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Scale-Up of Solids Distribution in Slurry, Stirred Vessels Based on Turbulence Intermittency G. Montante,† J. R. Bourne,‡ and F. Magelli*,† Dipartimento di Ingegneria Chimica, Mineraria e delle Tecnologie Ambientali, UniVersita` di Bologna, Viale Risorgimento 2, 40136 Bologna, Italy, and 1, King Stephen’s Mount, Worcester WR2 5PL, UK
Scale-up criteria for obtaining the same vertical concentration profile of particles in agitated, dilute suspensions are considered. Previous experiments in two sizes of high aspect ratio vessels stirred by multiple impellers (either pitched blade or Rushton turbines) gave n ) 0.93 in the scale-up relationship32 NDn ) constant. This result is intermediate between constant power per unit volume (n ) 2/3) and constant tip speed (n ) 1), although closer to the latter. It is now suggested that intermittent turbulent fluctuations greater than their average values play a part in maintaining vertical concentration gradients. An analysis of solids suspension by intermittent turbulence gives n ≈ 0.93. This value is also supported by additional experimental data as well as by new CFD results. Introduction Stirring operations in solid-liquid equipment and reactors are widespread in several sectors of the chemical, process, pharmaceutical, and mining industries. It is well recognized that the overall performance of this type of apparatus is greatly affected by the fluid dynamic behavior, which is usually described in terms of either particle off-bottom suspension or solids distribution throughout the vessel.1,2 This last aspect has been described in terms of either the height of the clear liquid layer above the suspension proper or the solids concentration profilessthe vertical gradients being much steeper than the radial ones.3-7 Indeed, the overall behavior of such a stirred medium is the result of the complex interaction occurring between solid particles and liquid turbulence:8 fluid turbulence anisotropy, large differences in eddy length scales, particle dynamics, and spatial distribution of these factors as well as their consequences for the local fluid-particle interactions and turbulence modulation make a detailed description almost impossible to achieve at present. Therefore, simplifications and schematisations are adopted, the nature of which depends on the specific objective. Nonhomogeneous solids distribution may be responsible for differences in conversion9 and selectivity10 of chemical transformation processes, may affect the solids concentration in the exit conduit in continuous flow processes,11,12 and may influence the behavior of crystallizers.13-15 In medium- to high-concentration systems, dramatic increases in liquid mixing time have also been reported, at least for specific operating conditions.16-18 Computational fluid dynamics (CFD) methods have been developed for the simulation of solid-liquid distribution in stirred vessels and entirely predictive Eulerian-Lagrangian and Eulerian-Eulerian models7,19-21 and large eddy simulation22,23 have been successfully employed in the past few years. For baffled vessels and dilute suspensions in particular, these methods have proved to be quite reliable and capable of predicting the actual equipment behavior. One important step in equipment design is vessel scale-up from the conditions established at the laboratory or pilot scale * To whom correspondence should be addressed. Tel.: +39 051 20 93147. Fax: +39 051 58 1200. E-mail:
[email protected]. † Universita` di Bologna. ‡ King Stephen’s Mount.
