Environmental Engineering Reference
In-Depth Information
EXAMPLE 9.5
near field
far field
(a) Compare the apparent diffusion coefficient of an oil
spill having a characteristic size of 50 m with an oil spill
whose size is 100 m. (b) Estimate how long it would take
for a small oil spill to grow to a size of 50 m.
h
Qc
e
Solution
L
u
a
(a) The apparent diffusion coefficient,
K
a
, is related to
the length scale,
L
, of the oil spill by
K
= 0 0103
.
L
1 15
.
cm /s
2
a
y
plume
When
L
= 50 m = 5000 cm,
K
a
=
0 0103 5000
.
(
)
.
1 15
=
185
cm /s
2
and when
L
= 100 m = 10,000 cm,
Figure 9.14.
Interface between near- and far-field models.
)
.
1 15
2
K
a
=
0 0103 10 000
.
(
,
=
410
cm /s
is equal to the length of the diffuser and the depth of
the plane area is equal to the plume thickness at the
trapping height. For surfacing plumes in unstratified
environments, the initial plume thickness can be taken
as 30% of the total depth. Assuming that the contami-
nant mass flux leaving the diffuser is equal to the con-
taminant mass flux across the source plane at the
trapping level, then
Therefore, when the oil spill doubles its size from
50 to 100 m, the apparent diffusion coefficient more
than doubles, going from 185 cm
2
/s to 410 cm
2
/s.
(b) The variance of the oil spill as a function of time
can be estimated by
2
2.34
2
σ
rc
=
0.0108
t
cm
or
Qc
=
u Lhc
0
(9.60)
e
a
σ
rc
= 0 104
1 17
.
t
.
cm
where
Q
is the effluent volume flux (l3T−1),
3
T
−1
),
c
e
is the
contaminant concentration in the effluent (Ml
−3
),
u
a
is
the ambient velocity (lT
−1
),
L
is the diffuser length (l),
h
is the height of the waste field (l), and
c
0
is the con-
taminant concentration crossing the source plane
(Ml
−3
). Equation (9.60) can be put in the more conve-
nient form
Defining the length scale,
L
, by
L
= 3
σ
rc
,
L
= 0 312
1 17
.
t
.
cm
When
L
= 50 m = 5000 cm,
5000
= .
0 312
1 17
t
.
cm
Qc
u Lh
e
c
=
(9.61)
0
which leads to
a
t
=
393 seconds
0
=
65 5 minutes
.
=
1 9 hours
.
0
Brooks (1960) proposed a simple far-field model that
assumes a constant ambient velocity,
u
a
, and neglects
both vertical diffusion and diffusion in the flow direc-
tion. These assumptions are justified by the observations
that the diffusive flux in the flow direction is usually
much smaller than the advective flux, vertical diffusion
is damped significantly by buoyancy, and the vertical
diffusion coefficient is much smaller than the transverse
diffusion coefficient. The governing steady-state
advection-diffusion equation is therefore given by
Therefore, a small oil spill will take approximately
1.09 hours to grow to a size of 50 m.
For multiport diffusers with closely spaced ports, far-
field models typically assume that the contaminant
source is a rectangular plane perpendicular to the mean
current at the trapping level of the plume, as shown in
Figure 9.14, where the width of the plane area (source)
Search WWH ::
Custom Search