Ind. Eng. Chem. Res. 1987, 26, 1372-1381
1372
U = average velocity, m/s U , = maximum velocity, m/s U , = superficial velocity, m/s W = width of the tube bank, m Y = distance of the elementary strip from the entrance, m 2 = height of the tube bank, m Greek Symbols (3 = tortuous path factor
= viscosity of the fluid, CP = density, kg/m3 7 = shear stress, kg f/m2 T~ = shear stress at wall, kg f/m2 A = difference t = void fraction
p p
Literature Cited Adams, D. Ph.D. Dissertation, Oklahoma State University, Stillwater, 1968. Adams, D.; Bell, K. J. Chem. Eng. Prog. Symp. Ser. 1968, 64(82), 133. Batra, V. K.: Fulford, G. D.: Dullien, F. A. L. Can. J . Chem. Eng. 1970, 48, 622. Bergelin, 0. P.; Brown, G. A.; Hull, H. L.; Sullivan, F. W. Trans. ASME 1950a. 72. 881. Bergelin, 0. P.; Brown, G. A.; Doberstein, S. C. Trans. ASME 1952, 74, 953.
Bergelin, 0. P.; Colburn, T.; Hull, H. L. Engineering Experimental Station Bulletin 2, 1950b; University of Delaware, Newark. Bergelin, 0. P.; Davis, E. S.; Hull, H. L. Trans. ASMe 1949, 71,369. Boucher, D. F.; Lapple, G. F. Chem. Eng. Prog. 1948,44, 117. Chand, P. M. Tech. Dissertation, Banaras Hindu University, Varanasi, India, 1979. Chilton, T. H.; Genereaux, R. P. J. AIChE 1933, 29, 161. Cruzen, C. G. M.S. Thesis, Oklahoma State University, Stillwater, 1964. Grimison, E. D. Trans. ASME 1937, 59, 583. Guntur, A. Y.; Shaw, W. A. Trans. ASME 1945, 67, 643. Huge, E. S. Trans. ASME 1936, 59, 573. Hughmark, G. A. J . AIChE 1972, 18, 1020. Kostic, Z. Presented at the International Seminar, Hercerg-Novi, Yugoslavia, 1969. Pierson, 0. L. Trans. ASME 1937, 50, 563. Tandon, S. K. M. Tech. Dissertation. Banaras Hindu University, Varanasi, India, 1976. Vossoughi, S.; Seyer, F. A. Can. J . Chem. Eng. 1974, 52, 666. Whitaker, S. J . AlChE 1972, 18, 361. Whitaker, S. Elementary Heat Transfer Analysis; Pergamon: New York, 1976. Zukauskas, A. Advances in Heat Transfer; Academic: New York, London, 1972; Vol. 8, p 93. Zukauskas, A.; Makarvicius, V.; Shalncianskas, Mintis, Vilnius, Lithuania, 1968. Received for review March 27, 1986 Accepted February 24, 1987
A Modified UNIFAC Model. 1. Prediction of VLE, hE, and y m U l r i c h Weidlicht and Jiirgen Gmehling" Lehrstuhl f u r Technische Chemie B, Universitat Dortmund, 0-4600 Dortmund 50, FRG
A modified UNIFAC model (mod. UNIFAC) has been developed which differs from the original UNIFAC method (orig. UNIFAC) in that it has a different combinatorial part and that temperature-dependent group interaction parameters and different van der Waals quantities have been introduced. The parameters were fitted simultaneously to a large number of VLE, ym,and hE data by using the Dortmund Data Bank. Compared to orig. UNIFAC, the modified model gave a relative improvement of 73% for the calculation of activity coefficients a t infinite dilution, 23% for the calculation of consistent binary vapor-liquid equilibria, and 70% for the calculation of binary excess enthalpies. The pure prediction of ternary VLE data gave an improvement of 11% , that of ternary hE data an increase in exactness of ca. 78% (in each case compared to orig. UNIFAC). 1. Introduction The most successful methods presently used for calculation of activity coefficients in the liquid phase are the group contribution methods, in which the liquid phase is considered to be a mixture of structural groups. This has the great advantage that any system of technical interest can be calculated by using a relatively small number of parameters which describe the interactions between the structural groups. The best-known and most successful of the group contribution methods so far proposed is the UNIFAC (UNIQUAC functional group activity coefficients) model (Fredenslund et al., 1977). It has already been used successfully in many areas (Gmehling, 1982), e.g., (1)for calculating vapor-liquid equilibria (Fredenslund et al., 1977), (2) for calculating liquid-liquid equilibria (Magnussen et al., 1981), (3) for calculating solid-liquid equilibria (Gmehling et al., 1978), (4) for determining activities in polymer solutions (Oishi and Prausnitz, 1978; Present address: Huls AG, ZB FE-Zentrale/Verfahrenstechnik, D-4370 Marl, FRG.
