Model-Based Optimal Strategies for Controlling Particle Size in

Model-Based Optimal Strategies for Controlling Particle Size in Antisolvent Crystallization Operations S. Mostafa Nowee,† Ali Abbas,† and Jose A. Romagnoli*,‡ School of Chemical and Biomolecular Engineering, UniVersity of Sydney, Australia, and Department of Chemical Engineering, Louisiana State UniVersity, Baton Rouge, Louisiana

CRYSTAL GROWTH & DESIGN 2008 VOL. 8, NO. 8 2698–2706

ReceiVed July 31, 2007; ReVised Manuscript ReceiVed January 11, 2008

ABSTRACT: In this paper, a model-based optimal strategy is presented for the control of particle size in antisolvent crystallization. Size is controlled on demand by dynamic optimization using a population balance based model. Knowledge of the ternary solute–solvent-antisolvent equilibrium and crystallization kinetics is crucial in this strategy, and these are thus incorporated in the model. The optimization is capable of determining the optimal antisolvent feed profile that achieves a desired particle size. Experimental validation of the strategy is carried out and presented herein. Such a strategy stands as a key solution to antisolvent operations ubiquitous in the pharmaceutical and fine chemicals industries.

1. Introduction Crystallization is a widely used technique in solid–liquid separation processes and has been referred to as an art rather than a science. The main objective of crystallization systems engineering is to develop advanced understanding of crystallization phenomena enabling the efficient bulk processing of high quality crystalline products. Crystallization, in many industries, is the most common way of production of high value chemicals with high purity and desired particle size and shape. Crude fixed-recipe methods are still being used in industrial operations to operate/control their crystallization processes forcing nonoptimal operations. This is more so in the pharmaceutical industry because of regulatory processing and product quality requirements. The American Food and Drug Administration (FDA) has recently come to pay attention to these pharmaceutical manufacturing issues, with a release of a regulatory framework guidance document for “innovative pharmaceutical development, manufacturing, and quality assurance”.1 This is a clear change toward more flexible operations, ones that favor the introduction of innovative process control methods. The driving force in crystal formation is supersaturation. The trend of supersaturation generation during the process has a direct and significant role on crystal characteristics such as size, morphology and purity. Crystal size is typically the most important factor to be controlled because of its close relation to the characteristics of the crystalline product, such as flowability and bulk density, while regulations and different applications may require particles of specific size or size range. Cooling and solvent evaporation are two ways of achieving supersaturation control. In the past decade, salting-out as a means to induce supersaturation has been drawing more attention. In this method a secondary solvent known as antisolvent or precipitant is added to the solution resulting in the reduction of the solubility of the solute in the original solvent and consequently supersaturation is generated. Antisolvent crystallization is an advantageous method where the substance to be crystallized (solute) is highly soluble, has solubility that is a weak function of temperature, is heat sensitive, or is unstable at high temperatures. This technique is an energy-saving alternative to evaporative crystallization, if the antisolvent can be separated at low (energy) costs. There have been a number of successful research investigations as evidenced in the literature on the control of the * Corresponding author. Tel: (225)-578-1377. Fax: (225)-578-1476. E-mail: [email protected]. † University of Sydney. ‡ Louisiana State University.

crystallization processes with most of these studies aiming to control the product crystal size or crystal size distribution (CSD). Researchers moving away from conventional operations have recently recognized the significant role a crystallization model can play in optimizing the manufacturing. Manipulated variables that would be candidates for optimization control variables are traditionally the cooling rate and seeding. Considerable effort has been made on particle size control on the batch cooling crystallization front.2 Most of these works consider supersaturation as the key variable to be optimized. This strategy which is based on the determination of the optimal temperature profile was first discussed by Mullin and Nyvlt (1971).3 It was though based on a key understanding, namely, that constant nucleation rate is required in order to keep the supersaturation low and within the metastable zone. Other works dealing with optimal cooling profile generation intended to keep supersaturation low in the beginning stages of the process inhibiting nucleation but promoting crystal growth. Other workers analyze, via simulations, the effect of seed load on product size characteristics.4 Their seeding loads are chosen arbitrarily providing empirical trends about the effect of seeding. They show that the product weight mean size and coefficient of variation may increase or decrease at increasing seed amounts attributing this variation to variations in kinetics. Only in recent years some experimental research appeared considering seeding as a factor to be optimized.5–10 In these investigations the seed chart was used to determine optimal seeding policies. These works demonstrated the impact seeding has on the crystallization showing that, even under uncontrolled profiles of cooling (e.g., natural cooling), the rate of nucleation could be kept low. Most of the research cited thus far is however overly experimental. A model-based dynamic optimization of seeded cooling crystallization was previously discussed.11 Seeding parameters in addition to supersaturation were optimized, and the objective function used was maximization of final mean size with minimum coefficient of variance. Three studies incorporated factorial experimental design using a surface response model in determining initial seed concentration and seed introduction time as well as the temperature profile.12–14 Their target was a specific crystal yield, size, shape and purity. Another investigation that dealt with model-based optimization of seeded cooling crystallizations determined the optimal seeding characteristics, mean size and seed mass, by maximization of product mean size and minimization of CSD coefficient of variance as objective functions, respectively.15 The simultaneous use of the two variables, temperature and seeding, is evident in recent work.11,16 More comprehensive studies solve the population

