SME
Environmental Engineering
Air pollution
Use this for “A stack emits 100 g/s of SO₂ into a 5 m/s wind; H = 60 m, σᵧ = 100 m and σz = 50 m at the receptor. Find the ground-level concentration.”
Diagram
C
=
Q
π × u × σᵧ × σz
×
e
−
H
2
÷ (2 ×
σz
2
)
=
(
100 ×
10
6
π × 5 × 100 × 50
)
×
0.487
=
620
μg/
m
3
at
ground
level
on
the
centreline.
The plume runs level at H = 60 m and spreads as it goes; at the receptor σz = 50 m, and the ground reflects the part that reaches it (the profile's lower bulge).
Heights are to scale; distance is not.
This is an example. Type your own values to start a new problem.
F
=
e
−
(
m
)
2
/
(
2
×
(
m
)
2
)
C (μg/m³)
=
Q (g/s)
×
π
×
u (m/s)
×
σᵧ (m)
×
σz (m)
Formulas
F
=
e
−
H
2
÷ (2 ×
σz
2
)
0.4868
=
e
−
60
2
÷ (2 ×
50
2
)
C
=
1,000,000
×
Q
×
F
π ×
u
× σᵧ × σz
619.8
=
1,000,000
×
100
×
0.4868
π × 5 × 100 × 50
Assumptions
•
A steady wind and a steady emission; the wind carries the gas faster than it spreads downwind.
•
The ground reflects the plume back up instead of absorbing it.
•
σᵧ and σz, the plume’s spreads, are read from the stability-class charts at the receptor’s distance.
Step-by-step
WE KNOW
Q = 100 g/s, u = 5 m/s, σᵧ = 100 m, σz = 50 m, H = 60 m
FIND
height factor (F), ground-level concentration (C)
Step 1 · Find height factor (F)
Use F = e−H² ÷ (2 × σz²)
The
ground
lies
H
below
the
centreline,
so
the
bell-shaped
profile
is
cut
by
this
factor
there.
F
=
e
−
60
2
÷ (2 ×
50
2
)
F
=
e
−
60
2
÷ (2 × 2,500)
F
=
e
−
60
2
÷ 5,000
F
=
e
−3,600 ÷ 5,000
F
=
e
−0.72
F
=
0.4868
Step 2 · Find ground-level concentration (C)
Use C = 1,000,000 × Q × F ÷ (π × u × σᵧ × σz)
The
emission
spreads
over
the
plume’s
cross-section
πσᵧσz,
carried
at
u
;
10
6
μg
make
1
g.
C
=
1,000,000
×
100
×
0.4868
π × 5 × 100 × 50
C
=
1,000,000
×
100
×
0.4868
500 × 50 × π
C
=
1,000,000
×
100
×
0.4868
25,000 × π
C
=
1,000,000
×
100
×
0.4868
78,540
C
=
100,000,000
×
0.4868
78,540
C
=
48,680,000
78,540
C
=
619.8
μg/
m
3
Check
Put the numbers back in. Both sides should match.
0.4868
=
e
−
60
2
÷ (2 ×
50
2
)
✓
619.8
=
1,000,000
×
100
×
0.4868
π × 5 × 100 × 50
✓
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