Modeling of Polar Systems with the Perturbed-Chain SAFT Equation of

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Ind. Eng. Chem. Res. 2005, 44, 6928-6938

Modeling of Polar Systems with the Perturbed-Chain SAFT Equation of State. Investigation of the Performance of Two Polar Terms Aleksandra Dominik and Walter G. Chapman* Department of Chemical and Biomolecular Engineering, Rice University, 6100 S. Main Street, Houston, Texas 77005

Matthias Kleiner and Gabriele Sadowski Universita¨ t Dortmund, Lehrstuhl fu¨ r Thermodynamik, Emil-Figge-Strasse 70, 44227 Dortmund, Germany

The Perturbed-Chain SAFT (PC-SAFT) equation of state is applied to model phase equilibria and the thermodynamic properties of ethers and esters. A systematic study of these two homologous series is conducted, and the performance of two different approaches for including the dipolar interactions in the equation of state is evaluated. Although both Polar PC-SAFT [dipolar contribution due to Jog and Chapman, Mol. Phys. 1999, 97, 307-319] and PC-SAFT + Fischer [dipolar contribution by Saager and Fischer, Mol. Simul. 1991, 6, 27-49] yield similar results for the considered systems, the parameters of Polar PC-SAFT are more physically meaningful than those of PC-SAFT + Fischer. Consequently, Polar PC-SAFT is considered to be more suitable for extrapolations, and for application to components that have multiple polar groups in the chain. 1. Introduction A molecule’s electrostatic moment has a strong effect on the phase behavior of the macroscopic system. In this work, we are concerned with dipolar interactions, which influence the phase behavior of numerous systems of industrial importance, such as mixtures containing ketones, ethers, and esters, as well as polar polymers and copolymers. Mixtures containing one polar and one nonpolar component often exhibit strong, positive deviation from ideality, resulting in azeotropic behavior. We can cite the mixture of 2-butanone with n-heptane as an example.1 The deviation from ideality is due to differences in intermolecular interactions. Dipolar interactions become increasingly important at low temperatures. For instance, in the polyethylene-dimethyl ether (PE-DME) system, the two-phase region extends to much higher pressures at low temperatures, where DME-DME polar interactions are favored over PEDME interactions.2 The phase behavior of polar polymer and copolymer solutions is determined by interactions of the multiple dipole moments present in the polymer chain. The effect of dipolar interactions on the phase behavior is difficult to model and predict accurately. In conventional equations of state (EOSs), the polar interactions are not taken into account explicitly. The nonideal behavior is modeled by fitting a large binary interaction parameter, which often is dependent on the temperature and composition of the system. Therefore, the predictive abilities of such models are often weak, because of the inability to extrapolate accurately with temperature and to multicomponent systems. Three methods could be considered for the incorporation of the polar contribution to the Helmholtz free * To whom correspondence should be addressed. Tel.: 713348-4900. Fax: 713-348-5478. E-mail: [email protected].

energy into an EOS. The first possibility would be to obtain the polar contribution from experimental results. An attempt to derive this contribution from experimental data has been made by Lee and Chao,3 who used an EOS for water and incorporated their expression into the BACK EOS.4 However, the intermolecular interactions of water are much more complex than a dipoledipole interaction, which makes the physical meaning of the polar contribution derived from an EOS for water unclear. Another pathway toward including the polar contribution would be to construct it on the basis of a theory. Several statistical mechanics-based perturbation theories have been developed for polar fluids; however, the u-expansion is the most widely applied theory.5,6 As stated by Gray and Gubbins,5 the u-expansion is valid only for mixtures of dipolar spheres. Attempts to extend the theory to nonspherical, chainlike molecules have proved difficult. Because no rigorous theory was available, the necessity of modeling nonspherical polar molecules for engineering applications has resulted in the development of approximate, often empirical, models. The most widely applied model represented the dipolar nonspherical molecule as a large sphere. The dipole moment was positioned at the center of the sphere, and the diameter of the sphere was chosen such that the molecular volume was preserved.7-14 However, the model is in poor agreement with the molecular simulation results.5 Moreover, this approach does not allow for modeling of molecules with multiple polar sites, such as diethers and triethers, diketones, and, more importantly, polar polymers and copolymers. Consequently, its use was limited to simple fluids and mixtures. Other approaches have used site-site perturbation theory, accounting for molecular shape through an interaction volume parameter. For instance, Walsh et al.15 modeled chainlike polar compounds, using the

10.1021/ie050071c CCC: $30.25 © 2005 American Chemical Society Published on Web 07/14/2005

