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Fluid Phase Equilibria 286 (2009) 162–169

Contents lists available at ScienceDirect

Fluid Phase Equilibria

journa l homepage: www.e lsev ier .com/ locate / f lu id

hase equilibria of carbon dioxide + poly ethylene glycol + water mixtures at highressure: Measurements and modelling

ngel Martín ∗, Huu Minh Pham, Andreas Kilzer, Sabine Kareth, Eckhard Weidnerehrstuhl für Verfahrenstechnische Transportprozesse, Ruhr Universität Bochum, Universitätsstr. 150, 44801 Bochum, Germany

r t i c l e i n f o

rticle history:eceived 15 May 2009eceived in revised form 13 August 2009ccepted 14 August 2009

a b s t r a c t

Phase equilibria of carbon dioxide + poly ethylene glycol (PEG) of average mol weight 6000 g/mol + watermixtures has been measured by the static method at conditions of interest for the development of Parti-cles from Gas Saturated Solutions (PGSS)-drying processes (pressure from 10 MPa to 30 MPa, temperature

vailable online 22 August 2009

eywords:upercritical carbon dioxidearticles from Gas Saturated SolutionsGSS-drying

from 353 K to 393 K). A thermodynamic model based on the Perturbed-Chain Statistical Associating FluidTheory (PC-SAFT) equation of state has been developed for correlating experimental data. The model isable to predict the composition of the liquid phase with an average deviation of 8.0%. However, the modeldoes not calculate correctly the concentration of PEG in the gas phase. The model is also capable of pre-dicting VLE data reported in the literature of PEG + CO2 mixtures with PEGs of molecular weights rangingfrom 1500 g/mol to 18500 g/mol as well as solid–fluid equilibrium of carbon dioxide + PEG mixtures at

C-SAFT equation of state pressures below 10 MPa.

. Introduction

Precipitation processes based on the use of supercritical car-on dioxide are versatile tools for producing solid formulations,articularly for polymers and pharmaceutical and food industrypplications. These processes have been successfully applied tobtain composites or encapsulates, which comprise an active com-ound loaded into a matrix of a carrier material [1,2], with theurpose of improving product preservation as well as controllinghe dissolution rate of the active compound. Several precipita-ion techniques have been developed in which the supercriticaluid has different functions, including amongst others the Rapidxpansion of Supercritical Solutions (RESS) process, the Supercriti-al Anti-Solvent (SAS) process and the Particles from Gas Saturatedolutions (PGSS) process. Recently, a modification of the PGSS pro-ess, denominated PGSS-drying, has been proposed. This techniquellows to process aqueous solutions to produce dry powders avoid-ng thermal degradation of the product that may be caused by otherrying processes, such as spray-drying. PGSS-drying has alreadyeen applied for producing a powder rich in green tea natural

ntioxidants [3]. It has also been used for producing particles ofoly ethylene glycol (PEG), a biocompatible polymer that can besed as a carrier material for developing formulations of naturalnd pharmaceutical compounds [4].

∗ Corresponding author. Tel.: +49 234 3226680; fax: +49 234 3214277.E-mail addresses: mamaan@iq.uva.es, angel.martin@vtp.ruhr-uni-bochum.de

Á. Martín).

378-3812/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.fluid.2009.08.010

© 2009 Elsevier B.V. All rights reserved.

Phase equilibrium data is necessary for determining suitableconditions for successfully performing the precipitation. It is alsoessential for any type of fundamental analysis of the process. In thecase of PGSS-drying of aqueous PEG solutions, the determination ofthe phase equilibrium of the ternary system CO2–PEG–H2O underhigh pressure is necessary.