to the industrial size. While general criteria have been established for mixing operations at large,24-27 that result in relationships of the form
NDn ) constant
(1)
rather scarce attention has been paid to the scale-up of solids concentration distribution. And when this has been done, the problem has been tackled mainly from an empirical point of view. Bourne and Hungerbuehler13 showed that a homogeneous suspension could be obtained in a continuous MSMPR crystallizer when the tip speed is kept constant (n ) 1). Buurmann et al.28 described the solids distribution quality in terms of the height of the homogeneous zone and arrived at the criterion n ) 0.78, although only limited experimentation was possible in the larger tank. They also explained differences in scale-up criteria available in the literature for off-bottom suspension as the result of different blade thickness. The application of turbulence theory to the description of solids distribution4 leads to the criterion n ) 1. Rieger et al.29 claimed two criteria, i.e., n ) 0.67 for “normal” operation and n ) 1 for highly homogeneous suspensions. In studies on the fluid dynamic behavior of vessels stirred with multiple radial or axial turbines, Magelli et al.30,31 proposed an empirical correlation for describing the vertical profiles, from which the criterion n ) 0.93 can be derived; the same criterion has been confirmed recently32 based on direct profiles comparison. An analysis of solids distribution in vessels stirred with a single pitched-blade turbine6,33 resulted in the value n ) 0.67. Approximately constant specific power (n ) 0.52-0.86) can be deduced as a criterion from the paper by Angst and Kraume.34 Ochieng and Lewis35 suggested that for low solids loadings these systems scale with impeller tip speed. Geisler et al.36 concluded that there is no simple and constant scale-up rule to describe the power input for a large range of suspension properties, vessel and impeller type, and sizesto get the same suspension quality with comparable criteria. Although the mentioned differences in the scale-up criterion do not appear to be great, they do have major effects on power consumption at significant scale changes, which suggests additional investigation to be performed. In this paper, an attempt is made to provide a theoretical framework for the scale-up of solid-liquid stirring equipment
10.1021/ie070339v CCC: $40.75 © 2008 American Chemical Society Published on Web 07/17/2007
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for keeping the same dimensionless solids concentration profile. Simplified fluid dynamic conditions are considered and the influence of turbulence intermittency is addressed as fundamental mechanism. The results of this theoretical analysis are checked with selected experimental data obtained in previous investigations as well as with the results of CFD simulations conducted in vessels of different scales. High aspect ratio vessels stirred with multiple impellers are adopted as a suitable benchmark in both approaches since they provide greater vertical gradients that allow easier and more significant comparison. Particle sizes in the inertial subrange and dilute suspensions are considered. Particle Settling Velocity and Implications for Scale-Up Solids distribution in a stirred vessel is the result of complex interaction between the particles and the fluid. At a local level, equilibrium between settling and suspension forces is relevant and for a single spherical particlesneglecting interphase forces other than dragsit reads as
πR2τ(dp) ) 4πR3(FS-FL)g/3
(2)
In the case of a still liquid, the drag on the particle has an unambiguous value resulting from the integration of the distributions of surface pressure and vorticity over the whole particle surface.37 Putting the expression
τ(dp) ) CDSVt2FL/2
(3)
in eq 2 gives the well-known expression of the steady-state “terminal” velocity of a particle in a still liquid. In a turbulent medium, the stresses are affected by the fluctuating nature of turbulence, and therefore, the particle experiences continual acceleration and deceleration due to the interaction with turbulent eddies, which depends on position and time. Even if considering average conditions of “free” turbulence, a reduction in the average settling velocity (and, thus, an increase in the drag coefficient) relative to the condition of still liquid has been reported in the literature for particles having higher density than the fluid in stirred vessels,30,38-41 in simple vertical conduits with grid-generated turbulence42 and with Couette-Taylor flow43 and oscillatory flow.44,45 Similar phenomena have been described for bubbles and drops rising in a denser liquid.46-49 Differences in these effects are likely to be found depending on the assumed turbulence structure, as has been thoroughly discussed for the case of liquid dispersion in a turbulent field.50-52 In the case of uniform “free” turbulence, the mean value of energy dissipation, 〈〉, can be simply evaluated for a dilute suspension as
FLN3D5 P ) NP ∝ NPN3D2 〈〉 ) F LV F LV
(4)
whence the average stress acting on the particle can be estimated as
Vs2 F ∝ CDT(〈〉dp)2/3FL τ(dp,NP) ) CDT 2 L
(5)
This is essentially the case where the applicability of Kolmogorov theory is assumed. The drag coefficient CDT is a constant, spatially uniform quantityshigher than that in a still liquid. Correlations for the average settling velocity, Vs, normal-
