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Simultaneous representation of excess enthalpy and vapor—liquid equilibrium data by the NRTL and UNIQUAC models
FIuid Phase Equilibria, 65 (1991) 111-133 Elsevier Science Publishers B.V., Amsterdam
111
Simultaneous representation of excess enthalpy and vapor-liquid equilibrium data by the NRTL and UNIQUAC models Ya$ar Demirel
and Hatice
Gecegiirmez
Faculty of Art and Sciences, University of Cukurova 01330 Adana (Turkey) (Received
May 29, 1990; accepted
in final form February
6, 1991)
ABSTRACT Demirel, Y. and Gecegijrmez, vapor-liquid equilibrium Equilibria, 65: 111-133.
H., 1991. Simultaneous representation data by the NRTL and UNIQUAC
of excess enthalpy and models. Fluid Phase
Using data for excess Gibbs energy, g”, and enthalpy of mixing, hE, temperature-dependent parameters of the NRTL and UNIQUAC models have been estimated for 44 systems of binary mixtures. Thirty-three of them include data for gE and hE at more than one different isotherm. The estimated parameters were later tested by predicting the total pressure, vapor-phase compositions and gE and hE data simultaneously and representing the effect of temperature on such data. System-dependency of the models, non-uniqueness in the parameters and the possibility of predicting partial miscibility for completely miscible non-ideal mixtures have been investigated.
INTRODUCTION
The NRTL (Renon and Prausnitz, 1968) and UNIQUAC (Abrams and Prausnitz, 1975) models have found extensive use in the correlation and prediction of fluid phase equilibria and chemical process calculations, such as distillation. They can predict the partial miscibility and are easily extendable to multicomponent mixtures using the parameters obtained from binary data alone. In a recent study, Cairns and Furzer (1988) pointed out that the limitations of these models must be considered in the design and retrofit of distillation columns, although there would appear to be a major difficulty in selecting the best model, since the choice may be system-dependent. This aspect needs to be investigated further in order to gain a critical and lasting evaluation on the behavior of the models. 037%3812/91/$03.50
Q 1991 Elsevier Science Publishers
B.V.
112
Renon and Prausnitz (1969) provided charts for estimating the binary adjustable energy parameters of the NRTL model from limiting activity coefficient data and from mutual solubilities. They suggested that, depending on the chemical nature of a mixture, a value of aij must be chosen. But as Tassios (1976) points out, the selection of proper values of clljj is ambiguous and difficult to apply, and the NRTL model performs best when the value of ‘Y;~is obtained by regression of the available experimental data. Flemr (1976) and McDermott and Ashton (1977) pointed out the theoretical inconsistency of the NRTL and UNIQUAC models and concluded that they should be treated only as empirical. Later Panayiotou and Vera (1981) and Iwai and Arai (1982) have discussed the statistical thermodynamic derivation of the UNIQUAC model. Tassios (1979) showed that multiple roots are possible for positive deviations from Raoult’s law for the NRTL model. However, he concluded that when (Y~,is allowed to vary, the multiplicity does not seem to represent a problem. Hanks et al. (1978) have performed a parametric analysis of the NRTL model to determine appropriate limits to its ability to represent gE and hE data simultaneously. This analysis showed that the NRTL model is not capable of correlating both gE and hE data for any system in which the value of hE exceeds a certain limit, A similar study performed by Demirel and Gecegiirmez (1989) indicates that the UNIQUAC model has a similar property. This limitation can be overcome by treating the parameters of the models as functions of temperature and using experimental heats of mixing along with vapor-liquid equilibrium (VLE) data (Murthy and Zudkevitch, 1979; Demirel and Gecegbrmez, 1989). The NRTL and UNIQUAC models have a moderate built-in temperature dependence. However, Murthy and Zudkevitch (1979) suggest that relying on the NRTL model for representing the effect of temperature on the activity coefficients can be unreliable. Both of the models perform better when the parameters vary with temperature as suggested in the literature (Anderson and Prausnitz, 1978; Demirel and McDermott, 1984). Pablo and Prausnitz (1988) suggest that for the NRTL model a strong temperature dependence is required to describe the coexistence curve in the critical region. In this study, temperature-dependent parameters for the NRTL and UNIQUAC models were estimated using gE and hE data simultaneously, and the accuracy in predicting gE, VLE and hE data by the equations has been investigated for various types of mixture including alcohols, carboxylic acids, water and esters.
113 HEAT OF MIXING
The rate of change of excess Gibbs energy, and hence the activity coefficients, y,, with respect to temperature is proportional to the excess enthalpy and is given by the Gibbs-Helmholtz equation
-=- ahEm 1 hE T2
aT
[
0)
P,x
Nagata and Yamada (1973) and Nagata et al. (1973) have shown that the NRTL model whose parameters are assumed to be expressed by a linear function of temperature is capable of representing both VLE and hE data with a single set of parameters. Renon and Prausnitz (1968) also assume that the NRTL parameters change with temperature in a linear way. g,, - g,, = ci + c2( T - 273.15)
(2)
g,, - g,, = c3 + c,( T - 273.15)
(3)
ai2 = c5 + cg( T - 273.15) With the NRTL becomes hE
parameters
(x,x2Gd
=
(Xl
+
(“hxl)
_
X2G2, >
(XlX2GlZ +(
1
x2 +
(XI I
w,2)
given in eqns. (2)-(4)
l-
+
(“12712X2 (x2
(c1
_
273
+
+I,)
>
( cg -
)
15~
.
x2G21)
the enthalpy
2
+
(4) of mixing
1 1
r,:X,C,RT2 Xl + XzG21
273.15~~) + $2y;,,2 1
12
(5)
where G2,
=
exp( -
a12721)
Gl2
=
exP( -
‘y12712)
721 =
(g21
-
&J/RT
712 =
(g,,
-
g22VRT
Abrams and Prausnitz (1975) state that when both VLE and liquid-liquid equilibrium (LLE) data are used to obtain the UNIQUAC parameters, they appear to be a smooth function of temperature. However, this is not the case for mixtures containing hydrogen bonding, such as water and alcohols (Anderson and Prausnitz, 1978; Murthy and Zudkevitch, 1979). In the present study, the effect of temperature on the characteristic energies is expressed as azl = d, + d/T
(6)
al2 = d, + d,/T
(7)
114
These are the expressions also used by Anderson and Prausnitz (1978). With the UNIQUAC parameters given in eqns. (6) and (7), the enthalpy of mixing becomes
where 72; = exp( -%1/T) G = exp( - +/T) 8j = Xjqi/Cxiqi The UNIQUAC model contains pure-component structural parameters r and q. Anderson and Prausnitz (1978) modified the UNIQUAC equation slightly and introduced new values of surface parameters, q’, for alcohols and water to be used in the residual part of the equation.
ESTIMATION
OF PARAMETERS
In estimating the temperature-dependent parameters, data for gE and hE were used simultaneously. The following objective function, which was also used by Nagata and Yamada (1973), was minimized:
where n and m are, respectively, the number of experimental gE and hE data points at a specified isothermal temperature. N is the number of isothermal system temperatures for the gE data and M is that for the hE data. For minimizing the function F, a package program called MINUIT (James, 1978) was used. The MINUIT program performs minimization and analysis of the shape of a multiparameter function, and incorporates the Fletcher and Simplex techniques. Nagata and Yamada (1973) suggest that the Simplex method is one of the most effective parameter seeking methods. Each technique may be used alone or in combination with the others, according to the behaviour of the function and the requirements of the user. Some global logic is built into the program so that, if one of the techniques fails another technique is automatically called to make another attempt.
115
An analytical method due to Tassios (1976) is used to determine the ability of the estimated parameters to predict the phase splitting. This consists of the following steps: (1) Regress the gE and hE data to obtain the temperature-dependent parameters of the models. (2) Introduce these values into the following expression for G”:
T.P
(3) Determine the value of xi = x0 for which G” value, by solving the following equation:
assumes
its minimum /
1,,
0 (11) = T,P (4) Insert the value of x1 into eqn. (10) and determine the value of G”. If G” > 0, no partial miscibility will be predicted for the estimated values of the parameters. If G” < 0, partial miscibility will be predicted. G
RESULTS AND DISCUSSION
Using data for gE and hE, temperature-dependent parameters of the NRTL and UNIQUAC models were estimated for 44 binary systems. These mixtures include hydrocarbon, ester, alcohol, ketone, carboxylic acid and aqueous components in various combinations. Only the values of the parameters c5 and cs, in the NRTL model, are forced to change between 0.1 and 0.7, and - 0.1 and 0.2, respectively. The estimated parameters for the NRTL and UNIQUAC models and variances of the fit, u, are given in Tables 1 and 2, respectively. The variance of the fit is obtained from u= (ffni’
NP 1
+
Ii4 (WNP)
) 02)
Here CNni and C”mi are the total number of data points for gE and hE, respectively, while NP is the number of parameters. The value of (I provides a measure of how well gE and hE data are represented simultaneously by the NRTL and UNIQUAC models. Temperature-dependent UNIQUAC parameters for the first 24 systems were given elsewhere (Demirel and Gecegormez, 1989).
