co
co
UJ
CD
Effective
Stack Height
Plume Rise
Sl:406
-------
SI406
Effective
Stack Height
Plume Rise
1974
UNITED STATES ENVIRONMENTAL PROTECTION AGENCY
Office of Air and Waste Management
Office of Air Quality Planning and Standards
Control Programs Development Division
Air PollutionTraining Institute
-------
This manual has been reviewed by the Control Programs
Development Division, EPA, and approved for publication.
Approval does not signify that the contents necessarily
reflect the views and policies of the Environmental
Protection Agency, nor does mention of trade names or
commercial products constitute endorcement or recommen-
dation for use.
-------
Objectives
EFFECTIVE STACK HEIGHT/PLUME RISE
OBJECTIVES
To introduce the student visually to elevated pollutant emissions
and to present the physical principles most important in determin-
ing the rise of a plume.
To present the background assumptions of several commonly used
equations for computing plume rise together with major advantages
and disadvantages or limitations of each.
To show visually several meteorological and topographical conditions
which limit the use of any plume rise equation.
To require the calculation of effective stack height by three
common equations so that the student realizes what input data is
needed; he can then compare the different answers obtained by
using the same data. The second exercise treats Briggs' equations
in more depth, while the third presents current EPA calculation
procedures.
To present an in-depth discourse on the development of the Briggs'
equations together with recent developments and modifications
suggested by Briggs and the Meteorology Laboratory, EPA.
-------
Introduction
EFFECTIVE STACK HEIGHT/PLUME RISE
Effective Stack Height/Plume Rise is a self-instructional package
designed by the Air Pollution Training Institute, Environmental Protection
Agency. An Air Pollution Training Institute Certificate of Completion will
be awarded if the learner achieves a satisfactory level on the problem
sets included in the package. The suggested involvement time is eight hours.
The package contains:
Plume Rise/Effective Stack Height; a work manual
Flume Rise; a text by Gary A. Briggs, Ph.D., Research Meteorologist
Atmospheric Turbulence and Diffusion Laboratory, NOAA
Plume Rise; an audio tape presentation by Briggs
Effective Stack Height; an audio-slide presentation by
James L. Dicke, Meteorologist, Meteorology Laboratory,
Air Pollution Training Institute, EPA
A complete listing of the components is on page 7.
The package consists of three exercises. Exercise one is made up
of a narrated slide series and an APTI article, both entitled Effective
Stack Height and both writen by Mr. Dicke. Exercise two is made up of
the text Plume Rise and an audio tape presentation by Dr.. Briggs with
accompanying lecture notes in the work manual. Exercise three contains
a summary of Dr. Briggs' lastest analyses and the current EPA calculation
procedures as stated by D. Bruce Turner, Environmental Applications Branch,
Meteorology Laboratory, EPA. Problem sets conclude each exercise.
Those who desire a certificate should complete each component in the
order of presentation in the exercise. Problem sets may be submitted
individually or the three sets may be submitted on completion of the
package. All problem sheets must be returned to receive credit. Adequate
space is provided for calculations. Extra paper may be used if additional
space is required, but these calculations must also be submitted to receive
credit toward a certificate.
-------
A critique form is provided in the appendix. It will take but a moment
to complete the basic questions. The Air Pollution Training Institute
welcomes evaluation. For extended comments, please use the back of the
critique form. As an effort towards maintaining a viable and current pack-
age, it will be necessary for the participant to complete the critique
before a certificate will be awarded.
Pre-addressed envelopes are provided. If these are not enclosed in the
package, please mail your problem sets and critique to:
Air Pollution Training Institute
National Environmental Research Center
Research Triangle Park, North Carolina 27711
Attention: Plume Rise Instructor SI - 406
-------
COMPONENTS
•THE PACKAGE PAGE
A. EFFECTIVE STACK HEIGHT 11
An audio-slide presentation by James L. Dieke
B, EFFECTIVE STACK HEIGHT 13
James L. Moke
C. PROBLEM SET OXE _, 27
D. PLUME RISE - , 37
Gary A. Briggs, Ph.D.
E. PLWiE RISE , 39
An audio presentation with supplementary
lecture materials bv G.A.
PROBLEM SET TWO
G. SOME RECENT ANALYSES OF PLUME RISE OBSERVATIONS 55
Gary A. Briggs
H. ESIIMAT10K OF PLUME RISE - 77
Bruce Turner
I. PROBLEM SET IHRE-E SI
APPESMX — 37
Critique fora
Effective Stack Height, a cued script
JLEARXSR RESPONSE TO THE PACKAGE
1. Submission of Problem Set One to AFII
2. Submission of Problem Set Two to APTI
3. Submission of Problem Set Three to APTI
Critique of the package (submit with Problem Set Three)
>AIR POLLUTICX 1RAIXIXG INSTITUTE
RESPONSIBILITY TO THE LEARXER
1. Instructor grading of the Problem Sets; return of
evaluation to the learner with set of correct answers
2, Instructor certifying satisfactory completion.
3. Registrar issuing Certificate of Completion to learner
-------
Exercise One
A. EFFECTIVE STACK HEIGHT
An audio-slide presentation by James L. Dicke
B. EFFECTIVE STACK HEIGHT
James L. Dicke
C. ' PROBLEM SET ONE
Please complete the exercise in the given order. Upon completion
of components A and B, please work the problems in set one and forward
to the Air Pollution Training Institute. A pre^addressed envelope is
provided. A critique of your calculations will be returned with a copy
of the correct.calculations. Please be certain that your name and address
is on each sheet. ALL CALCULATIONS MUST BE RETURNED TO RECEIVE CREDIT.
Air Pollution Training Institute
National Environmental Research Center
Research Triangle Park, North Carolina 27711
Attention: Plume Rise Instructor SI - 406
-------
A
EFFECTIVE STACK HEIGHT
James L. Dicke *
Component A, "Effective Stack Height", is an audio-slide presentation.
It is made up of 54 35mm slides with an accompanying 50 minute narration
on cassette tape. The slides and cassett2 tape are standard format and
should be suitable for any 35mm slide projector and audio cassette recorder-
playback unit. For your convenience, a cued script of the audio portion of
the presentation is included in the appendix.
When you have completed the audio-slide series, please turn your
attention to Mr. Dicke's article on effective stack height (component B).
Problem Set One, which may be forwarded toward credit for an Air Pollution
Training Institute certificate, completes the exercise.
*James L. Dicke, Meteorologist, Meteorology Laboratory and
Air Pollution Training Institute, Environmental Protection Agency
11
-------
EFFECTIVE STACK HEIGHT
James L. Dicke *
In any consideration of concentrations downwind from a source, it
is desirable to estimate the effective stack height, the height at
which the plume becomes level. Rarely will this height correspond
to the physical height of the stack.
A high velocity of emission of the effluents and a temperature
higher than that of the atmosphere at the top of the stack will
act to increase the effective stack height above the height of
the actual stack. The effect of aerodynamic downwash, eddies
caused by the flow around buildings or the stack, and also the
evaporative cooling of moisture droplets in the effluent may
cause lowering of the plume to the extent that it may be lower
than the physical stack height.
FFFECT OF EXIT VELOCITY AND STACK GAS TEMPERATURE
A number of investigators have proposed formulas for the esti-
mation of effective stack height under given conditions: Davidson
(1949), Sutton (1950), P.osanquet et al. (1950), Holland (1953),
Priestley (1956) .
A recent comparison of actual plume heights and calculations using
six of the available formulas was made by Moses and Strom (1961) .
The formulas used were Davidson-Bryant, Holland, Scorer, Sutton,
Bosanquet-Carev-Halton, and Bosanquet (1957). They found that
"There is no one formula which is outstanding in all respects."
The formulas of Davidson-Brvant, Holland, Bosanquet-Carey-Halton,
and Bosanquet (1957) appear to give satisfactory results for many
purposes. It must be pointed out that the experimental tests made
by Moses and Strom used stack gas exit velocities less than 15 m/sec
and that temperatures of the effluent were not more than that of the
ambient air.
Stewart, Hale, arid Crooks (1958) compared effective stack heights
for the Harwell reactor emitting radioactive Argon with computations
using the formula of Bosanquet et al. (1950). The temperature of
the gases was 50°C above that of the ambient air and stack gas
velocity was 10 m/sec. At low wind speeds, agreement between formula
and plume height were quite good. At wind speeds greater than 6 m/sec
and distances greater than 600 meters from the stack the formula under-
estimates effective stack height.
Two of the formulas for estimation of effective stack height are given
below. Both the Davidson-Brvant formula and Holland's formula fre-
quently underestimate the effective stack height. Therefore, a slight
safety factor is frequently made by using the following formulas.
*James L. Dicke, Meteorologist, Meteorology Laboratory, and
Air Pollution Training Institute, Environmental Protection Agency
13
-------
Effective Stack Height/Plume Rise
The Davidson-Bryant formula is:
I/A
/ v ^
AK =
Where:
AR = the rise of the plume above the
stack
d = the inside stack diameter
v = stack gas velocity
u = wind speed
AT = the stack gas temperature minus the
ambient air temperature (°K)
T = the stack gas temperature ( K)
Any consistent system of units for AH, d, v , and u may be used. It
is recommended that vs and u be in meters/sec and d in meters which
will give AH in meters.
The Holland stack rise equation is:
v d / AT
AH = I 1.5 + 2.68 X 10 p
T
s
Where:
AP = the rise of the plume above the stack
(meters)
v = stack gas velocity (m/sec)
d = the inside stack diameter (meters)
u = wind speed (m/sec)
p = atmospheric pressure (mb)
Tg = stack gas temperature (OK)
AT = as in equation (1) and 2.68 X 10~3 is a constant having
units of (m-1 mb~l).
14
-------
Effective Stack Height
(B)
It is recommended that the result from the above equation be used
for neutral conditions. For unstable conditions a value between
1.1 and 1.2 times that from the equation should be used for AH.
por stable conditions a value between 0.8 and 0.9 times that from
the equation should be used for AH.
Since the plume rise from a stack occurs through some finite distance
downwind, these formulas should not be applied when considering effects
near the stack. Note that these formulas do not consider the stability
of the atmosphere but only the ambient air temperature. Actually,
stability should have some effect upon the plume rise.
Lucas et al., 1963, have tested Priestley's theory on stack rise at
two power stations. These investigators write the formula for stack
rise:
(3)
Where:
max
effective rise above the top of
the stack (feet)
a = a variable affected by lapse rate
and topography (ft2 MW~lM sec~l)
Q = the rate of heat emission from the
stack in megawatts (MW)
u = the wind speed (ft/sec)
They determined a to be 4900 and 6200 for the 2 power stations under
neutral conditions. Clarke (1968) states that if the exit velocity
of the stack gases exceeds 2000 fpm (23mph), no rain will enter the
stack. As an example, he found from a ten year summary of U.S.
Weather Bureau wind data for Chicago that 98% of the days had a
maximum wind velocity <_ 20 mph.
EFFECT OF EVAPORATIVE COOLING
In the washing of effluent gases to absorb certain gases before re-
lease to the atmosphere, the gases are cooled and become saturated
with water vapor. Upon release from the absorption tower further
cooling is likely due to contact with cold surfaces of ductwork or
stack. This causes condensation of water droplets in the gas stream.
Upon release from the stack, the water droplets evaporate withdrawing
the latent heat of vaporization from the air and consequently cooling
15
-------
Effective Stack Height/Plume Rise
the plume, causing it to have negative buoyancy, thereby reducing
the stack height. (Scorer 1959).
The practice of washing power plant flue gases to remove sulfur
dioxide is practiced at Battersea and Bankside power stations near
London, where frequent lowering of the plumes to ground level is
observed.
EFFECT OF AERODYNAMIC DOWNWASH
The influence of the mechanical turbulence around a building
or stack may significantly alter the effective stack height. This
is especially true under high wind conditions when the beneficial
effect of high stack gas velocity is at a minimum and the plume
is emitted nearly horizontally. The region of disturbed flow
surrounds a building generally to twice its height and 5 to 10
times its height downwind. Most of the knowledge about the
turbulent wakes around stacks and buildings have been gained through
wind tunnel studies. Sherlock and Stalker (1940), (1941), Rouse
(1951)j Sherlock (1951), Sherlock and Leshner (1954), (1955), Strom
(1955-1956), Strom et al. (1957), and Halitsky (1961), 0-962), (1963).
By using models of building shapes and stacks the wind speeds re-
quired to cause downwash for various wind directions may be determined.
In the use of a wind tunnel the meteorological variables that may
most easily be taken into account are the wind speed and the wind
direction (by rotation of the model within the tunnel). The plant
factors that may be taken into consideration are the size and
shape of the plant building, the shape, height, and diameter of
the stack, the amount of emission, the stack gas velocity, and
perhaps the density of the emitted effluent. The study of the re-
leased plume from the model stack has been done by photography
(Sherlock and Lesher, 1954), decrease in light beam intensity (Strom,
1955) , and measurement of concentrations of a tracer gas (Strom
et al., 1957, Halitsky, 1963).
By determining the critical wind speeds that will cause downwash
from various directions for a given set of plant factors, the
average number of hours of downwash annually can then be calculated
by determining the frequency of wind speeds greater than the critical
speeds for each direction (Sherlock and Lesher, 1954). It is assumed
that climatological data, representative of the site considered,
are available.
It is of interest to note that the maximum downwash about a rec-
tangular structure occurs when the direction of the wind is at
an angle of 45 degrees from the major axis of the structureand
that minimum downwash occurs with wind flow parallel to the major
axis of the structure (Sherlock and Lesher, 1954;.
16
-------
Effective Stack Height (B)
It has been shown by Halitsky (1961), (1963) that the effluent
from flush openings on flat roofs frequently flows in a direction
opposite to that of the free atmosphere wind due to counter-flow
along the roof in the turbulent wake above the building. In
addition to the effect of aerodynamic downwash upon the release
of air pollutants from stacks and buildings, it is also necessary
to consider aerodynamic downwash when exposing meteorological
instruments near or upon buildings so that representative measure-
ments are assured.
In cases where the pollution is emitted from a vent or opening on
a building and is immediately influenced by the turbulent wake of
the building, the pollution is quite rapidly distributed within this
turbulent wake. An initial distribution may be assumed at the
source with horizonal and vertical variances of 6y2 and 62 in the
z
form of a binormal distribution of concentrations. These variances
are related to the building width and height.
The resulting equation for concentrations from this source has
(oy2 + 6y2)1/2 in place of a and (crz2 + <5Z2)'/2 in place of
az in the point source equations.
EFFECT OF VERY LARGE POWER PLANTS
A power generating plant in the range of 1000-5000 megawatt capacity
emits heat to such an extent that its own circulation pattern will
be set up in the air surrounding the plant. It is doubtful that
extrapolation of dispersion estimates from existing smaller sources
can be applied to these large plants. Fortunately, the effluent
plume will rise far above the ground and surface concentrations of
pollutants downwind will increase by only a rather small amount
most of the time. Such large plants will usually be engineered to
minimize the effects of the two preceding topics. The "2 1/2" rule
will tend to eliminate downwash and the "4/3's" rule for stack gas
velocity, i.e. v should exceed u by at least a third, will tend to
eliminate entrainment of the effluent into the wake of the stack.(Pooler, 1965)
Three weather conditions, however, can still bring ground level
fumigations: high winds, inversion breakup, and a limited mixing
layer with light winds. The climatolopy of these conditions will
determine the magnitude and frequency of the pollution problem.
Pooler (1965) has presented nomograms for estimating groundlevel
S02 concentrations for these three situations together with effective
stack height formulas. Bripgs (1965) has also presented a plume
rise model which has been compared with data from various TVA
power plants. Pooler (1967) introduced a slight modification to
one of Briggs's equation and suggests this latter equation be sub-
stituted for the Pooler (1965) inversion breakup fumigation equation.
Thus the basic equation for this important weather condition is:
17
-------
Effective Stack Height/Plume Rise
S dz
Where: u = wind speed (m/sec)
r = inside stack radium (m)
T = ambient air temperature (°K)
d6 = potential temperature lapse rate
dz
Other symbols are defined as in equation (2) above.
A correction term for the additive effect of multiple-stack sources
is also presented in Pooler (1967) .
Other industries are also utilizing tall stacks for pollutant dis-
persion and anticipate results similar to the low measured ground
level concentrations found near tall British power plant stacks.
A case in point is the 1250 foot stack constructed in 1971 at a
cost of $5.5 million to serve the Inco Copper Cliff smelter in the
Sudbury district of Ontario, Canada. This gigantic stack is 116
feet in diameter at the base, tapering to just under 52 feet in
diameter at the top. The interior diameter is 45 feet.
STATE OF THE ART
Considerable research is being conducted to further quantify the
dilution effects of tall stacks and to develop better models for
predicting the dispersion of power plant effluent in complex terrain
and meteorological regimes. An extensive series of field experiments
is being conducted, called the Large Power Plant Effluent Study
(LAPPES), near Indiana, Pennsylvania and a report has been published
by Schiermeier and Niemeyer (1968). Field measurements include de-
termining plume geometry by laser-radar, in-plume and ground level
S02 concentrations, vegetation damage and meteorological conditions
during the experiments.
A summary of recent European studies dealing with plume rise and
stack effluent dispersion is contained in the July 1967 issue of
the journal Atmospheric Environment.
In addition another recent publication which contains a specific
chapter on calculating effective stack height is the ASME Recommended
Guide for the Prediction of the dispersion of Airborne Effluents(1973;.
18
-------
Effective Stack Height (B)
mean velocity profile
*" potential
flow
Typical flow pattern around a cube with one face normal to the wind.
19
-------
Effective Stack Height/Plume Rise
A comprehensive literature survey in this field was conducted by
NAPCA (EPA) and incorporated into an annotated bibliography of
over 200 references. The publication, "Tall Stacks, Various
Atmospheric Phenomena and Related Aspectsvl (1969), includes
articles published through mid-1968.
A very recent series of EPA publications deal with air pollution
aspects of emission sources. Of particular interest is GAP Pub.
No. AP-96, "Electric Power Production - A Bibliography with Abstracts".
Section D on air quality measurements and Section E on atmospheric
interactions contain many specific references to plume rise deter-
mination, plume behavior, and pollutant concentrations associated with
this class of sources.
Briggs in his publication, Plume Rise (1969), has presented
both a critical review of the subject and a series of equations
applicable to a wide range of atmospheric and emission conditions.
These equations are being employed by an increasing number of
meteorologists and are used almost exclusively within EPA. An
important result of this study is that the rise of buoyant plumes
from fossil-fuel plants with a heat emission of 20 megawatts (MW) -
4.7 x 10° cal/sec - or more can be calculated from the following
equations under neutral and unstable conditions.
AH = 1.6 F1/3 u-1 x 2/3 (5)
AH = 1.6 F1/3 u V (10 h )2/3 (6)
where:
AH = plume rise
F = buoyancy flux
u = average wind at stack level
x = horizonal distance downwind
of the stack
h = physical stack height
Equation (5) should be applied out to a distance of 10 hs from the
stack; equation (6) at further distances.
The buoyancy flux term, F, may be calculated from:
= 3.7 x 10-5m4/sec3[Q (7)
J
we pT Leal/sec
20
-------
Effective Stack Height (B)
where:
g= gravitational acceleration
Q = heat emission from the stack, cal/sec
c = specific heat of air at constant
pressure
p= average density of ambient air
T= average temperature of ambient air
Alternatively, if the stack gases have nearly the same specific heat
and molecular weight as air, the buoyancy flux may be determined from:
J^ g v r2 (8)
T S
s
where the notation has been previously defined.
In stable stratification with wind equation (5) holds approximately
to a distance x = 2.4 u s~l/- where:
g 3e , a stability parameter (9)
T 3z
= lapse rate of potential temperature
Beyond this point the plume levels off at about
AH = 2.4 F 1/3 (10)
However, if the wind is so light that the plume rises vertically,
the final rise can be calculated from:
H = 5.0 F1/A s-3/8 (ID
For other buoyant sources, emitting less than 20 MW of heat, a con-
servative estimate will be given by equation (5) up to a distance of:
x = 3x (12)
where:
x*=0.52 fsec6/5 ] F2/5h 3/5 (13)
ft. 6/b
21
-------
Effective Stack Height/Plume Rise
This is the distance at which atmospheric turbulence begins to dominate
entrainment.
Anyone who is responsible for making plume rise estimates should
familiarize himself thoroughly with Briggs' work.
REFERENCES
1. Bosanquet, C. H., Carey, W. F. and Halton, E. M.
Dust Deposition from Chimney Stacks. Proc. Inst.
Mech. Eng. 162:355-367. 1950.
2. Bosanquet, C. H. The Rise of a Hot Waste Gas Plume.
J. Inst. Fuel. 30:197. 322-328. 1957.
3. Davidson, W. F. The Dispersion and Spreading of Gases
and Dust from Chimneys. Trans. Conf. on Ind. Wastes.
14 Annual Meeting, Industrial Hygiene Found. Amer. 38-55.
November 18, 1949.
4. Gifford, F. A., Jr. Atmospheric Dispersion Calculations
Using the Generalized Gaussian Plume Model. Nuclear
Safety. 2:56-59. December, 1960.
5. Halitsky, James. Diffusion of Vented Gas Around Buildings.
J. of APCA. 12:2. 74-80. February, 1962.
6. Halitsky, James. Wind Tunnel Model Test of Exhaust Gas
Recirculation at the NIH Clinical Center. New York Univ.
Tech. Report No. 785.1. 1961.
7. Halitsky, James. Some Aspects of Atmospheric Diffusion
in Urban Areas. Air Over Cities. R.obert A. Taft Sanitary
Engineering Center Technical Report A 62-5. 1962.
8. Halitsky, James, Gordon, Jack, Halpern, Paul, and Wu, Paul.
Wind Tunnel Tests of Gas Diffusion From a Leak in the Shell
of a Nuclear Power Reactor and From a Nearby Stack. New
York Univ. Geophysical Sciences Laboratory Report No. 63-2.
1963.