ized with the terminal velocity, Vt, have been set forth for this case.43,53 Needless to say, the stresses given by eq 5 are no more connected with translational velocity only. In the present context, fine-scale intermittency refers to fluctuations in the local instantaneous energy dissipation rate, , about its local mean value, 〈〉. Its recognition provides a more comprehensive description of the turbulence than the Kolmogorov theory used to derive eq 5 and resulting in n ) 2/3. Intermittency means that the particle is subject to random turbulent events. For those particles whose diameter falls within the inertial subrange (i.e., λK < dp < L), the fluctuating stresses associated with such events can be characterized by the scaling exponent, R. This last has been used to describe intermittency by means of a multifractal approach,54 so that the fluctuating local stresses are given by
τ(dp,R) ) CDT(〈〉dp)2/3FL(dp/L)2(R-1)/3
(6)
The parameter L represents the scale of large, energy-containing eddies and is of the same order as the integral scale of turbulence. For a stirred tank, it is proportional to the impeller diameter
L ) aD
(7)
where the proportionality constant a depends on the impeller type and the position. Similarly, the local mean turbulent energy dissipation rate, 〈〉, that affects particle behavior is proportional to the average specific dissipation rate
〈〉 ) b 〈〉
(8)
where b is a proportionality constant. A number of papers36,55-61 have discussed the spatial distribution of 〈〉, and the value of constant b is reported to depend on impeller type, the position, and D/T. Scale-up of the solids distribution in a stirred vessel implies the attainment of the same dimensionless vertical profiles in equipment of the same geometry (i.e., same a and b) and different size. The scale-up criterion then is
τ(dp,R) ∝ const
(9)
where for a given suspension (i.e., same values of dp, FS, FL, µ)
D2(R-1)/3 〈〉2/3
) ND(1-R/3) ) constant
(10)
For the most active turbulent fluctuations, the scaling exponent is in the range R ) 0.12-0.33,10 while in the nonintermittent case (Kolmogorov theory), it is R ) 1. For R ) 0.12 and R ) 0.33, the scale-up criterion described by eq 1 becomes, respectively,
ND0.96 ) constant
and
ND0.89 ) constant
(11)
with an average value equal to n ) 0.925. While for R ) 1 (that is, homogeneous dissipation rate), eq 10 gives
ND2/3) constant
(12)
Therefore, if the vertical solids concentration gradient is maintained by the strongest velocity fluctuations, the exponent n in eq 1 should be close to, but lower than 1, but definitely higher
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Figure 1. Geometrical configuration of the stirred vessels.
Figure 2. Comparison of experimental solid concentration profiles (dp ) 0.23 mm, Fs ) 2.46 kg/L, FL ) 0.996 kg/L) in T13 and T23 vessels (H/T ) 4, D/T ) 1/3). Horizontal lines: vertical position of the turbines. O, T13, N ) 25 s-1, µ ) 2.33 mPa‚s; b, T23, N ) 15.3 s-1, µ ) 2.02 mPa‚s.
than 2/3. Constant tip speed will then be a better approximation as the scale-up rule than constant power per unit volume. Analysis of the Profiles and Validation of Theory Detailed comparison of experimental, vertical solids concentration profiles obtained in two vessels (T ) 0.23 and 0.48 m) stirred with either Rushton or pitched-blade turbines has been effected by Montante et al.32 for dilute suspensions. The validity of the scale-up criterion given by eq 1 with n ) 0.93 was recognized for those conditions, which is practically identical to the mean (n ) 0.925) of the exponents in eq 11. To broaden the scope of the comparison and find additional evidence, two different approaches have been followed in this work: (i) analysis of data obtained in the past in vessels of two sizes30 and not examined in the above-mentioned paper and (ii) CFD simulation of solids distribution in vessels of three sizes. In both cases, nonstandard vertical, cylindrical vessels have been considered, similar to those used in the previous investigation.32 They had a flat bottom and a lid, were equipped with four vertical baffles of width equal to T/10, and were geometrically similar in all the other details. The aspect ratio was either H/T ) 4 or H/T ) 3, and accordingly, the number of impellers was 4 or 3. Agitation was provided in each vessel with sets of identical, evenly spaced Rushton turbines (T/D ) 3 and S/D ) 3) mounted on the same shaft; the lower impeller was at the distance S/2 from the base. A sketch of the vessel types is shown in Figure 1. Additional details regarding vessel and impeller size are given below. In both approaches dilute mean solids concentrations were considered. Particle size was always in the inertial subrange. Comparison of Additional Experimental Profiles. The program developed in a previous investigation30 was meant to determine vertical solids concentration profiles to be used for modeling purposes. Scale-up was of no interest in that context, so that experiments were not planned to be performed with the same suspensions at the two scales. Nevertheless, since over 170 experiments were conducted in the two vessels, it has been checked as to whether pairs of profiles existed that could validate further (or exclude) the criterion given by eq 11. The vessels used in that investigation were of two scales (T ) 0.132 m and T ) 0.236 m, T13 and T23 in shorthand), characterized by H/T ) 4 and stirred with four Rushton turbines (D/T ) 1/3, D ) 0.0434 m, and D ) 0.0787 m at the two scales, respectively). Though no pairs of data sets had been obtained with exactly the same suspension (i.e., particle size and density and liquid
Figure 3. Comparison of experimental solid concentration profiles (d p ) 0.23 mm, Fs ) 2.46 kg/L, FL ) 0.996 kg/L) in T13 and T23 vessels (H/T ) 4, D/T ) 1/3). Horizontal lines: vertical position of the turbines. (a) b, T23, N ) 18.8 s-1, µ ) 0.919 mPa‚s; O, T13, N ) 36.8 s-1, µ ) 0.776 mPa‚s. (b) b, T23, N ) 18.8 s-1, µ ) 0.919 mPa‚s; O, T13, N ) 30.8 s-1, µ ) 0.776 mPa‚s.
