Solids Concentration Distribution in Slurry Reactors Stirred with Multiple Axial Impellers F. MAGELLI,
and G. PASQUALI
Dipartimento di Ingegneria Chimica e di Processo, Bologna University, viale Risorgimento 40136 Bologna (Italy) V. MARISKO
and P. DITL
Department qf Chemical and Food Process Equipment Design, Czech Technical University, Suchbatarova 4, 16607 Prague 6 (Czechoslovakia) (Received May 18, 1990)
Abstract The solids concentration distribution in solid-liquid suspensions in batch, mechanically stirred, vessels was investigated. The vessels were characterized by a high aspect ratio and were stirred with various combinations of multiple axial impellers. The suspensions were made of glass beads of various sizes in water and aqueous solutions of polyvinylpyrrolidone (PVP). The solids concentration was measured along the vertical axis of the vessel by means of an optical technique. The suspension efficiency of the impellers was compared on the basis of the departure of the solids concentration from suspension homogeneity (given as the variance of the solids concentration along the axis with respect to the mean) versus power consumption per unit mass. The solids concentration profiles were also modelled with the one-dimensional sedimentation-dispersion model having the Pellet number as a single adjustable parameter. For each geometrical configuration the P&clet number was correlated with the ratio of the tip speed to particle settling velocity and a Reynolds number based on the Kolmogorov theory. Once the difference in aspect ratio is taken into account, a single correlation for all the geometrical configurations could be obtained.
1. Introduction Two different aspects must be considered when studying the performance of stirred, solid-liquid, contactors: c‘ omplete suspension’, that is, the minimum rotational speed for off-bottom suspension, and the way the solid particles are distributed throughout the apparatus. Whereas the first condition has been investigated extensively (e.g. refs. 1 and 2) the second has attracted much less interest. A list of relevant papers on the latter subject can be found elsewhere [ 31. A simple way to characterize the distribution is through an index giving the departure of the spatial solids distribution from the state of homogeneous suspension (e.g. the standard deviation of the distribution ). Recently, efforts have been directed towards a more detailed description of the solids distribution. Barresi and Baldi  have provided a general rational framework for the study of this problem and have shown that the mean turbulent P&let number is proportional to the ratio UJND. Tojo and Miyanami  assumed the onedimensional sedimentation-dispersion model for the 0255-2701/91/%3.50
description of the solids profiles in vessels stirred with either a propeller or a vibratory disk. A similar study was performed by Ayazi Shamlou and Koutsakos [A in a tank stirred with a pitch-blade turbine; a detailed investigation of the influence of several variables on the dimensionless parameter of the model was also made. Magelli et al. [ 31 used the same model to study the behaviour of vessels stirred with multiple Rushton turbines and characterized by a high aspect ratio; they put forward a correlation of the model parameter, which is also reliable for scale-up. In this paper the distribution of low concentration, mono-sized particles in vessels of high aspect ratio stirred with multiple axial impellers is studied. Attention is focused on geometrical configurations characterized by the use of either variable-pitch blades  or hydrofoil impellers. When analysed in terms of power consumption per unit mass the various systems exhibit the same performance for each suspension considered (i.e. equal standard deviation with respect to homogeneity). The solids profiles are also interpreted with the one-dimensional sedimentation-dispersion model. An overall correlation of the
29 (1991) 27 32
in The Netherlands
model parameter with both the operating conditions and the suspension characteristics is given; this holds good for all the systems considered.
2. Theory To interpret the solids distribution profiles in the vessels, the one-dimensional sedimentation-dispersion model is adopted . The steady-state mass balance for the solid phase in the stirred suspension within a batch vessel is
The z axis is directed upwards (i.e. z = 0 and z = H correspond to the base and the top of the vessel, respectively). With the conditions
4_=; ,”4(z) dz
the solution of eqn. (1) is
Pe PC exp( - Pe z/H) 1 exp( - Pe) d, where Pe = - U, HID, p. According to the system of coordinates adopted, U, is negative for ps > pL and the P&let number is positive. Suspension inhomogeneity can also be characterized by the standard deviation of the solids concentration with respect to the mean value 4,. If C#J represents the average concentration over the crosssection , then the standard deviation can be expressed as (4) It is intereiting to note that this parameter can be easily calculated from the model as 
Pe exp(2Pe) - 1 _ 1 ‘I2 C = [ Z [exp(Pe) - 112
Fig. 1. The geometrical configuration nos. 1 and 2; (b) geometry no. 3.
