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Prediction of ternary phase equilibria from binary data
-Thetmochimica Acta, 56 (1982) 43-57 Elsevier Scientific Publishing Company.
PREDICTION DATA
ISAMU
OF TERNARY
43 Amsterdam-Printed
in The Netherlands
PHASE
EQUILIBRIA
Kanazawa
Utlioersi[v,
FROM
BINARY
NAGATA
Department of Chemical lshikawa 920 (Jopan) (Received
8 December
Engineen’ng,
1-40-20.
Kodatswo,
Kamazuwcx,
1981)
ABSTRACT The two-parameter UNIQUAC equation is modified to give better results of vapor-liquid and liquid-liquid equilibria for a variety of binary systems. The proposed equation is easily extended to a multicomponent system without including any ternary (or multicomponent) parameters. The good capability of the equation in data reduction is shown by many illustrative examples for various kinds of strongly nonideal binary and ternary mixtures.
NOTATION
aij Bi.
gd P Pf
4i 4: ri R T AU-E” 11. Vk xi x* x!’ i, t
Yi 2
Extended UNIQUAC binary interaction parameter related to A,ji and 7ij Second virial coefficient for i-j interaction Excess molar Gibbs energy Total pressure Saturation pressure of pure component i Molecular-geometric area parameter for component i Molecular-interaction area parameter for component i Molecular volume parameter for component i Gas constant Absolute temperature Extended UNIQUAC binary interaction parameter Molar energy of mixing Molar liquid volume of pure component i Liquid-phase mole fraction of component i Liquid-phase mole fractions of component i in phase I and II Vapor-phase mole fraction of component i Lattick coordination number, here equal to 10
Greek letters Yi gi gij
rij
Activity coefficient of component i Area fraction of component i Local area fraction of,‘a molecule i about a central molecule j Extended UMQUAC binary parameter related to Al(ji
OO40-6031/82/tMOO-OOOO/SO2.75@ 1982 Elsevier Scientific Publishing Cornpan!
44
9i
+; 9;
Fugacity coefficient of component i Fugacity coefficient of pure component Segment fraction of component i
i at its saturation
pressure
INTRODUCTIOPU
The UNIQUAC equation, which was derived by Abrams and Prausnitz [l]. is particularly useful since it is applicable to a number of strongly nonideal liquid mixtures. This equation has two adjustable parameters per binary. To obtain better results with water or alcohols, Anderson and Prausnitz [2] have used the pure component molecular structural parameter q’. which is different from 4 for these components. in the residual term of the LJNIQUAC equation. Nagata and Katoh [3] have presented a modification of the UNIQUAC equation, called effective UNIQUAC. The effective UNIQUAC equation. like the UNIQUAC equation, usually requires an additional third parameter to correlate successfully ternary liquid-liquid equilibria. In this-work another modification of the UNIQUAC equation t0 predict simultaneously ternary vapor-liquid and liquid-liquid equilibria using binary parameters alone is proposed.
I3ASIC THERMODYNAM!C
RELATIONS
I present the following modified form of the two-parameter equation, called extended UNIQUAC. g E = gE (combinatorial)
UNIQUAC
+ gE (residual)
For a binary system gE (combinatoriai) RT g E (residual) RT
= x,ln
= -q+,ln(B,
+ &7;, _ _ _ _ > - qb,ln(
where Z is the coordination number and area fraction, 8, are given by
9, =
Xl41 “141 + x2q2
02 =
e2 + &)
equal to 10, and segment
fraction,
x242
x,q, +xX242
r, q and q’ are pure component molecular structural parameters. original UNIQUAC equation q ’ = q and in the effective UNIQUAC tion q’ = 1.
9,
(9 In the equa-
45
The extended UNIQUAC equation is easily derived if we assume that the energy of mixing, uE, for a binary mixture is given by lCE= q;x,e,,Au,,
+ q;x,e,zAu,,
(6)
and if we follow the procedure of Maurer and Prausnitz [4] used in the derivation of the three-parameter UNIQUAC equation. 4’ is a correction factor of interaction. The local area fraction is defined in terms of the area fraction, 0, and binary interaction parameter, Au. e,, = t32expl-Au21/RT)/[B,
+ &exp(-Au2,/RT)]
(7)
e 12= B,exp(--Au,,/RT)/[B,
+ B,exp( -Au,JRT)]
(8)
Values of q’ for water, nitromethane and alcohols have been empirically obtained to give a good fit to various systems containing these components: water, 0,96; nitromethane, 1.29; methanol, 0.99: ethanol, 0.92; propanols. 0.89; butanols, 0.88. For other components studied in this work I suggest 4 ’ = qoe2The two adjustable parameters, 112 and TV,! are related to binary interaction energies, AulZ and Au,,. 71X= exp( -Au,,/RT)
= exp( -a,,/T)
~~~= exp( -Au,,
= exp( -a?,
/RT)
(9)
/T)
(10)
The activity coefficients are given by the relation
where ni is the number of moles of component i, nT is the total number of moles (“r =Tl~i), and xi =ni/t~T. +‘I
lny,=ln$+l---
-he2
(
[(
4; z
5
92
1
_ie)
5 (
+2
lnF+l-7
q2 )
g” (combinatorial)
@I
11
‘(2)
- qlln(& + e2T2,)
(e2?&2 +2 )
i
qi I( 42 ii e, -Th,7,2 -I)
For a multicomponent gE and y as .follows. RT
(
ln$+l-F
)( ?$k-‘)
lny,=ln~+l-$-
+42e,
)
q1
(I,
i
(12)
- q;ln(o, + +,2)
32~2,
mixture the extended UNIQUAC
= z
-‘I]
-l)]
03)
equation provides
(14)
46 g ’ (residual)
(15)
RT 9.
