A Thermodynamic Analysis of Three-Phase Equilibria in Binary and

Supercritical fluids are being increasingly used as media for fine particles formation: the most important techniques are the rapid expansion of a sup...
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Ind. Eng. Chem. Res. 1997, 36, 5507-5515

5507

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 (SAS) Ireneo Kikic* and Michele Lora Dipartimento di Ingegneria Chimica, dell’Ambiente e delle Materie Prime, Universita` di Trieste, Piazzale Europa 1, I-34127 Trieste TS, Italy

Alberto Bertucco Istituto di Impianti Chimici, Universita` di Padova, via Marzolo 9, I-35131 Padova PD, Italy

Supercritical fluids are being increasingly used as media for fine particles formation: the most important techniques are the rapid expansion of a supercritical solution process (RESS), the particles from gas-saturated solutions process (PGSS), and the supercritical antisolvent recrystallization process (SAS). To verify the feasibility of such processes, and to optimize the choice of operative variables, it is important to understand the phase behavior of the systems involved. To perform a thermodynamic analysis, an equation of state has to be used to take into account the effects of pressure; moreover, a solid phase is involved, and the heavy component is usually a high molecular weight and poorly characterized compound. In this work the PengRobinson equation of state with classical mixing rules and one or two binary interaction parameters are used. The fugacity of the heavy component in the solid phase is calculated by means of a subcooled liquid reference state: only heat of fusion and melting temperature of the heavy component are needed. The aim of this work is to develop a thermodynamic model which allows to calculate solid-liquid-vapor (S-L-V) equilibria of binary (RESS and PGSS) and ternary (SAS) systems. In regard to binary systems, the knowledge of PUCEP and TUCEP allows the calculation of binary interaction parameters: then the P-T trace of S-L-V equilibrium and the solubility of the heavy component in the light supercritical fluid can be reasonably well predicted. For ternary systems available S-L-V and S-L1-L2-V equilibrium data are well correlated, so that an analysis on the effect of operating variables (P and T) on the SAS process can be performed. Introduction and Scope The use of supercritical fluids as media for fine particle formation is a rapidly developing field of research. The possibility of obtaining solvent-free microparticulate particles with a narrow size distribution curve makes this technology very attractive, especially in the pharmaceutical industry. Among others, micronization techniques of present interest are the rapid expansion of a supercritical solution (RESS), the supercritical antisolvent recrystallization (SAS), and the particles from gas-saturated solutions process (PGSS). The RESS technique (Matson et al., 1987; Chang and Randolph, 1989; Tom and Debenedetti, 1991; Mohamed et al., 1992) requires two steps: first the solute of interest has to be solubilized in a supercritical fluid, and then the expansion of the supercritical solution through a nozzle is performed in order to precipitate the solute: the uniform and high degree of supersaturation reached in this way leads to small particles with a narrow size distribution. At the base of the SAS technique (Gallagher et al., 1989; Yeo et al., 1993; Dixon et al., 1993; Berends et al., 1996) is the fact that fluids at a dense-gas state are generally soluble in organic solvents, and in solutions * Author to whom correspondence should be addressed. Tel: +39 (0)40 6763433. Fax: +39 (0)40 569823. E-mail: [email protected]. S0888-5885(97)00376-X CCC: $14.00

of these solvents with the substances which have to be crystallized. Gas dissolution causes a volumetric expansion of the solvent; finally it lowers its density, that is, its solvent power, and forces the solute to precipitate. Because of the high and uniform degree of supersaturation, small particles with a narrow size distribution curve are expected. Moreover, it is possible to extract all the solvent in the gas phase, and to obtain a solventfree product. Finally, the PGSS technique (Weidner et al., 1994) exploits the heavy substances melting point depression due to the pressurization with dense gases. The gas melts the solid, and a saturated liquid solution is formed. When expanding such a solution, the light gas evaporates and the temperature is decreased by a Joule-Thompson effect: due to the lowering of both pressure and temperature, high supersaturation occurs and small particles are likely to be formed. Although many parameters have to be accounted for in order to optimize these processes, it is clear that the most important one is the phase behavior of the systems at the conditions of interest. For example, it would be extremely helpful to be able to answer questions like “Is the heavy compound sufficiently soluble in the supercritical fluid (RESS)?” or “Is the dense gas sufficiently soluble in the liquid phase (PGSS)?” or again “Is the gas sufficiently miscible with the solvent and immiscible with the solute (SAS)?” © 1997 American Chemical Society

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Figure 1. P-T behavior of highly asymmetrical binary systems.

