Ecological Research Series
COOLING  TOWER  PLUME  MODEL
                    Environmental Research Laboratory
                   Office of Research and Development
                  U.S. Environmental Protection Agency
                           Corvallis, Oregon 97330

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                 RESEARCH REPORTING SERIES

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 Protection  Agency, have  been grouped into five series. These five broad
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This document is available to the public through the National Technical Informa-
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                                        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

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                              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.

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                          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.
                                   A. F. Bartsch
                                   Director
                                   Corvallis Environmental  Research
                                     Laboratory
                               m

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                               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.

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                               CONTENTS
Sections                                                         Page
   I           Introduction                                        ^
  II           Conclusions                                         2
 III           Recommendation                                      4
  IV           Pertinent Parameters                                c
   V           Basic Plume Principles                              6
  VI           Model Development                                  15
 VII           Comparison with Data                               -jg
VIII           References                                         39
  IX           Glossary                                           42
   X           Appendices                                         44

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                                FIGURES
 No.                                                              Page
 1          Direction  of momenta and forces.                         7
 2          Projected  area of  plume element.                         8
 3          Entrainment mechanisms.                                 10
 4          Difference between maximum entrainment
           assumptions.                                            12
 5          Lagrangian Puff Model predictions compared
           with jet data.                                          20
 6          Plume  sampling technique.                               21
 7a         Cooling tower plume data (Turkey Point) vs.
           model  predictions.  Average Runs 1 and 2,
           25 Feb 74, 125 m horizontal distance.                   22
 7b         Run 9, 23  Feb 74,  75 m horizontal distance.             24
 7c         Run 17, 23 Feb 74, 225 m horizontal distance.           25
 7d         Average Runs 3, 4, 5, 25 Feb 74, 200 m
           horizontal distance.                                   26
 7e         Run 10, 25 Feb 74, 240 m horizontal distance.          27
 7f         Run 9, 26  Feb 74,  125 m horizontal distance.           28
 7g         Run 5, 27  Feb 74,  130 m horizontal distance.           29
 8          Model  predictions  of buoyant temperature
           plume  in water.                                        31
 9          Multiple regression fit to Fan's data (Ref. 6).        32
 10         Multiple regression fit to EPA data.                .  34
 11         Comparison between model trajectory predictions
           and regression fit trajectories based on Fan.          35
 12         Comparison of Gaussian and linear profiles with
           average "top hat"  value.                               36
13         Isopleths of concentration after Fan (Ref. 6).         37
14        Comparison of Weil's model  and the basic
          Lagrangian Puff Model.                                  52
15        Moisture thermodynamics.                                55
                                vi

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

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                               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,

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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.

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                              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.

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                              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.

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                              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
          2.    Temperature
          3.    Humidity
          4.    Mass of condensed water (i.e.  fog droplets formed inside
               cooling tower)
          5.    Drift
               a.   Total drift emission
               b.   Size distribution

     B.   Ambient meteorological conditions (variation with altitude)

          1.    Wind speed
          2.    Wind direction
          3.    Temperature
          4.    Humidity

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                               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^bo 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.

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Momenta
                     Mixing
  Wind  Parcel
                        Plume Parcel
Mixed Parcel
 Forces
                           Buoyancy
                                 Pressure
                           Weight
    Figure 1 .    Direction of momenta and forces.

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       w
                 Side view.
                                       H  sine
                     Front view.
    I
H sine
                -b1
          bH sine
     2b"-
7T/2 b"  COS6
                                                              2b
ir/2 b'2 cose
        PROJECTED AREA = 2bH sin6+ (Tr/2)(b"2  -  b'2)  cos6

                       = 2bH sine + (Tr/2)(b"  +  b')(b"  - b1)  cose

                       = 2bH sine + irAb b cos6
            Figure 2.     Projected area of plume element.

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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.   CSee 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, Flatten 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).

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                                                                           V
ENTRAINMENT  =
                 ENTRAPMENT DUE TO
                 WIND IMPINGEMENT
              ENTRAINMENT  DUE TO PLUME
              SEGMENT MOVING WITH VELOCITY
              RELATIVE TO WIND (ASPIRATION)
                 P  projected
W
CLp7T2bH\ V-W cos 6\
                        Figure 3
      Entrainment mechanisms.

