Correlation and Prediction of Heats of Mixing of Liquid Mixtures lsamu Nagatal and Toshiro Yamada Department of Chemical Engineering, Kanazawa University, Kanazawa, Japan
The accuracy of the Wilson, Heil, and NRTL equations i s shDwn for calculations of heats of mixing for nonideal liquid mixtures. The binary parameters of the three equations are assumed to vary linearly with absolute temperature. Calculated results are compared with experimental data in four cases: (1) estimation of heats of mixing using binary parameters obtained from iscthermal vapor-liquid equilibria, (2) estimation of excess Gibbs free energies using binary parameters deterwined from heats of mixing, (3) simultaneous fit of excess Gibbs free energy and heat of mixing data, and (4) prediction of ternary heats of mixing from binary parameters. It i s found that the parameters obtained from either of the excess free energy and heat of mixing data are not necessarily suitable for representation of the other data, and a simultaneous fit of these data i s successful with the three equations. All the three equations are able to predict ternary heats of mixing with a good accuracy.
T h e process engineer often needs information on the equilibrium data of a nonideal liquid mixture. Recently published three excess free energy functions [Wilson, Heil, and nonrandom two-liquid (NRTL) equations] have been proved to be very useful in correlating binary vapor-liquid equilibria and predicting multicomponent equilibria from binary parameters. It is expected t h a t these equations should be applicable to calculations of heats of mixing of liquid mixtures using the well-known Gibbs-Helmholtz relation. Orye (1965) demonstrated t h a t the Wilson equation appears to give a reasonable but rough estimate of the heats of mixing especially for nonassociating solutions from phase equilibrium data along one isotherm only. Renon (1966) got a similar conclusion for the Heil and K R T L equations. Hanks et al. (1971) produced reliable x-y curves directly from measured heat of mixing data and pure component vapor pressure data for carefully selected six lionideal solutions using the Wilson and N R T L equations. I n these three investigations the built-in parameters in the excess free energy functions u ere assumed to be temperature independent. rlsselineau and Renon (1970) eytended the N R T L equation to have three temperature-dependent parameters to represent the properties of binary liquid mixtures over a certain range of temperature, and available data are vapor-liquid and liquidliquid equilibria, activity coefficients a t infinite dilution and heats of mixing. Duran and Kaliaguine (1971) showed the Wilson equation including temperature-dependent parameters gives a simultaneous representation of excess free energies and of heats of mixing for 11 completely miscible binary systems. Their approach concerning the parameters is different from our method discussed below. T o evaluate the relative merits of the three equations, this paper presents a systematic comparison of the calculation accuracy of the equations in the following cases: What calculation accuracy can be expected in estimation of heats of mixing from isothermal vapor-liquid equilibrium data? HOW accurately could excess free energy data be predicted from only To whom correspondence should be addressed. 574 Ind.
Eng. Chem. Process Des. Develop., Vol. 11,
No. 4, 1972
heat of mixing data? K h a t multicomponent prediction accuracy in heats of mixing can be obtained from binary parameters? Is the prediction accuracy dependent on the type of the equation used? Basic Equations
On the basis of the concept of local composition, the excess Gibbs free energy functions of the Wilson, Heil, and N R T L equations for expressing the nonideality of a binary liquid mixture are summarized as follows:
RT
=
x1 In yl
+ xz In y z =
-q[zl In
(2,
+ XZGZI)+ xz In + 1 + mGzi ~izGiz + ( + xzGz1 + xz---) (22
pz122 21
z1Glz)
21G1~
(1)
where g E is the excess Gibbs free energy. R , T , 2, and y are, respectively, the gas constant, the absolute temperature, the liquid mole fraction, and the activity coefficient. 712 = ( g l z - g2z)/RT;rzl = ( g z l - q d / R T ; G12 = PIZ exp ( - - ( Y I z ~ ; and GZ1 = pzl exp ( - c Y ~ z T ~ ~The ). Wilson, Heil, and X R T L equations are obtained by substituting for p , q , p12, and CYIZ the values listed in Table I as given by Renon and Prausnitz (1968). The activity coefficients are derived by appropriate differentiation of Equation 1.