0888-5885/87/2626-1372$01.50/0
Gottlieb and Herskowitz, 1981), (5) for determining vapor pressures of pure components (Jensen et al., 1981), (6) for determining the influence of solvent on reaction rate (Gmehling and Fellensiek, 1980; Lo and Paulaitis, 1981), ( 7 ) for determining flash points of solvent mixtures (Gmehling and Rasmussen, 1982), (8) for determining solubilities of gases (Nocon et al., 1983; Sander et al., 1983). Many comparisons made using experimental data for various combinations of substances have shown that the UNIFAC method is particularly well suited for calculating vapor-liquid equilibria. However, various publications (Kikic et al., 1980; Thomas and Eckert, 1984) have made clear that the results obtained for the calculation of activity coefficients at infinite dilution are in most cases unsatisfactory especially when systems with molecules very different in size are considered. However, the exact knowledge of this quantity is particularly important for separation techniques, since the number of plates can be particularly high for very dilute systems. The orig. UNIFAC model is also not able to calculate enthalpies of mixing and thus the temperature dependence of the Gibbs excess energy to the required degree of exactness. It was 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 1373 thus the purpose of the present work to modify the UNIFAC method in such a way that vapor-liquid equilibria, activity coefficients at infinite dilution, and enthalpies of mixing can be calculated sufficiently exactly using only one set of parameters. 2. Mod. UNIFAC The target which we set ourselves required the development of a modified form of the UNIFAC method (Weidlich, 1985). This required variation not only of the combinatorial part with respect to the form and size of the molecules but also of the residual part with regard to the description of the temperature dependence of the activity coefficients. Equations 1-3 show the relations for the combinatorial part in the orig. UNIFAC model. After
I
(3) some preliminary investigations we introduced an empirical 3/4-term,which was determined by simultaneously optimizing activity coefficients at infinite dilution for various alkane/alkane, alkane/alcohol, and alcohol/alcohol measurement series, into the combinatorial part. Thomas and Eckert (1984) obtained a similar result. Equation 1 was thus modified as
ri3/4
($/= 7rf14xj
(5)
The expressions for C#~I and Bi (see eq 2 and 3) remain unchanged. Temperature-dependent interaction parameters were introduced into the residual part. Equations 6 and 7 show the relationships for the group interaction parameters qnm in the orig. and mod. UNIFAC approach. The relative orig. UNIFAC: Qnm
exp(-anm/T)
(6)
mod. UNIFAC: qnm
= exp(-(a,,
+ bnmT+ cnmT2)/T)
(7)
van der Waals volumes and surfaces of the structural groups (Rkand Qkvalues) were not calculated from molecular parameters as in the orig. UNIFAC approach but fit together with interaction parameters (anm,b,, cnm)to the experimental values. In addition, special Rk and Qk values were introduced for the cyclic CH, and CH groups in order to permit a better description of systems containing cycloalkanes, which differ in some cases relatively strongly from the acyclic alkanes. Vapor-liquid equilibria for systems containing tertiary alcohols were not always satisfactorily described by UNIFAC (Weidlich et al., 1986). This is taken into account in the modified UNIFAC model by introducing separate van der Waals volumes and surfaces for the primary, secondary, and tertiary alcohol groups. Additional main groups are not introduced. The cyclic structure groups are
assigned to the main groups alkane (CH,) and the different alcohol groups to the main group alcohol (OH). A 6 X 6 parameter matrix was built up, which contains the following main groups: CH,, alkanes; C=C, alkenes; ACH, aromatics; ACCH,, substituted aromatics; OH, alcohols; CH,CO, ketones. Table I contains a survey of the fitted parameters and the substance combinations. Optimizations were carried out by using the Simplex-Nelder-Mead method (Nelder and Mead, 1965). The basis for the calculations was the Dortmund Data Bank (Gmehling and Onken, 1979; Gmehling, 1985), which at the time this work was carried out contained ca. 12 000 VLE and 4500 hE measurement series and 8000 ymvalues. At the moment, 18000 ymdata are stored. The majority of the vapor-liquid, mixing enthalpy and ymdata have been published in the DECHEMA Chemistry Data Series (Gmehling et al., 1977-1984; Christensen et al., 1984; Tiegs et al., 1986). 3. Results 3.1. Fitting the Mod. UNIFAC Interaction Parameters to Activity Coefficients at Infinite Dilution. Various preliminary investigations carried out prior to the fitting procedures and calculations to be described have shown that it is not possible to calculate activity coefficients at infinite dilution with a satisfactory degree of exactness when the parameters used are obtained only from vapor-liquid equilibrium data. The reason for this is that information on the behavior of the activity coefficient at extreme dilution is not available. The value of the activity coefficient may increase very rapidly when the concentration of the component considered lies between 0 and 5 mol %. However, in general there are either only very few data or none at all for this concentration range, so that large variations of the activity coefficient cannot be properly taken into account in the fitting procedure. On the other hand, it is of interest to investigate the degree of exactness with which vapor-liquid equilibria can be calculated when