Use of Calorimetry Model and Batch Control Technique for

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Use of Calorimetry Model and Batch Control Technique for Scale-Up of Unseeded Batch Cooling Crystallization of Poly(hydroxybenzophenone) Huiyong Kim,† Yookyung Bang,† Kwang Soon Lee,*,† and Dae Ryook Yang‡ Department of Chemical and Biomolecular Engineering, Sogang UniVersity, 1 Shinsoodong, Mapogu, Seoul 121-742, Korea, and Department of Chemical and Biological Engineering, Korea UniVersity, 1 Anamdong 5Ga, Seongbukku, Seoul 136-713, Korea

A scale-up method to design and implement a cooling profile for unseeded batch cooling crystallization has been investigated. The reduced ratio between the heat-transfer area and the volume during scale-up to a large crystallizer causes an increase in the temperature distribution in the reactor, which can trigger unwanted nucleation around the cold spots of the vessel wall. To address this problem, a method is proposed to design a cooling profile for a scaled-up crystallizer that can replicate the seed-generation stage of a small crystallizer by effectively suppressing the unwanted nucleation. The use of a batch control technique as method of implementing the cooling profile in a large crystallizer is also proposed. Experiments were conducted with a 100 mL reactor and a 5 L scaled-up reactor for the unseeded crystallization of poly(hydroxybenzophenone) (PHBP) to verify the performance of the proposed method. 1. Introduction Batch cooling crystallization is a widely employed industrial process, especially in the fine-chemicals and pharmaceuticals industries, because of its ability to produce pure and large crystals of narrow size distribution.1-3 The operation of batch cooling crystallization normally starts with initially supplied seed crystals of the appropriate size and amount so that only the seed crystals grow while the undesired nucleation is suppressed. When external seeding is inappropriate or hard to apply, the internal seeding method can be used, in which nucleation is induced for seed generation by intruding the labile zone to a predesignated degree, as shown in Figure 1. The most important step in the design of both seeded and unseeded batch cooling crystallization is to determine an appropriate cooling profile. The cooling profile is concerned only with crystal growth in seeded crystallization, whereas it is related to seed generation as well as crystal growth in unseeded crystallization. In the case of seeded crystallization, industrial processes have traditionally employed natural cooling, linear cooling, or controlled cooling as the cooling strategy.1,2 Determination of a true optimal cooling profile that maximizes (or minimizes) a certain mathematical objective function has also been continuously studied,4-8 and determination of an optimum supersaturation profile and methods for implementing it online have also gained increasing attention as a result of recent advancements in in situ supersaturation and crystal size distribution measurement techniques.9,10 The approaches mentioned above are useful for the crystal growth stage where the suppression of ancillary nucleation that produces undesired fines is considered to be an indispensible element of the optimizing objective. The fines created during crystal growth can be handled by alternative methods at the expense of additional costs such as increasing the amount of seed crystals,11 providing an external fines dissolution loop12 or an internal fines dissolution scheme,13 applying intermittent heating periods for fines dissolution,14 and so forth. Conversely, seed generation in unseeded crystallization * To whom correspondence should be addressed. Tel.: +82-2-7058477. Fax: +82-2-3272-0319. E-mail: [email protected]. † Sogang University. ‡ Korea University.

relies on only a cooling strategy, and the cooling profile should be designed more elaborately. Mathematical models of nucleation are not yet reliable enough to be used to derive a practicable optimal cooling profile for seed generation, despite continuous developments over the past eight decades or more15 since the pioneering contribution by Volmer and Weber16 on nucleation kinetics, and the cooling profile is generally determined through trial-and-error experiments for this reason. The experiments are usually conducted in a laboratory-scale crystallizer first and then transferred to a larger crystallizer for scaleup studies, and problems inevitably emerge in relation to the enlarged temperature distribution within the solution. The first problem is the decrease in the jacket temperature to compensate for the reduced ratio between the heat-transfer area and the volume. As a consequence, the supersaturation degree around the inner wall is locally increased compared to that at other locations in the vessel, which encourages unintended nucleation and deteriorates the control of product quality. This is indeed a well-known phenomenon, and various methods have been proposed for the removal of the fines generated during crystal growth;11-14 however, such methods are not pertinent to the unintended nucleation occurring during seed generation. The problem with the increasing temperature gradient is hard to treat rigorously, and few studies on finding an optimum cooling profile under such a situation have been published. Therefore, the emphasis of this study is on the nucleation stage, and a simple but useful idea is proposed to alleviate the temperature gradient problem by modifying the cooling profile

Figure 1. Operation patterns for (a) seeded crystallization and (b) unseeded crystallization.

10.1021/ie801039f CCC: $40.75  2009 American Chemical Society Published on Web 06/16/2009

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Figure 2. SEM images of crystal surface from (a) seeded crystallization and (b) unseeded crystallization.

using a calorimetry model such that the nucleation in a small reactor can be maximally replicated in a large reactor. Another problem is the decreasing accuracy in temperature control in a large reactor compared to that in a small crystallizer because of the increased thermal mass and heat of crystallization. To enhance the performance of temperature control, we propose to apply an ILC (iterative learning control) technique17 that can achieve asymptotically perfect tracking under model uncertainty and batchwise repeating disturbances through batchwise integral action. With the evolution of control theory and measurement techniques, advanced control strategies for crystallization processes have become an important research topic. For example, sophisticated optimal criterion-based nonlinear control18-20 has been investigated mainly in academia because of the intricate nonlinear characteristics of crystallization processes. Because this research involves only temperature control, an ILC technique is considered more appropriate for our purposes, as it is simpler and its ultimate performance is not dependent on model accuracy unless divergent. Seed generation of internal seeding designed in the laboratory will not be appropriately replicated in a commercial reactor unless the above two problems are correctly addressed. The objective of the present research was therefore to investigate a procedure for designing and implementing a cooling profile for unseeded batch cooling crystallization in a large-scale reactor. Special emphasis was focused on the design of the cooling profile for seed generation on the basis of experiments in a small crystallizer and calorimetry analysis. Using a simple thermal balance model, a method for generating a cooling profile that can replicate the results obtained with a small crystallizer in a large crystallizer has been proposed. As a method for implementing the cooling profile, we also propose to apply the BMPC (batch model predictive control) technique,21 an ILC technique that can perform real-time control as well as batchwise feedback. The material studied in this work is poly(hydroxybenzophenone) (PHBP), which is dissolved in a water/acetone cosolvent and grows to brittle needle-shaped crystals under cooling. A cooling profile for unseeded crystallization was designed through a series of experiments in a 100 mL crystallizer with the aim of scaling it up for a 5 L system to verify the proposed methods. The designed cooling profile was then applied to the 5 L crystallizer under temperature control with BMPC. 2. Description of Crystallization Process 2.1. Material. PHBP is a chemical used in optics-related applications. It forms brittle and thin needle-shaped crystals in water/acetone cosolvent under cooling operation. Samples were

prepared by dissolving 0.814, 1.403, 4.180, 8.371, and 18.71 g of PHBP in 100 g of 20:80 wt % acetone/water cosolvent at 35.1, 42.6, 47.7, 50.4, and 55.0 °C, respectively. 2.2. Why Unseeded Crystallization? PHBP forms brittle and thin needle-shaped crystals in water/acetone cosolvent under cooling. It can be grown under both seeded and unseeded conditions; however, the preparation of uniformly sized seed particles from grown crystals is not easy because the particles are easily crushed into fines during size reduction, which makes seed classification within a specific size range very difficult. In addition, seeds obtained by sieving or crushing tend to have rough surfaces, and particles with rough surfaces are prone to grow to thin threads that are easily agglomerated or entangled, as shown in Figure 2a. Unseeded crystallization with internal seeding was attempted to avoid such difficulties and to yield promising results. Figure 2b shows a scanning electron microscopy (SEM) image of crystals grown under unseeded crystallization, which manifests the advantage of unseeded crystallization with internal seeding. The products from unseeded crystallization can greatly facilitate post-treatment processes such as filtering, washing, and drying. 2.3. Experimental Setup. The experimental setup is shown in Figure 3. Two glass crystallizers of different volumes were used in the experiments: one with a volume of 100 mL and the other with a volume of 5 L. Both crystallizers had jackets through which cooling water from a constant-temperature bath was circulated. Temperatures at the jacket inlet and outlet and inside the reactor were monitored and selectively controlled by a PC interfaced with a National Instruments data acquisition and control system. To manipulate the jacket temperature sensitively, an in-line electric heater was installed before the jacket inlet and was manipulated by a controller (TC2) that was cascaded to the master controller (TC1). The master controller was designed to be switched between proportional-integral-derivative (PID) and BMPC modes depending on the application. PID control was used exclusively for the 100 mL reactor, whereas BMPC was used for the 5 L reactor. The bath temperature was regulated at roughly 10 °C below the set point of the crystallizer temperature. All temperatures were measured with K-type thermocouples that were recalibrated to within (0.1 °C. 3. Methodologies 3.1. Cooling Profile Design. For seed generation with internal seeding, we considered inducing nucleation under a constant temperature. Once nucleation occurred, the particles were held at a constant temperature for a certain period for stabilization through Ostwald ripening1 and were then subjected

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Figure 3. Experimental setup for the 100 mL and 5 L crystallizers.

Figure 4. Cooling-holding-cooling profile for unseeded crystallization.

to subsequent cooling for growth. A base-case cooling profile is shown in Figure 4. The design of the cooling profile includes determination of the initial cooling rate, holding temperature, and holding period. The holding temperature determines the amount of seed crystal generation under the assumption that the supersaturation completely turns to seed crystals. In a small crystallizer, the temperature difference, ∆Trj, developed between the jacket and reactor is negligible. As the size increases, however, ∆Trj should be larger to compensate for the reduced ratio between the heat-transfer area and the volume lowering the wall temperature. A sharp change in the cooling profile can lead to an even larger value of ∆Trj because of the requirement of instant change in heat transfer from a large mass of solution, which can initiate nucleation unexpectedly starting from the cold spots of the vessel near the wall. One way to alleviate this problem is to smooth the cooling profile to prevent the jacket temperature from decreasing below a certain limit where the nucleation is significantly accelerated. To illustrate this idea, consider a large crystallizer that has a time constant of 20 min. Under simplified assumptions such as no heat of nucleation, no jacket temperature distribution, and no heat loss, the heat balance of the crystallizer can be succinctly written as Mcp

dT dT ) UA(Tj - T) f τ + T ) T j, dt dt

τ ) 20 min (1)

Two cases of modified cooling profile depending on the degree of smoothing are compared in Figure 5 in terms of the resulting jacket temperatures calculated using eq 1. In the case of light smoothing, Tj is decreased by about 3 °C below the holding temperature, whereas Tj falls slightly below the holding temperature in the case of heavy smoothing. In the latter case, virtually nowhere inside the reactor would be colder than the holding temperature during the entire period of seed generation, and the possibility of spurious nucleation on the wall would be significantly reduced. As a result, it could be expected that the seed generation obtained in a small crystallizer could be more closely replicated in the large crystallizer. On the basis of the above reasoning, a procedure for cooling profile design for seed generation can be proposed as follows: (1) Set up a calorimetry model, which can be as simple as that in eq 1, of the scaled-up crystallizer. (2) Determine an appropriate cooling rate, holding temperature, and holding period through experiments in a small crystallizer from the viewpoint of the amount, size, and surface state of the seed crystals. (3) Smooth the cooling profile so that the jacket temperature estimate using the calorimetry model does not fall below a prespecified limit, at which the inner wall temperature is equal to or slightly higher than the holding temperature. (4) Verify the performance of the new cooling profile in the small crystallizer and make modifications if necessary. Because the purpose of profile smoothing is to keep the inner wall temperature higher than the holding temperature during cooling, the calorimetry model in eq 1 is thought to still be useful even though it is the simplest one. If a more precise calorimetry model is available, a tighter cooling profile can be obtained. The smoothing of the initial cooling-holding profile in Figure 4 can be achieved in various ways. In this study, the cooling rate, z(t) ≈ dT(t)/dt, was filtered using the third-order filter zF(s) )1/(as + 1)3z(s), where z(s) represents the Laplace transform of z(t), after z(t) has been shifted up so that z(0) ) 0. The function z(t) can be recovered by the inversion process. The smoothing of the holding-cooling profile for crystal growth was conducted by applying the same third-order

