United States Corvallis Environmental
Environmental Protection Research Laboratory
Agency Corvallis, Oregon 97330
A REVIEW OF THERMAL PLUME MODELING
SEPTEMBER 1978
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A REVIEW OF THERMAL PLUME MODELING
CERL-048
SEPTEMBER 1978
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EPA ERL-Corvallis Library
00002185
A REVIEW OF THERMAL PLUME MODELING
by
L. R. Davis
Oregon State University, and
Corvallis Environmental Research Laboratory
Corvallis, Oregon 97330
and
M. A. Shirazi
Corvallis Environmental Research Laboratory
Corvallis, Oregon 97330
Keynote Address Presented at 6th
International Heat Transfer Conference
Toronto, Canada August, 1978
ubrart
U S Environmental Protection Aomzey
Cofvftllw fun v nor.
200 S W 35th Stieot
CotvaUit, Oregoa 97330
CORVALLIS ENVIRONMENTAL RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
CORVALLIS, OREGON 97330
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A REVIEW OF THERMAL PLUME MODELING
L. R. Davis, Professor1
M. A. Shirazi, Environmental Scientist2
1 Oregon State University, Corvallis, Oregon
2 U. S. Environmental Protection Agency, Corvallis, Oregon
ABSTRACT
A review of the state-of-the-art of thermal plume modeling techniques is
presented with emphasis on power plant discharges. Models are considered for
single and multiple port discharges applicable to water and atmospheric
plumes. Representative examples of each type are presented with discusston.
References to other relevant works are cited when appropriate.
NOMENCLATURE
a. = coefficients in entrainment functions
b1 = characteristic radius of submerged plume, m
B = characteristic half width of surface plume, m
c = specific heat at constant pressure, J/kgK
CP = species concentration, kg/m3
D1 = jet diameter, m
= entrainment, m2/s
= g'U0R2, buoyancy flux used in Brigg's equations, mVs3
n = specific drag force, m3/s2
= U2R2, momentum flux in Brigg's equations, m4/s2
moo
= local Froude number defined in equation (23)
= discharge densimetric Froude number
° = U /(g'D)1/2(suubmerged discharge)
= U°/(g'H )V2(surface discharge)
P = b8dy foPce in equation (2)
g = acceleration ctf gravity, m/s2
g' = g(p, - p)/p = gAp/p, m/s2
K = specific surface heat transfer coefficient, m/s
i = Prandtl mixing length, m
L = spacing between discharge ports, m
n = horizontal surface plume coordinate, Figure 7
N = number of discharge ports
P = pressure, N/m2
P = plume perimeter in equation (37)
q = vapor content kg/m3
Q = volume flux, m3/s
r = radial coordinate
R = velocity ratio, U /U
-------
R = plume radius, m
s = plume coordinate along centerline, m
S = g(dT /dz)/T , stratification stability parameter, s"2
T = temperature, K
t = time, s
U = time average velocity in s direction, m/s
tU| = absolute plume velocity relative to ambient current
v = time average velocity normal to s direction
V = vector velocity
W = equivalent slot width, ttD2/(4L), m
x,y,z = rectangular coordinates
a = entrainment coefficient
A = excess above ambient value
r = adiabatic lapse rate, K/m
n = coordinate defined on Figure 6, m
0 = flow angle, 0! in horizontal plane relative to ambient current,
e2 in vertical plane relative to horizontal
v = eddy diffusivity, m2/s
c = coordinate defined on Figure 6, m
p = mass density, kg/m3
o = liquid moisture content, kg/m3
a , a = horizontal and vertical standard deviation of gaussian distribu-
y tion
<}> = energy dissipation function
Subscri pts
a = ambient conditions
c = plume centerline conditions
o = conditions at discharge
r = variable depends on coordinates s, n, and z
INTRODUCTION
The main by-product of electrical power generation is waste heat. For
every kW of electricity generated, about 2.0 kW of waste heat is produced.
If energy consumption continues as predicted (1_), by 1990 there will be the
equivalent of 1200-1OOOHIJ power plants required to generate the required
electricity. The cooling water passing through the condensors of these
plants will equal twice the combined flows of the Mississippi, Columbia,
and Saint Lawrence rivers, elevated 10 C above their natural temperatures.
The two main methods presently used to dispose of this waste heat are
once-through cooling and a closed loop cooling system using cooling towers.
In a once-through cooling system, water is withdrawn from a supply source
such as a river, lake, or the ocean, passed through condensors where it is
heated, and returned directly to the source. The method of returning this
heated water to the receiving body is of considerable interest because of
potential environmental effects that might result. Submerged discharges
through a single high velocity jet cause rapid mixing of the warm water with
the cooler receiving water. Thus, the higher temperatures occur in a rela-
2
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tively small mixing zone near the discharge but a large amount of receiving
water will be heated in excess of natural temperatures by a small amount.
This process can be enhanced further by using several smaller discharge
ports in a multiport diffuser. In contrast, discharge at the surface of the
receiving water through an open canal causes the heated water to float with
only moderate mixing with the receiving water. This produces elevated surface
temperatures and enhances the exchange of heat directly with the atmosphere.
In shallow water even a submerged discharge produces elevated surface temper-
atures due to inadequate mixing.
If direct discharge into the receiving water cannot be designed to meet
environmental restrictions, normally stated in terms of elevated surface tem
peratures, closed cycles are used. In these systems the cooling water is
recycled back and forth between the condensor, cooling tower, spray system,
or cooling pond whichever is deemed more appropriate.
Regardless of the method, the disposal of power plant waste heat usually
results in a large thermal plume either in water or in air, each with dis-
tinctive characteristics. The ability to accurately predict these character-
istics is essential in assessing the environmental effects of waste heat from
power plants. As a result, extensive theoretical and experimental work has
been done to develop and improve prediction methods. It is the object of
this paper to review the state-of-the-art of thermal plume prediction, pri-
marily from once-through cooling systems and closed systems using wet cooling
towers.
It is impossible to discuss all the literature that exists on this
subject in such a short paper. Therefore, the authors have limited this
paper to a few representative examples and refer without further discussion
to other works which may be equally as relevant for certain applications.
WATER PLUMES
Submerged Jets
The terms "jets" and "plumes" are often used interchangably, however, it
is generally understood that the term "jet" or "forced plume" applies to that
region where the momentum of the discharge cannot be neglected, usually near
the source. The term "plume" is applied to that region where the discharge
momentum contributes little to the total momentum of the plume. This usually
occurs some distance downstream. In this paper the terms plume, buoyant jet,
etc., are used to refer to the entire discharge system.