25-45
25-50
25-45
25-45
25-55
25-55
25-55
25-55
25-45
2. Methyl acetate(l)cyclohexane(2)
3. Methanol(l)ethyl acetate(2) 4. Ethanol(l)ethyl acetate(2) 5. 2-Propanol( l)ethyl acetate(2) 6. l-Propanol(l)ethyl acetate(2) 7. Ethyl formate( methanol(2) 8. Ethyl formate(l)ethanol(2) 9. Ethyl formate(l)1-propanol(2) 10. Ethyl formate(l)2-propanol(2)
25-50
Twin- T&w. (“C)
1. Methyl acetate(l)benzene(2)
System
- 124.44 1.84 - 467.26 - 11.75 717.79 - 3.26 705.93 3.47 568.81 - 2.14 617.11 - 2.52 166.92 - 4.70 309.02 - 4.07 357.15 - 1.55 405.62 - 2.27 383.90 - 3.30 440.42 - 3.86 386.82 - 2.36 766.33 12.72 738.33 - 1.93 788.27 - 1.84 355.10 -0.98 378.16 - 2.28 523.00 1.25 465.68 1.22 563.42 - 0.62 644.69 - 1.45 565.59 - 1.16 615.92 - 1.55
0.1507 0.5800 0.0762 - 0.4730 0.4704 0.1582 0.5027 0.2106 0.5931 -0.1185 0.5999 - 0.1243 0.2538 0.2909 0.1608 0.1986 0.5173 -0.3895 0.5685 -0.1994 0.5307 - 0.2120 0.5998 0.3554
g-1,
”(cal mol-’
::a1 mol-’ K-‘)
C5
C3 (cal mol-‘)
CI (cal mol-‘) K-‘)
Temperature-dependent parameters of the NRTL model and the variation of the fit
TABLE 1
1O-2 0.0678 1o-2 0.0718 1o-2 0.4400 10-Z 0.1249 1o-2 0.0411 10-2 0.0534 1O-2 0.1130 10-2 0.0898 10-3
1o-2 0.0438 1O-2 a
0.2549 10-2 a
(K-‘)
0V2
Nagata et al. (1976)
Nagata et al. (1976)
Nagata et al. (1976)
Nagata et al. (1976)
Nagata et al. (1975)
Nagata et al. (1975)
Nagata et al. (1975)
Nagata et al. (1975)
Nagata and Yamada (1973)
Nagata and Yamada (1973)
References
25-60
13. Ethanol(l)toluene(2) 14.2-Propanol(l)n-heptane(2) 15. n-Pentanol(l)n-hexane(2) 16. n-Pentanol(l)2,3-dimethyIbutane(2) 17. n-Pentanol(l)2_methylpentane(2) 18. IsopentanoI(l)n-hexane(2) 19. n-Pentanol(l)3-methylpentane(2)
20. n-Pentanol(l)2,2_dimethylbutane(2) 21. Acetonitrile(l)benzene(2) 22. Benzene(l)n-heptane(2) 23. Acetonitrile(l)n-heptane(2)
25-55
12. Methyl acetate(l)ethanol(2)
45
25-50
45-70
25
25
25
25
25
25-45
30-60
25-45
11. Methyl acetate(l)methanol(2) 463.88 0.69 493.99 0.27 26.36 - 1.28 478.63 -1.46 1459.60 - 5.97 1829.80 - 8.35 1466.90 -1.40 1727.10 -0.85 1741.60 - 0.02 1369.50 1.65 1447.90 2.88 1801.30 - 0.36 1760.50 - 0.72 229.62 1.18 - 172.11 -4.58 1233.60 - 17.63
444.03 - 2.94 488.52 - 2.91 767.46 - 2.08 577.71 - 3.25 591.59 0.09 864.35 - 0.69 319.91 - 1.39 544.81 1.65 529.40 1.57 184.05 - 1.97 85.29 - 2.74 636.60 1.80 557.93 1.65 487.52 - 2.88 771.67 3.81 979.36 10.81
0.5928 0.3566 0.6556 0.2030 0.1516 - 0.4945 0.5723 0.4540 0.4808 0.2046 0.4874 0.9453 0.4592 0.8791 0.5589 0.1410 0.5448 0.1297 0.3390 - 0.1072 0.3182 - 0.1198 0.5565 0.1459 0.5522 0.1362 0.6402 - 0.1085 0.3639 -0.1181 0.2358 - 0.2596 10-s 0.0878 10-4 0.1583 10-3 0.0316 10-s 0.0875 1o-2 0.1068 d 1o-2 0.1737 10-2 0.1678 1o-2 0.0991 d 10-2 0.0806 10-2 0.0873 10-l 0.1612 10-r 0.1805 10-2
1o-3 0.2000 ;0-3
0.0811 to-3
Nguyen and Ratcliff (1975a) Sayegh and Ratcliff (1976) Palmer and Smith (1972) Monfort (1983) Palmer and Smith (1972) Lundberg (1964) Palmer and Smith (1972)
Nguyen and Ratcliff (1975a) Sayegh and Ratcliff (1976) Nguyen and Ratcliff (1975a) Sayegh and Ratcliff (1976) Nguyen and Ratcliff (1975a) Sayegh and Ratcliff (1976) Nguyen and Ratcliff (1975a) Sayegh and Ratcliff (1976) Nguyen and Ratcliff (1975a) Sayegh and Ratcliff (1976)
Van Ness et al. (1967)
Van Ness et al. (1967)
Nagata et al. (1972)
Nagata et al. (1972)
40
28.1,4-Dioxane(l)acetonitrile(2) 29. Carbon tetrachloride(l)diethyl sulfide(2) 30. Chloroform(l)diethyl sulfide(2) 31. Toluene(l)1-chlorohexane(2) 32.1~Chlorohexane(l)ethylbenzene(2) 15-70
15-70
25
25
15-59
15-90
5-85
5-65
=ti, - =k, (“C)
27. n-Butanol(l)n-hexane(2)
24. Ethanol(l)cyclohexane(2) 25. Water(l)butyl glycol(2) 26. n-Butanol(l)n-heptane(2)
System
TABLE 1 (continued)
c4
335.69 -0.14 605.00 1.55 302.15 0.05 635.70 0.92 - 11.61 6.35 33.45 - 4.98 156.20 1.80 66.18 0.03
787.47 - 2.56 1536.00 0.53 739.85 1.23 - 151.06 0.10 - 148.28 0.11 - 576.05 7.68 - 213.03 0.27 - 112.53 0.53
( ca 1 mol-’ 956.01 - 0.50 654.50 1.81 487.05 - 1.99
K-l)
c3 (cal mol- ‘)
1772.50 - 1.19 802.63 3.73 1147.10 - 1.67
(calmol-’
c2
cl (Cal mol-‘) K-l) (K-l) 0.0812 d 1o-3 0.428 ’ 1O-2 0.2441 1O-2 ’ 1o-3 0.0808 1o-3 c 1o-3 0.2173 1O-3 0.2978 10-l 0.1395 1O-3 0.1405 10-r 0.1957 10 -’
0.4546 0.5746 0.1000 0.4269 0.4339 -0.2605 0.5225 0.2110 0.5381 0.8901 0.5131 0.3400 0.6960 0.8171 0.4380 0.5900 0.1570 -0.5564 0.1890 0.2100 0.6499 0.2936
er/r
g-l)
c5
Paul et al. (1988)
Francesconi and Commelh (1988) Gray et al. (1988a) Gray et al. (1988b) Gray et al. (1988a) Gray et al. (1988b) Paul et al. (1988)
Nguyen and Ratcliff (1975b) Berro et al. (1982)
Scatchard and Satkiewitcz (1964) Scatchard and Wilson (1964) Savini et al. (1965); Berro and Peneloux (1984) Nguyen and Ratcliff (1975b)
References
a b ’ d i
Parameters Parameters Parameters Parameters Parameters Parameters
obtained by obtained by obtained by which show which show obtained by
33. l-Chlorohexane(l)n-propylbenzene(2) 34. 1,3-Dioxolane(l)methylcyclohexane 35.1-Chloropentane(l)di-n-butyl ether(2) 36. 1,2-Dichloroethane(l)di-n-butyl ether(2) 37. l,l,l-Trichlorethane(l)di-n-butyl ether(2) 38. Ethanol(l)acetone(2) 39. Acetone(l)water(2) 40. Cyclohexane(l)methyl methactylate(2) 41. Isobutyric acid(l)cyclohexane(2) 42. Trimethylacetic acid(l)cyclohexane(2) 43. Isobutyric acid(l)n-heptane(2) 44. Trimethylacetic acid(l)n-heptane(2) - 87.55 0.95 878.00 - 0.54 - 22.39 - 1.34 - 118.84 - 1.11 464.84 2.31 260.36 - 4.07 653.56 9.27 227.63 - 1.83 914.38 2.76 566.26 1.31 1109.80 3.34 558.93 1.19 40.77 1034.30 - 0.84 69.18 1.52 576.79 0.22 - 496.65 - 1.59 398.08 1.61 657.75 - 1.71 600.09 - 1.48 417.11 0.51 255.71 1.11 508.12 0.83 277.79 1.24
- 0.56
0.1000 - 0.9886 0.5912 0.5490 0.1001 0.3295 0.6496 0.6090 0.1072 0.9571 0.4199 - 0.2058 0.6725 - 0.8256 0.4454 - 0.8637 0.5590 0.1042 0.4235 0.1868 0.6758 0.9379 0.3111 0.2009
0.4325 10-z 0.0915 10-z 0.2266 10-2 0.1893 10-s 0.2507 10-s 0.0454 10-2 0.1495 10-s 0.0358 10-s 0.0661 10-a 0.0868 10-2 0.0252 10-s 0.0370 10-2
Nagata and Yamada (1973) by using hE data and isothermal VLE data. Nagata et al. (1972) by using hE data and isothermal VLE data. Berro et al. (1982) by using hE data and isothermal VLE data. partial miscibility. complete miscibility. Berro and Peneloux (1984) by using hE data and isothermal VLE data.
25-45
25-45
25-45
25-45
25-75
50-200
25-150
10-70
5-77
5-50
40
25-90
Lark et al. (1987)
Lark et al. (1987)
Lark et al. (1987)
Nicolaides and Eckert (1978) Campbell et al. (1987) Villamanan and Van Ness (1984) Griswold and Wong (1949) Luo et al. (1987) Hull and Lu (1984) Lark et al. (1987)
Paul et al. (1988)
Paul et al. (1986)
Paul et al. (1986)
Castellari et al. (1988)
Paul et al. (1988)
120 TABLE 2 Temperature-dependent
parameters
System
of the UNIQUAC
Twin-Max (“C)
25. Water(l)butyl gfycol(2) 26. n-Butanol(l)n-heptane(2) 27. n-Butanol(l)n-hexane(2) 28. 1,4-Dioxane(l)acetonitrile(2) 29. Carbon tetrachloride(l)diethyl sulfide(2) 30. Chloroform(l)diethyl sulfide(2) 31. Toluene(l)1 -chlorohexane( 2) 32. l-Chlorohexane(l)ethylbenzene( 2) 33. l-Chlorohexane(l)n-propylbenzene(2) 34. 1,3-Dioxalane(l)methylcyclohexane(2) 35. l-Chloropentane(l)di-n-butyl ether(2) 36. 1,2-Dichloroethane(l)di-n-butyl ether(2) 37. l,l,l-Trichloroethane(l)di-n-butyl ether(2) 38. Ethanol(l)acetone(2) 39. Acetone(l)water(2) 40. Cyclohexane(l)methyl methacrylate(2) 41. Isobutyric acid(l)cyclohexane(2) 42. Trimethylacetic acid(l)cyclohexane(2) 43. Isobutyric acid(l)n-heptane(2) 44. Trimethylacetic acid(l)n-heptane(2)
5-85 15-90 15-59 40 25 25 15-70 15-70 25-90 40 5-50 5-77 10-70 25-150 50-200 25-75 25-45 25-45 25-45 25-45
model and the variation
do (K) d, (K2)
d, (K) d, (K’)
377.68 - 49434.0 - 940.63 767180.0 1196.8 2048.2 - 358.85 50510.0 -614.33 167990.0 737.21 - 204310.0 38.61 19241.0 - 155.44 21241.0 - 32.27 - 2524.3 240.83 771.0 56.77 652.8 33.91 - 7744.9 - 102.18 - 2312.1 156.34 69392.0 712.49 - 228200.0 - 64.94 15894.0 429.29 - 75521.0 316.51 - 54611.0 682.23 - 105450.0 184.01 - 24171.0
426.31 - 89183.0 - 389.75 62225.0 - 347.83 52997.0 781.02 - 106120.0 770.32 - 219510.0 - 487.55 109680.0 - 58.47 - 12424.0 173.15 - 26767.0 29.03 995.45 - 175.56 53095.0 -41.30 - 1115.6 - 19.95 4126.3 107.72 8799.0 - 277.82 52280.0 - 461.03 252910.0 6.36 24213.0 - 161.10 42030.0 - 126.18 29692.0 - 284.38 50801.0 - 18.96 5405.33
of the fit et/2
0.2017 0.2651 0.0943 0.2748 0.2668 0.1098 0.1592 0.2105 0.3107 0.1485 0.3055 0.3046 0.2368 0.1053 0.1638 0.0439 0.0620 0.0817 0.0666 0.0572
121
The NRTL model gives multiple roots for the systems 1, 2, 11, 12 and 19, as may be seen from Table 1. The NRTL parameters for the systems n-pentanol-2-methyl pentane and ethanol-cyclohexane predict partial miscibility, which is not, however, predicted by the UNIQUAC model. Also, although the system water-butyl glycol shows phase splitting, the NRTL parameters predict complete miscibility, whereas the UNIQUAC model predicts partial miscibility. Multiple roots also result when the UNIQUAC model is used. The parameters that yield the best variation of the fit have been tabulated. For the system acetonitrile-n-heptane, which shows phase splitting, the parameters of both equations predict partial miscibility. Tassios (1976) claims that by changing the initial values of parameters it may be possible to obtain the correct values for some systems, as is seen for the system n-pentanol-3-methylpentane, which is a highly non-ideal mixture. Simultaneous predictions of vapor-phase composition, P, gE and hE, by the models are shown in Table 3. The volume and surface area parameters of the UNIQUAC model are given in Table 4. The average absolute error, S, for gE, P and hE were calculated using
The root mean square deviation (RMSD) in predicting compositions was calculated from the following equation
the vapor
phase
The values of S( gE), S(P) and RMSD were calculated at each isotherm for gE. Pure-component vapor pressures were obtained from the Antoine equation and the necessary parameters were taken from Reid et al. (1977). The value of S( hE) was calculated at each isotherm for the hE data. The results indicate the sensitivity of the NRTL and UNIQUAC equations in representing gE, VLE and hE data simultaneously at various isotherms, and also show that the models are system-dependent. For a total of 947 data points for gE, the average absolute errors for gE and p calculations from the NRTL model are 7.4% and 3.3%, respectively, while the corresponding values from the UNIQUAC model are 7.2% and 2.9%. The values of RMSD for the NRTL and UNIQUAC models are 1.9% and 1.6%, respectively. For
10 10 11
55 55 55
45
45
50
45
Ethyl formate(l)methanol(2)
Ethyl formate(l)ethanol(2)
Ethyl formate(l)1-propanol(2)
Ethyl formate(l)2-propanol(2) 11
8
12
12
11
55
Methyl acetate(l)cyclohexane(2)
Methanol(l)ethyl acetate(2) Ethanol(l)ethyl acetate(2) 2-Propanol(l)ethyl acetate(2) l-Propanol(l)ethyl acetate(2)
12 9 11 8 9
n
30 40 50 35 40
(“(3
T(gE>
Methyl acetate(l)benzene(2)
System
and UNIQUAC
25 35 45 25 35 25 35 25 35 25 35 45 25 35 45 25 35 45 25 35 45 25 35 45
25 35
(“Cl
T(hE)
11 11 9 12 12 12 16 13 18 11 20 9 8 11 9 6 13 9 8 13 9 13 12 11
13 12
M
4.71
5.16
2.17
1.48
6.62
22.75
3.07
1.22
19.88 13.01 11.36 0.70 3.28
S(gE)
NRTL
0.59 0.46 1.16 4.07 4.10 1.75 3.10 3.79 4.70 3.15 3.42 1.86 1.85 3.14 1.46 2.93 3.54 1.68 0.845 2.36 2.66 1.04 1.29 1.75
2.96 8.73
S(hE)
-
1.83
1.59
4.64
1.08
1.33
6.21
1.34
1.56
0.34 1.06 0.41 2.50 2.25
S(P)
- _.
