Numerical modeling of jet hydrodynamics, mass transfer, and crystallization kinetics in the supercritical antisolvent (SAS) process

Numerical modeling of jet hydrodynamics, mass transfer, and crystallization kinetics in the supercritical antisolvent (SAS) process

J. of Supercritical Fluids 32 (2004) 203–219 Numerical modeling of jet hydrodynamics, mass transfer, and crystallization kinetics in the supercritica...

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J. of Supercritical Fluids 32 (2004) 203–219

Numerical modeling of jet hydrodynamics, mass transfer, and crystallization kinetics in the supercritical antisolvent (SAS) process A. Mart´ın, M.J. Cocero∗ Departamento de Ingenier´ıa Qu´ımica, Facultad de Ciencias, Universidad de Valladolid, 47011 Valladolid, Spain Received in revised form 30 January 2004; accepted 1 February 2004

Abstract A mathematical model for the supercritical antisolvent (SAS) process is presented and solved numerically. This model takes the main physical phenomena involved in this process into account, including jet hydrodynamics, mass transfer, phase equilibrium, as well as nucleation and crystal growth kinetics. The model allows to calculate the particle size distribution and the yield of the precipitation. The main innovation of this model concerns jet hydrodynamics, which is considered as the mixing of two completely miscible fluids forming a gaseous plume, and is modeled with a k–ε turbulence model. The model has been used to analyze the mechanism of particle formation in the SAS process, and to study the effects of the operating parameters on particle size and solid recovery. The comparison with experimental results shows good agreement in the trends. Particle size cannot be predicted accurately due to the lack of knowledge of some parameters of the model. © 2004 Elsevier B.V. All rights reserved. Keywords: Simulation; Micronization; Turbulence; ␤-Carotene; Ascorbic acid

1. Introduction Several micronization technologies take advantage of the physical properties of supercritical fluids. Among them, the gas or supercritical antisolvent (GAS or SAS) process and its variants, has received a considerable interest due to the wide range of materials that can be micronized with this technique [1,2], including explosives [3], polymers [4], superconductor precursors [5], pigments [6], pharmaceuticals [7,8], and natural compounds [9]. Most of the publications on the micronization of different materials with the SAS process are concerned with the experimental analysis of the effects of operating conditions on the size distribution and morphology of the particles. This type of analysis is often difficult, because of the interactions between operating parameters. A theoretical analysis of the process can be very useful for the interpretation of experimental results and the design of new experiments.

∗ Corresponding author. Tel.: +34-983-423-174; fax: +34-983-422-013. E-mail address: [email protected] (M.J. Cocero).

0896-8446/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.supflu.2004.02.009

In order to obtain a complete theoretical description of the SAS process, it is necessary to model all the physical phenomena that interact in this process. This includes the phase equilibrium, mass transfer, the fluid mechanics of the mixing between the organic solution and the supercritical antisolvent, as well as kinetics of particle nucleation and growth. A number of papers attempt to model each of these phenomena separately. However, the modeling of the process as a whole has not been achieved yet. The phase equilibrium in the SAS process is well known from a theoretical point of view, but there is a serious lack of experimental data for the multicomponent systems of interest for this process. Cubic equations of state have been used frequently to model the phase behavior in the SAS process [10]. For example, Shariati and Peters [11] used the Stryjek–Vera modification of the Peng–Robinson equation of state (PRSV) to model the phase behavior of the ternary system CO2 + 1-propanol + salicylic acid. More recently, Kikic et al. [12] applied the perturbed hard-sphere-chain (PHSC) equation of state [13] to describe the phase behavior in the SAS process. Those authors found that the PHSC EOS has significant advantages over the PR EOS when the critical properties of the solute are not known or are not well

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Nomenclature a b Cη , Cε1 , Cε2 d50 dP DAB g G H I j J k kB kij lij ms Mk n∗ N NA Ntot P Pk r r∗ R S Sc T v vs V V∗ w x Y z

coefficient in Peng–Robinson equation of state (Pa m6 mol−2 ) coefficient in Peng–Robinson equation of state (m3 mol−1 ) parameters in the k–ε turbulence model mean diameter in the particle size distribution function particle diameter (m) diffusivity (m2 s−1 ) gravity (m s−2 ) single molecule condensation rate in a particle (s−1 ) enthalpy (J mol−1 ) nucleation mass rate, Jn∗ MW3 /NA (kg m−3 s−1 ) mass flux density (kg m−2 s−1 ) nucleation number rate (m-3 s−1 ) turbulent kinetic energy Boltzmann constant (J K−1 ) interaction parameter in Peng–Robinson equation of state interaction parameter in Peng–Robinson equation of state mass of a molecule of solute (kg) kth moment of the particle size distribution (m3k m−3 ) number of molecules in the critical nucleus particle size distribution function Avogadro number total concentration of particles in particle size distribution (m−3 ) pressure (Pa) turbulent kinetic energy production radius and radial coordinate (m) radius of the critical nucleus (m) ideal gases constant (J mol−1 K−1 ) supersaturation Schmidt number temperature (K) velocity (m s−1 ); molar volume (m3 mol−1 ) volume of a molecule of monomer (m3 ) volume (m3 ) volume of the critical nucleus (m3 ) mass fraction mole fraction yield of the precipitation (wt.%) axial coordinate (m)