Analyses of buoyant jets cover a wide variety of discharge applications,
both in the atmosphere such as cooling tower and stack discharges and in
water. The main differences are the scale and nature of turbulence and
stratification encountered in aquatic and in atmospheric environments. Also
a major consideration distinguishing one medium from another is the use of an
appropriate equation of state and the consideration of water vapor in atmos-
pheric plumes and salinity in aquatic plumes.
3
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It has been found that one of the main scaling parameters in thermal
plume modeling is the densimetric Froude number which is the ratio of the
discharge momentum to buoyancy flux. It is expressed in various forms and
defined in this paper as F = U (g'D) 1/7 where U is the discharge velocity
through port diameter, D, afid g'0is the modified buoyancy gAp /p. A high
Froude number implies high momentum and a low Froude number implies high
buoyancy.
The general characteristics of the submerged discharge of a buoyant
fluid are shown in Figure 1. The profiles of velocity, concentration, tem-
perature, etc. at discharge are usually uniform or "top hat" in shape. The
shear layer separating this fluid from the ambient grows from a negligibly
small size to the full width of the plume along the zone of flow establish-
ment. The central core of uniform properties no longer exists beyond this
development length. At this point the zone of established flow beqins and
the profiles are bell shaped curves {2). Due to the different diffusion rates
of momentum and mass, velocity and concentration development lengths are
slightly different. A simplified analysis (3_) of the development length for
a round jet with no ambient current shows that the centerline excess tempera-
ture and density deficiency have already decreased to about 81% of their
discharge values at the end of the velocity development length. Beyond the
development zone the plume profiles will remain similar, spreading in width
and decreasing in centerline velocity and concentration. Interaction with
the free surface or other boundaries, merging of neighboring plumes, or the
encounter with ambient currents introduce changes in the decay rate and are
the cause of considerable analytical complications.
DRIFT ZONE —
SURFACE
TRANSITION ZONE.
FULLY DEVELOPED
PROFILES
ZONE OF
ESTABLISHED
FLOW
ZONE OF FLOW
ESTABLISHMENT
OUTFALL ^DEVELOPING PROFILES
Figure 1 General characteristics of subjerged buoyant discharges.
The surface encounter problem is not easy to handle. The most successful
methods have been through physical or numerical models. Most physical model
studies have been directed toward a particular installation and have included
4
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bottom topography and shore boundary effects. Examples of these can be found
in (4), (5), (6), (7). Other laboratory studies of discharges into shallow
water can be found in (8), (9), (10). Plume merging is handled in a variety
of different ways. Some of these are considered in the particular models
discussed below. Plume profile distortion due to ambient current is shown on
Figure 2. Instead of being Gaussian-profiles, twin vorticies are formed at
the edges of the plume due to the shearing action of the current. These
vorticies cause the plume to develop a horseshoe shaped profile (JJ_) and in
some cases to separate into two separate plumes. In most integral analyses,
this horseshoe shaped profile is ignored and Gaussian similarity is assumed
throughout. As a result, they only give predictions in the mean.
Buoyancy, stratification, and ambient current (wind) have considerable
effect on plume dynamics. Buoyancy always tends to bend the plume in the
direction of the buoyant force. Ambient stratification often causes the
plume to be trapped at a level of neutral buoyancy as shown in Figure 3.
This may occur with even the slightest stratification if discharge is deep
enough. Current always bends the plume downstream regardless of the magnitude
and can never be neglected in determining trajectory.
Submerged thermal plumes have been investigated for decades in such
early works as Reichardt (2), Schmidt (12J, Albertson, et al. (1_3), Rouse, et
al. (14), Brooks (15), Morton, et al. (16), Priestly and Ball (17), Abraham
(18) and many others. Among the more recent contributions are Fan (11),
Hoult, et al. (19), Fox (20), Hirst (21_), (22), Kannberg and Davis (23) (24),
Chen and Rodi (25) (26), Madni and Pletcher-(~27), Oosthuizen (28), Hwang and
PI etcher (29), Schatzmann (20), Kostovinos (3TT, Davis, et al.~T32). Excel-
lent reviews can be-found in (33), (21_), (25]7 Earlier work concentrated on
vertical discharges in an environment without ambient current. It was found
that for high Froude number discharges (momentum jets) the velocity and
concentration both decayed with the inverse of distance (13). For buoyant
piutnes with low discharge Froude number the velocity was found to decay as
x"1/3 and concentration as x"5/3 (14). In a recent study of vertical buoyant
discharges, Chen and Rodi (25), (26j correlated the data of several resear-
chers and found their correlations to agree with these earlier results. In
addition they added an intermediate zone and defined the limits of use of
each.
The governing equations which describe conditions throughout the dis-
charge field are the transport equations of mass, momentum, energy, and
species. These conservation equations can be written in vector form as:
Mass,
+ v-(pV) = 0
(1)
Momentum,
fr- + i v72 - tfx(vxtf) = "^P + vV2V
9t c p
(2)
5
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C - CENTERLINE
VALUE
0.25 C
SECTION - I-I
FROM FAN (JJ.)
Figure 2. Plume vortex motion caused by ambient current.
MAXIMUM HEIGHT
OF RISE
NEUTRAL BUOYANT
TRAPPED PLUME
STRATIFIED
AMBIENT
Figure 3. The effects of stratification on plume rise.
Energy,
+ -V.(-T) = J_ -.(k^T) + - T(|P) (v-V) (3)
pcp cp 31 p
6
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Species concentration,
|£ + V- (vC) = v.(DcvC) (4)
In addition there is the equation of state relating density to tem-
perature, species concentration, and pressure. Due to the difficulty in
solving these equations, a number of simplifying assumptions are usually
introduced. The most common assumptions are:
1) steady flow,
2) the flow field is fully turbulent, molecular diffusion is neglected,
3) density variations appear only in the buoyanct terms (Boussinesq
approximation).
4) pressures are purely hydrostatic,
5) boundary layer approximations may be applied in the jet.
Additional assumptions are applied when specific cases are considered.
Morton, et al. (1_6) in a now classic paper following the lead by Taylor (42)
integrated the simplified governing equations over the plume crosssection and
assumed that the fluid entrained into the plume to be linearly proportional
with the plume size, b, and centerline velocity, U . This entrainment hypo-
thesis has been successfully used by many others in approximating the exchange
mechanism between the plume and ambient.
