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Ind. Eng. Chem. Res. 2004, 43, 852-861
COSMO-RS Predictions in Chemical EngineeringsA Study of the Applicability to Binary VLE Oliver Spuhl and Wolfgang Arlt* Institut fu¨ r Verfahrenstechnik, Fachgebiet Thermodynamik und thermische Verfahrenstechnik, Technische Universita¨ t Berlin, Strasse des 17. Juni 135, 10623 Berlin, Germany
The applicability of the quantum-chemistry-based model COSMO-RS is investigated concerning the prediction of vapor-liquid equilibria, i.e., prediction of the activity coefficient of a component in a mixture. A broad range of systems was chosen to cover different and significant types of interactions in mixtures of alkanes, alkenes, cycloalkanes, alcohols, ethers, ketones, aldehydes, and alkyl benzenes. Predictions of activity coefficients and phase behavior are presented and compared with experimental data. Introduction Process synthesis and design in chemical engineering requires knowledge of the phase equilibrium. A significant amount of physical property data is necessary to predict or calculate phase behavior with the common tools of thermodynamics. Activity coefficient models such as NRTL,1 UNIQUAC,2 or the structure-interpolating UNIFAC3 and equations of state deliver the required information reliably and rapidly. In particular, equations of state can provide comprehensive information on the thermodynamic behavior of a pure or multicomponent system. The development of modern equations of state has made substantial progress through the application of theories derived from statistical mechanics.4-6 Still, there is a disadvantage. All models mentioned contain parameters that need to be fitted to experimental data. The acquisition of such data involves high financial and logistical expenses, as well as the consumption of human resources. Furthermore, these data will not be available in the case of new and uninvestigated substances. Aside from activity coefficient models and equations of state, the simulation of vapor-liquid equilibrium is an important task. In the past 15 years, a variety of methods have been proposed. An overview and a discussion can be found in refs 7-9. Models that predict and calculate physical properties solely on the basis of structural information of a molecule can be found in quantum chemistry. In the fields of highly specialized chemicals, pharmaceuticals, and biotechnology, such models are of particular interest. Quantum-chemistry-based models have not been widely applied to the design of processes. This lack has a number of reasons. First, there are no “easy accessible” models similar to common equations of state or activity coefficient models that deliver properties such as the free energy, an activity coefficient, or a vapor pressure. Second, the application of quantum chemical methods requires a fundamental knowledge in chemistry. Third, even today with rapidly increasing computer power, it is not possible to simulate several hundred molecules inside a box and calculate the interaction behavior by quantum methods. First attempts were published by Sum and Sandler10 for polar components and by Raabe * To whom correspondence should be addressed. E-mail:
[email protected].
and Koehler for nonpolar components.11 One of the first combinations of quantum chemical calculations, statistical thermodynamics, and multistage separation process calculations was done by Taylor et al. in 2002.12 They implemented the COSMO-RS model in the software package ChemSep and predicted column profiles. This implementation was possible because of the enormous progress in the field of solvation effects treated by quantum methods.13 One approach that connects methods from quantum chemistry with theories from statistical mechanics is the COSMO-RS model. This model allows for the true a priori prediction of thermophysical properties solely from information on the atomic