Scale-up criteria for the solids distribution in slurry reactors stirred with multiple impellers

Scale-up criteria for the solids distribution in slurry reactors stirred with multiple impellers

Chemical Engineering Science 58 (2003) 5363 – 5372 www.elsevier.com/locate/ces Scale-up criteria for the solids distribution in slurry reactors stir...

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Chemical Engineering Science 58 (2003) 5363 – 5372

www.elsevier.com/locate/ces

Scale-up criteria for the solids distribution in slurry reactors stirred with multiple impellers Giuseppina Montante, Davide Pinelli, Franco Magelli∗ DICMA—Department of Chemical, Mining and Environmental Engineering, University of Bologna, Viale Risorgimento 2, Bologna 40136, Italy Received 30 May 2003; received in revised form 1 September 2003; accepted 23 September 2003

Abstract Scale-up criteria for obtaining the same vertical concentration pro/les in agitated suspensions are discussed. The experiments were carried out in reactors of two scales (V = 39:6 and 261 l) characterised by a high aspect ratio and stirred with multiple, evenly spaced impellers of two di4erent types. The pro/les were determined under di4erent conditions: at constant tip speed, at constant speci/c power consumption, and at an intermediate condition (i.e., N ˙ D−0:93 ). The experimental pro/les were compared with di4erent approaches, namely on a qualitative basis, in terms of standard deviation and the parameter of the axial dispersion model with sedimentation. In all cases, the same criterion based on the aforementioned intermediate condition (closer to constant tip speed) was con/rmed. The experimental data were also examined in terms of e4ective particle settling velocity, which is a basic parameter for modelling, and fair agreement of the data obtained at the two scales with the di4erent impellers was obtained. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Mixing; Slurries; Dispersion; Multiphase reactors; Multiple impellers; Scale up

1. Introduction Scale-up is a signi/cant engineering activity that encompasses most of the knowledge needed to design and successfully run equipment at a larger scale (Bisio and Kabel, 1985; Zlokarnik, 2002; Tatterson, 1994, 2003) based on information taken at a small scale. Scale-down, which is usually performed with the purpose of investigating speci/c process features, can be viewed as a part of it. For mixing operations the scale-up criterion is customarily given as an overall equation NDn = const

or

Nr Drn = 1;

(1)

where the exponent n depends on the process(es) tackled. The basic ?uid-dynamic parameters scale in di4erent ways ([email protected] Wernesson and [email protected]@ard, 1999) and a speci/c criterion is strictly valid in a range of conditions corresponding to a well-de/ned regime. Mixing of suspensions is an important operation in the chemical and process industry. The so-called complete

∗ Corresponding author. Tel.: +39-51-2093147; fax: +39-051-581200. E-mail address: [email protected] (F. Magelli).

0009-2509/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2003.09.021

o4-bottom suspension and particle distribution in the bulk of the stirred medium are usually considered as separate tasks (Nienow, 1992), depending on the objective of the speci/c process under consideration. Complete o4-bottom suspension of solid particles has been the subject of many studies and a number of relationships have been given for the minimum impeller speed for particle suspension, which result in scale-up criteria of the type of Eq. (1) with the exponent n varying broadly. This has dramatic e4ects on speci/c power consumption and process productivity at the various scales as shown by Niesmak (1983). The di4erent scale-up criteria have been suggested to depend on di4erences in blade thickness (Buurman et al., 1986) and the speci/c mechanism involved in particle suspension, which are a4ected, in turn, by tank and impeller geometry and size, particle size and density, suspension concentration (Chudacek, 1986; Mak, 1999). Di4erent regimes have been invoked for particle– ?uid interaction (Voit and Mersmann, 1986; Molerus and Latzel, 1987a,b; Rieger and Ditl, 1985, 1994); an exponent variable with particle settling velocity (Myers et al., 1994) and a model based on mechanical energy balance that result in di4erent exponent values (Kraume and Zehner, 2002) have also been suggested. Much less information is available for solids distribution. Buurman et al. (1986) described solids distribution quality

