Ind. Eng. Chem. Res. 2003, 42, 4953-4961
4953
Vapor-Liquid Equilibrium of Systems Containing Alcohols Using the Statistical Associating Fluid Theory Equation of State Xiao-Sen Li and Peter Englezos* Department of Chemical and Biological Engineering, The University of British Columbia, 2216 Main Mall, Vancouver, B.C., Canada V6T 1Z4
The statistical associating fluid theory (SAFT) equation of state is employed for the correlation and prediction of vapor-liquid equilibrium (VLE) of binary mixtures of alcohols with water, carbon dioxide, butane, hexane, benzene, and other alcohols. In addition, ternary VLE for water/ 1,2-propanediol (propylene glycol)/1,2-ethanediol (ethylene glycol), carbon dioxide/methanol/ ethanol, and water/1,3 propanediol/1,2,3-propanetriol (glycerol) mixtures is predicted. In the SAFT equation, three molecular parameters, the Lennard-Jones (L-J) potential well depth, the soft-sphere diameter of the segments, and the number of segments of the molecule, are needed. These parameters are obtained from the thermodynamic properties of pure substances. For selfassociating pure substances, two additional parameters are needed, namely, the bonding volume and the association energy. The binary interaction parameters are fitted to experimental vaporliquid equilibrium data for binary systems. These binary parameters are used to predict the phase equilibria for ternary mixtures without any additional adjustment. The results were found to be in good agreement with the experimental data. 1. Introduction
Ares ) A - Aid ) Ahs + Achain + Adisp + Aassoc (1)
The statistical associating fluid theory (SAFT) is based on Wertheim’s first-order thermodynamic perturbation theory for associating fluids1 and has been developed very rapidly in recent years.2-5 Molecular-based equations of states with salient physically meaningful parameters are generally more reliable for extrapolation and prediction than empirical models. Consequently, SAFT has been used to model successfully a wide variety of thermodynamic properties and phase equilibria for industrially important fluids containing n-alkane mixtures,6 hydrogen fluoride,7 hydrogen chloride,8 alcohols,9-11 surfactants,12,13 and aqueous ethanolamine solutions.14 In this work, we applied the SAFT equation for the correlation and prediction of vapor-liquid equilibria of binary mixtures of alcohols with n-alkanes, water, carbon dioxide, and other alcohols. Furthermore, we employed SAFT to predict VLE in the ternary systems water/ 1,2-propanediol/1,2-ethanediol, carbon dioxide/methanol/ ethanol, and water/1,3 propanediol/1,2,3-propanetriol. 2. Equation of State In this paper, alcohols, n-alkanes, and carbon dioxide are described as homonuclear chainlike molecules. It is noted that this is an approximation because the actual molecules are heteronuclear. Therefore, here, a molecule of the chain or polar fluid is considered to be composed of m hard-spherical segments of diameter σ tangentially bonded together to form chains. The equation of state is written in terms of the Helmholtz free energy. The residual Helmholtz free energy for an n-component mixture of associating chain molecules can be expressed as the sum of hard-sphere repulsion, hard-chain formation, dispersion, and association terms * To whom correspondence should be addressed. Tel.: 1-604822-6184. Fax: 1-604-822-6003. E-mail: englezos@interchange. ubc.ca.
where Aid is the free energy of an ideal gas with the same density and temperature as the system; Ahs is the free energy of a hard-sphere fluid relative to that of the ideal gas; Achain is the free energy associated with the formation of chains from hard spheres; and Adisp and Aassoc are the contributions to the free energy of dispersion and association interactions, respectively. 2.1. Hard-Sphere Repulsion Term. The hardsphere term, Ahs, is calculated with the BoublikMansoori-Carnahan-Starling-Leland equation15,16 k
Ahs
ximi ∑ i)1
6 )
NkT
πFs
[
3ζ1ζ2 - ζ32/ζ23 1 - ζ3
ζ32 ζ23
]
ζ32/ζ23
+
ln(1 - ζ3) -
+
(1 - ζ3)2 k
ximi ln(1 - ζ3) ∑ i)1
(2)
where
π k ζl ) Fn ximidiil (l ) 0, 1, 2, 3) 6 i)1
∑
(3)
k
Fs ) Fn
ximi ∑ i)1
(4)
In the above equations, Fn is the total number density of molecules in the solution, and dii is the hard-sphere diameter of segment i. Its relationship to the soft-sphere diameter (σii) is based on the Barker-Henderson perturbation theory and is expressed by Cotterman et al.17 as follows
1 + 0.2977kT/ii dii ) (5) σii 1 + 0.331 63kT/ + 0.001 047(kT/ )2 ii ii where ii is the energy parameter of the L-J potential.
