2. 2
Why do we need them?
To predict the ambient air concentrations that will
result from any planned set of emissions for any
specified meteorological conditions, at any location,
for any time period, with total confidence in our
prediction.
The perfect model should match the reality, which is
impossible. Therefore the model is a simplification of
reality.
The simpler the model, the less reliable it is. The
more complex the model, the more reliable it is.
3. 3
General Form of Models
The models in this chapter are material balance models. The
material under consideration is the pollutant of interest.
The General material balance equation is:
Notes: 1) we need to specify some set of boundaries.
2) The model will be applied to one air pollutant at a time. In
other words, we cannot apply the model to air pollution
in general.
Accumulation rate = (all flow rates in)
– (all flow rates out)
+ (creation rate)
– (destruction rate)
4. 4
Three types of Models (in this chapter)
1. Fixed-Box Models
2. Diffusion Models
3. Multiple Cell Models
These models are called source-oriented models. We
use the best estimates of the emission rates of various
sources and the best estimate of the meteorology to
estimate the concentration of various pollutants at
various downwind points.
5. 5
1) Fixed-Box Models
The city of interest is assumed to be rectangular.
The goal is to compute the air pollutant concentration in this
city using the general material balance equation.
Fig. 6.1
De Nevers
6. 6
1) Fixed-Box Models
The city of interest is assumed to be rectangular.
The goal is to compute the air pollutant concentration in this
city using the general material balance equation.
Assumptions:
1. Rectangular city. W and L are the dimensions, with one
side parallel to the wind direction.
2. Complete mixing of pollutants up to the mixing height H.
No mixing above this height.
3. The pollutant concentration is uniform in the whole volume
of air over the city (concentrations at the upwind and
downwind edges of the city are the same).
4. The wind blows in the x direction with velocity u , which is
constant and independent of time, location, & elevation.
7. 7
… Assumptions
5. The concentration of pollutant in the air entering the city is
constant and is equal to b (for background concentration).
6. The air pollutant emission rate of the city is Q (g/s). The
emission rate per unit area is q = Q/A (g/s.m2). A is the
area of the city (W x L). This emission rate is assumed
constant.
7. No pollutant enters or leaves through the top of the box,
nor through the sides.
8. No destruction rate (pollutant is sufficiently long-lived)
8. 8
Now, back to the general material balance eqn
→Destruction rate = zero (from assumptions)
→Accumulation rate = zero (since flows are independent of time and
therefore steady state case since nothing is changing with time)
→ Q can be considered as a creation rate or as a flow into the
box through its lower face. Let’s say a flow through lower
face.
Accumulation rate = (all flow rates in)
– (all flow rates out)
+ (creation rate)
– (destruction rate)
9. 9
the general material balance eqn becomes:
The equation indicates that the upwind concentration is added
to the concentrations produced by the city.
To find the worst case, you will need to know the wind speed,
wind direction, mixing height, and upwind (background)
concentration that corresponds to this worst case.
0 = (all flow rates in) – (all flow rates out)
0 = u W H b + q W L – u W H c
Where c is the concentration in the entire city
uH
qL
b
c
10. 10
Example 6.1
A city has the following description: W = 5 km, L = 15 km, u
= 3 m/s, H = 1000 m. The upwind, or background,
concentration of CO is b = 5 μg/m3. The emission rate per
unit are is q = 4 x 10-6 g/s.m2. what is the concentration c
of CO over the city?
= 25 μg/m3
uH
qL
b
c
m
1000
m/s
3
m
15000
m
.
s
g
10
4
m
μg
5 2
6
3
c
11. 11
Comments on the simple fixed-box model
1) The third and the sixth assumptions are the worst (why?).
2) The fixed-box models does not distinguish between area
sources and point sources.
Area sources: small sources that are large in number and
usually emit their pollutants at low elevations; such as
autos, homes, small industries, etc.
Point sources: large sources that are small in number and
emit their pollutants at higher elevations; such as power
plants, smelters, cement plants, etc.
Both sources are combined in the q value. We know that
raising the release point of the pollutant will decrease the
ground-level concentration.
12. 12
… Comments on the simple fixed-box model
3) If you are laying out a new city, how would you lay it?
(page 125).
In light of this, would it be preferable to put your city in a
valley?
4) For an existing city, what actions would you take in order to
minimize air pollutant concentrations? (answer in words
that people can understand and act according to)
5) So far, the fixed-box model predicted concentrations for
only one specific meteorological condition. We know that
meteorological conditions vary over the year.