The work by Fan (lj_) was a significant contribution to the understanding
of buoyant discharges from a single source. He experimentally investigated
the effects of ambient current and stratification on buoyant plume character-
istics. He considered a wide range of discharge Froude numbers, ambient
velocities, and ambient density gradients. He also presented an integral
analysis similar to Morton et al. in which he assumed axisymmetric similar
profiles along the length of the developed zone of the plume. He considered
two cases: one where an ambient current caused the jet to bend over and a
second for discharge at an arbitrary angle into a stratified stagnant ambient.
A form of this latter model (3) was used to develop the nomograms for the
stagnant cases in reference (34). In these models the development zone
calculations were approximated by first assuming «a development length after
Albertson, et al. (13J of 6.2 jet diameters, tophat profiles at discharge,
gaussian profiles at the end, and ignoring buoyancy effects on momentum. The
models predict plume trajectory, size, and excess temperature decay. From
the trajectory information the maximum height of rise in a stratified ambient
can be determined. The stagnant ambient model predictions agreed well with
data, but the drag coefficient in the model with current had to.be varied in
order to achieve good trajectory predictions.
Hirst (21_) {22) developed an integral model for round, buoyant jets
discharged into flowing, stratified ambients. He used a two angle "natural"
coordinate system allowing for discharge at any horizontal or vertical angle
relative to the ambient current producing a three-dimensional trajectory.
Since the model considered current, stratification and varying discharge
angles, it proved to be very useful for many practical cases. It is limited,
however, to single plumes without bottom or surface encounter. The equations
in integral form become:
7
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Conservation of mass,
f
Jo
I Ordr = - lim (rv) = E (5)
' ~ ^-KO
Conservation of energy,
/> dT /o°
"D(T - V rdr - - f
0 " 0
I °°0(T - ?a) rdr = - j Ordr - lim rv'T1 (6)
Conservation of S-momentum,
a /°°° 02rdr = U E sin 0! cos 02 + f™g'rdr sin e2 - lim (ru'v1) (7)
ds Jo Jo r~°
The other momentum equations may be arranged to form the curvature
equations:
do? _T f °°g'rdr cos 02 - EU sin 0! sin 021 /q (8)
ds "LJ o J
do, _ EUa cos «1 (9>
ds q cos 02
where r is a radial coordinate perpendicular to the plume centerline, s is
the axial coordinate along the trajectory, the direction of plume flow, E is
the entrainment rate, 0 is the total (excess plus ambient) velocity in the s
direction, v is the velocity in the r direction, T is temperature, 02 is the
plume inclination relative to the horizontal, 0! is the plume horizontal flow
direction relative to the ambient current (ambient flow is at 90°), and g' is
the buoyancy flux term. The bars indicate time average values and the
primes indicate ambient fluctuating components. The term q is a momentum
term given by
1,
F2 7
0 rdr lim(r2v') (10)
r-Ko
Since the velocity term used in the continuity equation includes both
the excess velocity and the ambient current component in the s direction, the
entrainment function, E, accounts for both aspirated fluid entrainment as
well as forced entrainment caused by impingement with the ambient current.
Hirst developed his entrainment function by considering the mechanical energy
equation along with continuity and momentum to derive a form of the function
8
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which accounted for buoyancy effects on entrainment using the technique of
Fox (10) and then added terms to account for ambient velocity following the
ideas of Platten and Keffer (35) and Hoult, et al. (19). The resulting
function with suggested coefficients is given by,
E = (0.057 + 0.97 sin e2/FL) b [|U| + 9.0 U (1 - sin^ cos202)1/2] OD
where F.is a local Froude number and U is the absolute excess velocity in
the plume as compared to the ambient. Hirst evaluated the integrals in
equations (5)-(10) assuming Gaussian profiles in the zone of established flow
and a composite tophat central core attached to Gaussian shaped shear layers
on each side in the zone of flow establishment. The equations were then
solved numerically for plume growth, dilution and trajectory.
Since proven functions for the ambient turbulence terms were not avail-
able, they were eventually ignored. Hirst's model agrees fairly well with a
wide variety of experimental data both in the development zone and zone of
established flow, except for discharge in the direction of ambient flow (co-
flow discharge). In this -case slightly different trends are predicted by the
model than are observed (36). Ginsberg and Ades (37) show that a much better
prediction of trajectory is obtained with Hirst's model if the last coeffi-
cient in the entrainment function, instead of being constant at a value of
9.0, is made a function of the discharge Froude number and velocity ratio
according to:
0.195 -0352
a5 = 25.81 [F R ] - 10.33 (12)
where R = UQ/Ua.
Schatzmann (30) in a recent work presented a plume model patterned after
Hirst's in which several improvements are proposed. He does not invoke the
Boussinesq approximation and derives his working integral equations in what
he claims to be a more mathematically correct way than previous workers.
However, it is written with no Sj dependence which is set to 90° yielding a
two-dimensional trajectory. His entrainment is different from Hirst's with
the addition of a modifying term in the denominator. The resulting function
is,
P _ (a! + a2 sin e/F, )b
[ U + a4U sine]
K" a
where K' = 1 + [a3Ua cos e] /2{u| (13)
and the a's are empirically determined coefficients. In Schatzmann's analysis,
this entrained fluid only changes mass flow due to the excess velocity whereas
Hirst's includes the ambient current's contribution to total flow. The
difference between the predictions of Hirst's and Schatzmann's models is
small for most cases. A typical comparison with Fan's data is shown on
Figures 4 and 5 for trajectory and dilution.
9
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N
I
UJ
O
z
<
I—
cn
Q
<
o
i-
tr
LlI
>
1 1 1 r
--- SCHATZMANN MODEL
HIRST MODEL
0 FAN DATA
F0 = 40
>h R = 0.83
Figure 4
25 50 75 100 125 150
HORIZONTAL DISTANCE - X/D
A comparison of the model predictions for trajectory of Hirst
(21) and Schatzmann (30j as compared to Fan's data (11)
Multiple Port Discharges: The merging of multiple plumes has been handled by
either 1) assuming that a line of single round jets instantaneously becomes
a slot jet at some calculated merging location (3); 2) making an "equivalent
slot" of the multiple discharges at the source (9); or 3) by considering the
detailed dynamics of a gradual merging process using assumed merging profiles
(38), (24). The first technique proved to be a good approximation but to
satisfy conservation of mass, momentum, and energy through transition, the
centerline velocity and temperature had to be discontinuous (34).
o 0.25
o
\
V 0.20
o
£ 0.15
0
K 0.10
01
h-
w 0.05
o
z
o
o
°0 25 5 0 75 100 125 150 '
CENTERLINE DISTANCE - S/D
Figure 5 A comparison of the model predictions for dilution of Hirst (21)
and Schatzmann (30j as compared to Fan's data (11).
