Page 92 - Separation process principles 2

2.6 Activity-Coefficient Models for the Liquid Phase 57

1f (2-59) is combined with (2-93), an equation for the Rasmussen [50], Gmehling, Rasmussen, and Fredenslund
liquid-phase activity coefficient for a species in a multicom- [51], and Larsen, Rasmussen, and Fredenslund [52], has
ponent mixture is obtained: several advantages over other group-contribution methods:
(1) It is theoretically based on the UNIQUAC method;
C R
lnyi =lnyi +lnyi (2) the parameters are essentially independent of tempera-
C ture; (3) size and binary interaction parameters are available
= ln(qi/xi) + (212) qi ln(Bi/Bi) + li - (Oilxi) C xjlj for a wide range of types of functional groups; (4) predic-
j=l tions can be made over a temperature range of 275-425 K
-
C. combinatorial and for pressures up to a few atmospheres; and (5) extensive
comparisons with experimental data are available. All com-
ponents in the mixture must be condensable.
The UNIFAC method for predicting liquid-phase activity
coefficients is based on the UNIQUAC equation (2-97),
wherein the molecular volume and area parameters in the
R, residual combinatorial terms are replaced by

where -

For a binary mixture of species 1 and 2, (2-97) reduces to (6) k
in Table 2.9 for 2 = 10. where vf) is the number of functional groups of type k in
molecule i, and Rk and Qk are the volume and area parame-
ters, respectively, for the type-k functional group.
UNIFAC Model
The residual term in (2-97), which is represented by
Liquid-phase activity coefficients must be estimated for In y:, is replaced by the expression
nonideal mixtures even when experimental phase equilibria
data are not available and when the assumption of regular
k
solutions is not valid because polar compounds are present. '. (2-101)
For such predictions, Wilson and Deal [47] and then Den all functional groups in mixture
and Deal [48], in the 1960s, presented methods based on where rk is the residual activity coefficient of the functional
treating a solution as a mixture of functional groups instead group k in the actual mixture, and rf) is the same quantity
of molecules. For example, in a solution of toluene and ace- but in a reference mixture that contains only molecules of
tone, the contributions might be 5 aromatic CH groups, 1 type i. The latter quantity is required so that y: + 1.0 as
aromatic C group, and 1 CH3 group from toluene; and 2 CH3 xi + 1.0. Both rk and rf) have the same form as the resid-
groups plus 1 CO carbonyl group from acetone. Alterna- ual term in (2-97). Thus,
tively, larger groups might be employed to give 5 aromatic
CH groups and 1 CCH3 group from toluene; and 1 CH3
group and 1 CH3C0 group from acetone. As larger and
larger functional groups are used, the accuracy of molecular
representation increases, but the advantage of the group-
contribution method decreases because a larger number of
where 0, is the area fraction of group rn, given by an equa-
groups is required. In practice, about 50 functional groups
tion similar to (2-95),
are used to represent literally thousands of multicomponent
liquid mixtures.
To estimate the partial molar excess free energies, g:,
and then the activity coefficients, size parameters for each
where X, is the mole fraction of group rn in the solution,
functional group and binary interaction parameters for each
pair of functional groups are required. Size parameters can
be calculated from theory. Interaction parameters are back-
calculated from existing phase-equilibria data and then used
with the size parameters to predict phase-equilibria proper-
ties of mixtures for which no data are available.
and Tmk is a group interaction parameter given by an equa-
The UNIFAC (UNIQUAC Functional-group Activity tion similar to (2-96),
Coefficients) group-contribution method, first presented by
Fredenslund, Jones, and Prausnitz [49] and further devel- T~~ = exp (-F)
oped for use in practice by Fredenslund, Gmehling, and