Ind. Eng. Chem. Res., Vol. 44, No. 17, 2005 6929

Perturbed Anisotropic Chain Theory (PACT). PACT uses an interaction site perturbation theory with dipolar and multipolar interactions. The advantage of PACT is that it takes into account molecular shape through an interaction volume parameter. Very good results are obtained for mixtures of esters (µ ) 1.8 D, on average) and chloroalkanes with normal alkanes. Sear16 applied a technique similar to that of Wertheim to study the chain formation by dipolar hard spheres. This approach neglects long-range multibody interactions; therefore, it is applicable only at low temperature and low density. Finally, the polar contribution to the Helmholtz free energy can be constructed on the basis of computer simulation results. This approach has been followed by Fischer and co-workers17-19 for a fluid consisting of twocenter dipolar Lennard-Jones (2CLJ) molecules with fixed elongation, with the dipole embedded along the axis. Empirical expressions were fitted to the molecular simulation results to obtain a dipole-dipole term and a quadrupole-quadrupole term. It is important to systematically test these theories for mixtures in which the dipolar interactions (or “lack” thereof, as in ketone/alkane mixtures, for example) actually control the phase behavior and induce strong nonideality. To illustrate this point, Kraska and Gubbins12,13 have applied their Lennard-Jones SAFT (LJSAFT), which contains explicitly the dipole-dipole interaction contribution to the Helmholtz free energy, to mixtures of n-alkanes with n-alkanols and water. Hydrogen-bonding interactions are stronger than dipoledipole interactions, and they control the phase behavior for these mixtures, thus making the performance of the polar contribution to the EOS difficult to assess. A revealing test for the polar contribution is the application of the EOS, including the polar term, to systems containing one or several strongly polar components, such as ketones, ethers, and esters. The objective of this work is to evaluate the relative performance of two approaches proposed for incorporating the dipolar contribution into an engineering EOS: the dipolar theory developed by Jog and Chapman,20 and the dipolar term determined by Fischer and coworkers.17-19 The approach proposed by Jog and Chapman is based on Wertheim’s Thermodynamic Perturbation Theory offirst order (TPT1); the dipolar contribution to the Helmholtz free energy is calculated on the basis of a u-expansion. The theory has been successfully applied to ketones and their mixtures with alkanes,21,22 as well as to polar copolymers.21,23 Saager and Fischer, on the other hand, obtained the dipolar contribution to the free energy by fitting empirical expressions to the molecular simulation results, as previously mentioned. The theory has been applied to real substances,19,24 as well as to mixtures25,26 in the framework of the BACK EOS.4 To recommend the best approach to be used for complex systems including dipolar molecules, such as polymer and copolymer solutions, a systematic study of the homologous series of ethers and esters has been conducted in this work. A particular emphasis was given to molecules with multiple polar groups, to gain better understanding of the dependence of model parameters on the number of polar groups in the chain. Both dipolar terms are incorporated into the PC-SAFT EOS.27 The PC-SAFT model was chosen for this investigation, because of its excellent predictive abilities for polymer systems.28,29

2. Theory The molar residual Helmholtz free energy is given in terms of a perturbation expansion:

ares ) ahs + achain + adisp + apolar + aassoc

(1)

where ares is the Helmholtz free energy residual to an ideal gas at the same temperature and density as the fluid of interest. The hard-sphere contribution (ahs) is due to Carnahan and Starling,30 and the chain term (achain) was developed by Chapman and co-workers,31,32 on the basis of Wertheim’s TPT1. The species considered in this work are nonassociating; therefore, the association contribution (aassoc) to the residual Helmholtz free energy vanishes. The expression for the dispersion contribution (adisp) used in this work was obtained by Gross and Sadowski,27 who proposed the PC-SAFT EOS. These authors also conducted a thorough parametrization of the PC-SAFT EOS for nonpolar fluids. Research led by Jog and Chapman showed that the dipolar contribution to the Helmholtz free energy can be added to the sum as in eq 1 without modifying the hard-sphere, chain, and dispersion terms.20,33 Two approaches for expressing the dipolar contribution (apolar) are considered in this study; therefore, the basic assumptions and the fundamental equations for both dipolar terms are presented in this section. 2.1. Jog and Chapman Dipolar Term. Thermodynamic properties obtained using the dipolar term developed by Jog and Chapman20 on the basis of Wertheim’s TPT1 are in excellent agreement with molecular simulation for hard-sphere chains with a dipole moment oriented perpendicularly to the axis of the molecule. The model is realistic for many substances of great interest, such as ketones, esters, and polar polymers and copolymers. The change in free energy due to polar interactions is accurately obtained by dissolving all the bonds in a chain and then applying the u-expansion to the resulting mixture of polar and nonpolar spherical segments. The polar contribution written in the Pade´ approximate has the following form:

apolar )

a2 1 - a3/a2

(2)

where a2 and a3 are the second- and third-order terms in the perturbation expansion. Written for mixtures, and allowing for multiple dipolar segments, these terms have the following form:

a2 ) -

2π

F

9 (kT)2

∑i ∑j

µi2µj2 xixjmimjxpixpj I2,ij dij2

(3)

a3 ) 5 162

π2

F2

∑i ∑j ∑k

(kT)3

xixjxkmimjmkxpixpjxpk

µi2µj2µk2 I3,ijk dijdjkdik (4)

In the aforementioned equations, I2,ij and I3,ijk are the angular pair and triplet correlation functions, F is the number density of molecules, k is the Boltzmann constant, T is the temperature, dij is the average diameter of segments i and j, and µi is the dipole moment for component i. As shown by Jog et al.,21 they

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Ind. Eng. Chem. Res., Vol. 44, No. 17, 2005

are related to the corresponding pure fluid integrals by

j) I2,ij ) I2(η,m

(5)

j) I3,ijk ) I3(η,m

(6)

As mentioned previously, mi is the segment number of component i, and xi is the mole fraction of component i in the mixture. The universal parameters ci, ni, li, ki, oi, as well as R and γ, are given in the work of Mu¨ller et al.19 The reduced dipole moment µ/i in eq 9 is defined as

where

∑i ximi

m j )

(7)

and

η)

π 6

F

∑i ximidi3

(8)

