Scale-up of Continuous and Semibatch Precipitation Processes

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Ind. Eng. Chem. Res. 2000, 39, 2392-2403

Scale-up of Continuous and Semibatch Precipitation Processes Rudolf Zauner† and Alan G. Jones* Department of Chemical Engineering, University College London, London WC1E 7JE, United Kingdom

A segregated feed model linked to the population balance, which includes micromixing and mesomixing effects together with crystallization and agglomeration kinetics, is proposed for scaling up reactive precipitation processes. The model is solved using kinetic parameters extracted from laboratory-scale experiments together with local mixing parameters obtained from a computational fluid dynamics simulation. Predicted particle size characteristics are compared with experimental data collected on three different scales of operation (range 0.3-25 L) using the aqueous calcium oxalate system. The hybrid precipitation-mixing model accurately predicts mixing effects observed during the continuous mode of operation, including a maximum in the mean particle size and coefficient of variation with increasing power input on each of the three different scales. The influence of mixing on the mean particle size in semibatch operation is found to be more pronounced owing to the direct mixing of the feed solution with the other component already present in the reactor, and is also correctly predicted by the hybrid model. 1. Introduction

2. Literature

Scale-up in chemical engineering is a very difficult task in many cases as a variety of poorly defined and interacting factors can influence the quantity and quality of the final product of an industrial process. For many systems, such as precipitation, there is a lack of understanding as to what role certain physical and chemical processes play as the vessel capacity increases. Such operations are therefore scaled up empirically or by trial and error. In the simplest case, geometric, dynamic, kinematic, thermal, and/or chemical similarities are targets for scale-up. Similarity on different scales is fulfilled when the dimensionless groups (e.g., Reynolds number, Sherwood number, Damko¨hler number) have the same value on different scales. In most scale-up situations, however, total similarity on different scales cannot be established, as it is difficult or even impossible to maintain all the dimensionless groups that characterize the process. In these cases, a tradeoff between different dimensionless groups has to be found, with groups being “weighted” differently according to their expected influence on the process. Scale-up then loses its theoretical base and becomes empirical. Alternatively, a mathematical model can be developed that accounts for the physical and chemical processes considered important for scale-up. For this purpose, the process mechanisms have to be clearly understood and the model must not contain any phenomenological or unknown scale-dependent parameters, as it is not then suitable for quantitative scaleup predictions. Development of such a scale-up methodology would reduce development costs and shorten the time taken for the introduction of new products to the market.

Precipitation processes find widespread conventional and contemporary industrial application including production of fine chemicals, pharmaceuticals, and biotechnological and electronic materials. Particle characteristics (e.g., crystal size, shape, and degree of agglomeration) are often vitally important for both process efficiency and product quality, but unfortunately contradictory process scale-up criteria have been reported in the literature. One explanation for the variety of scaleup rules is the fact that when precipitation processes are scaled up it is not possible to keep all dimensional groups constant, as referred to above. For example, scale-up with a constant Reynolds number leads to different local Damko¨hler numbers and therefore different reaction and precipitation rates, whereas scaleup with constant specific power input leads to a different flow pattern and therefore different meso- and macromixing in the reactor. The mixing limitation can occur on the macroscale (macromixing limitation), mesoscale (mesomixing limitation), or microscale (micromixing limitation), or any combination of these. The influence of mixing on product quality and scaleup has been studied by a number of authors. Rice and Baud1 used constant specific power input as a scale-up criterion and found that this criterion can only be used for certain feed point locations. Moreover, they state that a localization of the reaction zone with scale takes place. To avoid this effect, the authors suggest changing the ratio of impeller to tank diameter. Bourne and Dell’Ava2 studied micromixing effects on different scales using competitive-consecutive azo coupling reactions in order to determine the segregation index and therefore obtain a measure of micromixing. They “excluded” any meso- and macromixing effects by feeding the reagents with feed rates much smaller than the circulation rate. For these micromixing-controlled conditions, they confirmed that constant specific power input is a suitable scale-up criterion. Mersmann and Laufhu¨tte,3 too, suggest scale-up with constant specific power input for micromixing-limited processes. The authors also provide

* To whom all correspondence should be addressed: a.jones@ ucl.ac.uk. † Present address: BASF AG, D-67056 Ludwigshafen, Germany.

10.1021/ie990431u CCC: $19.00 © 2000 American Chemical Society Published on Web 06/13/2000

