Environmental Transport and Fate
Chapter 8
–
Smokestack plumes
Benoit Cushman-Roisin
Thayer School of Engineering
Dartmouth College
A bit of history…
We tend to entertain romantic ideas of pre-industrial life as somehow healthier and
more environmentally conscious than our own today. In those days, nothing was
more valuable to the survival of the human race than the use of fire for warmth,
protection and early industry. But the same fire could do serious damage to human
health and the environment.
Until the invention of the chimney in Medieval
Europe in the 12th century, people had to
breathe the emissions from their own hearths.
The smoke from their indoor fires, although
vented through a roof opening, blackened the
inside of their homes with soot and presumably
the inside of their lungs, too.
For those people living in the plains and arid regions who lacked wood, animal dung
was the only
y source of fuel, adding
g disease vectors and odor p
problems to already
y
harsh living conditions. Living by the open flame was hardly idyllic.
Once the chimney was invented, its use became gradually universal,
and by the 18th century, once the industrial revolution got underway,
chimneys were common features at factories, mills and forges.
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Basic plume types
A morning coning plume
another coning plume
2
A coning plume in
Hanover, New Hampshire
A looping plume
(Scorer, 1997, page 390)
3
A looping plume
Plume from tannery in medina quarter of Fez, Morocco
Photographs taken on the evening of 4 April 2008
(Photographs courtesy of Betsy Dain-Owens, ENGS-43 student in Winter 2008)
Looping plume in Hanover, New Hampshire
(Photo by Benoit Cushman-Roisin)
4
A fanning plume
5
Examples of fumigation
Juarez Power Plant
http://windowoutdoors.com/FateXport/FateXport.html
Over Antarctica
photo by J. Dana Hrubes
dhrubes.home.att.net/march-05.html
Unknown location
http://www.cmar.csiro.au/airquality/index.html
The Gaussian model for smokestack plumes
It is assumed that the
structure of the plume is
Gaussian (bell curve) in both
cross-wind and vertical
directions.
IIn the
h d
downwind
i d di
direction,
i
a
highly advective situation is
assumed.
Note that the plume is
considered as originating at a
height H that may be greater
than the physical stack height
h, because of a possible
buoyancy rise.
(Masters, 1997, page 408)
6
Question:
How Gaussian is the
plume structure, really?
Snapshots reveal
comple shapes
complex
shapes.
But time averages over turbulent
fluctuations show much simpler
structures, with evident single
maximum and smooth tapering
away from it.
Question:
Is this tapering looking Gaussian?
Let’s plot to check…
7
In the vertical
In the horizontal
cross-wind direction
Looks pretty
Gaussian to me.
Don’t you agree?
With Gaussian distributions as the outcome, we view the problem as one of diffusion,
in three dimensions:
c
c
 2c
 2c
 2c
u
 Dx 2  D y 2  Dz 2  K c
t
x
x
y
z
steady
state
highly
advective
no
decay
Note that, although the problem is 3D, diffusion is only acting in 2D.
With x turned into travel time t = x/u, the solution is
c(t , y, z ) 

M
y2
z 2 
exp 

 4D t 4D t 
4 D y t 4 Dz t
y
z 

8
(Masters, 1997, p
page 407)
z=H
z=0
Then, we mind the impermeable ground surface by adding an image below ground.
With z = 0 at ground level, the actual source is at z = +H and the image is at z = –H:
c(t , y, z ) 

 ( z  H ) 2 
M
y 2    ( z  H ) 2 
  exp

exp 
exp 

 4D t 
4 Dz t 
4 D y t 4 Dz t
y 

 4 Dz t  

We are interested only in the ground-level concentration and set therefore z to zero:
cground (t , y )  c(t , y,0) 

2M
y2
H 2 
exp 

 4D t 4D t 
4 D y t 4 Dz t
y
z 

It is customary in this analysis to use  values instead of Dy and Dz values, because it
turns out that the latter ones are not constant (they tend to grow with the size of the
plume and to be affected by buoyancy and atmospheric conditions). So, we define
 y  2Dy t
 4 D y t  2 y2
 z  2 Dz t
 4 Dz t  2 z2
Both are functions of t and hence of x. The solution is now expressed as
cground ( x, y ) 
 y2
H2 
exp  2  2 


