EPA-660/2-/3-012
October 1973
Environmental Protection Technology Series
Negatively Buoyant Jets
In A Cross Flow
Office of Research and Development
U.S Environmental Protection Agency
Washington, D.C. 20460
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RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development,
U.S. Environmental Protection Agency, have been grouped into
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facilitate further development and application of environmental
technology. Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface
in related fields. The five series are:
1. Environmental Health Effects Research
2. Environmental Protection Technology
3. Ecological Research
4. Environmental Monitoring
5. Socioeconomic Environmental Studies
This report has been assigned to the ENVIRONMENTAL PROTECTION
TECHNOLOGY STUDIES series. This series describes research performed
to develop and demonstrate instrumentation, equipment and methodology
to repair or prevent environmental degradation from point and
non-point sources of pollution. This work provides the new or
improved technology required for the control and treatment of
pollution sources to meet environmental quality standards.
EPA REVIEW NOTICE
This report has been reviewed by the Office of Research and
Development, U.S. Environmental Protection Agency, and approved
for publication. Approval does not signify that the contents
necessarily reflect the views and policies of the U.S. Environmental
Protection Agency, nor does mention of trade names or commerical
products constitute endorsement or recommendation for use.
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EPA-660/2-73-012
October 1973
NEGATIVELY BUOYANT JETS IN
A CROSS FLOW
By
Jerry Lee Anderson
Frank L. Parker
Barry A. Benedict
Grant # R-800613
Project 16130 FDQ
Program Element 1BA032
Project Officer
Mr. Frank Rainwater
Pacific Northwest Water Laboratory
National Environmental Research Center
Corvallis, Oregon 97330
Prepared for
OFFICE OF RESEARCH AND DEVELOPMENT
U. S. ENVIRONMENTAL PROTECTION AGENCY
WASHINGTON, D.C. 20460
For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 - Price $2.50
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ABSTRACT
Negatively buoyant jets, or sinking jets, can be observed in many
problems of pollutant discharge. Any chamical waste that is heavier than
the receiving water into which it is discharged may act as a negatively
buoyant jet. In addition, when water is taken from the hypolimnion of
a deep lake or reservoir and used as cooling water, the temperature, and,
consequently, the discharge may behave like a negatively buoyant jet.
Two existing jet diffusion models have been utilized to predict
the trajectory and dilution of a positively buoyant jet, or a rising jet,
and have been modified to account for the sinking effect.
Twenty-four experimental investigations were conducted involving
different combinations of densimetric Froude number , velocity ratios,
and initial angle of discharge. Salt was used as the tracer, yielding
a fluid that was denser than the ambient receiving water and facilitated
measuring concentration profiles of the jet plume. The coefficient of
entrainment, the major mechanism of dilution, was determined as a func-
tion of the densimetric Froude number, velocity ratio, and initial
angle of discharge.
The redacted drag coefficient was chosen as zero for both models
since any other value would predict a trajectory whose rise would be
less than experimentally observed. For all angles of discharge the
entrainment coefficient increased with a decrease in the velocity ratio
and with an increase in densimetric Froude number. Additionally, there
was a marked decrease in the entrainment coefficient with a decrease in
the initial angle of discharge.
11
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TABLE OF CONTENTS
Page
ABSTRACT ii
LIST OF FIGURES v
LIST OF TABLES xi
ACKNOWLEDGMENTS xii
CHAPTER SECTIONS
I CONCLUSIONS 1
II RECOMMENDATIONS 3
III INTRODUCTION 6
IV REVIEW OF THE LITERATURE 10
V ANALYTICAL DEVELOPMENTS OF FAN'S AND ABRAHAM'S MODEL. . 36
VI METHODS AND MATERIAL 56
VII ANALYSIS FOR DATA AND PRESENTATION OF RESULTS 69
VIII SUMMARY AND CONCLUSIONS 109
IX LIST OF REFERENCES 121
X GLOSSARY - LIST OF NOTATIONS 125
XI APPENDICES
A SALINITY-DENSITY RELATIONSHIP 130
B COMPUTER PROGRAM - FAN'S MODEL 131
C COMPUTER PROGRAM - ABRAHAM'S MODEL 136
D COMPUTER PROGRAM - DRKGS 141
ill
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E CALIBRATION OF o.5 gpm ROTAMETER 145
F CALIBRATION OF 60° V-NOTCH WEIR 146
G COMPUTER PROGRAM - ANALYSIS 147
H OBSERVED VALUES AND THEORETICAL CURVES PREDICTED
BY FAN'S AND ABRAHAM'S MODEL 152
Iv
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LIST OF FIGURES
Figure
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
A SIMPLE JET (AXISYMMETRIC CASE)
PROFILE VIEWS OF SIMPLE PLUMES IN UNIFORM AND STRATIFIED
ENVIRONMENTS
PROFILE VIEWS OF BUOYANT JETS [AFTER FAN (3)]
SCHEMATIC DIAGRAM OF A PROFILE VIEW OF A ROUND BUOYANT
JET IN A UNIFORM CROSS STREAM OF HONOGENEOUS DENSITY
LENGTH OF ZONE OF ESTABLISHMENT VERSUS 1/k
JET WITH NEGATIVE BUOYANCY
MAXIMUM HEIGHT OF NEGATIVELY BUOYANT JETS [AFTER
CEDERWALL (5)]
SCHEMATIC DIAGRAM FOR THE ANALYSIS OF A ROUND BUOYANT
JET IN A CROSS STREAM
SCHEMATIC RELATIONSHIP BETWEEN INITIAL DISCHARGE POINT
AND END OF ZONE OF FLOW ESTABLISHMENT
DETAILS OF PROBE CONSTRUCTION
PHOTOGRAPH OF CONDUCTIVITY PROBE
BASIC MEASURING CIRCUIT [AFTER CLEMENT (35)]
SCHEMATIC OF CONDUCTIVITY MONITOR [AFTER CLEMENTS (34)]
CONDUCTIVITY MONITOR
ESTERLINE ANGUS RECORDER. . .
COMBINATION OF CONDUCTIVITY MONITOR AND RECORDER
POLYETHYLENE BARREL WITH FLOW AND TEMPERATURE CONTROL . .
JET TEMPERATURE CONTROL
PARTIAL CUTAWAY VIEW OF RECIRCULATING FLUME
Page
11
12
13
17
20
26
31
37
51
58
58
59
59a
60
61
61
62
64
66
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Figure Page
20 CONCENTRATION PROFILE AT s'/DQ = 3.54 FOR RUN NO. 34 .. 72
21 REPRESENTATIVE PROFILE VIEW OF A JET'S TRAJECTORY. ... 77
22 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 13 85
23 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 10 86
24 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 33 87
25 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 27 88
26 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 22 89
27 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 19 90
28 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 13 92
29 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 10 93
30 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 33 94
31 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 27 95
32 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 22 96
33 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 19 97
34 PHOTOGRAPH OF NEGATIVELY BUOYANT JET FOR RUN NO. 13,
F « 40, k = 10, 0' = 90° 98
35 PHOTOGRAPH OF NEGATIVELY BUOYANT JET FOR RUN NO. 10,
F = 10, k = 5, $' = 90° 99
36 PHOTOGRAPH OF NEGATIVELY BUOYANT JET FOR RUN NO. 33,
F z 40, k = 10, g» = 60°. 100
VI
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Figure Page
37 PHOTOGRAPH OF NEGATIVELY BUOYANT JET FOR RUN NO. 27,
F s 10, k * 5, B' = 60° ................ 101
38 PHOTOGRAPH OF NEGATIVELY BUOYANT JET FOR RUN NO. 22,
F = 40, k « 10, g' = 45° ............... 102
39 PHOTOGRAPH OF NEGATIVELY BUOYANT JET FOR RUN NO. 19,
F = 10, k * 5, B1 = 45° ................ 103
40 INVERSE VELOCITY RATIO, 1/k, VERSUS BQ/3^ ........ 115
41 VALUES OF a FOR EXPERIMENTAL COMBINATIONS OF F, k, and
117
42 DENSITY OF A SALT WATER AS A FUNCTION OF SALT
CONCENTRATION AND TEMPERATURE (PERRY'S CHEMICAL
ENGINEERS HANDBOOK, REF. 41) ............. 130
43 CALIBRATION OF 0.5 gpm ROTAMETER ............. 145
44 CALIBRATION OF 60° V-NOTCH WEIR ............. 146
45 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 18 ............... 152
46 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 13 ............... 153
47 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 12 . . . . ........... 154
48 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 11 ............... 155
49 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 16 ............... 156
50 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 10 ............... 157
51 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 9 ................ 158
52 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 15 ............... 159
53 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 34 ............... 160
VII
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Figure Page
54 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 33 161
55 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 32 162
56 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 28 163
57 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 30 164
58 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 27 165
59 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 29 166
60 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 31 167
61 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 26 168
62 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 22 169
63 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 21 170
64 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 20 171
65 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 24 172
66 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 19 173
67 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 23 174
68 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
FAN'S MODEL - RUN NO. 25 175
69 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 18 176
Vlll
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Figure Page
70 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 13 177
71 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 12 178
72 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 11 179
73 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 16 180
74 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 10 181
75 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 9 182
76 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 15 183
77 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 34 184
78 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 33 185
79 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 32 186
80 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 28 187
81 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 30 188
82 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 27 189
83 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 29 190
84 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 31 191
85 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 26 192
86 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 22 193
ix
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Figure page
87 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 21 194
88 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 20 195
89 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 24 196
90 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 19 197
91 OBSERVED VALUES- AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 23 198
92 OBSERVED VALUES AND THEORETICAL CURVES PREDICTED BY
ABRAHAM'S MODEL - RUN NO. 25 199
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LIST OF TABLES
Table Page
1 HEAVY VERTICAL JET EXPERIMENTS IN HOMOGENEOUS AMBIENT
FLUID 31
2 COMPARISON OF JET TEMPERATURE WITH AMBIENT FLUID
TEMPERATURE 63
3 COMBINATION OF DENSIMETRIC FROUDE NUMBER, VELOCITY
RATIO, AND INITIAL ANGLE OF DISCHARGE ACCORDING TO
FAN'S DEFINITIONS 73
4 COMBINATION OF DENSIMETRIC FROUDE NUMBER, VELOCITY
RATIO, AND INITIAL ANGLE OF DISCHARGE ACCORDING TO
ABRAHAM'S DEFINITIONS 74
5 SUMMARY OF NEGATIVELY BUOYANT JET EXPERIMENTS IN A
CROSS-FLOW FOR FAN'S MODEL 80
6 SUMMARY OF NEGATIVELY BUOYANT JET EXPERIMENTS IN A
CROSS-FLOW FOR ABRAHAM'S MODEL 82
XI
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ACKNOWLEDGMENTS
The successful completion of this investigation owes a great deal
to a great many - more individuals than can be mentioned here. The
undertaking of the laboratory investigation would have been virtually
impossible without the cooperation and assistance of the faculty, staff,
and students of the Environmental and Water Resources Engineering Depart-
ment at Vanderbilt University. Special thanks is due to several fellow
students who not only made helpful suggestions for successful completion
of this investigation, but aided in taking the laboratory data, especially
Ed Yandell, Bob Reimers, Eung Bai Shin, Greg Waggener, and Aaron Parker.
Thanks is also due Peggie Bush for superb typing and Larry Jones for his
drafting. Above all, the author wishes to express his appreciation to
his wife, Patsy, for her unwavering support, encouragement, and financial
assistance.
The investigations described herein were principally supported by
the National Center for Research and Training in the Hydrologic and
Hydraulic Aspects of Water Pollution Control in the Department of
Environmental and Water Resources Engineering at Vanderbilt University,
which was funded by the Environmental Protection Agency, Contract Number
16130 FDQ. The senior author was also supported for one year by an Air
Pollution Traineeship and for three years by a NASA Traineeship, Grate-
ful acknowledgment is made for the financial support from these groups.
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The authors are also indebted to Mr. Frank Rainwater, Chief,
National Thermal Pollution Research Program and Dr. Bruce Tichenor,
of the National Thermal Pollution Research Program for their helpful
discussions during the course of the work.
The investigation also served as partial fulfillment of the Ph.D.
requirements of the senior author.
Xlll
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I CONCLUSIONS
A laboratory investigation of negatively buoyant jets using two
models originally derived for the prediction of characteristics of a
positively buoyant jet has been completed. The results presented predict
trends rather than exact dilutions and jet trajectories. The author feels
that the utilization of Fan's model for the negatively buoyant jet is
theoretically more valid than the use of Abraham's model. Abraham's
model can also be used to predict the dilution of jet trajectory.
However, Abraham's model considers the direction of flow of the jet to
be parallel to the direction of flow of the ambient fluid at some
distance downstream from the discharge port. This is not the case for
a negatively buoyant jet, particularly one whose densimetric Froude
number is small, i.e., one in which the negative buoyancy term is large.
In the case of a jet with a small densimetric Froude number, the jet
will deflect downward towards the discharge level after reaching a
maximum height. However, Abraham's model is advantageous in that the
entrainment coefficients are constant and are not restricted to a fixed
relationship with the densimetric Froude number, velocity, ratio, and
initial angle of discharge.
Values of a, the entrainment coefficient, and C^, the reduced drag
coefficient, have been presented. It was found from fitting the pre-
dicted curves to the experimental data that the best fit occurred when
a value of C, equal to zero was used. Moreover, a relationship which
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will predict the entrainment coefficient used in the modified Fan's
model for negatively buoyant jets as a function -^\ densimetric Froude
number, velocity ratio, and initial angle of discharge has been
presented. Field studies of negatively buoyant jets in a cross-flow
are needed to verify appropriate values of a. This study has increased
the understanding and application of the integral theory of jet disper-
sion to situations other than positively buoyant jets in cross-streams.
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II RECOMMENDATIONS
The use of Fan's model fox prediction required a priori knowledge
of both the length of the zone of flow establishment and the reduced
angle of inclination at the end of the zone of flow establishment. Much
more information detailing the effects of a cross-stream is needed so
that one can adequately describe or predict the length of the zone of
flow establishment as a function of k, the velocity ratio. Heretofore,
investigation on the length of the zone of flow establishment as a
function of k and 3' has been limited to discharges of 90° for submerged
jets. Motz and Benedict (25) investigated the effect of the initial
angle of discharge and velocity ratio on the length of the zone of flow
establishment for a heated surface jet and found that the length of flow
establishment was strongly dependent upon the velocity ratio. This was
the approach taken by this author. However, since data was only avail-
able for a discharge of 90° for submerged jets, there should be further
investigation on the effect on the length of the zone of flow establish-
ment caused by a discharge angle other than 90°.
The reduced angle of inclination is also an important parameter for
the utilization of Fan's model. Scant data exists for the effect of
the velocity ratio on the value of the reduced angle of inclination.
Present information indicates that the ratio of the reduced angle of
inclination to the initial angle of discharge should decrease with a
decrease in velocity ratio; however; the data is very scattered as
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evidenced by Figure 39. Hence, no statement can be made regarding the
confidence with which one should use data now available.
Fan's and Abraham's models predict the dilution and jet trajectory
information for a single port injection system only. In practice, the
method of injection may vary from a single port system to a multiport
diffuser system. Hence, additional work is necessary to delineate the
effect of interference as the jet spreads and is intersected by an
adjacent jet.
Larsen and Hecker (39) investigated jet interaction for the case of
submerged diffusers with a heated effluent. They found that the surface
dilution due to jet interference was reduced for the case of a multi-jet
discharge. However, for a negatively buoyant jet, the case of jet
interference has not been investigated. The reduced dilution will
probably occur and give cause for some concern since the waste will
tend to sink back to the discharge level and subject the benthic
organisms to higher concentrations of the waste than were predicted by
a single jet discharge. Therefore, further investigations are needed to
characterize the extent of dilution for a multiport discharge system.
Another feature of the negatively buoyant jet model that should
receive additional attention is the maximum height of rise of the jet.
Holly and Grace (40) present data for the maximum height of rise of a
negatively buoyant jet in a flowing stream. However, no information is
available concerning the effects of the initial angle of discharge
since the studies conducted by Holly and Grace involved only a discharge
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angle of 90°. The equation developed by Holly and Grace utilized an
ambient densimetric Froude number which could possibly be related to a
velocity ratio.
The height of rise of the jet in a river or lake may be a limiting
criteria for the use of a particular initial angle of discharge. As
noted by Holly and grace (40), a primary consideration in designing an
outfall system for dense waste is deciding whether the dense plume will
be allowed to reach the surface, or whether it will be controlled so
that it remains submerged. One would like to be able to maximize the
dilution and still keep the jet submerged. The jet may have to be dis-
charged at some angle other than 90° to keep it submerged. Therefore, a
relationship difining the maximum height of rise of a negatively buoy-
ant jet as a function of the velocity ratio, jet densimetric Froude
number, and initial angle of discharge, is needed.
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CHAPTER I
III
INTRODUCTION
Jet dilution is one of the techniques that may be used to meet stream
quality standards in order to disperse dense waste waters. However, a
basic understanding of this flow phenomenon of a negatively buoyant jet
is needed to adequately control this type of pollution problem and meet
the stream quality standards. This thesis is expected to enhance the
general understanding of the flow phenomenon of a negatively buoyant jet.
Today, as never before, the nation's attention is focused on ecology
and the many parameters that affect an ecosystem. This is true whether
the ecosystem is found in a pond, lake, or in the very air we breathe.
This is exemplified by the increase in time, effort, and money being
utilized to alleviate some of the more pressing problems. A challenge to
all individuals was issued by President Johnson in 1968 in his assessment
of the Nation's water resources under the Water Resources Planning Act of
1965 when he said, "A nation that fails to plan intelligently for the
development and protection of its precious water will be condemned to
wither because of its shortsightedness" (1).
The Water Resources Council has reported that the conterminous
United States has a natural runoff averaging about 1,200 billion gallons
per day (bgd). Withdrawals have been estimated at 270 bgd in 1965 and
1,368 bgd in 2020. The large withdrawals estimated for 2020 in relation
to runoff indicate that even with increased inplant recycling, a large
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increase in reuse of water will be required. Hence, there will be an
ever-increasing need for increased investment in water development,
water conditioning, waste treatment, and water management to meet the
estimated requirements (1).
The ultimate disposal of man-made wastes is a major environmental
problem of today. Pollution of the receiving waters has to be controlled
effectively and reduced to such a level as to preserve the delicate
balance of the natural biological processes. This is true for all kinds
of waste for which an upper limit of pollutant can be defined. Where a
limit on the concentration of a particular pollutant exists, the most
advantageous method of discharging the waste would be one in which com-
plete-mixing was accomplished instantaneously. However, the method may
be limited due to the money available, hydraulic configuration of the
place of discharge, or other space restrictions. Yet a waste cannot
simply be discharged into a receiving stream in a manner in which its
dilution would be inhibited. Hence, a method of discharging a waste
must be found that lies between these two extremes, i.e., one that will
approach a completely-mixed concentration below the permitted upper
limit of the concentration of the particular pollutant within some
specified or known distance from the discharge point.
The use of jet diffusion has received much attention in the past as
a means of disposing of domestic sewage in marine areas. Brooks (2),
Fan and Brooks (3), and Cederwall (4,5) present solutions of a jet
diffusing into an ocean. The degree of treatment and the choice of
outfall site, as well as the design of the outfall structure, must be
carefully considered so that water quality requirements of the receiving
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water can be met. More and better information will be needed to
adequately design outfall sites and describe the dilution of waste water
discharges.
In the past, much work has been conducted on the trajectories and
dilution of positively buoyant jets in a flowing stream as exemplified
by the work of Fan (3), Abraham O), and Cederwall and Brooks (7).
Cederwall (5), Abraham (8), and Turner (9) present solutions for the
maximum height of rise of a vertical negatively buoyant jet in a stag-
nant environment and the concentration at its terminal height. However,
no work has been conducted on negatively buoyant jets discharged into
a flowing stream.
Negatively buoyant jets, or sinking jets, can be observed in many
problems of pollutant discharge. Any chemical waste heavier than the
receiving water may form a negatively buoyant jet. In addition, the
negatively buoyant jet is found in releases of cold hypolimnic waters to
the warmer receiving stream, and the release of a gaseous waste which
is heavier than the receiving ambient air. It is also expected that
this study on negatively buoyant jets will increase general under-
standing about mass and heat transfer across density gradients, a factor
of vital concern in many pollution problems.
This study will include a laboratory investigation of negatively
buoyant jets in a cross-flow. Various combinations of Velocity ratios,
densimetric Froude numbers, and initial angles of discharge will be
considered. Models for positively buoyant jets in a flowing stream
developed by Fan (3) and Abraham (6) will be modified and extended to
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include the negatively buoyant case. Values of a, the entrainment
coefficient, and, if applicable, C,, the drag coefficient, will be
determined.
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CHAPTER II
IV
REVIEW OF THE LITERATURE
Jets in General
In the past, more research attention has been devoted to the
problem of buoyant jets where the discharge fluid is lighter than the
receiving fluid and will rise. Fan (3) presents a comprehensive review
of the research heretofore accomplished. His review encompasses the
field of simple jets (Figures la and Ib), simple plumes (Figures 2a and
2b), vertical buoyant jets and inclined or horizontal buoyant jets
(Figures 3a and 3b). The variables found on the figures are defined as:
O1 = origin of the coordinate system (x',y'), point of jet
discharge
0 = origin of the coordinate systems (x,y), beginning of
the zone of established flow (The zone of flow estab-
lishment will be discussed in detail in a later section.)
0 = initial volume flux at the nozzle
xo
D = diameter of jet at orifice and orifice diameter
r = radial distance measured from the jet axis
x1 = coordinate axis in horizontal direction on the same
plane as jet axis with origin at 0'
y1 = coordinate axis in vertical direction, with origin
at 0'
U = ambient uniform velocity
£1
10
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VIRTUAL
SOURCE
a. CONCEPT OF A VIRTUAL SOURCE
0'
ZONE OF FLOW
ESTABLISHMENT
ZONE OF
ESTABLISHED FLOW
b. ZONE OF FLOW ESTABLISHMENT AND ZONE OF ESTABLISHED
FLOW
FIGURE 1 - A SIMPLE JET (AXISYMMETRIC CASE)
.
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VIRTUAL SOURCE
/J=CONST.
a A SIMPLE PLUME IN A UNIFORM ENVIRONMENT
VIRTUAL SOURCE
0'
b. A SIMPLE PLUME IN A LINEARLY STRATIFIED
ENVIRONMENT
FIGURE PROFILE VIEWS OF SIMPLE PLUMES IN UNIFORM
AND STRATIFIED ENVIRONMENTS
12
-------
Pa -• f (y)
a. AN INCLINED ROUND BUOYANT JET (FORCED PLUME) IN A
STAGNANT ENVIRONMENT WITH LINEAR DENSITY
STRATIFICATION
U,
yOa = constant
b A VERTICAL ROUND BUOYANT JET IN A UNIFORM HORIZONTAL
WIND (CROSS STREAM)
FIGURE 3 - PROFILE VIEWS OF BUOYANT JETS
[AFTER FAN (3)]
13
-------
U = initial jet discharge velocity
u = jet velocity at the centerline of the jet
p = density of the ambient fluid
3.
p = reference density taken as p (0)
O 3.