To better explain these points, some general trends of binary and ternary systems involved in these processes are discussed. Figure 1 shows the P-T behavior of a highly asymmetric binary mixture, that is, a mixture containing two substances with large differences in molecular size, structure, and intermolecular interactions: this is the case, for example, of a system with a light supercritical fluid (such as CO2, ethylene, or ethane) and a heavy hydrocarbon (either an aliphatic one, such as C-20, or an aromatic one, such as naphthalene or biphenyl); such kinds of binary systems are often involved in RESS and PGSS. Two main features are the triple-point temperature of the heavy component is much higher than the critical temperature of the light one; the solubility of the light component in the liquid phase is quite limited. These facts lead to a relatively small freezing point depression of the heavy component; accordingly, the S-L-V equilibrium takes place also at elevated pressures, and the S-L-V curve intersects the gas-liquid critical curve in two points: the lower critical end point (LCEP) and the upper critical end point (UCEP). At these two points, liquid and gas phases merge into a single fluid phase in the presence of excess solid. At temperatures between TLCEP and TUCEP a S-V equilibrium is observed. As the isotherm containing a vapor-liquid critical point must exhibit here a zero slope on a P-x diagram, the solubility of the heavy component in the gas phase increases very rapidly with pressure near the LCEP and the UCEP. The LCEP is usually close to the critical point of the light component: TLCEP is in general quite low; therefore, the solubility of the heavy component in the light one is limited. On the contrary, the temperature of the UCEP is much higher; this fact, together with the high value of ∂x/∂P|T close to the UCEP, leads to a high solubility of the heavy component in the supercritical fluid. More specifically, two kinds of behavior may occur: if the P-T trace of S-L-V equilibrium presents a decreasing temperature at increasing pressure, this is defined as a Class I system (curve I in Figure 1); if in the P-T trace a minimum appears when pressure is increased, this is a Class II system (curve II in Figure 1). Class I P-x diagrams at different values of temperature have been reported in McHugh and Krukonis

(1993). At temperature below TUCEP, only a solid-fluid equilibrium occurs; at temperature between TUCEP and the melting temperature of the heavy component, there is a range of pressure in which either V-L or S-V equilibrium can be observed, depending on the global composition of the mixture. A peculiar behavior is observed at TUCEP: as UCEP is a vapor-liquid critical point in the presence of excess solid, the corresponding P-x isotherm presents an inflection point at P ) PUCEP. The behavior of Class II systems is somehow different, as reported for example by McHugh and Krukonis (1993). At temperatures below the minimum in the S-L-V line (cf. Figure 1) a S-V equilibrium exists for each pressure. At temperatures between the minimum and TUCEP, an isotherm intersects the S-L-V line in two points: between the corresponding two values of pressure, either L-V or L-S equilibrium is observed. At TUCEP and pressure below PUCEP a vapor-liquid equilibrium occurs, but at PUCEP the two phases merge into one fluid phase, together with the appearance of a solid phase. A S-V equilibrium is again observed by increasing pressure. At T > TUCEP, the behavior is similar to Class I systems: depending on the general composition, either L-V or S-V equilibrium can be encountered. These features are interesting for both RESS and PGSS: in RESS it should be important to predict the behavior of the solubility in the supercritical fluid near the UCEP, in order to choose pressure and temperature values which give the maximum amount of solute in the supercritical solution without appearance of a liquid phase; in PGSS the knowledge of the P-T trace of the S-L-V equilibrium gives information on the pressure needed to melt the solute and form a liquid phase at a given temperature, and to calculate its composition. About ternary mixtures, which are mostly related to SAS, a thermodynamic study of the system solutesolvent-antisolvent is extremely useful to address the feasibility of the process and to exploit the effects of temperature and pressure. The antisolvent (dense gas) is miscible with the solvent, and its solubility increases with pressure, but it is immiscible with the solute: once a certain pressure (i.e., composition) is reached, the solution becomes supersaturated and the solute precipitates. For the description of these systems we refer directly to the Results and Discussion section where the behavior of the system CO2-toluene-naphthalene will be illustrated.