9. Halitsky, James. Gas Diffusion Near Buildings, Theoretical
Concepts and Wind Tunnel Model Experiments with Prismatic
Building Shapes. New York Univ. Geophysical Sciences
Laboratory Report No. 63-3. 1963.
22
-------
Effective Stack Height (B)
10. Holland, J. Z. A Meteorological Survey of the Oak Ridge
Area. AEC. Washington, Report ORO-99. 554-559, 1963.
11. Lucas, D. H., Moore, 0. J., and Spurr, G. The Pvise of Hot
Plumes from Chimneys. Inst. J. Air Wat. Poll. 7:473-500.
1963.
12. Moses, Harry, and Strom, Gordon H. A Comparison of Observed
Plume Rises with Values Obtained from Well-Known Formulas.
J. APCA. 11:10. 455-466. October, 1961.
13. Priestley, C. H. B. A Working Theory of the Bent Over Plume
of Hot Gas. Quart. J. Roy. Met. Soc. 82-352. 165-176.
1956.
14. Rouse, Hunter. Air-Tunnel Studies of Diffusion in Urban
Areas. On Atmospheric Pollution. Meteorol. Monogr.
1:4. 39-41. November, 1951.
15. Scorer, R.. S. Natural Aerodynamics. Pergamon. London.
186-217. 1958.
16. Scorer,~ R. S. The Behavior of Chimney Plumes. Int. J.
of Air Poll. 1:3. 198-220. January, 1959.
17. Sherlock, R. H., and Stalker, E. A. The Control of Gases
in the Wake of Smokestacks. Mech. Eng. 62:455. 1940.
18. Sherlock, R. H., and Stalker, E. A. A Study of Flow Phenomena
in the Wake of Smokestacks. Univ. of Mich. Eng. Res. Bulletin.
29: 1941. 49 pp.
19. Sherlock, R. H. Analyzing Winds for Frequency and Duration.
On Atmospheric Pollution. Meteorol. Monogr. 1:4. 42. 1951.
20. Sherlock, R. H., and Leshner, E. J. Role of Chimney Design
in Dispersion of Waste Gases. Air Repair. 4:2. 1-10.
August 1954.
21. Sherlock, R. H., and Lesher, E. J. Design of Chimneys to
Control Downwash of Gases. Trans. Amer. Soc. Mech. Enp.rs.
77:1. 1955.
22. Stewart, N. G., Gale, H. J., and Crooks, R. N. The
Atmospheric Diffusion of Gases Discharged from the
Chimney of the Harwell Reactor BEPO. Int. J. Air Poll.
1: 1/2. 87-102. 1958.
-------
Effective Stack Height/Plume Rise
23. Strom, G. H. Wind Tunnel Scale Model Studies of Air Pollution
from Industrial Plants. Ind. Wastes. September - October
1955, November - December 1955, January - February 1956.
24. Strom, G. H., Hackman, M., and Kaplin. E. J. Atmospheric
Dispersal of Industrial Stack Gases Determined by Con-
centration Measurements in Scale Model Wind Tunnel Ex-
periments. J. APCA. 7:3 November, 1957.
25. Sutton, 0. G. The Dispersion of Hot Gases in the Atmospheric.
J. Meteorol. 7:307-312. 1950.
26. Briggs, G. A. A Plume Rise Model Compared with Observations.
J. APCA 15:9. 433-438. 1965.
27. Pooler, F., Jr. Potential Dispersion of Plumes from Large
Power Plants, PHS Publication No. 99-AP-16. 1965.
28. Pooler, F., Jr. Derivation of Inversion Breakup Ground Level
Concentration Frequencies from Large Elevated Sources.
Air Resources Field Research Office, ESSA, NCAPC, Cincinnati,
Ohio, 1967. (Unpublished Manuscript)
29. Symposium on Plume Behavior. Air and Water Pollut. Int. J.
Vol. 10 Nos. 6/7, 393-409. 1966.
30. Moore, D. J. Physical Aspects of Plume Models. Air and Water
Pollut. Int. J. Vol. 10 Nos. 6/7, 411-417. 1966.
31. Nonhebel, G. British Charts for Heights of Industrial Chimneys
Air and Water Pollut. Int. J. Vol. 10 No. 3, 183-189.
32. Gartrell, F. E., Thomas, F. W., and Carpenter, S. B. Full-
Scale Study of Dispersion of Stack Gases - A Summary Report.
Chattanooga, Tennessee, 1964. (Reprinted by the U. S. Dept.
HEW, Public Health Service.)
33. Carson, J. E. and Moses, Harry. Calculation of Effective
Stack Height. Presented at 47th Annual Meeting. Amer.
Meteor. Soc. New York, Janaury 23, 1967.
34. Culkowski, W. M. Estimating the Effect of Buildings on
Plumes from Short Stacks. Nuclear Safety. 8: 257-259.
Spring 1967.
35. Symposium on Chimney Plume Rise and Dispersion. Atmos.
Environ. 1:351-440. July 1967.
24
-------
Effective Stack Height (B)
36. Recommended Guide for the Prediction of the Dispersion of
Airborne Effluents. 2nd Ed. M.E. Smith, Ed. ASME, 345 E.
47th St. New York, N. Y. 10017, 1973.
37. Clarke, John H. Effective Stack Design in Air Pollution
Control, Heating, Piping and Air Condition. 125-133,
March 1968.
38. Tall Stacks, Various Atmospheric Phenomena and Related
Aspects. National Air Pollution Control Administration,
Pub. No. APTD 69-12, May 1969.
39. Briggs, G. A. Plume Rise. AEC Critical Review Series.
1969. Avail, as TID-25075 from CFSTI, NBS, U.S. Department
Commerce, Springfield, Virginia 22151. $6.00.
40. Fay, J. A., Escudier, M., Hoult, D. P. A Correlation of
Field Observations of Plume Rise, JAPCA Vol. 20, No. 5,
pp. 391-397, June 1970.
41. TVA. Report on Full-Scale Study of Inversion Breakup at
Large Power Plants. Div. of Env. R&D Muscle Shoales,
Alabama. March 1970.
42. Schiermeier, F. A., Niemeyer, L. E. Large Power Plant
Effluent Study (LAPPES) Vol. 1 (1968). NAPCA Pub. No.
APTD 70-2, June 1970.
43. Air Pollution Aspects of Emission Sources: Electric Power
Production A Bibliography with Abstracts. EPA, Office
of Air Programs, Pub. No. AP-96, May 1971.
44. Moses, Harry and Kraimer, M. R. Plume Rise Determination -
A New Technique Without Equations. JAPCA Vol. 22, No. 8,
pp. 621-630, August 1972.
45. Bowman, W. A. and Biggs, W. G. Meteorological Aspects of
Large Cooling Towers. Paper 72-128 pres. APCA, June 1972.
46. Briggs, G. A. Some Recent Analyses of Plume Rise Observa-
tions, pp. 1029-1032 in Proceedings of the Second Interna-
tional Clean Air Congress. Ed. by H. M. Epland and W. T.
Berry. Academic Press, New York, 1971.
47. Briggs, G. A. Discussion on Chimney Plumes in Neutral and
Stable Surroundings. Atmos. Environ. 6:507-510, July 1972.
25
-------
Exercise Two
D. PLUME RISE
Gary A. Briggs
E. PLUME RISE
An audio presentation with supplementary
lecture materials by G.A. Briggs
F. PROBLEM SET TWO
Please complete the exercise in the given order. Upon completion
of components D and E, please work the problems in set two and forward
to the Air Pollution Training Institute. A pre-addressed envelope is
provided. A critique of your calculations will be returned with a copy
of the correct calculations. Please be certain that your name and address
is on each sheet. ALL CALCULATIONS MUST BE RETURNED TO RECEIVE CREDIT.
Air Pollution Training Institute
National Environmental Research Center
Research Triangle Park, North Carolina 27711
Attention: Plume Rise Instructor SI - 406
35
-------
D
PLUME RISE
Dr. Gary A. Briggs *
Component D, Plume Rise, is a separate 80 page text.
Briggs 1
5.7, p. 58:
Briggs has recommended the following correction be made in Eq.
The constant 2.9 should be 2.4
1. Briggs, G. A. 1972. Discussion on Chimney Plumes in Neutral and
Stable Surroundings. Atmos. Environ. 6:507-510, July 1972.
When you have completed the text, please turn to Component E, a taped
lecture by Dr. Briggs. The audio cassette is standard format and should
be suitable for any audio cassette recorder-playback unit. Please do not
attempt to use the lecture cassette without referring to the supplementary
written material.
Problem Set Two, which may be forwarded toward credit for an Air Pollution
Training Institute certificate, completes the exercise.
*Dr. Gary A. Briggs, Research Meteorologist
Atmospheric Turbulence and Diffusion Laboratory,
National Oceanic and Atmospheric Administration
37
-------
PLUME RISE
Gary A. Briggs *
Component E is an audio tape cassette of a lecture by Dr. Briggs. The
cassette is standard format and should be suitable for any audio recorder-
playback unit.
Please do not attempt to use the lecture cassette without referring to
the following supplementary material.
momentum Flux/ IT p
(po/p) w02 r02
buoyancy Flux/ TT p -t- good
g (1-pQ/p) w0 r02 -*- better
g (1-mo/m) w0 r02 (T/TO) -s- best*
+g QH/( TI Cp p T)
Avg. wind speed at plume height
Avg. wind speed at stack height
(g/T) 39/3 z
(g/T) (AT/Az + l°C/100m)
From top of stack to top of plume
QH m4 1013 mb
TT Cp p T secJ P MW
*Dr. Gary A. Briggs, Research Meteorologist
Atmospheric Turbulence and Diffusion Laboratory,
National Oceanic and Atmospheric Administration.
39
-------
Effective Stack Height/Plume Rise
Should heat of condensation be included in QH?
Usually not, but —
1. Calculate rise Ah on basis of no condensation.
2. Calculate max. volume flux Vmax = T u (0.5 Ah)2
(u ^0.2 Fl/4 sl/8 = 0.08 (F/sec)l/4 « 0.5 m/sec)
3. Calculate moisture capacity of entrained air:
Q = q , - q~ V
cap V sh a/ max
where q , is the saturation specific humidity at
the height of the plume (h = h + Ah) and "q" is the
S 3.
average specific humidity through the plume rise
layer, q , is a function of the ambient temperature
s ri
at height h.
4. Subtract Qcap from efflux rate of water vapor, Qwv
If remainder is positive, assume condensation adds
(Qwv ~ Qcap) L to heat emission QH, where L is the
heat -of fusion.
5. Recalculate buoyancy parameter F and new Ah.
6. Start at Step 2 again, using new value of Ah. Repeat
calculation until desired degree of convergence is
obtained.
40
-------
Plume Rise, Lecture Notes (E)
Should evaporative cooling be included in QJJ?
Yes!
To be conservative, assume complete evaporation of water droplets
in plume. Subtract Qwi L from the heat emissions, where Qwl
is the efflux rate of liquid water.
If the total buoyancy turns out to be negative, so is the plume
rise — the plume falls to the ground, and:
-------
Effective Stack Height/Plume Rise
'Rule of lowest plume rise": consequences for buoyant plumes
F \ 1/3 „
1. Ah = 5.0 FlM s -3/8 or Ah = 2.4
us
"Calm" formula gives lowest rise if u < ( T-Q" ) F1/^ sl/8
At most, upper limit is about 0.5 m/sec (F « Ifl3 3
s « 10-3 sec-2).
Therefore, for practical purposes, "calm" formula only
applies to zero wind speed.
2. Ah = 1.6 Fl/3 u-l (3x*)2/3 or Ah = 2.4
"Neutral" formula gives lowest rise if s
or if IT us "1/2 > 7x*.
us
42
-------
Plume Rise, Lecture Notes (E)
Buoyant plume rise in stable conditions
F I/3
Since Ah = 2.4 applies very widely to buoyant
plumes in stable air, it is seen that Ah is not very
sensitive to variations in u and s, particularly since
they tend to be negatively correlated: on windy nights,
the stability is small; wind speed increases with height,
but stability decreases. Hence, "ballpark" estimates of
Ah in such cases need not include u and s. On the
basis of TVA and Bringfelt data, I recommend:
Ah = 17 pl/3 (very stable or high wind)
Ah = 35 F1/3 (slightly stable or low wind)
... where Ah is in m, and F is in
43
-------
Effective Stack Height/Plume Rise
Buoyant plume rise in neutral conditions
"Plume Rise" recommends the "2/3 law" cut off at a distance
xmax as a simple approximation to buoyant plume rise in
neutral conditions, where xmax = 10 hs (only_ for fossil
fuel plants with QH > 20 MW) or xmax = 3x*. with x* being
a function of F and hs given by Eq. 4.35.
Recent examination shows that, within the range of present
data, a simple cut-off distance proportional to /]? works
just as well. This also avoids a problem in the case of
very small source heights. Specifically,
max
= 63 m
= 1250 ft /QH/(106 cal/sec)
This results in a "final" rise
Ah = 25 (sec-m -2/3) F2/3/u
Note:
the ~K =10
max
h formula
s
was NOT intended to be applied
to gas turbine plumes.
44
-------
Plume Rise, Lecture Notes
(E)
Comparison of 10 hs, 3x* (Eq. 4.35) and 1250 >'QH for Table 5.1 of
"Plume Rise".
SOURCE
Harwell
Bosanquet
Darmstadt
Duisburg
Tallawarra
Lakeview
Barley
Castle Donington
Northfleet
Shawnee
Colbert
Johnsonville
Widows Creek
Gallatin
Paradise
10hs (ft)
N.A.
N.A.
N.A.
N.A.
N.A.
4930
N.A.
2500
4250
4250
4900
4900
2500
3000
4000
5000
5000
5000
6000
3x* (ft)
1110
1450
1140
2100
2040
4890
1450
2300
4500
5100
4200
5000
2400
2920
4200
5700
5SOO
4400
7100
1250 ,'QH (ft)
1310
1550
1150
1700
2140
4250
1550
2700
4300
5000
3500
4300
2900
3250
4100
5100
5100
3650
5800
.45
-------
Exercise Three
G. SOME RECENT ANALYSES OF PLUME RISE OBSERVATIONS
Gary A. Briggs
H. ESTIMATION OF PLUME RISE
D. Bruce Turner
I. PROBLEM SET THREE
Please complete the exercise in the given order. Upon completion
of components G and H, please work the problems in set three and forward
to the Air Pollution Training Institute. A pre-addressed envelope is
provided. A critique of your calculations will be returned with a copy
of the correct calculations. Please be certain that your name and address
is on each sheet. ALL CALCULATIONS MUST BE RETURNED TO RECEIVE CREDIT.
Air Pollution Training Institute
National Environmental Research Center
Research Triangle Park, North Carolina 27711
Attention: Plume Rise Instructor SI - 406
53
-------
G
SOME RECENT ANALYSES
OF PLUME RISE OBSERVATIONS
Gary A. Briggs *
Introduction
Good estimates of plume rise are required to predict the dispersion of
continuous gaseous emissions having large buoyancy or a high efflux
velocity. The rise of such emissions above their source height often
account for a considerable reduction of the concentration experienced
at the ground .
According to a recent critical survey on the subject, several dozen
programs of plume rise observations have been carried out and the results
published. This alone does not solve the problem, however. The quality
of these observations varies considerably, and in some cases important
parameters were not measured. The picture is also blurred by the presence
of turbulence in the atmosphere, which causes the plume rise to fluctuate
rapidly in many situations. The great number of empirical plume rise
formulas in the literature reflect these uncertainties. Each formula is
based on an analysis of one or more sets of observations, but each time a
different style of analysis or a different collection of observations is
used, a different empirical formula results. When applied to new situa-
tions, the predictions of these formulas sometimes differ by a factor of
ten. Obviously, great care in the analysis of available observations is
required.
The present paper is a summary of some of the analyses of observations
made in my recent critical review on plume rise and in my doctoral disser-
tation. Both of these works include extensive comparisons of observations
with formulas; care was taken to categorize the data according to the type
of source and the meteorological conditions, and to weight the data accord-
ing to the quality and quantity of observations they represent.
:Dr. Gary A. Briggs, Research Meteorologist
Atmospheric Turbulence and Diffusion Laboratory,
National Oceanic and Atmospheric Administration.
Atmospheric Turbulence and Diffusion Laboratory Contribution Number 38.
This paper was presented at the 1970 International Air Pollution Conference
of the International Union of Air Pollution Prevention Associations and
portions have been extracted for publication by the Journal of the Air
Pollution Control Association.
55
-------
Effective Stack Height/Plume Rise
Momentum Conservation and the "1/3 Law" for Jets
One of the major findings of researchers in the field of plume rise is that
the radius of a plume bent over in a wind is approximately^proportional to
the rise of the plume centerline above its source height: - ' This is true
for a considerable distance downwind of the source, at least several stack
heights. Mathematically, we can express this by
where r is a characteristic plume radius, y is a constant (dimensionless),
and z is the rise of the plume centerline. Surprisingly, this simple re-
lationship accounts very well for the great bulk of observed plume rises,
when it is used with appropriate conservation assumptions.
For instance, no outside forces act on a non-buoyant plume (jet) rising
through unstratified surroundings, so we might expect that the total flux
of vertical momentum in the plume is conserved. If the plume is only
slightly inclined above the horizontal, is nearly the same density as the
ambient air, p-, and has a horizontal component of motion nearly equal to
the mean wind at that height, u, then the flux of mass is approximately
pirr2u. The flux of vertical momentum is then w(pirr2u) , where w = udz/dx,
the vertical velocity of the centerline of a plume segment moving down-
wind at a speed u; x is the distance downwind of the source. We then
have
2 222
w u r = u y z dz/dx = F = constant, (2)
m
where F is the initial vertical momentum flux divided by irp. For a jet
having the same density as the ambient air, which must be true if it is non-
buoyant, F is given by
22 22
F = w r = w D /4, (3)
m oo o ' ^ '
where WQ is the mean efflux velocity, r is the radius of the stack, and D
is its diameter, assuming a circular, vertical source. Integration of Equa-
tion 2 yields the prediction that
z3 = (3Fm / Y2u2)x
Ah = z = (ewV/4Y2u2) 1/3 x 1/3 W
Ah/D = (3 / 4Y2) 1/3 R 1/3 (x/D) 1/3 ,
where R = WQ / u, the ratio of the efflux velocity to the wind speed In
this paper, the plume rise", Ah, is identified with the height of the plume
centerline above the source (Ah = z).
56
-------
Some Recent Analyses of Plume Rise Observations (G)
The above prediction that the rise of a jet is proportional to the one-third
power of distance downwind, the "1/3 law", is very well confirmed by the
available observations on jets. These are plotted in Figure 1 for the data
in which R = 2, 4, 8, 16, and 40. (The code identifying the six experiments
is given in Reference 2.) Surprisingly, the dotted lines representing the
"1/3 law" give fair agreement with observations even in the upper-left part
of the figure, where the plumes are more nearly vertical than horizontal
(the derivation of Equation 4 utilizes the assumption that the plumes are
only slightly inclined). However, the data indicate a stronger dependence
on R than the two-thirds power. Specifically, the dotted lines represent
Equation 4 with
Y = 1/3 + R"
(5)
Thus, it appears that the "entrainment constant", y, varies with the ratio
of efflux velocity to wind speed for a jet. This turns out not to be true for
a buoyant plume. Substituting this expression for v into Equation 4, we
have finally
1+3R
Buoyancy Conservation and the "2/3 Law" for Buoyant Plumes
If a buoyant plume is rising through unstratified surroundings and it neither
gains nor loses buoyancy through radiation, ordinarily the total flux of buoy-
ancy is conserved (for an exemption, see Reference 5). Applying Newton's
Second Law to a segment of a bent-over plume moving downwind with the mean
speed of the wind, we find that the rate of vertical momentum flux increase
equals the buoyancy flux:
2 22
d(w u r )/dt = ud(w u r )/dx = b u r , (7)
2
where b is a characteristic buoyant acceleration of the plume and bur is the
buoyancy flux divided by irp. The initial value of bur is given by
F = g(l-p0 / P) worQ2 , (8)
where g is gravitational acceleration and p is the density of the effluent
at the stack. A better determination of F, that accounts for alteration of
buoyancy due to dilution with ambient air, is given by
F = g(l-mo / m) (T/To)worQ2 + gQH / (TTC pT) , (9)
57
-------
ORNL-DWG 69-13875
<1J
en
QJ
B
rH
PH
fl
W>
•H
0)
X
A:
a
n)
0.5 1 2 5 10 20 50
DISTANCE DOWNWIND/JET DIAMETER
100 200
Figure 1. Rise of jet centerlines versus distance downwind for R = 2, 4, 8, 16, and 40.
(dotted lines are Equation 6)(see Reference 2 for code identifying experiments)
-------
ORNL-DWG 68-12899
1000
-------
Effective Stack Height/Plume Rise
where m is the mean ambient molecular weight (28.9), T is the ambient absolute
temperature, c is the specific heat capacity of air (0.24 cal/gm - 1C), and
Qu is the heatpemission; subscript "o" denotes values for the efflux gas,
instead of the ambient air. The quantity g/ (ircppT) is just a constant,
3.7 10~5 (m4/cal-sec2), times the ratio of standard sea level pressure to
the ambient pressure. If the effluent has considerable latent heat due to
water vapor and condensation of the plume is likely to occur near the source,
as would be expected in cold or wet weather, this latent heat may be inclined
in the determination of QR; otherwise, only dry heat should be considered.
If the process producing the effluent is uniform, F is proportional to the
rate of production. For instance, for modern fossil fuel power plants, F
is about 1.5 m4/sec3 times the megawatts per stack generated.
2
When bur = F = constant, Equation (7) integrates to
wur2 = F + F x/u (10)
m
This relation implies that buoyancy becomes more important than the initial
momentum flux when x •» uFm / F. For a hot effluent with about the same heat
capacity and mean molecular weight as air, this occurs at x = uw / (g (T0/T-1)),
a distance of the order of 5 seconds times the wind speed for most hot plumes.
