Ind. Eng. Chem. Res. 2004, 43, 1803-1806
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RESEARCH NOTES Applying Association Theories to Polar Fluids Nicolas von Solms, Michael L. Michelsen, and Georgios M. Kontogeorgis* Centre for Phase Equilibria and Separation Processes (IVC-SEP), Department of Chemical Engineering, Technical University of Denmark, DK-2800 Lyngby, Denmark
The question of whether acetone can be treated as self-associating is considered. Motivations for considering ketones as self-associating are given, based partially on some of their purecomponent physical properties but especially on their phase behavior. It is shown that good vapor-liquid equilibria predictions can be obtained for the mixtures acetone-pentane and acetone-water by considering acetone as self-associating. The justification for this approach is the existence of a vast body of literature and experience in the modeling of associating fluids. Thus, we are able to use existing tools, methods, and models to predict phase behavior in fluids with complex behavior (although traditionally considered nonassociating) without resorting to the development of new models and extra terms. Introduction The statistical associating fluid theory (SAFT) equation of state1 is a highly successful and, by now, wellestablished model for predicting and correlating the phase equilibria of (especially) chain and associating fluids and fluid mixtures. In particular, the association contribution, based on the brilliant work of Wertheim,2-5 has provided a tool for the accurate modeling of phase equilibria containing such ubiquitous, yet complex, hydrogen-bonding fluids as water, alcohols, amines, and acids. This so-called “Wertheim term” has also been used with success together with a cubic equation of state in the cubic plus association (CPA) model.6 A new version of SAFT, known as the perturbed-chain SAFT (PC-SAFT), appeared recently7 and in the short time since then has received much attention. A simplified version of PC-SAFT was developed very recently in our group.8 Hydrogen bonding, or associating, fluids are modeled through the use of two extra parameters, association energy and association volume, which characterize an association site on a molecule rather than a molecule itself. This means a total of five pure-component parameters for an associating fluid (the other three are chain length, segment size, and segment dispersion energy parameters, and these are required by all compounds in SAFT, both associating and nonassociating). The presence of these extra parameters means that a decision must be made beforehand as to whether a compound is associating or not. This decision is usually fairly straightforward, based on some simple rules. Vinogradov and Linnell9 divide substances into four categories: (I) molecules with donor groups only (such as chloroform and acetylene); (II) molecules with acceptor groups only (such as ketones, ethers, and esters); (III) molecules with both (such as alcohols and * To whom correspondence should be addressed. Tel.: +45 45 25 28 59. Fax: +45 45 88 22 58. E-mail:
[email protected].
water), and (IV) molecules with neither (such as saturated hydrocarbons and carbon tetrachloride). While groups III and IV are fairly easily dealt with, problems arise with groups I and II, especially in mixtures containing one of each, such as the classic mixture chloroform-acetone, where cross-association possibly exists without self-association of either compound. The question of how to model cross-association in the absence of self-association has still not been adequately resolved. Part of the reason for this is that purecomponent association parameters are required when modeling cross-association. However, because there is no self-association, it is not possible to obtain these parameters only from pure-component vapor pressures and liquid densities. In this work, we suggest modeling of certain compounds from groups I and II, such as ketones, as if they were self-associating. This approach leads to improved phase equilibrium prediction without the need for additional model terms accounting for other postulated physical interactions, such as those due to polarity, for example. The model we have used throughout is the simplified version of PC-SAFT developed previously by our group.8 In the discussion that follows, we consider first the question of whether a substance is self-associating based on the physical properties of normal boiling point, enthalpy of vaporization at the boiling point, entropy of vaporization (Trouton’s constant), and water solubility. We then show some vapor-liquid equilibrium results for acetone mixtures, justifying the consideration of acetone as self-associating. Discussion To arrive at a definition of self-association that is based on “degree” rather than on “type”, some properties of lower molecular weight ethers, esters, ketones, alcohols, and water are shown in Table 1. The data in Table 1 are taken from Poling et al.10 A similar (less complete)
