fone plasticized membranes, but with them, permeation of the extraction agent (plasticizer) was apparently very small because the sulfones are relatively impermeable in vinylidene fluoride. Although the sulfone was not detected in the permeate, membrane deterioration with time would be expected because of a slow removal of the plasticizer. However, with the solvent membrane system the amount of extraction agent which permeates is significant. As a result, an additional separation of the permeate would be required although the amount of solvent in the product would probably be less than with liquid-liquid extraction. A detailed economic analysis would be required to determine whether or not this scheme would have any advantage over liquid-liquid extraction, however.
Literature Cited Crown Zellerbach Corp., Camas, Wash. 98607, "Dimethyl Sulfoxide," Technical Bulletin, page 19, 1966. McCandless, F. P., Ind. Eng. Chem., Process Des. Develop., 12, 354 (1973). Seibel, D. R . . McCandless, F. P., lnd. Eng. Chem., Process Des. Deve!op., 13, 76 ( 1 9 7 4 ) .
Department of Chemical Engineering Montana State Uniuersity Bozeman, Montana 59715
F. P. McCandless* David P. Alzheimer R. Bruce Hartman
Received for reuiezc August 13, 1973 Accepted January 25, 1974
Comparing Fugacity Coefficient Estimating Methods for Vapor-Liquid Equilibrium Data Reduction
The ability of two fugacity coefficient estimating methods is shown by using the Wilson equation in vapor-liquid equilibrium calculations for 30 binary and 6 ternary systems.
The Wilson equation (Wilson, 1964) for the excess Gibbs free energy of liquid mixtures has been discussed frequently to show a n accurate method for the correlation and prediction of vapor-liquid equilibria in completely miscible solutions (Cukor and Prausnitz, 1969; Eckert, et al., 1965; Hankinson, et al., 1972; Holmes and Van Winkle, 1970; Hudson and Van Winkle, 1970; Larson and Tassios, 1972; Nagata, 1973; Nagata and Ohta, 1969; Neretnieks, 1968; Orye and Prausnitz, 1965; Prausnitz, et d., 1967; Schreiber and Eckert, 1971; Tassios, 1971). In such calculations some authors assumed vapor phase ideality (Holmes and Van Winkle, 1970; Hudson and Van Winkle, 1970; Nagata and Ohta, 1969) and many other investigators took into consideration vapor phase nonideality. Recently, Nagata demonstrated that the vapor phase nonideality assumption usually improves multicomponent prediction accuracy obtained by the ideal vapor phase assumption. Neretnieks and Hankinson and others adopted the Redlich-Kwong equation of state to calculate the gasphase fugacity coefficients. Many investigators used the virial equation terminated after the second virial coefficient terms and the empirical correlation of O'Connell and Prausnitz (1967) to estimate the second virial coefficients. The virial equation method is easily extended to multicomponent systems, but it is not applicable to vapor mixtures involving strongly associated substances such as acids. To remove this disadvantage, Nothnagel, et al. (1973), presented a generalized method for estimating fugacity coefficients for a wide variety of mixtures including polar and strongly hydrogen-bonded components. It is our purpose to show here t h a t by using the Wilson equation for liquid phase nonideality and the method of Nothnagel and others for vapor phase nonideality a n improved representation of vapor-liquid equilibria can be established in comparison with results obtained by using the virial equation.
Binary Data Reduction The difference in capability of the two methods, the virial equation and the chemical theory of vapor imperfec312
Ind. Eng. Chern.. Process Des. Develop.,Vol. 13, No. 3, 1974
tions, was studied to fit binary data. Each set of the Wilson parameters was determined by a nonlinear leastsquare fitting program which minimizes the sum of squares of deviations in vapor mole fraction plus the sum of squares of relative deviation in pressure for all data points. Detailed calculation procedure was similar to the technique described by Prausnitz and others (1967). The chemical theory leads to give a good representation of the data for the ethyl alcohol-water system, better than that obtained by the virial equation. A similar trend is observed for the isopropyl alcohol-water system as shown in Table I, where the results of tests on 30 binary systems are listed. Nothnagel and others suggested that their correlation is probably less accurate than the virial equation for nonpolar components (typically hydrocarbons) based on the corresponding state theory, and t h a t it is probably more reliable for polar and associating vapors than the virial equation. Table I shows that the suggestion of Nothnagel and others is valid for alcohol-water systems only. However, for nonpolar components (benzene-cyclohexane and carbon tetrachloride-benzene) and polar associating vapors (acetone-methyl alcohol, ethyl alcohol-benzene, and methyl acetate-methyl alcohol, etc.) the least-square total pressure and vapor mole fraction fit described above was relatively insensitive to the fugacity coefficient estimating methods studied here. For vapor mixtures including carboxylic acids, only the chemical theory may be used and the virial equation must not be used, because the extent of acid dimerization is considerably large. Calculated results for the water-acetic acid system show that the smallest deviations of calculated values in relative pressure and vapor mole fraction from experimental data were found for the smoothed data of Sebastiani and Lacquaniti (1967).