Since PEG is frequently used as a carrier material, the informa-tion about phase behaviour of binary CO2–PEG systems is relativelyabundant. The solubility of CO2 into molten PEG at high pressurehas been measured by Lopes et al. [5] for low molecular weightPEGs, and by Wiesmet et al. [6] for high molecular weight PEGs. Thesolid–fluid equilibrium of PEG–CO2 mixtures has also been mea-sured by Weidner et al. [7]. However, information about the ternarysystems CO2–PEG–solvent that appears in several supercritical pre-cipitation processes is scarce. Donaldson et al. [8] measured theliquid–liquid equilibria of CO2–PEG systems with several organicsolvents. As far as the authors know, the phase equilibrium of theternary system CO2–PEG–H2O has not been previously reported.

Different approaches can be followed for modelling phasebehaviour of CO2–polymer–solvent mixtures at high pressures [9].Some of the most successful thermodynamic models for this typeof mixtures are the Sanchez–Lacombe lattice-fluid theory and theStatistical Associating Fluid Theory (SAFT) equation of state. In par-ticular, the Perturbed-Chain modification of the SAFT equation of

state (PC-SAFT EoS [10]) has been very successful for modellinggas–polymer systems at high pressures [11].

The aim of this work is to determine the phase behaviour of theternary system CO2–PEG–H2O at conditions where PGSS-dryingof aqueous PEG solutions is performed (pressure from 10 MPa to

Á. Martín et al. / Fluid Phase Equilibria 286 (2009) 162–169 163

ressur

3wps

2

twws

a

mabaocw[

dcptionm

ε

�

ε

Fig. 1. Flow diagram of the apparatus used for high p

0 MPa, temperature from 353 K to 393 K, mean PEG moleculareight 6000 g/mol). Furthermore, a thermodynamic model of thehase equilibrium of this system based on the PC-SAFT equation oftate is presented.

. The PC-SAFT equation of state

The PC-SAFT equation of state considers molecules to be consti-uted by chains of freely jointed spherical segments. This equationas developed in terms of the residual Helmholtz free energy ares,hich can be calculated as the sum of three contributions, as pre-

ented in Eq. (1):

res = ahc + adisp + aass (1)

In this equation, ahc accounts for the repulsion of the chain-likeolecule, using the hard-chain expression derived by Chapman et

l. [12], adisp accounts for the dispersion forces due to attractionetween temporarily induced dipoles, and aass accounts for thessociation between molecules, described by the association termf the original SAFT equation of state [13]. Full details about thealculation of these contributions to the residual Helmholtz energyere provided by Gross and Sadowski [10], and by Chapman et al.

13].Within this framework, non-associating molecules are

escribed by three parameters: the number of segments perhain m, the segment diameter �, and the depth of the pairotential ε/kb. For associating molecules, two more parameters,he association energy of interaction εAiBj/kb and the volume ofnteraction �AiBj between association sites A of molecule i and Bf molecule j, are required in addition to the specification of theumber of association sites and the association scheme of theolecule.Conventional mixing rules can be used for the parameters � and

/kb:

ij = 12

(�i + �j) (2)

ij = (εi · εj)0.5 · (1 − kij) (3)

e vapour–liquid measurements by the static method.

In case of mixtures with associating molecules, the followingmixing rules have been used for the association parameters [14]:

εAiBj = 12

(εAiBi + εAjBj) (4)

�AiBj = (�AiBi · �AjBj)0.5

((�i�j)

0.5

0.5(�i + �j)

)3

(5)

As shown in Eqs. (2)–(5), only the binary interaction parameterskij are required in order to apply the PC-SAFT EoS to mixtures.

3. Materials and methods

3.1. Materials

Poly ethylene glycol with an average mol mass of 6000 g/molwas obtained from Clariant (Burghausen, Germany) and used with-out further purification. Deionized water was used to preparesolutions. Carbon dioxide (purity: 99.5%) was supplied by Yara(Germany). Gases were of technical quality and were used as-obtained.

