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4579

Calculation of Solid-Liquid-Gas Equilibrium for Binary Systems Containing CO2 Jindui Hong, Hui Chen, and Jun Li* Department of Chemical and Biochemical Engineering, College of Chemistry and Chemical Engineering, Xiamen UniVersity, Xiamen 361005, Fujian, China

Henrique A. Matos and Edmundo Gomes de Azevedo Department of Chemical and Biological Engineering, Instituto Superior Te´cnico, AV. RoVisco Pais, 1049-001 Lisbon, Portugal

Two equations typically used for the pure-solid fugacity proved to be identical by selecting an appropriate relation for the pure-solid vapor pressure and the pure-liquid vapor pressure. On the basis of the pure-solid fugacity, a semipredictive model using solubility data (SMS) and a calculation model combining with GE models (CMG) were developed to calculate the solid-liquid-gas (SLG) coexistence lines of pure substances in the presence of CO2. For the SMS model, the Peng-Robinson equation of state (PR-EoS) with the van der Waals one-fluid mixing rule is used to correlate the solute solubility in CO2 to obtain the interaction parameter k12, which is further employed to predict the SLG coexistence lines by two methods: one adopts the fugacity coefficient of the solute in the liquid phase by an equation of state calculation (SMS-φ); the other uses the activity coefficient of the solute in the liquid phase calculated from the UNIFAC model (SMS-γ). For the CMG model, the PR-EoS with the linear combination of Vidal and Michelsen (LCVM) mixing rule, the Michelsen modified Huron-Vidal (MHV1) mixing rule, and a modified version (mLCVM) with the reevaluated parameter λ ) 0.18 are used. Results show that the SMS model can provide acceptable calculations of the SLG coexistence lines for most of the investigated systems. The predicted melting temperatures and solute compositions in liquid phase from a constant k12 are slightly better than those from the correlated one, while the predicted solute solubility data in CO2 from a constant k12 are worse than those from the correlated one. The CMG model with the mLCVM mixing rule calculates well the melting temperatures and solute compositions in liquid phase at SLG equilibrium and also gives acceptable calculations of the solute solubilities in supercritical CO2. Introduction Supercritical fluids (SCF) have been used in many different applications, namely to particle formation processes.1 A thorough knowledge of phase equilibrium, including the solidliquid-gas equilibrium (SLGE), can often provide important information that plays a key role in the understanding, operating, scale-up, and in general design of SCF-based particle formation processes.2-4 Some general rules for SLGE can be found in two recently published papers.5,6 In the available SLGE modeling studies,2,3,7-13 the approach using an equation of state (EoS) was widely employed to deal with the gas-liquid equilibrium (GLE) and the solid-gas equilibrium (SGE) or the solid-liquid equilibrium (SLE) to ultimately address the SLGE of a binary system. In line with this approach, it can be classified into two categories according to the process used to evaluate the solid solute fugacity: (1) using the pure-solid vapor pressure as the reference fugacity;7,8,13 (2) using a subcooled liquid to obtain a reference fugacity.2,3,9-12 The former method, originally proposed by McHugh et al.,7 was applied by Zhang et al.8 and by Uchida et al.,13 using either the experimental solid vapor pressure or that evaluated from the Antoine equation. The latter method was suggested by Kikic et al.2 that used the fugacity of a fictitious subcooled liquid together with the Peng-Robinson equation of state (PR-EoS) with two binary interaction parameters9 to investigate the SLGE curves of fats in supercritical CO2. Diefenbacher et al.3 did a * To whom correspondence should be addressed. E-mail: junnyxm@ xmu.edu.cn. Tel./Fax: (+86)-592 2183055.

