Modeling the Gas Antisolvent Recrystallization Process - Industrial

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Ind. Eng. Chem. Res. 2002, 41, 3566-3579

MATERIALS AND INTERFACES Modeling the Gas Antisolvent Recrystallization Process Gerhard Muhrer, Cheng Lin, and Marco Mazzotti* ETH Swiss Federal Institute of Technology Zurich, Institute of Process Engineering, Sonneggstrasse 3, CH-8092 Zurich, Switzerland

A mathematical model describing the gas antisolvent recrystallization process has been developed. In this process, precipitation of a solid from solution is triggered through the addition of a dense gas antisolvent under near-critical conditions. The possibility of controlling the particle size distribution of the produced microparticles makes it rather attractive for pharmaceutical and biotechnology related applications. Using the precipitation of phenantrene from toluene due to carbon dioxide addition as a model system, a parametric analysis of the system has been carried out. Significantly different effects have been observed experimentally, going from a major dependence of the average particle size on the antisolvent addition rate to no dependence at all. This work confirms such behavior and clarifies that this is controlled by the relative weight of secondary nucleation with respect to primary nucleation. Moreover, it provides useful hints on how to develop and optimize the gas antisolvent recrystallization process for a new application. 1. Introduction Particle formation techniques based upon the use of supercritical fluids have been given a lot of attention in the scientific literature over the past decade. The main reason for this is an increasing demand for particles with rather narrow particle size distributions in the micrometer and even nanometer ranges. Particles for applications in this size range include dyes, polymers, explosives, salts, superconductor and catalyst precursors, and pharmaceuticals, including proteins. Important aspects in the latter case are controlled drug delivery and direct drug delivery to the lung by inhalation. Processes utilizing supercritical fluids overcome most of the commonly known drawbacks of conventional particle size reduction techniques by taking advantage of the unique physicochemical properties of supercritical fluids. These include the tunability of system characteristics such as density or solvent power by variation of pressure (or temperature) and favorable properties such as relatively low viscosity coupled with rather high diffusivities when compared to liquids. Additionally and most importantly, microparticles with controlled particle size distribution and product quality may be obtained under mild and inert conditions because of the use of environmentally benign carbon dioxide as a supercritical fluid solvent or antisolvent. Processes utilizing supercritical fluids for particle production include rapid expansion of supercritical solutions (RESS), precipitation with compressed antisolvent (PCA), and gas antisolvent recrystallization (GAS). Numerous practical applications of these processes to the formation of microand nanoparticles of various materials have been reported and recently reviewed.1-5 Contrary to RESS, where binary mixtures of carbon dioxide and solubilized * To whom correspondence should be addressed. Phone: +41-1-6322456. Fax: +41-1-6321141. E-mail: mazzotti@ ivuk.mavt.ethz.ch.

solute are expanded across capillary nozzles from relatively high pressures, PCA and GAS exploit the very low solubility of many materials of practical interest in supercritical carbon dioxide. Carbon dioxide is used as an antisolvent for the solute initially solubilized in a conventional solvent, which is in turn completely miscible with CO2. Upon addition of CO2 to the system, a reduction of the solvent power of the mixed solvents (conventional solvent and carbon dioxide) occurs and the solute precipitates. In the GAS process, that we are considering in this work, this is accomplished by gradually adding compressed carbon dioxide to the initial solution. Despite the fact that many studies have reported about the applicability of these techniques to several different systems, only recently a few investigations have addressed the issue of how to control the final product quality through variations of the main operating parameters. Our group has investigated the effects of temperature, antisolvent addition rate, and initial solution volume on the final particulate product in the GAS recrystallization of a pharmaceutical intermediate precipitated from different organic solvents using carbon dioxide as an antisolvent.6 It was reproducibly shown that the average particle size of the final product decreased with increasing specific carbon dioxide addition rate (CO2 flow rate per unit volume of the initial solution) and decreasing temperature, which is in good agreement with findings by other researchers.7,8 The addition rate of the antisolvent was found to have the strongest impact on the mean particle size of the solid product; mean particle sizes could be reproducibly adjusted to values in a range 2 orders of magnitude wide. It is worth noting that, at intermediate addition rates, distinct bimodal particle size distributions (PSDs) were obtained, whereas relatively low or high rates yielded unimodal PSDs, as illustrated in Figure 1. Additionally, the choice of organic solvent was shown to have a decisive influence on morphology and crystal-

10.1021/ie020070+ CCC: $22.00 © 2002 American Chemical Society Published on Web 06/26/2002

Ind. Eng. Chem. Res., Vol. 41, No. 15, 2002 3567

Figure 1. Experimental results for the precipitation of an organic pharmaceutical compound from ethanol.6 Effect of relative carbon dioxide addition rate on average particle size at different operating temperatures: (3) 5 °C, (b) 25 °C, (0) 50 °C. Relative CO2 addition rate is defined as the ratio of the volumetric addition rate and the volume of initial solution loaded into the precipitation vessel. Bimodal distributions are represented by two circles, connected by a vertical dashed line. For the sake of clarity and to guide the eye, dashed lines connect the experiments at minimum and maximum addition rate at each temperature.

linity of the final product, which is again in good agreement with other studies.3 This analysis, together with reported results about protein precipitation,9,10 motivates further work in the direction of checking its generality. Among the numerous literature studies about antisolvent precipitation techniques, only a small number have addressed the important issue of process modeling.5 Most of these theoretical studies have dealt with the thermodynamics of the systems involved, whereas only a very few have covered mass transfer related issues. Three-phase equilibria in binary and ternary systems have been thoroughly analyzed for applications in all of the particle formation techniques utilizing supercritical fluids.11,12 The Peng-Robinson equation of state (PR-EOS) with conventional quadratic mixing rules was used, and the fugacity of the solute in the solid phase was calculated by means of a subcooled liquid reference state.13-15 Solid-liquid-vapor equilibrium calculations in the systems toluene-phenanthrenecarbon dioxide and toluene-naphthalene-carbon dioxide are in good agreement with experimental data.16 These findings have been extended to quaternary systems for separation and fractionation purposes.17 A large collection of experimental data and PR-EOS modeling of volume expansion in binary systems solventantisolvent is also available.18,19 Modeling of mass transfer phenomena in a supercritical antisolvent-based precipitation processes has mainly focused on the semicontinuous PCA spray process. A nonequilibrium thermodynamic model for the PCA process including jet hydrodynamics and mass transfer has been developed.20 This makes it possible to calculate the composition and flow rate profiles of the vapor and liquid phases along the precipitator provided that a thermodynamic model suitable to represent phase equilibria for the system of interest is available. The effects of process parameters are studied and thermodynamic criteria for selective precipitation from multicomponent mixtures are established. However, this model does not include particle formation kinetics. More recently, the mass transfer

