Page 354 - Air pollution and greenhouse gases from basic concepts to engineering applications for air emission control

332 11 Air Dispersion

• steady state ( oC ¼ 0)
ot
• x-direction transport by wind is much greater than that by eddy diffusion
2
o C
oC
(u D x ox 2 )
ox
Then Eq. (11.28) is further simplified as
2
2
oC o C o C
u ¼ D y þ D z ð11:29Þ
ox oy 2 oz 2
Integrating Eq. (11.29) with the following boundary conditions
C ¼ 0as x; y; z !1
C !1 at x; y; z ! 0
oC
D z ¼ 0at z ! 0 and x; y [ 0 ðwall boundaryÞ
oz
Z 1 1
Z
uCðy; zÞdzdy ¼ _ m at x [ 0 ðconservation of massÞ
1 0
we can get the air pollutant concentration at any point (x, y, z) as

2 2
_ m u y z
Cðx; y; zÞ¼ p ffiffiffiffiffiffiffiffiffiffiffi exp þ ð11:30Þ
2px D y D z 4x D y D z
If we define

2 2xD y 2 2xD z
r ¼ and r ¼ ð11:31Þ
y z
u u
where r y and r z are the dispersion coefficients in the transverse (y) and vertical (z)
direction, respectively.
Equation (11.30) can be rearranged as

" #
_ m 1 y 2 z 2
ð
Cx; y; zÞ ¼ exp ð11:32Þ
2pr y r z u 2r 2 2r 2
y z
In the preceding analysis, we have assumed the source of emission is at the
origin of the coordinate (z = 0). In reality, the actual emission source is at z =H,
therefore Eq. (11.32) shall be corrected as

" 2 #
_ m 1 y 2 ð z HÞ
ð
Cx; y; zÞ ¼ exp 2 2 ð11:33Þ
2pr y r z u 2r 2r
y z