where (921 - gI1), (91%- 9 2 2 ) ) and aiz are three adjustable parameters to be determined from experimental data. If
Table 1. Definition of pI q, Equation
P
9
Wilson Heil NRTL
0
1 1 0
a PI2
=
1 1
and
p12,
a12
PIZU
a12
:
1 1
V I P 2 VIIV2
1
I
t
a12
1/P21
these parameters are assumed to be temperature dependent, then the heats of mixing for a binary solution, hE, are obtained by differentiation of Equation 1 using the GibbsHelmholtz relation: - wO '
(4)
px1xz
XI 7 2 1 6 ' 2 1
7'21G21
[XI
+
xzG21
+
(XI
+
x2G2d2
7'1zG12 +
x2
+
+
aG12
+
1
( ~x2712G112 2 ~ 1 G 1 2 ) ~
(5)
where
Figure 1 . Temperature dependence of energy parameters A. Methyl acetate(1 )-methanol(Z) (Bekarek, 1968) 6. Acetone(1 )-chloroform(2) (Kudryavtseva and Surarev, 1 9 6 3 a ) C. Acetone(1 )-n-hexane(Z)(Kudryavtseva ond Surarev, 1 9 6 3 b )
Wilson
- gll) - 822)
(gzl (glz
and
Duran and Kaliaguine (1971) and Kaliaguine and Ramalho (1972) adopted Gij ( A i j in their papers) instead of (gu g j j ) as the Wilson parameters. It is convenient for the chemical engineer to use the Wilson parameters having a consistent form not only in vapor-liquid equilibrium calculations a t isobaric condition, but also in fit of gE and hE data a t isothermal condition. The temperature dependence of activity coefficients is important in design of distillation columns to be operated isobarically, because the temperature changes throughout the columns. When we use the Wilson equation for a n isobaric system (especially of a wide boiling range), the use of (gij - gjj) form is definitely preferable to the Aij form, because the latter should be valid a t only isothermal condition and enough equilibrium data to estimate the temperature dependence of Aij are not always available. I n other words the Aij form leads to less accurate data correlation of the isobaric system of large temperature range than the (gij - gjj) form. The same conclusion holds for prediction of isobaric multicomponent vapor-liquid equilibria from binary parameters as pointed out by Nagata (1971). Further, if experimental equilibrium data are available a t only one condition, the (gij - gjj) form is more useful for extrapolation of such data to another condition than the Aii form (Nagata and Ohta, 1969). Results
Estimation of Heats of Mixing from Vapor-Liquid Equilibrium Data. Nagata (1971) determined binary
----O-------@----
NRTL
Heil
---.-.A-...-_.__ A .__. --
-.
-.-o-.-
-.-e.-
parameters of the three equations for more than 600 completely miscible system by minimizing the sum of squares of deviations in vapor phase mole fraction plus the sum of squares of relative deviations in pressure for all d a t a points. The following equilibrium equation between a vapor and liquid phase was used in the calculation:
where $$, ut, Pis, and vi are, respectively, the vapor phase fugacity coefficient, vapor mole fraction, vapor pressure, and liquid molar volume of component i and P is the total pressure of the system in equilibrium. It is assumed t h a t the molar volume of pure liquid can replace the partial molar volume a t normal pressures. $ is calculated from the virial equation truncated after second term. Second virial coefficients are estimated from the correlation of O'Connell and Prausnitz (1967). It is observed t h a t the parameters of the three equations obtained from equilibrium data along various isotherms for a liquid mixture are usually temperature dependent as pointed out by Renon and Prausnitz (1968), Nagata and Ohta (1969), Asselineau and Renon (1970), and Duran and Kaliaguine (1971). Table I1 (deposited with the ACS Nicrofilm Depository Service) gives the values of the parameters of the three equations obtained from experimental isothermal vapor-liquid equilibrium data for seven nonideal systems. The systems listed here were selected as suitable samples to check the calculation accuracy of the method. The energy parameters, (gz3 g3J, calculated while making no assumption on the form of their variation with temperature, are plotted against temperature for three systems in Figure 1. A linear relationship beInd. Eng. Chem. Process Des. Develop., Vol. l l , No. 4, 1 9 7 2
575
300
0’ 0
0.2
0.4 0.6 XI (METHANOL)
I
I
0.8
1.0
Figure 2. Comparison of calculated and experimental heats of mixing for the methanol-benzene system at 35OC
------0
5
Exptl data at 35’C (Mrazek and Van Ness, 1961) Calcd data (Wilson)
.5000
02
___._.____ Calcd data (Heil)
09
0.6
08
3
X,(ACETONE)
----Calcd data (NRTL)
Figure 4. Comparison of calculated and experimental excess Gibbs free energies and fit of heats of mixing for the acetone-chloroform system A 0
Exptl g E data at 35’C (Kudryavtseva and Surarev, 19630) Exptl hE data at 25’C (Hirobe, 1926) Calcd hE data (Wilson, Heil, NRTL), hrvE,h H E , h.-qE Calcd g E data (Wilson), g w E Calcd g E data (Heil), g H E Calcd g E data (NRTL), g.VE
-----._._...__. -----
0
92
0.4
QB
0.8
1.0
X,