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in terms of the height of the homogeneous zone above the suspension rather than concentration pro/les proper: based on experiments conducted in vessels ranging from 0.24 to 4:26 m in diameter, they arrived at the criterion of Eq. (1) with n = 0:78. The application of turbulence theory to the description of solids distribution (Barresi and Baldi, 1987) leads to the criterion n = 1. Rieger et al. (1988) claimed two criteria, i.e. n = 0:67 for ‘normal’ operation and n = 1 for obtaining highly homogeneous suspensions. In a study on the behaviour of vessels 0.13 and 0:23 m in diameter stirred with multiple radial Rushton turbines, Magelli et al. (1990) arrived at the criterion n=0:93. Similar values for the exponent n were con/rmed for multiple axial impellers— although derived from correlations based on a single vessel size (Magelli et al., 1991). An analysis of solids distribution in vessels of diameter ranging from 0.61 to 2:67 m stirred with a single pitched-blade turbine (PBT) resulted in the criterion n = 0:67 (Mak and Ruszkowski, 1990; Mak et al., 1997). Although the mentioned di4erences in the scale-up criterion do not appear to be great, they do have major e4ects on power consumption at signi/cant scale changes. Therefore, an additional analysis is performed in this paper, aimed at extending the previous one on multiple impeller vessels (Magelli et al., 1990) to a bigger vessel size and a di4erent impeller type and analysing some reasons for the di4erences reported in the literature. 2. Description of the solids distribution in stirred vessels The spatial solids distribution in a stirred vessel is rather complex (Braginskii, 1968; Braginskii et al., 1968) and depends, in particular, on mixer geometry, impeller speed, the generated ?ow pattern as well as the suspension characteristics. In baLed vessels relatively lower solids concentration gradients prevail on horizontal planes than in the vertical direction (Yamazaki et al., 1986; Barresi and Baldi, 1987; Bilek and Rieger, 1990; Mak and Ruszkowski, 1990; Montante et al., 2002) especially above the agitator. This feature is reinforced in multiple impeller systems which can, therefore, be e4ectively characterised in terms of vertical solids concentration pro/les (Magelli et al., 1990; Montante et al., 2002). The analysis of these systems and the evaluation of the concentration pro/les can be conducted in various ways of di4erent complexity. A simple parameter to be used is the dimensionless standard deviation of the actual pro/le relative to vertical homogeneity (e.g., Bohnet and Niesmak, 1980; Barresi and Baldi, 1987)  0:5  2 = (Ci − 1) =i (2) i

or similar indices. It is a4ected by agitator type (Heywood et al., 1992; Pinelli et al., 2001), vessel aspect ratio (Magelli

et al., 1990) as well as suspension properties and operating conditions; although attractive for its immediacy, it lacks a direct physical signi/cance and, therefore, it cannot provide robust, full-purpose application. A more rational description can be attained by resorting to ?uid dynamic models. The unstructured, one-dimensional model with axial dispersion and sedimentation provides the vertical dimensionless pro/le C( ) =

PeS exp(−PeS ); 1 − exp(−PeS )

(3)

where the two factors a4ecting the solids distribution i.e., the gravitational force acting on the particles (through US ) and particle suspension due to stirring (through the dispersion coeOcient DeS )—are lumped into the single, dimensionless parameter PeS = −US H=DeS . Eq. (3) proved successful in the description of the solids concentration pro/les that establish in batch, baLed vessels stirred with multiple impeller of various types (Nocentini and Magelli, 1992; Pinelli et al., 2001), while being unable to describe the slight, yet well-de/ned departures between the actual and the theoretical monotonic pro/les. A simple relationship can be established between PeS and  (Magelli et al., 1990, 1991). A straightforward implementation of Eq. (3) also /ts the average pro/les in semibatch (Magelli et al., 1990) and through-?ow (Nocentini and Magelli, 1992) systems. According to a preliminary approach, the Peclet number can be calculated with empirical relationships of the form PeS = A(ND=Ut ) (3 =dP4 ) (H=T )2 ;