structure of a substance or mixture.14-16 The present study intends to provide a comprehensive overview of the abilities of COSMO-RS concerning the prediction of the vapor-liquid equilibria (VLE) of different classes of substances and their binary mixtures. The main focus is on substances that contain hydrogen, carbon, and oxygen and their mixtures, which are of interest in chemical engineering. The electrostatic interaction of a solute with the surrounding solvent can be modeled with the continuum solvation theory. In this way, solvent effects are incorporated into quantum chemical calculations. The solvent is treated as a continuous medium of dielectric constant . The solute is embedded inside an arbitrarily shaped cavity in the continuum. The interactions between solvent and solute are complex, particularly for arbitrarily shaped cavities. For that reason, Klamt and Schu¨u¨rmann17 developed a more efficient approach that replaces the dielectric medium of permeability with the scaled screening charges of a conductor. The much simpler boundary conditions of a conductor appear in the equations. This approximation is exact in the limit of ) ∞, within 0.5% accuracy for strong dielectric media, and within 10% accuracy for lower dielectric media compared to exact dielectric models. A COSMO calculation provides the screening charges on the surface of the cavity and is usually carried out at an adequate quantum level as provided by density functional theory. A COSMO calculation gives the energy, the geometry, and the screening charge density σ on the surface of a solute after quantum chemical self-consistency and geometry optimization loops. The transfer from the state of the molecule embedded in a virtual conductor to the real solvent is done by applying the COSMO-RS concept.14
10.1021/ie034009w CCC: $27.50 © 2004 American Chemical Society Published on Web 01/16/2004
Ind. Eng. Chem. Res., Vol. 43, No. 4, 2004 853
For this purpose, a fluid is considered to be an ensemble of closely packed and pairwise-interacting pieces of surface. This assumption is sufficient for liquids far from the critical state. Each piece of surface is characterized by its value of screening charge density σi. The interaction energy of the ensemble is then obtained by a statistically correct consideration of all possible pairs of pieces of surface. The composition of the ensemble that is needed to apply this procedure is delivered by the distribution function p(σ). This function describes the amount of surface in the ensemble having a screening charge density between σ and σ + dσ. Klamt denoted this distribution function as the σ profile. The representative σ profile of a mixture is the concentrationweighted average of the pure σ profiles. The σ profile of a component needs to be calculated only once. Because this is a time-consuming step, σ profiles are stored in a database. Aside from the electrostatic interactions, other intermolecular forces such as dispersive and repulsive interactions and hydrogen bonds also occur in fluid mixtures. They are merged in an energy concept within the COSMO-RS model. For further details, see the publications of Klamt.14,15 As a result, a COSMO-RS calculation provides the chemical potential of component i in the mixture. The activity coefficient γ of component i can be written as
(
)
µ0i
µi γi ) exp RT
(1)
where µi is the chemical potential of component i in the mixture and µ0i is the chemical potential of the pure ensemble, the reference state. In the study presented here, COSMO-RS is used as a GE model that provides the activity coefficient. All COSMO-RS calculations were performed using the latest COSMOtherm software package, version 1.2, release 07.02.
DEVP (%) )
|
DEVK (%) )
|
|
Pexp - Pcalc × 100 Pexp
|
calc Kexp 1 - K1
Kexp 1
× 100
(3)
(4)
The K factor can be expressed as
K1 )
y1 x1
where 1′ is the light-boiling substance in the mixture. Molecule Building and Quantum Chemistry