10.1021/ie030256o CCC: $25.00 © 2003 American Chemical Society Published on Web 09/06/2003
4954 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003
2.2. Hard-Chain Formation Term. The chain term, Achain, was derived by Chapman et al.3 as
A
chain
)
xi (1 - mi) ln[ghs ∑ ii (dii)] i)1
(6)
hs gseg ij (dij) ≈ gij (dij) )
3diidjj ζ2 1 + + 1 - ζ3 dii + djj (1 - ζ )2 3
(
)
Aassoc
ζ22
diidjj (7) 2 dii + djj (1 - ζ )3 3 2
Equation 7 for like segments becomes
3diiζ2 d2iiζ22 1 ) + + 1 - ζ3 2(1 - ζ )2 2(1 - ζ )3 3 3
(8)
ij ) (1 - kij)xiijj
(18)
disp
k
)
1
ximi ∑ T i)1
(Adisp + Adisp 1 2 /TR)
(9)
R
where
Adisp ) FR(-8.5959 - 4.5424FR - 2.1268F2R + 1 10.285F3R)
(10)
) FR(-1.9075 + 9.9724FR - 22.216F2R + Adisp 2 15.904F3R) (11) TR ) kT/x
(12)
6 ζ3 x2π
(13)
FR ) k
3
xσx )
k
∑ ∑ i)1 j)1 k
σx3 )
yiyjijσ3ij
yi )
(14)
k
yiyjσ3ij ∑ ∑ i)1 j)1 ximi k
NkT
)
[
1
(ln XA - XA /2) + Mi ∑i xi ∑ 2 A i
i
i
]
(19)
where Mi is the number of associating sites on molecule i. The term XAi is defined as the mole fraction of molecules i, in mixtures with other components, not bonded at site A and is given by
XAi ) (1 + NA
xjFXB ∆A B )-1 ∑j ∑ B j
i j
(20)
j
2.3. Dispersion Term. Dispersion forces are taken into account through the term Adisp, which is calculated using the expression of Cotterman et al.17 based on the Lennard-Jones potential. This expression has been found to be accurate by comparison with Lennard-Jones molecular simulation data.18
NkT
(17)
where kij is the binary interaction parameter. 2.4. Association Term. The Helmhotz energy due to association is calculated using the expression of Chapman et al.4
where
A
σij ) (σii + σjj)/2
k
NkT
ghs ii (dii)
the following combining rules
(15)
(16)
xjmj ∑ j)1 In the above equations, σij and ij are the cross parameters between different segments and are calculated by
where ∑Bj indicates a summation over all sites on molecule j, Aj, Bj, Cj, ...; ∑j indicates a summation over all components; F is the total molar density of molecules in the solution; and ∆AiBj is the associating strength and is given by
∆AiBj ) dij3gij(dij)segκAiBj[exp(AiBj/kT) - 1]
(21)
In eq 21, κAB is the bonding volume, and AB/k is the associating energy. For cross-associating mixtures, we use the combing rules4
κAjBi ) κAiBj ) (κAiBi + κAjBj)/2
(22)
AiBi AjBj AjBi ) AiBj ) (1 - kAB ) ij )x(
(23)
where kAB ij is the binary associating interaction parameter. The expressions for the chemical potentials of the components and for the compressibility factor needed in phase equilibrium calculation are given in Appendix A and Appendix B, respectively. In Appendix C, the procedure for the calculation of vapor-liquid phase equilibrium is presented. 