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Ind. Eng. Chem. Res. 1996, 35, 1438-1445
Prediction of Infinite-Dilution Activity Coefficients in Binary Mixtures with UNIFAC. A Critical Evaluation Epaminondas C. Voutsas† and Dimitrios P. Tassios* Laboratory of Thermodynamics and Transport Phenomena, Department of Chemical Engineering, National Technical University of Athens, 9 Heroon Polytechniou Str., Zographou Campus, 15780 Athens, Greece
Using an extensive data base, several UNIFAC-type activity coefficient models have been compared with respect to their accuracy in the prediction of infinite-dilution activity coefficients in binary mixtures. In the case of aqueous mixtures, the correlation of Pierotti, Deal, and Derr (PDD) has also been included in the comparison. For athermal alkane/alkane asymmetric mixtures the modified UNIFAC-type models perform very well, while the original UNIFAC ones give poor results. For nonaqueous polar mixtures, the modified UNIFAC of Gmehling et al. gives satisfactory results. For the highly nonideal aqueous systems, the PDD correlation is the best method. When the PDD correlation is not available, order of magnitude estimates can be obtained from the UNIFAC-LLE of Magnussen et al. Finally, for the hydrocarbon/water mixtures, the modified UNIFAC-type model of Hooper et al. should be used since it was specially developed for such systems. Introduction Activity coefficient measures the deviation from the ideality of a component in solution. At infinite dilution, a single solute molecule is completely surrounded by solvent. Therefore, infinite-dilution activity coefficients, γ∞, are useful because they give a measure of the greatest degree of the nonideality of a mixture. Interest in knowledge of infinite-dilution activity coefficients lies with their usefulness in the prediction of vapor-liquid equilibrium for the design of absorbers, strippers, distillation columns, etc. Similarly, limiting activity coefficients can be used to determine parameters for excess Gibbs free energy expressions, i.e., the NRTL, Wilson, UNIQUAC, or UNIFAC. For systems where limitedsor nosexperimental data are available, it is advantageous to use a predictive tool such as ASOG (Wilson and Deal, 1962; Derr and Deal, 1969, 1973) or the well-known UNIFAC model (Fredenslund et al., 1975, 1977). Several activity coefficient models have been tested in the literature for their accuracy in the prediction of infinite-dilution activity coefficients in nonaqueous mixtures (Thomas and Eckert, 1984; Gmehling et al., 1993; Kontogeorgis et al., 1994; Voutsas et al., 1995), where the values of the infinite dilution activity coefficient are generally low, typically well below 100. This is not the case with the very important industrial solvent water, which presents a unique behavior in solutions with other compounds (Shealy and Sandler, 1988). Due to this complex behavior, water exhibits great nonideality with organics. As a result, γ∞ values of organics in water are very large, especially for the higher molecular weight compounds; e.g., the γ∞ of n-decane in water is on the order of 108. In this work various UNIFAC-type models are tested with respect to their accuracy in the prediction of infinite-dilution activity coefficients in nonelectrolyte mixtures, with emphasis on aqueous ones. The correlations of Pierotti, Deal, and Derr (1959) have also been applied in the prediction of infinite-dilution activity coefficient in aqueous solutions due to their successful * To whom correspondence should be addressed. † e-mail:
[email protected].
