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Investigating Models for Associating Fluids Using Spectroscopy Nicolas von Solms,* Michael L. Michelsen, Claudia Pereira Passos, Samer O. Derawi,† 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
Two equations of state (PC-SAFT and CPA) are used to predict the monomer fraction of pure associating fluids. The models each require five pure-component parameters usually obtained by fitting to experimental liquid density and vapor pressure data. Here we also incorporate monomer fractions measured using spectroscopy, resulting in models that better predict the monomer fraction (fraction of molecules not participating in hydrogen bonding), without sacrificing the accuracy of the liquid density and vapor pressure correlations. Thus, it is clear that monomer fraction prediction depends on the way the parameters were obtained. The selection of appropriate association schemes is also investigated using spectroscopic data. For pure water a four-site scheme is shown to be the most appropriate scheme. In the case of pure alcohols, a three-site scheme is best for methanol; two- or three-site schemes perform about equally for ethanol; for higher alcohols a two-site scheme is preferred. This is in accordance with steric arguments. Some difficulties in the interpretation of spectroscopic data and their comparison with the predictions of association models are illustrated. Apparently anomalous data from different sources are shown to be consistent if interpreted correctly. 1. Introduction The SAFT family of equations comprises some of the very few equations of state which explicitly account for the hydrogenbonding interactions that exist in associating fluids such as water and alcohols. Two such equations are the “cubic plus association”1,2 (CPA) and the recently developed “perturbed chainSAFT” (PC-SAFT) equations of state.3 Both these equations of state require 5 pure-component parameters to be specified for each associating fluid. Three of these describe the molecule segment diameter, length, and segment dispersion energy (also required for nonassociating fluids) whereas the remaining two parameters characterize the hydrogen-bonding sites on the molecule. These five parameters are typically obtained by simultaneous fitting of the model parameters to experimental saturated liquid density and vapor pressure data. When so many parameters are required, the physical meaning of the parameters may be lost in the estimation procedure. It would thus be useful if further experimental data, specific to associating fluids, could be used when obtaining parameters for these fluids. Equations of state based on the Wertheim formalism4 (such as the SAFT variants) naturally calculate the degree of hydrogen bonding that exists in a pure fluid or mixture at given conditions. But IR and Raman spectroscopy can give us just such information experimentally. It is hoped that with use of extra experimental data, parameters that incorporate more of the physics of fluids and fluid mixtures may be obtained. While experimental spectroscopic data measuring the monomer fractions of associating fluids in mixtures with inert solvents is relatively plentiful,5 there is a comparative lack of data for pure components. This may be because these measurements are more difficult to do for pure saturated fluids over an extended temperature range. For this work, three sets of data were selected for comparison with model predictions. These were the data of * Corresponding author. Telephone: +45 45 25 28 67. Fax: +45 45 88 22 58. E-mail:
[email protected]. † Current address: Department of Chemical Engineering, Center for the Study of Polymer-Solvent Systems, Fenske Laboratories, Pennsylvania State University, PA 16801.
