Environmental Engineering Reference
In-Depth Information
The Fischer et al. (1979) relation given by Equation
(4.17) indicates that the actual downstream distance
required for complete cross-sectional mixing is
0.4
VT
w
= 0.4(1364) = 546 m.
)
2
V W L
(
−
L
′ =
0.1
(4.18)
w
ε
t
Clearly, cross-sectional mixing can be accelerated by
using a multiport diffuser in the center of a stream
rather than a single-port outlet on the side of a stream.
EXAMPLE 4.2
EXAMPLE 4.1
Estimate the distance downstream to where the waste-
water described in Example 4.1 is well mixed across the
stream if (a) the wastewater is discharged from the
center of the stream, and (b) the wastewater is dis-
charged through a 5-m-long multiport diffuser placed in
the middle of the stream.
A municipality discharges wastewater from the side of
a stream that is 10 m wide and 2 m deep. The average
flow velocity in the stream is 1.5 m/s, and the friction
factor is estimated to be 0.03 (calculated using the Cole-
brook equation). (a) Estimate the time for the wastewa-
ter to become well mixed over the channel cross section.
(b) How far downstream from the discharge location
can the effluent be considered well mixed across the
stream?
Solution
From the previous analysis:
ε
v
= 0.012 m
2
/s,
ε
t
= 0.11 m
2
/s,
and the time scale,
T
s
, for transverse mixing over a dis-
tance
s
is given by
Solution
(a) From the data given,
f
= 0.03 and
V
= 1.5 m/s.
Therefore, the shear velocity,
u
*
, is given by Equa-
tion (4.3) as
s
2
T
=
s
ε
t
In the previous example the wastewater was discharged
from the side of the channel, so the mixing width,
s
, was
the width of the channel,
w
.
f
V
0.03
8
u
=
=
(1.5)
=
0.092
m/s
*
8
(a) If the wastewater is discharged from the center of
the stream, the mixing width,
s
, for the wastewater
to become well mixed over the channel cross section
is given by
Since
d
= 2 m, the vertical and transverse diffusion
coefficients can be estimated as
ε
v
=
0.067
du
=
0.067(2)(0.092)
=
0.012
m /s
2
*
ε
t
=
0.6
du
=
0.6(2)(0.092)
=
0.11
m /s
2
w
10
2
*
s
=
=
=
5 m
2
The time scale for vertical mixing,
T
d
, is given by
and the corresponding time scale,
T
s
, is given by
d
2
2
0.012
2
T
d
=
=
=
333
seconds
=
5.6
minutes
ε
v
2
5
0.11
T
s
=
=
227
seconds
=
3.8 minutes
and the time scale for transverse mixing,
T
w
, is
given by
Since the flow velocity,
V
, is 1.5 m/s, the downstream
distance,
L
, for the wastewater to become well
mixed is
w
2
10
0.11
2
T
w
=
=
=
909
s
=
15
minutes
ε
t
L
=
0.4
VT
=
0.4(1.5)(227)
=
136
m
s
The discharge is well mixed over the channel cross
section when it is well mixed over both the depth
and the width, which in this case occurs after about
15 minutes.
(b) In a time interval of 15 minutes (= 909 seconds), the
discharged effluent travels a distance,
VT
w
, given by
(b) If the wastewater is discharged from a 5-m-long
diffuser centered in the stream, the mixing width,
s
,
for the wastewater to become well mixed over the
channel cross section is given by
w
−
5
10 5
2
−
s
=
=
=
2.5 m
2
VT
w
=
(1.5)(909)
=
1364 m
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