Modeling and Optimal Control of Solution Mediated PolymorphicTransformation of L‑Glutamic AcidEhsan Sheikholeslamzadeh and Sohrab Rohani*

Department of Chemical and Biochemical Engineering, Western University, London, Ontario, Canada, N6A 5B9

ABSTRACT: The solution-mediated polymorphic transformation (SMPT) of L-glutamic acid is modeled using the method ofmoments (MoM) with the addition of a dissolution term to account for the transformation of the metastable to the stablepolymorph. The numerical solution methodology involves the kinetics of nucleation, growth, and dissolution for the polymorphicsystem. The effects of the cooling profile, initial solute concentration, and seeding conditions on the product quality wereinvestigated. In supersaturated solutions with respect to both polymorphs, the natural cooling yielded the highest mass of themetastable form, while the nonlinear cooling resulted in the highest mass of the stable form (13.41 g/kg of solvent). The ratio ofthe stable to metastable form masses was higher with the higher cooling rate parameters. In solutions supersaturated with respectto the stable form and undersaturated relative to the metastable form, the dissolution of the metastable form favored theproduction of the stable form. The number-weight average size of the stable particles was 148.5 μm with the nonlinear coolingpolicy which was 51% and 134% more than those corresponding to the linear and natural cooling policies. Finally, nonlinearprogramming (NLP) was used in a dynamic mode to investigate the optimal control of the process with different objectivefunctions. It was shown that the optimal control policy had a favorable effect on the yield of the stable or metastable form as wellas the particle sizes at the end of the batch. The optimal control using an objective function to maximize the mass of themetastable form at the end of the batch resulted in 7.8 g of crystals/kg of solvent for metastable form which was 33% and 381%higher than the natural and linear cooling policies. For an objective function to maximize the mass of the stable form, the optimalcooling policy increased the mass of the stable form by 3.2% compared to the nonlinear cooling policy.

1. INTRODUCTIONThe crystallization process is an important unit operation in thechemical and pharmaceutical industries. This process involvestransport phenomena in which the driving force is super-saturation that causes the diffusion of mass from the solution tothe solid phase.1 The difference between the activity of acomponent in the bulk of a solution and the surface (saturationpoint) is the driving force for transport of the material. Thesupersaturation is mostly achieved by cooling, evaporation, orantisolvent addition.2 Advances in technologies of manufactur-ing pharmaceutical drugs have had much effect on reducing thetime of production and increasing the safety and profitability ofthe drugs.3 Many chemicals and pharmaceuticals have theability to crystallize in different lattice shapes, which is calledpolymorphism. The polymorphs of a substance have the sameproperties in the solution and gaseous phase, but they havedifferent behaviors in the solid state. There are some propertieswhich make the polymorphs differ from each other. Theseinclude solubility, dissolution rate, bioavailability, color,viscosity, density, crystal shape, and mechanical properties.4

For a pharmaceutical product, in the desired polymorph, thesize and the shape of the crystals are important.5 The efficiencyand performance of many downstream unit operations (such asfiltration, drying, and milling) in the particulate industriesdepend on the quality of the solids produced in thecrystallization operation. Most crystallization processes (espe-cially for the pharmaceuticals) are performed in batch units.Although the industrial batch processes for crystallization ofcomponents are well-understood, there are still some factors tostudy which can affect the size distribution of particles and thepolymorphic distribution of the product.6

The modeling theory for crystallization processes and,generally, particulate systems was first proposed by Hulburtand Katz.7 The methodology was pursued and developed byRandolph and Larson.8 The theory of particulate processes andpopulation balance equations with their analytical solutions wasextensively studied by Ramkrishna.9 Analytical solutions of thepopulation balance equation (PBE) are limited to simpleproblems. For more complex systems where there is more thanone form of crystal (such as polymorphic systems) and in thepresence of a dissolution process, the analytical solutions donot exist. There are some numerical methods of PBE solutions,namely, the method of moments,7−10 the method ofclasses,11,12 orthogonal collocation,13 Monte Carlo simula-tion,14 and the finite element method.15 Of the mentionedmethods, the method of moments (MoM) and the method ofclasses (MoC) are easier and faster than the other methodswhich can be used to optimize and control the crystallizationsystems. The ease of using the MoM is of importance whendealing with polymorphic transformation systems.The method of moments reduces the PBE to a set of

ordinary differential equations in terms of moments. The maindrawback of this method is that it cannot generate the crystalsize distribution.1 Another important note about this method isits inability to solve crystallization systems in which the kineticrates are nonlinear and size dependent. Some of the moments

