HONR 297
Environmental Models
Chapter 3: Air Quality Modeling
3.7: The Basic Plume Model
Puffs vs. Plumes
Consider two distinct
types of smoke stack
releases shown to the right
(Figure 3.15 from
Hadlock).
 The top part of the figure
shows a puff which is a
momentary release of
exhaust gases that forms a
discrete cloud.
 The puff can move
horizontally with any
existing wind and is free to
disperse or diffuse in three
dimensions (x- and yhorizontally, z-vertically).

Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment
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Puffs vs. Plumes
A plume, an example of
which is shown in the
bottom portion of Figure
3.15, is comprised of a
steady continuous release
of exhaust gases.
 The plume starts at the
stack and continues for an
indefinite distance in the
direction of any prevailing
wind (i.e. downwind).
 Concentration of the
exhaust material gradually
decreases the further one
gets from the smoke stack
source.

Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment
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Puffs vs. Plumes
It turns out that the second type
of smoke stack release is easier
to model!
 Here are some reasons why:

◦ Significant dispersion takes place in
only two directions – vertically, in the
z-direction and horizontally,
perpendicular to the axis of the
plume (y-direction).

For the figure at right, what would be
the y-direction in the plume release?
◦ Along the axis of the plume, in the
x-direction, change in concentration
is so slight and gradual that there is
only a small amount of dispersion in
this direction.
◦ Thus, to model the plume release,
we only need two input position
variables, instead of three for the
puff release!
Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment
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Gaussian Plume Model

For plume releases, here is the model that
has been developed for determining
concentration of exhaust material:
Equation (3) is known as the Gaussian
plume model!
 Let’s look at each term and constant in
the model …

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Gaussian Plume Model

C is the concentration at a point, measured
in amount of pollutant per volume.
◦ [C] = mass/length^3.

Q is the source term – it represents the
amount of pollutants emanating from the
stack per unit time.
◦ [Q] = mass/time.
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Gaussian Plume Model

u is the wind velocity in the prevailing
direction – it represents an average velocity
over the time period we are interested in
modeling a plume.
◦ [u] = length/time.
◦ Units of wind velocity are chosen to match the
desired concentration units for a given situation.
◦ For example, if we are interested in average onehour concentrations at a given point, we’d choose
an average wind velocity over one hour instead of
an instantaneous value for u.
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Gaussian Plume Model

y is the horizontal coordinate, measured in the
direction perpendicular to the axis of wind movement.
◦ [y] = length.

z is the vertical coordinate, which measures elevation
above the ground.
◦ [z] = length.

H is the effective stack height - which is the original
physical stack height h plus additional height which is
added to account for the fact that exhaust gases
exiting a smoke stack may move upwards a distance
vertically due to heat and momentum before starting
to move horizontally with the prevailing wind.
◦ [H] = length.
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Gaussian Plume Model

is a dispersion coefficient that accounts for dispersion in
the y-direction.
y
◦ This coefficient is analogous to the diffusion coefficients in the
one and two dimensional diffusion models we saw in Sections
3.5 and 3.6!
◦ [ y] = length.
◦ y is determined by current meteorological conditions.
◦ y also depends on downwind axial distance x, so it is a function,
not a constant.
◦ As we move downwind, one would expect the plume to be
wider, which in turn would correspond to a larger value of y.

is a dispersion coefficient that accounts for dispersion in the
z-direction.
z
◦ [ z] = length.
◦ z also is determined by meteorological conditions and axial
distance x!
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Gaussian Plume Model
Let’s look at the underlying ideas behind
equation (3)!
 First, using basic ideas from algebra (Laws
of Exponents, etc.), we can rewrite the
Gaussian plume model in the form:

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Gaussian Plume Model

Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment
In this form, (4), we see that the
concentration of pollutant C is a product
of three quantities:
◦ Source strength
◦ Diffusion effect in the y-direction.
◦ Diffusion effect in the z-direction.
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Source Strength
The first term in (4) measures the
strength of pollutant at the source, itself.
 It takes into account two factors:

◦ The rate Q at which pollutant is being
injected into the atmosphere.
◦ The wind speed u – higher wind speed will
lead to reduced concentration, due to larger
amounts of air mixing with the pollutant being
released.
 What about lower wind speed?
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Diffusion in the y-direction



