null  null
Chapter 8
LOCAL AIR POLLUTION
8.1
Introduction
Air pollution affects humans more than water pollution. Whereas we can always
treat the water before we drink it or use it, the air that we breathe must be
clean where we happen to be. Also, air quality tends to be worse in large cities
where there are more people and also more traffic and other unnatural emissions
to the atmosphere. Figures 8-1 and 8-2 show pictures of two nightmares of the
1950s, each in a large city.
In the face of such terrible situations as those in the 1950s, numerous governments have since responded by legislating air-quality standards, and the
result is that air quality is noticeably higher today than it was a few decades
ago. In the United States, Congress enacted the Clean Air Act with a series of subsequent amendments (in 1963, 1966, 1970, 1977 and 1990). Most
of the early regulations concerned point sources, such as smokestack emissions
from power plants, but with time regulations have been gradually extended to
encompass distributed sources, chiefly road traffic and, more recently, airport
emissions.
In the United States, air-quality legislation sets both emission standards and
National Ambient Air Quality Standards (NAAQS). Therefore, both emissions
(= source) and the air we breathe (= receptor) are regulated. The NAAQS are
divided into two categories: the toxic substances, for which extremely low levels
are tolerated, and six so-called criteria pollutants, which are chemicals that are
not immediately harmful except past a certain concentration and/or after a
certain time of exposure. These six criteria pollutants and their corresponding
standards are listed in Table 8-1.
181
182
CHAPTER 8. AIR POLLUTION
Figure 8-1. The Crystal Palace television mast protruding through the top
of the so-called “Great London Smog” of December 1952. [Courtesy Fairey Air
Surveys.]
Pollutant
CO
Carbon monoxide
Exposure duration
1 hour
Standard
35 ppm
Cause for concern
headaches
asphyxiation
decreased exercise tolerance
angina pectoris
8 hours
9 ppm
NO2
Nitrogen dioxide
1 year
0.053 ppm
aggravation of
respiratory disease
SO2
Sulfur dioxide
3 hours
1 day
1 year
0.50 ppm
0.14 ppm
0.03 ppm
shortness of breath
wheezing, odor
acid precipitation
damage to vegetation
O3
Ozone
1 hour
8 hours
0.12 ppm
0.08 ppm
eye irritation
interference with breathing
damage to materials and plants
Pb
Lead
3 months
1.5 µg/m3
blood poisoning
infant development
PM2.5
24 hours
1 year
65 µg/m3
15 µg/m3
lung damage
PM10
24 hours
1 year
150 µg/m3
50 µg/m3
visibility
respiratory disease
Table 8-1. The six chemicals designated as criteria pollutants by the US
Environmental Protection Agency and the corresponding National Ambient
Air Quality Standards.
183
8.1. INTRODUCTION
Figure 8-2. Riders of a local message-delivery company in Los Angeles being
outfitted with protective gas masks in the fall of 1955.
For particulate matter, one currently distinguishes between particles larger
than 10 µm (noted PM10 ), which mostly impact visibility, and particles larger
than 2.5 µm (noted PM2.5 ), which create respiratory disorders.
In atmospheric pollution, one needs also to distinguish between primary
pollutants, those directly emitted, and secondary pollutants, those chemically
created in the atmosphere from reactions among other pollutants. One example
is the conversion of sulfur dioxide (SO2 ) into sulfuric acid (H2 SO4 ), which
causes acid rain. Ground-level ozone (O3 ) is another example of a secondary
pollutant: There is virtually no emission of ozone to the atmosphere, but ozone
can be generated from nitrogen oxides in the process called photochemical fog,
for which the key reactions are an excitation by sunlight, the formation of
ozone, and a subsequent relaxation to the initial nitrogen oxide, according to:
NO2
+
O
NO
sunlight
−→
NO
+
+
O2
−→
O3
+
O3
−→
NO2
+
O
O2
Problems related to pollutant transport and diffusion in the atmosphere
exhibit different characteristics depending on their length and time scales. In
some broad way, we can distinguish five types in order of increasing scale, as
shown in Table 8-2.
184
CHAPTER 8. AIR POLLUTION
Type
Local
Example
plume from a
factory smokestack
Scales
up to 1 km in horizontal
several 10 m in vertical
fraction of 1 hour
Physics involved
near-ground air layer
subject to surface
roughness and convection
Urban
smog over
Los Angeles
several km in horizontal
several 100 m in vertical
several hours to days
local winds,
atmospheric boundary layer,
hills and mountains
Regional
acid rain
several 100 km in horizontal
10 km in vertical
several days
weather patterns,
cloud patterns
Continental
1986 Chernobyl
radioactive fallout
size of a continent
troposphere
days to weeks
weather patterns
Global
climate change
due to
greenhouse gases
size of earth
troposphere and above
decades and beyond
prevailing winds,
equator-to-pole
gradients
Table 8-2. Classification of situations involving atmospheric transport and
fate according to length and time scales.