10.1021/ie034243m CCC: $27.50 © 2004 American Chemical Society Published on Web 02/28/2004
1804 Ind. Eng. Chem. Res., Vol. 43, No. 7, 2004 Table 1. Some Properties of Lower Molecular Weight Ethers, Esters, Ketones, Alcohols, and Water from Poling et al.10 normal boiling point (K)
enthalpy of vaporization at the boiling point (kJ mol-1)
entropy of vaporization (Trouton’s constant) (J mol-1 K-1)
water solubility (mass %)
dipole moment (D)
dimethyl ether
248.3
21.5
86.6
7.12 (18 °C)
1.3
methyl methanoate methyl ethanoate ethyl methanoate
304.9 330.1 327.5
27.9 30.3 29.9
91.5 91.8 91.3
23 (25 °C) 24.5 (20 °C) 11.8 (25 °C)
1.8 1.7 2.0
acetone 2-butanone
329.2 352.7
29.1 31.3
88.4 88.7
∞ 25.9 (25 °C)
2.9 3.3
ethanol 1-propanol 2-propanol 1-butanol 2-butanol
351.8 370.9 355.4 390.9 372.7
38.6 41.4 39.9 43.3 40.8
109.7 111.6 112.3 110.8 109.5
∞ ∞ ∞ 7.4 (25 °C) 18.1 (25 °C)
1.7 1.7 1.7 1.8 1.7
water
373.15
40.7
109.1
Table 2. New PC-SAFT
1.8
Parametersa
methyl ethanoate (nonassoc.) methyl ethanoate (assoc.) acetone (nonassoc.) acetone (assoc.) water 2-propanol ethanol isobutane
M (g/mol)
m
σ (Å)
/k (K)
74.079 74.079 58.08 58.08 18.015 60.096 46.069 58.123
3.1421 3.8233 2.7741 3.0925 1.0656 3.0929 2.3827 2.2616
3.1888 2.959 3.2557 3.0848 3.0007 3.2085 3.1771 3.7574
235.75 172.45 253.41 168.32 366.51 208.43 198.24 216.53
κAB
AB/k (K)
1.9038
804.9
0.9639 0.0349 0.0247 0.0324
1321.2 2500.7 2253.9 2653.4
AAD % Psat
VsL
T range (K)
ref
1.1 0.83 0.99 0.26 1.88 0.7 0.99 0.55
0.95 0.38 1.95 0.26 6.83 1.25 0.79 1.47
200-500 200-500 250-480 200-500 273-647 185-508 230-516 113-407
this work this work 12 this work 16 16 16 7
a Experimental data are obtained from the DIPPR correlations.19 Associating acetone and methyl ethanoate are assumed to have two association sites and are modeled using the 2B association scheme of Huang and Radosz.20
table was used by Prausnitz et al.11 to illustrate a similar point. Rather than being a binary yes/no state (association exists/does not exist), association can be considered to exist in varying degrees, from very weak or none (as in alkanes) to very strong (as in acids). Generally, the boiling point is to some extent an indicator of association (because the association forces that exist between molecules in the liquid strengthen their mutual attraction and elevate the boiling point), although it also depends strongly on the molecular weight. So, for compounds of comparable molecular weight, the boiling point increases from ethers, through esters and ketones, to alcohols. Of course, water, the smallest molecule, has an astonishingly high boiling point. The enthalpy of vaporization is another guide to the degree of association. For the relatively small molecules shown in Table 1, we see that the alcohols and water have an enthalpy at the normal boiling point of around 40 kJ/mol, ketones and esters of around 30 kJ/mol, and ethers of around 20 kJ/mol. The entropy of vaporization (sometimes called Trouton’s “constant”, although it is not constant) is perhaps a more reliable guide to the degree of association. Here the alcohols and water have an entropy of vaporization of around 110 J/mol and the other substances of around 90 J/mol. However, an interesting trend is observed for the water solubility of the different classes of compounds. Acetone and alcohols up to propanol are fully miscible in water at around room temperature. However, the higher molecular weight species become less soluble as the chain length increases. However, the stronger crossassociation that exists between water and butanone renders it more soluble than either of the two butanol isomers. It is perhaps also noteworthy that the ketones are more polar than the other compounds, with a dipole
Figure 1. Vapor-liquid equilibrium of acetone (1)-pentane (2) at 298.15 K. The dotted line is the simplified PC-SAFT prediction when acetone is considered to be nonassociating (parameters from ref 12). The solid line is the prediction when acetone is considered to be self-associating (parameters in Table 2). Points are experimental data.17 Note that in both cases the binary interaction parameter kij ) 0. Pentane parameters are from ref 7.
