Crystallization Kinetics of Ampicillin

M. Ottens,*,† B. Lebreton,†,‡ M. Zomerdijk,† M. P. W. M. Rijkers,§O. S. L. Bruinsma,|,⊥ and L. A. M. van der Wielen†

Kluyver Laboratory for Biotechnology, Delft University of Technology, Julianalaan 67, 2628 BC Delft,The Netherlands, DSM Research, P.O. Box 18, 6160 MD Geleen, The Netherlands, and Laboratory for ProcessEquipment, Leeghwaterstraat 44, 2628 CA Delft, The Netherlands

The first results regarding the crystallization kinetics of the pure semisynthetic â-lactamantibiotic (SSA) ampicillin in water are presented. The experimentally obtained results at pH) 5 and T ) 298 K are described well by the population-based crystallization model presentedin this paper. The nucleation rate J (# m-3 s-1) is related to the supersaturation ratio of ampicillinS by J ) 2 × 107 exp[-3.383/(ln(S))2]. The surface energy γ of creating a nucleus of ampicillinmolecules therefore equals 5.83 mJ m-2. The growth rate G (m s-1) is determined with a newlydeveloped adapted single-crystal growth rate analysis method and is related to the supersatu-ration ratio of ampicillin S by G ) 4.57 × 10-8(S - 1)1.72.

Introduction

Increased product purity demands for pharmaceuticaland fine-chemical products as well as legislative andenvironmental reasons force pharmaceutical industriesto investigate new concepts either to optimize existingproduction operations or to develop new processes fornew products. In such new processes, a reduction of thenumber of process unit operations and waste materialstreams is paramount. Biotechnological operations, suchas enzymatic reactions applied in aqueous environ-ments, are becoming increasingly important for theproduction of pharmaceutical products such as penicillinderivatives.1 These new synthesis routes imply theapplication of appropriate separation techniques, whichplay an important role in the design of cost-effective unitoperations. In this respect, crystallization is a suitabletechnique for the recovery of pharmaceutical productsof relatively low solubility, such as â-lactam antibiotics.Crystallization is thus conducted in multicomponentsystems where the presence of solutes other than thetargeted product might significantly influence the kinet-ics of crystallization.2 It is known that foreign moleculessuch as degraded products and byproducts, additives,and other components can interfere with nucleation aswell as with the growth process.3 Crystal growth canbe disrupted by the incorporation of impurities into thecrystal lattice or by their adsorption at the surface ofthe crystal.4-6 The structure, size, and morphology ofthe final product might consequently be modified by thepresence of impurities.7 The present paper is part of alarger study focused on the identification of possibleeffects of impurities on the crystallization kinetics ofsemisynthetic penicillins and on the product quality.The impact of impurities will be monitored herein by

investigating the crystallization kinetics of the process,i.e., the induction time, desupersaturation rate, andgrowth rate.8

In this paper, the crystallization kinetics of pureampicillin from aqueous solution is determined. Fur-thermore, a model is developed that is capable ofdescribing the induction time, desupersaturation rateand growth rate. This model provides a frameworkwithin which the effects of impurities on semisyntheticantibiotic (SSA) crystallization can be accurately ana-lyzed and reported in future papers. The model pre-sented in this paper is an indispensable tool in furtherunderstanding the crystallization of antibiotics anddeveloping realistic mechanistic models for nucleationand growth.

As a relevant SSA model compound ampicillin (Ampi,Mw ) 0.349 41 kg mol-1) is considered. The structuralformula of this compound is

The solubility of pure ampicillin in water at 298 K isdepicted in Figure 1.9 Changing the pH of the aqueoussolution to the isoelectric point of ampicillin gives riseto a substantial decrease in the solubility and inducesthe supersaturation and subsequent crystallization ofthe semisynthetic antibiotic. The solubility is altered bythe introduction of salts.10

Model

To describe the crystallization process and to obtainrelevant kinetic parameters a crystallization model isused to simulate the solute concentration in the liquidphase as a function of time (the so-called desupersatu-ration curve) as well as the crystal size distribution(CSD). The model is based on a population balance forcrystals in a specific size class11,12

* Correspondence concerning this article should be ad-dressed to Marcel Ottens. E-mail: M.Ottens@tnw.tudelft.nl.