3. Pure-Component and Binary Interaction Parameter Estimation The SAFT equation requires three pure-component parameters for nonassociating fluids and five parameters for associating fluids. These parameters are the L-J potential well depth (/k), the soft-sphere diameter of the segments (σ), the number of segments in the molecule (m), the bonding volume (κAB), and the association energy between sites A and B (AB). These parameters can be estimated from the regression of saturated vapor pressure and liquid density data. In the case of mixtures, the SAFT equation uses van der Waals one-fluid mixing rules with the binary interaction parameter kij for the dispersion interactions and the parameter kAB ij for the associating interactions. These parameters are fitted to VLE data. A water molecule has two hydrogen atoms and two lone pairs of electrons that are able to form hydrogen bonds. As a result, each water molecule is capable of
Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4955 Table 1. Estimated Segment Parameters for Pure Fluids fluid
T (K)
m
σ (10-10 m)
/k (k)
AB/k (K)
κAB
water butane hexane benzene methanol ethanol ethylene glycol propylene glycol 1,3-propylene glycol glycerol DEG TEG
283-483 282-379 304-455 310-478 295-489 303-503 375-496 355-446 377-487 415-551 374-519 408-551
0.982 2.163 2.739 2.103 1.124 1.522 1.043 1.678 1.679 2.180 2.842 3.204
2.985 3.784 3.911 3.833 3.642 3.668 4.232 3.957 3.952 4.194 3.583 3.805
433.91 214.00 230.03 290.90 309.90 263.72 354.65 346.88 346.88 405.08 314.84 252.03
1195.20
0.038
2320.77 2616.52 2375.26 1910.97 1910.96 2195.15 1859.72 2470.02
0.019 0.012 0.020 0.021 0.037 0.004 0.031 0.061
RMSD Pa (%)
RMSD Fb (%)
0.39 2.72 5.04 5.37 1.01 0.25 0.99 1.29 1.89 0.95 2.10 3.67
0.59 0.51 0.52 1.01 0.79 0.94 0.50 0.49c 0.70c 0.47 0.73c 1.24c
x
N exp exp 2 b RMSD P (%) ) 100 ∑i)1 [(Pcal i -Pi )/Pi ] /(N-1), root-mean-square deviation for saturated vapor pressures. RMSD F (%) ) 100 N cal exp exp 2 c ∑i)1[(Fi -Fi )/Fi ] /(N-1), root-mean-square deviation of saturated liquid densities. Root-mean-square deviation of saturated vapor
a
x
densities.
Table 2. Binary Interaction Parameters Estimated from Regression of Binary VLE Experimental Data T (K)
kij
kAB ij
RMSD ya
RMSD P (%)
data source
water (1) + EG (2) water (1) + propylene glycol (2) water (1) + 1,3-propanediol (2) water (1) + glycerol (2) water (1) + TEG (2)
371.15-383.15 371.15-383.15 343.35-451.65 344.85-527.55 477.59
Water/Alcohol Systems 0.0302 0.0128 0.0214 -0.3398 + 119.85/T 0.3446
-0.0489 -0.0538 -0.0581 -0.1464 + 116.62/T 0.3473
0.0157 0.0078 0.0230 0.0079 0.0375
4.92 4.04 5.89 5.81 2.32
26 26 27 28 28
methanol (1) + ethanol (2) propylene glycol (1) + EG(2) EG (1) + glycerol (2) propylene glycol (1) + glycerol (2)
313.20 371.15-383.15 369.65-434.55 361.75-454.05
Alcohol/Alcohol Systems 0.1140 -0.0998 -0.0277 0.0227 0.0674 0.0145 0.1068
0.0289 0.0145 0.0104
2.86 3.57 4.24 3.84
29 26 29 29
butane (1) + methanol (2)
Alcohol/Alkane Systems 0.0009 0.0008 -0.0122 0.1390 -0.1651
0.0550 0.0426 0.0479 0.0235 0.0021
6.63 12.12 23.50 7.89 3.40
30
hexane (1) + TEG(2) benzene (1) + TEG(2)
373.15 323.15 273.15 473.00 354.95-428.75
31 32
CO2 (1) + methanol (2) CO2 (1) + ethanol (2)
313.20 313.20
Alcohol/CO2 Systems -0.1064 0.0330
0.0054 0.0059
8.03 4.53
33 33
system
a
RMSD y )
N exp 2 (ycal x∑i)1 i -yi ) /(N-1), root-mean-square deviation of component 1.