moment of around 3 D, compared with around 2 D for esters, alcohols, and water and around 1 D for ethers. We next consider the phase equilibrium behavior of systems containing acetone. Figure 1 shows vaporliquid equilibrium in the system acetone (1)-pentane (2) at 298.15 K. Both curves are predictions; i.e., the binary interaction parameter kij is set to zero. An excellent prediction is obtained if acetone is modeled as an associating fluid (solid line), whereas the prediction is qualitatively incorrect if acetone is modeled as a nonassociating fluid. The acetone parameters are given in Table 2 (considered as self-associating) and in ref 12
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Figure 2. Vapor-liquid equilibrium of acetone (1)-water (2) at 473.15 K. The dotted line is the simplified PC-SAFT prediction when acetone is considered to be nonassociating (parameters from ref 12). Water parameters are from ref 16. The solid line is the prediction when acetone is considered to be self-associating (parameters in Table 2). Points are experimental data.17 Note that in both cases the binary interaction parameter kij ) 0. Crossassociation is modeled using the Elliott rule, as discussed in the text.
(nonassociating). It should be mentioned that in both cases (associating and nonassociating acetone) the purecomponent parameters of acetone were obtained by fitting the parameters to pure-component liquid density and vapor pressure data. It may be mentioned that already, in 1992, Suresh and Elliott13 and later Elliott and Lira14 modeled acetone as self-associating without further discussion. An excellent prediction of the same acetone-pentane phase diagram (Figure 1) was obtained by Jog et al.15 by considering acetone as a polar compound. In this way, an additional polar term is introduced into the expression for the Helmholtz energy, involving two extra purecomponent parameters (because one of these is the dipole moment, which is readily available, only one additional parameter is adjusted to pure-component vapor pressure and liquid density data). An additional justification for modeling acetone as a self-associating fluid is shown in Figure 2. The dotted line is the vapor pressure prediction (kij ) 0) in the system acetone (1)-water (2) at 473.15 K, where acetone is modeled as a nonassociating fluid. Water is modeled as an associating fluid, and the parameters are from ref 16. Clearly, the prediction is way off, with the model predicting a large region of liquid-liquid immiscibility. The prediction with acetone modeled as self-associating is much better (solid line, kij still zero). Cross-association is modeled via the so-called Elliott rule,14 where the cross-association strength parameter is given as the geometric mean of the two self-association strength parameters:
∆AiBj ) x∆AiBi∆AjBj
(1)
The self-association strength parameters depend on the pure-component size and energy parameters in the usual way:8
[ ( ) ]
AiBj ∆AiBj ) dij3gij(d+ exp ij )κ
AiBj -1 kT
(2)
The reasons for this drastic improvement are twofold. First, as in Figure 1, an improved prediction is obtained
Figure 3. Vapor-liquid equilibrium in the system methyl ethanoate (1)-cyclohexane (2) at 1 atm. The dotted line is the prediction (kij ) 0) where methyl ethanoate is considered to be nonassociating; the solid line is the prediction (kij ) 0) where methyl ethanoate is considered to be associating. The ester parameters in each case are in Table 2; the cyclohexane parameters are from ref 7. Experimental data are from ref 18.
