WATER POLLUTION CONTROL RESEARCH SERIES •
16130FDQ03/71
HEATED SURFACE
JET DISCHARGED INTO A FLOWING
AMBIENT STREAM
U.S. ENVIRONMENTAL PROTECTION AGENCY
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WATffi POLLUTION CONTROL RESEARCH SIEIES
The Water Pollution Control Research Series describes the
results and progress in the control and abatement of pollution
in our Hation's waters. They provide a central source of
information on the research, development and demonstration
activities in the Sivironraental Protection Agency, through
inhouse research and grants and contracts with Federal,
State, and local agencies, research institutions, and
industrial organizations.
Inquiries pertaining to Water Pollution Control Research
Reports should be directed to the Chief, Publications Branch
(Water), Research Information Division, R&M, Efavironmental
Protection Agency, Washington, B.C. 20^60.
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HEATED SURFACE JET DISCHARGED INTO A
FLOWING AMBIENT STREAM
by
Louis H. Motz
Barry A. Benedict
National Center for Research and Training in
the Hydrologic and Hydraulic Aspects of
Water Pollution Control
Vanderbilt University
Nashville, Tennessee
for the
Environmental Protection Agency
Grant # 16130 FDQ
March, 1971
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EPA Review Notice
This report has been reviewed by the Environmental Protection
Agency and approved for publication. Approval does not
signify that the contents necessarily reflect the views and
policies of the Environmental Protection Agency nor does
mention of trade names or commercial products constitute
endorsement or recommendation for use.
c by the Siipermtcihlciit of Documents, I'.S. (iovcrnmrat Printing Ollicc, Washington. ]>.(.'. 204IK - Price *1.7~>
11
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ABSTRACT
The temperature distribution in the water body due to a discharge of
waste heat from a thermal-electrical plant is a function of the hydro-
dynamic variables of the discharge and the receiving water body. The
temperature distribution can be described in terms of a surface jet dis-
charging at some initial angle to the ambient flow and being deflected
downstream by the momentum of the ambient velocity. It is assumed that in
the vicinity of the surface jet, heat loss to the atmosphere is negligible.
It is concluded that the application of the two dimensional surface jet
model is dependent on the velocity ratio and the initial angle of discharge,
and the value of the initial Richardson number, as low as 0.22. Both
laboratory and field data are used for verification of the model which has
been developed. Laboratory data is used to evaluate the two needed
coefficients, a drag coefficient and an entrainment coefficient, as well
as the length of the zone of flow establishment and the angle at the end
of that zone. The drag coefficient and characteristics of the establish-
ment zone are found to be functions of the velocity ratio (ambient velocity/
jet velocity), while the entrainment coefficient is primarily a function
of geometry.
This report was submitted as a portion of the work under Grant
No. 16130 FDQ between the Federal Water Quality Administration, now in
the Environmental Protection Agency, and Vanderbilt University.
KEY WORDS
Cooling Water, Thermal Pollution, Jets, Turbulent Flow, Heated Water,
Thermal Power Plants, Thermal Stratification, Diffusion, Water Temperature,
Heat Exchange, Temperature
111
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ACKNOWLEDGMENTS
Special thanks are extended to Mr. E. M. Polk, Jr., who directed the
Vanderbilt University field surveys and made available much of the field
data used in this report.
Major thanks are also due to Dr. F. L. Parker for general technical
report guidance, to Mmes. Beverly Laird and Peggie Bush for typing the
report, and to the laboratory staff personnel for their outstanding
cooperation.
The investigations described herein were supported by the Federal
Water Quality Administration, of the Environmental Protection Agency,
through its establishment of the National Center for Research and Training
in the Hydrologic and Hydraulic Aspects of Water Pollution Control, con-
tract number 16130 FDQ. This support took the form of a research assis-
tantship, equipment, and computer time. During a portion of the work, the
senior author held a traineeship grant from the National Aeronautics and
Space Administration. Grateful acknowledgment is made for the financial
support from these groups.
Appreciation is also extended to the Tennessee Valley Authority for
their cooperation in conducting field surveys, supervised by Mr. Polk.
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TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS v
LIST OF TABLES ix
LIST OF ILLUSTRATIONS x
Chapter
I. INTRODUCTION 1
Effects of Heated Discharges 2
Temperature Distribution 2
Description of the Problem 5
II. PREVIOUS WORK 7
Submerged Jets 7
Surface Jets 13
Justification for Present Study 21
III. ANALYTICAL DEVELOPMENT 24
Assumptions 24
Shape of the Surface Jet 25
Conservation Equations 27
Velocity and Temperature Profiles 33
Solution of the Equations 37
Zone of Flow Establishment 46
Summary of Analytical Development 48
IV. LABORATORY EXPERIMENTS 50
Modeling 50
Laboratory Equipment and Procedure 53
V. RESULTS OF THE LABORATORY EXPERIMENTS 60
Relation of Circular Jet to Half-Width bo 60
Laboratory Measurements 62
Analysis of Data 64
Presentation of Results 75
VI. FIELD SURVEYS 87
Description of VU Surveys 87
VI1
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TABLE OF CONTENTS (Continued)
Chapter Page
Results of the Widows Creek Surveys 90
Results of the New Johnsonville Survey 114
Results of the Waukegan Survey 126
Comparison of Results 131
VII. DISCUSSION 133
Results of the Laboratory and Field Investigation 133
Possible Sources of Error 140
Application 142
Usefulness of the Proposed Model 144
VIII. SUMMARY AND CONCLUSIONS 145
Analytical Development 145
Laboratory Experiments 147
Field Surveys 147
Results of the Laboratory and Field Investigation 148
Application 149
Future Work 150
Appendix
A. LABORATORY LATERAL TEMPERATURE MEASUREMENTS, T'/TO 154
B. LOCATION OF LABORATORY TRAJECTORIES 159
C. FIGURES 46-73: TRAJECTORIES AND TEMPERATURE AND WIDTH PLOTS
FOR LABORATORY EXPERIMENTS 164
D. PRACTICAL APPLICATION EXAMPLE. 195
E. LIST OF SYMBOLS 200
LIST OF REFERENCES 203
vin
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LIST OF TABLES
Table
- 1. Laboratory Measurements
2. Laboratory Results ™
3. Results of Statistical Tests on Parameters 84
4. Relation of E to 6^ 86
5. Vertical Profile Statistics, VU Survey Number 1 101
6. Data, VU Survey Number 1 HO
7. Parameters, VU Survey Number 1 HI
8. Data, VU Survey Number 2 and TVA Survey 114
9. Parameters, VU Survey Number 2 and TVA Survey 115
10. Velocity Data, New Johnsonville Survey 120
11. Temperature Data, New Johnsonville Survey 122
12. Parameters, New Johnsonville Survey 123
13. Data, Waukegan Survey 129
14. Parameters, Waukegan Survey 129
15. Summary of Field Results 132
16. Assumed Values for Design Problem i94
IX
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LIST OF ILLUSTRATIONS
Figure Page
1. Vertical Entrainment versus Richardson Number 15
2. Temperature Rise Along Jet Axis at Water Surface 17
3. Temperature Concentration Along Jet Axis at Water Surface . . 19
4. Definition Sketch 28
5. Volume Flux and Momentum Flux versus Non-Dimensional Jet
Axis Distance 43
6. Temperature, Velocity, and Width Ratios versus
Non-Dimensional Jet Axis Distance 44
7. Effect of Reduced Drag Coefficient on Location of Jet
Trajectory 45
8. Zone of Flow Establishment 47
9. Jets and Temperature Probes 56
10. Surface Jet and Rotameter 56
11. Temperature Probes and Other Laboratory Equipment 57
12. Observed Temperature Distribution and Fitted Gaussian Curve,
Run 3-60-1 66
13. Temperature Distribution at s'/bo = 35.3, Run 3-60-1 67
14. Vertical Temperature Profiles, Run 1-60 68
15. Two-Dimensional Surface Jet, Run 3-90 69
16. Observed and Fitted Trajectories, Run 1-90 76
17. Observed Values and Fitted Curves for Temperature and Width,
Run 1-90 77
18. Observed Values and Fitted Curve for Length of Establishment
Zone versus Velocity Ratio 80
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LIST OF ILLUSTRATIONS (Continued)
Figure Page
19. Observed Values and Fitted Curve for Initial Angle versus
Velocity Ratio 81
20. Observed Values of Reduced Drag Coefficient versus Velocity
Ratio 82
21. Observed Values and Fitted Curve for Drag Coefficient
versus Velocity Ratio 83
22. Observed Values and Fitted Curves for Drag Coefficient
versus Velocity Ratio and Discharge Angle 85
23. Temperature Distribution, °F, at 1.0-foot Depth, Widows
Creek, VU1 92
24. Temperature Rise Along Jet Axis at Cross-Sections R-l to
R-3 93
25. Temperature Rise Along Jet Axis at Cross-Sections R-4 to
R-7 94
26. Temperature Distribution, °F, in Cross-Section R-l 95
27. Temperature Distribution, °F, in Cross-Section R-2 96
28. Temperature Distribution, °F, in Cross-Section R-5 97
29. Velocity Profiles at Cross-Section R-l 99
30. Velocity Profiles at Cross-Section R-2 106
31. Observed and Fitted Trajectories, Widows Creek, VU1 108
32. Observed Values and Fitted Curves for Temperature and Width,
Widows Creek, VU1 109
33. Temperature Distribution, °F, at 1.0-Foot Depth, Widows
Creek, VU2 112
34. Temperature Distribution, °F, at 0.5-Foot Depth, Widows
Creek, TVA 113
35. Observed and Fitted Trajectories, Widows Creek, VU2 116
36. Observed Values and Fitted Curves for Temperature and Width,
Widows Creek, VU2 117
XI
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LIST OF ILLUSTRATIONS (Continued)
Figure Page
37. Observed and Fitted Trajectories, Widows Creek, TVA 118
38. Observed Values and Fitted Curves for Temperature and
Width, Widows Creek, TVA 119
39. Observed Temperature Distribution, °F, at 1.0-Foot Depth,
New Johnsonville 121
40. Observed and Fitted Trajectories, New Johnsonville 124
41. Observed Values and Fitted Curve for Temperature, New
Johnsonville 125
42. Temperature Distribution, °F, at 1.0-Foot Depth, Waukegan. . 127
43. Observed Values and Fitted Curves for Temperature and
Width, Waukegan 130
44. Observed Field Values of Drag Coefficient versus Velocity
Ratio Plotted on Laboratory Curve 137
45. Observed Values of Drag Coefficient versus Reynolds Number . 139
46. Observed and Fitted Trajectories, Run 2-90 165
47. Observed Values and Fitted Curves for Temperature and
Width, Run 2-90 166
48. Observed and Fitted Trajectories, Run 3-90 167
49. Observed Values and Fitted Curves for Temperature and Width,
Run 3-90 168
50. Observed and Fitted Trajectories, Run 4-90
169
51. Observed Values and Fitted Curves for Temperature and Width,
Run 4-90 170
52. Observed and Fitted Trajectories, Run 5-90 171
53. Observed Values and Fitted Curves for Temperature and Width,
Run 5-90 I72
54. Observed and Fitted Trajectories, Run 1-60 173
55. Observed Values and Fitted Curves for Temperature and Width,
Run 1-60 ' 174
XII
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LIST OF ILLUSTRATIONS (Continued)
Figure Fa§e
56. Observed and Fitted Trajectories, Run 2-60 175
57. Observed Values and Fitted Curves for Temperature and Width,
Run 2-60 176
58. Observed and Fitted Trajectories, Run 3-60 177
59. Observed Values and Fitted Curve for Temperature and Width,
Run 3-60 178
60. Observed and Fitted Trajectories, Run 4-60 179
61 Observed Values and Fitted Curves for Temperature and Width,
Run 4-60 18°
62. Observed and Fitted Trajectories, Run 5-60 181
63. Observed Values and Fitted Curves for Temperature and Width,
Run 5-60 182
64. Observed and Fitted Trajectories, Run 1-45 183
65. Observed Values and Fitted Curves for Temperature and Width,
Run 1-45 184
66. Observed and Fitted Trajectories, Run 2-45 18S
67 Observed Values and Fitted Curves for Temperature and Width,
Run 2-45 186
68. Observed and Fitted Trajectories, Run 3-45
69 Observed Values and Fitted Curves for Temperature and Width,
Run 3-45 188
189
70. Observed and Fitted Trajectories, Run 4-45
71. Observed Values and Fitted Curves for Temperature and Width,
Run 4-45
191
72. Observed and Fitted Trajectories, Run 5-45
73. Observed Values and Fitted Curves for Temperature and Width,
Run 5-45
74. Predicted Values of Temperature and Width 197
75. Predicted Trajectory and Width 198
xiii
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CHAPTER I
INTRODUCTION
The determination of the temperature distribution of power plant
condenser cooling water discharges is of immediate interest. Almost one
half of all the water used in the United States is utilized for cooling
and condensing by the power and manufacturing industries. In 1964, the
cooling water intake was 50 x 1012 gallons, of which 80% was used by the
electric power generating industry, according to the Federal Water
Pollution Control Administration (FWPCA) (23). Presently, power is
generated by hydro- and steam-electric plants, with the latter requiring
the cooling'water for dissipation of waste heat. Since the remaining
sites which are suitable for hydro-electric plants are limited, steam-
electric plants will have to be increasingly relied on for future needs.
Of these, fossil-fueled plants operate at about 40% thermal efficiency,
and nuclear plants operate at about 33% efficiency. These low effi-
ciencies result in a large part of the heat produced by a steam-electric
plant being wasted to the atmosphere or into water bodies. Based on
the projected need for electric power, heat rejection from fossil- and
nuclear-fueled plants is expected to increase almost ninefold by the
year 2000. Thus, based on the present and future requirements for
cooling water, the effects of heated discharges should be examined.
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Effects of Heated Discharges
The addition of heated discharges to a water body can have chem-
ical, biological, and physical effects. The chemical and biological
effects of increased temperature on water quality, aquatic life, and
waste assimilative capacity have been noted by several authors. Ac-
cording to the FWPCA (23), which has defined the addition of waste heat
as thermal pollution, the dissolved oxygen may be reduced, chemical re-
actions increased, tastes and odors made more noticeable, and the rate
of oxygen depletion by organic wastes increased. According to Clark (15),
fish are particularly sensitive to changes in the thermal environment be-
cause, as cold-blooded animals, they are unable to regulate their body
temperature and can be harmed by an increase or decrease in their meta-
bolic rate. Cairns (11) has stated that large quantities of heat added
to a stream will cause all but the very tolerant forms of fish and other
aquatic life to disappear and may seriously impair the stream. Krenkel,
Thackston, and Parker (32) have cited evidence that an increase in stream
temperature due to an electric generating plant's heated discharge has
the same end result in terms of reduced waste assimilative capacity as
adding an equivalent amount of sewage or other organic waste to the river.
Since chemical and biological effects are a function of the temperature
distribution, the physical effects of heated discharges should be examined
Temperature Distribution
The physical effect, or the temperature distribution, is a function
of hydrodynamic and meteorological variables. The temperature distribu-
tion problem can be divided into two parts which are analyzed almost
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separately. According to Edinger (17) , the first part is the initial
mixing, or dilution, of the discharge, and the second is atmospheric
cooling. Edinger has stated that these two parts have different length
and time scales: mixing takes place in the immediate vicinity of the
discharge and affects a small portion of a water body, while atmospheric
cooling is relied on after mixing takes place and affects a larger por-
tion of a water body farther downstream.
Atmospheric Cooking
Heat transfer across the air-water interface requires a large sur-
face area and/or long time periods. Calculations, using standard heat
transfer rates and based on the work of Edinger and Geyer (19), indicate
that, for many water bodies, the downstream distance required for cooling
is on the order of miles. The distance is particularly large for a
relatively narrow stream whose width does not exceed 1000 feet or so,
and, thus, whose surface area is small. Edinger and Polk (20) have
described the initial mixing of heated discharges and have presented
data which tend to validate the assumption that atmospheric cooling can
be neglected in the vicinity of the discharge, at least as a first
approximation.
Mixing
The temperature distribution in the immediate vicinity of the dis-
charge can take several forms. Brooks (9) has noted two extremes: sur-
face spreading of hot water with minimal mixing, and extensive jet mixing
of the effluent with the receiving water. Churchill (14) has mentioned
that at several large Tennessee Valley Authority (TVA) steam plants, the
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stream velocity is not sufficiently high to mix the heated discharge,
and the discharge flows out across the receiving water as a density
overflow. Harleman and Elder (27) have noted that thermal stratification
can develop where part of the heated water intrudes upstream as a heated
layer and may be recirculated through the power plant unless a skimmer
wall is built at the intake channel. Bata (5) and Harleman (26) have
analytically described the formation of the upstream stratified layer.
Harleman (26) has also described the design of horizontal diffusers
placed across the bottom of the stream which completely mix the heated
discharge by entraining river water. Edinger and Polk (20) have studied
the lateral and vertical mixing in a uniform current from the point of
surface discharge to a completely mixed condition downstream. In most
literature dealing with mixing of surface discharges, it is assumed
that the mixing, if it occurs, is due to the velocity and associated
turbulence of the receiving water body. Also, most temperature dis-
tribution models based on the basic conservation of heat equation con-
sider only a uniform velocity field in the receiving water body.
Surface Jets
In some cases, the spatial distribution of temperature is a func-
tion of the velocity of the discharge as well as the velocity of the am-
bient stream. The velocity field of the ambient stream is no longer
uniform but is influenced by the velocity field of the cooling water in
the immediate vicinity of the discharge. In these cases, cooling water
discharged at the surface can have the characteristics of a surface jet
if the discharge flow possesses sufficient momentum. Some work on the
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problem of surface jets has already been done by Jen, Wiegel, and
Mobarek (29), Zeller (48), Carter (12), and others. However, this
problem needs more study in order to better analyze and predict the
spatial distribution of temperature in the vicinity of thermal-electric
generating plants.
Description of the Problem
When waste heat from a thermal-electric power plant is discharged
into a receiving water body, the temperature distribution in the water
body is a function of the hydrodynamic variables of the discharge and
the receiving water body. When the velocity field of the water body is
influenced by the velocity of the heated discharge, and the heat is
initially advected almost perpendicularly to the river flow, then the
temperature distribution can be described in terms of a surface jet
discharging at some initial angle to the ambient flow and being deflected
downstream by the momentum of the ambient velocity. The decrease in
temperature rise along the jet axis is due to the entrainment of colder
ambient water into the jet as the jet spreads laterally and, to some
extent, vertically.
Complete solution of the temperature distribution problem will not
be attempted in this study. The present investigation will consider
only the problem of surface jets. At some distance downstream from the
discharge, the jet velocity will have been decreased until it is equal
to the ambient velocity. The decrease in the remaining temperature excess
from this point could be determined from the temperature distribution
models which consider only the ambient, or river, velocity and ambient
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turbulence. However, the complexity of the two phenomena, neither of
which is at present completely understood, precludes immediate combina-
tion of the two in a model. At some farther distance downstream, after
mixing takes place, surface exchange of heat becomes important, but it
is assumed that, in the vicinity of the surface jet, heat loss to the
atmosphere is negligible.
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CHAPTER II
PREVIOUS WORK
A literature review by Krenkel and Parker (31) describing the many
aspects of the mechanisms and modeling of heated discharges is readily
available and will not be duplicated here. Instead, the previous work
done on submerged and surface jets will be examined in detail to provide
a basis for the development of an analytical model describing surface
jet discharges.
Subm erged Jets
Schlichting (41) has presented solutions for submerged plane and
axi-symmetric jets. These solutions require assumptions based on the
mixing length theories of Prandtl, von Karman, and Taylor.
Abramovich (2) has discussed solutions for simple jets and for jets
in a parallel ambient stream. For jets deflected by a cross-flow, ref-
erences are made to empirical relations, to superimposing the stream
functions of the jet and the external flow, and to a method which
balances the force caused by the pressure difference at the forward and
back surface of the jet by a centrifugal force.
Albertson, Dai, Jensen, and Rouse (4) have presented a model which
describes the behavior of a simple air-in-air jet. Analytic expressions
for the distributions of velocity, volume flux, and energy flux were
developed for the patterns of mean flow within submerged jets from both
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slots and orifices. A single experimentally-determined coefficient
was evaluated.
Entrainment
Taylor (44) made a simple transfer assumption of entrainment which
relates the inflow into the edge of a plume to a characteristic velocity
in the plume. This assumption retains the broad outline of the mechanics
of a rising, buoyant plume without the necessity for understanding in
detail how the turbulent eddies mix the heated and the ambient air. This
concept of entrainment has made possible the solution to many problems
describing buoyant and momentum jets which are difficult, if not impossible,
at present, to solve by other means.
Entrainment of a Buoyant Plume
Morton, Taylor, and Turner (34) used the concept of entrainment to
develop relations predicting the behavior of a buoyant plume rising
through a fluid with a linear density gradient. Three main assumptions
were made in deriving the prediction equations :
(1) The profiles of vertical velocity and buoyancy are similar
at all heights and are Gaussian, or, as shown in Equations 1 and 2,
u(x,r) = u(x) exp(-r2/b2) (1)
and
g[p - p(x,r)] g[p - p(x)]
exp (-rW) (2)
where 1 u(x,r) = the vertical velocity;
u(x) = the centerline velocity;
Dotation used in this chapter is unique to this chapter.
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x,r = the cylindrical coordinates with the x axis vertical;
b = the width of the plume;
p = the ambient density;
p(x,r) = the plume density;
p (x) = the centerline plume density; and
Pl = a reference density.
(2) The rate of entrainment of fluid at any height is proportional
to a characteristic velocity at that height, or, as shown in Equation 3,
-~^- - 2-rrbau (3)
dx
where Q = the plume flowrate; and a = the experimentally-determined en-
trainment coefficient.
(3) The fluids are incompressible, and local variations in density
are small compared to a reference density.
Based on the conservation of volume, momentum, and heat energy, a
system of ordinary differential equations was developed. The conservation
equations were written as shown in Equations 4-6,
Volume:
^- (>b2u) = 2irbau (4)
dx ,
Momentum:
~ (irb2u2) = 2TTb2g(p0 - P) (5)
Density Deficiency:
^— [Trb2u(pi - p)] = 2irbau(p1 - p0) (6)
Morton et al. (34) wrote the heat energy equation in terms of the density
difference instead of the temperature rise for a more unified treatment
of all types of convection problems. The temperature field is expressed
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10
indirectly in terms of the equivalent density deficiency.
Jets, Plumes, and Wakes
Morton (33) showed that jets, plumes, and wakes could be related by
means of a momentum-mass flux diagram. Using a simple model based on the
concept of entrainment, the relation was found from the solution of a
single, ordinary differential equation.
Morton's treatment was based on a common set of assumptions applied
to jets, plumes, and wakes. Mean cross-sectional profiles of velocity
were assumed similar along the axis. Longitudinal dispersion was assumed
negligible compared to lateral dispersion, making possible the usual
boundary layer assumptions for free turbulent shear flows. The flow was
assumed to be affected by density differences only in the form of
buoyancy forces. Entrainment, or turbulent mixing of the jet and the
ambient fluid, was represented by an inflow velocity across the jet
boundary, and this inflow was assumed to be proportional to the dif-
ference between a characteristic velocity along the axis and the velocity
of the ambient fluid.
Equations representing the conservation of mass, momentum, and den-
sity deficiency, similar to those used by Morton et_ al. (34), were written.
The non-dimensional variables, volume flux and momentum flux, and the den-
sity deficiency were defined, and then solutions representing a simple jet,
a jet in a uniform current, a buoyant jet, a simple plume in a stratified
environment, a buoyant jet projected along a uniform stream, a simple wake,
a forced wake, and a buoyant forced wake were presented in which the mo-
mentum flux, M, was a function of the volume flux, V. Morton (33) called
this M vs. V curve a momentum-mass diagram.
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Jet Trajectory
Bosanquet, Horn, and Thring (7) studied a negatively buoyant, sub-
merged jet flowing into a lighter, non-flowing ambient fluid. The jet
trajectory was predicted in terms of the initial angle of the discharge,
the initial velocity, the density ratio, and the nozzle diameter. The
predicted axis was shown to compare favorably with values obtained from
the water-in-water model.
