EPA 600/3-76-101
September 1976
Ecological Research Series
AN EXPERIMENTAL/ANALYTICAL
INVESTIGATION OF DEEP SUBMERGED
MULTIPLE BUOYANT JETS
Environmental Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Corvallis, Oregon 97330
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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 five series. These five broad
categories were established to facilitate further development and application of
environmental technology. Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface in related fields.
The five series are:
1. Environmental Health Effects Research
2. Environmental Protection Technology
3. Ecological Research
4. Environmental Monitoring
5. Socioeconomic Environmental Studies
This report has been assigned to the ECOLOGICAL RESEARCH series. This series
describes research on the effects of pollution on humans, plant and animal
species, and materials. Problems are assessed for their long- and short-term
influences. Investigations include formation, transport, and pathway studies to
determine the fate of pollutants and their effects. This work provides the technical
basis for setting standards to minimize undesirable changes in living organisms
in the aquatic, terrestrial, and atmospheric environments.
This document is available to the public through the National Technical Informa-
tion Service, Springfield, Virginia 22161.
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EPA 600/3-76-101
September 1976
AN EXPERIMENTAL/ANALYTICAL INVESTIGATION
OF DEEP SUBMERGED MULTIPLE BUOYANT JETS
by
L. D. Kannberg
L. R. Davis
Oregon State University
Corvallis, Oregon 97330
Grant No. R-800818
Project Officer
M. A. Shirazi
Assessment and Criteria Development Division
Corvallis Environmental Research Laboratory
Corvallis, Oregon 97330
U.S..ENVIRONMENTAL PROTECTION AGENCY
OFFICE OF RESEARCH AND DEVELOPMENT
CORVALLIS ENVIRONMENTAL RESEARCH LABORATORY
CORVALLIS, OREGON 97330
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DISCLAIMER
This report has been reviewed by the Corvallis
Environmental Research Laboratory, U. S. Environmental
Protection Agency, and approved for publication. Ap-
proval does not signify that the contents necessarily
reflect the views and policies of the U. S. Environ-
mental Protection Agency, nor does mention of trade
names or commercial products constitute endorsement
or recommendation for use.
11
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CONTENTS
LIST OF FIGURES iv
LIST OF NOMENCLATURE AND SYMBOLS x
ACKNOWLEDGMENTS xiv
SECTIONS
I INTRODUCTION 1
II SUMMARY 3
III CONCLUSIONS 5
IV HISTORICAL BACKGROUND 8
V EXPERIMENTAL WORK 11
Modeling Parameters 11
Apparatus and Data Acquisition 12
The Data and Its Treatment 28
Experimental Error Analysis 32
The Results 36
VI ANALYTICAL WORK 81
Introduction 81
The Analytical Problem 82
Employing Similar Profiles 94
Zone of Flow Establishment 98
Zone of Established Single Plume Flow 106
Zone of Merging Plumes 110
Boundary Turbulence Terms 120
Entrainment 123
Tuning the Model - Results 132
Some Comparisons and Predictions 170
Plume Width 178
VII REFERENCES 185
VIII APPENDIX A 189
111
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FIGURES
No. Page
1 Diffusers Used in the Experimental Work,
L/D's «* 10, 5, 2.5 14
2 Warm Water Constant Head Reservoir 16
3 Experimental Apparatus and Electronic Instrumentation 19
4 Plane of Traverse of the Sensor 22
5 Typical Visicorder Print of Temperature and Position 25
6 Data Treatment Process 26
7 Example of Typical Excess Temperature Data and It's
Representative Curve 29
8 Example of Typical Vertical Width Data and It's
Representative Curve 30
9 Example of Typical Trajectory Data and It's
Representative Curve 31
10 Confidence Interval for Typical Excess Temperature
Data 33
11 Confidence Interval for Typical Vertical Width Data 34
12 Confidence Interval for Typical Trajectory Data 35
13 Effect of Varying R on Excess Temperature Ratio for
L/D = 10., 9 = 15, F = 57.0. 38
14 Effect of Varying R on Width for L/D = 10., 0 = 15,
F = 57.0. 39
15 Effect of Varying R on Trajectory for L/D = 10.,
0 = 15, F = 57.0. 40
16 Effect of Varying R on Excess Temperature Ratio for
L/D = 10,, 6 * 90, F - 31.0. 41
17 Effect of Varying R on Width for L/D = 10., 6 = 90,
F = 31.0. 42
18 Effect of Varying R on Trajectory for L/D = 10.,
6 = 90, F = 31.0. 43
IV
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No. Page
19 Effect of Froude Number on Excess Temperature Ratio
for L/D = 10., 9 = 15, R - 0.103. 44
20 Effect of Froude Number on Width for L/D = 10., 6 = 15,
R = 0.103. 45
21 Effect of Froude Number on Trajectory for L/D = 10.,
0 = 15, R = 0.103. 46
22 Effect of Froude Number on Excess Temperature Ratio for
L/D = 10., 9 = 90, R = 0.250. 47
23 Effect of Froude Number on Width for L/D = 10., 0 = 90,
R = 0.250. 48
24. Effect of Froude Number on Trajectory for L/D = 10.,
0 = 90, R = 0.250. 49
25 Effect of Angle on the Excess Temperature Ratio for
L/D = 10., F = 31.1, R = 0.248. 51
26 Effect of Angle on Width for L/D = 10., F = 31.1,
R = 0.248. 52
27 Effect of Angle on Trajectory for L/D = 10., F = 31.1,
R = 0.248. 53
28 Effect of Angle on Dilution as Plotted with Trajectory
for L/D = 10., F = 31.1, R = 0.248. 54
29 Effect of L/D on Excess Temperature Ratio for 0 = 90,
F = 10.2, R = 0.10. 56
30 Effect of L/D on Width for 9 = 90, F = 10.2, R = 0.10. 57
31 Effect of L/D on Trajectory for 0 = 90, F = 10.2,
R = 0.10. 58
32 Effect of L/D on Excess Temperature Ratio for 0 = 90,
F = 11.0, R = 0.50. 59
33 Effect of L/D on Width for 0 = 90, F = 11.0, R = 0.50. 60
34 Effect of L/D on Trajectory for 0 = 90, F = 11,0,
R = 0.50 61
35 Effect of L/D on Excess Temperature Ratio for 0 = 90,
F = 54.5, R = 0.05. 62
36 Effect of L/D on Widths for 0 = 90, F = 54,5, R = 0,05. 63
37 Effect of L/D on Trajectory for 0 = 90, F = 54.5,
R = 0.05. 64
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No. Page
38 Effect of L/D on Excess Temperature Ratio for
6 = 90, F = 58.8, R = 0,50. 65
39 Effect of L/D on Width for 9 = 90, F = 58.8,
R = 0.50. 66
40 Effect of L/D on Trajectory for 0 = 90, F =* 58.8,
R = 0.50. 67
41 Effect of L/D on Excess Temperature Ratio for 9 = 45,
F = 10.7, R = 0.10. 68
42 Effect of L/D on Width for 9 = 45, F = 10.7, R = 0.10. 69
43 Effect of L/D on Trajectory for 9 = 45, F = 10.7,
R = 0.10. 70
44 Line of Traverse in a Current with Twin Vortex
Structure 71
45 Effect of Current to Discharge Velocity Ratio with
Angle and X/D as Predicted by the Regression Analysis 77
46 Effect of Angle with X/D at R = 0,10 as Predicted
by the Regression Analysis 78
47 Effect of Spacing with X/D as Predicted by the
Regression Analysis 79
48 The "Natural" Coordinate System Employed by Hirst 87
49 The Dominant Zones of Flow for Multiple Port Discharges 96
50 Comparison of the Gaussian and 3/2 Power Profiles 99
51 The Coordinate System for the Merging Plume Analysis 112
4
52 Model Prediction and the Morton, et al. Empirical
Curve for the Momentum Jet 134
53 Model Prediction and Experimental Data for Trajectory
of Single Port Discharges 135
54 Model Prediction of Trajectory of Single Port Discharges 136
55 Model Predictions and Experimental Data of Dilution for
Single Port Discharges (Original Graph by Cederwall3°) 137
56 Model and Experimental Crossflow Starting Lengths 139
57 Crossflow Model Prediction and Experimental Data
Trajectory Comparison 140
vi
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No,
Page
58 Concentration Profile for F = 20, and R = 0.125.
Ambient Flow Strikes Plume from Top of Figure,
(Taken from Fan25, Page 127) 141
59 Concentration Profiles for F = 40. and R = 0.125.
Ambient Flow Strikes Plume from Top of Figures
CTaken from Fan25) 141
60 Dilution for Crbssflow Discharge from a Single Port,
R = 0.0625, Compared to Fan25 142
61 Dilution for Crossflow Discharge from a Single Port,
R = 0.0825. 143
62 Dilution for Crossflow Discharge from a Single Port,
. R = 0.125 144
63 Dilution for Crossflow Discharge from a Single Port,
R = 0.25 I45
64 Trajectory Comparisons for Single Port Crossflow Dis-
charge, Model Includes a Curvature Term in the
Entrainment Function I49
65 Dilution Comparisons for Single Port Crossflow Dis-
charge. Curves Include Predictions by the Model
with a Curvature Term in the Entrainment Function,
R = 0.0625 15°
66 Dilution Comparisons for Single Port Cross Flow
Discharge. Curves Include Predictions by the Model
with a Curvature Term in the Entrainment Function,
R = 0.25 151
67 Co-flow Starting Length Comparison, Single Port Dis-
charge, Model Contains the Turbulence Terms 153
68 Velocity Dilution for Co-flow Single Port Discharge,
the Model Employs Turbulence Terms 154
69 Experimentally Obtained Co-flow Thermal Dilutions
of This Study for L/D = 10 and Various R's 156
70 The Value of Various Entrainment Models as Plotted
Against Plume Width b 158
71 Comparison of Model Predicted Trajectories with
Experimentally Obtained Trajectories for L/D = 2.5,
Crossflow Discharge 161
72 Comparison of Model Predicted Trajectories with
Experimentally Obtained Trajectories for L/D = 5.0,
Crossflow Discharge I62
vii
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No. Page
73 Comparison of Model Predicted Trajectories with
Experimentally Obtained Trajectories for L/D ^ 10,
Crossflow Discharge i63
74 Comparison of Experimental and Model Predicted
Excess Temperature for L/D = 2.5, R = 0,10,
Crossflow Discharge 164
75 Comparison of Experimental and Model Predicted
Excess Temperature for L/D = 2.5, R = 0.50,
Crossflow Discharge 165
76 Comparison of Experimental and Model Predicted
Excess Temperature for L/D = 5.0, R = 0.10,
Crossflow Discharge 166
77 Comparison of Experimental and Model Predicted
Excess Temperature for L/D = 5.0, R = 0.50,
Crossflow Discharge 167
78 Comparison of Experimental and Model Predicted
Excess Temperature for L/D = 10., R = 0.10,
Crossflow Discharge 168
79 Comparison of Experimental and Model Predicted
Excess Temperature for L/D = 10., R = 0.50,
Crossflow Discharge 169
80 Comparison of Excess Temperature Predicted by
Several Models and Experimental Data for L/D = 10,
R = 0.0-, F = 11, Horizontal Discharge 172
81 Comparison of Excess Temperature Predicted by
Several Models and Experimental Data for L/D = 10.,
R = 0.0, F = 30, Horizontal Discharge 173
82 Comparison of Excess Temperature Predicted by
Several Models and Experimental Data for L/D = 10.,
F = 55, R = 0.0, Horizontal Discharge 174
83 Comparison of Model Predicted Trajectories with
Experimental Data for L/D » 10, R = 0.0, Horizontal
Discharge 175
84 Trajectory and Dilution Prediction for Various Port
Spacings, F = 30, R, = 0.0, Horizontal Discharge 177
85 Comparison of Various Models and Experiment for
Merging Jets Excess Temperature, Emphasis on Comparison
of the "Entrainment Area" and '^Transition" Entrainment
Results 179
viii
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No.
86
87
88
Comparison of Momentum Jet Center-line Velocity
Predictions of Several Models and the Empirical
Curve of Morton, et al . "*
Comparison of Momentum Jet Half-Radii Predictions
of Several Models with Experimental Data and the
Empirical Curve of Morton, et al.1*
Comparison of the Width Predictions of the Koh and
Fan^3 Transition Model and the Davis Merging Model
with Experimental Data
181
182
184
IX
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LIST OF NOMENCLATURE AND SYMBOLS
A - Area
A - Entrainment surface area
entr
a - Entrainment coefficient
an , _ - Entrainment coefficients, also regression fit
u,i,^,... coefficients
a!. - Entrainment coefficient
41
al - Entrainment coefficient
4
a.. - (i,j - 1-5) Zone of flow establishment simultaneous
•* equation coefficients
B - Slot plume discharge point width
b - 3/2 power profile plume half width = .53b1
C - Species concentration
C - Concentration at port discharge
C - Ambient concentration
00
CD - Drag coefficient
c - Specific heat
c, ~ _ , - Entrainment coefficients for Davis model, zone of
estabiishment
D - Port diameter
D - Species diffusion coefficient
d. 0 - Coefficients defined for the zone of flow establishment
l — o
E - Entrainment
E - Round jet entrainment
r
E - Slot jet entrainment
o
e - Slot jet entrainment coefficient
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F - Froude Number = UQ / C ^— gD)1/2
F. - Plume local Froude Number = u / ( p — gb)
L c o
Fn - Drag force
1 - Time averaged quantity
f - Fluctuating quantity
f-, 2 - Defined quantities for the solution of simultaneous
' »•*• equations in the zone of flow establishment
G, - Flux quantities in the zone of single plume flow
1 , Z , o , 4
~g - Gravitational .force (without bar - it is the gravita-
tional constant
Hi •? •* A ~ plux quantities in the merging zone
1 , Z , O , 4
h, ~ - Incomplete integrals defined in the merging zone
1 , Z , O
i - Coefficients defined for the zone of flow establish-
k - Thermal conductivity
L - Distance between ports
N - Normal terms, employed in the drag force relation
, 1 , z , o
P - Pressure
P - Motion pressure [P-P^)
q~ - Defined by equation (39)
R - Towing ratio = U / U
0 oo O
r - Plume radius and radial coordinate
r - Species core radius for the zone of flow establish-
ment
r - Temperature core radius
r - Velocity core radius
r - Port radius = D/2
o
S,s - Distance along centerline and centerline coordinate
XI
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S - Starting length
T - Temperature
T ,1^ - Ambient temperature
T - Centerline temperature
c
ATC - CTC - TJ
T - Port discharge temperature
t - Time
t~ - An arbitrary point in time
U - Port discharge velocity
U^ - Ambient velocity
u - Velocity in the S direction
u - Centerline velocity in the S direction
C
Au - (u - U cos 9, sin 9,)
C C fc A
V - Vector velocity
v - Velocity in radial, r, direction
tf - Plume width, vertical or X-sectional
X - Horizontal downstream distance and coordinate
x - Dummy variable
Y - Vertical coordinate and height above ports
Z - Transverse coordinate along line of ports
a - Measure of merging = L/b
g - Coefficient of thermal expansion
F - A variable
Y - Coefficient of species concentration expansion
xii
-------
e - General eddy diffusivity
EC - Species eddy diffusivity
e, - Thermal eddy diffusivity
e - Momentum eddy diffusivity
£ - Merging coordinate along line of jet centerlines
r\ - Merging coordinate perpendicular to the - S plane
9? and 9 - Angle of plume centerline to the X - Z plane
9, - Angle of projection of the plume centerline on the
X - Z plane .from the Z-axis
e.
e
2
- 9 at the discharge point
- 9_ at the discharge point
K, and K9 - Curvatures of the plume centerline with respect to
91 and 92
X - Schmidt Number
v - Kinematic viscosity
p - Density
Ap - (P - Pj
Apc " Cpc ' p«)
p - Plume centerline density
p - Discharge density
p^ - Ambient density
a - Standard deviation
$ - Various quantities
<|> - Circumferential plume coordinate
Xlll
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ACKNOWLEDGMENTS
The authors are indebted to several people for the efforts
they extended in helping to reach a successful conclusion to this
study. Of particular mention is Dr. Mostafa Shirazi, whose
comments, criticisms, and encouragements were most helpful. We
are also indebted to Jim Shew, Jim Carr, G. Kranick, N. Kunz,
Barbara Gniewosz and Mary Holland for their assistance in the
experimental program. We are grateful for the assistance of the
staff at the U. S. Environmental Protection Agency's Corvallis
Environmental Research Laboratory, Corvallis, Oregon. And finally,
our thanks are extended to Chris Snow for her enthusiasm while
typing this work, and to the Environmental Protection Agency whose
financial support made this study possible.
xiv
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SECTION I
INTRODUCTION
Energy consumption in this country is doubling at a rate of
once every 15 years. As it now appears, fossil and nuclear elec-
tric generating plants will produce nearly all of the electricity
required to meet these demands. Thermodynamically, these plants
are 30 to 40% efficient meaning that 70 to 60% of the energy de-
veloped must be rejected. Considering the magnitude of this
energy release, the "waste" heat (or thermal) discharge emerges
as a legitimate environmental concern. Increased awareness of
the ecological effects of these waste heat discharges has resulted
in stringent state and federal regulation controlling it.
While several methods are available to discharge waste heat,
including the use of cooling devices and cooling ponds, the least
expensive method is once through cooling. There is naturally a
strong demand and competition for such use, which is consequently
regulated by local and federal guidelines. Since many state
regulations specify the maximum allowable temperature regime in
the neighborhood of an outfall, knowledge of the dilution charac-
teristics of various discharge systems is required before issuing
of a permit. Deep submerged thermal discharge has been recognized
as one that provides rapid dilution, thus causing small surface
temperature in the water body. While in some cases a simple
single port outfall may provide adequate dilution, many others re-
quire multiple port or slot diffusers to comply with the required
regulations.
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This report is concerned primarily with multiple discharges
and the effects on dilution of neighboring plumes interfering with
one another. In an effort to obtain quantitative information con-
cerning the dilution characteristics of merging thermal discharges
and in order to isolate these effects from others such as surface
and bottom interactions, deep submerged discharges were experi-
mentally and analytically investigated. The results of this in-
vestigation are presented in two parts. The first part is con-
cerned with the experimental program. The second part details a
recently advanced multiple port analysis ^ ^ and presents the
results of its application to the discharge conditions considered
in the experiment.
[1] Footnotes shall be indicated by a number in square brackets,
superscript numbers without brackets indicate References.
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SECTION II
SUMMARY
The results of an experimental and analytical s'tudy of deep
submerged multiple-port thermal discharges are presented. The
experimental results include the measured downstream thermal di-
lution, width, and centerline trajectory of the buoyant thermal
plume from multiport jets. Independent parameters for which
measurements were obtained include port spacing, discharge Froude
Number, discharge angle, and discharge to ambient velocity ratio.
Results indicate that decreasing port spacing greatly decreases
thermal dilution. Changing port spacing will also affect tra-
jectory to a small extent while only slightly changing plume width,
Altering the Froude Number appears to have little effect on down-
stream dilution, width, or trajectory when an ambient current is
present. By increasing discharge angle from the horizontal,
greater initial dilution may be obtained as well as greater widths
and higher trajectories. The effect of ambient current on dilu-
tion depends on the angle of discharge. For crossflow discharges
the thermal dilution at any point downstream decreased with in-
creasing ambient current, while for co-flow the reverse was ob-
served. The jets were bent over rapidly for crossflow discharges
particularly when large ambient currents were present.
The analytical portion of this report employs the lumped dif-
ferential model of Hirst as modified for merging multiple jets
by Davis. The essential features of the analysis are: 1) the
gradual transition of the profiles from simple axisymmetric pro-
-------
files to merging profiles and finally to fully merged, pseudo-
slot, two-dimensional profiles, and 2) an entrainment based on the
available entrainment surface.
Results indicate that the overprediction of plume character-
istics associated with "transition" or "equivalent slot" models
may be overcome using such an analysis and that suitable predic-
tion may be obtained.
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SECTION III
CONCLUSIONS
The experimental program provided results that offer impor-
tant information on the dilution, width, and trajectory of deep
submerged multiport discharges. This information may be summar-
ized as follows:
1) Increasing the velocity ratio, R, increased dilution
with downstream distance, x, except at steep dis-
charge angles (>60°). The trajectories were
dramatically affected by the towing rate even for
very small angles.
2) Froude No. had little if any effect on dilution,
width, or trajectory for cases with ambient cur-
rent with the possible exception of close spac-
ing and low R's where slightly lower dilution for
higher Froude No's, was observed.
3) Increasing the angle of discharge from the hori-
zontal up to about 60° increased the dilution; from
60° on, the general trajectories and dilution re-
mained about the same for cases with current.
4) Increasing the L/D decreased the thermal dilu-
tion dramatically, especially near a towing
ratio of R « 0.10. The trajectory appeared to rise
with decreased port spacing. However, the widths
showed little change with L/D variation.
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A model has been analyzed and implemented which attempts to
simulate multiple port thermal discharges. Agreement between the
model and experiment was generally quite good. The thermal di-
lutions and trajectories were predicted accurately for buoyant
single port jets of varying Froude Number, however, plume widths
and possibly centerline velocities were not predicted well for
high Froude Number discharges. Buoyant discharges into a co-
flowing stream were briefly considered. For co-flow it was found
that inclusion of the turbulence terms of the equations allowed
for prediction of dilution trends but the dilution could not be
accurately predicted. The field is in need of a more involved and
thorough examination of co-flow discharge.
Thermal dilution and trajectory for discharge into a cross-
flow were predicted reasonably well by the model. While the re-
sults deviate slightly for experiments for high or low ambient to
discharge velocity ratios, the prediction is quite good for moder-
ate velocity ratios. Evidence seems to support the need for an
additional entrainment term based on the drag induced curvature of
the jet.
The model advanced handled the merging of adjacent jets
in a manner that was more physically reasonable than any of the
other models thus far advanced for multiple port merging dis-
charges. By allowing the profiles of temperature, species, and
velocity to adjust naturally from the axisymmetric single plume
profiles to those approaching a two-dimensional slot profile,
the model avoided the arbitrary transition from one solution to
6
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another. Since the change in profiles was geared to the growth of
the jet, the transition was smooth and continuous. Davis sug-
gested an entrainment function which depends on the available en-
trainment area of the jet. This function was found to approach
a limit considerably less than the appropriate slot entrainment
value. Despite this, use of the "entrainment area" entrainment
function in the model provided predictions which agreed well with
the limited experimental data available.
When considering multiple-port crossflow discharge it was
found necessary to include the drag force on the plume. Unfortun-
ately, good agreement could not be obtained unless the drag coef-
ficient varied inversely with the ambient to discharge velocity
ratio.
The model predicted that changing port spacing would have a
significant effect upon dilution and trajectory.
When the multiple port discharges have merged to form a
pseudo slot jet, the entrainment remains between 50% and 70% less
than the normal slot entrainment value. This is true for at least
the first 10 or 15 port spacings along the plume centerline and
perhaps considerably further.
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SECTION IV
HISTORICAL BACKGROUND
It has only been in the last ten years that in-depth in-
vestigations of multiport diffusers have been performed, although
investigations of single port thermal plumes were carried out as
early as the 1930's. In that decade the study of wakes led to
treatments of free turbulence. An interesting paper by Reichardt
treated the diffusion of heat and momentum and provided one of
the first quantitative evaluations of the two diffusion processes.
2
Schmidt also considered the problem and employed the mixing
length theory to arrive at a solution for a point and line plume
that agreed quite well with his experimental results. In 1949 a
paper appeared by Albertson, et. al., which along with a later
4
paper by Morton, et. al., extensively documented the experimental
and theoretical treatments of those early years. Forstall and
38
Shapiro also provide excellent references for pre-1950 treat-
ments of slot and round discharges. Investigations by prominent
authors of the late 1950's and 1960's are summarized in Trent and
Welty5 and Hirst6.
The bulk of the work on multiport discharges has been in-
vestigations of discharges into confined environments. The pri-
mary aim has been to model a specific diffuser and site.
7
Jirka and Harleman published an extensive work concerning
multiport discharges into stagnant and flowing shallow ambients.
Q
Argue conducted a laboratory investigation of shallow multiport.
discharges into a flowing ambient at an angle of 20° from the
-------
9
horizontal. Larsen and Hecker performed experiments on multiple
port discharges into shallow ambients with the primary interest
on the free surface concentrations. All of the above were re-
stricted to discharges into confined environments. Such dis-
charges yield little information concerning the merging and mix-
ing of adjacent jets since in most cases boundary effects domin-
ated the hydraulics of the jet and necessarily influenced heavily
the mixing phenomenon.
Koh, et. al, investigated various diffuser configurations
(several staggered multiport diffuser manifolds) for discharge
into stagnant and flowing ambients (this was a basin model study
which included a specific geometry and site restrictions).
Liseth performed an experimental investigation of multiport
discharge into stagnant ambients from a diffuser with ports on
both sides of the manifold.
10
Iwasa and Yatsuzuka proposed a model (similar to the Hirst
single port treatment) and compared it with near surface concen-
trations taken from a system employing 8 radially discharging
ports from a vertical tube, each at a 45° circumferential dis-
placement. Acceptable success appeared to be obtained from this
technique, however, no attempt was made to account for merging
of the plumes which would occur in other geometries or closer
spacings. Essentially, little experimental work has been done
that focuses on the merging of adjacent jets or the effects of
spacing on dilution and trajectory.
Several analytical attempts have been made to account for
the merging of adjacent jets of a multiport thermal discharge.
9
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13
Koh and Fan formulated a mathematical model for analyzing a
multiport thermal discharge by matching single round port and slot
jet solutions at a desired transition point. The advantage of
this technique was its simplicity. However, the accuracy of this
model is questionable. A recent publication by Kannberg and
14
Davis compared data obtained for a multiple port discharge with
that predicted by the transition model. That comparison showed
the transition model overpredicting the dilution found experimen-
tally.
A slightly modified version of the Koh and Fan model was
employed by Shirazi and Davis ; however their work would be
subject to the same restrictions as the Koh and Fan work. Harleman
and Jirka cited an "equivalent slot" method for calculating di-
lution and trajectory. For the equivalent slot, the same dis-
charge per unit diffusion length and the same momentum flux per
unit length as the multiport discharge is required. This results
in a theoretical slot of width, B = D 7T/4L when D and L are the
actual port diameter and spacing respectively. This technique
was also found by Kannberg and Davis to overpredict the dilution
observed experimentally. To date, no theory has been advanced
which adequately handles merging multiport thermal discharges.
10
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SECTION V
EXPERIMENTAL WORK
MODELING PARAMETERS
In order to legitimately model the multiport thermal dis-
charge experimentally and theoretically, the laws of geometric and
dynamic similitude must be followed. Relations for similitude
may be obtained from a dimensional analysis. Such an analysis
yields the following independent parameters: 1) the densimetric
I/O
Froude No., F = U / l^-gDj , which is the ratio of inertial to
buoyant forces; 2) the current to jet discharge velocity ratio,
R - U /U ; 3) discharge port spacing, L/D; and 4) discharge angle
o ^°
relative to the current, 9. Since the plume is usually turbulent,
Reynolds Number (Re) effects are negligible (Re varied, 2100 to
6300).
The dependent variables are: 1) the ratio of local excess
temperature to the excess temperature at discharge, /T - T\/
(T - T \ = ATc/AT ; 2) dimensionless plume width, W/D (for a very
long diffuser and effects are small and the length of the plume is
ignored), and 3) plume centerline coordinates, X/D and Y/D.
In this investigation wide ranges of the independent variables
were considered. They were
L/D = 10,5,2.5
F = 10,30,58
6 = 0,15,30,45,60,90° from the horizontal
R = 0,0.05,.0.10,0.25,0.50.
Due to equipment limitations and lack of time, all combinations of
11
-------
these variables could not be considered. A parameter matrix show-
ing cases for which data was gathered is given in Table 1.
The data collection yielded excess plume centerline tempera-
ture, cross sectional width and position of maximum temperature
(tra j ectory).
APPARATUS AND DATA ACQUISITION
The experiments were conducted at the Hydraulics Laboratory
of the U. S. Environmental Protection Agency's Corvallis Environ-
mental Research Laboratory. Warm water was discharged into a tow-
ing channel (40* x 21 x 3') containing cool tap water. The dif-
fusers consisted of 2.54 cm. (1") O.D. thin wall tapered acrylic
manifolds with .635 cm. (1/4") I.D. round acrylic ports of approxi-
mately 10 cm. to 13 cm. length. There were 4 ports for the L/D =
10 diffuser; 6 ports for the L/D = 5 diffuser; and 8 ports for the
L/D = 2.5 diffuser. In each case the mass discharge rate from any
single port deviated less than 3.3% -from the average of all ports
and generally the deviation was much less. Figure 1 shows the dif-
fusers used in the study.
The flow rate tests were run at the nominal flow rates used
for the actual data. Very little deviation occurred with changes
in bulk flow rate. The measured deviation in temperature of the
F21
various ports varied less than .7% from port to neighboring port.1 J
The L/D = 10 diffuser ports had a 45° bend in them to allow for
measurements at an angle of 0* from the horizontal which were free
[2] Based on L/D = 10 where this deviation would be the greatest.
12
-------
TABLE 1 INDEPENDENT PARAMETER MATRIX OF EXPERIMENTAL CASES
Numbers in matrix indicate L/D's of experiments for indicated
F, R, and 9.
F = 1
R
9
0
15
45
90
1
0 0.05 0,10 0,25 0.50
10 10 10 10
10 10 10
2.5,5,10 10
2.5,5,10 2.5,5,10
F = 30
R 0 0.05 0.10 0.25 0.50
•e
0
15
30
45
60
90
10 10 10 10 10
10 10 10 10 10
10 10
10 10 10 10
10 10
10 10 10 10
F = 58
R 0 0.05 0,10 0,25 0.50
6
0
15
45
90
10 10 10 10 10
10 10 10 10
5,10 5,10
2.5,5,10 2.5,5,10 10 2.5,5,10
13
-------
Figure 1. Diffusers used in the experimental work, L/D's = 10, 5, 2.5.
14
-------
from manifold wake effects. Each of the other two had only
straight ports and did not allow measurements at small angles.
The manifold was connected to a warm water reservoir by supply
lines at both ends. The diffuser was mounted across the channel's
width and towed the length of the channel.
In order to hydraulically simulate an infinite string of
ports, image walls (°ne 1/16" Aluminum and one 1/8" plexiglass
plate) were placed at a distance L/2D outside the end ports of the
diffuser. These extended 15 cm. (23.6D) ahead of the
line of discharge and 125 cm. (198D) behind it. For the ex-
perimental program employed, the maximum boundary layer thickness
and displacement thicknesses developed on the image walls were
4.8D and 1.6D, respectively as calculated from flat plate boundary
layer theory35. The effect of the image walls on the dilution
was found to be negligible for the port spacing, L/D of 10; conse-
quently, for many of the runs performed at this spacing the image
walls were not used. However, the presence of the walls were
shown to decrease dilution by about 20% (compared to cases without
image walls) for a L/D of 5. Hence, they were incorporated for
L/D = 5 and L/D = 2.5.
The warm water reservoir was kept at constant head by bubbling
in air as the water was discharged. Figure 2 is a diagram of the
warm water reservoir.'- ^ As water is released from the reservoir,
air pressure pushes the water from the bubbling tubes until the
air escapes from the tubes into the reservoir. In this manner the
~[3] The use of this tank was originally suggested by Ken Loose,
formerly of EPA.
15
-------
„/
FILLING STOPPER
BUBBLING TUBES
SEALED TANK
ATMDSPHERIC PRESSURE LEVEL
MAIN DISCHARGE VALVE
AMBIENT WATER LEVEL
Figure 2. Warm water constant head reservoir.
-------
level of ambient air pressure is kept at the level of the bottom
of the bubbling tubes. Baffles were included in the construction
of the reservoir for the purpose of damping out waves that would
appear when the reservoir was being towed. The baffles proved to
be quite successful. The warm water reservoir was filled with
hot tap water from a conventional water heater of the desired
temperature prior to each run.
A main discharge valve was located at the outlet of the warm
water reservoir and acted as an on-off valve for the reservoir.
