Dispersion coefficient and settling velocity of the solids in agitated slurry reactors stirred with multiple rushton turbines

Dispersion coefficient and settling velocity of the solids in agitated slurry reactors stirred with multiple rushton turbines

Chemical Engineering Science 57 (2002) 1877 – 1884 www.elsevier.com/locate/ces Dispersion coe!cient and settling velocity of the solids in agitated ...

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Chemical Engineering Science 57 (2002) 1877 – 1884

www.elsevier.com/locate/ces

Dispersion coe!cient and settling velocity of the solids in agitated slurry reactors stirred with multiple rushton turbines M. Nocentini, D. Pinelli, F. Magelli ∗ DICMA-Department of Chemical, Mining and Environmental Engineering, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy Received 8 October 2001; accepted 22 February 2002

Abstract The feature of solids distribution in tanks stirred with multiple Rushton turbines was investigated. Both transient and steady-state experiments were performed in tanks of two scales with a variety of suspensions. The data were analysed with the axial sedimentation– dispersion model. The axial dispersion coe!cient of the solid phase was found not to di4er from that of the liquid by more than 20%. The e4ective particle settling velocity in the stirred medium was then determined. It is con8rmed that this parameter is di4erent from the terminal settling velocity. Their ratio exhibits the same dependence on Kolmogoro4 microscale and particle size as obtained previously with an indirect, approximate approach. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Chemical reactors; Mixing; Multiple Rushton turbines; Multiphase =ow; Particulate processes; Solids distribution

1. Introduction The knowledge of the spatial solids distribution in solid– liquid, agitated reactors is important for proper equipment design and reliable process operation—especially when through-=ow mode is adopted. The solids distribution is the result of two main competing actions, namely particle lift due to its interaction with liquid =ow and turbulence and particle tendency to settle. Several models have been proposed for calculating solids non-homogeneity that incorporate particle settling velocity (or particle drag coe!cient) and one or more parameters for describing macro-mixing phenomena. They span from the simple one-dimensional model with axial dispersion and sedimentation (e.g. Magelli, Fajner, Nocentini, & Pasquali, 1990) to two-dimensional (Penaz, Rod, & Rehakova, 1978; Sysova, Fort, Vanek, & Kudrna, 1984; Kudrna, Sysova, & Fort, 1986; Schempp & Bohnet, 1995) or multi-zone sedimentation–dispersion models (Yamazaki, Tojo, & Miyanami, 1986) and two- or three-dimensional network-of-zones models (Brucato & Rizzuti, 1988; Brucato, Magelli, Nocentini, & Rizzuti, 1991; McKee et al., 1994). Computational =uid dynamics tools have also started ∗

Corresponding author. Fax: +39-051-209-3147. E-mail address: [email protected] (F. Magelli).

being implemented with encouraging results (Bakker, Fasano, & Myers, 1994; Brucato, Ciofalo, Godfrey, Grisa8, & Micale, 1996; Brucato, Ciofalo, Grisa8, Magelli, & Micale, 1997; Decker & Sommerfeld, 1996; Barrue, Bertrand, Cristol, & Xuereb, 1999; Sha, Palosaari, Oinas, & Ogawa, 1999; Micale, Montante, Grisa8, Brucato, & Godfrey, 2000; and Montante, Micale, Brucato, & Magelli, 2000). The particle settling velocity in a stirred environment has been recognised to be usually lower than the terminal settling velocity (Magelli, et al., 1990; Brucato, Grisa8, & Montante, 1998; Pinelli, Nocentini, & Magelli, 2001). Though this 8nding is in good agreement with the results of more fundamental studies on the particle–liquid interaction (Schwartzberg & Treybal, 1968; Uhlherr & Sinclair, 1970; Levins & Glastonbury, 1972; Kuboi, Komasawa, & Otake, 1974; Nienow & Bartlett, 1974), it was arrived to in an indirect way—either by supposing that the dispersion coef8cient of the solid and of liquid phase are equal (Magelli et al., 1990; Pinelli, Nocentini, & Magelli, 1996) or as the result of parametric analyses of the whole phenomena (Nocentini & Magelli, 1992; Rousar, Van den Hakker, Ditl, & Havelkova, 1998). A more direct experimental approach, which is based on transient solids distribution analysis and permits to obtain the solids axial dispersion coe!cient, was adopted by this group (without publishing the results in learned journals)

