Solids distribution and rising velocity of buoyant solid particles in a vessel stirred with multiple impellers

Solids distribution and rising velocity of buoyant solid particles in a vessel stirred with multiple impellers

Chemical Engineering Science 63 (2008) 5876 -- 5882 Contents lists available at ScienceDirect Chemical Engineering Science journal homepage: w w w ...

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Chemical Engineering Science 63 (2008) 5876 -- 5882

Contents lists available at ScienceDirect

Chemical Engineering Science journal homepage: w w w . e l s e v i e r . c o m / l o c a t e / c e s

Solids distribution and rising velocity of buoyant solid particles in a vessel stirred with multiple impellers D. Fajner, D. Pinelli, R.S. Ghadge,1 , G. Montante, A. Paglianti, F. Magelli ∗ DICMA—Department of Chemical, Mining and Environmental Engineering, University of Bologna, via Terracini 28, 40131 Bologna, Italy



Article history: Received 14 May 2008 Received in revised form 12 August 2008 Accepted 26 August 2008 Available online 24 September 2008 Keywords: Mixing Multiphase flow Suspension Buoyant particles Multiple impellers Solids distribution


The distribution of buoyant solid particles in agitated suspensions has been studied. The investigation was carried out in a baffled vessel characterised by an aspect ratio equal to four and stirred with four radial impellers. Dilute suspensions of single-sized spherical particles of expanded polystyrene (density equal to 90.7 kg/m3 ) in water were used. Solid concentration was measured with a non-intrusive optical technique. Measurements were performed along the axis of the reactor to obtain steady-state vertical profiles (that increase from the vessel base to the top) as well as at fixed elevations to determine their transient after a pulse of solids injected at the bottom. Both the steady-state profiles and the transient concentration curves were interpreted in terms of the axial dispersion model with sedimentation. By data treatment the rising velocity in the agitated system could be determined, which proved to be significantly smaller than the rising velocity in a still liquid. The ratio of these two velocities is in reasonable agreement with a correlation of the ratio of the settling velocities for heavy particles with the ratio of the Kolmogorov microscale to particle diameter established in the past. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction Gas–liquid and solid–liquid agitated vessels and reactors are widely used for a variety of operations in the chemical and process industries and it is broadly recognised that rational design and reliable operation of this equipment depend greatly on proper modelling of their fluid dynamic behaviour. The fluid dynamics of multiphase stirred equipment is very complex and is usually described in terms of single parameters and phenomena. The main aspects are the following: just off-bottom particle suspension, power consumption, liquid mixing time, solids distribution and efficiency of sample withdrawal for solid–liquid systems (Nienow, 1992; Atiemo-Obeng et al., 2004; Nasr-El-Din et al., 1996) and power consumption, overall and local gas hold-up, ventilated cavities at the rear of the impeller blades, bubble size and bubble size distribution, mixing time, gas distribution in the vessel for gas–liquid systems (Middleton and Smith, 2004). Much less attention has been devoted to the behaviour of buoyant solid particles in agitated systems, the exception being the studies on solids drawdown and particle distribution (e.g., Joosten et al., 1977; Kuzmanic and Rušic,  1999;

Corresponding author. Tel.: +39 051 2090245; fax: +39 051 2090247. E-mail address: [email protected] (F. Magelli). On leave from: Chemical Engineering Division, Institute of Chemical Technology, University of Mumbai, Matunga, Mumbai 400019, India. 1

0009-2509/$ - see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2008.08.033