Received: October 2, 2012Revised: December 21, 2012Accepted: January 10, 2013Published: January 10, 2013

Article

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© 2013 American Chemical Society 2633 dx.doi.org/10.1021/ie302683u | Ind. Eng. Chem. Res. 2013, 52, 2633−2641

of a population density can be approximated experimentallythrough the use of sensors such as focused beam reflectancemeasurement (FBRM).16 In the current work we used themethod of moments with the dissolution term added toincorporate the transformation of the metastable form to thestable form. The entire concentration−temperature (C−T)surface was divided into subregions, and a slightly differentmodel was applied to each region. Because of the rapidcalculation of the proposed method, the optimal control andreal-time optimization of such systems can be very efficient.

2. PROCESS MODELING AND SIMULATION

The mathematical framework of a crystallizer consists of thepopulation balance equation, mass balance, and energybalance.7 The general one-dimensional form of the populationbalance equation for a batch and perfectly mixed system thatincludes the dissolution term is

δ

∂∂

+∂

∂−

∂∂

= − *

f t L

t

G t L f t L

L

D t L f t L

LB t L L L

( , ) [ ( , ) ( , )] [ ( , ) ( , )]

( , ) ( )

i i i i i

inucleation, (1)

where i refers to each polymorph in the system. f i(t,L), Gi(t,L),Di(t,L), Bnucleation,i, and δ are the number density function,growth rate, dissolution rate, and nucleation rate of polymorphi and the Dirac delta function, respectively. For the currentstudy we have made some assumptions in order to simplify themodeling. The breakage and agglomeration of particles werenot taken into account in the system, and the volume of thebatch is kept constant. The birth term is multiplied by the Diracdelta function (δ) to impose the nucleation process for particlesof size L*. The term for the dissolution rate D(t,L) is added toeq 1, as the process under study represents a polymorphictransformation system. It should be noted that for apolymorphic system, depending on where the processconcentration is located on the concentration−temperature(C−T) diagram (Figure 1), the dissolution term has to beadded or removed from eq 1.In addition to the PBE, we need to have mass and energy

balances in order to complete the model description. The massbalance for the batch crystallization system is

∫

∫

∑

∑

ρ α

ρ α

= −

+

∞

∞

C tt

G t L L f t L L

D t L L f t L L

d ( )d

3 ( , ) ( , ) d

3 ( , ) ( , ) d

ii i

ii i

c V0

2

c V0

2

i i

i i(2)

C, ρci, and αViare the solute concentration, particle density, and

volume shape factor of polymorph i, respectively. We added thesecond term on the right side of eq 2 to include the effect ofdissolution of polymorphs. The energy balance for a jacketedvessel is

∫∫

∑

∑

ρ ρ α

ρ α

= − Δ

+ Δ

− Δ

∞

∞

CT t

tH

G t L L f t L L

H D t L L

f t L L T

d ( )d

3

( , ) ( , ) d

3 ( , )

( , ) d UA

ii

i i

ii i

i

s s crystallization, c V

0

2

dissolution, c V0

2

lm

i i

i i

(3)

ρs, Cs, ΔHcrystallization,i, ΔHdissolution, UA, and Tlm are the solutiondensity, specific heat of the solution, heat of crystallization, heatof dissolution of polymorph i in solution, overall heat transfercoefficient, and log-mean temperature of the crystallizer,respectively. In the current study the temperature of thesystem can be controlled using a cascade controller. Therefore,the system temperature is adjusted to the proposed trajectoryand eq 3 need not be solved.

2.1. Numerical Solution of the Model. In order todistinguish the seeded and newly nucleated crystals, we dividethe particles into two parts:

• seeded particles with a growth governing process(although in rare cases the nucleation can also occuron the surface of the particles17)

• newly nucleated particles at high supersaturation levelsand then grown in size

With multiplication of eq 1 by Lj and after rearrangements,the following formulation will be derived for each polymorph inthe system:

μ=

tB

d

di

i

0

nucleation, (4)

μμ μ= − =− −

tjG jD j

d

d1, 2, ..., 5i

j

i ij

i ij1 1

(5)

Equations 4 and 5 are written for j = 0−5 to cover the zerothto fifth moments of each polymorph i present in the system. μi

j

is the jth moment of polymorph i. Based on the moments ofpolymorphs, the mean size of particles of each polymorph canbe estimated:1

∫∫

=

∞

∞LL f t L L

L f t L L[ ]

( , ) d

( , ) dm n i

mi

ni

,0

0 (6)

[L1,0] is the number-averaged mean size and [L4,3] is thevolume-averaged mean size for each polymorph. Having theinitial conditions, such as the seed size distribution,concentration, and temperature, eqs 4 and 5 with mass balancewill be solved for each polymorph over time. The seed sizedistribution in our study is assumed as18

Figure 1. Schematic representation of the polymorph solubility andthree distinct regions in between the curves for an arbitrarycomponent: (region A) nucleation and growth of stable andmetastable forms, (region B) nucleation and growth of stable formand dissolution of the metastable form, and (region C) dissolution ofboth forms.

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επ σ

μ

σ= = −

−⎡⎣⎢⎢

⎤⎦⎥⎥f L f L

L(0, ) ( )

2exp

2i ii

i

i

iseed,

seed,

seed,

seed,2

(7)

in which σseed,i and μseed,i are the standard deviation and averagesize of the seed of polymorph i and ε is a constant which is usedto set the mass of each polymorph seed. Also, the mass of eachpolymorph at the start of the process can be evaluated from

∫ρ α=∞

m f t L L L( , ) di i i iseed, seed, V,0 seed,

3(8)

ρseed,i and fseed,i are the density and number density ofpolymorph i in the seed. The Dormand−Prince method(Matlab, 2008) was used to solve the ordinary differentialequation (ODE) set. The moments for each polymorph arewritten for the nucleated particles and seeds, separately. For asystem of two polymorphs (which is the dominant case incrystallization systems), there would be a total of 25 ODEs thatneed to be solved, simultaneously. The initial conditions foreach set of equations are the moments of particles for eachpolymorph (from zeroth to fifth moments of seeds), initialtemperature, and concentration.2.2. Objective Functions for the Optimal Control of

the Process. For the optimal control of a polymorphictransformation crystallization system, we defined threeobjective functions which are of importance in industry:

• J1, maximize the mass of stable polymorph at the finalbatch time

• J2, maximize the average size of particles for the stableform at final batch time

• J3, maximize the mass of metastable form at the finalbatch time

The optimization procedure involves the discretization of thetime domain to finite time steps (t ∈ [0,tf]) and thenimplementing the constrained nonlinear programming (NLP)to optimize each segment of the control vector. In order toperform the dynamic optimization, we used the Karush−Kuhn−Tucker (KKT) method followed by conversion of theconstrained optimization function to a set of unconstrainednonlinear problems. This algorithm is efficient for thepolymorphic transformation system, in which there are highlynonlinear functions of growth, dissolution, and nucleation withmass balance equations. The mathematical formulation of theoptimal control for the process is

τJ x t u tmaximize ( ( ), ( ), )il

(9)

subject to

π=x t

tf x t u t

d ( )d

( ( ), ( ), )il

(10)

π=y t g x t u t( ) ( ( ), ( ), ) (11)

≤ ≤Ru t

tR

d ( )dmin max

(12)

≤ ≤U u t U( )min max (13)

=u fixed valuefinal (14)

where l and i denote the objective function number and thepolymorph, respectively. u(t) is the control vector (such asoperating temperature or antisolvent flow rate), and x(t) is thestate variable, which consists of moments, solute concentration,

etc. y(t) is any measurement vector such as mass or volume ofpolymorphs during the process time. π denotes all theparameters that are used in kinetic rates. Rmin and Rmax arethe minimum and maximum cooling rates. ufinal is the final valueof the control vector, which enforces the optimizationprocedure to bound the control at the final batch time. Theinitial guess for the temperature profile is a linear cooling,which for most cases provides a good start for the dynamicoptimization procedure.