between an isolated single droplet of organic solvent and a bulk compressed antisolvent, accounting for composition dependence of the diffusion coefficients, nonideality in the liquid-phase density due to the addition of an antisolvent, and corresponding changes in the droplet radius has been studied.21,22 It is apparent that a mathematical description of the process as a whole (i.e., including particle formation kinetics) is not yet available. A detailed model of GAS recrystallization should account for both physicochemical properties and transport phenomena. The former include near-critical thermodynamics of ternary systems and crystallization kinetics, including primary and secondary nucleation, and crystal growth. The latter involve mass transport during mixing of the compressed gas with the initial solution and migration of solute molecules from the bulk of the solution to the growing solid particles, as well as fluid mechanics aspects influencing mixing and aggregation and breakage of particles during growth. Although the development of such a detailed model constitutes our ultimate goal, we believe that, as a first step, a simplified model should be considered where only the essential features of the process are accounted for. It is our belief that filling the gap between experimental evidence and theoretical understanding through modeling and simulation will significantly enhance the possibility of designing and interpreting experiments and of exploiting the potential of a supercritical fluid based particle formation processes. The objective of this work is to develop a simplified model of the gas antisolvent recrystallization process, which accounts for two key aspects: (1) liquid-vapor equilibrium of solvent and antisolvent and its effect on the solubility of the solute, and (2) particle formation and growth. Such a model allows for a prediction of the final particle size distribution of the product, which is most important in determining product quality. Meaningful simulations require realistic model parameters, and this is a critical issue for crystallization in general and for GAS recrystallization in particular. Our approach is that of using experimental data when available and predictive methods when data are missing. Thus, in the following, we choose a model system whose thermodynamic behavior has been studied both theoretically and experimentally (i.e., the ternary system toluene-phenanthrene-carbon dioxide).11,16 An experimental study on the GAS precipitation of phenanthrene from toluene has been reported in the literature as well.23 In the case of particle formation kinetics under near- or supercritical conditions, little information is available in the literature, and predictive methods have been adopted. 2. Mathematical Model Let us consider the semibatch precipitation vessel with given overall volume, V, illustrated in Figure 2. It has one inlet, that is, the antisolvent (species A) feeding line, whereas the organic solution (constituted of solvent S and solute or product P) is loaded into the precipitator prior to antisolvent addition. Upon that, the volume of the liquid phase expands, eventually triggering solute precipitation. Antisolvent addition is completed when the volume of the liquid-phase reaches a prescribed final value; afterward, the system is allowed to achieve thermodynamic equilibrium. In our model, we assume that the gas and the liquid phases are always at the same pressure and that the temperature in the precipi-

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Ind. Eng. Chem. Res., Vol. 41, No. 15, 2002

mj )

∫0∞L jn dL

(6)

Mole fractions have to fulfill the stoichiometric relationships

xA + xS + xP ) 1

(7)

yA + yS ) 1

(8)

Liquid-vapor equilibrium of solvent and antisolvent is enforced through the isofugacity relationships

fA,L ) fA,V

(9)

fS,L ) fS,V

(10)

The total volume constraint can be cast as Figure 2. Schematic representation of the gas antisolvent recrystallization process.

tator is homogeneous in space and constant in time; hence, no energy balance is required. Both the liquid and gas phases in the reactor are well-mixed and mass transfer resistance is neglected, thus enabling instantaneous vapor-liquid equilibration upon antisolvent addition. The vapor phase is assumed to be free of solute as well as of any incondensable. The antisolvent addition rate is constant, and supersaturation is created in the liquid phase, where particle formation, through primary and secondary nucleation, and growth take place. Particle agglomeration and breakage are not accounted for. 2.1. Model Equations. The material balance equations for the precipitator can be written as follows:

d(NLxA + NVyA) ) QA dt

(1)

d(NLxS + NVyS) )0 dt

(2)

d(NLxP + NP) )0 dt

(3)

where all of the symbols are defined in the Notation section. The time-dependent population density function n(t, L) obeys the following population balance equation

∂n n d(NLvL) ∂n +G + )0 ∂t ∂L NLvL dt

(4)

where only growth and nucleation are accounted for, the latter through the boundary conditions given in the following discussion. The population density function is defined with reference to the volume of the liquid phase in such a way that n(t, L) dL gives at time t the number of crystals per unit volume of the solution in the size range between L and (L + dL). The amount of solute in the solid phase, NP, is given by

NLvLkvm3 NP ) vP

(5)

where m3 is the third moment of the density function, n, calculated according to the following general definition of the jth order moment of a distribution

V ) NLvL + NVvV

(11)

where the volume of the solid phase is neglected and the liquid and vapor molar volumes are calculated through proper equations of state

p ) p(vL, T, x)

(12)

p ) p(vV, T, y)

(13)

Solving the system of partial differential, ordinary differential, and algebraic equations 1-5 and 7-13 allows for the calculation of the 12 unknowns of the problem as a function of time (i.e., the molar hold-ups NL, NV, and NP; the mole fractions xA, xS, xP, yA, and yS; the molar volumes vL and vV; the pressure p; and the PSD n). Because the last unknown depends also on the particle size, L, through eq 4, the following boundary condition is given:

n(t, 0) )

B (t > 0) G

(14)

where B is the nucleation rate.24 Equation 14 implies the assumption that new particles are formed as nuclei with L ) 0. Details about the solution algorithm are given in the Appendix. 2.2. Initial Conditions. At the beginning of the experiment, t ) 0, the precipitator is loaded with a solution of solute P in the solvent S and is kept at atmospheric pressure by adding the necessary amount of antisolvent A. Thus, the initial conditions to be enforced at t ) 0 are as follows:

p ) patm

(15)

n(0, L) ) 0 (L g 0)

(16)

xSNL + ySNV ) NS0

(17)

xPNL ) NP0

(18)

where NS0 and NP0 are the initial molar amounts of solvent and solute, respectively. The initial values of the 10 remaining variables, namely, NL, NV, NP, xS, xP, xA, yS, yA, vL, and vV, are calculated by solving the system of eqs 5, 7-13, 17, and 18. 2.3. Constitutive Equations: Thermodynamics. The fugacity in the liquid and vapor phases is expressed


as

fi,R ) zi,Rφi,Rp (i ) A, S; R ) L, V)

(19)

The Peng-Robinson equation of state (PR-EOS) with quadratic mixing rules and one or two binary interaction parameters (kij, lij) is used for the computation of the fugacity coefficient φi,R according to the following equations:

p)

aR(T) RT (R ) L, V) (20) vR - bR vR(vR + bR) + bR(vR - bR) aR )

∑i ∑j zi,Rzj,Raij(T)

(21)

aij ) xai(T)aj(T)(1 - kij)

ln φk,R )

bR RT

aR(T)

bR )

