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Correlation of ternary liquid-liquid equilibrium data and prediction of quaternary liquid-liquid equilibrium data by means of the uniquac model
121
Thermochimica Acta, 140 (1989) 121-138 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
CORRELATION OF TERNARY LIQUID-LIQUID EQUILIBRIUM DATA AND PREDICTION OF QUATERNARY LIQUID-LIQUID EQUILIBRIUM DATA BY MEANS OF THE UNIQUAC MODEL
ISAMU NAGATA * and YUKIMASA
USUI
Department of Chemical Engineering and Division of Physical Sciences, Kanazawa Kodatsuno 2, Kanazawa 920 (Japan)
University,
(Received 22 March 1988)
ABSTRACT The UNIQUAC model is modified to include ternary parameters. Correction terms reduce to zero if a ternary system degenerates to a binary system. The proposed model greatly improves the correlation of ternary liquid-liquid equilibria for many systems. Predicted results for quatemary liquid-liquid equilibria confirm the promising ability of the proposed method.
INTRODUCTION
It is well known that commonly used local composition models for the excess Gibbs free energy can predict ternary (or multicomponent) vapour-liquid equilibria of non-electrolyte solutions from binary parameters alone; however, larger solubility envelopes than those observed experimentally are calculated in the prediction of type I ternary liquid-liquid equilibria. To obtain a satisfactory representation of such ternary liquid-liquid equilibria the binary parameters need to be adjusted using ternary tie-line data and this causes some loss of accuracy in the representation of the binary vapour-liquid equilibria [l]. Fuchs et al. [2] have proposed a method for calculating ternary vapour-liquid-liquid equilibria of acetonitrile-nheptane-benzene. Cha and Prausnitz [3] have presented a procedure which maintains thermodynamic consistency: the usual ternary expression for the excess Gibbs energy is multiplied by a composition-dependent correction factor C such that C = 1 whenever the ternary system degenerates to a binary system. This method gives a good representation of ternary vapour-liquid-liquid equilibria. Our preliminary study showed that this method is limited to ternary systems and does not work for quaternary liquid-liquid equilibrium calculations.
* To whom correspondence should be addressed. 0040-6031/89/$03.50
0 1989 Elsevier Science Publishers B.V.
122
We propose another thermodynamic method which can be applied to systems containing more than three components. The systems are studied at low or moderate pressures. In order to evaluate binary and ternary parameters in a newly modified UNIQUAC model, the following experimental values are necessary: vapour-liquid equilibrium data for completely miscible binary systems; mutual solubility data for partially miscible systems; tie-line data for ternary systems.
SOLUTION MODEL
According to the UNIQUAC model [l] the excess molar Gibbs free energy gE is given by the sum of the combinatorial and residual terms g E=
&mbinatorial
+
gE.sidual
0)
We retain the original form for g&binatoria, and correct the original expression of gEsidualby adding ternary empirical correction terms. The correction terms reduce to zero whenever the ternary system degenerates to a binary system.
where the coordination number 2 is set equal to 10, the segment fraction Oi, the area fractions 19,and 0: and the binary parameters 7ji are given by
0,
=
’
4ixi
(5)
Cqjxj i
(6)
Tkj are ternary parameters to be derived from ternary tie-line data; 5ki = rkji for i Z j f k, 7iii = riji = 7ijj = 5,; = 0 and aij ( = Auij/R) are binary energy parameters.
123
For a ternary mixture eqn. (3) is expressed by
eir,, + O;T,, + e,lft$~~) - qix, ln( eiT13 + eir,, + elT,, + e;e;T,23)
- q;x2 ln( 0i-ri2 +
(8)
Equation (8) is derived in the same way as used by Maurer and Prausnitz [4]. The excess energy of mixing uE for the ternary mixture can be given by UE = x1q1’81,Au,, + x2q;e,‘,Au,, +x14;e;iAu3,
+ xJq;e;,Au,,
+ x24;e;ZAu,,
+xi4;e;,,(Au,i
+ At+)
++4;ek(A+
+ Au,,)
+ +4;e;,Au,,
+ x24X,,(Au12
+ A%) (9)
where the local surface fractions O& and (&i are defined as
(11)
and the other t$; and B,ii parameters are similarly defined. $i are ternary coefficients. An expression for the excess molar Helmholtz energy is obtained using the relation
d(aE/T)= d(l/T)
(14
E
’
Integrating eqn. (12) from l/T, $
= i2Ed(
f
) + constant
to l/T, we obtain
of integration
(13)
At low pressures uE = gE and as l/T, + 0 we take the equation of Guggenheim [5] for athermal mixtures of molecules of various size and shape as a boundary condition of eqn. (12). Finally we obtain eqns. (2) and (8) by putting 7jki = GjkiTirki. The activity coefficient yi of component i is derived from
where nT is the sum of the number of moles of component
i. Substituting
124
eqns. (2) and (8) into eqn. component i in the mixture.
(14), we obtain
the
activity
coefficient
of
(15)
I
711 - e;e;r231+ e712+ K- e;e?h32 -4; 0,)3 2 3
+/13 + ce;- fl;e;bi23 3 3 C e,!T3 + e;e;r123
06)
j
In y2 is derived by cyclic advancement of the subscripts, i.e. 1 + 2, 2 + 3 and 3 + 1 and In y3 is expressed similarly. In a quaternary mixture ( gE/RT),,id”al and (In yl)residual are given by
4 C
eiql+ e;e;r231 + e;e;r241 + e;e;r341
i
eiT2+ e;e;r,32 + e;e;r,42 + e;e;r342
- q;x2 In i i j
- q;x, In
4 C l i
- q;x4In
4 C i i
4 (InYI
)residual
=
I
e;?3+ e;e;r,23 + e;e;r,43 + e;eir2,, I
eiT4+ e;eiT12, + e;e;r,34 + e;e.jr2,,
(17)
125
+
CALCULATED
0; [ 713
+
(8;
-
o;o;)
7123 +
toi
-
@x)
?43
-
e;eb2431
4
RESULTS
Correlation of ternary liquid-liquid
equilibria
Table 1 shows values of the pure component structural constants r, q and q’. The UNIQUAC binary parameters were obtained from binary experimental vapour-liquid and mutual solubility data. The fugacity coefficients for mixtures including acetic acid were calculated using a chemical theory of vapour imperfections and those for most other mixtures were obtained by the truncated volume-explicit form of the virial equation [6]. Pure and cross
TABLE
1
Pure component
molecular
structural
constants
a
Component
r
4
4’
Acetic acid Acetone Acetonitrile Benzene n-Butanol 2-Butanone n-Butyl acetate Chloroform Cyclohexane Ethanol Ethyl acetate n-Heptane Methanol Methyl acetate Methylcyclohexane Toluene Water
2.30 2.57 1.87 3.19 3.45 3.25 4.83 2.70 3.97 2.11 3.48 5.17 1.43 2.80 4.64 3.92 0.92
2.04 2.34 1.72 2.40 3.05 2.88 4.20 2.34 3.01 1.97 3.12 4.40 1.43 2.58 3.55 2.97 1.40
2.04 2.34 1.72 2.40 0.88 2.88 4.20 2.34 3.01 0.92 3.12 4.40 0.96 2.58 3.55 2.97 1.00
a Taken from ref. 6.