Ind. Eng. Chem. Res., Vol. 39, No. 7, 2000 2393

an overview of scale-up rules for mixing processes involving suspensions. In contrast to the authors mentioned above, Bourne and Yu4 found that scale-up with constant specific power input in the semibatch mode of operation is not suitable for parallel reaction systems. They explained the variance of the segregation index on different scales by the fact that the trajectory of the reaction zone and the spatial distribution of the power dissipation vary with scale even for homologous feed points. These effects are modeled using an experimental flow model. Wang and Mann5 showed that for mixing-limited consecutivecompetitive reaction systems, scale-up with constant tip speed leads to an increase in segregation with scale-up and therefore unsatisfactorily large-scale predictions. Tipnis et al.6 state that the single most important factor for scale-up is feed blending, which is influenced by the chemical species concentrations, the feed flow rates, the impeller geometry, and the feed pipe locations. The authors give no hard and fast rules for scale-up; however, a procedure is suggested with various scaleup criteria ranging from constant blending time to constant specific power input. Choplin and Villermaux7 investigated mixing phenomena in polymer reactors and concluded that successful scale-up is possible if only one of the multistep mixing mechanisms is controlling. For example, if the engulfment process is limiting, micromixing can be scaled up on the basis of constant power input per unit volume. For mesomixing limitation, a scale-up with constant tip speed is suggested. Publications concerning the scale-up of precipitation are very sparse in the literature. Momonaga et al.8 introduced a mixing-dependent scale-up factor based on equal power input per unit volume. They scaled up the batch precipitation process of methyl R-methoxyimino acetoacetate but did not carry out a study of the precipitation kinetics. Mahajan and Kirwan9 investigated micromixing and scale effects of the precipitation of Lovastatin in a two-impinging-jets (TIJ) precipitator. By means of the geometry of the TIJ mixer, it is possible to achieve very short micromixing times, i.e., in the range of the induction time, and therefore ensure homogeneous supersaturation throughout the reactor before nucleation starts. A scale-up criterion based on the Damko¨hler number is proposed. In conclusion, various scale-up criteria for the scaleup of mixing-limited processes have been proposed in the literature. Problems occur if more than one of these mixing processes controls the mixing quality and therefore the process. This is generally the case in precipitation processes owing to the very fast reaction kinetics, and therefore all these conventional scale-up criteria fail.

3. Mixing Model: Segregated Feed Model (SFM) The segregated feed model (SFM) is based on physically meaningful mixing parameters, diffusive micromixing time and convective mesomixing time, and was therefore selected to model the mixing effects during precipitation mentioned above. The SFM model was first used by Villermaux10 to investigate micromixing effects of consecutive-competitive semibatch reactions. Tosun11 subsequently applied it to predict the effects of mixing on semibatch polymerization and Marcant12 to model the semibatch precipitation of barium sulfate without

Figure 1. Segregated feed model (SFM).

accounting for agglomeration and disruption. The SFM was found to be particularly suitable for modeling mixing effects, as it combines the advantages of both a compartmental interaction by exchange with the mean (IEM) model and a physical model. In the SFM the reactor is divided into three zones: two feed zones f1 and f2 and the bulk b (Figure 1). The feed zones exchange mass with each other and with the bulk as depicted with the flow rates u1,2, u1,3, and u2,3, respectively, according to the time constants characteristic for micromixing and mesomixing. As imperfect mixing leads to gradients of the concentrations in the reactor, different supersaturation levels in different compartments govern the precipitation rates, especially the rapid nucleation process. Using the SFM, the influence of micromixing and mesomixing on the precipitation process and properties of the precipitate can be investigated. Mass and population balances can be applied to the individual compartments and to the overall reactor accounting for different levels of supersaturation in different zones of the reactor. The individual volumes of the feed plume 1 (Vf1) and feed plume 2 (Vf2) and the total volume of the precipitation reactor can be obtained from with

Vf1 dVf1 ) Qf1 dt tmeso1

(1)

dVf2 Vf2 ) Qf2 dt tmeso2

(2)

dVtot ) Qf1 + Qf2 - Qb dt

(3)

Vtot ) Vf1 + Vf2 + Vb

(4)

Mass balances for a species j in the three zones give

d(Vficj,fi) 0 - uj,fib - uj,f1f2 ) rj,fiVfi + Qficj,fi dt

(5)

d(Vbcj,b) ) rj,bVb - Qbcj,b + uj,f1b + uj,f2b dt

(6)

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Ind. Eng. Chem. Res., Vol. 39, No. 7, 2000

with the exchange flows between the compartments

uj,fib )

Vficj,fi Vfi(cj,fi - cj,b) + tmesoi tmicroi

(7)

uj,f1f2 )

(Vf1 + Vf2)(cj,f1 - cj,f2) t12

(8)

SFM Applied to Continuous Precipitation. Using the following assumptions, the SFM will be applied to continuous precipitation: kr (1) Instantaneous reaction; A + B 98 P. The reaction between the two feed solutions occurs instantaneously as soon as they are mixed. (2) Homogeneous conditions occur within the feed zones and the bulk. Within each compartment, there are no gradients in the field of supersaturation. (3) Nucleation is the only kinetic process occurring in the feed plumes. Nucleation, growth, agglomeration, and disruption take place in the bulk zone. (4) Direct diffusional mass exchange between the feed plumes is negligible as the feed points are too far apart and diffusion is too slow to play a significant role in the diffusional mass exchange between the feed streams, leading to t12 f ∞. volumes of compartments

Vfi ) Qfitmesoi

(9)

Vb ) Vtot - (Vf1 + Vf2)

(10)

from the literature and present experiments, respectively. One of the advantages of using a compartmental model is that the “full” population balance, including terms for size-dependent agglomeration and disruption, can be implemented in the model. Therefore, the population balance for the bulk becomes

∂nP,b ∂nP,b ) -Gb + B0b + Baggl + Bdisr - Daggl ∂t ∂L nP,bQb Ddisr ) 0 (16) Vtot with the birth terms for agglomeration and disruption

Baggl + Bdisr )