Greek letters α parameter in Peng–Robinson equation of state β coagulation coefficient (m3 s−1 ) ε turbulent kinetic energy dissipation rate η dynamic viscosity (kg m−1 s−1 ) κ parameter in Stryjek–Vera modification of PR EOS ν cinematic viscosity (m2 s−1 ) ρ density (kg m−3 ) σ interfacial tension (N m−1 ) σ standard deviation in particle size distribution σk, σε parameters in the k– turbulence model ϕ fugacity coefficient ω mass fraction; acentric factor Subscripts 1 2 3 c C f FM j r t z

component 1 (antisolvent) component 2 (solvent) component 3 (solute) critical property continuum regime fusion free molecular regime node index, radial direction radial direction turbulent axial direction

Superscripts L liquid M molecular n node index, axial direction S solid T turbulent

defined (e.g., polymers). Elvassore et al. [14] developed a group contribution method for estimating the parameters of the PHSC EOS. Werling and Debenedetti studied the mass transfer between a droplet of organic solvent and a compressed antisolvent, both at subcritical [15] and supercritical [16] conditions. This model considers the two-way mass transfer both into the droplet and into the antisolvent. However, the droplet is considered to be stagnant. Therefore, the only convective motion considered is that induced by the diffusion. In the last years, the hydrodynamics of the SAS process have been the subject of several papers. Most authors tackle this problem considering that the jet of organic solvent behaves like a liquid jet injected into a gas. This supposition allows to apply the classic theory of jet break-up. Some examples of this approach are the papers of Czerwonatis

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and Eggers [17], or of Kerst et al. [18], where the break-up length of the jet is studied and correlated as a function of the Reynolds and Weber numbers. Lengsfeld et al. [19] measured and calculated the jet break-up length. Their calculations included the variation of the surface tension with liquid and gas compositions. They concluded that the classic jet break-up theory could be applied successfully at subcritical conditions. However, at miscible conditions (above the critical point of the mixture organic solvent + CO2 ), the surface tension decreases to zero in a shorter distance than characteristic break-up lengths. Therefore, distinct droplets are never formed at supercritical conditions, and the jet spreads forming a gaseous plume. This theoretical result was confirmed experimentally by photographing the jet at different ambient pressures. Kerst et al. [18] and Chehroudi et al. [20] have published photographs of jets at different pressures, which show the same behavior: at subcritical conditions, the jet spreads forming droplets, but under miscible conditions, it forms a gaseous plume. Debenedetti [21] gave an expression for the homogeneous nucleation rate in supercritical fluids, which takes fluid-phase non-ideality into account. With this expression, attainable nucleation rates in the rapid expansion of supercritical solutions (RESS) process could be calculated. Subsequent publications [22] included particle growth by coagulation and condensation, applying the theory for particle growth in aerosol reactors developed by Pratsinis [23]. Helfgen et al. [24,25] modeled the RESS of benzoic acid, cholesterol and griseofluvin. Their model included particle growth by condensation and coagulation, and they solved the aerosol dynamic equation with the method of moments under the assumption of a continuous log-normal size distribution. They found that the main mechanism for particle growth was coagulation. Few publications address the particle nucleation and growth in the SAS process. Bristow et al. [26] studied the nucleation in the antisolvent precipitation of an ethanol solution of acetaminophen. These authors performed the precipitation under conditions of partial and complete miscibility, and measured the supersaturation of the solution and the effluent with an on-line UV detector. The parameters of the expression for the homogeneous nucleation rate were determined by linear regression of experimental data. The first attempt to model all the physical phenomena involved in the SAS process was presented by Lora et al. [27]. With this model, it was possible to calculate the crystallization yield, but not the particle size distribution. Later, Elvassore et al. [28] developed a model based on the mass transfer simulations of Werling and Debenedetti [15,16]. This model included the solute in mass transfer calculations. The droplet was still considered to be stagnant, and the diffusion flux in the ternary system solute–solvent–antisolvent was calculated with the generalized Maxwell–Stefan relations.

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The objective of this paper is to combine the description of all the physical phenomena relevant for the SAS process into a single model. This model pays special attention to the hydrodynamics of the process, which, according to experimental evidence, are modeled as a mixing process of completely miscible fluids under turbulent conditions. Mass transfer calculations also include turbulent diffusivities. It is considered that particles are formed by homogeneous nucleation, and grow by coagulation and condensation. With this model, it is possible to calculate the final particle size distribution and the yield of the precipitation.

2. Development of the model The model presented here is constituted by the conservation laws of mass, momentum, as well as turbulent kinetic energy and dissipation rate. All these conservation laws are written in cylindrical coordinates. Variations over the angular direction are neglected. Only steady-state calculations have been performed. It is assumed that temperature is constant in the precipitator; therefore, thermal effects have not been considered. This model can be applied in conditions of complete miscibility between organic solvent and CO2 (i.e., above the mixture critical point). In the following paragraphs, the equations of the model are described. First, the thermodynamics of the ternary system solute–solvent–antisolvent are addressed. Then, the continuity equation and momentum conservation laws that represent the hydrodynamics of the process are described. In this section, the modeling of turbulence is also included. Mass transfer under turbulent conditions is discussed. Finally, the expression for the homogeneous nucleation rate, and the general aerosol dynamic equation (GDE) that models particle growth will be presented. In Section 3, all these equations are put together, and the numerical methods used for solving them are described. 2.1. Thermodynamics An equation of state is needed to calculate phase equilibrium and fluid densities. The solutes considered in this work are pharmaceuticals, with well-defined chemical structure and critical properties. Therefore, the Peng–Robinson equation of state as modified by Stryjek and Vera with quadratic mixing rules has been chosen, because this equation has been used successfully to describe the thermodynamics of such systems [11]: P=

