Kinetics Estimation and Single and Multi-Objective Optimization of a

Feb 6, 2008 - The moments were calculated using the measured chord length distribution (CLD) generated by the ... rates using power law correlations...
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Kinetics Estimation and Single and Multi-Objective Optimization of a Seeded, Anti-Solvent, Isothermal Batch Crystallizer M. Trifkovic, M. Sheikhzadeh, and S. Rohani* Department of Chemical and Biochemical Engineering The UniVersity of Western Ontario London, Ontario, N6A 5B9, Canada

The nucleation and growth kinetic parameters of paracetamol in an isopropanol-water antisolvent batch crystallizer were estimated by nonlinear regression in terms of the moments of the crystal population density. The moments were calculated using the measured chord length distribution (CLD) generated by the FBRM. The measured supersaturation by ATR-FTIR spectroscopy was also used to calculate the nucleation and growth rates using power law correlations. Using the estimated kinetic parameters, the crystallization model based on the population and mass balance, was validated using the open-loop experimental particle size distribution and supersaturation results. Subsequently, the solution to the optimal antisolvent flow-rate profiles was obtained by applying nonlinear constrained single- and multiobjective optimization on the validated model. These profiles were implemented on the crystallizer and crystal-size distributions were compared with the openloop experiments. The bimodality in the particle size distribution (PSD), which was present in the open-loop experiments, was either minimized or completely eliminated with the optimal profile policies. The results of the multiobjective optimization showed an improvement of 27.5 µm and 3% in the volume weighted mean size and yield, respectively, in comparison to the best results obtained from the open-loop experiments. 1. Introduction Crystallization from solution is an industrially important operation because of its ability to provide high-purity separations. Batch and semibatch processes are of considerable importance in the fine chemicals and pharmaceutical industries. Most industrial crystallization batch processes operate in the open-loop manner. The main objective in a crystallization process is to improve the end-of-batch properties such as particle size distribution (PSD), purity, morphology, optical and polymorphic purity, yield, and batch time. An optimized process can be applied on a crystallization unit to achieve a desired PSD, which strongly influences the quality of the crystalline product. The evolution of in-line sensors for measuring key process variables such as supersaturation, size, and polymorphic form opened new directions in the crystallization research. Attenuated Total Reflectance - Fourier Transform Infrared (ATR-FTIR) spectroscopy has been extensively used for the in situ measurement of the concentration of the solute in the solution.1 In-line particle size measurement has been the subject of great interest in the crystallization community. Focused beam reflectance measurement (FBRM) and ultrasound probes have shown some success.2,3 The FBRM measures the chord length distribution (CLD) of particles. Conversion of these measurements to particle size distribution involves sophisticated mathematical algorithms.4 O’Sullivan et al.5 applied in situ ATR-FTIR and FBRM for monitoring the polymorphic transformation in the solution. Togakalidou et al.6 integrated the ATR-FTIR and FBRM measurements to estimate parameters for the nucleation and growth of particles in cooling crystallization. ATR-FTIR and FBRM have been also successfully applied for the feedback control of antisolvent crystallization.7,8 Off-line optimal control is one of the popular optimization methods that has been widely used for chemical processes. * To whom correspondence should be addressed. Tel.: (519) 6614116. Fax: (519) 661-3498. E-mail: [email protected].

Jones9 used Pontryagin’s minimum principle for the optimal control of batch crystallizers. Several other methods such as stochastic optimization and sequential quadratic programming have been applied to find the optimal temperature trajectory.10,11 A significant portion of research conducted thus far in this field dealt with the single objective optimization. However, in reality, crystallization processes involve several objectives that are often conflicting in nature. For example, it would be desirable to have a product PSD with a large mean size and a small coefficient of variation in a minimum time of operation. This article deals with the • Estimation of kinetic parameters using the measured growth and nucleation rates - by FBRM - and validated by measured supersaturation by ATR-FTIR. • Solving the population and mass balance simultaneously (using the estimated kinetic parameters) and validating the model using the open-loop measured PSD and supersaturation data. • Finding and implementing the optimal antisolvent flowrate profiles obtained from single and multiobjective optimization on the system. • Comparison of the results obtained from the optimal control with the open-loop experiments (using three different policies of antisolvent feeding) in terms of the final PSD and yield. 2. Materials and Experimental Methods The experiments were performed in a 1 L glass-jacketed vessel (Simular, Automated Chemical Reactors and Calorimeters; HEL Ltd., Vinelean, NJ). A Teflon-coated thermocouple was used for reading the temperature in the flasks. A metering pump was used for pumping water, which was used as an antisolvent in the crystallization process. The solution was stirred by a four-bladed propeller. An electronic balance (Oxford, B41002) was used for recording the amount of the antisolvent added to the solution. Moving average chord length distribution measurements were collected every 1 min using the in situ FBRM probe (Mettler Toledo, Redmond, WA). An in situ ATRFTIR (Hamilton Sundstrand, CA and DMD-270 diamond ATR

