A Parametric Study of Dipolar Chain Theory with Applications to

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Ind. Eng. Chem. Res. 2003, 42, 5687-5696

5687

A Parametric Study of Dipolar Chain Theory with Applications to Ketone Mixtures Sharon G. Sauer*,† and Walter G. Chapman‡ Department of Chemical Engineering, Rose-Hulman Institute of Technology, 5500 Wabash Avenue, CM-53, Terre Haute, Indiana 47803-3999, and Department of Chemical Engineering, Rice University, P.O. Box 1892, MS 362, Houston, Texas 77251-1892

Dipolar interactions, which traditionally have been very difficult to model or predict accurately, have a significant effect on the phase behavior of a plethora of systems of industrial importance such as mixtures containing ketones, aldehydes, ethers, and esters as well as polar polymers, copolymers, and a variety of biochemicals. Jog and Chapman proposed a model that accurately predicts the properties of chainlike molecules with single or multiple dipolar sites (Jog, P. K.; Chapman, W. G. Mol. Phys. 1999, 97, 307). In a more recent paper (Jog, P. K.; Sauer, S. G.; Blaesing, J.; Chapman, W. G. Ind. Eng. Chem. Res. 2001, 40, 4641), this theory was extended to mixtures of polar fluids. We continue the analysis and application of the method here in a parametric study using ketones and further demonstrate the ability of the model to accurately predict phase behavior for polar mixtures by applying the method to mixtures of alkanes and ketones of varying lengths. The ketone parameters are very well behaved in that they are smooth functions of molecular weight. The model is able to capture the physics of the phase behavior of a variety of alkane and ketone mixtures, including those of a diketone and alkane mixture. We utilize both the perturbed-chain and Chen-Krewlewski approaches to the dispersion term for the SAFT model, which demonstrates the inclusiveness of the polar term independent of the dispersion term. A comparison with UNIFAC shows the advantage of our polar chain theory for modeling systems containing molecules with multiple dipolar groups. 1. Introduction Dipolar interactions traditionally have been very difficult to model or predict accurately. Yet, they have a significant effect on the phase behavior of a plethora of systems of industrial importance such as mixtures containing ketones, aldehydes, ethers, and esters as well as polar polymers, copolymers, and a variety of biochemicals. Nonideal behavior in mixtures of a polar component with a nonpolar component is common. For example, the mixtures 2-butanone/n-heptane and 3-pentanone/n-heptane both show positive deviation from ideal solution behavior and form azeotropic mixtures, while a n-heptane/n-hexane mixture or a 2-butanone/ 3-pentanone mixture is nearly ideal. Deviation from ideal solution behavior results from differences in intermolecular interactions. In mixtures of polar and nonpolar molecules, the polar site has zero interaction with the nonpolar molecule. This means that there is a lack of attraction between the unlike components as compared to the attraction between the pure, or like, components. This lack of attraction results in positive deviation from ideal solution behavior. In conventional equations of state, this positive deviation from ideal solution behavior is modeled by fitting a large, possibly state-dependent, binary interaction parameter. To avoid fitting state-dependent mixture parameters, we have developed a method to explicitly * To whom correspondence should be addressed. Tel.: 812-877-8527. Fax: 812-877-8992. E-mail: [email protected]. † Rose-Hulman Institute of Technology. ‡ Rice University. Tel.: 713-348-4900. Fax: 713-348-5478. E-mail: [email protected].

include multiple long-range dipolar sites in an equation of state we call the polar SAFT equation of state.1-4 Polar SAFT is a statistical mechanics based approach that essentially separates the polar contribution from the shape contribution to the free energy. When the fact that the dipole is located on certain functional groups within a molecule (i.e., some segments are polar and others are nonpolar) is accounted for, chainlike molecules with multiple dipolar functional groups can be studied. Our polar chain theory has been validated through comparisons with molecular simulations,5 and the approach has been shown to be applicable to modeling phase equilibria for mixtures of dipolar fluids.4 A review of approaches to modeling polar fluids can be found in previous papers.4,5 In this work we present a parametric study of the polar SAFT approach using ketones as the model system. Two of the most commonly used SAFT dispersion terms have been used, and parameters for both are given. Additional applications of the model to a series of ketone/alkane mixtures and diketones are presented. In section 2, the equation of state is briefly described followed by a detailed parametric analysis of polar SAFT, based primarily on a homologous series of ketones from acetone to tridecanone, in section 3. Applications to binary mixtures are presented in section 4, and in section 5, the work is summarized and conclusions are stated. 2. The Equation of State The molar residual Helmholtz free energy is given in terms of a perturbation expansion:

ares ) ahs + adisp + apolar + achain + aassociation (1)

