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Ind. Eng. Chem. Res. 2010, 49, 12276–12283
Modeling of Vapor-Liquid Equilibria for CO2 + 1-Alkanol Binary Systems with the PC-SAFT Equation of State Using Polar Contributions† Luis A. Roma´n-Ramı´rez,‡ Fernando Garcı´a-Sa´nchez,*,§ Ciro H. Ortiz-Estrada,‡ and Daimler N. Justo-Garcı´a| Departamento de Ingenierı´a y Ciencias Quı´micas, UniVersidad Iberoamericana, Prolongacion Paseo de la Reforma 880, Lomas de Santa Fe, 01219 Me´xico, D.F., Me´xico, Laboratorio de Termodina´mica, Programa de InVestigacio´n en Ingenierı´a Molecular, Instituto Mexicano del Petro´leo, Eje Central La´zaro Ca´rdenas 152, 07730 Me´xico, D.F., Me´xico, and Departamento de Ingenierı´a Quı´mica Petrolera, ESIQIE, Instituto Polite´cnico Nacional, Unidad Profesional Adolfo Lo´pez Mateos, Zacatenco, 07738 Me´xico, D.F., Me´xico
In this work, we present the modeling of the vapor-liquid equilibria for the binary systems CO2 + 1-alkanol (from methanol to 1-nonanol) using the PC-SAFT equation of state as the thermodynamic model for the liquid and vapor phases. Four different versions of the PC-SAFT model were compared: the quadrupolar version for the CO2 molecule, the dipolar version for alkanols, a dipolar + quadrupolar version for CO2 and alkanols without considering cross-interactions, and the original version of the PC-SAFT equation. For the modeling of these systems, binary interaction parameters were fitted to experimental binary vapor-liquid equilibrium data of the form kij ) k0ij + k1ij/T. The results obtained showed that, for pure compounds, the inclusion of the dipolar term improved the predictions of the saturation region slightly, whereas for binary systems, the PC-SAFT model with dipolar and quadrupolar contributions yielded the best predictions of the experimental behavior, mainly because of the quadrupolar contribution for the CO2 molecule. The dipolar version of the PC-SAFT equation gave no better predictions than those obtained with the original PC-SAFT model with an association contribution. 1. Introduction Knowledge of phase equilibria and thermodynamic properties of pure components and their mixtures is very useful in relation to process design of materials or improvement of already existing processes in the chemical and petroleum industries. To calculate or predict such properties, it is necessary to employ thermodynamic models that can be applied over wide temperature and pressure ranges. Nowadays, the development of new processes at high pressures and temperatures, for example, enhanced oil recovery with high-pressure gas injection or separation processes with supercritical gases as solvents, has imposed the need of developing new thermodynamic models able to predict accurately the thermodynamic properties and phase behavior of the concerned mixtures over wide temperature and pressure ranges. In this aspect, the great progress in statistical thermodynamics has resulted in the development of new equations of state (EoS) such as the statistical associating fluid theory (SAFT) EoS, developed by Chapman et al.1 However, the perturbed-chain SAFT (PC-SAFT) EoS developed by Gross and Sadowski2 has proven to be reliable for the calculation of vapor-liquid equilibria for fluid mixtures with and without associating compounds.2,3 More recently, Gross,4 Gross and Vrabec,5 and Vrabec and Gross6 reported the corresponding terms for the quadrupolar, dipolar, and dipolar-quadrupolar contributions, respectively, and they showed the advantages of including these terms in the calculations of the phase diagrams over the original * To whom correspondence should be addressed. Tel.: +52 (55) 9175 6574. E-mail:
[email protected]. † Presented at the XXIX National Meeting of AMIDIQ, Puerto Vallarta, Jalisco, Me´xico, May 13-16, 2008. ‡ Universidad Iberoamericana. § Instituto Mexicano del Petro´leo. | Instituto Polite´cnico Nacional.