viscosity), which would have allowed straightforward comparison, a number of useful profiles was found. A means to overcome this constraint, though for limited viscosity differences (and same Fs and FL), is to calculate the impeller speed that would be necessary to have one of the profiles with the viscosity of the other by means of the empirical relationship N ∝ µ0.2, which can be easily obtained from the correlation of the Pe´clet number for a suspension stirred with Rushton turbines in a given vessel.31
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Figure 4. Comparison of CFD solid concentration profiles in T48 and T23. (a) ), T48; lines, T23 (solid line, n ) 1; dotted line, n ) 0.67); (b) ), T48; lines, T23 (solid line, n ) 0.96; dotted line, n ) 0.89). Horizontal lines: vertical position of the turbines.
Figure 5. Comparison of CFD solid concentration profiles in T48 and T100 vessels. (a) ), T48; lines, T100 (solid line, n ) 1; dotted line, n ) 0.67); (b) ), T48; lines, T100 (solid line, n ) 0.96; dotted line, n ) 0.89). Horizontal lines: vertical position of the turbines.
One example is shown in Figure 2 that refers to the following conditions. T13: N ) 25 s-1, dp ) 0.23 mm, Fs ) 2.46 kg/L, Cav ) 2.0 g/L, FL ) 0.996 kg/L, µ ) 2.33 mPa‚s. T23: N ) 15.3 s-1, dp ) 0.23 mm, Fs ) 2.46 kg/L, Cav ) 3.1 g/L, FL ) 0.996 kg/L, µ ) 2.02 mPa‚s. As is apparent, the fit of the two profiles is very good. Though the employed impeller speeds give n ) 0.83, it is to be noted that suspensions of slightly different viscosities were used at the two scales: in fact, lower viscosity at the smaller scale would have implied an higher rotational speed to maintain the same matchsthus a value of n higher than 0.83. With a 15% smaller viscosity in the smaller vessel, a 4% increase in N can be calculated by means of the above-mentioned normalization procedure, which yields n ) 0.92. Another example is shown in Figure 3 that refers to the following condition pairs. Figure 3a, T23: N ) 18.8 s-1, dp ) 0.23 mm, Fs ) 2.46 kg/L, Cav ) 1.8 g/L, µ ) 0.919 mPa‚s. T13: N ) 36.8 s-1, dp ) 0.23 mm, Fs ) 2.46 kg/L, Cav ) 3.1 g/L, µ ) 0.776 mPa‚s. Figure 3b, T23: N ) 18.8 s-1, dp ) 0.23 mm, Fs ) 2.46 kg/L, Cav ) 1.8 g/L, µ ) 0.919 mPa‚s. T13: N ) 30.8 s-1, dp ) 0.23 mm, Fs ) 2.46 kg/L, Cav ) 3.1 g/L, µ ) 0.776 mPa‚s. The calculated scale exponents are n ) 1.13 for the curve pair in Figure 3a and n ) 0.83 for the curves plotted in Figure 3b. Contrary to the previous case, a higher viscosity would have been required at the smaller scale, which would have implied lower rotational speed and lower n values for both cases. With the same viscosity in both vessels, by applying the abovementioned procedure, the exponents would be n ) 1.06 (for
Figure 3a) and n ) 0.76 (for the case in Figure 3b, exhibiting the poorer match). In no case was the constant specific power criterion (n ) 0.67) found to provide a good match of the profiles obtained at the two scales. Simulation of the Vertical Solids Concentration Profiles. Among the available approaches for the simulation of solidliquid mixing, the Eulerian-Eulerian model has been adopted in view of its simplicity and effectiveness.7,19,20 The stirred vessels were simulated by solving the RANS equations and using the sliding grid method. The computational domain consisted of a structured grid of about 207 900 cells (165 × 35 × 36 cells along the axial, radial, and angular directions, respectively) over an azimuthal extension of π, equivalent to 415 800 in the whole vessel. The continuous phase and the dispersed solid fraction were modeled with the “multifluid model”,20 according to which the phases are regarded as interpenetrating continua. For modeling the turbulent dispersion of the solids fraction, the eddy diffusivity hypothesis was adopted. The momentum transfer between the liquid and the solid phase was taken into account only by the drag force, thus neglecting Magnus lift, Saffman lift, Basset history, and added mass forces. The momentum transfer between the liquid and the solid fraction is dependent on the settling velocity, Vs, which was computed by adopting the following correlation that holds good for dilute suspensions53