L2 2

∫0L

Kaggln(Lu) n(Lv) dLu L2v

+

∫L∞ KdisrS′(Lu, Lv) n(Lu) n(Lv) dLu y

(17)

and the death terms for agglomeration and disruption

Daggl + Ddisr ) n(L)

∫0∞ Kaggln(Lu) dLu + Kdisrn(L)

(18)

The expression for the nucleation rate B0j in the compartment j is derived from the theory of primary nucleation and found to be (see, e.g., Mullin15)

[

B0j ) A exp -

mass balances e.g., for species A

]

(19)

) σj + 1

(20)

16πγ3ν2 3k3T3 ln2 Sj

Vf1cA,f1 Vf1(cA,f1 - cA,b) d(Vf1cA,f1) 0 ) Qf1cA,f1 dt tmeso1 tmicro1 Fc B0f1kvL30 V ) 0 (11) Mwc f1

with the supersaturation defined as

d(Vf2cA,f2) Vf2cA,f2 Vf2(cA,f2 - cA,b) ) dt tmeso2 tmicro2 Fc V ) 0 (12) B0f2kvL30 Mwc f2

The overall nucleation rate in the reactor becomes

Vf1cA,f1 Vf1(cA,f1 - cA,b) d(VbcA,b) + + ) -QbcA,b + dt tmeso1 tmicro1 Fc Vf2cA,f2 Vf2(cA,f2 - cA,b) + - B0bkvL30 V tmeso2 tmicro2 Mwc b Fc 1 V ) 0 (13) Gbkam2 2 Mwc b

( )

ν tmicro ) 17.3 loc tmeso ) Λ

1/2

(14)

( )

νA νB cA,j cB,j Sj ) Ksp

B0tot )

1/(νA+νB)

B0bVb + B0f1Vf1 + B0f2Vf2 Vtot

(21)

The dependence of the growth rate on supersaturation is modeled using the power law expression

Gb ) kgσ2b

(22)

Furthermore, the agglomeration and disruption kernels are also assumed to depend on the supersaturation in power law form (Zauner16):

Kaggl ) βagglf()σ2b

(23)

Kdisr ) βdisrg()σ-2 b

(24)

1/3

avg Q loc N4/3d

(15)

s

As mentioned above, the inverse of the time constants tmicroi (micromixing) and tmesoi (mesomixing) can be interpreted as transfer coefficients for mass transfer by diffusion and convection, respectively. The former are given by eqs 14 and 15 (Baldyga et al.13 and Baldyga et al.14) in which the coefficients 17.3 and Λ are obtained

Kinetic expressions for nucleation, growth, agglomeration, and disruption of calcium oxalate determined from laboratory-scale data (Zauner16) are listed in Table 1. The second moment of the particle size distribution used in the mass balances is obtained from

m2b )

∫0∞ nP,b(L)L2 dL

(25)


Figure 2. Dimensions of the 4.3 and 12 L large-scale DTB reactors. Table 1. Precipitation Kinetics for Calcium Oxalate (Zauner16) process

kinetic expression

nucleation ) 3.375 × 1015 exp52.09/(ln2 S) growth G ) 2.116σ2 agglomeration βaggl ) 5.431 × 10-17 (1 + 2.2961/2 - 2.429)S2.15 disruption βdisr ) 6.25 × 10-5 S-2.15 B0

SFM Applied to Semibatch Precipitation. Using similar assumptions as for the continuous mode, the SFM can be applied to semibatch precipitation: kr

(1) Instantaneous reaction; A + B 98 P. The reaction between the two feed solutions occurs instantaneously as soon as they are mixed. (2) Homogeneous conditions exist within the feed zone and the bulk. (3) Nucleation is the only kinetic process occurring in the feed plume. (4) Nucleation, growth, agglomeration, and disruption take place in the bulk zone. Determination of Mixing Times Using Computational Fluid Dynamics (CFD). Stirred tanks are the most common form of chemical reactors. Nevertheless, owing to high local gradients of the energy dissipation, the fluid dynamics are not well understood and depend to a large extent on the geometric details of the reactor. Different forms of impellers, baffles, and draft tubes can produce very different flow fields. As a stirred tank contains a moving impeller, the fluid cells surrounding the impeller are modeled as rotating blocks in CFD (Bakker et al.17). A sliding mesh technique was chosen to account for the movement of the rotating impeller grid relative to the surrounding motionless tank cells. Xu and McGrath18 compared the sliding mesh simulation results for a stirred tank with experimental laser doppler anemometry (LDA) data and found that the data corresponded very well. In this research, the stirred tank is modeled as a single-phase isothermal system; i.e., only the hydrody-

