Ind. Eng. Chem. Fundam., Vol. 17, No. 4, 1978 331
Optimal Representation of Binary Liquid Mixture Nonidealities George L. Nlcolaldes and Charles A. Eckert" Department of Chemical Engineering, University of Illinois, Urbana, Illinois 6 180 7
Recent advances in the measurement of solution properties permit a reevaluation of data representation for binary liquid niixtures. This work pursues the dual goal of determining how well available analytical expressions can correlate various types of mixture data (Le., limiting activity coefficients, vapor-liquid equilibria, enthalpies of mixing, and liquid-lliquid equilibria) and of suggesting an experimental approach to achieve optimal representation with minimum effort. New measurements of limiting activity coefficients are combined with data from the literature to investigate the utility of local-composition equations in extrapolating and cross-predicting the different types of binary data. The Wilson, UNIQUAC, and Zeta equations are found similar in their overall utility. No single experimental method can be! used to determine generally valid adjustable parameters in the equations, and the temperature dependence of the (excess Gibbs energy predicted by the models has limited applicability. Liquid-liquid equilibria cannot be predicted from any other data, but limiting activity coefficients at one temperature and a few heat of mixing points are sufficient to describe completely the nonideality of miscible systems over a reasonable temperature range.
Introduction Precise knowledge of excess thermodynamic properties of liquid mixtures is; essential to the design of many chemical engineering operations. Phase equilibria must be well understood in distillation, extraction, or extractive distillation. In addition, solution thermodynamics is becoming increasingly important to the prediction of kinetic solvent effects, a potentially major factor in the optimization of solvent composition. Efforts in solution thermodynamics are commonly centered around the prediction and correlation of activity coefficients, y L ,which are directly related to the molar excess Gibbs energy. gE = RTCx, In y L 1
Attempts to predict the excess Gibbs energy from pure component properties have been successful to some extent only for mixtures of simple, aprotic, nonpolar molecules to which regular solution theory is applicable (Hildebrand and Scott, 1962). Foir applications where more accurate information is necessary or for systems exhibiting greater nonideality, excess properties must be determined experimentally. The purposes of this paper are to consider (i) what physical measurements can yield the maximum amount of information with a minimum of experimental effort, and (ii) how the available data can be best correlated and extended. Activity coefficients have most often been correlated by fitting vapor--liquid equilibrium (VLE) data to semiempirical expressions for the excess Gibbs energy. Such expressions can be usually derived from a simple statistical mechanical model and incorporate several adjustable parameters. The well-known Margules and Van Laar equations are examples of this approach. Considerable improvement was achieved in 1964 with the Wilson equation (Wilson, 19164), the first of a series of semitheoretical entropic expressions based on the concept of local composition in a liquid mixture. With two adjustable parameters per binary only, the Wilson equation can be used to fit VLE data for miscible systems better than any previous model and, most important, it can be used to calculate accurately multicomponent equilibria from binary data only (Eckert et al., 1965; Orye and Prausnitz, 1965). A temperature dependence of the excess Gibbs energy is included in the Wilson equation which allows, with the adjustable parameters assumed constant, the extrapolation of VLE data over a small temperature range and suggests,
in principle, the possibility of cross-predicting VLE and heat of mixing data through the rigorous relation
Any expression whose form may be theoretically valid raises the question of what are the most readily accessible experimental data from which to estimate its parameters, and what other data can then be safely predicted. A number of local composition equations have been developed since 1964 (Orye, 1965; Heil and Prausnitz, 1966; Renon and Prausnitz, 1968; Bruin and Prausnitz, 1971; Morisue et al., 1972; Palmer and Smith, 1972; Hsieh and Eckert, 1974; Bradley, 1976; Abrams and Prausnitz, 1975). These equations are generally able to fit VLE binary data well, extend the binary data to multicomponent systems, and represent liquid phase immiscibility with two or three parameters per binary. They usually reduce to one another if well-defined changes are made in their derivations or they can be shown to be partial cases of generalized forms. No notable improvement over the Wilson equation has been claimed in the correlation of VLE data, the main purpose for which these equations have been developed, though efforts have been made to extend the utility of some equations to the prediction of ternary liquid-liquid equilibria (LLE) from binary data only (Renon and Prausnitz, 1968; Abrams and Prausnitz, 1975). New methods for measuring limiting activity coefficients (7") directly (Eckert et al., 1978) represent an advance in experimental efficiency just as the local-composition equations represented a significant analytical advance. For binary liquid mixtures where the volatilities of the components are not too far apart, y mdata can be obtained more easily and quickly than the y-x-P data generated by most traditional VLE methods. Besides, it has been known for some time (Shreiber and Eckert, 1971) that the y m values are just as effective in predicting VLE compositions as data taken over the entire concentration range, provided, as is most often the case, that the activity coefficients vary monotonically with the mole fraction. Using local composition equations as the basic computational tool, we suggest alternatives to the measurements of VLE a t every temperature of interest. Vaporliquid, liquid-liquid, and heat of mixing data from the literature were combined with activity coefficients a t infinite dilution, ym,and with additional, recently de-
0019-7874/78/1017-0331$01.00/00 1978 American Chemical Society
332 Ind. Eng. Chem. Fundam., Vol. 17, No. 4, 1978