Systems and Procedure A broad range of systems was chosen to meet the interests of chemical engineers. The main focus is on binary systems containing alkanes, alkenes, cycloalkanes, alcohols, alkyl benzenes, ketones, and ethers. All analyzed experimental data sets are isothermal P-x-y measurements. Table 1 lists the investigated mixtures. A thermodynamic consistency test following the proposal of Redlich and Kister98 and Herrington99 was performed on all experimental data. Only thermodynamically consistent VLE data were used to compare results from COSMO-RS with experimental data. VLE calculations were performed using the activity coefficient provided by COSMO-RS to describe the reality of the liquid phase. The pure components were considered to consist of one conformer. Associating interactions of molecules in the gas phase were neglected. Furthermore, the Pointing correction was set to unity, and ideal gas phase behavior was assumed. This leads to the widely used equation LV ) yiP xiγiPoi
on experimental data and is still the object of current research. The vapor pressures of the pure substances were determined in two ways. When experimental data sets contained pure-substance information, those values were taken and substituted into eq 2. Table 1 provides information on the availability of pure-component data. For data sets without pressure information for the pure compound, a vapor pressure correlation was used. In each case when the pure-substance vapor pressure was available, its value was compared to the vapor pressure correlation result. If the deviation was greater than 0.5%, the experimental value was used. This was done to demonstrate the ability of COSMO-RS to predict the reality of the liquid phase and keep differences in the system pressure between measurements and predictions at a minimum. The deviations between the calculated and experimental values of the system pressure as a function of concentration, of temperature, and also of the K factor were used to evaluate the ability of COSMO-RS to predict the phase behavior of the investigated mixtures. The deviations are given by
(2)
Although it is possible to predict the vapor pressure of a pure component with COSMO-RS, it was decided to use experimental data or results from vapor pressure correlations, as the accuracy of the values predicted by COSMO-RS is less than that of other methods based
All molecules were built with the aid of the molecular modeling software Chem 3D Ultra by CambridgeSoft. Chem 3D provides the starting geometry for geometry optimizations. In the concept of COSMO calculations, a cavity around the molecule is constructed for a given molecular geometry. The cavity is then discretized into small segments, so that, on each segment i, a constant screening charge density of σi can be assumed. At the end of quantum chemical self-consistency loops, the screening charge density σ and the dielectric interaction energy of the solute with the continuum are known. These calculations are combined with geometry optimizations. COSMO calculations and geometry optimizations were performed using the software package TURBOMOLE, version 5.4. A full TURBOMOLE BP RI-DFT COSMO optimization with the Ahlrichs-TZVP basis set100 was carried out. This combination of software, functional, and basis set was chosen following the recommendation of the COSMOtherm developers. They recommend the use of either TURBOMOLE or Gaussian03 to yield high-quality predictions of thermophysical data for chemical engineering.101 COSMOtherm will automatically convert the charge surfaces of COSMO files generated by Gaussian03 into a charge surface that is equivalent to the charge surface generated by TURBOMOLE. Beyond this, it is not possible to mix different quantum calculation methods for individual components
