Environmental Engineering Reference
In-Depth Information
Multiplying the previous two equations gives
L
mV
2
y
3
−
y
2
b
=
.
(3.44)
y
1
−
mx
0
−
For any number
P
of equilibrium stages, a general equation can be formed
L
mV
P
y
P
+
1
−
y
P
b
=
.
(3.45)
y
1
−
mx
0
−
To relate entrance and exit concentrations from the cascade, the previous equation for
P
equilibrium stages is summed over the entire cascade
L
mV
P
N
y
N
+
1
−
y
1
b
=
.
(3.46)
y
1
−
mx
0
−
1
Because
y
1
, the exit-phase concentration, is the unknown variable, we can rewrite this
equation so that
y
1
only appears in the numerator:
L
mV
P
1
N
y
N
+
1
−
y
1
b
=
P
.
(3.47)
L
mV
y
N
+
1
−
mx
0
−
1
N
1
+
The summation of the power series when
L
/
mV
<
1 leads to
L
mV
N
+
1
L
mV
−
y
N
+
1
−
y
1
b
=
.
(3.48)
L
mV
N
+
1
y
N
+
1
−
mx
0
−
1
−
Using the equilibrium relationship
y
1
to replace
mx
0
-
b
L
mV
L
mV
N
+
1
−
y
N
+
1
−
y
1
y
1
=
.
(3.49)
L
mV
N
+
1
y
N
+
1
−
1
−
This result is normally called the Kremser equation.
Note that, if
L
mV
)
N
/
mV
>
1, then Equation (3.47) can be divided by (
L
/
and the
same result is obtained for Equation (3.49).
In this form, the Kremser equation is useful for solving problems where
N
is fixed
and an exit composition needs to be calculated. When equilibrium is closely approached
(
y
1
→
y
1
)
,
Equation (3.49) becomes
y
1
y
N
+
1
−
y
1
−
1
−
(
L
/
mV
)
y
1
=
mV
)
N
+
1
.
(3.50)
1
−
(
L
/
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