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Figure 5. Tj profiles depending on the reference trajectory of T: (a) lightly smoothed and (b) heavily smoothed cases.

filter directly to the original profile and shifting the smoothed profile to the right so that the original holding period could be retained. 3.2. BMPC Technique21 Equation 1 shows that Tj is sensitively affected by T. When the reactor temperature is precisely tracked, Tj varies as predicted by the model; otherwise, Tj deviates from the predicted trajectory, reacting more sensitively to higher-frequency tracking errors. A batch control technique called BMPC21 was employed to meet this requirement. BMPC can, under some reasonable assumptions, achieve asymptotically perfect tracking under model uncertainty and batchwise repeating disturbances as the run number increases. This technique makes it possible not only for the designed cooling profile to be precisely pursued but also for the heat of nucleation, which occurs in a similar pattern in repeated runs and can be significant in a large crystallizer, to be effectively rejected. In addition, BMPC can handle operational constraints and, hence, can actively prevent Tj from decreasing below a certain predesignated limit.

BPMC was originally proposed as a multi-input/multi-output (MIMO) stochastic batch control technique. Because the experimental crystallizer can be represented by a single-input/ single-output (SISO) system that is not subject to any particular random disturbance, the original algorithm was tailored to a SISO deterministic version. BMPC for this study was based on a linear time-invariant batch process model described by y ) Gu + d

(2)

u ) [u(0) u(1) · · · u(N - 1) ]T y ) [y(1) y(2) · · · y(N) ]T d ) [d(1) d(2) · · · d(N) ]T

(3)

where

represent the input, output, and disturbance sequence vectors, respectively, of a SISO batch process over the batch horizon consisting of N sampling times.

G)

Figure 6. Cooling profiles in the 100 mL crystallizer under PID control with holding temperatures at 50.5, 51.0, and 51.5 °C.

[

g1 g2

0 g1

l l gN gN-1

··· 0 ··· 0 · ·· l g ··· 1

]

(4)

is the input-output map composed of the time-invariant impulse response coefficient, gi, that represents the output response at time i to a unit pulse input at time 0. For the crystallizer system in Figure 3, u(t) ) Tsp j (t) and y(t) ) T(t), where t represents the discrete-time index from this point onward. Let r denote the sequence of the reference trajectory, ek ) yk - rk, and the subscript k indicatee the kth batch run. The difference of eq 2 for the (k + 1)th and kth batches can be rearranged to yield a

Figure 7. Microscope images of final crystal with holding at (a) 50.5 and (b) 51.5 °C.

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and the same calculation is repeated at the next sampling time. Details of the BMPC algorithm can be found in Lee et al.21 4. Experiments

Figure 8. Designed cooling profile for the 5 L crystallizer.

recursive equation with respect to e, which is the tracking error transition equation over the batch index, as follows ek+1 ) ek + G∆uk+1

(5)

where ∆uk+1 ) uk+1 - uk. BMPC is constructed over eq 5 to calculate the control input at each sampling time t in the kth run by solving the following quadratic programming problem 1 min [ek(t + m|t)TQek(t + m|t) + ∆umk (t)TR∆umk (t)]

∆um k (t) 2

(6) subject to ek(t + m|t) ) ek(t|t) + G

(t)∆umk (t)

m

umin(t) e umk (t) e umax ∆umin e ∆umk (t) e ∆umax

(7) (8)