(4)

whose parameters A; ;  are to be found on experimental basis for each mixer con/guration (Magelli et al., 1991). In a more general way, the value of PeS to be used in Eq. (3) can be calculated once both parameters US and DeS are known. The former can be obtained from the empirical relationship (Pinelli et al., 2001) US =Ut = 0:4 tanh(16=dp − 1) + 0:6

for =dp ¡ 0:2 (5)

and the knowledge of the terminal particle settling velocity in a still liquid, Ut , while the dispersion coeOcient of the solid can be taken as equal to the axial dispersion coeOcient of the liquid (Nocentini et al., 2002). The last is, therefore, the only parameter that depends on mixer geometry and scale and can be determined experimentally (e.g., Pinelli and Magelli, 2000) or found in the literature. Additional details of this procedure are given elsewhere (Pinelli et al., 2001). It is also worth noting that, although being based on experiments conducted with Newtonian liquids in vessels stirred with Rushton turbines (Pinelli et al., 1996), Eq. (5) has a more general value: indeed, the same quantitative trend was con/rmed to hold good with hydrofoil impellers (Nocentini and Magelli, 1992) as well as with pseudoplastic liquids (Pinelli and Magelli, 2001).

G. Montante et al. / Chemical Engineering Science 58 (2003) 5363 – 5372

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Apart from several other ?ow models that are not considered here for the sake of brevity, the successful results obtained with computational ?uid dynamics, CFD, are to be mentioned. With an educated selection of the simulation approach, computational scheme and two-phase turbulence model, very good agreement can be obtained between the calculated and the experimental concentration pro/les for di4erent impeller systems (Montante et al., 2000, 2001, 2002) regardless of the adopted code (Montante and Magelli, 2002). It is worth mentioning that, among the fundamental parameters to be adopted in these simulations, an e4ective settling velocity in the stirred medium (which is usually lower than the terminal settling velocity, Ut ) is to be used for a successful prediction, for instance, as given by Eq. (5). 3. Experimental equipment and techniques 3.1. Equipment and experimental conditions The investigation was carried out in two vertical, cylindrical tanks (T =23:2 and T =48 cm in diameter and 0.0396 and 0:261 m3 in volume, respectively; in shorthand notation, T23 and T48 ) similar to those used in previous investigations (Magelli et al., 1990). The vessels were made of Perspex, had a ?at bottom and a lid and were equipped with four vertical T=10 baLes. The bigger vessel had 35 small lateral ports distributed over the whole height—in a narrow zone intermediate between adjacent baLes—for allowing the introduction of an optical probe for local solids concentration measurements (see below). The vessels were geometrically similar in all the details apart from the aspect ratio (H=T = 4 and H=T = 3 for T23 and T48 , respectively) and the number of impellers. Agitation was provided with sets of identical, evenly spaced turbines (4 and 3 in the T23 and T48 , respectively) mounted on the same shaft, namely: four-blade 45◦ pitched blade turbines (PBT) pumping downwards of diameter D = 9:4 cm in T23 and D = 19:5 cm in T48 (i.e., D=T = 0:41) and six-blade Rushton turbines of diameter D = 7:87 cm in T23 (D=T = 0:34) and either D = 19:2 cm or D = 15:1 cm in T48 (D=T = 0:40 or 0.31). The lowest impeller was always at the distance T=2 from the base, while impeller spacing was equal to vessel diameter T . Fig. 1 shows the sketch of the equipment of the H=T = 4 con/guration. The liquids used were water and aqueous solutions of polyvinylpyrrolidone (Newtonian behaviour, viscosity 0.9 and 6 mPa s). As the solids, spherical glass particles of narrow size distribution were used (dP =0:33 and 0:68 mm; S = 2450 kg=m3 ). The terminal settling velocity was in the range 1.3–10 cm=s. The suspensions used in the experiments were dilute (mean solids concentration in the range 5 –15 g=l). All the experiments were carried out at room temperature in batch conditions. The impeller speed was always higher than the just suspended condition Njs .