The present chapter deals with local forms of pollution.
8.2
Atmospheric Stability
Thermal stratification is often observed in the atmosphere, especially over the
diurnal cycle. When it occurs, stratification is characterized by colder, denser
air sinking below lighter, warmer air. This creates buoyancy forces, which form
an obstacle to turbulence and dispersion of contaminants. Whatever the air
contains by way of contaminants tends to remain at the level where it is instead
of being spread across the vertical. Because of this, it is important to have a
measure of stratification. This was done in Chapter 5 for lakes and reservoirs,
but it needs to be amended to take into consideration the compressibility of
the air in the atmosphere.
The neutral atmosphere
First, we need to understand the state of a well-mixed, neutral atmosphere.
This state is not trivial because the higher pressure on the ground compresses
the air more than the lower pressure aloft, leading to a decrease of density with
height that exists irrespectively of a thermal stratification.
To begin, let us consider a vertically stratified atmosphere with density ρ(z)
and at rest, and consider in it a small parcel contained between levels z and
z + dz, as depicted in Figure 8-3. This parcel is subject to pressure forces on
all sides from neighboring air parcels and to its own weight mg. If the crosssection of this parcel is A, its volume is Adz, its mass m = ρAdz and its weight
mg = ρgAdz.
185
8.2. PHYSICS
Figure 8-3. A small air parcel and the forces acting on it.
At equilibrium, the upward force from the supporting pressure below is
equal to the downward pressure at the top plus the weight:
pbelow A = pabove + mg,
which yields:
p(z + dz) − p(z) = − ρgdz.
In differential form, we write:
dp
= − ρg.
(8.1)
dz
In fluid mechanics, this relation is called the hydrostatic balance.
Next, we recall the equation of state. The atmosphere is a mixture of gases
(78% nitrogen, 21% oxygen and 1% other gases, in dry conditions), collectively
called air. From a physical point of view, the actual composition does not
matter, and within a high degree of accuracy, air can be considered as an ideal
gas, with a molecular weight intermediate between that of nitrogen and that of
oxygen. So, we can write the following equation linking its pressure p, density
ρ and temperature T :
p = RρT,
(8.2)
where T must be the absolute temperature (in degrees Kelvin), equal to the
temperature in ◦ C + 273.15. For air, the constant factor R is 287 J/(kg·K) =
287 m2 /s2 ·K.
If we take the z–derivative of this expression, we obtain:
dρ
dT
dp
= R
T + Rρ
.
(8.3)
dz
dz
dz
The next and final statement is more subtle. Let us seek the property
of a neutral atmosphere, namely one in which air parcels can be exchanged
among one another without altering the vertical profile of properties. If a
parcel of mass m is moved from level z to a higher level z + dz, its pressure
186
CHAPTER 8. AIR POLLUTION
drops and, according to (8.2), so must also its density or temperature or both.
Under a decompression, the parcel expands making its pressure do work against
surrounding parcels. This work, equal to pressure times change in volume
(pdV ), is spent at a cost to the internal energy (mCv T ) of the parcel. Thus,
the internal energy of the parcel drops according to:
mCv dT = − pdV.
Since density is mass per volume (ρ = m/V ), a division by m yields:
1
Cv dT = − pd
.
ρ
Dividing by the incremental height dz and effecting the derivative of 1/ρ, we
obtain:
Cv
p dρ
dT
= + 2
.
dz
ρ dz
(8.4)
The assumption that the air parcels are interchangeable allows us to consider the preceding changes of temperature and density as those in the vertical
in the atmosphere at rest. Equations (8.1), (8.3) and (8.4) form a 3–by–3 system for the three vertical derivatives dp/dz, dρ/dz and dT /dz. Solving for the
temperature gradient dT /dz, we obtain:
dT
= − g.
dz
Defining Cp = Cv + R for convenience, we arrive at:
(Cv + R)
g
dT
.
= −
dz
Cp
(8.5)
Thus, the temperature decreases with altitude at a constant rate. This
gradient
Γ =
g
,
Cp
(8.6)
is called by meteorologists the adiabatic lapse rate. Its value is (9.81 m/s2 )/(1005
m2 /s2 ·K) = 9.76 × 10−3 K/m, or about 1 degree for every 100 meters.
This drop of temperature with height explains why high mountains are
permanently covered with snow and why air is so cold at the altitude where jet
planes fly. A lower temperature causes a lower humidity saturation level, and
this is why clouds, which consist in fine water droplets, occur at some altitude
above the ground.
Thermal stratification
At this point, it is useful to derive again the expression for the natural
frequency of oscillations of a parcel in a stratified medium, now accounting
for the compressibility of the fluid. For this, consider an air parcel initially at
187
8.2. PHYSICS
level z with ambient temperature T (z) and displaced upward slightly by the
distance h. Once there, it is subject to a lesser pressure and expands (acquiring
a lower density) and loses temperature. The pressure difference is dp = −ρgh,
by virtue of (8.1), and the temperature drop is −gh/Cp , by virtue of (8.5).