termined (Nicolaides and Eckert, 1977) heats of mixing. A listing of the data sets used is given in Table I. The equations used are the Zeta equation (Hsieh and Eckert, 1974), the UNIQUAC equation (Abrams and Prausnitz, 19751, the Wilson equation (Wilson, 1964),and the Van Laar equation. The Wilson and Van Laar equations provide a good basis for comparison as, respectively, the first and still widely used entropic equation, and the most useful of the older enthalpic forms. The Zeta and UNIQUAC equations are two of the most recent local composition equations and incorporate the best-defined physical assumptions in their derivations. They have both been shown to include, as special cases, all major expressions developed earlier, and can be applied to systems that exhibit liquid-liquid immiscibility. Neither one has been previously used in connection with limiting activity coefficients or heat of mixing data. The Zeta equation is based on an extension of regular solution theory to include the concept of “free coordinates”, where the number of independently determinable neighbors for a given molecule lies closer to unity than the actual coordination number, depending on structure. In the UNIQUAC equation, the local area fraction is used as the primary concentration variable in connection with Guggenheim’s quasichemical analysis (Guggenheim, 1952). This investigation was restricted to binary systems only. Since multicomponent vapor-liquid equilibria can be calculated from binary data, part of the problem has been reduced to the determination of valid binary parameters a t any temperature of interest. These parameters can be obtained from different kinds of mixture data a t various temperatures and are not usually compatible, since the equations themselves are imperfect. The questions that arise are (i) what additional assumptions need to be made about the local composition parameters to generalize their applicability, (ii) what are the most useful data to fit, in terms of other data that can be predicted from the adjusted parameters, and (iii) which is the best equation to use in a particular application. These questions must be settled for binary systems before VLE multicomponent predictions can be made efficiently, and before the more difficult calculation of multicomponent LLE and h” data from binary data only can be successful. Considerable difficulties still exist with regard to forming a valid simultaneous representation of all binary mixture properties; fitting multicomponent data would complicate the problem further. Besides, any parameter determination method relying on multicomponent mixture data would require enormous experimental effort; it is preferable to use binary data only. Different aspects of the general problem posed in this work have also been investigated in previous studies. Schreiber and Eckert (1971) used activity coefficients a t infinite dilution with the Wilson equation to predict vapor-liquid equilibria; they estimated y m values by extrapolating classical VLE data. Duran and Kaliaquine (1971) and Tai et al. (1972) used the same equation to represent simultaneously VLE and hE data; Nagata and Tamada (1972) made a similar study with the Wilson, Heil, and NRTL equations. Asselineau and Renon (1970) studied the NRTL equation exhaustively with all data available a t the time. More recently, Bradley (1976) investigated the cross-prediction and simultaneous representation of phase equilibria and heats of mixing with several equations but a narrower data base; he has also reviewed all previous work in this area. Vonka et al. (1975) and Hanks et al. (1978) have examined the analytical a priori limitations of the Wilson and NRTL equations with
regard to the simultaneous representation of gE and hE. No other investigator has critically studied the Zeta or the UNIQUAC equation, or used directly measured activity coefficients a t infinite dilution. Equations Used The Van Laar equation is
where the constants A and B must be determined at every temperature. The Wilson equation has a temperature dependence built in gE
- = -xl In ( x ,
RT
+ Alzxz) - x 2 In (AZlxl+ x,)
(3)
(4) (5) as do other local composition equations. The UNIQUAC equation is gE
- = x1 In
RT
91 -+
xt
6, =
9 2
x 2 In -
+
X2
41x1 41x1 + 42x2
721
= ex.(
;
02
=
92x2
(8)
41x1 + 42x2
?iT-) - u11
- 4 1
and the Zeta equation is gE _ - -{(xlul + x2u2).[91In (a1+
f
RT
@l @Z a2In (a2+ L42191)]+ x1 In + x2 In (11)
X1
X2
(13)
A21
= exP(-
RT) c22 -- c12
S
{= -
(16)
UlUZ
The UNIQUAC and Wilson equations include two adjustable parameters per binary, the differences Xi, - Xii, and ui, - uii, respectively, which are initially assumed
Itid. Erig. Chem. Fundam.. Vol 17, No. 4, 1978
independent of temperature. The Zeta equation has two similar parameters, c,, - c, , and in addition a third parameter, s. Since the third parameter can be treated as a constant without limiting the applicability of the Zeta equation, the three equations can be compared on a common basis (Nicolaides, 1977). The parameters in 2111 four equations have approximate physical interpretations. The Van Laar parameters can be considered products of effective molecular volumes and an energetic interaction coefficient. The entropic equation parameters, A , - A,, etc., represent binary molecular interaction energies and, if assumed pairwise additive, can be used direct,ly in a ~nulticomponentmixture. This assumption of pairwise additivity has been repeatedly verified by accurate calculation of multicomponent vapor-liquid equilibria from binary data only. In addition, the two entropic equation parameters for a given binary can be related through configurational pure-component properties and reduce to one parameter only. Such a transformation can be very useful as it allows, the prediction of gE from one data point only, e.g., one ymvalue; it is an important potential application as well as one criterion for evaluating equations. Of even greater importance is the assumed temperature dependence of gE, as it determines both the possibility of extrapolating data t o different temperatures as well as the cross-prediction of VLE, LTX, and hE data. When this dependence fails to represent the data, the iiumber of parameters can be increased to four by assuming a linear temperature dependence in the original parameters. Any further increase in the number of parameters would essentially reduce the local composition equations to mathematical tools of no general physical validity. Adjustable parameters may be determined exactly at any temperature from limiting activity coefficient values, ym,or, for partially miscible systems, from mutual solubilities. VLE and tiE data reduction, on the other hand, requires nonlinear regression and several objective functions are possible, each of which may yield different parameter values. I11 this work, the root-mean-square percent error in the total pressure or the heat of mixing has beeii consistently used as the objective function to determine parariieters from VLE and 11” data, respectively, and all experimental points are given equal weight. The same quantities are used to evaluate the applicability, to a given data set, of parameters determined on a different basis.