Luck6 for methanol, ethanol, and water, the data of Lien7 for 1-propanol, and the data of Fletcher and Heller8 for 1-octanol. We have selected CPA and PC-SAFT as test models for analyzing spectroscopic data since there is extensive experience in our group in the area of phase equilibrium calculations using these equations of state. 2. Theory and Modeling Both CPA and PC-SAFT models account explicitly for association and can predict the monomer fraction of associating substances in pure fluids and mixtures (as can any of the SAFTtype models). By “monomer” is meant a molecule of an associating fluid not participating in any hydrogen bonding. The association contribution of the two models is based on Wertheim’s theory for associating fluids.4 Both PC-SAFT3 and CPA1,2 have been fully described in the literature and will not be discussed extensively in this work. Their differences lie in the way they describe repulsive and dispersive interactions, while the form of the association term is nearly identical for each model. The parameters of the association term may have different values in each model, although they have similar physical significance. According to both models, the monomer fraction (X1) of an associating compound in its pure fluid is given by the following equation
X1 )
XK ∏ K
(1)
where XK is the fraction of sites of type K that are not bonded, i.e., the “site” monomer fraction. XK is related to the association strength ∆AB (the key quantity in these models) between two association sites A and B as follows (for a pure component)
1
XA ) 1+F
∑B X ∆ B
(2)
AB
where the summation is over all sites which are permitted to
10.1021/ie051341u CCC: $33.50 © 2006 American Chemical Society Published on Web 06/22/2006
Ind. Eng. Chem. Res., Vol. 45, No. 15, 2006 5369 Table 1. Parameters Used in the CPA Equation of State; Association Schemes Refer to the Terminology of Huang and Radosz9
a
association scheme
reference
b (L/mol)
a0 (bar‚L2/mol2)
c1
AB (bar‚L/mol)
βAB
∆P (%)
∆F (%)
water
4C 3B
2 this work
0.014515 0.013164
1.2277 1.866570
0.67359 0.01070
166.55 280.00
0.0692 0.0149
0.79
0.47
methanol
2B 3B
2 22
0.030978 0.033392
4.0531 4.589750
0.43102 1.00676
245.91 160.70
0.0161 0.0344
0.56
0.48
ethanol
2B 3B
23 22
0.04911 0.050010
8.6716 8.575520
0.7369 1.05638
215.32 150.00
0.0080 0.0173
1.29
0.33
1-propanol
2B 3B
23 22
0.06411 0.065460
11.9102 12.75790
0.91709 0.98566
210.01 171.49
0.0081 0.0063
0.39
0.53
1-octanol
2B 3B
23 22
0.1485 0.148989
41.5822 41.90050
1.1486 1.05504
267.59 250.00
0.00014 0.00019
0.80
0.52
compound
a
∆P and ∆F are the average absolute deviations from experimental vapor pressure and saturated liquid density, respectively.
Table 2. Parameters Used in the PC-SAFT Equation of State; Association Schemes Refer to the Work of Huang and Radosz9
a
association scheme
reference
m
σ (Å)
(K)
AB (K)
κAB
∆P (%)
∆F (%)
water
2B 4C 4C 4C 4C 4C 4C 4C
3 this work this work this work this work this work this work this work
1.0656 2.0000 2.2500 2.5000 2.7500 3.0000 3.2500 3.5000
3.0007 2.3533 2.2462 2.1562 2.0794 2.0135 1.9570 1.9134
366.51 207.84 194.20 187.06 183.61 182.92 185.46 199.88
2500.7 1506.4 1479.6 1427.2 1354.1 1259.0 1128.8 839.0
0.034868 0.1550 0.2050 0.2646 0.3374 0.4287 0.5513 0.7901
1.88 0.70 0.66 0.55 0.44 0.43 0.55 0.73
6.83 1.66 1.28 1.08 1.05 1.12 1.20 1.24
methanol
2B 2B
3 18
1.5255 2.7921
3.2300 2.6510
188.90 186.60
2899.5 2090.2
0.035176 0.1460
2.36
2.01
ethanol
2B 2B
3 this work
2.3827 2.7710
3.1771 3.0123
198.24 194.84
2653.4 2472.9
0.032384 0.0468
0.99 0.15
0.79 0.17
1-propanol
2B 2B
3 15
2.9997 2.8148
3.2522 3.3085
233.40 236.34
2276.8 2370
0.015268 0.01457
0.85 0.18
1.71 0.57
1-octanol
2B
3
4.3555
3.7145
262.74
2754.8
0.002197
1.28
0.92
compound
a
∆P and ∆F are the average absolute deviations from experimental vapor pressure and saturated liquid density, respectively.