0009-2509/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 2 ) 0 0 0 8 7 - 8

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Table 1 Particle characteristics size (mm) Material

Density (kg l−1 )

Ut (cm s−1 )

In tank (s)

0.33 0.79 2.95 0.14 0.23 0.33 0.98

2.45 2.47 1.45 2.46 2.46 8.61 2.47

0.8; 4.2–5.3 3.3; 11.5 9.3; 18 1.2–1.7 3 2.3; 14.5 5;14

T23B ; T23A ; T13 T23B T23B T23A ; T13 T23A ; T13 T23B T23A

Glass Glass Plastic Glass Glass Bronze Glass

as well as by Brucato et al. (1998) and Montante, Grisa8, Micale, and Brucato (1998). Since the data published so far are rather limited, an additional investigation was conducted to broaden the knowledge in this area. The axial sedimentation–dispersion model was adopted for data analysis. The investigation was focused on the determination of the axial dispersion coe!cients of the solid phase and the subsequent calculation of the particle settling velocity in the stirred medium. The work was performed in two geometrically similar tanks of high aspect ratio agitated with multiple Rushton turbines, since this con8guration magni8es the vertical concentration gradients with respect to the radial ones. Thus, it allows an easier and more straightforward analysis of the dispersion phenomena and more reliable parameter estimate. Fig. 1. Geometrical con8guration of the tank and turbine location.

2. Experimental 2.1. Equipment and experimental conditions The investigation was carried out in vertical, cylindrical tanks of two scales characterised by aspect ratio H=T =4. The following tanks were used: a small one, T13 (T = 13:6 cm, V =7:04 l), and a bigger one, T23A (T =23:2 cm, V =7:04 l), which were identical to those used in previous investigations (Magelli et al., 1990; Nocentini & Magelli, 1992; Pinelli et al., 1996), see Fig. 1. The last one was substituted with one identical in all the dimensions except for the diameter, T23B (T = 23:2 cm, V = 39:6 l), after breakage. The T13 and T23A tanks were made of Pyrex, T23B was made of Plexiglas. They had four vertical T=10 baOes, =at bottom and a =at lid. Agitation was provided with four equal, evenly spaced Rushton turbines (D = 4:34 and 7:87 cm, respectively) mounted on the same shaft. The liquids used were water and aqueous solutions of polyvinylpyrrolidone (Newtonian behaviour, viscosity up to 16 mPa s). As the solids, mono-sized spherical particles of di4erent size and material were used, all with S ¿ L : their main characteristics are given in Table 1 with the prevailing terminal velocity. The mean solids concentration was in the range 2–5 g=l−1 for the glass particles, about 4 g=l−1 for the plastic ones.

The experiments were carried out at room temperature in batch conditions. The rotational speed was in the range 15 –25 s−1 in T23A and T23B and in the range 20 –37 s−1 in T13 , i.e. always higher than the “just suspended” condition. 2.2. Solids concentration measurement The solid concentration in the vessel was measured by means of the non-intrusive optical technique described by Fajner, Magelli, Nocentini, and Pasquali (1985). A laser diode and a silicon photo-diode, used as the light source and the receiver respectively, were mounted on a traversing system. The light beam passed through the vessel horizontally along a chord about 1 cm o4 the shaft, approximately midway between the vertical baOes. The measuring system was calibrated for each particle fraction. Since the radial solids concentration gradients are fairly limited in stirred tanks (Yamazaki, Tojo, & Miyanami, 1986; Barresi & Baldi, 1987; Bilek & Rieger, 1990; Mak & Ruszkowski, 1990) and, moreover, the aspect ratio of the tanks is such as to minimise the radial inhomogeneity relative to the vertical distribution, each measurement was representative of the solid concentration on the whole horizontal plane.

