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Study of ternary liquid—liquid equilibria and of multiphase equilibria
181
Fluid Phase Equilibria, 65 (1991) 181-207 Elsevier Science Publishers B.V., Amsterdam
Study of ternary liquid-liquid and of multiphase equilibria
equilibria
Hai Huang * Beijing Graduate School, University of Petroleum, P. 0. Box 902 Beijing 100083 (People’s Republic of China) (Received March 22, 1989; accepted in final form December 18, 1990)
ABSTRACT
Huang, H., 1991. Study of ternary liquid-liquid
equilibria and of multiphase equilibria.
Fluid
Phase Equilibria, 65: 181-207. In this study, we have calculated LLE and LLVE in ternary systems, using the PT equation of state with the mixing rules of Kurihara et al. Only two interaction parameters per binary system are used to correlate VLE and LLE data simultaneously and the results appear to be comparable with the activity coefficient approach using the same number of adjustable parameters. A wide range of systems has been studied. The multiphase P-T diagram for the system H,S-CO,-CH, is also plotted and compared with the experimental data.
INTRODUCTION
Liquid-liquid equilibria (LLE) form the basis of extractive unit operations. Due to the demand from chemical engineering, much work has already been done in this field, mainly using the NRTL (Renon and Prausnitz, 1968) and UNIQUAC (Abrams and Prausnitz, 1975) empirical equations to correlate and to predict the activity coefficient (AC) for LLE. Sorensen et al. (1979a,b) and Magnussen et al. (1980) have written excellent reviews. Sorensen and Arlt (1980) collected many LLE data and fitted them with NRTL and UNIQUAC equations. These works prove that NRTL and UNIQUAC equations are useful to calculate and to predict LLE under ordinary pressure. However, the main defect of the AC method is that the influence of pressure cannot be found from LLE by using this method. * Present address: Department of Chemical Engineering, University of Delaware, Newark, DE 19716, USA. 0378-3812/91/$03.50
0 1991 Elsevier Science Publishers B.V.
182
Popularization of computer technology has led to more and more methods of calculating and predicting multiphase equilibria being proposed. For instance, Michelsen (1980, 1982, and 1987) and Ohanomah and Thompson (1984) proposed complete algorithms to calculate multiphase equilibria, but we have not seen any report where the calculated multiphase P-T diagram agrees quantitatively with the experiment. Recently, a few reports about calculating LLE with equations of state (EOS) have been published. Among these, the work of Peng and Robinson (1976b) may be the pioneer; they calculated LLE with the PR EOS (Peng and Robinson, 1976a). Next, Mathias and Copeman (1983) calculated four binary systems of LLE with EOS (local composition). Leet et al. (1986) fitted data of LLE for several binary systems with the CCOR(I1) EOS. Luedecke and Prausnitz (1985) proposed a new EOS, and calculated LLE of binary systems with their new EOS. As regard to fitting data of LLE for ternary systems with EOS, we have found only one report in the literature, made by Hou and Hu (1988). Since in this case EOS involves ternary interaction parameters with insufficient experimental points, this method cannot be widely used without difficulty. In most cases, the interaction parameters used to describe LLE are correlated only with the LLE data. For example, in “Liquid-Liquid Equilibrium Data Collections” (Sorensen and Arlt, 1980) it was done in this way. Prausnitz et al. (1980) pointed out that “fitting ternary LLE data only may yield unrealistic parameters, that predict grossly erroneous results when used in regions not identical to those employed in data reduction”. Some authors try to predict ternary LLE only using pure component parameters and binary VLE data. For example, Ohta (1989) predicted ternary LLE in this way with three different methods, but generally, they cannot attain high accuracy in predicting LLE data. On the other hand, some workers have reported the method of fitting VLE data and LLE data with unique interaction parameters of AC, e.g. Bender and Block, 1975 and Anderson and Prausnitz, 1978, but very few such papers have been published. Prausnitz et al. (1980) reported some results related to ten groups of LLE data. Perhaps due to experimental difficulty, very few multiphase P-T diagrams plotted according to experiments have appeared in the literature. We find only one such report (Ng et al., 1985), in which the multiphase P-T diagram for the system methane(0.5)-carbon dioxide(O.l)-hydrogen sulfide(0.4) was studied experimentally. Before this experiment, Michelson (1982) had calculated the multiphase P-T diagram for the same system with the SRK EOS (Soave, 1972). The sketch of the phase boundary plotted according to his calculation seems like that found by experiment, but is too approximate for comparison in minute detail.
183 FITTING LLE DATA WITH PT EOS
In the preceding article (Huang 1990), we described how to modify the PT EOS (Pate1 and Teja, 1982) by means of a new mixing rule of Kurihara et al. (1987). There, we calculated VLE and LLE for binary systems, and predicted VLE for ternary systems from binary data. In the following, we shall apply the same device to calculate LLE for ternary systems. From the objective functions presented by Sorensen et al. (1979b), we choose three functions to fit the experimental data. The six binary interaction parameters of the ternary system are optimized in three steps. The three relevant objective functions are described below. (1) The sum of the squared differences between the logarithmic fugacities taken from a series of experiments: F=
fJ~(lnf,jl-lnhjIr)2 j=l
0)
j-1
where A4 is the number of experimental data, and fijk the fugacity calculated from EOS of the component i in the phase k under the temperature and pressure of the jth experiment. In the preceding article (Huang, 1990), we have given the expression for In & as a function of T, P and the composition. Now we choose In h.jr = In f;.jn instead of fiiI =fijrr as the criterion of phase equilibria, because in the criterion used later we shall have hi1 + 0 and fijrI + 0 during optimizing the interaction parameters of EOS. With the objective function F chosen above to adjust the interaction parameters of EOS, convergence is very fast. The parameters of EOS can be determined in advance. (2) The sum of the squared differences between the compositions obtained from experiment and from calculation. Here, the object of optimization is given by F=
5i f j=l
i=l
(x~~~-Z;~~)~,
(2)
k=l
where xijk denotes the mole fraction of component i of the phase k in the jth experiment, Tijk the corresponding value obtained in calculating conjugate phase compositions from the equimolar mixture of two experimental conjugate phases, and A4 the number of experimental data. The correlated interaction parameters of EOS found by this method are quite near to the optimal. Since there is no artificial intervention in the process of optimization, the convergence ought to be rapid.
184
(3) The sum of the squared differences between the compositions by experiment and those found by calculation optimally: P=
$ min{ i j=l
i=l
g
found
(xiik - T:jk)2}
(3)
k=l
where x”’ indicates line, i.e.
that the number
Z:kj is obtained
from the optimal
tie
would be a minimum. In order to determine the optimal tie line, we use the binodal curve in the phase diagram. The coordinate parameter of the tie line in the binodal curve is used as the sole variable in optimizing eqn. (4). For this purpose we use Michelsen’s computer program (Michelsen, 1980; Sorensen and Arlt, 1980). Since we choose In Air = In fijII as the criterion of phase equilibria, it is impossible to trace the binodal curve by starting from one side of the ternary phase diagram, where certain components may not exist, and their fugacity would be zero. Practically, we use EOS with the given interaction parameters to calculate the first and the second points of the binodal curve, and then step by step extend the curve with Michelson’s method. At this moment, EOS with binary interaction parameters intervenes. If the parameters used are too inaccurate, the initial points derived may be unsuitable for iteration. We have to change the initial points (artificial intervention) and slow down the speed of convergence. In practical calculations, the parameters are optimized in four steps: (1) Guessing the initial interaction parameters; (2) first adjustment of all six parameters by using objective function (1); (3) second adjustment of parameters by using objective function (2); (4) final optimization of all six parameters by using the objective function given by eqn. (3). Guessing the initial parameters, we use the binary interaction parameters from VLE. In the formula NRTL, the third parameter a is determined by the data of VLE but not optimized by that of LLE. In the following three steps, we always take as initial values the results of the preceding step. In the last step, the initial interaction parameters are then very near to the optimal ones, so that artificial intervention is often negligible. Our last step is analogous to the last step used by Sorensen et al. (1979b). When we make optimization, the relation between interaction parameters of EOS and the data of LLE is too complicated, and the iteration process of Newton’s method often diverges. So we utilize the simplex method to optimize the parameters and thus obtain satisfactory results.
185
For this purpose we take the average error as defined by Sorensen and Arlt (1980) Err = lOO(F/6/M)“2
(5)
where F is given by the eqn. (3). Here definition (5) is employed in order to compare our results with those obtained by the AC method.
TESTING
THE RESULTS
Our results of correlations are given in Table 1, which contains 10 ternary systems, 27 series of experimental data and 267 tie lines of equilibrium. All experimental data come from “Data Collection”, Vol. V edited by Sorensen and Arlt (1980). For comparison with the AC method, the interaction parameters in the NRTL and UNIQUAC equations are taken from the same source. The results obtained by the AC method are identical with those in the book of Sorensen and Arlt, but we reserve three effective figures to facilitate comparison. All experimental data of the same ternary system at the same temperature are compiled in one series of data, although they come from different authors. In fitting experimental data with the PT EOS, the pressure is assumed to be 2 atm. absolute. Here, we use UNIQUAC and NR-TL models to calculate the excess free enthalpy of the FT EOS. In fitting LLE data, the method of FT + UNIQUAC EOS is not satisfactory when used in some systems. That explains why we did not fit all data with this method alone. All interaction parameters used in the above four methods are shown in the Appendix. The PT EOS can provide the density of liquid phases, but a complete list of the densities for all experiments which we have done is too long, so we mention in the Appendix only the range of densities calculated from each series of data. In Fig. 1 we present in detail the results of our calculation for the ternary systems. In the figure, the binodal curve and the plait point are calculated from EOS, while the tie lines in Fig. 1 are plotted from experiments. This figure is similar to those in the book of Sorensen and Arlt.
FITTING VLE DATA AND LLE DATA WITH ONE SET OF INTERACTION ETERS
PARAM-
In this section, we fit six parameters altogether with LLE data only and obtain satisfactory results. However, numerical values of the parameters are not significant, due to the correlation between the parameters and the narrow composition range for some components. In many cases, they cannot
186 TABLE 1 Comparison of root-mean-square correlating liquid-liquid equilibria indicated). System
N
T (R)
deviation (mole percentage) with various methods for data (all data taken from Sorensen and Arlt (1980), as
NRTL
A) System: water(a)-ethanol(b)-benzene(c) Al 48 298.15 0.688 A2 8 299.15 0.340 6 293.15 0.824 A3 A4 6 308.15 0.498 5 318.15 0.544 A5 5 328.15 0.429 A6 5 337.15 0.585 A7
UNIQUAC
PTNRTL
Part 2, p. 350-360 0.832 0.779 0.578 0.649 0.829 1.029 0.611 0.510 0.582 0.348 1.112 0.349 0.709 0.434
B) System: water(a)-methanol(b)-benzene(c) Bl 13 303.15 1.313 13 B2 318.15 0.758 B3 15 333.15 0.757
Part 2, p. 121-123 1.381 1.439 0.879 0.479 1.205 0.713
C) System: Cl c2
n-hexane(a)-1-propanol(b)-water(c) 15 298.15 0.425 5 310.95 0.341
Part 2, p. 577-579 1.779 0.561 0.455 0.288
D) System: heptane(a)-benzene(b)-methanol(c) 5 279.95 Dl 0.636 D2 6 286.95 0.906 5 D3 305.95 0.663
Part 2, p. 118-120 0.507 0.840 0.759 1.040 0.622 0.613
E) System: water(a)-ethanol(b)-hexane(c) 8 El 293.15 1.020 E2 6 298.15 2.490
Part 2, p. 364-365 0.919 1.819
F) System: water(a)-acetic acid(b)-tetrachloromethane(c) Fl 10 298.15 0.664 0.640
Part 2, p. 4 0.590
Part 2, p. 287-296 0.568 0.436 0.878 0.485 0.793 0.620 0.289 0.331 0.860 0.860
H)System: HI H2
Part 3, p. 50-51 0.365 0.401 0.213 0.594
I) System: tetrachloromethane(a)-acetone(b)-water(c) I1 10 303.15 0.552 0.459 J) System: butanol(a)-ethanol(b)-water(c) Jl 10 298.15 0.364
2.405
2.423
8.714
3.755
0.723 0.832
G) System: water(a)-acetic acid(b)-toluene(c) 10 Gl 298.15 0.512 G2 8 303.15 1.009 13 G3 313.15 0.756 G4 5 328.15 0.233 G5 8 333.15 0.821 ethyl acetate(a)-butanol(b)-water(c) 5 273.15 0.660 6 293.15 0.492
PTUNIQUAC
2.593
Part 2, p. 7 0.506
Part 2, p. 340 0.356
0.388
0.229
187
R
40
k-
35
= u u
25
E
15
!?
10
30
z
20
5 0
0
1
2
10-l.
3 MOLE
4
5
6
PER
0 MOLE
5
10
PER
15
CENT
20 OF
25 121
8
OF
10
9 (31
PLRIT POINT EXPERIMENTRL
a 4
0
7
CENT
30
35
40
IN RIGHT
45
PHFlSE
Fig. 1. LLE of water(l)-ethanol(2)-benzene(3) (1980), Part 2, pp. 351, 354:
(T= 25°C). Data from Sorensen and Arlt
be used to predict VLE for the same system at the same temperature. For example, we obtain the interaction parameters for the system waterethanol-benzene in 318.15 K (see Appendix). With these parameters to predict VLE of binary water-ethanol and of binary ethanol-benzene, we would commit a gross error (for data source see Table 2). From experimental data, the RMS deviations are dp/p = 0.3175; 0.2975 and dy = 0.3953; 0.4095, respectively, which are obviously unacceptable. Therefore we try the method proposed by Prausnitz et al. (1980) to correlate the binary parameters, so as to fit both the LLE data and the VLE data. Our next object is to calculate the P-T diagram for multiphase equilibria, so that the ranges of temperature and pressure are large. It would be very difficult to determine the experimental standard deviations, if we use the maximum likelihood principle to set the objective function. In view of this situation, we use the classical method. For VLE data of binary system, our objective function is F VLE
=
f
(
(P,,cdpi,exp
-
‘)’
+
(Yi,exp
-
Yi,cal)‘)
i=l
where N denotes the total number of experimental points.