PI = initial jet density
g = gravitational acceleration
A simple jet (or ordinary momentum jet) is the turbulent flow
pattern generated by a continuous source of momentum. Albertson, Dai,
Jensen, and Rouse (10) have presented a model which describes the
behavior of a simple jet, whereas a simple plume is characterized by a
turbulent flow pattern generated by a continuous source of buoyancy.
The more common case that has received most of the attention in the
past is a steady release of heat. The plume has no initial momentum
flux. Hence, the main direction of flow of the plume is in the direction
of the buoyancy force. Due to continuous action of the buoyancy force,
the momentum flux of the plume increases with increasing height in the
case of a heated plume. A buoyant jet (or forced plume) is characterized
both by a steady release of mass, "momentum," and buoyancy from a source.
The source may be situated in either a uniform environment or stably
stratified fluid. Hence, the simple jet and simple plume are limiting
cases of buoyant jets.
Morton (11) and Fan (3) have presented solutions for buoyant jets.
Fan presents a solution for an inclined round buoyant jet in a stagnant
environment with linear density stratification and for a round buoyant
jet in a uniform cross-stream of homogeneous density. Development of
the round buoyant jet in a uniform cross-flow of homogeneous density
14
-------
will be discussed at length in the following pages.
Parker and Krenkel (12) present a more recent review of analytical
models of jet studies. In their report the integral approach of Morton,
Taylor, and Turner (13) is reviewed with Morton's (14) basic assumptions:
a. The fluids are incompressible
b. Flow is fully turbulent, implying no Reynolds number depen-
dence and that molecular diffusion is negligible compared to
turbulent diffusion.
c. Longitudinal diffusion is negligible compared to lateral
diffusion.
d. The largest variation of fluid density through the flow field
is small compared with the reference density. Hence, varia-
tions of density can be neglected when considering inertial
terms, but must be included in gravity terms. As Fan (3)
notes, this small density variation implies that the conserva-
tion of mass flux can be approximated by the conservation of
volume flux. The assumption of small density variation is
commonly called the Boussinesq assumption.
e. Velocity profiles are similar in consecutive transverse
sections of the jet in the zone of established flow. Fan (3)
also assumed a similarity for profiles of buoyancy and con-
centration of tracer. Buoyancy and concentration profiles
are given by the relation:
-r2/b2
pa - p*(s,r,*) = [Pa - PCs)] e'* /u (D
and
15
-------
c*(s.r.*) = c(s) e'r (2)
where p = density of the ambient fluid
3
p* = local density within a jet
s = distance along the jet axis from the zone of
establishment
r = radial distance measured from the jet axis on A
A = jet cross-section normal to the jet axis
= angular coordinate on a cross-section normal to
the jet axis
b = local characteristic length of half-width of the
jet
c* = local concentration value
c = concentration at the jet axis
Figure 4 shows a round buoyant jet discharging at a velocity
u into a uniform cross-stream of velocity U . In addition to
O a
the above variables, the other variables shown on Figure 4
are defined below:
6 = angle of inclination of the jet axis (with respect
to the x-axis)
The x-axis is formed by passing a plane through the origin, 0,
in the same direction as the flow of the ambient stream.
g1 = initial angle of inclination
3 = angle of inclination at the end of zone of flow
establishment
s' = length of zone of flow establishment
6
16
-------
WAKE REGION
ZONE OF FLOW ESTABLISHMENT
FIGURE 4 - SCHEMATIC DIAGRAM OF A PROFILE VIEW OF A ROUND BUOYANT JET
IN A UNIFORM CROSS STREAM OF HOMOGENEOUS DENSITY
-------
In addition to the above assumption, Fan made two additional
assumptions which are also appropriate in this study. They
are:
f. Within the range of variation, the density of the fluid is
assumed to be a linear function of either salt concentration
or heat content above the reference level. However, Morton
(13) did consider the density difference expressed as a linear
function of temperature difference and cubical expansion co-
efficient. This is a reasonable assumption as can be seen in
Appendix A. It is appropriate in this case since measurements
of the conductivity of the fluid were used to obtain the con-
centration of salt along the jet axis. The procedure will be
dealt with in a later chapter.
g. Curvature of the trajectory of the jet is small. That is, the
ratio of the local characteristic length to the radius of cur-
vature is small. Hence, the effect of curvature is neglected.
In the case of the negatively buoyant jet, for small Froude
numbers, the radius of curvature is small. Hence, the ratio
of the local characteristic length to the radius of curvature
may be large, but as a first approximation of the system, the
effect of curvature is also neglected.
Zone of Flow Establishment
It can be shown that a zone of flow establishment must exist
beyond the efflux section of either a two-dimensional jet or a three-
dimensional submerged jet (10). With reference to Figures (Ib) and (4),
18
-------
the fluid discharged from the boundary opening may be assumed to be of
relatively constant velocity. At the efflux section there will neces-
sarily be a pronounced velocity discontinuity between the jet and the
surrounding fluid. The eddies generated in the region of high shear
will immediately result in a lateral mixing which progresses both in-
ward and outward with distance from the efflux section. Fluid within
the jet is gradually decelerated; fluid from the surrounding region is
gradually accelerated and entrained into the jet. The limit of the
zone of flow establishment is reached when the mixing region has
penetrated to the centerline of the jet. Albertson, et al. (10),
reported that the zone of flow establishment for a three-dimensional
submerged jet in a stagnant non-stratified environment is X0/D0 = 6.2
where x = distance along the axis of the jet from the efflux (discharge)
o
section to point of established flow. In the case of a jet in a cross-
flow, x will be replaced by s^.
Fan (3) reported values obtained from Gbrdier (15), Jordinson (16),
and Keffer and Baines (17) for the length of the zone of flow estab-
lishment for different velocity ratios. Fan developed a plot of s^/DQ
versus k for k values of 4, 6, and 8 where
s1 = the distance along the jet axis from the discharge point
e
to the point of established flow and
k = velocity ratio = U /U
O d
where
U = jet discharge velocity and
U = ambient velocity.
a
19
-------
The equation of the data presented by Fan is
e o
as reported by Parker and Krenkel (12). Note that s'/D approaches
(4)
6.2, which is the same coefficient reported by Albertson, as 1/k
approaches zero.
Since the above equation was developed from using only 4 points,
it was felt necessary to seek additional sources of information. Pratte
and Baines (18) have presented results as shown in Figure 5.
10
8
0°
cn
i i i i
0.05
0.10 0.15
l/k
0.20
0.25
FIGURE 5 - LENGTH OF ZONE OF ESTABLISHMENT VERSUS l/k
These authors felt that the different diameters used in their
studies affected the length of the potential core (zone of flow
establishment). However, it is altogether possible that experimental
error could have just as easily been the villain, since the diameters
of the orifices used were all less than 1/2-inch. Hence, a least
20
-------
squares fit of the data was performed, and as a result the equation
s'/D = 5.91 e * was developed. Hence, this lends credence to the
e o r
equation developed by Parker and Krenkel (12) and will be used in this
report.
Mechanisms of Entrainment in Turbulent Flow
The mechanism of entrainment has received much attention over the
last 20 years in determining buoyant jet behavior. Tollmien (19) and
Schmidt (20) studied the problems of turbulent non-buoyant and buoyant
jets, issuing vertically upwards into a homogeneous fluid at rest.
It was noted that the similarity of velocity and concentration profiles
provided sufficient information to solve for a jet's trajectory with
either negligible or predominant influence of buoyancy. However, as
noted by Abraham (21), for jets which are characterized by a varying
influence of buoyancy effects within the field of motion, the entrain-
ment principle is necessary.
Morton, Taylor, and Turner (13) first proposed an entrainment
mechanism for the dilution of a maintained plume. They proposed the
equation
-j^- = 2 irotub (5)
dx
where -r*- represents the rate of change of volume flux and a is the
coefficient of entrainment. This equation states that the rate of
entrainment at the edge of the plume is proportional to some charac-
teristic velocity at that point. They noted that when a stream is in
contact with another stream the eddies which cause transfer of matter
21
-------
between them are characterized by velocities proportional to the
relative velocity of the two streams. They developed equations for
(a) a maintained plume in a uniform ambient stagnant fluid, (b) a point
source in a stratified fluid, and (c) a uniformly stratified fluid.
The entrainment coefficient was considered constant at about 0.093.
Morton (11) states that the structure of the turbulence within a
plume (jet in the case of a forced plume from a point source of
momentum) and the rate of entrainment at its mean edge depends only
on the difference in mean density and mean velocity between the plume
axis and the ambient fluid.
Fan (3) used this technique in the analysis of a turbulent round
buoyant jet in a flowing stream. The entrainment for a jet in a cross
stream is assumed to be represented by the equation
a£-21™*!^--uj (6)
where b is again a characteristic length defined by the assumed
velocity profile. The variables U. and U are defined by the following
J a
equations:
U. = 1 (U cos 8 + u) (7)
1 3-
IJ = t(U cos 6) + 7(U sin 0) (8)
3. 3. 3-
where i is a vector in the direction tangent to the jet axis and j is
a vector perpendicular to the jet axis. Hence, JU. - U | is the mag-
3 a
nitude of the vector difference in the two velocities. Fan assumed a
to be constant in the analysis.
Abraham (21) investigated the principle of entrainment and discussed
22
-------
its restrictions in solving problems of jets. It was maintained that
the entrainment coefficient, as defined above, was not constant.
Abraham introduced a new constant, E, which relates the rate at which
work is done by turbulent shear per unit time in a layer with some
thickness, dx, at some level, x, per rate of vertical flow, Q .
However, Fan and Brooks (22) in the discussion of Abraham (23),
considered the rate of entrainment to be proportional to the local
characteristic (or maximum) velocity and the local characteristic
radius of the jet or plume. They noted that the value of a for buoyant
plumes based on data of Rouse, Yih, and Humphreys (24) was 0.082, while
for momentum jets the value was 0.057 based on data by Albertson, Dai,
Jensen, and Rouse (10). Fan and Brooks recognized that the entrainment
coefficient could not be a universal constant, but varied as the
relative buoyancy or local Froude number changed. However, for sim-
plicity's sake, it was assumed constant. Fan (3) considered the en-
trainment coefficient to be constant along the jet trajectory for a
particular set of values of velocity ratio and densimetric Froude number,
but was a variable dependent on each different set of values of velocity
ratios and densimetric Froude number. The densimetric Froude number is
defined as:
U
F = ° (9)
/I AP
Vel^i
p o
where U = u + U cos B = initial jet discharge velocity
o o a o
U = ambient uniform velocity
a
u = jet discharge velocity at orifice
23
-------
also
8 = angle of inclination at the end of the zone of flow
establishment
g = gravitational constant
|£!| . Jj_L do)
a a
U u + "U cos g
k - — - ° a
K -
a
For the case under study, i.e., negatively buoyant jets, it is advan-
tageous to use the absolute value of the density difference. A more
explicit definition of the above variables is given in Figure 4.
Fan (5) found that values for a for a round buoyant jet in a cross
flow varied from 0.4 to 0.5 for a range of velocity ratios from 4 to
16 and a range of densimetric Froude numbers from 10 to 80. Benedict
and Motz (25) reported values of the entrainment coefficient varying
from 0.13 to 0.46 for heated surface jets discharged into flowing
ambient streams. Abraham (6) modified Fan's approach and considered
the solution of a round buoyant jet with two distinct regions of en-
trainment. This modification will be discussed in more detail in
Chapter III.
Abraham suggests that the jet velocity, at a sufficiently great dis-
tance downstream from the nozzle of the jet fluid, is about equal to the
velocity of the ambient fluid. Hence, the entrainment may be described
as if the jet was a cylindrical thermal. Richards (26) describes a
24
-------
cylindrical thermal as a body of fluid of cylindrical shape with its
horizontal axis moving through a stagnant surrounding fluid due to a
density difference between the surrounding fluid and the particular
body of fluid under consideration. Hence, Abraham describes the
volumetric flux and momentum flux of a round buoyant jet in terms of
the entrainment of a simple jet and the entrainment of a cylindrical
thermal. The values of the entrainment coefficients used for the simple
jet and cylindrical thermal were 0.057 and 0.5, respectively. These
values of the entrainment coefficient were considered constant for all
combinations of densimetric Froude numbers and velocity ratio.
It can be seen that there are many different values for the en-
trainment coefficient, dependent upon the case under study. However,
in most cases the value of the entrainment coefficient is unique for
a specific combination of densimetric Froude number and velocity ratio.
This approach is used in this study.
Negatively Buoyant Jets
The first attempt to deduce the path followed by a jet of initial
density different from its surroundings was reported by Groume-Grjimailo
(27). However, this formula neglected all consideration of viscosity
or entrainment and was really based on the parabolic path of a pro-
jectile.
Bosanquet, et al. (27), studied the effect of density difference
on the paths of jets. Equations were developed to predict the trajec-
tories. However, since these experimental tests were conducted in a
transparent box with baffles, they were essentially conducted in a
25
-------
stagnant environment. Hence, the applicability to a negatively-buoyant
jet in a cross-flow is limited.
Turner (9) studied jets and plumes with negative or reversing
buoyancy. In an attempt to explain the oscillation occurring at the
top of a cumulus cloud due to evaporation, Turner injected salt water
vertically into a stagnant basin. The most feasible explanation put
forth was that the evaporation produced a reversal of the buoyant force.
In these experiments the salt water was injected upwards.
Initially the pulse of fluid looked like a buoyant plume with a vortex-
like front and steady plume behind. The velocity decreased more rapidly
with height and instead of rising indefinitely with constant shape, the
whole plume broadened, came to rest, and then started to fall back.
The steady-state position was reached with the top at a lower height
than that attained initially, with an upflow in the center, and a
downflow surrounding this. Figure 6 is a sketch showing the relative
shape of a jet with negative buoyancy.
UPFLOW
I \ t / I
DOWNFLOW I / DOWNFLDW
FIGURE 6 - JET WITH NEGATIVE BUOYANCY
26
-------
The three phases distinguished were (a) a 'starting plume1,
advancing and growing by mixing over its top and sides, (b) the cap
stopped rising while it continued to grow; negative buoyancy is being
accumulated here, and (c) the cap collapsing. Turner studied two
different cases, a heavy jet injected upward and the plumes with re-
versing buoyancy. Turner's results concerning the heavy jet were of
particular interest to this study. For a heavy jet injected upwards,
two parameters, M (momentum flux) and F2 (buoyancy flux) were used to
define the flow from a small source (essentially a point source). The
developed equation took the form
yt = CM°-75F-°-5 (12)
where C = a constant
y = the mean vertical height of rise of the plume
M = —2-2- (13)
Pi - Pa ^ D ^_
Q 4F
Evaluation of the experimental data shows that C = 1.85. Hence, sub-
stituting the value of C and Equations 13 and 14 into Equation 12 and
dividing both sides of Equation 12 by DQf Equation 15 is developed.
=p = 1.74 F (15)
o
The main conclusion that can be drawn from the above research is
that jets in which the buoyancy force always act downward, and which
must be driven upward by momentum at the source, reach a steady height
27
-------
and fluctuate randomly about this height.
Abraham (8) has studied the problem of jets with negative buoyancy
in homogeneous fluids. He made a distinction between a zone with posi-
tive entrainment near the orifice and a zone with negative entrainment
near the ceiling level. Abraham developed an expression for the ceiling
level of a heavy jet injected upward, which is
~ = 1.94 F (16)
o
In the above Equations 15 and 16, the densimetric Froude number, F, is
calculated by dividing the density difference, Ap, by the density of
the jet fluid, p , which is different from the definition of F in
Equation 9. However, the resulting value of F will not be significantly
changed if it is divided by either p or p since p = p .
a i a l
Fan (28) used his model to theoretically predict the dilution and
trajectory of waste gas discharges from campus buildings. The mixing
of waste gas discharged from a vertical furaehood exhaust in a wind is
basically a phenomenon of turbulent jet mixing in a crosswind. The
waste gas jet bends over toward the downwind direction due to the action
of ambient wind motion. In the process, the jet entrains the ambient
air, growing in volume and width.
Fan also considered a sinking jet in a calm atmosphere. The
governing dimensionless parameters were velocity ratio and jet densi-
metric Froude number. The cases for a negatively buoyant jet considered
were for combinations of densimetric Froude numbers of 80, 40, and 20,
and velocity ratios of 4, 8, 12, 20, 24, and 32.
28
-------
In a calm atmosphere (U = 0 or k = °°) , a jet rises indefinitely
3.
if it has positive buoyancy or F > 0, but a jet with negative buoyancy
would eventually sink back to the discharge level and spread over the
roof or ground after reaching a maximum height. For a negatively
buoyant jet, the initial upward momentum of the jet is gradually reduced
by the constant downward action of the gravitational force. Using the
integral approach of Turner (9) , Fan made an estimate of the height of
rise of the jet as
-- = 1.9 F (17)
and the dilution ratio S at y = y as
L- U
s ~ F (for F
L- ' tJ
or
S «0.25 F (for a = 0.082) (19)
C-
Morton (11) investigated the height of rise of a negatively
buoyant forced plume issuing vertically into a stagnant environment.
He considered the effect of the discharge of momentum from a virtual
source of buoyancy and presented a solution in non-dimensional parameters,
An equation was developed for the maximum height of rise in terms of
the non-dimensional parameters. However, as noted by Morton, Turner,
and Taylor (13), a correction must be made for a jet or forced plume
emitted from an orifice when a point source is considered.
Cederwall (5) states that, in the case of a jet issued vertically
upwards into a homogeneous, lighter ambient fluid, the initial momentum
29
-------
and buoyancy force are opposing each other. However, the basic con-
siderations are essentially the same as for a positively buoyant jet,
as long as the vertical momentum of the jet is sufficient to maintain
positive entrainment. Equations 20 and 21 were developed to describe
the velocity and concentration profile for a negatively buoyant jet.
D -0.33
S~ (yr) = 6.2 (1 - 0.22---) (20)
o y DZF2
o
and
Do v» "°-33
= 5.6 -£ (1 - 0.22 -£— ) (21)
o y D2F2
o
Equation 22 was also developed to predict the maximum height of
rise for a negatively buoyant jet.
£=2.9F°'67 (22)
O
However, Cederwall noted that for small values of F, Equation 22 will
not hold. Figure 7 is a comparison of experimental data (Table 1)
obtained by Cederwall and is compared to the previously developed
Equations 15 and 16. Also, Cederwall compared Equation 23, from the
work of Priestley and Ball (29), in Figure 7 with data obtained at
Chalmers Institute of Technology (5).
~= 1.85 F (23)
o
The equations that Cederwall compared for the height of rise of a
negatively buoyant jet in a stagnant fluid are listed below:
30
-------
TABLE 1 - HEAVY VERTICAL JET EXPERIMENTS IN
HOMOGENEOUS AMBIENT FLUID
Run
1
2
3
4
5
6
F
9.3
10.5
20.4
28,8
31.8
20.6
D
o
14
17
30
32
43
26
D
o
Flowing Ambient
Fluid
--
--
--
54
58
38
FT
0
1.50
1.62
1.47
1.11
1.35
1.26
D F
0
Flowing Ambient
Fluid
--
--
--
1.88
1.82
1.84
3
2
LL.
\
cf
:£
~ 1
u(
,iiii
1 _-_ EQUATION 15
\ EQUATION 16
\ __ EQUATION 23
^ O
• ^ °o
0 STAGNANT AMBIENT STREAM
D FLOWING AMBIENT STREAM
, i i 1
) 10 20 30 40
DENSIMETRIC FROUDE NUMBER, F
^
m~
-
-
FIGURE 7 - MAXIMUM HEIGHT OF NEGATIVELY BUOYANT JETS
[AFTER CEDERWALL (5)]
31
-------
1. yt/DQ * 1.74 F (15)
2. yt/D = 1.94 F (16)
3. yt/DQ = 2.9 F°'67 (22)
4. yt/D = 1-85 F (23)
Cederwall noted that the equations developed by Abraham (Equation
16) and Priestley and Ball (Equation 23) gave better prediction of the
maximum height of rise of a negatively buoyant jet in a flowing stream.
However, the velocity of the flowing ambient stream [Cederwall (5)] is
not given, nor is the value of the jet discharge velocity given. Hence,
no statement can be made concerning the effect of the velocity ratio
or the relative strength of the jet on the terminal height of rise of
a negatively buoyant jet in a flowing ambient stream.
A review of the literature concerning negatively buoyant jets
indicates a paucity of information concerning the experimental verifi-
cation of any models which predict the dilution and trajectory of the
jet. However, Cederwall (5) noted that the theoretical considerations
for a negatively buoyant jet are essentially the same as these for a
positively buoyant jet. Cederwall noted that the only difference
between a negatively buoyant jet and a positively buoyant jet is that
the initial momentum and buoyancy force are opposing each other. Hence,
the models for a positively buoyant jet, with an appropriate modifi-
cation to the buoyancy force terms, should be able to predict the
dilution and trajectory of a negatively buoyant jet. Thus, the inte-
gral approach of Morton, Taylor, and Turner (13) applied to a negatively
32
-------
buoyant jet in the same manner as Fan (3) and Abraham (6) treated a
positively buoyant jet should yield a satisfactory and usable means of
evaluating the characteristics of a negatively buoyant jet. These two
models are developed in more detail in Chapter III.
33
-------
1. yt/D = 1.74 F (15)
2
. y /D = 1.94 F (16)
3. yt/Do = 2.9 F0-67 (22)
4. yt/D = 1.85 F (23)
Cederwall noted that the equations developed by Abraham (Equation
16) and Priestley and Ball (Equation 23) gave better prediction of the
maximum height of rise of a negatively buoyant jet in a flowing stream.
However, the velocity of the flowing ambient stream [Cederwall (5)] is
not given, nor is the value of the jet discharge velocity given. Hence,
no statement can be made concerning the effect of the velocity ratio or
the relative strength of the jet on the terminal height of rise of a
negatively buoyant jet in a flowing ambient stream.
Briggs (42) presents an excellent review of existing plume rise
observations and formulas. Nine formulas are reviewed and compared with
data from sixteen different sources. Briggs chose Equation 94 as the
best predictive equation. However, Equation 94 must be modified by
assuming that a ceiling height is reached at a distance of ten stack
heights downwind. Other equations are presented which are dependent upon
different stability conditions.
Ah = 1.6 F U"1(X1)'667 (94)
cL
in which Ah = plume rise above top of stack.
A review of the literature concerning negatively buoyant jets
indicates a paucity of information concerning the experimental verifi-
cation of any models which predict the dilution and trajectory of the
34
-------
jet. However, Cederwall (5) noted that the theoretical considerations
for a negatively buoyant jet are essentially the same as these for a
positively buoyant jet. Cederwall noted that the only difference
between a negatively buoyant jet and a positively buoyant jet is that
the initial momentum and buoyancy force are opposing each other. Hence,
the models for a positively buoyant jet, with an appropriate modification
to the buoyancy force terms, should be able to predict the dilution and
trajectory of a negatively buoyant jet. Thus, the integral approach of
Morton, Taylor, and Turner (13) applied to a negatively buoyant jet in
the same manner as Fan (3) and Abraham (6) treated a positively buoyant
jet should yield a satisfactory and usable means of evaluating the
characteristics of a negatively buoyant jet. These two models are
developed in more detail in Chapter III.