10.1021/ie034035u CCC: $25.00 © 2003 American Chemical Society Published on Web 10/01/2003

5688 Ind. Eng. Chem. Res., Vol. 42, No. 22, 2003

We also use the PC dispersion term based on BarkerHenderson perturbation theory applied to chainlike molecules proposed by Gross and Sadowski,12 which is given by disp Adisp adisp A1 2 ) + RT RT RT

Figure 1. Cartoon representation of the free-energy expansion for a polar-associating fluid. The polar and associating molecules are modeled by removing the association sites and covalent bonds to form a reference fluid of a mixture of polar and nonpolar segments. The theory builds the polar-associating molecules beginning with this reference fluid of polar segments, which is modeled as a hard-sphere fluid with the polar and dispersion contributions as a perturbation. The chain formation and association contributions are then added as perturbations to this reference fluid.

where the Helmholtz free energy is residual to an ideal gas at the same temperature and density as the fluid of interest. As depicted in Figure 1, the free energy of associating molecules with polar sites is calculated from the free energy of a reference fluid mixture of polar and nonpolar segments. This reference fluid of polar segments is modeled as a hard-sphere fluid with the polar and dispersion contributions as a perturbation. The theory builds the polar associating molecules by applying the chain and associating terms to this reference fluid mixture. The hard-sphere (ahs) and chain (achain) terms in the original SAFT equation of state due to Chapman et al.1-3 are unaffected by the presence of a dipole term as shown by Jog and Chapman.5 The species in this work are nonassociating so that the association (aassociation) contribution to the Helmholtz free energy is zero. Various approaches to the dispersion term (adisp) can be used. For example, CK-SAFT6,7 uses the Chen and Krewlewski8 (CK) dispersion term, LJ-SAFT9,10 uses a Lennard-Jones (LJ) potential for this contribution, and perturbed chain (PC)-SAFT11,12 uses a term based on site-site perturbation theory for chainlike molecules. In this work, the CK and PC forms of SAFT are used. In the CK form, molecular simulation results for the interaction energy of square-well spheres were originally fit to a power series by Alder et al.13 Chen and Kreglewski8 refit the constants in this power series (Dij) to experimental data for argon using the form

adisp RT

)m

( )( ) u

∑i ∑j Dij kT

i

η τ

j

(2)

For pure fluids, η is a reduced density given by η ) τFmv0, where v0 is the temperature-dependent segment molar volume, τ ) πx2/6, u is the interaction energy (i.e., the square-well depth), R is the universal gas constant, k is Boltzmann’s constant, T is temperature, and m is the number of segments in a molecule. For mixtures, the dispersion term is calculated using van der Waals one-fluid theory that equates the residual free energy of the mixture to that of a hypothetical pure fluid. The van der Waals one-fluid mixing rule requires the unlike pair interaction energy, uij, in addition to the pure fluid interaction energies, uii. The interaction energy between components i and j is given by the geometric mean of the pure fluid interaction energies, i.e., uij ) (1 - kij)xuiiujj, where kij is called the binary interaction parameter.

(3)

where Adisp and Adisp are given by 1 2

Adisp 1 RT

j) ) -2πFIdisp 1 (η,m

() ij

∑i ∑j xixjmimj kT σij3

(4)

and

Adisp 2 RT

(

hc

) -πFm j 1+Z +F

∂Zhc ∂F

)

-1

∑i ∑j

Idisp j) 2 (η,m

()

xixjmimj

ij

kT

2

σij3 (5)

with

(

1 + Zhc + F

)

8η - 2η2 )1+m j + (1 - η)4 20η - 27η2 + 12η3 - 2η4 (6) (1 - m j) [(1 - η)(2 - η)]2

∂Zhc ∂F

-1

ij ) (1 - kij)xiijj is the interaction energy between a molecule of type i and a molecule of type j, kij is the binary interaction parameter, σij ) (σii + σjj)/2 is the temperature-independent diameter between a molecule of type i and one of type j, and m j ) ∑iximi is the average disp disp chain length. I1 and I2 are power series in reduced density, where η ) (π/6)F∑iximidi3 and di is the temperature-dependent diameter for component i, and the coefficients of the series are functions of chain length, m j . The coefficients in the Idisp and Idisp expressions 1 2 were fit to a large set of experimental pure-component data for the n-alkanes. The parameters for PC-SAFT are σ, , and m for nonassociating fluids. Independent of the dispersion term used, the SAFT chain and association terms are those originally proposed by Chapman et al.1,9,10 For nonassociating, nonpolar fluids, SAFT has three pure-component parameters: the segment diameter or volume (v00), segment dispersion energy (u0/k), and chain length (m).1-3,6 To account for multiple dipolar segments within a molecule, the functional group dipole moment is required, and an additional parameter is necessary to account for the number of dipolar segments. Hence, xp is defined as the fraction of segments in a chain that are dipolar. As shown previously,5 the dipolar contribution to the Helmholtz free energy is accurately obtained by dissolving all of the bonds in a chain and then applying the u expansion to the resulting mixture of polar and nonpolar spherical segments. The polar contribution is written in the Pade´ approximate form of Rushbrooke et al.14