Then the effect of Fm quickly becomes negligible, and for the region in which
Equation (1) applies, we have
z3 = (3F/2Y2u3)x2
Ah = z = (3F/2Y2)1/3 iT1 x273 (11)
Ah/L = (3/2Y2)1/3 (x/L)2/3
Where L = F/u , a characteristic length for the rise of buoyant plumes.
Most of the observations available on buoyant plume rise approximate this
"2/3" of rise with distance downwind. This is illustrated in Figure 2,
which is a superposition of curves hand drawn through scatter diagrams of
Ah/L versus x/L for 16 individual sources1 (the sources are identified in
Reference 1). Only data for stable atmospheric conditions are omitted.
There is considerably greater scatter about the "2/3 law" in this figure
than there is about the "1/3 law" for jets in Figure 1. The difference is
probably due to the fact that all the experiments represented in Figure 1
were made under controlled conditions in wind tunnels or modeling channels,
while all the observations shown in Figure 2 were made on plumes from real
stacks in the atmosphere. This introduces the possibility of aerodynamic
effects caused by buildings near the stack and uneven terrain, and also
60
-------
LJ -
uo 10
CC
_l
<
z
z
UJ
5
o
o
10 " 2
^
X
PO
^E
_l
< 0.
• R= 8
A R= 4
R= EFFLU>
R- 16
• *
R = 8
A R=4
o
' V
A
EL
(
OC
n
^
IT
Y
/
WIND SPE
DO
*<>~<% '
—£*
A A
ED
'"3 —
A
°J"!2
» —
•or
4
nr
8<
2/
_
3
Q_
-c
A
L;
f
A
av
-«XJ
*.«LjP^-
^ _ 0^—0^
A ^-
V" FOR BU
*A*'
OYANT
-stf.
^
PLl
^
JME
r5
s
5^
9>
^
-f
01 0.02 0.05 0.1 0.2 0.5 1 2 5 10
Lx/L,
MOMENTUM FLUX ENHANCEMENT DUE TO BUOYANCY
en
o
S
rt>
rt
>
3
CD
0>
en
H-
0)
Figure 3. Nondimensionalized rise of model plume centerlines versus ratio of buoyancy-
induced momentum to initial momentum. (solid lines are Equation 13)
O
a'
01
(D
H
-------
Effective Stack Height/Plume Rise
assures greater fluctuations about the mean plume rise due to large turbulent
eddies in the atmosphere. Atmospheric turbulence should also lead to more
rapid mixing of plumes with ambient air, and therefore a downward departure
from the "2/3 law" should occur at some point downwind; however, Figure 2
offers no particular support for this expectation. This means either that
L does not correlate well with the distance of downward departure (leveling
off), or that "leveling off" occurs at greater values of x/L than those
measured up until now.
There is no evidence that y is dependent on R for buoyant plumes6, at least
when R > 1.2. Below this value, downwash of the plume into the low pressure
region in the wake of the stack is likely to occur. This reciprocal wind
speed relationship predicted by Equation 11 with y constant is well estab-
lished for buoyant plumes in neutral conditions. An analysis'of photo-
graphed plume diameters and concurrent plume rises of TVA plumes from single
stacks showed that y = 0.5. Bringfelt3obtained similar results, finding
an average value of y of 0.53 for eleven plumes in slightly stable or windy
conditions and 0.46 for ten plumes in strongly stable or weak wind conditions.
The behavior of buoyant plumes in stable conditions is well predicted by
y = 0.5, as will be shown, but the optimum fit' to the "2/3 law" at large
distances downwind in neutral and unstable conditions corresponds closer to
y = 0.6, or
Ah = 1.6 F1/3 u-1 x2/3 (12)
Transition to Buoyancy-Dominated Rise
Equation 10 implies that a transition from the "1/3 law" for momentum-
dominated rise to the "2/3 law" for buoyancy-dominated rise occurs as Fx/uFm
grows from small to large values. This prediction is confirmed by Figure 3,
which plots observations of plume rise modeled in a channel by Fan.7 The rises
are divided by 2/3 1/3 1/3 _2/3 f
L x = F u x
m
so there should be no variation with Fx/uF = Lx/I2 in the momentum-dominated
m
region. This is seen to be approximately true in the left-hand side of the
figure, where Lx/Lm < 0.3. However, there is a separation of the points
for different values^ of R in this region; this is easily accounted for by
letting y - 1/3 + R , as was done for jets. There is a clear upswing and
some convergence of the points in the right-hand side of the figure and
these appear to asymptotically approach the line representing the "2/3 law"
Equation 11, with y = 0.5.
The simplest way to describe this transition mathematically is to integrate
Equation 10, after substituting r = yz and w = udz/dx, and then substitute
the empirical value of y for jets in the momentum term (y = 1/3 + R~') and
62
-------
Some Recent Analyses of Plume Rise Observations (G)
the empirical value for y for buoyant plumes in the buoyancy term (y = 0.5).
The result is
(1/3 + R l)2 u2 2(0.5)2 u3
This equation is represented as solid lines in Figure 3, for R = 4, 8, and
16. It is seen to describe the transition region fairly well.
Stability-Limited Rise
When a plume rises in a stable environment, it entrains air and carries it
upward into regions of relatively warm ambient air. Eventually, the plume's
buoyancy becomes negative and its rise is terminated. If heat is conserved,
that is, the motion is adiabatic, the rate at which each plume element loses
temperature relative to the ambient temperature is just its rate of rise
times the ambient potential temperature gradient (potential temperature, 0,
is the temperature that air would acquire if it were compressed adiabatically
to a standard pressure, usually the mean pressure at sea level; the potential
temperature gradient is just the real temperature gradient plus the adia-
batic lapse rate, i.e., 30/3z = 3T/3z + 1°C/100m in our lower atmosphere).
The resulting decay of the buoyancy flux is expressed by
222
d(bur )/dt = ud(bur )/dx = -w sur (14)
where s =(g/T) 30/3z.
If we differentiate Equation 7 with respect to t and substitute in equation 14,
we find that
2222222 2
d (wur )/dt = u d (wur )/dx = -s(wur ) . (15)
-1/2
This equation establishes the fact that s is a characteristic time for
the decay of the momentum flux. If s is positive and approximately constant
with height, the momentum flux is a harmonic function of s/2 t:
2 „ _, 1/2^ , -1/2 „ _,_ ,1/2^
wur
= F cos(st) + s" F sin (St). (16)
m
Since r always increases, the plume centerline behaves like a damped harmonic
oscillator. If the wind speed is constant with height, a jet (F = 0) reaches
its maximum rise at x = ut = (rr/2) us /2 , and a buoyant plume (F = 0)
- I/, m
reaches its maximum rise at x = ITUS •< .
The above conclusions are based on conservation assumptions alone, and do not
63
-------
ORNL-DWG 68-12903A
0)
en
•H
PS
BO
•H
CD
O
id
4-1
cn
(TVA, CARPENTER 61/1 a/., 1967)
PLUME TOP
PLUME CENTER LINE
• THEORY
1 23456
x/x', NONDIMENSIONAL DISTANCE DOWNWIND
Figure 4. Nondimensionalized rise versus nondimensionalized distance downwind for single-
stack TVA plumes in stable conditions. (dotted line is Equation 18)
-------
ORNL-DWG 69-13872
LJ
CO
o
CO
UJ
Q
O
2
JT>~
\j
x
0
BRINGFELT,1968
PLUME CENTER LINE
THEORY
0
234567
x/x' ,NONDIMENSIONAL DISTANCE DOWNWIND
10
to
0>
O
cr
Figure 5. Nondimensionalized rise versus nondimensionalized distance downwind for plumes
measured by Bringfelt in stable conditions. (dotted line is Equation 18)
-------
Effective Stack Height/Plume Rise
,. «"---
point, as it would imply a shrinking plume. With u constant w = udz/dx, and
r = yz, Equation 16 can easily be integrated. For a jet we fmd that
/ • , /
Ah = z = | (sin (x/x'
2 1/2
Y us
(17)
/ Fm \l/3
= 3 (1 + 3R"1)" ( ) (maximum rise) ,
where x' = us 1 and y = 1/3 + R . For a buoyant plume we find that
3F \ 1/3 ,,,
Ah = z = ( — j (1- cos (x/x')) '
us
(18)
= 2.9 / F \ 1/3 = 2.9 L1/3 x'2/3 (maximum rise) .
There are sufficiently detailed data to verify Equation 18, which is shown
as dotted lines in Figure 4 and 5. Both these figures show plume rise,
divided by L x' , versus x/x'. To determine s, the measured potential
temperature gradients were averaged throughout the layer of plume rise
(from the top of the stack to the top of the plume). The first figure shows
centerlines of TVA plumes from single stacks that were observed to level
off in stable air. It also shows the observed rises of plume tops. The
second figure shows the longest plume centerlines observed in very stable
air by Bringfelt.3 Both of these figures give excellent support to the
prediction that the maximum rise is obtained at x = TTX' . There is only a
little evidence of oscillation beyond this point; evidently, most plume
centerlines experience considerable damping through continued mixing beyond
this point. The leveled-off plume rises, which range from 140 to 290m for
the TVA data and from 60 to 160m for the Bringfelt data, seem to be well
approximated by Equation 18 on the average. For the TVA data, the scatter
about the predicted rise seems to be greatest in the rising stage, which
66
-------
Some Recent Analyses ot Plume Rise Observations
(G)
Table 1.
Ratios of Calculated to Observed Plume Rises in Stable Conditions
Formula Bring
Holland 0.
Priestley ! 0.
Bosanquet
Schmidt, m = 0
Schmidt, m = 1/2
Equation 18
1.
0.
0.
0.
33
74
09
29
94
89
;felt :
± 73% : 0
± 221 0
±24% 1
± 072
± 27%
± 07%
0
0
0
TVA
.81 r
. 4-4 ±
. 22 ±
.28 r
.85 t
.96 ±
Bring felt
and TV A
07%
05%
12%
24";
25%
08%
0.72 =
0.47
1.20 ±
0.28
0.90 =
0.93 ±
39
35
18
16
27
-
?/
»
c,<
0 !
08%
approximates the "2/3 law" when x -- 2x'. There is less scatter in this
stage in the Bringfelt data, which utilized more representative wind speed
measurements and were taken during much more stable conditions.
Reference 2 also compares these observations with other formulas for buoy-
ant plume rise in stable conditions, namely; the Holland formula,9 minus
20% as suggested for stable conditions; Priestley's equation,10 reduced to
the case of a buoyant, point source; Bosanquet's formula,11 similarly re-
duced; and Schmidt's formula,1"1 with his parameter "m" set equal to 0 and 1/2.
The centerline plume rises at a standard distance x = 5x' were interpolated
and averaged for periods in which there were at least five photographs of
the plume at this distance. This yielded five periods each from the TVA and
the Bringfelt observations. The ratios of calculated to observed plume rises
were then calculated for each formula and each period. The median ratio and
the average percentage deviation from the median ratio for each formula are
shown in Table 1.
The TVA heat emissions were substantially higher than those observed by
Bringfelt, so the high percentage deviation exhibited by the Holland and
Priestley formulas may be because of too much and too little predicted de-
pendence of rise on heat emission, respectively. Clearly, Equation 18 for
maximum rise gives the most consistently good predictions for buoyant,
stability-limited plume rise.
Turbulence-Limited Rise
The simple plume rise model outlined in the preceding section, based on r = yz
and conservation assumptions, succeeds in predicting the approximate rise be-
havior of all available observations of plumes bent over in a wind. It is
very similar to the successful model for nearly vertical plumes suggested by
G.I. Taylor in 1945 and later developed by Morton, Taylor, and Turner. How-
ever as it stands, it predicts unlimited rise in neutral (s = 0) and unstable
environments (s < 0). This is contrary to the expectations of many plume rise
67
-------
Effective Stack Height/Plume Rise
observers, some of whom have assumed that the plume rise is the rise of the
plume at the point that it becomes hard to follow. Yet, no observations made
so far show any leveling-off tendencies, except in stable conditions.
Nevertheless, it is quite reasonable to expect more rapid growth of the plume
radius in neutral and unstable conditions, due to the presence of considerable
environmental turbulence. This, in turn, leads to a reduced rise velocity
and perhaps to a limited plume rise, at least in neutral conditions. The ques-
tion is how to account for the enhanced growth of the plume radius. One way
is to assume that only eddies of the same order of size as the plume radius
are effective at mixing ambient air into the plume, and that these eddies are
predominantly in the inertial subrange of the atmospheric turbulence spectrum.
This part of the spectrum is adequately characterized by the eddy energy
dissipation rate, e, and eddies of the order of r in size have velocities of
1/3 1/3
the order of e r . This suggests the relationship
dr/dt = Be1/3r1/3 , (19)
where 6 is a constant (dimensionless). This should not apply at small
distances, where r is small and w is large; Equation 1 gives a faster growth
rate at first (r = yz implies that dr/dt = yw). In References 1 and 2, I
developed a model identical to the one outlined so far, except for the
assumption that Equation 19 applies instead of Equation 1 beyond the dis-
tance at which Be r becomes equal to yw- Since this model is based
on an inertia! range atmospheric turbulence entrainment assumption, I call
this the "IRATE" plume rise model.2
2
For rise in neutral conditions, in which bur = F = constant, the "IRATE"
model predicts a very gradual leveling of the plume centerline beyond the
distance that 3e r becomes equal to yw> designated by x*. For a
buoyant plume, the "2/3 law", Equation 11, applies to the first stage of
rise. The distance of transition to the second stage is then given by
x* - (2/3)7/5 (YF)2/5 u3/5 (BE^) -y/> (20)
If E is approximately constant above the height at which this transition
occurs, the second stage rise (x > x*) is given by
(21)
Ah = (3F/2Y2) 1/3 u-1 x* 2/3 \H _ ^ (x/x* + 5/8)
6 32
While this is not a very simple formula, note that it is just the "2/3 law"
times a function of x/x*. A final rise, equal to 55/16 times the rise at
-------
Some Recent Analyses of Plume Rise Observations (G)
the transition point, is approached, but only at a great distance; 90% of
the asymptotic rise is achieved at x = 20x*, and x* can be as large as a
kilometer for a very buoyant or very high source. It is unlikely that the
maximum ground concentration occurs well before this point, especially since
Equation 19 predicts an extremely large radius at x = 20x*. If y = 0.5 and
the bottom of the plume is taken to be a distance (Ah - r) above the source
height, we find that the plume bottom begins to descend at about x = 2x*
and spreads down to the source height (r = Ah) at x = 5x*. Since the growth
of the plume radius is quite rapid at this point, the highest ground concen-
tration should ordinarily occur in this neighborhood. It therefore seems
prudent to use the rise at x = 5x*, 2.3 times the rise at the transition
point, as the "final" plume rise in neutral conditions. This rise is the
same as that given by the "2/3 law" with x = 3.5x*, and Equation 21 deviates
from the "2/3 law" by only 11% at x = 3.5x*. This suggests a much more
practical prediction procedure for buoyant plume rise in neutral conditions:
Ah = (3F/2 Y2) 1/3 u'1 x273 when x < 3.5x*
2 1 /T -1 ? /I
Ah = (3F/2 Y ) ' u (3.5x*) ' when x > 3 . 5x* . (22)
With this approach, in a simple way we recognize the observed fact that plume
rise is substantially a function of distance, yet we have a usable "final"
rise formula.
In order to use the formulas based on the "IRATE" model, an estimate of x*
is needed; this requires values of 6 and E for substitution into Equation
20. In Reference 1, B was conservatively estimated to be about unity, on
the basis of observations on the growth rates of puffs and particle clusters
(in order to infer the value of 3 from Equation 19, it was necessary to esti-
mate values of E, as this quantity was not measured in the puff and particle
O _
experiments). It is well known that e = 2.5u* /z in the neutral surface layer,13
but this layer extends only to a height of the order of 10 u*/f in neutral
conditions (u* is the friction velocity, z is the height above the ground,
and f is the Coriolis parameter); in mid-latitudes, this height is of the
order of 10 seconds times the wind speed. However, most plumes rise to heights
above this layer, where E is less dependent on wind speed and height.
In a convective mining layer, such as exists in the lowest few hundred to few
thousand meters on any sunny, non-windy day, the average value of e is about 2
(1/2) gH/(cppT), where H is the heat flux transported upward from the ground;
a fairly strong heat flux, 1 cal/cm2 -min, corresponds to e = 30 cm2 / sec3 ;
the eddy dissipation rate is relatively constant with height, except near
the ground, where the neutral surface layer expression dominates. Less is
known about the variation of E above the surface layer in neutral conditions.
69
-------
Effective Stack Height/Plume Rise
If e ceased to decrease with height at the top of this layer, it; would be
proportional to fu*2, which is approximately proportional to fu2. In Refer-
ence 1, measurements of E at heights from 15 to 1200m were shown to fit e<*u
slightly better than e<*u2 or e = constant (measurements made in stable or
convective conditions were excluded from this analysis). This result was
especially convenient for application to the IRATE plume rise model, as e«u
cancels out the wind speed in Equation 20; above the neutral surface layer,
x* is approximately independent of u. However, the great variation in E at
these heights, of the order of ±50%, can be expected to account for consider-
able variations in the plume rise.
In the above analysis, definite decrease of E values with height above the
ground was also noted. Above a height somewhere between 100 and 300m the
variation becomes much less. The best fit in the 15 to 300m range was given
by
e = 0.068 (m/sec)2 u/z . (23)
Substituted in Equation 20, this gives
x* = 2.16 m(F/m4/sec3) 2/5 (z/m) 3/5 . (24)
In References 1 and 2, it was suggested that conservative values of x* and
Ah would result by_ evaluating E at the source height, hs . Thus, z = hs ,
but no more than z = 300m, was substituted in Equation 24; for the present,
let us designate this estimation of x* by x* . A simpler estimate of x* is
suggested by a plot of total plume height, h1 + Ah, versus buoyancy flux.
Using TVA8, _CERL1,4'15 Bringfelt'6 and other observations, a quite conservative
value for z is given by
—
z = 22 m (F/m /sec ) ' < 100 m . (25)
The 100m maximum value of z may be overly conservative when the stack height
itself is greater than 100m, but consider also the fact that e does not diminish
very much with height above this elevation. The predicted final value for Ah
only depends on z °'4 , and the scatter in the few plume rise data at large
distances renders tentative any conclusion about the best evaluation of x* for
these purposes. Equations 23 and 25 give a particularly simple way to evaluate
x*, and can even be applied to ground sources without difficulty. Let us
designate this estimate of x* by x *:
X2* = 14m (F/m4/sec3) 5/8 when F < 55 m4/sec3
(26)
x2* = 34m (F/m4/sec3) 2/5 when F > 55 m4/sec3
70
-------
Some Recent Analyses of Plume Rise Observations (G)
A number of plume rise formulas for buoyant plumes in neutral conditions were
compared with all available observations in Reference 1; TVA8 and CERL14'15 data
each comprised about one-third of the data analyzed. The more recent obser-
vations by Bringfelt3-16 were added to the comparisons in "Reference 2. A simi-
lar analysis is summarized in Table 2. Since the relationship Ah<*u~ seemed
well verified1 for neutral conditions, the average value of uAh for each source
was calculated at the greatest distance downwind that was represented by at
least three 30-120 min periods of observations with at least five Ah deter-
minations each. The Bringfelt data had to be handled differently, because
there was only one period of observation for many of the sources; accord-
ingly, these are weighted only one-third as much as the "Reference 1" obser-
vations in the last column of Table 2. When appropriate measurements were
available, it was required that x < 2x', in order to exclude cases of sta-
bility-limited plume rise. Cases of probable downwash, terrain effects,
etc. were eliminated in "select'1 set of observations in Reference 1, and
agreement with almost every formula improved. Of 25 periods chosen by
Bringfelt for analysis,3 only 12 are selected here (periods 7, 8, lib, 15,
18, 27a, 27b, 31, 29, 41, A7a, and 27b); the periods rejected greatly in-
crease the scatter in the ratios of calculated to observed values for every
formula tested, tending to obscure the comparison. Table 2 shows the median
value of the ratio of calculated to observed plume rises, and the average
percentage deviation of these ratios from the median, for eight different
formulas of the Ah= u~' type. The last column combines the two sets of
select data, with appropriate weighting.
Of the first three formulas, which are empirical, it is seen that the Moses
and Carson18formula in which Ah ^Q,,''2 gives the most consistent fit to the
select data; the fit would be optimized by multiplying by a correction factor
of 2. The next three formulas are based on the Priestley10 model, the first
being the asymptotic prediction of the first stage19 that Ah ^QR
u x
This formula and the "2/3 law", Equation 11, are very similar, and neither
predict any "final" rise; yet, both of these formulas give better agreement
than the empirical formulas. The scatter in the Bringfelt data makes it
difficult to conclude that any one of the six formulas is superior, as about
± 15% seems to be the lowest possible scatter. The differences between the
last five formulas are also slight in the Reference 1 data, except in the
select set. In this set, as well as in the weighted select data, it is seen
that Equation 22 gives the best fit; this equation is simply the "2/3 law",
Equation 11, terminated at a distance x = 3.5x*. The second estimate for
x* x *, as given by Equation 26, seems to have a slight edge over x* = x *;
the amount of scatter and the scarcity of data at large values of x/x*
makes this comparison of x* = x * and x* = x* inclusive. Which value of x*
to be preferred is mostly just a matter of convenience.
When it is not clear whether plume rise is turbulence-limited or stability-
limited an analysis2 of the IRATE model with both factors included shows
that Equation 18 or Equation 22, whichever gives the lowest rise in a given
71
-------
Table 2.