If the atmosphere is not in a neutral state, then the ambient temperature
Ta = T (z + h) is not equal to the temperature Tp = T (z) + dT acquired by the
parcel. In other words, the parcel has adjusted its temperature according to
the adiabatic lapse rate but the ambient air does not follow the same rate of
temperature decrease with height. The difference is:
Ta − Tp
=
≃
g
T (z + h) − T (z) +
h.
Cp
g
dT
h.
+
dz
Cp
Because of this temperature difference, the displaced air parcel experiences
a buoyancy force not equal to its own weight and is subject to a net upward
force equal to:
F
=
=
buoyancy force − weight
ρa V g − ρp V g = (ρa − ρp )V g
p
p
Vg
−
RTa
RTp
p
(Tp − Ta )V g
RTa Tp
Tp − Ta
V g.
ρp
Ta
=
=
=
By virtue of Newton’s law, this force is equal to the mass times the acceleration:
d2 h
dt2
d2 h
ρp V 2
dt
m
=
F
=
ρp
Tp − Ta
V g.
Ta
Simplification of this last equation and replacement of the temperature difference by the preceding expression provide:
d2 h
dt2
=
=
Tp − Ta
Ta
dT
g
+ Γ h,
−
Ta
dz
g
(8.7)
188
CHAPTER 8. AIR POLLUTION
Figure 8-4. The various states of atmospheric stability according to the vertical variation of temperature.
where the adiabatic lapse rate Γ was defined earlier in (8.6).
The discussion now proceeds as for water in lakes. We define
g
dT
N2 =
+ Γ ,
T
dz
(8.8)
where T (z) is the ambient temperature profile, and conclude that the thermal
stratification is stable if the parcel is drawn back toward its original position,
which occurs when N 2 > 0, neutral if the parcel is at equilibrium at its new
location, which occurs when N 2 = 0, and unstable if the parcel moves further
away from its original position, which occurs when N 2 < 0. In sum, the
redefined N 2 above plays the same role as that defined in (5.6) for lake water.
Since the sign of N 2 depends on the deviation of the ambient temperature
gradient dT /dz from the adiabatic lapse rate Γ, we can state:
dT
dz
< −Γ
⇒
N 2 < 0 ⇒ Unstable atmosphere
dT
dz
= −Γ
⇒
N 2 = 0 ⇒ Neutral atmosphere
dT
dz
> −Γ
⇒
N 2 > 0 ⇒ Stable atmosphere.
The last case can be subdivided into two cases depending on the sign of the
temperature gradient:
−Γ <
dT
dz
≤ 0
⇒
Somewhat stable atmosphere
8.3. THE ATMOSPHERIC BOUNDARY LAYER
dT
dz
> 0
⇒
189
Very stable atmosphere.
In the first of these last two cases, the temperature decreases with height but
not as fast as the adiabatic lapse rate, while in the other, it actually increases
with height. This latter case is characterized by air so stable that little ventilation can take place, and, if it occurs near the ground, it creates an uncomfortable situation for people there, especially if there are sources of pollution.
Fortunately, such is not usually the case and, for that reason, meteorologists
have come to call this situation an inversion. The situation is recapitulated in
Figure 8-4.
8.3
The Atmospheric Boundary Layer
The diurnal cycle greatly affects the state of the lower atmosphere, with sunlight
heating the atmosphere during the day and cooling occurring at night. Because
sunlight consists mostly of radiation in the visible sprectrum and because air is
almost perfectly transparent, the solar radiation penetrates through the atmosphere with little absorption, and most of it is absorbed by the ground surface,
which is opaque. This delivers heat to the ground, which then re-radiates it in
longer, infrared wavelengths. Being relatively opaque to this other type of radiation, the atmosphere absorbs this second-hand radiation. In effect, therefore,
the atmosphere is heated from below, despite the fact that the sun is above!
Heating from below operates similarly to cooling from above, and we can
readily apply what we know of seasonal penetrative convection in lakes to
the daytime atmosphere: any thermal stratification is gradually eroded during
the hours of sun exposure. During the night, the ground surface cools in the
absence of sunlight, and this cooling from below creates a thermal stratification
that gradually extends upward. Figure 8-5 depicts a typical diurnal cycle of
the thermal stratification and convective mixing alternating between night and
day. As we can see, there periods of the day when the temperature profile with
height can exhibit or or two kinks.
The manner in which temperature varies with height determines zones of
stability and instability, which correspond to layers of active mixing and quiet
layers (Figure 8-6). Note that an air parcel acquiring speed in an unstable
zone, such as parcel 1 in Figure 8-6c will rise, with its temperature dropping
according to the adiabatic lapse rate, until it reaches ambient air of temperature
equal to itself, which may very well be inside a layer of stability. In other words,
a stable layer with thermal stratification can be partially eroded by the mixing
occurring across an adjacent zone of instability.