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From the previous discussion it appears that handling these systems is quite a difficult task: due to the high pressure, an equation of state must be used; since the solid phase has also to be considered, it must be able to calculate the fugacities in this phase. Some attempts have been made to provide a thermodynamic model of this kind (De Swaan Arons and Diepen, 1963; Chang and Randolph, 1990; Dixon and Johnston, 1991): see Kikic et al. (1997) for a brief discussion. In this work the Peng-Robinson equation of state (PREOS) is used to express fugacity coefficients, and the fugacity of the solute in the solid phase is calculated by means of a subcooled liquid reference state. Binary and ternary systems have been modeled and results compared with existing experimental data. Theoretical Approach 1. How To Handle Components in the Solid Phase. A conventional equation of state is not suitable to represent a solid phase, so that fugacities in it cannot be directly calculated. However, it is possible to relate the solid state with a fictitious liquid state; if we assume the solid phase to be pure, a relationship between the fugacity of the solid and that of the corresponding subcooled liquid can be found at given temperature and pressure. An equation developed by Prausnitz et al. (1986) is used, which is strictly valid at the triple point pressure of the solute:

ln

S f 10

f

L 10

∆hf1 ) RT

f 1

(

)

T f1 1T

of the exponential term, since the difference between the solid and liquid molar volumes is usually negligible. It is also possible to develop the model in a second way: it can be written as in eq 2, in which eq 1 is introduced to calculate the solid fugacity at P0:

[ ( )

S L f 10 (T,P) ) f 10 (T,P) exp

S P ν10 S S dP (T,P) ) f 10 (T,P0) exp P f 10 0 RT

∫

x 1φ j L1 P ) y1φ jG 1P

(7)

j L2 P ) y2φ jG x 2φ 2P

(8)

S ) x1φ j L1 P f 10

(9)

The calculation of the fugacity coefficients of the two components in the liquid and vapor phases is made through the well-known PREOS with classical mixing rules and either one (kij) or two (kij and lij) binary interaction parameters:

P)

a(T) RT v - b v(v + b) + b(v - b)

νL

10 dP ∫PPRT 0

(3)

S Pν10

∫P

0

L - ν10 dP RT

S L f 10 (T,P) ) f 10 (T,P) ×

[

(12)

( )]

S L ν10 - ν10 ∆hf1 T f1 dP + 1 0 RT T RT f 1

∑i ∑j xixjbij

(13)

bi + bj (1 - lij) 2

(14)

b) (4)

By introducing eq 1 into eq 4, and by solving with respect to the fugacity of the solid solute, eq 5 can be obtained:

∫PP

(11)

N N

S f 10 (T,P) ) L (T,P0) exp L f 10 f 10

exp

∑i ∑j xixjaij

aij ) (aiaj)0.5(1 - kij)

The ratio of eqs 2 to 3 leads to S f 10

(10)

N N

a)

L L (T,P) ) f 10 (T,P0) exp f 10

]

S T f1 (P - P0)ν10 + T RT

The expressions (5) and (6) are similar, and give practically identical results. On the other hand, they are quite easy to handle: only the heat of fusion, the melting temperature, and the fugacity of pure solute in the subcooled liquid phase (which can be calculated by the PREOS) are needed. Therefore it is convenient to use them in the equilibrium relations when the fugacity of the pure solid solute has to be calculated. 2. Binary Systems. In general, this case involves two components and three phases: therefore there is only one degree of freedom. The isofugacity equations for the two components (1, heavy component; 2, supercritical fluid) in the phases involved (indicated with G, L, and S) can be written as