10.1021/cg700720t CCC: $40.75  2008 American Chemical Society Published on Web 06/28/2008

Particle Size Control in Antisolvent Crystallization

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balance equation (PBE) for batch cooling crystallization systems and use temperature profile as well as seeding parameters as control variables.17–19 They also validate their optimization results experimentally. On the antisolvent crystallization front, there are numerous experimental works that look at the effect of various operating conditions on the process. The effect of different concentrations of aqueous and antisolvent solutions on crystal shape and distribution was previously investigated.20 The main variable studied was supersaturation. The researchers20 proposed a mechanistic formulation in finding the relation between nucleation and supersaturation. Other works considered antisolvent concentration and feedrate effects on final crystal habit.7,21–26 Size and morphology of benzoic acid crystals from ethanol– water system were investigated by varying the feed and bulk solution composition and feedrate.20 This study observed that supersaturation mainly governs the mean size while solvent composition has a significant effect on the crystal shape. Agglomeration and habit of paracetamol crystals were the focus of another study that varied the agitation speed and feedrate, which concluded that low agitation speed and high feedrate will result in excessive nucleation due to high local supersaturation leading to highly agglomerated product with lower mean size.26 A simulation study on seeded antisolvent crystallization of an active pharmaceutical ingredient (API) by water addition to original solution was previously presented.27 The model they incorporate neglects nucleation, breakage and agglomeration, and only considers a size independent growth kinetic derived from experimental studies. Their investigation avoids nucleation of undesired polymorph by keeping supersaturation at a certain level. Concentration controlled seeded antisolvent crystallization of a pharmaceutical compound was carried out which used an algebraic equation for the solubility as a function of % solvent.28 The main objective of that feedback concentration control system was to keep the supersaturation low and constant. For this, different constant supersaturation values were investigated and their influence over nucleation was discussed. In addition, that study28 presented simulation results investigating the antisolvent crystallization of paracetamol. Recent research trends in cooling crystallization, as evident in the literature, show an increase in the use of the dynamic crystallization model for achieving various optimal objectives. In spite of all the previous reports, very few other works implemented model-based optimization of antisolvent crystallization.29,30 This thus represents a big shortfall on one hand and opportunity on the other in the control of the product crystal properties to desired values. The work presented in this paper determines, via dynamic optimization, the antisolvent feed profile and subsequently validates this against experimental data. In this work, a population balance model for antisolvent mediated crystallization presented in a companion paper31 is implemented for these model-based dynamic optimization studies. Calculation of the optimal antisolvent feed trajectories for a range of objective functions was performed. The optimization studies were validated experimentally by implementation within an industrial distributed control system (DCS) environment.

2. Model Development The population balance formulation for particulate processes32 is used accounting for the evolution of crystal particles across temporal and size domains. For a crystallization system with crystal growth assumed to be nondispersed and independent of crystal size and where agglomeration and attrition are considered negligible, the PBE simplifies to ∂n(L, t) ∂n(L, t) n(L, t) dV ) -G +B∂t ∂L V dt

(1)

where n(L,t) is the number density of crystals, t is time, L is the characteristic crystal size and V is the suspension volume.

The nucleation (B) and growth rates (G) are described by the following constitutive relations:

B ) kb∆CbMT

( )

G ) kg

∆C C/

g

(2) (3)

where ∆C is the absolute supersaturation, C* is the saturated concentration, MT is the magma density, and kb and b are the nucleation parameters. The nucleation rate is equal to zero for all sizes but the first. In this growth rate power law model structure, the parameters kg and g are defined as functions of antisolvent mass fraction in solute free mixture (z3):

kg ) k0 + k1z3 + k2z32

(4)

g ) g0 + g1z3

(5)

The details of the model identification (kinetic sub models, eqs 2–5) which is used in this study are reported elsewhere.31 A set of operating equations for the ternary solvent–soluteantisolvent crystallization is needed. We first define the mass fraction of each component as

xi )

mi

∑ mi

(6)

where xi and mi are the mass fraction and mass of the ith component respectively. For our case, the number of components, i, is 3, and so x1, x2 and x3 are the primary solvent, solute and antisolvent mass fractions respectively and similarly m1, m2 and m3 are the primary solvent, solute and antisolvent masses respectively. Changes in suspension volume are evaluated by

dV dVL dVS ) + dt dt dt

(7)

In eq 7, the liquid phase volume, VL, is given by

VL )

∑ mi F

(8)

where F is the solution density, while the solute phase volume is given by

VS ) VRv

∫0∞ nL3 dL

(9)

where Rv is the crystal volume shape factor. Accordingly the numerator of eq 7 bears the antisolvent mass addition term as well as total solute mass in liquid phase. The mass balance, for the solute mass accumulation, is incorporated as33

dm2 ) -3VFcRvG dt

∫0∞ nL2 dL

(10)

where Fc is the crystal density. The concentration of the solute, C, is then defined as

C)

m2 VL

(11)

j , both Solution density, F, and volume mean size of particles, L being measured experimental variables are introduced to the model as

2700 Crystal Growth & Design, Vol. 8, No. 8, 2008

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Figure 1. Flow of calculations of the ternary antisolvent crystallization model. Bold box indicates the output from which the sizes of the particle population is derived.

L)

∫ n(L, t)L4 dL ∫ n(L, t)L3 dL

(12)

optimization. The mathematical description of the dynamic optimization problem (DOP) is, in the general form, given as

F ) (A0 + A1x2 + A2x3 + A3x2x3 + A4x2x23) × e(A5(x3 ⁄ x1)+(x3 ⁄ x1)2) (13) The density-concentration correlation (eq 13) is adopted from Galleguillos et al. (2003)34 who experimentally determined the set of constant coefficients (Ai) for the NaCl-water-ethanol system which is used in this study as the model antisolvent crystallization system. Knowledge of the equilibrium condition (solubility) of the crystallization system is crucial to the control of the particle size. We develop a model describing the solubility of the NaCl in the solvent-antisolvent mixture based on experimental solubility data.35 This data corroborates with other data presented elsewhere for the same ternary system.36 This model describing the saturated concentration dependence on solvent concentration in solute free mixture is shown in eq 14,

C/ )

(1 - z3) / C (K - z3) water

(14)

/ where C* is the molality of solute in the mixture and Cwater is molality of solute in pure water and has a value of 33.80 g/100 / g of water at constant temperature 25 °C. K, like Cwater , is constant at constant temperature and is equal to 0.83094. z3 is the mass fraction of antisolvent in solute-free mixture. The set of equations above representing the antisolvent crystallization model was solved in gPROMS environment (Process Systems Enterprise Ltd., U.K.) using backward finite discretization across the size domain, transforming the PBE into a set of ordinary differential equations. The calculation flow is summarized in the schematic of Figure 1.

3. Model-Based Optimization 3.1. Formulation. Once the optimal kinetic model is identified, one can proceed to the subsequent step of model-based dynamic

where Φ is a scalar objective function at the end of the optimization horizon, i.e. at t ) tf. F is the DAE system representing the process model differential (X), algebraic (Y) and input control (U) variables, with p being a time-independent parameter vector. This optimization formulation can be stated as finding the time-dependent control vector (U), the time-independent parameter vector (p) and the final time (tf) to minimize or maximize a performance measure or objective function (Φ). The control variables within vector U are allowed to vary according to a piecewise or polynomial functionality over the span of the time horizon t ∈ [0,tf]. The objective function of the DOP is subject to a set of constraints. Limiting the control and state variables are lower and upper bounds denoted by the superscripts min and max respectively. The constraints on the control variables are stated explicitly since modern optimization algorithms can handle them very efficiently. This is not generally the case of other types of constraints. Here the batch time (tf) is