"1 I 1 1 1—
SCHATZMANN MODEL
HIRST MODEL
G FAN DATA
R = 0.83
O o
10
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The equivalent slot concept is a useful approach in analyzing the charac-
teristics of certain multiport discharges, but as with most simplified analy-
ses, the limits of use should be understood. The fundamental premise behind
the equivalent slot is that beyond the point of merging, the plumes from a
row of equally spaced jets become the same as the plume from a slot jet
having the same mass and momentum flux per unit length as the multiple jets.
The width of the equivalent slot under these conditions becomes:
This presumes that up to the point of merging, the mixing mechanisms of the
two are the same, and of course they are not. The effect of this difference
on the ultimate diluation is substantial when a major portion of the dilution
occurs before merging. Otherwise, the approximation is useful for large
distances away from the point of discharge or for closely spaced discharge
ports.
The gradual merging approach developed by Davis (38) uses analysis due
to Hirst as a start and incorporates a merging process where neighboring
plumes are superimposed as shown in Figure 6. The profiles in this merging
zone are based on the requirement that along the line connecting plumes, the
profile should be smooth in all directions, the slope should be zero at ; =
0 and L/2, the profiles should be the superposition of the single profiles
where applicable with.no point allowed to exceed centerline properties, and
they should maintain the characteristics of similar profiles along the plume
centerline. For example the temperature profile during merging is given by,
where c2 = [b2 + ?2, &T is the centerline excess temperature at c = n = 0,
b is the plume half width, and L is the distance between ports. The coordin
ates n and x, are defined in Figure 6. Appropriate profiles of the same form
were assumed for velocity and concentration. The entrainment function used
is similar to that used by Hirst, but it also includes the effects of plume
competition before merging and variable entrainment surface during merging.
For example the entrainment function used during merging is given by,
W = ttD2/(4L).
2
2
and,
(15)
11
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A drag term was introduced into the model to account for the additional
bending observed in multiple port discharges as compared to single discharges.
A comparison of this model with limited data indicates good agreement and
smooth predictions throughout the merging zone (24).
All three of the above techniques for handling merging ignore end effects
and thus, are applicable to long diffusers. It has been observed (32^ 39)
that a plume from short diffusers does not retain its initial elongated
crosssection but becomes essentially circular within about four diffuser
lengths from discharge. This phenomonen is not handled in any of the existing
models and is at present best considered using physical models.
ENTRAINMENT
SURFACE
' !—!-U{
PLUME CROSS-SECTION
MERGED PROFILE
SINGLE PROFILES A I \ / \
vy v \
^ /\ AMBIENT LEVEL
\
I-I
Figure 6. Superposition of plumes used in the Davis (38) model
A truly two-dimensional diffuser is a line source or long slot jet. A
slot jet can be treated analytically using a two-dimensional version of round
jet theory with an appropriate entrainment function. Slot jets have been
studied both experimentally and analytically by numerous authors (9_>3]_> 40).
In a recent study, Kotsovinos (31_) performed a detailed investigation of slot
jet discharges giving velocity, temperature, and turbulence structure profiles
along the trajectory of the plume. Jirka and Harleman (9_) and Cederwall (40)
investigated slot discharges in shallow water.
Surface Discharge
The discharge of warm water from a canal to the surface of a receiving
body of water creates several complexities absent in submerged discharge.
One is the surface spreading phenomenon that causes the plume to be asymmetric.
The other is heat exchange with the atmosphere at the surface. Figure 7 is
a sketch of a typical surface plume with a plan, and cross-sectional view.
Buoyancy causes the central portion of the plume to rise more than the edges.
This creates buoyant spreading as well as different entrainment rates in the
vertical and horizontal directions.
12
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B-B
EXCESS
TEMPERATURE
A-A
PLAN
Figure 7 Sketch of typical surface plume - plan and cross-section.
Surface plumes can generally be divided into four regions as shown on
the elevation view in Figure 8. Zone I is the zone of flow establishment.
Zone II is the zone of fully developed jet type flow with high entrainment
and plume growth. Zone III is the zone of nearly constant depth where buoy-
ancy inhibits vertical entrainment, surface heat transfer and interfacial
shear play dominant roles. Zone IV is the zone of passive dispersion where
ambient turbulence and surface heat transfer are dominant. In two-dimensional
theory, an internal hydraulic jump may exist between regions II and III.
There is some question, however, whether an internal hydraulic jump will
exist in actual three-dimensional jets where lateral spreading is possible.
The dimensionless parameter that determines the magnitudes of these
opposing effects is again the densimetric Froude number, but in this case the
characteristic length is usually taken as the depth of the discharge canal,
Heat exchange at the surface does not affect the temperature of surface
plumes significantly in Zones I and II due to the dominance of entrainment.
As entrainment decreases and the plume surface area increases, surface heat
exchange becomes important. This heat exchange is due to both convection and
evaporation, Methods for calculating approximate values can be found in
references (41_, 43).
In the case of discharge in shallow basins the plume may be attached to
the bottom, eliminating vertical entrainment all together. In such cases,
vertical variations of temperature and velocity are often ignored and two-
dimensional models are constructed.
Various mathematical models of heated surface jets are available to
predict two- and three-dimensional plume configurations. Comprehensive
reviews are presented in references (44_) and (45) where many two- and three-
13
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I - DEVELOPMENT
SURFACE HEAT
EXCHANGE
11-NEAR
FIELD
JL
iii
IV-FAR
FIELD
INTERMEDIATE
PASSIVE
DISPERSION
ENTRAPMENT
Figure 8. Characteristic zones in surface discharge
dimensional models are classified and evaluated in detail. These models vary
from analytical solutions to the diffusion equation to numerical and integral
evaluations of the full governing equations in three dimensions. In addition,
there are phenomenological or semi-empirical models derived from the correla-
tion of large quantities of data.