0888-5885/96/2635-1438$12.00/0
performance. The remainder of the paper is organized as follows. A brief review of the activity coefficient models involved in the comparison is presented in the next section. The data base used for the evaluation of the above models follows. The results of the models are compared and discussed next. We close with our conclusions. UNIFAC-Type Models In the original UNIFAC model (Fredenslund et al., 1975, 1977) the activity coefficient is expressed as the sum of a combinatorial and a residual part:
ln γi ) ln γicomb + ln γires
(1)
The combinatorial part (ln γicomb) accounts for differences in the size and shape of the molecules, and the residual part (ln γires) accounts mainly for the effects which arise from energetic interactions between groups. For alkane/alkane mixtures, which exhibit almost athermal behavior, the residual contribution to the activity is expected to be close to zero. UNIFAC uses the same combinatorial expression as the UNIQUAC model (Abrams and Prausnitz, 1975), the so-called Staverman-Guggenheim (SG) one (Staverman, 1950):
ln γicomb ) ln
(
)
φi φi z φi φi + 1 - - qi ln + 1 xi xi 2 θi θi
(2)
In the above equation φi is the segment fraction of component i:
xiri φi )
∑j xjrj
(3)
where ri is the pure-component volume parameter. Also, in eq 2, θi is the surface area fraction of component i: © 1996 American Chemical Society
Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996 1439
θi )
xiqi
(4)
∑j xjqj
where qi is the pure-component area parameter and z is the coordination number (usually equal to 10). The volume and surface area parameters ri and qi are, in fact, the van der Waals volumes and surface areas (Vw, Aw) calculated by the group increments presented by Bondi (1968). The residual part is obtained by using the following relations:
ln γires )
[
∑k νk(i)(ln Γk - ln Γk(i)) θmΨmk
]
θm )
(6)
xiri2/3 φi )
∑j νm
Xm )
∑j ∑n
{[
(
exp -
xj (8)
(i)
(13) where T0 is an arbitrary reference temperature equal to 298.15 K. 4. Modified UNIFAC (Gmehling et al., 1993). First, the combinatorial term of this model is given by:
(
)
φi φi z φi φi + 1 - - qi ln + 1 (14) xi xi 2 θi θi
where
νn xj
xiri3/4
(9)
In order to improve the performance of the original UNIFAC model in the prediction of vapor-liquid equilibrium (VLE), liquid-liquid equilibrium (LLE), γ∞, and excess enthalpies, many modifications in the combinatorial, as well as the residual, term of this model have been suggested. In addition to the original UNIFAC model, with temperature-independent group-interaction parameters from Hansen et al. (1991), the following UNIFAC-type models will be considered and discussed further in this work: 1. Original UNIFAC with Temperature-Dependent Interaction Parameters (Hansen et al., 1992). In this model the group-interaction parameter Ψmn in the residual part of eq 9 is given by:
]}
amn + bmn(T - T0) T
φi )
∑j
(15)
xjrj3/4
and, second, the interaction parameter Ψmn in the residual part of eq 9 is given by:
{(
Ψmn ) exp -
)}
amn + bmnT + cmnT2 T
(16)
5. Modified UNIFAC Especially Suited for Infinite-Dilution Activity Coefficients (Bastos et al., 1988). This model uses the combinatorial term suggested by Kikic et al. (1980), which has the following form:
ln γicomb ) ln
(
)
φi φi z φi φi + 1 - - qi ln + 1 xi xi 2 θi θi
(17)
where
xiri2/3
(10)
where T0 is an arbitrary reference temperature equal to 298.15 K. 2. Original UNIFAC Especially Suited for LLE Calculations (Magnussen et al., 1981). This is a original UNIFAC-type model, where the interaction parameters have been obtained by fitting LLE experimental data.
]
)}
T0 + T - T0 T
T
ln γicomb ) ln
Ψmn ) exp[-(amn/T)]
{[
∑j xjrj
Ψmn )
Finally, in the original UNIFAC model the group interaction parameter Ψmn is given by:
Ψmn ) exp -
(12)
2/3
and the group mole fraction Xm by: (i)
(11)
where
amn + bmn(T - T0) + cmn T ln
(7)
∑n QnXn
φi φi +1xi xi
and, second, the interaction parameter Ψmn in the residual part of eq 9 is given by:
whereby the group area fraction θm is given by the following equation:
QmXm
()
ln γicomb ) ln
(5)
θmΨmk) - ∑ ∑ m m ∑n θnΨnm
ln Γk ) Qk 1 - ln(
3. Modified UNIFAC (Larsen et al., 1987). In this model two modifications with respect to the original UNIFAC model have been made. First, the combinatorial term is given by:
φi )
∑j
(18)
xjrj2/3
The residual part of this model has the same form as in the original UNIFAC one, where the interaction parameters have been obtained by fitting infinitedilution activity coefficient data.
