&EPA
United States
Environmental Protection
Agency
Air Pollution Training Institute
MD20
Environmental Research Center
Research Triangle Park NC 27711
EPA 450/2-81-077
October 1981
Air
APTI
Course 423
Dispersion of Air Pollution
Theory and Model
Application
Selected Readings Packet
-------
United States
Environmental Protection
Agency
Air Pollution Training Institute
MD20
Environmental Research Center
Research Triangle Park NC 27711
EPA 450/2-81-077
October 1981
Air
APTI
Course 423
Dispersion of Air Pollution —
Theory and Model
Application
Selected Readings Packet
Northrop Services, Inc.
P. O. Box 12313
Research Triangle Park, NC 27709
Under Contract No.
68-02-2374
EPA Project Officer
R. E. Townsend
United States Environmental Protection Agency
Manpower and Technical Information Branch
Office of Air Quality Planning and Standards
Research Triangle Park, NC 27711
-------
Notice
This is not an official policy and standards document. The opinions and selections
are those of the ^authors afnd not ^necessarily those of the Environmental Protection
Agency. Every attempt has 'been made to represent the present state of the art as
well as subject areas .-still under evaluation. Any mention of products or organiza-
tions does not constitute endorsement by the United States Environmental Protec-
tion Agency.
This document is issued by the Manpower and Technical Information Branch,
Control Programs Development Division, Office of Air Quality Planning and Stan-
dards, USEPA. It was developed for use.in training courses presented by the EPA
Air Pollution Training -Institute and others receiving contractual or grant support
from the Institute. Other organizations are welcome to use the document.
11
-------
Introduction
This package contains course material for your review. Before attending class, take
time to acquaint yourself with the items provided. An indepth study is not required
or intended. However, you should be familiar with the general order of items
involved and the overall ideas of each part.
The Workbook of Atmospheric Dispersion Estimates (WADE) and Plume Rise
will be used as the basis for homework assignments on Monday and Tuesday
nights. The following pages in WADE are recommended for study; pages
5 through 9 and page 17. The following pages in Plume Rise are also recom-
mended: page 44 and pages 57 through 60.
The student manual for Course 411, Air Pollution Meteorology, is enclosed for
those students who have not taken Course 411. You should review the manual to
familiarize yourself with the basic meteorology factors that influence air pollution
dispersion.
This material and the manual are an integral part of the course, so you must bring
them with you to class. Extra copies will not be available.
in
-------
Table of Contents
Page
1. Workbook of Atmospheric Dispersion Estimates by D. Bruce Turner. . . 1-1
2. Plume Rise by A. Briggs 2-1
3. Plume Rise from Multiple Sources by A. Briggs 3-1
4. Determination of Atmospheric Diffusion Parameters, 1976,
by R. R. Draxler 4-1
5. Atmospheric Dispersion Parameters in Gaussian Plume Modeling I and II,
1980, by Dr. S. P. S. Arya 5-1
6. Consequences of Effluent Release: Turbulent Diffusion Typing Schemes:
A Review by F. A. Gifford 6-1
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•TI
1-1
H
hiu
n
U.S. ENVIRONMENTAL PROTECTION AGENCY
-------
1-3
WORKBOOK OF
ATMOSPHERIC DISPERSION ESTIMATES
D. BRUCE TURNER
Air Resources Field Research Office,
Environmental Science Services Administration
ENVIRONMENTAL PROTECTION AGENCY
Office of Air Programs
Research Triangle Park, North Carolina
Revised 1970
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1-4
The AP series of reports is issued by the Environmental Protection
Agency to report the results of scientific and engineering studies,
and information of general interest in the field of air pollution.
Information presented in this series includes coverage of intramural
activities involving air pollution research and control technology
and of cooperative programs and studies conducted in conjunction
with state and local agencies, research institutes, and industrial
organizations. Copies of AP reports are available free of charge -
as supplies permit - from the Office of Technical Information and
Publications, Office of Air Programs, Environmental Protection
Agency, Research Triangle Park, North Carolina 27711, or from the
Superintendent of Documents.
7th printing January 1974
Office of Air Programs Publication No. AP-26
For sale by the Superintendent of Document!, U.S. Government Printing Office, Washington, D.C. 20402 - Price tl 00
Stock Number MC0-0016
ii
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1-5
PREFACE
This workbook presents some computational techniques currently used by scientists
working with atmospheric dispersion problems. Because the basic working equations are
general, their application to specific problems usually requires special care and judgment;
such considerations are illustrated by 26 example problems. This workbook is intended as an
aid to meteorologists and air pollution scientists who are required to estimate atmospheric
concentrations of contaminants from various types of sources. It is not intended as a com-
plete do-it-yourself manual for atmospheric dispersion estimates; all of the numerous compli-
cations that arise in making best estimates of dispersion cannot be so easily resolved.
Awareness of the possible complexities can enable the user to appreciate the validity of his
"first approximations" and to realize when the services of a professional air pollution mete-
orologist are required.
Since the initial publication of this workbook, air pollution meteorologists affiliated
with the Environmental protection Agency have turned to using the method of Briggs to de-
termine plume rise in most cases rather than using the plume-rise equation of Holland as set
forth in Chapter 4. The reader is directed to'
Briggs, Gary A. 1971 "Some Recent Analyses of Plume Rise Observations."
In Proceedings of the Second International Clean Air Congress. Academic Press,
New York, N. Y. pp 1029- 1032
and modified by
Briggs. Gary A. 1972; "Discussion, Chimney Plumes in Neutial and Stable
Surroundings." Atmospheric Environment, 6:507-510.
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1-6
ACKNOWLEDGMENTS
The author wishes to express his appreciation to Robert A. McCormick, Paul
A. Humphrey, and other members of the Field Research Office for their helpful dis-
cussions and review,- to Jean J. Sc'hueneman, Chief, Criteria and Standards Develop-
ment, National Center for Air Pollution Control, who suggested this workbook; to Phyllis
PoUand and Frank Sohiermeier, who checked the problem solutions; to Ruth Umfleet
and Edna Beasley for their add; and to the National Center for Air Pollution Control,
Public Health Service, and Air Resources Laboratory, Environmental Science Services
Administration, for their support.
IV
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1-7
CONTENTS
ABSTRACT ™
Chapter 1. INTRODUCTION 1
Chapter 2. BACKGROUND 3
Chapter 3. ESTIMATES OF ATMOSPHERIC DISPERSION 5
Coordinate System - 5
Diffusion Equations 5
Effects of Stability - 6
Estimation of Vertical and Horizontal Dispersion - 7
Evaluation of Wind Speed 7
Plots of Concentrations against Distance - 7
Accuracy of Estimates 7
Graphs for Estimates of Diffusion - 10
Plotting Ground-Level Concentration Isopleths 10
Areas Within Isopleths 17
Calculation of Maximum Ground-Level Concentrations 17
Review of Assumptions - 17
Chapter 4. EFFECTIVE HEIGHT OF EMISSION 31
General Considerations _ 31
Effective Height of Emission and Maximum Concentration 31
Estimates of Required Stack Heights _ 31
Effect of Evaporative Cooling 32
Effect of Aerodynamic Downwash 32
Chapter 5. SPECIAL TOPICS 35
Concentrations in an Inversion Break-up Fumigation 35
Plume Trapping 36
Concentrations at Ground Level Compared to Concentrations
at the Level of Effective Stack Height from Elevated Con-
tinuous Sources _ 36
Total Dosage from a Finite Release 37
Crosswind-Integrated Concentration - 37
Estimation of Concentrations for Sampling Times Longer
than a Few Minutes 37
Estimation of Seasonal or Annual Average Concentrations
at a Receptor from a Single Pollutant Source 38
Meteorological Conditions Associated with Maximum
Ground-Level Concentrations 38
Concentrations at a Receptor Point from Several Sources 39
Area Sources _ 39
Topography _ 40
Line Sources _ 40
Instantaneous Sources 41
Chapter 6. RELATION TO OTHER DIFFUSION EQUATIONS 43
Chapter 7. EXAMPLE PROBLEMS 45
Appendices: 57
1 — Abbreviations and Symbols 59
2 — Characteristics of the Gaussian Distribution 61
3 — Solutions to Exponentials 65
4 — Constants, Conversion Equations, Conversion Tables _ 69
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1-9
ABSTRACT
This workbook presents methods of practical application of the binormal con-
tinuous plume dispersion model to estimate concentrations of air pollutants. Estimates
of dispersion are those of Pasquill as restated by Gifford. Emphasis is on the estima-
tion of concentrations from continuous sources for sampling times of 10 minutes. Some
of the topics discussed are determination of effective height of emission, extension of
concentration estimates to longer sampling intervals, inversion break-up fumigation
concentrations, and concentrations from area, line, and multiple sources. Twenty-six
example problems and their solutions are given. Some graphical aids to computation
are included.
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1-11
Chapter 1 — INTRODUCTION
NOTE SEE PREFACE TO THE SIXTH PRINTING ON PAGE 111.
During recent years methods of estimating at-
mospheric1 dispersion have undergone considerable
re\isnm, primarily due to results of experimental
measurements. In most dispersion problems the
relevant atmospheric layer is that nearest the
ground, varying in thickness from several hundred
t'i a feu thousand meters. Variations in both
thermal and mechanical turbulence and in wind
\elocit\ are greatest in the layer in contact with
I he inrlace. Turbulence induced by buoyancy forces
in the atmosphere is closely related to the vertical
600
500
temperature structure. When temperature decreases
with height at a rate higher than 5.4 :F per 1000 ft
(1 C per 100 meters), the atmosphere is in un-
stable equilibrium and vertical motions are en-
hanced. When temperature decreases at a lower
rate or increases with height (inversion), vertical
motions are damped or reduced. Examples of typ-
ical variations in temperature and wind speed with
height for daytime and nighttime conditions are
illustrated in Figure 1-1.
234567
TEMPERATURE, °C
10 II 12
34567
WIND SPEED, m/sec
J
10 11
Figure 1-1. Examples of variation of temperature and wind speed with height (after Smith, 1963).
The transfer of momentum upward or down-
ward in the atmosphere is also related to stability;
\\hen the atmosphere is unstable, usually in the
daytime, upward motions transfer the momentum
"defmenr\ " due to eddy iriction losses near the
earth's surface through a relatively deep layer,
cau-ing the wind speed to increase more slowly
uith height than at nigh; (except in the lowest few
meters). In addition to thermal turbulence, rough-
nc^s elements on the ground engender mechanical
turbulence, which affects both the dispersion of
material in the atmosphere and the wind profile
variation of wind with height). Examples of these
eliei Is tin the resulting wind profile are shown in
Figure 1-2.
As wind speed increases, the effluent from a
continuous source is introduced into a greater vol-
ume of air per unit time interval. In addition to
this dilution by wind speed, the spreading of the
material (normal to the mean direction of trans-
port) by turbulence is a major factor in the dis-
persion process.
The procedures presented here to estimate at-
mospheric dispersion are applicable when mean wind
speed and direction can be determined, but meas-
urements of turbulence, such as the standard de-
viation of wind direction fluctuations, are not avail-
able. If such measurements are at hand, techniques
such as those outlined by Pasquill (1961) are likely
to give more accurate results. The diffusion param-
-------
1-12
eters presented here are most applicable to ground-
level or low-level releases (from the surface to about
20 meters), although they are commonly applied at
higher elevations without full experimental valida-
tion. It is assumed that stability is the same
throughout the diffusing layer, and no turbulent
transfer occurs through layers of dissimilar stability
characteristics. Because mean values for wind direc-
tions and speeds are required, neither the variation
of wind speed nor the variation of wind direction
with height in the mixing layer are taken into ac-
count. This usually is not a problem in neutral or
unstable (e.g., daytime) situations, but can cause
over-estimations of downwind concentrations in
stable conditions.
REFERENCES
Davenport, A. G., 1963: The relationship of wind
structure to wind loading. Presented at Int.
Conf. on The Wind Effects on Buildings and
Structures, 26-28 June 63, Natl. Physical Lab-
oratory, Teddington, Middlesex, Eng.
Pasquill, F., 1961: The estimation of the dispersion
of wind borne material. Meteorol. Mag. 90,
1063, 33-49.
Smith, M. E., 1963: The use and misuse of the at-
mosphere, 15 pp., Brookhaven Lecture Series,
No. 24, 13 Feb 63, BNL 784 (T-298) Brook-
haven National Laboratory.
600,—
URBAN AREA
GRADIENT WIND
SUBURBS
LEVEL COUNTRY
1-2. Examples of variation of wind with height over different size roughness elements (Tigures are percentages
of gradient wind); (from Davenport, 1963).
ATMOSPHERIC DISPERSION ESTIMATES
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1-13
Chapter 2 — BACKGROUND
For a number of years estimates of concentra-
tions were calculated either from the equations of
Sutton (1932) with the atmospheric dispersion
parameters C,, Cz, and n, or from the equations of
Bosanquet (1936) with the dispersion parameters
p and q.
Hay and Pasquill (1957) have presented experi-
mental evidence that the vertical distribution of
spreading particles from an elevated point is re-
lated tci the standard deviation of the wind eleva-
tion angle, ir,, at the point of release. Cramer (1957)
derived a diffusion equation incorporating standard
deviations of Gaussian distributions: <7, for the
distribution of material in the plume across wind
in the horizontal, and ..,. for the vertical distribution
of material in the plume. (See Appendix 2 for prop-
erties of Gaussian distributions.) These statistics
were related to the standard deviations of azimuth
angle,
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Chapter 3 — ESTIMATES OF ATMOSPHERIC DISPERSION
1-15
This chapter outlines the basic procedures to
he used in making dispersion estimates as sug-
gested hy Pasquill (1961) and modified by Gifford
M9G1).
«M>KI)IYYTK SYSTEM
In the system considered here the origin is at
ground level at or beneath the point of emission,
with the x-axis extending horizontally in the direc-
tion of the mean wind. The y-axis is in the hori-
xontal plane perpendicular to the x-axis, and the
/.-axis extends vertically. The plume travels along
or parallel to the x-axis. Figure 3-1 illustrates the
coordinate system.
becomes essentially level, and is the sum of the
physical stack height, h, and the plume rise, ^H.
The following assumptions are made: the plume
spread has a Gaussian distribution (see Appendix
2) in both the horizontal and vertical planes, with
standard deviations of plume concentration distri-
bution in the horizontal and vertical of a, and
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1-16
Any consistent set of units may be used. The most
common is:
,\ (g ITT ) or, for radioactivity (curies m"1)
Q (g see"') or (curies sec"1)
u (m sec"1)
-------
1-17
Some preliminary results of a dispersion experi-
ment in St. Louis (Pooler, 1965) showed that the
dispersion over the city during the daytime behaved
somewhat like types B and C; for one night experi-
ment ir, varied with distance between types D and E.
ESTIMATION OF VERTICAL AND
HORI/OMAL DISPERSION
Having determined the stability class from
Table 'i-1, one can evaluate the estimates of a-, and
", as a function of downwind distance from the
source, x. using Figures 3-2 and 3-3. These values
of " and .r, are representative for a sampling time
of aliout 10 minutes. For estimation of concentra-
tions for longer time periods see Chapter 5. Figures
.'i-2 and .'!-.'{ apply strictly only to open level country
and probably underestimate the plume dispersion
potential from low-level sources in built-up areas.
Although the vertical spread may be less than the
values lot class F with very light winds on a clear
night, quantitative estimates of concentrations are
nearly impossible for this condition. With very light
winds on a clear night for ground-level sources free
of topographic influences, frequent shifts in wind
direction usually occur which serve to spread the
plume horizontally. For elevated sources under
these extremely stable situations, significant con-
centrations usually do not reach ground level until
the stability changes.
A stable layer existing above an unstable layer
will have the effect of restricting the vertical diffu-
sion. The dispersion computation can be modified
lor this situation by considering the height of the
base of the stable layer, L. At a height 2.15 2,,.; x, is where n.,. --
The value of „„. 0.8 L
0.47 L
from any z from 0 to L
for x -'2 xL; x,, is where
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1-18
10,000
DISTANCE DOWNWIND, km
Figure 3-2. Horizontal dispersion coefficient as a function of downwind distance from the source.
ATMOSPHERIC DISPERSION ESTIMATES
-------
1-19
DISTANCE DOWNWIND, km
Figure 3-3. Vertical dispersion coefficient as a tunction of downwind distance from the source.
Estimates
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1-20
12345 6*10
CONC
SSOmeters
Figure 3-4. Variations in concentration in the vertical beneath a more stable layer
three cases (where CTZ can be expected to be within
a factor of 2) should be correct within a factor of 3,
including errors in o-y and u. The relative confidence
in the IT'S (in decreasing order) is indicated by the
heavy lines and dashed lines in Figures 3-2 and 3-3.
Estimates of H, the effective height of the plume,
may be in error because of uncertainties in the esti-
mation of AH, the plume rise. Also, for problems
that require estimates of concentration at a specific
point, the difficulty of determining the mean wind
over a given time interval and consequently the
location of the x-axis can cause considerable un-
certainty.
GRAPHS FOR ESTIMATES OF DIFFUSION
To avoid repetitious computations, Figure 3-5
(A through F) gives relative ground-level concen-
trations times wind speed (,\ u Q) against down-
wind distances for various effective heights of emis-
sion and limits to the vertical mixing for each sta-
bility class (1 figure for each stability). Computa-
tions were made from Eq. (3.3), (3.4), and (3.5).
Estimates of actual concentrations may be deter-
mined by multiplying ordinate values by Q/u.
PLOTTING GROUND-LEVEL
CONCENTRATION ISOPLETHS
Often one wishes to determine the locations
where concentrations equal or exceed a given mag-
nitude. First, the axial position of the plume must
be determined by the mean wind direction. For
plotting isopleths of ground-level concentrations,
the relationship between ground-level centerline
concentrations and ground-level off-axis concentra-
tions can be used:
(x,y,0;H)
(x,0,0;H)
= exp
(3.7)
The y coordinate of a particular isopleth from the
x-axis can be determined at each downwind dis-
tance, x. Suppose that one wishes to know the
off-axis distance to the 1CT' g m~' isopleth at an x
of 600 m, under stability type B, where the ground-
level centerline concentration at this distance is
2.9 x 10- s m- .
(x.y.O;H)
exp —
(x,0,0;H)
2.9 x 10
— = 0.345
10
ATMOSPHERIC DISPERSION ESTIMATES
-------
1-21
DISTANCE km
Figure 3-5A. xu/Q with distance for various heights of emission (H) and limits to vertical dispersion (L), A stability.
Estimates
11
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1-22
DISTANCE, km
Figure 3-5B. \u Q with distance for various heights of emission (H) and limits to vertical dispersion (I), B stability
12
ATMOSPHERIC DISPERSION ESTIMATES
-------
1-23
DISTANCE, km
Figure 3-5C. xirQ with distance for various heights of emission (H) and limits to vertical dispersion (L), C stability.
Estimates
13
-------
1-24
DISTANCE, km
Figure 3-5D. xu Q with distance for various heights of emission (H) and limits to vertical dispersion (L), D stability.
14
ATMOSPHERIC DISPERSION ESTIMATES
-------
1-25
10
100
DISTANCE, km
Figure 3-5E. xu 'Q with distance for various heights of emission (H) and limits to vertical dispersion (L), E stability.
Estimates
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1-26
10'
DISTANCE, km
Figure 3-5F. xu Q with distance for various heights of emission (H) and limits to vertical dispersion (L), F stability.
16
ATMOSPHERIC DISPERSION ESTIMATES
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1-27
Fi"H, Table A-l (Appendix 3) when exp
- 1.46
From Figure 3-2, for stability B and x = 600 m, <7V
92. Therefore y (1.46) (92) = 134 meters.
This is the distance of the 10"- isopleth from the
x-axis at a downwind distance of 600 meters.
can also he determined from:
v
-------
(NO
oo
H
O
a
M
S
O
C/3
"fl
W
C/3
H
CLASS A STABILITY
3 4
DOWNWIND DISTANCE («|, km
H
M
M
Figure 3-6A. Isopleths of ,xu Q for a ground-level source, A stability.
-------
M
C/l
a
3
CLASS B STABILITY
H = 0
ICT3 10"
3 4
DOWNWIND DISTANCE (x|. In
Figure 3-6B. Isopleths of ,\u Q for a ground-level source, B stability.
-------
3
o
en
•fl
1
M
PS
Cfl
M
O
PI
M
H
5
>
H
M
03
CLASS C STABILITY
Oo
O
DOWNWIND DISTANCE («),
Figure 3-6C. Isopleths of \u Q for a ground-level source, C stability.
-------
M
(n
St.
3
D
S
f>
w>
O
CLASS D STABILITY
H= 0
3 4
DOWNWIND DISTANCE («|. kn
Figure 3-6D. Isopleths of xu Q for a ground-level source, D stability.
-------
I 0
-i 0.5
¥
tst
IS*
O
CLASS E STABILITY
DOWNWIND DISTANCE (i), km
H
O
13
a
M
2
O
o
NH
C/l
"fl
s
M
O
M
7)
H
3 4
DOWNWIND DISTANCE («). km
W
c/s
Figure 3-6E, F. Isopleths of \u Q for a ground-level source, E and F stabilities.
-------
CLASS A STABILITY
= IOO
3 4
DOWNWIND DISTANCE («), km
Figure 3-7A. Isopleths or ,\u 'Q for a source 100 meters high, A stability.
00
00
-------
00
•fl
s
M
2
O
C/3
"0
M
SB
on
O
25
C/5
H
S
>
H
P3
Cfl
CLASS B STABILITY
H=IOO
1.7 x I O"5
3 *
DOWNWIND DISTANCE («), km
Figure 3-7B. Isopleths of ,\u Q for a source 100 meters high. B stability.
-------
CLASS C STABILITY
H = 100
3 4
DOWNWIND DISTANCE («), km
Figure 3-7C. Isopleths of \u Q for a source 100 meters high, C stability.
-------
00
H
O
s
PI
O
O
"0
M
7)
O
2;
B
H
CLASS D STABILITY
H= 100
tlftt
DOWNWIND DISTANCE (>). k
H
M
Figure 3-7D. Isopleths of \u Q for a source 100 meters high, D stability.
-------
CLASS E STABILITY
= IOO
3 4
DOWNWIND DISTANCE (x), km
CLASS F STABILITY
H=IOO
0 5
o LL
3 4
DOWNWIND DISTANCE (i ), I m
Figure 3-7E, F. Isopleths of \u Q for a source 100 meters high, E and F stabilities.
-------
1-38
10"
1
1
L ^
N.I
, M
E w
X
!N
i i
i
"•»
1
1
; ! ; !
;
~i J '
! p '
1 1 ' I
i :.; ! !
i ' •
i
j
I
I
;
i i '
i ; ;
! i ; i
;
i
1
• i !
i '
!,
i
1
1 ,
i
Xu
Q
n-2
Figure 3-8. Area within isopleths for a ground-level source (from Hilsmeier and Gifford).
Hilsmeier. W. F.. and F. A. Gilford, 1962: Graphs
liu1 estimating atmospheric (IjfTusion. ORO-54f>.
(.'.ik Kulue. Tenn. Atomic Knergy Commission.
1- pp.
List. R I . Kl.'il: Srnifh^unian Met<>tir'>lugii:al
1 •: ',.-. ^i\th Revved Edition, 497-.">05. Wash-
'ii4'.,.r,. !' ('.. Smithsonian Institution, .r>27 pp.
M.•*:..:. '' O. 19ti5: Persoiutl t-ommimication.
Pa<'null. K.. If1 til: The estimation of the dispersion
ei'sonal communication.
nm/'lrnrolii^v. i\Y\\ \nr\\.
of windhorne material. Meteorol. Ma^r.. !
1063, :):j-49.
Pooler, F., 19fio.
Sutlon, 0. G. 10.").'!: Mnr
McGraw-Hill .I.'J.'] pp.
Turner. D. B.. 1961: Relai ;onshi|js between L'4-
hour mean air quality measurement^ and tneie-
orological lactors in Nashville, Tenne^.^ee. J.
Air Poll. Cont. Assoc., li. 48.S-4MO.
28
ATMOSPHERIC DISPERSION ESTIMATES
-------
w
V)
§'
u
100
10"'
g Figure 3-9. Distance of maximum conca^'ration and maximum ,\u Q as a function of stability (curves) and effective height (meters) of emission
(numbers).
-------
1-41
Chapter 4 —EFFECTIVE HEIGHT OF EMISSION
GENERAL CONSIDERATIONS
In most problems one must estimate the effec-
tive stack height, H, at which the plume becomes
essentially level. Rarely will this height correspond
to the physical height of the stack, h. If the plume
is caught in the turbulent wake of the stack or of
buildings in the vicinity of the stack, the effluent
will be mixed rapidly downward toward the ground
(aeiodvnfmic down wash). If the plume is emitted
free of these turbulent zones, a number of emission
factors and meteorological factors influence the rise
of the plume. The emission factors are: velocity
of the effluent at the top of the stack, v.; tempera-
ture of the effluent at the top of the stack, Th; and
diameter ol (he stack opening, d. The meteorolog-
ical factor- influencing plume rise are wind speed,
u; temperature ol the air, T,,; shear of the wind
speed with height, du dz; and atmospheric sta-
bihf\ \o theoiy on plume rise takes into account
all ol 1he-,e variables; even if such a theory were
available, measurements of all of the parameters
would seldom be available. Most of the equations
thnl have been formulated for computing the ef-
fective height of emission are semi-empirical. For a
recent review of equations for effective height of
emission see Moses, Strom, and Carson (1964).
Moses and Strom (1961), having compared ac-
tual and calculated plume heights by means of six
plume rise equations, report "There is no one for-
mula which i* outstanding in all respects." The
formulas of Davidson-Bryant (1949), Holland
(1953). Bosanquet-Carey-Halton (1950), and Bo-
sanquel (1957) all give generally satisfactory re-
sults in the test situations. The experiments con-
ducted by Moses and Strom involved plume rise
from a stack of less than 0 5 meter diameter, stack
gas exit velocities less than 15 m sec"1, and effluent
temperature not more than 35 C higher than that
of the ambient air.
The equation of Holland was developed with
experimental data from larger sources than those
of Moses and Strom (stack diameters from 1.7 to
4.3 meters and stack temperatures from 82 to
20-1 C"); Holland's equation is used in the solution
of the problems given in this workbook. This equa-
tion frequently underestimates the effective height
of emission; therefore its use often provides a slight
"safety" factor.
Holland's equation is:
AH Vud- (1.5 + 2.68 x 10": p ~-~~— d) (4.1)
where:
AH the rise of the plume above the stack, m
VB ---= stack gas exit velocity, m sec '
d — the inside stack diameter, m
u -= wind speed, m sec"1
p = atmospheric pressure, mb
Ts = stack gas temperature, CK
Ta = air temperature, ' K
and 2.68 x 10~; is a constant having units of mb"1
m"1.
Holland (1953) suggests that a value between
1.1 and 1.2 times the AH from the equation should
be used for unstable conditions; a value between
0.8 and 0.9 times the AH from the equation should
be used for stable conditions.
Since the plume rise from a stack occurs over
some distance downwind, Eq. (4.1) should not be
applied within the first few hundred meters of the
stack.
EFFECTIVE HEIGHT OF EMISSION AND
MAXIMUM CONCENTRATION
If the effective heights of emission were the
same under all atmospheric conditions, the highest
ground-level concentrations from a given source
would occur with the lightest winds. Generally,
however, emission conditions are such that the ef-
fective stack height is an inverse function of wind
speed as indicated in Eq. (4.1). The maximum
ground-level concentration occurs at some inter-
mediate wind speed, at which a balance is reached
between the dilution due to wind speed and the
effect of height of emission. This critical wind speed
will vary with stability. In order to determine the
critical wind speed, the effective stack height as a
function of wind speed should first be determined.
The maximum concentration for each wind speed
and stability can then be calculated from Figure
3-9 as a function of effective height of emission
and stability. When the maximum concentration
as a function of wind speed is plotted on log-log
graph paper, curves can be drawn for each stability
class; the critical wind speed corresponds to the
point of highest maximum concentration on the
curve (see problem 14).
ESTIMATES OF REQUIRED STACK HEIGHTS
Estimates of the stack height required to pro-
duce concentrations below a given value may be
made through the use of Figure 3-9 by obtaining
solutions for various wind speeds. Use of this figure
considers maximum concentrations at any distance
from the source.
In some situations high concentrations upon the
property of the emitter are of little concern, but
Effective Height
31
-------
1-42
maximum concentrations beyond the property line
are of the utmost importance. For first approxima-
tions it can be assumed that the maximum concen-
tration occurs where \/7I VT = H and that at this
distance the a's are related to the maximum con-
centration by:
Q
-_ 0.117 Q
7T U 6 XM
(4.2)
Knowing the source strength, Q, and the concen-
tration not to be exceeded Xu««, one can determine
the necessary ay at for a given wind speed. Figure
4-1 shows a,
-------
1-43
Distance Downwind, km
Figure 4-1. The product of ^z as a function of downwind distance from the source.
Effective Height
33
-------
1-44
the height. Values other than 4.3 and 2.15 can be
used. When these values are used 97'/< of the dis-
tribution is included within these limits. Virtual
distances x, and x/ can be found such that at x},
", ", and at x,, a?, n Ind. Wastes, 14fh Ann. Meeting, Ind.
Hygiene Found. Amer., 38-55.
Halitsky. J.. 1961: Wind tunnel model test of ex-
haust gas recirculation at the NIH Clinical
Center. Tech. Rep. No. 785.1, New York Univ.
Halh^kv. •)., 1962: Diffusion of vented gas around
buildings. J. Air Poll. Cont. Assoc., 12, 2, 74-80.
Halitsky, J.. 1963: Gas diffusion near buildings,
theoretical concepts and wind tunnel model ex-
periments with prismatic building shapes. Geo-
physical Sciences Lab. Rep. No. 63-3. New
York Univ.
Hawkins, J. E., and G. Nonhebel, 1955: Chimneys
and the dispersal of smoke. J. Inst. Fuel, 28,
530-546.
Holland, J. Z., 1953: A meteorological survey of
the Oak Ridge area. 554-559 Atomic Energy
Comm., Report ORO-99, Washington, D.C.,
584 pp.
Moses, H., and G. H. Strom, 1961: A comparison
of observed plume rises with values obtained
from well-known formulas. J. Air Poll. Cont.
Assoc.. 11, 10, 455-466.
Moses, H., G. H. Strom, and J. E. Carson, 1964:
Effects of meteorological and engineering fac-
tors on stack plume rise. Nuclear Safety, 6, 1,
1-19.
Scorer, R. S., 1959: The behavior of plumes. Int.
J. Air Poll., 1, 198-220.
Sherlock, R. H., and E. J. Lesher, 1954: Role of
chimney design in dispersion of waste gases.
Air Repair, 4. 2, 1-10.
Strom, G. H., 1955-1956: Wind tunnel scale model
studies of air pollution from industrial plants.
Ind. Wastes, Sept. - Oct. 1955, Nov. - Dec. 1955,
and Jan. Feb. 1956.
Strom, G. H., M. Hackman, and E. J. Kaplin, 1957:
Atmospheric dispersal of industrial stack gases
determined by concentration measurements in
scale model wind tunnel experiments. J. Air
Poll. Cont. Assoc., 7, 3, 198-203.
34
ATMOSPHERIC DISPERSION ESTIMATES
-------
1-45
Chapter 5 — SPECIAL TOPICS
CONCENTRATIONS IN AN INVERSION
KRKAk-LP FUMIGATION
A surface-based inversion may be eliminated by
the upward transfer of sensible heat from the
ground surface when that surface is warmer than
the overlying air. This situation occurs when the
ground is being warmed by solar radiation or when
air flows from a cold to a relatively warm surface.
In either situation pollutants previously emitted
above the surface into the stable layer will be mixed
vertically when the}- are reached by the thermal
eddies, and ground-level concentrations can increase.
This process, called "fumigation" was described by
Hewson and Gill (1944) and Hewson (1945). Equa-
tions for estimating concentrations with these con-
ditions have been given by Holland (1953), Hew-
son (1955), Gifford (1960a), Bierly and Hewson
(1962), and Pooler (1965).
To estimate ground-level concentrations under
inversion break-up fumigations, one assumes that
the plume was initially emitted into a stable layer.
Therefore, ,r, and »,,. characteristic of stable condi-
tions must be selected for the particular distance
of concern. An equation for the ground-level con-
centration when the inversion has been eliminated
to a height h, is:
VK (x,y,0;H)
Q
exp ( — 0.5 p ) dp
\/2rr o-yp. U hi
exp
(5.1)
where p
h,—H
and IT, i.- is discussed below.
Values for the integral in brackets can be found in
most statistical tables. For example, see pages 273-
276, Bunngton (1953). This factor accounts for
the portion of the plume that is mixed downward.
If the inversion is eliminated up to the effective
stack height, half of the plume is presumed to be
mixed downward, the other half remaining in the
stable air above. Eq. (5.1) can be approximated
when the fumigation concentration is near its
maximum by:
(x,y,0;HK=
Q
27TU
exp —
H
2
during fumigation, for use in equation (5.2).
Eq. (5.4) should not be applied near the stack,
for if the inversion has been eliminated to a height
sufficient to include the entire plume, the emission
is taking place under unstable not stable conditions.
Therefore, the nearest downwind distance to be
considered for an estimate of fumigation concen-
trations must be great enough, based on the time
reqrired to eliminate the inversion, that this por-
tion of the plume was initially emitted into stable
air. This distance is x = utm, where u is the mean
Special Topics
35
-------
1-46
wind in the stable layer and tm is the time required
to eliminate the inversion from h, the physical
height of the stack to hi (Eq. 5.3).
tm is dependent upon both the strength of the
inversion and the rate of heating at the surface.
Pooler (1965) has derived an expression for esti-
mating this time:
Pa Cp
R
go
&z
(5.5)
where tm = time required for €he mixing layer to
develop from the top of the stack to the
top of the plume, sec
P,, --- ambient air density, g m~3
cp -= specific heat of air at constant pressure,
cal g-> °K-1
R
So
net rate of sensible heating of an air
column by solar radiation, cal m~- sec"1
—- — vertical potential temperature gradient,
'K m"1 ~
rate)
bz
r (the adiabatic lapse
h, height of base of the inversion sufficient
to be above the plume, m
h = - physical height of the stack, m
Note that hi — h is the thickness of the layer to be
heated and f — ^ — - j is the average height of the
layer. Although R depends on season, and cloud
cover and varies continuously with time, Pooler has
used n value of 67 cal m~- sec'1 as an average for
fumigation.
Hewson ( 1945) also suggested a method of esti-
mating the time required to eliminate an /inversion
to a height 2 by use of an equation of Taylor's
(1915, n. 8):
t
where:
z-
4 K
t
(5.6)
time required to eliminate the inver-
sion to height z, sec
z --= height to which the inversion has been
eliminated, m
K = eddy diffusivity for heat, m2 sec"1
Rewriting to compare with Eq. (5.5),
h,= — h=
4 K
(5.7)
Hewson (1945) has suggested a value of 3 m2 sec"1
for K.
PLUME TRAPPING
Plume trapping occurs when the plume is
trapped between the ground surface and a stable
layer aloft. Bierly and Hewson (1962) have sug-
gested the use of an equation that accounts for the
multiple eddy reflections from both the ground and
the stable layer:
X (x,0,z;H) =
Q
27TU
exp —
Hv1
„. ) \
exp -
N = J
+
1 / z — H — 2 NL
+ exp —
exp —
exp —
z + H — 2 NL
—H + 2 NL
z + H + 2 NL
r
(5.8)
where L is the height of the stable layer and J = 3
or 4 is sufficient to include the important reflec-
tions. A good approximation of this lengthy equa-
tion can be made by assuming no effect of the stable
layer until
-------
1-47
these is at the distance of maximum concentration
at the ground. As a rough approximation the maxi-
mum ground-level concentration occurs at the dis-
1
tance where u>. — —j^1 H. This approximation is
much hetter for unstable conditions than for stable
conditions. With this approximation, the ratio of
concentration at plume centerline to that at the
ground is:
,V (x, 0,H)
~.v
-------
1-48
Table 5-1 VARIATION OF CALCULATED CONCENTRATION
WITH SAMPLING TIME
Ratio of
Calculated Concentration
Sampling Time
3 minutes
15 minutes
1 hour
3 hours
24 hours
to 3-minute Concentration
1.00
0.82
0.61
0.51
0.36
This table indicates a power relation with time:
\ ot t~'MT. Note that these estimates were based
upon published dispersion coefficients rather than
upon sampling results. Information in the refer-
ences cited indicates that effects of sampling time
are exceedingly complex. If it is necessary to esti-
mate concentrations from a single source for the
time intervals greater than a few minutes, the best
estimate apparently can be obtained from:
(5.12)
where \s is the desired concentration estimate for
the sampling time, t»; \t is the concentration esti-
mate for the shorter sampling time, tk, (probably
about 10 minutes); and p should be between 0.17
and 0.2. Eq. (5.12) probably would be applied
most appropriately to sampling times less than 2
hours (see problem 19).
ESTIMATION OF SEASONAL OR AJNNUAL
AVERAGE CONCENTRATIONS AT A
RECEPTOR FROM A SINGLE POLLUTANT
SOURCE
For a source that emits at a constant rate from
hour to hour and day to day, estimates of seasonal
or annual average concentrations can be made for
any distance in any direction if stability wind "rose"
data are available for the period under study. A
wind rose gives the frequency of occurrence for
each wind direction (usually to 16 points) and wind
speed class (9 classes in standard Weather Bureau
use) for the period under consideration (from 1
month to 10 years). A stability wind rose gives the
same type of information for each stability class.
If the wind directions are taken to 16 points and
it is assumed that the wind directions within each
sector are distributed randomly over a period of a
month or a season, it can further be assumed that
the effluent is uniformly distributed in the hori-
zontal within the sector (Holland, 1953, p. 540).
The appropriate equation for average concentration
is then either:
X =
2 Q
X =
2.03Q
-------
1-49
2. For elevated sources maximum "instantaneous"
concentrations occur with unstable conditions
when portions of the plume that have undergone
little dispersion are brought to the ground.
These occur close to the point of emission (on
the order of 1 to 3 stack heights). These con-
centrations are usually of little general interest
because of their very short duration; they can-
not be estimated from the material presented in
this workbook.
3. For elevated sources maximum concentrations
for time periods of a few minutes occur with
unstable conditions; although the concentra-
tions fluctuate considerably under these condi-
tions, the concentrations averaged over a few
minutes are still high compared to those found
under other conditions. The distance of this
maximum concentration occurs near the stack
(from 1 to 5 stack heights downwind) and the
concentration drops off rapidly downwind with
increasing distance.
4. For elevated sources maximum concentrations
for time periods of about half an hour can occur
with fumigation conditions when an unstable
layer increases vertically to mix downward a
plume previously discharged within a stable
layer. With small AH, the fumigation can occur
close to the source but will be of relatively short
duration. For large AH, the fumigation will
occur some distance from the stack (perhaps 30
to 40 km), but can persist for a longer time
interval. Concentrations considerably lower than
those associated with fumigations, but of sig-
nificance can occur with neutral or unstable
conditions when the dispersion upward is se-
verely limited by the existence of a more stable
layer above the plume, for example, an inversion.
5. Under stable conditions the maximum concen-
trations at ground-level from elevated sources
are less than those occurring under unstable
conditions and occur at greater distances from
the source. However, the difference between
maximum ground-level concentrations for stable
and unstable conditions is only a factor of 2
for effective heights of 25 meters and a factor
of 5 for H of 75 m. Because the maximum
occurs at greater distances, concentrations that
are below the maximum but still significant can
occur over large areas. This becomes increas-
ingly significant if emissions are coming from
more than one source.
CONCENTRATIONS AT A RECEPTOR POINT
FROM SEVERAL SOURCES
Sometimes, especially for multiple sources, it is
convenient to consider the receptor as being at the
origin of the diffusion coordinate system. The
source-receptor geometry can then be worked out
merely by drawing or visualizing an x-axis oriented
upwind from the receptor and determining the
crosswind distances of each source in relation to this
x-axis. As pointed out by Gifford (1959), the con-
centration at (0, 0, 0) from a source at (x, y, H)
on a coordinate system with the x-axis oriented up-
wind is the same as the concentration at (x, y, 0)
from a source at (0, 0, H) on a coordniate system
with the x-axis downwind (Figure 5-2). The total
concentration is then given by summing the indi-
vidual contributions from each source (see problem
20).
SOURCE
U,y,H|
UPWIND
RECEPTOR
(0,0,01
DOWNWIND
Figure 5-2. Comparison of source-oriented and receptor-
oriented coordinate systems.
It is often difficult to determine the atmos-
pheric conditions of wind direction, wind speed, and
stability that will result in the maximum combined
concentrations from two or more sources; drawing
isopleths of concentration for various wind speeds
and stabilities and orienting these according to
wind direction is one approach.
AREA SOURCES
In dealing with diffusion of air pollutants in
areas having large numbers of sources, e.g., as in
urban areas, there may be too many sources of most
atmospheric contaminants to consider each source
Special Topics
39
-------
1-50
individually. Often an approximation can be made
by combining all of the emissions in a given area
and treating this area as a source having an initial
horizontal standard deviation, y,0;H) =
sin 0
(5.19)
This equation should not be used where & is less
than 45°.
40
ATMOSPHERIC DISPERSION ESTIMATES
-------
1-51
When estimating concentrations from finite line
sources, one must account for "edge effects" caused
by the end of the line source. These effects will of
course extend to greater cross-wind distances as
the distance from the source increases. For concen-
trations from a finite line source oriented cross-
wind, define the x-axis in the direction of the mean
wind and passing through the receptor of interest.
The limits of the line source can be defined as ex-
tending from y, to y., where y, is less than y.,. The
equation for concentration (from Button's (1932)
equation (11), p. 154), is:
-------
1-52
Gifford, F. A., 1959: Computation of pollution
from several sources. Int. J. Air Poll., 2, 109-
110.
Gifford, F. A., 1960a: Atmospheric dispersion cal-
culations using the generalized Gaussian* plume
model. Nuclear Safety, 2, 2, 56-59, 67-68.
Gifford, F. A., 1960b: Peak to average concentra-
tion ratios according to a fluctuating plume1 dis-
persion model. Int. J. Air Poll., 3, 4, 253-260.
Hewson, E. W., and G. C. Gill, 1944: Meteorolog-
ical investigations in Columbia River Valley
near Trail, B. C., pp 23-228 in Report submitted
to the Trail Smelter Arbitral Tribunal by R. S.
Dean and R. E. Swain, Bur. of Mines Bull 453,
Washington, Govt. Print. Off., 304 pp.
Hewson, E. W., 1945: The meteorological control
of atmospheric pollution by heavy industry.
Quart. J. R. Meteorol. Soc., 71, 266-282.
Hewson, E. W., 1955: Stack heights required to
minimize ground concentrations. Trans. ASME
77, 1163-1172.
Holland, J. Z., 1953: A meteorological survey of
the Oak Ridge area, p. 540. Atomic Energy
Comm., Report ORO-99, Washington, D. C.,
584 pp.
Nonhebel, G., 1960: Recommendations on heights
for new industrial chimneys. J. Inst. Fuel, 33,
479-513.
Pooler, F., 1965: Potential dispersion of plumes
from large power plants. PHS Publ. No. 999-
AP-16, 1965. 13 pp.
Singer, I. A., 1961: The relation between peak and
mean concentrations. J. Air Poll. Cont. Assoc.,
11, 336-341.
Singer, I. A., K. Imai, and R. G. Del Campos, 1963:
Peak to mean pollutant concentration ratios for
various terrain and vegetation cover. J. Air Poll.
Cont. Assoc., 13, 40-42.
Slade, D. H., 1965: Dispersion estimates from pol-
lutant releases of a few seconds to 8 hours in
duration. Unpublished Weather Bureau Report.
Aug. 1965.
Stewart, N. G., H. J. Gale, and R. N. Crooks, 1958:
The atmospheric diffusion of gases discharged
from the chimney of the Harwell Reactor BEPO.
Int. J. Air Poll., 1, 87-102.
Sutton, 0. G., 1932: A theory of eddy diffusion in
the atmosphere. Proc. Roy. Soc. London, A,
135, 143-165.
Taylor, G. I., 1915: Eddy motion in the atmos-
phere. Phil. Trans. Roy. Soc., A, 215, 1-26.
42
ATMOSPHERIC DISPERSION ESTIMATES
-------
1-53
Chapter 6 — RELATION TO OTHER DIFFUSION EQUATIONS
Most other widely used diffusion equations are
variant forms of the ones presented here. With re-
spect to ground-level concentrations from an ele-
vated source (Eq. 3.2):
X (x,y,0;H) =
1
Q
TT (7V a, u
exp —
exp —
2 V ^ J I (3.2)
Other well-known equations can be compared:
Bosanquet and Pearson (1936):
Q
(x,y,0;H) = =^-
y2?r pq x- u
exp —
y
qx
1 f H 1
I exp —
J L PXJ
(6.1)
where p and q are dimensionless diffusion coeffi-
cients.
Sutton (1947):
x (x,y,0;H) =
y-' , H;
2 Q
Cz x"
exp —
(6.2)
where n is a dimensionless constant and Cy and Cz
are diffusion coefficients in m"/2
Calder (1952):
x (x,y,0;H) =-
Q u
2 k- a vx- x-
exp
k vx x
(6.3)
where a --= —-. the ratio of horizontal eddy velocity
w
to vertical eddy velocity, k is von Karman's con-
k u
stant approximately equal to 0.4, and v, = p—
where z0 is a roughness parameter, m. z0
NOTE: Calder wrote the equation for the con-
centration at (x, y, z) from a ground-level source.
For Eq. (6.3) it is assumed that the concentration
at ground level from an elevated source is the same
as the concentraton at an elevated point from a
ground-level source.
Table 6-1 lists the expressions used in these
equations that are equivalent to ay and -_, ">
V2 '
Calder \ 2 a k v, x
u
V'2 p x
1 2'n
r °
y I k vx x
u
REFERENCES
Bosanquet, C. H., and J. L. Pearson, 1936: The
spread of smoke and gases from chimneys.
Trans. Faraday Soc., 32, 1249-1263.
Calder, K. L., 1952: Some recent British work on
the problem of diffusion in the lower atmos-
phere, 787-792 in Air Pollution, Proc. U. S.
Tech. Conf. Air Poll., New York, McGraw-Hill,
847 pp.
Sutton, 0. G., 1947: The problem of diffusion in
the lower atmosphere. Quart. J. Roy. Met Soc.,
73, 257-281.
Other Equations
43
-------
1-55
Chapter 7 — EXAMPLE PROBLEMS
The following 26 example problems and their
solutions illustrate the application of most of the
techniques and equations presented in this work-
book.
PROBLEM 1: It is estimated that a burning
dump emits 3 g sec"' of oxides of nitrogen.
What is the concentration of oxides of nitrogen,
averaged over approximately 10 minutes, from
this source directly downwind at a distance of
3 km on an overcast night with wind speed of
7 m sec"1? Assume this dump to be a point
ground-level source with no effective rise.
SOLUTION: Overcast conditions with a wind
speed of 7 m sec"1 indicate that stability class D
is most applicable (Statement, bottom of Table
3-1). For x = 3 km and stability D, <7, = 190 m
from Figure 3-2 and r B stability
and this effective height of 150 m is 7.5 x 10"".
vu Q 7.5 x 10"" x 151
= 2.8 x 10~4 g m"3 of S02
PROBLEM 5: For the power plant in problem 4,
at what distance does the maximum ground-
Example Problems
45
-------
1-56
level concentration occur and what is this con-
centration on an overcast day with wind speed
4 m sec""1?
SOLUTION: On an overcast day the stability
class would be D. From Figure 3-9 for D sta-
bility and H of 150 m, the distance to the point
of maximum ground-level concentration is 5.6
km, and the maximum xu/Q is 3.0 x 10~e.
Xm«i
3.0 x 10~8 x 151
= 1.1 xlO-'gnT1
PROBLEM 6: For the conditions given in prob-
lem 4, draw a graph of ground-level centerline
sulfur dioxide concentration with distance from
100 meters to 100 km. Use log-log graph paper.
SOLUTION: The frontal inversion limits the mix-
ing to L = 1500 meters. The distance at which
1 x'
' J g m-'
2.9 xlO-8
3.8 x 10-"
2.3 x 10-*
2.8 x ID"1
1.4 x 10-*
7.1x10-'
2.1 x 10-'
X'
g m~:
6.9x10-"
3.0 x 10-«
1.1 x 10-"
PROBLEM 7: For the conditions given in prob-
lem 4, draw a graph of ground-level concentra-
tion versus crosswind distance at a downwind
distance of 1 km.
SOLUTION: From problem 4 the ground-level
centerline concentration at 1 km is 2.8 x 10~*
g irT3. To determine the concentrations at dis-
tances y from the x-axis, the ground-level cen-
terline concentration must be multiplied by the
factor exp I — 1/2 "^
<7y = 157 meters at x = 1 km. Values for this
computation are given in Table 7-2.
Table 7-2 DETERMINATION OF CROSSWIND
CONCENTRATIONS (PROBLEM 7)
y,
m
± 100
±200
±300
±400
±500
y
0.64
1.27
1.91
2.55
3.18
'"K(i)1
0.815
0.446
0.161
3.87 x 10~2
6.37 x 10-'
x (*.y,o)
2.3 x 10-1
1.3x10-*
4.5 x 10-5
1.1x10-'
1.8x10-"
These concentrations are plotted in Figure 7-2.
PROBLEM 8: For the conditions given in prob-
lem 4, determine the position of the 10 "• g m~'
ground level isopleth, and determine its area.
SOLUTION: From the solution to problem 6, the
graph (Figure 7-1) shows that the 10~5 g m~'
isopleth intersects the x-axis at approximately
x = 350 meters and x = 8.6 kilometers.
46
ATMOSPHERIC DISPERSION ESTIMATES
-------
1-57
CROSSWIND DISTANCE ly) m
Figure 7-2. Concentration as a function of crosswind
distance (Problem 7).
The values necessary to determine the isopleth
half widths, y, are given in Table 7-3.
Table 7-3 DETERMINATION OF ISOPLETH WIDTHS
(PROBLEM 8)
x,
km
0.5
0.8
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
m
83
129
157
295
425
540
670
780
890
980
v 'CPnterlmel,
g m-
3.8x10-
2.3x10-'
28x10-'
14x 10-'
7.1 x ID-'"1
4. Ox 10- :'
2.4 x 10-'
1.8 x 10-
1.4x10--'
1.1 x 10-"
v (isopleth)
v (centerline)
0.263
4.35 x lO"2
3.53 x 10--'
7.14x10-=
1.42x10-'
0.250
0.417
0556
0.714
0.909
y/,,
1.64
2.50
2.59
2.30
1.98
1.67
1.32
1.08
0.82
0.44
y,
m
136
323
407
679
842
902
884
842
730
432
The orientation of the x-axis will be toward
225 close to the source, curving more toward
210 to 215 azimuth at greater distances be-
cause of the change of wind direction with
height. The isopleth is shown in Figure 7-3.
Since the isopleth approximates an ellipse, the
area may be estimated by ,. ab where a is the
semimajor axis and b is the semiminor axis.
a ----
8600 — 350
= 4125m
b = 902
A (rrr) = TT (4125) (902)
= 11.7 x 10KmJ
or A --= 11.7 km-
SOURCE
Figure 7-3. Location of the 10 6 g m ' ground-level iso-
pleth (Problem 8).
PROBLEM 9: For the conditions given in problem
4, determine the profile of concentration with
height from ground level to z = 450 meters at
x = 1 km, y = 0 meters, and draw a graph of
concentration against height above ground.
SOLUTION: Eq. (3.1) is used to solve this prob-
lem. The exponential involving y is equal to 1.
At x --= 1 km,
-------
1-58
Table 7-4 DETERMINATION OF CONCENTRATIONS FOR
VARIOUS HEIGHTS (PROBLEM 9)
b.
d.
f.
0—1.36
30—1.09
60-0.82
90—0.55
120—0.27
150
180
210
240
270
300
330
360
390
420
450
0.0
0
0
0
1
1.
1.
1.
2.
.27
.55
.82
.09
36
64
91
18
2.45
2.