of the vessels: (a) geometry
and four variable-pitch two-bladed turbines; for geno. 3: four evenly spaced A310 Lightnin impellers [ 121. The variable-pitch blade of the turbines used in the first two geometries is shown in Fig. 2 [ 10, 111. For each combination of either identical or mixed-type impellers, all the turbines had the same diameter. The power number for each geometrical configuration is also given in Table 1. Apparently, geometry no. 3 is rather different from ometry
3. Experimental The investigation was carried out in vertical, cylindrical, flat-bottomed vessels whose geometrical configuration is summarized in Fig. 1 and Table 1. The vessels were of fully closed design, had Pyrex walls and were equipped with four standard baffles of width 0.1 T. The suspensions were stirred with multiple impeller stirrers, consisting of a shaft with various combinations of axial impellers pumping downwards. The following impellers were used: for geometry no. 1 [ 10, 111: one backswept four-bladed turbine as the lowest impeller and four variable-pitch two-bladed turbines; for geometry no. 2: one variable-pitch, four-bladed turbine as the lowest impeller
Vessel diameter, T (m) Vessel height, H (m) Vessel volume, V (1) No. of impellers, per type: Variable-pitch 2-blade turbine’ Backswept Cblade turbine Variable-pitch Cblade turbine* Lightnin A310 Impeller diameter, D (m) Hs (m) % (m) Power number,b Np
of the vessels Geometry
0.236 0.545 23.8
0.236 0.545 23.8
0.236 0.944 40.1
4 1 0 0 0.124 0.103 0.042 2.3
4 0 1 0 0.124 0.106 0.065 2.1
0 0 0 4 0.096 0.236 0.118 1.2
BFor details of variable-pitch blades, see Fig. 2. bFor Re z l(r, overall value for equipment.
Fig. 2. The variable-pitch geometry nos. 1 and 2.
blade used in the axial turbines of
of the solids (glass
beads, p. = 2.45
0.14 0.33 0.66 0.98
11.3 0.7-4.7 1.7-9.7 3.3-15.0
geometry nos. 1 and 2 in aspect ratio and impeller spacing and size. The liquids used to prepare the model suspensions were water and dilute aqueous solutions of polyvinylpyrrolidone (PVP) exhibiting Newtonian behaviour (viscosity: 0.86-19 mPa s). Various fractions of glass beads of narrow size distribution were used as the solids; their characteristics are given in Table 2. The mean solids concentration was in the range 1.3-3.4 g/l. The experiments were carried out in a batch vessel at room temperature. For each fluid and particle size the rotational speed was roughly 0.8N,, N,, 1.2N, and 1.4N,. The solids profiles in the vessels were measured by means of the non-intrusive optical technique and the pertinent equipment described by Fajner et al. . A light-emitting diode and a silicon photodiode were used as light source and receiver, respectively. The light beam (about 4 mm in diameter) passed through the vessel horizontally along a chord about one centimetre off the axis, approximately midway between the vertical baffles. Thus, each measurement could be considered as the mean of the solids concentration C&(Z)over the whole horizontal section. For each experimental condition, the solids concentration was measured at S = 32 positions along the vertical axis of each vessel and the value of the dimensionless parameter Pe was found by fitting the experimental data with eqn. (3).
viscosity as well as a decrease in particle size tend to flatten the profiles. This general behaviour has already been reported for geometry no. 1  and is quite similar to that described for multiple Rushton turbines  and for pitch-blade turbines [7l. With geometry no. 3 a much higher impeller speed was necessary than with geometry nos. 1 and 2 to get similar profiles. No qualitative difference in behaviour was apparent for the conditions N > N, and N < N,. Obviously, for N c N, a certain amount of solids was stationary on the vessel bottom and C#J,in eqns. (3) and (4) was less than the amount charged to the vessel; its value was calculated with a discretization of eqn. (2b). At first the experimental profiles were analysed in terms of the standard deviation defined with eqn. (4). An example of the results is given in Fig. 3 in terms of u as a function of the power per unit mass: as one would expect, the departure from homogeneity decreases with increase in E (i.e. with increase in agitation speed) and with a decrease in particle size; different behaviour was also apparent for liquids of different viscosity. It is interesting to note that a single interpolating line can be obtained for all the geometries studied, for each particle size and liquid viscosity. This fact allows useful degrees of freedom for design optimization (in terms, for example, of torque, rotational speed, etc.). An example is given in Fig. 4, where the results obtained with Rushton turbines  are also plotted for the sake of comparison. The higher efficiency of the axial impellers in distributing the solids is apparent. The profiles were then interpreted with the onedimensional sedimentation-dispersion model; for each experiment the value of the Pe number that fitted the data best was obtained by means of eqn. (3). For each set of data the P&let values have been correlated with both the operating conditions and the suspension behaviour by means of a relationship of the form  Pe = Ai(U,/ND)ai(~3/~44)Bi
For any geometrical configuration, the solids concentration increases from the top to the bottom of the vessel. An increase in rotational speed and liquid
Fig. 3. Non-homogeneity in the solids distribution (expressed as the standard deviation) for geometry no. 2: 0, d, = 0.14 mm; 0, dp = 0.33 mm; A, dp = 0.66mm; Cl,dp = 0.98 mm (open symbols, y = 0.9 mPa s; solid symbols. fi = 18 mPa s).