Results and Discussion Miscible Systems. The temperature dependence of the excess Gihbs energy ]predicted by the local coinposition equations is often cited as one of their major advantages over the older enthallpic correlations, primarily for the purpose of predicting iisobaric vapor -liquid equilibria from isothermal data. This dependence was investigated by predicting isothermal VLE data with parameters determined by fittiiig data a t a different temperature, and by attempting to cross-predict VLE and heats of mixing. Typical results are shown in Table 11; complete tabulations can be found elsewhere (Nicolaides, 1977). There is a substantial iiicrease in the error of VLE prediction as the temperature differences between predicted and fit data sets increase. As expected, the local composition equations give better results than the Van Laar equation which contains no temperature-dependent term. Still, the temperature extrapolation of VLE data is seen as little more than an estimai,e. No improvement or7er the Wilson equation is seen with the more recent niodels and, indeed, the improvement of
333
all three over the Van Laar, while definite, is not as great as might have been expected. The temperature dependence of gEpredicted by the local composition models does little more than affect the VLE extrapolation in the right direction. In view of the above, little can be expected from a VLE-hE cross prediction, a more demanding undertaking. The results in Table I1 show that only the crudest hE prediction can be made from the VLE data while the reverse prediction is even less useful. The limitations of cross-predicting hE and VLE data with the Wilson equation have been known since 1965 (Orye, 1965) and were reconfirmed by Nagata and Yamada (1972) as well as by Branden (1976); the last two reached the same conclusion for other local composition equations. Our results for the [JNIQUAC and Zeta equations rule out the possibility that the measurement of the heat of mixing can, by itself, be considered an alternative to classical VLE measurements, with the theoretical tools available a t present. The opposite conclusion was reached by Hanks, Gupta, and Christensen (1971), who used heat of mixing data to predict VLE well for two binary systems with the Wilson equation and for four systems with the NRTL. Our results (Nicolaides, 1977) for the acetone -carbon tetrachloride system, one of the binaries they used with the Wilson equation, would support their conclusions but we found these results not to be typical for the much wider data base used in this study. Asselineau and Renon (19701, who were the first to study the possibility of cross-predicting thermodynamic data with the NRI‘T, equation, with an exhaustive data base, did not report any h”-VLE crosspredictive capability for the model while Nagata and Yamada (1972), who tested the Wilson, Heil, and NRTL equations in this regard with 13 systems, have suggested that “the method proposed by Hanks et al. (1971) should be limited to special cases”. More recently, Tan, Hanks, and Christensen (1977; 1978) used the NRTL equation and binary hE data to calculate isothermal ternary VLE data for two systems and isobaric VLE data for two binary and two ternary systems. Unlike heat of mixing data, activity coefficients a t infinite dilution (to provide an alternative to VLE measurements for inany systems. Two adjustable parameters can be determined from 7-values a t the same temperature. The data can be extrapolated to the temperatures of interest as described previously (Eckert et al., 1978). In Table 111,typical results are shown; the VLE are predicted very well although, as expected, no similar prediction can be made for the heats of mixing. This method cannot he applied to systems that exhibit extrema in their y vs. x curves, but such systems are relatively uncommon. A better view of the applicability and the siniilarities of the different entropic models can be gained by considering the interrelationship of the parameters obtained from different data sets. Figures 1 to 3 show parameter pairs from VLE and h” data fits, and from limiting activity coefficients. Parameter pairs are shown that give the best fits of VLE and 11” data, surrounded by regions in which any pair of parameters will fit the data within a small error-these regions are omitted for the Wilson fit of h” data where the minimum error is substantial. As shown, parameters determined from VLE data are unique, Whole regions of parameter values can predict the total pressure within the same small error, so that the “optimal” pair found is only incidental. The regions move with temperature and may overlap over a small temperature range. Parameter values may be changed to affect the equilibria predicted at one ternperature but still lie wholly within the
334
Ind. Eng. Chem. Fundam., Vol. 17, No. 4, 1978
Table I. References for Data Used system benzene( 1)-cyclohexane( 2)
benzene( 1)-heptane( 2)
acetone( 1)-benzene(2)