the parameters used are determined purely from ymdata. This becomes even more important when one realizes how rapidly and simply activity coefficients at infinite dilution can be measured by gas chromatography, for example. It thus appeared worthwhile to construct a parameter matrix by using the mod. UNIFAC method and a data base consisting solely of activity coefficients at infinite dilution. The basis for the parameter fitting procedure was formed by 791 measurement series with a total of 1773 values. The calculations yielded a mean percentage deviation of 5.3% , while that obtained with orig. UNIFAC was 21.1% (see Table 11, section 3.2). Vapor-liquid equilibria were subsequently calculated using only the interaction parameters of the mod. UNIFAC method fitted from ymdata. The results were again compared with those from the orig. UNIFAC method. Only those vapor-liquid equilibria from the Dortmund Data Bank were selected which passed both the consistency test of Van Ness et al. (1973) in the version suggested by Fredenslund et al. (1977) (the point test) and the Redlich-Kister area test (Redlich and Kister, 1948). Thus, almost only x , y, P, T data were used. Only in the cases of alkene/alcohol and alkene/ ketone systems were x , y, T data included which passed the Redlich-Kister area test. This was necessary because only very few complete consistent data sets are available for such system combinations. The calculations using mod. UNIFAC with 563 measurement series and 7690 values gave a mean absolute deviation of 0.0096 in the vapor-phase mole fraction, compared with 0.0105 for orig. UNIFAC (Table 111,section
1374 Ind. Eng. Chem. Res., Vol. 26, No. 7 , 1987 Table I. Fitted Parameters in Mod. UNIFAC Qk
Rk RCH3 RCH2 RCH
QCH~ QCH~ QCH
RC
Qc
RCH2,cyc
QcH~,~~~
RCH,cyc ROH.prim
QcH,~~ Q o H , ~ ~ ~ ~
ROH,eee ROH,tert
QOH,~.~
R C H ~ H RCH-CH
Q C H ~ H QCH=CH QCH* QCH=C
RCHp€ RCH-C
RC-c RACH RAC RACCH3
RACCH~ RACCH
interaction parameters ~ C H ~ ~ O H ~CH~IOH CCH2/OH
~ O H / C H ~ ~OHICH~ COH/CH,
systems alkanes/ alkanes alkanes/alcohols alcohols/alcohols
Q~~,telt
alkanes/alkenes alkenes/alkenes
"CH2IC-C bCH2/C==C
Qc-c alkanes/unsubst. aromatics unsubst. aromatics/unsubst. aromatics alkanes/subst. aromatics
QACH QAC QACCH~ QACCH:, QACCH
unsubst. aromatics/subst. aromatics subst. aromatics/subst. aromatics alkanes/ ketones ketones/ ketones alkenes/unsubst. aromatics alkenes/subst. aromatics
alkenes/alcohols alkenes/ ketones unsubst. aromatics/alcohols subst. aromatics/alcohols
~CH~CO/ACH
unsubst. aromatics/ ketones
~CH~COIACH CCH2CO/ACH
subst. aromatics/ ketones
aCH2CO/ACCH2
~CH~COIACCH~ CCH~COIACCH~ ~CH~COIOH
alcohols/ ketones
~CH~COIOH
Table 11. Deviation between Experimental and Calculated Activity Coefficients at Infinite Dilution mean % dev. measurement series values mod." mod.b orig. alkanes/alkanes 250 374 3.3 3.3 24.9 alkanes/alkenes 86 222 4.1 4.1 21.8 alkenes/alkenes alkanes/aromatics 93 228 4.2 4.3 10.7 aromatics/aromatics alkanes/ alcohols 88 234 8.3 8.7 28.6 alcohols/alcohols alkanes/ ketones 99 276 5.9 6.4 18.4 ketones/ ketones alkenes/aromatics 15 32 6.4 6.5 11.2 alkenes/alcohols 32 90 6.6 9.2 30.8 alkenes/ ketones 31 94 6.1 5.9 19.7 aromatics/alcohols 27 57 8.6 8.6 15.0 aromatics/ ketones 26 60 5.3 5.7 20.9 alcohols/ ketones 44 106 5.2 5.1 18.7 total 791 1773 5.3 5.6 21.1 Parameters from 7-data. *Parameters from 7- and VLE data.
3.2). This result is mainly due to an improved description of the alkane/alcohol and alcohol/ketone systems by mod.
UNIFAC: the introduction of a tertiary alcohol group is particularly advantageous. Systems containing tert-butyl alcohol are described much better by mod. than by orig. UNIFAC, while the description of systems containing primary and secondary alcohols is comparable. In some cases (alkene/alcohol, alkene/ ketone) the results obtained with mod. UNIFAC were worse than these from orig. UNIFAC: the reason for this is probably due to the structure of the data used in the fitting, since these consisted mainly of combinations of a molecule with a long carbon chain one with a short carbon chain. The VLE systems, however, contain only molecules with short chains. As a result of these calculations, we reach the following conclusions: when mod. UNIFAC parameters are fitted only to ymdata, vapor-liquid equilibria can be successfully calculated. The precondition is a broad spectrum of ym data; i.e., y m data should be available for systems containing only short or long and both short and long chain molecules. Using parameters of the mod. UNIFAC method fitted to ymdata, we then calculated the excess enthalpies of the various systems. The necessary thermodynamic relationships are collected in the Appendix section. The mean percentage deviation with respect to the size of the
Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 1375 Table 111. Deviation between Experimental and Calculated Vapor-Liquid Equilibria mean abs. dev of y measurement series values mod.a mod.b orip. alkanes/alkanes 36 408 0.0059 0.0048 0.0051 alkanes/alkenes 21 226 0.0048 0.0044 0.0056 alkenes/ alkenes 137 1959 0.0066 0.0046 0.0055 alkanes/aromatics aromatics/aromatics alkanes/alcohols 123 1769 0.0112 0.0095 0.0152 alcohols/alcohols alkanes/ ketones 56 769 0.0105 0.0110 0.0126 ketones/ ketones alkenes/aromatics 7 105 0.0060 0.0054 0.0054 11 alkenes/ alcohols 71 0.0344 0.0187 0.0200 alkenes/ ketones 5 47 0.0147 0.0122 0.0126 aromatics/alcohols 109 1455 0.0120 0.0094 0.0126 aromatics/ ketones 33 594 0.0078 0.0068 0.0071 alcohols/ ketones 25 287 0.0128 0.0110 0.0171 total 563 7690 0.0096 0.0079 0.0105 a
Parameters from 7-data.