(2)

and

RT f1

1-

(6)

(1)

S L where f 10 and f 10 are the fugacities of the solute in the solid and subcooled liquid phase and ∆hf1 and T f1 are the heat of fusion and the melting temperature of the solute. To take into account the effect of pressure it can be written:

∆hf1

(5)

A similar equation was obtained also by De Swaan Arons and Diepen (1963). It is possible to simplify eq 5 without loss of accuracy by dropping the first addendum

bij )

The fugacity of the heavy component in the solid phase S f 10 can be expressed by eq 5. 3. Ternary Systems. These systems involve three components (1, heavy solute; 2, supercritical antisolvent; 3, organic solvent) and either four (S, L1, L2, and G) or three (S, L, and G) phases. In any case, the solid phase is supposed to be a pure heavy component. With regards to the four-phase case, the system exhibits one degree of freedom; equilibrium equations

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can be written for each component in the two liquid phases and in the vapor phase:

xL1 j L1 jG 1 φ 1 P ) y1φ 1P

(15)

xL1 j L1 jG 2 φ 2 P ) y2φ 2P

(16)

j L1 jG xL1 3 φ 3 P ) y3φ 3P

(17)

L2 L2 j L1 j1 P xL1 1 φ 1 P ) x1 φ

(18)

L2 L2 j L1 j2 P xL1 2 φ 2 P ) x2 φ

(19)

L2 L2 xL1 j L1 j3 P 3 φ 3 P ) x3 φ

(20)

The equilibrium equation for the solute between the first liquid and the solid phase is given by S f 10 ) x1φ j L1 1 P

(21)

Fugacity coefficients are calculated by means of the PREOS with classical mixing rules, as outlined in section 2 for binary systems, whereas the fugacity of the solid solute is calculated by means of eq 5. If only three phases are present, the system exhibits two degrees of freedom; the previous equation system is simplified by considering only eqs 15, 16, 17, and 21. Results and Discussion In all calculations reported below the properties of pure compounds were taken from Reid et al. (1988). It is also important to keep in mind that the model is based on a cubic equation of state, and its parameters can be calculated starting from Tc, Pc, and vapor pressure (or acentric factor) of the pure components. But RESS, PGSS, and SAS can also deal with polymers: these are often poorly characterized compounds, and their critical constants cannot be calculated. In such cases our model is not applicable, and another approach has to be used, which does not need the knowledge of critical constants: equations like the PHCT EOS (Beret and Prausnitz, 1975) or the SAFT EOS (Huang and Radosz, 1990) could be a solution for this problem. 1. Binary Systems. The model has been tested on a number of binary systems (Diepen and Scheffer 1948; van Gunst et al., 1953; Diepen and Scheffer, 1953; Krukonis et al., 1984; McHugh and Yogan, 1984; Cheong et al., 1986; McHugh et al., 1988; Lemert and Johnston, 1989). The PREOS has been used either with one (kij) or with two (kij and lij) interaction parameters. In the first case, good results have been obtained in correlating the P-T trace of S-L-V equilibrium. Anyway, the position of the UCEP was not correctly calculated: in fact a liquid-vapor equilibrium was predicted at pressures well above the experimental observed UCEP pressure. Similar results were obtained also by Paulaitis (Paulaitis et al., 1983). By using two interactions parameters, better results have been obtained. It is now important to specify how interaction parameters have been calculated. As about Class I, the TUCEP isotherm presents an inflection point at P ) PUCEP: by setting to zero the first and second derivative at this point, two equations are

Figure 2. (a) P-T trace of solid-liquid-vapor equilibrium for the system ethylene-naphthalene; k12 ) 0.0093; l12 ) -0.0280; experimental data: Diepen and Scheffer (1948), van Gunst et al. (1953). (b) Solubility of naphthalene in supercritical ethylene; k12 ) 0.0093; l12 ) -0.0280; experimental data: Diepen and Scheffer (1953).