73
0.397
0.552
0.714
0.860
0.964
1.0
0.964
0.860
0.714
0.552
0.397
0.261
0.161
0.0929
0.0497
0.0241
1.36
1.64
1.91
2.18
2.45
2.73
3.00
3.27
3.54
3.82
4.09
4.36
4.64
4.91
5.18
5.45
0.397
0.261
0.161
0.0929
0.0497
0.0241
1.11
4.77
x
x
1.90 x
6.78
2.33
7.45
2.11
5.82
1.49
3.55
X
X
X
X
X
X
X
10-'
lO-3
io-3
10-'
10-'
10-'
irrs
10-"
io-°
10"
0.794
0.813
0.875
0.953
1.014
1.024
0.975
0.865
0.716
0.553
0.397
0.261
0.161
0.093
0.050
0.024
2.78 x
2.85 x
3.06 x
3.34 x
3.55
3.58
3.41
3.03
2.51
x
x
x
x
x
1.94x
1.39
9.14
5.64
3.26
x
x
x
X
1.75 x
8.40
X
10-'
10-'
10-'
10-'
10-'
10-'
10-'
10-'
10-'
10-'
10-'
10-"
10-"
10-'
10-'
10-"
These values are plotted in Figure 7-4.
500
OlO'5 10" 2«!0'4 3»IO'4 4MO'4
CONCENTRATION, g m->
Figure 7-4. Concentration as a function of height (Prob-
lem 9).
Verifying:
X (x,0,0) =
Q
y a, U
exp ^
exp —
151
2- 181 (136) 4
1 / 300
[1 / 0 Vl
--2-1-136-) J
11
= 2.44x
1.0 + exp -- -
= 2.44 x 10-' (1.0 + 8.70 x 10~-)
= 2.44 x 10-' (1.087)
= 2.7 x 10-' g nT3
PROBLEM 11: For the power plant in problem 4,
what will the maximum ground-level concentra-
tion be beneath the plume centerline and at
what distance will it occur on a clear night with
wind speed 4 m sec~l?
SOLUTION: A clear night with wind speed 4 m
sec"1 indicates E stability conditions. From Fig-
ure 3-9, the maximum concentration should
occur at a distance of 13 km, and the maximum
xu/Q is 1.7 x 10-"
Xmox =
Q
Q
u
= 6.4 x 10-= g
1.7 x 10-° x 151
' of S02
PROBLEM 12: For the situation in problem 11,
what would the fumigation concentration be the
next morning at this point (x = 13 km) when
superadiabatic lapse rates extend to include
most of the plume and it is assumed that wind
speed and direction remain unchanged?
SOLUTION: The concentration during fumiga-
tion conditions is given by Eq. (5.2) with the
exponential involving y equal to 1. in this prob-
lem.
XF (x,0,0;H) = _
_
\/2ir U o-yF hi
For the stable conditions, which were assumed
to be class E, at x = 13 km,
-------
1-59
(stable)
151
H 8 -•-• 520 + 19 = 539
\/2^4 (539) 330
= 8.5 x 10"r' g m"3 of SO.,
Note that the fumigation concentrations under
these conditions are about 1.3 times the maxi-
mum ground-level concentrations that occurred
during the night (problem 11).
PROBLEM 13: An air sampling station is located
at an azimuth of 203c from a cement plant at a
distance of 1500 meters. The cement plant re-
leases fine particulates (less than 15 microns
diameter) at the rate of 750 pounds per hour
from a 30-meter stack. What is the contribution
from the cement plant to the total suspended
particulate concentration at the sampling sta-
tion when the wind is from 30° at 3 m sec"1 on
a clear day in the late fall at 1600?
SOLUTION: For this season and time of day the
C class stability should apply. Since the sam-
pling station is off the plume axis, the x and y
distances can be calculated:
x= 1500 cos 7° = 1489
y -= 1500 sin 7° = 183
The source strength is:
Or cpp~l
Q = 750 Ib hr1 x 0.126 —° , _ = 94.5 g sec"1
Ib hr 1
At this distance, 1489 m, for stability C, a, =
150 m, 1
19'5 [15 1
- u H.5 1
19.5 (2.5)
'>]
-6 ( 10M 15
" "'6 ( 394 ) 1'5
2.6 (0.256) 1.5]
1.0]
= 48.8
u
The effective stack heights for various wind
speeds and stabilities are summarized in Table
7-5.
Table 7-5 EFFECTIVE STACK HEICffTS (PROBLEM 14)
u,
m sec"1
0.5
1.0
1.5
2
3
5
7
10
20
Class
AH,
m
97.6
48.8
32.6
24.4
16.3
9.8
7.0
4.9
2.4
D
h + AH,
m
127.6
78.8
62.6
54.4
46.3
39.8
37.0
34.9
32.4
Class B
1.15 AH, h -f
m
112.2
56.1
37.5
28.1
18.7
11.3
8.0
1.15 AH,
m
142.2
86.1
67.5
58.1
48.7
41.3
38.0
By use of the appropriate height, H, the maxi-
mum concentration for each wind speed and
stability can be determined by obtaining the
Example Problems
49
-------
1-60
maximum xu/Q as a function of H and stability
from Figure 3-9 and multiplying by the appro-
priate Q/u. The computations are summarized
in Table 7-6, and plotted in Figure 7-5.
i or'
7
5
i i r
i i i
0.5
2 34
WIND SPEED, m
20
Figure 7-5.
Maximum concentration as
wind speed (Problem 14).
a function of
Table 7-6 MAXIMUM CONCENTRATION AS A FUNCTION OF
WIND SPEED (PROBLEM 14)
Stability
Class
B
D
u,
m sec~'
0.5
1.0
1.5
2
3
5
7
0.5
1.0
1.5
2
3
5
7
10
20
H,
m
142.2
86.1
67.5
58.1
48.7
41.3
38.0
127.6
78.8
62.6
54.4
46.3
39.8
37.0
34.9
32.4
xu/Q..«-
m~J
8.0 x 10-«
2.0 x 10-'
3.1 x 10-=
4.1 x 10-'
5.7 x 10-"
7.8 x 10-''
8.7 x 10-'
4.4 x 10-°
1.42xlO-5
2.47x10-'
3.5 x 10-'1
5.1x10-'
7.3 x 10-"
8.2 xlO~5
9.4 x 10-3
1.1 x 10-«
Q/u,
g m-1
144
72
48
36
24
14.4
10.3
144
72
48
36
24
14.4
10.3
7.2
3.6
Xmax'
g m-3
1.15 xlO-3
1.44x10-"
1.49 xlO-3-*-
1.48 x 10-3
1.37 x 10-3
1.12xlO-3
8.96 x 10-«
6.34 x 10-'
1.02 x 10-3
1.19 xlO-3
1.26xlO-3«-
1.22 xlO-3
1.05 x 10-3
8.45 x 10-'
6.77 x 10-'
3.96 x 10-*
The wind speeds that give the highest maximum
concentrations for each stability are, from Fig-
ure 7-5: B 1.5, D 2.0.
PROBLEM 15: A proposed pulp processing plant
is expected to emit ^ ton per day of hydrogen
sulfide from a single stack. The company prop-
erty extends a minimum of 1500 meters from
the proposed location. The nearest receptor
is a small town of 500 inhabitants 1700 meters
northeast of the plant. Plant managers have
decided that it is desirable to maintain
concentrations below 20 ppb (parts per billion
by volume), or approximately 2.9 x 10~5 g m~ ,
for any period greater than 30 minutes. Wind
direction frequencies indicate that winds blow
from the proposed location toward this town
between 10 and 15 per cent of the time. What
height stack should be erected? It is assumed
that a design wind speed of 2 m sec"1 will be
sufficient, since the effective stack rise will be
quite great with winds less than 2 m sec"1.
Other than this stipulation, assume that the
physical stack height and effective stack height
are the same, to incorporate a slight safety
factor.
SOLUTION: The source strength is:
1000 Ib day'1 x 453.6 g Ib "'
y ~~ 86,400 sec day"1
FromEq. (4.2):
0.117 Q 0.117 (5.25)
= 5.25 g sec
(2.9 x 10-°) 2
= 1.06 x 104 m2
At a design distance of 1500 meters (the limit
of company property),
-------
1-61
.AH ==-
33.4
u
33.4
u
102
u
[1.5 + (2.46) 0.256 (2.44)]
(1.5 + 1.54)
The relation between as az and u is:
0.117 Q 0.117 (5.25) 2.12 x 104
17,
-------
1-62
PROBLEM 19: At a point directly downwind
from a ground-level source the 3- to 15-minute
concentration is estimated to be 3.4 x 10~3 g
m~J. What would you estimate the 2-hour con-
centration to be at this point, assuming no
change in stability or wind velocity?
SOLUTION: Using Eq. (5.12) and letting k = 3
min, s = 2 hours, and p = 0.2:
X •• l,..ur =
120
1
40
3.4 x 10
2.09
3.4 x 10-
(3.4 x 10-')
= 1.6 x 10-'g in-
Letting k 15 min, s = 2 hours, and p = 0.17
15
120
1
Q u
3.4 x 10
1.42
3.4 x 10-
(3.4 x 10-')
= 2.4 x 10~' gm~
The 2-hour concentration is estimated to be
between 1.6 x 10": and 2.4 x 10"' g m"'.
PROBLEM 20: Two sources of SO, are shown as
points A and B in Figure 7-6. On a sunny
summer afternoon the surface wind is from 60°
at 6 m sec"1. Source A is a power plant emitting
1450 g sec"1 SO, from two stacks whose physical
height is 120 meters and whose AH, from Hol-
land's equation, is AH (m) = 538 (m- sec'1) 'u
(m sec"1). Source B is a refinery emitting 126 g
sec"' SO. from an effective height of 60 meters.
The wind measured at 160 meters on a nearby
TV tower is from 70° at 8.5 m sec"1. Assuming
that the mean direction of travel of both plumes
is 245 , and there are no other sources of SO.,,
what is the concentration of SO, at the receptor
shown in the figure?
SOLUTION: Calculate the effective height of
Source A using the observed wind speed at 160
meters.
538
8.5
= 63.3
HA = 120 + 63 = 183 m
QA = 1450 g sec'1
HIt = 60 m
QB = 126 g sec"1
For a sunny summer afternoon with wind speed
6 m sec"1, the stability class to be expected is C.
The equation to be used is Eq. (3.2):
SOURCE A
i=24 6 km
,84km
RECEPTOR x.
SOURCE 6
. = 13.0 km
,= 4 0 km
SCALE, km
0 2 4
Figure 7-6. Locations of sources and receptor (Problem
20).
X (x,y,0;H) =
Q
-
TT crv <7Z U
exp
r i / H y]
exp I - -r(—J I
For Source A, x = 24.6 km, y = 8.4 km
a,. = 1810 m, o-z = 1120 m, u = 8.5 m sec"
1450
XA =
-r 1810 (1120) 8.5
8400 V
exp —0.5
1450
l20
exp [—0.5 (4.64)2]
5.42 x 10T
exp [—0.5 (0.164)2]
= 2.67 x 10~5) (2.11 x 10-') (0.987)
XA = 5.6x 10-10gnT:'
For Source B, x = 13.0 km, y = 4.0 km.
<7,. = 1050 m,
-------
PROBLEM 21: A stack 15 meters high emits 3 g
seer1 of a particular air pollutant. The sur-
rounding terrain is relatively flat except for a
rounded hill about 3 km to the northeast whose
crest extends 15 meters above the stack top.
What is the highest 3- to 15-minute concentra-
tion of this pollutant that can be expected on
the facing slope of the hill on a clear night when
the wind is blowing directly from the stack
toward the hill at 4 m sec"1? Assume that AH
is less than 15 m. How much does the wind
have to shift so that concentrations at this point
drop below 10~7 g m~J?
SOLUTION: A clear night with 4 m sec"1 indi-
cates class E stability. Eq. (3.4) for ground-
level concentrations from a ground-level source
is most applicable (See Chapter 5). At 3 km
for class E, a, = 140 m, = 1524'4.3 = 354. For class E the vir-
tual distance, x, = 8.5 km. For x = 1524 m,
a2 = 28.5. For x + xy =10,024 m,
-------
1-64
that it is 1600 on a sunny fall afternoon. What
is the concentration directly downwind from one
end of the source?
SOLUTION: Late afternoon at this tone of year
implies slight insolation, which with 3 m sec"1
winds yields stability class C. For C stability
at x = 400 m, = the
radius of the shell = 20 m o>0 = CTZ() == 9.3 m.
The virtual distances to account for this are:
xy = 250 m, x, = 560 m.
At x = 3000 m. x + xy = 3250 m, ^ = 100 m.
x + xz = 3560 m,
-------
1-65
2.7 x 10- (1.0) The decay of I"1 is insig-
nificant for 2 hours
Xi 2.7 x 10~K curies m~
PROBLEM 26: A spill estimated at 2.9 x 10'
grams of unsymmetrical dimethyl hydrazine
occurs at 0300 on a clear night while a rocket
is being fueled. A circular area 60 meters in
diameter built around the launch pad is revetted
into squares 20 feet on a side to confine to as
small an area as possible any spilled toxic liquids.
In this spill only one such 20- by 20-foot area is
involved. At the current wind speed of 2 m
sec ', it is estimated that the evaporation rate
will be 1100 g sec"1 The wind direction is pre-
dicted to be from 310: ± 15° for the next hour.
Table 7-8 gives the emergency tolerance limits
for UDMH vapor.
Table 7-8 EMERGENCY TOLERANCE LIMITS FOR UDMH
VAPOR VERSUS EXPOSURE TIME
Time,
minutes
5
15
30
60
Emergency Tolerance
Limits, g m~:l
1.2 x 10"'
8.6 x 10"-'
4.9 x 10~J
2.5 x 10--'
What area should be evacuated?
SOLUTION: From Table 3-1, the stability class
is determined to be Class F. This is not a point
source but a small area source. Allowing 4.3
-------
1-66
Table 7-10 DETERMINATION OF WIDTHS WITHIN
ISOPLETHS (PROBLEM 26)
X,
km
0.1
0.5
1.0
2.0
3.0
4.0
5.0
6.0
* + xy,
km
0.14
0.54
1.04
2.04
3.04
4.04
5.04
6.04
(Tv.
m
5.5
19
35
66
93
120
149
175
^ (center-line),
g m-'
3.
1.
7,
13.
1.
6x
.3x
.Ox
4.8 x
3
2
.5x
.7x
9
1
10-'
10-'
10-
10-
10-
10-
^ (isopleth)
^ (centerline)
1.8
2,27
6.94
1.92
3.57
5.20
7.14
9.26
X
X
X
X
X
X
X
X
10-
10-
10-
10-'
10-'
10-'
ID"1
10-'
y
ffy
3
.55
2.75
2.31
1,
1,
1.
0.
0.
.82
,44
.14
.82
.39
y,
m
20
52
80
120
134
137
122
68
145°
SCALE, km
1
Figure 7-8. Possible positions of the 2.5 x 1(TJ g m"
isopleth and the evacuation area (Problem 26).
56
ATMOSPHERIC DISPERSION ESTIMATES
-------
1-67
APPENDICES
339-301 O - 6
-------
1-69
Appendix 1: ABBREVIATIONS AND SYMBOLS
Abbreviation**
cal calorie
K
m
mb
sec
gram
degrees Kelvin
meter
millibar
second
Symbol
a ratio of horizontal eddy velocity to vertical
eddy velocity
c,, specific heat at constant pressure
C, Sutton horizontal dispersion parameter
Cz Sutton vertical dispersion parameter
d inside stack diameter at stack top
Di (x,y,0;H) Total dosage
e 2.7183, the base of natural logarithms
f (n,S,N) frequency of wind direction for a given
stability and wind speed class
h physical stack height
h, height of the base of an inversion
H effective height of emission
Hn effective height of emission for a particular
wind speed
k von Karman's constant, approximately equal
to 0.4
K eddy diffusivity
L two uses: 1. the height of an air layer that is
relatively stable compared to the
layer beneath it; a lid
2. the half-life of a radioactive
material
n Button's exponent
N an index for wind speed class
p three uses: 1. Bosanquet's horizontal disper-
sion parameter
2. atmospheric pressure
3. a dummy variable in the equa-
tion for a Gaussian distribution.
q two uses: 1. Bosanquet's vertical dispersion
parameter
2. emission rate per length of a line
source
Q emission rate of a source
Qi total emission during an entire release
R net rate of sensible heating of an air column
by solar radiation
s the length of the edge of a square area source
S an index for stability
tk a short time period
t,,, time required for the mixing layer to develop
from the top of the stack to the top of the
plume
t,, a time period
Ta ambient air temperature
Ts stack gas temperature at stack top
u wind speed
UN a mean wind speed for the wind speed class N.
V horizontal eddy velocity
v., stack gas velocity at the stack top
v.x a velocity used by Calder
w vertical eddy velocity
x distance downwind in the direction of the
mean wind
X,, design distance, a particular downwind dis-
tance used for design purposes
x,, the distance at which
-------
1-70
<3 the angle between the wind direction and a x» concentration measured over a sampling time,
line source tB
x concentration X relative concentration
Xcwi crosswind-integrated concentration y
Xd a ground-level concentration for design pur- xu ^^ concentration normalized for wind
P°ses Q speed
XF inversion break-up fumigation concentration x (Xiy)Z;H) concentration at the point (x, y, z)
Xk concentration measured over a sampling time, from an elevated source with effective
tk height, H.
x ^ maximum ground-level centerline concentra- x (x,e) the long-term average concentration at
tion with respect to downwind distance distance x, for a direction e from a source.
60 ATMOSPHERIC DISPERSION ESTIMATES
-------
Appendix 2: CHARACTERISTICS OF THE
GAUSSIAN DISTRIBUTION
The Gaussian or normal distribution can be de-
picted by the bellshaped curve shown in Figure A-l.
The equation for the ordinate value of this curve is:
1
exp I
(A.I)
Figure A-2 gives the ordinate value at any distance
from the center of the distribution (which occurs
at x). This information is also given in Table A-l.
Figure A-3 gives the area under the Gaussian curve
from — * to a particular value of p where p =
This area is found from Eq. (A.2):
Area (— v to p) =
1-71
exp (—0.5 p-) dp
(A.2)
Figure A-4 gives the area under the Gaussian
curve from —p to +p. This can be found from Eq.
(A.3):
Area (—p to +p) =
exp (—0.5 p-) dp
f
J -P
1
\/2rr
(A.3)
Figure A-l. The Gaussian distribution curve.
Appendix 2
61
-------
1-72
4 1.6 1.8 2.0 2.2 2.4
0,01 ''
00 02 04
3.8 4.0
Figure A-2. Ordinate values of the Gaussian distribution.
62
ATMOSPHERIC DISPERSION ESTIMATES
-------
1-73
4 0
35
3.0
2.5
2.0
1.5
1.0
0.5
0
-0.5
-1.0
-1 5
-2.0
-2.5
-3.0
-3 5
-4.0
m
:fi]
"3
m
-ttt Tj.
t H
if
I
:jgs
• *+T • -I-H
1
m
0.01 0.1 0.5 1 2 5 10
20 40 60 80 90 95 98 99
-^L- exp (-0.5 p2) dp
99.8 99.99
Figure A-3. Area under the Gaussian distribution curve from — - to p.
Appendix 2
63
-------
1-74
20 30 40 50 60 70 80
' -£r ««P <-0-5 PJ) dp
90 95 98 99 99.8 99.99
Figure A-4. Area under the Gaussian distribution curve between —p and +p.
64
ATMOSPHERIC DISPERSION ESTIMATES
-------
1-75
Appendix 3: SOLUTIONS TO EXPONENTIALS
Expressions of the form exp [—0.5 A-] where
A is H CTZ or y.'o-y frequently must be evaluated.
Table A-l gives B as a function of A where B = exp
[—0.5 A"]. The sign and digits to the right of the
E are to be considered as an exponent of 10. For
example, if A is 3.51, B is given as 2.11E — 03
which means 2.11 x 10~:(
Appendix 3 65
-------
Table A-l SOLUTIONS TO EXPONENTIALS B exp I—O.SA^l
The notation 2.16 E-l means 2.16 x 10 '
CT)
H
g
o
t/i
W
o
55
"o
M
5
2!
M
H
M
H
M
Cfl
A
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0- on
• o \j
0.90
1.00
1.10
1 70
1 . £ U
1.30
1.40
1.50
1.60
1.70
1.80
1.90
2.00
2.10
2.20
2.30
2.40
2.50
2.60
2.70
2.80
2.90
3.00
3.10
3.20
3.30
3.4"
3.50
3.60
3.70
3.80
3.90
4.00
4.10
4.20
4.30
4.40
4.50
4.60
4. 70
4.80
4.90
O.nO
B
I.OOE o
-y. 80E - )
y.ij^E _j
9!23E -1
6.83t -'
H.3SL -1
7.8*1: -1
6.67E -1
6. O^E - 1
<.!30K -1
J.7SE -1
3.2SE -1
2i7RE -1
2. 3ftE - 1
1 .9BE -1
1.6*1 -1
1.3
•..OIF -5
^.S<,E -5
l.ftOt -5
') . 9 1 E -ft
e.UL -ft
0.01
l.OOF 0
9.94E -1
9.7UF -1
9.S3E -1
9. 19F -1
S.78E -1
7|77F -1
6. ft IE -i
ft. OIF - 1
S.40F -1
4.24F -1
3.70F -1
3.20F. -1
2.74F -1
i . 32E - 1
1.94E -1
1 .61F -1
1.33F -1
l.OOF -1
«.70F -2
ft.94E -2
5.48F -2
4.29E -2
3.32F -2
2.54E -2
1.93F -2
I.45F -2
l.OUE -2
7.94E -3
S.79F -3
4.1RF -3
?.99F -3
7. HE -3
I.48F -3
1.03F. -3
7.05F -4
4.7'>r -<•
3.22F -4
?. 1 5C -4
1.42E -4
T, 25c -S
5.98E -5
3.83F -5
^ ,43F -5
1 .S2E -5
9.46F -ft
•>.B2F -ft
0.02
10.00E -\
9.93£ -1
9.51E -1
9.16E -1
8.74E -1
8.25E -1
7.77E -1
7 1 SF 1
6.5SE -1
5 94£ 1
5.34£ -1
4.1«E -1
3.6^E -1
3. 1"E -1
2.69E -1
2.2PE -1
1.91E -1
1.5PE -1
1.30E -1
l.OftE -1
8.51E -2
6.7PE -2
5.3KE -2
4.1«E -1
3.23E -2
2.47E -2
1.8P-E -2
1.41 E -2
1.0'E -?
7. 7r>E - •*
5.60E -*
4.04E -3
2.8"E -3
2.04E -3
1.43E -3
9.8°h -4
6.7°E -4
4.61E -4
3.in£ -4
2 .Oftt -4
1 . 3 ft E -'*
8.8ftE -•>
b.72E -^
3.6ftE -•*
<; . 3?E -s
1.45E -•>
9.0?L -ft
5.54£ -«.
0.03
10.00E -1
9|74E -1
0.47E -1
-V.12E -1
8.69E -1
H.20E -1
7.66E -1
' . 09E - 1
6.49E -1
5.28E -1
-.1 3E -1
3.60E -1
1.IOE -1
^.65E -1
2.24E -1
1.87E -1
1.55E -1
1.27E -1
1.04E -1
8.32E -2
ft .ft2E -2
^>.22E -2
4.07E -2
3.15E -2
2.41E -2
1.82E -2
1.37E -2
1.02E -2
'.46E -3
S.43E -3
3.91E -3
2.79E -3
1.97E -3
1.386 -3
9.53E -4
6.53E -<,
4.43E -4
2.«7E -4
' .98E -4
1.30E -4
«.49E -5
5.4RE -5
J.^OE -5
2.21E -5
1.39E -5
x .59E -ft
S .29E -6
0.04
9.99E -1
9.90E -1
I.72E -1
4.0HE -1
«.ft4E -1
H.15E -1
7.61E -1
'.03E - 1
S «2£ 1
•«.2?E -1
4.0HE -1
^.5^E -1
i.0b£ -1
2. ME -1
Z.2"E -)
1.84E -1
1 .S2E -1
1.2-JE -1
lioiE -1
h. 14£ -2
h.4f£ -2
•>.!"£ -2
i.97t -2
•1.07E -2
2.34E -2
1 .7 ^E -2
1 . 33£ -2
o.flSF .3
7 . 2 3 F - 3
b. 2^F -3
3.7HE -3
2,ftoE -3
1 .90E -3
I . 3 <£ -3
9. 1 HE -4
6.2HE -4
4.26E --
/ ^ R ^ (^ — H
1 <50 -^ • <»
) .?•>£ -'.
M . 1 3 £ -5
•).<;4E -s
}. 3<53E mlt
U68E -4
1.10E -4
7.13E -5
*.59E -5
2.92E -5
1.84E -5
1.15E -5
7.08E -6
4.33E -6
0.08
9.97E -1
9.84E -I
9.62E -I
9.30E -1
8.91E -1
8.45E .1
7.94E -1
7.38E -1
6.79E -1
6.19E -1
5.58E -
3.86E .
3.35E -
2.87E -
2.44E -
2.05E -
1.71E -1
1.41E -1
1.15E -1
9.29E -.2
7.43E »2
5.89E -2
«.62E -2
3.59E -2
2.76E -2
2.10E .2
1.58E -2
1.18E .2
8.71E -3
6.37E -3
4, big _3
3.31E -3
2.35E -3
1.65E -3
1.15E -3
7.89E -*
5.3flE -4
3.63E -4
2.43E -4
1.61E -4
1.05E -«
6.83E .5
2.79E .5
1.75E -5
1.09E -5
6.74E -6
4.12E -6
0.09
9.96E .1
9.82E .1
9.99E .1
9.27E .1
8.87E .1
8.40E .1
7.88E .1
7.32E .1
6.73E .1
6.13E -1
5.52E .1
4.93E .1
3.81E .1
3.30E -1
2.83E .1
2.40E -1
2.02E .1
1.68E .1
1.38E .1
1.13E .1
9.09E .2
7.27E .2
5.75E .2
O.S1E -2
3.49E .2
2.68E .2
2.04E .2
1.54E .2
1.15E .2
8.49E .3
6.17E -3
4.46E .3
3.20E .3
2.27E .3
1.S9E -3
1.11E .3
7.60E .4
5.18E .4
3.49E .4
2.33E .4
1.54E .4
1.01E .4
6.93E .5
4.19E .5
2.66E .5
1.67E .5
1.04E .5
6.42E .6
3.92E .6
-------
> Table A 1 (continued) SOLUTIONS TO EXPONENTIALS
•8
§
Q.
x" 5.00
w 5.10
5.20
5.30
5.40
5.50
5.60
5.70
5.80
5.90
6.00
6.10
6.20
6.30
6.40
6.50
6.60
6.70
6.80
6.90
7.00
7.10
7.20
7.30
7.40
7.50
7.60
7.70
7.80
7.90
8.00
8.10
8.20
8.30
8.40
8.50
8.60
8.70
8.80
8.90
9.00
9.10
9.20
9.30
9.40
9.50
-i 9.70
9.80
9.90
0.00
B
3. 7^E -6
2.25E -6
1.34C. -6
7. 9=ic -'
4.6«,E -7
2.7">t -7
l.5St -7
8.S1E -1
4.9«>E -B
2.7
7. l«j£ - 7
4.1PF -7
2.4?£ -7
1.39c -7
7.8AE -«
4.41E -^
2.4SE -R
1.3-5E -B
7.3ftE -9
3.97£ -9
2.12E -9
1.12E -9
5.8«E-10
3.0SE-10
1.56E-10
7.94E-11
4.0«E-ll
1.99E-11
9.81E-12
4.79E-12
2.32E-12
1.11E-12
5.25E-13
2.46E-13
1.14E-13
5.2(SE-l4
2.39E-14
1.08E-14
4.81E-1S
2. 13E-1*>
9.30E-1*.
4.03E-1'>
1.7?E-lft
7.33E-17
3.0«E-17
1.28E-17
5.28E-1R
2.15E-18
8.69E-19
3.47E-1"
1.37E-19
5.3PE-20
2.09E-20
8.0?E-21
3.0SE-21
1 . l^E-^ 1
4.2<\E-22
0.03
3.21E -6
1.9?E -6
I. t5E -6
ft. 7PE -7
1.96E -7
2.29E -7
1 .31E -7
'.42E -8
4.16E -8
Z.ME -8
1.27E -8
6,92E -9
3.73E -9
1.99E -9
1.05E -9
5.50E-10
2.85E-10
1.46E-10
7.42E-11
3.73E-11
1.86E-11
9.UE-12
4.46E-12
2.15E-12
1.03E-12
4.87E-13
2.28E-13
1.06E-13
4.86E-U
2.21E-14
9.96E-15
4.44E-15
1.96E-15
S.56E-16
3.70E-16
1.59E-16
6.72E-17
2.82E-17
1.17E-17
"•.83E-18
1.97E-18
'.93E-19
3. L7E-19
1.25E-19
".90E-20
1.90E-20
7.29E-21
Z.77E-21
1.04E-21
3.RRE-22
0.0*
3, U5E -6
t ,83E -6
L.09E -6
ft .'•IE • '
-J.7SE -7
^. 17E -T
1.24E -7
7.0LE -R
3.93E -8
Z.1BE -9
1.20E -8
6.51E -9
3.51E -9
1.87E -9
9.87E-10
S.16E-10
2.67E-10
1.37E-10
6.93E-11
3.4»E-ll
1 . 7^-11
slsiE-12
4.15E-12
2.00E-12
9.55E-13
4.52E-13
Z.llE-13
9.80E-14
4.50E-14
2.04E-14
9.19E-15
4.09E-15
l.POE-15
7.B7E-16
3.40E-16
1 .46E-16
6.17E-17
2.59E-17
1.07E-17
4.41E-18
1.80E-18
7.24E-19
2.89E-19
1.1*E-19
4.46E-20
1.73E-ZO
*>.%2E-21
2.51E-21
9.43E-22
3.51E-22
o.o1;
2.90E -6
1.74E -6
1.04E -6
6.09E -7
3.55F -7
2.05E -7
1 . 1 7E -7
6.62E -8
3.70E -8
2.0bE -8
1.13E -8
6.12E -9
3.29E -9
1.75E -9
9.25E-10
4.83F-10
2.50E-10
1.28E-10
6.47E-H
3.25E-11
1.61E-H
7.92F-12
3.86E-12
1.86E-12
8.87E-13
4.19E-13
1.96E-13
9.07E-14
4.16E-1*
1.89E-14
8.48E-15
3.77E-15
1.66E-15
7.24F-16
3.13E-16
1.34E-16
5.66E-17
2.37E-17
9.83E-18
4.04E-18
1.64E-18
6.61E-19
2.63E-19
1.04E-19
4.06E-20
1.57E-20
6.01E-21
2.28E-21
8.55E-22
3.18E-22
0.06
2.76E -6
1.65E -6
9.82E .7
5.77E -7
3.36E -7
1.94E -7
1.11E -7
6.25E -8
3.49E -R
1.94E -8
1.06E -8
5.76E -9
3.09E -9
1.65E -9
8.67E-10
4.52E-10
2.34E-10
1.19E-10
6.04E-11
3.03E-11
1.50E-11
7.38E-12
3.59E-12
1.73E-12
8.23E-13
3.88E-13
1.81E-13
8.39E-14
3.84E-14
1.74E-14
7.82E-15
3.48E-15
1.53E-15
6.66E-16
2.87E-16
1.23E-16
5.19E-17
2.17E-17
9.00E-18
3.69E-18
1.50E-18
6.03E-19
2.40E-19
9.46E-20
3.69E-20
1.43E-20
5.46E-21
2.07E-21
7.75E-22
2.88E-22
0.07
2.62E -6
1.57E -6
9.32E -7
5.47E -7
3.18E -7
1.83E -7
1.05E -7
5.90E -8
3.29E -8
1.82E -8
9.98E -9
5.41E -9
2.91E -9
1.55E -9
8.13E-10
4.24E-10
2.19E-10
1.12E-10
5.64E-11
2.82E-11
1.40E-11
6.87E-12
3.34E-12
1.60E-12
7.64E-13
3.60E-13
1.68E-13
7.77E-14
3.55E-14
1.61E-14
7.22E-15
3.20E-15
1.41E-15
6.13E-16
2.64E-16
1.13E-16
4.76E-17
1.99E-17
8.23E-18
3.37E-18
1.37E-18
5.50E-19
2.19E-19
8.61E-20
3.36E-20
1.30E-20
4.95E-21
1.87E-21
7.02E-22
2.60E-22
0.08
2.49E -6
1.49E -6
8.84E -7
5.19E -7
3. DIE -7
1.73E -T
9.87E -8
5.57E -8
3. HE -8
1.72E -8
9.39E -9
5.09E -9
2.73E -9
1.45E -9
T.62E-10
3.97E-10
2.04E-10
1.04E-10
5.27E-U
2.63E-H
1.30E-U
6.39E-12
3.10E-12
1.49E.12
7.09E-13
3.34E-13
1.56E-13
T.19E-1*
3.28E-1*
1.49E-1*
6.66E-15
2.95E-1'
1.30E-15
5.64E-16
2.43E-16
1.03E-16
4.3&E-17
1.82E-1T
T.53E-18
3.08E-18
1.25E-18
5.02E-19
1.99E-19
T.84E-20
3.05E-20
1.18E-20
4.50E-21
1.70E-21
6.36E-22
2.36E-22
0.09
2.37E -6
l.»2E .6
8.38E -7
4.91E -7
2. BSE -7
1.64E .7
9.32E .8
5.Z5E -8
2.93E .8
1.62E .8
8.84E .9
*.78E -9
2.56E .9
1.36E .9
7.14E-10
3.71E-10
1.91E-10
9.74E-H
4.92E-11
2.46E-11
1.226-11
5.95E-12
2.88E-12
1.38E-12
6.58E.13
3.09E-13
1.44E-13
6.65E.14
3.04E-14
1.J7E-14
6.14E-15
2.72E-15
1.19E-15
5.18E-16
2.23E-16
9.*9E-17
*.OOE-17
1.67E.17
6.89E.18
2.82E-18
1.14E-18
».58t-19
1.82E-19
7.14C-20
2.78E-20
1.07E-20
4.08E-21
1.54E-21
5.76E-22
2.13E-22
-------
1-79
Appendix 4
Appendix 4: CONSTANTS, CONVERSION
EQUATIONS, CONVERSION TABLES
Constants
e = 2.7183 —L- = 0.3679
e
TT = 3.1416 1 = 0.3183
2- = 6.2832 -1- = 0.1592
ZTT
\/27= 2.5066 —=• = 0.3989
2
V27T
-):/^= 15.75
0.7979
Conversion Equations and Tables
T(°C) =5/9 (T(°F) —32)
T(°K) =T(°C) + 273.16
T(°F) =» (9/5T(°C) ) +32
69
-------
-1
o
oo
o
CONVERSION FACTORS - VELOCITY
H
tal
flOSPHERIC DI
SPERSION
«>
i
H
B
WS
DESIRED UNITS METERS
PER SEC
GIVEN UNITS
METERS 1.0000
PER SEC E 00
FT 3.0480
PER SEC E-01
FT 5.0800
PER MIN E-03
KM 2.7778
PER HR E-01
MI(STAT) 4.4704
PER HR E-OI
KNOTS 5.1479
E-01
MMSTAT) 1.8627
PER DAY E-02
TO CONVERT A VALUE FROM A GIVEN
AND BENEATH THE DESIRED UNIT.
FT
PEP SEC
3.2808
E 00
1.0000
E 00
1.6667
E-02
9.1134
E-01
1.4667
E 00
1.6889
E 00
6.1111
E-02
UNIT TO A
NOTE THAT
FT KM
PER MIN PER HR
1.9685
E 02
6.0000
E 01
1.0000
E 00
5.4681
E 01
8.8000
E 01
1.0134
E 02
3.6667
E 00
3.6000
E 00
1.0973
E 00
1.8288
E-02
I. 0000
E 00
1.6093
E 00
1.8532
E 00
6.7056
E-02
MI(STAT)
PER HR
2.2369
E 00
6.8182
E-01
1.1364
E-02
6.2137
E-01
I. 0000
E 00
1.1516
E 00
4.1667
E-02
DESIRED UNIT, MULTIPLY THE GIVEN
E-xx MEANS 10 TO THE -xx POWER.
KNOTS
1.9425
E 00
5.9209
E-01
9.8681
E-03
5.39*9
E-01
8.6839
E-01
1.0000
E 00
3.6183
E-02
VALUE BY
MI(STAT)
PER DAY
5.3686
E 01
1.6364
E 01
2.7273
E-01
1.4913
E Oi
2.4000
E Oi
2.7637
E 01
1.0000
E 00
THE FACTOR OPPOSITE THE GIVEN UNITS
-------
>
•a
•a
n
a
a.
x"
CONVERSION FACTORS
GIVEN
GRAMS
PER
GRAMS
PER
KG
PER
KG
PER
LBS
PER
LBS
PER
LBS
PER
TONS
PER
TONS
PER
DESIRED UNITS
UNITS
SEC
MIN
HOUR
DAY
MIN
HOUR
DAY
HOUR
DAY
- EMISSION
GRAMS
PER SEC
1.0000
E 00
1.6667
E-02
2.7778
E-01
1.157*
E-02
7.5599
E 00
1.2600
E-01
5.2499
E-03
2.5200
E 02
1.0500
E 01
RATES
GRAMS
PER MIN
6.0000
E 01
1.0000
E 00
1.6667
E 01
6.9444
E-01
*.5359
E 02
7.5599
E 00
3.1499
E-01
1.5120
E 04
6.2999
E 02
KG
PER HOUR
3.6000
E 00
6.0000
E-02
1.0000
E 00
4.1667
E-02
2.7216
E 01
4.5359
E-01
1.8900
E-02
9.0718
E 02
3.7799
E 01
KG
PER DAY
8.6400
E 01
1.4400
E 00
2.4000
E 01
1.0000
E 00
6.5317
E 02
1.0886
E 01
4.5359
E-01
2.1772
E 04
9.0718
E 02
LBS
PER
1.3228
E-01
2.2046
E-03
3.6744
E-02
1.5310
E-03
1.0000
E 00
1.6667
E-02
6.9444
E-04
3,3333
E 01
1.3889
E 00
LBS
MIN PER HOUR
7.
E
1.
E
2.
E
9366
00
3228
-01
2046
00
9.1859
£.02
6.
E
1.
E
0000
01
0000
00
4.1667
E.02
2.
E
8.
E
0000
03
3333
01
LBS
PER
1.9048
E 02
3.1747
E 00
5.2911
E 01
2.2046
E 00
1.4400
E OS
2.4000
E 01
1.0000
E 00
4.8000
E 04
2.0000
E 03
TONS
DA" PER HOUR
3,9683
£.03
6.6139
£.05
1.1023
E-03
4.5930
E-05
3.0000
£.02
5.0000
£.04
2.0833
£.05
1.0000
E 00
4.1667
E-OZ
TONS
PER
9.5240
E-02
1.5873
E-03
2.6455
E-02
1.1023
E-03
7.2000
E-01
1.2000
E-02
5.0000
£.04
2.4000
C 01
1.0000
E 00
TO CONVERT A VALUE FROM & GIVEN UNIT TO A DESIRED UNIT, MULTIPLY THE GIVEN VALUE BY THE FACTOR OPPOSITE THE GIVEN UNITS
AND BENEATH THE DESIRED UNIT. NOTF THAT E-XX MEANS 10 TO THE -XX POWER.
oo
-------
oo
KILOMETER INCH
FOOT
YARD
MUE
"0
S
B
2
o
o
5)
>B
p]
JO
«
o
PI
73
H
i
>
H
cn
C/3
CONVERSION FACTORS - LENGTH
DESIRED UNITS METER CM MICRON
GIVEN UNITS
METER
CM
MICRON
KILOMETER
INCH
FOOT
YARD
MILE(STAT)
MilE (NAUT)
TO CONVERT A VALUE FROM A GIVEN UNIT TO A DESIRED UNIT, MULTIPLY THE GIVEN VALUE BY THE FACTOR OPPOSITE THE GIVEN UNITS
AND BENEATH THE DESIRED UNIT. NOTE THAT E-XX MEANS 10 TO THE -XX POWER.
1.0000
E 00
1.0000
E-02
1.0000
E-06
1.0000
E 03
2.9400
E-02
3.0480
E-01
9.1440
E-01
1.6093
E 03
1.8932
E 03
1.0000
E 02
1.0000
E 00
1.0000
E-04
1.0000
E 09
2.9400
E 00
3.0480
E 01
9.1440
E 01
1.6093
E 09
1.8932
E 09
1.0000
E 06
1.0000
E 04
1.0000
E 00
1.0000
E 09
2.9400
E 04
3.0480
E 09
9.1440
E 09
1.6093
E 09
1.8932
E 09
1.0000
E-03
1.0000
E-09
1.0000
E-09
1.0000
E 00
2.9400
E-09
3.0480
E-04
9.1440
E-04
1.6093
E 00
1.8932
E 00
3.9370
E 01
3.9370
E.01
3.9370
E-09
3.9J70
E 04
1.0000
E 00
1.2000
E 01
3.6000
E 01
6.3360
E 04
7.2962
E 04
3.2808
E 00
3.2008
E-0'2
3.28C8
E-06
3.2808
E 03
8<33?3
E-02
1.0000
E 00
3.0000
E 00
5*2800
E 03
6.0802
E 03
1.0936
E 00
1.0936
E-02
U0936
E-06
1.0936
E 03
2.7778
E-02
3.3333
E-Ol
1.0000
E 00
1.7600
E 03
2.0267
E 03
6.2)3',
E- '4
6*2137
E-06
6.2137
E-10
6r2l3t
£•01
1.97831
E.09
1.8939
£.04
9.6818
£.04
1.0000
E OfO
1.1516
E 00
9.3999
E-04
9.3999
E-06
9.3999
E-10
9.3959
e-oi
1.3706
E-05
1.6447
E-04
4.9340
E-04
8.6839
E-01
1.0000
E 00
-------
•o
•a
n
CONVERSION FACTORS - AREA
DESIRED UNIyS
GIVEN UNITS
so METER
SO KM
so CM
SO INCH
SO FOOT
SO YARD
ACRE
so STAT
MILE
SO NAUT
MllE
SO. METER
1,0000
E 00
1,0000
E 06
1,0000
E-04
6.4516
E-04
9,2903
E-02
8.3613
E-01
4.0469
E 03
2.5900
E 06
3.4345
E 06
SO KM
1,0000
E-06
1.0000
E 00
1.0000
E-10
6.4516
E-10
9.2905
E-08
8.3613
E-07
4.0469
E-03
2.5900
E 00
3.4345
E 00
SO CM
1.0000
E 04
1.0000
E 10
1.0000
E 00
6.4516
E 00
9.2903
E 02
8.3613
E 03
4.0469
E 07
2.5900
E 10
3.4345
E 10
SO INCH
1.5500
E 03
1.5500
E 09
1.5500
E-01
1.0000
E 00
1.4400
E 02
1.2960
E 03
6.2726
E 06
4.0145
E 09
5.3235
E 09
SO FOOT
1,0764
E 01
1,0764
E 07
1,0764
E-03
6.9444
E-03
1.0000
E 00
9.0000
E 00
4.3560
E 04
2.7878
E 07
3.6969
E 07
SO. YARD
1.1960
E 00
1.1960
E 06
1.1960
E-04
7,7160
E-04
1.1111
E-01
1.0000
E 00
4.8400
E 03
3.0976
E 06
4.1076
E 06
ACRE
2.4710
E-04
2.4710
E 02
2.4710
E-08
1.5942
E-07
2.2957
E-05
2.0661
E-04
1.0000
e oo
6.4000
E 02
8.4869
e 02
SQ STAT
1IIE
3,6610
E-07
3.8610
E-01
3,8610
E-ll
2.4910
E-10
3,5870
E.08
3.2283
£.07
1.5625
E-03
1.0000
E 00
1.3261
E 00
SO NAUT
MILE
2.9116
E-07
2.9116
E-01
2.9116
E-ll
1.8785
E-10
2.7Q50
E-08
2.4345
E-07
1.1783
E-03
7.5411
E-01
1.0000
E 00
TO CONVERT A VALUE FROM A GIVEN UNIT TO A DESIRED UNIT, MULTIPLY T*r GlyEN VALJt *Y THE FACTOR OPPOSITE THE GIVEN UNITS
AND BENFATH THE DESIRED UNIT. NOTE THAT E-XX MEANS 10 TO THE -XX f =.
oo
oo
-------
oo
irSTRFO LT'ITS CD "
GtVEM HNITS
LITE"
cu
cu STAT cj MAUT j s FLUID u s QUART u s GALLON
MILE MILE OUNCE
>
H
S
O
0)
*0
a
a
2
o
O
55
TJ
M
JO
VI
0
•z
W
Cfl
H
g
>
H
M
VI
cu METED i.oooo $.9997
F 00 E
-------
•o
•a
a
&
X
*>.
CONVERSION FACTORS
DESIRED UNITS
GIVEN UNITS
GRAM
MICROGRAM
KILOGRAM
METRIC TON
SHORT TON
LONG TON
GRAIN
OUNCE
(AVOP)
LB (AvDP)
- MASS
GRA^
1.0000
E 00
1.0000
E-06
1. 0000
E 03
1,0000
E 06
9.0718
E 05
1.0160
E 06
6.4799
E-02
2.8349
E 01
4.5359
E 02
MICROGRAM
1.0000
E 06
1.0000
E 00
1.0000
E 09
1.0000
E 12
9.0718
E 11
1.0160
E 12
6.4799
E 04
2.8349
E 07
4.5359
E 08
KILOGRAM
1.0000
E-03
1.0000
E-09
1.0000
E 00
1.0000
E 03
9.0718
E 02
1.0160
E 03
6.4799
E-05
2.8349
E-02
4.5359
E-01
METR1C TON
1.0000
E-06
1.0000
E-12
1.0000
E-03
1.0000
E 00
9.0718
E-01
l.OUO
E 00
6.4799
E-08
2.8349
E-05
4.5359
E-04
SHORT TON
1.1023
E-06
1.1023
E-12
1.1023
E-03
1.1023
E 00
1.0000
E 00
1,1200
E 00
7.1428
E-08
3.1250
E-05
5.0000
E-04
LONG TON
9.B421
E-07
9.8421
E-13
9.8421
E-04
9.8421
E-01
8,9286
E-01
1.0000
E 00
6.3775
E-08
2.7902
E-05
4.4643
E-04
GRAIN
1.5432
E 01
1.5432
E-05
1.5432
E 04
1.5432
E 07
1.4000
E 07
1.5680
E 07
1.0000
E 00
-,3750
E 02
',0000
E 03
OUNCE
(AVDP)
3.5274
E-02
3.5274
E-08
3.5274
E 01
3.5274
E 04
3.2000
E 04
J.5840
E 04
2.2857
E-03
1.0000
E 00
1.6000
E 01
IB (AV
2.2046
E-03
2.2046
E-09
2.2046
E 00
2.2046
E 03
2,0000
E 03
2.2400
E 03
1.4286
E-04
0.^500
E-02
1.0000
E 00
-. ••: FACTOR OPPOSITE THE GIVEN UNITS
00
-------
oo
CT)
CONVERSION FACTORS - FLO*
DESIRED UNITS CU METER CU METER LITER
LITER
LITER
CU FT
CU FT
CU FT
CU
GIVEN UNITS
PER SEC
PER HR
PER SEC PER MIN
PER HR
PER SEC PER MIN PER HR
PER SEC
J,
H
s
o
"5
X
a
2
o
o
33
t
0
z
PJ
Cfl
H
i
H
PI
CC
cu METER
PER SEC
cu METER
PER HR
LITER
PER SEC
LITER
PER' MIN
LITER
PER HR
CU FT
PER SEC
cu FT
PER MIN
cu FT
PER HR
cu CM
PER SEC
TO CONVERT A
AND BENEATH
1.0000
E 00
2.7778
E-04
1.0000
E-03
1.6667
E-05
2.7779
E-07
2.8317
E-02
4.7195
E-04
7.8658
E-06
1.0000
E-06
VALUE FROM A GIVEN
THE DESIRED UNIT.
3.6000
E 03
1.0000
E 00
3.6001
E 00
6.0002
E-02
1.0000
E-03
1.0194
E 02
1.6990
E 00
2.8317
E-02
3.6000
E-03
UNIT TO A
NOTE THAT
9.9997
E 02
2.7777
E-01
1.0000
E 00
1.6667
E-02
2.7778
E-04
2.8316
E 01
4.7194
E-01
7.8656
E-03
9.9997
E-04
DESIRED
5.9998
E 04
1.6666
E 01
6.0000
E 01
i.oooo
E 00
1.6667
E-02
1.6990
E 03
2.8316
E 01
4.7194
E-01
5.9998
E-02
UNIT, MULTIPLY
E-XX MEANS 10 TO THE -
3.5999
E 06
9.9997
E 02
3.6000
E 03
6,0000
E 01
1,0000
E 00
1.0194
£ 05
1.6990
E 03
2.8316
E 01
3.5999
E 00
THE GIVEN
XX POWER.
3.5314
E 01
9.8096
E-03
3.5315
E-02
5.8859
E-04
9.8098
E-06
1.0000
E 00
1.6667
E-02
2.7778
E-04
3.5314
E-05
VALUE BY
2.1189
E 03
5.8857
E-01
2.1189
E 00
3.5315
E-02
5.8859
E-04
6.0000
E 01
1.0000
E 00
1.6667
£.02
2.1189
E-03
THE FACTOR
1.2713
E 05
3,5314
E 01
1,2714
£ 02
'.1189
E 00
3,5315
E-02
3.6000
E 03
6.0000
E 01
1.0000
E 00
1.2713
E-01
OPPOSITE
1.0000
E 06
2.7778
E 02
1.0000
E 03
1.6667
E 01
2.7779
E-01
2.8317
E 04
4.7195
E 02
7.8658
E 00
1.0000
E 00
THE GIVEN UNITS
-------
13
•a
3
B.
x'
*•.
CONVERSION FACTORS - CONCENTRATION, DENSITY
DESIRED
GIVEN UNITS
GRAM PER
CU METER
MG PER
CU METER
MICROGRAM
PER CU M
MICROGRAM
PER LITER
GRAIN PER
CU FT
OUNCE PER
CU FT
LB PER
CU FT
GRAM PER
CU FT
LB PER
CU METER
UNITS GRAM PER
CU METER
1,0000
E 00
I. 0000
E-03
1.0000
E-06
9.9997
E-04
2.2883
E 00
1.0011
E 03
1.6018
E 04
3.5314
E 01
4.5359
E 02
MG PER
CU METER
1.0000
E 03
1.0000
E 00
1,0000
E-03
9.9997
E-01
2.2883
E 03
1.0011
E 06
1.6018
E 07
3.5314
E 04
4.5359
E 05
MICROGRAM
PER CU M
1.0000
E 06
1.0000
E 03
1.0000
E 00
9.9997
E 02
2.2883
E 06
1.0011
E 09
1.6018
E 10
3.5314
E 07
4.5359
E 08
MICROGRAM
PER LITER
1.0000
E 03
1.0000
E 00
1.0000
E-03
1.0000
E 00
2.2884
E 03
1.0012
E 06
1.6019
E 07
3.5315
E 04
4.9360
E 05
GRAJN PER
CU FT
4,3700
E-01
4,3700
E-04
4,3700
E-07
4,3699
E-04
1.0000
E 00
4.3750
E 02
7.0000
E 03
1.5432
E 01
1.9822
E 02
OUNCE PER
CU FT
9.9885
E-04
9.9885
E-07
9,9885
E-10
9.9883
E-07
2.2857
E-03
1.0000
E 00
1.6000
E 01
3.5274
E-02
4.5307
E-01
LB PER
CU
6.2428
£.05
6,2428
E-08
6,2428
E-ll
6,2427
E-08
1.4286
E-04
6.2500
E-02
1.0000
E 00
2.2046
E-03
2.8317
E-02
GRAM PER
FT CU FT
-------
oo
00
CONVERSION FACTORS - DEPOSITION RATF
GIVEN UNITS
(SHORT TON ,STAT. MILE)
>
H
S
O
(/)
"0
K
M
S
0
O
53
>fl
M
so
«
o
55
m
rf.
t/J
H
5
N*
>
H
M
C/l
GM PER SO
M PER MO
KG PER SO
KM PER MO
MG PER SO
CM PER MO
TON PER SO
MI PER MO
02 PER 50
FT PER MO
LB PER
ACRE PERMO
GM PER SO
FT PER MO
MG PER SO
IN PER MO
TO CONVERT A
AND BENEATH
1.0000
E 00
1.0000
E-03
1.0000
E 01
3.5026
E-01
3.0515
E 02
1.1208
E-01
1.0764
E 01
1.5500
E 00
VALUE FROM A GIVEN
THE DESIRED UNIT.