10.1021/ie071125g CCC: $40.75 © 2008 American Chemical Society Published on Web 02/06/2008

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Figure 1. Experimental setup.

Figure 2. Solubility data for paracetamol in a water-isopropanol mixture.

immersion probe) was used for in-line collecting of infrared spectra. The FTIR spectra were related to the solute concentra-

tion in the solution using the calibration curve developed by Hojjati and Rohani.12 Particle size distribution of the final product was measured by Malvern 2000 SCIRRCCO (Malvern Instruments Ltd., UK). Sheikhzadeh et al.13 presented a more detailed explanation of the experimental setup and the connections between the Simular-HEL reactor, FBRM, and ATR-FTIR software with Matlab (Figure 1). A typical experimental procedure consisted of the following: The crystallizer was charged with a given amount of paracetamol (acetaminophen or 4-acetamidophenol) and a solvent mixture of isopropanol and water. The batch temperature was raised to 24 °C and kept constant for half an hour to ensure that all of the crystals were dissolved. The temperature was then lowered to 20 °C to start the experiment. The initial concentration was equal to or slightly greater than the solubility point at 20 °C. The solubility curve, as well as the operating region, is shown in Figure 2. The crystallizer was seeded with a predetermined amount of seeds. The initial mass of the solution in all of the experiments was 400 g, and an additional 400 g of

Table 1. Experimental Conditions and Constraints variable

experimental conditions

constraints

a

name

kv mIPA+H2O rc L0 C0 C

shape factor initial mass of solvent crystal density nuclei size initial concentration concentration

C* T

solubility temperature

FRmax FRgradient

minimum flow-rate constraint maximum rate of change of antisolvent flow-rate initial mass fraction of water final mass fraction of water concentration constraint final batch time

mw initial mw final C > C* T

FTIR peak height at different wavelegths.

value 1.5 400 1293 0 g0.132 -0.3319 [P(1515) - P(1030)] + 2.1587 [P(1245)-P(1030)] + 0.0308a 20

units g kg/m3 µm gsolute/gsolvent(IPA+H2O) gsolute/gsolvent(IPA+H2O) gsolute/gsolvent(IPA+H2O) °C

0.5 1

g/min g/min

0.6 0.8 gsolute/gsolvent(IPA+H2O) 2

hr

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The mean particle size in the suspension can be defined as the ratio of the first moment to the zeroth moment.14

L h j + 1,j (t) )

µ1 (t)

) L1,0,k

µ0 (t)

(4)

The rate of change of the mean particle size with respect to time can be taken as the growth rate G.15

G(t)exp )

Table 2. Estimated Kinetic Parameters (Confidence Interval ) 95%)

N(t) )

kinetic parameters (R ) 0.05)

value × 104

g1 g2 g3 g4 g5

2.67 ( 5.55 × 10-2 0.89 ( 7.85 × 10-2 - 1.39 ( 2.71 × 10-2 91.75 ( 11.5083 991.93 ( 93.8337

b0 b1

682.92 ( 26.06 14.99 ( 2.494

growth

nucleation

antisolvent was added in 2 h. The addition of 400 g of antisolvent (water) was initiated 30 min after seeding the crystallizer. The speed of the stirrer was set in all of the experiments at 200 rpm to ensure complete suspension of solid particles and no air entrainment throughout the experiments. At the end of each experiment, the suspension was withdrawn, filtered through a 1 µm filter, air-dried, and its size distribution was measured. The experimental conditions and constraints are listed in Table 1.