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The basic idea of assuming one entrainraent 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
elemental plume surface in order to find the volume entrainment (see
Figure 4).

          W sin e 2irbAs 3                                             (1)

where 3 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 2bAs sin 0                                                (2)

Note that 2bAs sin 0 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
TrB.

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 (a:0.10).  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 plume.  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,
it does this via the pressure field.

                                11

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Entrainment
  wind
2 p b h w
TT & 2 p b h w = 1.88 (2 p h b w)

for a typical 3 = .6
shear or aspirative       2 a p TT b h w
                                     2 a  p TT  b h w
 Figure  4.  Difference between maximum entrainment assumptions:
             a.  wind projected area and aspirative entrainment, and
             b.  effective entrainment due to a typical entrainment
             assumption.

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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
pressure
force
mass flow
through
projected
area
    aspirated
+   mass
    flow
wind
velocity
horizon-
- tal plume
velocity
Finally, after algebraic manipulation, (see Appendix D) there results:
Entrainment
 mass flow
 through
 projected
 area
 aspirated
 mass flow
                                                 Horizontal plume velocity
                                                      wind velocity
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

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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 train,  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

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                              SECTION VI

                           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 b .
HQ is the length of the plume mass and is chosen to be comparable to b


          HQ = VQAt                                                   (4)


The increment in the plume mass is evaluated from the assumed rate of
entrainment:
          DM = z (rate of entrainment)^ At                            (5)


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:
DM: TT  (P    (2bH sin 0 + TrbAb cos 0) WAt + cnr2bHp ^JV-W cos 0|}At   (6)
             approximate project area        shear entrainment


where an estimate for Ab is   H, or
                            o S
                                15

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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 hut further refinements might be made
relative to the particular model.  As a first approximation a may be taken
to be :0.10 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.

      +.LA +   M^ + DM-U  + horizontal pressure force X At
     Ut+At = - * - 1 - - -            (8)
                                IT+DM

Using the momentum hypothesis discussed earlier, the equation for the
horizontal velocity can be written without explicitly defining the
horizontal pressure as:


     Ut+At = (MtUt+patm(2bH sin 9 + ubdb cos 0) W2At                  (9)


             +a p    ^2bH|V-W cos e|WAt)/(Mt+DM)
Note, however, that the assumption for the horizontal pressure force
from which the particular entrainment hypothesis was derived is:

Horizontal
pressure = (pa.  (2bH sin 0 + Trbdb cos 0) W + ap .   irZbHlV-W cos 0|)(W-U)  (10)
force        atm                                a m

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:
     Vertical pressure force :  - horizontal pressure force
Physically this term tends to zero as tan e 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 = 0.1.

Similarly, the new vertical velocity due to entrainment is:


     Vt+At 3 IO  + (vertical pressure force) At                      (11)

                             Mt+DM


                               16

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

     The new plume mass is:


          Mt+At = M* + DM                                             (12)


     The new plume temperature is:

                  M*! + DM T .
                  - - atm    _    (amb1ent  lapse) AZ
The new plume density is evaluated from an equation of state


e.g.,     P =                                                       04)
              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
II.

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:


     a : Patm - PP1ume g                                              (15)

             p plume


This vertical acceleration modifies the new vertical  velocity by


     AV = a At                                                        (16)


Therefore:


     Vt+At = Vt+At + AV



The new location (trajectory of the plume puff) is:


      t+4t    *
X
             X  +             At                                      (18)


                              17

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           . zt + y*
                      2       ttu

The speed of the plume puff along the trajectory is:


     V = ^,2^2"                                                       (2Q)



The sine of the angle of inclination is:


     sin 0 =    V                                                     (21)

              U2+V2


The average radius of the plume puff can be found:
           - M                                                        (22)

       b   = (M/puH)1/2
Where the elemental plume puff length H is


     Ht+At = Ht +  (vt _ Vt-At} Rt At
                   /(dX)2+(dY)2
Time is updated


     t = t+At                                                         (24)
and the procedure is repeated.   The Lagrangian Puff models for water and
air are included in Appendix C.
                               18