∑i ∑j zi,Rzj,Rbij

(23)

bij )

bi + bj (1 - lij) 2

(24)

( ) [ ]

bk pvR

2

- 1 - ln

∑i zi,Raik

-

aR

2x2bRRT

(22)

p(vR - bR)

bk

bR

(

RT

ln

B)

vR + (1 + x2)bR vR + (1 - x2)bR

(k ) A, S; R ) L, V) (25) For given T, p and composition of phase R, zi,R, eq 20 yields the molar volume vR, and eq 25 gives φk,R explicitly. Equation 20 is used to obtain the right-hand side of eqs 12 and 13 with R ) L and R ) V, respectively. Because the solute is assumed to be nonvolatile (yP ) 0), all of the summations in these equations involve all three species in the liquid phase but only solvent and antisolvent in the vapor phase. Contrary to solvent and antisolvent, the solute partition between the liquid and solid phases is governed by the kinetics of particle formation and growth rather than by thermodynamics. The driving force for the former is supersaturation, defined as

S)

fP,L(x, p, T)

(26)

fP,P(p, T)

Here, the numerator is calculated using eq 19 at the composition, pressure, and temperature of the liquid phase in the system. The denominator is calculated by applying the Poynting correction to the fugacity of the solid at a reference pressure p0

fP,P )

fP,P0 exp

[∫

]

[

]

vP vP(p - p0) dp ) fP,P0 exp p0 RT RT

(27)

fP,P0 ) fP,L(x0, p0, T)

(28)

p

At a given temperature T, the three-phase system considered here has only one degree of freedom. Hence, at any given pressure (e.g., p or p0 as in eq 28), the composition of all phases can be calculated by solving the system of eqs 7-10, 12, 13, and 26, where S ) 1, and eqs 19-25 as well as 27 and 28 are used. It is worth noting that the results about the gas antisolvent process presented in the following sections are independent of the specific choice of the thermodynamic model. In this case, an EOS-based model is used because this works well for the model system selected for this work, as discussed in the following sections. However, such a choice does not affect the generality of our conclusions. 2.4. Constitutive Equations: Particle Formation and Growth. Contrary to the case of phase equilibrium, relationships for nucleation and growth kinetics of the systems considered in gas antisolvent recrystallization studies have not been established yet. Therefore, rather general rate equations will be used in the following discussion. As discussed next, model parameters will be chosen with the only constraint of being physically realistic. Particle formation is due to both primary nucleation, with rate B′, and to secondary nucleation, with rate B′′. Hence, the overall nucleation rate is defined as the sum of these two contributions

where

{

B′ + B′′ (S > 1) (S e 1) 0

(29)

Primary nucleation does not require the presence of crystals or particles of the solute. It can be homogeneous or heterogeneous, when triggered by particles of different nature (e.g., by dust) or by solid parts in the precipitator. Because our main goal is to analyze and explain the effect of process parameters on the final particle size distribution, classical nucleation theory has been adopted.25-27 Preliminary simulations have shown that the functional dependence of B′ on the solute concentration, cP ) xP/vL, and on the supersaturation, S, given by eq 26 is crucial in determining the process evolution. In the frame of classical nucleation theory, such dependence is the same for homogeneous and heterogeneous primary nucleation rates, the difference between the two being limited to the coefficients in the preexponential term and in the exponent of the rate expression.25 Therefore, in this context, where nucleation rate parameters are only physically reasonable values, we consider only homogeneous primary nucleation, thus using the following relationship for B′:

B′ ) 1.5D(cPNA)7/3

x

γ vP × kT NA

[

exp -

( ) ( ) (ln1S) ] (30)

16π γ 3 vP 3 kT NA

2

2

It is worth noting that particles are assumed to be spherical and that the interfacial tension γ is a function of temperature. Such a dependence is quite debated and plays a minor role in the isothermal process studied here; hence, it is omitted. It is worth noting that eq 30 and the theory behind it contain no assumptions about the crystal structure of

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Table 1. List of Model Parameters parameter

value

description

10-4 m3/mol

1.512 × 0.09 0.12 0 0 3 × 10-4 Pa‚s 3.46 × 10-9 m2/s 1.75 × 10-2 J/m2 3.75 × 10-6 m/s 2 10-14/10-24

vP k12 k23 k13 lij η D γ kg g R′′

molar volume of phenanthrene crystals interaction parameter CO2-toluene (PR-EOS, ref 31) interaction parameter CO2-phenanthrene (PR-EOS, ref 32) interaction parameter toluene-phenanthrene (PR-EOS) interaction parameters (PR-EOS) dynamic viscosity solute diffusion coefficient solid-liquid interfacial tension parameter in the growth rate model exponent in the growth rate model coefficient in the secondary nucleation rate model

the formed particles. As a matter of fact, it applies also to the nucleation of amorphous particles. Secondary nucleation occurs in the presence of particles of the solute. If it follows a surface mechanism, its rate B′′ is proportional to the surface area of the particles that are already present, and the functional dependence on supersaturation is significantly different from that of B′ 25

B′′ )

R′′avD dM4

[ ( ) ]

γdM2 2 1 exp -π kT ln S

as in eq 1). The final pressure, where the antisolvent addition is terminated, is 58.5 bar. For the sake of simplicity, but without the loss of generality, particles are assumed to have spherical shape (i.e., kv ) π/6 and ka ) π). The physical properties of solid phenanthrene used in the model are summarized in Table 1. For a pure fluid, the constant b in the PR-EOS is given by

(31)

b ) 0.077 80

RTc pc

(34)

The specific surface area, av, is accounted for through the second moment of the particle size distribution, m2, defined in eq 6

where the subscript c denotes critical values. On the other hand, a(T) is given by

av ) kam2

a(T) ) a(Tc)R(T)

(32)

All parameters in eqs 30 and 31 are physical properties except for R′′. This is an empirical coefficient accounting for the fraction of fragments formed by secondary nucleation events that evolve to nuclei and then particles. It can be shown that, according to eqs 30 and 31, B′′ grows faster than B′ at low supersaturation values but slower at large supersaturation levels. Several relationships have been proposed and used to describe particle growth. We have verified that the functional form of the growth rate is not critical in determining the model behavior; hence, the following general empirical correlation has been adopted:

G)

{

kg(S - 1)g (S > 1) (S e 1) 0

(33)