Ratios of Calculated to Observed Plume Rises in Neutral Conditions
Formula
Holland (9)
Stumke (17)
Moses and Carson (18)
Priestley (10, 19)
Lucas, et al. (14)
Lucas (20)
Equation 11 *
Equation 22 * (x* = x *)
Equation 22 * (x* = x *)
Reference 1
(all data)
0.44 ± 37%
0.79 ± 27%
0.54 ± 34%
1.44 ± 26%
1.36 ± 21%
1.18 ± 20%
1.17 ± 23%
1.12 ± 21%
1.12 ± 17%
Reference 1
(select)
0.47 ± 26%
0.72 ± 24%
0.48 ± 19%
1.41 ± 18%
1.24 ± 22%
1.16 ± 14%
1.17 ± 12%
1.13 ± 08%
1.13 ± 06%
Bringfelt
(select)
0.26 ± 32%
0.82 ± 35%
0.53 ± 28%
1.42 ± 15%
1.36 ± 13%
1.12 ± 17%
1.08 ± 13%
0.99 ± 15%
1.00 ± 13%
Weighted,
Select Data
0.40 ± 35%
0.74 ± 29%
0.48 ± 23%
1.41 ± 17%
1.35 ± 19%
1.16 ± 15%
1.17 ± 13%
1.11 ± 10%
1.11 ± 09%
60
•H
Q)
o
OJ
M-l
* with y = 0.5
-------
Some Recent Analyses of Plume Rise Observations
situation, offers a good approximation of the very complicated prediction
that results when both e and s are greater than zero. In unstable conditions,
there is no strong evidence that the average plume rise differs much from its
value at the same wind speed in neutral conditions, but the rise is much more
variable.1'2
Summary and Conclusions
Direct analysis of plume rise observations and several comparisons of obser-
vations with a number of empirical and theoretical formulas have shown that
very satisfactory predictions of plume rise are given by a rather spare phy-
sical-mathematical model. This model was briefly outlined here and is more
rigorously developed in Reference 2; it basically consists of the assumptions
that momentum, buoyancy, and potential temperature are conserved, that the
horizontal component of motion of plume elements is essentially equal to the
mean wind speed, u, and that r = yz in a first stage of rise and dr/dt =
Be r in a second stage of rise (r is the characteristic plume radius,
z is the rise of the plume centerline above the source height, e is the eddy
dissipation rate of ambient atmospheric turbulence, and y and 0 are dimen-
sionless constants). Empirical guidance is used in evaluating y, 6, and e.
The assumptions that r = yz and that momentum is conserved in a non-buoyant
plume (jet) in unstratified surroundings lead to a simple "1/3 law" of rise
that fits a large variety of observed jet center lines:
/ R \2/3 1/3
Ah/D = 1.89( j (x/D) /J , (6)
^ 1+3R
where x is the distance downwind, D is the stack diameter, and R is the ratio
of efflux velocity to wind speed. To derive Equation 6, one must assume that
y = 1/3 + R . On the other hand, there is no evidence that y is a function
of R for buoyant plumes. The assumptions that buoyancy is conserved and that
the initial plume momentum is negligible for a very buoyant plume in unstratified
surroundings lead to the often-cited "2/3 law" of rise:
Ah = 1.6F1/3 u-1 x 2/3 , (11)
where F is the initial buoyancy flux divided by irp; complete expressions for
F are given by Equations 8 and 9. The constant in Equation 11 is based on the
best fit to data shown in Table 2, and corresponds to y = 0.6. Only equations
that include the second stage entrainment assumption that dr/dt = Be r
give a better fit to observations of the rise of hot plumes in near-neutral
conditions. For plumes in which both momentum and buoyancy are significant,
Equation 13 gives a semi-empirical transition between Equations 6 and 11.
73
-------
Effective Stack Height/Plume Rise
Buoyancy becomes the dominant factor for most hot plumes at a distance down-
wind of the order of five seconds times the wind speed.
The assumption that the potential temperature of entrained air is conserved
leads to the prediction that a buoyant plume attains a maximum rise at a
—1/2
distance x = ITUS in stable air (s = (g/T)39/3z, g is gravitational
acceleration, T is the absolute ambient temperature, 6 is the ambient
potential temperature, and 38/3z = 3T/3z + 1° C/100m) . This prediction is
very well confirmed by plots of plume rise versus distance in stable con-
ditions (38/3z and u are averaged from the top of the stack to the top of
the plume). These plots also indicate that the "2/3 law" is approximated
-1/2
when x < 2us and that the plume centerlines level off at a height
Ah = 2.9 (-) (18)
This corresponds to the maximum rise given by r = jz with y = 0.5. Equation
18 gives substantially better agreement with observations than other formulas
tested for buoyant, stability-limited rise. It should be noted that in very
light winds the well-proven'-2 formula of Morton, Taylor, and Turner best
applies if it gives a lower plume rise than Equation 18:
Ah = 5.0 F1M a'378 <27>
In neutral conditions, a limited rise results only after the second stage en-
entrainment assumption is utilized. A good approximation to the complete
prediction for buoyant plumes in neutral conditions is given by
1/3 -1 2/3
Ah = 1.6 F ' u x ' when x < 3.5x*
(22)
1/3-1 9/1
Ah = 1.6 F /J u (3.5x*)Z/J when x > 3.5x* ,
where x* is the distance of transition from the first stage to the second
stage of rise. This equation gives a somewhat better fit to observations
than any other formula tested when x* is estimated by:
x* = 14m (F/mA/sec3) 5/8 when F < 55 m4/sec3
(26)
x* = 34m (F/m /sec ) when F > 55 m
74
-------
Some Recent Analyses of Plume Rise Observations (G)
This equation for x* should be considered tentative, since it is based on
limited empirical determinations of 6 and e, and there is too much scatter
in the few observed plume rises at large values of x/x* to make any strong
conclusions about x*. Equations 22 and 26 apply satisfactorily to the mean
rise in unstable conditions as well, and also in slightly stable conditions
if they give a lower rise than Equation 18.
REFERENCES
1. Briggs, G.A., Plume Rise, AEC Critical Review Series, TID-25075 (1969)
2. Briggs, G.A., "A Simple Model for Bent-Over Plume Rise", Doctoral
Dissertation, The Pennsylvania State University (1970).
3. Bringfelt, B., "A Study of Buoyant Chimney Plumes in Neutral and
Stable Atmospheres", Atmos. Environ. 3: 609-623 (1969).
4. Scorer, R.S., Natural Aerodynamics, pp. 143-217, Pergamon Press, Inc.,
New York (1958).
5. Gifford, F.A., "The Rise of Strongly Radioactive Plumes", J. Appl.
Meteorol. 6:644-649 (1967).
6. Fay, J.A., Escudier, M. and Hoult, D.P., "A Correlation of Field
Observations of Plume Rise", Fluid Mechanics Lab. Pub. No. 69-4,
Massachusetts Institute of Technology (1969).
7. Fan, L., "Turbulent Buoyant Jets into Stratified or Flowing Ambient
Fluids", Califoria Institute of Technology, Report KH-R-15 (1967).
8. Carpenter, S.B. et al.,"Report on a Full Scale Study of Plume Rise
at Large Electric Generating Stations", Paper 67-82, 60th Annual
Meeting of the Air Pollution Control Association, Cleveland, Ohio
(1967).
9. U.S. Weather Bureau, "A Meteorological Survey of the Oak Ridge Area:
Final Report Covering the Period 1948-1952", USAEC Report ORO-99,
pp. 554-559 (1953).
10. Priestley, C.H.B., "A Working Theory of the Bent-Over Plume of Hot
Gas", Quart. J. Roy. Meteorol. Soc., 82:165-176 (1956).
11. Bosanquet, C.H., "The Rise of a Hot Waste Gas Plume'1, J. Inst. Fuel,
30:322-328 (1957).
12. Schmidt, F.H., "On the Rise of Hot Plumes in the Atmosphere", Int.
J. Air Water Pollut., 9:175-198 (1965).
75
-------
Effective Stack Height/Plume Rise
13. Lumley, J.L. and Panofsky, H.A., The Structure of Atmospheric
Turbulence, pp. 3-5, John Wiley & Sons, Inc., New York (1964).
14. Lucas, D.H., Moore, D.J. and Spurr, G., "The Rise of Hot Plumes
from Chimneys" Int. J. Air Water Pollu., 7:473-500 (1963).
15. Hamilton, P.M., "Plume Height Measurements at Two Power Stations",
Atmos. Environ., 1:379-387 (1967).
16. Bringfelt, B., "Plume Rise Measurements at Industrial Chimneys",
Atmos. Environ., 2:575-598 (1968).
17. Stumke, H., "Suggestions for an Empirical Formula for Chimney
Evaluation", Staub, 23:549-556 (1963); translated in USAEC Report
ORNL-tr-997, Oak Ridge National Laboratory.
18. Moses, H. and Carson, J.E., "Stack Design Parameters Influencing
Plume Rise", Paper 67-84, 60th Annual Meeting of the Air Pollution
Control Association, Cleveland, Ohio (1967).
19. Csanady, G.T., "Some Observations on Smoke Plumes", Int. J. Air
Water Pollut., 4:47-51 (1961).
20. Lucas, D.H. "Application and Evaluation of Results of the Tilbury
Plume Rise and Dispersion Experiment", Atmos. Environ., 1:421-424
(1967).
21. Morton, B.R., Taylor, G.I., and Turner, J.S., "Turbulent Gravita-
tional Convection from Maintained and Instantaneous Sources, Proc.
Roy. Soc. (London), Ser. A., 234:1-23 (1956).
76
-------
H
ESTIMATION OF PLUME RISE
D. Bruce Turner *
The Environmental Application Branch has used the equations of ,
Dr. Gary Briggs for a number of years to estimate plume rise.
Dr. Briggs has revised the equation several times.(1,2,3)
The following procedures are consistent with the way in which the Meteorology
Laboratory calculates Briggs1 plume rise:
The following symbols are used:
IT A Constant - 3.14
g Graviational Acceleration = 9.80 m sec
T Ambient Air Temperature, ^K
u Average wind speed at stack level, m sec
v Stack gas exit velocity, m sec
d Top inside stack diameter, m
T Stack gas exit temperature, K
S 3-1
V Stack gas volume flow, m sec
4-3
F Buoyancy flux parameter, m sec
x Distance at which atmospheric turbulence begins to dominate
entrainment, m
AH Plume rise above stack top, m.
x Downwind distance from the source, m.
x,- Distance downwind to final rise, m.
38/3z Vertical potential temperature gradient of atmosphere, K m
s Restoring acceleration per unit vertical displacement for
adiabatic motion in the atmosphere - a stability parameter,
O
sec-
If T is not given, we have been using:
T = 293°K (20°C) for design calculations
V = —^ vsd~ = 0.785 v,d
f
-T \ / T -T \ ,„.
s \ (2)
*D. Bruce Turner, Chief, Environmental Applications Branch
Meteorology Laboratory, Environmental Protection Agency
May, 1973
-------
Effective Stack Height/Plume Rise
unstable or neutral conditions:
x" = 14 F5//8 For F less than 55 (3)
A ? / S
x = 34 F For F greater than or equal to 55 (4)
The distance of the final rise is: x = 3.5 x* (5)
The final plume rise:
1.6 F (3.5 x--) ,,,
AH = (6)
u
For x less than the distance of final rise:
n , ,-,1/3 2/3
1.6 F x . .
AH = ( '
u
For stable conditions, 36/3z is needed
If 86/3z is not given use:
0.02 °K m l for stability E
0.035 °K m'1 for stability F
= 9.806
Calculate
AH = 2.4 /jM 1/3
\us
(9)
78
-------
Estimation of Plume Rise (H)
and
1/4
AU _ 5 F (plume rise for calm conditions) (10)
s 3/8
Use the smaller of these two AH's
This is the final rise.
The distance to final rise is:
= 3.14 u
Xf 1/2 (11)
If you want to calculate rise for a downwind distance x less than
x , this is given by
AH - 1-6 F x 2/3 (12)
which is the same equation used for unstable and neutral
conditions.
Although (under stable conditions) the plume begins to rise
according to the 2/3 power with distance, it does not continue the
same rate of rise to the distance of final rise, xc, given by
equation (11). Therefore equation (12) will give a iH higher
than the final rise at distances beyond about 2/3 xf. It is
therefore recommended that when using equation (12), the result
be compared with the final rise and the smaller value used.
In effect then, for determining the plume rise at a distance,
x, (during stable conditions) the minimum value of the three
values of AH determined by equations (9), (10) and (12) should
be used.
Problem set three (component I) follows this article. An Air Pollution
Training Institute certificate will be awarded upon satisfactory completion
of the three problem sets and the return of a completed critique. All
calculations must be returned to the Air Pollution Training Institute. A
set of answer sheets will be returned to the learner.
79
-------
Effective Stack Height/Plume Rise
REFERENCES
1. Briggs, Gary A., 1969: Plume Rise.USAEC Critical Review
Series TID-25075, National Technical Information Service,
Springfield, Va. 22151
2. Briggs, Gary A., 1971: Some Recent Analyses of Plume Rise
Observation pp. 1029-1032, in Proceedings of the Second
International Clean Air Congress, edited by H. M. England and
W. T. Berry. Academic Press, New York.
3. Briggs, Gary A., 1972: Discussion on Chimney Plumes in Neutral
and Stable Surroundings. Atmos. Environ. 6, 507-510 (Jul 72)
80
-------
Appendix
CRITIQUE FORM
EFFECTIVE STACK HEIGHT SCRIPT
87
-------
EFFECTIVE STACK HEIGHT
A cued script of James L. Dicke's audio tape in Component A.
The topic I am going to discuss is effective stack height and its
calculation. Before proceeding any further, we should define exactly
what is meant by effective stack height. 2Effective stack height is the
height at which the plume centerline from a smoke stack becomes essentially
level. To make sure that everyone understands, let's go through that again.
The effective stack height is the height above ground at which the plume
centerline becomes essentially level. 3^^ j_s true whether we are talk-
ing about the plume from a large stack or ^the emission from a small pipe
or vent near rooftop, with an effective stack height not much above the
top of the building.
^Effective stack height, which we will denote by capital H, is equal
to the physical height of the stack, small h, plus the rise of the plume
above the stack, delta h. &In this diagram, the effective stack height is
reached near the right end of the slide, where the centerline of the plume
is leveling out. When an effective stack height is specified, a virtual
origin for the plume has been assumed. It is assumed that the plume is
not emitted from the stack, but from some point above and possibly upwind
of the stack. This virtual origin is such that a plume with dimensions
similar to those of the real plume results at the distance where the
effective stack height is reached. 7Here you can see the virtual origin
is slightly behind the stack. It is at height H, which is equal to the
physical stack height, small h, plus the plume rise, delta h.
®Next, we should answer the question, "Why do we need to know the
effective stack height?" Primarily, the effective stack height is required
because of its effect on the ground-level concentration of contaminants.
Its practical use in the diffusion equation can be demonstrated to you in
other sessions. As the effective stack height increases, the concentration
at ground level becomes less due to the fact that there is more atmosphere
in which to dilute the pollutants. Also, as effective stack height increases,
the distance to the maximum concentration is moved further downstream from
91
-------
the source and the maximum concentration becomes smaller.
9As you can see from the example, if we assume that emissions from
both stacks are equivalent, the concentration at some point downwind from
the lower stack will be greater than that from the taller stack, since the
emissions from the taller stack have a greater volume of the atmosphere in
which to dilute before they significantly affect the ground. Also, the
maximum concentration, due to the lower stack, will be greater and closer
to the source than that for the taller stack, which will have a smaller
maximum concentration with the point of maximum concentration off the slide
some place to the right. ^A practical example is considered in this slide
taken near Brilliant, Ohio. Suppose that all three sets of stacks have
equivalent emissions. It is apparent that the largest concentration will
be caused by the middle plant. The plant on the right will cause the
smallest concentrations since it has the greatest effective stack height.
" Again, effective stack height is equal to the physical stack height
plus plume rise. Usually, the physical stack height can be measured or
is already known. Therefore, we will be mainly concerned with calculating
the plume rise.
stacks have some plume rise, though there are cases where plume
rise is zero or negative. The momentum and buoyancy of gases in a plume
cause the plume to rise, sometimes to several times the physical stack
height. As you can see in this slide, the effective stack height, which
is near the top of the slide, is at least three tines the physical stack
height. '3 With no wind the plume might rise straight up until it
reaches a level where there is some wind, and then level out quite rapidly,
or expending all its buoyancy and momentum, it just remains stationary,
forming a cloud. When there is some wind, the plume will gradually slope
upward and over until it levels out. 14In any case, we are interested
priinarily in the height at which the plume becomes level at some point
downstream from the source.
92
-------
15If there is no buoyancy or momentum, the plume beccrres level almost
iimediately, especially if wind speeds are moderate. As you can see in
this case, the plume does not rise at all after leaving the stack, but
becomes level immediately. 16Here plume rise is zero, and effective stack
height is equal to the physical stack height. ^Haaever, in cases of very
strong winds or inproper stack design, aerodynamic downwash on the lee side
of the stack can cause the plume to lower below the top of the stack. ^In
fact, where there are short stacks, or the effluent is being emitted right
at rooftop, aerodynamic downwash on the lee side of the building is ccmmon.
^Here plume rise can be thought of as being negative and, as a result, the
effective stack height is less than the physical stack height. Another
factor which can contribute to the lowering of a plums below the physical
stack height is evaporative cooling of the moisture droplets in the plume.
We will consider mainly cases where there is some plume rise, but we will
discuss aerodynamic downwash and evaporative cooling briefly near the end
of this session.
20we are interested in cases where there is rise of the plume. 21 Most
of the plume-rise equations are obtained either empirically or through a
theoretical, derivation. The empirical equations are obtained by arranging
the stack and atmospheric parameters so that visual observations or physical
measurements of plume rise will be duplicated by the equation. Theoretical
equations are derived from physical principles including dimensional analysis.
However, both types of equations are dependent on data collected in a labora-
tory, a wind tunnel or in the field, since the theoretical equations usually
include at least one constant which is dependent on sampled data.
All the equations which have been derived to date are subject to certain
criticisms. These criticisms boil down to the fact that there is no one
equation which is universally accurate and reliable. No one equation applies
to all sizes of sources and under all atmospheric conditions. Some equations
give a generally good estimate of plume rise, but only when the equation is
used with the specific stack and under the atmospheric conditions for which
93
-------
the plurte-rise equation was derived. Also, there is a general lack of plume-
rise data on a large variety of sources and atmospheric conditions; this
makes it difficult to cone up with a widely applicable equation. Also, even
when sufficient data are available for obtaining an equation, there is still
the problem of little independent data with which to check the equation.
22in the past 20 years there have been many attempts at expressions for
plume-rise - twenty equations and more. The first two references which you
see give a very good evaluation of plume-rise equations and demonstrate whid:
equations give the most reasonable results. Some additional factors which
affect plume rise are also discussed. The third reference, by Stem, gives
a good review of plume-rise equations to 1968. These references are among
those listed in Part B.
rise is a result of the buoyancy and momentum of the stack
effluent and the manner in which they are affected by the atmosphere.
4ihe individual parameters which affect buoyancy and momentum with regard
to the stack are: the stack diameter at the top of the stack in meters;
stack gas velocity, meters per second; stack gas temperature, degrees
Kelvin; and the heat content of the effluent, frequently expressed as
calories per second. We will consider units only in the MGS system.
5The individual atmospheric parameters which affect the buoyancy and
momentum are: wind speed, meters per second; potential temperature lapse
rate - degrees Kelvin per 100 meters; atmospheric stability - whether
unstable, neutral' or stable; atmospheric temperature - degrees Kelvin; and
atmospheric pressure in millibars. In this session we will assume the mean
molecular weight of the atmosphere and the stack gas are essentially the
same.
26Various combinations of these elements contribute individually to the
momentum and buoyancy. The parameters which contribute to momentum are the
stack gas exit velocity, the stack diameter at the top, and the atmospheric
wind speed. The parameters which contribute to buoyancy are the stack gas
exit temperature, the atmospheric temperature, potential temperature lapse
94
-------
rate, and the heat content of the effluent. Usually, the buoyancy term is
expressed either as a temperature difference between the gas and the atmos-
phere or as the heat content of the effluent.
2'Two of the plume-rise equations which frequently appear are the
Bryant-Davidson and the Holland equations. These are of interest because
they are widely used; they provide simple computation schemes; and they are
conservative, thus providing a safety factor. By conservative, I nean that,
if anything, they underestimate the amount of plume rise; this allows an
over-calculation of the pollutant concentration at the ground. Thus, by
using these equations, a ground level concentration will never be under-
estimated if the source conditions are similar to those for which the
equation was derived. Fran the health and safety standpoint this is the
desirable situation.
The Bryant-Davidson equation has, as you see here, the momentum term
on the left and the buoyancy term on the right. The momentum term is based
on wind tunnel experiments and the buoyancy term was added later to take
buoyancy into consideration. The authors state that this equation should
be applied only to stacks of moderate or great height; however, the stacks
referred to are actually rather small by today's standards. Other problems
with the equation are, there is no allowance for different atmospheric
stabilities, and the equation is based on wind tunnel data rather than on
data collected in the field.
28The Holland equation, which we will go through rather thoroughly, is
presented here. The first portion of the equation on the right is the momen-
tum term, the other, with the temperature difference, is the buoyancy term.
As you can see, there is a correction allowable for unstable and for stable
conditions. The equation, as we see it here, is for neutral stability con-
ditions. To apply it to unstable atmospheric conditions, it is appropriate
to multiply the answer by 1.1 or 1.2; for stable atmospheric conditions,
the effective stack height should be multiplied by 0.9 or 0.8. The data
on which the Holland equation are based are: physical stack heights from
95
-------
approximately 50 to 60 meters, diameters in the range of 2 to 4 meters,
stack gas exit velocities from 2 to 20 raps, exit temperatures from about
350 to 450°K, atmospheric wind speeds from approximately 0.5 to over 9 raps,
and all atmospheric stabilities. The criticisms of this equation, which, I
might mention, apply to many of the equations we have seen, are: it is
empirical; the coefficient 2.68 x 10~ has units which are per meter per
millibar; the data used in obtaining the equation are limited; and the
equation is conservative. The last item, as I stated earlier, is actually
desirable.
Bosanquet equation is the complex expression which you see here.
The terms A, X, and X require separate calculations and the functions f,
and £„ are evaluated from tables. As you might imagine, this equation is
rather difficult and time consuming to use. Although it gives a generally
good approximation of plume rise, the approximations are consistently on
the high side.