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                              SECTION VII

                         COMPARISON WITH DATA


The model predictions have been compared with three different types of
data:

      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 +
0.2C.  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 (9.8C/km).
                                19

-------
             o

             c
            O
            i i
            o
            UJ
                                                 2  O
                                                 4  O
                                                 6  A
                                                 6  a
                                                IO  e


                                 CHAN 8 KENNEDY   4.24 
                                                6.33 *
                                                8.77 
                                Lagrangian Puff

                              Model Predictions
                      HORIZONTAL DISTANCE
                        (in source diameters)
Figure 5.     Lagrangian Puff Model predictions compared  with jet data.
                                       20

-------
                 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

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    en

    O)
                                      Predicted average value
                                              measured value
                                                                 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

-------
ro
u>
          PARAMETERS
Initial temperature C
*Efflux velocity m/sec
Initial liquid water kg/kg
Source height m
Source diameter m
Pressure mb
13 m wind (Uw)jj)/sec
Wind lapse sec
13 m temperature C
Temperature lapse C/m
Ambient mixing ratio kg/kg
Source mixing ratio kg/kg
                                    TABLE 1.   INPUT CONDITION USED IN FIGURE 7a - 7g

                                   7a           7b           7c           7d           7e
7f
30.0
8.4
0.0
13.0
8.0
1000.0
5.4
-0.011
14.5
-.009
0.0093
33.2
8.4
0.0
13.0
8.0
1000.0
4.3
-0.033
25.3
-.012
0.0142
33.5
8.4
0.0
13.0
8.0
1000.0
3.9
0.0
26.0
-.02
0.0164
30.0
8.4
0.0
13.0
8.0
1000.0
5.4
0.0
17.5
-.012
0.0094
9p turatprl
31.1
8.4
0.0
13.0
8.0
1000.0
5.8
-0.03
23.2
-.014
0.0100
25.8
8.4
0.0
13.0
8.0
1000.0
7.0
0.0
17.5
-.014
0.0054
25.0
8.4
0.0
13.0
8.0
1000.0
5.9
+0.024
15.2
-.015
0.0067
         *The volumetric air flow, determined by averaging several  traverses, was 422 m /sec

-------
    .c
    en
    O
              25    26    27

            Temperature (C)
                                            Predicted average value
                                                    measured value
1
1
1


15

20
Mixing ratio (gm/kg)
Figure 7b.   Run 9, 23 Feb 74, 75m horizontal distance.
                                        24

-------
     cn
     t
     OJ
              25    26    27


            Temperature  (C)
                                              j    Predicted  average  value
                                                         measured  value
15
20
    Mixing ratio (gm/kg)
Figure 7c.  Run 17, 23 Feb 74, 225m horizontal distance.
                                        25

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-------
                                               Predicted average value
              22
23
                                                       measured value
15
            Temperature ( C)
                     Mixing ratio (gm/kg)
Figure 7e.  Run 10, 25 Feb, 240m horizontal  distance.
                                       27

-------
     O
              17    18    19
            Temperature (C)
                                                Predicted  average  value
                                                       .measured  value
                        10
Mixing ratio (gin/kg)
Figure 7f.   Run 9,  26 Feb 74,  125m horizontal  distance.
                                        28

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     O)
     o>
                                                 Predicted average value
                                                         measured value
14    15
                          16
                               \
            Temperature ( C)
5                           10
    Mixing ratio (gm/kg)
Figure 7g.  Run 5, 27 Feb 74, 130m horizontal  distance.
                                        29

-------
 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/KQ-  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 are* 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

-------
OJ
          o
          Q
            40
             35
             30
             25
          UJ
          I-
          co
          Q


          -J
          LJ
             20
15
             I0
                              20     30     40     50     60      70     80     90     100

                                HORIZONTAL  DISTANCE  (X/D0)	-
               Figure 8.    Model predictions of  buoyant  temperature plume in  water.