Discharge water flowing from this main discharge valve divided into
the two manifold supply lines (1/2" acrylic flexible tubing). A
control valve on each supply line was adjusted to give control of
both bulk flow rate and individual supply line flow rate. The
flow rate out of the port in whose wake the temperatures were moni-
tored was assumed to be the average of all the ports and was com-
puted by measuring the bulk flow out of the reservoir during a
given time and dividing by the number of ports. This was done
for each run.
For F = 10 the nominal discharge velocity, U , was 25 cm./sec.
and the nominal difference between discharge temperature and am-
bient water temperature was about 33°C (depending on ambient temp-
erature). Cases at F = 30 had a nominal U of 50 cm./sec. and had
a nominal difference between discharge temperature and ambient
temperature of about 18°C (depending on ambient temperature). The
F = 58 runs maintained a nominal discharge velocity of 75 cm./
sec. "• •" and nominal difference between discharge temperature and
17
-------
ambient temperature of about 14°C'- -" Cdepending on ambient temp-
erature). Ambient temperatures varied from a low of 11.30°C to
a high of 24.44*0 with the seasons.
The temperature of the discharge was measured at the point
where the warm water was discharged into the cooler channel water.
A Hewlett-Packard Quartz Thermometer was used to measure all re-
ference temperatures (ambient and discharge temperatures were
measured to 0.01°C).
A conical hot film sensor [TSI, model 12-30W) with a Thermal
Systems, Inc. constant temperature anemometer was used to record the
excess temperatures in the field of the jet. The sensor was
mounted on a rod that traversed vertically through the plume. The
vertical motion was motorized and its direction and speed controlled
remotely. The sensor was fixed at some downstream position X/D
relative to the line of discharge for each run. * During each
run the sensor would be moved up and down through the plume sev-
eral times. In this manner the vertical temperature profile could
be obtained at a single downstream distance. The temperature sig-
nal of the anemometer and a potentiometric position signal were
recorded on a Honeywell Visicorder. The experimental apparatus is
illustrated in Figure 3.
At a later date the signals were examined and a value placed
on the maximum mean temperature in the vertical profile and its
position. The points where the bottom and top of the jet were en-
countered were also determined. During many runs more than one
[4]For L/D = 5 and L/D = 2.5 these values were about 60 cm,/sec.
and 11°C, respectively.
[5] Runs were made with different X/D positions of the probe.
18
-------
POTENTIOMETER GEARED
TO VERTICAL
HEIGHT
DC DRIVE MOTOR
I— MAIN DISCHARGE VALVE
WARM WATER RESERVOIR
SIFPLY LINE VALVES
DIFFUSER mNIFOLD
BALL SCREW DRIVE
POTENTIOMETER SIGNAL
o o o o o o o
I I I I I I I
•ODOO
•D D
ooo
THERMD SYSTEMS INC,
CONSTANT \
TE/vPERATURE ANEMOMETER —^
LIGHT SENSITIVE PRINT
HONEYWELL VISICORDER -*
Figure 3. Experimental apparatus and electronic instrumentation.
19
-------
traverse was performed (1 to 12). This provided more information
since each traverse was evaluated for the above items and plotted.
In order to have reasonable confidence in the final curves multi-
ple points were obtained at each downstream position. In some
cases this required additional runs. Due to the multitude of runs
taken, duplication of conditions was impossible without enormous
loss of time. The standard deviation from the desired values
was about 8% on Froude Number and R, the towing ratio. The L/D
and angle of discharge were reasonably exact.
Considerable noise in the electronics due to the proximity
of large power equipment and use of flourescent lights was noted.
Contamination of the sensor also offered some trouble. Occasion-
ally, high gains were necessary for small temperature differences
which amplified the noise and the normally slight drift of the
signal.
These factors compound the analysis of the signal to noise
ratio. The predominant noise element, the AC 60 Hz. noise, was
appreciable in some cases. While generally the 60 Hz. noise was
on the order of 12% of the signal, it reached 30% for some cases
requiring high gain. The random noise, however, was at most 6%
of the signal. The measurements requiring high gains were ordin-
arily in regions where the jet turbulence was on the order of
2 Hz. or 3 Hz. Near the discharge point lower gains were needed
and higher signal to 60 Hz. noise ratios were evident. Often in
this region the 60 Hz. noise was only about 6% of the signal ampli-
tude. Near the discharge the jet turbulence was on the order of
50 Hz. Hence, where the noise was the greatest it was most easily
20
-------
recognized. It should be mentioned that the jet turbulence near
the centerline was such that the signal contained turbulent os-
cillations about the mean of anywhere from 25% to 100% (generally
about 70% of the mean signal). In general one may say that while
the occasionally large 60 Hz. noise impaired the precision of the
measurements it did not detract from the accuracy.
The mechanism to move the sensor vertically for traversing
the plume employed a double ball screw drive powered by a remotely
controlled D.C. motor (see Figure 3). The sensor was positioned
laterally on a rod such that it followed the vertical centerplane
of the jet chosen for measurements, as shown in Figure 4. Once
in position the sensor was fixed so that only vertical motion
occurred. The sequence of events, called a run, which formed the
basic experimental test is enumerated as follows:
1) Calibrate the T.S.I, anemometer using an overheat
ratio of 1.075 and obtain a temperature versus
voltage line (always linear but of slightly varying
slope).
2) Prepare and align the traversing mechanism for the
particular downstream distance, X/D, then align and
position image walls as necessary.
3) Fill reservoir with warm water for the desired
temperature.
4) Check for ambient stratification (if stratified
then mix; an ambient stratification of 0.05°C was
the maximum allowed. Generally it was about +_ 0.02°C).
5) Obtain ambient temperature in the channel water.
21
-------
SENSOR
CENTER PLANE OF PLUME
K)
PLANE OF TRAVERSE
LINE OF TRAVERSE
\
EDGE OF CHANNEL
Figure 4. Plane of traverse of the sensor.
-------
6) Measure initial probe height.
7) Open the main valve and allow the water to issue
from the ports. (Prior to this the supply line
control valves will have been adjusted to give a
balanced flow rate near the desired value.)
8) Allow the discharge temperature to reach equilibrium
and record this value.
9) Initiate tow (if required) and begin traversing the
jet with the sensor. Particular emphasis was placed
on the region of maximum temperature during the tra-
verse. The traverse was often stopped in and near
the point of maximum temperature so that an accurate
record of the temperature there was obtained. (The
frequency response of the sensor was well above the
50 to 60 Hz. maximum fluctuation rate of the turbulent
eddying jet. The eddy structure was certainly evi-
dent in the Visicorder print, however, no attempt was
made to analyze this.)
10) After passing through the plume several times or at
the conclusion of the tow, the final sensor height was
determined and the port discharge temperature again
measured. (The reservoir water cooled slightly during
the run and as such the average of the before and after
port discharge temperatures were used. The difference
between these two temperatures never exceeded 2.5%
and was generally less than 1.5%.) It is conceiv-
able that the discharge temperatures were depressed
23
-------
during the towing due to the forced convection on
the supply lines. However, since the supply lines
were made of thick wall acrylic tubing the depression
would not be inordinate and would be compensated for
by the "after" port discharge temperature measurement.
11) The volumetric flow rate for the discharge was measured
by timing the change in water level in the reservoir,
from this the average port discharge velocity was
calculated.
12) The main valve was shut off and the test ended. If
a tow was made, the average speed of two was computed.
(Care was taken to use only that portion of the tow-
ing channel that was uniform in its towing speed.)
A typical visicorder plot of the temperature and position
signals is shown in Figure 5. The temperature plots obtained on
the visicorder were examined and values ascribed for the maximum
mean temperature, its vertical position and the top and bottom of
the vertically traversed plume.
These values were estimated by visual scrutiny of the visi-
corder plots. Little can be said to describe this process except
that runs were eliminated where the position and quantity of the
mean temperature were indistinguishable from the rest of the record.
The determination of these values thus was somewhat subjective.
Visual scrutiny was also employed when estimating curves through
log-log plots of the data values obtained as described above.
These values were normalized and reduced with the aid of the
computer to the forms AT /ATQ, Y/D, and W/D. Figure 6 shows the
24
-------
T.S.I, TEMPERATURE SIGNAL
INCREASING
TEMPERATURE
AND PROBE
HEIGHT
POTENTIOMETER POSITION SIGNAL
Figure 5. Typical Visicorder print of temperature and position
25
-------
ESTIMATE THE MEAN VALUE OF THE WXIMUM PROFILE TEMPERATURE AND
THE POSITIONS OF 'BOTTOM' AND 'TOP OF PLUME BY EYE,
RECORD THE VALUES AS WELL AS
OTHER IMPORTANT INFORMATION,
ENTER THE INFORMATION ON
DATA CARDS FOR REDUCTION,
THE NORMALIZED DATA IS
PLOTTED ON LOG-LOG GRAPH
AND A LINE DRAWN BY EYE
THROUGH THE APPROXIMATE
MEAN OF ALL THE DATA
GROUPS,
THE DATA IS NORMALIZED BY COMPUTER
AND DIMENSIONLESS PARAMETERS COMPUTED,
THE NORMALIZED DATA IS GIVEN IN
APPENDIX Av WITH VERTICAL WIDTHS,
Figure 6. Data treatment process.
26
-------
stages of processing the data. The edges of the plume were speci-
fied as those points where the mean temperature began to deviate
from the ambient. These points were usually obvious because one
generally encountered "eddying balls" of warm fluid rather than an
indistinct merging of the plume temperature into that of the
ambient and were observed both at the top and bottom of the plume.
The position values were used to determine the vertical width and
then with the trajectory the cross section widths.
In general the apparatus operated as desired and had accept-
able error. The channel was well suited for the type of work per-
formed, however, its potential for offering an ambient free of tur-
bulence was not used to the full extent. The major drawback of the
towing channel was its short length. Indeed some of the fastest
towing speeds allowed for only about 17 seconds of run time thus
requiring numerous runs. The instrumentation was good although
microthermocouples might have offered a more noise free response
than the T.S.I., had it been feasible to employ them.
Sensor residence time at or near the centerline was about
4 sec. although residence times ranged from 2 sec. to 15 sec. The
measured time constant associated with the signal was a = 7.675.
Thus the signal would go from 0 to .67 of the step change value in
.14 sec. and .9 of the step value in .3 sec. The sensor residence
times were sufficient to allow for reasonable approach to the true
mean signal. The longer residence times were necessary for slower
towing speeds where the scale of turbulence was larger.
Implicit in the discussion and in the measurements have been
several concepts. First, it was assumed that the jet possesses a
27
-------
single maximum mean temperature and that this maximum may be
measured as the mean of the signal; i.e., a single maximum mean
temperature exists (at least vertically) and is measurable.
Second, it has been assumed that all processes affecting the
measurements and their treatment may be considered random. With
the exception of the 60 Hz. noise, no examinations were made to
verify either of these statements or that the values reported
here are anything but true mean values.
THE DATA AND ITS TREATMENT
The experimental plan called for measurements of jet excess
temperature ratio, trajectory, and width downstream from the points
of discharge for various values of port spacing (L/D) , discharge
angle (6), Froude No. (F), and velocity ratio (R). These measure-
ments were performed as cited in the previous section. At each
downstream distance (X/D) several values for each of the above
measurements were obtained. These were then tabulated and plotted
for all the downstream distances. An example of the plots and
several data points for temperature, width and trajectory are
shown on Figures 7, 8, and 9. Some of the data points have been
shifted off the true X/D value in order to clarify the plot. The
lines drawn through the data are intended to be the average values .
An attempt to fit the data with a least square curve fit was found
to be undesirable in some cases and at best not significantly dif-
ferent from the "eye" fits shown in the figures. Had there been
measurements taken at more downstream positions, the least squares
method would have worked better.
28
-------
1,0
0,5
g,
I
o
0,2
0,1
,05
,02
,01
1 I
L/D = 10
0 = 15
F = 30,52
R = 0,500
A EXCESS TEMPERATURE
DATA POINTS
I \ I I I I I I
I
I I I I I I I I
I
1II I I I I-
I I I I I I I
2 5 10 20 50 KD
HORIZONTAL DIST/WCE - X/D
Figure 7. Example of typical excess temperature data and its' representative curve.
-------
100
BO
20
JO
L/D
9
F
R
ID
15
30,52
0,500
VERTICAL WIDTH
DATA POINTS
I I I I I I I I
I I I
D
I I I I I
5 ID 20
HORIZONTAL DISTANCE - X/D
50
Figure 8. Example of typical vertical width data and its' representative curve.
-------
100
90
20
r^^^
I 5
^ n
LTD = 10
9 = 15
F = 30,52
R= 0,500
O TRAJECTORY DATA POINTS
I I I I I I
II I I I I-
o o
I I I I I I I
5 ID 20
HORIZONTAL DISTWCE - X/D
50
Figure 9. Example of typical trajectory data and its' representative curve.
-------
The data gathered in this study is detailed in Appendix A.
The information given in that appendix includes the measured
plume information and the experimental discharge conditions.
While the vertical widths measured in the experiments are given
in Appendix A, the discussion here is restricted to the plume
widths in a plane perpendicular to the centerline trajectory,
i.e., the cross-sectional widths. The cross-section widths were
generated by plotting the experimental centerline and plume edge
data and measuring the widths at various points downstream on a
line estimated to be perpendicular to the local centerline tra-
jectory. Curves are presented in Reference 41, for all of the in-
formation pres.ented in Appendix A.
EXPERIMENTAL ERROR ANALYSIS
An error analysis for the data was undertaken. Using a method
for small sample data groups outlined by Benedict, for obtaining
the estimated 95% probability confidence interval. Employing that
technique on the typical data curves offered earlier (Figures 7, 8,
and 9) the confidence interval may be drawn to illustrate the
quality of the data. Figures 10, 11 and 12 shows graphically the
results of such an illustration.
As a general rule when considering all of the data, the 95%
probability confidence intervals may be said to be of the follow-
ing dimensions:
32
-------
1,0
0,5
0,2
0,1
,05
,02
,01
^ I
L/D=10
0 = 15
F= 30,52
R = 0,500
A EXCESS TEMPERATURE
DATA POINTS
1 I ! I I I I I
1 \ I I I I H
INDIVIDUAL GROUP CALCULATION
N FOR 95% PROBABILITY
20% RULE FOR 95% PROBABILITY
I I I I I I I I
I
1 I 1 I I I I
5 10 20
HORIZONTAL DIST/NCE - X/D
50
100
Figure 10. Confidence interval for typical excess temperature data.
-------
100
50
20
ID
L/D = 10
0 = 15
F- 30,52
R- 0,500
D VERTICAL WIDTH
DATA POINTS
INDIVIDUAL GROUP CALC,.
FOR 95% PROBABILITY
V
20% RULE FOR 95% PROBABILITY
J L
J I L
5 10 20
HORIZONTAL DISTANCE - X/D
50
Figure 11. Confidence interval for typical vertical width data.
100
-------
1DO
20
ID
5-
"1I1II I I I I I
170 = 10
9 = 15
F= 30,52
R-0,500
O TRAJECTORY DATA POINTS
1\I I I I I-
30% RULE FOR 95% PROBABILITY
INDIVIDUAL GROUP CALC, H
FOR 95% PROBABILITY
I I I I 1 I I
I Kill
5 ID 20
HORIZONTAL DISTWCE - X/D
100
Figure 12. Confidence interval for typical trajectory data.
-------
For Excess Temperature Ratio: 95% C.I, encloses approximately
20% of the value of the Excess
Temperature Ratio.
For Cross-Section Width : 95% C.I. encloses approximately
20% of the value of the X-Sec.
Width.
For Vertical Height
(Trajectory) : 95% C.I. encloses approximately
30% of the value of the Verti-
cal Height.
As is obvious from the general statement above, the quality
of the trajectory data appears to be slightly less accurate than
the excess temperature ratio and the cross-section width data.
This should be kept in mind when examining the data presented and
when considering the discussion of the results in the next session.
Another measure of the quality of the data is given by the
correlation coefficient of a least squares curve fit. While this
basically relates a proposed curve equation to the data, it also
implies qualitatively how much of the data variation follows legiti-
mate trends and how much is really random scatter. At the close
of the next section curve fits are offered which include correla-
tion coefficients.
THE RESULTS
The effects of F, R, 8, and L/D on dilution, plume width and
trajectory are of major concern. The results are best demonstrated
by the plots of ATc/ATQ, W/D, and Y/D plotted against X/D for the
various combinations of F, R, 6, and L/D as given in Appendix A.
36
-------
The effect of the R at low discharge angles is similar to that
reported in Reference 14 for co-flow. The dilution was greater for
increased towing speed. This observation is supported by both the
excess temperature ratio and the width as shown in Figures 13, 14,
and 15 for 9 = 15°. However, for 6 = 90° the trends are distinctly
different. Figures 16, 17, and 18 illustrate that in this case the
dilution is greater for slower towing rates, when compared for
various distances downstream. The trajectories are dramatically
affected by towing rates even for very small angles as seen in
Figures 15 and 18. The results quoted are typical of the results
for other conditions examined including other L/D's.
The effect of Froude Number is very minor. The information
offered in Reference 14 indicated that the dilution increased with
decreasing Froude Number. The data presented there to support that
conclusion indicated a very minor effect. Figures 19-24 show
Froude Number effects on temperature, width, and trajectory for two
different combinations of 6 and R for L/D = 10. It can be seen
that there is little if any change in dilution, widths, and tra-
jectory for current cases at an L/D of 10. The same trends were
observable for other conditions tested with the exception of low
R's for L/D = 5 and L/D = 2.5 where there was slightly lower dilu-
tion for higher Froude Numbers. In general, it may be stated that
Froude Number variation has very little effect on discharges into
an ambient current.
The effect of angle of discharge on dilution, widths, and tra-
jectory for the cases with current for an L/D of ten, F = 30 and
37
-------
1,0
0,5
o
i
00
0,2
0,1
,05
I
L/D = 1Q
0 = 15
F = 57,0
R-
0,051
- 0,101
B—-0,253
G 0,503
I I I I I I I I
T I I I I I h
,02
.01
1 1 I I I I 11
I
I 1 I I I I I
5 10 20
HORIZONTAL DIS1ME - X/D
50
100
Figure 13. Effect of varying R on excess temperature ratio for L/D=10., 9=15, F=57.0.
-------
CM
3DO
90
20
I I I I I I I I
L/D-1D
0 = 15
F = 57,0
R =
3 0,051
A-— 0,101
Q 0,253
Q 0,503
I IT I M-
J L
5 ID 20
HORIZONTAL DISTANCE - X/D
50
]J90
Figure 14. Effect of varying R on width for L/D=10., 9=15, F=57.0.
-------
100
50
20
to 10
5-
"•—i—r
iyi) = ]0
0-15
F = 57,0
R -
G 0,051
A -0,101
Q 0,253
O 0,503
I I I I I H
i I I I. I i I I
I I I I I I I
5 10 20
HORIZONTAL DISTANCE - X/D
50
100
•Figure 15, Effect of varying R on trajectory for L/D=10., 9=15, F=57.0,
-------
1,0
0,5
0,2
0,1
,05
,02
,01
^ I
L/D=1D
0 = 90
F = 31,0
R =
G 0,050
A-_ 0,098
- 0,247
— 0,499
1 I I I I 1 I I
1 I I I I I H
I I I I I I i
I
I I I I I I I
5 10 20
HORIZONTAL DISTONCE - X/D
50
100
Figure 16. Effect of varying R on excess temperature ratio for L/D=10., 9=90, F=31.0.
-------
100
BO
20
ID
K)
170 = 10
9 = 90
F - 31,0
R-
0,050
-- 0,098
0,247
O— 0,499
0
5 ID
HORIZONTAL DIS1ME -
20
90
100
Figure 17. Effect of varying R on width for L/D=10., 9=90, F=31.0,
-------
50
9 = 90
F = 31,0
I I I I I II I
0,050
20
A- 0,(
Q 0,247
--0,499
5-
I I I I H
i I I I I I I I
I
1 I I I I
2 5 10 20 50
HORIZONTAL DIST/NCE - X/D
Figure 18. Effect of varying R on trajectory for L/D=10., 9=90, F=31.0.
100
-------
1,0
0,5
o
0,2
0,1
,05
,02
,01
r |
L/D-10
9 = 15
R = 0,103
F-
A 10,39
Q 32,40
O 57,38
I I 1 I MM
I I I I I I I I
1 1 1 1 1 1 1-
1
1 1 1 1 1 1 1
5 ]fl 20
HORIZONTAL DISIME - W
50
100
Figure 19. Effect of Froude Number on excess temperature ratio for L/D=10., 9=15, R=0.103.
-------
100
20
10
tn
L/D=1Q
9 = 15
R = 0,103
F =
A- 10,39
B 32,40
O 57,38
I I I I I I I
1 I I I I I H
I I I I I I I I
5 10 20
HORIZONTAL DISTANCE - X/D
50
100
Figure 20. Effect of Froude Number on width for L/D=10., 9=15, R=0.103.
-------
100
20
fc 10
T -
L/D-JD
9 = 15
R = 0,
T—I I I
10,39
32,10
O --- 57,38
1
T—I I I I I
X -
I I I I I I I I
1
I I I I I i I
2 5 10 20 50 100
HORIZONTAL DISTAKE - X/D
Figure 21. Effect of Froude Number on trajectory for L/D=10., 6=15, R=0.103.
-------
1,0
0,5
r
o
0,2
0,1
,05
,02
,01
I
0 = 90
R = 0,250
n _
10,74
30,03
0 --- 58,31
1 I I I I I I I
I I I I I I I I
1II I I I I-
I
I I I I I I I
5 10 20
HORIZONTAL DIST/NCE - X/D
50
100
Figure 22. Effect of Froude Number on excess temperature ratio for L/D^IO., 6=90, R-0.250,
-------
100
50
20
^ I
LD-iD
e = 90
R = 0,250
F-
I I I I I I I!
- 30,03
-58,31
I II llh
00
I I I I I I I I
1 I I
II
5 ID 20
HORIZONTAL DISTANCE - X/D
50
Figure 23. Effect of Froude Number on width for 1/0=10., 0=90, R=0.250.
100
-------
J1JU
50
§ 20
y
£ 10
s
> »lt
1 5
2
i
— ' 1 1
- LTD = 10
~~ 9 = 90
R= 0,250
F =
- A- - - 10,74
Q 30,03
O -58.31
—
—
—
—
1 1
1 1 1 1 1 1 1 ' i 1 1 1 1 1 1 h
—
—
—
^ -A
-- - T-^V^^ —
- -^"Z^^- ^ Z
—
—
1 1 1 1 1 1 1 , ( 1 1 1 1 1 1 1
2 5 ID 20 50
HORIZOWTAL DIST/^CE - X/D
Figure 24. Effect of Froude Number on trajectory for L/D=10., 9=90, R=0.250.
]DO
-------
R = . 25 are recorded on Figures 25-27. As can be seen, the increase
in the angle of discharge increases the dilution. This seems ap-
propriate since greater initial dilution occurs in the 90° dis-
charge as compared to the 15° discharge. A more informative graph
is offered in Figure 28 which is a combined trajectory-temperature
plot for F = 30, R = 0.25 and L/D = 10. As one notices, the 90° and
60° dilution and trajectory are very similar. It is interesting
to note that after X/D = 35 the 15° results show less dilution than
the 0° results. Using the results from this graph, attempts were
made to predict the results at 45° for cases at extremes of tow-
ing ratio and Froude No. using 15° and 90° data previously obtained.
While the higher towing rates were predicted quite accurately, the
results at lower towing rates were not well predicted. Accordingly,
additional runs were taken at 45° for the slower towing rates.
Complete data exists for angles of 0°, 15°, and 90° as well as
partial data at 45° for L/D = 10. However, only selected runs
exist at 90° and 45° for the other L/D's.
Of all the parameters of interest the port spacing seems to
be the most critical. It is this variable that will most affect
construction costs, and it is probably the most important thermal
design parameter related to the siting of a plant. The comparisons
offered in this discussion on L/D comprise nearly all the experi-
mental cases. Four cases are offered at 90° and one at 45° for
the comparison of L/D effects.
The excess temperature ratio illustrated in Figures 23, 32,
35, 38, and 41 seems to be markedly dependent on L/D. The trend
appears to be decreasing dilution with decreasing port spacing.
50
-------
1,0
0,5
r
o
0,2
0,1
,05
,02
,01
I
I I I I I I I I
0 = 0,35,30,45,60,90
F = 31,1
R = 0,248
I I I 1 I 1 LI
1 I I I I I H
I
I I I I I I I
5 JO 20
HORIZONTAL DISTWCE - X/D
50
100
Figure 25. Effect of angle on the excess temperature ratio for L/D=10., F=31.1, R=0.248.
-------
100
50
n I I I I
I7D=1D
8 = 0,15,30,45,60,90
F = 31,l
R = 0,248
1I I I I I-
20
tn
I I I I I I I I
I
I I I I 1 I I
5 ID 20
HORIZONTAL DIST/WCE - X/D
50
1DO
Figure 26. Effect of angle on width for L/D=10., F=31.1, R=0.248,
-------
100
50
1 I I
170=10
0 = 15,30,45,60,90
F=31,l
R = 0,248
I I I I I I I
i I i rn i-
w
01
20
10
5
J L
I I 1 I I I
5 ID 20
HORIZONTAL DISIME - X/D
50
100
Figure 27. Effect of angle on trajectory for L/D=10., F=31.1, R=0.248.
-------
in
32,5
ID,
AVAT0 VALUES
A 0,195
00,150
©0,100
Q 0,075
00,065
I
20 30 10 50 60
HORIZONTAL DISTANCE - X/D
Figure 28. Effect of angle on dilution as plotted with trajectory for L/D=10., F=31.1, R=0.248.
-------
That Figures 29 and 41 show more dramatic trends may be attribut-
able to the fact that the towing ratio is 0.10. As is common with
plumes discharged into a current, twin vortices often occur. In
the experiments performed, this twin vortex structure seemed to
be more defined at R = 0.10. Since the sensor was located in the
centerplane of the discharging port, the measurements were actually
taken between the two vortices as illustrated in Figure 44, and as
cited in Hirst , the maximum temperatures often occur near the
center of each of these vortices. It is probable that for the
distance downstream that the twin vortex structure was maintained
our sensor did not pass directly through the region of hottest
discharge but rather very close to it. No attempt was made to
search for the hottest parts of the discharge except to verify
that they were indeed slightly off the center plane.
The vortex effect may increase the lateral entrainment. This
effect would be most obvious in comparisons of L/D effect since
the jets must compete for lateral entrainment with the competition
getting more intense for closer spacings.
Referring to Figures 30, 33, 36, 39, and 42 which present
L/D effects on width, the dramatic differences noticed before in
excess temperature are not extended to widths. In fact, no clear
trend exists in widths. It appears that in all cases though, the
width of the jet for L/D = 2.5 is slightly greater than that for
the other L/D's. It can be concluded that the width of the jet
cannot be used as a measure of dilution for close spaced jets when
comparing to larger jet spacings.
55
-------
1,0
0,5
0-90
F = 10,2
R-0,10
On
ON
0,2
0,1
,05
,02
,01
A- -- 10
- 5
• 2,5
1 I I I I I I ' 1 1 1 -I I I I I-
I I I I I I I I
I I I I II I
5 ]0
HORIZONTAL DIS1ME -
20
50
100
Figure 29. Effect of L/D on excess temperature ratio for 9=90, F=10.2, R=0.10.
-------
100
50
20
ID
^ I I
8 = 90
F=JD,2
R=0,IO
L/D =
A ID
G 2,5
I I I I I I I
1 I I I I I H
tn
I I I I 1 Ul
5 ID 20
HORIZDNT71 DIST/WCE - X/D
50
300
Figure 30. Effect of L/D on width for 9=90, F=10.2, R=0.10.
-------
in
oo
§
1
1
1
3
i
JJJU
90
20
10
5
2
i
H ' 1 I
- 0 = 90
_ F = 10,2
R = 0,10
L/D =
A 10
~~ 3 2,5
—
—
—
—
1 1
1 1 1 1 I 1 1 '1 1 1 1 1 1 1 1-
—
—
^- — '** ~"
e-^--" '-^-' ~^r^ ' ~
.. — — "" g . — ' "
__
1 1 1 1 1 1 1 , 1 1 1 1 1 1 1 1
5 10 20
HORIZONTAL DIST/NCE - X/D
50
100
Figure 31. Effect of L/D on trajectory for 0=90, F=10.2, R=0.10.
-------
1,0
0,5
<£>
0,2
0,1
,05
,02
,01
- ' 1 III
- 9 = 90
"~ F = 11,0
R = 0,50
L/D =
- A- 10 V
\
~~ O 2,5
—
—
, i iii
MM ' 1 1 1 1 1 1 Ih
—
.^^
^^-^__
^"^x>T" -
x ^
in IN
1
2
5 ]Q
HORIZONTAL DIST/^CE -
20
50
100
Figure 32. Effect of L/D on excess temperature ratio for 9=90, F=11.0, R=0.50.
-------
100
90
cz r
20
9 = 90
F- 11,0
R = 0,50
L/D"
A---— 3D
5
— 2,5
i i i -i i
ID
5 ]Q 20
HORIZONTAL DISTANCE - X/D
90
100
Figure 33. Effect of L/D on width for 0=90, F=11.0, R=0.50.
-------
50
20
10
2
5 —
I
9-90
F-11,0
R-0,50
L/D =
A- - - 10,
3 2,5
I I Mill
II I I I-
1 i I I i »i I
1
J I Mill
1
5 ID 20
HORIZONTAL DISTM! - X/D
50
100
Figure 34. Effect of L/D on trajectory for 0=90, F=11.0, R=0.50
-------
ON
1,0
0,5
5
0,2
0,1
,05
,02
1 I
9 = 90
F= 54,5
R = 0,05
1 \ I I I 1 I I
1II I I I I-
10
- 5
3 2,5
,01
I I I I I I I I
I
I i I i I I I
5 ID 20
HORIZONTAL DIST/WCE - W
50
]00
Figure 35. Effect of L/D on excess temperature ratio for 9=90, F=54.5, R=0.05.
-------
100
20
C/J
"^ I
0 = 90
F = 54,5
R = 0,05
L/D =
A- jfl
0 2,5
I I I I I I I
1I
I I I I I I I
5 ID 20
HORIZONTAL DISTME - X/D
50
1DO
Figure 36. Effect of L/D on widths for 9=90, F=54.5, R=0.05.
-------
100
50
20
y
ca
- •
T—i I II I I I
T—r
9 = 90
R=0,05
A JO
- 5
1 1
5 ID 20
HORIZOMTAL DIST/^CE - W
50
Figure 37, Effect of L/D on trajectory for 9=90, F=54.5, R=0.05,
-------
a\
tn
1,0
0,5
I
I
o
0,2
0,1
,05
^ I
9 = 90
F=58,8
R = 0,50
L/D =
A 10
- 5
— 2,5
I I I I I I I I
I I I I I II-
,02
.01
_L
I I I I I i i
5 10
HORIZONTAL DIST/NCE -
20
50
100
Figure 38. Effect of L/D on excess temperature ratio for 9=90, F=58.8, R=0.50.
-------
100
90
20
9 = 90
F=58,8
R=0,50
L/D =
10
- 5
2,5
10
5 10 20
HORIZONTAL DISTMCE - X/D
50
100
Figure 39, Effect of L/D on width for 9=90, F=58.8, R=0,50.
-------
Figure 39, Effect of L/D on width for 0=90, F=58.8, R=0,50.
1DO
I—I I I I H
90-
20
y
10
s
^™^^
15
0 = 90
F = 58,8
R = 0,50
17D =
A 10
- 5
— 2,5
T—I I I I I I
I I I I I i I I
l l I I I I I I
5 10 20
HORIZONTAL DIST/NCE - X/D
50
100
Figure 40. Effect of L/D on trajectory for 0=90, F=58.8, R=0.50,
-------
oo
0,5
J
-------
100
I
I I I I I I I I
III 1 I I-
50
20
5
F=1D,7
R = 0.1D
A --- JO
- 5
— 2,5
I I I I IJ IJ
I I I I M I
5 ID 20
HORIZONTAL DISTANCE - X/D
50
IDO
Figure 42. Effect of L/D on width for 9=45, F=10.7, R=0.10.