In eqs 3 and 4, xpi is the fraction of dipolar segments in component i, whereas xi is the mole fraction of component i in the mixture, and mi is the segment number of component i. In the case of a molecule with one dipolar function in the chain, xp should be equal to 1/m. The PC-SAFT model extended to polar systems, and including the dipolar term proposed by Jog and Chapman, will hereafter be referred to as Polar PC-SAFT. The parameters of the model are the segment number m, the segment diameter σ, the segment dispersion energy /k, and the fraction of polar segments in the chain xp. 2.2. Fischer’s Dipolar Term. The dipolar term of Fischer and co-workers17-19 was obtained on the basis of molecular simulations by fitting empirical expressions to the simulation data. For this purpose, simulations with two-center Lennard-Jones (2CLJ) molecules of a fixed elongation L* ) L/σ ) 0.505 were performed for different temperatures, densities, and dipole moments. The dipolar EOS was applied to real fluids24 and mixtures.25,26 One restriction of the model in its original form is the fixed elongation, which makes it nontrivial to apply to strongly asymmetric mixtures. Another limitation is that only one polar group per molecule is considered. As a result, the original model does not account for multiple functional groups with different polarity within a molecule explicitly in the EOS. To consider the nonspherical shape and multiple dipolar functional groups explicitly, the polar term was modified following a methodology similar to that proposed by Jog and Chapman.20 As in the previously presented approach, the parameter xp is introduced in this model. The dipolar term can then be written as

apolar )

∑i ∑j

28

xixjmimjxpixpj

( ) () [ ( )] T

∑ cq 1.13T q)1

F

nq/2

lq/2

F0

(µ/i µ/j )kq/8 exp -oq

×

F

F0

2

(9)

with

( )

R + (1 - R) T π F ) Fd 3 F0 6 ij 0.1617 T0

γ

(10)

and

T0 )

()

ij 1.268 k

(11)

µ/2 i )

µi2 idi3

(12)

where µi is the molecular or group dipole moment, which can be obtained from experiments or quantum chemical calculations. By applying this modified dipolar term of Saager and Fischer with PC-SAFT (hereafter referred to as PC-SAFT + Fischer), the EOS has four adjustable pure-component parameters as Polar PC-SAFT: m, σ, , xp. The goal of this work is to choose the best methodology for modeling components with multiple dipolar groups, and eventually to propose a methodology for modeling polar polymers and copolymers in the framework of PC-SAFT. The relatively small number of systems previously examined with both approaches is not sufficient for determining which term will perform better for polar polymers. A more-detailed study of polar systems is conducted here to evaluate the relative performance of both approaches. 3. Model Parameters and Regression Method The PC-SAFT model including either of the dipolar terms considered in this work has four adjustable purecomponent parameters: the segment number (m), the segment diameter (σ), the segment dispersion energy (/k), and the fraction of polar segments in a chain (xp). These parameters can be regressed by fitting purecomponent vapor pressure and liquid density data. For a molecule with a single polar function in the chain, the experimental value of the dipole moment of the molecule can be used. Alternatively, if experimental values are not readily available, the value of the dipole moment can be obtained from quantum mechanics. For molecules with multiple polar sites, the value of the functional group’s dipole moment is to be used in the model. Ideally, the value of xp should be k/m, where k is the number of polar functions in the chain and m is the number of segments of the molecule. In the framework of SAFT, molecules are depicted as chains of tangentially connected segments, which is somewhat oversimplified. Thus, SAFT reflects averages, and not the exact shape and structure of the molecules. Consequently, it is not realistic to expect that the value of xp can be fixed to k/m for all molecules with k polar functions. The value of the polar segment fraction is dependent on the strength of the dipole moment of the molecule and its chain length, as will be demonstrated later in this article. Therefore, xp will be an adjustable parameter in this model. One of the objectives of this work, however, is to establish the relationship between the strength of the dipole moment, the number of polar functions, the chain length of the molecule, and the value of the xp parameter, to obtain a predictive model for polar polymers and copolymers. The four parameters were obtained for the homologous series of ethers, including symmetric, nonsymmetric, and polyfunctional ethers, and the homologous series of esters. The pure-component data used in this


study are from the Daubert and Danner pure-component data compilation.34 EOS parameters are usually regressed by fitting vapor pressure and liquid density data, applying a Levenberg-Maquart minimization algorithm. The objective function Fobj is defined as

(

nexp

F

obj

)

∑ i)1

)

fexp - fcalc i i fexp i

2

(13)

In eq 13, f ∈ [Pvap, Fliq], and nexp is the total number of experimental data points. In the case of nonpolar PCSAFT, parameters obtained from this method give satisfactory results for mixtures,27 because the minimum of the objective function is narrow and the parameters are unique. When the polar contribution is included, it has been observed that there exist several sets of parameters (m, σ, /k, xp) that reproduce purecomponent experimental data with good accuracy. Yet, only one set of parameters gives results in good agreement with mixture data. This behavior of the objective function was observed for Polar PC-SAFT and PCSAFT + Fischer. Consequently, the method for obtaining pure-component parameters for both dipolar PCSAFT models has been modified. To obtain a unique set of parameters, one set of binary mixture data must be included in the regression. However, the choice of the second component of the mixture should be made carefully. It is important to choose a component for which pure-component parameters have been determined and tested, and that does not contain any functionality (polar group, self-associating components, quadrupolar moment, cyclic components). Therefore, mixtures of ethers and esters with alcohols, water, carboxylic acids, cyclic alkanes, and chloroalkanes and fluoroalkanes have been excluded from the parameterfitting procedure. Only mixtures of ethers and esters with normal alkanes are considered for the regression. The values of the parameters, as well as the average absolute deviations (AADs) over the considered temperature range for vapor pressures and saturated liquid densities are reported in Table 1 for Polar PC-SAFT and in Table 2 for PC-SAFT + Fischer. In the case of Polar PC-SAFT, it has been observed that, after the value of the product xpm that gives good results for both binary and pure-component data is determined for one component, the same value of xpm can be retained for other components with the same molecular dipole moment and the same number of dipolar groups in the chain. For example, all the ethers with a single polar group have a dipole moment of 1.2 D. The product of xpm has been determined for diethyl ether, for which good-quality pure-component data, as well as a large selection of binary mixture data, were available. The value of xpm has been maintained constant for the entire series, and the results obtained for other mixtures of monofunctional ethers are in very good agreement with experimental data. The same behavior has been observed for the homologous series of ketones22 and esters (this work). This remarkable property of Polar PC-SAFT demonstrates the fact that the model captures, to a certain extent, the molecular structure of fluids. Moreover, only one set of binary data is needed for parameter regression for a homologous series of polar components. For PC-SAFT + Fischer, the values of the product xpm range between 2.07 and

