Optimization of Initial Conditions for Preferential Crystallization

Dec 13, 2005 - either with p for the preferred (seeded) or with c for the counter. (unseeded) ..... model, the GSL C library is used.21 For each of th...
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Ind. Eng. Chem. Res. 2006, 45, 759-766

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Optimization of Initial Conditions for Preferential Crystallization I. Angelov,*,† J. Raisch,†,‡ M. P. Elsner,§ and A. Seidel-Morgenstern§,| System- und Regelungstheorie and Physikalisch-Chemische Grundlagen der Prozesstechnik, Max-Planck-Institut fu¨r Dynamik komplexer technischer Systeme, Sandtorstrasse 1, 39106 Magdeburg, Germany, Lehrstuhl fu¨r Systemtheorie technischer Prozesse, Otto-Von-Guericke-UniVersita¨t, Postfach 4120, 39016 Magdeburg, Germany, and Lehrstuhl fu¨r Chemische Verfahrenstechnik, Otto-Von-Guericke-UniVersita¨t, 39016 Magdeburg, Germany

The presented paper addresses the optimization of isothermal preferential crystallization for the separation of pure enantiomers from racemic mixtures. For the optimization, a rigorous process model is used. A sequence of stochastic and deterministic minimization algorithms is applied. The initial values of the mass of the racemate, the mass of the seeds, and the mean seed size are interesting parameters being optimized. The approach is illustrated by a case study for the amino acid L-/D-threonine. A promising cyclic operation mode is discussed. Although the results are not yet validated experimentally, the aim of this work is to increase the applicability of the process. 1. Introduction Enantiomers are symmetric organic molecules with identical chemical and physical properties but with different properties regarding metabolism.1,2 Chemical synthesis typically provides these molecules in the form of an equal-percentage racemate. Because of their identical properties, the separation of enantiomers is a complicated process. A possibility for separation is preferential crystallization. “Preferential crystallization” is of practical interest because of its simplicity. It is applicable for systems which crystallize as conglomerates. From a chemical engineering point of view, this process is very often recipe-based, and rigorous models allowing methodical investigations are rare; predictions are rarely used. This results in a restricted use of preferential crystallization in the chemical industry.3,4 2. Batch Model for Preferential Crystallization In the following, the concept of preferential crystallization is briefly reviewed. We shall also discuss a dynamic model, which is predominantly taken from ref 5. Detailed treatises of preferential crystallization can be found in the literature.4,6,7 An initially unsaturated racemic solution at a temperature T2 becomes supersaturated but remains clear if it is rapidly cooled to a lower temperature T1 within the metastable zone. Retarded, its composition will change in order to reach thermodynamic equilibrium. In the equilibrium state, the liquid phase possesses a racemic composition and the solid phase consists of a mixture of crystals of both enantiomers. However, after seeding with homochiral crystals, it can be observed that the system does not reach equilibrium directly but moves along a specific * To whom correspondence should be addressed. Phone: +49-3916110-372. E-mail: [email protected]. † Arbeitsgruppe: System- und Regelungstheorie, Max-Planck-Institut fu¨r Dynamik komplexer technischer Systeme. ‡ Lehrstuhl fu¨r Systemtheorie technischer Prozesse, Otto-vonGuericke-Universita¨t. § Arbeitsgruppe: Physikalisch-Chemische Grundlagen der Prozesstechnik, Max-Planck-Institut fu¨r Dynamik komplexer technischer Systeme. | Lehrstuhl fu¨r Chemische Verfahrenstechnik, Otto-von-GuerickeUniversita¨t.

trajectory. Thus, under specific conditions and in a restricted time interval, it is possible to preferentially produce crystals of just one of the enantiomers. The homochiral surface area offered initially and the specific driving forces occurring during the process are responsible for successful operation of this process. Regarding productivity, a cyclic operation mode enables a “quasi-continuous” enantioseparation, which is obviously much more attractive than a single batch mode. An applicable configuration can consist of two crystallizers connected in series. In each of these, the separation of an enantiomer is performed (Figure 1). Initially, a supersaturated solution of the racemate is in the first vessel. After the addition of homochiral seeds of type E1, only the seeded enantiomer is crystallizing within a limited time period. The trajectory of change is A f B. To harvest this enantiomer as a high purity product, the process has to be stopped before the undesirable counter-enantiomer nucleation becomes significant. For harvesting the pure solid product, a filtration device is located between the two crystallization vessels. After removal of the product, the solution is transferred to the second vessel. To ensure a quasi-continuous enantioseparation process, a certain amount of racemate has to be added in the second vessel (shifting from B to C, Figure 1). Consequent seeding with counter-enantiomer E2 crystals will result in the crystallization of this enantiomer in the following half cycle, causing a shift from C to D. After stopping this process at a certain time, the product is harvested and racemate is added again (shifting from D to A), which completes the cycle. A simplified dynamic model of an ideally mixed batch crystallizer is obtained by assuming isothermal conditions and overall growth rates, G(k), and nucleation rates, B(k), for each of the two components k. For each process, k will be substituted either with p for the preferred (seeded) or with c for the counter (unseeded) enantiomer. Below, these rates are considered to depend on the supersaturation and not on the particle dimensions. Under these assumptions, the resulting population balance is8

