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Computational Fluid Dynamics Modeling of the Precipitation Process in a Semibatch Crystallizer Hongyuan Wei,*,† Wei Zhou,‡ and John Garside§ BHR Group Limited, Cranfield, Bedfordshire MK43 0AJ, U.K., Fluent Incorporated, 10 Cavendish Court, Centerra Resource Park, Lebanon, New Hampshire 03766-1442, and Department of Chemical Engineering, UMIST, P.O. Box 88, Manchester M60 1QD, U.K.
To better understand the complicated phenomena in a precipitation process, computational fluid dynamics (CFD) is utilized to account for reactions, crystallization, mixing, and their interactions. The moment transformations of the population balances taking into account the chemical and crystallization kinetics are integrated into a CFD solver to describe the generation and transportation of the crystal phase. The current work applies this CFD tool to simulate the reactive precipitation process in a semibatch crystallizer, which is widely used in chemical and pharmaceutical industries. BaSO4 precipitation is employed as an example system. The influence of hydrodynamics in the stirred tank, as characterized by the impeller speed and feed location, on the distribution of supersaturation and the subsequent crystal size distribution is investigated. The numerical predictions are validated with measurements and can be used as an aid in the optimization of semibatch precipitator design. 1. Introduction Precipitation in semibatch crystallizers is widely used in the fine chemical and pharmaceutical industries to produce well-characterized substances because of their ease of operation, the reduced significance of heattransfer effects, and the reduction of byproducts. In the past decades, much modeling effort has been devoted to understanding the performance of semibatch crystallizers,1,2 but little work has been done to study the influence of hydrodynamics in such units. The use of computational fluid dynamics (CFD) to predict the performance of chemical processes has risen dramatically in the past 10 years. CFD is now not only the software for fluid flow calculations but also a tool to simulate more complicated systems involving heat transfer, chemical reactions, and multiphase phenomena. It has become an important tool in the chemical and pharmaceutical industries to assess process design, optimization, and scale-up. Some pioneering CFD studies have been done to understand the behavior of simplified precipitators.3-5 In 1997, Wei and Garside6 also reported the modeling of the precipitation process in a realistic stirred vessel. In their work, the influence of the agitation speed and feed concentration on product crystal size distribution (CSD) has been systematically investigated. In a further study, Wei and Garside7 looked at the scale-up issue for precipitation in a continuous reactor. Al-Rashed and Jones8 studied gas-liquid reactive precipitation in a batch reactor using a two-dimensional CFD model, in which transient calculations were involved. More re* Corresponding author. Tel: ++44 (0)1234 750422. Fax: ++44 (0)1234 750074. E-mail:
[email protected]. † BHR Group Limited. ‡ Fluent Incorporated. Tel: ++1 (603) 643-2600. Fax: ++1 (603) 643-3967. E-mail:
[email protected]. § UMIST. Tel: ++44 (0) 161 2004011. Fax: ++44 (0) 161 2377219. E-mail:
[email protected].
cently, Piton et al.9 implemented the four-enivironment generalized micromixing (4-EGM) model into their CFD simulations for the precipitation of barium sulfate in a tubular reactor, to resolve the subgrid-scale concentration fluctuations. They found that the effect of micromixing on the CSD is primarily through the nucleation rate. Crystal mean size and mean number density are relatively insensitive to micromixing at the given inlet conditions. A similar study has also been done by Marchiso et al.10 in a Couette precipitator. Instead, they used a β-PDF model to describe the subgrid micromixing effect. In this paper, a three-dimensional semibatch precipitator is employed as a benchmark case to study the influence of flow characteristics in this operating mode on the crystal properties. The transient behavior of the distributions of CSD and supersaturation are investigated and compared to the results derived from experiments and perfect mixing models. 2. Modeling Approach In a precipitation system, solid particles are formed from solutions because of supersaturation. A rigorous approach to solve this system would use a multiphase model to predict the solid-liquid interactions. However, because the particle size is very small (typically smaller than 10 µm) in most reactive precipitation, the impact of solids on the flow field is negligible. Therefore, a single-phase flow model is used here, with the particles considered to follow the flow. In this study, a semibatch precipitator is modeled by using the finite volume method (FLUENT5). Flow fields are solved from the Navier-Stokes equations. A timedependent solver technique is utilized to resolve the transient behavior of a semibatch reactor. The RNG k- turbulence model is chosen for strong swirling flows. The three-dimensional computational domain is dis-
10.1021/ie001123v CCC: $20.00 © 2001 American Chemical Society Published on Web 07/28/2001