3.2. Methods

Phase equilibrium of CO2–PEG–H2O mixtures has been mea-sured by the static method. A flow diagram of the apparatus used forthis purpose is presented in Fig. 1. The main item of this apparatusis a stainless-steel high pressure equilibrium cell with an internalvolume of approximately 4.3 L, which is able to withstand pressuresof up to 40 MPa at temperatures in the range 298–573 K. This cell isequipped with a displaceable piston, which is driven by hydraulicoil (product code W.-Nr. 1.4571, Witzenmann, Germany) pressur-ized with two piston pumps (Sitec, Switzerland). By displacing thispiston, it is possible to vary the pressure when the equilibrium cell isclosed, and to maintain a constant pressure when a sample is being

retrieved from the cell. The equilibrium cell is also equipped with astirrer which allows improving the contact between phases, there-fore reducing the time required for equilibration. The cell is heatedelectrically. Pressure and temperature inside the equilibrium cellare measured with a manometer and a Ni–Cr–Ni thermocouple,

164 Á. Martín et al. / Fluid Phase Equi

Table 1VLE data of CO2–H2O mixtures obtained in this work.

T (K) P (MPa) w1CO2

(wt%) wgCO2

(wt%)

353 10 4.11 ± 0.12 99.68353 20 5.14 ± 0.12 99.64

was

iww6wstctapwwosastew1Ts

TV

353 30 5.5 ± 0.9 99.60393 10 3.2 ± 0.5 98.70393 30 5.8 ± 0.4 98.73

ith accuracies of ±0.3% and ±0.1 K, respectively. PID controllersctuate on the hydraulic oil pump and on the electrical resistanceo as to maintain constant temperature and pressure inside the cell.

The following experimental procedure was used: after preheat-ng the equilibrium cell to the desired temperature, it was rinsed

ith distilled water and subsequently emptied and degassed. After-ards, approximately 1 L of aqueous solution with a predefined PEG

000 concentration was pumped into to the cell via P1. Then CO2as fed to the cell, compressing it up to the desired operating pres-

ure by means of the membrane pump P3, and preheating it upo the operating temperature with heater H1. After this, all valvesonnected to the cell were closed, the mixer was switched on, andhe mixture inside the cell was maintained at constant temperaturend pressure during a period of at least 6 h in order to ensure thathase equilibrium has been achieved. The duration of this periodas set by performing preliminary experiments in which samplesere periodically taken and analyzed, until no significant variation

f concentration with time was observed (with respect to mea-uring accuracy), which indicates that phase equilibrium had beenttained. These preliminary experiments indicated that 4 h wereufficient to attain equilibrium; in the experiments, the fluid mix-ure was maintained inside the cell during 2 additional hours tonsure that equilibrium was indeed reached. After this, the mixer

as switched off and the mixture was left inside the cell duringh to ensure that phase separation by decantation was complete.hen, samples of gas and liquid phases were taken, adding to a totalample volume of approximately 100 mL. Pressure inside the cell

able 2LE data of CO2–PEG mixtures. Data obtained in this work and literature data [19].

T (K) P (MPa) w1CO2

(wt%) this work w1CO2

(wt%) Ku

353 10 14.4 ± 0.3 14.4353 20 22 ± 2 22.4353 30 24.0 ± 0.6 23.3393 10 – 11.9393 20 – 17.6393 30 22 ± 2 22.9

Fig. 2. VLE data of CO2 + H2O mixture, comparison between

libria 286 (2009) 162–169

was maintained constant during the sampling period by the actionof the hydraulic oil circuit. The samples were collected in the gasand liquid probes, GP and LP. The composition of these samples wasdetermined as follows: the amount of carbon dioxide was obtainedby measuring the volume of gas released from the samples upondepressurization, using gas meters with an accuracy of ±0.001 L.The amount of water and PEG in the samples was determined bythe difference in weight between the filled and empty probes, usinga balance with an accuracy of ±0.0001 g. Finally, the amount ofwater in the mixture was obtained by measuring the loss of weightafter 2 h of drying at 378 K, according to the procedure establishedin the norm DIN EN 827 [15]. The global accuracy of the composi-tion measurement was calculated on the basis of the accuracy of thedetermination of CO2, H2O and PEG amounts according to standardstatistic procedures [16].

