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Ind. Eng. Chem. Res. 2005, 44, 1610-1624
Performance of a Conductor-Like Screening Model for Real Solvents Model in Comparison to Classical Group Contribution Methods Hans Grensemann and Ju 1 rgen Gmehling* Department of Industrial Chemistry, University of Oldenburg, D-26111 Oldenburg, Germany
Group contribution methods, such as modified UNIFAC (Do) are powerful tools for the reliable prediction of phase equilibria and excess properties, which are necessary information for the development, design, and optimization of separation processes and other applications of industrial interest (Gmehling, J. Present Status of Group-Contribution Methods for the Synthesis and Design of Chemical Processes Fluid Phase Equilib. 1998, 144, 37-48). Despite the efficiency and simple applicability, group contribution methods have the disadvantage that they need at least a limited number of experimental data for fitting the required group interaction parameters; however, the necessary data are sometimes missing so that predictions for these systems cannot be performed. The conductor-like screening model for real solvents (COSMO-RS) developed by Klamt et al. (Klamt, A. Conductor-Like Screening Model for Real Solvents: A New Approach to the Quantitative Calculation of Solvation Phenomena. J. Phys. Chem. 1995, 99, 2224-2235. Klamt, A.; Jonas, V.; Bu¨rger, T.; Lohrenz, J. C. W. Refinement and Parametrization of COSMORS J. Phys. Chem. A 1998, 102, 5074-5085. Klamt, A.; Eckert, F. COSMO-RS: A Novel and Efficient Method for the A Priori Prediction of Thermophysical Data of Liquids. Fluid Phase Equilib. 2000, 172, 43-72) has attracted much attention in the chemical engineering community. From the results of quantum chemical calculations for a single molecule, the COSMO-RS model is used to compute the required activity coefficients and, therefore, provides a possible route for a priori prediction of phase equilibria. Since for this new method, up to now no comprehensive validation of the predicted results exists, a comprehensive comparison with the results of different group contribution methods was carried out using a large database (Dortmund Data Bank) (Dortmund Data Bank and DDB Software Package, DDBST GmbH, Oldenburg, Germany, 2004 (www.ddbst.de)). Not surprisingly, the comparison showed that much more reliable results were obtained using group contribution methods, in particular, modified UNIFAC. For example, in the case of VLE (database: >1360 thermodynamically consistent data sets), only a 0.35 mole % higher mean absolute deviation (0.94 mole % instead of 0.59 mole %) was observed for the vapor phase composition than for the correlation method UNIQUAC. For COSMO-RS(Ol)*, 5 times higher mean deviations (1.86 mole % (2.45 mole % instead of 0.59 mole %)) are observed. However, by an empirical modification of the COSMO-RS(Ol)* model for systems with either ethers or water, the mean deviations could be distinctly reduced (COSMO-RS(Ol)). Particularly poor results for the COSMO-RS(Ol) model were obtained for aqueous systems, for example, amine-water systems. Furthermore, it could not be confirmed that the weaknesses of group contribution methods (isomer and proximity effects) can be reduced by using the new approach, for example, a miscibility gap still is predicted for tert-butyl alcohol with water. From the VLE results of hexane with perfluorohexane, the deficiency of the COSMO-RS approach caused by the unconsidered dispersive forces become apparent. But despite these partly negative remarks, the authors recommend the application of the COSMO-RS approach for process development in the different areas, in particular, when the required group interaction parameters of modified UNIFAC (Do) or other group contribution methods are missing. Introduction Reliable knowledge of thermodynamic properties is a prerequisite for the synthesis, design, and optimization of different processes, in particular, separation processes in the chemical, pharmaceutical, food, gas processing, and petroleum industries. Especially during the development of the various processes, predictive methods are valuable tools for the estimation of the required mixture properties. The most successful predictive gE models are group contribution * Corresponding author. Tel: ++49 (0) 441 798 3831. Fax: ++49 (0) 441 798 3330. E-mail: gmehling@ tech.chem.uni-oldenburg.de.
