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In-Depth Information
6.2 PARAMETERS OF ACTIVITY COEFFICIENT EQUATIONS
Activity coefficient equations are derived from the general formulation of excess free
energy,
∂G
E
∂n
i
RT
ln
γ
i
=
(6.12)
T,P,n
j
=
i
where
n
i
is the number of moles of component
i
,and
n
j
=
i
indicates any component
except
i
. Applying equation (6.12) to a model of
g
E
, the total excess free energy per
mole of mixture produces the activity coefficient equation. This is illustrated by the
application of equation (6.12) to
g
E
RT
=−
x
1
ln
(x
1
+
12
x)
−
x
2
ln
(x
2
+
21
x
1
)
(6.13)
the two-component formulation of
g
E
for the Wilson equation, and results in
RT
ln
γ
1
=−
x
1
ln
(x
1
+
12
x)
−
x
2
12
x
x
+
12
x
2
−
21
21
x
1
+
x
2
(6.14)
where the activity coefficient for component 1 is
1
,and
12
and
21
are parameters
which are determined by regressing experimental data. When used in multicomponent
calculations, binary parameters representing all permutations of the components are
used in the multicomponent form of the Wilson equation:
γ
⎛
⎝
⎞
⎠
+
1
−
N
N
x
k
ki
j
x
j
kj
ln
γ
i
=−
ln
x
j
ij
(6.15)
j
k
The Aspen Plus implementation of the Wilson parameter
ij
includes the user's choice
of terms for temperature dependency and is shown in
b
ij
T
+
c
ij
ln
T
e
ij
T
2
ln
A
ij
=
a
ij
+
+
d
ij
T
+
(6.16)
where in Aspen Plus's terminology,
A
replaces
.
When regressing data the user may select which parameters of equation (6.16) are
to be regressed from the Properties/Regression/R-1/Input form parameter tab, shown
in Figure 6.2. On each column of the central part of the display reading downward,
the parameter is identified as binary, the name of the activity model used, the element
number [which corresponds to the coefficients in equation (6.16), e.g.,
a
ij
is 1], and
the components are identified in order; for example, ethanol is 1 and water is 2, so
column 1 refers to parameter
a
12
and column 2 refers to parameter
a
21
. The remainder
of the input permits the user to omit, fix the value of, regress the parameter and assign
an initial value, define the bounds of the regression, and set a scale factor such that all
parameters have comparably scaled contributions to the regression. The last entry on
the column is used for regression of equations of state.