(6)
188 TABLE 2 Fitting VLE data and LLE data with unique interaction parameters. Data liquid-vapor equilibria system and of ternary liquid-liquid equilibria system
of binary
T(K)
P (atm)
Source
A Heptane(a)-benzene(b)-methanol(c) (a)-(b VLE 15 (b)-(c) VLE 9 (a)-(c) LLE 1 (a)-(b)-(c) LLE 6
318.15 308.15 305.15 305.95
0.1664-0.2962 0.2675-0.3366 2 2
Palmer (1972) Scatchard (1946) Wittrig (1977) Wittrig (1977)
B Water(a)-ethanol(b)-benzene(c) (a)-(b) VLE 13 (b)-(c) VLE 12 (a)-(c) LLE 1 (a)-(b)-(c) LLE 12
313.15 318.15 298.15 298.15
0.0989-0.1761 0.3566-0.2675 2 2
Mertl(1972) Brown (1954) Ssrensen (1979a) Bancroft (1942)
C Heptane(a)-benzene(b)-acetonitrile(c) (a)-(b) VLE 15 (b)-(c) VLE 12 (a)-(c) LLE 1 (a)-(b)-(c) LLE 9
318.15 318.15 318.15 318.15
0.1664-0.2962 0.3566-0.2675 2 2
Palmer Brown Palmer Palmer
D Cyclohexane(a)-benzene(b)-nitromethane(c) (a)-(b) VLE 13 298.15 (b)-(c) VLE 12 318.15 (a)-(c) LLE 1 298.15 (a)-(b)-(c) LLE 3 298.15
0.1251-0.1430 0.1542-0.2979 2 2
Tasic (1978) Sanuders (1961) Serensen (1979b) Week (1954)
E 2,2,4-Trimethylpentane(a)-cyclohexane(b)-furfural(c) (a)-(b) VLE 7 308.15 (b)-(c) LLE 1 298.15 (a)-(c) LLE 1 298.15 (a)-(b)-(c) LLE 8 298.15
0.1258-0.1878 2 2 2
Battind (1966) Ssrensen (1979~) Ssrensen (1979d) Henty (1964)
F n-Hexane(a)-methylcyclopentane(b)-aniline(c) (a)-(b) VLE 9 333.15 (b)-(c) LLE 1 298.15 (a)-(c) LLE 1 298.15 (a)-(b)-(c) LLE 7 298.15
0.6847-0.7538 2 2 2
Beyer (1965) Ssrensen (1979e) Serensen (1979e) Darwent (1943)
Type
N
(1972) (1954) (1972) (1972)
G 2,2,4-Trimethylpentane(a)-benzene(b)-furfural(c) (a)-(b) VLE 7 308.15 (b)-(c) VLE 10 355.85427.85 (a)-(c) LLE 1 298.15 (a)-(b)-(c) LLE 9 298.15
O.l2OO-0.1956 1
Weissman (1960) Thorton (1951)
2 2
Serensen (1979~) Henty (1964)
H Cyclohexane(a)-benzene(b)-furfural(c) (a)-(b) VLE 11 (b)-(c) VLE 10
0.1251-0.1430 1
Tasic (1978) Thornton (1951)
2 2
Ssrensen (1979d) Henty (1964)
(a)-(c) (a)-(b)-(c)
LLE LLE
1 11
298.15 355.85427.85 298.15 298.15
189
For LLE data of a ternary function is defined as F LLE
system,
as in the last section,
(7)
=
Assembling the binary the objective function is F = Fl,2,“LE/Nl,2,“LE
the objective
VLE data and the ternary
+ F2,3,“LE/N2,3.VL.E
LLE data as a whole,
+ F,,2,3,LLE/N~,2,3,LLE
where the subscripts represent the kind of equilibria and the components involved. In eqn. (8), the two binaries (1,2) and (2,3) are completely miscible and the binary (1,3) is partially miscible. For LLE of type II, in which only the binary (1,2) is completely miscible, the term containing F2,3,VLE, corresponding to the second term of eqn. (8), is missing. Now, the NRTL equation is used to correlate the free enthalpy (Huang, 1990) defined by the F’T EOS. To fit binary VLE data in a binary system, three binary interaction parameters have to be determined. The third parameter aij is fixed according to the binary VLE data. For the pair of immiscible components, their third interaction parameter (yij is determined with the ternary LLE data. Hence while correlating the binary interaction parameters of a ternary system with eqn. (8), there are six parameters to be correlated. We take the objective function (8) and employ the simplex method to correlate the six parameters.
Notes to Table 2 Data references Bancroft, W.D. and Hubard, S.S., (1942), J. Am. Chem. Sot., 64: 347. Battind, R., (1966) J. Phys. Chem., 70: 3408. Beyer, W., Schuberth, H. and Leibnitz, E., (1965), J. Prakt. Chem., 27: 276. Brown, I. and Smith, F., (1954) Aust. J. Chem., 7: 264. Darwent, B. De B. and Winkler, C.A., (1943) J. Phys. Chem., 47: 442. Henty, C.J., McManamey, W.J. and Prince, R.G.H., (1964) J. Appl. Chem., 14: 148. Mertl, I., (1972) Collect. Czech. Chem. Commun., 37: 366. Palmer, D.A. and Smith, B.D., (1972), J. Chem. Eng. Data, 17: 71. Scatchard, G., Wood, S.E. and Mochel, J.M., (1946), J. Am. Chem. Sot., 58: 1960. Saunders, D.F. and Spaull, A.J.B., (1961) Z. Phys. Chem. (Frankfurt), 28: 332. Sorensen, J.M. and Arlt, W., (1979), Liquid-liquid Equilibrium Data Collection, (Binary System) Chemistry Data Series, Vol. V, Part 1, 1979a, p. 341; 1979b, p. 33; 1979c, p. 268; 1979d, p. 258; 1979e, p. 367. Tasic, A., Djordjevic, B. and Grozdanic, D., (1978) Chem. Eng. Sci., 33: 189. Thornton, J.D. and Garner, F.H., (1951) J. Appl. Chem., 1: ~61. Week, HI. and Hurt, H., (1954) Ind. Eng. Chem., 46: 2521. Weissman, S. and Wood, S.E., (1960), J. Chem. Phys., 32: 1153. Wittrig, T.S., (1977), B.S. Degree Thesis, University of Illinois, Urbana.
80.910 2543.1
- 32.570 60.465
HP T
- 138.84 22326.00
141.01 - 51.605
335.25 - 1699.5
245.42 66.032
GP T
FP T
EP T
DP T
CP T
573.61 - 127.67
BP T
88.260 - 13.641
- 27.130 - 2061.9
162.13 20708.00
- 112.66 50.633
- 173.60 1840.2
- 135.93 - 51.210
- 163.72 - 289.58
- 55.810 - 152.16
0.010
0.028
0.140
15.00
0.010
2.000
1.654
0.500
41.170 - 27.658
- 4.9800 - 35.009
410.08 646.98 354.83 919.71
34.820 79.614
- 4.9800 - 35.009
105.01 -41.765
23.710 675.56
2057.42 1004.25
2.4900 - 2461.18
A21
283.76 631.42
410.08 646.98
517.19 1132.3
545.71 649.72
115.13 249.30
1419.32 4203.10
A12
122.21 191.78
(a)-(c) a12
A,,
A21
(a)-(b)
AP T
System
0.120
0.190
0.527
0.190
0.068
0.333
0.039
0.038
@42
12.000 - 101.22
12.000 - 512.03
228.71 1093.0
354.83 919.71
82.200 - 126.20
89.570 121.97
- 149.34 218.65
1284.21 696.073
42
(b)-(c)
71.000 292.74
71.000 874.11
54.360 - 504.33
41.170 - 27.658
73.790 431.28
60.280 225.85
1131.13 571.350
0.150
0.150
0.057
0.120
0.086
0.700
0.527
0.450
a12
and of NRTL (P,
- 86.890 81.382
A21
Fitting VLE data and LLE data with unique interaction parameters. Binary interaction parameters of UNIQUAC Prausnitz’s work (UNIQUAC); T, this work (PI+ NRTL).
TABLE 3
191
In Table 2, there are eight ternary systems, which were treated before by Prausnitz et al. (1980) with the AC method (UNIQUAC). The binary VLE data (or miscibility data) and ternary LLE data involved are listed in Table 2, and the majority are selected from the work of Prausnitz et al. The binary interaction parameters of PT + NRTL EOS used in this work are shown in Table 3. For comparison, the UNIQUAC binary parameters of the AC method used by Prausnitz et al. are tabulated side by side. In fitting these data, the deviations between calculation and experiment are shown in Table TABLE 4 Fitting VLE data and LLE data with unique interaction parameters. Comparison of deviation LLE
VLE
System
(a-b)
(b-c)
(a-c)
(a-b-c)
dp/p
dy
dp/p
dy
dx
dx
Type
A
Prausnitz This work
0.0130 0.0151
0.00575 0.00948
0.0689 0.0116
0.04476 0.01997
0.00376 0.02205
0.00850 0.01757
I
B
Prausnitz This work
0.0169 0.0238
0.01296 0.01466
0.0544 0.0572
0.01644 0.02543
0.00255 0.00119
0.01120 0.02923
I
C
Prausnitz This work
0.0206 0.0153
0.01308 0.00945
0.0277 0.0064
0.01582 0.01126
0.00954 0.00560
0.01074 0.00583
I
D
Prausnitz This work
0.0175 0.0407
0.01887 0.02043
0.0242 0.0140
0.00800 0.01021
0.00437 0.00865
0.01009 0.00962
I
E
Prausnitz This work
0.0171 0.0043
0.00500 0.00652
0.01721 0.00573
(LLE)
0.00264 0.00566
0.00745 0.00163
II
F
Prausnitz This work
0.0158 0.0093
0.00740 0.00309
0.04676 0.03428
(LLE)
0.01687 0.02857
0.01363 0.01100
II
G
Prausnitz This work
0.0244 0.0252
0.00795 0.01023
0.0359 0.0305
0.01864 0.00762
0.00264 0.00566
0.00850 0.01494
I
H
Prausnitz This work
0.0087 0.0428
0.00582 0.01836
0.0359 0.0343
0.01864 0.00731
0.01721 0.00573
0.01838 0.01029
I
Root-mean-square deviation: l/2
forLIE:dx= f E 5 [Xi.j,m.cal - x~,,,~,~~~I~/~/N/M i i=l j=l m-1 l/2 forVLE dp/p= ? [ Pi(cal)/Pi(cxp) - l] ‘/N I=1
I
l/2
dy =
t [Y,.;w i=l
-
~,.i~ex~~l*/N
192
4. In the same table, the deviations of the AC approach is also given, which are calculated with the parameters already known. From this table we can see, for fitting VLE data and LLE data simultaneously, the accuracy of PT + NRTL EOS is near to that of the AC approach. The results of fitting VLE data and LLE data of ternary systems are shown in Fig. 2. It seems that, in fitting binary VLE data, the deviations mainly come from the PT EOS, which is somewhat inaccurate to describe the saturated vapor pressure of pure fluid. In the present work, the special parameters F and E, of the PT EOS are treated as constant, and are obtained by fitting saturated vapor pressure and density over a very large range of temperature (from the triple point to the critical point). However, in fitting VLE data with the AC approach, we use the empirical equation proposed by Prausnitz et al. (1980), in comparison to which the PT EOS is naturally approximate for saturated vapor pressure at lower reduced temperature. If the special parameters F and E, are correlated over a small range of temperature, the accuracy of fitting VLE data with PT EOS would increase.
2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.01: 0
1
Xl -
2
3
4
Y1110
1,
5
i 6
MOLE
: 7
: 8
: 9
FRRCTION
I 10 I
28 27 26 25 24
kl
23 22 21 201+--G. 0 1 Xl -
2 Yl(l0
3
4 1.
5 MOLE
6
7
8
FRRCTION
9
10 I
Fig. 2. (a) P-x-y diagram of binary system n-heptane(l)-benzeneJ2) (T = 318.15 K). (b) P-x-y diagram of binary system acetonitrile(l)-benzene(2) (T= 318.15 K). (c) LLE diagram of ternary system n-heptane(l)-benzene(2)-acetonitrile(3) (T = 318.15 K; P = 2 atm.).
193 20 18 16 14 12 10 a 6 a 2 I Ki
2
3
4
5
MOLE
10-l.
6
PER
20 19
o ox
.16 17
7
CENT
8 OF
9
10
(31
PT-NRTL PLAIT POINT EXPERIMENTRL
16 15 14 13 12 11 10 m -
0 MOLE
2 PER
4
6
8
10
CENT
OF
(21
12
14
16
IN RIGHT
18
20
PHRSE
Fig. 2. (continued).
In this work, the majority of special parameters F and cc in the F’T EOS were taken from Georgeton et al. (1986). Nevertheless, for some components, we could not find such parameters in the literature, which were correlated with the saturated vapor pressure and density of pure fluid. They are now shown in Table 5. In preparing this table, the data of saturated vapor pressure and of density are taken from Daubert and Danner (1985). In the PT EOS, all critical constants are taken from Reid et al. (1977), the
TABLE 5 PT EOS parameters Compound
Formula
F
%
Acetonitrile Aniline Furfural Methylcyclopentane Nitromethane 2,2,4_Trimethylpentane
C,H,N GH,N C,H,4 C6H,, CH,NO, CsH,,
0.451753 0.914014 0.947067 0.767125 0.647310 0.855657
0.22043 0.30216 0.29730 0.31611 0.25928 0.31428
194
only exception is furfural, for which the critical constants are taken from Daubert and Danner (1985).
CALCULATION
AND PREDICTION OF MULTIPHASE
EQUILIBRIA
From the above results we see that under lower pressure and at lower reduced temperature, the PT EOS with modified mixing rule does not have any evident advantage in comparison with the AC method, in fact both methods give the same accuracy, even when fitting VLE data and LLE data. However, under high pressure or near the critical point, the situation is different. In such a region, the AC method is not applicable to describe phase equilibrium, but it is reasonable to expect the EOS to be valid. Therefore, using our method it is possible to calculate and predict quantitatively the pressure-temperature diagram for multiphase equilibria. Dealing with the PT + NRTL EOS, we have utilized Michelsen’s algorithm (1982) to calculate the multiphase equilibria and its P-T diagram. Like Michelsen, the criterion of phase stability is that the inequality g(Y) = CY,(ln yj + In &(Y) - hi) >, 0
(9)
i
holds true for any Y( yi, y,, . . . , y,) ( yi a 0 and Cr, = 1). Here h, = In zi + In $+(Z), where the vector Z( zl, z2,. . . , z,) denotes the composition of the fluid in the tested phase. Hence the test for phase stability is reduced to determining the global minimum gmin of g(Y) in eqn. (9). If the problem of stability is for a system involving two liquid phases, Michelsen proposed to take the pure component as the initial value for minimizing g,in. However, we find that the initial values so chosen are not reliable for many LLE systems. For example, the system composed of heptane-methanol-benzene at 298.15 K near the plait point is unstable (it splits into two liquid phases). However, if either one of pure heptane, pure methanol and pure benzene is used to get the initial value, we would reach the false conclusion that the system is stable, i.e. gti,, 2 0. If the components (I) and (3) in a ternary system are partially miscible or immiscible, the influence of concentration of the component (2) on the distribution ratio of two phases is important. Therefore, if we calculate the binodal curve, we can correlate the parameters Ai,k( i = 1, 2, 3, 4; k = 1, 2, 3) in the following equations: In
Ki,I
In
Ki,H
=
=
Al,i
A3,t
+
A*.iXZ,I
+
A4,iX2.11
(10)
(11)
195
Then, the initial value for determining the minimum of g(Y) in eqn. (9) can be calculated by setting yi = ziKi,, (and yi = zi/Ki,rI), where K,,,(i = 1, 2, 3; m = I, II) are calculated from eqns. (10) and (11). In this manner we find that the test of phase stability for LLE is fruitful. However, this device is applicable only to ternary systems. For calculating the phase-split and multiphase boundary, Michelsen’s method is used throughout without any change.