RT a α(T) − v − b v(v + b) + b(v − b)

where





α(T) = 1 + κ 1 −



T Tc

(1)

0.5 2 (2)

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Stryjek and Vera modified the temperature dependence of the α term as follows:   0.5    T T κ = κ0 + κ1 1 + 0.7 − (3) Tc Tc with κ0 = 0.378893 + 1.4897153ω − 0.17131848ω2 + 0.0196554ω3

(4)

Since the PRSV equation of state is not able to represent the behavior of solid phases, the fugacity of the solid is obtained from the fugacity of a reference sub-cooled liquid with Eq. (5). It is possible to simplify this equation by neglecting the first addendum of the exponential term, because the difference between solid and liquid molar volumes is usually very small. For details, one is referred to elsewhere [10,29]:    P νS − νL 1 Hf 1 10 10 S L ϕ3 = ϕ3 exp − dP + (5) RT R Tf T P0

In turbulent flow, the actual velocity can be regarded as the sum of a mean value plus the fluctuations. In a system with variable density, it is convenient to use the Favre, or mass weighted, decomposition [32]. Introducing this decomposition in Eqs. (6) and (7), and taking a time average over a large number of fluctuations, the so-called time-smoothed equations of motion are obtained. The motion equation cannot be solved unless a model is provided that ties the Reynolds stress tensor in the time-smoothed equations of motion to the mean velocity in a physically consistent fashion. In this work, the Launder–Sharma model has been used [33]. This is a two-equation k–ε model, which consists of two semi-empirical equations for the turbulent kinetic energy k, and the turbulent kinetic energy dissipation rate ε:  

1 ∂ ηt ∂ ¯ ∂ ¯ ¯ ¯ε {ρ¯ kv¯ z } + r ρ¯ kv¯ r − η + k = P¯ k − ρ¯ ∂z r ∂r σk ∂r (8)  

∂ 1 ∂ ηt ∂ {ρ¯ ¯ εv¯ z } + r ρ¯ ¯ εv¯ r − η + ε¯ ∂z r ∂r σε ∂r ε¯ ρ¯ ¯ ε2 = Cε1 P¯ k − Cε2 k¯ k¯

2.2. Hydrodynamics

(9)

As discussed in Section 1, it has been observed experimentally that under conditions of complete miscibility between CO2 and organic solution (that is, above the critical point of the mixture CO2 –solution), the solution injected into the precipitator does not form droplets and behaves as a jet [18–20]. Therefore, hydrodynamics have been modeled considering this flow pattern, and applying standard computational fluid dynamics (CFD) techniques. The continuity equation in cylindrical coordinates for a system with variable density in steady state reads:

where the production term is given by:      2 ∂¯vz 2 ∂¯vr 2 v¯ r P¯ k = 2ηt + + ∂z ∂r r  2 ∂¯vz ∂¯vr + ηt + ∂r ∂z

∂ 1 ∂ (ρvz ) + (rρvr ) = 0 ∂z r ∂r

σk = 1,

(6)

The equation of motion must take into account that density and viscosity are variable. This equation can be greatly simplified with the boundary-layer approximation [30,31]. This approximation can be applied to those systems in which the convective motion occurs mainly in one single direction. This is the case in the jet flow, in which the axial velocity is some orders of magnitude higher than the radial velocity. Under these conditions, it is possible to neglect the second derivatives in the direction of flow, against the second derivatives normal to the direction of flow. The entire equation of motion in radial direction can also be neglected. Thus, the equation of motion in axial direction is:       ∂vz ∂vz 1 ∂ ∂vz ρ vr + vz =− r η + ρg (7) ∂r ∂z r ∂r ∂r Eqs. (6) and (7), along with the contour conditions of symmetry around the axial axis, and zero velocity at the walls, can be used to calculate the axial and radial velocities in the precipitator.

(10)

And the parameters of the model are: Cη = 0.09,

Cε1 = 1.44, σε = 1.3

Cε2 = 1.92, (11)

With this model, the turbulent or eddy viscosity can be calculated with the following equation: ηt =

Cη ρk2 ε

(12)

The boundary conditions for Eqs. (8) and (9) state that k and ε must be symmetrical around the axial axis. 2.3. Mass transfer Mass transfer calculations consider the effect of turbulence. Turbulent diffusivity has been calculated from turbulent cinematic viscosity using the Reynolds analogy, an empiricism that has been verified experimentally [30]: ScT =

νT ≈1 DT

(13)

Turbulent diffusivity is a property of the flow, not of the substance, and it has the same value for all the chemical species in the solution. Eq. (14) is the equation of continuity

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for a chemical species a. This equation has been simplified by neglecting the diffusive flux in the axial direction, against the convective flux in this direction:   ∂ωa ∂ωa 1 ∂ (14) ρ vr + vz =− (rjar ) + ra r ∂r ∂r ∂z Since the system under study is multicomponent, the diffusive flux jar should be calculated with the generalized Maxwell–Stefan equations. However, since the turbulent diffusivity DT is several orders of magnitude higher than the molecular diffusivity DM , all the binary diffusivities DAB will have almost the same numerical value (D12 ∼ = D13 ∼ = D23 ∼ = DT ). Therefore, it is possible to simplify the Maxwell–Stefan relations to the well-known Fick law for binary mixtures [30]: jar = −ρDT

∂ωa ∂r

(15)

r∗ = 2



σvs kB T



207

1 ln S − Kxe3 (S − 1)

(20)