2.67
2.41
6.06
0.70
0.62
5.14
3.96
1.28
0.76 0.38 0.58 2.10 2.71
_.
4.48
5.26
2.58
1.68
6.63
22.76
2.61
1.23
20.05 11.66 14.50 1.98 3.21
S(gE)
-
._ - ..-
4.75 3.82 4.36 4.53 5.22 4.06 4.04 3.25 4.59 3.36 2.13 2.42 10.42 4.71 7.82 10.54 4.59 7.85 10.04 3.10 8.35 8.09 1.43 8.46
5.22 6.58
S(hE)
UNIQUAC
-
15.83
1.39
4.79
1.15
1.35
6.23
1.32
1.52
0.37 1.13 0.48 2.01 1.88
S(P)
8.43
1.00
0.31
0.48
0.98
23.43
0.90
0.48
0.71 0.40 0.51 2.31 2.86
RMSD
models using the temperature-dependent
RMSD
Average absolute error, S x 100 and RMSD x 100
Simultaneous representation of hE and isothermal VLE data by the NRTL parameters
TABLE 3
8
9 9 9 9 9 19 11
25 25 25 25 45 70 45
n-Pentanol(l)2,3-dimethylbutane(2) n-Pentanol(l)2_methylpentane(2) Isopentanol(l)n-hexane(2) n-Pentanol(l)3_methylpentane(2) n-Pentanol(l)2,2dimethylbutane(2) Acetonitrile(l)benzene(2) Benzene(l)n-heptane(2)
n-Pentanol(l)n-hexane(2)
2-Propanol(l)n-heptane(2)
30 45 60 30 45 60 25 30 45 25
Ethanol(l)toluene(2) 18 18 17 17 17 16 9
10 13
45 55
Methyl acetate(l)ethanol(2)
13 13
35 45
Methyl acetate(l)methanol(2)
2.75
9
13 14 8 4
45 25 50
9
9
9
9
1.48 1.18 1.01 2.57 2.25 1.87 2.26
7.60 3.79 9.19
2.81
0.35 0.22 2.38
0.83 1.03 0.72 0.36 1.05 1.97 1.22
0.07
2.85
0.13
0.04
7.72
2.54
7.12
2.63
3.24
7.21
3.42
0.03
0.09
2.52
2.75
1.34 0.87 0.69 2.18 1.46 1.05 0.03
1.17 1.44
2.63 3.04 1.67 1.08 0.70 2.29 1.30 0.76 2.06
0.38 0.68
0.75 1.17
3.58
1.54 0.93 1.92 3.80 3.71 3.41 9.06 5.91 5.05 14.67 11.65 10.63 15.30 15.59 15.66 2.76
3.16
8.78 12.88
4.89 4.15
14 16 10 11 12 7 26 23 26 24 22 21 9
45
25
25
25
25
25
25 35 45 25 35 45 25 45 60 30 45 60 25
7.65 4.26 8.70
2.55
2.58
3.24
2.51
2.69
1.97 2.41 2.78 3.23 3.16 2.95 2.14
9.13 12.90
4.94 4.07
2.72 3.08 2.55 2.64 0.71 1.04 0.61
4.67 5.18 4.89 4.02 2.11 1.40 1.60 1.30
2.31
4.21
2.79 3.02 1.32 1.27 1.33 3.89 2.98 2.22 5.52
0.77 1.14
3.58 1.25 4.02 6.54 2.55 6.26 10.43 6.20 8.20 11.95 9.14 13.86 3.30
1.31 0.78 0.49
0.05
0.11
0.19
0.11
0.08
0.91 0.78 0.80 2.30 2.11 2.08 0.20
1.17 1.37
0.41 0.71
n-Butanol(l)n-hexane(2) l+Dioxane(l)acetonitrile(2) Carbon tetrachIoride(l)diethyl sulfide ChIoroform(l)diethyl sulfide(2) Toluene(l)1-chIorohexane(2)
n-Butanol(l)n-heptane(2)
Water(l)butyl gIycol(2)
Acetonitrile(l)n-heptane(2) Ethanol(l)cycIohexane(2)
System
TABLE 3 (continued)
5 20 35 50 65 5 25 45 65 85 30 45 55 55 15 40
9 9 9 9 9 8 8 8 8 8 19 22
23 12 10 9 12 12
5 20 35 50 65 5 25 45 65 85 60 90
59 40 25 25 50 70
15 25
25
25
45
T (hE) (“C)
8
n
45
VgE) (“C)
15.68 5.92
11 10
8.00 5.84
9.60
10
19 17
1.97
1.71 1.54 2.02 1.66 1.81 26.46 26.46 25.98 22.42 22.38 6.47 3.59
7.09
S(gE)
NRTL
10 10 10 10 10 8 8 8 8 8 17 17 10 10 10
13
m
Average absolute error, S
4.31 7.43
2.86
3.56
6.90
9.53 3.90 3.82 4.60 4.94 31.42 17.11 21.78 37.83 30.52 14.89 14.48 22.58 8.58 4.13
2.68
0.23 0.35
1.26
0.50
1.32
2.90
1.60 2.56 2.45 2.24 2.70 15.44 14.48 14.95 15.34 13.73 2.05 1.44
11.65
S(P)
100 and RMSD
S(hE)
x
100
0.22 0.26
0.75
0.76
1.28
0.77
0.98 1.36 1.60 1.81 1.83 0.94 1.88 2.08 2.61 2.00 0.82 0.92
8.73
RMSD
x
9.75 12.31
5.93
15.46
12.68
3.40
5.35 4.35 3.84 4.03 4.74 9.50 8.90 8.88 8.54 8.65 4.47 2.14
7.31
1.85 2.26
2.90
5.60
12.30
12.73 7.65 7.32 7.77 2.11 11.95 13.00 13.16 10.62 12.87 18.60 22.00 18.00 23.00 5.10
2128
S(hE)
UNIQUAC S(gE)
0.22 0.22
1.27
0.49
1.74
2.76
2.50 1.73 1.52 1.67 1.87 7.53 6.29 6.75 6.31 6.88 0.51 0.92
12.34
S(P)
0.19 0.09
0.75
0.55
1.64
0.56
0.98 1.03 0.92 1.04 1.30 0.55 0.39 0.49 0.70 1.04 0.83 0.83
9.22
RMSD
G
Isobutyric acid(l)cyclohexane(2) Trimethylacetic acid(l)cyclohexane(2) Isobutyric acid(l)n-heptane(2) Trimethylacetic acid(l)n-heptane(2)
Cyclohexane(l)methyl methacrylate(2)
Acetone(l)water(2)
Ethanol(l)acetone(2)
l,l,l-Trichloroethane(l)di-n-butyl ether(2)
1,2-Dichloroethane(l)di-n-butyl ether(2)
l-Chlorohexane(l)ethylbenzene(2) l-Chlorohexane(l)n-propylbenzene(2) 1,3-Dioxalane(l)methylcyclohexane l-Chloropentane(l)di-n-butyl ether( 2) 12 12
40 50
100 125 150 100 150 200 45 60 75 25 45 25 45 25 45 25 45 9 11 10 14 14 14 17 17 16 12 11 11 12 12 12 10 10
13 12 13 12 13
26
40
57 77 97 50 70
10 14 8
50 70 90
9 9 9 8
35 35 35
11
19
11 12 11 12 12
20 19 19 19 19
17 19 19 19 16
35
25
50
25 10 35 25 50
5 25 40 5 25
15 25 25 15 40
3.00 3.06 1.51 11.27 11.02 5.95 2.32 2.55 2.55 1.74 1.50 5.95 5.22 0.83 0.80 2.36 2.75
14.84 10.11 14.05 24.05 12.15
18.09 12.84
6.39
16.53 9.30 17.31
0.29
1.16
0.76
3.32
0.38
1.77
2.95 3.00 1.13 1.97 1.76
9.49 3.81 4.01 4.30 2.11
7.67 10.14 3.11 4.13 2.00
1.01 1.42 2.98 3.62 3.27 4.14 0.26 0.24 0.46 2.88 2.34 4.38 3.60 1.47 1.44 3.86 3.72
0.75 0.36 0.31 0.43 1.08
19.59 18.69
3.99
0.25 0.25 0.41
1.77 1.17 0.83 2.50 3.43 3.69 0.51 0.50 0.45 3.96 4.66 0.98 2.51 4.90 4.85 1.63 2.28
0.46 0.19 0.25 0.08 0.32
3.66 2.82
4.79
0.20 0.18 0.50
2.41 4.72 5.14 8.00 8.80 5.79 3.39 2.55 2.75 2.93 3.25 5.64 4.45 5.00 4.36 3.28 3.89
16.39 9.21 15.05 20.44 19.93
17.39 15.02
10.93
22.12 12.04 18.10
1.97
1.70
2.42
2.58
1.70
8.90
12.89 8.31 12.71 7.80 7.80
17.80 10.46 30.33 5.71 4.36
3.77 5.18 3.75 4.93 2.20
0.60 1.32 3.29 5.03 3.22 3.22 0.34 0.26 0.24 3.53 3.01 4.35 2.96 4.45 3.37 4.36 4.54
0.80 0.26 0.47 0.74 1.26
19.65 18.72
2.38
0.33 0.09 0.38
1.49 1.08 0.95 3.32 3.05 2.99 0.44 0.48 0.49 3.98 4.71 0.61 2.16 4.98 4.99 1.40 2.35
0.43 0.17 0.25 0.14 0.34
3.67 2.82
3.34
0.17 0.12 0.48
W
126 TABLE 4 Volume and surface area parameters
Methyl acetate Benzene Cyclohexane Methanol Ethyl acetate Ethanol 2-Propanol 1-Propanol Ethyl formate Toluene n-Heptane n-Pentanol n-Hexane 2,3-Dimethylbutane 2-Methylpentane Isopentanol 3-Methylpentane 2,ZMethylpentane Acetonitrile Water Butyl glycol n-Butanol l+Dioxane Carbon tetrachloride Diethyl sulfide Chloroform I-Chlorohexane Ethylbenzene n-Propylbenzene 1,3-Dioxolane Methylcyclohexane I-Chloropentane Di-n-butyl ether 1,2-Dichloroethane l,l,l-Trichloroethane Acetone Methyl methacrylate Isobutyric acid Trimethylacetic acid
for the UNIQUAC
model
r
4
4’
2.800 3.190 3.970 1.430 3.480 2.110 3.249 3.249 2.817 3.920 5.170 4.597 4.500 4.500 4.499 5.923 4.499 4.498 1.870 0.920 4.697 3.450 3.185 3.330 3.902 2.700 5.064 4.600 5.272 2.511 4.640 4.390 6.093 2.880 3.541 2.570 3.922 3.550 4.224
2.580 2.400 3.010 1.430 3.120 1.970 3.124 3.128 2.576 2.970 4.400 4.208 3.860 3.860 3.396 4.516 3.396 3.932 1.724 1.400 4.556 3.050 2.917 2.620 3.296 2.340 4.272 3.510 4.048 2.100 3.550 3.732 5.176 2.520 3.032 2.340 3.564 3.148 3.768
2.580 2.400 3.010 0.960 3.120 0.920 0.890 0.890 2.576 2.970 4.400 1.150 3.860 3.860 3.396 1.150 3.396 3.932 1.724 1.000 4.556 0.880 2.917 2.620 3.296 2.340 4.272 3.510 4.048 2.100 3.550 3.132 5.176 2.520 3.032 2.340 3.564 3.148 3.768
a total of 1101 data points for hE, the average absolute errors for hE calculations from the NRTL and UNIQUAC models are 5.8% and 6.‘7%, respectively. The predictions of the UNIQUAC model compare well with
127
those of the NRTL model, although the latter has six estimated parameters, and the effect of temperature on g E, VLE and hE data is well represented by both models. This shows that eqns. (2)-(4), (6) and (7) are capable of expressing the temperature dependency of the models for various types of mixture, including ones with hydrogen bonding. Using the error matrix, the off-diagonal elements of the correlation coefficient matrix, which is explained elsewhere in detail (Demirel and Gecegormez, 1989), is calculated as pjj =
Uij/(
p;; =
I) J
uiiujj)1’2
1
Pij
=
Pji
(17)
TABLE 5 The elements of correlation coefficient matrix, v: p,, = 1; pi, = p,, P IJ
System a 2
3
27
22
38
43
25
0.461 0.354 0.645 0.707 0.087 0.482 0.777 0.671 0.794 0.723 0.596 0.330 0.456 0.476 0.722
0.271 0.203 0.789 0.230 0.566 0.326 0.247 0.624 0.838 0.119 0.348 0.293 0.328 0.736 0.428
0.304 0.262 0.236 0.274 0.041 0.938 0.168 0.303 0.894 0.842 0.530 0.042 0.800 0.859 0.864
0.744 0.946 0.535 0.758 0.970 0.531 0.336 0.419 0.153 0.530 0.693 0.922 0.530 0.820 0.357
0.348 0.335 0.016 0.568 0.937 0.100 0.573 0.167 0.499 0.464 0.568 0.029 0.306 0.306 0.909
0.942 0.579 0.611 0.414 0.546 0.862 0.580 0.616 0.938 0.796 0.232 0.034 0.420 0.717 0.457
0.778 0.589 0.951 0.904 0.598 0.449 0.001 0.014 0.002 0.027 0.459 0.893 0.987 0.312 0.007
0.923 0.899 0.718 0.935 0.811 0.979
0.892 0.137 0.424 0.096 0.427 0.985
0.976 0.993 0.944 0.997 0.987 0.984
0.960 0.969 0.873 0.923 0.937 0.986
0.999 0.985 0.989 0.974 0.983 0.995
0.982 0.990 0.988 0.991 0.989 0.983
NRTL CL21 P31 p32 P41 IL42 P43 P51 p52 t453 CL54
P61 CL62 IL63 p64 p65
UNIQUAC PZI P31 p32 1141 p42 IL43
0.273 0.356 0.935 0.829 0.355 0.315
a 2, methyl acetate-cyclohexane; 3, methanol-ethyl acetate; 27, n-butanol-n-heptane; 22, benzene-n-heptane; 38, ethanol-toluene; 43, isobutyric acid-n-heptane; 25, water-butyl gIyco1.