Mixing height
Because the heat source during the day is at ground level, an important
characteristic of the temperature profile is the vertical height over which an
unstable parcel of air taking off from the ground will rise. The vertical excursion
190
CHAPTER 8. AIR POLLUTION
Figure 8-5. Typical variation of the temperature profile over the diurnal cycle.
Nighttime cooling generates a progressively thicker inversion, which is gradually
eroded in the course of the following day. Sustained convection during the day
is usually able to break through the inversion built during the previous night,
leading to a neutral atmosphere in the late afternoon.
of such air parcel is called the mixing height. Graphically, it is constructed by
tracing a line on the temperature–altitude plot from the temperature value
at ground level upward with backward slope equal to the adiabatic lapse rate
until it intersects the ambient temperature profile, as shown in Figure 8-6c. For
pollution sources on the ground, such as road traffic, this height determines how
high the pollutants will mix at that time of day.
Because of the diurnal cycle (recall Figure 8-5), the mixing height varies
over the 24-hour period. It is inexistent during the night because of the ground
inversion, unless there is another source of heat, such as an urban area, then
develops starting at sunrise to increase steadily during the day hours. When
the sun sets, it collapses quite dramatically. An example is provided in Figure
8-7, which traces the diurnal evolution of the mixing height over the city of
Schenectady, in the state of New York, during the summer of 1994. By reaching
over 1500 m, this mixing height is relatively large, and the reason is that it was
measured in the heat of the summer. Generally, mixing heights reach several
hundred meters during the morning and may attain 1000 m at their peak in
late afternoon.
8.4
Smokestack Plumes
Various types of plumes
The state of the atmosphere in the lower 1000 meters or so greatly affects
the dispersion of emissions. A most important situation is that of smokestack
plumes. Depending on the stability or instability of the lower atmosphere, a
8.4. SMOKESTACK PLUMES
191
Figure 8-6. Layers of stability and instability in a compound temperature
profile and determination of the thickness of the mixed layer, also called mixing
height: (a) local stability determined from temperature gradient, (b) parcels 1
and 2 are on the move, and (c) the overshoot of the ground parcel 1 sets the
mixing height. [From Masters, 1997]
plume may behave very differently, as shown in Figure 8-8.
If the atmosphere is in a neutral state (Figure 8-8a), the plume gradually expands in the vertical direction, symmetrically upward and downward. It simultaneously expands laterally (in the direction transverse to the wind), although
that aspect is usually not perceived by an observer standing on the ground
some distance away and therefore looking sideways at the plume. Because it
forms an expanding cone, the plume is said to be coning. If the atmosphere is
unstable (Figure 8-8b), active convection sends thermals of warm air upward
while colder air descends. This activity disrupts the plume, alternatively raising
some sections and bringing down others. The result is a plume that meanders
in the vertical, and for this reason it said to be looping. Such situation is often
observed on warm summer afternoons. By contrast, if the atmosphere is stably
stratified, at least up to the level of the smokestack (Figure 8-8c), the low level
of turbulence prevents any significant growth of the plume in the vertical. The
192
CHAPTER 8. AIR POLLUTION
Figure 8-7. Mean and standard deviations of the mixing height averaged over
nine summer days with clear sky over Schenectady, New York. [From Berman
et al., 1997]
plume still expands in the horizontal direction transverse to the wind and is
therefore shallow but opening wide, like a fan, and is said to be fanning. Such
situation is most typical of winter mornings.
Combination of stable and unstable atmospheric conditions are possible and
cases of special interest arise when the level between stable and unstable regions (which varies through the day according to Figure 8-5) reaches the height
of the smokestack. During daytime, when new convection gradually erodes the
inversion built during the previous night (right panel of Figure 8-5), the situation is one with turbulence on the lower side and calm on the upper side.
In such unfortunate case, the emissions from the stack are drawn downward
far more effectively that they are taken aloft. The result is fumigation of the
ground (Figure 8-8d). This is a very unfavorable situation. During nighttime,
the opposite situation may occur, with an inversion below some neutral layer
remaining from the previous day (left panel of Figure 8-5). The plume is more
readily entrained upward than downward, and the result is a much healthier
situation with a lofting plume (Figure 8-8e).
The Gaussian model
The concentration distribution of a pollutant emitted by a smokestack can
be appropriately described by a diffusion model. The situation is highly advective because the Peclet number is high, an assertion to be verified a posteriori.
It may also be considered as steady, that is, when one considers an average
over the many turbulent fluctuations and assumes that the plume responds
relatively quickly to changes in the atmospheric structure. Turbulent diffusion
8.4. SMOKESTACK PLUMES
193
Figure 8-8. A selection of plume types according to ambient thermal stratification. [Adapted from Masters, 1997]
194
CHAPTER 8. AIR POLLUTION
Figure 8-9. How dilution increases proportionally to wind speed. [Adapted
from Stern et al., 1984]
occurs in both cross-wind and vertical directions, but at uneven paces. The diffusion is thus two-dimensional and anisotropic. The model equation governing
the distribution of the concentration c(x, y, z) of the pollutant is:
∂c
∂2c
∂2c
+ Dz
,
(8.9)
= Dy
2
∂x
∂y
∂z 2
where u is the wind velocity at the height of the smokestack, the x–direction
is taken downwind, y is the other horizontal direction, transverse to the wind,
z is vertical upward, and Dy and Dz are the diffusivities in the y– and z–
directions, respectively.