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Table 1. Investigated Systems by Class system n-pentane + n-hexane n-hexane + 2,2-dimethylbutane n-hexane + n-heptane n-heptane + n-octane n-octane + 2-methylpentane n-hexane + 3-methylpentane n-butane + 1-butene n-hexane + 1-hexene n-heptane + 1-heptene n-octane + 1-heptene isobutane + methanol isobutane + 1-propanol n-pentane + methanol n-pentane + ethanol n-pentane + 1-butanol n-pentane + 2-butanol n-pentane + 1-pentanol n-hexane + methanol n-hexane + ethanol n-hexane + 1-butanol n-hexane + 2-butanol n-hexane + isobutanol n-hexane + 1-pentanol n-heptane + methanol n-heptane + ethanol n-heptane + 1-propanol n-heptane + 2-propanol n-heptane + 1-butanol n-heptane + isobutanol n-heptane + tert-butyl alcohol n-heptane + 1-pentanol n-heptane + 2-pentanol n-heptane + 3-pentanol n-heptane + 2-methyl-1-butanol n-heptane + 3-methyl-1-butanol n-octane + 1-butanol 224-trimethylpentane + ethanol 224-trimethylpentane + 1-propanol 224-trimethylpentane + 1-hexanol n-pentane + benzene n-hexane + benzene n-heptane + benzene n-heptane + toluene n-heptane + p-xylene n-heptane + ethylbenzene n-octane + benzene n-octane + toluene n-octane + p-xylene n-decane + toluene n-decane + p-xylene methanol + ethanol methanol + 2-propanol methanol + 2-methyl-1-propanol methanol + 3-methylbutanol ethanol-1-propanol ethanol + 1-butanol ethanol + 2-methyl-1-propanol ethanol + 3-methylbutanol 1-propanol + 2-methyl-1-propanol 1-propanol + 3-methylbutanol 1-butanol + 2-butanol 1-butanol + tert-butyl alcohol 1-butanol + 2-methyl-1-propanol 2-butanol + tert-butyl alcohol 2-butanol + 2-methyl-1-propanol tert-butyl alcohol + 2-methyl-1-propanol 2-methyl-1-butanol + 3-methylbutanol n-butane + acetone n-hexane + acetone n-heptane + 2-pentanone alkane-ether n-hexane + tert-amyl methyl ether decane + methyl tert-butyl ether n-pentane + n-propanal n-heptane + n-butanal n-heptane + isobutanal cycloalkane-alkane n-hexane + cyclohexane n-hexane + methylcyclopentane n-heptane + cyclohexane
temperature range
data points
alkane-alkane 298.15 10 298.15 11 323.15 9 328.15 14 293.15-293.15 12 303.15-313.15 21 alkane-alkene 310.93 10 333.15 11 328.15 15 328.15 12 alkane-alcohol 373.15 13 340.8 8 372.7-422.6 30 372.7-422.6 31 303.2 14 303.15 14 303.15 14 298.15-333.15 55 333.15 8 323.15-332.53 39 348.15 14 332.53 22 323.15 14 298.15 30 298.15-363.15 83 303.15-333.15 84 303.15-333.15 54 313.15 19 298.15-313.15 36 313.15 20 348.15-368.15 58 348.15-368.15 73 348.15-368.15 68 348.15-368.15 60 348.15-368.15 74 373.15 19 333.15 17 345.15 20 313.15 12 alkane-alkyl benzene 318.15 13 298.15-333.15 114 298.15-313.15 60 298.15-313.15 46 313.15 13 313.15 14 313.15 14 313.15-333.15 28 313.15 14 313.15 12 313.15 14 alcohol-alcohol 298.15 15 328.15 20 323.15-343.15 40 323.15-343.15 30 313.15 11 323.15-363.15 24 323.15-353.15 40 232.15-353.15 40 343.15-353.15 20 353.15 10 313.15 15 313.15 17 313.15 18 313.15 17 313.15 20 313.15 21 353.15 10 alkane-ketone 273.2-293.2 18 283.2-328.2 48 363.15 11
ref(s)
data at xi ) 0 and/or xi ) 1
29 29 86 38 55 27
yes/yes yes/yes yes/yes yes/yes no/no no/no
37 37 38 38
yes/yes no/no yes/yes yes/yes
53 94 89 25 72 72 72 47, 75 54 26, 44 19 26 72 47 47, 67, 72, 80, 96 83, 95 83 97 44, 97 97 58 92 91 92 58 46 45 45 44
yes/yes yes/no yes/yes yes/yes yes/yes yes/yes yes/yes yes/yes yes/yes yes/yes yes/yes yes/yes yes/yes no/no yes/yes yes/yes yes/yes no/no no/no no/no yes/yes yes/yes yes/yes yes/yes yes/yes yes/yes no/no no/no no/no
85 24, 43, 60, 74 43, 57 43, 79 43 43 43 22, 43 43 43 43
no/no yes/yes yes/yes yes/yes yes/yes yes/yes yes/yes yes/yes yes/yes yes/yes yes/yes
40 41 41 41 96 40 41 41 41 41 35 35 35 35 35 35 35
yes/yes no/no yes/yes yes/yes no/no no/no yes/yes yes/yes yes/yes yes/yes yes/yes yes/yes yes/yes yes/yes yes/yes yes/yes yes/yes
59 21, 51 76
yes/yes yes/yes yes/yes
298.15 22 308.15-328.15 59 alkane-aldehyde 313.15 25 318.15 37 335 16
20a 20b
yes/yes yes/yes
70 70 78
yes/yes yes/yes yes/yes