namics of the reactor are simulated. In the model equations of the turbulence, the k- model was used, assuming that turbulence is isotropic. The k- model offers a good compromise between computational economy and accuracy of the solution. Ranade19 has used it successfully to model stirred tanks under turbulent conditions. Manninen and Syrja¨nen20 modeled turbulent flow in stirred tanks and tested and compared different turbulence models. They found that the standard k- model predicted the experimentally measured flow pattern best. One of the great advantages of CFD is that local data for the fluid velocity and energy dissipation can be obtained. As the local energy dissipation is a measure of the degree of local micromixing in the reactor, the micromixing time can be calculated directly from this parameter using eq 14. To account for the rotation of the impeller, the zone surrounding the impeller was modeled using a sliding mesh approach. Grid independence was established for all the problems solved in this research by stepwise and/ or local refining of the grid until the solution for the energy dissipation of a grid cell close to the impeller no longer changed by more than 1%. Two different types of impeller, a Rushton turbine and a propeller, were defined as groups and implemented in the geometry. To find the micromixing times in the feed zones and the mesomixing time characteristic of blending of the feed solutions with the bulk in the segregated feed model, it is essential to determine the local distribution of the specific energy dissipation in the reactor. It is possible to obtain this local distribution using CFD and find the diffusive and convective exchange parameters. The micromixing and mesomixing times were determined from the local energy dissipation at the feed points. The micromixing time and mesomixing time vary between 0.004 and 2.05 s and between 0.002 and 1.02 s, respectively, depending on the geometry, scale, stirrer speed, and feed point position. Therefore, the volume of the feed zones Vf1 and Vf2 covers a range from 0.0002

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Figure 3. Experimental setup for continuous scale-up experiments.

to 0.1% of the total volume, and the reaction and the fast nucleation process take place in a very limited volume around the feed zones, while the slower kinetic processes occur in the bulk zone. This result clearly shows the importance of mixing in systems with very fast or instantaneous reactions. Because the local energy dissipation is different on different scales of operation with the same mean specific power input, the micromixing (diffusion) and mesomixing (convection) times on different scales differ. Consequently, different diffusive and convective mass transfer between the feed zones and the bulk zone leads to different levels of supersaturation on different scales and therefore to different precipitation kinetic rates and mean crystal sizes with scale-up. 4. Experimental Setup and Procedure Continuous Experiments. Calcium oxalate was precipitated from reacting supersaturated solutions of calcium chloride CaCl2 and sodium oxalate Na2C2O4. The geometries of the 4.3 and 12 L precipitation reactors are shown in Figure 2. The draft tubes of the reactor are hollow in order to allow cooling or heating fluid to circulate. The 300 mL laboratory-scale reactor (1) (ds ) 35 mm, D ) 65 mm) is geometrically similar to the larger reactors. A marine-type impeller (propeller) provides a smooth and even flow field throughout the reactor. The feeding and withdrawing of solution and product from the feed tanks (2) and withdrawal tank (3), respectively, is established by peristaltic pumps (6) controlled by a PCPump unit (Spectrum Ltd, UK) (9) as shown in Figure 3. Balances (4) connected to the PCPump system are used to determine the speed of the pumps and therefore the desired feed and withdrawal rates. This technique ensures high accuracy and a constant residence time. A Spectrum PCLab unit (8) controls the temperature in the reactor and measures process variables on-line. The reactor jacket is thermostated by either cooling or heating water, according to

Table 2. Experimental Conditions for Continuous Experiments condition

set 1

set 2

concentration residence time feed point position

0.01 M 11 min outside DT (od)

0.04M 7.5 min inside DT (id)

Table 3. Dimensions of Semibatch Reactors scale (L)

d (mm)

D (mm)

h (mm)

H (mm)

s (mm)

1L 5L 25 L

40 65 110

105 180 300

35 60 100

110 190 320

5 8.5 15

the desired reactor temperature, using a Haake thermostatic bath (7). The particle size distribution is determined with a Coulter Counter Multisizer II. The specific power input was varied between 0.0024 and 8.09 W/kg in order to investigate the influence of mixing on scale-up. The other experimental conditions are listed in Table 2. Semibatch Experiments. The experimental setup used for the semibatch experiments is similar to that used for the continuous experiments. The particle size distribution is measured using a Coulter Counter Multisizer II and/or a Sympatec Helos laser scattering analyzer. The detailed geometry of the precipitation reactor with a Rushton turbine as stirrer is shown in Figure 4. The vessel is equipped with four baffles but in contrast to the reactor used for the continuous experiments does not contain a draft tube. The dimensions for the 1, 5, and 25 L reactors with different impellers (Rushton turbine and marine-type propeller) are given in Table 3. The conditions chosen for the 1, 5, and 25 L experiments are given in Table 4. 5. Model Predictions To solve the model eqs 1-25 as a system of differential and algebraic equations, the population balance (partial differential equation, eq 16) was dis-

Ind. Eng. Chem. Res., Vol. 39, No. 7, 2000 2397 Table 4. Experimental Conditions for Semibatch Experiments condition

from

to

reference

end concentration feed time specific energy input feed point position

0.008 M 10 min 2.5 × 10-4 W/kg impeller region

0.04 M 40 min 21.6 W/kg surface region

0.008 M 40 min impeller region

Table 5. Input and Output Variables for Segregated Feed Model (Continuous) input variables τ L0 Vtot ka, kv Mw ν avg loc cA0, cB0 Ksp dS N

residence time particle size (1st size interval) reactor volume shape factor molecular weight kinematic viscosity specific power input local specific power input initial concentrations solubility product diameter of stirrer stirrer speed

output variables tmicro tmeso Vi Li ni σj mk L10 L32 L43

micromixing time mesomixing time volume of compartment i particle size at interval i population density at Li supersaturation in compartment j kth moment of the distribution number mean size area mean size volume mean size

Figure 5. Scale-up of continuous calcium oxalate precipitation: volume mean size L43 (300 mL, 4.3 L, and 12 l; 0.04 M, 7.5 min, id).