bounded since it may be necessary to have the batch operating within a certain time frame. In effect, the DOP would minimize the batch time. Control variables U(t) may have lower and upper bounds imposed by the physical limitations of the manipulated control action such as maximum flow rate of a pump or max cooling rate available. Time-independent parameters p represent those variables that are fixed throughout the time horizon [0,tf]. Other constraints (w) are defined and can be of several types including the end-point type, which usually represents certain conditions that the process must satisfy at the end of the optimization horizon. For convenience, end-point constraints are divided into inequality (wei) and equality (wee) constraints. Although the latter are a special case of the former, differentiating them simplifies the definition of some optimization problems. The interior-point type (wI) are used to enforce process variables to lie within the defined upper and lower bounds at some distinct time or times during the operation, and finally we have the inequality path constraints (wp) which must be satisfied at all times during the operation. In this particular antisolvent crystallization application, ten piecewise-constant control intervals were implemented and the time horizon and system control variable are subject to the following bounds:

time horizon: ethanol feedrate:

1800 e tf e 16000 s

(15)

0.375 e Q(t) e 125 mL/min (16)

The following end-point constraints are imposed:

final total volume: final yield:

100 e V e 500 mL

(17)

20 e yield%

(18)

Depending on the control target, different size control related optimization objectives may be formulated as (1) keeping the supersaturation constant, (2) minimizing nucleation, (3) maximizing final mean size and (4) minimizing the span of the size distribution. The results reported here correspond to maximization of final mean size for two cases, each with an additional end-point constraint specification in the final size range required, and a third case targeted at maximizing particle size via a minimization of nucleation objective. The results reported here thus correspond to the three specific objective functions: Max j Case 1: Maximization of final mean size u(t),t∈[0,tf] L(tf). Case 2: Achievement of desired final mean size of 100 µ Max j (tf). u(t),t∈[0,tf] L Case 3: Minimization of nucleation rate throughout the Min process u(t),t∈[0,tf] B(t). Accompanying the first two of the above three objectives are additional end-point constraint specifications in the final size range required, respectively written as

150 e L(tf) e 500 µm

(19)

90 e L(tf) e 100 µm

(20)

j (as defined in eq 12), was The volume weighted mean size, L used in the objective functions of cases 1 and 2, since this parameter is what is measured by the available particle size analyzer. This thus serves well our goal of experimental validation. One should note that the computed optimal profiles would be expected to change significantly if the number mean size was used instead. This is because the volume mean size is highly insensitive to fine density at the end of the batch as pointed out by previous studies.37,38

Figure 2. Experimental Setup. (A) Crystallization vessel. (B) Temperature control system. (C) Pt100 thermocouple. (D) Ethanol addition line. (E) Pump. (F) Ethanol reservoir. (G) FBRM probe. (H) Analog output card. (I) FBRM control computer. (J) DCS station. (K) Controller I/O terminals. (L) DCS server. (M) Feed profile to pump.

Figure 3. Optimal feedrate and experimental validation of mean size and concentration results from optimal trajectory case 1.

3.2. Experimental Setup. Figure 2 shows the schematic of the experimental apparatus, instrumentation and control system used. In all experiments, purified water by a Milli-Q system was used. The purities of sodium chloride (NaCl) salt (Merck) and ethanol (Merck) used in the experiments were 99.5% and 99.9% respectively. Ethanol was added to the aqueous NaCl solution using a calibrated digital dosing pump (Grundfos, Denmark). Temperature was controlled, at 25 °C for all experiments, using a Pt100 thermocouple connected to heating/ cooling circulator (Lauda, Germany). Particle chord length was measured online every 2 s using focused beam reflectance measurement (FBRM) probe (Mettler-Toledo Lasentec Products, USA). Infrequent samples were removed isokinetically from the crystallizer for particle size and density measurements. Particle size was measured using Mastersizer 2000 particle size analyzer


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Figure 4. Optimal feedrate and experimental validation of mean size and concentration results from optimal trajectory of case 2.

Figure 7. Comparison of optimal feeding policies under different initial concentrations of solute.

Figure 5. Optimal feedrate and experimental validation of mean size and concentration results from optimal trajectory of case 3.

Figure 8. Mean size growth trends from optimal feeding policies under different initial concentrations of solute.