-------
CO
                             20     30     4O    50     60     70
                                HOR/ZONTAL  DISTANCE  (X/D0)	
80     90
100
                    Figure ft      Multiple regression  fit to Fan's data (Ref.  6)

-------
A similar set of regression fit curves derived from measurements reported
by Chasse and Wi.niars.ki (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

-------
     20     30     40     50     60    70
       HORIZONTAL  DISTANCE  (X/D0)	
80     90    100
Figure 10.    Multiple regression fit to EPA data.

-------
 00
 en
           O
          Q
          \

          M



          LJ
          O
           O

           h-
           cr
           LU
                               ^T"^"-~^ "* "^            REGRESSION FIT TO FAN'S DATA
                                                            MODEL  PREDICTIONS

                                                            I       I       I
                               20     30     40    50     60     70

                               HORIZONTAL DISTANCE  (X/DQ)
100
Figure 11.     Comparison between model trajectory predictions and regression  fit trajectories  based on  Fan.

-------
                                                           3x  average
Figure 12.      Comparison  of Gaussian  and  linear  profiles with
               average "top hat"  value.
                                        36

-------
A.    Plan view.  Zero isopleth is not shown.
      concentration is (c).
The center!ine peak
B.    Perspective view.
Figure 13.    Isopleths of cpncentration after Fan (Ref: 6),

                                           37

-------
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

-------
                             SECTION VI.U

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

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

3.   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.

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

5.   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.

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

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

8.   EG&G, Inc.  Potential Environmental Modifications  Produced  by Large
     Evaporative Cooling Tower.  Federal Water Quality  Administration,
     Washington, D.C.   Report 16130DNH01/71.  January  1971.

9.   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.

10.   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

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

12.  Hewett, T. A., 0. A. Fay, and D.  P- Hoult.   Laboratory Experiments
     of Smokestack Plumes in a Stable  Atmosphere.   Atmospheric Environ-
     ment 5:767-789, 1971.

13.  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.

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

15.  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.

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

17.  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.

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

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

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

21.  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).

22.  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.

23.  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-
     TP0808.  1968.
                              40

-------
24.  Shirazi, M. A., L.  R.  Davis,  and 1C.  V-  Byrara.   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.

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

26.  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.

27.  Taylor, 6. 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.

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

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

30.  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.

31.  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.
                                41

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                              SECTION IX
                               GLOSSARY
A         area
a         vertical acceleration of plume puff
b         local plume radius
C         specific heat
DM        entrained incremental mass
d         moisture parameter
e         base of natural logarithm
F         buoyancy flux
F.jt       buoyancy flux including latent heat
Fim       buoyancy flux subtracting liquid moisture
g         gravitational acceleration
H         length of plume segment
L         latent heat of vaporization
lb        buoyancy scale length
M         mass of plume element
P         pressure
q         mixing ratio
R         gas constant
S         stability parameter
s         distance along plume center!ine
T         temperature
T         virtual  temperature
t         time
U         horizontal  velocity component of plume puff
V         vertical  velocity component of plume puff
V         total velocity of plume puff
                               43

-------
W         wind velocity

X         horizontal distance

Z         vertical distance

a         shear entrainment coefficient
3         wind impingement coefficient
r         lapse rate
p         density
a         liquid water mixing ratio

Subscripts

atm       atmospheric
ad        adiabatic
as        moist adiabatic
ms        maximum, saturated
md        maximum, dry
                              44

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                               SECTION X



                              APPENDICES





Appendi-x                                                         Page



A         Review of Pertinent Plume Models                       45



B         Moisture Computation                                   54



C         Program Listing                                        57



D         Entrainment Computation                                61
                                45

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                              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 in a
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

-------
 not model the plurae where roost of the significant changes occur.  However,
 it is often convenient to use a Gaussian model to solve 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 C2).  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 F1/3 M"1/3 X2/3      (neutral conditions)           (25)


  Maximum AZ = 2.9 ( )1>/3              (uniform stratification)       (26)
                    WS

          AZ = 5.0 F1/4 S"3/8           (calm conditions)              (27)


 Where:

          AZ = height of rise
           W = wind speed
           X - distance1 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:
          AZ . (3  }l/3 Fl/3 w-l/3 x2/3                               (

                23

where:

           3 = entrainment constant

This is the same as Brigg's relation because the median value of 3
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 3.  They suggest using a value of
                                  48

-------
 8 = 0.81 for  the  region where  th.e  plume  is  still  rising.  To  find  the
 height to which the  plume will  rise  in a  stabile  stratified atmosphere,
 they suggest  a value of s =*  0.55.

