The basic idea of assuming one entrainment mechanism based on the wind
component normal to the plume and another based on the difference
between the jet velocity and the velocity component of the external
stream parallel to the jet trajectory is physically very appealing.
(See Figure 3). The parallel shear is like the velocity shear when a
jet is discharged into a quiescent media. The coefficient (a) of this
self-induced entrainment or aspiration is fairly well established.
However, a physical interpretation of the entrainment raises a serious
question due to the normal component of the wind.

It is generally assumed (e.g. 4, 13, 15) that the component of the wind
normal to the plume is multiplied by the total cylindrical area of the
Figuee a) plume surface in order to find the volume entrainment (see
Figure 4).

W sin © 2nbds 8 (1)
where 8 is an entrainment coefficient.

Physically it seems that maximum entrainment due to the direct action of
the wind (i.e., not including aspiration) would occur when all of the
mass crossing the projected area was mixed with the plume mass and
carried up with the plume. This would yield

W 2bds sin 0 (2)

Note that 2bas sin © is simply the projected area of a cylindrical plume
segment in the direction of the wind. If consistent definitions of the
plume radius are used it would appear by comparing these relationships
that this often used entrainment assumption is too large by the factor
mB.

There is evidence to indicate that a large part of the wind impinging on
the plume projection tube is actually carried around and entrained in
the down wind side of the plume. The wind imparts a double rolling
vortex motion to the plume cross section. Chan and Kennedy's data (4)
show this to occur near the orifice. An integration of pressure around
the plume would imply a drag coefficient.

The hypothesis that will be used in this report is that the aspiration
type entrainment can be calculated via the parallel velocity shear (see
Figure 3) where the coefficient is assumed (a20.10). This is similar

to several of the models mentioned earlier (e.g. 3, 12). However, the
maximum additional mass that can be entrained is simply that which
passes through the projected area. Generally, this is less than the
maximum because some of the mass passing through the projected area is
deflected around the plume and may never enter the piume. Note, however,
that a portion of the deflected mass will still be entrained from the
back side of the plume by means of the double vortex motion. As mentioned
previously, the deflected mass still imparts momentum to the plume, but
it does this via the pressure field.

11


ce

(2/Do) ————>-

VERTICAL DISTANCE

40

35

T T T T T T T T
20, 24 -
LEGEND boo
+ R/oe 8 no L . -_
Bee '6 at —!
cae

I i _| J | | {

10 20 30 40 isi) 60 70 80 90 100
HORIZONTAL DISTANCE (X/Do) ———~—

Figure &

Multiple regression fit to Fan's data (Ref. 6).


150

o
=)
H Predicted average value
Ee '
= i
e
s, |
oOo
s
2 o H __—measured value
8 --

7 1B _/\ 5 10

Temperature (°C) Mixing ratio (gm/kg)

Figure 7f. Run 9, 26 Feb 74, 125m horizontal distance.

28




SECTION IV
PERTINENT PARAMETERS

Parameters that must be specified in order to predict plume behavior
are:

A. Source parameters

1. Air flow rate

a. Diameter of tower

b. Air velocity

Temperature

Humidity

Mass of condensed water (i.e. fog droplets formed inside
cooling tower)

Drift

a. Total drift emission

b. Size distribution

oO Pwr

B. Ambient meteorological conditions (variation with altitude)

Wind speed
Wind direction
Temperature
Humidity

Pon


8 = 0.81 for the region where the plume is still rising. To find the
height to which the plume will rise in a stable stratified atmosphere,
they suggest a value of g = 0.55.

A slightly more general form of the plume rise equation is shown by
Hewett, Fay and Hoult (12):

2/3 = (3, L-cos(t# 3 (29)
b Q b
where

1, = (30)

3

=[7

The maximum plume rise (the plume oscillates about a mean) is found at
X/1,S = a, where:

z
m ~ 7 6 y1/3

9737 (5) (31)
1,9°/3 68

Fay (1970) gives the rise in the leveled off region as:

Z X

r = 1.53 (3)/3 at Ts ° 15 (32)
or

Z., 2/3

T= 2.27 (S) for B = 0.55 (33)

b

VAPOR PLUMES

There is some disagreement in the scientific literature with regard to
predicting the difference in trajectory between dry plumes and vapor
plumes. Csanady (6) states that the direct dynamic effects of evapora-
tion and condensation on plume path are minor. His examples show the
vapor plume rising only slightly faster than dry plumes. Hosler (14)
presents examples where the ultimate height of a vapor plume is several
hundred feet lower then that of a comparable dry plume.

49


APPENDIX A
REVIEW OF PERTINENT PLUME MODELS

There have been several reviews of plume models:

(1) Briggs (2).

(2) The review included in Chan & Kennedy's report (4).
(3) Aynsley and Carson (unpublished) (1).

(4) ASME review (draft, to be published) (21).

This section is not intended to give a complete survey of all past work
that has been done, rather it is intended to highlight some of specific
past work and point out differences among models.

Present day theories of moist plume behavior rely heavily on previous
theories of smoke plume rise and are subject to the same uncertainties.
Numerous analytical methods have been proposed. In 1969 Briggs (2)
indicated that at the time over 30 models were available.

Much work has been done since then, and several methods have been
modified in an effort to account for the effects of moisture. Also,
there has been considerable related work in the analysis of submerged
buoyant jet discharges in water. Roughly speaking most of the analytical
work falls into these categories:

Gaussian Plume Models
Buoyant Plume Models
Numerical Plume Models

GAUSSIAN

Gaussian plume models assume that the concentration of material ina
plume cross section has a Gaussian distribution. The rate of growth of
the plume cross section as indicated by the rate of increase of the
standard deviations of the vertical and horizontal distributions is
usually expressed as an exponential function of downwind distance. The
usual assumption is that these functions can be related to the turbulent
diffusion of the atmosphere which is categorized under different classes
of stability. Generally, the Gaussian plume models are used after the
plume has assumed to be leveled off. The height of plume rise needs to
be determined by other methods.

There are several Gaussian models in the literature. One of the most

widely used is incorporated in Turner's Workbook on Atmospheric Disper-
sion (28). The main difficulty with the Gaussian models is that they do

47


Note that in the case of an isothermal atmosphere, the method of either
Fay, Hanna or Weil predict a maximum moist plume rise roughly 20 to 30
percent higher than the dry plume. However, when the lapse rate was -
0.0055 °K/m, Hanna's method predicted moist plume rise almost 2 1/2
times the dry plume rise, while Weil's method predicted about twice the
dry plume rise. Unfortunately, the plume trajectories plotted in Figure
3 of Weil's paper can be misleading. Integrating Weil's equations for
the 30 meter tower radius listed, results in a dry plume trajectory
considerably lower than indicated in the paper. A comparison with
Weil's published trajectory is shown in Figure 14. As a check on the
solution to Weil's equation, also shown is an independent trajectory
obtained from a Lagrangian finite difference model. It compares closely
with the numerical solution of Weil's equation. It appears that Weil's
trajectories have resulted from considering a point source emission
rather than a finite tower diameter.

As discussed above, the difference in the various plume rise estimates
are significant. Hopefully field data will lead additional credibility
to a given method.

NUMERICAL PLUME MODELS

Several numerical cooling tower plume models have been developed.
Unfortunately they are generally proprietary and only brief descriptions
are available in the literature. An exception is the EG&G model (8)
which was derived under an EPA contract from a cumulus cloud model
developed at Penn State by Weinstein and Davis. This model has detailed
cloud physics relationships, and the entrainment is assumed inversely
proportional to the plume radius. This assumption may be reasonable for
a cloud or a nearly leveled off plume puff traveling with the wind, but
jit is questionable for a plume in a moderately high wind.

The SAUNA (17) computer program appears to be a one dimensional model
but a diffusion like spread of matter is assumed to progress downwind
from each segment of the plume. This gives rise to a deep "wake"
underneath the plume. Some versions use an empirical entrainment
mechanism which is more complicated than the EG&G model. An attempt has
been made to parameterize a turbulent type of entrainment along with the
downwash from the stack.

One of the more sophisticated plume models reported in the literature is
that of Systems Science and Software (26). This model calculates a two-
dimensional plume cross-section using vorticity relationships, and is
able to predict a bifurcation of the plume into two separate parts.
However, the horizontal momentum equation is neglected. Entrainment is
computed from self-induced "turbulence". The cost of running this
proprietary program tends to prohibit its use in prototype design trade-
offs.

53


MIXING RATIO: Q

SATURATION MIXING
RATIO CURVE

Dry mixing line

SLOPE: (7-1,)/(0-0,) INITIAL STATE (T) .Q))

MIXED STATE (13.9)

BEFORE CONDENSATION CONDENSATION PATH

(SLOPE: ~¢p/t)

TANGENT LINE TO
POINT (T, .Q,)

TEMPERATURE: T ——————-»~

Figure 15 Moisture thermodynamics.