-------
100
1
1 1 1 I I 1 1
i i i i i -
20
F=10,7
R-OJD
L/D-
A--- 10
Q 5
-2,5
I I I I I I I I
i
1 1 1 1
5 10 20
HORIZONTAL DIST/NCE - X/D
50
100
Figure 43. Effect of L/D on trajectory for 9=45, F=10.7, R=0.10.
-------
TWIN VORTICES
PLANE OF TRAVERSE
LINE OF TRAVERSE
PORT OF
DISCHARGE
NEASUREf^ENT
DISCHARGE MANIFOLD
\
s
EDGE OF CHANNEL
Figure 44. Line of traverse in a current with twin vortex structure,
-------
The trajectory exhibits some rather strange results. In
general, one would anticipate that the trajectory would be shifted
upward with decreasing L/D. And indeed such seems to be the case
when considering only the data for L/D's of 5 and 2.5. But as
Figures 31, 34, 37, 40, and 43 show, in all the cases the tra-
jectory for L/D = 10 was between the trajectories for L/D = 5 and
L/D = 2.5. The most plausible explanation for this result lies in
the fact that the image walls were not used for many runs at L/D =
10. As was mentioned earlier, runs with and without image walls
were performed to determine the effect of their presence; the
effect measured being the dilution (Excess Temperature Ratio) with
no comparisions between trajectories. While the dilution was not
significantly affected, the trajectory may have been.
By Regression Curve Fits
In order to offer a homogeneous and unbiased examination of
the data collected, a program of regression analysis was performed
on the data (except discharge into stagnant ambients). Employment
of the Statistical Interactive Programming System (SIPS) available
at Oregon State University provided least-squares regression fits
where the curve fit provided is in algebraic form. If Y is the
dependent variable and X. (i = l,2,...m) are the independent vari-
ables the regression analysis provides the coefficient, a., in an
equation of the type
m
Y = a + I a. X.;
0 x x
72
-------
a. being those coefficients which give the best least-squares fit.
By letting Y above be the natural logarithm of a measured dependent
ATc W Y
variable -. T > n" > or 77 » an<^ ^' a^ove ^e tne natural logarithms
0 I* X
of the independent variables, g- , 0 , F , and =• , the algebraic
equation may be written as
m
InCY) = a + £ a. In (X.) where
0 i=l 1 x
AT r w Y L X
Y = j~ , % , or p- and X4 = ^ , 9 , Fy , R , and ^ .
o
This would provide a more suitable final relation of the type
an T ai a2 a3 a4 X a5
Y " e F 6 Fr R F
The regression analysis was carried out with the natural log's of
the dependent and independent variables resulting in a weighted
least squares fit. It is likely that the logarithmic weighting
provided a better fit than might otherwise be obtained since the
magnitudes of the variables are rendered with less absolute varia-
tion. Shown in Table 2 are the results of the regression analysis
The coefficients, a. (i = 0-5) of the curve fits are given for
AT
~ , - , and ^ , at several angles, and for the entire data set
(except 6 = 0°, and R = 0). The correlation coefficients are also
given for each curve as well as the number of observations con-
sidered in each curve fit. One is reminded that the correlation
coefficient is for the log.-log. curve fit. The regression analy-
sis results for dilution are shown graphically in Figures 45, 46,
73
-------
TABLE 2 COEFFICIENT MATRIX FOR MULTIPLE REGRESSION ANALYSIS
format * = e ,°(|j) (9) (F^ (R) (^-)
For $ =
ATc/ATo
0° ao ftl
0° +.35796 N/A*
15° +.84039 N/A*
45° +.66400 -.36808
90° -.10230 -.41294
all***
less 0° +.54328 -.46867
all
less 0 +152258 -.45247
modified
a ***
a4
• 9 -in radians
a2 a3 a4 a5 R+** N
N/A .04842 -.22908 -.90020 .93576 611
N/A .083518 -.024894 -.94828 .97363 448
N/A .12021 -.010037 -.78424 .93461 338
N/A .06786 +.11936 -.50626 .91683 534
-.43947 .077242 +.030846 -.67315 .90656 1385
-.26781 +.068075 -.13853 X -.67836 .91470 1385
(1.-. 96832 X 0)
N - number of observations
N/A - either Not Applicable or Not Available
* All of the same L/D C^ = 10).
** R+ is the value of the correlation coefficient
*** 9 is in radians
-------
TABLE 2 (contQ
a L a, a.~
format $ = e (=•} (9) (F,
For $ =
W/D
•
-vl
en
Fox $ =
Y/D
6° a0
0 -.39863
15° -.31603
45° +.39420
90° +1.0502
all
less 0 .62596
15° -.52972
45° -.22073
90° +1.1766
all
less 0 .43287
a_ a4 „ a_
) (R) (77) - 9 is in radians
al a2 a3 &4 a5
NA* +.028992 -.19104 ,54616
NA* +.052352 -.20669 .52352
-.031263 -.011977 -.30276 .42466
-.8972 +.01337 -.55311 .25288
-.088409 .30780 +.036148 -.36893 .34930
NA* -.16358 -.47332 .54447
+.067745 -.07233 -.54100 .48679
-.22950 -.095152 -.79071 .21150
-,051189 .53654 -.10060 -.63491 , .36350
R+**
.91638
.86959
.87379
.90388
.84752
.90734
.94061
.93038
.91619
N
611
448
338
534
1385
447
338
534
1384
N - number of observations
N/A - either Not Applicable ox Not Available
* all at same L/D (jj = 10)
** R+ is the value of the correlation coefficient
-------
and 47. The figures are nomographs showing the isolated effect of
several independent variables: towing ratio, angle of discharge,
and port spacing,
21
Shirazi, et. al., have pointed out that experimental evi-
dence indicates that for co-flow the dilution increases with in-
creasing towing ratio while for crossflow the dilution decreases
with increasing towing ratio. They also point out that dilution
decreases with increasing Froude No. for crossflow discharge while
dilution increases with increasing Froude No. for co-flow discharge
The curve fits offered for the data collected in this study support
the change of dilution trend with towing ratio for the co-flow and
cross flow discharge but not the change in effect of Froude No.
In fact, the curve fits here suggest very little Froude No. effect.
The small and unchanging Froude No. effect is supported by the
in
[6]
curve fits of Chasse and Winiarski, as is the change in sign of
the towing ratio exponent, a. with angle of discharge.
An effort was made to include the variation of the towing
rate effect on dilution with discharge angle by placing a factor
(1.-. 968320) (see Table 2) in the exponent. Inclusion of this par-
ticular factor was suggested by matching the exponents for co-flow
and cross flow cases. With this modified exponent the correlation
coefficient increased, but only by about 1%.
The effect of decreasing ^ is to markedly decrease dilution.
However, the trajectory and plume width are not nearly so greatly
effected by changes in r .
[6]The co-flow of Chasse and Winiarski, however, employed a false
bottom which slightly distorted the trajectory and probably
the dilution for that case when compared to the results of
this study.
76
-------
1,0
,50
,20
.
,02
,01
I I I I I I I I
DISCHARGE AT 0 = 15
DISCHARGE AT 0 = 90
I I I I I 1 I i
1II I I I I-
R = 0,05
= 0,1D
= 0,25
= 0,50
5 10 20
HORIZONTAL DISTANCE - X/D
I I I I 1 I I
50
100
Figure 45. Effect of current to discharge velocity ratio with angle and X/D as predicted by the
regression analysis.
-------
1,0
,50
VJ
oo
-20
,10
,05
< ,02
,0.
I I I I I I I I I
1
9 = 60
L J I 1 I I I I
T I I 1 I I I H
I
0 = 30
I I I I I I I
5 10 20
HORIZONTAL DISTANCE - W
50 ]JOO
Figure 46. Effect of angle with X/D at R=0.10 as predicted by the regression analysis.
-------
1,0
,50
to
CD
,20
,10
,05
CD
< ,01
I
I I I I I I I I
UD-5
L/D =
L 1 I 1 1 I I I
T I 1 I I I H
I
I I 1 1 I I I
2 5 10 20 90 100
HORIZONTAL DISTANCE - X/D
Figure 47. Effect of spacing with X/D as predicted by the regression analysis.
-------
The number of observations is 1931 for cases with current;
there were additional stagnant runs which were not considered here.
Care should be used when employing the regression curves.
The regression fits for each angle are in general superior to the
curve fits which include angle as an independent variable. Care
should also be exercised to insure that the case under considera-
tion lies within the experimental data upon which the regression
analysis operated. Table 1, given earlier in the text, provides
limits for application of the regression curves. It should also be
noted that data for port spacings of 5, and 2.5 exist only for
crossflow and 45° angle discharge and that there are considerably
fewer observations at these smaller spacings than at p- = 10.
The regression analysis offers curves which give a least
squares curve fit to the independent variables. These "fits" are
by nature one-dimensional (i.e., one regression coefficient per
independent variable) and are effectively weighted so that the
most deviant cases have the greatest effect. For these reasons,
secondary trends such as change of angle effect with changes in -^ ,
or changes in Froude No. effect with ^ are not available from such
an analysis. The effects at considerably different values of an
independent variable are averaged with weight being thrown to the
most deviant cases.
These factors should be kept in mind when dealing with the
regression equations.
80
-------
SECTION VI
ANALYTICAL WORK
INTRODUCTION
The first portion of this thesis has been devoted to the dis-
cussion of experimental data describing dilution, trajectory and
plume width of multiport thermal discharges. The second portion
will be devoted to the description of a mathematical and computer
model for predicting these quantities and the method used to de-
termine the necessary coefficients in this model.
Several models have been put forth to describe successively
more complex discharge conditions. The first studies were aimed
at describing the simple momentum jet. Such studies were carried
out primarily in the 1940fs and are well documented in References
5, 6, and 38. The buoyant jet in stagnant water was next to be
treated followed by the buoyant jet discharged into a flowing
stream.
The governing differential equations for these cases involved
turbulent terms and were coupled. The treatment undertaken then
and which continues now was to use the axisymmetric boundary layer
and Boussinesque assumptions and cross sectional jet integrals.
Transverse velocity and species profiles were estimated from ex-
perimental data and used in the equations. The result was a series
of partially coupled, nonlinear, ordinary differential equations in
which the streamwise direction was the independent variable. The
dependent variables then became the pertinent characteristic
measures of the similar profiles, i.e., centerline excess velocity,
81
-------
centerline excess species, plume half width, and relative growth
rate of centerline velocity and temperature. A similar procedure
will be described in this work when an attempt is made to include
the effects of neighboring plumes.
The model to be presented is a submerged multiport version of
the multiple cooling tower plume model proposed by Davis . This
model uses the Hirst single port program as a starting point.
The multi-port computer program was completed in the present effort
and coefficients for entrainment and drag were determined that
gave the best agreement with the experimental data presented in
this study. The fundamentals of the Hirst and Davis models are
presented here for completeness.
THE ANALYTICAL PROBLEM
A model is to be constructed which will determine the plume
characteristics of the turbulent discharge of heated water from
a single line series of round ports into either a quiescent or a
uniform, unconfined ambient. The orientation of the discharge, the
spacing between ports and the relative velocity of the ambient
fluid are variable. The ambient may be stratified and the dis-
charge diameter, velocity, temperature and species concentration
are variable.
The equations which describe conditions throughout the dis-
charge field of the jet are the transport equations of mass, mo-
39
mentum energy, and species. These are, conservation of mass,
IS- + V-(pV) = 0 , (1)
82
-------
conservation of energy,
|I + V- (VT) = O- V- (kVT) + £- » - T(|£) (V. V) , (2)
conservation of momentum,
-Pl + vv*y . (3)
and conservation of species,
|£ + v- cvc) = V-(DCVC) . (4)
Represented above are six equations with the unknowns being
three velocity components, pressure, temperature and species con-
centration (C) . The pressure gradient may be written as
VP = Poog~ + VP+ , (5)
where p^ is the hydrostatic force and ^P is the motion pressure
force. The body force term is due to gravitational action on the
jet fluid and may be written
pF = pg~ . (6)
The final equation needed to completely define the equations is an
equation of state, i.e.
p = p(T,C,P) (7)
which may be considered a seventh equation.
83
-------
The equations as they appear in (1) - (7) are in their most
general form,^ ^ less considerations for turbulence. However, one
must note that these equations are three-dimensional, nonlinear and
coupled. Because of these qualities, they are extremely difficult
to solve.
By making the following assumptions the equations may be
simplified with only a minimal loss in generality and accuracy.
The assumptions are (c.f. Hirst ):
1) steady flow in the mean,
2) fully turbulent jet flow, molecular diffusion
is neglected,
3) incompressible flow; density variations appear
only in the buoyancy terms (Boussinesque approxi-
mation) ,
4) all other fluid properties are constant,
5) fluid velocities are low (low Eckert Number)
enough to neglect frictional heating,
6) the motion pressure gradient is small so that the
only significant pressure variation is purely
hydrostatic,
7) changes in density are small enough to assume a
linear equation of state (as will be seen, the
equation of state may be written as a double
Taylor series expansion, here it is assumed to be
of linear fashion),
~[7]Exception taken for v, which has been taken as constant.
84
-------
8) the jet flow is axisymmetric and
9) the flow within the jet is that of the boundary
layer type and the boundary layer approximations
are valid.
As cited in the experimental discussion, the discharge normal to a
current may be decidedly non-axisymmetric for a significant portion
of the downstream distance. For this reason, 8) above, must cer-
tainly be questionable. However, for the purposes of a general
model, axisymmetric flow is assumed (the model will later modify
this to account for merging of the plumes) .
With the above assumption, the governing equations C1) - (4)
and (7) are written as:
continuity,
V-V =0 (8)
energy,
V- (VT) = 0 > O)
species,
V- (VC) = 0 , (io)
momentum,
fW2 - Vx(VxV) = "gPp + P'g . (ID
o
and p=p(T,C,P) can be written as,
state,
= PO(I - B(T - TO) - y(
c -
has been assumed
T,C
zero .
85
-------
Equation (12) may be rewritten as
P - P,
T(C. - c)
, CIS)
and then incorporated into £11") as
- Vx(VxV) =
- T)
- c
Hirst took these equations and employed generalized coordinates to
transform them into the so called "natural" coordinate system.
Such a system is shown in Figure 48.
Employing the axisymmetric assumption reduces equations (8) -
(10) and (14) to:
continuity,
I
_Crv) . 0
, (15)
energy,
3T 3T
species
, C17)
s-momentum,
U
3u_
3s
- p
« n
Sin 9
. CIS)
y-momentum
(uff * v||
sin
cos
86
-------
00
Figure 48. The "natural" coordinate system employed by Hirst
-------
and x-momentum,
X (KI cos QI cos 02 - <2 sin QI sin 92) , (20)
where u = component of velocity in the s direction,
v = component of velocity in the r direction,
K. = curvature of s with respect to 9.,
and K- = curvature of s with respect to 92«
By writing the equations above in terms of the fluctuating and
steady quantities and taking the time average, i.e.
f = f~ + f '
where
f = lim <0 . f 2 fdg ( the time averaged quantity
-»2t''-t
and f is the fluctuating component (note that
the turbulence effects can be included in the model. Hirst did this
(3 9 \
u»v, and T^^y^i to arrive
at the following relations:
continuity,
3u + !_ 3(rv) f21,
3s r 3r » I J
energy,
88
-------
species
3s
3r
r 3r
s-momentum,
-If
y-momentum,
U— + V — 1
3s 3r /
5ln
and x-momentum,
v
cos
cos 92 =
cos
6. sin
1
6
-
X
1 3 (ruk v1) _
F 3^ ~ sin 92
d T
sin
V
cos
cos 9. cos
1
6
, (26)
where
q+ = U2
2
3v'
One notes that the assumption of axisymmetric flow has been ex-
tended to the turbulent fluctuations in the tj> direction so that
F8l
those terms do not appear. The remaining Reynolds Stress termL J
is present, u"v', as are the turbulent convective terms v'l1 and
v'C1. The inclusion of the turbulent convective terms and the
Reynolds Stress term make the equations significantly more
complex than would be obtained for laminar flow. The six
equations (21) - (26) contain the six terms u, v, C, T, 9^ -and Q^
[8] Other terms have been dropped via the boundary layer assump-
tions .
89
-------
but they also contain u'v' , v1 , v'T' , and v'C1 . In order to
obtain closure, we assume the following for treatment of these
turbulent terms:
, , 3u
u v = -€
I?
and v1v* = 0
Often the relations are further simplified by em = eh = ec = £* the
Reynolds Analogy where e is a general eddy viscosity. Even with
these simplifications the relations remain difficult to solve, for
although they are now of only two dimensions, they are still non-
linear and coupled.
The initial conditions for u, v, T, and C etc., for these para-
bolic differential equations must be specified at s = 0 and will be
the discharge conditions of the jet. The boundary conditions are:
u -»• U sin 9, = cos 9, as r -*• °° >
OO I £
31 fT -
^f ^^ T^^^~ *^~ I CXI v L 3 S JT ~ ^^ 9
OS TJQ
or v •*• — where E =
T -*• T as r •*•
CO
as r -»•
90
-------
A 3u 3v 9TI 3C
and ~
At this point Hirst reduces the complexity of the equations
by another degree with the formal integration of the equations
with respect to r. As he states^ *, the process of integration
represents an averaging process which obscures some of the infor-
mation contained in the differential equations. The obvious in-
tent of integrating the governing equations is to lump the problem
in the r-direction and thereby avoid solving the boundary value
type problem obtained above. Not only will integration reduce the
dimensions of the problem but it will provide a purely parabolic,
albeit coupled, problem in six variables. However, it will be
necessary to provide u, T and (T profiles in r for the integration.
The profiles of u, T and C cannot be expected to be constant in
s. However, by judicious expression of these profiles, one can
create profiles which will be similar in shape at all s and whose
only differences (in s) will be the changing of certain character-
istic measures of the jet. Most often these characteristic measures
are the centerline values of velocity, temperature, species and
width of the jet.
It shall be assumed that such similar profiles exist although
they are yet to be specified. The integration proceeds as:
continuity,
[10] Reference 6, page 11.
91
-------
energy;
OO OO OO
f fifl rdr + f v|I rdr = - f I
A 3s Jn 3r /, r
rdr , (28)
species,
—
rdr
r%z. rdr = - I -
, (29)
s-momentum,
a2
ds J_ 2
OO '
/" -3« j f
I V-K— rdr = f
f v-^ rar /
p
grdr sin 6.
OD
" J r
rdr , (30)
y-momentum,
d_
ds
+ / V-K— rdr
sin 9.
•/
o Po
grdr
OO
f\
9r
, (31)
and x-momentum,
rdr
" rdr
cos 9 cos
= q*(K1 sin
cos 02 +
cos Q sin
-£
aru v
or
, (32)
92
-------
where q*
With the evaluation of certain terms, requiring the use of inte-
gration by parts and the continuity relation, and employing the
truncated equation of state, it is possible to render the equations
(27) - (32) in their most useful forms. These are given below.
Conservation of Mass,
oo
J >•
-j— I Urdr = - lim (rv) = E , (33)
Conservation of Energy,
^ f u (f - Tw) rdr = - j^- f iirdr - lim frv^T1") , (34)
^O •' O
Conservation of Species,
. r °° ,_ dC^ f °°
£- / u C - C rdr = - -j-^ I urdr - lim frv^C"1") , (35)
05 •/ ' IS JQ r-»-oo '
and Conservation of S-Momentum,
d
ds"
00 00
/ u2rdr = U E sin 9. cos 9- + / g
Jo co 1 2 J0
B(T - T.)
- c
rdr sin 6_ - lim (ru1 v' \ . (36)
T-+-CO
OOj _
KOO
Now, the other two momentum equations may be put in the form
K, = ... and <2 = ... by simple rearrangement of the equations.
If one divides (31) by sin 92 and subtracts (30) from the result,
one obtains
93
-------
cos
sin ei cos
"
f Poo - P J COS 92
V —I grdr sin 6 = 0
•'o P 2
or
C - C rdr cos
/ I
m sin 9T sin 99J / q . (37)
\
j
Likewise, if one divides (32) by cos QI cos 92 and subtracts (30)
from the result one obtains
(K, sin 9, cos 92 + KJ cos 9j sin
EU sin 9, cos 9.
— grdr sin
'° PO
cos 9, cos 9.
or, rearranging and substituting for <>
ds q cos 92
where
d9 EU^ cos Ql
~ • C38)
00 / _ \
= f G2rdr . |i - lim (r2v' )
./O 4 ->o> X '
(39)
r->o>
These then become the final two differential equations of our six
problem equations.
EMPLOYING SIMILAR PROFILES
The processes which characterize the buoyant jet lead to a
natural separation of the jet into several regimes. Tn the past
94
-------
these have been given as; CI] The zone of flow establishment near
the discharge port, CUD The zone of fully developed velocity, temp-
erature and species profiles, CHI] The transition region at the
free water surface or the maximum height of rise in stratified
environments and (IV) The region of drift flow after transition.
For multiport discharges there is a fifth region where neighboring
plumes merge due to entrainment and plume growth. This fifth
zone CV) can start anywhere along the plume depending on the dis-
tance between discharge ports, current, Froude Number, etc. These
zones are illustrated in Figure 49.
The zone of flow establishment is usually only a few dis-
charge diameters long and is characterized by jet type flow where
velocity, temperature and species profiles change from top-hat
shapes at the point of discharge to bell-shaped profiles at the end
of the zone. Zone II is characterized by a continuation of similar
bell-shaped profiles.
The zone of merging plumes is characterized by a gradual change
from a series of axisymmetric plumes to a long, two-dimensional
slot plume. This region may exist for a considerable distance
along the plume before two-dimensional slot flow is realized. In
zone III the flow changes from a rising plume to a drifting layer
of zone IV.
For deep submerged buoyant jets, the zone of flow establishment,
the zone of established flow and the merging zone are where most
of the dilution occurs. Only these three zones are of concern in
the problem of present interest.
95
-------
<£>
ON
ZONE OF ESTABLISHED
SINGLE PLUME FLOW
I - ZONE OF FLOW
ESTABLISHMENT
II -
III - TRANSITION REGION
IV - REGION OF DRIFT FLOW
V - ZONE OF MERGING
Figure 49. The dominant zones of flow for multiple port discharges
-------
Equations (33) - C38) may be reduced to simple differential
equations by employing similar profiles for u, T~, and C as mentioned
above. However, the similar profiles employed differ according to
the characteristic zone of the jet; i.e., the zone of flow establish-
ment requires a different set of profiles than required for the
zones of established flow or merging. The method of modeling the
discharge is to employ the applicable profiles in each successive
region. The equations for zone I are solved numerically using the
port discharge conditions as initial conditions. The solution
advances until zone II is reached. The conditions at the end of
zone I are used as initial conditions to zone II. The zone II
equations are solved (with modified profiles) successively as the
solution continues on in s until merging begins. Here slightly
different profiles are employed and the solution to the equations
continues until a desired limit is reached. In this manner the
differential equations are approximately solved in each of the
characteristic regions.
The profiles often employed are the Gaussian profiles in which
excess velocity, excess temperature, and excess species are written
-(ir)2 W -
Lu <* e V l' , AT « e V L/ , and AC « e
where X is a measure of the relative spreading of temperature,
17
species and velocity profiles. However, Davis found that em-
ployment of these profiles in the integral equations was not pos-
sible for merging plumes. This was due to the Gaussian profiles
97
-------
extending all the way to infinity. It was, therefore, necessary
to adopt another basic profile,40 of the 3/2 power profile used
successfully by Stolzenbach and Harleman to facilitate merging,
the same profile is assumed for temperature, velocity and species.
They are written as:
/ /T.\3/2\2 _ / /T.vV2'\2
AU oc h _ flj I f AT « li - jlj 1 , and
(• • i«n'
AC
where b differs from bj given in the Gaussian profile. A compari-
son of these two basic profiles is shown in Figure 50. Employing
the 3/2 power profile, it is now possible to carry out the inte-
grations in r and arrive at the final nonlinear, coupled, ordinary
differential equations for the various zones of interest. The
development will be curtailed slightly by not giving the species
equation. The species equation is identical to the energy equation
with changes of T to C, T to C , and T to C .
" oo oo' O O
ZONE OF FLOW ESTABLISHMENT
The similar profile relationships for the zone for flow
F121
establishment are1 J:
u = U , r < r ; (40)
O ~™"~ U
/ /r _ r vV2\2
u ' (Uo ' U~ cos 62 sin 9l) I1 - ( b U) /
+ U^ cos 62 sin 6^ rlru ; (41)
[12]All terms are time averaged.
98
-------
to
0
,1
Figure 50. Comparison of the Gaussian and 3/2 power profiles.
-------
T - T = T - T , rrt • (43)
If -5 and -x— are the same, T and C will grow at the same rate.
d Z d Z
Employing these relationships, equations (33) - (38) become:
continuity,
energy,
j—— i —^ • «*..«, • „ -„ i - lim (rv T ) » (45)
s-momentum,
.£_ = EU cos 6,, sin 6,
00 ^ 1
and curvatures,
d6., - EU sin 9, sin
dT~
g sin 62(BAToi51 + YACoi52) - lim (ru^) , (46)
3H-_ 3-- lim U2V } , (47)
100
-------
d6l EUoo cos 6
> C48)
d . _ lim r2v cos
where
d1 =- .45UQ + .55U cos 92 sin 9;l
d2 = . 25714UQ + .74286Uoo cos 62 sin 6X
d3 = .31558Uo + .13442Uro cos 82 sin QI
d. = .13352U + .12362U cos 90 sin 6,
4 O °° 2 1
dg = .31558Uo2 + .26885UoUoo cos 62 sin QI
d6 = .13552UQ2 + .24724UoUOT cos 02 sin 6X
r 5 * 2
isl = -^- + .45rtb + .25714 |-
^2 ' ¥+ '45rcb + -25714 T-
Taking the implied differentials and holding 6- constant for
(44) - (46) [13^ one obtains:
continuity,
Uoruru' + dl(rub< + bru') + d2bb' = E
energy,
U AT r^r/ + AT d-fbr,.' + r.b')
oott o3\t t/
+ T^
o
dT /U
IT
and s-momentum,
"[TI] Hirst argues that this will be of small contribution and will
simplify the algebra. While one might question the argument
he offers, it seems well worth its exclusion since its inclu-
sion would further couple the equations.
101
-------
Uo2ruru' + ds(bru' + rub') + d6bb'
C°S 9 Sin 9
+ g sin 62 (BATQi51 + yACoi52 ) - lim (ru'v') . (51)
The variables to be solved for are TU, rt> b, Q^, and 92. The
solution technique for these equations is: 1) get simultaneous
solution where the equations are setup in the form:
a
1m
r '
u
rt'
•S
=
k «
fl
f2
fs
(the species equation may be added as required)
where
35
51
a
45
a52 = 0
; (52)
a
1 1
a!2
a!3
a!4
E15
a31
a32
a__
a~.
= U r + d,b
0 U 1
= 0
= Vu + d2b
= 0
= 0
= U 2r •»• dcb
o u 5
= 0
• d5ru + d6b
- o
a21 = °
/I
a22 - Wt <
a23 = ATod3rt J
a24 ' °
a25 " °
a41 = °
a42 .= 0
a43 = °
a44 = 1
H AT d.b
O 3
y AT bd.
o 4
102
-------
a
53
55
3.T
- d2
))-
'/
and £
cos 9 sin 8 + g sin 0
- lira rrv ' u ' \
cos
- lim
cos 6.
(-EUro sin
sin
U 2r 2
o u
d5bru
then 2) use a Hamming Predictor-Corrector scheme
"• ^
to solve
equations (51) for r , r , r , b, 6,
and
The calculation continues until either r , r or r reaches
zero. At that point the equations must be changed since from then
on the centerline values of the quantities whose core radii have
already reached zero will begin to diminish.
In most cases r will reach zero before r since the scalar
properties diffuse more rapidly in ambients where great scalar
gradients do not exist. In this event the relations that change
[14] The simultaneous solution scheme is called from the IBM
Hamming Predictor-Corrector subprogram employed.
103
-------
are ij-j> ^59* energy and species. If r reaches zero before r
the new relations are
isl = i52 = 0.12857b*
and energy,
-2 •
- lira rv'T
ira (rv'T1) . (53)
The process remains as before, simultaneous solution and integration
but with the variables now being r , AT , AC , b, Q,, and 9».
U C t- X ^-
If the velocity core expires before the scalar cores, the
equations must be changed as well and Auc (Au£ = UG - U^ cos Q^
sin 0,) replaces r as a variable. However, all the relations
change. The new equations are:
continuity,
.12855b2Au ' + (.2571Au b + .4858bUro cos 6, sin 8,\ b1 = E , (54)
c \ c °° £ i/
energy,
+ U cos 90 sin 6,) T + d_b r
: oo z i/ T J
r 2 2]
-4- + .31588r.b + .13552^- Au '
£. t * J *•
- fj- (.2571AU
c
2
4858U cos 6 sin 6 + iuc * U,, cos 62 sin 8j) -j-
' C55)
104
-------
s-momentum,
.13352b2Auc + .2S714b2\Jm cos 92 sin ej Auc'
+ b' .13352bAuc2 + . 51824bAu(;Uoo cos Q sin 6
+ bU^2 cos 202 sin 261| = EU^ cos 92 sin QI
+ g Sin 82 (BAV51 + YACoi52) - lim ('^^) , (56)
\ / r-*-°°
and curvatures,
d6.
ds
cos 9
l)^ ~ (•13352Auc2 + -51428AucUoo cos 92 sin e
cos 292 sin 29^ - |i - lim (r2^)! cos 92 , (57)
J
and
de.
-EU^ sin 92 sin 91 +
|^-(.13352Au 2 + ,51428Au U cos 6n sin 9,
[ ^ \ c c °° 2 1
+ U^2 cos 292 sin 291) - |1 - lim (r2v^j , (58)
where
d? = .31558Auc + -4511^ cos 92 sin BI
da = .13352Au + .25714U cos 90 sin 0.
8 c °° 2 1
and is- and i52 are the same as those first given in the general
development zone analysis.
The initial conditions for the zone of flow establishment are
simply the conditions of jet discharge:
r = .50 , r. = .5D , b = 0.0 , 0, = 6, , and 99 = 9,
u t i IQ / ^Q
105
-------
The equations and solution techniques are now completed for the zone
of flow establishment with the exception of the entrainment, E,
which will be discussed later.
ZONE OF ESTABLISHED SINGLE PLUME FLOW
The calculation of plume properties has proceeded through the
zone of flow establishment, according to the differential equations
given in the previous section, until r and r are all zero. Gener-
ally, ATc/AT is less than 1.0 and often Auc/AU is less than 1.0
because of the definition of Au and AU . At this time the plume
co r
width will be about 2.6 port diameters and merging will not begin
if the port spacing is greater than this. In such a case the plume
will continue growing with the geometry and character of a single,
fully developed, buoyant jet until merging begins. In this region
the profiles remain axisymmetric and similar. The characteristic
variables of the jet are Au , AC , AT , b, 0,, and 0_. The similar
C C C ± £,
profiles adopted for this region are:
u = Au + U cos 00 sin 0,
oo 2 1
where
Au = Au
c
and AT = AT
c
(the species will have the same profile as temperature).
With these profiles the integrals in equations (33) - (38) can be
evaluated to yield:
continuity,
%- (.12857AU b2 + U cos 0, sin 0. ^ ) = E , (59)
as\ c °° z i ^ /
106
(• -(
(• •(
-------
energy
d
5- (.066758Au AT b2 - .12857AT U cos 90 sin 9,b5
ds \ cc c°° 2 1
dToo / ,2 V / _, ,-\
= • dT~ M2857Aucb + S- uoo cos 92 sin 61 - lim (r^n r6(n
\ / "I^^^OO
s-momentum,
^- b2 (.066758Auc2 + .25714AucUoo cos 92 sin
cos 292 sin 261) = EU^ cos 92 sin 9^^
+ .12857b2g sin 92(BATc + YACC ) ~ lim (rv'u1) , (61)
and curvatures,
d8
ds
- = EU cos 9, /( b2 (. 066758Au 2 + .25714Au U cos 90 sin 9.