Cross Flows
Keffer and Baines (30) studied the flow of a vertical air-in-air
jet directed normally to a uniform, steady ambient current for velocity
ratios (jet/ambient velocity) of 2, 4, 6, 8, and 10. The integrated
X.
equations of continuity and motion along the deflected jet axis were made
non-dimensional after the general method of Morton (33). Entrainment
was defined in terms of an inflow velocity, v., which was assumed pro-
portional to the difference between the centerline jet velocity, U, and
the ambient velocity, U . The entrainment relation was written as shown
a
in Equation 7,
v = E(U -U ) (7)
J. d
Keffer and Baines observed that the ambient flow was decelerated
at the upstream surface of the jet, creating a positive pressure region,
and that separation occurred at the rear, creating a negative pressure
region.
They also studied the effects of the velocity ratio on the zone of
flow establishment, which is defined as the region in which the velocity
distribution changes from a uniform distribution at the jet nozzle to a
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12
fully developed Gaussian distribution at the beginning of the zone of
established jet flow. They observed that at low velocity ratios (2 and 4)
the establishment zone was deflected downstream by the pressure field,
and the beginning of the established flow region was at an angle less
than 90° to the ambient current. At these low ratios, the jet was ob-
served to cling to the wall, and entrainment may have been restricted by
the proximity of a solid surface. For velocity ratios greater than 4,
the beginning of the established flow region was observed to be approx-
imately above the center of the jet orifice. For these larger ratios,
the effect of the pressure field was mainly to change the cross-section
from a circular shape at the orifice to a distorted kidney shape at the
end of the zone of flow establishment. In all of the cases studied, a
pair of vortices was observed along the jet axis. Keffer and Baines
felt that the vortices were caused by the separation of the ambient flow.
Keffer and Baines observed that the velocity excess for the jet in
an ambient current decreased much more rapidly than that reported by
Albertson _et_ al. (4) for a simple jet where the ambient velocity is zero,
and that the rate of decrease increased with distance from the source.
They felt that the entrainment was augmented by the twin vortices, which
do not exist in the free jet where the ambient velocity is zero. Also,
they observed that the lateral spreading did not appear to be affected
by the vortices. They reported that the entrainment coefficient, E,
which was determined by fitting the observed velocities measured with a
hot-wire anemometer to the predicted values, varied along the jet axis
from 0.3 to 1.6 and also varied as a function of the velocity ratio.
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13
Buoyant Jet in a Cross Flow
Fan (22) used Morton's work (33) as a basis for studying an in-
clined, round buoyant jet in a stagnant environment with linear density-
stratification and for studying a round buoyant jet in a uniform cross
stream of homogeneous density. Extending the integral technique of
analysis, for the case of a uniform cross stream, Fan included the effect
of an initial angle of discharge at the end of the zone of flow establish-
ment that was not 90° to the ambient flow, presented empirical relations
describing the zone of flow establishment, assumed an entrainment
mechanism based on the vector difference between a characteristic jet
velocity and the ambient velocity, and approximated the effect of the
pressure gradient across the jet parallel to ambient current by a drag
coefficient. From laboratory experiments in which the jet Froude Number
was varied from 10 to 80 and the velocity ratio (jet/current) varied
from 4 to 16, the entrainment coefficient and the drag coefficient were
found to vary from 0.4 to 0.5 and from 1.7 to 0.1, respectively. Fan
determined the entrainment coefficient from the jet centerline dilution
ratios and noted that the cross-sectional concentration profiles were
horse-shoe shaped with the maximum concentration at two sides of the
plane of symmetry. The maximum concentrations were from 1.60 to 1.80
times greater than the values at the jet axis, or centerline.
Surface Jets
Jets discharged at the surface of a heavier fluid have been studied
by several authors. The jet fluid, usually water, was the same as the
ambient fluid except that its density was less. This was accomplished
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14
by heating the jet fluid or by increasing the density of the ambient
water by adding salt.
Vertical Entrainment and the Richardson Number
Ellison and Turner (21) studied the entrainment process in strati-
fied flows and found that vertical entrainment could be expressed as a
function of the Richardson Number. In the case of a surface jet with
the ambient velocity zero, the inflow velocity into the turbulent jet
region was assumed to be proportional to the velocity of the surface jet.
The constant of proportionality was called the entrainment coefficient, E.
Ellison and Turner performed a series of laboratory experiments
using a salt solution to increase the density of the ambient fluid. With
no density difference, the flow formed a simple, two-dimensional half-jet,
the depth of which increased nearly linearly with distance. When the
ambient fluid was heavier than the jet fluid, the rate of increase of
depth became smaller and at some distance downstream, depending on the
density difference and the rate of flow, the jet depth changed very-
little with distance. These experiments also demonstrated that, in most
practical cases, the surface layer will attain an equilibrium state in
which the Ri Number does not vary with distance downstream. The observed
relation between E and the Ri Number is shown in Figure 1.
Small Richardson Numbers
Jen, Wiegel, and Mobarek (29) performed laboratory studies on
the mixing of heated, buoyant jets discharging horizontally at the
surface of a large body of initially quiescent receiving water. The
-------
15
FIGURE 1.--VERTICAL ENTRAINMENT VERSUS RICHARDSON NUMBER
[After Ellison and Turner (21)]
-------
16
initial Ri Number of the jets was defined as shown in Equation 8,
Apgd'
Ri = - ° (8)
where Ap = the density difference;
g = the acceleration of gravity;
p = the ambient density;
d = the jet diameter; and
U = the initial jet velocity.
In these studies, the Ri Number was small, i.e., 5.0 x 10~5 < Ri <
o — o —
3.0 x 10~3.
Jen jrt &\_. observed that the jet mixed less with depth than it did
laterally by a factor of about two. They attributed this difference to
the buoyancy force, which they felt inhibits the vertical component of
the turbulent velocity fluctuations but does not inhibit the lateral
component. An empirically-determined equation for the temperature along
the jet axis was presented.
Larger Richardson Numbers
Taraai, Wiegel, and Tornberg (43) studied the same type of buoyant
jets as Jen et_ al_. (29) except that the Ri Number was larger, i.e.,
1.0 x 10" 2 _<_ Ri ^ 2.0 x 10" l. A comparison of the data taken at these
higher Richardson Numbers with the empirical equation of Jen _e_t al. ,
plotted in Figure 2, showed that the temperature rise along the jet
centerline was generally greater than predicted for values of Ri
approaching 2 x lO'1.
-------
1.0
0.5
T/TO
o.i
.05
.01
TJ
o o
A A
A
O
0.16
0.09
0.15
0.008
BEST FIT LINE
JEN, WIEGEL, & MOBAREK (29)
TAMAI, WIEGEL, & TORNBERG (43)
RU
10
x/d
50
100
500 1000
o
FIGURE 2.--TEMPERATURE RISE ALONG JET AXIS AT WATER SURFACE
[After Tamai, Wiegel, and Tornberg (43)]
-------
18
Approximate Theory
Hayashi and Shuto (28) also studied the diffusion of warm water jets
discharged horizontally at the surface of an initially quiescent water
body. The initial temperature was from 0.2°C to 28°C higher than the
receiving water, and the Ri Number of the jets varied from 0.004 to 0.54.
An approximate theory was developed in which the non-linear inertial
terms representing the square of the jet velocity in the equation of
motion were neglected, and an approximate solution was presented for the
case of no vertical entrainment. The experimental results were compared
with the approximate theory, and the effect of the Richardson Number on
vertical entrainment was in agreement with the work of Ellison and
Turner (21). Figure 3 shows a comparison between the approximate theory
and one set of experimental data of Hayashi and Shuto.
Similarity Criteria
Stefan (42), using the dimensionless equation of motion, continuity,
and heat transfer, presented similarity criteria for the flow of heated
water over a stagnant and colder body of water. He felt that turbulent
flow in the model, matching of reduced gravitational forces and surface
heat losses as well as sufficient size of the model were the most es-
sential modeling requirements.
Integral Technique
Zeller (48) developed a two-dimensional mathematical model based
on the work of Morton (33) and Fan (22) to describe the observed tempera-
ture distribution offshore from a steam-electric generating plant which
discharged heated water through two outfalls 640 feet apart. According
-------
1.0
0.5
T/T,
o
o o
« u
SS88
THEORETICAL CURVE FOR NO
VERTICAL ENTRAPMENT
— O
8
o
0.2
o.i
0
1000
2000
3000
4000
5000
6000
7000
FIGURE 3.--TEMPERATURE CONCENTRATION ALONG JET AXIS AT WATER SURFACE
[After Hayashi and Shuto (28)]
-------
20
to Zeller, field surveys indicated that, beyond a region close to shore,
the warm water from each outfall spread as a two-dimensional surface jet
about 3 feet thick and could be followed for several thousand feet out
into the lake. Measurements of the temperature distribution were made
by traversing the jets in a boat, and jet velocities were determined by
tracking the paths of drogues.
Zeller developed a system of two-dimensional mass, momentum, and
heat energy conservation equations. Based on the work of Ellison and
Turner (21), vertical entrainment was assumed negligible because of the
stability of the surface jet at high Richardson Numbers. Lateral en-
trainment was assumed to be represented by the expression shown in
Equation 9,
|J = EzU (9)
which equates the increase in flow rate along the jet axis to the product
of the entrainment coefficient, E, the depth, z, and the jet velocity, U.
Zeller reported that the value of the entrainment coefficient varied from
0.127 to 0.993 for 22 field surveys.
Slot Jet
Carter (12) developed a mathematical model to describe the be-
havior of a two-dimensional slot jet discharged perpendicularly into a
flowing ambient stream. Equations representing the change in momentum
along the jet axis were developed, and the decrease in temperature rise
along the axis was described in terms of an empirically-measured dilution.
The solution to the equations predicted the location of the jet trajectory
in terms of empirically-determined coefficients and the dilution. The
-------
21
zone of flow establishment and the pressure gradient that exists across
the jet parallel to the ambient flow were both considered in the mathe-
matical model.
Justification for Present Study
A survey of the literature indicates that further study on surface
jets is justified. Previous work does not contain a completely suitable
method for analyzing and predicting the spatial temperature distribution
in the vicinity of power plants located on rivers when the discharge
velocity is important.
Laboratory Studies
The results of laboratory studies by Jen, et al. (29), and Tamai,
et al. (43) are applicable primarily to jets discharging into ambient
water bodies that have no appreciable velocity. The effect of a cross
flow on a submerged jet has been discussed by Keffer and Baines (30),
and this effect must be considered in the present case of a surface jet.
Theory
None of the theoretical models examined is completely satisfactory.
The model of Hayashi and Shuto (28) neglects the inertial terms in the
equation of motion, which is permissible only for velocities much less
than those normally encountered at a power plant discharge. Also, their
model is applicable only to a quiescent water body.
Zeller's (48) definition of entrainment, shown in Equation 9,
§ - EzU (9)
-------
22
is different from that used by Morton (33), Fan (22), and Keffer and
Baines (30) , all of whom have defined the entrainment in terms of the
difference between some characteristic jet velocity and the velocity of
the moving ambient current. Also, Zeller's model does not consider the
effect of the pressure gradient that exists across the surface jet
parallel to the ambient current in predicting the trajectory of the jet,
nor does it consider the zone of flow establishment.
The theoretical model developed by Carter (12) differs from others
in several respects. The decrease in temperature along the jet axis is
described in terms of an empirically-measured dilution, but the entrain-
ment mechanism causing the dilution is not specifically considered. The
equation of volume continuity is not used, and its integrated form,
the volume flux equation, does not appear. Since the continuity equation
is not used, in order to integrate the combined momentum equation which
contains two dependent variables and one independent variable, an
assumption has to be made concerning one of the dependent variables. In
order to graphically integrate the momentum equation, it is assumed that
the component of the jet velocity in the same direction as the ambient
velocity is equal to the ambient velocity everywhere in the zone of
established jet flow. The validity of this assumption, according to
Carter, is "...it appears from the data that the pressure field, the
length of the zone of flow establishment, and the velocity field may
adjust themselves so that at (x ,y ) [the beginning of the zone of
C G
established jet flow] the x-component of the jet velocity is equal to
the u [the ambient velocity]." In addition, the length of the zone of
3.
flow establishment is assumed constant with respect to the velocity
-------
23
ratio, which differs from the results reported by Fan (22), Keffer and
Baines (30), Pratte and Baines (36), and others. Finally, the increase
in the width of the jet cannot be predicted by the solution presented
by Carter.
Present Study
Therefore, using the work of Morton (33), Fan (22), and Zeller (48)
as a base, a more refined model of surface jets has been developed which
is able to describe certain cases of heated power plant discharges.
-------
CHAPTER III
ANALYTIC DEVELOPMENT
The present study, which is based on the Morton technique of
integral analysis, develops a system of ordinary differential equations,
which, when solved numerically, predicts the jet trajectory, width,
velocity, and temperature distribution for the case of a two-dimen-
sional surface jet.
Assumptions
Certain assumptions are made in the development of the mathe-
matical model:
(1) The jet is assumed to be two-dimensional.
(2) Profiles of velocity and temperature normal to the jet
axis are similar along the length of the jet axis.
(3) The flow regime is completely turbulent, which means that
molecular diffusion can be neglected, and that the flow is independent
of the Reynolds Number.
(4) Changes in density are small compared to a reference den-
sity. Thus, inertial forces due to density gradients are negligible,
and mass flux terms can be replaced by volume flux terms. This is
commonly called the Boussinesq assumption.
(5) Turbulent mixing into the jet can be represented by entrain-
ment, or an inflow velocity across the jet boundary.
24
-------
25
(6) Separation of the ambient flow around the jet can be repre-
sented by a drag force.
(7) Temperature losses in the region of the jet are small com-
pared to losses farther downstream and can be neglected.
Shape of the Surface Jet
Observation of warm water surface jets indicates that a surface
jet may be either two- or three-dimensional. For a jet to be two-
dimensional, only the width should increase, and the change in depth
along the axis should be zero. For a jet to be three-dimensional,
both the depth and the width should increase along the jet axis.
Primarily, two forces determine the extent of vertical spreading. The
inertial force, or the difference between the axial velocity of the jet
and the ambient velocity, creates the shearing force which entrains
ambient fluid, causing the jet boundary to spread laterally and
vertically. The buoyancy force due to the density difference between
the heated surface jet and the heavier ambient fluid opposes the vertical
spreading of the jet. When the inertial forces dominate, the jet will
spread equally in the vertical and lateral directions. However, when
the buoyancy is large, vertical spreading is suppressed.
Richardson Number
Quantitatively, vertical spreading, or vertical entrainment, is
an inverse function of the Richardson Number of the jet. When the
ambient velocity is not zero, the Richardson Number can be defined as
shown in Equation 10, or
-------
26
Apgd'
Ri = - ° - (10)
where p = density of the jet;
Ap = density difference between the jet and the ambient flow;
g = acceleration of gravity;
d = the diameter of the jet;
U = axial velocity of the jet; and
U = ambient velocity.
3.
When the inertial force is large, or the Richardson Number much less
than 1.0, vertical entrainment takes place. However, when the buoyancy
force is large, or the Richardson Number approximately 1.0 or greater,
vertical entrainment is small.
Field Cases
In many field cases, the density difference due to a heated dis-
charge is relatively large, and the square of the velocity difference is
relatively small, particularly when the initial jet velocity is not much
greater than the ambient velocity. Thus, the initial Richardson Number
of the jet is 1.0, or very close to 1.0. Theory and laboratory experi-
ments by Ellison and Turner (21) and field observations by Zeller (48)
indicate that the Richardson Number along the jet axis changes rapidly
from an initial value less than 1.0 to a value equal to or greater than
1.0, and that vertical entrainment decreases rapidly almost to zero
along the axis.
Thus, if the initial Richardson Number of the jet is 1.0, or close
to 1.0, vertical entrainment can be neglected, and the decrease in
-------
27
temperature along the jet axis can he approximately described in terms
of a two-dimensional surface jet deflected downstream by the ambient
velocity.
Conservation Equations
The differential equations for conservation of volume flux, momen-
tum flux, and heat energy can be integrated over the cross-sectional area
of a two-dimensional momentum jet to obtain integral equations based on
the method of Morton (33). A definition sketch is shown in Figure 4.
Conservation of Volume Flux
Integrating the volume continuity equation over the cross-section
gives the relation between the rate of change of volume flux along the
jet axis and the flow entrained into the jet across the outer edge as
shown in Equation 11,
v.dC (11)
where u = the jet velocity directed along the s axis;
v. = the inflow velocity;
dA = the differential area; and
C = the circumference through which entrainment takes place.
The inflow, or entrainment, velocity is assumed in this study to
be proportional to the magnitude of the difference between a characteris-
tic velocity along the jet and the parallel component of the ambient
velocity. If this entrainment is expressed as an equality, then the
inflow velocity can be written as shown in Equation 12,
-------
28
7
FIGURE 4.--DEFINITION SKETCH
-------
29
v. = E(U - U cos (3) (12)
1 3.
where U = the centerline velocity;
U cos B = the parallel component of the ambient velocity, U ;
a a
8 = the angle between the jet and the ambient current; and
E = the experimentally-determined entrainment coefficient.
Substituting this expression for entrainment, Equation 12, into
the continuity relation, Equation 11, gives the integral form of the
volume continuity equation, Equation 13,
d
j u dA = CE(U - U^cos 3) (13)
A
This definition of entrainment is different from the definition
used by Zeller (48), who assumed that entrainment was related to only
the jet velocity, as shown in Equation 9,
|f = EzU (9)
Zeller's definition is really applicable only when the jet is at
90° to the ambient flow, or when the jet velocity is an order of magni-
tude greater than the ambient velocity. Realistically, as the jet is
deflected, the parallel component of the ambient velocity becomes more
and more important. When the jet and ambient velocities are nearly
equal and parallel, mixing due to entrainment becomes quite small, be-
cause the velocity field is nearly uniform. Zeller's model does not
take this into account.
-------
30
Conservation of Momentum
The momentum equation along the jet axis can be integrated over
the cross-section to give the relation between the rate of change of
jet momentum flux, the rate of entrainment of ambient momentum flux,
and the force exerted by the pressure gradient across the jet. For
convenience, the resulting momentum equation is resolved into its
longitudinal and lateral components along the x and y axes, respectively.
The pressure gradient, due to the separation of the ambient flow, is
assumed to be represented by a drag force.
In the x-direction, the rate of change of momentum flux is equal
to the rate of entrainment of ambient momentum flux and to the x-com-
ponent of the drag term, or as in Equation 14,
^ U2dA
cos
= CE(U - U cos B) U + F sin & (14)
3. 3-D
where the drag term, F , is assumed to be related to the ambient velocity,
U , as shown in Equation 15,
a
CnU2 z sin
and where C is defined as the experimentally-determined drag coefficient,
In the y-direction, since all the ambient flow is oriented along
the x axis, the rate of change of momentum flux decreases due to the
y-component of the drag term, or as in Equation 16,
(16)
To compare the drag term used in the present study with the drag
terms used by Carter (12) for a two-dimensional slot jet and by Fan (22)
d
ds
u2dA sin 6
= -FD cos
-------
31
for a submerged ax i- symmetric jet, Equation 15 can be written in terms
of the drag force normal to the jet axis as shown in Equation 17,
Cp U2(zds) sin B
= a a - (17)
Carter defined the drag force, dP, normal to the jet axis in terms
of the ambient density, a drag coefficient, the ambient velocity, and
the projected area of the jet as shown in Equation 18,
C p U2zds
dp = U a a - (18)
Upon examining the definition sketch, Figure 4, it can be seen that at
distances downstream, where 3 approaches zero, the y-component of
Equation 18, or dP cos 3, would continue to deflect the jet by con-
tinuing to change the y- mom en turn flux. Thus, the trajectory of the jet
would be an ellipse. This is not realistic, since all the ambient
momentum is in the x-direction, and, as 3 approaches zero, all the jet
momentum is also in the x-direction. Thus, the y-momentum of the jet
could not continue to change, and the jet should remain parallel to the
ambient flow.
In comparison, the y-component of Equation 17, or (p F ds) cos 3,
a L)
becomes zero as the jet becomes parallel to the ambient flow, because
the projected area normal to the ambient current, or (zds sin 6) in
Equation 17, goes to zero. The addition of the sin 3 term makes the
drag force representation in this study more realistic, since the jet
remains parallel to the ambient flow as (3 goes to zero.
Fan used a sin23 term in his formulation of the drag force
normal to the jet axis as shown in Equation 19,
-------
32
Cnp U2sin2B (2/2 rds)
(F.ds) -
where r = radius of the axi- symmetric jet. The U2sin2B term is the
square of the ambient velocity component normal to the projected area.
This formulation differs from Equation 17, in which the component of
the projected area normal to the ambient velocity is considered.
According to Prandtl and Tietjens (35), pressure drag depends more on
the form of the body and on separation at the rear of the body than on
conditions at the front of the body. Thus, it is felt that the sin 3
term used in the present study is adequate to describe the conditions at
the front of the jet, and that Fan's sin26 term may be an unnecessary
refinement .
Conservation of Temperature
If the temperature can be treated as a conservative property,
which is reasonable considering the relatively small size of the jet
surface area, then the integrated temperature equation can be written as
in Equation 20,
±- | | uT' dA j = 0 (20)
where T is the temperature rise at any point in the cross-section. The
temperature rise is the difference between the jet and the ambient
temperatures .
Equations 13, 14, and 20 represent the general integrated forms
of the equations of volume, momentum, and heat flux.
-------
33
Velocity and Temperature Profiles
Profile Shape
The Morton technique of integral analysis requires an assumption
regarding the lateral distribution of velocity and temperature. Ac-
cording to Morton (33), the result of assuming similar profiles and the
form of the inflow velocity is to suppress analytic solution of the
details of the lateral structure of the jet. Thus, any reasonable pro-
file shape can be assumed in the theoretical model. One such shape
assumed by Morton is the laterally-averaged, or top-hat, profile in which
the temperature and velocity have constant values across the entire width
of the jet. However, the top-hat profile is not the most realistic
assumption, since profiles almost Gaussian in shape in turbulent jets
have been reported by Wiegel, Mobarek, and Jen (46), Cederwall (13), and
Samai (40) among others. Thus, the Gaussian profile seems to more
accurately describe the details of the lateral distribution and is the
shape assumed in this study.
This study also assumes that the lateral spreading of heat is the
same as the lateral spreading of momentum. For some cases of jets, de-
tailed velocity and temperature measurements indicate that the assumption
is not completely valid. Rouse, Yih, and Humphreys (39) report that the
spread of temperature is greater than the spread of velocity by a factor
of 1.16 for the case of a vertical buoyant jet. However, for other cases,
detailed velocity and temperature measurements are lacking. For example,
Fan (22) assumes that the spreading ratio is unity for the case of a jet
discharged into a flowing stream. Since no detailed velocity measurements
-------
34
were made in the present study, the same assumption —that the spreading
ratio is unity --is made here.
Continuity Equation
If the velocity distribution of the jet is approximated by a
Gaussian profile, then the velocity, u, at any point in the cross-
section can be related to the centerline velocity, U, by Equation 21, or
u = U exp(-n2/2o2) (21)
where o = the standard deviation; and
n = distance along the axis perpendicular to the s axis.
Substituting Equation 21 into the integral on the left side of the
continuity equation, Equation 13, gives Equation 22,
+b -t-00
U exp (-n2/2a2) zdn = Uz exp (-n2/2o2) dn (22)
-b -°°
where b = the jet half-width;
dA = zdn; and
z = jet depth.
The integral on the right side of Equation 22 is a definite integral
whose value is a/2rT . Equating this result to the right side of the
continuity equation, Equation 13, gives, for the continuity equation,
Equation 23,
|- (Uza/2T ) = 2zE(U - U cos 6) (23)
G.S 3-
where 2z = C, the circumference through which entrainment takes place.
If the half-width b is defined as in Equation 24,
b = a/2 (24)
-------
35
and this relation, Equation 24, substituted into equation 23, then the
continuity equation can be written in terms of the centerline velocity
and jet width as in Equation 25,
i- (Ub) = — (U - U cos 8) (25)
QS j a
after dividing both sides by z.
Momentum Equations
The square of the velocity, U, is approximated using Equation 21
and is written as in Equation 26,
u2 = U2 exp(-n2/a2) (26)
Upon substituting this relation, Equation 26, into the general
form of the x-momentum equation, Equation 14, and using the relations
dA = zdn and C = 2z; the definition of the drag term, Equation 15; and
the definition of the half-width, Equation 24; then the x-momentum
equation can be written in terms of the centerline velocity and the jet
width as in Equation 27,
^- (U2b cos 0) = /2 — (U - Ua cos 3) Ua + — -2-2 (27)
VTT /IT
after dividing both sides by z.
