Page 93 - Separation process principles 2

58 Chapter 2 Thermodynamics of Separation Operations

where amk # ak,,. When m = k, then amk = 0 and Tmk = 1.0.
For rf), (2-102) also applies, where 0 terms correspond to the
pure component i. Although values of Rk and Qk are different
for each functional group, values of a,k are equal for all sub-
groups within a main group. For example, main group CH2
consists of subgroups CH3, CH2, CH, and C. Accordingly,

Thus, the amount of experimental data required to obtain
values of amk and ak,, and the size of the corresponding bank
of data for these parameters is not as large as might be
expected.
The ability of a group-contribution method to predict
liquid-phase activity coefficients has been further improved
I I I I
by introduction of a modified UNIFAC method by Gmehling 0 0.2 0.4 0.6 0.8 1 .O
[51], referred to as UNIFAC (Dortmund). To correlate data x,, mole fraction methanol in liquid
for mixtures having a wide range of molecular size, they Figure 2.20 Equilibrium curves for methanoUcyclohexane
modified the combinatorial part of (2-97). To handle temper- systems.
ature dependence more accurately, they replaced (2-105) [Data from K. Strubl, V. Svoboda, R. Holub, and J. Pick, Collect. Czech.
with a three-coefficient equation. The resulting modification Chem. Commun., 35,3004-3019 (1970).]
permits reasonably reliable predictions of liquid-phase ac-
tivity coefficients (including applications to dilute solutions
xl = 0.8248 to 1 .O and for methanol-rich mixtures ofxl = 0.0
and multiple liquid phases), heats of mixing, and azeotropic
to 0.1291. Because a coexisting vapor phase exhibits only a
compositions. Values of the UNIFAC (Dortmund) parameters
single composition, two coexisting liquid phases prevail at
for 5 1 groups are available in a series of publications starting
opposite ends of the dashed line in Figure 2.20. The liquid
in 1993 with Gmehling, Li, and Schiller 1531 and more re-
phases represent solubility limits of methanol in cyclohexane
cently with Wittig, Lohmann, and Gmehling [54].
and cyclohexane in methanol.
For two coexisting equilibrium liquid phases, the relation
Liquid-Liquid Equilibria yi(lf)xi(') = y,(L2)~(2) must hold. This permits determination of
the two-phase region in Figure 2.20 from the van Laar or
When species are notably dissimilar and activity coefficients other suitable activity-coefficient equation for which the
are large, two and even more liquid phases may coexist at constants are known. Also shown in Figure 2.20 is an equi-
equilibrium. For example, consider the binary system of librium curve for the same binary system at 55°C based on
methanol (1) and cyclohexane (2) at 25°C. From measure- data of Strubl et al. [56]. At this higher temperature,
ments of Takeuchi, Nitta, and Katayama [%], van Laar con- methanol and cyclohexane are completely miscible. The
stants are A12 = 2.61 andAZ1 = 2.34, corresponding, respec- data of Iser, Johnson, and Shetlar [57] show that phase in-
tively, to infinite-dilution activity coefficients of 13.6 and stability ceases to exist at 45.75"C, the critical solution tem-
10.4 obtained using (2-72). These values of A12 and AZ1 can perature. Rigorous thermodynamic methods for determining
be used to construct an equilibrium plot of yl against xl as- phase instability and, thus, existence of two equilibrium liq-
suming an isothermal condition. By combining (2-69), uid phases are generally based on free-energy calculations,
where K; = yi /xi, with as discussed by Prausnitz et al. [4]. Most of the empirical
and semitheoretical equations for the liquid-phase activity
coefficient listed in Table 2.9 apply to liquid-liquid systems.
The Wilson equation is a notable exception.
one obtains the following relation for computing yi from xi:
2.7 DIFFICULT MIXTURES
The equation-of-state and activity-coefficient models pre-
sented in Sections 2.5 and 2.6, respectively, are inadequate
Vapor pressures at 25°C are Pf = 2.452 psia (16.9 kPa) and for estimating K-values of mixtures containing: (1) both polar
P," = 1.886psia (13.0 kPa). Activity coefficients can be com- and supercritical (light-gas) components, (2) electrolytes,
puted from the van Laar equation in Table 2.9. The resulting and (3) both polymers and solvents. For these difficult
equilibrium plot is shown in Figure 2.20, where it is observed mixtures, special models have been developed, some of
that over much of the liquid-phase region, three values of y, which are briefly described in the following subsections.
exist. This indicates phase instability. Experimentally, single More detailed discussions of the following three topics are
liquid phases can exist only for cyclohexane-rich mixtures of given by Prausnitz, Lichtenthaler, and de Azevedo [4].