2  y  z
 2 y 2 z 
2M
It remains to determine the value of M.
9
M
amount
mass
mass / time
emission rate S




missing dimension x  length x  length / time
wind speed
u
Sketch shows how the wind acts as a diluting mechanism.
It makes sense therefore that the wind speed u should be in the denominator.
The solution now takes the form:
cground ( x , y ) 
 y2
H2

exp  
2
 2
 u y  z
2 z2
y

S




in which
cground = ground concentration distribution (in mg/m3)
S = emission rate from the smokestack (in mg/s)
u = wind speed at height H (in m/s)
y = horizontal transverse dispersion coefficient (in m)
z = vertical dispersion coefficient (in m)
[Both y and z are functions of downwind distance x.]
y = cross-wind distance (in m)
[Take y = 0 for downwind direction]
H = h + h = effective stack height (in m)
h = physical stack height (in m)
with
h = height adjustment (in m)
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Some of the values depend on the state of the atmosphere.
Traditional classification of common atmospheric conditions (Turner, 1970)
Surface
wind speed
( m/s)
Day
solar
strong
moderate
Night
cloudiness
radiation
slight
overcast
cloudy
clear
D
D
E
F
D
D
E
F
C
D
D
D
E
C–D
D
D
D
D
D
D
D
D
D
D
D
<2
A
A–B
B
2–3
A–B
B
C
3–5
B
B–C
5–6
C
>6
C
A = very unstable
B = moderately unstable
C = slightly unstable
D = neutral
E = slightly stable
F = stable
overcast
Notes:
- Surface wind is measured 10 m above ground.
- A “cloudy night” is one with more than half cloud cover.
- A “clear night” is one with less than half cloud cover.
Wind velocity profile and rotation
u( z  H )
 H 


u ( z  10 m)  10 m 
Stability class
A
B
C
D
E
F
p
Exponent p
0.15
0.15
0.20
0.25
0.40
0.60
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Pasquill curves to obtain the pair of dispersion coefficients
(Source: Masters, 1997, page 412)
If you don’t want to use the graphs (ex. in creating a Matlab code), you may rely on:
 y  a x 0.894
 z  c xd  f
with x in kilometers and  values obtained in meters
12
Or, if you are lazy and want a value from a table…
We have yet to determine the effective smokestack height H, the height at which the
plume appears to be originating.
H  h  h
h  buoyancy
y y rise  ppossible downwash
 hb  hd
Downwash occurs in strong wind (u) and with weak gas
ejection velocity (ws) and is caused by low pressure in the
wake of the smokestack.
Rule:

w 
hd  4 r 1.5  s 
u
(
H)

if ws < 1.5 u(H) and
in which r = inner stack radius
13
A way to avoid downwash
Helix on outside of the
stack forces wind to rise
upon approaching stack
To determine the buoyancy rise hd (more common than downwash), we first need
to calculate the buoyancy flux F:

T 
F  g r 2 ws 1  air 
 T

ffumes 

in which
F = buoyancy flux (in m4/s3)
g = 9.81 m/s2, the earth’s gravitational acceleration
r = inner stack radius at tip (in m)
ws = fumes vertical ejection velocity (in m/s)
Tair = absolute ambient temperature (in K), at stack height
Tfumes = absolute temperature of gas fumes (in K)
(Recall: Absolute temperature in degree Kelvin = temperature in oC + 273.15)
14
Then, we need to distinguish whether the plume is “bent-over” or “vertical”.
1) Bent-over plume:
for stability classes A, B, C and D (unstable and neutral states)
If
F  55 m 4 / s 3
then
x f  49 F 5 / 8
If
F  55 m 4 / s 3
then
x f  119 F 2 / 5
hb  1.6
F 1/ 3 x 2f / 3
u
xf
Distance over which plume rises
2) Vertical plume:
for stability classes E and F only (stable states)
N2 
g  dTair

 ,

Tair  dz

with  
g
Cp
1/ 4
If u  0.275 ( FN )1/ 4
 F 
then hb  4.0  3 
N 
If u  0.275 ( FN )1/ 4
 F 
then hb  2.6  2 
N u
1/ 3
15
Vertical plume in Hanover on a
cold winter morning
For y = 0
(downwind direction)
(Source: Masters,
1997, page 416)
16
Graph to determine maximum ground concentration and its distance from the stack
3
(Source: Masters, 1997, page 417)
9 x 10-6
Capping by inversion
First determine the distance xL over which the capping inversion is reached:
XL
then
is such that  z  0.47 ( L  H ) at
c( x, y  0) 
S
2 u  y L
for
x  XL
x  2 XL
17
Watch out for the sea breeze!
Danger of fumigation in a sea-breeze
18
Downdraft in wake of building
Kuwait oil fires of 1st Gulf War (1991)
About 600 naturally pressurized oil wells
were set on fire in Kuwait by Saddam
Hussein’s retreating army in late
February 1991, injecting massive
quantities of smoke, unburned
hydrocarbons, sulfur dioxide and
nitrogen oxides into the atmosphere.
19
Note how single plumes merge to
make super-plumes.
These profiles give an idea of how high the
pollution reached in the atmosphere.
http://materialstechnology.tms.org/edu/article.aspx?articleID=2535
THE END
http://www.legendsofnasca
ar.com/HudsonSmokeStack.jpg
http://buckfifty.org/2009/05/12/sunday-february-26-1950/
http://www.panoramio.com/photo/25258369
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