Environmental Engineering Reference
In-Depth Information
(a)
(b)
Condenser
accumulator
D
,
x
D
Condenser
D
,
x
D
Q
R
V
n
+1
L
n
B
,
x
B
Reboiler
VL
B
,
x
B
Q
R
Reboiler
Figure 4.1
(a) Single-staged, and (b) multistaged batch distillation processes.
where
x
F
is the feed mass fraction,
x
B
,
f
is the final mass fraction in the bottoms, and
x
D
,
avg
is the average mass fraction drawn from the distillate. Both
F
and
x
F
are known quantities
and normally either
x
B
,
f
or
x
D
,
avg
is specified. The above two balances combined with the
Raleigh equation are used to solve for the three unknowns,
D
,
B
and either
x
B
,
f
or
x
D
,
avg
.
The Raleigh equation, presented below, is a differential mass balance that assumes the
hold-up in the accumulator and column is negligible, such that only hold-up in the bottom
stage or reboiler is significant. Then, if a differential quantity d
B
of composition
x
D
is
removed from the system, the mass balance becomes:
−
x
D
d
B
=−
d(
Bx
B
)
.
(4.4)
Expanding and solving the Raleigh equation yields
ln
B
F
x
F
d
x
B
x
D
−
=−
x
B
.
(4.5)
x
B
,
f
Performing this integration requires that
x
D
be related to
x
B
.Insingle-stage distillation
cases where constant relative volatility can be assumed, the equilibrium relationship is:
α
x
y
=
1)
x
,
(4.6)
1
+
(
α
−
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