Ind. Eng. Chem. Res., Vol. 42, No. 22, 2003 5689

Apolar 2 a RT ) RT Apolar 3 1 - polar A2 polar

(7)

For mixtures, Apolar and Apolar , allowing for multiple 2 3 dipolar segments, give

Apolar 2

)-

RT

F

2π

9 (kT)2

∑i ∑j

µi2µj2 xixjmimjxpixpj I2,ij (8) dij3

and

Apolar 3

)

RT

F2 π2 162 (kT)3 5

∑i ∑j ∑k

µi2µj2µk2 xixjxkmimjmkxpixpjxpk I3,ijk (9) dijdjkdik

where I2,ij and I3,ijk are the angular pair and triplet correlation functions, respectively, and are given by

I2,ij )

3dij3 HS gij (r,F*)r-6 dr ) 3 4π

∫

/ / -4 dr/ij ∫1∞gHS ij (rij,F*)rij

(10)

and

I3,ijk )

192 14π 1/2 π 5 5

∫|(σ(σ /σ/σ )r)r ij

jk

ij

jk

( )

∫0∞dr/12 r/12-2∫0∞dr/13 r/13-2

/ +(σ /σ )r/ ik jk 13 12 / -(σ /σ )r/ | ik jk 13 12

dr/23 r/23-2gijk(r/12,r/23,r/13) ψ222 (R1,R2,R3) (11)

where the function ψ222(R1,R2,R3) is a well-known function of angles R1, R2, and R3 of the triangle formed by the centers of the three molecules.15 These integral equations, I2,ij and I3,ijk, are cast in terms of a reduced radius (r/ij ) rij/dij). In this reduced form, we assume that the integrals are independent of components at a given reduced density. Therefore, the integrals I2,ij and I3,ijk in eqs 8 and 9 can be related to the corresponding pure fluid integrals, I2 and I3, by

I2,ij ) I2(Fdx3)

(12)

I3,ijk ) I3(Fdx3)

(13)

and

where dx3 ) ∑imixidii3. Another approach that should give similar results would be to use a van der Waals one-fluid theory approximation for dx. The four pure-component parameters, the segment size, the segment interaction energy, the number of segments per chain (m), and the fraction of polar segments in a chain (xp), are fitted to the saturated liquid density and vapor pressure data of the pure components. For molecules with a single polar site, the experimental value for the dipole moment of either the molecule or the functional group can be used. If experimental values are not available, the dipole moment of the functional group calculated from quantum mechan-

ics can also be used. For components with a single polar site and in a homologous series, we use the average value for the dipole moment of the series. For systems with multiple dipolar groups, we use the value of the functional group dipole moment. A detailed description of the development of the polar term can be found in Jog and Chapman5 for pure fluids and in Jog et al.4 for mixtures. Other thermodynamic quantities such as pressure and chemical potential can be obtained by differentiation of the Helmholtz free energy given by eq 1. 3. Parametric Study We have fitted the four polar SAFT parameters for the homologous series of ketones, from acetone to tridecanone. The equation of state parameters are fit to pure-component vapor pressure and liquid density data by minimizing the squared percent deviation between the experimental values and the model predicted values for each data point. The pure-component data used in this study are from the compilation of Smith and Srivastava.16 Each vapor pressure and saturated liquid density data point is weighted equally in the fitting. Average absolute deviations over the data set of less than 5% in the pure-component vapor pressure are needed in order for the model to yield reasonable results for vapor-liquid equilibria in most binary systems. The values of the four model parameters along with the average absolute deviations are listed in Table 1 for polar CK-SAFT and in Table 2 for polar PC-SAFT. Note that the average absolute deviations in vapor pressures for 2-octanone and 4-heptanone are significantly greater than the other compounds. This occurs with both dispersion terms. Concern regarding these differences led to an investigation of the consistency of the reported experimental data. In Figure 2a, the vapor pressure data for the 2-alkanones (butanone through undecanone) are plotted as the natural log versus the inverse of the temperature. A comparison of the vapor pressure data, along with linear fits, of each compound in the homologous series is presented. Note that the magnitudes of the slopes for the majority of the curves increase with increasing molecular weights of the compounds. However, the magnitude of the slope for the 2-octanone data is greater than those of 2-heptanone and 2-hexanone. In Figure 2b, the results of 2-octanone predicted by the polar CKSAFT model are shown to have a trend similar to those of the other 2-alkanones. Thus, the 2-octanone parameters presented in Table 1 do agree well with data for the other alkanones, which gives reason to question the accuracy of the experimental vapor pressure data for 2-octanone. Similar results were obtained for 4-heptanone using the ketones with the carbonyl positioned on the center carbon for comparison. On the basis of this analysis and the ability of these parameters to accurately represent binary systems as demonstrated in the next section, the parameters for 2-octanone and 4-heptanone appear to be reliable. It is impressive that SAFT could not be correlated to the questionable data using physically realistic parameters. Parametric Analysis. We have examined the equation of state parameters in regards to their physical representation of molecular characteristics. Because the change in molecular size from one member to the next in a homologous series is that of a methylene group, the molecular volume, as well as the chain length, should