57


N

av4eReR

wind velocity
horizontal distance
vertical distance

shear entrainment coefficient
wind impingement coefficient
lapse rate

density

liquid water mixing ratio

Subscripts

atm
ad
as
ms
md

atmospheric
adiabatic

moist adiabatic
maximum, saturated
maximum, dry

44


SECTION YI
MODEL DEVELOPMENT

The calculation procedure is extremely simple. The trajectory of a
group of plume particles (a plume puff) is traced in time. Hence, the
method could be basically classified as a Lagrangian formulation. The
plume puff gains mass as ambient fluid is entrained and mixed within it,
but once entrained the new mass becomes an indistinguishable part of the
plume puff. In the simplest version, the plume is assumed to be essen-
tially a cylindrical segment whose radius grows as mass is entrained.

The initial plume mass is identified as the mass issuing from the tower
with radius by-

2
Mo = emby Hy (3)

Hy is the length of the plume mass and is chosen to be comparable to bo:
Ho = Vat (4)

The increment in the plume mass is evaluated from the assumed rate of
entrainment:

DM = 5 (rate of entrainment) ; At (5)
i

For instance, assuming the total entrainment is a function of the
horizontal wind and a shearing action of the plume relative to the wind
as mentioned earlier:

OM: Y foam (26H sin 6 + mbdb cos 0) Wat + an2bHy,,,|V-W cos of}at (6)
approximate project area shear tra inment

where an estimate for Ab is a H, or

be (pt_ptSty 4

axe+az2 (7)

15


A similar set of regression fit curves derived from measurements reported
by Chasse and Winiarski (5) is shown in Figure 10. The trajectories of
Figure 9 and Figure 10 are roughly similar. Figure 10 was based on
temperature measurements whereas Fan's measurements were based on salinity
concentrations. The order of magnitude is comparable between Figure 9
and Figure 10. However it should be emphasized that neither Figure 9

nor Figure 10 can be used as an absolute standard in as much as different
sets of data yield somewhat different regression fitted curves.

A comparison between the trajectories predicted by the model and the
trajectories from the regression fit to Fan's data is shown in Figure

11. With respect to the entire range of parameters shown, the trajectory
correlation is reasonable. The differences between the model prediction
and the regression fitted trajectories are of the order of the experi-
mental uncertainty of the measurements.

Comparison between the predicted temperature ratios in Figure 8 and the
regression fit temperature curves in Figure 9 is more complicated
because Fan's temperature data are based on the "peak" temperatures of
vertical temperature profiles made through the center of the plume,
whereas the predicted temperature ratio in Figure 8 is the average over
a plume cross section with finite width. If the average excess tempera-
ture were redistributed axisymmetrically with either a linear profile
over the same predicted width or a Gaussian profile with the width
encompassing about 95 percent of the distribution equal to the predicted
width, the peak excess temperature would be about three times the average
excess temperature (see Figure 12).

The difficulty in making an absolute comparison is that the plume
cross-section and temperature distribution are not always axisymmetric.

A few detailed concentration surveys over a complete plume cross-section
show that the profile may be double-humped with twin peaks on either

side of the centerline. Judging from Fan's (9) cross-sectional isopleths,
Figure 13, these off-center peaks can be quite high, sometimes 70 percent
greater than the centerline peak. In this particular example, the ratio
of the centerline peak excess temperature to the average excess tempera-
ture is on the order of two.

In other words, to estimate centerline peak values from Figure 8 to

compare with Figure 9 one might multiply the average excess temperature
ratio by a factor of 3 if the local cross-section were Gaussian axisymmet-
ric or somewhat lower than 3 were the profile bimodel. Again, the
regression fitted curves should not be regarded as absolute values. A
regression fit to another data set cf. (Figure 10) show the temperature
lines to slope more like that predicted by the model. With these considera-
tions in mind, the predicted temperature ratios are in reasonable agree-
ment with the regression fitted curves.

It is very difficult to try to compare plume "widths" between Figures 8
and 9. The predicted width can best be interpreted as the average
radius of a plume cross-section. However, the plume cross-section may
sometimes be elliptical or horseshoe-shaped.

33


FOREWORD

Effective regulatory and enforcement actions by the Environmental
Protection Agency would be virtually impossible without sound
scientific data on pollutants and their impact on environmental
stability and human health. Responsibility for building this data
base has been assigned to EPA's Office of Research and Development
and its 15 major field installations, one of which is the Corvallis
Environmental Research Laboratory.

The primary mission of the Corvallis laboratory is research on the
effects of environmental pollutants on terrestrial, freshwater,
and marine ecosystems; the behavior, effects and control of pollu-
tants in lake systems; and the development of predictive models on
the movement of pollutants in the biosphere.

This report describes the development of a model for predicting
cooling tower plumes and the comparison of the model predictions
with laboratory and field data.

LA Pile Cr
A. F. Bartsch
Director

Corvallis Environmental Research
Laboratory


150

Predicted average value

100

__measured value

Hetght (m)

50

;
t
1
!
. '
LN
16 W7 18 10 15

Temperature (°C) Mixing ratio (gm/kg)

Figure 7d. Average Runs 3,4, and 5, 25 Feb 74, 200m horizontal distance.

26


2L

oaee

Entrainment
ae ind 2pbhw 178 2pbhw= 1.88 (2 ph b w)
for a typical 8 = .6
shear or aspirative 2apaTbhw 2apmbhw

Figure 4. Diff€rence between maximum entrainment assumptions:
a. wind projected area and aspirative entrainment, and
b. effective entrainment due to a typical entrainment
assumption.


t ttt
tet jt, very

2

z at

The speed of the plume puff along the trajectory is:

Vev¥2..2

The sine of the angle of inclination is:
Vv

“yeay?

sino =

The average radius of the plume puff can be found:

omb-H = M

b= (M/onh)!/?

Where the elemental plume puff length H is

yttat yt yt yteaty yt a

“(ax) (av)?

Time is updated

t= ttat

(19)

(20)

(21)

(22)

(23)

(24)

and the procedure is repeated. The Lagrangian Puff models for water and

air are included in Appendix C.

18


MOISTURE
Moisture affects the plume in several ways:

1. The presence of water vapor in the plume makes it less dense
than dry air at the same temperature.

2. If water vapor condenses, latent heat is released. This
raises the temperature of the vapor, water and air mixture
slightly.

3. Similarly, if liquid water evaporates the temperature of the
plume mixture cools slightly.

4. The presence of liquid water (fog drift) increases the average
density of the plume mixture.

5. If liquid water falls out of the plume (rain, or drift fall
out) the average plume density should decrease.

The calculation of moisture effects is complicated and sensitive to
small differences. This is particularly noticeable when both the
atmosphere and the plume are close to saturation. In this case, the

extent of the visible part of the plume is very sensitive to the ambient
humidity.

In attempting to analyze moisture effects a compromise must generally be
made between computation time and accuracy. The method which will be
used for evaluating moisture effects is shown in the appendix.

14


150

\ Predicted average value
t

i=]
o
2
U
t
e \—smeasured value
~~ 1
» 1
=
D
2
a
=
oO
wo

22 23 24 \) 10 15

Temperature (°c) Mixing ratio (gm/kg)

Figure 7e. Run 10, 25 Feb, 240m horizontal distance.

27


The plume also induces some mass to be entrained due to the difference
between the plume velocity and the wind speed. This can be visualized
as an aspiration or shear type entrainment. (See Figure 3). When there
is no wind this is the only entrainment. It is not certain how much
momentum this aspirated mass adds to the plume. The hypothesis to be
used here is that on the average this aspirated mass has a horizontal
velocity component equal to the free stream velocity. Generally, the
momentum entrained by this mechanism will be less than the momentum
entrained by the wind.

ENTRAINMENT

In any detailed calculation of plume behavior, a knowledge of how the
plume takes in or mixes with ambient air is critical. However, as is
pointed out by Lin (20), the various entrainment mechanisms that have
been proposed make it immediately apparent that there is still no
consensus regarding the nature of the entrainment mechanism or the
correct formulation for the jet trajectory.

It is difficult to compare entrainment and drag coefficients used in
different models because the formulation of the entrainment terms are
different. Generally, the entrainment term involves the product of an
entrainment coefficient and some "characteristic velocity". However,
there is no general agreement on what velocity is appropriate. For
example, Keffer and Baines (18) assume an entrainment mechanism based
upon the scalar difference between the averaged velocity taken over the
jet cross-section and the external stream velocity. Their experimental
results show the entrainment coefficient to be a variable along the jet
trajectory.

In order to account for entrainment due to the pair of vortices in the
wake of a jet, Platten and Keffer (23) introduced another entrainment
function and therefore another entrainment coefficient.

Fan (9) assumed entrainment mechanics based on the vector sum of the jet
velocity and the velocity component of the external stream parallel to
the jet trajectory. However, he also included a drag force as though
the jet were a solid body.

Hoult, Fay, and Forney (15) assumed two entrainment mechanisms, one due
to the difference between the jet velocity and the velocity component of
the external stream parallel to the jet trajectory, and the other due to
the component of the external stream normal to the jet trajectory. They
do not use a drag term.

