DeS ) Pe´clet number for the solid Re ) ND2FL/µ or ND2FL/ηav ) rotational Reynolds number ReP ) dPnUt2-nFL/ks , dPUt*FL/ηav or dPUSFL/ηav ) particle Reynolds numbers ReP* ) particle Reynolds number based on particle radius (see eq 9) t ) time (T) UL ) superficial velocity of the liquid (L T-1) US ) settling velocity of solid particles in stirred liquid (L T-1) Ut ) settling velocity of solid particles in quiescent liquid (terminal velocity) (L T-1) Ut* ) terminal settling velocity of solid particles calculated with ηav (L T-1) V ) tank volume (L3) z ) vertical coordinate (L) Xi(n) ) drag coefficient correction factor, eqs 8, 10, and 12 γ ) shear rate (T-1)  ) P/FLV ) average power consumption per unit mass (L2 T-3) ζ ) z/H ) dimensionless vertical coordinate ηav ) average apparent viscosity, eq 13 (M T-1 L-1) ϑ ) tDeL/H2 ) dimensionless time λ ) (ν3/)0.25 ) Kolmogoroff microscale (L) µ ) dynamic viscosity (Newtonian liquid) (M T-1 L-1) ν ) µ/FL or ηav/FL ) kinematic liquid viscosity (L2 T-1) FL ) liquid density (M L-3) FS ) solid density (M L-3) τ ) shear stress (M L-1 T-2)

Acknowledgment This work was financially supported by the University of Bologna and the Italian Ministry of University and Research (PRIN 1998). The collaboration of Dr. D. Fajner and Messrs. F. Orlandini and G. Diletti in carrying out the experimental program is gratefully acknowledged. Literature Cited (1) Nienow, A. W. Suspension of solids in liquids. In Mixing in the Process Industries, 2nd ed.; Harnby, N., Edwards, M. F., Nienow, A. W., Eds.; Butterworth-Heinemann: London, 1992; Chapter 16. (2) Schwartzberg, H. G.; Treybal, R. E. Fluid and particle motion in turbulent stirred tanks. Ind. Eng. Chem. Fundament. 1968, 7, 6. (3) Levins, D. M.; Glastonbury, J. R. Particle-liquid hydrodynamics and mass transfer in a stirred vessel. Part I: Particleliquid motion. Trans. Inst. Chem. Eng. 1972, 50, T32. (4) Nienow, A. W.; Bartlett, R. The measurement and prediction of particle-fluid slip velocities in agitated vessels. In Proceedings

