Environmental Engineering Reference
In-Depth Information
V
,
y
R
L
,
x
1
b
,
x
A, b
Figure 3.32
Bottom stage of a cascade.
derived for the cascade described above [11]. A constant value of the relative volatility,
α
AB
(see Equation (3.23)), is assumed for each stage.
Consider the bottom of any equilibrium-stage process as shown in Figure 3.32, where
L
is the liquid stream exiting the cascade,
b
is the product stream, and
V
is the stream which
is vaporized and fed back into the cascade.
From the definition of a separation factor (Equation (2.9)), one can write
y
A
y
B
R
=
α
R
x
A
b
,
(3.52)
x
B
where
A
and
B
denote the two components of a binary mixture. A mass balance yields
Vy
R
=
Lx
1
−
bx
A
,
b
.
(3.53)
The case of minimum number of equilibrium stages corresponds to infinite flow from and
back into the cascade, such that no product is removed. In this case
=
=
.
V
L
and
b
0
The mass balance, then, reduces to
y
R
=
x
1
.
(3.54)
Substitution back into the separation factor equation gives
x
A
x
B
1
=
α
R
x
A
b
.
(3.55)
x
B
To relate equilibrium Stage 2 to equilibrium Stage 1 and the final stage
R
, one can write
x
A
x
B
2
=
α
1
x
A
1
=
α
1
α
R
x
A
R
.
(3.56)
x
B
x
B
This development can be followed to the top of the cascade for
N
equilibrium stages
x
A
x
B
N
=
α
N
α
N
−
1
...α
1
α
R
x
A
b
,
(3.57)
x
B
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