4. Results and discussion

4.1. Binary subsystems

The vapour–liquid equilibrium (VLE) of CO2 + water andCO2 + PEG 6000 mixtures has been measured in order to validatethe experimental procedure by comparison with data previouslyreported by other authors. The results obtained are presented inTables 1 and 2, respectively. In these tables, compositions areexpressed as mass fractions. Fig. 2 presents a comparison of theVLE data measured in this work and data reported in the litera-ture [17,18] for binary mixtures CO2 + H2O. It can be seen that thedata obtained in this work agrees well with literature data. Thesame observation can be made with respect to binary CO2–PEG6000 mixtures [19], as depicted in Fig. 3.

4.2. CO2–PEG 6000–H2O system

In this work, VLE data of CO2–PEG 6000–H2O mixtures havebeen measured in the pressure range from 10 MPa to 30 MPa

kova [19] wgCO2

(wt%) this work wgCO2

(wt%) Kukova [19]

99.95 99.999.96 99.999.97 –

– 99.9– 99.8

99.96 99.9

literature data [17,18] and data obtained in this work.

Á. Martín et al. / Fluid Phase Equilibria 286 (2009) 162–169 165

Fa

armlwPtetc

Table 4PC-SAFT EoS pure component properties considered in this work [10,14].

m � (Å) ε/kb (K) �AiBj ε/kb (K)

TE

ig. 3. VLE data of CO2 + PEG 6000 mixture, comparison between literature data [19]nd data obtained in this work.

nd at temperatures ranging 353–393 K. Experimental results areeported in Table 3. In this table, compositions are expressed asass fractions. Experimental data are also represented in triangu-

ar diagrams in Figs. 4–8. These diagrams show a two-phase regionhich encompasses two of the binary subsystems (water + CO2 and

EG + CO2), and two single phase regions: a gas phase in whichhe majority compound is CO2, and a liquid phase. As it may bexpected, experimental data shows that the solubility of CO2 inhe liquid phase increases when pressure is increased, when theoncentration of polymer in the liquid is increased, or when tem-

able 3xperimental VLE data of CO2 + PEG 6000 + H2O mixtures.

T (K) P (MPa) w1CO2

(wt%) w1PEG (wt%) w

353 10 14.4 ± 0.3 85.7 ± 0.3353 10 11.6 ± 0.5 79.9 ± 0.4353 10 8.0 ± 0.4 65.1 ± 0.3 2353 10 5.9 ± 0.5 47.50 ± 0.13 4353 10 4.4 ± 0.5 28.53 ± 0.13 6353 10 4.1 ± 0.4 9.69 ± 0.08 8353 10 4.1 ± 0.1 0 9

353 20 22 ± 2 78 ± 2353 20 18 ± 2 74.6 ± 1.8353 20 10.2 ± 0.5 64.157 ± 0.4 2353 20 6.9 ± 0.9 47.0 ± 0.3 4353 20 5.5 ± 1.9 28.4 ± 1.2 6353 20 5.3 ± 1.0 9.5 ± 0.8 8353 20 5.14 ± 0.12 0 9

353 30 24.0 ± 0.6 76.0 ± 0.6353 30 21.5 ± 0.5 71.6 ± 0.4353 30 12.5 ± 1.4 62.9 ± 0.7 2353 30 8.3 ± 0.4 46.49 ± 0.14 4353 30 6.9 ± 1.0 28.1 ± 0.4 6353 30 6.7 ± 0.6 9.8 ± 0.3 8353 30 5.5 ± 0.9 0 9

393 10 11.9 ± 1.7 88.1 ± 0.8393 10 6.0 ± 0.5 67.3 ± 0.3 2393 10 5.1 ± 0.5 48.7 ± 0.8 4393 10 4.3 ± 1.7 28.9 ± 0.4 6393 10 3.4 ± 1.5 10.7 ± 0.9 8393 10 3.2 ± 0.5 0 9

393 30 22 ± 2 79 ± 2393 30 19 ± 2 74.4 ± 1.9393 30 13 ± 2 63.2 ± 0.8 2393 30 9.7 ± 0.2 45.3 ± 0.4 4393 30 7.9 ± 1.3 36.8 ± 0.5 5393 30 8.0 ± 1.6 34.1 ± 0.9 5393 30 5.8 ± 1.5 0 9

CO2 2.0729 2.7852 169.21 – –PEG 0.0506·MW 2.9326 236.13 0.02380 2405.0H2O 1.0953 2.8898 365.96 0.03487 2515.7

perature is decreased. The aforementioned characteristics allowconcluding that this system shows a type II triangular phase dia-gram [20].