°° I/ I \ c c oo 2 1
1
9 / — "-rj \ I
cos 292 sin 29j - |- - lim (r2v' ) cos 9 (62)
r-*-<» J
and
ds
= /- EUro sin 92 sin Oj + .12857b2g (3A?c + YACc) cos 62J/
.066758Au 2 + ,25714Au U cos 99 sin 9.
c c °° *•
cos 292 sin 261 - \ lim (r2v' j .. (63)
r->°°
107
-------
These may be written as:
Sj = E , (64)
d G - dT~G
d? G2 - - dl~ Gi
d_
ds
G_ = EU sin 6. cos 9_ + G.g sin 90 - lim (rv'u1) » (66)
3 °° 1 24° 2 \ /
= EU_ cos 9, /|(G, - ^- - lim (r2v' )} cos
r-»-«>
/[(•• -t- -'•
and
i8_
=/- EU sin 90 sin 9. + g cos
\ ~ 2 1
- - lim r . (68)
These equations may be integrated for G. (i = 1,2,3,4) using the
Hamming Predictor-Corrector method. Then the variables Au , AC .
c c
AT , and b (8, and 6_ will already have been obtained) may be
obtained once G. (i = 1,2,3,4) are known, from the relations below:
G. = .12857Au b2 + 5- U cos 60 sin 9. , C69)
1 c 2 °° • 2 - 1
G- = .066758AU AT b2 - .12857b2AT U cos 90 sin 6. , C70)
i C C C °° 2 1
G_ = .066758Au 2b2 + .25714b2Au U cos 60 sin 9.
3 c c °° 2 1
+ Y~ U«2 cos 292 sin 261 , (71)
108
-------
and
, (72)
Put in a more direct form, from (69) and (71),
( \1 /o
BB /BBZ 2 cos 292 sin ^ , (74)
BB = 2G..U cos 90 sin 9, - G, - 4.0386G..U cos 6- sin 0. , (75)
Joo 2 13 1°° 2 l'xy
and
CC = 4.0386G12 . (76)
With b determined, Au is found to be
v 2
G, - -2. U cos 9_ sin 61
A 12°° 2 1 fT,^
Au = , (77)
c .12857b2
and likewise
AT = — 2 . C78)
C b2f.066758Au - .12857U cos 9_ sin 9.)
\ C °° 2 I/
G. may be determined with (78) (and its species equivalent when
necessary) above.
The quantities G. (i = 1,2,3,4) are the local mass flux,
energy flux, species flux (when used), momentum flux and density
109
-------
difficiency respectively. The integration of these "secondary"
variables (G.) as opposed to the "primary" variables, Au , AC ,
3. C C
AT , and b, is advantageous in two ways. First, the fluxes are
of the natural properties of the jet. Second, and more importantly,
the integration of the secondary variables does not involve the
matrix solution of simultaneous equations, as does integration
of the equations employing the primary variables.
Employing the values of Au , AC , AT and b calculated at
C C C
the end of the zone of flow establishment as the initial values
for the governing equations in the zone of established single plume
flow, one may proceed using the equations developed above.
ZONE OF MERGING PLUMES
At some point, the edges of the adjacent plumes will begin
to merge. HThen this occurs, the discharge loses its axisymmetry
and the profiles become dependent on the angle with respect to
the neighboring plume. This does not invalidate the lumping inte-
gral, process nor is the concept of similarity threatened. However,
certain adjustments must be made.
The original equations (33) - (38) could have included an
asymmetric quality if the integrals had been considered as area
integrals rather than line integrals, with the surface integration
being
i»2n>°°
Jo Jo rCr,
Now when axisymmetry is assumed, as was done earlier, the integral
in <}> can be brought outside the integral in r, evaluated to be
2ir, and divided out from both sides to obtain the relations
iio
-------
(38). However, now that ajxisymmetric profiles no longer exist
it is necessary to include this integral into the equations, i.e.,
where one hasjor(r,s)rdr in (33) - (38) one now has
f f r(r,4>,s) rdrd<() (79)
•'o •'o
In this form the equations are quite general and assuming one
knows the complete profiles of velocity, temperature, and species,
the integrals could be carried out numerically if not in closed
form.
Rather than deal with profile integrals of the (79) type,
it is more convenient to employ a different coordinate system
when merging begins. This new coordinate system is shown in
Figure 51. In the new coordinate system £ lies through the axis of
a line of adjacent jets and n is perpendicular to the £-s plane
and hence is perpendicular to the line of jets. If the profiles
are symmetric with respect to both these axes (to « in n and to
L/2 in £), then the integration is simplified greatly. However,
this case is only attainable when the adjacent plumes have the
same velocity, temperature, and species profiles; are of equal
spacing from the origin (n = C = 0) jet and all have centerlines
lying on the £-axis. This is tantamount to saying that the jets
are line discharged normal to a uniform free stream (if one exists)
in a common hydrostatic plane and experience the same dilution
and ambient history. The most likely deviation from these conditions
would be a line discharge into a non-normal uniform free stream.
It is not likely that small deviations from the normal would
111
-------
MERGING JETS
Figure 51. The coordinate system for the merging plume analysis
-------
severely affect the calculations, but no experimental evidence
exists with which to determine the allowable extent of the devia-
tion.
If symmetry exists only with respect to £, then the develop-
ment becomes more difficult since the integration of the profiles
are more complex. However, this is probably the only way to include
other multiple discharge geometries once merging is initiated.
In the final analysis it may be necessary to forgo any symmetry
conditions and treat the most complex of multiple discharge con-
figurations three dimensionally. The analysis presented here (as
in Reference 17) will restrict itself to the case where symmetry
exists with respect to both ri and £.
Experiments made in this study indicate that the temperature
at £ = L/2, ri = 0 is approximately twice that of the temperature
at £ = 0, n = L/2 during merging. This suggests that the profiles
may be assumed to simply be the superposition of adjacent single
plume profiles. Considering this, the merging profiles should
satisfy the following:
a) the profiles should be smooth in all directions,
b) the slopes should be zero at £ = 0, n = 0, and r,
= L/2, n = 0,
c) when the plumes just begin to merge they should
retain their single plume profiles,
d) the profiles should be the superposition of the
single plume profiles (where applicable) with no
point allowed to exceed centerline properties.
113
-------
In keeping with the similar profiles developed prior to
this, and the discussion above, the following are the assumed
profiles in the zone of merging:
u = Au + U cos 0_ sin 0.
00 ^ A
, (80a)
Au = Au = Au l -
n £ I \c
» C80b)
= iul -
for 0<5 C (^ c ^ C
at C = L/2, then Ar = AT .
? c
114
-------
In order to make the treatment homogeneous, the governing
equations will be divided by 2ir, as appropriate to the area inte-
grals. This was also done in the first two zones. With the pro-
files described above, the governing relations C33) - (38) may be
written as follows.
The continuity equation becomes,
cos e sin ed • C82)
where
- H - E
dT Hl ~ E
Hl = ^b2Auch!(a) + F~uoo cos 92 sin
a = b
hl(C° =
(a - X)3/2) dX , (84)
and
, I /I _(|) + sin -i(|) . (85)
Now after Au = Au at C = L/2, then Au - Au and one obtains,
£ c t> c
H, = — f.45Au + U cos 9_ sin eJh_(a) . (86)
ITT[ c°° 2 1J2
115
-------
In the equations a = L/b represents the degree of merging since
for a = 2 . , b = L/2 and the plumes are just beginning to merge
and for a - 1., b = L and the plumes are nearly merged.
For the energy equation,
L/2
|/O /o'K
cos
d_ .
ds 2
dTf
dT
/ -..-,. \
H - lim frv'T1 )
f (87)
where
H2 = -63y6b2ATcAuch3(a) +
cos
sin QI ^ (a) , C88]
and where
a/2
U - X
dx
3/2
)2(l--(» -
l - (a -
. (89,
When AT = AT at £ = L/2, AT = AT . With the profile descrip-
L, C (, C
tion used, Au approaches Au at £ = L/2 to the same degree that
\> c
AT approaches AT . Hence, when AT = AT , Au = Au . So with
Q C C, C C, C
AT,. = AT and Au,, = Au one obtains,
£ c £ c
H0 = —AT /.31558AU + .45U cos 00 sin 9.)h-fa) . (90)
2TTC\ C 0° 2 1/2
The species equation would take on the same form as the energy
equation under the same assumptions that in this region T, C, and
u profiles are the same.
116
-------
The s-momentum equation takes the following form for the zone
of merging:
u + U^ cos 92 sin 9J dr|dC
H, = EU cos 90 sin 9, + H.g sin 6_ - lira (ru* vr) (91)
A oo 7 1 4 2. \ I ' ^ '
ds 3 °° 2
where
cos 6 sin
cos 292 sin 291 h2(a) , (92)
and
.9,
After Au = Au , equations (92) and (93) become
b2h9(a)
H = 1 /.31558Auc2 + .9Au(,Uoo cos 92 sin
2(
. (93)
cos 262 sin 6 , (94)
and
. (95)
For the zone of merging the curvature equations take the
forms ,
d6 EU cos 9,
~ , (9<3
ds
.im Irv1 /) cos 9
/ E2
\H3 "5— lim
117
-------
and
— - " EU°° Sln 92 Sin 6l + H4g 2
-
^^
H3 - f- - lim (r'v1 )
The functions hj (a) > "2 t°0 and h3 C0^ are incomplete inte-
grals in a. These integrals are not solved in closed form at
present but may be readily solved numerically. In fact, the re-
sults of this numerical integration show that the integrals may be
adequately represented by a simple interpolating scheme between
various values of the evaluated integrals. The computer program
evaluates the integrals in this form.
The quantities H. (i = 1,2,3,4) are the same as G. (i = 1,2,
3,4) in the previous subsection. They represent the local mass
flux, energy flux, momentum flux, and density deficiency. The
advantages of using these secondary variables was pointed out
earlier.
In the process of solving the differential equations it is
necessary (as in the previous subsection) to solve for the primary
variables. Once H. (i = 1,2,3,4) are known, the relations for H.
= H. (Au , AC , AT , b, 0,, 0,, h, ~ ,(a)) may be used to obtain
JL C C C X ^ JL ) £ ) 3
Au , AC , AT , and b. The following equations are the result of
the simultaneous solution of those equations. Taken in sequence
for the region after merging is initiated,
/BB2 CC V'
\/ » A * 2 AA /
V 4AA /
b = 1- HT n/^-r- f^) , C98)
118
-------
where
rh (a)h3(a)
77921 11 (-99)
i. 2 /• „. i I
BB = 2H1Ua> cos 92 sin
h (a)h (a)
- I.5584H.U cos 69 sin 6. ——-= H, , (100)
1 h 2(a) 3
and
CC = 2.4480H 2
h1(o)
,2
H, - —U cos 00 sin 0, h (a)
1 TTOQ X 1 x
AU = -i 1 —i , (102)
c .28648b2h1(a)
and
ATc = __ . ± . (103)
^(.63116Au h_(a) + .9U cos 6_ sin.6 h, (a))
TT\ C3 °° 2. ll/
When the primary variables at £ = L/2 sum to the centerline values,
due to merging, the relations above change to the following:
I BB
" \~ 2AA "
1/2
4AA
where
- - £C . (104)
2 AA
AA = .1777h2(a)Uoo2 cos 292 sin 2Q1 , C105)
BB = - 1.1168H U cos 60 sin 0. - H. » (106)
1 °o 2 1 6
119
-------
and
V
= 4.8959 -i- ; (107)
Auc = 6.9813H1 — 2.22220^ cos 62 sin 6 ; (108)
and finally
AT = — n -2 . (109)
b h0(a) l.31558Au + .45U cos 8,, sin 6,)
^ \ c <» 2 I/
The initial conditions for the merging zone equations are the
values of G. (i = 1,2,3,4) when a = 2.
The appropriate governing differential equations have now
been established. During the discussions suitable initial con-
ditions have been specified. No boundary conditions remain to be
specified with the possible exception of those boundary turbulence
terms, of the type lim (rvkT'), and the entrainment, E = lim(rv).
Discussion follows on these subjects.
BOUNDARY TURBULENCE TERMS
Most analytical discussions deal with discharges where the
boundary turbulence terms are neglected, since describing these
terms is somewhat difficult. The only experimental work found
that has included such a description in relation to submerged dis-
19
charges has been that of McQuivey, Keefer, and Shirazi , supple-
mented by their summarization in Reference 20, The information pro-
21
vided by those reports was used by Shirazi, Davis, and Byram
in the Hirst single port plume model in order to model co-flow
120
-------
discharge. While their results were less than conclusive, it
appears necessary to include these terms Cat least for co-flow)
2 0
but at higher magnitudes than suggested by experiment . The
model presented here employs the same boundary turbulence model
that Shirazi, Davis, and Byram used.
If it is assumed that
—i—r dC —i i du —i i , em
v c = -e -5— , v u - -e -5— , v T , and e, = e = T— .
c dr ' m dr ' ' h c X '
where
e = momentum eddy diffusivity,
e, = thermal eddy diffusivity,
e = species eddy diffusivity,
and X = turbulent Schmidt Number (approximately 1.13); and if
Gaussian profiles are assumed, the turbulence terms may be written
as
>j \ O O
lim
7*fr\-l "h H4
^TT~) ~ IFITMTI
/ \ v » * \
and
- 2
/- I r U C / ~ \r U 'l - ~ '^r 7 ' ^112^
T-*- \/2b, \ o o o/ V o c
v i \ /
The v1v1 term is assumed to be,
lim /Tz^rvr\ lim /r2i
121
-------
By suitable approximation to the downstream interval and making
use of the empirical relations developed in Reference 20, an
approximate relationship may be developed for e. in terms of
/ r i si
Froude Number, ambient discharge velocity ratio, and v u1u*/U L
The boundary turbulence terms can then be written as
lim,
[/ i | | \ 1-1 82 2.1 8 2 1.1 3 1"]
.192/-SLg-S-J R F
4Xe
-2
U
/ab^Vc
im/r7TTr\
u Uu C I
bi\ ° °/
1.1 8 2 2.182 1.131
R '
, 1.1 8 2 2.1 8 2 1.1 3 1
R F
-2
X2 ATc
e — , (115)
-2
4 X2 utc
AT
f— > (H6)
lim
/^bj
R2. .5618_
. (117)
21
Shirazi, et.al. , reported that employing the above relationships
in the equations had little if any effect on the dilution in
crossflow but had a significant effect on co-flow. However, the
values employed for (u' /U^j ' were significantly larger than
19
those measured by McQuivey, et.al. . As is noted by Shirazi, et.
21
al, , the entrainment term is considerably larger than the boundary
turbulence terms. Therefore, the inclusion of turbulent effects
in entrainment would probably give more satisfying results.
[15]Personal communication with M. A. Shirazi, Research Engineer,
U. S. Environmental Protection Agency, Corvallis Environ-
mental Research Laboratories, Corvallis, OR.
122
-------
Shirazi, et.al. found it necessary to change ( u' /U
depending on towing ratio, R, and Froude Number in order to get
satisfactory agreement with data.
ENTRAINMENT
The mathematical definition of entrainment has been given as
lim Crv) = E. Physically, entrainment is the rate of ambient
r-H»
fluid brought into the jet by virtue of the shear or turbulent
transport action at the jet boundary. By continuity, it is also
the local rate of change of-total mass flux through the cross-
section of the jet. The entrainment is important in the develop-
ment of a suitable model since it controls the growth of the jet,
and via the governing equations determines the dilution and tra-
jectory. Unfortunately, this term has eluded explicit definition
and thus appears in the form of a postulated function. How this
function is postulated determines the value of the model as a
predictive tool.
Lack of an explicit definition for entrainment is a mixed
blessing. On the one hand, some of the information lost through
integration may be returned to the model in the entrainment function
On the other hand, the roles of the various physical elements must
be estimated and weighted into the entrainment function. Deter-
mining which physical actions should be included and to what
degree is enlightened guesswork at best, especially when the dis-
charges are physically complex. If the model is physically sound,
any conceptual errors in the entrainment function will plague
efforts to match model predictions to experiment.
123
-------
The entrainment for the simple plume was first discussed by
22
Taylor who was seeking to simplify the entrainment concept by
using the bulk properties of the plume to describe entrainment
rather than the mixing length concepts used previously. Morton,
4
et.al. employed the assumption that entrainment was proportional
to the relative plume centerline velocity and the local character-
istic width measure, of the plume, i.e.
E = ab,Au (118)
1 c
This type of entrainment term remains today as the most basic
entrainment function form. Most models employing it yield reason-
able prediction of dilution and trajectory for discharge into
deep quiescent stratified or unstratified ambients. In the
above equation "a" is the entrainment coefficient. For Gaussian
distributions (i.e. b. is the characteristic width measure) the
best values for "a" have been found to be 0.05?'- •* for the simple
momentum jet and 0.085 •* for the low Froude Number discharges.
Observing that different values were necessary for different
23
Froude Numbers, Fox developed an entrainment function of the
form' -
C119)
r a,i
= k * pf
L i-i J
by arguing that consistency among the differential equations and
similar profiles (extended to u*vzas well) requires the above re-
lationship. This entrainment function implies that the local
buoyancy influences entrainment processes. The concept is rather
weak when one considers that the dominant processes are primarily
[16] Albertson, et . al . 3
[17] Abraham, G . 2 *
124
-------
turbulent and are in fluids of only average Prandtl Number (0.7 to
about 7). Perhaps the most compelling reason for inclusion of the
local Froude Number term is that Hirst found this function to
give better prediction than the Morton, et.al.4 relation in his
model. With equation CH9) and the proper entrainment coefficient
Cabout 0.80) the dilution could be predicted for discharge of a
buoyant jet at all angles with acceptable agreement.
Discharges to flowing ambients have hot been modeled yet
with complete satisfaction. Agreement has been obtained between
models and experiment, but in reaching such agreement the models
require entrainment functions whose coefficients vary with the
25
discharge conditions. Fan conducted experimental research into
buoyant jet discharge into a crossflow which he used to determine
the entrainment coefficients of his computer model. He employed
the same entrainment function as Morton, et.al. but with the
vector ambient-to- j et velocity difference. Fan included the
drag force due to the pressure variation around the jet in his
horizontal and vertical momentum equations. His vertical and hori
zontal momentum equations were
=
( u2rdr cos e = EU^ + FD sin 92 , U20)
and
) = -f ^ rgdr - Fn
/ Jo P D
- u2rdr sin 8, = - rgdr - Fn cos 9. . (121)
2/ P D 2
With this and the drag force equation,
FD = CpUj sin 292 yzbj , C122)
125
-------
the coefficients left to be determined by matching model to ex-
periment were, the entrainment coefficient, "a", and the drag
coefficient, Cn. Unfortunately, Fan could not obtain good agree-
ment except by varying "a" and Cn with the discharge conditions.
The variation was considerable for Cn, ranging from 0.1 to 1.7,
while "a" ranged from 0.4 to 0.5.
O £ 0-7
Flatten and Keffer as well as Hoult, et.al. , chose to
alter the entrainment function rather than employ the drag force.
Flatten and Keffer dealt only with non-buoyant jets in a cross-
flow. The entrainment function they employed had two terms,
E = a,b(u_ - U_ sin 6,\ + a_bUM [sin 6, - sin 6, ) - (123)
tlb(uc - U^ sin 6.JJ + a2bUw (si
The first term is the familiar entrainment term due to jet tur-
bulence. The second term is a function included to "account for
the vortex shear inflow." As has already been mentioned, the
crossflow discharge induces twin vortices to form in the jet,
which persist downstream for some distance. The influence of these
vortices on entrainment is supposedly modeled by the second term
of Flatten and Keffer's entrainment function. Despite the in-
clusion of this second term and second entrainment coefficient,
they were unable to get agreement without varying both coefficients
2 7
Hoult, et.al. also employed an entrainment function with
two terms;
E = a1b|uc - U^ cos 62| + a2bU00 sin 92 , "(124)
the first term being jet turbulence entrainment, the second one
associated with forced entrainment due to a normal external
126
-------
velocity. The agreement with experiment obtained by Hoult, et.al.
was better than that of Flatten and Keffer, since the entrainment
"constants" did not vary as much. Flatten and Keffer's entrain-
ment coefficients ranged over a factor of 3, while the values
given by Hoult, et.al., a = 0.12 and a» ranging from 0.6 to 0.9
(in a leter report, a, = 0.11 and a- = 0.6), varied to a smaller
degree.
Hirst employed an entrainment function of the form
E =
b'uc - uco sin ei cos
U^b v/1 - (sin QI
cos
This is seen to be a combination of Fox's entrainment function and
a generalized Hoult, et.al. entrainment function. Hirst compared
computer runs for a Gaussian profile model with data from four-
teen other authors for conditions of crossflow and co-flow (as
well as 62 = 135° and 45°) discharge into stratified and unstrati-
fied, flowing and stagnant ambients. The values he obtained for
entrainment coefficients were a^ = 0.057, a2 = 0.97, and a3 = 9.0
for Gaussian profiles; when altered for the 3/2 power profile these
values become 0.029, 0.51, and 4.8 respectively.
Since the computer routine used to solve the 3/2 power pro-
file merging equations presented here is essentially that of Hirst
(with appropriate profile and merging changes) an examination of
the results of his modeling effort might prove valuable.
Hirst obtains excellent agreement for the simple momentum
jet, (a = 0.057, Gaussian). However, when buoyancy is also con-
127
-------
sidered, the agreement is not as good. He notes that the pre-
dicted trajectories lie considerably below the experimental results
25
of Fan .especially for higher Froude Numbers. By placing a =
0.082 better agreement was obtained, but this value gave less
acceptable predictions for many other flows. For co-flow cases
Hirst's predictions are less dilute than corresponding experi-
ments, especially for higher RTs. This he attributes to the longer
29 6 F181
starting lengths proposed by Abramovich (c.f. Hirst LJ) than
supposedly really exist. For the discharges into a crossflow Hirst
obtains good agreement (trajectory) for R = 0.125. However, the
predicted trajectories are slightly lower than experiment for
R<0.10 and are higher than experiment for R>0.10. In general, the
dilutions Hirst obtained for crossflow were greater than that
predicted from experiment. His results for stratified ambient
discharges gave good agreement with experiment.
In summary, Hirst's work involved the use of constant entrain-
ment coefficients but did not give exceptional agreement with ex-
perimental data for all discharge conditions. Inability to match
the data for all discharge conditions may imply that the true en-
trainment is not accurately modeled by the proposed entrainment
function; it may also imply that assumptions made during the
development of the model render the model less universal than
hoped. The predictions of Hirst give adequate agreement for a
*~
moderate range of all the parameters with, constant entrainment
coefficients.
[18]Hirst modifies the original starting length function of
Abramovich to obtain agreement with Albertson, et.al.3,
for R = 0.0.
128
-------
Hirst discusses what elements an entrainment function should
include. These are:
1) local mean flow conditions within the jet, u and b;
2) local buoyancy within the jet, F ;
L
3) velocity ratio, R;
4) initial jet orientation, 0, and 6- ;
o o
and 5) ambient turbulence.
Or A
The entrainment function of Fan and Morton, et.al. include only
23
1). Fox's entrainment function includes 1) and 2). Flatten and
Keffer's entrainment function employs only terms due to 1), 3),
27
and 4). Hoult, et.al. , used an entrainment function having only
1), and 3), with 4) included somewhat implicitly. The Hirst entrain-
ment function contained 1), 2), and 3) with 4) involved implicityly.
None of the entrainment functions contain effects of ambient tur-
bulence (although the terms in the governing equations accounting
for ambient turbulence are in the Hirst program but set to zero).
Koh and Fan were one of the first to deal with the case of
merging adjacent plumes. They used a computer routine which would
begin with a single round port solution and at some point switch
to a slot solution. Two criteria were given for determining the
transition point (when b = L/2, and Eround = Esiot^' however,
the two criteria gave essentially the same solution. While this
was a way of handling the multiport case, it did not model the
merging region.
Harleman and Jirka approached the problem slightly differently.
They stated that the multiport case could be adequately modeled
by an "equivalent slot" solution. By making the multiport discharge
129
-------
momentum and mass fluxes per unit length equal to those of a slot
discharge, an equivalent slot discharge width may be defined.
Combining this with a newly defined slot Froude Number provides
sufficient information so that the standard slot solutions may
be used to predict the trajectory and dilution of the multiport
discharge. However, a recent report on deep submerged multiple
port discharges into stagnant and coflowing ambients CKannberg
14
and Davis ) seems to dispute the acceptability of an equivalent
slot solution. According to that report, both the transition
model and the equivalent slot model over-predicted dilution. In
each of these cases no attempt has been made to include merging
effects in the entrainment function, except to switch from a round
jet entrainment function to a slot jet entrainment function.
14
Kannberg and Davis speculate that the entrainment model
should be sensitive to the area of entrainment which diminishes
as the plumes merge. And indeed, effects of adjacent plumes may
be evident long before the jet boundaries touch, since the jets
are always competing for common entrainment fluid. In this light
Davis proposed that the entrainment function contain an additional
term to allow for effects of competition and reduction of the
entrainment surface. Before boundary contact, the form of the en-
trainment function is given as
+. a3UJ> sin eJ ; (126)
130
-------
while after boundary contact (when b>^ L/2) it takes the form,
E = a + bu - u cos 9
- I cos lTs + a3Uoo| sin e . (127)
The change in entrainment functions is due to the change in en-
trainment area. Before merging, the entrainment area was IT on a
side, but after merging begins, the entrainment area becomes ap-
proximately Aentr = b (TT - 2 cos ~. [L/(2b)]) on each side.
Ideally there should be no difference between a, and a ' since
44
at b = L/2 the two entrainment functions are the same. The en-
trainment equations of Davis are the same as Hirst's except for
the modification due to merging, and like Hirst's include elements
1-) , 2), 3) and 4) (implicityly). As Davis mentions, the entrain-
ment coefficients other than a. (a4*) in his entrainment function
should be approximately those of Hirst. Since the Davis entrain-
ment function is similar to the Hirst function, it should suffer
the same deficiencies for single port discharges if the same co-
efficients are used as Hirst recommended.
The entrainment function(s) adopted initially in the present
modeling effort were those of Davis. Since the model here includes
the zone of flow establishment, the Davis entrainment function,
—|- = Cj .0204 +'.0144|- | 1 - R cos 92|
o o L ° J L
_° + C.R sin 6 1 + / , (128)
131
-------
is used in that zone.
The values suggested by Davis for use in the entrainment
function employed in this study are essentially those of Hirst,
altered to the different plume width definition;
c, = 1.05, c_ P 34., c, = 4.3,
a.^ = 0.029, a2 = 0.51, and a_ = 4.8.
One notes that the terms in the entrainment function appear
in linear combination. This speaks for the simplicity of the en-
trainment models presently available.
TUNING THE MODEL - RESULTS
The governing differential equations have been determined,
the entrainment specified, and expressions provided for boundary
turbulence. All that remains is to determine the best entrainment
coefficients. The calculations were carried out on an IBM 370/158
computer operated by Optimum Systems, Inc., of Bethesda, Maryland.
The computer code employed was originally for the Hirst model.
Extensive revision of the code was performed in order to accomodate
the different profiles and the merging process. The code was then
used in the present study by tuning the entrainment coefficients.
Alterations were made as necessary to examine the influence of
various entrainment and turbulence terms. The terms of the en-
trainment function allow for a successive evaluation of the en-
trainment coefficients. For the case of the momentum jet in a
quiescent ambient, only c.. and a. are involved. Therefore, to
tune the model for these coefficients, various values of c^ and
a., were used in the model. The results were compared with the data
132
-------
4
of Morton, et.al. , with the conclusion that c, = 1.06, and a. =
0.029 gave the best fit. The model prediction and results of Morton,
4
et.al. , are shown in Figure 52. As can be seen, excellent agree-
ment results. The coefficients are very nearly those suggested
by Hirst when converted to the definition of plume width used here.
For the case of the buoyant jet, the coefficients c_ and a?
are the additional terms to be determined. No attempt was made
to tune c?; the value, 34., given by Hirst was considered adequate.
However, problems arose when attempts were made to tune a~ . Use of
values near those suggested by Hirst resulted in trajectories
considerably below those of experiment, primarily for moderate
Froude Numbers (30-100). In order to reach acceptable agreement
with experiment, a_ had to be set to zero and a raised to 0.05,
rather than 0.029 as given for the momentum jet. Model predicted
trajectories (a- = 0.0, a.^ = 0.05) are compared with experimental
trajectories and predicted trajectories from other models in Figures
53 and 54. The comparison of dilution for experiments and models
r 191 i
are given in the CederwallL J type graph of Figure 55. As is evi-
dent from these graphs, the coefficients suggested give good
agreement at low and moderate Froude Numbers but less satisfactory
results at high Froude Numbers. It would seem that there is some
Froude Number effect that is not included in the entrainment
function. However, since no rational explanation exists for how
and why such a term should be included, its inclusion is not
justified.
~[T9] Originally attributed to Cederwall30 (c.f. Fan25).
133
-------
1,0
,5
,2
=P
,05
,02
,01
1 TT
MOMENTUM JET
THIS STUDY, F = 999
& » 0,029
i i i MM—•—i—i i i i II-M
MORTON,ET AL/,CURVE, F = <*>
i I 1 I I I I II .1 i I I I I I II i I I I I I I I I
5 10 20 50
HORIZONTAL DISTANCE - X/D
100 200
500 103
fl
Figure 52. Model prediction and the Morton, et al. empirical curve for the momentum jet.
-------
01
60
50
30
20
10
3/4 r- -,-,
O ANWAIfO F=ll
— FWF
HIRST"
THIS STUDY, di =0.05
eL2=o.o
or /
O FAN0, F = 33 /
20
30 40 50 60 70
HORIZONTAL DISTANCE - X/D
80
90
Figure 53. Model prediction and experimental data for trajectory of single port discharges,
-------
320
100
I 60
o\
20
0
1
HIRST"
THIS STUDY,a-|=0,05^2=0.0
0 20 40
// , f
'/ /
60 80 100 120
HORIZONTAL DISTANCE - X/D
Figure 54. Model prediction of trajectory of single port discharges.
160 JBO
-------
OKI
10
1
X—N
iC'
,5
I I
I I I 11II ~
T)
CEDERWALL^ ANALYSIS
ABRAHA^ ANALYSIS
FAN AND BROOKS5' ANALYSIS
THIS STUDY, a.jO.05
O
I I I I I I II
I I I I I 1 I-
,2
,1
O DANISH ISOTOPE CENTER
D FRANKEL AND CUNNING
D> CEDERWALL
I
I I I I I I I I
I I
MULTIPLE PORT DISCHARGE L/D=10
O KANNBERG AND DAVIS, F=55
• KANNBERG AND DAVIS, R=30
O KANNBERG AND DAVIS, F=H
I I I I I II . I I I I I I
I I
,2
10,
20,
50,
,1
Figure 55. Model predictions and experimental data of dilution for single port discharges
30,
1.0 2,0 5,
(Y/D) / F
100,
(original
graph by Cederwall ).
-------
Crossflow discharge provides the means of tuning the best
value for c3 and a_. The primary source of experimental data con-
25
cerning this discharge is Fan . c_ was obtained by tuning to
match predicted starting lengths with the starting length curve
offered by Hirst18, S = 6.2 De~3'4R. This curve is the same one
6
25 31
obtained by Fan and given graphically for the data by Gbrdier
(c.f. Fan). The best value found for c_ was 6.0. With this c
J O
value, the starting lengths are as shown in Figure 56.
Considerable difficulty was encountered when trying to tune
for a_. The final value determined for a_ was 11.5 and, as is
pointed out in Figure 57, agreement with Fan's trajectory data
for high and low current ratios is not exceptionally good.
The comparisons of dilution are somewhat hindered by the fact
that experimental values were taken in the cleavage between the twin
vortices observed. Hence the measured dilutions are depressed
below the true profile maximums located at the centers of each
vortex. The measured maximum concentration may be depressed as
much as 65% from the vortex center concentration according to
Fan's measurement. Liberty was taken to reproduce two figures
presented by Fan in Figures 58 and 59. The profiles amply testify
to the twin vortex structure and the depressed centerline values.