† Delft University of Technology.‡ Current address of B. Lebreton: Genentech, 1 DNA way

(MS#75), South San Francisco, CA 94080.§ DSM Research.| Laboratory for Process Equipment.⊥ Current address of O. S. L. Bruinsma: SASOL Center for

Separation Technology, Private Bag X6001, Potschefstroom,2520, RSA

4821Ind. Eng. Chem. Res. 2001, 40, 4821-4827

10.1021/ie0101238 CCC: $20.00 © 2001 American Chemical SocietyPublished on Web 10/05/2001

where V is the compartment volume in m3, n is thenumber of particles per volume per size class in # m-4,t is the time in s, x is the length coordinate in m, G isthe growth rate in m s-1, b is the birth function in acertain crystal size class in # m-4 s-1, d is the deathfunction in a certain crystal size class in # m-4 s-1, Φkis the flow rate of the kth stream containing crystalsentering or leaving the compartment in m3 s-1.

Equation 1 reduces under the constraints of a constant-volume batch operation (no input and output) and underthe assumptions of no agglomeration, no breakage, andno death; birth only in the lowest particle class; and size-independent growth to the following hyperbolic partialdifferential equation (PDE)

To solve this PDE, boundary and initial conditions arerequired. The boundary condition at x ) 0 is given bythe commonly applied equation

where J is the nucleation rate in # m-3 s-1. The initialcondition at t ) 0 is given by

The desupersaturation is obtained by solving the fol-lowing component mass balance

where Mw is the molar mass of the crystallizing com-ponent. The mass of crystals formed (M) is given byintegration over all crystal size classes

where kv is the crystal shape factor and Fv is the crystaldensity in kg m-3. The nucleation rate is described by12

The growth rate is given by12

Both the nucleation rate and the growth rate aredependent on the supersaturation ratio of the crystal-lizing component

Solving the model contained in eqs 2-9 essentiallymeans solving the hyperbolic partial differential equa-tion (PDE). A numerical solution method is used to solvethe discretized PDE. A first-order single step upwarddiscretization in space and in time is used, leading tothe following Eulerian scheme

with index i indicating the space coordinate and indexj indicating the time coordinate. The choice of the valuesof the time step and space step governs the numericalstability of the system. Care should be taken in selectingthese step sizes. In general the so-called Courant-Fredrich-Levy number should be smaller than 0.5,13

or in this case

To obtain a mechanistic model for the growth rate andthe nucleation rate, the comprehensive crystallizationmodel can be used and tested for different relationsbetween the nucleation and growth rates. However, thekinetic parameters can also be determined by plottingthe results from the crystallization experiments accord-ing to eqs 7 and 8.

Nucleation

A useful lumped parameter for monitoring the nucle-ation mechanism is the induction time, which is definedas the period of time that elapses between the achieve-ment of supersaturation and the appearance of crystalshaving a “detectable” size. The induction time dependsnot only on the initial supersaturation but also on thedetection method. If, for example, a concentrationmeasurement is used, the induction time depends onthe conversion, whereas for light reflection detection,it depends instead on the crystal surface area produced.In general, one can simplify this problem as follows:once the detectable value of a moment of the CSD, mi,det,is exceeded, the induction period has elapsed (where iis the order of the moment)

or

Figure 1. Solubility of ampicillin as function of the pH of thesolution at 298 K and 1 bar.