Iny,=ln~+‘-~i
(
I
--q(l*(Be,71i)
4
)
+qiC
j
qj
(21
(
ln2+1-2 1
'j -4if:
I 1
(16)
'q
j
k
where segment Siri $Bsi= 2 sjrj
fraction.
$. and area fraction.
6. are given by (17)
j 6; =
$T;,
(18)
.I .I
.i
The vapor-liquid described by
equilibrium
relationship
for any component
i [S] is
Qi)‘i P = YiSi~~Pi” exp[ ~L( P - P:)/RT]
(1%
where J is the vapor-phase mole fraction. P is the total pressure, P” is the pure component vapor pressure, and uL is the pure component molar liquid volume. The fugacity coefficient. +. is calculated by
(20) The pure-component and cross virial coefficients method of Hayden and O’Connell [6]. The equation of multicomponent liquid-liquid phases 1 and II, for each component i, is (YiSi)’
= (yiX.i)”
are calculated equilibria
by the
between
two
(21)
RESULTS
The optimum parameters were found by satisfaction of a statistical criterion based on the maximum likelihood principle. Computer programs used in this work are similar to those described -in the monograph of Prausnitz et al. [S]. r, q, and other parameters required to estimate the second virial coefficients are listed in the same monograph. The value of q' is independent of binary data after a single_value is fixed by examining several binary systems. Table 1 lists a number of binary parameters. Figures l-3
47 TABLE
!
Binary interaction parameters for extended System(l)-(2)
No. of points
UNIQUAC
Temp. (“C)
equation
Energy parametks,
Ethanol-acetonitrile Ethanol-cyclohexane Tetrachloromethaneacetonitrile Tetrachloromethanecyclohexane Benzene-cyclohexane Acetone-acetonitrile Acetone-cyclohexane Ethanol-water
Acetonitrile-2-propanol Methyl acetate-cyclohexane Ethanol-benzene Methanol-tetrachloromethane Methyl acetate-methanol Methanol-benzene Benzene-nitromethane Benzene-furfural Benzene-2.2,~trimethylpentane Cyclohexane-2,2,4-tri niethylpentane Trichlororgethanebenzene. Methyl acetate-benzene Methyl acetate-kchloromethane
Variance of fit
Ref.
9.20 1.90 4.27 3.5 1 3.82 7.63 1.53
7 7 8 8 9 10 10
a21
a12
Acetonitrile-benzene Benzene-n-heptane Ethanol- n-octane
K
11 15 17 19 14 7 7
45 45 45 75 40 35 50
- 8.32 69.5 1 266.97 243.76 286.07 202.70 197.24
13
45
641.58
78.08
4.22
11
9 9 7 10 12 9 10 13 13 15
40 70 39.99 45 25 45 25 40 55 50
-66.31 27.50 - 16.01 -61.63 123.68 90.11 155.60 78.55 48.38 70.27
91.69 - 2.87 126.69 120.93 435.40 431.88 30.48 153.28 243.38 356.68
0.08 0.09 0.76 0.67 8.78 10.40 8.47 12.68 6.25 2.69
12 12 12 13 14 15 16 17 17 18
8 9 11 12
35 45 25 45
182.19 163.07 75.30 82.40
250.13 25 1.65 917.18 768.33
13.72 16.17 3.11 1.52
19 19 20 21
9 6 7 7 9 9 10 7
35 55 20 30 35 55 25 25
122.16 138.36 371.89 356.37 85.33 80.04 166.91 358.80
1161.86 1067.35 55.80 27.63 836.10 825.10 105.83 - 17.88
1.97 13.16 7.40 14.30 4.54 5.45 8.55 1.78
22 23 24 24 22 22 25 26
7 8
35 45
2.77 1.40
133.53 129.44
0.37 0.07
27 27
8
45
- 20.96
30.70
1.79
28
19 17
50 50
- 93.94 242.47
13.46 - 164.78
5.03 5.00
29 29
16
50
- 185.65
12.47
6.65
29
304.46 61.71 1228.39 1129.06 171.03 1306.77 1155.43
48
TABLE 1 (continued) No. of points
System( l)-(2)
Temp. (“0
Energy parameters, al?
Ttichloromethanemethanol Acetone- trichlor+ methane Acetone-methanol Water-methanol Methanol-ethanol Ethanol-toluene 2-Propanol-cyclohexane n-Hexane-ethanol 2-Propanol-benzene I -Propanol-benzene Benzene-I-butanol Nitromethane-benzene Nitromethane-tetra chloromethane Ethanol-2-butanone 2-Butanone-benzene Tetrachloromethanebenzene n-Hexane-nitroethane Acetonitrile-n-heptane Benzene-water Acetonitrile-cyclohexane
Methanol-cyclohexane Furfural-cyclohexane
Furfural-2,2,4-t& methylpentane Nitromethane-cyclohexane ’ S = Solubility
25
50
933.25
29
50 50 25
- 178.99
K
Variance of fit
Ref.
- 101.41
16.36
30
0.73 - 26.70
II.69
532.56 616.40 100.09 206.08
12.39 2.15 29.37 1.74 1.74 6.63 9.07 1.63 0.45 0.68 2.41
30 30 16 16 31 31 32 9 33 34 34 35
a21
3.5 10 11 10 10 9 16 I2 11 9 12
45 45 45 45
266.93 - 67.54 - 260.48 78.4’7 84.08 137.84 1324.64 122.05 96.05 543.70 69.5 1
12 12 10
45 25 25
132.06 30.76 - 177.36
542.52 297.37 328.99
5.66 2.18 1.96
35 20 20
8 13 Sa S
40 45 45 25
- 4.54 381.66 391.74
0.08 5.33
1447.40
12.68 159.44 959.88 906.28
36 37 7 38
25 25
375.11 373.36 363.20 347.73 36 1.42
93 1.03 909.36 876.18 1349.8 752.69
18 39 18 18 40
25
342.76
822.9 1
40
25
443.80
786.79
40
25 35 55 50 40
40 45 50
85.58 253.80 862.56 774.14 916.74 23 1.27
data.