2.19 for ethers with one polar function, as opposed to Polar PC-SAFT, for which xpm is 1.0. This indicates that the parameter xp is not as physically meaningful in the Fischer and Saager approach as it is in the Jog and Chapman approach. This variation of xpm could be explained by the fact that, for the model of Fischer and Saager, simulation data of nonspherical molecules were used. If the segment appproach is applied to that model, the dipole moment ranges over more than one (spherical) segment, which may result in higher values of the product xpm, compared to Polar PC-SAFT. The obtention of Polar PC-SAFT parameters is more complex for components with multiple dipolar groups in the chain. In this work, we focus on the series of ethers, because no experimental data are available for molecules with multiple dipolar groups from other series of polar components. For polyfunctional ethers, only the value of their molecular dipole moment is available.35,36 Yet, the value of the dipole moment of the polar segment is needed for the calculations. This value could be obtained from quantum calculations. In this study, however, the value of the polar segment dipole moment is set to 1.2 D for all polyfunctional ethers when applying Polar PC-SAFT. For PC-SAFT + Fischer, it turned out that the experimental dipole moments of the molecules must be used as described in the next section. 4. Results 4.1. Monofunctional Components. A thorough parametrization was conducted for the homologous series of ethers and esters in the framework of both Polar PC-SAFT and PC-SAFT + Fischer (see Table 1 for pure-component parameters from Polar PC-SAFT, and Table 2 for parameters of PC-SAFT + Fischer). Both approaches give an excellent representation of the pure-component properties of dipolar components with one dipole moment on the chain. As previously mentioned, however, the value of xpm for Polar PCSAFT is constant for any given homologous series, which greatly facilitates parameter regression. A comparison of results obtained for mixtures including a normal alkane, and a component with one polar function in the chain reveals that both Polar PC-SAFT and PCSAFT + Fischer describe the phase behavior of mixtures very accurately, without the need to adjust a binary interaction parameter. This is illustrated in Figure 1 for the system of 1,2-propylene oxide in n-hexane, and in Figure 2 for the butyl acetate-cyclohexane system. A few parameters were regressed for ethers, and an almost-complete study for esters was conducted by Gross and Sadowski27 with the PC-SAFT model without accounting for dipolar interactions. Consequently, it is possible to compare the binary results obtained for binary mixtures using the original, nonpolar PC-SAFT, with the results from PC-SAFT including a polar contribution. The comparison is performed here for Polar PC-SAFT, because the performances of both polar terms for mixtures including monofunctional polar molecules are more or less identical. In Figure 3, the results for a mixture of dimethyl ether and propane at four different temperatures are presented. Polar PCSAFT represents the binary data with good accuracy at all considered temperatures, whereas PC-SAFT does not capture the azeotropic behavior of the mixture. The PC-SAFT results are expected, because the EOS is unaware of the effect of the polar groups on the phase behavior. The nonideality is caused by a lack of attrac-

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Table 1. Pure-Component Parameters for the Polar PC-SAFT Model AAD% Pvap

xp

temperature range [K]

F

Monoethers 215.98 220.59 236.19 235.89 236.95 217.84 230.20 237.92 240.24 231.49 221.35

0.4977 0.3474 0.2947 0.2322 0.1931 0.4007 0.3639 0.3276 0.2900 0.3248 0.2966

200-400 200-466 250-520 300-580 350-610 280-420 220-460 200-500 250-530 300-490 250-500

0.80 0.72 0.47 0.63 0.36 0.35 0.92 0.28 0.36 0.75 3.32

3.5850 3.5973 3.5556 3.5319

Diethers 234.98 239.59 221.20 223.34

0.8602 0.9451 0.5955 0.6806

250-470 350-520 300-510 300-520

4.6265 6.6346 7.8270

3.5194 3.4077 3.4184

Polyethers 225.62 214.38 225.24

0.9078 0.7401 0.7082

44.05 58.08 58.08 72.11 72.11

1.4953 2.0105 1.9227 2.1875 2.2846

methyl formate ethyl formate propyl formate butyl formate methyl acetate ethyl acetatef propyl acetate butyl acetate methyl propionate ethyl propionate propyl propionate

60.053 74.08 88.11 102.13 74.08 88.11 102.13 116.16 88.11 102.13 116.16

1.8742 2.2246 2.4498 2.9234 2.3967 2.7481 3.1606 3.6600 2.7813 3.1522 3.4836

3.5425 3.6793 3.8039 3.7940 3.5477 3.6511 3.6866 3.6751 3.5969 3.6463 3.7476

acetone 2-butanone 2-pentanone 3-pentanone 2-hexanone 3-hexanone 2-heptanone 4-heptanone 2-octanone 3-octanone 2-nonanone 5-nonanone 2-undecanone 6-undecanone 2-tridecanone

58.078 72.1 86.13 86.13 100.16 100.16 114.19 114.19 128.22 128.22 142.24 142.24 170.3 170.3 198.35

2.221 2.418 2.826 2.812 3.232 3.202 3.704 3.651 4.002 4.134 4.55 4.448 5.441 5.399 6.25

component

molecular weight, Mw [g/mol]

m

dimethyl ether diethyl etherf di-n-propyl ether di-n-butyl ether di-n-pentyl ether methyl ethyl ether methyl n-propyl ether methyl n-butyl ether methyl n-pentyl ether ethyl n-propyl ether diisopropyl ether