∫0∞n(t)dr ≈

FBRMlast channel

∑

i)1

Ni,k ) Nk

3.1. Parameter Estimation Methodology. In this study, the chord length distributions measured by FBRM were used to estimate the moments of the particle population density. The growth and nucleation rates were calculated from the moments directly, without converting the CLD measurements to the PSD. The CLD information from 1 to 1000 µm was collected in 90 channels. The moments of the CLD were calculated as

∫

∞ j r n(r, 0

FBRMfinal channel

∑ i)1

t)dr ≈

rave,ij Ni(rave,i, k) (1)

where ri and Ni are the chord length and number of particles in channel i, respectively, and k is the discrete time counter. rave,i is expressed as

rave,i )

ri + ri - 1 2

(2)

A family of possible particle sizes can be generated by the ratio of (j + 1)st to the jth moment of distribution. Therefore,

L h j + 1,j (t) )

µj + 1 (t) µj(t)

∫0∞r j + 1n(r, t)dr ≡ ∫0∞r jn(r, t)dr

(3)

(6)

The nucleation rate was calculated as the difference in the total number of particles in each sampling time per mass of solvent.

Bexp(t) )

1 ∆Nk 1 dN(t) ≡ ) Bexp,k Mk ∆t M(t) dt

(7)

where Bexp,k and Mk are the nucleation rate and total mass of solvent in the kth time interval, respectively. In an antisolvent crystallization, the total mass of solvent changes throughout the experiment as follows:

Mk ) Minitial + Mantisolvent,k

(8)

On the other hand, nucleation and growth rates can also be calculated using correlations in terms of the supersaturation, which is defined as,

∆Ck ) Ck - C*k

3. Theory

µj(t) ≡

(5)

The total number of crystals in the whole FBRM size range (1 to 1000 µm) was used to compute the nucleation rate. The total number of particles at time t is defined as

Figure 3. Antisolvent flow-rate policies.

rate

h 1,0(t)) d(L h 1,0) ∆(L x ) Gexp,k dt ∆t

(9)

where Ck and C*k are solute concentration and saturation concentration (solubility) at time k, respectively. The solute concentration can be measured directly from in situ ATR-FTIR spectroscopy, and solubility is function of mass percent of antisolvent in solution (Table 1). There are numerous correlations for nucleation and growth rates in batch and semibatch crystallization processes. For an antisolvent crystallization, kinetic parameters are usually functions of the mass of antisolvent.14 In this study, the following nucleation and growth correlations were used.

( ) ( ( )) () ln

Bpred,k ) b0 exp -b1

ln

Fs

C/k Ck

2

(10)

C/k

Gpred,k ) kg∆Ckg

(11)

kg ) g1mwk2 + g2mwk + g3

(12)

g ) g4mwk + g5

(13)

where b0 and b1 are nucleation and g1 to g5 are growth constants, mw is antisolvent mass percent, and Fs is solid crystal density. In this study, the measured supersaturation by ATR-FTIR and

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Figure 4. Comparison between the measured rate and the estimated nucleation rate for three sets of experiments.

Figure 5. Comparison between the measured rate and the estimated growth rate for three sets of experiments.

correlations given by eqs 10-13 were used to calculate the predicted nucleation and growth kinetics. However, the calculated supersaturation by the model may be used instead, and therefore the entire algorithm will need FBRM data alone. The main drawback of such an approach is the poor accuracy in kinetic parameter estimates because they are strongly dependent on the solution concentration. The objective function of the nonlinear parameter estimation algorithm at the end of the batch, F(tfinal), was set as the minimization of the difference of the sum-of-squares of growth and nucleation rates calculated by eqs 5 and 7 (Bexp,k and Gexp,k) and those given by eqs 10-13 (Bpred,k and Gpred,k) at each sampling interval k. K

FB(tfinal) )

(Bexp,k - Bpred,k)2 ∑ k)1

(14)

K

FG(tfinal) )

(Gexp,k - Gpred,k)2 ∑ k)1

(15)

where K is an integer equal to tfinal/∆t. The microscopic inspection and the sieve analysis of the final product showed that breakage and agglomeration were low and the crystal shape remained unchanged. The measured supersaturation, nucleation, and growth rates data from three independent experiments using various open-loop antisolvent feeding policies - were used to validate the accuracy of the estimated kinetic parameters. 3.2. Crystallization Model Formulation. A mathematical framework suited to modeling crystallization processes is the