No attempt was made to tune to these depressed measurements.
However, as is seen in Figures 60-63, the centerline concentrations
predicted by the model range from 45% to 100% higher than measured
values depending on R. It can then be assumed that the program
predicts the approximate local maximum concentrations likely to
occur downstream from the crossflow discharge.
138
-------
04
0>
CO
7 hD
HIRST EQUATION, Se = 6.2D6
•71
GORDIERS^1 DATA (c,F,
FLATTEN AND KEFFER'S DATA (C,F, HIRST18)
RESULTS OF THIS STUDY WITH Cj= 6,0
1
1
1
1
1
0,2 0,3 0,4 0,5 0,6
TOWING RATIO - R
0,7
0,8 0,9
Figure 56. Model and experimental crossflow starting lengths.
-------
60 -
50
BROKEN LINES - CURVES THROUGH EXPERIMENTAL DATA OF
SOLID LINES - THIS STUDY WITH d= 11,5
EXPERIMENTAL DATA FROM THIS STUDY FOR !M),50, L/EKLO,
30
20
10
0
0
10 20 30 40 50 60 70
HORIZONTAL DISTANCE - X/D
Figure 57. Crossflow model prediction and experimental data trajectory comparison.
-------
+ MEASURED POINTS
Figure 58. Concentration profile for F=20. and R=0.125.
Ambient flow strikes plume from top of figure,
25
(taken from Fan , pg. 127)
S/D =32.5
C = 0.05S
c
D =.76cm.
MEASURED POINTS
SCALE
0 cm.
Figure 59. Concentration profile for F=40. and R=0.125. Ambient
25
flow strikes plume from top of figure, (taken from Fan )
141
-------
K)
1,0 rr
0,5 -
0,2
0,1
,05
,02
,01
I I
9 = 90
R= 0,0625
I I I I I I I
III I I I I-
MODEL PREDICTION, THIS STUDY WITH & - 11,5
CURVE THROUGH FAN'S25 DATA
F = 20
ESTIMATED MAXIMUM PLUME EXCESS
TEMPERATURE (1,65 TIMES FAN'S
MEASUREMENTS)
CURVE THROUGH FAN's" DATA
I I I I I I I I I
J.
' I I I I I I
5 3D 20
HORIZONTAL DIST/^CE - X/D
50
100
Figure 60. Dilution for crossflow discharge from a single port, R-0.0625, compared to Fan
25
-------
1,0
0,5
0,2
0,1
.05
,02
,01
I I
0 = 90
R = 0,0825
I I I I I I I
CURVE THROUGH FAN's^ DATA
F=20
1 I I I I I H
MODEL PREDICTION, THIS STUDY WITH
a3 = 11,5, F = 30
/- ESTIMATED MAXIMUM PLUME EXCESS
.TEMPERATURE (1,65 TIMES ;
FAN'S
CURVE THROUGH
I I I I I I I I
I I I I I I I
5 10
HORIZONTAL DISTM! -
20
50
3DO
Figure 61. Dilution for crossflow discharge from a single port, R=0.0825.
-------
1,0
0,5
<1 0,2
o
i
HJ 0,1
,05
,02
,01
I I I II MM
9 = 90
R = 0,125
1 I I I I I H
CURVE THROUGH FAN'S25 DATA
F = 20
MODEL PREDICT I ON, THIS STUDY WITH
= U,5, F = 30
ESTIMATED MAXIMUM PLUME
EXCESS TEMPERATURE (1,65;
TIMES FAN'S MEAS.)
CURVE THROUGH FAN'S25 DATA
-L
i
I I I I I I I I
J_
I I I I I I
2 5 10 20 50
HORIZONTAL DISWNCE - X/D
Figure 62. Dilution for crossflow discharge from a single port, R=0.125.
100
-------
1,0
0,5
en
0,2
0,1
,05
,02
,01
1 I I I I I I I
9 = 90
R = 0,25
CURVE THROUGH FAN'S DATA
I I I I I I I I
I I lilt H
•MODEL PREDICTION, THIS STUDY WITH
a3 = n,5, F = 30
ESTIMATED MAXIMUM EXCESS —
PLUME TEMPERATURE (1,65
TIMES FAN'S MEASJ
or
CURVE THROUGH FAN's" DATA
F=20
I I I I I 1 I
5 10 20
HORIZONTAL DIST/NCE - X/D
50
100
Figure 63, Dilution for crossflow discharge from a single port, R=0.25,
-------
The inability to match the trajectory for several current
ratios initiated a search for methods that would give better agree-
ment. The drag force on the crossflow discharge was included (as
was done by Fan), but the results were not an improvement. Like-
wise, several alterations were considered in the R sin 62 term of
the entrainment function, notably, raising R to exponents other
than one. While exponents of 3/2 and 2 gave good trajectory and
dilution fits, there was no physical explanation as to why powers
other than one should be employed. However, an argument based on
the curvature and vortex action of the jet might provide better
agreement and involve physical insight into the entrainment pro-
cesses.
The twin vortices are probably generated by a combination of
bending and edge shearing on the jet. The fluid at the edge of
the jet is of lower axial momentum and hence is easily sheared
downstream by the flowing ambient. As is well known, when fluid
in a circular conduit is forced through a bend, the high velocity
center fluid resists bending and pushes to the top of the bending
conduit thereby forcing the slower fluid around the edge of the
conduit to the bottom. Continued action of this sort results in
twin vortices in the conduit and an increase in momentum loss
due to viscous shear at the walls. In conduits the twin vortices
have been observed to persist as far as 50 to 75 pipe diameters
downstream Csee Reference 32). The strength of the vortex may be
measured by the curvature of the pipe.
The same action occurs in the jets discharged to a crossflow.
Here the jet is bent over by the oncoming free stream rather than
146
-------
the confines of the pipe, with the result being the formation of
the twin vortices, increased entrainment and increased dilution.
It is important to note that the increased entrainment and increased
dilution are a result of the curvature induced vortex initiation
and hence is not represented by the R sin 92 entrainment term.
Although inclusion of the effects of curvature and shear
induced vortices in the entrainment function certainly seems de-
sirable in light of the above discussion, care must be exercised.
Certainly when curvature is high and a significant amount of warm,
high velocity, centerline fluid is being pumped into immediate
contact with the ambient, entrainment is going to be enhanced. But
what about downstream? It has been stated that the vortex structure
persists far downstream for pipe flow, and indeed for free twin
vortices, as are formed in the wake of aircraft, the vortices
decay as x'1/3. However, Brown33 notes that vortex wakes are found
to grow at much slower rates than those of non-rotating wakes. It
is now known that the rotation produces a certain stabilizing effect
on the system and appears to cause a reduction in turbulent eddy
diffusivity at least in the radial direction. Apparently, an en-
trainment term modeling curvature and shear induced vortices should
be short lived axially.
An attempt was made to add a term to the entrainment function
d99
of the form a,.R -3—- , the thought being that additional entrainment
j CL S
due to initial vortex action would be proportional to the curvature
as is indicated from pipe flow Csee Reference 33).
The entrainment function would then be,
147
-------
de
+ a3bUw sin 92 + a^bR^ . (129)
The attempt failed due to instabilities in the predictor-
corrector integration probably attributable to coupling between
entrainment and the curvature equation. It is possible that another
integrating scheme (Runge-Kutta for example) might not be plagued
75 ^2o~^2
by these difficulties. A function, ~ -- = — which approximates
d9 R s2
-T —
to within 60% at all ambient to current ratios (0.05 to 0.50)
-|— ranges over three orders of magnitude for high towing rates]
.)
was tried. The trajectories and dilutions obtained, for cases
where a_ was 45 and a, was 5.0 are shown in Figures 64, 65, and 66.
o o
While agreement is quite good (remembering the depressed experi-
mental concentration values) the term given above is an imprecise
d92
alternative for the -3— term and without more refinement renders
as
the results mere speculation. But the results using this term
are promising. Until further investigation of initial entrainment
in crossflow discharges, the curvature entrainment term remains
unproven and therefore was not included in subsequent tuning work.
Co-flow discharge has presented prediction and tuning problems
18
to nearly all modelers. Hirst was able to get acceptable starting
lengths (greater than experiment but very close to the curve given
28 21
by Abramovich ). However, as pointed out by Shirazi, et.al. , the
dilution trends are opposite those of experiment for R and Froude
2 1
Number. Shirazi, et.al. proceed to include the boundary turbu-
148
-------
60
50
I
a
30
20
10
0
0
BROKEN LINES - CURVES THROUGH THE EXPERIMENTAL DATA OF
SOLID LINES - THIS STUDY
F=30,a3=U,5
"'F=20}lH).25
I
10
20
70
90
30 40 50 60
HORIZONTAL DISTANCE - X / D
Figure 64. Trajectory comparisons for single port crossflow discharge, model includes a curvature
term in the entrainment function.
-------
1,0
0,5
tn
O
0,2
0,1
,05
,02
,01
I I
9 = 90
R = 0,0625
I I I I I I I
III I I I I-
ESTIMATED MAXIMUM PLUME EXCESS
TEMPERATURE ( 1,65 TIMES FAN'S
MEASUREMENTS)
MODEL PREDICTION
QLJ= 11,5, F = 30
CURVE THROUGH FAN's" DATA
F=20
•MODEL PREDICTION, £3 = 5,0,5.5 =
F=30
CURVE THROUGH FAN's" DATA
F-40
I I I I I I I I | | I l_L
5 10 20
HORIZONTAL DIST/WCE - X/D
50
100
Figure 65. Dilution comparisons for single port crossflow discharge. Curves include predictions
by the model with a curvature term in the entrainraent function, R=0.0625.
-------
1,0
0,5
r
o
0,2
0,1
,05
,02
,01
1 I 1 I I I I I I
9 = 90
R = 0,25
MODEL PREDICTION, d = 5,
1—I I I I I H
MODEL PREDICTION,^ = 11,5, F = 30 —
ESTIMATED MAXIMUM EXCESS
PLUME TEMPERATURE (1,65
TIMES FAN'S MEAS.)
or
CURVE THROUGH FAN V^ DATA
or
CURVE THROUGH FAN'S 3 DATA
= 20
I I I I I I I I
I
I i I I I i I
5 10 20
HORIZONTAL DIST/NCE - X/D
50
100
Figure 66. Dilution comparisons for single port cross flow discharge. Curves include predictions
by the model with a curvature term in the entrainment function, R=0.25.
-------
lence terms discussed earlier and obtain acceptable agreement
although artificially high turbulence values were needed. Without
turbulence terms the merging model predicts starting lengths which
greatly exceed both Hirst's predictions and experiment. It was
necessary to include the boundary turbulence terms discussed earlier
in order to diminish starting lengths and get the proper dilution
trends with R. The effects of including boundary turbulence terms
on the starting length for the merging model are shown in Figure
67. The higher the values ofvu* /U^, the greater the boundary
turbulence. Included in Figure 67 are correlations and experimental
data of other authors. While there appears to be some disagree-
ment in the exact values of the starting length, there is little
doubt of its trend with R. This trend makes it quite difficult
to match experimental data beyond the starting length since the
starting length increases with R while the downstream concentration
decreases with R. The best results (shown in Figure 68) are not
very satisfactory, however, they are an improvement over results
without turbulence terms and it appears that the values ofv u1 /Vm
which give the improved fit are near those obtained experimentally
19 / i ^
by McQuivey, et.al. (McQuivey, et.al., foundv u /U^ to be about
0.033 for discharge to a smooth walled flume, the value giving
"ball park" dilutions for the model is 0,025 Calthough comparison
is rather difficult without accurate starting lengths).
3 8
The results of Forstall and Shapiro , also shown in Figure
68, are somewhat misleading since the dilution is reduced with in-
20
creasing velocity ratio, R. The studies offered by Shirazi, et.al.
152
-------
On
oo
I
32-
11-
10-
9-
8-
CO-FLOW STARTING LENGTHS
BROKEN LINES - RESULTS OF
THIS STUDY WITH SHIRAZI,
ET AL, TURBULENCE
TERMS
"'2/u
»/^
uu = 0,03
7
6
5
4
3
2
1
n
- x-^-- — A :^-^ ~^^ /
*1 ^^c^^ \
J^^\ ^- L
^szZf-' ^ Y ^FORSTALL AND SHAPIRO"*3 CURVE
/ ABRAMOVICH RELATION"-^ c //, . -IODNT,
/ OQ — kH T LL".)u
_ / ue ,27 1 - R 1/.214 + 1.44R
/ ^P,
l__ "ADJUSTED" ABRAMOVICH o_goDl+R 1, A FORSTALL AND SHAPIRO^ DATA
~~ CORRELATION (c,F, HIRST10") °0 ""^ ^ ~ ^HA + 1-18R
1 1 1 1 1 1 1
0
,05
,1
,15
,2 ,25 ,3
TOWING RATIO - R
,35
,45
,5
Figure 67. Co-flow starting length comparison, single port discharge, model contains the turbulence terms,
-------
1,0
0,5
0,2
,05
,02
,01
R = 0,05
BROKEN LINES - EMPIRICAL
EQUATION FOR DATA BY
PORSTALL AND SHAPIRO™, F
T~i—i i 11 U-H
= 0,25 - THIS STUDY
WITHOUT TURBULENCE
TERMS
CD
AUQ X/D
SOLID LINES - FROM THIS STUDY USING
TURBULENCE TERMS OF SHIRAZI, ET AL,,
F = 30, uu = 0,025
I I I I I 11II .1 I I I I 11II .1 i I I I 111
500 105
10 20 50 100
HORIZONTAL DISTANCE - X/D
200
Figure 68. Velocity dilution for co-flow single port discharge, the model employs turbulence terms,
-------
7 /-
as well as Chassi and Winiarski give the opposite trend for co-
flow discharge. Some results of the experimental work presented
earlier in this text are given in Figure 69. These results
definitely show the dilution increasing with increasing velocity
ratio, R.
21
Shirazi, et.al. speculate that more satisfactory model pre-
diction would be obtained if the turbulence were included in the
entrainment function. The results given in the present study seem
to support this speculation. However, too many unsupported assump-
tions are necessary in the formulation of the boundary turbulence
terms used here to provide any confidence in the method. In the
present study, for R>0.10, the velocity decayed faster than the con-
centration in the zone of flow establishment. This may be due
to the scaling of the turbulent terms, however, there may be many
other causes.
In any event, co-flow discharges cannot as of yet be accurately
predicted, although ball park numbers and trends may be duplicated.
The only coefficients remaining to be tuned are a^ Ca4') and
c. associated with the effect of merging plumes. It is reasonable
to expect that the entrainment should be near that for a single
round port before merging begins and near that of a slot jet after
a long period of merging. If one takes the single round port
entrainment to be E - 2iraAu b (a is the entrainment coefficient)
A -f
as suggested by Morton, et.al. , one sees that -^- « b. If the slot
jet entrainment is taken as Ee = 2Le Au as suggested by Fan and
5 5 C-
Brooks37, and Koh and Fan13 as well as others, with e = 0.16, then
155
-------
1,0
0,5
en
ON
0,2
0,1
,05
,02
,01-
I I I I Mil f—1 1 I I I Mil
| - 1 I I I Mi,
9 = 0
R = 0,10
= 0,50
EXPERIMENTAL DATA FROM THIS STUDY, STRAIGHT LINE
FIT BY INSPECTION
R=0,05
R - 0,97
SOLID LINE - F = 60
BROKEN LINE - F = 11
I I I I I I I II
\
R = 0,25
I I I I I I Ml
III I II I I
10 20 50 100 200
HORIZONTAL DISTANCE - X/D
500 103
Figure 69, Experimentally obtained co-flow thermal dilutions of this study for L/D=10 and various R's
-------
E
« L and is constant for any L. It would seem plausible that
* *
the entrainment should move smoothly from the round port case to
the slot case as merging progresses. The value of these functions
may be graphed along with that of the Davis "entrainment surface"
entrainment function. Such a graph is shown in Figure 70. Here
a = 0.043, e = 0.16 and L is taken to be 5. As can be seen, the
round port entrainment continues to grow as the entraining area
27rb grows. The slot jet entrainment remains constant since its
entrainment area is constant and the Davis "entrainment area"
function (which starts at b = L/2) diminishes to a constant since
the available entrainment area of the merging jets diminishes to a
constant. It is noticed that the Davis entrainment function is
only about 27% of the slot entrainment function in the limit as
b approaches infinity.
The Davis entrainment function was employed with a. and c.
tuned to give the most satisfactory agreement with experimental
data. When tuning the model for the merging jets, the model pre-
dictions were matched against the crossflow experimental data. The
values found most suitable were a. = 0.2 although the results did
not allow for good comparison of c. values and a. = 0.0 would
probably yield nearly identical results over the range of comparison
When tuning to crossflow discharges, it was found necessary to
25
include the drag force analysis previously used by Fan . Since
no such drag force was necessary for the single port case, the drag
force was written in the following manner:
CD
Zone of Flow Establishment, FQ = -Q^ Uoo2N/2 , (130)
157
-------
tn
00
I
L/D = 5
I I I I 111
E
I I I I I-H
AU xQ-rrcos
- THE DAVIS
ENTRAPMENT
AREA" FUNCTION -
I I I I I I I
L I I I I I II
I I I I I I I
10 20
pufEwimu
500 105
Figure 70, The value of various entrainment models as plotted against plume width b.
-------
where
N = N.J2 + N.22 + N32
and
N. = - cos 299 sin 91 cos 0, ,
N2 = sin 262 + cos 292 cos 29, ,
N = - sin 99 cos 99 sin 9, ;
O &* £* J.
Zone of Single Plume Established Flow,
Cnb! „
FD - -T- »- IT • tl31)
where N, NL, N2, and N, are the same as above;
and the Zone of Merging Plumes,
F = C LUTO2 V-
where N, N,, N_, and N_ are the same as above.
J. £ O
The only equations which change are the curvature equations.
These now become
(si
,„ i Jsin 292 + cos 292 cos 29, - sin 261 sin 292
= IEU cos 9, + F,
, — IHUUU3U.T1- n
ds I °° ID cos 9,
/q cos 92 , (133)
and
Is" = IK/ KT • T~) + Y(C " C~)| rdr c°s 92
sin 0. sin 091/ q , C134)
159
-------
In the expression for FD, the drag coefficient, C , must be
determined. The values of CD which gave reasonable agreement were
CD = 3.0 for R = 0.10, and CD = 0.70 for R = 0.50. The trajectories
obtained with these values are given in Figures 71, 72, and 73,
for L/D values of 2.5, 5., and 10. respectively. As can be seen,
the model matches the data for the most part, especially for what
would be moderate Froude Numbers. However, there appears to be
a significant change in trajectory with Froude Number for the model
predictions. The effect of Froude Number is not nearly as notice-
able in the experimental data. One also notes that the plume
seems to follow a straight line trajectory after the jet is initially
bent over. The straight line represents a balance between the drag
force, buoyancy force, and added vertical momentum due to entrain-
ment. Additional experimental data further downstream would have
been useful in assessing this effect.
The dilution comparisons are given in Figures 74-79 for various
L/D's. In general, the predicted excess temperature concentrations
match the data quite well. At L/D = 10 the dilution is greater
in experiment than prediction. But this is probably due to the
depressed temperature measurements made between the twin axial
vortices. It is expected that as merging proceeds this vortex
structure would be broken down. The improved thermal dilution
agreement at closer spacings tends to support this idea.
The results of tuning the merging coefficients would be much
more satisfactory if the drag coefficient were a single value for
25
all flows. The model employed by Fan also utilized a drag coef-
ficient, and the value he found necessary to match experiment also
160
-------
60
50
40
S 30
20
10
0
L/D = 2,5
SOLID LINES - MODEL PREDICTIONS
BROKEN LINES - EXPERIMENTAL DATA OF
THIS STUDY
0 10 20 30 40
HORIZONTAL DISTANCE - X/P . . .
Figure 71. Comparison of model predicted trajectories with experimentally obtained trajectories
for L/D=2.5, crossflow discharge.
-------
NJ
60
50
S 30
g
t—t
I 20
10
\
0
0
L/D - 5,0
SOLID LINES - MODEL PREDICTIONS
BROKEN LINES - EXPERIMENTAL DATA OF
THIS STUDY
a 3=11,5
a/rO,16
C=K),70
10 20 30 40 50 60
HORIZOMTAL DISTANCE - X/D
70
90
Figure 72. Comparison of model predicted trajectories with experimentally obtained trajectories
for L/D=5.0, crossflow discharge.
-------
60 - L/D = 10, 0 = 90
40
20
10
0
SOLID LINES - MODEL PREDICTIONS
BROKEN LINES - EXPERIMENTAL DATA OF
THIS STUDY
0
10
20
70
30 40 50 60
HORIZONTAL DISTANCE - X/D
Figure 73. Comparison of model predicted trajectories with experimentally obtained trajectories
for L/D=10, crossflow discharge.
-------
1,0
0,5
o
i
0,2
0,1
,05
,02
,01
1 I I I I I I
I I I I I I H
L/D = 2,5
9 = 90
R = 0,10
SOLID LINES - MODEL PREDICTIONS WITH d^lLS, dfjO.16, C =3,0
_ BROKEN LINES - EXPERIMENTAL DATA FROM THIS STUDY
1
I I I I I I I I
I
I I I I I II
L 2 5 ]0 20 50
HORIZONTAL DIST/WCE - X/D
Figure 74. Comparison of experimental and model predicted excess temperature for L/D=2.5,
R=0.10, crossflow discharge.
100
-------
ON
cn
1,0
0,5
-------
1,0
0,5
0,2
0,1
,05
,02
,01
^ I
L/D = 5,0
8 = 90
R = 0,10
I I I I I I I
1 I I I I I H
SOLID LINES - MDDEL PREDICTIONS WITH d J=1L5,
BROKEN LINES - EXPERIMENTAL DATA FROM THIS STUDY
I
I 1 I I 1 I I I
I
I I I I I I I
2 5 10 20 50
HORIZONTAL DIST/NCE - X/D
Figure 76. Comparison of experimental and model predicted excess temperature for L/D=5.0,
R=0.10, crossflow discharge.
300
-------
1,0
0,5
$
I
o
0,2
ON
0,1
,05
,02
,01
1 I
LTD = 5,0
0 = 90
R = 0,50
1 I I I I I I I
"I I I I I I h
SOLID LINES - MODEL PREDICTIONS WITH d=11,5,
CD=0,70
BROKEN LINES - EXPERIMENTAL DATA FROM THiS STUDY
I
I I I I I I I I
I
I I I I I 1
50
2 5 10 20
HORIZONTAL DISTANCE - X/D
Figure 77. Comparison of experimental and model predicted excess temperature for L/D=5.0,
R=0.50, crossflow discharge.
100
-------
1,0
0,5
OV
00
0,2
0,1
,05
,02
,01
1
1—T~
L/D = 10
0 = 90
R = 0,10
1 I I I I I I
I I I I I H
SOLID LINES - MODEL PREDICTIONS WITH d5=11,5
a^.16, CD=3,0
BROKEN LINES - EXPERIMENTAL DATA CURVES OF THIS STUDY
1
I I I I I I I I
I
1 J J I I 1 I
5 10 20
HORIZONTAL DISTANCE - X/D
50
100
Figure 78. Comparison of experimental and model predicted excess temperature for L/D=10,,
R=0,10, crossflow discharge.
-------
1,0
0,5
< 0,2
CD
•—«
fe
S 0,1=
,05
,02
,01
1 I I I I I I I I
1 I I I I I H
LTD = 10
0 = 90
R=0,50
— SOLID LINES - MODEL PREDICTIONS WITH d3=11.5, SL^O.IG, CD=0,70
^m
BROKEN LINES - EXPERIMENTAL DATA FROM THIS STUDY
I
1 I I I I I I 1
1
I I I I I I I
2 5 10 20 50 100
HORIZONTAL DISTANCE - X/D
Figure 79. Comparison of experimental and model predicted excess temperature for L/D=10.,
R=0.50, crossflow discharge.
-------
varied (from 0.1 at R = 0,0625 to 1,7 at R = 0.25). However,
while his coefficient increased with increasing R, the results for
this study required that CD decrease with increasing R. This trend
is more in line with the results for flows around cylinders and
spheres.
It should be noted that the drag is also dependent on spacing.
Inherent in the expressions offered here for drag force is the
variation of the effective CD with spacing. For wide spacings the
drag force is very small. The drag force grows during plume growth
and merging of the jets. Once the width of the jet is the same as
the port spacing, the drag force is independent of width of the jet
Setting aside the questions and problems raised in these at-
tempts to tune the model to experimental data; the model is now
complete. All of the necessary entrainment coefficients are speci-
fied and the model predicts trajectory and dilution with acceptable
accuracy for a wide variety of discharge conditions. The co-flow
discharge, even with the turbulence parameters discussed earlier,
will not, however, give exceptional prediction.
The final recommended entrainment coefficient values are:
Cj = 1.06, c2 = 34., c3 = 6.0, c4 = 0.20,
ax = 0.05, a2 = 0.0, a3 = 11.5, a4 = 0.16,
CD = 3.0 at R = 0.10, and CD - 0.70 at R = 0.50.
SOME COMPARISONS AND PREDICTIONS
The purpose of the analytical development of the model pre-
sented in this study was to obtain a predictive tool to handle
multiple port discharges. It was important that this model did
170
-------
not suffer from the same difficulties as the Koh and Fan13 "transi-
tion" model or the Jirka and Harleman "equivalent slot" model.
The major problem of these models was that they over-predicted
dilution. The transition model had some additional difficulties.
When the transition point was reached, several of the plume charac-
teristics underwent step changes in value in order to accomodate
the shift from the round port solution to the slot solution and
still maintain a conservation of momentum, energy and mass flow.
By referring to Reference 14 we may reproduce the predictions
of these two models, experimental data and the "merging" model dis-
cussed here for horizontal discharge into a quiescent ambient of
a multiport diffuser with an L/D of 10. The comparison of the
predictions is offered in Figures 80, 81, and 82 (the predictions
of Jirka and Harleman were taken from Figure 2.4 of Reference 7
with the aid of experimental trajectories from this study). The
Davis model of this study with the entrainment coefficients
already arrived at, accurately predicts the dilution of the multiple
port discharge for the cases presented. The plume characteristics
of the Davis solution remain smooth and continuous functions during
merging. As shown in Figure 83, the trajectories of the experimental
data are matched quite well by the predictions of the Davis merging
model, although the Koh and Fan transition model also matched the
data.
With the model developed exhibiting the desired merging proper-
ties Csroooth, continuous, and accurate), certain aspects of the
merging process .may be explored. The notable interest is in the
effect of port spacing on trajectory and dilution.
171
-------
1,0
0,5
< 0,2
i
o
0,1
,05
,02
,01
T 1—TTT
1 1—I I I I I H
LTD = 10
0 = 0
F=H
R-0,0
•ROUND PORT SOLN,, d~ 0,065
JJRKA AND HARLEMAN
EQUIVALENT SLOT MODEL
EXPERIMENTAL DATA FROM THIS STUDY
DAVIS MERGING MODEL
KOH AND FAN TRANSITION MODEL-
I
I I LI 1 J I 1
I
I I I I I I I
50
100
12 5 10 20
AXIAL (INTERLINE DISTANCE - S/D
Figure 80, Comparison of excess temperature predicted by several models and experimental data
for L/D=10, R=0.0, F=ll, horizontal discharge.
-------
1.0
0.5
< 0,2
I
CD
0,1
,05
1
CD
,02
,01
1—I—I I I I 111
6 = 0
F = 30
R = 0,0
KOH AND FAN TRANSITION MODEL
— <•> EXPERIMENTAL DATA FROM THIS STUDY
JIRKA AND HARLEMAN EQUIVALENT SLOT MODEL
DAVIS MERGING MODEL, F=30 —
1 I I I I I I
1
I I I I I I I
2
50
100
5 10 20
AXIAL (INTERLINE DISTANCE - S/D
Figure 81. Comparison of excess temperature predicted by several models and experimental data
for L/D=10., R=0.0, F=30, horizontal discharge.
-------
1,0
0.9
,0.
0,]-
,05-
1
0 = 0
F = 55
R = 0,0
I I I FT II I
I I I I I I I-
DAVIS MERGING MODEL, F=60 —
KOH AND FAN TRANSITION MODEL
EXPERIMENTAL DATA FROM THIS STUDY
JIRKA AND HARLEMAN EQUIVALENT SLOT MODEL
_L
I I I I I I 111 I \ I I I I I 11
50
100
1 25 10 20
AXIAL (INTERLINE DISTANCE - S/D
Figure 82. Comparison of excess temperature predicted by several models and experimental data
for L/D=10., F=55, R=0.0, horizontal discharge.
-------
120-
100
00
S
20
0
LTD = 10
9 = 0
R=0,0
SOLID LINES - DAVIS MERGING MODEL
1
F = 60
EXPERIMENTAL DATA
O F = 54,6
O F = 29,6
O F = 10,0 FROM DYE
STUDIES
1
i
1
0
160 180
80 100 120
HORIZONTAL DISTANCE - X/D
Figure 83. Comparison of model predicted trajectories with experimental data for L/D=10, R=0.0,
horizontal discharge.
-------
In Figure 84 the trajectory and dilution points for hori-
zontal discharge into a quiescent ambient is given for port
spacings of 10, 5, and 2.5. The results demonstrate that with
the merging model having the Davis "entraining area" entrainment
function, the port spacing has a large affect on both the tra-
jectory and the excess temperature concentration. No attempt
was made to compare this with experimental data that might be
available for comparison for discharge from close spaced jets
(L/D = 2.5) into a quiescent ambient.
It is tempting to wonder about the effect of raising the en-
trainment value for the fully merged plume from the Davis "en-
training area" value,
E = 0.05 LAU /it
to that of the slot jet,
E = 0 . 16LAu /IT *• •*
s c
The entrainment increase would be greater than three-fold. With
such an increase, it is expected that the effect of reducing port
spacing would be less than that illustrated in Figure 84.
In an attempt to satisfy such speculation, an entrainment
function similar to that used in the Koh and Fan transition model
was used in the merging model developed here. The entrainment was
allowed to grow with b, as does the round jet entrainment, until
some limit was reached, after which it remained constant. The
entrainment function before reaching the limit was
[20] The entrainment functions are divided by 2ir.
176
-------
60-
50
30
20
10
LTD =10
O 0,200
<>0,]20
Q 0,098
F = 30
9 = 0
R = 0,0
~0 1Q20304050607080 90
HORIZONTAL DISTANCE - X/D
Figure 84. Trajectory and dilution prediction for various port spacings, F-30, R=0.0, horizontal discharge.
-------
E - O.OSb AucCl. - 2a
when the value for E reached
E = 0.16L Au (1. - a;.)/7r
5 C T" X
the entrainment remained constant at that value of E . The above
entrainment functions would be the same as those employed by the
Koh and Fan transition model if a^. were 0.0, and the value 0.05
were 0.043. The model, employing a' = 0.5, seemed to give good
agreement with experimental data for crossflow (L/D = 2.5, 5, 10)
and stagnant horizontal discharge (L/D = 10). It is interesting
that with a' =0.0 the dilution was greatly over-predicted result-
ing in excess temperatures much lower than experimentally measured.
Figure 85 shows the dilutions as predicted by the merging model
employing the entrainment functions considered above with a' =
0.50, for stagnant horizontal discharge and L/D = 10. It is seen
that for the case offered, agreement is good for dilution.
From the consideration of this alternate entrainment function
it is apparent that the merging model of Davis provides a physically
accurate treatment of the merging process and that in the limit
(as b approaches infinity) the entrainment does not seem to approach
the value obtained for the slot jet but rather approaches a value
between 0.0159L and 0.025lJ21-'.
