Environmental Engineering Reference
In-Depth Information
Mass balance on system:
In
=
Out
⇒
O
N
+
1
+
V
0
=
O
1
+
V
N
=
M
.
(5.5)
We know that
M
must lie on a line connecting
O
1
and
V
N
, and
O
N
+
1
and
V
0
(these lines
are
not
tie-lines). Using the lever-arm rule, we can locate
M
.Inother words, we can write:
M
=
O
1
+
V
N
M
is on this lever arm;
(5.6)
M
=
O
N
+
1
+
V
0
M
is also on this lever arm.
(5.7)
Therefore, we can use the lines
O
N
+
1
MV
0
and
O
1
MV
N
to get the values of the composi-
tions of the streams exiting/entering the column.
[
Note:
O
N
+
1
and
V
0
are
not
on the solubility envelope because they are entering streams
and are not necessarily in equilibrium.
O
1
and
V
N
,however,
are
on the solubility envelope
since they are exit streams and leave the column in equilibrium with each other.]
Mass balance on Stage 1:
V
0
+
O
2
=
O
1
+
V
1
O
1
−
V
0
=
O
2
−
V
1
=
.
(5.8)
Mass balance on Stages 1 and 2:
V
0
+
O
3
=
O
1
+
V
2
O
1
−
V
0
=
O
3
−
V
2
=
.
(5.9)
Mass balance on Stage
j
:
O
1
−
V
0
=
O
j
+
1
−
V
j
=
.
(5.10)
Mass balance on
N
stages (entire column):
O
1
−
V
0
=
O
N
+
1
−
V
N
=
.
(5.11)
Therefore, net flow
∆
is constant through cascade
.
These
mass balan
ce equations show that
must lie on the line
V
0
O
1
and also on
the line
V
N
O
N
+
1
. The p
roced
ure for plotting the
point is illustrated in Figure 5.13.
O
1
V
0
[Remember that
V
0
/
is positive, so that
the net flow is toward the rich end of the column, i.e., the extract (
V
) phase.]
Now let
V
N
,
V
0
, and
O
N
+
1
stay the same while
O
1
(raffinate) moves up and to the left.
This corresponds to a “less pure” raffinate. As
O
1
is allowed to move up and to the left,
the ratio
V
0
/
O
1
=
.Inthe previous figure,
V
0
>
O
1
and
O
1
increas
es since
goes further to the left of the graph. When the lines
O
1
V
0
and
O
N
+
1
V
N
are parallel,
V
0
/
O
1
=∞
/
∞=
1 and
=
0.
When
O
1
is allowed to move even further to the left,
will be to the right of the graph
instead of to the left of it, and
O
1
>
is negative. This means that the net flow
would be toward the lean end of the cascade, i.e., the raffinate (
O
) phase.
V
0
, and
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