3. MODEL COMPOUND AND THE SIMULATIONUSING THE CONVENTIONAL COOLING POLICIES

One of the interesting model compounds to study in the fieldof polymorphic transformation is L-glutamic acid, which has astable form (α) and a metastable form (β).19 Scholl et al.conducted extensive experiments on polymorphic trans-formation of L-glutamic acid and found the kinetic parametersrepresenting the important behaviors during the process.20

Cornel et al. studied the effect of process parameters (such asstirring rate and impurity) on the transformation rate ofmetastable form to stable form at 45 °C.21 Ono et al. usedRaman spectroscopy to quantify the fraction of stable form tometastable form during crystallization of L-glutamic acid atdifferent temperatures.17 They concluded that the temperaturehas a strong effect on the transformation process. The kineticparameters which are used in our study have been found byScholl et al.20 Also, the data on the solubility of the two formswere found from the literature and fitted.19 All possible cases ofcrystallization scenarios based on the initial concentration ofsolute and its position with respect to the solubility of twopolymorphs have been studied. For all cases the volume of thesystem is kept constant and the temperature profile is changed.The following are the nucleation, growth, and dissolutionkinetic equations implemented in our model:20

= −α α αα

α

⎡⎣⎢

⎤⎦⎥B k S

KS

explnn

7/3 n2

(15)

= − −−α α α

α

α

⎡⎣⎢

⎤⎦⎥G k S

K

S( 1) exp

1g5/6 g

(16)

= −α α αD k S(1 )d (17)

μ= + −β β ββ

ββ α

β

β

⎡⎣⎢⎢

⎤⎦⎥⎥

⎡⎣⎢⎢

⎤⎦⎥⎥B k S

K

Sk

K

Sexp

lnexp

lnn7/3 n

2 s2 s

(18)

= − −−β β β

β

β

⎡⎣⎢⎢

⎤⎦⎥⎥G k S

K

S( 1) exp

1g5/6 g

(19)

B, G, and D are birth, growth, and dissolution rates,respectively. In eqs 15−19, supersaturation is defined as theratio of solute concentration to the solubility concentration at agiven temperature. The kinetic parameters which were used inthis study are shown in Table 1.20,22

3.1. Case 1. Supersaturation with Respect to BothForms (Region A in Figure 1). This case has been studied byCornel et al., experimentally.21 In their study they performeddifferent cases of polymorphic transformation with initialconcentrations high enough to start nucleation and growth ofboth polymorphs. All experiments were conducted at aconstant temperature of 45 °C. In order to examine the

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behavior of the crystallization process, we assumed that bothstable and metastable forms were present, initially. The seedsize distribution of both polymorphs is given by eq 7, withparameters given in Table 2. The product quality indices are of

high importance in the particulate industry, such as the ratio ofthe mass of the two forms and the ratio of the seeded volume tothe newly nucleated volume for each polymorph.In this case of study we examined the effect of the cooling

profile on the supersaturation profile. The cooling profile wasimplemented from the following relation:23,24

= − −⎛⎝⎜

⎞⎠⎟T t T T T

tt

( ) ( )P

o o fbatch (20)

where To, Tf, tbatch, and P are initial temperature, finaltemperature, batch time, and a power number of the coolingpolicy. The value of P can be changed according to the desiredpolicy which will be implemented on the system. It should benoted that the cooling policy given in eq 20 can result indifferent conventional cooling policies such as natural, linear,and near optimal. The three following values of P were selectedto cover three common cooling conditions:

• P = 0.1, which corresponds to an approximation ofnatural cooling

• P = 1.0, which represents linear cooling• P = 3.0, which can be considered as a nonlinear

approximate optimal cooling

The initial solute concentration and temperature are 43 g ofsolute/kg of solvent and 45 °C, respectively. According to theresults shown in Figure 2a, the solute concentration dropsrapidly using natural cooling. Therefore most of the super-saturation is consumed initially, yielding a rich content of fineparticles. For the higher values of P, the concentration curveshows a sudden decrease around 0.25 h, and then there is amoderate supersaturation until nearly the end of the process.Therefore, the profile with a higher value of P will result in abetter quality of product. It should be noted that for all threeconditions we assumed simultaneous nucleation and growth ofboth polymorphs (region A in Figure 1).Figure 2c shows that the ratio of the stable form to the

metastable form for all three cooling policies drops in the first0.1 h of the process and then increases until the end of thebatch time. However, the values of this ratio are lower than