Figure 4. Semibatch reactor with Rushton turbine (dimensions in Table 3).

cretized in the particle size domain. The discretization method of Kumar and Ramkrishna21,22 was chosen for this purpose as it presents a solution to population balance problems without restricting the choice of the grid of the discretized length scale. The set of differential equations of the population balance was solved using the NAG subroutine D02EAF, which is particularly suitable for stiff systems of first-order ordinary differential equations. This variable-order, variable-step method implements the backward differentiation formulas (BDF) and is of an explicit type (Schuler23). The ODEs are integrated over a time range of 10 residence times (continuous mode of operation), assuming that after that time steady state has been achieved, or over the feeding time (semibatch mode of operation). Subroutines calculate values for the change of population due to nucleation, growth, agglomeration, and disruption. Another NAG subroutine solves simultaneously the

coupled algebraic equations for the level of supersaturation in the different compartments and the exchange of mass and particles by micromixing and mesomixing. The population density of each class in the different compartments is calculated from the number of particles in each size class. The input and output variables of the SFM for continuous modeling are listed in Table 5. Continuous Mode of Operation. To determine the mesomixing time, a least-squares fit of the 300 mL continuous calcium oxalate (CaOx) precipitation results for the number mean size and nucleation rate was performed. From these calculations, the factor Λ in eq 15 was obtained as 17.7. Using the kinetic parameters determined from the laboratory-scale continuous experiments (Zauner16), the large-scale experiments were simulated with the segregated feed model (SFM) and compared with the experimental findings. Volume Mean Size L43. In Figure 5, the volume mean size is plotted versus the specific power input for the 300 mL, 4.3 L, and 12 L reactors and under the conditions according to set 1 in Table 2. Both the experimental results and model predictions show a maximum of L43 for a power input between 0.4 and 0.5 W/kg. Such a maximum was observed earlier for the precipitation of barium sulfate (Kim and Tarbell24). The steep decrease of the volume mean size for specific power inputs greater than 1 W/kg is probably due to very high disruption rates under these vigorously

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Figure 6. Scale-up of continuous calcium oxalate precipitation: volume mean size L43 (300 mL, 4.3 L, and 12 L; 0.01 M, 11 min, od).

agitated conditions. The model predicts the experimental data satisfactorily, but no clear trend with scale-up can be determined. Under this particular set of conditions, scale-up with constant mean specific power input could be a reasonably good scale-up criterion. In this case, the curves on different scales would thus merge into one curve. The ideal MSMPR model, in which it has been assumed that the nucleation rate and the growth rate are constant and that secondary nucleation does not occur, is not capable of predicting any of the experimentally observed effects of power input on mean size, as one of the assumptions for an MSMPR crystallizer is well-mixedness. The results of the second set of conditions (0.01 M, 11 min, od) are depicted in Figure 6. Surprisingly, the volume mean size obtained from the 300 mL small-scale experiments is substantially larger than that from the 4.3 and 12 l scales. Without adjusting any parameters and using only small-scale kinetic data, the segregated feed model (SFM) correctly predicts both the larger mean sizes observed on the 300 mL scale and the smaller mean sizes on the 4.3 and 12 L scales. As already observed above, the ideal MSMPR model cannot account for any of the mixing effects and under- or overpredicts the volume mean size. One explanation for the smaller values of L43 on the larger scales could be that even in geometrically similar locations in the reactors different mixing conditions prevail owing to the different shear rate gradients in reactors of different sizes. In the 300 mL reactor, the energy dissipation at the feed point is higher than that in the same region of the larger reactors, even though the feed point is remote from the impeller zone (feed point position od outside the draft tube). Steeper gradients in the shear rate distribution lead to lower levels of specific energy dissipation in regions remote from the impeller zone on the larger scales. As the precipitation rates and consequently the mean size of the product are different, different mixing times on different scales account for this effect. The conventional scale-up criteria based on constant specific power input, constant tip speed, and constant stirrer speed fail. Coefficient of Variation CV. The coefficient of variation CV, which is a measure of the width or spread of a particle size distribution, is plotted versus the specific power input in Figure 7. For an ideal MSMPR, a CV of 0.5 independent of the mixing conditions (assumption of perfect mixedness) is obtained. In the experiments

Figure 7. Scale-up of continuous calcium oxalate precipitation: coefficient of variation, CV (300 mL, 4.3 L, and 12 l; 0.01 M, 11 min, od).