∂F(k) N (t, x) ∂ ) -G(k) (F(k) (t, x)), k ∈ {p, c} ∂t ∂x N

(1)

where t and x represent time and the characteristic particle size,

10.1021/ie050673w CCC: $33.50 © 2006 American Chemical Society Published on Web 12/13/2005

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Figure 1. Batch cycles for preferential crystallization.

G(k) is the growth rate, and F(k) N is the number density function. For enantiomer k, the mass balance for the solution is given by

dm(k) (liq) dt

) -3FskvG

∫

(k)

∞ 2 (k) x FN (t, 0

x) dx

(2)

For the number density functions, the following boundary and initial conditions apply:

F(k) N (t, 0) )

B(k) G(k)

(k) F(k) N (0, x) ) FN,seeds(x)

(3) (4)

where B(k) is the nucleation rate. The number density function (k) FN,seeds (x) of seeded crystals can be determined, e.g., by microscopy. Phenomena like agglomeration, abrasion, and breakage are neglected in this mathematical approximation. More sophisticated models should take these phenomena into account. This distributed model (1)-(4) can be greatly simplified by converting it into a moments model.9 From partial differential equation (PDE) 1 with boundary condition 3, a set of ordinary differential equations (ODEs) for the moments of the crystal size distribution (CSD) can be derived. The moments are defined by

µ(k) i (t) )

∫0∞xiF(k)N (t, x) dx,

i ) 0, 1, 2, ...

(5)

The zeroth moment µ0 corresponds to the overall number of crystals, and the third moment µ3 is proportional to the volume of the crystalline material in the crystallizer. By partial integration, it follows from (1) and (3) that

dµ(k) 0 (t) ) B(k) dt dµ(k) i (t) (k) (t), i ) 1, 2, ... ) iG(k)µi-1 dt

(6)

In the model, only the first four moments for each enantiomer are considered. The third moments are needed for calculation of the purity restriction (section 3.4). There are a variety of mechanisms of crystal growth, which may take place in competition.10 Although more detailed models cited in the literature give a good insight into the mechanisms of crystal growth, they imply many parameters which are difficult to determine or predict. Below, a simple power-law equation is chosen to quantify crystal growth:

G(k) ) kg(S(k) - 1)g

(7)

The term kg is an overall crystal growth rate constant, and S(k) denotes the actual degree of supersaturation for enantiomer k, i.e.

S(k) )

w(k) w(k) eq

(8)

The term w(k) eq is the equilibrium mass fraction for each of the components. The mass fraction is defined as

w(p) :)

m(p) (p)

m w(c) :)

+ m(c)+ mW m(c)

m(p) + m(c) + mW

(9)

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where m(p) and m(c) are the masses of the two dissolved enantiomers and mW is the mass of the solvent (here water). If crystals are already dispersed in the crystallizing medium, secondary nucleation can occur at supersaturation levels which are significantly lower than those at which primary nucleation takes place.10 Phenomena like microbreakage and abrasion might result in the formation of secondary nuclei. On the basis of previous experiments published in refs 11 and 12, it is highly probable that for the preferred (seeded) enantiomer, i.e., k ) p, secondary nucleation is significant. Its rate can be described by an overall power-law expression (p)

(p) B(p) ) k(p) - 1)b µ(p) b (S 3

(10)

where kb is generally assumed to be related to the stirring power. For the case of the process that is not interrupted, primary (heterogeneous) nucleation is assumed to initiate the crystallization of the counter enantiomer, i.e., k ) c. To quantify this effect, the following semiempirical expression was suggested10 and used below: -b B(c) ) k(c) b e

(c)/ln2S(c)

(11)