M. Nocentini et al. / Chemical Engineering Science 57 (2002) 1877–1884

3. The model and data treatment 3.1. Solids distribution To interpret the solids concentration distribution in the tank, the one-dimensional sedimentation–dispersion model is adopted, which is regarded here just as a phenomenological model. With the z-axis directed downwards (so that z =0 and H correspond to the top and the base of the vessel, respectively), the mass balance for a non-reacting solid phase in the stirred suspension for batch systems is @CS @ 2 CS @CS − US = DeS : (1) 2 @t @z @z According to the system of coordinates adopted, US is positive for S ¿ L . The dimensionless solution of Eq. (1) under unsteady state for the mentioned experimental conditions is (see the Appendix for the details) exp(PeS ) C( ; ) = PeS + 2 exp(PeS =2 ) exp(PeS ) − 1

80

elevation (cm)

Two kinds of experiments were performed for each experimental condition, namely transient and steady-state runs. According to the 8rst technique, a certain amount of solid particles was injected at the top of the vessel ( = 0) in a very short time and the local solids concentration change with time was detected at selected elevations ( =0:14, 0.42, 0.66, 0.91 in T23B , = 0:14, 0.90 in T23A and = 0:33, 0.83 in T13 ). Higher accuracy was reached for the T23B measurements as a consequence of standardisation and automation of the experimental procedure. With the steady-state technique the solids concentration was measured at 32 vertical elevations to obtain the vertical concentration pro8les. Details on the objective of each technique and data treatment are given below. Each measurement, in either transient or steady-state conditions, was performed at least twice to check for reproducibility—often the dynamic measurements were repeated up to 8ve times.

1879

60

40

20

0 0.0

2.0

4.0

6.0

8.0

-1

concentration (g L )

Fig. 2. Example of a solids concentration pro8le obtained under steady—state conditions in tank T23B . Horizontal lines: (—·—) turbine location; (- - - - -) vertical elevation for the transients experiments. Pro8les: (symbols) experimental values; (line) best-8t theoretical pro8le.

was available for the same experimental condition. Thus, this last parameter was calculated by matching Eq. (3) to the solid concentration pro8le obtained with the steady-state measurements (Magelli et al., 1990). As shown in previous work, the experimental vertical pro8les are characterised by singularities in correspondence with the turbines and midway between the turbines (Barresi & Baldi, 1987; Magelli et al., 1990; Bilek & Rieger, 1990; Pinelli, Nocentini, & Magelli, 2001) provided that the particles are not too small (Yamazaki et al., 1986). This fact suggested to chose the elevations where to detect the transient curves in such a way as to minimise the departure between the experimental and the theoretical pro8les. One example of an actual concentration pro8le is shown in Fig. 2, with the best 8t theoretical curve as well as the positions

∞  (n)2 cos{n(1 − )} − n(PeS =2) sin{n(1 − )} × exp[ − {(PeS =2)2 + (n)2 }]; [(PeS =2)2 + (n)2 ] cos{n}

(2)

n=1

where PeS = US H=DeS and  = tDeS =H 2 . It is worth noting that at steady state (that is when  → ∞), Eq. (2) reduces to the simple dimensionless pro8le (Magelli et al., 1990) exp(PeS ) C( ) = PeS : (3) exp(PeS ) − 1

of the turbines and of the selected heights for the transient experiments. 4. Results and discussion

3.2. Data treatment

4.1. Generic remarks

Each experimental solid concentration curve obtained from the transient experiments at every selected elevation was normalised and compared with the theoretical one, Eq. (2). The parameter DeS was then determined with a best-8t procedure. This calculation required that the value of PeS

Figs. 3a–c show three pairs of transient curves obtained at three di4erent elevations for the same operating condition. As is apparent, the curve shape depends markedly on . And so does the asymptotic value C( ; ∞). In all cases, the agreement between the experimental and the

M. Nocentini et al. / Chemical Engineering Science 57 (2002) 1877–1884 4.5

0.20

experim. curve

3.5

best fit curve

2.5 2

1.5

DeS/ND

normalised conc. (-)

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0.5

-0.5

-5

0

5

10

normalised conc. (-)

25

0.00

1.8

experim. curve

1.4

best fit curve

1'000

10'000

100'000

1'000'000

Re

1.0

Fig. 4. Dimensionless axial dispersion coe!cients for glass particles at the two scales: (◦; •) dp = 0:33 mm; ( ; ) dp = 0:23 mm; (void symbols) T23A ; (solid symbols) T13 .