Strictly speaking, Eqs. (17)–(20) are valid only for dilute, binary solutions, but they have been used successfully to fit experimental nucleation rates in the SAS process [26]. The interface energy σ is dependent upon the orientation of the surface planes relative to the solid’s internal stresses. In this work, as in most studies of homogeneous nucleation of solids, σ has been considered a parameter, without modifying the form of the equations. The condensation rates in the continuum regime GC (particle size mean free path of the gas) and in the free molecular regime GFM (particle size mean free path of the gas), are given by Eqs. (21) and (22), respectively [23,25]. To get an expression for the entire regime, the harmonic mean of GC and GFM is used (Eq. (23)). The effect of particle curvature on condensation rate (Kelvin effect) has been neglected:

Boundary conditions state the symmetry around the axial axis. The continuity equation for a chemical species must be solved for the solute and the organic solvent. For the organic solvent, ra = 0, whereas for the solute, ra is the sum of nucleation and condensation rates:   ∞ MW3 Jn∗ + G(V) N(V) dV (16) ra = NA V∗

GC (V) = (48πV)1/3 D(Ntot − Neq )

Equations for nucleation and condensation rates are presented in Section 2.4.

2.5. Particle growth

2.4. Nucleation and condensation kinetics

The general dynamic equation for simultaneous particle nucleation, condensation and coagulation is [23,25]:

It is assumed that the particles are formed by primary nucleation. The number of critical nuclei formed per unit time in a unit of volume can be calculated with the following rate expression, which is based on the classical nucleation theory [21]: 

σv2s

0.5

Px3 (2πms kB T)0.5 kB T   3  2  1 16π σvs 2/3 × exp − 3 kB T ln S − Kxe3 (S − 1)

J = 2Ntot

(17) In this equation, the term Kxe3 (S − 1) takes into account the fluid-phase non-idealities. The factor K can be calculated approximately with the following equation [22]: K=

ϕ(T, P, 0) 1 ln x3 ϕ(T, P, x3 )

(18)

Assuming the nucleus to be spherical, its critical size is given by:  2/3 3  3 1 σvs 32π ∗ n = (19) 3 kB T ln S − Kxe3 (S − 1)

 GFM (V) = (36π)

1/3

kB T 2πMW3 /NA

(21) 0.5

× V 2/3 (Ntot − Neq )

(22)

1 1 1 = + G(V) GC (V) GFM (V)

(23)

∇ · (ρuN) = J(V ∗ ) δ(V − V ∗ )    1 + 2 

nucleation V



β(V − V¯ , V) N(V − V¯ ) N(V¯ ) dV¯  

0

coagulation

 − N(V) 

∞ 0

∂(Gn) β(V, V¯ ) N(V¯ ) dV¯ −  ∂V    

coagulation

condensation

(24) The GDE can be solved with the method of moments assuming that the particle size distribution follows a log-normal size distribution (Eq. (25)) [23–25]. This assumption restricts the form of the solution; however, experimental results indicate that the particle size distribution during coagulation fits a log-normal function [34]. And even if the experimental particle size distribution is not truly log-normal, it is customary to present experimental results with equivalent log-normal parameters:     nP 1 ln(V/V50 ) 2 1 N(V) = exp − 18 ln σ V 3(2π)1/2 ln σ (25)

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The results of applying the method of moments assuming a log-normal size distribution function, are three differential equations for the zeroth, first and second moment. Expressions for these equations were given by Pratsinis [23]. The three parameters of the particle size distribution function (number concentration of the particles, mean particle diameter, and standard deviation), can be calculated from the three moments, as described by Pratsinis [23].

which may result in a different behavior during the crystallization; and because both systems have been studied experimentally [35,36]. The physical properties used in this work are shown in Table 1. The parameters needed for the calculations are phase equilibrium data, and solid–fluid interfacial tension. Phase equilibrium data can be found in the literature, but usually the interfacial tension is not known. Typical values of solid–fluid interfacial tensions are around 0.01 N m−1 . Values used in this work are 0.005 N m−1 for ␤-carotene, and 0.03 N m−1 for ascorbic acid. A smaller value has been used for ␤-carotene, because the solubility of ␤-carotene in CO2 and dichloromethane is very low, and this usually means that the interfacial tension should be small. Values used in this work are only rough estimations, and it should be noted that interfacial tension varies with fluid composition, and therefore, with the concentration of organic solvent in the fluid. With these values of the interfacial tensions, typical values of the pre-exponential factor in the nucleation rate equation (Eq. (17)) are 10.8 for ␤-carotene and 74.4 in the case of ascorbic acid, which are similar to the values reported by Bristow et al. [26] for the crystallization of acetaminophen. Since small variations in σ can lead to large variations in the predicted particle size, the calculations shown here are descriptive rather than predictive. For example, if σ is varied from 0.005 to 0.006 N m−1 with the operating conditions shown in case A in Table 2, the mean particle size varies from 1.4 to 3.5 ␮m. The reason is that with higher interfacial tensions, nucleation is slower, and therefore particles become bigger. A similar or even larger effect of the interfacial tension on the predicted particle size has been reported for the RESS process [22]. However, σ does not affect the trends of change in particle size with the operating parameters, which are the main subject of the present work. The operating conditions used for the calculations are summarized in Table 2. These are the same conditions as used in experimental work [35,36]. Case A is the base case. In cases B–N, the following operating parameters are varied around the base case conditions, one at a time: concentration of the initial solution, temperature, pressure, solution flow rate, and CO2 flow rate. Case O shows the optimized operating conditions. In cases P–R, different configurations of the nozzle are studied. Cases S and T are the simulations