Figure 6. PSD with three flow-rate policies.

population balance, which describes the state of the particle size distribution. The model is developed under the assumptions that the crystallizer is well mixed, crystal agglomeration and breakage phenomena are neglected, there is no growth dispersion, and crystals are born at size zero. Crystal growth is also considered independent of size for the sake of simplicity of the model and predictions of growth rates. Experimental observations justified this assumption as well. Under these abovementioned assumptions, the 1D evolution of the crystal-size

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Figure 7. Predicted PSD for antisolvent flow-rate policies. (a) Policy 1, (b) policy 2, and (c) policy 3.

distribution with a characteristic length L and the solute balance can be described by the following set of equations:16

∂n(L, t) ∂n(L, t) + G(t) )0 ∂t ∂L

(16)

dC ) -3FkvG(t)µ2(t) dt

(17)

The solution of eqs 16 and 17 requires an appropriate boundary and initial conditions, which are given as,

n(L, 0) ) n0(L)|t ) 0 n(0, t) )

B(t) | G(t) L ) 0

C(0) ) C0

(18) (19)

{

}

0.0032(Lfinals - Ls)(Ls - Linitals) Linitials e Ls e Lfinals 0 otherwise (21)

where the superscript s stands for the seed crystals.

n(L1, t1)∆L1 ) n(L2, t1, ∆t)∆L2

(22)

which results in

n(L2, t1 + ∆t) ≈

n(L1, t1) 1 + (∂G(L, t1)/∂L)|L ) L1∆t

(23)

(20)

where C0 is the initial concentration of the solute in the solvent and no is the population density of seeds at time zero that was assumed to have a parabolic distribution:17

n0(L)|t ) 0 ) n(Ls, 0) )

By solving the above eqs 16-21, time evolution of the crystal-size distribution and solute concentration can be obtained for an isothermal batch operation. A new methodology, which converts eq 16 to a set of algebraic equations, has recently been presented by Hu et al.17 In the absence of aggregation, breakage, and nucleation, the population densities at times t1 and t1 + ∆t are related by eq 22,

By assuming that the secondary nuclei have the same growth rate as of the seed crystals, the population density of new crystals can be obtained. If the growth rate is size-independent, then n(L2,t1 + ∆t) ) n(L1,t1). In this case, the shape of the seed distribution does not change and simply moves along the L axis at a rate equal to G. More details on the solution of the population balance can be found in Hu et al.17,18,19 To evaluate the accuracy of the model, simulation, and experimental results of the final particle, size distribution and supersaturation corresponding to three different open-loop antisolvent feeding policies were compared. The validated model

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Figure 8. Supersaturation profiles obtained experimentally and by model (a) policy 1, (b) policy 2, and (c) policy 3.

was used to compute the optimal antisolvent feeding profile for single-objective (S/O) and multiobjective (M/O) optimizations. 4. Result and Discussion 4.1. Parameter Estimation and Model Validation. The estimated nucleation and growth kinetic parameters, as well as their confidence intervals, are listed in Table 2. The validation of the model was performed prior to the offline optimization to ensure the accuracy of the model prediction. It consisted of performing the open-loop experiments with three different antisolvent flow-rate policies, and ensuring that the final size distribution and the concentration profile generated by the model were close to the experimental results. The least-squares fitting and calculating the confidence intervals for the parameters was performed using Levenberg-Marquardt method, which is built in the lscurVefit routine of the MATLAB Optimization toolbox. The typical characteristic curves for three antisolvent feeding policies used in the open-loop experiments are shown in Figure 3. All three modes can provide constructive information at high, medium, and low levels of supersaturation at the initial, intermediate, and the final stage of the crystallization. For three policies, the seed loading, defined as the ratio of mass of the seed crystals to the theoretical yield of product, was set at 0.04 and the batch time was 2 h. Figures 4 and 5 show the comparison between the estimated and measured nucleation and growth rates, respectively. As seen from these figures, the magnitudes of the values for the predicted nucleation and growth rates are close to those obtained experimentally. The discrepancy in the shapes between modeled and experimental curves for growth and nucleation rates can be attributed to the fact that the model does not account for agglomeration and breakage. As stated above,