343.15 333.15 298.15-333.15
37 37 37
no/no yes/yes yes/yes
7 10 24
Ind. Eng. Chem. Res., Vol. 43, No. 4, 2004 855 Table 1. Continued system
temperature range
data points
cyclohexane + 2-propanol cyclohexane + 2-butanol cyclohexane + tert-butyl alcohol methylcyclohexane + 2-propanol methylcyclohexane + 1-butanol
cycloalkane-alcohol 323.15-333.15 323.15-338.15 328.2-343.3 323.15-333.15 363.15
cyclohexane + benzene cyclohexane + toluene cyclohexane + m-xylene
19 29 33 23 19
ref(s)
data at xi ) 0 and/or xi ) 1
63 19 82 63 77
no/no yes/yes yes/yes no/no no/no
cycloalkane-alkyl benzene 313.05-423.15 73 318.15 11 298.15 13
52, 56, 90, 93 34 93
yes/yes yes/yes no/no
cyclohexane + methyl tert-butyl ether cyclohexane + tert-amyl methyl ether cyclohexane + acetone cyclohexane + methyl ethyl ketone
cycloalkane-ether 313.15 298.15-318.15 313.15 323.15
22 72 25 31
56 20c 28, 66 28
yes/yes yes/yes yes/yes yes/yes
cyclohexane + cyclohexanone methylcyclohexane + methyl-isobutyl-ketone
323.15 353.15
13 16
23 77
no/no no/no
cyclohexane + n-propanal
cycloalkane-aldehyde 318.15 17
48
yes/yes
49 30 30 30 30, 84 61 42 52 33 33 87 32
no/no yes/yes yes/yes yes/yes yes/yes no/no yes/yes yes/yes yes/yes yes/yes yes/yes yes/yes
31, 40, 41, 88 41 41 40, 41 35 39
yes/yes no/no yes/yes yes/yes no/no no/no
cycloalkane-ketone
alcohol-ether
methanol + diethyl ether methanol + n-butyl-ethyl ether methanol + di-isopropyl ether methanol + methyl-n-butyl ether methanol + methyl tert-butyl ether ethanol + di-ethyl ether ethanol + di-n-butyl ether ethanol + di-isopropyl ether 1-propanol + di-n-propyl ether 2-propanol + di-n-propyl ether 2-propanol + di-isopropyl ether tert-butyl alcohol + methyl tert-butyl ether
303.15 315-335 320-330 310-330 298.15-325 273.2-323.2 323.2-352.2 344.17-363.2 278.15-323.15 278.15-323.15 330-340 323.2
methanol + acetone methanol + 2-butanone ethanol + acetone ethanol + 2-butanone 2-propanol + acetone 1-butanol + acetone
alcohol-ketone 293.15-372.8 323.15 305.15-321.15 298.15-328.15 328.15 298.15
methanol + benzene methanol + toluene ethanol + toluene 1-propanol + 135-trimethylbenzene 2-propanol + benzene 2-propanol + toluene 2-propanol + 135-trimethylbenzene 1-butanol + benzene 1-butanol + toluene 2-butanol + benzene tert-butyl alcohol + benzene isobutanol + benzene 1-pentanol + toluene 3-methyl-1-butanol + toluene
alcohol-alkyl benzene 318.15 32 318.15 11 303.15-333.15 60 328.15 13 323.15-343.15 38 298.15-313.15 42 353.15 16 298.15 9 363.15 17 298.15 9 298.15 11 298.15 12 303.15-383.15 101 353.15-380.15 43
81 62 83 18 63 79 18 71 77 71 71 71 73 68
yes/yes yes/yes yes/yes no/no no/no yes/yes no/no yes/yes no/no yes/yes yes/yes yes/yes no/no yes/yes
acetone + diisopropyl ether
ketone-ether 343.26-363.29
52
yes/yes
acetone + toluene methyl ethyl ketone + benzene methyl ethyl ketone + toluene 2-butanone + ethylbenzene 2-pentanone + toluene 4-methyl-2-pentanone + toluene
ketone-alkyl benzene 308.15-328.15 44 313.15-333.15 51 323.15-348.15 86 328.15-348.15 45 323.15 27 323.15 28
36 36, 65 36, 64 36 36 36
no/no yes/yes yes/yes no/no yes/yes yes/yes
methyl-tert-butyl-ether + 1-hexene
313.15
24
57
yes/yes
pentanal + toluene
aldehyde-alkyl benzene 313.15 13
50
yes/yes
15
36
yes/yes
12
35
no/no
17
41
yes/yes
12
37
no/no
33
ketone-aldehyde 318.15
2-butanol + 1-hexene
333.15
cyclohexane + 1-octene
86 13 42 27 14 6
alkene-ether
methyl ethyl ketone + n-propanal
propanal + methanol
10 27 34 33 44 114 54 30 112 91 48 11
alcohol-alkene alcohol-aldehyde 318.15 cycloalkane-alkene 313.15
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Ind. Eng. Chem. Res., Vol. 43, No. 4, 2004
Figure 1. Deviations for compounds with data from three or more systems. Table 2. Deviations between Calculated and Experimental Values of the Pressure and the K factor mean dev in K (%)
mean dev in P (%)
number of data points