5690 Ind. Eng. Chem. Res., Vol. 42, No. 22, 2003 Table 1. Parameters for the Polar CK-SAFT Model for Ketones avg abs dev (%) ketone

MW (g/mol)

temp (K)

v00 (mL/mol)

u0/k (K)

m

Mu (D)

xp

vapor pressure

liquid density

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

58.08 72.107 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

275-478 270-374 282-353 285-347 299-332 299-324 306-416 275-384 319-420 316-360 340-422 301-356 309-427 303-352 341-427

13.24 14.31 14.66 14.46 14.53 14.53 14.51 14.49 14.45 14.46 14.41 14.59 14.36 14.32 14.20

232.97 238.58 239.40 239.45 239.58 237.85 237.99 232.91 231.50 232.79 233.94 232.38 232.68 228.68 231.03

2.944 3.370 3.81 3.81 4.350 4.33 4.945 4.91 5.59 5.57 6.200 6.12 7.4 7.43 8.580

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

0.1695 0.1465 0.1315 0.13125 0.11621 0.1147 0.1015 0.1001 0.0891 0.0893 0.08065 0.08305 0.06736 0.0692 0.0614

2.261 2.404 1.546 1.060 0.569 0.266 0.942 9.377 10.303 2.994 1.569 2.132 3.415 2.418 2.813

1.440 2.145 0.619 0.892 2.713 2.423 1.180 1.882 1.254 0.587 1.156 0.821 1.236 0.693 0.941

Table 2. Parameters for the Polar PC-SAFT Model for Ketones avg abs dev (%) ketone

MW (g/mol)

temp (K)

v00 (mL/mol)

u0/k (K)

m

Mu (D)

xp

vapor pressure

liquid density

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

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

20.00 21.38 21.75 21.67 22.11 22.15 22.12 22.37 22.50 21.93 22.34 22.74 22.33 22.38 22.68

259.99 270.00 264.97 265.83 263.80 262.10 259.17 255.72 257.07 254.12 256.81 256.25 253.80 251.65 253.34

2.221 2.418 2.826 2.812 3.232 3.202 3.704 3.651 4.002 4.134 4.550 4.448 5.441 5.399 6.250

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

0.2258 0.2074 0.1770 0.1774 0.1543 0.1560 0.1360 0.1350 0.1254 0.1200 0.1101 0.1125 0.0919 0.0924 0.0799

1.338 0.498 0.033 0.197 0.013 0.010 0.638 9.703 12.330 3.671 0.155 0.013 0.144 1.305 0.177

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

be a linear function of the molecular weight. In Figure 3, the molecular volume, which is given by the product of the number of segments per molecule and the segment volume (mv00), is shown as a function of the molecular weight for the CK parameter set. The results of the ketones for polar SAFT are compared with results of the alkanes and alkanols for nonpolar SAFT. (Parameters for the alkanes and alkanols are those fit by Huang and Radosz.6) Note that the molecular volumes of the alkanols and alkanones are very similar as a function of the molecular weight. This is expected because the molecular formulas differ only by two hydrogen atoms. Also, as the chain length increases, the effect of the single oxygen atom on the molecular volume is expected to become less and less important and, at sufficiently high molecular weight, the ketone molecular volume should approach that of the alkanes, which it appears to do. For polar PC-SAFT, a linear relationship between the molecular volume and molecular weight also exists; however, the limiting behavior of approaching the molecular volume of the alkanes is not observed. For each method and series considered, the chain length is also a linear function of the molecular weight. Because the change in the chain length and molecular volume from one compound to the next in a homologous series represents the addition of a methylene group to the chain, we expect this value to be very nearly constant. One exception is perhaps for the compounds with carbon numbers of 3 or less, where the fluid behavior is known to differ significantly from the higher molecular weight components in a series.