Hirst (13) uses similar ideas but modifies the first entrainment coeffi-
cient so that it is a slight function of Froude number. He does not
allow for drag. A tabular comparison of some of the different entrainment
and drag formulations is included in the report by Chan and Kennedy (4).


NUS Corporation has developed a plume model which uses both entrainment
and drag terms (19), The drag is calculated like the drag on a solid
body. A drag coefficient of 0.3 is used. There may be some ambiguity
in the entrainment calculation. Apparently, an entrainment coefficient
of 4a is assumed, where a is on the order of 0.1. However, equation 21
of (19) seems to imply that there is no change in volume flux along a
stream line. This would mean no entrainment. Unfortunately. complete
details about the model are not available, and the effect of this
apparent inconsistency is not known. The computational framework of the
model is reported to be similar to the Weinstein and Davis cumulus
convection model. Both the NUS model and the EG&G model assume a
Gaussian dispersion after the maximum plume elevation is reached.

55


SECTION I

INTRODUCTION

This report will describe an approach to cooling tower plume modeling
which gives predictions which compare favorably to data without requiring
specific adjustment of empirical coefficients.

Using basic principles, the fundamental conservation laws are applied to
a finite parcel of a cooling tower plume in such a way that it is not
necessary to solve simultaneous partial differential equations. Rather,
the basic physics can be applied using average properties at a plume
cross section. The result is simple algebraic equations that can
quickly be solved step by step on a computer. This approach enables one
to follow more closely the effect of different entrainment and drag
hypotheses. Based upon numerical experiments with this model, a logical
Physical formulation for the entrainment and drag mechanisms was found
which yielded reasonable agreement with laboratory and field data.

Prior to discussion the details of the computational procedure, the
following questions are answered.

1. What is meant by a cooling tower plume?
2. What environmental problems are associated with cooling tower
plumes?

3. What parameters are required for making plume predictions?
4. What are the basic physical principles and how are they
applied in the model?

Following this, the computational procedure of the model can be easily
explained because of its close correlation with the physical principles.
Finally, a comparison of model predictions with data are given.

The appendix includes a brief technical review of recent plume models,
including discussions of the similarities and differences among various
models.

In recent years there has been an increasing concern about the potential
environmental effects of large cooling tower plumes. These cooling

tower plumes consist largely of heated air and water vapor. The water
vapor may condense into small droplets, and these pure water droplets
form a fog making part of the plume visible. In addition to the condensed
water droplets, there is a small amount of circulating water which is
carried over with the air rushing through the tower. This water is
called drift and has about the same chemical composition as the cooling
water in the tower. The visible plume from the cooling tower is chiefly
pure water which generally should not be considered harmful. However,


fo)
wo
-

Predicted average value

measured value

Height (m)

25 26 «27 15 : 20
Temperature (°c) Mixing ratio (gm/kg)

Figure 7b. Run 9, 23 Feb 74, 75m horizontal distance.

24


DSIG= DTEM*CPD/EL

T= T+ DTEM

SIG= (SIG#PM +SIGA#DM)/SUM + DSIG

IF (SIG. LTe ZERO) SIG= ZERO

Q = ESO #EXP(EL/RV#((T-T273)/T273/T))/10006 #P622
111 CONTINUE

DENA=P/R/TA/ (ONE+SIXI#QA) # (ONE+SIGA)

DEN =P/R/T /(ONE*SIXI#Q )#(ONE*SIG)

TERM= (EINS+ZWET) * (UW=U) #U/V/SUM

IF (U/V .GT. 10.) TERM= ZERO

V= (PM#V SUM +(DENA=DEN) #G/DEN®DT/TWO -TERM
Pm= SUM

Viz VEL

VEL= SORT (U#U+ V¥V)

DZ= V¥DT

Dx= U*DT

DD= SORT(DZ*D0Z+ DX*DX)
H= H+ (VEL-V1)/DD*H#DT
BSAVE= 38

B= SQRT(PM/(DEN*PI#H) )
OB= B-BSAVE

OT= DT+ DIT
X= X+ DX

Z= 2402

S= S+ DD

IF (J «LE. 2) GO TO 96
IF (X .GT. DESIRED] «AND. X eL Te DESIRED2) GO TO 98
IF (J7150 -(J-1)7150 .NE. 1) GO To 99
98 RATIOZ= Z/80
RATIOS= S/BO
RATIOR= B/BA
RATIOX= X/B0
WRITE (61913) Als RATIORsHsPMsDMsZWEI sUBsTsUsV9 VEL
LsRATIOS9Q9SIG
99 CONTINUE
999 CONTINUE
CALL EXIT
END

62


TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)

1, REPORT NO. 2. 3. RECIPIENT'S ACCESSION NO.
EPA-600/3-76-100
4. TITLE AND SUBTITLE iS. REPORT DATE
. September 1976
Cooling Tower Plume Model 6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S) 8. PERFORMING ORGANIZATION REPORT NO.

Lawrence D. Winiarski and Walter E. Frick

9. PERFORMING ORGANIZATION NAME AND ADDRESS eae 10. PROGRAM ELEMENT NO,
Assessment and Criteria Development Division FHE252
Corvallis Environmental Research Lab 77, CONTRACT/GRANT NO.

200 SW 35th Street
Corvallis, Oregon 97330

12, SPONSORING AGENCY NAME AND OPron A 13, TYPE OF REPORT AND PERIOD COVERED
U.S. Environmental Protection Agency in-house
Corvallis Environmental Research Center 74, SPONSORING AGENCY CODE

Corvallis, Oregon 97330
EPA/ORD

75. SUPPLEMENTARY NOTES

16, ABSTRACT
A review of recently reported cooling tower plume models yields none
that is universally accepted. The entrainment and drag mechanisms and
the effect of moisture on the plume trajectory are phenomena which are
treated differently by various investigators. In order to better
understand these phenomena, a simple numerical scheme is developed which
can readily be used to evaluate different entrainment and drag assumptions.
Preliminary results indicate that in moderate winds most of the entrain-
ment due to wind can be accounted for by the direct impingement of the
wind on the plume path. Initially, the pressure difference across the
plume is found to produce a substantial drag force. Thus, it is likely
that a certain portion of the plume bending is due to these pressure
forces, and artificially increasing wind entrainment to fit experimental
data is unnecessary.

7. KEY WORDS AND DOCUMENT ANALYSIS.
fa. DESCRIPTORS b.IDENTIFIERS/OPEN ENDED TERMS |c. cCosati Field/Group
coolint towers, plumes
plume computer programs .
plumes trajectories rel d ae
plumes cooling towers group
plumes, thermal analysis
plumes, atmospheric diffusion
13, DISTRIBUTION STATEMENT 19, SECURITY CLASS (This Report) 21. NO. OF PAGES
Release unlimited unclassified
20. SECURITY CLASS (This page) 22, PRICE
unclassified
EPA Form 2220-1 (9-73) 6.

# U S. GOVERNMENT PRINTING OFFICE: 1976-698-026/129 REGION 10



SAN N QTL WNL

Figure 6. Plume sampling technique.

Relative humidity was the humidity parameter actually measured; however,
it was deemed more appropriate to use the temperature data along with

the relative humidity data to calculate the actual vapor content. The
plume can be discerned better by examining a profile of absolute humidity
which, in a well mixed environment, is more nearly uniform. Relative
humidity, however, would change with temperature even if the water
content is constant. The wind conditions at the site together with the
measured lapse rate indicated that for most of the data runs the atmos-
phere could be considered well mixed.

The data used to plot the curves shown were discrete points tabulated
from a strip chart recorder. On the order of 50 data points were used
for each run, roughly 10 per minute. The data plotted show the signifi-
cant fluctuation but not all the fluctuations that were recorded.

21


Sections

I

II
Ill
IV

Vv

VI
VII
VII
IX

CONTENTS

Introduction
Conclusions
Recommendation
Pertinent Parameters
Basic Plume Principles
Model Development
Comparison with Data
References

Glossary

Appendices

15
19
39
42
44


ABSTRACT

A review of recently reported cooling tower plume models yields none
that is universally accepted. The entrainment and drag mechanisms and
the effect of moisture on the plume trajectory are phenomena which are
treated differently by various investigators. In order to better
understand these phenomena, a simple numerical scheme is developed which
can readily be used to evaluate different entrainment and drag assumptions.
Preliminary results indicate that in moderate winds most of the entrain-
ment due to wind can be accounted for by the direct impingement of the
wind on the plume path. Initially, the pressure difference across the
plume is found to produce a substantial drag force. Thus, it is likely
that a certain portion of the plume bending is due to these pressure
forces, and artificially increasing wind entrainment to fit experimental
data is unnecessary.

This report is submitted by the Pacific Northwest Environmental Research

Laboratory of the Environmental Protection Agency. Work was completed
as of February 1975.

iv


ENTRAINMENT DUE TO ENTRAINMENT DUE TO PLUME i

WIND IMPINGEMENT + SEGMENT MOVING WITH VELOCITY
RELATIVE TO WIND (ASPIRATION)

ENTRAINMENT =

PAorojected W Ap 2bH|V-W cos 6|

Figure 3 Entrainment mechanisms.