¨ Takahashi and Sasaki, 1999; Bao et al., 2005; Ozcan-Ta skin,  2006). Though the knowledge about the fluid dynamics of multiphase systems has significantly advanced in the last decade, many aspects are still to be fully understood—the drag exerted by the carrier liquid on the dispersed phase being the most relevant one addressed in this paper. The effective particle settling velocity in agitated, dilute suspensions was shown to be smaller than that in a still liquid (Schwartzberg and Treybal, 1968; Levins and Glastonbury, 1972; Nienow and Bartlett, 1974; Kuboi et al., 1974; Magelli et al., 1990; Brucato et al., 1998; Pinelli et al., 2001; Doroodchi et al., 2008) and correlations for its calculation have also been proposed. With this correction taken into account, the vertical solids distribution calculated with computational fluid dynamics (CFD) matches the experimental data fairly well (Montante and Magelli, 2005) and the effect of vessel bottom roughness on the particle just suspended condition can be explained adequately (Ghionzoli et al., 2007). A similar velocity reduction has been invoked to model the behaviour of gas–liquid systems (Bakker and van den Akker, 1994; Lane et al., 2002, 2005; Khopkar et al., 2003, 2004; Khopkar and Ranade, 2006; Kerdouss et al., 2006; Scargiali et al., 2007; Montante et al., 2007), but greater uncertainty exists in this case to get experimental evidence due to the difficulty to deal with these complex systems. The purpose of this paper is to analyse the behaviour of a particulate system where the dispersed phase is made of solids lighter than the agitated liquid: apart from direct interest for some

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fermentation and biotechnological processes, food and mineral processing, waste water treatment and polymerisation reactors and polymer processing, with this model system it is possible to mimic some aspects of the behaviour of a gas–liquid system albeit without bubble formation, deformation, breakage and coalescence. The investigation is based on the measurement of vertical solid concentration profiles at steady-state conditions as well as of local concentration transients. By interpreting the data with a simple phenomenological model along the same lines followed in the past with settling solids (Nocentini et al., 2002; Pinelli et al., 2004), the rising velocity of the particles was determined. The study has been performed in a vessel of high aspect ratio agitated with multiple impellers as this configuration allows to magnify the vertical concentration gradients and, thus, to make the analysis more reliable. 2. Materials and methods 2.1. Experimental set-up and conditions The investigation was carried out in a vertical, cylindrical vessel (diameter T = 23.2 cm, volume V = 39.6 L) characterised by an aspect ratio H/T = 4—identical to those used in previous studies (Magelli et al., 1990; Nocentini et al., 2002; Pinelli et al., 2004). The vessel was made of Plexiglas, equipped with four vertical T/10 baffles and had a flat bottom. The top of the vessel was closed by a lid, whose presence prevented from sucking air into the stirred slurry as gas bubbles would have biased the solid concentration measurements. Agitation was provided with four equal, evenly spaced Rushton turbines mounted on the same shaft (D = 7.87 cm, D/T = 0.34). A schematic diagram of the experimental vessel is shown in Fig. 1(a). The liquid used for the slurries was deionised water. As the solids, spherical particles of expanded polystyrene were used: expandable particles were sieved to get a narrow particle size distribution ( ≈ 0.65 mm), expanded at 80 ◦ C for half an hour and sorted mechanically again. In spite of the experimental care, it was not possible to attain a sharp dimensional and density cut for the particles; their average properties were dP = 1.95 mm and S = 90.7 kg m−3 . The calculated rising velocity of these particles in a still liquid is equal to 19 cm s−1 , which is in the range of the rising velocities of gas bubbles in clean water. Dilute suspensions were employed, in the concentration range Cavg = 0.18–1.26 g L−1 , corresponding to 0.002–0.014 volumetric fraction. The experiments were carried out at room temperature in batch conditions. The rotational speed was in the range 10.5–25 s−1 and the conditions were fully turbulent flow; these values are higher than the just drawdown speed, which for the suspension under study was in the range 8–9 s−1 depending on the mean solids concentration (Takahashi and Sasaki, 1999). The solids concentration in the vessel was measured by means of the non-intrusive optical technique described by Fajner et al. (1985). A laser diode and a silicon photo-diode were used as the light source and the receiver, respectively, and were mounted on a traversing system. The light beam passed through the vessel horizontally along a chord about one centimetre off the shaft, approximately midway between consecutive vertical baffles; each measurement could be considered as representative of the solid concentration on the whole horizontal plane. The measuring system was calibrated for the selected particle fraction. Two kinds of experiments were performed, namely steady state and transient runs. With the former technique, the solids concentration was measured at either 60 or 32 elevations to obtain the vertical solids concentration profile. For the transient technique, a certain amount of solid particles was injected at the base of the vessel ( = 0) in a very short time with a macrosyringe (Fig. 1(b), Pinelli et al., 2004) and the local concentration change with time was detected at selected elevations ( = 0.10, 0.23–0.30 and 0.8).

