1. Air Pollution Control EENV 4313
Chapter 6
Air Pollutant Concentration Models

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. 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. Three types of Models (in this chapter)
Fixed-Box Models
Diffusion Models
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. 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 De Nevers

6. 1) Fixed-Box Models Assumptions:
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:
Rectangular city. W and L are the dimensions, with one side parallel to the wind direction.
Complete mixing of pollutants up to the mixing height H. No mixing above this height.
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).
The wind blows in the x direction with velocity u , which is constant and independent of time, location, & elevation.

7. … Assumptions
The concentration of pollutant in the air entering the city is constant and is equal to b (for background concentration).
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.
No pollutant enters or leaves through the top of the box, nor through the sides.
No destruction rate (pollutant is sufficiently long-lived)

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. 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

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

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. … 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. 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. …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

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. …example 6.2 cont. For u = 6 m/s and H = 1000 m →
Note that L is now 5km, not 15km
For u = 6 m/s and H = 1000 m →

17. Graphical Representation of the Fixed-Box Model Equation (Fig. 6
Graphical Representation of the Fixed-Box Model Equation (Fig. 6.2 in your textbook)
Slope = (L/uH)
Ambient air concentration, c
Emission rate, q, g/s.km2

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 q1 q2 c2 c1 b

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 q1 q2 c2 c1 b

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 q1 q2 c2 c1 b

21. 2) Diffusion Models
Called as diffusion models. However, they are actually dispersion models.
Such models usually use the Gaussian plume idea.
Fig De Nevers

22. Problem Statement
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.

23. Description of Situation in Fig. 6.3
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?

24. … cont. Description of Situation in Fig. 6.3
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

25. The Basic Gaussian Plume Equation
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:

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)
SO2 concentration at the center line of the plume
SO2 concentration at a point 60 m to the side of and 20 m below the centerline
2) Diffusion Models

27. … solution of example 6.4
2) Diffusion Models

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

29. Horizontal dispersion coefficient
Figure 6.7 De Nevers

30. Vertical dispersion coefficient
Figure 6.8 De Nevers

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. 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

33. Stability Classes
Table 3-1 Wark, Warner & Davis
Table 6-1 de Nevers

34. Example 6.5

35. Some Modifications of the Basic Gaussian Plume Equation
The effect of the ground
Mixing height limits and one dimensional spreading

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

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!)

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

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.

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.
This is the ground-level modification of equation It takes reflection into account.

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.

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. … 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
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
2) Diffusion Models