-------
CHAPTER III
V
ANALYTICAL DEVELOPMENT OF FAN'S AND ABRAHAM'S MODEL
In this chapter both Fan's and Abraham's models will be reviewed
and normalized so that they can be used to describe negatively buoyant
jets.
Fan's Model for a Round Buoyant Jet in a Uniform Cross Stream
Fan (3) applies the integral approach of Morton, Taylor, and
Turner (13) to solve the problem of a round buoyant jet in a cross
stream. Figure 8 shows a round buoyant jet discharging at a velocity
u into a uniform cross stream of velocity U . The densities of the
o 'a
discharged fluid and the ambient fluid are p and p , respectively.
1 3-
The flow becomes fully developed at a short distance s' from the nozzle,
C
O1 is taken at the beginning of the zone of flow establishment, and 0
is taken at the beginning of the zone of established flow. 9 is the
angle of the inclination of the trajectory with respect to the horizon-
tal x-axis.
For the case of a positively buoyant jet, there are two reasons
for the deflection of the jet toward the downstream direction - the low
pressure region established behind the jet and the entrainment of
ambient horizontal momentum as the jet entrains the fluid of the cross
stream. Yet, in the case of the negatively buoyant jet, the jet will
reach a maximum height and then bend over toward the region of y = 0
36
-------
'
i
AMBIENT
VELOCITY
DRAG FORCE
FIGURE 8 - SCHEMATIC DIAGRAM FOR THE ANALYSIS OF A ROUND BUOYANT JET
IN A CROSS STREAM
-------
or towards the discharging level or lower. The jet will rise due to its
vertical momentum flux, but at the same time the negatively acting
buoyancy flux will tend to reduce the jet's initial momentum in such a
manner that it bends over and returns towards the discharge level.
Fan found by dimensional analysis that the flow of a positive
buoyant jet was characterized by the densimetric Froude number (Equation
9) and by a velocity ratio which represents the relative strength of the
jet into a cross flow as defined by Equation 3.
Basic Assumptions
The assumptions adopted by Fan are outlined below.
1. Velocity profiles are assumed to be similar and Gaussian above
the component of the ambient velocity U cos 6
u*(s,r,) = U cos 9 + u(s) e"r /b (24)
e*
where u* = velocity at a local point and b = local
characteristic length.
2. The entrainment relation for a jet in a cross stream is
assumed to be represented by the equation
- 2 TT ab |U. - Uj (6)
where Q is the volumetric flux.
3. Buoyancy profiles are assumed to be Gaussian
p - p*(s,r,<>) = [p - p(s)] e'
<& »
where p* = local density within a jet,
4. Concentration profiles of a certain tracer are assumed to be
38
-------
Gaussian
T2/h2
c*(r,s,4) = c(s) e"r /b
where c* = local concentration value.
5. The effect of the presence of the pressure field can be lumped
into a gross drag term proportional to the square of the veloc
ity component of the ambient stream normal to the jet axis.
The drag coefficient is assumed constant.
Peyel_opme_nt__of_ Equations
Fan makes use of the equation of conservation of mass, the equa-
tion of conservation of momentum, and geometric equations to describe
the round buoyant jet in a cross stream. The equations developed are
outlined below.
1. Conservation equations - Equation 6 is integrated to attain
an expression for the continuity of fluid.
~^f ~~ ^11 IAL/ 1 U • ~ U 1
ds ' j a1
where Q is the volumetric flux.
Q = I u* dA (25)
where A is jet cross section normal to the jet axis.
Fan defines the boundary of this jet as /2b. Then, substituting
Equation 24 for u* into Equation 25 and integrating between the limits
of r = 0 and r = -/2b, Equation 25 becomes
r -r2/b2
Q = 2irr(U cos 6 + ue ) dr
J,. a
39
-------
After the substitution of the limits
Q =
-r2/b2
cos 9 + ue ' ) dr
which can be approximated as
Q 2TT
r U cos 9 dr + ure dr
-r2/b2
After integration,
Q = irb2(2U cos 6 + u)
ol
Hence,
§= |- E"b2(2U cos 6 + u)] = 2irab|U. - "u,
QS 3. J c
(26)
Thus, |U. - U | is the vector difference of the jet velocity and
J a
the ambient velocity.
U. = i(U cos 8 + u)
J a
and
U = i U cos 6 + j U sin 6
a a J a
Therefore,
JU. - U | = J(U cos 9 + u - U cos 9)2 + (0 - U sin 6)2
a V a a a
Hence,
|U - U" | = V (u2 + U 2 sin2 6)'
J a, a
Dividing Equation 26 by TT,
^-[b2(2U cos 6 + u)] = 2ab J(U 2 sin2 6 + u2) (27)
OS di V cL
40
-------
2. The momentum equations can be written by assuming a gross drag
coefficient, C,, In the x-direction, the rate of change of
momentum flux is equal to the rate of entrainment of ambient
momentum flux plus the drag force acting on the jet.
4- f p*u*(u* cos 8)dA = 2™b p U |U. - if | + F sin 6 (28)
dsj. a a j a u
where Fn is the drag force per unit length assumed to be
p U2 sin2 6
FD * Cd -^-~ 2 V^b (29)
The left side of Equation 28, after substitution of Equation 24
and making use of Boussinesq's assumption that p*~g p&, becomes
r2/b2 2
§- 1 p 2irr(U cos 6 + ue ' ) cos 9 dr
ds JQ a a
After integration,
~l
p „ * r b2 cos e + u)2 cos e C30)
Ma ds L 2 a -I
Substituting Equation 29 and 30 into Equation 28 and dividing by pair,
the x-momentum equation becomes
_ cos
ds L 2 a
0 + u)2 cos 9 1= 2cxb U J(U2 sin2 6
J a. t A
- U2b sin 3 6 (31)
•TT a
3. In the y-direction, the rate of change of the momentum flux
is equal to the gravity force acting on the jet cross-section
minus the y-component of the drag force. For a negatively
buoyant jet, the buoyant force will be acting in a manner to
41
-------
reduce momentum in the y-direction. Hence, a negative sign
correction to the buoyancy term will be made on Fan's equation
for y-momentum so that the absolute value of the Froude number
can be used later.
-
ds A
p*u*(u* sin e)dA = -
' '
sin e)dA = - g(p - p*)dA - Fn cos 9 (32)
The left-hand side of the equation is treated in the same manner
as the left-hand side of Equation 28, as shown by Equation 30. The
first term of the right-hand side of the equation is treated as below
substituting Equation 1.
g(p - P*)dA = - I g(p - p)e"r
A a
g(P - P*)dA
-I
g(p - p*)dA = - irb2g(p - p)
A a
Therefore,
COS
u)2 sin 6J = -nl
p U2 sin2 6
(33)
Dividing Equation 33 by p TT yields, for the y-momentum equation,
ct
1 . (pa "
cos 9 + u)2 sin el = -b2g • a
pa
Cd-/2'
- bU2 sin2 6 cos 6 (34)
42
-------
4. The density deficiency flux induced is conserved since the
ambient fluid is homogeneous, so
IF f "*(Pa - P*)dA = 0 (35)
•'A
Substituting Equations 1 and 24 into Equation 35 yields
d -r2/b2 -r2/h2
~ 2Trr(U cos 6 + ue r /D ) (p - p)e r /D dr = 0
ds a a
d ' b-U cos Of + — uf
a cos (.Pa pj + ^ u^Pa ~ PJ
= 0
cos 6 •»- u) (p - p) =0 (36)
d
Therefore, Equation 36 yields
4- [b2(2U cos 6 + u) (p - p)] = 0 (37)
Uo A a.
5. The flux of any specific tracer of concentration, c*, contained
in the jet flow will be conserved in a fashion similar to the
density deficiency, as shown below.
4-f u* c* dA = 0 (38)
ds J
Substituting Equations 2 and 24 into Equation 38 and inte-
grating yields
4~ [b2(2U cos 6 + u)c] = 0 (39)
dS cl
6. There exist two geometric relationships due to the jet
trajectory which are utilized.
^- x = cos 9 (40)
ds
and
43
-------
- y = sin 6 (41)
Hence, there are 7 unknowns - u, p - p, c, b, 6, x, and y -
3.
that can be solved for with seven simultaneous ordinary differ-
ential equations - Equation Numbers (27), (31), (34), (37),
(39), (40), and (41).
Initial Condition for Fan's Model
Fan considers his model applicable after the zone of flow estab-
lishment. The initial conditions at the origin, 0, at the end of the
zone of flow establishment are given below.
u(0) = UQ
b(0) = bo
P(0) = PI
0(0) = eo = po (42)
c(0) = CQ
x(0) = 0
y(0) = 0
at s = 0
where u = initial jet discharge velocity at orifice
b = jet half-width at the end of the zone of flow establishment
as defined in Equation 24
Pt = jet density
c = initial concentration at the end of the zone of flow estab-
o
lishment
44
-------
x = horizontal coordinate measured from the end of the zone of
flow establishment
y = vertical coordinate measured from the end of the zone of flow
establishment
s = parametric distance along jet axis measured from the end of
the zone of flow establishment
By considering the conservation of the flux of density deficiency
between the two cross-sections at point 0 and 0',
cos 6 * u)
0
'2U cos 6 + u
a o o
2U cos 6 + u
where k' = —^ ~ = k + cos GO (43)
a
. bo • Do -
In addition, the initial angle, QQ, is the reduced angle of in-
clination. It will always be less than the initial angle of discharge
due to deflection in the zone of flow establishment. Fan developed a
functional relationship involving k to express the value of 9Q. However,
for the negatively buoyant jet, no functional relationship exists. So,
in each case, the jet trajectory is plotted, the zone of establishment
is measured along the axis using values calculated from Equation 4, and
then the angle at the end of the zone of flow establishment is measured.
Hence, when Fan uses 9 in his derivation, it is equivalent to 3Q in
45
-------
this report. In addition, £' is the initial angle of inclination at the
orifice.
Dimensionless Parameters
Fan's equations are normalized by defining dimensionless parameters
as follows, using initial values.
Volumetric flux parameter,
b2(2U cos 9 + u)
(45)
b2(2U cos 8 + u )
o^ a o o'
momentum flux parameter,
b2(2U cos 6 + u)2
m = - - - (s-direction) (46)
bo(2Ua COS 8o * V2
h = m cos 6 (x-direction) (47)
v = m sin 6 (y-direction) (48)
velocity ratio,
u + U cos 8 U
v _
k
a a
buoyancy parameter,
f = b0 g/^k.zuz) (49)
a
k1 = k + cos 6 (43)
o ^ J
(50)
i_ — — .» _i
where F = densimetric Froude number,
coordinates,
46
-------
2c*
x : n = r— x
b
o
(51)
= —
o
where a = coefficient of entrainment.
Normalization of Fan's Equations
Fan's equations are normalized using the dimensionless parameters
on pages 39 and 40, resulting in Equations 52-56, 60, and 61. Fan's
equations 27, 31, 34, 37, 39, 40 and 41:
dri 1 u Q
k'2 v m
C1
3. -77 = - f — - — sin2 6 cos 6 (54)
4. dJI= cos 9 (55)
5. i= sin 9 (56)
d^
r -i-5
where ^ = I sin2 6 + (k' 2. . 2 cos 6) 2 J (57)
and C' = C , -/2/a TT (58)
d a
6. The equations of continuity of a tracer, 38, and density
deficiency, 37, may be solved immediately. The continuity of a tracer
is expressed as
b2(2U cos e + u)c = constant (59)
3.
Therefore,
47
-------
r b2(2U cos 6 + u ) . .
c_ = pi a o oi_ = I (6Q)
o b2(2U cos 9 + u) y
a
7. The conservation of density deficiency .ntegrated as
b2(2U cos 6 + u) (p - p) = constant (61)
a a
Hence, values of both c and (p - p) can be determined easily from
cl
the solutions of the terms b2(2U cos 6 + u). This leaves only five
a
unknowns contained in five simultaneous ordinary differential equations.
The initial conditions for £ = 0 are
P(0) = 1
m(0) = 1
o (62)
n(0) = 0
= 0
Abraham's Model for Round Buoyant Jets Issuing
Vertically into a Flowing Stream
Abraham (6) has modified Fan's model by using entrainment coeffi-
cients that are constant, i.e., they are not functions of either F, k, or
8'. For Fan's model the value of the entrainment coefficient and the
o
drag coefficient is a function of the velocity ratio and densimetric
Froude number. Abraham recognizes that this may preclude the use of
Fan's model beyond the range of conditions covered by the experiments.
Basic Assumptions
Abraham considers two regions of the jet. In the vicinity of the
nozzle the trajectory of a round buoyant jet in a cross flow may be
48
-------
described as a vertical line, provided that the cross flow is weak and
that the initial momentum of the jet is strong. The region near the
nozzle may be described as a jet primarily influenced by its initial
momentum, and thus the volumetric flux may be expressed as
- = a 2-nbu (63)
' mom J
where a = coefficient of entrainment for jets primarily influenced
mom j r
by initial momentum
a = 0.057 according to Albertson, et al. (10)
mom & '
The variables u and b are the same as described earlier.
The second region is a great distance downstream from the nozzle
where the velocity of the jet fluid is about equal to the velocity of
the ambient fluid. Here, entrainment may be described by the relation-
ship for a cylindrical thermal in a stagnant fluid as described by
Richards (26). The entrainment satisfies the relationship,
dV
-5-9-= a , 27T B_ (64)
dy' th th
where y' = vertical coordinate indicating position of center of
thermal jet
V = quantity of fluid contained in a thermal jet per unit of
length
B , = radius of thermal defined by Equation 66
th (65)
a = coefficient of entrainment for thermals moving through
th
stagnant ambient fluid [0.5 according to Richards (26)]
The entrainment of a cylindrical thermal may be described by
Equation 64 when the concentration of a tracer carried by the thermal
49
-------
is expressed as
c*(s',r,«fr) = c(s') e "' (66)
This expression is equivalent to the one used by Fan shown in Equation
2. Abraham assumes the same velocity profile as described by Fan in
Equation 24,
_T2/b2
u*(s',r,<|>) = U cos 0 + u(s') e ' (24)
£L
Abraham combines Equations 63 and 64 and describes the volumetric
flux as
r
u*dA = 2irb(a u + a., U sin 0 cos 6) (67)
*—, i u \*rv ~~ f» ii u i \* fc* • **-_i_ **
ds' I. mom th a
J A
r\
where dy1 has been replaced by sin 6 ds1.
The cosine function in Equation 67 was arbitrarily introduced in
the term involving a , to avoid this term contributing to entrainment in
the initial region near the orifice where the jet is primarily a momentum
jet. In addition, tMe prime notation is used to indicate the coordinates
of the jet from the discharge point. Hence, comparing Fan's notation
with Abraham's notation, the relationship for the coordinates is as
follows:
s' = s1 + s
e
x' = x1 + x (68)
y1 = y1 + y
where s', x1, and y1 are the distances from the discharge point to the
end of the zone of flow establishment [Figure (9)].
50
-------
e
jO I^VNN WW>.NNXXN"\\.\.X\xVvV\V
FIGURE 9 - SCHEMATIC RELATIONSHIP BETWEEN INITIAL DISCHARGE POINT AND
END OF ZONE OF FLOW ESTABLISHMENT
Development of Equations
The following equations were developed by Abraham using Equation
67 to describe the volumetric and momentum flux equations . Where the
equations are the same as developed by Fan, they will receive the same
number. The equations are outlined below.
1. Continuity,
[b2(2Ua cos 6 + u)] =
ds
2. x-momentum,
d 1 b2
u
U& sin 6 cos 6) (69)
ds
r MjL (2U cos 6 + u)2 cos 8
L ^ 3-
s\
2bU (a u + a , U sin 9 cos 6) + —— U2 b sin3 6 (70)
a mom th a n a
3. y-momentum - The equation for y-momentum is essentially the
same as developed by Fan as shown by Equation 34 with
51
-------
modification for the buoyancy force in the negative y-direction.
r- (2Ua cos e + u)2 sin e
P - P C,j2
j =
,
-b2g -2 --- _ u2 b sin2 6 cos 6 (34)
pa * a
4. Continuity of tracer,
[b2(2U cos 9 + u)c] = 0 (39)
ds a
5. Density deficiency,
^-r[b2(2U cos 0 + u)(pa - P)] = 0 (37)
ds a a
6. Geometric relationships,
3-T x1 = cos 8 (40)
ds1
3JT /' = sin 9 (41)
It is important to note that the values of x1, y', and s? for
Abraham's equation are measured from the orifice rather than the end of
the zone of flow establishment and are defined as
x1 = horizontal coordinate, measured from orifice
y1 » vertical coordinate, measured from orifice
s1 = distance along jet axis, measured from orifice
Initial Conditions
There are seven unknowns for Abraham's model (u, b, c, p -p, 9,
3.
x1, and yr) with seven equations (69, 70, 34, 39, 37, 40, and 41). The
initial conditions are:
u(0) = UQ
c(0) = c
52
-------
P(0) = PI
b(0) = DQ/2
x'(0) = 0
(71)
y'(0) = 0
6(0) = eo = $•
at s' = 0
where c = initial concentration,
o
u = initial jet discharge velocity,
B' = initial angle of inclination at discharge point, and
D = diameter of orifice.
o
Dimensionless Parameters
Abraham's equations are normalized in the same manner as Fan's
equations with the following exceptions:
Buoyancy parameter:
2U) (72)
f = (k/k')2/2F2 (73)
coordinates :
<; • r ' = — s'
S . (, b
O
x : n' = ^-x' C74)
y : V =f-y'
o
53
-------
Normalization of Abraham's Equation
Abraham's Equations 69, 70, 34, 40, and 41 are normalized using the
dimensionless parameters listed on page 46, resulting in Equations 75-
79.
1. ~- = a Jm -- V— (2 a - or, sin 6)cos 9 (75)
d?' mom , , /— mom th
K'vm
- :rV = — — |2o (— - 2 cos 0) + 2a . sin e cos 9
C '2 ^
— - co a .
VnT k'2 mom y
3 e] (76)
sn
3. jV = -f - - C, — ^ sin2 6 cos e (77)
4. -p-f n1 = cos 9 (78)
5. -4r V = sin 9 (79)
where C, = - (80)
da TT
The density deficiency and continuity of tracer are solved in the
same manner as Fan's equation for density deficiency and continuity of
tracer.
The initial conditions for ?' = 0 are:
v(0) = 1
m(0) = 1
6(0) - g^ (81)
n'(0) = 0
C'(0) = 0
54
-------
Solution of Equations
A system of simultaneous differential equations can be solved using
a Fourth-Order Runge Kutta technique. Such a program is available at
the Vanderbilt University Computer Center in the Scientific Subroutine
library. The listing of the computer program used to solve Fan's model
is included in Appendix B, and the computer program used to solve
Abraham's model is included in Appendix C. The subroutine that contains
the Fourth-Order Runge Kutta solution to solve a system of simultaneous
ordinary differential equations is the same for both Fan's model and
Abraham's model and is included in Appendix D. However, the main
programs, which contain the individual equations for the two different
models, are different.
55
-------
CHAPTER IV
VI
METHODS AND MATERIAL
Introduction
The objective of this research was to measure the vertical profile
of a negatively buoyant jet at points downstream from the jet discharge
point in order to obtain dilution data, jet half-width data, and jet
trajectory data for a negatively buoyant jet. Salt solutions of dif-
ferent concentrations were used to model the jet for two reasons.
First, a salt solution is heavier than water and can easily be used to
model a negatively buoyant jet. Secondly, the vertical profiles of the
jet could be measured very easily by using a conductivity probe with
appropriate monitoring devices.
Conductivity Probe
The conductivity probe was similar to the one used by Krenkel (30)
and Burdick (31) with a few modifications. The electrode consisted of
a 16-gage blunt stainless steel hypodermic needle as the outer electrode
and a length of 0.01 in. diameter platinum wire as the inner electrode.
The platinum wire was insulated from the hypodermic needle by a small
capillary of plastic tubing into which it was threaded. Both ends of
the electrode were sealed with a non-conducting epoxy cement. The
bottom portion of the needle and the exposed portion of the platinum
were then plated with platinum black to provide a large surface area.
56
-------
However, the hypodermic needle was first plated with copper as shown by
Krenkel (32) so that the platinum black would adhere more strongly.
Copper was plated on the stainless steel by using a length of cleaned
copper wire as one electrode and the needle as the other electrode. A
5-volt power supply with variable output was used as the driving force
in the plating operation. A voltage output of 1.5 volts was used. This
circuit was completed by using 1 N copper sulfate as the electrolyte.
The copper wire was helically wound to increase the surface area in the
copper sulfate solution. After the copper plating had been thoroughly
washed, the needle was placed in a 100-milliliter solution containing
3 grains of platinic chloride and 0.02 grams of lead acetate. The needle
leads were then connected to about three volts direct current as shown
by Clements (33). The direction of current was reversed every 15
seconds until a uniform coating was deposited on both electrodes.
(Two or three minutes is suggested.) The electrodes were then rinsed
in running tap water for half an hour and stored in distilled water
when not in use. It was necessary to keep the probe wet at all times
in order to keep the platinum black coating in proper condition.
The probes vere finally covered with a non-conducting rubber coating
to insure insulation and to give the soldered leads more strength.
Detailed views of the probe are shown in Figure 10 and 11.
Conductivity Monitor and Recorder
The conductivity monitor was designed as described by Clements
(31). The basic premise of construction involved using a series
resistance arrangement with an a.c. source having a low internal
57
-------
i.
I1
16
PLASTIC
EPOXY RESIN
PLATINUM
RUBBER COATING
DETAILS OF PLATINIZED TIP
FIGURE 10 - DETAILS OF PROBE CONSTRUCTION
FIGURE 11 - PHOTOGRAPH OF CONDUCTIVITY PROBE
58
-------
impedance as shown in Figure 12. This type of circuit provided an output
CELL
EC
FIGURE 12 - BASIC MEASURING CIRCUIT
[AFTER CLEMENTS (35)]
voltage, E , which was nearly linearly proportional to the conductivity
G, of the test solution, where E. was a constant supply voltage of 5
volts and G was the reciprocal of probe resistance, R . The linearity
may be seen by considering Equation 82.
E.R
If GR " .01, E is directly proportional to G. This type of monitor
was chosen for the following characteristics.
1. Low cost - approximately $25.00 for parts.
2. Drift was low - about 0.12 percent per hour with normal
ventilation.
3. Calibration was linear over a large range.
4. Polarization effects at the conductivity cell electrodes were
negligible due to the 1 kHz excitation frequency.
59
-------
01
vo
IIOv.
LINE
47
.01 k
.01
NE-51
100 ma.
800 p.Lv. 470
22 Iw.
——s/VA/^ J
£
250 v.
20
350v.
,50k
10 w.
S
r
eASII FIL.
STANCOR
PS-8416
.001
7hy.
36k 36 k 50 ma.
2w. 2w.
RESISTANCES IN OHMS
k=IOOO
CAPACITANCES IN MICROFARADS
47k < = = .5
D.C. OUTPUT
I
INI9I A.C. OUTPUT
.001 .001
COND. CELL
68 kS 68k> 68 k,
FIGURE 13 - SCHEMATIC OF CONDUCTIVITY MONITOR [AFTER CLEMENTS (34)]
-------
5. Response was fast, with an average time constant of 0.025
second.