46.069 74.123 102.177 130.23 158.28 60.1 74.123 88.15 102.177 88.15 102.177

2.0090 2.8787 3.3930 4.3064 5.1782 2.4953 2.7480 3.0529 3.4480 3.0791 3.3721

3.4343 3.5549 3.7425 3.7429 3.7474 3.4580 3.6183 3.6801 3.7199 3.7021 3.7856

dimethoxymethane 1,2-dimethoxyethane diethoxymethanef 1,2-diethoxyethane

76.095 90.122 104.149 118.17

2.4994 2.9626 3.6103 4.1143

DEG DMEb,f TRG DMEc TEG DMEd

134.17 178.23 222.282

ethylene oxide 1,2-propylene oxidef 1,3-propylene oxide 1,2-butene oxide tetrahydrofuran

µa [D]

xpm

0.90 0.90 1.14 0.71 2.06 1.30 1.12 1.28 0.67 1.30 1.84

1.3 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

0.98 0.38 0.25 0.56

0.43 1.92 1.86 1.30

1.2 1.2 1.2 1.2

2.15 2.8 2.15 2.8

350-600 300-620 400-680

2.32 0.39 0.30

1.36 2.18 3.66

1.2 1.2 1.2

4.2 4.91 5.54

Epoxies and Cyclic Ethers 3.5860 288.93 0.5350 3.6095 258.82 0.3979 3.5990 289.54 0.4161 3.7903 275.31 0.3657 3.6241 279.07 0.33502

170-455 310-470 260-510 200-510 200-530

3.01 2.21 2.37 1.66 0.44

1.55 1.87 1.68 2.62 1.40

1.7 2.0 1.8 1.7 1.75

0.8 0.8 0.8 0.8 0.8

Esters 232.19 251.54 252.77 249.41 238.87 236.99 238.87 237.43 240.92 236.76 241.51

0.8004 0.6743 0.6123 0.5131 0.6259 0.5458 0.4746 0.4098 0.5393 0.4759 0.4306

200-470 210-495 215-550 215-550 200-480 215-485 200-530 230-560 200-515 215-545 260-560

1.66 2.06 1.91 2.25 1.29 1.13 1.41 0.78 1.41 0.32 2.09

1.36 1.46 2.75 3.69 1.66 1.64 1.41 0.53 2.70 0.71 2.78

1.75 1.96 1.91 1.90 1.7 1.84 1.86 1.86 1.7 1.75 1.79

1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5

Ketonese 3.607908253 259.99 3.689051518 270 3.710210701 264.97 3.705656195 265.83 3.730568917 263.8 3.732817265 262.1 3.731131258 259.17 3.745135035 255.72 3.752375796 257.07 3.720417659 254.12 3.743460108 256.81 3.765670418 256.25 3.742901466 253.8 3.745693011 251.65 3.762355566 253.34

0.2258 0.2074 0.177 0.1774 0.1543 0.156 0.136 0.135 0.1254 0.12 0.1101 0.1125 0.0919 0.0924 0.0799

253-463 270-374 282-353 285-347 299-332 299-324 306-416 275-384 319-420 316-360 340-422 301-356 309-427 303-352 341-427

4.726 0.647 0.235 0.137 0.025 0.036 0.556 0.864 1.523 0.327 0.099 0.053 0.047 0.183 0.022

1.338 0.498 0.033 0.197 0.013 0.01 0.638 9.703 12.33 3.671 0.155 0.013 0.144 1.305 0.177

2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.7

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

σ [Å]

/k [K]

a Values of the segment dipole moment used in the calculations. In the case of monofunctional molecules, it is the experimental value of the molecular dipole moment. b Diethylene glycol dimethyl ether. c Triethylene glycol dimethyl ether. d Tetraethylene glycol dimethyl ether. e Parameters for ketones taken from ref 22. f The following binary systems were used in the fitting procedure: diethyl ether + n-hexane (from ref 47), diethoxymethane + n-heptane (from refs 45 and 46); 1,2-propylene oxide + n-hexane (from ref 38); DEG DME + n-decane (from ref 45); and ethyl acetate + n-heptane (from ref 42).

tion between the unlike components, as compared to the attraction between the pure, or like, components. When the EOS does not explicitly account for dipolar interactions, this lack of attraction cannot be predicted. A relatively high value of the binary interaction parameter (kij ) 0.03) is necessary for PC-SAFT to reproduce the nonideal behavior accurately, despite the weak dipole moment of dimethyl ether. Moreover, it is likely that the interaction parameter is temperature-dependent

outside of this temperature range. On the other hand, the polar term captures the temperature dependence well within this range of temperatures and is expected to yield good results at temperatures outside the range. For esters, results for PC-SAFT and Polar PC-SAFT are compared for the binary mixture of ethyl acetate with heptane in Figure 4. The mixture exhibits azeotropic behavior, which is very accurately reproduced by polar PC-SAFT with a binary interaction parameter

Ind. Eng. Chem. Res., Vol. 44, No. 17, 2005 6933 Table 2. Pure-Component Parameters for Ethers and Esters Obtained for PC-SAFT + Fischer AAD% Pvap

xp

temperature range [K]

F

Monoethers 3.5652 223.76 3.6768 230.82 3.7797 241.81 3.4899 223.44 3.6414 211.26

0.7550 0.6145 0.5170 0.8545 0.5749

200-450 250-520 300-580 280-420 250-500

0.29 0.35 2.71 0.46 2.18

3.1970

Diethers 3.4819 225.81

0.6397

300-500

134.175 222.282

4.7104 7.9200

Polyethers 3.4839 225.81 3.3661 227.43

0.6397 0.325

1.2-propylenoxide 1.3-propylenoxide tetrahydrofuran

58.08 58.08 72.12

2.1341 1.9754 2.2631

methyl formate ethyl formate propyl formate ethyl acetate n-propyl acetate n-butyl acetate methyl propionate ethyl propionate propyl propionate