Similarly, the y-momentum equation, Equation 16, can be written in
terms of U and b as in Equation 28,
, /=• CnU sin B cos 6
|- (U2b sin 3) = - — -5-2 (28)
ds ^ 2
-------
36
Temperature Equation
If the temperature rise distribution is approximated by a Gaussian
profile, then the temperature rise, T' , in the cross-section is related
to the centerline temperature rise, T, by Equation 29,
T* = T exp (-n2/2o2) (29)
Substituting Equations 21 and 29 into the general form of the
temperature equation, Equation 20, gives the conservation of temperature
equation in terms of U, T, and b, or Equation 30,
(UTb) = 0 (30)
Summary of the Equations
The integrated mass, momentum, and temperature rise equations
give a system of 4 equations and 4 unknowns:
Four equations:
continuity:
^- (Ub) = — (U - U cos 6) (25)
dS v^ a
x- component momentum:
U2sin23
?P /?
(U2b cos 3) = /2 — (U - U cos 6) U + —
a &
/TT /IT
(27)
y- component momentum:
, /=• CU2sin 3 cos g
^ (U2b sin 6) = - ^ JLS - (28)
/if ^
temperature rise:
(UTb) = 0 (30)
-------
37
Four unknowns :
U, T, b, 6
The geometry of the jet trajectory as shown in Figure 4 gives two
additional equations and unknowns,
Two geometry equations:
~ = cos B (31)
&. = sin 6 (32)
ds
Two unknowns :
x, y
Thus, there are six unknowns--U, T, b, B, x and y--and six
equations, all functions of the jet axis distance, s. In addition,
there are two experimentally-determined coefficients- - C and E.
Solution of the Equations
No n- Dimensional Volume Flux and Momentum Flux
The volume continuity equation and the two momentum equations are
ma-1' non-dimensional by defining V as the non-dimensional volume flux
and M as the non-dimensional momentum flux as shown in Equation 33,
(33)
o o
o o
Using Equation 33, then Equations 25, 27, and 28 take the re-
spective forms:
-------
38
volume continuity:
f. - S - A COS „ (34,
x- component of momentum flux:
d(M If g) = /2 A [f - A cos 8 + A C; sin2e] (35)
y- component of momentum flux:
d(M ^n g) = -/2 A (A C^ sin 3 cos 6) (36)
where S = the non-dimensional jet axis distance;
S = (2E//i? b )s (37)
A = the ratio of the ambient velocity to the initial jet velocity;
A = Ua/Uo (38)
i
and CL = the reduced drag coefficient;
p = CD/4E (39)
Geometry Equations
The geometry equations, Equations 31 and 32, can be written in
non-dimensional form as shown in Equations 40 and 41,
3! = cos 3 (40)
(41)
where
X= (2E/bo)x ; Y= (2E/vbo)y (42)
Temperature Equation
The temperature equation, Equation 30, can be integrated immediately
to give Equation 43,
w0-= l (43)
-------
39
where U T , and b = the initial values of the velocity, temperature
o o o
rise, and jet half-width, respectively.
Solution in Terras of S, and X and Y
There are now five differential equations, Equations 34, 35, 36,
40, and 41, and five unknowns, or V, M, B, X, and Y, all functions of
the jet axis distance, S. By solving this set of differential equations
for M and V, then the velocity ratio, U/UQ, the temperature rise ratio,
T/T , and the half-width ratio, b/bQ, can be solved as functions of S.
Using the definition of the volume flux and the momentum flux,
or Equation 33, the ratio U/U is related to M and V as shown in
Equation 44,
U/U = M/V (44)
o
Using the definition of V, Equation 34, and the integrated tempera-
ture equation, Equation 43, the ratio T/TQ is related to V as shown in
Equation 45,
T/T = 1/V (45)
o
Using Equation 34, the ratio b/bQ is related to V and M as in
Equation 46,
b/b = V2/M (46)
o
The ratios U/U , T/T , and b/b are functions of S, since M and
O O \j
V are functions of S.
Next, by solving the set of differential equations for 3, then
U/U , T/T , and b/b can also be expressed as functions of the rectangular
coordinates, X and Y. As shown in Equation 47 and 48, these coordinates
-------
40
can be located by integrating Equations 40 and 41 along the jet axis,
S, or
S
X = f cos B dS (47)
o
Y = sin B dS (48)
o
Numerical Solution
A numerical integration method was used to solve the five equations
for the five unknowns--V, M, B, X, and Y -- as functions of S. Then,
at each step of the integration, values of U/U , T/T , and b/b were
calculated from the values of M and V and expressed as functions of
S, and X and Y.
The system of differential equations was solved using General
Electric's FORTRAN subprogram RKPBX$, which is available in the time-
sharing service program library. This subprogram integrates a system
of first-order differential equations by the fourth-order Runge-Kutta
method.
The five equations — the volume flux equation, the two momentum
flux equations, and the two geometry equations — were rewritten in a
form suitable for the subprogram. Equations 34, 35, 36, 40, and 41
became, respectively,
continuity:
1/2
dV _ (L2 + F2) AL (49)
dS " V "
-------
41
x-component of momentum:
dL
dS
(L2 + F2)
1/2
AL
(L2
\CF
ALD
(L2 + F2)
(50]
y-component of momentum:
ACDFL
(L
(51)
geometry equations:
dX
dS
(L2 + F2)1/2
(52)
dY
dS
(L
T7T
where L = M cos 6 ;
F = M sin B ;
M = (L2 + F2)1/2; and
3 = arctan (F/L)
(53)
(54)
(55)
(56)
(57)
-------
42
Example Solution
The numerical solution was used to solve for values of V, L, F,
X, and Y and then M, T/T , U/U , and b/b at each increment of
dS = 0.1 along the jet axis, S, for given values of the coefficients A
and C*
For example, consider a surface jet discharging into an ambient
stream at an initial angle of 6 = 60.0 . The initial value of the
volume flux is V =1.0 from Equation 33. The initial value of L =0.50
o M o
from Equation 54, and the initial value of F = 0.86603 from Equation 55,
since the initial value of the momentum flux is M =1.0 from Equation 33.
o
The initial values of the rectangular coordinates are X =0.0 and
Y =0.0. For this example, the value of the initial jet velocity is
assumed to be four times greater than the ambient velocity, or A = 0.25
from Equation 38. The reduced drag coefficient, Cn, is assigned the
values 0.0, 0.5, and 1.0 to illustrate its effect on the location of
the jet trajectory.
After integrating the system of equations, the result can be shown
graphically. The volume flux, V, and the momentum flux, M, are plotted
as functions of the jet axis distance, S, in Figure 5. The temperature
rise, velocity, and width ratios -- T/T , U/U , and b/b -- are plotted
as functions of S in Figure 6. The location of the jet trajectory as a
function of the reduced drag coefficient, C*, is plotted in Figure 7.
The value of T/T along the jet trajectory is not significantly changed
by the different values of C . For example, at S = 100.0, the value of
the temperature rise is T/T = 0.0597, 0.0582, and 0.0572 for the re-
spective values of Cl = 0.0, 0.5, and 1.0.
-------
FIGURE 5.--VOLUME FLUX AND MOMENTUM FLUX VERSUS NON-DIMENSIONAL JET AXIS DISTANCE
-------
0.01
50.0 100.
FIGURE 6.--TEMPERATURE, VELOCITY, AND WIDTH RATIOS VERSUS NON-DIMENSIONAL JET AXIS DISTANCE
-------
40
30 -
Y
20 _
10 _
0
10
80
FIGURE 7.--EFFECT OF REDUCED DRAG COEFFICIENT ON LOCATION OF JET TRAJECTORY
-------
46
Zone of Flow Establishment
In practical applications, the zone of flow establishment of the
jet, illustrated in Figure 8, must be determined. This zone is a mixing
region in which turbulent mixing changes the uniform temperature and
velocity profiles at the jet origin to fully-developed turbulent profiles
which are Gaussian in shape at the beginning of the established flow
region.
Previously, when the zone of establishment has been determined for
jets in a flowing stream, it has been done by measuring the length, sg,
and the initial angle, 3 , at the end of the zone of establishment.
Fan (22), for example, empirically related the length of the establish-
ment zone and the initial angle, (3 , to the ratio of the ambient velocity
and the initial jet velocity, or A = U /U . Based upon Gordier's data (25),
cl O
the length of the establishment zone was found by Fan to be related to
A as shown in Equation 58,
—— = 6.2 exp (-3.32 A) (58)
d1
o
The initial angle at the end of the zone, 8 , was found to be related
to A as shown in Equation 59,
(3 = 90° - 110° A (59)
o
Unfortunately, Fan's results in Equations 58 and 59 are not
directly applicable to the present study. In cases where the initial
angle of the discharge is 60° or even 45°, Fan's results can not be used
directly since the initial angle, 8*, in Equation 59 is 90°. In cases
-------
u
U = U0 T - T0
U )
0
To
— >
(U
TF
1
0'
f f^&
J 0
/ \
i 7
\ ^.DO
X
FIGURE 8.--ZONE OF FLOW ESTABLISHMENT
-------
48
of two-dimensional surface jets where the range of the velocity ratio, A,
is anticipated to be 0. 20 <_ A <_ 0.80 or higher, Fan's results should not
be directly applied even when & = 90° since his results were developed
for a different range of A, or 0.125 <_ A <_ 0.25. Therefore, it will be
necessary to extend Fan's work by empirically determining how the length
of the zone of flow establishment, s', and how the initial angle at the
end of the zone of establishment, B , are related to the velocity ratio,
A = U /U , for cases in the range of 0.20 <_ A <_ 0.80 and at the same
a o
time for cases where the initial angle of the discharge, 6 , is not
90° to the ambient flow.
Summary of Analytical Development
By considering the basic volume, momentum, and heat equations, a
model has been developed describing a two-dimensional surface jet de-
flected downstream by a flowing ambient current. By using a numerical
solution, values of temperature rise, velocity, and width can be pre-
dicted as functions of a non-dimensional jet axis distance. The location
of the jet trajectory can be predicted in terms of non-dimensional
rectangular coordinates. The model contains two experimentally-determined
coefficients, the entrainment coefficient and the drag coefficient. Since
the non-dimensional coordinates S, X, and Y are defined as functions of
the entrainment coefficient, E, in Equations 37 and 42, and since the
location of the jet trajectory depends on the drag coefficient, C , these
coefficients must be evaluated by laboratory experiments before the model
can be used to predict temperature distributions at field sites. Also,
the zone of flow establishment of the jet must be determined from laboratory
-------
49
experiments before the mathematical model can be used in practical
situations.
-------
CHAPTER IV
LABORATORY EXPERIMENTS
The laboratory experiments were designed to functionally relate
the entrainment and drag coefficients and the zone of flow establishment
to the velocity ratio, A, and the initial angle of discharge, 3 .
Modeling
The complete modeling of a heated discharge would require modeling
several phenomena: the inertial mixing due to the jet velocity, the
buoyancy due to the temperature difference between the jet and the
ambient current, the advection of the heat by the ambient current, the
turbulent diffusion in the ambient stream, and the evaporative heat loss
across the air-water interface. According to Ackers (3) and other in-
vestigators, it is impossible to accurately reproduce all the phenomena
in the same model. Thus, it is necessary to identify the dominant forces
present in a given situation.
Dominant Forces in a Surface Jet
As assumed in the development of the mathematical model, the three
dominant forces involved in a surface jet are the inertial, viscous, and
buoyancy forces. The ratio of the difference between the inertial force
of the jet and the ambient current to the buoyancy force due to the
temperature difference between the jet and the ambient current is the
Richardson Number and can be written as in Equation 10. The ratio of
50
-------
51
the inertial force of the jet to the viscous force is the jet Reynolds
Number. In addition, the ratio of the inertial to the viscous forces
in the stream, or the ambient Reynolds Number, must be considered.
Jet Reynolds and Richardson Numbers
Two of the criteria considered in modeling a surface jet are the
equivalence of the model and prototype jet Reynolds and Richardson
Numbers. Unfortunately, exact equivalence cannot be achieved in the
laboratory.
Equivalence of the initial Richardson Numbers (RiQ Number) can be
achieved by reducing the velocity and velocity differences in the model
This is shown in Equation 60,
Kl -
O
Apgd^
p(U - U )2
^ o a
m
Apgd^
>
-------
52
where v = kinematic viscosity. This is because the diameter, d', of the
model is still much smaller than that of the prototype, and the kinematic
viscosity, v, is approximately the same in both the model and the proto-
type.
Thus, approximate equivalence of the Ri Numbers can be achieved
only at the expense of very small jet Re Numbers in the model. Other
investigators, such as Abraham (1) and Burdick and Krenkel (10), have
assumed that, if the model jet flow is turbulent, as in the prototype,
i.e., if the Re Number is greater than some critical value, then the
criterion for exact equivalence between model and prototype jet Re
Numbers can be relaxed. Then, the criterion for similarity between
model and prototype Ri Numbers can be achieved.
Ambient Reynolds Numbers
The third criterion, exact equivalence of prototype and model
ambient Reynolds Numbers, also cannot be achieved in the laboratory.
It is assumed that the requirements for exact equivalence can be relaxed
if the flow regime in the laboratory flume is turbulent.
Model Surface Jet
The model surface jet is thus an approximation of the prototype.
Equivalence of model and prototype Ri Numbers is chosen in preference
to equivalence of Re Numbers. Prototype Ri Numbers can be calculated
based on conditions representing typical field sites, and the model Ri
Numbers should be on the same order of magnitude. Since prototype Re
Numbers indicate a turbulent flow regime in the jet and in the ambient
stream, the model Re Numbers should also indicate a turbulent flow regime.
-------
53
Laboratory Equipment and Procedure
The laboratory experiments were performed in the hydraulics
laboratory of the Department of Environmental and Water Resources En-
gineering at Vanderbilt University.
Flume
The flow system used was a rectangular flume 60 feet long, 2.0 feet
wide, and 1.0 foot deep. The flume has glass sides and a painted steel
bottom, and is mounted on a truss system so that the slope can easily
be varied.
Jets
Circular jets were selected for laboratory use for two reasons --
convenience and the object of the experiments. Using circular jets re-
quired only a minimum of commercially-available fittings, while using
square or rectangular orifices would have required extensive fabrication
of special shapes and equipment. The object of the experiments -- re-
lating the entrainment and drag coefficients and the zone of establish-
ment to the velocity ratio, A, and the initial angle, $Q -- was re-
stricted to what were considered the significant factors influencing the
spatial distribution of temperature. Understanding these factors, the
influences of A and g', more fully was felt to be necessary before ^
considering the many possible variations of orifice shape.
The design of the jets was based on the desire to maintain con-
stant Re Numbers in the jet and in the flume. It was decided to main-
tain a constant jet velocity and jet ReQ Number and vary the velocity
ratio, A = U /U , by varying the ambient velocity, U , in the flume.
a o'
-------
54
Since it was decided to use the maximum flowrate, Q - 0.15 cfs, this
d
meant that the flume, or ambient, Reynolds Number could be kept constant
by inversely changing the ambient depth, z , with changes in the flume
velocity, U, . The ambient Reynolds Number can be written as shown in
cl
Equation 62,
a
v
where Re = the ambient Re Number; and
a
z = the ambient depth.
a
U z
Re = _JLJL_ (62)
Thus, for each desired value of the velocity ratio, A = ua/uo> a dif~
ferent value of U was indicated, since U was constant. For each value
a o
of U , then, a different value of z was indicated from Equation 62,
3. **
since Re and Q were to be kept constant. This resulted in the jets
a a
being designed to enter the flume at different heights so that the top
of the jet discharge for each jet when in use would coincide with the
ambient water surface.
The 1.0- inch diameter jets were built using PYC pipe and Plexiglass,
One of the glass panels near the upstream end of the flume was removed,
and a Plexiglass panel with a circular opening approximately 0.7 feet in
diameter was sealed in its place. Into this opening could be placed a
Plexiglass "plug" drilled to hold a vertical row of four jets mounted so
thpt they would be flush with the inside wall of the flume. The jets
were designed so that each one could be sealed when not in use. Since
it was decided to study the effect of three different jet angle orienta-
tions, or g' = 90°, 60°, and 45° to the ambient flow, three of these
plugs were constructed, each containing a vertical row of jets oriented
-------
55
at one of the angles, 3 , mentioned above. Figure 9 shows the three
plugs with the jets offset horizontally for clearance, while Figure 10
shows one of the jets in use with the other three sealed off.
Temperature Determination
The temperature distribution was determined using probes small
enough to produce minimum disturbance in the flow pattern, a manually-
operated switchbox, and a digital thermometer. Point measurements were
made by using a single Cole-Parmer No. 8432-1 probe mounted on a traveling
point gauge apparatus. This probe had a length of 2 1/2 inches, and the
temperature-sensitive bulb at the tip of the probe had a diameter of
1/8 inch. Lateral temperature measurements were determined using
eleven Cole-Parmer No. 8434 stainless steel probes with a length of
4 1/2 inches and a diameter of 5/32 inches. These probes were spaced
laterally on a movable platform which was suspended over the water sur-
face. Preliminary investigation indicated that the thermal sensitivity
of these probes was limited to the lower 1/2 inch or so of the tip.
Thus, when the tips were immersed below the water surface, a vertically-
averaged temperature could be read over this depth. A manually-operated
YSI twelve-point switchbox was used to route the output of the probes to
a Digitec Model 1515 digital thermometer. Preliminary investigation also
indicated that the time-varying turbulent fluctuations of temperature
along the centerline of the surface jets were sufficiently small to allow
average centerline temperatures to be determined directly from the read-
out scale of the digital thermometer, precluding the necessity of re-
cording apparatus. Figure 11 shows the temperature probes, the switch-
box, and the digital thermometer.
-------
6
PIGURE 9.--JETS AND TEMPERATURE PROBES
FIGURE 10.--SURFACE JET AND ROTAMETER
-------
ill*
?: •
FIGURE 11.--TEMPERATURE PROBES AND OTHER LABORATORY EQUIPMENT
-------
58
Other Equipment
Other laboratory equipment was also used in performing the ex-
periments. A 60° V-notched weir installed at the upstream end of the
flume was used to measure the ambient flowrate, Q . A 2.00-gpm capacity
cL
rotameter was used to measure the initial jet flowrate, Q . The
traveling point gauge was used to measure the flume depth, z , which
3.
was controlled by means of a perforated baffle installed at the down-
stream end of the flume. The spatial location of the temperature probe
attached to the traveling point gauge apparatus relative to the origin
of the jet was determined by measuring vertically with the point gauge,
laterally by means of a scale mounted on the lateral arm of the
traveling cart, and longitudinally by means of a tape stretched along
the upper edge of the flume wall. The longitudinal location of the
platform containing the eleven laterally-spaced probes was determined
by means of the same tape. The jet discharge was heated by mixing
about 150 gallons of water in a tank into which steam could be admitted
at a flowrate adjusted to maintain the desired elevated temperature.
The heated water, dyed with Pontacyl Pink to allow visual observation of
the jet, was continuously pumped into a 10.0-foot constant heat tank,
then allowed to flow through the rotameter and into the flume, entering
at the ambient water surface.
Procedure
The procedure followed in performing the laboratory experiments
was similar for all the runs. Selection of parameters such as ambient
depth, z , and the initial temperature of the heated discharge for each
3.
run was based on preliminary calculations of the desired value of the
-------
59
velocity ratio, A, and the desired values of the Ri and Re Numbers.
o
Uniform ambient flow was obtained by adjusting the slope of the flume
before each run. Steady-state conditions were insured by allowing the
jet to discharge into the ambient flow for at least 15 minutes before
data were taken. Lateral temperature distributions were measured at
predetermined intervals starting at the downstream end of the flume.
Point temperature measurements were made to accurately locate the jet
centerline, to determine the zone of flow establishment, and to measure
vertical temperature distributions. The initial temperature of the jet
was measured with the point probe in the mouth of the jet flush with
the inside wall of the flume. The flume water was not recirculated,
since preliminary calculations had indicated that the ambient tempera-
ture would be increased during the estimated time for each run by the
heated jet by about 5°F, which was significant compared to the initial
temperature rises planned.
-------
CHAPTER V
RESULTS OF THE LABORATORY EXPERIMENTS
The results of the laboratory experiments were analyzed in terms
of the parameters in the mathematical model.
Relation of Circular Jet to Half-Width b
— - • -- — -- o
The two-dimensional mathematical model predicts the temperature
distribution along the jet axis in terms of the initial jet half-width,
b , located at the end of the zone of flow establishment as seen in
o
Equation 37,
S = (2E/A b )s (37)
In order to plot the observed trajectories and temperature data in
terms of non-dimensional coordinates, it was necessary to calculate the
i
value of b , which can be determined from the value of the diameter, d ,
and the value of the equivalent half-width at the origin of the jet, b .
Circular and Square Orifices
According to Yevdjevich (47), the entrainment characteristics of a
square jet approximate the entrainment characteristics of a circular jet
if the cross-sectional areas of the two orifices are equal. If the
circular orifice diameter is d , and the square orifice half-width is b ,
then equating the cross-sectional areas of the two shapes gives
Equation 63,
= 2.26 b^ (63)
60
-------
61
This equation is established by Yevdjevich for a vertical three-
dimensional jet but is assumed in this study to be applicable to a
horizontal surface jet as well.
t
Relation of b to b
o o
The value of the initial jet half-width, b , at the end of the
!
zone of flow establishment can be related to the half-width, b , at the
origin by considering the conservation of heat flux between the two
cross-sections at 0' and 0 shown in the definition sketch of the zone
of flow establishment, Figure 8. Using the assumed Gaussian velocity
and temperature profiles, Equations 21 and 29, and integrating the con-
servation of heat flux relation, Equation 30, at 0' and 0, and then
equating the resulting heat flux at O1 to the flux at 0 gives Equation 64,
2 U T z b' = UTzb /n~//2 [64)
O O O 0 O
Since the temperature rise, T , the jet velocity, U , and the depth, z ,
at 0' are equal to the respective centerline values of T and U and the
value of z at 0 in Equation 64, then b can be related to b', as shown
M ' o o'
in Equation 65,
b = 1.60 b' (65)
oo • J
Relation of b to d
o o
The two equations, Equations 63 and 65, express d' and b as
functions of b . Solving the equations to eliminate b' relates the
initial half-width at the end of the zone of flow establishment, b ,
to the diameter, d', of a circular jet as shown in Equation 66,
bo = 0.708 d^ (66)
-------
62
In the laboratory experiments the diameter, d^, of the jets was equal
to 1.0 inch. Therefore, from Equation 66, the value of the half-width
was b = 5.90 x 10-2 feet.
o
Laboratory Measurements
A complete summary of the data is presented in Appendices A and B.
Velocities
The ambient velocity, U , and the initial jet velocity, UQ, for
each run were calculated from the measured flowrates, since no attempt
was made to directly measure the ambient velocities or the spatially-
varied velocity distribution in the jet flow field. The value of the
velocity ratio, A = U /U , was calculated from these values of U and
a o £*
b' . Then the initial jet Reynolds Number was calculated using the
o
diameter, d , and Equation 61, and the ambient Reynolds Number was cal-
culated using measured values of the ambient depth, z , and Equation 62.
These values are presented in Table 1.
Temperature
The initial jet Richardson Number was calculated from Equation 60,
using the initial jet and ambient temperatures, which are presented in
Table 1, and standard tables for the density of water. These values are
presented in Table 1.
Geometry
The values of the initial angle of the discharge, &', are also
presented in Table 1.
-------
TABLE 1
LABORATORY MEASUREMENTS
Run
1-90
2-90
3-90
4-90
5-90
1-60
2-60
3-60
3-60-1
4-60
5-60
1-45
2-45
3-45
4-45
5-45
Ua
ft/sec
0.38
0.23
0.16
0.12
0.10
0.31
0.23
0.16
0.17
0.12
0.10
0.34
0.22
0.16
0.12
0.10
U0
ft/sec
0.52
0.52
0.52
0.52
0.52
0.52
0.52
0.52
0.52
0.51
0.52
0.52
0.52
0.52
0.52
0.52
A
Eq.38
0.73
0.44
0.30
0.23
0.20
0.67
0.44
0.30
0.32
0.23
0.19
0.66
0.42
0.30
0.23
0.18
Re0
Eq.61
5240
5240
5240
5240
5240
5240
5240
5240
5240
5240
5240
5240
5240
5240
5240
5240
Rea
Eq.62
5370
5340
5370
5300
4590
5300
5340
5.370
5400
5300
4300
5370
5300
5340
5370
4300
za
ft
0.20
0.33
0.48
0.63
0.62
0.21
0.33
0.49
0.46
0.64
0.63
0.22
0.34
0.49
0.63
0.62
Ambient
Temp.