bond with site A. Equation 2 can be solved explicitly for XA if the bonding scheme is known and the expressions have been presented by Huang and Radosz.9 The strength of association between two sites is given below for the two models. For CPA
[ ( ) ]
∆AB ) g(F) exp
AB - 1 bβAB RT
(3)
where g(F) is the radial distribution function of a reference fluid of hard spheres, AB is the association energy, b is the co-volume parameter from the Soave-Redlich-Kwong (SRK) equation of state, and βAB is the association volume. For PC-SAFT
[ ( ) ]
∆AB ) g(d)seg exp
AB - 1 κABd3 kT
(4)
where d is the diameter of a chain segment on a molecule and κAB is the association volume (corresponding to the βAB of CPA). Tables 1 and 2 show the values of the parameters for the different species tested in this work, as well as the association schemes employed. Various association schemes have been proposed for different compounds and will be tested in this work. The three schemes usually employed for water and alcohols are the so-called 2B, 3B, and 4C schemes. This terminology is taken from the work of Huang and Radosz.9 The 2B scheme (two association sites) has one proton donor and one proton acceptor per molecule. The 3B scheme (three association sites) has either two proton donors and one proton acceptor or one proton donor and two proton acceptors per molecule. The 4C scheme (four association sites) has two proton donors and two proton acceptors per molecule. In the cases of 2B and 4C all sites are considered equivalent. However, in the
case of 3B only two of them are equivalent. Table 3 illustrates these bonding schemes in the SAFT formalism for alkanols and water from Huang and Radosz.9 For alkanols in the 3B formalism, sites A and B correspond to oxygen lone pairs, while site C corresponds to a hydrogen atom. Due to the asymmetry of the association, the fraction of nonbonded hydrogen atoms (XC) is not equal to the fraction of nonbonded lone pairs (XA or XB). Even for a fully hydrogen-bonded system, XA ) XB ) 0.5. For alkanols in the 2B formalism, the two lone-pair oxygens are considered to be a single site. In the case of water in the 4C formalism, the bonding symmetry means that all nonbonded site fractions are equal. For water in the 3B formalism, either the two-lone pair electrons on the oxygen atom are considered to be a single site, or else the two hydrogen atoms are lumped together into a single site, labeled C. For 2B water, the oxygen sites are lumped together as a single site B and the hydrogens as a single site A. In all cases, the fraction of monomers (completely nonbonded molecules, X1) is equal to the product of the fractions of all nonbonded site types. In cases where bonding is symmetrical (2B and 4C), the fraction of nonbonded sites is assumed to be equal for all types of sites. This is a statistical consideration that all site types on a molecule are equally likely to participate in hydrogen bonding. Thus, in the usual SAFT-type models such effects as hydrogen bond cooperativity10 are ignored. Formally, we can convert from site fraction to monomer fraction using eq 1. Thus, for 4C water (see Table 3) we have
X1 ) XAXBXCXD ) (XA)4
(5)
If alcohol is assumed to be a rigorous 3B model (see Table 3), then the fraction of nonbonded hydroxyl hydrogen atoms is XC.
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Table 3. Bonding Types in Alcohols and Water
Figure 1. Fraction of free -OH groups (filled squares) and fraction of monomeric molecules (filled diamonds) for water, both obtained from experiment. The filled data points are the digitized data of Luck.6 The fraction of monomeric molecules can be calculated (open diamonds) from the spectroscopically determined fraction of free -OH groups. Similarly, the fraction of free -OH groups can be calculated (open squares) from the experimentally determined monomeric fraction using eq 5.
Converting to monomer (fully dissociated) fraction can be done using the relation
X1 ) XAXBXC
(6)
In the 3B scheme, sites A and B have an equal probability of bonding; thus, XA ) XB. The fraction of nonbonded C sites XC can be given in terms of XA as follows: for every C site that bonds, either an A or a B site must also bond. Thus, the fraction of molecules bonded at C is twice the fraction of sites bonded at A (or B). Symbolically,
authors’ results were based on a neutron diffraction with isotope substitution method. Chatzidimitriou-Dreismann12 supports the conclusion for water, calculating it to be a 3.5 site molecule using neutron diffraction. Guo et al.13 are in disagreement with the results of Dixit et al.11 Their results based on X-ray emission spectroscopy support a picture of water with a much greater degree of hydrogen bonding, implying a picture with more than 3.6 sites per molecule. In the case of CPA, all three schemes have been employed for water, but only the 2B and 3B for alcohols, while the calculations with PC-SAFT are mostly based on literature parameters using the 2B scheme. However, new 4C parameters for water were developed as part of this work. 3. Discussion
(1 - XC) ) 2(1 - XA)
(7)
which can be rearranged to give
XA ) XB ) (XC + 1)/2
(8)
Equation 6 then becomes
X1 ) (XA)2(2XA - 1)
(9)
Finally, for a 2B model
X1 ) XAXB ) (XA)2
(10)