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cretized by unstructured hexahedral cells. Hexahedral cells introduce less numerical diffusion errors than do tetrahedral cells. A sliding mesh technique is used to provide a time-accurate solution for flows driven by rotating impellers, which cannot be expressed by the steady-state conventional methods such as the momentum source model or snapshot approach. The technique explicitly models the interactions between impellers and baffles without any a priori description of impeller boundary conditions. In the sliding mesh approach, two cell zones are used. The two cell zones move relative to each other along the grid interface during the calculations. As the rotation takes place, alignment of the two grids along the grid interface is not required because the solver interpolates solution values at grid points in one zone to provide solution values at grid points in another zone. CSD is one of the most important properties describing the product qualities. To predict the CSD, the particle population balance equation needs to be solved. However, direct solution of the partial differential equation for the population balance requires extensive computational time. Instead, the moment transformation approach is an alternative that avoids expensive computational effort. The moments of the distribution represent the average and total properties of the solid phase, and the method solves transport equations for these moments. This approach is sufficient to provide information useful for engineering and design purposes. The transient expression of the moment transportation equations can be written as11
∂mj + ∇(u bmj) ) ∇(Γj,eff∇mj) + 0jJ + jGmj ∂t j ) 0, 1, 2, 3 (1) where J is the nucleation rate and can be represented as12 2
J ) A exp(-B/ln S)
(3)
The key parameter for determining these kinetics is the local supersaturation, S, which reflects the thermodynamic driving force for crystallization and is defined as13 1/v vS ) (cv+ + c- /Ksp)
mass, Mt, and the mass-averaged crystal coefficient of variation, cv, as follows: N
d32 )
(4)
The detailed descriptions of the nucleation and growth kinetics for the BaSO4 system have been fully discussed previously by Wei and Garside.5 The transport equations of the moments are solved as scalar transport equations in FLUENT5. Because the RNG k- model is used to describe the turbulence, the transport equations involved in this work are Reynolds-averaged. The effect of micromixing is neglected. Agglomeration is also neglected in this modeling. In practice, this can be an important kinetic effect in some specific systems, particularly at high supersaturation. The moments are used to calculate the mass-averaged crystal mean size, d32, the mass-averaged total crystal
m3Vi ∑ i)1
(5)
N
m2Vi ∑ i)1 where Vi is the volume of cell i N
Mt ) kVFc
m3Vi ∑ i)1
(6)
N
Vi ∑ i)1 where Fc is the crystal density, and
(2)
and G is the growth rate and is usually expressed by the relationship13
G ) kg(S - 1)g
Figure 1. Dimensions of the stirred vessel.
cv )
3. Case Studies
x
N
N
m2Vi∑m0Vi ∑ i)1 i)1 N
(
-1
(7)
m1Vi)2 ∑ i)1
The sample case used in this study is created based on the experimental work done by Chen et al.14 in a semibatch precipitator. The semibatch tank is of laboratory scale and has four vertical baffles. The fluid in the vessel is agitated by a six-blade Rushton turbine. The principal dimensions of the stirred tank can be found in Figure 1. BaSO4 precipitation is selected as a sample system. The reaction in this precipitation system can be represented as
BaCl2 + Na2SO4 f 2NaCl + BaSO4V
(8)
A total of 2.46 L of the reactive solution of sodium sulfate (B) is initially present in the tank. A total of 40 mL of a barium chloride solution (A) is fed continuously into the vessel over the given batch period of 180 s at a given flow rate. Two representative feed locations (k and m) are studied (Figure 1). The feed concentration of A is fixed at 1.0 mol/L and the initial concentration of B at 0.016 26 mol/L. Three impeller speeds, 150, 300, and 600 rpm, are studied, which correspond to impeller
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Figure 2. Flow pattern in the precipitator at 150 rpm.
Reynolds numbers (FND2/µ) of 6250, 12 500, and 25 000, respectively. The fluid domain is meshed with 201 626 hexahedral cells. With this grid size, the velocity fields in such a scale tank can be predicted with quantitative accuracy.15 The feed pipe is not physically meshed. Instead, the momentum source and scalar sources are introduced at the feed location to mimic the injection. The equivalent feed pipe diameter used in CFD simulations is over 4 times larger than that in the experiments because of the limitation of the mesh density. Because the feeding mass of A is negligibly small ( 50 s), where constant C1 varies with the agitation speed and is determined by the mixing intensity and constant C2 is independent of the agitation speed. It is also noticed that the mean crystal size in the tank with the agitation speed of 600 rpm increases in a manner that is almost identical with that predicted by the perfect mixing model. This observation can be explained by the improvement in mixing with increasing agitator speed. Therefore, the product quality predicted in the tank with high agitation speed (>600 rpm) is similar to that predicted by the perfect mixing model.
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Figure 3. Supersaturation distribution in the tank at different operating times (N ) 150 rpm, feed point at m).
Figure 4. Variations of crystal number concentrations with the operating time.