In Figure 4, the change in the molecular volume (v00mCn - v00mCn-1) between sequential members of a homologous series is presented as a function of the carbon number, where we consider n ) 4 and higher. For CK-SAFT, the average change in the molecular volume is 7.7 ( 1.6 mL/mol for the alkanes and 8.4 ( 0.6 mL/mol for the alkanols. Using polar CK-SAFT, the average change in the molecular volume is 8.3 ( 0.7 mL/mol for the ketones. Note that the change in the molecular volume for the ketones and alkanols are statistically the same as would be expected because they have approximately the same molecular formula. For CK-SAFT, we can deduce that the segment volume of a methylene group is this average change in the molecular volume in the homologous series of alkanes (7.7 ( 1.6 mL/mol). By subtraction of the molecular volume of an alkane of carbon number n - 1 from that of an alkanone of carbon number n, the result is an estimate of the segment volume of a carbonyl group (CdO). Using values from 2-butanone to 2-nonanone, the average segment volume of a carbonyl group is 13.2 ( 1.5 mL/mol. Similarly, using values from 1-butanol to 1-decanol, an estimate for the segment volume for an alcohol group (CH2OH) is 12.7 ( 0.9 mL/mol. The range in the series considered in estimating these group values omits the lower molecular weight compounds because, as stated earlier, they tend to deviate from the behavior of the heavier compounds in a series. The upper end of the range considered is truncated at the last sequential compound for which we have parameters. For PC-SAFT, similar calculations yield a value

Ind. Eng. Chem. Res., Vol. 42, No. 22, 2003 5691

Figure 4. Change in the molecular volume between two members in a homologous series. CK-SAFT results are for alkanes6 and alkanols,6 and polar CK-SAFT results are for ketones.

Figure 2. (a) Natural log of the vapor pressure of the 2-ketones as a function of the inverse temperature. (b) Same as part a but with the 2-octanone results replaced by the predicted results from polar CK-SAFT. The experimental data are from Smith and Srivastava.16 The curves are linear fits to the data to guide the eye. Figure 5. Factor of a xpm fit for each ketone as a function of the molecular weight for the ketones.

Figure 3. Molecular volume as a function of the molecular weight. CK-SAFT results are for alkanes6 and alkanols,6 and polar CKSAFT results are for ketones.

of 10.3 ( 0.8 mL/mol for the segment volume of a methylene segment and 9.8 ( 1.2 mL/mol for a carbonyl group. While these results are preliminary, this method can be used to develop a group contribution based parameter set for SAFT. To accomplish this, an exceptionally thorough, consistent, and systematic parametric study of a wide range of compounds is critical. The segment dispersion energy, which accounts for the van der Waals type attraction between segments, appears to approach a limiting value at high molecular weight. For the PC-SAFT model, the ketones reach the

limiting value of the alkanes at a molecular weight of approximately 200 g/mol.17 The polar contribution to the Helmholtz free energy is related to the product of the fraction of polar segments (xp) and the square of the dipole moment (µ2). In this study, the average molecular dipole moment of the homologous series of the ketones (2.7 D)18 is used. For a homologous series of molecules with a single polar site, we expect that the product xpm will be constant. It was found that for several of the ketones the product xpm equals one-half. Therefore, this constant value was used for this homologous series and held for both the CK and PC dispersion models. As shown in Figure 5, the onehalf value is adhered to within (0.03 throughout the series. While it appears that the polar CK-SAFT parameters may be diverging at the higher molecular weight, use of this factor in predicting the parameters for 8-pentadecanone, which is discussed in the Parameter Predictions section below, is successful. During the fitting process for the four pure-component parameters in this model, it was noted that a broad minimum exists such that a wide range of parameters will accurately represent the experimental pure-component data. However, the ability of the pure-component parameters to predict the phase behavior of mixtures, which are sensitive to the parameter values, provides a more rigorous test of the effectiveness of the parameter values. We demonstrate the reliability of these ketone parameters by applying this more rigorous examination in section 4.