PC-SAFT EoS. The complete perturbed-chain polar statistical associating fluid theory (PCP-SAFT) model, in which the polar contribution consists of quadrupole-quadrupole, dipole-dipole, and dipole-quadrupole terms, was used very recently by Tang and Gross7 to model successfully the phase equilibria of hydrogen sulfide and carbon dioxide in mixture with hydrocarbons and water. Karakatsani and Economou8 also extended the PC-SAFT EoS to dipolar and quadrupolar fluids based on the perturbation theory for polar fluids developed by Larsen et al.9 In this new simplified polar PC-SAFT model, the so-called truncated PCSAFT (tPC-SAFT) model, appropriate expressions are included fordipole-dipole,quadrupole-quadrupole,anddipole-quadrupole interactions, and induced dipole interactions are calculated explicitly as well. The tPC-SAFT EoS was used by Karakatsani et al.10 to model the vapor-liquid equilibria of a wide variety of highly nonideal polar systems such as binary systems of methanol + n-alkane (from methane to n-heptane) and water + 1-alkanol (from methanol to 1-butanol), as well as to predict the liquid-liquid equilibria of the methanol + water + cyclohexane ternary system at different temperatures. These authors concluded that the tPC-SAFT model calculations are in good agreement with the experimental data and that this model can be used for multicomponent, multiphase predictions over wide temperature and pressure ranges, as was demonstrated by Karakatsani and Economou,11 who used this equation of state to calculate the phase equilibria for binary, ternary, and quaternary polar fluid mixtures. In a recent study, Al-Saifi et al.12 predicted vapor-liquid equilibria in water-alcohol-hydrocarbon systems with dipolar PC-SAFT EoS. They presented a very nice comparison of different PC-SAFT dipolar terms, namely, the PC-SAFT-JC (Jog-Chapman),13-16 PC-SAFT-GV (Gross-Vrabec),5 and PCSAFT-KSE (Karakatsani-Spyriouni-Economou)17 approaches, to evaluate the performance of these dipolar terms in a variety
10.1021/ie901992k 2010 American Chemical Society Published on Web 10/27/2010
Ind. Eng. Chem. Res., Vol. 49, No. 23, 2010
of binary systems containing water, alcohol, and hydrocarbon compounds. According to their results, these authors concluded that, in general, the PC-SAFT-JC dipolar term was superior to the other two dipolar terms, but its predictive capability for some systems such as methanol and ethanol with benzene and toluene was not as good. In the case of CO2 + alcohol systems, although several studies have focused on modeling the phase behavior17-19 and critical points20 of these systems with the PC-SAFT EoS, only Karakatsani et al.17 considered the dipolar moments of methanol and ethanol for modeling the vapor-liquid equilibria of CO2 + methanol and CO2 + ethanol binary systems. These complex systems were also studied recently by Lo´pez et al.21 and Gutierrez et al.,22 who modeled the vapor-liquid equilibria of CO2 + alcohol binary systems with the PR (Peng-Robinson) equation of state.23 In the first case, Lo´pez et al. obtained the binary interaction parameter of the second virial coefficient and the two nonrandom two-liquid (NRTL)24 model parameters for 10 CO2 + 1-alkanol (from methanol to 1-decanol) binary systems using the PR EoS with the Wong-Sandler25 mixing rules. In the second case, Gutierrez et al. used the PR EoS with quadratic mixing rules and two adjustable parameters to correlate their measured isothermal vapor-liquid equilibrium data for the systems CO2 + 1-propanol, CO2 + 2-methyl-1-propanol, CO2 + 3-methyl-1-butanol, and CO2 + 1-pentanol. In both cases, the agreement between experimental and calculated values was, on the whole, satisfactory; however, these authors had to use two and three parameters to represent the vapor-liquid equilibrium data for each isotherm correctly. The complexity of modeling CO2 + alcohol systems consists mainly in the fact that they are highly nonideal because of the presence of polar moments and association phenomena in the form of hydrogen bonds, so that these systems develop complex fluid phase diagrams of types II, III, and IV, according to the van Konynenburg and Scott classification.26 The aim of the present work was to model the vapor-liquid equilibria of CO2 + 1-alkanol binary systems by including in the PC-SAFT EoS the contributions due to the quadrupolar moment of CO2 and the dipolar moments of the alcohols, to show that such contributions improve the predictions of the vapor-liquid equilibria for the systems studied. Toward