Environmental Engineering Reference
In-Depth Information
A
I
is the interfacial area between the liquid and vapor phases,
x
A
I
and
y
A
I
are the interfacial
mole fractions,
x
A
and
y
A
are the bulk-phase mole fractions, and
k
y
are the individual mass
transfer coefficients for the liquid and vapor phases, respectively.
Since the interfacial area,
A
I
,isdifficult to determine accurately, these rate equations
may be rewritten as:
Rate
Volume
=
k
y
a
(
y
A
−
y
A
I
)
=
k
x
a
(
x
A
I
−
x
A
)
,
(6.10)
where
a
is the specific surface of the column packing. It is also difficult to determine
a
accurately, so values are correlated into lumped term mass transfer correlations for
k
y
a
and
k
x
a
.Itisalso possible to write the equation without the interfacial mole fractions by
using overall coefficients
K
y
and
K
x
:
Rate
Volume
=
y
A
)
K
x
a
(
x
A
−
K
y
a
(
y
A
−
=
x
A
)
.
(6.11)
Here,
x
A
is the liquid mole fraction that would be in equilibrium with
y
A
, and
y
A
is the
vapor mole fraction that would be in equilibrium with
x
A
.
Since the mass transfer occurs in series, we can derive an overall mass transfer coefficient
in terms of mass transfer in the gas phase:
1
K
y
a
=
m
k
x
a
+
1
k
y
a
.
(6.12)
Similarly, in terms of mass transfer in the liquid phase:
1
K
x
a
=
1
k
x
a
+
1
mk
y
a
.
(6.13)
This form should look familiar; it is the sum-of-resistances model from Chapter 3. If
m
is small,
K
y
≈
k
x
and
the liquid-phase resistance controls. Remember what a small or large value of
m
implies
for the solubility of a solute in the liquid phase relative to the gas-phase concentration.
Another approach to understanding the controlling resistance is to rearrange the equation
for the rate
k
y
and the gas-phase resistance is the dominant. If
m
is large,
K
x
≈
/
volume in each phase:
k
x
a
k
y
a
=
y
A
−
y
A
I
−
x
A
I
.
(6.14)
x
A
−
This can be rewritten as
1
/
k
y
a
y
A
−
y
A
I
gas-phase mass transfer resistance
liquid-phase mass transfer resistance
.
−
k
x
a
=
x
A
I
=−
(6.15)
1
/
x
A
−
If gas-phase resistance is negligible,
y
A
≈
y
A
I
.
Similarly,
x
A
≈
x
A
I
if the liquid-phase
resistance can be neglected.
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