Fig. 1. The experimental set-up: (a) the geometry of the stirred vessel; (b) the injection device (A: inner surface of the lid; B: breech in the closed position; C: breech in the open position).

Intermediate positions  = 0.6–0.7 were avoided as producing inaccurate DeS estimates due to the curve shape (Nocentini et al., 2002). Each local measurement—either at steady state to construct a vertical profile or transient—was affected by considerable noise due to low solid concentration and relatively big particle size. Therefore, 1000 data points were acquired at each elevation during the steadystate experiments and averaged, while transient measurements were performed at least three times to determine more reliable average trends and values. The steady-state vertical profiles exhibit singularities in correspondence with the impellers as well as midway between them (Pinelli et al., 2001) that result in local departures between the actual experimental profile and the average one given by Eq. (2) below. The above-mentioned vertical positions where to detect the transient curves were chosen at the elevations where these singularities were minimal and the departures of the experimental profile relative to the model curve were small (Nocentini et al., 2002). 2.2. The model for solid particle distribution and experimental curve treatment The flow field in a stirred liquid is recognised to be highly complex in standard vessels and, even more so, in multiple impeller systems. Very complex is also the motion of heavy solid particles (Levins and Glastonbury, 1972; Kuboi et al., 1974; Nouri and Whitelaw, 1992; Guiraud et al. 1997; Montante and Lee, 1999; Virdung and Rasmuson, 2008), while detailed information about the behaviour of buoyant particles is relatively scant apart from the overall behaviour already referenced to in the Introduction.


D. Fajner et al. / Chemical Engineering Science 63 (2008) 5876 -- 5882

The solids distribution in baffled, stirred vessels is usually characterised by fairly limited radial concentration gradients (Yamazaki et al., 1986; Barresi and Baldi, 1987; Bilek and Rieger, 1990; Mak and Ruszkowski, 1990; Guiraud et al., 1997; Montante et al., 2002): consequently, the vertical concentration profiles are their main distinctive feature, especially in vessels of high aspect ratio agitated with multiple impellers. Two opposing factors affect the solids distribution, namely particle settling and the solids dispersion coefficient. Accordingly, the vertical solids concentration distribution in the stirred vessel can be interpreted with a simple one-dimensional sedimentation-dispersion model (Magelli et al., 1990). With the z axis directed upwards (so that the dimensionless coordinates  = 0 and 1 correspond to the base and the top of the vessel, respectively) and for S < L (so that US is positive with buoyant particles), the dimensionless solids concentration under unsteady conditions after an instantaneous solids injection (schematised as Dirac function) at  = 0 is given by (Nocentini et al., 2002) exp(PeS ) + 2 exp(PeS /2) exp(PeS ) − 1 ∞  (n)2 cos{n(1 − )} − n(PeS /2) sin{n(1 − )}

C ∗ (, ) = PeS ×


[(PeS /2)2 + (n)2 ] cos{n} 2

× exp[−{(PeS /2) + (n)2 }]

3. Results and discussion 3.1. Steady-state experiments Due to the lower density of the solid particles with respect to the liquid, the steady-state, vertical concentration profiles increase from the base to the top of the vessel and, thus, are symmetrical to those obtained in the case of S > L (Magelli et al., 1990). A few examples of these profiles are shown in Fig. 2. It can be noticed that the influence of the impeller rotational speed on curve slope is significant: the higher the rotational speed, the flatter is the curve. On the other hand, the shape of the dimensionless vertical concentration profiles is not affected by the mean solids hold-up (Fig. 3), at least in the investigated range. The vertical profiles were fitted with Eq. (2) as discussed above and the resulting PeS values are plotted in Fig. 4. The significant sensitivity of curve fitting to the experimental error at high and low impeller speeds (i.e., solids accumulation at the vessel top and the role of local discrepancies from the model curve especially for nearly vertical profiles) limited the investigated range of rotational speeds. Apart from this, the plot shows that the higher the impeller speed, the smaller the Péclet number, which, in view of Eq. (2), corresponds to flatter curves.