0.6 0.2 -0.2 -5

0

5

10

15

20

25

time (s)

(b) normalised conc. (-)

20

time (s)

(a)

(c)

15

0.10

4.5 experim. curve

3.5

best fit curve

2.5 1.5 0.5 -0.5

-5

0

5

10

15

20

25

time (s)

Fig. 3. Comparison of experimental and theoretical dimensionless curves for transient experiments at selected elevations in tank T23B . dp =0:79 mm; water; (a) = 0:14; (b) = 0:42; and (c) = 0:91.

theoretical curves was fairly good. In passing, this fact justi8es the use of the simple =uid dynamic model adopted in this study. Two further aspects are worth mentioning. The overshoot exhibited at = 0:14 was always (slightly) higher for the theoretical curve than for the experimental one: this seems to depend on the way the solids are actually introduced into the reactor (“point” injection) and on the non-in8nite value of the radial dispersion mechanism and coe!cient. As regards the choice of the measurement positions, apart from the above-mentioned requirement that the real-steady-state concentration does not deviate from the theoretical pro8le signi8cantly, values close to 0.6 – 0.7 are to be avoided. Indeed, at these elevations the curves feature a smoothed peak that cannot be easily recognised in the experimental curve due to data noise, thus making parameter evaluation rather inaccurate. For this reason, the curves at these elevations were not considered. 4.2. Axial dispersion coe8cient The values of the axial dispersion coe!cient for the solid phase, DeS , were determined as outlined before. They were usually slightly lower at = 0:14 than at = 0:41 and 0.91,

which fact is probably to be attributed to the experimental technique. However, since this deviation was always within the experimental error (±15%), they can be considered as independent of the measurement position (the only exceptions were the experiments with bronze particles, characterised by U = L ≈ 7:6, in which case the di4erences could be as high as ±30%). Therefore, the average of the measured values was retained and used for subsequent treatment. As already reported (Magelli, Fajner, Nocentini, & Pasquali, 1986; Montante, Grisa8, Micale, & Brucato, 1998), the DeS values were proportional to the rotational speed. The values at the two scales were di4erent and could be reconciled by dividing them by ND2 , as shown in Fig. 4 for meaningful conditions. Thus, the behaviour of the axial dispersion coe!cients for the solids is quite consistent with that for the liquid (Magelli et al., 1990). The axial dispersion coe!cients for the solids were then compared with the dispersion coe!cients of the liquid. These last values were taken from the literature (Magelli et al., 1990) and the possible solids in=uence on DeL values was disregarded due to the dilute suspensions used. For comparison, the ratio DeL =DeS is plotted as a function of =dp in Fig. 5—this last parameter being the ratio between Kolmogoro4 microscale and particle size, which has been used in a 8rst attempt to describe the particle–turbulence interaction (Magelli et al., 1990). The microscale  was calculated as an average value from literature power consumption data by neglecting local di4erences in speci8c power dissipation, , (Barthole et al., 1982; LaufhVutte & Mersmann, 1985; Fort, Machon, & Kadlec, 1993; Zhou & Kresta, 1996). As is apparent from Fig. 5, the average DeS and DeL values do not di4er by more than 20% from each other for any of the conditions tested or the two-scale tanks (actually, the scattering drops to ±10% for the experiments conducted in the T23B with the optimised experimental procedure). This fact is in agreement with previous results obtained in equipment stirred with non-standard, high D=T turbines (Montante et al., 1998) and it is also consistent with most recent

M. Nocentini et al. / Chemical Engineering Science 57 (2002) 1877–1884

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1.4 1.2

DeL/DeS

1.0 0.8 0.6 0.4 0.2 0.0 0.001

0.010

0.100

1.000

λ /dp dp=0.33mm, T23 dp=0.79mm, T23 dp=2.95mm plastic, T23 dp=0.13mm, T23

dp=0.23mm, T23 dp=0.98mm, T23 dp=0.23mm, T13 dp=0.13mm, T13

dp=0.30mm, T13 DeL/DeS=1 +20% -20%

Fig. 5. Comparison of the axial dispersion coe!cients (average data) for the liquid and the solid phase: (◦; •) dp = 0:33 mm (all materials); ( ; ) dp = 0:23 mm; (; 4) dp = 0:13 mm; (♦) dp = 2:95 mm; (+) dp = 0:98 mm; (×) dp = 0:79 mm; (void symbols) T23A and T23B ; (solid symbols): T13 . (Dashed lines) 1 ± 20%.

with dp increase—the in=uence of the rotational speed being marginal. Whether this derives from a change in the particle–turbulence interaction (Tsuji, Morikawa, & Shiomi, 1984) cannot be stated on the basis of the present analysis.

