Comparing Fugacity Coefficient Estimating Methods for Vapor-Liquid

Comparing Fugacity Coefficient Estimating Methods for Vapor-Liquid Equilibrium Data Reduction. Isamu Nagata, Shuhei Yasuda. Ind. Eng. Chem. ... Publis...
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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