Typical two-dimensional models are those of Hoopes, et al. (46), Carter
(47), riotz and Benedict (48_), Wada (49), Edinger and Polk (59), Csanady (51),
and Sundaram, et al. (52J. Among the three-dimensional models are those of
Stolzenbach and Harleman (53J, Stefan, et al.(54_), Edinger and Polk (50),
Prych (56J, Shirazi and Davis (57_), and Odgaard (58). The most notable
phenomenological models are those of Pritchard (59j, (60j. In addition,
there are several numerical finite difference models discussed in the next
section.
The three-dimensional models are of main interest here because they
provide the most information and because two-dimensional models can be usually
derived from them. The governing equations (l)-(4) and usual simplifications
apply to surface discharges as well as submerged discharges when appropriate
surface boundary conditions are specified. The horizontal momentum equations
are usually separated into an s-direction equation and a lateral buoyant
spreading equation. Vertical momentum in surface plumes is usually ignored.
The buoyant spreading assumptions employed have been key factors in deter-
mining the accuracy of model predictions. The Stolzenbach-Harleman model is
probably the most sophisticated integral model with a good treatment of the
development zone. Lateral buoyant spreading is a function of the temperature
gradient in that direction. It predicts the near field with zero or low
cross currents fairly well, but runs into computational difficulties at
higher cross currents and over predicts plume widths in the intermediate and
far fields. As a result, this model has not been used extensively.
14
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The Prych model is simpler and only includes an approximate analysis of
the development zone based on an assumed development length, but it runs
without difficulty for most cases, including cross currents. The integral
form of the governing equations used by Prych can be expressed as,
Conservation of Mass:
3?/u rdA = *bE (,6>
Conservation of Energy:
r
3s JUrTrdA = "2J KTr(n- 0> s)dn <17)
Conservation of Momentum in X and Y directions:
3? f l^Acose - Sfx + FpSine cose (18)
y°U2dAsine = Sf^ - FDcose - jjrsin e (19)
where P is the integrated hydrostatic pressure force due to density differ-
ences within the plume and ambient, F. is a drag force normal to the plume,
Sf and Sf are interfacial shear forces in the X and Y directions, the
coordinate^system used is shown on Figure 7. The velocity and temperature
profiles used were expressed as,
T = aT exp(-n2/B2) • exp(-Z2/H2) (20)
I v
U = AU exp(-n2/B2) • exp(-Z2/H2) + U, cose (21)
r C a
where B and H are the characteristic half width and depth of the olume respec-
tively.
For closure, expressions for the entrainment and lateral spreading
rates, dB/ds are needed. Prych determines entrainment from Norton-type
entrainment functions for each horizontal and vertical layer within the
plume. Vertical entrainment is modified to account for local buoyancy effects.
In addition, he adds ambient turbulent entrainment using vertical and hori-
zontal eddy diffusion coefficients.
The lateral spreading, dB/dS, is determined by considering non-buoyant
spreading and adding to that, spreading due to buoyancy. The non-buoyant
spreading is determined by considering vertical and horizontal entrainment
and momentum in the absence of buoyancy effects. To account for buoyant
spreading, Prych used a buoyant spreading velocity equal to that caused by a
density wave of an immiscible film spreading over water. The resulting
buoyant spreading rate was expressed as,
15
-------
(§; * [2(F£ -i)]"1/2
b
where F^ is a local Froude number given by
fl = -yi (23)
L iiU g1 AT H3Bz
With the assumed profiles and selected model coefficients, equations (16)(19)
and (22) were evaluated with a fourth order Rung-Kutta integration in the S
direction to yield plume trajectory, centerline temperatures, width and deoth
and surface isotherms. Unfortunately the solution has a singularity at a
local Froude number, F,, equal to 1.0 and the buoyant spreading assumptions
cause over prediction of plume width.
The Odgaard model is a modified version of the Prych model with pro-
visions for bottom attachment. This is done by limiting the vertical entrap-
ment and modifying the plume cross-sectional shape when attached.
The Shirazi-Davis model is also based on Prych1s, however, several
modifications were made to improve it. The development zone calculations
were modified by introducing an empirically determined development length
which was a function of the discharge aspect ratio and Froude number. Time
of travel calculations were introduced. The biggest modification, however,
was in the buoyant spreading assumptions where interfacial shear and a non-
abrupt density front were accounted for. The expression for buoyant spreading
developed was,
(jjf) = 1.4 (§¦ F* - 1)'1/2 (24)
Although this expression is quite similar to the one used by Prych, the
modifications were sufficient to cause a much better prediction of plume
growth. The coefficients within the Shirazi-Davis model were systematically
tuned using the correlations obtained by Shirazi (61_) from a variety of
surface plume data. Like Prych's model, the Shirazi-Davis also has computa-
tional singularities. These were avoided in reference (52) by using an
extrapolation routine to yield usable data in special cases where the singu-
larities were encountered.
Pritchard's integral/phenomenological model (60), is basically a non-
buoyant jet analysis with toDhat profiles and special interpretations to
account for plume spreading and energy loss. The plume depth is assumed
constant except for a linearly varying region close to the outfall. Ambient
current is superimposed on the no-current solution to obtain trajectory, thus
limiting it to low ambient velocities. Centerline temperatures are determined
from entrainment assumptions and are corrected to account for environmental
conditions. Once the distance to a particular isotherm is determined, the
surface area within the full isotherm is determined by assuming "bullet"
16
-------
shaped areas after an initial spreading region as shown on Figure 9. The
width of the "bullet" is made a function of the centerline distance to the
isotherm using field observations as a guide. This model relies heavily on
the engineering judgment of the user and can give very good or very bad
answers depending on the user.
SMOOTH CURVE
ASSUMED ISOTHERM SHAPE
DISTANCE - S
WIDTH = 0.25 S
0.33 S
PARABOLA
0.75 S
Figure 9. "Bullet" shaped surface isotherm area used by Pritchard.
In the detailed comparison of surface plume models to field data by
Dunn, et al. (45), it was concluded that the Shirazi-Davis integral model and
the Pritchard integral/phenomenological model (60) were superior in simulating
surface plume properties than the other models considered. None of the
models, however, were able to accurately predict all the cases considered all
the tine. Bottom and shoreline encounter cannot be handled in general inte-
gral and empirical models. In addition, wind shear and current distortion of
plume profiles are not handled properly in existing models. For a discussion
on these subjects refer to (63J and (64).