Parameters from 7-and VLE data.
maximum value of the corresponding measurement series was (for 745 data sets and 12 360 values) 36.2% using mod. UNIFAC and 42.1% using orig. UNIFAC. The results show that not only a large data base but also different thermodynamic quantities should be used for fitting the required parameters. Only in those cases for which activity coefficients at infinite dilution are exactly measurable with a clear temperature dependence can these be successfully used as a basis for calculation of hE data (alkane/alcohol, alkane/ ketone). 3.2. Fitting the Mod. UNIFAC Interaction Parameters to Activity Coefficients at Infinite Dilution and to Vapor-Liquid Equilibrium Data. The quality of the calculation of vapor-liquid equilibria can be improved when, in addition to the 7-values, VLE data are included in the fitting of the interaction parameters of the modified UNIFAC method. The basis for this fitting is provided by 791 7"-measurement series with 1773 values and 563 thermodynamically consistent binary VLE measurement series with 7690 values. Table 11, in which the mean percentage deviations for the calculation of the 7" data using mod. UNIFAC (parameters from y" and VLE data) and orig. UNIFAC are collected, also shows that the quality of the calculation of 7-data is only slightly worse than that described in section 3.1. The mean percentage deviation was 5.6%. The vapor-liquid equilibrium data were described with a mean absolute deviation of the molar fractions in the vapor phase of 0.79 mol % . Table I11 shows the results compared with calculations using orig. UNIFAC and mod. UNIFAC (pa-
rameters from 7" data). Table I v contains the results of the excess enthalpy calculations. In summary, we can conclude that mod. UNIFAC gives extremely good results both for the calculation of activity coefficients at infinite dilution and for vapor-liquid equilibria when both ymand VLE data are included in the optimization process. However, the determination of enthalpies of mixing is unsatisfactory. 3.3. Fitting to Activity Coefficients at Infinite Dilution as well as Vapor-Liquid Equilibria and Excess Enthalpy Data. In this third variant, the van der Waals dimensions and the group interaction parameters were fitted to three different types of data: to activity coefficients at infinite dilution, vapor-liquid equilibria, and excess enthalpies, whereby in each case only binary data sets were considered. Table V contains a survey of the final van der Waals values of the structural groups in the mod. UNIFAC method. The new interaction parameter matrix is to be found in Table VI. Table VI1 shows the results of the y" calculations with mod. UNIFAC, taking into account the parameters obtained from y",VLE, and hE data. The mean percentage deviation for the 1773 values in 791 measurement series is 5.7% and is thus only slightly worse than those in sections 3.1 (5.3%) and 3.2 (5.6%). The results of the VLE calculations using the new parameters are collected in Table VIII. The total mean absolute deviation of the molar fraction in the vapor phase is 0.81 mol % (563 measurement series with 7690 values). In comparison, the calculations in sections 3.1 (parameters from 7-data) and 3.2 (parameters from ymand VLE data) gave mean absolute deviations of the vapor-phase concentrations of 0.96 and 0.79 mol %, respectively. The inclusion of hE data in the optimization lead to a considerable improvement in the calculation of this thermodynamic quantity. In 745 series, 12 360 values deviate from the experimental data on the average by 12.6% in relation to the maximum hE value of the corresponding measurement series. The results for the various substance combinations are collected in Table IX. Particularly good results were obtained for alkanes/ketones (4.5% 1, alcohols/ketones (7.3%), and alkanes/aromatics (7.4%). The largest deviations are shown by alkane/alkene systems with 21.9%, aromatic/aromatic systems with 32.1% , and aromatic/ketone systems with 39.8%. When the absolute deviations for alkane/alkene systems (25.1 J/mol) or aromatic/aromatic systems (14.1 J/mol) are considered, these are found to be only small. Both substance combinations exhibit only very low enthalpies of mixing, so that relatively small absolute deviations lead to large percentage deviations. Aromatics and ketones
Table IV. Deviation between Experimental and Calculated Excess Enthalpies mod. UNIFAC" values
AhE/hE,,, 70
AhE, Jim01
40 276
387 4434
135.0 51.3
205 90
4121 1452
8 80 20 26 745
measurement series alkanes/alkenes alkanes/aromatics aromatics/aromatics alkanes/alcohols alkanes/ ketones ketones/ ketones alkenes/aromatics aromatics/alcohols aromatics/ ketones alcohols/ ketones total "Parameters from
7-
mod. UNIFACb
AhE/hEmaa, 70
AhE, J/mol
153.0 123.0
110.0 46.7
17.8 14.9
128.0 95.1
85 1225 324 332
11.6 18.9 133.0 15.7
58.4 228.0 158.0 241.0
12360
36.2
data. bParameters from 7- data and VLE data.
orig. UNIFAC %
AhE, J/mol
127.0 81.5
38.1 69.9
47.7 334.0
14.8 14.2
105.0 88.3
14.7 38.2
110.0
15.2 22.9 238.0 8.1
71.9 264.0 258.0 115.0
48.3 26.6 87.7 42.5
238.0 338.0 99.0 648.0
35.6
AhE/hE,,,,
42.1
264.0
1376 Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987
tained in comparison with these for binary systems may result from the fact that all the measurement series, and not only consistent data sets, were used for the comparison. Calculation of the enthalpies of mixing with mod. UNIFAC gave a mean percentage deviation with respect to the maximum value of the system considered of 11.2% (orig. UNIFAC 50.2%). It should be noted that these ternary data were not included in the determination of the van der Waals values and group interaction parameters. Thus, the results described are based purely on a prediction carried out with the help of the parameters fitted from binary data.