methods such as UNIFAC6,7 and modified UNIFAC (Do),9,10 which can reliably be used for estimating the required phase equilibria and excess properties.7,10 In the case of group contribution equations of state, such as PSRK11,12 and VTPR,13-15 not only subcritical systems but also systems with supercritical compounds, as required, for example, for processes such as absorption, supercritical extraction, etc., can be predicted. Group contribution methods subdivide a molecule into functional groups. The physical view of the interactions in the mixture then changes from a collection of pairwise interacting molecules to a collection of pairwise interacting groups,8 which are assumed to be independent of each other. By a thermodynamic cycle, it can be
10.1021/ie049139z CCC: $30.25 © 2005 American Chemical Society Published on Web 02/04/2005
Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005 1611
shown that the thermodynamic properties of the mixture are obtained by summing up contributions from all the pairwise interacting groups.8 This fragmentation of molecules greatly reduces the number of required parameters.16 Once the group interaction parameters have been determined from available experimental data, they can be used to predict the activity coefficients of all systems consisting of these functional groups. In addition to these advantages, all group contribution methods have common drawbacks, which limit their applicability. First, they need at least a few experimental data for fitting the required group interaction parameters. Sometimes the necessary database, for example, for reactive systems such as isocyanatesalcohols, is missing. This means that the mixture properties for these systems cannot be predicted. Second, due to the fragmentation of the molecules into functional groups, group contribution methods cannot distinguish among isomers. Furthermore, the change of the electronic structure by neighbors (proximity effect, e.g., R-halogen carboxylic acids) is disregarded. This means, for example, that these methods give identical results for the activity coefficients of the three isomeric xylenes, for example, in alcohols, ketones, etc. Since the phase equilibrium behavior strongly depends on the pure component properties of the isomeric molecules, such as vapor pressure, melting point, heat of fusion, critical temperature, and pressure, VLE, SLE, etc., are nearly correctly described. In most cases, the different pure component data of the isomers have a stronger influence on the results than the slightly changed activity coefficients. With the improved performance of computers, a new class of predictive models for thermophysical properties based on quantum chemical methods has been developed. These predictive models may be used for calculating properties of molecules in the ideal gas phase using molecular orbital (MO) calculations or density functional methods (DFT). For fluid phase calculations, dielectric continuum solvation models have turned out to be powerful tools.17 Klamt proposed an extension of these dielectric continuum solvation methods, the conductorlike screening model for real solvents (COSMO-RS)3,4 which has attracted much attention in the chemical engineering community. Starting from the solvation of molecules in a conductor, the chemical potential (activity coefficients) of any species in a mixture is determined from ab initio solvation calculations with the COSMO-RS model. For the calculation of the chemical potentials (activity coefficients) using this new model, analogously to group contribution methods, the interactions between the molecules are taken into account. But instead of dividing the molecules into functional groups, as in group contribution methods, the molecule surface is subdivided into segments with equal area, and the chemical potential is derived from the interactions between the surface segments in the system. The advantage of the COSMO-RS model, as compared to group contribution methods, is that theoretically, only a few basic element-specific parameters as well as parameters for the calculation of the interaction energies have to be fitted to a limited database, which can then be used for all further calculations. A further advantage of COSMO-RS, for which only a COSMO calculation has to be accomplished for each compound considered is, that theoretically, a distinction
of the isomeric compounds should be possible. Furthermore, it seems to be possible that proximity effects can be taken into account by the COSMO-RS model. This model was originally applied for the prediction of vapor pressures and partition coefficients, resulting in an accuracy within a factor of 2. Later, it was shown18,19 that COSMO-RS may also be applied to the prediction of phase equilibria; however, a few drawbacks are wellknown. In the COSMO approach, primarily electrostatic forces are considered. For a more reliable description, dispersive, polarization, cavitation, and hydrogen bonding energies have to be taken into account quantitatively. Up to now, however, an extensive examination of the COSMO-RS approach with the results of group contribution methods has only been performed for activity coefficients at infinite dilution.20 A comprehensive comparison with VLE and excess enthalpies is missing. Therefore, using over 1362 consistent binary VLE data, more than 14 550 binary activity coefficients at infinite dilution, and ∼8200 binary enthalpy of mixing data sets, all taken from the Dortmund Data Bank (DDB),5 an extensive comparison of the experimental data with the results of the different predictive methods has been performed to get an idea about the reliability of the new predictive model. Theory The concept for the description of the molecular interactions is based on the physical view of pairwise interacting surface segments. The difference between the screening charge densities σ and σ′ of a contact pair is a measure for the misfit of the screening charge situation of both segments to the situation inside the ideal conductor. The resulting misfit energy Emf of this electrostatic interaction can then be calculated following eq 1 using the effective contact surface area aeff; the net screening charge densities, σm and σn, of the two segments; and an energy factor, R′.2,3 This factor can be calculated from electrostatic theory, but was fitted to improve the results.