3. Formulation of the Optimization Problem 3.1. Introduction. A typical process goal is to obtain the maximum amount of crystalline product with desired properties (purity and crystal size). As we investigate the isothermal case of preferential crystallization, the variables which influence the process are ms (mass of the seeds), mr (mass of the racemate, i.e., of the solid feed mixture), and the seed size distribution FN,seeds(x). In this approach, a normal distribution is used to describe the seed size distribution, which is characterized by its mean xj and its variance σ2. Three free parameters are specified in the optimization process: ms, mr, and xj. The variance σ2 is considered fixed; moreover, its influence upon the process is obvioussthe bigger the variance, the higher the number of small crystals in FN,seeds(x). This results in bigger surface areas of the crystal seeds, which will lead to a faster crystallization process. Preferential crystallization is carried out in the metastable zone, i.e., the zone in which a certain degree of saturation will not necessarily result in immediate crystallization. There is a zone width valid for homogeneous solutions (solutions free from crystals and impurities). When crystals of the same substance are placed in a supersaturated solution within this zone, they begin to grow, and the solution will eventually reach equilibrium when the concentration equals the saturation concentration. The degree of supersaturation (influenced by mr) increases the rate at which seeds are growing, but it also increases the rate of nucleation, which will decrease the purity of the product. The initial conditions for the crystallization process are determined by the mass of the seed crystals ms and their mean size xj. A higher amount of invested crystal seeds implies higher purity of the crystalline product at the end of the process but also a lower yield. The above discussion reveals possibilities for influencing the quality of the preferentially crystallized product. An extra restriction comes from the fact that this process is performed in cycles (Figure 1). The difference between the concentrations of the two enantiomers (enantiomeric excess) in the liquid phase depends on the previous batch process. In the beginning of each new batch process, the two concentrations can be increased by adding racemate, but an initial excess will always remain before the next batch starts.

In conclusion, in the beginning of every batch process, a decision should be made regarding the optimal amount of added racemate (mr) and seeds (ms) and the optimal mean size of the seed crystals (xj). Optimality is defined with respect to a cost function (section 3.4) subject to a given purity constraint. Because batch crystallization systems are nonlinear dynamic systems, optimization based on deterministic methods is likely to be inefficient and less likely to find the optimal values. For optimization problems in which local optima exist, the solutions found by gradient algorithms depend on the initial parameter values.13 3.2. Optimization Method. Optimization methods based on stochastic techniques are promising when the structure of the search space is not smooth or well understood. Often, the goal is to find a point in the optimization space where a real valued cost function is minimized. This allows the simulation and optimization routines to be implemented separately; that is, the simulation routine is a “black box” for the optimization process.14 Another property of stochastic algorithms is that there is no need to know the gradient of the cost function. The algorithms converge probabilistically to the global minimum under several restrictions.15 In this work, the algorithm of simulated annealing is used, which is based on an analogy with the physical process of annealing of metal. In this process, the metal is heated to a high temperature at which its atoms possess high energy. By cooling, this energy is reduced. The cooling speed determines if the atoms will form a crystalline solid or a glass. In the case of slow cooling, when the melt spends a long time at temperatures near the freezing zone, the result is a crystalline solid material. Then, its atoms have minimum energy. On the other hand, if the cooling is fast, the resulting crystals may have many defects or the solid will possess a glasslike structure. The energy of the atoms in that case will be much higher than the minimum energy in the slow cooling case. 3.3. Outline of the Algorithm. The simulated annealing introduced by Metropolis et al.16 to solve optimization problems is similar to the metal annealing process described above. For each new perturbation of the parameters (“random walk” through the parameter space), the cost function E(i+1) is calculated and compared with the cost function of the previous set of parameters resulting in E(i). If the value of E(i+1) is smaller than E(i), then this state is accepted and the evaluated parameters are updated. Otherwise, the difference ∆E ) E(i+1) - E(i) is compared using a uniform random generator U ∈ [0, 1) with a Maxwell-Boltzmann distribution. If

e-∆E/Tcost > U

(12)

holds, then the new parameters are accepted for the next iteration. In (12), Tcost is an optimization parameter that plays a role similar to that of the physical temperature in the metal annealing process. While the temperature is still high, the probability for accepting a point with a worse cost is also high. When Tcost becomes low, (12) is less likely to hold. This increases the possibility of finding the global optimum. An extension of simulated annealing is the algorithm of adaptiVe simulated annealing (ASA).17 The advantages of this algorithm are observed mainly in multidimensional optimization cases. The ASA algorithm (previously also known as very fast simulated annealing) can use different annealing temperatures for a multidimensional parameter space. In this way, the algorithm covers the different sensitivities in the different parameter dimensions. A further improvement comes from the reannealing implementation, which allows extending the range