Table 1. Kinetic Parameters with Their Values and Units

symbol value unit

knα 8.0 × 105 number/m3 sKnα 1.0 × 10−1 −kgα 2.5 × 10−7 m/sKgα 9.0 × 10−2 −kdα 3.5 × 10−5 m/sknβ 5.4 × 104 number/m3 sKnβ 1.5 × 101 −kgβ 6.0 × 104 number/m2 sKgβ 1.0 × 10−3 −kgβ 6.5 × 10−8 m/sKgβ 1.6 × 10−1 −

Table 2. Seed Size Distribution Parameters for Stable (α)and Metastable (β) Forms Used in Eq 7

polymorph ε μ (m) σ (m) αV ρc (kg/m3)

stable 1 × 1010 50 × 10−6 2 × 10−6 0.031 1540metastable 1 × 1010 30 × 10−6 2 × 10−6 0.48 1540

Figure 2. (a) Supersaturation profile, (b) three cooling policies implemented on the crystallization system, (c) ratio of stable to metastable masses,and (d) mass of metastable form of L-glutamic acid over time (for case 1).

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unity for the whole process and for all three cooling policies.The decrease in the ratio in the first part of the process is dueto the rapid growth and nucleation of the metastable form incomparison to the stable form. The cooling policy with P = 3.0has the highest value of ratios and favors the production of thestable form. From Figure 2d it can be seen that the mass of themetastable form is increased for all three policies; however, forP = 0.1 the increase in metastable production is the highest.Under these conditions, if the objective is to produce themaximum amount of the metastable form, the cooling policy ofP = 0.1 should be selected to favor the formation of that form.3.2. Case 2. Supersaturation with Respect to the

Stable Form and Undersaturation with Respect to theMetastable form. If the initial solute concentration lies inbetween the solubility curves of the stable and metastableforms, then the metastable form dissolves. This increases thesolute concentration and thus, the supersaturation with respectto the stable form is raised. The metastable form dissolves andthe stable form is produced. The transformation from themetastable to the stable form will last until all of the metastableform is dissolved. In this case, we considered three scenarios:

• unseeded condition• seeded with only the stable form• seeded with both polymorphs

3.2.1. Case 2.1. Unseeded Condition (Region B, Figure 1).At 45 °C the solubilities of the stable and metastable forms are16.46 and 23.01 g of solute/kg of solvent, respectively. In orderto start the crystallization process with an unseeded condition,there should be a large supersaturation initially and bemaintained during the process. Therefore, we selected theinitial supersaturation with respect to the stable form as muchas possible to induce the nucleation of this form. It should benoted that the process operating point at start is in region B ofFigure 1. Because of the small distance of the starting point ofthe process to the metastable solubility curve, it is more likelythat the C−T operating curve crosses the metastable solubilitycurve and produces the metastable form to some extent. Thebatch time for the unseeded case was set to 15 h. This isbecause of the kinetic behavior of nucleation for L-glutamic acidwhich enforces the nucleation to start with much delay. Basedon Figure 3, the supersaturation is a strong function of thecooling rate. For the sudden drop in temperature at the earlystages of the process, there is a sudden increase insupersaturation and then a decrease, instantaneously. However,the supersaturation profiles for other cooling rates are different.As the power number for cooling policy (P, eq 20) increases,the average supersaturation is higher and thus, the productquantity and quality will be changed.For the natural cooling profile, the average size of the stable

form increases until nearly 4 h of the process, as thesupersaturation is high in the early stages of the operation.The average size (number-weight) remains constant from 5 hof the process until the end of the batch. This behavior is alsothe same for the case of linear cooling; however, in the first 4 hof the process, the average size is nearly zero. For the case of P= 3.0, the size remains nearly zero for the first 7.5 h of theoperation. The second part of the process (for P = 3.0) showsthe growth in size. During this second period, the temperatureis suddenly reduced to 25 °C. Because of this sudden change,the nucleation occurs rapidly followed by growth and anincrease in the size of the stable form. It is worth noting thatthe average size of the particles is based on the number of