this end, four different versions of the PC-SAFT model were compared: the quadrupolar version for the CO2 molecule (PCSAFT-Q), the dipolar version for alkanols (PCSAFT-D), the dipolar + quadrupolar version for CO2 and alkanols without considering cross-interactions (PCSAFT-QD), and the original version of the PC-SAFT equation. In all cases, the associating term was used for the alcohols. In this work, we performed the modeling of vapor-liquid equilibria for nine CO2 + 1-alkanol (from methanol to 1-nonanol) binary systems with the PC-SAFT EoS using quadrupolar-quadrupolar and dipolar-dipolar contributions for CO2 and the 1-alkanols, respectively. For this, we used the polar terms as given by Gross4 and by Gross and Vrabec5 for the quadrupolar and dipolar contributions, respectively. Dipolar-quadrupolar contributions, as given by Vrabec and Gross,6 were not taken into account in the calculations because they did not give any substantial improvement in the representation of the data and, in some cases, the representation of the data was worse when this polar term was used. All computer programs used in this work were programmed in MATLAB27 for educational purposes. These programs were
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tested by reproducing the vapor-liquid phase behavior of pure components and binary systems containing nonassociating, associating, and polar components, previously reported in the literature.2-5 Details concerning the algorithms for the calculation of vapor-liquid equilibria and the estimation of the binary interaction parameters for the different versions of the PC-SAFT equation of state with MATLAB library routines can be found elsewhere.28 2. Thermodynamic Model In the PC-SAFT equation of state,2 the molecules are conceived to be chains composed of spherical segments in which the pair potential for the segment of a chain is given by a modified square-well potential.29 Nonassociating pure components are characterized by three molecular parameters: the (temperature-independent) segment diameter σ, the depth of the potential ε, and the number of segments per chain m. For associating components, two additional association parameters are required for their characterization: the association energy εAB and volume κAB for each site-site interaction. When the molecules exhibit various attractive interactions, the whole equation of state is given as the sum of the ideal-gas contribution (id), a hard-chain term (hc) connecting the spherical segments, a contribution for the dispersive attraction (disp), a term for associating interactions (assoc), and contributions due to polar interactions (polar). The form of the PC-SAFT equation of state used in this work, written in terms of the Helmholtz energy for an N-component mixture of molecules, can be expressed in terms of reduced quantities as a˜res ) a˜hc + a˜disp + a˜assoc + a˜polar ) a˜hc + a˜disp + a˜assoc + (a˜QQ + a˜DD) (1) where the terms in the expansion of a˜res correspond to the hard-chain reference contribution, the dispersion contribution to account for the attractive interactions, the association contribution, and the contribution due to polar interactions, respectively. In this work, only dipolar-dipolar and quadrupolar-quadrupolar interactions were taken into account for modeling the CO2 + 1-alkanol binary systems considered. Dipolar-quadrupolar interactions were not taken into account in the modeling of these binary systems because they did not improve the representation of the data and, in some cases, the results were worse when this polar contribution was introduced in the calculations. The hard-chain reference contribution and the dispersion contribution to the Helmholtz energy are given in ref 2, so the interested reader can consult this reference for more details on the PC-SAFT EoS. The contribution to the Helmholtz energy from association is given by
[
∑ x ∑ (ln X N
a˜assoc )
Ai
i
i)1
-
Ai
)
XAi 1 + Mi 2 2
]
(2)
where Mi is the number of bonding sites per molecule; xi is the mole fraction of component i; and XAi is the fraction of A sites on molecule i that do not form associating bonds with other active sites, which is given by
[
XAi ) 1 + NAv
∑ ∑FX
Bj
j
j
Bj
]
∆AiBj
-1
(3)
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where ∑Bj is a summation over all sites (starting with A) on molecule i, ∑j is a summation over all components, Fj ) xjF is the molar density of component j, and ∆AiBj is a measure of the association strength between site A on molecule i and site B on molecule j. ∆AiBj is, in turn, a function of the association volume κAiBj, the association energy εAiBj, and the radial distribution function as follows