of the First European Conference on Mixing and Centrifugal Separation (Cambridge, Sep 9-11, 1974); BHRA: Cranfield, U.K., 1974; p B1.1. (5) Kuboi, R.; Komasawa, I.; Otake, T. Fluid and particle motion in turbulent dispersion. II. Influence of turbulence of liquid on the motion of suspended particles. Chem. Eng. Sci. 1974, 29, 651. (6) Nouri, J. M.; Whitelaw, J. H. Particle velocity characteristics of dilute to moderately dense suspension flows in stirred reactors. Int. J. Multiphase Flow 1992, 18, 21. (7) Uhlherr, P. H. T.; Sinclair, C. G. The effect of free-stream turbulence on the drag coefficient of spheres. In Proceedings of CHEMECA 1970; Butterworths: Melbourne, Australia, 1970; Vol. 1, p 1. (8) Brucato, A.; Grisafi, F.; Montante, G. Particle drag coefficients in turbulent fluids. Chem. Eng. Sci. 1998, 53, 3295. (9) Magelli, F.; Fajner, D.; Nocentini, M.; Pasquali, G. Solid distribution in vessels stirred with multiple impellers. Chem. Eng. Sci. 1990, 45, 615. (10) Cutter, L. A. Flow and turbulence in a stirred tank. AIChE J. 1966, 12, 35. (11) Okamoto, Y.; Nishikawa, M.; Hashimoto, K. Energy dissipation rate distribution in mixing vessels and its effects on liquid-liquid dispersion and solid-liquid mass transfer. Int. Chem. Eng. 1981, 21, 88. (12) Barthole, J. P.; Maisonneuve, J.; Gence, J. N.; David, R.; Mathieu, J.; Villermaux, J. Measurement of mass transfer rates, velocity and concentration fluctuations in an industrial stirred tank. Chem. Eng. Fundament. 1982, 1, 17. (13) Laufhu¨tte, H. D.; Mersmann, A. Laser-Doppler velocimetry as a suitable measuring technique for the determination of flow behaviour in stirred fluids. Ger. Chem. Eng. 1985, 8, 371. (14) Wu, H.; Patterson, G. K. Laser-Doppler measurements of turbulent-flow parameters in a stirred mixer. Chem. Eng. Sci. 1989, 44, 2207. (15) Fort, I.; Machon, V.; Kadlec, P. Distribution of energy dissipation rate in an agitated gas-liquid system. Chem. Eng. Technol. 1993, 16, 389. (16) Zhou, G.; Kresta, S. M. Distribution of energy between convective and turbulent flow for three frequently used impellers. Chem. Eng. Res. Des. 1996, 74, 379. (17) Ng, K.; Yianneskis, M. Observations on the distribution of energy dissipation in stirred vessels. Chem. Eng. Res. Des. 2000, 78, 334. (18) Magelli, F.; Fajner, D.; Nocentini, M.; Pasquali, G.; Ditl, P.; Marisko, V. Solids concentration distribution in slurry reactors stirred with multiple axial impellers. Chem. Eng. Process. 1991, 29, 27. (19) Brucato, A.; Magelli, F.; Nocentini, M.; Rizzuti, L. An application of the network-of-zones model to solids suspension in multiple impeller mixers. Chem. Eng. Res. Des., 1991, 69, 43. (20) Nocentini, M.; Magelli, F. Solid distribution in slurry reactors stirred with multiple impellers: continuous flow systems. In Fluid Mechanics of Mixing: Modelling, Operations and Experimental Techniques; King, R., Ed.; Kluwer Academic Publishers:, Dordrecht, The Netherlands, 1992; p 73. (21) Montante, G.; Grisafi, F.; Micale, G.; Brucato, A. Particle Settling Velocities in Turbulent Stirred Tanks. Proceedings of the 6th International Conference on Multiphase Flow in Industrial Plants (Milan, Italy, Sep 24-25, 1998); ANIMP: Milan, Italy, 1998; p 311. (22) Brucato, A.; Ciofalo, M.; Grisafi, F.; Magelli, F.; Micale, G. On the simulation of solid particle distribution in multiple impeller agitated tanks via computational fluid dynamics. AIDIC Conference Series; AIDIC: Milan, Italy, 1997; Vol. 2, p 287. (23) Micale, G.; Montante, G.; Grisafi, F.; Brucato, A.; Godfrey, J. CFD simulation of particle distribution in stirred vessels. Chem. Eng. Res. Des. 2000, 78, 435. (24) Montante, G.; Micale, G.; Brucato, A.; Magelli, F. CFD simulation of particle distribution in a multiple-impeller highaspect-ratio stirred vessel. In Proceedings of the 10th European Conference on Mixing (Delft, The Netherlands, July 2-5, 2000); van den Akker, H. E. A., Derksen, J. J., Eds.; Elsevier: Amsterdam, 2000; p 125. (25) Rousar, I.; van den Akker, H. E. A.; Ditl, P.; Havelkova, D. Turbulence modeling of solid dispersion in agitated tank. Presented at the 13th CHISA Congress, Prague, Czech Republic, Aug 1998; Paper G3.3. (26) Tunstall, E. B.; Houghton, G. Retardation of falling spheres by hydrodynamic oscillations. Chem. Eng. Sci. 1968, 23, 1067.

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Received for review December 5, 2000 Revised manuscript received July 3, 2001 Accepted July 3, 2001 IE0010518