∂Vn(t,x)

∂t) -

∂Vn(t,x) G(t,x)

∂x+

Vb(t,x) - Vd(t,x) - ∑k

Φknk(t,x) (1)

∂n(t,x)∂t

) -∂[n(t,x) G(t)]

∂x(2)

n(t,0) )J(t)G(t)

(3)

n(0,x) ) 0 (4)

MwVdC(t)

dt+

dM(t)dt

) 0 (5)

M(t) ) ∫0

∞kvFvn(t,x)x3 dx (6)

J(t) ) J0 exp(- Bln(S(t))2) (7)

G(t) ) kg(S(t) - 1)n (8)

S(t) )C(t)CS

(9)

nij+1 ) ni

j - ∆t∆x

(Gij ni

j - Gi-1j ni-1

j ) (10)

Gij∆t

∆x< 0.5 (11)

mi,det ) ∫0

G(t)tind J(t)G(t)

xi dx )J(t)

(i + 1)G(t)[G(t)tind]

i +1

(12)

4822 Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001

In terms of initial supersaturation S0, using eqs 7 and8, this becomes

Because the exponential term will dominate, a plotof ln(tind) versus [ln(S0)]-2 for different crystallizationexperiments will yield B/(i + 1) as the slope. Then, thetype of detector determines i; in our case, laser reflectionis used for detection, so i ) 2.

The nucleation in this study is considered to beprimary nucleation. The factor B in eq 7 is given by12

and from B the value for the surface energy γ isobtained. Ultimately, from the desupersaturation ex-periments, the value of J0 is obtained.

The primary nuclei are assumed to be spherical. Foranisotropic crystals the surface energy of each face mustbe different as this is the cause of anisotropy. However,whether a nucleus is already faceted or is spherical isa question that has not been solved. It seems that acertain size is needed before a spherical crystal startsto develop facets. The shape of the crystals becomesneedlelike during crystal growth. This is due to thedifference in growth velocities for the different crystalfaces. No different face-growth velocities are used in thismodel, but this phenomenon is taken into account byusing a shape factor based on the aspect ratio of thecrystals.

In eq 14, the relation between the induction time andthe (initial) supersaturation is given. This relationprovides essential information for the nucleation kinet-ics, i.e., the factor B. To assess the value of B, it is goodpractice to calculate the associated surface energy γ,ensuring that the rate equation is correct and that theexperimental surface energy is of the right order ofmagnitude. Finally, γ can be used to decide whether thedominant nucleation is homogeneous or heterogeneous.

Growth

In the case of the growth rate equation, the param-eters kg and n in eq 8 can be obtained from growth rateexperiments as the abscissa and the slope, respectively,of plots of ln(G) vs ln[S(t) - 1)]. Because ampicillin is arather large molecule, the crystallization rate is rela-tively low, and the desupersaturation period turns outto be long enough to determine the growth kinetics fromthe second part of the batch experiment (see Batch-Crystallization Experiments section below).

Experimental Setup and Techniques forDetermination of Kinetic Parameters

Materials. Ampicillin trihydrate (Ampi) was providedby DSM (Geleen, The Netherlands).

Batch-Crystallization Experiments. Ampicillinwas crystallized from freshly prepared solutions. Theexperiments were conducted at pH 5.0 and T ) 25 °C

with different initial supersaturations S0 ) 2.37 (Xa),3.29 (Xb), 2.45 (Xc), and 1.72 (Xd).

Batch-crystallization experiments were conducted ina 2-L jacketed reactor containing three baffles and twointer-MIG II impellers, with a stirring speed of 300 rpm.The temperature was set to 25 °C. Pure material wasfirst dissolved under acidic conditions at pH 1.9 for 20min, using HCl as the solvent (2 M). The pH wassubsequently raised to a value of 5.0 using ammonia inwater (12 M). The final solubility of ampicillin in thissolution of 18.9 mol L-1 is higher than the solubility ofampicillin in pure water (see Figure 1) due to thepresence of the salt (NH4Cl). This “salting-in” effect isdescribed elsewhere.10 Temperature and pH were keptat their set values. A laser probe (Delft Unversity ofTechnology, Delft, The Netherlands) was inserted intothe reactor to monitor the changes of turbidity by laserlight reflection. Temperature, pH, and laser signal werecontinuously recorded using a Biodacs system (App-likon, Schiedam, The Netherlands); see Figure 2.

Samples were carefully extracted at various timeintervals. Aliquots of the samples were filtered usingnylon membranes (0.2 µm; Gelmann Sciences), and thefiltrates were appropriately diluted for subsequentreverse-phase chromatography analysis, while the re-maining samples were used for crystal growth ratedetermination. The final product of each experimentwas dried at 40 °C and subsequently analyzed for crystalsize distribution. Its crystal size distribution (CSD) wasestimated from the image analysis of a minimum of 500particles observed via light microscopy (Leica Q500IW,Leica, Cambridge, U.K.). Image analysis appeared to bethe most suitable and reliable technique for CSDdetermination because of the needlelike shape of thecrystals. Scanning electron microscopy (SEM) imageswere acquired to characterize the final product of eachcrystallization using a scanning electron microscope(JSM-5400; JEOL, Tokyo, Japan). Samples were coatedwith gold for 3 min using an ion sputtering gun (JFC-1100E; JEOL, Tokyo, Japan).

Reverse-Phase Chromatography. Samples (con-taining Ampi) were separated and analyzed by reverse-phase chromatography using a HPLC Waters systemcomprising a Waters 996 PDA detector, a Waters 910Wisp injector, and a Waters 590 pump. The reverse-phase column was a Zorbax SB-C18 column (4.6 × 75mm with a pore size of 3.5 µm; Hewlett-Packard, PaloAlto, CA). The buffer consisted of 8 mmol L-1 tetrabu-

Figure 2. Experimental setup for ampicillin crystallizationexperiments with the adapted single-crystal growth rate analysistechnique (ASCGRA).

tind ) [(i + 1)mi,det

J(t) G(t)i ]1/(i+1)

(13)

tind ∝ [(S0 - 1)n]-1/(i+1) exp[ B(i + 1)(ln(S0))

2] (14)

B ) 16πν2γ3

3(kT)3(15)

Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001 4823

tylammonium bromide, 10 mmol L-1 Na2HPO4, and15% (v/v) acetonitrile and was brought to pH 6.6 withH3PO4. The elution profile was isocratic, and the ab-sorbance was measured at 230 nm. In combination withthe saturation concentration, the supersaturation couldthus be calculated.

Crystal Growth Rate Determination via AdaptedSingle-Crystal Growth Rate Analysis (ASCGRA).A sample was sucked from the batch crystallizer andintroduced into a glass cell. The crystal growth was thenmonitored using a phase contrast microscope (Olympus,Tokyo, Japan) by acquiring images at various timeintervals (ranging between 20 and 200 s). A few crystalswere selected from the large number of crystals observedper image, and their lengths were measured via imageanalysis. The growth rate was thus determined. Theoperation was repeated for a series of samples corre-sponding to distinct supersaturations, decreasing intime during the batch experiment.

Results

Crystal Morphology. SEM images of the pureampicillin crystal obtained from experiment Xc areshown in Figure 3 at two magnifications. It can be seenthat the ampicillin crystals formed are boatlike crystalsthat are tapered and have mixed blunt and sharp ends.Furthermore, they show some limited twinning andlimited breakage. The mean aspect ratio AR variesbetween the experiments performed. For experimentsXa, Xb, and Xc, the AR values were 7.74, 5.44, and 8.17,respectively. The corresponding shape factors kv thenbecome 0.0250, 0.0507, and 0.0225, respectively.

Desupersaturation Curve. It can be seen fromFigure 4 that the initial supersaturation S0 has apronounced effect on the crystallization rate. The higherthe initial supersaturation S0 the faster the desuper-saturation and the crystallization. The inset of Figure4 also shows that the crystallization actually needs aspecific induction time to obtain a substantial rate. Theinduction times measured with the laser reflectionmethod are indicated in Figure 4.