R′ Emisfit(σ, σ′) ) aeffemisfit(σ, σ′) ) aeff (σ + σ′)2 (1) 2 If strongly polar compounds, such as alcohols or water, are present, then in addition to the electrostatic energy, Emf, hydrogen bond interactions, Ehb, have to be taken into account. Klamt and Eckert proposed a simple expression for this kind of interaction by defining hydrogen bond donors as segments with a screening charge density, σdon, smaller than a negative cutoff value, σhb, and acceptors as those with a screening charge density, σacc, larger than σhb. Hydrogen bond interaction only occurs between the donor and acceptor segments.3
Ehb(σ, σ′) ) aeffchbmin {0, max[0, σacc σhb]min[0, σdon + σhb]} with
σacc ) max[σ, σ′] and σdon ) min[σ, σ′] where chb is an adjustable coefficient.
(2)
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Since the kinetic energy of the molecules will increase with increasing temperature, the formation of directional H bonds is reduced at higher temperatures. Therefore, the hydrogen bond interaction energy is also temperature-dependent, whereby Klamt proposed the following simple relation:
(
chb(T) ) chbmax 0, 1 - cThb + cThb
298.15 K T
)
(3)
where the coefficient cThb is an adjustable factor. Finally, the summation over all the possible combinations of pairwise interacting surface segments allows a description of the interactions in the system. Since only the ideal screening charge density is considered as descriptor of the surface segments, for the description of the interactions in the ensemble, only the screening charge density distribution pS(σ) (called σ profile) of the system S is required. In COSMO-RS, the σ profile is the most important characteristic property for each compound. As shown later, the activity coefficient is calculated solely from this information. In mixtures, the probability pS(σ) of finding a segment with screening charge density σ is obtained from the mole fraction xi weighted sums of the σ profiles pi(σ) of all pure components, i, in the system.
pS(σ) )
∑i xipi(σ) ∑i ∫ xi
(4)
pi(σ′) dσ′
µiS )
[∫
pS(σ′) exp
(6)
introduced in The combinatorial contribution µi,comb S eq 6 takes into account the difference in shape and size of the molecules and is equivalent to the combinatorial contribution in classical activity coefficient models. In our investigation, for the combinatorial part, the following modified combinatorial expression (Guggenheim22 and Stavermann23) was used:
ln γiC ) 1 - Vi + ln(Vi) + 1 -
Vi Vi + ln Fi Fi
(7)
For the calculation of Vi and Fi, the following relations were developed,
Vi )
Fi )
( ) ( ) ri
∑j
(8)
rjxj
qi
∑j
β(1-rmin,i/rmax,i)
λ(1-qmin,i/qmax,i)
(9)
qjxj
with β and λ as adjustable parameters. The minimal and maximal molecular volume and molecular area used in the exponential term, (1 - rmin,i/ rmax,i) and (1 - qmin,i/qmax,i), respectively, are given by the following equations,
To obtain the chemical potential of a surface segment with the screening charge density σ in the ensemble S, Klamt derived the following self-consistent equation,2,3
µS(σ) ) -RT ln
∫ pi(σ)µS(σ) dσ + µi,comb S
rmin,i ) min(ri, r#i )
(10)
rmax,i ) max(ri, r#i )
(11)
with
{
} ]
µS(σ′) - aeffe(σ, σ′) dσ′ RT
with
r#i ) (1 - xi)
∑ j*i x
(12)
j
and
e(σ, σ′) ) emisfit(σ, σ′) + ehb(σ, σ′) ) Emisfit(σ, σ′) + Ehb(σ, σ′) (5) aeff where µ′(σ) is the normalized chemical potential of a surface segment with charge density σ in an ensemble S of independent surface segment pairs characterized by the σ profile pS(σ). Since the chemical potential of one surface segment is influenced by the chemical potential of all other surface segments in the ensemble S, an iterative solution of eq 5 is necessary, but with µS(σ) ) 0 as starting value, it converges rapidly, and µS(σ) can be obtained within milliseconds. Instead of the chemical potential of one surface segment, the chemical potential of a molecule i in a given mixture (characterized by a probability distribution pS(σ)) is the important key value for industrial applications. The integration of the σ potential µS(σ) in the ensemble S over all the surface segments in the molecule i given by the σ profile pi(σ) delivers the desired chemical potential of molecule i in the mixture.