0.887 0.956 0.612 0.998 0.846 0.931 0.999
c3
0.833 0.900 0.985 0.983 0.819 0.972 0.999
c4
CS
0.964 0.961 0.977 0.841 0.958 0.955 0.055
0.729 0.977 0.983 0.988 0.951 0.950 0.999
acetate; 27, n-butanol-n-heptane;
0.902 0.981 0.984 0.997 0.953 0.971 0.997
0.916 0.947 0.954 0.999 0.966 0.999 0.954
d,
22, benzene-n-heptane;
0.793 0.969 0.947 0.999 0.967 0.999 0.947
d,
C2
Cl
0.842 0.838 0.913 0.991 0.855 0.957 0.997
UNIQUAC
NRTL
P:
a 2, methyl acetate-cyclohexane; 3, methanol-ethyl isobutyric acid-n-heptane; 25, water-butyl glycol.
2 3 27 22 38 43 25
System ’
The global correlation coefficients
TABLE 6
0.786 0.983 0.988 0.999 0.974 0.999 0.988
d4
38, ethanol-toluene;
0.916 0.983 0.985 0.999 0.963 0.999 0.985
d3
43,
129
where uii represents the elements of the error matrix. If v is positive definite, p < 1 for all elements. If p = 0, then the parameters are uncorrelated and if p = 1, the par ameters are completely correlated. For a given parameter, the global correlation coefficient is given by
/.A;= 1 -
[ UkkVLk]
-l
(18)
and is the correlation between it and that linear combination of the other parameters most highly correlated with it. All such coefficients should be between zero and one for a positive definite error matrix. The values of error matrix correlations and the global correlation coefficients for some of the systems are given in Tables 5 and 6, respectively. The parameters for some of the systems are highly correlated, but for such parameters it is not possible to specify unique values from a given set of data. Such problems have arisen when the same free energy function is used for two of the phases, as in LLE, at points where the mole fractions in two phases are identical.
CONCLUSIONS
Although the simultaneous fit of Gibbs energy and enthalpy of mixing data for 44 binary systems is satisfactory with the NRTL and UNIQUAC models, choice of the best model is mainly system-dependent. The use of temperature-dependent parameters improves the performance of the models, but care should be exercised in the selection of the best model for a given system, and in the estimation of the parameters for especially non-ideal mixtures.
ACKNOWLEDGMENT
The authors thank the Computer Centre of Cukurova University for the computation facilities provided.
LIST OF SYMBOLS
aij Cl,
c3
c29 c4
UNIQUAC binary interaction parameter related to riT values of (g2, - gir) and (g,, - g,,) at 0°C (cal mol-‘) coefficients of temperature change of (g,, - g,,) and (g,, - g,,) (cal mol-’ K-‘)
130
value of (Y,~at 0 “C coefficient of temperature change of ai2 (K-l) ‘6 UNIQUAC parameters related to ajj (K) d,, d, UNIQUAC parameters related to ajj (K’) 6 d, F objective function as defined by eqn. (9) excess molar Gibbs energy (cal mol-‘) gE constant in eqn. (5) Gi Gh Gibbs energy of mixing (cal mol-‘) G”, G “’ second and third derivative of Gibbs energy of mixing with respect c5
excess enthalpy of mixing (cal mol-‘) number of experimental hE and gE data points, respectively, at a specified isothermal temperature number of isothermal system temperatures for hE and gE data, respectively number of parameters molecular geometric area parameter for component i molecular interaction area parameter for component i total pressure (Pa) molecular volume parameter for pure component i gas constant (cal mall’ K-‘) root mean square deviation absolute temperature (K) average absolute error (eqns. (13), (14), (15)) elements of error matrix liquid-phase mole fraction of component i vapor-phase mole fraction of component i
hE m, n M, iv NP 4; 4,! P r, R RMSD T s ‘ij xi Yi
Greek letters a12
; u 7ij 7,; pij p,,
non-randomness constant for binary l-2 interaction activity coefficient of component i area fraction of component i in residual contribution coefficient variance of the fit NRTL binary parameter UNIQUAC binary parameter elements of correlation coefficient matrix (eqn. (17)) elements of global correlation vector (eqn. (18))
Subscripts exptl calcd
experimental calculated
to the activity
131
4 .i max min
component maximum minimum
Superscript
E
excess
REFERENCES Abrams, D.S. and Prausnitz, J.M., 1975. Statistical thermodynamics of liquid mixtures: A new expression for the excess Gibbs energy of partly or completely miscible systems. AIChE J., 21: 1166128. Anderson, T.F. and Prausnitz, J.M., 1978. Application of the UNIQUAC equation to calculation of multicomponent phase equilibria 1. Vapour-liquid equilibria. Ind. Eng. Chem. Proc. Des. Dev., 17: 552-560. Berro, C. and Peneloux, A., 1984. Excess energies and excess volumes of 1-butanol-n-heptane and 2-methyl-1-propanol-n-heptane binary systems. J. Chem. Eng. Data, 29: 206214. Berro, C., Rogalski, M. and Peneloux, A., 1982. Excess Gibbs energies and excess volumes of 1-butanol-n-hexane and 2-methyl-l-propanol-n-hexane binary systems. Ind. Eng. Chem. Process Des. Dev., 27: 352-355. Cairns, B.P. and Furzer, I.A., 1988. Sensitivity testing with the predictive thermodynamic models NRTL, UNIQUAC, ASOG and UNIFAC in multicomponent separations of methanol-acetone-chloroform. Chem. Eng. Sci., 43: 495-501. Campbell, S.W., Wilsak, R.A. and Thodos, G., 1987. Vapor-liquid equilibrium measurements for the ethanol-acetone systems at 372.7, 397.7 and 422.6 K. J. Chem. Eng. Data, 32: 357-362. Castellari, C., Francesconi, R. and Comelli, F., 1988. Vapor-liquid equilibrium, excess Gibbs energy and excess enthalpy of 1,3-dioxolane-methylcyclohexane at T = 313.15 K. Can. J. Chem. Eng. 66: 131-135. Demirel, Y. and Gecegormez, H., 1989. Simultaneous correlation of excess Gibbs energy and enthalpy of mixing by the UNIQUAC equation. Can. J. Chem. Eng., 67: 455-461. Demirel, Y. and McDermott, C., 1984. Prediction of phase behaviour and bubble point of partially miscible systems. Chim. Acta Turc. 12: 173-188. Flemr, V., 1976. A note on excess Gibbs energy equations based on local composition concept. Coll. Czech. Chem. Commun., 41: 3347-3349. Francesconi, R. and Comelli, F., 1988. Excess thermodynamic properties for the binary system 1,4-dioxane-acetonitrile at 40°C. J. Chem. Eng. Data, 33: 80-83. Gray, D.F., Pasco, N.F. and Williamson, A.G., 1988a. Enthalpies of mixing of diethyl sulfide+ carbon tetrachloride and diethyl sulfide+ chloroform at 25 o C. J. Chem. Eng. Data, 33: 333-334. Gray, D.F., Pasco, N.F. and Williamson, A.G., 1988b. Liquid-vapor equilibria in mixtures of carbon tetrachloride and chloroform with dimethyl sulfide and diethyl sulfide. J. Chem. Eng. Data, 33: 335-337. Griswold, J. and Wong, S.Y., 1949. Phase equilibria of the acetone-methanol-water system from 100 o C into the critical region. I. Chem. E. Symp. Ser., No. 3, 48: 18-34.