nonassociating alkane-alkane 6 alkane-alkene 4 alkane-alkyl benzene 11 cycloalkane-alkane 3 cycloalkane-alkyl benzene 3 alkane-ketone 3 alkane-ether 2 alkane-aldehyde 3 cycloalkane-ether 2 cycloalkane-ketone 4 cycloalkane-aldehyde 1 ketone-alkyl benzene 6 ketone-ether 1 alkene-ether 1 aldehyde-alkyl benzene 1 ketone-aldehyde 1 cycloalkane-alkene 1
1.70 1.22 3.16 0.95 3.04 3.64 0.50 4.14 1.53 3.26 6.58 2.70 4.19 0.61 0.72 1.21 0.31
1.20 1.16 4.52 0.90 3.39 6.74 3.99 4.02 0.91 5.69 4.33 3.35 5.21 0.78 0.86 0.51 1.13
77 48 342 41 97 77 81 78 94 98 17 310 30 23 12 14 12
self-associating 29 5 14 12
4.14 4.42 2.87 8.98
7.39 5.22 5.87 9.25
1005 123 414 608
cross-associating 6 6.46 17 2.25
8.74 2.25
188 368
3.80
4157
system group
alkane-alcohol cycloalkane-alcohol alcohol-alkyl benzene alcohol-ether alcohol-ketone alcohol-alcohol sum
number of systems
136
2.98
of the mixture, e.g., if one component is too complex for a full DFT optimization. Results All results are summarized in Table 2 and Figure 1. Table 2 lists the investigated groups of systems with their deviations between calculated and experimental values. Furthermore, the number of systems and the
number of data points are given. The systems are grouped as nonassociating, self-associating, and crossassociating mixtures. As can be seen in Table 2, the overall deviation is below 4% in pressure and K factor. Figure 1 contains only results for groups with three or more data sets to allow a general conclusion. The deviation is smaller for nonassociating mixtures than for associating mixtures. Nonpolar molecules and their mixtures show deviations of lower than 4% (alkanes, cycloalkanes, alkenes, ethers). The phase behavior is dominated by van der Waals interactions, and the activity coefficients are in the region of unity. This result might be considered to be trivial, but the fact that the COSMO-RS energy concept is applicable to nonpolar mixtures shows that this combination of quantum chemistry and statistical mechanics describes the phase behavior for even this kind of mixture. Nonassociating molecules having a polar functional group exhibit higher deviations in mixtures with nonpolar components. The stronger interactions are reproduced by COSMO-RS within a deviation of 7%. Systems containing self-associating components show the highest deviations in this study. Particularly, the phase behavior of mixtures containing alcohols and ethers cannot be predicted with the same accuracy as attained for the other mixtures. A reason for this behavior might be the linear mixing of the purecomponent σ profiles to represent multicomponent mixtures. In an ensemble of formerly nonassociating and self-associating molecules, the mixing of surface pieces without using information on the parts of surface that would be more likely to be interacting could provide an incorrect value for the ensemble interaction energy. See below for a detailed description of the phase behavior in mixtures with self-associating components. Mixtures of associating components show diverse behavior. Activity coefficients in alcohol-alcohol systems are close to unity. This is reproduced by COSMO-
Ind. Eng. Chem. Res., Vol. 43, No. 4, 2004 857
Figure 2. Deviations in pressure in the systems containing n-alkanes and alkyl benzenes.
RS with a deviation close to 2%. Alcohols in mixtures with ketones cause a cross-associating interaction of the hydroxyl group with the keto group. This results in higher activity coefficients than COSMO-RS predicts, leading to deviations at finite dilution of around 10%. Klamt and Eckert15 reported a difference between the calculated chemical potential and the experimental value of 1.7 kJ/mol. An evaluation of the error is possible at the state of infinite dilution where the relation ∞ γi,COSMO-RS ∞ γi,experimental
( )
∆µli ) exp RT
Figure 3. Activity coefficients as a function of the mole fraction of benzene in the system benzene + n-heptane at 313.15 K (experimental data from ref 40, Table 1).