Figure 8. Scale-up of continuous calcium oxalate precipitation: nucleation rate B0 (300 mL, 4.3 L, and 12 L; 0.04 M, 7.5 min, id).

and simulations, a CV larger than 0.5 was found under almost the whole range of conditions, indicating a wider spread than in the ideal scenario. The reason for this increase in spread could be the occurrence of agglomeration on one hand and on the other the inhomogeneity of the field of supersaturation resulting in locally different rates. At very high specific power inputs, a coefficient of variation smaller than 0.5 was found. This could provide further evidence of the importance of particle disruption, which narrows the distribution leading to a smaller CV. The model predictions and experimental data correspond well. Nucleation Rate B0. As mentioned above, nucleation is the first kinetic process occurring during precipitation and depends largely on the level of supersaturation. In Figure 8, the overall mean nucleation rate in the reactor is shown, plotted versus the specific power input. Depending on the scale of operation, a maximum occurred between 0.03 and 0.1 W/kg. The experimental and modeling results exhibit the same trend and cover the same order of magnitude; however, no pronounced effect of scale-up on the nucleation could be found. The decrease in B0 at high power inputs could be attributed to shorter mixing times and therefore a more uniform level of supersaturation throughout the reactor. The decrease of B0 at low power inputs might be due to insufficient mesomixing (blending) leading to very small zones where nucleation can occur. In addition, Figure 8 shows that the model predicts larger nucleation rates on the 4.3 L scale than on the 300 mL and 12 L scales.


Figure 9. Mean particle size for reference conditions (CaOx, Rushton turbine, 40 min feed time, feed point position close to the impeller, total concentration 0.008 M).

This could be an indication that both micromixing (diffusion) and mesomixing (convection) can become limiting, depending on the scale. Semibatch Mode of Operation. Volume Mean Size L43. In comparison with the continuous mode of operation, the mean size was found to depend to a greater degree on the mixing conditions on all scales in the semibatch mode. In Figure 9, the results for the reference conditions (Rushton turbine, 40 min feed time, feed point position close to the impeller, total concentration 0.008 M) for calcium oxalate confirm this observation. By changing the energy input, the volume mean size was varied over a wide range from 7 to 26 µm. For the two reactor scales of 1 and 5 L, scale-up with constant specific power input seems appropriate, while for the 25 L scale smaller particle sizes are obtained in the industrially important range from 0.1 to 1 W/kg. The kinetic parameters, which were determined from laboratory-scale continuous experiments as a function of the energy input and/or supersaturation, were applied to the semibatch mode of operation without any adjustments or parameter fitting. The segregated feed model (SFM) slightly underestimates the mean particle size in the range between 0.01 and 1 W/kg but correctly predicts the smaller particle size obtained experimentally for the 25 L reactor. On the same scale, the model also predicts a lesser degree of dependence of the particle size on the specific power input owing to the interactions of mixing and the precipitation kinetics. This behavior has also been observed experimentally in this research. For a different stirrer type, i.e., the marine-type propeller, a different range of specific energy dissipations is covered with the same range of stirrer speeds (Figure 10). The propeller stirrer was operated between 100 rpm ( ) 10-3 W/kg, Re ) 6.3 × 104) and 1500 rpm ( ) 4.67 W/kg, Re ) 1.3 × 106). Analogously to the scale-up experiments with a Rushton turbine, the mean particle size on the 25 L scale is smaller than that on the smaller scales. This could be explained by the lower level of energy dissipation around the feed point, which is caused by an increase in gradient in the shear rate with scale-up. Consequently, the local energy dissipation in the feed region decreases on a larger scale, and the mixing times increase. As already pointed out for the Rushton turbine, the model correctly predicts this trend for the propeller too. Small particle sizes obtained at low energy inputs are probably a result of local zones with very high levels of

Figure 10. Mean particle size for marine-type impeller (CaOx, 40 min feed time, feed point position close to the impeller, total concentration 0.008 M).

Figure 11. Mean particle size for feed point position close to surface (CaOx, Rushton turbine, 40 min feed time, total concentration 0.008 M).

supersaturation and therefore high nucleation rates. At high energy inputs, in contrast, breakage might act as a size-reducing process, leading to smaller particles. When the feed point position is moved to a position further away from the impeller in a zone with lower local energy dissipation, the particle size drops by about 25% for the whole range of conditions (Figure 11). Under such conditions, the effect of producing smaller mean particle sizes on the 25 L scale is even more pronounced. Again, all curves show a maximum; the influence of the power input on particle size is weakest on the largest scale, however, causing a shallower maximum than on the 1 and 5 L scales. At a higher feed rate (10 min feeding time instead of 40 min), the model slightly underestimates the volume mean particle size. The particle size is approximately 30% smaller than that for the reference conditions, as a higher level of supersaturation prevails in the reactor owing to the faster feeding (Figure 12). Therefore, moving the feed point position to a zone with lower local energy dissipation and increasing the feed rate have the same effect. An increase in feed rate, however, reduces the feed time and therefore the batch time and increases the efficiency of the process. Number Density of Particles in the First Size Interval n(L0). As the nucleation rate in a semibatch precipitation reactor is strongly time-dependent, an alternative

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Figure 12. Mean particle size for high feed rate (CaOx, Rushton turbine, feed point position close to the impeller, total concentration 0.008 M).

Figure 14. Scale-up with constant stirrer speed (CaOx, semibatch precipitation).

Scale-up with Constant Stirrer Speed. This scaleup criterion is based on achieving a constant pumping rate per unit volume with scale-up and therefore leads to similar macromixing on different scales, as the circulation time in the reactor remains constant.