°F
43.5
45.5
46.0
47.0
47.7
43.5
45.6
46.2
43.9
47.1
46.8
44.7
45.5
46.4
46.6
47.0
Initial
Jet Temp.
°F
111.0
106.0
91.5
84.2
70.5
101.0
106.0
93.0
93.0
83.0
74.5
101.0
107.0
92.5
84.0
73.0
BO
90.0
90.0
90.0
90.0
90.0
60.0
60.0
60.0
60.0
60.0
60.0
45.0
45.0
45.0
45.0
45.0
(RicOm
Eq.60
1.17
2.61X10'1
9.55xlO-2
6.90xlO-2
3.20xlO-2
6.32X10-1
2.65X10-1
l.OSxlO"1
8.74xlO"2
6.15xlO-2
3.88xlO-2
5. 92X10'1
2.48X10'1
1.04X10-1
6.11xlO-2
3.83xlO"2
(Ri0)p
Eq.60
4.12
3. 55x10' 1
l.llxlCT1
5,07xlO-2
3.57xlO-2
2.51
3.67X10-1
1.07X10'1
1.37X10-1
5.13xlO-2
3.04xlO-2
2.27
3.16X10'1
l.OSxlO'1
5.32x10-2
2.82xlO-2
-------
64
Analysis of Data
Reynolds Number
The Re Numbers of the jet and ambient flow, shown in Table 1, were
all Re = 5,000, which indicates that the Re Numbers were constant, and
that the flow regime was almost fully turbulent.
Comparison of Model and Prototype Ri
The initial Richardson Number for each value of A in Table 1 can
be compared to a representative prototype RiQ Number calculated from
Equation 60 by assuming a 15° temperature rise from 70°F to 85°F, a
10.0-foot discharge depth, and an ambient velocity of Ua = 1.0 ft/sec.
These prototype values were based upon anticipated conditions at several
Tennessee Valley Authority generating plants. The model and prototype
Ri Numbers for corresponding values of A = U /U in Table 1 are seen
o a o
to all be of the same order of magnitude.
Gaussian and Two-Dimensional Assumptions
The temperature data were examined to determine the validity of
the Gaussian and two-dimensional assumptions made in the mathematical
model.
Gaussian Assumption.-- The assumption made in Equation 29 that the
lateral temperature distribution perpendicular to the jet axis is
Gaussian in shape was examined for a representative run. Surface
temperature measurements were made with the point probe at a distance
far enough downstream from the origin of the jet so that the cross-
section, measured laterally across the flume, was perpendicular to the
-------
65
axis of the jet. The surface temperature measurements were plotted, and
then a Gaussian curve was fitted to the distribution as shown in
Figure 12. The mean of this distribution is 0.56 feet, and the standard
deviation of the sample is 0.27 feet. The correlation coefficient for
the fitted Gaussian distribution is equal to 0.90, which indicates that
the Gaussian distribution is adequate to describe the temperature pro-
file. The bimodal distribution observed by Fan (22) for a submerged jet
was not observed in this study of a surface jet.
Two-Dimensional Assumption. - - The assumption that the surface jet
spreads laterally much greater than it does vertically was examined by
measuring cross-sectional and vertical temperature distributions. A
cross-section drawn from measurements made with the point probe is
shown in Figure 13. In terms of the 0.05 temperature rise contour in
Figure 13, the jet has spread laterally from y'/bQ = 1.25 to y'/bQ = 20.0,
while vertically the jet has spread from z/bQ = 1.25 to z/bQ = 2.75.
Thus, for this representative cross-section, the rate of lateral
mixing is 7.3 times greater than the rate of vertical mixing, which is
approximately a full order of magnitude.
Vertical temperature profiles were measured with the point probe
at several points along the jet centerline for each run. The repre-
sentative profiles shown in Figure 14 indicate that vertical entrainment
is small, since the jet does not spread vertically along the jet axis.
If vertical entrainment were significant, then the temperature profiles
shown in Figure 14 would indicate more and more vertical mixing with
distance along the jet axis. Visual observation also indicated the
limited extent of vertical mixing, as seen in Figure 15.
-------
0.3
0.2 _
0.1 -
0.0
y , feet
FIGURE 12.--OBSERVED TEMPERATURE DISTRIBUTION AND FITTED GAUSSIAN CURVE
AT s'/bQ = 35.3, RUN 3-60-1
-------
0
1 _
2 -
3 -
4 -
1 1 1 1 \
FIGURE 13.--TEMPERATURE DISTRIBUTION AT s'/b = 35.3, RUN 3-60-1, T/T
-------
1 1 1 T
0
z 2
i I I I L
s'/b0=17.0
I I I L
.0 .1 .2 .3 .4 .5 .6
.0
T/T0
.1 .2 .3 .4 .5.0 .1 .2
FIGURE 14.--VERTICAL TEMPERATURE PROFILES, RUN 1-60
-------
FIGURE 15.--TWO-DIMENSIONAL SURFACE JET, RUN 3-90
•
-------
70
The vertical temperature profiles in Figure 14 also indicate that
temperature measurements made just below the water surface with the
point probe should agree with measurements taken just below the surface
with the larger set of probes. The larger probes measure average
temperature values over a vertical distance of approximately z/b = 0.5.
The reason for this agreement is that the slope of the vertical profiles
over this depth is almost zero. As a result, the centerline values of
temperature reported in this study were made with the larger probes
mounted on the movable platform and with the point probe. These center-
line values, then, are the maximum temperatures, laterally and vertically,
in each of the cross-sections.
Geometry Effects
The laboratory flume was not wide enough for the jet to spread
completely unaffected by the sides of the flume. At some point down-
stream, the jet would intersect the sides of the flume, and the location
of the trajectory and the width of the jet would no longer be functions
of only the relative momenta of the jet and the ambient current. As had
been anticipated, this effect was particularly apparent for those runs
where the value of A - U /U was relatively small and the angle g' of the
a o ' o
discharge equal to 90°, and for those runs where A was relatively large
and 3' equal to 45°.
o M
Temperature measurements taken with the larger probes mounted on
the movable platform at x'/h>o = 33.8 (x' = 2.0 feet), x'/bQ = 50.7
(x = 3.0 feet), and farther downstream showed a tendency for the
trajectories to converge along the sides of the flume. Also, the exact
-------
71
lateral location of the maximum temperature rise in the cross-sections
from the measured lateral distributions was difficult to determine. As
a result, data for the observed trajectories at x'/b = 33.8 and beyond
were not used in locating the jet centerline or in determining the drag
coefficient, Cn. However, the decrease in the temperature rise due
to lateral mixing as far downstream as x'/b = 84.5 (x =5.0 feet) seemed
to be relatively unaffected by the sides of the flume, and these data,
taken with the larger probes, were used, along with values taken from
x'/b = 0.0 to x'/b = 16.9 with the point probe, in determining the
value of the entrainment coefficient, E.
Width Measurements
The width of the jet was determined from measurements made with
the eleven probes mounted on the movable platform. At each of several
points along the jet, measurements were made of the lateral temperature
distribution, and then the standard deviation of the temperature rise
in terms of l'/T was computed. Then, values of b/bQ were calculated
using b =0.71 inches and Equation 24,
(24)
Determination of the Zone of Flow Establishment
and the Entrainment and Drag Coefficients
Graphical methods were used to determine the parameters and co-
efficients from the laboratory data and from numerical solutions of the
mathematical model.
-------
72
Trajectories and Temperature Plots.-- The jet trajectory for each
run was determined from the measured location of the maximum temperature
in each cross-section taken along the jet axis. These data points were
plotted relative to the jet origin in terms of x'/b and y /b . A smooth
curve was fitted through the observed points on the jet centerline by
using an n-th order polynomial fitting routine. Next, the temperature
data were plotted on log-log plots in terms of T/T vs. s'/b , where
s'/b had been measured for each data point along the fitted y /b vs.
x /b curves.
Zone of Flow Establishment.-- The length of the zone of flow
establishment in terms of s'/b for each run was measured by examining
e o '
the data points on the T/T vs. s /b curve and interpolating to the
i
largest value of s /b where T/T was still equal to 1.0. This measured
i
value of s /b was the length of the zone of flow establishment, or
s'/b .
e o
The angle, 3 , at the end of the zone of flow establishment was
determined for each run bv using the value of s /b from the T/T vs.
0 e o o
f
s /b plot and measuring this distance along the fitted trajectory curve
from x /b = y'/b = 0.0. The angle at this value of s /b was measured,
giving the value of the angle at the beginning of the established jet
flow, or 6 , The rectangular coordinates xe/b and y /b were also
measured at this value of s'/b .
e o
Entrainment Coefficient.-- To determine the value of the entrainment
coefficient, numerical solutions of T/T vs. S were computed and plotted
on a log-log scale for each observed value of A and 3 . A preliminary
-------
73
value of E for each run was determined by over-laying the predicted
T/T vs S curve with the observed T/T vs. s'/b data plot. A set of
o o o
corresponding values of S and s'/bQ was noted after sliding the T/TQ vs.
s'/b plot horizontally until the best fit of the data and the theoreti-
cal curve was obtained. Then, Equation 37 was solved for E,
E = (^ S/2) / (s/bQ) (37)
Next, the value of S* , which is the length of the establishment
zone in non-dimensional form, was calculated from the observed value
of s'/b , the preliminary value of E, and Equation 67,
e o c
s . (67)
The T/T vs. S plot, which is referenced to the beginning of the
jet flow region, was then redrawn as T/TQ vs. S* , referenced to the jet
origin, using the value of S and Equation 68,
S' = S' + S (68)
e
A set of corresponding values of S* and s /bQ was noted after sliding
the T/T vs. s'/b plot horizontally over the T/TQ vs. S* curve until
the best fit of the data and the theoretical curve was obtained. Then,
Equation 69 was solved for the final value of E,
E = CAT s'/2) / (s'/bj
Drag Coefficient.-- To determine the value of the drag coefficient,
C , for each run, numerical solutions of X and Y were computed along
-------
74
with the T/T solutions for each value of A and 6 • For each run, at
o o
least two numerical solutions for two different assumed values of the
reduced drag coefficient, CL, were computed. Each of the predicted
trajectories was plotted in terms of x'/b and y'/b , using Equation 42
to calculate x/b and y/b ,
X = C2E//JT b )x ; Y = (2E//JT bQ)y (42)
and then using Equations 70 and 71 to calculate x /b and y /bQ,
*'/bo = x/bo + Xe/bo (70)
y'/b = y/b + v'/b (71)
J o ' o • e' o
This was possible since the value of E, and x'/b and y'/b had already
r e o e o
been determined from the temperature plots.
The observed value of the reduced drag coefficient for each run
was determined by over-laying the predicted trajectories with the
observed jet trajectory. The value of CL was chosen by interpolating
between the predicted trajectories to the observed trajectory. For
example, if the values CL = 0.5 and 1.0 were assumed in obtaining two
predicted trajectories, and if the observed trajectory were located
exactly half-way between the two predicted trajectories, then the
observed value would be C = 0.75.
The value of the drag coefficient, C , for each run was obtained
from the observed values of E and C and Equation 39,
Cp = CD/4E (39)
-------
75
h'uitli of the J_e£
The predicted and observed values of b/b were then plotted. The
numerical solution obtained for each run predicted b/b vs. S. Predicted
t
alue? of b/b vs. s /b were calculated and plotted as smooth curves
by using the observed values of E and S* determined for each run from the
centerline data and Equations 68 and 69,
S' = S* + S (68)
"
v
bo)s (69)
Then, the observed values of b/b vs. s'/b were plotted on the same
o o
figure.
Presentation of Results
Trajectories, and Temperature and Width Plots
The trajectories and temperature plots, from which the values of
C and E were determined, and the width plots are presented in Figures
16 and 17 for a representative laboratory run and in Appendix C for the
other cases. The data points which indicate the observed trajectory
for each run are plotted on the same figure as the fitted trajectory
computed from the numerical solution. The observed T/T vs. s'/b data
oo
for each run are plotted on the same figure as the fitted T/T0 curve com-
puted from the numerical solution. The observed b/b vs. s'/b data for
o o
each run are plotted on the same figure as the predicted b/b curve also
computed from the numerical solution.
-------
A =0.73
Cn= 0.4
V4EV°-7
0
FIGURE 16.--OBSERVED AND FITTED TRAJECTORIES, RUN 1-90
-------
1.0
0.5 -
0.1 -
0.05 -
0.01
0.1
100.0
- 50.0
10.0
5.0,
1.0
0.5 1.0
5.0 10.0
50.0 100.
s'/b
o
1'lCURli 17.--OBSHRVHD VALUHS AND IMTTHi) CURVliS 1'OR TJ-MPliRATURli AND WIDTH, RUN 1-90
-------
78
Parameters
The values of the various parameters as determined from the
laboratory data are presented in tabular form. The values of A, 3Q,
s'/b , x'/b , and y'/b are presented in Table 2. The corresponding
e o e o -V o ^
best-fit values of E and C^, the values of S^ calculated from Equation 67
and X* and Y* calculated from Equation 72,
e e
X* = (2E//rF b )x' ; Y* = (2E/v/if bo)/g (72)
and values of C calculated from Equation 39 are also presented in
Table 2.
Relation of Parameters to A and g
The object of the laboratory experiments was to determine how the
establishment zone and the entrainment and drag coefficients are related
to the velocity ratio and to the discharge angle. The experimentally-
determined values of s'/b and 6/6* are related to A as shown in
e o o o
Figures 18 and 19, respectively. The values of C' are plotted as a
function of A in Figure 20, and the values of the drag coefficient, CD,
are related to A as shown in Figure 21. The correlation coefficients of
the relations and the correlation coefficient at the 1% level of
significance are shown in Table 3. These correlation coefficients were
obtained using a regression analysis routine, which gives the correlation
coefficients and predicted values for assumed arithmetic, semi-log, and
log-log relations between the dependent and independent variables. The
assumed relation which gave the highest correlation is the relation
presented in each of Figures 18, 19, and 21. The regression analysis
-------
TABLE 2
LABORATORY RESULTS
A
0.73
0.44
0.30
0.23
0.20
0.67
0.44
0.30
0.23
0.19
0.66
0.42
0.30
0.23
0.18
>;
90.0
90.0
90.0
90.0
90.0
60.0
60.0
60.0
60.0
60.0
45,0
45.0
45.0
45.0
45.0
3
o
r ~
34.5
47.5
50.0
62.0
71.5
13.5
36.5
40.0
41.0
45.0
25.0
23.0
26.5
33.5
35.0
s'/b
e o
0.9
1.4
1.3
1.7
1.9
0.7
1.3
1.5
2.0
2.3
0.7
1.2
1.7
2.1
2.3
x'/b
e o
0.6
0.6
0.6
0.4
0.1
0.6
0.9
1.2
1.4
1.4
0.6
1.1
1.5
1.6
1.8
y'/b
e o
0.8
1.3
1.3
1.7
2.0
0.3
0.8
1.0
1.3
1.9
0.4
0.6
0.8
1.1
1 .3
E
0.46
0.39
0.31
0.47
0.44
0.24
0.13
0.19
0.25
0.29
0.19
0.13
0.21
0.27
0.22
s'
e
0.47
0.62
0.46
0.90
0.94
0.24
0.19
0.32
0.56
0.75
0.15
0.18
0.40
0.64
0.57
t
X
e
0.31
0.26
0.21
0.21
0.05
0.19
0.13
0.26
0.39
0.46
0.13
0.16
0.36
0.49
0.45
Y'
e
0.41
0.57
0.45
0.90
0.99
0.16
0.12
0.21
0.37
0.62
0.08
0 . 09
0.19
0.33
0.52
c;
0.4
1 .0
2.8
2.0
2.0
0.08
1.5
3.0
2.7
3.4
1.5
1.0
1.0
1.0
2.0
cn
0.7
1.6
3.5
3.8
3.5
0.1
0.8
2.3
2.7
3.9
1.1
0.5
0.8
1.1
1 .7
-------
80
5.0
2.0
Ji
bo
1.0
0.5
0.1
I I I I I
O 90.00
D 60.0°
A 45.00
O -
ID \-
I I I I I I
0.2
0.5
1.0
FIGURE 18.--OBSERVED VALUES AND FITTED CURVE FOR LENGTH OF
ESTABLISHMENT ZONE VERSUS VELOCITY RATIO
-------
1.0
I I
'o
0.5
0.0
O 90.0°
D60.0°
A45.00
A
D
0.5
A
FIGURE 19.--OBSERVED VALUES AND FITTED CURVE FOR INITIAL
ANGLE VERSUS VELOCITY RATIO
1.0
-------
82
4.0, 1 1 1 1 | | | |
D
D
D O
2.0 I- AO O
1.0 h- A A
CD
0.5
0.2
0 90.0 °
a 60.0 °
A 45.0 °
1 1 I I I I 1 I
0.1 0.2 0.5 1.0
FIGURE 20.--OBSERVED VALUES OF REDUCED DRAG COEFFICIENT
VERSUS VELOCITY RATIO
-------
83
FIGURE 21.--OBSERVED VALUES AND FITTED CURVE FOR DRAG
COEFFICIENT VERSUS VELOCITY RATIO
-------
84
also indicated, as shown in Table 3, that the entrainment coefficient,
E, does not appear to be a function of A over the range of A investigated
in this study.
TABLE 3
RESULTS OF STATISTICAL TESTS ON PARAMETERS
Observed Correlation Coefficient
Correlation at 1% Level of
Relation Coefficient Significance
s;/bo
t
CD vs
E vs.
CD)VS
CD(VS
( O —
IP ™~
f O —
vs. A 0.95
vs. A 0.87
. A 0.72
A 0.01
. A
90.0°) 0.98
. A
60.0°) 0.97
. A
45.0°) 0.99
0.64
0.64
0.64
0.64
0.96
0.96
0.96
Examination of Figure 21 indicates that C may be a function of
B', the initial angle of discharge, as well as a function of A. This
possible relation was tested, and the results are shown in Figure 22.
The correlation coefficients, which are presented in Table 3, are good.
However, because of the small number of data points, the confidence
limits of each of the curves plotted in Figure 22 tend to overlap, and
t
the relation of C to 8 as well as to A cannot be conclusively es-
tablished from this study.
-------
85
0.3
\ \
°
o
\ \
\
\
I I v
2.0 f- \ \ V
CD ^o =OU-U-A V?
• \ \
A \ \
\ \ \
\ \
\ \ \
0.5
I l\ I I K (
0.1 0.2 0.5 1.0
FIGURE 22.--OBSERVED VALUES AND FITTED CURVES
FOR DRAG COEFFICIENT VERSUS VELOCITY
RATIO AND DISCHARGE ANGLE
-------
86
A dependence of E on the initial angle of discharge was indicated
by an F test, which was used to compare the variances of the experi-
mentally-determined values of E for each of the three discharge angles.
The results of the F test are presented in Table 4, and they show that
the mean value of E at 6* = 90° is significantly different from the mean
o
of the values obtained at g' = 60° and 45°. Thus, E appears to be a
o
function of 8*.
o
TABLE 4
RELATION OF E TO g'
e' E
o
Level of
F Ratio Significance
90° 0.41
23.1 0.1%
60° 0.22
0.20 None
45° 0.20
-------
CHAPTER VI
FIELD SURVEYS
Data were obtained from five surveys at three different steam-
electric generating plants. Data from three of the surveys were col-
lected by Vanderbilt University (VU) personnel (37), and data from the
other two surveys were taken from reports written by Churchill (45)
and Beer and Pipes (6) .
Description of VU Surveys
Flowrates
The field surveys were made on regulated river systems over a
period of time when plant discharge and river flowrate were maintained
very near to steady-state conditions. Stream flowrates were not
measured in the field but were determined from data provided by the
regulating agency. Plant flowrates were determined from pumping records
routinely kept by plant personnel.
Geometry
River and discharge channel geometries were determined from field
measurements and from maps. Ground control was established using a
transit and a stadia rod to lay out a base line which paralleled the
river along one bank for a distance extending several thousand feet
downstream. From this base line, stations were located which were
perpendicular to the river centerline and, if possible, to the jet
87
-------
88
centerline. At each of these stations, the cross-sectional area of the
channel was determined using a Raytheon portable recording fathometer
carried in one of the survey boats. From the base line measurements,
the longitudinal location of each of the cross-sections could be plotted
on a scale drawing traced from a U. S. Geological Survey (USGS) 1:24000
scale map of the area. In addition, the initial angle, 8^, of the
discharge channel was measured from these maps.
Velocity Data
Measurements were made to determine the magnitude and direction of
the initial jet velocity, U , and the ambient velocity, U .
o a
The magnitudes of U and U were determined by two methods. By
the first method, the magnitudes of the velocities were measured at a
cross-section in the discharge channel and at a river cross-section
just upstream from the discharge. The measurements were made with a
Price current meter mounted on one of the survey boats. Lateral
stations within each cross-section were located using anchored buoys
whose lateral distance from the bank had been measured either with a
transit on the bank and a stadia rod held in the boat or by means of
timed runs from the bank in the boat. Vertical distances were measured
using the cable to which the velocity meter was attached. Ideally, this
first method should have given detailed velocity profiles from which
spatially-averaged ambient and initial jet velocities could be determined.
By the second method, magnitudes of the velocities were cal-
culated indirectly using the cross-sectional areas obtained from the
depth-sounding records and the flowrates obtained from gauging station
-------
89
and power plant records. This second method yielded average values of
U and U which were in close agreement with the values obtained by the
o a
first method.
The directions of U and U were measured from the mar* of the
o a
area. This method was chosen because the velocity vectors could not be
measured directlv.
Temperature Distribution
Temperature measurements were made at each of the cross-sections
using Whitney temperature probes carried in the survey boats. Lateral
stations were located using the anchored buoys, while vertical measure-
ments were made using the cable to which each of the probes was attached.
Approximately 100 measurements were made in each of the cross-sections.
Procedure
The normal procedure for the field surveys involved most of two
days spent on preliminary work and most of a third day spent obtaining
the temperature and velocity data. The preliminary work involved laying
out the base line, locating the cross-sections, making a depth-sounding
record across the channel at each of the cross-sections, and setting out
the buoys. The distance from the base line to each of the buoys was
then measured. The temperature and velocity data were measured on the
last day during the time when steady plant and river discharges were
maintained. Normally, one boat and at least three men were required for
the preliminary work, while three boats -- one to measure velocities and
two to measure temperatures -- and as many as eight or nine men were re-
auired on the third day. The field surveys were quite extensive in scope
-------
90
and recorded considerable data, which were used in other studies in
addition to this present study.
Results of the Widows Creek Surveys
Introduction
Data were collected by VU and USGS personnel on November 20 and 21,
1968, and by TVA personnel on August 30, 1967, at TVA's Widows Creek
Steam Plant. The data for the VU surveys were obtained from a report
by Polk (37) and for the TVA survey from the report by Churchill (45) .
At this plant, cooling water from the condensers is discharged into a
small coal-barge harbor and flows directly onto the river surface with
a minimum of mixing. Velocity measurements taken at several stations
in the river indicated that the velocity field of the receiving ambient
water body was influenced by the discharge. Since the discharged heat
was initially advected across the river almost perpendicularly to the
river flow, its spatial distribution can be described in terms of a jet
discharging at some initial angle to the ambient flow. Temperature
measurements indicated that the rate of lateral spreading for the first
5000 feet downstream was approximately 10 times greater than the rate
of vertical spreading due to the buoyancy of the heated discharge.
Thus, the application of the two-dimensional surface jet theory to de-
scribe a large, initial portion of the mixing zone is considered
reasonable.
VU Survey Number 1, November 20
Temperature Measurements.-- The results of the temperature
measurements are shown in Figures 23-28. The temperature contours at the
-------
91
1.0-foot depth and the location of the cross-sect ions are shown in
Figure 23. The vertical temperature profiles along the jet centerline,
which is assumed to be located at the point of the maximum temperature
in each cross-section for a distance of about 7000 feet downstream,
are shown in Figures 24 and 25. The temperature contours for three of
the downstream cross-sections are shown in Figures 26-28. The jet
spread laterally from an initial discharge width on the order of 100
feet to a width of approximately 1200 feet, while it spread vertically
from an initial depth on the order of 10 feet to almost 15 feet. Thus,
the rate of vertical spreading was almost an order of magnitude less
than lateral spreading and can be considered negligible for the first
five downstream cross-sections, which is approximately 5000 feet.