 A slightly more general form of the  plume rise equation is shown by
 Hewett, Fay and Hoult  (12):
    ,^2/3  =  { i Cl-cosCys)]}1/3                                    (29)
     'bb         ^         V

where


                                                                       (30)
The maximum plume rise  (the plume oscillates about a mean) is found at
X/l^S = ir, where:

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



     f-  =1.53 (|)2/3     aty. > 1.55                              (32)
      'b          6             'b^

or
         = 2.27 (S)2/3     for B = 0.55                               (33)
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.  Hosier (14)
presents examples where the ultimate height of a vapor plume is several
hundred feet lower then that of a comparable dry plume.
                                   49

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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 terras of the moist adiabatic lapse rate rather
than the dry adiabatic lapse rate, that is:


           S = ^-   (r _ -r)                                          (34)
               T      as
                atm

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., r > r  ),
the maximum height Z  , for a condensed plume is related to the maximum
height, Z  ., for a dry plume:


          Zms _ 
-------
where

          T  = the virtual temperature (T+.61qT)
           q = raiding ratio
           L = latent heat of vaporization

For a given set of conditions, the ratio of Manna's, moist plume rise to
the maximum dry plume rise is:
          Zms , (lit) 1/3                                              (38)

          Zmd    Fid


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 tower:  IL = 5m/sec, b. = 30m, AT. = 10 C, Aq, = .008,
A a. = 3 x 10"5.       1111

where


     a- = mass of liquid water/mass of air


Using Hanna's methodology, and a slightly more approximate form of:



          F.. = V.b.2g [1-  +.61 Aq + ^	  Aq]                      (39)
           it    i i     T             c T
                          e             p e


Weil calculates:


          Zms
          Zmd
              =1.44 (Hanna's first method independent of stratification)
                                 51

-------
           * 1.3Q (Manna's second method, dT /dz * Q).
          T- A
           rod
           ms

          Zmd
=2.45 (Banna's second method,  dTQ/dz = -.0055  K/ro)
Weil maintains that in the case of a saturated plume in a stable saturated
atmosphere, the equations for Z   and X   are identical for the dry
plume problem.  The only real oranges are that the definitions of the
initial buoyancy flux F.  and the stability parameter include the
initial vapor (Aq^) and water (A a.) differences.
That  is:


                        AT,
          Fim * V1bi29 <-^T  +-611  -Ai'                          <40>
and:
                      r)                                              (41)
where rm 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
          Jni  =  -,.-,8        (_e  = Q)                               (42)

          Zmd                  dz


          Zms                  dTe            K
          Y^-  =  1.93        C-  =-0.0055)                      (43)
           md                  d              m
                                  52

-------
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 plurae.  However, when the lapse rate was -
0.0055 K/m, Banna'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
it 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

-------
     o  2
     o
     JD
    N
                                     Weil's Dry Plume Solution foe Point Source



                                     Weil's Dry Plume Equation  integrated for

                                               Finite  Source



                                    	 Lagrangian Finite Difference Model
3


X
                                          LbdSd
Figure 14        Comparison of Weil's  model  and  the  basic Lagragian
                  Puff Model

-------
NUS Corporation has, developed a plume model which uses. both, entrainroent
and drag terms 091,  The drag is. calculated like the drag on a solid
body.  A drag coefficient of Q.3 is used.  There may b_e 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 09} 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

-------
                              APPENDIX a

                         MOISTURE COMPUTATION
For sub-saturated conditions moisture is treated as a simple conservative
property, i.e.
                 (Q, M + Q  DM)
                     M + DM
                                                                       (44)
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
(T-i, q-i).  (Note:  When first saturating (T, , q-, ) may not actually be on
the saturation line.  However, if the time increments 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 (T2, q2).  At this
point qo is compared to the saturated mixing ratio, q , evaluated for
T2-  Using the integrated Clausius - Clapeyron equation:
1000
                             exp
             [k
                                        273 T
                                                                      (45)
Since the condensation path slope is nearly horizontal, the temperature
rise AT caused by condensation can be approximated by:
-.622
                            (q  q )
                              '   i
                                                                      (46)
The change in liquid water mixing ratio is then:
                e  AT
          A a =
                                                 (47)
                               56