3. Numerical calculation The model is constituted by a system of coupled partial differential equations (PDEs). Those equations have been solved numerically with the DuFort–Frankel finite difference method [31]. With this method, the system of PDEs is transformed into a system of algebraic equations, which can be solved sequentially at each height of the precipitator. In order to obtain the solution at a certain height n + 1, the DuFort–Frankel method uses the solutions at two previous heights, n and n − 1. Therefore, a different numerical method must be used to calculate the solution at n = 2. A simple explicit finite difference method has been used for this purpose. A computational grid with variable spacing between points have been used, in order to concentrate more points near r = 0 and z = 0, where more accuracy in the calculations is required. The size of the grid depends on the particular problem to be solved; a typical size is 950 points in the radial direction (5 mm), and 1500 points in the axial direction (50 mm). Mass balances were checked at the end of each simulation. Deviations were always less than 2%. Boundary conditions for the velocities at z = 0 (exit of the nozzle) are the velocity profiles for laminar flow inside a tube. For concentrations, a sigmoidal profile has been used at z = 0 instead of the more rigorous step profile, in order to avoid numerical problems.

4. Results The model has been applied to study the micronization of two solutes: ␤-carotene dissolved in dichloromethane and ascorbic acid dissolved in ethanol. These substrates have been chosen because they have different chemical structures, Table 1 Physical properties of materials in this study

CO2 Dichloromethane Ethanol ␤-Carotene Ascorbic acid a b c

MW (g mol−1 )

Tc (K)

Pc (bar)

ω

κ1

Hf (J mol−1 )

Tf (K)

vs (m3 mol−1 )

44.010 84.993 46.069 536.87 176.13

304.10 510.0 513.92 1028.4a 790.91b

73.8 60.8 61.4 11.88 44.19b

0.239 0.199 0.644 0.926a 1.57b

0.04285 0.0746 −0.03374 0.015c 0.0014c

– – – 21052c 11500c

– – – 633c 464c

– – – 5.37 × 10−4 9.59 × 10−5

Estimated with the Cholakov method and fitted to experimental phase equilibrium data [42]. Estimated with the Constantinou–Gani group contribution method [38]. Fitted to experimental phase equilibrium data.

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Table 2 Summary of calculations Case

Substance

T (K)

P (MPa)

C0 (g l−1 )

Fsolution (kg h−1 )

FCO2 (kg h−1 )

d50 (␮m)

Y (wt.%)

A B C D E F G H I J K L M N O Pa Qc Rd S T

␤-Carotene ␤-Carotene ␤-Carotene ␤-Carotene ␤-Carotene ␤-Carotene ␤-Carotene ␤-Carotene ␤-Carotene ␤-Carotene ␤-Carotene ␤-Carotene ␤-Carotene ␤-Carotene ␤-Carotene ␤-Carotene ␤-Carotene ␤-Carotene Ascorbic acid Ascorbic acid

308 308 308 303 313 318 308 308 308 308 308 308 308 308 303 308 308 308 318 318

15 15 15 15 15 15 16 17 12 10 15 15 15 15 10 15 15 15 11.5 11.5

5 4.5 5.5 5 5 5 5 5 5 5 5 5 5 5 5.5 5 5 5 6.5 29

0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.2 0.5 0.3 0.3 0.3 0.3 0.3 0.3 0.436 0.436

4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 3 7.5 3.9 6 6 4.5 4.5 4.5 15 15

1.4 1.5 1.1 1.0 1.8 2.3 1.3 1.3 1.2 1.1 1.4 1.1 1.4 1.2 1.0 9.2 1.2 2.2 2.7 1.3

67 59 70 80 41 15 65 64 69 73 67 66 66 65 79 22b 67 67 88 94

a b c d

Nozzle without CO2 through the concentric tube. Precipitation not finished after 10 cm of precipitator. Nozzle diameter 0.1 mm. Nozzle diameter 0.5 mm.

with ascorbic acid. The range of temperatures considered in the calculations with ␤-carotene is narrow due to the thermal instability of this substance [37]. The mean diameter and yield obtained in each case are shown in Table 2. In this table, yield is defined as the ratio between the mass of precipitated substrate and the mass of substrate introduced in the precipitator.

In the calculations, the mixer is a coaxial nozzle, in which the solution flows through the inner tube, and CO2 flows through the coaxial annulus. This type of mixer is used frequently in real pilot plants. The dimensions of the nozzle considered in this work are: 0.2 mm nozzle diameter and 2 mm concentric tube diameter in the case of ␤-carotene, and 0.067 mm nozzle diameter and

Fig. 1. Solubility of ␤-carotene in supercritical CO2 , T = 313 K.

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Table 3 Binary interaction parameters System

k12

k13

k23

l12

l13

l23

CO2 (1) + dichloromethane (2) + ␤-carotene (3) CO2 (1) + ethanol (2) + ascorbic acid (3)

0.0646 0.066

0.1165 −0.074

−0.0234 0

0.0886 0.005

0.0588 0.15

0 0

2 mm concentric tube diameter in the case of ascorbic acid. 4.1. Phase equilibrium The systems of interest for the SAS process are ternary systems. Experimental data of the phase behavior of such systems is very scarce. However, it is possible to use measured data for the binary subsystems to fit the binary interaction parameters of the PRSV EOS, and then use this equation to predict the phase behavior of the ternary system. With this approach, it is expected to have relatively large deviations between calculated and experimental data in the region of low concentrations of organic solvent. With higher concentrations of organic solvent, calculated phase equilibrium should be more reliable. Several authors have studied the solubility of ␤-carotene in supercritical CO2 [38,40–42]. Fig. 1 presents the measured solubilities at 313 K. Solubilities calculated with the PRSV EOS are also presented in this figure. Reaves et al. [43] measured the critical points of CO2 –dichloromethane mixtures, and Jay and Steytler [44] reported that the solubility of ␤-carotene in dichloromethane at 20 ◦ C and 1 bar is 0.667% (w/w). With these data, it is possible to obtain the binary interaction parameters of all pairs in the system CO2 –dichloromethane–␤-carotene. Results are presented in Table 3.