I. 0000
E 03
1.0000
E 00
1.0000
E 04
3.5026
E 02
3.0515
E 05
1.1208
E 02
1.0764
E 04
1.5500
E 03
UNIT TO A
NOTE THAT
1.0000
E-01
1.0000
E-04
1.0000
E 00
3.5026
E-02
3.0515
E 01
1.1208
E-02
1.0764
E 00
1.5500
E-01
2.8550
E 00
2.8550
E-03
2.8550
E 01
1.0000
E 00
8.7120
E 02
3.2000
E-01
3.0731
E 01
4.4252
E 00
DESIRED UNIT, MULTIPLY
E-XX MEANS 10 TO THE -
3.2771
E-03
3.2771
E-06
3.2771
E-02
1.1478
E-03
1.0000
E 00
3.6731
E-04
3.5274
E-02
5.0795
E-03
THE GIVEN
XX POWER.
8.9218
E 00
8.9218
E-03
8.9218
E 01
3.1250
E 00
2.7225
E 03
1.0000
E 00
9.6033
E 01
1.3829
E 01
VALUE BY
9.2903
E-02
9.2903
E.05
9.2903
E-01
3.2541
E-02
2.8349
E 01
1.0413
E-02
I. 0000
E 00
1.4400
E.Ol
THE FACTOR
6.4516
E-01
6.4516
E-04
6.4516
E 00
2.2598
£-01
1.9687
E 02
7.2313
E-02
6.9444
E 00
1.0000
E 00
OPPOSITE THE GIVEN UNITS
-------
•B
13
5
CONVERSION FACTORS - PRESSURE
DESIRED UNITS MILLIBAR BAR
GIVEN UNITS
ATMOSPHERE fJYrSc*.
-------
CONVERSION FACTORS - TIME
DESIRED UNITS SECOND
GIVEN UNITS
MINUTE
HOUR
WEEK
MONTH <28> MONTH 001 MONTH on YEAR (365) YEAR O66)
>
H
2
o
M
*B
sc
2
D
en
-0
H
73
V)
0
z
P3
'A
'J*
H
2
•4
>
H
•/.
SECOND 1.0000
E 00
MINUTE 1.6667
E-02
HOUR 2.7778
E-04
WEEK 1.6534
E-06
MONTH (28) 4.1336
E-07
MONTH (30) 3.8580
E-07
MONTH (31) 3.7336
E-07
YEAR (365) 3.1710
E-08
YEAR (366) 3.1623
E-08
TO CONVERT A VALUE FROM A GIVEN
AND BENEATH THE DESIRED UNIT.
6.0000
E 01
1.0000
E 00
1.6667
E-02
9.9206
E-05
2.4802
E-05
2.3148
E-05
2.2401
E-05
1.9026
E-06
1.897*
E-06
UNIT TO A
NOTE THAT
3.6000
E 03
6.0000
E 01
1. 0000
E 00
5.9524
E-03
1.4881
E-03
1.3889
E-03
1.3441
E-03
1.1416
E-04
1.1384
E-04
DESIRED . __ _ _
6.0480
E 05
1.0080
E 04
1.6800
E 02
1.0000
E 00
2.5000
E-01
2.3333
E-01
2.2581
E-01
1.9178
E-02
1.9126
E-02
UNIT, MULTIPLY
2.4192
E 06
4.0320
E 04
6.7200
E 02
4.0000
E 00
1.0000
E 00
9.3333
E-01
9.0323
E-01
7.6712
E-02
7.6503
E-02
THE GIVEN
2.5920
E 06
4.3200
E 04
7.2000
E 02
4.2857
E 00
1.0714
E 00
1.0000
E 00
9.6774
E-01
8.2192
E-02
8.1967
E-02
VALUE BY
2.6784
E 06
4.4640
E 04
7.4400
E 02
4.4286
E 00
1.1071
E 00
1.0333
E 00
1.0000
E 00
8.4932
E.02
8.4699
E.02
THE FACTOR
3.1536
E 07
5.2560
E 05
8.7600
E 03
5.2143
E 01
1.3036
E 01
1.2167
E 01
1.1774
E 01
1.0000
E 00
9.9727
E.01
OPPOSITE
E-XX MEANS 10 TO THE -xx POWER.
3.1622
E 07
5.*704
E 05
8.78*0
E 03
5.4286
E 01
1.3071
E 01
1.2200
E 01
1.1806
E 01
1.0027
E 00
1.0000
E 00
THE GIVEN UNITS
-------
•o
n
3
0.
x"
*.
CON\/FKSlCj o
- -11
l.U.^00
F 13
1 .0000
E 06
4. lH7iS
F 10
1 .7^SB
F 01
2.9113
E-01
9.9061
F-01
9.9QH1
F-01
7.4b86
F 02
< I 1 O-iA ' r -.1FG°ftAT T
' rn i i INT i
i .oo'ia i.o^oo
r.-0-l E-06
1.0000 1.0000
E 0'! F-'-13
1.00'tO 1.0'lUO
L 01 f 'JO
4. Id '6 4. 1«7<>
E-01 F-Oft
1.75H8 1.7-J83
E-0-07
Q.99H1 9.90H1
E-0<» E-07
-).99«1 9.9^51
c.t]t» F-07
7.t,5»6 7.4*186
E-U1 E-04
CAL (IMT) bTU
"FR 5rC PEP ^1
2.3
-------
<£>
K)
ABS JOULE CAL (INT) CAL (15) INT KW-HR ABS KW-HR BTU
O
Cfl
t
a
B
B
»
Cfl
O
CONVERSION FACTORS - ENtRGY, WORK
DESIRED UNITS ERG DYNE-CM
GIVEN UNITS
ERG
DYNE-CM
ABS JOULE
CAL (INT)
CAL (15)
INT KW-HR
ABS KW-HR
BTU
TO CONVERT A VALUE FROM A GIVEN UNIT TO A DESIRED UNIT, MULTIPLY THE GIVEN VALUE BY THE FACTOR OPPOSITE THE GIVEN UNITS
AND BENEATH THE DESIRED UNIT. NOTE THAT E-Xx'MEANS 10 TO THE -XX POWER.
1.0000
E 00
1.0000
C 00
1.0000
E 07
4.1866
E 07
4.1855
P 07
3.6007
E 13
3.6000
E 13
1.0551
E 10
1.0000
E 00
1.0000
E 00
1.0000
E 07
4.1868
E 07
4.1855
E 07
3.6007
E 13
3.6000
E 13
1.0551
E 10
1.0000
E-07
1.0000
F-07
1.0000
E 00
4.1868
E 00
4.1855
E 00
3.6007
E 66
3.6000
E 06
1.0551
E 03
2.3884
E-08
2.3884
E-08
2.3884
E-01
1.0000
E 00
9.9968
E-oi
8.6QOO
E 05
8.5984
E 05
2.5200
E 02
2.3892
E-08
2.3892
E-08
2.3892
E-01
1.0003
E 00
1.0000
E 00
8.6027
E 05
8.6011
E 05
2.5208
E 02
2.7773
E-14
2.7773
E-14
2.7773
E-07
1.1628
E-06
1.1624
E-06
1.0000
E 00
9.9981
E-01
2.9302
E-04
2.7778
E-14
2.7778
E-14
2.7778
E-Ot
1.1630
E-06
1.1626
E-06
1.0002
E 00
1.0000
E 00
2.9307
E-04
9.4781
£•11
9.4781
E-ll
9.4781
E-04
3.9683
E-03
i.9671
E-03
3.4128
E 03
5. 4121
E 03
1.0000
E 00
t/1
-------
•o
•o
re
I
X
CONVERSION FACTORS - ENERGY PER UNIT AREA
DESIRED UNITS LANStEY
GIVEN UNITS
CAL (15) BTU INT KW-HR ABS JOULES
PER SO CM PER SO FT PER SO M PER SO CM
LANGLEY
CAL (15)
PER SO CM
BTU
PER SO FT
INT KW-HR
PER SO M
ABS JOULES
PER SO CM
1.0000
E 00
1.0000
E 00
2.7133
E-01
8.6029
E 01
2.3692
E-01
1.0000
E 00
1.0000
E 00
2.7133
E-01
8.6029
E 01
2.3892
E-01
3.6855
E 00
3.6855
E 00
1.0000
E 00
3.1706
E 02
8.8054
E-01
1.1624
E-02
1.162*
E-02
3.1540
E-03
1.0000
E 00
2.7772
E-03
4.1855
E 00
4,1855
E 00
1.1357
E 00
3.6007
E 02
1.0000
E 00
TO CONVERT A VALUE FROM A GIVEN UN,T TO A DESIRED UNIT, MULTIPLY THE GIVEN VALUE BY THE FACTOR OPPOSITE THE G.VEN UNITS
AND BENEATH THE DESIRED UNIT. NOTE THAT E-XX MEANS 10 TO THE -XX POWER.
00
CO
00
-------
<£>
CONVERSION FACTORS - POWER PER UNIT AREA (CAL ARE 19 DEG)
DESIRED UNITS CAL PER ?Q CAL PER 50
Dt ' M PER SEC CM PER "I IN
GIVEN UNITS
LAN«LEY CAL PER SO BTU PER SO BTU PER SQ ABS
PER MIN CM PER DAY FT PER MIN FT PER DAY PER SQ CM
»
G
H
M
C/l
CAL PER SO
M PER SEC
CAL PER SO
CM PER MIN
LANGLEY
PER MIN
CAL PER SO
CM PER DAY
BTU PER SO
FT PER MIN
BTU PER SO
FT PER DAY
ABS WATT
PER SO CM
TO CONVERT A
AND BENEATH
1.0000 6
E 00
1.6667 1
E 02
1.6667 1
E 02
1.1574 6
E-01
4.5222 2
E 01
3.1404 1
E-02
2.3892 1
E 03
VALUE FROM A GIVEN
THE DESIRED UNIT.
.0000
E-03
.0000
E 00
.0000
E 00
.9444
E-04
.7133
E-01
.8843
E-04
.4335
E 01
UNIT TO A
NOTE THAT
6.0000
E-03
1.0000
E 00
1.0000
E 00
6.9444
E-04
2.7133
E-01
1.8843
E-04
1.4335
E 01
8.6400
E 00
1.4400
E 03
1.4400
E 03
1.0000
E 00
3.9072
E 02
2.7133
E-01
2.0643
E 04
2.2113
E-02
3.6855
E 00
3.6855
E 00
2.5594
E-03
1.0000
E 00
6.9445
E-04
5.2833
E 01
DESIRED UNIT, MULTIPLY THE GIVEN
E-XX MEANS 10 TO THE -XX POWER.
3.1843
E 01
5.3071
E 03
5.3071
E 03
3.6855
E 00
1.4400
E 03
1.0000
E 00
7,6079
E 04
VALUE BY
4.1855
E.04
6.9758
E-02
6.9758
E-02
4.8443
E-05
1.8928
E-02
1.3144
E-05
1.0000
E 00
THE FACTOR OPPOSITE THE GIVEN UNITS
-------
2-1
W. ©TV.
Air Resources Atmospheric Turbulence and Diffusion Laboratory
Environmental Science Services Administration
Oak Ridge, Tennessee
Prepared for
Nuclear Safety Information Center
Oak Ridge National Laboratory
U.S. ATOMIC ENERGY COMMISSION
Office of Information Services
1969
-------
2-2
Available'as TID-25075 for $6.00 from
National Technical Information Service
U. S. Department of Commerce
Springfield. Virginia 22151
Library of Congress Catalog Card Number: 72-603261
Printed in the United State of America
USAEC Technical Information Canter. Oak Ridge. Tennessee
November 1969: latest printing. April 1974
-------
2-3
FOREWORD
Scientists and technologists have been concerned in recent years about the
"explosion" of original literature engendered by the staggering volume of research and
development being undertaken throughout the world. It has proved all but impossible
for scientific workers to keep up with current progress even in quite narrow fields of
interest. Automated retrieval systems for identifying original literature pertinent to the
interests of individuals are being developed. These systems are only a partial solution,
however, because the original literature is too large, too diverse, too uneven in quality,
to fully satisfy by itself the information needs of scientists.
In this situation of vastly expanding knowledge, there is increasing recognition of
the valuable role that can be played by critical reviews of the literature and of the
results of research in specialized fields of scientific interest. Mr. Briggs's study, the
third published in the AEC Critical Review Series, is an excellent example of this
genre.
This review is also significant as a further step in the unceasing effort of the AEC
to assure that nuclear plants operate safely. Plume Rise is a much needed addition in a
field in which a meteorologist must choose from over 30 different plume-rise formulas
to predict how effluents from nuclear plants are dispersed into the atmosphere. Mr.
Briggs presents and compares all alternatives, simplifies and combines results whenever
possible, and makes clear and practical recommendations.
The Atomic Energy Commission welcomes any comments about this volume,
about the AEC Critical Review Series in general, and about other subject areas that
might beneficially be covered in this Series.
iii
-------
2-5
CONTENTS
FOREWORD iii
SYNOPSIS 1
?. INTRODUCTION 2
2.BEHAVIOR OF SMOKE PLUMES 5
Downwash and Aerodynamic Effects 5
Plume Rise 8
Diffusion 11
3. OBSERVATIONS OF PLUME RISE 16
Modeling Studies 16
Atmospheric Studies 18
-------
2-6
4. FORMULAS FOR CALCULATING
PLUME RISE 22
Empirical Formulas 22
Theoretical Formulas 25
5. COMPARISONS OF CALCULATED
AND OBSERVED PLUME BEHAVIOR 38
Neutral Conditions 38
Stable Conditions 50
6. CONCLUSIONS AND RECOMMENDATIONS 57
APPENDIX A 61
Effect of Atmospheric Turbulence on Plume Rise
APPENDIX B 65
Nomenclature
APPENDIX C 67
Glossary of Terms
REFERENCES 69
AUTHOR INDEX 77
SUBJECT (NDEX 80
-------
2-7
SYNOPSIS
The mechanism of plume rise and dispersion is described in qualitative1 terms with
emphasis on possible aerodynamic, meteorological, and topographical effects.
Plume-rise observations and formulas in the literature are reviewed, and a relatively
simple theoretical model is developed and compared with other models. All available
data are used to test the formulas for a number of idealized cases.
The inverse wind-speed relation, Ah <* u ', is shown to be generally valid for the
rise of a hot plume at a fixed distance downwind in near-neutral conditions. Nine
formulas of this type are compared with data from sixteen different sources, and the
best agreement is obtained from the "% law," Ah = 1.6F* u"'x", modified by the
assumption that a ceiling height is reached at a distance of ten stack heights
downwind. The term F is proportional to the heat emission. In uniform stratification
buoyant plumes are seen to follow the % law until a ceiling height of 2.9 (F/us)* is
approached, where s is proportional to the potential temperature gradient. In calm
conditions the formula Ah = 5.OF* s" * is in excellent agreement with a wide range of
data.
Formulas of a similar type are recommended for nonbuoyant plumes on the basis
of much more Limited data.
-------
2-8
1
/NTRODUCT/ON
The calculation of plume rise is often a vital consideration in predicting dispersion of
harmful effluents into the atmosphere, yet such a calculation is not straightforward.
The engineer or meteorologist must choose from more than thirty different plume-rise
formulas, and a casual search through the literature for help in choosing is likely to be
confusing. The purpose of this survey is to present an overall view of the pertinent
literature and to simplify and combine results whenever possible, with the objective of
setting down clear, practical recommendations.
The importance of stack height and buoyancy in reducing ground concentrations
of effluents has been recognized for at least 50 years.1 In a 1936 paper Bosanquet and
Pearson* showed that under certain conditions the maximum ground concentration
depends on the inverse square of stack height, and experience soon confirmed this
relationship.3 Later the stack height in this formula came to be replaced by the
"effective stack height," which was defined as the sum of the actual stack height and
the rise of the plume above the stack. Since smoke plumes from large sources of heat
often rise several stack heights above the top of the stack even in moderately high
winds, plume rise can reduce the highest ground concentration by an order of
magnitude or more.
In spite of the importance of plume rise in predicting dispersion, there is much
controversy about how it should be calculated. A recent symposium on plume
behavior,4 held in 1966, summarizes the current state of affairs. Lucas expressed a
desire for better agreement between empirical results and stated flatly, "There are too
many theoretical formulae and they contradict one another!" Spun lamented, "The
argument for and against different plume rise formulae can be discussed clinically by
-------
2-9
INTRODUCTION
physicists and theoreticians, but the engineer who has to apply the formulae is obliged
to make a choice." He then compared five recent formulas for a specific example and
concluded that the results varied by a factor of 4 in the calculated maximum ground
concentration. Even worse examples were given in the same symposium.
There are reasons for the lack of agreement. Different techniques for measuring
plume height and wind speed can account for some of the disparity in the data, but
the differences in the results are due primarily to the different concepts of what
constitutes effective stack height. A recent paper by Slawson and Csanady states:
With an ostrich-like philosophy, the effective stack height is often defined to be
the point where the plume is just lost sight of. It is then not very surprising to find
that the observed thermal rise of the plume depends, for example, on a power of the
heat flux ranging from */4 to 1.0, influenced by i number of factors including,
presumedly, the observer's eyesight.|
It was natural for early plume-rise observers to assume that a smoke plume leveled off
in all conditions and that the plume was near the height of leveling off when it was
inclined only slightly above the horizontal; subsequent observations suggest otherwise.
The early formula of Holland,6 sometimes called the Oak Ridge formula, was based on
photographic data that followed the plumes only 600 ft downwind,7 yet recent data
of the Tennessee Valley Authority (TVA) show plumes still rising at 1 and even 2
miles downwind. Over this distance even a small inclination above the horizontal
becomes important. The plume height normally of greatest concern is that above the
point of maximum ground concentration, and it seems logical to define this as the
effective stack height, as suggested by Lucas.4 A major difficulty with this definition is
that none of the present observations goes that far downwind. In practice we must
choose formulas for plume rise on the basis of agreement with data on hand and, at
the same time, be aware of the limitations of the data.
General plume behavior, which is discussed briefly in the next chapter, has been
described in greater detail in other publications. The textbook by Sutton8 first
reviewed all aspects of diffusion, including plume rise. Pasquill9 surveyed the subject
in considerably more detail and on the basis of more data than was previously
available. The first edition of Meteorology and Atomic Energy1 ° adequately covered
the qualitative aspects of plume rise and diffusion, but the new edition11 is
quantitatively more up-to-date. An excellent survey by Strom12 reviewed all aspects
of plume behavior, including the potential for modeling dispersion. Smith briefly
reviewed the main qualitative considerations in plume rise and diffusion13 and more
recently discussed the practical aspects of dispersion from tall stacks.14 The practical
experience of TVA has been described by Thomas,1 s by Gartrell,1' and by Thomas,
Carpenter, and Gartrell.17 The British experience with diffusion from large power
plants and their tall-stack policy has been analyzed by Stone and Clark.1 *
Several attempts have been made at setting down definite procedures for
calculating diffusion, including the plume rise. The first, primarily concerned with dust
fRef. S.page 311.
-------
2-10
INTRODUCTION
deposition, was by Bosanquet, Carey, and Halton.19 Hawkins and Nonhebel'7
published a procedure based on a revised formulation for plume rise by Bosanquet.20
More recently, Nonhebel3 ' gave detailed recommendations on stack heights, primarily
for small plants, based on the Bosanquet plume-rise formula and the Sutton diffusion
formula.8'9 Many of these recommendations were adopted in the British Memoran-
dum on Chimney Heights.11 which has been summarized by Nonhebei.23 Scorer and
Barrett2 A outlined a simple procedure applicable to long-term averages. A
CONCAWEf publication25'26 presented a method for determining stack height for a
plant built on flat, open terrain with a limited range of gas emissions; this method
included a formula for plume rise based on regression analysis of data. The American
Society of Mechanical Engineers (ASME)21 has prepared a diffusion manual with
another formula for plume rise. The implications of this formula and the CONTAWh
formula are discussed in Ref. 28. Further discussions of plume-rise questions can be
found in Refs. 29 to 33.
•fCONCAWE (Conservation of Clean Ail and Water, Western Europe), a foundation established
by the Oil Companies' International Study Group for the Conservation of Clean Air and Water.
-------
2-11
2
BEHAVIOR
OF SMOKE PLUMES
Plume dispersion is most easily described by discussing separately three aspects of
plume behavior (11 aerodynamic effects due to the presence of the stack, buildings,
and topographical features, (2) rise relative to the mean motion of the air due to the
buoyancy and initial vertical momentum of the plume, and (3) diffusion due to
turbulence in the air. In reality all three effects can occur simultaneously, but in the
present state of the art they are treated separately and are generally assumed not to
interact. This is probably not too unrealistic an assumption. We know that undesirable
aerodynamic effects can be avoided with good chimney design. Clearly the rise of a
plume is impeded by mixing with the air, but there is not much agreement on how
important a role atmospheric turbulence plays. It is known that a rising plume spreads
outward from Us center line faster than a passive plume, but this increased diffusion
rate usually results in an only negligible decrease of ground concentrations.
The following sections discuss the three aspects of plume diffusion. Symbols and
frequently used meteorological terms are defined in Appendixes B and C.
DOWNWASH
AND AERODYNAMIC EFFECTS
Downwash of the plume into the low-pressure region in the wake of a stack can
occur if the efflux velocity is too low. If the stack is too low, the plume can be caught
in the wake of associated buildings, where it will bring high concentrations of effluent
-------
2-12
BEHAVIOR OF SMOKE PLUMES
r '
(a] STACK DOWNWASH
(/>) BUILDING DOWNWASH
(c) TERRAIN DOWNWASH
Fig. 2.1 Undesirable aerodynamic effects.
to the ground and even inside the buildings. A similar effect can occur in the wake of a
terrain feature. These three effects are illustrated in Fig. 2.1.
The wind-tunnel studies of Sherlock and Stalker34 indicate that downwash is
slight as long as w0 > l.Su, where W0 is the efflux velocity of gases discharging from
the stack and u is the average wind speed at the top of the stack. These results are
consistent with elementary theoretical considerations: when w0 > 1.8u, the upward
momentum of the stack gases should overcome the downward pressure gradient
produced by the wind blowing around the stack on the basis of the pressure
distribution around an infinite circular cylinder in a crosswind given by Goldstein;35
when w0 <0.8u, the smoke can be sucked into the lower pressure region across the
entire back of the chimney. If the plume is very buoyant, i.e., if the efflux Froude
number, Fr, is 1.0 or less, the buoyancy forces are sufficient to counteract some of the
adverse pressure forces, and the preceding criterion for w0 could be relaxed This
factor probably abates downwash at the Tailawarra plant, cited in Table 5.1, where
Experiments are still needed to determine quantitatively the effect of the efflux
Froude number on the abatement of downwash, unfortunately, the experiments of
-------
2-13
DOWNWASH AND AERODYNAMIC EFFECTS
Sherlock and Stalker involved only high values of Fr, and thus buoyancy was not a
significant factor.
Nonhebel2 ' recommends that w0 be at least 20 to 25 ft/sec for small plants (heat
emission less than 106 cal/sec) and that w0 be in the neighborhood of 50 to 60 ft/sec
for a large plant (e.g., with a heat emission greater than 107 cal/sec). Larger efflux
velocities are not necessary since such high winds occur very rarely; in fact, much
higher velocities may be detrimental to the rise of a buoyant plume because they are
accompanied by more rapid entrainment of ambient air into the plume. Scorer36
reports that, when efflux velocity must be low, placing a horizontal disk that is about
one stack diameter in breadth about the rim of the chimney top will prevent
downwash.
One of the most enduring rules of thumb for stack design was a recommenda-
tion37 made in 1932 that stacks be built at least 2.5 times the height of surrounding
buildings, as illustrated in Fig. 2.2. If such a stack is designed with sufficient efflux
velocity to avoid downwash, the plume is normally carried above the region of
downflow in the wake of the building. If the stack height or efflux velocity is slightly
Fig. 2.2 Flow past a typical power plant.
lower, in high winds the plume will get caught in the downflow and be efficiently
mixed to the ground by the increased turbulence in the wake of the building. If the
stack is less than twice the building height, at least part of the plume is likely to be
caught in the cavity of air circulating in the lee of the building; this can bring high
concentrations of effluent to the ground near the building and even into the building.
The streamlines in Fig. 2.2 also illustrate the advantage of constructing a chimney on
the side of the building facing the prevailing wind, where the air is still rising.
Still, this is only a rough rule, because the air-flow pattern around a building
depends on the particular shape of the building and on the wind direction. Details on
these effects are given by Halitsky.38 Also, for sources with very small emissions, the
rule for stacks 2.5 times higher than nearby buildings may be impracticable. Lucas39
suggested a correction factor for smaller stacks, and this has been incorporated into
the British Memorandum on Chimney Heights.22 The correction factor is also
reported by Ireland40 and Nonhebel.23 The behavior of effluents from very short
stacks has been discussed by Barry,4' Culkowski,43 and Davies and Moore.43 For such
sources plume rise is probably negligible.
-------
2-14
BEHAVIOR OF SMOKE PLUMES
It is much more difficult to give any rules about the effect of terrain features,
partly because of the great variety of possibilities. Fortunately the general effect of
terrain and buildings on a plume can be fairly well modeled in a wind tunnel, such as
the one at New York University or at the D.S.l.R. (National Physics Laboratory,
England). Stumke44-45 gives a method for correcting effective stack height for a
simple step in the terrain, but only streamline flow is considered.
A curious aerodynamic effect sometimes observed is bifurcation, in which the
plume splits into two plumes near the source. This is discussed by Scorer,36 and a
good photograph of the phenomenon appears in Ref. 46. Bifurcation arises from the
double-vortex nature of a plume in a crosswind, but it is not clear under what
conditions the two vortices can separate. However, bifurcation is rare and appears to
occur only in light winds.
Scriven47 discusses the breakdown of plumes into puffs due to turbulent
fluctuations in the atmosphere. Scorer46 discusses the breakdown into puffs of
buoyant plumes with low exit velocity and includes a photograph. The process appears
to be associated with a low efflux Froude number, but a similar phenomenon could be
initiated through an organ-pipe effect, e.g., if the vortex-shedding frequency of the
stack corresponds to a harmonic mode of the column of gas inside the stack.
PLUME RISE
Although quantitative aspects of plume rise are the concern of the bulk of this
report, only the qualitative behavior is discussed in this section. More detailed
discussions can be found in a paper by Batchelor48 and a book by Scorer.46 It is
assumed that the plume is not affected by the adverse aerodynamic effects discussed in
the previous section since these effects can be effectively prevented.
The gases are turbulent as they leave the stack, and this turbulence causes mixing
with the ambient air; further mechanical turbulence is then generated because of the
velocity shear between the stack gases and the air. This mixing, called entrainment, has
a critical effect on plume rise since both the upward momentum of the plume and its
buoyancy are greatly diluted by this process. The initial vertical velocity of the plume
is soon greatly reduced, and in a crosswind the plume acquires horizontal momentum
from the entrained air and soon bends over.
Once the plume bends over, it moves horizontally at nearly the mean wind speed
of the air it has entrained; however, the plume continues to rise relative to the ambient
air, and the resulting vertical velocity shear continues to produce turbulence and
entrainment. Measurements of the mean velocity distribution in a cross section of a
bent-over plume show the plume to be a double vortex, as shown in Fig. 2.3. Naturally
the greatest vertical velocity and buoyancy occur near the center of the plume, where
the least mixing takes place. As the gases encounter ambient aii above the plume,
vigorous mixing occurs all across the top of the plume. This mixing causes the plume
diameter to grow approximately linearly with height as it rises.
-------
2-15
PLUME RISE
If the plume is hot or is of lower mean molecular weight than air, it is less dense
than air and is therefore buoyant. If the heat is not lost and the atmosphere is well
mixed, the total buoyant force in a given segment of the moving plume remains
constant. This causes the total vertical momentum of that segment to increase at a
constant rate, although its vertical velocity may decrease owing to dilution of the
momentum through entrainment.
Fig. 2.3 Cross section of mean velocity distribution in a bent-over plume.
At some point downwind of the stack, the turbulence and vertical temperature
gradient of the atmosphere begin to affect plume rise significantly. If the atmosphere
is well mixed because of vigorous turbulent mixing, it is said to be neutral or adiabatic.
In such an atmosphere the temperature decreases at the rate of 5.4°F per 1000 ft. This
rate of decrease, which is called the adiabatic lapse rate (P), is the rate at which air
lifted adiabatically cools owing to expansion as the ambient atmospheric pressure
decreases. If the temperature lapse of the atmosphere is less than the adiabatic lapse
rate, the air is said to be stable or stably stratified. Air lifted adiabatically in such an
environment becomes cooler than the surrounding air and thus tends to sink back. If
the temperature actually increases with height, the air is quite stable. Such a layer of
air is called an inversion. If the temperature lapse of the atmosphere is greater than the
adiabatic lapse rate, the air is said to be unstable or unstably stratified. Air lifted
adiabatically in such an environment becomes warmer than the surrounding air, and
thus all vertical motions tend to amplify.
The potential temperature, 6, is defined as the temperature that a sample of air
would acquire if it were compressed adiabatically to some standard pressure (usually
1000 millibars). The potential temperature is a convenient measure of atmospheric
stability since
•"•
where F = 5.4°F/1000 ft = 9.8°C/km. Thus the potential temperature gradient is
positive for stable air, zero for neutral air, and negative for unstable air.
-------
2-16
10 BEHAVIOR OF SMOKE PLUMES
If the ambient air is stable, i.e., if dO/Sz > 0, the buoyancy of the plume decays as
it rises since the plume entrains air from below and carries it upward into regions of
wanner ambient air. If the air is stable throughout the layer of plume rise, the plume
eventually becomes negatively buoyant and settles back to a height where it has zero
buoyancy relative to the ambient air. The plume may maintain this height for a
distance of 20 miles or more from the source. In stable air atmospheric turbulence is
suppressed and has little effect on plume rise.
If the atmosphere is neutral, i.e., if 60/6z = 0, the buoyancy of the plume remains
constant in a given segment of the plume provided the buoyancy is a conservative
property. This assumes no significant radiation or absorption of heat by the plume or
loss of heavy particles. Since a neutral atmosphere usually comes about through
vigorous mechanical mixing, a neutral atmosphere is normally turbulent. Atmospheric
turbulence then increases the rate of entrainment; i.e., it helps dilute the buoyancy
and vertical momentum of the plume through mixing.
If the atmosphere is unstable, i.e., if 60/6z < 0, the buoyancy of the plume grows
as it rises. Increased entrainment due to convective turbulence may counteract this
somewhat, but the net effect on plume rise is not well known. The few usable data for
unstable situations seem to indicate slightly higher plume rise than in comparable
neutral situations. On warm, unstable afternoons with light wind, plumes from large
sources rise thousands of feet and even initiate cumulus clouds.
Measurements are made difficult by fluctuations in plume rise induced by
unsteady atmospheric conditions. On very unstable days there are large vertical
velocity fluctuations due to convective eddies that may cause a plume to loop, as
shown in Fig. 2.5d. Figure 2.4 illustrates the large variations in plume rise at a fixed
distance downwind during unstable conditions. On neutral, windy days the plume
trajectory at any one moment appears more regular, but there still may be large
fluctuations in plume rise due to lulls and peaks in the horizontal wind speed. Since
the wind is responsible for the horizontal stretching of plume buoyancy and
momentum, the wind strongly affects plume rise. In stable conditions there is very
little turbulence, and plume rise is also less sensitive to wind-speed fluctuations. This
can be seen in Fig. 2.4. In this case the plume leveled off in stable air, and its rise
increased in a smooth fashion as the air became less and less stable owing to insolation
at the ground.
One might ask whether plume rise is affected by the addition of latent heat that
would occur if any water vapor in the stack gases were to condense. This is an
important question because there may be as much latent heat as there is sensible heat
present in a plume from a conventional power plant. It is true that some water vapor
may condense as the plume entrains cooler air, but calculations show that in most
conditions the plume quickly entrains enough air to cause the water to evaporate
again. Exceptions occur on very cold days, when the air has very little capacity for
water vapor, and in layers of air nearly saturated with water vapor, as when the plume
rises through fog. Observations by Serpolay49 indicate that on days when cumulus
clouds are present condensation of water from entrained air may increase the
buoyancy of the plume and enhance its ability to penetrate elevated stable layers.
-------
2-17
DIFFUSION
11
1000
800
— 600
400
200
TIME
Fig. 2.4 Fluctuations of plume rise with time (Gallatin Plant, Tennessee Valley Authority).
Ordinarily only the sensible heat of the plume should be used in calculations.
One might also ask whether thermal radiation can significantly alter the heat
content of a plume, i.e., its buoyancy. Not much is known about the radiative
properties of smoke plumes, but crude calculations show that radiation is potentially
important only for very opaque plumes some thousands of feet downwind and should
have little effect on clean plumes from modern power plants or on plumes from
air-cooled reactors. Plumes from TVA plants have been observed to maintain a
constant height for 20 miles downwind in the early morning; thus there appears to be
negligible heat loss due to radiation.
DIFFUSION
Detailed diffusion calculations are beyond the scope of this review, but the main
types of diffusion situations should be discussed with regard to plume rise. On a clear
-------
2-18
12
BEHAVIOR OF SMOKE PLUMES
'ViCTUfiL TEMPERATURE PROFILE
\ADIABATIC LAPSE RATE
(0) FANNING
nl
(b) FUMIGATION
_ v/"
= = "=-= ---.-'
(c) CONING
(d) LOOPING
Jf
TEMPERATURE-*- i«i LOFTIMG
Fig. 2J Effect of temperature profile on plume rise and diffusion.
-------
2-19
DIFFUSION 13
night the ground radiates heat, most of which passes out into space. In this process the
air near the ground is cooled, and an inversion is formed. The stable layer may be
several thousand feet deep; so most plumes rising through it lose all their buoyancy
and level off. This behavior is called fanning and is pictured in Fig. 2.5a. When the sun
comes up, convective eddies develop and penetrate higher and higher as the ground
warms up. As the eddies reach the height at which the plume has leveled off, they
rapidly mix the smoke toward the ground while the inversion aloft prevents upward
diffusion. This phenomenon, called fumigation, can bring heavy concentrations of
effluent to the ground (Fig. 2.5b). Just after an inversion has been broken down by
convective eddies or in cloudy, windy conditions, the atmosphere is well mixed and
nearly neutral. Then the plume rises and diffuses in a smooth fashion known as coning
(Fig. 2.5c). As the heating of the ground intensifies, large convective eddies may
develop and twist and fragment the plume in a looping manner (Fig. 2.5d). Diffusion is
then more rapid than in a neutral atmosphere. The convection dies out as the sun gets
lower, and an inversion again starts to build from the ground up. This ground inversion
is weak enough at first that the plume can penetrate it, and the plume diffuses upward
but is prevented by the stability below from diffusing downward. This lofting period
(Fig. 2.5e) is the most ideal time to release harmful effluents since they are then least
likely to reach ground.
The meteorological conditions that should be considered in stack design depend on
the size of the source, the climatology of the region, and the topography. In
reasonably flat terrain, high wind with neutral stratification usually causes the highest
ground concentrations since there is the least plume rise in these conditions. The mean
concentration of the effluent in the plume is reasonably well described by a Gaussian
distribution, for which the maximum ground concentration is given by
o, 2Q Q
X=—-—2= 0.164— (2.2)
oy neuh uh
where Q is the rate at which pollutant is emitted, u is the mean wind speed at the
source height, and h is the effective stack height (defined as the sum of the actual
stack height, hj, and the plume rise, Ah); oz/oy is the ratio of the vertical dispersion to
the horizontal dispersion and is equal to about 0.7 in a neutral atmosphere for an
averaging period of 30 min.25 Variation with distance has been neglected in deriving
Eq. 2.2. This equation is valid only when the atmosphere is neutral from the ground
up to at least twice the effective stack height. Inversions may exist below this height
even in windy conditions. A diffusion model for this case is given by Smith and
Singer.50 If the plume reaches the height of the inversion and penetrates it, as can be
predicted by Eq. 4.30,t none of the effluent reaches the ground. If the plume does not
penetrate, the inversion acts as an invisible ceiling and prevents upward diffusion.
A good measure of the efficiency of the diffusion process on a given occasion is
tSce "Basic Theory Simplified" in Chapter 4.
-------
2-20
14 BEHAVIOR OF SMOKE PLUMES
Q/X, the effective ventilation, which has the dimensions of volumetric flow rate (/3/t).
For the case just described,
-=6.1uh2=6.1u(hs + Ah)2 (2.3)
Naturally the effective ventilation is large for extremely high wind speeds, but it is also
large at low values of u because of very high plume rise. It is at some intermediate
wind speed that Q/x attains a minimum, i.e., x attains a maximum; this wind speed is
called the critical wind speed. If the dependence of Ah on • is known, Eq. 2.3 can be
differentiated and set equal to zero to find the critical wind speed. The result can be
substituted into the plume-rise equation and into Eq. 2.2 to find the highest expected
ground concentration for the neutral, windy case, Xmux
There is evidence that fumigation during calm conditions may lead to the highest
ground-level concentrations at large power plants. This type of fumigation can occur
near the center of large slow-moving high-pressure areas in so-called "stagnation"
conditions. Such high-pressure systems usually originate as outbreaks of cold,
relatively dense air, and, as these air masses slow down, they spread out much in the
manner of cake batter poured into a pan. Since the air underneath the upper surface of
these air masses is appreciably colder than the air above it, a subsidence inversion forms
and presents a formidable barrier to upward mixing; such an inversion normally occurs
1500 to 4000 ft above the ground.51 In combination with a near-zero wind speed, a
subsidence inversion severely limits atmospheric ventilation, and the b'ttle ventilation
that occurs is due to convective mixing from the ground up to the inversion.
Fortunately such circumstances are rare except in certain geographical areas. The
southeastern United States, one such region, averages 5 to 15 stagnation days a year
with the higher figure occurring in the Carolina* and Georgia.5' Nevertheless, there is
only one outstanding case of fumigation during stagnation in all the years of
monitoring SO2 around TVA power plants. In this instance ground concentration near
an isolated plant was 50% higher than the maximum observed in windy, neutral
conditions, and this condition continued for most of one afternoon. The wind speed
was 0 to 1 mph, and the effective ventilation, as defined above, was 1.5 x 10* cu
ft/sec (4.3 x 10* m3/sec). This value is adequate for a simall plant but too small for a
large plant. There is not much hope of improving the effective ventilation in this rare
condition, for a stack would have to be thousands of feet high to ensure that the
plume could penetrate a subsidence inversion. The only way to reduce ground
concentrations in this case seems to be to reduce the emission of pollutants;
accordingly, TVA stockpiles low-sulfur coal for use when the Weather Bureau predicts
stagnation conditions.
Similar conditions occur under marine inversions, such as are found along the
Pacific coast of the United States. The inversions there are sometimes less than 1000 ft
above the ground,*' and plumes from high stacks can often penetrate them. Such
penetration can be predicted by equations presented in later chapters.
-------
uhW
(2.4)
where u is the average velocity of the along-valley drainage flow at night, h is the
effective stack height at night, and W is the average width of the valley up to height h.
An elevated plateau can also be subjected to intensified fumigation if during an
inversion the plume rises slightly higher than the plateau and drifts over it. This has
occurred at a plant on the Tennessee River where the river cuts a 1000-ft-deep gorge
through the Cumberland Plateau.17 Careful consideration should be given to this
possibility at such a site. Topographic effects are discussed by Hewson, Bierly, and
Gill."
2-21
DIFFUSION 15
Fumigation associated with inversion breakdown may be serious when topography
is prominent. If the plume does not rise out of a deep valley during the period of the
nighttime inversion, the pollutant will mix fairly uniformly down to the ground during
fumigation; therefore concentration is given by
-------
2-22
3
OBSERVATIONS
OF PLUME RISE
Dozens of plume-rise observations have been made, and each is unique in terms of type
of source and technique of measurement. Observations have been made in the
atmosphere, in wind tunnels, in towing channels, and in tanks. Brief descriptions of
these experiments are given in this chapter.
MODELING STUDIES
Plume rise is a phenomenon of turbulent fluid mechanics and, as such, can be
modeled; i.e., it can be simulated on some scale other than the prototype. There are
obvious advantages to modeling plume rise. For example, the model plume can be
measured much less expensively than the real plume since it is not necessary to probe
high above the ground, and the variables can be controlled at will. The main difficulty
is in ensuring that the behavior of the model plume essentially duplicates that of a real
smoke plume. The most obvious requirements are that all lengths be scaled down by
the same factor and that the wind speed and efflux velocity be scaled down by
identical factors. For exact similarity the Reynolds number has to be the same in
model and in prototype. The Reynolds number is defined by
Re=^ (3.1)
where v is a characteristic velocity, / is a characteristic length, and v is the kinematic
viscosity of air or the fluid in which the model is measured. Exact similarity is seldom
16
-------
2-23
MODELING STUDIES 17
possible in modeling since Re is of the order of 106 for a real plume. Fortunately fully
turbulent flow is not very dependent on Reynolds number so long as it is sufficiently
high. In most experiments Re is at least 103 on the basis of efflux velocity and stack
diameter, but the adequacy of this value is not certain.
For buoyant plumes the Froude number must be the same in model and in
prototype. Since we are unable to scale down gravity, which is a prerequisite for the
existence of buoyancy, the basic requirement is that
'
(3.2)
model ' 'prototype
provided the temperature or density ratios are kept unchanged.
Numerous measurements have been made on the simple circular jet.53'54
Schmidt55 first investigated the heated plume with zero wind. Yih56 studied the
transition from laminar to turbulent flow in a heated plume. Later, Rouse, Yih, and
Humphreys57 studied the detailed distribution of vertical velocity and temperature in
a fully turbulent hot plume from a gas flame near the floor of an airtight,
high-ceilinged room. They measured temperature with a thermocouple and velocity
with a 1 Vin. vane on jeweled bearings. The important result of all these investigations
is that both jets and hot plumes are cone shaped in calm, unstratified air. The
half-cone angle is smaller for the heated plume than for a jet, and the decreases of
temperature and velocity with distance above the source are consistent with heat and
momentum conservation principles. Also, the cross-sectional distributions of vertical
velocity and temperature excess are approximately Gaussian except close to the
source. The characteristic radius describing the temperature distribution in a heated
plume is 16% greater than that for the velocity distribution.
Several modeling studies have been made on heated plumes rising through a stable
environment. Morton, Taylor, and Turner58 confirmed predictions by using measured
releases of dyed methylated spirits in a 1-m-deep tank of stratified salt solution.
Crawford and Leonard59 ran a similar experiment with a small electric heater to
generate a plume on the floor of an ice rink. The invisible plume was observed with the
Schlieren technique, and convection thermocouples were used to measure the
temperature profile of the air above the ice. Their results are, in fact, in good
agreement with those of Ref. 58, although they miscalculated the constant in the
equation of Ref. 58 by a factor of 2V4. Vadot60 conducted experiments with an
inverted plume of heavier fluid in a tank of salt solution. His inversions were quite
sharp in contrast to the smooth density gradients used in the preceding studies.
A number of wind-tunnel investigations of jets in a crosswind have been made. The
early study of Rupp and his associates" has been used as the basis for a momentum
contribution to plume rise by several investigators. Callaghan and Ruggeri6 2 measured
the temperature profile of heated jets in experiments in which the efflux velocities
were of the order of the speed of sound. Keffer and Baines* 3 measured rise for only
four stack diameters downwind and obtained some velocity and turbulence intensity
measurements within the jets. Halitsky44 and Patrick65 summarized the work of
-------
2-24
1g OBSERVATIONS OF PLUMh RISL
previous investigators. In addition, Patrick presented new measurements to about
20 slack, diameters downstream, including detailed profiles of velocity and concentra-
tion of a tracer (nitrous oxide).
The effect of buoyancy on plume rise near the stack was studied by Bryant and
Cowdrey66'67 in low-speed wind in a tunnel. Vadot60 made a study of buoyancy
effects in a towing channel with both stratified and unstratified fluids. This study was
unusual in that the ambient fluid was at rest and the effect of crosswind was
incorporated by towing the source at a constant speed down the channel. This is a
valid experimental technique since motion is only relative. However, Vadol's source
was a downward-directed stream of dense fluid. There is some question whether a
bent-over plume from such a source behaves as a mirror image of a bent-over plume
from an upward-directed stream of light fluid. Subtle changes in the entrainment
mechanism could take place owing to centrifugal forces acting on the more dense fluid
inside the plume. The recent treatment by Hoult, Fay, and Forney68 of past modeling
experiments tends to confirm this. The bent-over portion of a hot plume behaves
much like a line thermal, which was modeled for both dense and light plumes by
Richards,69 who found that the width of the thermals increased linearly with vertical
displacement from their virtual origins, just as had been observed for jets and plumes
that were not bent over. The line thermal was also modeled numerically by Lilly.70
Lilly did not have enough grid points to reach the shape-preserving stage found in
laboratory thermals, but, as larger computers are developed, numerical modeling
should be quite feasible. Extensive experiments made recently by Fan71 in a modeling
channel included plume rise both into a uniform crossflow and into a calm stream with
a constant density gradient. In the latter case mosi of the plumes were inclined; i.e..
the stacks were not vertical. Although the buoyancy of these plumes was varied, they
were momentum dominated for the most part. The behavior of plumes with negative
buoyancy in a crosswind was modeled by Bodurtha.72
ATMOSPHERIC STUDIES
The first full-scale plume-rise data were given in an appendix to the Bosanquet,
Carey, and Halton paper" of 1950. The center lines o'f plumes from four chimneys
were traced from visual observation onto a Perspex screen. The observations were
carried only as far as 800 ft downwind of the stacks, where apparently the visibility of
the plumes was lost. These observations also appear in a paper by Priestley.73
Holland6 published some of the details of the observations that he used in deriving the
Oak Ridge formula, but the distance of observation was not mentioned. According to
Hawkins and Nonhebel7 the plume heights were measured at only two or three stack
heights downwind and were obtained from photographs. Holland found only a small
correlation between plume rise and the temperature gradient, which was measured
near the ground. However, the plume is affected only by the temperature gradient of
the air through which it is rising, and the gradient near the ground is not a good
-------
2-25
ATMOSPHERIC STUDIES 19
measure of the gradient higher up. Stewart, Gale, and Crooks74'75 published a survey
of plume rise and diffusion parameters at the Harwell pile. Vertical surveys of the
invisible plume were conducted by mounting up to ten Geiger counter units on the
cable of a mobile barrage balloon. The stack was a steady, known source of radioactive
argon (" ' Ar), and the Geiger units were arranged to measure the disintegration of beta
particles, which have a maximum range of only 3 m in air. Again, the temperature
gradient was measured well below plume level except for a few runs that were made in
neutral conditions. Most of the wind-speed measurements were also made at a height
well below the plume height. Since wind speed generally increases with height, the
reported wind speeds are probably too low for such runs.
Ball76 made measurements on very small plumes from lard-pail-type oil burners.
The heights were estimated at 30 and 60 ft downwind by visual comparison with 10-ft
poles and were averaged over 2 or 3 min. There was some tendency for the burning
rate to increase with wind speed. Moses and Strom77 ran experiments on a source with
about the same heat emission, but here the effluent was fed into an 111-ft
experimental stack with a blower. Plume-rise data at 30 and 60 m downwind were
obtained photographically and averaged over 4 min. Wind speed was interpolated at
plume level from measurements from a nearly 150-ft meteorological tower. The
temperature gradient was measured between the 144- and 5-ft levels of the tower. This
provided only a fair measure of the actual gradient at plume level since the gradient
above 111 ft may be quite different from that near the ground. In only 2 of the
36 runs, the plume appeared to level off owing to stable conditions. These data tend to
be dominated by momentum rise.
Danovich and Zeyger78 published some plume-rise data obtained from photo-
graphs. However, the effective rise was assumed to occur when the plumes were still
inclined at 10 to 15° above horizontal, and plumes have been observed to rise many
times the height at this point. Some interesting data were obtained from exhaust
plumes of rocket motors by van Vleck and Boone,7' including some runs with
complete temperature profiles furnished. The sources ranged up to 1000 Mw, which is
about ten times the heat-emission rate of a large power plant stack. However, they
were not true continuous sources since burning times varied from 3 to 60 sec.
Extensive plume photography was carried out at two moderate-size power plants
in Germany by Rauch.80 Plume center lines were determined for 385 runs at Duisburg
and for 43 runs at Darmstadt. Each determination consisted of two or three time
exposures of about 1 min each, together with five instantaneous pictures taken at set
time intervals. The horizontal speed of the plume was calculated by following irregular
features of the plume from one negative to the next. This method should provide a
good measure of the wind speed experienced by the plume. In most of the
photographs, the plume center line could not be determined for a distance downwind
of more than 1000ft, although a few could be determined out to 3000ft. The
accuracy of the temperature-gradient measurements was such that only general
stability classifications could be made. In practice no measurements in very unstable
conditions were made because of looping, and no measurements in stable conditions
were made far enough downwind to show the plume leveling off. In fact, not one of
-------
2-26
20 OBSERVATIONS OF PLUME R1SL
the 428 plume center lines leveled off. It would therefore seem that Rauch's
extrapolation of these center lines to a final rise is rather speculative.
Much more extensive observations, consisting of about 70 experiments on more
than 30 smoke stacks in Sweden, were recently made by Bringfelt,8' and some of the
preliminary data have been reported by Hogstrb'm.82 Each experiment consisted in
taking about one photograph a minute for 30 to 60 min. The center lines were
measured up to 9000 ft downwind, and wind speed and temperature gradient were
measured at the plume level.
Some observations of plume rise at a small plant were reported by Sakuraba and
his associates.83 The best fit to the data was given by Ah « u"", but downwash was
likely at the higher wind speeds since the wind speeds exceeded the efflux velocity.
The temperature gradient and distance downwind were not given. More observations
were carried out by the Central Research Institute of Electric Power Industry,
Japan,84 in which temperature and wind profiles were measured, as well as the vertical
profile of SO2 concentration at 1 km downwind.
Several groups have shown continuing interest in plume-rise measurements. The
Meteorology Group at Brookhaven National Laboratory has conducted several
programs by burning rocket fuel on the ground near their well-instrumented 420-ft
meteorological tower. Limited data85 were published in 1964 from tests in which
there was some difficulty in obtaining a constant rate of heat release. This problem has
been overcome, and more detailed data are available.86
Csanady published plume-rise observations87 from the Tallawarra power station in
New South Wales in 1961. Plume rise was measured photographically, and wind speed
was determined from displacement of plume features in a succession of photographs
Csanady has been conducting a continuing program of plume-rise and dust-deposition
research at the University of Waterloo in Ontario since 1963. More-elaborate
photographic measurements of plume rise made at the Lakeview Generating Station
were published by Slawson and Csanady.5l88 Tank, wind-tunnel, and small-scale
outdoor studies are now in progress.89'90
The Central Electricity Research Laboratory in England has been conducting
plume-rise studies for some time. In 1963 they published results from the Earley and
Castle Donington power stations.9 ' The measurements were unique in that the plumes
were traced a long distance downwind by injecting balloons into the base of the
chimney.'2 The balloons were observed to stay within'the plumes when the plumes
were purposely made visible, but there may have been systematic errors due to the
relative inertia and buoyancy of the balloons. Although some of the balloons
continued to rise beyond 2 miles downwind, the reported rises were in the range
3600 to 6000 ft downwind. The motion of the balloons provided a convenient
measure of wind speed. More recently measurements were made by Hamilton9 3 at the
Northfleet Power Station by using lidai to detect the plume. Lidar is an optical radar
that uses a pulsed ruby laser. It measures the range and concentration of
light-reflecting particles and can detect smoke plumes even when they become invisible
to the eye.94'95 Some searchlight determinations of the height of the Tilbury plant
plume are also given in Refs. 93 and 96.