ACKNOWLEDGEMENT

The assistance, advice and consultation supplied by Mr. James Chasse
is gratefully acknowledged.

vii


The values of a in the literature depend on how the plume width, and
characteristic velocity are defined and whether the jet is buoyant. The
order of magnitude is known but further refinements might be made
relative to the particular model. As a first approximation a may be taken
to be 20.10 based on experimental studies of submerged jet, (4, 13).

The new horizontal momentum of the plume is simply the old horizontal
momentum + the horizontal momentum of the entrained mass + impulse of
horizontal pressure (i.e., drag) forces on the plume. The new horizontal
velocity (U) of the plume is simply the new horizontal momentum divided
by the new plume mass.

mtyt + DM-U., + horizontal pressure force X At

yttat ; (8)
M°+DM
Using the momentum hypothesis discussed earlier, the equation for the
horizontal velocity can be written without explicitly defining the
horizontal pressure as:
uttat 2 (utube, (2bH sin @ + mbdb cos 0) Wat (9)

atm

40 atm T2bH|V-W cos o|Wat)/(M*4DM)

Note, however, that the assumption for the horizontal pressure force
from which the particular entrainment hypothesis was derived is:

Horizontal
pressure = (xm (2bH sin © + mbdb cos ©) W + ao t m2bH|V-W cos @|)(W-U)
force atm

Assuming the total pressure force vector acts normal to the plume it is
possible to approximate a vertical pressure force component (provided
the plume is not close to horizontal) as:

_ horizontal pressure force
tan 0

Vertical pressure force =

Physically this term tends to zero as tan © tends to zero but because of
the numerical treatment this term has to be set to zero; in this case it
is set to zero when tan © = 0.1.

Similarly, the new vertical velocity due to entrainment is:

ytrat . mtyt + (vertical pressure force) at (11)
wtsom

16

(10)


Ecological Research Series



10.

SECTION VIIL
REFERENCES

Aynsley, E. and J. E. Carson. Environmental Effects of Water
Cooling for Power Plants: A Status Report. Argonne National
Laboratory, Argonne, Illinois. Unpublished. 1973.

Briggs, G. A. Plume Rise. U. S. Atomic Energy Commission, Oak
Ridge, Tennessee. Critical Review Series TID-25075. November
1969.

Briggs, G. A. and S. R. Hanna. Comments on a Comparison of Wet and
Dry Bent-over Plumes. Journal of Applied Meteorology 11:1386-1387,
December 1972.

Chan, T. L. and J. F. Kennedy. Turbulent Nonbuoyant or Buoyant
Jets Discharged into Flowing or Quiescent Fluids. University of
Towa, Iowa City. Iowa Institute of Hydraulic Research, Report No.
140. August 1972.

Chasse, J. P. and L. D. Winiarski. Laboratory Experiments of
Submerged Discharges with Current. U. S. Environmental Protection
Agency, Corvallis, Oregon. Pacific Northwest Environmental Research
Laboratory, Working Paper No. 12. June 1974.

Csanady, G. T. Bent-over Vapor Plumes. Journal of Applied Meteor-
ology 10:36-42, February 1971.

Davies-Jones, Robert P. Discussion of Measurements Inside High-
Speed Thunderstorm Updrafts. Journal of Applied Meteorology
13:710-717, September 1974.

EG&G, Inc. Potential Environmental Modifications Produced by Large
Evaporative Cooling Tower. Federal Water Quality Administration,
Washington, D.C. Report 16130DNHO1/71. January 1971.

Fan, L. N. Turbulent Buoyant Jets Into Stratified or Flowing
Ambient Fluids. California Institute of Technology, Pasadena. W.
M. Keck Lab of Hydraulics and Water Resources, Report No. KH-R-15.
June 1967.

Fay, J. A. M. Escudier, and D. P. Hoult. A Correlation of Field

Observations of Plume Rise. Journal of the Air Pollution Control
Association 20:391-397, June 1970.

39


SECTION II
CONCLUSIONS

A numerical model for a cooling tower plume was developed and used to
evaluate various entrainment assumptions. It was found that most of the
plume bending is due to the momentum of the wind mass that passes through
the projected area of the plume. The momentum of the wind mass is
imparted to the plume in two ways: (1) The mass entrained gives its
Momentum directly to the plume; (2) The deflection of some of the wind
yields a strong horizontal pressure force near the source. Furthermore,
the vertical acceleration of a plume parcel due to buoyant forces must
take into account the mass of the wind that has to be displaced.

An entrainment and drag hypothesis was developed that gives reasonable
agreement with actual field test data and with laboratory test data for
air and water plumes over a wide range of conditions without having to
adjust empirical coefficients.

The model is believed to accurately predict average plume properties
which are internally consistent and in agreement with fundamental
conservation laws.


there is reason for concern if the cooling tower plume reduces visibility
or contributes to icing at a nearby highway or airport. Drift water
could also be a contributing factor to local icing but currently there
is also concern about the adverse effects of materials carried with the
drift. For example, if seawater is used in the tower the salt deposition
might harm nearby vegetation. Chemicals used to control fouling or
inhibit corrosion also might be objectionable when spread with the

drift. If contaminated water is used, the possibility of spreading
bacteria or virus with the drift should be considered. Finally, the
question has been raised as to whether cooling towers can affect the
weather due to the large releases of both heat and water vapor.

As the questions concerning the possible effects of cooling towers

become more detailed, it is desirable to have acceptable mathematical
models that have sufficient generality to handle a wide range of possible
conditions. A sensitivity analysis with such mathematical models will
indicate how carefully one must specify input conditions (ambient meteor-
ology and tower performance) in order to get certain details about

cooling tower plumes. There is a possibility that a knowledge of the
sensitivity of the plume behavior with respect to input conditions might
indicate which environmental questions would warrant most careful considera-
tion and which are not likely problem areas at a given site.


3 —
L
2
aye 2e
aod
2
a)
NN
N =
‘+= Weil's Dry Plume Solution for Point Source
Ls ~-~— Weil's Dry Plume Equation integrated for
Finite Source
Lagrangian Finite Difference Model
i ! ! i l !
! 2 3 4 5 6
X/ LygSg
Figure 14 Comparison of Weil's model and the basic Lagragian
Puff Model

54


BS
te}

Oo
oa

ow
jo)

iw)
oa

20

Ge

VERTICAL DISTANCE (Z/D,) ———

5 aa

K=2 =——— — —— REGRESSION FIT TO FAN'S DATA
MODEL PREDICTIONS

| | i | i 1 ! | 1
10) 10 20 30 40 50 60 70 80 90 100

HORIZONTAL DISTANCE (X/DQ) ———*

Figure 11. Comparison between model trajectory predictions and regression fit trajectories based on Fan


£2

TABLE 1. INPUT CONDITION USED IN FIGURE 7a - 7g

PARAMETERS 7a 7b 7c 7d 7e 7f 7g
Initial temperature °C 30.0 33.2 33.5 30.0 31.1 25.8 25.0
*Efflux velocity m/sec 8.4 8.4 8.4 8.4 8.4 8.4 8.4
Initial liquid water kg/kg 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Source height m 13.0 13.0 13.0 13.0 13.0 13.0 13.0
Source diameter m 8.0 8.0 8.0 8.0 8.0 8.0 8.0
Pressure mb 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0
13 m wind (U,,) _A/sec 5.4 4.3 3.9 5.4 5.8 7.0 5.9
Wind lapse sec -0.011 -0.033 0.0 0.0 ~0.03 0.0 +0.024
13 m temperature °C 14.5 25.3 26.0 17.5 23.2 17.5 15.2
Temperature lapse °C/m -.009 -.012 ~.02 -.012 -.014 -.014 -.015
Ambient mixing ratio kg/kg 0.0093 0.0142 0.0164 0.0094 0.0100 0.0054 0.0067
Source mixing ratio kg/Kg wert n nn nnn een n nn ene nn reer n nner nnn Saturated ---------------------------------------

*The volumetric air flow,

determined by averaging several traverses, was 422 m/sec


VW.

12.

13.

14,

15.

16.

7.

18.

19.

20.

21.

22.

23.

Hanna, S. R. Rise and Condensation of Large Cooling Tower Plumes.
Journal of Applied Meteorology 11:793-799, August 1972.

Hewett, T. A., J. A. Fay, and D. P. Hoult. Laboratory Experiments
of Smokestack Plumes in a Stable Atmosphere. Atmospheric Environ-
ment 5:767-789, 1971.

Hirst, E. A. Analyses of Round, Turbulent Jets Discharged to
Flowing Stratified Ambients. Oak Ridge National Laboratory, Oak
Ridge, Tennessee. Report ORNL-4685. June 1971.