prediction of nucleation and growth, in the present study, was based on the measured concentration and solubility data obtained from the experiment. However, if in-line concentration measurement is not available, the solution of mass balance (eq 17) can be used. The particle size distributions for the final products of the three open-loop experiments, measured by the Malvern 2000 SCIRRCCO instrument, are shown in Figure 6. As seen from the plot, policy 3 resulted in the largest particles. However, with this mode of antisolvent addition, the nucleation events were also more significant and resulted in a bimodal particle size distribution. This could be attributed to the higher supersaturation peak and the resulting secondary nucleation. The simulated PSD for policy 3 is also presented in Figure 6, and it shows a good agreement between the final seed size distribution and the experimental data. However, the size distribution of the newly nucleated crystals has a relatively big offset in comparison to the experimentally obtained PSD. The predicted PSDs for the three open-loop policies are also shown in Figure 7. It can be observed that policy 3 resulted in large particles as well as the greatest number of the newly nucleated particles. The model simulation of policy 1 resulted in the smallest particles, which agreed with the data obtained experimentally. This can be explained due to the fact that, at the beginning of the experiment, the flow-rate of the antisolvent was very low and led to low supersaturation and low nucleation rates. However, toward the end of the batch the amount of added antisolvent reached its peak that resulted in producing the highest supersaturation at the end of the experiment. Consequently, there was not enough time for the growth of particles until the end of the experiment. The above observations can also be deduced from the measured supersaturation profiles. It can be also noted that the increase in supersaturation for policy 1 toward the end of the experiment caused the simultaneous occurrence of

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Figure 9. Optimal profiles for S/O-1 with two different initial guesses.

Figure 11. Concentration profiles with three optimal profiles. (a) S/O-1, (b) S/O-2, and (c) M/O.

Figure 10. Optimal flow-rate profiles for S/O-2 and M/O problems.

nucleation and growth of crystals after 60 min (part a of Figure 4 and part a of Figure 5). From this point onward, the competition between these two kinetics continued until the end of the experiment and subsequently led to a product with the largest amount of fine crystals. The predicted (modeled) concentration profiles were calculated using eqs 16-23. The measured and predicted supersaturations for the three antisolvent feeding policies are shown in Figure 8. It is noted that the modeled supersaturation profiles have trends similar to the experimental profiles. It can also be observed that the maximum increase in supersaturation, with respect to the initial supersaturation point, was the highest for policy 3 (0.012 g solute/g solvent) and the lowest for policy 1 (0.006 g solute/g solvent). In policy 3, the initial flow rate of the antisolvent was very high and resulted in a higher supersaturation peak. Both the width of the PSD and the concentration profiles indicated that the model and the estimated nucleation and growth rate parameters were acceptable and allowed us to proceed with the optimization step and to find the optimal antisolvent profile. 4.2. Offline Optimal Control. The objective function was expressed in terms of the properties associated with the product quality, such as, large crystal size, narrow PSD, and high yield. Several objective functions were considered, and the resulting PSD distributions were compared. Constraints reported in Table 1 were used in all of the objective functions. The solutions of the optimal antisolvent profiles were obtained by applying constrained nonlinear singleobjective and multiobjective optimization algorithms to the

crystallization model (eq 16-23). The MATLAB optimization function fmincon was used for solving the single-objective optimization problem, and the fminimax optimization function was used for the multiobjective optimization problem. Both functions are based on a sequential quadratic programming and require the initial guess for the antisolvent flow-rate feeding policy. fminimax optimization function minimizes the worstcase value of a set of multivariable functions, starting with an initial estimate. Case I - Single-Objective 1 (S/O-1). The first objective function was defined in terms of the nucleation and growth rates. The minimization of this objective resulted in suppressing nucleation and enhancing the growth of the existing seeds. The control variable was the antisolvent flow-rate profile (FR).