similar modeling work which did show that the size and morphology of particles produced by the rapid expansion of supercritical solutions (RESS) process were strongly influenced by the solid-liquid-gas (SLG) phase behavior. In another work,11 the perturbed-hard-sphere-chain (PHSC) EoS was used to calculate the SLG coexistence lines that determined the appropriate conditions to control the solid particles or liquid droplets generated by a particle formation from gas-saturated solutions process (PGSS). Consequently, if the solid vapor pressure is measured experimentally or calculated from the Antoine equation, both methods lead to different results.14,15 However, we will show later that these two methods are identical in nature when an appropriate calculation of the solid vapor pressure is adopted. It is well-known the limitations of EoS to describe liquid solutions; when the solid vapor pressure is measured experimentally or calculated from the Antoine equation, the abovementioned approach has typically little success to predict accurately the normal melting points.16 Lemert et al.17 proposed the application of the regular solution theory (RST) to describe the SLE while the GLE was still dealt with by an EoS. Results indicated that this approach could provide good correlations for the SLG coexistence lines of binary and ternary (with a cosolvent) systems, especially under low pressures. Later, Li et al.16 used another activity coefficient model (NRTL) instead of RST to describe the SLE, where the two NRTL parameters were adjustable and another interaction parameter k12 was obtained from the correlation of solute solubility data for

10.1021/ie801179a CCC: $40.75 2009 American Chemical Society Published on Web 04/02/2009

4580 Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009

thermodynamic consistency; results showed that the correlations were quite satisfactory with an average AADT (as defined in Table 2) of 0.39 K for the investigated binary systems. Because most of the models mentioned for the SLGE modeling are correlative, developing predictive models is both attractive and important when scarce experimental melting data are available or difficult to measure. Kikic et al.2 used binary interaction parameters regressed from the upper critical end point (UCEP) to predict the SLG coexistence lines; this approach did not require any melting data, and therefore is predictive. Yet, UCEP points for many systems may actually not be easily obtained. Recently, Bertakis et al.14 developed a model called universal-mixing-rule Peng-Robinson-UNIFAC (UMR-PRU) to calculate the SLG coexistence curves, with relatively good success. Yet, the authors only presented results for the naphthalene/CO2 and phenanthrene/CO2 systems, and at high pressures the calculated P-T projection deviates notoriously from the experimental points. In the present work, we first identify the two expressions for the solid solute fugacity that are typically used and then we present two new models: a semipredictive model using solubility data (SMS) and a calculation model combining with GE models (CMG). Finally, we compare and discuss the calculated results for nine binary systems involving different types of molecules, including large fatty acids. Modeling Solid Solute Fugacity. There are two equations used frequently to calculate the fugacity of a solid solute. Equation 17,8,13 uses the saturation pressure of the pure-solid solute and a Poynting correction for the pressure effect. Equation 22,3,9-12,18 considers the pure-solid fugacity as that of a subcooled liquid at system’s temperature.

[

S,sat f S20(T, P) ) PS,sat 20 φ20 exp

[

(

VS20(P - PS,sat 20 ) RT

)

]

(1)

Tm ∆fusH 1+ RTm T L S,sat S (PL,sat P(VS20 - VL20) 20 V20 - P20 V20) + (2) RT RT

f S20(T, P) ) f SCL 20 (T, P) exp

]

In the equations above, subscript 2 denotes the solute, subscript 0 indicates a pure substance’s property, and superscript S L,sat represents solid phase; PS,sat 20 and P20 are, respectively, the vapor pressure of the pure-solid solute and that of the subcooled liquid S L and V20 are the solute’s solid at the system’s temperature T; V20 S,sat is the fugacity and liquid molar volumes, respectively; φ20 coefficient of the solid solute at its saturation pressure at temperature T, PS,sat 20 . Because the saturation pressure is normally S,sat SCL can be assumed to be equal to 1. f 20 (T,P) is very low, φ20 the fugacity of the hypothetical subcooled liquid, and ∆fusH is the enthalpy of fusion of the pure solute at its normal melting point, Tm. S,sat is obtained either from experimental data or Typically P20 calculated from the Antoine equation; this can not guarantee the equality of the pure solid fugacities from eq 1 and eq 2. Equation 2 is relevant to the melting point pressure dependence and therefore it is widely used in modeling the SLGE. The L,sat relation between PS,sat 20 and P20 is expressed by the SLE equation

