Page 365 - Air pollution and greenhouse gases from basic concepts to engineering applications for air emission control

11.3 Gaussian-Plume Dispersion Models 343

To take both mixing height and the ground surface effects into consideration, the
term for vertical dispersion is further refined in the Gaussian dispersion model and it
leads to

" #
_ m y 2
Cx; y; zÞ ¼ exp
ð
2pr y r z u 2r 2 y
ð11:54Þ
( " # " #)
j¼1 2 2
X ð z H þ 2jz mix Þ ð z þ H þ jz mix Þ
exp þ exp
2r 2 2r 2
j¼ 1 z z
where z mix ¼ mixing height; in practice, Eq. (11.54) can be limited to j ¼ 1; 0; þ1
for values from r z to z mix . When r z [ z mix , the plume reflects between the mixing
height and the ground surface. After a few reflections, the air can be considered as
completely mixed along vertical direction. Then we can estimate the perfectly
mixed concentration by

" #
_ m y 2
Cx; yÞ ¼ p ffiffiffiffiffiffi exp 2 for r z [ z mix : ð11:55Þ
ð
2pr y z mix u 2r y

11.4 Gaussian Puff Models

We have assumed in the Gaussian models that the wind speed is constant (or used
average wind speed) and the emission source is continuous and steady. When the
wind is variable or the emissions source is not steady, we have to employ another
type of model for air dispersion analysis. It is called Gaussian Puff Model. Unlike
the plume models, puff models work for low wind speed conditions. They also can
handle the change of wind directions over time by coupling with wind rose.
The Gaussian puff models can be derived in a Lagrangian frame reference, a
frame that is attached to the center for the puff. Consider a cubic control volume
along the center of the plume (z ¼ H). Equation (11.25x) becomes
oC oC

DJ x ¼ D x D x DyDz ð11:56Þ
ox ox
x xþDx
Equations (11.25y) and (11.25z) remain the same. With Eqs. (11.25y), (11.25z),
and (11.56), Eq. (11.28) becomes
2
2
2
oC o C o C o C
¼ D x þ D y þ D z ð11:57Þ
ot ox 2 oy 2 oz 2