Chemistry Reference
In-Depth Information
X
x
2
p;j
¼
1
(29)
j
which is valid in each of the phases. Rearranging (
28
), summing up over all
polymer components, and using (
29
) one obtains:
X
I
2
j
x
I
2
x
I
2
x
I
2
p;j
'
1
¼
(30)
II
2
j
'
j
For given temperature, pressure, and polymer distribution in one phase (
x
I
2
p;j
),
equations (
27
) and (
30
) can be used to determine the unknowns
x
I
1
and
x
I
1
.
Depending on the expressions for the fugacity coefficients
'
I
1
and
'
I
2
p;j
, additional
unknowns, e.g.,
m
I
2
¼
P
j
x
I
2
p;j
m
2
p;j
will have to be determined. The
m
2
p,j
are the
segment numbers of the various polymer species (which are of course the same in
the two phases). Additional equations can easily be obtained by multiplying (
28
),
e.g., with
m
2
p,j
, again rearranging, summing up over all polymer species, and using
(
29
) to eliminate the unknown molecular weight distribution in the second phase:
X
I
2
j
x
I
2
x
I
2
x
I
2
p;j
m
2
p;j
'
m
I
2
¼
(31)
II
2
j
'
j¼
1
Applying the approach given by (
29
) (
31
), rather than the classical approach of
considering only (
27
) and (
28
), has the advantage that the number of equations to be
solved numerically is independent of the number of considered polymer species.
After solving (
27
), (
30
), and (
31
), the missing concentrations of the single
polymer species (molecular weight distribution) in the second phase can easily be
obtained from rearranging (
28
):
I
2
j
x
I
2
p;j
'
x
I
2
x
I
2
x
I
2
p;j
¼
(32)
II
2
j
'
A very elegant version of this approach is the so-called continuous thermody-
namics [
56 59
]. It can be considered as a reformulation of the classical thermody-
namic relationships that allows for using continuous molecular weight distribution
functions
W
(
M
) rather than the mole fractions of discrete polymer components.
Using this approach, e.g., (
30
) becomes:
ð
x
I
2
x
I
2
I
Þ
'
2
ð
M
Þ
1
¼
W
ð
M
dM
(33)
II
'
2
ð
M
Þ
M
For certain combinations of analytical molecular weight distributions (e.g.,
Schulz Flory distribution) and fugacity-coefficient expressions, the integral in
(
33
) can be solved analytically and no (time-consuming) summation is required.
