Environmental Engineering Reference
In-Depth Information
environmental applications are shown in Figure 3.19.
Solutions describing the steady-state concentrations
downstream of these sources are discussed separately
below.
M
AV
kx
V
y V
D x
y
c x y
( ,
)
=
exp
erfc
(3.126)
2
2
where
A
is the area over which the mass flux
M
occurs.
Since the source is theoretically of infinite extent, then
M
/
can be considered as a single parameter that
describes the mass flux density (ML−2T−1)
−2
T
−1
) across the
plane source.
Semi-Infinite Source.
A semi-infinite source occurs at
x
= 0 and covers the planar region −∞ <
y
< 0 and
−∞ <
z
< ∞. Assuming that the longitudinal advective
flux is much greater than the longitudinal diffusive flux
(i.e., Pe =
Vx
/
D
x
>> 1), the advection-diffusion equation
describing the steady-state concentration distribution
simplifies to
EXAMPLE 3.11
The confluence of a small contaminated stream and a
wide clean stream is illustrated in Figure 3.20, where the
contaminated stream is 3 m wide and 2 m deep. The
merged stream has an average velocity of 10 cm/s, and
longitudinal and transverse diffusion coefficients are
estimated as 5 m
2
/s and 0.1 m
2
/s, respectively. If the flux
of a conservative contaminant from the contaminated
stream is 0.5 kg/s, estimate the concentration 9 m from
the left bank of the stream at a section 15 m down-
stream of the confluence. How does this concentration
compare with the concentration 9 m from the left bank
immediately downstream of the confluence? Neglect
the influence of images in accounting for the sides of the
stream.
2
∂
∂
c
x
∂
∂
c
V
=
D
−
kc
(3.125)
y
y
2
The solution of Equation (3.125) for a continuous mass
flux,
M
(MT
−1
), is given by
b
/2
b
/2
rectangular source
semi-in
nite source
z
z
Solution
y
y
From the given data:
w
= 3 m,
d
= 2 m,
V
= 10 cm/s
= 0.1 m/s,
D
x
= 5 m
2
/s,
D
y
= 0.1 m
2
/s,
M
= 0 . kg/s
,
y
= 9 m − 3 m = 6 m, and
x
= 15 m. Since the contami-
nant is conservative,
k
= 0. Substituting these data into
Equation (3.126) gives
x
x
V
V
(a)
(b)
M
AV
kx
V
y V
D x
y
c x y
( ,
)
=
exp
erfc
2
2
Figure 3.19.
Plane sources with two-dimensional diffusion.
left bank
right bank
merged stream
stream 2
stream 1
2 m
contaminated
stream
clean
stream
3 m
Figure 3.20.
Confluence of two streams.
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