-------
                          SATURATION MIXING
                          RATIO CURVE
         Dry mixing line
         SLOPE:  (T-Ta)/(Q-Qa)
MIXED STATE  (T2,Q2)
BEFORE CONDENSATION
INITIAL STATE
 :ONDENSATION PATH
 (SLOPE: -C/L)
                             TAMGENT LINE TO
                             POINT  (T.Q)
                    TEMPERATURE: T
      Figure 15        Moisture thermodynamics.
                                      57

-------
The adjusted amount of liquid water mixing ratio becomes:
                   ff, dm
                  -3-  +  A. a                                      (.48)
At this point the vapor mixing ratio must be corrected.   Simply subtract-
ing A a 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 (Tp, q?) starts to fall below the saturation line, evaporation
begins.  This mechanism is handled  in the same way,  except that AT and
A a become negative.
                              58

-------
                              APPENDIX C


      PROGRAM H20MODEL
      DATA  (R = 287.)  (G=9.B)  (TWO=2.)  (PI =3. 14 16)  * (P622=.622>  (tL=2500.)
     1, (RV = .<*61)  (7273 = 273.)  
-------
      00= VEL*DT
      BSAVE= B
      B= SQRT(PM/(O
      DB= (B-8SAVE)
      OH= (V'EL-V1)/DO*H*DT
      h= H+  DH
      X= X*  DX
      L- Z + DZ
      OT= DT+DTT
      IF (J  .EQ. 1) GO TO 98
      IMJ/200-(J-1)/200 .NE. 1) GO TO 99
      IF (RATIOZ .GT. ^0. .OR. RATIOX .GT. 100.) GO  TO 999
98    RATIOX= X/BO
      RATIOZ= Z/BO
      RATIOR= B/BA
      WRITE(6l13) KATIOX,RATIOZRATIORHPMDM,ZWI  UBTUVVEL
99    CONTINUE
999   CONTINUE
      CALL EXIT
      END
                                      60

-------
11
13
                                                   (P622=.622) * (EL=2500.)
                                                   (CPD= 1.003)
      PROGRAM AIRMODEL
      DATA (R=287.)  (6=9.8)  (TWO=2.)  (PI =3. 1416)
     1, (RV=.461)  (T2 73=273.)  (50=6.11) , (ZERO=0.)
     2 (ADI A = . 0098)  (ONE = 1.)   (SIX I = .61)
      FORMAT (2F7.13F7.2,2E10.24F7.4,2F7.2,2F8.4,2F7.3)
      FORMATUX3F7.3F7.b4El0.2,F7.23F8.4F7.2F8.62E8.1)
      DO  99^ KAY= 1 5
      FORMAT (9F8.5I8)
      RE AD ( 60  6)  V,UWT,TAHAE,BDTLUL
      IF  (EOF(60) )  CALL EXIT
      READ (606)  U,DTOQUA  SIG SIGA PDUW
      READ (606)  DESIRED1DESIRED2
      Q      =  ESO  *EXP(EL/RV*( ( T-T273) /T273/T ) )/1000.  *P622
      AK= SORT (y/*V*U*U)/UW
      DEN =P/R/T  /(ON*SIXI*U )*(ON+SIG)
      DENA=P/R/TA/(ONE+SIXI*UA)*(ON*SIGA)
      FR= V/SQRT ( (DENA-DEN) /0N*TWO*B*G)
      DT= ONE/SORT (VEL^VEL*AK*AK/29.*UW^UI) *AK/200.
      DTT= DT/30.
      Z=  X=  DZ=S                                     =ZERO
      BA= B
      B0= B*TWO
      A=  .057
      WRITE  (61.7)
      FORMAT (" ********^********tt*tt*^*ft***********-tt'tt'tt'
     1"0        T      TA      U      V     UW       B         H
     2   SIG   SIGA     QA       Q      K     FR      DTO      OUW
     3    E")
      WRITE(6111  )T,TA0,VUW,8 ,H,DTSIGSIGAQAQ,AK,FRDTODUw,AE
      WRITE  (618)
      FORMAT (" ----- SUBSEQUENT PLUME  VALUES ----- "/"     X/D     Z/L)     ti/
     ID   THICK      MASS  UEL MASS      ZwEl      DEL  6    TEMP  HOR-VEL VER
     ^R-VEL  TOT-VEL    S/D  Mix  R. LIQ HHO      ")
      VEL=
                                                                      DT
                                                                      A
      DO 99 J= 1LUL
      UW= UW-t- DUW-^DZ
      TA= TA- DTO*DZ
      DP= -
      P= P+
      EINS=
                                                       VL*H/DD)
110
      DM= (EINS + ZWED^O/UW
      SUM= PM--DM
      U= (PM*U + INS"*UW)/SUM*  ZWEI*U//SUM
      QSS   =  ESO  *EXP(EL/KV*( ( T-T273) /T273/T ) )/1000.
      TS= T
      T= (PM^T* OM-TA)/SUM -ADIA*DZ
      QSl   =  ESO  J>EXP(EL/RV*( ( T-T273) /T273/T ) )/1000.
      Q= (Q*PM*QA*OM)/SUM
      IF (Q .GT. QSl  .OR.  SIG  ,GT.  ZERO)  GO TO 110
      GO TO 111
      DTM= ( (T-TA)/(Q
     1QSS-Q)
                                   61