-------
2-27
ATMOSPHERIC STUDIES 21
The Tennessee Valley Authority has also conducted plume-rise measurements over
many years. The plume-rise and dispersion results97'98 published in 1964 were based
on helicopter probes of S02 concentrations in the plume. The helicopter also
measured the temperature gradient up to the top of the plume. Plumes in inversions
were observed to become level and maintain a nearly constant elevation as far as
9 miles downwind. Much more detailed studies at six TVA plants have recently been
completed.99 Heat emissions ranged from 20 to 100 Mw per stack with one to nine
stacks operating. Complete temperature profiles were obtained by helicopter, and
wind profiles were obtained from pibal releases about twice an hour. Such intermittent
sampling of wind speed does not provide a good average value, however, and may
account for some of the scatter in the results. After several different techniques were
tried, with good agreement among them, infrared photography was used to detect the
plume center line. Complete plume trajectories as far as 2 miles downwind were
obtained from the photographs.
There have been a few atmospheric studies concerned particularly with plume rise
in stable air. Vehrencamp, Ambrosio, and Romie100 conducted tests on the Mojave
Desert, where very steep surface inversions occur in the early morning. The heat
sources were shallow depressions, 2.5, 5, 10, and 20 ft in diameter, containing ignited
diesel oil. Temperature profiles were measured with a thermocouple attached to a
balloon, and the dense black plumes were easily photographed. Davies101 reported a
10,000-ft-high plume rise from an oil fire at a refinery in Long Beach, Calif. The heat
release was estimated to be of the order of 10,000 Mw;102 i.e., about 100 times the
heat emission from a large power plant stack. Observations of plume rise into multiple
inversions over New York City were presented recently b> Simon and Proudfit.103
The plumes were located with a fast-response SO2 analyzer borne by helicopter, and
temperature profiles were also obtained by helicopter.
-------
2-28
4
FORMULAS
FOR CALCULATING
PLUME RISE
There are over 30 plume-rise formulas in the literature, and new ones appear at the
rate of about 2 a year. All require empirical determination of one or more constants,
and some formulas are totally empirical. Yet the rises predicted by various formulas
may differ by a factor greater than 10. This comes about because the type of analysis
and the selection and weighting of data differ greatly among various investigators.
Emphasis is given here on how the formulas were derived and on the main features
of each. Complicated formulations are omitted since readers may check the original
references. For convenience all symbols are defined in Appendix B.
EMPIRICAL FORMULAS
Formulas for Buoyant Plumes
Of the purely empirical plume-rise formulas, the first to be widely used was that
suggested by Holland6 on the basis of photographs taken at three steam plants in the
vicinity of Oak Ridge, Tenn. The observed scatter was large, but the rise appeared to
be roughly proportional to the reciprocal of wind speed. Holland used the wind-tunnel
results of Rupp and his associates61 for the momentum-induced part of the rise and.
by assuming a linear combination of momentum and buoyancy rises, found the best fit
to the data with
-^) D + 4.4X 10-" \-^F± ^ (4
11 ' ' cal/sec j u l
22
-------
2-29
EMPIRICAL FORMULAS 23
The dimensions of constants are given in brackets. Thomas1 s found that a buoyancy
term twice as large as that in Eq. 4.1 gave a better fit to observations at the TVA
Johnsonville plant, and Sturnke'04 recommended a rise nearly three times that given
by Eq. 4.1 on the basis of comparisons with many sets of observations.
Another early empirical formula was suggested by Davidson105 in 1949 on the
basis of Bryant's66 wind-tunnel data:
Equation 4.2, although a dimensionally homogeneous formula, is physically over-
simplified in that the buoyancy term (AT/TS) does not take into consideration the
total heat emission or the effect of gravity, without which buoyancy does not exist.
The main weakness of Eq. 4.2 is that it is based on data obtained at only seven stack
diameters downwind and often greatly underestimates observed rises.
Berlyand, Genikhovich, and Onikul1 °6 suggested
/w0\ F
Ah= 1.9 ( — I D+5.0 — (4.3)
\ u / u
where F is a quantity that is proportional to the rate of buoyancy emission from the
stack. This formula is dimensionally consistent, but few details are given about the
observations on which it is based. The constant in the buoyancy term, 5.0, is curiously
almost two orders of magnitude smaller than the constants recommended b>
Csanady,87 by Briggs,1 ' •' °7 and by the new ASME manual.27
On the basis of data from four stacks, namely, the Harwell stack,74-75 Moses and
Strom's experimental stack,77 arrd the two stacks reported by Rauch,80 Stumke108
derived the formula
'.-< ,4.4,
The argument for omitting emission velocity from the buoyancy term is not clear.
The constants and exponents for the various terms resulted from applying the method
of least squares to the observed and calculated rises.
Lucas, Moore, and Spurr9 ' fitted observed plume rises at two of their plants with
-
[(cal/sec)NJ
OS-
(4.5)
The heat emissions varied from 4 to 67 Mw, and the plumes were traced to about a
mile downwind by releasing balloons in the stacks (see "Atmospheric Studies" in
Chap. 3). The formula is based on a simplification of Priestley's theoretical plume-rise
model.73 The best values for the constant in Eq. 4.5 differed by 25% at the two
-------
2-30
24 FORMULAS FOR CALCULATING PLUME RISL
plants, and further variations have been observed at other plants.93 Lucas109 noted
sonic correlation with stack height and suggested a modification of Eq. 4.5:
Recently a CONCAWE working group25'26 developed a regression formula based
on the assumption that plume rise depends mainly on some power of heal emission
and some power of wind speed. The least-squares fit to the logarithms of the
calculated-to-observed plume-rise ratios was
fft-( ft/sec)* 1 QH
Ah =1.40 - - ^— (4-7>
L(cal/sec)* J u*
Data from eight stacks were used, but over 757< of the runs came from Rauch's80
observations at Duisburg, i.e., from just one stack. Most of these data fall into a small
range of OH and of u- and therefore it is difficult to establish any power-law relation
wuh confidence.
Even more recently Moses and Carson1 ' ° developed a formula of the same type as
Eq. 4.7 with data for ten different stacks, but again the Duisburg observations were
heavily weighted. A momentum term of the type that appears in the formulas of
Holland6 Berlyand and his associates,106 and Stumke108 was included, bui the
optimum value of the constant turned out to be very small. The least-squares fit was
given by
OH (4.8)
Actually, changing the exponent of QH to V3 or \ increased the standard error very
little. This insensitivity is due partly to the small range of QH into which the bulk of
the data fell. Another shortcoming of this analysis, as well as of the analysis by
Stumke, is that absolute values of the error in predicted rises were employed. This
tended to weight the analysis in favor of situations with high plume rise; cases with
high wind speed counted very little since both the predicted and the observed rises,
and hence their differences, were small. Relative or percentage error, such as used by
CONCAWE by means of logarithms, results in more even weighting of the data.
Formulas for Jets
One of the first empirical relations for the rise of pure jets was given by Rupp et
al 6 ' This relation was determined from photographs of a plume in a wind tunnel. The
investigators found the height of the jet center line at
Ah =
(4.9)
-------
2-31
THEORETICAL FORMULAS 25
the point at which the plume became substantially horizontal, i.e., when its inclination
was only 5 to 8°
Subsequent investigators have all given empirical relations for the jet center line as
a function of downwind distance. The results are summarized in Table 4.1 for the case
in which the density of the jet is the same as that of air. A theoretical formula to be
given later in the chapter is included for comparison.
Table 4.1
COMPARISON OF EMHRJCAL RESULTS FOR JET
CENTER LINES AS A FUNCTION OF DOWNWIND DISTANCE
Investigator
Eq.4.33
Ruppetal.
Callaghan and
Rugger*62
Gordier (b\
Patrick")
Shandorov (by
AbramovKh" ')
Patrick65
Concentration
Velocity
Range of
R = (WO/M)
2 to 31
2 tu 22
6 to 45
8to54
Maximum
x/D
47
81
22
34
Ah/D
1.44R°-67(x/D)°-33
1.91R°-6I(x/D)°-30
1.31R°-74(x/D)°-37
0.84 R°-78(x/D)0-3'
1.00ROIS(x/D)°-34
1.00R°-85(x/D)°-38
Ah/D at 5.7°
Inclination
3.2 R1'00
>1.5R'-°°
4.0 R0-87
3.3R1'17
1.8R1'28
1.9 R1'29
2.3 R'-37
The early Callaghan and Ruggeri62 experiments involved heated, supersonic jets in
a very narrow wind tunael; so application of their results to free, subsonic jets is
questionable. Since the penetration was determined as the highest point at which the
temperature was 1°F above the free-stream temperature, the rises given represent the
very top of trw plume and are noticeably higher than in other experiments. The
Gordier formula was obtained from total-head traverses in a water tunnel as reported
by Patrick.65 The formub attributed to Shandorov by Abramovich''' was based on
experiments that included various angles of discharge and density ratios. The Patrick6 s
formula* were based bo* on the height or maximum concentration of nitrous oxide
tracer and on the height of maximum velocity as determined by a pitot tube.
THEORETICAL FORMULAS
There are many theoretical approaches to the problem of plume rise, and some of
them are quite complex. To reproduce them all here would be tedious and of little
help to most readers. Instead, the various theories are compared with a relatively
simple basic plume-rise theory based on assumptions common to most of the theories.
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2-32
26 FORMULAS FOR CALCULATING PLUME RIS1
It will be shown later that this basic theory in its simplest form gives good agreement
with observations.
Basic Theory
In most plume-rise theories, buoyancy is assumed to be conserved; i.e., the motion
is considered to be adiabatic. This means that the potential temperature of each
element of gas remains constant. It is also assumed that pressure forces are small and
have little net effect on the motion, that they merely redistribute some of the
momentum within the plume. Molecular viscosity is also negligible because the ptume
Reynolds number is very high, and local density changes are neglected. These
assumptions lead to three conservation equations:
^ • pp Vp = 0 (continuity of mass) (4.10)
—- = 0 (buoyancy) (4.11)
dt
__E = — 8 Ic (momentum) (4.12)
where vp = the local velocity of the gas in the plume
pp = the local gas density
0p = the local potential temperature
6'= 6f - 8 = the departure of the potential temperature from the temperature of the
environment at the same height
Ic = the unit vector in the vertical direction (buoyancy acts vertically)
Equations 4.10, 4.11, and 4.12 can be transformed to describe the mean motion of
a plume by integrating them over some plane that intersects the plume. It is most
convenient to integrate over a horizontal plane because then the mean ambient values
of potential temperature (6), density (p), and velocity (v^) can be considered constant
over the plane of integration and are assumed to be functions of height only.
Furthermore, if v, is assumed to be horizontal, the vertical component of Vp, denoted
by w', is due entirely to the presence of the plume. Thus w'is a convenient variable
with which to identify the plume.
A further simplification results from assuming that the vertical velocity and the
buoyancy are everywhere proportional to each other in a horizontal section of the
plume since it is then unnecessary to assume any specific distribution of either. This
assumption is approximately true for measured cross sections of vertical plumes.57
Admittedly it does not hold near the height of final rise in a stable-atmosphere,
because buoyancy decays more rapidly than vertical velocity in such a situation.
A steady state is assumed. To obtain Eq. 4.13, we combine Eq. 4.10 times 6 'with
Eq. 4.11 tunes pp and integrate the resulting equation over a horizontal plane,
-------
2-33
THEORETICAL FORMULAS 27
assuming that the vertical velocity and the buoyance are everywhere proportional to
each other. Similarly, to obtain Eq.4.14, we combine Eq. 4.10 times vp with Eq.4.12
times pp and integrate the resulting equation over the same horizontal plane. The
plane of integration must completely intersect the plume so that 8 '= 0 around the
perimeter of the plane. The resulting equations for the nel buoyancy flux and
momentum flux in a plume are
£--!V (4.131
ago. &£„.«.*,
//Ppw dxdX
where
(vertical volume flux) (4.15)
(stability parameter) (4.16)
//(g/T)flppw'dxdy
F2 = - - - (buoyancy flux) (4.17)
W = JJ V*Pw'dxdy (momentum flux) (4.18)
up
The vertical volume flux of the plume, as defined in Eq. 4.15, is the total vertical
mass flux divided by irp, where p is the environmental density. The stability
parameter, s, can be interpreted as the restoring acceleration per unit vertical
displacement for adiabatic motion in a stratified atmosphere (either stable or
unstable); in an unstable atmosphere, s is negative; Fz is the vertical flux of the
buoyant force divided by np; 3 is an average plume velocity at a given height, as
defined by the total velocity field at that height weighted by the normalized vertical
mass flux; w is the vertical component of v" and is the velocity of plume rise at any
given height. The drag term in Eq.4.14 is not written out since it will be dropped
later, but it can be interpreted as the net horizontal advection of momentum deficit
across the boundary of the plane of integration.
The initial conditions are
(4.19a)
FmC (4.19b)
-------
2-34
28 FORMULAS FOR CALCULATING PLUME RISt
and
Fz= (l-£)gw0rJ = F (4.19c)
For a hot source
cal/sec
i3.7x 10'5 " ' QH (420)
[ cal/sec J
where cp is the specific heat of air at constant pressure.
Equations 4.13 and 4.14 can be solved for the mean motion of a plume through
any atmosphere, including one with stability varying with height and wind shear.
However, the equations cannot be solved until some specific assumption is made about
the growth of volume flux with the height (dV/dz). This assumption, called an
entrainment assumption, is necessary to describe the bulk effect of turbulence in
diffusing momentum and buoyancy in a plume.
Basic Theory Simplified
It is desirable to reduce the basic theory to the simplest form that works. To be
more specific, we would like to derive from the basic theory simple formulas that
agiee with data. To do this, we must make the simplest workable entrainment and
drag-force assumptions, assume simple approximations for the atmosphere, treat the
stack as a point source, and treat the plume as being either nearly vertical or nearly
horizontal, i.e., ignore the complicated bending-over stage.
When the wind speed is sufficiently low, a plume rises almost vertically, and the
drag force and mechanically produced atmospheric turbulence are negligible. The
turbulence that causes entrainment of ambient air is generated within the plume b> the
shear between the vertical plume motion and the almost stationary environment. The
simplest workable entrainment hypothesis for this case is that the entrainment
velocity, or the average rate at which outside air enters the plume surface, is
proportional to the characteristic vertical velocity (w) at any given height. This
assumption, based on dimensional analysis, will be called the Taylor entrainment
assumption after the author"2 who suggested it in 1945. If (V/w)* is defined as a
characteristic plume radius, the rate at which the volume flux grows in a given
increment of height is then 27r(V/w)'4 aw, where a is called the entrainment constant
-------
2-35
THEORETICAL FORMULAS 29
and is dimensionless. The complete set of equations governing the vertical plume are
then
dz
which was given as Eq. 4.13,
d(wV)_
W (4.21)
and
-7-= 2a(wV)^ (4.22)
This set of equations is equivalent to the relations given by Taylor112 in 1945 and
further developed in 1956 in a classic paper58 by Morton, Taylor, and Turner, who
found that a value of 0.093 for the entrainment constant gave the best fit to observed
profiles of heated plumes. Briggs1' 3 found that a = 0.075 gives the best predictions of
the height of the top of stratified plumes in stable air, based on the height at which the
buoyancy flux decays to zero. The latter value is used here. The direct empirical
determination of entrainment in jets by Ricou and Spalding1'4 yields a comparable
value of 0.080.
The case of a bent-over plume, in which the vertical velocity of the plume is much
smaller than the horizontal velocity, is simpler. Both the total plume velocity and its
horizontal component are then very close to the ambient wind speed, u, which is
assumed constant; wind shear is neglected. It is more reasonable in this case to
integrate Eqs. 4.10 to 4.12 over a vertical plane intersecting the plume since a vertical
plane is more nearly perpendicular to the plume axis. When this is done, the resulting
equations are identical to Eqs. 4.13 and 4.21, provided that s is constant over the
plane of integration, that Fz, V, and wV are defined as fluxes of plume quantities
through a vertical plane, and that the drag term is zero. Measurements by Richards69
of the mean streamlines near horizontal thermals suggest that the drag term is zero
provided the chosen plane of integration is large enough. This is also intuitively evident
since one would not expect a vertically rising plume to leave a very extensive wake
underneath it.
In the initial stage of rise of a bent-over plume, the self-induced turbulence
dominates the mixing process, and the Taylor entrainment hypothesis can be used
again. The main difference from a vertical plume is that in this case the velocity shear
is nearly perpendicular to the plume axis, rather than parallel to it. This apparently
results in more efficient turbulent mixing since the entrainment constant for a
bent-over plume is about 5 times as large as that for a vertical plume. With a
characteristic plume radius defined as (V/u)**, the rate at which the volume flux grows
-------
2-36
30 FORMULAS FOR CALCULATING PLUMt RISh
in a given increment of axial distance is 2-n(V/u)^ >w, where 7 is the entrainment
constant for a bent-over plume. If this is transformed to vertical coordinates, the
plume rise is governed by Eqs. 4.13, 4.21, and
dV u
— = 27(uV)* (4.23)
which is comparable to Eq. 4.22. Since u is a constant, Eq. 4.23 can readily be
integrated. For a point source this yields a characteristic radius equal to -yz. The
relation is confirmed by modeling experiments of Richards69 and by photographs of
full-scale plumes made by TVA99 (see Fig. 4.1). On the basis of these photographic
plume diameters, 7 = 0.5.
1400
1200
(000
800
Z 600
400
200
° GALLATIN
~a PARADISE ~
• WIDOWS CREEK
zoo
4OO
600 BOO (OOO
PLUME RISE (ft)
1200 1400
Fig. 4.1 Photographic plume depth (top to bottom) vs. plume rise (center line) at TVA plants.
Atmospheric turbulence is small in a stable environment and can be neglected, in
which case Eq. 4.23 is valid up to the point where a bent-over plume reaches its
maximum rise. However, in a neutral or unstable atmosphere, turbulence is vigorous
enough to eventually dominate the entrainment process. This occurs some distance
-------
2-37
THEORETICAL FORMULAS 31
downwind of the stack when the vertical velocity of the plume becomes small
compared with ambient turbulent velocities. The simplest measure of the effective
intensity of atmospheric turbulence is the eddy energy dissipation, e , because it
adequately describes the part of the turbulence spectrum thai is most effective at
diffusing the plume relative to its axis, i.e., the inertial subrange. The characteristic
radius of the plume, (V/u)\ determines the range of eddy sizes that most efficiently
diffuse the plume, if these two terms are adequate enough to characterize
entrapment, the effective entrainment velocity must be given by (te^V/u) *, where (3
is a dimensionless constant; the exponents of the terms result from dimensional
considerations. Since the entrainment velocity in the initial stage of plume rise iS7w,
for the simplest model of a bent-over plume an abrupt transition to an entrainment
velocity of 0e ** (V/u) " is assumed to occur when >w = (3eii(V/u)v'
The solution for the bending-over stage of a plume in a crosswind is less certain
because both shear parallel to the plume axis and shear perpendicular to the axis are
present. Both mechanisms operate at once to cause turbulent entrainment. Drag force
could contribute to the bending over of the plume since there could be an extensive
wake downwind of the plume in this case, but the drag force will have to be neglected
at present owing to insufficient knowledge. In the early stage of bending over, the
vertical-plume model is applicable except that there is a perpendicular shear velocity
nearly equal to u. If the two contributions to entrainment can be summed in the
manner of vectors, the resultant entrainment velocity becomes (a2w2 + 72u2)\and
the plume center line is given by Eqs. 4.13, 4.21 , and
-) (a'w2 +7V)* (4.24)
w
Before applying models of the verticaJ plume and bent-over plume to specific
cases, some approximations about the source can be made. Usually it is reasonable to
assume that either the initial vertical momentum or the buoyancy dominates the rise.
In the former case the plume is called a jet, and we set F equal to 0. Unheated plumes
composed mostly of air are in this category. Most hot plumes are dominated by-
buoyancy, and we can neglect the initial vertical momentum flux, Fm. At a sufficient
distance from the stack, e.g., beyond 20 stack diameters downwind, we can neglect the
finite size of the source and treat the stack merely as a point source of momentum
flux or buoyancy flux.
Some of the approximations that come out of the simplified theory are given in
Eqs. 4.25 to 4.34. Vertical plumes are indicated by the term "calm" and bent-over
plumes by "wind." For rise into stable air in which s is constant, we have
Ah = 5.0FV* (buoyant, calm) (4.25)
Ah = 2.4 (L\% (buoyant, wind) (4.26)
\us/
-------
2-38
32 FORMULAS FOR CALCULATING PLUML RISL
Ah = 4f ( — ] (jet. calm) (4.27)
Ah= 1.5 f s~ (jet, wind) (4.28)
In the calm case, Eq. 4.25 gives the height at which the buoyancy goes to zero. In
the windy cases for a bent-over plume, the equations are integrated to the point where
w = 0, and the plume is assumed to fall back to the level at which the buoyancy is zero
with no further mixing. More details are given by Briggs.1 ' 3 The plume will penetrate
a ground-based inversion or stable layer if the preceding formulas predict a rise higher
than the top of the stable air. If the air is neutrally stratified above this level, a
buoyant plume will continue to rise since it still has some buoyancy. A jet will fall
back and level off near the top of the stable air because it acquires negative buoyancy
as it rises.
The model predicts penetration of a sharp, elevated inversion of height z, through
which the temperature increases by ATj if
Zj<7.3F0-« b?'6 (buoy ant, calm) (4.29)
Zj<2.o(- (buoyant, wind) (4.30)
z,<1.6t /Ln.^ (jet, calm) (4.31)
where b, = g ATj/T. The buoyant plume is assumed to penetrate if its characteristic
temperature excess, given by (T/g)Fz/V, exceeds AT, at the height of the inversion.
For the first stage of rise, the bent-over model predicts plume center lines given by
Ah = l.SF^if'x* (buoyant, wind) (4.32)
Ah = 2.3F*u~'sx* (jet, wind). ^33,
For the general case where s is positive and constant, Eqs. 4.13 and 4.21 can be
combined with the transform dz = (w/u) dx to give
,
dx2
This is the equation of a simple harmonic oscillator. Since V always increases, the
plume center line behaves like a damped harmonic oscillator (the author has observed
t Empirical; numerical value difficult to determine from present model.
-------
2-39
THEORETICAL I ORMULAS 33
such behavior at a plant west of Toronto in the early morning). Since V ~ u-/2z2, the
preceding expression can be integrated and satisfies the initial conditions when
[(72/3)us*] Ah3=Fm sin(xs*/u)+Fs')Ml -cos(xs*/u)]
This equation is valid only up to the point of maximum rise because beyond this point
a negative entrainment velocity would be implied. According to this equation a jet
(F = 0) reaches its maximum height at x = (ir/2) us"*4 and a buoyant plume (Fm = 0)
reaches its maximum height at x = jrus'*4. At much smaller distances the plume center
line is approximated by
From this equation it is seen that the ratio Fx/Fmu is a general criterion of whether a
bent-over plume is dominated by buoyancy or by momentum at a given distance
downwind. It, in fact, represents the ratio of buoyancy-induced vertical momentum to
initial vertical momentum.
For the buoyant bent-over plume in neutral conditions, the first stage of rise is
given by Eq. 4.32 up to the distance at which atmospheric turbulence dominates the
entrainment. The complete plume center line is given by Eq. 4.32 when x < x* and by
Ah=1.8Fsu-1x'* |f+2i^ + y (f*) U'+JIT*/ (434)
when x >x*, where x* is the distance at which atmospheric turbulence begins to
dominate entrainment. This distance is given by
Results from puff and cluster diffusion data and from measurements of eddy energy
dissipation rates, given in Appendix A, show that 0 = 1 is acceptable as a somewhat
conservative approximation. In the surface layer of the atmosphere defined by
constant stress, e.g., the lowest 50 ft or so, it is well established1'5 that € = u*3/0.4z,
where z is the height above the ground and u* is the friction velocity. If we
approximate z by z= Ah, the final plume rise given by Eq. 4.34 is Ah = 4.5 F/uu*2;
since u = u* and changes only gradually with height in the neutral surface layer, this
result is similar to those of earlier theories36'46•' °7 that predict Ah « p/u3
Unfortunately, this clear relation between e and u* breaks down at heights more
typical of smoke plumes. In Appendix A. data from 50 to 4000 ft above the ground
give more support to the empirical relation
,,0.73 Kb
[sec2] z
-------
2-40
34 FORMULAS FOR CALCULATING PLUME RISE
up to z = 1000 ft, then becoming constant with height. If we conservatively
approximate t with the stack height, the resulting estimate for x* becomes
' = 0.52 I-^H F*h? (hs<1000ft)
•'" (hs>1000ft) (4.35)
Other Theories
There is such a variety of plume-rise theories in the literature that only the briefest
discussion of each must suffice. One can only be amazed, and perhaps perplexed, at
the number of different approaches to the solution of this fascinating fluid-dynamics
problem. The theories will be discussed chronologically, first for the calm case and
then for the crosswind case.
The first theoretical treatment was of a jet in calm surroundings and was given by
Tollmien1 " in 1926. Rather than making an entrainment assumption, he used the
Prandtl mixing-length hypothesis to derive a specific velocity-profile law that agrees
quite well with data. A similar approach was used for heated plumes in calm air by
Schimdt55 in 1941. Rouse, Yih, and Humphreys57 treated the same problem by
assuming eddy viscosity diffusion of the buoyancy and momentum by a process
analogous to molecular diffusion. They determined experimentally that the mean
temperature and velocity profiles are approximately Gaussian with the characteristic
plume radius growing linearly with height. Yih56 also considered the case of a laminar
plume, which does not apply to full-scale plumes.
Batchelor48 considered the same problem in 1954 by dimensional analysis. He
included the case of a stratified environment and found power-law expressions for the
mean plume velocity and temperature as functions of height in an unstable atmosphere
whose potential temperature gradient is also approximated by a power law. The first
theoretical model for a vertical plume rising through any type of stratification was
given by Priestley and Ball11? in 1955. Their equations are similar to the preceding
equations for the vertical plume except that the entrainment assumption, Eq. 4.22, is
replaced by an energy equation involving an assumption about the magnitude and
distribution of the turbulent stress. Vehrencamp, Ambrosio, and Romie100 were the
first to apply the results from an entrainment model to final rise in stable air by using
the Taylor entrainment assumption. A general model involving this assumption and
complete with experimental verification was put forth by Morton, Taylor, and
Turner58 in 1956. This model is called the M,T,&T model in the discussion that
follows. The M,T,&T model is virtually identical to the vertical-plume model presented
in the section "Basic Theory Simplified" of this chapter and differs from the Priestley
and Ball1! 7 model mainly by predicting a wider half-cone angle for jets than for
buoyant plumes. This is actually observed in the laboratory. Both the M.T.&T model
and the Priestley and Ball model predict a linear increase of radius with height in the
-------
2-41
THEORETICAL FORMULAS 35
unstratified case and give similar results for the final plume height but disagree
somewhat on the values of the numerical constants. Estoque1'8 further compares
these two theories.
Morton1' 9 extended the numerical integrations of the M,T,&T model to the case
of a buoyant plume with nonnegligible initial momentum and concluded thai
increasing the efflux velocity can actually lessen rise in stable conditions because of
increased entrainment near the stack level. In another paper,120 he extended the
theory to include augmented buoyancy due to the condensation of moisture of the
entrained air. Hino121'122 made further calculations with the M,T,&T model,
including the effects of a finite source radius. Turner1 J3 coupled the M,T,&T model
with a vortex ring model to predict the speed of rise for a starting plume in neutral
surroundings. Okubo124 expanded the M,T,&T model to the case of a plume rising
through a salinity gradient in water.
A generalized theory for steady-state convective flow incorporating several of these
solutions was given by Vasil'chenko.12S Recently Telford126 proposed another type
of entrainment assumption in which the entrainment velocity is proportional to the
magnitude of turbulent fluctuations in the plume as calculated from a turbulent
kinetic energy equation. Telford's results are similar to those of the M,T,&T model for
a buoyant plume, except near the stack, but his model predicts too-rapid growth for a
jet. This happens because the model is, in effect, based on the assumption that the
scale of the energy-containing turbulent eddies is proportional to the plume radius, but
this is not true for a jet, because most of the turbulent energy is generated while the
jet radius is relatively small. Morton127 has further criticized Telford's model in a
recent note.
Lee128 developed a model for a turbulent swirling plume. He used the Prandtl
mixing-length hypothesis. Still another problem was explored by Fan,71 who
extended the M,T,&T theory to the case of nonvertical emissions and tested the result
in a modeling tank with linear density stratification.
One of the earliest theories for a bent-over buoyant plume was given by Bryant66
in 1949. A drag-force assumption was included, and the entrainment assumption was
in the form of a fairly complicated hypothesis about how the plume radius grows with
distance from the source along its center line. Eventually the radius in this model
becomes proportional to x" , which is too small a growth rate compared with
subsequent observations.
In 1950 Bosanquet, Carey, and Halton' 9 published a well-known theory that was
later revised by Bosanquet.20 The entrainment assumptions were similar to those
made in the simplified theory here except that the same entrainment constant was
applied to both the vertical and the bent-over stages of plume development, i.e., 7 = a.
In addition, a contribution to the entrainment velocity due to environmental
turbulence was assumed that was proportional to the wind speed. This assumption
eventually led to a linear growth of plume radius with distance downwind and resulted
in a final height for a bent-over jet and rise proportional to log x for a buoyant plume.
The theory tends to underestimate rise at large distances downwind (see Fig. 5.3 in the
next chapter).
-------
2-42
36 FORMULAS FOR CALCULATING PLUME RISE
About the same time, Sutton129 developed a simple theory for a buoyant plume
in a crosswind which was based on Schmidt's55 result for a vertical plume, i.e., w a
(F/z)*4. Sutton replaced z in this relation with the distance along the plume center line
and took the horizontal speed of the plume to be equal to u. The expression is
dimensionally correct and, at large distances, approaches the form given by Eq. 4.32.
Priestley73 adapted his and Ball's vertical-plume model to the bent-over case. The
average radius of a horizontal section was assumed to grow linearly with height, and
the entrainment constant was modified by a factor proportional to u1* Thus the
equations of rise were identical to those for a vertical plume except for the
entrainment constant modification. Priestley coupled this first-phase theory with a
second phase in which atmospheric turbulence dominates the mixing. This latter phase
is complicated and yields some unrealistic results, as was mentioned by Csanady.87
The first phase leads to an asymptotic formula identical to Eq. 4.32 times a factor
proportional to (F/x)''/" ; namely,
Ah = 2.7 [^J jFVx* (4.36)
Lucas, Moore, and Spurr91 were able to simplify Priestley's theory considerably. For
the first stage of rise, they obtained a plume rise 15% greater than that given by
Eq. 4.36, and, for the atmospheric-turbulence-dominated stage, they obtained
where x, is the distance of transition to the second stage. It was estimated that x, =
660 ft, in contrast to the transition distance x* given by Eq. 4.35, which depends on
both the source strength and the height in the atmosphere.
Scorer36'1 30 introduced a simple plume-rise model for which he assumed that the
plume radius grows linearly with height (see Fig. 4.1). The constant governing the
growth rate depends on whether the plume is nearly vertical or bent over and also on
whether it is dominated by momentum or by buoyancy in a given stage. Scorer
considered all the separate possibilities and then matche'd them at the bend-over point
to get a complete set of formulas for rise in neutral conditions. The predictions for
transitional rise, the plume center line before final height is reached, are similar to
those given by Eqs. 4.32 and 4.33. In addition, he postulated thai the active rise
terminates when the vertical velocity of the plume reduces to the level of atmospheric
turbulence velocities, which he took to be some fraction of the wind speed. This led to
the prediction that Ah a F/u3 for a very buoyant plume. This type of formula has
been given by many authors, but the leveling off of the plume in neutral conditions
has not yet actually been observed. Furthermore, it now appears that atmospheric
turbulence velocities are less strongly related to wind speed at typical olume
heights.131
-------
2-43
THEORETICAL FORMULAS 37
A great variety of work has been done in the last 6 years. Lilly70 constructed a
numerical model of the two-dimensional vortex pair seen in a vertical cross section of a
bent-over buoyant plume. Keffer and Baines63 presented a model for the bending-over
stage of a jet with an entrainment assumption similar to the one in the bending-over
model given in this review except that only the horizontal shear was included.
Danovich and Zeyger78 developed a theory along the lines of the Priestley theory for
the first stage but with the second-stage dynamics determined by the diffusion of
buoyancy by atmospheric turbulence. The type of diffusion assumed was essentially
the same as that observed for total diffusion of gases in a passive plume. However,
total diffusion includes the meandering of the plume axis caused by shifts in wind
direction, whereas the action of buoyancy on the plume is affected only by the
diffusion of buoyancy relative to the plume axis. Only relative diffusion should be
used. The same criticism applies to a theory developed by Schmidt,1 32 which is based
on the assumption that the spread of material equals that given by the total diffusion
of a passive plume. There is also the criticism that the diffusion of a rising plume,
especially in its early stages, is not the same as for a passive plume, because the rising
plume generates its own turbulence in addition to the ambient turbulence. These
problems were also pointed out by Moore.1 33
Equations 4.25 through 4.28 and Eqs. 4.32 and 4.33 wen proposed by Briggs107
on the basis of rather elementary dimensional analysis as an extension of Batchelor's'18
and Scorer's"6 approaches. Bnggs1 '3 recently considered in some detail the
penetration of inversions by plumes of all types by using a model based on the
simplified theory given here. Gifford1 34 extended this type of model to the case of a
bent-over plume whose total buoyancy flux increases linearly with time as it moves
away from the source, again using the Taylor entrainment assumption. Modeling
experiments of Turner135 with thermals of increasing buoyancy support this
assumption.
A model by Csanady13' for the bent-over buoyant plume included the effect of
eddy-energy dissipation and of inertial subrange turbulence in the relative diffusion of
plume buoyancy. In a later paper by Slawson and Csanady,5 a three-stage model was
proposed. In the first stage, self-generated turbulence dominates, and the governing
equations are in fact the same as those given in the bent-over plume model here. The
second stage is dominated by inertial subrange atmospheric turbulence, and, in the
third stage, the plume is supposed to be large enough for the eddy diffusivity to be
essentially constant, as is the case for molecular diffusion. This model yields a radius
proportional to x* and a constant rate of rise in the final stage rather than any
limiting height of rise.
Very recently a model along the lines of the basic theory presented here was
developed by Hoult, Fay, and Forney,137 in which entrainment velocity depends on
the longitudinal and transverse shear velocities. This theory is more elaborate than the
simplified theory presented here, in that y may be a function of w0/u and the Froude
number at the stack but does not take into account the effect of atmospheric
turbulence.
-------
2-44
5
COMPARISONS
OF CALCULATED
AND OBSERVED
PLUME BEHAWOR
NEUTRAL CONDITIONS
Buoyant Plumes in Neutral Conditions
Some previous comparisons of plume-rise formulas with data for the case of hot
bent-over plumes in near-neutral conditions were reviewed by Moses, Strom, and
Carson13 8 and are only summarized here. Moses and Strom139 compared a number of
formulas with data from their experimental stack. However, there was much scatter in
the data, and only the absolute differences between observed and calculated values
were used in the analysis, rather than their ratios. The results of the comparisons were
rather inconclusive. Rauch80 made a brief comparison of the Holland6 formula and
that of Lucas, Moore, and Spurr" with his own data and found the latter formula,
multiplied by a factor of 0.35, to be a better fit. Stu'mke10* made more extensive
comparisons between 8 different formulas and the data of Bosanquet, Carey, and
Halton,1' of Stewart, Gale, and Crooks,75 and of Rauch.80 By computing the ratios
of calculated to observed rises, Stiimke concluded that the Holland formula,
multiplied by a factor of 2.92, works best.
Since these comparisons were made, a number of new formulas have appeared,
including those by Sturnke,'08 Moses and Carson,110 CONCAWE,25'26 the modified
Lucas formula,109 and Eq. 4.34, published for the first time here. In addition, more
data are now available, especially the data from three Central Electricity plants and six
TVA plants; therefore comparisons can now be made over a much wider range of
conditions.
38
-------
2-45
NEUTRAL CONDITIONS 39
First, a simple wind-speed relation would be convenient since this would allow
some reduction of a large amount of data that covers a wide range of wind speeds,
source strengths, and measuring distances. Many formulas, both empirical and
theoretical, suggest that plume rise is inversely proportional to wind speed, at least at a
fixed point downwind. In Fig. S.I, data from a large number of sources tend to
confirm this. In each graph the plume rise at one or more fixed distances is plotted
against wind speed on logarithmic coordinates so that Ah «if1 is represented by a
straight line with a slope of -1; such lines are indicated for reference.
For most of the sources, Ah a u"' is the best elementary relation. It would be
difficult to make a case for Ah <* if* , as appears in the CONCAWE2 s l2 6 formula. A
better fit would result only for the Duisburg data, upon which the CONCAWE formula
is very largely based. A few of the sources, in particular Shawnee and Widows Creek,
show a greater decrease of Ah with increasing u, which probably indicates some form
of downwash at higher wind speeds. However, the Davidson-Bryant105 prediction
that Ah is proportional to u"1 '* would not fit most of the data.
With the inverse wind-speed law reasonably well established for neutral conditions,
we can now average the product of plume rise and wind speed for all wind speeds to
greatly reduce the volume of data. Such a presentation was first employed by
Holland.6 In Fig. 5.2, u Ah is plotted as a function of x for all available data sources.
The average heat efflux per stack, in units of 10* cal/sec, is given in parentheses
following each identification code, along with the number of stacks if more than one.
The key to the code is given in Table 5.1. In system A at Harwell (HA), wind
measurements were at a height of 27 m, whereas in system B (HB) the measurements
were at 152m. which is much closer to the height of the plume. A considerable
amount of data is presented in Fig. 5.2. A general criterion was that each point plotted
should represent at least three periods of 30 to 120 min duration each and that each
period should be represented by at least five samples of plume rise or some equivalent
amount of data.
The outstanding feature of Fig. 5.2 is that all the plume center lines continued to
rise as far as measurements were made; there is no evidence of leveling off. In general,
the plume center lines approximate a 2/3 slope, as predicted by the "2/3 law" in
Eq. 4.32. This means that the final rise has not definitely been measured in neutral
conditions, and therefore we will have to find some other way of defining effective
stack height.
The same data as in Fig. 5.2, along with the data of Ball,76 are plotted in Fig. 5.3.
Both the rise and the distance downwind are made nondimensional by means of the
length L= F/u3. The result is a somewhat entangled family of curves that lie between
1.0 and 3.0 times F^if'x*. Rise for a buoyant plume according to the Bosanquet
theory20 and the asymptotic plume rises according to Csanady87 in 1961 and to
Briggs' °7 in 1965 are shown. They all underestimate rise at large values of x/L.
The Bosanquet20 formulation underestimates plume rise when x/L> Ifr3 The
CONCAWE2 s l26 relation that Ah is proportional to u"* and the Davidson-Bryant1 °5
relation that Ah is proportional to u"1 * are not valid for most data sources. Formulas of
the type Ah a L = F/u3 are difficult to test because they apply only to final rise in
-------
2-46
40
CALCULATED AND OBSERVED PLUME BEHAVIOR
0H = 1.0 X 104 coi/sec
• «=30H A« = 60 It
~ 20
MJ
«/>
K
UJ
2
=1 ,0
a
2
30O
- ZOO
UJ
in
a:
£ 100
3 80
a
60
50
\ XAA
\? i Y*
'•'•V>|
•*»•
*'
,.
A
A
!'
^
8
g
^ A ** 4
^V»* . . •,
5 1O 16
BALL
WIND S
OH = 1.6
OH = 1.1 X 106 coiAec • . = 20
• > = 1260 to 2150 It A « = 60
> :
.\->-
•:\
• •- •
\
•
00
00
00
DO
DO
60
2
PEE
< I07
0 II
0 (1
OH = 1.2 X 107 col/sec
• « = 1000 (i
A « * 3OOO li
"*A
V
•
\
*
0
LA
D (ft/
colAtc
LUU XA 1
X
100 s* *
80 ^^
60
X
A^
\
• -
y~
IT
S
^
\
V
40
KEVIEW
ec)
1200
IOOO
800
600
,
100
?00
2 20 30 40 12 20 30 40
HARWELL BOSANOUET
6
•
A
\
*
200
100
80
60
40
D
WIDOWS
JOHNSC
coi/sec
V
y
s
-»
K =
-
10
*
S
s
30
OH - 1.2 X 106 to
2.9 X 1O« coiAec
• « - 330 It
A » = 820 (i
\
«•
— •'*•?
•
A
V
A,
>f
•
i
2 20 3
DUISBURG
CREEK,
X 107 calAec
JNVILLE, OH = 11 X 10
X 2 STACKS
y A
\
\^
A
A
6 8 10 20
WIDOWS CREEK AND
JOHNSONVILLE
WIND SPEED (ft/sec)
Fig. 5.1 Plume rise vs. wind speed in near-neutral conditions.
-------
2-47
NLUTRAL CONDITIONS
41
e SHAWNEE, OH -- 55 X 106
col/sec X 8 STACKS
A COLBERT. QH = 67 X I06
col/sec X 3 STACKS
700
500
UJ
<2 300
or
UJ
§ 200
_j
O-
100
qo
*<
X
.
1*
»_
s
0
A
^ I
\
\
•
DO 11 5<
y
7 6 9 10 20 30
SHAWNEE AND COLBERT
1OOO
800
6OO
400
200
OH = 1 7 X I07 col/sec TOTAL
O » = 1000 ft, 1 STACK
• > = 1000 (I, 2 STACKS
A » = 2500 ft, 1 STACK
100
8 10
i
Ajij.
°\ *
o\°°
°u\'
s
D
\
^
O
o
H
20 30 40
GALLATIN
WIND SPEED (ft/sec)
2.1 x to7 col/sec of slock
= 1000 ll. 1 STACK
' 1000 It, 2 STACKS
= 3000 tt, .1 STACK
= 3000 ft. 2 STACKS
-- 5000 ft
120
2000
1000
600
6OO
40O
200
20 3O 40 5O 60
PARAD'SE
i EARLEY. OH = I.O X 106 to 5.1 X 106
col/sec X 2 STACKS. « = 3600 to
6000 ft
CASTLE DONINGTON, 0H = 0.8 X
I07 to 16 X I07 col/sec X 2
STACKS. « = 3600 to 6000 ft
NORTHFLEET. 0M = 08 X I07 to
1.2 X I07 col/sec X 2 STACKS.
i = 4000 to 8000 11
X
S
\
\
10 2O 3d 40 5O 60
EARLE-. CASTLE DOMINGTON',
WIND SPEED Kt/sec)
-------
2-48
CALCULATED AND OBSERVED PLUML BEHAVIOR
20.00O
50O IOOO 2OOO
DISTANCE DOWNWIND (ft)
5000
IO.OCJO
Fig. 5.2 Plume rise times wind speed vs. downwind distance in near-neutral conditions The
average heal efflux per stack, in units of 106 cal/scc, and the number of stacks, if more than one,
arc given in parentheses. See Table 5.1 for identification of sources and for additional data.
neutral conditions, which has not yet been clearly observed. Therefore only relations
of the type 4h.cc u"' have been chosen for the comparison shown in Table 5.1. Data
are given for the plume rises at the maximum distance downwind for which there was
sufficient information to meet the data criterion set-up for Fig. 5.2. The ratio of
calculated to observed plume rises times wind speed was computed for each source and
each formula, and the results were analyzed on a one-source one-vote basis. The
exceptions to this rule were plants that were run both with one stack and with two
stacks emitting (Paradise and Gallatin) and plants at which there were substantial
amounts of data for different rates of heat emission (Earley, Castle Donington.
Northfleet). The median value of the ratio was also computed for each plume-rise
formula, along with the average percentage deviation from the median. The same
computation was repeated for a selected set of data that excluded the following data
sources Ball, source very small, Harwell A, wind speed measured much below plume
and obvious!) lower than that measured with system B: Bosanquet. no stack heights
indjcated and length of runs uncertain; Darmstadt, low efflux velocity and insufficient
-------
5 10 20 50 IOO 200 5OO (000 2000 50OO IO.OOO
«/L, NONDIMENSIONAL DISTANCE DOWNWIND
Fig. 5.3 Nondimensional plume rise vs. nondimensional distance downwind in near-neutral
conditions. Rise for a buoyant plume according to Bosanquet and the asymptotic plume rises
according to Csanady and Briggs are shown. See Table 5.1 for identification of other curves
and for additional data.
O
O
z
D
O
<£>
-------
Table 5.1
COMPARISON OF CALCULATED VALUFS WITH OHSl RVATIONS FOR NEUTRAL CONDITIONS
Code
II
IIA
Illl
IK)
DS
Mil
1
1
1
1:
CD
CD
N
N
S
c
J
we
G
0
P
P
Source
H.illt
lljrwcll A}
Harwell II
llnvaiii|iu-tl
Danmlaill t
Dinsliuru
1 allawarra^
l.aki'viewf
CI-.GH plants
1 arley
!• arlcy
Castle Donington
Castle Donington
NorthflcetJ:
NorthlltetJ
TVA plants
ShawnccJ
Colbert*
Johnsonville
Widows Creek f
Gallatin
Gallatin
Paradise
Paradise
Reference
76
75
75
73
80
HO
H7
H8
91
91
91
91
93
93
99
Number
of
stacks
2
2
2
2
2
2
8
3
2
1
1
2
1
2
hs.
ft
200
200
246
410
2HH
493
250
250
425
425
492
492
250
300
400
500
500
500
600
600
D,
fl
11.3
1 1 3
6.5
7.5
11.5
20.5
19.5
12.0
12.0
23.0
23.0
19.7
19.7
14.0
16.5
14.0
20.8
25.0
25.0
26.0
26.0
*o.
ft/sec
32.6
32.6
31.9
15.7
28.0
12 0
65.0
18.3
56.0
40.9
54.7
46.3
70.0
48.7
42.9
94.8
71.5
52.4
23.7
51.3
57.2
Range of u.
ft/sec
2 to 14
14 to 30
17 to 38
14 to 33
16 to 25
15 to 29
20 to 23
25 to 49
14 to 35
14 to 35
10 to 26
10 to 35
13 to 52
13 to 52
8 to 29
10 to 17
6 to 22
8 to 21
7 to 34
5 to 39
6 to 55
12 to 34
On/stack,
10 cal/sec
0.0096
1.10
1.10
1.54
0.855
1.88
2.93
11.6
1.54
4.72
11.95
16.0
7.9
11.95
5.45
6.74
10.8
16.8
16.9
8.55
20.2
21.9
xV
fl
145
370
370
4851
380
705
680
1630
485
760
1510
1700
1400
1660
805
975
1400
1910
1920
1460
2300
2380
x.
ft
60
2950
1900
600
820
1 150
1000
3250
4800
4800
4800
4800
5900
5900
2500
1000
2500
2500
3000
2000
4500
4500
u Ah,
ftft/sec
112
4,430
3,980
2,450
2.150
3,400
5.500
22,100
5,580
8,150
14,800
18,600
10,900
11,150
6,210
7,200
10,100
8,000
14,250
7.850
21,200
20,000
fCalculated from Eq. 4.35.
JNot included in selected data.
5 Height
H Height
chosen for computing x
chosen for computing x
• = 20 ft.
• = 250 ft.
-------
Table 5.1 (Continued)
Ratio of calculated to observed values of u Ah
Code
It
IIA
MB
BO
I)S
I)B
T
I
!.
1
CD
CD
N
N
S
c
J
we
c.
G
r
F
Source
Ball
Harwell A
Harwell K
Bosanquct
Darmstadt
Duisburg
Talluwarra
l.akevicw
(T.GB plants
l-.arley
Karlcy
Castle Donington
Castle Donington
Northneet
Northflect
TVA plants
Shawnee
Colbert
Johnsonville
Widows Creek
Gallatin
Gallatin
Paradise
Paradise
Median for
Median for
Reference
76
75
75
73
80
80
87
88
91
91
91
91
93
93
99
all data
selected data
Moses and
Carson110
1.59
0.43
0.48
0.92
0.78
0.74
0.57
0.28
0.40
0.48
0.43
0.39
0.47
0.56
0.68
0.66
0.59
0.94
0.53
0.68
0.39
0.42
0.54 ± 34%
0.48 ± 19%
Stiimke'08
0.74
0.83
0.75
1.04
1.12
1.53
0.41
0.72
0.57
0.74
0.62
0.79
0.84
0.90
0.96
0.66
1.32
0.90
1.53
0.65
0.70
0.7 9 ±27%
0.72 ± 24%
Holland6
0.04
0.23
11.25
0.44
0.25
0.38
0.30
0.31
0.18
0.37
0.44
0.47
0.44
0.65
0.54
0.55
0.66
1.18
0.64
0.58
0.51
0.58
0.44 + 37%
0.47 ± 26%
Priestley"'87
(first phase)
1.31
2.00
1.60
1.19
1.49
1.47
0.91
0.78
2.49
2.26
1.57
1.34
2.24
2.43
1.88
0.86
1.37
1.94
1.35
1.41
1.19
1.28
1.44 ±26%
1 .4 1 ± 18%
Lucas, Moore,
and Spun"
.51
.70
.59
.38
.69
.62
.02
0.62
1.59
1.44
1.01
0.86
1.24
1.35
1.69
0.96
1.24
1.76
1.05
1.37
0.80
0.85
1.36 ±21%
1.24 ±22%
Lucas109
0.78
1.27
1.19
1.12
1.36
1.60
0.87
0.68
1.30
1.16
1.01
0.86
1.35
1.47
1.36
0.82
1.21
1.92
1.15
1.50
0.96
1.03
1 . 1 8 ± 20%
1.16 ± 14%
Eq. 4.32
("2/3 law")
0.86
1.40
1.17
0.98
1.13
1.17
0.76
0.66
1.73
1.71
1.29
1.13
1.75
1.96
1.53
0.77
1.19
1.73
1.10
1.21
1.03
1.11
1.17 ±23%
1.17± 12%
Eq.4.34
0.72
0.95
0.93
0.98
1.09
1.16
0.75
0.64
1.05
1.25
1.17
1.05
1.46
1.73
1.40
0.77
1.17
1.72
1.09
1.20
1.00
1.09
1 .09 ± 1 9%
1 .09 ± 7%
-------
2-52
46 CALCULATED AND OBSERVED PLUME BEHAVIOR
data; TaJlawara and Lakeview, much higher rise than comparable sources in Fig. 5.2,
possibly due to lakeshore effect; Widows Creek, downwash, possibly due to a 1000-fi
plateau nearby, shown in Figs. 5.1 and 5.2; Northfleet, terrain downwash reported by
Hamilton93 and rise much lower than at Castle Donington at same emission;Colbert
and Shawnee, many stacks. The results in Table 5.1 help justify the exclusion of these
data, since with the selected data the average deviation from the median is
considerably reduced for seven of the eight formulas.
The first three formulas tested in Table 5.1 are completely empirical and do nol
allow for the effect of distance of measurement on plume rise as the remaining five
formulas do; consequently, these three formulas give poorer agreement with data. The
Holland6 formula (Eq. 4.1) in particular shows a high percentage of scatter. The
formula of Stiimke108 (Eq. 4.4) is perhaps slightly preferable to that of Moses and
Carson"0 (Eq. 4.8), although the latter shows less scatter in comparison with the
selected data. All three of these formulas underestimate plume rise, but this
shortcoming can be corrected by multiplying the formulas by a constant that
optimizes the agreement.
The next three formulas are based on the Priestley73 theory. The first is the
asymptotic formula for the first-phase theory87 (Eq. 4.36), which predicts a rise
proportional to xV Even though this is a transitional-rise formula, which does not
apply to a leveling off stage of plume rise, it shows less scatter compared with
observations than the three empirical final-rise formulas. The next formula (Eq. 4.37),
by Lucas, Moore, and Spurr," includes both a transitional- and a final-rise stage and
gives a little better agreement with data. When Eq. 4.37 is multiplied by the empirical
stack-height factor suggested by Lucas,109 i.e., 0.52 + 0.00116 h,, the agreement is
considerably better. However, one should be cautious about applying this formula to
plants with heat emission less than 10 Mw, because it predicts continued plume rise to
almost 1 km downwind regardless of source size. For instance, for the very small
source used by Ball,'" the predicted final rise is 12 times the rise measured at 60ft
downwind, it seems unlikely that such a weakly buoyant plume so close to the ground.
where turbulence is stronger, will continue to rise over such a long distance.