These equations are essentially independent of the SAFT formalism: eq 5 is based on the assumption that all sites have an equal probability of bonding, and eq 8 results simply from the fact that for every C site that bonds, only half as many A or B sites bond. A and B sites are indistinguishable in the 3B or 4C schemes. There is some debate in the literature about exactly how associating fluids bond in liquids. While it is clear from their structures that water is a “rigorous” 4C molecule and alcohols are “rigorous” 3B molecules, Dixit et al.11 calculate 3.6 hydrogen bonds per water molecule (a 3.6 site molecule) and 1.8 hydrogen bonds per methanol molecule (a 1.8 site molecule). These
Figure 1 shows the fraction of free -OH groups (filled squares) and fraction of monomeric (completely nonbonded) molecules (filled diamonds) for water. The filled data points are the digitized data of Luck.6 In the discussion in Luck,6 water is assumed to be a 4C molecule and alcohols 3B molecules. This is the correct interpretation in terms of a “rigorous” atomic description of the molecules. Furthermore, in Luck6 the data are reported as a fraction of free -OH groups. From a careful reading of this article6 it is clear that this is equivalent to the fraction of nonbonded hydrogen atoms in the hydroxyl group. If we assume a 4C model for water, according to the formalism of Table 3, then the fraction of free -OH groups corresponds to the fraction of nonbonded C or D sites (XC or XD). The fraction of monomeric molecules can then be calculated (open diamonds) from the spectroscopically determined fraction of free -OH groups using eq 5. Conversely, the fraction of free -OH groups can be calculated (open squares) from the experimentally determined monomeric fraction. The agreement between these two unrelated sets of data when applying the transformation between nonbonded sites and monomeric molecules already strongly suggests that pure liquid water behaves as a four-site (4C) molecule. Figure 2 shows the effect of parametrization on association. A total of seven sets of PC-SAFT parameters were obtained for water as a 4C molecule (see Table 2 for the parameters).
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Figure 2. Effect of parametrization on association. A total of seven sets of PC-SAFT parameters were obtained for water as a 4C molecule (see Table 2). The figure compares predictions of the percent of free -OH groups with those obtained spectroscopically by Luck.6 The parameter sets were obtained by fixing the chain length parameter m at equally spaced values: m ) 2.00 (dashed line); m ) 3.00 (solid line); m ) 3.25 (dot-dashed line); m ) 3.50 (dotted line). For comparison the XA predicted using the 4C model for water with CPA is shown as well (double dot-dashed line).
The figure compares predictions of the percent of free -OH groups with those obtained spectroscopically by Luck.6 The parameter sets were obtained by fixing the chain length parameter m at equally spaced values from m ) 2.00 to m ) 3.50 and then allowing the remaining four parameters to vary in order to minimize the error between the model predictions and the experimental data for saturated vapor pressure and saturated liquid density. As can be seen from Table 2, the errors for all the parameter sets are comparable, with a possible optimum around m ) 2.75-3.00. All sets describe the degree of hydrogen bonding reasonably well, with a similar temperature dependence, and a rapid increase in XA as the critical temperature (374 °C) is approached. For comparison the XA predicted using the 4C model for water with CPA is shown as well. Because of the difficulty of comparing predictions with different schemes, only predictions using the “rigorous” 4C scheme are shown. While Figure 2 suggests that the 4C scheme is appropriate for pure water, it also shows that pure-component parameters have a marked effect. All 7 sets of parameters for PC-SAFT would be considered acceptable in terms of the agreement between the model and the experimental liquid density and vapor pressure. However, not all sets predict the same fraction of free hydrogen bonds. It is in such cases that spectroscopic data can provide an additional experimental aid in arriving at purecomponent parameters which yield the most meaningful physical behavior (for which experimental data are available) for the pure component. Of course this is still no guarantee of the best phase equilibrium behavior in mixtures. It should be mentioned here that the value of m for water is higher than the expected value for a nearly spherical molecule, i.e., m ) 1. One possible reason for this is that the high chainlength m (which can be thought of as a shape factor) somehow accounts for the polarity of water, which is otherwise not considered in the model. Karakatsani et al.14 recently presented a polar SAFT model, where m ) 1, and extra terms and parameters then account for water’s polarity. Figure 3 is a comparison of the predictions of various models and schemes for the monomer fractions of pure water. Because of the different association schemes that may be proposed (see Table 3), it is difficult to compare model predictions based only
Figure 3. Comparison of the predictions of various models and schemes for the monomer fraction of pure water. It is seen that the PC-SAFT (4C, m ) 3.50) (upper solid line) and CPA 3B (upper dotted line) schemes perform best here. The PC-SAFT 2B (dashed line) tends to overpredict the monomer concentration at lower temperature. The PC-SAFT (4C, m ) 2.00) (lower solid line) and CPA 4C (lower dotted line) tend to underpredict the monomer concentration over the temperature range. The data are digitized from Luck.6 For both PC-SAFT 4B parameter sets it can be seen that the monomer fraction increases very rapidly as the critical temperature predicted by the model is approached.