if acetone is modeled as self-associating. Second, and probably more importantly, because acetone now has associating parameters, the cross-association that certainly exists between acetone and water is naturally taken into account in the model. Thus, the fraction of unbonded sites on water molecules (which enters into the Helmholtz energy expression for association) is influenced not only by the degree of self-association in water but also by the degree of cross-association of water and acetone. It is, of course, possible to model cross-association without self-association, although, as mentioned above, the question of how to obtain the association parameters of a non-self-associating compound has not yet been satisfactorily resolved. Many of the substances higher up on Table 1 also exhibit azeotropic behavior in mixtures with hydrocarbons. Examples are the systems methyl methanoatehexane, methyl ethanoate-pentane, methyl ethanoatecyclohexane, ethyl methanoate-cyclohexane, and diethyl ether-pentane. None of these azeotropes can be predicted when all species are modeled as nonassociating fluids, although good correlations (nonzero kij) can usually be obtained. Figure 3 shows a temperaturemole fraction diagram at a pressure of 1 atm for the system methyl ethanoate (1)-cyclohexane (2). The prediction (kij ) 0) for methyl ethanoate modeled as a nonassociating fluid is shown as a dotted line. The prediction (kij ) 0) for methyl ethanoate modeled as an associating fluid is shown as a solid line. The experimentally observed azeotrope is predicted at the correct concentration, although the azeotropic temperature is overpredicted by about 2 K. Figure 4 shows a plot of the log of the association strength (∆AB from eq 2, excluding the radial distribution function gij) as a function of the reciprocal temperature for ethanol (solid line), 2-propanol (dot-dashed line), acetone (dashed line), and methyl methanoate (dotted line) in the temperature range 220-500 K. It is clear from Figure 4 that the temperature dependence of the association strength is greater for the alcohols than for the ketone or ester. This is due to the fact that the association strength depends much more strongly on the association energy parameter () than on the
1806 Ind. Eng. Chem. Res., Vol. 43, No. 7, 2004
both experience and literature on the modeling of associating fluids, it seems useful to benefit from this experience and the modeling tools that accompany it. In this way improved prediction of the phase equilibrium can be achieved with the knowledge and tools we already have at hand, without the need for extra terms (and the extra model development that accompanies them) to account for postulated physical interactions. Literature Cited
Figure 4. Log of the association strength (∆AB from eq 2, excluding the radial distribution function gij) as a function of reciprocal temperature for ethanol (solid line), 2-propanol (dotdashed line), acetone (dashed line), and methyl ethanoate (dotted line) in the temperature range 220-500 K.
association volume parameter (κ). We have found surprisingly high values of κ for acetone (about 30 times that of ethanol) and methyl ethanoate (about 60 times that of ethanol), although the association energy parameters are correspondingly lower (acetone is about half that of ethanol, and methyl ethanoate, about a third). It is difficult to compare these parameters in isolation; rather the overall association strength (∆) should be considered, as is done in Figure 4, which shows expected trends and magnitudes. Finally, perhaps a word on parameter estimation is in order. The usual way to obtain pure-component parameters (and the way which we have employed here) is to regress all parameters simultaneously against experimental saturated liquid density and vapor pressure data for the pure component (which, in practice, usually means regressing against the DIPPR correlations19 for a compound). In the case of an associating compound, this means simultaneously regressing five pure-component parameters. When attempting to obtain as many as five pure-component