60.053 74.08 88.11 88.11 102.13 116.16 88.11 102.13 116.16

2.3829 2.5818 2.9067 3.1878 3.2563 3.6102 3.0235 3.4309 3.7225

component diethyl ether di-n-propyl ether di-n-butyl ether methyl ethyl ether diisopropyl ether 1,2-dimethoxyethane DEG DMEb TEG DMEc

molecular weight, Mw [g/mol]

m

74.12 102.177 130.23 60.1 102.177

2.8311 3.5577 4.0882 2.4254 3.6883

90.122

µa [D]

xpm

1.43 1.47 5.04 1.47 3.28

1.22 1.21 1.2 1.22 1.26

2.14 2.17 2.11 2.07 2.12

1.04

1.21

1.71

2.73

300-500 410-680

0.8 2.73

1.18 2.63

1.97d 2.45d

3.00 2.57

Epoxies and Cyclic Ethers 3.4739 252.95 0.916 3.5336 290.97 0.936 3.62159 282.42 0.86

310-470 280-510 250-500

3.92 2.68 0.78

1.62 0.18 0.75

2 1.8 1.75

1.95 1.85 1.95

Esters 239.57 233.64 241.65 221.85 237.44 240.05 238.65 231.56 234.61

180-480 200-500 220-520 190-510 263-548 200-560 220-520 200-540 260-560

1.13 0.67 0.38 0.43 1.86 0.8 0.84 0.67 2.5

0.99 1.85 1.61 1.72 1.44 2.65 1.96 0.97 2.52

1.75 1.96 1.91 1.84 1.86 1.86 1.7 1.75 1.79

2.01 2.32 2.44 2.68 2.78 2.95 2.59 2.85 2.93

σ [Å]

3.2219 3.4424 3.5409 3.4284 3.5692 3.6721 3.4717 3.5421 3.5877

/k [K]

0.8431 0.8988 0.8399 0.8406 0.8537 0.8169 0.8561 0.8297 0.787

a Molecular dipole moments used in the calculations. Values taken from ref 49. b Diethylene glycol dimethyl ether. c Tetraethylene glycol dimethyl ether. d Values taken from ref 50. e The following binary systems were used in the fitting procedure: diisopropyl ether + n-heptane (from ref 45); 1,2-dimethoxyethane + n-heptane (from ref 45); 1,2-propylene oxide + n-hexane (from ref 38); DEG DME + n-decane (from ref 45); TEG DME + n-heptane (from ref 43); and ethyl acetate + n-heptane (from ref 42).

Figure 1. Vapor-liquid equilibrium for the 1,2-propylene oxidehexane system. The solid line represents Polar PC-SAFT, and the dashed line represents PC-SAFT + Fischer; the binary interaction parameter was set to zero in both cases. Symbols are experimental data by Bougard and Jadot.38

Figure 2. Vapor-liquid equilibrium for the butyl acetatecyclohexane system. The solid line represents Polar PC-SAFT, and the dashed line represents PC-SAFT + Fischer; the binary interaction parameter was set to zero in both cases. Symbols are experimental data from the Dechema compilation.39

set to zero. This is not the case for the original PCSAFT, which again does not capture the nonideal behavior of the mixture. The case of mixtures containing two polar components was also examined, and good results were observed with both polar terms. Such mixtures show almost-ideal behavior, because of the similarity of both components; therefore, they are not as challenging a test for the polar terms as the strongly nonideal mixtures that contain polar and nonpolar components.

4.2. Difunctional Components. The true challenge of this work is to choose xp for components with multiple polar groups, and to be able to predict values of xp for polar polymers and copolymers. Ideally, the product xpm should be equal to k, where k is the number of polar groups in the chain. For monofunctional molecules, however, this equation holds only for weakly polar ethers in the case of Polar PC-SAFT, and it is not applicable for PC-SAFT + Fischer for any components considered in the study. Only one molecule with two

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Figure 3. Vapor-liquid equilibria of the dimethyl ether-propane system. Solid lines are the results from Polar PC-SAFT, dashed lines are the results from nonpolar PC-SAFT, and symbols are experimental data from Horstmann et al.40 and Giles and Wilson.41 Binary interaction parameters are set to zero.

Figure 4. Vapor-liquid equilibria of the ethyl acetate-heptane system. Dashed line represents nonpolar PC-SAFT, and the solid line represents polar PC-SAFT. Binary interaction parameters set to zero. Symbols are experimental data from Shealy and Sandler.42

polar sites (2,4-pentadione) has been previously considered in the framework of Polar PC-SAFT. The results obtained by Sauer and Chapman22 for the 2,4-pentadione + cyclohexane system were very good, but applying the model to only one system that contains a component with more than one polar site is not sufficient for proposing a methodology for modeling systems that contain components with multiple polar groups. In this work, parameters were obtained for two difunctional and several polyfunctional ethers (see Tables 1 and 2). The results obtained from both models for systems including polyethers reveal the differences between Polar PC-SAFT and PC-SAFT + Fischer.