Figure 6. FBRM counts in the 1–5 µm range over time for the three antisolvent crystallization optimization cases.

(Malvern Instruments, U.K.) while density was measure using a density meter (Anton Paar DMA 5000, Austria). Samples were filtered using a syringe microfilter prior to density measurements. The concentration of the solution was inferred from the density reading using an empirical correlation.34 The FBRM signal was converted to 4–20 mA signals using an 8-channel analog output PCI card inserted in the FBRM computer. This was then connected to the DCS for data monitoring and archiving.

Figure 9. Relative supersaturation trends from optimal feeding policies under different initial concentrations of solute.

The ethanol addition profile was implemented and controlled from within a distributed control system (DCS) environment (Honeywell, USA). This work is geared toward the closed-loop



Figure 10. Growth and nucleation rates from optimal feeding policies under different initial concentrations of solute.

Figure 13. Relative supersaturation trends from optimal feeding policies under different antisolvent feed concentrations. Figure 11. Comparison of optimal feeding policies under different antisolvent feed concentration.

Figure 12. Mean size growth trends from optimal feeding policies under different antisolvent feed concentrations.

optimal control of crystal size where we, after determining the optimal policies (antisolvent flowrate), can use these profiles as real-time moving set-points that direct the regulatory control. This involves the definition and implementation of multilayer

control architecture within the DCS environment.39 This advanced strategy is discussed briefly here to show the reader how the optimization work presented in this paper can be, in general, exploited in full real-time optimizing crystallization control. In a higher order layer, off-line optimization uses the crystallizer model and process optimization to calculate optimal CSD properties which become the set-point for an intermediate layer. Model predictive control, in the intermediate layer, determines the real-time optimal antisolvent flowrate set-point trajectory of the plant given the production requirements (CSD) and operational constraints, and keeps the process operating near optimum efficiency by constantly adjusting the set-points and responding to plant disturbances while keeping constraints within given bounds. This layer utilizes the process model to guide its actions which consequently become the set-points to the lower layer. In the lower layer, regulatory control takes action, where the process variables, namely the crystallizer antisolvent feedrate are kept at their set-points. In essence the role of the regulatory layer is to follow the directives (set-points) of the upper control layers. The flexibility in the design of this control architecture allows the lower level regulatory control set-point to be input from either one of the following three sources: (1) Manual set-point: This is used when the antisolvent feedrate is required to be set at a constant value. This is pure regulatory control.


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Figure 14. Growth and nucleation rates from optimal feeding policies under different antisolvent feed concentrations. Table 1. Comparison of Yields and Total Added Antisolvent Amounts for All Optimal Cases initial solute concn (g/100 g of water)

feed antisolvent concn (wt %)

yield (%)

total ethanol used (mL)

33.980 32.281 30.582 33.980 33.980

100 100 100 90 75

97.9 99.9 99.9 80.5 33.0

296.5 313.2 313.2 288.5 248.5

(2) Ramp-soak set-point: This is a set-point varying with time, and it changes according to an operator built-in and predefined antisolvent feedrate profile (this study). This constitutes openloop control. (3) External set-point: This is used when an external program is used to provide the set-point. A model predictive controller, for example, could be used as the external set-point source and thus provides closed loop control. 3.3. Optimal Strategies/Recipes and Experimental Validation. The idea that any final mean size could be obtained using the framework presented here was tested in the first two cases. In the first case (case 1), the objective function was to maximize the final mean size in an open range (eq 19). The dynamic optimization output is displayed in Figure 3. The antisolvent feeding profile is shown to be slow initially with gradual increase toward the middle of the batch. This corroborates with other results40 and is explained as follows. A slow addition rate leads to lower nucleation levels initially, until the accumulation of sufficient surface area for crystal growth to become dominant. This also underlines the outcome of our previous simulation study31 which indicates that low antisolvent feeding policy at early stages of the process followed by increment in the addition rate will result in smaller number of particle formation and consequently larger mean size of final product. To demonstrate the capability of this model-based optimal approach, we tested it using a different objective (case 2), that of achieving a final mean particle size inside a specific and narrower range (eq 20). The optimal feed profile, unlike that in case 1, resulted in a higher initial rate of antisolvent addition followed by a similar trend of addition observed for case 1 (Figure 4). This could be explained by the optimization moving to initiate a higher rate of nucleation at the beginning of the process than the previous case. This, for the subsequent process time, proceeds with a mean size maximizing policy which resembles the trajectory achieved from case 1.