rj
qmin,i ) min(qi, q#i )
(13)
qmax,i ) max(qi, q#i )
(14)
with
q#i ) (1 - xi)
qj
∑ j*i x
(15)
j
The combinatorial part used (eq 7) is different from the expression used by Klamt et al.2-4 The molecular volume, ri, and molecular area, qi, of the compounds required for the calculation of Vi and Fi can be obtained directly from the calculated COSMO surfaces. A critical examination of the different combinatorial parts was already carried out by Lin et al.24,25 The performance of our combinatorial part is shown in Figure 1 for the activity coefficient at infinite dilution of n-heptane in n-alkanes, together with published experimental data and the results obtained by modified UNIFAC (Do) respective to COSMO-SAC.52
Ind. Eng. Chem. Res., Vol. 44, No. 5, 2005 1613 Table 1. Parameters for COSMO-RS(Ol) parameter
value
aeff R′ chb cThb σhb (ref 4) λ β rav (ref 3)
6.31 Å2 30.741 kJ/mol 36.516 kJ/mol 1.552 0.0082 e/Å2 0.773 0.778 0.5 Å
Table 2. Used Cavity Radii for COSMO-RS(Ol)
Figure 1. Example for the performance of the new developed combinatorial part of COSMO-RS(Ol) for symmetric and asymmetric n-heptane/alkane systems (b, experimental γ∞ values of n-heptane taken from ref 5; s, modified UNIFAC (Do); ‚‚‚ COSMORS(Ol)); - - - COSMO-SAC (ref 52).
The activity coefficient of a molecule i can then be calculated using the following equation,
(
γiS ) exp
)
i µiS - µS)i RT
(16)
where µiS=I is the chemical potential of molecule i in the pure liquid, i, and µiS is the chemical potential of molecule i in the mixture S. (Please note that following the solvation thermodynamics of Ben-Naim,26 the chemical potential from eq 6 is a “pseudo chemical potential”, which is the standard chemical potential minus RT ln xi.)
element
cavity radius (Å)
H C N O F S Cl Br I
1.30 2.00 1.83 1.72 1.72 2.16 2.05 2.16 2.32
were taken into account. In a conformational study, it was found that often, one conformer gives the best results for VLE, but the enthalpy of mixing of the same binary system was described much better by another conformer. Therefore, no multiple conformations were taken into account for the comprehensive comparison. The COSMO screening charge densities σ* obtained from TURBOMOLE are first of all averaged using the following equation proposed by Klamt et al.3
∑µ σ/µRµν
σν )
∑µ
Computational Details Quantum Chemical Calculations. The molecular structures of more than 2 500 compounds are directly taken from the chemical structure database (ChemDB) of the DDB, which at present contains the structures for ∼18 000 compounds. To start from a reliable 3D geometry for the COSMO calculations, first of all, a molecular mechanics optimization using the Tinker27 program package has been performed for every compound. Additionally, a semiempirical quantum chemical energy minimization on the AM128 level using the GAMESS29 program package was performed for every compound. At the end, 1073 compounds were used for the comprehensive model comparison presented here. The COSMO method implemented in TURBOMOLE30 is used for the quantum chemical calculation, because it is an efficient and fast program package to produce accurate COSMO screening energies from the density functional theory (DFT). The COSMO calculations have been carried out using the DFT level combined with the BP functional31,32 and the triple-ζ valence basis set (TZVP).33 For the cavity construction in the COSMO calculation, the given atomic radii of the TURBOMOLE program package (Table 2) were used directly, since Klamt et al.4 found no indications that these radii are very sensitive to the quantum mechanical method used. Following Klamt et al.,34 the recommended COSMO radii should be ∼1.17((0.02) times larger than the Bondi van der Waals radii.35 After the COSMO calculation, all molecule structures where visually checked for physical feasibility. For the comprehensive comparison, no multiple conformations
(17)
Rµν
whereby the parameter Rµν is given as
Rµν )
rµ2rav2
(
exp 2
rµ2 + rav
dµν2 rµ2 + rav2
)
(18)
In eq 18, dµν is the distance between the segments µ and ν, rµ is the mean radius of segment µ calculated from the area of segment µ, and rav is an adjustable parameter. The characteristic σ profile of the molecule is obtained by counting the amount of surfaces in the ensemble with a specific screening charge density between σ and σ + ∆σ, with ∆σ ) 0.001 e/Å2. After that, every interval p′(σn) of this histogram was averaged using the following equation to obtain the final σ profile, p(σ).
p(σn) )
1n+1
∑ p′(σn)
3n-1
(19)
Thermodynamic Calculations. For the comprehensive model test, we did not use the commercially available COSMOtherm program package. Instead, on the basis of the various publications for the COSMORS model, our own program was developed and directly integrated into the already existing and extensively tested thermodynamic phase equilibrium computation routines and data banks. This offers the advantage that within the planned further model development, modifications can be directly integrated.