Fig. 1. The geometrical con/guration of vessel T23 .

3.2. Solids concentration measurement All the experiments were performed in steady-state conditions. The solids concentration was measured at several elevations by means of an optical attenuation technique. A laser diode and a silicon photo-diode were used as the light source and the receiver, respectively, and the solids concentration was determined from the measured light attenuation by means of a calibration curve. In the smaller vessel, T23 , the non-intrusive technique used by Fajner et al. (1985), was applied that is based on traversing the stirred suspension with a light beam along a chord about 1 cm o4 the axis, approximately midway between vertical baLes. Each measurement was considered as representative of the solids concentration on the whole horizontal plane and a vertical pro/le was determined from the measurements at 32 or 64 elevations.

G. Montante et al. / Chemical Engineering Science 58 (2003) 5363 – 5372 1.00

0.75 z/4T(--)

For the T48 vessel, where the whole optical path along a chord was too long so that almost complete light attenuation resulted even with very dilute suspensions, a probe was used that could be inserted into the vessel thus providing local measurements (Montante et al., 2002). It consisted of a tube (0:8 cm in diameter) that encased two optical /bres connected to the diode and the photodiode; a small mirror was /xed to one end perpendicular to the axis, at a distance of 2:4 cm from the /bre tips, thus de/ning the measurement volume. The probe could be inserted horizontally into the vessel at the various elevations through the lateral ports. Four radial positions at each elevation were analysed between the vessel axis and the wall (except at the turbine level where only the two outer positions were accessible); the concentration values at each height were almost constant especially above each turbine (as reported for a variety of vessels stirred with several types of agitators) so that their average could be retained and a vertical pro/le could be obtained as with T23 .

0.50

0.25

0.00 0.0

1.0

0.0

1.0

0.0

1.0

(a)

C (--)

2.0

3.0

2.0

3.0

2.0

3.0

1.00

0.75 z/4T (--)

5366

0.50

0.25

4. Results and discussion (b)

C (--)

1.00

0.75 z/4T (--)

The vertical concentration pro/les obtained at both scales exhibit the same trend discussed previously for baLed vessels (Magelli et al., 1990; Pinelli et al., 2001). Namely, the solids concentration increases from the top to the bottom of the vessel and slight departures from this monotonic trend are apparent at the turbine level (for each PBT) or midway between the turbines (for the Rushton turbines). The dimensionless pro/le shape is essentially the same at each turbine along the vessel height regardless of the speci/c turbine. Two di4erent approaches were followed for assessing the scale-up criteria to attain the same vertical concentration pro/le in reactors of di4erent scale. Firstly, the value of the exponent n in the relationship NDn = constant was tested by simply comparing the pro/le measured in T48 with those determined with the same suspension in T23 under selected agitation conditions—in particular, those corresponding to n = 1 (constant tip speed), n = 23 (constant speci/c power consumption), n = 0:93 (intermediate value, to be justi/ed below). An example of such a direct, qualitative comparison of the three mentioned criteria is shown in Figs. 2(a) – (c) for PBT mixers. The pro/les are plotted in dimensionless form and, because of the di4erent vessel aspect ratio, they do not extend to the same height. As is apparent  fromthe /gures, the criterion of constant speci/c power n = 23 is inadequate since the pro/les exhibit greater discrepancies; out of the other two, the one on n = 0:93 is slightly preferable. It is worth noting that even the pro/le discontinuities at the turbines are reproduced quite well at the two scales. This fact can be better revealed in Fig. 3, where the dimensionless pro/les relevant to each turbine (or ‘module’) of T48 are compared (Fig. 3a) with the average module pro/le of T23 —the last being the

0.00

0.50

0.25

0.00

(c)

C (--)