acetone( 1)-carbon tetrachloride( 2)
nitromethane( 1)-benzene( 2 )
acetonitrile( 1)-benzene( 2 )
carbon tetrachloride( 1)-acetonitrile( 2 )
nitromethane( 1 -carbon tetrachloride( 2) chlorobenzene( )-1-nitropropane( 2)
acetone( 1)-ethanol( 2 )
ethanol( 1)-benzene(2)
ethanol( 1)-toluene( 2)
ethanol(1)-heptane(2)
ethanol(1)-chlorobenzene( 2)
data
temp (K) or press. (mmHg) 283 K 313 K 343 K 298 K 352-311 293-351 333 K 353 K 298 K 331-366 304-329 318 K 318 K 293-350 304-329 318 K 318 K 296-347 304-328 318 K 318 K 318 K 359-371 318 K 318 K 318 K 318 K 318 K 318 K 293-353 318 K 318 K 318 K 348 K 393 K 298 K 323 K 353 K 354 K 305 K 313 K 321 K 298 K 323 K 322-348 306-327 318 K 318 K 342-350 346 K 308 K 328 K 298 K 333 K 293 K 318-349 303 K 323 K 343 K 283 K 303 K 348 K 293 K 319-348 298 K 298 K 323 K 323-348
K K
K K K K K K
K
K
K K K
K
K
K
reference Boublik (1963) Scatchard e t al. (1939) Scatchard et al. (1939) Stokes et al. (1969) Nicolaides (1977) Nicolaides (1977) Brown and Ewald (1951) Brown and Ewald (1951) Lundberg (1964) Nicolaides (1977) Nicolaides (1977 ) Brown and Smith (1957) Brown and Fock (1957) Nicolaides (1977) Nicolaides (1977) Brown and Smith (1957) Brown and Fock (1957) Nicolaides (1977) Nicolaides (1977) Brown and Smith (1955) Brown and Fock (1956) Nicolaides (1977) Nicolaides (1977) Brown and Smith (1955) Brown and Fock (1956) Nicolaides (1977) Nicolaides (1977) Brown and Smith (1954) Brown and Fock (1956) Nicolaides (1977) Nicolaides (1977) Brown and Smith (1955) Brown and Fock (1956) Lacher et al. (1941) Lacher e t al. (1941) Nicolaides and Eckert 1978 Nicolaides and Eckert (19781 Nicolaides (1977) Nicolaides (1977) Gordon and Hines (1946) Gordon and Hines (1946) Gordon and Hines (1946) Nicolaides and Eckert 1978 Nicolaides and Eckert 11978) Nicolaides (1977) Nicolaides (1977) Brown and Smith (1954) Brown e t al. (1956) Nicolaides (1977 ) Nicolaides (1977) Kretchmer and Wiebe (1949) Kretchmer and Wiebe (1949) Van Ness et al. (1967) Van Ness e t al. (1967) Nicolaides (1977) Nicolaides (1977) Smyth and Engel (1929) Smyth and Engel (1929) Smyth and Engel (1929) Van Ness e t al. (1967) Van Ness et al. (1967) Van Ness et al. (1967) Nicolaides (1977) Nicolaides (1977) Schulze (1956) Nicolaides and Eckert 1978 Nicolaides and Eckert I1978\ Nicolaides (1977)
Ind. Eng. Chem. Fundam., Vol. 17, No. 4, 1978
335
Table I (Continued) system
data
ethanol( 1)-hexane( 2)
VLE VLE VLE
hE hE LLE VLE VLE VLE hE
cyclohexane( 1)-aniline( 2 )
hE cyclohexane( 1 )-aniline( 2) hexane( 1)-nitroethane( 2)
LLE
hexane( 1)-nitrobenzene( 2)
LLE V LE VLE VLE
hE hE LLE VLE hE
methylcyclohexane( 1)-furfural( 2 )
7-W
cyclohexane( 1)-furfural( 2)
LLE V LE hE 7-W
heptane( 1 )-furfural( 2 )
LLE V LE ym(*)
isooctane( 1)-furfural( 2 )
_____
0
LLE VLE
temp ( K ) or press. (mmHg) 273 K 303 K 328 K 303 K 318 K 273-304 308 K 323 K 392 K 308 K 323 K 293 K 308-351 300-305 318 K 308 K 323 K 293 K 305-338 266-293 294 K 298 K 308 K 298 K 323 K 287-343 760 m m 348 K 347-373 298-338 760 mm 348 K 337-349 294-364 760 mm 332-369 278-364 760 m m
Isii (1935) Isii (1935) H o and Lu (1963) Savini e t al. (1965) Savini e t al. (1965) Angelescu and Giusca (1942) Abello e t al. (1968) Abello et al. (1968) Kortuem and Freier (1954) Nicolaides and Eckert 1978 Nicolaides and Eckert 119781 Nicolaides (1977) Nicolaides (1977) Snyder and Eckert (1973) Edwards (1962) Nicolaides and Eckert 1978 Nicolaides and Eckert 11978) Nicolaides (1977) Nicolaides (1977) Snyder and Eckert (1973) Neckel and Volk (1964) Neckel and Volk (1964) Neckel and Volk (1964) Nicolaides and Eckert 1978 Nicolaides and Eckert 11978) Pennington and Marwil (1953) Thornton and Garner (1951) Nicolaides and Eckert (1978) Nicolaides (1977) Pennington and Marwil (1953) Thornton and Garner (1951) Nicolaides and Eckert (1978) Nicolaides (1977) Pennington and Marwil (1953) Thornton and Garner (1951) Nicolaides (1977) Pennington and Marwil (1953) Thornton and Garner (1951)
K
K K
K K
K
K K
K K K K
Region of 0 5 % error in tolol pressure
-
Paraimeters from y m data 01 iicreasing ternoeratures Minirwn error in VLE
reference
Region of
0 5% error
Region of 5 % error in
in toto1 pressure the
heot of miring
Parameters from y m data c t increasing ternwratdres
600 'vlin,rnum error in h E
I I L
-200
A
200
A l2 - Xli
400
600
I
I cal/molei
Figure 1. Wilson equation parameters determined from vapor-liquid equilibrium, heat of mixing, and limiting activity coefficient data for the benzene(l)-cyclohexane(2) system.
same error region for another temperature. Parameters from VLE and hE do not overlap and the failure of the cross-prediction is not due, as one might have hoped, to the nonuniqueness of the parameters, in which case a better parameter estimation procedure would have been
VLE134315"Kli>-\
- 4oc
-200
0
200
'\
400
1
uzl- u l A (cal/mole)
Figure 2. UNIQUAC equation parameters determined from vapor-liquid equilibrium, heat of mixing, and limiting activity coefficient data for the benzene(l)-cyclohexane(2) system.