on the percent of free -OH groups, as was done in Figure 2. However, one can compare predictions of monomer concentration for various models and association schemes by applying eqs 5-10 where appropriate. It is seen that the PC-SAFT (4C, m ) 3.50) (upper solid line) and CPA 3B (upper dotted line) schemes perform best here. The PC-SAFT 2B model (dashed line) tends to overpredict the monomer concentration at lower temperature. The PC-SAFT (4C, m ) 2.00) (lower solid line) and CPA 4C model (lower dotted line) tend to underpredict the monomer concentration over the temperature range, although both are better than PC-SAFT 2B at lower temperatures (less than 300 °C). The data are digitized from Luck.6 This figure suggests that either 4C or 3B is the appropriate scheme for pure water. Since it is clear that parametrization plays an important role, it may be possible to improve the prediction for 4C water using CPA. In addition to modeling pure-component properties, a further, usually sterner, test of the pure parameters is how they perform when used in predicting mixture properties (such as liquid-liquid equilibrium in water-hydrocarbon mixtures). Figure 3 also demonstrates one of the difficulties in finding purecomponent parameters when many parameters need to be found (in this case five) using only data for saturated liquid density and vapor pressure. It is difficult to say in this case that any one of the seven sets of 4C water parameters in PC-SAFT is “best”. It is in just such a case that extra data, such as that obtained by spectroscopy, may be a guide in finding suitable pure-component parameters. Figure 4 shows the monomer mole fraction for 1-propanol. The experimental data are from the thesis of Lien7 where they are reported as monomer fraction. To compare different association schemes (2B and 3B) for alcohols (see Table 3), the results have been converted to monomer fraction using eqs 5-10 in Figures 4-8. CPA 2B (dashed line) performs best here, followed by PC-SAFT (2B, our parameters15) (dot-dashed line). PC-SAFT (2B, Gross and Sadowski3) and CPA 3B do not perform as well. It seems that the 2B scheme is most appropriate for this alcohol. Figure 5 shows monomer mole fraction for 1-octanol. The experimental data are monomer fraction calculated from the data
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Figure 4. Monomer fraction for 1-propanol. The experimental data are from the thesis of Lien7 where they are reported as monomer fraction. To compare different association schemes (2B and 3B) for alcohol (see Table 3), the results have been converted to monomer fraction using eqs 5-10. CPA 2B (dashed line) performs best here, followed by PC-SAFT 2B (our parameters,15 dot-dashed line). PC-SAFT 2B (Gross and Sadowski3) and CPA 3B do not perform as well.
Figure 5. Monomer fraction for 1-octanol. The experimental data are monomer fraction calculated from Fletcher and Heller.8 To compare different association schemes (2B and 3B) for alcohol (see Table 3), the results have been converted to monomer fraction using eqs 5-10. The PC-SAFT results (solid line) suggest that the 2B scheme is most appropriate, while for CPA, the 3B scheme (dotted line) gives better agreement than the 2B scheme (dashed line).
of Fletcher and Heller.8 The PC-SAFT results (solid line) suggest that the 2B scheme is most appropriate, while for CPA, the 3B scheme (dotted line) gives better agreement than the 2B scheme (dashed line). Figure 6 shows the monomer mole fraction for methanol. The experimental data are digitized from Luck.6 The lattice fluid hydrogen-bonding theory (LFHB) results (dashed line) are converted from nonbonded site fraction values reported in Gupta et al.16 In LFHB theory, the parameter which is compared with the experimental results is(1 - N11/N). This is not the fraction of monomeric alkanol, as reported,16 but corresponds exactly with the nonbonded site fraction XC. In Gupta et al.16 N11 is the number of existing hydrogen bonds and N is the total possible number of bonds. The equivalence of (1 - N11/N) and XC is easily verified and is shown clearly in the work of Economou and Donohue.17 In the LFHB theory of Gupta et al.16 alcohols are treated as 2B molecules. In that work, the fraction of nonbonded hydroxyl hydrogens is compared with the same param-
Figure 6. Monomer fraction for methanol. The experimental data are digitized from Luck.6 To compare different association schemes (2B and 3B) for alcohol (see Table 3), the data have been converted to monomer fraction using eqs 5-10. The LFHB results (dashed line) are converted from nonbonded site fraction values reported in Gupta et al.16 Parameters used in the CPA model are given in Table 1 (CPA 3B, dash-dotted line; CPA 2B, dotted line) and for PC-SAFT 2B in Table 2 (ref 18, solid line; ref 3, dash-double dotted line).