RESULTS OF CALCULATION
We have calculated the P-T diagram of the ternary system CH,-CO,H,S together with the part of the 3-phase equilibrium. The results are compared with those of Ng et al. (1985) found by experiment. Firstly, the binary interaction parameters are correlated with the binary VLE data, and the objective function is given in eqn. (6). The interaction parameters of CH,-H,S are adjusted with the experimental LLE data of Ng et al. Table 6 provides the VLE data, which we employed to correlate the interaction parameters of Table 7. For the system CH,-H,S there are two groups of parameters, the adjusted and the original. Only the adjusted parameters are used here. We use the binary interaction parameters to calculate the phase diagram of the ternary system CH,-C02-H,S at 6.06 MPa and 211.45 K ( - 61.7 ’ C), from which the parameters Akqi(k = 1, 2, 3, 4; i = 1, 2, 3) of eqns. (10) and (11) are correlated. They are listed as Table 8, and are used in checking the stability of each phase.
TABLE 6 Binary interaction parameters of the system CH,-CO,-H,S. binary data
Vapor-liquid
equilibrium
System
Type
N
T (R)
P (atm)
Source
CH,-CO, H,S-CO, CH,-H,S
VLE VLE VLE
89 91 61
153.15-270.15 254.07-366.44 277.59-344.36
6.8- 84.1 20.0- 80.0 11.5-132.6
Knapp (1982a) Knapp (1982b) Sage (1955)
Data references Knapp, H., Doting, R., Oellrich, L., Plocker, U. and Prausnitz, J., (1982), Vapor-Liquid Equilibrium Mixtures of Low Boiling Substances, Chemistry Data Series, Vol. VI, 1982a, p. 401, p. 405; 1982b, pp. 583-584. Sage, B.H. and Lacey, W.N., (1955) Some Properties of the Lighter Hydrocarbons, Hydrogen Sulfide and Carbon Dioxide, American Petroleum Institute, p. 56-57.
196 TABLE 7 Binary interaction
parameters
of PT+ NRTL EOS for the system CH,-CO,-H,S
Binary Interaction
System
CH,-CO, CO,-H,S CH,-H,S CH, -H,S a Adjusted
Deviations
parameters
A,,
AZ1
aI2
dP/P
dy
58.962 127.62 - 75.011 - 53.933
116.97 163.35 248.12 222.37
0.270 0.300 0.390 0.180
0.01747 0.01804 0.04674 0.05360
0.01190 0.00977 0.01839 0.02168 a
according
to the liquid-liquid
’
equilibrium.
TABLE 8 The parameters
Al, AZ, A3 and A4 of eqns. 10 and 11 for the ternary
system
CH,(l)-
C%(2)-H2S(3)
Methane Carbon dioxide Hydrogen sulfide
Al
A2
A3
A4
9.834 - 2.5185 - 11.7801
- 1.8745 0.5100 2.2149
9.3178 - 2.3479 - 11.2586
- 2.0178 0.5419 2.3986
With the parameters given in Tables 6-8, we calculated the 3-phase equilibrium of this system under three different conditions, and then compared the results with the experimental data of Ng et al. as shown in Table 9, where the mole fraction of this system for each component is also given. The composition which we used was taken from the data of Ng et al. after rounding off. We also calculated the VLE of the system under the conditions used by Ng et al., and then compared the results of experiment, as shown in
3’ 1.8
/. 2.0
2.2’
2.4
TEMPERRTUREI
Fig. 3. Pressure-temperature drogen sulfide(3).
2.6 10-2.
2.8
1 3.0
Kl
diagram
of ternary
system methane(l)-carbon
dioxide(2)-hy-
6.06
-61.7
5.75
4.75
H2S
co2
CH,
H2S
co2
CH,
H,S
co2
0.2961 0.1202 0.5837
0.2564 0.1383 0.6053
0.1901 0.1395 0.6704
talc.
0.2877 0.1224 0.5899
0.2542 0.1275 0.6183
0.2063 0.1319 0.6618
0.6708 0.0830 0.2462
0.7188 0.0890 0.1922
0.8175 0.0806 0.1019
Exp.
(L2)
0.7248 0.0884 0.1868
0.7237 0.0920 0.1843
0.7431 0.0915 0.1653
Calc.
CH, Liquid
Composition of the system (mole fraction): Calculation: CH,: 0.4989; C02: 0.0988; H,S: 0.4023. Experiment: CH,: 0.4988; C02: 0.0987; H,S: 0.4022; impurities, trace. Root-mean-square deviation: dp/p: 0.0512; dx: 0.0261; du: 0.0221.
5.03
- 70.5
3.51
Exp.
3.68
Exp.
H,S Liquid (Ll)
CL-I,
Component
(MPa)
Calc.
Pressure
- 83.0
(C)
Temp.
Comparison with experimental data for three-phase equilibria
TABLE 9
..-.
0.8971 0.0518 0.0511
0.9225 0.0399 0.0376
0.9383 0.0287 0.0330
Exp.
(v)
Vapor
._
0.9054 0.0428 0.0519
0.9357 0.0326 0.0317
0.9623 0.0217 0.0160
talc.
36.5
27.1
17.0
Exp.
(Ll)
36.3
28.8
21.4
Calc.
Volume Percent
12.3 51.2
12.4 60.5
13.9 69.1
Exp.
(L2)(v)
13.8 49.9
12.2 59.0
13.4 65.2
talc.
T
(C)
9.0 10.0 9.50 -18.0 - 18.2 - 18.2 -45.6 -45.6 -45.7 - 74.0 - 74.0 - 14.5 12.2 - 12.9
P
(MPa)
5.95 8.24 10.55 3.51 6.81 9.06 2.83 4.86 7.78 1.50 2.86 4.20 11.84 7.13
8.8 16.32 26.11 5.69 16.41 25.83 6.26 14.04 29.41 4.31 9.71 18.93 35.85 37.16
Exp.
CH,
0.15 -0.04 -0.95 0.57 0.52 1.74 0.83 1.15 5.01 0.79 2.28 4.44 - 3.22 - 12.2
Diff. 8.27 8.97 9.85 1.69 10.76 10.76 10.80 12.73 10.20 13.13 14.88 14.24 9.83 13.63
Bxp.
co2
Liquid phase (mole %a)
hp. 82.88 74.71 63.38 86.62 72.83 63.41 82.94 73.23 60.39 82.56 75.41 66.83 54.32 49.21
Diff. - 0.94 0.02 0.08 0.28 0.26 0.42 0.71 0.24 1.12 1.03 0.22 - 0.29 0.17 - 2.48
H2S
0.18 0.02 0.87 -0.85 -0.78 -2.16 - 1.54 - 1.39 -6.13 - 1.82 -2.50 -4.15 3.05 14.65
Diff.
Comparison with experimental data for two-phase equilibria
TABLE 10
58.04 61.02 59.59 68.08 75.63 74.12 82.26 86.22 82.18 90.02 93.17 94.10 50.66 70.75
Exp.
‘334
0.31 0.09 0.76 0.50 -0.19 -0.02 0.10 - 0.05 -4.02 0.37 0.50 0.40 4.98 - 1.91
Diff. 10.34 9.96 9.58 10.71 8.82 8.47 8.67 6.74 6.45 6.48 4.30 3.38 9.80 8.99
Exp.
co2
Vapor phase (mole W)
ExP. 31.62 29.02 30.83 21.21 15.55 17.41 9.07 7.04 11.37 3.50 2.53 2.52 39.54 20.26
Diff. 0.08 0.22 0.28 -0.01 0.17 0.00 - 0.02 - 0.09 0.80 - 0.47 - 0.45 - 0.35 0.04 - 0.07
H2S
-0.39 -0.31 -1.04 -0.49 0.02 0.02 -0.08 0.14 3.22 0.10 -0.05 0.05 - 5.02 1.98
Diff. 5.95 8.24 10.55 3.51 6.81 9.06 2.83 4.86 7.78 1.50 2.86 4.20 11.84 7.13
9.0 10.0 9.50 - 18.0 -18.2 -18.2 -45.6 -45.6 -45.1 -14.0 - 74.0 -74.5 12.2 -72.9
phlPa) (‘c,
3.3 7.6 14.7 3.3 12.3 27.3 4.2 11.3 48.9 3.0 1.1 18.4 3.8 38.7
Exp.
3.1 7.8 15.5 3.3 13.7 28.2 4.9 12.9 49.4 3.1 8.5 21.0 16.3 37.5
Calc.
Liquid volume (8)
199 7.5 7.0 ; IL 5
6.5
s
5.5
z
5.0
:
4.5
A EXPERIMENTAL
6.0
4.0 3.5 18.5
19 19.5
20
20.5
21 21.5
22 22.5
TEMPERRTIJREI1O-l.Kl
Fig. 4. Pressure-temperature diagram of ternary system methane(l)-carbon drogen sulfide(3) (part of the three-phase equilibrium).
dioxide(2)-hy-
Table 10. The pressure-temperature diagram of this system is given in Fig. 3, where the experimental data are marked as well. For clearness, the part of the 3-phase equilibrium is enlarged and shown in Fig. 4. Tables 8 and 9 and Figs. 3 and 4 clearly prove that the calculated results agree with the experimental ones. Within our knowledge such a comparison has never been reported before.
CONCLUSION
AND DISCUSSION
This paper is the first one to report fitting LLE data with EOS, which involves only two interaction parameters per binary system, and has an accuracy equal to that of the AC approach using the same number of adjustable parameters. A wide range of systems containing fluids far from ideal have been studied. Moreover its effectiveness is not limited by temperature and pressure. That means, we can use it to calculate multiphase equilibria in a P-T diagram. The calculation of multiphase P-T diagrams which agree quantitatively with the experimental data, as we have done here, has not been described before. For describing fluids far from ideal with EOS, we have done only the first step, and the following problems might be important. (1) In the preceding article (Huang, 1990), we find it is more accurate to describe VLE with PT + UNIQUAC EOS, but for many systems, PT + UNIQUAC EOS diminishes the accuracy when used to describe LLE. We guess the UNIQUAC parameters of area and of volume may be inaccurate. Indeed, if all interaction parameters are set to zero, PT + UNIQUAC EOS cannot be used to describe accurately the VLE of hydrocarbon-hydrocarbon binary systems, although it is possible for the PT EOS or PT + NRTL
200
EOS to do that. For taking UNIQUAC area and volume parameters (see Gmehling and Onken, 1977), the reference fluid is the ideal solution (Abrams and Prausnitz, 1975), which does not adapt to our mixing rule (Huang, 1990). We doubt whether the accuracy of describing LLE may be increased by changing these parameters. (2) For VLE and for LLE, only binary and ternary systems are involved here. For testing phase stability of LLE by assigning initial’ values, our method is applicable only to ternary systems. When applied to systems consisting of four or more components, correctness of our conclusion seems subject to further examination. (3) We find in the present work that the interaction parameters are sensitive to temperature. For describing pressure-temperature diagrams of multiphases more accurately, it seems important to study the relation between parameters and temperature. (4) In the present form, the Patel-Teja EOS cannot be used to calculate accurately the critical volume of pure fluids. To describe in our manner the phase behavior in the critical region for a mixing fluid, it seems necessary to make some modifications to the PT EOS.
ACKNOWLEDGEMENT
The author thanks Professor C.-C. Hwang (Nanjing University, Nanjing, China) for language revision of the manuscript.
LIST OF SYMBOLS
A F
7 G K A4
N P
z! r
T V
parameter of interaction between two molecules parameter in the Patel-Teja equation of state fugacity in zero-pressure reference state fugacity free enthalpy phase equilibrium constant total number of components in fluid number of experimental points pressure surface area parameter (in UNIQUAC eqn.) gas constant volume parameter (in UNIQUAC eqn.) temperature phase mole fraction
201
Y, Z vector of composition mole fraction in liquid phase x mole fraction in vapor phase Y z total mole fraction in mixture Greek letters
Iz
parameter or of NRTL model fugacity coefficient parameter of PT equation of state
Superscripts E
excess property
Subscripts critical property component i and j 4 j m phase m Other symbols which we have used are explained first time. C
where they occur for the
REFERENCES Abrams, D. and Prausnitz, J.M., 1975. Statistical thermodynamics of liquid mixtures: A new expression for the excess Gibbs energy of partly of completely miscible systems. AIChE J., 21: 116-128. Anderson, T.F. and Prausnitz, J.M., 1978. Application of the UNIQUAC Equation to calculation of multicomponent phase equilibria. 2. Liquid-liquid equilibria. Ind. Eng. Chem. Proc. Des. Dev., 17: 561-567. Bender, E. and Block, U., 1975. Thermodynamic calculation for liquid-liquid extraction. Verfahrenstecknik, 9: 106. Dauber& T.E. and Danner, R.P., 1985. Data Compilation Tables Of Properties Of Pure Compounds, American Institute of Chemical Engineers, New York. Hou, Y.C. and Hu, W., 1988. Calculation of ternary liquid-liquid equilibrium by equation of state. Proc. Chem. Ind. Eng. Sot. China, May 30-June 2,1988, p. 65-74. Huang, H., 1990. A new mixing rule for the Patel-Teja equation of state. Study of vapor-liquid equilibria. Fluid Phase Equilibria, 58: 93-115. Georgeton, G.K., Smith, R.L. and Teja, A.S., 1986. Application of cubic equations of state to polar fluids and fluid mixtures, in K.C. Chao, and J. Robinson (I%.),, Equations of State: Theories and Applications. ACS Symp. Ser. 300, American Chemical Society, Washington D.C., p. 434.