[ ( )]

L,sat PS,sat 20 ) P20 exp

Tm ∆fusH 1RTm T

(3)

Moreover,

[

L L,sat f SCL 20 (T, P) ) f 20(T, P20 ) exp

VL20(P - PL,sat 20 ) RT

L,sat L,sat f L20(T, PL,sat 20 ) ) P20 φ20

]

(4) (5)

L L,sat L,sat where f 20 (T,P20 ) and φ20 are, respectively, the fugacity and the fugacity coefficient of the liquid (or subcooled liquid) at L,sat L,sat S,sat , and φ20 ) 1 just as φ20 . By introducing eqs 3-5 into P20 eq 1, we obtain eq 2. This indicates that these two expressions for the solid solute fugacity are the same, and we can therefore use either eq 1 or eq 2, satisfying to eqs 3-5, to establish the predictive models that the following sections present. SMS Model. The SLGE of a binary system is expressed by the following phase equilibrium equations:

f G1 (T, P) ) f L1 (T, P)

(6)

f S2 (T, P) ) f G2 (T, P)

(7)

f S2 (T, P) ) f L2 (T, P) ) x2φL2 P

(8)

f S2 (T, P) ) f L2 (T, P) ) x2γ2 f SCL 2 (T, P)

(9)

where subscripts 1 and 2 represent the solvent and the solute, respectively. Assuming that the solid phase contains solute only (that consequently implies that the presented models can not explain the SLG curves with a temperature maximum and a S (T,P), and then eq 1 or temperature minimum6), f 2S(T,P) ) f 20 eq 2 can be combined into eqs 7-9. Equation 6 is a GLE, whereas eq 7 is a SGE, and eq 8 and eq 9 are two forms of SLE. As indicated elsewhere,16 in addition to two known relations (x1 + x2 ) 1 and y1 + y2 ) 1), only three additional equations are required. Therefore either eq 8 or eq 9 is needed to conveniently model a binary system. When eq 7 is chosen to correlate the experimental solubility data of a solute in supercritical CO2, the original PR-EoS19 and the van der Waals one-fluid (vdW-1) mixing rule are used to obtain the binary interaction parameter k12. Then eqs 6, 7, and 8 can predict the SLG coexistence lines with the calculated k12. This is a fugacity coefficient approach (hereafter referred to as the SMS-φ model) because all the fugacities in the liquid and the gas phases should be expressed by the corresponding fugacity coefficients and calculated from the PR-EoS. When eq 9 is selected instead of eq 8, it is an activity coefficient approach (denoted later as the SMS-γ model), in which the activity coefficient can be obtained from a convenient model, namely UNIFAC.20 Linear temperature dependent UNIFAC interaction parameters are used for the gas-containing mixtures.21 Tables A1 and A2 (see Appendix) show the UNIFAC parameters, namely the group area (Qk), the volume (Rk) parameters, and the group interaction parameters. In this work, both SMS-φ and SMS-γ were tested. The calculation algorithm is similar to those presented by Li16 and by Lemert,17 and the calculation program was coded in C++ computer language. CMG Model. The SMS model is limited by the availability of solute solubility data in supercritical fluids. As it is wellknown, EoS/GE predictive models can be used to calculate the GLE, liquid-liquid equilibrium (LLE), vapor-liquid-liquid equilibrium (VLLE), and SGE.21-25 On the SMS-γ model, we can use the mixing rules that are usually adopted in the EoS/ GE models, such as the Michelsen-modified Huron-Vidal

Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009 4581

Figure 1. Schematic representation of the various steps involved in the models presented in this work. Table 1. Physical Properties of the Pure Compounds compound