Figure 5. Vapor-liquid equilibria of the di-n-propyl etherheptane and diethoxymethane-heptane systems from Polar PCSAFT. Binary interaction parameters set to zero. Symbols are experimental data from Treszczanowicz et al.45,46

Figure 6. Vapor-liquid equilibria of the 1,2-dimethoxyethaneheptane and 1,2-diethoxyethane-heptane systems from Polar PCSAFT. Binary interaction parameters set to zero. Symbols are experimental data from Treszczanowicz et al.45,46

In the case of Polar PC-SAFT, the segment dipole moment is used in the calculations of thermodynamic properties for components that have more than one polar function in their chain. Results obtained for mixtures of diethers with n-alkanes are in excellent agreement with experimental data, as shown in Figures 5 and 6. The experimental molecular dipole moments differ for the diethers considered in this study. Two of them (1,2-dimethoxyethane and 1,2-diethoxyethane) have a molecular dipole moment equal to 1.71 D. Diethoxymethane and dimethoxymethane have molecular dipole moments of 1.22 and 1.0 D,36 respectively. However, in the parameter regression, the value of 1.2 D used for all ethers has been retained for the segment dipole moment. Therefore, to obtain satisfactory results for binary mixtures of diethers from Polar PC-SAFT,


Figure 7. Vapor-liquid equilibrium of the 1,2-dimethoxyethaneheptane system from PC-SAFT + Fischer. Binary interaction parameters set to zero. Dashed line: segment dipole moment of 1.2 D used in the calculations. Solid line: experimental molecular dipole moment of 1.71 D used in the calculations. Symbols are experimental data from Treszczanowicz and Lu.45

Figure 8. Vapor-liquid equilibria of the DEG DME-decane system at 120 °C. Solid lines are the results from Polar PC-SAFT, dashed lines are the results from PC-SAFT + Fischer, and dotdashed lines are results from UNIFAC. Binary interaction parameters of the PC-SAFT versions were set to zero. Symbols are experimental data from Treszczanowicz.46

different values of the product xpm are necessary for the strongly polar molecules on one hand (1,2-dimethoxyethane and 1,2-diethoxyethane), and the weakly polar components on the other hand (diethoxymethane and dimethoxymethane). The parametric study of the PC-SAFT + Fischer model, in the case of diethers, revealed that the experimental molecular dipole moments must be used when the PC-SAFT + Fischer model is applied to get a good parameter set that describes binary mixtures, as shown in Figure 7. The calculations of the binary system of 1,2-dimethoxyethane and heptane show exemplarily that it is not possible to predict the azeotropic behavior with the parameters fitted using the group dipole moment of ethers of 1.2 D. In contrast, the results using the parameters fitted with the molecular dipole moment of 1.71 D for 1,2-dimethoxyethane are in good agreement with the experimental data. The results of the study of diethers with both Polar PC-SAFT and PC-SAFT + Fischer demonstrate that both models are able to represent the pure-component properties, as well as the binary mixture phase equilibria, with good accuracy. When Polar PC-SAFT is used, however, the segment dipole moment is used, which is consistent with the theoretical derivations of the dipolar contribution (see the reports by Jog and coworkers20,21). To get satisfactory agreement with experimental data for mixtures in the case of PC-SAFT + Fischer, the experimental molecular dipole moment had to be used for diethers. This potentially could cause problems for polar components with multiple polar groups, for which the molecular dipole moment is not readily available. 4.3. Polyfunctional Molecules. The results obtained with Polar PC-SAFT and with PC-SAFT + Fischer for diethylene glycol dimethyl ether (DEG DME) in n-decane are compared to results from modified UNIFAC in Figure 8. The parameters for DEG DME for both Polar PC-SAFT and PC-SAFT+Fischer were obtained using the method described in the previous

section of this article (mixture data was also considered in the regression). Both models accurately describe the binary vapor-liquid equilibrium, as well as the pure component thermodynamic properties of DEG DME. The group contribution parameters from Gmehling et al.37 were used for the modified universal quasichemical functional group activity coefficient (UNIFAC) calculations. Although PC-SAFT with either polar term describes the vapor-liquid equilibrium of the system very accurately, the UNIFAC calculation overestimates the azeotropic pressure. This comparison shows that, for the considered mixtures of polyfunctional molecules with alkanes, any of the applied PC-SAFT versions including the dipolar contribution had superior results, compared to modified UNIFAC calculations. A similar result was obtained earlier for a mixture of a diketone with an alkane. In this case, modified UNIFAC predicts a heterogeneous azeotrope, while the system actually exhibits a homogeneous azetorope.22 The polar parameter xp exhibits a consistent monotonic variation with the number of polar functions in the chain in the case of Polar PC-SAFT (see Table 1). This behavior is not observed with the PC-SAFT + Fischer model, which corroborates the assertion that the xp parameter is not as physically meaningful in the PC-SAFT + Fischer model as it is in Polar PC-SAFT. Finally, systems that contained tetraethylene glycol dimethyl ether (TEG DME) with various n-alkanes were investigated. Polar PC-SAFT gives a satisfactory description of both pure component properties (see Table 1) and phase behavior of the mixture (see Figure 9). To reproduce the liquid-liquid equilibria of TEG DME with various alkanes, very small values of the binary interaction parameter are sufficient. The values of the product xpm for TRG DME and TEG DME were obtained by fitting a simple correlation to the values of xpm regressed for shorter ethers, thus taking advantage of the monotonic variation of the polar parameter in Polar PC-SAFT. To propose a correlation of the polar parameter with the number of polar groups in the chain that

6936


Figure 9. Liquid-liquid equilibria of TEG DME and various n-alkanes. Results shown are from Polar PC-SAFT. Symbols are experimental data from Mozo et al.43 and Treszczanowicz and Cies´lak.44

Figure 11. Excess enthalpies of the binary mixtures of polyethers with n-dodecane. Comparison of experimental data from Burgdorf et al.48 with Polar PC-SAFT predictions using kij ) 0.0 for DEG DME + n-dodecane, kij ) 0.004 for TRG DME + n-dodecane, and kij ) 0.005 for TEG DME + n-dodecane.