202 Gmehling, J. and Onken, U., 1977. Vapor-Liquid Equilibrium Data Collection, Vol. 1, DECHEMA Chem. Data Set, DECHEMA, Frankfurt, Germany. Kurihara, K., Tochigi, K. and Kojima, K., 1987. Mixing rule containing regular solution and residual excess free energy. J. Chem. Eng. Jpn., 20: 227-231. Leet, W.A., Lin, H.-M. and Chao, K.-C., 1986. Cubic chain-of-rotators equation of state II for strongly polar substances and their mixtures. Ind. Eng. Chem. Fundam., 25: 695-701. Luedecke, D. and Prausnitz, J.M., 1985. Phase equilibria for strongly nonideal mixtures from an equation of state with density-dependent mixing rules. Fluid Phase Equilibria, 22: 1-19. Mathias, P.M. and Copeman, T.W., 1983. Extension of the Peng-Robinson equation of state to complex mixtures: equation of the various forms of the local composition concept. Fluid Phase Equilibria, 13: 91-108. Magnussen, T., Sorensen, J.M., Rasmussen, P. and Fredenslund, A., 1980. Liquid-liquid equilibrium data: their retrieval, correlation and prediction. Part III prediction. Fluid Phase Equilibria, 4: 151-163. Michelsen, M.L., 1980. Calculation of phase envelopes and critical points for multicomponent mixtures. Fluid Phase Equilibria, 4: l-10. Michelsen, M.L., 1982. The isothermal flash problem. Parts I and II. Fluid Phase Equilibria, 9: l-40. Michelsen, M.L., 1987. Multiphase isenthalpic and isentropic flash algorithms. Fluid Phase Equilibria, 33: 13-27. Ng, H.-J., Robinson, D.C. and Leu, A.-D., 1985. Critical phenomena in a mixture of methane, carbon dioxide and hydrogen sulfide. Fluid Phase Equilibria, 19: 273-286. Ohanomah, M.O. and Thompson, D.W., 1984. Computation of multicomponent phase equilibria. Parts I, II and III. Comput. Chem. Eng., 8: 147-170. Ohta, T., 1989. Prediction of ternary phase equilibria by the PRSV2 equation of state with the NRTL mixing rule. Fluid Phase Equilibria, 47: 1-15. Patel, N.C. and Teja, A.S., 1982. A new cubic equation of state for fluids and fluid mixtures. Chem. Eng. Sci., 37: 463-473. Peng, D.-Y. and Robinson, B., 1976a. A new two-constant equation of state. Ind. Eng. Chem. Fundam., 15: 59-64. Peng, D.-Y. and Robinson, B., 1976b. Two- and three-phase equilibrium calculations for systems containing water. Can. J. Chem. Eng., 54: 595. Prausnitz, J.M., Anderson, T.F., Grens, E.A., Eckert, C.A., Hsieh, R. and O’Connell, J.P., 1980. Computer Calculations for Multicomponent Vapor-Liquid and Liquid-Liquid Equilibria. Prentice-Hall, Englewood Cliffs, NJ. Reid, R.C., Prausnitz, J.M. and Sherwood, T.K., 1977. The Properties of Gases and Liquids. McGraw-Hill, New York, 3rd edn. Renon, H. and Prausnitz, J.M., 1968. Local compositions in thermodynamic excess functions for liquid mixtures. AIChE J., 14: 135. Soave, G., 1972. Equilibrium constants from a modified Redlich-Kwong equation of state. Chem. Eng. Sci., 27: 119771203. Sorensen, J.M., Magnussen, T., Rasmussen, P. and Fredenslund, A., 1979a. Liquid-liquid equilibrium: their retrieval, correlation and prediction. Part I. Fluid Phase Equilibria, 2: 297-309. Sorensen, J.M., Magnussen, T., Rasmussen, P. and Fredenslund, A., 1979b. Liquid-liquid equilibrium: their retrieval, correlation and prediction. Part II. Fluid Phase Equilibria, 3: 47-82. Sorensen, J.M. and Arlt, W., 1980. Vapor-liquid Equilibrium Data Collection, Vol. V, Part 1-3, DECHEMA Chem. Data Ser., DECHEMA, Frankfurt, Germany.
AIJ
- 549.02 855.59 - 335.53 - 565.31 927.73 - 100.65 - 584.98 900.30 -314.65
For system A6 (a)-(b) 353.51 (a)-(c) 1996.1 (b)-(c) - 44.688
For system A7 (a)-(b) 269.98 (a)-(c) 1788.6 (b)-(c) 4.2625
- 237.79 646.08 - 24.511
- 216.57 652.57 - 4.4733
- 226.14 631.49 - 1.1154
For system Bl System: Water(a)-methanol(b)-benzene(c) (a)-(b) 1347.9 - 693.61 - 284.24 266.87 (a)-(c) 2106.1 1391.5 315.18 753.90 (b)-(c) 508.70 118.62 -51.480 428.62
- 42.947 309.88 - 81.437
- 66.208 301.54 - 69.009
- 99.331 316.75 - 62.773
- 193.32 619.94 153.44
For system A5 (a)-(b) 142.11 (a)-(c) 1575.4 (b)-(c) 151.46
- 137.53 326.15 - 150.19
- 585.19 875.62 48.459
For system A4 (a)-(b) 442.95 (a)-(c) 1957.1 (b)-(c) 4.4456
AJI
System: water(a)-ethanol(b)-benzene(c) - 441.74 266.93 - 266.09 986.99 249.80 807.82 118.04 - 73.352 256.iO
AIJ
UNIQUAC
For system Al, A2, A3 (a)-(b) 376.33 (a)-(c) 2797.7 (b)-(c) 87.744
AJI
NRTL (a = 0.2)
Parameter
(R)
Binary interaction parameters used to calculate data in Table 1
TABLE Al
APPENDIX
8705.4 - 253.15 7.1397
19999.00 693.55 842.64
9999.9 609.74 869.01
8096.1 545.44 1231.3
5159.8 347.44 928.45
4404.8 152.92 1048.3
AIJ
PT-NRTL
- 407.86 1456.0 714.87
- 931.79 927.22 - 662.67
- 947.99 826.72 - 630.47
- 977.42 716.35 - 627.88
- 953.38 772.76 - 474.13
T 894.28 657.14 - 327.30
AJI
0.3 0.3 0.3
0.3 0.3 0.3
0.3 0.3 0.3
0.3 0.3 0.3
0.3 0.3 0.3
0.3 0.3 0.3
a
- 19781.00 103.15 - 18586.00
810.34 - 17912.00 331.63
122.52 209.09 - 1468.5
AIJ
PT-UNIQUAC
658.36 - 11475.00 159.00
503.93 139.60 61.433
525.49 - 999.68 205.11
AJI
UNIQUAC
399.28 823.76 - 5.1997
- 126.27 184.93 165.09
92.194 678.79 201.68
- 199.65 9.9500 - 143.65
heptane(a)-benzene(b)-methanol(c) 266.12 55.852 - 90.662 516.13 689.14 8.8469 98.175 215.71 - 71.223
- 521.56 540.07 - 61.370
System: 27.085 453.96 - 34.497
For system D2 (a)-(b) 172.78 (a)-(c) 424.15 (b)-(c) - 28.354
For system Dl (a)-(b) (a)-(c) (b)-(c)
- 241.39 1758.1 1044.1
309.11 6793.7 - 263.63
1184.0 6768.8 - 246.44
3441.5 284.28 - 1336.7
6329.8 - 160.46 156.60
6694.2 - 157.91 4370.9
For system C2 (a)-(b) 885.17 (a)-(c) 771.92 (b)-(c) - 287.82
240.38 633.85 364.15
PT-NRTL AIJ
- 890.56 - 472.06 - 1259.3
- 334.07 283.21 - 85.755
- 703.07 1161.3 47.990
144.84 583.44 495.96
AJI
For system Cl System: n-hexane(a)-I-propanol(b)-water(c) (a)-(b) 141.04 164.15 20.673 53.282 (a)-(c) 1060.0 1763.6 894.71 286.10 (b)-(c) - 390.96 1192.9 - 116.97 321.73
- 140.47 331.19 - 56.442
AIJ
- 648.68 1190.2 109.02
AJI
NRTL (a = 0.2)
AIJ
For system B2 (a)-(b) 1303.5 (a)-(c) 2827.9 (b)-(c) 535.76 For system B3 (a)-(b) 1356.6 (a)-(c) 3008.7 (b)-(c) 590.53
(K)
Parameter
TABLE Al (continued)
- 142.52 - 4462.0 467.50
- 139.00 -4481.7 264.72
- 239.76 19997.00 3601.2
602.04 1826.3 4050.9
- 404.57 1353.3 569.51
- 464.30 1211.8 553.05
AJI
2.0 0.02 0.50
2.0 0.02 0.50
0.2 0.2 0.2
0.2 0.2 0.2
0.3 0.3 0.3
0.3 0.3 0.3
a
PT-UNIQUAC
- 80.934 - 14660.00 - 1929.8
AIJ
AJI
195.74 291.94 88.072
- 125.53
(b)-(c)
(a)-(b)
154.75
154.75
473.17
2102.8 - 92.441
1267.2 29.822
525.57 60.388
861.16
For system G3
G+(b)
- 94.072
401.73
- 274.93
- 274.26
1160.3
(b)-(c)
- 237.84
(a)-(c)
2173.5
For system G2 (a)-(b) - 108.27
- 162.60
- 35.149
433.38
- 323.46
- 127.84
- 44.159
797.83
- 60.466
For system GI System: water(a)-acetic acid(b)-toluene(c) (a)-(b) - 108.27 - 231.84 - 323.46 - 60.466 (a)-(c) 2173.5 1160.3 433.38 797.83 (b)-(c) 401.73 - 274.26 - 35.149 - 44.159
(b)-(c)
(a)-(c)
337.78
System: water(a)-acetic acid(b)-tetrachloromethane(c) 185.08 - 374.61 - 238.97 - 48.556
- 80.747
649.69
- 72.446
337.78
649.69
- 72,446
For system Fl (a)-(b)
248.20
495.06
- 380.36
- 80.747
495.06
- 380.36
300.20
1267.6
- 514.60
248.20
1267.6
- 514.60
144.39
3.4314
216.66
(b)--(c)
2054.0
For system E2 (a)-(b)
(a)-(c)
300.20
2054.0
(b)-(c)
(a)-(c)
639.76
- 108.03
78.022 water(a)-ethanol(b)-hexane(c)
- 169.07
568.52
- 590.50
System:
324.56
For system El
157.35
(a)-(c)
For system D3
(a)-(b)
- 9460.3
- 952.87
970.01
- 16589.00
- 613.62
1174.8
- 4333.0
640.77
566.13
307.94
- 1116.5
590.18
- 2713.2
- 859.09
1826.9
- 1677.4
- 438.19
6288.7
- 335.22 - 189.78
- 1480.2
- 16155.00
307.32
- 1454.5
- 3912.1
866.91
- 1320.4
- 308.01
379.22
- 1308.4
916.68
- 260.53
17995.00
1002.4
- 571.17
18426.00
841.97
- 4222.3
2.0
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.1
0.1
0.1
0.1
0.1
0.1
0.5
0.02
- 19999.00
13.468
- 12612.00
4324.1 176.12
- 11274.00
- 15.932 1698.0 591.44
1660.5 621.12
- 164.03 643.54 - 438.58
- 348.44 915.46 - 387.18
939.20 - 398.95
58.624 462.76 134128
- 141.19 363.26 29.713
388.83 37.507
AIJ
UNIQUAC
- 243.34 308.83 - 199.44
System: butanol(a)-ethanol(b)-water(c) 270.88 - 450.93 190.76 - 311.68 1579.4 - 23.464 - 35.903 - 180.00 - 16.989
For system Jl (a)-(b) (a)-(c) (b)-(c)
134.54 105.05 317.95
System: tetrachloromethane(a)-acetone(b)-water(c) 770.01 - 359.03 368.93 - 141.50 1516.6 2074.8 1003.8 595.35 382.16 227.04 463.04 - 105.87
0.53079 340.08 - 34.955
97.241 69.843 256.57
- 193.77 523.85 - 9.1953
- 145.31 682.77 - 3.5008
720.22 - 14.202
AJI
For system 11 (a)-(b) (a)-(c) (b)-(c)
581.53 1021.5 1543.3
System: ethyl acetate(a)-butanol(b)-water(c) - 370.24 366.27 2.6032 180.70 893.17 378.72 - 326.91 1277.3 - 20.089
For system H2 (a)-(b) - 304.96 (a)-(c) 126.70 (b)-(c) - 329.35
For system Hl (a)-(b) (a)-(c) (b)-(c)
For system G5 (a)-(b) -111.72 (a)-(c) 1767.9 (b)-(c) 783.45
For system G4 (a)-(b) (a)-(c) (b)-(c)
(b)-(c)
(a)-(c)
AIJ
09
AJI
NRTL (a = 0.2)
Parameter
TABLE Al (continued)
1470.8 - 550.09 - 118.48
329.72 - 187.98 - 873.42
- 925.37 - 1337.5 - 505.66
- 840.70 - 1279.4 -491.15
- 8295.1 1161.4 - 874.62
- 6883.7 1108.9 - 886.87
1140.9 - 965.07
AIJ
PT-NRTL
5053.2 731.01 - 141.99
67.818 1679.7 270.42
- 3602.8 368.86 - 119.95
- 3209.3 516.23 - 310.58
- 1359.8 417.72 - 7830.2
- 1481.7 538.94 - 6396.5
715.47 - 8930.2
AJI
0.1 1.0 1.4
0.3 0.05 0.505
0.2 0.4 1.0
0.2 0.4 1.0
0.2 0.2 0.2
0.2 0.2 0.2
0.2 0.2
0!
901.64 445.75 254.62
AIJ
- 19895.00 11.929 1244.2
AJI
PT-UNIQUAC
E
207 TABLE A2 Density (g cmP3) calculated for the ternary systems in Table 1 System
Density of left phase
Density of right phase
Al, A2, A3 A4 A5 A6 A7
1.0 -0.855 0.985-0.861 0.983-0.875 0.980-0.872 0.980-0.870
0.867-0.854 0.859-0.848 0.850-0.841 0.840-0.832 0.831-0.823
Bl B2 B3
1.01 -0.845 1.00 -0.830 0.994-0.817
0.863-0.846 0.849-0.830 0.834-0.814
Cl c2
0.709-0.883 0.6440.769
0.948-0.914 0.984-0.943
Dl D2 D3
0.708-0.723 0.694-0.731 0.682-0.699
0.786-0.782 0.777-0.762 0.746-0.729
El E2 Fl
0.988-0.774 0.909-0.743 1.03 -1.20
0.662-0.679 0.658-0.710 1.59 -1.37
Gl G2 G3 G4 GS
1.04 1.05 1.03 1.02 1.01
0.874-0.937 0.882-0.969 0.866-0.946 0.852-0.920 0.861-0.937
Hl H2 I1 Jl
0.951-0.899 0.947-0.867 1.58 -1.00 0.840-0.874
-1.07 -1.10 -1.08 -1.06 -1.07
1.03 -1.02 1.02 -1.01 1.02 -0.898 0.998-0.963
Fluid Phase Equilibria, 65 (1991) 181-207 Elsevier Science Publishers B.V., Amsterdam
Study of ternary liquid-liquid and of multiphase equilibria
equilibria
Hai Huang * Beijing Graduate School, University of Petroleum, P. 0. Box 902 Beijing 100083 (People’s Republic of China) (Received March 22, 1989; accepted in final form December 18, 1990)
ABSTRACT
Huang, H., 1991. Study of ternary liquid-liquid
equilibria and of multiphase equilibria.
