Environmental Engineering Reference
In-Depth Information
plume, and the length scale
l
m
measures the distance to
where the ambient current begins to become more
important than the jet momentum in controlling the
motion of the plume. The length scale
l
M
is sometimes
referred to as the
jet/plume transition length scale
, and
l
m
is sometimes referred to as the
jet/crossflow length scale
.
Defining the plume dilution,
S
, by the relation
2
1
perpendicular
0.5
45
o
parallel
zero current speed
0.2
c
c
0.1
e
(9.34)
S
=
0.5 0.1 0.2
0.5
1
2
5
10
20
50 100
F
=
u
a
3
/
b
0
the functional relationship given by Equation (9.30) can
be written in the form
Figure 9.10.
line plume dilution characteristics.
Source
:
Roberts et al. (2010).
Sq
u y
l
l
l
l
y
l
= ′
0
M
M
f
,
,
(9.35)
5
Sq
u y
y
l
a
Q
m
m
= ′
0
f F
,
(9.40)
7
a
m
In most sewage outfalls, the port geometry has a rela-
tively minor influence on the dilution of the effluent
plume. Under these circumstances, the dilution becomes
insensitive to the value of
l
Q
, and the functional expres-
sion for the plume dilution, Equation (9.35), can be
written as
Experiments to determine an empirical equation for the
relationship given by Equation (9.40) were conducted
by Roberts (1979) and Tian et al. (2004a) for cases
where the initial momentum of the jet is negligible.
Under these circumstances, the effluent momentum flux
is negligible, which means that
m
0
is not included in the
dimensional analysis and Equation (9.40) becomes
Sq
u y
l
l
y
l
= ′
0
M
f
,
(9.36)
6
a
m
m
Sq
u y
0
= ′
f F
8
( )
(9.41)
This relationship can be further reduced by considering
the physical meaning of the length-scale ratio
l
M
/
l
m
.
Using the definitions of
l
M
and
l
m
given by Equations
(9.32) and (9.33),
a
This functional relationship, as derived experimentally
by Roberts (1979) and Tian et al. (2004a), is shown in
Figure 9.10, where
θ
indicates the direction of the
current relative to the diffuser discharge. The results
shown in Figure 9.10 clearly indicate that currents per-
pendicular to the diffuser discharge produce the great-
est dilutions. For example, when
F
≈ 100, the
perpendicular alignment (
θ
= 90°) results in a dilution
that is four times greater than that for a parallel
alignment (
θ
= 0°). In cases where
F
is small, the
ambient flow has a minimal effect compared with the
effect of buoyancy on the plume dynamics. Under these
circumstances (
F
< 0.1) the plume behavior is approxi-
mately the same as for a stagnant ambient and, accord-
ing to dimensional analysis, can be expressed in the
form
2 3
/
l
l
u
b
3
=
(9.37)
M
a
m
0
which measures the relative importance of the ambient
flow and buoyancy on the dynamics of the plume motion.
An ambient/discharge Froude number,
F
, that is com-
monly used in practice and is defined by
3
u
b
F
=
(9.38)
0
and the ratio
l
M
/
l
m
can be expressed in terms of
F
by the
relation
Sq
u y
0
=
C F
− /
1 3
(9.42)
l
l
LS
M
=
2 3
F
/
(9.39)
a
m
where
C
lS
is a (line) plume dilution coefficient under
stagnant conditions. Values of
C
lS
found by various
investigators are given in Table 9.6. When the ambient
and therefore the functional expression for the plume
dilution becomes
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