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of more insensitive parameters. The routine has been implemented in C by Lester Ingber and other contributors and is detailed in ref 18. 3.4. Choosing the Cost Function. Our preferential crystallization model consists of mass balances for the liquid phase (eq 2), a set of four moments (eq 6, where i ) 1-3), and growth and nucleation terms (eqs 7-11) for each enantiomer. The solutions of these equations give mathematical descriptions for the averaged properties of the batch process. Furthermore, because of the assumption that the growth rate does not depend on the crystal length, the CSD of the seeded enantiomer at the end of the batch process can be split into two parts:

F(p)(tend, x) )

{

F(p) n (tend, x) for x < ∆x F(p) s (tend, x) for x g ∆x

(13)

The first part, F(p) n (tend, x), is formed by nucleation during the batch process.19 This part is strongly influenced by the initial supersaturation level. The second part, F(p) s (tend, x), consists of grown seed crystals. Assuming that there is no breakage and agglomeration, it is identical to the seed CSD but “shifted” by a certain size ∆x, which is the size gained during the process. According to the model assumptions, the shape of this part of the distribution cannot be influenced by mr and ms. The physical constraints of the process are given by

ms,min e ms e ms,max mr,min e mr e mr,max xjmin e xj e xjmax

(14)

where the masses of seeds and racemate added before each batch process have to be restricted to a meaningful positive range. There are more requirements which should be taken into account. The main goal of preferential crystallization is to obtain a pure product from the racemate. The following purity constraint is used:

m(p) end,n (c) m(p) end,n + mend

g Pmin

(15)

(c) where m(p) end,n and mend are the masses of the seeded and the unseeded crystalline enantiomers at the end of the batch resulting from nucleation. Evidently, for the unseeded enantiomer, m(c) end ) m(c) end,n. In (15), Pmin is the minimum acceptable purity at the end of the process related to the nucleated crystals. The overall product purity including the seeds always exceeds this value. Another restriction which plays an important role in this process is related to the achieved mean crystal length of the seeds at the end of each batch process (eq 16). This restriction is formulated as

jx(p) s,end jx(p) s

g Qmin

(16)

where xj(p) s,end is the mean length of grown seed crystals at the end of the batch and xj(p) s is the average length of the seed crystals. Because of the competitive nature of the preferential crystallization, the purity restriction (eq 15), and the initial excess, constraint 16 can usually be limited between 1.2 and 2.0.

A suitable cost function CF (to be minimized) is (c) CF :) m(p) liq,end - mliq,end

(17)

(c) where m(p) liq,end and mliq,end are the masses of the seeded and the unseeded enantiomers in the liquid phase at the end of the batch process. This cost function automatically maximizes a “process yield” for the desired enantiomer, which is equivalent to maximization of the enantiomeric excess in the liquid phase of the batch process. By minimizing this cost function, the optimization parameters ms, mr, and xj are chosen to provide a small value for m(p) liq,end (p) and a large value for m(c) liq,end. A small value of mliq,end reflects the fact that a large amount of the seeded enantiomer is transformed into the solid phase by nucleation and growth. On the other hand, a large value for m(c) liq,end means that nucleation and growth for the unseeded enantiomer are small, which improves the purity of the crystalline product.

4. Case Study: Amino Acid L-/D-Threonine In this work, a quantification of the preferential crystallization process was attempted based on available experimental data for the test system L-/D-threonine in water. This amino acid crystallizes as a conglomerate. On-line polarimetry and density measurements as well as off-line surface area measurements of solid samples by microscopic investigation were used for the estimation of crystal growth kinetics based on the method of moments. The kinetics parameters were determined from the laboratory experiments using a nonlinear least squares approach20 and are presented in Table 4. More details regarding the experimental setup can be found in ref 5. The first cycle starts with a solution containing equal parts of both enantiomers. This leads to a smaller amount of crystalline product at the end compared to the second and later batch processes starting with nonracemic solutions. This difference is advantageous in order to suppress the nucleation of the unseeded enantiomer. In this study, for numerical integration of the dynamical model, the GSL C library is used.21 For each of the batch processes, the cost function (eq 17) is minimized. The constraints in eq 14 are implemented as boundaries for the search space. Constraints 15 and 16 can be implemented as a penalty in the cost function, but in this case study, they are passed to the ASA algorithm as an additional flag. The parameters obtained from stochastic optimization are used as initial guesses for deterministic sequential quadratic programming (SQP) in order to further improve the cost function. No problems were encountered with the SQP algorithm as the initial guesses found by ASA are very close to the optimal values. As it has been observed experimentally that, depending on the initial conditions, the process usually reaches its maximum enrichment of the counter enantiomer in the solution within 300-400 min,5 the time for each batch process was limited to 360 min. The purity constraint (eq 15) was set to 95%. The equilibrium mass fractions are calculated from the following empirical dependencies for the seeded (k ) p) and the unseeded (k ) c) enantiomers, based on experimental data:

w(p) eq ) 0.0985 -

0.0072 (c) m 0.0913 liq

w(c) eq ) 0.0986 -

0.0073 (p) m 0.0913 liq

(18)