Fluid
Phase Equilibria, 65: 181-207. In this study, we have calculated LLE and LLVE in ternary systems, using the PT equation of state with the mixing rules of Kurihara et al. Only two interaction parameters per binary system are used to correlate VLE and LLE data simultaneously and the results appear to be comparable with the activity coefficient approach using the same number of adjustable parameters. A wide range of systems has been studied. The multiphase P-T diagram for the system H,S-CO,-CH, is also plotted and compared with the experimental data.
INTRODUCTION
Liquid-liquid equilibria (LLE) form the basis of extractive unit operations. Due to the demand from chemical engineering, much work has already been done in this field, mainly using the NRTL (Renon and Prausnitz, 1968) and UNIQUAC (Abrams and Prausnitz, 1975) empirical equations to correlate and to predict the activity coefficient (AC) for LLE. Sorensen et al. (1979a,b) and Magnussen et al. (1980) have written excellent reviews. Sorensen and Arlt (1980) collected many LLE data and fitted them with NRTL and UNIQUAC equations. These works prove that NRTL and UNIQUAC equations are useful to calculate and to predict LLE under ordinary pressure. However, the main defect of the AC method is that the influence of pressure cannot be found from LLE by using this method. * Present address: Department of Chemical Engineering, University of Delaware, Newark, DE 19716, USA. 0378-3812/91/$03.50
0 1991 Elsevier Science Publishers B.V.
182
Popularization of computer technology has led to more and more methods of calculating and predicting multiphase equilibria being proposed. For instance, Michelsen (1980, 1982, and 1987) and Ohanomah and Thompson (1984) proposed complete algorithms to calculate multiphase equilibria, but we have not seen any report where the calculated multiphase P-T diagram agrees quantitatively with the experiment. Recently, a few reports about calculating LLE with equations of state (EOS) have been published. Among these, the work of Peng and Robinson (1976b) may be the pioneer; they calculated LLE with the PR EOS (Peng and Robinson, 1976a). Next, Mathias and Copeman (1983) calculated four binary systems of LLE with EOS (local composition). Leet et al. (1986) fitted data of LLE for several binary systems with the CCOR(I1) EOS. Luedecke and Prausnitz (1985) proposed a new EOS, and calculated LLE of binary systems with their new EOS. As regard to fitting data of LLE for ternary systems with EOS, we have found only one report in the literature, made by Hou and Hu (1988). Since in this case EOS involves ternary interaction parameters with insufficient experimental points, this method cannot be widely used without difficulty. In most cases, the interaction parameters used to describe LLE are correlated only with the LLE data. For example, in “Liquid-Liquid Equilibrium Data Collections” (Sorensen and Arlt, 1980) it was done in this way. Prausnitz et al. (1980) pointed out that “fitting ternary LLE data only may yield unrealistic parameters, that predict grossly erroneous results when used in regions not identical to those employed in data reduction”. Some authors try to predict ternary LLE only using pure component parameters and binary VLE data. For example, Ohta (1989) predicted ternary LLE in this way with three different methods, but generally, they cannot attain high accuracy in predicting LLE data. On the other hand, some workers have reported the method of fitting VLE data and LLE data with unique interaction parameters of AC, e.g. Bender and Block, 1975 and Anderson and Prausnitz, 1978, but very few such papers have been published. Prausnitz et al. (1980) reported some results related to ten groups of LLE data. Perhaps due to experimental difficulty, very few multiphase P-T diagrams plotted according to experiments have appeared in the literature. We find only one such report (Ng et al., 1985), in which the multiphase P-T diagram for the system methane(0.5)-carbon dioxide(O.l)-hydrogen sulfide(0.4) was studied experimentally. Before this experiment, Michelson (1982) had calculated the multiphase P-T diagram for the same system with the SRK EOS (Soave, 1972). The sketch of the phase boundary plotted according to his calculation seems like that found by experiment, but is too approximate for comparison in minute detail.
183 FITTING LLE DATA WITH PT EOS
In the preceding article (Huang 1990), we described how to modify the PT EOS (Pate1 and Teja, 1982) by means of a new mixing rule of Kurihara et al. (1987). There, we calculated VLE and LLE for binary systems, and predicted VLE for ternary systems from binary data. In the following, we shall apply the same device to calculate LLE for ternary systems. From the objective functions presented by Sorensen et al. (1979b), we choose three functions to fit the experimental data. The six binary interaction parameters of the ternary system are optimized in three steps. The three relevant objective functions are described below. (1) The sum of the squared differences between the logarithmic fugacities taken from a series of experiments: F=
fJ~(lnf,jl-lnhjIr)2 j=l
0)
j-1
where A4 is the number of experimental data, and fijk the fugacity calculated from EOS of the component i in the phase k under the temperature and pressure of the jth experiment. In the preceding article (Huang, 1990), we have given the expression for In & as a function of T, P and the composition. Now we choose In h.jr = In f;.jn instead of fiiI =fijrr as the criterion of phase equilibria, because in the criterion used later we shall have hi1 + 0 and fijrI + 0 during optimizing the interaction parameters of EOS. With the objective function F chosen above to adjust the interaction parameters of EOS, convergence is very fast. The parameters of EOS can be determined in advance. (2) The sum of the squared differences between the compositions obtained from experiment and from calculation. Here, the object of optimization is given by F=
5i f j=l
i=l
(x~~~-Z;~~)~,
(2)
k=l
where xijk denotes the mole fraction of component i of the phase k in the jth experiment, Tijk the corresponding value obtained in calculating conjugate phase compositions from the equimolar mixture of two experimental conjugate phases, and A4 the number of experimental data. The correlated interaction parameters of EOS found by this method are quite near to the optimal. Since there is no artificial intervention in the process of optimization, the convergence ought to be rapid.
184
(3) The sum of the squared differences between the compositions by experiment and those found by calculation optimally: P=
$ min{ i j=l
i=l
g
found
(xiik - T:jk)2}
(3)
k=l
where x”’ indicates line, i.e.
that the number
Z:kj is obtained
from the optimal
tie
would be a minimum. In order to determine the optimal tie line, we use the binodal curve in the phase diagram. The coordinate parameter of the tie line in the binodal curve is used as the sole variable in optimizing eqn. (4). For this purpose we use Michelsen’s computer program (Michelsen, 1980; Sorensen and Arlt, 1980). Since we choose In Air = In fijII as the criterion of phase equilibria, it is impossible to trace the binodal curve by starting from one side of the ternary phase diagram, where certain components may not exist, and their fugacity would be zero. Practically, we use EOS with the given interaction parameters to calculate the first and the second points of the binodal curve, and then step by step extend the curve with Michelson’s method. At this moment, EOS with binary interaction parameters intervenes. If the parameters used are too inaccurate, the initial points derived may be unsuitable for iteration. We have to change the initial points (artificial intervention) and slow down the speed of convergence. In practical calculations, the parameters are optimized in four steps: (1) Guessing the initial interaction parameters; (2) first adjustment of all six parameters by using objective function (1); (3) second adjustment of parameters by using objective function (2); (4) final optimization of all six parameters by using the objective function given by eqn. (3). Guessing the initial parameters, we use the binary interaction parameters from VLE. In the formula NRTL, the third parameter a is determined by the data of VLE but not optimized by that of LLE. In the following three steps, we always take as initial values the results of the preceding step. In the last step, the initial interaction parameters are then very near to the optimal ones, so that artificial intervention is often negligible. Our last step is analogous to the last step used by Sorensen et al. (1979b). When we make optimization, the relation between interaction parameters of EOS and the data of LLE is too complicated, and the iteration process of Newton’s method often diverges. So we utilize the simplex method to optimize the parameters and thus obtain satisfactory results.
185
For this purpose we take the average error as defined by Sorensen and Arlt (1980) Err = lOO(F/6/M)“2
(5)
where F is given by the eqn. (3). Here definition (5) is employed in order to compare our results with those obtained by the AC method.
TESTING
THE RESULTS
Our results of correlations are given in Table 1, which contains 10 ternary systems, 27 series of experimental data and 267 tie lines of equilibrium. All experimental data come from “Data Collection”, Vol. V edited by Sorensen and Arlt (1980). For comparison with the AC method, the interaction parameters in the NRTL and UNIQUAC equations are taken from the same source. The results obtained by the AC method are identical with those in the book of Sorensen and Arlt, but we reserve three effective figures to facilitate comparison. All experimental data of the same ternary system at the same temperature are compiled in one series of data, although they come from different authors. In fitting experimental data with the PT EOS, the pressure is assumed to be 2 atm. absolute. Here, we use UNIQUAC and NR-TL models to calculate the excess free enthalpy of the FT EOS. In fitting LLE data, the method of FT + UNIQUAC EOS is not satisfactory when used in some systems. That explains why we did not fit all data with this method alone. All interaction parameters used in the above four methods are shown in the Appendix. The PT EOS can provide the density of liquid phases, but a complete list of the densities for all experiments which we have done is too long, so we mention in the Appendix only the range of densities calculated from each series of data. In Fig. 1 we present in detail the results of our calculation for the ternary systems. In the figure, the binodal curve and the plait point are calculated from EOS, while the tie lines in Fig. 1 are plotted from experiments. This figure is similar to those in the book of Sorensen and Arlt.
FITTING VLE DATA AND LLE DATA WITH ONE SET OF INTERACTION ETERS
PARAM-
In this section, we fit six parameters altogether with LLE data only and obtain satisfactory results. However, numerical values of the parameters are not significant, due to the correlation between the parameters and the narrow composition range for some components. In many cases, they cannot
186 TABLE 1 Comparison of root-mean-square correlating liquid-liquid equilibria indicated). System
N
T (R)
deviation (mole percentage) with various methods for data (all data taken from Sorensen and Arlt (1980), as
NRTL
A) System: water(a)-ethanol(b)-benzene(c) Al 48 298.15 0.688 A2 8 299.15 0.340 6 293.15 0.824 A3 A4 6 308.15 0.498 5 318.15 0.544 A5 5 328.15 0.429 A6 5 337.15 0.585 A7
UNIQUAC
PTNRTL
Part 2, p. 350-360 0.832 0.779 0.578 0.649 0.829 1.029 0.611 0.510 0.582 0.348 1.112 0.349 0.709 0.434
B) System: water(a)-methanol(b)-benzene(c) Bl 13 303.15 1.313 13 B2 318.15 0.758 B3 15 333.15 0.757
Part 2, p. 121-123 1.381 1.439 0.879 0.479 1.205 0.713
C) System: Cl c2
n-hexane(a)-1-propanol(b)-water(c) 15 298.15 0.425 5 310.95 0.341
Part 2, p. 577-579 1.779 0.561 0.455 0.288
D) System: heptane(a)-benzene(b)-methanol(c) 5 279.95 Dl 0.636 D2 6 286.95 0.906 5 D3 305.95 0.663
Part 2, p. 118-120 0.507 0.840 0.759 1.040 0.622 0.613
E) System: water(a)-ethanol(b)-hexane(c) 8 El 293.15 1.020 E2 6 298.15 2.490
Part 2, p. 364-365 0.919 1.819
F) System: water(a)-acetic acid(b)-tetrachloromethane(c) Fl 10 298.15 0.664 0.640
Part 2, p. 4 0.590
Part 2, p. 287-296 0.568 0.436 0.878 0.485 0.793 0.620 0.289 0.331 0.860 0.860
H)System: HI H2
Part 3, p. 50-51 0.365 0.401 0.213 0.594
I) System: tetrachloromethane(a)-acetone(b)-water(c) I1 10 303.15 0.552 0.459 J) System: butanol(a)-ethanol(b)-water(c) Jl 10 298.15 0.364
2.405
2.423
8.714
3.755
0.723 0.832
G) System: water(a)-acetic acid(b)-toluene(c) 10 Gl 298.15 0.512 G2 8 303.15 1.009 13 G3 313.15 0.756 G4 5 328.15 0.233 G5 8 333.15 0.821 ethyl acetate(a)-butanol(b)-water(c) 5 273.15 0.660 6 293.15 0.492
PTUNIQUAC
2.593
Part 2, p. 7 0.506
Part 2, p. 340 0.356
0.388
0.229
187
R
40
k-
35
= u u
25
E
15
!?
10
30
z
20
5 0
0
1
2
10-l.
3 MOLE
4
5
6
PER
0 MOLE
5
10
PER
15
CENT
20 OF
25 121
8
OF
10
9 (31
PLRIT POINT EXPERIMENTRL
a 4
0
7
CENT
30
35
40
IN RIGHT
45
PHFlSE
Fig. 1. LLE of water(l)-ethanol(2)-benzene(3) (1980), Part 2, pp. 351, 354:
(T= 25°C). Data from Sorensen and Arlt
be used to predict VLE for the same system at the same temperature. For example, we obtain the interaction parameters for the system waterethanol-benzene in 318.15 K (see Appendix). With these parameters to predict VLE of binary water-ethanol and of binary ethanol-benzene, we would commit a gross error (for data source see Table 2). From experimental data, the RMS deviations are dp/p = 0.3175; 0.2975 and dy = 0.3953; 0.4095, respectively, which are obviously unacceptable. Therefore we try the method proposed by Prausnitz et al. (1980) to correlate the binary parameters, so as to fit both the LLE data and the VLE data. Our next object is to calculate the P-T diagram for multiphase equilibria, so that the ranges of temperature and pressure are large. It would be very difficult to determine the experimental standard deviations, if we use the maximum likelihood principle to set the objective function. In view of this situation, we use the classical method. For VLE data of binary system, our objective function is F VLE
=
f
(
(P,,cdpi,exp
-
‘)’
+
(Yi,exp
-
Yi,cal)‘)
i=l
where N denotes the total number of experimental points.