-------
      DSIG= DTEM*CPO/EL
      T= T + DTEM
      SIG= (SIG*PM +SIGA*DM)/SUM + DSIG
      IF (SIG. LT. ZERO)  SIG= ZERO
      Q     =  ESO  *EXP(EL/RV*T-T273)/T273/T) )/1000. *P622
111   CONTINUE
      DENA=P/R/TA/
-------
                              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        +         (pA W + aspiration) (H-U)


      horizontal momentum                     pressure
     flux of the entrained                     force
             fluid                           assumption


              PA W2 + (aspiration) W                                  (49)


           total horizontal momentum flux
              assumed to be available
therefore:


     entrainment = (pA W + aspiration) jj-                              (50)
                                63

-------
                                   TECHNICAL REPORT DATA
                            (Please read Instructions on the reverse before completing)
 1. REPORT NO.
         EPA-600/3-76-100
                                                            3. RECIPIENT'S ACCESSIOr+NO.
 4. TITLE AND SUBTITLE

  Cooling Tower Plume Model
              5. REPORT DATE
                  September  1976
                                                            6. PERFORMING ORGANIZATION CODE
 7. AUTHOR(S)

  Lawrence D.  Winiarski and Walter E.  Frick
                                                            8. PERFORMING ORGANIZATION REPORT NO
 9. PERFORMING ORGANIZATION NAME AND ADDRESS
  Assessment  and Criteria Development Division
  Corvallis Environmental Research  Lab
  200 SW 35th Street
  Corvallis,  Oregon 97330
              10. PROGRAM ELEMENT NO.

               EHE252
              11. CONTRACT/GRANT NO.
 12. SPONSORING AGENCY NAME AND ADDRESS
  U.S.  Environmental Protection  Agency
  Corvallis Environmental Research Center
  Corvallis, Oregon 97330
              13. TYPE OF REPORT AND PERIOD COVERED
                 in-house    .	
              14. SPONSORING AGENCY CODE
                                                               EPA/ORD
 15. 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.
                                KEY WORDS AND DOCUMENT ANALYSIS
                  DESCRIPTORS
                                              b.IDENTIFIERS/OPEN ENDED TERMS
                                                                         c. COS AT I Field/Group
  coolint towers, plumes
  plume computer programs
  plumes trajectories
  plumes cooling towers
  plumes, thermal analysis
  plumes, atmospheric  diffusion
                             Field 14
                             group 131 +
  Release unlimited
                                              19. SECURIT1
 unclassified
                                                        CLASS (This Report)'
21. NO. OF PAGES
   72
20. SECURITY CLASS (This page)
 unclassified
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