Page 344 - Fundamentals of Air Pollution 3E

III. Estimating Concentrations from Point Sources 299

This infinite series converges rapidly, and evaluation with N varying from
- 4 to + 4 is usually sufficient. These equations are used when evaluating
by computer, as the series # 3 can easily be evaluated.
When estimates are being made by hand calculations, Eq. (19-2) is fre-
quently applied until <r z = 0.8L. This will cause an inflection point in a
plot of concentrations with distance.
By adding Eq. (19-4), which includes multiple eddy reflections, and
changing the criteria for the use of Eq. (19-3) to situations in which <j z is
evaluated as being greater than 1.6L, a smooth transition to uniform mixing,
Eq. (19-3), is achieved regardless of source or receptor height. By differenti-
ating Eq. (19-2) and setting it equal to zero, an equation for maximum
concentration can be derived:

and the distance to maximum concentration is at the distance where
5
cr 2 = H/(2)° , This equation is strictly correct only if the cr 2/cr y ratio is con-
stant with distance.

B. Alternate Coordinate Systems for the Gaussian Equations
For estimating concentrations from more than one source, it is convenient
to use map coordinates for locations. Gifford (4) has pointed out that the
resulting calculated concentration is the same whether the preceding axis
system is used or whether an origin is placed at the ground beneath the
receptor, with the #-axis oriented upwind, the z-axis remaining vertical,
and the y-axis crosswind.
This latter axis system is convenient in assessing the total concentration
at a receptor from more than one source provided that the wind direction
can be assumed to be the same over the area containing the receptor and
the sources of interest.
Given an east-north coordinate system (R, S) the upward distance x and
the crosswind distance y of a point source from a receptor are given by

where R p/ S p are the coordinates of the point source; R r, S r are the coordi-
nates of the receptor; and 0 is the wind direction (the direction from which
the wind blows). The units of x and y will be the same as those of the
coordinate system R, S. In order to determine plume dispersion parameters,
distances must be in kilometers or meters. A conversion may be required
to convert x and y above to the appropriate units.