Fig. 2 presents the calculated solubility of ␤-carotene at 308 K and 15 MPa in mixtures of dichloromethane and CO2 , as a function of the concentration of CO2 in the fluid phase. This figure also shows the variation of composition along the axial coordinate. These profiles have been calculated starting at different radial positions at the outlet of the nozzle, and following the stream lines from there. A remarkable characteristic of the composition profiles, is that at different radial positions in the jet, almost the same composition profile is followed. But the rate at which this profile is followed will be different for each radial position inside the jet. For example, the region of supersaturation will be reached later in the center of the jet. So at this time, some amount of particles produced in the borders of the jet will have diffused to the center of the jet, and therefore condensation rate will be larger than when the border reached the region of supersaturation. This explains that in the center of the jet, solute concentration decreases faster with CO2 concentration. The solubility of ascorbic acid in supercritical CO2 has been measured by Cortesi et al. [39]. The phase equilibrium of CO2 –ethanol mixtures have been studied by several authors, for example, by Chiehming et al. [45]. The solubility of ascorbic acid in ethanol has not been found in the literature; therefore, the interaction parameter of the pair ethanol-ascorbic acid has not been fitted. The binary interaction parameters of the system CO2 –ethanol–ascorbic acid are presented in Table 3.

Fig. 2. Solubility of ␤-carotene in mixtures of dichloromethane and CO2 as a function of the composition of the fluid and composition profiles, T = 308 K, P = 15 MPa.

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Fig. 3. Axial velocity. Operating conditions as in case A of Table 3.

4.2. Velocity and concentration profiles The main features of velocity and concentration profiles in the jet are described in this section. All the calculations described in this section have been performed with the operating conditions of case A in Table 2. Fig. 3 shows the axial component of the velocity in the jet. It can be seen that the jet slows down and widens along the axial coordinate, and draws in fluid from the surrounding mass of fluid. This can be seen better in the stream lines (Fig. 7). Since the inlet velocity of the jet is much higher than the inlet velocity of CO2 , the jet drags the CO2 that enters through the annulus. This hydrodynamic effect is the main mechanism of mixing between CO2 and organic solvent. Fig. 4 shows the concentration of organic solvent in the precipitator. The concentration profile follows closely the velocity profile, thus indicating that convection is the main mechanism of mass transfer. Fig. 5 shows the supersaturation. The highest supersaturations, and therefore the fastest particle formation, appear in the border of the jet, which is the region of contact of the solution with the CO2 . At a height of about 10 mm, CO2 reaches the center of the jet, and particle formation occur in the entire cross-section of the jet. Supersaturation decreases as the solute crystallizes, and at a height of 35 mm, its value is around 1, and therefore the crystallization is finished. This can also be seen in Fig. 8, which shows the variation of the

precipitation yield. It can be seen that after 35 mm of precipitator, the yield is almost constant. This height corresponds to a residence time of about 0.05 s. The highest value of supersaturation is 3.7, which is a relatively small value compared to the supersaturations of 10–15, which are reached in the RESS process [24]. Another important conclusion that can be extracted from Figs. 5 and 8, is that most of the precipitation takes place in the first 20 mm of precipitator. As can be seen in Fig. 5, in this region the mixing between the organic solution and CO2 is still incomplete. Therefore, the kinetics of the mixing affects particles formation, and precipitation occurs in an environment with large variations in composition. Fig. 9 depicts the nucleation and the condensation rate over the width of the precipitator at different heights. It can be seen that nucleation always occurs in the border of the jet, which is the region of highest supersaturation, as can be seen in Fig. 5. In the first 1–2 mm, nucleation and condensation are competitive processes, with comparable rates. After 5 mm of precipitator, nucleation is almost negligible. The reason can be found in Fig. 5: after this height, supersaturation decreases to values below 3. In the beginning of the precipitator, condensation also occurs in the borders of the jet, which is the region in which particles are being formed. However, after 3 mm of precipitator, condensation moves to the center of the jet. The condensation rate increases with supersaturation, but also with

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Fig. 4. Concentration of organic solvent. Operating conditions as in case A of Table 3.

Fig. 5. Supersaturation. Operating conditions as in case A of Table 3.

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Fig. 6. Mean particle diameter. Operating conditions as in case A of Table 3.

the concentration of solute in the fluid phase, as can be seen in Eqs. (21) and (22). Thus, when particles reach the canter of the jet dragged by turbulence, elevated condensation rates are found there, because there are high solute concentrations in the canter of the jet. In Fig. 9, it can be seen that condensation rate is rather high. In fact, condensation is the main mechanism of particle growth, while coagulation plays a minor role. In contrast, in the RESS process, coagu-

lation is the main mechanism of growth, while condensation is almost negligible [24,25]. The relatively small supersaturations found in the SAS process, and the relevance of condensation in this process, agree with experimental results, which show that relatively big, and usually crystalline particles are obtained with the SAS process, compared with the smaller, non-crystalline particles, which are obtained with the RESS process. Fig. 6 shows the mean particle diameter. It can be seen that the biggest particles are found in the border of the jet, because in this region particles are formed first and grow for a longer time, and also because condensation rate is higher in this region. 4.3. Effect of operating parameters

Fig. 7. Stream lines. Operating conditions as in case A of Table 3.