Numerical Models
In the past few years there has been considerable interest in the devel-
opment of advanced numerical models capable of predicting thermal plume
properties including surface, bottom, and shoreline restrictions, wind effects
and transient environmental conditions which cannot be handled with integral
or phenomenological models. Although numerical methods are capable of greater
generality, they are considerably more costly to run and in most cases require
very large computers. As a result, they are usually used for specific appli-
cations rather than parametric studies.
The complete equations (l)-(4) are often used in numerical modeling.
They are expressed in finite difference form and applied to a two- or three-
17
-------
dimensional grid covering the region of interest. The equations are solved
by one of a number of numerical schemes with appropriate initial and/or
boundary conditions. Some use implicit methods while others use explicit.
Some use the full parabolic equations while others use elliptic expressions.
Various assumptions are made to reduce computation time and make a solution
tractable. These include variations of transient or steady state solutions,
completely free surface or the rigid lid approximation, and often that pressures
are hydrostatic only. In addition, most models employ the Boussinesq approximation
and assume turbulent flow ignoring molecular diffusion. One of the more
difficult problems in numerical modeling is the simulation of turbulence and
the resulting eddy diffusion coefficients.
The rigid lid approximation assumes no vertical motion at the free
surface. It greatly reduces run time since small tine steps are required for
stability in handling free surface gravity waves. A surface oressure varia-
tion, however, must be calculated to account for surface elevations in the
plume due to buoyance. This pressure variation must be calculated accurately
since the slope of the surface is directly related to dispersion rates.
The hydrostatic approximation removes vertical advection and diffusion
terms from the governing equations. This makes pressure a function of gravity
only and greatly simplifies calculations. This is a reasonable assumption
for surface discharges at low Froude numbers but it is usually not used in
models capable of simulating submerged discharges.
Eddy diffusion models for mass and heat vary considerably. Excellent
discussions of the various types can be found in (05), and (65J. The most
common expressions used in thermal plume modeling are based on Prandtl's
mixing length theory where the apparent eddy diffusivities are given by
where I is the mixing length. The expressions used to determine the mixing
length are as numerous as expressions for entrainment. A second type of ex-
pression uses the turbulent kinetic energy along with the mixing length or
energy dissipation function. These are calculated following the suggestions
of Kolmogorov (66j and Prandtl {§]_) from one or more independent differential
equations.
Typical numerical models can be found in references (68-87). Since the
authors have not had firsthand experience with any of these models, they
hesitate to pass judgment. Excellent reviews of these and other models can
be found in (45J, and (88). The authors of these reviews conclude that
integral and phenomenological models are developed about as far as possible
and that numerical models, although having greater potential for development,
are disappointing in their present predictive capabilities.
18
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ATMOSPHERIC DISCHARGES
Power plant discharges into the atmosphere resulting in thermal plumes
fall into two main areas: moist plumes from wet cooling towers and dry
plumes from stacks. Since dry cooling tower plumes are just a special case
of wet plumes, it is felt that they can be easily handled using appropriate
wet plume models and are therefore not considered in this paper.
Stack Plumes
The prediction of the rise and dispersion of stack plumes has been the
subject of study for decades (89-92) and many papers have been written on the
subject (93). The concern is not only for the particulate matter discharged
from these stacks but also for the products of combustion (primarily S02)
that might form acid mist or acid rain causing damage to vegetation and
property. Many models have been proposed to predict plume rise from stacks
but probably the most widely used are Gaussian plume models or simple plume
rise equations developed from basic conservation equations.
Gaussian Plume Models: Gaussian plume models are probably the oldest but
least accurate method of predicting plume properties. This method is essen-
tially empirical in that experimentally determined values are used to deter-
mine the standard deviations of Gaussian-shaped plume property profiles
super-imposed on a separately calculated plume height. For example, the
expression for concentration is given by:
Ci = 2*y^Ua {1 + exp^" 2 ^ ^
where C- is the concentration of the variable of concern, Z is the plume
height, Q. is the variable source strength at discharge, U is the mean wind
speed, ana cz, a are the vertical and horizontal cross-wind standard devia-
tions of the Gaussian profiles. These standard deviations are usually deter-
mined from empirical expressions as functions of downwind distance and atmos-
pheric stability or from curves as found in Pasquill (94) or Gifford (95).
Since plume rise must be calculated, this method is usually used in conjunc-
tion with plume rise models or where changes in plume height are not important
such as after reaching maximum height of rise. Probably the widest appli-
cation of this method can be found in Turner (96).
Plume Rise Formulas: Briggs (93) presents a critical review of existing
empirical and theoretical modeTs and compares them to one of his own derived
from the conservation of mass, buoyancy, and momentum. The simplified formu-
las developed from these equations have achieved wide acceptance in predicting
plume rise of vertical and bent-over plumes. In the development of these
formulas, Briggs assumes: 1) a virtual source, 2) negligible wind shear and
plume curvature, 3) horizontal velocities equal to the wind speed (bent-over
assumption), 4) tophat profiles, 5) a homogeneous atmosphere, and 6) a Taylor-
type entrainment model (42) with one coefficient based on a rising thermal.
He considers two types of discharge, one with buoyancy and no initial momentum
and the other with initial momentum and no buoyancy. The resulting equations
are:
19
-------
For maximum height of rise into a stable atmosphere:
Buoyant bent-over plume with wind,
Zn,ax = 2'4 rF/
Buoyant vertical plume without wind
Zmax = 5-° F1/4S"3/8 (28)
Nonbuoyant bent-over jets with wind,
W ^ 1/3s"V6 (29)
Nonbuoyant vertical jets without wind,
Z„ax = "
-------
Wet Cooling Tower Plumes
The main environmental concerns of cooling tower plumes are those of
fogging, icing and drift. Since the discharge from wet cooling towers is
near saturation, a large visible plume is created under many environmental
conditions.
This is best shown by considering the thermo-dynamics of air-water vapor
mixtures. Figure 10 is a psychrometric chart for moist air showing the
saturation curve of absolute humidity versus temperature. Point A is typical
of a wet cooling tower discharge at a saturated state. Points B, C, and D
represent different atmospheric conditions. The lines connecting point A
with points B, C, and D are approximate mixing lines. The resulting mixture
of tower effluent with ambient air will have states along the mixing line,
starting at A and moving along the line as dilution increases. If the mixed
state falls to the left of the saturation curve, supersaturation exists where
condensation occurs and small liquid droplets are formed causing a visible
cloud. States to the right of the saturation curve are unsaturated and the
plume is invisible. Thus, if the effluent is mixed with air of state B, the
plume would initially be visible and would remain so until state E is reached
at which point the plume would disappear. For atmospheric conditions at
point C the plume would be very long and could possibly merge with existing
clouds. For State D, the plume would be very short if visible at all.