(1) 3.2. Transient experiments

where C* = C/Cavg , PeS = Us H/DeS and  = tDeS /H2 . According to this definition, the PeS value is positive for the buoyant particles. At steady state (i.e., when → ∞ ), Eq. (1) reduces to exp(PeS ) exp(PeS ) − 1


The values of the parameters PeS and DeS for each condition were determined by matching these equations to the experimental profiles by best fit techniques. For the steady-state experiments, PeS was obtained by comparing Eq. (2) to each experimental, dimensionless vertical profile. For the transient experiments, each experimental transient concentration curve at a selected elevation  was normalised with mean solids concentration and the parameter DeS was then determined by fitting the theoretical curve given by Eq. (1) to the normalised experimental one. Parametric model analysis has shown that the DeS value is mainly affected by the ascending part of the curve for a given PeS value. The PeS value obtained from steady-state measurement under the same experimental conditions was used in Eq. (1) for this last calculation. In both cases, the fit between the theoretical and the experimental curves was always fairly good. Since the injection time of the solids in transient experiments was short (about 2–4 s) but not instantaneous, the influence of this parameter on the determined DeS value was investigated. For this purpose the theoretical curves with an imperfect pulse were recalculated by means of the convolution procedure adopted previously for critical situations (Pinelli et al., 2004). Despite limited improvement in the quality of curve fitting and a very small change in DeS value, this technique was used under all conditions. It is worth mentioning that the errors in the estimates of parameters were not negligible. These are due to (i) residual polydispersity of the solid particles (in size and density) after sorting of the solid fraction to be used in the experiments, (ii) very low solids concentration in the transient experiments due to the limited syringe volume and (iii) relatively big particle size with respect to the light beam. A few of these error sources were reduced, but they could not be avoided. Some of these aspects will be addressed further in the Results section.



0.6 z/H

C ∗ (, ) = PeS

The fit between a single experimental and theoretical transient curves is strongly affected by the experimental noise (Fig. 5), which



0 0


2 C/Cavg



Fig. 2. Dimensionless steady-state vertical concentration profiles: influence of the rotational speed. (The associated power consumption per kg of suspension is also given in round brackets). Cs ,av = 0.76 g L−1 ; 䊉: 630 rpm ( = 1.7 W kg−1 ); : 850 rpm ( = 4.2 W kg−1 ); +, 920 rpm ( = 5.4 W kg−1 ); : 1240 rpm ( = 13.3 W kg−1 ); : 1500 rpm ( = 23.8 W kg−1 ).

D. Fajner et al. / Chemical Engineering Science 63 (2008) 5876 -- 5882





C/Cavg (--)





0.4 z/H

experimental data theoretical curve, z/H=0.80

0.0 0



15 t (s)





3.0 experimental data theoretical curve, z/H=0.30

2.5 0.2

C/Cavg (--)


0 0


2 C/Cav


1.5 1.0



Fig. 3. Dimensionless steady-state vertical concentration profiles: influence of mean solids concentration Cs ,av . N = 850 rpm; ×: 0.18 g/L; : 0.36 g/L; : 0.76 g/L; +: 1.26 g/L.

0.0 0






t (s) 8.0

Fig. 5. Example of transient solids concentration curves. N = 1050 rpm; (a)  = 0.30; (b)  = 0.80. Symbols: experimental data; lines: best fit curves with common DeS and PeS values (DeS = 155 cm2 /s, PeS = 2.04).