In the above expressions, ∆umk (t) )[∆uk(t) ∆uk(t + 1) · · · ∆uk(t + m - 1)]T represents a future input difference sequence, ek(t) represents ek when ∆uk(t) ) ∆uk(t + 1) ) · · · ) ∆uk(N - 1) ) 0, ek(t|t) represents an estimate of ek(t) on the basis of the measurements up to t in the kth batch, and ek(t + m|t) represents the prediction of ek when ∆umk (t) is applied to the process. Equation 7 is the output prediction equation, and eq 6 corresponds to the regular model predictive control (MPC) criterion except that the runwise input difference is penalized instead of the temporal input difference. Just as the temporal input difference penalty creates an integral action over time in the regular MPC technique, the runwise input difference penalty term provides a batchwise integral action in the BMPC technique that can remove tracking offset as the run number increases, despite the model uncertainty and run-invariant disturbance. Once ∆umk (t) has been computed, the first element ∆uk(t) is implemented to the process such that uk(t) ) uk-1(t) + ∆uk(t),

4.1. Solution Preparation. A saturated solution at 55 °C was prepared by dissolving due amount of 99.7 wt % PHBP in the solvent (acetone/water ) 20:80 by weight), filtering the solution using a 2 µm-pore filter, and stabilizing it at 65 °C for 1 h. 4.2. Measurement of Crystal Size Distribution (CSD). PHBP crystals grow rapidly along the longitudinal direction. They can grow to lengths of a few centimeters or more if there is no breakage. Because the crystal length is heavily affected by breakage under agitation, the crystal size was characterized by the thickness, which is considered to be a more appropriate representation of the crystal growth conditions. The thickness distribution of the product crystals was obtained from microscope images of around 400 randomly sampled particles with the aid of an image analysis program. 4.3. Base-Case Cooling Profile. Preliminary experiments in the 100 mL crystallizer showed that the following cooling strategy yields satisfactory results: (1) linear cooling from 65.0 to 51.5 °C at -15.0 °C/h, (2) holding at 51.5 °C for 1.8 h, (3) linear cooling at -15.0 °C/h to 40 °C. The cooling rates were determined by taking into account the ∆Trj estimate in the 5 L system. According to the procedure described in subsection 3.1, the cooling profile for the 5 L crystallizer was generated. 4.4. Model Identification and Controller Setup. The operation of the 100 mL crystallizer was carried out under PID control. PID parameters were initially determined using the internal model control (IMC) tuning rule and were adjusted during operation. The process model for IMC tuning was found from the process reaction curve.22 BMPC was employed in the operation of the 5 L crystallizer. After cooling profile smoothing, N was found to be 480 from 65 to 40 °C with a sampling time of 0.5 min. For BMPC design, it is necessary to identify G in eq 5. Because G represents the map between u(t) ) Tsp j (t) and y(t) ) T1(t) and model error within a certain bound does not hamper the ultimate tracking performance, G was identified with the crystallizer filled with the solvent not carrying out crystallization. An identification experiment was conducted at around 50 °C using a PRBS (pseudorandom binary sequence) excitation in Tsp j (t). The input-output data were fit to an ARX (autoregressive and exogenous input) model, and impulse response coefficients for G construction were derived. The time constant, τ, for eq 1 was estimated from the identified model and

Figure 9. BMPC temperature profiles for batch numbers (a) 1 and 3 and (b) 5 and 7.

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Figure 10. Microscope images of crystals grown in the 5 L crystallizer along the filtered cooling trajectory.

found to be 15 min. Also, the following constraints were imposed on the input variable -2 °C/min e u(t) - u(t - 1) e 3 °C/min holding temp - 0.5 °C e u(t) during seed generation

(9)

