Environmental Engineering Reference
In-Depth Information
and the Raleigh equation can be integrated analytically to obtain:
ln
B
F
1
ln
x
B
,
f
(1
ln
1
1
α
−
−
x
F
)
−
x
F
=
x
F
1
x
B
,
f
+
.
(4.7)
1
−
x
B
,
f
−
When constant relative volatility cannot be assumed, either graphical or numerical tech-
niques, such as Simpson's rule, must be applied [2].
4.3.1
Multistaged batch distillation
In multistaged systems,
x
D
and
x
B
are not in equilibrium and, hence, integration of Equa-
tion (4.5) requires a different relationship between these product compositions. This is
obtained by stage-by-stage calculations. If one assumes negligible hold-up at each stage
and at the condenser and accumulator, mass balances can be written for any time
t
during
the batch operation,
V
n
+
1
=
L
n
+
D
(4.8)
V
n
+
1
y
n
+
1
=
L
n
X
n
+
Dx
D
,
(4.9)
where
L
,
V
, and
D
are the liquid and the two vapor flowrates, respectively.
These balances are the same as for the rectifying section of a continuous column, except
that they are time dependent. If constant molal overflow is assumed,
L
and
V
become
constant, and the operating line is
1
x
D
.
L
V
x
n
+
L
V
y
n
+
1
=
−
(4.10)
/
At any time during column operation, this is a line with slope
L
V
that intersects the
diagonal at
x
D
,
V
slope, or
x
D
will vary during
operation, such that the operation line will be constantly changing. If the reflux ratio
is varied, McCabe-Thiele analysis can be applied on a stage-by-stage basis to find the
relationship between
x
D
and
x
B
. The operating line is drawn for a number of
x
D
values
and a specified number of equilibrium stages is stepped off to find the corresponding
x
B
.
Given
x
B
values for each
x
D
, Equation (4.5) can be solved by either numerical integration,
such as Simpson's rule, or graphical techniques [2].
x
D
. Either the reflux ratio, and thus
L
/
4.3.2
Operating time
Because it is not a continuous process, the operating time is an important consideration
in batch processes. Since the distillate is usually the product, the operating time can be
given as a ratio of the total quantity of distillate collected to the distillate flowrate:
D
total
D
,
t
=
(4.11)
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