The combination of 1-kHz excitation frequency and the large surface
area exposed by the platinum black made the polarization effects at the
conductivity probe neglible. Figure 13 is a diagrammatic sketch of the
monitor and Figure 14 is a photograph of the conductivity monitor.
FIGURE 14 - CONDUCTIVITY MONITOR
The recorder chosen was an Esterline-Angus Model T171B, Port-A-
Graph recorder. The input resistance for the recorder was 1.8 to 2
mega ohms. The wide range of 2,5,10,20,500,200 MV, 0.5, 1,2,5,10,20,50
V afforded the needed flexibility for measuring various outputs with the
needed sensitivity. Figures 15 and 16 show the Esterline-Angus Recorder
and the combination of conductivity monitor and recorder, respectively.
60
-------
FIGURE 15 - ESTERLINE-ANGUS RECORDER
FIGURE 16 - COMBINATION OF CONDUCTIVITY MONITOR AND RECORDER
61
-------
Injection and Flow Control
A commercial grade salt was used as the trac material. The salt
was well-mixed using a 1500-gpm stirring apparatus in a 50-gal poly-
ethylene barrel.
The barrel was equipped with internal helically wound copper coils
to keep the tracer material at approximately the same temperature as the
ambient fluid (Figure 17). This was accomplished by passing water from
the same source as the ambient fluid through approximately 60 feet of
copper tubing. This approach worked very well as may be seen in Table
2.
FIGURE 17 - POLYETHYLENE BARREL WITH FLOW AND
TEMPERATURE CONTROL
62
-------
TABLE 2 - COMPARISON OF JET TEMPERATURE WITH
AMBIENT FLUID TEMPERATURE
Run No.
9
10
11
12
13
15
16
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
Temperature
Ambient (°F)
64.6
66.3
66.3
66.8
67.6
68.7
68.5
70.5
73.3
73.3
73.2
73.2
75.5
76.0
76.0
72.2
76.2
76.2
77.6
77.6
77.6
76.1
76.1
75.5
Temperature
Jet (°F)
65.4
67.5
67.5
67.6
68.5
69.0
69.1
71.0
73.7
73.7
73.4
74.4
75.6
76.0
76.0
72.2
76.4
76.4
77.8
77.8
77.2
75.9
76.6
75.9
63
-------
A student's t-test for the comparison of the means shows that there
is no significant difference in the two populatic--•-. Hence, the conclu-
sion can be drawn that the data obtained came from the same population.
Figure 18 shows how temperature control was obtained.
FIGURE 18 - JET TEMPERATURE CONTROL
The salt solution was then pumped through a PVC piping system by a
12-gpm pump through a 0.5-gpm Fisher-Price rotameter to measure the flow
rate. The excess was recirculated back into the mixing chamber. The
rotameter calibration curve is presented in Appendix E. The desired
flow-rates were obtained by adjustment of valves located in the piping
network. The piping network was then connected to the injection port
located in the bottom of the laboratory flume.
64
-------
Laboratory Flume
The flume used is located in the hydraulics laboratory of the
Department of Environmental and Water Resources Engineering at Vanderbilt
University. The flume is 60 feet long, 1.0 foot deep, and 2.0 feet wide.
The channel walls consist of glass panels, and the steel bottom was
painted with an epoxy-based paint to prevent chemical corrosion. The
entire flume system is supported at two points, one of which has a
mechanical screw jack mechanism for changing the channel slope. However,
all experiments were made with a horizontal slope. A variable-speed
recirculating pump is located at the lower end of the flume by which
water may be recirculated if it is necessary. At times it was necessary
to recirculate to obtain the desired ambient flow rate in the flume at
the required depth. The maximum flow rate with recirculation was 0.48
cfs. The maximum flow rate available for a single pass through the
flume was 0.15 cfs. The flow rate was measured by a 60° V-notch weir
installed at the upstream end of the flume. A calibration curve of
flows over the weir was used which appears in Appendix F and is very
close to the theoretical equation of
2.5
0 = 1.4076 H
xa w
where Q = flow, cfs
H = head of water above apex of notch, feet
w
A bank of ripple siding was installed near the upstream end to
aid in straightening the flow. A point gauge mounted on a traversing
mechanism was used to measure the flume depth and local positions of
the jet with an accuracy of 0.001 foot. The required depth of flow was
65
-------
controlled by means of a perforated baffle installed at the downstream
end of the flume. A diagrammatic sketch of the flume and appurtenances
may be seen in Figure 19.
60 FEET
A. PROPELLER PUMP
B. PERFORATED BAFFLE
C. INJECTION PORT
D. PIVOT FOR SLOPE
ADJUSTMENT
E. POINT GAGE
F. BAFFLE
G. SHARP-CRESTED
WEIR
FIGURE 19 - PARTIAL CUTAWAY VIEW OF RECIRCULATING FLUME
The injection of the salt solution was obtained either by using a
length of standard 3/8-inch copper tubing as the port or injecting
vertically from a port in the bottom of the flume. For the 90° jet,
the salt water was injected vertically from the bottom of the flume using
a previously installed withdrawal tap. The diameter of the jet at this
point is 0.80 centimeter. The 45° jet was a standard 45°-3/8-inch copper
ell inserted into the withdrawal port. This essentially raised the
elevation of the jet 0.106 feet above the bottom of the flume.
66
-------
The diameter of the 45° jet was 0.95 cm. The 60° jet was constructed
by bending a length of 3/8-inch copper tubing to an angle of 60° from
the horizontal. The jet was then inserted into the withdrawal port,
thus raising the elevation of the jet 0.081 feet above the bottom of the
flume. The diameter of the 60° jet was 0.72 cm. The location of the
withdrawal tap is in the center of the flume 1.0 feet from each wall.
Procedure for Obtaining Correct Salt Concentration
The limiting factors for determining the various parameters during
the experiments were the ambient flow rate and the depth of the water
in the flume. The usual depth of water in the flume was between 10 and
12 inches. A depth of 10 inches and a maximum flow rate for a single
pass of 0.15 cfs was used for preliminary calculations. This set the
preliminary ambient velocity. Then, using the required velocity ratio,
the preliminary jet velocity was determined. Knowing the required jet
velocity and densimetric Froude number, the preliminary density differ-
ence was calculated for a particular jet diameter. The salt was added
to water and sufficient time was allowed for complete mixing. The
density of the solution was then determined using previously weighed
specific gravity bottles. After the density difference was accurately
determined, the jet velocity was again determined using the required
densimetric Froude number. Then, using the required velocity ratio,
the ambient velocity was calculated. Using the maximum flow rate of
0.15 cfs, the depth of flow in the flume was calculated.
If recirculation was required to obtain the desired combination of
parameters, a salt solution was initially mixed and then the density
67
-------
difference was determined as for a single pass. Then, the required jet
velocity and ambient velocity were calculated. Using a depth of 12
inches, the ambient flow rate was calculated. The required flow rate
was then obtained using the recirculating pump.
68
-------
CHAPTER V
VII
ANALYSIS OF DATA AND PRESENTATION OF RESULTS
The objective of this research was to ascertain the applicability
of two different jet models which describe the dilution, jet half-width,
and jet trajectory of positively buoyant jets in flowing streams to the
case of a negatively buoyant jet in a flowing stream. The specific
objectives are outlined below:
1. Determine if Fan's model could be used to predict the dilution,
jet half-width, and jet trajectory for a negatively buoyant
jet.
2. Determine if Abraham's model could be used to predict the
dilution, jet half-width, and jet trajectory of a negatively
buoyant jet.
3. Determine the range of values of the coefficient of entrain-
ment, a, and drag coefficient, C^.
4. Seek the functional relationship of a and Cd as a function of
the velocity ratio, k, and the densimetric Froude number, F,
and the initial angle of discharge, p^.
5. If both models can be used, determine the best model.
A chronological schedule for obtaining and evaluating the data is
outlined below:
1. Determine the actual conditions for a particular laboratory
investigation.
a. jet velocity
69
-------
b. ambient velocity
c. density of salt water
2. Obtain probe calibration curve of millivolt output versus
concentration of salt in jet.
3. Obtain concentration profile data measured in millivolt output
from laboratory investigation.
4. Reduce millivolt output data describing the profile to actual
concentration data using the calibration curve found in step 2
above.
5. Determine the centerline concentration, the vertical position,
y, and the standard deviation, o, , for a particular profile
which is located at a known distance downstream, x, from the
discharge point.
6. Measure the length of the zone of flow establishment along the
axis of the jet trajectory using Equation 4.
7. Determine 6 from plotted jet trajectory data obtained using
the profile data.
8. Obtain theoretical dilution, jet half-width, and jet trajectory
for initial guesses of a and C,.
9. Fit theoretical data to experimental data to obtain the best
values of a and C,.
A laboratory investigation was used to answer the specific objec-
tives of this research. Experiments were conducted for various combina-
tions of k, F, and 6 , with values of k ranging from 5 to 20, F ranging
from 10 to 40, and initial angles of discharge of 90°, 60°, and 45°. A
laboratory run consisted of an experimental investigation of one
70
-------
particular combination of the above parameters, e.g., k ~ 5, F ~ 10, and
g1 = 90°. The exact combinations are found in Table 3 for Fan's defini-
o
tions and in Table 4 for Abraham's definitions. The laboratory conditions
were set up as discussed in Chapter IV.
Procedure for Obtaining Salt Concentration Profiles
For each laboratory run, a calibration curve of millivolts output
versus salt concentration was made for various dilutions. Hence, there
was no need for a temperature correction, since the probe was calibrated
each time a run was made. Concentration profile data was initially
taken starting at the downstream end of the jet. This procedure was
followed so that any disturbance created by the probe would be felt only
downstream of the profile under investigation. Figure 20 is a typical
example of the data taken over one cross-section.
The strip charts of millivolt output at each sampling point were
analyzed for each individual run. Values of the average millivolt output
at a point in a cross-section for various time increments, At, were
estimated by eye from the trace on the strip chart. These were then
averaged according to Equation 83 to obtain the time-averaged millivolt
output for that point .
iV-i.j.k""!'
• ' ^— - t83)
I At.
where m. , = time-averaged millivolt output at location j in cross-
J »k
section, k
m. . , = millivolt output at location (j,k) over time increment
71
-------
S'/D
x'/D
c/c
3.54
2.00
0.48
0.55
i >
FIGURE 20 - CONCENTRATION PROFILE AT sVDQ = 3.54 FOR RUN NO. 34
-------
TABLE 3 - COMBINATION OF DENSIMETRIC FROUDE NUMBER,
VELOCITY RATIO, AND INITIAL ANGLE OF DISCHARGE
ACCORDING TO FAN'S DEFINITIONS
Run
No.
9
10
11
12
13
15
16
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
Froude
No., F
10.3
10.9
20.4
21.1
40.7
10.1
20.2
44.2
11.7
21.6
23.5
43.2
10.9
20.8
10.4
46.9
11.3
21.3
10.6
20.6
10.3
22.5
42.7
45.4
Velocity
Ratio
k = U /U
o a
10.3
5.5
10.2
5.3
10.2
20.1
20.2
5.5
5.9
10.8
5.9
10.8
10.9
20.8
20.9
5.9
5.7
10.6
10.6
20.6
20.6
5.6
10.6
5.7
Initial Angle
of
Discharge
*o
90
90
90
90
90
90
90
90
45
45
45
45
45
45
45
45
60
60
60
60
60
60
60
60
Angle at End of
Zone of Flow
Establishment
3o
74
62.5
78
74.8
76.3
81.8
78.3
58.3
31.8
36.5
28.0
36.3
28.0
34.5
29.3
30.0
48.8
51.3
52.0
53.0
53.8
52.0
53.8
47.8
Reynolds
No.
R
e
1650
1190
2200
1050
2030
1110
2240
4700
1730
3200
1600
3140
2540
4980
1590
4580
1180
2220
1870
3636
1035
1080
2040
2940
73
-------
TABLE 4 - COMBINATION OF DENSIMETRIC FROUDE NUMBER,
VELOCITY RATIO, AND INITIAL ANGLE OF DISCHARGE
ACCORDING TO ABRAHAM'S DEFINITIONS
Run No.
9
10
11
12
13
15
16
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
Froude No.
F
10.0
10.0
20.0
20.0
40.0
10.0
20.0
40.0
11.4
21.4
22.8
42.8
10.7
20.7
10.4
45.7
11.0
21.0
10.5
20.5
10.3
22.0
42.0
44.0
Velocity
Ratio
k - U /U
o a
10.0
5.0
10.0
5.0
10.0
20.0
20.0
5.0
5.7
10.7
5.7
10.7
10.7
20.7
20.7
5.7
5.5
10.5
10.5
20.5
20.5
5.5
10.5
5.5
Initial Angle
of
Discharge
B;
90
90
90
90
90
90
90
90
45
45
45
45
45
45
45
45
60
60
60
60
60
60
60
60
Reynolds
Number
R
e
1580
1090
2180
1900
2000
1110
2220
4250
1700
3170
1555
3110
2440
4930
1580
4320
1150
2200
1840
3610
1000
1060
2030
2850
74
-------
i = index number of time period
j = index number denoting a given location in a given cross-
section, k
k - index number denoting a given cross-section
n = number of time increments used
At. = time period used in time-averaging
The time-averaged millivolt outputs were then used as input data
for a computer program called Analysis. A listing of the computer
program, Analysis, is included in Appendix G. This program used the
calibration curve to convert the time-averaged millivolt output, uij^*
to a time-averaged concentration, c. , . Knowing c. ,, which was the
j ,K j j^
concentration at location (j,k), yR, the y-location of the mean value
of the concentration profile, ck, the mean values of the concentration
profile, and the standard deviation, a^, were calculated using
Equations 84, 85, and 86, respectively.
N
, ,
y ,
y
k
k
'
(84)
c =
k
N
(85)
75
-------
2
N
f r • v1
j,k yj,k
(y- J2
E c. k
ok-|/ l±L_i! (86)
E c. .
*,i J'k
where N = number of j location in a given cross-section, k
y! . = the vertical distance from the discharge point
J >K
CI = Ay, the incremental distance between each y. ,
It is important to note that c. , and c, are equivalent to
J »K K
c*(s,r,) and c(s), respectively, where c*(s,r,4>) is a local con-
centration value for any given y-distance from the center line and
for any given cross -sect ion, k, and c(s) is the center line con-
centration for any given cross-section, k. This is equivalent to
the nomenclature used in Equation 2 for Fan's model and Equation
66 for Abraham's model. The values of c,, y/, and a, are then
used as input to the computer programs for comparison with the
theoretical predictions. Using Equation 24, the relationship between
o, and b, the jet half -width, is defined as
b = /2 ak (87)
76
-------
Hence, the centerline concentration, c, , was calculated. For
each c,, there was both an x'-coordinate and a y'-coordinate. The
x'-coordinate represented the horizontal distance from the discharge
point, whereas the y'-coordinate was the vertical location of the
centerline concentration measured from a horizontal plane passed through
the discharge point. Figure 21 gives a more explicit description of
the location of a jet's trajectory.
FIGURE 21 - REPRESENTATIVE PROFILE VIEW OF A JET'S TRAJECTORY
Jet Trajectories
Using the x'- and y'-coordinates, the jet trajectory of each run
was plotted. Twenty-four runs were made. Table 3 gives the various
combinations of densimetric Froude number, velocity ratio, and initial
angle of discharge according to Fan's definitions. The angle at the
end of the zone of flow establishment and the jet Reynolds number are
also included, where the jet Reynolds number is defined as
77
-------
U D
R = -2—2. (88)
e v
where R = jet Reynolds number
C
v = kinematic viscosity of salt water
Table 4 gives the various combinations of densimetric Froude number,
velocity ratio, and initial angle of discharge according to Abraham's
definitions. Table 4 also includes the jet Reynolds number.
The difference between the two tables arises from the different
definitions of the initial jet velocity. Fan defines UQ, the initial
jet velocity at the end of the zone of flow establishment, as
U = u -i- U cos 0 (89)
o o a o
whereas Abraham defines U as
U = u + U cos B1 (90)
o o a o
For both Fan's model and Abraham's model, the value of UQ is the same,
but the addition of the vector component of the ambient velocity with
two different initial angles give rise to two values for the component
vector. The process of obtaining the value of BQ will be discussed in
detail later, whereas g' is simply the horizontal angle that the discharge
orifice makes with the ambient flow. By reviewing the initial conditions,
Equations 42 and 52, this difference in the initial angle is illustrated.
Thus, using Equation 4, and knowing the value of the velocity ratio,
k, the length of the zone of flow establishment, s^, can be determined.
This length is then measured along the jet axis. Once the end of the
zone of flow establishment is determined, the x^ and y^ coordinates are
determined. This is only necessary for the use of Fan's model, since
78
-------
Abraham does not consider the zone of flow establishment. Once the end
of the zone of flow establishment is determined, the initial angle of
inclination, B , of the jet is measured. Hence, all of the unknown
variables are known and the theoretical solution using Fan's and
Abraham's model can be sought.
Only six runs will be discussed in the main text. However, the
plots of the dilution, jet half-width, and trajectory for all 24 runs
and for both models can be found in Appendix H. Tables 5 and 6 give
the pertinent location, dilution, and jet half-width data for the six
combinations discussed using Fan's model and Abraham's model, respectively,
Fitting of Data to Theoretical Curves
Trial computer solutions were made for the fitting of the data to
Fan's model. Initial guesses of a and C, were made based on experience.
However, it was soon found that, for any value of Cd, the jet trajectory
was under-predicted. The theoretical considerations concerning the
behavior of C. in the development of a positively buoyant jet apparently
d
are not applicable when applied to the case of a negatively buoyant jet.
The buoyancy forces that tend to retard the vertical momentum are
apparently greater than the drag forces. Hence, in an effort to force
the model to fit the data, the value of Cd = 0.00 was used for all runs.
Hence, only an initial guess for a was needed.
Several trial computer solutions were made using different values
of a. Theoretical values of the jet dilution and jet half-width were
plotted on the same graph as the experimental data. Experimental values
of jet dilution and jet half-width were both used to select an appro-
priate value of a which predicted the best fit. In most cases, a best
79
-------
TABLE 5 - SUMMARY OF NEGATIVELY BUOYANT JET EXPERIMENTS
IN A CROSS-FLOW FOR FAN'S MODEL
Run
No.
13
10
33
8
po
90°
90°
60°
F
40.7
10.9
42.7
k
10.2
5.5
10.6
gi
po
76.3
62.5
53.8
s1
e
D
4.45
3.19
4.45
X1
e
D
0.75
1.10
2.43
"i
D
4.38
3.07
3.75
s
D
9.75
15.75
21.20
31.56
56.81
112.50
156.94
207.02
5.81
11.38
16.39
21.52
36.42
41.62
51.69
76.88
4.06
10.42
20.90
31.22
41.39
51.39
101.53
201.58
X
D
4.25
9.25
14.25
24.25
49.25
104.88
149.25
199.25
3.90
8.90
13.90
18.90
33.90
38.90
48.90
73.90
2.57
7.57
17.57
27.57
37.57
47.57
97.57
197.57
£
D
8.71
12.05
14.20
16.72
19.45
19.86
19.12
18.09
4.28
6.60
7.52
5.98
5.90
4.56
3.84
1.96
3.14
6.97
10.01
11.94
13.14
12.33
14.83
11.57
c
o
c
9.13
14.60
17.80
30.97
36.86
62.34
87.71
87.04
9.99
11.59
14.33
17.56
20.25
35.37
36.74
40.70
4.01
7.43
16.76
25.46
31.97
35.13
60.06
77.11
b_
D
o
4.47
4.15
5.19
6.42
6.81
9.61
11.03
11.68
2.14
3.24
3.56
2.95
4.97
5.35
5.06
4.11
2.20
3.58
4.91
4.96
6.44
7.23
9.12
13.50
00
o
-------
TABLE 5 - Continued
Run
No.
27
22
19
6°
60°
45°
45°
F
11.3
43.2
11.7
5.7
10.8
5.9
Bl
0
48.8
36.3
31.8
s;
D
3.19
4.45
3.19
x'
e
D
1.94
3.45
2.68
yl
~D
2.57
2.93
1.81
s
D"
3.97
9.44
19.58
29.75
39.86
50.00
75.49
2.00
7.68
18.26
28.58
38.84
48.86
99.10
2.58
8.00
18.05
38.10
48.21
68.36
x
D"
3.06
8.06
18.06
28.06
38.06
48.06
73.06
1.54
6.55
16.55
26.55
36.55
46.55
96.55
2.32
7.32
17.32
27.32
47.32
67.32
y
D
2.43
4.75
5.93
7.04
6.00
4.78
0.00
1.18
3.95
7.54
10.08
11.99
13.41
17.52
1.18
3.06
4.08
4.15
3.69
0.96
c
o
c
4.34
7.44
11.65
18.62
21.28
23.66
45.05
2.56
4.11
7.44
12.25
14.77
19.58
29.82
3.67
5.56
10.12
17.82
25.08
31.70
b
D
o
1.68
2.61
3.24
3.90
4.48
4.50
5.26
1.74
2.57
3.39
4.52
5.18
5.63
6.62
1.72
2.20
3.15
4.30
5.40
5.42
00
-------
TABLE 6 - SUMMARY OF NEGATIVELY BUOYANT JET EXPERIMENTS
IN A CROSS-FLOW FOR ABRAHAM'S MODEL
00
NJ
Run
No.
13
33
22
B
po
90°
60°
45°
F
40
42
42.8
k
10
10.5
10.7
sf
D
14.20
20.20
25.65
36.01
61.26
116.95
161.39
211.48
8.50
14.86
25.35
35.67
45.83
55.83
105.97
206.03
6.45
12.14
22.72
33.03
43.29
53.32
103.56
xf
D
5.00
10.00
15.00
25.00
50.00
105.62
150.00
200.00
5.00
10.00
20.00
30.00
40.00
50.00
100.00
200.00
4.99
10.00
20.00
30.00
40.00
50.00
100.00
zl
D
13.15
16.49
18.64
21.16
23.89
24.30
23.56
22.52
6.89
10.72
13.76
15.69
16.89
16.08
18.58
15.32
4.10
6.87
10.46
13.01
14.92
16.34
20.45
c
o
c
9.13
14.60
17.86
30.97
36.86
62.54
87.71
87.04
4.01
7.03
16.76
25.46
31.97
35.13
60.06
77.11
2.56
4.11
7.44
12.25
14.77
19.58
29.82
b
D
o
4.47
5.15
5.19
6.42
6.81
9.61
11.04
11.63
2.20
3.58
4.91
4.96
6.44
7.23
9.13
13.50
1.70
2.57
3.39
4.56
5.18
5.63
6.63
-------
TABLE 6 - Continued
00
Run
No.