If the origin of the axes is taken on the ground at the foot of the smokestack,
then the emission source S (mass of pollutant emitted per time) is located at
x = 0, y = 0 and z = H, where H is the height of the smokestack. Because
the ground is an impermeable boundary, a virtual source must be imagined at
z = −H to negate the fux at z = 0. From what we learned in Chapter 2, the
solution is then:
u
c(x, y, z) =
M
y2
p
×
exp −
√
4Dy t
4πDy t 4πDz t
(z − H)2
(z + H)2
exp −
+ exp −
,
4Dz t
4Dz t
(8.10)
195
8.4. SMOKESTACK PLUMES
where time t is actually the travel time to distance x, namely t = x/u, and M
is the amount released per missing dimension, which is:
M =
amount released
amount released
time
S
=
×
=
. (8.11)
x − length
time
x − length
u
That M has the velocity u in the denominator is a result of dilution, as explained on Figure 8-9.
We are interested almost exclusively in pollution at ground level, where
people breathe. Thus, we set z to zero, to obtain:
y2
H2
2M
p
exp −
−
√
cground (x, y) =
4Dy t
4Dz t
4πDy t 4πDz t
Because observational data are typically captured in terms of standard deviations by fitting observed concentration sections to bell-curve distributions,
we eliminate the
z in favor of the corresponding standard dep use of Dy and D√
viations σy = 2Dy t and σz = 2Dz t, in keeping with (2.19). The revised
expression is, after a simplification by a factor 2 in the front fraction:
S
y2
H2
cground (x, y) =
exp −
−
.
(8.12)
πuσy σz
2σy2
2σz2
At this stage, the x variable hides inside the expressions of σy and σz ,
which are growing with distance from the origin. The question of how they
vary with the downwind distance x is a complicated one, and tradition has
it to rely here almost exclusively on empirical evidence. For this, the most
common atmospheric situations are classified in six categories, labeled from A
to F, according to Table 8-3.
Surface
wind
speed
(m/s)
<2
2–3
3–5
5–6
>6
Day
solar insolation
strong
A
A–B
B
C
C
moderate
A–B
B
B–C
C–D
D
slight
B
C
C
D
D
Night
cloudiness
overcast
D
D
D
D
D
overcast
D
D
D
D
D
cloudy
E
E
D
D
D
clear
F
F
E
D
D
Table 8-3. Classification of the most common atmospheric conditions, according to Turner (1970): A is very unstable, B moderately unstable, C slightly
unstable, D neutral, E slightly stable, and F stable. “Surface” wind speed is
measured 10 m above the ground. A “cloudy night” is one with more than half
cloud cover, whereas a “clear” night is one with less than half cloud cover.
The next step is to use the charts provided on Figure 8-10, which provide
the values of σy and σz as functions of the distance x from the base of the
196
CHAPTER 8. AIR POLLUTION
Figure 8-10. Horizontal and vertical standard deviations, σy (top panel)
and σz (bottom panel), as function of downwind distance x and atmospheric
stability class. [From Turner, 1970]
197
8.4. SMOKESTACK PLUMES
smokestack and of the atmospheric stability class. Alternatively, one may use
numbers from the Table 8-4.
x
(km)
0.2
0.4
0.6
0.8
1.0
2
4
8
16
20
A
51
94
135
174
213
396
736
1367
2540
3101
B
37
69
99
128
156
290
539
1001
1860
2271
σy
C
25
46
66
85
104
193
359
667
1240
1514
D
16
30
43
56
68
126
235
436
811
990
E
12
22
32
41
50
94
174
324
602
735
F
8
15
22
28
34
63
117
218
405
495
A
29
84
173
295
450
1953
B
20
40
63
86
110
234
498
1063
2274
2904
σz
C
D
14
9
26
15
38
21
50
27
61
31
115
51
216
78
406 117
763 173
934 196
E
6
11
15
18
22
34
51
70
95
104
F
4
7
9
12
14
22
32
42
55
59
Table 8-4. Standard deviations σy and σz (in meters) for the various atmospheric stability classes and for selected distances x (in kilometers) downwind
from the smokestack.
We are now in a position to confirm whether the initial assumption of a
highly advective situation was indeed correct. For this, we construct the Peclet
number [see Equation (2.63)], which consists in a ratio with three quantities.