[ ( ) ]
∆AiBj ) σij3gij(dij)segκAiBj exp
εAiBj -1 kT
(5)
∑ (a 4
JQQ 2,ij )
QQ n,ij
κ
) √κ
(
AiBi AjBj
κ
2√σiiσjj σii + σjj
a˜
)
QQ
(7)
1 - a˜QQ ˜ QQ 3 /a 2
where a˜2QQ and a˜3QQ are the quadrupolar second-order and thirdorder perturbation terms, respectively. For linear and symmetric molecules, the second- and third-order terms for the tangentsphere framework can be written as 32 F 4
()
a˜QQ 2 ) -π
∑∑ i
εii εjj σii5σjj5 n n Q*2Q*2JQQ kT kT σ 7 Q,i Q,j i j 2,ij
xixj
j
ij
(8) and a˜QQ 3 )
π 33 F 3 4
()
( )( )
∑∑ i
xi xj
j
nQ,inQ,jQ*i 3Q*j 3JQQ 3,ij +
εii kT
2
3/2
4π 3 F 3 4
3 2
()
σii5σjj5σkk5
εjj kT
3/2 σ 15/2σ 15/2 ii jj σij12
∑∑∑ i
j
k
n n n Q* Q*j Q*k 2JQQ 3,ijk 3 Q,i Q,j Q,k i
σij3σik3σjk
2
2
where a˜2DD and a˜3DD are the dipolar second- and third-order perturbation terms, respectively. For linear and symmetric molecules, these perturbations can be written as a˜DD 2 ) -πF
εii εjj σii3σjj3
∑ ∑ x x kT kT i j
i
σ3ij
j
and a˜DD 3 ) -
4π2 2 F 3
εii εjj εkk σii3σjj3σkk3 × σijσikσjk
∑ ∑ ∑ x x x kT kT kT i j k
i
j
k
nµ,inµ,jnµ,kµ*i 2µ*j 2µ*k 2JDD 3,ijk
(14) where F is the molecular number density; xi is the mole fraction; µ*i 2 ) i2/(miεiiσii3) is the dimensionless squared dipolar moment; nµ,i is the number of dipole moments per molecule; and the combining rules are σij )(σi + σj)/2, and εij ) (εiεj)1/2 for the segment size and segment energy parameters, respecDD DD and J3,ijk denote integrals over the tively. The symbols J2,ij reference-fluid pair correlation function and over three-body correlation functions, which are evaluated as power functions, as
∑ (a
DD n,ij
+ bDD n,ij
n)0
)
εij n η kT
(15)
4
JDD 3,ijk )
∑c
DD n n,ijkη
(16)
n)0
(9)
where F is the molecular number density, Q*i 2 ) Qi2/(miεiiσii5) is the dimensionless squared quadrupolar moment, and nQ,i is the number of quadrupolar moments per molecule. For the case of the CO2, there is only one quadrupolar moment of strength Q*i 2, so that nQ,i ) 1. The combining rules used for crossinteractions is the arithmetic mean for the segment size parameters, σij )(σi + σj)/2, and the geometric mean for the segment energy parameters, εij ) (εiεj)1/2.
nµ,inµ,jµ*i 2µ*j 2JDD 2,ij
(13)
JDD 2,ij )
εii εjj εkk × xixjxk kT kT kT
(12)
1 - a˜DD ˜ DD 3 /a 2
4
×
(11)
a˜DD 2
(6)
a˜QQ 2
QQ n n,ijkη
QQ whereas J3,ij is set equal to zero, so that only the second term on the right-hand side of eq 9 is taken into account to evaluate QQ the third-order perturbation term. In these expressions, aQQ n,ijk, bn,ij , and cQQ are coefficients that depend on chain length m, as given n,ijk by Gross.4 The contribution to the Helmholtz energy from dipolar interactions is given by5
3
for the association volume. It is worth mentioning that, within the scope of this investigation, no binary correction parameters are introduced in these combining rules, so that only the binary interaction parameter kij, given by eq 24 in ref 2, is adjusted from experimental binary vapor-liquid equilibrium data at each temperature level to correct the dispersive interactions. The contribution to the Helmholtz energy from quadrupolar interactions is given by4
(10)
n)0
a˜DD )
)
)
εij n η kT
4
∑c
JQQ 3,ijk )
for the association energy and AiBj
+ bQQ n,ij
n)0
(4)
where σij ) (σi + σj)/2 and gij(dij)seg ≈ gij(dij)hs is given by eq 8 in ref 2. The combining rules for cross-association used here for the CO2 + 1-alkanol systems are those previously applied by Gross and Sadowski3 for association systems, namely 1 εAiBj ) (εAiBi + εAjBj) 2
QQ QQ and J3,ij denote integrals In eqs 8 and 9, the symbols J2,ij QQ over the reference-fluid pair correlation function, and J3,ijk denotes an integral over three-body correlation functions. The QQ QQ and J3,ijk are evaluated as power functions, as integrals J2,ij
DD DD DD where an,ij , bn,ij , and cn,ijk are coefficients that depend on chain length m, as given by Gross and Vrabec.5 The density for a given system pressure psys is determined iteratively by adjusting the reduced density η (packing fraction) until pcal ) psys. For a converged value of η, the number density of molecules F, given in Å-3, is calculated from
F)
(∑ )
6 η π
-1
N
ximidi3
i)1
(17)
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a
Table 1. Pure-Component Parameters of the PCSAFT-D Equation of State for 1-Alkanols component ib Mi (g/mol) methanol ethanol 1-propanol 1-butanol 1-pentanol 1-hexanol 1-heptanol 1-octanol 1-nonanol