In this section, the effect of the operating parameters on the mean particle diameter and the precipitation yield is discussed. The operating parameters considered are the initial concentration of the solution, temperature, pressure, solution flow rate, and CO2 flow rate, which are varied as shown in cases B–N of Table 2. Results are summarized in Fig. 10. After this, the results of this section are used to propose an optimization of the operating parameters, with the objective of minimizing the mean particle diameter. An increase in the initial concentration of the solution has two opposed effects: on one hand, with higher concen-

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Fig. 8. Precipitation yield vs. length of precipitator. Operating conditions as in case A of Table 3.

trations, it is possible to achieve higher supersaturations, which tend to diminish the particle size; and on the other hand, condensation is directly proportional to the concentration of solute, and the increase of the condensation rate with higher concentrations tends to increase the particle size. As can be seen in Fig. 10, the first effect prevails under the operating conditions considered in this work, and smaller particles are obtained with higher concentrations. As far as the yield is concerned, it is determined by the amount of solute that remains dissolved in the fluid phase. Since this amount does not vary with the initial concentration, higher yields are obtained with higher concentrations, because in this conditions the amount dissolved is a smaller fraction of the total amount of solute. Temperature has a strong effect on the solubility of ␤-carotene at 15 MPa: the solubility increases with temperature in the entire range of organic solvent concentrations. Thus, with higher temperatures, supersaturation is reduced,

and bigger particles are formed. The variation of particle size with pressure is more complex: first the particle size increases with pressure, it goes through a maximum at a pressure of about 15 MPa, and then decreases slightly. The reason is shown in Fig. 11: with higher pressures, the solubility of ␤-carotene in CO2 is higher, but the solubility in the organic solvent is lower. Therefore, at higher pressures, the increase of the supersaturation in the solvent-rich region can balance the decrease in the CO2 -rich region. In Fig. 10, it can be seen that temperature and pressure, which are the parameters that influence phase equilibrium, are the parameters with the strongest influence in the mean particle size. The yield of the precipitation increases when the solubility of ␤-carotene in the final fluid phase is reduced, so higher yields are obtained with lower pressures and temperatures. If the solution flow rate is increased, the mixing between the jet and the surroundings is better due to the stronger turbulence. This leads to higher supersaturations in the jet,

Fig. 9. Nucleation and condensation rates over the radius of the precipitator at different lengths. Operating conditions as in case A of Table 3.

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Fig. 10. Effect of operating parameters on the mean particle diameter (d50 ). Data shown in cases A–N of Table 3.

as can be seen in Fig. 12, which shows the supersaturation inside the jet for two different flow rates of solution. Therefore, smaller particles are obtained with higher solution flow rates. However, the effect of this parameter is relatively small compared to that of the parameters which affect phase equilibrium. This agrees with experimental results for several substances. Solvent flow rate does not have a significant effect on yield. CO2 flow rate has a small influence on the mixing: with higher flow rates, turbulence increases, and higher supersaturations are reached in the border of the jet. Nevertheless, the main influence of CO2 flow rate on the process is that it

determines the composition of the bulk fluid, that results of the complete mixing between CO2 and solvent. If the flow rate of CO2 is increased, the bulk fluid has a smaller amount of organic solvent, the solubility of the solid in this fluid is smaller, and therefore smaller particles are obtained. The increase of CO2 flow rate may increase or decrease the yield, because it is possible that the decrease in solubility does not compensate the increase in the total amount of fluid that is introduced in the precipitator. With the results of this section, an optimized set of operating parameters has been proposed, as shown in case O of Table 2. The optimized pressure and temperature are

Fig. 11. Solubility of ␤-carotene on mixtures of dichloromethane and CO2 as a function of the composition of the fluid, T = 308 K, P = 10–17 MPa.

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Fig. 12. Supersaturation in the jet (r = 0.5Rnozzle ) for different solution flow rates. Operating conditions as in cases K and L of Table 3.

relatively low, with the objective of reducing the solubility of the solid, while staying in the region of complete miscibility between organic solvent and CO2 . Once reached the region of complete miscibility, a further increase in pressure does not have a significant effect on the effectiveness of mixing, and it decreases the yield of the crystallization. The results of the model advise to use high initial concentrations, and high CO2 flow rates. With the optimized parameters, the particle mean diameter has been reduced a 29%, with respect to the base case, and yield has been increased a 12%. 4.4. Effect of the design of the nozzle The influence of the dimensions and the design of the nozzle will be discussed in this section. First we will analyze the case P of Table 2. It can be seen that the operating conditions in this case are the same as used in the base case. The differ-

ence is that in this case, CO2 is not introduced through the concentric annulus, but through a different nozzle, which is placed relatively far from the nozzle of the organic solution. This arrangement is used frequently in pilot plants, and has the advantage of being simpler than the concentric nozzle. Since the inlet velocity of CO2 is much lower than the inlet velocity of the solution, this flow has a relatively small influence on hydrodynamics and mixing. However, if CO2 is not introduced through the annulus, the fluid that diffuses into the jet is no longer almost pure CO2 , but fluid from the bulk fluid phase, which has some amount of organic solvent. This greatly reduces the supersaturation, as can be seen in Fig. 13. Therefore, bigger particles are formed (9.2 ␮m compared to 1.4 ␮m in the base case). Particle formation is also much slower than in the base case. In fact, the maximum nucleation mass rate achieved in this case is I = 7.3 × 10−6 kg m−3 s−1 , compared to I = 322 kg m−3 s−1

Fig. 13. Supersaturation over the radius of the precipitator at different heights, nozzle without CO2 through the concentric tube, and operating conditions as in case P of Table 3.

