Ind. Eng. Chem. Res. 1993,32, 3150-3161
3150
Calculation of Vapor-Liquid Equilibria from Infinite-Dilution Excess Enthalpy Data Using the Wilson or NRTL Equation A r t h u r S.Gow Department of Chemistry and Chemical Engineering, University of New Haven, 300 Orange Avenue, West Haven, Connecticut 06516-1999
A study was undertaken to determine the potential for using infinite-dilution excess enthalpy data with activity coefficient models to predict vapor-liquid equilibria for binary systems. Expressions for the infinite-dilution reduced excess enthalpies in a binary liquid mixture derived from the Wilson or NRTL equation with temperature-independent parameters are solved for two unknown model parameters given limiting values of reduced excess enthalpy (partial molar excess enthalpy) H E / x 1 x 2 . The Wilson equation is modified using the van der Waals covolume b in place of the pure-liquid molar volume, and an average value of the nonrandomness factor is chosen to obtain a two-parameter NRTL equation. Vapor-liquid equilibria were predicted for 19 isothermal and 8 isobaric binary systems. Good results are generally obtained for nearly ideal systems using either model; however, the Wilson equation is superior for moderately nonideal systems. The temperatureindependent Wilson model fails to accurately describe vapor-liquid equilibria for strongly nonideal mixtures with highly unsymmetric excess enthalpy profiles. The proposed method appears to be particularly well suited for estimating phase behavior of environmentally safe refrigerant mixtures. Introduction Phase equilibria are essential for chemical separation process design. Experimental data are frequently unavailable over the range of conditions of interest; hence, the required data must be estimated using one of a variety of approaches. Among the most popular techniques is the so-called “phi-gamma” approach in which activity coefficients are estimated by the UNIFAC (Fredenslund et al., l975,1977a,b) or ASOG (Kojima and Tochigi, 1979) group contribution methods, and vapor-phase fugacity coefficients are calculated from a corresponding-states correlation such as proposed by Prausnitz (Prausnitz, 1969; Smith and Van Ness, 1987). A popular alternative approach is to use a cubic equation of state such as Soave’s modification of Redlich and Kwong’s equation (Soave, 1972) or the Peng-Robinson (1976) equation to calculate species fugacity coefficients in all phases. A major drawback of this “phi-phi” approach is the necessity of knowing binary interaction constants for each pair of species present in the mixture. These parameters are not available “a priori” and must be found by correlating some experimental equilibrium data for the system of interest. Generally, experimental data which describe the nature of molecular interactions must be known for each binary system, regardless of the approach employed. The fundamental questions are “what data” and “how much” are needed? Group contribution methods such as UNIFAC and ASOG can be used to calculate phase equilibria for a very large number of multicomponent mixtures from data for a relatively small number of functional groups. Another type of approach uses minimal experimental data for a binary system at extremes of composition to predict phase equilibria over the entire composition range. Infinitedilution activity coefficients have been quite successfully used in this endeavor (Renon and Prausnitz, 1969;Eckert and Schreiber, 1971; Nicolaides and Eckert, 1978a). The success of this approach and the capability to accurately determine dilute solution calorimetric data suggest that calculation of vapor-liquid equilibria (VLE) from infinitedilution excess enthalpy data should also be investigated. Several approaches based on the Wilson or NRTL equation using excess enthalpy data over the entire 0888-5885/93/2632-3150$04.00/0
composition range have been advanced. Hanks and coworkers (Hanks et al., l971,1978a,b; Tan et al., 1977,1978; Garcia Calzon et al., 1986) developed local composition schemes to predict VLE for a wide variety of nonideal binary systems. In many cases, the NRTL equation with temperature-independent parameters provided a simple HE versus composition function. Parameters were obtained from HE data, and VLE predictions were made for totally miscible binary hydrocarbon-hydrocarbon mixtures (Hanks et al., 19711,totally miscible ternary systems (Tan et al., 1977, 1978), and partially miscible binary systems (Tan et al., 1977). Fair to excellent results were obtained for totally miscible binary and ternary systems while the method yielded poor predictions of the binodal solubility curve for partially miscible systems. The method was also successful in predicting phase equilibria for 10 binary hydrocarbon mixtures at higher temperatures than at which HE data were available (Hanks et al., 1978b). Garcia Calzon et al. (1986) investigated the prediction of VLE from HE data in nearly ideal mixtures of n-hexane and n-octane with the hexane isomers. They were fairly successful using various models for GE considering the magnification of errors in HE data for mixtures exhibiting only minor deviations from Raoult’s law. Others have also investigated VLE-HE cross-prediction using local composition models. Duran and Kaliaquine (1971) and Tai et al. (1972) successfully used the Wilson equation to simultaneously correlate VLE and HE data. Vonka et al. (1975) and Hanks and co-workers (1978a) have examined the limitations of applying the Wilson and NRTL equations to prediction of phase equilibria from excess enthalpy data. Bradley (1976)presents an excellent review of early work. The approach presented in this paper parallels the use of infinite-dilution activity coefficients with local composition models developed by Eckert and colleagues (Eckert and Schreiber, 1971;Nicolaides and Eckert, 1978a). Approach Liquid mixture models for description of low to moderate pressure phase equilibria are generally formulated in terms of the excess Gibbs energy. The rigorous property 0 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32, No. 12,1993 3151 relation for the excess Gibbs energy is