N1 ) N2

Figure 13. Number density of nuclei for reference conditions (CaOx, Rushton turbine, 40 min feed time, feed point position close to the impeller, total concentration 0.008 M).

quantity, the number density of particles in the first size interval n(L0), was chosen to characterize nucleation. In Figure 13 the results for the reference conditions on the 1 and 5 L scales confirm the high nucleation rates and therefore large number of particles expected at low power inputs. Up to about ) 1 W/kg, a decrease of n(L0) was found experimentally, and at higher power inputs the data scattered substantially. As the number density was calculated from a volume size distribution, even small deviations in the mass distribution result in large errors in the number distribution. Therefore, its sensitivity to experimental errors is very high. The SFM predicts the trend of the decrease but deviates where the data start to scatter with increasing power input. At even higher power inputs, the model predicts an increase in n(L0). This increase has been experimentally verified for experiments carried out with a propeller impeller. 6. Scale-up Recommendations For many unit operations in chemical engineering, theoretical or empirical scale-up rules have been established. In processes, however, where kinetic rates (e.g., reaction or precipitation rates) are controlled by the degree of mixing, these simple scale-up criteria often fail. The approach toward scale-up developed in this research takes into account these mixing effects on different scales and has thus been successfully applied to continuous and semibatch precipitation processes.

(26)

In Figure 14, the mean particle size is plotted versus the stirrer speed for semibatch precipitation of calcium oxalate. The data scatter substantially with scale-up; this criterion is therefore not suitable for scaling up precipitation processes. This result is as expected, since it has been demonstrated (Zauner16) that macromixing is not the controlling process for the geometries under investigation as the residence time distribution (RTD) obtained suggests ideal macromixing. Scale-up with Constant Tip Speed. If the tip speed of the impeller blades is kept constant with scale-up, i.e.

N1πds1 ) N2πds2 ) const

(27)

the criterion for scale-up becomes

N2 ) N1

() V2 V1

-1/3

(28)

This criterion has been applied to the same set of data as above, with the result that the data on different scales are less scattered than for scale-up with constant stirrer speed (Figure 15). Scale-up with constant tip speed, which implies constant shear in the impeller region, can be considered an approximation to scale-up with constant mesomixing, as the blending of the incoming reactant with the bulk or second reactant is closely linked to the shear field in the mixing zone. As Oldshue25 pointed out, however, with larger scales, the maximum shear rate in the impeller zone increases while the average shear rate in this zone decreases. Thus the shear rate distribution changes with scaleup, and this criterion is therefore only an approximation, based on mean shear rates in the impeller zone. Scale-up with Constant Power Input per Unit Volume. The scale-up criterion that is probably most widely used for mixing-limited unit operations is based on constant power input per unit volume according to (Harnby et al.26)


Figure 15. Scale-up with constant tip speed (CaOx, semibatch precipitation).

)

kpowerd5s N3 ) const V

(29)

This leads to

()

V2 N2 ) N1 V1

Figure 16. Precipitation process scale-up methodology.

-2/9

(30)

The importance of the local energy dissipation and thus the specific power input for micromixing has already been referred to, where the micromixing time was related to the Kolmogoroff length scale of mixing (eq 14). Even though it was possible to predict properties on different scales using this criterion, it failed with respect to other conditions. Figure 6 shows the experimental results of scale-up of continuous precipitation, where the application of ) constant was not successful. In Figure 10, widely scattering results were obtained with this criterion for semibatch precipitation too. The failure of “conventional” criteria may be due to the fact that it is not only one mixing process that can be limiting; rather, for example, an interplay of micromixing and mesomixing can influence the kinetic rates. Thus, by scaling up with constant micromixing times on different scales, the mesomixing times cannot be kept constant but will differ, and consequently the precipitation rates (e.g., nucleation rates) will tend to deviate with scale-up. Scale-up with the Segregated Feed Model (SFM). To account for both micromixing and mesomixing effects, a mixing model for precipitation based on the segregated feed model has been developed and applied to continuous and semibatch precipitation. The model can be extended by establishing a network of ideally macromixed reactors if macromixing plays a dominant role. The model is able to predict the influence of mixing on particle properties and kinetic rates on different scales for a continuously operated reactor and a semibatch reactor with different types of impellers and under a wide range of operational conditions. From laboratoryscale experiments, the precipitation kinetics for nucleation, growth, agglomeration, and disruption have to be determined (Zauner and Jones27). The fluid dynamic parameters, i.e., the local specific energy dissipation around the feed point, can be obtained either from computational fluid dynamics (CFD) or from laser doppler anemometry (LDA) measurements. In the compartmental SFM, the population balance is solved and

the particle properties of the final product are predicted. As the model contains only physical and no phenomenological parameters, it can be used for scale-up. 7. Conclusions Precipitation Kinetics. Reliable kinetic data are of paramount importance for scale-up. Apart from nucleation and growth, secondary processes such as agglomeration and breakage occur and cannot be neglected as they have a strong influence on the particle size distribution and product properties. If necessary, even aging and ripening processes have to be investigated experimentally and implemented in the population balance. Why Do Conventional Scale-up Criteria fail? The conventional scale-up criteria “scale-up with constant stirrer speed”, “scale-up with constant tip speed” and “scale-up with constant specific energy input” are all based on the assumption that only one mixing process is limiting. If, for example, the specific energy input is kept constant with scale-up, the same micromixing behavior could be expected on different scales. The mesomixing time, however, will change with scale-up; as a result, the kinetic rates and particle properties will be different and scale-up will fail. Scale-up Methodology. The methodology of how to scale up a precipitation process is depicted in Figure 16. For large-scale predictions, kinetic information can be obtained from laboratory-scale experiments. On the large scale, only CFD simulations (or LDA measurements) need to be carried out to obtain the local energy dissipation in the feed zone(s). Using this information, the segregated feed model predicts the particle properties on the large scale. Scale-up thus becomes possible without time-consuming and costly large-scale precipitation experiments. This methodology is very efficient as it combines the advantages of both a CFD and a population balance approach without having to solve the equations together, which is currently still impracticable owing to computational demand and simulation time required. The results of the model have been successfully validated with experiments under a wide range of

2402


conditions for the precipitation of calcium oxalate for both continuous and semibatch precipitation.