forming up to four hydrogen bonds. Thus, the three-site model and the four-site model are generally used for the water molecule. Economou and Tsonopoulos19 and Yakoumis et al.20 showed that the four-site model performs better with SAFT. Consequently, this approach is used in this study. Each hydroxyl group on an alcohol molecule has one hydrogen and two lone pairs of electrons that are able to form hydrogen bonds. Thus, each hydroxyl group can generate up to three hydrogen bonds. Here, a two-site model is used for each hydroxyl group on the alcohol, as was also done by Muller et al.21 and Gupta et al.22 In this work, the segment parameters (/k, σ, m, κAB, AB ) for water, n-butane, n-hexane, benzene, methanol, ethanol, glycerol, ethylene glycol (EG), propylene glycol (1,2-propanediol), 1,3-propanediol, diethylene glycol (DEG), and triethylene glycol (TEG) are obtained by simultaneously fitting the experimental saturated vapor pressures and liquid (or vapor) densities available in the literature.23,24 It is noted that, for propylene glycol, 1,3-propanediol, DEG, and TEG, because experimental data on their saturated liquid densities could not be found in the literature, saturated vapor pressures and saturated vapor densities were used to estimate the segment parameters. In addition, with regard to steric
hindrance effects, for simplicity, we neglected the association of the oxygens of the oxyethylene units on the DEG molecule and the TEG molecule in this work. The regressed segment parameters for the pure fluids and the deviations in vapor pressures and liquid (or vapor) densities are reported in Table 1. The fit was found to be excellent. These segment parameters for pure fluids are used for the estimation of the interaction parameters for the corresponding binary systems. Finally, it is noted that the optimization is carried out using the simplex method.25 4. Phase Equilibria of Binary Mixtures This work considers the following binary mixtures: methanol/butane, TEG/hexane, TEG/benzene, ethylene glycol/water, propylene glycol/water, 1,3-propanediol/ water, glycerol/water, TEG/water, ethylene glycol/propylene glycol, ethylene glycol/glycerol, propylene glycol/ glycerol, carbon dioxide/methanol, carbon dioxide/ ethanol, and methanol/ethanol. Using the mixing rules of eqs 17, 18, 22, and 23, the binary interaction parameters are obtained by fitting vapor-liquid phase equilibrium experimental data. The results are shown in Table 2.
4956 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003
Figure 1. Calculated results of vapor-liquid equilibrium for the water (1)/ethylene glycol (2) system at 371.15 K, 383.15 K and 395.15 K.
Figure 3. Predicted results of vapor-liquid equilibrium for the water (1)/ethylene glycol (2) system at 6.67 KPa.
Figure 2. Calculated results of vapor-liquid equilibrium for the water (1)/propylene glycol (2) system at 371.15, 383.15, and 395.15 K.
Figure 4. Calculated results of vapor-liquid equilibrium for the carbon dioxide (1)/methanol (2) system at 313.2 K.
Table 3. Predicted VLE Results for Water/EG, Water/ Propylene Glycol, and Propylene Glycol/EG Systems system water (1) + EG (2)
T (K)
RMSD RMSD data y P (%) source
395.15 0.0231 311.50-406.50 0.0166 water (1) + propylene 395.15 0.0129 glycol (2) propylene glycol (1) 395.15 0.0180 + EG (2)
5.17 4.98 5.05
26 29 26
3.56
26
Figures 1 and 2 present composition diagrams for water/ethylene glycol and water/propylene glycol at 371.15, 383.15, and 395.15 K. Only the data at 371.15 and 383.15 K were used for parameter estimation. Using the binary parameters obtained from the fitting of these data, we employed SAFT to make prediction at 395.15 K, as seen in Table 3. The agreement between the data and the predicted values is excellent. For the water/ ethylene glycol system, we also made additional predictions, which are listed in Table 3 and shown in Figure 3. The results for other water/alcohol systems are not shown but were found to be equally good.
Figure 4 shows the results for carbon dioxide/ methanol system, and Figure 5 shows the results for ethylene glycol/glycerol. The results for propylene glycol/ glycerol are not shown but were equally satisfactory. In all cases, there is very good agreement between the data and the calculated values. Only at the critical point of CO2/methanol were the calculations found to be inadequate. It is noted that carbon dioxide/ethanol and other alcohol/alcohol systems exhibited equally good agreement between data and calculated values. The segment parameters of carbon dioxide [the L-J potential well depth (/k), the soft-sphere diameter of the segments (σ), and the number of segments in the molecule (m)] were obtained from fitting the experimental data for the vapor-liquid phase equilibrium of the carbon dioxide/methanol mixture at 313.2 K. These regressed parameter values are /k ) 125.80 K, σ ) 3.144 × 10-10, and m ) 1.833. This approach was followed because the temperature is above the carbon dioxide critical temperature, and therefore, it is not reasonable to use the segment parameters of pure carbon dioxide obtained from fitting the experimental saturated vapor pressures and liquid densities at subcritical conditions.
Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4957
Figure 5. Calculated results of vapor-liquid equilibrium for the ethylene glycol (1)/glycerol (2) system at 1.33 KPa.
Figure 7. Calculated results of vapor-liquid equilibrium for the hexane (1)/triethylene glycol (2) system at 473.00 K.
Figure 6. Calculated results of vapor-liquid equilibrium for the butane (1)/methanol (2) system at 273.15, 323.15, and 373.15 K.
Figure 8. Liquid phase stability function for the hexane (1)/ triethylene glycol (2) system at 473.00 K.