10
27
19
B°
90°
60°
45°
10
11
11.4
5
5.5
5.7
s'
D
9.00
14.56
19.58
24.71
39.61
44.81
54.88
80.06
7.17
12.64
22.78
32.94
43.06
53.19
78.68
5.77
11.19
21.24
41.29
51.40
71.55
X1
D
5.00
10.00
15.00
20.00
35.00
40.00
50.00
75.00
5.00
10.00
20.00
30.00
40.00
50.00
75.00
5.00
10.00
20.00
40.00
50.00
70.00
y1
~D
7.34
9.66
10.59
9.04
8.96
7.62
6.90
5.01
5.00
7.32
8.50
9.61
8.97
7.35
2.57
2.99
4.87
5.89
6.06
5.50
2.77
c
o
c
9.99
11.59
14.35
17.56
20.25
35.37
36.74
40.70
4.34
7.44
11.65
18.62
21.28
23.68
45.05
3.67
5.36
10.12
17.82
25.08
31.70
b
D
0
2.15
3.24
3.36
2.95
4.97
5.35
5.05
4.11
1.68
2.61
3.24
3.90
4.48
4.50
5.26
1.72
2.20
3.15
4.30
5.40
4.42
-------
fit curve pertaining to a particular value of a was obvious. However,
in several cases two or more values of a could fit the data, i.e., one
value of a would cause the theoretical value of jet dilution to best
fit the experimental data of dilution, and a different value of a would
cause the theoretical value of jet half-width to best fit the experi-
mental data of jet half-width. Since the equations of jet dilution and
jet half-width are coupled, a technique using the squared difference
between the theoretical and experimental data was used to select the
best fit value of a. The squared difference between a theoretical
point and an experimental point was calculated both for the dilution
and jet half-width. Then all the squared differences for each value of
a were summed. The value of a giving the smallest value of squared
difference was used.
Figures 22, 23, 24, 25, 26, and 27 show the theoretical fit of
Fan's model and the experimental results for six of the twenty-four
combinations of k, F, and 3' in question. Lines on the figures represent
the theoretical values of jet dilution, jet half-width, and jet
trajectory. A circle, o, represents an experimental value of dilution,
and a triangle, A, represents an experimental value of jet half-width
at a location s/D on the jet axis. On the trajectory curve, a circle,
o, represents an experimental location (x/D, y/D) of its respective
centerline dilution.
The fitting of the theoretical curves to experimental data for
Abraham's model was much easier than the fitting of the data to Fan's
model, since the entrainment coefficient for Abraham's model is
assumed constant, i.e., it is not a function of k, F, and B'. Again,
84
-------
60
40-
I 1 T
20
-o
J L
J , L
0
IOO
40
80 120
x/D
160
200
KXX>
s/D
FIGURE 22 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 13
85
-------
30
4O
50 60
x/D
70 80 90 100
100
IO
F * 10.9
K =5.5
is90°
* = 62.5°
a" « 0.7
. DILUTION
b/D
' '
' 11
-\—i—i i 111
Oo
X
i I i i i i
i i i I I i I I
10
s/D
100
IOOO
FIGURE 23 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 10
86
-------
60
F «42.7
- K « 10.6
s/D
1000
FIGURE 24 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 33
87
-------
30
60 70 80 90
K> 20 30 40
100
lOOc
1000
FIGURE 25 - OBSERVED VALUES AND THEORETICAL CURVE
PREDICTED BY FAN'S MODEL - RUN NO. 27
88
-------
30-
0 K> 20 3O 40 50 60 70 80 90 100
1000
s/D
FIGURE 26 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 22
89
-------
30-
20
K)
10 20 30 4O 50 60 70 80 9O 100
100
£0«3I.8°
a « 0.3
8/0
1000
FIGURE 27 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 19
90
-------
the value of the drag coefficient, C,, was assumed equal to zero. For
each run, only the values of the densimetric Froude number and velocity
ratio are needed to obtain the theoretical results as outlined in Table
4.
Figures 28, 29, 30, 31, 32, and 33 show the theoretical values from
Abraham's model and the experimental results for the six combinations
under consideration. In addition, lines on the figures represent the
theoretical values of jet dilution, jet half-width, and jet trajectory
predicted using Abraham's model. A circle, o, represents an experi-
mental value of dilution, and a triangle, A, represents an experimental
value of jet half-width at some location s'/D on the jet axis. On the
trajectory curve, a circle, o, represents an experimental location
(x'/D, y'/D) of its respective centerline dilution. In addition,
Figures 34, 35, 36, 37, 38, and 39 are photographs of the jet in actual
laboratory conditions.
Discussion of Results - Fan's Model
For all angles of discharge (90°, 60°, and 45°), the best fit
occurred at values of high densimetric Froude numbers and low velocity
ratios, e.g., F = 46.9, k = 5.9, and B^ = 90°. For the sake of dis-
cussion and comparison of the "goodness-of-fit" of the data to the
theoretical model for the different combinations of F, k, and B^, the
value of U will be considered constant in determining the values of F
o
and k as defined by Equation 9 and 11, respectively. At a high
densimetric Froude number, e.g., F = 46.9, the difference between the
initial density of the jet and the ambient density is very small.
91
-------
60
40
§
20
4
8C
120
160
200
100
1000
s'/D
FIGURE 28 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 13
92
-------
30
10 20 30 40 50 60 70 80 90 100
F =10.0
K =5.0
DILUTION
100
1000
S'/D
FIGURE 29 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 10
93
-------
60
40
§
20
40^ ^0 120 160 20O
X'/D
s'/D
100
1000
FIGURE 30 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 33
94
-------
15
10
10
X-/D
40
50
IOOO
FIGURE 31 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 27
95
-------
30
20
I
•»
s
10
10 20 30 40 50 60 70 80 9O 100
x'/D
100
1000
S'/D
FIGURE 32 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 22
96
-------
30
I
20
10 20 30 40 50 60 70 80 90 100
lOOr
F =11.4
ILK =5.7
IOO
1000
s'/D
FIGURE 33 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 19
97
-------
FIGURE 34 - PHOTOGRAPH OF NEGATIVELY BUOYANT JET
FOR RUN NO. 13, F * 40, k = 10, B1 = 90°
98
-------
FIGURE 35 - PHOTOGRAPH OF NEGATIVELY BUOYANT JET
FOR RUN NO. 10, F * 10, k * 5, B = 90°
99
-------
FIGURE 36 - PHOTOGRAPH OF NEGATIVELY BUOYANT JET
RUN NO. 33, F = 40, k * 10, 6 = 60°
100
-------
4IO 31
I/, e
FIGURE 37 - PHOTOGRAPH OF NEGATIVELY BUOYANT JET
FOR RUN NO. 27, F * 10, k ~ 5, 3' = 60°
101
-------
I
FIGURE 38 - PHOTOGRAPH OF NEGATIVELY BUOYANT JET
FOR RUN NO. 22, F * 40, k = 10, 6' = 45°
102
-------
FIGURE 39 - PHOTOGRAPH OF NEGATIVELY BUOYANT JET
FOR RUN NO. 19, F = 10, k - 5, 3 = 45°
103
-------
Consequently, the effect of the negatively buoyant force which tends
to retard the vertical momentum is diminished. The relationship of
the buoyancy parameter, f, in Equation 54 as a function of the
densimetric Froude number is defined in Equation 50. In addition to
the small density difference for high densimetric Froude numbers, at
low velocity ratios, i.e., k = 5.9, the strength or relative velocity
of the ambient fluid is greater. Considering the same value of U , at
low velocity ratios the ambient velocity, U , is greater than at
3.
higher velocity ratios. Hence, the component of the jet velocity due
to the ambient velocity will be a larger percentage of the jet velocity.
Also, the normal component of the ambient velocity, U sin 6, will be
cl
larger. Referring to Equation 6 and substituting Equations 7 and 8,
Equation 91 can be developed.
^= 2irab(U2 sin2 6 + u2)0'5 (91)
Q-S <£
Hence, the rate of dilution of the pollutant, salt in this case, will
be greater than at higher velocity ratios. This can be envisioned as
a larger normal component of the ambient velocity acting to entrain
more of the ambient fluid into the jet to increase its dilution, thus
reducing the effect of the negatively buoyant force which is acting to
retard the vertical momentum of the jet. This larger normal component
will necessarily increase the vorticity in the region of the jet, as
outlined by Flatten and Keffer (38); thus, it will increase the rate of
dilution or mixing. The cross-flow component, U sin 8, decreases
3.
slowly relative to the growth in cross-sectional area. Hence, the
cross-flow shearing force acts over an increasing jet circumference and
104
-------
the rate of change of vortex inflow velocity would increase with
distance as discussed by Flatten and Keffer (38).
Also, the case of a high densimetric Froude number and low
velocity ratio more nearly approximates Fan's basic assumptions as
outlined in Chapter II. The radius of curvature for this combination
of densimetric Froude number and velocity ratio is large. Hence, there
will be no overlapping or interacting of the concentration profiles.
In addition, the cases of low densimetric Froude number, i.e.,
the density difference between the initial jet density and ambient
density is large, and low velocity ratios, e.g., F = 10.9, k = 5.5,
$' = 90°, can be fitted using Fan's model. However, the fit is not as
good as for the previously discussed case of a high densimetric Froude
number and low velocity ratio. Yet the ability to fit the data with
the model for low densimetric Froude numbers and low velocity ratios
is attributable to the greater relative strength of the ambient velocity
where the normal component of the ambient velocity is large. Hence,
the dilution of the pollutant occurs at a faster rate for the same
reasons advanced earlier. Consequently, the effect of the negatively
buoyant force is diminished.
Yet for the cases of low densimetric Froude numbers and high
velocity ratios, the ability to predict the jet dilution, jet half-
width, and jet trajectory is not very good. The cases of a low
densimetric Froude number are the ones with a larger density difference.
Hence, the negatively buoyant force tends to retard the vertical
momentum and the advancement of the y-component of the jet axis to a
greater degree than in the case of a high densimetric Froude number.
105
-------
Once more, for high velocity ratios, the normal component of ambient
velocity will be small compared to the jet velocity. Hence, the dilu-
tion capacity of the ambient stream is lessened. Consequently, the
effect of the negatively buoyant force is not readily diminished.
Also, the radius of curvature for this case is very small with the
result being an overlapping and interacting of profiles which will
inhibit the ability to adequately describe the concentration profiles.
Discussion of Results - Abraham's Model
The fitting of Abraham's model to the experimental results was much
easier than for Fan's model, for reasons outlined earlier. The same
type of situation was found fox Abraham's model as for Fan's model. The
cases of high densimetric Froude number and low velocity ratios, e.g.,
F = 40.0, k = 10.0, and &' = 90°, were fitted with the best results.
All cases of high densimetric Froude number would fit the experimental
data for all angles of discharge. This is due, in part, to the small
density differences. The radius of curvature is very large, hence the
dilution due to the action of the cylindrical thermal is more closely
approximated. This is true for this case because the jet axis is nearly
parallel to the direction of the ambient velocity. In addition, for low
velocity ratios, the strength of the ambient velocity is greater than at
high velocity ratios. Actually, for a high densimetric Froude number,
the jet may be treated as a simple jet consisting of only momentum forces,
The first term on the right-hand side of Equation 70 accounts for the
region of the jet which is affected by its initial momentum.
In the cases of low densimetric Froude numbers and low velocity
106
-------
ratios, e.g., F = 10.0, k = 5.0, and g' = 90°, Abraham's model begins to
deviate from the experimental data after the jet reaches its maximum
height. Apparently, in the initial reaches where the jet behaves like
a momentum jet, the model will predict the jet dilution and jet tra-
jectory satisfactorily. However, after the jet reaches its maximum
height and is deflected towards the discharge level, the predicted data
begin to deviate from the experimental data. Similarly, as discussed
above, the assumption was made that the direction of flow is parallel
to the jet axis at distances far downstream. This assumption is violated.
However, at a distance far downstream, the confidence with which the
centerline dilutions are measured and reported is open to question, due
to techniques of measuring the concentration profiles. In addition, the
jet dilution is under-predicted. As a result, the model will under-
predict the jet trajectory due to the larger negatively buoyant force
which will hinder the jet from increasing its y-component.
In the cases of low densimetric Froude numbers and high velocity
ratios, e.g., F = 10.0, k = 20.0, and 6^ = 90°, the fit of Abraham's
model to the experimental data is poor. Due to the large density
difference, the vertical momentum is retarded to such an extent that the
jet trajectory is grossly under-predicted. The assumption that the jet
centerline is parallel to the direction of flow at distances far down-
stream is flagrantly violated as indicated in the example cited.
Hence, when the density differences are small and the strength of
the ambient velocity is large with respect to the jet velocity,
Abraham's model can be used to predict the jet dilution, jet half-width,
107
-------
and jet trajectory. For the cases of a low densimetric Froude number
and high velocity ratio, Abraham's model should not be used.
108
-------
CHAPTER VI
VIII
SUMMARY AND CONCLUSIONS
Introduction
Two existing jet models have been analyzed to determine the
feasibility of their use to predict the dilution, jet half-width, and
jet trajectory of a negatively buoyant jet. An equation has been
developed to predict the coefficient of entrainment, a, used in Fan's
model as a function of the jet densimetric Froude number, velocity
ratio, and initial angle of discharge. The limits of application,
sources of errors, and recommendations for further research are found
in the discussion that follows.
Fan's Model
Fan's model (3) was developed to predict the dilution and buoyant
jet trajectory in a cross-stream with a uniform velocity. An integral
approach was used to develop equations of (a) continuity, (b) momentum,
(c) conservation of tracer and density deficiency, and (d) two geometric
equations. The same approach could be modified to predict the behavior
of a negatively buoyant jet in a cross-stream. The main difference
would be in the derivation of the y-momentum equation where the direction
of the buoyancy term is simply reversed (i.e., given a negative sign) in
order for the analogy of a negatively buoyant jet to be appropriate.
Twenty-four experiments were completed in the laboratory flume at
Vanderbilt University. Eight combinations of densimetric Froude number
109
-------
and velocity ratio were utilized for each angle of discharge, i.e., 90°,
60°, and 45°. The data was analyzed using the seven equations
developed in the integral approach. Values wer- ruined for a, the
coefficient of entrainment, and C,, the reduced drag coefficient, by
trial and error fitting of the experimental data to the theoretical
solutions. The value of the drag coefficient, C,, was assumed to be
zero because, for any value of C,, the jet trajectory was under-pre-
dicted. Fan (3) and Motz and Benedict (25) stated that the value of
C, was predominately used in determining the best fit for the jet
trajectory and played an insignificant role in its effect on the dilu-
tion of the salt water tracer along the jet axis. Hence, only values
of a were obtained from the fitting of the experimental dilution and
jet half-width data along the jet axis to theoretical dilution and jet
half-width data.
In all cases (90°, 60°, and 45°), the best fit of experimental
and predicted data occurred at values of high densimetric Froude numbers
and low velocity ratios. At high densimetric Froude numbers, the effect
of the density difference was small. In addition, at low velocity
ratios, the momentum and velocity of the ambient fluid is greater.
Hence, dilution of the salt water is attained at a faster rate. Con-
sequently, the sinking of the jet after attaining its maximum height is
much less for jets with low velocity ratios compared to jets with a
high velocity ratio. In addition, for jets with small densimetric
Froude numbers (large density differences) and small velocity ratios,
the prediction for the dilution and the jet trajectory was also better
than for the same densimetric Froude number and high velocity ratios.
110
-------
This is due to the higher relative effect of the ambient velocity.
However, due to the higher density differences for low values of den-
simetric Froude number, sinking of the jet after the jet reaches its
maximum height is more pronounced than for jets investigated with high
values of densimetric Froude number. For jets with small densimetric
Froude numbers, there is an inflection in both the theoretical dilution
and jet half-width curves. By comparing the jet trajectory curve and
the dilution curve, this inflection occurs at the point where the jet
has reached its maximum height. Consequently, it is at the same point
along the jet axis that the angle, 9, changes from a positive value to
a negative value.
In most cases, Fan's model can be used to predict the dilution
and jet trajectory. One of the basic assumptions for the development
of the equations was that the radius of curvature was large. However,
for a negatively buoyant jet with a low densimetric Froude number, this
is not the case. The radius of curvature of the jet axis is small for
low values of densimetric Froude number due to the greater density
differences after the maximum height has been reached, which causes a
more rapid downward deflection and sinking of the jet axis than for
jets with high values of densimetric Froude number. In addition,
accuracy in measuring the real profile may be limited due to the
interacting or overlapping of sections of profiles along the jet axis.
Nevertheless, if a value of a is known for some combination of densi-
metric Froude number, velocity ratio, and initial angle of discharge,
Fan's model can be used to predict the dilution and jet trajectory. A
mathematical expression for obtaining a as a function of F, k, and 3;
111
-------
will be evaluated later.
Abraham's Model
Abraham's model is a modification of Fan's model. Abraham con-
siders the coefficient of entrainment constant, not a function of the
densimetric Froude number, velocity ratio, or initial angle of dis-
charge. Abraham considered the equation of continuity to consist of
two terms, a term related to a simple jet in a region near the discharge
point and a term related, to a cylindrical thermal for a region at some
distance downstream from the discharge point. In addition, the equa-
tion governing the momentum in the x-direction was different from that
used in Fan's model due to the above consideration. Moreover, Abraham
does not consider a zone of flow establishment in the development of
his model.
The use of Abraham's model has one advantage over Fan's model in
that the coefficients of entrainment are set a priori. Hence, the
prediction of the dilution, jet half-width, and jet trajectory curves
rely only on a knowledge of the densimetric Froude number, velocity
ratio, and initial angle of discharge. In this study, Abraham's model
was used to predict dilution, jet half-width, and trajectory curves
for the experimental conditions. For values of high densimetric Froude
number and low velocity ratios, the fitting of the jet trajectory
curves was excellent.
Albertson, et al. (10), developed two separate equations to predict
the dilution of a simple momentum jet. One predicted the dilution
(volumetric flux ratio) in the zone of flow establishment while the
112
-------
other predicted the dilution in sections beyond the zone of flow estab-
lishment. Abraham used a relationship developed by Albertson to express
the dilution of the jet tracer near the discharge when, in reality,
the equation can actually be used only to describe the dilution in the
region of established flow. Hence, the prediction of the dilution in
the region of zone of flow establishment is not valid, but this does
not preclude the use of the model to predict dilution at cross-sections
beyond the zone of flow establishment.
In the case of a high densimetric Froude number and low velocity
ratio, the jet is approaching the case of a simple jet. The case of a
high densimetric Froude number and low velocity ratio approaches the
case for which Abraham's assumptions are valid. Basically, the region
near the nozzle is a momentum (simple) jet, and in the region where the
cylindrical thermal approximation is valid, the predominant direction
of flow of the jet is parallel to the direction of flow of the flowing
ambient stream. Hence, for these cases, Abraham's model fit the
experimental data exceptionally well. However, for the cases with
low values of densimetric Froude number and high velocity ratios, the
fit of the trajectory curve was not as good as discussed above. In all
cases the predicted height of rise of the jet was underestimated with
respect to the actual height of rise observed in the experiments.
The assumption that at some distance downstream from the discharge point
the direction of flow is parallel to the direction of flow of the ambient
stream is violated. Hence, the theoretical prediction of the jet tra-
jectory after the deflection in a downward direction is not valid.
113
-------
Angle at the End of the Zone of Flow Establishment
One of the more important parameters in the utilization of Fan's
model was the reduced angle of inclination or angle at the end of flow
establishment, 8 • There was no data existing for a priori prediction
of B for a negatively buoyant jet. However, in the region in which
this initial deflection occurs, Abraham considered the jet to act as a
simple jet since the initial momentum forces are greater than the
buoyancy forces. With this in mind, data for the reduced angle of in-
clination of a simple jet in a cross-flow does exist. Fan (3) pre-
sented results from Gordier (15), Jordinson (16), and Keffer and Baines
(17) in his study of a round, positively-buoyant jet in a cross-flow.
For the above studies the angle of discharge was 90°. Fan used Equa-
tion 92 to predict the reduced angle of inclination at the end of the
zone of flow establishment as
8n
g£- = 90° - 110/k (92)
po
Motz and Benedict (25) presented results obtained for 8/6* in a
study of a heated surface jet into a cross-flow. In the present study,
the reduced angle of inclination was obtained as discussed earlier.
Figure 40 represents the data found in Fan's (3), Motz and Benedict's
(25) and the present study of the angle at the end of the zone of flow
establishment versus the inverse of the velocity ratio, 1/k. The
value of 8 /81 seems to decrease with an increase in 1/k. With an in-
o o
crease in 1/k, the relative value of the ambient velocity is higher.
Consequently, the deflection of the jet will be more due to the in-
creased amount of entrained ambient momentum in the x-direction.
114
-------
r o
1.0
0.9
0.8
0.7
0.6
0.5
0.0
0.1
O - 90°
A - 45°
O- 60°
• - 90° ~ MOTZ - BENEDICT (25)
A - 45° '
• -60°- "
• - 90° ' FAN (3)
0.2 0.3
l/k
0.4
FIGURE 40 - INVERSE VELOCITY RATIO, l/k, VERSUS
-------
However, there is quite a bit of scatter in the data, particularly for
the 45° jet. For the 60° and 90° jets at low 1/k values, the values of
8 /81 are clustered closely around values of 8 /B' = .89 for 1/k - 20
o o o o
and 8/6' = .845 for 1/k ~ 10. However, the rest of the data exhibits
a high degree of scatter. Hence, no statement can be made for a general
predictive equation relating the value of the angle at the end of zone
of flow establishment as a function of the inverse of the velocity
ratio, 1/k.
Coefficient of Entrainment
The most important parameter for the prediction of the dilution,
jet half-width, and jet trajectory curves is the coefficient of en-
trainment, regardless of whether it is a positively or negatively
buoyant jet. In Fan's model the coefficient of entrainment is constant
for a given combination of initial angle of discharge, densimetric
Froude number, and velocity ratio.
Fan (3) presents data for the prediction of the coefficient of
entrainment, a, for a positively buoyant jet. The present study presents
data for an a priori prediction of a for a negatively buoyant jet. The
values of a were obtained for the 24 combinations of densimetric Froude
number, velocity ratio, and initial angle of discharge. Figure 41 shows
the relationship between the values of a for the various combinations.
A multiple linear regression analysis was performed on the data. The
basic equation related a as a function of Log F, Log (k), and sin (8^)•
Equation 93 was developed and the correlation coefficient was 0.91.
a = -0.107 + 0.104 Log(F) - 0.553 Log(k) + 1.05 sin (6') (93)
116
-------
DENSIMETRIC FROUDE NUMBER, F
FIGURE 41 - VALUES OF a FOR EXPERIMENTAL COMBINATIONS OF F, k, and 3'
-------
A test of significance of the correlation coefficient shows that, with
the null hypothesis, there is no relationship between the variables
which can be rejected at the 1% level. Hence, Equation 92 will give a
value of a for any combination of densimetric Froude number, velocity
ratio, and initial angle of discharge within the ranges of these
parameters used in this study.
The values of a ranged from 0.15 to 0.9. The value of a = 0.15
was found for the combination of F - 10, k - 20, and 3' = 45°, and the
value of a = 0.9 was found for the combination of F - 40, k ~ 5, and
3' = 90°. The value of ot decreases as the case of a coflowing stream
is approached.
Due to geometric limitation, the degree that ambient fluid is
entrained is less for discharge of 45° than for a discharge of 90°.
Hence, as 01 is decreased the volume of ambient fluid available for
in-flow and dilution of the jet on the side of the jet near the boundary
is decreased. Motz and Benedict (25) obtained similar results for a
heated surface jet. When the discharge angle was changed from 90° to
45°, the value of the entrainment coefficient was changed from 0.4 to
0.2. Equation 92 will predict a decrease in ot for a decrease in the
initial angle of discharge. Also, the value of a tends to decrease as
the velocity ratio increases. As the velocity ratio increases, the
effect of the ambient current decreases (i.e., the dilution caused by
the ambient current is less). Consequently, the values of the entrain-
ment coefficient are less. One would expect lower values of a for this
situation, since the jet is approaching the case of a simple jet issuing
into a stagnant environment with a = 0.057.