18 8 13 7 31 12 10 12 1 1 1 14
1 12 11 7 15 17 9 10 24 12
VL VL VL VL VL VL VL VL MS MS MS VL
MS VL VL VL VL VL VL VL VL VL
Acetic acid- n-butanol Acetic acid- n-butyl acetate Acetic acid-chloroform Acetone-acetic acid Acetone-chloroform Acetone-cyclohexane Acetonitrile-acetone Acetonitrile-benzene Acetonitrile-cyclohexane Acetonitrile-cyclohexane Acetonitrile- n-heptane Acetonitrile-methyl acetate Acetonitrile-methylcyclohexane Acetonitrile-toluene Benzene-acetone Benzene-cyclohexane Benzene- n-heptane Benzene-methyl acetate Benzene-toluene n-Butanol- n-butyl acetate n-Butanol-chloroform n-Butanol-ethanol
Number of data points
Type a
results of phase equilibrium
System (l-2)
,TABLE 2 Binary calculated
25 25 45 40 45 50 82-105 50-54 62-115 80-115
116-120 118-122 65-106 40 35 25 45 45 25 45 45 50
Temp. (“C)
data reduction
45.16 - 22.88 174.00 - 11.20 19.07 - 143.88 - 60.03 - 187.14 - 268.29 - 197.11
546.68 - 146.29 - 98.44 522.62 - 140.77 32.35 119.29 - 51.54 74.49 46.41 23.71 - 99.26 566.26 255.70 - 108.79 70.13 31.35 203.46 69.93 516.85 785.66 364.39
- 296.30 356.52 346.43 - 277.64 28.38 181.66 - 88.43 247.13 556.70 516.57 545.79 186.16
a21
(K)
1112
= c c ’
c
’
’
’
’
parameters
(K)
UNIQUAC
0.27 0.97 1.52 0.55
0.50
0.44
1.44 0.42
0.64
8.39
0.02 0.01 0.10 0.04
0.00
0.02
0.00 0.01
0.02
0.48
6T (K)
Root-mean-square SP (Torr)
0.2 1.1 1.5 0.7
0.1
0.1
0.2 0.2
0.7
1.5
1.4 9.8 5.3 5.0
13.9 1.5
8.7
7.6
SY (~10~)
deviations 6x (X103)
0.31 0.84 0.72 1.77 5.70 0.56 15.97 12.58 5.10
0.46
10.54 228.43 385.19 13.46 9.17 22.55 0.66 9.74
Variance of fit b
WI 1211 WI 1231 [I71 1241 1251 1261 1271 WI
WY [ill WI 1131 P41 v51 WI 1171 WI WI 1171 [I91
Reference
1 1 14 14 1 1 1 15 7 1 7 1 8
13 10 14 10 10 11 10 10 13 1 1
MS MS VL VL MS MS MS VL VL MS VL MS VL
VL VL VL VL VL VL VL VL VL MS MS 25 25 40 35 25 25 80 20 25 40 25
20 25 80-111 55 20 25 25 35 45 25 35 25 35 56.75 50.60 489.63 486.03 594.91 578.34 935.54 982.17 - 58.04 391.90 1369.41 1350.4 97.91
0.3 1.3 0.2 0.0
0.0 0.0
0.5
2.1 0.5
1.5 0.8
= F/( number
0.01 0.00 0.01 0.00
0.38 0.67 0.90 0.13
0.06
1.68 0.00 0.00
0.00 0.04
2.20 1.32
0.04 0.13
0.13 0.06
2.07 1.82
of freedom)
- 35.76 12.23 594.60 ’ 1086.98 164.24 32.75 36.24 555.01 c 636.17 ’ 618.91 1453.2
927.37 958.78 - 215.61 - 146.56 - 57.25 - 49.48 106.22 - 203.29 32.55 1250.4 - 90.10 11.03 59.68
of degrees
40.55 36.24 - 148.29 - 134.05 49.20 - 20.72 12.23 - 258.51 - 128.35 58.58 341.45
a VL. vapour-liquid equilibria; MS, mutual solubilities. L L, Variance of fit = (sum of squared, weighted residuals)/(number ’ Taken from ref. 6.
n-Butanol-water n-Butanol-water 2-Butanone- n-butanol 2-Butanone-ethanol 2-Butanone-water 2-Butanone-water n-Butyl acetate-water Chloroform-ethanol Chloroform-toluene Chloroform-water Cyclohexane-ethanol Cyclohexane-methanol Cyclohexane-methyl acetate Cyclohexane-methylcyclohexane Cyclohexane-toluene Ethanol-ethyl acetate Ethanol-toluene Ethanol-water Methanol-ethanol Toluene-methylcyclohexane Water-acetic acid Water-acetone Water-ethyl acetate Water-toluene 0.01 0.00 3.09 1.47 7.22 3.80 0.00 1.18 22.07
17.66
5.77
16.93 4.29
17.56 14.01
of data points-number
2.9 5.7 4.5
0.2
8.8
6.9 2.1
4.6 7.6
of parameters).
1391 [401 1411 [421 [431 1431 WI 1451 [461 [411 1301
[301 1331 [341 [351 1301 [361 1371 1381
1321 v91
[301 1311
1291
results
deviation
between
the experimental
Acetonitrile-acetone-cyclohexane Acetonitrile-benzene-cyclohexane Acetonitrile-benzene-cyclohexane Acetonitrile-benzenen-heptane Acetonitrile-methyl acetate-cyclohexane Acetonitrile-toluene-cyclohexane Acetonitrile-toluene-methylcyclohexane Methanol-ethanol-cyclohexane Water-acetic acid- n-butanol Water-acetic acid-n-butyl acetate Water-acetic acid-chloroform Water-acetone-chloroform Water-acetone-chloroform Water- n-butanol- n-butyl acetate Water- n-butanol-chloroform Water-2-butanonen-butanol Water-2-butanonen-butanol Water-chloroform-toluene Water-ethanoln-butanol Water-ethanol-2-butanone Water-ethanol-chloroform Water-ethanol-ethyl acetate Water-ethanol-toluene
System (l-2-3)
a Root-mean-square
I I I I I I I I I I I I II II II II II I I I I I
I
Type
Ternary calculated
TABLE 3
-
-
.-
-
equilibrium
compositions.
0.1205 0.2128 0.1621 0.1643 - 0.2527 0.3239 0.2270 - 0.0825 0.4190 - 0.4998 - 2.8468 - 0.3923 0.1260 - 0.2778 0.0851 0.3802 0.2040 -0.0119 1.1175 0.8726 0.5837 0.6051 - 0.2516
r123
parameters
0.0073 0.2752 0.0551 0.0742 0.1011 0.2100 0.2040 0.0253 -0.8117 - 0.6487 - 0.4392 0.7391 0.1382 0.6041 0.1800 1.8098 2.6930 - 0.0163 0.5560 7.5817 0.4615 1.5703 1.1673
r132
ternary
liquid-liquid
0.0432 0.3161 0.4065 0.2997 0.5789 0.1776 0.1509 0.2699 1.5079 0.7861 0.4747 0.1030 0.0358 0.7934 0.1458 0.1033 0.4646 0.0018 0.5313 0.0758 0.2231 0.0002 0.4635
T231
UNIQUAC
and calculated
25 25 45 45 25 25 25 25 25 25 25 25 25 25 25 20 25 25 25 25 25 40 25
(“C)
Temp.
0.39 0.36 0.24 0.74 0.46 0.44 0.50 0.21 0.42 i.51 1.10 0.68 0.26 0.50 0.12 0.58 0.13 0.01 0.49 0.21 0.87 0.72 0.67
RMS dev. a (mol%)
II81 1331 [331 1491 1501 1491 1331 [5Il [291 ~521 [531 1511 [521 1511 1411 [531
PO1
1171 1481 1181
1471
WI WI
Reference
;;
00
129
fraction
Mole
MfiXHYL
CYCLOHEXCE
ACETATE
0.2 iCETONlTRlLE
C
WATER
W:\I‘IX
Mole
0. 2
0.1
,raLt,o”
-
V
V
0. 6
0.8
Molr fract,on
N ItITI'AwI.
Fig. 1. Calculated and experimental liquid-liquid equilibria for type I ternary systems. Calculated: - - -, without correction; with correction. Experimental tie-line (O- .- .-0): A, methanol-ethanol-cyclohexane it 25’ C [18]; B, acetonitrile-methyl acetate-cyclohexane at 25OC [48]; C, water-ethanol-ethyl acetate at 40 “C [41]; D, water-ethanoln-butanol at 25 o C [51].