The last two formulas are based on the simplified theory given in the section,
"Basic Theory Simplified" in Chapter 4. The "2/3 law" (Eq. 4.32), another transi-
tional-rise formula, agrees about as well with these data.as the Lucas109 formula just
discussed. Equation 4.34, which includes both a transitional-rise and a final-rise stage.
gives both improved numerical agreement and much less percentage of scatter. Clearl>
it is the best of the eight formulas tested in Table 5.1 and is the one recommended for
buoyant plumes in neutral conditions (for optimized fit it should be divided by 1.09).
Eq. 4.34 should not be applied beyond x = 5x*, because so few data go beyond
thus distance. In some cases the maximum ground concentration occurs closer to the
source than this, and Eq. 4.34 applied at the distance of the maximum gives the besi
measure of effective stack height. (Beyond this distance plumes diffuse upward, and
the interaction of diffusion with plume rise cannot be neglected.) One conservative
approach is to set x = 10 hj, which is about the minimum distance downwind at which
maximum ground concentration occurs. For the fossil-fuel plants of the Central
-------
2-53
NEUTRAL CONDITIONS 47
Electricity Generating Board (CEGB) and TVA in Table 5.1, at full load this distance
turns out to be in the range 2.5 < (x/x*) < 3.3. At x/x* = 3.3, Eq. 4.34 gives a plume
rise only 10% lower than Eq. 4.32, but at twice this distance the plume rise is
increased by only 27%. This suggests a rule of thumb that Eq. 4.34 can be
approximated by Eq. 4.32, the "2/3 law," up to a distance of 10 stack heights, beyond
which further plume rise is neglected, i.e.,
Ah = 1.8 F* u'1 x* (x<10hs)
Ah = 1.8FV(JOhs)* (x>10hs)
For other sources a conservative approximation to Eq. 4.34 is to use Eq. 4.32 up to a
distance of x = 3x* and then to consider the rise at this distance to be the final rise.
Surprisingly, Eq. 5.1 compares even better with the data in Table 5.1 than the
recommended Eq. 4.34. Excluding Ball's data, which were for a ground source, the
median ratio of calculated to observed plume rises is about 1.13, and the average
deviations are ±17% for all data and ±4% for the selected data. Because of the nature
of the approximation used in Eq. 5.1 and the scarcity of data beyond x = 5x*, Eq. 5.1
is recommended as an alternative to Eq. 4.34 only for fossil-fuel plants with a heat
emission of at least 20 Mw at full load.
For multiple stacks the data show little or no enhancement of plume rise over that
from comparable single stacks in neutral conditions. Observations at the Paradise
Steam Plant were about equally split between one-stack operation and two-stack
operation with about the same heat emission from the second stack. In Fig. 5.1 the
plume rises in these two conditions can be seen to be virtually indistinguishable.
However, the same figure shows a clear loss in plume rise at Gallatin for the cases in
which the same heat emission was split between two stacks. In Table 5.1 average
plume rises for plants with two stacks are somewhat less than those for plants with one
slack, at least in comparison with Eq. 4.34. Colbert, with three stacks, seems lo have
an enhanced rise, but Shawnee, with eight or nine stacks operating, has a lower rise
than would be expected for a single stack. This may be due to downwash, as noted in
the discussion of Fig. 5.1. In summary, the observations do not clearly support any
additional allowance for plume rise when more than one stack is operating. It is
beneficial to combine as much of the effluent as possible inlo one stack to get the
maximum heat emission and the maximum thermal plume rise. This has been the trend
for large power plants both in England and in the United States.
Few data are available to evaluate plume rise in unstable conditions. Slawson88
found a just slightly higher average rise in unstable than in neutral conditions, as well
as more scatter, as might be expected owing to convective turbulence. The same
general features are evident in the TVA data. The buoyancy flux of the plume
increases as it rises in unstable air, but there is also increased atmospheric turbulence;
it is not clear which influence has the greater effect on the plume. However, because of
lack of empirical evidence, it is possible only to recommend for unstable conditions
the same formulas that apply in neutral conditions, specifically Eq. 5.2.
-------
2-54
48
CALCULATED AND OBSERVED PLUME BEHAVIOR
Jets in Neutral Conditions
Most data for jets in a crosswiiul do not extend very far downwind; so in Fig. 5.4
they are compared with the bending-over plume model in "Basic Theory Simplified,"
Chapter 4; Ah/D is plotted as a function of R = w0/u for two different distances
60
50
a 20
LU
f-
UJ
S
5
t-
UJ
o '0
z
D
O
o
5 10 20
RATIO OF EFFLUX VELOCITY TO CROSSWIND VELOCITY
50 60
Fig. 5.4 Plume rise of jets in crosswind compared with values for bending-over plume model
1> art A f Bvitn«4 •»«<* /"<,«...«•_. * ' VI 1 /-• K- . i ^,1 6 5
B and C, Bryant and Cow dry
Cand R.Callaghan and
Ruggeri62
F,Fan71
J.Jordinson65
K and B, Keffer and
Baines63
N and C, Norster and Chapman*
P-C, concentration profiles,
P-S, Schlieren photographs.
Patrick65
P-V, velocity profiles,
Patrick65
-------
2-55
NCUTRAL CONDITIONS
downwind, x = 2D and x = 15D. The two families of curves group together rather well,
considering the variety of experiments jnd measurement techniques, which include the
photographic center lines b> Bryant and Cowdry67 (B and C), the temperature survey
by Morster and Chapman65 (N and C). the velocity survey by Keffer and Baines63 (K
and B), the total pressure measurements by Jordinson65 (J), the top of the
temperature profile measured by Callaghan and Ruggeri62 (C and R), the photo-
graphic measurements by Fan71 (F), and the three different sets of measurements
made by Patrick.65 i.e., concentration profiles (P-C), velocity profiles (P-V). and
SchJieren photographs (P-S). The data are fit rather well by the dashed line that
represents the formula given by the bending-over plume model (Eqs. 4.14 and 4.24);
the resultant formula is probably not of practical value since it applies only near the
source and, being unwieldy, is not written out. This is just a test of the entrainment
assumption. Only the Callaghan and Rugged data do not fit the pattern. A number of
reasons are possible, one being that the jet velocities were near supersonic and another
being that this jet was more nearly horizontal, the distance downwind being about
twice the rise. The main reason this curve is higher is probably that it represents the
lop of the jet rather than the center line.
A comparison of values from Eq. 4.33 with the few sets of data that go as far as
100 or 200 stack diameters downwind is shown in Fig. 5.5. Equation 4.33 does fairly
49
20
Ri~O
50 100
D'STi-gCE DOWNWIND TO JET DIAMETER
200
Pi,1?. S.5 Plume rise of jets in crosswind compared with values from Eq. 4.33. (R = «o/u. asterisks
denote Ah D= 3 OR )
-------
2-56
50 CALCULATED AND OBSERVED PLUML BEHAVIOR
well even when the plume is more vertical than horizontal (Ah > x) and works quite
well when the plume is more horizontal. The exception is that it overestimates the rise
measured by Fan at the lower value of R = w0/u, specifically at R = 4. This lends some
credence to the suggestion made by Hoult, Fay, and Forney68 that the entrainmeni
constant •> may be a function of R although the particular function that they suggest
works poorly in the present model. It should be noted that Fan's plumes were partially
buoyant, but these effects are minimized by rejecting data for which Fx/Fmu, the
ratio of buoyancy-induced momentum flux to initial momentum flux, is greater than
0.5.
As for the final rise of a jet, again it appears that none has been measured, but the
asterisks in Fig. 5.5 at Ah/D = 3.0R (see Table 4.1) indicate a reasonable value for
maximum observed rise; i.e.,
Ah = 3^ D (5.2)
u
This is twice the value given by Eq. 4.9, the often-cited formula of Rupp and his
STABLE CONDITIONS
Penetration ef Elevated Inversions
A hot plume will penetrate an inversion and continue to rise if at that elevation the
plume is warmer than the air above the inversion, i.e., if its temperature excess exceeds
AT,. A jet, on the other hand, must have enough momentum to force its way through
an inversion, and then it must eventually subside back to the level of the inversion
since it is cooler than the air above. For the case of no wind, the simplified vertical
model with boundary conditions implies that penetration ability is a function of b,, z,,
Fm , and F. Then conventional dimensional analysis predicts penetration when
zibp-6F-°-4
-------
STABLE CONDITIONS
10
n.~
g liJ O
to Z 'u_
0 " b, -°'6
0.2
1 10
NONDIMENSIONAL MOMENTUM FLUX
(F b.°-eFH-2)
40
Fig. 5.0 Maximum nondimensional inversion height for penetration by plume vs. nondimensional
momentum flux (based on data from Vadot ).
2-57
51
proportionality is roughly 1.6, as given in Eq. 4.31. As a simple, conservative criterion
for a vertical plume, Vadot's experiments suggest penetration when
(5.4)
A bent-over buoyant plume rising through neutrally stratified air should penetrate
an inversion at height Zj if, as expressed by Eq. 4.30,
This equation (Eq. 4.30) was derived from the simplified bent-over plume model,
which gives a characteristic temperature excess of the plume of
T F
-^
g uz'
(5.5)
for a plume rising through neutral air. Eq. 5.5 is easier to apply to cases where there
are two or more inversions separated by neutral stratification. Initially Fz = F, and 6'
decreases with the inverse square of the height above the source until the plume
reaches the first inversion. As the plume rises through the inversion, its potential
temperature is unaffected, but the potential temperature of the ambient air increases
by ATji thus d' is reduced by ATj. If 6' remains positive, the plume is buoyant and
continues to rise with 8' proportional to z"J until it reaches the height of the next
inversion. The same procedure is repeated until the plume reaches an inversion it
cannot penetrate, i.e., until 0'
-------
2-58
52 CALCULATED AND OBSERVED PLUME BEHAVIOR
The results obtained by applying this procedure to the data of Simon and
Proudfit103 from the Ravenswood plume in New York City, which include plume
penetrations of multiple inversions, are shown in Table 5.2, along with the
temperature excesses of the plume relative to the air above the inversion as calculated
by subtracting ATj from Eq. 5.5 applied at the top of the inversion. It can be seen that
every one of the eight nonpenetrations is predicted by a negative calculated 6'. In one
case penetration is questionable because the plume center line ascended only 10m
higher than the inversion; so the lower part of the plume was undoubtedly below the
inversion. Only one of the five penetrations was not predicted, and that was with a
negative 8' of only 0.2°C, near the limits of the accuracy of temperature
measurements. The procedure given in the discussion following Eq. 5.5 appears to be a
good predictor but, perhaps, just slightly conservative.
Rise Through Uniform Temperature Gradient
Also of particular interest is the case in which the plume rises through air with a
fairly uniform temperature gradient. In this case we can approximate s as a constant.
For the calm case the simple vertical model predicts that the buoyancy of a hot plume
decays to zero according to Eq. 4.25. This formula was derived by M,T,&T58 from
virtually the same model, and a similar formula was derived by Priestley and Ball.1 '1
The ability of Eq. 4.25 to predict the final height of the tops of plumes is shown in
Fig. 5.7. Data are plotted from the modeling experiment in stratified salt solution by
M,T,&T,58 from the modeling experiment in air near the floor of an ice nnk of
Crawford and Leonard,59 from the experiments of Vehrencamp, Ambrosio. and
Romie100 on the Mojave Desert, and from the observation by Davies1 01-1 °2 of the
plume from a large oil fire. Equation 4.25 correctly approximates the top of the
massive smoke plume that billowed out of the Surtsey volcano in 1963.")0 The rate
of thermal emission was estimated to be of the order of 100,000 Mw,'4' or about a
thousand times greater than the heat emission from a large stack. For the average lapse
rate observed in the troposphere (6.5°C/km), Eq. 4.25 gives a rise of 5 km, or about
16,000 ft; the observed cloud top ranged from 3 to 8 km.
As the nondimensional momentum flux is increased, Morton's1'' numerical
solution indicates lessened plume rise, just as inversion penetration ability was seen to
decline in Fig. 5.6. There are no data to show this, but three experiments with vertical
plumes by Fan71 indicate gradual enhancement of rise over that given by Eq. 4.25
when Fm sH/F > 1.8. Dimensional analysis of the vertical model indicates that
Ah = CF*s-* (5.6)
for a pure jet, where C is a constant. The values of C that correctly describe Fan's
plumes, which were momentum dominated but not pure jets, are 4.53, 4.43, and 4.18.
A value of C = 4 is suggested as an approximation, as in Eq. 4.27.
-------
M
H
Table 5.2
INVERSION PENETRATION AT THE RAVENSWOOD PLANTf
Date
May 25
July 20
July 21
September 8
September 9
Time
1825
0552-0559
0617-0820
0600-0724
0828
0648-0930
1000-1020
0640-0705
0747-0850
0930-1000
OH,
I07 cal/sec
1.97
0.98
1.11
1.13
1.64
1.66
1.77
1.20
1.54
2.13
u,
in/sec
9.0
10.5
7.3
4.3
2.7
7.5
5.4
9.6
9.1
9.6
Plume
height.
m
295
350
360
360
510
410
560
350
370
390
Inversion height, m
Bottom
145
325
255
365
540
410
240
360
360
620
360
260
370
420
Top
180
475
275
395
580
450
280
410
400
650
400
300
410
530
ATj,
0.2
0.7
0.3
2.0
1.9
0.6
0.6
0.4
0.8
0.4
2.1
0.7
1.6
1.8
Calculated
e',
°c
15
-0.5
0.05
-2.0
-1.9
-0.45
1.7
0.0
-0.6
-0.3
-2.0
-0.2
-1.6
-1.7
O
O
0
O
to
Penetration
Yes
No
Yes
No
No
No
Yes
Yes
7
No
No
Yes
No
No
t Stack height, 155 m.
K)
-------
2-60
54
CALCULATED AND OBSERVED PLUME BEHAVIOR
<0,000
1000
=• 100
UJ
tn
IT
u
2
-i 10
a
0<
I
"DAVIES (LONG BEACH)
/%EHR
/. AMRR
VEHRENCAMP,
AMBROSIO. WOMIE
(MOJAVE DESERT)
^CRAWFORD AND LEONARD
(ICE RINK)
MORTON, TAYLOR, TURNER '
(TANK)
10
100
1000
10,000
Fig. 5.7 Rise of buoyant plumes in calm, stable air.
For the case of a bent-over plume rising through stable air with constant s, the
quasi-horizontal model can be applied both to a buoyant plume and to a jet to yield
Eqs. 4.26 and 4.28, respectively. There are no data to test Eq. 4.28, but Eq. 4.26 and
several other formulas can be compared with data from buoyant plumes released in
stable air. These data include nine runs made at Brookhaven86 with 15-sec ignitions of
rocket fuel, six runs by TV A99 with large single stacks, and seven runs by Van Vleck
and Boone79 with 60-sec firings of horizontal rocket motors. Admittedly the plumes
were not continuous in two of these experiments, and the plume rises were defined
somewhat differently in each case. In each case the ratios of the calculated to observed
rises were computed. The resulting median values of this ratio and mean deviation
from the median are
Holland6 0.44 ±131%
Priestley73 0.42 ± 43%
Bosanquet2 ° 1.22 ± 26%
Briggs, Eq. 4.26 0.82 ±13%
Holland6 suggested that Eq. 4.1 be reduced by 20% to predict rise in stable conditions,
but this may be seen to work poorly. The Priestley73 and Bosanquet20 theoretical
formulations are both complex; so they were simplified to the case for a buoyant
-------
2-61
STABLE CONDITIONS
5
in
IT
z
o
5
o
o
z
55
PLUME TOP
PLUME CENTER LINE
2 34567
), NONDIMENSIONAL DISTANCE DOWNWIND
Fig. 5.8 Rise of buoyant plumes in stable ail in crosswind at the TVA Paradise and Gallatin
plants.
point source. Clearly Eq. 4.26 gives the most consistent agreement, and on the average
it slightly underestimates rise. A constant of 2.4/0.82 = 2.9 works best, i.e.,
(5.7)
Ah =2.9« —
A further test of the simplified theory for bent-over plumes is shown in Fig. 5.8
for six periods of TVA data, which include the complete trajectories of the plume
center lines and plume tops in stable air. The center lines follow the "2/3 law" in the
first stage of rise with a fairly typical amount of scatter and reach a maximum in the
neighborhood of x = n us'14 as is predicted by theory. There is less scatter in the
final-rise stage, where four of the six trajectories almost coincide. The actual final
heights range from 450 to 1500 ft. The plume tops level out at
Ah = 4.0
(-)*
Vus/
(5.8)
When two or three stacks were operating at the TVA plants, there was some
evidence of enhanced final rise in stable conditions. The maximum enhancement that
-------
2-62
56 CALCULATED AND OBSERVED PLUME BEHAVIOR
could be expected according to Eq. 5.7 would be 26 and 44% for two and three stacks,
respectively, if the total heat emission could simply be lumped together in computing
F. The averaged observed enhancement relative to Eq. 5.7 was +20% with two stacks
operating and +30% with three stacks operating except that when the wind was
blowing along the line of three stacks at Colbert the enhancement was +40%.
Enhancement also depends on stack spacing since the plumes can hardly be expected
to interact with each other if they are too far apart, especially if the wind is
perpendicular to the line of stacks. In the preceding cases the stacks were spaced less
than 0.9(F/us)*, or about one-fourth of the plume rise apart.
-------
2-63
There is no lack of plume-rise formulas in the literature, and selection is complicated
by the fact thai no one formula appb'es to all conditions. For a given situation many
different predictions emerge, as is shown in Table 5.1. The variety of theoretical
predictions follows from the great variety of assumptions used in the models; the
disagreement among empirical formulas is due to the different weighting of data used
in their formulations and to variability among the data. Another factor is the frequent
disregard of the dependence of plume rise on distance downwind of the stack. In the
formulas recommended in the following paragraphs, aU symbols are given in Appendix
B, and the constants in the formulas are optimized for the best fit to data covered by
this survey. Readjustment of the constants in previously cited equations is indicated
by primes on the equation numbers.
An important result of this study is that buoyant plumes are found to follow the
"2/3 law" for transitional rise for a considerable distance downwind when there is a
wind, regardless of stratification; i.e.,
Ah=1.6FV'x" (4.32')
The bulk of plume-rise data are fit by this formula.
In neutral stratification Eq. 4.32' is valid up to the distance x/x*= 1, beyond
which the plume center line is the most accurately described by
(4.34,
57
-------
2-64
58 CONCLUSIONS AND RECOMMENDATIONS
where
x* = 0.52 fe£ Fv'h* (hs<1000ft)
(4.35)
x* = 331^1 F* (h.>1000ft)
Equation 4.35 is the best approximation of x* at present for sources 50ft or more
above the ground; for ground sources an estimated plume height can be used in place
of hs. Equation 4.34' applies to any distance such that x/x* > 1, but owing to lack of
data at great distances downwind x/x* = 5 is suggested as the maximum distance at
wjvich it be applied at present. Even though Eq. 4.34' is the best of the dozen or so
formulas considered, the average plume rise at a given plant may deviate from the
value given by Eq. 4.34' by ±10% if the site i& flat and uniform and by ±40% if a
substantial terrain step or a large body of water is nearby. Furthermore, normal
variations in the intensity of turbulence at plume heights at a typical site cause x¥ to
vary by about ±20% on the average, with corresponding variations in Ah. For
fossil-fuel plants with a heat emission of 20 Mw or more, a good working
approximation to Eq. 4.34' is given by
Ah= 1.6FHu-' x" (x<10ht) ,
Ah = 1.6FV1 (lOh,)* (x>10hs) (''
For other sources, a conservative approximation to Eq. 4.34 'is to use Eq. 4.32 'up to
a distance of jc = 3x", then to consider the rise at this distance to be the final rise.
Equations 4.34' and 5.1' are also recommended for the mean rise in unstable
conditions although larger fluctuations about the mean should be expected (see
Fig. 2.4).
In stable stratification Eq. 4.32' holds approximately to a distance x = 2.4us~\
beyond which the plume levels off at about
(5.7)
as illustrated in Fig. 5.8. The top of the stratified plume is about 38% higher than that
predicted by Eq. 5.7, which describes the plume center line. Although no significant
increase in transitional rise is found when more than one stack is operating, some
enhancement of the final rise in stable conditions is observed provided the stacks are
close enough. If the wind is so light that the plume rises vertically, the final rise is
given accurately by
Ah = 5.0FV (4.25)
-------
2-65
CONCLUSIONS AND RECOMMENDATIONS 59
In computing s for Eqs. 4.25 and 5.7, an average potential temperature gradient is
calculated for the stable layer or for the layer expected to be traversed by the plume.
A buoyant plume will penetrate a ground inversion if both Eq. 5.7 and Eq. 4.25
give a height higher than the top of the inversion. The plume will penetrate an elevated
inversion if the top of the inversion lies below both Eq. 5.4 and Eq. 4.30, i.e.,
zj<4F°-'V-6 (calm) (5.4)
* (wind) (4'30)
All the preceding formulas apply to buoyant plumes, which include most plumes
from industrial sources, and they are fairly well confirmed by observations. Because of
a relative lack of data, it is more difficult to make firm recommendations of formulas
for jets. It appears that in neutral, windy conditions the jet center line is given by
(4.33)
\ u / w/
at least up to the point that
M-j "0 j--v /r f\\
= 3 — D (5.2)
u x '
as long as w0/u > 4. It can be only tentatively stated that in windless conditions the
jet rises to
Ah A~J (5.6)
where 4 is used as the value of C. This is on the basis of only three experiments. If
there is some wind and the air is stable, the minimum expected theoretical rise is
Ah= l.SI-^L) s'* (4.28)
Unfortunately there are no published data for this case, and it would be presumptuous
to recommend any formula without testing it. However, since Eq. 4.28 is based on the
same model, we should not use Eq. 5.6 or Eq. 5.2 if it gives a higher rise than
Eq. 4.28 does. The most conservative of the three formulas is the one that best
applies to a given situation. The same can be said of Eqs. 4.34', 5.7, and 4.25 for a
buoyant plume.
Obviously more experiments are needed to complete our basic understanding of
plume rise. In particular they are needed for jets at large distances downwind for all
-------
2-66
60 CONCLUSIONS AND RECOMMENDATIONS
stability conditions and for buoyant plumes at distances greater than ten stack heights
downwind in neutral conditions. Once the fundamental results are complete, it will be
worthwhile to study in detail the effect of the finite source diameter, the bending-over
stage of plume rise, the effect of wind shear and arbitrary temperature profiles, the
interaction of plumes from more than one stack, and the interaction of plume-rise
dynamics with diffusion processes.
-------
2-67
As discussed in "Basic Theory Simplified," in Chapter 4, entrainment of ambient air
into the plume by atmospheric turbulence is due mostly to eddies in the inertia!
subrange; so, for a bent-over plume or a puffin a neutral atmosphere, the entrainment
velocity, or velocity of growth, is given by
dr/dt=0e*r* (A.I)
where 0 is a dimensiordess entrainment constant, e is the eddy energy dissipation rate,
and r is a characteristic radius defined as (V/u)* for a bent-over plume. To apply this
entrainment assumption, some simple method of estimating e at plume heights is
needed, and (3 must be determined.
Ideally e would be related in some simple way to wind speed (u) and height above
the ground (z). In the neutral surface layer, e.g., the lowest 50 ft or so, such a relation
is well described by the expression1'5 e = u*3/0.4z, where u* is the friction velocity
and is proportional to the wind speed at some fixed height. Unfortunately, at typical
plume heights no such simple relation is found to exist. The turbulence becomes more
intermittent and is affected more by departures from neutral stability and by terrain
irregularities over a wide area. Still, enough data exist to estimate mean values of e
along with the amount of variability that should be expected.
Recent estimates of e were made by Hanna,142 who used vertical-velocity spectra
measured in a variety of experiments, and by Pasquill,1*3 who used high-frequency
standard deviations of wind inclination measured with a lightweight vane mounted on
captive balloons at Cardington, England. Hanna used data from towers at Round Hill,
31
-------
2-68
62 APPENDIX A
Mass.,144 and Cedar Hill, Tex.,145 from aircraft measurements made over a great
variety of terrain by the Boeing Company,146 and from several low-level installations
(below 50 ft). These values of e are used in Table A.I to test the relation'e* <* u">
by computing the median value of e* u"m and the average deviation from the median
value for m = 0, l/3, %, and 1 at each height of each experiment. Because e is sensitive
to atmospheric stability, only runs in which —1.0 < Ri < 0.15 were used from the
Round Hill and Cedar Hill data, where Ri is the local Richardson number; the Boeing
runs during very stable conditions and Pasquill's measurements above inversions were
omitted. Also omitted were the few runs made during very low wind speeds, i.e., less
than 2m/sec.
Table A.I shows that the excellence of the fit is rather insensitive to increasing the
value of m, especially at Round Hill and Cedar Hill. The best overall fit is with m = '/3;
the average percentage deviation from the median is lowest with m = */3 for four of the
eight sets of data and, on the average, is only 9% greater than the minimum value of
percentage deviation (indicated by t in Table A.I). This is fortunate because the
expression for x", the distance at which atmospheric turbulence begins to dominate
entrainment, turns out to be independent of wind speed when eSau** (see
Eq. 4.35 and the preceding discussion in Chap. 4). It is therefore very desirable to
adopt this approximation, keeping in mind the scatter about the median values shown
in the table.
It is evident in Table A.I that e^/uH decreases with height. With a power law
relation of eK/u* « I"11, the optimum value of n depends on which data are used. The
best least-squares fit to log e* /u* = constant-n log z is n = 0.29 for all the data but
n = 0.37 if the Pasquill data at 4000 ft are omitted. At Round Hill n = 0.31 between 50
ft and 300 ft, and at Cedar Hill n = 0.39 between 150 ft and 450 ft, but in Pasquill's
data n is only 0.15 between 1000 ft and 4000 ft. These values are roughly consistent
with the following three published conclusions: (1) Hanna143'147 confirmed the
relation e* = 1.5 ow Xm*for a wide variety of data, where <*w is the variance of
vertical velocity and Xm is the wavelength of maximum specific energy in the
vertical-velocity spectra; (2) data compiled in a note by Moore131 indicate almost no
dependence of ow on height from about 100 to 4000 ft except for very high wind
speeds (u > 10 m/sec); (3) Busch and Panofsky14* conclude that \m « z near the
ground and reaches a maximum or a constant value somewhere above z = 200 m. The
simplest expression consistent with all of the preceding evidence is eH /uH oc z" H up
to a height of the order of 1000 ft and then becomes constant with height. In the last
column of Table A.I, an expression of this type is compared with the data. The best
estimate of energy dissipation appears to be
e» = 0.9 [ft* /sec* ] us T * (z < 1000 ft)
(A.2)
e* =0.09 [ft* /sec"] u* (z > 1000 ft)
There remains the problem of how to determine the value of the dimensionless
constant 0, particularly when no observations of plume, puff, or cluster growth include
-------
2-69
EFFECT OF ATMOSPHERIC TURBULENCE
Table A.I
ENERGY DBSSIPATION VS. WIND SPEED AND HEIGHT
Source
Height, Number
ft of runs
f 1*4 /sec
frH
Round Hill 50
Round Hill 150
Cedar Hill 150
Round Hill 300
Cedar HlU 450
Boeing 750
Pasquill 1000
Pasquill 4000
8
11
9
4
6
22
31
10
0.636 17% 0.2661 M%| 0.103116% 0.042118% 0.98
0.495 11% 0.177110% 0.0631 8%t 0.022110% 0.94
0.457 20% 0.159118% 0.057 16%t 0.020119% 0.84
0.470 11% 0.1511 7%t 0.049 7%t 0.01717%t 1.01
0.331 9%t 0.1041 9%t 0.034 11% 0.010117% 0.80
0.256 20%t 0.083124% 0.028 34% 0.009153% 0.75
0.269 38%t 0.097144% 0.042 46% 0.018153% 0.97
0.172 49% 0.079 1 42%1 0.030 47% 0.011159% 0.79}
| Minimum value of percentage deviation.
}? = 1000ft.
simultaneous, independent measurements of e. The approach used in this review is to
assume the validity of Eq. A.2 at the time and place of diffusion experiments and to
compare the results with Eq. A.I.
Frenkiel and Katz149 used two motion-picture cameras to photograph smoke
puffs released above an island in the Chesapeake Bay. The puffs were produced by
small detonations of gunpowder from an apparatus on the cable of a tethered balloon.
The radii of the puffs were calculated from their visible areas at 1-sec intervals. The
values of fa* shown in Table A.2 were calculated from the first 2 sec of puff growth
by using Eq. A.I as a finite difference equation, i.e., by setting dr/dt = Ar/A t. Smith
and Hay150 published some data from several experiments on the expansion of
clusters of particles. In their short-range experiments, Lycopodium spores were
released at a height of 2 m and were collected on adhesive cylinders lined up
perpendicular to the wind at 100 tn downwind, yielding a lateral standard deviation of
particle distribution (ov). 'n their medium-range experiments, fluorescent particles
were released from an airplane at heights of 1500 to 2500 ft several miles upwind of a
sampling apparatus mounted on the cable of a captive balloon, yielding a vertical
standard deviation of particle distribution (oz). The values of 0e* shown in Table A.2
for the Smith and Hay experiments were calculated from the integral of Eq. A.I for a
point source, namely,
63
Interpreting the effective radius of a rising plume in terms of oy or az is difficult, but
in this case it was assumed that az = ay and that r = 2)ioy, as is true in the "top hat"
model equivalent to a Gaussian plume in the Morton, Taylor, and Turner58 theory.
The last column of Table A.2 shows the value of 0e^ inferred from the diffusion
data divided by the value of e1* calculated from Eq. A.2. The values of/3 inferred from
this calculation range from 0.62 to 0.82, a remarkably small range considering the
-------
2-70
64 APPENDIX A
Table AJ
GROWTH RATE OF PUFFS AND PARTICLE CLUSTERS
Source
Smith and Hay
Runs 1-5
Runs 7-10
Frenkiel and Katz
z = 15 to 22 m
z = 39 to 61 m
Smith and Hay
(May 7, 1959)
Number
of runt
5
4
6
7
4
z,ft
14 = Oy
13 = Oy
58
164
2500
u, tt/tec
18
30
19
52
16
Se^ft^/aec
0.60 ± 7%
0.96 ±18%
0.40+ 7%
0.48 ± 23%
0.17 ± 17%
calculated
0.62
0.82
0.64
0.78
0.74
indirectness of this approach and the wide range of variables involved. Note that the
short-range experiments of Smith and Hay were probably carried out within the
surface layer, where Eq. A.2 is not actually valid; nevertheless, the error in estimating e
is not large for moderate wind speeds at these heights. Table A.2 suggests that 0 ~ 0.7,
but, considering the small number of data and the indirectness of this analysis, the
more conservative value of 0 = 1.0 is recommended.
It should be cautioned that the characteristic plume radius, r, that appears in
Eq. A.I is not necessarily the same as the visible radius or other measures of size of a
passive puff or plume, and so the evaluation of 0 made in Appendix A is not directly
applicable to diffusion problems other than plume rise.
-------
2-71
Dimensions of each term are given in brackets: / = length, t = time, r = temperature,
m = mass.
bj Inversion parameter = g ATj/T [//t2 ]
CD Drag coefficient [dimensionless]
D Internal stack diameter [/]
F Buoyancy flux parameter [f/t3]', see Eqs. 4.19c and 4.20
Fm Momentum flux parameter [/*/t2 ]; see Eq. 4.19b
Fz Vertical flux of buoyant force in plume divided by up [f /t3] ;see Eq. 4.17
Fr Froude number = WQ/[g(AT/T)D] [dimensionless]
g Gravitational acceleration [//t2 ]
h Effective stack height = \ + Ah [/]
hs Stack height [/]
Ah Plume rise above top of stack [/]
k" Unit vector in the vertical direction [dimensionless]
L Characteristic length for buoyant plume in crosswind = F/u3 [/]
Q Emission rate of a gaseous effluent [m/t]
QH Heat emission due to efflux of stack gases [mf /t3 ]
R Ratio of efflux velocity to average windspeed = wc/u [dimensionless]
r Characteristic radius of plume or puff, defined as (V/u)*4 for a bent-over
plume [/]
T0 Internal stack radius [/]
-------
2-72
APPENDIX B
s Restoring acceleration per unit vertical displacement for adiabatic motion in
atmosphere [f5]; see Eq. 4.16
T Average absolute temperature of ambient air [r]
Ts Average absolute temperature of gases emitted from stack [T]
AT Temperature excess of stack gases = Ts — T[r]
ATj Temperature difference between top and bottom of an elevated inversion
[r]
3T/3i Vertical temperature gradient of atmosphere [r/l]
t Time [t]
u Average wind speed at stack level [//t]
u* Friction velocity in neutral surface layer [//t); see Ref. 115
V Vertical volume flux of plume divided by TT [/3/t] ;see Eq. 4.15
$ Average velocity of plume gases [//t] ;see Eq. 4.18
v" Velocity excess of plume gases = tfp — ?e [//t]
?,, Average velocity of ambient air [//t]
vp Average local velocity of gases in plume [//t]
w Vertical component of v1 = kj 3 [l/l]
w' Vertical component of Vp = k • Vp [l/l]
w0 Efflux speed of gases from stack [l/l]
x Horizontal distance downwind of stack [/]
x* Distance at which atmospheric turbulence begins to dominate entrainmenl
[/]; see Eq. 4.34.
y Horizontal distance crosswind of stack [/]
z Vertical distance above stack (/]
z Height above the ground [/]
z, Height of penetratable elevated inversion above stack [/]
o Entrainment constant for vertical plume [dimensionless]; see Eq. 4.22
0 Entrainment constant for mixing by atmospheric turbulence [dimensionless];
see Eq. A.2
r Adiabatic lapse rate of atmosphere = 5.4°F/1000 ft [r/l]
y Entrainment constant for bent-over plume [dimensionless] ; see Eq. 4.23
6 Eddy energy dissipation rate for atmospheric turbulence [/2/t3]; see
Ref. 115
8 Average potential temperature of ambient air [r]
6' Potential temperature excess of plume gases = 8p — 6 [T]
Op Average potential temperature of gases in plume [T]
3fl/3z Vertical potential temperature gradient of atmosphere [T//] ; see Eq. 2.1
p Average density of ambient air [m//3 ]
Po Density of gases emitted from stack [m//3 ]
pp Average density of gases in plume [m//3 ]
oz/ay Ratio of vertical dispersion to horizontal dispersion [dimensionless]
X Concentration of a gaseous effluent [m//3)
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Adiabatic lapse rate The rate at which air lifted adiabatically cools owing to the drop
of pressure with increasing height, S.4°F/1000 ft in the earth's atmosphere.
Advection The transport of a fluid property by the mean velocity field of the fluid.
Buoyant plume A plume initially of lower density than the ambient fluid after the
pressure is adiabatically brought to equilibrium. Usually, the term "buoyant
plume" refers to a plume in which the effect of the initial momentum is small, and
the term "forced plume" refers to a plume with buoyancy in which the effect of
the initial momentum is also important.
Convection Mixing motions in a fluid arising from the conversion of potential energy
of hydrostatic instability into kinetic energy. It is more precise to term this motion
"free convection" to distinguish it from "forced convection," which arises from
external forces.
Critical wind speed In the context of this critical review, the wind speed at the height
of an elevated plume for which the maximum ground concentration is highest in
neutral conditions.
Diffusion The mixing of a fluid property by turbulent and molecular motions within
the fluid.
Downwash The downward motion of part or all of a plume due to the lower pressure
in the wake of the stack or building or due to g downward step of the terrain.
Effective stack height Variously defined. The three most common definitions are: (1)
the height at which a plume levels off, which has been observed only in stable
conditions; (2) the height of a plume above the point of maximum ground
concentration; (3) the virtual height of plume origin based on the diffusion pattern
67
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68 APPENDIX C
at large distances downwind of the stack. Definition 1 is the easiest to apply in
stable conditions; definition 2 is the most practical in neutral and unstable
conditions; definition 3 is comprehensive but difficult to apply.
Efflux velocity The mean speed of exiting stack gases.
Entrainment The dilution of plume properties due to mixing with the ambient fluid.
Final rise The total plume rise after leveling off, if this occurs, especially as opposed
to the term "transitional rise."
Froude number The ratio of pressure forces to buoyant forces. The efflux Froude
number of a stack may be defined as Wo/[g(AT/T)D].
Fumigation The downward diffusion of pollutants due to convective mixing
underneath an inversion that prevents upward diffusion.
Inversion A layer of air in which temperature increases with height. Such a layer is
also stable.
Jet A nonbuoyant plume.
Lapse rate The rate at which temperature drops with increasing altitude; the negative
of the vertical temperature gradient.
Neutral In hydrostatic equilibrium. A neutral atmosphere is characterized by an
adiabatic lapse rate, i.e., by potential temperature constant with height.
Plume rise The rise of a plume center line or center of mass above its point of origin
due to initial vertical momentum or buoyancy, or both.
Potential temperature The temperature that a gas would obtain if it were adiabati-
cally compressed to some standard pressure, usually 1000 mb in meteorological
literature.
Stable Possessing hydrostatic stability. A stable atmosphere has a positive potential
temperature gradient.
Stratification The variation of potential temperature with height. Usually the term
"stratified fluid" refers to a fluid possessing hydrostatic stability, as does the
atmosphere when the potential temperature gradient is positive.
Temperature gradient In meteorology, usually the vertical gradient of mean tempera-
ture.
Transitional rise The rise of a plume under the influence of the mean wind and the
properties of the plume itself; i.e., the rise before atmospheric turbulence or
stratification has a significant effect.
Turbulence Three-dimensional diffusive motions in a fluid on a macroscopic scale.
According to Lumley and Panofsky,115 turbulence is also rotational, dissipative,
nonlinear, and stochastic.
Unstable Possessing hydrostatic instability. An unstable atmosphere has a negative
potential temperature gradient.
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33: 647-661 (1966).
129. O. G. Sutton, The Dispersion of Hot Gases in the Atmosphere, / Meteorol, 7: 307-312
(1950).
130. R. S. Scorer, The Rise of a Bent-Over Plume, Advance! in Geophysics, Vol. 6, pp. 399-411,
F. N. Frenkiel and P. A. Sheppaid (Eds.), Academic Press, Inc., New York, 1959.
131. D. J. Moore, Discussion of Paper: Variation of Turbulence with Height, Atmos Environ.,
1:521-522(1967).
132. F. H. Schmidt, On the Rise of Hot Plumes in the Atmosphere, Int. J. Air Water Pollut.,
9: 175-198(1965).
133. D. J. Moore, On the Rise of Hot Plumes in the Atmosphere, Int. J. Air Water Pollut.,
9: 233-237 (1965).
134. F. A. Gifford, The Rise of Strongly Radioactive Plumes, /. Appl. Meteorol.. 6:644-649
(1967).
135. J. S. Turner, Model Experiments Relating to Thermals with Increasing Buoyancy, Quart. J.
Roy. Meteorol. Soc., 89: 62-74 (1963).
-------
2-81
REFERENCES 75
136. G. T. Csanady, The Buoyant Motion Within a Hot Gas Plume in a Horizontal Wind,/ Fluid
Mech., 22: 225-239(1965).
137. D. P. Hoult, J. A. Fay, and L. J. Forney, A Theory of Plume Rise Compared with Field
Observations, Paper 68-77, 61st Annual Meeting of the Ail Pollution Control Association,
June 23-28, 1968, St. Paul, Minn.
138. H. Moses, G. H. Strom, and J. E. Carson, Effects of Meteorological and Engineering Factors
on Stack Plume Rix.Nucl. Safety. 6(1): 1-19 (1964).
139. H. Moses and G. H. Strom, A Comparison of Observed Plume Rises with Values Obtained
from Well-Known Formulas,/ AirPollut. Contr. Ass.. 11: 455-466 (1961).
140. R. Anderson et al., Electricity in Volcanic Clouds, Science, 148(3674): 1179-1189 (1965).
141. S. Thorarinsson and B. Vonnegut, Whirlwinds Produced by the Eruption of Surtsey Volcano,
Bull. Amer. Meteorol. Soc., 45: 440443 (1964).
142. S. Hanna, A Model of Vertical Turbulent Transport in the Atmosphere, Ph. D. Thesis, The
Pennsylvania State University, 1967.
143. F. Pasquill. The Vertical Component of Atmospheric Turbulence at Heights up to 1200
Metres, Atmos. Environ., 1: 441-450 (1967).
144. F. Record and H. Cramer, Turbulent Energy Dissipation Rates and Exchange Processes Above
a Non-homogeneous Surface, Quart. J. Roy. Meteorol. Soc.. 92: 519-532 (1966).
145. J. Kaimal, An Analysis of Sonic Anemometer Measurements from the Cedar Hill Tower,
Report AFCRL-66-542, Air Force Cambridge Research Laboratory, 1966.
146. Boeing Company, Low Level Critical Air Turbulence, Technical Progress-Monthly Report,
Contract No. AF33(615)-3724, Doc. No. 83-7087-11 and 83-7087-16 (1967).
147. S. Hanna, A Method of Estimating Vertical Eddy Transport in the Planetary Boundary Layer
Using Characteristics of the Vertical Velocity Spectrum, /. Atmos. Sci.. 25: 1026-1033
(1968).
148. N. E. Busch and H. A. Panofsky, Recent Spectra of Atmospheric Turbulence, Quart. / Roy.
Meteorol. Soc.. 94: 132-148 (1968).
149. F. N. Frenldel and I. Katz, Studies of Small-Scale Turbulent Diffusion in the Atmosphere,/
Meteorol.. 13: 388-394 (1956).
150. F. B. Smith and J. S. Hay, The Expansion of Clusters of Particles in the Atmosphere, Quart. J.
Roy. Meteorol. Soc.. 87: 82-101 (1961).
-------
2-83
AUTHOR INDEX
Abersold,J.N.,2
Abramovich, G. N., 25
Ambrosio, A., 21, 34, 52
American Society of Mechanical
Engineers (ASME), 4, 23
Anderson, R., 52
Baines, W. D., 17,37,48,49
Ball,F. K., 19,34,39,44,46,52
Barrett, C. F.,4
Barry, P. J., 7
Batchelor,G.K.,8,34, 37
Beall.S.E., 17,22,24, 25,50
Berlyand, M. Ye.,23,24
Best, A. C., 4
Bierly.E. W., 15
Bodwitha, F. T., 18
Boeing Company, 62
Boone.F.W., 19,54
Bomwasser, L. P., 17,22,24,25, 50
Bosanquet, C. H., 2,4,18, 35, 38, 39,
43,54
Briggs, G. A., 23, 29, 32, 37, 39,43, 54
Bringfelt, B., 20
Brummage, K. G.,4
Bryant, L. W., 18, 23, 35, 39, 48, 49
Busch,N.E.,62
Callaghan, E. F., 17,25,48,49
Carey, W. F.,4, 18, 35,38
Carpenter, S. B.,3, 15,21, 30,44,54
Carson, J.E., 24, 38,45,46
Chapman, C. S., 48, 49
Clark, A. J., 3
C ONCAWE (see footnote, page 4),
4,24,38,39
Cowdrey.C. F., 18,48, 49
Cramer, H., 62
Crawford,!. V., 17, 52
Crooks, R.N., 19, 23,38, 44
Csanady.G. T.,3,20,23, 36,37,
39,43,44,46
Culkowski.W. M.,7
Danovich.A. M.,19,37
Davidson, W. F., 23, 39
Da vies, I., 20
Da vies, P.O., 7
Davies,R.W.,21,52
77
-------
2-84
78
EsJoque, M. A., 35
Fan, L, 18, 35,48,49, 52
Fay, J. A., 18, 37,50
Forney, L.J., 18,37,50
Frenkiel.F. N.,63
FrizzoIa,J.A.,20, 54
Gale, H.J., 19,23,38,44
Gartrell.F. E.,3,15,21
Genikhovich, Ye. I., 23,24
Gifford, F. A., 37
Gill.G. C., 15
Goldstein, S., 6
Halitsky.J., 7, 17
Halton.E.M., 4, 18,35,38
Hamilton, P.M., 20,24,44,46
Hanna.S. R.,61,62
Hawkins,J.E.,4, 18
Hay,J.S.,63
Hewson, E. W., 15
Hill.G. R.,2
Hino,M.,35
Ho-gstrtim.V., 20
Holland, J. Z., 3,18, 22, 24, 38, 39,
45,46,54
Hosier, C. R., 14
Hoult.D. P., 18,37, 50
Humphreys, H.W., 17,34
Ireland, F. E., 7
James, K. W., 20
Johnson, D. F., 17, 22, 24, 25, 50
Jordinson,R.,48,49
Kaimal,J.,62
Katz,I.,63
Keffer.T. F., 17,37,48,49
Lee, S., 35
Leonard, A. S., 17,52
Lilly, O.K., 18,37
Lucas, D. H., 2, 3, 7, 20, 23, 24,36,
38,44,45,46
Lumley,J.L.,61
Moore, D. J., 7; 20,23, 36, 37,38,44,
45,46,52,62
Moriguchi.M., 20
Morton, B. R., 17, 29, 34, 35, 52,63
AUTHOR
Moses, H., 19, 23, 24,38, 45,46
Nonhebel,G.,4,7, 18
Norstor, E. R., 48, 49
Okubo, A., 35
Gnikul, R. I., 23, 24
Pai.S. I..17
Panofsky.H. A., 61,62
Pasquill, F,, 3,61
Patrick, M. A., 17,25,48,49
Pearson, J. L., 2
Priestley, C. H. B., 1 8, 23, 34, 36, 44,
45,46,52,54
Proudfit, W., 2, 52
Rauch.H., 19, 23,24,38,44
Record, F., 62
Richards, R. S., 18,29
Ricou. F. P., 29
Romie.F. E.,21,34,52
Rouse, H., 17,34
Ruggeri, R. S., 17, 25,48,49
Rupp, A. F., 17,22,24, 25,50
Sakuraba, S., 20
Sato, J., 20
Schlichting, H., 17
Schmidt, F. H., 37
Schmidt, W., 17, 34, 36
Scorer, R. S.,4,7, 8, 36,37
Scriven, R. A., 8
Serpolay.R., 10
Sherlock, R. H., 6
Simon, C., 21, 52
Singer, I. A., 13,20, 54
Slawson, P. R., 3,20,37,44,47
Smith, F. B., 63
Smith, M. E.,3, 4, 13,20,23,54
Spalding, D. B., 29
Spurr, G., 2, 20, 23, 36, 38, 44, 45, 46
Stalker, E. A., 6
Stewart, N. G., 19,23,38,44
Stone, G. N., 3
Strom, G. H.,3, 19,23, 38
Stumke, H., 8, 23, 24, 38, 45, 46
Sutton.O. G., 3, 36
Taylor, G. I.. 17,29, 34, 52,63
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2-85
AUTHOR INDEX
Telford, J. W., 35 Vasil'chenko, 1. V., 35
Thomas, F. W., 3, 15,21,23 Vehrencamp, J. E., 21, 34, 52
Thomas, M. D., 2 Vonnegut, B., 52
Thorarinsson.S., 52 Wells, A. E., 2
Tollmien.W., 34 Williams, F., 20
Turner, J.S., 17,29,34,35,37,52,63 Yamazi, I., 20
Vadot, L., 17,18, 50, 51 Yih, C. S., 17, 34
Van Vleck, L. D., 19, 54 Zeyger, S. G., 19, 37
79
-------
2-86
SUBJECT INDEX
Bifurcation, 8
Brookhaven National Laboratory,
20,54
Building effects, 7
Buoyancy, 6, 8-9, 10, 11, 18,22-24,
26,31-33,36,50,52
Buoyancy flux, 23, 27-28, 47
Central Electricity Research
Laboratories, 20, 38
Condensation of plume, 10, 35
Coning, 12-13
Diffusion, 2-4, 11-15,37,46,63-64
effect of temperature profile on, 12
Dispersion (see Diffusion)
Downwash, 5-8,39
Drag force on plume, 27, 28, 29,31,
35
Efflux velocity, 5-7, 8, 35
Entrapment, 8, 28-31, 33, 34, 37,49
Entrainment velocity, 28, 31, 35, 37
Fanning, 12-13
Froude number, 6, 8, 17
Fumigation, 12-15
Inversions, 9, 13, 14-15, 17,21,37,
50-53, 59
Jets, 17-18,24-25,29,37,48-50,
52,59
Lofting, 12-13
Looping, 1 2-13
Modeling studies, 16-18
Momentum, 6, 8, 26, 27, 31, 33, 35,
36,50
Momentum flux, 27, 50, 52
Multiple stacks, 47, 55-56, 58
Plume radius, growth of, 8, 30, 34, 36
Plume rise, aerodynamic effects on,5-8
definition of, 3, 39,46-47
effect on diffusion, 2,13-15
fluctuations in, 10, 11, 58
measurement of, 18-21
modeling of, 16-18
in neutral air, 10,17-21, 33, 38-50,
51,57-58
qualitative description of, 8-11
in stable air,10, 17, 18,19,21,29,
31-32,50-56,58-59
80
-------
2-87
SUBJECT INDLX
near stack (first stage), 32, 36. 55,
57,59
in unstable air, 10, 47, 58
Plume rise formulas, empirical, 22-25
recommended, 57-59
theoretical, 31-33,36
Plume rise model, bending-over
plume, 31
bent-over plume, 29-31
vertical plume, 28-29, 34
Plumes, dense, 17, 18
downwash of, 5-8
inclined, 18,35
looping of, 12-13
puffing of, 8, 12
Potential temperature, 9, 26, 51
Potential temperature gradient,
9-10,59
(See also Stability)
Radiation, thermal, 11
Reynolds number, 16-17.26
Stability, effect on plume, 9-10, 13
measurement cf, 19-21
81
Stack height, determination of, 3-4, 7,
13-15
effect on plume rise, 24, 34, 46-47
Stratification (see Stability)
Taylor entrainment hypothesis, 28-30.
34
Temperature gradient, 9, 52
(See also Stability)
Temperature inside plume, 8. 17, 26
Tennessee Valley Authority (TVA),
3,14,21,30,38,54
Terrain effects, 8, 15,46
Turbulence, atmospheric, 9, 30-31.
33,35,61-64
inertial subrange, 31, 37
self-induced, 8, 17, 28,29,35,37
Two-thirds law of rise, 32, 37, 42-47,
55,57
Velocity inside plume, 8, 9, 17, 26-27,
29
Volume flux of plume, 27, 28, 29-30
Wind speed, effect on plume, 8, 17-18,
29,35.36,39-12,61-64
measurement of, 19-21
NOTICE
This book, was prepared under the sponsorship of the United States Government.
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any of their employees, nor any of their contractors, subcontractors, or their
emp'oyees, makes any warranty, express or implied, or assumes any legal iiabi'"v
or responsibility for the accuracy, completeness or usefulness of any information.
apparatus, product or process disclosed, or represents that us use would no'.
infringe privately owned rights.
-------
2-88
NUCLEAR SAFETY INFORMATION CENTER
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P. 0. Box Y
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-------
3-1
PLUME RISE FROM MULTIPLE SOURCES
Gary A. Briggs
Air Resources
Atmospheric Turbulence and Diffusion Laboratory
National Oceanic and Atmospheric Administration
Oak Ridge, Tennessee
-------
3-3
Abstract
A simple enhancement factor for plume rise from multiple sources is
proposed and tested against plume rise observations. For bent-over, buoyant
plumes, this results in the recommendation that multiple source rise be
calculated as [(N+S) / (1+S)]1'3 times the single-source rise, Ah^, where
N is the number of sources and S = 6 (total width of source configuration
/ N^'3 Ah^)^' - For calm conditions, a crude but simple method is suggested
for predicting the height of plume merger and subsequent behavior, based on
the geometry and velocity variations of a single, buoyant plume. Finally,
it is suggested that large clusters of buoyant sources might occasionally
give rise to concentrated vortices, either within the source configuration
or just downwind of it.
-------
3-5
Introduction
In spite of extensive literature on the subject of plume rise,
there are many questions of practical consequence still unaswered.