3. Constants in eqns. (6), (8) and (9)
Geometry no. 1 (i = 1) Geometry no. 2 (i = 2) Geometry no. 3 (i = 3) All data (i = 0), eqn. (9) Rushton turbines [ 31
536 3208 2410 211 330
1.40 1.89 1.58 1.65 1.17
0.20 0.24 0.22 0.22 0.095
0.986 0.971 0.987 0.969 0.962
0.90 0.91 0.90 0.90 0.93
8 E (W/kg)
Fig. 4. Influence of power consumption per unit mass on solids distribution homogeneity for d,, = 0.33 mm: A, geometry no. 1; 0, geometry no. 2; V, geometry no. 3; Kl, Rushton turbines  (open symbols, b = 0.9 mPa s; solid symbols, p = 7- 11 mPa s).
This equation was shown to be suitable for correlating the data for multiple Rushton turbines  and is consistent with the theoretical treatment given by Barresi and Baldi . It is worth recalling that the last term in the equation is the ratio between the particle size and the mean Kolmogorov microscale; the latter parameter seems to be a suitable length scale, since only eddies which are sufficiently larger than the microscale can be effective in the eddy-particle interaction, by which the solids distribution is produced [ 31. For each of the three geometrical arrangements studied the values of the parameters Ai, cli and /Iiin eqn. (6) were determined through multiple linear regression of the data; the resulting values and the regression coefficient Ri are given in Table 3. As an example, the correlation for geometry no. 1 is also shown in Fig. 5. Based on eqn. (6), the following relationship between N and D can be deduced for a given suspension and the same value of Pe (i.e. the same dimensionless profile) : ND”1 = constant
Fig. 5. Correlation of the data for geometry no. 1 in terms of eqn. (6): V, d,=0.33mm; A, dr,=0.66mm; 0, dp=0.98mm (open symbols, p = 0.93-1.0 mPa s; solid symbols, p = 9.2-16.7 mPa s; half symbols, 5.3-6.0 mPa s).
Although no direct investigation was performed during this study about the effect of varying D (by changing either the equipment scale or D/T)? eqn. (7) also seems acceptable as a scale-up rule in vrew of previous results on multiple-impeller [ 3, 131 and single-impeller equipment [7, 141. Due to the form of eqn. (6), it can easily be shown that ai + 2bi ni=m
The values of n, (also given in Table 3) are essentially the same for all the arrangements; this value (n s 0.90) is in between those provided by the simple criteria of constant tip speed (n = 1) and constant power per unit volume (n = 0.67). Comparison of the three arrangements is not straightforward if performed merely in terms of the sets of parameters given in Table 3. In any case, for any particle size and liquid viscosity, very similar profiles (and thus Pe values) are obtained with geometry nos. 1 and 2 which are characterized by close values of Np. On the contrary, the Pe values provided by the correlation for geometry no. 3 are about three times higher than those for the other two geometries. Indeed, this can be attributed to the major difference in vessel aspect ratio. The following equation can then be proposed to correlate all the data: Pe = A,(HfT)6(U,lND)aO(v3/~dp4)~~
The values of 6 and POin eqn. (9) have been assumed a priori, while A, and a, were determined through linear regression. The reason for this choice (as opposed to the possibility of taking all of them as adjustable parameters) was to get a final equation where the functional dependence from the several parameters is the same as already found in simpler cases, rather than resorting to purely statistical techniques. Accordingly, the following assumptions were made: 6 = 2, /I, = 0.22. The first value was based on results obtained in two vessels stirred with Rushton turbines and characterized by H/T = 4 and H/T = 3 [ 151. The experimental profiles were interpreted with the sedimentation-dispersion model and the results (Pee._) were compared with the correlation obtained for four Rushton turbines  implemented with the ratio of the two different aspect ratios, namely Pecalc= 330(Ut/ND)‘~‘7(v3/~dP4)o~w5(3/4)2 The parity plot given in Fig. 6 supports the choice of 6 = 2. The value of B, is simply the mean of the 8, values obtained with each geometrical configuration
31 1.5 PeEXP
Fig. 8. Comparison of the prediction given by eqn. (5) (solid line) with the data points: 0, geometry no. 1; A, geometry no. 2; 0, geometry no. 3; V, Rushton turbines .