Ternary Systems Ternary vapor-liquid equilibria are calculated from only binary data by following the methods of Prausnitz and others (1967). For comparison here, bubble pressure calculations were made for six ternary systems. Table I1
Table I. Comparison of Correlation of Binary Vapor-Liquid Equilibrium Data
Root-mean-square deviations Relative pressure
Pressure, mm ( X 1000) ( X 1000) Hg
Wilson parameters (cal/mol)
No.
6 7 8
9 10 11'
12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27 28 29 30
System, 1-2 Acetone-acetic acidc Acetone-acetic acidd Acetone-acetic acidd Acetoneacetic acidd Acetone-isopropyl alcohole Acetone-methyl alcohole Acetone-water1 Benzene-cyclohexane0 Benzene-cyclohexane* Benzene-isopropyl alcohol% Carbon tetrachloride-benzene7 Carbon tetrachloride-benzene j Cyclohexane-isopropyl alcoholk Diethyl etheracetic acidd Diethyl etheracetic acidd Diethyl etheracetic acidd Ethyl alcoholbenzene0 Ethyl alcoholcyclohexane0 Ethyl alcohol-isopropyl alcoholi Ethyl alcoholwater' Isopropyl alcohol-wateri Methyl acetatebenzenem Methyl acetatemethyl alcohol" Methyl alcoholbenzene" Methyl alcoholbenzene0 Methyl alcoholcarbon tetrachloridep Methyl alcohol-isopropyl alcohole Water-acetic acidq Water-acetic acid' Water-acetic acid'
Temp or press.
No.of data points
760 mm 30°C 55 "C
9 7 7 7 14
500
55OC
28
760 mm 760 mm
II b
Ia
(xi? - xI1)
- A??)
(xl? -
Vapor mole fraction
~~-
ill) (A?,, - A?.)
I
I1
I
I1
I1
0
580 2186 1709 1216 405
100
11
8
- 126
546
- 64
542
3
5
7
6
2
3
10 5
383 170
1472 80
373 126
1502 132
5 1
4 2
8 4
7 3
4 1
3 1
760 mm
12
149
135
186
92
2
3
2
2
2
2
760 mm
19
121
1056
202
1001
8
5
5
5
6
4
70°C
8
- 190
323
- 187
319
3
3
3
3
2
2
4OoC
8
- 291
490
- 291
490
3
3
4
4
1
1
760 mm
12
220
1500
255
1550
12
11 11
12
9
8
5O0C 40OC
14
I
- 200 - 834 - 799 - 704
29 13
8
16
18
13
21 9
1
10 3 2 1
0
5
4
596.4 mm
7
545
10
34
16
20
487.9 mm
7
490
40
25
16
12
389.7 mm
7
450
50
32
20
12
760 mm
5
1485
70
1370
170
5
6
8
6
4
5
760 mm
5
1956
319
1977
380
6
8
4
7
5
6
760 mm
10
- 60
20
- 60
106
6
3 1 0
9
5
3
760 mm
19
442
901
393
952
19
11
9
7
14
8
760 mm
18
776
1222
760
1255
18
14 12
11 14
11
760 mm
12
670
- 320
506
- 210
9
10
10
4
8
760 mm
14
- 63
731
30
710
10
5
760 mm
18
1740
100
1703
199
1 2 1 0 9
55OC
9
1827
141
1820
210
5
55OC
6
2292
156
2300
200
10
11 11
10
6
7
55°C
20
48
11
0
170
1 1 1 0 6
6
4
3
760 mm 760 mm 760 mm
19 10 8
1000
- 450
11
831 1031
-215 - 534
7 13
6
8
7
8
1
0 8
1
7 9
0
3
16 13 9
4 7 3
8
5 10
I = Fugacity coefficients were obtained by the physical theory proposed by O'Connell and Prausnitz. I1 = Fugacity coefficients were calculated by the chemical theory of vapor imperfections of Nothnagel, et a/. Data of Othmer (1943). Data of Meehan and Murphy (1965). e Data of Freshwater and Pike (1967). 1 Data of Ochi and Kojima (1971).0 Data of Morachevskii and Zharov (1963). Data of Nagata (1962b). Data of Nagata (1964a). Data of Scatchard, et al. (1940). Data of Nagata (1964b). 1 Data of Kojima, et al. (1968). 'n Data of Nagata (1962a). Data of Nagata (1969). Data of Scatchard, et a/. (1946). 7' Data of Scatchard, et al. (1952). Data of Brown and Ewald (1950). Data of Sebastiani and Lacquaniti (1967). ' Data of York and Holmes (1942). )L