Figure 6 shows the results for butane/methanol at three temperatures, 273.15, 323.15, and 373.15 K. The agreement is satisfactory except at 273.15 K, where the deviation is 23.5%, as seen in Table 2. All other alcohol/ alkane results were good except for TEG/hexane, as shown in Figure 7. The problem here is not the rootmean-square deviation (RMSD) but the fact that the calculated values show erroneous phase separation. For a binary liquid mixture to be stable and remain as a single liquid phase, the following condition has to be satisfied
stability condition is satisfied for this system. However, it is satisfied at the cost of a large calculated deviation. Such situations are often encountered with correlated VLE data for alkane/alcohol systems.34 The problem can be eliminated via a constrained parameter estimation approach described by Englezos and Kalogerakis.35
SF )
∂ ln f L1 >0 ∂x1
(24)
where f L1 is the fugacity of component 1 in the liquid (L) phase and SF stands for stability function. Here, the function SF is evaluated at all experimental data points at 473 K for the hexane/TEG mixture, and the results are shown in Figure 8. As can be seen, SF becomes negative between x ) 0.53 and 0.88 at kij ) 0.1390, and hence, the condition for liquid-phase stability is violated. SF becomes positive at kij ) 0.0349, and thus, the
5. Phase Equilibria of Ternary Mixtures The vapor-liquid phase equilibria of the ternary mixtures can be predicted with the binary interaction parameters of the constituent binary subsystems and the molecular parameters of the pure components. Here, three ternary systems were considered. Table 4 summarizes the predicted results. 5.1. Water/Propylene Glycol/Ethylene Glycol. The ternary composition diagrams of this mixture at 371.15, 383.15, and 395.15 K were calculated. The results at 371.15 K are shown in Figure 9. The dashed tie lines are experimental results, whereas the solid tie lines are calculated. Because vapor-phase compositions mainly concentrate on the right lower corners of the phase diagrams, it is quite difficult to compare the calculated values with the experimental data. To make
4958 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 Table 4. Predicted VLE Results for Ternary Vapor-Liquid Equilibria system water (1) + propylene glycol (2) + EG (3) CO2 (1) + methanol (2) + ethanol (3) water (1) + 1,3-propanediol (2) + glycerol (3) a
RMSD y )
T (K)
P (kPa)
y1
RMSD ya y2
y3
371.15 383.15 395.15 313.2 313.2 313.2 372.25-397.95
5.3-34.7 9.3-42.7 14.7-49.1 2.0 × 103 4.0 × 103 6.0 × 103 30.0
0.0336 0.0379 0.0327 0.0016 0.0005 0.0038 0.0366
0.0226 0.0202 0.0263 0.0026 0.0008 0.0016 0.0360
0.0120 0.0188 0.0090 0.0011 0.0011 0.0020 0.0015
data source 26 33 27
N exp 2 (ycal x∑i)1 i -yi ) /(N-1), root-mean-square deviation of component 1, 2, or 3.
Figure 9. Predicted results of vapor-liquid equilibrium for the water (1)/propylene glycol (2)/ethylene glycol (3) system at 371.15 K.
Figure 10. Comparison between calculated and experimental vapor mole fractions for the water (1)/propylene glycol (2)/ethylene glycol (3) system at 371.15 K.
an effective comparison, the experimental vapor-phase mole fractions can be plotted versus the calculated values for two of the three components, namely, propylene glycol and ethylene glycol. The results at 371.15 K are shown in Figure 10. As can be seen in this figure, the predictions are quite satisfactory. Similar predictions were found at the other temperatures 5.2. Carbon Dioxide/Methanol/Ethanol. Predicted and experimental phase diagrams of this mixture at 2, 4, and 6 MPa and 313.2 K were obtained. The results at 2 MPa are shown in Figure 11. Again, the plots of the calculated vapor-phase mole fractions versus the experimental values are presented to better illustrate the agreement between the calculated values and experimental data, as shown in Figure 12. As can be seen, the agreement is good. 5.3. Water/1,3-Propanediol/Glycerol. The binary parameters obtained from fitting the experimental VLE
Figure 11. Predicted results of vapor-liquid equilibrium for the carbon dioxide (1)/methanol (2)/ethanol (3) system at 313.2 K and 2.0 MPa.