For the velocity scale U , we take the wind speed u, for the length scale L
the arbitrary distance x, and for the horizontal diffusivity Dx the same value
as the cross-wind diffusivity Dy , since dispersion presumably occurs at the
same p
rate in both downwind and crosswind, horizontal directions. Then, with
σy = 2Dy t, we can write:
UL
ux
2utx
UL
=
= 2
= 2
=
Pe =
Dx
Dy
σy /2t
σy2
x
σy
2
,
in which we have also used the fact that ut = x. Thus, the estimation of the
Peclet number reduces to comparing the standard deviation σy to the distance
x. From the top panel of Figure 8-10 (or the numbers listed in Table 8-4),
we note that σy ranges between x/40 and x/4, which implies that the Peclet
number ranges between 32 and 3200. The Peclet number is therefore always
at least one order of magnitude greater than unity, and the situation can be
considered as highly advective.
Effective smokestack height
The gases emitted from a smokestack are typically hot and have therefore a
certain buoyancy that tends to raise the plume a certain height above the top
of the stack. In addition, it is not unusual to blow the gases at high speed to
further increase this height in order to reduce the ground effect further. This
creates the necessity to distinguish between the physical height h of the stack
from the level H from where the gases actually disperse. This height is called
the effective stack height and is to be computed from
198
CHAPTER 8. AIR POLLUTION
Figure 8-11. Difference between a bent-over plume and a vertical plume.
[Adapted from Lyons and Scott, 1990]
H = h + ∆h,
(8.13)
where ∆h is called the plume rise. In practice this can be a significant portion
of h, if not larger. Needless to say, it is the effective stack height H that needs
to be used in Equations (8.10)–(8.12).
To determine the plume rise, there is a series of steps. First, one determines
the buoyancy flux parameter F , which is defined as:
Ta
F = gr2 ws 1 −
,
Ts
(8.14)
where r is the inner radius of the stack at its exit section, ws the upward vertical
velocity with which the fumes blow out of the stack, Ta the ambient temperature at stack height (in absolute degrees Kelvin), and Ts the exit temperature
of the fumes (in absolute degrees Kelvin).
Next, one needs to distinguish among two types of plumes, as indicated in
Figure 8-11. A bent-over plume occurs during stability class A to D, and a
vertical plume during stability class E and F.
For a bent-over plume, one calculates the distance xf over which the plume
rises, using one of the following empirical formulas:
199
8.4. SMOKESTACK PLUMES
Figure 8-12. An example of downwash.
If F < 55 m4 /s3
4
If F ≥ 55 m /s
3
then
then
xf = 49 F 5/8
xf = 119 F
(8.15)
2/5
(8.16)
with xf obtained in meters when F is expressed in m4 /s3 . Once this distance
is determined, one can finally calculate the buoyancy rise
2/3
∆hb = 1.6
F 1/3 xf
,
(8.17)
u
which, unlike the previous formula, is a dimensionally consistent equation.
For a vertical plume, it is necessary to determine first the stratification
parameter N 2 , which was defined in (8.8):
dTa
g
+ Γ ,
N2 =
Ta
dz
before calculating the buoyancy rise as follows:
14
If u < 0.275(F N )1/4
then
∆hb = 4.0
F
N3
If u ≥ 0.275(F N )1/4
then
∆hb = 2.6
F
N 2u
13
(8.18)
(8.19)
In addition to the buoyancy rise, one needs to worry about the so-called
downwash if the wind is strong. Downwash is caused by the formation of a
vortex in the wind wake behind the stack, and, if the wind speed is large while
the gas vertical velocity is weak, the low pressure inside this vortex is capable
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CHAPTER 8. AIR POLLUTION
of pulling the fumes downward, below the rim of the stack before they can rise
again under their buoyancy (Figure 8-12). This explains why sometimes the
tip of a smokestack is blackened by repeated exposure to smoke.
Remedies against downwash are: blowing the fumes at a higher speed,
building a taller stack, and avoid emissions during strong winds. Another way
of alleviating downwash, if the stack is metallic, is to place heliocidal ribs on
the outer surface of the stack near its tip of the stack in order to induce a
vertical component in the air flow.
The downwash value is calculated as follows: It is nil if ws ≥ 1.5u, but if
ws < 1.5u, it is equal to
ws ∆hd = 4r 1.5 −
.
(8.20)
u
The overall plume rise is finally obtained from
∆h = ∆hb − ∆hd .
(8.21)
Maximum ground concentration
We are interested in the location on the ground where the concentration is
highest. Obviously this will be in the downwind direction, and we can set y to
zero:
S
H2
cground,downwind(x) =
.
(8.22)
exp −
πuσy σz
2σz2
Finding the location x where this function reaches a peak value is very difficult
because the parameters σy and σz are complicated functions of x. To alleviate
the task, the problem was solved for many possibilities and the results gathered
in a chart, reproduced here as Figure 8-13. For a given stability class and stack
height, the chart provides the distance xmax to the maximum and the value of
the ratio ucmax /S, in which cmax is the maximum concentration.