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To be able to distinguish the different versions of the COSMO-RS approach, our version is hereafter called COSMO-RS(Ol). The required new model parameters were obtained by fitting the parameters simultaneously to a comprehensive database consisting of 112 binary thermodynamically consistent VLE data sets, 95 hE data sets, 173 γ∞ data points, 31 SLE data sets of eutectic systems, and some azeotropic data, since only parameters obtained by a simultaneous fit to a comprehensive database can provide a reliable description of the real behavior across the whole composition and temperature range. Within the last 20 years, much experience has been gained in fitting parameters of UNIFAC and modified UNIFAC (Do). Before the fitting procedure was started, the selected mixture data was evaluated using thermodynamic consistency (Redlich-Kister area and GibbsDuhem rule) and plausibility tests. To fit the required COSMO-RS(Ol) model parameters, the following objective function was used.
F ) (aeff, R′, chb, cThb, σhb, λ, β, rav) ! min )
(20)
In the objective function, F, the deviations between the experimental and calculated activity coefficients (∆VLE, ∆γ∞, ∆SLE, ∆AZD) and the deviations between the experimental and calculated enthalpy of mixing (∆hE) were summed up. This means that with the exception of the γ∞ values, the phase equilibrium data with the help of the pure component properties had to be converted to activity coefficients.
F ) wVLE
∑ ∆VLE + wEh ∑ ∆hE + wγ∞ ∑ ∆γ∞ + wSLE ∑ ∆SLE + wAZD ∑ ∆AZD (21)
By varying the weighting factors, wi, for the different thermodynamic properties in the objective function, an optimal result could be achieved for the selected data sets. The new optimized model parameters for COSMORS(Ol) are given in Table 1. Since the optimized model parameters σhb and rav were nearly identical with the previously published parameters,3,4 the published values were used in the final optimization. For the comprehensive model comparison, in addition to the modified UNIFAC (Do) and original UNIFAC, the modified UNIFAC (Ly)36 and the analytical solutions of groups (ASOG)37,38 model were also included, since all these models are used for industrial applications. Modifications of COSMO-RS(Ol). The occurrence of hydrogen bondings in the COSMO-RS theory is only controlled by the cutoff value, σhb in eq 2. This means that hydrogen bonds are formed regardless of the involved atoms when donor and acceptor segments are available in the solution. However, hydrogen bondings in real solutions only occur between polarized hydrogen atoms connected to strongly electronegative atoms, such as oxygen, nitrogen, or other strong electronegative atoms. This has been taken into account in COSMORS(Ol) by introducing a second σ profile, phb(σ), only for surface segments of atoms where hydrogen bondings are physical meaningful. Since the chemical potential of one surface segment is influenced by the chemical potential of all the other surface segments in the ensemble S, the two σ profiles had to be combined. This was achieved by replacing the
Figure 2. VLE prediction of COSMO-RS(Ol)* for the system diisopropyl ether (1)/2-propanol (2) (2, b experimental values taken from ref 44).
screening charge density, σ, with the following new descriptor, η, which is a simple combination of the screening charge density σ and the type δ of the bonding (eq 22).
η ) (σ, δ)
(22)
The resulting contact energy E(η, η′) between nonhydrogen bonding surface segments is given by the following equation,
E(η, η′) ) Emisfit(η, η′)
(23)
and for hydrogen bonding surfaces segments, it is given by eq 24.
E(η, η′) ) Emisfit(η, η′) + Ehb(η, η′)
(24)
To obtain the chemical potential of a surface segment with the screening charge density σ and the bonding type δ in the ensemble S, the self-consistent eq 5 was replaced by the following equation:
µS(η) ) -RT ln
[∫
{
pS(η′) exp
} ]
µS(η′) - aeffe(η, η′) dη RT (25)
Introducing the new descriptor η into eq 5, the chemical potential of a molecule i in a given mixture S has the following form:
µiS )
∫ pi(η)µS(η) dη + µi,comb S
(26)
Throughout the evaluation of the different predictive models, very large deviations for the COSMO-RS(Ol) model were observed for alcohol-ether systems. In Figure 2, the result for the azeotropic system 2-propanol-diisopropyl ether is shown. It can be seen that the predicted vapor-liquid equilibrium behavior using modified UNIFAC (Do) is in good agreement with the experimental data, whereas nearly ideal behavior is predicted by COSMO-RS(Ol). Figure 3 shows the normalized σ profiles of the two components. Whereas for the negative side, 2-propanol shows a broad peak at -0.013 [e/Å2] resulting from the
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Figure 3. Normalized σ profiles for diisopropyl ether and 2-propanol.
