M E 433
Professor John M. Cimbala
Lecture 17
Today, we will:
•
Continue discussing the Gaussian plume model – prediction of hazardous area downwind
of a plume, what to do about temperature inversions
Example: Gaussian plume
Given:
A buoyant plume emitting air pollution, under the following conditions:
Stack height = 80 m
•
Buoyant plume rise = 40 m above stack exit
•
Clear summer day with Sun high in the sky (early afternoon)
•
The ground reflects (does not absorb) the air pollutant
•
Average wind speed = 5.5 m/s
•
The stack emits the air pollutant at a rate of 110 g/s
To do:
Calculate the downwind location that has the maximum ground concentration.
Solution:
•
•
• First, use Table 20.1 to determine the atmospheric stability class:
At U = 5.5 m/s in the daytime with strong incoming solar radiation, this is Class C.
• Next, use Table 20.2 to obtain the coefficients for
• calculation of dispersion coefficients:
For Class C, we have a = 104, b = 0.894, c = 61.0, d = 0.911, and f = 0.
• At a given x location, calculate the dispersion coefficients:
cx d + f , with x in units of km and σy and σz in units of m.
σ y = axb , σ=
z
• The effective stack height is H = hs + δh = 80 + 40 = 120 m.
• Use the reflecting ground Gaussian plume equation at y = 0 (centerline) and z = 0
(ground) to calculate the maximum ground concentration at various values of x:
2
2 
2 

 
   2 
m j ,s 
z + H   
 1  y   z − H   
 1 y
=
cj
exp −   + 
  + exp −   + 
 
 

2p U ssssss
2
2  y  




  
y z
y 
z
z




 
 


Table to be filled in during class: Note: U = 5.5 m/s, H = 120 m, and m j ,s = 110 g/s .
x (km)
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
cj (µg/m3)
x (km)
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
cj (µg/m3)
Filled table (from Excel, reflecting ground):
Note: U = 5.5 m/s, H = 120 m, and m j ,s = 110 g/s .
x (km)
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
cj (µg/m3)
18.6
82.1
144.9
180.2
191.7
188.9
179.0
166.1
Plot of ground concentration:
x (km)
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
cj (µg/m3)
152.6
139.6
127.4
116.3
106.4
97.4
89.5
82.4
Summary of Gaussian plume model for an absorbing ground with an elevated temperature
inversion: We added an image source, mirror imaged about the bottom of the elevated
temperature inversion:
z
Bottom of elevated temperature inversion
H
HT
x
So, we add this image source to our Gaussian plume equation for an absorbing ground. After
some algebra [using the mathematical fact that ea+b = eaeb ], we get:
Gaussian plume model for an absorbing ground with an elevated temperature inversion
where the bottom of the inversion (the “lid”) is at z = HT:
 1  y 2  
 1  z − ( 2 H − H ) 2  
 1  z − H 2 
m j ,s

T
 + exp  − 
exp  −    exp  − 
cj
=
 

  
2p U sssss
2
2
2




y z
z
z
 
  
 
 
  y   
Next, consider the same case but with a reflecting ground: