HONR 297
Environmental Models
Chapter 3: Air Quality Modeling
3.2: Physical Principles
Initial Assumptions
For our investigation of air quality models, we will
focus on point sources of air pollution – the
main reason is to allow us to work with simpler
mathematical models.
 We will also assume that the point sources are in
a steady state, i.e. the amount of material being
released from a point source doesn’t change over
the time period we are considering.
 A standard example of a point source in steady
state to keep in mind is a smoke stack that is
releasing a steady amount of smoke or gases at its
top!

2
Smoke Stacks




Think of all the smoke
stacks you have ever
seen, ranging from large
factory or power plant
stacks to a single
chimney on a house.
Essentially all such stacks
are the same.
In this section, we will
look at the various
processes that affect the
way smoke disperses
once released.
With this in mind …
3
Smoke Stacks – Emissions Plume
What mental images
do you have of the
emissions plume
from a smoke stack?
 Let’s take a few
moments to think
about this …

4
Smoke Stacks – Emissions Plume

Does the plume
◦ Go directly upward?
◦ Turn to one side?
◦ Spread out in the
vertical direction?
◦ Stay confined to a
single limited level?
◦ Climb higher and
higher?
◦ Move down towards
the ground?
◦ Do anything else?
5
Smoke Stacks – Emissions Plume

Sample plume behavior, from
Hadlock, Figure 3.1, p. 62.
Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment
6
Smoke Stacks – Emissions Plume



Exhaust emitted from a
smoke stack can take on
many different patterns.
One big factor will be
wind, but there are many
others, as we will see.
Suppose we want to
study plume behavior
from an existing smoke
stack or want to predict
behavior for exhaust
from a new building
before it is built!
7
Mathematical Models

To make these types of
predictions about plume
behavior, we can use
mathematical models that
take into account factors
such as
◦ Nature of the proposed (or
existing) site.
◦ Historical or typical weather
patterns in the area.
◦ Physical characteristics of
the smoke stack.
◦ Etc.

This is what is done by
both plant/site designers
and regulatory agencies!
8
Pollutant Dispersal Processes

1.
2.
3.
4.
The key physical processes that control the way pollutants
disperse into the atmosphere can be classified as follows:
The propensity of gaseous materials to move from areas of
higher concentration to areas of lower concentration by
means of a simple diffusion process.
The patterns of movement of the surrounding air, including
both horizontal movement (wind) and vertical movement.
The physical characteristics of the exhaust gases
themselves, such as temperature, momentum, and perhaps
other properties.
Other aspects of the surroundings, such as special
topographic features or changes in elevation.
9
Diffusion






What happens if a small amount of perfume is placed on a cloth or
sprayed in room?
Shortly thereafter, the perfume will have spread throughout the room.
The reason is that the gas molecules generated by evaporation of the
perfume start out at a high concentration near the point of origin and
tend to spread out to areas of lower concentration, eventually
permeating the whole room.
The same happens if gasoline is spilled – the gasoline vapors will move
from the area of higher concentration near the spill to areas of lower
concentration throughout the room.
Another example is a lump of sugar in a cup of coffee or tea –
dissolved sugar molecules will gradually spread out throughout the
liquid, even without any stirring – again, the sugar molecules are
moving from areas of higher concentration of sugar to areas of lower
concentration.
All of these situations are examples of a process known as diffusion!
10
Diffusion

From Wikipedia:
◦
◦
◦


Diffusion is one of several transport
phenomena that occur in nature. A
distinguishing feature of diffusion is that it
results in mixing or mass transport without
requiring bulk motion. In Latin [the] word
"diffundere" means "to spread out".
(Wikipedia link) (YouTube video)
YouTube Video – Diffusion
Diffusion is caused by kinetic energy (i.e.
energy due to motion) present in
individual molecules which causes them
to bounce around in random patterns
until they are more evenly distributed –
this type of motion is often referred to
as a random walk.
This process is similar to shuffling a
deck of cards – if one starts with all
hearts together in one place and
shuffles the cards a number of times (a
randomizing process), the hearts will
eventually be mixed into the deck in a
more uniform and spread out fashion.
11
Three-Dimensional Diffusion