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Figure 2. Optimal parameters for the first batch process.

Figure 4. Cost function for the first batch process.

Figure 3. Gained masses for the first batch process.

Figure 5. Optimization results for the first batch process (Qmin ) 1.4).

4.1. First Batch Process (A f B). The sequence of batch processes starts from a solution containing equal masses of both enantiomers which is seeded with the L-enantiomer (k ) p). In Figure 1, the trajectory for this batch process would roughly consists of the second half of the line AB, as the L-enantiomer is denoted with E1 and the D-enantiomer with E2. The optimal amount of racemate mr for each goal constraint is added and dissolved in the solution which later is saturated by cooling to the working temperature T ) 33 °C. Figure 2 presents the optimal parameters for different values of constraint 16. An increase of the Qmin restriction implies that the growth should be favored, and this has several consequences: the added racemate mass mr becomes smaller in order to suppress the nucleation of the unseeded enantiomer. In this way, the dissolved enantiomer is being depleted predominantly for crystal growth. As a result of the combination of lower supersaturation and a higher Qmin constraint, the mean crystal seed size xj is decreasing and the seed mass ms is also decreasing (Figure 2). Thus, a compromise should be made between increasing the crystal size of the final product and the process yield, defined as the gained crystal mass (m(p) g ): (p) mg(p) ) m(p) end - ms

(19)

Figure 3 presents the influence of Qmin on the obtained crystalline mass at the end m(p) end, the yield mg, and the ratio mg/ ms. Higher values of Qmin result in smaller masses of the crystalline product m(p) end for the seeded enantiomer, but as the

needed seed mass also becomes lower (Figure 2), the yield mg is not changed significantly. An important consequence is that the ratio mg/ms becomes larger than one for higher values of Qmin, which means that the yield mg is larger than the invested seed crystal mass ms. Thus, larger values for Qmin imply smaller masses of the final product. But as the mean size of the seeds distribution is not easily modified, the obtained results should serve as an estimation for Qmin based on the available xj for seeds. Figure 4 presents the minimal values for the cost function (eq 17) for the different values of Qmin. The CF values are negative because at the end of the batch process the mass fraction of the unseeded enantiomer in the liquid phase is larger than the mass fraction of the seeded one. Figure 5 shows transients of the optimal profiles of the dissolved enantiomers for Qmin ) 1.4 to present the overall tendency of this initial batch process. The mass in the liquid phase of the seeded enantiomer decreases because of crystal nucleation and growth, while the concentration of the unseeded enantiomer is almost constant, which ensures the purity of the final product. For this first batch process, the optimized parameters, initial conditions, and the resulting final values of the crystalline masses are summarized in Table 1. The optimal amount of racemate mr is added to initial enantiomer masses in the (c) liquid phase (m(p) liq ) 0.091 kg and mliq ) 0.091 kg). This forms (p) the initial masses of the dissolved enantiomers (mliq,t)0 and (c) mliq,t)0 ). Then, the process starts by adding seed crystals with optimized mass (ms) and mean size (xj). The masses of the

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Table 1. First Batch Processa,b

a

Qmin

mr (kg)

(p) mliq,t)0 (kg)

(c) mliq,t)0 (kg)

ms (kg)

xj (m)

m(p) end (kg)

m(c) end (kg)

1.2 1.3 1.4 1.5 1.6

0.011481 0.010850 0.010562 0.010405 0.010302

0.096741 0.096425 0.096281 0.096202 0.096151

0.096741 0.096425 0.096281 0.096202 0.096151

0.008001 0.006432 0.005528 0.004985 0.004633

0.001998 0.001212 0.000874 0.000687 0.000567

0.014560 0.012757 0.011753 0.011154 0.010764

0.000078 0.000040 0.000029 0.000024 0.000021

See section 4.1. b Here, the L-enantiomer is seeded (index p) and the D-enantiomer is unseeded (index c).