The tuning factors were chosen as m ) min[20,(N - t], Q ) I, and R(t) ) 0.1((N - t)/m)I with N ) 480. 5. Results and Discussion In Figure 6, the solution temperature is shown for three different holding temperatures of 50.5, 51.0, and 51.5 °C. The heat evolution during nucleation appears earlier and with a larger magnitude as the holding temperature is lowered, which is manifested by the increased temperature oscillation under PID control. Microscope images of the crystals grown from the seeds generated at 50.5 and 51.5 °C are shown in parts a and b, respectively, of Figure 7. The superior crystal quality shown in Figure 7b compared to that in Figure 7a in terms of size uniformity and surface cleanness can be attributed to the difference in the quality of seed crystals because they were both grown under the same cooling trajectory. Repeated experiments showed that the seed crystals generated at a smaller supersaturation degree grew to crystals of better quality in general. The holding temperature was chosen to be 51.5 °C based on these preliminary experiments, With the results obtained from the 100 mL crystallizer, a cooling profile for the 5 L crystallizer was generated by smoothing the base-case profile and estimating the jacket temperature using the calorimetry model in eq 1. The resulting

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cooling profile and the associated jacket temperature estimate are shown in Figure 8. In the smoothing, Tj was allowed to go down to 51.0 °C. Plots of the performance of BMPC in the 5 L crystallizer, initialized with PID control of jacket temperature in the first batch, are shown in Figure 9. As the batch number increased, a quite precise tracking could be achieved over the entire batch horizon except in a short period where the heat of nucleation was salient. The jacket temperature trajectories in Figures 8 and 9 closely resemble each other except for a slightly higher jacket temperature in Figure 9 because of the heat loss to the ambient and some undershoot at around 43 min caused by the heat of nucleation. A microscope image of PHBP crystals grown in the 5 L crystallizer under BMPC is shown in Figure 10. Also, the number densities of the thickness distributions of the product crystals from the 100 mL and 5 L crystallizers are compared in Figure 11. The crystal shape and surface state in Figure 10 are similar to those in Figure 7b, and the mean thicknesses from the two reactors are quite close. Although the crystal thickness is more broadly spread in the 5 L product, the results in Figures 10 and 11 are considered to verify the appropriateness of the proposed method. Comparison of Figure 6 with Figure 9 shows that the time when the heat of nucleation becomes prominent is significantly earlier in the 5 L reactor than in the 100 mL reactor. Many factors could contribute to these phenomena; however, the particles precipitated at the gas-liquid interface, which was more active in the 5 L vessel because of its larger mouth, are thought to incite and expedite the burst of nucleation. In addition, the nucleation time was observed to move back and forth by up to 10 min, depending on the batch run. Although quite precise tracking could be attained in the 5 L reactor, BMPC has limitations in handling such a variation because, in the present algorithm, the input action for the next run is calculated by assuming that the disturbance in the present run will be repeated in the next run. The control aspect would be further improved if an adaptation scheme that detected the moment of nucleation heat evolution and exerted a precomputed compensating input signal were developed and incorporated into the BMPC algorithm. 6. Conclusions A scale-up procedure to design and implement a cooling profile for unseeded batch cooling crystallization was proposed. The base-case cooling profile was assumed to consist of three parts: initial cooling, holding for the induction of nucleation, and subsequent cooling for crystal growth. The profile was

Figure 11. Crystal thickness distributions of the products from (a) 100 mL and (b) 5 L crystallizers.

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modified for a scaled-up reactor on the basis of the experimental results in a small test reactor and a calorimetry model of the large reactor. Special focus was placed on the nucleation stage so that the seed generation in the test reactor could be replicated in the scaled-up reactor. Extensive experiments were conducted in 100 mL and 5 L reactors for the crystallization of PHBP, considering them as a test reactor and a scaled-up reactor, respectively. The proposed procedure and control technique were shown to work satisfactorily, yielding similar qualities of grown crystals in the two reactors. The ideas behind the present research are believed to be useful in handling the problems related to uneven temperature distributions in the generation of seed crystals in unseeded batch cooling crystallization in a large reactor. One of the future improvements is to equip the BMPC algorithm with an adaptation scheme to detect the onset time of nucleation, which drifts depending on the run, and exert a feed-forward control action learned from the previous operation at the correct moment. Acknowledgment The authors acknowledge financial support from KOSEF (R01-2006-000-11377-0) and also from KETEP (20065045). Supporting Information Available: BMPC algorithm and MATLAB program code embedded in the Labview vi file used in the experiments. This material is available free of charge via the Internet at http://pubs.acs.org. Literature Cited (1) Myerson, A. S., Ed. Handbook of Industrial Crystallization; Butterworth-Heinemann: London, 1993. (2) Mullin, J. W. Crystallization, 4th ed.; Butterworth-Heinemann: Oxford, U.K., 2001. (3) Jones, A. G. Crystallization Process Systems; Butterworth-Heinemann: Oxford, U.K., 2002. (4) Mullin, J. W.; Nyvlt, J. Programmed cooling of batch crystallizers. Chem. Eng. Sci. 1971, 26, 369–377. (5) Jones, A. G.; Mullin, J. W. Programmed cooling crystallization of potassium sulphate solutions. Chem. Eng. Sci. 1974, 29, 105–118. (6) Lang, Y.-D.; Cervantes, A. M.; Biegler, L. T. Dynamic optimization of a batch cooling crystallization process. Ind. Eng. Chem. Res. 1999, 38, 1469–1477.