(6)
188 TABLE 2 Fitting VLE data and LLE data with unique interaction parameters. Data liquid-vapor equilibria system and of ternary liquid-liquid equilibria system
of binary
T(K)
P (atm)
Source
A Heptane(a)-benzene(b)-methanol(c) (a)-(b VLE 15 (b)-(c) VLE 9 (a)-(c) LLE 1 (a)-(b)-(c) LLE 6
318.15 308.15 305.15 305.95
0.1664-0.2962 0.2675-0.3366 2 2
Palmer (1972) Scatchard (1946) Wittrig (1977) Wittrig (1977)
B Water(a)-ethanol(b)-benzene(c) (a)-(b) VLE 13 (b)-(c) VLE 12 (a)-(c) LLE 1 (a)-(b)-(c) LLE 12
313.15 318.15 298.15 298.15
0.0989-0.1761 0.3566-0.2675 2 2
Mertl(1972) Brown (1954) Ssrensen (1979a) Bancroft (1942)
C Heptane(a)-benzene(b)-acetonitrile(c) (a)-(b) VLE 15 (b)-(c) VLE 12 (a)-(c) LLE 1 (a)-(b)-(c) LLE 9
318.15 318.15 318.15 318.15
0.1664-0.2962 0.3566-0.2675 2 2
Palmer Brown Palmer Palmer
D Cyclohexane(a)-benzene(b)-nitromethane(c) (a)-(b) VLE 13 298.15 (b)-(c) VLE 12 318.15 (a)-(c) LLE 1 298.15 (a)-(b)-(c) LLE 3 298.15
0.1251-0.1430 0.1542-0.2979 2 2
Tasic (1978) Sanuders (1961) Serensen (1979b) Week (1954)
E 2,2,4-Trimethylpentane(a)-cyclohexane(b)-furfural(c) (a)-(b) VLE 7 308.15 (b)-(c) LLE 1 298.15 (a)-(c) LLE 1 298.15 (a)-(b)-(c) LLE 8 298.15
0.1258-0.1878 2 2 2
Battind (1966) Ssrensen (1979~) Ssrensen (1979d) Henty (1964)
F n-Hexane(a)-methylcyclopentane(b)-aniline(c) (a)-(b) VLE 9 333.15 (b)-(c) LLE 1 298.15 (a)-(c) LLE 1 298.15 (a)-(b)-(c) LLE 7 298.15
0.6847-0.7538 2 2 2
Beyer (1965) Ssrensen (1979e) Serensen (1979e) Darwent (1943)
Type
N
(1972) (1954) (1972) (1972)
G 2,2,4-Trimethylpentane(a)-benzene(b)-furfural(c) (a)-(b) VLE 7 308.15 (b)-(c) VLE 10 355.85427.85 (a)-(c) LLE 1 298.15 (a)-(b)-(c) LLE 9 298.15
O.l2OO-0.1956 1
Weissman (1960) Thorton (1951)
2 2
Serensen (1979~) Henty (1964)
H Cyclohexane(a)-benzene(b)-furfural(c) (a)-(b) VLE 11 (b)-(c) VLE 10
0.1251-0.1430 1
Tasic (1978) Thornton (1951)
2 2
Ssrensen (1979d) Henty (1964)
(a)-(c) (a)-(b)-(c)
LLE LLE
1 11
298.15 355.85427.85 298.15 298.15
189
For LLE data of a ternary function is defined as F LLE
system,
as in the last section,
(7)
=
Assembling the binary the objective function is F = Fl,2,“LE/Nl,2,“LE
the objective
VLE data and the ternary
+ F2,3,“LE/N2,3.VL.E
LLE data as a whole,
+ F,,2,3,LLE/N~,2,3,LLE
where the subscripts represent the kind of equilibria and the components involved. In eqn. (8), the two binaries (1,2) and (2,3) are completely miscible and the binary (1,3) is partially miscible. For LLE of type II, in which only the binary (1,2) is completely miscible, the term containing F2,3,VLE, corresponding to the second term of eqn. (8), is missing. Now, the NRTL equation is used to correlate the free enthalpy (Huang, 1990) defined by the F’T EOS. To fit binary VLE data in a binary system, three binary interaction parameters have to be determined. The third parameter aij is fixed according to the binary VLE data. For the pair of immiscible components, their third interaction parameter (yij is determined with the ternary LLE data. Hence while correlating the binary interaction parameters of a ternary system with eqn. (8), there are six parameters to be correlated. We take the objective function (8) and employ the simplex method to correlate the six parameters.
Notes to Table 2 Data references Bancroft, W.D. and Hubard, S.S., (1942), J. Am. Chem. Sot., 64: 347. Battind, R., (1966) J. Phys. Chem., 70: 3408. Beyer, W., Schuberth, H. and Leibnitz, E., (1965), J. Prakt. Chem., 27: 276. Brown, I. and Smith, F., (1954) Aust. J. Chem., 7: 264. Darwent, B. De B. and Winkler, C.A., (1943) J. Phys. Chem., 47: 442. Henty, C.J., McManamey, W.J. and Prince, R.G.H., (1964) J. Appl. Chem., 14: 148. Mertl, I., (1972) Collect. Czech. Chem. Commun., 37: 366. Palmer, D.A. and Smith, B.D., (1972), J. Chem. Eng. Data, 17: 71. Scatchard, G., Wood, S.E. and Mochel, J.M., (1946), J. Am. Chem. Sot., 58: 1960. Saunders, D.F. and Spaull, A.J.B., (1961) Z. Phys. Chem. (Frankfurt), 28: 332. Sorensen, J.M. and Arlt, W., (1979), Liquid-liquid Equilibrium Data Collection, (Binary System) Chemistry Data Series, Vol. V, Part 1, 1979a, p. 341; 1979b, p. 33; 1979c, p. 268; 1979d, p. 258; 1979e, p. 367. Tasic, A., Djordjevic, B. and Grozdanic, D., (1978) Chem. Eng. Sci., 33: 189. Thornton, J.D. and Garner, F.H., (1951) J. Appl. Chem., 1: ~61. Week, HI. and Hurt, H., (1954) Ind. Eng. Chem., 46: 2521. Weissman, S. and Wood, S.E., (1960), J. Chem. Phys., 32: 1153. Wittrig, T.S., (1977), B.S. Degree Thesis, University of Illinois, Urbana.
80.910 2543.1
- 32.570 60.465
HP T
- 138.84 22326.00
141.01 - 51.605
335.25 - 1699.5
245.42 66.032
GP T
FP T
EP T
DP T
CP T
573.61 - 127.67
BP T
88.260 - 13.641
- 27.130 - 2061.9
162.13 20708.00
- 112.66 50.633
- 173.60 1840.2
- 135.93 - 51.210
- 163.72 - 289.58
- 55.810 - 152.16
0.010
0.028
0.140
15.00
0.010
2.000
1.654
0.500
41.170 - 27.658
- 4.9800 - 35.009
410.08 646.98 354.83 919.71
34.820 79.614
- 4.9800 - 35.009
105.01 -41.765
23.710 675.56
2057.42 1004.25
2.4900 - 2461.18
A21
283.76 631.42
410.08 646.98
517.19 1132.3
545.71 649.72
115.13 249.30
1419.32 4203.10
A12
122.21 191.78
(a)-(c) a12
A,,
A21
(a)-(b)
AP T
System
0.120
0.190
0.527
0.190
0.068
0.333
0.039
0.038
@42
12.000 - 101.22
12.000 - 512.03
228.71 1093.0
354.83 919.71
82.200 - 126.20
89.570 121.97
- 149.34 218.65
1284.21 696.073
42
(b)-(c)
71.000 292.74
71.000 874.11
54.360 - 504.33
41.170 - 27.658
73.790 431.28
60.280 225.85
1131.13 571.350
0.150
0.150
0.057
0.120
0.086
0.700
0.527
0.450
a12
and of NRTL (P,
- 86.890 81.382
A21
Fitting VLE data and LLE data with unique interaction parameters. Binary interaction parameters of UNIQUAC Prausnitz’s work (UNIQUAC); T, this work (PI+ NRTL).
TABLE 3
191
In Table 2, there are eight ternary systems, which were treated before by Prausnitz et al. (1980) with the AC method (UNIQUAC). The binary VLE data (or miscibility data) and ternary LLE data involved are listed in Table 2, and the majority are selected from the work of Prausnitz et al. The binary interaction parameters of PT + NRTL EOS used in this work are shown in Table 3. For comparison, the UNIQUAC binary parameters of the AC method used by Prausnitz et al. are tabulated side by side. In fitting these data, the deviations between calculation and experiment are shown in Table TABLE 4 Fitting VLE data and LLE data with unique interaction parameters. Comparison of deviation LLE
VLE
System
(a-b)
(b-c)
(a-c)
(a-b-c)
dp/p
dy
dp/p
dy
dx
dx
Type
A
Prausnitz This work
0.0130 0.0151
0.00575 0.00948
0.0689 0.0116
0.04476 0.01997
0.00376 0.02205
0.00850 0.01757
I
B
Prausnitz This work
0.0169 0.0238
0.01296 0.01466
0.0544 0.0572
0.01644 0.02543
0.00255 0.00119
0.01120 0.02923
I
C
Prausnitz This work
0.0206 0.0153
0.01308 0.00945
0.0277 0.0064
0.01582 0.01126
0.00954 0.00560
0.01074 0.00583
I
D
Prausnitz This work
0.0175 0.0407
0.01887 0.02043
0.0242 0.0140
0.00800 0.01021
0.00437 0.00865
0.01009 0.00962
I
E
Prausnitz This work
0.0171 0.0043
0.00500 0.00652
0.01721 0.00573
(LLE)
0.00264 0.00566
0.00745 0.00163
II
F
Prausnitz This work
0.0158 0.0093
0.00740 0.00309
0.04676 0.03428
(LLE)
0.01687 0.02857
0.01363 0.01100
II
G
Prausnitz This work
0.0244 0.0252
0.00795 0.01023
0.0359 0.0305
0.01864 0.00762
0.00264 0.00566
0.00850 0.01494
I
H
Prausnitz This work
0.0087 0.0428
0.00582 0.01836
0.0359 0.0343
0.01864 0.00731
0.01721 0.00573
0.01838 0.01029
I
Root-mean-square deviation: l/2
forLIE:dx= f E 5 [Xi.j,m.cal - x~,,,~,~~~I~/~/N/M i i=l j=l m-1 l/2 forVLE dp/p= ? [ Pi(cal)/Pi(cxp) - l] ‘/N I=1
I
l/2
dy =
t [Y,.;w i=l
-
~,.i~ex~~l*/N
192
4. In the same table, the deviations of the AC approach is also given, which are calculated with the parameters already known. From this table we can see, for fitting VLE data and LLE data simultaneously, the accuracy of PT + NRTL EOS is near to that of the AC approach. The results of fitting VLE data and LLE data of ternary systems are shown in Fig. 2. It seems that, in fitting binary VLE data, the deviations mainly come from the PT EOS, which is somewhat inaccurate to describe the saturated vapor pressure of pure fluid. In the present work, the special parameters F and E, of the PT EOS are treated as constant, and are obtained by fitting saturated vapor pressure and density over a very large range of temperature (from the triple point to the critical point). However, in fitting VLE data with the AC approach, we use the empirical equation proposed by Prausnitz et al. (1980), in comparison to which the PT EOS is naturally approximate for saturated vapor pressure at lower reduced temperature. If the special parameters F and E, are correlated over a small range of temperature, the accuracy of fitting VLE data with PT EOS would increase.
2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.01: 0
1
Xl -
2
3
4
Y1110
1,
5
i 6
MOLE
: 7
: 8
: 9
FRRCTION
I 10 I
28 27 26 25 24
kl
23 22 21 201+--G. 0 1 Xl -
2 Yl(l0
3
4 1.
5 MOLE
6
7
8
FRRCTION
9
10 I
Fig. 2. (a) P-x-y diagram of binary system n-heptane(l)-benzeneJ2) (T = 318.15 K). (b) P-x-y diagram of binary system acetonitrile(l)-benzene(2) (T= 318.15 K). (c) LLE diagram of ternary system n-heptane(l)-benzene(2)-acetonitrile(3) (T = 318.15 K; P = 2 atm.).
193 20 18 16 14 12 10 a 6 a 2 I Ki
2
3
4
5
MOLE
10-l.
6
PER
20 19
o ox
.16 17
7
CENT
8 OF
9
10
(31
PT-NRTL PLAIT POINT EXPERIMENTRL
16 15 14 13 12 11 10 m -
0 MOLE
2 PER
4
6
8
10
CENT
OF
(21
12
14
16
IN RIGHT
18
20
PHRSE
Fig. 2. (continued).
In this work, the majority of special parameters F and cc in the F’T EOS were taken from Georgeton et al. (1986). Nevertheless, for some components, we could not find such parameters in the literature, which were correlated with the saturated vapor pressure and density of pure fluid. They are now shown in Table 5. In preparing this table, the data of saturated vapor pressure and of density are taken from Daubert and Danner (1985). In the PT EOS, all critical constants are taken from Reid et al. (1977), the
TABLE 5 PT EOS parameters Compound
Formula
F
%
Acetonitrile Aniline Furfural Methylcyclopentane Nitromethane 2,2,4_Trimethylpentane
C,H,N GH,N C,H,4 C6H,, CH,NO, CsH,,
0.451753 0.914014 0.947067 0.767125 0.647310 0.855657
0.22043 0.30216 0.29730 0.31611 0.25928 0.31428
194
only exception is furfural, for which the critical constants are taken from Daubert and Danner (1985).
CALCULATION
AND PREDICTION OF MULTIPHASE
EQUILIBRIA
From the above results we see that under lower pressure and at lower reduced temperature, the PT EOS with modified mixing rule does not have any evident advantage in comparison with the AC method, in fact both methods give the same accuracy, even when fitting VLE data and LLE data. However, under high pressure or near the critical point, the situation is different. In such a region, the AC method is not applicable to describe phase equilibrium, but it is reasonable to expect the EOS to be valid. Therefore, using our method it is possible to calculate and predict quantitatively the pressure-temperature diagram for multiphase equilibria. Dealing with the PT + NRTL EOS, we have utilized Michelsen’s algorithm (1982) to calculate the multiphase equilibria and its P-T diagram. Like Michelsen, the criterion of phase stability is that the inequality g(Y) = CY,(ln yj + In &(Y) - hi) >, 0
(9)
i
holds true for any Y( yi, y,, . . . , y,) ( yi a 0 and Cr, = 1). Here h, = In zi + In $+(Z), where the vector Z( zl, z2,. . . , z,) denotes the composition of the fluid in the tested phase. Hence the test for phase stability is reduced to determining the global minimum gmin of g(Y) in eqn. (9). If the problem of stability is for a system involving two liquid phases, Michelsen proposed to take the pure component as the initial value for minimizing g,in. However, we find that the initial values so chosen are not reliable for many LLE systems. For example, the system composed of heptane-methanol-benzene at 298.15 K near the plait point is unstable (it splits into two liquid phases). However, if either one of pure heptane, pure methanol and pure benzene is used to get the initial value, we would reach the false conclusion that the system is stable, i.e. gti,, 2 0. If the components (I) and (3) in a ternary system are partially miscible or immiscible, the influence of concentration of the component (2) on the distribution ratio of two phases is important. Therefore, if we calculate the binodal curve, we can correlate the parameters Ai,k( i = 1, 2, 3, 4; k = 1, 2, 3) in the following equations: In
Ki,I
In
Ki,H
=
=
Al,i
A3,t
+
A*.iXZ,I
+
A4,iX2.11
(10)
(11)
195
Then, the initial value for determining the minimum of g(Y) in eqn. (9) can be calculated by setting yi = ziKi,, (and yi = zi/Ki,rI), where K,,,(i = 1, 2, 3; m = I, II) are calculated from eqns. (10) and (11). In this manner we find that the test of phase stability for LLE is fruitful. However, this device is applicable only to ternary systems. For calculating the phase-split and multiphase boundary, Michelsen’s method is used throughout without any change.