{

B Minimize J1 ) FR(t) G subject to dFR e 0.5 g/min c1 : dt c2: C > C* c3: tfinal ) tmax c4: FRmin < FR < FRmax

}

(24)

The equality and inequality constraints were chosen on the basis of the knowledge gained from the open-loop experiments. The final-time constraint, c3, was specified so that there is enough time allowed for the growth of particles. Case II - Single Objective 2 (S/O-2). In this case, minimization of the ratio of the third moment of the newly

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Figure 12. PSD of optimal profiles and policy 3.

nucleated crystals to the seed crystals in each optimization segment was attempted,

{

Minimize FR(t)

J2 )

µn3 µs3

subject to c1, c2, c3, c4

}

(25)

where µ3n and µ3s are the third moments of the PSD of the newly nucleated and the seed crystals, respectively. Case III - Multi-Objective (M/O). The objective for the third case, J3, was a vector consisting of three single objective functions. The first two objective functions dealt with the minimization of the nucleation, whereas the third one dealt with the minimization of the coefficent of variation of the seed’s particle size distiribution. The first objective functions attributed to the elimination of the PSD for the newly born crystals, and thus the third objective function was considered only for the minimization of the coefficient of variation for the seeded crystals.

{

Minimize FR(t) Minimize FR(t) Minimize FR(t) subject to c1, c2, c3, c4

J3,1 )

µn3

µs3 B J3,2 ) G

J3,3 )

x

µs3

µs5 (µs4)2

-1

}

(26)

The worst case scenario of these three objective functions was chosen in each optimization segment. The first step in the optimization was to test the sensitivity of the optimal results in terms of different initial guesses. Figure 9 shows the optimal profiles for case I obtained using two completely different initial guesses. As seen from the plot, the resulting optimal antisolvent flow rates were very similar, although the intial guesses were entirely different. This indicates that the optimization step was robust. The optimal profiles for case 2 (S/O-2) and the M/O case are depicted in Figure 10. For the case of M/O, the optimal profile resulted in a smooth exponential curve with the initial flow rate being lower in comparisson to the S/O case. The concentration profiles for three objective functions are depicted in Figure 11. The second single objective function (S/O-2) resulted in the highest drop in the solubility, which correponded to the drop in the flow-rate profile for this objective function. Among the open-loop experiments, policy 3 resulted in the highest product yield and the largest final product mean size (Figure 6). Thus, the PSDs obtained with optimal profiles were compared with the results of policy 3. Figure 12 shows the PSD of the seed, the three optimal profiles, and that of policy 3. It can be observed that all of the optimal profile policies resulted in larger particles and a narrower PSD compared to the best open-loop experiment. In addition, the bimodality in the PSD of policy 3 was either minimized or completely eliminated with the optimal profile policies. The optimization problems, which dealt with the minimization of nucleation and the maximization of the growth rate (S/O-1 and M/O), resulted in the PSD without secondary nucleation and a larger mean particle size.

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Table 3. Comparison between End of Batch Properties for Different Experiments

yield (%) surface weighted mean (µm) volume weighted mean (µm)

seed

policy 3

S/O-1

S/O-2

M/O

75.8 290.5

48.5 74.2 342.1

48.7 83.2 362.4

49.5 74.2 343.0

50.1 87.1 369.6

The best PSD result was obtained from the M/O optimal profile, which was expected considering that the objective function dealt with minimization of the coefficient of variation and supressing the nucleation simultaneously. The shape of the PSD corresponding to the multiobjective optimization case showed a complete shift of the seed PSD to larger sizes, which indicated good growth of particles with minimal breakage and agglomeration. The comparison between the yield, surfaceweighted mean, and the volume-weighted mean between the above-mentioned profiles is given in Table 3. These results show the superiority of the M/O profile with respect to the all of the tested criteria. 5. Conclusions The kinetic parameters of paracetamol in IPA and water were estimated from the CLD data measured by the FBRM probe. The information on the solute concentration provided by the FTIR probe was used to calculate the predicted nucleation and growth rates. A crystallization model was developed by solving the population and mass balance simultaneously and incorporating the estimated nucleation and growth rate kinetic constants. The simulated results were compared with the open-loop experiments to test the accuracy of the model. The single and multiobjective nonlinear constrained optimization was performed on the validated model to obtain the optimal profiles that applied to the system. The multiobjective optimization function resulted in the best product properties in terms of the final PSD and product yield. Notations b ) nucleation constants B ) nucleation rate (number/gsolvent min) c ) optimization constraints C ) concentration (gsolute/gsolvent) CLD ) chord length distribution CSD ) crystal size distribution F ) least squares objective function for nucleation and growth estimation FR ) flow rate (g/min) g ) growth constants G ) growth rate (m/min) J ) objective for optimal control kg ) growth constant kv ) shape factor K ) total number for sampling intervals L ) crystal size (m) mw ) anti-solvent mass percent M ) mass of solvent (g) n ) crystal population density (number/gsolvent m) N ) total number of particles P ) ATR-FTIR peak PSD ) particle size distribution r ) chord length (m)