Beyond this point, it appears that the ambient turbulence of the stream
can no longer be neglected, because vertical mixing began to take place.
Determination of Ambient Velocity.-- The value of the ambient
velocity, U , was determined from the ambient flowrate, cross-sectional
£L
areas, and velocity measurements. The river flowrate at Widows Creek
was estimated to be 26,170 cfs, which was the average flowrate during
the period of the survey at Nickajack Dam, located about 20 miles up-
stream. The minimum flowrate at Nickajack was 26,000 cfs and the max-
imum approximately 27,000 cfs. The flow at Guntersville Dam, located
approximately 60 miles downstream, was also maintained steady during the
period of the field survey. Thus, the value of Q^ = 26,170 cfs is con-
sidered a reasonable estimate of the river flowrate. The value of the
ambient flowrate past the discharge was calculated to be Q = 23,800 cfs
-------
R-l R-2 R-3 R-4 R-5 R-6 K-i
I I I I
FIGURE 23.--TEMPERATURE DISTRIBUTION, °F, AT 1.0-FOOT DEPTH, WIDOWS CREEK, VU 1
i
-------
AT CROSS-SECTION R-l
L
j
I
AT CROSS-SECTION R-2
J
I
1
I
AT CROSS-
SECTION R-3
0
46 8 10 12 0246802
TEMPERATURE RISE, °F
FIGURE 24.--TEMPERATURE RISE ALONG JET AXIS AT CROSS-SECTIONS R-l TO R-3
-------
AT R-4
1 L
AT R-5
J I L
AT R-6
AT R-7
J L
024602
TEMPERATURE RISE, °F
0
FIGURE 25.--TEMPERATURE RISE ALONG JET AXIS AT CROSS-SECTIONS R-4 TO R-7
-------
0
200 400 600 800 1000 1200
DISTANCE FROM LEFT BANK, FEET, FACING DOWNSTREAM
FIGURE 26.--TEMPERATURE DISTRIBUTION, °F, IN CROSS-SECTION R-l
1400
•
-------
CO
LU
CO
D-
LU
Q
0
2
4
6
8
10
12
14
16
18
20
22
24
0 200 400 600 800 1000 1200
DISTANCE FROM LEFT BANK, FEET, FACING DOWNSTREAM
FIGURE 27.--TEMPERATURE DISTRIBUTION, °F, IN CROSS-SECTION R-2
•;
' i
-------
0
2
uj 4
U_
6
8
10
12
CO
o_
16
18
20
22
24
I
I
I
1
0 200 400 600 800 1000 1200
DISTANCE FROM LEFT BANK, FEET, FACING DOWNSTREAM
FIGURE 28. --TEMPERATURE DISTRIBUTION, °F, IN CROSS-SECTION R-5
-------
98
by subtracting from Q = 26,170 cfs the value of the diverted plant
flowrate, Q = 2370 cfs.
Cross-sectional areas were measured for each of the downstream
cross-sections. The average value of the first five cross-sections was
22,400 ft2. This downstream reach is considered to be the extent of
the influence of the two-dimensional surface jet. The value of the
ambient velocity was found to be U =1.06 ft/sec by dividing the am-
3
bient flowrate by the average cross-sectional area.
To check this method of obtaining U , velocity measurements made
3
during the survey by USGS personnel were used. Velocity measurements
taken in the undisturbed ambient velocity field at cross-section R-l
are shown in Figure 29. The value of the velocity obtained after ver-
tically averaging each of the profiles shown in Figure 29 using a
planimeter and then averaging these values was equal to 1.05 ft/sec at
cross-section R-l. Dividing the value of Q = 23,800 cfs by the value
3
of the cross-sectional area at R-l, or 23,200 ft2, gave a comparable
value of U =1.03 ft/sec. Thus, using the flowrate and the cross-
3
sectional area gave the same value of ambient velocity as did detailed
velocity measurements for cross-sections as well. Since the cross-
sectional areas varied from 23,900 ft2 to 18,800 ft2, it is felt that
using the flowrate Q = 23,800 cfs and the average value of the cross-
3
sectional areas of the first five downstream stations gave the best
determination of the ambient velocity along the reach of stream affected
by the jet discharge.
-------
0
2
~- 4
1l
LU
a:
^ «
to o
io
214
LU
QQ
£16
£318
o
20
22
24
STATION 1
MB
I I I I
STATION 2
i I L I I
I I
STATION 3
I I 1 L
i i
STATION 4
I I I I
STATION 5
.0
1.0.0
.5 1.0 .0
.5 1.0
FT/SEC
.0 .5 1.0 .0 .5 1.0
FIGURE 29.--VELOCITY PROFILES AT CROSS-SECTION R-l
to
<£>
-------
100
Initial Jet Width.-- The value of the width of the discharge was
estimated to be 187 feet from the scale drawing traced from the USGS
map of the Widows Creek area. The value of the half-width at the origin
was half of this, or b' = 93.5 feet. The value of bQ, the initial half-
width at the beginning of the zone of established jet flow, was found to
be b = 150.0 feet from Equation 65,
b = 1.60 b' (65)
o o
Depth of the Jet.-- It was originally intended to theoretically
calculate the depth of the jet at the mouth of the discharge channel.
According to Harleman (26), when the density difference between the dis-
charge water and river water is sufficiently great, the colder ambient
water will intrude into the discharge channel under the lighter con-
denser water. The interface between the discharge water and ambient
water can be described in terms of two-layer stratified flow theory.
Upon assuming that a critical Froude Number occurs at the junction of
the discharge channel and the river, the depth of the upper layer, or
the depth of the jet, can be predicted based on Harleman's work in terms
of the discharge flowrate, the channel width, and the density difference.
However, the predicted depth of the upper layer using the observed
field values for the plant flowrate, the discharge width, and the density
difference was on the order of 15.0 feet, or almost the entire depth of
the ambient stream. The vertical temperature profiles in Figure 24 and
25 indicate that the average depth of the jet in the ambient stream was
considerably less than 15.0 feet. Therefore, it was concluded that the
conditions necessary to theoretically predict the jet depth were not
-------
101
met at Widows Creek.
Therefore, the depth of the jet was empirically estimated from the
field data in a manner which is consistent with the definition of the
half-width, b, in Equation 24,
b = /2 G (24)
The jet depth, which is assumed constant compared to the jet width, was
estimated from the vertical temperature data shown in Figures 24 and 25
by using a similar definition, shown in Equation 73,
z = /2 a (73)
o z
where o is the standard deviation of the vertical temperature data.
z
The standard deviation and sample variance for each of the first
five sets of vertical temperature data are shown in Table 5. The es-
timated jet depth was found by calculating the standard deviation from
the average of the variances. The standard deviation was 4.24 feet, and
the jet depth was found to be z =6.0 feet by using Equation 73.
TABLE 5
VERTICAL PROFILE STATISTICS
VU SURVEY NUMBER 1
Location
R-l
R-2
R-3
R-4
R-5
Standard
Deviation
3.71
3.07
5.07
4.31
4.75
Sample
Variance
13.7
9.4
25.7
18.6
22.6
-------
102
Initial Jet Velocity.-- The value of the initial jet velocity was
calculated to be U = 2.12 ft/sec from Equation 74,
Q = U z 2 b' (74)
0 0 O 0
where Q = 2370 cfs;
z =6.0 feet; and
b' = 93.5 feet.
o
Velocity Ratio.-- The value of the velocity ratio was found to be
A = 0.50 using U =1.06 ft/sec and U =2.12 ft/sec and Equation 38,
3. O
A = U /U (38)
a o
Initial Angle of Discharge.-- The initial angle, 3 , of the dis-
charge was found by assuming that the centerline axis of the river
passed through the center of cross-sections R-l and R-5, which are
shown in Figure 23. This reach of the river between R-l and R-5 was
assumed to be a straight segment, with the direction of the ambient
velocity along this axis. The x -axis was parallel to this centerline
axis. The origin of the x and y axes was assumed to be located, in
terms of x , in the middle of the discharge channel, and, in terms of
y , at the edge of the ambient stream. After locating the x1 and y1
axes in this way, the initial angle of the discharge was measured,
giving g' = 85.0°.
Zone of Flow Establishment.-- From Figure 18, the length of the
zone of flow establishment was estimated to be s'/b = 1.05, using
e o 6
A = 0.5. From Figure 19, the initial angle at the beginning of the
-------
103
zone of established jet flow was estimated to be 3 =46.7°, using
A = 0.5 and B = 85.0°.
Jet trajectory.-- The distance along the jet trajectory was found
from Figure 23 by measuring the distance s /b along a smooth curve from
the origin to the location of the maximum temperature in each cross-
section. Then, the value of T/T at each of the cross-sections was
calculated.
Entrainment Coefficient.-- Numerical solutions for A = 0.5 and
g = 46.7° were used to determine the value of E. First, a preliminary
o
value of E = 0.2 was determined by sliding the T/T vs. s'/b data plot
horizontally over the T/T vs. S curve, which was computed from S = 0.0.
When the best fit of the data and the theoretical curve was obtained, a
corresponding set of values of s'/b and S was noted. Equation 37 was
then solved for the preliminary value of E,
E - (v^ S/2) / (s/bQ) (37)
Next, Equation 67 was solved for s', using the value of sg/bo = 1.05
and E = 0.2,
S' = (2E/A"b )s' (67)
Q O c
The T/T vs. S plot was then redrawn as T/TQ vs. S*, referenced to the
jet origin, using the value of S and Equation 68,
S' = S' H- S (68)
e
A set of corresponding values of s' and s'/bQ was noted after sliding
the T/T vs. s'/b plot horizontally over the T/TQ vs. s'/bQ curve until
the best fit of the data and the theoretical curve was obtained. Then,
-------
104
Equation 69 was solved for the final value of E,
E = (/S" s'/2) / (s'/b) (69)
0
Drag Coefficient.-- To determine the value of the drag coefficient,
C , numerical solutions of X and Y were computed along with the T/TQ
solutions with A = 0.50 and g = 46.7°. Two different values of the
o
reduced drag coefficient, C* = 0.5 and 1.0, were assumed. Each of the
predicted trajectories was plotted in terms of x /b and y'/b ,
X - (2E//Fb )x ; Y = (2E//F bQ)y (42)
, i
In the field cases, the values of x /b and y /b were considered to be
e o e o
negligible compared to the values of x/b and y/b . Thus, it was
assumed that x'/b = x/b , and y /b = y/b .
o o o o
The value of the reduced drag coefficient was determined by over-
laying the predicted trajectories with the observed jet trajectory. For
this field case, interpolation between the predicted trajectories to the
observed trajectory was not necessary, because the assumed value of
C* = 1.0 gave the best fit.
The value of the drag coefficient was found to be CL = 0.6 from
t
the observed values of E = 0.16 and CL = 1.0 and Equation 39,
Cp = CD/4E (39)
Width Ratio.-- The observed values of the width ratio b/b were
obtained by calculating the standard deviation of the temperature values
at each of the cross-sections and then using Equation 24,
-------
105
(-4)
and the value of b =150.0 feet.
Initial Ri Number.-- The value of the initial Richardson Number
of the jet was calculated to be RiQ = 0.22 using the values
AD/P = 1.29 x 10~3, U =2.12 ft/sec, U =1.06 ft/sec, ZQ = 6.0 feet,
and Equation 60.
Check of Initial Jet Velocity.-- The value of UQ =2.12 ft/sec
was obtained by using Equation 74,
Q = U z 2 b' (74)
xo o o o
To check this method of obtaining U , the velocity measurements
made by USGS personnel were used. Since the velocity measurements made
at the discharge near cross-section R-l were inconsistent and scattered,
presumably due to turbulence caused by the presence of several large
piers used by coal barges, it was necessary to use the measurements
taken at R-2. These velocity measurements, shown in Figure 30, in-
dicated that the jet discharge created a non-uniform velocity field
when compared to the unaffected, uniform part of the velocity field at
R-l, shown in Figure 29.
The observed velocity and temperature rise at R-2 was examined to
see if the values of U and A used in the numerical solution were con-
sistent with the observed values. As seen in Figure 30, the maximum
value of the velocity in cross-section R-2 was 1.7 ft/sec. At this
same cross-section, the value of the temperature rise was T/TQ = 0.49.
The numerical solution using A = 0.50 and 6Q = 46.7° predicted that
-------
0
2
B 4
l_l_
6
T—r
' '
1 I ~T
I I I
I I
o
a
DC:
00
8
10
12
14
16
18
22
STATION 4
24
I I I I I I
STATION 5
STATION 6
I I I I I I
STATION 7
I I I I I I
I I I I I I
.0 .5 1.0 .0 .5 1.0 .0 .5 1.0 .0 .5 1.0
FT/ SEC
FIGURE 30.--VELOCITY PROFILES AT CROSS-SECTION R-2
-------
107
T/T = 0.49 at S* = 28.0. At this same value of s' , the numerical solu-
tion also predicted a value of U/U =0.8. If U = 1.7 ft/sec, then the
initial jet velocity based on data taken at this one cross-section should
have been U = 2.13 ft/sec, which compares favorably with the value cal-
culated from Equation 74, or U =2.12 ft/sec, which is based on data
taken at five cross-sections.
Presentation of Results.-- The data for the November 20 survey are
presented in Table 6, and the parameters are presented in Table 7. The
trajectory and temperature plots, from which the values of CL and E were
determined, are presented in Figures 31 and 32. The data points which
indicate the observed centerline of the jet are plotted on Figure 31,
along with the fitted trajectory computed from the numerical solution.
The observed T/T vs. s'/b data are plotted on Figure 32, along with
the fitted T/TQ curve also computed from the numerical solution. The
observed values of b/b are plotted on Figure 32 along with predicted
b/b curve computed from the numerical solution. The observed value of
E = 0.16 determined from the centerline temperature data was used to
calculate the values of s'/b and from S, using Equations 68 and 69,
S' = S' + S (68)
e
(2E/A" b )s' (69)
-------
20.0 -
0.0
10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0
x'/b0
FIGURE 31.--OBSERVED AND FITTED TRAJECTORIES, WIDOWS CREEK, VU 1
00
-------
1.0
0.5
0.1
0.05
0.01
0.1
I I
A = 0.50
E - 0.16
I I
0.5 1.0
5.0 10.0
50.0 100.
s'/br
FIGURE 32.--OBSERVED VALUES AND FITTED CURVES FOR TEMPERATURE AND WIDTH, WIDOWS CREEK, VU 1
-------
110
TABLE 6
DATA, VU SURVEY NUMBER 1
Cross-Section
R-l
R-2
R-3
R-4
R-5
R-6
R-7
VU Survey Number
Temperature
S'/bo
4.46
14.0
24.3
34.5
45.2
54.2
63.5
2 and TVA Survey
Measurements . - - The
T/TO
0.82
0.49
0.38
0.33
0.31
0.099
0.13
temperature
b/bo
1.00
2.11
2.02
2.80
3.16
--
--
measurements taken
November 21 by VU personnel and August 30 by TVA personnel at Widows
Creek showed that the temperature distribution could again be described
in terms of a two-dimensional surface jet. The surface temperature
contours for these two surveys are shown in Figures 33 and 34. The
location of the cross-sections was the same as for the first survey.
Presentation of Result!;.-- The procedure for analyzing the data
was the same as that used for the first survey. The numerical solution
was used with different values of A and 3Q since the velocity ratio,
A = U /U , was different in each case from the first survey. The data
a o
-------
Ill
TABLE 7
PARAMETERS, VU SURVEY NUMBER 1
Observed Values
T , Initial Temperature Rise, °F
Q , River Flowrate, ft3/sec
Q , Plant Flowrate, ft3/sec
b', Discharge Half-Width, ft
B0, Discharge Angle, °
14.2
26,170
2,370
93.5
85.0
Calculated Values
Qa, Ambient Flowrate, ft3/sec
Rio, Initial Ri Number
Average Cross-Sectional Area, ft2
Ua, Ambient Velocity, ft/sec
b0, Initial Jet Half-Width, ft
zo, Jet Depth, ft
Uo, Initial Jet Velocity, ft/sec
A, Velocity Ratio
se/b0, Establishment Zone
3 , Initial Jet Angle, °
23,800
0.22
22,400
1.06
150
6.0
2.12
0.50
1.05
46.7
Observed Results
E, Entrainment Coefficient
Cp, Reduced Drag Coefficient
CD, Drag Coefficient
0.16
1.0
0.6
-------
FIGURE 33.--TEMPERATURE DISTRIBUTION, °F, AT 1.0-FOOT DEPTH, WIDOWS CREEK, VU :
-------
FIGURE 34.--TEMPERATURE CONTOURS, °F, AT 0.5-FOOT DEPTH, WIDOWS CREEK, TVA
-------
114
for VU Number 2 and TVA surveys are presented in Table 8, and the
parameters are presented in Table 9. The trajectories and temperature
plots are presented in Figures 35-38. The observed and predicted
values of b/b are presented in Figures 36 and 38. For the VU survey,
the observed values were E = 0.16, CD = 0.9, and CD = 0.6. For the
TVA survey, the observed values were E = 0.16, CD = 0.4, and CD = 0.3.
TABLE 8
DATA, VU SURVEY NUMBER 2 AND TVA SURVEY
Cross- t
Section s /b
R-l
R-2
R-3
R-4
R-5
R-6
R-7
4.46
14.0
24.3
34.5
45.2
54.2
63.5
VU No.
0.81
0.56
0.40
0.42
0.25
0.15
--
2
b/b
o
--
1.66
2.03
2.48
2.58
--
--
TVA
T/T
o
0.74
0.50
0.47
0.41
0.28
0.07
0.08
b/b
o
--
1.28
1.98
2.55
2.88
4.01
4.15
Results of the New Johnsonville Survey
Introduction
Data (37) were collected by VU personnel on May 29, 1969, at
TVA's New Johnsonville Steam Plant located on Kentucky Lake. This
plant is similar to Widows Creek in that condenser water is discharged
into a small coal barge harbor and then flows onto the river surface.
The spatial distribution of the discharged heat is again described in
-------
115
TABLE 9
PARAMETERS, VU SURVEY NUMBER 2 AND TVA SURVEY
Observed Values
T0, °F
Qr, ft3/sec
Q0, ftVsec
b0, ft
30,
VU No. 2
13.0
26,170
1,840
93.5
85.0
TVA
14.2
47,000
2,200
93.5
85.0
Calculated Values
Qa, ft3/sec
Rio
Average Cross-
Sectional Area, ft2
Ua, ft/sec
b0, ft
z0, ft
U0, ft/ sec
A
1
Bo. °
VU No. 2
24,330
0.64
22,400
1.09
150
6.0
1.64
0.67
0.84
36.5
TVA
44,800
1.21
26,250
1.71
150
5.2
2.27
0.75
0.75
32.3
Observed Results
E
c;
D
CD
VU No. 2
0.16
0.9
0.6
TVA
0.16
0.4
0.3
-------
i i i i i r
i r
20.0
15.0
o
•O
10.0
A = 0.67
- 0.9
5.0
0.0
5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0
x'/bQ
FIGURE 35.--OBSERVED AND FITTED TRAJECTORIES, WIDOWS CREEK, VU 2
-------
1.0
0.5
0.1
0.05
0.01
0.1
r
Q = 36.5°
A = 0.67
E = 0.16
0.5 1.0
s'/b
o
5.0 10.0
100.0
50.0
10.0
5.0
1.0
50.0 100.0
FIGURE 36.--OBSERVED VALUES AND FITTED CURVES FOR TEMPERATURE AND WIDTH, WIDOWS CREEK, VU 2
-------
20.0 -
15.0
o
10.00-
5.0 -
0.0
5 0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0
x'/D0
FIC.URH 37.--OBSl-;RVH1> AND 1MTTHI) TRA.JKCTORTT.S, W1POWS CRIiliK, TVA
-------
1.0
0.5
0.1
0.05
0.01
0.1
r
ft =36.5(
A = 0.75
E = 0.16
0.5 1.0
5.0 10.0
s'/b
100.0
50.0
10.0
5.0
1.0
50.0 100.0
o
Flf.URi; 38.--OBS1-RV1:!) VALUES AND FTTTHD CURVES FOR TllMPHRATURP. AND WIDTH, WIDOWS CRTJiK, TV A
-------
120
terms of a two dimensional surface jet deflected by the ambient current.
At the discharge, the width of the ambient water body is only four times
larger than the width of the discharge. As a result, the entrainment
characteristics and the spatial location of the jet were affected by the
narrow boundaries. The surface temperature contours are shown in
"igure 59.
Depth of the Jet
At New Johnsonville, detailed velocity measurements were made at
the mouth of the discharge channel. These velocity measurements were
used to estimate the depth of the jet at this field site. The sample
standard deviation of the velocity data shown in Table 10 was calculated,
and then the depth of the jet was found to be ZQ = 5.95 feet, using
Equation 73,
(73)
z
o
VELOCITY DATA,
Depth, ft
1.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
22.0
= v'2 a
z
TABLE 10
NEW JOHNSONVILLE SURVEY
Velocity, ft/sec
0.99
0.93
0.73
0.10
0.14
0.01
0.10
0.14
0.0
0.0
-------
0 1000 2000 FEET
i i i i ^
OLD RIVER
CHANNEL
FIGURE 39.--OBSERVED TEMPERATURE DISTRIBUTION, °F, AT 1.0-FOOT DEPTH, NEW JOHNSONVILLE
-------
122
Presentation of Results
Except for the depth determination discussed above, the procedure
for analyzing the data was the same as for the Widows Creek surveys.
The numerical solution was used with A = 0.57 and 8Q = 30.0°. The
temperature data are presented in Table 11 and the parameters in
Table 12, while the trajectory and temperature plots are presented in
Figures 40 and 41, respectively. The width ratio, b/bQ, is not cal-
culated. The distortion caused by the narrow boundaries precluded
measuring with any degree of accuracy the location of the lateral
temperature rise perpendicular to the jet axis. The observed values
of E = 0.04 and C* = 3.0 reflect the influence of the geometry.
TABLE 11
TEMPERATURE DATA, NEW
JOHNSONVILLE SURVEY
0.63 1.00
5.62 0.91
12.5 0.82
15.7 0.73
-------
123
TABLE 12
PARAMETERS, NEW JOHNSONVILLE SURVEY
Observed Values
T , °F
Qr, ftVsec
Q0, ft3/sec
bO> ft
C °
11.0
26,500
2,180
250
60.0
Calculated Values
Qa, ft3/sec
Ri0
Average Cross-
Sectional Area, ft
Ua, ft/sec
b0, ft
z0, ft
U0, ft/sec
A
V
se/b
B0' °
24,320
3.30
58,182
0.42
400
5.95
0.73
0.57
0.93
30.0
Observed Results
E
CD
CD
0.044
3.0
0.5
-------
4o.--oBsuRvi-;i) AND MTTI;U TRAJiiCTORii-s, NHW
-------
1.0
0.5 -
T/T,
o
0.1
0.05
0.01
3 =30.0
o
A = 0.57
E = 0.04
J L
J L
1
0.1
0.5 1.0
5.0 10.0
s'/b
o
50.0 100.
FIGURE 41.--OBSERVED VALUES AND FITTED CURVE FOR TEMPERATURE, NEW JOHNSONVILLE
-------
126
Results of the Waukegan Survey
Introduction
Data obtained from a report by Beer and Pipes (6) at Waukegan
generating station located on Lake Michigan were analyzed in terms of a
two-dimensional surface jet. The cooling water from the plant was dis-
charged into a 2000-foot long channel and then into the lake. There was
no appreciable ambient velocity, and, therefore, the velocity ratio was
equal to zero, or A = U /U =0.0. This case of a jet discharging into
3. O
a stagnant environment ordinarily could not be described as two-
dimensional, since the Richardson Number is quite low, and vertical en-
trainment would be significant. However, the shallow depth of the lake
near the shore inhibits vertical entrainment to some degree. Over a
longitudinal distance of 3600 feet, the jet spread laterally from an
initial width on the order of 250 feet to about 2500 feet and spread
vertically from about 6 feet to a depth on the order of 15 feet. Thus,
the rate of lateral mixing was approximately five times greater than
vertical mixing. Even though the rate of lateral mixing was not a full
order of magnitude greater than vertical mixing, it is felt that appli-
cation of the two-dimensional model is reasonable, if only in order to
examine a case in which the two-dimensional assumption is not fully met.
Temperature Measurements
The temperature contours at the 1.0-foot depth are shown in
Figure 42.