Table V. Main Groups and Subgroups and the Corresponding van der Waals Quantities for the Mod. UNIFAC Method group 1 2 3 4 5 6
CH; CH
c:
c-CH~ C-CH
0.6325 0.6325 0.6325 0.6325 0.7136 0.3479
1.0608 0.7081 0.3554 0.0 0.8635 0.1071
1.2832 1.2832 1.2832 1.2832 0.3763 0.3763
1.7094 1.5315 1.3660 1.0432 0.4321 0.2113
7 8 9 10
CH,=CH CH=CH CH,=C CH=C
C=C
11 12
ACH AC
ACH
13 14 15
ACCHi
0.9100 0.9100 0.9100
0.9490 0.7962 0.3769
16 17 18
ACCH, ACCHZ ACCH OH, prim OH, sec OH, tert
OH
0.8927 0.8789 0.7826
19 20
CH,CO CH&O
CH&O
1.2302 1.0500 1.0500 1.7048 1.7048
4. Discussion The calculations were based on a total of 2099 series of measurements with 21 823 binary data and 47 series with 4935 ternary data. These were in detail
1.6700 1.5542
data
series
binary y m binary VLE binary h E ternary VLE ternary h E
791 563 745 102 45
values 1773
7690 12360 3457 1478
Only binary data sets were used in the optimization process. The mod. UNIFAC approach is, in contrast to the original model, able to describe activity coefficients at infinite dilution, vapor-liquid equilibria, and excess enthalpies well. The necessary precondition is that all three types of data are used for fitting the parameters. Table X gives an impression of the broad spectrum of data which (thanks to the presence of the Dortmund Data Bank) were available for the optimization process. The table contains all substance combinations and types of data (y", VLE, hE) and the temperature range covered by the data used. In addition, the numbers of carbon atoms in the shortest and longest molecular chains are given for each group of substances to give an idea of the size distribution of the molecules used for the calculations: these vary between C2 and CS6.The lowest temperature is -40 "C?the highest +180 "C. Calculations using the mod. UNIFAC method gave the following results, represented for the y mdata by the percentage, for the binary VLE data by the mean absolute deviation of the molar fractions in the vapor phase, and for the binary hE data by the mean percentage deviation
sometimes show exothermic and sometimes endothermic behavior on mixing. This is reproduced qualitatively by mod. UNIFAC. 3.4. Calculation of Vapor-Liquid Equilibria and Excess Enthalpies of Ternary Systems. The final parameter set obtained using y m ,VLE, and hE data was used to calculate ternary vapor-liquid equilibria and excess enthalpy data; these were compared with experimental values. The same calculations were also carried out by using the original UNIFAC method. The data base was again the Dortmund Data Bank, from which 102 VLE measurement series with 3457 values and 45 hF' measurement series with 1478 values were checked. According to the parameter matrix developed, the components belonged to the substance groups alkane, alkene, substituted and unsubstituted aromatic, alcohol, and ketone. By use of 3457 measured VLE values, the mod. UNIFAC method gave a mean absolute deviation of the mole fraction in the vapor phase of 0.0134, while that obtained with orig. UNIFAC was 0.0151. The slightly less exact results obTable VI. Mod. UNIFAC Interaction Parameter Matrix CHZ
3
ACH ACCH? OH CHzCO
c=c
4 5 6 bnln
1
CHZ C=C ACH ACCH, OH CHqCO
2 3 4
5 6 Cnm
1
2 3 4
5 6
CH2
c=c
ACH ACCH, OH CH,CO
-100.0 16.07 47.20 1606.0 199.0
2 175.1 0.0 96.74 329.9 101.1 -321.9
1
2
0.0 0.1256 -0.2998 0.3575 -4.746 -0.8709
-0.3252
1 0.0
anm
1 2
1
0.0 0.0 0.0 0.0
0.9181 X 0.0
2 0.0 0.0 0.0 0.0 0.0 0.0
4
7.339 -189.9 139.2 0.0 2559.0 -146.6
3 0.0933 0.4102
0.0
-0.4971 -0.7991 0.2541 0.5349
3
114.2 -55.89 0.0 -45.33 2032.0 -57.53
0.0
699.7
4 -0.4538 0.2671 -0.6500 0.0 -10.40 0.2419
0.0
0.4223 -4.917 1.212
3 0.0 0.0 0.0 0.0 0.3120 X -0.3715 X lo-'
5 2777.0 3663.0 2123.0 -318.9
4 0.0 0.0 0.0 0.0
0.9456 X lo-' 0.1133 X
6 433.6 633.6 146.2 1001.0 208.3 0.0
5
6
-4.674 -7.863 -3.804 8.000
0.1473 -0.7255 -1.237 -1.871 -0.6099 0.0
0.0
-1.102 5 0.1551 x lo-* 0.0
0.1746 X 0.2056 x 10-1 0.0 0.0
6 0.0 0.0 0.4237 X IO-* 0.2390 X 0.0 0.0
Ind. Eng. Chem. Res., Vol. 26, No. 7 , 1987 1377 Table VII. Deviation between Experimental and Calculated Activity Coefficients at Infinite Dilution Using Mod. UNIFAC (Parameters from Y, VLE, and h E Data) mean % dev. measurement values mod. orig. series alkanes / alkanes 250 374 3.2 24.9 4.6 22.3 alkanes/alkenes 73 189 alkenes/alkenes 13 33 4.7 19.3 3.8 10.7 221 alkanes/aromatics 88 7 7.1 13.6 aromatics/aromatics 5 alkanes/alcohols 82 216 9.0 30.5 alcohols/alcohols 6 18 2.5 6.6 alkanes/ ketones 89 5.8 18.8 247 ketones/ ketones 10 29 10.5 14.7 alkenes/aromatics 15 32 7.4 11.2 alkenes/alcohols 32 90 9.7 30.8 alkenes/ ketones 31 94 5.8 19.7 aromatics/alcohols 27 57 9.4 15.0 aromatics/ ketones 26 60 6.5 20.9 alcohols/ ketones 44 5.7 18.7 106 total 791 1773 5.7 21.1 Table VIII. Deviation between Experimental and Calculated Vapor-Liquid Equilibria Using Mod. UNIFAC (Parameters from y-, VLE, and h E Data) mean abs. dev. measurement of Y series values mod. orig. alkanes/alkanes 36 408 0.0047 0.0051 alkanes/alkenes 15 174 0.0049 0.0060 alkenes/alkenes 6 52 0.0021 0.0044 alkanes/aromatics 132 1915 0.0047 0.0055 aromatics/aromatics 5 44 0.0021 0.0042 alkanes/alcohols 95 1445 0.0103 0.0169 alcohols/alcohols 28 324 0.0085 0.0078 alkanes/ ketones 53 734 0.0104 0.0128 ketones/ ketones 3 35 0.0072 0.0081 alkenes/aromatics 7 105 0.0057 0.0054 alkenes/alcohols 11 71 0.0182 0.0200 alkenes/ ketones 5 47 0.0114 0.0126 aromatics/alcohols 109 1455 0.0103 0.0126 aromatics/ ketones 33 594 0.0063 0.0071 alcohols/ ketones 25 287 0.0119 0.0171 total 563 7690 0.0081 0.0105
with respect to the maximum hE value (results using orig. UNIFAC in parentheses): 7-
VLE
hE
5.7% 0.81 mol % 12.6%
(21.1%) (1.05m o l %) (42.1%)
In the case of the activity coefficients at infinite dilution, this represents an improvement of 73%, in the case of the vapor-liquid equilibria 23%, and for the excess enthalpies 70%.