132 Hanks, R.W. and Christensen, J.J., 1979. The prediction of multicomponent vapor-liquid equilibria from binary heats of mixing. I. Chem. Eng. Symp. Ser., No. 56, 1: 29-49. Hanks, R.W., Tan, R.L. and Christensen, J.J., 1978. In: Hanks and Christensen (1979) p. 34. Hull, D.M. and Lu, B.C.-Y., 1984. Vapor-liquid equilibria in mixtures of cyclohexane and methyl methacrylate. J. Chem. Eng. Data, 29: 417-419. Iwai, Y. and Arai, Y., 1982. Note on the local area fraction. Fluid Phase Equilibria, 9: 201-202. James, F., 1978. Interpretation of the errors on parameters given by MINUIT. Data handling division, Cem, Geneva. Lark, B.S., Banipal, T.S. and Singh, S., 1987. Excess Gibbs energy for binary mixtures containing carboxylic acids. 3. Excess Gibbs energy for isobutyric acid and trimethylacetic acid + cyclohexane and + n-heptane. J. Chem. Eng. Data, 32: 402-406. Luo, B., Hamam, S.E.M., Benson, G.C. and Lu, B.C.-Y., 1987. Excess volumes and excess enthalpies for binary mixtures of methyl methacrylate with cyclopentane and with cyclohexane. J. Chem. Eng. Data, 32: 81-82. Lundberg, G.W., 1964. Thermodynamics of solutions. XI. Heats of mixing of hydrocarbons. J. Chem. Eng. Data, 9: 193-198. McDermott, C. and Ashton, N., 1977. Note on the definition of local composition. Fluid Phase Equilibria, 1: 33-35. Monfort, J.P., 1983. Vapor-liquid equilibria for benzene-acetonitrile and toluene-acetonitrile mixtures at 343.15 K. J. Chem. Eng. Data, 28: 24-27. Murthy, A.K.S. and Zudkevitch, D., 1979. Effect of heat of mixing and vapor-liquid equilibrium on design, performance and economics of distillation. I. Chem. E. Symposium Series, No. 56, 1: 51-78. Nagata, I. and Yamada, T., 1973. On the parameter-seeking method of local composition equations. J. Chem. Eng. Jpn., 6: 215-218. Nagata, I., Ohta, T. and Takahashi, T., 1972. Excess thermodynamics functions of methyl acetate-methanol and methyl acetate-ethanol systems. J. Chem. Eng. Data, 5: 227-231. Nagata, I., Ohta, T., Takahashi, T. and Gotoh, K., 1973. Thermodynamic properties of methyl acetate-benzene and methyl acetate-cyclohexane mixtures. J. Chem. Eng. Jpn., 6: 129-133. Nagata, I., Yamada, T. and Nakagawa, S., 1975. Excess Gibbs free energies and heats of mixing for binary systems ethyl acetate with methanol, ethanol, 1-propanol and 2-propanol. J. Chem. Eng. Data, 20: 271-275. Nagata, I., Ohta, T., Ogura, M. and Yasuda, S., 1976. Excess Gibbs free energies and heats of mixing for binary systems: ethyl formate with methanol, ethanol, 1-propanol, 2-propanol. J. Chem. Eng. Data, 21: 310-313. Nicolaides, G.L. and Eckert, C.A., 1978. Experimental heats of mixing of some miscible and partially miscible nonelectrolyte systems. J. Chem. Eng. Data, 23: 152-156. Nguyen, T.H. and Ratcliff, G.A., 1975a. Heats of mixing of binary systems of iso-pentanol and n-pentanol with hexane isomers at 25°C: measurement and prediction by analytical group solution models. J. Chem. Eng. Data, 20: 256-258. Nguyen, T.H. and Ratcliff, G.A., 197513. Heats of mixing of n-alcohol-n-alkane systems at 15 and 55 o C. J. Chem. Eng, Data, 20: 252-255. Pablo, J.J. and Prausnitz, J.M., 1988. Thermodynamics of liquid-liquid equilibria including the critical region. AIChE J., 34: 1595-1606. Palmer, D.A. and Smith, B.D., 1972. Thermodynamic excess property measurements for the acetonitrile-benzene-n-heptane system at 45’C. J. Chem. Eng. Data, 17: 71-76. Panayiotou, C. and Vera, J.H., 1981. Local compositions and local surface area fractions: a theoretical discussion, Can. J. Chem. Eng., 59: 501-505.
133 Paul, HI., Krug, J., Gutsche, B. and Knapp, H., 1986. Measurements of VLE, hE and uE for binary mixtures of dibutyl ether with 1-chloropentane, 1,2-dichloroethane, and l,l,l-trichloroethane. J. Chem. Eng. Data, 31: 448-456. Paul, HI., Krug, J. and Knapp, H., 1988, Measurements of VLE, oE and hE for binary mixtures of 1-chlorohexane with three n-alkylbenzenes: toluene, ethylbenzene, n-propylbenzene. J. Chem. Eng. Data, 33: 453-454. Reid, R., Prausnitz, J.M. and Sherwood, T.K., 1977. The Properties of Gases and Liquids. McGraw-Hill, New York. Renon, H. and Prausnitz, J.M., 1968. Local compositions in thermodynamic excess functions for liquid mixtures. AIChE J., 14: 135-144. Renon, H. and Prausnitz, J.M., 1969. Estimation of parameters for the NRTL equation for excess Gibbs energies of strongly nonideal liquid mixtures. Ind. Eng. Chem. Process Des. Dev., 8: 413-419. Savini, C.G., Winterhalter, D.R. and Van Ness, H.C., 1965. Heats of mixing of some alcohol-hydrocarbon systems. J. Chem. Eng. Data, 10: 168-171. Sayegh, S.G. and Ratcliff, G.A., 1976. Excess Gibbs energies of binary systems of iso-pentano1 and n-pentanol with hexane isomers at 25“C: measurement and prediction by analytical group solution model. J. Chem. Eng. Data, 21: 71-74. Scatchard, G. and Satkiewitcz, F.G., 1964. Vapor-liquid equilibrium. XII. The system ethanol-cyclohexane from 5 to 65°C. J. Am. Chem. Sot., 86: 130-133. Scatchard, G. and Wilson, G.M., 1964. Vapor-liquid equilibrium. XIII. The system waterbutyl glycol from 5 to 85°C. J. Am. Chem. Sot. 86: 133-137. Tassios, D., 1976. Limitations in correlating strongly nonideal binary systems with the NRTL and LEMF equations. Ind. Eng. Chem. Process Des. Dev., 15: 574-578. Tassios, D., 1979. The number of roots in the NRTL and LEMF equations and the effect on their performance. Ind. Eng. Chem. Process Des. Dev., 18: 182-186. Van Ness, H.C., Soczek, C.A., Peloguin, G.L. and Machado, R.L., 1967. Thermodynamic excess properties of three alcohol-hydrocarbon systems. J. Chem. Eng. Data, 12: 217-224. Villamanan, M.A., Van Ness, H.C., 1984. Excess Thermodynamic properties for water/ acetone. J. Chem. Eng. Data 29: 429-431.
111
Simultaneous representation of excess enthalpy and vapor-liquid equilibrium data by the NRTL and UNIQUAC models Ya$ar Demirel
and Hatice
Gecegiirmez
Faculty of Art and Sciences, University of Cukurova 01330 Adana (Turkey) (Received
May 29, 1990; accepted
in final form February
6, 1991)
ABSTRACT Demirel, Y. and Gecegijrmez, vapor-liquid equilibrium Equilibria, 65: 111-133.
H., 1991. Simultaneous representation data by the NRTL and UNIQUAC
of excess enthalpy and models. Fluid Phase
Using data for excess Gibbs energy, g”, and enthalpy of mixing, hE, temperature-dependent parameters of the NRTL and UNIQUAC models have been estimated for 44 systems of binary mixtures. Thirty-three of them include data for gE and hE at more than one different isotherm. The estimated parameters were later tested by predicting the total pressure, vapor-phase compositions and gE and hE data simultaneously and representing the effect of temperature on such data. System-dependency of the models, non-uniqueness in the parameters and the possibility of predicting partial miscibility for completely miscible non-ideal mixtures have been investigated.
INTRODUCTION
The NRTL (Renon and Prausnitz, 1968) and UNIQUAC (Abrams and Prausnitz, 1975) models have found extensive use in the correlation and prediction of fluid phase equilibria and chemical process calculations, such as distillation. They can predict the partial miscibility and are easily extendable to multicomponent mixtures using the parameters obtained from binary data alone. In a recent study, Cairns and Furzer (1988) pointed out that the limitations of these models must be considered in the design and retrofit of distillation columns, although there would appear to be a major difficulty in selecting the best model, since the choice may be system-dependent. This aspect needs to be investigated further in order to gain a critical and lasting evaluation on the behavior of the models. 037%3812/91/$03.50
Q 1991 Elsevier Science Publishers
B.V.
112
Renon and Prausnitz (1969) provided charts for estimating the binary adjustable energy parameters of the NRTL model from limiting activity coefficient data and from mutual solubilities. They suggested that, depending on the chemical nature of a mixture, a value of aij must be chosen. But as Tassios (1976) points out, the selection of proper values of clljj is ambiguous and difficult to apply, and the NRTL model performs best when the value of ‘Y;~is obtained by regression of the available experimental data. Flemr (1976) and McDermott and Ashton (1977) pointed out the theoretical inconsistency of the NRTL and UNIQUAC models and concluded that they should be treated only as empirical. Later Panayiotou and Vera (1981) and Iwai and Arai (1982) have discussed the statistical thermodynamic derivation of the UNIQUAC model. Tassios (1979) showed that multiple roots are possible for positive deviations from Raoult’s law for the NRTL model. However, he concluded that when (Y~,is allowed to vary, the multiplicity does not seem to represent a problem. Hanks et al. (1978) have performed a parametric analysis of the NRTL model to determine appropriate limits to its ability to represent gE and hE data simultaneously. This analysis showed that the NRTL model is not capable of correlating both gE and hE data for any system in which the value of hE exceeds a certain limit, A similar study performed by Demirel and Gecegiirmez (1989) indicates that the UNIQUAC model has a similar property. This limitation can be overcome by treating the parameters of the models as functions of temperature and using experimental heats of mixing along with vapor-liquid equilibrium (VLE) data (Murthy and Zudkevitch, 1979; Demirel and Gecegbrmez, 1989). The NRTL and UNIQUAC models have a moderate built-in temperature dependence. However, Murthy and Zudkevitch (1979) suggest that relying on the NRTL model for representing the effect of temperature on the activity coefficients can be unreliable. Both of the models perform better when the parameters vary with temperature as suggested in the literature (Anderson and Prausnitz, 1978; Demirel and McDermott, 1984). Pablo and Prausnitz (1988) suggest that for the NRTL model a strong temperature dependence is required to describe the coexistence curve in the critical region. In this study, temperature-dependent parameters for the NRTL and UNIQUAC models were estimated using gE and hE data simultaneously, and the accuracy in predicting gE, VLE and hE data by the equations has been investigated for various types of mixture including alcohols, carboxylic acids, water and esters.
113 HEAT OF MIXING
The rate of change of excess Gibbs energy, and hence the activity coefficients, y,, with respect to temperature is proportional to the excess enthalpy and is given by the Gibbs-Helmholtz equation
-=- ahEm 1 hE T2
aT
[
0)
P,x
Nagata and Yamada (1973) and Nagata et al. (1973) have shown that the NRTL model whose parameters are assumed to be expressed by a linear function of temperature is capable of representing both VLE and hE data with a single set of parameters. Renon and Prausnitz (1968) also assume that the NRTL parameters change with temperature in a linear way. g,, - g,, = ci + c2( T - 273.15)
(2)
g,, - g,, = c3 + c,( T - 273.15)
(3)
ai2 = c5 + cg( T - 273.15) With the NRTL becomes hE
parameters
(x,x2Gd
=
(Xl
+
(“hxl)
_
X2G2, >
(XlX2GlZ +(
1
x2 +
(XI I
w,2)
given in eqns. (2)-(4)
l-
+
(“12712X2 (x2
(c1
_
273
+
+I,)
>
( cg -
)
15~
.
x2G21)
the enthalpy
2
+
(4) of mixing
1 1
r,:X,C,RT2 Xl + XzG21
273.15~~) + $2y;,,2 1
12
(5)
where G2,
=
exp( -
a12721)
Gl2
=
exP( -
‘y12712)
721 =
(g21
-
&J/RT
712 =
(g,,
-
g22VRT
Abrams and Prausnitz (1975) state that when both VLE and liquid-liquid equilibrium (LLE) data are used to obtain the UNIQUAC parameters, they appear to be a smooth function of temperature. However, this is not the case for mixtures containing hydrogen bonding, such as water and alcohols (Anderson and Prausnitz, 1978; Murthy and Zudkevitch, 1979). In the present study, the effect of temperature on the characteristic energies is expressed as azl = d, + d/T
(6)
al2 = d, + d,/T
(7)
114
These are the expressions also used by Anderson and Prausnitz (1978). With the UNIQUAC parameters given in eqns. (6) and (7), the enthalpy of mixing becomes
where 72; = exp( -%1/T) G = exp( - +/T) 8j = Xjqi/Cxiqi The UNIQUAC model contains pure-component structural parameters r and q. Anderson and Prausnitz (1978) modified the UNIQUAC equation slightly and introduced new values of surface parameters, q’, for alcohols and water to be used in the residual part of the equation.
ESTIMATION
OF PARAMETERS
In estimating the temperature-dependent parameters, data for gE and hE were used simultaneously. The following objective function, which was also used by Nagata and Yamada (1973), was minimized:
where n and m are, respectively, the number of experimental gE and hE data points at a specified isothermal temperature. N is the number of isothermal system temperatures for the gE data and M is that for the hE data. For minimizing the function F, a package program called MINUIT (James, 1978) was used. The MINUIT program performs minimization and analysis of the shape of a multiparameter function, and incorporates the Fletcher and Simplex techniques. Nagata and Yamada (1973) suggest that the Simplex method is one of the most effective parameter seeking methods. Each technique may be used alone or in combination with the others, according to the behaviour of the function and the requirements of the user. Some global logic is built into the program so that, if one of the techniques fails another technique is automatically called to make another attempt.