130
WATER
N-BlJlYL
ACETATE
N-BLPTANOL
B
WATER
Mole
fraction
CHLOROFORM
C
Fig. 2. Calculated and experimental liquid-liquid Calculated: - - -, without correction; (o- . -. -0): A, water- n-butanol- n-butyl acetate ai form at 25O C [51]; C, water-2-butanone-n-butanol
equilibria for type II ternary systems. with correction. Experimental tie-line 25 o C 1331; B, water- n-butanol-chloroat 25°C [52].
131
second virial coefficients were estimated by the correlation of O’Connell [7]. The Poynting correction included the pure volume calculated from the modified Rackett equation [8]. Parameter estimation, based on the maximum likelihood described by Prausnitz et al. [6], was performed by minimizing function
N
F=C i
[
(P-Q2 + (T-f.)’ + (x*i-P*;)2+ (J+;-j$;)* * OP
2 (JT
d
U)?
Hayden and liquid molar principle as the objective
09)
I
where a circumflex indicates the most probable calculated value corresponding to each measured point and the standard deviations for the measured variables were set as up = 1 Torr for pressure, ur = 0.05 K for temperature, a, = 0.001 for liquid-phase mole fraction and r+ = 0.003 for vapour-phase mole fraction. Table 2 gives the binary calculated results. The computer program used in fitting the proposed model to ternary experimental tie-lines was similar to that as described in the supplement to ref. 9. Ternary calculated results are summarized in Table 3, which contains 18 type I and 5 type II systems. In type I, one binary system is partly miscible and the other two binary systems are completely miscible; in type II, two binary systems are partially miscible and one binary system is completely miscible. Figures 1 and 2 compare calculated results with experimental values for typical systems. It can be seen that the calculated results based on binary parameters alone deviate appreciably from the experimental data in many systems. However, when ternary parameters are included in the calculations, the calculated results are much improved. Figure 3A shows calculated ternary liquid-liquid equilibria for the acetonitrile-benzene-cyclohexane system at 45 o C. We estimated ternary liquid-liquid equilibria at 25 o C using the same ternary parameters and the same binary parameters for the acetonitrile-benzene and benzene-cyclohexane systems and changed only the binary parameters for the acetonitrile-cyclohexane system. Figure 3B compares predicted results with experimental values. If the ternary liquid-liquid equilibrium prediction obtained using binary parameters only is extremely poor, the use of three ternary parameters may be insufficient for reasonable quantitative agreement with experiment. Prediction of quaternaly liquid-liquid equilibria
The binary and ternary parameters given in Tables 2 and 3 were used to predict four type I and five type II quaternary liquid-liquid equilibria. Table 4 shows the absolute arithmetic mean deviations between the experimental and calculated liquid-liquid equilibrium compositions. Ruiz and Gomis [55] calculated liquid-liquid equilibria for quaternary aqueous sys-
calculated
Acetonitrile(1) -cyclohexane(2) -benzene(3) -acetone(4)
Acetonitrile(1) -cyclohexane(2) -benzene(3) -methyl acetate(4)
Acetonitrile(1) -cyclohexane(2) -benzene(3) -toluene(4)
I
I
0.83 0.79 0.22 0.23
0.91 0.86 0.11 0.34
0.36 0.38 0.04 0.04
15
16
17
0.51 0.52 0.21 0.15 0.46 0.56 0.12 0.32 0.37 0.39 0.04 0.04
1.02 0.99 0.13 0.38
0.47 0.48 0.06 0.05
AAM (mol%)
1
0.45 0.45 0.05 0.05
0.66 0.80 0.13 0.36
0.65 0.74 0.23 0.18
RMS (mol%)
Component poor phase
0.97 0.89 0.26 0.26
AAM = RMSb (mol%) (mol%)
;;;nes
1
Component rich phase
Number of
results at 25 o C
I
Type System
Quaternary
TABLE 4
Water(l) -ethanol(2) - n-butanol(3) -chloroform(4) Water(l) -ethanol(2) - n-butanol(3) -chloroform(4)
II
[48]
[54] II
System
Water(l) -ethanol(2) -chloroform(3) - toluene(4)
Type
[47] II
Ref.
4.99 1.97 2.17 1.40
2.57 1.02 1.25 0.71 0.95 0.39 0.71 0.32
44
38 =
1.26 0.48 0.80 0.42
1.11 0.57 0.44 0.57 17
0.89 0.49 0.36 0.26
1 RMS (mol%)
Component rich phase AAM (mol%)
;;;y
Number of
3.30 0.49 1.84 1.02
4.60 1.10 2.22 1.31
1.48 0.71 0.46 1.02
AAM (mol%)
1
[531
I511
I511
5.91 2.02 2.87 1.70 3.63 0.68 2.17 1.23
Ref.
2.72 1.07 0.94 1.42
RMS (mol%)
Component poor phase
E
17 =
Acetonitrile(1) -cyclohexane(2) -methylcyclohexane(3) -toluene(4) Water(l) -acetone(2) -acetic acid(3) -chloroform(4)
0.64 0.27
0.46 0.14 1.43 0.56 1.22 0.55
0.34 0.10 1.08 0.42 0.88 0.32
2.22 0.92
1.10 0.43
0.52 0.22
5.49 2.64
2.49 1.08
0.56 0.09 1.39 0.36 0.87 1.79
0.86 0.29
1.29 0.42
2.78 1.14
0.70 0.12 2.11 0.59 1.23 2.34
1.05 0.40
2.27 0.92
5.55 2.66
[49]
[20]
[20]
II
II
II
-2-butanone(3) - n-butanol(4) Water(l) -ethanol(2) -2-butanone(3) - n-butanol(4)
- n-butanol(3) - n-butyl acetate(4) Water(l) -ethanol(2)
Water(l) -acetic acid(2)
10 c
11
40
a AAM = absolute arithmetic mean deviation between the experimental and calculated liquid-liquid mean square deviation. ’ Rejected tie-lines for which the calculations did not show phase separation.
32
20
Acetonitrile(1) -cyclohexane(2) -methylcyclohexane(3) -toluene(4)
equilibrium
0.49 1.44 1.04 0.40 0.37 0.77
0.87 1.97 0.48
0.66
2.07 0.68
compositions.
0.67 2.48 1.10 0.50 0.44 0.35
0.19 3.55 0.63
0.11 1.95 0.53 0.44 0.97 1.02 0.45 0.32 0.26
0.29
0.70 0.53
0.25
0.49 0.39
1521
1521
1331
b RMS = root-
0.78 2.59 1.36 0.43 0.61 0.86
1.19 3.62 0.58
0.92
2.78 1.07
134 BENZENE A
ACFTOSITRILE
A
Mole fraction
CYCLOHEXWE
BENZEhE
B
ACUI'OONITR ILE
Mole fraction
CYCLOHEXANE
Fig. 3. Calculated and experimental liquid-liquid equilibria for acetonitrile-benzene-cyclohexane. Experimental tie-line (O-.-.-O): A, 45 ’ C; B, 25 ’ C [18]. Calculated: - - -, with correction; - - - - - -, with correction using binary parameters without correction; for acetonitrile-benzene and benzene-cyclohexane and ternary correction parameters identical with those used at 45’C.
terns by establishing the parameters of the ordinary UNIQUAC model [l] representing the binary and ternary systems involved in the quaternary systems. The ordinary UNIQUAC model has 12 adjustable binary interaction parameters in the correlation of quaternary data sets. Some of these parameters can be fixed in fitting the model to the binary and ternary liquid-liquid equilibrium data included in the quaternary system. For type I quaternary systems as shown in Fig. 4A, three options are available. (1) The parameters a,, and u2i from the binary system 1-2, ai4, and ui3, u3i, u23 and u32 u419 (224 and a42 from the ternary system l-2-4 from the ternary system l-2-3 are obtained independently and the two . . remammg parameters u34 and u43 are adjustable. (2) The parameters ui2 and u21 from the binary system l-2 are obtained and the other 10 parameters are fitted. (3) All 12 parameters are fitted. For type II quaternary systems as shown in Fig. 4B, four options are available. (1) The parameters ui2, uzi, u13 and uj2 from the two binary systems l-2 and 2-3, ui3 and u3i from the ternary system l-223 and ui4, u419 u24, u42, u34 and u43 from the simultaneous correlation of the ternary systems l-2-4 and 2-3-4 are obtained. There are no more adjustable parameters. This simultaneous correlation is poorer in correlating the ternary
135
A
B
Fig. 4. Quatemary examples of liquid-liquid
TABLE
equilibria for type I (A) and II (B) systems.