3. Model System and Model Parameters The objective of this work is to analyze the effect of the antisolvent addition rate on the particle size distribution of the particles formed. As a model system, we have selected the precipitation of phenanthrene from toluene using carbon dioxide as an antisolvent. For this system, detailed thermodynamic information is available, as discussed in this section, and educated guesses of some of the particle formation and growth rate parameters can be made. Process temperature and volume of the precipitator have been kept constant at T ) 298 K and V ) 4 × 10-3 m3 in all of the simulations. Typical initial amounts of solvent and solute are 35 and 15 g, respectively, unless otherwise indicated; this corresponds to an initial supersaturation level according to eq 26 of S0 ) 0.9. The antisolvent addition rate, QA, has been varied over more than 2 orders of magnitude, between 1 and 200 g/min (note that, for convenience, in the following discussion, we will use mass flow rates instead of molar flow rates

(35)

with

a(Tc) ) 0.457 24

(RTc)2 pc

(36)

and

R(T) ) [1 + β(1 - xT/Tc)]2

(37)

β is the following function of the Pitzer acentric factor, ω:

β ) 0.374 64 + 1.542 26ω - 0.269 92ω2

(38)

The interaction parameters used in eqs 22 and 24 are taken from the literature and summarized in Table 1, along with the corresponding references. The accuracy of the thermodynamic model used is illustrated in Figure 3. Here, the experimental values of the liquidphase mole fractions of carbon dioxide and phenanthrene at T ) 298 K are plotted as a function of pressure under conditions where the three phases coexist at equilibrium.16 The curves in the same figure are obtained by solving the thermodynamic model equations discussed in Section 2.3; the agreement between experimental data and model results is rather satisfactory. As already stated, the values of the parameters in the particle formation and growth rate models are chosen so as to be physically realistic. The diffusion coefficient of the solute, D, appearing in eqs 30 and 31 is assumed to be constant and estimated through the StokesEinstein relation

D)

kT 2πηdM

(39)


Figure 3. System toluene-phenanthrene-CO2: composition of the liquid phase at equilibrium with the gas phase and with solid phenantrene at T ) 298 K, as a function of pressure.16 The symbols are the experimental data, whereas the solid lines are the model results.

In eq 39, the dynamic viscosity is assumed to be constant because calculated and measured dynamic viscosities for mixtures of toluene and carbon dioxide neglecting the presence of the solute exhibit only a slight decrease with increasing xA.28 The molecular diameter dM entering in eqs 31 and 39 is estimated as26 3

dM )

x

vP NA

(40)

The interfacial tension of the growing crystals is assumed to be constant and given in Table 1, along with the parameters used in the growth model. The empirical parameter, R′′, accounting for the effectiveness of secondary nucleation is varied in a rather large range, as discussed in the next section. 4. Tuning the Final Particle Size Distribution 4.1. Experimental Evidence and Simulation Results. Several experimental investigations of the gas antisolvent recrystallization process have highlighted the importance of the antisolvent addition rate in determining the particle size distribution of the final product. This possibility was already discussed and exploited in the work where this process was proposed for the first time.7 On the basis of a qualitative physical argument, it was shown that increasing addition rate would lead to smaller particles and possibly also to multimodal particle size distributions. In a later contribution, experimental observations on two different systems confirmed this earlier analysis.8 On one hand, the carbon dioxide-induced gas antisolvent recrystallization of L-asparagine from ethanol-water mixtures at two different carbon dioxide addition rates yielded several times smaller average particle size at the larger rate; the particle size distribution was unimodal in both cases. On the other hand, the recrystallization of ascorbic acid from ethanol yielded a bimodal particle size distribution at the larger carbon dioxide addition rate. The more recent results illustrated in Figure 1 were consistent with the previous works and have confirmed the original physical argument through a

rather systematic experimental analysis.6 It was shown that bimodal particle size distributions can be reproducibly attained and that the key parameter is the specific addition rate (i.e., the ratio between the addition rate and the initial volume of the solution). Rather different results have been obtained in the case of the precipitation of different proteins. In all of the reported experimental investigations, the effect of the carbon dioxide addition rate on the average particle size was minor.9,10,29 Finally, it is worth noting that the results reported on the recrystallization of phenantrene from toluene (i.e., our model system) cannot be compared with the others since a rather complex pressurization policy has been adopted in that case.23 Which property of the investigated materials and systems is responsible for such different patterns of behavior, whereby, in the case, of proteins the average particle size varies of only a few percent when changing the antisolvent addition rate and, in the case of the organic pharmaceutical illustrated in Figure 1, the average particle size can be tuned in a range between 300 nm and 8 µm? An analysis based on model simulations is particularly suited to answer this question. Starting from the set of parameters used to characterize the model system toluene-phenanthrene-carbon dioxide and presented in the previous section, we have carried out a parametric analysis where the physical parameters have been significantly varied with respect to the base values reported in Table 1 and the addition rates have been changed between 1 and 200 g/min (i.e., in a physically realistic range when considering experimental results). This has shown that the one and only parameter controlling the qualitative effect of the antisolvent addition rate on the particle size distribution of the final product is the empirical secondary nucleation rate parameter R′′. As mentioned earlier, R′′ is a measure of the effectiveness of the physical events leading to secondary nucleation and ultimately determines which of the two phenomena between primary and secondary nucleation is predominant. It is worth noting that this is the only parameter in the relationships describing particle formation and growth mechanisms that cannot be estimated based on physical properties of the system considered. In particular, three different patterns of behavior corresponding to three different ranges of R′′ values have been identified, as will be discussed in detail in the following section. With the only exception of the parameter R′′, the same set of parameters, namely, those in Table 1, has been used in all of the simulations reported in section 4, and the same range of values of the antisolvent addition rate has been explored. In discussing the simulation results, we will analyze the particle size distribution of the formed particles, namely, both its whole shape and its statistical properties. These are defined next in terms of the moments of the distribution given by eq 6: the mean of the distribution, L h , providing the average particle size

L h ) m1/m0

(41)

the coefficient of variation, cv, accounting for the size variation of the particles relative to the mean size

cv ) xm0m2/m12 - 1

(42)

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Figure 4. Lower range of R′′ values. Average particle size (mean of the PSD) of precipitated phenanthrene particles as a function of the carbon dioxide addition rate. For the three antisolvent addition rate values indicated, the number density distributions of the particle populations are drawn in the insets.

the skewness, γ1, which is a measure of the symmetry of the distribution

γ1 )

m02m3 - 3m0m1m2 + 2m13 (m0m2 - m12)3/2

(43)