115
An analytical method due to Tassios (1976) is used to determine the ability of the estimated parameters to predict the phase splitting. This consists of the following steps: (1) Regress the gE and hE data to obtain the temperature-dependent parameters of the models. (2) Introduce these values into the following expression for G”:
T.P
(3) Determine the value of xi = x0 for which G” value, by solving the following equation:
assumes
its minimum /
1,,
0 (11) = T,P (4) Insert the value of x1 into eqn. (10) and determine the value of G”. If G” > 0, no partial miscibility will be predicted for the estimated values of the parameters. If G” < 0, partial miscibility will be predicted. G
RESULTS AND DISCUSSION
Using data for gE and hE, temperature-dependent parameters of the NRTL and UNIQUAC models were estimated for 44 binary systems. These mixtures include hydrocarbon, ester, alcohol, ketone, carboxylic acid and aqueous components in various combinations. Only the values of the parameters c5 and cs, in the NRTL model, are forced to change between 0.1 and 0.7, and - 0.1 and 0.2, respectively. The estimated parameters for the NRTL and UNIQUAC models and variances of the fit, u, are given in Tables 1 and 2, respectively. The variance of the fit is obtained from u= (ffni’
NP 1
+
Ii4 (WNP)
) 02)
Here CNni and C”mi are the total number of data points for gE and hE, respectively, while NP is the number of parameters. The value of (I provides a measure of how well gE and hE data are represented simultaneously by the NRTL and UNIQUAC models. Temperature-dependent UNIQUAC parameters for the first 24 systems were given elsewhere (Demirel and Gecegormez, 1989).
25-45
25-50
25-45
25-45
25-55
25-55
25-55
25-55
25-45
2. Methyl acetate(l)cyclohexane(2)
3. Methanol(l)ethyl acetate(2) 4. Ethanol(l)ethyl acetate(2) 5. 2-Propanol( l)ethyl acetate(2) 6. l-Propanol(l)ethyl acetate(2) 7. Ethyl formate( methanol(2) 8. Ethyl formate(l)ethanol(2) 9. Ethyl formate(l)1-propanol(2) 10. Ethyl formate(l)2-propanol(2)
25-50
Twin- T&w. (“C)
1. Methyl acetate(l)benzene(2)
System
- 124.44 1.84 - 467.26 - 11.75 717.79 - 3.26 705.93 3.47 568.81 - 2.14 617.11 - 2.52 166.92 - 4.70 309.02 - 4.07 357.15 - 1.55 405.62 - 2.27 383.90 - 3.30 440.42 - 3.86 386.82 - 2.36 766.33 12.72 738.33 - 1.93 788.27 - 1.84 355.10 -0.98 378.16 - 2.28 523.00 1.25 465.68 1.22 563.42 - 0.62 644.69 - 1.45 565.59 - 1.16 615.92 - 1.55
0.1507 0.5800 0.0762 - 0.4730 0.4704 0.1582 0.5027 0.2106 0.5931 -0.1185 0.5999 - 0.1243 0.2538 0.2909 0.1608 0.1986 0.5173 -0.3895 0.5685 -0.1994 0.5307 - 0.2120 0.5998 0.3554
g-1,
”(cal mol-’
::a1 mol-’ K-‘)
C5
C3 (cal mol-‘)
CI (cal mol-‘) K-‘)
Temperature-dependent parameters of the NRTL model and the variation of the fit
TABLE 1
1O-2 0.0678 1o-2 0.0718 1o-2 0.4400 10-Z 0.1249 1o-2 0.0411 10-2 0.0534 1O-2 0.1130 10-2 0.0898 10-3
1o-2 0.0438 1O-2 a
0.2549 10-2 a
(K-‘)
0V2
Nagata et al. (1976)
Nagata et al. (1976)
Nagata et al. (1976)
Nagata et al. (1976)
Nagata et al. (1975)
Nagata et al. (1975)
Nagata et al. (1975)
Nagata et al. (1975)
Nagata and Yamada (1973)
Nagata and Yamada (1973)
References
25-60
13. Ethanol(l)toluene(2) 14.2-Propanol(l)n-heptane(2) 15. n-Pentanol(l)n-hexane(2) 16. n-Pentanol(l)2,3-dimethyIbutane(2) 17. n-Pentanol(l)2_methylpentane(2) 18. IsopentanoI(l)n-hexane(2) 19. n-Pentanol(l)3-methylpentane(2)
20. n-Pentanol(l)2,2_dimethylbutane(2) 21. Acetonitrile(l)benzene(2) 22. Benzene(l)n-heptane(2) 23. Acetonitrile(l)n-heptane(2)
25-55
12. Methyl acetate(l)ethanol(2)
45
25-50
45-70
25
25
25
25
25
25-45
30-60
25-45
11. Methyl acetate(l)methanol(2) 463.88 0.69 493.99 0.27 26.36 - 1.28 478.63 -1.46 1459.60 - 5.97 1829.80 - 8.35 1466.90 -1.40 1727.10 -0.85 1741.60 - 0.02 1369.50 1.65 1447.90 2.88 1801.30 - 0.36 1760.50 - 0.72 229.62 1.18 - 172.11 -4.58 1233.60 - 17.63
444.03 - 2.94 488.52 - 2.91 767.46 - 2.08 577.71 - 3.25 591.59 0.09 864.35 - 0.69 319.91 - 1.39 544.81 1.65 529.40 1.57 184.05 - 1.97 85.29 - 2.74 636.60 1.80 557.93 1.65 487.52 - 2.88 771.67 3.81 979.36 10.81
0.5928 0.3566 0.6556 0.2030 0.1516 - 0.4945 0.5723 0.4540 0.4808 0.2046 0.4874 0.9453 0.4592 0.8791 0.5589 0.1410 0.5448 0.1297 0.3390 - 0.1072 0.3182 - 0.1198 0.5565 0.1459 0.5522 0.1362 0.6402 - 0.1085 0.3639 -0.1181 0.2358 - 0.2596 10-s 0.0878 10-4 0.1583 10-3 0.0316 10-s 0.0875 1o-2 0.1068 d 1o-2 0.1737 10-2 0.1678 1o-2 0.0991 d 10-2 0.0806 10-2 0.0873 10-l 0.1612 10-r 0.1805 10-2
1o-3 0.2000 ;0-3
0.0811 to-3
Nguyen and Ratcliff (1975a) Sayegh and Ratcliff (1976) Palmer and Smith (1972) Monfort (1983) Palmer and Smith (1972) Lundberg (1964) Palmer and Smith (1972)
Nguyen and Ratcliff (1975a) Sayegh and Ratcliff (1976) Nguyen and Ratcliff (1975a) Sayegh and Ratcliff (1976) Nguyen and Ratcliff (1975a) Sayegh and Ratcliff (1976) Nguyen and Ratcliff (1975a) Sayegh and Ratcliff (1976) Nguyen and Ratcliff (1975a) Sayegh and Ratcliff (1976)
Van Ness et al. (1967)
Van Ness et al. (1967)
Nagata et al. (1972)
Nagata et al. (1972)
40
28.1,4-Dioxane(l)acetonitrile(2) 29. Carbon tetrachloride(l)diethyl sulfide(2) 30. Chloroform(l)diethyl sulfide(2) 31. Toluene(l)1-chlorohexane(2) 32.1~Chlorohexane(l)ethylbenzene(2) 15-70
15-70
25
25
15-59
15-90
5-85
5-65
=ti, - =k, (“C)
27. n-Butanol(l)n-hexane(2)
24. Ethanol(l)cyclohexane(2) 25. Water(l)butyl glycol(2) 26. n-Butanol(l)n-heptane(2)
System
TABLE 1 (continued)
c4
335.69 -0.14 605.00 1.55 302.15 0.05 635.70 0.92 - 11.61 6.35 33.45 - 4.98 156.20 1.80 66.18 0.03
787.47 - 2.56 1536.00 0.53 739.85 1.23 - 151.06 0.10 - 148.28 0.11 - 576.05 7.68 - 213.03 0.27 - 112.53 0.53
( ca 1 mol-’ 956.01 - 0.50 654.50 1.81 487.05 - 1.99
K-l)
c3 (cal mol- ‘)
1772.50 - 1.19 802.63 3.73 1147.10 - 1.67
(calmol-’
c2
cl (Cal mol-‘) K-l) (K-l) 0.0812 d 1o-3 0.428 ’ 1O-2 0.2441 1O-2 ’ 1o-3 0.0808 1o-3 c 1o-3 0.2173 1O-3 0.2978 10-l 0.1395 1O-3 0.1405 10-r 0.1957 10 -’
0.4546 0.5746 0.1000 0.4269 0.4339 -0.2605 0.5225 0.2110 0.5381 0.8901 0.5131 0.3400 0.6960 0.8171 0.4380 0.5900 0.1570 -0.5564 0.1890 0.2100 0.6499 0.2936
er/r
g-l)
c5
Paul et al. (1988)
Francesconi and Commelh (1988) Gray et al. (1988a) Gray et al. (1988b) Gray et al. (1988a) Gray et al. (1988b) Paul et al. (1988)
Nguyen and Ratcliff (1975b) Berro et al. (1982)
Scatchard and Satkiewitcz (1964) Scatchard and Wilson (1964) Savini et al. (1965); Berro and Peneloux (1984) Nguyen and Ratcliff (1975b)
References
a b ’ d i
Parameters Parameters Parameters Parameters Parameters Parameters
obtained by obtained by obtained by which show which show obtained by
33. l-Chlorohexane(l)n-propylbenzene(2) 34. 1,3-Dioxolane(l)methylcyclohexane 35.1-Chloropentane(l)di-n-butyl ether(2) 36. 1,2-Dichloroethane(l)di-n-butyl ether(2) 37. l,l,l-Trichlorethane(l)di-n-butyl ether(2) 38. Ethanol(l)acetone(2) 39. Acetone(l)water(2) 40. Cyclohexane(l)methyl methactylate(2) 41. Isobutyric acid(l)cyclohexane(2) 42. Trimethylacetic acid(l)cyclohexane(2) 43. Isobutyric acid(l)n-heptane(2) 44. Trimethylacetic acid(l)n-heptane(2) - 87.55 0.95 878.00 - 0.54 - 22.39 - 1.34 - 118.84 - 1.11 464.84 2.31 260.36 - 4.07 653.56 9.27 227.63 - 1.83 914.38 2.76 566.26 1.31 1109.80 3.34 558.93 1.19 40.77 1034.30 - 0.84 69.18 1.52 576.79 0.22 - 496.65 - 1.59 398.08 1.61 657.75 - 1.71 600.09 - 1.48 417.11 0.51 255.71 1.11 508.12 0.83 277.79 1.24
- 0.56
0.1000 - 0.9886 0.5912 0.5490 0.1001 0.3295 0.6496 0.6090 0.1072 0.9571 0.4199 - 0.2058 0.6725 - 0.8256 0.4454 - 0.8637 0.5590 0.1042 0.4235 0.1868 0.6758 0.9379 0.3111 0.2009
0.4325 10-z 0.0915 10-z 0.2266 10-2 0.1893 10-s 0.2507 10-s 0.0454 10-2 0.1495 10-s 0.0358 10-s 0.0661 10-a 0.0868 10-2 0.0252 10-s 0.0370 10-2
Nagata and Yamada (1973) by using hE data and isothermal VLE data. Nagata et al. (1972) by using hE data and isothermal VLE data. Berro et al. (1982) by using hE data and isothermal VLE data. partial miscibility. complete miscibility. Berro and Peneloux (1984) by using hE data and isothermal VLE data.