118
-------
Jet Reynolds Number
Values of the jet Reynolds number as defined by Equation 87 for the
cases presently under study are listed in Tables 3 and 4. The values
range from 1000 to 5000. Rawn, Bowerman, and Brooks (35) stated that,
for R > 2,000, the jet flow will usually be turbulent, but turbulence
6
is probably not fully developed until R reaches 10,000 or 20,000.
G
Frankel and Cummings (36) stated that, for a fully turbulent jet, the
Reynolds number is sufficiently large to have no appreciable influence
on the dilution. Rouse (37) noted that, for low Reynolds numbers, the
expansion in a lateral direction will be due solely to mixing caused by
molecular diffusion, i.e., viscous shear. However, at high Reynolds
numbers, eddies will be generated in the zone of maximum instability,
and the expansion process will occur much more rapidly. The eddies
forming in this region will not only produce the deceleration and
acceleration of the respective zones, but will themselves penetrate
farther and farther into each zone. Burdick (31) concluded that, as
long as the jet could be considered turbulent, the Reynolds number
could be neglected as a significant parameter.
The runs with jet Reynolds numbers greater than 2000 had the best
agreement of predicted data to experimental data. For experimental runs
with jet Reynolds numbers less than 2000, the rate of decay of the
potential core will be less than for the fully turbulent case. Hence,
the length of the potential core will be longer. Consideration of the
longer length of the potential core would cause the experimental values
of the y-coordinate to be decreased. This would imply a better fit of
the jet trajectory since the jet trajectory was under-predicted using
119
-------
the previously determined values of the y-coordinate. Yet this effect
may be offset by the possibility of a smaller reduced angle of inclina-
tion, B , for a longer potential core. A smaller BQ would imply a
smaller initial vertical momentum component which would tend to reduce
the height of rise of the jet and to again cause under-pediction of
the jet axis. In addition, there will be less dilution in the cases of
jet Reynolds numbers less than 2000 due to the decreased turbulence.
Consequently, values of a for the experimental runs with Rg > 2000 may
be too low in considering application to a fully turbulent jet. Hence,
o should be higher to reflect the increased dilution due to turbulent
flow conditions. A higher a would also imply a better fit of the
trajectory data. Hence, the difficulty of fitting the trajectory data
could possibly have been due to violation of the model assumption of
fully turbulent flow.
The reasons for the low jet Reynolds number are twofold. The
maximum flow in the laboratory flume was restricted and the size of the
flume was also restricted. Hence, in some cases the jet velocity had
to be so low to meet the above restrictions that the criterion that the
jet Reynolds number be greater than 2,000 had to be violated.
120
-------
IX
LIST OF REFERENCES
1. Water Resources Council, "The Nation's Water Resources," Washington,
D. C., 1968.
2. Brooks, N., and Koh, R., "Discharge of Sewage Effluents from a Line
Source into a Stratified Ocean," International Association for
Hydraulic Research, Xlth Congress, Leningrad, September, 1965, pp.
1-8.
3. Fan, L. N., "Turbulent Buoyant Jets into Stratified or Flowing
Ambient Fluids," Technical Report No. KH-R-15, W. M. Keck Laboratory
of Hydraulics and Water Resources, California Institute of Tech-
nology, Pasadena, California, June, 1967.
4. Cederwall, K., "Jet Diffusion: Review of Model Testing and Com-
parison with Theory," Hydraulics Division, Chalmers Institute of
Technology, Gb'teborg, Sweden, February, 1967.
5. Cederwall, K., "Hydraulics of Marine Waste Water Disposal," Report
No. 42, Hydraulic Division, Chalmers Institute of Technology,
Gb'teborg, Sweden, January, 1968.
6. Abraham, G., "The Flow of Round Jets Issuing Vertically into Ambient
Fluid Flowing in a Horizontal Direction," Proceedings of the 5th
International Water Pollution Research Conference, San Francisco,
July-August, 1970, pp. Ill 15/1 - III 15/7.
7. Cederwall, K., and Brooks, N., "A Buoyant Slot Jet into a Stagnant
or Flowing Environment," Report No. KH-R-25, W. M. Keck Laboratory
of Hydraulics and Water Resources, California Institute of Tech-
nology, Pasadena, California, March, 1971.
8. Abraham, G., "Jets with Negative Buoyancy in Homogeneous Fluid,"
Journal of Hydraulic Research, Vol. 5, No. 4, 1967, pp. 235-248.
9. Turner, J. S., "Jets and Plumes with Negative or Reversing
Buoyancy," Journal of Fluid Mechanics, Vol. 26, Part 4, 1966,
pp. 779-792~T~
10. Albertson, M. L., Dai, Y. B., Jensen, R. A., and Rouse H
"Diffusion of Submerged Jets," Transactions^ American Society o±
Civil Engineers, Vol. 115, 1950, pp. 639-697.
11. Morton, B. R., "Forced Plumes," Journal of Fluid Mechanics, Vol. 5,
Part I, 1959, pp. 151-163.
12 Parker F L. and Krenkel, P. A., "Thermal Pollution: Status of
the Si "Report No. 3, Naiional Center for Research and Training
£ t£ Hydrotgic and Hydraulic Aspects of Water Pollution Control,
Vanderbilt University, Nashville, Tennessee, December, 1969.
121
-------
13. Morton, B. R., Taylor, G. I., and Turner, J. S., "Turbulent
Gravitational Convection from Maintained and Instantaneous Sources,"
Proceedings, Royal Society of London, Vol. 234A, No. 1196, January,
1956, pp. 1-23.
14. Morton, B. R., "On a Momentum-Mass Flux Diagram for Turbulent Jets,
Plumes, and Wakes," Journal of Fluid Mechanics, Vol. 10, 1961, pp.
101-112.
15. Gordier, R. L., "Studies on Fluid Jets Discharging Normally Into
Moving Liquid," Technical Report No. 28, Series B, St. Anthony
Falls Hydraulic Laboratory, University of Minnesota, Minneapolis,
Minnesota, August, 1959.
16. Jordinson, R., "Flow in a Jet Directed Normal to the Wind," Aero.
Research Council, Report and Memo No. 3074, 1956.
17. Keffer, J. F., and Baines, W. D., "The Round Turbulent Jet in a
Cross-Wind," Journal of Fluid Mechanics, Vol. 15, 1963, pp. 481-
496.
18. Pratte, B. D., and Baines, W. D., "Profiles of the Round Turbulent
Jet in a Cross Flow," Journal of the Hydraulics Division, American
Society of Civil Engineers, Vol. 92, No. HY6, Proc. Paper 5556,
November, 1967, pp. 53-64.
19. Tollmien, W., "Strahlverbreiterung," Zeitschr. Angew. Math, und
Mech., 1926, pp. 468-478.
20. Schmidt, W., "Turbulente Ausbreitung eines Stromes Erhitzter Luft,"
Zeitschr. Angew. Math, und Mech., Vol. 21, 1941, pp. 265-278,
pp. 351-363.
21. Abraham, G., "Entrainment Principle and Its Restriction to Solve
Problems of Jets," Journal of Hydraulic Research, Vol. 3, No. 2,
1965, pp. 1-23.
22. Fan, L. N., and Brooks, N. H., Discussion of "Horizontal Jets in a
Stratified Fluid of Other Density," by G. Abraham, Journal of the
Hydraulics Division, American Society of Civil Engineers, HY2,
March, 1966, pp. 423-429.
23. Abraham, G., "Horizontal Jets in a Stagnant Fluid of Other Density,"
Journal of the Hydraulics Division, American Society of Civil
Engineers, HY4, July, 1965, pp. 139-154.
24. Rouse, H., Yih, C. S., and Humphreys, H. W., "Gravitational Con-
vection from a Boundary Source," Tellus, Vol. 4, 1952, pp. 201-210.
122
-------
25. Motz, L. H., and Benedict, B. A., "Heated Surface Jet Discharged
into a Flowing Ambient Stream," Report No. 4, National Center for
Research and Training in the Hydrologic and Hydraulic Aspects of
Water Pollution Control, Vanderbilt University, August, 1970.
26. Richards, J. M., "Experiments on the Motion of Isolated Cylindrical
Thermals Through Unstratified Surroundings," International Journal
of Air and Water Pollution, Vol. 7, 1963, pp. 17-34.
27. Bosanquet, C. H., Horn, G., and Thring, M. W., "The Effect of
Density Differences on the Path of Jets," Journal of Fluid Mechanics,
Vol. 5, Part I, January, 1959, pp. 340-352.
28. Fan, L. N., and Brooks, N. H., "Dilution of Waste Gas Discharge
from Campus Buildings," Technical Memorandum 68-1, W. M. Keck Lab-
oratory of Hydraulics and Water Resources, California Institute of
Technology, Pasadena, California, January, 1968.
29. Priestley, C. H. B., and Ball, F. K., "Continuous Convection from
an Isolated Source of Heat," Quarterly Journal of the Royal
Meteorological Society of London, England, Vol. 81, 1955.
30. Krenkel, P. A., and Orlob, G. T., "Turbulent Diffusion and the
Reaeration Coefficient," Journal of the Sanitary Engineering
Division, American Society of Civil Engineers, Vol. 88, SA2,
March, 1962.
31. Burdick, J. C., and Krenkel, P. A., "Jet Diffusion Under Stratified
Flow Conditions," Technical Report No. 11, Environmental and Water
Resources Engineering, Vanderbilt University, 1967.
32. Krenkel, P. A., "Turbulent Diffusion and Kinectics of Oxygen
Absorption," Ph.D. Dissertation, University of California, Berkeley,
California, 1960.
33. Clements, W. C., "Pulse Testing for Dynamic Analysis. Part I.
An Investigation of Computational Methods and Difficulties in
Pulse Testing. Part II. Application of Pulse Testing to Flow and
Extraction Dynamics," Ph.D. Dissertation, Vanderbilt University,
Nashville, Tennessee, June, 1963, pp. 91-93.
34. Clements, W. C., "Electrical Conductivity Monitor," Instruments
and Control Systems, Vol. 41, November, 1968, pp. 97-98.
35. Rawn, A. M., Bowerman, R. F., and Brooks, N. H., "Diffusers for
Disposal of Sewage in Sea Water," Journal of the Sanitary Engineering
Division, American Society of Civil Engineers, Vol. 86, March, 1960.
36. Frankel, R. M., and Gumming, J. D., "Turbulent Mixing Phenomena of
Ocean Outfalls," Journal of the Sanitary Engineering Division,
American Society of Civil Engineers, Vol. 91, April, 1965.
123
-------
37. Rouse, H., Engineering Hydraulics, Wiley and Sons, Inc., New York,
1950.
38. Flatten, J. L., and Keffer, J. F., "Entrainment in Deflected
Axisymmetric Jets at Various Angles to the Stream," Technical
Publication No. 6808, Department of Mechanical Engineering,
University of Toronto, June, 1968.
39. Larsen, J., and Hecker, G. E., "Design of Submerged Diffusers and
Jet Interaction," Preprint No. 1614, presented at the American
Society of Civil Engineers National Water Resources Engineering
Meeting, Atlanta, Georgia, January, 1972.
40. Holly, F. M., and Grace, J. L., "Model Study of a Dense Fluid in a
Flowing Fluid," Preprint No. 1587, presented at the American Society
of Civil Engineers National Water Resources Engineering Meeting,
Atlanta, Georgia, January, 1972.
41. Perry's Chemical Engineers' Handbook, Fourth Edition, McGraw-Hill,
New York, New York, 1969.
42. Briggs, G. A., "Plume Rise," AEC Critical Review Series, No. 26,
1969.
124
-------
GLOSSARY
LIST OF NOTATIONS
-------
X
LIST OF NOTATIONS
A - Jet cross-section normal to jet axis, L2
b - Local characteristic length of half-width of the jet, L
b - Jet half-width at the end of the zone of flow establishment, L
o
B , - Radius of thermal, L
th
c - Concentration at the jet axis
c* - Local concentration value
c - Initial concentration at the end of the zone of flow establish-
o
ment, i.e., at 0
CI - Ay, the incremental distance between each y' ^,L
C, - Drag coefficient for Fan's model
C', - Reduced drag coefficient for Fan's Model
d
C - Drag coefficient for Abraham's model
da
c. - Local concentration at location (j,k)
c - Mean value of the concentration profile
D - Diameter of jet at orifice and orifice diameter, L
o
E. - Constant supply voltage
E - Internal voltage of conductivity monitor
o
f - Buoyancy parameter
F - Densimetric jet Froude number
F - Drag force per unit length, L T-
125
-------
g - Gravitational acceleration, LT 2
G - Conductivity of salt solution
h - Dimensionless horizontal or x-momentum flax parameter
H - Head of water above apex of notch, L
W
i - Vector in the direction tangent to the jet axis or index number
of time period
j - Vector perpendicular to the jet axis or index number denoting
a given location in a cross-section, k
k - Velocity ratio or index number denoting a given cross-section
k' - k + cos
o
m - Momentum flux parameter
m.
m. . ,
1,3,k
- Time-averaged millivolt output at location j in cross-section, k
J >*
Millivolt output at location (j,k) over time increment A^
n - Number of time increments
N - Number of j location in a given cross-section, k
0 - Origin of the coordinate system (x,y), beginning of the zone of
flow establishment
0' - Origin of the coordinate system (x',y?), beginning at the point
of jet discharge
Q - Volumetric flux, L3!"1
Q - Flow in laboratory flume, L3T-1
3.
Q - Initial volume flux at the nozzle, L T
r - Radial distance measured from the jet axis, L
R - Initial jet Reynolds number
126
-------
s - Distance along the jet axis from the zone of flow establishment,
L
s' - Distance along the jet axis from the jet discharge point, L
s1 - Jet axis length of the zone of flow establishment, L
6
S - Dilution ratio at terminal height of rise, y
At. - Time period used in time-averaging, T
u - Jet velocity at the centerline of the jet, LT 1
u - Jet discharge velocity at orifice, LT'1
u* - Jet velocity at a local point, LT 1
U - Ambient uniform velocity, LT
a
"1
'1
U - Initial jet discharge velocity, LT
v - Dimensionless vertical or y-momentum flux parameter
V - Quantity of fluid contained in a thermal jet per unit of
length, L2
x - Horizontal coordinate measured from the end of the zone of
flow establishment, L
x1 - Coordinate axis in horizontal direction on the same plane as
jet axis with origin at 0', L
x1 - Horizontal distance from jet discharge point to the end of the
e
zone of flow establishment, L
y - Vertical coordinate measured from the end of the zone of flow
establishment, L
y' - Coordinate axis in vertical direction, with origin at 0', L
y' - Vertical distance from the jet discharge point to the end of
Je
the zone of flow establishment, L
127
-------
y£ - y-location of the mean value of the concentration profile, L
y! , - Vertical distance of local point at location (j ,k) from jet
J >K
discharge point, L
y - Mean vertical height of rise of the plume, L
a - Coefficient of entrainment
a - Coefficient of entrainment for jets primarily influenced by
initial momentum
a , - Coefficient of entrainment for thermal moving through a
stagnant ambient fluid
£ - Dimensionless vertical distance (y)
£' - Dimensionless vertical distance (y1)
n - Dimensionless horizontal distance (x)
r\' - Dimensionless horizontal distance (x1)
C - Dimensionless distance along s-axis (s)
C1 - Dimensionless distance along s-axis (s')
8 - Initial angle of inclination at the end of the zone of flow
establishment, degrees
g' - Initial angle of inclination at jet discharge point, degrees
9 - Angle of inclination of the jet axis (with respect to the
x-axis), degrees
Q - Initial angle of inclination (with respect to x-axis) (6 =
6 for Fan's model and 6 = g' for Abraham's model), degrees
o o o
M - Volumetric flux parameter
v - Kinematic viscosity, L2?"1
128
-------
p - Density of the ambient fluid, FT2L k
3.
p - Reference density taken as p (0), FT2L~k
O 3.
p - Initial jet density, FT2L~1+
p* - Local density within a jet, FT2L ^
- Angular coordinate on a cross-section normal to jet axis,
degrees
a, - Standard deviation at cross-section k, L
k
Ah - Plume rise above top of stack
129
-------
XI
APPENDIX A
SALINITY-DENSITY RELATIONSHIP
-------
o
10
>-
TEMPERATURES IN °C
0.5 1.0 1.5 2.0 2.5 3.0 3.5
SALT CONCENTRATION IN WEIGHT PER CENT
FIGURE 42 - DENSITY OF SALT WATER AS A FUNCTION OF SALT CONCENTRATION AND TEMPERATURE
(PERRY'S CHEMICAL ENGINEERS HANDBOOK, REF. 41)
-------
APPENDIX B
COMPUTER PROGRAM - FAN'S MODEL
-------
C****THI S PROGRAM CONTAINS THE MAIN PROGRAM* AMD
C TWO SUBROUTINES NEEDED TO PROVIDE INPUT AMD
C OUTPUT CONTROL TO SOLVE THE FIVE EQUATIONS IN
C FANS MODEL. THE MAIN PROGRAM CONTAINS THE
C NECESSARY INPUT INFORMATION WHILE SUBROUTINE FCT HAS
C THE EQUATIONS FOR INTEGRATION AND SUBROUTINE OUTP
C PROVIDES THE NECESSARY OUTPUT CONTROL. THE OUTPUT
C VARIOUS DISTANCES ARE IN TERMS OF DIAMETERS. THE
C OTHER OUTPUT VALUES ARE SELF-EXPLANATORY.
C IN ADDITION* SUBROUTINE DRKGS MUST ALSO BE USED AS
C OUTLINED IN THE MAIN PROGRAM.
IMPLICIT REAL*3C A-£)
INTEGER NDIM* IHLF* I*NDATA*MI
REAL OUTP* FCT
REAL*3 AUX<<3* 5)*Y( 5)*DERYC 5)*PRMTC5)* SDATAC25)*
ABDATAC25)*XDATA( 25) * Y DAT AC 25) * CONC 1 ( 25) * VARC 90)
COMMON /INPUT/ K, BETA1* BOUPAR, CD, KK* BO* DI AM* ALPHA
EXTERNAL FCT* OUTP
DATA DERY/5*.2DO/
DO 973 MI = 1*10
C******INPUT THE VALUE OF INDEX***********************
C******INDEX = 1 ALLOWS ONE TO INPUT EXPERIMENTAL DATA
C******AND THEN MAKE AN INITIAL GUESS AT A VALUE OF
C****** ALPHA. IF ANOTHER VALUE OF ALPHA IS DESIRED
C****** AFTER THE INITIAL GUESS SIMPLY FOLLOW THE
C******INI TI AL DATA FOR INDEX=1 WITH A CARD WITH INDEX=2
C****** AND THEN ANOTHER CARD WITH THE SECOND GUESS AT
C******ALPHA. THIS MAY BE DONE FOR AS MANY TIMES AS
C******DESIRED UP TO 10- IF YOU DESIRE THE CALCULATION
C******TO STOP. INSERT A CARD WITH INDEX = 3«
C
C
READC5*974) INDEX
I FC INDEX .EG). 2) GO TO 975
I FC INDEX .EG. 3) GO TO 976
C**TITLE OF PROJECT OR RUN UNDER INVESTIGATION**********
READC 5*804)< VAR( I )* 1= 1*20)
C******INPUT INITIAL JET LOCATION* JET TERMINAL DISTANCE*
O******INCREMENTAL DISTANCE* AND ALLOWABLE ERROR
READ(5*800)(PRMTCI )* 1 = 1*4)
C******PARAMETERS NECESSARY FOR OPERATION
C******
C 1 FROUDE NUMBER
C 2 ALPHA-COEFFICI ENT OF ENTRAINMENT
C 3 K- VELOCITY RATIO
C 4 BETAI -INITIAL ANGLE OF DISCHARGE-DEGREES
C 5 CD-DRAG COEFFICIENT
READC 5* 8 79) FROUDE* ALPHA* X* BETAI * CD
C****INPUT INITIAL CONDITIONS****************************
131
-------
C 1. YC 15 = 1.0
C 2. YC2)=1.0
C 3. YC3) = ANYTHING—NEVER USED
C 4. YC4)=0»0
C 5. YC5)=0.0
READC5*801)CYCI5*1=1,5)
READC5>802) NDIM
C****INPUT REDUCED ANGLE OF INCLINATION AT END OF ZONE
C*** OF FLOW ESTABLISHMENT
READC5*973)BETA1
BETA1 = BETA1/57.29578
C
C*******************************************************
C
C INPUT THE NUMBER OF CROSS-SECTIONS ALONG THE
C JET AXIS CNDATA), THE DIAMETER OF THE JET
C CDIAM) FOR THIS PARTICULAR RUN.
C INPUT THE X-DISTANCE*Y-DI STANCE* S-DI STANCE*
C CONCENTRATION RATIO* AND JET HALF-WIDTH DATA FOR A
C PARTICULAR CROSS-SECTION DOWNSTREAM.