One of these is the question of plume rise at large distances down-
wind in neutral conditions, where the few available data show no
leveling; one can "fit" a linear, a power law, or an asymptotic
exponential curve to these data, depending on which data are selected,
how they are weighted, and, to some extent, on one's personal pre-
judices. On the other hand, there are several special cases for
which simple power law approximations have been confirmed both by
full scale observations of plume rise and by physical modeling; this
holds especially for buoyant plumes in the close-in rising stage,
where wind velocity is the only atmospheric variable of consequence,
2
and for final rise in stable conditions.
One inadequately answered question is whether single source
plume rise is augmented by the presence of nearby plumes. This
question is of decreasing importance to the tall stack problem,
since the trend has been to combine as much effluent as possible
into one or two tall stacks, which assures the maximum possible plume
rise. However, large cooling towers are frequently paired and clusters
of up to 30 towers are being considered. Even with plume rise only
depending on the 1/3 power of buoyancy flux, it is possible that
the plume from such a cluster will combine and rise three times as
high as the plume from a single tower isolated from the cluster.
A General Approach
Obviously, if the sources are very close to each other the plumes
will combine and if they are very far apart the plumes will rise
separately. It seems reasonable to assume that the resultant rise
will be the single-source rise times some function of the number of
sources and the ratio of spacing between the sources to the single-
source rise (this assumes sources of approximately equal magnitude):
f (N, s/Ab^) , (1)
-------
3-6
where Ah^ is the rise from N sources and s is the center-to-center
spacing between the sources. An alternative that suggests itself
in the case of a line of sources is to replace s with the spacing
perpendicular to the wind direction, s, (s, is zero if the wind
o
is parallel to the line of sources). As previously noted, at one
TVA power plant greater rise enhancement was observed when the wind
was parallel to the line of three stacks.
The rise enhancement is not necessarily a monotonically
decreasing function of s/Ah.. . As can be visualized with the
help of Figure 1, inbetween the uncombined stage of rise (Ah < s)
and the fully combined stage (Ah » s), there is an intermediate
stage where the double vortex flows associated with isolated, bent-
4
over plumes may interact in a complex way, possibly causing
increased entrainment and decreased plume rise. However, this is a
transient stage, and given the normal scatter observed in the behavior
of turbulent plumes even in quiescent surroundings , it is
likely that the rise enhancement will appear to be a monotonic
function anyway.
COMBINED
\ S^—^>^ STAGE
oco
ENVELOPE OF
AREA SWEPT BY
A SINGLE PLUME
CONFLICT AND
REORGANIZATION
UNCOMBINED STAGE
VIRTUAL ORIGINS
Figure 1. Cross-sections of two adjacent bent-over plumes showing
geometry of flow at three distinctly different stages
of rise.
-------
3-7
It was decided to try fitting data from multiple sources to
a simple monotonic function of s/Ah.. that has the correct asymptotes,
namely
AhN TN + S
1 + S
(2)
where E will be called the "enhancement factor," S is a nondimensional
spacing factor, and Ah is proportional to the n power of source
strength.
Several possibilities were tried for S, the most obvious of
which was S « s/Ah.. . The other possibilities were based on the notion
that if one has a line of evenly-spaced sources and N ->• °°, the number
of effectively combined plumes would be proportional to the resultant
rise divided by the spacing:
1/n = N + S B ENAhl (3)
N 1 + S s
This assumption leads to E = (Ah /s) , which in turn leads to
S <* N (s/AhO ^1~n^ if N» S» 1, i.e., if the number of
effectively combined plumes is much less than N but much greater than 1,
An alternative formulation, equally valid when N » 1, is
s « [(N-Dsf-11 (4)
I n I
N Ah,
L iJ
In this case S ~n has a simple interpretation; it is the ratio of
total line width to thd total possible rise if all the plumes com-
bined. This has the advantage of being easily applied to a cluster,
as well as a line, by merely using the largest dimension across the
cluster in place of the total line width.
-------
3-8
These formulations assume that n < 1. If n >_ 1, it would be
possible for E^ to "jump" from E = 1 to E = N below some critical
value of s/Ah1. For a large (yN» 1), homogeneous two-dimensional
array of sources the term on the right side of Equation (3) would
be squared, which results in an equation similar to Equation (4)
except the exponent becomes 2/(l-2n). This leads to similar behavior
of the rise enhancement except for a more abrupt rise as s/Ab...
decreases, with a "jump" in E^ if n >_ 1/2.
Application to Buoyant, Bent-Over Plumes
The best data available to test the above approach are extensive
observations made by the Tennessee Valley Authority (TVA) in 1963
to 1965 . These include many observations with one or two stacks
operating (at two sites both N = 1 and N = 2 cases are available)
and some observations with lines of 3, 4, 8, and 9 stacks operating.
These plumes are buoyancy dominated beyond a distance of about 5
seconds times the wind speed, and in the great majority of cases are
bent-over (x > Ah, where x is the distance downwind of the stack).
Therefore I undertook a comparison of these data with two well-
2
proven formulas for buoyant, bent-over plumes , namely
Ah = Cx F1/3 u'1 x 2/3 and (5)
1/3
where C^ and C2 are dimensionless constants, u is the mean wind
speed at plume height, F is the flux of buoyant force in the plume
divided by IT times the ambient density p, and G is the restoring
acceleration per unit vertical adiabatic displacement in stable air.
More specifically,
G = £ li = £ [11 + -1^1 m
T 3z T [3z 100m J (7)
-------
3-9
where g = gravity, T is the ambient absolute temperature, and 36/3z
is the ambient potential temperature gradient. Experience has shown
that best results obtain when 39/8z is averaged between the source
height and the top of the plume. For the isothermal case (3T/3z = 0),
-1 2
G = 3000 sec . Also, for plumes in which buoyancy is due to sensible
heat flux, Q , we can write
n
QH (8)
IT c pT
p
A
•« 8.9-S-j QR / (MW)
sec
4
- 3.7 -S-j QR / (103 cal/sec)
sec
where c is the specific heat capacity of air. The approximations
are for sea level; F is inversely proportional to ambient pressure.
Reported values of C.. range from 1.2 to 2.6 and of C9 range
2
from 1.8 to 3.1 when applied to the plume centerline . The wide
range is partly due to different measurement techniques, greatly
different scales of sources, and in some cases to extraneous local
effects. Since the apparent enhancement factor E^ is going to
directly depend on what is accepted as the correct single-source
values of C- and C_, it seems most appropriate to establish them
on the basis of the same data set, especially since it includes many
single-stack experiments.
The distance x = 1000 ft was chosen to test Equation (5), the
"2/3 law," because this distance is well into the buoyancy-dominated
region of rise, is well short of the distance at which atmospheric
turbulence might diminish the rise, and is well represented by the
available data. To limit the extent of stability effects, periods
-1/2
in which 1000 ft > 2uG ' were not used, since the 2/3 law rise is
-------
3-10
diminished by more than 10% in such cases . On the other hand
o
since experiments have substantiated the theoretical prediction
-1/2
that a buoyant plume reaches its maximum rise at x « TruG , I
-1/2
chose x = 4uG~ as the "standard distance" for testing the pre-
diction of Equation (6). Beyond this distance, the number of
available data diminished rapidly. Some of the periods of obser-
vation were suitable for testing both formulas, containing data
as far as x = 6000 ft or more. In most cases, however, the distance
-1/2
x = 4uG was not reached, the stratification being close to
neutral (G -»• 0). Consequently, fewer data were available to test
Equation (6).
Some additional periods to test Equation (6) for 3 and 4 stacks
were found in some 1957 observations made at the Colbert power plant
o
by TVA . In these observations the plume top and bottom elevations
were determined by S02 sampling with a helicopter at 1/2, 3/4, 1, and 2
miles downwind and, in some cases, at further distances. In this
analysis the average of the rises at these four distances was used,
except in one case there was no determination at 3/4 mile and in
-1/2
another case 1/2 mile was less than 4uG
Few plume rise data are free of extraneous effects, and in
some cases the data do not make sense if these effects are ignored.
For instance, on the one day at the Widows Creek power plant that
the winds came from the southeast quadrant, the observed values of
C.^ were much lower than those observed on the other three days.
However, there is an unusual topographic feature at this site,
namely, a plateau escarpment 900 ft above grade about 7000 ft to
the southeast (the plateau runs southwest-northeast). In wind
tunnels the cavity region of such drops is observed to end at
roughly 10 times the height of the drop downwind, with pronounced
subsidence in this area. It seems likely that the plume was imbedded
in such an area of terrain induced subsidence on this day, so these
-------
3-11
three periods were eliminated. Some form of downwash is also
suspected at the Shawnee plant, since the observed values of
C.^ are very low except when the wind speed is less than 12 ft/sec.
This suspicion is reinforced by the fact that in most of these
cases the bottom of the plume was observed to drop below the stack
top, i.e., the reported plume depth was greater than twice the
centerline rise. This plant is situated in very flat terrain and
the stacks are 2 1/2 times the building height, but it may be that
the line of ten stacks itself forms a vigorous wake (the stacks
average 19 ft outside diameter and are spaced 83 ft apart). To
eliminate such cases of likely downwash, periods were omitted when
the ratio of plume depth to centerline rise >_ 1.6 (the median
value at x = 1000 ft for single plumes was 0.85, with an average
deviation of +22%). Since the bottom of a plume is more susceptible
than the top to stack- or building-induced downwash, as a further
precaution the rise of the plume top above stack height was
used in this analysis instead of the centerline rise; in fact,
comparison showed that the scatter resulting in observed values
of C1 and C_ was less or unchanged in every case.
In addition, three periods were eliminated from the comparison
with Equation (6) because the measured temperature profiles did not
extend to the top of the plume (Gallatin, 3/18/64, and Shawnee,
4/10/65). This left only one suitable period of data at Shawnee,
with 9 stacks operating and the stratification only slightly stable.
Unfortunately, the observed value of C~ was a little less than the
average for single stacks, so the 9-stack data seemed altogether
inadequate for the present purpose.
When the data were compiled by TVA they were divided into
periods mostly ranging from 30 to 180 minutes duration, averaging
about 90 minutes; the period length chosen depended on the relative
-------
3-12
constancy of meteorological conditions and the temporal spacing of
helicopter soundings to measure temperature profiles and pibal
releases to measure wind profiles. Within these periods, the number
of photographs of the plume at the distances specified above ranged
from zero to more than 30. It was arbitrarily decided to require
A "observations" (photographs) or more at that distance for a period
to be used in this analysis, in order that it be adequately represented.
With such a range of period duration and number of observations
per period, how to weight the data was problematic. The more periods,
the aore likely that the wide range of possible meteorological con-
ditions is well represented. The longer the period duration, the
better it is represented by temperature profiles (usually one per
hour) and wind profiles (usually two per hour). The larger the
number of observations per period, the better the plume rise is
represented. There is also the question of whether to use average or
median plume rises. The former is more commonly employed, but in
a nonlinear relationship, the average of the function is not necessarily
the function of the average argument; it probably is not. On
the other hand, if the relationship is monotonic the median of the
function is given by the function of the median argument. Perhaps
more clearly, we can write.
average x (Ah) ^ x (average Ah), but
median X ( h) = x (median Ah),
provided the relationship is monotonic. This condition is satisfied
in the case of Gaussian plume diffusion models, provided that X is
the ground concentration at any point. Furthermore, when the number
of data are few the median is less affected by an anomalous datum,
although it may be more erratic if the distrubution of values is
bimodal.
-------
Number
of
Stacks
1
2
2
3
9
Number
of
Stacks
1
2
3
3*
4*
Table
Number
of
Periods
53
13
13
5
3
Table
Number
of
Periods
10
10
4
4
4
ICaJ.
Range
Mulc±ple source rise compared to Ah "• C F u
Values of C.. shown in parentheses
of s *
1/3 -1 2/3
u x Periods
.94 -
.29 -
.35 -
.26 -
Kb).
Range
(F/us)
.40 -
.70 -
.66 -
.51 -
(2.
1.30 1.
.87 1.
.58 1.
.29 1.
11)
17
29
53
62
Averages
Dura-
tions
(2.20)
1.11
1.22
1.46
1.51
L x2/3
By Medians By
Observa-
tions
(2.18)
1.12
1.16
1.47
1.51
Multiple source rise compared to
Values
of s *
1/3
Per
(4.
.99 1.
.91 1.
.79 1.
.59 1.
of
iods
30)
15
24
13
32
C» shown
Averages
Dura-
tions
(4.66)
1.05
1.17
1.04
1.22
Periods
(2.14)
1.10
1.21
1.43
1.51
Ahl - C2
Dura-
tions
(2.17)
1.06
1.19
1.41
1.49
Observa-
tions
(2.14)
1.08
1.15
1.43
1.51
Average
of
Averages
(2.16)
1.13
1.22
1.48
1.55
Median
of
Medians
(2.14)
1.08
1.21
1.43
1.51
(F/us)1/3
in parentheses
By
Observa-
tions
(4.23)
1.14
1.29
1.15
1.34
Medians By
Periods
(3.96)
1.25
1.39
1.27
1.43
Dura-
tions
(4.70)
1.01
1.17
1.07
1.21
Observa-
tions
(3.81)
1.25
1.45
1.32
1.49
Average
of
Averages
(4.40)
1.12
1.23
1.10
1.29
Median
of
Medians
(3.96)
1.20
1.39
1.27
1.43
* 1957 data (by SO- sampling)
00
frw
00
-------
3-14
The criticality of the weighting technique employed is illustrated
by Table 1. Reading down any column, we find the expected trend of
plume rise with the number of stacks and the spacing factor. Reading
across any row, however, we note large disparities In observed values
of Ah^/ h , depending on which kind of average or median is used. In
general, median values are lower than average values in the case of
the 2/3 law (Equation 5), indicting that anomalous rises tend to be
higher than expected; it may be that the measured wind speeds are
too high in these cases, due to inadequate sampling. The same is
true for the stable rise formula (Equation 6) for one stack, except
that wind speed is not such a strong determining factor in this case.
Curiously, the two "high rise" periods here are also the two periods
of the greatest wind direction shear (105° and 170°). The median and
the average rises compared to Equation (6) are in good agreement for N > 1,
but unfortunately substantial differences appear when these rises are
divided by the single stack rise computed by the two methods. This
emphasizes the importance of obtaining good base values of C.. and C_
for single sources for comparing with multiple source values. Unfortunately
only 10 periods were suitable for -determining C.. C was determined from
53 periods, and shows good agreement between the average values and the
median values.
Table 2 shows the resulting values of s/Ah* and the observed values
of the dimensionless spacing factor S calculated by two different methods
r /T./ \l/3
F u x or (F/us) was used to nondimensionalize data for
each period). The values designated (avg.) were calculated using
the average of averages of s/Ah*. Ahj/Ah*, and A^/Ah* = C± or C,,.
The values designated (med) were calculated using the median of
medians for the same quantities. Such values are shown for (Al^/Ah*) 4 C
and C2 in the last two columns of Table l(a) and l(b). S was calculated *
from the relation S - (N - £/)/(£/ - 1) with EN . (,,^ , or
-------
3-15
It is readily seen that the type of calculation used makes little dif-
ference with regard to s/Ah*, but greatly affects S, particularly in
the stable case.
Table 2 - Nondimensional spacing factors
Equation
5
5
5
5
6
6
6
6
N
2
2
3
9
2
3
3*
4*
s/Ah*(avg)
1.16
.66
.46
.26
.68
.76
.72
.55
s/Ah*(med)
1.18
.70
.46
.26
.64
.70
.725
.55
s(avg)
1.22
0.21
0(EN>Nl/3)
1.95
1.59
1.31
4.79
1.58
S(med)
2.90
0.33
0.04
2.28
0.39
0.19
0.91
0.55
1957 data (by SO sampling)
-------
3-16
Table 3. Error in predicting Ah-jAb* for various estimates of S
Plume
Rise
Equation
5
5
5
5
6
6
6
6
Number
of
Stacks
2
2
3
9
2
3
3*
A*
Using Averages
Eq.(9) Eq.(lO)
5^3.19
-2%
-7%
-13%
+22%
S1=12.2
-1%
-4%
+7%
0%
S2=1.83
-1%
-4%
-11%
+16%
S2-11.2
+1%
-5%
+6%
-2%
Eq.(ll)
S3-=4.00
+1%
-3%
-12%
+7%
S3=22.6
+3%
-6%
+6%
-5%
Using
Eq.(9)
5^5.08
0%
-8%
-13%
+20%
5^3.08
-1%
-5%
+4%
+2%
Medians
Eq.(lO)
S2=2.80
+1%
-5%
-11%
+12%
S2=2.72
0%
-6%
+3%
0%
Eq.(ll)
S3=6.50
+2%
-4%
-12%
+2%
S3=5.52
+1%
-6%
+2%
-2%
(Medians)
Eq.(ll)
V6
+3%
-4%
-12%
+3%
V6
+1%
-7%
+1%
-3%
* 1957 data (by SO sampling).
-------
3-17
Finally, both estimates of S were used to develop optimum
approximations of S based on N and s/Ahj (for the latter, (s/Ah*) * CL
or C? was used). Three formulas were tried in each case:
(s/Ah ) (9)
N (s/Ah1)3/2 (10)
c - e f^-1) s 1 3/2
b — £>0 I —T~T-^— TilI v-L-U
3 |y/3 Ahx J
as discussed in an earlier section (in both Equation (5) and Equation
(6), n = 1/3). Optimum values of S1, S, and S_ were computed from
1 ' * J
the above formulas and the values of S, s/ h , C. and C_ shown in
Tables 1 and 2. Within each group the S. values ranged considerably,
so an overall "optimum" value was chosen using a weighted geometric
average. The weighting factor was the number of periods per subgroup
times - d(in E )/d(£n S) at the observed S value, as calculated from
Equation (2). This derivative indicates in a rough way the sensitivity
of the plume rise prediction to a compromised value of S, deviating
from the specific optimum.
Table 3 shows how well these "optimum" estimates of S predict
the average or median plume rise (Ah^/Ah*) for each subgroup of data.
It is interesting to note that the use of medians instead of averages
improves the performance of all three estimates for S for both plume
rise equations. For the data compared with plume rise Equation (5),
Equation (10) for S works better than Equation (9) and Equation (11)
works best of all. For the data compared with Equation (6), all
three estimates for S perform about the same. Another interesting
feature of the calculations with medians is that the optimum values
of S1, S_, and S« turn out to be about the same with either plume
rise formula, in contrast to the calculations using averages. This
is a very desirable result, as it permits a "universal" approximation
for the nondimensional spacing factor, namely Equation (11) with
C. = 6.
'^ *r
r»
I-1
-------
3-18
(see the last column of Table 3). This seems the best choice since
Equation (11) clearly works best for the "2/3 law" of rise, has a
simple interpretation, and is easily adapted to clustered sources
as well as line sources. For n = 1/3 this estimate for S can be
readily substituted in Equation (2) to get the enhancement factor
over single-stack plume rise.
To adapt Equation (12) to clustered sources, simply replace
(N-l)s by the greatest distance across the cluster. This seems a
fairly safe procedure, since this equation is not based on a wind
direction-dependent spacing factor, such as sd- The data fairly
indiscriminately include cases of wind parallel, perpendicular,
and diagonal to the line of stacks. Since Equation (12) is valid
whether the plumes overlap each other vertically or flow together
side-by-side, it seems likely to work satisfactorily in mixed cases,
although it is possible that very different types of plume inter-
action could occur. As a factor of conservatism, the coefficient
S_ - 6 does more severely underpredict rises than it overpredicts
them in Table 3.
Incidentally, I did make similar calculations with the directional
spacing s^. In comparisons with Equation (5), it worked much better
than s for the 3 and 9 stack data, but did poorly for the 2-stack
data (the highest values of Ah2/Ah*, tended to occur with the larger
values of sd/Ah*, contrary to expectations). In comparisons with
Equation (6), sd again worked poorly for the 2-stack data, although
sd/Ah* was remarkably well correlated with Ah3/Ah*, as was previously
noted. In view of these mixed results, the limited applicability
of sd (to lines of sources only), and the presence of large wind
direction shears with height at times, the emphasis in this paper
is on s instead of s,.
d
One may be tempted to further generalize Equation (12) for other
values of n, such as might apply for final rise in neutral or unstable
conditions, by putting C3 inside the parentheses, changing the exponent
to l/(l-n), and replacing N1/3 with Nn. This would give S1'11 =15
times the maximum horizontal dimension of the source configuration
divided by the rise for the fully combined plumes. It seems intuitively
-------
3-19
reasonable that substantial combustion will occur if the rise is
15 times the total source diameter, but still I would not recommend
this procedure as it is too speculative. It would be safer to just
apply Equation (12) at the distance where the "2/3 law" rise terminates
for a single source (see ref. 7) and use
Ah,,
E = — - =
N Ahx
as for the "2/3 law." The termination distance is probably extended
when the plumes combine in neutral and unstable conditions, but no
data exist to confirm this. The termination distance in stable con-
ditions does not depend on the source strength, which is why n is
the same for Equations (5) and (6).
Technique for Buoyant, Vertical Plumes
The generalized approach described earlier could also be used
to predict multiple source rise in nearly calm conditions, when buoyant
plumes rise vertically until they reach a limiting height in stable air.
For rise in uniformly stratified stable air, one would use Equations
(2) and (A) with n = 1/4, as the tops of single plumes are found at
(14)
(ref. 1). Unfortunately, no data is on hand to test this approach
for vertical plumes. Furthermore, any results for bent-over plumes
can not be adapted to vertical plumes because the geometry of the flow
is quite different. For instance, in the rising stage the plume
radius R = 0.5z for a bent-over plume and =» O.lz for a vertical plume,
where z is the height above the virtual point source.
There is a simple alternative approach, however, based on what is
known about single plume vertical velocity profiles. According to labor-
atory measui
is given by
9
atory measurements on buoyant, vertical plumes the vertical velocity
w . 6.9 (F/z)1/3 e~96
-------
3-20
where r is the distance off-axis. This gives a volume flux 0.226 F z
-1/2
and the effective plume radius is at least R = (96) z = O.lz
(this is for a top-hat profile with the vertical velocity anywhere
within 'the plume equal to the measured axial velocity) . Imagining N
sources clustered in an area of maximum dimension (center-to-center)
D, the plumes must combine at z £ ^96 (D/2) / VN" = 5 D/ /N^ or else
2
the total plume cross-sectional area exceeds (ir/A)D . Considering
the effect of inflow velocity at the circumference on bending the
peripheral plumes toward the center, plume merger occurs at less
than z = 4D/ V"N" (this was done by assuming horizontal inflow, dif-
ferentiating the volume flux to get entrainment rate at each level,
assuming conservation of entrained radial momentum for the peripheral
plumes, and assuming the axial value for vertical velocity). This
result suggests that plume merger always occurs if the rise exceeds
2.8D (for a "cluster" of 2 sources), and if the rise exceeds even
ID when N_>_ 16.
If the single-source rise exceeds the plume merger height calulated
above, one could treat the sources as a single one, but there is no
ready model for handling the dynamics of the transition region.
One crude way to handle this would be to consider the plumes to be
separate below the height of merger z and to be one plume above
z = zm' ^^ a source strength (NF) and a virtual point source height
at z = (1 - VrN)zm, i.e., below the actual source height. This results
in the same total cross-sectional area of the plume (s) at z = z and
a disparity in the axial velocities equal to N1/6. This disparity
would diminish above z = z^, as the plume adjusts to being unified.
Of course, if this technique results in less total rise than for a
single plume, which can happen if n < 1/2, then it would be more
realistic to use the single- source rise.
-------
3-21
Multiple Source Behavior with Vorticity
There is an interesting, and possibly important, question about
the behavior of the rising plume from a multiple source in the presence
of vorticity. Under the right combination of large-scale horizontal
vorticity, and vertical driving force (such as buoyancy) one or more
areas of concentrated vorticity can develope.
The exact mechanism of vortex formation is complex and not fully
understood, but can be described roughly as follows. In an area of
steady-state rbtation, in the absence of friction the centripetal
acceleration of a fluid is just balanced by a radial pressure gradient.
Now if such an area of rotation with a vertical axis is located over a
horizontal surface, the centripetal acceleration is zero at the surface,
since friction causes the velocity to approach zero there. The radial
pressure gradient then induces a horizontal inflow near the surface
until it is balanced by frictional forces. This horizontal convergence
can be maintained if some continuous removal mechanism is available, such
as suction from above or a continuous supply of buoyancy. Unless over-
whelmed by turbulent friction, the angular momentum of converging fluid
tends to be conserved, which leads to a concentration of vorticity and
greatly increased tangential velocity near the surface around the center
of the inflow.
This'general phenomenum occurs with a wide range of scales and
intensities in nature. The hurricane developes from large-scale vorticity
due to the earth's rotation and maintains itself with buoyancy
generated by latent heat release from convective showers around
the eye wall. Tornadoes may be induced by "suction" from low pressure
created aloft in the "tornado cyclone," which again may derive from
vertical instability created by latent heat release. In contrast,
dust devils feed on buoyancy generated by dry heat that is most
intense at the ground. Their vorticity may derive from topographically-
induced eddies, surface roughness inhomogeneity, mechanical shear,
or from convective eddies. Such convective eddies are present
throughout any day that is not heavily overcast, with vertical and
horizontal scales of the order of 1 km and velocities of the order
-------
3-22
of 1 m/sec. Waterspouts may generate In a similar fashion, with most
of the buoyancy due to the lower molecular weight of water vapor,
rather than to dry heat. Vortices have also been observed dangling
from smoke plumes from volcanoes.10 In the case of Surtsey volcano,
wind speeds of the order of 90 m/sec were estimated in one of the
vortices.10 The vortex field could derive from either the wake of
the volcano or the wake produced by the plume itself.
Concentrated vortices have also been produced in the atmosphere
by man's activities. Of considerable interest to the present paper,
in France a multiple source consisting of 100 oil burners generated
11 12
dust devils near the burners "with intensities equal to small tornadoes."
This array, producing a total of 700 MW of heat, was built as an experiment
to artificially induce cumulus convection. "Fire whirlwinds" have
Occurred over fire-bombed cities, large oil fires, and over natural
and intentional timber burns. One of the latter produced a 1200 ft.
diameter whirl which lifted 30 inch by 30 ft. logs. Similar vortices
(although less intense) have been produced by relatively modest burns
14
of less than 100 acres and by experimental bonfires releasing only
100 MW of heat. Interestingly, I have not seen any vortex phenomena
reported for a compact source of heat, such as a chimney plume or a
cooling tower plume.
The conditions necessary for the formation of concentrated vortices
are poorly defined by present knowledge. The best presently available
tool for exploring these conditions is physical modeling. A number of
laboratory experiments on vortices have been performed over smooth plates
(see, for instance, refs. 15-19). Fitzjarrald15 used particularly simple
boundary conditions, namely a uniformly heated, circular plate surrounded
by plexiglass vanes tilted uniformly with respect to the local tangent.
bepending on the relative temperature elevation of the plate and the
degree of tilt, five qualitatively different flow regimes were observed.
The type of flow regime depends generally on the ratio of buoyancy-
induced velocity to tangential velocity at the periphery of the heated
region. When this ratio is large a pure plume (no vortex) developes
and when this ratio is low pure swirl developes. Well formed vortices
-------
3-23
are generated at intermediate values. It is difficult to apply the
results of a smooth plate (laminar boundary layer) experiment directly
to the atmosphere, but one interpretation of the above experiment is that
natural convection is likely to produce vortices (namely, "dust 'devils")
but a single cooling tower or chimney plume is much too buoyancy-
2
dominated to do so. However, a large ( 1 km ) multiple source might
sometimes produce a vortex; if it does, the vortex velocities are likely
to be about three times those occurring in natural dust devils (see
Appendix A).
The smooth-plate laboratory studies on vortices beg many questions
about their applicability to multiple buoyancy source behavior in a
field of vorticity. For instance, what are the effects of ground
roughness, the "spottiness" of the buoyancy source, the source structure,
the above-ground heat release, and the presence of a "lid"? These
questions could be at least partially answered by performing similar
laboratory experiments with models simulating prototype geometry.
The "dangling vortices" observed on the downwind side of the Surtsey plume,
to my knowledge, have not been modeled yet; however, since the phenomenum
is more likely to have off-site effects, it would be prudent to model
a large multiple source in a low wind speed (atmospheric boundary layer)
wind tunnel, as well as for the zero crosswitid case. In either case,
it is important to scale the imposed velocities (either the tangential
velocity of imposed circulation or the crosswind velocity) to the
1/3
scale velocity for buoyancy, (F/R) (F is the total buoyancy flux
parameter for the complex of sources and R is the complex radius).
In other words, characteristics Froude numbers must be the same in
model and prototype.
Summary and Recommendations
It seems reasonable to expect enhanced plome rise over that of
a single source when two or more sources are in close proximity.
A simple enhancement factor was postulated that had the correct
asymptotes for large spacing (EL^ = 1, no enhancement) and very close
-------
3-24
spacing (EL, = 11°, where N is the number of sources and the single-
source rise Ah. is proportional to the single-source strength to the
nth power). This was compared with TVA observations of plume rise from
lines of stacks (N = 1, 2, 3, A, and 9) for three different assumed
forms of the nondimensional spacing factor and for two different plume
rise equations:
1 ~ 1 u and
r F i173
Ah = C -| (6)
L J
The "2/3 law," Equation 5, was applied at x = 1000 ft. when 1000 ft.
1/2
was less than 2 u G , and the stable rise law, Equation 6, was applied
-1/2
at x - A u G . A few periods of observations were excluded because
of strong reason to suspect downwash or because of too few plume
photographs (less than 4).
The nondimensionalized observed rises, and hence the observed
values of EL. = Ah^/Ah , varied significantly depending on whether
the data were weighted by number of periods, number of photographs,
or duration of periods, and also depending on whether averages or
medians were employed. Medians resulted in the best ordering of
data, and also resulted in nearly the same optimum nondimensional
spacing factor for both plume rise formulas. Recommended is
1/3
i+S | with S = 6 | -i^iii \ (13) and (12)
where S is the spacing between adjacent sources.
While this empirical enhancement was developed using line source
data, it is suggested that it could be conservatively applied to
clusters of sources by replacing (N-l)s with the maximum diameter of
the cluster. For "final" rise in neutral conditions, for which no
adequate data exist, the above formulas can again be recommended
since n - 1/3 is conservative for this case.
-------
S-25
No suitable data from multiple sources were found for calm con-
ditions, but from basic knowledge of the geometry and dynamics of
buoyant, point-source plumes, one can infer that plume merger will
occur at a height z = 4 D/ fil , where D is the diameter of a cluster
of sources. The characteristics of the merged plume could be roughly
predicted by assuming a single, combined source at a virtual origin
z = (1 - /N)z .
m
Finally, it is suggested that large clusters of buoyant sources, in
the presence of. vorticity fields due to natural convection or due to
the wake of the source and plume itself, may be capable of producing
concentrated vortices. Certainly they would release as much energy
as many natural sources which have produced strong vortices, and many
concentrated vortices have been observed to develope over man made
area sources of heat as well. The results of one smooth-plate laboratory
experiment, if applicable to multiple sources of buoyancy in the real
atmosphere, imply that a large source of that type could occasionally
produce a vortex on the scale of a large dust devil and with velocities
about three times those in dust devils. Certainly such an extrapolation
is not conclusive, but accurate modeling of specific source condigurations
in the presence of a vorticity field is strongly recommended when any
substantial jump in source size is proposed.
-------
3-26
Appendix A: An Interpretation of a Convective Vortex Experiment
To apply the results of Fitzgerrald's experiment to multiple
sources of buoyancy (rather than a uniformly heated surface), it is
necessary to estimate the total heat flux removed from the plate by
the flow. Only the plate temperature excess, AT , the tangential
velocity V at the periphery (R - 50 cm), and the vane tilt angle 6
00
are reported. We assume an inflow velocity at the periphery U^ = V^
cot 6. Outside the vortex, a constant inflow angle = 6 can be assumed with
a local velocity V = V^ esc 6(R/r); this satisfies both conservation
of angular momentum and continuity. Outside the vortex, the flow
appears laminar, so a rough estimate of the local heat transfer rate
can be made by using the rate for uniform, laminar flow over a flat
plate using (R - r) sec 6 for the travel distance and the local total
20
velocity in place of the velocity at infinity. The resulting total
1/2
F •= 1.322 g(AT /T) (U v R) R, where v is the kinematic viscosity.
P °°
Defining a characteristic buoyant velocity V = (F/R) , we discover
B
that the observed inflow veloctity U^ = 0.6 VR with only 6% average
deviation, i.e., it depends mostly on the buoyancy flux calculated
above and not on the inflow angle. Thus V /V =0.6 tan 6 in this
00 B
experiment.
The observed flow regimes depended strongly on tan 6, and much
less strongly on AT (ref. 15, Fig. 12). Since it is difficult to
specify an equivalent flat plate AT for a multiple source, I will
interpret the results for variations of tan 6 = 1.7 V^/V only,
evaluated at average AT « AO°C. When the ratio V^/V is less than
0.15, there is no significant vortex. Above this value and below 0.35,
a one-cell (all upward motion) vortex forms. Above 0.35 a two-cell
vortex (inner core of subsiding air) with much stronger circulation
forms, but above 0.6 this two-cell vortex becomes more turbulent and
diffuse. When V^ exceeds 0.9, the whole flow swirls, with no con-
centration of vorticity. At smaller values of AT , these transition
values of VjV^ shift downward, being proportional to AT 1/2
P
-------
3-27
ToT< apply this result to phenomena in the mixing layer of the
atomosphere, we need to estimate possible values of V available from
natural convection. This requires mulitple-point field data. In lieu
21
of this, I used the results of a numerical experiment by Deardorff
to estimate that the maximum possible V 0.8 (Hz.) , where z. is
i i i
the height of the mixing layer (usually of the order of 1 km) and
H = g HQ/(c p T), where HQ is the average sensible heat flux at the
ground (note that H is defined similarly to F; for an area of radius
2 —2 2 3
R, F = H R ). For a strongly convective day (H = 10 m /sec and
3
z =10 m), the maximum V 1.7 m/sec. This occurs on a scale 2R z ,
1/3
about 1 km. On much smaller scales the maximum V^ 6(Hz.) (R/z.).
Now apply these estimates to three very different kinds of sources:
1. A large cooling tower with R = 25 m, a total heat rejection of 2000 MW
and a sensible heat rejection of 400 MW. With sensible heat only (no
4 3 1/3
condensation), we have F = 3500 m/sec , V = (F/R) = 5.2 m/sec.
-223 3
On a strongly convective day H = 10 m /sec and z = 10 m,
1/00 *
so V^ <_ 6(10) ' (25/10 )m/sec = 0.32 m/sec and VjV^ <_ 0.06.
Thus such sources would not be expected to produce a vortex.
This is even more true of smaller cooling towers and hot plumes
from stacks, since presumedly the available V is smaller with
smaller radius.
2. Twenty (20) of the above towers clustered over an area with R = 500 m.
I O
with sensible heat only, F = 71,000 m /sec , V£ « 5.2 m/sec again,
and V /V« < 1.7/5.2 = 0.33. Vortices, especially the single-cell
•» . B —
type, are possible on this scale due to the larger available V^.
t, 243
3. Strong «atural convection on the same scale as above. F * HR « 2500 m /sec ,
VD - 1.7 m/sec, V /VD < 1.7/1.7 - 1.0. All types of vortices are possible,
B °° B —
depending on the magnitude of V^. Scales smaller than z^^ are probably
favored, since V /V^ becomes smaller for them. The threshold for dust-
00 B 1/3 1/3
devil formation is the scale R (.15/6)z±(HR) /(Hz±) - 0.004 z± 4m,
quite small.
-------
3-28
Thus, a very large multiple, buoyant source in a field of natural
convection may occasionally produce a vortex on the scale of a large
dust devil. However, if velocities in such vortices scale roughly to
V,., the velocities in a vortex produced by the source in example (2)
B
above would be about three times those of a large, natural dust devil.
Acknowledgements
This research was performed under an agreement between the Atomic
Energy Commission and the National Oceanic and Atmospheric Administration.
-------
3-29
References
1. G. A. Briggs, Plume Rise. TID-25075 (1969).
2. G. A. Briggs, Discussion of Chimney Plumes in Neutral and Stable
Surroundings, Atmos. Environ., _6:507-510 (1972).
3. Reference 1, pp. 55-56.
4. Reference 1, pp. 8-9.
5. R. S. Richards, Experiment on the Motions of Isolated Cylindrical
Thermals through Unstratified Surroundings, Int. J. Air Water Pollut.,
7^:17-34 (1963).
6. S. B. Carpenter, F. W. Thomas, and F. E. Gartrell, Full-Scale Study
of Plume Rise at Large Electric Generating Stations, Tennessee Valley
Authority (TVA), Muscle Shoals, Ala.(1968) (additional data was obtained
in the form of computer print-out by personal communication).
7. G. A. Briggs, Some Recent Analyses of Plume Rise Observations, 2nd Inter-
national Air Pollution Conference, Washington, D. C. (1970).
8. F. E. Gartrell, F. W. Thomas, and S. B. Carpenter, Full Scale Study of
Dispersion of Stack Gases, Tennessee Valley Authority, Chattanooga (1964).
9. H. Rouse, C. S. Yih, and H. W. Humphreys, Gravitational Convection
from a Boundary Source, Tellus, 4^:201-210 (1952).
10. S. Thorarinsson and B. Vonnegut, Whirlwinds Produced by the Eruption of
Surtsey Volcano, Bull. Amer. Meteor. Sec., 45:440-444 (1964).
11. J. Dessens, Man-Made Thunderstorms, Discovery, 25.'40-43 (1964).
12. J. Dessens, Man-Made Tornadoes, Nature, 193:13-14 (1962).
13. H. E. Graham, Fire Whirlwinds, Bull. Amer. Meteor.. J36:99-102 (1955).
14. D. A. Haines and G. H. Updike, Fire Whirlwind Formation over Flat
Terrain, U.S.D.A. Forest Service research paper NC-71 (1971).
15. D.E. Fitzjarrald, A. Laboratory Simulation of Coavectiye Vortices,
J. Atmos. Sci.. _30:894-895 (1973).
16. C. A. Wan and C. C. Chung, Measurement of Velocity Field in a
Simulated Tornado-Like Vortex Using a Three-Dimensional Velocity Probe,
J. Atmos. Sci.. ^9:116-127 (1972).
17. N. B. Ward, The Exploration of Certain Features of Tornado Dynamics
Using a Laboratory Model, J. Atmos. Sci.. 2?_:1194-1204 (1972).
18. B. R. Morton, The Strength of Vortex and Swirling Core Flowd, J. Fluid
Mech.. 38:315-333 (1969).
19. B. R. Morton, Geophysical Vortices, Chapter 6 of Progress in Aeronautical
Sciences, Vol. T_* Pergamon Press, Oxford, London, Edinburgh, N. Y., Paris,
Frankfurt (1966).
-------
3-30
20. W. H. Giedt, Principles of Engineering Heat Transfer. D. Van Nostrand
Co., Inc., Princeton, Toronto, London, N. Y. (1957) (see Eq. 7.38 on p. 147)
21. J. W. Deardorff, Numerical Investigation of Neutral and Unstable
Planetary Boundary Layers, J. Atmos. Sci., 2^:91-115 (1972) (see especially
Fig. 22).
-------
4-1
Determination of Atmospheric
Diffusion Parameters
R. R. Draxler
-------
4-3
4i«iur*-.'« Em iroitmrni \ol 10. pp 99-10." Pergimon Pieii 1976 Printed in Oral Bnum
DETERMINATION OF ATMOSPHERIC
DIFFUSION PARAMETERS
R. R. DRAXLER
The Pennsylvania State Universit). Pennsylvania, U.S.A.*
(Firs; received 21 Fehruary 1975 and in final form 16 June 1975)
Abstract—Manx methods used to predict the concentration of effluents from a continuous point source
require the diffusion parameters a, and a.. There are several methods available to determine them
various forms of power laws, averaging wind records, and graphs Based on field experiments, a general
equation is developed thai approaches the correct theoretical limits This equation leads to a method
for estimation of a, and a. which gives satisfactory estimates in all cases except for vertical diffusion
from ground sources during unstable stratification and vertical diffusion from elevated sources during
stable stratification, presumably because of lack of vertical homogeneity Alternate techniques are sug-
gested for these cases
Frequently in air pollution problems it is necessary
to determine the concentration of contaminates
downwind from a continuous point source. Appli-
cation of the Gaussian diffusion equation requires a
knowledge of the vertical and horizontal growth of
the plume This growth is usually expressed in terms
of the standard delation of the concentrations in the
lateral and vertical directions. ay and o,. respectively.
It would be desirable that the method used to
determine these diffusion parameters should be con-
sistent with the theory of diffusion developed by Tay-
lor. He showed that for an ensemble average of par-
ticle displacements during conditions of stationarity
and homogeneity the diffusion parameters may be
written as:
rf f
Jo^o
(1.1)
where /?(;) is the Lagrangian autocorrelation of the
appropriate component of the wind velocity fluctua-
tions; (riyTp are the variances of the lateral or vertical
components of the wind velocity, respectively; T is
the diffusion time. One would use F7 for horizontal
diffusion and vT7 for vertical diffusion. The autocorre-
lation starts at 1.0 and approaches zero for large dif-
fusion times Therefore, near the source the growth
of the plume is linear with respect to diffusion time.
At large times the growth becomes proportional to
the square root of time.
Exact knowledge of the behavior of the autocorre-
lation at intermediate times is difficult to obtain for
routine use in air pollution problems. Several
methods have been suggested to determine ar and
<7_ which do not require the autocorrelation function.
These methods vary considerably in their develop-
ment and application. Some rely more on empirical
data than others. Separate methods are often recom-
* Currently with NOAA—Air Resources Laboratories.
8060 13th Street. Silver Spring. MD 20910. L'.S.A
mended for elevated and ground sources Frequently
the different techniques in practical use are inconsis-
tent with each other.
It would be worthwhile to have a method which
is consistent with Taylor's theory, easy to apply, and
applicable to as many situations as possible. Pasquill
(1971) suggested a relationship for the diffusion para-
meters derived from Taylor's equation.
af = a,.Tf,(TiL). (1.2)
a. = a. T/: (T;rL). (1.3|
where tL. the Lagrangian time scale, is defined by:
(:)d: (1.4)
The S.D of the horizontal and vertical wind com-
ponents are given by a, and a,, respectively, and
/! and /2 are universal functions subject to the same
restrictions as equation (1.1). It is then necessary to
find the specific forms of the unknown functions /,
and/j. ar>d to design a practical method for the esti-
mation of tL. in order to obtain a general technique
for estimation of of and a..
ANALYSIS OF EXPERIMENTAL DATA
Since arT ^ e,X and «r.T s o^X one can rewrite
equations (1.2 and 1.3) as:
o.
and
a.
(2.1)
(2.2)
where o, and ot are the S.D. of the azimuth and
elevation angles, respectively, and X is the downwind
distance. The nature of functions /, and /; may be
determined by evaluating data from field diffusion ex-
periments where the variables specified in equations
(2.1 and 2.2) may be obtained.
99
-------
4-4
100
R. R. DKAXLER
Table I. Summary of ground source diffusion experiments
Site
Meteorological Data
Diffusion Data
Terrain/Tracer
Ocean Breeze u, o- at J-7 •
(Haugen and
Fuquay, 1963) AT (16. S • - 1.8 •)
Dry Oulch u, cg at 3.7 •
(Haugen and
Puquay, 1963) AT (16.5 B - 1.8 m)
PrairieGrass u, oe, o. at 2 m
(Barad, 1958) *
(Haugen, 1J59) AT (16 a - 2 m)
Green Glow u, oe at 2.1 »
(Fuquay .Simpson
and Hinds, 196li Rl (15.2 n - 2.1 «)
N.R.T.S. u, o., o. at « m
(Islltzer and Rl. IT (16 m - 1 m)
Dunbauld, 1963) Rl , AT (8 in - 1 m)
Oy (1200, Z400, 4800 •)
30 mln release
0, (853,1500,2301,4715,
5665 m)
30 Bin release
Concentrations along arcs
(50,100.200,100,800 m)
10 mln release
Ov (200,600,1600,3200,
15800,25600 m)
30 mln release
0, (100,200.
-------
4-5
Determination of •tmospheric diffusion parameters
101
Rr a bulk Richardson number, was used in most
cases It is defined by:
(2.4)
S2AZ
where g is gravity; AZ is the height difference between
the heights of the temperature sensors, AT" is the tem-
perature difference across AZ; u is the mean wind
speed at level Z.; y4 is the dry adiabatic lapse rate.
The stratification was divided into three categories,
(Rf < 0—unstable; Rf = 0—neutral; Rp > 0—stable),
but finally sorted by only two, stable or unstable. Not
enough data were in the neutral case to justify an ad-
ditional classification. In some cases not enough data
were available to compute Rp But in all cases suffi-
cient information was available to classify the stratifi-
cation during the trials as either stable or unstable.
The bivane data for Prairie Grass (Haugen, 1959)
list a^ the S.D. of the vertical velocity. Then:
* = £- (2-5)
There are no wind fluctuation data for Hogstrom's
experiments. We assume that up to at least the first
sampling arc the growth of the plume is linear, so
that:
"... = °-f (2.6)
Presentation of the diffusion data
All of the diffusion data from the various exper-
iments were separated according to source height.
stable or unstable stratification, and for horizontal
or vertical diffusion The data may then be plotted
with the right side of equations (2.1 or 2.2) as the
ordinate. The abscissa will be some function of T/tL.
The true Lagrangian time scale, tL. a constant,
could not be determined from the data presented for
most diffusion experiments. Instead, it is possible to
20
normalize the abscissa by a quantity 7",. another con-
stant, proportional to tL. Taylor showed that /, ind
/2 must stan at 1.0 for small T and decrease with
increasing T. We can then define 7; as the diffusion
time, T; required for/, or/2 to become equal to 0.5.
The relationship between tL and T, is derived in the
Appendix.
For each trial of an experiment the log of equations
(2.1 or 22) was plotted against log T and the linear
regression equation coefficients determined The value
of 7 when/, or/, were equal to 0.5 was then com-
puted, this being equal to 7" for that trial Also all
trials for a particular experimental site were plotted
together in the same way and then a mean 7| for
that site was determined This value of 7" was then
used for those trials for which the regression equation
could not be computed
The combined results for the four diffusion cate-
gories for ground and elevated sources are shown on
Figs. 1 and 2. Not included on Fig 1 is vertical diffu-
sion during unstable stratification and Fig 2 lacks
vertical diffusion during stable stratification.
Determination of the specific form of f , and f:
The data from Fig. 1 for horizontal diffusion from
a ground source were replotted as k>g(/",-l) on the
ordinate and log (T/TJ on the abscissa From this plot
a simpler equation was derived, which provides a
satisfactory fit, and fits Taylor's limit for large T.
This curve is drawn on Figs. 1 and 2 so that it may
be compared to the data. The assumption that /, = /2
can be justified in all categories except for vertical
diffusion in unstable stratification from a ground
source and vertical diffusion during stable stratifica-
tion from an elevated source. The theory is not satis-
1 '
+
3» *
1 1 1 1 1 1 __
9 10
I » T/T,
I • /i.j for diffusion frpm a ground tource Vertical diffusion during unstable stratification is not
included.
-------
4-6
102
R R DKAXLEK
20
i e
I 6
14
I 2
1-0
OS
0-6
0«
02
00
i—i—r
t - ,
;--
I I I L
5 6 7 e 9 C
T/T,
Fig 2. /i 2 for diffusion from an elevated source. Vertical diffusion during stable stratification is nol
included
factory in these categories due to the vertical inhomc-
geneit) Otherwise, considering the scatter on Figs. 1
and Z the vertical diffusion and horizontal diffusion
characteristics are satisfactorily described by the same
equation, equation (2.7). In two excluded categories
other curves were fitted to minimize errors of esti-
mation of a., although they do not satisfy Taylor's
limits
There was too much scatter of the data during ver-
tical diffusion from a ground source during unstable
stratification to pick any particular form of /2 As
pointed out before, there was considerable depletion
of the tracer noted at N.RT.S.. and a. at Prairie
Grass can only be confirmed to 100m from the
source Depletion would result in an overestimate of
a, when computed by continuity. Since the least dep-
letion and scatter occurs in the Prairie Grass data.
they shoud be relatively representative. By computing
a mean /2 at each of the arcs. /, takes on a parabolic
shape for Prairie Grass An approximation of that
curve is given by;
, 0.3(7/7, - 0.4)2
0.16
+ 0.7.
(2.8)
and a, or at. Estimated
-------
Determination of atmospheric diffusion parameter*
Table 3. Summary of T, and equations used for /, and /2
103
Horizontal Diffusion Vertical tJffuslon
4-7
Stratification
Stable
Unstable Stable Unstable
Ground Sources
TI (s)
f1.2
Elevated Sources
TI (s)
f1.2
300
s)2.7
(T>550s)3.3
1000
2.7
300 50
2.7 2.7
1000 100
2.7 2.9
100
2.8
500
2.7
T, for each category has some merit considering its
simplicity. The final
-------
4-8
104
R. R. DHOCLEK
source height Table 3 is entered and a 7; and equation
for/, , is selected. This in combination with equation
(3.1) is sufficient to compute arJ.
COMPARISON TO OTHER PREDICTION FORMULAS
A comparison of the predictions of a, and a, by
the equations indicated in Table 3 is given in the
first column of Table 4 which is broken down into
the eight diffusion categories. The top number in each
group is the geometric mean of the ratio of the com-
puted a, or a. to the one observed in the field exper-
iments The lower number is the standard deviation
of the log of the ratios. A better prediction is indi-
cated b> a smaller S.D and a ratio closer to one.
A farriiK of curves of a, and a, for varying stability
and distance is given in Turner (1967) and Gtfford
(1968). The curves are separated according to stabi-
lity, based on of or on a table of insolation and wind
speed, devised by Pasquill (1962). The curves were
derived from ground source experiments with a, aver-
aged over ten minutes. B> determining the stability
from ot. as suggested b> Gifford (1968), some of the
experimental data were compared to this prediction
method These results are also summarized in Table
4. For the elevated source the computed J«ur
**rtlc«) • Bttklt
srrts *£s.,.
ffct MF mMfttr !• ibt !*•
te (fw ont eftttrwa In th
• undjrd Attlauar of tta
.0?
.H
• t»
.51
>0
• K
11
H
• n
')*
.01
1}1
•rtrl
O.Bt
1.77
o.le
1.6*
1 .*£
0.49
1 13
l'i?
O.tl
1 12
t MV »r ik
14 tlp*rla»n
1C .«.
0.711
0.93
1.51
1.11
1.71
OK
1.05
1.60
l.OE
1 29
1.02
l.«
1.12
1 33
• rstio or tkt t«vuitl tv
data as a function of T/Tt one can make the exponen-
tial fit the data at short diffusion times by defining
1L = ty6.36. Then when T/Tj « l./,.j - 1- Equation
(4.2) may now be used as an equation for /,.j in con-
junction with the Tfs given in Table 3 to make predic-
tions of the diffusion parameters These results are
also shown in Table 4. The predictions by this
method are satisfactory where the conditions of
stationary and homogeneous turbulence exist. Fitting
the exponential to the data at short diffusion times
is acceptable for the range of atJ's encountered in
the diffusion experiments.
Acknowledgements—The author wishes to thank Dr F.
Pasquill for his suggestions and Dr. H Panofsk) for his
help in preparation of the manuscript
REFERENCES
Barad M. L. (Editor) (1958) Project prairie grass, a field
program in diffusion Geophysical Research Papers. No
59, Vols I and II. Air Force Cambridge Research
Center.
Barad M L and Haugen D A (1959) A preliminary eva-
luation of Sutton's hypothesis for diffusion from a con-
tinuous point source J. Meieorol 16. 12-20.
Bowne N (1960) Measurements of atmospheric diffusion
from an elevated source 6th A.E.C. Air Cleaning Confer-
ence. 7-9 July 1959. TID-7593, pp. 76-88
Elderlun C. E.. Hinds W T. and NuUej N. E. (1963) Dis-
persion from elevated sources Hanford Radiological
Sciences Research and Development Annual Report for
1963 HAV-81746, pp 1.29-1.36.
Fuquay J J., Simpson C L. and Hinds W. T. (1964) Predic-
tion of environmental exposures from sources near the
ground based on Hanford experimental data. J. appl
Meieorol 3. 761-770.