5
Comparison of the present calculated results with those of Ruiz and Gomis for quaternary liquid-liquid equilibria Type
I
System
Water -acetone -acetic acid -chloroform Water -ethanol -chloroform -toluene Water -ethanol - n-butanol -chloroform Water -acetic acid - n-butanol - n-butyl acetate Water -ethanol -2-butanone - n-butanol
Ruiz and Gomis
This work Number of tie-lines used a
AAM b (mol%)
Number of tie-lines used ’
Number of fitted parameters to quatemary data
AAM (mol%)
32
0.89
25
2 10 12
1.14 0.55 0.51
17
0.71
31
44 38 d
1.85 1.13
34
40
0.69
34
11 10 d
1.03 0.58
29
0 6 8 12 0 6 8 12 0 6 8 12 0 6 8 12
3.33 2.00 1.97 1.88 1.66 1.22 1.07 1.03 0.98 0.97 0.86 0.64 1.23 1.04 0.64 0.56
a Included only quatemary tie-lines. b Absolute arithmetic mean deviation between experimental and calculated liquid-liquid equilibrium compositions. ’ Included some binary, ternary and quaternary tie-lines. d Rejected tie-lines for which the calculations did not show phase separation.
136
systems because the parameters obtained for the 2-4 interaction are very different when the two ternary systems l-2-4 and 2-3-4 are correlated separately. Our new approach does not have this disadvantage. (2) The parameters ui2, aZ1, uZ3 and u3* from the binary systems l-2 and 2-3 and ui3 and usi from the ternary system l-2-3 are obtained and the other six parameters are fitted. (3) The parameters u12, uzl, al3 and us2 from the binary systems l-2 and 2-3 are obtained and the eight remaining parameters are fitted. (4) All 12 parameters are adjustable. In Table 5 the absolute arithmetic mean deviations (mol%) for the five aqueous systems obtained in this study are compared with the results of Ruiz and Gomis. The number of tie-lines used by Ruiz and Gomis in fitting the UNIQUAC binary parameters to quaternary liquid-liquid equilibria is less than the total number of quaternary tie-lines reported in their experimental data. The tie-lines used by them include selected quaternary tie-lines plus some binary and ternary tie-lines. The number of quaternary tie-lines used in both methods is not equal. Our method based on ternary information gives better results than the corresponding values of Ruiz and Gomis. Ruiz and Gomis have stated that they do not recommend the use of parameters obtained from ternary liquid-liquid equilibrium data because they lead to substantial increases in the mean deviation between experimental and calculated molar fractions due to the small physical significance of some parameters obtained in this way. Our method does not suffer from this disadvantage. Our method based on the UNIQUAC model which includes additional quaternary parameters improves the prediction of the quaternary liquid-liquid equilibria for most of the systems studied here. Further work is in progress.
LIST OF SYMBOLS
‘ij
UE
F
gE P
4;
4: i T
Auij UE xi
UNIQUAC
binary interaction parameter related to Auij and rij
excess molar Helmholtz energy objective function as defined by eqn. (19) excess molar Gibbs energy total pressure molecular-geometric area parameter for pure component i molecular-interaction area parameter for pure component i molecular-geometric volume parameter for pure component i universal gas constant absolute temperature UNIQUAC binary interaction parameter excess molar energy of mixing liquid-phase mole fraction of component i
137
Yi
z
vapour-phase mole fraction of component i lattice coordination number, here equal to 10
Greek letters
e;, e;jk *P, uT
uXT uY ‘ij ‘ijk ‘i
activity coefficient of component i ternary coefficient area fraction of component i in combinatorial contribution to the activity coefficient area fraction of component i in residual contribution to the activity coefficient local surface fractions standard deviations for the pressure and temperature standard deviations for the liquid and vapour phase compositions UNIQUAC binary parameter UNIQUAC ternary parameter segment fraction of component i
REFERENCES 1 T.F. Anderson and J.M. Prausnitz, Ind. Eng. Chem. Process Des. Dev., 17 (1978) 552, 561. 2 R. Fuchs, M. Giper and J. Gaube, Fluid Phase Equilibria, 14 (1983) 325. 3 T.-H. Cha and J.M. Prausnitz, Ind. Eng. Chem. Process Des. Dev., 24 (1985) 551. 4 G. Maurer and J.M. Prausnitz, Fluid Phase Equilibria, 2 (1978) 91. 5 E.A. Guggenheim, Mixtures, Clarendon, Oxford, 1952. 6 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, NJ, 1980, Chapters 3, 4, 6 and Appendices C, D. 7 J.G. Hayden and J.P. O’Connell, Ind. Eng. Chem. Process Des. Dev., 14 (1975) 209. 8 C.F. Spencer and R.P. Danner, J. Chem. Eng. Data, 17 (1972) 236. 9 J.M. Sorensen, T. Magnussen, P. Rasmussen and A. Fredenslund, Fluid Phase Equilibria, 3 (1979) 47. 10 A. Rius, J.L. Otero and A. Macarron, Chem. Eng. Sci., 10 (1959) 105. 11 N.G. Andreeva, L.F. Komarova, Yu.V. Garber and V.S. Anikeev, Zh. Prikl. Khim. (Leningrad), 49 (1976) 1161. 12 J.J. Conti, D.F. Othmer and R. Gilmont, J. Chem. Eng. Data, 5 (1960) 301. 13 G.F. Meehan and N.F. Murphy, Chem. Eng. Sci., 20 (1965) 757. 14 J. Zawidzki, Z. Phys. Chem., 35 (1900) 129. 15 S.P. Pm-i, J. Polak and A.J. Ruether, J. Chem. Eng. Data, 19 (1974) 87. 16 I. Brown and F. Smith, Aust. J. Chem., 13 (1960) 30. 17 D.A. Palmer and B.D. Smith, J. Chem. Eng. Data, 17 (1972) 71. 18 I. Nagata and T. Ohta, J. Chem. Eng. Data, 28 (1983) 256. 19 D.P. DiElsi, R.B. Patel, M.M. Abbott and H.C. Van Ness, J. Chem. Eng. Data, 23 (1978) 242. 20 I. Nagata and S. Nakamura, J. Chem. Thermodyn., 17 (1985) 1103.