We can think of diffusion
as a process that is three
– dimensional, two –
dimensional, or even one
– dimensional, depending
on a given physical
situation!
Using the perfume
example, imagine a puff
of perfume is sprayed in
the center of a room, as
shown to the right
(Figure 3.2 of Hadlock, p.
63).
Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment
12
Three-Dimensional Diffusion



Initially, all of the
perfume is concentrated
near the center of the
room.
As time progresses, the
perfume molecules will
spread out in a uniform
fashion, in all directions!
The driving mechanism
is the kinetic energy in
the perfume molecules,
spreading out via
random walks!
Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment
13
Three-Dimensional Diffusion

If we measure the
concentration in the
x-, y-, or z- direction at
times t1, t2, and t3,
where t1 < t2 < t3, we
will find that the
concentration (as a
function of distance
from the center of the
room will have shapes
similar to those given
in Hadlock as Figure
3.3 on p. 64.
Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment
14
Two-Dimensional Diffusion
Diffusion can also occur in two
dimensions – imagine for
example a drop of ink placed in
the center of a white
handkerchief or a thin layer of
gelatin.
 Again, the ink molecules are
moving from an area of higher
concentration to lower
concentration via random walks!
 Two dimensional diffusion can
also be used to model perfume
sprayed at the center of a “lowceilinged room” or a uniform
vertical column of perfume in the
middle of a room, as shown to
the right (Figure 3.4 in Hadlock,
p. 65).

Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment
15
Two-Dimensional Diffusion



For each of these cases, the idea to
keep in mind is that every vertical level is
assumed to be identical to the levels above
and below – there is no difference in
concentration as we move vertically, only as
we move horizontally does the
concentration change!
In this case, we’d expect concentration
graphs similar to those in Figure 3.3, at
instances in time, measured as a
function of distance from the source, as
we move in the x- or y-direction!
What would the concentration look like
as we move in the z-direction from the
center in the second situation, at
instances in time?
◦

Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment
Constant concentration, which will
reduce over time …
Note: In the second situation shown to
the right, we call the column of perfume
a line source.
16
One-Dimensional Diffusion






If we spray perfume in a tube filled with air or
have a situation in which a wall running down
the middle of a room has been painted, we
can model the process by which the perfume,
or paint fumes spread out as one-dimensional
diffusion.
In this case, at any instant in time,
concentration of the perfume or paint smell
depends only on the distance along the tube
from the center or perpendicular distance
from the wall.
Hadlock provides figures to illustrate this –
see Figure 3.5 on p. 66.
In this case, we’d expect concentration graphs
similar to those in Figure 3.3, at instances in
time, measured as a function of distance from
the source, as we move in the x- direction!
In the y- or z- direction, the concentration
would be constant at any given instant in time!
Note: in this case, we call the wall of paint a
“planar source”.
Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment
17
Air Movement

So far, we’ve been discussing the first
physical process that controls the way
pollutants disperse in the atmosphere,
namely,
◦ The propensity of gaseous materials to move from
areas of higher concentration to areas of lower
concentration, which is achieved via diffusion!

The second factor we need to look at
when dealing with pollutant dispersal in
the air is the movement of the air itself!
18
Air Movement

Question:
◦ Suppose we live one mile
from a power plant and the
wind is blowing directly
from the power plant
towards our home.
◦ Further, suppose there is
some level of contaminants
in the air near our house
as a result.
◦ What would happen if we
had the same situation, but
the wind is blowing twice as
hard?
◦ Think about this for a few
minutes …
19
Air Movement

Possible answers:
◦ The wind is harder, so the
air moves faster, which
implies that pollutants
travel to our house more
quickly, with less time to
diffuse, so the pollution
concentration levels near
our house increase.
◦ The wind is blowing harder,
so the air is stirred up
more, which increases the
diffusion rate, so the
pollution concentration
levels near our house
decrease.
20
Air Movement – Wind Speed
Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment
Wind speed is not constant at all heights above the ground.
 Friction of the air mass along the ground causes air near the earth’s
surface to move more slowly than air higher up.
 This is illustrated in Figure 3.6, p. 67 of Hadlock, shown above – longer
arrows indicate greater wind speed.
 This difference in wind speed as we move vertically upwards from the
earth’s surface, known as a “wind-speed gradient”, causes mixing between
adjacent layers of air, with a net result of vertical transport of pollutants!