25-45
25-45
25-45
25-45
25-75
50-200
25-150
10-70
5-77
5-50
40
25-90
Lark et al. (1987)
Lark et al. (1987)
Lark et al. (1987)
Nicolaides and Eckert (1978) Campbell et al. (1987) Villamanan and Van Ness (1984) Griswold and Wong (1949) Luo et al. (1987) Hull and Lu (1984) Lark et al. (1987)
Paul et al. (1988)
Paul et al. (1986)
Paul et al. (1986)
Castellari et al. (1988)
Paul et al. (1988)
120 TABLE 2 Temperature-dependent
parameters
System
of the UNIQUAC
Twin-Max (“C)
25. Water(l)butyl gfycol(2) 26. n-Butanol(l)n-heptane(2) 27. n-Butanol(l)n-hexane(2) 28. 1,4-Dioxane(l)acetonitrile(2) 29. Carbon tetrachloride(l)diethyl sulfide(2) 30. Chloroform(l)diethyl sulfide(2) 31. Toluene(l)1 -chlorohexane( 2) 32. l-Chlorohexane(l)ethylbenzene( 2) 33. l-Chlorohexane(l)n-propylbenzene(2) 34. 1,3-Dioxalane(l)methylcyclohexane(2) 35. l-Chloropentane(l)di-n-butyl ether(2) 36. 1,2-Dichloroethane(l)di-n-butyl ether(2) 37. l,l,l-Trichloroethane(l)di-n-butyl ether(2) 38. Ethanol(l)acetone(2) 39. Acetone(l)water(2) 40. Cyclohexane(l)methyl methacrylate(2) 41. Isobutyric acid(l)cyclohexane(2) 42. Trimethylacetic acid(l)cyclohexane(2) 43. Isobutyric acid(l)n-heptane(2) 44. Trimethylacetic acid(l)n-heptane(2)
5-85 15-90 15-59 40 25 25 15-70 15-70 25-90 40 5-50 5-77 10-70 25-150 50-200 25-75 25-45 25-45 25-45 25-45
model and the variation
do (K) d, (K2)
d, (K) d, (K’)
377.68 - 49434.0 - 940.63 767180.0 1196.8 2048.2 - 358.85 50510.0 -614.33 167990.0 737.21 - 204310.0 38.61 19241.0 - 155.44 21241.0 - 32.27 - 2524.3 240.83 771.0 56.77 652.8 33.91 - 7744.9 - 102.18 - 2312.1 156.34 69392.0 712.49 - 228200.0 - 64.94 15894.0 429.29 - 75521.0 316.51 - 54611.0 682.23 - 105450.0 184.01 - 24171.0
426.31 - 89183.0 - 389.75 62225.0 - 347.83 52997.0 781.02 - 106120.0 770.32 - 219510.0 - 487.55 109680.0 - 58.47 - 12424.0 173.15 - 26767.0 29.03 995.45 - 175.56 53095.0 -41.30 - 1115.6 - 19.95 4126.3 107.72 8799.0 - 277.82 52280.0 - 461.03 252910.0 6.36 24213.0 - 161.10 42030.0 - 126.18 29692.0 - 284.38 50801.0 - 18.96 5405.33
of the fit et/2
0.2017 0.2651 0.0943 0.2748 0.2668 0.1098 0.1592 0.2105 0.3107 0.1485 0.3055 0.3046 0.2368 0.1053 0.1638 0.0439 0.0620 0.0817 0.0666 0.0572
121
The NRTL model gives multiple roots for the systems 1, 2, 11, 12 and 19, as may be seen from Table 1. The NRTL parameters for the systems n-pentanol-2-methyl pentane and ethanol-cyclohexane predict partial miscibility, which is not, however, predicted by the UNIQUAC model. Also, although the system water-butyl glycol shows phase splitting, the NRTL parameters predict complete miscibility, whereas the UNIQUAC model predicts partial miscibility. Multiple roots also result when the UNIQUAC model is used. The parameters that yield the best variation of the fit have been tabulated. For the system acetonitrile-n-heptane, which shows phase splitting, the parameters of both equations predict partial miscibility. Tassios (1976) claims that by changing the initial values of parameters it may be possible to obtain the correct values for some systems, as is seen for the system n-pentanol-3-methylpentane, which is a highly non-ideal mixture. Simultaneous predictions of vapor-phase composition, P, gE and hE, by the models are shown in Table 3. The volume and surface area parameters of the UNIQUAC model are given in Table 4. The average absolute error, S, for gE, P and hE were calculated using
The root mean square deviation (RMSD) in predicting compositions was calculated from the following equation
the vapor
phase
The values of S( gE), S(P) and RMSD were calculated at each isotherm for gE. Pure-component vapor pressures were obtained from the Antoine equation and the necessary parameters were taken from Reid et al. (1977). The value of S( hE) was calculated at each isotherm for the hE data. The results indicate the sensitivity of the NRTL and UNIQUAC equations in representing gE, VLE and hE data simultaneously at various isotherms, and also show that the models are system-dependent. For a total of 947 data points for gE, the average absolute errors for gE and p calculations from the NRTL model are 7.4% and 3.3%, respectively, while the corresponding values from the UNIQUAC model are 7.2% and 2.9%. The values of RMSD for the NRTL and UNIQUAC models are 1.9% and 1.6%, respectively. For
10 10 11
55 55 55
45
45
50
45
Ethyl formate(l)methanol(2)
Ethyl formate(l)ethanol(2)
Ethyl formate(l)1-propanol(2)
Ethyl formate(l)2-propanol(2) 11
8
12
12
11
55
Methyl acetate(l)cyclohexane(2)
Methanol(l)ethyl acetate(2) Ethanol(l)ethyl acetate(2) 2-Propanol(l)ethyl acetate(2) l-Propanol(l)ethyl acetate(2)
12 9 11 8 9
n
30 40 50 35 40
(“(3
T(gE>
Methyl acetate(l)benzene(2)
System
and UNIQUAC
25 35 45 25 35 25 35 25 35 25 35 45 25 35 45 25 35 45 25 35 45 25 35 45
25 35
(“Cl
T(hE)
11 11 9 12 12 12 16 13 18 11 20 9 8 11 9 6 13 9 8 13 9 13 12 11
13 12
M
4.71
5.16
2.17
1.48
6.62
22.75
3.07
1.22
19.88 13.01 11.36 0.70 3.28
S(gE)
NRTL
0.59 0.46 1.16 4.07 4.10 1.75 3.10 3.79 4.70 3.15 3.42 1.86 1.85 3.14 1.46 2.93 3.54 1.68 0.845 2.36 2.66 1.04 1.29 1.75
2.96 8.73
S(hE)
-
1.83
1.59
4.64
1.08
1.33
6.21
1.34
1.56
0.34 1.06 0.41 2.50 2.25
S(P)
- _.
2.67
2.41
6.06
0.70
0.62
5.14
3.96
1.28
0.76 0.38 0.58 2.10 2.71
_.
4.48
5.26
2.58
1.68
6.63
22.76
2.61
1.23
20.05 11.66 14.50 1.98 3.21
S(gE)
-
._ - ..-
4.75 3.82 4.36 4.53 5.22 4.06 4.04 3.25 4.59 3.36 2.13 2.42 10.42 4.71 7.82 10.54 4.59 7.85 10.04 3.10 8.35 8.09 1.43 8.46
5.22 6.58
S(hE)
UNIQUAC
-
15.83
1.39
4.79
1.15
1.35
6.23
1.32
1.52
0.37 1.13 0.48 2.01 1.88
S(P)
8.43
1.00
0.31
0.48
0.98
23.43
0.90
0.48
0.71 0.40 0.51 2.31 2.86
RMSD
models using the temperature-dependent
RMSD
Average absolute error, S x 100 and RMSD x 100
Simultaneous representation of hE and isothermal VLE data by the NRTL parameters
TABLE 3
8
9 9 9 9 9 19 11
25 25 25 25 45 70 45
n-Pentanol(l)2,3-dimethylbutane(2) n-Pentanol(l)2_methylpentane(2) Isopentanol(l)n-hexane(2) n-Pentanol(l)3_methylpentane(2) n-Pentanol(l)2,2dimethylbutane(2) Acetonitrile(l)benzene(2) Benzene(l)n-heptane(2)
n-Pentanol(l)n-hexane(2)
2-Propanol(l)n-heptane(2)
30 45 60 30 45 60 25 30 45 25
Ethanol(l)toluene(2) 18 18 17 17 17 16 9
10 13
45 55
Methyl acetate(l)ethanol(2)
13 13
35 45
Methyl acetate(l)methanol(2)
2.75
9
13 14 8 4
45 25 50
9
9
9
9
1.48 1.18 1.01 2.57 2.25 1.87 2.26
7.60 3.79 9.19
2.81
0.35 0.22 2.38
0.83 1.03 0.72 0.36 1.05 1.97 1.22
0.07
2.85
0.13
0.04
7.72
2.54
7.12
2.63
3.24
7.21
3.42
0.03
0.09
2.52
2.75
1.34 0.87 0.69 2.18 1.46 1.05 0.03
1.17 1.44
2.63 3.04 1.67 1.08 0.70 2.29 1.30 0.76 2.06
0.38 0.68
0.75 1.17
3.58
1.54 0.93 1.92 3.80 3.71 3.41 9.06 5.91 5.05 14.67 11.65 10.63 15.30 15.59 15.66 2.76
3.16
8.78 12.88
4.89 4.15
14 16 10 11 12 7 26 23 26 24 22 21 9
45
25
25
25
25
25
25 35 45 25 35 45 25 45 60 30 45 60 25
7.65 4.26 8.70
2.55
2.58
3.24
2.51
2.69
1.97 2.41 2.78 3.23 3.16 2.95 2.14
9.13 12.90
4.94 4.07
2.72 3.08 2.55 2.64 0.71 1.04 0.61
4.67 5.18 4.89 4.02 2.11 1.40 1.60 1.30
2.31
4.21
2.79 3.02 1.32 1.27 1.33 3.89 2.98 2.22 5.52
0.77 1.14
3.58 1.25 4.02 6.54 2.55 6.26 10.43 6.20 8.20 11.95 9.14 13.86 3.30
1.31 0.78 0.49
0.05
0.11
0.19
0.11
0.08
0.91 0.78 0.80 2.30 2.11 2.08 0.20
1.17 1.37
0.41 0.71
n-Butanol(l)n-hexane(2) l+Dioxane(l)acetonitrile(2) Carbon tetrachIoride(l)diethyl sulfide ChIoroform(l)diethyl sulfide(2) Toluene(l)1-chIorohexane(2)
n-Butanol(l)n-heptane(2)
Water(l)butyl gIycol(2)
Acetonitrile(l)n-heptane(2) Ethanol(l)cycIohexane(2)
System
TABLE 3 (continued)
5 20 35 50 65 5 25 45 65 85 30 45 55 55 15 40
9 9 9 9 9 8 8 8 8 8 19 22
23 12 10 9 12 12
5 20 35 50 65 5 25 45 65 85 60 90
59 40 25 25 50 70
15 25
25
25
45
T (hE) (“C)
8
n
45
VgE) (“C)
15.68 5.92
11 10
8.00 5.84
9.60
10
19 17
1.97
1.71 1.54 2.02 1.66 1.81 26.46 26.46 25.98 22.42 22.38 6.47 3.59
7.09
S(gE)
NRTL
10 10 10 10 10 8 8 8 8 8 17 17 10 10 10
13
m
Average absolute error, S
4.31 7.43
2.86
3.56
6.90
9.53 3.90 3.82 4.60 4.94 31.42 17.11 21.78 37.83 30.52 14.89 14.48 22.58 8.58 4.13
2.68
0.23 0.35
1.26
0.50
1.32
2.90
1.60 2.56 2.45 2.24 2.70 15.44 14.48 14.95 15.34 13.73 2.05 1.44
11.65
S(P)
100 and RMSD
S(hE)
x
100
0.22 0.26
0.75
0.76
1.28
0.77
0.98 1.36 1.60 1.81 1.83 0.94 1.88 2.08 2.61 2.00 0.82 0.92
8.73
RMSD
x
9.75 12.31
5.93
15.46
12.68
3.40
5.35 4.35 3.84 4.03 4.74 9.50 8.90 8.88 8.54 8.65 4.47 2.14
7.31
1.85 2.26
2.90
5.60
12.30
12.73 7.65 7.32 7.77 2.11 11.95 13.00 13.16 10.62 12.87 18.60 22.00 18.00 23.00 5.10
2128
S(hE)
UNIQUAC S(gE)
0.22 0.22
1.27
0.49
1.74
2.76
2.50 1.73 1.52 1.67 1.87 7.53 6.29 6.75 6.31 6.88 0.51 0.92
12.34
S(P)
0.19 0.09
0.75
0.55
1.64
0.56
0.98 1.03 0.92 1.04 1.30 0.55 0.39 0.49 0.70 1.04 0.83 0.83
9.22
RMSD
G
Isobutyric acid(l)cyclohexane(2) Trimethylacetic acid(l)cyclohexane(2) Isobutyric acid(l)n-heptane(2) Trimethylacetic acid(l)n-heptane(2)
Cyclohexane(l)methyl methacrylate(2)
Acetone(l)water(2)
Ethanol(l)acetone(2)
l,l,l-Trichloroethane(l)di-n-butyl ether(2)
1,2-Dichloroethane(l)di-n-butyl ether(2)