138 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55
0. Muthu, P.J. Maher and B.D. Smith, J. Chem. Eng. Data, 25 (1980) 163. I. Brown and F. Smith, Aust. J. Chem., 10 (1957) 423. G. Scatchard, S.E. Wood and J.M. Mochel, J. Am. Chem. Sot., 62 (1940) 712. I. Nagata and H. Hayashida, J. Chem. Eng. Jpn., 3 (1970) 161. H. Schuberth, J. Prakt. Chem., 6 (1958) 129. Yu.N. Sheinker and E.M. Peresleni, Zh. Fiz. Khim., 26 (1962) 1103. B.V.S. Rao and C.V. Rao, Chem. Eng. Sci., 17 (1962) 574. L. Gay, Chim. Ind. (Milan), 18 (1927) 187. F. Ruiz, M. Galan, D. Prats and A. Ancheta, J. Chem. Eng. Data, 29 (1984) 143. J.M. Sorensen and W. Arlt, Liquid-Liquid Equilibrium Data Collection, Binary Systems, Chemistry Data Series Vol. V, Part 1, DECHEMA, Frankfurt am Main, 1979. W.K. Park, B.Y. Chung and H.S. Shim, Hwahak Konghak, 10 (1972) 281. I. Nagata, T. Ohta and S. Nakagawa, J. Chem. Eng. Jpn., 9 (1976) 276. F. Ruiz, D. Prats, V. Gomis and P. Varo, Fluid Phase Equilibria, 18 (1984) 171. G. Scatchard and CL. Raymond, J. Am. Chem. Sot., 60 (1938) 1278. T. Ohta, T. Kinoshita and I. Nagata, J. Chem. Eng. Data, 28 (1983) 36. G. Scatchard and F.G. Satkiewicz, J. Am. Chem. Sot., 86 (1964) 130. I. Nagata and K. Katoh, Thermochim. Acta, 39 (1980) 45. I. Nagata, T. Ohta, T. Takahashi and K. Gotoh, J. Chem. Eng. Jpn., 6 (1973) 129. J.B. Ott, K.T. Marsh and R.H. Stokes, J. Chem. Thermodyn., 12 (1980) 1139. Z. Kh. Anisimova and E.G. Konakbaeva, Teplofiz. Svoistva Veshchestv Mater., 7 (1973) 167. I. Mertl, Collect. Czech. Chem. Commun., 37 (1972) 366. C.B. Kretschmer and R. Wiebe, J. Am. Chem. Sot., 71 (1949) 1793. D.J. Hall, C.J. Mash and R.C. Pemberton, NPL Report Chem. No. 95, January 1979 (U.K. National Physical Laboratory, Division of Chemistry Standards). G. Schneider, Z. Phys. Chem. N.F., 27 (1961) 171. MS. Lazeeva and N.P. Markuzin, Zh. PrikI. Khim. (Leningrad), 46 (1973) 360. W.G. Beare, G.A. McVicar and J.B. Ferguson, J. Phys. Chem., 34 (1930) 1310. I. Nagata, J. Chem. Eng. Data, 31 (1986) 70. I. Nagata, Thermochim. Acta, 101 (1986) 205. F. Ruiz and D. Prats, Fluid Phase Equilibria, 10 (1983) 77. W.D. Bancroft and S.D. Hubard, J. Am. Chem. Sot., 64 (1942) 347. F. Ruiz, D. Prats and V. Gomis, J. Chem. Eng. Data, 29 (1984) 147. F. Ruiz, D. Prats and V. Gomis, An. Quim., 82 (1986) 393. F. Ruiz, D. Prats and V. Gomis, J. Chem. Eng. Data, 30 (1985) 412. I. Nagata, Fluid Phase Equilibria, 24 (1985) 259. F. Ruiz and V. Gomis, Ind. Eng. Chem. Process Des. Dev., 25 (1986) 216.
Thermochimica Acta, 140 (1989) 121-138 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
CORRELATION OF TERNARY LIQUID-LIQUID EQUILIBRIUM DATA AND PREDICTION OF QUATERNARY LIQUID-LIQUID EQUILIBRIUM DATA BY MEANS OF THE UNIQUAC MODEL
ISAMU NAGATA * and YUKIMASA
USUI
Department of Chemical Engineering and Division of Physical Sciences, Kanazawa Kodatsuno 2, Kanazawa 920 (Japan)
University,
(Received 22 March 1988)
ABSTRACT The UNIQUAC model is modified to include ternary parameters. Correction terms reduce to zero if a ternary system degenerates to a binary system. The proposed model greatly improves the correlation of ternary liquid-liquid equilibria for many systems. Predicted results for quatemary liquid-liquid equilibria confirm the promising ability of the proposed method.
INTRODUCTION
It is well known that commonly used local composition models for the excess Gibbs free energy can predict ternary (or multicomponent) vapour-liquid equilibria of non-electrolyte solutions from binary parameters alone; however, larger solubility envelopes than those observed experimentally are calculated in the prediction of type I ternary liquid-liquid equilibria. To obtain a satisfactory representation of such ternary liquid-liquid equilibria the binary parameters need to be adjusted using ternary tie-line data and this causes some loss of accuracy in the representation of the binary vapour-liquid equilibria [l]. Fuchs et al. [2] have proposed a method for calculating ternary vapour-liquid-liquid equilibria of acetonitrile-nheptane-benzene. Cha and Prausnitz [3] have presented a procedure which maintains thermodynamic consistency: the usual ternary expression for the excess Gibbs energy is multiplied by a composition-dependent correction factor C such that C = 1 whenever the ternary system degenerates to a binary system. This method gives a good representation of ternary vapour-liquid-liquid equilibria. Our preliminary study showed that this method is limited to ternary systems and does not work for quaternary liquid-liquid equilibrium calculations.
* To whom correspondence should be addressed. 0040-6031/89/$03.50
0 1989 Elsevier Science Publishers B.V.
122
We propose another thermodynamic method which can be applied to systems containing more than three components. The systems are studied at low or moderate pressures. In order to evaluate binary and ternary parameters in a newly modified UNIQUAC model, the following experimental values are necessary: vapour-liquid equilibrium data for completely miscible binary systems; mutual solubility data for partially miscible systems; tie-line data for ternary systems.
SOLUTION MODEL
According to the UNIQUAC model [l] the excess molar Gibbs free energy gE is given by the sum of the combinatorial and residual terms g E=
&mbinatorial
+
gE.sidual
0)
We retain the original form for g&binatoria, and correct the original expression of gEsidualby adding ternary empirical correction terms. The correction terms reduce to zero whenever the ternary system degenerates to a binary system.
where the coordination number 2 is set equal to 10, the segment fraction Oi, the area fractions 19,and 0: and the binary parameters 7ji are given by
0,
=
’
4ixi
(5)
Cqjxj i
(6)
Tkj are ternary parameters to be derived from ternary tie-line data; 5ki = rkji for i Z j f k, 7iii = riji = 7ijj = 5,; = 0 and aij ( = Auij/R) are binary energy parameters.
123
For a ternary mixture eqn. (3) is expressed by
eir,, + O;T,, + e,lft$~~) - qix, ln( eiT13 + eir,, + elT,, + e;e;T,23)
- q;x2 ln( 0i-ri2 +
(8)
Equation (8) is derived in the same way as used by Maurer and Prausnitz [4]. The excess energy of mixing uE for the ternary mixture can be given by UE = x1q1’81,Au,, + x2q;e,‘,Au,, +x14;e;iAu3,
+ xJq;e;,Au,,
+ x24;e;ZAu,,
+xi4;e;,,(Au,i
+ At+)
++4;ek(A+
+ Au,,)
+ +4;e;,Au,,
+ x24X,,(Au12
+ A%) (9)
where the local surface fractions O& and (&i are defined as
(11)
and the other t$; and B,ii parameters are similarly defined. $i are ternary coefficients. An expression for the excess molar Helmholtz energy is obtained using the relation
d(aE/T)= d(l/T)
(14
E
’
Integrating eqn. (12) from l/T, $
= i2Ed(
f
) + constant
to l/T, we obtain
of integration
(13)
At low pressures uE = gE and as l/T, + 0 we take the equation of Guggenheim [5] for athermal mixtures of molecules of various size and shape as a boundary condition of eqn. (12). Finally we obtain eqns. (2) and (8) by putting 7jki = GjkiTirki. The activity coefficient yi of component i is derived from
where nT is the sum of the number of moles of component
i. Substituting
124
eqns. (2) and (8) into eqn. component i in the mixture.