obtained which allow the calculation of the two binary parameters. It is worth noting that only the values of PUCEP and TUCEP are needed to apply this method. As it can be seen from Figure 2a, the P-T trace of S-L-V equilibrium is represented acceptably well, and the position of the UCEP is accurately predicted. Furthermore, with the same values of the parameters, the solubility curves of the heavy component in the light supercritical fluid can be predicted: it is clear from Figure 2b that calculated lines are in good agreement with experimental data. It is important to notice that the significant increase of solubility near the UCEP is predicted with reasonable accuracy. The approach is slightly different for Class II systems: the TUCEP isotherm presents a horizontal peak at PUCEP, and neither first nor second derivatives can be defined or calculated; only left-side and right-side first derivatives can be defined, and both are set to zero to evaluate the binary interaction parameters. Some results for the system CO2-biphenyl are shown in Figures 3 and 4: a remarkable improvement has been obtained with respect to the results of Paulaitis et al. (1983), who used a similar model, but only one parameter. Again, the UCEP position is predicted quite well (Figure 3a), as well as the P-T trace of S-L-V equilibrium and the compositions of liquid and vapor phases at the S-L-V equilibrium (Figure 3b). Also the prediction of the heavy compound solubility at various

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Figure 3. (a) P-T trace of solid-liquid-vapor equilibrium for the system CO2-biphenyl; k12 ) 0.0808; l12 ) -0.0188; experimental data: McHugh and Yogan (1984), Paulaitis et al. (1983). (b) Composition of liquid and vapor phases at S-L-V equilibrium for the system CO2-biphenyl; k12 ) 0.0808; l12 ) -0.0188; experimental data: Cheong et al. (1986).

temperatures using the calculated interaction parameters provides good results (Figure 4a,b). It is interesting to note that the P-y isotherms pass continuously from a S-V through a V-L to a S-V equilibrium. Similar results (summarized in Table 1) were obtained for the systems: xenon-naphthalene, ethanenaphthalene, CO2-naphthalene, ethylene-biphenyl, and ethane-biphenyl. 2. Ternary Systems. The model has been tested on available data for the systems CO2-toluenephenanthrene, CO2-toluene-naphthalene (Dixon and Johnston, 1991), and CO2-toluene-β-carotene (Chang and Randolph, 1990): good results for all systems in the correlation of the liquid phase composition have been obtained (Kikic et al., 1997). The most important features are exhibited by CO2toluene-naphthalene. Two kinds of data are available for this system: 1. Composition of the liquid phase at S-L-V equilibrium as a function of pressure at T ) 298 K (Dixon and Johnston, 1991); 2. P-T data of the S-L1-L2-V equilibrium, of the critical L1)L2-V locus (liquid-liquid critical points in presence of vapor), and of the critical L1-L2)V locus (liquid-vapor critical points in presence of another liquid) (Hong and Luks, 1992). It is important to represent the data for the system CO2-toluene-naphthalene in a ternary diagram, to

Figure 4. (a) Solubility of biphenyl in supercritical CO2; T ) 322.65 K; experimental data: McHugh and Paulaitis (1980). (b) Solubility of biphenyl in supercritical CO2; T ) 328.15 K; experimental data: McHugh and Paulaitis (1980), Paulaitis et al. (1983). Table 1. Percent Root-Mean-Square Deviation between Experimental and Calculated P-T Traces of S-L-V Equilibrium and Values of Binary Interaction Parameters for Some Binary Systems system

number of exp. points

rmsd% (on T)

kij

lij

xenon-naphthalene ethane-naphthalene CO2-naphthalene ethylene-biphenyl ethane-biphenyl