32.042 46.069 60.069 74.123 88.150 102.177 116.203 130.230 144.257
εAiBi/k (K) ∆pv (%) ∆FL (%) T range (K) µc(D) state or solventc T range(K)c
mi
σi (Å)
εi/k (K)
κAiBi
2.7175 2.5378 3.3459 2.9709 3.9587 3.3470 4.2543 4.3229 4.8130
2.6741 3.1009 3.1050 3.4980 3.3390 3.7445 3.5891 3.7195 3.6842
175.26 188.23 209.60 243.10 238.92 261.57 255.75 262.69 261.17
0.143749 0.040491 0.041726 0.013441 0.012529 0.012047 0.002114 0.002668 0.001125
2102.4 2566.9 2167.3 2425.5 2108.2 2402.5 2725.6 2718.4 2976.6
1.41 0.32 1.37 0.95 0.48 0.65 0.68 0.70 0.55
a Vapor pressure and saturated liquid density data taken from Daubert et al.31 data taken from McClellan.32
Using Avogadro’s number and appropriate conversion factors, F produces the molar density in different units such as kmol · mol-3. Equations for the compressibility factor are derived from the relation Z)1+η
( ) ∂a˜res ∂η
T,xi
) 1 + Zhc + Zdisp
(18)
The pressure p can be calculated in units of Pa ) N · m-2 by applying the relation
(
p ) ZkTF 1010
Å m
3
)
(19)
The expression for the fugacity coefficient of component i is given by ln φi )
[
res
∂(na˜ ) ∂ni
]
+ (Z - 1) - ln Z
(20)
F,T,nj*i
where
[
∂(na˜res) ∂ni
]
) a˜res +
F,T,nj*i
( ) ∂a˜res ∂xi
N
-
F,T,xj*i
∑ k)1
[( ) ] xk
∂a˜res ∂xk
F,T,xj*k
(21)
In eq 21, partial derivatives with respect to mole fractions are N xi ) 1. calculated regardless of the summation restriction ∑i)1 3. Results and Discussion 3.1. Pure Components. To apply the PC-SAFT model to the vapor-liquid equilibrium calculation in the different systems of study by considering the polar moments of the alcohols, it was necessary to carry out a regression of the pure-component parameters for the alcohols. To summarize, for each pure component, three molecular parameters are needed, namely, the (temperature-independent) segment diameter σ in angstroms, the interaction energy ε/k in kelvin, and the number of segments per chain molecule m, along with two association parameters, namely, the association energy εAiBj/k in kelvin and the association volume κAiBj (dimensionless), for each site-site interaction. In this work, the five parameters were derived simultaneously from vapor pressure and density data for pure components by applying the simplex procedure30 for minimizing the following objective function Ndata
S1 )
∑ i)1
[(
cal pexp v,i - pv,i
pexp v,i
) ( 2
+
cal Fexp L,i - FL,i
Fexp L,i
)] 2
(22)
b
0.89 0.29 0.53 0.57 0.84 0.37 0.79 0.49 1.55
200-508 230-509 240-531 230-557 250-580 250-605 265-626 280-646 280-667
1.692 1.708 1.669 1.67 1.66 1.65 1.73 1.65 1.61
gas gas gas gas benzene benzene benzene benzene benzene
343-503 353-498 273-503 383-488 298 298 293 298 298
Two association sites assumed for all substances. c Dipolar moments
where pv,i and FL,i are the vapor pressure and saturated liquid density, respectively, from experiment i and Ndata is the total number of experimental data. In this equation, the superscripts exp and cal denote the experimental and calculated properties, respectively. The vapor pressure and saturated liquid density data used in the regression were taken from Daubert et al.31 In all cases, 50 experimental data, from a temperature just above the triple point and up to 0.99 times the critical temperature of the fluid, were used in the adjustment of these parameters. Table 1 presents the values of the fitted parameters for the nine 1-alkanols studied for the dipolar PC-SAFT (PCSAFT-D) model. This table also includes the absolute relative deviations obtained for vapor pressure 1 ∆pv ) Npv
Npv
∑ i)1
|
cal pexp v,i - pv,i
pexp v,i
|
(23)
|
(24)
and saturated liquid density ∆FL )
1 NFL
NFL
∑ i)1
|
cal Fexp L,i - FL,i
Fexp L,i
as well as the temperature range considered and the experimental value of the dipolar moment reported by McClellan32 for each alcohol. All experimental moments for the alcohols presented in Table 1 are in good agreement with those reported in the Handbook of Chemistry and Physics,33 including the state or solvent in which they were measured. For all components, two association sites were assigned. An examination of Table 1 shows that the absolute relative deviations of the PCSAFT-D equation of state from vapor pressure data are, in the majority of cases, below 1%, whereas those from liquid density data are also below 1%, with the exception of the value for 1-nonanol. Figure 1 shows the vapor-liquid equilibria of three alkanols (ethanol, 1-hexanol, and 1-nonanol) in a temperature-density (T-F) diagram. This figure indicates that the densities of the liquid