The second term in (4) represents the
diffusion effect in the y-direction.
This term is very similar to terms we saw in
the one-dimensional or two-dimensional
diffusion formulas, from Sections 3.5 and 3.6
(equations (1) and (2)).
Are there any differences?
◦ Yes – equations (3) and (4) are steady-state
formulas – they assume the contaminant
concentration doesn’t change over time, due to
continuous output at the source.
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Diffusion in the z-direction

The third term in (4) represents the diffusion
effect in the z-direction.
◦ Note that the Gaussian plume model, actually uses
the vertical distance from the effective stack height H,
z-H, this appears in the first exponent involving z.
◦ The second exponent involving z, z+H, is an error
correction factor used to account for vertical diffusion
being blocked in the downward direction once
material from the plume reaches ground level.
◦ This can be significant in short smoke stacks, but is
less significant in taller stacks (why?).
 For large H values, the second term will be small!
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“Natural” Questions about the
Gaussian Plume Equation
1. Why doesn’t x show up in equations
(3) or (4)? Shouldn’t concentration C
depend on axial distance downwind from
the source?
 2. How do we calculate dispersion
coefficients y and z?
 3. Where does the plume equation come
from? How is it derived?

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1. Why doesn’t x show up in (3) or
(4)?
Recall that dispersion coefficients y and
z are functions of the axial distance x
from the source, so concentration C does
depend on x!
 Thus, we could write y(x) and z(x) in
(3) and (4) if we wish, but this makes the
equation more complicated.
 Also, note that y and z are functions of
atmospheric stability class (see p. 71, Table
3.1).

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2. How do we find
y and
z?
Meteorologists have made extensive studies
of the effects of atmospheric stability class
and other factors such as wind speed.
 Using both experimental and theoretical
mathematical models, they have come up
with a set of well-accepted values of
dispersion coefficients y and z as
functions of axial downwind distance x and
atmospheric stability class.
 Values can be read off of the graphs in
Figures 3.16 and 3.17 (on the next two
slides) for y and z, respectively!

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Horizontal dispersion coefficient y
as a function of x and stability class
Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment
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Vertical dispersion coefficient z as
a function of x and stability class
Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment
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Vertical Dispersion Coefficients
Note that the scales on the graphs are
logarithmic, so each subdivision on an axis
scale represents a multiplier for the base
to the left.
 For example, the marks between 100 and
1000 correspond to 2x100, 3x100, 4x100,
… , 9x100.
 A point halfway between 100 and 1000
would correspond to a number between
300 and 400.

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3. Where does (3) or (4) come
from?





The mathematics leading to the Gaussian plume
equation involves partial derivatives and statistics!
For more details see Chapter 6.
Section 6.2 shows how the Gaussian normal
distribution from statistics is related to the
diffusion equations we have been studying – it
turns out that there is a direct relationship!
This is why equation (3) or (4) is called the
Gaussian plume equation.
Recall that diffusion results from molecules taking
“random walks”!
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3-D Plume Coordinate System and
Dispersion Effects
Figure 3.18 (on the next slide) gives a threedimensional representation of both the plume
spreading out and the contaminant concentration
as a function of vertical and horizontal distance.
 The elliptical cross-sections represent the
physical plume spreading out as x increases.
 The bell-shaped curves represent the fact that
contaminant concentration decreases as one
moves away from the plume’s center axis.
 Also note that the effective stack height H and
stack height h are shown on this graph!

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3-D Plume Coordinate System and
Dispersion Effects
Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment
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An Applied Example!





You live two miles due east of a
coal-fired utility power plant that
produces electricity for your city.
The stack on the plant is 350 feet
high, and the ground is level.
On a given day, the sun is shining
brightly and the wind is blowing
from the southwest to the
northeast at 10 mph.
Measurements at the plant stack
of the concentration of nitrogen
oxides in the exhaust gases show
that such pollutants are being
released at the rate of 80 lb/min.
What would you expect the
concentration of nitrogen oxides
(NOx) to be at your residence?
Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment
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An Applied Example!
Image Courtesy Charles Hadlock: Mathematical Modeling in the Environment
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An Applied Example!