30i"he Lucas equation was developed by the Central Electricity Research
Laboratory in Great Britain. It is the first of several equations we will
consider that does not have a momentum term, only a buoyancy term. In these
equations it is assumed that the upward momentum of a plume is negligible
compared to its buoyancy. The Lucas equation, as you can see, is expressed
as the heat content of the effluent divided by wind speed multiplied by an
empirical coefficient, where the coefficient depends on the source being
considered. This coefficient is variable for different source types and also
can vary for sources of the same type. In other words, two power plants may
have different coefficients. The approximations given by this equation are
consistently on the high side; also this equation is not very sensitive to
changes in stack diameter.
3 1 New, having covered the background, we are prepared to discuss some
of the more commonly used plume-rise equations and to look at some of their
desirable and undesirable qualities. The Stumke equation is a supposed
improvement of the Holland equation. The first term on the right hand side
96
-------
of the equation can be considered the momentum term, stack gas exit velocity
times diameter divided by wind speed; and the second term, which expresses
the temperature difference between the gas and the atmosphere, can be thought
of as the buoyancy term. This equation gives a good approximation of plume
rise, but it predicts a bit high for some of the observed data. However, for
a widely applicable equation, it probably gives as good a result as any other
equation.
G3NCAWE equation also has a European background. It was developed
by a group concerned with emissions from oil refineries. The equation gives
reasonably good estimates of plume rise, but it is not recommended for use
with large power plants.
"Briggs has derived an equation from dimensional considerations. The
general equation states that the plume rise is equal to the buoyancy flux,
F, to the 1/3 power, times the downward distance, to the 2/3 power, divided
by the wind speed measured at stack height. The first equation should be
applied out to distances up to ten times the physical stack height. At
greater distances the plume is assumed to have leveled off and if there is
no change in wind speed or stability, the plume centerline will remain at
an essentially constant height. In addition, these two equations should be
applied only to sources emitting at least 20 megawatts of heat or at least
5 million calories per second.
Under stable conditions a stability parameter, s, is introduced, pro-
portional to the potential temperature lapse rate/
If there is no wind, the above equations become meaningless and thus
the last equation should be used.
3^In this slide we can see how to calculate the buoyancy flux and sta-
bility parameter terms. The buoyancy flux is equal to the temperature
difference between the stack gases and the atmosphere divided by the stack
gas temperature times the acceleration of gravity, the stack gas exit velocity
and the square of the stack radius , or it may be approximated by a constant
97
-------
times the heat emission expressed in calories per second. The stability
pararreter 's' is equal to the acceleration of gravity divided by the atmos-
pheric temperature at stack height multiplied by the lapse rate of potential
temperature through the layer in which the plume is dispersing. Anyone who
is responsible for making plume rise estimates should be thoroughly familiar
with Briggs' work.
The last three equations we have considered, the Lucas equation, the
CCNCAWE equation, and the Briggs equation, have been getting quite a bit of
discussion recently. The series of Briggs equations are considered to be
the most up-to-date equations and those most applicable to large power plants,
which are the source of many air pollution problems. However, as I indicated,
they are not all-encompassing, and do have certain deficiencies.
35Qne other thing we should consider with regard to plume-rise equations
is some of the work done by Moses and Carson. They have taken the basic form
of many plume-rise equations with a momentum term to some power times a co-
efficient plus a buoyancy term to some power times a coefficient plus a
constant. They have taken this equation, used much of the aval .Table plume-
rise data, and by using regression techniques, have determined values for
the coefficient. They have done this in three ways for the equation as you
see here, with C,- = 0, and with just the buoyancy term. In most cases the
plume-rise equations which resulted are accurate to within approximately 30
meters. One other item that should be noted about this technique is that
for some of their evaluations they got a negative coefficient for C-, this,
in effect, says that the momentum of a plume detracted from the plume rise,
which may be unrealistic. This indicates the problems involved in deriving
an equation where it is fitted only to describe the data, without proper
consideration of the physical realities.
36Now, let us take a look at the results that several of these equations
give in predicting plume rise. We will consider the Lucas, the Briggs, and
the Holland equations, as applied to heat emission data from power plants.
As you can see, the Lucas equation overpredicts for most of the data. The
98
-------
Briggs equation goes through approximately the middle of the data. The
Holland equation underpredicts for all the data except for sources with very
large heat emissions. Remember, however, the original data used to develop
the equation did not include heat emissions of this magnitude. Therefore,
if the Lucas equation .is used, you would be predicting more plume rise than
actually occurred, the Briggs equation would give a good average, and with
the Holland equation, you would be obtaining an underestimate except for
very large sources. Also, another problem with plume rise data is demon-
strated here. These plume-rise observations are supposedly for the same
stack under similar atmospheric conditions, and you can see the wide range
of plume rises which are obtained.
3'Now, let us consider an example in which we can use the Holland equation
to calculate plume rise. 3oTnese data are from the TVA's Shawnee power plant
near Paducah, Kentucky. The stack gas exit velocity is 14.7 mps, the diameter
of the stack at the top is 4.27 meters, the height of the stack is 77 meters,
its exit temperature is 416 degrees Kelvin. The atmospheric parameters are
wind speed five meters per second, atmospheric temperature 288 degrees Kelvin,
and pressure 1,000 millibars. In evaluating this equation you can see the
computed plume rise is 63 meters. The effective stack height is equal to the
physical stack height plus plume rise. The physical stack height was 77 meters
and the calculated plume rise was 63 meters, giving us an effective stack
height for this plant and these atmospheric conditions of 140 meters. ^^A
similar calculation using the Briggs equation results in a plume rise of 158
meters. This is 2 and 1/2 times the rise we calculated using the Holland
equation and, based on our previous discussion, is about what you might expect.
have discussed various ways to estimate effective stack height
through plume-rise computations. It should be pointed out that there are
adverse effects of meteorology and terrain which can make these plume-rise
calculations unrealistic. The conditions which can adversely affect plume
rise are elevated inversions, irregular terrain, and changing thermal
regimes .
99
-------
41If -there is an inversion based at sane level above the ground where
the temperature of the air starts increasing with height, the inversion
base will act as a lid on the rise of the effluent, not allowing it to
penetrate through the inversion. 42Some actual scenes in the New York City
area when there were data to indicate an elevated inversion based at 1,500
feet, shows the effect of inversions on plume rise. Hare we see a plume
which appears to be rising and gradually leveling out. Then, all of a
sudden, its top becomes level, as if sheared off, or as if somebody put
their hand down on top of it. ^Another good example is this slide where,
in the upper left hand corner, the plume, which is relatively small, can
be seen to be rising gradually, then suddenly flattens out.
44In cases of complex terrain, under reasonably stable flow on the wind-
ward side of a hill, the plume actually rises over the hill instead of im-
pacting into the side of the hill. This is where a plume-rise calculation
would indicate that the plume should impact on the side of the hill where,
in fact, it rises over the hill. On the downwind side, the turbulence
induced by the terrain causes the plume to be downwashed, so an effective
stack height calculation here would be of no use since the plume is actually
lowered by induced turbulence and is not allowed to rise naturally.45jn
cases of complex terrain and highly unstable meteorological conditions, as
you can see here, a calculation of an effective stack height would be of no
use in calculating the impact on the hill to the right, since the plume is
looping and has a much stronger impact on the hill than a calculation using
an effective stack height would indicate. The plume impinges right on the
hill.
46In this case, we have a plume emitted over grassy terrain behind the
hangers. An effective stack height is established then, as the plume moves
out over the concrete runways where the convective turbulence is much
greater, the plume seems to be lifted to a new effective stack height.
47Now we will consider what can be called negative plume rise, consisting
of evaporative cooling of the effluent and aerodynamic downwash. A good
100
-------
example of the negative plume rise caused by evaporative cooling is what
happened to a power plant in Great Britain several years ago. In this
case the effluent was put through a spray tower to absorb gases. The
gases were thus cooled and saturated with water vapor. Contact with the
cold surfaces of the duct work caused further cooling and condensation of
the water vapor. When the effluent was released into the atmosphere, the
condensed water droplets evaporated, withdrawing the latent heat of vapori-
zation from the surrounding plume and air, thus causing the plume to cool
below the atmospheric temperature. The plume thus had negative buoyancy and
the effective stack height was reduced to below the physical stack height.
In this case, the plume actually came right down to the ground. For this
plant the adverse effect caused by evaporative cooling resulted in ground-
level concentrations which were greater than before anything had been done
to the stack effluent.
48 With regard to aerodynamic dcwnwash, eddies which are the result of
mechanical turbulence around a building or low stack can affect the effective
stack height. They cause the plume to be downwashed. This is especially so
when the wind speed is high, momentum is small, and the plume is emitted
horizontally. Most knowledge about this situation has come from wind tunnel
studies such as those conducted by Halitsky at New York University. The
results of these investigations show that maximum downwash around a rectan-
gular building occurs when the wind direction is at a 45° angle to the major
axis of the structure and is a minimum when it is parallel to this axis.
Maximum downwash would occur when the wind blows from one corner of the
building to the opposite corner, and the least downwash would be when the
wind was blowing parallel to the building. ^Also, it has been shown that
effluents from flush openings on rooftops frequently flow in a. direction
opposite to the wind, due to counterflow induced by turbulence along the
roof in the turbulent wake above the building. ->OThe region of disturbed
flow extends up to twice the building height and 5 to 10 times its height
downwind as indicated by the streamlines. 51 TWO rules of thumb which are
101
-------
commonly used to prevent downwash are (1) the stack gas exit velocity should
be 1.5 times the average wind speed to prevent dcwnwash in • the wake of the
stack, and (2) a stack should be 2.5 times the height of the building
adjacent to the-stack, to overcome building turbulence. These items
should be kept in mind whenever one deals with a situation where down-
wash could be taking place, although there is no really good quantitative
way to handle all situations. 52jjere is an example of a power plant which
has short stacks certainly less than two times the height of the adjacent
building, and you can see the inferior stack which is subject to aerodynamic
dcwnwash. ^IXMnwash is not readily apparent here but I have seen the situ-
ation when the effluent came down directly on top of the river, flowed across
the river right above its surface and up the hill on the opposite side.
->T?Vgain, in this instance, to assume an evalated emission from this source
would, be a mistake.
102
-------
-------
Hum©
G
,
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
-------
Available as TID-25075 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 States of America
USAEC Technical Information Center, Oak Ridge, Tennessee
November 1969; latest printing, December 1972
-------
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.
in
-------
CONTENTS
FOREWORD iii
SYNOPSIS 1
1. INTRODUCTION 2
2.BEHAV/OR 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
-------
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 INDEX 80
VI
-------
SYNOPSIS
The mechanism of plume rise and dispersion is described in qualitative 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 "2/3 law," Ah = 1.6F'6 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 2/3 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 A 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.
-------
1
INTRODUCTION
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
Pearson2 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!" Spurr lamented, "The
argument for and against different plume rise formulae can be discussed clinically by
-------
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 % to 1.0, influenced by a number of factors including,
presumedly, the observer's eyesight.j
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 Q 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,15 by Gartrell,1 6 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 8
Several attempts have been made at setting down definite procedures for
calculating diffusion, including the plume rise. The first, primarily concerned with dust
fRef. 5, page 311.
-------
INTRODUCTION
deposition, was by Bosanquet, Carey, and Halton.19 Hawkins and Nonhebel
published a procedure based on a revised formulation for plume rise by Bosanquet.
More recently, Nonhebel2 ' 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 Chimnev Heights,22 which has been summarized by Nonhebel.23 Scorer and
Barrett24 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)27 has prepared a diffusion manual with
another formula for plume rise. The implications of this formula and the CONCAWE
formula are discussed in Ref. 28. Further discussions of plume-rise questions can be
found in Refs. 29 to 33.
fCONCAWli (Conservation of Clean Air and Water, Western Europe), a foundation established
by the Oil Companies' International Study Group for the Conservation of Clean Air and Water.
-------
2
BEHAVIOR
OF SMOKE PLUMES
Flume dispersion is most easily described by discussing separately three aspects of
plume behavior: (1) aerodynamic effects due to the presence of the stack, buildings,
and topographical teatures; (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 b\ mixing with the an, but there is not much agreement on how
important a role atmospheric tuibuleuce plays. It is known that a rising plume spreads
outward from its center line fastei than a passive plume, but this increased diffusion
rate usually icsults in an only negligible decrease of ground concentrations.
The following sections discuss the three aspects ol plume dit fusion. Symbols and
frequently used metcoiological leims are delined in Appendixes B and C.
DOWIMWASH
AND AERODYNAMIC EFFECTS
Oownwash of the plume into the low-pressure region in the wake of a stack can
occui if the efflux velocity is too low. If the stack is too low, the plume can be caught
m the wake of associated buildings, where it will bring high concentrations of effluent
-------
BEHAVIOR OF SMOKE PLUMES
(a] STACK DOWNWASH
(t>) BUILDING DOWNWASH
n.
(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 down wash 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 > l.Su, 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 Tallawarra plant, cited in Table 5.1, where
Fr =
g(AT/T)D
= 0.5
Experiments are still needed to determine quantitatively the effect of the efflux
Froude number on the abatement of downwash, unfortunately, the experiments of
-------
OO\\\\\ASH AND \1 RODYNA.MIC 111 TLCTS
Sherlock and Stalker involved only high values of Fr. and thus buoyancy was not a
significant factor.
Nonhcbel'' recommends that \v0 be at least 20 to 25 ft/sec for small plants (heat
emission less than 10" cal/sec) and that \v0 be in the neighborhood of 50 to oO ft/sec
tor 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. Scorer3"
reports that, when efflux velocity must be low, placing a hori/ontal disk that is about
one stack diameter in breadth about the rim of the chimney top will prevent
down wash.
One of the most enduring rules of thumb for stack design was a recommenda-
tion3'7 made in 1^52 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
velocitx to avoid down wash, 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 I lo\\ past a u pical po\\cr plant.
lower, in high winds the plume will get caught in the downflow and be efficiently
mixed to the ground b\ 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 aii 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, \\here 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.3^ 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 Mcm^rjnJmu su Chimney Heights.22 The correction factor is also
reported b\ Ireland'10 and Nonhebel.23 The behavior of effluents from very short
stacks has been discussed by Barry."*' Culkowski.42 and Davies and Moore.43 For such
sources plume rise is probably negligible.
-------
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.I.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, 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 air 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.
-------
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 (F), 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, 8, 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.
-------
10 BEHAVIOR OF SMOKE PLUMES
If the ambient air is stable, i.e., if 5d/8z> 0, the buoyancy of the plume decays as
it rises since the plume entrains air from below and carries it upward into regions of
warmer 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 50/5z = 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 50/5z < 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
-------
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
-------
12
BEHAVIOR OF SMOKE PLUMES
\/ACTUAL TEMPERATURE PROFILE
,ADIABATIC LAPSE RATE
(a) FANNING
(b) FUMIGATION
(c) CONING
(d) LOOPING
TEMPERATURE—"- (e) LOFTING
Fig. 2.5 Effect of temperature profile on plume rise and diffusion.
-------
DIH'USION 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, convcctive 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
convectivc 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
2Q Q
= 0.164— (2.2)
ay Treuh 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, hs, and the plume rise, Ah); aijay 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
Hq. 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
"Basic Theory Simplified" in Chapter 4.
-------
14 BEHAVIOR OF SMOKE PLUMES
Q/X, the effective ventilation, which has the dimensions of volumetric flow rate (/ /t).
For the case just described,
-=6.1uh2 =6.1u(hs +Ah)2 (2.3)
A.
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 u 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, Xmax-
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 Little 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 Carolinas and Georgia.5: Nevertheless, there is
only one outstanding case of fumigation during stagnation in all the years of
monitoring S02 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 108 cu
ft/sec (4.3 x 106 m3/sec). This value is adequate for a small 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,5l and plumes from high stacks can often penetrate them. Such
penetration can be predicted by equations presented in later chapters.
-------
DIFIUSION 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
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.52
-------
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 =17 (3.1)
where v is a characteristic velocity, I 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
-------
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
Schmidt5 s first investigated the heated plume with zero wind. Yih5 6 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 '/4-in. 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 Leonard5 9 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 21/,. Vadot50 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 associates6 ' has been used as the basis for a momentum
contribution to plume rise by several investigators. Callaghan and Rugged62 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 Baines63 measured rise for only
four stack diameters downwind and obtained some velocity and turbulence intensity
measurements within the jets. Halitsky64 and Patrick65 summarized the work of
-------
18 OBSERVATIONS OF PLUME RISE
previous investigators. In addition, Patrick presented new measurements to about
20 stack 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, Vadot'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,6 9 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 most 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 paper19 of 1950. The center lines of 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
-------
ATMOSPHERIC STUDIES 19
measure of the gradient higher up. Stewart. Gale, and Crooks74'73 published a survey
ot plume rise and ditfusion 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 (4 J 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 t>0 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 Strom7 7 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 t>0 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
3b runs, the plume appeared to level off owing to stable conditions. These data tend to
be dominated b\ momentum rise.
Danovich and Zeyger'8 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,'9 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.50 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 1000 ft, although a few could be determined out to 3000 ft. 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
-------
20 OBSERVATIONS OF PLUME RISE
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 l and some of the
preliminary data have been reported by Hogstrom.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 observations8 7 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.5'88 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 1 The measurements were unique in that the plumes
were traced a long distance downwind by injecting balloons into the base of the
chimney.92 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 lidar 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.
-------
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;1 °2 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 by Simon and Proudfit.103
The plumes were located with a fast-response S02 analyzer borne by helicopter, and
temperature profiles were also obtained by helicopter.
-------
4
FORMULAS
FOR CALCULAT/NG
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 associates6 ' 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
Ah = 1 .5 D + 4.4 X 10- H (4
u [ cal/sec l
22
-------
EMPIRICAL FORMULAS 23
The dimensions of constants are given in brackets. Thomas1 5 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 Stumke104 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) docs 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 ( — ) D+ 5.0— (4.3)
V 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 by
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 and the two stacks reported by Rauch,80 Stumke108
derived the formula
AH = 1.5 > D+118 o l+1 (4.4)
\ u / [ sec
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
°" (4.5)
u v '
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
-------
24 FORMULAS FOR CALCULATING PLUME RISE
plants, and further variations have been observed at other plants.93 Lucas noted
some correlation with stack height and suggested a modification of Eq. 4.5:
Ah = (134 + 0.3 [ft'1 ] h.)
v s
Recently a CONCAWE working group2 5 '2 6 developed a regression formula based
on the assumption that plume rise depends mainly on some power of heat emission
and some power of wind speed. The least-squares fit to the logarithms of the
calculated-to-observed plume-rise ratios was
[ft-(ft/sec)3'4l Q$
Ah =1.40 — H -J- (4-7)
L (cal/sec) * J u/4
Data from eight stacks were used, but over 75% 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 QH and of u, and therefore it is difficult to establish any power-law relation
with confidence.
Even more recently Moses and Carson11 ° 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
Holland,6 Berlyand and his associates,106 and Stumke108 was included, but the
optimum value of the constant turned out to be very small. The least-squares fit was
given by
•rir- -
|_ (cal/sec) <* J u v
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 =1.5 -^
(4.9)
-------
THEORETICAL TORMULAS 25
the point at which the plume became substantially horizontal, i.e., when its inclination
was only 5 to S1"'
Subsequent investigators have all given empirical relations for the jet center line as
a (unction 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 EMPIRICAL RESULTS FOR JET
CENTER LINES AS A FUNCTION OF DOWNWIND DISTANCE
Range of
Investigator R = (.WQ/U)
Eq. 4.33
Rupp et al.61 2 to 31
Callanhan and
n '62
Ruggen
Gordier (b\
Patrick65)
Shanelorov (by
Abramovich11 ') : to 22
Patrick65
Concentration 6 to 45
Velocity S to 54
Maximum
x/D Ah/D
1.44 R°'6V,
47
SI 1.91 R0-61 1.\,
1.31R°'74ix
0.84 R°'7S(v
22 1.00R°-85(.x.
34 1.00R°-S5(.x
Ah/D at 5.7°
Inclination
.•Dl0-33
^Dl0'30
Dl0-37
,D)0.39
D10.34
D)°-3S
^.:RI
>1.5 R1
4.0 R°
3.3 R1
l.S R1
1.9 R1
2.3 R!
.00
.00
.8 7
.1 7
.2 S
.2 9
.37
The early Callaghan and Rugged62 experiments involved heated, supersonic jets in
a very narrow wind tunnel; 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 the 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."5 The formula attributed to Shandorov by Abramovich1 i' was based on
experiments that included various angles of discharge and density ratios. The Patrick""
formulas were based both on the height or maximum concentration ot nitrous oxide
tracer and on the height of maximum velocity as determined by a pilot tube.
THEORETICAL FORMULAS
There are main 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 theon, based on assumptions common to most of the theories.
-------
26 FORMULAS FOR CALCULATING PLUME RISE
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 plume
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)
d0p
= 0 (buoyancy) (4.11)
dt
^2 = ! 0' £ (momentum) (4.12)
where vp = the local velocity of the gas in the plume
pp = the local gas density
6p = the local potential temperature
6'= 9p — 6 = the departure of the potential temperature from the temperature of the
environment at the same height
fc = 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 (0), density (p), and velocity (ve) can be considered constant
over the plane of integration and are assumed to be functions of height only.