present examples of typical sets of binary data which I have correlated. The original UNIQUAC equation predicts phase separation for the ethanol-noctane’system [2]. The extended UNIQUAC equation correlates this system with good accuracy (Fi g. 1) and works well for many types of mixtures including partially miscible ones. Table2 represents the predicted results of vapor-liquid equilibria for six representative. ternary systems. Calculated
380
340 m t E _ 300 z 2 : : 260
220
0.0
0.4 0.2 Mole fraction
.0.6 0.8 of component 1
1.0
0.2 Nolo
0.0
0.4 fraction
of
0.6 0.8 compcment 1
1.0
Fig. 1. Representation of vapor-liquid equilibria for (a) the tetrachloromethane( I ), Calculated: 0. acetonitrile(2) and (b) the ethanol( I)-tt-octane(2) systems at 45OC. experimental. Data of Brown and Smith [I If for tetra~h~oromethane-acetonit~~e; data of Boublikova and Lu [S] for ethanol-pr-octane.
I
,
I
(b)
1
~ 300 = B . e 260 : z L; 220
180 7
-. 0.0
0.2 Mole
0.4 fraction
0.6 0.8 of component 1
1.0
0.0
0.2 Mole
0.4 fraction
0.6 0.S of component 1
1.0
of vapor- liquid equilibria for (a) the methanol{ 1) Fig. 2. Representation tetrach~orome~~~2) and (b) the me~~ol( 1) -benzene(2) systems at 35’ C. t Calculated; 4#, experimental. Data of Scatchard et al. [22].
lb)
351
3oc
i50
50
I
5 3 1 0.0
0.3 0.6 0.2 0.6 Xolc fracticn of coqmncnt 1
1.0
1
,
I
0.2
0.4
,
0.6
i
1
1.0
0.9
b:olc fraction of component
1
equilibria for (a) the nitromcthane( I)Fig. 3. Representation of vapor- liquid . tctrachloromethane(2) and (b) the rr-hexane( I)-nitroethane(2) systems at 45°C. Data of Brown and Smith [35] for nitromethane8. experimental. Calculated: tetrachloromcthane; data of Edwards [37] for n-hcxane-nitroethnnc.
CYCLOHEXANE
4
FURFURAL
2,2,4-TRIMETHYLPENTANE
Fig. 4. Predicted ternary liquid-liquid equilibria for the furfural-cyclohexanc-2.2,4trimethylpcntane system at 25OC. e- -, Tie-line data of Henty et al. [41]; calculated. Concentrations are in mole fractions.
9
51
TABLE 2 Prediction
of ternary
System
vapor-liquid Temp. (“Cl
equilibria No. of points
Absolute Vapor compn (molt %)
Acetonitrilebenzenen-heptane Watermethanolethanol Ethanol2-butanone-
Ref.
Press. (mm Hg)
PXSS. (76)
3.6
1.24
7
1.3
1.88
16
I .o
0.90
20
1.4
0.20
23
3.9
0.92
29
85
I.57
30
0.48
45
51
0.40 0.43 0.42
25
37
25
33
55
8
50
66
50
30
benzene MethanoItetrachloromethanebenzene Methyl acetatetrichloromethanebenzene Acctonetrichloromethanemethanol
arith. mean dev.
0.95 0.94 0.32 0.24 0.36 0.25 0.15 0.13 0.32 0.29 0.32 0.95 1.11 1.05
BENZENE
FURNRAL
%%+TRIMETHYLPENTANE
Fig. 5. Predicted ternary liquid-liquid equilibria for the furfural-benzene-2,2,4-trimethylpen, calculated. tane system at 25OC e--, Tie-line data of Henty et al. [41]: Concentrations are in mole fractions.
52
EXANE
Fig. 6. Predicted temq liquid-liquid equilibria for the furfurzl-benzene-cyclohexane sys, calculated. Concentratem at 25°C. O- - -, Tie-line data of Henty et 21. [41]: tions are in mole fractions.
vapor-phase compositions and pressures agree well with experimental values. Especially excellent agreement is obtained for the methanoltetrachloromethane-benzene system, probably because the binary data of superior quality are adequately correlated with the extended UNIQUAC equation.
ETHANOL
ACETO
EXANE
Fig. 7. Predicted ternary liquid-liquid equilibria for the acetonitrile-ethanol-cyclohexane , calculated. system at 40°C. 9- -, Tie-line data of Nagata 2nd Katoh [IS]; Concentrations are in mole fractions.
53 i-PROPANOL
ACETO
EXANE
Fig. 8. Predicted ternary liquid-liquid equilibria for the acetonitrile-2-propanol-cyclohcsanc system at 5O’C. 8- -, Tie-line data of Nagata and Katoh [IS]: . calculated. Concentrations are in mole fractions.
For systems where two binaries are partially miscible and the third binary is entirely miscible, it is rather easy to predict the ternary equilibria as
pointed
out
by several
authors.
For
systems
where
only
one
binary
is
partially miscible, than vapor-liquid
liquid-liquid equilibrium calculation is more sensitive equilibrium calculation to small errors in the predicted
activity
values. Figures 4- 14 present
coefficient
several representative
exam-
ETHANOL
BENZENE
WATER
Fig. 9. Predicted ternary liquid-liquid equilibria For the benzene-ethanol-water system at Tie-line data of Bancroft and Hubard [42]; , calculated. Con25°C. Q- -‘-) centrations are in mole fractions.