Fig. 6. Experimental results obtained with three Rushton turbines in a vessel with aspect ratio H/T = 3. (Experimental conditions: dr, = 0.33 mm, p = 0.73-7.4 mPa s.)
Fig. 7. Correlation of the overall data: 0, geometry no. 2; Cl, geometry no. 3.
ues for the Rushton turbines that are roughly twice as high as those obtained with the axial impellers, thus stressing the lower efficiency of radial impellers with respect to axial ones. The slight difference in the value of n between the axial impellers and the Rushton turbines cannot be explained either. It is fair to note that the concentration profiles [ 14, 161 obtained experimentally, especially for higher concentrations and larger particles, show a distinct deviation from a monotonic decrease in solids concentration from bottom to top, namely, a local, significantly larger concentration, just above the impellers. However, concentration profiles measured with multiple impellers at low solids concentrations of fine particles were found to be reasonably monotonic [ 31.
geometry no. I; A,
(Table 3). The results for the overall data are given in Table 3 as A,, u, and b, and are shown in Fig. 7. Apparently, the correlation proves quite satisfactory, thus supporting the approach proposed. The parameters 0 and Pe obtained for the three geometries studied in this paper as well as those obtained with Rushton turbines  are plotted against each other in Fig. 8. The solid line gives the theoretical prediction of ~7 as provided by eqn. (5). By assuming a power dependence of 0 on Pe, the best-fit curve in the Pe range O-6 becomes 0 = 0.29Pe0.92
which is much simpler than eqn. (S), thus providing a useful means of getting d from Pe values. The correlation parameters found previously for multiple Rushton turbines  are also given in Table 3 for comparison. Their values are rather different from those found in the present study. Although no explanation for this difference seems possible at present, for equal conditions (i.e. particle size, liquid viscosity, tip speed) the correlations provide Pe val-
The solids distribution in mechanically stirred suspensions has been studied. Various combinations of multiple axial impellers were used in vessels characterized by two different values of the aspect ratio. The performance of various combinations of multiple axial impellers in terms of solids distribution capability has been studied. For each suspension, the departure of the solids concentration profiles from homogeneity depends on the power consumption per unit mass. The variance of the experimental profiles is independent of the specific geometric configuration as far as axial impellers are concerned. A full description of the profiles can be given in terms of the one-dimension sedimentation-dispersion model having a single adjustable parameter (the P&let number). A correlation of the results in terms of this parameter with operating conditions and suspension characteristics has been provided, for each geometrical configuration studied and for the overall data. The present results have also been compared with those obtained previously in equipment stirred with multiple Rushton turbines . Both analyses (i.e. the study of suspension efficiency in terms of solids concentration variance and power consumption per unit mass, and the interpretation of the results with
the model mentioned and their correlation with eqns. (6) and (9)) quantitatively confirm the superiority of axial impellers with respect to radial Rushton turbines for solids distribution in stirred equipment.
kinematic liquid viscosity, m’/s suspension density, kg/m3 liquid density, kg/m3 solid particle density, kg/m3 variance of solids concentration profile, eqn. (4) volumetric solids concentration in crosssection (average value at vertical height z) overall average volumetric solids concentration
Acknowledgements Thanks are due to the Ministries of Foreign Affairs of Italy and Czechoslovakia and to the Czechoslovak Commission for Research, Investment and Development, for granting one of us (VM) a fellowship for carrying out the experimental programme in Italy. The research was supported financially by the Italian Ministry of Education (funds ‘MPI 40%’ and ‘MPI 60%‘). The valuable collaboration of G. P Barbieri in carrying out the experimental programme is gratefully acknowledged. An outline of these results was presented at the 12th Conference on Mixing, organized by the Engineering Foundation and held at Potosi, MO, U.S.A., August 6-l 1, 1989. Nomenclature
‘4 D D d,” ’ H x NP NS
Pe R S T t us u, V Z
air 6 &
constant in eqns. (6) and (9) impeller diameter, m solids dispersion coefficient, m2/s solid particle diameter, m vessel height, m index defining geometrical configuration rotational speed, s- ’ power number rotational speed for ‘just suspended condition’, s- ’ exponent in eqn. (8) power consumption, W = - u,HIQ., p, solids P&let number regression coefficient number of measurement points for each profile vessel diameter, m time, s settling velocity of solid particles in stirred medium, m/s settling velocity of solid particles in quiescent liquid [ 171, m/s vessel volume, m3 vertical coordinate, m exponents in eqns. (6) and (9) exponent in eqn. (9) = P/p V, average power consumption per unit mass, W/kg viscosity, Pa s
referring to geometry no. i referring to overall data together)
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