(
)
Vs 16λK ) 0.4 tanh - 1 + 0.6 Vt dp
(13)
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In this equation, the settling velocity decrease (and drag coefficient increase) relative to the value in a still fluid is conceived to be affected by the strength of particle-eddies interaction;30 inertial effects are not considered in this equation. The Kolmogorov microscale, λK, is the length scale to which the size of the intervening energetic eddies are compared and was evaluated from the average power dissipation, 〈〉, based on the experimental power number. In turn, the particle terminal velocity, Vt, was computed using a standard correlation.62 It is worth noting that eq 13 allows one to take into account the effect of free turbulence on the particle settling velocity in an average way. Due to their definition, the ratio Vs/Vt is correlated simply to the square root of the ratio of the drag coefficients in the still and the turbulent liquid, (CDT/CDS)0.5. For the RANS equations closure, the “homogeneous” k- model was employed. For each simulation, periodic boundary conditions were adopted on the azimuthal direction, and the calculations were started from still fluid and homogenously distributed particles in the vessel. No-slip boundary conditions via conventional “wall functions” were adopted at the walls. The other details on the numerical solution procedure do not differ from those adopted in previous works20 and, therefore, are omitted here. For the solution, the commercial CFD code CFX-4.3 was used. To obtain the vertical solid concentration profiles, the mean of the radial values at selected heights were calculated. Three vessels of different scales (T ) 0.23, 0.48, and 1.00 m) were considered for the simulations, characterized by H/T ) 3 and stirred with three Rushton turbines (D/T ) 1/3). The suspensions were made of spherical solid particles (dP ) 0.33 mm, FS ) 2.45 kg/L) and a Newtonian liquid (µ ) 5 mPa‚s, FL ) 1 kg/L). As impeller speeds, N ) 500 rpm was selected for the T48 vessel and different values in the other two vessels in order to check the validity of the scale-up criteria n ) 1, 0.96, 0.89, and 0.67. In each case, N was greater than the just suspended condition. The results of the simulations are shown in Figures 4a,b and 5a,b for the comparison of the dimensionless, vertical concentration profiles of T23 versus T48 and T100 versus T48, respectively. It is clear that the criteria given by eq 11 are better than the other two tested cases. In particular, the checked criterion of constant specific power (n ) 0.67) is definitely inadequate, while constant tip speed can be considered a satisfactorily good approximation. Conclusions Measured and calculated vertical concentration distributions of fully suspended solid particles in dilute conditions in geometrically similar vessels, having high aspect ratios and multiple impellers, correspond when n ≈ 0.93, where n is the exponent on the vessel size in the scale-up relationship NDn ) constant. A simple balance between a particle’s weight and the turbulent drag acting upward on a particle, including a multifractal formulation for eddies of different intensities, leads to values of n ≈ 0.9, when suspension is attributed to the more vigorous eddies. This result is closer to the well-known scaleup rule of constant tip speed (n ) 1), rather than to constant power input per unit volume (n ) 2/3). Acknowledgment This work was financially supported by the Italian Ministry of University and Research (MIUR) and the University of Bologna under project PRIN 2005.