5692 Ind. Eng. Chem. Res., Vol. 42, No. 22, 2003 Table 3. Predicted Polar CK-SAFT Parameters avg abs dev (%) compound

MW (g/mol)

temp (K)

v00 (mL/mol)

u0/k (K)

m

Mu (D)

xp

vapor pressure

liquid density

8-pentadecanone 2,4-pentadione

226.4 100.11

443-555; 312-351 289-359; 283-359

14.38 14.53

228.5 239.6

9.68 4.02

2.7 2.7

0.05165 0.171

7.60 7.83

1.60 0.795

Figure 6. Vapor pressure curve for 8-pentadecanone. Comparison of experimental data16 and polar CK-SAFT prediction.

Parameter Predictions. By the development of correlations of the model parameters with molecular weight, parameters for other members of the ketone homologous series can be predicted. When the correlations for the molecular volume and chain length are used and xpm is set to one-half, the parameters for 8-pentadecanone are predicted. For polar CK-SAFT, the value of the segment dispersion energy (u0/k) is estimated based on the approach of a limiting value exhibited by the higher molecular weight ketones. The results for the vapor pressure curve, predicted and experimental, are shown in Figure 6. The parameter values, along with the range of temperatures of the experimental data with which the percentage differences are calculated, are listed in Table 3. Note that there is a significant difference in temperature ranges for the liquid density and vapor pressure data that were available, which may account for the differences in predicted and experimental values for the liquid densities (1.6%) and the vapor pressures (7.6%). Thus, the evaluation of the reliability of these parameters based solely on the deviation from experimental data is inconclusive. Mixture data were not found but would certainly provide a more rigorous test of the parameters. The parameters for 2,4-pentadione were also estimated. The segment volume and interaction energy parameters for 2,4-pentadione were assumed to be the same as those for 2-hexanone, the ketone of similar molecular weight. The fraction of polar segments, xp, and chain length, m, were adjusted to correlate with the pure-component 2,4-pentadione vapor pressure and liquid density experimental data. Because 2,4-pentadione has two carbonyl groups, we expect the chain length to be larger than that of the five-carbon molecule 3-pentanone but not as long as the six-carbon molecule 3-hexanone. A chain length of the average of these two compounds was determined to be appropriate. It was found that a value of 0.69 for xpm fit the pure-component experimental vapor pressures and liquid densities well. This is a 37% increase over the effective number of polar segments for 3-pentanone. While we might expect the number of polar segments to double for a diketone, the two carbonyl groups in 2,4-pentadione tend to orient

Figure 7. Two possible conformations of 2,4-pentadione. Built and optimized using Molecule 3D.23

Figure 8. Vapor pressure curve for 2,4-pentadione. Comparison of experimental data16 and polar CK-SAFT prediction.

such that the effects of the dipoles are partially canceled by each other, as can be seen from the molecular structures shown in Figure 7. Hence, for this case, the effect is increased but not doubled, which exemplifies the complexity of polar systems and potential difficulties in establishing a universal group contribution approach. The predicted vapor pressure curve and experimental data for 2,4-pentadione are shown in Figure 8, and the parameters are listed in Table 3. An investigation using a series of diketones and triketones would certainly be of interest. However, the availability of pure-component and mixture data for these is limited. 4. Application to Phase Equilibria in Binary Systems In a recent publication,4 we demonstrated the ability of the polar SAFT equation of state to predict the effect of the alkane solvent chain length on the phase behavior of binary alkane/acetone mixtures. Here, the effects of the molecular size of the ketones, temperature, and multiple polar groups are considered. In Figures 9-11, the predictions of CK-SAFT, polar CK-SAFT, and polar PC-SAFT for the P-x-y diagrams of several ketone/

Ind. Eng. Chem. Res., Vol. 42, No. 22, 2003 5693 Table 4. Binary Interaction Parameters for Ketone/ Alkane Mixtures system 2-butanone/heptane 2-butanone/decane 3-pentanone/heptane 3-pentanone/heptane 4-heptanone/hexane 5-nonanone/hexane 2,4-pentadione/ cyclohexane

Figure 9. 2-Butanone in heptane at 45 °C. Comparison of experimental data24 and model predictions: (a) CK-SAFT; (b) polar CK-SAFT; (c) polar PC-SAFT.