1.6 1.4 1.2

DeL/DeS

1.0 0.8

4.3. Settling velocity

0.6 0.4 0.2 0.0 0.00

0.02

0.04

0.06

0.08

λ /dp

Fig. 6. Comparison of the axial dispersion coe!cients for the liquid and the solid phase: scattering of the data before averaging: (◦) = 0:91; (×), = 0:41; (•) = 0:14; dp = 0:33 (glass only), 0.79 and 2:95 mm particles in water; T23B vessel.

conclusions regarding turbulent di4usivities of the liquid and the solid phases in a stirred tank (Montante & Lee, 1999; Sessiecq, Mier, Gruy, & Cournil, 1999). In Fig. 6, the values are also reported of the not averaged parameters from the measurements e4ected at various heights in the T23B vessel with water. An identical trend was obtained with liquids of higher viscosity. It is interesting to note that the scatter of these raw data increases

Once PeS and DeS were known for a given operating condition, the particle settling velocity in the stirred tank could be simply calculated from PeS de8nition: US = PeS DeS =H . Then US was compared with the terminal settling velocity, Ut , this last being calculated with Turton and Levenspiel’s correlation (1986). Fig. 7 shows that US is usually lower than Ut , thus con8rming previous 8ndings obtained in the same equipment under the simplifying assumption DeS = DeL (Magelli et al., 1990) as well as with a direct approach in Couette–Taylor =ow (Brucato et al., 1998) and mechanically stirred systems (Montante et al., 1998). Also, the values of the ratio US =Ut are in good agreement with the correlation given by Pinelli et al. (1996), which is reported as a solid line in the 8gure. The classi8cation of the data points is similar to that of the data on which the mentioned correlation is based on: whether this speci8c fact depends on inadequacy of the adopted =ow model, oversimpli8cation of the envisaged particle–turbulence interaction or both factors is unclear at this stage.

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Acknowledgements

1.4 1.2

Us /U t

1.0 0.8 0.6 0.4 0.2 0.0 0.001

0.01

0.1

1

λ /dp

Fig. 7. Ratio of the settling velocity in the stirred system, US , and the terminal settling velocity, Ut . Comparison of the present data with a correlation based on DeS = DeL (Pinelli et al., 1996). Symbols: see Fig. 5; lines: ±30%.

Finally, it is interesting to note that the US values parallel those of the slip velocity in stirred suspensions (Montante & Lee, 1999; Sedivy, Ditl, & Rieger, 1999).

5. Conclusions The dispersion coe!cients of the solid phase were measured and analysed at two scales—reference being made to the sedimentation–dispersion model. From their values, the particle settling velocities in the stirred system were also evaluated. The dispersion coe!cients for the solid phase were found not to di4er by more than 20% from those of the liquid. And the ratio between the settling velocity in the stirred system and that in a quiescent liquid, US =Ut , is con8rmed to be usually less than unity, the dependence on =dp being equal to that found in the past on the basis of simplifying assumptions. With the pair of the mentioned parameters, the solid distribution in tanks of high aspect ratio and stirred with multiple Rushton turbines can be estimated via the simple sedimentation–dispersion model. Although the extrapolation of the correlation US =Ut vs. =dp to more general cases and, in particular, its use in more sophisticated models are questionable conceptually, the consistency of this 8nding with the results of various analyses is certainly supportive to this end (see Brucato, Magelli, Nocentini, & Rizzuti, 1991, Brucato et al., 1996, 1997, 1998; Micale et al., 2000; Montante et al., 1998; Nocentini & Magelli, 1992; Rousar et al., 1998 for di4erent models and di4erent equipment con8gurations). This study also suggests the need for a better, more fundamental understanding of the particle–turbulence interaction to explain the overall, complex phenomenon of solids dispersion.