4.3. PC-SAFT EoS parametrization and results

Pure component properties required for application of the PC-SAFT EoS are summarized in Table 4. The parameters of waterand carbon dioxide have been obtained from the literature [10,14].In case of water, a 2B association scheme according to the clas-sification of Chapman et al. [13] has been considered. It is wellknown that the 4C scheme provides a more physically realisticrepresentation of the molecule of water than the 2B scheme. How-ever, it has been demonstrated [14] that the application of the 2Bscheme within PC-SAFT allows obtaining a similar accuracy as the4C scheme, with a considerable reduction in model complexity andcomputational requirements. In case of PEG molecules, a 2B asso-ciation scheme has been considered to account for association by

the terminal hydroxyl groups. Based on previous results [6], it isexpected that the contribution of the association term will be neg-ligible for high molecular weight PEGs such as PEG 6000. However,the contribution of association may become important for lowermolecular weight PEGs. The parameters of PEG have been obtained

1H2O (wt%) wg

CO2(wt%) wg

PEG (wt%) wgH2O (wt%)

0 99.95 0.048 08.525 ± 0.014 99.83 0.029 0.146.97 ± 0.11 99.82 0.026 0.166.6 ± 0.4 99.8 ± 0.2 0.025 ± 0.009 0.21 ± 0.187.1 ± 0.4 99.67 0.01 0.326.17 ± 0.12 99.64 0.005 0.365.89 ± 0.12 99.68 0 0.32

0 99.96 0.04 07.1 ± 0.4 99.81 0.033 0.165.64 ± 0.07 99.79 0.035 0.186.1 ± 1.1 99.70 ± 0.13 0.034 ± 0.014 0.27 ± 0.126.1 ± 0.8 99.65 0.025 0.325.2 ± 0.9 99.62 0.011 0.374.86 ± 0.12 99.64 0 0.36

0 99.97 0.032 06.92 ± 0.09 99.77 0.053 0.174.6 ± 0.7 99.73 0.054 0.225.2 ± 0.3 99.55 ± 0.15 0.041 ± 0.005 0.41 ± 0.155.0 ± 1.3 99.55 0.028 0.423.5 ± 0.7 99.56 0.013 0.434.5 ± 0.9 99.60 0 0.40

0 99.95 0.5 06.7 ± 0.3 99.77 0.028 0.2066.3 ± 0.4 99.75 0.022 0.2266.8 ± 1.4 99.65 0.009 0.3425.9 ± 0.6 99.63 0.003 0.3666.8 ± 0.5 98.70 0 1.3

0 99.96 0.036 06.80 ± 0.09 99.19 0.028 0.7783.5 ± 0.2 98.94 0.026 1.0344.4 ± 0.6 98.9 ± 0.2 0.022 ± 0.011 1.09 ± 0.25.3 ± 0.8 98.70 0.02 1.288.0 ± 0.7 98.68 0.007 1.3124.2 ± 1.5 98.73 0 1.264

166 Á. Martín et al. / Fluid Phase Equilibria 286 (2009) 162–169

F 353 Ke

bpKPvo

Fe

Fe

ig. 4. Triangular phase equilibrium diagram of CO2 + PEG6000 + H2O mixtures (T =nlargement of the gas phase region is shown.

y simultaneous correlation of PVT experimental data of molten

oly ethylene glycol [19] and VLE data of CO2 + PEG reported byukova [19] and in this work (Table 2), because the results of theC-SAFT EoS regarding PVT properties are very insensitive to thealue of some pure component properties (particularly the depthf the pair potential), which advises against correlation solely based

ig. 5. Triangular phase equilibrium diagram of CO2 + PEG 6000 + H2O mixtures (T = 353 Knlargement of the gas phase region is shown.

ig. 6. Triangular phase equilibrium diagram of CO2 + PEG 6000 + H2O mixtures (T = 353 Knlargement of the gas phase region is shown.