m
nGE n p nHE d(m) = dP- d T = x l n y i dni
(1)
RP
from which the partial differential equations Substitution of eqs lOand 11into eqs 12 and 13respectively followed by differentiation according to eq 3 (assuming constant a12 and u21) leads to (3)
[gIxl4 = + (& b l ) exp( 2) (14) a12
and (4) are obtained. Equations 3 and 4 are useful for predicting low to moderate pressure phase equilibria, where pressure has a negligible effect on excess Gibbs energy. These equations may be applied to any GE model to provide relations for the liquid excess enthalpy and species activity coefficients. Wilson (1964) proposed an expression for the excess Gibbs energy based on “local composition” in a liquid mixture. Amodified form of Wilson’s equation was chosen for the present study. Wilson’s expression for the excess Gibbs energy of a binary mixture is
GE
= -xl ln(x,
+ x,A,,)
- x , ln(x,
+ xlA2,)
+ ul,(
%)exp( 2) (15)
Equations 14 and 15 express the infinite-dilution reduced excess enthalpy (limiting partial molar excess enthalpy) of components 1 and 2 respectively in a binary mixture provided the binary constants a12 and a21 are temperatureindependent. Similar expressions may be obtained using other models for the excess Gibbs energy of a binary mixture. Renon and Prausnitz (1968) introduced a local composition model based on the “two-liquid” theory. The NRTL equation for the excess Gibbs energy is
(5) where
with modified binary parameters given here by Aij =
= u2,
a21
(2) 2)
GI, = exp(-arlz) and G,, = exp(-arzl)
exp(
(17)
and
aij is an interaction parameter which is assumed here to be independent of temperature and composition, and bi and bj are the van der Waals excluded volumes for components i and j , respectively, which are expressed in terms of fluid critical temperature Tcjand critical pressure Pi as
m
b21 and r,, = RT The activity coefficients are given by r12=
bl,
(18)
(7) Wilson’s (1964) original formulation of eq 6 uses purefluid molar volumes at an arbitrary temperature. The use of hi's provides a good estimate of the relative size ratio of molecules Vi/Vj while being completely independent of temperature. The van der Waals constant bj is readily computed from critical constants (eq 7). Expressions for the activity coefficients of components 1and 2 in a binary liquid mixture (obtained by applying eq 4 to eq 5) are In y1 = -ln(z,
+ x2h12)+ x2(
4 2
x1+
XZA12
xz + X l A Z l
(8)
At infinite dilution of component i in component j , eqs 8 and 9 become respectively In 7,- = -In A12 + 1- Azl
(10)
+ 1- A,,
(11)
In y2- = -In A,, However, by definition
Evaluation of eq 19 for x 1 = 0, xz = 1 and eq 20 for x 2 = 0, XI = 1yields expressions for the infinite-dilution activity coefficients of components 1and 2 which when differentiated with respect to temperature according to eq 3 (assuming blz and bzl are temperature-independent) lead to the following expressions for the infinite-dilution reduced excess enthalpies: = b,,
+ (b,, -
s)2)
= b,,
+ (b,,
g) m)
[3,
-
exp( exp(
(21)
- 4 1
(22)
Equations 14 and 15 or 21 and 22 may be solved for the two binary parameters in either the Wilson or NRTL equation given the two infinite-dilution reduced excess enthalpies HIE”and HzE“for a binary liquid mixture at a particular temperature, the pure-component critical constants for the approach using Wilson’s equation, and a value of the nonrandomness factor a for the NRTL
3152 Ind. Eng. Chem. Res., Vol. 32, No. 12, 1993 Table I. van der Waals Constants, Infinite Dilution Reduced Excess EnthalDies. and Calculated Binarv Parameters. system benzene (1)-n-heptane (2) 298.15 benzene (1)-n-hexadecane (2) 298.15 benzene (1)-isooctane (2) 298.15 benzene (l)-octene-l (2) 298.15 benzene (l)-cyclohexane (2) 298.15 benzene (l)-decalin (2) 298.15 benzene (1)-tetralin (2) 298.15 cyclohexane (1)-n-heptane (2) 298.15 cyclohexane (1)-n-hexadecane (2) 298.15 cyclohexane (l)-isooctane(2) 298.15 cyclohexane (l)-octene-1(2) 298.15 cyclohexane (1)-rn-xylene (2) 298.15 cyclohexane (l)-decalin (2) 298.15 cyclohexane (1)-tetralin (2) 298.15 n-heptane (1)-n-hexadecane (2) 298.15 n-heptane (1)-octene-1 (2) 298.15 n-heptane (Wtoluene (2) 298.15 n-heptene (1)-rn-xylene (2) 298.15 n-heptane (ll-decalin (2) 298.15 n-heptane (1)-tetralin (2) 298.15 isooctane (1)-n-hexadecane (2) 298.15 octene-1 (1)-n-hexadecane (2) 298.15 toluene (1)-n-hexadecane (2) 298.15 decalin (1)-n-hexadecane (2) 298.15 tetralin (1)-n-hexadecane (2) 298.15 isooctane (l)-n-octane (2) 298.15 methylcyclohexane (1)-n-heptane (2) 298.15 1-chlorobutane (1)-carbon tetrachloride (2) 313.15 1,2-dichloroethane (l)-carbon tetrachloride (2) 313.15 1,2-dichloroethane (1)-1-chlorobutane (2) 313.15 cyclopentane (1)-tetrachloroethylene (2) 298.15 cyclohexane (1)-aniline (2) 308.15 323.15 acetone (l)-ethanol(2) 298.15 323.15 acetonitrile (1)-benzene (2) 318.15 acetonitrile (1)-n-heptane (2) 318.15 methyl acetate (1)-benzene (2) 298.15 308.15 methyl acetate (1)-cyclohexane 298.15 308.15 318.15 methanol (l)-ethyl acetate (2) 298.15 308.15 ethanol (l)-ethyl acetate (2) 298.15 308.15 n-propanol (1)-ethyl acetate (2) 298.15 308.15 318.15 2-propanol (l)-ethyl acetate (2) 298.15 308.15 methanol (1)-benzene (2) 298.15 308.15 318.15 methanol (1)-toluene (2) 298.15 308.15 318.15 methanol (1)-ethyl benzene (2) 298.15 308.15 318.15 ethanol (1)-benzene (2) 298.15 308.15 318.15 ethanol (Wtoluene (2) 298.15 308.15 318.15 ethanol (l)-ethylbenzene (2) 298.15 308.15 318.15 2-propanol (1)-benzene (2) 298.15 308.15 318.15 2-propanol (1)-toluene (2) 298.15 308.15 318.15
119.46 119.46 119.46 119.46 119.46 119.46 119.46 141.31 141.31 141.31 141.31 141.31 141.31 141.31 204.89 204.89 204.89 204.89 204.89 204.89 220.52 224.40 149.62 229.97 212.54 220.52 171.09 152.74 108.57 108.57 117.93 141.31