alkane mixtures are compared with experimental data. Overall, the polar SAFT model is able to capture the physics of these systems in which the phase behavior is dominated by the polar effects. However, nonpolar SAFT does not give acceptable results, with a zero value for the binary interaction parameter, k12. Because of the neglect of polar effects, nonpolar SAFT predicts nearly ideal solution behavior. The mixtures in this study show strong positive deviation from ideality due to the lack of polar interactions between the two components, as discussed in the Introduction. By adjustment of k12 (see Table 4), a more accurate fit is obtained for each model for most systems. On the whole, polar SAFT predicts the shape of the pressure-composition curves quite well, showing qualitative agreement with zero k12 and quantitative agreement with binary interaction parameters of typically 4-5 times less than that required for nonpolar SAFT for these systems.

temp (°C) 45.0 70.0 40.05 80.0 65.0 60.0 25.0

CK-SAFT 0.050 0.050 0.035 0.035

polar CK-SAFT

polar PC-SAFT

0.009 0.012 0.003 0.006 0.000 0.000 0.000

0.012 0.012 0.000 0.012 0.000 0.000

Parts a-c of Figure 9 compare the results of the three models for 2-butanone in heptane at 45 °C. The nonpolar CK-SAFT predicts nearly ideal solution behavior. Because the pure-component vapor pressure of 2-butanone predicted by CK-SAFT using the Huang and Radosz parameters does not agree with the experimental data, a k12 of 0.05 gives only qualitative agreement for the 2-butanone-rich phases. In contrast, with a zero binary interaction parameter (k12), both polar CK-SAFT and polar PC-SAFT (Figure 9b,c) predict the nonideal behavior and the formation of an azeotrope. Without adjustment of k12, the agreement is qualitative. Small k12’s are needed for each polar method (∼0.01) to obtain very accurate results. Similar trends in model capability were observed for 2-butanone in decane, a longer-chain alkane.17 The difficulty of accurately modeling polar mixtures is well demonstrated by the systems in Figures 10 and 11. Here we have 3-pentanone in heptane at two temperatures showing nonideal behavior. Because the pure-component vapor pressures are so similar, they form azeotropic mixtures over a very small pressure range. In Figure 10a-c, the results of 3-pentanone in heptane at 40.05 °C are shown for k12 of zero. Nonpolar CK-SAFT predicts a very slight positive deviation from ideality and does not give the correct pure-component vapor pressure for 3-pentanone or heptane (Figure 10a). Thus, while adjustment of k12 will give the correct form of nonideal behavior, it does not accurately predict the compositions of the 3-pentanone-rich phases. Both forms of polar SAFT predict positive deviations from ideality and the formation of an azeotrope. For polar CK-SAFT (Figure 10b), a very small adjustment of k12 (0.003) is needed to obtain excellent agreement with the data. Polar PC-SAFT (Figure 10c) requires no adjustment of the binary interaction parameter and gives an excellent representation of the heptane-rich phase and slightly overpredicts the compositions in the 3-pentanone-rich phases. The P-x-y representation of the 3-pentanone/heptane mixture at 80 °C is presented in Figure 11a-c. As in the 40 °C case, CK-SAFT predicts nearly ideal solution behavior with a value of zero for the binary interaction parameter and does not accurately reproduce the pure-component vapor pressures for either component. Using the same value of k12 as that for the 40 °C case, the nonpolar model does again predict the correct form of the nonideal behavior but does not accurately predict the composition of either phase. Both forms of polar SAFT again predict positive deviations from ideality and the formation of an azeotrope. For polar CK-SAFT (Figure 11b), a small adjustment of k12 (0.006) gives the correct azeotropic composition but overpredicts the vapor pressures for the 3-pentanonerich phases. Polar PC-SAFT (Figure 11c), which accurately represented the pure-component vapor pres-

5694 Ind. Eng. Chem. Res., Vol. 42, No. 22, 2003

Figure 10. 3-Pentanone in heptane. Comparison of experimental data25 and model predictions at 40.05 °C: (a) CK-SAFT; (b) polar CK-SAFT; (c) polar PC-SAFT.

sures, needs an adjustment to the geometric mean of the pure fluid interaction energies of k12 ) 0.01 to give accurate representation of the azeotrope. However, a value of k12 that gives excellent agreement with the experimental data for the heptane- and 3-pentanonerich phases was not determined. A thermodynamic consistency test19 of the experimental data for the 3-pentanone in heptane at 80 °C revealed that the data appear to be adequate but not of a high degree of consistency, whereas that at 40.05 °C is of high consistency. Because of the small pressure range for this system, the data could be very difficult to measure. Longer-chain alkanones were also considered. Polar SAFT predictions for 4-heptanone at 65 °C and for 5-nonanone at 60 °C in hexane are in very good agreement with experimental data but show nearly

Figure 11. 3-Pentanone in heptane. Comparison of experimental data26 and model predictions at 80.0 °C: (a) CK-SAFT; (b) polar CK-SAFT; (c) polar PC-SAFT.