sought. The hE-determined parameters are better defined but were plotted for a larger allowed error to consider the possibility of overlap. The narrow, similarly oriented regions from the VLE data suggest that the two parameters are not independent but may be related linearly. One-parameter versions of
338 Ind. Eng. Chem. Fundam., Vol. 17, No. 4, 1978 Table 11. Binary Vapor-Liquid Equilibrium and Heat of Mixing Data Fit to Two-Parameter Excess Gibbs Energy Equations and Cross-Redicted Equations rms R error in total pressure or heat of mixing
system benzene-cyclohexane
equation Wilson
UNIQUAC
data set fit (i) (ii j (iii) (iv) (i) (ii) (fii) (1v)
0)
Zeta
Van Laar
(ii) (iii) (iv) (i 1 (ii) (iii)
(i)
Wilson
UNIQUAC
Zeta
Van Laar
0) (ii) (iii) (iv) (VI 0) (ii) (iii) (iv) (VI (i) (ii) (iii) (iv) (v)
0)
(ii) (iii)
343 K
(VLE)
(VLE)
(VLE)
0.2 1.8 3.2 29.1 0.2 1.7 3.1 22.0 0.2 1.7 3.1 23.3 0.2 2.7 4.6
1.7
2.9 1.3
UNIQUAC Zeta Van Laar
fi) (ii j (i) fii) '(ij (ii) (i)
30.0 2.8 1.2 0.0 21.9 2.8 1.2
0.0
1.3 23.5 2.9
0.0
0.0
23.2 5.1 2.1
2.0
0.0
(ii)
(iv)
298 K (hE 1 59.2 64.4 68.9 9.2 58.2 63.6 68.4 0.5 58.1 63.5 68.6 0.2
0.0
1.3 30.0 1.7 0.0 1.3 22.2 1.7
(iii)
313K
321 K
(VLE)
(VLE)
(VLE)
0.2 1.9 2.4 49.4 55.6 0.2 1.9 2.4 59.9 74.4 0.2 1.9 2.4 63.8 79.0 0.3 2.4 3.3
1.8 0.1 0.5 48.8 54.7 1.8 0.2 1.0 58.2 71.5 1.8 0.2 1.0 61.9 75.8 2.3 0.2 0.9
2.2 0.5 0.2 46.3 51.8 2.1 0.5 0.2 54.5 66.6 2.1 0.5 0.2 57.8 70.5 3.2 0.9 0.2
(i)
Wilson
0.0
305K
318K acetonitrile-benzene
(iii)
313 K
0)
acetone-ethanol
(ii)
283 K
(iv)
(VI
298K (hE 1 59.4 63.2 63.8 30.9 31.2 56.8 61.2 61.9 0.6 7.7 57.5 62.5 63.1 0.9 7.4
323K (hE 1 62.2 65.5 66.1 32.7 32.4 60.4 64.1 64.8 7.0 0.7 60.8 60.8 65.5 6.8 1.2
(ii)
318K
(VLE)
W E1
0.3 5.2 0.4 3.7 0.4 7.0 0.4
20.5 3.0 19.0 3.8 20.4 3.8
Table 111. Binary Vapor-Liquid Equilibria and Heats of Mixing Calculated from Limiting Activity Coefficients rms 7'% error in total pressure or heat of mixing system benzene-cyclohexane
acetone-e thanol
acetonitrile- benzene
VLE, 283 K VLE, 313 K VLE, 343 K
h E , 298 K VLE, 305 K VLE, 313 K VLE, 321 K h E , 298 K h E , 323 K VLE, 318 K h E , 318 K
the local composition equations are based on such a relationship, established through pure component properties. The uncertainty in the parameters also shows that VLE data may be less useful, for purposes of cross-prediction,
Wilson
UNIQUAC
0.6
0.5 0.1 0.1 61.9 1.5 2.3 1.7 52.5 61.3 1.1 19.5
0.1 0.1
62.8 1.2 2.1 1.5 56.3 63.2 0.74 22.0
Zeta
0.5 0.1 0.1 61.8 1.5 2.3 1.7 53.1 61.0 1.2 23.0
Van Laar
0.5 0.1 0.1 1.6
2.4 1.8 1.4
than either hE or y mdata, since the exact values obtained would depend heavily on parameter estimation techniques. They may also be the easiest to predict, since a large number of parameters are equally suitable. Laborious
Ind. Eng. Chem. Fundam., Vol. 17, No. 4, 1978 337
Reqton of 5% error i n the heal of rnlx~ng Mimmurr error in VLE
I
C,,-C,
(cal/mole)
Figure 3. Zeta equation parameter determined from vapor-liquid equilibrium, heat of mixing, and limiting activity coefficient data for the benzene(l)-cyclohexime(2) system.
VLE experimental methods probably yield the least useful information for the effort expended. In contrast to the VLE regions, single lines are obtained from the y mdata. These lines were drawn through a few points, each representing one parameter pair at one temperature. The lines are compatible with the VLE results and are more easily determined with the recently developed techniques (Eckert et al., 1978). Obviously, any two VLE data points could be used to obtain a single pair of parameter values a t any given temperature but it must be stressed that the accuracy required of such data points increases sharply with their distances from the two composition limits. All three equations give similar patterns, indicating that they are variations of the same essential model. A simple modification of the local composition equation that enhances its appllicability considerably is the increase of parameters from two to four per binary through the inclusion of a linear temperature dependence in each energy parameter. That the "constant" parameters are actually functions of temperature is clear from the results of the previous section; it has also been suggested by several authors. The linear dependence is not necessarily the only possibility but there are insufficient data available to permit a more complex choice. Typical results are shown in Table IV. First, for each binary system and each equation used, all VLE data sets are fitted simultaneously with four parameters and then the error in the prediction of each VLE and each hE set, from the common parameters, is tabulated. Second, the hE data are fitted and similar tabulation made, and finally, all VLE and hE are fitted together, weighted as shown in the table. Clearly, four parameters per binary are sufficient to correlate VLE data a t different temperatures. The data are represented almost as well as when they are fit separately a t each temperature with two parameters. The prediction of the heat of mixing from VLE data only is also improved slightly for most systems when four parameters are used, which suggests that the temperature dependence introduced may have some theoretical validity. The uncertainty in the four parameters determined from VLE data is considerably greater than the uncertainty in the two parameters adjusted previously, but parameter regions cannot now be mapped easily.
al
3
3
d
0
3
0
9 al
3 3 Q
338
Ind. Eng. Chem. Fundam., Vol. 17, No. 4, 1978
Table V. Binary Vapor-Liquid Equilibria and Heats of Mixing Calculated from y a and hE Data Reduced Simultaneously rms % error in total pressure of heat or mixing system
Wilson
UNIQUAC ~~
benzenecyclohexane
acetoneethanol
acetonitrilebenzene
VLE, 283 K VLE, 313 KQ VLE, 343 K h E , 298 K VLE, 305 K VLE, 313 K4 VLE, 321 K h E , 298 Kb h E , 323 K VLE, 318 K h E , 318 K
Temperature of set used.
ym
data.