204.89 525.33 220.52 224.40 141.31 229.97 212.54 204.89 525.33 220.52 224.40 180.83 229.97 212.54 525.33 224.40 149.62 180.83 229.97 212.54 525.33 525.33 525.33 525.33 525.33 238.36 204.89 126.80 126.80 152.74 144.54 136.83
112.35
84.08
117.89 117.89 112.28
119.46 204.89 119.46
112.28
141.31
65.77
141.98
84.08
141.98
107.95
141.98
110.93
141.98
65.77
119.46
65.77
149.62
65.77
177.81
84.08
119.46
84.08
149.62
84.08
177.81
110.93
119.46
110.93
149.62
3220 5707 3482 2 155 3159 2793 518 832 1935 594 816 2039 240 1696 379 150 2 682 1811 401 2 156 896 595 2 447 762 2 334 102 103 26.1 2 649 853 931 9 717 9 641 4 517 5 273 1666 11343 1789 1492 8 938 8 806 8 896 6 412 6 142 8 046 8 395 8 197 7 819 8 317 8 199 9 037 15 700 16 608 17 538 16 515 17 212 17 887 17 096 17 910 18 608 15 933 16 864 17 910 16 631 17 096 17 492 17 678 18 375 19 073 16 864 17 352 17 608 17 678 18 027 18306
4916 9474 5420 3783 3513 4003 800 1400 3287 784 1482 2875 -112 2482 387 175 1980 1537 271 1 994 1040 695 4 967 1268 4 026 102 169 -24.9 2 829 1785 979 15 352 15 114 5 070 5 294 2 700 13 433 1 767 2 023 8000 7 975 9 281 3 632 4 833 5 486 7 296 7 181 6 289 7 106 7 673 7 574 1512 1 768 2 175 1791 2 186 2 605 2 373 2 768 3 233 1570 1942 2 384 1 500 2000 2 547 1919 2 431 3 012 3 826 4 454 5 094 3 745 4 454 5706
2720 5610 3040 1670 2; 3 2320 442 395 1940 512 413 1350 2030 1170 1920 -1.55 1600 1090 774 1 460 1400 1370 2 200 773 1980 143 -201 682 1 880 263 688 9 670 9 580 3 530 4 170 726 11300 1160 722 8 660 8 490 8 650 5 990 5 730 7 640 8 100 7 860 7 350 7900 7 890 8 700 15 300 16 100 17 000 16 100 16 800 17 500 16 800 17 600 18 300 15 400 16 200 17 200 16 200 16 600 16 900 17 300 17 900 18 600 16 100 16 700 16 900 17 100 17 400 17800
3360 6910 3770 2180 2440 2250 142 912 -9.47 134 927 1870 -1570 1380 -1880 177 1370 916 -365 1150 -856 -1 150 1790 -24.4 1820 -43.5 431 -461 1750 1450 340 15 100 14 900 4 430 4 630 2 140 13 200 994 1440 7 670 7 590 8 870 2 480 3 510 4 890 6 720 6 750 5 740 6 580 7 250 7 200 1450 1710 2 130 1 740 2 130 2 550 2 320 2 720 3 180 1 530 1900 2 350 1460 1950 2 500 1 880 2 400 2 980 3 800 4 430 5 060 3 720 4 430 5 680
3750 8630 4180 3110 2350 2990 1190 1660 2750 954 1830 2 220 -1 300 2 040 250 423 781 608 -436 1030 842 674 4 090 1580 3 250 63.7 835 -787 1780 2 260 622 16 500 16 100 3 740 3 870 2380 14900 976 1670 8240 8000 9380 2580 3630 5020 7040 6980 5680 6640 7550 7630 3380 3780 4330 3820 4320 4840 4520 5040 5610 3490 4010 4620 3550 4110 4710 4 140 4750 5440 6000 6710 7400 6030 6810 8140
1920 5 830 2 240 824 1890 1460 -358 -250 622 -163 -321 799 1980 498 142 -235 2 040 1 280 880 1360 208 20.7 1190 -290 1000 38.7 -580 956 1 500 -434 393 11 900 11 800 3 220 3 850 3410 13200 1020 390 8920 8600 9090 5110 4780 6970 7850 7730 6840 7540 7930 8700 14400 15300 16200 15200 15900 16600 15900 16800 17500 14600 15500 16600 15300 15800 16200 16400 17200 18000 16100 16700 17100 16900 17400 18000
Ind. Eng. Chem. Res., Vol. 32, No. 12, 1993 3163 Table I. (Continued) bi
T (K) (cm3/mol)
system
298.15 308.15 318.15 298.15 308.15 318.15 298.15 308.15 318.15 298.15 308.15 318.15 298.15 308.15 318.15 298.15 308.15 318.15 298.15 308.15 318.15 298.15 308.15 318.15 298.15 308.15 318.15 298.15 308.15 318.15 278.15 288.15 298.15 308.15 318.15 323.15
2-propanol (l)-ethylbenzene (2) n-propanol (1)-benzene (2) n-propanol (1)-toluene (2) n-propanol (1)-ethylbenzene (2) n-butanol (1)-benzene (2) n-butanol (1)-toluene (2) n-butanol (Wethylbenzene (2) n-amylalcohol (1)-benzene (2) n-amyl alcohol (1)-toluene (2) n-amyl alcohol-ethylbenzene (2) acetone (1)-water (2)
b2
110.93
(cm3/mol) 177.81
107.95
119.46
107.95
149.62
107.95
177.81
132.43
119.46
132.43
149.62
132.43
177.81
154.13
119.46
154.13
149.62
154.13
177.81
112.35
30.51
Hi' (J/mol)
(J/mol)
HzE"
a12 (J/mol)
aai (J/mol)
(J/mol)
(J/mol)
17 492 17 887 18 306 17 096 17 096 17 096 16 980 17 236 17 445 18 027 18 189 16 306 18 027 18 027 18 207 17 608 17 608 17 608 18 748 18 748 18 748 18 027 18 027 18 027 16 980 16 980 16 980 17 608 17 608 17 608 -12 487 -11 338 -10 193 -8 903 -6 960 -7 225
4 059 4 815 5 489 2 105 2 524 3 035 1942 2 477 3 047 2 198 2 768 3 408 2 291 2 721 3 187 2000 2 512 3 128 2 268 2 814 3 431 2 442 2 814 3 222 1965 2 466 2 966 2 163 2 687 3 256 3 947 4 144 4 105 4 533 5 860 4 881
17 OOO 17 400 17 900 16 300 16 200 16 200 16 300 16 600 16 700 17 500 17 600 17 700 17 OOO 17 OOO 17 OOO 16 800 16 800 16 800 18 100 18 OOO 18 OOO 16 900 16 800 16 800 16 100 16 OOO 16 OOO 16 800 16 800 16 800 -12 500 -11 300 -10 200 -8 900 -6 960 -7 230
4 030 4 780 5 460 2 080 2 490 3000 1910 2 440 3 010 2 170 2 740 3 370 2 270 2 700 3 160 1980 2 480 3 090 2 250 2 790 3 400 2 430 2 800 3 200 1940 2 440 2 930 2 140 2 660 3 222 755 OOO 354 OOO 173 OOO 82 600 32 100 33 800
6 330 7 170 7 920 4 240 4 650 5 140 4 060 4 620 5 220 4 470 5 080 5 740 4 570 5 010 5 470 4 210 4 720 5 330 4 630 5 200 5 830 4 730 5 100 5 510 4 080 4 570 5 040 4 380 4900 5 470 169 OOO 118 OOO 81 500 32 900 17 500 17 600
16 800 17 400 18 OOO 15 900 15 900 15 900 15 700 16 OOO 16 300 16 800 17 100 17 300 16 900 16 900 16 900 16 400 16 400 16 500 17 600 17 600 17 700 16 900 16 900 16 900 15 700 15 700 15 800 16 400 16 400 16 500 -12 500 -11 300 -10 200 -6 940 -4 570 -4 820
biz
b21
a Data sources for He values. All 27 hydrocarbon (1)-hydrocarbon (2) systems: Lundberg (1964). Chlorobutane (1)-carbon tetrachloride (2), 1,2-dichloroethane(1)-carbon tetrachloride (2), and 1,2-dichloroethane(l)-chlorobutane (2): Munoz Embid et al. (1990). Cyclopentane (1)-tetrachloroethylene (2): Polak et al. (1970). Cyclohexane (1)-aniline (2) and acetone (1)-ethanol (2): Nicolaides and Eckert (1978b). Acetonitrile (1)-benzene (2) and acetonitrile (l)-n-heptane (2): Palmer and Smith (1972). Methyl acetate(1)-benzene (2) and methyl acetate (1)-cyclohexane (2): Nagata et al. (1973). All four alcohol (Wethy1 acetate systems: Nagata et al. (1975). All 18 alcohol (1)-hydrocarbon (2) systems: Mrazek and Van Ness (1961). Acetone (1)-water (2): French (1989).