Fig. 2. Comparison of solid pro/les measured in T48 ( ) and T23 (4) stirred with PBT agitators. dp = 0:33 mm; = 0:9 mPa s; N = 8:07 s−1 in T48 . (a) N = 17 s−1 in T23 (n = 1:02); (b) N = 15:5 s−1 in T23 (n = 0:90); (c) N = 13 s−1 in T23 (n = 0:65).

average of data points at the same elevation in each module of the smaller vessel (Fig. 3b). Similar results were obtained for all the other cases. The only exception was the case of the bigger particles in water agitated by the PBTs for which pro/le superposition was imperfect in correspondence with the singularities at the impellers, probably because of separation tendency of the particles from the main ?ow structures. Another example of data matching is shown in Figs. 4a and b, where the pro/les obtained for the Rushton turbines for the condition n = 0:93 are plotted. The single experimental pro/les were then /tted with Eq. (3) to determine the dimensionless parameter PeS .

G. Montante et al. / Chemical Engineering Science 58 (2003) 5363 – 5372 0.25

1.00

0.75

0.15 ζ (--)

(z-(n-1)T)/H(--)

0.20

0.10

0.00

0.00 0.0

0.5

(a)

1.0 C* (--)

1.5

2.0

0.0

1.0

0.25

0.25

0.20

0.20

0.15 0.10 0.05

2.0

3.0

C (--)

(a)

(z-(n-1)T)/H(--)

(z-(n-1)T)/H(--)

0.50

0.25

0.05

0.15 0.10 0.05

0.00 0.0

(b)

5367

0.5

1.0 C* (--)

1.5

0.00

2.0

0.0

(b)

Fig. 3. Dimensionless pro/les for each PBT (‘module’) in the stirred vessels: dp = 0:33 mm; = 0:9 mPa s. (a) T48 (N = 8:07 s−1 ): , lowest turbine; , middle turbine; , upper turbine; —, average T23 pro/le (N = 15:5 s−1 ); (b) T23 (N = 15:5 s−1 ): , lowest turbine; 4, second turbine from bottom; , third turbine from bottom; , uppermost turbine.



This last parameter was correlated with the operating conditions and suspension characteristics by means of Eq. (4). Figs. 5a and b, show this correlation for the two impeller systems. For broader comparison, data collected in previous investigations have been reworked and added in the plots to those of the present study—in particular, data obtained in T23 with the PBTs (Montante et al., 2001) as well as in both T13 and T23 with the Rushton turbines (Magelli et al., 1990; Pinelli et al., 1996; Nocentini et al., 2002). For both agitation systems, all the data points were /tted by the correlation reasonably well with the following parameters:  = 1:404 and  = 0:161 for the PBTs and  = 1:195 and  = 0:101 for the Rushton turbines, respectively. Thus, Eq. (4), which was originally proposed as a means to correlate the data at a given scale (Magelli et al., 1991), is also adequate for the scale-up. A single equation with  = 1:211 and  = 0:103 can correlate the whole data, although with slightly bigger scatter. It can be easily calculated from Eq. (4) that the condition for having the same PeS value at the two scales with the same suspension for both systems reduces to ND0:93 = constant—the value of the exponent being that tested in the /rst approach. (Actually, the value of the exponent is slightly di4erent for the two geometries: n = 0:91 for the PBTs and 0.93 for the Rushton turbines.)

0.5

1.0 C* (--)

1.5

2.0

Fig. 4. Solid concentration pro/les in the vessels stirred with Rushton turbines D=T = 1=3; suspensions dp = 0:33 mm; = 0:9 mPa s. (a) Comparison of the pro/les measured in T48 ( ) and T23 ( ) at N = 8:49 s−1 and N = 15:5 s−1 (n = 0:93), respectively; (b) T48 (N = 8:49 s−1 ): , lowest turbine; , middle turbine; , uppermost turbine; —, average T23 pro/le (N = 15:5 s−1 ).