Gifford F. A. (1968) An outline of theories of diffusion
in the lower layers of the atmosphere Meteorology and
Atomic Energy (edited by Slade D H.) National Techni-
cal Information Service, T1D-21490.
Haugen D. A. (Editor) (1959) Project prairie grass, a field
program in diffusion. Geophysical Research Papers, No.
59. Vol. III. Air Force Cambridge Research Center
Haugen D A and Fuquay J. J. (Editors) (1963) The Ocean
Breeze and Dry Gulch Diffusion Programs, Vol. 1. Air
Force Cambridge Research Laboratories and Hanford
Atomic Products Operation, HW-78435.
Hay J. S. and Pasquill F. (1957) Diffusion from a fixed
source at a height of a few hundred feet in the atmos-
phere J. Fluid Mech 2. 299-3)0.
Hilst G R and Simpson C L (1958) Observations of verti-
cal diffusion rates in stable atmospheres. J. Meieorol.
15. 125-126
Hogstrom U. (1964) An experimental study on atmospheric
diffusion tellus 16. 205-251
hlitzer N. F. (1951) Sbort-tange atmospheric dispersion
measurements from an elevated source. J. Meieorol. 18.
443-450.
Islhzer N. F. and Dumbauld R. K. (1963) Atmospheric
diffusion-deposition, studies over flat terrain. Ini. J. Air
Wat. Pollul 7. 99-1022.
Pasquill F. (1962) Atmospheric Diffusion Van Noctrand.
New York
Puqufll F. (1969) The influence of the turning of wind
with height on crosswind diffusion. Phil Trans. R. Soc
(Land). A365, 173-181.
Pasquill F. (1971) Atmospheric dispersion of pollution Q
J. R. met. Soc 9J, 369-395.
-------
4-9
Determination or atmospheric diffusion parameters
105
Smith F. B. (1%5) The role of wind shear in horizontal
diffusion of ambient panicles. Q J. R. met. Soc. 91. 318-
329.
Smith M E. (1956) The variation of effluent concentrations
from an elevated point icurce A.M.A. Arch Ind Hth
14.56-68
Turner D. B (1967) Workbook of Atmospheric Dispersion
Estimates Public Health Service. 999-AP-26.
Walker E. R. (1965) A paniculate diffusion experiment.
J. Appl Meteorol. 4. 614-621.
APPENDIX
Relation between T, and the Lagrangian scale
The prediction method outlined in Table 3 works well
in all diffusion categories as compared to other popular
methods The exponimial form of the autocorrelation gives
comparable results in those diffusion categories where
homogeneous conditions are present. There is some ques-
tion as to whether the exponential function drops off too
'quick!) to be used at longer distances.
The constant of proportionality between IL and 7j can
be extracted as follows: from Taylor's equation it can be
ihown that as T approaches infinity.
Equation (2.7) at large 7" becomes
J'-3 0.9
Combining equations (5 1 and 5.2) yields:
1L = TV 1. 64
(5.1)
(5.2)
(5.3)
Fig 5. The autocorrelation for the exponential (solid lines)
and equation (5.4) (dashed line) The solid lines starting
from the top down represent tL = T,. tL •= T,,'1.64 and
tL = 7, '6.36.
Form of r/ir Lagrangian correlation function
The autocorrelation function for equation (2.7) may be
obtained by solving Taylor's equation for/, ,. then substi-
tution of equation (2.7) and double differentiation yields
~- 2(1 + o.9v r/T;»2
(54)
Equations (5.4 and 4.1 ) are plotted on Fig 5. With tL - TV
1.64 equation (4.1). the exponential, approaches the data.
represented by equation (5.4). only at large distances. When
the exponential is fitted to the data close to the source
as before so that IL •= T/6.36. the autocorrelation
approaches zero much too soon to properly describe diffu-
sion at large distances. The predictions of diffusion using
the exponential form when compared to experimental data
were satisfactory because practically all field experiments
were conducted no farther than several kilometers from
the source.
-------
5-1
Atmospheric Dispersion
Parameters in Gaussian
Plume Modeling
Dr. S. P. S. Arya
-------
5-3
ATMOSPHERIC DISPERSION PARAMFTERS IN GAUSSIAN PLUME MODELING
igf Outline.
I. THEORETICAL BASIS OF THE GAUSSIAN PLUME MODELING AND DISPERSION
PARAMETERS
1. Conservation of Mass - Diffusion Equation
2. Gradient Transport Theories
3. Statistical Theories of Diffusion
4. LaGrangian Similarity Theories
5. Contemporary Numerical Models of Dispersion
II. EXPERIMENTAL EVALUATIONS OF STABILITY AND DISPERSION PARAMETERS
1. Stability Parameters and Typing Schemes
2. Diffusion Measurement Techniques
3. Plume Diffusion Experiments
4. Empirical Sigma Schemes
5. Accuracy of Dispersion Estimates
This outline follows more detailed material presented on the following
pages. Use as a guide.
-------
5-5
I, THEORETICAL EASES OF THE GAUSSIAN PUJE MODELING AND DISPERSION P/WETE
1, CONSERVATION OF MASS - DIFFUSION EQUATION
(A) JNSTANTANFms PIFHJSION EQUATION.
CONSIDERATION OF MASS CONSERVATION OF ANY CONTAMINANT IN AN
ELEMENTAL FLUID VOLUTE, USING AN EULERIAN REFERENCE COORDINATE
SYSTEM, YIELDS
u irr "*• v -^~ •*•
^ X <
WHERE,
^ = INSTANTANEOUS CONCENTRATION OF THE CONTAMINANT [ML J
Uy V, W = INSTANTANEOUS VELOCITY COMPONENTS IN X, Y AND Z DIRECTIONS
FX; Fy, FZ = INSTANTANEOUS FLUXES OF THE CONTAMINANT IN X, Y AND Z
DIRECTIONS [ML^T"1]
ASSUMPTIONS: (i) NO SOURCES OR SINKS WITHIN THE ELEMENTAL VOLUME
(ll) NO CHEMICAL TRANSFORMATIONS
(ill) NO WASHOUT AND GRAVITATIONAL SETTLING
(B) REYNOLDS - AVERAGED EQUATION
JEYNOLDS' DECOMPOSITION: u = u + u'
v = v + v'
w = w + w'
%=Z+X'
HERE D REPRESENTS MEAN AND (') THE FLUaUATION, AFTER SUBSTITUTING IN
I)/ AVERAGING, AND NEGLECTING THE MOLECULAR DIFFUSION TERMS, ONE OBTAINS
^ REYNOLDS-AVERAGED EQUATION OF DIFFUSION:
-------
5-6
,
WHERE.,
"V,, 'V', 'W' ARE COVARIANCES OR TURBULEf^ FLUXES OF THE CONTAMINANT IN
X, Y AND Z DIRECTIONS,
(c) THE CLOSURE PROBLEM OF TURBULENCE
ALTHOUGH THE REYNOLDS-AVERAGED EQUATIONS OF FLOW AND DIFFUSION
ARE MUCH SIMPLER (WELL-BEHAVED) THAN THE INSTANTANEOUS EQUATIONS,
THE AVERAGED EQUATIONS ARE NOT A CLOSED SET AS THEY ALWAYS HAVE
MORE UNKNOWNS THAN THE NUMBER OF EQUATIONS, THIS PROBLEM IS
REFERRED TO AS THE CLOSURE PROBLEM OF TURBULENCE,
2. GRADIENT TRANSPORT THEORIES
(A) EDDY DIFFUSIVITY HYPOTHESIS
IN THE GRADIENT-TRANSFER APPROACH IT IS ASSUMED, ON THE BASIS
OF ANALOGY WITH MOLECULAR TRANSFER PROCESSES, THAT TURBULENT
TRANSFER (FLUX) OF MATERIAL IS DOWN THE GRADIENT OF ITS MEAN
CONCENTRATION, AT A RATE WHICH IS PROPORTIONAL TO THE GRADIENT,
THE EDDY DIFFUSIVITY HYPOTHESIS IMPLIES THE RELATIONS
-------
5-7
k 2*
^ ^
WHERE y AND ARE EDDY DIFFUSIVITIES IN X, Y AND Z DIRECTIONS,
AFTER SUBSTITUTING FROM (3) INTO (2), THE MEAN DIFFUSION EQUATION BECOMES
WHICH CAN BE SOLVED IF THE MEAN VELOCITY FIELD AND K^ Ky AND 1^ CAN BE
SPECIFIED,
(B) THE FICKIAN' DIFFUSION THEORY
BASIC ASSUMPTIONS:
(l) 1^, Ky AND ^ ARE CONSTANTS AND DO NOT VARY IN SPACE AND
TIME,
(ll) THE FLUID MEDIUM IS MOVING AT A CONSTANT VELOCITY WHICH
IS INDEPENDENT OF SPATIAL POSITION
THEN, THE SIMPLIFIED DIFFUSION EQUATION FOR SUCH A 'HOMOGENEOUS'
MEDIUM, MOVING AT A UNIFORM VELOCITY U, IS
-------
5-8
WHICH HAS BEEN SOLVED FOR THE VARIOUS SOURCE CONFIGURATIONS
AND DIFFERED BOUNDARY CONDITIONS (SEE SuTTON, 1953), WE DISCUSS
HERE ONLY A FEW OF THESE SOLUTIONS,
(C) gfWF, f I t^FNTARY SOLUTIONS OF THE FlCKIAN DIFFUSION EOUATIOM
(i) SOLUTION FOR AN INSTANTANEOUS POINT RELEASE:
AN IMPORTANT ELEMENTARY SOLUTION OF (5) IS FOR THE CASE
OF AN INSTANTANEOUS RELEASE AT A POINT (x', Y', Z') IN
AN INFINITE MEDIUM, WITH THE COORDINATE SYSTEM MOVING
WITH THE MEDIUM AT SPEED U,
WHERE,
Qj = THE INSTANTANEOUS SOURCE STRENGTH (M)
T = THE TIME AFTER THE RELEASE
THIS SOLUTION FIRST OBTAINED BY ROBERTS (1923) IS A
FUNDAMENTAL BUILDING BLOCK OF THE FlCKIAN DIFFUSION THEORY,
ITS INTEGRATION ALONG ONE, TV/0 AND TKREE DIMENSIONS IN SPACE
YIELDS SOLUTIONS FOR THE LINE, AREA AND VOLUME SOURCES
RESPECTIVELY, INTEGRATION WITH RESPECT TO TIME GIVES THE
CONTINUOUS POINT SOURCE SOLUTION, WHICH MAY IN TURN BE
INTEGRATED WITH RESPECT TO THE SPATIAL DIMENSIONS TO GIVE
SOLUTIONS FOR THE CONTINUOUS LINE AND AREA SOURCES. IN
DOING THIS, ONE IS ESSENTIALLY USING THE PRINCIPLE OF
SUPERPOSITION, ACCORDING TO WHICH CONCENTRATION FIELDS OF
MULTIPLE SOURCES ARE SIMPLY ADDITIVE SINCE DIFFUSION IS
BASICALLY A LINEAR PROCESS,
-------
5-9
(n) SOLUTION FOR A CONTINUOUS POINT SOURCE:
ROBERTS (1923) WAS ALSO THE FIRST TO GIVE THE SOLUTION
OF THE FlCKIAN DIFFUSION EQUATION FOR A CONTINUOUS POINT
SOURCE IN AN INFINITE (UNBOUNDED) MEDIUM, IN A FIXED
COORDINATE SYSTEM WITH THE SOURCE LOCATED AT THE ORIGIN,
*.,. -..- Q „
WHICH, WITH THE SLENDER-PLUME APPROXIMATION VALID
SUFFICIENTLY FAR AWAY ROM THE SOURCE, GIVES THE SIMPLER
EXPRESSION
THIS IS ALSO THE EXACT SOLUTION OF THE FlCKIAN DIFFUSION
EQUATION WITH THE DIFFUSION IN THE X-DIRECTION NEGLECTED,
(D) GAUSSIAN DTSPERSION PARAMETERS
Al_L THE SOLUTIONS OF THE FlCKIAN DIFFUSION EQUATION ARE FOUND TO
GIVE CONCENTRATION DISTRIBUTIONS, WHICH ARE GAUSSIAN, THESE
CAN BE EXPRESSED IN THE STANDARD FORM FOR A GAUSSIAN OR NORMAL
DISTRIBUTION, FOR EXAMPLE,
-------
5-10
FOR THE INSTANTANEOUS POINT SOURCE INITIALLY LOCATED AT THE ORIGIN,
WHERE,
FOR THE CONTINUOUS POINT SOURCE AT THE ORIGIN,
/y
/J "
, Oy A^Da ARE THE STANDARD DEVIATIONS OF THE GAUSSIAN
DISTRIBUTION FUNCTION IN X, Y AND Z DIRECTIONS, THEY SERVE AS
CONVENIENT LENGTH SCALES OF THE WIDTHS OF THE DISTRIBUTION AND
ALSO MEASURES OF THE WIDTHS OF PUFF OR PLUME (ACTUALLY, THE
SO-CALLED 10%-WIDTH OF A PUFF OR PLUME IS ^,3 TIMES THE STANDARD
DEVIATION). FOR THIS REASON, THESE ARE ALSO CALLED THE DISPERSION
PARAMETERS,
EdUATIONS (10) AND (11) ARE THE FlCKIAN DIFFUSION THEORY RELATIONS
FOR THE GAUSSIAN DISPERSION PARAMETERS AS FUNCTIONS OF TRAVEL TIME
OR DISTANCE AND EDDY DIFFUSIV1TIES, NOTE THAT THE THEORY PREDICTS
THE AVERAGE WIDTH OF THE PUFF OR PLUME FROM A POINT SOURCE GROWING
IN PROPORTION TO THE SQUARE-ROOT OF TIME AFTER RELEASE OR DISTANCE
FROM THE SOURCE.
r ^ — 1 OU
- - - 2
^ ^ 0~ a "2-^ J
-------
5-11
(E) ftBSQBEiiffliAND REFLECTION AT THE BOUNDARIFS
FOR DIFFUSION PURPOSES, THE LOWER ATMOSPHERE, SPECIALLY THE
ATMOSPHERIC BOUNDARY LAYER (ABL), CANNOT BE CONSIDERED.INFINITE
OR UNBOUNDED. ThlE CONCENTRATION FIELD IS CONSIDERABLY ALTERED
BY THE PRESENCE OF THE BOUNDARIES, SUCH AS THE GROUND SURFACE
BELOW AND A CAPPING INVERSION ABOVE, BOTH OF WHICH ACT AS
BARRIERS TO THE DIFFUSING MATERIAL. THE CONTAMINANT MAY (l)
STICK TO THE BOUNDARY AND BE ABSORBED, (II) BOUNCE BACK OR BE
REFLECTED FROM IT, OR (III) PARTIALLY ABSORBED AND PARTLY REFLECTED,
FROM A PURELY MATHEMATICAL POINT OF VIEW, THE SIMPLEST BOUNDARY
CONDITION IS THAT OF A PERFECTLY REFLECTING SURFACE, FOR WHICH
THE METHOD OF IMAGES CAN BE USED, THE PROCEDURE IS TO INTRODUCE
AN IMAGE OF THE SAME STRENGTH AS THE REAL SOURCE AND LOCATED AT
THE IMAGE POINT OF THE REAL SOURCE IF THE BOUNDARY WERE A REFLECTING
MIRROR, THEN THE PRESENCE OF THE BOUNDARY is IGNORED AND THE
MEDIUM IS CONSIDERED INFINITE, FOR AN ELEVATED SOURCE IN THE
ABL BOTH THE GROUND SURFACE AND THE INVERSION BASE MAY BE
CONSIDERED AS REFLECTING BOUNDARIES, To REPLACE THEM, MULTIPLE
IMAGE SOURCES RESULTING FROM MULTIPLE REFLECTIONS FROM THE TWO
BOUNDARIES MAY HAVE TO BE CONSIDERED,
(F) THF GAUSSIAN PLUME DIFFUSION MODEL FOR AN ELEVATED SOURCE
USING THE METHOD OF IMAGES AND THE PRINCIPLE OF SUPERPOSITION,
EQ, (11) CAN BE USED TO GIVE THE CONCENTRATION FIELD DUE TO AN
ELEVATED POINT SOURCE, FOR SHORT DISTANCES FROM THE SOURCE TO
WHICH DIFFUSION MAY NOT BE RESTRICTED BY THE CAPPING INVERSION
OR THE TOP OF THE ABL,
-------
5-12
WHERE,
H = THE EFFECTIVE SOURCE HEIGHT
Z = THE HEIGHT ABOVE THE GROUND SURFACE
Y = THE LATERAL DISTANCE FROM THE PLUME CENTERLINE
THE G.L.C, AT THE PLUME CENTERLINE IS GIVEN BY
06 -
0 TT
-------
5-13
(5) AN INFINITE SPEED OF MOVEMENT OF THE POLLUTANT IS IMPLIED,
3. STATISTICAL THEORIES OF DIFFUSION
(A) LAGRANGIAN FRAME OF REFERENCE
Z
A VI'
(X,Y,Z)
U'
WITH THE ORIGIN MOVING ALONG WITH THE MEAN FLOU,
(X.Y.Z) = THE COORDINATES OF A TAGGED PARTICLE AT TIME T
GJ',V',U') = THE LAGRANIGAfvl VELOCITY COMPONENTS OF THE TAGGED
PARTICLE
NOTE THAT
U -
-------
5-14
(B) TAYIOR'S THEORY
SIMPLIFYING ASSUMPTIONS MADE BY TAYLOR (1921) WERE:
(1) FIELD OF TURBULENCE is HOMOGENEOUS IN ALL THREE DIRECTIONS
AND ALSO STATIONARY.
(2) ENSEMBLE AVERAGES TAKEN OVER AN INFINITE NUMBER OF REALIZATIONS
(TAGGED PARTICLES)
(3) DIFFUSING PARTICLES ARE PASSIVE AND NONBUOYANT,
IT CAN BE SHOWN FROM KINEMATIC CONSIDERATIONS ALONE, THAT
VELOCITY VARIANCE.
WiERE,
5 FIT
_ Uy , tit.
MEAN-SQUARE PARTICLE DISPLACEMENTS
0
is The LAGRANGIAN AUTOCORRELATION FUNCTION.
»d
LAGRANGIAN INTEGRAL TIME SCALE, -T - \P/£ W
-------
FOR SUALL. ISABEL
.IlftS \^*-' L )
V ~
I *""""
,
t
-------
5-16
10-
0.1
•INITIAL STAGE
TRANSITION-&^-«
STAGE I
FINAL STAGE
0.1
1.0
t
10
LIMITS OF THE TRANSITIONAL OR INTERMEDIATE STAGE OF DISPERSION AND THE
ACTUAL SHAPE OF THE ABOVE GRAPH IN THIS REGION DEPENDS ON THE SHAPE OF
f^ , ALTHOUGH'THESE ARE NOT OVERLY SENSITIVE TO IT, fbRE IMPORTANT
TO KNOW IS THE TIME SCALE , WHICH IS RELATED TO THE SIZE OF THE
LARGE EDDIES. A GOOD APPROXIMATION IS:
-------
5-17
RELATIONSHIP TO EULERIAN STATISTICS
DIFFERENT THEORETICAL APPROACHES, AS WELL AS A LIMITED NUMBER
OF DIRECT OBSERVATIONS INDICATE THAT
L
WHERE ~JC - THE EllLERIAN INTEGRAL TIME SCALED
«* o,4
" /
A,
= a,4
INTENSITY OF TURBULENCE ^-^7
U
HAY AND PASQUILL (1959) ALSO SUGGESTED A SIMPLE RELATIONSHIP
FOR
Q f ^ \ r^ / "*v \ UUCN *r~ &C
WHICH IMPLIES SIMILARITY OF THE SHAPES OF THE LAGRANGIAN AND
EULERIAN AUTOCORRELATION FUNCTIONS,
15
-------
5-18
(D)
BASED ON A SIMPLE HYPOTHESIS,
VHERE
f> = \ H
;
HAY AND PASQUILL (1959) DERIVED THE FOLLOWING RESULTS, IRRESPECTIVE
OF THE DISTANCE FROM THE SOURCE:
x
WHEREO^ IS THE S,D, OF THE WIND DIREaiON MEASURED IN RADIANS,
&
THE FIRST SUBSCRIPT REFERS TO THE TOTAL SAMPLING PERIOD AND THE
SECOND SUBSCRIPT REFERS TO THE RUNNING AVERAGE INTERVAL.
OTHER EXTENSIONS OF THE STATISTICAL THEORY HAVE BEEN PROPOSED,
SPECIALLY IN CONJUNCTION WITH THE LAGRANIGIAN SIMILARITY THEORY.
(E) RELATIONSHIP BETWEEN THE PARTICLF_DTSPFRSTON PARAMETERS AND
CONCENTRATION DISTRIBUTION IN A PLUME
UN THE EASIS OF STATISTICAL RANDOM-WALK MODELS THE 3-D PROBABILITY
FUNCTION OF PARTICLE DISPLACEMENTS IN A STATIONARY AND HOMOGENEOUS
FIELD OF TURBULENCE IS EXPECTED TO BE GAUSSIAN, THIS DISTRIBUTION
IS COMPLETELY SPECIFIED BY ITS SECOND MOMENTS W, Y* AND T~.
SINCE CC-FIELD IS DIRECTLY RELATED TO THE PARTICLE DISTRIBUTION
IN SPACE, IT FOLLOWS THAT ^-DISTRIBUTION MUST BE GAUSSIAN TOO,
-------
5-19
FURTHERMORE,
x1 =
WHERE E ) y" FOR
V \-=~ )
THESE MAY BE COMPARED WITH THE K-THEORY RELATION
> VT
FOR ALL X
WHICH IS SIMILAR TO THE STATISTICAL THEORY RESULT ONLY FOR
LARGE X, THUS THE STATISTICAL THEORY PROVIDES A BASIS FOR
SPECIFYING FOR LARGE X: — B
"" /
£ v
17
-------
5-20
(G) i IMITATIONS OF TAYLOR'S STATISTICAL THEORY
(1) ASSUMES THE TURBULENCE FIELD TO BE HOMOGENEOUS, WHICH
CERTAINLY IS NOT TRUE IN THE VERTICAL DIRECTION, SPECIALLY
IN THE LOWER LAYERS OF THE ATMOSPHERE,
(2) ASSUMES UNIFORM MEAN FLOW WITH NO VERTICAL AND DIRECTIONAL
SHEARS,
•5) THE RESULTS STRICTLY APPLY WHEN EXPRESSED IN TERMS OF THE
LAGRANGIAN STATISTICS, WHICH is DIFFICULT TO GET, CONVERSION
TO EtlLJERIAN STATISTICS INVOLVE SOME APPROXIMATIONS AND
HYPOTHESES,
([0 TURBULENCE MEASUREMENTS USING SOPHISTICATED INSTRUMENTATION
HAVE TO BE MADE IN ORDER TO SPECIFY (Ty AS A FUNCTION OF
00 AND TE ,
U':,KANI--IAN SIMILARITY THEORIES
(A) fe,..SjlllLARIIYJJrEORY OF DIFFUSION IN THE SuRFACF lAYFR
THIS IS AN EXTENSION OF THE MoNIN-QBUKHOV SIMILARITY THEORY
TO THE DISPERSION OF MATERIAL IN THE SURFACE LAYER, THE BASIC
4S3IMPTIONS ARE:
(i) LAGRANGIAN PARTICLE VELOCITIES ARE DETERMINED UNIQUELY BY
Z, U, H/PC AND
T0
0PC
J
WHERE,
U# = THE FRACTION VELOCITY
HQ = THE SURFACE HEAT FLUX
TQ = THE REFERENCE TEMPERATURE
(n) THE SOURCE is LOCATED AT OR NEAR THE SURFACE,
(nO TRAVEL TIMES OR DISTANCES FROM THE SOURCE UNDER CONSIDERATION
ARE SMALL ENOUGH SO THAT THE PLUME STAYS WITHIN THE SURFACE
LAYER,
-------
5-21
(B) A-THEORETicAi .FXPRFSSTON FOR T^ PLUME CENTERi INF
FOLLOWING RATCHELOR (1959), THE BASIC LAGRANGIAN SIMILARITY
RELATIONS ARE:
w
if
d't
L
u -
= U (c2
WHERE b £^ 0, 4 AND ^ —
FURTHER ASSUMPTIONS ARE REQUIRED FOR THE SIMILARITY FUNCTION
(j) (Z/L) IN ORDER TO DESCRIBE THE PLUME CENTERLINE, FOR THE
NEUTRAL CASE, HOWEVER, (j) = 1, AND ONE OBTAINS THE RESULT
2*
2-f 4?(Wnc)
Z '
SfV\LL TE°J1 FOR
PLUTt CENTERLINE
19
-------
5-22
(c) THFORFTICAL EXPRESSIONS FOR THE CONCENTRATION FIELD
FROM SIMILARITY RELATIONS FOR THE PROBABILITY OF A PARTICLE
REACHING THE POINT (X,Y,Z) AFTER A TRAVEL TIME T, ONE CAN DERIVE
"-A-X v *-z
i
i
WHERE F IS A UNIVERSAL SIMILARITY FUNCTION, WHICH REMAINS TO
BE FULLY DETERMINED FROM OBSERVATIONS,
FOR THE G.L.C, AT THE PLUME CENTERLINE (Y=Q, 2=0)
60
~
0
/; c §
«. 3
2
i __ i
/./
cli
EVEN THE SIMPLER FUNCTION FQ HAS NOT BEEN COMPLETELY DETERMINED
FROM OBSERVATIONS,
-------
5-23
NEUTRAL CASE (- = 0) J """ y ^
USING THE SIMILARITY RELATION £- ^"/
1
= bU
SO THAT
AND FURTHER TRANSFORMATION OF VARIABLES,
AFTER SOME APPROXI^TIONS AND SUBSTITUTIONS (BATCHELOR, 1959,
a Q
b
A
WHERE
IS GIVEN BY
II ' "
feb
)-l
tr X
21
-------
5-24
(D) PENSION TO STRATIFIED CONDITIONS
MOM IN (1959) EVALUATED j) ([) FROM THE TURBULENT KINETIC ENERGY
(TKE) EQUATION AND DETERMINED THE PLUME CENTERLINE FOR .A
CONTINUOUS LINE SOURCE AS A FUNCTION OF STABILITY,
GlFFORD (1962) ASSUMED THAT THE FORM OF ^--DISTRIBUTIONS IS
INDEPENDENT OF Z/L AND THAT THEY ARE SCALED BY Z ONLY, ON THIS
I
BASIS, HE WAS ABLE TO EXPRESS G.L.C. AS A FUNCTION OF DISTANCE
AND STABILITY, THE EXPONENT IN
P IS FOUND TO BE GREATER THAN 2 FOR UNSTABLE CONDITIONS AND
SUBSTANTIALLY LESS THAN 2 IN STABLE CONDITIONS,
CFRMAK (.1963) AND OTHERS HAVE PROVIDED LABORATORY DATA TO
TEST THESE RESULTS,
FURTHER EXTENSIONS AND APPLICATIONS OF THE LAGRANGIAN SURFACE-
LAYER SIMILARITY THEORY HAVE BEEN REPORTED BY THE FOLLOWING
- INVESTIGATORS:
CHATWIN (1968)
HORST (1979)
HUNT AND WEBER (1979)
(E) LIMITATIONS OF THE SURFACE-LAYER SIMILARITY THEORY
1. IT DOES NOT DEAL WITH DIFFUSION IN Y-DIRECTION,
2, ONLY NEAR-GROUND-LEVEL SOURCES CAN BE CONSIDERED,
3. COMPARATIVELY SHORT DISTANCES FROM THE SOURCE ARE CONSIDERED
SO THAT THE PLUME STAYS WITHIN THE SURFACE LAYER,
4, SIMILARITY FUNCTIONS HAVE NOT BEEN FULLY DETERMINED
FROM EXPERIMENTS,
-------
5-25
(F) IHE SIMILARITY THFORY OF DIFFUSION IN THE MIXED LAYER
DEARDORFF (1972), DEARDORFF AND WILLIS (1974), ETC, HYPOTHESISED
THAT IN AN UNSTABLE OR CONVECTIVE MIXED LAYER (Z £. - U-»
TURBULENCE AND DISPERSION ARE SCALED BY
LENGTH SCALE: H(MIXED-LAYER DEPTH)
VELOCITY SCALE: W (CONVECTIVE VELOCITY)
TIME SCALE:
K
DIMENSIONAL CONSIDERATIONS SUGGEST THAT
cr
h
Jl\ _
h
23
-------
5-26
(G) DETERMINATION OF SIMILARITY FUNCTIONS
SIMILARITY FUNCTIONS Fy AND FZ HAVE BEEN DETERMINED ROM
LABORATORY, (WlLLIS AND DEARDORFF, 1976, 1978) AND NUMERICAL
MODELING EXPERIMENTS (LAMB, 1978, 1979). RESULTS ARE:
1, THE RATES OF LATERAL AND VERTICAL SPREAD FROM AN ELEVATED
SOURCE (H > 0,1 H) ARE CONSIDERABLY DIFFERENT FROM THOSE
OF A SURFACE SOURCE (H < 0,1 H).
2, THE LXUS OF MAXIMUM % IN AN ELEVATED SOURCE PLUME FOLLOWS
A DESCENDING PATH THAT INTERCEPTS THE GROUND AT A DISTANCE
x = 2HG7VI*
3. THE LOCUS OF MAXIMUM X FOR A SURFACE SOURCE ASCENDS
BEGINNING AT X = HU/W*
4, THE MAGNITUDE OF THE MAXIMUM G.L.C, FROM ELEVATED SOURCES IS
*y i o Q . v HD
^n MAY = 1.2 — AND IT OCCURS AT A DISTANCE X.,.Y = 2 '
HHU W*
(H) LIMITATIONS OF THE MIXED LAYER SIMILARITY THEORY
1, APPLIES ONLY WHEN THE DISPERSION IS DOMINATED BY BUOYANCY
GENERATED TURBULENCE, I.E., -H/L > 10, -H/L > 1,
2. THE MEAN VELOCITY IN THE MIXED LAYER LIES IN THE RANGE
1,2 Ww < u < 6 W^, IF U is TOO SMALL TAYLOR'S HYPOTHESIS,
ON WHICH THE TRANSFORMATION T = X/U IS BASED, MAY NOT BE
VALID, IF U IS TOO LARGE, THE WIND SHEAR IS LIKELY TO
BECOME IMPORTANT,
3, ONLY NONBUOYANT SOURCES HAVE BEEN CONSIDERED,
5, CONTEMPORARY NUMERICAL MDDELS OF DISPERSION
WE SHALL ONLY BRIEFLY MENTION HERE SOME OF THE BETTER KNOWN
CONTEMPORARY MODELS OF DISPERSION IN THE ABL, WHICH CAN BE USED
FOR ESTIMATING THE VARIOUS PARAMETERS APPEARING IN SIMPLER MODELS,
IF NOT FOR THE MORE ROUTINE APPLICATIONS AT THIS STAGE,
-------
5-27
Ci nsuRE MODELS
FOLLOWING THE SUCCESSFUL DEVELOPMENT OF SECOND-ORDER CLOSURE
MODELING OF MEAN FLOW AND TURBULENCE IN THE ABL THE SAME
TECHNIQUE IS NOW BEING USED FOR MODELING DISPERSION (LfWELLEN
AND TESKE, 1975 A,B), SUCH MODELS ARE BASED ON THE REYNOLDS-
AVERAGED EQUATIONS OF SECOND MOMENTS OR REYNOLDS' FLUXES
(VARIANCES AND COVARIANCES OF VELXITY AND CONCENTRATION
FLUCTUATIONS), IN SPITE OF THE LARGE NUMBER OF CLOSURE
ASSUMPTIONS THAT HAVE TO BE MADE, SUCH MODELS APPEAR TO BE
MORE PROMISING FOR HANDLING THE VARIOUS COMPLEX TERRAIN
SITUATIONS COMMONLY ENCOUNTERED IN NATURE, AS COMPARED TO THE
SIMPLER CLASSICAL THEORIES OF DISPERSION, THESE ARE ALSO
CAPABLE OF HANDLING THE LARGE RANGE OF STABILITY CONDITIONS.
(B) THREE-DIMENSIONAL LARGE-EDDY SIMULATION MODELS
THESE ARE THE BRUTE FORCE ATTEMPTS AT NUMERICALLY SOLVING THE
NAVIER-STOKES EQUATIONS AND THE DIFFUSION EQUATION USING THE
LARGEST AVAILABLE COMPUTER CAPACITY AND STORAGE (HOTCHKISS,
1972; DEARDORFF, 1972, 1974), ONLY THE EDDIES LARGER THAN THE
GRID SIZE AND SMALLER THAN THE COMPUTATION BOX CAN BE SIMULATED
REALISTICALLY; THE SUB-GRID SCALE MOTIONS HAVE TO BE PARAMETERIZED
USING AN EDDY DIFFUSIVITY OR SECOND-ORDER CLOSURE APPROACH, WHEN
MOST OF THE TURBULENT KINETIC ENERGY IS LIKELY TO BE CONTAINED IN
SUBGRID SCALE MOTIONS, AS HAPPENS IN STABLY STRATIFIED CONDITIONS,
SUCH MODELS BECOME VERY UNRELIABLE, GOOD SIMULATIONS HAVE BEEN
MADE, HOWEVER, OF THE NEUTRAL, UNSTABLE AND CONVECTIVE BOUNDARY
LAYERS, RECENTLY, LAMB (1978, 1979) HAS COMBINED THE RESULTS OF
DFJ^RDORFF'S (1974) NUMERICAL MODEL WITH THE LAGRANGIAN MIXED-LAYER
SIMILARITY THEORY FOR A BETTER SIMULATION OF DISPERSION FROM AN
ELEVATED POINT SOURCE IN THE CONVECTIVE ABL,
25
-------
5-28
(c) APPLICABILITY OF VARIOUS THEORIES TO THE DETERMINATION OF THE
GAUSSIAN PLUME DISPERSION PARAMETERS IN THF HOMOGENEOUS ABL
CLASS OF THEORY
GRADIENT TRANSPORT
STATISTICAL
SURFACE-LAYER SIMILARITY
MIXED-LAYER SIMILARITY
COMBINATION OF THE ABOVE
SECOND-ORDER CLOSURE
DlSPERS
°v
No
YES
No
No
YES
YES
>ION PARAMETER PREDICTED
rz
YES, FOR LONG RANGE DIFFUSION
IN NEUTRAL AND UNSTABLE CON-
DITIONS ONLY
YES, FOR SHORT RANGE DIFFUSION
FROM ELEVATED SOURCES ONLY
YES, FOR SHORT RANGE DIFFUSION
FROM GROUND SOURCES ONLY
YES, FOR SHORT RANGE DIFFUSION
FROM ELEVATED SOURCES ONLY
YES
YES
II, EXPERIMENTAL EVALUATIONS OF STABILITY AND DISPERSION PARWETERS
1, STABILITY PARAMETERS AND TYPING SCHEMES
(A) STATIC STABILITY AND TEMPERATURE GRADIFNT
STATIC STABILITY OF THE LOWER ATMOSPHERE is CHARACTERIZED BY
THE TEMPERATURE GRADIENT OR LAPSE RATE, THE ATMOSPHERE IS CALLED
STABLE, WHEN
NEUTRAL, WHEN
UNSTABLE, WHEN
>r
"
OT
OY
> O
a
^ n ,
'2>2r
-------
5-29
THE MAGNITUDE OF aT/az OR 30/2Z is A QUANTITATIVE MEASURE OF
STATIC STABILITY. SlNCE, IT USUALLY VARIES WITH HEIGHT IN THE
BOUNDARY LAYER., IT SHOULD BE SPECIFIED AT SOME STANDARD.-REFERENCE
LEVEL, PREFERABLY IN THE SURFACE LAYER,
(B) DYNAMIC STABILITY. RICHARDSON NUMBERS, FTC.
STATIC STABILITY DOES NOT INCLUDE THE EFFECT OF WIND SHEAR OR
MECHANICAL TURBULENCE AND, AS SUCH, IS NOT A GOOD MEASURE OF THE
STRENGTH OF TURBULENCE AND MIXING, TURBULENCE AND DIFFUSION IN
AN ATMOSPHERIC LAYER DEPEND ON ITS DYNAMIC STABILITY, WHICH
DEPENDS ON BOTH THE WIND SHEAR AND TEMPERATURE GRADIENT, IT IS
USUALLY CHARACTERIZED BY THE GRADIENT RlCHARDSCN NUMBER
—j- o ±
O
IN WHICH 3V/9Z IS THE MAGNITUDE OF WIND SHEAR,
AN ALTERNATIVE AND, PERHAPS, MORE CONVENIENT STABILITY PARAMETER
IS THE BULK RlCHARDSON NUMBER
V
WHEREAS IS THE POTENTIAL TEMPERATURE DIFFERENCE BETWEEN THE
TWO SPECIFIED LEVELS, AND V IS THE AVERAGE WIND SPEED AT SOME
HEIGHT IN BETWEEN THE ABOVE TWO LEVELS,
RICHARDSON NUMBERS MAY NOT BE APPROPRIATE MEASURES OF STABILITY
IN THE BULK OF THE CONVECTIVELY MIXED LAYER ABOVE AN UNSTABLE
SURFACE LAYERj THEY MAY HAVE EVEN WRONG SIGN THERE, V/HEN
DIFFUSION FROM AN ELEVATED SOURCE IN THE MIXED LAYER IS OF
27
-------
5-30
INTEREST, THE MOST APPROPRIATE STABILITY PARAMETER MIGHT BE THE
RATIO OF THE MIXED LAYER (BOUNDARY LAYER) DEPTH H TO THE OBUKHOV
LENGTH L DEFINED AS
3
Uv
|_ _
k ^
H/L IS PROBABLY THE BEST MEASURE OF STABILITY AS IT AFFECTS
TURBULENCE AND DIFFUSION IN THE WHOLE BOUNDARY LAYER UNDER
VARIOUS STABILITY CONDITIONS,
FOR DIRECT EVALUATION OF L ONE NEEDS TO MEASURE THE MOMENTUM
AND hEAT FLUXES IN THE SURFACE LAYER, HOWEVER, IT CAN BE
ESTIMATED INDIRECTLY FROM SIMPLER MEASUREMENTS OF THE BULK
RICHARDSON NUMBER IN THE SURFACE LAYER, USING THE ESTABLISHED
FLUX-PROFILE RELATIONS
(c) STABILITY AND TURBULENCE TYPING SCHEMES
EMPIRICALLY DERIVED DISPERSION PARAMETERS FROM DIFFUSION
EXPERIMENTS HAVE BEEN REPRESENTED IN TERMS OF VARIOUS
STABILITY CLASSES OR TURBULENCE TYPES, WE SHALL MENTION HERE
ONLY SOME OF THE BETTER KNOWN CLASSIFICATION OR TYPING SCHEMES,
(i) PASQUILL'S STABILITY CATEGORIES:
PASQUILL (1951) INTRODUCED THE FOLLOWING STABILITY CATEGORIES,
DETERMINATION OF WHICH REQUIRED ONLY QUALITATIVE OBSERVATIONS
OF SURFACE WIND SPEED, INSOLATION AND CLOUDINESS
A - EXTREMELY UNSTABLE D - NEAR NEUTRAL
B - MODERATELY UNSTABLE E - SLIGHTLY STABLE
C - SLIGHTLY UNSTABLE F - MODERATELY STABLE
-------
5-31
SURFACE
WIND SPEED
(M/S)
<=2
2-3
3-5
5-6
-6
DAYTIME INSOLATION
STRONG
A
A-B
B
C
C
MODERATE
A-B
B
B-C
C-D
D
SLIGHT
B
C
C
D
D
NIGHTTIME CONDITIONS
CLOUDINESS
^
__
E
D
D
D
CLOUDINESS
— o
_
F
E
D
D
(n) TURNER'S MODIFICATION OF THE PASQUILL SCHEME:
TURNER (1961., 1964) INTRODUCED A VERSION OF PASQUILL
STABILITY SCHEME IN WHICH THE SOLAR INSOLATION IS CLASSIFIED
IN TERMS OF SOLAR ELEVATION ANGLE, CLOUD AMOUNT AND HEIGHT,
THE PROCEDURE IS OBJECTIVE AND INVOLVES METEOROLOGICAL
QUANTITIES THAT ARE KNOWN FOR MOST LOCATIONS, TURNER
LABELLED HIS STABILITY CLASSES NUMERICALLY FROM 1
(EXTREMELY UNSTABLE) TO 7 (EXTREMELY STABLE),
(in) THE BROOKHAVEN NATIONAL LABORATORY (BNL) TURBULENCE TYPES:
THE BNL TURBULENCE TYPES (SINGER AND SMITH, 1966) ARE BASED
ON THE RANGE OF FLUCTUATIONS OF THE HORIZONTAL WIND DIRECTION
TRACE AS RECORDED BY A BENDIX-pRIEZ AEROVANE LOCATED AT THE
108 M LEVEL OF THE BNL TOWER, THE FIVE CATEGORIES ARE
DEFINED AS:
A, PEAK TO PEAK FLUCTUATIONS OF WIND DIRECTION ^90°,
82- FLUCTUATIONS RANGING FROM 40P TO 90°,
B}, FLUCTUATIONS RANGING FROM 15° TO 45°,
C, FLUCTUATIONS GREATER THAN 15° DISTINGUISHED BY
UNBROKEN SOLID CORE OF THE TRACE,
D, THE TRACE APPROXIMATES A LINE; SHORT-TERM FLUCTUATIONS
DO NOT EXCEED 15°,
29
-------
5-32
NOTE THAT THE BNL STABILITY CATEGORIES OR GUSTINESS CLASSES ARE
SITE SPECIFIC AND WIND TRACE CHARACTERISTICS REFER TO A HEIGHT
OF 108 M, WHICH USUALLY LIES ABOVE THE SURFACE LAYER,
(iv) CRAMER'S CLASSIFICATION BASED ON STANDARD DEVIATIONS OF WIND
DIRECTION,
CRAMER (1957) SUGGESTED THE FOLLOWING STABILITY CLASSIFICATION
SYSTEM BASED ON MEASUREMENTS OF THE STANDARD DEVIATIONS OF THE
HORIZONTAL AND THE VERTICAL WIND DIRECTIONS G^, AND (Ti :
CRAMER'S TURBULENCE CLASSES
STABILITY DESCRIPTION
EXTREMELY UNSTABLE
NEAR-NEUTRAL (ROUGH
SURFACE; TREES., BUILDINGS)
NEAR-NEUTRAL (SMOOTH GRASS)
EXTREMELY STABLE
(Jl (DEG.)
o
30
15
6
3
0$ (DEG,)
10
5
2
1
THIS SYSTEM WAS BASED ON THE ROUND HILL AND THE PROJECT
PRAIRIE GRASS DATA AND is EXPECTED TO BE SITE SPECIFIC TOO,
(v) THE TVA STABILITY CLASSIFICATION SCHEME:
THIS SCHEME USED BY CARPENTER EL AL (1971) IS BASED ON THE
MEAN POTEhTTIAL TEMPERATURE GRADIENT AT THE PLUME HEIGHT,
WHICH VARIED FROM ABOUT 150 M TO 500 M, THEIR SCHEME DEFINES
SIX STABILITY CATEGORIES AS FOLLOWS: _
STABILITY CATEGORY :=—-
NEUTRAL
SLIGHTLY STABLE
STABLE
ISOTHERMAL
MODERATE INVERSION
STRONG INVERSION
0,00 K/1QO M
l!64
//
;/
//
/;
-------
5-33
SINCE THE TEMPERATURE GRADIENT REFERS TO THE PLUME HEIGHT,
WHERE NEAR ZERO OR SLIGHTLY POSITIVE tfO/dZ MAY BE OBSERVED
EVEN DURING UNSTABLE AND CONVECTIVE CONDITIONS, SUCH
CONDITIONS ARE INCLUDED IN MASKED FORM AMONGST THE NEUTRAL
AND SLIGHTLY STABLE CLASSES,
(D) RELATION AMONG STABILITY TYPING SCHEMES AND PARAMFTFRS
BECAUSE OF THE DIFFERED CRITERIA AND DATA BASES INVOLVED IN THE
EMPIRICAL FORMULATIONS OF THE VARIOUS STABILITY AND TURBULENCE
TYPING SCHEMES DISCUSSED IN THE PREVIOUS SECTION, PRECISE
QUANTITATIVE RELATIONS AMONG THEM ARE DIFFICULT TO OBTAIN,
BASED ON EXPERIENCE, APPROXIMATE CORRESPONDENCE OF THE PASQUILL
STABILITY CLASSES WITH OTHER CLASSIFICATION SCHEMES IS SHOWN IN
THE FOLLOWING TABLE (GlFFORD, 1976):
STABILITY
DESCRIPTION
VERY UNSTABLE
MODERATELY UNSTABLE
SLIGHTLY UNSTABLE
NEAR NEUTRAL
MODERATELY STABLE
VERY STABLE
PASQUILL
TYPE
A
B
C
D
E
F
TURNER
TYPE
1
2
3
i|
6
7
BNL
TYPE
%
B-j
Bl
C
D
(DEG)
25
20
15
10
5
2,5
EVEN MORE DIFFICULT IS TO RELATE QUALITATIVE STABILITY CATEGORIES
LIKE THOSE OF PASQUILL TO SOME OF THE QUANTITATIVE STABILITY
PARAMETERS, LAPSE RATE OR TEMPERATURE GRADIENT HAS PROVED TO BE
AN UNCERTAIN DISCRIMINATOR, BUT AN APPROXIMATE CORRESPONDENCE
HAS BEEN ESTABLISHED BETWEEN THE PASQUILL STABILITY TYPES AND
THE OBUKHOV LENGTH OR RICHARDSON NUMBER CORRESPONDING TO SOME
HEIGHT IN THE SURFACE LAYER (GOLDER, 1972j PASQUILL AND SMITH,
1971). 31
-------
5-34
RELATIONS BETWEEN PASQUILL TYPE AND STABILITY PARAMETERS RA
AND L OVER A SHORT GRASS SURFACE, ZQ = 1 CM, ACCORDING TO PASQUILL
AND SMITH (1971),
PASQUILL
TYPE
A
B
C
D
E
F
Rj AT 2 M
-1.0 TO -0,70
-0,50 TO -0,40
-0.17 TO -0,13
0
0.03 TO 0,05
0.05 TO 0.11
L M
-2 TO -3
-4 TO -5
-12 TO -15
C70
35 TO 75
8 TO 35
f
FOR GOLDERS CURVES AND OTHER DISCUSSION, SEE GlFFORD (1971).
IN A COMPARATIVE STUDY OF THE VARIOUS QUANTITATIVE STABILITY
PARAMETERS FOR DESCRIBING DISPERSION IN THE ATMOSPHERIC
BOUNDARY LAYER, V/EBER EL AL (1977) HAVE CONCLUDED THAT THE USE
OF RA OR I/I AT SOME HEIGHT IN THE SURFACE LAYER GIVES THE
BEST RESULTS. THEY DID NOT CONSIDER, HOWEVER, THE DISPERSION
FROM ELEVATED SOURCES IN THE MIXED LAYER, FOR WHICH H/L IS THOUGHT
TO BE A MORE APPROPRIATE PARAMETER. IF DIFFUSION ESTIMATES FROM
GAUSSIAN MODELS ARE TO BE IMPROVED, THE SIMPLE QUALITATIVE
STABILITY TYPING SCHEMES MUST BE REPLACED BY THE ABOVE MENTIONED
QUANTITATIVE PARAMETERS IN SPECIFYING THE DISPERSION PARAMETERS.
EXPERIMENTAL DETERMINATION OF RA OR L REQUIRES ONLY THE SIMPLE
MICROMETEOROLOGICAL MEASUREMENTS OF THE TEMPERATURE DIFFERENCE
BETWEEN THE TWO LEVELS AND THE WIND SPEED AT AN INTERMEDIATE
LEVEL IN THE SURFACE LAYER, ASSUMING THAT THE ROUGHNESS PARAMETER
CAN BE ROUGHLY ESTIMATED FROM THE CHARACTER OF THE SURFACE.
-------
5-35
2, DIFFUSION MEASUREMENT TECHNIQUES
(A) AEROSOL AND GASEOUS TRACER TECHNIQUES
(B) EADQACTIVE TRACER TECHNIQUES
(c) BALLOON TRACKING TECHNIQUES
(D) REMOTE SENSING TECHNIQUES
FOR DESCRIPTIONS OF THESE READ:
"METEOROLOGY AND ATOMIC ENERGY - 1968", D, SLADE (ED,), PP. 293-300,
"ATMOSPHERIC TECHNOLOGY", NCAR, NUMBER 7, 1975,
"LECTURES ON AIR POLLUTION AND ENVIRONMENTAL IMPACT ANALYSES",
D, HAUGEN (ED,), CHAPTER 8,
3. PLUME DIFFUSION EXPERIMENTS
(A) EARLY FIELD EXPERIMENTS
(B) RECENT FIELD EXPERIMENTS
;}
(c) LABORATORY EXPERIMENTS
FOR DESCRIPTIONS OF THESE READ:
"METEOROLOGY AND ATOMIC ENERGY - 1958", D, SLADE (ED,), CHAPTER 4
"ATMOSPHERIC DIFFUSION", BY F. PASQUILL, CHAPTER 4
"A SUFWRY OF RECENT ATMOSPHERIC DIFFUSION EXPERIMENTS", BY
R, DRAXLER, NOAA TECH, MEMO, ERL/ARL-78,
4, EMPIRICAL SIGMA SCHEMES
(A) PASQUILL-GIFFORD SCHEME
(B) THE BNL SCHEME
(c) THF TVA SCHEME
(D) URSAN SIGMA SCHEME
(E)' M^RKEE'S DISPERSION CURVES
(F) TRC RURAL DISPERSION CURVES
(G) BRIGGS' INTERPOLATION SCHEME
33
-------
5-36
FOR A DETAILED DISCUSSION AND INTER COMPARISON OF THESE READ:
"ATMOSPHERIC DISPERSION PARAMETERS IN GAUSSIAN PLUME MODELING,
PART I" BY A, H, WEBER, EPA-€OQ/n-7b-030, 1976,
'TURBULENT DIFFUSION-TYPING SCHEMES: A REVIEW" BY F, A, GIFFORD,
NUCLEAR SAFETY, ]L No, 1, 1976,
d) FACTORS RESPONSIBLE FOR DIFFERENCES IN VARIOUSCT-SCFEMES
DIFFERENCES IN THE FOLLOWING SITE, SOURCE AND BOUNDARY
LAYER PARAMETERS:
(1) SURFACE ROUGHNESS AND OTHER CHARACTERISTICS
(2) STABILITY INDEX OR PARAMETER
(3) SAMPLING TIME
M) HEIGHT OF THE SOURCE AND PLUME BUOYANCY
(5) MIXED LAYER HEIGHT
OTHER FACTORS ARE:
(6) INAPPROPRIATE AND UNWARRANTED EXTRAPOLATIONS
(7) METHOD OF DETERMINING
-------
5-37
(A) LATERAL DISPERSION PARAMETERS
ESTIMATION OF
or =g-
YO V
YO
-I
U
BASED ON THE WORK OF NIEUSTADT
UNSTABLE CONDITIONS,
= 1 -j-fl F
1 Y
WHERE,
H
VAN DUUREN (1979). FOR
ALSO NOTE
A - 1,78, B = 0,059, c -- 2,5, D = ,0013.
FOR STABLE CONDITIONS, DRAXLER (1976) GIVES
-1
T0 = 1000 s,
35
-------
5-S8
ESTIMATION OF cr
ON THE WORK OF BRIGGS (1969. 1979)
WHERE AH = PLUME RISE
ESTIMATION OF CTY2
PASQUILL (1974, 1976)
CTY2 * 0,174 A©
WHERE A0 IS THE CHANGE IN THE WIND DIRECTION (iN RADIANS)
OVER THE VERTICAL EXTENT OF THE PLUME,
TOTAL DISPERSION -crY
T FY)2 +
-------
5-39
VERTICAI DTSPFRSTON PARAMETERS
^Q:
ACCORDING TO THE SIMILARITY THEORY
'zo
-^ Fzfc.f 1
W /. VJ. 1_ H
WHERE FZ IS A UNIQUE FUNCTION. FOR THE SAKE OF CONVENIENCE,
FZ CAN BE EVALUATED SEPARATELY FOR UNSTABLE AND STABLE CONDITIONS,
FOR CONVECTIVE CONDITIONS (- L>:>l) A SATISFACTORY DETERMINATION
OF FZ IS MADE FROM A LIMITED FIELD DATA (DRAXLER, 1976)
LABORATORY EXPERIMENTS (WlLLIS & DEARDORFF. 1976. 1978) AND
NUMERICAL EXPERIMENTS (LAMB. 1978, 1979). SEE FIGURES 1 AND 2
OF IRWIN (1979).