-------
FIGURE 42.--TEMPERATURE DISTRIBUTION, °F, AT 1.0-FOOT DEPTH, WAUKEGAN
[AFTER BEER AND PIPES (6)]
-------
Zone of Establishment
The length of the zone of establishment is not estimated from
Figure 18 because the laboratory experiments did not include the case of
A = 0.0. Instead, the length was taken to be s /d = 6.2 from the work
of Albertson ejt al_. (4). Expressed in terms of b , the length is found
to be s'/b = 4.4 bv using Equation 66,
e o ' '
b = 0.708d' (66)
o o
Results of Analysis
The procedure for analyzing the data was the same as for the other
surveys. The numerical solution was used with A = 0.0. The temperature
data are presented in Table 15 and the parameters in Table 14, while the
temperature plot is presented in Figure 43. Values of b/bQ estimated
from Figure 42 are also presented in Figure 43 along with the predicted
b/b curve. The value of the entrainment coefficient was found to be
o
E = 0.44.
The temperature and width plots, Figure 43, indicate that the two-
dimensional model can be matched to the observed data reasonably well in
the zone of established flow. However, in the zone of establishment,
the predicted values are greater than the observed data. For a distance
of s'/b =4.4, the curve predicts T/TQ =1.0, while over the same
distance, the values of the data decrease from T/TQ = 0.8 to 0.7.
The initial temperature rise, T = 15.0°F, was computed from the
difference of the reported condenser temperature, 60°F, and the lowest
ambient temperature in the region of the jet, or 45°F. Examination of
the surface contour plot, Figure 42, indicates that the temperature at
-------
129
TABLE 15
DATA, WAUKECAX SURVEY
>'»„
0.
0.
0.
1.
1.
3 .
4.
5 .
6.
7
9.
11.
14.
16.
0
39
48
29
95
41
34
60
89
99
65
6
7
8
T/T
o
1
0
0
0
0
0
0
0
0
0
0
0
0
0
.0
.80
.80
.80
.6~
.6~
.67
.60
.55
.40
.40
. 55
. 53
.20
b/b
o
--
--
--
--
--
--
2.0
2.75
3.5
4.75
5.25
5.0
--
--
TABLE 14
PARAMETERS, IVAUKEGAN SURVEY
Observed Values
T0, °F 15.0
Q0, ft3/sec 1,690
b'Q, ft 125
Calculated Values
Ri0 0.01
b0, ft 200
U0, ft/sec 2.6
A 0.0
Observed Results
0.44
-------
1.0
0.5
0.1
0.05
0.01
0.1
• 90.0
A =0.0
E = 0.44
I I
o o o
0.5 1.0
T/l
b/b
o
D
I I
5.0 10.0
s'/b
o
I
100.0
50.0
10.0
5.0
1.0
50.0 100.0
O
FIGURE 43.--OBSERVED VALUES AND FITTED CURVES FOR TEMPERATURE AND WIDTH, WAUKEGAN
-------
131
the origin of the jet ma}' have been on the order of 58°F, and that the
initial temperature rise may have been 13°F, instead of 15°F. If an
initial temperature rise of T = 15.0°F could be assumed, then all the
po
ints in the zone of establishment would be closer to the predicted
value of T/T =1.0.
o
Comparison of Results
The results from the three Widows Creek surveys, the New Johnsonville
survey, and the Waukegan survey are summarized in Table 15. The values
of the entrainment coefficient at Widows Creek were consistently E = 0.16
for three different field surveys with different values of A = ua/l]Q-
The small value of E = 0.04 at New Johnsonville indicates the influence
of the relatively narrow boundaries, which apparently greatly inhibited
lateral entrainment. The value of E = 0.4 at Waukegan seems reasonable,
since there were no boundaries to inhibit the lateral entrainment of
colder water into the jet. However, this value may indicate the effects
of vertical entrainment, since the rate of lateral mixing was not a full
order of magnitude greater than vertical mixing at the Waukegan site.
Because of the scatter of the data, it is difficult to determine the
significance of this effect.
The values of the drag coefficient, C , ranged from 0.3 to 0.6, as
A varied from 0.75 to 0.50. Over this narrow range of A, at least, it
appears that C is a function of A under field conditions as well as
under laboratory conditions.
-------
132
TABLE 15
SUMMARY OF FIELD RESULTS
Widows Creek
VU No. 1 0.
VU No. 2 0.
TV A 0 .
New Johnsonville 0.
U'aukegan 0.
A
50
67
75
57
00
;' Ri
0 0
85.0 0
85.0 0
85.0 1
60.0 5
0
T 1
.64
.21
.30
.01
E
0.
0.
0.
0.
0.
CD
16 1.0
16 0.9
16 0.4
04 5.0
44
Cn
0.
0.
0.
0.
6
6
5
5
-------
CHAPTER VII
DISCUSSION
Results of Laboratory and Field Investigations
Establishment Zone
Laboratory results describing how the zone of flow establishment
is related to the velocity ratio are presented in Figures 18 and 19.
These relations were used in analyzing the field data in order to obtain
initial values of the jet axis distance and the initial angle, - from
o
which to obtain the numerical solution. The laboratory results were
used because detailed field measurements of the establishment zone could
not be obtained due to time and equipment limitations. It is felt that
the empirical curves in Figures 18 and 19 are adequate to describe the
observed field conditions, as can be seen by examining the temperature
and trajectory plots of the field data.
The length of the zone of flow establishment is, in general,
slightly less than values found in the literature. A possible explana-
tion is that the relations in this study were derived for the cases of
the jet origin coinciding with the flume wall, while the relation used
by Fan (22), for instance, was developed for a jet placed away from the
boundary into a region of more nearly uniform flow. The effect of the
velocity gradient near the boundary, it is felt, is to shorten the
length of the establishment zone. Since the discharge coincides with
133
-------
134
the boundary at most field sites, the relations derived in the present
study are felt to be more realistic when applied to field sites.
Entrainment Coefficient
The entrainment mechanism, or the inflow velocity, is assumed to
represent the turbulent mixing of the jet. The laboratory results
presented in Tables 2, 3, and 4 indicate that E = 0.4 when PQ = 90.0°,
and that E = 0.2 when t' = 60.0° and 45.0°. Thus, E appears to be a
o
function of the initial angle of discharge of the jet. Apparently, as
;;' is decreased, the volume of colder water available for inflow and
o
dilution of the heated jet on the side of the jet near the boundary is
also decreased. Thus, the smaller values of E at PQ = 60.0° and 45.0°
indicate the effect of the near boundary on entrainment.
The values of E at the two field sites located on rivers are lower
than the laboratory results for comparable values of j3\ These lower
values apparently indicate the effect of the narrow boundaries on lateral
inflow on both sides of the jet, since the ratios of the ambient width
to the discharge width of 7.5 at Widows Creek and 3.5 at New Johnsonville
are lower than the ratio of 24.0 used in the laboratory experiments.
The values of E at Widows Creek are consistently equal to 0.16 for
three different field survevs with different values of A = U /U . Thus,
d O
it appears that, for a given site, the entrainment coefficient is
reasonably constant.
For cases where A = 0.0, other investigators, such as Fan (22),
have reported laboratory values of the entrainment coefficient on the
order of 0.10. The value of E = 0.44 at the Waukegan field site is
-------
135
areater and possibly indicates the effects of ambient turbulence and
£>
vertical mixing, the effects of which are not considered in the
analytical model.
The entrainment coefficients determined in this study differ
slightly from values found in the literature. Fan (221 has reported
values on the order of 0.45 determined from measurements along the axis
of a buoyant jet discharged vertically into a uniform, ambient current.
However, his values would be smaller if a more realistic, bimodal pro-
file had been assumed, since the maximum cross-sectional concentrations
on both sides of the jet axis were 60 to 80% greater than concentrations
along the jet axis. Keffer and Baines (30) have reported values of the
entrainment coefficient which varied along the jet axis from 0.5 to
1.6 for air-in-air jets subject to a steady cross-wind. Zeller (48)
presented values which varied from 0.127 to 0.993, not along the jet
axis, but as determined from 22 sets of data for two surface jets at a
field site. The values of E determined in the present study are not
unreasonable, however, when the variation in the values of E reported by
these other investigators is considered.
The entrainment coefficient appears to be a function of the
assumed profile shape and other factors, the effects of which are not
yet completely understood. For example, although some investigators
suggest that E is a function of A, the results of the present study
indicate that boundary geometry may be more significant when relatively
narrow channels and discharge angles less than 90° are considered.
-------
156
Drag Coefficient
The laboratory results presented in Tables 2 and 3, and in
Figures 21 and 22 indicate that C varies with the velocity ratio and,
possibly, with the angle of discharge. Values of C_. vs. A have also
been reported by Fan (22) and Carter (12).
The range and magnitude of C from C.I to 3.8 are greater than
would be expected based on the work of Fan, who reported a range of C
from 0.1 to 1.7, and Carter, who reported values of CD ranging from 0.74
to 1.36. However, as shown in Equations 17-19, the analytical form of
the drag force used in this study is different from the forms used by
Fan and Carter. Thus, a direct comparison of C values cannot be made.
The possible dependence of C on 3 is indicated in Figure 22.
The values of C are seen to decrease as S* decreases from 90° to 60°
D o
and 45°. In the analytical model, the drag force is assumed to represent
the effects of the pressure gradient caused by the separation of the
ambient flow behind the jet. At o' = 60.0° and 45.0°, the jet is closer
and closer to the near boundary, the presence of which may inhibit the
ambient flow and reduce the separation of the flow. Thus, the drag force
mav be reduced at 6' = 60.0° and 45.0°, which would account for the
o
smaller values of C observed at 3' = 60.0° and 45.0°.
The field values of C are very close to the values that would
have been predicted based on the laboratory results, as shown in
Figure 44. In this figure, the observed field values of C vs. A, de-
termined over the range of A from 0.5 to 0.75, are plotted along with
the empirical laboratory curve, which was shown in Figure 21. Unfor-
tunately, the lack of field values of CQ determined in the range of
-------
137
1.0 -
0.5
'D
0.2
0.1
i—rr
LABORATORY CU
D
WIDOWS CREEK
O VU1
D VU 2
0 TVA
NEW JOHNSONVILLE
o V
I I I I I I
0.2
0.5
1.0
FIGURE 44.--OBSERVED FIELD VALUES OF DRAG COEFFICIENT
VERSUS VELOCITY RATIO PLOTTED ON LABORATORY CURVE
(From Figure 21)
-------
138
A from 0.5 to 0.2 or lower restricts the usefulness of Figure 44 at the
present time.
The comparison of the observed field values of C with the laboratory-
determined curve in Figure 44 implies that C is not a function of the
Reynolds Number but only a function of A. However, since Reynolds Num-
bers in the field may be two or more orders of magnitude greater than in
the laboratory, it may not be possible to merely extrapolate the C vs. A
relation from the laboratory to field sites as is done in Figure 44.
Literature references, such as Prandtl and Tietjens (35), present
C values for solid objects in a moving stream as functions of a Reynolds
Number defined in terms of the ambient velocity and viscosity and some
characteristic length, or diameter, of the solid body. In the present
study, the jet is treated as a solid body in terms of the drag force.
Therefore, it is possible to define a Reynolds Number, ReD, in terms of
the ambient velocity and viscosity and the depth of the jet as shown in
Equation 75,
U z
Re = __§_ (75)
D v
a
A plot of Cn vs. Re , shown in Figure 45, indicates that C may
have the same characteristics as the drag coefficient usually associated
with flow around solid bodies. The laboratory values of CD decrease as
the Re Number increases in the range of 102 < ReQ < I0k, and the field
values of C become relatively constant as the Re Number approaches
values of Re - 106. A quantitative relation between CD and ReD in
Figure 45 cannot be derived, since there are not enough data points, par-
ticularly in the range of 104 < ReQ < 105. However, a possible relation
-------
10
5
. J
i—r
D
A
A
D
LABORATQRY RESULTS
A
90^
o
O
D 60.0
45.0°
i i 1—r
FIELD RESULTS
WIDOWS CREEK
O VU 1
D VU 2
Q TVA
A NEW JOHN-
SONVILLE
.1
D
o
1 1
J L
J L
103
Re
D
10
10
FIGURE 45.--OBSERVED VALUES OF DRAG COEFFICIENT VERSUS REYNOLDS NUMBER
-------
140
between C and Re is suggested by Figure 45, and this relation should
be studied further.
Possible Sources of Error
Determination of E and Cn
Values of E and C were obtained by fitting the theoretical curves
to the observed data. Since the solutions were in numerical rather
than analytical form, best-fit values were obtained by inspection. Some
of the scatter in the reported results may be due to the graphical method
of solution, but it is felt that this source of error is negligible when
reasonable care is taken in determining E and Cp.
Ambient Turbulence
The effects of mixing due to ambient turbulence are not considered
in the analytical model. It is assumed that the decrease in the tempera-
ture is due to the entrainment of ambient fluid, which is caused by the
difference between the jet and the ambient velocities. As long as the
jet velocity is greater than the parallel component of the ambient
velocity, the decrease in temperature can be described by the model.
However, as the jet velocity approaches the same direction and magnitude
as the ambient velocity, the model predicts that entrainment and dilution
decrease as the velocity difference decreases. Beyond this point, the
observed decrease in the temperature would have to be accounted for by
turbulent diffusion in the ambient stream.
Temperature measurements smaller than T/TQ =0.10 were not used in
determining the values of the entrainment coefficient. It was felt that
the let velocities beyond this point had approached the same magnitude
-------
141
and direction as the ambient velocity, and that the temperature decreases
beyond this point were caused by the ambient turbulent diffusion.
Density Differences
Changes in density along the jet axis could, in some cases, affect
the characteristics of the jet. However, since the maximum density dif-
ference was approximately A,?/r = 0.9°o in the laboratory and approximately
Ar/o = 0.1°! in the field, changes in density are felt to have been
negligible compared to a reference density, c .
Temperature Losses
Heat losses across the jet surface can be considered negligible, as
demonstrated by the following calculations based on the work of Edinger
and ("ever (19) . The decrease in temperature rise along a stratified
stream can be approximated by Equation 76,
T/T = expC-Kx/c C U z ) ("61
o J v o p o o
where K = the thermal exchange coefficient; and
C = the heat capacity of water.
P ^
Applying this equation to a surface jet actually exaggerates the de-
crease in temperature rise due to surface heat exchange, since lateral
mixing of the jet will also reduce the temperature rise and, consequently,
reduce the driving force for surface heat exchange.
Selecting representative values of K = 200 Btu/ft-i-day-°F
(=2.32 x 10'3 Btu/ft2-sec-°F), c = 62.4 lb/ft3, and C = 1.0 Btu/lb °F,
then values of T/T versus x/z can be computed for laboratory and field
o o ^
conditions from Equation 76. For example, using the laboratory value of
U = 0.5 ft/sec, Equation 76 predicts T/T = 0.98 at x/z = 200. In the
a o o
-------
14;
laboratory, values on the order of T/TQ =0.10 were measured at X/ZQ = 200,
The observed decrease in T/T due to both lateral mixing and surface heat
o
exchange was therefore on the order of 0.90, while the predicted decrease
in T/T due to surface heat exchange alone from Equation 76 is only about
o
0.02. Thus, the decrease in T/T due to lateral mixing and surface heat
exchange is more than an order of magnitude greater than the decrease due
to surface heat exchange alone.
Using the field value of U =1.06 ft/sec, Equation 76 predicts
3.
T/T = 0.965 at x/z = 1000, or x/b = 40. In the field, values on the
o o o
order of T/T = 0.32 were measured at x/b = 40. The observed decrease
o o
in T/T due to lateral mixing and surface heat exchange was therefore on
o
the order of 0.7, while the predicted decrease due to surface heat ex-
change alone from Equation 76 is about 0.04. Therefore, since the de-
crease in T/T due to lateral mixing and surface heat exchange is more
o
than an order of magnitude greater than the decrease due to surface heat
exchange alone in the laboratory and in the field, then the effects of
surface heat exchange can reasonably be neglected in the present study.
Applicat ion
The usefulness of the two-dimensional surface jet model in
analyzing field data appears to be dependent on the velocity ratio and
the discharge angle, the value of the initial Richardson Number, and the
intensity of the ambient turbulence.
Velocity Ratio and Discharge Angle
For the ranges of A and s' considered in the laboratory and in the
field, the results of the present study indicate that, even when the
-------
143
discharge velocity is on the same order of magnitude as the ambient veloc-
ity, the effects of the discharge velocity should be considered, as long
as the initial direction of the discharge velocity is significantly dif-
ferent from the direction of the ambient current. The effect of changes
in the discharge velocity on the spatial temperature distribution can be
seen by examining the observed laboratory and field trajectories. As the
velocity ratio is changed, the location of the trajectory is also changed.
The analytical model, which considers the non-uniform velocity field
caused by the surface jet, can be used to describe how these changes in
the velocity ratio affect the temperature distribution.
Initial Richardson Number
The value of the initial Ri Number can be used to indicate whether
or not the jet will be two-dimensional. Jen et_ al_. (29) report that the
rate of lateral mixing is only twice the vertical rate for the range of
5.0 x 1CT5 < Ri < 3.0 x 10"3, while the results of Tamai et al. (43)
o
indicate that the lateral rate is somewhat greater when the Ri Number
& o
is in the range of 1.0 x 10~2 < Ri < 2.0 x 10'l. The results of the
— o —
present study indicate that the two-dimensional assumption is valid when
the Ri Number is as low as Ri =0.22, which is the lowest of the values
o o
determined at Widows Creek, where the rate of lateral mixing was found
to be about 8.0 times the vertical rate. This is reasonable, since
Ellison and Turner (21) and Zeller (48) have all reported for the case of
a surface jet that the value of the Ri Number along the jet axis rapidly
increases to 1.0 or greater, and that vertical mixing becomes negligible.
-------
144
The two-dimensional assumption also appears to be valid at even lower
values of the Ri Number in cases where vertical entrainment is inhibited
by shallow depths.
Ambient Turbulence
The effects of ambient turbulence have not been investigated in
the present study. Based on the laboratory and field results, the
analytical model seems adequate to describe the decrease in the tempera-
ture rise to values as low as T/TQ •=• 0.10 as far downstream as x/bQ = 100.
In highly turbulent streams, the temperature decrease may be even more
dependent on the intensity of the ambient turbulence.
Usefulness of the Proposed Model
When a heated discharge has the characteristics of a two-dimensional
surface jet, then the location of the jet trajectory and the changes in
temperature and width along the trajectory can be determined by the
model developed in the present study. The model developed by Zeller (48)
does not consider the pressure gradient across the jet. The importance
of the pressure gradient on the location of the jet can be seen by con-
sidering the magnitude of the CD values determined in the present study.
The model developed by Carter (12) cannot predict the change in the jet
width along the jet trajectory.
It is felt that the present study represents an advance in the
analysis and prediction of temperature distributions. The usefulness
of the proposed model should increase as more cases of field data are
analyzed to determine the values of the entrainment and drag coefficients
over a wide range of field conditions.
-------
CHAPTER VIII
SUMMARY AND CONCLUSIONS
A survey of the literature indicated that further study of surface
lets was justified. Previous work did not contain a completely suitable
method for quantitatively describing the spatial temperature distribution
in the vicinity of power plants located on rivers where the discharge and
the ambient velocities should both be considered. The analytical and
experimental studies performed by Jen, Wiegel, and Mobarek (29), Tamai,
Wiegel, and Tornberg (43), and Hayashi and Shuto (28) are applicable
primarily to jets discharging into ambient water bodies that have no
appreciable velocity. The two-dimensional surface jet model developed
by Zeller (48) does not consider the pressure gradient that exists across
the surface jet parallel to the ambient current, while the model developed
by Carter (12) cannot predict the change in the width of the jet along
the axis. Therefore, a model is developed which is able to better de-
scribe certain cases of heated power plant discharges.
Analytical Development
The present study, which is based on the previous work of Morton
(54), Fan (22), and Zeller (48), develops a system of ordinary differen-
tial equations, Equations 25, 27, 28, 30, 31, and 32, which, when solved
numerically, predicts the jet trajectory, width, velocity, and temperature
distribution for the case of a two-dimensional surface jet.
145
-------
146
The entrainment mechanism is assumed to represent the turbulent
mixing of the jet. The inflow velocity, which is assumed to be propor-
tional to the difference between the centerline velocity of the jet and
the parallel component of the ambient velocity, is written as in
Equation 12,
v = E(U - U cos 3) (
i a
The coefficient E is defined as the experimentally-determined entrain-
ment coefficient.
The pressure gradient, due to the separation of the ambient flow,
is assumed to be represented by the drag force normal to the jet axis,
as shown in Equation 17,
C n U2(z ds) sin 6
Da a
The coefficient, C , is the experimentally-determined drag coefficient.
As the jet becomes parallel to the ambient flow, the drag force goes to
zero, because the projected area normal to the ambient current, or
(z ds sin i?) , goes to zero.
In practical applications, the zone of flow establishment of the
jet, illustrated in Figure 8, must be determined. As in previous work on
jets in a non-parallel stream, it was necessary to empirically determine
how the length of the zone of flow establishment and how the initial
angle at the end of the zone are related to the velocity ratio and to
the initial angle of the discharge.
-------
147
Laboratory Experiments
The laboratory experiments were designed to study the entrainment
and drag coefficients and the zone of flow establishment. The object was
to functionally relate these coefficients and the zone of establishment
to the velocity ratio and to the initial angle of discharge. The velocity
ratio, A. = U /U , was varied from 0.180 to 0.727, and lets with discharge
a o
angles of a' = 45.0°, 60.0°, and 90.0° were used.
The values of the entrainment coefficient are presented in
Tables 2, 5, and 4. The results indicate that E = 0.4 when |3^ = 90.0°,
i
and that E = 0.2 when [3 = 60.0° and 45.0°. Thus, it appears that E is
a function of boundary geometry. For the range of A used in the present
study, E was not found to be a function of A.
The results presented in Tables 2 and 5 and Figures 21 and 22
show that C ranged from 0.1 to 3.8. It appears that CQ varies with the
velocity ratio, as shown in Figure 21, and, possibly, with the initial
angle of discharge, as shown in Figure 22.
The length of the zone of establishment and the initial angle at
the end of the zone were related to the velocity ratio. The empirically-
determined relations describing s'/bo vs. A and PO/BO vs. A are shown
in Figures 18 and 19, respectively.
Field Surveys
Data obtained from five field surveys at three different steam-
electric generating plants were analyzed to see how well the two-dimen-
sional surface jet model describes the observed temperature distribu-
tions .
-------
148
The values of E and C from the field surveys are presented in
Table 15. The values of E at Widows Creek are consistently equal to
0.16 for three different field surveys with different values of
\ = U /U . The snail value of E = 0.04 at New Johnsonville reflects
a o
the influence of the relatively narrow boundaries, which apparently
greatly inhibit lateral entrainment. The value of E = 0.4 at Waukegan
seems reasonable, since there are no boundaries to inhibit the lateral
entrainment of cooler water into the jet. However, this last value may
indicate the effects of ambient turbulence and vertical entrainnent, the
effects of which the analytical model does not consider. The scatter
of the data at this site makes it difficult to determine how great these
effects are.
The values of C are very close to the values that would have been
predicted based on the laboratory results, as shown in Figure 44.
The laboratory-derived relations describing the zone of flow
establishment were used in analysing the field data. The laboratory
results were used because detailed field measurements of the establish-
ment zone could not be obtained due to time and equipment limitations.
Results of the Laboratorv and Field Investigations
Results indicate that the value of the entrainment coefficient is
a function of boundary geometry. The results of the laboratory studies
indicate that lateral inflow, or dilution, is inhibited along the near
boundary when the discharge angle is decreased: the value of E decreased
from 0.4 to 0.2 when •' was decreased from 90° to 60° and 45°.
o
-------
149
The results of the field studies indicate that lateral inflow on
both sides of the jet is inhibited by a narrow channel. It appears
that E decreases as the ratio of the channel width to the discharge
width decreases, because values of E = 0.16 were determined at Widows
Creek, where the ratio is about 7.5, and a smaller value of E = 0.04
was determined at New Johnsonville, where the ratio is about 3.5.
The field values of C closely agree with the empirically-determined
laboratory curve, as shown in Figure 44. However, the possible dependence
of C on the Reynolds Number is also considered. A plot of CD vs. the
Reynolds Number, shown in Figure 45, indicates that C may have the same
characteristics as the drag coefficient usually associated with flow
around solid bodies. The laboratory values of C decrease as the ReQ
Number increases in the range of 102 < Re < 10\ and the field values
of C become relatively constant and equal to about 0.5 at values of
Re - 106. It is felt that this possible relation between CD and the
Re Number should be studied further.