phase are welldescribed by the PCSAFT-D model. Results of the PC-SAFT equation of state for associating components obtained with purecomponent parameters determined by Gross and Sadowski3 are given for comparison. As can be seen from Figure 1, the PCSAFT-D and PC-SAFT models are very similar in representing the experimental vapor pressure and liquid density data, and no essential difference was found in using either approach to model these properties for the pure 1-alkanols considered in this work. 3.2. Mixtures. The binary interaction parameter kij (defined in eq 24 of ref 2) for correcting the cross-dispersive interactions of the PC-SAFT equation of state with its different contributions to the Helmholtz energy for representing the experimental
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Table 2 gives the values of the parameters k0ij and k1ij for the different CO2 + 1-alkanol binary systems studied. This table also includes the temperature ranges considered, the numbers of points used, and the references from which the experimental data were taken. The agreement between the calculated and experimental values was established through the absolute relative deviations in pressure, ∆p, and absolute deviations in mole fraction for the vapor phase, ∆y, of the lightest component (CO2 in this case) ∆p )
∆y ) Figure 1. Experimental and calculated saturated liquid and vapor densities for ethanol, 1-hexanol, and 1-nonanol.
vapor-liquid equilibrium data of each CO2 + 1-alkanol system was estimated by minimizing the sum of squared relative deviations of bubble-point pressures and the sum of squared deviations in vapor-phase mole fractions. The simplex optimization procedure30 was used in the computations by searching the minimum of the objective function Npts
S2 )
∑ i)1
[(
pexp - pcal i i pexp i
)
]
2
2 + (yexp - ycal i i )
(25)
exp where (pexp - pcal - ycal i i ) and (yi i ) are the differences between the experimental and calculated values of bubble-point pressures and vapor-phase mole fractions, respectively, for experiment i, and Npts is the total number of experiments. In eq 25, the calculated values of pressure and vapor-phase composition were obtained by solving the equilibrium conditions (using the PCSAFT EoS for the calculation of the fugacity coefficients, as given by eq 20) through the application of the bubble-point pressure method; that is, for a given temperature and liquidphase composition, find the pressure and vapor-phase composition. For the representation of the CO2 + 1-alkanol systems, we used the pure-component parameters reported by Gross and Sadowski3 for the PC-SAFT equation and those reported by Gross4 for the PCSAFT-Q equation using the literature value of the quadrupolar moment of Q ) 4.4 D · Å for CO2. In the case of the PCSAFT-D and PCSAFT-QD equations, the purecomponent parameters reported in Table 1 were used for performing the calculations. The individual binary interaction parameters, adjusted from the regression of isothermal vaporliquid equilibrium data, were correlated as functions of temperature of the form kij ) k0ij + k1ij/T, where T is the absolute temperature.
Npts
|
pexp - pcal i i
1 Npts
∑
1 Npts
∑ |y
i)1
pexp i
Npts
exp i
- ycal i
i)1
|
|
(26)
(27)
which were obtained with the optimal values of the binary interaction parameters. The values are reported in Table 3. This table shows the correlative capabilities of the PCSAFTQ, PCSAFT-D, PCSAFT-QD, and PC-SAFT equations obtained using the van der Waals one-fluid mixing rules and a temperature-dependent binary interaction parameter for each of the CO2 + 1-alkanol binary systems. Overall, it can be said that the quality for correlating the experimental vapor-liquid equilibrium data of these binary systems with the PCSAFT-Q and PCSAFT-QD equations is superior to that obtained with the other equations, showing the effect due to the quadrupolar moment of CO2 (Q ) 4.4 D · Å) on the representation of these systems. Because the results presented in Table 3 follow similar modeling trends, only the fits of these equations for three selected CO2 + 1-alkanol systems are shown in Figures 2-4. Figure 2 shows that both the PCSAFT-Q and PC-SAFT equations fit the data well for the system CO2 + methanol at 352.6 K at low and moderate pressures, but it is less satisfactory when pressure increases, in particular, close to the system critical point. The fact that predictions at higher pressures are not precise indicates that the interaction parameters for these equations are unable to adequately