Figure 7. Monomer fraction for ethanol. The experimental data are digitized from Luck.6 To compare different association schemes (2B and 3B) for alcohol (see Table 3), the data have been converted to monomer fraction using eqs 5-10. The LFHB results (dashed line) are converted from nonbonded site fraction values reported in Gupta et al.16 Parameters used in the CPA model are given in Table 1 (CPA 3B, dash-dotted line; CPA 2B, dotted line) and for PC-SAFT 2B in Table 2 (this work, solid line; ref 3, dash-double dotted line).
eter from Luck.6 Both the data and the LFHB model predictions are converted to monomer fractions for an unambiguous comparison: the data are converted using the 3B scheme assumed by Luck, whereas the LFHB predictions are converted using the 2B scheme of the model. The predictions of LFHB are good: only the CPA 3B scheme is better (dot-dashed line). For the PC-SAFT 2B scheme, our parameters18 (solid line) perform better than those of Gross and Sadowski3 (double dotdashed line), while the CPA 2B scheme lies between the PCSAFT predictions (dotted line). We could conclude from this figure that the 3B association scheme is most appropriate for methanol. Figure 7 shows monomer mol fraction for ethanol. The experimental data are digitized from Luck.6 Once again, the LFHB results (dashed line) are converted from nonbonded site fraction values reported in Gupta et al.16 The LFHB results now compare very favorably with the experimental data. Both sets
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Figure 8. Monomer fraction for different alcohols with CPA. All alcohols are modeled as 3B molecules. The expected trend is observed, whereby the monomer fraction increases as carbon number increases. The steric hindrance resulting from the longer hydrophobic chain on the alcohol means less association between molecules at the same temperature. The figure is representative, only showing one model (CPA) with one association scheme (3B). This trend is followed consistently by the other models and schemes.
of PC-SAFT 2B parameters (this work, solid line and Gross and Sadowski3, double dot-dashed line) perform less satisfactorily, but still adequately, especially at lower temperatures. Both sets of CPA parameters result in overpredictions at lower temperatures, with the 2B scheme (dotted line) better than the 3B scheme (dot-dashed line). The shape of the curve is also better for CPA 2B than CPA 3B. It can be concluded that the 2B scheme is probably better for ethanol than the 3B scheme. Figure 8 shows predictions of monomer fraction for different alcohols with CPA. All alcohols are modeled as 3B molecules for consistency in the comparison. The expected trend is observed, whereby the monomer fraction increases as carbon number increases. There is a volume dilution effect of the increasingly larger nonassociating groups, meaning that the concentration of associating sites on larger alcohols is lower. Primary alcohols are little affected by steric hindrance, owing to the orientational specificity of hydrogen bonding, as discussed for example in Liu and Elliott.19 The figure is representative, only showing one model (CPA) with one association scheme (3B). This trend is followed consistently by the other models and schemes and by the experimental data. In Moorthi and Nagata20 the experimental data from Luck6 for methanol and ethanol, from Lien7 for 1-propanol, and from Fletcher and Heller8 for 1-octanol are modeled using a dimerization theory. However, these authors interpret all the experimental data as a fraction of free OH groups (XC). This is correct for the data of Luck, but the data of Lien and of Fletcher and Heller are reported as monomer fraction. This leads to the apparent anomaly that ethanol is less bonded than 1-propanol at the same temperature. There is a certain degree of confusion in the literature between nonbonded site fractions and fully nonbonded (monomeric) molecules. Figure 9 illustrates another example. Figure 9 shows both site and monomer fractions for methanol modeled as a 3B molecule using CPA. This figure indicates the pitfalls of an asymmetric association scheme (3B). The nonbonded site fractions XA and XB are shown as solid lines. It is clear from Table 3 that XA and XB can never be less than 0.5. Nonbonded hydroxyl hydrogens (XC) are shown as a dashed line, and the overall methanol monomer fraction calculated from eqs 6 and 8 as a dotted line. In Luck,6 it is XC that is reported. Kahl and Enders21 compared the predictions of XA for water as a 3B model
Figure 9. Site and monomer fractions for methanol modeled as a 3B molecule in CPA. This figure indicates the pitfalls of an asymmetric association scheme. The nonbonded site fractions XA and XB are shown as solid lines. It is clear from Table 3 that XA and XB can never be less than 0.5. Nonbonded hydroxyl hydrogens (XC) are shown as a dashed line and the overall methanol monomer fraction calculated from eq 6 as a dotted line. In Luck,6 it is XC that is reported.