More complicated situations
The preceding Gaussian model applies only if the atmospheric structure is
uniform over the vertical extend of the plume. Should there be an inversion
aloft, as it is often the case in mid-morning and around noon, the upward dispersion of the plume will be capped at the base of the inversion. The situation
is depicted in Figure 8-14, which also displays the rule to apply.
An important parameter is the distance xL over which the top side of the
plume reaches the base of the inversion, that is, where the plume’s half width
2σz equals the height L − H from the tip of the stack to the base of the
inversion. Thus, xL is such that σz = 0.50(L − H). [Figure 8-14 advocates
taking a slightly more conservative coefficient of 0.47.] The situation is quite
complex between xL and 2xL , because of the presence of two boundaries, the
inversion and the ground surface, but beyond 2xL the plume can be considered
as well mixed over the vertical, and the problem is reduced to one with a single
201
8.4. SMOKESTACK PLUMES
Figure 8-13. Chart to determine the maximum ground concentration from
a smokestack. To use this chart, select the curve corresponding to the atmospheric stability class, and on this curve select the point corresponding to the
effective smokestack height H. Then, move across to determine the distance
xmax (in km), where the maximum occurs, and down to determine the value of
ucmax /S (in 1/m2 ). [From Turner, 1970]
dimension, namely diffusion in the crosswind direction. With M now equal to
S/(uL), the ground concentration is given by:
cground (x, y)
=
=
M
y2
p
exp −
4Dy t
4πDy t
y2
S
√
exp −
.
2σy2
2πuLσy
(8.23)
Figure 8-15 illustrates fumigation caused by a sea breeze. Over land in the
proximity to a coastline (seashore or lakeshore), there exists during daytime
a temperature difference between the warmer ground and the colder water,
which generates an inversion over the sea and a wind carrying this inversion
landward. But, upon encountering the roughness of the landscape, a turbulent
boundary layer is created that grows from the shore inland, as depicted in
Figure 8-15. If a smokestack is placed near the shore, the chance is that its
tip will lie above this boundary layer, but its downwind plume, which expands
downward toward the boundary layer growing upward, will eventually be drawn
into it. Unfortunately, energetic dispersion below an inversion is the recipe
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CHAPTER 8. AIR POLLUTION
Figure 8-14. The effect of an inversion on the dispersion of a smokestack
plume. [Adapted from Masters, 1997]
Figure 8-15. How a sea breeze can create fumigation. The upper panel depicts
the scenario, while the lower panel provides the rule to determine the footprint
of the plume at ground level. Here, h is the local height of the inversion.
[Adapted from Lyons and Scott, 1990]
8.5. URBAN POLLUTION
203
Figure 8-16. How the proximity to a building can create a downwash.
[Adapted from Lyons and Scott, 1990]
for fumigation (see Figure 8-8d). The morale is an advice: Never locate a
smokestack between a coastline and a residential area!
Among the countless possible situations that affect the behavior of smokestack
plumes, we mention here only one more because of its frequency of occurrence.
The situation is that of a smokestack near a building, such as a building of the
factory or power plant that needs the smokestack. Wind flow around and on
top of the building creates low pressure zones that tend to suck the air downward, and the result is a downwash of the plume over the building (Figure
8-16). Also, the building make create a certain shelter against fumes on the
downwind side.
The rules used in practice to estimate the vertical drop to the plume’s
mid-line are quite cumbersome and unreliable, because building come in many
different forms and sizes. Wind-tunnel experimentation is to be preferred during the design stage.
Erecting a smokestack at least 2.5 times the height of the highest proximate
building will usually overcome any problem.
8.5
Urban Pollution
The problem
The need for electric power and road traffic in and around cities contribute
to cause local air pollution problems. At this scale, we are no longer concerned
204
CHAPTER 8. AIR POLLUTION
Figure 8-17. A bird’s eye view of Los Angeles, as seen from above the ocean.
[From an article in Scientific American]
by the plume of a single smokestack but by the aggregate effect of several
stacks, the distributed sources of road traffic, and the countless other forms of
exhaust that activities in a modern city generate.
The problem can be usually analyzed from the perspective of a materials
balance over a specific piece of the atmosphere around the city, called the
airshed. In most instances, this volume cannot be defined with much precision,
since an airshed is in no way as contained as water in a lake. An airshed
may not always be consistently defined, because its existence is at the mercy
of atmospheric conditions, such as the origin of winds and the presence of an
inversion. Nonetheless, it is a useful concept.
Los Angeles
Los Angeles provides a nearly perfect example of an airshed. This city
is litterally cornered between the Pacific Ocean and an arc of mountains and
ridges on the opposite side (Figure 8-17). With winds blowing most frequently
from the ocean, over the city and up above the mountains and a frequent
inversion aloft, the airshed is quite well defined (Figure 8-18). The air over
the city is blocked below by land, to the East by a range of mountains, to the
West by the incoming wind (including the sea breeze), and above by a frequent
inversion.