13. 13
Modifications to improve the fixed-box model
1) Hanna (1971) suggested a modification that allows one to
divide the city into subareas and apply a different value of q
to each. (since variation of q from place to place can be
obtained; q is low in suburbs and much higher in industrial
areas).
2) Changes in meteorological conditions (comment #5) can be
taken into account by
a. determine the frequency distribution of various values of
wind direction, u, and of H
b. Compute the concentration for each value using the
fixed-box model
14. 14
…Modifications to improve the fixed-box model
c. Multiply the concentrations obtained in step b by the
frequency and sum to find the annual average
ies
meteorolog
all
over
y
meteorolog
that
of
occurrence
of
frequency
y
meteorolog
for that
ion
concentrat
ion
Concentrat
Average
Annual
15. 15
Example 6.2
For the city in example 6.1, the meteorological conditions
described (u = 3 m/s, H = 1000 m) occur 40 percent of the
time. For the remaining 60 percent, the wind blows at right
angles to the direction shown in Fig. 6.1 at velocity 6 m/s
and the same mixing height. What is the annual average
concentration of carbon monoxide in this city?
First we need to compute the concentration resulting from each
meteorological condition and then compute the weighted
average.
For u = 3 m/s and H = 1000 m → c = 25 μg/m3
16. 16
…example 6.2 cont.
For u = 6 m/s and H = 1000 m →
3
3
3
ies
meteorolog
all
over
m
μg
15
0.6
m
μg
8.33
0.4
m
μg
25
ion
Concentrat
Average
Annual
y
meteorolog
that
of
occurrence
of
frequency
y
meteorolog
for that
ion
concentrat
ion
Concentrat
Average
Annual
3
2
6
3
m
g
8.33
m
1000
m/s
6
m
5000
m
.
s
g
10
4
m
μg
5
c
c
Note that L is now
5km, not 15km
17. 17
Graphical Representation of the Fixed-Box
Model Equation (Fig. 6.2 in your textbook)
Ambient
air
concentration,
c
Emission rate, q, g/s.km2
18. 18
Example
A pollutant concentration was calculated to be c1 with
emission rate q1. If the Environmental Authority wishes to
reduce the concentration to c2, compute the new allowable
emission rate (q2)
We can use graphical interpolation:
OR
Note this can be done only when the meteorological
parameters are constant
b
c
b
c
q
q
1
2
1
2
q1
q2
c2
c1
b
result
same
get the
to
(1)
by
(2)
Divide
2
2
1
1
L
uH
b
c
q
L
uH
b
c
q
19. 19
Example 6.3 (fractional reduction in emission rate)
The ambient air quality standard for particulates (TSP) in the
USA in 1971 was 75 μg/m3 annual average. In 1970 the
annual average particulate concentration measured at one
monitoring station in downtown Chicago was 190 μg/m3.
The background concentration was estimated to be 20
μg/m3. By what percentage would the emission rate of
particulates have to be reduced below the 1970 level in
order to meet the 1971 ambient air quality standard?
c1 = 190 μg/m3 , c2 = 75 μg/m3
q
c2
c1
b
20. 20
Example 6.3 (fractional reduction in emission rate)
c1 = 190 μg/m3 , c2 = 75 μg/m3
OR: you can use interpolation from
the graph
%
67
67
.
0
20
190
75
190
rate
emission
in
reduction
Fractional
q1
q2
c2
c1
b
b
c
c
c
b
c
b
c
q
q
q
q
q
1
2
1
1
2
1
2
1
2
1
1
1
rate
emission
in
reduction
Fractional
21. 21
2) Diffusion Models
Called as diffusion models. However, they are actually
dispersion models.
Such models usually use the Gaussian plume idea.
Fig. 6.3
De Nevers
22. 22
2) Diffusion Models
Point source (smoke stack) located at (0, 0, H) that steadily
emits a pollutant at emission rate of Q (g/s)
The wind blows in the x-direction with velocity u.
The goal is to compute the concentration due to point source
at any point (x, y, z) downwind.
Problem Statement
23. 23
2) Diffusion Models
The origin of the coordinate system is (0, 0, 0), which is the
base of the smoke stack.
Plume is emitted form a point with coordinates (0, 0, H)
H = h + Δh , h = physical stack height
Δh = plume rise
Plume rises vertically at the beginning (since it has higher
temperature and a vertical velocity), then levels off to travel in
the x-direction (wind direction).
As the plume travels in the x-direction, it spreads in the y and
z directions.
The actual mixing mechanism is the turbulent mixing; not the
molecular diffusion.
What will happen if the molecular diffusion was the only mechanism?