Furthermore, if ve 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.5 7
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 times pp and integrate the resulting equation over a horizontal plane,
-------
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. 4.1 2
times pp and integrate the resulting equation over the same horizontal plane. The
plane of integration must completely intersect the plume so that 9'=0 around the
perimeter of the plane. The resulting equations for the net buoyancy flux and
momentum flux in a plume are
(4.13)
where
v = /j_Pp_w dxdV (vertical volume flux) (4.15)
?rp
g 30
s = = ;r— (stability parameter) (4.16)
//(g/T)<9'p w dxdy
Fz= - - - - '(buoyancy flux) (4.17)
•np
(momentum flux) (4.18)
The vertical volume flux of the plume, as defined in Eq. 4.15, is the total vertical
mass flux divided by rrp, 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; v 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 tf 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
v=w0£ (4.19a)
=Fm£ (4.19b)
-------
28 FORMULAS FOR CALCULATING PLUME RISE
and
FZ= (l-^)gWor^F (4.19c)
For a hot source
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
agree 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 by 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 author1 12 who suggested it in 1945. If (V/w)1'4 is defined as a
characteristic plume radius, the rate at which the volume flux grows in a given
increment of height is then 2rr(V/w)^ aw, where a is called the entrainment constant
-------
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,
(4.21)
and
— = 2a(wV)H (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 Spalding114 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
-------
30
FORMULAS FOR CALCULATING PLUME RISE
in a given increment of axial distance is 27r(V/u)^ ?w, where 7 is the entrapment
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 . ,,n
— = 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
1000
Q_
UJ
O
800
600
400
200
0 200 400 600 800 1000 1200 1400
PLUME RISE (ft)
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
-------
THLORET1CAL I-ORMULAS 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 that 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
entrainment, the effective entrainment velocity must be given by /3e'^(V/u) '6 . where 0
is a dimensionless constant; the exponents of the terms result from dimensional
considerations. Since the entrainment velocity in the initial stage of plume rise is 7W,
for the simplest model of a bent-over plume an abrupt transition to an entrainment
velocity of (3e ^ (V/u) 1/6 is assumed to occur when yw = (3e^(V/u)'f
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)1/4, and
the plume center line is given by Eqs. 4.13, 4.21, and
=2 («2w2+72u2)H (4.24)
Before applying models of the vertical 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 = 5.0FV3(! (buoyant, calm) (4.25)
Ah =2A(L\* (buoyant, wind) (4.26)
\us/
-------
32 FORMULAS FOR CALCULATING PLUME RISE
(jet, calm) (4.27)
Ah=1.5 s Get, 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 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 Zj through
which the temperature increases by ATj if
F 6
Zi<7.3F°'4 b"j°-6 (buoyant, calm) (4.29)
(buoyant, wind) (4.30)
Zi<1.6t (FjnY Get, calm) (4.31)
Ui /
where b; = g AT,/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-54 (buoy ant, wind) (4.32)
Ah = 2.3Fmu~ V* (jet, wind) (4.33)
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
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
tEmpirical; numerical value difficult to determine from present model.
-------
THEORETICAL FORMULAS 33
such behavior at a plant west of Toronto in the early morning). Since V ~ U72z2 , the
preceding expression can be integrated and satisfies the initial conditions when
[(72/3)us^j Ah3 = Fm sin (xs'^/u) + Fs~^ [1 — 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 = (n/2) us" ^ and a buoyant plume (Fm = 0)
reaches its maximum height at x = TTUS"^. At much smaller distances the plume center
line is approximated by
Fx \*
)"
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=1.8FV'x*« |+^4+ i(-*) 1+1-^ (4.34)
LJ ^J x j \x / J \ j x /
when x >x*, where x* is the distance at which atmospheric turbulence begins to
dominate entrainment. This distance is given by
x* = 0.43F!503~3ue~1)3/5
Results from puff and cluster diffusion data and from measurements of eddy energy
dissipation rates, given in Appendix A, show that j3 = 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 established11 s that e = 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 cc u* and changes only gradually with height in the neutral surface layer, this
result is similar to those of earlier theories36'46'107 that predict Ah °c F/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
sec
-------
34 FORMULAS FOR CALCULATING PLUME RISE
up to z ^ 1000 ft, then becoming constant with height. If we conservatively
approximate z with the stack height, the resulting estimate for x* becomes
x* = 0.52 p F'*h* (hs<1000ft)
x* = 33 f-^1 F'« (hs > 1 000 ft) (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
Tollmien116 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. Yih5 6 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 Ball117 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 Ball117 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
-------
THEORETICAL FORMULAS 35
unstratified case and give similar results for the final plume height but disagree
somewhat on the values of the numerical constants. Estoque118 further compares
these two theories.
Morton11 9 extended the numerical integrations of the M,T,&T model to the case
of a buoyant plume with nonnegligible initial momentum and concluded that
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. Turner123 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.1 2S 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,7 ] 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 Bosanquct, Carey, and Halton1 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).
-------
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 oc
(F/z)^ . 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 u^ . 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.8 7
The first phase leads to an asymptotic formula identical to Eq. 4.32 times a factor
proportional to (F/x)~'/lz ; namely,
Ah = 2.7 [(jL)M] FV'x3/< (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 xt is the distance of transition to the second stage. It was estimated that x1 =
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 matched 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 that 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 cc p/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 plume
heights.131
-------
I'M! ORMK'AL 1 OKMUL AS 37
A groat variety of work has been done in the last 0 \eais. Lilly70 constiueted a
lumiencal model ol the two-dimensional vortex pair seen in a vertical cross section of a
bent-over buoyant plume. KetTei and Barnes"'1 presented a model for the bonding-over
stage ot a jot with an entrammont assumption similar to the one in the bonding-over
model given in this review except that only the hori/ontal shear was included.
Oanovich and 7e\'ger7h developed a theory along the lines of the Priestle\ theory tor
the first stage but with the second-stage dynamics determined by the diffusion of
buoyancy by atmosphenc turbulence. The typo 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 ot buoyancy relative to the plume axis. Only relative diffusion should be
used. The same criticism applies to a theory developed b\ Schmidt,1'': which is based
on the assumption that the spread of material equals that given by the total diffusion
ot 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 tuibulonco in addition to the ambient turbulence. These
problems were also pointed out by Moore.1
Fquations 4.25 through 4.28 and Fqs. 4.32 and 4.33 were proposed by Briggs107
on the basis of rather elemental}, dimensional analysis as an extension of Batcheloi s4 ^
and Scorer's4'1 approaches. Briggs113 recentK considered in some detail the
penetiation of inversions by plumes of all types by using a model based on the
simplified theory given here. Clifford1'14 extended this type of model to the case of a
bent-over plume whoso total buoyancy flux increases linearly \\ith time as it mo\es
away trom the source, again using the l'a> lor entrainment assumption. Modeling
experiments of Turner1 ''^ with thermals of increasing buov anoy support this
assumption.
A model by Csanady1 3p for the bent-ovei buoyant plume included the effect ot
eddy-oneigy dissipation and of ineitial subrange turbulence in the relative diffusion of
plume buoyancy. In a later paper by Slawson and Csanadv / 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 ineitial 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 moleculai diffusion. This model \ields a radius
pioportional to x1^ and a constant rate of rise in the final stage rathei than an\
limiting height of rise.
Very recentK a model along the lines of the basic theor\ presented here was
developed by Hoult, Fay, and Forney,1'1' in which entrainment velocity depends on
the longitudinal and transverse shear velocities. This theor\ is more elaborate than the
simplified theorv presented hoio. in that ) ma\ be a function of vv0 u and the Fronde
numbei at the stack but does not take into account the effect of atmospheric
turbulence.
-------
5
COMPAR/SONS
OF CALCULATED
AND OBSERVED
PLUME BEHAVIOR
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 Strom1 3 9 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 Spurr9 1 with his own data and found the latter formula,
multiplied by a factor of 0.35, to be a better fit. Stu'mke104 made more extensive
comparisons between 8 different formulas and the data of Bosanquet, Carey, and
Halton,19 of Stewart, Gale, and Crooks,75 and of Rauch.80 By computing the ratios
of calculated to observed rises, Stu'mke 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 Stu'mke,108 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 mad- over a much wider range of
conditions.
38
-------
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. 5.1, 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 <* u~! is represented by a
straight line with a slope of — 1; such lines are indicated for reference.
For most of the sources, Ah ^ u~: is the best elementary relation. It would be
difficult to make a case for Ah °= u^, as appears in the CONCAWE25 '26 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—Bryant1 °5 prediction
that Ah is proportional to u"1 A 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 106 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 mat 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^u^x'4. Rise for a buoyant plume according to the Bosanquet
theory20 and the asymptotic plume rises according to Csanady87 in 1961 and to
Briggs1 °7 in 1965 are shown. They all underestimate rise at large values of x/L.
The Bosanquet20 formulation underestimates plume rise when x/L>103 The
CONCAWE2 5 '2 6 relation mat Ah is proportional to u~'4 and the Davidson-Bryant1 °5
relation that Ah is proportional tou'1'" are not valid for most data sources. Formulas of
the type Ah oc L = F/u3 are difficult to test because they apply only to final rise in
-------
40
CALCULATED AND OBSERVED PLUME BEHAVIOR
35
30
£
20
UJ
CO
a:
LJ
5
5 to
0-
8
6
QH = 1.0 X to4 cal/sec
• x = 30 ft A x = 60 ft
\
•
IN
»
•^
AA
1%
A
'S
N
— • —
I <
It
A
A
L
>4
»-
><
I
t
J-
>_ 5
BALL
i
J
A
H
AA
>t—
F
?
r~ ' "
10 1
1000
800
600
400
200
160
6 2
QH = 1.2 X 107 col/sec
• x = 1000 ft
A x = 3000 ft
^A
^
•
\
*
\
N
\
s
\
0 40 6
LAKEVIEW
200
100
80
60
40
0 1
QH = 1.2 X 106 to
2.9 X 106 cal/sec
• x = 330 ft
A x = 820 ft
_^
* 1
•
(
_\A
^
\
i
2 20 3
DUISBURG
WIND SPEED (ft/sec)
300
UJ
ir
^ 100
3 80
a.
60
50
1
QH = 1.1 X 106 cal/sec
• x = 1260 to 2t50 ft
s
• \
•
.
*\
1
2 20 30 40
HARWELL
200
100
80
30
1
OH = 1.6 X 107 cal/sec
• x = 200 ft
A x = 600 ft
N
^p
\
\»
A\
\
A^
•
\
\
k^
^
1200
1000
800
400
200
• WIDOWS CREEK,
QH = 1.7 X 107 cal/sec
A JOHNSONVILLE, QH = 1.1 X 107
cal/sec X 2 STACKS
N
.
•
^
s
-9
~
10
A
^S
0(
\ A
\w
\^
^\
3ft 4
i
\
N
(A
'• —
2 20 30 40 68 10 20
BOSANQUET WIDOWS CREEK AND
JOHNSONVILLE
WIND SPEED (ft/sec)
Fig. 5.1 Plume rise vs. wind speed in near-neutral conditions.
-------
NEUTRAL CONDITIONS
41
900
700
500
^ 300
ce
ZOO
• SHAWNEE, OH = 5.5 X 106
col/sec X 8 STACKS
A COLBERT, QH = 6.7 X 10s
col/sec X 3 STACKS
100
90
1 '
-X
f. -
•
>_
\
0
A
A ^
\
\
\
•
00 ft 5
•S
3ft 67
fl
7 8 9 10 20 30
SHAWNEE AND COLBERT
0H = 1.7 X (O7 cal/sec TOTAL
O x = 1000 ft, 1 STACK
• i = 1000 ft, 2 STACKS
A x = 2500 ft, 1 STACK
800
600
400
200
100
£
0
A
\ A
°\ ^
^\ o o
" (J
\
•)
\
^
o
0
]\
3 10 20 30 40
GALLATIN
WIND SPEED (ft/sec)
• EARLEY, OH = 1.0 X 106 lo 5.1 X 106
col/sec X 2 STACKS, x = 3600 to
6000 ft
1200
1000
800
LU
in
cc 400
UJ
5
0_
200
i?n
0H = 2.1 X I07
O x = 1000 ft,
• x = 1000 ft,
Ax = 3000 ft,
Ax = 3000 ft,
V x = 5000 ft
A \ V
4^
Nj*
s
"\ • A .
i^
col/sec per stock
STACK
2 STACKS
1 STACK
2 STACKS
V
\
A V\(
A A
V A A
^A
,
^*d
^
0
\
^\
NA^
K
• o
0
\
- A-
\
N
O
A
O
2000
1000
800
600
400
POO
T CASTLE DONINGTON, 0H = 0.8 >
107 to 1.6 X 107 cal/sec X 2
STACKS, x =3600 to 6000 ft
A NORTHFLEET, 0H = 0.8 X 107 to
1.2 X 107 col/sec X 2 STACKS,
x = 4000 to 8000 ft
\
1
T \
1
\
V s
\
»
r\ _
V
\
\A
\
V
x
\
^\x
>s
A
10
20 30 40 50 60
PARADISE
10 20 30 40 50 60
EARLEY, CASTLE DONINGTON,
AND NORTHFLEET
WIND SPEED (ft/sec)
-------
42
CALCULATED AND OBSERVED PLUME BEHAVIOR
20,000 h
500
100
200
500 1000 2000
DISTANCE DOWNWIND (ft)
500O
10,000
Fig. 5.2 Plume rise times wind speed vs. downwind distance in near-neutral conditions. The
average heat efflux per stack, in units of 106 cal/sec, and the number of stacks, if more than one,
are 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 Ah oc if1 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 obviously lower than that measured with system B; Bosanquet, no stack heights
indicated and length of runs uncertain; Darmstadt, low efflux velocity and insufficient
-------
1000
500
-------
Table 5.1
COMPARISON OF CALCULATED VALUES WITH OBSERVATIONS FOR NEUTRAL CONDITIONS
Code
B
HA
HB
BO
DS
DB
T
L
E
E
CD
CD
N
N
S
c
J
we
G
G
P
P
Source
Ball*
Harwell A}
Harwell B
Bosanquet^:
Darmstadt:):
Duisburg
Tallawarra:):
LakeviewJ
CEGB plants
Earley
Earley
Castle Donington
Castle Donington
Northfleett
North fleet:f
TVA plants
ShawneeJ
Colbert^
Johnsonville
Widows Creek:f
Gallatin
Gallatin
Paradise
Paradise
Reference
76
75
75
73
80
80
87
88
91
91
91
91
93
93
99
Number
of
stacks
1
1
1
1
1
1
1
2
2
2
2
2
2
8
3
2
1
1
2
1
2
Us,
ft
200
200
246
410
288
493
250
250
425
425
492
492
250
300
400
500
500
500
600
600
D,
ft
11.3
11.3
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
w0,
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
Qn/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
x* •(•
ft
14§
370
370
485H
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
1150
1000
3250
4800
4800
4800
4800
5900
5900
2500
1000
2500
2500
3000
2000
4500
4500
u Ah,
ft- ft/ 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
•("Calculated from Eq. 4.35.
±Not included in selected data.
§ Height chosen for computing x* = 20 ft.
51 Height chosen for computing x* = 250 ft.
-------
Table 5.1 (Continued)
Code
B
HA
HB
BO
DS
DB
T
L
E
L
CD
CD
N
N
S
c
J
we
G
G
P
F
Source
Ball
Harwell A
Harwell B
Bosanquet
Darmstadt
Duisburg
Tallawarra
Lakeview
CEGB plants
Earley
Earley
Castle Donington
Castle Donington
Northflcet
North fleet
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
,, 110
Carson
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%
Stumke108
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
0.79 ± 11%
0.72 ± 247,
Ratio
Holland6
0.04
0.23
0.25
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%
af calculated to
n • a 73,87
Priestley
(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 ± 1 8%
Dbserved values (
Lucas, Moore,
and Spurr91
1.51
1.70
1.59
1.38
1.69
1.62
1.02
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%
jf u Ah
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
1.18 ± 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
.73
.71
.29
.13
.75
.96
1.53
0.77
1.19
1.73
1.10
1.21
1.03
1.11
1.17 ±23%
1 . 1 7 ± 1 2%.
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 ± 19"',,
1.09 ±7%
-------
46 CALCULATED AND OBSERVED PLUME BEHAVIOR
data; Tallawara and Lakeview, much higher rise than comparable sources in Fig. 5.2,
possibly due to lakeshore effect; Widows Creek, down wash, possibly due to a 1000-ft
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 not
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
Carson110 (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 x3/(. 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,91 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 hs, 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,76 the predicted final rise is 12 times the rise measured at 60 ft
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. Clearly
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
this 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 best
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 hs, which is about the minimum distance downwind at which
maximum ground concentration occurs. For the fossil-fuel plants of the Central
-------
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 FHu~'x% (x<10hs)
Ah= 1.8 FV1(10hs)Ss (x>!0hs)
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 prume 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
stack, at least in comparison with Eq. 4.34. Colbert, with three stacks, seems to have
an enhanced rise, but Shawnce, with eight or nine stacks operating, has a lower rise
than would be expected for a single stack. This may be due to down wash, as noted in
the discussion of Fig. 5.1. In summary, die 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 into 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.
-------
48
CALCULATED AND OBSERVED PLUME BEHAVIOR
Jets in Neutral Conditions
Most data for jets in a crosswind 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
10
C£>
a:
UJ
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.
B and C, Bryant and Cowdry
C and R, Callaghan and
Ruggeri62
F.Fan71
J, Jordinson
K and B, Keffer and
63
61
N and C, Norster and Chapman6
P-C, concentration profiles,
Patrick65
P-S, Schlieren photographs,
Patrick65
P-V, velocity profiles,
Patrick65
-------
NEUTRAL CONDITIONS
49
downwind, x = 2D and x = 15D. The two families of curves group together rather well,
considering the variety of experiments and measurement techniques, which include the
photographic center lines by Bryant and Cowdry67 (B and C), the temperature survey
by Norster 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
Schlieren 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 Ruggeri 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
top 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
80
-------
50 CALCULATED AND OBSERVED PLUME 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 entrainment
constant 7 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 ^2 D (5.2)
This is twice the value given by Eq. 4.9, the often-cited formula of Rupp and his
STABLE CONDITIONS
Penetration of 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
ATj. 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 bj, zi;
Fm , and F Then conventional dimensional analysis predicts penetration when
zibP-6F-°-4
-------
STABLE CONDITIONS
51
g uj o
if) -1- 'u_
s oS ° -
5 tt: -°
2 uj
O > "
0
0
7.
.2
, . r
6 h
?
14
b,
-n
R
c
O
A_
o
^-
.
^-
.9
u
i
^1
^**^ o
- / u *V3
m/b;)
0
>-^
^
4(
NONDIMENSIONAL MOMENTUM FLUX
, 0.8 ,--1.2,
Fig. S.b Maximum nondimensional inversion height for penetration by plume vs. nondimensional
momentum flux (based on data from Vadot ).
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
K-0.6
(5.4)
A bent-over buoyant plume rising through neutrally stratified air should penetrate
an inversion at height Z; 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
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 AT,; thus 6' is reduced by AT;. If 6' remains positive, the plume is buoyant and
continues to rise with 6' proportional to z~2 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 6' < AT,.
-------
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 AT; 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 6' 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.11 7
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 rink of
Crawford and Leonard,59 from the experiments of Vehrencamp, Ambrosio, and
Romie100 on the Mojave Desert, and from the observation by Davies1 01'102 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 1963.14° The rate
of thermal emission was estimated to be of the order of 100,000 Mw,141 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's119 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 s^/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.
-------
H
Table 5.2
INVERSION PENETRATION AT THE RAVENSWOOD PLANTt
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
QH,
107 cal/sec
1.97
0.98
1.11
1.13
1.64
1.66
1.77
1.20
1.54
2.13
u,
m/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
°C
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
F
6 ,
°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
m
n
o
o
H
O
z
Penetration
Yes
No
Yes
No
No
No
Yes
Yes
t
No
No
Yes
No
No
fStack height, 155 m.
-------
54
CALCULATED AND OBSERVED PLUME BEHAVIOR
10,000
(000
- 100
(0
O.I
/•
,rff^RA
/'*
MORTON, T
(
0
/
iVFORD AND
(ICE RIN
AYLOR.TUR
TANK)
AVIES (LON
/
A
/
/ VEHRE
/i AMBRC
(MOJAV
LEONARD
<)
NER
3 BEACHlT
NCAMP,
SIO, ROMIE
E DESERT)
y
O.I
10 100
5F'/4s-V8(ft)
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%
Bosanquet20 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
-------
STABLU CONDITIONS
5
in
or
d
2
O
a
z
o
55
PLUME TOP
PLUME CENTER LINE
23456
x/(us-'/2), NONDIMENSIONAL DISTANCE DOWNWIND
l<"ig. 5.8 Rise of buoyant plumes in stable air 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.,
Ah
/F Y*
= 291 — 1
Vus/
(5.7)
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 = 7rus~^ 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 —
(5.8
When two or three slacks were operating at the TVA plants, there was some
evidence of enhanced final rise in stable conditions. The maximum enhancement that
-------
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.
-------
6
CONCLUS/ONS
AND RECOMMENDATIONS
There is no lack of plume -rise formulas in the literature, and selection is complicated
by the fact that no one formula applies 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, all 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
(434',
57
-------
58 CONCLUSIONS AND RECOMMENDATIONS
where
F2/5h|! (hs<1000ft)
(4.35)
F% (hs>1000ft)
Equation 4.35 is the best approximation of x* at present for sources 50 ft 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
which 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 is 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= 1.6 F^u"1 x% (x<10hs) ,
Ah= 1.6 FV1 (10hs)*> (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*, 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)
-------
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.,
zi<4F°-4b^°-6 (calm) (5.4)
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)
at least up to the point that
Ah=3^D (5.2)
as long as w0/u > 4. It can be only tentatively stated that in windless conditions the
jet rises to
Ah = 4 HH
(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 = 1.5 flll 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
-------
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.
-------
A
APPENDIX:
EFFECT OF A7WIOSPHER/C
TURBULENCE
ON PLUME RISE
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 inertial
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 =j3el'rl> (A.I)
where p1 is a dimensionless 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 p1 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! s 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 ol e
along with the amount of variability that should be expected.