parameters through regression, it is usually possible to find that multiple parameter sets yield an acceptable error. However, the final five parameters that were chosen were those that resulted in the smallest errors in saturated liquid density and vapor pressure (see Table 2). No attempt was made to “manipulate” the pure-component parameters in order to obtain improved mixture predictions. We have found that for acetone and methyl ethanoate a larger association volume parameter (κ), combined with a smaller association energy parameter () (compared to, say, ethanol), results in excellent agreement with pure-component experimental data while also providing good predictions for binary vapor-liquid equilibrium data. It may be that this description of association (lower association energy with a larger range) provides a good model of the effect of polarity. Conclusion We have found that the prediction of phase equilibrium in systems containing polar compounds such as ketones and esters is improved if they are modeled as associating compounds. Because there is a wealth of
(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. (2) Wertheim, M. S. Fluids with Highly Directional Attractive Forces. I. Statistical Thermodynamics. J. Stat. Phys. 1984, 35, 19. (3) Wertheim, M. S. Fluids with Highly Directional Attractive Forces. II. Thermodynamic Perturbation Theory and Integral Equations. J. Stat. Phys. 1984, 35, 35. (4) Wertheim, M. S. Fluids with Highly Directional Attractive Forces. III. Multiple Attraction Site. J. Stat. Phys. 1986, 42, 459. (5) Wertheim, M. S. Fluids with Highly Directional Attractive Forces. IV. Equilibrium Polymerization. J. Stat. Phys. 1986, 42, 477. (6) Kontogeorgis, G. M.; Voutsas, E. C.; Yakoumis, I. V.; Tassios, D. P. An Equation of State for Associating Fluids. Ind. Eng. Chem. Res. 1996, 35, 4310. (7) Gross, J.; Sadowski, G. Perturbed-Chain SAFT: An Equation of State Based on a Perturbation Theory for Chain Molecules. Ind. Eng. Chem. Res. 2001, 40, 1244. (8) von Solms, N.; Michelsen, M. L.; Kontogeorgis, G. M. Computational and Physical Performance of a Modified PC-SAFT Equation of State for Highly Asymmetric and Associating Mixtures. Ind. Eng. Chem. Res. 2003, 42, 1098. (9) Vinogradov, S. N.; Linnell, R. H. Hydrogen Bonding; van Nostrand Reinhold: New York, 1971. (10) Poling, B. E.; Prausnitz, J. M.; O’Connell, J. P. The Properties of Liquids and Gases, 5th ed.; McGraw-Hill: New York, 2001. (11) Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular Thermodynamics for Fluid-Phase Equilibria, 3th ed.; Prentice-Hall: Englewood Cliffs, NJ, 1999. (12) Kouskoumvekaki, I. A.; von Solms, N.; Michelsen, M. L.; Kontogeorgis, G. M. Application of the Perturbed Chain SAFT Equation of State to Complex Polymer Systems Using Simplified Mixing Rules. Fluid Phase Equilib. 2004, 215, 71. (13) Suresh, S. J.; Elliott, J. R., Jr. Multiphase Equilibrium Analysis via a Generalized Equation of State. Ind. Eng. Chem. Res. 1992, 31, 2783. (14) Elliott, J. R., Jr.; Lira, C. T. Introductory Chemical Engineering Thermodynamics; Prentice-Hall: Englewood Cliffs, NJ, 1999. (15) Jog, P. K.; Sauer, S. G.; Blaesing, J.; Chapman, W. G. Application of Dipolar Chain Theory to the Phase Behavior of Polar Fluids and Mixtures. Ind. Eng. Chem. Res. 2001, 40, 4641. (16) Gross, J.; Sadowski, G. Application of the Perturbed-Chain SAFT Equation of State to Associating Systems. Ind. Eng. Chem. Res. 2002, 41, 5510. (17) Gmehling, J.; Onken, U.; Rarey, J. R. Vapor-Liquid Equilibrium Data Collection: Ketones; Chemistry Data Series; DECHEMA: Frankfurt, Germany, 1993; Vol. 1, Part 3b. (18) Nagata, I. Vapor-Liquid Equilibria for the Ternary System Methyl Acetate-Benzene-Cyclohexane. J. Chem. Eng. Data 1962, 7, 461. (19) DIPPR Tables of Physical and Thermodynamic Properties of Pure Compounds; AIChE: New York, 1998. (20) Huang, S. H.; Radosz, M. Equation of State for Small, Large, Polydisperse and Associating Molecules. Ind. Eng. Chem. Res. 1990, 29, 2284.
Received for review November 10, 2003 Revised manuscript received February 9, 2004 Accepted February 16, 2004 IE034243M