, P = 10 MPa), experimental data (squares) and model results (continuous line). An

on PVT data. The effective number of segments m of PEG has been

considered directly proportional to the mean molecular weight ofthe polymer. With this approach, described by Gross and Sadowski[21] and others, it is possible to predict the phase behaviour of mix-tures with polymers of different molecular weights using the sameset of parameters. With the parameters reported in Table 4, the

, P = 20 MPa), experimental data (squares) and model results (continuous line). An

, P = 30 MPa), experimental data (squares) and model results (continuous line). An

Á. Martín et al. / Fluid Phase Equilibria 286 (2009) 162–169 167

Fig. 7. Triangular phase equilibrium diagram of CO2 + PEG 6000 + H2O mixtures (T = 393 K, P = 10 MPa), experimental data (squares) and model results (continuous line). Anenlargement of the gas phase region is shown.

F 393 Ke

PPd

A

Cpfeltara

ig. 8. Triangular phase equilibrium diagram of CO2 + PEG 6000 + H2O mixtures (T =nlargement of the gas phase region is shown.

C-SAFT predicts the experimental density data of PEG 1500 andEG 18500 reported by Dee et al. [22] with an average absoluteeviation (AAD%, Eq. (6)) of 0.1%.

AD % = 1ndata

·ndata∑i=1

∣∣Yexp − Ycalc∣∣

Yexp · 100 (6)

The value of the interaction coefficient between water andO2 has been calculated by correlation of experimental VLE dataublished by Bamberger et al. [17]. The correlation has been per-ormed by minimizing the average absolute deviation betweenxperimental and calculated bubble point pressures. The calcu-ated interaction coefficients range from k13 = −0.0162 at T = 323 K

o k13 = −0.0086 at T = 353 K showing linear variation with temper-ture over this temperature range (R2 = 0.995). The obtained linearelationship is shown in Table 5. With these values of the inter-ction coefficient, the PC-SAFT EoS predicts experimental bubble

Table 5Correlated PC-SAFT EoS binary interaction parameters.

kij

CO2 (1) + PEG (2) −30.042/T + 0.0899CO2 (1) + H2O (3) −28.782/T + 0.0731PEG (2) + H2O (3) −121.39/T + 0.1939

, P = 30 MPa), experimental data (squares) and model results (continuous line). An

point pressures with AAD% = 2.3%. A graphical comparison betweenexperimental and calculated data is shown in Fig. 9. It can be seenthat certain small deviations occur in calculations of gas phasecompositions (AAD% = 13%). It must be noted that because the con-centration of water in the gas phase is very small, the experimentaldetermination of this concentration is more prone to experimentalerrors than the measurement of liquid phase composition. Indeed,in Fig. 9 it can be observed that experimental gas phase composi-tions show more dispersion than liquid compositions. Moreover,because the concentration of water is small, small absolute errorsin the calculation of gas composition result in large AAD% by appli-cation of Eq. (6). This justifies in part that the deviations betweenmodel and experimental results are larger in the gas phase than inthe liquid phase. The equation of state model is also able to repro-duce bubble point pressure data of CO2 + H2O mixtures reported byBermejo et al. [23], which was not used to calculate model parame-ters and was determined in the pressure range from P = 1.55 MPa to8.34 MPa and in the temperature range from T = 296.7 K to 369.7 K.In this case the deviation between experimental data and modelresults is AAD% = 5.8%.