shows results. The chemical theory is superior to the virial equation, giving minor improvement in deviations from the experimental vapor mole fractions for the methyl alcohol-isopropyl alcohol-water and acetone-methyl alcoholisopropyl alcohol systems. Again, it is shown t h a t the use of the chemical theory of vapor imperfections does not in-
troduce appreciable improvement in the ternary prediction of vapor-liquid equilibria for the remaining systems. In conclusion, the combining use of the chemical theory of vapor imperfections and the Wilson equation with two parameters for each binary pair can give more improved representation of data for binary alcohol-water systems Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 3,1974
313
Table 11. Calculations for Ternary Systems Based on Binary Data Deviation in vapor mole fractiona ( X 1000)
System
Temp or press.
Acetone (1) Methyl alcohol (2) Isopropyl alcohol (3)
55 "C
Benzene (1) Cyclohexene (2) Isopropyl alcohol (3)
760mm
Ethyl alcohol (1) Benzene (2) Cyclohexene (3)
760mm
Ethyl alcohol (1) Isopropyl alcohol (2) Water (3)
760mm
Methyl acetate (1) Methyl alcohol (2) Benzene (3)
760mm
Methyl alcohol (1) Carbon tetrachloride (2) Benzene (3)
Naof data points
27
70
19
53
18
Arithmetic
8
Rootmean square
~
~
I d
116
I
I1
Relativeb ( X 1000) - -I I1 I I1
(Freshwater and Pike, 1967) -2 0 7 5 8 7 -5 -6 10 9 13 11 7 6 9 7 11 8 (Nagata, 1964) -3 -3 4 4 4 4 1 1 5 5 7 7 2 2 4 6 6 7 (Morachevskii and Zharov, 1963) 5 6 12 1 2 18 18 -3 -4 4 5 6 6 -2 - 2 11 12 17 17 (Kojima, et al., 1968) -1 -1 12 9 14 11 6 4 12 9 15 12 -5 -3 8 7 10 9 (Nagata and Ohta, 1971) 1 1 4 3 5 4 1 0 5 4 6 6
-7
-9f
3 4
5 27
0 4 5
(1)-(2) (1)-(3) (2)-(3)
10 13
7
7
10
9
5
6 8 9
0 7 9
4 6
9 12 -6
-3
0
-5
6
6"
-3 9 12
6
Binary No. of combina- binary tion system (1)-(2) (1)-(3) (2)-(3)
0 9 13
-2
7
9
(1)-(2) (1)-(3) (2)-(3)
17
(1)-(2) (1)-(3) (2)-(3)
19 20 21
18 8
8
6 7
6 6
(1)-(2) (1)-(3) (2)-(3)
23 22 24
6 6 6
7 7 7
4 4 4
(1)-(2) (1)-(3) (2)-(3)
26 25 11,12
11 11 11
3
-4' 4u
0 5 7
5
4
-3 6 7
4 6 8
(Scatchard and Ticknor, 1952) 6 5 6 5 7 6 -6 -6 6 6 12 1 2 0 1 7 6 11 11
3
(mm Hg) I I1
100 14"
7
-1
Absolutec
12 14
8 9
-2 55'C
Deviation in pressure
Absolute arithmetic
-3
= (Pc%~ -c Pexpt~)/Pc,,,tl. ,~ AP = Pcnle,l - PexPti. I = Fugacity coefficients were obtained a AY = 3ic1lcd - ye\,,t1. A P r e ~ by the physical theory proposed by O'Connell and Prausnitz. e I1 = Fugacity coefficients were calculated by the chemical theory of vapor imperfections of Nothnagel, et al. f Arithmetic. 0 Absolute arithmetic. Root-mean square.