54
NITROMETHANE
CYCLOHEXANE
Fig. 10. Predicted ternary liquid-Iiquid equilibria for the cyclohexanc-benzene-nitromethane systems at 25°C. Experimental data of Week and Hunt [43], 0 (solubility). 9, calculated. Concentrations are in mole fractions. (tie-line):
-
to indicate successful predictions of ternary liquid-liquid equilibria by using only binary parameters. Anderson and Prausnitz {2] have much improved ternary prediction of liquid-liquid equilibria for mixtures containing methanol from binary data with their modified UNIQUAC equation, and have used a calculation method which utilizes some ternary data in the
pies
MEWYL
MET
ACETATE
EXANE
for the methanol-methyl Fig. 11. Predicted ternary liquid-liquid equilibria cyclohexane system at 25% BP- - -, Tie-line data OF Sugi et al. [44]; -, Concentrations are in mole fractions.
acetatecalculated.
55 IETRACHLOROMETHANE
Fig. 12. Predicted ternary liquid-liquid cyclohexane system at 25T. 9--,
equilibria for the methanol-tetrachloromethaneTie-line data of Yasuda et al. [45];
calculated. Concentrationsare in mole fractions. parameter
estimation
to represent
,
well the ternary systems. In most cases
only a single tie-line is required. Inclusion of the single ternary tie-line into the method of data reduction gives satisfactory ternary representation but does so at a high price of some loss of accuracy in the representation of vapor-liquid equilibria for component binaries as shown by results for the acetonitriie-benzene-It-heptane system. On the contrary, the present method BENZENE
Fig. 13. +Aicted ternary liquid-liquid equilibria for the acetonitrile-benzene-rr-heptane Tie-Line data of Palmer and Smith [?‘I; system at 45T. @I- --, , calculated. Concentrations are in mole fractions.
56 TETRACHLOROMETHANE
ACETONITRILE
CYCLOHEXANE
Fig. 14. Predicted ternary liquid-liquid equilibria for the acetonitrile-tetrachloromethanecgclohexanc system at 45’C. @I#,- - -. Tie-line data of Nagata et al. [46]: calculated. Concentrations are in mole fractions.
.
can predict considerably well ternary liquid-liquid equilibria without loss of accuracy in the representation of the binary vapor-liquid equilibria. The extended UNIQUAC equation as well as other common models for the excess Gibbs energy cannot properly describe the flat region near the plait point.
REFERENCES I
D.S. Abrams
2 T.F. Anderson 3
4 5
6 7 8 9 10 11 12 13 14 !5
and J.M. Prausnitz. AIChE J.. 21 (1975) I 16. and J.M. Prausnitz. Ind. Eng. Chem.. Process
Des. Dev., 17 (1978) 552, 561. I. Nagata and K. Katoh, Fluid Phase Equilibria, 5 (198 1) 225. G. Maurcr and J.M. Prausnitz, Fluid Phase Equilibria, 2 (1978) 9 I. J.M. Prausnitz, T.F. Anderson, E.A. Grens, C.A. Eckert, R. Hsieh and J.P. O’Connell, Computer Calculations for Multicomponent Vapor-Liquid and Liquid-Liquid Equilibria, Prentice-Hall. Englewood Cliffs, i 980. J.G. Hayden anrt J.P. O’Connell, Ind. Eng. Chem., Process Des. Dev.. 14 (1975) 209. D.A. Palmer and B.D. Smith, J. Chem. Eng. Data, 17 (1972) 7 1. L. Boublikova and B.C.-Y. Lu. J. Appl. Chem. (London). 19 (1969) 89. H. Sugi and T. Katayama, J. Chem. Eng. Jpn., 11 (1978) 167. G. Scatchard and F.G. Satkicwicz. J. Am. Chcm. Sot.. 86 (1964) 130. I. Brown and F. Smith, Aust. J. Chem.. 7 (1954) 269. G. Scatchard, SE. Wood and J.M. Mochel, J. Am. Chem. Sot., 61 (1939) 3206. I. Erown and F. Smith, Aust. 3. Chem., 13 (I 960) 30. P.S. Put-i, J. Polak and J.A. Ructher, J. Chem. Eng. Data, 19 (1974) 87. A.N. Marinichev and M.P. Susarev, Zh. Prikl. Khim., 38 (1965) 378.