In Figure 3.19, choose an xy-coordinate
system with the x-axis along the plume
centerline through the power plant.
Thus, the x-axis makes a 45-degree angle
with a line from the power plant to your
house, since the wind is blowing from
southwest to northeast.
The y-axis is perpendicular to the x-axis –
choose the origin of the xy-coordinate
system to be located at the power plant,
since this is the source point.
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An Applied Example!



Using the 45-45-90 right triangle formed,
with the hypotenuse corresponding to the
side between the power plant and your
house, we can find the coordinates of the
house!
The hypotenuse has length 2 miles.
Since the hypotenuse of a 45-45-90 right
triangle is √2 times the length of the triangle
legs, it follows that the triangle legs have
length 2/√2 = √2 miles 1.414 miles!
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An Applied Example!
It follows that the xy-coordinates of your
house are (in miles)
(x, y) = (√2, -√2) (1.414, -1.414).
 To use the Gaussian plume equation (3)
(or (4)), we need to know Q, u, y, z, and
diffusion coefficients y and z.

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An Applied Example!







Here’s what we know:
Q = 80 lb/min
(source term)
u = 10 mi/hr
(average wind speed)
x = √2 mi
(downwind distance)
y = -√2 mi
(horizontal displacement)
z = 0 mi
(vertical elevation)
Use Table 3.1 on p. 71 to find the atmospheric
stability class!
◦ Since we have bright sun and a wind of 10 mi/hr, it
follows from Table 3.1 that the atmospheric stability
class is B.
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An Applied Example!



From Figure 3.16, we can find the dispersion
coefficient y using the downwind distance x
and atmospheric stability class.
Note that the graph in Figure 3.16 has units
of meters on each axis and our length units
for Q and stack height are given in feet.
Thus, we need to convert our x value to
meters, read off y in meters, and convert
y back into feet!
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An Applied Example!
x = √2 mi 1.414 mi = (1.414 mi)*(5280
ft/mi)*(0.3048 m/ft) = 2275.956 m.
 Atmospheric stability class is B.
 Thus, y
300 m = (300 m)*(3.28084
ft/m) = 984.252 ft.

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An Applied Example!

To find z, using the same x (in m) and
stability class values with the graph in
Figure 3.17, we find that
230 m = (230 m)*(3.28084 ft/m) =
z
754.593 ft.
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An Applied Example!
Finally, we need an H – value for the
effective stack height.
 Since we are only given physical stack
height h = 350 ft, we will choose this for
H.
 Note that this is a more conservative
estimate, because using a smaller value for
H will raise the concentration at ground
level.

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An Applied Example!
Using Mathematica along with equation (3), we find
that after converting miles to feet and hours to
minutes,
C = 1.12 x 10-20 lb/ft3.
 Air contaminant concentrations are usually measured
in parts per million (ppm) or parts per billion (ppb)
instead of lb/ft3, so given that 1 ppm of NOx
corresponds to 1.1 x 10-7 lb/ft3, it follows that
 C = (1.12 x 10-20 lb/ft3)*(1 ppm NOx)/(1.1 x 10-7 lb/ft3
NOx) = 1 x 10-13 ppm.
 Since contaminants in air are undetectable at
concentration levels less than 1 ppb = 10-3 ppm, it
follows that the level of NOx concentration is 10-10
ppb, well below detection limits – hence we are safe!

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PLUME Spreadsheet



Our textbook author has included a
spreadsheet program, PLUME, that will work
in Excel!
If you don’t have a copy that came with your
book, a copy can be downloaded from our
class web page – see the Hadlock Textbook
Floppy Disk Files (ZIP) link.
PLUME includes built-in formulas to
compute the dispersion coefficients y and
z!
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PLUME Spreadsheet
Using PLUME, reproduce the work done above in
the Applied Example (Hadlock, p. 95 problem #1).
Be sure to enter quantities with the correct units
within PLUME.
 Compare your results within PLUME to those we
found above – do they agree?
 If not, justify or resolve any discrepancies.
 Using PLUME, try Hadlock problems # 3 and 4 on
p. 95. Hint: For problem #3, part (d), the longterm average concentration will be
0.25*(concentration when wind blows towards
home) + 0.75*(concentration when wind does
not blow towards home).

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Resources

Charles Hadlock, Mathematical Modeling
in the Environment – Chapter 3, Section 7

Figures 3.15, 3.16, 3.17, 3.18, and 3.19 used with
permission from the publisher (MAA).
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