Figure 12. Comparison between calculated and experimental vapor mole fractions for the carbon dioxide (1)/methanol (2)/ethanol (3) system at 313.2 K and 2.0 MPa.
data of 1,2-propanediol/glycerol were used for the 1,3propanediol/glycerol system, because experimental VLE data for 1,3-propanediol/glycerol could not be found in the literature. It is noted that 1,2-propanediol and 1,3propanediol have similar structures, and as seen in Table 1, the values of their corresponding molecular parameters are almost the same. Table 4 shows the rootmean-square deviations (RMSDs) between the predicted results and experimental data in the vapor-phase compositions of this ternary mixture at 30 kPa and temperatures from 372.25 to 397.95 K. The predictions compare very well with the experimental data. 6. Conclusions The SAFT equation of state has been employed to predict the phase equilibria for three ternary systems,
Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4959
water/propylene glycol/ethylene glycol/, carbon dioxide/ methanol/ethanol, and water/1,3-propanediol/glycerol. Satisfactory agreement has been found between predicted values and experimental data. Pure-compound parameters were obtained from the correlations of saturated vapor pressures and liquid (or vapor) densities. The molecular parameters of carbon dioxide at temperatures above its critical temperature were obtained by fitting the vapor-liquid equilibrium experimental data for carbon dioxide/methanol system. Binary parameters were estimated from the regression of the vapor-liquid equilibrium data for binary mixtures of n-alkane/alcohol, alcohol/water, alcohol/alcohol, and carbon dioxide/alcohol. The quality of the computed binary phase equilibrium is also very good, as was found from a comparison with the data. Acknowledgment The financial support of the National Science and Engineering Research Council of Canada (NSERC) is greatly appreciated. Nomenclature A ) Helmholtz free energy, J d ) hard-sphere diameter, 1 × 10-10 m g ) radial distribution function k ) Boltzmann constant, J K-1 m ) effective number of segments M ) number of associate sites N ) number of molecules NA ) Avogadro’s number, 6.022 17 × 1023 mol-1 R ) ideal gas constant, 8.3143 J mol-1 K-1 T ) absolute temperature, K x ) mole fraction in the liquid phase xi ) mole fraction of component i XAi ) mole fraction of molecule i not bonded at site A y ) mole fraction in the vapor phase Z ) compressibility factor Greek Letters β ) 1/kT /k ) energy parameter of dispersion, K AB/k ) energy parameter of association between sites A and B, K κAB ) bonding volume ∆AB ) association strength between sites A and B µ ) chemical potential F ) molar density, mol/m-3 Fn ) number density, m-3 σ ) soft-sphere diameter, 1 × 10-10 m
sion, and association terms. The expression is as follows hs chain µres + µdisp + µassoc i ) µi + µi i i
The hard-sphere contribution to the chemical potential, µhs i , can be expressed as
3mid3iiζ1ζ2 + 3mid2iiζ22/ζ23 2mid3iiζ32/ζ23 mid3iiζ32/ζ23 - 2mid3iiζ32/ζ33 + + (1 - ζ3)3 (1 - ζ3)2 +
3mid2iiζ22ζ3- 2mid3iiζ32 ζ33
Appendix A. Expression for the Chemical Potentials of Components We discuss the chemical potential terms in the order used to present the corresponding Helmholtz energy terms, that is, in the order hard-sphere, chain, disper-
ln(1 - ζ3) - mi ln(1 - ζ3) (A2)
The chain contribution to the chemical potential, µchain , can be expressed as i
µchain i RT
) (1 - mi) ln[ghs ii (dii)] + k
F
∑ l)1
where
[ ] ∂ghs ll (dll) ∂Fi
[ ]
xl(1 - ml) ∂ghs ll (dll) ghs ll (dll)
∂Fi
[
2(1 - ζ3)3
+
(A3)
T,V,Fj*i
