Chemistry Reference
In-Depth Information
volume increment
v
0, etc. Each volume increment
v
at
each stage
m
is considered to form liquid liquid equilibrium (
mv
) between the sol
phase I and the gel phase II.
The gel phase II, which is coated on the surface of the small glass beads, is
stationary, i.e., it remains at the same stage
m
during the progress of time. However,
the moving sol phase I always remains in the same volume increment
v
. Figure
5
depicts this situation. Starting the fractionation, the total polymer is assumed to be
precipitated at stage
m
¼
1 occupies stage
m
¼
¼
m
P
¼
0 or to be distributed evenly among the
m
P
+1
stages from
m
¼
0to
m
¼
m
P
. The temperature gradient is expressed by [
50
]:
T
m
¼
T
0
m
<
m
P
(43)
T
m
¼
T
0
ð
m
m
P
ÞD
Tm
>
m
P
:
T is the constant temperature
difference between neighboring stages. The segment fraction Z of the solvent in
the solvent + nonsolvent mixture supplied to the entry (*) of the column,
Z
v;
0
,is
assumed to be given by [
50
]:
Here
T
m
is the temperature of stage
m
and
D
v
v
Z
0
;v
¼
Z
0
;
0
þ
Z
1
exp
;
(44)
where
Z
0
;
0
,
Z
, and
v
*
are the parameters of this function. The polymer fractions
are obtained from the sol phase I of the last stage.
The suggested theory is based on the model described above, which subdivides
the column fractionation procedure into many local phase equilibria (Fig.
5
). In this
way, the phase equilibrium relation presented above can be applied. The considered
volume increment
v
and the considered column stage
m
are indicated as subscripts
of the corresponding quantities. The feed quantities for every LLE can be calculated
by applying the above model. However, it has to be taken into account that the feed
phase is not a homogenous phase. The feed phase (
m
+1,
v
+1)
F
is the sum of the
mobile phase (
m
,
v
+1)
I
and the stationary phase (
m
+1,
v
)
II
. After equilibrium, the
sol phase (
m
+1,
v
+1)
I
and the gel phase (
m
+1,
v
+1)
II
are formed. Therefore,
the mass balance for the copolymer reads:
D
m;vþ
1
þ
mþ
1
;v
þ
0
;
0
X
I
W
I
tlf
X
II
W
II
y
X
F
W
F
l 1
ð
f
Þ
ð
r
;
y
Þ
ð
r
;
y
Þ
ð
r
;
y
Þ
mþ
1
;vþ
1
;
l
X
F
W
F
¼
ð
r
;
y
Þ
(45)
where the parameters
, t, and y are given by:
¼
0
for
m
þ
1
¼
0
¼
1
for
m
þ
1
>
0
t
¼
0
for
v
þ
1
¼
0
t
¼
1
for
v
þ
1
>
0
(46)
y
¼
0
for
m
þ
1
>
m
P
y
¼
0
for
m
þ
1
<
m
P
and
v
þ
1
¼
m
:
