Environmental Engineering Reference
In-Depth Information
Thus, if the inlet and outlet compositions are known, then
V
O
can be determined. For
an extraction process, this corresponds to the required solvent
/
/
diluent flow ratio. The
ratios
O
F
could be determined with the appropriate substitution in one of the
component balances. [
Remember
: these ratios are determined by mass balances around
the control volume. Nothing has been stated about thermodynamic equilibrium within the
control volume.]
Setting these equations equal to each other and rearranging gives:
z
B
o
−
/
F
and
V
/
x
B
i
y
B
i
−
z
B
o
x
A
i
=
z
A
o
.
(3.31)
z
A
o
−
y
A
i
−
The left-hand side of this equation is the slope from
F
to
O
, and the right-hand side is the
slope from
F
to
V
. The equation itself is in the three-point form of a straight line. Hence,
the points (
z
A
o
,
y
B
i
) are all on a straight line.
Using the similar triangles in the figure:
z
B
o
), (
x
A
i
,
x
B
i
), and (
y
A
i
,
FO
VF
=
FB
VA
=
z
B
o
−
x
B
i
z
B
o
.
(3.32)
y
B
i
−
The right-hand side of this equation was already shown to be equal to
V
/
O
, therefore:
V
O
=
FO
VF
=
lever-arm rule
.
(3.33)
When applied to extraction problems, the two feed streams
V
and
O
are equivalent to the
incoming feed and solvent streams. The stream
F
would represent a two-phase mixture,
which would separate into the raffinate and extract phases. The component
A
is usually the
solute and the component
B
is usually the diluent, although the lever-arm rule will work
no matter how the axes of the diagram are arranged.
When solving extraction problems
graphically, it is really useful to remember equations:
Vy
A
i
+
Ox
A
i
Vy
B
i
+
Ox
B
i
z
A
o
=
or
z
B
o
=
+
+
O
that help to plot the mixing point without having to re-solve the mass balances. Either one
of these co-ordinates is sufficient to plot the mixing point, since it is already known to be
colinear with the points of the two inlet streams. Remember that the lever-arm rule will
work in any part of the diagram, whether it is two-phase or not (but extraction cannot
perform a separation in the single-phase region)
.[
Remember
: The end points of the line
correspond to the compositions of the streams, while the ratios of lengths can then be used
to find their flowrates.]
V
O
V
Summary
1 The lever-arm rule is a
grap
hical alternative to solving a mass balance
.
2 The points on the line
OFV
correspond to the compositions of the various streams.
3 This method is useful for solving:
(a) for a ratio of stream flowrates (
V
O
or any other);
(b) for a stream composition (
x
A
i
, etc.).
/
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