Induction Time Experiments. The plot of ln(tind)versus [ln(S0)]-2 for the different crystallization experi-ments gives a straight line, as shown in Figure 5. Theslope of the curve equals B/3 and has a value of 1.127,so B ) 3.383. From eq 15, the value for the surfaceenergy is calculated. The molar volume of ampicillin isobtained from the UNIQUAC volume parameter R )10.587 (R ) V/VCH4, where VCH4 is the reference volume

of methane ) 15.17 cm3 mol-1 9) giving 160.61 cm3

mol-1. The volume of one molecule of Ampi, then, is Vdivided by Avogadro’s number Na, i.e., ν ) 2.667 × 10-28

m3. The surface energy of creating a nucleus γ iscalculated from eq 15 via

with T ) 298 K. γ then is 5.83 mJ m-2, whichcorresponds to heterogeneous primary nucleation.14

Ex Situ Growth Rate Measurements. From theslope of the curve in Figure 6, the power in the crystalgrowth relation n is obtained and has an overall valueof 1.72. The intercept in Figure 5 gives a value for thegrowth rate coefficient kg of 4.57 × 10-8 m s-1. It canbe seen that, for the different series, a different slopecan be identified. It seems that the growth rate isdependent on the initial super saturation S0.

Crystal Size Distribution Measurement. Themeasured CSD can be used to validate the experimen-tally determined kinetic parameters. Only for experi-ment Xc was a reliable CSD measured. During the first

Figure 3. SEM image of the needle shaped final product resulting from batch-crystallization of pure ampicillin (experiment Xc). Theexperiment was conducted with a initial supersaturation ratio of 2.45 (see Experimental Setup and Techniques for Determination ofKinetic Parameters). (Left) Magnification 1 cm ) 100 µm. (Right) Details of a crystal needle tip at magnification 1 cm ) 10 µm.

Figure 4. Desupersaturation curve for amplicillin crystallizationat different initial supersaturations. Xa (circles), Xb (squares), Xc(triangles), and Xd (diamonds) pure ampicillin crystallization atpH ) 5 and T ) 298 K. See Experimental Setup and Techniquesfor Determination of Kinetic Parameters for precise conditions. Sbased on concentrations; see eq 9.

γ ) x33B(kT)3

16πν2

4824 Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001

experiments, i.e., Xa and Xb, because the computercrystal-selecting method for size measurement overse-lected particles from the lower size classes, these CSDshad crystal size values that were too small. For Xd, noCSD was measured. An extra crystallization experimentXe was performed with a more accurately determinedCSD. This was done to check the model and the obtainedkinetic parameters by comparing the model CSD withthe experimental CSD.

Model-Based Validation. The obtained kinetic datawere validated using the model introduced previously,and consequently, the model results are compared withexperimental results acquired as described in the previ-ous section. The model is “fed” with the obtained kineticparameters that are used in the calculations, namely,B ) 3.383, n ) 1.72, and kg ) 4.57 × 10-8 m s-1. Bycomparing the model and experimental desupersatura-tion profiles, the value of the preexponential nucleationrate factor J0 can be obtained. The time and space stepsizes used in the calculations are mostly ∆t ) 40 s and

∆x ) 5 × 10-6 m, respectively, except for Xb for highernumerical stability. The results and conditions areshown in Table 1.

The value of the aspect ratio AR of 14.9 in Xe differsfrom the previous ones. It was obtained from a manualimage analysis and is more accurate than the othervalues. However, the influence of the AR value on thecalculated desupersaturation, nucleation and growthkinetics, and CSD is limited. The optimized values forJ0 from the desupersaturation curves vary slightly withS0 (see Table 1). However, the desupersaturation curvesand nucleation and growth kinetics can properly bedescribed with a single value for J0, i.e., J0 ) 2.0 107 #m-3 s-1.

The results are shown in the Figures 7-10. For theevolution of the crystal size distribution in time, theessential elements of the crystallization are well cap-tured by the model. At S0, the crystal growth is fast,whereas after a while, the decreased supersaturationcauses the growth rate to decrease and, finally, to reachzero. Figure 7 shows a comparison between the modeleddesupersaturation and experimental data set Xc, andFigure 8 shows the modeled nucleation and growth ratesduring crystallization, together with the experimentalgrowth rate data. These figures clearly show the abilityof the model to describe and predict the crystallizationbehavior of ampicillin as given by the desupersaturationand the nucleation and growth rates. The inductionperiod is very well captured with the model and theobtained kinetic data. In Figure 9, the calculated andexperimental final scaled crystal size distributions are

Figure 5. Induction time experiments for the determination ofthe nucleation rate. Xa, Xb, Xc, and Xd pure ampicillin crystal-lization at pH ) 5 and T ) 298 K. See Experimental Setup andTechniques for Determination of Kinetic Parameters for preciseconditions. S based on concentrations; see eq 9.