0.3 0.1 0.4 0.5
1.1 2.1
1.6 2.2 2.2
0.7 2.9
0.3 0.1 0.3 1.1 1.5 2.3 1.8 5.5 5.2 1.1 4.4
Zeta ~
0.3 0.1 0.4 1.0 1.3 2.3 1.9 3.8 3.5 1.2 3.8
Temperature of hE data
Heats of mixing can be correlated well with four parameters per binary and there is a substantial change from the two-parameter fit shown in Table IV. There is no corresponding improvement in the prediction of VLE data from the heats. In the two-parameter case, the incompatibility of parameters obtained from hE and VLE data was established by mapping regions within which the data could be predicted with small error. No such incompatibility necessarily exists when four parameters are used, and a simultaneous fit of VLE and hE data is generally successful as shown in Table IV. For miscible systems, the remaining question is how to determine the four parameters most efficiently. From the results shown, it is clear that (i) limiting activity coefficients can supplant classical VLE data, and (ii) heats of mixing are not sufficient to determine the parameters, nor can they be predicted from other kinds of data alone. Thus, two ympoints a t one temperature and one hE data set can be considered the minimum data required to determine four adjustable parameters per binary, as is illustrated in Table V. The choice of equation is not critical. Details of the data reduction procedure can be found elsewhere (Nicolaides, 1977). Partially Miscible Systems. Liquid-liquid equilibria can be correlated by the same expressions for gE as those for vapor-liquid equilibria or heats of mixing. There are several reasons for which the correlations of LLE data are important, apart from their direct application to extractions or other separations. The measurement of liquidliquid equilibria by conventional techniques is relatively simple for systems that are partially miscible a t ordinary temperatures and may be an attractive alternative to any other technique mentioned in this work. Two adjustable parameters can be calculated for every binary LLE data point and since every point is at a different temperature, the temperature dependence of the parameters can be determined. In principle, then, any other data could be calculated from the adjusted parameters. The rigor required of any model used for the prediction of VLE from LLE data is comparable to that required for the temperature extrapolation of VLE since it is the same activity coefficients, as functions of temperature, that are needed. The unequivocal determination of the parameters should simplify the problem further. Predicting hE is, again, a more severe test of the model. The reverse process, computing LLE from VLE data, is less likely since very accurate parameter values are necessary. In liquid phase separations the very existence of two phases depends on the nonideality of the solution,
____ 270
0
01
02
03
04
2-Parameter Correlotlons 4-Porometer Correlations 05
06
07
08
09
Mole Fraction of Cyclohexane
Figure 4. Correlation of liquid-liquid equilibria for the cyclohexane-aniline system with the UNIQUAC and Zeta equations in 2and 4-parameter forms.
and the particular equilibrium compositions are determined by small changes in the shape of the Gibbs energy of mixing curve; predicted mole fractions are thus very sensitive to parameter values. In contrast, vapor-liquid equilibria exist for ideal mixtures too and nonideality imposes a correction only on the equilibrium compositions. It is this sensitivity that makes the correlation of LLE data a stringent test for excess Gibbs energy expressions and, in particular, for the temperature dependence of gE predicted. LLE data were correlated in this work with the UNIQUAC and the Zeta equations. Two values of the third Zeta parameter, s, were used to see whether a three-parameter version of the equation increases its versatility. First, parameter pairs at every temperature were calculated exactly for each partially miscible binary. Some parameter pairs were used separately to predict liquid-liquid compositions over the entire temperature range of interest on the basis of the temperature dependence of the equation only. Then, all parameter pairs were fitted to linear functions in temperature and the resulting four-parameter equations were used to predict the LLE again. Typical results are shown in Figure 4. The two-parameter equations are wholly inadequate to fit the data; they predict nearly constant compositions with temperature, approximately equal to the compositions used to obtain the parameters. The four-parameter versions fit most of the data well but fail near the critical solution temperature, where the temperature dependence is strongest. Indistinguishable results are obtained with all equations. The parameter pairs computed from LLE a t the lowest and highest available temperatures, and the four parameters fitted to all pairs, were then used to predict VLE and heats of mixing. Typical results are shown in Table VI. VLE data are predicted badly with a few exceptions and there is no definite pattern as to which set of parameters is preferable. Heats of mixing are predicted even less correctly and, surprisingly, the error increases when four parameters are used. In Figures 5 and 6, parameters of the UNIQUAC and the Zeta equations determined from LLE and VLE data are plotted as functions of temperature for a typical system. When the temperature range is substantial, as for this cyclohexane-aniline system, the linear extrapolations of LLE parameters above the critical solution temperature are unrelated to the values determined from VLE data; particular pairs of parameters though may accidentally
Ind. Eng. Chem. Fundam., Vol. 17, No. 4, 1978 330 Table VI. Liquid Phase Nonidealities Calculated From Liquid-Liquid Equilibrium Data Fit t o Excess Gibbs Energy Expressions rms % error in total pressure and heat of mixing, or average error in LLE
cy clohexane-aniline equation
UNIQUAC
Zeta (s = 2.0)
Zeta (s = 5 . 0 )
no. of parameters
temp of LLE data used
308K (VLE)
323K (VLE)
392K (VLE)
308K
323K
(hE)
(hE )
2 2 4 2 2 4 2 2 4
273.8 K 303.6 K 273.8-303.6 K 273.8 K 303.6 K 273.8-303.6 K 273.8 K 303.6 K 273.8-303.6 K
9.7 10.1 10.9 9.2 8.7 9.3 10.9 10.6 11.7
11.0 9.9 20.2 11.3 7.9 17.7 12.5 10.3 21.4
17.8
44.9 49.0 200.8 50.3 52.6 110.9 50.6 52.7 136.0
46.5 50.8 187.0 51.6 54.1 125.2 51.8 54.4 150.4
1.8 43.3 19.1 2.2 41.1 19.6 2.3 44.6
LLE
0.022
0.023
0.022
\
I 3 00 350 4 00
TEMPERATURE
(OK)
Figure 5. UNIQUAC equation parameters from LLE and VLE data as functions of temperature for the cyclohexane(l)-aniline(2) system.