Figure 4. Improvement achieved for VLE by an empirical handling of hydrogen bondings for the oxygen atom inside the ether group of COSMO-RS(Ol) for the system diisopropyl ether (1)/2-propanol (2) (2, b experimental values taken from ref 44).
polar hydrogen atom of the hydroxyl group, diisopropyl ether shows nearly no segments with a negative polar surface density. On the positive side, 2-propanol also shows a broad peak at +0.015 [e/Å2] resulting from the oxygen atom. The peak of diisopropyl ether in the same range is just one-half as large as the 2-propanol peak. Since the two characteristic σ profiles differ clearly in shape, the poor prediction of the COSMO-RS(Ol) model is surprising. Since no extra hydrogen bonding occurs in pure diisopropyl ether, because the necessary donor segments (segments with a more negative surface density than σhb) are not available, it was decided to exclude all acceptor surface segments of the oxygen atom inside the ether group from hydrogen bonding so that except for the electrostatic interaction energy, no extra hydrogen bond energy is taken into account for such surface segments when in a mixture donor segments are available from the other compounds. The result of this empirical modification of COSMORS(Ol) is shown in Figure 4 for the system 2-propanoldiisopropyl ether. As can be seen, much better results are obtained; for example, azeotropic behavior is predicted, although the azeotropic composition is not in
Figure 5. Results for 139 consistent VLE data sets for alcohol/ ether systems.
Figure 6. Achieved improvement of the VLE prediction of COSMO-RS(Ol) for the system ethanol (1)/water (2) (2, b experimental values taken from ref 45) by an empirical correction of the σ profile of the water molecule (dashed line, COSMO-RS(Ol)*; dotted line, COSMO-RS(Ol)).
total agreement with the experimental findings. In Figure 5, the improvement achieved by the empirical handling of hydrogen bondings for the oxygen atom inside the ether group of COSMO-RS(Ol) for 139 consistent VLE data for ether/alcohol systems is shown. The mean deviations are reduced by ∼40% when going from COSMO-RS(Ol)* to COSMO-RS(Ol). Please note that COSMO-RS(Ol)* means our COSMO-RS(Ol) model without the empirical modifications introduced for systems with ethers, respective to water (see next section). Poor VLE results were also observed for aqueous binary systems. For example, often a miscibility gap was predicted for homogeneous systems, as shown in Figure 6 for the system ethanol-water. The reason is that the predicted activity coefficient at infinite dilution of ethanol in water is 4 times overestimated. To improve the predicted results for aqueous systems, for water, an empirical correction of the COSMO densities was introduced. During the calculation of the σ profile, a scaling factor of 0.905 was used for the screening charge densities of all the surface segments of the water molecule. From Figure 6, it can be seen that after the empirical correction, much better results (e.g., no mis-
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For the different models, in addition to the deviations in vapor phase composition (∆y), the deviations in system pressure (∆P) and temperature (∆T) were also determined using the following equations.
∆P )
∆y )
1
cibility gap) are achieved, although the results are still poor, when compared with the results of modified UNIFAC. In Figure 7, the improvement obtained by this empirical correction of COSMO-RS(Ol) for 106 consistent VLE data sets of homogeneous aqueous systems is shown. As can be seen, the obtained mean deviations are reduced by ∼40% when going from COSMO-RS(Ol)* to COSMO-RS(Ol). However, at the same time, it can be seen that the deviations of the COSMO-RS(Ol) models for aqueous systems are still very large (∆y ) 3.94 mol % for COSMO-RS(Ol), ∆y ) 6.45 mol % COSMO-RS(Ol)*), when compared to modified UNIFAC (Do), with a mean deviation of 1.37 mol %. Unfortunately, however, in a few cases, the uncorrected version provides better results. To avoid the few worse predictions, work on a more sophisticated correction of the COSMO profile for water is in progress. Results for Different Thermodynamic Properties Vapor-Liquid Equilibria. When the different separation processes are compared, great advantages become apparent for distillation processes.39 However, for the synthesis and design of distillation processes, reliable vapor-liquid equilibria are required. To simplify for the VLE calculations, ideal vapor phase behavior was assumed at the moderate pressures (in most cases,