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Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996
6. Modified UNIFAC (Hooper et al., 1988). This model is applied only to correlate liquid-liquid equilibria for water/hydrocarbon (and some hydrocarbon derivatives) mixtures. It uses the same combinatorial expression as in the modified UNIFAC model of Larsen et al., and the interaction parameter Ψmn in the residual part of eq 9 is given by:
{(
Ψmn ) exp -
)}
amn + bmnT + cmnT2 T
(19)
Correlation of Pierotti, Deal, and Derr (PDD) (1959) In the work of Pierotti et al. (1959) experimental phase equilibria data for many structurally related sets of binary systems have been reduced to activity coefficients of the solute at infinite dilution in the solvent and correlated on the basis of systematic effects on the logarithm for infinite-dilution activity coefficients caused by systematic changes in solute and/or solvent structure. For methylene homologs R1X1, at infinite dilution in methylene homologs R2X2, where R1 and R2 are hydrocarbon radicals and X1 and X2 are the solute and solvent functional groups, the general expression of the solute infinite-dilution activity coefficient is given by:
n1 C1 1 + D(n1 - n2)2 + F2 (20) log γ ) A1,2 + B2 + n2 n1 n2 where A1,2 ) coefficient which depends on the nature of solute and solvent functional groups X1 and X2, B2 ) coefficient which depends only on the nature of solvent functional group X2, C1 ) coefficient which depends only on the nature of solute functional group X1, D ) coefficient independent of both X1 and X2, F2 ) coefficient which essentially depends only on the nature of solvent functional group X2, n1, n2 ) numbers of carbon atoms in hydrocarbon radicals R1 and R2, respectively. Correlation constants for infinite-dilution activity coefficients of various combinations of homologous sets of solutes and solvents are given in the paper of Pierotti et al. (1959). In the present work this predictive scheme is applied only in the prediction of infinite-dilution activity coefficient in aqueous mixtures, since it is considered to be a useful tool for these kinds of mixtures. Data Base A very difficult part of this work was the selection of the data base used for the evaluation. This is due to the fact that the infinite-dilution activity coefficient values published by different authors are often contradictory, especially for the strongly nonideal aqueous mixtures. For example, the datum of Pescar and Martin (1966) for heptane infinitely dilute in water at 20 °C was 8050, while the value obtained from mutual solubility data was 1.9 × 106. Wherever possible, the data were checked by consideration of other sources or through plots versus carbon number, e.g., for the alcohol/n-alkane mixtures. The data base used for the evaluation of the models consists of more than 600 data points for the following systems: (a) alkane/alkane nearly athermal mixtures; (b) nonaqueous polar mixtures containing particular compound classes; (c) aqueous mixtures containing organic solutes infinitely diluted in water and the water/ alkane and water/1-alkanol ones. The complete data bank of the experimental infinite-dilution activity coef-
Table 1. Mean Percent Absolute Deviations (% AAD) between Experimental and Calculated Infinite-Dilution Activity Coefficients of Short-Chain Alkane Solutes in Various Long-Chain Alkane Solvents solute/solventa
ref
T-range (K)
n-C5/n-C16,20,22,28 n-C6/n-C16,18,20,22,28 n-C7/n-C16,18,20,22,24,28,30,32,34,36 n-C8/n-C18,20,22,24,28,30,32,34,36 n-C10/n-C20,22,24,28,30,32,34,36 c-C6/n-C16,18,20,24,28,32 3-C5/n-C18,20,22,24,28,30,32,34,36,36 2,3-C4/n-C18,20,22,24,28,30,32,34,36 2,2,4-C5/n-C20,22,24,28,30,32,34,36