For the sake of convenience, another measure of the average particle size will be adopted, namely, the mode of the distribution (i.e., the particle size where n exhibits its maximum). 4.2. Lower Range of r′′ Values. Figure 4 illustrates the typical behavior of the lower range of R′′ values, particularly where R′′ ) 10-24. The main diagram in this figure reports the average particle size as a function of the antisolvent addition rate, in a log-log plot. As average particle size, the mode of the population density function n is adopted. It can readily be observed that this is a straight line and that the average particle size decreases from about 2 mm at the lowest addition rate of 1 g/min to less than 30 µm at the highest addition rate of 200 g/min. The particle size distribution of the product is unimodal at all addition rates and rather narrow. This can be seen in the smaller diagrams, where the population density function, n, of the produced particles is plotted for three particular values of the

antisolvent addition rate, namely, QA ) 3, 15, and 70 g/min. For these three simulations, Table 2 reports the statistics of the particle size distribution. It is worth noting that the skewness of these particle populations ranges between 0.87 and 0.91; this indicates a moderate right tailing of the distribution. The observed behavior can be given a conceptual explanation by considering that, during volume expansion, there are three phenomena, which are competing. Two of them, namely, particle nucleation (either primary or secondary) and particle growth, consume supersaturation, whereas on the contrary the third phenomenon (i.e., antisolvent addition) creates supersaturation. At the start of the process, when particles are not yet present, only primary nucleation and antisolvent addition compete. The larger the addition rate, the higher the supersaturation level where nucleation is triggered and, hence, the larger the nucleation rate itself and the number of nuclei and particles formed. This is confirmed by Figure 5, where the nucleation rate is drawn as a function of run time for the same three simulations considered in Figure 4. As expected, at the addition rate of 70 g/min, the nucleation rate is more than 4 orders of magnitude larger than at the 3 g/min addition rate. Moreover, the larger the addition rate, the shorter the time when nucleation occurs. After nucleation has

Ind. Eng. Chem. Res., Vol. 41, No. 15, 2002 3573 Table 2. Model Parameters, Moments, and Statistics of the PSD, for the Particle Populations, Whose PSDs Are Shown in Figures 4, 6, and 8 R′′ lower range

10-24

intermediate range

10-19

upper range

10-14

QA [g/min]

m0 [1/m3]

m1 [m/m3]

m2 [m2/m3]

m3 [m3/m3]

L h [µm]

cv

γ1

3 15 70 1 5 7 20 100 3 15 70

6.92 × 107 3.84 × 109 1.68 × 1011 1.57 × 108 4.66 × 108 8.12 × 108 8.11 × 109 3.85 × 1011 1.06 × 1013 1.36 × 1013 2.73 × 1013

6.26 × 104 9.02 × 105 1.12 × 107 9.15 × 104 2.04 × 105 3.05 × 105 1.48 × 106 1.94 × 107 1.53 × 108 1.79 × 108 2.81 × 108

5.86 × 101 2.20 × 102 7.82 × 102 6.50 × 101 1.02 × 102 1.27 × 102 2.82 × 102 1.01 × 103 2.64 × 103 2.83 × 103 3.61 × 103

5.69 × 10-2 5.58 × 10-2 5.66 × 10-2 5.63 × 10-2 5.75 × 10-2 5.72 × 10-2 5.62 × 10-2 5.54 × 10-2 5.48 × 10-2 5.46 × 10-2 5.74 × 10-2

906 235 67 582 438 376 183 50 14.4 13.1 10.3

0.18 0.19 0.20 0.47 0.39 0.33 0.22 0.18 0.44 0.46 0.49

0.91 0.87 0.87 1.48 0.17 -0.12 0.34 0.85 1.51 1.65 1.59

occurred, the particles formed can grow, thus consuming supersaturation and at the same time building up the surface area necessary to make secondary nucleation possible. However, under the conditions of the simulations illustrated in Figures 4 and 5 (i.e., in the lower range of R′′ values), the rate of secondary nucleation remains negligible during the whole process (cf. Figure 5), and no new particles are formed. Thus, the whole particle population is produced during the single nucleation event illustrated in Figure 5, and the final particle size distribution is unimodal and narrow as shown in Figure 4, with a cv of about 0.2 (cf. Table 2). With reference to the particle size distributions reported in the latter figure, let us consider also the number of particles obtained with a size corresponding to the mode of the distribution. It can be seen that, as expected, such numbers for the three addition rates considered are in the same proportion as the maximum calculated nucleation rate (cf. Figure 5). The inverse of the cube of the values of the mode of the three distributions scale similarly, because the overall amount of solute loaded in the precipitator, as well as the overall volume of the solid product at the end of the process, are the same. 4.3. Upper Range of r′′ Values. Figure 6 illustrates the typical behavior within the upper range of R′′ values, which is exemplified by R′′ ) 10-14. There is a significant difference with respect to the lower range case in that the average particle size exhibits a rather small varia-

Figure 5. Dynamics of the gas antisolvent recrystallization process in the lower range of R′′ values. Nucleation rates as a function of the run time for the three simulations whose corresponding final PSD is illustrated in Figure 4. (Solid line) primary nucleation rate; (dashed line) secondary nucleation rate. In these simulations, B′′ is smaller than the lower bound of the vertical scale; hence, it is not shown.

tion in the range of antisolvent addition rates considered. In particular, the average particle size decreases only from about 10 µm at 1 g/min addition rate to about 6 µm at QA ) 200 g/min (i.e., a decrease of less than 50% as compared to an almost 2 orders of magnitude change in Figure 4). Also, in the upper range, the particle size distribution of the product is unimodal at all addition rates, even though it exhibits a longer tail toward the large particle sizes. This can be observed in the smaller diagrams in Figure 6, where the population density function of the formed particles is plotted for three particular values of the antisolvent addition rate, namely, QA ) 3, 15, and 70 g/min. This is quantified in Table 2, where the statistics of these three particle populations are reported, and it can be observed that their skewness is significantly larger than in the lower range, namely, between 1.51 and 1.65. Figure 7 shows the nucleation rate as a function of the run time for the same three addition rate values. It can be readily observed that the behavior is just the opposite of that illustrated in Figure 5 for the lower range. In fact, in this case, secondary nucleation reaches much higher rates than primary nucleation and determines the final particle size distribution. In fact, the primary nucleation event forms enough particles and enough surface area to trigger secondary nucleation, whose rate under these conditions is so large that the secondary nucleation event forms much more particles than the first one. Because the secondary nucleation rate is less sensitive to the antisolvent addition rate than the primary nucleation rate, so are the number of particles formed and their average particle size. The final particle size distribution exhibits a tail because two distinct nucleation events take place; in fact, the tail is constituted of particles formed during the primary nucleation burst that have more time to grow than those formed afterward. It is worth noting that, in comparing Figures 4 and 6, attention is to be paid to the trends, particularly to how average particle size changes as a function of antisolvent addition rate, rather than to the absolute figures (e.g., the absolute value of the average particle size). In fact, in our analysis, we use always the same expression for the primary nucleation rate, with the same parameters, whereas we change the weight given to secondary nucleation. This implies that when increasing R′′ the overall nucleation rate, B, given by eq 29 increases as well and the average particle size decreases. This is not physically significant because it is a consequence of how the parameteric analysis is carried out. On the contrary, what matters here is the comparison among situations where the weights of

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Figure 6. Upper range of R′′ values. Average particle size (mean of the PSD) of precipitated phenanthrene particles as a function of the carbon dioxide addition rate. For the three antisolvent addition rate values indicated, the number density distributions of the particle populations are drawn in the insets.