Figures 1-4 give an impression of the exactness of the calculations. Figure 1 contains y-x diagrams for 16 different alkane/alcohol systems. Figure 2 contains the same number of diagrams for 16 different alkane/ketone systems. The values calculated by using mod. UNIFAC are represented by the solid line, the experimental values by the dots. The experimental data lie-with minimal deviations-on the calculated curves; thus, the azeotropic points are also extremely well described. Figures 3 and 4 contain hE-x diagrams for 16 different alkane/aromatic and alkane/ketone systems, respectively. The diagrams in Figures 3 and 4 contain not only the experimental valued and those calculated from mod. UNIFAC (which are represented by dots and solid lines) but also the data obtained from orig. UNIFAC (dashed line). The curves show that mod. UNIFAC gives much better results for enthalpies of mixing than does wig. UNIFAC. It should be noted that the same four group interaction ~ a r a m e t e r ~ - a ~ ~ ~, C/ H~~~C O, ~/ C~H~ ,~C, HCH,CO, bcH&o,cH-were used for the calculation of the y-x 6igure 2) and h%-x (Figure 4) diagrams of the alkane/ketone systems. Calculations of ternary vapor-liquid equilibria using mod. UNIFAC gave an improvement of 11%relative to orig. UNIFAC (mod. UNIFAC 1.34 mol 70,orig. UNIFAC 1.51 mol %). The determination of ternary excess enthalpies by means of mod. UNIFAC lead to an increase exa. tness of ca. 78% (mod. UNIFAC 11.2%, orig. UNIFAC 50 %). Part of the improvement is due to the introduction of separate structure groups for primary, secondary, and tertiary alcohols and of special CH2 and CH groups for cyclic compounds. These changes make themselves particularly felt for calculations involving systems containing tert-butyl alcohol. Another part of the improvementespecially for the calculation of enthalpies of mixing-is due to the introduction of temperature-dependent interaction parameters. The same model has also been used for the prediction of gas solubilities, liquid-liquid equilibria, retention data, and flash points (Tiegs et al., 1987). However, some shortcomings should also be mentioned. Mod. UNIFAC, like orig. UNIFAC, is not able to calculate excess enthalpies for alkane/alkane systems. The calculation of the activity coefficients of these systems is confined to the combinatorial part, since in alkane mixtures only one main group (CH,) is present. The residual part is thus equal to zero, since according to the definition there are no interactions between main groups of the same type (anm.= 0). The combinatorial part is however temperature-independent, so that it makes no contribution to the excess enthalpy. The excess enthalpies are thus also zero, independent of their composition: this does not agree with the experimental facts.
Table IX. Deviation between Experimental and Calculated Exce'ss Enthalpy Data Using Mod. UNIFAC (Parameters from y-, VLE, and h E Data) mod. UNIFAC orip. UNIFAC measurement .IhE/hE,,,, AhE, AhE/hE,,, AhE, % tJ /mol 70 ,J /mol series values alkanes/alkenes 21.9 25.1 40 387 38.1 47.7 alkanes/aromatics 210 3407 7.4 49.8 49.0 393.0 66 1027 aromatics/aromatics 32.1 14.1 139.0 135.0 205 4121 14.7 alkanes/alcohols 11.9 87.7 110.0 alkanes/ ketones 60.6 60 854 29.9 391.0 4.5 10.3 30 598 50.1 ketones/ ketones 7.9 82.6 36.9 8 85 238.0 48.3 alkenes/aromatics 8.5 80 1225 aromatics/alcohols 12.5 154.0 338.0 26.6 52.1 20 324 87.7 aromatics/ ketones 39.8 99.0 648.0 112.0 26 332 42.5 alcohols/ ketones 7.3 12.6 total 745 12360 42.1 0
1378 Ind. Eng. Chem. Res., Vol. 26, No. 7 , 1987 1.00
0,60 0, 60
0.L O
0.20 0.00 0.80 0,bO
0.LO 0, 20
0.00 0.80 0 60 0,LO
0.20 0.00
0 80 0 bo 0. LO
0.20
0.00
x1
-
x1
- - x1
x1
Figure 1. Experimental and (from mod. UNIFAC) calculated vapor-liquid equilibria for binary alkane/ alcohol systems. 1.00
0.00
Ob0 O.LO
0.20 0.03 0.80
0.60 0.60
0.20 0.00
o,eo 0.60 0,LO
0.20 Cyclonexanone 0.00
080 0.60 0.10
0.20 0.00
ow
- - - -
0.20 O,M Xl
0.60
om om
0.20 OM x1
0.60
o.ea
om
OM
0.20
x1
ow o.ao om
020 0.~0 050 0 . 0
1.00
x1
Figure 2. Experimental and (from mod. UNIFAC) calculated vapor-liquid equilibria for binary alkane/ketone systems.
Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 1379
,.. . .. Elhylb.nzene
I
T.25.0'C TOI".".
Cycloh.dane
I
T=25DC n-Wan. Toluene
H.lhylcycloh*xane Ethyl benzene
I
o-H.ptan. Xylene-11 .&I
800
OD0
020 0.40
0.60
0.80 0.W
0.20
Ob0
0.M) 0.M
OD0
I
Ti35.O.C Toluene n-Drcane
Oh0
0.60
n
I
n-Hrptonr Ethylbenzene
0.20
Ta250C Toluene
- H*xad.can.
I
xy1.ne-11 41
n-Drconr
0.20
0.00 0.00
0.40 Ob0
0.W
1.00
mod. UNIFAC, (- - -) orig. UNIFAC.
Figure 3. Experimental and calculated mixing enthalpies of binary alkane/aromatic systems: (-)
t
hE
T
hE
.
75
E
7
I
1 hE
T
hE 0.W
0.20 Ob0
x,-
0.60 0.80
0.00
020
0.40 050
x1
OEC
Om
-
0.20
xt
0,40 040
OB0
0.00
Figure 4. Experimental and calculated mixing enthalpies of binary alkane/ketone systems: (-)
Further shortcomings occur when systems containing two alcohols are to be treated, in particular systems such
0.20
0.40
0.60
0.80 I,W
x1
mod. UNIFAC, (- - -) orig. UNIFAC.
as propanol (l)/propanol(2),butanol (l)/butanol(2), and butanol (l)/tert-butyl alcohol (2). The introduction of
1380 Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 Table X. Temperature Ranges and Size Molecules kind of temp range, data “C alkanes / alkanes 20-115 YVLE 25-75 20-120 alkanes/alkenes 7alkenes/alkenes VLE 38-55 hE 7-
10-50 alkanes/aromatics aromatics/aromatics
10-100
VLE
hE 7-
20-180
7-120 alkanes/alcohols alcohols/alcohols
20-120
VLE
0-90
hE
7-65
7-
alkanes/ ketones ketones/ ketones
0-120
VLE
-35-95
hE
-40-45
7,-
alkenes/aromatics
VLE
0-55
hE Y-
25-50 alkenes/alcohols
alkenes/ ketones
aromatics/alcohols
20-108 25-110
VLE
hE 7-
0-90 35-75
VLE 7-
20-100 55-95
VLE 7-
20-75
10-60 aromatics/ ketones
20-155
VLE
25-60
hE
25-45
7-
alcohols/ ketones
20-100
VLE
25-55
hE
20-50
Distribution of no. of C atoms
alkanes: alkenes: alkanes: alkenes: alkanes: alkenes:
c5-c36
C5-CZo c4-cl4
c&e
c6-c12
c,&g
alkanes: aromatics: alkanes: aromatics: alkanes: aromatics:
c5-c36
alkanes: alcohols: alkanes: alcohols: alkanes: alcohols:
C5-C30 CZ-Cz2 C5-CI1 C&
alkanes: ketones: alkanes: ketones: alkanes: ketones:
c5-c36 c&,g
alkenes: aromatics: alkenes: aromatics: alkenes: aromatics:
C5-CZo Ce-Cg
Ce-ClO
increase the number of parameters further and so lessen the advantages of the UNIFAC group contribution method, which is that it requires only a small number of group interaction parameters. Nevertheless work is going on to improve this situation. It should be mentioned that a similar model was also developed in the group of Prof. Fredenslund (Larsen, 1986). To avoid duplicate work, it is the goal of the two groups to work together on the further development of a unified model.
Supplement A listing of a subroutine for the prediction of VLE, ym, hE,and cPEusing mod. UNIFAC may be obtained from the authors.
c5-cl4
ce-cg
C5-C22 Ce-c10
c&16
C2-CI4
C5-ClO C3-C7 c4-c16
C3-Cll
c647
CB-C7 ce-cs c6-c8
alkenes: alcohols: alkenes: alcohols:
C5-C18 CZ-Cz2 c6-cs Cz-C4
alkenes: ketones: alkenes: ketones: aromatics: alcohols: aromatics: alcohols: aromatics: alcohols: aromatics: ketones: aromatics: ketones: aromatics: ketones: alcohols: ketones: alcohols: ketones: alcohols: ketones:
C5-CZo C3-C19 C5-C7 C3-C4 ce-cs C2-C22 c&9
C2-C1, cs-cg C2-C13 CB-cB C3-clg CS-cB c3-c6
CB-Cs C3-cB C2-Czz C3-CI9 c2-c6
C3-C,
c&
Acknowledgment We thank Dr. U. Onken for his supporting interest in this work and “Deutsche Forschungsgemeinschaft und Arbeitagemeinschaft Industrieller Forschungsvereinigunen” for financial assistance.