Figure 10. Monomer fraction of methanol in the binary mixture methanolcarbon tetrachloride at 20 °C. The solid line is the prediction of CPA using a 3B association scheme for methanol; the dashed line is for a 2B scheme. The agreement with the data found in Prausnitz et al.24 is excellent for the 3B scheme.
(the first 3B model for water in Table 3) with the measurements of Luck.6 Since XA is always greater than 0.5 for this scheme, the model apparently performs poorly (the data for free OH groups are shown in Figure 1 and approach about 10% as the temperature goes to freezing). However, converting both the data and the predictions to a monomer basis using the appropriate expressions from eqs 5-10 shows that the agreement is actually much better. Finally, Figures 10 and 11 show predictions for the mixture methanol-carbon tetrachloride. Figure 10 shows the monomer fraction of methanol in the mixture, while Figure 11 shows phase behavior (vapor-liquid equilibrium) in the same system. It is clear that the 3B association scheme works best in the mixture. It may be noted that both results in Figure 11 are pure prediction; i.e., no binary interaction parameters have been applied. The azeotropic composition and pressure are very well predicted when methanol is modeled as a 3-site molecule. These two figures provide further support for the premise that methanol be treated as a 3-site molecule. Once again, it should be
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“Advanced Thermodynamic Tools for Computer-Aided Product Design”. Georgios Folas is acknowledged for providing the CPA 3B parameters for water, methanol, ethanol, 1-propanol, and 1-octanol prior to publication. Nomenclature and Abbreviations CPA ) cubic plus association LFHB ) lattice fluid hydrogen bonding PC-SAFT ) perturbed chain-statistical associating fluid theory X1 ) monomer concentration XA ) fraction of molecules not bonded at site A Literature Cited
Figure 11. Pressure-composition diagram in the system methanol-carbon tetrachloride at 35 °C. The solid line is the prediction of CPA using a 3B association scheme for methanol and the dashed line is for a 2B scheme. The agreement with the data of Scatchard et al.25 is very good for the 3B scheme, particularly near the azeotrope. No binary interaction parameters have been applied (kij ) 0).
emphasized that the parameters for 3B methanol were obtained solely on the basis of pure component experimental liquid densities and vapor pressures coupled with spectroscopic data. Using density and vapor pressure data alone cannot help choose among association schemes. 4. Conclusions We have tested the ability of PC-SAFT and CPA to predict the monomer fraction of pure associating fluids. Both models can in general give monomer fractions reasonably well. The correct trends are also observed: the monomer fraction is higher as alcohol chain length goes up. For association models with 5 parameters we can use spectroscopic data to tune the model, resulting in a model with better predictions for monomer fraction, without sacrificing the accuracy of the liquid density and vapor pressure correlations. Thus, it is clear that monomer fraction prediction depends on the way the parameters were obtained. Based on our results, for pure water we feel that 4C is the most appropriate scheme. For pure alcohols it is difficult to distinguish between 2B or 3B. Probably 3B is best for methanol, 2B or 3B seem about equal for ethanol, and for higher alcohols 2B is preferred. This is in accordance with steric arguments. Care should be taken when doing calculations in mixtures, however, since the association schemes which are appropriate for the pure liquid may not necessarily apply, although we have shown that association schemes chosen based on purecomponent spectroscopic data can be used successfully to predict both association and phase behavior in mixtures. Care should also be taken both when using spectroscopic data and when studying association, especially within the SAFT formalism. It is important to distinguish between monomer fraction and site fraction and be aware of how they relate to each other. Acknowledgment The authors are grateful to the Danish Technical Research Council for financial support of this work as part of the project
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ReceiVed for reView December 1, 2005 ReVised manuscript receiVed May 11, 2006 Accepted May 22, 2006 IE051341U