C
C
C******************************************************
READC 5*9 705NDATA* DIAM*(XDATACI) , YDATAC I )* SDATAC I ) ,
4CONC1CI)*BDATACI)>I=1*NDATA)
GO TO 953
975 READC5*973)ALPHA
953 CONTINUE
CORR = YC2)
Y(2)=CORR*DCOS(BETA15
YC 35 = CORR*DSIN(BETA15
KK=K+DCOSCBETA1>
BOUPAR=-C((K/KK>**2.5)/((DSQRTC2.DO))*ALPHA))/FROUDE**
$2
C****WRITE THE WORKING PARAMETERS************
WRI TEC 6*905)
WRITEC6*504)(VARCI)*1=1*20)
WRITEC 6* 900)FROUDE* K* ALPHA* BOUPAR* CD* BETAI
IFCINDEX .GT. 15GO TO 977
BETA5=BETA1* 5 7.29 5 78
WRITEC 6*910)BETAS
WRITEC6*911)DIAM
BO = DIAM*CDSQRTCK/(2.DO*KK))5
WRITEC6*969)K*KK*BO
WRITEC6*905)
WRITEC 6*968)
968 FORMATC1X* 'EXPERIMENTAL f* Tl6* 'EXPERIMENTAL'* T32*
& 'EXPERIMENTAL'* T45* 'EXPERIMENTAL '* T59* 'EXPERI
«MENTALf/T6* 'JET'*T16* 'CONCENTRATION ', T36* 'JETST46
** 'X-DISTANCE'*T60> 'Y-DI STANCE VT5* 'AXIS'*T33* 'HA
132
-------
&LF- WIDTH '/T6* 'S
& '//)
DO 20 I=1*NDATA
'B/D'* T49 , 'X/D1* 162, 'Y/D
SDATACI) = SDATACI )/DI AM
XDATACI) = XDATACI )/DI AM
YDATACI) = YDATACI )/DI AM
BDATACI) = BDATACI ) *DSQRTC 2 . DO 3 /DI A>M
COMCKI) = l.DO/CCONCK I )*.01DO)
WRITEC6*966) SDATAC I )>COMCI< I )*BDATA< I )* XDATACI ) , YD AT AC
$1 )
20 COMTIMUE
977 WRI TEC 6*904)
CALL DRKGSCPRMT*Y^DERY*NDIM* IHLF* FCT^OUTP^ AUX)
YC 1)=1.0
YC2)=1-0
YC 55 = 0-0
97B CONTINUE
IFCIHLF.EO. 1 1) GO TO 1 1
IFCIHLF.EO. 1?)GO TO 1 2
WRITEC6j901) IHLF
11 WRI TEC 6*902) IHLF
12 WRITEC6..903) IKLF
STOP
800 FORMATC3F10. 1,D10.4>
801 FORMATC5F5-1)
802 FORMATCI 1)
804 FORMATC20A4)
879 FORMATCF10.5*F5-2* 3F5. 1)
900 FORMATC///T88* 'FAN ' '5 MODEL ' // ' DEN SI METR1 C FROUDE MO.'
&, F6.3/ 'VELOCITY RATI 0 ' * T33* f= ' * T39* F4. I/ "ENTRAIMMEMT C
SOEFFI CIENT C
&ALPHA) % T33> '= '* T38* F5. 3/ 'BUOYANCY PARAMETER *> T33* '= '*
$T35*FS-5/ 'DR
&AG COEFFICIENT', T33> '= ' , T3B , F5 . P/ ' I MI TI AL AMGLEOF T)l S
SCHARGEST33*
*'= ST39,F3.0//)
901 FORMATC/ '****ERROR --- SI GNC PRMTC 3) ) -ME. SI GNCPRMTC
&2)-PRMTC 1) ) . '/ 'IKLF = t*I2*')V)
902 FORMATC/ '****ERROR --- IN3ITIAL INCREMENT BISECTED
&MORE THAN 10 TIMES1/ ' C IHLF = SI2j')f/)
903 FORMATC/ «****ERROR --- PRMTC 3) = O'/'CIHLF = SIS-1)1/)
904 FORMATCT?> MET%T18* ' VOLtMETRI C '> T 32, 'CONCENTRATI OM '-
&T49> 'MOMENTUM %T65* ' VELOCI TY f > T3 3-» 'JETST99* 'XST1 14*
&'Y'*T126> 'BETAVT7* 'AXISf*T20> »FLUX'>T51, 'FLUXf,T66*
A'RATIO'*T79> 'K ALF- VI DTHf • * T9 5, 'DI STAMCE ' , T 1 1 0, 'DI STA
ftNCEV/TS* 'S/D',T65* 'UJ/UO STS3* 'B/D'*T98* 'X/DST1 13*
&'Y/D'>T124* 'DEGREES'//)
905 FORMATC 1H1)
133
-------
910 FORMATC/'BETAS = '^FS.2/5
911 FOR^JATC/'DI AM = **F5.2>
969 FORM ATC T21, 'Kt*T26, '= ' * T30» F8 . 5/T2 1 * 'K * • STS6* '= '» T
970 FORMATCI2-. F3.2/C5Fi0.5»
973 FORWATCF10.5)
974 FORMATC 111)
976 END
C
C
C*****SUBROUTI!ME FCTCX* Y» DERY)
C COMTAIMS THE DIFFERENTI Al, EQUATIONS THAT ARE
C TO BE INTEGRATED AMD ARE LISTED AS DERYC I ) .
C
C
SUBROUTINE FCTCX, Y* DERY)
IMPLICIT REAL*3
REAL*B YC5)^ DERYC5)
COM-MOM /INPUT/ K>BETA1*BOUPAR*CD*KK,BO,DI AM, ALPHA
M=DSCRT(Y(2)**2-»-Y( 3>**2>
BETA=DATAMCYC3)/Y(2) )
IFCBETA) 1>2*2
1 IFCCD)2*2*3
3 CD = -CD
PRINT 901^ CD
2 COMTIMUE
SS=DSIM(BETA>**2
SSS=DSIM( BETA)**3
CS=DCOSCEETA)
TJSW=YC D/DSGRTCM)
MU=M/Y( 1>
UM=Y( 1>/M
CHI=DSORT( SS+CKK*MU-2.DO*CS>**2>
DERYC 1)=(USM*CHI )/KK
DERY(2)=(USM/KKK)*(2.DO*CHH-CD*SSS)
DERYC 3) = BOUPAR*UM-( CD*USM*SS*CS/KKK)
DERYC4)=CS
DEPYC 5)=DSIM(BETA)
901 FORMAT( lX*F5-25
RETURN
END
C
C
O****SUBROUTINE OUTPCX* Y* DERY, IHLF*MDIM, PRMT)
C
C
SUBROUTIME OUTP(X>Y> DERY*
IMPLICIT REAL*8CA-Z>
134
-------
INTEGER *JDIM, IHLF
REAL*3 YC5)* DERYC5)* PRMTC5>
COMMON /INPUT/ K*BETA 1,BOUPAR,CD,KK,BO,DIAM, ALPHA
IFCDABSC(X*!O.DO)-IDIMTC
BETA = CDATANCYC 3)/YCS» )
R=R*80/DIAM
URATIO = CYC1)*KK*BQ**S/R**2>-DCOSCBETA>
BETA = BET A* 5 7. 29 5 78
XX=X*BO/CDIAM*2.DO*ALPHA)
YC^)*BO/C DIAM*2.DO*ALPHA)
Y( 5)*BO/CDI AM*2.DO*ALPrIA)
WRITE(6*901)XXJY< 1 ) > COMC, M^ URATI 0* R, XKKX* YYYY* BF.TA
901 FORMATC
RETURM
135
-------
APPENDIX C
COMPUTER PROGRAM - ABRAHAM'S MODEL
-------
C****THIS PROGRAM CONTAINS THE MAIN PROGRAM, AMD
C TWO SUBROUTINES MEEDED TO PROVIDE INPUT AND
C OUTPUT CONTROL TO SOLVE THE FIVE EQUATIONS IN ABRA-
C HAMS MODEL. THE MAIM PROGRAM CONTAIN THE
C NECESSARY INPUT INFORMATION WHILE S',DERY<5>,?RMTC5),SDATAC25),
&BDATAC 25),XDATAC25>,YDATAC25), CONCH 2 5), VARCRO)
COMMON /INPUT/ K,KK,BOUPAR,CD, AMOM, ATHERM,BO,DI AM
EXTERNAL FCT, OUTP
DATA DERY/5*.2DO/
C** TITLE OF PROJECT OR RUN UNDER INVESTIGATION**********
RF.AD(5,S04)C VARCI ), 1 = 1,20)
C******r\jprjT INITIAL JET LOCATION, JET TERMINAL DISTANCE,
C*******INCREMENTAL DISTANCE* AND ALLOWABLE ERROR
READC 5,300>(PRMT(I ), 1= 1,4)
C******PARAMETEES NECESSARY FOR OPERATION
f ******
C 1 FROUDE NUMBER
C ?. K- VELOCITY RATIO
C 3 BETAI-INI TI AL ANGLE OF DI SCHARGE- DEGREES
READC 5,£?03)FFOUDE, K, BETAI
C*** THF. VALUES OF THE EMTRAINMENT COEFFICIENT
C*** A\in T.P.^-r- COEFFICIENT ARE CONSTANT FOR ABRAHAM'S
C*** MODEL. HENCE THE VALUES CAN BE IDENTIFIED
C*** FOR ALL COMBINATIONS WITHIN THE
COMPUTER PROGRAM.
£MOM=0.057DO
0« 5DO
NP'JT INITIAL CONDITIONS****************************
C I . Y ( 1 ) = 1 . 0
C ?. YC?)=1.0
f 3. Y(3)= ANYTHING- -NEVER USED
C 4. YC4)=0-0
C 5- V(5) = 0.0
PEAD< 5,^01)C Y< I) , 1=1, 5)
READ(5,^Of5) NLIM
BETAI = BFT^I/57.2957R
C***
136
-------
c***
C*** IT IS NECESSARY TO INPUT THE VALUES OF THE
C*** RESPECTIVE LENGTHS FOR THE ZONE OF
C*** FLOW ESTABLI SHMENT* I .E. XESTAB= THE X
C*** DISTANCE WITHIN THE ZONE OF LOW ESTABLISHMENT
YESTAB= THE Y DISTANCE WI THIN THE ZONE OF FLOW
ESTABLI SHM EN T* AND SESTAB= THE S DI STANCE
WITHIN THE ZONE OF FLOW ESTABLISHMENT. THE
C*** VALUES OF XDATACI), YDATAC I ) , AND SDATAC I ) THAT
C*** FOLLOW MUST HAVE THESE VALUES OF XESTAB* YESTABj
C*** AND SESTAB ADDED SO THAT THE DISTANCES WILL BE WITH
C*** RESPECT TO THE DISCHARGE POINT-
C***
READC 5,8031XESTAB, YESTAB, SESTAB
C
c* ******************************************************
c
C INPUT THE NUMBER OF CROSS- SECTI ONS ALOWG THE
C JET AXIS CNDATA), THE DIAMETER OF THE JET
C CDIAM) FOR THIS PARTICULAR RUN.
C INPUT THE X-DI STANCE* Y- DISTANCE* S-DI STANCE*
C CONCENTRATION RATIO* AND JET HALF- WIDTH DATA FOR A
C PARTI CULAR CRO SS- SECTION DOWNSTREAM.
C
C
c************************* *****************************
READC 5*970)!MDATA., DI AM* (XDATAC I >*YDATAC I ) * SDATAC I ) >
&CONC1C I )*BDATACI )*!= 1*NDATA)
CORR = YC2)
YC2)=CORR*DCOSCBETAI )
YC 3)=CORR*DSINCBETAI )
KK=K+DCOSCBETAI)
BQUPAR=-K**2/C 2. DO*KK**2*FROUDE**2)
PETAI=BETAI*57-2957S
VRITEC 6*905)
C****WRITE T4E VOFKING PARAMETERS************
VRITEC 6*804) C VAR< I )* 1= 1* 20)
VRI TEC 6* 900) FROUDE* '«* AMOM* A THERM* BOUPAR* CD* BETAI
CD=CD*DSQRTC 2.DO)/PI
WRITEC 6*91 1 )DI AM
BO = DI A>1/ 2. DO
VRI TEC 6*9£9)K*KX*BO
DO 20 I=1*NDATA
SDATAC I) =C SDATAC I )+ SESTAB)
KDATAC I ) =CXDATAC I )+XESTAB)/DI AM
YDATAC I ) = C YDATAC D+YESTAe)/DI AM
BDATAC I ) = BDATAC I )*DSQRTC2.DO)/DI
CONS1CI) = l.DO/CCONCKI>*.01DO>
137
-------
WRITEC 6*966)SDATACI> , CONC 1 C I ) * BOATAC I ) * XDATAC I ) * YDATAC
SI)
20 CONTINUE
WRI TEC 6*904)
CALL DRKGSC PRMT* Y* DERY* NDIM*IHLF*FCT* OUTP* AUX)
IFCIHLF.EQ.11)GO TO 11
IFCIHLF.EQ.12)GO TO 12
WRI TEC 6*90 DIHLF
11 WRITEC 6*902)IHLF
12 WRITEC 6*903)IHLF
STOP
800 FORMATC3F10.1*D10.4)
801 FORMATC5F5.1)
802 FORMATCI1)
803 FORMATC3F10.5)
804 FORMATC20A4)
900 FORMATC///T26* 'ABRAHAM "S MODEL '// 'DENSIMETRI C '*
&' FROUDE NO. f*T33* '=f*T37* F6.3/'VELOCITY RATIO'
&*T33*'='*T39*F4.I/'MOMENTUM COEFFICIENT'/
*'OF
&' THERMAL'/'COEFFICIENT OF
4T3S*F5.3/ 'BOUYANCY PARAMETER'* T33* '= '*T35*F3.5/
A'DRAG COEFFICIENT'*T33* '='*T38*F5.2/'I NITIAL
&' OF DISCHARGE'*T33* '= ' * T40* F3-0)
901 FORMATC/'****ERROR SIGNCPRMTC3)).NE.SIGNCPRMTC
&2)-PRMTC 1)). VIHLF = '*I2>')'/)
902 FORMATC/'****ERROR INITIAL INCREMENT BISECTED
&MORE THAN 10 TIMES V'CIHLF = '*I2*')'/>
903 FORMATC/ »****ERROR PRMTC3) = O'/'CIHLF = '*I2*f)V
904 FORMATC///' JET VOLUMETRIC CONCEN- MOMENTUM'*
&' JET'* T46* fX'*T56* 'Yf* T66* 'BETA'/' AXIS'*
«T10* 'FLUX'*T 18* 'TRATION'*T28* 'FLUX'* T36* 'HALF'*
ST43* 'DISTANCE DISTANCE DEGREES VT36* ' WI DTH '
&/ ' S/D'*T1 1* *U'*T29* 'M'*T37* 'B/D'*T45* 'X/D'*
&T56*'Y/D'///)
905 FORMATC1H1)
911 FORMATC/'DIAM = ',F5»2>
966 FORMATC T5* F8 . 4* T20* FS • 4* T34* F8 - 4, T47* F8 . 4* T60* F8 • 4 )
968 FORMATCIX*'EXPERIMENTAL'*T16*'EXPERIMENTAL'*T32*
A'EXPERIMENTAL',T45*'EXPERIMENTAL'*T59* 'EXPERIMENTAL'
&/T6* 'JET'*T16* 'CONCENTRATION'*T36* 'JET'*T46*
&'X-DI STANCE'* T60* 'Y-DI STANCEVT5* 'AXIS'* T33*
&'HALF-WIDTH'/T6* 'S/D'* T36* 'B/D'*T49* 'X/D'*T68*
4*Y/DV/)
969 FORMATCT21* 'K'*T26* '= '* T30* F8 . 5/T2 1* 'K" '*
*T26* '= '*T30*F8.5/T21* 'BO **T26* '= '* T30* F8 . 5)
970 FORMATCI2*F3.2/C5F10.5))
END
C
C
138
-------
C***** SUBROUTINE FCTCX* Y* DERY)
C CONTAINS THE DIFFERENTIAL EQUATIONS THAT ARE
C TO BE INTEGRATED AND ARE LISTED AS DERYCI).
C
C
SUBROUTINE FCTC X* Y* DERY)
IMPLICIT REAL*8
REAL*5 Y<5)* DERYC5)
COMMON /INPUT/ K* KK* BOUPAR* CD* AMOM* AT4ERM* BO* DI AM
M=DSQRTCYC2)**2+YC 3)** 2)
BETA=DATANCYC 3)/YC2) )
IFCBETA) 1*2*2
1 IFC ATHERM)2*2* 3
3 CD = -CD
ATHERM=-ATHERM
WRITEC6*901) CD*ATHERM
2 CONTINUE
S=DSINCBETA)
SS=DSINCBETA)**2
SSS=DSIN C BETA) ** 3
C5=DCOS(BETA)
USM=YC 13/D50RTCM)
MU=M/YC 1)
UM=YC 1)/M
DERYC 1)=4. DO*C AtMOM*DSQRTCM)-C USM/KK) * ( 2. DO* AMQM*CS- AT4
SERM*S*CS) )
DEPYC 2> = C 4. DO*USM/XKK5*C 2. DO*AMOM*C KX*MU-2. DO*CS)
&+2.DO*ATHERM*S*CS+CD*SSS>
DERYC 3> = 4.DO*CBOUFAR*UM-CCD*USM*SS*CS/K.KK) )
DERY(45=CS
DERYC 5) = S
901 FORMATC 1X*F5«2* 7X* F5.2)
RETURN
END
C
C
C***** SUBROUTINE OUTPCX* Y* DERY* IHLF*MDI M* PFMT)
C
C
SUBROUTINE OUTPCX,Y, DERY* I !-rLF*NDIM* PRMT)
IMPLICIT REAL*3CA-?:>
INTEGER NDIMj I 4LF
REAL*3 YC5)-> DERYC 5)* PRMTC 5>
COMMON"/ I MPUT/ X*KK* BOUPAR, CD* AMOM* A THERM* BO* DI AM
IFCDABSC (X*10.DO)-IDINTCCX+ lD-6>* 10. D0» > • GT.
& .001 DO) RET URN
IFCX .LT. l.ODO) GO TO 10
IFCDABSCX-IDINTCX+ ID- 6) > - GT. . 00 1 DO) RETURN
IFCX .LT. 40- 01) GO TO 10
139
-------
I FCDAPSC (XX 10.DCD-IDIMTCCX+lD-6)/10.DO> ) . GT,
& .001)RETURN
10 CONG = l.BO/YC1)
M=DSCRT(YC 2
BETA = CDATAM
IvRI TE(6*901)XX>YC 1 > , COMC^M, R, XXXX, YYYY* BETA
901 FORMATC IX* F6. 2.. T8* F7. 8* T 18* F7. 5^ T27, F7. 2* T36*
&F6.P»T44,F7.S,T52* FB . 2* T65* F6- S)
RETURN
EMD
140
-------
APPENDIX D
COMPUTER PROGRAM - DRKGS
-------
c
c
C THIS ISA SCIENTIFIC LIBRARY PROGRAM TO INTEGRATE A
C SYSTEM OF SIMULTANEOUS ORDINARY DIFFERENTIAL
C EQUATIONS USING A FOURTH-ORDER RUNGE-KUTTA TECHNIQUE.
C THIS PROGRAM IS AVAILABLE A MOST COMPUTER CENTERS-
G THE INPUT VARIABLES ARE EXPLAINED IN THE PROGRAMS
C FAN AND/OR ABRAHAM.
C
C
C
SUBROUTINE DRKGSC PRMT, Y* DERY^NDIM, IHLF* FCT* OUTPj AUX)
DIMENSION YC 1)>DERYC 1>* AUXC8> 1 ) > AC 4) , BC 4) * CC4>*PRMTC 1)
DOUBLE PRECISION PRMT> Y,» DERY* AUX.. A., B> C^X^XEND* H, AJ, BJ,
1DELT
DO 1 I = 1*NDIM
1 AUXC8* I ) = . 066666666666666667DO*DERY( I )
X=PRMTC 1)
XEND=PRMTC2)
H=PRMT(3)
PRMT(5)=O.DO
CALL FCTCX*Y*DERY)
C
C ERROR TEST
I FCH*CXEND-X> ) 38* 37* 2
C
C PREPARATIONS FOR RUNGE-KUTTA METHOD
2 A( 1>».5DO
A(2)=. 2928932188 1345248 DO
A< 3)= 1-707 10678 1 1865475DO
A<4>=. 16666666666666667DO
B< 1)=2.DO
B< 3) = 1
B(4>=2.DO
C( 1) = .5DO
C(2)=. 2928932188 1345248 DO
CC 35=1.707106781186547500
C(4)=.5DO
C
C PREPARATIONS OF FIRST RUNGE-KUTTA STEP
DO 3 I = 1*NDIM
AUXC 1*I)=YCD
AUXC2, I)=DERYCI )
AUX(6*I)=0»DO
IREC=0
IHLF=-1
141
-------
ISTEP=0
IEND=0
C
C
C START OF A RUMGE-KUTTA STEP
4 IF< CX+H-XEND)*H> 7*6* 5
5 H=XEND-X
6 IEND=1
C
C RECORDING OF INITIAL VALUES OF THIS STEP
7 CALL OUTP
BJ=BCJ>
CJ=CCJ)
DO 11 I=1*NDIM
R1=H*DERYCI)
R2=AJ*I > + R2-CJ*Rl
IFCJ-4)12*15*15
12 J=J+1
IFCJ-3)13*14*13
13 X=X+.5DO*H
14 CALL FCTCX*Y*DERY)
GOTO 10
C END OF INNERMOST RUNGE-KUTTA LOOP
C
C TEST OF ACCURACY
15 IFC I TEST) 16* 16*20
C
C IN CASE ITEST=0 THERE IS NO POSSIBILITY FOR TESTING OF
C ACCURACY
16 DO 17 I=1*NDIM
17 AUX(4*I)=Y(I )
ITEST= 1
I STEP= I STEP+1 STEP-2
18 IHLF=IHLF+1
X=X-H
H=.5DO*H
DO 19 I=1*NDIM
Y(I)=AUX(1*1)
DERYCI)=AUX(2*I)
142
-------
19 AUXC6* I ) = AUXC3* I )
GOTO 9
C
C IN CASE ITEST=1 TESTING OF ACCURACY IS POSSIBLE
20 IMOD=ISTEP/2
IF(ISTEP-IMOD-IMOD)81*S3*81
21 CALL FCTCX*Y*DEKY)
DO 22 I=1*NDIM
AUX<5*I)=YU)
22 AUXC7*I)=DERY(I)
GOTO 9
C
C COMPUTATION OF TEST VALUE Dh.LT
23 DELT=O.DO
DO 24 I=1*NDIM
24 DELT=DELT+AUXC8*I)*DARSCAUXC4*I)-Y
IFCDELT-PRMTC4))28*28* 25
C
C ERROR IS TOO GREAT
25 IFCIHLF-10)26*36, 36
26 DO 27 I=1*NDIM
27 AUXC4, I) = AUX( 5* I )
ISTEP=ISTEP+I STEP-4
X = X-H
IEND= 0
GOTO 18
C
C RESULT VALUES ARE GOOD
28 CALL FCTCX*Y*DERY)
DO 29 I=1*NDIM
AUXC1*I)=YCI)
AUXC2*I)=DERY(I)
AUXC3*I)=AUX(6*I)
Y
IF(PRMTC 5))40* 30* 40
30 DO 31 I = 1*NDIM
YCI)=AUX(1*I)
31 DERY(I) = A.UX(2* I )
IREC=IKLF
I FCI END) 32* 32* 39
C
C INCREMENT GETS DOUBLED
32 IHLF=IHLF-1
ISTEP=ISTEP/2
H=H+H
IFCIHLF)4* 33* 33
33 I MOD=I STEP/2
IFCISTEP-I MOD-I MOD)4* 34*4
143
-------
34 IFCDELT-.02DO*PRMT<4»35> 35>4
35 IHLF=IHLF-1
ISTEP=ISTEP/2
H=H+H
GOTO 4
C
C
C RETURNS TO CALLING PROGRAM
36 IHLF=11
CALL FCT(X*Y*DERY)
GOTO 39
37 IHLF=12
GOTO 39
38 IHLF=13
39 CALL OUTPCX*Y.»DERY*IHLF>NDIM.»PRMT>
40 RETURN
END
144
-------
APPENDIX E
CALIBRATION OF 0.5 gpm ROTAMETER
-------
100
C/1
5
UJ
O.I
0.2 0.3 0.4 0.5
FLOWRATE, GALLONS PER MINUTE
FIGURE 43 - CALIBRATION OF 0.5 gpm ROTAMETER
0.6
-------
APPENDIX F
CALIBRATION OF 60° V-NOTCH WEIR
-------
0.7
0.6
1 0.5
a:
0.4
8 0.3
oc
u
| 0.2
fe
I
CD
x
O.I
i i i i i i
0.01
0.02 0.03 0.040.05 O.I
FLOWRATE, CUBIC FEET PER SECOND
0.2 0.3
FIGURE 44 - CALIBRATION OF 60° V-NOTCH WEIR
-------
APPENDIX G
COMPUTER PROGRAM - ANALYSIS
-------
c
C COMPUTER PROGRAM TO CONVERT RAW LABORATORY DATA
C TO JET AXIS DATA.