0.80
0.70
0.90
Application of the Proposed Approach Isothermal heat of mixing data were obtained at various temperatures for 60 binary systems exhibiting a wide range of liquid solution behavior. Infinite-dilution reduced excess enthalpies, HxE"and Hz"", were determined for each binary system. Most data sources provided correlations for HEIx1x2 versus XI which facilitated calculation of the limiting values. In other cases, the experimental heat of mixing data were plotted as H E I x 1 x 2 versus x1 and . + [HE/ fit to a polynomial to obtain [ H E / X I X ~ I ~ ~and
Stokes and &rfitt
16
14
~ 1 ~ 2 l X ~ ~ l . 12
Van Ness (1961)
10 0.00
0.01
0.02
-
0.03
0.04
1000 0.05
xl Figure 1. Reduced excess enthalpy versus mole fraction of ethanol for ethanol (1)-benzene (2) mixtures at 298.15 K. The solid curves were regressed to experimental data (open circles) and yielded values for HIE" andH2E' nearly identical to those reported in another study (solid circles).
equation. In either case, two nonlinear equations are solved by successive substitution for two unknown parameters.
This graphical estimation procedure is illustrated in Figure 1for the system ethanol (1)-benzene (2) at 298.15 K for which detailed heat of mixing data in both dilute regions have been measured (Stokes and Burfitt, 1973). HE/x1x2 changes rapidly in the hydrocarbon-rich region for alcohol-hydrocarbon mixtures; thus, such systems provide a particularly stringent test of the ability to precisely determine infinite-dilution reduced excess enthalpies. The plots in Figure 1 yield infinite-dilution reduced excess enthalpies of 15 700 and 1560 J/mol for components 1 and 2, respectively. These values are in excellent agreement with the values of HI""and &E" (15 933 and 1570J/mol, respectively) reported by Mrazek and Van Ness (1961) from a correlation of data over the full range of composition. Similar good results were observed for other systems.
3154 Ind. Eng. Chem. Res., Vol. 32, No. 12, 1993 Table 11. Summary of Results for Calculation of Isothermal Binary TVLE
system benzene (l)-octene-l (2)
chlorobutane (l)-carbon tetrachloride (2)
1,2-dichloroethane(1)-carbon tetrachloride (2)
1,2-dichloroethane(1)-chlorobutane (2)
cyclopentane (1)-tetrachloroethylene (2)
cyclohexane (1)-aniline (2)
acetonitrile (l)-benzene (2) acetonitrile (1)-n-heptane (2) benzene (1)-n-heptane (2)
methyl acetate (1)-benzene (2)
methyl acetate (l)-cyclohexane (2)
methanol (l)-ethyl acetate (2) ethanol (l)-ethyl acetate (2) n-propanol (l)-ethyl acetate (2) 2-propanol (1)-ethyl acetate (2) methanol (1)-benzene (2) ethanol (1)-benzene (2)
(K) 283.15 283.15 283.15 283.15 303.15 303.15 303.15 303.15 323.15 323.15 323.15 323.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 298.15 298.15 298.15 298.15 308.15 308.15 308.15 323.15 323.15 323.15 318.15 318.15 318.15 318.15 318.15 318.15 318.15 318.15 318.15 318.15 303.15 303.15 303.15 323.15 323.15 323.15 308.15 308.15 308.15 313.15 313.15 313.15 328.15 328.15 328.15 328.15 328.15 328.15 328.15 328.15 328.15 328.15 328.15 328.16 308.15 308.15 308.15 328.15 328.15 328.15
model Raoult’s law Wilson eq NRTL eq UNIFAC Raoult’s law Wilson eq NRTL eq UNIFAC Raoult’s law Wilson eq NRTL eq UNIFAC Raoult’s law Wilson eq NRTL eq UNIFAC Raoult’s law Wilson eq NRTL eq UNIFAC Raoult’s law Wilson eq NRTL eq UNIFAC Raoult’s law Wilson eq NRTL eq UNIFAC Wilson eq NRTL eq UNIFAC Wilson eq NRTL eq UNIFAC Wilson eq NRTL eq UNIFAC Wilson eq NRTL eq UNIFAC Raoult’s law Wilson eq NRTL eq UNIFAC Wilson eq NRTL eq UNIFAC Wilson eq NRTL eq UNIFAC Wilson eq NRTL eq UNIFAC Wilson eq NRTL eq UNIFAC Wilson eq NRTL eq UNIFAC Wilson eq NRTL eq UNIFAC Wilson eq NRTL eq UNIFAC Wilson eq NRTL eq UNIFAC Wilson eq NRTL eq UNIFAC Wilson eq NRTL eq UNIFAC
VLE for Selected Systems. TPARM (K) N 8 298.15 8 298.15 8 8 8 8 298.15 8 298.15 8 8 298.15 8 298.15 8 8 22 313.15 22 313.15 22 22 22 313.15 22 313.15 22 22 22 313.15 22 313.15 22 22 15 15 298.15 15 298.15 15 14 308.15 14 308.15 14 323.15 15 323.15 15 15 318.15 11 11 318.15 11
318.15 318.15 298.15 298.15 298.15 298.15
9 9 9 15 15 15 15 16 16 16
308.15 308.15
12 12 12
308.15 308.15
8 8 8 9 9 9
318.15 318.15 308.15 308.15
11 11 11
308.15 308.15
11
318.15 318.15 308.15 308.15
11 11 12 12 12
12 12 12
308.15 308.15 318.15 318.15
9 9 9 9 9 9
~ca~cllN &exp