coincide. When the temperature range is smaller, as for hexane-nitroethane, parameters may be generally compatible. This is not surprising considering that if the VLE data were a t a temperature within the region of partial miscibility, values from LLE a t the same temperature would lie in the parameter space of the former; the accuracy of the data would be the limiting factor since the ability of the equatio:ns to correlate the composition dependence of the activity coefficients is well established. From the results shown above it is clear that, with existing models, liquid-liquid equilibria must be measured separately; they also have very limited applicability to the prediction of any other kind of data. Still, these data constitute an unequivocal test of the temperature dependence of gE predicted by a new model and, if this dependence were established, they could be an alternative to VLE measurements for partially miscible systems.
Conclusions and Recommendations A method has been presented for minimizing the experimental effort needed to describe the excess thermodynamic properties of binary liquid mixtures of nonelectrolytes. Different manifestations of nonideality were viewed collectively arid models compared not merely by their ability to fit data but to extrapolate and cross-predict them. The new experimental techniques for measuring activity coefficients a t infinite dilution directly offer an excellent experimental alternative, especially when used with a few heat of mixing data. The data base formed can be used to evaluate the applicability of liquid models other than the ones discussed here. T h e theoretical validity of the local-composition equations, reflected i n their ability to cross-predict different types of binary liquid mixture properties or extrapolate with respect to temperature, is severely limited. Similarly limited is the physical significance of their parameters, which cannot be consistently interrelated
1 300
350
TEMPERATURE (OK)
Figure 6. Zeta (s = 2.0) equation parameters from LLE and VLE data as functions of temperature for the cyclohexane(l)-aniline(2) system.
through pure-component properties. The only substantial improvement since the development of the Wilson equation is the ability of the more recent equations to represent liquid-phase separation with two parameters per binary only. Liquid-liquid equilibria and at least a few heat of mixing data points must be measured separately if such information is needed. Vapor-liquid equilibrium measurements can be replaced for most systems by limiting activity coefficients on both sides of a binary system either a t several temperatures or close to the temperature of interest. Alternatively, ymdata a t one temperature can be used with hE points to obtain a good representation of nonidealities in miscible systems over a modest temperature range. Experimental limitations on the types of systems for which ymdata can be obtained have been discussed elsewhere (Eckert et al., 1978). Whether the hE--y" measurements can be made faster and a t less expense than VLE measurements at different temperatures depends, of course, on the equipment a t hand and the particular VLE method used. It is important, however, to be aware of these new alternatives. Prediction of multicomponent vapor-liquid equilibria from binary data only is well established for most local composition equations. If a valid representation of the excess Gibbs energy temperature dependence can be established for binary systems, it will be possible to predict other information reliably from even fewer data than the
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Ind. Eng. Chem. Fundam., Vol. 17, No. 4, 1978
minimum necessary found in this study. To this end, it does not seem likely that deriving a few more localcomposition equations will be effective; the questions of binary temperature dependence and binary-multicomponent generalization could well be separated. The prediction of multicomponent liquid-liquid equilibria and heats of mixing from binary data is not established and may depend largely on the correct determination of binary parameters a t the temperature of interest; the applicability of liquid mixture models t,o binary-multicomponent calculations will become clearer when binary syst,ems are better understood. Nomenclature A = Van Laar equation parameter B = Van Laar equation parameter c . . = molecular interaction parameter in Zeta eqiiation g' = excess molar Gibbs energy hE = molar heat of mixing ni = number of moles of component i in moisture P = total pressure yi = area parameter for component,i in IJNIQrJAC equation R = gas content ri = volume parameter for component i in IJNIQUAC equation s = parameter in Zeta equation ?' = absolute temperature u,, = molecular interaction parameter in IJNlQr JAC equation ui = liquid molar volume of component i xi = liquid mole fraction of component i 2 = liquid coordination number yi = activity coefficient of component i yim= activity coefficient at component i at infinite dilution { = parameter in Zeta equation Bi = area fraction in UNIQIJAC equation iiij = Wilson equation parameter or Zeta equation parameter Xi, = molecular interaction parameter in Wilson equation ri, = UNIQUAC equation parameter @i = segment fraction in UNIQUAC equation or voliime fraction in Zeta equation Literature Cited Abello, L., Servais, B., Kern, M., Pannetier, G., Bull. Soc. Chim. Fr., 11, 4361
(1968). Abrams, D. S., Prausnitz, J. M., AIChE J . , 21, 116 (1975). Angelesou, E., Giusca, R., 2.Phys. Chem. A , 145 (1942). Asselineau, L., Renon, H., Chem. Eng. Scj., 25, 1211 (1970). Boublik, T,, Collect. Czech. Chem. Commun., 28, 1771 (1963). Bradley, J. T., Ph.D. Dissertation, The University of New South Wales, 1976. Brown, I., Ewaid. A. H., Aust. J . Sci. Res. (Ser. A ) , 4, 198 (1951). Brown, I., Smith, F., Aust. J . Chem., 7,264 (1954). Brown, I., Smith, F., Aust. J . Chem., 7,269 (1954). Brown, I., Smith, F., Aust. J . Chem., 8,62 (1955). Brown, I., Smith, F., Aust. J . Chem., 8 , 501 (1955). Brown, I., Fock, W., Aust. J . Chem., 9, 180 (1956). Brown, I., Fock, W., Smith, F.. Aust. J . Chem., 9, 364 (1956).