PLUME WIDTH
One consideration may have become conspicuous due to its
absence. All of the comparisons and careful tuning of the model
[21]Multiply these values by 2ir for the entrainment associated
with models where the governing equations are not divided
by 2?r. 178
-------
1,0
0,5
<
i
o
0,2
0,1
,05
,02
,01
I
LTD = 10
9 = 0
KOH AND FAN TRANSITION MODEL
I Mill
I I I If H
DAVIS MERGING MODEL WITH
"ENTRAPMENT AREA" ENTRAPMENT H
FUNCTION, d^ = 0,16
DAVIS MERGING
MODEL/TRANSITION" EN-_J
TRAINMENT FUNCTION
EXPERIMENTAL DATA FROM THIS STUDY
JIRKA AND HARLEMAN EQUIVALENT SLOT MODEL
I I I I I I I
5 10 20
HORIZONTAL DISTANCE - X/D
50
100
Figure 85. Comparison of various models and experiment for merging jets excess temperature,
emphasis on comparison of the "entrainment area" and transition" entrainment results
-------
dealt with the thermal dilution and trajectory; no comparisons have
been made between model predictions and experimental values of
plume width or centerline velocity. Part of the reason is that
these quantities are more difficult to measure than plume tempera-
ture and trajectory. Very little velocity data exists for complex
flow conditions or widely varying discharge parameters. Plume
width has been difficult to define experimentally although the
profile half-radii (radius to the point where axial velocity is
one half of the centerline value) may be determined if accurate
velocity or temperature profiles are known. The bulk of the ex-
perimental data then is for concentration and trajectory down-
stream from the discharge.
The centerline velocity and plume half-radii for several models,
4
the experimental work of Morton, et.al. , and some experimental work
from this study are given in Figures 86 and 87. Figure 86 reveals
that the centerline velocities predicted by the Fan model (using
a.. = 0.043, where a, has been adjusted to the plume definition used
here) and the model discussed in this study (using a-1 = 0.05) are
less than the Morton, et.al., values or the predictions of the model
offered in this study when &1 = 0.029. With &l = 0.029 the results
are very similar to those of the Hirst model for the momentum jet.
When one examines Figure 87 it is seen that the plume width
agreement between prediction and experiment for the Fan model
Ca = 0.043) and the tuned model given here with &1 = 0.029 is good
Cagain this is similar to the Hirst model). However, with al = 0.05
the model developed in this text over-predicts plume width for high
Froude Number single port discharges.
180
-------
=P
OO
LOr—T—[• p-T-r-
; THIS STUDY WITH
O.sU di= 0,029
F = 999
0,2
0,1
,05
,02
,01
1 I I I Mil ' I 1 I I I M-H
NORTON, ET AL.7 CURVE,
UQ X/D
THIS STUDY, Oi= 0,05
F=60 \
iq /
FAN (C.F. SHIRAZI AND DAVIS^^ F = °° , d-^ = 0.043—
I II II L_J I I I I III
10 20 50 100
AXIAL CENTERLIE DISTANCE - S/D
200
500
Figure 86, Comparison of momentum jet centerline.velocity predictions of several models and
the empirical curve of Morton, et al. ,
-------
oo
I I I I I I III
I I Mill-
MORTON,ET AL,4, F =
--0,0951-*-)
FROM MANIPULATION OF EQUA,
47 & 48 OF REFERENCE 4
FAN (C,F, SHIRAZI AND DAVIS^)
F = 60, d = 0,0435
EXPERIMENTAL DATA, THIS STUDY, REDUCED
TO HALF-RADII, F=55, I7EKLO
I I I I I I I II .1 I I I I I I II .1 I I I I I I I
10 20 50 TJOO 200 500 K)5 2X1D5 SxlO3
AXIAL (HTERLINE DISTANCE - S/D
Figure 87, Comparison of momentum jet half-radii predictions of several models with experimental
data and the empirical curve of Morton, et al. .
-------
The tuning process recommendation was that a1 = 0.05. It was
felt that accuracy in thermal concentrations and trajectories was
more ecologically useful than accurate plume widths. Therefore,
the modeling here of single port discharges is deficient in that
the model predicts exaggerated widths for high Froude Number single
port discharges. However, for multiple port discharges the pre-
dicted plume widths compare quite nicely with experimental values.
Shown in Figure 88 are the width predictions of the Davis merging
model and the Koh and Fan transition model for horizontal multiple
port discharge (L/D = 10) into a quiescent ambient. Also offered
in Figure 88 are reduced data from this study (measured widths
divided by 0.8). As can be seen, the Davis merging model provides
excellent agreement for the widths of these buoyant discharges.
The Koh and Fan model, however, overpredicts the widths. This is
probably due to the use of the slot entrainment function which is
significantly larger than the value employed in the Davis entrain-
ment model when merging approaches completion. The lack of a smooth,
continuous transition to the slot jet flow may also contribute
to this. One may conclude that the width predictions are accurate
for the merging model when multiple port discharges are considered
but are not as accurate for single port high Froude Number discharges
183
-------
oo
100
50
20
1 10
5
L/D = 10
9 = 0
1 I I I INI
R = 0,0
KOH AND FAN TRANSITION MODEL'
T I I I I I 111 x'1
I I I I ll-M
o
DAVIS
p=3Q-/ MERGING
MODEL
O F=30 REDUCED EXPERIMENTAL WIDTHS_
O H50 OF THIS STUDY
I I I I I I I II .1 I I I I I I II . I I I I I I I 1
10 20 50 10
AXIAL CEMTERLINE DISTANCE - S/D
20
50 100
13
Figure 88. Comparison of the width predictions of the Koh and Fan transition model and the
Da"vis merging "model with experimental data.
-------
SECTION VII
REFERENCES
1. Reichardt, H. , Impuls- Und Warmeaustausch in freier Turbulenz,
Z. agnew. Math. Mech-. , Volume 24, num. 5, pp. 268-72, 1944.
2. Schmidt, W. , Z. Agnew. Math. Mech., Vol. 21, pp. 265, 351, 1941.
3. Albertson, M. L., Y. B. Dai, R. A. Jensen, and Hunter Rouse,
D iffusion of Submerged Jets, A.S,C,E., Dec. 1948, pp. 1571-96.
4. Morton, B. R., Sir Geoffrey Taylor, J. S. Turner, Turbulent
Gravitational Convection from Maintained and Instantaneous
Sources. Proceedings of the Royal Society of London, Ser. A,
254; 1-23.
5. Trent, Donald S., and J. R. Welty, Numerical Thermal Plume Model
for Vertical Outfalls in Shallow Water, Environmental Protection
Technology Series, EPA-R2-73-162, March 1973.
6. Hirst, E. A.,- Analysis of Round Turbulent, Buoyant Jets Dis-
charged to Flowing Stratified Ambients. Oak Ridge, Oak Ridge
National Laboratory, Dept. No. ORNL-4685, 36 p., 1971.
7. Jirka, G., and D. R. F. Harleman, The -Mechanics of Submerged
Multiport Diffusers for Buoyant Discharges in Shallow Water,
MIT Ralph M Parsons Laboratory for Water Resources and Hydro-
dynamics, Report No. 169, March 1973.
8. Argue, J., The Mixing Characteristics of Submerged Multiple
Port Diffusers for Heated Effluents in Open Channel Flow,
University of Iowa, Masters Thesis, May 1973.
9. Larsen, J. and G. E. Hecker, Design of Submerged Diffusers and
Jet Interactions, ASCE National Water Resources Engineering
Meeting, Jan. 24-28, 1972, Atlanta, GA, Reprint No. 1614.
10. Koh, R. C., N. H. Brooks, E. H. Wolanski, and E. J. List, Basin
Model Studies of Diffusers, SCE Report No. 4, W. M. Keck Hy-
draulics Laboratory, CIT, May 1973.
11. Liseth, P., Mixing of Merging Buoyant Jets from a Manifold in
Stagnant Receiving Water of Uniform Density, Hydraulic Engineer-
ing Laboratory Report HEL 23-1, University of California, Berke-
ley, November, 1970,
12. Iwasa, Yoshiaki, and Mashio Yatsuzuka, Spread of Heated Waters
from Multiport Diffuser, Proceedings of the U. S. - Japan Joint
Seminar on Engineering and Environmental Aspects of Waste Heat
Disposal, Paper #9, Tokyo, Japan, April 15, 1974.
185
-------
13. Koh, R. C. and L. N, Fan, Mathematical Models for the Pre-
diction of Temperature Distributions Resulting from the Dis-
charge of Heated Water in Large Bodies of Water, E.P.A. Water
Pollution Control Research Series Report No. 1613DW)1)/70,
October 1970.
14. Kannberg, L. D. and L. R. Davis, Experimental Investigation
of Deep Submerged Multiple Buoyant Jets into Stagnant and Co-
moving Ambients, Paper presented at Thermal Pollution Analysis
Conference held at VPL $ SU, Blacksburg, Virginia, May 1974.
15. Shirazi, M. A., and L. R. Davis, Workbook of Thermal Plume
Prediction Vol. I: Submerged Discharge, E.P.A. Environmental
Protection Technology Series, EPA-R2-72-005a, August, 1972.
16. Benedict, R. P., Engineering Analysis of Experimental Data,
Transaction of ASME, Journal of Engineering for Power, January
1969, pp. 21-30.
17. Davis, L. R. , Analysis of Multiple Cell Mechanical Draft Cool-
ing Towers, E.P.A. Natl. Envir. Res. Cntr., Corvallis, OR
Ecological Res. Series, EPA-660/3-75-039, June 1975.
18. Hirst, E. A. Analysis of Buoyant Jets Within the Zone of
Flow Establishment, Oak Ridge National Lab., Report No. ORNL-
TM-3470, August 1971.
19. McQuivey, R. S., T. N. Keefer, and M. A. Shirazi, Basic Data
Report on the Turbulent Spread of Heat and Matter, U. S. De-
partment of the Interior, Geological Survey, and the U. S.
Environmental Protection Agency, Open-file Report, Fort Collins
Colo., August 1971.
20. Shirazi, M. A., R. S. McQuivey, and T. N. Keefer, Heated Water
Jet in Coflowing Turbulent Stream, Journal of the Hydraulics
Division, A.S.C.E., Vol. 100, No. HY7, Proc. Paper 10661, July
1974, pp. 919-934.
21. Shirazi, M. A., L. R. Davis, K. V. Byram, An Evaluation of
Ambient Turbulence Effects on a Buoyant Plume Model, Proceed-
ings of the 1973 Summer Computer Simulation Conference, July
17, 18, 19, Montreal, P,Q., Canada.
22. Taylor, Sir Geoffrey, Dynamics of a Mass of Hot Gas Rising in
Air, U. S. Atomic Energy Commission, MDDC 919, LADC 276, 1945.
23. Fox, D. G., Forced Plume in a Stratified Fluid, J. Geophys.
Res., Vol. 75, No, 33, pp. 6818-35, 1970.
24. Abraham, G., Horizontal Jets in Stagnant Fluid of Other Den-
sity, J. Hyd. Div., A.S.C.E,, Vol. 91, No. HY4, 1969, pp. 139-
153.
186
-------
25. Fan, L-N., Turbulent Buoyant Jets into Stratified or Flowing
Ambient Fluids, Keck Lab of Hyd, and Water Resources, California
Inst. of Tech.., Rept No, KH-R^-15, June 1967.
26. Flatten, J. L., and J. F. Keffer, Entrainment in Deflected
Axisymetric Jets at Various Angles to the Free Stream, Univ.
of Toronto, Mech. Engr. Dept., UTME-TP-6808, 1968.
27. Hoult, D. P., J. A. Fay, and L. J. Forney, A Theory of Plume
Rise Compared with Field Observations, J. Air Pollut. Cntrl.
Asso., Vol. 19, No. 9, pp. 585-90, 1969.
28. Hoult, D. P., and J. C. Weil, Turbulent Plume in a Laminar
Cross Flow, MIT Fluid Mechanics Lab., Pub. No. 70-8, 1970.
29. Abramovich, G. N., The Theory of Turbulent Jets, Translation by
Scripta Technica, M.I.T. Press, 1963.
30. Cederwall, K., Jet Diffusion: Review of Model Testing and
Comparison with Theory, Hyd. Div., Chalmers Inst. of Tech.
Goteborg, Sweden, Feb. 1967.
31. Gordier, R. L., Studies on Fluid Jets Discharging Normally
into Moving Fluids, St. Anthony Falls Hyd. Lab., Tech. Rpt.
28, Ser. B., Univ. of Minn., 1959.
32. Rouse, Hunter, Fluid Mechanics for Hydraulic Engineers, Dover,
1961; McGraw Hill, 1938; Copyright-United Engineering Trustees,
Inc.
33. Brown, C. E., Aerodynamics of Wake Vortices, AIAA Journal,
Vol. II, No. 4, pp. 531-536, Apr. 1973.
34. Anwar, H. 0., Behavior of Buoyant Jet in Calm Fluid, ASCE
J. Hydraulics Div., Vol. 95, No. HY4, pp. 1289-1303, 1969.
35. Schlichting, H., Boundary-Layer Theory, translated by J. Kestin,
McGraw-Hill Co., 6th Edition, 1968.
36. Chasse, J. P., and L. Winiarski, Laboratory Experiments of
Submerged Discharges with Current, Environ. Prot. Agency,
Pacific NW. Environ. Research Lab., Working Paper #12, June 1974
37. Fan, L-N, and N. H. Brooks, Numerical Solution of Turbulent
Buoyant Jet Problems, W. M. Keck Laboratory, Calif. Inst. of
Tech., Report No. KH-R-18, January 1969.
38. Forstall, W., and A. H. Shapiro, The Turbulent Mixing of Co-
axial Gas Jets, J. Appl. Mech,, Vol. 17, pp. 399-408, 1950.
39. Bird, R. B., W. E. Stewart, and E. N. Lightfoot, Transport
Phenomena, John Wiley & Sons, Inc., New York, 1960.
187
-------
40. Stalzenbach, K. D., Harleman, D. R. F., An Analytical and
Experimental Investigation of Surface Discharge of Heated
Water, EPA Water Pollution Control Series 16130 DJV 02/71,
February 1971.
188
-------
SECTION VIII
APPENDIX A
Appendix A contains a complete list of the normalized experimental
V
data. Data is listed successively according to L/D, 9, F, R, and •=-
with the discharge velocity and temperature and ambient temperature
also given. In this listing
L/D = port spacing, L, in port diameters, D,
9 = vertical angle of discharge from the horizontal
downstream direction in degrees
V
= Discharge densimetric Froude
/ p - p Number,
__..„
R = Discharge to ambient velocity ratio,
X/D = Distance downstream, X, in port diameters, D,
-v^0- = c " a = Dimensionless normalized thermal
m° Io " la concentration,
w
=r = Vertical jet width, W, in port diameters, D,
Y
— = Vertical distance, Y, to jet thermal centerline
in port diameters, D,
V = port discharge velocity in cm/sec.,
T = port discharge temperature in degrees Celsius,
and
T = ambient water temperature in degrees Celsius.
189
-------
L/D
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
e F
0 11.15
11.89
11.50
11.8
11.0
0 11.09
0 10.70
0 10.68
0 10.84
o 10.30
0 11,66
0 11.32
0 11.38
0 10.83
APPENDIX A.
R I/D
0
0
0
0
0
0.093
0.107
0.093
0.102
0.100
0.102
0.098
0.088
0.254
10
20
30
40
50
20
30
40
50
60
80
100
140
20
TABULATED DATA
AWATo W/D T/D V0
0.41
0.18
0.12
0.086
0.045
0.188
0.195
0.197
0.203
0.134
0.123
0.130
0.112
0.114
0.102
0.082
0.086
0.072
0.057
0.065
0.066
0.046
0.040
0.043
0.053
0.055
0.033
0.041
0.032
0.040
0.037
0.039
0.042
0.045
0.191
0.199
0.196
0.199
6.7
8.3
12.
15.
18.
4.5
5.0
4.9
7.0
6.3
7.1
8.3
9.1
8.1
9.0
10.2
7.5
8.6
11.0
12.8
13-1
12.7
13.5
12.8
13.7
13.8
12.3
12.1
9.8
12.1
23.3
12.3
13.5
13.8
5.3
5.7
5.2
5.7
i.O* 25.00
6.4* 26.50
20.0* 26.4
38.0* 26.7
65.0* 26.4
26.50
25.44
26.5
26.2
23.18
26.38
26.80
27.50
25.77
TO
45.05
45.12
46.06
46.16
48.42
44.31
44.04
45.73
44.78
41.67
41.90
43.78
44.67
44.42
*a
20.00
20.40
21.01
21.08
20.84
5.96
5.75
5.4
5.64
5.23
5.10
4.97
4.76
7.88
*As determined from dye studies.
190
-------
APPENDIX A (continued). TABULATED DATA
L/D
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
9 F R X/D ATc/A^o
0 11.26 0.251 30 0.128
0.131
0.121
0 10.96 0.248 40 0.125
0.106
0.117
0 10.81 0.244 50 0.083
0.072
0.069
0 10.77 0.255 60 0.063
0.054
0.054
0 10.62 0.257 80 0.044
0.043
0.049
0 11.07 0.251 100 0.040
0.041
0.043
0 10.59 0.459 20 0.148
0.154
0.124
0.147
0.143
0.121
0 11.07 0.501 30 0.091
0.093
0.119
0.101
0.090
0 10.82 0.486 40 0.091
0.079
0.077
0.066
0.064
0.067
0 10.38 0.511 50 0.065
0.064
0.072
0.059
0.059
0.062
W/tf T;
5.9
4.8
5.0
5.8
6,2
5-7
8.9
6.2
6.3
9.7
5-9
7.3
7.3
8.3
7.4
10.9
7.4
12.9
4.7
4.5
3.8
5.4
4.4
4.2
5.4
5.0
5.4
5.2
5.8
7.0
7.4
5.5
4.5
7.7
6.0
8.5
8.6
8.3
8.0
6.4
10.2
to v0 T0 ?a
25.7 42.65 7»79
25.3 43.16 8.03
25.79 44.55 8.02
25.58 44.39 8.13
25.16 44.32 8.24
25.71 43.47 8.30
26.82 47.50 9.57
25.44 43.29 9.49
25.5 44.27 9.3
24.44 44.27 9.36
191
-------
APPENDIX A (continued). TABULATED DATA
L/D
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
e
0
0
0
0
0
0
0
0
0
0
0
F
11.03
10.57
11.25
10.25
10.59
10.67
10.49
10.14
10.70
29.0
29.0
28.8
29.8
29.6
29.2
30.6
30.7
25.73
R
0.485
0.486
0.500
0.962
0.962
0.944
0.946
0.946
0.955
0
0
0
0
0
0
0
0
0.057
I/D ATo//^o
60
80
100
20
40
60
80
100
140
10
20
30
40
50
60
72
80
20
0.040
0.049
0.043
0.043
0.051
0.042
0.040
0.042
0.046
0.048
0.042
0.019
0.018
0.017
0.057
0.030
0.018
0.014
0.013
0.011
0*420
0.196
0.160
0.088
0.076
0.071
0.069
**
0.177
0.158
0.166
0.178
0.163
V/D
9.4
9.3
7.7
10.5
8.3
9.2
6.5
8.1
9.0
9.6
10.1
12.3
12.1
11.2
4.0
5.4
7.5
8.9
10.5
12.9
4.0
10.
13.
16.
19.5
23
26.4
33.
7.3
5.4
6.3
7.3
6.5
T/D V0
25.59
25.13
25.69
25.73
25.72
25.70
25.68
25.66
25.66
0.0* 49.4
1.5* 49.4
4.0* 49.4
9.3* **9.fc
15.3* 49.6
19.5* 49.5
43. * 49.5
** 49.5
48.72
TO
43.60
44.58
42.87
47.35
45.80
45.43
46.11
48.00
45.53
36.00
36.60
37.75
37.00
37.30
37.20
36.13
36.09
34.19
T*
9.11
8.99
8.84
13.34
13.24
13.19
13.04
12.96
12.94
21.11
21.28
21.41
21.34
21.51
21.74
21.74
21.75
11.52
192
-------
L/D
e
APPENDIX A (continued). TABULATED DATA
F R 2/D
10.
10.
10.
10.
10.
10.
10.
10.
0 30.36 0.050 20 0.219
0.210
0.240
0.217
0.228
0.203
0.212
0 29.29 0.045 30 0.150
0.128
0.132
0 29.20 0.054 30 0.145
0.135
0.141
0.139
0.130
0.134
0 30.01 0.050 40 0.108
0.122
0.112
0.118
0.116
0 30.50 0.054 40 0.112
0.113
0.115
0.118
0 28.93 0.050 50 0.089
0.080
0.093
0.081
0.085
0 30.29 0.053 50 0,086
0.087
0.074
0.092
0.079
0.079
0 29.66 0.050 60 0.060
0.057
0.057
0.059
0.057
5.6
7.2
6.0
6.0
6.2
5.6
7.5
8.0
8.9
9.9
7.7
7.7
8.1
8.4
7.3
7.0
10.2
10.6
10.8
10.3
10.7
10.5
11.1
11.0
9.2
12.0
11.7
12.5
12.2
12.3
12.3
11.0
11.1
13.4
11.2
12.3
11.4
14.2
12.6
11.1
11.5
50.48 32.10 9.85
50.33 33.15 11.73
48.66 32.23 10.06
49.64 31.80 12.09
49.87 31.67 10.30
50.31 33.16 11.90
49.55 31.74 10.20
49.37 32.04 12.25
193
-------
APPENDIX A (continued). TABULATED DATA
L/D
10.
10.
10.
10.
10.
10.
10.
10.
e F R i/v ATc/AT0
0 31.30 0.049 60 0.068
0.074
0.077
0.068
0.072
0 31.82 0.050 70 0.051
0.052
0.051
0.054
0.059
0 30.81 0.054 80 0.068
0.061
0.065
0.059
0.070
0 29.89 0.047 80 0.066
0.066
0.057
0.058
0.069
0 29.79 0.045 100 0.048
0.048
0.053
0.051
0.057
0 29.96 0.050 140 0.046
0.050
0.054
0.047
0.048
0 31.10 0.095 20 0.209
0.213
0.207
0 31.59 0.104 20 0.200
0.230
0.228
0.193
0.230
0.221
0.228
w/n iy
13.2
14.9
14.7
12.2
15.2
11.4
16.4
15.9
13.9
16.1
18.7
13.6
12.1
14.9
13.2
13.4
15.8
14.9
15.3
14.1
15.8
18.0
17.3
15.6
14.9
24.4
23.5
19.1
25.3
18.9
5.2
6.6
5.1
7.2
5.4
5.8
6.1
5.3
7.8
4.5
Yip m
O O **
50.10 31.20 10.43
52.06 31.67 12.59
48.47 31.39 14.35
49.61 32.28 10.57
49.70 32.45 10.66
49.70 32.34 10*82
52.53 31.87 10.53
49.39 32.13 14.40
194
-------
APPENDIX A (continued). TABULATED DATA
L/D
10.
10.
10.
10.
10.
10.
10.
10.
10.
OF R X/D ATC/ATO
0 31.04 0.096 30 0.221
0.206
0 31.53 0.107 30 0.152
0.128
0.141
0.141
0.154 '
0 29.45 0.105 40 0.114
0.101
0.105
0.082
0.093
0 30.94 0.104 40 0.112
0.108
0.110
0.110
0.118
0 31.15 0.111 50 0.066
0.059
0.071
0.073
0.062
0 30.96 0.110 50 0.089
0.103
0.090
0.094
0 30.17 0.119 60 0.051
0.052
0.054
0.060
0 31.73 0.095 60 0.085
0.088
0.088
0.082
0.080
0 31.60 0.094 80 0.046
0.053
0.057
W/D
5.8
5.9
7.3
5.8
6.4
5.7
6.9
9.2
6.9
8.3
6.9
7.9
9.2
6.9
8.9
9.3
7.7
8.7
11.3
7.6
8.9
8.1
7.5
9.1
7.5
8.0
10.5
12.6
12.4
9.1
10.4
12.8
12.8
13.3
11.9
11.7
11.5
14.3
Y/D V0 T0 Ta
49.11 30.98 10.30
47.94 31.44 14.43
47.77 32.50 15.03
49.78 32.80 14.27
48.80 31.53 14.70
50.92 32.60 13.91
47.60 32.35 14.80
50.92 32.60 13.91
50.96 32.17 14.62
195
-------
APPENDIX A (continued). TABULATED DATA
L/D
10.
10.
10.
10.
10.
10.
10.
10.
10.
e F R X/D ATO/ATO
0 30.32 0.100 80 0.055
0.054
0.062
0.058
0.054
0 31.8? 0.095 100 0.051
0.044
0.046
0.051
0 29.37 0.113 100 0.059
0.050
0.052
0.048
0 31.87 0.100 140 0.046
0.057
0.038
0 28.92 0.104 140 0.043
0.043
0.036
0.041
0.042
0 30.67 0.250 20 0.14?
0.160
0.195
0.200
0 30.58 0.254 20 0.172
0.198
0.203
0.174
0.201
0.190
0 20.6? 0.24? 30 0.120
0.128
0 32.21 0.243 30 0.146
0.130
0.144
W/D I
15.9
15.0
12.5
13.9
11.5
17.1
15.7
15.8
14.8
10.5
12.8
14,8
14.3
20.1
15.8
18.7
24.5
22.9
16.4
17.5
14.8
5.2
4.7
5.0
5.4
4.0
4.8
4. ft
4.6
5.5
5.6
7.8
8.6
6.6
6.3
6.3
/D V0 T0 Ta
48.73 32.59 13.79
50.30 31.46 14.32
48.15 33.05 13.62
49.60 32.25 14.23
48.15 33.45 13-55
50.42 32.15 10.72
48.58 33.00 15.41
50.50 32.34 11.02
50.75 32.75 15.33
196
-------
APPENDIX A (continued). TABULATED DATA
L/n
10.
10.
10.
10.
10.
10.
10.
10.
e F R I/D ATC/ATO
0 30.64 0.249 40 0.093
0.083
0.078
0.102
0 31.26 0.247 40 0.063
0.074
0.067
0.067
0.082
0.078
0 31.85 0.239 50 0.066
0.061
0.054
0.056
0 33.03 0.239 50 0.053
0.068
0.061
0.063
0 31.82 0.241 60 0.048
0.042
0.043
0 31.83 0.254 60 0.055
0.060
0.052
0.056
0.056
0.049
0 31.36 0.239 80 0.050
0.052
0.047
0 29.98 0.260 80 0.046
0.046
0.045
0.042
0.044
0.041
W/D Y,
9.4
9.5
8.6
8.6
7.4
7.1
5.9
6.9
6.6
6.7
10.7
7.7
7.5
9.5
8.5
7.5
8.3
7.9
8.7
9.7
7.6
7.7
7.9
9.3
7.7
8.3
9.0
8.2
10.8
10.4
10.7
10.7
9.2
12.9
11.0
8.6
f* V0 T0 Ta
52.08 32.30 11.20
49.60 33.02 15.52
52.60 31.97 11.79
51.98 32.83 15.57
53.20 31.52 11.50
48.84 33.26 15.70
52.60 32.13 11.92
48.21 33.^2 15.64
10. 0 30.25 0.248 100 0.045 13.5 50.09 31.81 12,05
0.033 12.0
197
-------
APPENDIX A (continued). TABULATED DATA
L/D
10.
10.
10.
10.
10..
10.
10.
10.
10.
10.
e F R X/D ATC/AT0
0 32.44 0.243 100 0.046
0.042
0.041
0.040
0.039
0 28195 0.262 140 0.040
0.025
0.036
0 31.09 0.261 140 0.027
0.026
0.022
0.022
0 34.2 0.450 20 0.131
0.145
0.140
0 30.89 0.494 20 0.127
0.135
0.121
0.131
0 28.64 0.518 30 0.083
0.088
0.076
0 29.65 0.506 30 0.097
0.088
0.099
0.074
0.075
0 28.32 0.510 40 0.092
0.091
0.097
0 30.66 0.495 40 0.074
0.086
0.065
0.081
0 28.72 0.517 50 0.058
0.046
0.067
0.046
198
W/D I
12.0
11.0
10.9
10.0
10.8
14.0
15.4
11.4
10.4
12.0
9.6
12.1
5.6
4.1
5.5
3.7
6.1
6.0
4.9
7.2
7.4
5.0
4.4
6.7
6.2
6.8
3.3
6.0
5.6
7.8
5.9
6.1
5.1
6.0
8.3
9.6
8.5
6.7
•/D V0 T0 Ta
51.52 33.16 15.77
47.53 31.63 12.17
49.01 33.02 15.88
55.55 31.63 12.97
51.05 33.95 15.21
48.58 32.80 12.90
49.58 34.25 15.16
48.97 33.18 12.75
50.34 33.65 14.93
48.58 32.70 12.82
-------
APPENDIX A (continued). TABULATED DATA
L/D
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
e F
0 28.68
0 29.32
o 29.05
0 30.99
0 26.13
0 29.53
0 28.78
o 30.65
0 29.66
0 54.4
56.7
54.4
51.*
56.4
54.9
54.9
53.8
R
0.520
0.500
0.509
0.470
0.512
0.506
0.511
0.499
0.516
0
X/D
50
60
60
80
80
100
100
140
140
10
20
30
40
60
80
100
140
ATC/AT0
0.053
0.048
0.059
0.062
0.064
0.048
0.064
0.055
0.046
0.051
0.049
0.048
0.046
0.038
0.048
0.036
0.036
0.032
0.048
0.033
0.029
0.034
0.028
0.029
0.029
0.033
0.024
0.033
0.018
0.021
0.017
0.497
0.277
0.181
0.121
0.090
0.063
0.048
0.042
W/D Y/D
6.4
7.3
6.7
4.9
6.6
6.7
8.8
8.5
7.8
9.5
5.6
6.5
8.9
8.3
9.4
9.2
9.1
11.2
11.9
10.9
10.0
8o5
12.9
10.2
9.1
12.6
13-5
7.9
13.7
10.3
11.8
4.0* 0.0*
8.0 0.8*
11. 1.2
17. 2.8
21. 7.3
28. 12.0
38. 18.7
** 44.
vo
48.09
48.97
49.13
51.86
44.21
49.00
49.37
49.60
48.66
76.4
73.6
76.3
75.03
70.96
70.2
70.3
74.6
To
34.28
32.28
34.44
32.36
34.41
32.09
34.77
31.37
33.50
29.26
29.16
30.53
32.51
30.10
30.16
30.28
30.29
Ta
15.04
12.67
14.87
12.75
14.77
12.64
14.69
12.54
14.64
18.03
17.97
17.72
19.58
19.75
19.BO
20.05
17.69
199
-------
APPENDIX A (continued). TABULATED DATA
L/D
10.
10.
10.
10.
10.
10.
10.
10.
6 F R X/D ATC/AT0
0 54.87 0.051 20 0.196
0.210
0.210
0.224
0.217
0 58.05 0.052 20 0.242
0.236
0.231
0.230
0.221
0.224
0 53.06 0.052 30 O.U56
0.160
0.154
0.151
0.155
0 53.90 0.055 20 0.192
0.237
0.247
0.251
0 53.41 0.052 30 0.153
0.132
0.160
0.158
0.153
0 49.75 0.055 50 0.091
0.098
0.097
0.092
0.091
0 52.16 0.055 30 0.145
0.148
0.148
0.150
0.149
0 52.68 0.0512 40 0.113
0.103
0.110
0.107
0.115
W/D I
7.7
6.5
6.3
7.5
7.7
6.5
5.9
7.1
6.0
6.6
5.5
8.1
7.2
8.3
8.7
8.9
6.5
6.5
7.0
6.5
7.2
5.7
5.9
8.7
8.6
11.0
11.4
12.5
9.1
14.2
8.3
9.0
6.7
7.4
7.8
8.4
11.0
10.4
10.2
10.5
/D V0 T0 Ta
70.48 26.25 13.07
76.06 28.10 14.50
70.05 26.85 13.17
74.27 26.52 10.04
74.32 26.69 9.81
69.15 26.65 9.80
68.97 28.35 14.62
74.05 26.85 9.6l
200
-------
APPENDIX A (continued). TABULATED DATA
L/D
10.
10.
10.
10.
10.
10.
10.
10.
10.