0.009 0.008 0.007 0.006 0.044 0.052 0.051 0.003 0.018 0.014 0.025 0.009 0.011 0.005 0.006 0.008 0.001 0.004 0.001 0.003 0.008 0.002 0.007 0.077 0.010 0.080 0.392 0.019 0.036 0.092 0.111 0.003 0.039 0.038 0.004 0.028 0.028 0.002 0.099 0.212 0.007 0.156 0.285 0.011 0.094 0.137 0.004 0.153 0.255 0.011 0.106 0.179 0.009 0.155 0.253 0.016 0.068 0.210 0.012 0.065 0.166 0.014
-
W e x p Kcad/ KexpIIN
0.029 0.028 0.022 0.020 0.134 0.158 0.157 0.007 0.053 0.039 0.071 0.025 0.014 0.007 0.007 0.010 0.001 0.004 0.001 0.003 0.008 0.002 0.047 0.272 0.030 0.120 0.582 0.032 0.090 0.249 0.300 0.007 0.110 0.105 0.012 0.064 0.064 0.005 0.154 0.319 0.010 0.437 0.723 0.025 0.257 0.388 0.010 0.378 0.647 0.021 0.819 1.515 0.033 0.570 0.987 0.036 0.202 0.589 0.027 0.164 0.453 0.034
ZI(psxp
-
PdIWN 0.104 0.185 0.274 0.024 0.087 0.180 0.263 0.014 0.075 0.172 0.251 0.008 0.038 0.035 0.030 0.029 0.116 0.116 0.113 0.017 0.060 0.026 0.054 0.013 0.084 0.020 0.029 0.039 0.124 0.525 0.110 0.247 1.003 0.096 0.032 0.185 0.008 0.161 2.199 0.024 0.093 0.352 0.373 0.005 0.081 0.076 0.022 0.068 0.068 0.027 0.353 0.961 0.045 0.516 1.504 0.034 0.313 0.429 0.005 0.565 1.138 0.016 0.525 0.971 0.011 0.647 1.306 0.036 0.160 1.491 0.012 0.179 0.686 0.011
Ind. Eng. Chem. Res., Vol. 32, No. 12,1993 3155 Table XI. (Continued) system ethanol (1)-toluene (2)
acetone (1)-water (2)
TVLE (K) 308.15 308.15 308.15 328.15 328.15 328.15 373.15 373.15 373.15 473.15 473.15 473.15
model Wilson eq NRTL eq UNIFAC Wilson eq NRTL eq UNIFAC Wilson eq NRTL eq UNIFAC Wilson eq NRTL eq UNIFAC
TPARM (K) 308.15 308.15 318.15 318.15 323.15 323.15 323.15 323.15
UYeIP -
N 10 10 10 10 10 10 22 22 22 25 25 25
YdCW 0.062 0.165 0.021 0.064
0.142 0.011 0.134 0.158 0.018 0.076 0.099 0.056
-
Z I ( K q - KmdI K 4 N
PdIWN
0.120 0.318 0.041 0.110 0.242 0.019 0.329 0.371 0.059 0.233 0.292 0.165
0.200 1.465 0.036 0.219 1.035 0.012 0.249 0.253 0.024 0.099 0.123 0.104
El(P-
0 Data sources for isothermal VLE. Benzene (l)-octene-1 (2): Vera and Prausnitz (1971). Chlorobutane (l)-carbon tetrachloride (2), 1,2-dichloroethane (l)-carbon tetrachloride (2), and 1,2-dichloroethane (l)-chlorobutane (2): Munoz Embid et al. (1990). Cyclopentane (1)-tetrachloroethylene (2): Polak et al. (1970). Cyclohexane (1)-aniline (2): Abello et al. (1968). Acetonitrile (1)-benzene (2), acetonitrilen-heptane (2), and benzene (1)-n-heptane (2): Palmer and Smith (1972). Methyl acetate (1)-benzene (2) and methyl acetate (l)-cyclohexane (2): Nagata et al. (1973). All four alcohol (l)-ethyl acetate (2) systems: Nagata et al. (1975). Methanol (1)-benzene (2): Scatchard et al. (1946). Ethanol (1)-benzene (2): Ho and Lu, (1963). Ethanol (1)-toluene (2): Kretachmer and Wiebe (1949). Acetone (1)-water (2): Griewold and Wong (1952).
Table 111. Summary of Results for Calculation of Isobaric Binary
benzene (l)-cyclohexane (2)
353.25,353.89
acetonitrile (1)-benzene (2)
354.93,353.25
acetonitrile (1)-n-heptane (2)
354.93,371.58
benzene (1)-n-heptane (2)
353.25,371.58
methyl acetate (1)-benzene (2)
330.76,353.25
methanol (l)-ethyl acetate (2)
337.90,349.87
ethanol (l)-ethyl acetate (2)
351.48,349.87
methanol (1)-benzene (2)
337.91,353.25
Raoult’s law Wilson eq NRTL eq UNIFAC Wilson eq NRTL eq UNIFAC Wilson eq NRTL eq UNIFAC Raoult‘s law Wilson eq NRTL eq UNIFAC Wilson eq NRTL eq UNIFAC Wilson eq NRTL eq UNIFAC Wilson eq NRTL eq UNIFAC Wilson eq NRTL eq UNIFAC
VLE for Selected Systems at 760 m m H p
298.15 298.15 318.15 318.15 318.15 318.15 298.15 298.15 308.15 308.15 308.15 308.15
9 9 9 9 5 5 5 8 8 8 5 5 5 5 12 12 12 14 14 14
308.15 308.15
11 11 11
318.15 318.15
15 15 15
0.027 0.079 0.078 0.005 0.013 0.074 0.004 0.078 0.314 0.065 0.028 0.088 0.109 0.007 0.037 0.037 0.007 0.079 0.115 0.013 0.164 0.276 0.014 0.038 0.155 0.040
0.076 0.188 0.187 0.016 0.041 0.206 0.007 0.135 0.509 0.109 0.061 0.248 0.319 0.022 0.115 0.115 0.013 0.127 0.189 0.025 0.540 0.924 0.026 0.067 0.285 0.074
0.007 0.020 0.019 0.000
0.003 0.016 0.001 0.016 0.070 0.011
0.006 0.022 0.026 0,001
0.006 0.006 0.002 0.022 0.028 0.000
0.035 0.057 0.003 0.008 0.040 0.004
a Data sources for isobaric VLE. Benzene (l)-cyclohexane (2): Thornton and Garner (1951). Acetonitrile (1)-benzene (2), acetonitrile (1)-n-heptane (2), and benzene (1)-heptane (2): Tripathi and Asselineau (1975). Methyl acetate (1)-benzene (2): Nagata (1962). Methanol (l)-ethyl acetate (2) and ethanol (l)-ethyl acetate (2): Zandijcke and Verhoeye (1974). Methanol (1)-benzene (2): Williams et al. (1948).