57 16 D.J. Hall, C.J. Mash and R.C. Pemberton, Natl. Phys. Lab. Rep. Chem., 95 (1979). 17 I. Mertl, Collect. Czech. Chem. Commun.. 37 (1972) 366. 18 I. Nagata and K. Katoh, Thermochim. Acta, 39 (1980) 45. 19 I. Nagata, T. Ohta, T. Takahashi and K. Gotoh, J. Chem. Eng. Jpn., 6 (1973) 129. 20 T. Ohta, J. Koyabu and I. Nagata, Fluid Phase Equilibria, 7 (1981) 65. 21 I. Brown and F. Smith, Aust. J. Chem., 7 (1954) 264. 22 G. Scatchard, S.E. Wood and J.M. Mochel. J. Am. Chem. Sot., 68 (1946) 1960. 23 G. Scatchard and L.B. Ticknor, J. Am. Chem. Sot., 74 (1952) 3724. 24 V. Bektiek, Collect. Czech. Chem. Commun., 33 (1968) 2608. 25 D.F. Saunders and A.J.B. Spaull, Z. Phys. Chem. N-F., 28 (1961) 332. 26 J. Kenny, Chem. Eng. Sci., 6 (1957) I 16. 27 S. Weissman and S.E. Wood, J. Chem. Phys., 32 (1960) 1153. 28 R. Battino, J. Phys. Chem., 70 (1966) 3408. 29 I. Nagata and H. Hayashida, J. Chem. Eng. Jpn., 3 (1970) 16 I. 30 W.H. Sevems, Jr., A. Sesonske, R.H. Perry and R.L. Pigford, AIChE J., 1 ( 1955) 401, 3 1 C.B. Kretschmer and R. Wiebe, J. Am. Chem. Sot., 7 1 ( 1949) 1793. 32 I. Nagata, T. Ohta and Y. Uchiyama, J. Chem. Eng. Data, 18 (1973) 54. 33 I. Brown, W. Fock and F. Smith, Aust. J. Chem., 9 (1956) 364. 34 I. Brown and F. Smith, Aust. J. Chem., 12 (1959) 407. 35 I. Brown and F. Smith, Aust. J. Chem., 8 (i955) 501. 36 G. Scatchard, S.E. Wood and J.M. Mochel, J. Am. Chem. Sot., 62 ( 1940) 7 12. 37 J.B. Edwards, Ph.D. Dissertation, Georgia Institute of Technology, 1962. 38 J. Polak and B.C.-Y. Lu, Can. J. Chem., 51 (1973) 4018. 39 I. Nagata, Y. Nakamiya, K. Katoh and J. Koyabu, Thermochim. Acta, 45 (1981) 153. 40 J.M. SBrensen and W. Arlt, Liquid-Liquid Equilibrium Data Collection, Binary Systems, DECHEMA Chem. Data Ser., Vol. V, Part 1, Frankfurt/M, 1979. 41 C.J.. Henty, W.J. McManamey and R.G.H. Prince, J. Appl. Chem. (London), 14 ( 1964) 148. 42 W.B. Bancroft and S.S. Hubard, J. Am. Chem. Sot., 64 (1942) 347. 43 HI. Week and H. Hunt, Ind. Eng. Chem., 46 (1954) 2521. 44 H. Sugi, T. Nitta. and T. Katayama, J. Chem. Eng. Jpn., 9 (1976) 12. 45 M. Yasuda, H. Kawade and T. Katayama, Kagaku Kogaku Ronbunshu, I (1975) 172. 46 I. Nagata, K. Katoh and J. Koyabu, Thermochim. Acta, 47 (1981) 225.
PREDICTION DATA
ISAMU
OF TERNARY
43 Amsterdam-Printed
in The Netherlands
PHASE
EQUILIBRIA
Kanazawa
Utlioersi[v,
FROM
BINARY
NAGATA
Department of Chemical lshikawa 920 (Jopan) (Received
8 December
Engineen’ng,
1-40-20.
Kodatswo,
Kamazuwcx,
1981)
ABSTRACT The two-parameter UNIQUAC equation is modified to give better results of vapor-liquid and liquid-liquid equilibria for a variety of binary systems. The proposed equation is easily extended to a multicomponent system without including any ternary (or multicomponent) parameters. The good capability of the equation in data reduction is shown by many illustrative examples for various kinds of strongly nonideal binary and ternary mixtures.
NOTATION
aij Bi.
gd P Pf
4i 4: ri R T AU-E” 11. Vk xi x* x!’ i, t
Yi 2
Extended UNIQUAC binary interaction parameter related to A,ji and 7ij Second virial coefficient for i-j interaction Excess molar Gibbs energy Total pressure Saturation pressure of pure component i Molecular-geometric area parameter for component i Molecular-interaction area parameter for component i Molecular volume parameter for component i Gas constant Absolute temperature Extended UNIQUAC binary interaction parameter Molar energy of mixing Molar liquid volume of pure component i Liquid-phase mole fraction of component i Liquid-phase mole fractions of component i in phase I and II Vapor-phase mole fraction of component i Lattick coordination number, here equal to 10
Greek letters Yi gi gij
rij
Activity coefficient of component i Area fraction of component i Local area fraction of,‘a molecule i about a central molecule j Extended UMQUAC binary parameter related to Al(ji
OO40-6031/82/tMOO-OOOO/SO2.75@ 1982 Elsevier Scientific Publishing Cornpan!
44
9i
+; 9;
Fugacity coefficient of component i Fugacity coefficient of pure component Segment fraction of component i
i at its saturation
pressure
INTRODUCTIOPU
The UNIQUAC equation, which was derived by Abrams and Prausnitz [l]. is particularly useful since it is applicable to a number of strongly nonideal liquid mixtures. This equation has two adjustable parameters per binary. To obtain better results with water or alcohols, Anderson and Prausnitz [2] have used the pure component molecular structural parameter q’. which is different from 4 for these components. in the residual term of the LJNIQUAC equation. Nagata and Katoh [3] have presented a modification of the UNIQUAC equation, called effective UNIQUAC. The effective UNIQUAC equation. like the UNIQUAC equation, usually requires an additional third parameter to correlate successfully ternary liquid-liquid equilibria. In this-work another modification of the UNIQUAC equation t0 predict simultaneously ternary vapor-liquid and liquid-liquid equilibria using binary parameters alone is proposed.
I3ASIC THERMODYNAM!C
RELATIONS
I present the following modified form of the two-parameter equation, called extended UNIQUAC. g E = gE (combinatorial)
UNIQUAC
+ gE (residual)
For a binary system gE (combinatoriai) RT g E (residual) RT
= x,ln
= -q+,ln(B,
+ &7;, _ _ _ _ > - qb,ln(
where Z is the coordination number and area fraction, 8, are given by
9, =
Xl41 “141 + x2q2
02 =
e2 + &)
equal to 10, and segment
fraction,
x242
x,q, +xX242
r, q and q’ are pure component molecular structural parameters. original UNIQUAC equation q ’ = q and in the effective UNIQUAC tion q’ = 1.