correct the cross-dispersive interactions of this system when calculations are made over a wide range of pressures. From this figure, it can also be seen that predictions obtained with the PCSAFT-D and PCSAFT-QD models are less satisfactory at this temperature. In this case, it seems that the dipolar contribution to the Helmholtz energy does not improve the predictive capability of the PC-SAFT equation. Figure 3 presents the pressure-composition phase diagram of the CO2 + 1-pentanol system at 426.9 K. Here, the PCSAFTQD equation yielded the best predictions of the vapor-liquid equilibrium behavior at this temperature compared to the other equations. This equation is able to represent correctly this behavior up to 16 MPa but overpredicts the critical region. The
Table 2. Interaction Parameters kij0 and kij1 to Correct Cross-Dispersive Interactions for CO2 (i) + 1-Alkanol (j) Systems equation of state PCSAFT-Q
PCSAFT-D
PCSAFT-QD
PCSAFT
j
kij0
kij1
kij0
kij1
kij0
kij1
kij0
kij1
Np
T range (K)
ref(s)
methanol ethanol 1-propanol 1-butanol 1-pentanol 1-hexanol 1-heptanol 1-octanol 1-nonanol
-0.0588 -0.0624 -0.0830 -0.1142 -0.0095 -0.0154 -0.0759 0.0313 -0.0793
2.6208 2.6758 18.8287 34.1128 2.6570 5.4733 23.3089 -8.1646 29.4824
-0.0361 -0.0436 -0.0513 -0.2443 -0.0817 -0.0788 -0.1144 0.1169 -0.3144
3.3424 13.7833 31.1837 101.5360 40.7833 49.3932 63.4329 -9.7858 130.4468
-0.1375 -0.0687 -0.0918 -0.1431 -0.0810 -0.1331 -0.0959 0.0257 -0.0718
8.8646 -2.2802 18.5909 39.8215 21.4782 41.7545 30.5765 -6.9398 25.9270
0.0354 0.0610 -0.0580 -0.2136 -0.09526 -0.0502 -0.0921 0.1212 -0.3050
-5.8339 -20.2448 37.5173 95.0851 45.8355 39.3207 56.3772 -10.8430 128.5285
42 56 41 55 39 41 18 51 34
230-478 291-373 308-337 293-427 283-427 303-433 375-432 308-453 308-328
34-36 37, 38 39, 40 41-43 44, 45 46, 47 47 37, 48, 49 37
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Figure 2. Experimental and calculated pressure-composition phase diagram for the CO2 (1) + methanol (2) system at 352.6 K.
Figure 3. Experimental and calculated pressure-composition phase diagram for the CO2 (1) + 1-pentanol (2) system at 426.9 K.
PCSAFT-Q equation overpredicts the phase diagram along the whole CO2 mole fraction range, whereas both the PC-SAFT and PCSAFT-D equations underpredict the phase diagram up to about 0.7 CO2 mole fraction and 18 MPa and overpredict it at higher pressures. Figure 4 presents the performances of the different PCSAFT models in predicting the vapor-liquid phase behavior of the CO2 + 1-octanol system at 328.2 K. This figure shows that the PCSAFT-Q and PCSAFT-QD models underpredict this phase behavior up to about 14 MPa, whereas the PCSAFT and
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Figure 4. Experimental and calculated pressure-composition phase diagram for the CO2 (1) + 1-octanol (2) system at 328.2 K.
PCSAFT-D models underpredict it up to 12 MPa; however, the predictions obtained with the first two models are closer to the experimental data. An inspection of Figures 2-4 shows that the mixture critical point is overpredicted for all isotherms studied. This is because the SAFT models, and essentially all equations of state, are poor in the critical region. Moreover, because CO2 + 1-alkanol binary systems are highly nonideal because of the presence of polar moments (quadrupolar for CO2 and dipolar for alcohols) and because alcohols exhibit association phenomena in the form of hydrogen bonds, modeling the phase behavior of these systems is difficult with any equation of state because the systems develop complex fluid phase diagrams, according to the classification of van Konynenburg and Scott.26 Specifically, the CO2 + 1-alkanol binary systems exhibit phase transitions of types II (CO2 with methanol, ethanol, and 1-propanol), III (CO2 with 1-hexanol, 1-heptanol, 1-octanol, and 1-nonanol), and IV (CO2 with 1-pentanol). This is why the modeling results are sometimes good for low and heavy alcohols but not for the intermediate ones. For the CO2 + 1-alkanol (from methanol to 1-nonanol) systems studied in this work, the agreement between calculation and experiment was good for all isotherms at the lowest pressures, because this region is removed from the mixture critical point and the inclusion of binary parameters has a relatively small effect. However, at the highest pressures, the calculations indicate that the systems are beginning to approach the mixture critical region where binary parameters have a large effect, so that the mixture critical point is difficult to correlate.