RESULTS OF CALCULATION
We have calculated the P-T diagram of the ternary system CH,-CO,H,S together with the part of the 3-phase equilibrium. The results are compared with those of Ng et al. (1985) found by experiment. Firstly, the binary interaction parameters are correlated with the binary VLE data, and the objective function is given in eqn. (6). The interaction parameters of CH,-H,S are adjusted with the experimental LLE data of Ng et al. Table 6 provides the VLE data, which we employed to correlate the interaction parameters of Table 7. For the system CH,-H,S there are two groups of parameters, the adjusted and the original. Only the adjusted parameters are used here. We use the binary interaction parameters to calculate the phase diagram of the ternary system CH,-C02-H,S at 6.06 MPa and 211.45 K ( - 61.7 ’ C), from which the parameters Akqi(k = 1, 2, 3, 4; i = 1, 2, 3) of eqns. (10) and (11) are correlated. They are listed as Table 8, and are used in checking the stability of each phase.
TABLE 6 Binary interaction parameters of the system CH,-CO,-H,S. binary data
Vapor-liquid
equilibrium
System
Type
N
T (R)
P (atm)
Source
CH,-CO, H,S-CO, CH,-H,S
VLE VLE VLE
89 91 61
153.15-270.15 254.07-366.44 277.59-344.36
6.8- 84.1 20.0- 80.0 11.5-132.6
Knapp (1982a) Knapp (1982b) Sage (1955)
Data references Knapp, H., Doting, R., Oellrich, L., Plocker, U. and Prausnitz, J., (1982), Vapor-Liquid Equilibrium Mixtures of Low Boiling Substances, Chemistry Data Series, Vol. VI, 1982a, p. 401, p. 405; 1982b, pp. 583-584. Sage, B.H. and Lacey, W.N., (1955) Some Properties of the Lighter Hydrocarbons, Hydrogen Sulfide and Carbon Dioxide, American Petroleum Institute, p. 56-57.
196 TABLE 7 Binary interaction
parameters
of PT+ NRTL EOS for the system CH,-CO,-H,S
Binary Interaction
System
CH,-CO, CO,-H,S CH,-H,S CH, -H,S a Adjusted
Deviations
parameters
A,,
AZ1
aI2
dP/P
dy
58.962 127.62 - 75.011 - 53.933
116.97 163.35 248.12 222.37
0.270 0.300 0.390 0.180
0.01747 0.01804 0.04674 0.05360
0.01190 0.00977 0.01839 0.02168 a
according
to the liquid-liquid
’
equilibrium.
TABLE 8 The parameters
Al, AZ, A3 and A4 of eqns. 10 and 11 for the ternary
system
CH,(l)-
C%(2)-H2S(3)
Methane Carbon dioxide Hydrogen sulfide
Al
A2
A3
A4
9.834 - 2.5185 - 11.7801
- 1.8745 0.5100 2.2149
9.3178 - 2.3479 - 11.2586
- 2.0178 0.5419 2.3986
With the parameters given in Tables 6-8, we calculated the 3-phase equilibrium of this system under three different conditions, and then compared the results with the experimental data of Ng et al. as shown in Table 9, where the mole fraction of this system for each component is also given. The composition which we used was taken from the data of Ng et al. after rounding off. We also calculated the VLE of the system under the conditions used by Ng et al., and then compared the results of experiment, as shown in
3’ 1.8
/. 2.0
2.2’
2.4
TEMPERRTUREI
Fig. 3. Pressure-temperature drogen sulfide(3).
2.6 10-2.
2.8
1 3.0
Kl
diagram
of ternary
system methane(l)-carbon
dioxide(2)-hy-
6.06
-61.7
5.75
4.75
H2S
co2
CH,
H2S
co2
CH,
H,S
co2
0.2961 0.1202 0.5837
0.2564 0.1383 0.6053
0.1901 0.1395 0.6704
talc.
0.2877 0.1224 0.5899
0.2542 0.1275 0.6183
0.2063 0.1319 0.6618
0.6708 0.0830 0.2462
0.7188 0.0890 0.1922
0.8175 0.0806 0.1019
Exp.
(L2)
0.7248 0.0884 0.1868
0.7237 0.0920 0.1843
0.7431 0.0915 0.1653
Calc.
CH, Liquid
Composition of the system (mole fraction): Calculation: CH,: 0.4989; C02: 0.0988; H,S: 0.4023. Experiment: CH,: 0.4988; C02: 0.0987; H,S: 0.4022; impurities, trace. Root-mean-square deviation: dp/p: 0.0512; dx: 0.0261; du: 0.0221.
5.03
- 70.5
3.51
Exp.
3.68
Exp.
H,S Liquid (Ll)
CL-I,
Component
(MPa)
Calc.
Pressure
- 83.0
(C)
Temp.
Comparison with experimental data for three-phase equilibria
TABLE 9
..-.
0.8971 0.0518 0.0511
0.9225 0.0399 0.0376
0.9383 0.0287 0.0330
Exp.
(v)
Vapor
._
0.9054 0.0428 0.0519
0.9357 0.0326 0.0317
0.9623 0.0217 0.0160
talc.
36.5
27.1
17.0
Exp.
(Ll)
36.3
28.8
21.4
Calc.
Volume Percent
12.3 51.2
12.4 60.5
13.9 69.1
Exp.
(L2)(v)
13.8 49.9
12.2 59.0
13.4 65.2
talc.
T
(C)
9.0 10.0 9.50 -18.0 - 18.2 - 18.2 -45.6 -45.6 -45.7 - 74.0 - 74.0 - 14.5 12.2 - 12.9
P
(MPa)
5.95 8.24 10.55 3.51 6.81 9.06 2.83 4.86 7.78 1.50 2.86 4.20 11.84 7.13
8.8 16.32 26.11 5.69 16.41 25.83 6.26 14.04 29.41 4.31 9.71 18.93 35.85 37.16
Exp.
CH,
0.15 -0.04 -0.95 0.57 0.52 1.74 0.83 1.15 5.01 0.79 2.28 4.44 - 3.22 - 12.2
Diff. 8.27 8.97 9.85 1.69 10.76 10.76 10.80 12.73 10.20 13.13 14.88 14.24 9.83 13.63
Bxp.
co2
Liquid phase (mole %a)
hp. 82.88 74.71 63.38 86.62 72.83 63.41 82.94 73.23 60.39 82.56 75.41 66.83 54.32 49.21
Diff. - 0.94 0.02 0.08 0.28 0.26 0.42 0.71 0.24 1.12 1.03 0.22 - 0.29 0.17 - 2.48
H2S
0.18 0.02 0.87 -0.85 -0.78 -2.16 - 1.54 - 1.39 -6.13 - 1.82 -2.50 -4.15 3.05 14.65
Diff.
Comparison with experimental data for two-phase equilibria
TABLE 10
58.04 61.02 59.59 68.08 75.63 74.12 82.26 86.22 82.18 90.02 93.17 94.10 50.66 70.75
Exp.
‘334
0.31 0.09 0.76 0.50 -0.19 -0.02 0.10 - 0.05 -4.02 0.37 0.50 0.40 4.98 - 1.91
Diff. 10.34 9.96 9.58 10.71 8.82 8.47 8.67 6.74 6.45 6.48 4.30 3.38 9.80 8.99
Exp.
co2
Vapor phase (mole W)
ExP. 31.62 29.02 30.83 21.21 15.55 17.41 9.07 7.04 11.37 3.50 2.53 2.52 39.54 20.26
Diff. 0.08 0.22 0.28 -0.01 0.17 0.00 - 0.02 - 0.09 0.80 - 0.47 - 0.45 - 0.35 0.04 - 0.07
H2S
-0.39 -0.31 -1.04 -0.49 0.02 0.02 -0.08 0.14 3.22 0.10 -0.05 0.05 - 5.02 1.98
Diff. 5.95 8.24 10.55 3.51 6.81 9.06 2.83 4.86 7.78 1.50 2.86 4.20 11.84 7.13
9.0 10.0 9.50 - 18.0 -18.2 -18.2 -45.6 -45.6 -45.1 -14.0 - 74.0 -74.5 12.2 -72.9
phlPa) (‘c,
3.3 7.6 14.7 3.3 12.3 27.3 4.2 11.3 48.9 3.0 1.1 18.4 3.8 38.7
Exp.
3.1 7.8 15.5 3.3 13.7 28.2 4.9 12.9 49.4 3.1 8.5 21.0 16.3 37.5
Calc.
Liquid volume (8)
199 7.5 7.0 ; IL 5
6.5
s
5.5
z
5.0
:
4.5
A EXPERIMENTAL
6.0
4.0 3.5 18.5
19 19.5
20
20.5
21 21.5
22 22.5
TEMPERRTIJREI1O-l.Kl
Fig. 4. Pressure-temperature diagram of ternary system methane(l)-carbon drogen sulfide(3) (part of the three-phase equilibrium).
dioxide(2)-hy-
Table 10. The pressure-temperature diagram of this system is given in Fig. 3, where the experimental data are marked as well. For clearness, the part of the 3-phase equilibrium is enlarged and shown in Fig. 4. Tables 8 and 9 and Figs. 3 and 4 clearly prove that the calculated results agree with the experimental ones. Within our knowledge such a comparison has never been reported before.
CONCLUSION
AND DISCUSSION
This paper is the first one to report fitting LLE data with EOS, which involves only two interaction parameters per binary system, and has an accuracy equal to that of the AC approach using the same number of adjustable parameters. A wide range of systems containing fluids far from ideal have been studied. Moreover its effectiveness is not limited by temperature and pressure. That means, we can use it to calculate multiphase equilibria in a P-T diagram. The calculation of multiphase P-T diagrams which agree quantitatively with the experimental data, as we have done here, has not been described before. For describing fluids far from ideal with EOS, we have done only the first step, and the following problems might be important. (1) In the preceding article (Huang, 1990), we find it is more accurate to describe VLE with PT + UNIQUAC EOS, but for many systems, PT + UNIQUAC EOS diminishes the accuracy when used to describe LLE. We guess the UNIQUAC parameters of area and of volume may be inaccurate. Indeed, if all interaction parameters are set to zero, PT + UNIQUAC EOS cannot be used to describe accurately the VLE of hydrocarbon-hydrocarbon binary systems, although it is possible for the PT EOS or PT + NRTL
200
EOS to do that. For taking UNIQUAC area and volume parameters (see Gmehling and Onken, 1977), the reference fluid is the ideal solution (Abrams and Prausnitz, 1975), which does not adapt to our mixing rule (Huang, 1990). We doubt whether the accuracy of describing LLE may be increased by changing these parameters. (2) For VLE and for LLE, only binary and ternary systems are involved here. For testing phase stability of LLE by assigning initial’ values, our method is applicable only to ternary systems. When applied to systems consisting of four or more components, correctness of our conclusion seems subject to further examination. (3) We find in the present work that the interaction parameters are sensitive to temperature. For describing pressure-temperature diagrams of multiphases more accurately, it seems important to study the relation between parameters and temperature. (4) In the present form, the Patel-Teja EOS cannot be used to calculate accurately the critical volume of pure fluids. To describe in our manner the phase behavior in the critical region for a mixing fluid, it seems necessary to make some modifications to the PT EOS.
ACKNOWLEDGEMENT
The author thanks Professor C.-C. Hwang (Nanjing University, Nanjing, China) for language revision of the manuscript.
LIST OF SYMBOLS
A F
7 G K A4
N P
z! r
T V
parameter of interaction between two molecules parameter in the Patel-Teja equation of state fugacity in zero-pressure reference state fugacity free enthalpy phase equilibrium constant total number of components in fluid number of experimental points pressure surface area parameter (in UNIQUAC eqn.) gas constant volume parameter (in UNIQUAC eqn.) temperature phase mole fraction
201
Y, Z vector of composition mole fraction in liquid phase x mole fraction in vapor phase Y z total mole fraction in mixture Greek letters
Iz
parameter or of NRTL model fugacity coefficient parameter of PT equation of state
Superscripts E
excess property
Subscripts critical property component i and j 4 j m phase m Other symbols which we have used are explained first time. C
where they occur for the
REFERENCES Abrams, D. and Prausnitz, J.M., 1975. Statistical thermodynamics of liquid mixtures: A new expression for the excess Gibbs energy of partly of completely miscible systems. AIChE J., 21: 116-128. Anderson, T.F. and Prausnitz, J.M., 1978. Application of the UNIQUAC Equation to calculation of multicomponent phase equilibria. 2. Liquid-liquid equilibria. Ind. Eng. Chem. Proc. Des. Dev., 17: 561-567. Bender, E. and Block, U., 1975. Thermodynamic calculation for liquid-liquid extraction. Verfahrenstecknik, 9: 106. Dauber& T.E. and Danner, R.P., 1985. Data Compilation Tables Of Properties Of Pure Compounds, American Institute of Chemical Engineers, New York. Hou, Y.C. and Hu, W., 1988. Calculation of ternary liquid-liquid equilibrium by equation of state. Proc. Chem. Ind. Eng. Sot. China, May 30-June 2,1988, p. 65-74. Huang, H., 1990. A new mixing rule for the Patel-Teja equation of state. Study of vapor-liquid equilibria. Fluid Phase Equilibria, 58: 93-115. Georgeton, G.K., Smith, R.L. and Teja, A.S., 1986. Application of cubic equations of state to polar fluids and fluid mixtures, in K.C. Chao, and J. Robinson (I%.),, Equations of State: Theories and Applications. ACS Symp. Ser. 300, American Chemical Society, Washington D.C., p. 434.