Tc (K)

Pc (MPa)

ω

V L (dm3/mol)

V S (dm3/mol)

Tm (K)

∆fusH (kJ/mol)

PS,sat

myristic acida palmitic acid e stearic acid e benzoic acid f naphthalene f biphenylg phenanthrene f ibuprofenh tripalmitini carbon dioxide f

841.6 776.0 799.0 751.0 748.4 788.95 890.0 756.3 889.1 304.19

1.635 1.510 1.360 4.470 4.050 3.840 3.250 2.180 0.51 7.382

0.9612 1.061 1.084 0.6039 0.3020 0.3640 0.4290 0.7490 1.8200 0.2276

0.2644b 0.301f 0.337 f 0.112 0.131 0.1558 0.167 0.2162c 0.93

0.2235b 0.252 f 0.283 f 0.0928 0.112 0.1310 0.155 0.1875 0.87

325.45c 335.66 f 342.49 f 395.52 353.43 342.65 372.38 345.2c 337.4

45.10c 53.711 f 61.210 f 18.075 19.318 18.582 16.463 25.47c 121

d d d d d d d d d

a Tc, Pc, and ω from ref 27. b From ref 28. c Determined in this work. d Calculated from eq 3 with PL,sat estimated from the Ambrose-Walton method.29 e Tc, Pc, and ω from ref 24. f From ref 14. g From ref 17. h Tc, Pc, ω, and VS from ref 30. i Tc, Pc, ω, and ∆fusH from ref 16.

(MHV1),26 linear combination of Vidal and Michelsen (LCVM)22 mixing rules to replace the vdW-1 mixing rule in the SMS-γ model and then construct a model combining with GE models (CMG) for the SLG coexistence lines. For the original LCVM mixing rule, the attractive term parameter a of the PR-EoS can simply be expressed by eq 10, together with the commonly used linear mixing rule for the covolume parameter b: a ) RbRT

(10)

parameters, their calculations do not require any further experimental data. In eq 12, C1, LCVM and C2, LCVM are calculated from eqs 14 and 15 with λ ) 0.36, AV ) -0.623 and AM ) -0.52. When parameter λ is zero, the original LCVM (λ ) 0.36) is reduced to MHV1 (λ ) 0), that was also used in this work and compared to other options described before. In this paper, a modified LCVM mixing rule (mLCVM) was introduced by fixing the parameter λ at λ ) 0.18. Figure 1 summarizes the calculation scheme including the options of all models presented above.

where Results and Discussion

nc

∑xb

b)

(11)

i i

i)1

R)

nc GE0 b + C2,LCVM xi ln + C1,LCVM RT b i i)1

∑

1

GE0 ) RT

nc

xiai

∑ b RT i)1

(12)

i

nc

∑ x ln γ i

(13)

i

i)1

(

1 λ 1-λ ) + C1,LCVM AV AM C2,LCVM )

1-λ AM

)

(14)

(15)

In the equations above, the summations are over the number of components, nc. Similarly to eq 9, ln γi in eq 13 can be obtained from the UNIFAC model. Since the activity coefficients in eq 9 and eq 13 are calculated from the functional group interaction

The models described in the previous section were tested for the SLGE calculation of nine solutes with CO2. Table 1 lists the physical properties of the pure compounds used in this work. Results from the SMS Model. A solubility correlation for nine solutes (see Table 2) in supercritical CO2 from the PREoS with the vdW-1 mixing rule was implemented. This correlation describes well the SGE for most of the systems investigated with an average AARDy (see Table 4) of 20.6% except for the tripalmitin/CO2 system. This exception is likely related to the uncertainty of the tripalmitin’s critical parameters, estimated by a group contribution method.16 In the correlations, we used 13 solubility data points for myristic acid,31,32 27 for palmitic acid,31-33 25 for stearic acid,34,35 15 for benzoic acid,36 54 for naphthalene,37 39 for biphenyl,37 101 for phenanthrene,38 29 for ibuprofen,39 and 25 for tripalmitin.31,40 Table 2 summarizes the prediction results by the SMS model using the k12 obtained from the correlations. For all systems investigated here except myristic acid/CO2, ibuprofen/CO2, and tripalmitin/CO2, the average AADTs (see Table 4) are 5.0 and 7.4 K for the SMS-φ and SMS-γ models, respectively, indicat-