2.5 4.0

2.0 DeS/DeL

Pe (--)



0.0 400



1000 N (rpm)




Fig. 4. The PeS values as a function of rotational impeller speed. Data points, experimental values; line, data correlation.

is higher at the higher elevation probably due to the locally higher solids concentration. The values of the axial dispersion coefficient for the solid phase, DeS , determined with a best fit procedure were lower for  = 0.10 and 0.23–0.30 than at  = 0.80. This deviation, that was not as limited as that noticed in previous investigations with settling solids (Nocentini et al., 2002; Pinelli et al., 2004), and the error associated to PeS determination result in appreciable differences in DeS estimates. The mean of the coefficients measured at two elevations was retained for the subsequent calculations. The average dimensionless parameter DeS /ND2 in the whole investigated range was found to be equal to 0.17 ( ± 50% error).

1.5 1.0 0.5 0.0 50000


100000 Re (--)



Fig. 6. Ratio of the axial dispersion coefficients for the solid and the liquid phases. (average values for the whole vessel).

The dispersion coefficients for the solids were then compared with those of the liquid phase, DeL /ND2 = 0.13 determined in the past for the same equipment configuration and impeller arrangement in the turbulent regime (Pinelli et al., 2001). The possible influence of the solids on DeL can be disregarded in this study due to the very dilute suspensions (Pasquali et al., 1985). The ratio of the dispersion coefficients of the two phases is plotted in Fig. 6. In spite of the extent of the experimental error and the rather limited number of data


D. Fajner et al. / Chemical Engineering Science 63 (2008) 5876 -- 5882


relationship was obtained:


US /Ut = 0.32 tanh[19(/dp )(/ L )0.5 − 1] + 0.6



1.0 0.8 0.6 0.4 0.2 0.0 0.001


0.1 λ/dp



Fig. 7. Ratio of the settling/rising velocity, US , in stirred systems to the terminal value, Ut . 䊉: buoyant particles; other data points: settling particles (+ × : Rushton turbines of three scales; ♦ : PBT of two scales; : A310 impellers); solid line: correlation for settling solids; dashed lines: ± 30% of the correlation line.

points, the conclusion that these coefficients are (roughly) equal can be extended to the case of buoyant particles, as has been possible in a variety of other cases, like solids settling in vessels stirred with multiple impellers (Brucato et al., 1998; Nocentini et al., 2002; Pinelli et al., 2004), bubble columns (Kato et al., 1972) and loop reactors (Liang et al., 1996) and is expected as a limiting behaviour for particulate systems (Soo, 1978). This order of magnitude result is anyhow sufficient for the following developments. 3.3. Particle rising velocity The particle rising velocity in the stirred tank were calculated from PeS definition, US = PeS DeS /H, and compared with the terminal rising velocity, Ut . This was calculated with the Turton and Levenspiel's correlation (1986). Though affected by rather high error, US is significantly lower than Ut , similar to the findings brought out for settling solids in vessels stirred with various impellers (Fajner et al., 1985; Magelli et al., 1990; Nocentini et al., 2002; Pinelli et al., 2004), in Couette–Taylor flow (Brucato et al., 1998), in a turbulent generator device (Doroodchi et al., 2008). The values of the ratio US /Ut were then compared with one empirical correlation established for solids settling in a dilute, turbulent medium (Pinelli et al., 2001) which is independent of impeller type, vessel scale and liquid viscosity: US /Ut = 0.4 tanh(16/dp − 1) + 0.6


where /dp is the ratio between the Kolmogorov microscale length,  = (3 /)0.25 , and the particle diameter. The influence of solid-toliquid density was disregarded in the mentioned study since the difference in these parameters were fairly limited. An implementation to account for the inertial effects associated to the particle–fluid interaction was suggested in fundamental studies on particle inertia effects in isotropic turbulence (Oesterlé and Zaichik, 2006; Doroodchi et al., 2008) and the analysis of gas bubble motion in a turbulent medium (Porte and Biesheuvel, 2002). A term in the form ( /L )0.5 was deduced from this last study (Scargiali, 2007) and has been implemented in Eq. (3) as a means to improve it.. The correlation constants in Eq. (3) were tuned with the original Pinelli et al.'s data (2001): this results in slight reduction in the average error (about 20% lower variance) and the confirmation that the optimal exponent value in the correcting term is equal to 0.5. The following