23 16 46 26 14

0.22 0.11 0.07 0.17 0.16

0.025 0.026 0.079 0.007 0.024

-0.023 -0.031 -0.037 -0.012 -0.020

take into account the influence of overall composition (see Figure 5). In Figure 5a equilibrium without liquid-liquid immiscibility is presented. Point a corresponds to the composition of the liquid phase in equilibrium with vapor and solid and is determined once P and T are set. If P is raised at constant T, point a goes toward pure CO2. At a temperature between about 315 and 325 K, raising pressure lets a second liquid phase to appear (Figure 5b): depending on the overall composition, either L1-L2-V or S-L2-V equilibrium may occur. If pressure is raised above L1)L2-V critical point, two different behaviors are evidenced: 1. At a temperature between about 320 and 325 K the second liquid phase appears at a L1-L2 critical point (c ) d); at increasing pressure, d goes toward e until the two phases d and e merge at a liquid-vapor critical point, and the triangle

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a

b

c

d

Figure 5. Ternary diagrams for the system CO2-toluene-naphthalene: (a) typical S-L-V equilibrium at T and P assigned; (b) S-L1-V and L1-L2-V equilibrium occurring at a temperature between 315 and 325 K and pressure between 82 and 98 bar. (c) S-L1-L2-V equilibrium occurring at a temperature between 315 and 320 K and pressure between 82 and 91 bar. (d) S-L1-V and S-L1-L2 equilibrium occurring at a temperature between 315 and 320 K and pressure above that of S-L1-L2-V equilibrium at each temperature.

c-d-e degenerates; 2. At a temperature between about 315 and 320 K the second liquid phase appears at a liquid-liquid critical point, as in the first case. By an increase in pressure, the two pairs of points a-c and b-e approach and merge: a four-phase equilibrium (SL1-L2-V) is obtained (Figure 5c). By another increase in pressure, the S-L1-L2-V area splits into two triangles having a common vertex (Figure 5d: the splitting occurs in point d of Figure 5c, from which d1 and d2 originate). Our model has been used to correlate equilibrium data by Dixon and Johnston (1991); we have also tried to represent the P-T trace of S-L1-L2-V equilibrium and the compositions of L1, L2, and V phases. About the interaction parameters, for the pair CO2toluene they have been taken from the literature (kij ) 0.09, lij ) 0; Ng and Robinson, 1978); for the pair toluene-naphthalene, it has been checked that kij ) 0, lij ) 0 reproduce well the solubility data of naphthalene in pure toluene at the temperature of interest; for the pair CO2-naphthalene either one (kij) or two (kij and lij) interaction parameters have been used, as outlined in section Theoretical Approach. When one interaction parameter is used, experimental data are correlated acceptably (see Figure 6a); anyway, an unexpected behavior is observed: solubility curves at high pressure show an “S” shape, so that there is a range where three values of composition correspond to a single pressure value. Such a behavior means the presence of a second liquid phase, which is not the case at 298 K (see Hong and Luks, 1992). Hence, one interaction parameter is not enough to represent experimental data.

By using two interaction parameters, the correlation of data is of course much better, and S-like behavior does not occur anymore (Figure 6b); note that binary interaction parameters for CO2-naphthalene were regressed directly from ternary data, because binary data at 298 K were not available. To represent S-L1-L2-V equilibrium, interaction parameters for CO2-naphthalene were regressed from binary data of solubility of naphthalene in CO2 at 318 K and kept constant with temperature. The P-T trace of the S-L1-L2-V equilibrium is represented well (Figure 7). In Figure 8 the solubility of naphthalene in the liquid phase at various temperature is represented, with more attention to the four-phase area: the heavy-faced lines represent the composition of L1, L2, and V phases at equilibrium. The critical behavior of the system at the bottom and top pressure of S-L1-L2-V equilibrium is evident; moreover, it is important to notice that, at high temperature, the solvent power of CO2 on naphthalene becomes high, also because toluene is acting as an entrainer; at increasing pressure, liquid and vapor phases merge into a critical point, above which a S-V equilibrium occurs. This behavior is clearly represented in a ternary diagram: with reference to Figure 5a and increasing pressure, the binary CO2-toluene pair becomes fully miscible (dotted line); afterward, the L+V area gets narrow until L and V phases merge. In Figure 9a the influence of temperature on the solubility of the solute (normalized on its value at low pressure) is evidenced: the higher the temperature, the higher the pressure required to obtain the sudden solubility drop. But the solubility of the solute can be suitably represented also as a function of CO2 mole

Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997 5513

Figure 8. Mole fraction of naphthalene in the liquid phase for the system CO2-toluene-naphthalene; T ) 316, 317, 318.5, 320, and 321 K.

a

Figure 6. (a) Composition of the liquid phase at solid-liquidvapor equilibrium for the system CO2-toluene-naphthalene; T ) 298.15 K; k12 ) 0.121; experimental data: Dixon and Johnston (1991). (b) Composition of the liquid phase at solid-liquid-vapor equilibrium for the system CO2-toluene-naphthalene; T ) 298.15 K; k12 ) 0.094; l12 ) -0.024; experimental data: Dixon and Johnston (1991).

Figure 7. P-T trace of S-L1-L2-V equilibrium for the system CO2-toluene-naphthalene; experimental data: Hong and Luks (1992).

fraction (Figure 9b). In such a diagram very close curves are obtained; hence the solubility of the solute depends only on the CO2 mole fraction in the liquid phase: whatever the temperature, the solute precipita-

b

Figure 9. (a) Solubility of naphthalene in the liquid phase normalized on its value at low pressure as a function of P; T ) 290, 298, 306, and 314 K; k12 ) 0.094; l12 ) -0.024. (b) Solubility of naphthalene in the liquid phase normalized on its value at low pressure as a function of xCO2P; T ) 290, 298, 306, and 314 K; k12 ) 0.094; l12 ) -0.024.

tion is influenced only by the amount of CO2 which has dissolved in the liquid. Finally it is interesting to compare the new model with previous ones (Chang and Randolph, 1990; Dixon and Johnston, 1991). With respect to the Chang and Randolph model (Figure 10), the comparison is carried out for the system

5514 Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997

Figure 10. Solubility of phenanthrene in the liquid phase (system CO2-toluene-phenanthrene): comparison between the proposed model and the Chang and Randolph model; T ) 298 and 304 K; k12 ) 0.136; experimental data: Dixon and Johnston (1991).

CO2-toluene-phenanthrene, and the improvement in the correlation of experimental data is evident, especially because first, the new model is able to predict the horizontal asymptote of the solubility at high pressure, while the other one is not. Second, the Chang and Randolph model considers only the binary system solvent-antisolvent, so that there is no solute dependence: instead, experimental data make evident the different behavior of different solutes (Kikic et al., 1997), which is well represented by our model. The inadequacy of the Chang and Randolph model was recently underlined also by Peters (Peters et al., 1997). With respect to the Dixon and Johnston model, it provides similar results as our model, but it is more complicated because it needs the evaluation of activity coefficients, that is, solubility parameters. Conclusions A thermodynamic model has been developed for representing solid-liquid-vapor equilibrium of binary and ternary systems. For binary systems the model allows the calculation of interaction parameters k12 and l12 to be used in the Peng-Robinson equation of state from PUCEP and TUCEP only, and the prediction of the solubility curves and of the S-L-V equilibrium with good accuracy. For ternary systems our model is an improvement with respect to the ones by Chang and Randolph (1990) and Dixon and Johnston (1991). Available S-L-V equilibrium data are represented satisfactorily, as well as S-L1-L2-V data for the system CO2-toluenenaphthalene. Acknowledgment The authors acknowledge the M.U.R.S.T. (Ministero dell’Universita` e della Ricerca Scientifica e Tecnologica) for financial support. Literature Cited Berends, E. M.; Bruinsma, O. S. L.; de Graauw, J.; Van Rosmalen, G. M. Crystallization of Phenanthrene from Toluene with Carbon Dioxide by the GAS Process. AIChE J. 1996, 42, 431. Beret, S.; Prausnitz, J. M. ‘Perturbed Hard Chain’ Theory: an Equation of State for Fluids Containing Small or Large Molecules. AIChE J. 1975, 21, 6.

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Received for review May 29, 1997 Revised manuscript received August 5, 1997 Accepted August 9, 1997X IE970376U X Abstract published in Advance ACS Abstracts, October 15, 1997.