Figure 6. Ex situ crystal growth rate determination by the flowcell method. Xa (triangles), Xb (diamonds), Xc (squares), and Xd(circles) pure ampicillin crystallization at pH ) 5 and T ) 298 K.See Experimental Setup and Techniques for Determination ofKinetic Parameters for precise conditions. S based on concentra-tions; see eq 9.

Table 1. Model-Based Validation Results for J0 andNumerical Settings

experi-ment S0 AR

J0(# m-3

s-1) N Mlength

(m)time(s)

∆t(s)

∆x(m)

Xa 2.37 7.74a 1 × 107 1000 200 1 × 10-3 4 × 104 40 5 × 10-6

Xb 3.29 5.44a 4 × 107 2000 100 1 × 10-3 2 × 104 10 1 × 10-5

Xc 2.45 8.17a 2 × 107 1000 200 1 × 10-3 4 × 104 40 5 × 10-6

Xd 1.72 10b 4 × 107 4000 400 2 × 10-3 16 × 104 40 5 × 10-6

Xec 2.64 14.9 2 × 107 1000 200 1 × 10-3 4 × 104 40 5 × 10-6

a Value based on size classes that are too low. b Not available,interpolated value based on Xa-Xc. c Only S0, AR, and the finalCSD are measured (particle class step size ) 50 µm).

Figure 7. Model-based desupersaturation curve comparison withexperimental values from data set Xc. Initial supersaturation ratioof 2.45. Density of crystals used in calculation Fv ) 1500 kg m-3.Lines are the model; points are experiments. In addition, themodeled amount of crystals formed (M) is given by the increasingline.

Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001 4825

shown for the data set Xc for final verification of thekinetic parameters. We see that the average crystal sizeis predicted reasonably well. To validate these kineticparameters more extensively, we used Xe with anaccurately measured final CSD, as shown in Figure 10.The number-average particle sizes from the experimen-tal and modeled CSDs are calculated according to

where M is the number of particle size class intervals.The relative difference between the modeled and ex-perimental average particle size values is given by

For experiment Xc, with an experimental dp,av of 126.5µm and a modeled dp,av of 193.4 µm, this yields a ∆ valueof 34.6%. For experiment Xe, with an experimental dp,avof 207.2 µm and a modeled dp,av of 227.5 µm, this yieldsa ∆ value of 8.9%. Thus, the average size is actuallypredicted quite accurately for Xe. It can be seen thatthe shapes of the experimental and model curves aresimilar but with a slight overprediction of the averagecrystal size. This could be due to several factors, oneimportant one of which is the breakage of the needlelikecrystals.

Discussion

Different growth mechanisms can be identified bydifferent values for the power number n in the growthrate equation. The birth and spread mechanism resultsin n ) 5/6; kinetic roughening results in n ) 1, and spiralgrowth in n ) 2 (e.g., Sohnel and Garside14). In thiswork, the power n in the growth rate equation was closeto 2 (1.72) for a wide range of experiments. This wasalso observed for the growth rate of similar molecules,such as aspartame, and indicates a spiral growthmechanism15 (the so-called Burton-Cabrera-Frank(BCF) model16). Thus, surface integration, rather thanmass transfer, is the dominant mechanism duringcrystallization of ampicillin under the conditions re-ported in this paper.

The shape of the experimental CSD in Figure 9 differsfrom the model CSD. This is largely because of the peaklying at ∼25 µm. This point arises as a result of theselection technique for the crystal size measurements.By shifting to manual selection, a much better picturearises in Figure 10. Furthermore, the front edges of themodeled CSD curves are not completely steep, whichcan be attributed to numerical diffusion. The averagepredicted crystal size in Figure 10 is somewhat too large.This can be related to breakage during crystallizationand sample preparation. The drying step during thissample preparation process results in breakage of thecrystals because of their relative fragility. This gives asmaller average particle diameter. Another issue is thenumber of crystals in the image analysis needed for

Figure 8. Nucleation (dotted line) and growth rate (solid line).Comparison with experimental data set Xc. Conditions are thesame as in Figure 7.