In case of the CO + PEG binary subsystem, the interaction coef-

2ficient has been calculated by correlation of VLE data measured byWiesmet et al. [6] for CO2 + PEG 4000 mixtures in the temperaturerange from T = 328 K to 373 K. The obtained interaction coefficientsvary from k12 = −0.0028 at T = 328 to k12 = 0.0098 at T = 373 K, with a

168 Á. Martín et al. / Fluid Phase Equilibria 286 (2009) 162–169

ntal d

l(apFotttc8ts4

ErfmPtticoe

Fr

Fig. 9. VLE of CO2 + H2O mixtures, experime

inear relationship between k12 and T over this temperature rangeR2 = 0.93, Table 5). The average deviation between experimentalnd calculated bubble point pressures is AAD% = 9.0%, and a com-arison between experiments and model results is depicted inig. 10. As shown in Table 4, the EoS model can be applied to PEGsf different molecular weights, because the parameter m is propor-ional to the molecular weight of the polymer and it accounts forhe effects of a variation in chain length. Indeed, the application ofhe model with the parameters reported in Tables 4 and 5 for cal-ulating the VLE data of mixtures of CO2 + PEG 1500 and CO2 + PEG000 results in AAD% of 14.8% and 12.4% respectively, with respecto the experimental data presented by Wiesmet et al. [6], which areimilar values of AAD% as those obtained with the system CO2 + PEG000 used to correlate model parameters.

In addition to the VLE equilibrium of CO2 + PEG mixtures, theoS model can also be used for calculating the solid–fluid equilib-ium of this mixture. The calculation of this property is interestingor analysis of precipitation processes, because it allows to deter-

ine whether a solid or a molten product can be obtained. Since theC-SAFT EoS is not capable of representing solid phases, an addi-ional model for calculating the fugacity of solid phase is required. In

his work that calculation has been performed with Eq. (7), accord-ng to the procedure proposed by Shariati and Peters [24], whichonsists in calculating the fugacity of the solid phase as a functionf the fugacity of a sub-cooled reference liquid. For applying thisquation the temperature and enthalpy of fusion of the polymer

ig. 10. VLE of CO2 + PEG 4000 mixtures, experimental data [6] (symbols) and modelesults (lines).

ata [17] (symbols) and model results (lines).

Tfus and �fusH as well as the molar volume of the solid phase vs

must be known. The values of these physical properties have beenobtained from the literature [22,25] (for PEG 4000, Tfus = 331.8 Kand �fusH = 742.5 kJ/mol).

ln ϕis = ln ϕi

l + �fusH

R

(1

Tfus− 1

T

)+ �i

s

RT(P − P0) (7)

Fig. 11 shows the results obtained with this procedure forCO2 + PEG 4000 mixtures. Comparison with experimental data [7]demonstrates that the model is able to predict the melting pointat pressures below 10 MPa. At higher pressures, experimental datashow an increase in melting temperature due to the effect of staticpressure, a phenomenon that cannot be predicted with the EoSmodel. This phenomenon may have been caused by several rea-sons, including a change in the crystalline structure of PEG particleswhen pressure is increased, or a selective extraction of low molecu-lar weight fractions of the polymer by supercritical carbon dioxide.In any case, none of these effects is described by the model and thediscrepancy between model results and experimental data is prob-ably due to the limitations of the simple model used to calculatethe fugacity of the solid phase (Eq. (7)).

The value of the interaction coefficient between PEG and H O

2reported in Table 5 has been calculated by correlation of the VLEdata of the ternary system CO2 + PEG + H2O presented in this work.The values obtained are k23 = −0.150 at T = 353 K and k23 = −0.115at T = 393 K. The triangular diagrams presented in Figs. 4–8 show

Fig. 11. Solid–Liquid–Vapor equilibria of CO2 + PEG 4000 mixtures, experimentaldata [7] (symbols) and model results (line).

e Equi

rtcoettptCwcprtdttnai

5

hsIbc

hlb2Ccacv

ss(ii

ms

La

a

a

aA�kkm

[[

[[

[[

[

[[

[

[[[[

libr. 238 (2005) 220–228.[24] A. Shariati, C.J. Peters, J. Supercrit. Fluids 23 (2002) 195.