4582 Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009 Table 2. Calculation Results from the SMS Model with Correlated k12 and Fixed k12

solutea myristic acid41 palmitic acid14 stearic acid14 benzoic acid14 naphthalene14 biphenyl42 phenanthrene14 ibuprofen30,41 tripalmitin16

AADT AADT AADT AADT k12 from (K)b,c from (K)c from (K)d from (K)d from SMS-γ SMS-φ SMS-γ correlation SMS-φ 4.6 4.1 6.2 5.0 5.4 4.7 -

0.038 0.104 0.0890 0.0265 0.0971 0.0853 0.131 0.0564 -

1.9 2.3 9.6 10.3 12.6 7.5 -

1.9 4.4 4.8 3.6 4.4 2.0 9.7 1.6 5.9

1.5 2.2 1.2 2.4 9.4 7.1 13.4 5.6 1.7

a Source of melting data. b The absolute average deviation for the solute melting temperatures at SLGE is defined as: AADT ) 1/ n n∑i)1 |T icalcd - T iexpt|, where n is the number of melting temperature data. c Correlated k12 were used. d k12 ) 0.1.

Table 3. Calculated Average Absolute Deviations for the Several Systems with λ-Optimal, MHV1 (λ ) 0), LCVM (λ ) 0.36), and mLCVM (λ ) 0.18) solute

AADT (K) (λ-optimal)

AADT (K) (λ ) 0)

AADT (K) (λ ) 0.36)

AADT (K) (λ ) 0.18)

myristic acid palmitic acid stearic acid benzoic acid naphthalene biphenyl phenanthrene ibuprofen tripalmitin

0.9 (0.29) 1.9 (0.20) 1.1 (0.22) 0.8 (0.35) 2.2 (0.05) 1.9 (0.12) 1.8 (0.14) 1.5 (0.09) -

4.6 6.4 6.7 5.5 2.7 3.7 3.5 3.9 2.9

2.8 6.7 6.7 0.8 6.4 -

2.1 2.0 2.3 3.0 3.4 2.9 2.2 4.2 -

ing that the former model is slightly better, in particular for the well-studied systems such as benzoic acid/CO2, naphthalene/ CO2, biphenyl/CO2, and phenanthrene/CO2. Figure 2 shows that, for most systems, both SMS-φ and SMS-γ with correlated k12 are capable to predict the trend of SLG coexistence lines; yet, they can not predict the trend for the myristic acid/CO2 (Figure 2a) and ibuprofen/CO2 systems over about 7 MPa (not shown). For these two systems, a larger k12 is expected for a good prediction of SLG data. It has been reported that the PR-EoS with the vdW-1 mixing rule could give an acceptable SGE prediction using a constant k12 (such as k12 ) 0.1 for many solids except steroids and hydroxyl-aromatic acids).24 The average AARDy is 62.9% for all the investigated systems (58.2% for systems, except tripalmitin/CO2). In addition, k12 ) 0.1 could also provide trend predictions of the SLGE data for all the investigated systems using both the SMS-φ and SMS-γ models as Table 2 summarizes. The average AADTs obtained from both models using a constant k12 are, respectively, 4.3 and 4.9 K for all the investigated systems; the average AADTs are 4.8 and 6.0, respectively, for systems except myristic acid/CO2, ibuprofen/