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in the base case. This can affect the yield of the precipitation. As can be seen in Table 2, only the 22% of the solid has precipitated after 10 cm of precipitator, which is a reasonable length for a pilot scale precipitator. Due to the low concentration of particles, condensation rate is also small in this case. For this reason, while in the previous cases, condensation kept supersaturation in the center of the jet in small values, in this case comparatively high supersaturations are found in the center of the jet, as can be seen in Fig. 13. Therefore, in this case particle formation does not occur mainly in the borders of the jet, but in all the section of the jet with similar rates. Cases Q and R of Table 2 show the effect of the diameter of the nozzle. The influence of this parameter, and the reasons that underlie this trend, are similar to those of the solution flow rate: if nozzle diameter is reduced for a given solution flow rate, the increase in velocity causes a stronger turbulence. This improvement of the mixing leads to higher supersaturations and smaller particles. 4.5. Calculations with ascorbic acid The production of particles of ascorbic acid with the SAS process, with two different initial concentrations, as shown in cases S and T of Table 2, has been simulated. The operating conditions shown in this table are the same used in experimental work with this substance [36]. The objective of this section is to make clear the validity of the numerical results presented in this work. When ascorbic acid is precipitated from a 0.9 wt.% solution in ethanol, needle-like crystals of about 30 ␮m are obtained experimentally. At higher solution concentrations of 4 wt.%, the product consists of small spherical particles of about 1–5 ␮m. The mean particle sizes obtained with the model are 2.7 and 1.3 ␮m, respectively. The larger discrepancy in the first result is partly due to the change in the morphology of the solid, from spheres to needles, which cannot be predicted by the model. So it can be concluded that the model cannot predict the mean particle size accurately (and of course, it cannot predict the change in the morphology), but it predicts the trend of variation with initial concentration correctly, and gives a reasonable estimation of particle size. The reason for this, as explained before, is that some parameters of the model, as the interfacial tension, are not known. Precipitation yields obtained experimentally are of about 90%. Calculated results are 88% for the diluted solution, and 94% for the concentrated solution. It can be seen that the model predicts the yield accurately. This is not surprising, since the yield depends only on phase equilibrium, which can be modeled with reliability.

5. Conclusions A mathematical model for the supercritical antisolvent crystallization process has been developed. This model rep-

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resents simultaneously phase equilibrium, mass transfer, jet hydrodynamics and crystallization kinetics. According to experimental evidence, jet hydrodynamics has been modeled as the mixing of a gaseous plume under turbulent flow. A two-equation k–ε model of turbulence has been used for this purpose. The effect of turbulence has been considered in mass transfer calculations. Phase equilibrium has been modeled with the Stryjek–Vera modification of the Peng–Robinson equation of state. The particle size distribution resulting from this process has been calculated considering that nucleation can be described with the classical theory, and including the mechanisms of particle growth by condensation and coagulation. Results show that the most relevant aspect of jet hydrodynamics, is that the jet draws in fluid from the surrounding mass of fluid. This is the main mechanism of mixing between the organic solution and the antisolvent. Particle formation occurs mainly in the border of the jet, which is the region of highest supersaturation. The main mechanism of particle growth is condensation. The effect of the operating parameters of the process on the particle size distribution and the precipitation yield has been analyzed. Results show that with higher initial concentrations, smaller particles are formed, because it is possible to achieve higher supersaturations. Higher solution flow rates increase turbulence and improves mixing, leading to smaller particles. With higher CO2 /organic solution ratios, smaller particles are obtained. It has been shown that the parameters that have more influence in the particle size, are those that affect phase equilibrium: temperature and pressure. Any variation in those parameters which reduces the solubility of the solute in the fluid phase, lead to higher supersaturations and smaller particles. Therefore, it has been suggested that is convenient to use low temperatures, and pressures slightly above the mixture critical point. The influence of the dimensions and the design of the nozzle has been discussed. It has been shown that with smaller nozzle diameters, the increase in the inlet velocity of the solution results in a better mixing and smaller particles, in the same way that an increase in the solution flow rate affects those parameters. It has been concluded that the operation with a nozzle without a concentric tube for the CO2 does not have a significant effect on hydrodynamics. However, with this configuration of the nozzle, the fluid that diffuses into the jet is not pure CO2 , but a fluid with a considerable amount of organic solvent. This causes a drastic reduction in supersaturation, and bigger particles are obtained. The velocity of the precipitation is also reduced, so it may not finish inside the precipitator. The reliability of the calculations presented in this work has been tested by comparing the results of the model to experimental results with ascorbic acid. It has been concluded that the model predicts correctly the variation of particle size with the operating parameters, but fails to predict the value of the mean particle size. The reason is that the interfacial

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tension between the solid and the fluid phase is not known. Therefore, the results shown here are descriptive of the main trends rather than predictive. The model developed in this work is a powerful tool that can contribute to the understanding of the SAS process. This model can help in the interpretation of experimental results, and can be applied in the scale-up of the process.

Acknowledgements This project has been financed by the Spanish Ministry of Science and Technology, project PPQ 2003-07209.

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