Nomenclature unm= UNIFAC group interaction parameter between groups n and m b,, = UNIFAC group interaction parameter between groups n and m c, = UNIFAC group interaction parameter between groups n and m gE = molar Gibbs excess energy hE = molar excess enthalpy P = total pressure q i = relative van der Waals surface of component i Qk = relative van der Waals surface of structural group k R = gas contstant ri = relative van der Waals volume of component i Rk = relative van der Waals volume of structural group k T = temperature, K xi = mole fraction of component i in the liquid phase y i = mole fraction of component i in the vapor phase Greek Symbols rk = group activity coefficient of group k in the mixture r k ( i ) = group activity coefficient of group k in the pure substance y6,= activity coefficient of component i ukL = number of structural groups of type k in molecule i qnm = UNIFAC group interaction parameters between groups n and m (see eq 6 and 7) Indexes C = combinatorial part E = excess quantity i = component i R = residual part = at infinite dilution
-
Appendix: Thermodynamic Relationships for Determining Excess Enthalpies Using UNIFAC The Gibbs-Helmholtz equation
C3-C4
primary, secondary, and tertiary alcohol groups is not sufficient to improve the results compared to orig. UNIFAC. However, since the systems referred to (alkane/ alkane, alcohol/alcohol) have enthalpies of mixing which are very small, the disadvantages discussed above are not of great importance for the use of the method in technical chemistry. An improvement might be made by introducing a cyclic alkane main group and main groups for the different OH groups (primary, secondary, tertiary); however, this would
allows the calculation of the molar excess enthalpy, hE, with the help of a model for the Gibbs excess energy, gE. The following expressions apply for the UNIFAC model:
_yiR + In yic) RT - Cxi(ln i gE
In yiR = Cvk(i)(lnrk - In k
rk(i))
(A-2) 64-31
Ind. Eng. Chem. Res., Vol. 26, No. 7 , 1987 1381 Since the combinatorial part is temperature-independent, only the residual part is used for calculating the excess enthalpy.
The above statement also supplies to the expression
Literature Cited
rk is the group activity coefficient of group k in the mixture and rkcO is the group activity coefficient of group k in the pure substance. For the orig. and mod. UNIFAC models, the following relations then apply: orig. UNIFAC:
A similar expression applies for
The quantities are then referred to the pure substance and not to the mixture as in eq A-5. mod. UNIFAC:
Christensen, C.; Gmehling, J.; Rasmussen, P.; Weidlich, U. Heats of Mixing Data Collection; DECHEMA Chemistry Data Series; DECHEMA Frankfurt, 1984;Vol. 111, 2 parts. Fredenslund, A.; Gmehling, J.; Rasmussen, P. Vapor-Liquid Equilibria Using UNIFAC; Elsevier; Amsterdam, 1977. Gmehling, J. Habilitationsschrift;University of Dortmund: Dortmund, 1982. Gmehling, J. CODATA Bulletin 58 (Thermodynamic Databases); Pergamon: Oxford, 1985;p 56. Gmehling, J.; Anderson, T. F.; Prausnitz, J. M. Ind. Eng. Chem. Fundam. 1978,17,269. Gmehling, J.; Fellensiek, J. 2.Phys. Chem. NF 1980,122,251. Gmehling, J.; Onken, U. The Proceedings of the Sixth International CODATA Conference; Pergamon: Oxford, 1979;p 163. Gmehling, J.; Onken, U.; Arlt, W.; Weidlich, U.; Grenzheuser, P.; Kolbe, B. Vapor-Liquid Equilibrium Data Collection; DECHEMA Chemistry Data Series; DECHEMA: Frankfurt, 1977-1984; Vol. I, 13 parts. Gmehling, J.; h m u s s e n , P. Ind. Eng. Chem. Fundam. 1982,21,186. Gottlieb, M.; Herskowitz, M. Macromolecules 1981,14,1468. Jensen, T.; Fredenslund, A.; Rasmussen, P. Ind. Eng. Chem. Fundam. 1981,20,239. Kikic, I.; Alessi, P.; Rasmussen, P.; Fredenslund, A. Can. J . Chem. Eng. 1980,58,253. Larsen, B. L. Ph.D. Dissertation, Danmarks Tekniske Hajskole, Lyngby, 1986. Lo, H. S.; Paulaitis, M. E. AIChE J. 1981,27,842. Magnussen, T.; Rasmussen, P.; Fredenslund, A. Ind. Eng. Chem. Process Des. Dev. 1981,20,331. Nelder, J. A.; Mead, R. Comput. J. 1965, 7,308. Nocon, G.; Weidlich, U.; Gmehling, J.; Menke, J. Onken, U. Fluid Phase Equilib. 1983,13,381. Oishi, T.; Prausnitz, J. M. Ind. Eng. Chem. Process Des. Deu. 1978, 17,333. Redlich, 0.;Kister, A. T. Ind. Eng. Chem. 1948,40, 341. Sander, B.; Skjold-Jmgensen, S.; Rasmussen, P. Fluid Phase Equilib. 1983,11, 105. Thomas, E. R.; Eckert, C. A. Ind. Eng. Chem. Process Des. Deu. 1984,23, 194. Tiegs, D.; Gmehling, J.; Medina, A.; Soares, M.; Bastos, J.; Alessi, P.; Kikic, I. Activity Coefficients at Infinite Dilution; DECHEMA Chemistry Data Series; DECHEMA: Frankfurt, 1986;Vol. IX,2 parts. Tiegs, D.; Holderbaum, T.; Menke, J.; Gmehling, J., submitted for publication in Ind. Eng. Chem. Res. 1987. Van Ness, H. C.; Byer, S. M.; Gibbs, R. E. AIChE J . 1973,19,238. Weidlich, U. Ph.D. Dissertation, University of Dortmund, Dortmund, 1985. Weidlich, U.; Berg, J.; Gmehling, J. J . Chem. Eng. Data 1986,31, 313.
Received f o r reuiew April 1, 1986 Accepted March 19,1987