b b b, c b, c c b b, c b, c c
313-343 303-343 303-393 308-393 353-393 298-343 308-393 308-393 353-393
overall
d
e
f
g
31.1 9.4 6.9 13.1 23.5 5.0 3.1 8.2 24.1 3.6 2.6 7.5 21.8 3.0 2.3 6.3 17.3 2.4 2.1 4.7 17.3 11.6 12.0 4.4 28.3 3.5 2.7 8.2 28.4 3.7 3.0 8.3 25.0 6.8 3.6 9.1 24.1
4.9
3.8
7.5
a
2,3-C4 ) 2,3-dimethylbutane, 2,2,4-C5 ) 2,2,4-trimethylpentane, 3-C5 ) 3-methylpentane, c-C6 ) cyclohexane. b Kniaz (1991). c Parcher et al. (1975). d Original UNIFAC. e Modified UNIFAC (Larsen et al., 1987). f Modified UNIFAC (Bastos et al., 1988). g Modified UNIFAC (Gmehling et al., 1993). Table 2. Mean Percent Absolute Deviations (% AAD) between Experimental and Calculated Infinite-Dilution Activity Coefficients of Long-Chain Alkane Solutes in Various Short-Chain Alkane Solvents (Experimental Data from Kniaz (1991)) solute/solvent
T-range (K)
b
c
d
e
n-C12,16,20,22,32/n-C6 n-C16,18,20,24,26,36/n-C7 n-C16,22,28/c-C5 n-C12,16,18,19,20,22,24,26/c-C6 n-C16,18,20,22,28/2,2-C4 n-C16,22,28/2,3-C4 n-C16,22/2-C5 n-C16,18,20,22/3-C5
288-293 262-293 218-274 257-305 256-285 254-282 250-283 259-278
47.5 46.4 66.3 48.7 56.2 57.1 53.5 43.9
11.7 9.5 20.1 7.7 29.2 20.3 25.7 10.4
8.2 5.4 20.5 7.7 19.5 15.2 21.8 5.7
17.0 15.4 8.9 10.1 28.8 26.2 29.5 14.9
51.1
15.0
11.3
17.4
overall a
2,2-C4 ) 2,2-dimethylbutane, 2-C5 ) 2-methylpentane, c-C5 ) cyclopentane. b Original UNIFAC. c Modified UNIFAC (Larsen et al., 1987). d Modified UNIFAC (Bastos et al., 1988). e Modified UNIFAC (Gmehling et al., 1993).
ficients, which was used for the evaluation of the models, as well as the predicted values with the UNIFAC-type models is included as supporting information. Results and Discussion Case A. Alkane/Alkane Mixtures. In Tables 1 and 2 infinite-dilution activity coefficient prediction results for short-chain alkanes in long-chain ones and longchain alkanes in short-chain ones are tabulated, respectively. Typical results are shown graphically in Figures 1 and 2. The models are evaluated using the mean percent absolute deviations (% AAD) of the activity coefficient predictions, defined as:
% AAD )
1
NP
∑
NPi)1
|
|
γicalc - γiexp γiexp
× 100
(21)
where NP is the total number of data points. The three modified UNIFAC-type models of Larsen et al., Bastos et al., and Gmehling et al. perform fairly satisfactorily in the prediction of infinite-dilution activity coefficients of short-chain alkanes in long-chain ones, while the original UNIFAC one gives poor results, especially for the more asymmetric mixtures. When infinite-dilution activity coefficients of long-chain alkanes in short-chain ones are considered, the modified UNIFAC-type models of Larsen et al., Bastos et al., and
Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996 1441 Table 3. Mean Percent Deviations (% AAD) between Experimental and Calculated Infinite-Dilution Activity Coefficients for Eight Classes of Solutes in a Variety of Nonaqueous Mixtures solute compound class
NP
a
b
c
d
e
f
alkanes alkenes alcohols ketones acids halogens nitro compounds aromatics
168 30 32 65 8 42 11 106
31.6 24.7 25.7 13.1 31.2 13.6 136.6 13.7
29.5 21.6 20.5 29.3 35.9 10.8 54.1 13.7
32.0 14.3 59.3 16.1 85.8 38.7 9.5 32.4
37.5
37.1 40.6 35.7 19.8
16.3 14.6 18.8 10.2 23.6 3.0 9.8 5.8
22.8 15.8 33.7 11.6 20.8
9.8 78.6 30.6
a
Original UNIFAC (Hansen et al., 1991). b Original UNIFAC (Hansen et al., 1992). c Original UNIFAC (Magnussen et al., 1981). d Modified UNIFAC (Larsen et al., 1987). e Modified UNIFAC (Bastos et al., 1988). f Modified UNIFAC (Gmehling et al., 1993).