Hosler, C. L. Wet Cooling Tower Plume Behavior. Presented at
American Institute of Chemical Engineering 68th National Meeting,
Houston, Texas. March 2, 1971.

Hoult D. P., J. A. Fay and L. J. Forney. A Theory of Plume Rise
Compared with Field Observations. Journal of the Air Pollution
Control Association 19:585-590, 1969.

Hoult, D. P. and J. C. Weil. Turbulent Plume in a Laminar Cross
Flow. Atmospheric Environment 6:513-531, 1972.

Junod, A., R. J. Hopkirk, D. Schneiter, and D. Hashke. Meteorologi-
cal Influences of Atmospheric Cooling Systems as Projected in
Switzerland. Presented at Symposium on the Cooling Tower Environment
University of Maryland, Baltimore. March 4-6, 1974.

Keffer, J. F. and W. D. Baines. The Round Turbulent Jet in a Cross
Wind. J. Fluid Mech. 15:481-496, 1963.

Lee, Jiin. Lagrangian Vapor Plume Model, Version 3. NUS Corpora-
tion, Rockville, Maryland. Report No. NUS-TM-5-184. July 1974.

Lin, J. T. Three Theoretical Investigations of Turbulent Jets.
University of Iowa, Iowa City. Iowa Institute of Hydraulic Research,
Report No. 127. January 1971.

McVehil, George E. and K. E. Heikes. Cooling Tower Plume Modeling
and Drift Measurement, A Review of the State-of-the-Art.” Prepared
for ASME Contract G-131-1, by Ball Bros. Research Corp., Boulder,
Colorado. October 1974 (to be published).

Morton, B. R., G. I. Taylor, and J. S. Turner. Turbulent Gravita-
tional Convection from Maintained and Instantaneous Sources. Proc.
Roy. Soc. London 234 A, 1 January 1954.

Platten, J. L. and J. F. Keffer. Entrainment in Deflected Axisymmet-
ric Jets of Various Angles in the Stream. University of Toronto,
Canada. Department of Mechanical Engineering, Report No. UTME-
TPO808. 1968.

40


150

1
8 ' Predicted average value
'
=>
is
=
2
s measured value
‘oe
=x
i=]
wo

14 15 Vv 10 15

Temperature (°C) Mixing ratio (gm/kg)

Figure 7a. Cooling tower plume data (Turkey Point) vs. model
predictions. Average Runs 1 and 2, 25 Feb 74, 125m
horizontal distance. See Table I for input data.

22


il
13

6

110

PROGRAM AIRMODEL

DATA (R=2876) » (G=408) » (TWO=26) » (PI=341416) 9 (P622=.622) » (EL=2500.)
19 (RV=.461) 9 (T273=2736) 9 (ESO=6011) » (ZERO=06) 9 (CPD=1.003)
29 (ADIA=.0098) 9 (ONE=1e)9 (SIXI=.61)

FORMAT (OF Tels 3F7e292El10.294F 7649 2F 7o292F 8049 2F 703)
FORMAT (1X9 BF 7o39F 7eS9GEL0 e29F 7029 3F Be49F Tele be692EBel?
DO 999 KAY= 195

FORMAT (9F 8.5918)

READ (6096) VsUWsTsTAsHsAsEsBoOTsLUL

IF (EOF (60)) CALL EXIT

READ (6096) UsDTOsGsWAs SIG sSIGA »PsDUW

READ (6096) DESIREDI»sDESIRED2

Q = E50 *EXPCEL/RV#((T-T273)/T273/T) 1/1000. #P622
AK= SQRT (V®V+U#U) /UW

DEN =P/R/T /(ONE*SIXTFQ )*(ONE+SIG)

DENA=P/R/TAZ (ONE*+SIXI*QA) * (ONE+SIGA)

FR= V/SQRT ( (OENA-DEN) ZDEN* TWO#B#G)

DT= ONE/SQRT (VEL#VELFAKFAK/29.+UW#UW) #AK/ 2006

DIT= DT/30-

Z= A= OZ=S =ZERO
BA= B

Bi BFTWO

A= 2057

WRITE (61+7)

FORMAT (10 SSSR EE a7

1"0 T TA U v Uw Bb H oT
é SIG SIGA Qa Q K FR oTO DUW A
3 E")

WRITE(61LoLL )ToTAsUsV9UW9B sHsDTsSIGsSIGAsQAs Qe AK sFReDTUsOUWs ASE
WRITE (6198)

FORMAT (") ---== SUBSEQUENT PLUME VALUES=----"7" X70 2/0 Bs
10) THICK MASS VEL MASS ZwE] DEL 8B TEMP HOR-VEL VER
eR-VEL TOT-VEL S/D) MIX Re LIQ HHO ")

VEL= SURT(U#U+ Viv)

PM= PI#B#B#HFDEN

DO 99 Je IeLUL

Uw= UW+ DUWFDZ

Ta= TA- DTO#*DZ

DP= -DENA*G*DZ

P= P+)P

EINS= E*DENA#UW#DT# (TWOFBRHEV/VEL HPI *#BeDBHU/ VEL#H/DD)
ZWEL= A#DENA#TWO*P TL #6 *H*®DT#AKS (UW#U/VEL@VEL)
OM= (EINS+ZWET) *U/UW

SUM= PM+DM

U= (PM#U+EINS*#UW) /SUM+ ZWEL#UW/SUM

Qss = ESO #EXP(EL/RV#((T-7T273)/T273/T)) 710006 *PO22
Ts= T

T= (PM#T+ DM®TA)/SUM ~ADTA#DZ

Qs = ESO EXP (EL/RV® ((T-T273)/T273/T) 71000. #POZ2

Q= (Q#PM+QA*DM) ZSUM
IF (Q .GT. QS1 .OR, SIG .GT. ZERO) GO TO 110
Gu TO lll

OTEM= ((T-TA)/(Q-QA) ~PO22#RVH(T#TS) #(T#TS) /46/QS 1ZELZ (USS +P 622) ) #(
10SS-Q@)

61


7a

7b
7c
7d

Te
7f
79

10
1

12

13
14

15

FIGURES

Direction of momenta and forces.
Projected area of plume element.
Entrainment mechanisms.

Difference between maximum entrainment
assumptions.

Lagrangian Puff Model predictions compared
with jet data.

Plume sampling technique.

Cooling tower plume data (Turkey Point) vs.
model predictions. Average Runs 1 and 2,

25 Feb 74, 125 m horizontal distance.

Run 9, 23 Feb 74, 75 m horizontal distance.
Run 17, 23 Feb 74, 225 m horizontal distance.

Average Runs 3, 4, 5, 25 Feb 74, 200 m
horizontal distance.

Run 10, 25 Feb 74, 240 m horizontal distance.
Run 9, 26 Feb 74, 125 m horizontal distance.
Run 5, 27 Feb 74, 130 m horizontal distance.

Model predictions of buoyant temperature
plume in water.

Multiple regression fit to Fan's data (Ref. 6).

Multiple regression fit to EPA data.

Comparison between model trajectory predictions

and regression fit trajectories based on Fan.

Comparison of Gaussian and linear profiles with

average "top hat" value.
Isopleths of concentration after Fan (Ref. 6).

Comparison of Weil's model and the basic
Lagrangian Puff Model.

Moisture thermodynamics.

vi

~
Sc ea N
@

20
21

22
24
25

26
27
28
29

31

34

35

36

52
55


The reasoning that will be used here is that if the local horizontal
pressure force is known or can be estimated, this pressure force can be
subtracted from the total horizontal momentum flux to yield the horizontal
momentum flux added to the plume by entrainment.

This momentum flux divided by the wind velocity yields the local entrain-
ment. Assuming that the local horizontal pressure force can be approxi-
mated as the force needed to decelerate the available mass flux to the
velocity of the plume results in the following relationship:

horizontal mass flow aspirated wind horizon-

pressure = | through + mass x | velocity - tal plume

force projected flow velocity
area

Finally, after algebraic manipulation, (see Appendix D) there results:

mass flow aspirated . :
Entrainment = {through + mass flow] x “onion ta) plume velocity

projected y

area

Note that this method of calculation allows for the trends that Chan and
Kennedy (4) show in their report. Their data indicates that the initial
horizontal pressure force (before a plume parcel has acquired an appreci-
able horizontal velocity) may be quite large, whereas, the entrainment
is initially small, but grows larger. These drag and entrainment hypo-
theses have been tested using a physical integration scheme that has
proved to be simple and direct. A comparison with several data sets is
shown in Section 6. Note that there has been no tuning or adjustment of
coefficients to match specific data runs. The basic hypotheses in the
model account for a variety of conditions.

BUOYANCY

As long as the density of the plume is less than the density of the
surrounding air, a net upward force is exerted on the plume parcel. The
magnitude of this force can be estimated to be the weight of an equiva-
lent volume of ambient air minus the weight of the plume parcel. This
force imparts an acceleration to the plume parcel, but it is not clear
how to calculate this acceleration in as much as it is not clear how
much mass is involved. It appears that when the plume parcel has a
vertical motion into undisturbed ambient air, a mass of ambient air
corresponding to the displacement of undisturbed ambient air must be
accelerated. In cloud physics work, (7). experiments have also indicated
that an equivalent virtual mass must be added to the cloud mass. The
virtual acceleration then is the net upward force divided by the total
mass involved.