Table 3. Absolute Relative Deviations in Pressure, ∆p, and Absolute Deviations in Vapor-Phase Mole Fraction, ∆y, for CO2 (i) + 1-Alkanol (j) Systems equation of state PCSAFT-Q
PCSAFT-D
PCSAFT-QD
PCSAFT
j
∆p (%)
∆y (mol %)
∆p (%)
∆y (mol %)
∆p (%)
∆y (mol %)
∆p (%)
∆y (mol %)
methanol ethanol 1-propanol 1-butanol 1-pentanol 1-hexanol 1-heptanol 1-octanol 1-nonanol average deviations
7.0 15.0 6.9 10.1 19.4 18.5 4.1 8.4 5.2 10.5
2.90 2.10 0.18 0.90 0.83 1.30 0.82 0.44 0.37 1.09
12.7 17.1 12.6 12.8 18.1 19.0 13.3 17.2 15.0 15.3
2.97 2.00 0.25 1.50 4.10 2.40 0.40 0.37 0.36 1.59
11.4 15.0 8.2 10.3 11.7 13.0 4.6 8.9 5.1 9.8
3.31 2.10 0.27 1.00 2.20 2.10 0.76 0.47 0.36 1.40
7.9 15.9 12.9 12.1 20.4 18.0 11.6 17.0 15.2 14.6
2.73 2.00 0.28 1.40 4.50 2.10 0.44 0.36 0.37 1.58
T range (K) 230-478 291-373 308-337 293-427 283-427 303-433 375-432 308-453 308-328
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The results of the modeling of vapor-liquid equilibrium data for a series of binary systems containing CO2 and 1-alkanols (from methanol to 1-nonanol) presented herein indicate the limitations of the PC-SAFT model (with and without polar contributions) to represent satisfactorily the phase behavior of this type of system with a single interaction parameter to correct for cross-dispersive interactions. Nevertheless, a possible improvement of the representation of the vapor-liquid equilibrium data for these complex systems could be obtained if, for instance, a second binary interaction parameter were introduced in the dispersive energy contribution, as was done by Tang and Gross7 for modeling the system hydrogen sulfide + methane, to improve the representation of the data at high pressures. At present, we are in progress with these calculations. 4. Conclusions In this work, we have investigated the capability of the PCSAFT equation of state to represent the vapor-liquid equilibrium data of CO2 + 1-alkanol (from methanol to 1-nonanol) binary systems by considering three different combinations of the polar terms, namely, PCSAFT-Q, PCSAFT-D, and PCSAFT-QD, and their comparison with the original PC-SAFT EoS. In all of these equations, the van der Waals one-fluid mixing rules were used to represent the measured vapor-liquid equilibrium data of the nine CO2 + 1-alkanol binary systems studied by adjusting a single temperature-dependent interaction parameter for each equation to correct the cross-dispersive interactions of these systems. In general, the four versions of the PC-SAFT equation correlated the data satisfactorily at low pressures for most of the systems studied; however, at high pressures, the correlation was rather poor, and only slight advantages were found in using the PCSAFT-Q and PCSAFTQD equations to fit the data over the whole temperature and pressure range studied for each CO2 + 1-alkanol binary system, mainly because of the effect of the quadrupolar term to improve the representation of these complex systems. Thus, according to the results obtained from the correlation of the vapor-liquid equilibrium data for the nine CO2 + 1-alkanol binary systems studied, it can be concluded that the PC-SAFT equation with and without polar contributions is unable to model satisfactorily the vapor-liquid equilibria of these systems over the whole temperature and pressure range considered when it uses a single interaction parameter to correct only the cross-dispersive interactions, even if this parameter is temperature-dependent. Acknowledgment This work was supported partially by the Molecular Engineering Research Program of the Mexican Petroleum Institute under Research Project D.00484 and by the National Council for Science and Technology (CONACYT) of Mexico under Project 83842. L.A.R.-R. gratefully acknowledges the CONACYT of Mexico for their financial support through an M.Sc. fellowship (Grant 225307). The authors are thankful to the anonymous reviewers for their comments, suggestions, and criticisms, which helped very much in the revision of the manuscript. Literature Cited (1) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. New Reference Equation of State for Associating Liquids. Ind. Eng. Chem. Res. 1990, 29, 1709–1721.
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ReceiVed for reView December 15, 2009 ReVised manuscript receiVed October 4, 2010 Accepted October 11, 2010 IE901992K