Furthermore, we proceed with another optimization study targeting a different type of objective function being the minimization of nucleation rate throughout the process (case 3) which is another way of expressing the maximum mean size objective. The optimal addition feeding policy of antisolvent (Figure 5) like in case 1 begins with a slow feedrate which obeys the idea of low nucleation rate for the primary stages of the process. Toward the middle of the process, where sufficient active surface area of particles is attained, there is a high feeding rate to lead toward the end-product target of maximum mean size. Toward the end of the process time the optimizer continues with a low feedrate in order to derive the most possible precipitation from the solution. It is noted that the final mean size achieved in this case (∼150 µm) is very close to the one achieved in case 1. The strategies obtained from optimization were experimentally validated. During the experiments the optimal feeding profiles calculated for the above three optimization cases and shown in Figures 3–5 were actually implemented and the crystal mean size and the solution density were the monitored states. For these feeding profiles, the experimental results are plotted against the model predictions of mean size and concentration in Figures 3–5. It could be observed that there is a satisfactory agreement between the model predictions of the system behavior with the real experimental measurements. The FBRM chord-length counts for the size range 1–5 µm for the three optimization cases are shown in Figure 6. There is a clear trend in the data which is consistent with previous knowledge. The optimization of case 2 exhibits a count profile that deviates from the other two optimization cases especially in the initial stage of the batch. Cases 2 and 3 almost overlap in the first 30 min of the batch but diverge later. These count profiles can be explained in correlation with the antisolvent feedrate profiles. We observe a steep rise in number counts in the first few minutes for case 2. This represents a nucleation burst associated with the higher antisolvent feedrate during that early period of the batch (compare feedrate profile of Figure 4 to those of Figures 3 and 5). A higher feedrate as calculated by the optimization is expected when the objective is to achieve a lower size of particles i.e. ∼100 µm in this case. This higher feedrate translates into higher supersaturation creation inducing higher nucleation rates early on. Thus there would be more supersaturation diverted toward creation of more particles of smaller sizes.


We observe that the counts measured in cases 1 and 3 have delayed nucleation evidenced by the first steep rise in numbers being observed about 30 min into the batch. This suggests that the optimizations targeted at maximizing size (cases 1 and 3) proceed into the metastable region very gradually, so as to reduce any nucleation bursts. Hence a smaller number of nuclei would be present and supersaturation would then be consumed by the growth on the surfaces of this smaller population of particles. These trends underline the significant role nucleation and its onset play in the control of particle size in antisolvent crystallization. Understanding this process better will surely lead to improved control of particle size. Toward the later stages of the batches, the count trends seem to stabilize explained by the depletion of the supersaturation driving force. The key message here is that the model-based optimization is in essence calculating the temporal distribution of the supersaturation driving force between nucleation and growth. This competition for superstaturation between nucleation and growth is an essential element in this size control problem, and its coordination is nontrivial in the absence of this model-based strategy. One consistent trend across the three cases is the decline of the FBRM chord length count over time. The supersaturation profile generated under the different feeding rates has a big role to play here. Supersaturation was not directly measured here, but its trend is expected to closely match the trends observed by the FBRM counts. More nuclei form in the beginning due to primary nucleation. As supersaturation is depleted steadily, primary nucleation rate reduces and the nuclei initially formed grow out of the size range 1–5 µm. This and the possibilities of agglomeration becoming more probable as the mass of crystals increases in the process are responsible for the decline in the nuclei counts.

4. Optimization Sensitivity to Operating Conditions In addition to the optimization cases studied in the previous section, the effects of two different operating variables, initial concentration of solute and antisolvent concentration in the feed, were studied. This is to determine the effect these variables have on the optimization calculation of the antisolvent feed profile. Here, the maximum mean size objective was used. 4.1. Effect of Initial Solute Concentration on the Optimization Calculation. Three different initial concentrations were investigated, all being close to saturated brine at 25 °C (30.582, 32.281 and 33.980 g/100 g of water). In all cases maximization of mean size was the objective and constraint 1 (eq 19) was chosen as the additional end-point constraint for mean size. The calculated optimal feedrate profiles are plotted in Figure 7. The results show the vast impact the initial conditions have on the outcome of the optimization. The changes are reflected in both the magnitude of feedrates and the process time. Consequently different mean size temporal trends are achieved (Figure 8), demonstrating that with lower initial concentrations higher mean sizes are attainable. Additionally the relative supersaturation levels are different (Figure 9). Since the feedrates are higher at early stages of the process in cases of low initial concentrations, high rates of supersaturation are expected which result in high nucleation and growth rates (Figure 10); thus a burst of nucleation followed by rapid growth occurs. It is interestingly noted here that all these resulting optimal supersaturation profiles are not constant throughout the process time, contrary to the traditional strategies aiming to keep supersaturation at a constant value for maximizing particle size. On the other hand in Figure 10, the solid line which represents the trend