Application
The results of the laboratory and field investigations indicate
that the application of the two-dimensional surface jet model is dependent
on the velocity ratio and the initial angle of discharge, the value of
the initial Richardson Number, and, possibly, the intensity of the am-
bient turbulence.
Even when the jet velocity is on the same order of magnitude as
the ambient velocity, it appears that the temperature distribution can
still be described in terms of a surface jet, as long as the initial
-------
150
direction of the discharge velocity is significantly different from the
direction of the ambient current.
The two-dimensional assumption appears valid when the initial
Richardson Number is as low as 0.22, which is the lowest of the values
determined at Widows Creek, where the rate of lateral mixing was found
to be about 8.0 times greater than the vertical rate.
The effect of ambient turbulence is not considered in the proposed
model. However, the model seems to adequately describe the decrease in
the temperature rise as far downstream as s/b - 100, unless the ambient
stream is highly turbulent.
Future Work
Entrainment and Drag Coefficients
The accurate prediction of values of E and C for design purposes
depends upon additional work. More field data should be obtained and
compared to the proposed model in order to determine the values of E and
C at different field sites under widely varying conditions.
The effects of boundary geometry on the temperature distribution
are reflected in the empirically-determined values of E. According to
Rouse (58), at some distance downstream in a narrow channel, lateral
inflow becomes negligible, and the volume flux of the jet becomes con-
stant. Thus, boundary effects could be treated theoretically by
modifying the system of equations so that lateral entrainraent goes to
zero at some point downstream.
The effects of the ratio of the channel width to the discharge
width could be studied in the laboratory by varying the width of the
-------
151
channel or the size of the jet. Also, these effects would have to be
considered in modeling a specific field site.
Vertical Entrainment
Future work should consider the effects of vertical entrainment,
which causes a greater decrease in the temperature rise that the two-
dimensional model predicts. Vertical entrairunent occurs when the
shearing force of the jet velocity is greater than the opposing buoyancy
force, which is due to the density difference between the jet and the
heavier ambient fluid. When the initial Richardson Number is very small,
the rate of vertical spreading is no longer negligible compared to the
lateral rate and must be considered.
It appears that the rate of vertical spreading could be related
to the lateral rate in terms of the Richardson Number, which, according
to Ellison and Turner (21), increases with distance along the jet axis.
Along the jet axis, the vertical rate of spreading would decrease as a
function of the increasing Richardson Number. A three-dimensional
analytical model would have to be formulated in which the decrease in
temperature rise would be greater than that predicted by a two-dimensional
model but less than the decrease predicted by an axi-symmetric model.
Fietz and Wood (24) have studied the case of a three-dimensional density
current along a sloping floor. However, considerable work remains to be
done in order to develop a useful model for cases of surface jets.
Ambient Turbulence
Two approaches to the problem of temperature prediction in the
mixing zone downstream from a power plant are currently in use. The
-------
152
first approach, illustrated by the present study, is to assume that the
heated discharge can be described in terms of a momentum jet. The
second approach, illustrated by the work of Edinger and Polk (20), is to
assume that mixing can he determined by considering the diffusion due to
ambient turbulence. However, in some cases, as noted by Csanady (16) and
others, both jet momentum and ambient turbulence should be considered.
Future investigators should consider combining the two approaches.
Pratte and Baines (36) note the importance of the relative size of the
turbulent eddies compared to the size of the jet cross-section. Some
suggestions are offered by Briggs (8), who has studied plume rise and
dispersion in the atmosphere. He feels that the point at which the plume
velocity becomes small compared to the ambient velocity can be charac-
terized by the eddy dissipation, which is a measure of the ambient tur-
bulence, and some characteristic radius of the plume. Such an approach
might be applied to the case of a surface jet.
Establishment Zone
The present derivation of the establishment zone assumed a uniform
temperature and velocity distribution at the point of discharge. However,
at many field sites, the temperature and velocity profiles are likely to
be non-uniform at the mouth of the discharge channel, and the establishment
zone is likely to be shorter than would be predicted by the present
relations. More consideration should also be given to predicting the
initial jet depth in terms of the cold water wedge that, according to
Harleman (26), may intrude into the discharge channel.
-------
155
Surface Heat Exchange
The effects of surface heat exchange should be considered for cases
where a large surface area is available for cooling. To more accurately
describe conditions on lakes and wide rivers, the equation for the con-
servation of heat could be rewritten to include a "sink" term, which
would represent the loss of heat across the air-water interface. Based
on the work of Rdinger and Geyer (19) and Edinger, Puttwciler, and
Geyer (18), the heat loss term would be a function of the equilibrium
temperature and the thermal exchange coefficient.
Analytical Solution
The present study solves a system of equations by numerical inte-
gration and then evaluates the entrainment and drag coefficients by in-
spection. If an analytical solution describing the location of the jet
trajectory and the decrease in the temperature rise could be obtained,
then the coefficients could be evaluated, for instance, by a least-
squares technique. This would eliminate some of the subjectivity inherent
in the present method. At present, however, analytical solutions for
all but the simplest cases of jets are limited to cases of irrotational
flow. Gordier (25), for instance, studied a two-dimensional slot jet
in terms of free streamline analysis and conformal mapping. The useful-
ness of ideal flow theory would depend on realistically approximating
the turbulent entrainment of the jet.
These suggestions for future work should help to further refine the
analysis and prediction of the temperature distribution in the vicinity of
thermal-electric power plants. It is hoped that the present study is
adequate to serve as a base for future research.
-------
APPENDIX A
LABORATORY
LATERAL TEMPERATURE MEASUREMENTS, T/T
Run
1-90
1-90
1-90
1-90
1-90
2-90
2-90
2-90
2-90
2-90
3-90
3- 90
3-90
3-90
3-90
\^bo
xl/v\
84.5
67.7
SO. 7
33.8
16.9
84.5
67.7
50.7
33.8
16.9
84 . 5
67.7
50 . 7
33.8
16.9
1.41
0.08
0.08
0.10
0.11
0.19
0.04
0.04
0.03
0.03
0.06
0.03
0.02
0.02
0 . 03
0.05
4.51
0.08
0.09
0.12
0.16
0.26
0.04
0.05
0.06
0.11
0.23
0.03
0.03
0.04
0.07
0.15
7.61
0.09
0.10
0.11
0.14
0.07
0.07
0.07
0.10
0.13
0.24
0.05
0.05
0.09
0.13
0.24
10.7
0.10
0.09
0.07
0.05
0.01
0.09
0.10
0.10
0.15
0.16
0.08
0.10
0.12
0.13
0 . 1 9
13.8
0.04
0.04
0.04
0.00
0.00
0.08
0.11
0.12
0. 17
0.10
0.12
0.12
0.13
0.14
0.16
16.9
0.04
0.04
0.01
0.00
0.00
0.09
0.10
0.02
0.09
0.03
0.09
0.08
0.10
0.11
0.07
20.0
O.OJ
0.01
0.00
0.00
0.00
0.06
0.07
0.10
0.05
0.00
0.07
0.07
0.05
0.11
0.00
23.1
0.00
0.00
0.00
0.00
0.00
0.06
0.07
0.04
0.02
0.00
0.07
0.07
0.05
0.08
0.00
26.2
0.00
0.00
0.00
0.00
0.00
0.0.3
0.03
0.01
0.01
0.00
0.05
0.07
0.07
0.08
0.00
29.3
0.00
0.00
0.00
0.00
0.00
0.04
0.03
0.01
0.00
0.00
0.05
0.05
0.08
0.01
0 . 00
32.4
0.00
0.00
0.00
0.00
0.00
0.06
0.04
0.01
0.01
0.00
0.08
0.03
0.01
0.01
0.00
h/bo
7.9
7.4
6.7
5.2
4.2
12.6
11.7
9.8
8.1
6.5
12.5
11.5
11.4
5.0
3.5
-------
APPENDIX A -- Continued
Run
4-90
4-90
4-90
4-90
4-90
5-90
5-90
5-90
5-90
5-90
1 - 60
1-60
1-60
1-60
1-60
1-60
\^bo
x'/bX
o N^
84.5
50.7
33.8
16.9
8.45
84.5
50.7
33.8
16.9
8.45
84.5
67.7
50.7
33.8
16.9
8.45
1.41
0.03
0.01
0.01
0.01
0.02
0.02
0.02
0.01
0.01
0.02
0.09
0.10
0.12
0.18
0.34
0.53
4.51
0.03
0.02
0.04
0.09
0.22
0.02
0.03
0.02
0.04
0.14
0.13
0.18
0.23
0.28
0.20
0.01
7.61
0.03
0.05
0.07
0.17
0.30
0.02
0.07
0.07
0.14
0.25
0.17
0.20
0.17
0.09
0.02
0.00
10.7
0.04
0.06
0.13
0.21
0.24
0.07
0.06
0.11
0.24
0.30
0.09
0.09
0.07
0.03
0.01
0.01
13.8
0.06
0.10
0.15
0.08
0.16
0.07
0.10
0.15
0.20
0.06
0.04
0.04
0.06
0.00
0.00
0.00
16.9
0.09
0.13
0.17
0.02
0.02
0.10
0.10
0.19
0.08
0.01
0.04
0.02
0.01
0.01
0.01
0.01
20.0
0.09
0.11
0.13
0.00
0.01
0.09
0.14
0.17
0.02
0.00
0.01
0.01
0.00
0.00
0.00
0 . 00
23.1
0.13
0.08
0.11
0.00
0.01
0.10
0.14
0.03
0.00
0.00
0.01
0.00
0.00
0 . 00
0.00
0.00
26.2
0.12
0.06
0,05
0.00
0.00
0.11
0.13
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
29.3
0.15
0.06
0.03
0.00
0.00
0.11
0.06
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
32.4
0.13
0.06
0.01
0.00
0.00
0.11
0.00
0.00
o.oc
0.00
0.00
0.00
0.00
0.00
0.00
0.00
b/h0
12.2
11.1
9.4
5.8
6.5
11.6
10.2
7.3
5.5
4.4
7.6
6.9
6.9
6.1
5.7
6.1
-------
APPHNIHX A -- Continued
Run
2- 60
2-60
2-60
2-60
2-60
3-60
3-60
3- 60
3-60
3-60
4-60
4-60
4-60
4-60
4-60
Q\*
84 . 5
67.7
50.7
33.8
16.9
84.5
67.7
50 . 7
33.8
16.9
84.5
50. 7
33.8
16.9
8.45
1.41
0.06
0.06
0.06
0 . 04
0.07
0.05
0.04
0.04
0.04
0.03
0.07
0.05
0 . 04
0 . 03
0.12
4.51
0.04
0.04
0.06
0.13
0.37
0.03
0.04
0 . 06
0.09
0.25
0.05
0 . 0-1
0.05
0. 22
0.40
7.61
0.06
0.07
0. 14
0 . 30
0.24
0.04
0.05
0.11
0.22
0.22
0.04
O.OS
0.15
0.27
0.04
10.7
0.13
0.17
0.23
0.24
0.06
0.09
0.15
0.19
0.21
0.05
0.07
0.14
0.18
0.08
0.02
13.8
0 . 1 5
0 . 20
0.21
0.15
0.01
0 . 1 7
0.21
0.25
0.19
0.04
0.08
0.18
0.18
0.04
0.01
16.9
0.15
0. 15
0.12
0.06
0.01
0 . 1 5
0.17
0.10
0.08
0.01
0.11
0.15
0.12
0 . 03
0.01
20.0
0.10
0.07
0.06
0.01
0.00
0.15
0.11
0.05
0.04
0.00
0.11
0.15
0.04
0.00
0.00
23.1
0 . 0 6
0.07
0.05
0.01
0.00
0.13
0.10
0.03
0.17
0.00
0.11
0.07
0.04
0.00
0.00
26.2
0.06
0.04
0.02
0 . 00
0.00
0.06
0.05
0.01
0.00
0.00
0.08
0.05
0.01
0.00
0.00
29.3
0.04
0.04
0.01
0.00
0.00
0.06
0.03
0.00
0.00
0.00
0.07
0.04
0.01
0.01
0.01
32.4
0.06
0.01
0.01
0 . 00
0 . 00
0.08
0.02
0.01
0.01
0.00
0.14
0.03
0.00
0.01
0.01
h/b
o
11.5
9.8
8.6
6.7
6.1
1 I . 3
9.6
7.8
7.5
13.4
10.4
8.2
6.8
6.8
-------
APPF.NinX A -- Continued
Run
5-60
5-60
5-60
5-60
5-60
1-45
1-45
1-45
1-45
1-45
1-45
2-45
2-45
2-45
2-45
2-45
\^bo
X'/V,
84.5
50,7
33 . 8
16.9
8.45
84.5
67.7
50.7
33.8
16.9
8.45
84.5
67.7
50 . 7
33.8
16.9
1.41
0.05
0.03
0.03
0.02
0.03
0.09
0.09
0.11
0.13
0.38
0.56
0.10
0.12
0.11
0. 12
0.15
4.51
0.02
0.04
0.04
0.15
0.22
0.13
0.17
0. 25
0.38
0.17
0.02
0.11
0.09
0.11
0. 14
0.43
7.61
0.04
0.08
0.08
0.22
0 . 35
0.18
0.23
0 . 23
0.09
0.01
0.01
0.09
0.11
0.18
0 . 34
0.21
10.7
0.05
0.10
0.15
0.19
0.08
0.17
0.16
0.09
0.04
0.01
0.01
0.15
0.20
0.30
0. 29
0.07
13.8
0.04
0.15
0.19
0.11
0.01
0.06
0.06
0.04
0.01
0.01
0.01
0.18
0 . 22
0.25
0.13
0.01
16.9
0.08
0.15
0.19
0.01
0.01
0.07
0.04
0.02
0.01
0.01
0.01
0.]9
0.21
0.17
0.07
0.01
20.0
0.11
0.11
0.04
0.01
0.00
0 . 04
0.01
0.01
0.01
0.01
0.01
0.13
0.13
0.08
0.02
0 . 00
23.1
0.11
0.08
0.04
0.01
0.00
0.01
0.01
0.01
0.01
0.01
0.01
0.11
0.10
0.06
0.01
0 . 00
26.2
0.11
0.04
0.01
0.00
0.00
0.01
0.01
0.01
0.01
0.01
0.0]
0.08
0.06
0.02
0.00
0 . 00
29.3
0.11
0.04
0.01
0.00
0 . 00
0.01
0.01
0.01
0.01
0.01
0.01
0.06
0.06
0.01
0.00
0 . 00
32.4
0.15
0.03
0.01
0.01
0.0.1
0.01
0.01
0.01
0.01
0.01
0.01
0.09
0.02
0 . 0 1
0.00
0 . 00
b/h0
12.6
10.4
8.1
7.4
6.2
9.2
8.6
8.5
8.6
9.4
9.4
12.3
10.8
8.8
7.0
6.1
-------
APPHNDIX A -- Continued
Run
3-45
3-45
3-45
3-45
3-45
4-45
4-45
4-45
4-45
4-45
5-45-
5-45
5-45
5-45
5-45
X^o
x^v\
84.5
67.7
50.7
33.8
16.9
84.5
50 . 7
33.8
16.9
8.45
84.5
50.7
33.8
16.9
8.45
1.41
0.06
0.07
0.07
0.07
0.11
0.06
0.05
0.05
0.04
0.10
0,06
0.04
0.03
0.01
0.04
4.51
0.03
0.06
0.07
0.11
0.27
0.05
0.05
0 . 09
0.21
0.51
0.06
0.08
0.06
0.19
0.42
7.61
0.05
0.07
0.12
0.25
0.08
0.05
0.08
0.16
0.33
0.02
0.04
0.10
0.17
0.29
0.06
10.7
0.10
0.14
0.20
0.17
0.05
0.09
0.13
0.24
0.05
0.01
0.08
0.11
0.21
0.21
0.02
13.8
0.09
0.17
0.16
0.05
0.00
0.09
0.17
0.21
0.01
0.00
0.08
0.15
0.13
0.06
0.00
16.9
0.14
0.15
0.08
0.03
0.00
0.12
0.17
0.14
0.01
0.00
0.11
0.17
0.15
0.08
0.00
20.0
0.13
0.12
0.05
0.00
0.00
0.09
0.12
0.05
0.00
0 . 00
0.10
0.11
0.10
0.02
0.00
23.1
0.07
0.06
0.02
0.00
0.00
0.09
0.06
0.04
0.00
0.00
0.10
0.10
0.08
0.00
0.00
26 . 2
0.06
0.05
0.00
0.00
0.00
0.08
0.02
0.01
0.00
0.00
0.10
0.06
0.02
0.00
0.00
29.3
0.06
0.03
0.00
0.00
0.00
0.08
0.02
0.00
0.00
0.00
0.08
0.04
0.00
0.00
0.00
32.4
0.03
0.01
0.01
0.01
0.00
0.10
0.02
0.01
0.00
0.00
0.11
0.01
0.01
0.00
0.00
b/b0
11.5
10.4
8.3
6.9
6.6
13.2
10.0
7.8
5.8
5.8
13.2
10.3
8.7
6.6
4.9
-------
APPENDIX R
'.OCATION Ol; LABORATORY TRAJIiCTORIKS
•-D
Run 1-1)0
x'/b
o
0.07
0.49
0. 71
1.41
2.82
4.23
5 . 64
7.05
8.46
8.46
12.7
16.9
16.9
33.8
50.7
67.7
84.5
v'/b
0
0.28
0.63
0 . 99
1.41
1.97
2.26
2.26
2.93
3.03
3.03
3.81
4.09
4.09
4.51
4.51
7.61
10.7
*'"•„
0.29
0.80
1.21
1 .96
3.41
4.81
6.2
7.6
9. 1
9.1
13.3
17.6
17.6
34.5
51.4
68.4
85.2
1 .00
1 .00
0.97
0.85
0.75
0.63
0.54
0.48
0.4:
0.47
0.33
0.26
0.26
0.16
0.1?
0. 10
0.10
)
7
7
)
)
)
)
)
)
x'/b
o
0.01
0.01
0.14
0.28
0.71
0.71
1 .41
2. 82
4.2.3
5 . 64
7.05
8.46
12.7
16.9
16.9
33.8
50.7
67.7
84.5
Run
v'/b
0
0.14
0.71
0.71
0.99
1.41
1.55
2.11
3.17
3.67
3.88
4.72
4.44
5.64
5.99
5.99
13.8
13.8
13.8
10.7
2-90
s'/b
o
0.14
0.71
0.72
1 .03
1 .58
1 . 70
2.71
4.15
5 . 79
7 . 09
8.40
9.81
14.2
18.4
18.4
35.3
52.2
69.2
86.0
[ T/T
o
0.97
0.97
1 .00
0.98
0.97
0.97
0.83
0.67
0.57
0.47
0.41
0.37
0.27
0.22
0.24
0.17
0.12
0.11
0.09
x'/h
0
0.01
0.35
0.42
0.35
0.71
0.49
1.41
2.82
4.23
5.64
7.05
8.46
12.7
16.9
16.9
33.8
50.7
67.7
84.5
Run
v'/b
o
0.13
0.78
0.99
1 .06
1.20
1 .41
2.33
2.95
3.95
4.37
4.93
4.79
5.77
6.35
6.34
13.8
13.8
13.8
13.8
3-90
s'/b
o
0.13
0.85
1.07
0. 12
1 . 39
1 .49
2.65
4 . 1 5
5.85
7.25
8.65
10.1
14.4
18.5
18.5
35.3
52.3
69 . 3
86.1
T/T
o
0.99
1 . 00
0.98
1 . 00
0.96
0.97
0.79
0 . 64
0.57
0.53
0.47
0.44
0 . 29
0.25
0.24
0.14
0.13
0.12
0.12
-------
APPENDIX B -- Continued
x'/b
o
0.01
0.01
0.01
0.28
0.71
0.71
1.41
2.82
4.23
5.64
7.05
8.45
8.45
16.9
33.8
50.7
84.5
Run ^
y'/bo
0.28
0.85
1.13
1.55
1.69
2.11
3.17
4.79
4.79
5.50
6.35
7 . 33
7.61
10.0
16.9
16.9
29.3
1-90
s'/b
o
0.28
0.85
1.13
1.57
1.83
2.22
3.35
5.10
6.55
8.09
9.4
11.4
11.4
20.2
37.1
54.0
87.8
T/T
o
1.00
1.00
0.99
0.99
0.96
0.87
0.70
0.52
0.43
0.40
0.28
0.30
0.30
0.21
0.17
0.13
0.16
x'/bo
0.01
0.01
0.01
0.01
0.01
0.71
1.41
2.82
4.23
5 . 64
7.05
8.46
12.7
16.9
33.8
50.7
84.5
Run .