particles present in the system, which is different from thevolume average size. This type of size was considered here, asthe dominant process is nucleation, instead of growth. All threecurves of average sizes in Figure 3 show a maximum pointduring the process. This happens because of the rate of changeof the first moment of particles is faster with respect to thezeroth moment of particles at a specific time of the process.However, after that time, the size of the particles tends to beconstant. The mass of the stable form produced minus the massof the metastable form for a range of cooling policies from anear-to-flat cooling policy during the considered batch time (P= 4.0) to a sudden drop at initial (P = 0.01) is illustrated inFigure 4. The batch time for all of the power numbers is 2 h.On the basis of the results shown in Figure 4, two

conclusions can be drawn:

Figure 3. Time evolution of supersaturation with respect to solubilityof the stable form (top) and average size of the stable form based onnumber-weighted definition (bottom) in case of unseeded operation.

Figure 4. Difference of the mass of stable form produced from themetastable form in a wide range of cooling policies for three differentinitial concentrations.

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• For small P values up to 0.2, the stable form production isinversely related to the initial concentration of batch systemand this behavior will be reversed for power numbers higherthan 0.2.• For each initial concentration, there is an optimum point at

which the difference between the stable and metastable formmasses is maximized. For three different initial concentrations(while all other process and seeding parameters are keptconstant), the optimum power number is related to the initialconcentration and its value ranges from 0.17 to 0.47 for initialconcentration from 17 to 21 g of solute/kg of solvent.It can be concluded that in order to achieve as much stable

form as possible for the cooling policies given by eq 20, one hasto select a near-to-natural cooling policy. However, this case isonly valid when the process is conducted in an unseededmanner and the batch time is short.3.2.2. Case 2.2. Seeded Only with the Stable Form (Region

B in Figure 1). Here, we seed the crystallizer with the stablepolymorph in the absence of the metastable form. The initialsolute concentration for all conditions was set to 20 g of solute/kg of solvent, 45 °C was the initial temperature, and 20 °C wasthe final temperature. The seed parameters were taken fromTable 1, except the average size (μ) and ε, which were variedover three values. At first, the ε value was adjusted to maintainthe same mass of 0.6 g of seeds/kg of solvent for all threeconditions and μ was changed from 50 to 150 μm. Theoperation time for this case was set to 5 h and the linear coolingpolicy was adopted for all three seed conditions. We studied theeffect of changing the seed size on the process performance.From Figure 5 it is found that no metastable form is producedduring the process because of undersaturation with respect tothe metastable form, while the mass of the stable form increasesduring the run. It should be noted that the mass of the stableform at the end of the batch for all three seed sizes was almostthe same (12.48 g/kg of solvent), which implies the

independency of product mass with seed size. This fact is inagreement with the findings by Cornel et al.22 For the seeds ofhigher mean size, the precipitation process will be more rapidthan that for the seeds with smaller mean size. As a result, thesupersaturation will be higher for the case of smaller seed size.The effect of changing the seed mass of the stable form on

the process output is shown in Figure 6. The seed parameters

are the same as the ones in Table 1 except for the ε, whichdefines the mass of the seed. The average size of the seed inthree conditions was 50 μm. The initial concentration, initialand final temperatures, and other conditions were the same asabove.As the mass of seeds increases, the concentration of solution

is dropped more rapidly. This is because of the presence ofmore particles that are used as precipitation sites forcrystallization from the solution. Therefore, as depicted inFigure 6, the supersaturation for the case of seed mass of 0.6 g/kg of solvent has the highest value, while the curve shows thelower values for the higher mass of seeds. Because finally theconcentration is the same for three cases, the same mass ofcrystals will be precipitated on the seed surfaces. As a result,lower seed loading (or lower numbers of particles) results inlarger average size (Figure 6).