13


“ns 1.30 (Hanna's second method, dT,/dz = 0).

thd

ans

—= = 2.45 (Hanna's second method, dT /dz = -.0055° K/m)
Z

md

Weil maintains that in the case of a saturated plume in a stable saturated
atmosphere, the equations for Zn and X__ are identical for the dry

plume problem. The only real c Snges all& that the definitions of the
initial buoyancy flux F and the stability parameter include the

initial vapor (aq; ) and'Water (4 3;) differences.

That is:

2. ATi
Fay = Vyby9 (= +.61q; -A 93) (40)

i

and:

d
S= r (ry?) (41)
a

where T is a reference lapse rate and d is a moisture parameter.
Equations for these are given in Weil's paper (29).

Using these definitions and the same initial conditions as before, Weil
calculates:

Z, dT
aS. 1.18 (—& = 0) (42)
and qd,
Z dT
ms e °K

= 1.93 (— = -0.0055 -) 4
Tad d m (43)

Zz

52


13
li

APPENDIX C

PROGRAM H2OMODEL

DATA (R=2874) 9 (G=9.8) 9 (TWO=2.4) 5 (PI=361416)
Lo (RV=6461) 9 (T273=2736) 9 (ESO=6611) » (ZERO=04) » (CPD=1.003) + (D0=1.)
(BASET=1.5)

29 (HHOC=200525) » (UNE THOU=10000) 9

DU 999 K= 1912

FORMAT (1X9 3F 7e39F 7e594E 10 e2eF 729 BFR e4sF 7e2s 3EBe 1)

FURKMAT (12E9.294F 6.2)
FORMAT (9F 8.5918)

READ (60 96) VeUWs Ts TAsHst x Asbo DT sLUL

IF (EOF (60)) CALL EXIT

Z= K= TIME=0Z=06=6SAVE=ZWETI=DH=S =U

BA= 6
BO= B¥TWO
WRITE (6197)

9 (P622=.622) 9 (EL=25006)

= ZERO

FORMAT (1 FHP RH UHH HHH HEE EE RRS

VER VEL

LO TEMP AM TEMP HOR VEL
emMe STEP
VEL= SQRT(UFU+ viv)

DENA=ONE THOU-HHUC#* (TA-bASET) * (TA-BASET)

DEN= ONETHOU-nHOC* (T-BASET) * (T-BASET)

PM= PMG= PLl¥#3#6*#H*DEN
IF ( Uw WWE. ZERO) AK= VEL/UW

FR= VEL/SORT ( (OLNA=DEN) JDEN*TWO*#B#G)

DT= 164/5URT (VELFVEL #AKFAK/20
OTT= DT/100.

+UwFUW) ZI 300% AK/ 106

WI
K

ND

DIA THICK#TI

FROUDE "y

2.

WRITE (61911 ) Ts TAsUs Vo UW se bOoHs DT sZERVs ZEROs ZEROs AK oFR

WRITE (61948)

FORMAT (" =-==-SUBSCQUENT PLUME VALUES-----"7"

iD THICK MASS DEL MASS Z2wET vEL 8
2R-VeL TOT-VeL s/D ")
ZwEL= ZERO

LUL= 10000

DO 99) J= ds Lub

K/0 £/v os
TEMP HUR-VEL VER

EINS= E*DENA#UW#OT# (TWO#BeHFV/VEL +P I #68 #0BHU/ VEL HA/ DD)
ZWEL= A*DENA#TWOFP LRBeHFOTHABS (UWFU/VEL=VEL)

UM= (EINS+ZWEL) #U/UW
SUM= PM+DM

U= (PM*®U4+E INS#UW) /SUM+ ZWe T*UW/SUM

T= (PM#T+ DM#PA)75UM

DEN= ONETHOU~HHOC# (T-BASET) #(T-BASET)
TERM= (EINS+2wel) *(UW-U) FU/V/SUM

IF (U/V 6GT2106) TERM=ZERO

V=PM#V/SUM+ (UENA-DEN) #G/UEN®OT/TWO -TERM

Vi= VEL
PmM= SUM
VEL= SORT (U#U+ VEV)
DZ= V*¥DT
Dx= U*DT

59


Atmospheric conditions were generally fairly steady during the approxi-
mately five minutes it took to make a traverse. However, even under
these conditions, the plume may exhibit transient behavior. In some
cases it is desireable to put together a composite profile by averaging
several short-term observations. In Figure 7a the plots are made by
averaging together those data runs which were made at the same downwind
location under nearly the same environmental conditions. Figure 7a
yields the best definition of the invisible plume.

It is desirable to check the model under a variety of conditions.
Unfortunately good field data are very scarce and difficult to obtain.
Most of the data available are for cases where the wind bends the plume
over within a rather short distance. In order to check out a plume
model, it is necessary to look at how the model predicts for a wide
range of conditions. Figure 8 shows how the model predicts over a range
of both velocity and Froude numbers. It is difficult to find a compre-
hensive set of consistent data to check this range of parameters in
order to make such an overall comparison.

The format of Figure 8 facilitates an overall comparison of how different
source and ambient conditions affect plume trajectory, temperature and
width. The plume trajectories for four different Froude numbers (F = 5,
10, 20, 50) are indicated by the long dashed or solid lines that represent
the center lines of the plumes as they bend over in the wind. (The
Froude number is the ratio of inertia force to bouyant force, and thus
is an inverse measure of bouyancy.) The effect of wind on the plume may
be seen by comparing the trajectories for the three different K values
shown K = 2, 5, 10. (Recall that K is the ratio of source velocity to
wind velocity and hence is an inverse measure of the wind effect). In
any given set of K values, isopleths of temperature excess ratios (solid
lines) and isopleths of width ratios (short dashed lines) are shown.

The ratio of widths is the width of the plume at that point divided by
the initial diameter: equal to R/R.. The temperature ratio excess is
the difference between the local average plume temperature and the
ambient temperature divided by the difference between the initial average
plume temperature and the ambient temperature. The plume temperature
always decreases because it is mixing with a cooler environment so the
temperature excess ratios are less than one. The values of the tempera-
ture excess ratio isopleths and the "width" ratio isopleths ar@ indicated
at opposite ends of their respective lines. A small legend indicating
the meaning of all the lines is shown in the upper left hand corner of

Figure 8.

The curves in Figure 9 have been generated by fitting exponential
functions of the dimensionless parameters to Fan's (9) data set, a
manner patterned after the analysis by Shirazi et al., (24). This
provides a means of interpolating and extrapolating Fan's data for a

range of parameters.

30


Le

(Z/00) ———>-

VERTICAL DISTANCE

40 T T Td T

ae. T T [49 D T
EGEND ‘ Set
L 3 \ oS —_ ste
BEL R/Ro 8 NOL an ~sa a

° 10 20 30 40 SO 60 70 80
HORIZONTAL DISTANCE (X/Do) ———»>

Figure 8. Model predictions of buoyant temperature plume in water.


not model the plume where most of the significant changes occur. However,
it is often convenient to use a Gaussian model to solye for the effects
of atmospheric diffusion further downwind where the dynamics of the

plume are unimportant.

BUOYANT

Buoyant plume rise equations generally have a form similar to the
empirical equations obtained by Briggs (2). These can be related to the
entrainment concepts suggested by Taylor (27) and Morton, Taylor and
Turner (22). Similar derivations are reported by Slawson and Csanady
(25) Briggs (2), Hoult (15). Briggs (2) formulations are the simplest:

az = 1.6 FS wll? 2/3 (neutral conditions) (25)

Maximum AZ = 2.9 (Ey /3 (uniform stratification) (26)
WS
az = 5.0 FI/4 5-9/8 (calm conditions) (27)
Where:

AZ = height of rise

W = wind speed

X = distance downwind

F = buoyancy flux which is proportional to the heat flux

S = stability parameter which is proportional to the

potential temperature difference

The coefficients can be shown to be related to the rate at which air is
entrained into the plume. Based on entrainment assumptions suggested by
Taylor (27) and Morton, Taylor, and Turner (22), similar equations have
been derived by Slawson and Csanady (25), Briggs (2). Hoult (15). The
formulation used by Hoult for neutral conditions can be written:

4Z = ay'8 pl/3 yo V/3 x2/3 (28)
2B
where:
8 = entrainment constant
This is the same as Brigg's relation because the median value of 8
determined from laboratory tests is reported to be about 0.6 by Fay et

al. (10) and Hoult and Weil (16). However, Fay et al. state that field
data indicate a higher value for g. They suggest using a value of

48


SECTION VII
COMPARISON WITH DATA

ne model predictions have been compared with three different types of
lata:

1. Laboratory tests with air jets.
2. Actual field data taken on a large single cell cooling tower.
3. Laboratory tests with plumes in water.