(14), we obtain
the
activity
coefficient
of
(15)
I
711 - e;e;r231+ e712+ K- e;e?h32 -4; 0,)3 2 3
+/13 + ce;- fl;e;bi23 3 3 C e,!T3 + e;e;r123
06)
j
In y2 is derived by cyclic advancement of the subscripts, i.e. 1 + 2, 2 + 3 and 3 + 1 and In y3 is expressed similarly. In a quaternary mixture ( gE/RT),,id”al and (In yl)residual are given by
4 C
eiql+ e;e;r231 + e;e;r241 + e;e;r341
i
eiT2+ e;e;r,32 + e;e;r,42 + e;e;r342
- q;x2 In i i j
- q;x, In
4 C l i
- q;x4In
4 C i i
4 (InYI
)residual
=
I
e;?3+ e;e;r,23 + e;e;r,43 + e;eir2,, I
eiT4+ e;eiT12, + e;e;r,34 + e;e.jr2,,
(17)
125
+
CALCULATED
0; [ 713
+
(8;
-
o;o;)
7123 +
toi
-
@x)
?43
-
e;eb2431
4
RESULTS
Correlation of ternary liquid-liquid
equilibria
Table 1 shows values of the pure component structural constants r, q and q’. The UNIQUAC binary parameters were obtained from binary experimental vapour-liquid and mutual solubility data. The fugacity coefficients for mixtures including acetic acid were calculated using a chemical theory of vapour imperfections and those for most other mixtures were obtained by the truncated volume-explicit form of the virial equation [6]. Pure and cross
TABLE
1
Pure component
molecular
structural
constants
a
Component
r
4
4’
Acetic acid Acetone Acetonitrile Benzene n-Butanol 2-Butanone n-Butyl acetate Chloroform Cyclohexane Ethanol Ethyl acetate n-Heptane Methanol Methyl acetate Methylcyclohexane Toluene Water
2.30 2.57 1.87 3.19 3.45 3.25 4.83 2.70 3.97 2.11 3.48 5.17 1.43 2.80 4.64 3.92 0.92
2.04 2.34 1.72 2.40 3.05 2.88 4.20 2.34 3.01 1.97 3.12 4.40 1.43 2.58 3.55 2.97 1.40
2.04 2.34 1.72 2.40 0.88 2.88 4.20 2.34 3.01 0.92 3.12 4.40 0.96 2.58 3.55 2.97 1.00
a Taken from ref. 6.
18 8 13 7 31 12 10 12 1 1 1 14
1 12 11 7 15 17 9 10 24 12
VL VL VL VL VL VL VL VL MS MS MS VL
MS VL VL VL VL VL VL VL VL VL
Acetic acid- n-butanol Acetic acid- n-butyl acetate Acetic acid-chloroform Acetone-acetic acid Acetone-chloroform Acetone-cyclohexane Acetonitrile-acetone Acetonitrile-benzene Acetonitrile-cyclohexane Acetonitrile-cyclohexane Acetonitrile- n-heptane Acetonitrile-methyl acetate Acetonitrile-methylcyclohexane Acetonitrile-toluene Benzene-acetone Benzene-cyclohexane Benzene- n-heptane Benzene-methyl acetate Benzene-toluene n-Butanol- n-butyl acetate n-Butanol-chloroform n-Butanol-ethanol
Number of data points
Type a
results of phase equilibrium
System (l-2)
,TABLE 2 Binary calculated
25 25 45 40 45 50 82-105 50-54 62-115 80-115
116-120 118-122 65-106 40 35 25 45 45 25 45 45 50
Temp. (“C)
data reduction
45.16 - 22.88 174.00 - 11.20 19.07 - 143.88 - 60.03 - 187.14 - 268.29 - 197.11
546.68 - 146.29 - 98.44 522.62 - 140.77 32.35 119.29 - 51.54 74.49 46.41 23.71 - 99.26 566.26 255.70 - 108.79 70.13 31.35 203.46 69.93 516.85 785.66 364.39
- 296.30 356.52 346.43 - 277.64 28.38 181.66 - 88.43 247.13 556.70 516.57 545.79 186.16
a21
(K)
1112
= c c ’
c
’
’
’
’
parameters
(K)
UNIQUAC
0.27 0.97 1.52 0.55
0.50
0.44
1.44 0.42
0.64
8.39
0.02 0.01 0.10 0.04
0.00
0.02
0.00 0.01
0.02
0.48
6T (K)
Root-mean-square SP (Torr)
0.2 1.1 1.5 0.7
0.1
0.1
0.2 0.2
0.7
1.5
1.4 9.8 5.3 5.0
13.9 1.5
8.7
7.6
SY (~10~)
deviations 6x (X103)
0.31 0.84 0.72 1.77 5.70 0.56 15.97 12.58 5.10
0.46
10.54 228.43 385.19 13.46 9.17 22.55 0.66 9.74
Variance of fit b
WI 1211 WI 1231 [I71 1241 1251 1261 1271 WI
WY [ill WI 1131 P41 v51 WI 1171 WI WI 1171 [I91
Reference
1 1 14 14 1 1 1 15 7 1 7 1 8
13 10 14 10 10 11 10 10 13 1 1
MS MS VL VL MS MS MS VL VL MS VL MS VL
VL VL VL VL VL VL VL VL VL MS MS 25 25 40 35 25 25 80 20 25 40 25
20 25 80-111 55 20 25 25 35 45 25 35 25 35 56.75 50.60 489.63 486.03 594.91 578.34 935.54 982.17 - 58.04 391.90 1369.41 1350.4 97.91
0.3 1.3 0.2 0.0
0.0 0.0
0.5
2.1 0.5
1.5 0.8
= F/( number
0.01 0.00 0.01 0.00
0.38 0.67 0.90 0.13
0.06
1.68 0.00 0.00
0.00 0.04
2.20 1.32
0.04 0.13
0.13 0.06
2.07 1.82
of freedom)
- 35.76 12.23 594.60 ’ 1086.98 164.24 32.75 36.24 555.01 c 636.17 ’ 618.91 1453.2
927.37 958.78 - 215.61 - 146.56 - 57.25 - 49.48 106.22 - 203.29 32.55 1250.4 - 90.10 11.03 59.68
of degrees
40.55 36.24 - 148.29 - 134.05 49.20 - 20.72 12.23 - 258.51 - 128.35 58.58 341.45
a VL. vapour-liquid equilibria; MS, mutual solubilities. L L, Variance of fit = (sum of squared, weighted residuals)/(number ’ Taken from ref. 6.
n-Butanol-water n-Butanol-water 2-Butanone- n-butanol 2-Butanone-ethanol 2-Butanone-water 2-Butanone-water n-Butyl acetate-water Chloroform-ethanol Chloroform-toluene Chloroform-water Cyclohexane-ethanol Cyclohexane-methanol Cyclohexane-methyl acetate Cyclohexane-methylcyclohexane Cyclohexane-toluene Ethanol-ethyl acetate Ethanol-toluene Ethanol-water Methanol-ethanol Toluene-methylcyclohexane Water-acetic acid Water-acetone Water-ethyl acetate Water-toluene 0.01 0.00 3.09 1.47 7.22 3.80 0.00 1.18 22.07
17.66
5.77
16.93 4.29
17.56 14.01
of data points-number
2.9 5.7 4.5
0.2
8.8
6.9 2.1
4.6 7.6
of parameters).