202 Gmehling, J. and Onken, U., 1977. Vapor-Liquid Equilibrium Data Collection, Vol. 1, DECHEMA Chem. Data Set, DECHEMA, Frankfurt, Germany. Kurihara, K., Tochigi, K. and Kojima, K., 1987. Mixing rule containing regular solution and residual excess free energy. J. Chem. Eng. Jpn., 20: 227-231. Leet, W.A., Lin, H.-M. and Chao, K.-C., 1986. Cubic chain-of-rotators equation of state II for strongly polar substances and their mixtures. Ind. Eng. Chem. Fundam., 25: 695-701. Luedecke, D. and Prausnitz, J.M., 1985. Phase equilibria for strongly nonideal mixtures from an equation of state with density-dependent mixing rules. Fluid Phase Equilibria, 22: 1-19. Mathias, P.M. and Copeman, T.W., 1983. Extension of the Peng-Robinson equation of state to complex mixtures: equation of the various forms of the local composition concept. Fluid Phase Equilibria, 13: 91-108. Magnussen, T., Sorensen, J.M., Rasmussen, P. and Fredenslund, A., 1980. Liquid-liquid equilibrium data: their retrieval, correlation and prediction. Part III prediction. Fluid Phase Equilibria, 4: 151-163. Michelsen, M.L., 1980. Calculation of phase envelopes and critical points for multicomponent mixtures. Fluid Phase Equilibria, 4: l-10. Michelsen, M.L., 1982. The isothermal flash problem. Parts I and II. Fluid Phase Equilibria, 9: l-40. Michelsen, M.L., 1987. Multiphase isenthalpic and isentropic flash algorithms. Fluid Phase Equilibria, 33: 13-27. Ng, H.-J., Robinson, D.C. and Leu, A.-D., 1985. Critical phenomena in a mixture of methane, carbon dioxide and hydrogen sulfide. Fluid Phase Equilibria, 19: 273-286. Ohanomah, M.O. and Thompson, D.W., 1984. Computation of multicomponent phase equilibria. Parts I, II and III. Comput. Chem. Eng., 8: 147-170. Ohta, T., 1989. Prediction of ternary phase equilibria by the PRSV2 equation of state with the NRTL mixing rule. Fluid Phase Equilibria, 47: 1-15. Patel, N.C. and Teja, A.S., 1982. A new cubic equation of state for fluids and fluid mixtures. Chem. Eng. Sci., 37: 463-473. Peng, D.-Y. and Robinson, B., 1976a. A new two-constant equation of state. Ind. Eng. Chem. Fundam., 15: 59-64. Peng, D.-Y. and Robinson, B., 1976b. Two- and three-phase equilibrium calculations for systems containing water. Can. J. Chem. Eng., 54: 595. Prausnitz, J.M., Anderson, T.F., Grens, E.A., Eckert, C.A., Hsieh, R. and O’Connell, J.P., 1980. Computer Calculations for Multicomponent Vapor-Liquid and Liquid-Liquid Equilibria. Prentice-Hall, Englewood Cliffs, NJ. Reid, R.C., Prausnitz, J.M. and Sherwood, T.K., 1977. The Properties of Gases and Liquids. McGraw-Hill, New York, 3rd edn. Renon, H. and Prausnitz, J.M., 1968. Local compositions in thermodynamic excess functions for liquid mixtures. AIChE J., 14: 135. Soave, G., 1972. Equilibrium constants from a modified Redlich-Kwong equation of state. Chem. Eng. Sci., 27: 119771203. Sorensen, J.M., Magnussen, T., Rasmussen, P. and Fredenslund, A., 1979a. Liquid-liquid equilibrium: their retrieval, correlation and prediction. Part I. Fluid Phase Equilibria, 2: 297-309. Sorensen, J.M., Magnussen, T., Rasmussen, P. and Fredenslund, A., 1979b. Liquid-liquid equilibrium: their retrieval, correlation and prediction. Part II. Fluid Phase Equilibria, 3: 47-82. Sorensen, J.M. and Arlt, W., 1980. Vapor-liquid Equilibrium Data Collection, Vol. V, Part 1-3, DECHEMA Chem. Data Ser., DECHEMA, Frankfurt, Germany.
AIJ
- 549.02 855.59 - 335.53 - 565.31 927.73 - 100.65 - 584.98 900.30 -314.65
For system A6 (a)-(b) 353.51 (a)-(c) 1996.1 (b)-(c) - 44.688
For system A7 (a)-(b) 269.98 (a)-(c) 1788.6 (b)-(c) 4.2625
- 237.79 646.08 - 24.511
- 216.57 652.57 - 4.4733
- 226.14 631.49 - 1.1154
For system Bl System: Water(a)-methanol(b)-benzene(c) (a)-(b) 1347.9 - 693.61 - 284.24 266.87 (a)-(c) 2106.1 1391.5 315.18 753.90 (b)-(c) 508.70 118.62 -51.480 428.62
- 42.947 309.88 - 81.437
- 66.208 301.54 - 69.009
- 99.331 316.75 - 62.773
- 193.32 619.94 153.44
For system A5 (a)-(b) 142.11 (a)-(c) 1575.4 (b)-(c) 151.46
- 137.53 326.15 - 150.19
- 585.19 875.62 48.459
For system A4 (a)-(b) 442.95 (a)-(c) 1957.1 (b)-(c) 4.4456
AJI
System: water(a)-ethanol(b)-benzene(c) - 441.74 266.93 - 266.09 986.99 249.80 807.82 118.04 - 73.352 256.iO
AIJ
UNIQUAC
For system Al, A2, A3 (a)-(b) 376.33 (a)-(c) 2797.7 (b)-(c) 87.744
AJI
NRTL (a = 0.2)
Parameter
(R)
Binary interaction parameters used to calculate data in Table 1
TABLE Al
APPENDIX
8705.4 - 253.15 7.1397
19999.00 693.55 842.64
9999.9 609.74 869.01
8096.1 545.44 1231.3
5159.8 347.44 928.45
4404.8 152.92 1048.3
AIJ
PT-NRTL
- 407.86 1456.0 714.87
- 931.79 927.22 - 662.67
- 947.99 826.72 - 630.47
- 977.42 716.35 - 627.88
- 953.38 772.76 - 474.13
T 894.28 657.14 - 327.30
AJI
0.3 0.3 0.3
0.3 0.3 0.3
0.3 0.3 0.3
0.3 0.3 0.3
0.3 0.3 0.3
0.3 0.3 0.3
a
- 19781.00 103.15 - 18586.00
810.34 - 17912.00 331.63
122.52 209.09 - 1468.5
AIJ
PT-UNIQUAC
658.36 - 11475.00 159.00
503.93 139.60 61.433
525.49 - 999.68 205.11
AJI
UNIQUAC
399.28 823.76 - 5.1997
- 126.27 184.93 165.09
92.194 678.79 201.68
- 199.65 9.9500 - 143.65
heptane(a)-benzene(b)-methanol(c) 266.12 55.852 - 90.662 516.13 689.14 8.8469 98.175 215.71 - 71.223
- 521.56 540.07 - 61.370
System: 27.085 453.96 - 34.497
For system D2 (a)-(b) 172.78 (a)-(c) 424.15 (b)-(c) - 28.354
For system Dl (a)-(b) (a)-(c) (b)-(c)
- 241.39 1758.1 1044.1
309.11 6793.7 - 263.63
1184.0 6768.8 - 246.44
3441.5 284.28 - 1336.7
6329.8 - 160.46 156.60
6694.2 - 157.91 4370.9
For system C2 (a)-(b) 885.17 (a)-(c) 771.92 (b)-(c) - 287.82
240.38 633.85 364.15
PT-NRTL AIJ
- 890.56 - 472.06 - 1259.3
- 334.07 283.21 - 85.755
- 703.07 1161.3 47.990
144.84 583.44 495.96
AJI
For system Cl System: n-hexane(a)-I-propanol(b)-water(c) (a)-(b) 141.04 164.15 20.673 53.282 (a)-(c) 1060.0 1763.6 894.71 286.10 (b)-(c) - 390.96 1192.9 - 116.97 321.73
- 140.47 331.19 - 56.442
AIJ
- 648.68 1190.2 109.02
AJI
NRTL (a = 0.2)
AIJ
For system B2 (a)-(b) 1303.5 (a)-(c) 2827.9 (b)-(c) 535.76 For system B3 (a)-(b) 1356.6 (a)-(c) 3008.7 (b)-(c) 590.53
(K)
Parameter
TABLE Al (continued)
- 142.52 - 4462.0 467.50
- 139.00 -4481.7 264.72
- 239.76 19997.00 3601.2
602.04 1826.3 4050.9
- 404.57 1353.3 569.51
- 464.30 1211.8 553.05
AJI
2.0 0.02 0.50
2.0 0.02 0.50
0.2 0.2 0.2
0.2 0.2 0.2
0.3 0.3 0.3
0.3 0.3 0.3
a
PT-UNIQUAC
- 80.934 - 14660.00 - 1929.8
AIJ
AJI
195.74 291.94 88.072
- 125.53
(b)-(c)
(a)-(b)
154.75
154.75
473.17
2102.8 - 92.441
1267.2 29.822
525.57 60.388
861.16
For system G3
G+(b)
- 94.072
401.73
- 274.93
- 274.26
1160.3
(b)-(c)
- 237.84
(a)-(c)
2173.5
For system G2 (a)-(b) - 108.27
- 162.60
- 35.149
433.38
- 323.46
- 127.84
- 44.159
797.83
- 60.466
For system GI System: water(a)-acetic acid(b)-toluene(c) (a)-(b) - 108.27 - 231.84 - 323.46 - 60.466 (a)-(c) 2173.5 1160.3 433.38 797.83 (b)-(c) 401.73 - 274.26 - 35.149 - 44.159
(b)-(c)
(a)-(c)
337.78
System: water(a)-acetic acid(b)-tetrachloromethane(c) 185.08 - 374.61 - 238.97 - 48.556
- 80.747
649.69
- 72.446
337.78
649.69
- 72,446
For system Fl (a)-(b)
248.20
495.06
- 380.36
- 80.747
495.06
- 380.36
300.20
1267.6
- 514.60
248.20
1267.6
- 514.60
144.39
3.4314
216.66
(b)--(c)
2054.0
For system E2 (a)-(b)
(a)-(c)
300.20
2054.0
(b)-(c)
(a)-(c)
639.76
- 108.03
78.022 water(a)-ethanol(b)-hexane(c)
- 169.07
568.52
- 590.50
System:
324.56
For system El
157.35
(a)-(c)
For system D3
(a)-(b)
- 9460.3
- 952.87
970.01
- 16589.00
- 613.62
1174.8
- 4333.0
640.77
566.13
307.94
- 1116.5
590.18
- 2713.2
- 859.09
1826.9
- 1677.4
- 438.19
6288.7
- 335.22 - 189.78
- 1480.2
- 16155.00
307.32
- 1454.5
- 3912.1
866.91
- 1320.4
- 308.01
379.22
- 1308.4
916.68
- 260.53
17995.00
1002.4
- 571.17
18426.00
841.97
- 4222.3
2.0
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.1
0.1
0.1
0.1
0.1
0.1
0.5
0.02
- 19999.00
13.468
- 12612.00
4324.1 176.12
- 11274.00
- 15.932 1698.0 591.44
1660.5 621.12
- 164.03 643.54 - 438.58
- 348.44 915.46 - 387.18
939.20 - 398.95
58.624 462.76 134128
- 141.19 363.26 29.713
388.83 37.507
AIJ
UNIQUAC
- 243.34 308.83 - 199.44
System: butanol(a)-ethanol(b)-water(c) 270.88 - 450.93 190.76 - 311.68 1579.4 - 23.464 - 35.903 - 180.00 - 16.989
For system Jl (a)-(b) (a)-(c) (b)-(c)
134.54 105.05 317.95
System: tetrachloromethane(a)-acetone(b)-water(c) 770.01 - 359.03 368.93 - 141.50 1516.6 2074.8 1003.8 595.35 382.16 227.04 463.04 - 105.87
0.53079 340.08 - 34.955
97.241 69.843 256.57
- 193.77 523.85 - 9.1953
- 145.31 682.77 - 3.5008
720.22 - 14.202
AJI
For system 11 (a)-(b) (a)-(c) (b)-(c)
581.53 1021.5 1543.3
System: ethyl acetate(a)-butanol(b)-water(c) - 370.24 366.27 2.6032 180.70 893.17 378.72 - 326.91 1277.3 - 20.089
For system H2 (a)-(b) - 304.96 (a)-(c) 126.70 (b)-(c) - 329.35
For system Hl (a)-(b) (a)-(c) (b)-(c)
For system G5 (a)-(b) -111.72 (a)-(c) 1767.9 (b)-(c) 783.45
For system G4 (a)-(b) (a)-(c) (b)-(c)
(b)-(c)
(a)-(c)
AIJ
09
AJI
NRTL (a = 0.2)
Parameter
TABLE Al (continued)
1470.8 - 550.09 - 118.48
329.72 - 187.98 - 873.42
- 925.37 - 1337.5 - 505.66
- 840.70 - 1279.4 -491.15
- 8295.1 1161.4 - 874.62
- 6883.7 1108.9 - 886.87
1140.9 - 965.07
AIJ
PT-NRTL
5053.2 731.01 - 141.99
67.818 1679.7 270.42
- 3602.8 368.86 - 119.95
- 3209.3 516.23 - 310.58
- 1359.8 417.72 - 7830.2
- 1481.7 538.94 - 6396.5
715.47 - 8930.2
AJI
0.1 1.0 1.4
0.3 0.05 0.505
0.2 0.4 1.0
0.2 0.4 1.0
0.2 0.2 0.2
0.2 0.2 0.2
0.2 0.2
0!
901.64 445.75 254.62
AIJ
- 19895.00 11.929 1244.2
AJI
PT-UNIQUAC
E
207 TABLE A2 Density (g cmP3) calculated for the ternary systems in Table 1 System
Density of left phase
Density of right phase
Al, A2, A3 A4 A5 A6 A7
1.0 -0.855 0.985-0.861 0.983-0.875 0.980-0.872 0.980-0.870
0.867-0.854 0.859-0.848 0.850-0.841 0.840-0.832 0.831-0.823
Bl B2 B3
1.01 -0.845 1.00 -0.830 0.994-0.817
0.863-0.846 0.849-0.830 0.834-0.814
Cl c2
0.709-0.883 0.6440.769
0.948-0.914 0.984-0.943
Dl D2 D3
0.708-0.723 0.694-0.731 0.682-0.699
0.786-0.782 0.777-0.762 0.746-0.729
El E2 Fl
0.988-0.774 0.909-0.743 1.03 -1.20
0.662-0.679 0.658-0.710 1.59 -1.37
Gl G2 G3 G4 GS
1.04 1.05 1.03 1.02 1.01
0.874-0.937 0.882-0.969 0.866-0.946 0.852-0.920 0.861-0.937
Hl H2 I1 Jl
0.951-0.899 0.947-0.867 1.58 -1.00 0.840-0.874
-1.07 -1.10 -1.08 -1.06 -1.07
1.03 -1.02 1.02 -1.01 1.02 -0.898 0.998-0.963