21
Air Movement – Solar Insolation
Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment
Solar insolation (i.e. sunlight) warms the ground, which in turn warms the
air above it.
 This makes the air near the earth’s surface less dense, causing it to rise
(similar to how a hot air balloon works), as in Hadlock’s Figure 3.7 above.
 As the warmer air rises, air moves in from the sides or above, again causing
vertical mixing of air, hence vertical mixing of pollutants if present!

22
Exhaust Gas Characteristics
Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment

A third factor we must consider when working with dispersal of pollutants
in the air is physical characteristics of the exhaust gases, themselves!
◦ Exhaust gas is hot, so it will rise!
◦ Exhaust gases will be moving at a high velocity, so they exit rapidly from the top of a
smoke stack and continue to move upward vertically for some distance (they have
momentum), until they mix with or are slowed down by the surrounding air mass.
◦ The net effect of hot gas and momentum is that initially the exhaust gases move
upwards in a column, so we can think of this as an imaginary extension of the stack
height, possibly from two to ten times the actual stack height!
◦ See Hadlock, Figure 3.8 shown above.
23
Surface Characteristics

Finally, a factor that contributes to the
dispersal of pollution in the air is
characteristics of the earth’s surface,
including
◦ Changes in elevation (topography), which can
cause deflection of exhaust plumes.
◦ Surface roughness (trees vs. buildings vs. open
fields vs. water), which can cause vertical
mixing in a fashion similar to that discussed
above for wind velocity gradients.
24
Surface Characteristics
Images Courtesy Charles Hadlock: Mathematical Modeling in the Environment
25
Summary
In summary, the four main factors that affect
dispersion of air pollutants are
1. Tendency of pollutants to Diffuse, causing
movement from high to low concentration
2. Air Movement and Mixing
3. Exhaust Gas Characteristics
4. Surface Characteristics
 Of these, the first and second will be the most
important for our models.
 It will turn out that the third and fourth factors
can be incorporated into the models we will
use in a simple fashion.

26
Stability


Finally, for our mathematical models, we
will need some additional background
material related to air movement and
mixing, namely the concept of stability.
Definition:
◦
◦



We say an air mass is stable if there is
relatively little vertical mixing per unit of
horizontal distance traveled.
An air mass is said to be unstable if the
amount of vertical mixing per unit of
horizontal distance is relatively high.
We expect a plume of exhaust gas to
disperse less in stable atmospheric
conditions and disperse more in
unstable atmospheric conditions!
Thus, we can think of a stable air mass
as corresponding to a stable plume
and an unstable air mass as
corresponding to an unstable plume!
See Hadlock, Figure 3.11 at right.
Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment
27
Stability

What characteristics
might affect plume
stability (think of
what might affect air
stability in general)?
◦ Wind speed.
◦ Solar insolation.
Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment
28
Pasquill-Gifford Stability Categories
Meteorologists have studied various combinations
of atmospheric conditions and have developed a
scheme for classifying air stability!
 The scheme (which is used by industry and
regulatory agencies) is summarized in Table 3.1 on
p. 71 of Hadlock.
 Air stability is classified from A (most unstable) to
F (most stable), as a function of

◦ Wind speed (measured 10 meters above the ground).
◦ Incoming solar radiation (see table for details).
◦ Cloud cover at night (see table for details).
29
Resources

Wikipedia (diffusion definition)
◦ http://en.wikipedia.org/wiki/Diffusion
◦ You Tube Videos
 http://www.youtube.com/watch?v=n5nubvwJJQM
 http://www.youtube.com/watch?v=6zysNwl6Zuo
 http://www.youtube.com/watch?v=_oLPBnhOCjM

Charles Hadlock, Mathematical Modeling
in the Environment, Section 3.2
◦ Figures 3.1 – 3.11 used with permission from
the publisher (MAA).
30