νtip ) tip speed (m s-1) Vj ) subvolume of compartment j (m3) Vtot ) reactor volume (m3)

Acknowledgment

Greek

The authors thank the Training and Mobility Programme of the European Commission for the financial support that enabled R.Z. to spend time at the Large Scale Facility COPES, Clausthal Centre of Process Engineering SystemssDesign and Research, where part of the work was carried out. R.Z. was supported by the EPSRC.

βaggl ) agglomeration kernel βdisr ) disruption kernel γ ) surface energy (J m-2) ) specific power input (W kg-1) Λ ) mesomixing coefficient (eq 15) µ ) dynamic viscosity (kg m-1 s-1) ν ) kinematic viscosity (m2 s-1) νi ) stoichiometric coefficient of component i F ) solution density Fc ) crystal density (kg m-3) σj ) supersaturation in compartment j τ ) residence time (s)

Nomenclature A ) nucleation rate constant (m-3 s-1) b ) bulk compartment in SFM b(ν, xk) ) breakage function B ) birth rate (m-4 s-1) B0 ) nucleation rate (general) (m-3 s-1) B0i ) nucleation rate in compartment i (m-3 s-1) c ) concentration (mol m-3) c* ) equilibrium concentration (mol m-3) cj ) concentration of reactant j (mol m-3) c0 ) initial concentration (mol m-3) c0j ) feed concentration of reactant j (mol m-3) CV ) coefficient of variation dS ) diameter of stirrer (m) D ) reactor diameter (m) D ) death rate (m-4 s-1) f1, f2 ) feed compartments in SFM g ) growth rate order G ) growth rate (m s-1) k ) Boltzmann constant (J K-1) k ) kinetic energy (m2 s-2) ka ) surface area shape factor kg ) growth rate coefficient (m s-1) kpower ) power coefficient kpump ) pumping rate coefficient kv ) volumetric shape factor Kaggl ) agglomeration function (m3 s-1) Kap ) activity product (mol2 m-6) Kdisr ) disruption function (s-1) Ksp ) solubility product (mol2 m-6) L ) particle size (m) L43 ) volume mean size (m) mj ) jth moment of distribution (mj-3) Mwj ) molecular weight of component j (kg mol-1) n ) nucleation rate order n ) population density (general) (# m-4) nk ) population density of stream k (# m-4) nP ) population density of product P (# m-4) n0 ) population density of nuclei (# m-3) N1, N2 ) stirrer speed on scales 1 and 2, respectively (s-1) Ni ) number of particles in size class i (# m-3) P ) power input (W) Po ) power number Qk ) flow rate of stream k (m3 s-1) Qpump ) pumping capacity of impeller (m3 s-1) rj,k ) reaction rate of component j in compartment k S ) supersaturation (general) S′ ) disruption function t12 ) time constant for direct mixing of feed streams (s) tmesoj ) time constant for mesomixing in compartment j (s) tmicro,j ) time constant for micromixing in compartment j (s) T ) temperature (K) uj,ik ) exchange flow of component j between compartments i, k (m3 s-1) ν ) velocity (m s-1)

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Ind. Eng. Chem. Res., Vol. 39, No. 7, 2000 2403 (21) Kumar, S.; Ramkrishna, D. On the solution of population balance by discretization I. A fixed pivot technique. Chem. Eng. Sci. 1996, 51, 1311-1332. (22) Kumar, S.; Ramkrishna, D. On the solution of population balance by discretization II. A moving pivot technique. Chem. Eng. Sci. 1996, 51, 1333-1342. (23) Schuler, H. Prozeβsimulation (in German); VCH-Verlag: Weinheim, 1996. (24) Kim, W.-S.; Tarbell, J. M. Micromixing effects on barium sulfate precipitation in an MSMPR reactor. Chem. Eng. Commun. 1996, 146, 33-56. (25) Oldshue, J. Y. Fluid Mixing Technology; McGraw-Hill: New York, 1983.

(26) Harnby, N.; Edwards, M. F.; Nienow, A. W. Mixing in Process Industries; Butterworth & Co.: Markham, ON, 1985. (27) Zauner, R.; Jones, A. G. Determination of nucleation, growth, agglomeration and disruption kinetics from experimental precipitation data: The calcium oxalate system. Chem. Eng. Sci., in press.

Received for review June 14, 1999 Revised manuscript received April 13, 2000 Accepted April 17, 2000 IE990431U

Scale-up of Continuous and Semibatch Precipitation Processes

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