(5)
can be applied. Hence, the given difference in the chemical potentials leads to a maximal difference of a factor of 2 between the calculated and measured values of the activity coefficient at infinite dilution. Nonassociating Mixtures. Mixtures of alkanes show an almost ideal behavior. In all cases, this is predicted by COSMO-RS. Mixtures of n-alkanes and alkyl benzenes follow slightly nonideal behavior. Of this type of mixture, systems containing benzene, toluene, p-xylene, and ethylbenzene were studied. The chain lengths of the alkanes varied from C5 to C10. Figure 2 shows the deviations in pressure for the investigated systems. Generally, it can be seen that the longer the chain length of the n-alkane, the higher the deviation in pressure, except for the case of the ethylbenzene mixture. In this case, keeping the chain length of the alkane constant and adding more alkyl groups to the aromatic structure causes the deviation to decrease (see, e.g., Figure 2, n-heptane + alkyl benzene). Except for the n-pentane + benzene system, all pure-component pressures were measured and provided by the individual authors. Consequently, an error associated with calculating this property using a correlation can be ruled out. The deviations are produced by the activity coefficients, which were predicted too high. Figure 3 shows the finite activity coefficients in the system benzene + n-heptane as a function of the mole fraction of benzene. Similar results were found for all other systems. The ketone-alkyl benzene mixtures are the third nonassociating group of mixtures investigated, but they have one component, the ketones, with a polar part within the molecule. Figure 4 shows the deviations in pressure for four different ketones with three alkyl benzenes. It can be seen that, in mixtures with toluene, the deviations in pressure decrease when the alkyl part of the ketone increases. Considering methylethyl ketone
Figure 4. Deviations in pressure in the systems containing ketones and alkyl benzenes.
in mixtures with different alkyl benzenes, the deviation is almost constant at around 5%. Looking at the activity coefficients at finite concentration, COSMO-RS predicts them within 10% accuracy. In contrast to n-alkanes, activity coefficients of ketones in mixtures with alkyl benzenes are predicted lower than experimental data indicate. This is true for both of the components in the mixture. Whereas COSMO-RS predicts the activity coefficients to be close to unity, in reality, they are around 1.2-2.0 in the considered systems. Still, this is in the range of the given accuracy of COSMO-RS, which was mentioned above. Self-Associating Systems. Alcohols in mixtures with alkanes show strongly nonideal behavior. The overall deviation in pressure between experimental data and predicted values was found to be 7.4%. Figure 5 shows the individual results. The uncertainty in predicting the VLE cannot be characterized by a trend in terms of varying chain length of either the alcohol or the alkane. A more clear conclusion can be made when activities at infinite dilution are considered. Figures 6 and 7 illustrate the predicted activity coefficients at infinite dilution. Figure 6 represents data for five n-alkanes in solution with three different alcohols. The predicted values are always lower than the experimental values. The average difference is 23%. This behavior can be found in almost all investigated alcohol-alkane mixtures. Whereas predicted activity coefficients at
858
Ind. Eng. Chem. Res., Vol. 43, No. 4, 2004
Figure 5. Deviations in pressure in the systems containing alcohols and alkanes.
Figure 6. Activity coefficients of n-alkanes at infinite dilution in different alcohols: filled symbols, experimental data;102 open symbols, COSMO-RS predictions.
infinite dilution of the alkanes are lower than experimental data, the predicted values of the alcohols are always higher. This can be seen in Figure 7, where data for ethanol in n-alkanes are shown. The average difference between predictions and measurements is 44%. This overestimation of the activity coefficients leads to the reported deviations in pressure. Nevertheless, the azeotropic, heteroazeotropic, or close-boiling behavior of all of the mixtures can be reproduced by COSMO-RS. The same behavior can be observed in alcohol-alkyl benzene systems. Whereas the activity coefficient of the alcohol is overestimated, the activity coefficient of the aromatic component is underestimated. Nevertheless, COSMO-RS predicts the general behavior, which is dominated by strong interactions of the polar alcohol molecules with their own species. The accuracy can be summarized to a deviation of about 3% in pressure and includes the prediction of the azeotropic or close-boiling behavior, as well as the location of the azeotropic point. A different quality of prediction is found in mixtures of alcohols and ethers. Figure 8 shows the deviations in
Figure 7. Activity coefficients of ethanol at infinite dilution in different alcohols: filled symbols, experimental data;102 open symbols, COSMO-RS predictions.