Figure 10. Liquid-liquid equilibria of TEG DME and various n-alkanes. Results from PC-SAFT + Fischer. Symbols are experimental data from Mozo et al.43 and Treszczanowicz and Cies´lak.44

Figure 12. Excess enthalpies of the binary mixtures of polyethers with n-dodecane. Comparison of experimental data from Burgdorf et al.48 with PC-SAFT + Fischer predictions using kij ) 0.0 for DEG DME + n-dodecane and kij ) 0.008 for TEG DME + ndodecane.

is valid for extrapolation, investigation of longer chain molecules is necessary. We note that the number of polar segments increases with the number of polar functions but is not doubled or tripled, as compared to a monofunctional molecule, as one might have expected. This is due to the partial cancellations by one another of the dipole moments in the chain, and illustrates the difficulties encountered for modeling of polar systems. Remarkably, Polar PC-SAFT is able to predict the excess enthalpies of mixtures of polyethers with nalkanes, as shown in Figure 10. The binary interaction parameter of the system DEG DME + n-dodecane was set to zero for this calculation. The binary interaction parameter of the system TEG DME + n-dodecane was obtained by assuming a linear dependence of kij between TEG DME and n-alkanes on the molecular weight of n-alkanes. Recall that binary interaction parameters

were obtained for TEG DME and three n-alkanes from phase equilibrium data, as shown in Figure 9. Finally, no binary phase equilibrium data were considered in this study for TRG DME. Consequently, the value of kij for the TRG DME + n-dodecane system was interpolated from the values of kij used for DEG DME and TEG DME in n-dodecane. A linear dependence of kij on the molecular weight of the polyether was assumed in this case. The good agreement with experimental data and the ability to predict the value of the binary interaction parameter for polyfunctional molecules illustrate the predictive abilities of Polar PC-SAFT. PC-SAFT + Fischer is able to represent the pure component properties of TEG DME, as well as its liquid-liquid-phase equilibria with n-alkanes (see Figure 11). The excess enthalpies of mixtures of DEG DME


+ n-dodecane and TEG DME + n-dodecane are in good agreement with experimental data (see Figure 12). The binary parameter for the TEG DME + n-dodecane system was obtained by assuming a linear dependence on the molecular weight of n-alkanes, as it is done for Polar PC-SAFT. Because no monotonic dependence of the product xpm is observed in the case of PC-SAFT + Fischer, the value of xpm must be adjusted to the binary data in this case. However, the product xpm for TEG DME is detemined to be even smaller than that for DEG DME (see Table 2), which is not physically meaningful. 5. Conclusions The PC-SAFT model extended to polar systems was applied to model phase equilibria of ethers and esters. The results obtained from PC-SAFT, including two different approaches for accounting for dipolar interactions, were compared, and a parametric study of both Polar PC-SAFT and PC-SAFT + Fischer was conducted. The study revealed that both models yield very similar results for all systems considered. However, it was observed that the polar parameters obtained from Polar PC-SAFT are physically more meaningful than those regressed using PC-SAFT + Fischer. Therefore, the extrapolative capabilities of Polar PC-SAFT, with respect to parameter estimation of complex molecules, are expected to be superior to those of PC-SAFT + Fischer. Acknowledgment The authors gratefully acknowledge the financial support of the Consortium of Complex Fluids. Literature Cited (1) Takeo, M.; Nishii, K.; Nitta, T.; Katayama, T. Fluid Phase Equilib. 1979, 3, 123. (2) Hasch, B.; Lee, S.-H.; McHugh, M. Strenghts and limitations of SAFT for calculating polar copolymer-solvent phase behavior. J. Appl. Polym. Sci. 1994, 59, 1107. (3) Lee, M.; Chao, K. Augmented BACK EOS for polar fluids. AIChE J. 1988, 34, 825-833. (4) Chen, S.; Kreglewski, A. Applications of augmented van der Waals theory of fluids. I. Pure fluids. Ber. Bunsen-Ges. 1977, 51, 1048. (5) Gray, C.; Gubbins, K. Theory of Molecular Fluids; Clarendon Press: Oxford, U.K., 1984; Vol. 1. (6) Twu, C.; Gubbins, K. Thermodynamics of polyatomic fluid mixtures. II. Polar, quadrupolar and octopolar molecules. Chem. Eng. Sci. 1978, 33, 879. (7) Vimalchand, P.; Donohue, M. Thermodynamics of quadrupolar molecules: the perturbed-anisotropic-chain theory. Ind. Eng. Chem. Res. 1985, 24, 246. (8) Gubbins, K.; Gray, C. Perturbation Theory for the Angular Pair Correlation Function in Molecular Fluids. Ind. Eng. Chem. Res. 1972, 23, 187-191. (9) Cotterman, R.; Schwartz, B.; Prausnitz, J. Molecular thermodynamics of fluids at low and high densities. Part I: Pure fluids containing small and large molecules. AIChE J. 1986, 32, 17871798. (10) Cotterman, R.; Schwartz, B.; Prausnitz, J. Molecular thermodynamics of fluids at low and high densities. Part II: phase equilibria for mixtures containing components with large differences in molecular size or potential energy. AIChE J. 1986, 32, 1799-1812. (11) Xu, K.; Li, Y.-G.; Liu, W.-B. Application of perturbation theory to chain and polar fluids: pure alkanes, alkanols and water. Fluid Phase Equilib. 1998, 142, 55-66.

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Received for review January 18, 2005 Revised manuscript received May 17, 2005 Accepted June 7, 2005 IE050071C

Modeling of Polar Systems with the Perturbed-Chain SAFT Equation of

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