Equations 14 and 15 were next solved for the Wilson parameters a12 and a21 and eqs 21 and 22 were solved with a = 0.30 for the NRTL parameters b12 and bzl for each of the systems studied. Table I summarizes the infinitedilution reduced excess enthalpies and binary parameters for both the Wilson and NRTL equations at several temperatures for all 60 systems. The van der Waals constants b for each of the pure components are also listed. The data in Table I clearly demonstrate that the parameters are not truly temperature-independent for either the Wilson or NRTL equation. For example, the Wilson parameters a12 and a21 vary up to 2000 J/mol over a temperature range of 20 K (Le., from 298.15 to 318.15 K) for 18 binary alcohol-aromatic hydrocarbon systems. This constitutes a considerable percentage change, especially for the parameter a21. Similar trends are observed for other systems for which heat of mixing data are available a t multiple temperatures. This finding is not
surprising, since it has been suggested by several workers (Duran and Kaliaquinene, 1971;Nagata and Yamada, 1974; Vonka et al., 1975; Prausnitz et al., 1986) that these parameters should be considered to be functions of temperature. Nonetheless, it is useful to examine for what types of systems eqs 14 and 15 or 21 and 22 are approximately valid. It should be emphasized that these equations in their given form cannot be derived with temperature-dependent parameters. Isothermal VLE for 19 systems a t several temperatures were calculated using four bubble-point temperature schemes (i.e., Raoult’s law, Wilson-Prausnitz, NRTLPrausnitz, and UNIFAC-Prausnitz). Similar schemes were used to calculate the bubble-point temperature and vapor composition for eight isobaric binary systems. The vapor phase model Prausnitz refers to the simple corresponding-states correlation (Prausnitz, 1969; Smith and Van Ness, 1987)for the vapor-phase fugacity coefficients.
3156 Ind. Eng. Chem. Res., Vol. 32, No. 12, 1993 1.oo Raoult’s law was used as a basis of comparison for mixtures containing only nonpolar and/or slightly polar species, and the UNIFAC method provided a benchmark for Wilson 0.80 or NRTL model performance for moderately and strongly nonideal systems. VLE prediction results for the 20 selected systems are 0.60 presented in Tables I1 and 111. Table I1 summarizes the results for isothermal systems. The table lists the temx perature at which isothermal VLE were calculated, T ~ E ; 0.40 the model used (i.e., Raoult’s law, Wilson-Prausnitz, NRTL-Pruasnitz, or UNIFAC-Prausnitz); the temper; ature at which parameters were obtained, T p m ~the 0.20 number of mixture experimental data points N,the average error in vapor-phase mole fraction per point, Clyexp -ycalcl/ N, the average percent error in component 1distribution I 0.00 coefficient per point, CI(K1exp - K1 calc)/K1exp)l and the 0.00 0.20 0.40 0.60 0.80 1.00 average percent error in absolute total pressure per point, ~ l ( P ~ x ~ - P ~ ~ c ) / P e xTable p l / ~ .I11gives similar information xl for the isobaric systems studied. Results for four representative systems are now examined in detail, and various issues including effects of system type and the method of I VLE data representation on model performance are addressed. 0 (1970) The system cyclopentane (1)-tetrachloroethylene (21, for which isothermal VLE and heat of mixing data were available at 298.15 K (Polak et al., 19701, illustrates the degree to which the proposed approach works for simple nonpolar mixtures using three conventional means of representing isothermal binary VLE. Wilson and NRTL parameters were determined from the limiting partial molar excess enthalpies HIEmand H2E”, and isothermal VLE were predicted at 298.15 K using each of the four schemes described above. Figure 2 presents the experimental VLE and predictions using the Wilson-Prausnitz bubble-point pressure routine. The results for Raoult’s 0.00 0.20 0.40 0.60 0.80 1.00 law are also shown for comparison. Figure 2a depicts the vapor-liquid composition ( y l - x l ) XI. Y l diagram. Description of y1 versus x 1 is excellent for the Wilson-Prausnitz method. Average errors in vapor mole fraction per point range from 0.005 for Wilson-Prausnitz to 0.011 for Raoult’s law. The major improvement over 0.80 Raoult’s law is evident in the description of the bubblepoint curve shown in Figure 2b (about 3% average error in P versus x1 for the Wilson or NRTL equations as 0.40 compared to 8.4 % error for Raoult’s law). Figure 2c shows equilibrium K values (Ki= yi/xi) plotted against system total pressure. Ki versus P is a standard representation of isothermal VLE for simple relatively nonpolar mixtures such as hydrocarbon mixtures. Again, the deficiency of - Wilson -0.40 K2 ’\ Raoult’s law is clearly apparent. Mixtures containing moderately polar species are often -0.80 fairly well described by the Wilson model but are generally poorly described by the NRTL model. Heat of mixing I and VLE for the acetonitrile (1)-benzene (2) system at -1.20 0 60 120 180 240 300 318.15 K were obtained by Palmer and Smith (1972).HIE“ and H2E” were determined and used to predict isothermal P (mm Hg) VLE for this system at 318.15 K. A comparison of the Figure 2. Experimental and calculated isothermal VLE for the performance of the Wilson and NRTL BUBLP approaches cyclopentane (1)-tetrachloroethylene (2) system at 298.15 K: (a) is given in Figures 3. Results from the UNIFAC method yl-xl diagram;(b) P-xl,yl diagram;(c)K-Pdiagram. Results for the are also shown for comparison. Wilson-Prausnitz approach are superior to Raoult’s law in all three The yl-xl diagram for this system is presented in Figure plots for this simple mixture. 3a. The Wilson-Prausnitz and UNIFAC-Prausnitz methods yield comparable results in this case (0.010 versus 0.007 of the NRTL model is much more pronounced in deaverage deviation in vapor mole fraction for 11data points scription of the P-x1,y1 diagram (Figure 3b). The system for the Wilson and UNIFAC methods, respectively); total pressure is only slightly underestimated using the however, results for the NRTL-Prausnitz method with a Wilson-Prausnitz routine. In contrast, the NRTLPrausnitz routine significantly overestimates the total = 0.30 are considerably less satisfactory (0.077 average pressure (2.7 % average error in P for Wilson’s equation deviation in vapor mole fraction). The poor performance 7
’
/c \ c 1
1 \ ’
Ind. Eng. Chem. Res., Vol. 32, No. 12,1993 3157 1.00 I
a
0.80 .