-16 0 +16 +32
TEMPERATURE - C
Figure 10 Psychrometric chart for air-water vapor mixutre with possible
mixing lines of cooling tower plume with ambient air.
21
-------
In addition to being of concern to air traffic, this visible plume may
under downwash conditions cause fog and, in some cases, ice. Also, the water
spray drops carried away from the tower in the form of drift are of particular
concern since they carry with them the dissolved solids, chemicals, and
sometimes bacteria and virus from the recirculating cooling water.
As a result of these concerns, extensive work has been done in develop-
ing models of varying complexity to predict the characteristics of cooling
tower plumes. Models interested in the deposition of cooling tower drift
usually have a drift model superimposed on a plume model. Since this paper
is interested mainly in thermal plumes, these drift models will not be consi-
dered. Persons interested in drift deposition should consult references (97-
101).
Several good survey reports have been published on cooling tower plumes.
The most notable are references (102) and (103). The first is a compilation
of the proceedings of a symposium held at the University of Maryland and
includes several very good papers. The second is a (1975) state-of-the-art
review prepared for the ASME Research Committee on Atmospheric Emissions and
Plume Behavior from Cooling Towers.
There is such a variety of plume models in the literature that only a
few of the typical ones can be considered in this paper. In general, cooling
tower plume models can be classified as a) Gaussian, b) closed form bent-over
plume models, c) integral models, and d) numerical models.
Gaussian and Bent-Over Cooling Tower Plume Models: Gaussian and bentover
models applied to cooling tower plumes are essentially the same as forostacl<
plumes except that the concentration of water content is also considered.
From the water content and thermodynamic relations, the existence of a visible
plume is determined. Examples of Gaussian models used for cooling towers can
be found in (104, 105, 106).
The models of Hanna (107), Wigley and Slawson (108), and Csanady (109)
are typical of bent-over cooling tower plume models. Since entrainment is
assumed to be constant, the plume radius and dilution can be expressed as
(103),
R/Ro = (Uo/Ua)1/2 + aZ (34)
Q/Qq = [1 + a (Ua/Uo)1/2Z/RQ]2 (35)
where a is the entrainment coefficient, R is the initial radius, and 0 is
the initial volume flux. The end of the Visible plume is taken when th£
relative humidity falls below 100%. The entrainment coefficient is subject
to interpretation and could best be classified as an empirical constant which
has different values depending on local effects. Hanna (1]_0) suggests a
value of a = 0.5 while others (103), (111) suggest a value of 0.3.
22
-------
These bent-over plume models have the advantage of being simple and are
a considerable improvement over Gaussian plume models. They have been found
to yield satisfactory results for many conditions in the intermediate region
of the plume, especially for natural draft towers that have little vertical
momentum at discharge. They should obviously not be used near the tower
where initial momentum affects dilution and trajectory, or far away from the
tower where ambient turbulence dominates, or for non-homogeneous atmospheres.
Accuracy is improved by numerically integrating the equations with variable
boundary conditions and selecting coefficients suitable for these conditions.
Hanna (107) uses a successive approximation method to partially account for
latent heat effects on plume rise. Meyer et al. (113) patch a Gaussian plume
model to a bent-over plume model after it has reached its maximum height of
rise under stable conditions.
Integral Models: Integral models have the advantage of satisfying the
governing conservation equations as well as including many of the effects
ignored in the simplified bent-over plume models such as plume bending, wind
shear, drag, finite discharge radius, plume cross-sectional profiles, plume
merging from multiple sources, and varying atmospheric conditions. Typical
of these are the models of Wigley and Slawson (114), Slawson, et al. (Hi),
Hanna (115), Meyer, et al. (113), Weil (116), Tsai and Huang (117), Winiarski
and Frick (118), Lee (119), Wu and Koh (120), Davis (38), and Macduff and
Davis (121). These vary in complexity from the step-by-step integration of
simple bent-over plume models to complete models that include all of the
above effects.
The integral form of the conservation of mass and momentum are given by
equations (5) and (7). A drag term is sometimes added to the momentum equa-
tion. The conservation of energy and total moisture can be written in the
following form (116), (120), (121) assuming that condensed water droplets
travel with the plume:
The conservation of energy:
In the above q is the vapor phase specific humidity, a is the liquid phase
moisture content, R is the condensation (evaporation) rate, r is the adiabatic
lapse rate, and h^ is the latent heat of vaporization (condensation). If
thermodynamic equilibrium is assumed, the saturated specific humidity can be
determined as a function of temperature from the Clausius-Clapeyron equation
for tabulated values.
^ Jo(T - Ta)dA = [ rsine] J OdA + dA
The conservation of moisture (liquid and vapor):
(36)
(37)
23
-------
These equations can be integrated once appropriate profiles for U, T, q,
and o are assumed. The actual shape of the assumed profiles is not important
if similarity is assumed throughout the length of the nlume, including the
zone of flow establishment, since the integrated values would differ only by
a constant. If the change in profiles through the zone of flow establishment
and merging zones between plumes is to be accounted for and if the distribution
of properties across the plume is desired, the assumed profiles become impor-
tant. ['lost models use "top hat" profiles with Davis (38_) and Macduff and
Davis (121) being the exception where polynomial profile was used to facilitate
smooth merging of multiple plumes. The entrainment function, E, is an impor-
tant part of the integral plume analysis since it determines the rate of
dilution bending and plume growth. Most are based on the Morton entrainment
assumption with variations to account for wind impingement, buoyancy, velocity
deficit, and plume merging. When combinations of these various effects are
considered, additional empirical entrainment coefficients must be included
within the function. Thus the more complicated versions have as many as five
empirical coefficients.
Meyer et al. (113) present two different models. One is a simple bent-
over plume following the lines of Briggs (93), Hanna (_76), Wiglev and Slav/sen
(114) and the other more complete model that includes plume bending similar
to the models of Weil (116), Wu and Koh (120) and Winiarski and Trick (118).
These later models do not make the bent-over plume assumption but rather
calculate the plume trajectory from both the vertical and horizontal momentum
equations step-by-step as integration proceeds. Meyer uses a simple entrain-
ment function that includes coefficients for wind sheer only. They handle
the merging of multiple plumes by considering three regions: one when the
plumes are separate; the second when plumes are merging and the circumference
of the plumes used in the entrainment function is a function of the degree of
merging: and third when the plumes are completely merged where a rectangular
shaped cross-section is assumed.
