Environmental Engineering Reference
In-Depth Information
where
2
D
solute
v
p
L
v
p
c
3
u
o
σ
σ
o
=
v
o
=
(9.48)
o
h
and
h
=
half-width of channel
x
=
axial distance
ε
o
=
h
/
x
.
An important limit in concentration polarization is complete rejection (
R
=
1,
C
p
=
0).
v
p
).
For organics, this can lead to gel layer formation which controls the permeation rate (
For this limiting case, the equation for concentration polarization becomes
C
g
C
b
=
e
+
v
p
/
k
(9.49)
C
g
=
gel layer concentration.
Rearranging and solving for
v
p
,
k
ln
C
g
C
b
v
p
=
.
(9.50)
Mass transfer correlations have been derived that are used for membranes of various
configurations [16]. For flow through
tubes
, the following are used (see Appendices A and
B for details on dimensionless numbers and mass transfer correlations, respectively).
Turbulent flow
The Sherwood number (
Sh
) is given by:
023
Re
0
.
8
Sc
0
.
33
Sh
=
0
.
.
(9.51)
Laminar flow
86
ReSc
d
h
L
0
.
33
=
.
,
Sh
1
(9.52)
where
d
h
=
hydraulic diameter
4(cross-sectional area)
wetted perimeter
=
(9.53)
and
L
axial length of tube.
An alternative correlation for
k
uses the shear rate ( ˙
=
γ
).
0
.
33
D
0
.
67
L
−
0
.
33
k
=
0
.
816 ˙
γ
,
(9.54)
where
8
d
γ
=
˙
(for tubes)
(9.55)
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