0.60 *
0.60
r
r
r
x
>,
'
-
A
Scatchard et 81. (1946)
0.40
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n nn W."W
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.
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1 1 1
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o
250 150
200 0.00
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b
0.80
0.60
......
Scatchard et al. (1946)
-
UJIFAC
100
0.20
0.40 XI.
0.60
0.80
1.00
yl
Figure 3. Experimental and calculated isothermal VLE for the acetonitrile (1)-benzene (2) system at 318.15 K. (a) yl-xl diagram; (b) P - z ~ Jdiagram. ~ A good description of the experimentaldata is achievedusing the Wilson-Prausnitz approach;however the NRTLPrausnitz approach fails to produce satisfactory results.
versus 19.5% average error in Pfor the NRTL equation). The UNIFAC method (average error of 0.8% in P for 11 points) is only slightly better than the Wilson model in this case. Methanol (1)-benzene (2) is representative of a large class of highly nonideal binary mixtures which contain one very polar species such as an alcohol which undergoes significant hydrogen bonding and a relatively (but not entirely) inert component such as an aliphatic or simple aromatic hydrocarbon. Mrazek and Van Ness (1961) measured isothermal heats of mixing for 18binary alcoholaromatic hydrocarbon systems from 298.15 to 318.15 K using a novel isothermal dilution calorimeter. Included in their measurements are excess enthalpy data for the methanol (1)-benzene (2) system at 308.15 K. Scatchard et al. (1946)measured isothermal VLE for the same system at 308.15 K. These data are presented in Figure 4 along with VLE for this system calculated using the WilsonPrausnitz BUBLP routine. Results from the UNIFAC method are shown for comparison. Wilson model description of the yl-xl diagram for this system (Figure 4a) is fairly good. The average error of 0.069 in vapor composition is biased by considerable deviations in the benzene-rich region ( X I < 0.10). Pversus X I (Figure 4b) shows considerably greater error (Le., an
0.00
0.20
0.40 XI.
0.60
0.80
1.00
yl
Figure 4. Experimental and calculated isothermal VLE for the methanol (1)-benzene (2) system at 308.15 K (a)yl-rl diagram; (b) P-rlyl diagram.The Wilson-Prausnitz approachproduces only fair results for this strongly nonideal system.
average of 17.5% per point). In contrast, errors for the UNIFAC method are only 0.012 in y1 and 1.2% in P. It is also noted that the NRTL-Prausnitz BUBLP approach completely fails to describe the phase behavior of this system (0.210 error in y1 versus X I and 152.7 3'% error in P versus X I ) . Similar results are observed for other alcoholhydrocarbon systems included in the present study, and an interpretation based on the inability of Wilson parameters to sufficiently back-calculate excess enthalpy data for alcohol-hydrocarbon systems is given later in this paper. Isobaric VLE were also calculated for eight systems at atmospheric pressure. HE data correlation and VLE prediction were done at the same temperature for each of the three previous isothermal systems discussed; hence, it is useful to examine cases where this is not true (isobaric systems) to try to ascertain the effects of differences between correlation and prediction temperatures. The isobaric yl-xl diagram for the system methyl acetate (1)-benzene (2) at 760 mmHg (Nagata, 1962) is shown in Figure 5a. The Wilson model overcorrects for positive deviations from Raoult's law (average deviation in y1 is 0.037 for the Wilson-Prausnitz routine and 0.007 for the UNIFAC-Prausnitz routine). This overcorrection is more apparent in the temperature-composition diagram (Figure 5b). The average errors of 0.6% and 0.2% reported in
3158 Ind. Eng. Chem. Res., Vol. 32, No. 12, 1993
.oo
1
a
0.80
0.60 r
x
6
0.40
-
0.20
V.W
Nagata ( 1962) Wilson UNIFAC
.
0.80
1.00
~~
0.00
0.20
0.60
0.40
xl 360
I
I
b
Nagata 0
-
350
' +
(1962) Wilson
.
340
I
330
320
'
0.00
I 0.20
0.60
0.40 XI.
0.80
1.00
yl
Figure6. Experimentaland calculated isobaric VLE for the methyl acetate (1)-benzene (2) system at 760 mm Hg: (a)yl-rl diagram;(b) T - q y l diagram.A good qualitativedescriptionof the data is produced using thewilson-Prausnitz approach.
Table I11 for the Wilson and UNIFAC methods respectively are artificially low because the temperature range for this system is small compared to absolute temperatures. The VLE system temperatures for this data set vary from about 330 to 355 K whereas model-parameters were determined from heat of mixing data a t 308.15 K. It is likely that the temperature variation has some impact, but it is hard to isolate this effect considering that similar errors in vapor mole fraction and total pressure are observed for calculation of isothermal VLE for methyl acetate (1)-benzene (2) a t 303.15 K from HE data at 298.15 K (see Table 11). It is also interesting to note that excellent results are achieved using the Wilson model to predict isobaric VLE for acetonitrile (1)-benzene (2) at atmospheric pressure (bubble-pointtemperatures in the vicinity of 350 K)from isothermal heat of mixing data at 318.15 K. The degree to which model parameters can be used to back-calculate HE data appears to be a much better performance indicator. A comprehensive discussion of this and other important issues is now presented.