y'/bo
0.07
0.56
0.99
1.13
1.55
1.97
3.38
5.78
5.99
6.91
8.11
9.16
11.1
14.2
16.9
23.1
29.3
5-90
''/"o
0.07
0.56
0.99
1.13
1.55
2.09
3.64
6.30
7.35
9.05
10.4
12.2
16.8
22.0
38.9
55.8
89.6
T/T
o
1.00
1.00
0.98
0.98
0.95
0.85
0.60
0.50
0.40
0.36
0.32
0.31
0.23
0.18
0.19
0.15
0.11
x'/bo
0.07
0.42
0.71
1.41
2.82
4.23
5 . 64
I
7.05
8.46
8.46
12.7
16.9
16.9
33.8
50.7
67.7
84.5
Run
y'/bo
0.13
0.25
0.35
0.49
0.77
0.83
1.20
1.41
1.62
1.76
2.26
2.54
2.54
4.51
4.51
7.61
7.61
1-60
''">,,
0.15
0.49
0.79
1.45
2.86
4.30
5.71
7.15
8.55
8.56
12.8
17.0
17.0
33.9
50.8
67.8
84.6
T/T
o
1 .00
0.98
0.96
0.85
0.67
0.59
0.57
0.57
0.55
0.55
0.51
0.41
0.41
0.28
0.23
0.20
0.17
-------
APPHNDIX B -- Continued
Run 2-60
x'/b
o
0.71
1.20
1.41
2.82
4 . 23
5.64
7. OS
8.46
12.7
16.9
16.9
35. 8
SO . 7
67.7
84. S
_Z^
0.56
0.85
1.06
1.90
2.68
2.82
3.24
3.67
5.36
6.35
4.51
7.61
10.7
13.8
13.8
**o
0.90
1.65
1.95
3.60
5.15
6 . 60
8.05
9 . 50
13.9
18.3
18.3
35.2
52.1
69 . 1
85.9
T/T
o
1.00
0.99
0.90
0.84
0.80
0.73
0.69
0.65
0.57
0 . 50
0.37
0.31
0. 23
0. 21
0.16
Run 3-60
x'/b y/b s'/b
o J o o
0.28 0.13 0.31
0.85 0.44 0.96
0.49 0.40 0.63
0.71 0.80 1.07
1 .41 1.2 1 .79
2.82 2.1 3.55
4.23 2.8 5.20
5.64 3.5 6.65
7.05 3.9 8.15
8.46 4.0 9.51
12.7 5.5 13.9
16.9 6.3 18.4
16.9 6.3 18.4
16.9 4.5 18.4
53.8 7.6 35.3
50.7 13.8 52.2
67.7 13.8 69.2
8-1.5 15.8 86.0
T/TO
1.00
0.98
0.99
0.94
0.91
0.77
0.66
0.53
0.49
0.38
0 . 34
0.27
0.31
0.25
0.22
0.25
0.21
0.17
Run 4-60
x'/b
o
0.28
0.71
1.41
1 .69
2.82
4.23
5.6'l
7.05
8.45
8.45
16.9
1 6 . 9
33. 8
50 . 7
84 . S
y'/bo
0.23
0.69
1.10
1.49
2.61
3.07
3.98
4 . 09
5.23
4.51
7.47
7.61
12.3
15.8
20.0
^0
0 . 36
0.99
1.79
2.25
3.80
5.30
6 . 90
8.25
9.90
9 . 90
18.7
18.7
3 5 . 6
52.5
86.3
T/TO
1 .00
1 .00
0.99
0.97
0.75
0.55
0.53
0.43
0.46
0.40
0.30
0.28
0.18
0.18
0 . 1 4
-------
APPENDIX B -- Continued
Run 5-60
x'/bo
0 01
0 71
0.71
141
2.82
4.25
5.64
7 05
8.46
8 45
12 7
16 9
169
^3 8
50.7
84.5
y'/bn
0.13
1.01
1.16
1.96
3.10
4.02
4.51
5.64
5.91
7.61
7.47
8.32
7.61
15.5
15.5
20.0
s''bo
0.13
1.23
1.36
2.35
4.10
5.90
7.35
9.10
10.5
10.5
15.0
19.2
19.2
36.1
53.0
86.8
T/To
0.98
1.00
0.99
1.00
0.60
0.45
0.44
0 . 39
0.37
0.35
0.29
0.24
0.22
0.19
0.15
0.15
Run 1-45
x'/bo
0.01
0.14
0.71
1.41
2.82
4.23
5.64
7.05
8.45
8.46
12.7
16.9
16.9
33.8
50.7
67.7
84.5
y'/b0
0.13
0.17
0.42
0.63
0.85
1.13
1.27
1.34
1.41
1.41
1.83
2.68
1.41
4.51
4.51
7.61
7.61
s'/bo
0.13
0.20
0.82
1.45
2.85
4.30
5.72
7.10
8.49
8.50
12.8
17.1
17.1
34.0
50.9
67.9
84.7
T/To
1.00
0.98
0.88
0.76
0.67
0.65
0.61
0.57
0.56
0.52
0.50
0.49
0.38
0.38
0.25
0.23
0.18
Run 2-45
xVb
o
0.28
0.71
1.41
2.82
4.23
5.64
7.05
8.46
12.7
16.9
16.9
16.9
33.8
50.7
67.7
y'/bo
0.14
0.42
0.85
1.55
1.83
2.40
2.68
3.10
3.95
4.79
4.79
4.51
7.61
10.7
13.8
s'/bo
0.31
0.82
1.60
3.15
4.55
6.10
7.50
8.95
13.2
17.5
17.5
17.5
34.4
51.3
68.3
T/To
1.00
1.00
0.90
0.82
0.78
0.72
0.69
0.63
0.53
0.46
0.49
0.43
0.34
0.30
0.22
-------
APPENDIX B -- Continued
Run 3-45
x'/bo
0.71
1.41
1.62
2.82
4.23
5.64
7.05
8.46
12.7
16.9
16.9
16.9
33.8
50 . 7
67.7
84.5
yVb0
0.30
0.90
0.70
1.41
2.0
2.7
3.1
3.4
5.1
5.5
5.5
4.5
7.6
10.7
13.8
16.9
s'/bo
0.77
1.69
1 . 80
3.25
4.75
6.30
7.75
9.20
13.7
17.9
17.9
17.9
3.48
53.7
68 . 7
85.5
T
00
92
98
90
66
58
51
46
40
34
33
27
25
20
17
14
x'/bo
0.71
1.41
1.41
2.12
2.82
4.23
5.64
7.05
8.45
8.46
12.7
16.9
16.9
33.8
50 . 7
94 . 5
Run
yVbo
0.41
0.97
1 . 00
1.25
1 ,83
2.95
5.52
3.95
4.51
4.65
6 . 49
6 . 63
6.63
35.7
52.6
96 . 4
4-45
s'/bo
0.81
1.75
1.75
2.75
3.65
5.35
6.90
8.40
9 . 90
9.80
14.5
18.8
]8.8
35.7
52.6
96 . 4
T/T
o
1.00
0.99
0.97
0.97
0.92
0.72
0.57
0.51
0.51
0.44
0 . 36
0.31
0.33
0.24
0.17
0.12
x'/b
o
0.21
0 . 49
0.71
0.63
1.41
2.82
4.23
5.64
7. 05
8.45
8.46
12.7
16.9
1 6 . 9
33.8
50.7
84.5
Run
v'/b
o
0.07
0.13
0.44
0.63
I . 11
2.11
2.82
3.38
4.51
4.94
4.51
6.63
7.61
7.61
10.7
16.9
16.9
5-45
o
0.22
0.51
0.83
0.89
1.85
3 . 60
5.25
6.70
8 . 50
9.90
9.90
14.4
18.7
.18.7
35.6
52.5
86.5
_^o___
0.98
1 . 00
1 .00
0.94
0.98
0.91
0.78
0 . 63
0,56
0.42
0.49
0 . 4 2
0 . 34
0 . 29
0.21
0. 15
0. 11
-------
APPENDIX C
FIGURES 46 - 73
TRAJECTORIES AND TEMPERATURE AND WIDTH PLOTS
FOR LABORATORY EXPERIMENTS
164
-------
FlGURli 46.--OBSHRVKD AND FITTliU TRAJfcCTORH-S, RUN 2-90
-------
1.0
0.5
0.1
0.05
0.01
0.1
- 47.5
A - 0.44
E = 0.39
0.5 1.0
5.0 10.0
s'/b
100.0
50.0
10.0
5.0
1.0
50.0 100.0
o
FIGURE 47.--OBSERVED VALUES AND FITTED CURVES FOR TEMPERATURE AND WIDTH, RUN 2-90
-------
FIGURE 48.--OBSERVED AND FITTED TRAJECTORIES, RUN 3-90
-------
A - 0.30
E = 0.31
100.0
- 50.0
-= 10.0
- 5.0
1.0
50.0 100.0
FIGURE 49.--OBSERVED VALUES AND FITTED CURVES FOR TEMPERATURE AND WIDTH, RUN 3-90
-------
10
8
r~
V62.0
A =0.23
(V2.0
0
2
4
8 10
x'/b0
12
14 16
18 20
PIGURH 50.--OBSKRVHU ANU FITTHU TRAJKCTORIHS, RUN 4-90
-------
1.0
0.5
0.1
0.05
0.01
0.1
A =0.23
E = 0.47
i i
0.5 1.0
5.0 10.0
100.
50.0
10.0
5.0
1.0
50.0 100.
s'/br
FIGURE 51.--OBSERVED VALUES AND FITTED CURVES FOR TEMPERATURE AND WIDTH, RUN 4-90
-------
0
8
10
x'/b
12
14
16
18
20
o
FIGURE 52.--OBSERVED AND FITTED TRAJECTORIES, RUN 5-90
-------
1.0
jo—oo-o
A- 0.20
E= 0.44
0
5.0 10.0
50.0 100.
s'/b,
FIGURE 53.--OBSERVED VALUES AND FITTED CURVES FOR TEMPERATURE AND WIDTH, RUN 5-90
-------
10
i r
i i i r
8
6
o
"V,
-------
1.0
A -0.67
E = 0.24
0.01
50.0 100.
FJGURIi 55.--OBSJ;RVHD VALUHS AND F ITT1-D CURVHS FOR THMPHRATIIRH AND WIDTH, RUN 1-60
-------
lOi—
0
18 20
•IGURIi 56.—OBSI-RVI-D AND FITTI-U TRAJECTORIHS, RUN 2-60
-------
1.0
0.5
0.1
0.05
0.01
0.1
V36-5
A =0.44
E-0.13
I
Oi O
I
0.5 1.0
100.
- 50.0
s'/b,
5.0 10.0
10.0
- 5.0
1.0
50.0 100.
FIGURE 57.--OBSERVED VALUES AND FITTED CURVES FOR TEMPERATURE AND WIDTH, RUN 2-60
-------
10
8
6
O
.0
V40'0'
A =0.30
CD-3.o
0
6
8
10
14
16
18
20
FIGURE 58.--OBSERVED AND FITTED TRAJECTORIES, RUN 3-60
-------
1.0
0.5
0.1
0.05
0.01
0.1
A =0.30
E=0.19
0.5 1.0
I I
s'/b
5.0 10.0
I
100.
50.0
10.0
5.0
1.0
50.0 100.
00
FIGURE 59.--OBSERVED VALUES AND FITTED CURVES FOR TEMPERATURE AND WIDTH, RUN 3-60
-------
10
8
O
_0
T r
i r
V41-0
A =0.23
CD-2J
i r
0
8
10 12 14 16 18 20
x'/b
o
FIGURE 60.--OBSERVED AND FITTED TRAJECTORIES, RUN 4-60
'.£>
-------
1.0
0.5
0.1 -
0.05
0.01
0.1
0.5 1.0
s'/b
5.0 10.0
1.0
50.0 100.
o
FIGURE 6l.--OBSliRVIiU VALUI-S AND FITTliD CURVIiS FOR TBlPIiKATURli AND WIDTH, RUN 4-60
-------
A = 0.19
Cn =3.4
FiCURli 62.--OBSI:;RVL;i) AND H1TTLD TRAJECTORILiS, RUN 5-60
CO
-------
A = 0.19
E - 0.29
50.0 100.
•JGURl- 03.--OBSURV1-L) VAl.Ul-S AND !•'ITTIiD CURVliS FOR Tl;Ml'l:RATURli AND WIDTH, RUN 5-bO
-------
10
8
O
.O
1 I I I
A =0.66
Cp-1.5
I I
18 20
FIGURE 64.--OBSERVED AND FITTED TRAJECTORIES, RUN 1-45
c»
-------
1.0
0.5
0.1
0.05
0.01
0.1
V25'0'
A =0.66
E = 0.19
0.5
50.0 100.
00
FIGURE 65.--OBSERVED VALUES AND FITTED CURVES FOR TEMPERATURE AND WIDTH, RUN 1-45
-------
A-0.42
cIT i.o
I;ir,URK bb.--OBSl:RVLiU AND FITTED TRAJliCTOR ILiS, RUN 2-45
CO
Ui
-------
A =0.42
E - 0.13
50.0 100.
co
-------
10
8 -
6 -
o
4 _
2 -
0
16
18 20
HGUKI- b8.--OBSl:RVm) AND l:l'lTi;L) TRAJLCTORlliS, RUN 3-45
-------
1.0
0.5 -
0.1 -
0.05 -
0.01
0.1
0.5 1.0
5.0 10.0
50.0 100.
s'/b
o
FJGURl: 69.--OBSLiRVI;D VALUHS AND F1TTHD CURVliS FOR TliMPLiRATURIi AND WIDTH, RUN 3-45
-------
10
8 -
6 -
o
.a
0
ft-33-5
A =0.23
2 _
FIGURE 70.--OBSERVED AND FITTED TRAJECTORIES, RUN 4-45
-------
1.0
0.5
0.1
0.05
0.01
0.1
$,-33.5
A-0.23
E-0.27
0.5 1.0
5.0 10.0
s'/b
100.
50.0
10.0
5.0
1.0
50.0 100.
o
FIGUKi; 71 .--OBSI'KViiU VALLIliS AND FiTTFU CURVIiS FOR TBllMiRATURH A.N'D IV1DT1I, RUN
-------
10 _
8 -
6 _
o
JQ
0
2 -
18 20
FIGURE 72.--OBS1-RV1-U AND FITTliU TRAJliCTORIl-S, RUN 5-45
-------
50.0 100.
FIGURE 73.--OBSliRVliD VALUES AND FITTED CURVES FOR TEMPERATURE AND WIDTH, RUN 5-45
-------
APPENDIX D
PRACTICAL APPLICATION EXAMPLE
Representative values of field parameters have been chosen to il-
lustrate the practical application of the model developed in the present
study.
The assumed values of the parameters are summarized in Table 16.
The assumed plant flowrate, QQ = 2600 ft3/sec, is subtracted from the
assumed river flowrate, OF = 31,900 ft3/sec, to obtain the ambient flow-
rate, 0 = 29.300 ft3/sec. The ambient width is assumed to be 2000 feet,
d *
and the ambient depth is assumed to be 15 feet, giving a cross-sectional
area of 30,000 ft2. Dividing the ambient flowrate by the area gives
the ambient velocity, U =0.98 ft/sec.
3.
The jet discharge is assumed to have a flowrate of Q = 2600 ft3/sec,
a half-width of b = 100 feet, and a depth of 10 feet. Dividing the value
of Q by the cross-sectional area, or 2000 ft2, gives the initial jet
velocity, U =1.3 ft/sec. The value of b , the initial half-width at
o o
the beginning of the zone of established jet flow, is calculated to be
b = 160 feet from Equation 65,
b0 = 1.60 t/ (65)
The value of the velocity ratio is found to be A = 0.75, using
U =0.98 ft/sec and U =1.3 ft/sec and Equation 38,
a o
A - Ua/UQ (38)
193
-------
194
TABLE 16
ASSUMED VALUES FOR DESIGN PROBLEM
Parameter Value
Q , River Flowrate, ft3/sec 31,900
Q , Plant Flowrate, ft3/sec 2,600
Q , Ambient Flowrate, ft3/sec 29,300
3
Ambient Width, feet 2,000
Ambient Depth, feet 15.0
Ambient Cross-sectional Area, ft2 30,000
U , Ambient Velocity, ft/sec 0.98
3.
b', Discharge Half-Width, feet 100.0
z , Discharge Depth, feet 10.0
Discharge Cross-sectional Area, ft2 2,000
U , Initial Jet Velocity, ft/sec 1.3
b , Initial Jet Half-Width, feet 160.0
o
A, Velocity Ratio 0.75
B , Initial Discharge Angle, ° 90.0
E, Entrainment Coefficient 0.25
CD, Drag Coefficient 0.5
C , Reduced Drag Coefficient 0.5
T , Initial Temperature Rise, °F 15.0
Ri , Initial Richardson Number >1.0
-------
195
The initial angle of discharge is assumed to be 6 =90.0°. From
Figure 18, the length of the zone of flow establishment is estimated
to be s'/b = 0.75, using A = 0.75. From Figure 19, the initial angle
e o
at the beginning of the zone of established jet flow is estimated to be
B = 34.2°, using A = 0.75 and p1 = 90.0°.
Q Q
Values of the entrainment and drag coefficients are chosen based
on the laboratory and the limited field results of the present study.
The entrainment coefficient is assumed to be E = 0.25, since the dis-
charge width is one- tenth as large as the ambient width, while the drag
coefficient is assumed to be C = 0.5, based upon the field results
shown in Figures 44 and 45. The value of the reduced drag coefficient
is found to be C' = 0.5 from Equation 39,
A numerical solution for T/T and b/b vs. S and for X and Y can
o o
be computed, using the values of A = 0.75, (3 - 34.2°, and C* = 0.5.
Using the values s'/b = 0.75 and E = 0.25, the value of S? is
e o e
found to be equal to 0.2 from Equation 67,
S; = C2E//F bo) S; (67)
The values of T/T and b/b are then referenced to the discharge
in terms of S?, using the value of s' and Equation 68,
G
s' = S* + S (68)
e
Using the value of E = 0.25 and solving Equation 69 for s'/b ,
-------
196
s'/b = (vV S?/2E) (69)
then, the predicted values of T/T and b/b vs. s /b can be plotted
as shown in Figure 74.
Using the value of E = 0.25 and solving Equation 42 for x/b and
x/bQ = (vV X/2E) ; y/bo = (A~ Y/2E) (42)
then, the predicted location of the trajectory can be plotted as shown
in Figure 75. The width of the surface jet in terms of b/b is also
shown in Figure 75. It is assumed that x /b = x/b , and y'/b = y/b ,
o o o o
since the values of x'/b and y'/b were considered to be negligible
when compared to the values of x/b and v/b in the field cases of the
o ' o
present study.
The application of the model depends on the magnitude of the
initial Richardson Number, accurate prediction of the initial jet depth,
and the intensity of the ambient turbulence.
Assuming a 15°F initial temperature rise from 70°F to 85°F, the
initial Richardson Number is found to be greater than 1.0, using the
values Ap/p = 2.6 x 10~3, U = 1.3 ft/sec, LI = 0.98 ft/sec, and
O 3.
z = 10.0 feet. Thus, the two-dimensional model should be applicable,
since the value of the Ri Number > 1.0 indicates a lack of vertical
o
mixing.
Since the initial jet velocity is calculated from the value of
the initial jet depth, accurate prediction of the depth is necessary
before the proposed model can be used. At present, this depth is
-------
1.0
0.5 -
0.1 -
0.05 -
0.01
0.1
0.5 1.0
5.0 10.0
s/b
50.0 100.
o
FIGURE- 74.-- PREDICTED VALUES OF TEMPERATURE AND WIDTH
-------
20.0 -
15.0 _
10.0 ~
5.0
0
5.0 10.0 15.0 20.00 25.0 30.0 35.0 40.0 45.0 50.0
x'/b,
o
FIGURE 75.-- PRF.niCTF.n TRAJECTORY AND WIDTH
'30
-------
199
difficult to predict, particularly at field sites where a cold water
wedge would be expected to intrude into the discharge channel. Harleman's
work (26) could be used as a guide, hut more research must be done con-
cerning those field sites where the conditions necessary to apply
Harleman's two-layer stratified flow theory are not met.
In some cases, ambient turbulence can be expected to cause a
greater decrease in the temperature rise than the two-dimensional jet
model predicts. Vertical mixing due to ambient turbulence was noted at
some of the field sites in the present study. The effects of ambient
turbulence must be studied further before they can be quantitatively
included in the surface jet model.
-------
APPENDIX R
LIST OF SYMBOLS
a - Subscript pertaining to ambient fluid
A - Ratio of the ambient velocity to the initial jet velocity,
U /U , M°L°T0
a o
b - Half-width of jet, L
C - Circumference through which entrainment takes place, L
C - Drag coefficient, M°L°T°
C' - Reduced drag coefficient, MCL°T°
dA - Differential area of jet cross-section, L-
d' - Initial let diameter, L
o
e - Subscript pertaining to establishment zone
E - Entrainment coefficient, M°L°T°
exp - Exponential function
F - Drag term, L3T~-
Fr - Froude Number, M°L°T°
a - Acceleration of gravity, LT~^
M - Non-dimensional momentum flux
o - Subscript denoting initial values
0 - Origin of the coordinate system (x,y), beginning of zone of
established flow
O' - Origin of the coordinate system (x',v'), point of jet discharge
Q - Flowrate, L:'T~l
T - Subscript pertaining to river flow
Re - Revnolds Number, M°L°T°
200
-------
201
APPENDIX E--Continued
Ri - Richardson Number, M°L°T°
s - Coordinate axis of jet referenced to beginning of zone of
established flow, L
s - Coordinate axis of jet referenced to point of jet discharge, L
s - Length of zone of flow establishment, L
S - Non-dimensional distance along jet axis referenced to beginning
of zone of established flow
S! - Non-dimensional distance along axis referenced to point of jet
discharge
S - Non-dimensional length of zone of flow establishment
e
I - Centerline temperature rise, °F
T' - Temperature rise at any point in the jet cross-section, °F
u - Velocity along the jet axis, LT~-
U - Centerline jet velocity, LT~:
U - Ambient velocity, LT~L
3-
U0 - Initial jet velocity, LT"2
Y - Non-dimensional x'olume flux
v. - Inflow velocity, LT~ ^
x - Longitudinal axis referenced to beginning of zone of established
flow, L
x - Longitudinal axis referenced to point of discharge, L
x' - Longitudinal length of establishment zone, L
e
X - Non-dimensional longitudinal axis referenced to beginning of
zone of established flow
X1 - Non-dimensional longitudinal axis referenced to point of
discharge
x' - Non-dimensional longitudinal length of establishment zone
-------
202
APPENDIX E--Continued
y - Lateral axis referenced to beginning of zone of established
flow, L
y' - Lateral axis referenced to point of discharge, L
y' - Lateral length of establishment zone, L
c
Y - Non-dimensional lateral axis referenced to beginning of zone
of established flow
Y - Non-dimensional lateral axis referenced to point of discharge
Y* - Non-dimensional lateral length of establishment zone
z - Depth of jet, L, subscript pertaining to vertical coordinate
B - Angle between jet and ambient current, degrees
n - Distance along axis perpendicular to s axis, L
v - Kinematic viscosity, L2!'1
p - Density, FT2L-k
a - Standard deviation, L
-------
LIST OF REFERENCES
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Journal of the Hydraulics Division, Am. Soc. of Civil Eng.,
Vol. 91, No. HY4, Proc. Paper 4411, July, 1965, pp. 139-154.
2. Abramovich, G. N., The Theory of Turbulent Jets, M.I.T. Press,
Cambridge, Mass., 1963.
3. Ackers, P., "Modeling of Heated-Water Discharges," Engineering
Aspects of Thermal Pollution, edited by F. L. Parker and
P. A. Krenkel, Vanderbilt Univ. Press, Nashville, Tenn.,
1969, pp. 172-212.
4. Albertson, M. L., Dai, Y. B., Jensen, R. A., and Rouse, H.,
"Diffusion of Submerged Jets," Transactions, Am. Soc. of
Civil Eng., Vol. 115, 1950, pp. 639-697.
5. Bata, G. L., "Recirculation of Cooling Water in Rivers and Canals,"
Journal of the Hydraulics Division, Am. Soc. of Civil Eng.,
Vol. 93, No. HY3, Proc. Paper 1265, June, 1967.
6. Beer, L. P., and Pipes, W. 0., "Environmental Effects of Condenser
Water Discharge in Southwest Lake Michigan," Consulting En-
gineering Report, Environmental Sciences Industrial Bio-Test
Laboratory, Inc., 106 pp.
7. Bosanquet, C. H., Horn, G., and Thring, M. W., "The Effect of
Density Differences on the Paths of Jets," Proceedings, Royal
Soc. of London, Vol. 263A, No. 263, September, 1961, pp. 340-352.
8. Briggs, G. A., "Plume Rise," Air Resources Atmospheric Turbulence
and Diffusion Laboratory, Environmental Science Services
Administration, Oak Ridge, Tenn., September, 1969, 81 pp.
9. Brooks, N. H., Discussion of "Mechanics of Condenser-Water Dis-
charge from Thermal-Power Plants, " by D. R. F. Harleman,
Engineering Aspects of Thermal Pollution, edited by F. L.
Parker and P. A. Krenkel, Vanderbilt Univ. Press, Nashville,
Tenn., 1969, pp. 165-172.
10. Burdick, J. C. Ill, and Krenkel, P. A., "Jet Diffusion Under
Stratified Flow Conditions," Tech. Report No. 11, Sanitary
and Water Resources Engineering, Vanderbilt Univ., Nashville,
Tenn., 1967, 100 pp.
203
-------
204
11. Cairns, John, Jr., "Effects of Heat on Fish," Industrial Wastes,
Vol. 1, No. 5, May-June, 1956, pp. 180-183.
12. Carter, H. H., "A Preliminary Report on the Characteristics of a
. Heated Jet Discharged Horizontally into a Transverse Current,
Part I - Constant Depth," Tech. Report No. 61, Chesapeake Bay
Institute, The Johns Hopkins Univ., Baltimore, Md., November,
1969, 38 pp.
; i
13. Cederwall, K., "Jet Diffusion: Review of Model Testing and Com-
parison with Theory," Hydraulics Division, Chalmers Institute
of Technology, Goteborg, Sweden, February, 1967, 28 pp.
14. Churchill, M. A., "Effects of Density Currents in Reservoirs on
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1 Accession Number
w
n 1 Subject Field & Group
05D, 02E
SELECTED WATER RESOURCES ABSTRACTS
INPUT TRANSACTION FORM
Organization
Department of Environmental and Water Resources Engineering,
Vanderbilt University, School of Engineering, Nashville, Tennessee
HEATED SURFACE JET DISCHARGED INTO A FLOWING AMBIENT STREAM
10
Aathorfs)
Motz, Louis H,
Benedict, Barry A.
|£ I Project Destination
FWQA Contract No. 16130FDQ
21J
Note
221
2 I Citation
Federal Water Quality Administration Water Pollution Control Research Series,
Office of Research and Development, Report based on Contract No. 16130 FDQ, March,1971
23
Descriptors (Starred First)
*Cooling Water, *Thermal Pollution, *Jets, Turbulent flow, *Heated water
*Thermal power plants, Thermal stratification, Diffusion, Water temperature,
Heat exchange, Temperature
25
Identifiers (Starred first)
*Heat discharge, Density differences, Buoyant Jets, Ambient Fluids,
Temperature profiles, Widows Creek, New Johnsonville, Waukegan Surveys
27
Abstract
The temperature distribution in the water body due to a discharge of waste
heat from a thermal-electrical plant, is a function of the hydrodynamic variables
of the discharge and the receiving water body. The temperature distribution can be
described in terms of a surface jet discharging at some initial angle to the ambient
flow and being deflected downstream by the momentum of the ambient velocity. It is
assumed that in the vicinity of the surface jet, heat loss to the atmosphere is
negligible. It is concluded that the application of the two dimensional surface jet
model is dependent on the velocity ratio and the initial angle of discharge, and the
value of the initial Richardson number, as low as 0.22. Both laboratory and field
data are used for verification of the model which has been developed. Laboratory
data is used to evaluate the two needed coefficients, a drag coefficient and an
entrainment coefficient, as well as the length of the zone of flow establishment and
the angle at the end of that zone.
Abstractor
Institution
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