of nucleation and growth rates for the high initial concentration calculation shows that a phase of secondary nucleation happens, so with the presence of these fine particles the final mean size in this case is lower compared to the other two optimizations. The results from this optimization study highlights the importance of the initial conditions of the batch, in this scenario that solute initial concentration can play a significant role with respect to maximization of mean size, thus it is worth considering this parameter as a time-invariant control variable which basically can be formulated as an initial condition optimization along with the determination of optimal feedrate control variable. 4.2. Effect of Feed Antisolvent Concentration on the Optimization Calculation. In another series of optimization sensitivity studies, the antisolvent concentration in the feed was varied and its effect on the calculated optimal feeding trajectory was investigated. Different ethanol concentrations of 75, 90 and 100 wt % in the feed were implemented. In all of these cases the initial solute concentration was kept constant at 33.980 g/100 g of water. The calculated antisolvent feedrate profiles are displayed in Figure 11. As in the initial solute concentration study presented in the previous section, we again observe a significant effect imparted by this new parameter on the optimization results. The resulting trends indicate that increasing the dilution of the antisolvent feeds result in higher feedrates at the beginning of the process. This trend and the shorter calculated process durations are similar to the ones observed when the initial solute concentration was diluted and resulted in higher initial feedrates and shorter process times (Figure 7). Larger final mean sizes were achieved more rapidly under more diluted feeds as shown in Figure 12. From the three feeds studied it is suggested that the feed containing 90 wt % of antisolvent is the better dilution if higher final mean size is desired. A lower concentration of ethanol (50 wt %) was also investigated. However due to low antisolvent and high solvent concentrations in the feed no supersaturation was achieved, consequently leaving the solution particle free. For maximization of final mean size, low initial solute concentrations and dilute feeds result in higher calculated feedrates which in turn lead to initial conditions of higher supersaturation (Figures 9 and 13). This has previously been explained as being an internal seeding strategy where the system is forced to nucleate by the initial antisolvent rush, and growth subsequently occurs on the surfaces of these first nuclei.7 However, simulation results under these feeding recipes also indicate that high initial growth rates exist (Figures 10 and 14), driven by the high supersaturation present. When bringing the final crystal yield and total amount of antisolvent added into consideration, a clearer picture is drawn especially if the economic objective of the manufacturing is a strong function of yield and antisolvent usage rate. When comparing these optimal recipes on this basis, it is shown that using dilute feeds results in lower yields (Table 1) due to lower total amount of antisolvent added and due to the restriction imposed by the endpoint constraint on total volume (eq 17). On the other hand for the rest of the cases where pure antisolvent is fed to the system, high yields are achieved, though at the expense of using more antisolvent. It follows that an economic objective for the optimization of this process is worthy of future examination.

5. Conclusions The model-based approach presented here allowed the systematic and rapid process development of optimal operation of antisolvent crystallization. A mechanistic model previously


developed and identified was instrumental in calculating optimal antisolvent feeding recipes via dynamic optimization and targeting different crystal end-product size objectives. Experimental data corroborated the theoretical optimization calculations. Significantly, it was successfully demonstrated via this method how the particle size of the end-product can be directly controlled on demand. Further and via dynamic optimization studies, it was possible to learn that the initial conditions of initial solution concentration as well as the feed antisolvent concentration play significant roles in the outcomes of the crystallization batch and its crystalline particulate end-product. Such a strategy is generic to be readily implemented for control of particle size of pharmaceuticals or fine chemical compounds and falls in line with the initiatives toward innovative manufacturing advocated by modern industry as well as (recently) regulatory bodies, Viz. FDA. Acknowledgment. Experimental work was carried out at Nanyang Technological University (Singapore) with the support of the School of Chemical and Biomedical Engineering.

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