ideal solution behavior.17 This type of behavior is expected because of the large ratio of pure-component vapor pressures and the dilution effect of the polar group in longer-chain alkanones. The results for the two dispersion terms (CK and PC) show comparable agreement with the experimental data for the systems studied here. This could be indicative of the important role that the long-range polar interactions play for these systems. The results do indicate that the long-range polar interactions can be included explicitly independently of the dispersion term utilized. To test the idea of using a functional group dipole moment, application to mixtures in which one component contains more than one carbonyl group is needed. While data for these systems are limited, a comparison of polar CK-SAFT predictions, using the parameters determined in this work (see section 3) and experimental data for 2,4-pentadione in cyclohexane at 25 °C, is presented in Figure 12. The polar model captures the

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were able to capture the minute details of the phase behavior of several binary systems. This indicates both the predictive capability of the model and the importance of explicitly accounting for the polar interactions directly in the theory. In the case of the cycloalkane/ diketone mixture, UNIFAC was found to give inadequate results while polar SAFT gave an excellent prediction of the phase behavior. The analysis of mixtures of two unlike polar components will be the subject of further work. In future work, we will also include quadrupolar and polarizability terms. We expect the power of this approach to be of greatest importance in the modeling of polar copolymers and biochemicals.4,5,22 Acknowledgment

Figure 12. 2,4-Pentadione in cyclohexane. Comparison of experimental data27 and model predictions for polar CK-SAFT (s) and UNIFAC20,21 (- -) at 25.0 °C.

physics of the system giving excellent quantitative agreement with the experimental data with no adjustments to the binary interaction parameter. A comparison with UNIFAC,20 using the ASPEN Properties21 software, for this system is included in Figure 12. UNIFAC predicts liquid-liquid equilibrium with a heterogeneous azeotrope. Sufficient data for diketones and higher are not available at present for further testing. Comparison of the results in Figures 9-11 with the UNIFAC model indicates that the polar SAFT model with a zero value for the binary interaction parameter is comparable to the UNIFAC model for the alkane/ ketone systems. However, the UNIFAC model significantly overestimates the nonideality when attempting to model the diketone/cycloalkane system predicting a nonexistent liquid-liquid equilbrium. This shows the difficulty in modeling systems with multifunctional groups. We expect that the further application of the polar SAFT model will continue to show significant improvement over existing models and extend the range of polar systems that can be accurately modeled. 5. Conclusions An in-depth parameter study of polar SAFT, using both the CK and PC dispersion terms, is presented. On the basis of the parameter analysis, the reliability of the experimental data for the vapor pressures of 4-heptanone and 2-octanone is determined to be questionable. The polar SAFT parameters are shown to be wellbehaved functions of the molecular weight. Moreover, it is shown that the explicit inclusion of polar effects provides a systematic set of parameters for a homologous series of ketones, from acetone to 2-tridecanone. Correlations of these parameters based on the molecular weight can be used to predict parameters for longer chains within the series and for some diketones. We have shown that the polar SAFT equation of state for mixtures accurately predicts the effect of dipolar groups and molecular shape on the phase behavior of binary mixtures of polar and nonpolar components. We applied polar SAFT to binary alkane/ketone and cycloalkane/diketone systems. In the ketone/alkane systems, in which the chain length of the ketone was allowed to vary, polar CK-SAFT and polar PC-SAFT

We thank the Robert A. Welch Foundation for their financial support of this work and NSF for a graduate fellowship for S.G.S. Nomenclature A ) individual contributions to the Helmholtz free energy a ) molar Helmholtz free energy Dij ) matrix of parameters for the CK dispersion term dii ) segment diameter dx ) average segment diameter gHS ij ) hard-sphere pair correlation function k ) Boltzmann constant kij ) binary interaction parameter m ) chain length or number of segments within a molecule m j ) average number of segments per molecule N ) number of molecules NAV ) Avogadro’s number R ) gas constant T ) temperature (K) u11 ) molecular interaction energy u0/k ) segment interaction energy v00 ) segment volume x ) mole fraction xp ) fraction of polar segments Greek Symbols  ) potential well depth for modified square well potential η ) packing fraction µ ) dipole moment (D) π ) pi F ) molecular density σ ) temperature independent diameter τ ) πx2/6 Subscripts i, j, k ) indices Superscripts * ) a reduced quantity disp ) dispersion term hs ) hard sphere polar ) dipolar property res ) residual property

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Resubmitted for review August 1, 2003 Accepted August 6, 2003 IE034035U