than that obtained by the virial equation. Another advantage of the chemical theory may be applicable to vapor mixtures involving carboxylic acids for which the virial equation gives erroneous results. Both methods give nearly similar data reduction for the other binary systems studied here. Calculations are straightforward and extended to ternary systems, with results similar to those found for the binary vapor-liquid equilibria. Acknowledgment The authors are grateful to the Data Processing Centers, Kyoto and Kanazawa Universities, for the use of their facilities. Nomenclature X = pair interaction energy in Wilson equation, cal/mol Literature Cited Brown, I . , Ewald. A . H.. Aust. J. Sci. Res., Ser. A, 3, 306 (1950) Cukor. P. M . , Prausnitz, J. M . , Chem. Eng. Symp. Ser., No. 32, 3, 88 (1969). Eckert, C. A . , Prausnitz, J. M., Orye, R. V., O'Connell. J. P., AlChE Ind. Chem. Eng. Symp. Ser., 1 , 75 (1965) Freshwater, D. C , Pike. K. A . , J. Chem. Eng. Data, 12, 179 (1967) Hankinson, R. W., Langfitt, E. D., Tassios, D. P., Can. J , Chem. Eng., 50, 511 (1972). Holmes. M. J., Van Winkle, M . , Ind. Eng. Chem., 6 2 ( 1 ) , 21 (1970). Hudson. J. W., Van Winkle, M., lnd. Eng. Chem., Process Des. Develop., 9, 466 (1970). Kojima, K . , Ochi, K., Nakazawa. Y . , Kagaku Kogaku, 32,441 (1968) Larson. C. D., Tassios. D. P., lnd. Eng. Chem., Process Des. Develop., 11, 35 (1972)
314
Ind. Eng. Chem., Process Des. Develop.,Vol. 13, No. 3 , 1974
Meehan, G. F , Murphy, N. F., Chem. Eng. Sci., 20, 757 (1965) Morachevskii, A . G . , Zharov, V T.. Zh. Prikl. Khim., 36, 2771 (1963) Nagata, I,, Can. J. Chem. Eng., 42, 82 (1964a). Nagata, I , Mem. Fac. Tech., Kanazawa Univ., 3, 1 (1964b) Nagata, I., J. Chem. Eng. Data, 7 , 360 (1962a). Nagata, I . ,J . Chem. Eng. Data, 7 , 461 (1962b). Nagata, I . ,J. Chem. Eng. Data, 14. 418 (1969) Nagata, I., J. Chem. Eng. Jap., 6, 18 (1973). Nagata, I., Ohta, T., J , Chem. Eng. Data, 16, 164 (1971). Nagata, I., Ohta, T.. Kagaku Kogaku, 33, 263 (1969) Neretnieks, I . , Ind. Eng. Chem., Process Des. Develop., 7 , 335 (1968) Nothnagel, K. H.. Abrams. D.S., Prausnitz, J. M . , Ind. Eng. Chem., Process Des. Develop., 1 2 , 25 (1973). Ochi. K . , Kojima, K., Kagaku Kogaku, 35, 583 (1971) O'Connell. J. P.. Prausnitz, J. M . , Ind. Eng. Chem.. Process Des. Develop.. 6, 245 (1967). Orye, R . V., Prausnitz, J. M., lnd. Eng. Chem., 5 7 ( 5 ) , 18 (1965). Othmer, D. F . , lnd. Eng. Chem., 35, 614 (1943) Prausnitz. J M.. Eckert. C A , . Orve. R. V.. O'Connell. J. P.. "Cornouter Ciculations for Multicomponeit Vapor-Liquid Equilibria," PrenticeHall, Englewood Cliffs, N . J., 1967. Scatchard. G., Ticknor, L B., J. Amer. Chem. SOC.,74, 3724 (1952). Scatchard, G., Wood, S. E., Mochel. J. M . , J. Amer. Chem. SOC., 62, 712 (1940). Scatchard, G.. Wood, S. E., Mochel, J. M., J. Amer. Chem. Soc., 68, 1960 ( 1946). Schreiber. L. E.. Eckert, C. A , , Ind. Enq. Chem., Process Des. Develop., 10,572 (1971). Sebastiani, E., Lacquaniti, L., Chem. Eng. Sci., 22, 1155 (1967) Tassios, D. P.,AIChEJ., 17, 1367 (1971). Wilson, G. M . . J. Amer. Chem. SOC.,86, 127 (1964) York, R.. Holmes, R . C., Ind. Eng. Chem., 34, 345 (1942).
Department of Chemical Engineering Kanazawa University Kanazawa, 920, Japan
Isamu Nagata* Shuhei Yasuda
Received f o r review December 28, 1973 Accepted March 6,1974