Figure 7 shows the transient behavior of the coefficient of variation (cv) of crystals at different agitation speeds in CFD simulations as well as the prediction from the perfect mixing model. The values of cv are in the range of 0.4-0.7 for all operating conditions considered in this study. Further study on the cv as a function of the impeller speed and feed locations will be conducted in the future work. 4.3. Effect of the Impeller Speed. Several experimental studies have reported that the effect of the impeller speed on CSD is very complicated.14,17,18 This
Figure 5. Variations of crystal mass concentrations with the operating time.
complex effect reflects the interplay among three parties: reaction, crystallization, and mixing. The CFD simulation study of Wei and Garside6 also revealed the same features in a continuous precipitator. In this paper, three impeller speeds (150, 300, and 600 rpm) are investigated. Figure 8a shows the mean crystal size at 180 s as a function of the impeller speed. The mean crystal size decreases from 13 µm at 150 rpm to about 8.5 µm at 300 rpm before increasing to 11.2 µm at 600 rpm. A similar trend was found by Chen et al.14 in their experimental work, although the measured crystal sizes are about 4 times smaller than CFD
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Figure 6. Variations of mean crystal sizes d32 with the operating time.
Figure 8. Effect of impeller speeds on mean crystal size, d32: (a) CFD predictions; (b) experimental measurements.
Figure 7. Variations of the crystal cv values with the operating time.
predictions. One contribution to this size difference may come from the different feed pipe diameters used in the experiments and CFD simulations (section 3). Kim and Tarbell18 found that bigger feed pipe diameters resulted in a larger crystal average size in their BaSO4 precipitator. Also, the supersaturation level around the feed pipe is underestimated by CFD simulations, because using larger feed pipes evens out the supersaturation level. Consequently, the mean crystal size is overpredicted in CFD simulations. The perfect mixing model, on the other hand, cannot be used to predict the influence of the impeller speed because the model assumes 100% uniformity of mixing irrespective of the impeller agitation. In both experiments14,19 and CFD simulations, a critical value of the impeller speed, Ncrit, is observed in the relation between d32 and the impeller speed. At Ncrit, the mean crystal size is a minimum. This phenomenon can be explained by the interactions between mixing and the crystal nucleation and growth kinetics. The mean crystal size is determined by the competition between nucleation and growth, both of which depend on supersaturation. In general, very high supersaturation favors nucleation over growth, resulting in a large number of smaller crystals. This is especially true above the critical value of supersaturation at which homogeneous nucleation starts to predominate over heterogeneous nucleation. On the other hand, in a semibatch precipitator in which there is circulation through regions of high and low supersaturation, the residence
time in the supersaturated zone also has an influence. Clearly, the impeller speed affects both the peak level of supersaturation and the extent of the supersaturated zone, as can be seen in Figure 9. The mean residence time in the supersaturated zone can therefore be expected to influence the mean crystal size in a complex manner. As noted earlier, agglomeration is neglected in the simulations presented here. Agglomeration can occur in the BaSO4 system, and this will introduce some uncertainty when comparing experimental data with simulations. 4.4. Effect of the Feed Location. Two feed locations, m and k, are studied in the CFD simulations (Figure 1). The effects of the feed location on the mean crystal size and cv are clearly shown in Figure 10. If the feed is added at point k, where mixing is relatively poor, the mean crystal size is increased while the crystals may be more widely spread in size (as demonstrated by a larger cv). Poor mixing encourages local high supersaturation, and particles remain in this region relatively longer; therefore, some particles could grow very large. On the other hand, poor mixing results in large spatial differences in supersaturation; therefore, crystals could be formed with a wider range of size. 5. Conclusions The modeling of a semibatch precipitator using CFD techniques has been successfully developed. The transient behavior of CSD and supersaturation distribution is discussed in detail to help in the better understanding of the interplay between reaction, crystallization kinetics, and mixing. CFD simulations reveal the following: (1) The reactive precipitation only occurs in an effective zone that occupies a small part of the working
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Figure 9. Supersaturation distribution at the time of 10 s with different impeller speeds.
(4) Supersaturation distributions are also greatly influenced by the impeller speed, because the effective reaction zone within the precipitator varies with the agitation conditions. (5) The feed location is another important operating parameter controlling CSD. Feeding into the region where mixing is poor may lead to a larger average crystal size with a more widely spread distribution. This study shows how CFD modeling could be a useful tool in guiding the design and optimization of semibatch precipitators. Acknowledgment The authors acknowledge Dr. Simon Leefes from BHR Group Limited who provided valuable inputs for the draft of the paper. Notation
Figure 10. Effect of the feed locations on (a) mean crystal size, d32, and (b) cv.
volume that is relatively constant in size over the operating period. (2) The volume-averaged mean crystal size in the vessel increases linearly with the operation time. The linear profiles have the same slope at different impeller speeds. (3) The impeller speed significantly affects the mean crystal size. CFD predictions of this effect show trends similar to those obtained from experimental investigation.