A comparison with non-bouyant air jet data is shown in Figure 5. Note
that two independent sets of centerline data points are shown. The
model centerline predictions, indicated by solid lines, compare closely
without having to "tune" any coefficients. The parameter K, shown on
Figure 5, is the ratio of exit velocity to wind velocity.

A comparison with actual field data taken by the Environmental Protection
Agency using the technique shown in Figure 6 is given in Figures 7a-7g.

A lightweight (130 gram) radiosonde transmitter was attached to the tail
of an 8-foot long blimp which was tethered to the cooling tower and
allowed to "wind vane" downward. The blimp was used in preference to a
spherical balloon because it generated aerodynamic lift as the wind
increased, and flew much like a kite, rising higher with stronger winds.
A tethered spherical balloon would have practically no aerodynamic lift
and could be blown close to the ground in a strong wind.

An operator, positioned underneath the balloon, traversed the balloon
vertically through the plume by means of a lightweight monofilament line
hanging from the balloon (see Figure 6). Simultaneous sightings of the
balloon from a theodolite and a transit spaced a known distance apart
were correlated with the temperature and humidity recorded by the
radiosonde receiving station. The temperature sensor in the radiosonde
had a time constant of approximately 2.5 seconds with an accuracy of +
0.2°C. The humidity sensor was a premium carbon hygristor with an
accuracy of about 5 percent and a time constant similar to the tempera-
ture sensor's.

The data plotted in Figures 7a-~7g show the temperature and absolute
humidity (or mixing ratio), grams of water per kilogram of air, as a
function of the height above the ground at a given distance from the
tower. The dotted lines show the predicted width, height, average
temperature and average absolute humidity for the same conditions. In
order to facilitate comparison of the atmospheric gradient (lapse rate)
with the adiabatic or neutral gradient, the temperature profile is
plotted between parallel lines whose slope is equal to the adiabatic
lapse rate (9.8°C/km).

19


where

the virtual temperature (T+.61qT)
mixing ratio
latent heat of vaporization

ra
uuu

For a given set of conditions, the ratio of Hanna's moist plume rise to
the maximum dry plume rise is:

ins (F4t)1/3 (38)

Ta = Fia

This ratio is independent of ambient temperature and humidity stratifica-
tion. Hanna also gives a second method of calculating plume rise. For
this approach, one first calculates the dry plume rise, then computes

the fraction of the initial excess vapor that would condense at this
height and adds the latent heat released by the condensed vapor to the
dry plume buoyancy flux. A new plume is calculated and the procedure
iterated until the plume rise converges to a constant value.

Weil (29) has made a comparison of the saturated plume trajectory with

the dry plume trajectory for some conditions typical of a large natural

draft cooling Bower: Uj = 5m/sec, b; = 30m, aT; = 10° C, 4q; = .008,
= 3x10

where
o; = mass of liquid water/mass of air
Using Hanna's methodology. and a slightly more approximate form of:
AT; L

= V,b;2a [— +.61 ag +
Ty Cale

Aq] (39)

Weil calculates:

ons,

Z

= 1.44 (Hanna's first method independent of stratification)
md

51


98

99
999

O0= VEL*DT
BSAVE= B

B= SORT (PM (DEN*PI#H) )

DBb= (8-BSAVE)

OH= (VEL=V1) /DDFH*DT

H= H+ OH
X= X+ OX
2= Z+02Z
DOT= DT+DTT

IF (J .EQ. 1)

GO TO 98

Ir (J7200-(J-1)7200 sNE~ 1) GO TO 99
IF (RATIOZ «GTe 406 ORs RATIOX .GTe 1006) GO TO 999

RATIOX= X/BO
RATIOZ= Z/B0
RATIOR= B/HA
WRITE (61913)
CONTINUE
CONT INUE
CALL EXIT
END

RATIOXsRATIUZsRATIORsHs PMs DMs ZWEI

60

sDBsTsUsVeVEL


DISCLAIMER

This report has been reviewed by the Corvallis Environmental Research
Laboratory, U.S. Environmental Protection Agency, and approved for

Publication. Mention of trade names or commercial products does not
constitute endorsement or recommendation for use.

Vi


Furthermore, the half width used in Figure 9 was defined as the width
where the concentration was half the centerline peak measurement. Thus,
the width curves are not really very comparable. Preliminary modifica-
tions to the model which provide a mechanism to allow for the flattening
of the plume to an ellipsoidal profile indicate that "width" curves in
Figure 8 based on the minor axis would tend to be more nearly vertical.
The temperature curves remain about the same, because the average proper-
ties which adhere to the fundamental conservation principles remain
basically the same.

38


3x average

Figure 12. Comparison of Gaussian and linear profiles with
average "top hat" value.

36


APPENDIX D
ENTRAINMENT COMPUTATION

The total entrainment (impingement + aspiration) is computed such that
the horizontal momentum flux of the entrained mass plus the assumed
pressure force equals the sum of the horizontal momentum of the wind

impinging on the projected area plus the horizontal momentum that the
aspirated wind mass carries with itself.

(entrainment) W + (oA,W + aspiration) (W-U)

horizontal momentum

pressure
flux of the entrained force
fluid assumption

oA W + (aspiration) W

(49)
total horizontal momentum flux
assumed to be available
therefore:
entrainment = (oA,W + aspiration) 7 (50)

63


SECTION V
BASIC PLUME PRINCIPLES

In order to be able to predict plume behavior with any degree of confi-
dence, one must have a good understanding of certain basic phenomena.
These phenomena are as follows:

1. Momentum transfer from the wind to the plume.

2. Entrainment or dilution of plume properties due to mixing of
ambient air.

3. Buoyancy acceleration.

4. Moisture effects.

MOMENTUM TRANSFER
The wind can impart horizontal momentum to a plume in two ways:

1. By direct entrainment.
2. By pressure differences (drag hypothesis).

It has not been determined how much momentum transfer is due to each
mechanism. There is no consensus about either the formulation of the
terms or the coefficients. Some (e.g. 13, 15) maintain that all plume
bending is due to entrainment of the wind particles by the plume. This
results in essentially the inelastic collision problem exemplified in
basic physics. (See Figure 1).

It is important to distinguish plume bending due to entrainment from
plume bending due to pressure. Numerical experiments were performed
which indicate that the amount of entrainment necessary to achieve
observable plume bending by entrainment alone would result in excessive
dilution of plume properties.

Based on these numerical experiments, it is the hypothesis of this
report that the transfer of horizontal momentum from the wind “to the
plume (thus causing the plume bending) results primarily from the
momentum of the wind that passes through the projected area of the
plume. (See Figure 2). The mass going through this area imparts its
momentum to the plume in two ways. Close to the source, most of the
mass is deflected around the jet. This results in a strong pressure
force. However, a short distance away from the source, the wind mass
begins to penetrate the plume thereby adding momentum by direct entrain-
ment. It is still not certain how to divide the momentum transfer
between these two mechanisms, but the hypothesis here is that their sum
is always equal to the momemtum in the wind mass passing through the
projected area of the plume.


Wigley and Slawson (31) make a comparison of wet and dry plumes using
essentially the same equation as equation A-5 except that the stability
parameter is defined in terms of the moist adiabatic lapse rate rather
than the dry adiabatic lapse rate, that is:

s=% (r.. -r) (34)

as
Ta tm

According to this theory, wet plumes "see" the atmosphere as being more
unstable than do dry plumes, and, as a consequence, will rise higher
given identical stack parameters (initial temperature excess and efflux
velocity) and environmental conditions. Furthermore, if the vapor plume
changes between liquid and gaseous phases, it is possible for the
"stability" to be different along different parts of the trajectory.

If both moist and dry plumes behave in a stable manner (i.e., Tr >T,.),
the maximum height Z_., for a condensed plume is related to the max? ium
height, Lad? for a dh plume :

tas (Pag ap) V/2 (35)

1/2
2nd (Tas T)

The asymptotic heights of rise will also be in the same ratio. For the
case of an isothermal atmosphere:

Zns

—> = 1.26 (36)
thd

Briggs and Hanna (3) argued that Wigley and Slawson's simple replacement
of the dry by the moist adiabatic lapse rate was not justified.

Hanna (11) has suggested a simple method for estimating the rise Of a
moist, buoyant plume in a stable atmosphere. He indicates that if
complete condensation of all the initial excess vapor in the plume
occurred, then the initial buoyancy flux should include the latent heat
released by the excess vapor. The buoyancy flux would then be given by:

T,-T
2 Vv Yatm L
Fag = Vyby°9 [= + (q,-9,,,.) J (37)
it Vi T i ‘atm CT

v Pp

50


APPENDIX B
MOISTURE COMPUTATION

For sub-saturated conditions moisture is treated as a simple conservative
property, i.e.

DM
_ (Gm +a, DM) aay

Q
2 M+ OM

Saturation and condensation are encountered when the mixing ratio of the
mixed parcel is larger than the saturation mixing ratio for the corre-
sponding temperature. Figure 15 describes the pertinent mechanisms.