3.2.3. Case 2.3. Seeded with Both Polymorphs (Region B,Figure 1). In the final case we studied the effect of the presenceof stable and metastable form seeds together. In order to seethe effect of dissolution and cooling policy on the productquality, three cooling scenarios were selected. The seeding forall three cooling policies was the same, with an average seedsize of 40 μm and standard deviation of 2 μm. The initial soluteconcentration for all conditions was set to 20 g of solute/kg ofsolvent, with 45 °C as the initial temperature and 20 °C as thefinal temperature. The seeding mass of metastable and stable

Figure 5. Concentration evolution (top) and supersaturation(bottom) over time for three different seeding average sizes (μm).

Figure 6. Supersaturation profile (top) and average size of particles(bottom) for three different masses of stable form seeds.

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forms was 2.39 and 0.03 g of seed/kg of solvent, respectively.The reason for setting a much higher value for the metastableseeding mass compared with the stable form is the dissolutionprocess of the metastable form, which will be transferred to thestable form (depending on the cooling policy adopted). Theprocess variables are shown in Figure 7.Figure 7a illustrates the average size of the metastable form

(L1,0) for the cooling scenarios. The natural cooling policy (P =0.1) results in the largest mean size over the entire process (thisis due to the short batch time employed). The sharp increase inthe average size is a result of the fast cooling process, whichenforces the C−T curve to cross the metastable solubility curve,and results in the formation of the metastable polymorph. Thenext highest average size is for linear and nonlinear (P = 3.0)scenarios. It is seen that the curves for the size of the metastableform for these two procedures lie on each other until nearly 0.5h after the start of the process. The reason for this behavior isthat the metastable solubility curve has an approximately equaldistance with the C−T curve for both forms and thus, the sameundersaturation. However, after this time, the size of themetastable form differs for the two policies, as the cooling pathshows its effect on the process output.Figure 7b shows the average size of the stable form over the

process time. Due to the reduction in size of the metastableform and its dissolution, the stable form will nucleate and grow.It is interesting to see that the rapid cooling makes the stableform grow in size for nearly 2.5 h of the process and thenincrease very slightly. For the case of linear cooling, the size ofthe stable form rapidly grows (even higher than P = 3.0) andthen approaches a constant value. The last cooling policy (P =3.0) shows a near-to-linear growth of the stable form, anddecreases in slope in the last 1 h of the process. This behavior is

a result of keeping a high and nearly constant supersaturationthroughout the run.In Figure 7c the supersaturation curves for the three methods

are displayed. As expected for the rapid cooling policy, thesupersaturation sharply increases and then decreases. The linearcooling has a lower slope in the C−T curve, but still has highsupersaturation in the first 2.5 h of the process. However, for P= 3.0, because of (1) lower slope in the temperature profile and(2) dissolution of the metastable form at the early stage of theprocess, the supersaturation is maintained at a nearly constantlevel. After the whole dissolution of the metastable form, andon the other hand, cooling of the process (the decrease intemperature starts nearly 2 h after the start of the process),again the supersaturation is increased. The supersaturation levelis nearly constant and high with an average value of 1.3.Finally, in Figure 7d the masses of the stable and metastable

forms over time for the three cooling policies are illustrated.The rapid cooling leads to a high production of the metastableform compared with the other two cooling methods. The finalmass of the stable form for the linear and nonlinear cooling (P= 3.0) is almost the same, while the rapid cooling gives thelowest mass of the stable form and results in the highest mass ofthe metastable form. The initial mass of the stable seed form isnearly zero, while the mass of the metastable form seed isaround 2.4 g of seed/kg of solvent. At the end of the batchprocess, the rapid cooling gives nearly 3 times the metastableproduct mass as was seeded initially. For the other twoprocedures, almost all of the metastable form is transformed tothe stable form.

Figure 7. Polymorphic transformation in the presence of stable and metastable form seeds using three cooling policies: (a) average size of metastableform, (b) average size of stable form, (c) supersaturation with respect to stable solubility, and (d) masses of stable and metastable forms.

