Figure 3: Typical Velocity, Temperature Profile, and Plume Shapes.
The general equation to calculate the steady
state concentration of an air contaminant in the ambient air resulting
from a point source is given by:
where
c(x,y,z) = contaminant concentration at the
specified coordinate [ML^{3}],
x = downwind distance [L],
y = crosswind distance [L],
z = vertical distance above ground [L],
Q = contaminant emission rate [MT^{1}],
sy
= lateral dispersion coefficient function [L],
sz
=
vertical dispersion coefficient function [L],
u = wind velocity in downwind direction [L
T^{1}],
H = effective stack height [L].
The effective stack height, H, is equal to the physical stack height
plus the plume rise. The plume rise under different conditions can be obtained
by empirical equations derived by many researchers and the choice is based
on the prevailing conditions at the site. Plume rise calculations
reported in this chapter primarily use Brigg's equation.
In the above equation sy
, the lateral dispersion coefficient function and sz
, the vertical dispersion coefficient functions depend on the downwind
distance and the atmospheric stability class. These coefficients in meters
can be obtained from the equations utilized by the Industrial Source Complex
(ISC) Dispersion Model developed by USEPA (1995):
sy = 465.11628 (x) tan(TH)
where
TH = 0.01745 [ c  (d) ln (x)]
sz = a x^{b}
Table 1: Constants a,b,c,d depend on Pasquill Stability categories
defined by Turner (1995).
Surface wind speed at
10 m (m/s) 
Day 
Night 
Incoming Solar radiation 
Cloud Cover 
Strong 
Moderate 
Slight 
Thinly Overcast 
Mostly Cloudy 
< 2 
A (s = 1) 
AB 
B (s = 2) 


23 
AB 
B 
C (s = 3) 
E (s = 5) 
F (s = 6) 
35 
B 
BC 
C 
D 
E 
56 
C 
CD 
D (s = 4) 
D 
D 
>6 
C 
D 
D 
D 
D 
The values for these constants for various stability classes are reported
by USEPA. This concept is covered in most undergraduate classes on Air
Pollution and students are expected to master this model. Teaching modules
described in this chapter were developed on this concept and applications
for different scenario were included.
Dispersion coefficients can also be obtained using PasquillGiffordTurner
estimates shown in the equations below.
where
s = an integer [16] representing the atmospheric stability shown in
Table 1
kx,x = empirical constants, values for each of
the stability class can be obtained from Green et al. (1960).
Vertical and crosswind dispersion coefficients as functions of downwind
distances for different stability classes are shown Figures 4 and 5.
Figure 4: Vertical Dispersion Coefficient as a Function of Downwind
Distance from Source.
Figure 5: Horizontal Dispersion Coefficient as a Function of Downwind
Distance from Source.
A Gaussian dispersion example is for both cases with/without reflection.