Nomenclature a, b ) constants in eqs 7 and 8 C ) local mass concentration of solids (kg m-3) Cav ) average mass concentration of solids (kg m-3) CDS ) drag coefficient, in still liquid (-) CDT ) fluctuating drag coefficient, in turbulent liquid (-) CDT ) average drag coefficient, in turbulent liquid (-) D ) impeller diameter (m) dp ) particle diameter (m) g ) gravity acceleration (m s-2) H ) vessel height (m) L ) scale of large eddies (m) N ) agitation speed (s-1) NP ) power number (-) n ) exponent in eq 1 (-) P ) power consumption (W) R ) particle radius (m) S ) turbine spacing (m) T ) tank diameter (m) V ) vessel volume (m3) Vs ) particle settling velocity in a turbulent liquid (m s-1) Vt ) particle terminal velocity in a still liquid (m s-1) z ) axial coordinate (m) Y ) off-bottom turbine clearance (m) Greek Symbols R ) scaling exponent (-) ) local, instantaneous dissipation rate (W kg-1) 〈〉 ) local, mean dissipation rate (W kg-1) 〈〉 ) volume average dissipation rate (W kg-1) λK ) (ν3/〈〉)1/4, Kolmogorov length scale (m) µ ) liquid viscosity (Pa s) FL ) liquid density (kg m-3) Fs ) particle density (kg m-3) τ ) stress (Pa) Literature Cited (1) Gray, J. B.; Oldshue, J. Y. Agitation of Particulate Solid-Liquid Mixtures. in Mixing-Theory and Practice; Uhl, V. W., Gray, J. B., Eds.; Academic Press: New York, 1986; Vol. III, Chapter 12, pp 1-61. (2) Nienow, A. W. Suspension of Solids in Liquids. In Mixing in the Process Industries, 2nd ed.; Harnby, N., Edwards, M. F., Nienow, A. W., Eds.; Butterworth-Heinemann: London, 1992; Chapter 16, pp 364-365. (3) Yamazaki, H.; Tojo, K.; Miyanami, K. Concentration Profiles of Solids Suspended in a Stirred Tank. Powder Technol. 1986, 48, 205. (4) Barresi, A.; Baldi, G. Solid Dispersion in an Agitated Vessel. Chem. Eng. Sci. 1987, 42, 2949. (5) Bilek, P.; Rieger, F. Distribution of Solid Particles in a Mixed Vessel. Collect. Czech Chem. Commun. 1990, 55, 2169. (6) Mak, A. T. C.; Yang, S.; Ozcan-Taskin, N. G. The Effect of Scale on the Suspension and Distribution of Solids in Stirred Vessels. Re´ cent Progre` s en Ge´ nie des Proce´ de´ s; Lavoisier: Paris, 1997; Vol. 11 (52), pp 97-104. (7) Montante, G.; Pinelli, D.; Magelli, F. Diagnosis of Solids Distribution in Vessels Stirred with Multiple PBTs and Comparison of two Modelling Approaches. Can. J. Chem. Eng. 2002, 80, 665. (8) Crowe, C.; Sommerfeld, M.; Tsuji, Y. Multiphase Flows with Droplets and Particles; CRC Press: Boca Raton, FL, 2001. (9) Nocentini, M.; Pasquali, G. A Theoretical Evaluation of Catalyst Distribution on the Performance of non Ideal Slurry Reactor. Chem. Eng. Commun. 1989, 77, 195. (10) Bałdyga, J.; Bourne, J. R. Turbulent Mixing and Chemical Reactions; Wiley: New York, 1999. (11) MacTaggart, R. S.; Nasr-El-Din, H. A.; Masliyah, J. H. Sample Withdrawal from a Slurry Mixing Tank. Chem. Eng. Sci. 1993, 48, 921. (12) Barresi, A.; Kuzmanic, N.; Baldi, G. Continuous Sampling of a Slurry from a Stirred Vessel: Analysis of the Sampling Efficiency and
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ReceiVed for reView March 6, 2007 ReVised manuscript receiVed May 28, 2007 Accepted May 30, 2007 IE070339V