Referring to the diagram (Figure 15), the parcel is initially in state
(Ty, 4,). (Note: When first saturating (T,, q,) may not actually be on
thd saturation line. However, if the time Incrdments are sufficiently
small the error introduced is negligible. Furthermore, after this first
step the initial state will be on the saturation curve.) Condensation
presumably occurs continuously, equivalently we can suppose that the
parcel mixes without condensation and reaches state (To, ). At this
point q. is compared to the saturated mixing ratio, q.7 evaluated for
Fng the integrated Clausius - Clapeyron equation:

Tp. Us
q (6.11)(.622) pk { T9273), 4s)
s tooo | PR, 273 T,

Since the condensation path slope is nearly horizontal, the temperature
rise AT caused by condensation can be approximated by:

2

1 -T R, oT
at=[ 4% -.622-2 —1_} (a,-a)) (46)
1 ‘a La, q1+-622

The change in liquid water mixing ratio is then:

C. aT
a: (47)

We

ao

56


~

Side view. Front view.

Lo» | A.

che Ls (+) -—20" (-)
el Py —

b

ss

bH sind 1/2 b"= cose 1/2 b'® cose

2bH siné+ (1/2)(b"@ - b'*) cose
2bH sin@ + (7/2)(b" + b')(b" - b') cosé
2bH sin@ + 7Ab b cosé

PROJECTED AREA

"

w

Figure 2. Projected area of plume element.


EPA-600/3-76-100
September 1976

COOLING TOWER PLUME MODEL

by

Lawrence D. Winiarski and Walter F. Frick

Assessment and Criteria Development Division

Corvallis Environmental Research Laboratory
Corvallis, Oregon 97330

CORVALLIS ENVIRONMENTAL RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
CORVALLIS, OREGON 97330


150

i=]
So
- Predicted average value
ud
!
< ! Measured value
e
s 7
ono
s H
=x
iJ
wo
=
=
v
t
|

4 15 16 \ 5 10

Temperature (°c) Mixing ratio (gm/kg)

Figure 7g. Run 5, 27 Feb 74, 130m horizontal distance.

29


Note that for a horizontal wind, the entrained mass does not carry any
vertical momentum with it.

The new plume mass is:

wert = yt py (12)

The new plume temperature is:

+.
teat T+ OM Tat

T - (ambient lapse) AZ (13)

wteom

The new plume density is evaluated from an equation of state

e.g., p = (14)

ia
RT

The inclusion of concentrations of other parameters such as water vapor,
water droplets, salt drift, etc., requires slight modifications in the
temperature and density calculations, but the basic philosophy is the
same. A routine for calculating phase change is explained in Appendix
Il.

The change in the density of the plume relative to the atmosphere
results in a buoyant force imparting a vertical acceleration to the
plume mass equal to:

e _P
a: atm plume g (15)

2 Polume

This vertical acceleration modifies the new vertical velocity by

AV = a At (16)
Therefore:

yttat . yttat yay (17)
The new location (trajectory of the plume puff) is:

that, yt Us ttt ee at (as)

7


RESEARCH REPORTING SERIES

Research reports of the Office of Research and Development, U.S. Environmental
Protection Agency, have been grouped into five series. These five broad
categories were established to facilitate further development and application of
environmental technology. Elimination of traditional grouping was consciously

planned to foster technology transfer and a maximum interface in related fields.
The five series are:

1. Environmental Health Effects Research
2. Environmental Protection Technology
3. Ecological Research

4, Environmental Monitoring

5. Socioeconomic Environmental Studies

This report has been assigned to the ECOLOGICAL RESEARCH series. This series
describes research on the effects of pollution on humans, plant and animal
species, and materials. Problems are assessed for their long- and short-term
influences. Investigations include formation, transport, and pathway studies to
determine the fate of pollutants and their effects. This work provides the technical
basis for setting standards to minimize undesirable changes in living organisms
in the aquatic, terrestrial, and atmospheric environments.

This document is available to the public through the National Technical Informa-
tion Service, Springfield, Virginia 22161.


24.

25.

26.

27.

28.

29.

30.

31.

Shirazi, M. A., L. R. Davis, and K. VY. Byram. Effects of Ambient
Turbulence on Buoyant Jets Discharged into a Flowing Environment.
U.S. Environmental Protection Agency, Corvallis, Oregon. Pacific
Northwest Environmental Research Laboratory, Working Paper No. 2.
June 1974,

Slawson, P. R. and G. T. Csanady. On the Mean Path of Buoyant,
Bent-over Chimney Plumes. J. Fluid Mech. 28:311-322, 1967.

Taft, J. A Numerical Model for the Investigation of Moist Buoyant
Cooling Tower Plumes. Systems Science and Software, La Jolla,
California. Report No. S55-R-74-2110. February 1974.

Taylor, G. I. Dynamics of a Mass of Hotgas Rising in the Air.
U.S. Atomic Energy Commission, Los Alamos, New Mexico. Los Alamos
Scientific Laboratory, Report No. MDDC-919 (LADC-276). 1945.

Turner, D. Bruce. Workbook of Atmospheric Dispersion Estimates.
U.S. Public Health Service, Washington, D.C. Publication No. 999
AP-26. Revised 1969.

Weil, J. C. The Rise of Moist, Buoyant Plumes. Journal of Applied
Meteorology 13:435-441, June 1974.

Wigley, T. M. L. and P. R. Slawson. A Comparison of Wet and Dry
Bent-over Plumes. Journal of Applied Meteorology 11:335-340, 1972.

Wigley, T. M. L. and P. R. Slawson. On the Condensation of Buoyant,

Moist, Bent-over Plumes. Journal of Applied Meteorology 10:253-
259, April 1971.

4)


150

t
!
Predicted average value

bh<

100

measured value

Height (m)

50

25 26 27 A/- 15 20

Temperature (°C) Mixing ratio (gm/kg)

Figure 7c. Run 17, 23 Feb 74, 225m horizontal distance.

25


ve

(2/00) ————>-

VERTICAL DISTANCE

ph
to}

ol
an

a
fo)

nN
a

T T T T T T T
LEGEND

FROUOE

R/Ro 2 3 NO,

710
Kae
=>

“or 50
DV/e7.

10 20 30 50 60 70
HORIZONTAL DISTANCE (X/Do) ————~-

Figure 10. Multiple regression fit to EPA data.

100


VERTICAL DISTANCE (in source diameters)

ar rq 4
KEFFER & BAINES 2 ©
4 0
6 b
8 Oo
10 @
Ue a CHAN@ KENNEDY 4240
oO 633 4
. B77 5
Lagrangian Puff
Model Predictions
n n { 1
° 1 2 3 4

HORIZONTAL DISTANCE
(in source diameters)

Figure 5. Lagrangian Puff Model predictions compared with jet data.

20


SECTION III
RECOMMENDATIONS

The entrainment and drag hypotheses developed here should be checked
with more detailed data, including a better measure of plume properties
over the plume cross section. The basic single cell model should be
modified so that it can predict a non-circular cross section. There is
some laboratory evidence to indicate that this may occur. Preliminary
modifications of the model to predict non-circular cross sections
indicate that the model has reached a level of development where this
effect can be noticable. More data could also be acquired on multicell
towers with the wind coming at various angles to the tower axis. This
would aid in understanding the entrainment and drag mechanism for
multiple sources. The model should be adapted to handle this situation.


The adjusted amount of liquid water mixing ratio becomes:

+
a, + a, qd,

M+d)

=

° + Ao (48)

At this point the vapor mixing ratio must be corrected. Simply subtract-
ing 4 o will, however, lead to iteration errors. Instead the adjusted
temperature T + AT is used in the integrated Clausius - Clapeyron equa-
tion to find the new mixing ratio. Note then that this method will
overestimate the amount of vapor condensed.

If (To» %) starts to fall below the saturation line, evaporation

begin his mechanism is handled in the same way, except that AT and
A o become negative.

58


a

a)

ag og
=

o

it
im

am

a

tAH YUN DB 2

<i< ¢

SECTION LX
GLOSSARY
area
vertical acceleration of plume puff
local plume radius
specific heat

entrained incremental mass
moisture parameter

base of natural logarithm

buoyancy flux
buoyancy flux including latent heat

buoyancy flux subtracting liquid moisture
gravitational acceleration
length of plume segment

latent heat of vaporization
buoyancy scale length

mass of plume element
pressure

mixing ratio

gas constant

stability parameter
distance along plume centerline

temperature

virtual temperature
time

horizontal velocity component of plume puff

vertical velocity component of plume puff
total velocity of plume puff

43


Appendix

0 °o0UmDWDlUD

SECTION X
APPENDICES

Review of Pertinent Plume Models
Moisture Computation
Program Listing

Entrainment Computation

45


Momenta

Mixing A Mixed Parcel
'
- ae!

[
©) Plume Parcel

Wind Parcel

Forces
Buoyancy

Pressure

Weight

Figure 1. Direction of momenta and forces.