37
-------
5-40
FOR NELTTRAL AND STABLE CONDITIONS, ON THE BASIS OF OBSERVATIONS
ANALYSED BY DRAXLER (1976) AND RESULTS OF A SECOND-ORDER CLOSURE
MODEL (LEWELLEN AND TESKE, 1975)
1 u
o
Fz =
1 + 0,945 (I
V'o
j FOR H = 0
0,806
, FOR > 0,1
WHERE TQ IS A FUNCTION OF EFFECTIVE SOURCE HEIGHT H,
ACCORDING TO PASQUILL (1976)
-------
5-41
REFERENCES
1. Csanady, G.T., Turbulent Diffusion in the Environment,
Kluwer Boston, Inc, Boston, 247 pages. 1973
2. Gifford, F.A., Turbulent Diffusion - Typing Schemes: A
Review, Nuclear Safety. 17, pages 68-86. 1976
3. Irvn'n, J.S., Schemes for Estimating Dispersion Parameters
as a Function of Release Height, U.S. EPA Report EPA-
600/4-79-062, 91 pages. 1979
4. Pasquill, F., Atmospheric Diffusion, 2nd Ed., John Wiley
and Sons, New York, 429 pages. 1974
— Atmospheric Dispersion Parameters in Gaussian Plume
Modeling, Part II. Possible Requirements for Change
in the Turner Workbook Values, U.S. EPA Report EPA-
600/4-76-030b, 53 pages. 1976
5. Slade, D.H. (Ed), Meteorology and Atomic Energy, U.S. Atomic
Energy Commission, Oak Ridge, Tennessee, pages 13-188.
1968
6. Turner, D.B., Workbook of Atmospheric Dispersion Estimates,
U.S. EPA, RTP, NC, 84 pages. 1970
7. Weber, A.H., Atmospheric Dispersion Parameters in Gaussian
Plume Modeling, Part I. Review of Current Systems and
Possible Future Developments, U.S. EPA Report EPA-600/4-
76-030a, 69 pages. 1976
-------
6-1
Consequences of
Effluent Release
F. A. Gifford
-------
68
Consequences of
Effluent Release
Edited by R. L. Shoup
Turbulent Diffusion-Typing Schemes: A Review
By F. A. Grfford*
6-3
Abstract: Recent environmental concerns have greatly in-
creased the need for turbulent typing schemes in atmospheric
diffusion calculations. The standard methods by Brookhaven
National Laboratory. Pasquill, the Tennessee Valley Authority,
and others are reviewed, and differences, inconsistencies, and
modifications to the basic schemes are discussed. Various
exceptional flows occur to which existing turbulence typing
schemes should not be applied directly: diffusion in near-calm,
ven stable conditions, diffusion over cities, water bodies, and
irregular terrain, and diffusion in building wakes and near
highways Possible modifications to typing schemes in these
cases are discussed. In all such exceptional cases, many more
obsenational data are needed before reliable diffusion esti-
mates can be madn.
Pollutants are released from various sources near the
earth's surface, and the resulting ground-level air
concentration patterns have to be estimated. This
information is needed for a wide variety of air-pollu-
tion analyses and forecasts required by various provi-
sions of the National Environmental Policy Act
(NEPA), as well as for facility siting and design and
many other industrial and social planning purposes.
For instance, average values of pollutant concentra-
tions must be calculated over periods ranging from an
hour or less to a year in order to satisfy various current
legal requirements for environmental control.
'Franklin A. Gifford is Director of the Atmospheric
Turbulence and Diffusion Laboratory of the National Oceanic
and Atmospheric Administration, Oak Ridge, Term. He re-
ceived the B.S degree in meteorology from New York
University in 1947 and the M.S. and Ph.D. degrees from Penn
State University in 1954 and 19S5, respectively. He spent
5 years with Northwest Airlines (1945-1950) and 16 years
with the U. S. Weather Bureau (1950-1966) before assuming
his present position in 1966. He has been a member of the
Advisory Committee on Reactor Safeguards from 1958 to
1969 and a consultant to the Committee since 1969.
Air concentration patterns are controlled by atmo-
spheric diffusion, a process that depends on the state
of the atmospheric turbulence at any location and
time; however, atmospheric turbulence is difficult and
expensive to measure directly. Consequently it is useful
to be able to describe the boundary-layer turbulence in
terms of routine measurements of the mean values* of
meteorological quantities and their vertical gradients,
principally the average temperature, the horizontal
wind, and the vertical gradients of wind and tempera-
ture. The theory of the relation between these quanti-
ties and the turbulence has been worked out in
considerable detail for the lower part of the boundary
layer and is, by and large, quite successful. Detailed
summaries were given in a recent workshop on micro-
meteorology' and in a review by Panofsky? However,
the relation between the quantities and atmospheric
diffusion is much less well understood. Therefore it has
been found necessary to develop empirically based,
more-or-less qualitative, turbulence typing schemes in
order to handle practical atmospheric diffusion prob-
lems.
Probably the most widely used typing system is
based on the scheme propoved by Pasquil].3 A closely
related method, the Brookhaven National Laboratory
(BNL) turbulence types, is also in fairly widespread
use. The recent surge of activity in the area of
air-pollution analysis, in the wake of NEPA and such
court cases as the Sierra Club ruling on environmental
nondegradation, has highlighted the need for such
turbulence typing systems. By emphasizing the often
considerable social and economic issues that ride on
diffusion calculations, the current need has led to a
•Averaged over a time period cf the ordei of 30 to 60 min.
NUCLEAR SAFETY, Vol 17. No 1, J*nu»ry-Fttiru»rv 1976
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6-4
CONSEQUENCES OF EFFLUENT RELEASE
69
large number of applied studies involving various
developments and modifications of the original typing
v.hemes. These have not always been entirely in
: greement, either with one another or with the original
Titent. It seems useful to try to sort out much of this
material with the object of bringing out interrelations
•nd emphasizing, if only qualitatively, the straight-
:>rward physical reasoning that underlies all these
yping schemes.
TURBULENCE TYPING SCHEMES
IBNL Turbulence Types
An early attempt to categorize turbulence was
made by Giblett,4 who was concerned with the
dimensions of eddies as they affected the mooring and
ground handling of large airships. He distinguished
categories of atmospheric turbulence based on the
character of the wind as measured continuously by a
sensitive recorder (Dines anemograph) and the ac-
companying vertical temperature gradients. This sys-
tem consisted of four types, ranging from type 1
(unstable, gusty, cumulonimbus clouds present) to
type IV (strong temperature inversion; anemograph
trace shows practically no turbulence).
The BNL turbulence typing scheme, as originally
presented by Smith,5 is quite similar to Giblett's
four-category scheme. The BNL scheme has been
refined, developed, and summarized in a series of
papers.6"10 The types are based on the range of
fluctuations of the (horizontal) wind-direction trace as
recorded by a Bendix-Friez aerovane located at the
108-m level of the BNL tower. It was found desirable
to expand the original four-category scheme, and the
BNL types now have .the following definitions:'
A. Fluctuations (peak to peak) of the horizontal
wind direction exceeding 90°.
Bj. Fluctuations ranging from 40 to 90°.
B,. Fluctuations similar to A and B2 but confined
to IS and 45° limits.
C. Fluctuations greater than 15° distinguished by
the unbroken solid core of the trace.
D. The trace approximates a line; short-term fluctu-
ations do not exceed 15°.
(Fluctuations are recorded over a 1-hr period.)
This system was applied to the analysis of extensive
air concentration data in the form of measurements of
the dispersion of oil-fog plumes from a source 108 m
high. The BNL types were related to the observed
horizontal standard deviations of the plume concentra-
tion distribution. Values of the vertical plume standard
deviations oz were calculated on the assumption of a
Gaussian form for the concentration distribution. As in
Sutton's1' diffusion theory, the power laws for verti-
cal and horizontal spread as a function of downwind
distance, or(x) and o,.(x), were assumed to have equal
indices. These results are summarized in Table 1, and
curves of oy and oz vs. downwind distance x are
reproduced in Fig. 1.
The BNL scheme provides for categorizing turbu-
lence by means of reasonably simple measurements and
relating the categories to atmospheric dispersion esti-
mates derived from data. Note that the categories are
site specific, applying strictly to conditions equivalent
to those found at BNL. The diffusion data are for a
nonbuoyant plume released at 108 m, and the wind
speeds and trace characteristics refer to that height. All
measurements refer to average values over a period of
the order of 1 hr (wind averaged over 1 hr, concentra-
tions averaged over 30 to 90 min).
Pasquill's Turbulence Types
Pasquill3 proposed a simple scheme of turbulence
typing that has been widely applied. Information on
this scheme has been included in earlier papers by
Table 1 Properties of the BNL Turbulence Types
Type
A
B,
B,
C
D
Seasonal
frequency, %
1
3
42
14
40
A 77 AT
pa H3m,°C
-1.25 t T
-1.6± 0.5
-1.2 ±0.65
-0.64 ± 0.52
+2.0 ± 2.6
Average wind
tpeedat
108 tn, m/sec
1.8 ± 1.1*
3.8 i 1.8
7.0 * 3.1
10.4 t 3.1
6.4 t 2.6
Ou « m
0.40x*'"
0.36X°"
0.32X°"
0.31x° "
Oyt tn n
0.41xe-" 0.19
0.33x'-" 0.28
0.22X'-" 0.45
0.06X*-1" 0.58
cy
0.56
0.50
0.45
0.44
cz
0.58
0.46
0.32
0.05
Average wind
speed at
9m, m/sec
2.5
3.4
4.7
1.9
•Standard deviation.
NUCLEAR SAFETY. Vol. 17. No 1. .tonuvy-Ftbrucry 1976
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6-5
70
CONSEQUENCES OF EFFLUENT RELEASE
10"
to'
10'
10
104
10
10J
10
FTTTI
TT
! \\l\m
= 111!
TT
IN
ITTffi
101 10' 103 10' 105
x, DISTANCE DOWNWIND (ml
Fig. 1 Curves of Oy and o2 for BNL turbulence types from
Singer and Smith.9 Letters refer to BNL stability types in
Table 1.
Meade 1S'13 that were based on a still earlier unpub-
lished note by Pasquill. Pasquill presented information
on the lateral spreading 6 and the vertical spreading h
of diffusing plumes in the form of a graph for the latter
and a table for the former as functions of six
atmospheric stability classes designated A to F These
were arranged so that class A corresponds to extremely
unstable conditions and class F to stable conditions.
The quantities h and 6 mark the 10% points of the
plume concentration distribution relative to its mean
centerline value. The applicable stability category is
chosen by reference to a table relating these to
observed wind speed, cloud cover, and isolation condi-
tions (Table 2). These weather elements are widely
observed routinely all over the world.
Gifford14 described this turbulence typing scheme
in a review article based on the earlier presentations of
Pasquill's h and 8 values by Meade and converted the
plume spreading data into families of curves of the
standard deviations, ay and az, of the plume concen-
NUCLEAR SAFETY. Vol. 17. No. 1. J»nuwv-F«bru«rv 1976
nation distribution (Fig. 2).* This was done partly
because the standard deviation is a very commonly
used statistic and partly to emphasize that the method
could readily be used with the Gaussian plume for-
mula. A plume formula of this type had been used as a
convenient interpolation formula for diffusion data by
Cramer,15 Hay and Pasquill," and others. Pasquill's
typing scheme has almost always been used and quoted
in the form of these or similar graphs of ay and o2
which, for this reason, are frequently called the
Pasquill-Gifford (PC) curves. Although grateful for
the association, the writer would like to reemphasize
that the idea behind this useful scheme is attributed to
Pasquill.
Turner17'18 introduced a version of Pasquill's
scheme in which the incoming solar radiation is
classified in terms of elevation angle and cloud amount
and height. The procedure is objective and involves
meteorological quantities (i.e., cloud cover and height
and solar angle) that are known for most locations.
Thus it is well adapted to air-pollution studies and has
been widely used.
Turner expressed his Oy and oz curves as functions
of travel time, t = x/u, rather than downwind distance
x. Curves were labeled numerically: 1 for extreme
instability, 4 for neutral conditions, to 7 for extreme
stability, etc. Turner pointed out that curves for
classes 1 to 5 are essentially identical to PC curves A to
E. It seems clear that Turner intended this correspon-
dence and that his use of numbers rather than letters to
designate the stability types was fortuitous. However,
Colder,20 who studied large amounts of micro-
meteorological and diffusion data, including the
Kerang (Australia), Round Hill (Mass.), O'Neill (Nebr.),
Hanford (Wash.), and Cape Kennedy (Fla.) data sets.
calculated both Pasquill and Turner classes and found
some differences. Colder concluded that the best
conversion is provided by A to 1, B to 2, C to 3, D to
4, E to 6, and F to 7.
KlugJ 1 developed a typing scheme very similar to
Pasquill's. It differs primarily in that Table 2 is
The curves of h published by Pasquill9 were slightly
modified as compared with the earlier versions, those presented
by Meade,13'1' and in the earlier note by Pasquill. The o:
curves of Fig. 2 are based on the later h curves and conse-
quently differ slightly from those in Cifford,1' which were
necessarily based on the earlier version. The principal dif-
ference is that the A and B curves of Fig. 2 bend upward less
rapidly for x greater than about 200 m. This is in accord with
theoretical results on the free-convection limit of boundary-
layer turbulence, which affects the A and B categories and, as
will be seen, is closer to recently proposed interpolation
formulas by Briggs.' *
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6-8
CONSEQUENCES OF EFFLUENT RELEASE
Table 2 Meteorological Conditions Defining Pa squill Turbulence Types
D: Neutral conditions*
E: Slightly stable conditions
F: Moderately (table conditions
A: Extremeiy unstable conditions
B: Moderately unstable conditions
C: Slightly unstable conditions
Daytime insolation
Nighttime conditions
(peed, m/sec
<2
2
4
6
>6
Strong
A
A-B
B
C
C
Moderate
A-B
B
B-C
C-D
D
Slight
B
C
C
D
D
cloudiness^
E
D
D
D
"'•
cloudiness
F
E
D
D
•Applicable to heavy overcast day or night.
tThe degree of cloudiness is defined as that fraction of the sky above the local
apparent horizon that is covered by clouds.
71
ID4
103
ID-
I INI
ml
I I I iliTT
ID3
- i M iiini i f\ i iniii j^TiiiKi
/ - /
/ / / -
i" / // !
~~ » * * ^
10-' 10° ID1
«. DISTANCE DOWNWIND (km)
103
ID'1 10° 10'
«. DISTANCE DOWNWIND (km)
ID2
Fig. 2 Curves of ay and o2 for Pasquill's turbulence types based on Pasquffl.' See also Gifford,'
Sbde,>« and Turner.' •
replaced by a more detailed set of rules relating
cloudiness, wind speed, time of day, and season.
Pasquill's types, which were subjectively chosen,
appear to be approximately linearly related to turbu-
lence intensity, which is a desirable property. Luna and
Church" showed that the total change in median
turbulence intensity (at Augusta, Ga.) as the category
changes from A through F is equal to about an order of
magnitude and occurs approximately linearly. How-
ever, attempts to relate the types to various objective
stability criteria (such as lapse rate and bulk Richard-
son number) have been characterized by considerable
scatter.
Use of Measurements of Wind-Direction
Standard Deviation
Cramer15'53 suggested a method of classifying
turbulence for the purpose of diffusion estimation
based on the standard deviation of the wind measured
by bidirectional wind vanes. By correlating observa-
tions of 04 and 0£, the azimuth and elevation angle
standard deviations, for a range of stabilities with
NUCLEAR SAFETY. Vol. 17. No. 1. J^iuwy-Ftbriwry 1976
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6-7
72
CONSEQUENCES OF EFFLUENT RELEASE
simultaneously measured horizontal plume spreading
data, he set up a table of correspondences between a^
and OE values and the plume standard deviations using
a four-category system (Table 3).*
Cramer's system was based on the Round Hill and
Prairie Grass experimental data. Studies of these and
later experimental diffusion data, summarized by
Islitzer and Slade,24 generally supported the kind of
values proposed by Cramer (e.g., the summaries of
Idaho Falls data by Islitzer25 and of Hanford data by
Fuquay etal.).36 These were also similar to the
experimental values presented by Hay and Pasquill.1'
On the basis of these and related studies, hlitzer and
Slade34 proposed correspondences between 04 values
and the PC curves. These are summarized in Table 4,
together with the conversions to the Turner and the
BNL types.
It seemed that, at least in principle, plume standard
deviations could be estimated by measuring either the
lapse rate — A/VAr or the standard deviation of the
horizontal wind direction OA as well as mean wind
speed. For this reason relations among the Pasquill
types, lapse rates, and 04 values have frequently been
adopted as standards (e.g., U.S. Nuclear Regulatory
Commission (KRC) Regulatory Guide 1.21].3' This
method is satisfactory at any particular site; see, for
example, the study by Vogt and Geiss32 of dispersion
at Julich. However, the relation of turbulence type to
lapse rate has generally proved to be too variable from
site to site to be very useful, for reasons given below.
TVA Experience
Carpenter et al.33 summarized 20 years of Ten-
nessee Valley Authority (TVA) experience with the
measurement of concentration patterns and related
values of meteorological parameters. The emissions in
this case were all in the form of buoyant plumes from
tall stacks. Stack heights ranged from about 75 to
250 m, and the effective stack height (i.e., the stack
height plus buoyant plume rise) was rarely less than
twice that figure. TVA used a six-category typing
scheme, ranging from neutral to strong inversion, based
on lapse rate. The resulting families of a curves are
•It was pointed out by Holland3' and verified by
Market" that there is » simple convenient rufe of thumb
relating the wild-direction standard deviation for a sample of
the order of an hour and the range of wind-direction
fluctuations over the period; namely, 04 = Umax ~ <4min)/6,
where A is measured in degrees. Thus 04 can easily be found
directly from the tract of A (I), i.e., the chart record of a wind
vane, by inspection.
NUCLEAR SAFETY. Vol. 17. No. 1. January-February 1976
Table 3 Cramer's Turbulence Classes
Stability description 04, deg a
Extremely unstable
Near neutral (rough surface;
trees, buildings)
Near neutral (very smooth grass)
Extremely stable
30
15
6
3
10
5
2
1
Table 4 Relations Among Turbulence Typing Methods
Stability
description
Very stable
Moderately unstable
Slightly unstable
Neutral
Moderately stable
Very stable
Pasquill
A
B
C
D
E
F
Turner*
1
2
3
4
6
7
BNLt
B:
B,
B,
C
D
°A<
degj
25
20
15
10
5
2.5
•Colder.30
fPhiladelphia Electric Company."
tSlade."
reproduced in Fig. 3, together with the lapse-rate
values measured at plume height, to which they apply.
Further details of the TVA approach can be found in
Islitzer and Slade.3* It should be noted that the TVA
plume samples refer to an effective averaging time of
about 2 to 5 min, which is somewhat shorter than that
for the other schemes.
MODIFICATIONS OF THE
BASIC SCHEMES
The preceding section is a brief, essentially histori-
cal, account of the major turbulence typing systems
now in use. Because they reflect different diffusion-
data bases and, to a certain extent, were at least
originally addressed to. different applied problems,
these schemes might be expected to differ from each
other, and they do. Comparison of Figs. 1 to 3 reveals
major disagreements; the curves do not have the same
shape. The PC curves of oz have larger values and more
sharply increasing upward curvature with distance for
unstable conditions and conversely for stable condi-
tions, although the difference is in that case less
pronounced. PC curves of ay are slightly steeper than
the BNL curves for all stability conditions but more so
for stable. These differences have been discussed by
several workers; see, for example, Strom.34 The TVA
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CONSEQUENCES OF EFFLUENT RELEASE
73
,
10* -
: A
icr
10
Average potential temperature gradient
with height
— (°K/100m)
Neutral
Sightly stable
Suble
Isothermal
Moderate inversion
Strong inversion
0.00°KnOOm
0.27 °K/100m
0.64 °K/100m
1.00°K/100m
1.36°K/100m
1.73°K/100m
to-
10'
101
lI TT TTTT]
I I 1 I T
102 103 104
>, DOWNWIND DISTANCE Im)
10s
Fig. 3 Curves of Oy and o2 for TV A data from Carpenter
etal." Average potential temperature gradients with height
refer to plume height.
curves differ from both the BNL and the PC curves.
Not only are the shapes of the TVA curves rather
different, particularly for shorter distances, but also
the range of itmospheric stabiJity conditions en-
countered for these elevated plumes is much narrower
and includes no unstable conditions at plume height.
This is in contrast to the wide range of near-surface-
level stability conditions encountered for releases near
the ground.
Briggs's Interpolation Formulas
This situation has been discussed and resolved to
the extent possible by Briggs." The diffusion-data
bases for the various typing systems have the following
characteristics. The PC curves were developed pri-
marily with the aid of diffusion measurements made to
a distance of 800 m using a passive (i.e., nonbuoyant)
tracer gas that was released near the surface. The BNL
curves also reflected nonbuoyant-plume-dispersion data
but from an elevated (108-m) source. Ground con-
centration values were obtained out to several kilome-
ters, but only rarely were measurements made within
800 m of the source. On the other hand, TVA data
reflected still greater effective heights, from 150 to
600 m or more, and downwind distances of up to tens
of kilometers. Moreover, the rate of spreading of
plumes from sources of this type primarily reflects
buoyancy and entrainment effects on plume behavior
rather than ambient atmospheric turbulence properties
to considerable distances downwind, of the order of 5
to 10 source heights. According to Briggs,35 the
diffusion of a plume from such a source is quite
different from that of passive diffusion from a ground-
level source (i.e., the PC curves).
This led Briggs to propose a series of interpolation
formulas for a curves that would have the following
properties: they would agree with PC curves given by
Cifford,14 Slade,30 and Turner" in the range
100 m 100 m. The a curves in the ASME guide
reflect primarily BNL experience. Other than at small
distances, where the TVA curves display strong plume-
buoyancy effects, the TVA and BNL curves agree
reasonably well with one another and, except for A
and B conditions as noted, with the PC curves at about
10km. Beyond that distance, TVA curves are less
steeply inclined. Briggs's recommendations apply up to
10 km and could perhaps be extended to 20 or 30 km,
although he does not recommend this. Few plume-
dispersion values have been reported for distances
beyond 10 km. Differences among the various sets of
curves probably reflect the uncertainty of the data
fairly well. However, as pointed out recently by
Draxler,38 there are systematic differences in oy and
oz values computed from the various sets of diffusion
data related to release height. Briggs's recommended
interpolation formulas are summarized in Table 5 and
shown in Fig. 4. These are intended primarily for use in
NUCLEAR SAFETY. Vol. 17. No. 1. J»nu«rv-*«truwv 1976
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74
CONSEQUENCES OF EFFLUENT RELEASE
Table 5 Formulas Recommended by Briggs1'
foroy(x)and oz(x), ]07 < x < 10* m,
Open-Country Conditions
Puquill
type
A
B
C
D
E
F
0.22*0
0.16*0
0.11*0
0.08*0
0.06*O
0.04* (1
o», m
+ 0.0001*)"
+ o.oooi*)-
+ o.oooi*r
+ o.oooi*)-
+ o.oooi*)-
+ o.oooi*)-
H
s
H
s
H
*
°i
0.20*
0.12*
0.0&x(l +
0.06* (1 +
0.03* (1 +
0.016*0
,,m
0.0002*)-*
0.0015*)"*
0.0003*)"'
+ 0.0003*)'1
calculating ground-level concentrations, in particular
the maximum values of these quantities for plumes
from elevated stack sources. Consequently these values
reflect diffusion data for a higher source at greater
downwind distances.
Use of Power-Law Interpolation Formulas
Many authors have proposed power-law formulas
for the type oy = ax^, oz = cx^ for use in diffusion
formulas. The parameters of these expressions have
been tabulated in terms of each of the standard typing
schemes by various authors. Values of a, b. c, and d
have been given for Pasquill's turbulence types by
Tadmor and Cur,39 Fuquay etal.,40 Martin and
Tickvart,41 and Eimutis and Konicek.42 The BNL
curves have been approximated as power laws by
Singer and Smith.9 Smith,37 and Islitzer andSlade.'4
Values of power-law parameters for Cramer's scheme
are contained in his paper and in the summary by
Islitzer and Slade.74 In addition, values of the Sut-
ton1' stability parameter n and diffusion coefficients
Cy and Cz, based on data comparisons, have been given
by Yanskey et al.,43 as well as in Table 1 and in several
of the foregoing references. In Sutton's work, n defines
the exponent of a power law for a values. Finally, TVA
power-law interpolation formulas have been given by
Montgomery et al.44
A genera] limitation of all these results is that no
single power law can fit diffusion data over all
downwind distance ranges. This point was first made
clear by Barad and Haugen45 and is obvious from
Figs. 1 to 3. Moreover, a single power exponent for
both horizontal and vertical spreading, as in Table 1, is
now known to be inadequate. The elevated-source
diffusion observations of oz reported by Hogstrbm46
10*
102 103 10*
x, DOWNWIND DISTANCE (ml
103
— 101
10°
I I I I
10s 102 103 104
.. DOWNWIND DISTANCE (ml
105
Fij. 4 Curves of Oy utd o2 bised on interpolation formulas by Briggs'' for flow over open country
(*e Table 5); from Hosker.''
NUCLEAR SAFETY. Vol. 17, No. 1. Jinu.ry-Ftbrujry 1976
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CONSEQUENCES OF EFFLUENT RELEASE
76
and Discussed by Pasquill*7 show that ay varies as
distance to a power in the neighborhood of 0.85, as
compared with about 0.55 for or, in neutral condi-
tions. Briggs's equations (Table 5) are the simplest
interpolation formulas that give reasonable approxima-
tions to the various diffusion types over the range
100 < x < \ Q* m. The use of simple power laws in
diffusion equations has some purely mathematical
advantages, and, for some people, this seems to
outweigh the problem of the limited distance over
which they apply. For this reason, such power laws will
probably always be used to some extent. As long as the
distance range is suitably, restricted, this practice is
acceptable, although Briggs's formulas are preferable.
On the other hand, interpolated values of the
parameters in power-law formulas for ay and oz have
been quoted to three and occasionally four significant
figures in some of the papers referred to above. This
gives a quite false impression of the degree of precision
involved. Studies such as those by Luna and Church52
and Colder20 indicate that estimates of a values by
these turbulence typing methods have considerable
scatter. Pasquill48 concludes that estimates of pollu-
tion concentration based on typing methods may be
accurate to within 20% for long-term averages, given
good quality emissions and meteorological data, but
may exceed a factor of 2 for short-term values.
Relation of Empirical Stability Categories
to Boundary-Layer Turbulence Criteria
From studies by Luna and Church,75 >49 Colder,50
and others, it is known that qualitative stability
categories like those of Pasquill correspond generally to
direct measurements of boundary-layer turbulence
intensity but that there is considerable scatter. Lapse
rate has also proved to be an uncertain discriminator,
partly because material dispersing from surface sources
experiences a much wider range of lapse-rate condi-
tions compared with those experienced by elevated
emissions. The lapse rates corresponding to the data of
Fig. 2 reflect surface-level emissions, whereas those
shown in Fig. 3 are based on elevated emissions and are
measured at plume height. But variations in surface
roughness and thermal properties (soil type and mois-
ture content) from site to site, not specifically allowed
for in the simple typing schemes originally proposed,
should also have an effect, particularly on the vertical
dispersion. This situation has led various workers to
examine relations between stability types and theo-
retical criteria, or indices, of boundary-layer turbulence
that specifically account for these factors.
Islitzer15 gave Richardson numbers for the Pasquill
types ranging from -0.26 for type A to 0.046 for type
F. The values were calculated from micrometeorologi-
cal profile data measured oh a 45-m mast at the
National Reactor Testing Station (now the Idaho
National Engineering Laboratory) in Idaho Falls. The
Richardson number (Ri) is defined by
(1)
where g = gravitational acceleration
T = absolute temperature
8 - potential temperature
36/3z = minus the vertical gradient of potential
temperature lapse rate
3u/3z = wind shear
Thus the Richardson number contains information of
the required kind; however, it varies with height in the
steady-state boundary layer. A more useful index of
the state of the boundary-layer turbulence is the
Monin—Obukhov length
L = -(UlcppD/kgH (2)
where cp = specific heat at constant pressure
p = density
k = von Karman's constant
H - vertical heat flux
u»~ friction velocity as determined from the
surface shear stress ut = (r/p)^
As a rule, all these parameters can be assumed to be
constants or to vary only slowly in a steady-state
boundary layer. Therefore it seems likely that L should
bear a convenient relation to turbulence types.
Gifford50 estimated order-of-magnitude relations
between stability classes and L ranging from ±103 m
for near neutral conditions to +10 m for very stable
and —10m for very unstable conditions. These values
were chosen arbitrarily, based on qualitative indica-
tions provided by studies of boundary-layer wind
profiles in conditions of varying stability. Pasquill and
Smith,51 guided by detailed atmospheric diffusion
experiments with accompanying micrometeorological
profile data, provided more refined estimates, specifi-
cally tailored to the Pasquill stability categories, for the
case of flow over a fairly smooth surface (short grass,
zc " 1 cm). These are summarized in Table 6.
Colder,10 using the five detailed micrometeorologi-
cal data sets referred to previously,calculated! values
and Pasquill stability classes to derive the relation
Aown in Fig. 5. He also gives nomograms relating Ri to
NUCLEAR SAFETY. Vol. 17. No. 1. Jmuvy-February 1976
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CONSEQUENCES OF EFFLUENT RELEASE
Table 6 Relations Between PasquQI Type
wtd TurbuJence Criteria Ri and L for Flow over Short
Grass,z0 e 1 cm, According to Pasqufll and Smith5'
Puquill
type
A
B
C
D
E
f
Ri(a(2m)
-1.0- -0.7
-0.5 - -0.4
-0.17 - -0.13
0
0.03 - 0.05
0.05-0.11
L
-2
-4
-12
35
8
, m
--3
--5
--15
•D
-75
-35
z/Zo and to the more easily measured bulk Richardson
number B (Lettau and Davidson52), which he defines
is follows:
g
B~T
(3)
Since Ri is analytically related to L, Eq. 3 and Fig. 5
provide the means for determining Pasquill's categories
over various surfaces, given values of meteorological
quantities usually available. The required measure-
ments are made, by regulation, at all nuclear power-
reactor sites. In principle, this method should provide
stability class estimates exhibiting less scatter than the
lapse-rate method because it accounts for variations in
thermal and mechanical turbulence parameters from
site to site.
Diffusion Categories for Great Distances
The foregoing schemes for classifying turbulent
diffusion are all specifically restricted to distances up
to 10 km or several tens of kilometers at most because
the experimental data base of these essentially qualita-
tive and empirical schemes is very scanty for downwind
distances beyond a few kilometers. However, this
restriction has not always been observed in applica-
tions. Nonetheless, many urgent environmental prob-
lems require consideration of diffusion at great
distances from sources.
Diffusion beyond a few kilometers from a source,
even in the relatively straightforward case of open
country that is assumed in typing schemes, is com-
plicated by a number of effects that are not particu-
50
-0.12
-0.10
0.06
0.08
Fig. 5 Curves by Colder1 ° showmg Pasquill's turbulence types is a function of the Monin-Obukhov
(lability length and the aerodynamic roughness length.
NUCLEAR SAFETY, vol. i?. NO. i. j»nuw-Fibruw 1976
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CONSEQUENCES OF EFFLUENT RELEASE
77
•jrly important at short distances. The underlying
. Use of the mixing depth as a limit of oz
i recommended in nearly all typing schemes. As travel
.ime (downwind distance) increases, diurnal changes in
governing such parameters as stability become im-
portant.
Considering these problems, Smith54 (see also
Pasquill47) enlarged on Pasquill's original scheme as
follows. He obtained numerical solutions to the diffu-
sion equation for downwind distances up to 100km,
using wind-speed and diffusivity values based on actual
experience over a range of stability conditions. He then
used these results to define ot values based on (l)the
stability of the lower layers, as ordinarily determined
in Pasquill's method, and (2) the overall stability of the
planetary 'boundary layer. Provision is also made to
introduce the "typical" roughness length over the
plume path, the incoming solar radiation, the upward
heat flux, the mixing depth, and the variation of
stability along the path. The method is not yet
complete (curves for ay have not yet been published),
10
iov
1,0*
e
10'
i i in
1 i 11 utt
but it will ultimately provide a way to extend the basic
typing scheme to distances up to 100 km. Curves of oz
computed by Hosker5 5 according to Smith's procedure
are shown in Fig. 6.
DIFFUSION CATEGORIES FOR
EXCEPTIONAL FLOWS
Various flows occur in the planetary boundary
layer which, from the viewpoint of the standard
turbulence categories, must be considered exceptional
despite their practical importance in applications.
Estimates of spread based on Pasquill's categories are
intended to be applied in specifically limited situations
only: u > 2 m/sec, nonbuoyant plumes, and flow over
open country. This is because boundary-layer turbu-
lence is conceived, for the purpose of turbulence
typing, as consisting of a mechanical component
created by frictional wind shear at the >urface and a
thermal component arising from vertical boundary-
layer heat flux. Their relative importance in any
particular situation determines the turbulence type;
e.g., type A is low mechanical and high thermal
content, type D is all mechanical, etc. However, flows
exist for which the turbulence is not generated, solely
102 103 104
x. DOWNWIND DISTANCE (ml
10=
10
10=
>. DOWNWIND DISTANCE (ml
Fig. 6 Curvet of oz (i, * 10 cm) and oz (l, = 100 cm) bucd on Smith*i method;'' aftei Hodcer.''
NUCLEAR SAFETY. Vol- t7. No 1. Jcnuary-February 1976
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78
CONSEQUENCES OF EFFLUENT RELEASE
by these two mechanisms, e.g., flows over cities, flows
over large bodies of water, flows with buoyancy, wake
flows (flows behind obstacles), and very light-wind,
ttable flows (calm, clear nights). These all clearly lie
outside the limits of Pasquill's basic system and were
specifically excluded by him. Attempts have neverthe-
less been made to apply PC curves to diffusion
estimates in these situations, a Procrustean approach
which understandably always fails. This is not tiie fault
of the typing system, of course, but of the application,.
In some cases, reasonable modifications can be sug-
gested, as will be described below, and in other cases
^his is not yet possible. In all these exceptional cases,
much more research is needed.
Diffusion Categories in Near-Calm,
Very Stable Conditions
Beattie56 determined the frequency of occurrence
of Pasquill classes at eight British meteorological
stations, and results for others were reported by
Bryant.57 Others have since repeated this exercise at
various locations. The results are similar, as a rule,
although there is some variation with locality. Cate-
gories A and B provide around 10%, C and D around
60%, and E about 10%; category F applies in the
remaining 20% of the time, at least at the British
stations.
However, included in the latter 20% are a number
of near-calm situations typically occurring on clear
nights with frost or heavy dew. Such conditions were
specifically excluded by Pasquill from the original
categories because the diffusing plume could be ex-
pected to be very variable with "little definable travel."
Since these conditions occurred some 5 to 8% of the
time in Beattie's study, they have considerable prac-
tical importance. Beattie assigned them the designation
G without proposing any o curves.
On the not unreasonable assumption that actual
diffusion under category G conditions would be less
than that under F conditions, users have arbitrarily
assigned diffusion values; see, for instance, NRC
Regulatory Guide 1.21 (Ref. 31), which indicates that
category G diffusion has been assumed to be ap-
preciably slower than category F. Atmospheric diffu-
sion experiments reported by Sagendorf48 suggest that
under category G conditions the plume is subject to a
good deal of irregular horizontal "meander," or swing-
ing. The applicable value of o^ , instead of being the
small value indicated in Regulatory Guide 1.70, was
found to be greater than 8° and at times equaled 20°
or more. When averaged over 1 hi, the resulting
concentration values at a point are much lower than
was at first assumed under these conditions. Nickola,
Clark, and Ludwick,59 on the basis of results of two
low-wind (l.Sm/sec) diffusion experiments in which
the tracer was released for 30min from a point quite
near the ground, came to similar conclusions. In the
test run under stable conditions, varying between types
E and G, the averaged concentration values cor-
responded approximately to category C. In the test run
under unstable conditions, varying between types A and
D, the average concentration values were found to be a
factor of 2 below category A values.
A review of several sets of diffusion data for such
light-wind, stable conditions by Van der Hoven60
indicates that the effective a values can correspond to
anything between categories A and F. This supports
Pasquill's original assertion that diffusion under these
conditions will be very irregular and indefinite. In
dealing with these conditions at any site, it will clearly
be necessary as a minimum to have measurements or
estimates of 04 , as well as the usual quantities required
to define the turbulence type.
Diffusion over Cities
Diffusion over citier is enhanced, compared with
that over open country, not only because the surface
roughness is greatly increased but also because of the
great heat capauty of the cities. Thus both mechanical
and thermal turbulence are increased. The net increase
in turbulence intensity is evidently about 40%, as
compared with open country, according to Bowne,
Ball, and Anderson.*1 This study and other material
on atmospheric transport and dispersion over cities
were summarized by Gifford.62
Estimates of turbulence types of urban diffusion
have been based on the series of observations of
diffusion over St. Louis reported by McElroy and
Pooler.63 On the basis of these data, Pasquill64
compared diffusion types in open country and over a
city (Table 7). Johnson etal.6S analyzed additional
urban tracer experiments and presented revised esti-
mates of at. Considering these data and analyses,
Briggs" proposed the urban ay and o2 curves shown
in Fig. 7 and described in Table 8. These are based on
Figs. 9 and 10 of the paper by McElroy and Pooler.*3
The o2 curves are in essential agreement with those of
Johnson et al.
Diffusion over Water
Flow over bodies of water has long been known to
be characterized by greatly reduced turbulence in-
NUCLEAR SAFETY. Vol 17. No. 1. January-Ftbriary 1976
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CONSEQUENCES OF EFFLUENT RELEASE
79
Table 7 Vertical Diffusion oz over St. Louis
Compared with Diffusion over Open Country6 3 '*4
Downwind
distance,
km
1
10
Location
City'
Gtyi
Open country
City'
Cityt
Open country
Ratio of o2 to value in
neutral conditions for
fUbility categories
B C D E-F
4.5 2.7 1.7 0.7
4.0 1.4 1.5 0.6
3.2 1.9 1.0 0.5
9 3.4 1.0 0.3
11 4.1 1.2 0.4
6 2.4 1.0 0.3
•Using McElroy and Pooler's cum for B = ±0.01 in theii
Fig. 2.
tUsing data for bulk Richardson number B = ±0.01 in
evening conditions only.
Table 8 Formulas Recommended by Briggs1' for
0y(x)and oz(x); 103 mHosker."
NUCLEAR SAFETY. Vol. 17. No. 1, J»ou«ry-Ftbri»rv 1976
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80
CONSEQUENCES OF EFFLUENT RELEASE
tensity66 and a correspondingly decreased diffusion
rate.67 According to Kjtaigorodskii,68 in order to
describe the neutral boundary-layer wind profile over a
water surface, account has to be taken of the fact that
the waves are in motion relative to the air. Conse-
quently they do not act as ordinary, land-surface, fixed
roughness elements except in the initial stages of wave
development.
In engineering terms the roughness length 20, which
serves to characterize the wind profile and turbulence,
depends on an "equivalent sand roughness" of the sea
surface, hs, which is in turn a function of the stage of
wave development. The exact form of this dependence
is determined by the Reynolds number of the sur-
face,68 which may be either aerodynamically
"smooth," or "fully rough" (see Schlichting69). A
simple expression characterizing hs is not available. In
order to evaluate the surface Reynolds number and to
compute z0 from hs, Kitaigorodskii68 considers the
flow over individual waves of all possible phase
velocities and determines hs as a function of 5(oj), the
frequency spectrum of the waves, and the root-mean-
square rms wave height a.
The wave frequency co and the phase velocity c are
related, for deep-water gravity waves, by c = £/cj.70
The frequency spectrum will typically have a peak at
some frequency o;0 corresponding to a phase speed c0.
Using the experimentally and theoretically supported
assumption that only "steep" waves (i.e., those with
w>w0) can contribute to the drag, Kitaigorodskii68
finds
ifu0ut/g> 1
(4)
o.3Bul/g
These equations may be interpreted as follows. When
WO"«/£ = K*/CO ** ]' so l^at M* is mucn Breater than
the phase speeds of all the waves that contribute to the
drag, the waves all behave as immobile roughness
elements, and so the equivalent sand roughness of the
sea surface is approximately equal to the rms value of
the wave heights. This corresponds to the very early
stages of wave development. At the other extreme,
where u0ujg = ujc0 < 1, hs is independent of the
state of wave development and is determined only by
the aerodynamic quantity u,. For these fully devel-
oped waves, note that hs is quite small; if i/» =
50 cm/sec, a fairly large but realistic value, hs is less
than 1 cm. For the intermediate stages of wave
development, corresponding to usually observed situa-
NUCLEAR SAFETY. Vol. 17, No. 1. January -February 1976
tions, u0ujg= ujc0 °- 0.01 to 1.0, and hs depends
on the wave spectrum parameters a and u>o as we^ as
on u». Hence hs can be expected to vary with such
factors as fetch and duration of the wind. In these
cases of intermediate wave development, hs can be
much smaJler than the rms wave height. Therefore it
seems quite possible that, even for large waves on a
rough sea, the surface may not be fully rough in the
usual aerodynamic sense. This may be the reason why
diffusion observations over the sea, such as those
reported recently by Raynor etal.,71 show little
spreading and marked departure from the standard PG
curves.
The effect of mechanical roughness can be intro-
duced into the marine boundary layer as outlined
above, although the details are somewhat complex as
compared with the situation over land. Another major
difference arises from the intense evaporation of water
that takes place from the sea surface most of the time.
Density stratification over water is controlled by the
heat flux, as over land, but also depends on the
water-vapor flux. (The water-vapor flux may well exert
an important degree of control on the turbulence type
over heavily vegetated land as well. This point deserves
more consideration than it has received.) If fluctua-
tions of virtual temperature are considered, rather than
those of temperature as ordinarily defined (Lumley
and Panofsky,72 p. 95), the vapor flux can be taken
into account. This leads (see, for example, Monin73) to
a redefinition of the stability parameters Ri and L for
overwater flows, as follows:
(5)
where m = 0.61cp0/C,£ being the latent heat of
vaporization (for 6 ~ 300°K, m •*• 0.075) and
Lw = L (1 -r m/B0Jl
(6)
where Lw is the Monin—Obukhov length over water,
Rf is the usual flux form of Richardson number, and
B0 is the Bowen ratio:
B0=(cpIK)(8a-ew)/(ea-ew)
(7)
where e is specific humidity. Over the ocean, \B0\
usually range between '/4 and '/10, so that the term
m/B0 is quite significant.74>7!
The above presents at least a general framework for
including the complexities present in flows over water
in the determination of the characteristics of turbulent
diffusion. Pasquill's turbulence types could, in prin-
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6-16
CONSEQUENCES OF EFFLUENT RELEASE
ciple al least, be determined for overwater flows by
calculating the appropriate roughness and stability
length and then referring to nomograms of Colder20 or
Smith.54
Diffusion in the Lee of Flow Obstacles
Most sources of airborne contaminants are located
on or near buildings or other structures, such as cooling
towers. Isolated tall stacks, which, when properly
designed, do as a practical matter approximate the
point source assumed in diffusion theory, are the
exception rather than the rule among pollutant
sources. Thus it is curious and disturbing to find that
so little is known about the properties of diffusion in
ftie wakes that exist in the atmosphere downwind of
such structures.
A wake is a region of,low-speed flow that extends
downwind from a flow obstacle. Within the wake the
flow is turbulent, having properties at first strongly
conditioned by the size and shape of the obstacle. The
lowered wind speed in the wake creates shear at the
boundary, and the resulting fine-scale turbulence en-
trains air from the ambient atmospheric flow into the
wake, gradually expanding it, reducing the velocity
deficit, and ultimately dissipating the wake. Thus
dilution downwind of a source like a roof vent or a
building leak is strongly influenced by the building
nearby and then farther downwind c
-------
CONSEQUENCES OF EFFLUENT RELEASE
6-17
400
200
too
~K 50
20
10
10
20
50 100
. DOWNWIND DISTANCE (ml
200
400
Fig. 8 Johnson's*' companion of vertical diffusion based on observations of concentrations
perpendicular to a highway vs. distance from the highway for various stability categories. Conventional
PC curves are indicated for comparison.
NUCLEAR SAFETY. Vol. 17. No. 1. Januwy-Februwy 1976
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6-18
CONSEQUENCES OF EFFLUENT RELEASE
83
Diffusion near Highways
The wakes generated by vehicles are even less
understood than the wakes behind buildings. Dabbert,
Cagliostro, and Meisel84 conducted a literature survey
and concluded that there is very little definite informa-
tion on vehicle-wake properties. Several studies8 5>&6
have recently been reported on diffusion near road-
ways. Not surprisingly, it has been found that concen-
trations measured near highways do not conform to
the standard PG curves. Johnson's comparison of
curves of oz vs. downwind distance (from a highway),
inferred from concentration data, is reproduced in
Fig. 8. The figure shows that there is little if any
organization of the data by stability classes and the
vertical diffusion is considerably enhanced over the
usual PG curves. This should certainly be interpreted as
a wake effect, although of a more complicated kind,
involving penetration and interaction of successive
vehicle wakes.
The strength and the distance from the highway to
which this complex effect dominates diffusion and
beyond which presumably the ordinary PG curves,
suitably adjusted for initial wake diffusion, will then
apply will be determined by extension of the above
and related studies, such as those summarized by
Ludwig el al.87
Diffusion in Irregular and Rugged Terrain
As previously mentioned, Pasquill's typing scheme
is designed only to account for mechanically and
thermally generated boundary-layer turbulence. Flows
in rugged terrain have irregular, often turbulent,
features that originate otherwise than with boundary-
layer turbulence and heat transfer [e.g., drainage
(katabatic) winds, vortices shed from terrain obstacles,
channeling effects, and flow separations of various
kinds]. None of these features were contemplated in
the original typing systems, and so departures under
such conditions can and do occur.
Methods of calculating diffusion over hills and
terrain obstacles, based on the assumption of potential
flow of the mean motion, have been discussed by
Stumke88-'0 and Berlyand.91'92 As to diffusion
categories under such flow conditions, several papers at
the American Meteorological Society Symposium on
Atmospheric Diffusion and Air Pollution, Santa
Barbara, Calif., Sept. 9-13, 1974, touched on this
topic. Start, Dickson, and Hicks93 reported results of a
series of diffusion measurements conducted in a deep,
steep-walled canyon system in southern Utah. They
found that diffusion rates are systematically greater
within these deep canyons, implying departures from
the usual Pasquill categories. These departures resulted
in lower concentrations, compared with those cal-
culated from the usual PG curves. The differences
ranged from a factor of 1.4 in category B conditions to
4 in weak lapse to near-neutral conditions to 15 in
category F conditions. The authors state that most of
the phenomena mentioned earlier (i.e., greatk en-
hanced roughness, density flows, wake flows, and
channeling effects) were probably operating. Similar
results were reported by Hovind, Spangler, and Ander-
son.94 Start et al. believe that their results represent a
fairly extreme example of the terrain effect on
diffusion categories and speculate that less-rugged
terrain should lead to departures intermediate between
these results and the open-country values. More experi-
mental work clearly is needed.
SUMMARY AND CONCLUSIONS
Recent environmental concerns have greatly in-
creased the need to calculate air concentrations down-
wind from pollutant sources of various kinds. Because
concentration depends on diffusion and hence on
atmospheric turbulence, which is difficult and expen-
sive to measure, qualitative turbulence typing schemes
have been devised. These attempt to relate certain
average properties of the planetary boundary layer
(including wind speed, stability, insolation, surface
roughness, and heat flux) to atmospheric diffusion.
The most widely used of several turbulence typing
schemes is that proposed by Pasquill3 for diffusion
from low-level, nonbuoyant sources over open country.
Its relation to other typing schemes is shown in
Table 4. Modifications of Pasquill's scheme have been
proposed to account for elevated and buoyant sources
(Table 5), theoretical boundary-layer stability criteria
(Table 6 and Fig. 5), and diffusion at great distances
downwind (10 to 100 km).
There are various boundary-layer flows that can be
classed as exceptional, in that they involve sources of
turbulence (and hence diffusion) additional to the
mechanical friction and thermal buoyancy that are the
basic mechanisms in Pasquill's original scheme. The
turbulence categories have been extended in attempts
to account for (1) diffusion in near-calm, very stable
conditions, (2) diffusion over cities; (3) diffusion over
water; (4) diffusion in the lee of flow obstacles
(wakes); (5) diffusion near highways, and (6) diffusion
in irregular and rugged terrain. Available guidelines on
these exceptional cases, summarized previously, should
be used whenever applications require them, however.
NUCLEAR SAFETY. Vol. 17. No 1. J»nu««v Fet*o«'V
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CONSEQUENCES OF EFFLUENT RELEASE
not all details have been worked out. More research
and, in particular, more careful experimental studies
are needed to resolve several important problem areas.
ACKNOWLEDGMENTS
The writer wishes to thank R. P. Hosker for
clarifying the role of water waves as aerodynamic
roughness elements and for many helpful suggestions.
S.D. Swisher supplied preliminary reference material
which materially aided the preparation of this review.
This work was done under an agreement between the
U. S. Energy Research and Development Administra-
tion and the National Oceanic and Atmospheric
Administration.
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logical Society, Boston, Mass.
NUCLEAR SAFETY, Vol. 17. No. 1, Jinuwy-Febfuary 1976
-------
TECHNICAL REPORT DATA .
/Please read Juslructions on the reverse before completing!
4,
7.
9.
RtPUHFNO rj J. HtUI
EPA-450/? RI 077
TITLE ANDTuiTrrTi 5 REPO
APTI Course 4^3
Dispersion of Air Pollution — Theory and Model6'PERF
Application Selected Readings Packet
AUTHOR(S) 8. PERF
D. R. Bullard, Editor
PERFORMING ORGANIZATION NAME AND ADDRESS 10. PRC
Northrop Services, Inc.
P.O. Box 12313 11. CON
Research Triangle Park, NC 27709
12. SPONSORING AGENCY NAME AND ADDRESS 13. TYP
US Environmental Protection Agency Se]
Manpower and Technical Information Branch H.SPO
Air Pollution Training Institute E
Research Triangle Park, NC 27711
15.SUPPLEMENTARY NOTES
Project Officer for this packet is R. E. Townsend,
MD-17, RTP, NC 27711
16
17
18.
RT DATE
1981
ORMING ORGANIZATION CODE
ORMING ORGANIZATION REPORT NO.
GRAM ELEMENT NO.
B 18A2C
TRACT/GRANT NO.
58-02-2374
E OF REPORT AND PERIOD COVERED
-ected Readinps Pa^k^t
NSORING AGENCY CO6E
:PA-OANR-OAQPS
EPA-ERC,
.ABSTRACT
The Selected Readings Packet is to be used with Course 423,
"Dispersion of Air Pollution Theory and Model Application," as
designed and presented by the EPA Air Pollution Training Institute
(APTI). The Selected Readings Packet contains articles to supple-
ment the course text.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS b. IDENTI F IE RS/OPEN ENDE
Air Pollution Training Cou
Air Quality Modeling Selected Rea
Dispersion Packet
DISTRIBUTION STATEMENT Unlimited avail- 19. SECURITY CLASS fivi«/
, , „ _T , . , m . i T .0 Unclassif le
ahiei from National TerhnT^Ri InfT-
..t ion Service 5285 Port Roya! „„.»"! ^™^
Springfield. VA 22161 L
DTE RMS c. COSATI Field/Group
rse 13B
dings 51
68A
Report) 21. NO. OF PAGES
1
age) 22. PRICE
1
EPA Form 2220-1 (9-73)
6-23
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