1391 [401 1411 [421 [431 1431 WI 1451 [461 [411 1301
[301 1331 [341 [351 1301 [361 1371 1381
1321 v91
[301 1311
1291
results
deviation
between
the experimental
Acetonitrile-acetone-cyclohexane Acetonitrile-benzene-cyclohexane Acetonitrile-benzene-cyclohexane Acetonitrile-benzenen-heptane Acetonitrile-methyl acetate-cyclohexane Acetonitrile-toluene-cyclohexane Acetonitrile-toluene-methylcyclohexane Methanol-ethanol-cyclohexane Water-acetic acid- n-butanol Water-acetic acid-n-butyl acetate Water-acetic acid-chloroform Water-acetone-chloroform Water-acetone-chloroform Water- n-butanol- n-butyl acetate Water- n-butanol-chloroform Water-2-butanonen-butanol Water-2-butanonen-butanol Water-chloroform-toluene Water-ethanoln-butanol Water-ethanol-2-butanone Water-ethanol-chloroform Water-ethanol-ethyl acetate Water-ethanol-toluene
System (l-2-3)
a Root-mean-square
I I I I I I I I I I I I II II II II II I I I I I
I
Type
Ternary calculated
TABLE 3
-
-
.-
-
equilibrium
compositions.
0.1205 0.2128 0.1621 0.1643 - 0.2527 0.3239 0.2270 - 0.0825 0.4190 - 0.4998 - 2.8468 - 0.3923 0.1260 - 0.2778 0.0851 0.3802 0.2040 -0.0119 1.1175 0.8726 0.5837 0.6051 - 0.2516
r123
parameters
0.0073 0.2752 0.0551 0.0742 0.1011 0.2100 0.2040 0.0253 -0.8117 - 0.6487 - 0.4392 0.7391 0.1382 0.6041 0.1800 1.8098 2.6930 - 0.0163 0.5560 7.5817 0.4615 1.5703 1.1673
r132
ternary
liquid-liquid
0.0432 0.3161 0.4065 0.2997 0.5789 0.1776 0.1509 0.2699 1.5079 0.7861 0.4747 0.1030 0.0358 0.7934 0.1458 0.1033 0.4646 0.0018 0.5313 0.0758 0.2231 0.0002 0.4635
T231
UNIQUAC
and calculated
25 25 45 45 25 25 25 25 25 25 25 25 25 25 25 20 25 25 25 25 25 40 25
(“C)
Temp.
0.39 0.36 0.24 0.74 0.46 0.44 0.50 0.21 0.42 i.51 1.10 0.68 0.26 0.50 0.12 0.58 0.13 0.01 0.49 0.21 0.87 0.72 0.67
RMS dev. a (mol%)
II81 1331 [331 1491 1501 1491 1331 [5Il [291 ~521 [531 1511 [521 1511 1411 [531
PO1
1171 1481 1181
1471
WI WI
Reference
;;
00
129
fraction
Mole
MfiXHYL
CYCLOHEXCE
ACETATE
0.2 iCETONlTRlLE
C
WATER
W:\I‘IX
Mole
0. 2
0.1
,raLt,o”
-
V
V
0. 6
0.8
Molr fract,on
N ItITI'AwI.
Fig. 1. Calculated and experimental liquid-liquid equilibria for type I ternary systems. Calculated: - - -, without correction; with correction. Experimental tie-line (O- .- .-0): A, methanol-ethanol-cyclohexane it 25’ C [18]; B, acetonitrile-methyl acetate-cyclohexane at 25OC [48]; C, water-ethanol-ethyl acetate at 40 “C [41]; D, water-ethanoln-butanol at 25 o C [51].
130
WATER
N-BlJlYL
ACETATE
N-BLPTANOL
B
WATER
Mole
fraction
CHLOROFORM
C
Fig. 2. Calculated and experimental liquid-liquid Calculated: - - -, without correction; (o- . -. -0): A, water- n-butanol- n-butyl acetate ai form at 25O C [51]; C, water-2-butanone-n-butanol
equilibria for type II ternary systems. with correction. Experimental tie-line 25 o C 1331; B, water- n-butanol-chloroat 25°C [52].
131
second virial coefficients were estimated by the correlation of O’Connell [7]. The Poynting correction included the pure volume calculated from the modified Rackett equation [8]. Parameter estimation, based on the maximum likelihood described by Prausnitz et al. [6], was performed by minimizing function
N
F=C i
[
(P-Q2 + (T-f.)’ + (x*i-P*;)2+ (J+;-j$;)* * OP
2 (JT
d
U)?
Hayden and liquid molar principle as the objective
09)
I
where a circumflex indicates the most probable calculated value corresponding to each measured point and the standard deviations for the measured variables were set as up = 1 Torr for pressure, ur = 0.05 K for temperature, a, = 0.001 for liquid-phase mole fraction and r+ = 0.003 for vapour-phase mole fraction. Table 2 gives the binary calculated results. The computer program used in fitting the proposed model to ternary experimental tie-lines was similar to that as described in the supplement to ref. 9. Ternary calculated results are summarized in Table 3, which contains 18 type I and 5 type II systems. In type I, one binary system is partly miscible and the other two binary systems are completely miscible; in type II, two binary systems are partially miscible and one binary system is completely miscible. Figures 1 and 2 compare calculated results with experimental values for typical systems. It can be seen that the calculated results based on binary parameters alone deviate appreciably from the experimental data in many systems. However, when ternary parameters are included in the calculations, the calculated results are much improved. Figure 3A shows calculated ternary liquid-liquid equilibria for the acetonitrile-benzene-cyclohexane system at 45 o C. We estimated ternary liquid-liquid equilibria at 25 o C using the same ternary parameters and the same binary parameters for the acetonitrile-benzene and benzene-cyclohexane systems and changed only the binary parameters for the acetonitrile-cyclohexane system. Figure 3B compares predicted results with experimental values. If the ternary liquid-liquid equilibrium prediction obtained using binary parameters only is extremely poor, the use of three ternary parameters may be insufficient for reasonable quantitative agreement with experiment. Prediction of quaternaly liquid-liquid equilibria
The binary and ternary parameters given in Tables 2 and 3 were used to predict four type I and five type II quaternary liquid-liquid equilibria. Table 4 shows the absolute arithmetic mean deviations between the experimental and calculated liquid-liquid equilibrium compositions. Ruiz and Gomis [55] calculated liquid-liquid equilibria for quaternary aqueous sys-
calculated
Acetonitrile(1) -cyclohexane(2) -benzene(3) -acetone(4)
Acetonitrile(1) -cyclohexane(2) -benzene(3) -methyl acetate(4)
Acetonitrile(1) -cyclohexane(2) -benzene(3) -toluene(4)
I
I
0.83 0.79 0.22 0.23
0.91 0.86 0.11 0.34
0.36 0.38 0.04 0.04
15
16
17
0.51 0.52 0.21 0.15 0.46 0.56 0.12 0.32 0.37 0.39 0.04 0.04
1.02 0.99 0.13 0.38
0.47 0.48 0.06 0.05
AAM (mol%)
1
0.45 0.45 0.05 0.05
0.66 0.80 0.13 0.36
0.65 0.74 0.23 0.18
RMS (mol%)
Component poor phase
0.97 0.89 0.26 0.26
AAM = RMSb (mol%) (mol%)
;;;nes
1
Component rich phase
Number of
results at 25 o C
I
Type System
Quaternary
TABLE 4
Water(l) -ethanol(2) - n-butanol(3) -chloroform(4) Water(l) -ethanol(2) - n-butanol(3) -chloroform(4)
II
[48]
[54] II
System
Water(l) -ethanol(2) -chloroform(3) - toluene(4)
Type
[47] II
Ref.
4.99 1.97 2.17 1.40
2.57 1.02 1.25 0.71 0.95 0.39 0.71 0.32
44
38 =
1.26 0.48 0.80 0.42
1.11 0.57 0.44 0.57 17
0.89 0.49 0.36 0.26
1 RMS (mol%)
Component rich phase AAM (mol%)
;;;y
Number of
3.30 0.49 1.84 1.02
4.60 1.10 2.22 1.31
1.48 0.71 0.46 1.02
AAM (mol%)
1
[531
I511
I511
5.91 2.02 2.87 1.70 3.63 0.68 2.17 1.23
Ref.