Description of Situation in Fig. 6.3
24. 24
2) Diffusion Models
If we place a pollutant concentration meter at some fixed
point in the plume, we would see the concentration oscillating
in an irregular fashion about some average value (snapshot in
Fig. 6.4). This is another evidence of the turbulent mixing.
This average value is the value that the Gaussian plume
model calculates
The model does not calculate the instantaneous concentration value. It
only calculates the average value.
Therefore, results obtained by Gaussian plume calculations should be
considered only as averages over periods of at least 10 minutes, and
preferably one-half to one hour.
The Gaussian plume approach calculates only this average
value
… cont. Description of Situation in Fig. 6.3
25. 25
2) Diffusion Models
Where σy = horizontal dispersion coefficient (length units)
σz = vertical dispersion coefficient (length units)
The name “Gaussian” came from the similarity between the
above equation and the Gauss normal distribution function
used in statistics.
The previous equation can also be written the following form:
The Basic Gaussian Plume Equation
2
2
2
2
2
)
(
2
exp
2 z
y
z
y
H
z
y
u
Q
c
2
2
2
2
2
exp
2
exp
2 z
y
z
y
H
z
y
u
Q
c
26. 26
Example 6.4
Q = 20 g/s of SO2 at Height H
u = 3 m/s,
At a distance of 1 km, σy = 30 m, σz = 20 m (given)
Required: (at x = 1 km)
a) SO2 concentration at the center line of the plume
b) SO2 concentration at a point 60 m to the side of and 20 m
below the centerline
2) Diffusion Models
27. 27
… solution of example 6.4
2) Diffusion Models
3
3
0
2
2
2
2
g/m
1770
g/m
00177
.
m)
m)(20
m/s)(30
(3
2
(g/s)
20
2
1
0
0,
centerline
At the
a)
2
)
(
2
exp
2
z
y
z
y
z
y
u
Q
c
e
z - H
y
H
z
y
u
Q
c
3
3
2
2
2
2
g/m
145
)
0818
.
0
)(
g/m
(1770
)
20
(
2
)
20
(
)
30
(
2
60
exp
m)
m)(20
m/s)(30
(3
2
(g/s)
20
m
20
-
m,
0
6
CL
the
below
m
20
and
side
the
to
60m
point
a
At
b)
c
z - H
y
28. 28
What about σy and σz? (Dispersion coefficients)
σy ≠ σz ►► Spreading in the two directions are not equal
Most often σy > σz ►► Elliptical contour concentration at a
given x .
Symmetry is disturbed near the ground.
To determine σy > σz , use figures 6.7 and 6.8
2) Diffusion Models
31. 31
Notes on Figures 6.7 and 6.8
Both σy & σz are experimental quantities. The derivations of
equations 6.24 and 6.25 do not agree with reality.
We will only use figures 6.7 and 6.8 to find σy & σz.
Plotted from measurements over grasslands; i.e. not over
cities
However, we use them over cities as well since we have nothing
better
Measurements were made for x ≤ 1 km. Values beyond 1 km
have been extrapolated.
2) Diffusion Models
32. 32
What are the A to F categories?
A to F are levels of atmospheric stability (table 6.1).
Explanation:
For a clear & hot summer morning with low wind speed, the sun heats
the ground and the ground heats the air near it. Therefore air rises and
mixes pollutants well.
►► Unstable atmosphere and large σy & σz values
On a cloudless winter night, ground cools by radiation to outer space
and therefore cools the air near it. Hence, air forms an inversion layer.
►► Stable atmosphere and inhibiting the dispersion of pollutants
and therefore small σy & σz values
2) Diffusion Models
35. 35
Some Modifications of the Basic Gaussian Plume
Equation
a) The effect of the ground
b) Mixing height limits and one dimensional spreading
36. 36
a) The Effect of the Ground
Equation 6.27 assumes that the dispersion will continue
vertically even below the ground level! The truth is that
vertical spreading terminates at ground level.
To account for this termination of spreading at the ground
level, one can assume that a pollutant will reflect upward
when it reaches the ground
ground
from
upward
reflected
ion
concentrat
itself
plume
to
due
ion
concentrat
c
37. 37
… the Effect of the Ground
This method is equivalent to assuming that a mirror-image
plume exists below the ground.
The added new concentration due to the image plume uses
z+H instead of z – H . (draw the plume to check!)
