Environmental Engineering Reference
In-Depth Information
A similar model is obtained from the liquid phase:
x
A
in
L
k
x
aA
c
d
x
A
x
A
−
=
x
A
I
=
H
L
N
L
.
(6.21)
x
A
out
To find
N
G
and
N
L
,wecan return to:
k
x
a
k
y
a
=
y
A
−
y
A
I
x
A
I
.
This equation is the slope of a line from (
y
A
,
−
x
A
−
x
A
I
), the bulk and interfacial
mole fractions, respectively. The interfacial mole fractions are in equilibrium, and so the
point (
y
A
I
,
x
A
I
)isonthe equilibrium line. The point (
y
A
,
x
A
)isonthe operating line.
From any point on the operating line (
y
A
,
x
A
), a line with slope
x
A
)to(
y
A
I
,
k
y
a
will intersect
the equilibrium line at the corresponding (
y
A
I
,
x
A
I
). This can be done for a number of
points, and a plot of 1
−
k
x
a
/
/
(
y
A
I
−
y
A
)vs
y
A
can be generated. The area under the curve
between
y
A
in
and
y
A
out
will be
N
G
for the section.
N
L
can be found similarly by plotting
1
/
(
x
A
−
x
A
I
).
It would be useful to eliminate the need to determine the interfacial compositions and
just use the bulk concentrations in each phase. This can be done by using an overall
mass transfer coefficient along with the appropriate driving force. The same derivation
procedure results in
y
A
in
V
K
y
aA
c
d
y
y
A
−
=
y
A
=
H
OG
N
OG
.
(6.22)
y
A
out
A complete listing of these terms is shown in Table 6.3 (see p. 176).
If we assume linear phase equilibrium (
y
=
mx
) and a linear operating line, an analytical
expression can be obtained for
N
OG
:
y
A
in
d
y
N
OG
=
mx
A
in
.
(6.23)
(1
−
mV
/
L
)
y
+
y
A
out
(
mV
/
L
)
−
y
A
out
Integrating and using the definition of the absorption factor
A
:
(
A
y
A
in
−
−
1)
mx
A
in
1
A
ln
+
A
y
A
out
−
mx
A
in
N
OG
=
.
(6.24)
(
A
−
1)
A
An analogous derivation based on liquid-phase concentrations yields:
=
H
OL
N
OL
L
K
x
aA
c
H
OL
=
(6.25)
(1
x
A
in
−
A
y
A
in
/
m
ln
−
A
)
+
x
A
out
−
y
A
in
/
m
N
OL
=
.
(1
−
A
)
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