As recalled in Section 2, pro/le forecast by means of the above-mentioned model, Eq. (3), is accomplished more rationally by calculating PeS with the relevant DeS and US values. Because this last parameter has been successfully correlated to suspension properties and operating conditions through Eq. (5), it is then useful to ascertain whether data obtained at di4erent equipment scales reconcile with this correlation. The simple procedure adopted to this end is as follows: by taking DeS = DeL (Nocentini et al., 2002)—with the DeL values calculated from published correlations (Pinelli et al., 2001) or determined experimentally as discussed for instance by Pinelli and Magelli (2000)—it is possible to calculate US as −PeS · DeL =H for each experimental condition; once the terminal settling velocity Ut has been determined by means of the literature correlations (e.g., Turton and Levenspiel, 1986), the ratio US =Ut can be calculated and plotted as a function of =dP . The values obtained with this procedure in T48 with both mixer con/gurations are shown in Fig. 6 together with those referring to the Rushton turbines in T13 and T23 ; Eq. (5) is also drawn for reference. As is apparent, the same plot accommodates the data of the three scales fairly well. The PBTs data for T23 also /t in very well, but have been omitted for the sake of clarity. Therefore, it is

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Pe Returb-beta

100.0

10.0

1.0

0.1 1

10

(a)

100

1000

100

1000

ND/Ut

Pe Returb-beta

100.0

10.0

1.0

0.1 1

10

(b)

ND/Ut

Fig. 5. Empirical correlation of Peclet number with the operating conditions and suspension characteristics: (a) Rushton turbines and (b) PBTs. +; T13 ; ×; T23 .

1.2

Us /Ut

48 ;

Table 1 Comparison of solids distribution for the pair of tanks in terms of  values (dP = 0:327 mm; = 0:85 mPa s)

1.4

1.0

Impellers

T (cm)

D (cm)

N (s−1 )



0.8

PBT PBT PBT PBT Rushton Rushton Rushton Rushton

48 23.2 23.2 23.2 48 23.2 23.2 23.2

19.5 9.4 9.4 9.4 15.1 7.87 7.87 7.87

8.07 13.0 15.5 17.0 8.49 13.1 15.5 16.3

0.38 0.52 0.38 0.32 0.64 0.75 0.61 0.56

0.6 0.4 0.2 0.0 0.001

•; T

0.01

0.1

1

Comments N ˙ D−0:65 N ˙ D−0:90 N ˙ D−1:02 N ˙ D−0:66 N ˙ D−0:93 N ˙ D−1:0

λ/d p



Fig. 6. US =Ut as a function of =dP . , data for T48 (both Rushton turbines and PBTs); ×; T23 (Rushton turbines only); +; T13 (Rushton turbines only). —, Eq. (5); · · · · · ·, lines relative to ±30% error.

concluded that Eq. (5) is independent of equipment scale and mixer con/guration, at least for the conditions investigated.

It is worth noting that Eq. (5) suggests di4erent scale-up criteria depending on the considered =dP range. For either =dP ¿ 0:2 or =dP ¡ 0:01, an almost constant tip speed can be deduced as a scale-up rule, while in the intermediate region a complex situation prevails which results in n ¡ 1. Indeed, the existence of di4erent regimes for

G. Montante et al. / Chemical Engineering Science 58 (2003) 5363 – 5372 Table 2 Relative standard deviation for the single turbines (PBT, dP = 0:327 mm; = 0:85 mPa s) Condition T48 ; N T23 ; N Same, Same,

= 8:07 s−1 = 13:0 s−1 N = 15:5 s−1 N = 17:0 s−1

I

II

III

IV

average

0.07 0.09 0.05 0.04

0.16 0.18 0.12 0.11

0.11 0.18 0.13 0.12

— 0.19 0.14 0.13

0.11 0.16 0.11 0.10

I refers to the lowest turbine, III and IV to the uppermost turbines.