2
2
2
2
2
2
2
2
2
exp
2
exp
2
2
exp
2
exp
2
z
y
z
y
z
y
z
y
H
z
y
u
Q
H
z
y
u
Q
c
2
2
2
2
2
2
2
exp
2
exp
2
exp
2 z
z
y
z
y
H
z
H
z
y
u
Q
c
38. 38
Example 6.6 (effect of ground)
Q = 20 g/s of SO2 at Height H
u = 3 m/s,
At a distance of 1 km, σy = 30 m, σz = 20 m (given)
Required: (at x = 1 km)
SO2 concentration at a point 60 m to the side of and 20 m
below the centerline: a) for H = 20 m
b) for H = 30 m
2) Diffusion Models
2
2
2
2
2
2
2
exp
2
exp
2
exp
2 z
z
y
z
y
H
z
H
z
y
u
Q
c
39. 39
… example 6.6
a) For H = 20 m i.e. the concentration at the ground level itself (z = 0)
(z – H)2 = (-H)2 = H2
(z + H)2 = (H)2 = H2
Therefore the answer will be exactly twice that in the 2nd part of example 6.4
c = (145 μg/m3) × 2 = 290 μg/m3 .
b) For H = 30 m
i.e. about 22 % greater than the basic plume equation (since the basic
plume eqn does not take ground reflection into account.
2
2
2
2
2
2
)
20
(
2
40
exp
)
20
(
2
20
exp
)
30
(
2
)
60
(
exp
20)
(
30)
(
3)
(
2
20
c
2
exp
2
1
exp
2
exp
00177
.
0
c
3
6
g/m
10
177
1
.
0
00177
.
0
135
.
0
605
.
0
135
.
0
00177
.
0
c
40. 40
Ground-Level Equation
Set z = 0 in the equation accounting for the ground effect:
This is the most widely used equation because it applies directly to the
problem of greatest practical interest, which is the ground-level
concentration.
2
2
2
2
2
2
2
exp
2
exp
2
exp
2 z
z
y
z
y
H
z
H
z
y
u
Q
c
2
2
2
2
2
2
2
exp
2
exp
2
exp
2 z
z
y
z
y
H
H
y
u
Q
c
2
2
2
2
2
exp
2
2
exp
2 z
y
z
y
H
y
u
Q
c
2
2
2
2
2
exp
2
exp
z
y
z
y
H
y
u
Q
c
This is the ground-level
modification of equation 6.27. It
takes reflection into account.
41. 41
Using Figure 6.9 to estimate Ground-Level
concentration
Figure 6.9 in your text book describes a way of finding the
concentration at the line on the ground directly under the
centerline of the plume.
→ z = 0 and y = 0
cu/Q can be plotted against “x” to obtain figure 6.9. Note that
the right-hand side depends on H, σy, and σz . Therefore, we
should have a group of H curves. Also figure 6.9 is for
category C stability only.
2
2
2
2
2
exp
2
exp
z
y
z
y
H
y
u
Q
c
2
2
2
exp
1
z
z
y
H
Q
cu
42. 42
Example 6.8 (using figure 6.9)
Q = 100 g/s at Height H = 50 m
u = 3 m/s, and stability category is C
At a distance of 1 km, σy = 30 m, σz = 20 m (given)
Required:
Estimate the ground-level concentrations directly below the
CL of the plume at distances of 0.2, 0.4, 0.5, 1, 5, 10 km
downwind
2) Diffusion Models
43. 43
… example 6.8 (using figure 6.9)
Using figure 6.9, we can read the values of cu/Q at each distance
The third column is obtained by multiplying the 2nd column by Q/u
It is obvious that one of the benefits of figure 6.9 is that one can
know the maximum ground level concentration & its distance
downwind by inspection only
2) Diffusion Models
Distance (km) cu/Q (m-2) c (μg/m3)
0.2 1.7×10-6 57
0.4 4.4×10-5 1467
0.5 5.3×10-5 1767
1 3.6×10-5 1200
5 2.7×10-6 83
10 7.8×10-7 24
44. 44
Plume Rise
This equation is only correct for the
dimensions shown.
Correction is needed for stability classes other than C:
→ For A and B classes: multiply the result by 1.1 or1.2
→ For D, E, and F classes: multiply the result by 0.8 or 0.9
2) Diffusion Models
s
a
s
s
T
T
T
PD
u
D
V
h 3
10
68
.
2
5
.
1
Δh = plum rise in m
Vs = stack exit velocity in m/s
D = stack diameter in m
u = wind speed in m/s
P = pressure in millibars
Ts = stack gas temperature in K
Ta = atmospheric temperate in K
46. 46
Multiple Cell Models
Complex simultaneous reaction rate expressions (Figure 1.2)
Multiple cell modeling is used. The Urban Airshed Model
UAM is an example this modeling type.