C
C
REAL MC100)
DIMENSION CC100)>SC100)*DC100)*X3C100)*CXDC100)*CXDSQC
$100),C1C100
&)*CALC2C100)*XDISTC25)*DIF2C100)*YMEAN1C25)*CONMAXC25)
f *STDEV1C25)
C YCEPT AND SLOPE ARE THE PERTINENT PARAMETERS TO
C DEFINE THE CALIBRATION CURVE. YCEPT AND SLOPE ARE
C DIFFERENT FOR EACH RUN. THE CALIBRATION CURVE ISA
C LINEAR EXPRESSION WHICH RELATES THE MILLIVOLT OUTPUT
C TO THE CONCENTRATION OF SALT.
C
READC 5*950)YCEPT*SLOPE
C IRUNNO = THE EXPERIMENTAL RUN NO.
C PGTV = IS THE RELATIVE DEPTH OF THE MEASURING DEVICE
C TO TOP OF THE WATER
C. PGBV = IS THE RELATIVE DEPTH OF THE MEASURING
C DEVICE TO THE BOTTOM OF THE WATER OR THE
C BOTTOM OF THE LABORATORY FLUME
C PROBTW = IS THE RELATIVE LOCATION OF THF, CONDUCTIVITY
C PROBE WITH RESPECT TO THE MEASURING DEVICE
C AND TOP OF THE WATER
C ELEJET = THE HEIGHT OF THE JET DISCHARGE POINT FROM
C BOTTOM OF THE FLUME
C THE ABOVE VARIABLES ARE NEEDED TO TRANSFORM THE
C LABORATORY DATA OF THE MEASURED DEPTHS TO ACTUAL
C DEPTHS WITH RESPECT TO THE JET DISCHARGE POINT.
READC 5*903)1RUNNO* PGTW*PGBW*PROBTW* ELEJET
READC 5*920)FROUDE
C FROUDE = DENSIMETRIC FROUDE NUMBER FOR THIS EX PERI -
C MENTAL RUN
WRITEC6*802)IRUNNO
DEPTH =PGTW-PGBW
DEPINC = PGTW-PROBTW
ELEINC = ELEJET - PGBW
WEI TEC 6* 907)PGTW* PGBW* PPOBTW* DEPTH* DEPINC* ELEINC
C N = THE NUMBER OF CROSS-SECTIONS ALONG THE JET AXIS
C Xl= THE RELATIVE LOCATION OF THE JET DISCHARGE POINT
C IN THE X-DIRECTION. THIS VALUE MAYBE ZERO OR
C SOME OTHER RELATIVE X VALUE
C DIAM = THE JET DIAMETER
READC5*900)N*X1*DIAM
READC5*912)AK*BETA
C AK = THE VELOCITY RATIO FOR THIS EXPERIMENTAL RUN
C BETA = THE INITIAL ANGLE OF DISCHARGE
WRITEC6*801)N*X1*DI AM
147
-------
DO 10 J=1,N
C Nl = NUMBER OF Y LOCATIONS ON THE CROSS-SECTION OF THE
C JET AXIS THAT WERE MEASURED
C X = THE RELATIVE X LOCATION OF THE CROSS-SECTION WITH
C RESPECT TO THE JET DISCHARGE POINT.
C CI = THE DELTA Y VALUE OR THE INCREMENTAL Y DISTANCE
C ALONG THE CROSS-SECTION
C MCI) « THE AVERAGE MILLIVOLT OUPUT AT THIS PARTICULAR
C Y LOCATION
C DC I) = THE RELATIVE Y LOCATION AT WHICH THE MILLIVOLT
C READINGS WERE TAKEN
READC5,910)N1*X,CI,CMCI),DCI ),I=1*N1)
CI=CI*30.48
S02PI a 2.506628274631
DIF22 = 0.0
CNSUM1 =0.0
CNSUM2 * 0-0
FREO a 0.0
X2=CX-X1)/DI AM
XDI STCJ)=X2*DI AM
WRI TEC 6* 901) J> X, X2
WRITEC6,902)
C
C
C CALCULATES THE MEAN Y-DISTANCE* THE STANDARD DEVI A-
C TION* AND VARIANCE OF THE PARTICULAR CROSS-SECTION
C
C
DO 20 I=1*N1
DCI)=DCI)-PGBW+DEPINC-ELEINC
DO) = DCI)*30.48
C(I)=YCEPT + SLOPE*MCI)
IFCCCI) -GT. 0.000000)00 TO 21
CCI) = 0-00000000
SCI) = 10000000.00
GO TO 22
21 S(I)=1./CCI)
22 CKI) = CCI)*100.00
FREO = FREQ + CKI)
CXDCI ) = CKI) * DCI)
CXDSQCI) - CXDCI) * DCI)
CNSUM1 = CNSUM1 + CXDCI)
CNSUM2 = CNSUM2 + CXDSQCI)
20 CONTINUE
YN1EAN = CN SUM 1/FREO
STD = SQRTCCCNSUM2 - CCCNSUM1**2)/FREQ))/FREQ)
YMEAN1CJ)=YMEAN
STDEV1CJ)=STD
C
C
148
-------
C CALCULATES THE CONCENTRATION AT THE MEAN Y-DI STANCE
C
C
CONMAXC J>=CFREQ*CI >/CSTD*SQ2PI )
VAR = STD**2
DO 40 I = 1,N1
WRI TEC 6,8 00) MCI), SC I ) , DC I > , CC I ) , C 1 C I )
40 CONTINUE
WRI TEC 6*81 1)FREQ
WRITEC6,812)YMEAN, STD, VAR
WRITEC6>814>
C
C
C CALCULATES THE GAUSSIAN DISTRIBUTIONS USING THE ABOVE
C INFORMATION AND THE COMPARES THE CALCULATED VALUE
C WITH THE OBSERVED VALUE AND CALCULATES THE
C SQUARED DIFFERENCE.
DO 60 I = 1,N1
X3CI) = DCI> -YMEAN
CALC2CI) = C(FREQ*CI >/ C STD+SQ2PI » *EXPC -CX3C I ) **2> /C 2.
$*STD**2) )
DIF2CI) = **2
DIF22 = DIF22 + DIF2CI)
VRI TE( 6, 8 1 3) DC I ) ,X3C I ) > C 1 C I ) , C ALC2C I ) , DI F2C I )
60 CONTINUE
VRITEC6*815)DIF22
NPTS = Nl
XMAX = 60-00
C
£
C XMAX IS A SCALING FACTOR FOR THE PLOT IN SUB-
C ROUTINE PLOT2D.IT SHOULD HOWEVER BE SOME
C INTEGER VALUE OF 12 SINCE 120 COLUMNS ARE
C USED IN THE OUTPUT
WRITEC6*957MRUNNO,FROUDEJ AK*XDISTC J)
CALL FLO T2D< NPTS* XMAX, D*Cl*CfiLC2)
10 CONTINUE
WRITEC6,958) ^Amj
C WRITES OUT THE PERTINENT INFORMATION FOR EACH
C CROSS- SECTION, X- DISTANCE, MEAN Y- DISTANCE, MAXI-
C MUM CONCENTRATION, AND THE STANDARD DEVIATION
WRI TEC 6,959 )XDI STC J3 , YMEAN 1 C J5 , CONMAXC J> , STDEV1 C J)
, I 2/T 1 1, 0
801 FORMATCT11,'NO. OF OBSERVATIONS '
, F5-2,' CENTIMETERS VT11, -DIAMETER OF
SETS T33, '= '>
&T36,F5-2, ' CENTIMETERS1///)
149
-------
802 FORMATC1H1/////T33*'RUN NO. «*I2///)
811 FORMATCT58*' '/T46*'SUMMATION = SF1I.7)
812 FORMATC T26* 'Y-MEANST48* '= »*T52>F8-4/T26* 'STANDARD DEV
$IATION'*T48
** '='*T54*F8.6/T26* 'VARIANCE'*T48* •« '*T54*F8«6)
8 13 FORMATC T9*F6. 3* T23*F6. 3*T35*F1 1. 1, T49*F1 1. 7* T73* Fl 1 . 7)
814 FORMATC1H3T9*'OBSERVED'*T22*'DISTANCE'*T37*'OBSERVED',
ST51*'CALCULA
ATEDST72* ' DI FFERENCE VT1 1* 'DEPTH '*T24* 'FROM'*
4T34, fCONCENTRATION',T53* •SQUARED VT11> 'CCM) S T24*
A'MEAN',T38* 'X100'*T54* fX100'jT72* 'FOR NORMAL '/
&T24*•CCM>'*T51-'NORMAL CURVE*,T75*'CURVE')
815 FORMATCT73*' VT60* 'SUMMATION = '*
*T76*F1 1-7)
900 FORMATCI5*F5.2»F5.3)
901 FORMATC1H1/////T26*'CROSS-SECTION NO. '*I2//T4,'DISTAN
$CE FROM DISC
4HARGE POINT '>T45*'='*T48*F6.2*' CENTIMETERS'/T4,'X/D
S(NO. OF JET
ADI AMETERS DOWNSTREAM) '*T45> '= ST48*F6-2>
902 FORMATCT27* *Y- DISTANCE VT28> 'FROM JET V IX* 'M1LLI VOLT',
$T14>'DILUTIO
*N'*T28* 'DISCHARGE'*T41* 'CONCENTRATI ON'*T5S* 'CONCENTHAT
JION*
S/T30* 'POINT'* T62* 'X100VT30* '(CM) ')
903 FORMAT(I2*4F5.3)
907 FORMATCT7* 'POINT GAGE TOP OF WATER'*T35* '='*T39*F5.3*T
$47* 'FEETVT7
ft*'POINT GAGE BOTTOM OF WATER'*T35*'='*T39*F5.3*T47*'FE
$ET'/T7*'PROB
&E TOP OF WATER'*T35* '=**T39*F5.3* T47,'FEET VT7* 'DEPTH
JOF WATER**T3
A5* '='*T39*F5.3*T47* 'FEETVT7, 'DEPTH INCREMENT'* T35* '= '
$*T39*F5-3*T4
A7^ 'FEET VT7j. 'LOCATION OF JET ABOVE VT7* 'BOTTOM OF FLUM
$E',T35* '='*T
A39*F5.3*T47*'FEET'///>
910 FORMATCI5*F5.2*F5-3/C6F10.2))
912 FORMATC2F5.2)
920 FORMATCF5.2)
950 FORMATC2F15.9)
957 FORMATC1HIT53*'RUN NO. = '*T64*I2//T50*'FROUDE NO. = '
$*T64*F5.2//T
A50*'VELOCITY'/T52*'RATIOCK) = '*T64*F5.2//T50* 'X-DISTA
$NCE ='»T63*F
A8.4*T72*•CENTIMETERS'////)
958 FORMATC 1H1T3* 'X-DI STANCE'* T20* 'Y-DI STANCE'* T41 * 'MAXIMU
$M'*T54* 'STAN
ADARDVT39* 'CALCULATED'* T54* 'DEVIATION'/
AT38* 'CONCENTRATIONVT2* '< CENTIMETERS) '*T21* 'CMEAN DI ST
$.) V
150
-------
&T20*'(CENTIMETERS)'//)
959 FORMATCT5* F8 • 4, T22., F 7. 3.. T4 1 * F9 . 5* T56-. F 7. 4)
END
C
c
C
C PLOTS THE OBSERVED AMD CALCULATED VALUES ON THE SAME
C PLOT
C
C
SUBROUTINE PLOT2DCNPTS* KM AX.. D* Cl* CALC2)
INTEGER BLANK/' '/*O/ '0 ' /*G/*G'/
INTEGER BAR/"C•/>MINUS/'-'/
DIMENSION D< 100)*C1C100)>CALC2(100)*LINE(121)*XTEMPC13
$)
DELTY = D(1>-DC2>
DO 100 I=1^NPTS
DO 101 J=1 * 121
101 LINE(J)=BLANK
IFCABS(D(I)> .GT. DELTY/2.5GO TO 200
DO 102 K=l,121
102 LINECK)=MINUS
200 DO 104 J=1>I2l> 10
104 LINECJ)=BAR
N=«CALC2CI )*120«>/XMAX)+ 1.5
LINECN)=G
N=CCC1CI )*120.)/XMAX)+l-5
LINECN>=0
100 WRITE(6^900)DCI )*LINE
X = 0-0
DELTX=XMAX/12.
DO 103 J=l*13
XTEMP(J)=X
103 X=X+DELTX
WRITEC 6^901)XTEMP
900 FORMATCIX*F9.4^2X>121A15
901 FORMATC 12X,13C 'C %9X)/Tl1, 13(F6.2
RETURN
END
151
-------
APPENDIX H
OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S AND ABRAHAM'S MODEL
-------
30
20
K> 20 30 40
60 70 80 90 IOO
1000
s/D
FIGURE 45 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 18
152
-------
60
40
20
-o
0
40
80 120
x/D
160
8/D
200
FIGURE 46 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 13
153
-------
60
40
20
40
80 120
x/D
160
200
F =21.1
=5.3
ff. * 90
1000
FIGURE 47 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 12
154
-------
60
40
T~ "I 1 1 r
20
J L
0
100
40
,n
x/D
160
200
1111
F =20.4
K. * 10.2
ft s 78.0
a «0.5
10
DILUTION
I -- 1 - 1 — [MM
0 0
1
i I I 1 11 r
y
, , , , , M
10 100
8/D
1000
FIGURE 48 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 11
155
-------
30
TO K> 20
30 40 50 60
x/D
70 80 90 IOO
100
F *20.2
K -20.2
BOO
3/D
FIGURE 49 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 16
156
-------
30
K> 20
30 40 5O 60
x/D
70 80 90 100
100
F « 10.9
UK -5.5
*>•
1000
FIGURE 50 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 10
157
-------
100
10
1
• T 1 1 | 1 1 1 1
F «I0.3
LK «io.3
. 0;«90°
£*«74.0°
a «0.4
: o
— 1 1 1 [Mil
y
DILUTION"-/
o0/
o 7
•—Tar-"** A
A
—I— TT
i i i 1 i i ij^
10 100 KX
X)
s/D
FIGURE 51 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 9
158
-------
x/D
100
10
~~ T
- F « 10.1
Z- K * 20.1
0* 81.8°
a «0.2
1 _ 1 _ 1 1 III!
1 -- 1 - 1 | 1 1 1 1
DILUTION-;
10
50
1 1 1 | 1 1 1 L
i i i i i i i
s/D
100
1000
FIGURE 52 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 15
159
-------
30
0 10 20 30 40 50 60 70 80 90 100
100
KXX>
FIGURE 53 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 34
160
-------
80 120
x/0
KX>
200
F «42.7
K« 10.6
• 60°
53.8°
a -0.3
3/D
KXX>
FIGURE 54 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 33
161
-------
30
1 r
— —I 1 r
10 20 30 4O 50 60 70 8O 9O IOO
X/D
F «22.5
K « 5.6
•60°
A, « 52.0°
a «0.4
s/D
100
KXX>
FIGURE 55 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 32
162
-------
60
T~ 1 1 T
40
2O
o o
100
160
ZOO
F =21.3
K= 10.6
1000
s/D
FIGURE 56 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 28
163
-------
30
90 100
100
[—
F «20.6
. K • 20.6
. 53 0«
a «0.2
s/D
FIGURE 57 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 30
164
-------
30
T 1 1 T
Q
>,
20
10
60 70 80 90 100
s/D
1000
FIGURE 58 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 27
165
-------
K>
100
10
I -- 1 — I
F «I0.6
IK «I0.6
520-
0.3
-i—i—i i i 111
DILUTION 7
b/De
1_L_L
T -- 1
10
3/D
100
U
I « I I I I 11
1000
FIGURE 59 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 29
166
-------
10
5
K>
100
20
,r.
x/D
40
50
1000
FIGURE 60 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 31
167
-------
30
20
10
-i 1 1 1 1 1 i r
j - 1
-I L
"90 IOC
_
K> 20 3O 40
50
x/D
60 70 80
lOO
F " 46.9
IK «5.9
- A-30LO-
a «0.3
3/D
FIGURE 61 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 26
168
-------
K> 2O 30 40 50 60 70 80 90 TOO
s/D
FIGURE 62 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 22
-------
60
40
-i r
-T— —r
20
o .
F = 23.5
K =5.9
8/D
KXX)
FIGURE 63 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 21
170
-------
60
T T
I T
40
20
80 120
x/D
160
200
lOOc
F =21.6
. K = 10.8
3/D
FIGURE 64 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 20
171
-------
30
60 70 80 90 100
K) 20 30 40
100
KXX>
3/D
FIGURE 65 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 24
172
-------
30
20
10
K> 20 30 40 50 60 70 80 90 100
100
3/D
FIGURE 66 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 19
173
-------
10
K) 20 30 40 50
x/D
KXX)
FIGURE 67 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 23
174
-------
15
K>
10
20 x/D 3°
40
50
100
10
1 1 1
F * 10.4
K =20.9
= 29.3°
a « 0.15
J I L
10
./o
1 1 1 1 1 1 1
100
IOOO
FIGURE 68 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY FAN'S MODEL - RUN NO. 25
175
-------
30
10 20 30 40 50 60 70 80 90 100
too
F *40.0
K -5.0
K>
DILUTION
o
#
A
^/
,
I
T I I
10 100
8'/0
1000
FIGURE 69 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 18
176
-------
60
40
20
~i r
•o-
o
45-
x'/D
120
160
200
100
s'/D
IOO
1000
FIGURE 70 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 13
177
-------
6O-
40
>
h
S
20
T 1 1 r
o o
x'/D
20
160 200
100
F «20.0
K -5.0
1000
FIGURE 71 -• OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 12
178
-------
60
T 1 1 r
40
§
20
~SO~ 120 160 200
x'/D
F « 20.0
K , - 10.0
' * 90°
s'/D
100
1000
FIGURE 72 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 11
179
-------
30
10 20 30 40 50 60 70 80 90 100
100
F
K
10
^ "~
20.0
20.0
90°
DILUTION
I I I I
I I
10 100
s'/D
ITT
1000
FIGURE 73 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 16
180
-------
30
10 20 30 40 50 6O 70 80 90 100
DILUTION
1000
s'/D
FIGURE 74 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 10
181
-------
15
10
20
x'/D
30
100
4O
50
1000
S'/D
FIGURE 75 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 9
182
-------
15
KVD
100
1 1 II
F * 10.0
K =20.0
• 90°
10
I -- 1 - 1 MM
DILUTION ?
/a
40
50
1000
FIGURE 76 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 15
183
-------
30
20
10
i i
o o
K> 20 30 4O 50 60 70 80 90 IOO
X'/O
F «44.0
K -5.5
" 60°
s'/D
FIGURE 77 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 34
184
-------
60
40
§
T T
^^ * L^
:
o
F -42.0
K -10.5
s'/D
IOO
1000
FIGURE 78 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 33
185
-------
30
I
20
10
10 2O 3O 4O
5O
x'/O
60 70 80 90 100
s'/D
100
1000
FIGURE 79 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 32
186
-------
60
x'/D
200
1000
FIGURE 80 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 28
187
-------
30
10 20 30 40 50 60 70 80 90 100
20.5
20.5
60°
DILUTION
s'/D
IOO
IOOO
FIGURE 81 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 30
188
-------
15-
10
20 ,^ 3°
x'/D
100
10
MM
F « 11.0
1 K « 5.5
DILUTION
1 -- 1 - 1 MM
10
i i i i i i i i
40
50
I I' T { 1 1 1 I.
s'/D
100
1000
FIGURE 82 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 27
189
-------
x-/D *>
50
lOOr
S'/D
FIGURE 83 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 29
190
-------
15
10
10
100
10
F «I0.3
K -20.5
- 60*
DILUTION
20
x'/D
30
I I I I 11
10
s'/D
100
40
50
1000
FIGURE 84 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 31
191
-------
O 10 20 30 40 50 60 70 80 90 100
X'/D
100
87D
FIGURE 85 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 26
192
-------
30
20
10
"0 ~ 10 20 3O 40 50 60 70 80 9O 100
X'/D
1000
FIGURE 86 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 22
193
-------
60
T— T
4O
§
>»
20
1_ l_
160
200
1000
FIGURE 87 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 21
194
-------
60
40
40
x-/o
160
200
- DILUTION
S'/D
100
1000
FIGURE 88 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 20
195
-------
30
20
I
*
•»
10
10 10 20 30 40 50 60 70 80 9O 100
x'/D
100
1000
FIGURE 89 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 24
196
-------
3O
20
10
1000
FIGURE 90 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 19
197
-------
15
10
10
x'/D
A f IS
100
4O
50
F -I0.7
K -I0.7
' • 45°
1000
f'/D
FIGURE 91 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 23
198
-------
15
100
10
T 1—r~ I i i 11
F -10.4
_K -20.7
- DILUTION
i ' L
/
50
I [MIL
10
s'/D
100
1000
FIGURE 92 - OBSERVED VALUES AND THEORETICAL CURVES
PREDICTED BY ABRAHAM'S MODEL - RUN NO. 25
. GOVERNMENT PBINTING OfFICE:1974 546-319/385
199
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SELECTED WATER
RESOURCES ABSTRACTS
INPUT TRANSACTION FORM
Kt-
w
Negatively Buoyant Jets in a Cross Flow
Anderson, J.L., Parker, F.L., & Benedict, B.A.
". P- 'iroth "~JZ"C •
A ~-rtK,..
16130 FDQ
Vanderbilt University
Environmental & Water Resources Engineering
Nashville, Tennessee
i? ;v°'--"-W3 r-MI* ti Environmental Protection Agency
Environmental Protection Agency No. EPA-660/2-73-012, October 1973,
nd
200 words
Modification of Fan's and Abrahams jet diffusion models were used to predict the
trajectory and dilution of a negatively buoyant jet. Such jets can occur when a
chemical waste is discharged into a less dense ambient water or when cool, hypo-
limnetic water is used for condenser cooling water and discharged into less dense
surface waters, then a sinking jet would result. Experimental investigations
were conducted involving different combinations of densimetric Froude number, velocity
ratio, and initial angle of discharge. Salt was used as the tracer, yielding a fluid
that was denser than the ambient receiving water and facilitated measuring concentratioi
profiles of the jet plume. The experimental data was then fitted with predicted jet
dilution, trajectory, and cross sectional values for each model. The values of the
entrainment coefficient were chosen as the one which best fit the experimental
data for the particular combination of densimetric Froude number, velocity ratio,
and initial angle of discharge. The value of the drag coefficient was chosen as
zero for both models since any other value would predict a trajectory whose rise
would be less than experimentally observed. Typically, for all angles of discharge
the value of entrainment increased with a decrease in the velocity ratio and with an
increase in densimetric Froude number. Additionally, there was a marked decrease
in the entrainment coefficient with a decrease in the initial angle of discharge.
17a. Descriptors
Thermal Pollution, Thermal Power Plants, Density Currents, Entrainment
17h. Identifiers
Jet Discharge, Near Field, Jet Trajectory, Negative Buoyant Jet, Waste Dilution,
Densimetric Froude Number
05B
9, K '.city -ss..
20. Sccur y Cfass.
'
?.}.
. of
Send To:
WATER RESOURCES SCIENTIFIC INFORMATION CENTER
U.S. DEPARTMENT OF THE INTERIOR
WASHINGTON. D. C. 2O24O
Dr. Frank L. Parker
Vanderbilt University, Nashville, Tenn.
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