Environmental Engineering Reference
In-Depth Information
z
y
x
z
s
z
1
FIGURE 6.30
Emission of a plume of air pollutant from a tall chimney.
The simplified expression is then
D
y
∂
2
D
z
∂
2
U
∂
]
∂x
=
[
A
[
A
]
[
A
]
+
.
(6.143)
∂y
2
∂z
2
Upon integration and rearranging, the following solution results:
exp
y
2
σ
,
z
s
)
2
Q
s
1
2
(z
−
[
A
]
(x
,
y
,
z)
=
−
y
+
(6.144)
z
π
U
σ
y
σ
z
σ
y
=
z
=
where
2
D
z
x/U
are the respective squares of the mean stan-
dard deviations in
y
- and
z
-directions for the concentrations. The above equation is
only applicable for the concentration in the downwind direction up to the point in
the
x
-direction where the ground-level concentration (
z
σ
2
D
y
x/U
and
σ
0) is significant. After this
“reflection” of pollutants will occur, because soil is not a pollutant sink and material
will diffuse back to the atmosphere. This gives rise to a mathematical equivalence
of having another image source at a distance
z
s
(Figure 6.31). A superposition of the
two solutions gives the final general equation for pollutant concentration anywhere
downwind of the stack. The concentration of vapor in the air at any point from a
plume of effective height
z
s
is given by
=
e
−
((z
s
−
z)
2
/
2
σ
z
)
e
−
( y
2
/
2
σ
Q
s
y
)
.
z
)
e
−
((z
s
+
z)
2
/
2
σ
[
A
]
(x
,
y
,
z)
=
+
(6.145)
π
U
σ
y
σ
z
The concentration is given in g/m
3
if
Q
s
is expressed in g/s. It can be converted to the
more common units of
g/m
3
or ppmv using appropriate conversions.
μ
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