complete o4-bottom suspension has been demonstrated by several authors (Rieger and Ditl, 1985, 1994; Voit and Mersmann, 1986; Molerus and Latzel, 1987a,b): although the speci/c mechanisms involved for solids suspensions at the vessel bottom and solids distribution in the bulk do not coincide, a link exists between the two processes (Walton, 1995). In fact, the same scale-up criterion has been arrived at regardless of the regime. The criterion derived in the foregoing is much closer to that of constant tip speed than of constant speci/c power. This result reinforces the commonly accepted idea that circulation plays a major role in solids distribution. By no means is this to be viewed as contradictory with respect to turbulent eddy–particle interaction as the basic mechanism invoked for the particle dispersion process. Apart from the fact that the turbulent theory results in n = 1 when applied to a whole vessel (Barresi and Baldi, 1987), apparently the two mechanisms act in such a way that the local phenomena are controlled by turbulence, while strongly supported by bulk circulation and convective mixing. The common result arrived at by means of the abovementioned approaches, i.e. that an overall scale-up rule

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Nr ˙ Dr−n holds for solids distribution with n equal to about 0.93, is in disagreement with that presented in similar analyses conducted by Mak and Ruszkowski (1990) and Mak et al. (1997). Since these authors evaluated their solids distributions in terms of the relative standard deviation, , our pro/les have been reworked to calculate this parameter also. For the T23 vessel, only the pro/le portion corresponding to three turbines (the lowest or the uppermost ones) has been considered—in either case the results were almost identical for the dP = 0:33 mm particles and very similar for the 0:675 mm particles. Table 1 shows the results for two speci/c conditions: the reliability of a scale-up criterion of about n = 0:93 is con/rmed. The results for the other cases are very similar. The same analysis has also been extended to each turbine (or ‘module’) of the reactors, while calculating the average concentration CS; av; j for each pertaining zone. The resulting values of the relative standard deviation are rather variable from module to module (up to about ±50%) relative to their average (see Table 2 and Fig. 3). No /rm conclusion can be drawn from this analysis except that the scale-up criterion is intermediate between constant speci/c power and constant tip speed. In addition, the standard deviation for the lowest turbine is seen to be lower than that of the upper ‘modules’. Indeed, the intrinsic error in the experimental technique and the relatively low number of data points per module do not permit to extend this statistical analysis further and to strictly rely upon its results. A review of Mak et al.’s experimental technique, methodology and investigated conditions did not reveal reasons that could justify the di4erent scale-up criteria arrived at, but the fact that a number of experiments were conducted at N ¡ Njs (Mak, 1999), a condition where the evaluation of the mean solids concentration of the suspension and,

Table 3a Comparison of Mak’s equipment and conditions with those of the present paper

Vessel size (diameter) Particle size and material Base shape No. of impellers Agitation conditions Analysis of distribution No. of sample points Mean concentration

Mak’s papers

This work

T = 0:61; 1:83; 2:67 m 0.15 –0:21 mm sand Torispherical bottom One N ¿ Njs as well as N ¡ Njs  5 vertical (×4 in T183 ) Up to 30% wt

T = 0:13; 0:23; 0:48 m 0:13; 0:33; 0:68; 1:1; 3:0 mm Flat bottom Multiple N ¿ Njs only Several methods 32 to 64 vertical (×4 in T48 ) 1% wt (up to 10% wt)

Table 3b Comparison of Mak’s conditions with this paper: parameters and regimes Parameter Ar dP =T =dP a Mostly,

b Mostly,

Critical value 40 ≈ 3 × 10−3 — Ar ¿ 40. below the critical value.

