ESTIMATION OF PARAMETERS FOR THE NRTL EQUATION FOR EXCESS GIBBS ENERGIES OF STRONGLY NONIDEAL LIQUID MIXTURES H E N R l
R E N O N ’
A N D
J .
M. P R A U S N I T Z
Department of Chemical Engineering, University of California, Berkeley, Calif. 94720 Charts are given for estimating NRTL (nonrandom, two-liquid) parameters from limiting activity coefficient data or from mutual solubilities. Calculated and experimental results are given for binary and ternary systems.
FORquantitative studies on nonideal properties of liquid mixtures it is convenient to express these properties with excess functions. For liquid mixtures a t modest pressures remote from critical conditions, the excess functions are insensitive to pressure; a t constant temperature, therefore, the excess functions of such mixtures depend only on liquid composition. The excess Gibbs energy is the excess function of primary interest in chemical engineering and numerous proposals have been made for relating the excess Gibbs energy to liquid composition; the best known, perhaps, are those by Margules and van Laar. A generalization of such proposals was made over 20 years ago by Wohl (1946), but more recently it has been shown that some useful relations for the excess Gibbs energy could not be obtained from Wohl’s generalization. I n particular, Wilson’s “local composition” concept (1964) resulted in an equation qualitatively different from those proposed earlier. The local composition concept provides a convenient method for introducing nonrandomness into the liquid-mixture model. Wilson (1964) and Orye and Prausnitz (1965) showed that Wilson’s equation is in many respects more useful and more directly applicable to strongly nonideal mixtures than any other two-parameter equation, and a particular advantage of Wilson’s equation for binary systems follows from its straightforward generalization to multicomponent mixtures without need for ternary (or higher) parameters. However, as pointed out by Wilson (1964) and Orye and Prausnitz (1965), Wilson’s equation also contains some undesirable features; especially important is its inapplicability to liquid mixtures with only partial miscibility. I n an effort to overcome this disadvantage while retaining the useful local composition concept, we have developed a new equation for the excess Gibbs energy based on Scott’s two-liquid model of mixtures (1956). This new equation, called the N R T L (nonrandom, twoliquid) equation, was described in detail by Renon (1968), who showed that it appears to be applicable to a wide variety of mixtures for calculating vapor-liquid and liquidliquid equilibria. Robert L. Pigford suggested that the practical utility of the N R T L equation could be enhanced if we made available convenient charts for estimating ’Present address, Institut Francais du Petrole, Rueil-Malmaison, France.
NRTL parameters from limited experimental data. A number of such charts are presented here. NRTL Equation
The NRTL equation for the molar excess Gibbs energy of a binary mixture is a function of mole fractions x1 and x 2 :
where
with g 1 2
= gz,
Equation 1 contains two temperature-dependent parameters, (g12 - gZ2)and (921 - gll), in addition to a nonrandomness parameter, a 1 2 ,which, to a good approximation, does not depend on temperature and can often be estimated with sufficient accuracy from the nature of components 1 and 2 (Renon and Prausnitz, 1968). For some systems it appears that (g~, - gll) and ( g 1 2 - g22) are linear functions of temperature. From Equation 1 the activity coefficients, yl and y2, are obtained by differentiation (see Equations 21 and 22 of Renon and Prausnitz, 1968). The N R T L equation is readily generalized to multicomponent mixtures; the result for such mixtures contains only binary parameters. When experimental phase-equilibrium data are available over a range of composition, the optimum parameters (gzl - g11) and (g12 - gZ2)can be determined by a leastsquares fit. [Such a calculation is tedious but can be performed easily with an electronic computer, as shown by Prausnitz et al. (1967).] However, it is also possible to estimate these two parameters from limiting activitycoefficient data, or, for partially miscible systems, from mutual solubilities. VOL. 8 N O . 3 J U L Y 1 9 6 9
413
NRTL Parameters from limiting Activity Coefficients
and
The limiting activity coefficients are designated by rl" and y Z . I n many cases these can be obtained by a minimum experimental effort, and for a large variety of mixtures they can be estimated from the correlation of Pierotti et al. (1959). At a given temperature T and a fixed a 1 2 , values of yl"(T) and r Z ( T ) uniquely determine values of parameters 712 and 7 2 1 . T o obtain these parameters from rl" and y;, let
s
E
%(721
+
712)
(2)
D
E
?4(721
- 712)
(3)
and I t then follows that
712
=S -D
I n Figures 1 to 5, for given values of ( ~ 1 2 , we present the relations between rl",7;) and S and D. To use Figures 1 to 5 we proceed as follows. First, experimental or estimated values of rl" and y2m a t the same temperature must be available. Second, depending on the chemical nature of the mixture, a value of a12 must be chosen; the values recommended by Renon and Prausnitz (1968) provide a convenient guide. Upon calculating loglo (yPy2m) and log,, (yl"/y2m), Figures l to 5 give values of S and D, which, in turn, yield 7z1 and r12as given by Equations 4 and 5. T o illustrate such a calculation, we consider three systems: ethanol-hexane a t 55" C., methyl ethyl ketonehexane a t 60°C., and propanol-water a t 60°C. For each
Figure 1. Parameters in NRTL equation from activity coefficients a t infinite dilution for a12 = 0.20
Figure 2. Parameters in NRTL equation from activity coefficients at infinite dilution for a12
414
=
0.30
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(5)
Figure 3. Parameters in NRTt equation from activity Coefficients at infinite dilution for ai2 =
0.40
Figure 4. Parameters in NRTL equation from activity coefficients a t infinite dilution for a12
=
0.47
Figure 5. Parameters in NRTL equation from activity coefficients at infinite dilution for a12 = 0.30 Small or negative deviations from Raoult's law
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415
Table 1. NRTL Parameters from Limiting Activity Coefficients
T.,
System Ethanol(l)-n-hexane(Z) Methyl ethyl ketone (1)-n-hexane(2) l-Propanol(l)-water(2) a
D
g12 - g22, Cal./Mok
gzi - gll, Cul./Mole
1.940
0.386
1010
1515
0.8
1.2
0.669 1.327
0.286 1.300
253 18
630 1735
0.7 1.7
0.8 2.1
OC.