9,
(9 In the equa-
45
The extended UNIQUAC equation is easily derived if we assume that the energy of mixing, uE, for a binary mixture is given by lCE= q;x,e,,Au,,
+ q;x,e,zAu,,
(6)
and if we follow the procedure of Maurer and Prausnitz [4] used in the derivation of the three-parameter UNIQUAC equation. 4’ is a correction factor of interaction. The local area fraction is defined in terms of the area fraction, 0, and binary interaction parameter, Au. e,, = t32expl-Au21/RT)/[B,
+ &exp(-Au2,/RT)]
(7)
e 12= B,exp(--Au,,/RT)/[B,
+ B,exp( -Au,JRT)]
(8)
Values of q’ for water, nitromethane and alcohols have been empirically obtained to give a good fit to various systems containing these components: water, 0,96; nitromethane, 1.29; methanol, 0.99: ethanol, 0.92; propanols. 0.89; butanols, 0.88. For other components studied in this work I suggest 4 ’ = qoe2The two adjustable parameters, 112 and TV,! are related to binary interaction energies, AulZ and Au,,. 71X= exp( -Au,,/RT)
= exp( -a,,/T)
~~~= exp( -Au,,
= exp( -a?,
/RT)
(9)
/T)
(10)
The activity coefficients are given by the relation
where ni is the number of moles of component i, nT is the total number of moles (“r =Tl~i), and xi =ni/t~T. +‘I
lny,=ln$+l---
-he2
(
[(
4; z
5
92
1
_ie)
5 (
+2
lnF+l-7
q2 )
g” (combinatorial)
@I
11
‘(2)
- qlln(& + e2T2,)
(e2?&2 +2 )
i
qi I( 42 ii e, -Th,7,2 -I)
For a multicomponent gE and y as .follows. RT
(
ln$+l-F
)( ?$k-‘)
lny,=ln~+l-$-
+42e,
)
q1
(I,
i
(12)
- q;ln(o, + +,2)
32~2,
mixture the extended UNIQUAC
= z
-‘I]
-l)]
03)
equation provides
(14)
46 g ’ (residual)
(15)
RT 9.
Iny,=ln~+‘-~i
(
I
--q(l*(Be,71i)
4
)
+qiC
j
qj
(21
(
ln2+1-2 1
'j -4if:
I 1
(16)
'q
j
k
where segment Siri $Bsi= 2 sjrj
fraction.
$. and area fraction.
6. are given by (17)
j 6; =
$T;,
(18)
.I .I
.i
The vapor-liquid described by
equilibrium
relationship
for any component
i [S] is
Qi)‘i P = YiSi~~Pi” exp[ ~L( P - P:)/RT]
(1%
where J is the vapor-phase mole fraction. P is the total pressure, P” is the pure component vapor pressure, and uL is the pure component molar liquid volume. The fugacity coefficient. +. is calculated by
(20) The pure-component and cross virial coefficients method of Hayden and O’Connell [6]. The equation of multicomponent liquid-liquid phases 1 and II, for each component i, is (YiSi)’
= (yiX.i)”
are calculated equilibria
by the
between
two
(21)
RESULTS
The optimum parameters were found by satisfaction of a statistical criterion based on the maximum likelihood principle. Computer programs used in this work are similar to those described -in the monograph of Prausnitz et al. [S]. r, q, and other parameters required to estimate the second virial coefficients are listed in the same monograph. The value of q' is independent of binary data after a single_value is fixed by examining several binary systems. Table 1 lists a number of binary parameters. Figures l-3
47 TABLE
!
Binary interaction parameters for extended System(l)-(2)
No. of points
UNIQUAC
Temp. (“C)
equation
Energy parametks,
Ethanol-acetonitrile Ethanol-cyclohexane Tetrachloromethaneacetonitrile Tetrachloromethanecyclohexane Benzene-cyclohexane Acetone-acetonitrile Acetone-cyclohexane Ethanol-water
Acetonitrile-2-propanol Methyl acetate-cyclohexane Ethanol-benzene Methanol-tetrachloromethane Methyl acetate-methanol Methanol-benzene Benzene-nitromethane Benzene-furfural Benzene-2.2,~trimethylpentane Cyclohexane-2,2,4-tri niethylpentane Trichlororgethanebenzene. Methyl acetate-benzene Methyl acetate-kchloromethane
Variance of fit
Ref.
9.20 1.90 4.27 3.5 1 3.82 7.63 1.53
7 7 8 8 9 10 10
a21
a12
Acetonitrile-benzene Benzene-n-heptane Ethanol- n-octane
K
11 15 17 19 14 7 7
45 45 45 75 40 35 50
- 8.32 69.5 1 266.97 243.76 286.07 202.70 197.24
13
45
641.58
78.08
4.22
11
9 9 7 10 12 9 10 13 13 15
40 70 39.99 45 25 45 25 40 55 50
-66.31 27.50 - 16.01 -61.63 123.68 90.11 155.60 78.55 48.38 70.27
91.69 - 2.87 126.69 120.93 435.40 431.88 30.48 153.28 243.38 356.68
0.08 0.09 0.76 0.67 8.78 10.40 8.47 12.68 6.25 2.69
12 12 12 13 14 15 16 17 17 18
8 9 11 12
35 45 25 45
182.19 163.07 75.30 82.40
250.13 25 1.65 917.18 768.33
13.72 16.17 3.11 1.52
19 19 20 21
9 6 7 7 9 9 10 7
35 55 20 30 35 55 25 25
122.16 138.36 371.89 356.37 85.33 80.04 166.91 358.80
1161.86 1067.35 55.80 27.63 836.10 825.10 105.83 - 17.88
1.97 13.16 7.40 14.30 4.54 5.45 8.55 1.78
22 23 24 24 22 22 25 26
7 8
35 45
2.77 1.40
133.53 129.44
0.37 0.07
27 27
8
45
- 20.96
30.70
1.79
28
19 17
50 50
- 93.94 242.47
13.46 - 164.78
5.03 5.00
29 29
16
50
- 185.65
12.47
6.65
29
304.46 61.71 1228.39 1129.06 171.03 1306.77 1155.43
48
TABLE 1 (continued) No. of points
System( l)-(2)
Temp. (“0
Energy parameters, al?