Figure 7. Dynamics of the gas antisolvent recrystallization process in the upper range of R′′ values. Nucleation rates as a function of the run time for the three simulations whose corresponding final PSD is illustrated in Figure 6. (Solid line) primary nucleation rate; (dashed line) secondary nucleation rate.

primary and secondary nucleation are different. In fact, this makes indeed a significant difference in determining the effect of antisolvent addition rate on the final particle size distribution. 4.4. Intermediate Range of r′′ Values. The behavior exhibited by systems where the value of R′′ is in the intermediate range, namely, R′′ ) 10-19 in the following

simulations, is completely different. The key difference is illustrated in the diagrams of Figure 8, where the population density function, n, of the formed particles is plotted for five different antisolvent addition rates (i.e., 1, 5, 7, 20, and 100 g/min). For intermediate values of the addition rate, a bimodal particle size distribution is obtained, whereas this is unimodal at very low and


Figure 8. Intermediate range of R′′ values. Average particle size (mean of the PSD in the case of unimodal distributions, and height of the peak maxima in the case of bimodal distributions) of precipitated phenanthrene particles as a function of the carbon dioxide addition rate. For the five antisolvent addition rate values indicated, the number density distributions of the particle populations are drawn in the insets.

very large addition rates. In the main diagram of Figure 8, the particle size corresponding to the maxima of the distribution n are reported as a function of the antisolvent addition rate. This corresponds to the mode of the distribution when this is unimodal, whereas two values are plotted in the case of bimodal distributions. Likewise, in the lower range of R′′ values, the average size of the particles decreases sharply with increasing addition rate, but contrary to that case, there is a range of addition rate values where bimodal distributions occur. The mechanism through which bimodal particle size distributions are obtained is clear when the time evolution of nucleation rates, primary and secondary, is analyzed. These are illustrated in Figure 9 in a loglog plot for the five simulations at different addition rates already highlighted in Figure 8. For the sake of clarity, the same diagram, but in linear scale, is drawn in Figure 10 for the case where QA ) 5 g/min. Moreover, the maximum value of primary, B′, and secondary, B′′, nucleation rates and the corresponding times where these are achieved (i.e., t′ and t′′) are reported in Table 3. As already observed in discussing the lower and the upper ranges, the primary nucleation rate is more sensitive to the antisolvent addition rate than the secondary nucleation rate. Thus, at a very high addition rate (e.g., where QA ) 100 g/min), primary nucleation

is much faster than secondary nucleation, whose contribution to the final unimodal particle size distribution is negligible; this is the same behavior observed in the lower range (cf. Figures 4 and 5). The opposite situation occurs at a very low addition rate (e.g., where QA ) 1 g/min). Under these conditions, primary nucleation has a minor effect, and the behavior is similar to that observed in the upper range. These analogies are further confirmed by the skewness of the two distributions reported in Table 2. This is 1.48 at QA ) 1 g/min and 0.85 at QA ) 100 g/min (i.e., values in the same range as those observed in the upper and in the lower ranges, respectively). At intermediate values of the antisolvent addition rate (e.g., where QA ) 5 g/min) (cf. Figure 10), the maximum primary and secondary nucleation rates attain the same order of magnitude, as illustrated in Figure 9 and reported in Table 3. Therefore, the two nucleation events produce a comparable number of particles. Because the two families of particles (i.e., those formed during the first and the second burst of nucleation) are originated at distinct times, these grow for a longer and for a shorter time, respectively (i.e., to a larger and to a smaller size, respectively). As a consequence, the final particle size distribution is bimodal, as illustrated in Figure 8. It is also worth noting, in the same figure, that at larger addition rates the dependence of the average

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Figure 9. Dynamics of the gas antisolvent recrystallization process in the intermediate range of R′′ values. Nucleation rates as a function of the run time for the five simulations whose corresponding final PSD is illustrated in Figure 8. (Solid line) primary nucleation rate; (dashed line) secondary nucleation rate.

Figure 10. Dynamics of the gas antisolvent recrystallization process in the intermediate range of R′′ values. Nucleation rates (linear scale) as a function of the run time for the simulation in Figures 8 and 9, where QA ) 5 g/min. Table 3. Time Where the Primary and Secondary Nucleation Rates Attain Their Highest Value, Respectively, and Corresponding Maximum, for the Five Simulations Runs Illustrated in Figures 8 and 9 QA [g/min]

t′ [s]

Bmax′ [#/m3s]

t′′ [s]

Bmax′′ [#/m3s]

1 5 7 20 100

7297 1490 1071 380 78

9.3 × 104 4.4 × 107 1.4 × 108 4.8 × 109 1.0 × 1012

7347 1508 1082 385 79

9.1 × 106 3.0 × 107 4.3 × 107 1.3 × 108 6.9 × 108

particle size on QA is significant, as in the lower range case, whereas it is much less pronounced at low addition rates (i.e., similarly to what is observed in the upper range). This is nicely consistent with the explanation provided previously about the transition from unimodal, to bimodal, to unimodal particle size distributions again, when increasing the value of the antisolvent addition rate. 4.5. Discussion and Conclusions The analysis presented in the previous section has led to the identification of three different patterns of behavior in terms of the effect of the antisolvent addition

rate on the characteristics of the final particle size distribution. These correspond to three ranges of values of the secondary nucleation rate parameter R′′. When secondary nucleation is slow (i.e., in the lower range of R′′ values), the final particle size distribution is unimodal and narrow, and the average particle size can be tuned over orders of magnitude when changing the antisolvent addition rate in a realistic range of values (cf. Figure 4). When secondary nucleation is fast with respect to primary nucleation (i.e., in the upper range), the final particle size distribution is again unimodal, but the average particle size is much less sensitive to variations of the antisolvent addition rate, as illustrated in Figure 6. Finally, in the intermediate range of R′′ values, bimodal final particle size distributions are obtained for intermediate values of the addition rate (see Figure 8). The analysis of the process dynamics in the three different cases has allowed a mechanistic explanation of these different patterns of behavior to be given. When comparing these simulation results with the experimental findings, striking similarities can be recognized. On one hand, the recrystallization of Lasparagine from ethanol-water mixtures using carbon dioxide as an antisolvent8 follows a pattern of behavior similar to that exhibited by systems in the lower range of R′′ values. On the other hand, the recrystallization of ascorbic acid from ethanol yielding a bimodal particle size distribution8 resembles what is observed previously for systems in the intermediate range. This is even more convincingly the case for the gas antisolvent recrystallization of an organic pharmaceutical from ethanol that is illustrated in Figure 1.6 In fact, both the experimental data in Figure 1 and the simulation results in Figure 8 span a range of addition rates broad enough to include cases featuring both the bimodal distributions at intermediate addition rates and the unimodal distributions at larger and smaller addition rate values. Finally, the recrystallization of proteins, particularly of lisozyme,9,10,29 where a negligible effect of addition rate on the average particle size of the final precipitate, follows the same pattern of behavior of systems in the upper range of R′′ values. These theoretical findings, together with their qualitative agreement with the experimental results, demonstrate the importance of the antisolvent addition rate


Figure 11. Effect of the initial concentration of the solute on the average particle size of the formed particles, at constant overall amount of solute and antisolvent addition rate. All the other parameters are the same as those in the simulations in the lower range of R′′ values.