Mak’s papers 105

≈3× 8 × 10−5 to 3 × 10−4 ¿ 0:12

This work 106a

1 to 2 × 6 × 10−4 to 1:4 × 10−2b 0.06 –0:03, see Fig. 6

References Molerus and Latzel, 1987a,b Rieger and Ditl, 1994 Magelli et al., 1990

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therefore, of the relative standard deviation is rather critical. Three main di4erences regarding the analysed systems mark their and our studies (Table 3a) and deserve some comments: bottom shape (torispherical vs. ?at), the use of one vs. multiple impellers, the size of the vessels (0.61, 1.83 and 2:67 m vs. 0.13, 0.23 and 0:48 m). Flat-bottomed vessels usually scale with slightly higher n values for complete suspension than pro/led-based ones (Chudacek, 1986). However, the above-mentioned result of Barresi and Baldi (1987) obtained in a dished-bottom vessel weakens this possible explanation for the discrepancy. Moreover, our result is qualitatively in agreement with Buurman et al.’s /ndings, based on the study of dished-bottom vessels, according to whom on a large scale homogeneity is attained at a higher speci/c power input than on a small scale. On the other hand, it is likely that for impellers other than the bottom one, turbulence mechanisms (Barresi and Baldi, 1987) remain the sole responsible factors, as they simply reinforce the liquid–particle interaction without any direct e4ect on o4-bottom suspension (Armenante and Li, 1993), thus implying n = 1. As for the di4erence in vessel sizes, only speculations can be made. If T ≈ 0:5 m would be a critical dimension discriminating di4erent regimes, then no real con?ict existed between the results under discussion: but the occurrence of di4erent regimes, as de/ned by a number of criteria suggested in previous studies on solids suspension, does not seem to play a role in this case; indeed, the relevant parameters are mostly or completely in the same region (see Table 3b) in both studies.

C∗ CS CS; av CS; av; j dp D DeL DeS H j n N Njs P PeS Returb T US Ut V z

CS ( )=CS; av; j , dimensionless solids concentration in each ‘module’, dimensionless solids concentration, ML−3 average solids concentration, ML−3 average solids concentration of ‘module’ j (from −T=2 to +T=2 relative to each turbine), ML−3 solid particle diameter, L turbine diameter, L dispersion coeOcient for the liquid, L2 T−1 dispersion coeOcient for the solid, L2 T−1 vessel height, L number of turbines; also: turbine and ‘module’ number from the base, dimensionless exponent in Eq. (1), dimensionless rotational speed, T−1 minimum impeller speed for solids suspension, T−1 power consumption for a single impeller, ML2 T−3 −US H=DeS , PTeclet number for the solid, dimensionless dP4 =3 = (dP =) 4 , turbulence Reynolds number, dimensionless vessel diameter, L settling velocity of solid particles in a stirred liquid, LT−1 settling velocity of solid particles in quiescent liquid (“terminal” velocity) LT−1 tank volume, L3 vertical coordinate, L

Conclusions

Greek letters

The solids distribution in vessels of di4erent scales stirred with multiple PBTs and Rushton turbines was investigated for de/ning a suitable scale-up rule for obtaining equal solids concentration pro/les. Di4erent approaches were followed, all leading to a criterion N ˙ D−n with n equal to about 0.93. The discrepancy with the result of a similar study (Mak and Ruszkowski, 1990; Mak et al., 1997), where N ˙ D−0:67 was found, cannot be easily explained. Possible reasons have been evidenced and discussed, but a /rm conclusion on this matter is not possible at present. In view of the slight di4erence in the value of the exponent found in this study, the conservative criterion of constant tip speed is, therefore, recommended.



Notation Ar C

dp3 L gS= 2 , Archimedes number, dimensionless CS ( )=CS; av , dimensionless solids concentration, dimensionless

  L 

jP=L V , average power consumption per unit mass, L2 T−3 z=H , dimensionless vertical coordinate, dimensionless (3 =)0:25 , Kolmogorov microscale, L dynamic liquid viscosity, MT−1 L−1 kinematic liquid viscosity, L2 T−1 liquid density, ML−3 relative standard deviation of the vertical solids distribution, Eq. (2), dimensionless

Subscripts r

relative to the ratio of parameters at large and small scale, dimensionless I,II,III,IV referring to each turbine starting from the base (modules of height T ), dimensionless

Acknowledgements This work was /nancially supported by the University of Bologna and the Italian Ministry of University and

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