6 F R X/D ATC/AT0
0 52.59 0.050 40 0.104
0.112
0.108
0.113
0.104
0 55.29 0.054 50 0.098
0.094
0.098
0.098
0 54.00 0.051 50 0.086
0.091
0.087
0.091
0 54*03 0.052 60 0.070
0.075
0.089
0.081
0.075
0 53.67 0.048 60 0.076
0.080
0.071
0.079
0.066
0 53.^ 0.052 80 0.060
0.065
0.057
0.064
0 52.94 0.055 80 0.58
0.056
0.048
0.057
0.050
o 53.76 0.052 100 0.053
0.061
0.065
0.066
0.056
o 52.38 0.^)50 100 0.054
0.051
0.047
0.050
0.048
W/D Y
9.6
10.9
8.9
12.7
8.6
14.0
11.7
11.8
10.4
11.3
11*2
12.4
12.8
13.4
12.0
11.6
11.6
12.2
14.1
11.8
13.0
13.5
12.6
18.6
17.2
18.9
18.8
15.2
18.2
15.1
16.9
15.9
13.1
23.1
19.8
18.3
20.1
12.7
18.7
19.9
19.2
17.9
/D V0 T0 Ta
71.07 28.87 14.74
73.14 26.95 13.29
71.96 28.57 14.73
75.42 26.64 9.46
72.39 28.86 14.80
75.76 26.94 9.25
70.23 28.54 14.86
73.96 26.20 9.10
70.88 29.00 14.94
201
-------
APPENDIX A (continued). TABULATED DATA
L/D
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
0 F R I/D ATo/ATo
0 54.39 0.056 140 0.055
0.047
0.050
0.038
0.050
0 56.80 0.049 140 0.042
0.040
0.039
0.038
0 54.21 0.101 20 0.186
0.185
0.216
0 52.54 0.104 20 0.262
0.253
0.251
0 54.28 0.105 30 0.148
0,154
0.154
0 52.63 0.104 30 0.152
0.167
0.154
0.146
0.155
0 55.04 0.100 40 0.113
0.091
0.107
0.100
0 52.68 0.110 40 0.121
0.124
0.127
0 54.14 0.110 50 0.086
0.083
0.081
0 54.16 0.108 50 0.090
0.092
0.098
0.094
W/D I
18.3
24.6
24.0
23.8
20.9
24.6
23.6
24.0
20.7
4.6
5.4
6.7
5.2
6.7
6.6
8.6
7.1
7.3
6.9
7.8
8.0
8.9
5.3
8.6
8.5
9.1
7.1
6.7
8.7
8.9
10.2
12.5
10.5
11.1
9.1
11.2
11.4
/D V T
' t> AO
75.05 26.20
75.14 28.40
74.53 26.53
71.96 27.05
74.29 26.46
70.31 26.43
75.97 26.73
69.52 28.65
73.43 26.29
71.10 28.57
Ta
8.85
14.70
10.22
9.92
10.31
9*80
10.53
15.27
10.40
15.32
202
-------
APPENDIX A (continued). TABULATED DATA
L/D 6 P R 2/D AVAT0 W/D Y/D Vo To
10. 0 54.16 0.099 60 0.081 8.9 77.42 26.71 10.61
0.084 9.9
0.080 9.5
10. 0 54.06 0.109 60 0.088 8.4 70.07 28.55 16.15
0.088 11.0
0.087 8.7
0.078 11.2
10. 0 54.37 0.102 80 0.062 13.8 74..53 26.64 10.73
0.052 10.9
0.057 13-6
10. 0 54.52 0.105 80 0.060 15.4 70.9 28.27 15.12
0.055 11.5
0.062 13.5
0.058 13.3
0.060 15.5
10. 0 53.31 0.107 100 0.047 15-8 73.46 26.79 10.82
0.013 15.5
0.057 14.4
10. 0 58.50 0.098 100 0.0^1 17.2 76.4 28.31 15.03
0.049 12.7
0.050 14.1
0.053 15.6
10. 0 52.68 0.105 140 0.047 19.6 72.42 26.76 10.90
0.046 19.5
10. 0 55.72 0.100 140 0.039 13.0 76.90 26.82 10.79
0.044 15.1
0.043 18.2
10. 0 58*51 0.098 140 0.046 18.5 77.16 28.47 14.96
0.045 19.1
0.039 19.2
0.039 21.0
10. 0 56.83 0.254 20 0.200 5.2 73.96 26.39 11.17
0.207 5.3
0.194 5.3
10. 0 51.38 0.264 20 0.204 5*9 71.18 27.37 10.06
0.200 4.3
0.190 6.5
0.193 5.4
203
-------
APPENDIX A (continued). TABULATED DATA
L/D
10.
10.
10.
10.
10.
10..
10.
10.
10.
10.
OP R I/D ATc/A*o
0 55.75 0.260 30 0.119
0.129
0.120
0.119
0 54.40 0.250 30 0.140
0.140
0.1*5
0.142
0 55.03 0.260 40 0.08?
0.098
0.099
0.088
0 51.55 0.257 40 0.103
0.113
0.112
0.102
0 56.77 0.228 50 0.065
0.073
0.079
0.081
0 51.12 0.260 50 0.072
0.086
0.072
0 57.96 0.240 60 0.076
0.071
0.063
0.070
0 50.17 0.266 60 0.060
0.071
0.066
0.071
0 58.74 0.241 80 0.058
0.051
0.054
0 52.43 0.261 80 0.049
0.057
0.047
0.047
W/D T,
6.1
5.6
6.0
6.6
6.9:
6.1
6.9
7.0
7.8
7.6
7.3
6.1
7.1
7.7
9.2
8.2
9.4
8.5
7.8
6.4
9.5
7.4
8.7
8.3
6.8
9.4
6.2
9.4
7.2
7.1
7.5
12.4
9.7
10.8
10.1
10.2
11.4
8.0
/D V0 T0 Ta
72.1 26.22 1U10
72.07 26.43 10.22
71.74 26.38 11.05
71.85 27.65 10.42
71.9 25.76 11.05
70.72 27.43 10.31
76.01 26.48 10.98
69.52 27.53 10.53
76.26 26.22 10.89
70.31 26.85 10.67
204
-------
APPENDIX A (continued). TABULATED DATA
L/D 0 F R X/D ATc/ATo W/D Y/D V<, To
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
0 59.83 0.242 100 0.045
0.055
0.043
0.043
0 57.92 0.250 100 0.043
0.045
0.040
0.043
0 56.58 0.0250 140 0.04?
0.037
0.040
0 56.56 0.249 140 0.035
0.028
0.033
0.032
0 55.07 0.451 20 0.133
0.130
o 54.96 0.525 20 0.105
0.115
0.093
0 57.39 0.499 30 0.091
0.083
0.092
0.082
0 57.62 0.496 30 0.091
0.087
0.091
0 55.84 0.520 40 0.058
0.056
0.053
0.070
0.057
0 56.59 0.517 40 0.082
0.071
0.066
0 52.37 0.545 50 0.043
0.045
0.040
9.2
11.1
12.7
12.5
10.8
10.7
10.2
10.1
12.9
12.4
16.5
18.7
14.6
17.1
17.4
4.6
3.7
5.1
4.9
5.2
6.5
6.0
5.5
5.7
5.9
5.3
5.8
7.2
6.1
6.4
7.2
6.8
5.1
5.3
5.8
7.8
6.3
6.6
76.26 25.78
75.57 27.34
74.70 25.91
76.1 28.03
69.50 25.71
71.88 26.04
75.43 26.69
75.58 27.71
72.76 26.58
75.56 27.22
69.00 26.78
10.78
13.20
10.96
13.24
11.22
10,02
11.37
13.74
11.55
13.64
11.45
205
-------
APPENDIX A(continued). TABULATED DATA
L/D
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
0 F R 1/D AT0/AT0 W/D
0 55.9* 0.516 50 0.0*1
0.052
0.050
0 50.90 0.566 60 0.0*7
0.037
0.0*0
0 56.29 0.510 60 0.0*6
0.0*2
0.0**
0 53.59 0.553 80 0.035
0.030
0 55.63 0.516 80 0.035
0.028
0.023
0 55.58 0.536 100 0.019
0.025
0.023
0.027
0 56.15 0.516 100 0.031
0.029
0.023
15 10.50 0.110 10 0.36*
0.372
0.393
O.*17
15 10.55 0.106 20 0.193
0.187
0.185
0.185
0.186
0.188
15 9.77 0.110 20 0.173
0.187
0.192
0.172
0.20*
0.187
0.190
9.3
7.5
7.7
10.9
5.5
8.6
8.2
7.7
9.*
12.7
9.1
9.3
8.5
13.6
13.2
6.8
8.0
6.1
10.3
11.8
11.8
3.*
3.9
3.8
3.7
6.5
*.9
5.8
6.6
5.5
6.9
*.9
6.7
7.1
5.8
7.7
6.1
*.9
Y/D
3.2
3.3
3.2
2.8
6.9
6.*
*.9
6.1
5.9
6.2
5.7
6.9
6.3
5.8
6.3
5.5
6.2
V0 T0 TI
73.08 27.03 13*69
66.73 26.75 11.6*
73.70 27.60 13.5*
68.52 26.23 11.67
72.73 27.5* 13.*8
70.27 26.05 11.79
73.17 27.** 13.*1
2*.08 *5.26 16.60
22.60 *2.60 16.35
23.73 *8.59 19.27
206
-------
APPENDIX A (continued). TABULATED DATA
L/D e F R X/D ATO/ATO M/D I/D v
10. 15 9.50 0.106
10.. 15 10.63 0.099
10. 15 10.30 0.105
10. 15 10.56 0.102
10. 15 9.32 0.108
10. 15 10.05 0.105
10. 15 10.10 0.106
10. 15 11.07 0.109
10. 15 12.20 0.108
30 0.109
0.116
0.127
0.115
0.125
0.128
30 0.122
0.112
0.114
0.120
0.120
0.121
40 0.080
0.079
0.080
0.079
40 0.078
0.087
0.084
0.089
0.080
50 0.056
0.054
0.055
0.054
50 0.064
0.059
0.056
0.058
60 0.041
0.041
0.041
0.040
60 0.040
0.043
0.036
0.043
60 0.051
0.049
0.048
8.2
6.3
7.1
6.1
9.6
10.6
8.0
8.2
8.3
6.5
8.7
6.7
10.5
8.8
11.5
13.1
9.6
7.4
10.5
7.4
8.2
13.7
9.9
8.7
11.4
12.4
13.5
15.4
11.5
15.0
16.5
16.7
13.0
10.6
9.4
12.5
13.1
10.4
8.8
10.8
9.2 23,28 48,90
9.5
8.7
8.7
9.4
9.3
8.9 25.49 47.97
8.8
9.4
9.0
9.5
10.2
10.6 24.77 48.08
10.6
10.1
10.1
10.3 25.11 47.54
10.7
10.3
11.5
12.2
11.8 23.40 49.79
13.1
14.2
13.9
15.3 23.96 47.60
12.3
15.4
12.4
14.9 23.67 46.92
12.9
13.5
17.4
9.0 23.48 43.14
11.5
11.5
12.2
14.6 23.66 40.13
11.9
12.5
19.19
19.09
18.99
18.84
18.80
18.68
18.69
18.56
18.52
207
-------
APPENDIX A (continued). TABULATED DATA
L/D 8 F R I/D AVAT0 */D Y/D vo *o Ta
10.
10.
10.
10.
10.
10.
10.
10.
10*
10.
10.
10.
15
15
15
15
15
15
15
15
15
15
15
15
10.56
10.3*
10.72
10.80
11.80
ll.*7
11.85
11.40
10.9*
10.99
11.07
11.31
0.106
0.260
0.251
0.2*7
0.237
0.240
0.248
0.251
0.259
0.252
0.25*
0.2*8
100
10
10
20
20
20
30
30
*0
40
50
50
0.041
0.03*
0.3*6
0.35*
0.3*2
0.360
0.359
0.3*3
0.183
0.177
0.179
0.163
0.17*
0.168
0.171
0.18*
0.168
0.188
0.125
0.117
0.112
0.128
0.1*1
0.100
0.110
0.097
0.092
0.08*
0.093
0.065
0.066
0.068
0.071
0.076
0.077
0.087
21.9
24.3
*.3
2.9
3.*
*.o
3.2
3.7
6.3
5.0
*,*
5.0
5.2
5.9
5.1
5.7
5.5
6.1
5.5
6.0
5.6
6.9
6.*
8.7
8.0
9.6
6.9
7.6
7.9
7.*
8.5
7.9
9.0
7.5
8.5
9.8
20.*
20.*
2.3
2.9
2.5
2.7
2.5
2.6
*.l
*.2
*.3
5.1
*.9
5.1
*.3
*.0
*.6
*.7
5.9
5.7
5.6
5.8
5.3
5.*
6.2
6.8
6.2
5.8
6.*
7.6
7.5
7.6
6.9
6.3
6.5
7.6
2*. 23
2*,40
25.35
25.39
26.14
25.85
25.20
25.20
24.40
24.95
2*.91
25.09
*5.29
*5.99
46.13
45.88
*3.55
**.25
*2.*0
*2.99
*3.39
44.10
*3.76
*3.21
16.71
15.*3
15.52
15.53
15.53
15.65
i*.01
14.16
14.26
14.29
14.37
14.41
208
-------
APPENDIX A (continued). TABULATED DATA
L/D 8 F R X/D AT0/AT0 W/D */*> Vo To Ta
10.
10.,
10.
10.
10.
10.
10.
10.
10..
10.
10.
10.
10.
10.,
10.
10.
10.
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
11.48
11.10
10.65
10.60
11.13
10.08
9.99
10.14
10.66
10.92
10.44
10.41
10.59
10.04
9.94
10.03
11.12
0.242
0.258
0.497
0.500
0.475
0.495
0.501
0.495
0.489
0.478
0.500
0.499
0.483
0.505
0.504
0.514
0.464
60
60
10
10
10
20
20
20
30
30
30
40
40
40
50
50
50
0.055
0.058
0.068
0.066
0.062
0.314
0.325
0.329
0.328
0.293
0.294
0.151
0.151
0.183
0.166
0.180
0.174
0.147
0.141
0.114
0.119
0.124
0.119
0.117
0.098
0.096
0.097
0.084
0.080
0.069
0.083
0.079
0.086
9.1
7.6
13.4
9.7
10.9
2.8
2.6
2.8
3.5
2.5
3.2
3.8
4.0
4.8
3.9
4.8
5.0
5.1
6.6
4^.8
5.2
4.6
5.1
4.3
7.2
6.4
6.3
8.1
8.2
7.7
5.8
6.1
6.9
7.3
7.5
7.9
7.2
7.7
1.4
1.4
1.8
1.7
2.0
1.7
2.1
2.5
2.1
2.6
2.9
2.5
3.1
3.4
3.4
2.8
3.3
3.3
4.3
3.6
3.8
3.5
4.1
4.6
3.8
4.2
3.1
4.2
25.53
23.70
25.38
25.26
16.06
25.49
25.20
25.30
25.39
26.13
25.00
25.35
25.97
24.83
24.94
24.84
27.00
43.35
41.82
46.53
46.37
45.81
49.45
49.33
48.86
46.91
47.06
47.07
47.72
48.02
48.33
48.97
48.38
47.51
14.47
14.53
15.80
15.82
15.76
17.1
17.01
16.90
16.88
16.80
16.72
16.67
16.63
16.52
16.54
16.47
16.37
209
-------
APPENDIX A (continued). TABULATED DATA
L/D 6 F R
10. 15 11.44 0.488
10. 15 11.10 0.513
10. 15 11.38 0.496
10. 15 11.12 0.498
10. 15 11.63 0.476
10* 15 10.98 0.501
10.. 15 30.44 0
10. 15 30.42 0
10. 15 30.86 0
VD ATC/ATO
60 0.068
0.066
60 0.079
60 0.075
0.075
60 0.066
0.068
80 0.063
0.060
80 0.059
10 0.382
0.410
0.396
0.402
0.392
0.390
0.384
20 0.174
0.192
0.180
0.183
0.194
0.192
0.172
30 0.140
0.140
0.153
0.143
0.154
0.150
0.143
W/D
8.1
6.2
11.0
7.8
6.7
6.6
7.1
8.4
7.6
5.9
6.6
5.1
5.0
6.1
5.5
4.2
4.9
6.1
7.8
11.8
10.9
10.8
8.4
7.7
13.6
14.1
13.1
18.6
13.2
12.8
12.5
I/D
4.4
3.8
3.4
3.4
4.0
3.9
4.1
4.2
6.1
5.4
4.2
4.1
3.3
4.2
4.7
4.2
4.2
9.6
8.1
9.6
7.8
7.7
7.2
8.1
12.2
12.0
12*3
12.0
9.8
10.5
10.4
V T
»o *•<>
25.62 44.08
24.83 44.00
25.62 44.21
25.42 44.77
26.57 44.75
25.19 44.88
49.54 36.82
50.75 37.30
50.53 36.60
Ta
15.86
15.78
15.67
15.60
15.50
15.44
21.52
21.29
20.88
0.154 13.6 9.6
0.157 12.9 11.2
10. 15 30.48 0 40 0.089 27.9 22.3 50.28 36.63 20.61
0.084 26.2 19.4
0.099 31.9 18.6
0.094 28.7 17.0
0.086 25.4 19.0
0.099 25.8 15.6
0.095 21.3 16.4
210
-------
APPENDIX A (continued). TABULATED DATA
L/D
e
R
ATC/AT0 W/D I/D Vc
10. 15 30.10 o
10.. 15 30.83 0
10. 15 36.94 0.01*8
10. 15 34.12 0.050
10.. 15 32.41 0.057
10. 15 35.46 0.047
10. 15 29.47 0.057
10. 15 36.24 0.053
50 0.089
0.077
0.084
0.077
0.085
0.086
0.090
60 0,07^
0.068
0.066
0.072
0.071
0.072
10 0.406
0.408
0.415
O.M8
20 0.209
0.220
0.201
0.220
0.213
0.208
30 0.141
0.147
0.148
0.142
40 0.098
0.095
0.099
0.107
0.098
0.098
50 0.089
0.084
0.079
0.080
60 0.067
0.071
0.071
0.071
32*1
32.1
24.0
27.5
30,4
22.1
27.0
38.2
23.3
31.9
30.5
35.9
32.1
6.0
4.8
3.9
4.6
6.0
6.3
6.4
7.3
6.4
6.4
9.4
8.7
8.5
9.0
11.9
9.3
10.5
11.3
13.6
10.8
13.2
14.2
15.0
12.4
11.9
15.5
14.4
17.4
21.3 49.83 36.63
22.2
22.9
23.9
21.0
20.5
18.8
30.5 50.45 36.22
32.5
31.7
31.7
30.9
28.4
2.7 56.75 32.83
2.8
2.7
2.9
4.9 53.30 33-18
5.0
4.7
4.8
4.6
5.6
7.3 52.83 34.26
6.9
6.7
7*2
8.7 57.30 34.23
10.1
8.8
9.5
8.9
8.9
10.8 47..30 33.78
10.3
9.4
9.7
9.6 54.44 32.10
10.3
11.0
11.9
20.45
20.23
16.76
16.66
16.66
16.60
16.51
16.41
211
-------
APPENDIX A (continued). TABULATED DATA
L/D 6 F R I/D ATC/AT0 W/D I/D Vo To Ta
10. 15 29.59 0.107
10. 15 32.56 0,096
10. 15 29.6? 0.103
10. 15 29.06 0.106
10. 15 33.32 0.105
10.. 15 32.2 0.104
10. 15 33.81 0.098
10. 15 34.11 0.099
10. 15 32.61 0.104
10.. 15 32.83 0.095
10 0.309
0.333
0.349
0.331
0.346
0.373
20 0.195
0.200
0.186
0.184
20 0.178
0.223
0.184
0.191
0.195
0.203
20 0.202
0.214
0.207
0.202
30 0.125
0.138
0.127
0.130
30 0.122
0.135
0.124
0.136
40 0.099
0.105
0.095
40 0.104
0.104
0.092
0.083
40 0.101
0.096
0.096
50 0.079
0.077
5.2
4.4
5.2
4.6
4.8
4.1
7.7
6.2
7.2
6.9
6.3
6.5
6.8
7.4
5.9
6.7
6.9
5.9
7.8
6.7
9.8
9.7
9.6
9.6
7.8
9.4
8.7
7.3
8.9
9.2
9.5
11.2
10.6
9.1
12.5
11.9
10.0
9.8
10.8
10.7
2.9 45.52 31.25
2.1
2.0
2.7
2.1
2.3
3.9 49.97 30.-39
4.5
3.6
3.3
4.1 47.50 31.56
4.2
3.7
3.9
4.7
3.7
3.7 46.50 31.59
3.6
3.7
4.3
4.8 48.75 29.07
5.4
5.5
5.5
5.2 47.40 29.29
5.9
5.9
5.2
6.0 51.20 29.81
6.3
7.6
7.3 51.35 29.71
6.6
7.3
6.7
7.6 49.20 29.81
6.6
7.3
8.0 50.50 30.04
7.2
13.34
11.34
11.44
11.58
11.06
11.22
10.70
10.85
10.97
10.20
*•
212
-------
APPENDIX A (continued). TABULATED DATA
L/D 6
F
R
¥/D I/D Vc
10o 15 35.45 0.096 50 0.072 10.6 6.0 51.05 28.49 10.57
0.074 11.4 6.7
0.068 13.4 6.1
10. 15 33.64 0.091 60 0.061 11.9 io.l 52.80 30.53 10.04
0.060 12.5 10.5
0.066 13.6 10.6
10. 15 30.07 0.252
10, 15 29.98 0.250
10. 15 30.20 0.251
10. 15 31-13 0.239
10. 15 28.85 0.256
10. 15 31.84 0.250
10. 15 30.10 0.251
10. 15 30.08 0.249
10. 15 31.94 0.231
10. 15 34.21 0.262
10. 15 32.83 0.253
10
10
10
20
30
30
40
40
50
50
60
0.277
0.275
0.284
0.281
0.349
0.331
0.325
0.218
0.206
0.209
0.219
0.132
0.127
0.138
0,136
0.109
0.109
0.106
0.101
0.102
0.103
0.107
0.099
0.097
0.077
0.079
0.071
0.068
0.069
5.0 3.0 42.05 29.36 14.22
4.0 3.1
4.2 2.3 41.60 29.21 14.27
3.7 2*0
3.4 2.5 41.66 29.08 14.27
4.0 2.5
4.0 3.8
5.9 3.1 43.88 29.52 14.20
5.5 3.4
5.5 2.8
6.3 3.4
6.5 4.0 41.12 29.76 14.16
6.0 4.1
7.0 4.0 41.92 27.97 14.05
7.3 3.4
9.0 4.7 41.66 29.08 14.10
7.7 3.4
7.2 5.6
7.9 5.7 42.11 29.33 14.11
8.5 4.3
8.6 3.8
8.7 5.1 43.03 28.48 14.10
8.7 5.0
8.3 4.2
10.4 4.8 48.98 29.71 13.81
8.4 4.2
10.7 3.6 49.20 30.77 13.7*
9.8 5.2
9.4 5.5
213
-------
L/D 6
APPENDIX A (continued). TABULATED DATA
P R X/D ATc/ATo W/B I/D
10.
10.
10.
10.
10..
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
33.05
31.00
31.1*9
31.43
31.20
31.61
31.39
30.81
30.43
30.7*
31.05
31.17
30.07
30.62
31.53
30.91
29.79
0.259
0.25*
0.252
0.252
0.248
0.246
0.242
0.246
0.247
0.244
0.244
0.244
0.259
0.252
0.251
0.248
0.252
60
10
10
20
20
20
30
30
30
40
40
40
50
50
50
30
10
0.065
0.072
0.343
0.340
0.349
0.174
0.180
0.188
0.190
0.157
0.115
0.116
0.126
0.129
0.124
0.124
0.080
0.080
0.101
0.096
0.101
0.077
0.067
0.079
0.075
0.064
0.069
0.117
0.110
0.315
0.304
12.6
10.3
4.1
4.8
3.7
6.2
5.9
5.4
7.0
7.5
8.5
7.2
8.3
7.8
7.8
10.8
7.6
7.4
9.8
8.4
7.1
12.4
10.1
9.1
7.9
9.7
10.7
7.3
8.2
5.4
4.0
5.0
4.0
3.8
3.3
3.4
4.6
4.5
4.0
3.8
4.7
5.2
5.3
4.7
5.4
5.3
5.0
7.2
6.8
5.6
5.9
6.0
7.2
6.2
7.0
6.8
6.8
5.3
5.7
4.2
3.6
3.7
49.10
49.80
49.80
50.72
50.50
51.14
51.23
51.00
50.83
51.06
51.75
51.67
48.64
49.88
50.18
51.23
49.80
30.52
36.61
36.33
36.84
36.96
37.02
36.24
36.64
36.86
36.82
37.07
37.05
36.61
36.82
36.34
37.39
37.67
13.65
21.70
21.85
21.91
22.00
22.10
20.39
20.46
20.53
20.63
20.92
21.13
21.42
21.48
21.62
21.73
21.85
214
-------
L/D
e
APPENDIX A (continued). TABULATED DATA
F R 3/D AT0/AT0 W/D Y/D Vc
10.
10.
10.
10.
10.
10.
10.
10.
10.
10..
10.
10.
10.
10.
10.
10.
10.
10.
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
30.35
30.18
30.89
29.69
28.61
29.00
30.91
30.24
30.25
30.80
31.29
30.75
30.99
30.30
30.41
30.64
30.5*
31.28
0.497
0.502
0.496
0.506
0.505
0.506
0.506
0.512
0.511
0.500
0.492
0.504
0.496
0.506
0.499
0.490
0.487
0.493
10
10
10
20
20
20
30
30
30
30
40
40
40
40
50
50
60
60
0.273
0.334
0.353
0.335
0.173
0.176
0.184
0.170
0.168
0.123
0.122
0.149
0.137
0.120
0.127
0.119
0.136
0.141
0.125
0.098
0.104
0.096
0.091
0,097
0.095
0.085
0.078
0.083
0.077
0.067
0.064
0.083
215
3.2
3.6
3.3
3.8
5.0
6.1
5.0
6.3
5.3
5.6
4.9
5.3
5.8
6.9
6.1
5.8
5.6
6.7
4.8
6.2
7.5
8.4
6.9
6.0
5.7
9.3
6.9
7.1
6.0
8.0
8.7
7.8
1.3
1.6
1.6
0.8
1.6
1.9
2.1
2.1
2.1
3.3
3.1
2.4
2.9
3.2
2.0
2.0
2.5
3.0
3.2
2.2
2.1
3.2
2.1
3.1
3.0
2.5
2.4
2.7
3.7
2.4
2.9
3.7
41.48
41.10
41.79
40.78
40.30
40.78
50.01
49.28
49.21
50.34
51.44
49.94
50.75
49.72
50.09
50.67
50.92
50.75
28.88
28.82
28.69
29.03
29.59
29.57
33.50
33.68
33.61
33.71
33.83
33.48
33.71
33.71
33.78
33.83
34.05
33.27
14.28
14.32
14.32
14.36
14.35
14.40
15.47
15.44
15.38
15.33
15.26
15.19
15.23
15.11
15.04
14.89
14.98
14.79
-------
APPENDIX A (continued). TABULATED DATA
L/D 9 F R I/D ATC/AT0 W/D T/D Vo To Ta
10.. 15 55.73 0.051
10 15. 56.73 0.050
10.. 15 56.26 0.049
10. 15 55.62 0.050
10.. 15 56.30 0.056
10. 15 62.01 0.049
10.. 15 62.13 0.049
10 0.438
0.459
0.451
0.470
0.435
0.427
0.436
20 0.218
0.224
0.216
0.212
0.207
30 0.148
0.148
0.156
0.153
0.155
0.153
0.1*4
30 0.14?
0.14?
0.150
0.150
40 0.103
0.110
0.113
0.109
oai3
0.116
50 0.083
0.091
0.086
0.083
0.083
0.086
60 0.071
0.076
0.081
0.074
0.075
5.0
5.7
4.5
6.0
4.9
5.0
5.7
6.5
6.8
8.3
8.1
8.1
10.5
11.2
10.5
9.9
8.2
8.5
9.6
9.7
10.8
8.5
12.1
9.2
12.1
12.2
11.3
12.4
10.8
12.2
10.6
13.4
15.8
12.6
14.2
14.4
12.8
12.6
13.1
13.8
3.6 79.00 30.58
3.7
3.6
4.0
3.4
3.7
3.6
5.4 80.40 30.52
5.1
5.2
6.1
6.2
8.5 80.56 30.67
9.1
8.4
9.1
7.8
9.3
8.2
7.8 80.50 30.97
8.4
7.9
8.8
9.7 75.25 29.20
9.1
10.4
10.6
10.3
10.6
11.4 84.09 20.40
10,4
1119
11.8
11.1
11.0
12.6 84.09 29.34
12.8
12.4
11.9
11.7
16.05
15.94
15.78
15.89
15.80
15.62
15.58
216
-------
APPENDIX A (continued). TABULATED DATA
L/D 6 F R
10o. 15 55.84 0.103
10.. 15 56.81 0.098
10. 15 56.3^ 0.099
10.. 15 57.90 0.099
10. 15 58.38 0.100
10. 15 56.95 0.103
10. 15 58.59 0.100
10. 15 58.22 0.101
10.. 15 56.30 0.103
10.. 15 58.4? 0.100
I/D ATC/AT0
10 0.423
0.405
0.462
0.389
0.396
10 OJ^35
0.438
0.425
20 0.223
0.226
0.207
0.217
20 0.208
0.197
0.195
0.192
0.179
30 0.142
0.135
0.118
0.111
0.112
30 0.129
0.132
0.131
0.119
40 0.110
0.123
0.116
0.102
0.113
40 0.092
0.104
0.111
0.111
50 0.084
0.093
0.090
50 0.076
0.086
0.081
0.074
W/D
4.6
4.3
4.8
4.6
4.6
4.8
4.8
4.2
6.7
6.8
7.3
7.8
6.9
7.-3
8.0
8.2
6.4
8.9
8.8
8.9
10.3
9.8
8.4
8.2
7.4
10.5
10.4
10.7
9.2
10.9
9.4
10.9
12.4
10.8
9.2
8.1
11.4
13.0
9.7
13.5
12.5
12.1
*fl> *o T0
4.2 77.68 30.22
4.1
4.2
4.5
3.5
3.6 78.34 30.06
3.9
3.4
6.0 77.74 30.10
5.6
6.0
5.6
5.2 76.65 26.86
5.4
5.7
5.5
5.4
6.0 77.99 27.10
5.7
6.0
5.0
5.5
6.3 76.24 27.19
5.4
6.8
4.9
6.8 77.68 27.01
7.3
8.1
6.4
7.9
7.0 77.18 27.06
7.8
6.9
7.1
7.1 74.62 27.07
6.7
7.7
8.3 76.00 26.74
6.9
8.5
6.6
».
16.15
14.18
16.24
11.40
11.49
11.58
11.67
11.79
11.89
12.04
217
-------
APPENDIX A (continued). TABULATED DATA
L/D
10.
10.
10.
10.
10.
10..
10.
10.
10.
10..
10..
10.
10.
e
15
15
15
15
15
15
15
15
15
15
15
15
15
F
57.38
57.38
57.46
57.12
55.57
55.17
57.04
55.05
53.79
54.72
60.39
57.35
58.09
R
0.102
0.102
0.249
0.249
0.257
0.259
0.251
0.254
0.258
0.252
0.244
0.256
0.252
X/D
60
60
10
10
10
20
20
20
30
40
40
50
50
AT0/AT0
0.069
0.070
0.070
0.072
0.069
0.062
0.073
0.072
0.075
0.421
0.356
0.380
0.422
0.425
0.209
0.205
0.223
0.231
0.210
0.187
0.212
0.195
0.142
0.144
0.143
0.093
0.096
0.101
0.098
0.102
0.078
0.083
0.080
0.089
0.091
W/D
12.0
12.4
11.1
9.4
13.9
10.5
13.1
14.6
10.9
3.9
4.2
4.9
4.1
4.9
5.4
6.4
5.9
5.1
5.1
5.8
5.2
5.9
7.1
7.0
7.5
9.5
9.6
0.5
9.7
8.8
9.6
7.7
9.4
10.1
9.8
T/D
7.7
7.2
7.8
7.4
8.5
7.3
8.7
6.3
9.9
2.3
2.3
2.4
2.6
2.6
3.7
3.5
4.5
4.0
4.0
4.2
3.9
3.6
4.5
4.2
4.1
5.3
4.7
5.5
4.8
5-3
5.1
5.2
4.4
5.8
4.6
V0
74.64
74.78
76.25
76.13
74.28
74.04
76.68
74.86
74.02
75.72
77.64
74.18
75.18
TO
26.81
26.90
29.16
29.22
29.24
29.29
29.31
29.34
29.75
29.77
26.94
26.98
26.97
T*
12.15
12.25
16.01
15.96
15.88
15.83
15.80
15.72
15.69
15.50
12.99
12.78
12.73
218
-------
APPENDIX A (continued). TABULATED DATA
L/ff
10.