Figure 6. Optimal parameters for the second batch process, starting with the initial enantiomeric excess obtained after the first batch optimized for Qmin ) 1.4.

Figure 8. Obtained mg/ms ratio for the second batch process.

Figure 9. Mass of the seeded crystals for the second batch process. Figure 7. Gained crystal mass for the second batch process.

crystalline product at the end of the batch process are m(p) end for the seeded enantiomer and m(c) for the unseeded enantiomer. end 4.2. Second Batch Process (C f D). In the previous batch process (A f B), the crystallization of the seeded enantiomer was predominant. This is reflected in a final excess of the unseeded enantiomer in the liquid phase. In Figure 1, this is denoted by point B. The racemate addition shifts this point to C where the certain excess is preserved. At this point, seeds of the D-enantiomer (k ) p) are added and a second process in the cycle starts (determined by mass trajectory C f D). Different values for Qmin than those used in the previous batch process might be favorable. Figure 6 presents the optimal parameters for the second batch process (assuming a final excess after the first batch process resulting from optimized parameters for Qmin ) 1.4) for different values of the restriction Qmin. We observe the same tendency as in the first batch process, but because of the initial excess, the quantity of the added racemate is much higher. The yield

after the second batch process is illustrated in Figure 7. This is the mass of the crystals formed by nucleation and growth (eq 19). The presented results are for different initial conditions resulting from different constraints Qmin for the first batch process. Figure 8 presents the ratio mg/ms for the second batch process, which is much higher than the same ratio for the first batch (Figure 3) due to the enantiomeric excess with which this process starts. The overall tendency for the crystalline product of the seeded enantiomer from the first batch process (Figure 3) is preserved (Figure 9). Also, an important dependency is observed: the minimization of CF ensures the largest value of the enantiomeric excess in the liquid phase at the end of the process, which benefits the next crystallization. In this way, as the best achieved value for the cost function for the first batch process was for Qmin ) 1.2, the resulting enantiomeric excess will have its largest value after exactly this process. Then, a second batch starting after a first process with Qmin ) 1.2 is expected to result in a better value for the cost function than if it had started after another possible first batch with a larger constraint for Qmin. This is observed in Figure 10.

Ind. Eng. Chem. Res., Vol. 45, No. 2, 2006 765 Table 2. Optimized Second Batch Process, Starting with the Initial Enantiomeric Excess Obtained after the First Batch Optimized for Qmin ) 1.4a,b

a

Qmin

mr (kg)

(p) mliq,t)0 (kg)

(c) mliq,t)0 (kg)

ms (kg)

xj (m)

m(p) end (kg)

m(c) end (kg)

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

0.016379 0.016306 0.016238 0.016178 0.016126 0.016081 0.016042 0.016008 0.015979

0.104442 0.104405 0.104371 0.104341 0.104315 0.104293 0.104273 0.104256 0.104242

0.098245 0.098209 0.098175 0.098145 0.098119 0.098097 0.098077 0.098060 0.098045

0.001568 0.001492 0.001430 0.001372 0.001325 0.001284 0.001250 0.001221 0.001196

0.003608 0.002391 0.001783 0.001421 0.001180 0.001009 0.000880 0.000781 0.000702

0.016193 0.016081 0.015983 0.015894 0.015819 0.015754 0.015700 0.015653 0.015613

0.000713 0.000683 0.000656 0.000633 0.000614 0.000597 0.000583 0.000571 0.000561

See section 4.2. b Here, the L-enantiomer is unseeded (index c) and the D-enantiomer is seeded (index p).

Table 3. Third, Fourth, and Fifth Batch Processes with Qmin ) 1.4a

a

batch no.

mr (kg)

(p) mliq,t)0 (kg)

(c) mliq,t)0 (kg)

ms (kg)

xj (m)

m(p) end (kg)

m(c) end (kg)

3 4 5

0.017050 0.017062 0.017063

0.106044 0.106090 0.106092

0.098343 0.098347 0.098348

0.000891 0.000881 0.000878

0.001971 0.001975 0.001976

0.017118 0.017155 0.017154

0.000784 0.000787 0.000787

See section 4.3. Table 4. Used Parametersa description

symbol

value

growth coefficient growth exponent nucleation coefficient (preferred) nucleation exponent (preferred) nucleation coefficient (counter) nucleation exponent (counter) density of crystals volume shape factor mass of water variance of the seeds distribution

kg g k(p) b b(p) k(c) b b(c) F kv mW σ2

2.2864 × 10-7 7.2533 × 10-1 5.6962 × 108 2.3463 5.6451 × 102 2.8653 × 10-2 1250 0.0288 0.8017 5.3705 × 10-4

a

Figure 10. Cost function for the second batch process, cf. Figure 4.

