Environmental Engineering Reference
In-Depth Information
disperse in the longitudinal direction. This is commonly
referred to as
longitudinal dispersion
or
shear dispersion
and is illustrated in Figure 3.24. The term “shear” is used
to characterize dispersion because velocity gradients
are associated with shear. Consider the longitudinal
section shown at the top of Figure 3.24, where disper-
sion is caused by the vertical variation in velocity. under
these conditions, a tracer that is initially distributed uni-
formly over the vertical (at time
t
1
) at a later time (
t
2
) is
distributed such that the tracer nearer to water surface
has moved farther downstream than the tracer that is
nearer the bottom of the channel. vertical mixing is also
enhanced by increased vertical concentration gradients
(an hence increased vertical diffusion) caused by veloc-
ity shear. A similar mechanism occurs in response to
transverse variations in velocity across the channel as
shown at the bottom of Figure 3.24. In this case, the
tracer nearer the center of the channel moves down-
stream faster than the tracer nearer a side of the channel,
thereby inducing longitudinal dispersion and enhanced
lateral mixing in response to increased lateral concen-
tration gradients.
As a further illustration, consider the case shown in
Figure 3.25 where the a fluid flows between two bound-
aries, where
x
and
y
are the coordinates in the longitu-
dinal and transverse directions,
u
(
y
) is the longitudinal
flow velocity that varies in the transverse direction, and
W
is the distance between the two boundaries. If we
assume that the flow is turbulent and that turbulent dif-
fusion is much greater that molecular diffusion, then
∞
∑
∫
( )
( )
L
=
R s ds
≈
R s
∆
s
11
11
11
0
=
( )( . )
1 0 5
+
( .
0 49 0 24 0 084 0 022
+
.
+
.
+
.
+
0 0044
.
+
0 0018 0 00004 0 00004 1
.
−
.
+
.
)( )
+
( .
0 00004 0 5
)( . )
=
0 13
.
m
Therefore, the correlation length scale,
L
11
, in the flow
direction is equal to 0.13 m. The Lagrangian time scale,
T
11
, is approximated by
L
0 13
0 25
.
.
11
T
≈
=
=
0 54
.
second
11
V
These results indicate that the longitudinal velocity fluc-
tuations remain correlated for distances on the order of
0.13 m, and for times on the order of 0.54 second. The
estimated longitudinal diffusion coefficient,
ε
11
, in the
flow direction is then given by
L
11
ε
≈
σ
2
=
σ
2
T
=
(
225 0 54
)( .
)
=
122
cm /s
2
11
v
v
11
1
V
1
3.6 DISPERSION
Dispersion
is defined as mixing caused by spatial varia-
tions in the mean velocity. A frequently encountered
case of dispersion occurs in open channels where trans-
verse variations in the mean velocity cause a tracer to
z
water surface
vertical velocity distribution
tracer at
t = t
1
tracer at
t = t
2
x
z
side of channel
transverse velocity distribution
tracer at
t = t
1
tracer at
t = t
2
x
Figure 3.24.
Dispersion processes.
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