Figure 1. Experimental and predicted infinite-dilution activity coefficients of n-hexane in long-chain n-alkanes (T ) 373 K). Experimental data from Parcher et al. (1975).
Figure 3. Infinite-dilution activity coefficient of n-alkanes in ethanol (T ) 298 K) as a function of the number of carbon atoms in n-alkane. Experimental data from Pierotti et al. (1959).
Figure 2. Experimental and predicted infinite-dilution activity coefficients of long-chain n-alkanes in n-hexane (T ) 269-293 K). Experimental data from Kniaz (1991).
Gmehling et al. give lower values than the experimental values, but still satisfactory ones, with the one of Bastos et al. being somewhat better (Figure 2). On the other hand, the original UNIFAC model significantly underestimates the experimental data, giving average absolute errors, typically higher than 40%. Case B. Nonaqueous Polar Mixtures. Table 3 compares the infinite-dilution activity coefficient predictions in nonaqueous mixtures. Detailed results are presented in the supporting information (Table S-1). The model of Gmehling et al. performs satisfactorily with typical errors below 10-15%, except in the case of the strongly associating acid/solvent mixtures. On the other hand, the other models perform erratically, giving occasionally very large errors. The model of Gmehling et al. gives also better results than the other UNIFAC-type ones as the size difference in the mixture increases. An example is shown in Figure 3 for the infinite-dilution activity coefficients of
n-alkanes in ethanol. Moreover, due to better estimation of hE values with the model of Gmehling et al. (1993), a more reliable description of the temperature dependence of the activity coefficients is obtained by this model. An example is presented in Figure 4, where infinite-dilution activity coefficient values of acetone in methanol are plotted versus temperature. Case C. Aqueous Mixtures. In Figures 5-13 infinite-dilution activity coefficient prediction results of both organic solutes in water and water in organic solvent systems are shown graphically. Detailed results are presented in the supporting information (Table S-2). For the case of n-alkanes in water mixtures (Figures 5 and 6) only the model of Hooper et al., developed specifically for these types of systems, performs satisfactorily, while the other UNIFAC models strongly underestimate the experimental data. The PDD model gives also fairly satisfactory results. On the other hand, in the case of water in n-alkane systems (Figure 6), as the carbon number in the n-alkane increases, the performance of the model of Hooper et al. becomes progressively poorer, giving an absolute error of 130% in the predicted γ∞ of water in n-octane, while the original UNIFAC model (Hansen et al., 1991) and the UNIFAC-LLE one of Magnussen et al. become progres-
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Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996
Figure 4. Infinite-dilution activity coefficient of acetone in methanol as a function of temperature. Experimental data from Tiegs et al. (1986).
Figure 6. Experimental and predicted infinite-dilution activity coefficients of water in n-alkanes at T ) 298 K. Experimental data are smoothed values from Sørensen and Arlt (1979).
Figure 5. Experimental and predicted infinite-dilution activity coefficients of n-alkanes in water at T ) 293 K. Experimental data from McAuliffe (1966).
Figure 7. Experimental and predicted infinite-dilution activity coefficients of cycloalkanes in water at T ) 298 K. Experimental data from McAuliffe (1966) and Sørensen and Arlt (1979).
sively better. As for cycloalkanes in water systems (Figure 7), only the model of Hooper et al. gives relatively satisfactory results. In Figure 8 prediction results for aromatic hydrocarbons in water are presented. All models give satisfactory results except for that of Larsen et al., which gives increasingly larger errors with increasing carbon number, and that of Hansen et al. (1992), which fails. In Figures 9 and 10 prediction results for both 1-alkanols in water and water in 1-alkanols are presented. For the case of 1-alkanol/water mixtures only the model of Pierotti et al. remains successful even for the higher 1-alkanols. The PDD model performs also successfully in the prediction of γ∞ of water in 1-alkanols. Finally, in Figures 11-13 results for 2-ketones in water, n-aldehydes in water, and n-acids in water mixtures are shown. It is evident that in all cases the
PDD model performs well even at high asymmetries, while that of Magnussen et al. provides poorer but relatively satisfactory results. All the other models fail, since they strongly underestimate the experimental data. Conclusions Various UNIFAC-type group-contribution models have been compared with respect to their performance in the prediction of infinite-dilution activity coefficients using an extensive data base of more than 600 experimental points for both nonaqueous and aqueous mixtures. In the case of water-containing mixtures the predictive expressions of Pierotti, Deal, and Derr have also been included in the comparison. For the alkane/alkane nearly athermal mixtures, the modified UNIFAC-type models perform very well, with the one of Bastos et al. being generally superior. As
Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996 1443
Figure 8. Experimental and predicted infinite-dilution activity coefficients of aromatic hydrocarbons in water at T ) 298 K. Experimental data from Dutt and Prasad (1989).