Ttichloromethanemethanol Acetone- trichlor+ methane Acetone-methanol Water-methanol Methanol-ethanol Ethanol-toluene 2-Propanol-cyclohexane n-Hexane-ethanol 2-Propanol-benzene I -Propanol-benzene Benzene-I-butanol Nitromethane-benzene Nitromethane-tetra chloromethane Ethanol-2-butanone 2-Butanone-benzene Tetrachloromethanebenzene n-Hexane-nitroethane Acetonitrile-n-heptane Benzene-water Acetonitrile-cyclohexane
Methanol-cyclohexane Furfural-cyclohexane
Furfural-2,2,4-t& methylpentane Nitromethane-cyclohexane ’ S = Solubility
25
50
933.25
29
50 50 25
- 178.99
K
Variance of fit
Ref.
- 101.41
16.36
30
0.73 - 26.70
II.69
532.56 616.40 100.09 206.08
12.39 2.15 29.37 1.74 1.74 6.63 9.07 1.63 0.45 0.68 2.41
30 30 16 16 31 31 32 9 33 34 34 35
a21
3.5 10 11 10 10 9 16 I2 11 9 12
45 45 45 45
266.93 - 67.54 - 260.48 78.4’7 84.08 137.84 1324.64 122.05 96.05 543.70 69.5 1
12 12 10
45 25 25
132.06 30.76 - 177.36
542.52 297.37 328.99
5.66 2.18 1.96
35 20 20
8 13 Sa S
40 45 45 25
- 4.54 381.66 391.74
0.08 5.33
1447.40
12.68 159.44 959.88 906.28
36 37 7 38
25 25
375.11 373.36 363.20 347.73 36 1.42
93 1.03 909.36 876.18 1349.8 752.69
18 39 18 18 40
25
342.76
822.9 1
40
25
443.80
786.79
40
25 35 55 50 40
40 45 50
85.58 253.80 862.56 774.14 916.74 23 1.27
data.
present examples of typical sets of binary data which I have correlated. The original UNIQUAC equation predicts phase separation for the ethanol-noctane’system [2]. The extended UNIQUAC equation correlates this system with good accuracy (Fi g. 1) and works well for many types of mixtures including partially miscible ones. Table2 represents the predicted results of vapor-liquid equilibria for six representative. ternary systems. Calculated
380
340 m t E _ 300 z 2 : : 260
220
0.0
0.4 0.2 Mole fraction
.0.6 0.8 of component 1
1.0
0.2 Nolo
0.0
0.4 fraction
of
0.6 0.8 compcment 1
1.0
Fig. 1. Representation of vapor-liquid equilibria for (a) the tetrachloromethane( I ), Calculated: 0. acetonitrile(2) and (b) the ethanol( I)-tt-octane(2) systems at 45OC. experimental. Data of Brown and Smith [I If for tetra~h~oromethane-acetonit~~e; data of Boublikova and Lu [S] for ethanol-pr-octane.
I
,
I
(b)
1
~ 300 = B . e 260 : z L; 220
180 7
-. 0.0
0.2 Mole
0.4 fraction
0.6 0.8 of component 1
1.0
0.0
0.2 Mole
0.4 fraction
0.6 0.S of component 1
1.0
of vapor- liquid equilibria for (a) the methanol{ 1) Fig. 2. Representation tetrach~orome~~~2) and (b) the me~~ol( 1) -benzene(2) systems at 35’ C. t Calculated; 4#, experimental. Data of Scatchard et al. [22].
lb)
351
3oc
i50
50
I
5 3 1 0.0
0.3 0.6 0.2 0.6 Xolc fracticn of coqmncnt 1
1.0
1
,
I
0.2
0.4
,
0.6
i
1
1.0
0.9
b:olc fraction of component
1
equilibria for (a) the nitromcthane( I)Fig. 3. Representation of vapor- liquid . tctrachloromethane(2) and (b) the rr-hexane( I)-nitroethane(2) systems at 45°C. Data of Brown and Smith [35] for nitromethane8. experimental. Calculated: tetrachloromcthane; data of Edwards [37] for n-hcxane-nitroethnnc.
CYCLOHEXANE
4
FURFURAL
2,2,4-TRIMETHYLPENTANE
Fig. 4. Predicted ternary liquid-liquid equilibria for the furfural-cyclohexanc-2.2,4trimethylpcntane system at 25OC. e- -, Tie-line data of Henty et al. [41]; calculated. Concentrations are in mole fractions.
9
51
TABLE 2 Prediction
of ternary
System
vapor-liquid Temp. (“Cl
equilibria No. of points
Absolute Vapor compn (molt %)
Acetonitrilebenzenen-heptane Watermethanolethanol Ethanol2-butanone-
Ref.
Press. (mm Hg)
PXSS. (76)
3.6
1.24
7
1.3
1.88
16
I .o
0.90
20
1.4
0.20
23
3.9
0.92
29
85
I.57
30
0.48
45
51
0.40 0.43 0.42
25
37
25
33
55
8
50
66
50
30
benzene MethanoItetrachloromethanebenzene Methyl acetatetrichloromethanebenzene Acctonetrichloromethanemethanol
arith. mean dev.
0.95 0.94 0.32 0.24 0.36 0.25 0.15 0.13 0.32 0.29 0.32 0.95 1.11 1.05
BENZENE
FURNRAL
%%+TRIMETHYLPENTANE
Fig. 5. Predicted ternary liquid-liquid equilibria for the furfural-benzene-2,2,4-trimethylpen, calculated. tane system at 25OC e--, Tie-line data of Henty et al. [41]: Concentrations are in mole fractions.