Figure 11. Dissolved enantiomers for the second batch process (Qmin ) 1.4), starting with the initial enantiomeric excess obtained after the first batch optimized also for Qmin ) 1.4.

The mass profiles for restriction Qmin ) 1.4 of the second batch process (initial conditions again resulting from first batch with Qmin ) 1.4) are shown in Figure 11. For this batch process, the mass of the seeds is smaller and the mean crystal size of the seeds (Figure 2) is larger than those in the first batch process for the same restriction Qmin (Figure 6). This is reflected in a smaller initial rate of crystallization for the seeded enantiomer (Figures 5 and 11). Again, the mass of the unseeded enantiomer

unit m‚s-1 m-3s-1 s-1 kg‚m-3 kg m2

See section 4.

in the liquid phase is almost constant, which ensures high purity of the final product. Table 2 contains optimized parameters and the resulting initial enantiomer masses for the liquid phase and the final masses for the solid phase if the initial enantiomeric excess is obtained after a first batch process optimized for Qmin ) 1.4. 4.3. Batch to Batch Stability (A f B)|(C f D). An important question for this cyclic process is whether the optimal quantity of the racemate added after each batch process converges to a constant value (if there are no disturbances). Such batch to batch stability would provide an equal rate of production for both enantiomers after each crystallization cycle. Table 3 presents the optimal values for three consequent batch processes (each of them is optimized for Qmin ) 1.4, and the third batch follows after second optimized for the same restriction). The initial enantiomeric excess (in the liquid phase) for each crystallization cycle is almost equal to the final enantiomeric excess (in the liquid phase) with an opposite sign with respect to the enantiomer type (L or D). The enantiomeric excess Ex can be calculated as (p) (c) (c) Ex :) (mliq,t)0 - m(p) end + ms) - (mliq,t)0 - mend)

(20)

The optimal parameters for the added racemate mass (mr), the mass of the seeds (ms) and the mean seeds size (xj) converge to certain values, and therefore, the yield becomes almost constant. Conclusions This article presented an approach for the optimization of an effective and comparatively cheap technology of enantioseparation, i.e., preferential crystallization. It is based on a

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nonlinear model for the process. This model is an adequate representation of the process for crystallizers where the assumption for ideal mixing is satisfied. A cost function is constructed based on the difference of the masses of the seeded and unseeded enantiomers in the liquid phase. As during the process of preferential crystallization the mass in the liquid phase of the unseeded enantiomer should be kept, as much as possible, constant, the minimization of this function results in a maximal amount of enantiomer transferred from the liquid to the solid phase for the seeded enantiomer and a minimal amount of crystallized unseeded enantiomer. The further minimization of this function leads to the maximization of the yield at the end of the batch process. If the main goal of the preferential crystallization process is assumed to be to increase the mass of produced enantiomer, then a constraint regarding increasing the size of seeds, Qmin, should be set to lower values (where higher yields are achieved). As the second and later batch processes start with an enantiomeric excess in the liquid phase, the yield obtained from them is higher than the one in the first batch process, which starts with equal parts of the two enantiomers in the liquid phase. Moreover, as a minimization of the proposed cost function means a larger enantiomeric excess in the liquid phase at the end of the process, this favors the following batch, where the counter enantiomer is seeded. In this way, after few batch processes, the values of the optimal parameters and the final enantiomeric excess in the liquid-phase converge to a certain value, which is referred to as batch to batch stability. In this work, the combination of stochastic optimization by simulated annealing and a subsequent use of a deterministic approach (SQP), which uses the obtained optimal parameters from the stochastic optimization as initial guesses, is shown to be a technique capable of finding optimal values for the isothermal case of the preferential crystallization process. A further increasing of the purity could be achieved if the final product can be sieved, and in this way, the crystals originating from nucleation will be separated. As the nucleated crystals of the counter enantiomer will be in the same size range, this will leave crystals grown from the seeds with purity higher than specified. The sieved lower fraction (originated from nucleation) could be used for preparing a saturated solution in which the crystals from the upper seeds fraction could be reseeded, and in this way, their length can be further increased. Notation Latin Symbols B ) nucleation rate, s-1 Ex ) enantiomeric excess, kg FN ) number distribution function, m-1 G ) (global) crystal growth rate, m‚s-1 m ) mass, kg Q ) gained crystal length ratio S ) supersaturation degree t ) time, s w(k) ) weight fraction (mass of component k per total mass), kg‚kg-1 x ) effective particle length, m Greek Symbols µi ) ith moment, mi F ) density, kg‚m-3 Subscripts and Superscripts c ) counter (unseeded) enantiomer