Winiarski and Frick divide entrainment into two components, one due to
the direct impingement of wind which is strictly a function of geometry and
thus requires no coefficient and a second that is due to velocity differences
between the plume and ambient with the usual coefficient. Drag is also
handled in this model in a fundamental way by considering the entrained fluid
and the difference between the ambient and plume horizontal velocities. They
also include a virtual mass in their vertical momentum claiming that in order
for the plume to rise due to buoyancy, it must move an equal amount of mass
of ambient air above it.
The model of Wu and Koh was developed to handle the merging of multiple
moist plumes having an arbitrary source configuration. It is therefore not
restricted to linear multi-port discharges. They use the full governing
equations (5, 7, 36, 37) including drag and propose an entrainment model
which includes the effects of shear, buoyancy and ambient turbulence. Their
entrainment function is:
24
-------
E = P {ai U + a2U, sine cose + a3U'}
a a
where P is the plume perimeter as determined by plume geometry described
below, U is the net velocity in the plume relative to the ambient velocity,
U and U' is a turbulent intensity function. The constants als a2, and a3
afe entrlinment coefficients. The first accounts for asperation type entrain
ment, the second for buoyant thermal type entrainment, and the third is for
ambient turbulence.
The value of a: is suggested to be 0.1116 for the round portion of plume
from tower discharges with low discharge Froude numbers and 0.198 for the
slot portion of the plume. The values of a3 and ai+ were taken as 0.3536 and
1.0 respectively. The merging of two round plumes of different size is
accomplished by forming two half round plumes connected by a slot section as
shown on Figure 11. The plumes are merged when the-area of the trapezoid
between the straight sections equals the sum of the areas of the two half
circles thus conserving f-low area. The perimeter used in the entrainment
function is that formed by this modified cross-section. The merging of two
modified shapes is accomplished in a similar manner as shown in Figure 12
with the new plume again having two half round ends of radius B1 and B2
connected by a slot section of length A inclined at an angle 0. Various
combinations of merging are considered in the model. This model has not been
compared to data so its accuracy is not known at present.
Figure 11 Cross-section of round plumes before and after merging as used in
The cooling tower plume model developed by Davis (38) is a version of
the gradual merging multiple plume model (23J discussed in the submerged
water plume section that contains moisture effects. In Macduff and Davis
Z
BEFORE
MERGING
AFTER MERGING
the Wu-Koh model (120)
25
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(121) the model is extended to include the possibility of merging within the
zone of flow establishment. Total moisture (liquid and vapor) is assumed to
have the same 3/2 power profiles as temperature. The Clausius - Clapeyron
relation is used to calculate vapor content in the saturated portion of the
plume. The difference between total moisture and vapor content yields the
liquid moisture content as shown on Figure 13.
BEFORE
MERGING
AFTER MERGING
Figure 12 Cross-section of two modified plumes before and after merging as
used in the Wu-Koh model (120).
When liquid phase is present, the plume is assumed to be visible. Since
the edge of the plume always approaches the ambient unsaturated vapor content,
the portion of the plume cross-section that is visible varies depending on
the degree of mixing, ambient conditions, etc. It is possible then to have
average plume properties, as predicted by top-hat profiles, that are unsatura-
ted and thus invisible while in actuality there is still a portion of the
plume that is visible. The Macduff-Davis model will properly handle this
problem, but it is a complicated model, especially when the plumes are merging
in the zone of flow establishment. Although the merging process proposed in
this model has been shown to work well with submerged diffusers (23), it has
not been compared to sufficient cooling tower data to know if it improves
prediction over simplier models.
Briggs (122) handles multiple cooling tower plumes and the effects of
merging by simply adjusting the plume rise of a single plume using an "enhance-
ment coefficient" given by,
Z =
m
ZS<1
N + S
1/3
+ S
)
(39)
26
-------
£ 0.8
LIQUID,
-------
merits of plume characteristics in the field using tethered balloons (118)
(132). Laboratory studies are valuable in studying individual variables
under controlled conditions as well as structural and local terrain effects.
Many of the empirical constants used in plume modeling are best determined
using laboratory experiments. It is difficult, however, to simulate actual
atmospheric stability conditions in the laboratory and care must be taken to
insure that correct scaling laws are followed. Good discussions of scale
effects in cooling tower model studies are found in references (1_33) and
(134).
Field measurements are difficult to obtain. However, photographic
studies of visible plume length have been quite satisfactory. Aircraft
flythrough measurements of dilution are difficult to evaluate due to instru-
ment lag and the effects of large scale ambient turbulence on plume motion.
Tethered balloon techniques using suspended instrument packages seem to work
well in giving time average values, however, the data collected by this
method to date is minimal. This method has the advantage of being able to
measure the properties in the invisible portion of the plume but positioning
the instrument package in the plume is sometimes limited by local terrain,
power lines, buildings, etc. In addition, the correlation between different
traverses of the instrument package through the plume only have meaning if
the wind is holding fairly constant.
The collection of field data should be continued for a wide variety of
ambient and discharge conditions so that plume models can be evaluated from a
firm data base.
Numerical Cooling Tower Plume Models: Numerical cooling tower plume models
have the same advantages and suffer from the same disadvantages as numerical
water plume models. There are several in the literature but due to the
proprietary nature of most, only a few give details of the models. The EG&G
model developed by Weinstein and Davis (135) is probably the best documented,
having been developed under an EPA contract. It has a fairly good treatment
of plume thermodynamics and cloud physics but the entrainment assumptions are
questionable and only calm atmospheric conditions are modeled. As a result,
this model has mainly been used as a stepping stone to better models. These
include the models of Stephen and Moroz (136), the Swiss-developed SAUNA and
KUMULUS (137, 138) models, and the Systems Science and Software model (139).
These later models are more sophisticated and from what can be found in the
literature, they appear to consider nearly all the factors influencing plume
characteristics including droplet formation and plume vorticity. An indepen-
dent evaluation of them, however, has not been conducted to the authors
knowledge.
ACKNOWLEDGEMENT
The authors wish to acknowledge the information personally provided them
by various researchers. They especially express their appreciation to Mr.
Maurice Ades, Department of Nuclear Engineering, Oregon State University,
whose many hours of unselfish assistance made this paper possible.
28
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