Figure 9. Experimental and predicted infinite-dilution activity coefficients of 1-alkanols in water at T ) 298 K. Experimental data from Pierotti et al. (1959), Sørensen and Arlt (1979), and Hwang et al. (1992).
expected, original UNIFAC greatly exaggerates the degree of nonideality of these mixtures, strongly underestimating the experimental data especially at larger asymmetries. For nonaqueous polar mixtures, where the infinitedilution activity coefficients do not reach very large values (typically well below 100), the model of Gmehling et al. (1993) is found to give satisfactory results and is recommended over the other UNIFAC-type models which often lead to large errors. Finally, for the case of the strongly nonideal watercontaining mixtures, where infinite-dilution activity coefficientssespecially of the organic solutesreach very large values, only the model of Pierotti, Deal, and Derr is found to perform satisfactorily even for the high asymmetric mixtures, i.e., a large number of carbon atoms in the organic compound. Since, however, the
Figure 10. Experimental and predicted infinite-dilution activity coefficients of water in 1-alkanols at T ) 298 K. Experimental data from Pierotti et al. (1959).
Figure 11. Experimental and predicted infinite-dilution activity coefficients of 2-ketones in water at T ) 298 K. Experimental data from Pierotti et al. (1959) and Hwang et al. (1992).
PDD model is applicable to a limited number of cases, the model of Magnussen et al. provides at least order of magnitude estimates. All the other models yield γ∞ values much lower the experimental ones. An exception is the case of alkane/water mixtures where the model of Hooper and Prausnitz gives the best results, since it was specifically developed for these types of systems. Acknowledgment E.C.V. acknowledges the National Scholarship Foundation of Greece for their financial support. Supporting Information Available: Two tables including experimental and predicted, with various UNIFAC-type models, infinite-dilution activity coefficients of various solutes in both nonaqueous and aqueous binary mixtures (16 pages).
1444
Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996 γi ) activity coefficient of component i Θm ) surface fraction of group m in the liquid phase ν(i)k ) number of structural groups of type k in molecule i Ψnm ) UNIFAC group-interaction parameter between groups n and m Superscripts comb ) combinatorial term res ) residual term Acronyms ASOG ) analytical solution of group theory/model UNIFAC ) universal functional activity coefficient model
Literature Cited
Figure 12. Experimental and predicted infinite-dilution activity coefficients of n-aldehydes in water at T ) 298 K. Experimental data from Pierotti et al. (1959) and Hwang et al. (1992).
Figure 13. Experimental and predicted infinite-dilution activity coefficients of n-acids in water at T ) 298 K. Experimental data from Pierotti et al. (1959) and Hwang et al. (1992).
Nomenclature anm ) UNIFAC group-interaction parameter between groups n and m (K) bnm ) UNIFAC group-interaction parameter between groups n and m cnm ) UNIFAC group-interaction parameter between groups n and m (K-1) hE ) excess enthalpy (J/mol) qi ) relative van der Waals surface area of component i Qk ) relative van der Waals surface area of subgroup k ri ) relative van der Waals volume of component i T ) absolute temperature (K) xi ) mole fraction of component i in the liquid phase Xm ) group mole fraction of group m in the liquid phase Greek Symbols Γk ) group activity coefficient of group k in the mixture Γ(i)k ) group activity coefficient of group k in the pure substance
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Received for review June 14, 1995 Revised manuscript received November 17, 1995 Accepted December 12, 1995X IE9503555
X Abstract published in Advance ACS Abstracts, February 15, 1996.