Recent estimates of e were made by Hanna,14: who used vertical-velocity spectra
measured in a variety of experiments, and by Pasquill,'43 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,
61
-------
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 relation1^ oc um
by computing the median value of e^> u"m and the average deviation from the median
value for m = 0, V3, 2/3, 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 PasquilPs 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 = V3;
the average percentage deviation from the median is lowest with m = l/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 f 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 e^u^ (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^/u^ decreases with height. With a power law
relation of e^/u^ <* z~", 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) Hanna142'147 confirmed the
relation e* = 1.5 aw A^ 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 Moore1 31 indicate almost no
dependence of CTW on height from about 100 to 4000 ft except for very high wind
speeds (u > 10 m/sec); (3) Busch and Panofsky148 conclude that Xm oc 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 e^/u^ « 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%]uH z~ * (z< 1000 ft)
(A.2)
e1* = 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
-------
EFFECT OF ATMOSPHERIC TURBULENCE
63
Table A.I
ENERGY DISSIPATION VS. WIND SPEED AND HEIGHT
Source
Round Hill
Round Hill
Cedar Hill
Round Hill
Cedar Hill
Boeing
Pasquill
Pasquill
Height,
ft
SO
150
150
300
450
750
1000
4000
Number
of runs
8
11
9
4
6
22
31
10
e*.
ftH/sec
0.636 ± 17%
0.495 ± 11%
0.457 ±20%
0.470 ±11%
0.331 ± 9%t
0.256 + 20%+
0.269 + 38%|
0.172 ±49%
elS/uii,
ft W /sec $4
0.266 ± 14%t
0.177 ± 10%
0.159 ± 18%
0.151 ± 7%t
0.104± 9%f
0.083 ± 24%
0.097 ±44%
0.079 ±4274
eS/u?s,
sec" ^
0.103 + 16%
0.063 ± 874
0.057 ± 16%t
0.049 ± 7%f
0.034 + 11%
0.028 ± 34%
0.042 ± 46%
0.030 + 47%'
ftt/u,
ft-H
0.042 + 18%
0.022 + 10%.
0.020 ±19%.
0.017 ±774
0.010 + 17%
0.009 ±53%,
0.01 8 ±53%
0.01 1 +59%
(fZ/u)H,
ft *s /sec*'
0.98
0.94
0.84
1.01
0.80
0.75
0.97
0.79]:
"[Minimum value of percentage deviation.
±z = 1000 ft.
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 /3e^> 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 m downwind, yielding a lateral standard deviation of
particle distribution (ay). In 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 (az). The values of /3ew 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,
Interpreting the effective radius of a rising plume in terms of ay or az is difficult, but
in this case it was assumed that CTZ = ay and that r = 2^ ay, as is true in the "top hat"
model equivalent to a Gaussian plume in the Morton, Taylor, and Turner5 8 theory.
The last column of Table A.2 shows the value of (3e^ inferred from the diffusion
data divided by the value of e* calculated from Eq. A.2. The values of 0 inferred from
this calculation range from 0.62 to 0.82, a remarkably small range considering the
-------
64 APPENDIX A
Table A.2
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 runs
5
4
6
7
4
z, ft
14 = 0y
13 = Oy
58
164
2500
u, ft/sec
18
30
19
52
16
(3e^, ft % /sec
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 |3 ~ 0.7,
but, considering the small number of data and the indirectness of this analysis, the
more conservative value of/3 = 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 /3 made in Appendix A is not directly
applicable to diffusion problems other than plume rise.
-------
B
APPENDIX:
NOMENCLATURE
Dimensions of each term are given in brackets: / = length, t = time, T = temperature,
m = mass.
bj Inversion parameter = g AT;/T [l/t2 ]
CD Drag coefficient [dimensionless]
D Internal stack diameter [/]
F Buoyancy flux parameter [/4/t3] ; see Eqs. 4.19c and 4.20
Fm Momentum flux parameter [I4 /t2 ] ; see Eq. 4.19b
Fz Vertical flux of buoyant force in plume divided by Tip [I4/t3]; see Eq. 4.17
Fr Froude number = Wo/[g(AT/T)D] [dimensionless]
g Gravitational acceleration [//t2 ]
h Effective stack height = hs + 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 [m/2/t3]
R Ratio of efflux velocity to average windspeed = w0/u [dimensionless]
r Characteristic radius of plume or puff, defined as (V/u) ^2 for a bent-over
plume [/]
r0 Internal stack radius [/]
65
-------
66 APPENDIX B
s Restoring acceleration per unit vertical displacement for adiabatic motion in
atmosphere [t"2]; see Eq. 4.16
T Average absolute temperature of ambient air [r]
Ts Average absolute temperature of gases emitted from stack [r]
AT Temperature excess of stack gases = Ts — T[T]
AT; Temperature difference between top and bottom of an elevated inversion
M
Vertical temperature gradient of atmosphere [r/l]
t Time [t]
u Average wind speed at stack level [l/t]
u* Friction velocity in neutral surface layer [l/t] ; see Ref. 115
V Vertical volume flux of plume divided by TT [/3 /t]; see Eq. 4.15
v Average velocity of plume gases [l/t]; see Eq. 4.18
v" Velocity excess of plume gases = \?p — ve [l/t]
ve Average velocity of ambient air [l/t]
vp Average local velocity of gases in plume [l/t]
w Vertical component of v = k • \f [l/t]
w' Vertical component of Vp = k • vp [l/t]
w0 Efflux speed of gases from stack [/ /t]
x Horizontal distance downwind of stack [/]
x* Distance at which atmospheric turbulence begins to dominate entrainment
[/]; see Eq. 4.34.
y Horizontal distance crosswind of stack [/]
z Vertical distance above stack [I]
z Height above the ground [/]
Z; Height of penetratable elevated inversion above stack [/]
a Entrainment constant for vertical plume [dimensionless] ; see Eq. 4.22
/3 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]
7 Entrainment constant for bent-over plume [dimensionless] ; see Eq. 4.23
e Eddy energy dissipation rate for atmospheric turbulence [/2/t3] ; see
Ref. 115
6 Average potential temperature of ambient air [T]
6' Potential temperature excess of plume gases = 6p — 0 [T]
dp Average potential temperature of gases in plume [T]
Vertical potential temperature gradient of atmosphere [r/l]; 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 ]
az/ay Ratio of vertical dispersion to horizontal dispersion [dimensionless]
X Concentration of a gaseous effluent [m//3 ]
-------
c
APPENDIX:
GLOSSARY OF TERMS
Adiabatic lapse rate The rate at which air lifted adiabatically cools owing to the drop
of pressure with increasing height, 5.4°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 a 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
-------
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.
-------
REFERENCES
1. A. E. Wells, Results of Recent Investigations of the Smelter Smoke Problem, Ind. Eng. Chem.,
9: 640-646 (1917).
2. C. H. Bosanquet and J. L. Pearson, The Spread of Smoke and Gases from Chimneys, Trans.
Faraday Soc., 32: 1249-1264(1936).
3. G. R. Hill, M. D. Thomas, and J. N. Abersold, Effectiveness of High Stacks in Overcoming
Objectionable Concentration of Gases at Ground Level, 9th Annual Meeting, Industrial
Hygiene Foundation of America, Nov. 15-16, 1944, Pittsburg, Pa., Industrial Hygiene
Foundation, Pittsburgh, Pa.
4. Symposium on Plume Behavior,//;?. /. Air Water Pollut., 10: 343-409 (1966).
5. P. R. Slawson and G. T. Csanady, On the Mean Path of Buoyant, Bent-Over Chimney Plumes,
J. FluidMech., 28: 311-322 (1967).
6. U. S. Weather Bureau, A Meteorological Survey of the Oak Ridge Area: Final Report Covering
the Period 1948-1952, USAEC Report ORO-99, pp. 554-559, Weather Bureau, 1953.
7. J. E. Hawkins and G. Nonhcbel, Chimneys and the Dispersal of Smoke, J. Inst. Fuel,
28: 530-545 (1955).
8. O. G. Sutlon,Micrometeorology, McGraw-Hill Book Company, Inc., New York, 1953.
9. F. Pasquill, Atmospheric Diffusion, D. Van Nostrand Company, Inc., Princeton, N. J., 1962.
10. U. S. Weather Bureau, Meteorology and Atomic Energy, USAEC Report AECU-3066, 1955.
11. Environmental Science Services Administration, Meteorology and Atomic Energy —1968,
USAEC Report TID-24190, 1968.
12. G. H. Strom, Atmospheric Dispersion of Stajk Effluents, Air Pollution, Vol. 1, pp. 118-195,
A. C. Stem (Ed.), Academic Press, Inc., New York, 1962.
13. M. E. Smith, Atmospheric Diffusion Formulae and Practical Pollution Problems,/. Air Pollut.
Contr. Ass., 6: 11-13 (1956).
14. M. E. Smith, Reduction of Ambient Air Concentrations of Pollutants by Dispersion from
High Stacks, USAEC Report BNL-10763, Brookhaven National Laboratory, 1966.
15. K W. Thomas, TVA Air Pollution Studies Program, Air Repair, 4: 59-64 (1954).
69
-------
70 REFERENCES
16. F. E. Gartrell, Transport of S02 in the Atmosphere from a Single Source, in Atmospheric
Chemistry of Chlorine and Sulfur Compounds, pp. 63-67, American Geophysical Union
Monograph 3, 1959.
17. F. W. Thomas, S. B. Carpenter, and F. E. Gartrell, Stacks—How High?, /. AirPollut. Contr.
Ass., 13: 198-204 (1963).
18. G. N. Stone and A. J. Clarke, British Experience with Tall Stacks for Air Pollution Control on
Large Fossil-Fueled Power Plants, in Proceedings of the American Power Conference, Vol. 29,
pp. 540-556, Illinois Institute of Technology, Chicago, 111., 1967.
19. C. H. Bosanquet, W. F. Carey, and E. M. Halton, Dust Deposition from Chimney Stacks, Proc.
Inst. Mech. Eng., 162: 355-367 (1950).
20. C. H. Bosanquet, The Rise of a Hot Waste Gas Plume,/. Inst. Fuel, 30: 322-328 (1957).
21. G. Nonhebel, Recommendations on Heights for New Industrial Chimneys (with discussion),/.
Inst. Fuel, 33: 479-511(1960).
22. Ministry of Housing and Local Government, Clean Air Act, 1956: Memorandum on Chimney
Heights, S. O. Code No. 75-115, Her Majesty's Stationery Office, London.
23. G. Nonhebel, British Charts for Heights of New Industrial Chimneys, Int. J. Air Water Pollut.,
10: 183-189 (1966).
24. R. S. Scorer and C. F. Barrett, Gaseous Pollution from Chimneys, Int. J. Air Water Pollut.,
6: 49-63 (1962).
25. CONCAWE, The Calculation of Dispersion from a Stack, Stichting CONCAWE, The Hague,
The Netherlands, 1966.
26. K. G. Brummage, The Calculation of Atmospheric Dispersion from a Stack, Atmos. Environ.,
2: 197-224(1968).
27. M. E. Smith (Ed.), Recommended Guide for the Prediction of the Dispersion of Airborne
Effluents, 1st ed., American Society of Mechanical Engineers, New York, May 1968.
28. G. A. Briggs, CONCAWE Meeting: Discussion of the Comparative Consequences of Different
Plume Rise Formulas, Atmos. Environ., 2: 228-232 (1968).
29. Discussion of Paper on Chimneys and the Dispersal of Smoke (see Ref. 7), /. Inst. Fuel,
29: 140-148(1956).
30. C. H. Bosanquet and A. C. Best, Discussion,/. Inst. Fuel, 30: 333-338 (1957).
31. Symposium on the Dispersion of Chimney Gases, Dec. 7, 1961, Int. /. Air Water Pollut.,
6: 85-100(1962).
32. Round Table on Plume Rise and Atmospheric Dispersion, Atmos. Environ., 2: 193-196 and
2: 225-250(1968).
33. Symposium on Chimney Plume Rise and Dispersion: Discussions, Atmos. Environ.,
1: 425-440(1967).
34. R. H. Sherlock and E. A. Stalker, A Study of Flow Phenomena in the Wake of Smoke Stacks,
Engineering Research Bulletin 29, University of Michigan, Ann Arbor, Mich., 1941.
35. S. Goldstein (Ed.), Modern Developments in Fluid Dynamics, p. 430, Dover Publications,
Inc., New York, 1965.
36. R. S. Scorer, The Behavior of Chimney Plumes, Int. J. Air Pollut., 1: 198-220 (1959).
37. Committee Appointed by the Electricity Commissioners, The Measures Which Have Been
Taken in This Country and in Others to Obviate the Emission of Soot, Ash, Grit and Gritty
Particles from the Chimneys of Electric Power Stations, Her Majesty's Stationery Office,
London, 1932.
38. J. Halitsky, Gas Diffusion Near Buildings, Meteorology and Atomic Energy—1968, USAEC
Report TID-24190, pp. 221-255, Environmental Science Services Administration, 1968.
39. D. H. Lucas, Symposium on the Dispersion of Chimney Gases, Royal Meteorology Society,
Dec. 7, 1961,Int. J. AirWater Pollut., 6: 94-95 (1962).
40. F. E. Ireland, Compliance with the Clean Air Act: Technical Background Leading to the
Ministry's Memorandum on Chimney Heights,/. Inst. Fuel, 36: 272-274 (1963).
-------
REFERENCES 71
41. P. J. Barry, Estimates of Downwind Concentration of Airborne Effluents Discharged in the
Neighborhood of Buildings, Canadian Report AECL-2043, 1964.
42. \V. M. Culkowski, Estimating the Effect of Buildings on Plumes from Short Stacks, Nud.
Safety, 8(3): 257-259(1967).
43. P. O. Davies and D. J. Moore, Experiments on the Behavior of Effluent Emitted from Stacks
at or near the Roof of Tall Reactor Buildings, Int. J. Air Water Pollut., 8: 515-533 (1964).
44. H. St'umke, Consideration of Simplified Terrain Types in the Calculation of the Turbulent
Propagation of Chimney Gases, Staub, 24: 175-182 (1964); translated in USAEC Report
ORNL-tr-981, Oak Ridge National Laboratory.
45. H. Stu'mke, Correction of the Chimney Height Due to an Influence of the Terrain, Staub,
24: 525-528 (1964); translated in USAEC Report ORNL-ti-997, Oak Ridge National
Laboratory.
46. R. S. Scorer, Natural Aerodynamics, pp. 143-217, Pergamon Press, Inc., New York, 1958.
47. R. A. Scriven, On the Breakdown of Chimney Plumes into Discrete Puffs, Int. J. Air Water
Pollut., 10: 419-425 (1966).
48. G. K. Batchelor, Heat Convection and Buoyant Effects in Fluids, Quart. J. Roy. Meteorol.
Soc., 80: 339-358 (1954).
49. R. Serpolay, Penetration d'une couche d'inversion par un cumulus industriel, Int. J. Air Water
Pollut., 8: 151-157 (1964).
50. M. E. Smith and I. A. Singer, An Improved Method for Estimating Concentrations and
Related Phenomena from a Point Source Emission,/. Appl. Meteorol., 5: 631-639 (1966).
51. C. R. Hosier, Climatological Estimates of Diffusion Conditions in the U. S., Nud. Safety,
5(2): 184-192 (1963).
52. E. W. Hewson, E. W. Bierly, and G. C. Gill, Topographic Influences on the Behavior of Stack
Effluents, in Proceedings of the American Power Conference, Vol. 23, pp. 358-370, 1961.
53. H. Schlichting, Boundary Layer Ttieory, pp. 607-613, McGraw-Hill Book Company, Inc., New
York, 1960.
54. S. I. Pai, Fluid Dynamics of Jets, D. Van Nostrand Company, Inc., New York, 1954.
55. W. Schmidt, Turbulente Ausbreitung eines Stromes Erhitzter Luft, Z. Angew. Math. Mech.,
21: 265-278 and 21: 351-363 (1941).
56. C. S. Yih, Free Convection Due to a Point Source of Heat, in Proceedings of First U. S.
National Congress of Applied Mechanics, pp. 941-947, American Society of Mechanical
Engineers, 1951.
57. H. Rouse, C. S. Yih, and H. W. Humphreys, Gravitational Convection from a Boundary
Source, Tellus, 4: 201-210 (1952).
58. B. R. Morton, G. I. Taylor, and J. S. Turner, Turbulent Gravitational Convection from
Maintained and Instantaneous Sources, Proc. Roy. Soc. (London], Ser. A, 234: 1-23 (1956).
59. T. V. Crawford and A. S. Leonard, Observations of Buoyant Plumes in Calm Stably Stratified
Air, /. Appl. Meteorol, 1: 251-256 (1962).
60. L. Vadot, Study of Diffusion of Smoke Plumes into the Atmosphere (in French), Centre
Interprofessionnel Technique d'Etudes de la Pollution Atmospherique, Paris, 1965.
61. A. F. Rupp, S. E. Beall, L. P. Bornwasser, and D. F. Johnson, Dilution of Stack Gases in Cross
Winds, USAEC Report AECD-1811 (CE-1620), Clinton Laboratories, 1948.
62. E. E. Callaghan and R. S. Ruggeri, Investigation of the Penetration of an Ail Jet Directed
Perpendicularly to an Air Stream, Report NACA-TN-1615, National Advisor,' Committee for
Aeronautics, 1948.
63. T. F. Keffer and W. D. Baines, The Round Turbulent Jet in a Crosswind, /. Fluid Mech.,
15: 481-497 (1963).
64. J. Halitsky, A Method for Estimating Concentration in Transverse Jet Plumes, Int. J. Air
Water Pollut., 10: 821-843 (1966).
65. M. A. Patrick, Experimental Investigation of the Mixing and Penetration of a Round
Turbulent Jet Injected Perpendicularly into a Traverse Stream, Trans. Inst. Chem. Eng.
(London), 45: 16-31 (1967).
-------
72 REFERENCES
66. L. W. Bryant, The Effects of Velocity and Temperature of Discharge on the Shape of Smoke
Plumes from a Funnel or Chimney: Experiments in a Wind Tunnel, National Physical
Laboratory, Great Britain, Adm. 66, January 1949.
67. L. W. Bryant and C. F. Cowdrey, The Effects of Velocity and Temperature of Discharge on
the Shape of Smoke Plumes from a Tunnel or Chimney: Experiments in a Wind Tunnel, Proc,
Inst. Mech. Eng. (London), 169: 371-400(1955).
68. D. P. Hoult, J. A. Fay, and L. J. Forney, Turbulent Plume in a Laminar Cross Wind,
Massachusetts Institute of Technology, Department of Mechanical Engineering, 1967.
69. R. S. Richards, Experiment on the Motions of Isolated Cylindrical Thermals through
Unstratified Surroundings, Int. J. Air Water Pollut., 1: 17-34 (1963).
70. D. K. Lilly, Numerical Solutions for the Shape-Preserving Two-Dimensional Thermal
Convection Element, J.Atmos. Scl, 21: 83-98 (1964).
71. L. Fan, Turbulent Buoyant Jets into Stratified or Flowing Ambient Fluids, California
Institute of Technology, Report KH-R-15, 1967.
72. F. T. Bodurtha, Jr., The Behavior of Dense Stack Gases, /. Air Pollut. Contr. Ass.,
11: 431-437 (1961).
73. C. H. B. Priestley, A Working Theory of the Bent-Over Plume of Hot Gas, Quart. J. Roy.
Meteorol. Soc., 82: 165-176 (1956).
74. N. G. Stewart, H. J. Gale, and R. N. Crooks, The Atmospheric Diffusion of Gases Discharged
from the Chimney of the Harwell Pile (BEPO), British Report AERE HP/R-1452, 1954.
75. N. G. Stewart, H. J. Gale, and R. N. Crooks, The Atmospheric Diffusion of Gases Discharged
from the Chimney of the Harwell Reactor BEPO,/«r. /. Air Pollut., 1: 87-102 (1958).
76. F. K. Ball, Some Observations of Bent Plumes, Quart. J. Roy. Meteorol. Soc., 84:61-65
(1958).
77. H. Moses and G. H. Strom, A Comparison of Observed Plume Rises with Values Obtained
from Well-Known Formulas, J. Air Pollut. Contr. Ass., 11: 455-466 (1961).
78. A. M. Danovich and S. G. Zeyger, Determining the Altitude of Rise of a Heated Contaminant
in the Atmosphere, JPRS-28,188, pp. 52-66; translated from Trudy, Leningradskii
Gidrometeorologischeskii Institut, Vypusk 18 (Proc. Leningrad Hydrometeorol. Inst.,
Vol. 18).
79. L. D. Van Vleck and F. W. Boone, Rocket Exhaust Cloud Rise and Size Studies, Hot Volume
Sources, paper presented at 225th National Meeting of the American Meteorology Society,
January 29-31, 1964, Los Angeles, Calif.
80. H. Rauch, Zu'r Schornstein-Uberhohung, Beitr. Phys. Atmos., 37: 132-158 (1964); translated
in USAEC Report ORNL-tr-1209, Oak Ridge National Laboratory.
81. B. Bringfelt, Plume Rise Measurements at Industrial Chimneys, Atmos. Environ., 2: 575-598
(1968).
82. V. Hbgstrom, A Statistical Approach to the Air Pollution Problem of Chimney Emission,
Atmos. Environ., 2: 251-271 (1968).
83. S. Sakuraba, M. Moriguchi, I. Yamazi, and J. Sato, The Field Experiment of Atmospheric
Diffusion at Onahama Area (in Japanese with English figure headings), /. Meteorol. Res.,
19(9): 449-490(1967).
84. Diffusion Subcommittee, Report on Observations and Experiments on Smoke Rise Patterns
During Formation of Inversion Layer (in Japanese), Tech. Lab., Cent. Res. Inst. Elec. Power
Ind., (Japan), Mar. 12, 1965.
85. I. A. Singer, J. A. Frizzola, and M. E. Smith, The Prediction of the Rise of a Hot Cloud from
Field Experiments,/. Air Pollut. Contr. Ass., 14: 455-458 (1964).
86. J. A. Frizzola, I. A. Singer, and M. E. Smith, Measurements of the Rise of Buoyant Clouds,
USAEC Report BNL-10524, Brookhaven National Laboratory, April 1966.
87. G. T. Csanady, Some Observations on Smoke Plumes, Int. J. Air Water Pollut 4-47-51
(1961).