Figure 9. Comparison of the modeled final crystal size distribu-tion curves of the product after amplicillin crystallization at aninitial supersaturation of 2.45 (Xc) with experimental results(circles, 525 crystals, size classes of 10 µm). Pure ampicillincrystallization at pH ) 5 and T ) 298 K. See Experimental Setupand Techniques for Determination of Kinetic Parameters forprecise conditions. Conditions are the same as in Figure 7.

dp,av )

∑i)1

M

dp,ini

∑i)1

M

ni

∆ ) 100%{dp,avmod - dp,av

exp

dp,avmod }

Figure 10. Comparison of the modeled final crystal size distribu-tion curves of the product after amplicillin crystallization at aninitial supersaturation of 2.64 (Xe) with experimental results(circles, 172 crystals, size classes of 50 µm). Pure ampicillincrystallization at pH ) 5 and T ) 298 K. See Experimental Setupand Techniques for Determination of Kinetic Parameters forprecise conditions.

4826 Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001

reliable CSD determination. We chose a minimal num-ber of 500, because of the long analysis time, but thisnumber might actually not be sufficient. Therefore, anextra crystallization experiment was performed, Xe, inwhich the AR and CSD were accurately measured forcomparison with the model and final validation of thekinetic parameters. Overall, it can be stated that themodel gives a good prediction of the CSD and theaverage crystal size, indicating the validity of theobtained kinetic parameters.

Conclusions

The kinetics of pure ampicillin crystallization in anaqueous solution has been determined. These data canbe used to design crystallization systems for the separa-tion of ampicillin. The adapted single-crystal growthrate analysis method proves useful in determininggrowth rates for the relatively slow ampicillin crystal-lization. Furthermore, a valuable modeling tool isdeveloped for analyzing the experimental crystallizationdata of ampicillin and more general of SSAs. The modelcovers the experimental data well, thereby verifying thekinetic relation obtained from experiments. Agglomera-tion and breakage terms could be inserted, but the needfor such a step should be paramount; otherwise, it wouldonly unnecessarily complicate the model. The model canbe used to test different relations for growth G andnucleation J and to apply these G and J relations tothe entire experimental database. A parametric opti-mization should be performed, but that falls beyond thescope of the present paper and is work for a futurepaper. The model can be extended by connecting thecrystallization model to an accurate thermodynamicmodel for a description of the solubilities of the SSAsand their precursors, as developed in another paper.10

This should refine the model predictive capability tocapture the specific solubility behavior of the SSAs andtheir precursors.

Acknowledgment

The authors thank DSM, Geleen, The Netherlands,for the chemicals and Chemferm and the Dutch Min-istry for Economical Affairs for financial support. R.Grimbergen and T. van der Does from DSM are kindlyacknowledged for their valuable input. M. Hoeben isthanked for performing experiment Xe and S. H. vanHateren for performing the image analysis.

Symbols

AR ) aspect ratiob ) birth function, # m-4 s-1

C ) concentration, mol m-3

d ) death function, # m-4 s-1

dp ) crystal length, mG ) growth rate, m s-1

J ) nucleation rate , # m-3 s-1

k ) Boltzmann constant ) 1.83 × 10-23 J K-1

kg ) growth rate coefficient, m s-1

kv ) shape factorM ) mass, kgM ) number of space steps, #n ) number of particles per volume per size class, # m-4

N ) number of time steps, #S ) supersaturation ratioT ) temperature, Kt ) time, s

V ) volume, m3

x ) length, m

Greek Letters

Φ ) flow rate, m3 s-1

Fv ) crystal density, kg m-3

γ ) interfacial free energy, J m-2

ν ) molecular volume, m3

Subscripts

av ) averageAMPI ) ampicillin0 ) initialS ) saturated

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Received for review February 7, 2001Revised manuscript received July 10, 2001

Accepted July 18, 2001

IE0101238

Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001 4827