dii π ) mid2ii + 6 (1 - ζ3)2 T,V,Fj*i
3dll(1 - ζ3 + 2diiζ2)
]
d2ll(2ζ2 - 2ζ2ζ3 + 3diiζ22) 2(1 - ζ3)4
(A4)
The dispersion contribution to the chemical potential, µdisp , can be expressed as i
[ ]
miAdisp Fs ∂Adisp µdisp miAdisp i 1 2 1 + + ) 2 RT TR T ∂F T R i R
-
Superscripts assoc ) association interaction chain ) hard-sphere chain disp ) dispersion interaction hs ) hard sphere A, B ) association site res ) residual term
mid3iiζ0 + 3mid2iiζ1 + 3midiiζ2 3mid2iiζ22/ζ32 + 2mid3iiζ32/ζ33 - mid3iiζ32/ζ23 1 - ζ3
µhs i ) RT
Subscripts i, j, k ) components
(A1)
(
FsAdisp 1 TR2
)[ ]
FsAdisp ∂TR 2 3 ∂Fi T
+2
R
+
T,V,Fj*i
[ ]
Fs ∂Adisp 2 2 ∂F T i R
T,V,Fj*i
(A5)
T,V,Fj*i
where
[ ] ∂Adisp 1 ∂Fi
mid3ii ) (-8.5959 - 9.0848FR x2 T,V,Fj*i 6.3807FR2 + 41.1416FR3) (A6)
[ ] ∂Adisp 2 ∂Fi
T,V,Fj*i
)
mid3ii (-1.9075 + 19.9449FR x2 66.648FR2 + 63.616FR3) (A7)
4960 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 k
[ ] ∂TR ∂Fi
k
k
xlmlσil3 - ∑xlmlσil3il) ∑ l)1 l)1
2miF(x )
chain
Z (A8)
β2x σ3x F2s
T,V,Fj*i
)
∑ i)1
ghs ii (dii)
{∑(
) RT
Ai
[ ( ) ( )]} )
1 1 ln XAi - XAi + Mi + 2 2 Aj
T,Fk*i
1
2
X Aj
( ) ∂Fj
) -(XAj)2NA
{∑
( )
T,Fj*i
[ ( ) ]}
Bi
+ XBk
∂∆
∂Fi
T,Fl*i
[∂∆AjBk/∂F
The expression for et al.4
i]T,Fj*i
(A10)
T,Fl*i
]
∂(A/NkT) ∂F
[
6ζ1ζ2ζ3 - 3ζ1ζ2 - 2ζ32/ζ + 2ζ32/ζ23
Z
xipφLi ) yipφVi
3
ln φi )
(1 - ζ3)3
]
)
∑ i)1
ximi(Zdisp 1 /TR
µres i - ln Z RT
(C4)
yi φLi Ki ) ) V xi φ
(C5)
i
2
3ζ2 /ζ3 - ζ2 /ζ3
(C3)
where the superscripts L and V denote the vapor phase and the liquid phase, respectively; P is the pressure; fi is the fugacity of component i; and φi is the fugacity coefficient of component i. The relation between φi and the residual chemical potential of component i is as follows4
(1 - ζ3)2 (B2)
k
disp
(C2)
where µres and Z are the residual chemical potential of i component i and the compressibility factor, respectively. and Z can be obtained from eqs A1 and B1, µres i respectively. The phase equilibrium constant, Ki, can be obtained from eq C3 as
3 2 3 m 6ζ1ζ2 - ζ2/ζ3 - ζ2/ζ3 + ζ0ζ3 ζ0 1 - ζ3
3
+
f Li (T,x,FL) ) f Vi (T,y,FV) (i ) 1, 2, ..., n)
(B1)
where
+
(C1)
T,N
) 1 + Zhs + Zdisp + Zchain + Zassoc
Zhs )
P L(T,x,FL) ) PV(T,y,FV)
is given by Chapman
The compressibility factor, Z, can be calculated from the Helmholtz free energy, A, via
[
The system in vapor-liquid phase equilibrium at a certain temperature has the following relation between the two phases
Equation C2 can be expressed as
Appendix B. Expression for the Compressibility Factor
Z)F
(B7)
AjBk
∂X
∂Fi
j ki
k
Bk
- Aassoc] ∑i xiµassoc i
[
(B6)
Appendix C. Calculation of Vapor-Liquid Equilibrium
xk ∆A B ∑k ∑ B
XBi∆AjBi + F
RT
(1 - ζ3)4
(A9)
where
∂XAj
1
]
1.5d2iiζ22ζ3
The parameters µassoc and Aassoc are given by eqs A9 i and 19, respectively.
j
-
+
(1 - ζ3)3 Zassoc )
∑j xjF∑ A
1
∂X
∂Fi
+
(1 - ζ3)2
3diiζ2ζ3 + d2iiζ22
The association contribution to the chemical potential, , can be expressed as µassoc i
µassoc i
[
xi(1 - mi) ζ3 + 1.5diiζ2
+
2 Zdisp 2 /TR)
(B3)
where
Zdisp ) FR(-8.5959 - 9.0848FR - 6.3807F2R + 1 41.1416F3R) (B4) Zdisp ) FR(-1.9075 + 19.9449FR - 66.648F2R + 2 63.616F3R) (B5)
which can be transformed using eqs C4 and C5 into
Ki )
[( ) ( ) ]
µres i ZV exp RT ZL
L
-
µres i RT
V
(C6)
In this work, the vapor-phase compositions and pressures of the system are calculated using the SAFT equation, given the liquid-phase compositions and the temperatures of the system. The calculation procedure is as follows: 1. Obtain the initial values of the regressed parameters (kij, kAB ij ). 2. Obtain the initial values of the system pressure and the mole fraction of the component in the vapor phase (P0, y0).
Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4961
3. Obtain the initial values of FV,0 and FL,0, and carry out iterative calculations using the compressibility factor expression (eq B1), with T, P0, x, and y0 as known values. Finally, obtain the actual values, FV and FL, and calculate ZL and ZV. 4. Calculate Ki using eq C6. 5. Calculate yi (yi ) Kixi). 6. Determine whether s ) ∑ni yi equals 1. If it equals 1, go to step 8. Otherwise, go to next step. 7. Normalize yi. Determine the result as new y0new, V,0 choose new P 0new, FL,0 new, and Fnew, and then return to 0 0 step 3. Here, Pnew ) 0.618P + 0.382P0s. 8. Determine whether the objective function reaches a minimum. If so, the calculation ends. Otherwise, return to step 2 after adjusting the regressed parameters. Literature Cited (1) Wertheim, M. S. Thermodynamic Perturbation Theory of Polymerisation. J. Chem. Phys. 1986, 87, 7323. (2) Huang, S. H.; Radosz, M. Equation of State for Small, Large, Polydisperse and Associating Molecules. Ind. Eng. Chem. Res. 1990, 29, 2284. (3) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. Equation of State Solution Model for Associating Fluids. Fluid Phase Equilib. 1989, 52, 31. (4) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. New Reference Equation of State for Associating Liquids. Ind. Eng. Chem. Res. 1990, 29, 1709. (5) Huang, S. H.; Radosz, M. Equation of State for Small, Large, Polydisperse, and Associating Molecules: Extensions to Mixtures. Ind. Eng. Chem. Res. 1991, 30, 1994. (6) Voutsas, E. C.; Boulougouris G. C.; Economou, I. G.; Tassios, D. P. Water/Hydrocarbon Phase Equilibria Using the Thermodynamic Perturbation Theory. Ind. Eng. Chem. Res. 2000, 39, 797. (7) Visco, D. P.; Kofke, D. A. A Comparison of Molecular-Based Models to Determine Vapor-Liquid Phase Coexistence in Hydrogen Fluoride. Fluid Phase Equilib. 1999, 160, 37. (8) Galindo, A.; Florusse, L. J.; Peters, C. J. Prediction of Phase Equilibria for Binary Systems of Hydrogen Chloride with Ethane, Propane and n-dodecane. Fluid Phase Equilib. 1999, 160, 123. (9) Pfohl, O.; Pagel, A.; Brunner, G. Phase Equilibria in Systems Containing o-Cresol, p-Cresol, Carbon Dioxide, and Ethanol at 323.15-473.15 K and 10-35 MPa. Fluid Phase Equilib. 1999, 157, 53. (10) Hasch, B. M.; Maurer, E. J.; Ansanelli, L. F.; McHugh, M. A. (Methanol Plus Ethene)sPhase Behavior and Modeling with the SAFT Equation of State. J. Chem. Thermodyn. 1994, 26, 625. (11) Zhang, Z.-Y.; Yang, J.-C.; Li, Y.-G. Prediction of Phase Equilibria for the CO2-C2H5OH-H2O System Using the SAFT Equation of State. Fluid Phase Equilib. 2000, 169, 1. (12) Li, X.-S.; Lu, J.-F.; Li, Y.-G.; Liu, J.-C. Studies on UNIQUAC and SAFT Equations for Nonionic Surfactant Solutions. Fluid Phase Equilib. 1998, 153, 215. (13) Li, X.-S.; Lu, J.-F.; Li, Y.-G. Study on SAFT Equation Incorporated with MSA for Ionic Surfactant Solutions. Fluid Phase Equilib. 2000, 168, 107. (14) Button, J. K.; Gubbins, K. E. SAFT Prediction of VaporLiquid Equilibria of Mixtures Containing Carbon Dioxide and Aqueous Monoethanolamine or Diethanolamine. Fluid Phase Equilib. 1999, 160, 175. (15) Boublik, T. Hard Sphere Equation of State. J. Chem. Phys. 1970, 53, 471. (16) Mansoori, G. A.; Carnahan, N. F.; Starling, K. E.; Leland, T. W. Equilibrium Thermodynamic Properties of the Mixture of Hard Spheres. J. Chem. Phys. 1971, 54, 1523.
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Received for review March 24, 2003 Revised manuscript received June 30, 2003 Accepted July 2, 2003 IE030256O