2.72 1.07 0.94 1.42
RMS (mol%)
Component poor phase
E
17 =
Acetonitrile(1) -cyclohexane(2) -methylcyclohexane(3) -toluene(4) Water(l) -acetone(2) -acetic acid(3) -chloroform(4)
0.64 0.27
0.46 0.14 1.43 0.56 1.22 0.55
0.34 0.10 1.08 0.42 0.88 0.32
2.22 0.92
1.10 0.43
0.52 0.22
5.49 2.64
2.49 1.08
0.56 0.09 1.39 0.36 0.87 1.79
0.86 0.29
1.29 0.42
2.78 1.14
0.70 0.12 2.11 0.59 1.23 2.34
1.05 0.40
2.27 0.92
5.55 2.66
[49]
[20]
[20]
II
II
II
-2-butanone(3) - n-butanol(4) Water(l) -ethanol(2) -2-butanone(3) - n-butanol(4)
- n-butanol(3) - n-butyl acetate(4) Water(l) -ethanol(2)
Water(l) -acetic acid(2)
10 c
11
40
a AAM = absolute arithmetic mean deviation between the experimental and calculated liquid-liquid mean square deviation. ’ Rejected tie-lines for which the calculations did not show phase separation.
32
20
Acetonitrile(1) -cyclohexane(2) -methylcyclohexane(3) -toluene(4)
equilibrium
0.49 1.44 1.04 0.40 0.37 0.77
0.87 1.97 0.48
0.66
2.07 0.68
compositions.
0.67 2.48 1.10 0.50 0.44 0.35
0.19 3.55 0.63
0.11 1.95 0.53 0.44 0.97 1.02 0.45 0.32 0.26
0.29
0.70 0.53
0.25
0.49 0.39
1521
1521
1331
b RMS = root-
0.78 2.59 1.36 0.43 0.61 0.86
1.19 3.62 0.58
0.92
2.78 1.07
134 BENZENE A
ACFTOSITRILE
A
Mole fraction
CYCLOHEXWE
BENZEhE
B
ACUI'OONITR ILE
Mole fraction
CYCLOHEXANE
Fig. 3. Calculated and experimental liquid-liquid equilibria for acetonitrile-benzene-cyclohexane. Experimental tie-line (O-.-.-O): A, 45 ’ C; B, 25 ’ C [18]. Calculated: - - -, with correction; - - - - - -, with correction using binary parameters without correction; for acetonitrile-benzene and benzene-cyclohexane and ternary correction parameters identical with those used at 45’C.
terns by establishing the parameters of the ordinary UNIQUAC model [l] representing the binary and ternary systems involved in the quaternary systems. The ordinary UNIQUAC model has 12 adjustable binary interaction parameters in the correlation of quaternary data sets. Some of these parameters can be fixed in fitting the model to the binary and ternary liquid-liquid equilibrium data included in the quaternary system. For type I quaternary systems as shown in Fig. 4A, three options are available. (1) The parameters a,, and u2i from the binary system 1-2, ai4, and ui3, u3i, u23 and u32 u419 (224 and a42 from the ternary system l-2-4 from the ternary system l-2-3 are obtained independently and the two . . remammg parameters u34 and u43 are adjustable. (2) The parameters ui2 and u21 from the binary system l-2 are obtained and the other 10 parameters are fitted. (3) All 12 parameters are fitted. For type II quaternary systems as shown in Fig. 4B, four options are available. (1) The parameters ui2, uzi, u13 and uj2 from the two binary systems l-2 and 2-3, ui3 and u3i from the ternary system l-223 and ui4, u419 u24, u42, u34 and u43 from the simultaneous correlation of the ternary systems l-2-4 and 2-3-4 are obtained. There are no more adjustable parameters. This simultaneous correlation is poorer in correlating the ternary
135
A
B
Fig. 4. Quatemary examples of liquid-liquid
TABLE
equilibria for type I (A) and II (B) systems.
5
Comparison of the present calculated results with those of Ruiz and Gomis for quaternary liquid-liquid equilibria Type
I
System
Water -acetone -acetic acid -chloroform Water -ethanol -chloroform -toluene Water -ethanol - n-butanol -chloroform Water -acetic acid - n-butanol - n-butyl acetate Water -ethanol -2-butanone - n-butanol
Ruiz and Gomis
This work Number of tie-lines used a
AAM b (mol%)
Number of tie-lines used ’
Number of fitted parameters to quatemary data
AAM (mol%)
32
0.89
25
2 10 12
1.14 0.55 0.51
17
0.71
31
44 38 d
1.85 1.13
34
40
0.69
34
11 10 d
1.03 0.58
29
0 6 8 12 0 6 8 12 0 6 8 12 0 6 8 12
3.33 2.00 1.97 1.88 1.66 1.22 1.07 1.03 0.98 0.97 0.86 0.64 1.23 1.04 0.64 0.56
a Included only quatemary tie-lines. b Absolute arithmetic mean deviation between experimental and calculated liquid-liquid equilibrium compositions. ’ Included some binary, ternary and quaternary tie-lines. d Rejected tie-lines for which the calculations did not show phase separation.
136
systems because the parameters obtained for the 2-4 interaction are very different when the two ternary systems l-2-4 and 2-3-4 are correlated separately. Our new approach does not have this disadvantage. (2) The parameters ui2, aZ1, uZ3 and u3* from the binary systems l-2 and 2-3 and ui3 and usi from the ternary system l-2-3 are obtained and the other six parameters are fitted. (3) The parameters u12, uzl, al3 and us2 from the binary systems l-2 and 2-3 are obtained and the eight remaining parameters are fitted. (4) All 12 parameters are adjustable. In Table 5 the absolute arithmetic mean deviations (mol%) for the five aqueous systems obtained in this study are compared with the results of Ruiz and Gomis. The number of tie-lines used by Ruiz and Gomis in fitting the UNIQUAC binary parameters to quaternary liquid-liquid equilibria is less than the total number of quaternary tie-lines reported in their experimental data. The tie-lines used by them include selected quaternary tie-lines plus some binary and ternary tie-lines. The number of quaternary tie-lines used in both methods is not equal. Our method based on ternary information gives better results than the corresponding values of Ruiz and Gomis. Ruiz and Gomis have stated that they do not recommend the use of parameters obtained from ternary liquid-liquid equilibrium data because they lead to substantial increases in the mean deviation between experimental and calculated molar fractions due to the small physical significance of some parameters obtained in this way. Our method does not suffer from this disadvantage. Our method based on the UNIQUAC model which includes additional quaternary parameters improves the prediction of the quaternary liquid-liquid equilibria for most of the systems studied here. Further work is in progress.
LIST OF SYMBOLS
‘ij
UE
F
gE P
4;
4: i T
Auij UE xi
UNIQUAC
binary interaction parameter related to Auij and rij
excess molar Helmholtz energy objective function as defined by eqn. (19) excess molar Gibbs energy total pressure molecular-geometric area parameter for pure component i molecular-interaction area parameter for pure component i molecular-geometric volume parameter for pure component i universal gas constant absolute temperature UNIQUAC binary interaction parameter excess molar energy of mixing liquid-phase mole fraction of component i
137
Yi
z
vapour-phase mole fraction of component i lattice coordination number, here equal to 10
Greek letters
e;, e;jk *P, uT
uXT uY ‘ij ‘ijk ‘i
activity coefficient of component i ternary coefficient area fraction of component i in combinatorial contribution to the activity coefficient area fraction of component i in residual contribution to the activity coefficient local surface fractions standard deviations for the pressure and temperature standard deviations for the liquid and vapour phase compositions UNIQUAC binary parameter UNIQUAC ternary parameter segment fraction of component i
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