t ) time (min) tfinal ) final batch time (min) Greek Symbols ∆C ) supersaturation (gsolute/gsolvent) ∆t ) sampling interval (min) µ ) moments of size distribution with dimension F ) solid density (kg/m3) Superscripts and Subscripts 0 ) initial * ) saturation exp ) experimental g ) growth i ) FBRM channel jth ) moment of the population density k ) sampling time counter n ) new crystals pred ) predicted s ) seed crystals w ) water Literature Cited (1) Togkalidou, T.; Fujiwara, M.; Patel, S.; Braatz, R. D. Solute Concentration Prediction Using Chemometrics and ATR-FTIR Spectroscopy. J. Cryst. Growth 2001, 231, 534-543. (2) Mougin, P.; Wilkinson, D. In Situ Measurement of Particle Size during the Crystallization of L-Glutamic Acid under Two Polymorphic Forms: Influence of Crystal Habit on Ultrasonic Attenuation Measurement. Cryst. Growth Des. 2002, 2, 227-234. (3) Higgins, J. P.; Arrivo, S. P.; Tharau, G.; Green, R. L.; Bowen, L.; Lange, A.; Templeton, A. C.; Thomas, D. L.; Reed, R. A. Spectroscopic Approach for On-Line Monitoring of Particle Size during Processing of Pharmaceutical Nanoparticles. Anal. Chem. 2003, 75, 17771785. (4) Tadayyon, A.; Rohani, S. Determination of PSD by ParTec 100: Modeling and Experimental Results. Part. Part. Syst. Charact. 1998, 15, 127-135. (5) O’Sullivan, B.; Glennon, B. Application of in Situ FBRM and ATRFTIR to the Monitoring of the Polymorphic Transformation of D-Manitol. Org. Process Res. DeV. 2005, 9 (6), 884-889. (6) Togkalidou, T.; Tung, H.; Sun, Y.; Andrews, A. T.; Braatz, R. D. Parameter Estimation and Optimization of a Loosely Bound Aggregating Pharmaceutical Crystallization Using in Situ Infrared and Laser Backscattering Measurements. Ind. Eng. Chem. Res. 2004, 43, 61686181. (7) Zhou, G. X.; Fujiwara, M.; Woo, X. W.; Rusli, E.; Tung, H.; Starbuck, C.; Davidson, O.; Ge, Z.; Braatz, R. D. Direct Design of Pharmaceutical Antisolvent Crystallization through Concentration Control. Cryst. Growth Des. 2006, 6, 892-898. (8) Sheikhzadeh, M., Trifkovic M., Rohani S. Adaptive MIMO NeuroFuzzy Logic Control of Seeded and Unseeded Anti-Solvent Semi-Batch Crystallizer. Chem. Eng. Sci. Accepted, 2007. (9) Jones, A. G. Optimal Operation of a Batch Cooling Crystallizer. Chem. Eng. Sci. 1974, 29, 1075-1087. (10) Rohani, S.; Bourne, J. R. A Simplified Approach to the Operation of a Batch Crystallizer. Can. J. Chem. Eng. 1990, 68, 795-806. (11) Miller, S. M.; Rawlings, J. B. Model Identification and Control Strategies for Batch Cooling Crystallizer. AIChE J. 1994, 40, 13121327. (12) Hojjati, H.; Rohani, S. Measurement and Prediction of Solubility of Paracetamol in Water-Isopropanol Mixture. Part 1: Measurement. Org. Process Res. DeV. 2006, 10(6), 1101-1109. (13) Sheikhzadeh, M.; Trifkovic M.; Rohani S. Fuzzy Logic and Rigid Control of a Seeded Semi-Batch, Anti Solvent, Isothermal Crystallizer. Chem. Eng. Sci. Accepted, 2007. (14) Mersmann, Ed.A. Crystallization Technology Handbook; Marcell Dekker Inc.: New York, 1995.

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ReceiVed for reView August 16, 2007 ReVised manuscript receiVed November 19, 2007 Accepted November 28, 2007 IE071125G