l-Chlorohexane(l)ethylbenzene(2) l-Chlorohexane(l)n-propylbenzene(2) 1,3-Dioxalane(l)methylcyclohexane l-Chloropentane(l)di-n-butyl ether( 2) 12 12
40 50
100 125 150 100 150 200 45 60 75 25 45 25 45 25 45 25 45 9 11 10 14 14 14 17 17 16 12 11 11 12 12 12 10 10
13 12 13 12 13
26
40
57 77 97 50 70
10 14 8
50 70 90
9 9 9 8
35 35 35
11
19
11 12 11 12 12
20 19 19 19 19
17 19 19 19 16
35
25
50
25 10 35 25 50
5 25 40 5 25
15 25 25 15 40
3.00 3.06 1.51 11.27 11.02 5.95 2.32 2.55 2.55 1.74 1.50 5.95 5.22 0.83 0.80 2.36 2.75
14.84 10.11 14.05 24.05 12.15
18.09 12.84
6.39
16.53 9.30 17.31
0.29
1.16
0.76
3.32
0.38
1.77
2.95 3.00 1.13 1.97 1.76
9.49 3.81 4.01 4.30 2.11
7.67 10.14 3.11 4.13 2.00
1.01 1.42 2.98 3.62 3.27 4.14 0.26 0.24 0.46 2.88 2.34 4.38 3.60 1.47 1.44 3.86 3.72
0.75 0.36 0.31 0.43 1.08
19.59 18.69
3.99
0.25 0.25 0.41
1.77 1.17 0.83 2.50 3.43 3.69 0.51 0.50 0.45 3.96 4.66 0.98 2.51 4.90 4.85 1.63 2.28
0.46 0.19 0.25 0.08 0.32
3.66 2.82
4.79
0.20 0.18 0.50
2.41 4.72 5.14 8.00 8.80 5.79 3.39 2.55 2.75 2.93 3.25 5.64 4.45 5.00 4.36 3.28 3.89
16.39 9.21 15.05 20.44 19.93
17.39 15.02
10.93
22.12 12.04 18.10
1.97
1.70
2.42
2.58
1.70
8.90
12.89 8.31 12.71 7.80 7.80
17.80 10.46 30.33 5.71 4.36
3.77 5.18 3.75 4.93 2.20
0.60 1.32 3.29 5.03 3.22 3.22 0.34 0.26 0.24 3.53 3.01 4.35 2.96 4.45 3.37 4.36 4.54
0.80 0.26 0.47 0.74 1.26
19.65 18.72
2.38
0.33 0.09 0.38
1.49 1.08 0.95 3.32 3.05 2.99 0.44 0.48 0.49 3.98 4.71 0.61 2.16 4.98 4.99 1.40 2.35
0.43 0.17 0.25 0.14 0.34
3.67 2.82
3.34
0.17 0.12 0.48
W
126 TABLE 4 Volume and surface area parameters
Methyl acetate Benzene Cyclohexane Methanol Ethyl acetate Ethanol 2-Propanol 1-Propanol Ethyl formate Toluene n-Heptane n-Pentanol n-Hexane 2,3-Dimethylbutane 2-Methylpentane Isopentanol 3-Methylpentane 2,ZMethylpentane Acetonitrile Water Butyl glycol n-Butanol l+Dioxane Carbon tetrachloride Diethyl sulfide Chloroform I-Chlorohexane Ethylbenzene n-Propylbenzene 1,3-Dioxolane Methylcyclohexane I-Chloropentane Di-n-butyl ether 1,2-Dichloroethane l,l,l-Trichloroethane Acetone Methyl methacrylate Isobutyric acid Trimethylacetic acid
for the UNIQUAC
model
r
4
4’
2.800 3.190 3.970 1.430 3.480 2.110 3.249 3.249 2.817 3.920 5.170 4.597 4.500 4.500 4.499 5.923 4.499 4.498 1.870 0.920 4.697 3.450 3.185 3.330 3.902 2.700 5.064 4.600 5.272 2.511 4.640 4.390 6.093 2.880 3.541 2.570 3.922 3.550 4.224
2.580 2.400 3.010 1.430 3.120 1.970 3.124 3.128 2.576 2.970 4.400 4.208 3.860 3.860 3.396 4.516 3.396 3.932 1.724 1.400 4.556 3.050 2.917 2.620 3.296 2.340 4.272 3.510 4.048 2.100 3.550 3.732 5.176 2.520 3.032 2.340 3.564 3.148 3.768
2.580 2.400 3.010 0.960 3.120 0.920 0.890 0.890 2.576 2.970 4.400 1.150 3.860 3.860 3.396 1.150 3.396 3.932 1.724 1.000 4.556 0.880 2.917 2.620 3.296 2.340 4.272 3.510 4.048 2.100 3.550 3.132 5.176 2.520 3.032 2.340 3.564 3.148 3.768
a total of 1101 data points for hE, the average absolute errors for hE calculations from the NRTL and UNIQUAC models are 5.8% and 6.‘7%, respectively. The predictions of the UNIQUAC model compare well with
127
those of the NRTL model, although the latter has six estimated parameters, and the effect of temperature on g E, VLE and hE data is well represented by both models. This shows that eqns. (2)-(4), (6) and (7) are capable of expressing the temperature dependency of the models for various types of mixture, including ones with hydrogen bonding. Using the error matrix, the off-diagonal elements of the correlation coefficient matrix, which is explained elsewhere in detail (Demirel and Gecegormez, 1989), is calculated as pjj =
Uij/(
p;; =
I) J
uiiujj)1’2
1
Pij
=
Pji
(17)
TABLE 5 The elements of correlation coefficient matrix, v: p,, = 1; pi, = p,, P IJ
System a 2
3
27
22
38
43
25
0.461 0.354 0.645 0.707 0.087 0.482 0.777 0.671 0.794 0.723 0.596 0.330 0.456 0.476 0.722
0.271 0.203 0.789 0.230 0.566 0.326 0.247 0.624 0.838 0.119 0.348 0.293 0.328 0.736 0.428
0.304 0.262 0.236 0.274 0.041 0.938 0.168 0.303 0.894 0.842 0.530 0.042 0.800 0.859 0.864
0.744 0.946 0.535 0.758 0.970 0.531 0.336 0.419 0.153 0.530 0.693 0.922 0.530 0.820 0.357
0.348 0.335 0.016 0.568 0.937 0.100 0.573 0.167 0.499 0.464 0.568 0.029 0.306 0.306 0.909
0.942 0.579 0.611 0.414 0.546 0.862 0.580 0.616 0.938 0.796 0.232 0.034 0.420 0.717 0.457
0.778 0.589 0.951 0.904 0.598 0.449 0.001 0.014 0.002 0.027 0.459 0.893 0.987 0.312 0.007
0.923 0.899 0.718 0.935 0.811 0.979
0.892 0.137 0.424 0.096 0.427 0.985
0.976 0.993 0.944 0.997 0.987 0.984
0.960 0.969 0.873 0.923 0.937 0.986
0.999 0.985 0.989 0.974 0.983 0.995
0.982 0.990 0.988 0.991 0.989 0.983
NRTL CL21 P31 p32 P41 IL42 P43 P51 p52 t453 CL54
P61 CL62 IL63 p64 p65
UNIQUAC PZI P31 p32 1141 p42 IL43
0.273 0.356 0.935 0.829 0.355 0.315
a 2, methyl acetate-cyclohexane; 3, methanol-ethyl acetate; 27, n-butanol-n-heptane; 22, benzene-n-heptane; 38, ethanol-toluene; 43, isobutyric acid-n-heptane; 25, water-butyl gIyco1.
0.887 0.956 0.612 0.998 0.846 0.931 0.999
c3
0.833 0.900 0.985 0.983 0.819 0.972 0.999
c4
CS
0.964 0.961 0.977 0.841 0.958 0.955 0.055
0.729 0.977 0.983 0.988 0.951 0.950 0.999
acetate; 27, n-butanol-n-heptane;
0.902 0.981 0.984 0.997 0.953 0.971 0.997
0.916 0.947 0.954 0.999 0.966 0.999 0.954
d,
22, benzene-n-heptane;
0.793 0.969 0.947 0.999 0.967 0.999 0.947
d,
C2
Cl
0.842 0.838 0.913 0.991 0.855 0.957 0.997
UNIQUAC
NRTL
P:
a 2, methyl acetate-cyclohexane; 3, methanol-ethyl isobutyric acid-n-heptane; 25, water-butyl glycol.
2 3 27 22 38 43 25
System ’
The global correlation coefficients
TABLE 6
0.786 0.983 0.988 0.999 0.974 0.999 0.988
d4
38, ethanol-toluene;
0.916 0.983 0.985 0.999 0.963 0.999 0.985
d3
43,
129
where uii represents the elements of the error matrix. If v is positive definite, p < 1 for all elements. If p = 0, then the parameters are uncorrelated and if p = 1, the par ameters are completely correlated. For a given parameter, the global correlation coefficient is given by
/.A;= 1 -
[ UkkVLk]
-l
(18)
and is the correlation between it and that linear combination of the other parameters most highly correlated with it. All such coefficients should be between zero and one for a positive definite error matrix. The values of error matrix correlations and the global correlation coefficients for some of the systems are given in Tables 5 and 6, respectively. The parameters for some of the systems are highly correlated, but for such parameters it is not possible to specify unique values from a given set of data. Such problems have arisen when the same free energy function is used for two of the phases, as in LLE, at points where the mole fractions in two phases are identical.
CONCLUSIONS
Although the simultaneous fit of Gibbs energy and enthalpy of mixing data for 44 binary systems is satisfactory with the NRTL and UNIQUAC models, choice of the best model is mainly system-dependent. The use of temperature-dependent parameters improves the performance of the models, but care should be exercised in the selection of the best model for a given system, and in the estimation of the parameters for especially non-ideal mixtures.
ACKNOWLEDGMENT
The authors thank the Computer Centre of Cukurova University for the computation facilities provided.
LIST OF SYMBOLS
aij Cl,
c3
c29 c4
UNIQUAC binary interaction parameter related to riT values of (g2, - gir) and (g,, - g,,) at 0°C (cal mol-‘) coefficients of temperature change of (g,, - g,,) and (g,, - g,,) (cal mol-’ K-‘)
130
value of (Y,~at 0 “C coefficient of temperature change of ai2 (K-l) ‘6 UNIQUAC parameters related to ajj (K) d,, d, UNIQUAC parameters related to ajj (K’) 6 d, F objective function as defined by eqn. (9) excess molar Gibbs energy (cal mol-‘) gE constant in eqn. (5) Gi Gh Gibbs energy of mixing (cal mol-‘) G”, G “’ second and third derivative of Gibbs energy of mixing with respect c5
excess enthalpy of mixing (cal mol-‘) number of experimental hE and gE data points, respectively, at a specified isothermal temperature number of isothermal system temperatures for hE and gE data, respectively number of parameters molecular geometric area parameter for component i molecular interaction area parameter for component i total pressure (Pa) molecular volume parameter for pure component i gas constant (cal mall’ K-‘) root mean square deviation absolute temperature (K) average absolute error (eqns. (13), (14), (15)) elements of error matrix liquid-phase mole fraction of component i vapor-phase mole fraction of component i
hE m, n M, iv NP 4; 4,! P r, R RMSD T s ‘ij xi Yi
Greek letters a12
; u 7ij 7,; pij p,,
non-randomness constant for binary l-2 interaction activity coefficient of component i area fraction of component i in residual contribution coefficient variance of the fit NRTL binary parameter UNIQUAC binary parameter elements of correlation coefficient matrix (eqn. (17)) elements of global correlation vector (eqn. (18))
Subscripts exptl calcd
experimental calculated
to the activity
131
4 .i max min
component maximum minimum
Superscript
E
excess
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