Information Technology Reference
In-Depth Information
TABLE 6.2
Infinite Dilution Activity Coefficients for Various Equations
γ
1
γ
2
Equation
ln
and ln
ln
γ
1
=
A
12
;
ln
γ
2
Van Laar
=
A
21
ln
γ
1
=
1
−
ln
12
−
21
;
ln
γ
2
Wilson
=
1
−
ln
21
−
12
ln
γ
1
= τ
21
+ τ
12
exp
(
−α
12
τ
12
)
;
ln
γ
2
NRTL
= τ
12
+ τ
21
exp
(
−α
12
τ
21
)
ln
γ
1
Uniquac
=
ln
(r
1
/r
2
)
+
[5 ln
(q
1
r
2
/q
2
r
1
)
−
ln
τ
21
+
1
− τ
12
]
+
I
1
−
(r
1
/r
2
)I
2
ln
γ
2
=
ln
(r
2
/r
1
)
+
[5 ln
(q
2
r
1
/q
1
r
2
)
−
ln
τ
12
+
1
− τ
21
]
+
I
2
−
(r
2
/r
1
)I
1
the 0.0 and 1.0. values of
x
1
. These are related to the parameters by various algebraic
expressions, as shown in Table 6.2.
The adjustable parameters are
A
ij
for van Laar,
ij
for Wilson,
τ
ij
and
α
12
for
NRTL, and
τ
ij
for Uniquac. The
r
,
q
,and
I
parameters are pure component values
which can be found in the topic by Poling et al. (2000).
An alternative approach is to recognize that a single data point yields two equilibrium
equations with the parameters
a
ij
as unknowns. Solution of these equations yields good
estimates of the
a
ij
for a reasonable data point.
Both of these approaches can be implemented easily using the Aspen Plus regression
system. The former approach would involve using the entire data set for the regression
with the initial values set to zero. If this produces good results, nothing more need be
done; if not, the experimental activity coefficients are listed under regression/results/tab
profiles and can be displayed with Aspen Plus's plotting capability and extrapolated
by eyeball.
The latter approach can also be implemented with Aspen Plus by regressing a single
data point. Subsequently, the regression with the full data set can be done with the
estimated values of the
a
ij
pair as starting values. If one decides to regress the
b
ij
pair, an initial value can be obtained by noting that
a
ij
=
b
ij
/T
in the absence of a
four-parameter fit.
Aspen Plus provides a very useful Plotting Wizard which has many built-in plots
related to regression of equilibrium data (see Figure 6.4, as well as the ability to
select which variables to plot from a variety of Aspen Plus data sources. The Wizard
is invoked by selecting Plot from the main display menu. All results shown may be
obtained by selecting Properties/Regression/R1 with tab Profiles and may be plotted
by selecting the column to be plotted and associating it with the desired axis available
in the Plot menu. Figure 6.5 shows an example of a gamma(1) and gamma(2) versus
x
1
plot.
As an example, Table 6.3 gives the
T
-
P
-
x
-
y
data for the system cyclohexane
(CH) -isopropyl alcohol (IPA) of Nagata (1963). A plot of the activity coefficients of
both components versus the mole fraction of component 1 (CH) is shown in Figure 6.5.
Extrapolation of the curves produces infinite dilution activity coefficients of about 4.8
for CH and 9.5 for IPA.
Applying the infinite dilution form of the van Laar equation to the estimated infinite
dilution coefficients yields 1.57 and 2.25 for
a
12
and
a
21
, respectively. Calculating
b
12
and
b
21
and noting that the original van Laar parameter
A
12
is modified to
b
12
T
A
12
=
a
12
+