Model Description:
The airspace above the
city is divided into multiple
cells. Each cell is normally
from 2 to 5 km each way and
is treated separately from the
other.
Four or six layers in the
vertical direction, half below
the mixing height and half
above.
47. 47
How does the Model Work?
Mass balance for each cell. To start the simulation, we should
have initial distribution of pollutants.
The program calculates the change in concentration of the
pollutant for a time step (typically 3 to 6 minutes) by
numerically integrating the mass balance equation (eqn 6.1)
Complex computations requiring data on:
Wind velocity and direction
Emissions of the ground-level cells
Solar inputs
48. 48
…How does the Model Work?
The concentrations from the end of the previous time step are
used to first compute the changes in concentration due to
flows with the winds across the cell boundaries, and then
compute the changes due to chemical reactions in the cell.
These two results are combined to get the concentration in
each cell at the end of the time step.
Therefore, the model needs subprograms for the chemical
transformations during the time step in any cell and
subprograms for deposition of the pollutant from the ground-
level cells.
49. 49
…How does the Model Work?
Complex computations requiring data on:
Wind velocity and direction
Emissions of the ground-level cells
Solar inputs
The previous data are needed to simulate a day or a few days
in an urban area. What if such data are not available?
→The program has ways of estimating them. The following is a common
procedure:
Choose a day on which the measured pollutant concentration was the
maximum for the past year.
The model is run using the historical record of the wind speeds and
directions, solar inputs, and estimated emissions for that day.
The model’s adjustable parameters are modified until the calculated
concentrations match well with the observed ambient concentrations
for that day.
50. 50
…How does the Model Work?
Then the model is re-run with different emission rates
corresponding to proposed (or anticipated) future situations
and the meteorology for that day.
In this way the model performs a prediction of the worst day
situation under the proposed future emission pattern.
51. 51
Receptor-Oriented Models
The previous models are called source-oriented models. We use
the best estimates of the emission rates of various sources and
the best estimate of the meteorology to estimate the
concentration of various pollutants at various downwind points.
In receptor-oriented models, one examines the pollutants
collected at one or more monitoring sites, and from a detailed
analysis of what is collected attempts to determine which sources
contributed to the concentration at that receptor. This source
differentiation is not an easy process; for example:
If the pollutant is chemically uniform (e.g. CO, O3, SO2), then there is no
way to distinguish between sources.
If the pollutant is not chemically uniform; i.e. consisting of variety of
chemicals within the pollutant itself (e.g. TSP, PM10, PM2.5), one can
analyze their chemical composition and make some inferences about the
sources. (Aluminum and silicon example in page 148)
52. 52
Receptor-Oriented Models
When results of both types disagree significantly, we tend to
believe the receptor-oriented model because we have more
confidence in chemical distribution data than we have in the
meteorological data.
If the goal is to estimate the effects of proposed new sources
(e.g. for permitting issues), source oriented models are used.
Receptor-oriented models cannot be used in such cases.
Therefore receptor-oriented models are mostly used to
test the estimates made by source-oriented models
Simultaneously test the accuracy of the emissions estimates that are
used in source-oriented models
53. 53
Building Wakes & Aerodynamic Downwash
When the wind flows over the building, a plume may get sucked
and trapped into low-pressure wake behind the building. This
will lead to high local concentration.
A simple rule of thumb for avoiding this problem is to make
the stack height at least 2.5 times the height of the tallest
nearby building.
Another simple rule of thumb:
downwash unlikely to be a problem if:
hs hb + 1.5 Lb
hs : stack height
hb : building height
Lb : the lesser of either building height or maximum
projected building width.
57. Structure Influence Zone (SIZ): For downwash analyses
with direction-specific building dimensions, wake effects
are assumed to occur if the stack is within a rectangle
composed of two lines perpendicular to the wind direction,
one at 5L downwind of the building and the other at 2L
upwind of the building, and by two lines parallel to the
wind direction, each at 0.5L away from each side of the
building, as shown below. L is the lesser of the height or
projected width. This rectangular area has been termed a
Structure Influence Zone (SIZ). Any stack within the
SIZ for any wind direction shall be included in the
modeling.
57
59. For US EPA regulatory applications, a building is considered
sufficiently close to a stack to cause wake effects when the
distance between the stack and the nearest part of the
building is less than or equal to five (5) times the lesser of
the building height or the projected width of the building.
Distancestack-bldg <= 5L
59
60. Figure 4.6: GEP 360° 5L and Structure Influence Zone (SIZ) Areas of
Influence (after U.S. EPA).
60