A, B ) constants in the nucleation rate equation c-, c+ ) concentration of the cation and anion expressed in molars, kmol/m3 cv ) coefficient of variation D ) impeller diameter, m d32 ) mean crystal size G ) growth rate, m/s kg ) rate constant of the growth rate Ksp ) thermodynamic solubility product, kmol2/m6 kV ) volume shape factor mj ) jth moment Mt ) total crystal mass concentration, kg/m3 N ) agitation speed, rpm S ) supersaturation t ) time, s u ) velocity, m/s Vi ) ith cell volume, m3 Greek Symbols F ) fluid density, kg/m3 Fc ) crystal density, kg/m3 µ ) viscosity, m2/s Γeff ) effective diffusion coefficient
Ind. Eng. Chem. Res., Vol. 40, No. 23, 2001 5261 Subscripts ν ) number of ions into which a molecule dissociates c ) crystal crit ) critical point eq ) equilibrium
Literature Cited (1) Tavare, N. S.; Garside, J. Simulation of reactive precipitation in a semi-batch crystallizer. Trans. Inst. Chem. Eng. 1990, 68A, 115. (2) Tavare, N. S.; Garside, J. Silica precipitation in a semi-batch crystallizer. Chem. Eng. Sci. 1993, 48 (3), 475. (3) Seckler, M. M.; Bruinsma, O. S. L.; Van Rosmalen, G. M. Influence of hydrodynamics on precipitation: A computational study. Chem. Eng. Commun. 1995, 135, 113. (4) Van Leeuwen, M. L. J.; Bruinsma, O. S. L.; Van Rosmalen, G. M. Influence of mixing on the product quality in precipitation. Chem. Eng. Sci. 1996, 51 (11), 2595. (5) Wei, H.; Garside, J. Application of CFD modelling to precipitation system. Trans. Inst. Chem. Eng. 1997, 75A, 719. (6) Wei, H.; Garside, J. CFDsSimulation of precipitation processes in stirred tanks. Proceedings of the Institute of Chemical Engineers, Research Event, Nottingham, U.K., Institution of Chemical Engineers, U.K.; 1997; p 517. (7) Wei, H.; Garside, J. Simulations and scale-up of precipitation using CFD techniques. Proceedings of the 3rd International Symposium on Mixing in Industrial Processes, Osaka, Japan, 1999; The Society of Chemical Engineers, Japan; p 45. (8) Al-Rashed, M. H.; Jones, A. G. CFD modelling of gas-liquid reactive precipitation. Chem. Eng. Sci. 1999, 54, 4779. (9) Piton, D.; Fox, R. O.; Marcant, B. Simulation of fine particle formation by precipitation using computational fluid dynamics. Can. J. Chem. Eng. 2000, 78, 983.
(10) Marchiso, D. L.; Barresi, A. A.; Baldi, G.; Fox, R. O. Comparison of different modelling approaches to turbulent precipitation, Proceedings of the 10th European Conference on Mixing, Delft, The Netherlands, 2000; Elsevier Science; p 77. (11) Randolph, A. D.; Larson, M. A. Theory of Particulate Processes, 2nd ed.; Academic Press: New York, 1988. (12) Nielsen, A. E. Kinetics of precipitation; Pergamon Press: Oxford, U.K., 1964. (13) Sohnel, O.; Garside, J. Precipitation: Basic Principles and Industrial Applications; Butterworth-Heinemann: Oxford, U.K., 1992. (14) Chen, J.; Zheng, C.; Chen, G. Interaction of macro and micromixing on particles size distribution in reactive precipitation. Chem. Eng. Sci. 1996, 51, 1957. (15) Ng, K.; Fentiman, N. J.; Lee, K. C.; Yianneskis, M. Assessment of sliding mesh CFD predictions and LDA measurements of the flow in a tank stirred by a Rushton impeller. Trans. Inst. Chem. Eng. 1998, 76A, 737. (16) Villermaux, J. A simple model for partial segregation in a semibatch reactor. AIChE Annual Meeting, San Franciso, 1989; Paper 145a. (17) Fitchett, D. E.; Tarbell, J. M. Effect of mixing on the precipitation of barium sulfate in an MSMPR reactor. AIChE J. 1990, 36 (4), 511. (18) Kim, W.-S.; Tarbell, J. M. Micromixing effects on barium sulfate precipitation in an MSMPR reactor. Chem. Eng. Commun. 1996, 146, 33. (19) Pohorecki, R.; Baldyga, J. The use of a new model of micromixing for determination of crystal size in precipitation. Chem. Eng. Sci. 1983, 38, 79.
Received for review December 18, 2000 Revised manuscript received June 6, 2001 Accepted June 6, 2001 IE001123V