The result of data treatment for the buoyant particles is plotted in Fig. 7 and compared with this correlation and the previous data regarding settling solids. The Kolmogorov microscale length, , was calculated from overall power consumption data for single impellers multiplied by the impeller number, by neglecting the local ¨ differences in power dissipation (Barthole et al., 1982; Laufhutte and Mersmann, 1985; Fort et al., 1993; Bourne, 1994; Zhou and Kresta, 1996; Ng and Yianneskis, 2000) as well as impeller interaction (Smith et al., 1987; Hudcová et al., 1989; Montante and Magelli, 2004). In spite of the high error associated to the above-mentioned uncertainties, the new data confirm the significant role of turbulence in reducing the rising velocity of the buoyant particles and exhibit the same behaviour as the settling ones. 4. Conclusions The goal of this investigation was to study the distribution of buoyant solid particles in stirred vessels as well as the features of the parameters that affect it. Experiments were conducted to determine both the vertical, steady-state solids concentration profiles and the transient concentration response at selected elevations to a pulse injection of solids at the vessel base. The vertical profiles and the response curves were interpreted with the sedimentation–dispersion model and the analysis was focused to determine the particle rising velocity. In spite of some experimental problems and intrinsic errors in the measurements, the following conclusions have been obtained: • The experimental vertical solid concentration profiles are significantly affected by impeller speed but provide consistent PeS values. • The experimental transient curves are fitted fairly well with the theoretical ones calculated with the simple sedimentation– dispersion model. However, the axial dispersion coefficients of the solid phase determined at different elevations are different so that taking a single value for the whole system implies a relatively high error in the coefficient. • The axial dispersion coefficient for the solid phase can be considered equal to that of the liquid phase only to a rough approximation (within ± 50% error). • Based on the dispersion coefficients for the solid phase, the particle rising velocities in the stirred systems were also evaluated. The ratio between the rising velocity in the stirred system and that in a quiescent liquid, US /Ut , is confirmed to be as low as 0.15. • A correlation of US /Ut with /dp , established previously for settling particles characterised by small density difference relative to the liquid, has been implemented with a particle-to-liquid density term for broader validity. • Though limited in number and affected by relatively high experimental error, the data of the buoyant particles fit in with this correlation. These results are considered to be useful for the prediction of solids distribution in processes where floating particles are to be dispersed in the liquid. And even if, strictly, they apply only to buoyant particles, they can also help describe the behaviour of gas–liquid systems under idealised conditions (viz, no bubble deformation, breakage and coalescence and low bubble concentration). In both process categories, the remarkable reduction in particle/bubble rising velocity relative to the unstirred case plays a significant role in the spatial distribution of the dispersed phase and, thus, in process performance. In view of the error affecting the experiments, confirmation of these results with alternative techniques would be useful.

D. Fajner et al. / Chemical Engineering Science 63 (2008) 5876 -- 5882

Notation C C* = C/Cavg Cavg D DeL DeS dp H N P PeS = Us H/DeS Re = ND2 L / t T US Ut V z

local volumetric solids concentration, ML−3 dimensionless local solids concentration, dimensionless average volumetric solids concentration, ML−3 turbine diameter, L dispersion coefficient for the liquid, L2 T−1 dispersion coefficient for the solids, L2 T−1 solid particle diameter, L vessel height, L impeller rotational speed, T−1 power consumption, ML2 T−3 Péclet number for the solids, dimensionless rotational Reynolds number, dimensionless time, T tank diameter, L rising/settling velocity of buoyant/settling particles in stirred liquid, LT−1 rising/settling velocity of solid particles in a quiescent liquid (“terminal” velocity), LT−1 tank volume, L3 vertical coordinate (directed upwards), L

Greek letters

 = P/L V  = z/H  = tDeL /H2  = (3 /)0.25   L S 

average power consumption per unit mass, L2 T−3 dimensionless vertical coordinate (directed upwards), dimensionless dimensionless time, dimensionless Kolmogorov micro length scale, L dynamic viscosity, MT−1 L−1 kinematic liquid viscosity, L2 T−1 liquid density, ML−3 solid density, ML−3 |S −S |, density difference, ML−3

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