Á. Martín et al. / Fluid Phas

esults of calculations with the EoS model. As it can be seen inhese figures, the agreement between experimental liquid phaseomposition and model results is good, with an average deviationf AAD% = 8.0%. However, the EoS model is unable to reproduce thexperimental gas phase compositions and particularly the concen-ration of PEG 6000 in the gas phase. According to experiments,here is a measurable (albeit very small) amount of PEG in the gashase, while the EoS model predicts an almost zero PEG concentra-ion in this phase. The solubility of some polymers in supercriticalO2 can be unusually high considering the very high moleculareight of these substances. These unusually high solubilities can be

aused by specific interactions between the functional groups of theolymer, or by a high flexibility of the polymer chain, amongst othereasons [26]. It appears that the PC-SAFT EoS model developed inhis work is unable to describe these phenomena. Nevertheless, thisiscrepancy is more relevant for theoretical aspects than for prac-ical development of PGSS-drying processes. In these processes,he low solubility of the polymer in the gas phase can be safelyeglected, and a good description of the liquid phase, which can beccomplished with the EoS model presented in this work, is moremportant.

. Conclusions

The fluid phase equilibrium of CO2 + PEG 6000 + H2O mixturesas been measured at temperatures of 353 K and 393 K and pres-ures of 10 MPa, 20 MPa and 30 MPa by the static method. TypeI triangular phase diagrams have been obtained, with CO2 solu-ilities in the liquid phase which increase when pressure or PEGoncentration is increased and when temperature is decreased.

A thermodynamic model based on the PC-SAFT equation of stateas been developed. The model correctly describes fluid phase equi-

ibria of the binary subsystems, with average absolute deviationsetween experimental and calculated bubble point pressures of.3% in the case of CO2 + H2O mixtures and of 9.0% in the case ofO2 + PEG 4000 mixtures. The model also predicts correctly theomposition of the liquid phase in the ternary system, with anverage deviation of 8.0%. However, model results yield PEG con-entrations in the gas phase that are much lower than experimentalalues.

By combining the equation of state with a simple model forolid phase fugacity, it is also possible to describe certain regions ofolid–fluid equilibria of CO2 + PEG mixtures at moderate pressuresP < 10 MPa). At higher pressures, the model for solid phase fugac-ty fails to represent the effect of static pressure that results in anncrease of melting temperature with increasing pressure.

The information obtained with this work is useful for deter-ining the conditions with which the PGSS-drying process can

uccessfully produce dry PEG particles.

ist of symbolsass association contribution to the residual Helmholtz free

energy (J/mol)dis dispersive contribution to the residual Helmholtz free

energy (J/mol)hc hard-chain contribution to the residual Helmholtz free

energy (J/mol)res residual Helmholtz free energy (J/mol)

AD% average absolute deviation (Eq. (6))fusH enthalpy of fusion (J/mol)

b Boltzmann’s constant (J/K)ij interaction coefficient

effective number of segments (chain length)

[

[

libria 286 (2009) 162–169 169

ndata number of data pointsP pressure (Pa)R ideal gas constant (J/mol K)T temperature (K)Tfus melting temperature (K)vi

S solid molar volume of component i (m3/mol)wl liquid composition (mass fraction)wg gas composition (mass fraction)x liquid composition (mol fraction)y gas composition (mol fraction)Yexp experimental valueYcalc calculated valueε/kb dispersion energy of interaction between segments (J)εAiBj/kb association energy of interaction between site A of

molecule i and site B of molecule j (J)ϕ fugacity coefficient�AiBj volume of interaction� Lennard–Jones segment diameter (Å)

Acknowledgements

Authors thank Clariant (Germany) and Yara (Germany) for pro-viding materials used in this work. A. Martín thanks the Alexandervon Humboldt foundation (Germany) for a postdoctoral researchfellowship.

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