52
EXANE
Fig. 6. Predicted temq liquid-liquid equilibria for the furfurzl-benzene-cyclohexane sys, calculated. Concentratem at 25°C. O- - -, Tie-line data of Henty et 21. [41]: tions are in mole fractions.
vapor-phase compositions and pressures agree well with experimental values. Especially excellent agreement is obtained for the methanoltetrachloromethane-benzene system, probably because the binary data of superior quality are adequately correlated with the extended UNIQUAC equation.
ETHANOL
ACETO
EXANE
Fig. 7. Predicted ternary liquid-liquid equilibria for the acetonitrile-ethanol-cyclohexane , calculated. system at 40°C. 9- -, Tie-line data of Nagata 2nd Katoh [IS]; Concentrations are in mole fractions.
53 i-PROPANOL
ACETO
EXANE
Fig. 8. Predicted ternary liquid-liquid equilibria for the acetonitrile-2-propanol-cyclohcsanc system at 5O’C. 8- -, Tie-line data of Nagata and Katoh [IS]: . calculated. Concentrations are in mole fractions.
For systems where two binaries are partially miscible and the third binary is entirely miscible, it is rather easy to predict the ternary equilibria as
pointed
out
by several
authors.
For
systems
where
only
one
binary
is
partially miscible, than vapor-liquid
liquid-liquid equilibrium calculation is more sensitive equilibrium calculation to small errors in the predicted
activity
values. Figures 4- 14 present
coefficient
several representative
exam-
ETHANOL
BENZENE
WATER
Fig. 9. Predicted ternary liquid-liquid equilibria For the benzene-ethanol-water system at Tie-line data of Bancroft and Hubard [42]; , calculated. Con25°C. Q- -‘-) centrations are in mole fractions.
54
NITROMETHANE
CYCLOHEXANE
Fig. 10. Predicted ternary liquid-Iiquid equilibria for the cyclohexanc-benzene-nitromethane systems at 25°C. Experimental data of Week and Hunt [43], 0 (solubility). 9, calculated. Concentrations are in mole fractions. (tie-line):
-
to indicate successful predictions of ternary liquid-liquid equilibria by using only binary parameters. Anderson and Prausnitz {2] have much improved ternary prediction of liquid-liquid equilibria for mixtures containing methanol from binary data with their modified UNIQUAC equation, and have used a calculation method which utilizes some ternary data in the
pies
MEWYL
MET
ACETATE
EXANE
for the methanol-methyl Fig. 11. Predicted ternary liquid-liquid equilibria cyclohexane system at 25% BP- - -, Tie-line data OF Sugi et al. [44]; -, Concentrations are in mole fractions.
acetatecalculated.
55 IETRACHLOROMETHANE
Fig. 12. Predicted ternary liquid-liquid cyclohexane system at 25T. 9--,
equilibria for the methanol-tetrachloromethaneTie-line data of Yasuda et al. [45];
calculated. Concentrationsare in mole fractions. parameter
estimation
to represent
,
well the ternary systems. In most cases
only a single tie-line is required. Inclusion of the single ternary tie-line into the method of data reduction gives satisfactory ternary representation but does so at a high price of some loss of accuracy in the representation of vapor-liquid equilibria for component binaries as shown by results for the acetonitriie-benzene-It-heptane system. On the contrary, the present method BENZENE
Fig. 13. +Aicted ternary liquid-liquid equilibria for the acetonitrile-benzene-rr-heptane Tie-Line data of Palmer and Smith [?‘I; system at 45T. @I- --, , calculated. Concentrations are in mole fractions.
56 TETRACHLOROMETHANE
ACETONITRILE
CYCLOHEXANE
Fig. 14. Predicted ternary liquid-liquid equilibria for the acetonitrile-tetrachloromethanecgclohexanc system at 45’C. @I#,- - -. Tie-line data of Nagata et al. [46]: calculated. Concentrations are in mole fractions.
.
can predict considerably well ternary liquid-liquid equilibria without loss of accuracy in the representation of the binary vapor-liquid equilibria. The extended UNIQUAC equation as well as other common models for the excess Gibbs energy cannot properly describe the flat region near the plait point.
REFERENCES I
D.S. Abrams
2 T.F. Anderson 3
4 5
6 7 8 9 10 11 12 13 14 !5
and J.M. Prausnitz. AIChE J.. 21 (1975) I 16. and J.M. Prausnitz. Ind. Eng. Chem.. Process
Des. Dev., 17 (1978) 552, 561. I. Nagata and K. Katoh, Fluid Phase Equilibria, 5 (198 1) 225. G. Maurcr and J.M. Prausnitz, Fluid Phase Equilibria, 2 (1978) 9 I. J.M. Prausnitz, T.F. Anderson, E.A. Grens, C.A. Eckert, R. Hsieh and J.P. O’Connell, Computer Calculations for Multicomponent Vapor-Liquid and Liquid-Liquid Equilibria, Prentice-Hall. Englewood Cliffs, i 980. J.G. Hayden anrt J.P. O’Connell, Ind. Eng. Chem., Process Des. Dev.. 14 (1975) 209. D.A. Palmer and B.D. Smith, J. Chem. Eng. Data, 17 (1972) 7 1. L. Boublikova and B.C.-Y. Lu. J. Appl. Chem. (London). 19 (1969) 89. H. Sugi and T. Katayama, J. Chem. Eng. Jpn., 11 (1978) 167. G. Scatchard and F.G. Satkicwicz. J. Am. Chcm. Sot.. 86 (1964) 130. I. Brown and F. Smith, Aust. J. Chem.. 7 (1954) 269. G. Scatchard, SE. Wood and J.M. Mochel, J. Am. Chem. Sot., 61 (1939) 3206. I. Erown and F. Smith, Aust. 3. Chem., 13 (I 960) 30. P.S. Put-i, J. Polak and J.A. Ructher, J. Chem. Eng. Data, 19 (1974) 87. A.N. Marinichev and M.P. Susarev, Zh. Prikl. Khim., 38 (1965) 378.
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