Brown, I., Fock. W.. Aust. J . Cbem., 10, 417 (1957). Brown, I., Smith, F., Aust. J . Chem., IO, 423 (1957). Bruin, S.,Prausnitz. J. M., Ind. Eng. Process Des. Dev., 10, 562 (1971). Duran, J. L., Kaliaguine, S.,Can. J , Chem. Eng., 49, 278 (1971). Eckert, C. A., Newman, B. A.. Nicolaides, G. L., Long, T. C., AIChE J., in press,
1978. Eckert, C. A., Prausnitz, J. M., Orye, R. V., O'Connell, .I. P., AIChE-lnst. Chem. Eng. Symp. Ser., 1, 75 (1965). Edwards, J. B., Ph.D. Dissertation, Georgia Institute of Technology, 1962. Gordon, A. R., Hines, W. G., Can. J . Res., 248, 254 (1946). Guggenheirn, E. A., "Mixtures", Clarendon Press, Oxford, 1952, Hanks, R . W., Gupta, A. C., Christensen, J. J., Ind. Eng. Chem. Fundam., 10,
504 (1971). Hanks, R. W., Tan, R. L., Christensen, J. J., Therm. Acta, 23, 41 (1978). Heil, J. F., Prausnitz, J. M., AIChE J., 12, 678 (1966). Hildebrand, J. H., Scott, R. L.. "Reaular Solutions". Prentice.Hall, Enalewood Cliffs, N.J., 1962. Ho, J. C. K., Lu. B. C. Y., J . Chem. Eng. Data, 8, 549 (1963). Hsieh, C. K. R., Eckert, C. A., "The Zeta Equation-A New Solution to the Gibbs-Duhern Equation", presented at the 71st National Meeting of AIChE, Tulsa. Okla.. 1974. Isii, N., 2. SO;. Chem. Ind. ~ p n .38, , 661 (1935). Kortuern, G., Freier, H. J., Chem. Ing. Techn., 28, 670 (1954). Kretchmer, C. B.. Wiebe, R.. J . Am. Chem. Soc., 71, 1793 (1949). Lacher, J. R., Buck, W. B., Parry, W. H., J. Am. Chem. Soc., 83, 2422 (1941). Lundberg, G. W., J . Chem. Eng. Data, 9, 193 (1964). Morisue, T., Noda, K., Ishida, K., J . Chem. Eng. Jpn., 5, 219 (1972). Nagata. I., Yamada, T., Ind. Eng. Chem. Process Des. Dev.. 11. 574 (1972). Neckel, A., Volk, H., Monatsh. Cbem., 95, 822 (1964). Nicoiaides, G. L., Eckert, C. A., J . Chem. Eng. Data, 23, 152 (1978). Nicolaides, G. L., Ph.D. Dissertation, University of Illinois, 1977. Orye, R. V.. Prausnitz, J. M., Ind. Eng. Chem., 57 (5),19 (1965). Orye, R. V.. Ph.D. Dissertation, University of California, Berkeley, 1965. Palmer, D. A., Smith, B. D., Id.€4,Chem. Process Des. Dsv., 11, 114 (1972). Pennington, E. N., Marwil, S. J., Ind. Eng. Chem., 45, 1371 (1953). Renon, H., Prausnitz, J. M., AIChE J . , 14, 135 (1968). Savini, C. G., Winterhalter. D. R., Van Ness, H. C., J . Chem. Eng. Data, 10,
168 (1965). Tai, T. B., Ramalho, R. S., Kaliguine, S., Can. J . Chem. Eng., 50, 771 (1972). Scatchard, G., Wood, S. E.. Mochei, J. M., J . Phys. Chem., 43, 119 (1939). Schreiber, L. B., Eckert, C. A., Ind. Eng. Chem. ProcessDes. Dev., 10, 572
(1971). Schulze, W., Z . Phys. Chem. (Frankfurt am Main), 315 (1956). Srnyth, C. P., Engel, E. W., J . Am. Chem. SOC.,51,2660 (1929). Snyder, R. B., Eckert, C. A,, J . Chem. Eng. Data, 18, 282 (1973). Stokes, R. H., Marsh, K. N., Tomlins, R. P., J . Chem. Therm., 1, 211 (1969). Tan, R. L., Hanks, R. W., Christensen, J. J.. Therm. Acta, 21, 157 (1977). Tan, R. L., Hanks, R. W., Christensen, J. J., Therm. Acta, 23, 29 (1978). Thornton, J. D.. Garner, F. H., J . Appl. Chem., I , Suppl. Issue No. I , S61
(1951). Thornton, J. D.. Garner, F. H., J . Appl. Chem., I , Suppl. Issue No. I . S74
(1951). Van Ness, H. C., Soczek, C. A., Peloquin, G. L., Machado, R. L., J. Chem. Eng. Data. 12, 217 (1967). Van Ness, H. C., Soczek, C. A,, Kochar, N. K., J. Chem. Eng. Data, 12, 346
(1967). Vonka, P., Novak, J. P., Suska, J., Pick, J., Chem. Eng. C o m u n . , 2, 51,(1975). Wilson, G. M., J . Am. Chem. Soc., 86, 127 (1964).
Received for reuiew September 2, 1977 Accepted July 25, 1978
The authors gratefully acknowledge the financial support of the Phillips Petroleum Company and the National Science Foundation.
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