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4. OPTIMAL CONTROL FOR POLYMORPHICTRANSFORMATION IN THE PRESENCE OFDISSOLUTION PROCESS

With the conventional cooling discussed above, some coolingpolicies resulted in higher masses of the stable or metastableform, while the average size of products was not favorable. Onthe other hand, some policies achieved a low mass of productwith a desirable average size. However, using an optimal controlpolicy with the objectives defined in section 2.2, it is possible tocompromise between the yield of the desirable form and itsquality in terms of the average crystal size. All cases are for theseeded crystallization with both the stable and metastable formswith the initial concentration between the two solubility curves.The initial solute concentration, initial temperature, and finaltemperature were selected as 20 g of solute/kg of solvent, 45°C, and 20 °C, respectively. The batch time was selected as 4 hfor all of the conditions. Rmin and Rmax were taken as 0.0 and 2.0°C/min, respectively. The initial masses of the stable andmetastable forms were 0.03 and 2.39 g of seed/kg of solvent,and the average size of the stable and metastable form seedswas 40 μm with a standard deviation of 2 μm.Figure 8 shows the optimal cooling curves in addition to the

three conventional cooling policies. Figure 8a shows the sixcooling policies for the crystallization process. For the firstobjective function (J1, maximizing the mass of the stable format the end of the batch), the curve of cooling is nearly acombination of the two nonlinear and linear curves. Theoptimal cooling curve for the second objective function (J2,maximizing the average size of the stable form at the end of theprocess) lies nearly on the cooling policy of P = 3.0 until 2 hfrom the process start. The slope will be higher as the processcontinues to the end of the batch time. The optimal curve for

the third objective function (J3, maximizing the mass of themetastable form at the end of the process) is very different fromthose of the other cooling policies. Until 1 h into the process,the curve is nearly the same as that for the nonlinear mode (P =3.0). From 1 to 2 h of the process, the temperature suddenlydrops to the value of 20 °C and is kept constant at thistemperature until the end of the process.The average size of the stable form is illustrated in Figure 8b.

This shows that the optimal cooling policy will reach an evenlarger final product size of particles in comparison to thenonlinear cooling. It is worth noting that other values of Phigher and lower than P = 3.0 were tried to confirm that theoptimal policy has the highest value of mean size. Parts c and dof Figure 8 show the masses of the stable and metastable forms,respectively. It can be seen from Figure 8d that the optimalcooling policy for the third objective function, J3, produces thehighest mass of the metastable product, even higher than therapid cooling. The numerical values of the objective functionsfor the optimal and regular policies are shown in Table 3.

Figure 8. Optimal trajectories of (a) temperature for different objective functions, (b) average size of stable form (J2), (c) mass of stable form (J1),and (d) mass of metastable form (J3).

Table 3. Optimal Values of the Mass of Stable Form, Mass ofMetastable Form, and Size of Stable Form with Their Valuesfrom Conventional Cooling Policies

objective function

naturalcooling(P = 0.1)

linearcooling(P = 1.0)

nonlinearcooling(P = 3.0)

optimalcooling

J1 (g/kg of solvent) 8.71 13.02 13.41 13.85J2 (μm) 65.54 98.60 148.50 153.83J3 (g/kg of solvent) 5.87 1.62 0.67 7.80

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5. CONCLUSIONS

In this study, the modeling of a batch polymorphiccrystallization process of L-glutamic acid with the dissolutionterm of the metastable form was developed. The model wassolved using the method of moments. The polymorphictransformation process was studied for solutions supersaturatedwith respect to both forms and undersaturated with respect tothe metastable form. The effects of cooling policies, seedaverage size, and loadings were studied. The method is soflexible that it can handle any kind of process conditions andseeding parameters. The optimal control policy was used withthree single-objective functions. The optimal profiles showbetter performance in comparison with the conventionalcooling policies. Because of changing the stability of thepolymorphic forms beyond a specific temperature in enantio-tropic systems (transition temperature), the modeling andoptimal control of such processes is of importance and interest.

■ AUTHOR INFORMATION

Corresponding Author*E-mail: srohani@uwo.ca.

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

The authors of this paper gratefully thank the Natural Sciencesand Engineering Research Council of Canada (NSERC) forproviding the funds for the project.

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■ NOTE ADDED AFTER ASAP PUBLICATIONThis paper was published on the Web on February 7, 2013,with errors at the beginning of eq 3. The corrected version wasreposted on February 8, 2013.

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