8.5. URBAN POLLUTION
205
Figure 8-18. The boundaries of the Los Angeles airshed.
The low ventilation through its airshed is not conducive to transporting
emissions of the city over a broad region and diluting them by dispersion.
Rather, local emissions tend to remain trapped near their place of origin. It
is no surprise, therefore, that the city of Los Angeles has been plagued with
poor air conditions for a long time (recall Figure 8-2). What exacerbates the
pollution problem is the fact that Los Angeles is a sunny city. Indeed, sunlight
can induce chemical reactions upon hitting certain chemicals, and the primary
pollutants turn into secondary pollutants, which may compound the problem.
An important and particularly problematic occurence is photochemical smog.
The city of Los Angeles is notorious for its episodic smog. The combination
of processes leading to smog conidtions is:
1. The wind blows from the Pacific Ocean, over the city and toward the
East,
2. There is an inversion aloft,
3. The sun shines, and
4. Motor traffic is heavy,
Photochemical smog
Smog is a word formed by the association of fog and smoke. It occurs when
smoke is trapped locally under a combination of winds and orography, and is
further made visible by condensation at some level (fog). The chemical soup
at the origin of smog consists in nitrogen oxides (NOx ) and volatile organic
coumpounds (nicknamed VOCs), coming from combustion and unburnt hydrocarbons, respectively. The chemical reactions to which these substances are
subjected are extremely complex, and more than hundred chemical reactions
206
CHAPTER 8. AIR POLLUTION
have been identified to date, by in-situ measurements and laboratory experimentation. At the core are the following reactions.
First, the most stable form of nitrogen oxide is NO2 , and the other two
forms (NO and N2 O) are rather quickly oxidized into that form. But, under
sunlight NO2 can be broken up:
NO2 + sunlight energy −→ NO + O•
The atomic oxygen (O•) is extremely reactive (as indicated by the heavy dot
placed after its symbol) and quickly reacts with one of the abundants species
in air, namely oxygen (O2 ) to form ozone (O3 ):
O• + O2 + M −→ O3 + M•
Here, M stands for any neighboring molecule (usually N2 ) that absorbs the
extra energy. Without this bumper action, the ozone would have too much
energy and spontaneously dissociate. Ozone is oxygen with an extra atom,
which it is ready to shed on another molecule that would be more stable with it,
such as NO. Thus, whenever an ozone molecule encounters a nitrogen monoxide
molecule, the two spontaneously exchange an oxygen atom:
O3 + NO −→ O2 + NO2
This third reaction forms the relaxation that returns the nitrogen dioxide and
oxygen consumed in the first two, closing a loop that can be repeated as long as
sunlight strikes. The result is an equilibrium between certain concentrations of
nitrogen dioxide, nitrogen monoxide and ozone. (The concentration of oxygen
is kept nearly at the constant 21% of the atmosphere.)
This equilibrium can be greatly perturbed, however, by the presence of
volatile organic compounds. These VOCs typically have a –CH3 radical, namely
the molecule terminates on one side with a link to a carbon atom, which is itself
linked to three hydrogen atoms. A large category of hydrocarbons and other
forms of VOCs can therefore be denoted as R–CH3 , where R is whatever else
the molecule contains.
The chemical brew continues as follows. Atomic oxygen occasionally reacts
with a water molecule, which is present as long as there is moisture in the air:
O• + H2 O −→ 2 OH•
where the so-called hydroxyl radical OH• is extremely reactive. Upon encountering a VOC, it rips one of its hydrogen
OH• + R–CH3 −→ H2 O + R–CH2 •
leaving the excited radical do the following, systematically passing the excitation forward
R–CH2 • + O2 −→ R–CH2 O2 •
R–CH2 O2 • + NO −→ R–CH2 O• + NO2
8.5. URBAN POLLUTION
207
in which we note the disappearance of one NO molecule. The chain reaction
continues:
R–CH2 O• + O2 −→ R–CHO + HO2 •
The new VOC (R–CHO) is less reactive, but the radical HO2 • carries the
process forward one more step
HO2 • + NO −→ NO2 + OH•
and a second NO is converted into NO2 . Moreover, the hydroxyl radical OH•
is regenerated, acting therefore as a catalyst. The sequence of reactions can
thus be repeated as long as VOCs are present. In the process, more NO is
converted in NO2 , reducing the chance of an encounter between NO and O3
while increasing the level of NO2 ready to be dissociated by sunlight to form
new ozone. The net effect is to maintain the ozone concentration at a higher
level than if VOCs were not present. Put in other words, VOCs are doubly
harmful: Besides being carcinogenic, they are also responsible for increased
ozone levels.
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CHAPTER 8. AIR POLLUTION
Chapter 9
NEXT CHAPTER
THIS IS TO ENSURE THAT CHAPTER 8 ENDS ON AN EVEN PAGE SO
THAT CHAPTER 9 CAN BEGIN ON AN ODD PAGE.
209
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