-------
REFERENCES 73
88. P. R. Slawson, Observations of Plume Rise from a Large Industrial Stack, Research Report 1,
USAKC Report NYO-3685-7, University of Waterloo, Department of Mechanical Engineering,
1966.
89. G. T. Csanady (Project Supervisor), Diffusion in Buoyant Chimney Plumes, Annual Report,
1966, USA EC Report NYO-3685-10, University of Waterloo, Department of Mechanical
Engineering, January 1967.
90. G. T. Csanady (Project Supervisor), Research on Buoyant Plumes, Annual Report, 1967,
USAKC Report NYO-3685-13, University of Waterloo, Department of Mechanical Engineer-
ing, January 1968.
91. D. 11. Lucas, D. J. Moore, and G. Spurr, The Rise of Hot Plumes from Chimneys, Int J. Air
Water Pollut., 1: 473-500 (1963).
92. D. H. Lucas, G. Spurr, and !•'. Williams, The Use of Balloons in Atmospheric Pollution
Research, Quart. J. Roy. Meteorol. Soc., 83: 508-516 (1957).
93. P. M. Hamilton, Plume Height Measurements at Two Power Stations, Atnios. Environ.,
I: 379-387 (1967).
94. P. M. Hamilton, The Use of Lidar in Air Pollution Studies, Int. J. Air Water Pollut..
10: 427-434 (1966).
95. P. M. Hamilton, K. W. James, and D. J. Moore, Observations of Power Station Plumes Using a
Pulsed Ruby Laser Rangcfmder, Nature. 210: 723-724 (1966).
96. D. H. Lucas, K. W. James, and I. Davies, The Measurement of Plume Rise and Dispersion at
Tilbury Power Station, ,4tnios. Environ.. I: 353-365 (1967).
97. I1'. E. Gartrell, I'. W. Thomas, and S. B. Carpenter, An Interim Report on Full Scale Study of
Dispersion of Stack Gases,/. A ir Pollut. Con tr. Ass.. 11: 60-65(1961).
98. !•'. E. Gartrell, E. W. Thomas, and S. B. Carpenter, Ful] Scale Study of Dispersion of Stack
Gases, Tennessee Valley Authority, Chattanooga, 1964.
99. S. B. Carpenter et al.. Report on a Full Scale Study of Plume Rise at Large Electric
Generating Stations, Paper 67-82, 60th Annual Meeting of the Air Pollution Control
Association, Cleveland, Ohio, 1967.
100. J.E. Vehrencamp, A. Ambrosio, and F. E. Romie, Convection from Heated Sources in an
Inversion Layer, Report 55-27, University of California, Los Angeles, Department of
Engineering, 1955.
101. R. W. Davies, Large-Scale Diffusion from an Oil Fire, in Advances in Geophysics. Vol. 6,
pp. 413-415, F. N. Frenkiel and P. A. Sheppard (Eds.), Academic Press, Inc., New York,
1959,
102. R. W. Davies, Jet Propulsion Laboratory. Pasadena, Calif., personal communication, 1966.
103. C. Simon and W. Proud fit. Some Observations of Plume Rise and Plume Concentration
Distributed Over N.Y.C., Paper 67-83, 60th Annual Meeting of the Air Pollution Control
Association, Cleveland, Ohio, June 11 — 16, 1967.
104. 11. St'u'mke, Zur Berechnung Der Aufstiegshohe von Rauchfahnen, VDI Forschungsh., 483,
Ausg. B. 27: 38-48, 1961.
105. W. 1-'. Davidson, The Dispersion and Spreading of Gases arid Dusts from Chimneys, in
Transactions of Conference on Industrial Wastes. 14th Annual Meeting of the Industrial
Hvgicnc Foundation of America, pp. 38-55, Industrial Hygiene Foundation, Pittsburgh, Pa.,
1949.
106. M. Ye. Berlyand, Ye. L. Genikhovich, and R. 1. Onikul. On Computing Atmospheric Pollution
by Discharge from the Stacks of Power Plants, in Problems of Atmospheric Diffusion and Air
Pollution. JPRS-28, 343: 1-27 (1964); translated from Tr. Gl. Geofi:. Obser., No. 158.
107. C. A. Briggs, A Plume Rise Model Compared with Observations,/ Air Pollut. Contr. Ass.,
15: 433-438 (1965).
108. H. St'u'mke, Suggestions for an Empirical Formula for Chimney Elevation, Staub, 23: 549-556
(1963); translated in USAEC Report ORNL-tr-977, Oak Ridge National Laboratory.
109. 1). 11. Lucas, Application and Evaluation of Results of the Tilbury Plume Rise and Dispersion
Experiment, Atnws. Environ.. 1: 421-424 (1967).
-------
74 REFERENCES
110. H.Moses and J. E. Carson, Stack Design Parameters Influencing Plume Rise, Paper 67-84,
60th Annual Meeting of the Air Pollution Control Association, Cleveland, Ohio, 1967.
111. G. N. Abramovich, The Theory of Turbulent Jets, The M.I.T. Press, Cambridge, Mass., 1963.
112. G. I. Taylor, Dynamics of a Mass of Hot Gas Rising in the Air, USAEC Report MDDC-919
(LADC-276), Los Alamos Scientific Laboratory, 1945,
113. G. A. Briggs, Penetration of Inversions by Plumes, paper presented at 48th Annual Meeting of
the American Meteorological Society, San Francisco, 1968.
114. F. P. Ricou and D. B. Spalding, Measurements of Entrainments by Axisymmetrical Turbulent
Jets,/. FluidMech., 11: 21-32 (1961).
115. J. L. Lumley and H. A. Panofsky, The Structure of Atmospheric Turbulence, pp. 3-5, John
Wiley & Sons, Inc., New York, 1964.
116. W. Tollmien, Berechnung der Turbulenten Ausbreitungsvorgange, Z. Angew. Math. Mech.,
4: 468-478 (1926).
117. C. H. B. Priestley and F. K. Ball, Continuous Convection from an Isolated Source of Heat,
Quart. J. Roy. Meteorol Soc., 81: 144-157 (1955).
118. M. A. Estoque, Venting of Hot Gases Through Temperature Inversions, Geophysical Research
Directorate Research Note 3, Report AFCRC-TN-58-623 (AD-160756), Air Force Cambridge
Research Center, 1958.
119. B. R. Morton, The Ascent of Turbulent Forced Plumes in a Calm Atmosphere, Int. J. Air
Pollut., 1: 184-197 (1959).
120. B. R. Morton, Buoyant Plumes in a Moist Atmosphere,/ Fluid Mech., 2: 127-144 (1957).
121. M. Hino, Ascent of Smoke in a Calm Inversion Layer of Atmosphere; Effects of Discharge
Velocity and Temperature of Stack Gases, Tech. Lab., Cent. Res. Inst. Elec. Power Ind.
(Japan), Report TH-6201, May 1962.
122. M. Hino, Limit of Smoke Ascent in a Calm Inversion Layer of Atmosphere, Tech. Lab., Cent.
Res, Inst. Elec. Power Ind. Rep. (Japan), (text in Japanese; figure headings in English),
14(1): 9-43 (1963).
123. J. S. Turner, The 'Starting Plume' in Neutral Surroundings, /. Fluid Mech., 13:356-368
(1962).
124. A. Okubo, Fourth Report on the "Rising Plume" Problem in the Sea, USAEC Report
NYO-3109-31, Johns Hopkins University, 1968.
125.1. V. Vasil'chenko, On the Problem of a Steady-State Convection Flow, Tr. Gl. Geofiz.
Observ., No. 93, 1959.
126. J. W. Telford, The Convective Mechanism in Clear Air,/. Atmos. Sci., 23: 652-665 (1966).
127. B. R. Morton, On Telford's Model for Clear Air Convection (with reply), /. Atmos. Sci,
25: 135-139(1968).
128. S. Lee, Axisymmetrical Turbulent Swirling Natural-Convection Plume, /. Appl. Mech.,
33:647-661(1966).
129. O. G. Sutton, The Dispersion of Hot Gases in the Atmosphere, /. Meteorol., 1: 307-312
(1950).
130. R. S. Scorer, The Rise of a Bent-Over Plume, Advances in Geophysics, Vol. 6, pp. 399-411,
F. N. Frenkiel and P. A. Sheppard (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).
-------
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 Air 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 Rise.M/ci Safety, 6(1): 1-19 (1964).
139. H. Moses and G. H. Strom, A Comparison of Observed Plume Rises with Values Obtained
from Weil-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: 440-443 (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, J. Atmos. Sci., 25: 1026-1033
(1968).
148. N. E, Busch and H. A. Panofsky, Recent Spectra of Atmospheric Turbulence, Quart. J. Roy.
Meteorol. Soc., 94: 132-148 (1968).
149. F. N. Frenkiel 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).
-------
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
Bornwasser, 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
Bnngfelt,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. !., 3, 20, 23, 36, 37,
Culkowski.W. M.,7
Danovich, A. M., 19,37
Davidson, W. F., 23, 39
Da vies, I., 20
Davies,P. 0., 7
Davies,R. W.,21,52
77
-------
78
AUTHOR INDEX
Estoque, 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
Frizzola, 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
HOgstro'm, 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
Moses, H., 19,23,24,38,45,46
Nonhebel,G.,4, 7, 18
Norster,E. R.,48,49
Okubo,A.,35
Onikul, 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., 18, 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,!. 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,0. G.,3,36
Taylor, G. I., 17,29, 34, 52,63
-------
AUTHOR INDEX
Telford, J. W., 35 Vasil'chenko, I. 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
-------
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)
Down wash, 5-8, 39
Drag force on plume, 27, 28, 29, 31,
35
Efflux velocity, 5-7,8,35
Entrainment, 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, 12-13
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
-------
SUBJECT INDEX
81
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 of, 19-21
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,
29,35,36,39-42,61-64
measurement of, 19-21
NOTICE
This book was prepared under the sponsorship of the United States Government.
Neither the United States nor the United States Atomic Energy Commission, nor
any of their employees, nor any of their contractors, subcontractors, or their
employees, makes any warranty, express or implied, or assumes any legal liability
or responsibility for the accuracy, completeness or usefulness of any information,
apparatus, product or process disclosed, or represents that its use would not
infringe privately owned rights.
-------
NUCLEAR SAFETY INFORMATION CENTER
Plume Rise was originally prepared for the Nuclear Safety Information Center,
one of the U. S. Atomic Energy Commission's specialized information
analysis centers. Established in 1963 at the Oak Ridge National Labo-
ratory, the Nuclear Safety Information Center serves as a focal point for
the collection, storage, evaluation, and dissemination of nuclear safety
information. The subject coverage, which is comprehensive in the nuclear
safety field, includes such primary subject areas as
General Safely Considerations Fission-Product Transport
Plant Safety Features Reactor Operating Experiences
Consequences of Activity Release Instrumentation, Control,
Accident Analysis and Safety Systems
The Nuclear Safety Information Center publishes periodic staff studies,
bibliographies, and state-of-the-art reports; disseminates selected informa-
tion on a biweekly basis; answers technical inquiries as time is available;
provides counsel and guidance on nuclear safety problems; and cooperates
in the preparation of Nuclear Safety, a bimonthly technical progress review
-sponsored by the AEC.
Services of the Nuclear Safety Information Center are available to govern-
ment agencies, research and educational institutions, and the nuclear
industry. Inquiries are welcomed.
J. R. Buchanan, Assistant Director
Nuclear Safety Information Center
Oak Ridge National Laboratory
-------
AEC
CRITICAL
REVIEW
SERIES
As a continuing series of state-of-the-art studies published by the AEC Office of
Information Services, the AEC Critical Reviews are designed to evaluate the existing
state of knowledge in a specific and limited field of interest, to identify significant
developments, both published and unpublished, and to synthesize new concepts out of
the contributions of many.
SOURCES OF TRITIUM AND ITS
BEHAVIOR UPON RELEASE
TO THE ENVIRONMENT
December 1968 (TID-24635)
D. G.Jacobs
Oak Ridge National Laboratory
REACTOR-NOISE ANALYSIS
IN THE TIME DOMAIN
April 1969 (TID-24512)
Nicola Pacilio
Argonne National Laboratory
and Comitato Nazionale
per I'Energia Nucleare
PLUME RISE
November 1969 (TID-25075)
G. A. Briggs
Environmental Science Services
Administration
ATMOSPHERIC
TRANSPORT PROCESSES
Elmar R. Reiter
Colorado State University
Part 1 : Energy Transfers
and Transformations
December 1969 (TID-24868)
Part 2: Chemical Tracers
January 1971 (TID-25314)
Part 3: Hydrodynamic Tracers
May 1972 (TID-25731)
THE ANALYSIS OF
ELEMENTAL BORON
November 1970 (TID-25190)
Morris W. Lerner
New Brunswick Laboratory
AERODYNAMIC CHARACTERISTICS
OF ATMOSPHERIC BOUNDARY LAYERS
May 1971 (TID-25465)
Erich J. Plate
Argonne National Laboratory
and Karlsruhe University
NUCLEAR-EXPLOSION
SEISMOLOGY
September 1971 (TID-25572)
Howard C. Rodean
Lawrence Livermore Laboratory
BOILING CRISIS AND CRITICAL
HEAT FLUX
August 1972 (TID-25887)
L. S. Tong
Westinghouse Electric Corporation
NEPTUNIUM-237
PRODUCTION AND RECOVERY
October 1972 (TID-25955)
Wallace W. Schulz and Glen E. Benedict
Atlantic Richfield Hanford Company
Available from the National Technical Information Service, U. S. Department of Commerce,
Springfield, Virginia 22151.
-------
m m
-------
ALL Problem Sheets and ANY additional calculations
m¥ust be returned to APTI to receive credit.
c
EFFECTIVE STACK HEIGHT PROBLEMS
SET ONE
Submit all calculations leading to and involving final answer(s) to each
problem.
•H
60
H
cd
01
u
f,y
A power plant with an 80-meter stack 3.5 meters in diameter,
emits effluent gases at 93°C with an exit velocity of 15
meters/sec. What is the effective stack height when the
wind speed is 4 meters/sec, using the Bryant-Davidson stack
rise equation? Assume air temperature is 20°C.
3.:
5.
ex
•H
) (/. z
2^.7 /».
26-7
CO
•H
a)
oo
cd
tx
co
co
a) cu
cu M 4->
0 id
Cd T3
B
-------
USE THIS SIDE FOR CALCULATIONS BEFORE USING ADDITIONAL SHEETS
28
-------
ALL Problem Sheets and ANY additional calculations
must be returned to APTI to receive credit.
Problem Set One (C)
<*-<
C
•H
M
IB
4-1
C
T3
0)
4J
rt
M
o
14-1
O.
U)
01
•H
Using the above" conditions and atmospheric pressure of 1010 mb,
what is the effective stack height calculated from the Holland
equation for neutral stability?
x*
(/.s
ui
U)
a) a)
29
-------
USE THIS SIDE FOR CALCULATIONS BEFORE USING ADDITIONAL SHEETS
30
-------
ALL Problem Sheets and ANY additional calculations
must be returned to APTI to receive credit.
f-J
O
•H
M
t-i
a
CO
•H
H
Problem Set One (C)
3. Briggs has published generalized plume rise equations which EPA
is incorporating into dispersion calculations involving elevated
emission sources. A simplified working equation is given by:
i—i
cd
> 1/3 _i ?/3
g Ah = 1.6 F ' u x ' ; (x < 10 h )
3 s
•^ 1/3-1 ?/3
S Ah = 1.6 F ' u (10 h ) ' • (x > 10 h )
tu s s
Where:
Ah = plume rise (m)
cu AT ? 4 3
•£ F = buoyancy flux = — gvsrz (m /sec )
•H = £^(?'%'"(j^) /Ss*i&<>^ {s ?~: •'<-• 1<" • d
lu u = wind speed (m/sec) • '
4-J
0)
o x = downwind distance (m)
h = physical stack height (m) - <> O
S
v_ = exit velocity (m/sec)
Pu
•H
C-J
CO
CO
QJ QJ
" (U J_l 4-1
_ g T3 W
r = stack radius (m) 3-^> ^/^ / 7^-/r, ^,~3\ ^
For the power plant in Problems #1 and #2, assuming ambient air temperature
of 20°C, what is the plume rise at: a) 350 m and b) 1750 m downwind?
31
-------
USE THIS SIDE FOR CALCULATIONS BEFORE USING ADDITIONAL SHEETS
32
-------
ALL Problem Sheets and ANY additional calculations
-must be returned to APTI to receive credit.
rt
o
0)
oo
M
rt
0)
0)
4J
m
M
o
M-l
M
•H
H
Problem Set One (C)
Discuss very briefly whether or not this simplification of the Briggs
equation should be used for this power plant.
en
w
m m
cu
g -a
a T3
33
-------
USE THIS SIDE FOR CALCULATIONS BEFORE USING ADDITIONAL SHEETS
34
-------
ALL Problem Sheets and ANY additional calculations
giust be returned to APTI to receive credit.
V
e.
F
EFFECTIVE STACK HEIGHT PROBLEMS
SET TWO
Submit all calculations leading to and including final answer(s) to each
problem.
Using the Colbert power plant data in Table 5.1, p. 44, of Plume Rise
by Briggs, calculate the expected plume rise under the following
stability conditions:
1. Neutral and Unstable u = 5 m/s, Eq. 4.20 for F, Eq. 5.1' for Ah
a. Ah at 800 m /^ * 3OO -F+ = 9V.VJ/5-,
J = /o'.SfV - 5 03 m
^
b. Ah at 8000 m ^j A iff q ft'yic. - /3. o^*
37x/o-£/r'»%*.3 7^
^ ^-/>fec J P«
" C3()^tre- X * '' ®
-------
USE THIS SIDE FOR CALCULATIONS BEFORE USING ADDITIONAL SHEETS
48
-------
ALL Problem Sheets and ANY additional calculations
be returned to APTI to receive credit.
rt
QJ
t
a)
a
10
•H
Problem Set Two (F)
SQ —2
2. Stable u = 2 m/s, T = 280°K, — = 2 x 10
a Z
-1/2
a. At what distance is x = 2.4 us ; why is this calculation
important?
SJ b. Ah at 800 m
o
/ ,-, (--_>>, t> C /" /* ,T r ' "f" '«':>,-,
l^ T *i / J~ /7-y C-* / /n -w=—,-,—*»-rw -^ > -•( o ^ ,) / ^
2-0 A
2
49
-------
USE THIS SIDE FOR CALCULATIONS BEFORE USING ADDITIONAL SHEETS
50
-------
ALL Problem Sheets and ANY additional calculations
>r.ust be returned to APTI to receive credit.
Problem Set Two (F)
3. "No wind' Ah from Eq. 4.25; Assume top of surface-based inversion at
500 m.
tH
ed
to
rt
'rl
60
13
(U
o.
CO
(U
t>0
CO
•H
£1
*
5,00
t ft r\*~f
51
-------
USE THIS SIDE FOR CALCULATIONS BEFORE USING ADDITIONAL SHEETS
52
-------
ALL Problem Sheets and ANY additional calculations
must be returned to APTI to receive credit.
EFFECTIVE STACK HEIGHT PROBLEMS
I
SET THREE
Submit all calculations leading to and including final answer(s) to each
problem.
Refer to Part H of this package, which reflects current usage of the
Briggs' equations in the Meteorology Laboratory, EPA.
Under Unstable or Neutral Conditions:
1. Using the Colbert plant data, what is x*?
•H
NJ
60
,
//
>^}( i~c> 3 '",'
0)
Ml
ctf
to
•H
-77
3.
\
}
/
0)
I
81
-------
USE THIS SIDE FOR CALCULATIONS BEFORE USING ADDITIONAL SHEETS
82
-------
ALL Problem Sheets and ANY additional calculations
must be returned to APTI to receive credit.
Problem Set Three (I)
Under Unstable or Neutral Conditions:
2. What is the distance of the final plume rise, xf?
ra
cfl
a)
•H
00
(-1
n)
B
a
•H
T3
a)
4-1
n)
M
o
M-l
l-l
01
a
CO
(n
QJ a)
bo B T3 a)
rt I n) "3 -"-1
83
-------
USE THIS SIDE FOR CALCULATIONS BEFORE USING ADDITIONAL SHEETS
84
-------
ALL Problem-sheets and AJM£ additional calculations
must be returned to APTI to receive credit.
Problem Set Three (I)
Under Unstable or Neutral Conditions:
3. What is the plume rise Ah, that can be expected a mile (1500 m)
from the plant if the wind speed is 5 m/sec?
o
0)
M
>.
ra
o.
0) -H
M
H
o
M-l
C
•H
60
M
oj
a
•H
T3
0)
4-J
B)
M
O
>W
M
0)
CO
a) a)
•H a) M -u
g x) n)
a) n) 73 *J
oo 2; <; c/3
td
a.
to
•H
H
85
-------
USE THIS SIDE FOR CALCULATIONS BEFORE USING ADDITIONAL SHEETS
86
-------
EFFECTIVE STACK HEIGHT/PLUME RISE
STUDENT CRITIQUE
PLEASE CIRCLE A RESPONSE.
1. How many hours did you spend completing this package?
<2 2-4 4-6 6-8 8-10 • >10
2. In terms' of coverage of the topic, how would you rate this package?
Too Narrow 123-4-567 Too Broad
Too Elementary
Too Little
Material
1
1
2
2
3
3
.4
4
5
5
6
6
7
7
Too Advanc
Too Much
Material
3. How would you rate this package in terms of overall value to you?
Not Worth 1 2 3 4 ' 5 6 7. Significantly
the Time Improve My Work
4. What responsibility do you have'-for calculating or reviewing effective
stack height estimates?
None at Currently Will Assume None Planned
Present Involved Shortly (Within next two years)
5. Additional Comments:
A. Package Contents
B. Administrative Aspects (grading, Certificate, etc.)
C. General Suggestions
if,
1 .'D 1)
II i-i u
e -a a
A CRITIQUE MUST 1TE SUBMITTED BEFORE A CERTIFICATE CAN j
BE AWARDED BY THE AIR POLLUTION TRAINING INSTITUTE. \
89
------- |