Environmental Engineering Reference
In-Depth Information
Rearranging,
α
x
y
=
applies to any stage.
1
+
x
(
α
−
1)
Substituting into the overall mass balance:
α
x
n
−
1
L
V
x
n
+
D
V
1)
=
x
D
;
D
=
V
−
L
;
1
+
x
n
−
1
(
α
−
L
V
=
L
V
=
D
V
=
1
−
1
−
R
V
=
R
V
x
n
+
(1
−
R
V
)x
D
;
R
V
=
internal reflux ratio.
x
n
x
n
−
1
+
ax
n
+
bx
n
−
1
+
C
=
0 (Ricatti difference equation);
1
α
−
(
α
−
1)
x
D
(1
−
R
V
)
−
α
x
D
(1
−
R
V
)
a
≡
1
;
b
≡
;
c
≡
1)
.
R
V
(
α
−
1)
R
V
(
α
−
Substitution: let
x
n
=
h
(this is a coordinate transformation which shifts the system
to the intersection of the operating and equilibrium lines):
z
n
+
(
z
n
+
h
)(
z
n
−
1
+
h
)
+
a
(
z
n
+
h
)
+
b
(
z
n
−
1
+
h
)
+
c
=
0
h
2
z
n
z
n
−
1
+
z
n
(
a
+
h
)
+
z
n
−
1
(
b
+
h
)
+
+
h
(
a
+
b
)
+
c
=
0
set
=
0
4
c
(
a
1
2
b
)
2
h
=
−
(
a
+
b
)
±
+
−
⇒
becomes defining equation for
h
.
Now divide by z
n
z
n
−
1
:
1
z
n
−
1
1
z
n
1
+
(
a
+
h
)
+
(
b
+
h
)
=
0
.
1
z
n
Let v
n
=
v
n
(
b
+
h
)
+
v
n
−
1
(
a
+
h
)
=−
1
v
(
h
)
n
n
=
c
β
n
(
b
n
−
1
(
a
β
+
h
)
+
β
+
h
)
=
0
a
+
h
β
=−
b
+
h
1
v
(
P
)
n
=−
a
+
b
+
2
h
C
1
n
a
+
h
1
1
z
n
=
1
x
n
−
v
n
=
−
−
2
h
=
h
.
b
+
h
a
+
b
+
Rearrange and substitute to get the equation for
x
n
:
1
x
n
=
h
+
C
1
n
.
a
+
h
1
−
−
b
+
h
a
+
b
+
2
h
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