This work was 8nancially supported by the Italian Ministry of University and Research and the University of Bologna (PRIN 1998) and by CNR (Grant 84.01156.95). The collaboration of Dr. D. Fajner and Messrs. F. Orlandini, L. Giamperi, F. Magri and S. Milan in carrying out the experimental programme is gratefully acknowledged. Part of the results given in this paper were anticipated at the ‘World Conference III of Chemical Engineering’ (Tokyo, September 1986) and at the ‘Third International Symposium on Mixing in Industrial Processes’ (ISMIP-3, Osaka, September 1999). Appendix. The aim of this appendix is to derive the response of the sedimentation–dispersion model to a Dirac impulse of solids at z = 0 in batch conditions. A classical analytical approach is adopted. The following initial and boundary conditions apply to Eq. (1) for the considered situation: CS (z; 0) = 0 0 6 z 6 H;  dCS  + US CS |z=H = 0; −DeS d z z=H  dCS  + US CS|z=0 = CS; av H(t): −DeS d z z=0

(A.1) (A.2) (A.3)

The L-transforms of Eq. (1) with the condition (A.1) and the transforms of Eqs. (A.2) and (A.3) are d2 G dG − US − SG = 0; d z2 dz  dG  − US G|z=H = 0; DeS d z z=H  dG  − US G|z=0 = CS; av H: DeS d z z=0

DeS

(A.4) (A.5) (A.6)

The solution of Eq. (A.4) has the form G = K1 exp(1 z) + K2 exp(2 z);

(A.7)

where the characteristic roots are √ US ± (US2 + 4DeS S) 1; 2 = 2DeS √ 2 = Pe=2 ± (Pe =4 + H 2 S=DeS ) and the constants K1 and K2 can be speci8ed by means of the boundary conditions as K1 = −CS; av H exp(2 H )=(US − DeS 1 ) [exp(1 H ) − exp(2 H )]; K2 = CS; av H exp(1 H )=(US − DeS 2 ) [exp(1 H ) − exp(2 H )]:

M. Nocentini et al. / Chemical Engineering Science 57 (2002) 1877–1884

By substitution of these expressions into Eq. (A.7), the transformed concentration is obtained as follows: CS; av H G(z; S) = exp(1 H ) − exp(2 H )  ×

exp(1 H ) exp(2 z) exp(2 H ) exp(1 z) − US − DeS 2 US − DeS 1

References

 ;

(A.8) whence

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Bakker, A., Fasano, J. B., & Myers, K. J. (1994). E4ects of =ow pattern on solid distribution in a stirred tank. Institution of Chemical Engineers Symposium Series, 136, 1–8. Barresi, A., & Baldi, G. (1987). Solid dispersion in an agitated vessel. Chemical Engineering Science, 42, 2949–2956. Barrue, H., Bertrand, J., Cristol, B., & Xuereb, C. (1999). Eulerian simulation of dense solid–liquid suspension in multi-stage stirred vessel. Proceedings of the third international symposium of mixing in industrial processes Osaka, 19 –22 September (pp. 37– 44). Tokyo: Society of Chemical Engineers Japan.

G(z; S) H exp(US z=2DeS )[ cos{(H − z)} − US =2DeS sin{(H − z)}] (A.9) = CS; av S sin{H } √ with n = [ − (US =2DeS )2 − Sn =DeS ]. Barthole, J. P, Maisonneuve, J., Gence, J. N., David, R., Mathieu, J., & The poles of the denominator of (A.9) are Villermaux, J. (1982). Measurement of mass transfer rates, velocity Sn = −US2 =4DeS − n2 2 DeS =H 2 :

(A.10)

Anti-transformation of Eq. (A.9) with the method of the residues gives Eq. (2). Notation CS C CS; av D DeL DeS dp H N P PeS t US Ut V z 

  ! L S

local volumetric solids concentration, ML−3 =CS =CS; av , dimensionless local solids concentration, dimensionless number average volumetric solids concentration, ML−3 turbine diameter, L dispersion coe!cient for the liquid, L2 T−1 dispersion coe!cient for the solids, L2 T−1 solid particle diameter, L tank height, L rotational speed, T−1 power consumption, ML2 T−3 =US H=DeS , PZeclet number for the solids, dimensionless number time, T settling velocity of solid particles in stirred liquid, LT−1 settling velocity of solid particles in a quiescent liquid (“terminal” velocity), LT−1 tank volume, L3 vertical coordinate (directed downwards), L =P= V , average power consumption per unit mass, L2 T−3 =z=H , dimensionless vertical coordinate (directed downwards), dimensionless number =tDeL =H 2 , dimensionless time, dimensionless number =( 3 =)0:25 , Kolmogoro4 microscale, L dynamic viscosity, MT−1 L−1 kinematic liquid viscosity, LT−1 liquid density, ML−3 solid density, ML−3

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