be better thanPor Tversus x l using either method (Wilson or NRTL). The Wilson model produces good-to-excellent yl-xl diagrams for the four systems discussed earlier; however, system pressure- or temperature-composition profiles for the same systems show greater deviation. The distribution coefficient for species 1, K1 ( y ~ l x l )is an alternative measure of system composition. Errors in K1 versus x1 (reported in Tables I1 and 111) follow a trend similar to errors in vapor-phase mole fraction. The use of infinite-dilution reduced excess enthalpies with Wilson's equation generally performs much better than with the NRTL equation for most of the systems studied. The apparent limitation of the NRTL equation in this work is that the nonrandomness factor a must be arbitrarily specified (a = 0.30 here) to obtain a twoparameter model. The general failure of the NRTL equation is quite unfortunate, since the NRTL equation is capable of describing liquid-liquid equilibria whereas Wilson's equation is not. Perhaps a more systematic method for choosing a value of the nonrandomness factor according to type of binary system would produce better results using the NRTL-Prausnitz approach. Model predictive performance for a given system should depend to a large extent on the ability to "back-calculate" heat of mixing data over the full liquid composition range. Such calculations were made with the Wilson model for the four highlighted systems, and the results are presented in Figures 6. Description of the excess enthalpy profiles for cyclopentane (1)-tetrachloroethylene (2) at 298.15 K and for acetonitrile (1)-benzene (2) at 318.15 K is excellent, suggesting that the temperature-independent Wilson model is useful for describing phase behavior of nonpolar mixtures and nonideal systems with moderately unsymmetric excess functions. However, only fair agreement between calculated and experimental heat of mixing data is observed for the system methyl acetate (1)-benzene (2) at 308.15 K (Figure 6b), while the results for methanol (1)-benzene (2) at 308.15 K (Figure 6b) are very poor. The results from Figure 6 are in accord with the results from Figures 2-5, indicating that the capacity of estimated parameters to describe heat of mixing data is paramount in determining VLE predictive performance. One possible reason for the failure in HE representation for alcohol-hydrocarbon systems is that there are solutions to eqs 14 and 15 which are unattainable by the method of successive substitution. An alternative explanation is that the temperature dependence of (1.12and (1.21 is great enough to make eqs 14and 15inaccurate for suchsystems. Finally, vapor-phase association must play a role for alcoholcontaining systems; however, UNIFAC predictions are so good that no reasonable amount of unaccounted for alcohol association could offset large Wilson model errors. Nevertheless, future work should employ a vapor-phase model (Prausnitz et al., 1980) which properly accounts for association. Some additional insight is available for alcohol-hydrocarbon mixtures. The Wilson parameters a12 and a21 exhibit strong trends with HIE" and HzE", respectively. The data for the 18 binary alcohol-hydrocarbon systems in Table I are plotted as a12 versus H l E mand (1.21 versus H2E"in Figure 7a and 7b, respectively. A clear linear trend is evident in each case with a near perfect linear relationship between a21 and HzE". These plots yield the relations
Discussion Some general conclusions can be reached from the results of this study. First, predictions of y1 versus x1 appear to
a12= -217.2 and
+ 0.974H:"
(23)
Ind. Eng. Chem. Res., Vol. 32, No. 12, 1993 3159 500
19
a
,,B'-QQ
b
\
Y\
I
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\
z
Palmer and '\ Smith (1972) '\
0
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(1961) Calculated
5000
1
0
-.
4000
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Calculated
F
%
400
200
3000
2000
0 0.00
1000
0.20
0.40
0.60
0.80
1.00
1000 2000
xl
Figure 6. Back-calculated heat of mixing for the systems: (a) cyclopentane (1)-tetrachloroethylene (2) at 298.15 K (Polak et al., 1970) and acetonitrile (1)-benzene (2) at 318.15K (Palmerand Smith, 1972); (b) methanol (1)-benzene (2) at 308.15 K (Mrazek and Van Ness, 1961) and methyl acetate (1)-benzene (2) at 308.15 K (Nagata et al., 1973). Results are excellent for both systems in (a), fair for methyl acetate-benzene, and poor for methanol-benzene.
= -43.8 + 1.003H,Em (24) where all quantities are expressed in J/mol. These results can be cautiously explained in terms of the vastly different behavior of alcohol-rich versus hydrocarbon-rich mixtures. In the infinitely dilute alcohol region the value of HIE" is approximately equal to the enthalpy of hydrogen bond formation for the alcohol, assuming the hydrocarbon solvent to be inert. Thus, it can be seen from eq 23 that the Wilson parameter 012 is nearly equal to the hydrogen bond energy of the alcohol. However, this physical significance is very imprecise considering the poor results in Figure 6b. Of practical importance is that the correlations given by eqs 23 and 24 can be used to provide rough estimates of the Wilson parameters a12 and u21. The approach presented in this paper may be useful for new systems for which few experimental thermodynamic data are available. One highly potential application is the preliminary identification of chlorofluorocarbon (CFC) alternative mixtures with low ozone depletion potentials. Results from calorimetric studies in the dilute regions of such mixtures (likely to consist of various hydrofluorocarbons) could be used to identify azeotropes with excellent
3000
4000
5000 6000
HE2inf (J/mol)
Figure 7. Correlation of Wilson parameters with infinite- dilution reduced excess enthalpies for binary alcohol-aromatic hydrocarbon systems: (a) the parameter a12 versus HIE";(b) the parameter (121 versus HzE'.
potential for CFC substitution. In contrast, it is unlikely that the proposed approach in its present form will find any great use for strongly nonideal systems.
~2~
Conclusions An approach is presented using the Wilson or NRTL equation with limiting partial molar excess enthalpies to calculate isothermal or isobaric VLE for several systems over limited ranges of temperature. Analysis of several factors including the type of mixture, temperature difference between parameter estimation and VLE calculation, ability to back-calculate heat of mixing, and vaporphase effects suggests that the Wilson equation may be useful in cross-predictingVLE from infinite-dilution excess enthalpy data for nearly ideal and moderately nonideal systems. However, reasonable estimates of vapor-liquid equilibria are also possible for strongly nonideal systems. The proposed modellapproach may be particularly useful in reducing experimental effort in exploratory research for new mixtures.
Nomenclature aij = Wilson binary parameter (J/mol) bi = van der Waals constant for species i (cms/mol) bjj = NRTL binary parameter (J/mol)
3160 Ind. Eng. Chem. Res., Vol. 32, No. 12, 1993
GE = excess Gibbs energy (J/mol) Gij = NRTL binary parameter EIE = excess enthalpy (J/mol) Hpm= infinite-dilution reduced excess enthalpy of species i (J/mol) Ki = distribution coefficient for species i n = total moles (mol) ni = moles of species i (mol) N = number of data points P = absolute total pressure (mmHg) PCdc= calculated total pressure (mmHg) Pci = critical pressure of species i (bar) PSI,,= experimental total pressure (mmHg) PisAT = vapor pressure of species i (mmHg) R = gas constant (J/(mol K)) or (cm3bar/(mol K)) T = absolute temperature (K) Tci = critical temperature of species i (K) TisAT = saturated temperature of species i (K) T p m ~= temperature at which H P were determined (K) T ~ = temperature E at which isothermal VLE were predicted
(K)
VE = excess volume (cm3/mol) Vi = liquid molar volume of species i (cm3/mol) = liquid-phase mole fraction of species i y d c = calculated vapor-phase mole fraction of species i yexp= experimental vapor-phase mole fraction of species i yi = vapor-phase mole fraction of species i xi
Greek Letters a = nonrandomness factor in NRTL equation yi = activity coefficient of species i yim = infinite-dilution activity coefficient of species i Aij
= Wilson binary parameter NRTL binary parameter
qj=
Subscripts c = critical value calc = calculated value exp = experimental value i = species i j = species j nj = constant moles of species j P = constant pressure T = constant temperature x = constant composition x,"o = infinitely dilute species i Superscripts E = excess property SAT = saturated value m = infinite-dilution value
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Received for review February 24, 1993 Revised manuscript received August 13, 1993 Accepted September 7, 1993. Abstract published in Advance ACS Abstracts, November 1, 1993.