Thus summarizing, this study provides an explanation for different experimental observations in the gas antisolvent recrystallization of a variety of compounds. Interestingly enough, it has been shown that substantially different patterns of behavior can be explained by invoking differences just in the relative weight of the primary and of the secondary nucleation rates. Unfortunately, no criterion that allows for the a priori prediction of the behavior of a new system has been provided. However, the findings discussed previously make two things of practical importance possible. On the one hand, the operating conditions of a few experiments can be selected easily to check what pattern of behavior is followed by the system under examination. On the other hand, once the system is classified within one of the three categories discussed previously, a rational approach toward the identification of the optimal operating conditions can be followed. Of course, the model developed in this work can be a rather useful tool to design and optimize gas antisolvent recrystallization processes once all of the physical phenomena involved have been quantitatively characterized. Appendix

Figure 12. Effect of the initial amount of solute loaded in the precipitator on the average particle size of the formed particles, at constant initial concentration and antisolvent addition rate. All the other parameters are the same as those in the simulations in the lower range of R′′ values.

to control the final particle size distribution and to tune it in accordance with the product specifications. When possible (i.e., in the case of systems following the lower and intermediate range of R′′ values), the average particle size can be tuned over up to 2 orders of magnitudes by selecting the antisolvent addition rate within a physically realistic range. It is worth noting that other operating parameters that are as easy to vary as the addition rate do not allow the achievement of the same objective. To show this, the effect of changing the initial solute concentration (i.e., the initial saturation level S0) at constant overall amount of solute and that of changing the overall amount of solute at constant S0 are illustrated in Figures 11 and 12, respectively; in both cases, a system having the same set of parameters as that considered in the simulations reported in Figure 4 and for a constant value of the addition rate (i.e., QA ) 20 g/min) has been considered. It can be readily observed that both the initial saturation level and the overall amount of solute do not allow for a tuning of the average particle size but in a rather small range of values as compared to what can be achieved by changing the antisolvent addition rate, as shown in Figure 4.

To solve the population balance equations, we use the software package Parsival, which uses a Galerkin h,pmethod based on a generalized finite-element scheme with a self-adaptive grid.30 Because implementing the complex phase equilibrium equations in Parsival is not straightforward, we have splitted the computation in two stages. Stage 1. First, the volume expansion process is simulated, under some simplifying assumptions about the behavior of the solute. The equations to be solved are eqs 1, 2, 7-10, and 11-13. Equations 15 and 17 are used as initial conditions. Because there are 10 unknowns, namely, NL, NV, p, vL, vV, yA, yS, xA, xS, and xP, another specification is necessary. There are two possibilities corresponding to two limit cases. In the first case, we assume that the solute has no influence on the volume expansion process; hence, we set xP ) 0 and solve the nine equations listed. In the second limit case, we assume that the solute precipitates instantaneously, reaching saturation of the liquid phase under the current conditions. This can be enforced by using eq 26, with S ) 1, as the 10th equation in this system. In either case, the simulation yields the time evolution of all of the unknowns involved during volume expansion. Of particular interest are the molar volume of the liquid phase, vL, and the molar fluxes of antisolvent, FA, and of solvent, FS, which are exchanged between the vapor and the liquid (these are defined as positive when the net flux is from the vapor to the liquid phase). While FA is positive and significant, because it represents the effect of the addition of antisolvent to the precipitator, the quantity FS is typically negative and very small and can in practice be neglected. We have carried out the volume expansion simulation, in the two limit cases mentioned, under different operating conditions and always verified that the values of vL and FA differ by no more than a few percent. Therefore, we conclude that either result can be used to describe the expansion process also in the presence of particle formation. Stage 2. Parsival is used to solve the equations describing the liquid and solid phase in the precipitator, where the volume of the liquid phase changes as calculated in Stage 1. The equations to be solved are

3578


the following material balances for solvent, antisolvent, and solute:

d(NLxA) ) FA(t) dt

(44)

d(NLxS) ) FS(t) ≈ 0 dt

(45)

d(NLxP + NP) )0 dt

(46)

together with eqs 4, 5, 7, and 12. The input parameters are the molar volume vL(t) and the molar fluxes FA(t) and FS(t). The initial conditions are given by eqs 1518, and the boundary condition for the population balance equation is eq 14. The seven equations allow the seven unknowns NL, NP, p, n, xS, xA, and xP to be calculated. The supersaturation S needed in the equations for B and G is calculated through eqs 26-28, where the expression for fP,L(x, p, T) contains the fugacity coefficient φP,L that is easily calculated through eq 25, using the current values of vL, p, and liquid-phase composition. It can be observed that following this approach the pressure evolution during the precipitation process is calculated twice (i.e., during each stage of the solution algorithm). We have verified that these two profiles differ by only a few percent (i.e., the approximation introduced by decoupling volume expansion and particle formation is acceptable). Notation a ) parameter in the PR-EOS, [J‚m3/mol2] av ) specific surface area of the particles in the dispersion, [m2/m3] b ) parameter in the PR-EOS, [m3/mol] B ) nucleation rate, [#/(m3s)] ci ) molar concentration of component i in the liquid phase, ci ) xi/vL, [mol/m3] cv ) coefficient of variation of the PSD dM ) molecular diameter, [m] D ) diffusion coefficient of the solute in the liquid phase, [m/s] fi,R ) fugacity of component i in the R phase, [N/m2] Fi ) molar flow rate of component i from the vapor to the liquid phase, [mol/s] g ) exponent in the growth model G ) growth rate, [m/s] k ) Boltzmann’s constant, k ) 1.38 × 10-23 J/K, [J/K] ka ) surface shape factor kg ) rate constant in the growth model, [m/s] kij ) interaction parameter in the PR-EOS kv ) volume shape factor lij ) interaction parameter in the PR-EOS L ) particle characteristic length, [m] L h ) mean of the PSD, [m] mj ) jth order moment of the PSD, [mj/m3] n ) population density function, [m-4] NA ) Avogadro’s number, NA ) 6.022 × 1023 mol-1, [mol-1] NR ) molar hold-up in the R phase, [mol] p ) pressure, [N/m2] Qi ) molar flow rate of component i, [mol/s] R ) universal gas constant, R ) 8.3144 J/(mol‚K), [J/(mol‚ K)] S ) supersaturation t ) time, [s] T ) temperature, [K] vR ) molar volume of the R phase, [m3/mol]