Figure 8. Deviations in pressure in the systems containing alcohols and ethers.
pressure. The values are plotted for mixtures with increasing numbers of carbon atoms within the linear ether chain. Additionally, two nonlinear ethers, the diisopropyl ether and MTBE are shown. The deviation
Ind. Eng. Chem. Res., Vol. 43, No. 4, 2004 859
experimental values. These results are in the given limit of accuracy. Conclusion
Figure 9. Deviations in pressure in the systems containing alcohol + alcohol.
We have tested the applicability of COSMO-RS to predict binary vapor-liquid equilibria when it is used as a GE model. One hundred thirty-six binary systems were chosen, covering the most important substances in chemical engineering. It was found that COSMORS predicts the phase behavior of binary VLE in good agreement with experimental data. The deviations for many classes of systems are at or below 5% for both the K value and the system pressure. Mixtures of nonassociating components are predicted nearly exactly. When components with different polar functional groups interact with each other, the phase behavior of the resulting mixtures cannot predicted with the same accuracy. Still, the nonideal behavior is reproduced, and azeotropic and close-boiling effects are predicted. At the present, we have found COSMO-RS to be a helpful tool in predicting the general mixing behavior of two or more components. Components are not restricted in structure, except that the number of atoms must not exceed 60. This restriction comes from the quantum chemical calculations. Structure optimizations using the DFT method will not converge for molecules with more than 60 atoms. Acknowledgment The authors gratefully thank Dr. A. Klamt for providing us the COSMOtherm program and for his assistance in the quantum chemical calculations.
Figure 10. Activity coefficients as a function of the mole fraction of acetone in the system acetone + methanol at 308.15 K (experimental data from ref 25, Table 1).
decreases with the number of carbon atoms of the ether when mixtures with linear ethers are considered. It then rises when one compares the values of di-n-propyl ether + ethanol and diisopropyl ether + ethanol with a more branched structure. The reason for the more inaccurate prediction, compared to all other systems, can be found in the activity coefficients at finite dilution, which are, for both of the components, always lower in prediction than in experiment. Cross-Associating Systems. As mentioned above, COSMO-RS predicts the VLE behavior of alcohol + alcohol mixtures within a mean deviation of 2% in pressure. Figure 9 gives a detailed overview of the investigated systems. Mostly, the deviation scatters around 3%. No trend can be observed in terms of the different sizes and shapes of the molecules in the mixtures. The interaction behavior is reproduced within the error of measurements, and it can be concluded that, for this class of mixtures together with a known vapor pressure of the pure component, predictions with COSMO-RS deliver accurate results. As observed in the mixtures of alcohol + ether, the activity coefficients at finite dilution predicted by COSMO-RS are lower than the experimental values in the alcohol + ketone mixtures also. This causes a negative deviation of the system pressure. Figure 10 shows the predicted and measured data in the system acetone + methanol at 308.15 K. Investigations of activity coefficients at infinite dilution show a maximum deviation of 50% between COSMO-RS predictions and
List of Symbols ) permeability (A s V-1 m-1) σ ) screening charge density (A s m-2) γi ) activity coefficient of component i in a mixture µi ) chemical potential of component i (J/mol) µ0i ) chemical potential of the pure liquid component i (J/ mol) R ) universal gas constant (J mol-1 K-1) T ) temperature (K) GE ) excess Gibbs free energy (J/mol) P ) pressure (bar) xi ) mole fraction of component i in the liquid phase yi ) mole fraction of component i in the gas phase
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Received for review July 17, 2003 Revised manuscript received December 2, 2003 Accepted December 8, 2003 IE034009W