end ) at the end of the batch process i ) number for the moment k ) number specifying the enantiomer (k ∈ {p, c}) liq ) liquid phase n ) resulting from nucleation p ) preferred (seeded) enantiomer s ) resulting from seed seeds ) seeds tot ) total Literature Cited (1) Collins, A. N.; Sheldrake, G. N.; Crosby, J. Chirality in industry: the commercial manufacture and applications of optically actiVe compounds; John Wiley & Sons: New York, 1994. (2) Collins, A. N.; Sheldrake, G. N.; Crosby, J. Chirality in industry II: the commercial manufacture and applications of optically actiVe compounds; John Wiley & Sons: New York, 1997. (3) Grabowski, E. J. J. Enantiopure drug synthesis: from aldomet to imipenem and beyond. Presented at the 16th International Symposium on Chirality (ISCD 16), New York, 2004. (4) Jacques, J.; Collet, A.; Wilen, S. H. Enantiomers, racemates and resolution; Krieger: Malabar, FL, 1991. (5) Elsner, M. P.; Ferna´ndez Mene´ndez, D.; Alonso Muslera, E.; SeidelMorgenstern, A. Experimental study and simplified mathematical description of preferential crystallisation. Chirality 2005, 17, 183-195. (6) Collet, A. Separation and purification of enantiomers by crystallisation methods. Enantiomer 1999, 4, 157-172. (7) Sheldon, R. A.; Hulshof, L. A.; Bruggink, A.; Leusen, F. J. J.; van der Haest, A. D.; Wijnberg, H. Crystallization techniques for the industrial synthesis of pure enantiomers. Proceedings Chiral 90 Symposium Manchester, Spring Innovations Ldt.: Stookport, England, 1990; 101-107. (8) Ramkrishna, D. Population Balances: Theory and Applications to Particulate Systems in Engineering; Academic Press: New York, 2000. (9) Randolph, A. D.; Larson, M. A. Theory of Particulate Processes; Academic Press: New York, 1988. (10) Mersmann, A. Crystallization Technology Handbook; Marcel Dekker: New York, 2001. (11) Alvarez-Rodrigo, A.; Lorenz, H.; Seidel-Morgenstern, A. Online Monitoring of Preferential Crystallization of Enantiomers. Chirality 2004, 16, 499-508. (12) Elsner, M. P.; Lorenz, H.; Seidel-Morgenstern, A. Preferential crystallisation for enantioseparationsNew experimental insights indispensable for a theoretical approach and an industrial application. Proceedings of the 10th International Workshop on Industrial Crystallization BIWIC 2003, Coquerel, G., Ed.; University of Rouen: Mont Saint-Aignan, 2003; ISBN 3-86130-198-9. (13) Li, L.; Barry, D. A.; Morris, J.; Stagnitti, F. CXTANNEAL: an improved program for estimating solute transport parameters. EnViron. Modell. Software 1999, 14, 607-611. (14) Kirkpatrik, S.; Gellat, C. D., Jr.; Vecchi, M. P. Optimization by Simulated Annealing. Science 1983, 220, 671-680. (15) Locatelli, M. Convergence of a Simulated Annealing Algorithm for Continuous Global Optimization. J. Global Optimization 2000, 18, 219233. (16) Metropolis, N.; Rosenbluth, A.; Rosenbluth, M.; Teller, A.; Teller, E. Equation of state calculations by fast computing machines. J. Chem. Phys. 1953, 21, 1087-1092. (17) Ingber, L. Very fast simulated reannealing. Math. Comput. Modell. 1989, 12, 967-973. (18) Ingber, L. Adaptive Simulated Annealing (ASA). http://www. ingber.com/ (accessed on May 31, 2005). (19) Vollmer, U.; Raisch, J. Control of batch cooling crystallisers based on orbital flatness. Int. J. Control 2003, 1635-1643. (20) Rawlings. J.; Miller, S.; Witkowski, W. Model identification and control of solution crystallization processes: a review. Ind. Eng. Chem. Res. 1993, 32 (7), 1275-1296. (21) GSL Reference Manual. http://www.gnu.org/software/gsl/ (accessed on May 31, 2005).

ReceiVed for reView June 9, 2005 ReVised manuscript receiVed September 29, 2005 Accepted November 2, 2005 IE050673W