V ) volume of the precipitator, [m3] xi ) mole fraction of component i in the liquid phase yi ) mole fraction of component i in the vapor phase zi,R ) mole fraction of component i in the R phase Greek Letters R ) parameter in the PR-EOS, R ) R(T) R′′ ) secondary nucleation rate parameter β ) parameter in the PR-EOS γ ) interfacial tension, [J/m2] γ1 ) skewness of the PSD η ) dynamic viscosity of the liquid phase, [Pa‚s] φi,R ) fugacity coefficient of component i in the R phase ω ) Pitzer acentric factor Superscripts and Subscripts ′ ) primary nucleation ′′ ) secondary nucleation 0 ) reference or initial state atm ) atmospheric pressure A ) antisolvent c ) critical parameter i, j, k ) running parameters L ) liquid phase P ) product, and product (solid) phase S ) solvent V ) vapor phase R ) phase R

Literature Cited (1) Subramaniam, B.; Rajewski, R.; Snavely, K. Pharmaceutical processing with supercritical carbon dioxide. J. Pharm. Sci. 1997, 86, 885-890. (2) Bungert, B.; Sadowski, G.; Arlt, W. Separations and materials processing in solutions with dense gases. Ind. Eng. Chem. Res. 1998, 37, 3208-3220. (3) Reverchon, E. Supercritical antisolvent precipitation of micro- and nanoparticles. J. Supercrit. Fluids 1999, 15, 1-21. (4) Jung, J.; Perrut, M. Particle design using supercritical fluids: Literature and patent survey. J. Supercrit. Fluids 2001, 20, 179-219. (5) Thiering, R.; Dehghani, F.; Foster, N. R. Current issues relating to anti-solvent micronization techniques and their extension to industrial scales. J. Supercrit. Fluids 2001, 21, 159-177. (6) Mu¨ller, M.; Meier, U.; Kessler, A.; Mazzotti, M. Experimental study of the effect of process parameters in the recrystallization of an organic compound using compressed carbon dioxide as antisolvent. Ind. Eng. Chem. Res. 2000, 39, 2260-2268. (7) Gallagher, P. M.; Krukonis, V. J.; Botsaris, G. D. Gas antisolvent (GAS) recrystallization: Application and particle design. In Particle Design via Crystallization; Ramanarayanan, R., Kern, W., Larson, M., Sidkar, S., Eds.; AIChE Symposium Series 284; American Institute of Chemical Engineers: New York, 1991; Vol. 87, pp 96-103. (8) Tschernjaew, J.; Berger, T.; Weber, A.; Ku¨mmel, R. Herstellung von Mikropartikeln durch Kristallisation mittels hochkomprimierter Gase. Chem.-Ing.-Tech. 1997, 69, 670-674. (9) Yeo, S.-D.; Lim, G.-B.; Debenedetti, P. G.; Bernstein, H. Formation of microparticulate protein powders using a supercritical fluid antisolvent. Biotechnol. Bioeng. 1993, 41, 341-346. (10) Thiering, R.; Dehghani, F.; Dillow, A.; Foster, N. R. The influence of operating conditions on the dense gas precipitation of model proteins. J. Chem. Technol. Biotechnol. 2000, 75, 2941. (11) Kikic, I.; Lora, M.; Bertucco, A. A thermodynamic analysis of three-phase equilibria in binary and ternary systems for applications in rapid expansion of a supercritical solution (RESS), particles from gas-saturated solutions (PGSS), and supercritical antisolvent crystallization (SAS). Ind. Eng. Chem. Res. 1997, 36, 5507-5515. (12) Kikic, I.; Lora, M.; Bertucco, A. A thermodynamic analysis of three-phase equilibria in binary and ternary systems for applications in rapid expansion of a supercritical solution (RESS), particles from gas-saturated solutions (PGSS), and supercritical

Ind. Eng. Chem. Res., Vol. 41, No. 15, 2002 3579 antisolvent crystallization (SAS), correction. Ind. Eng. Chem. Res. 1998, 37, 1577. (13) Peng, D. Y.; Robinson, D. B. A new two constant equation of state. Ind. Eng. Chem. Fundam. 1976, 15, 59-64. (14) De Swaan Arons, J.; Diepen, G. A. M. Thermodynamic study of melting equilibria under pressure of a supercritical gas. Recl. Trav. Chim. Pays-Bas 1963, 82, 249-256. (15) Prausnitz, J. M.; Lichtenthaler, R. N.; Gomes de Azevedo, E. Molecular thermodynamics of fluid-phase equilibria; Prentice Hall: Upper Saddle River, NJ, 1999. (16) Dixon, D. J.; Johnston, K. P. Molecular thermodynamics of solubilities in gas antisolvent crystallization. AIChE J. 1991, 37, 1441-1449. (17) Bertucco, A.; Lora, M.; Kikic, I. Fractional crystallization by gas antisolvent technique: Theory and experiments. AIChE J. 1998, 44, 2149-2158. (18) Kordikowski, A.; Schenk, A. P.; Van Nielen, R. M.; Peters, C. J. Volume expansion and vapor-liquid equilibria of binary mixtures of a variety of polar solvents and certain near-critical solvents. J. Supercrit. Fluids 1995, 8, 205-216. (19) De la Fuente Badilla, J. C.; Peters, C. J.; De Swaan Arons, J. Volume expansion in relation to the gas-antisolvent process. J. Supercrit. Fluids 2000, 17, 13-23. (20) Lora, M.; Bertucco, A.; Kikic, I. Simulation of the semicontinuous supercritical antisolvent process. Ind. Eng. Chem. Res. 2000, 39, 1487-1496. (21) Werling, J. O.; Debenedetti, P. G. Numerical modeling of mass transfer in the supercritical antisolvent process. J. Supercrit. Fluids 1999, 16, 167-181. (22) Werling, J. O.; Debenedetti, P. G. Numerical modeling of mass transfer in the supercritical antisolvent process: miscible conditions. J. Supercrit. Fluids 2000, 18, 11-24. (23) Berends, E. M.; Bruinsma, O. S. L.; de Grauw, J.; van

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Received for review January 23, 2002 Revised manuscript received May 15, 2002 Accepted May 21, 2002 IE020070+

Modeling the Gas Antisolvent Recrystallization Process - Industrial

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