loglor;
log10y2"
a12
S
55
1.322
1.013
0.47
60 60
0.565 1.147
0.507 0.533
0.20 0.30
6P/Pb 6y'
From Pierotti's correlation. * R.m.s. relative deviation in total pressure x 100. e R.m.s. absolute deviation in vapor male fraction x 100.
1.0
I
1
I
- CALCULATED ,
075
0
,
I Or
EXPERIMENTAL
I
--
CALCULATED , o EXPERIMENTAL
I
-
300
075
I" E E
J
K W
8
200
050 a
> W L K
%
J
2 too
0 25
I 0 50
I
OY
025
0
I 0.75
I o
00
0.50 .
0.2 L 5
X~~~~~~~~
Figure 6. Vapor-liquid equilibria for ethanol-n-hexane
at
55" c.
I."
n,
< O
,
1.0 O
10
X~~~~~~~
I
0.75 A
e
positions and total pressures (Kudryavtseva and Susarev, 1963, Figure 6; Hanson and Van Winkle, 1967, Figure 7; Murti and Van Winkle, 1958, Figure 8). Agreement between calculated and experimental results is good for all three systems; mean deviations between calculated and observed vapor compositions and total pressures are reported in Table I.
I
CALCULATED EXPERIMENTAL
/ '
0.75b
Figure 8. Vapor-liquid equilibria for 1-propanol-water at 60" C.
NRTL Parameters from Mutual Solubilities
300
I/
0 1 0
I 0.25
I 0.50
I 0.75
E
I o
1.0
X~~~~~~
Figure 7. Vapor-liquid equilibria for methyl ethyl ketonen-hexane at 60"C.
of these three binary systems we estimate yl" and 7; from Pierotti's correlation, with results shown in Table I. Using the recommended values of a I 2 ,we then obtain, for each system, S and D and finally (g12- g2?) and - gil), as shown in Table I. With these estimated NRTL parameters, y-x and P-x diagrams were calculated over the entire composition range (Figures 6, 7, and 8). Also shown, for comparison, are experimental vapor com416
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For binary mixtures with only partial miscibility, it is possible to estimate NRTL parameters from mutual solubilities a t the same temperature. Let x7 be the mole fraction of component 1 in phase a, where i t is smaller, and let x i be the mole fraction of component 2 in phase b, where it is smaller, such that x?/x! 5 1. For a fixed value of aI2 we may obtain S and D from Figure 9, 10, or 11; the NRTL parameters are then found from Equations 4 and 5. Figures 9 to 11 were prepared by solving simultaneously the equilibrium equations 7% = 7:xih
(6)
and a
a
b
b
72x2 = 72x2
and the stoichiometric relations x; x; = 1
+
and
x? + x; = 1
(7)
I5
- I3 -- ;-
D N
0
D
i
'
05
00 6
I O
Figure
I5
20
25 - L o * , o l x g x;1
30
5
4
8
6
9. Parameters in NRTL equation from mutual solubilities for
a12
IO
I2
IO
I2
= 0.2
15
IO I
-N
-..
0 I
0 rn i
05
1
0
IO
06
I5
30
25
20
-LoPlo(xp
5
4
8
6
x;)
Figure 10. Parameters in NRTL equation from mutual solubilities for
=
0112
0.3
I3
-
10
DN
x
0 \-
-9 01
05
00 6
IO
15
20
25
30
4
5
6
8
10
I2
Figure 11. Parameters in N R T L equation from mutual solubilities for aiz = 0.4
T o illustrate the use of mutual solubility data we consider the system n-hexane-nitroethane. Using mutual solubility data a t 5" and 25°C. reported by Hwa, Techo, and Ziegler (1963) we obtain the parameters shown in Table 11. Assuming linear temperature dependence, we obtain a t 45" C.: (g12
- gzz)
= 474 cal./mole
and
(gZ1- gll) = 680 cal./mole
Calculated and experimental vapor-liquid equilibria for this system a t 45OC. are shown in Figure 12. The r.m.s. (absolute) deviation in vapor-phase mole fraction is 0.002 and the r.m.s. (relative) deviation in total pressure is 0.006. Experimental results are taken from Edwards (1962). Ternary Systems
For an additional illustration of Figures 9 t o 11 we consider the ternary system n-octane-isooctaneVOL. 8 N O . 3 JULY 1 9 6 9
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