(7) Hu, Q.; Rohani, S.; Jutan, A. Modelling and optimization of seeded batch crystallizers. Comput. Chem. Eng. 2005, 29, 911–918. (8) Sarkar, D.; Rohani, S.; Jutan, A. Multi-objective optimization of seeded batch crystallization processes. Chem. Eng. Sci. 2006, 61, 5282– 5295. (9) Miller, S. M.; Rawlings, J. B. Model identification and control strategies for batch cooling crystallizers. AIChE J. 1994, 40, 1312–1327. (10) Nagy, Z. K.; Chew, J. W.; Fujiwara, M.; Braatz, R. D. Comparative performance of concentration and temperature controlled batch crystallizations. J. Process Control 2008, 18, 399–407. (11) Kubota, N. Seeding policy in batch cooling crystallization. Powder Technol. 2001, 121, 31–38. (12) Jones, A. G.; Chianese, A. Fines destruction during batch crystallization. Chem. Eng. Commun. 1987, 62, 5. (13) Ooshima, H.; Shan, G.; Igarashi, K. Production of large crystals with narrow crystal size distribution by a novel WWDJ batch crystallizer. Chem. Eng. J. 2002, 85, 2–3. (14) Moscosa-Santillan, M.; Bals, O.; Fauduet, H.; Porte, C.; Delacroix, A. Study of batch crystallization and determination of an alternative temperature-time profile by on-line turbidity analysis: Application to glycine crystallization. Chem. Eng. Sci. 2000, 55, 3759–3770. (15) Kashchiev, D. Nucleation: Basic Theory with Application; Butterworth-Heinemann: Oxford, U.K., 2002. (16) Volmer, M.; Weber, A. Keimbildung inubersattigten gebilden. Z. Phys. Chem. 1926, 119, 277–301. (17) Xu, J.-X.; Panda, S. K.; Lee, T. H. Real-time IteratiVe Learning Control: Design and Applications; Advances in Industrial Control Series; Springer: New York, 2009. (18) Eaton, J. W.; Rawlings, J. B. Feedback control of chemical processes using on-line optimization techniques. Comput. Chem. Eng. 1990, 14, 469–479. (19) Tadayyon, A.; Rohani, S. Extended Kalman Filter-based Nonlinear Model Predictive Control of a Continuous KCl-NaCl Crystallizer. Can. J. Chem. Eng. 2001, 79, 255–262. (20) Nagy, Z. K.; Braatz, R. D. Robust Nonlinear Model Predictive Control of Batch Processes. AIChE J. 2003, 49, 1776–1786. (21) Lee, K. S.; Chin, I.-S.; Lee, H. J.; Lee, J. H. Model predictive control technique combined with iterative learning for batch processes. AIChE J. 1999, 45, 2175–2187. (22) Seborg, D. E.; Edgar, T. F.; Mellichamp, D. A. Process Dynamics and Control, 2nd ed.; John Wiley and Sons: New York, 2003.

ReceiVed for reView July 4, 2008 ReVised manuscript receiVed May 21, 2009 Accepted May 22, 2009 IE801039F