CO2, and tripalmitin/CO2. Compared to those obtained from a correlated k12, the results show that, for most systems, a constant k12 ) 0.1 provides better SLGE results. Figure 3 compares the experimental42 with the calculated solute compositions (x2) in liquid phase at SLGE for the naphthalene/CO2 and biphenyl/CO2 systems. Results indicate that SMS-predictions for φ are slightly better than the results for SMS-γ, while the predictions for SMS-φ using a constant k12 are better, in particular for the biphenyl/CO2 system, than those using a correlated k12 at high pressures. Results from the CMG Model. For the CMG model, a first approach was adopted using the MHV1 (λ ) 0) mixing rule and the LCVM (λ ) 0.36) mixing rule. The calculated results are shown and compared in Figure 4. When using LCVM, it is clear that the calculated SLG coexistence lines are far below the experimental data, in particular for the ibuprofen/CO2 system (not shown). On the contrary, when using MHV1, the calculated SLG coexistence lines are usually far above the experimental data, in particular for acid-containing systems. We determined the optimal λ by minimizing the difference between calculated and experimental melting data for each system with the LCVM mixing rule. Table 3 lists the correlated λ values and the corresponding AADTs. The average of these correlated λ is 0.18, which is precisely the average λ at λ ) 0 (MHV1) and at λ ) 0.36 (LCVM). This λ ) 0.18 serves as the revaluated parameter in the modified LCVM mixing rule (mLCVM) to be applied to the CMG model. The calculated results from the CMG model with the mLCVM mixing rule are summarized in Table 3, together with the values with the optimal λ. On the basis of these values and in Figure 4 representation when the recommended value of λ ) 0.18 is used, the calculation accuracy is much improved: for most systems AADT is about 3 K, and the average AADT for all systems is 2.6 K (see Table 4), indicating that the CMG model gives reasonable calculations. Nevertheless, the CMG model did not give a satisfactory calculation for the tripalmitin/ CO2 system, just as the SMS model did not, which may again be attributed to the uncertainty inherent to the estimation of the tripalmitin’s critical constants. Figure 5 compares the calculated solute composition (x2) in liquid phase at SLGE with the experimental data for the naphthalene/CO2 and biphenyl/CO2 systems. The same figure indicates that the CMG model with the mLCVM mixing rule performs better than that with the MHV1 and LCVM mixing rules in liquid composition calculations. In the case of the calculation of the solute solubility data (y2) in supercritical CO2, Table 4 shows that the CMG model with LCVM performs better than with MHV1 and mLCVM. It is noted that the CMG model with the LCVM mixing rule calculates well the y2 data, but on the contrary fails by a significant margin the calculation of the solute melting temperatures and

Table 4. Calculation Results from the New Models Presented in This Work SMS models AADT (K)a AADT (K)b AARDy (%)a,c AARDx (%)d

CMG

SMS-φ correlated k12 SMS-γ correlated k12 SMS-φ constant k12 SMS-γ constant k12 MHV1 λ ) 0 LCVM λ ) 0.36 mLCVM λ ) 0.18 5.0e 6.8e 20.6 13.4

7.4e 11.3e 20.6 16.4

4.1 5.9 58.2 6.1

5.4 7.7 58.2 14.7

4.6 6.1 74.3 31.9

4.7 f 8.7 f 42.7 -

2.6 4.3 62.8 11.7

n calcd expt |1 - y2,i /y2,i |) AADT for systems under all investigated pressures. b AADT for systems under pressures larger than 10 MPa. c AARDy ) (1/n∑i)1 × 100%, where y2 is the solute mole fraction in supercritical CO2, n is the number of data points, and superscripts “expt” and “calcd” represent n calcd expt experimental data and calculated values, respectively. d AARDx ) (1/n∑i)1 |1 - x2,i /x2,i |) × 100%, where x2 is the solute mole fraction in liquid phase at SLGE, n ) 31 is the total number of data points for the naphthalene/CO2 and biphenyl/CO2 systems. e AADT of the investigated systems except f myristic acid/CO2 and ibuprofen/CO2. AADT of the investigated systems except naphthalene/CO2, biphenyl/CO2 and ibuprofen/CO2. g Tripalmitin/CO2 is excluded in calculating AADT and AARDy. a

Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009 4583

Figure 2. SLG coexistence lines of some typical binary systems: (a) myristic acid/CO2; (b) palmitic acid/CO2; (c) naphthalene/CO2; (d) biphenyl/CO2. (2) Experiment; (s) SMS-φ with a correlated k12; (---) SMS-γ with a correlated k12; ( · · · ) SMS-φ with k12 ) 0.1; (- · - · ) MS-γ with k12 ) 0.1.

Figure 3. Mole fractions in liquid phase for naphthalene/CO2 (left) and biphenyl/CO2 (right) at SLGE: (9) experiment; (s) SMS-φ with a correlated k12; ( · · · ) SMS-γ with a correlated k12; (---) SMS-φ with k12 ) 0.1; (- · - · ) SMS-γ with k12 ) 0.1.

of the x2 data for the naphthalene/CO2 and biphenyl/CO2 systems at high pressures. Model Selection. The calculation results including AADT and AARDx at SLGE and AARDy for the investigated systems from all the presented models in this work are compared in Table 4. According to this table, the CMG model with mLCVM and SMS-φ with k12 ) 0.1 are recommended for quantitative calculations when only the melting temperature at SLGE is concerned or when we need both the melting temperature and x2 at SLGE. When both the melting point and x2 and y2 data are required, SMS-φ and SMS-γ with k12 ) 0.1 and CMG with mLCVM are recommended for trend calculations. If there are available solute solubility data in supercritical CO2, the SMS-φ model with a correlated k12 is recommended for acceptable predictions of the melting temperature and x2 at SLGE, but it

may fail predictions for some simple solutes as Table 2 indicates. Table 4 also shows AADTs for the investigated systems under pressures larger than 10 MPa, a pressure range usually available for particle formation processes with supercritical fluids, indicating slightly worse predictions of the proposed models at high pressures. Conclusions In this work, two models, the SMS model and the CMG model, were developed for modeling the SLGE of binary systems containing high pressure CO2. Results from the models were compared, and the following conclusions can be drawn: (1) The two traditionally applied equations for the fugacity of solid solute are proved to be identical when it is used an

4584 Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009

Figure 4. SLG coexistence lines of some typical binary systems: (a) myristic acid/CO2; (b) palmitic acid/CO2; (c) naphthalene/ CO2; (d) biphenyl/CO2. (2) Experiment; (s) CMG with MHV1 (λ ) 0); (---) CMG with LCVM (λ ) 0.36); ( · · · ) CMG with mLCVM (λ ) 0.18).

Figure 5. Mole fractions in liquid phase for naphthalene/CO2 (left) and biphenyl/CO2 (right) at SLGE: (9) experiment; (s) CMG with MHV1 (λ ) 0); (---) CMG with LCVM (λ ) 0.36); ( · · · ) CMG with mLCVM (λ ) 0.18).

appropriate relation between the pure-solid vapor pressure and the pure-liquid vapor pressure; therefore, any of them can be used to implement appropriate SLGE calculation models. (2) Two methods, SMS-φ and SMS-γ, applied to the SMS model give good correlations of the solute solubility in supercritical CO2; they also give relatively accurate calculations of the melting temperatures and the solute compositions in the liquid phase at SLGE. (3) Two methods, SMS-φ and SMS-γ, with k12 ) 0.1 can be used for trend predictions of the solute solubility in supercritical CO2 and the melting temperatures and the solute compositions in liquid phase at SLGE. (4) The CMG model shows that the mLCVM mixing rule with a re-evaluated λ ) 0.18 provides good calculations of the melting temperatures and of the solute compositions in liquid

phase at SLGE, and also acceptable calculations of the solute solubility data in supercritical CO2. Acknowledgment For financial support, the authors are grateful to SRF for ROCS, SEM, NCET of Fujian Province, NSFC, China (Project 20876127), European Union Programme FEDER, and FCT, Lisbon (Project POCI/EQU/55911/2004). Appendix UNIFAC Parameters. Table A1 gives the group area and volume parameters for the original UNIFAC model; Table A2 gives the group interaction parameters for the original UNIFAC model.

Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009 4585 Table A1. Group Area (Qk) and Volume (Rk) Parameters for the Original UNIFAC Model23,29 group number 1

group CH2

3

ACH

20 41 56

COOH COO CO2

subgroup

Rk

Qk

CH3 CH2 CH ACH AC COOH COO CO2

0.9011 0.6744 0.4469 0.5313 0.3652 1.3013 1.3800 1.29623

0.848 0.540 0.228 0.400 0.120 1.224 1.200 1.26123

Table A2. Group Interaction Parameters for the Original UNIFAC Model23,29 group m

group n

Amn [K]

Anm [K]

Bmn

Bnm

CO2 CO2 CO2 CO2 CH2 CH2 CH2 ACH

CH2 ACH COO COOH ACH COOH COO COOH

110.60 -26.80 168.6 218.57 61.13 663.5 387.1 537.4

116.70 187.00 -31.7 358.13 -11.12 315.3 529.0 62.32

0.5003 -1.2348 -2.676 -0.7217 0 0 0 0

-0.9106 1.0982 6.977 -0.3666 0 0 0 0

a Interaction parameters between CO2 and groups are from ref 23. Interaction parameters between groups of pure compounds are from ref 29.

b

List of Symbols a, b ) equation of state parameters AADT ) absolute average deviation of temperature AARD ) absolute average relative deviation CMG ) calculation model combining with GE models EoS/GE ) equation of state combined with an excess Gibbs model f ) fugacity ∆fusH ) enthalpy of fusion (kJ mol-1) GLE ) gas-liquid equilibrium k12 ) binary interaction parameter between components 1 and 2 LCVM ) linear combination of Vidal and Michelsen mixing rules LLE ) liquid-liquid equilibrium mLCVM ) LCVM mixing rule modified in this work MHV1 ) Michelsen modified Huron-Vidal mixing rule n ) number of data points NRTL ) nonrandom two-liquid activity coefficient model P ) pressure (MPa) PGSS ) particles from gas-saturated solutions PR-EoS ) Peng-Robinson equation of state R ) gas constant (J K-1 mol-1) RESS ) rapid expansion of supercritical solutions RST ) regular solution theory SCF ) supercritical fluids SGE ) solid-gas equilibrium SLE ) solid-liquid equilibrium SLG ) solid-liquid-gas SLGE ) solid-liquid-gas equilibrium SMS ) semipredictive model with solubility data SMS-φ ) semipredictive model using the fugacity coefficient method SMS-γ ) semipredictive model using the activity coefficient method T ) temperature (K) Tm ) melting temperature (K) UCEP ) upper critical end point V ) molar volume (dm3 mol-1) vdW-1 ) van der Waals mixing rule (one interaction parameter) VLE ) vapor-liquid equilibrium

VLLE ) vapor-liquid-liquid equilibrium x ) mole fraction in liquid phase y ) mole fraction in vapor phase Greek letters λ ) parameter of the LCVM mixing rule γ ) activity coefficient φ ) fugacity coefficient ω ) acentric factor Superscripts calcd ) calculated expt ) experimental G ) gas phase L ) liquid phase sat ) saturation S ) solid phase SCL ) subcooled liquid phase Subscripts 0 ) pure substance 1 ) solvent 2 ) solute c ) critical property fus ) fusion i,j ) components m ) melting

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ReceiVed for reView July 31, 2008 ReVised manuscript receiVed February 26, 2009 Accepted February 26, 2009 IE801179A