19.
10.
10..
10.
10..
10..
10.
10.
10.
10.
10.
10.
10.
10.
10.
10..
10.
10.
10.
10..
10.
0
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
F
58.44
57.21
57.98
58.88
56.96
54.45
55.90
56.82
57.34
56.52
55.78
54.43
56.08
56.42
54.35
55.43
55.30
55.81
54.63
55.02
55.68
55.33
R
0.250
0.258
0.251
0.250
0.489
0.510
0.499
0.504
0.494
0.502
0.504
0.510
0.502
0.505
0.523
0.516
0.502
0.499
0.509
0.498
0.498
0.500
X/D
50
60
60
60
10
10
10
10
10
20
10
20
20
30
30
30
40
40
40
40
50
50
AVAT0
0.083
0.085
0.078
0.073
0.075
0.083
0.076
0.074
0.073
0.266
0.263
0.251
0.300
0.274
0.175
0.166
0.172
0.134
0.130
0.123
0.112
0.104
0.089
0.096
0.083
0.080
0.083
W/D
10.7
9.9
10.0
10.3
8.9
9.3
8.8
10.6
9.4
5.5
4.0
4.0
5.1
4.0
6.0
7.1
5.2
4.4
7.2
6.6
5.7
7.1
8.3
8.2
8.0
6.4
8.8
I/D
5.6
5.4
4.8
5.1
6.2
5.4
5.4
7.0
5.4
2.7
1.9
2.4
2.3
2.2
2.1
2.7
2.4
1.9
2.3
3.5
3.8
3.4
4.5
3.0
3.8
3.4
3.6
Vo
75.82
73.97
74.98
75.82
76.52
73.62
75.38
74.24
75.62
73.80
74.22
72.84
74.64
74.64
71.74
73.09
74.67
75.30
73.80
73.99
74.52
74.07
TO
26.98
26.82
26.79
26.77
29.44
29.59
29.58
28.98
29.20
29.03
29.44
29.57
29.51
29.41
29.40
29.44
29.93
29.96
30.02
29.94
29.87
29.92
T*
12.63
12.46
12.38
12.52
16.07
16.10
16.18
16.24
16.31
16.37
16.41
16.48
16.53
16.57
16.63
16.74
16.76
16.85
16.90
16.93
16.97
17.04
219
-------
L/D
APPENDIX A (continued). TABULATED DATA
6 F B X/D
10. 15 54.76 0.509
10. 15 57.9* 0.497
10. 15 55.98 0,508
10. 15 57.46 0.485
10. 15 55.58 0.497
10. 30 32.50 0
10.. 30 30.77 0
10. 30 30.61 0
10. 30 30.32 0
50 0.090
60 0.051
60 0.054
60 0.052
60 0.060
10 0.348
0.358
0.381
0.358
0.378
0.382
0.381
0.379
0.381
0.374
0.378
20 0.163
0.152
0.166
0.167
0.175
0.171
0.167
0.170
0.178
0.184
0.179
30 0.109
0.108
0.102
0.105
0.117
0.103
0.110
0.116
40 0.072
0.072
0.069
6.6
7.8
8.6
10.0
9.0
8.8
6.7
7.6
8.1
6.9
5.9
8.5
7.6
6.8
6.4
7.2
10.6
13.6
17.3
16.4
11.1
9.5
11.8
13.6
14.0
11.5
12.0
19.9
15.9
12.7
23.7
21.1
27.1
19.5
21.6
33.9
39.5
32.5
4.0
3.5
2.5
3.9
4.0
8.1
7.3
7.1
7.1
7.1
7.2
7.3
7.2
7.2
7.2
7.1
17.1
18.0
17.8
17.0
16.7
16.8
17.1
16.7
16.7
15.5
15.3
16.7
27.8
26.5
25.7
27.8
26.1
26.3
25.7
52.1
57.5
50.7
73.15 29.90 17.08
75.98 28.95 16.06
73.4 28.99 16.13
77.07 29.48 16.18
75.38 29.71 16.21
50.13 34.67 20.00
50.42 36.20 20.13
50.51 36.45 20.26
50.99 37.14 20.64
220
-------
L/D e
APPENDIX A (continued). TABULATED DATA
F R I/O ATC/AT0 W/D I/D
10.
10.,
10.
10.
10.
10.
10.
10.
10.
10.
10..
10..
10.
10.
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30.12
32.59
30.49
31.5*
32.30
32.60
32.37
32.93
33.89
32.07
31.53
31.02
30.82
30.68
0
0
0
0.255
0.250
0.248
0.249
0.252
0.241
0.244
0.239
0.249
0.245
0.245
50
40
60
~
50
50
40
40
30
30
30
20
20
10
10
0.071
0.069
0.073
0.062
0.068
0.064
0.072
0.057
0.066
0.057
0.055
0.056
0.062
0.061
0.057
0.059
0.063
0.061
0.070
0.070
0.087
0.082
0.097
0.094
0.089
0.091
0.151
0.151
0.137
0.134
0.263
0.259
0.265
0.264
0.279
32*5
27.2
26.4
25.5
22.9
28.9
27.2
33.8
35.5
30.*
38.9
41.4
11.0
10.8
9.0
10.7
9.1
9.4'
10.0
9.2
7.6
7.4
8.*
7.6
7.3
718
8.0
8.2
8.0
6.1
5.0
5.1
4.6
3.7
6.7
50.2
33.6
31.0
31.9
35.3
32.3
32.7
54.8
54.8
53.1
56.9
53.1
8.8
8.6
8.3
7,9
8.7
6.8
8.9
8.3
7.3
7.9
9.0
7.3
7.7
7.7
6.9
6.3
6.3
6.4
4.9
4.6
5.3
4.5
5.0
49.32
51.78
49.63
49.87
51.02
51.32
50.63
50.63
51.20
50.70
51.42
50.72
50.48
50.38
36.53
35.94
36.63
36.02
36.05
36.00
35.91
35.54
35.19
36.24
37.01
37.10
37.16
37.24
20.75
21.01
21.18
21.33
21.43
21.46
21.57
21.61
21.66
21.71
21.77
21.80
21.83
21.87
221
-------
APPENDIX A (continued). TABULATE) DATA
L/D
10.
10.
10..
10.
10..
10.
10.
10..
10.
10.
10.
e F
30 30.52
30 30.59
30 29.38
30 29.96
45 11.18
45 10.61
45 11.50
45 10.13
45 10.34
45 9.96
45 10.40
R I/D AT0/AT0
0.256 10 0.259
0.245 20 0.136
0.136
0.251 40 0.083
0.083
0.248 50 0.069
0.069
0.101 10 0.196
0.208
0.202
0.195
0.214
0.194
0.101
0.104 20 0.112
0.106
0.103
0.109
0.111
0.096 20 0.099
0.097
0.10?
0.103
0.098
0.107 30 0.070
0.072
0.072
0.102 30 0.076
0.076
0.073
0.072
0.105 40 0.050
0.049
0.052
0.053
0.102 40 0.053
0.048
0.045
W/D
5.1
7.0
7.3
8.7
7.2
9.4
9.3
8.5
7.6
6.9
6.2
7.8
7.1
8.4
6.3
9.4
9.4
9.1
10.7
8.8
8.8
10.4
8.1
7.*
11.2
7.1
7.0
12.8
12.6
11.9
10.6
17.1
16.4
11.9
12.5
12.2
11.4
11.3
I/D
4.7
7.0
6.6
10.2
8.1
9.6
9.2
10.0
10.0
9.0
10.5
10.0
10.0
9.1
12.8
13.3
13.5
12.3
13.3
12.7
12.1
14.0
13.4
14.3
14.7
15.0
14.9
16.2
16.6
15.6
15.5
15.1
16.6.
17.1
15.9
17.5
16.9
17.7
V0
50.28
50.32
49.14
50.33
25.28
24.70
25.82
23.89
24.73
23.90
24.57
TO
37.26
37.33
37.80
37.91
45.00
45.54
44.65
46.04
46.60
46.63
45.98
*a
21.77
21.95
22.05
22.14
17.33
15.68
17.16
15.54
15.50
15.32
15.18
222
-------
L/D
10..
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
e
45
45
45
45
45
45
45
45
45
45
45
45
45
45
45
45
45
«M A 40MJ
F
10.22
11.53
10.64
10.63
11.08
10.6?
11.10
11.00
11.2
10.88
10.96
10.88
10.62
10.78
11.17
10.75
11.15
tf fc*» *n& ^ w i
R
0.104
0.101
0.496
0.506
0.489
0.511
0.502
0.501
0.506
0.486
0.493
0.502
0.513
0.494
0.499
0.509
0.502
LM l ir^aiv
X/D
50
50
10
10
10
20
20
20
30
30
30
40
40
40
50
50
50
ATC/AT0
0.043
0.044
0.041
0.036
0.151
0.161
0.160
0.190
0.202
0.102
0.108
0.188
0.182
0.109
0.112
0.092
0.082
0.079
0.081
0.083
0.086
0.078
0.073
0.067
0.068
0.072
0.080
0.073
0.074
0.065
0.067
0.060
0.064
W/D
18.3
16.9
15.6
13.9
4.1
3.9
4.8
4.2
5.2
5.1
5.6
6.6
6.2
5.9
6.6
6.5
6.2
8.1
5.7
6.9
7.8
7.9
7.6
9.1
8.8
7.5
5.8
7.8
6.9
8.8
9.1
9.4
7.6
I/D
19.0
18.3
17.0
18.1
4.1
3.6
4.0
3.9
4.1
4.3
4.2
4.7
4.4
5.1
5.1
5.4
5.1
5.3
5.5
5.2
5.1
5.6
5.5
4.8
4.8
5.7
5.8
6.5
6.5
6.5
5.5
5.8
5.5
V0
24.24
24.62
25.20
24.79
25.49
24.14
25.14
24.96
24.86
25.46
25.28
24,90
24.27
25.21
24.98
24.57
24.66
TO
46.08
41.93
46.85
46.27
45.75
45.13
45.20
45.29
44.52
46.05
45'.53
45.25
45.22
46.16
44.43
45.33
44.11
*a
15.02
14.82
17.40
17.51
17.56
17.56
17.68
17.74
17.80
16.57
16.68
16.75
16*83
16.38
17.00
17.13
17.22
223
-------
APPENDIi A (oontinaed). IAHJLATH) DATA
L/D
10.
10.
10.
10..
10.
10.
10.
10..
10.
OF R
45 32.41 0
45 30.84 0
45 31.34 0
45 31.60 0
45 31.72 0
45 32.31 0
45 31.44 0
45 32.20 0
45 32.19 0
VD AT0/AT0
40 0.078
0.079
0.076
40 0.085
0.074
40 0.066
0.075
0.075
50 0.068
0.057
50 0.075
0.076
0.070
0.072
30 0.096
0.099
0.095
30 0.081
0.086
0.087
20 0.146
0.144
0.138
0.136
0.148
10 0.274
0.301
0.310
0.293
0.294
0.299
0.290
0.294
0.294
V/D
37.1
40.5
24.0
36.0
34.7
35.7
32.5
32.9
26.2
27.9
22.8
26.3
20.3
27.9
16.4
18.8
22.2
15.6
22.4
20.1
9.8
10.2
12.4
11.0
14.5
8.5
10.2
10.2
10.4
12.3
11.1
9.9
10.2
11.1
*/D V0 TO
45.5 50.63 32.80
49.7
45.5
44.6 51.10 34.70
47.4
47.3 50.63 34.10
47.4
47.3
64.5 51.10 34.18
76.1
6l.l 50.93 34.04
56.3
66.2
59.0
34.3 50.43 33.42
34.5
35.9
32.0 50.97 34.46
32.8
30.9 *
20.9 50.71 33.74
19.8
21.3
21.4
21.8
11.8 50.30 33.64
10.1
10.4
10.6
9.5
9.7
10.6
10.3
10.1
*•
15.90
16.70
16.85
16.93
17.02
17.18
17.35
17.39
17.56
224
-------
APPENDIX A (continued). TABULATED DATA
t/D 0 F R I/D AT0/AT0 W/D Y/D Vo
10. 45 32.71 0.052
10. 45 33.35 0.048
10. 45 34.82 0.049
10. 45 34.55 0.050
10. 45 34.47 0.049
10 0.316
0.291
0.295
0.327
0.316
0.311
0.315
0.310
0.331
0.318
0.317
20 0.148
0.156
0.159
0.138
0.139
0.136
0.133
0.127
0.125
0.143
0.144
30 0.091
0.095
0.089
0,091
0.093
0.095
0.092
0.094
0.093
40 0.063
0.063
0.065
0.067
50 0.065
0.063
0.063
0.065
0.070
0.073
0.063
0.062
6.3
7.5
7.1
7.1
7.3
7.6
7.3
5.8
6.8
7.6
6.7
11.7
12.8
11.5
10.9
14.9
12.3
11.5
10.3
12.4
9.6
9.9
10.7
13.*
12.6
13.0
14.9
12.3
18.1
11.5
9.6
16.1
15.4
18.9
15.3
15.7
16.9
17.5
17.1
18.0
20.8
19.0
18.6
8.5 49.71 31.96
7.8
8.6
8.4
7.8
8.3
8.9
8.3
8.0
7.8
8.6
14.9 50.47 31.82
16.8
15.1
13.8
15.5
14.4
15.6
15.0
13.2
15.1
15.6
18.9 50.47 30.78
17.5
20.4
20.9
20.3
19.1
19.2
12.2
17.9
23.7 50.60 30.98
24.2
24.1
24.1
23.5 50.56 30.98
22.5
26.0
27.6
25.6
25.6
27.2
20.0
15.50
15.49
15.44
15.36
15.28
225
-------
APPENDIX A (continued). TABULATED DATA
L/D 0 F R
10. 45 35.49 0.051
10. 45 34.67 0.096
10. 45 33.22 0.099
10.. 45 33.79 0.099
10. 45 34.35 0.098
10. 45 32.95 0.100
10. 45 32.65 0.103
10. 45 32.33 0.248
2/D ATo/AT0
60 0.051
0.055
0.050
0.056
0,054
0.061
0.061
10 0.280
0.270
0.264
0.269
0.241
0.276
20 0.139
0.123
0.125
0.128
0.132
30 0.083
0.084
0.085
0.079
0.080
0.086
40 0.070
0.067
0.060
0.068
0.069
50 0.059
0.062
0.059
0.062
60 0.047
0.050
0.045
10 0.202
0.175
0.185
W/D
10.3
18.0
24.7
24.2
19.7
17.5
21.0
6.4
7.8
9.7
6.8
7.0
5.8
10.2
9.8
9.7
9.8
9.8
13.4
14.6
13.1
12.8
10.0
9.5
9.3
5.8
9.7
10.6
10.7
12.0
12.4
14.8
14.1
12.2
11.7
15.4
6.9
6.1
5.6
*/D V0 T0
26.6 50.25 30.11
26.5
27.5
28.7
26.9
26.9
26.9
7.9 52.37 31.84
8.0
7.6
8.3
7.8
7.7
13.3 50.03 31.79
13.8
13.2
13.7
12.0
14.8 50.50 31.63
14.2
15.1
14.2
13-8
14.8
16.3 50.69 31.35
15.8
15.7
18.6
15.5
16.6 49.46 31.77
18.0
18.7
18.7
22.7 49.00 31.77
21.3
24.1
5.5 50.13 33.54
4.6
5.7
T*
15.18
15.62
15.66
15.70
15.74
15.78
15.81
17.72
226
-------
APPENDIX A (continued). TABULATED DATA
L/D 9 F R X/D AT0/AT0 W/D T/D Vo T0 T.
10.
10..
10.
10.
10,
10.
10.
10..
10..
10.
10.
10.
10.
10..
45
45
45
45
45
45
45
45
45
45
45
45
45
45
32.61
32.37
32.46
32.2?
31.69
31-45
32.A8
32.23
31.92
29.97
31.35
31.10
31.16
31.40
0.245
0.246
0.246
0.241
0.243
0.246
0.242
0.250
0.243
0.245
0.244
0.243
0.249
0.249
10
10
20
20
30
30
40
40
50
50
50
40
30
20
0.191
0.193
0.217
0.228
0.109
0.114
0.118
0.102
0.104
0.102
0.090
0.096
0.079
0.085
0.072
0.067
0.063
0.064
0.059
0.056
0.058
0.054
0.067
0.059
0.078
0.073
0.082
0.085
0.123
0.127
6.5
5.5
5.3
5.5
6.8
7.9
7.2
7.0
7.0
6.9
9.0
7.6
8.3
7.3
9.1
10.1
11.1
9.6
11.9
11.5
11.0
9.2
10.4
9.2
8.8
7.9
9.0
8.0
8.6
7.7
6.5
5.5
6.5
5.6
6.9
6.7
6.9
7.6
6.8
7.0
9.1
8.1
7.9
8.3
9.5
9.7
10.0
7.5
11.3
9.8
10.9
8.1
9.6
9.0
10.0
8.3
8.9
8.7
8.4
7.7
50.72
50.19
50.69
51.32
50.80
50.56
51.93
51.49
50.58
49.88
51.56
50.97
50.71
51.02
33.68
33.66
34.31
34.82
35.06
35.15
35.05
35.06
34.88
36.16
35.95
35.93
35.81
35.83
17.85
17.95
18.84
18.98
19.05
19.08
19.15
19.21
19.25
19.33
19.51
19.64
19.74
19.84
227
-------
L/D e
APPENDIX A (continued). TABULATED DATA
F R X/D AVAT0 W/D T/D Vc
10. 45 55.29 0.050
10. 45 54.28 0.051
10. 4$ 55.0? 0.050
10. 45 53.50 0.051
10. 45 54.15 0.051
10. 45 5^.57 0.052
10. 45 54.59 0.101
10.. 45 54.20 0.100
10 0.309
0.310
0.330
0.320
0.267
0.300
0.276
0.305
20 0.136
0.139
0.138
0.144
0.149
0.144
30 0.101
0.101
0.105
0.111
0.096
0.104
0.107
40 0.072
0.070
0.072
0.071
50 0.049
0.055
0.055
0.039
0.051
60 0.055
0.045
0.043
0.048
10 0.256
0.251
0.289
0.257
0.263
20 0.132
0.124
0.122
0.116
8.6
9.2
8.7
9.1
9.1
8.8
8.5
8.5
12.6
11.2
12.1
11.9
11.2
11.2
14.8
13.4
12.5
12.5
13.2
14.8
14.1
17.1
18.6
22.5
18.8
20,4
20.9
20.9
20.1
19.3
22.2
18.9
19.6
19.6
7.3
8.1
7.7
6.7
7.1
11.6
10.9
10.7
12.0
8.9 75.61
8.9
9.4
9.5
8.6
8.9
8.9
8.7
13.9 74.42
16.4
14.2
15.1
15.1
14.6
18.9 75.50
18.3
18.6
18.8
19.0
18.5
18.5
24.3 73.92
23.0
21.4
23.0
25.8 75.13
25.8
25.9
27.1
24.6
26.9 74.52
26.3
26.3
27.6
8.5 73.92
7.9
8.2
8.4
8.1
11.9 74,59
12.5
11.8
11.2
27.85 12.02
27.85 11.89
27.81 11.78
27.94 11.66
27.96 11.49
27.52 11.30
27.71 12.22
28.13 12.30
228
-------
APPENDIX A (continued). TABULATED DATA
L/D e F a
10. 45 53.65 0.101
10. 45 57.80 0.105
10. 45 55.06 0.105
10. 45 57.46 0.100
10. 60 29.72 0
10. 60 30.68 0
10. 60 30.69 0.252
10. 60 30.70 0.250
10. 60 30.70 0.2*3
V» -
30
40
50
60
6.6
8.5
12.8
18.1
20.3
29.8
35.0
5.0
6.3
7.6
10.8
13.9
16.4
23.9
27.6
3*.9
10
10
20
&TC/AT0
0.091
0.090
0.080
0.082
0.083
0.080
0.070
0.065
0.06*
0.057
0.055
.0.0*8
0.052
0.050
0.048
0.051
0.047
0.404
0.297
0.186
0.123
0.084
0.060
0.047
0.39*
0.360
0.261
0.219
0.164
0.129
0.082
0.060
0.054
0.172
0.157
0.156
0.154
0.080
0.084
W/D
10.9
10.9
9.9
11.8
10.2
9.7
13.0
11.0
13.0
15.0
14.8
14.6
1*.5
16.2
15.7
14.8
13.5
6.0
6.5
6.3
6.8
7.4
6.8
T/D Y0
15.9 74.30
15.2
16.0
15.2
14.6
14.6
15.3 72.7*
15.0
15.9
16.0 72.00
18.1
16.3
16.7
18.6 75.13
18.5
19.3
18.7
10.3 *9.86
14.4
22.6
35.6
*3.3
58.1
68.0
10.1 *8.8*
11.8
15.1
19.6
26.0
33.7
43.3
58.1
73.2
7.0 49.7*
6.6
6.7 50.40
6.5
7.8 50.48
8.4
T T
ro *•
28.29 12.37
27.53 1*.95
28.35 15.07
28.40 15*15
38.52 23.14
36.60 22.04
37.7* 23.23
38.10 23.33
38.15 23.3*
229
-------
APPENDIX A (continued). TABULATED DATA
L/D
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10..
10.
10..
e
60
60
60
60
60
60
60
60
60
60
60
90
90
F
31.04
29.70
30.50
30.19
30.70
30.82
31.30
30.98
30.37
30.43
31.17
10.06
10.57
R
0.247
0.257
0.244
0.246
0.245
0.253
0.251
0.251
0.254
0.254
0.243
0.092
0.091
X/D
30
40
50
50
50
40
30
30
20
20
10
40
40
AVATo
0.057
0.064
0.058
0.046
0.050
0.036
0.040
0.047
0.049
0.057
0.056
0.051
0.063
0.068
0.043
0.050
0.099
0.094
0.083
0.074
0.156
0.038
0.037
0.044
0.039
0.046
0.047
0.044
0.048
0.053
W/D
8.3
7.7
9.4
11.7
11.0
10.2
11.6
7.5
8.4
8.7
9.4
8.9
9.3
8.5
9.3
8.3
6.9
7.7
7.9
7.0
5.6
19.9
18.8
13.6
30.4
26.6
12.3
26.2
23.3
22.6
I/O
10.7
9.0
8.5
9.8
8.9
12.9
13.5
11.5
9.9
11.6
10.5
10.7
9.4
9.3
9.4
9.6
9.2
8.0
9.4
8.9
6.6
21.4
26.0
22.3
24.1
22.2
19-7
21.2
20.1
19.7
Yo
50.99
49.64
50.97
50.47
50.89
49.64
50.05
50.13
49.64
49.64
50.84
25.20
25.79
TO
38.15
38.74
38.79
38.84
38.71
38.09
37.99
38.64
38.59
38.60
38.64
49.85
48.79
*a
23.38
23.66
23.75
23.82
23-93
24.03
24.13
24.72
24.28
24.36
24.43
19.25
19.35
0.041 18.6 26.3
0.040 20.9 23.7
10.. 90 10.59 0.099 30 0.056 10.9 17.4 25.30 48.00 19.45
0.057 12.6 14.8
0.052 14.0 17.5
0.048 21.6 18.9
230
-------
APPENDIX A (continued). TABULATED DATA
L/D
10.
10..
10.
10.
10,
10.
10.
10.
10.
10.
10.
10.
10.
e
90
90
90
90
90
90
90
90
90
90
90
90
90
F R
10.37 0.101
9.84 0.102
9.60 0.106
11.03 0.259
11.59 0.249
10.90 0.257
10.97 0.238
11.19 0.246
11.24 0.244
8.88 0.266
11.06 0.257
10.98 0.256
10.87 0.246
X/D
20
10
5
20
20
10
10
10
5
30
30
30
5
AT0/AT0
0.050
0.046
0.050
0.049
0.043
0.066
0.067
0.072
0.069
0.081
0.079
0.106
0.097
0.124
0.118
0.111
0.069
0.074
0.079
0.077
0.109
0.128
0.120
0.113
0.136
0.143
0.165
0.068
0.061
0.061
0.064
0.065
0.058
0.059
0.169
0.178
V/D
12.3
16.0
13.0
13.8
11.6
7.0
8.4
12.2
9.1
7.6
11.2
11.8
8.5
10.1
9.8
10.5
6.3
7.2
5.0
8.8
6.0
6.4
6.7
7.3
7.1
5.4
7.2
6.8
9.8
7.9
8.5
9.*
8.6
6.6
5.1
4.8
I/ff
16.6
17.2
21.9
22.4
22.1
15.9
15.7
15.8
16.9
17.4
16.3
12.0
10.9
13*4
13.4
12.8
9.7
7.8
9.9
8.9
9.1
8.3
9.3
6.8
7.2
6.7
5.9
12.1
9.5
12.1
10.9
11.3
10.6
11.1
6.6
7.0
Vo
24.66
24.53
24.10
25.25
26.30
24.63
24.90
25.11
24.86
22.53
24.74
24.82
25.21
TO
47.84
49.80
50.15
46.02
45.76
45.62
45.83
45.42
44.92
50.04
45.09
45.52
46.01
T*
19.5^
19.60
19.70
18.78
18.88
18.94
19.02
19.12
19.19
18.30
18.59
18.70
17.36
231
-------
L/D
6
APPENDIX A (continued). TABULATED DATA
F R X/ET AT0/AT0 W/D T/D
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10..
10..
10.
10.
10.
10.
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
10.66
10.30
11.06
10.61
10.1*
10.45
10.61
11.06
11.06
11.10
11.51
11.23
10.91
10.97
11.2*
11.2*
10.98
0.2*6
0.262
0.2*3
0.2*6
0.253
0.2*8
0.237
0.*99
0.*85
0.*89
0.*96
0.494
0.509
0.507
O.*90
0.*93
0.50*
5
10
10
20
20
30
30
40
40
30
30
20
10
10
10
10
20
0.172
0.177
0.115
0.122
0.125
0.132
0.07*
0.076
0.080
0.081
0.056
0.055
0.0*7
0.050
0.061
0.059
0.057
0.058
0.068
0.070
0.086
0.108
0.102
0.1*2
0.137
0.136
0.151
0.1*7
0.101
0.10*
5.0
*.9
7.7
6.2
6.8
6.*
6.9
8.0
7.7
6.8
10*9
6.*
7.8
7.2
9.3
8.0
6.6
5.9
8.*
6.9
*.3
6.0
*.7
5.1
5.1
*.8
*.2
*.2
7.7
6.*
7.2
6.5
7.7
8.1
7.5
6.*
10.6
10.2
9.0
9.0
6.*,
9.8
11.6
10.*
5.8
5.8
5.*
5.*
6.2
3.8
3.5
6.2
*.5
5.6
5.6
*.o
*.*
*.*
5.8
*.7
25.21
2*.57
26.0*
24.80
23.6*
24.60
25.*
2*.6l
25.0*
25.32
24.90
24.92
2*.65
2*.73
25.50
25.5*
2*.90
*6.83
*7.22
*6.70
46.50
46.46
*6.87
*7.*6
**.7*
*5.*0
*5.7*
*3.80
**.7*
*5.*3
*5.52
*5.63
*5.67
*5.59
17.50
17.61
17.69
17.90
18.0*
18.08
18.15
18.22
18.27
18.3*
18.*1
18.48
18.67
18.73
18.61
18.70
18.53
10. 90 10.99 0.502 20 0.099 8.0 *.8 2*.97 *5.*8 18.50
232
-------
APPENDIX A (continued). TABULATED DATA
L/D 6 F R X/D ATC/AT0 W/D T/D *o To
10
10.
10..
10.
10.
10..
10..
10.
10.
10.
10.
10.
10..
90
90
90
90
90
90
90
90
90
90
90
90
90
10.2*4.
10.51
10.66
10.29
10.09
10.40
9.85
9.70
10.24
10.02
9.98
10.58
10.19
0.498
0.495
0.476
0.491
0.510
0.462
0.511
0.513
0.507
0.509
0.510
0.483
0.489
5
5
5
5
5
10
10
10
20
20
30
30
40
0.262
0.257
0.246
0.269
0.212
0.223
0.257
0.243
0.216
0.159
0.158
0.139
0.146
0.151
0.130
0.108
0.110
0.089
0.098
0.081
0.090
0.088
0.060
0.057
4.2
*.9
*.3
3.8
5.0
5.0
4.5
4.1
3.6
5.5
6.1
4.4
6.2
5.4
4.7
5.6
5.6
6.8
6.0
7.9
5.5
7.4
8.2
7.8
3.9
2.8
4.3
3.1
3.9
1.6
3.9
4.1
2.7
4.8
4.0
4.7
2.6
4.5
3.8
6.3
4.0
4.9
4.1
7.0
6.6
4.8
6.2
6.0
24.43
24.50
25.39
24.44
24.11
25.60
24.30
24.30
24.73
24.86
24.37
26.00
25.11
47.63
46.73
47.63
47-55
47.80
49.01
49.10
49.80
48.27
49.42
48.80
49.07
49.20
18.72
18.77
18.83
18.92
18.97
18.92
18.99
19.05
19.06
19.15
19.21
19.27
19.3^
10. 90 29.63 0.050 40 0.054 28.3 ^.4 50.08 39.40 24.30
0.054 25.4 46.0
0.046 32.2 42.9
0.049 28.5 45.4
10.. 90 31.79 0.0486 30 0.057 24.7 33.8 51.65 38.33 24.08
0.070 36.8 40.7
0.056 37.2 38.2
0.065 29.8 37.6
0.063 24.5 36.5
233
-------
APPENDIX A (continued). TABULATED DATA
L/D 0 F R X/D AT0/AT0 W/D T/D Vo
10. 90 30.97 0.050 20 0.057 28.5 32.6 50.63 38.47 24.08
0.056 29.7 34.1
0.061 29.3 32.6
0.056 27.6 32.6
0.057 29.7 34.8
0.055 26.8 34.1
10. 90 30.98 0.049 10 0.107 22.2 27.2 50.80 38.57 24.12
0.099 26.9 24.9
0.095 16.7 26.6
0.098 25.2 26.5
0.094 24.2 27.0
0.100 26.8 27.7
0.103 26.1 28.3
10. 90 30.99 0.050 5 0.124 29.5 22.9 50.45 38.39 24.12
0.110 18.9 20.2
0.111 28.2 19.7
0.120 21.9 20.6
0.119 18.6 21.6
10. 90 30.94 0.095 5 0.140 10.0 12.2 50.80 34.23 16.21
0.137 10.4 13.4
0.138 9.4 13.5
10. 90 30.66 0.099 5 0.157 10.1 12.9 49.52 33.85 16.30
0.148 9.7 11.6
0.159 11.4 12.1
10. 90 32.26 0.099 10 0.093 9.0 15.0 51.10 33.40 16.40
0.103 8.9 14.0
0.098 9.7 14.7
10. 90 32.77 0.096 10 0.095 11.7 16.8 51.49 33.22 16.46
0.085 10.4 14.3
0.086 10.1 14.9
10. 90 31.68 0.096 20 0.062 17.7 18.3 51.67 34.39 16.90
0.056 15.5 18.9
0.057 16.0 16.6
10. 90 31.63 0.097 30 0.064 18.6 14.8 51.93 34.75 17.27
0.061 17.5 14.4
10. 90 30.75 0.101 30 0.052 16.8 18.4 50.38 34.76 17.40
0.052 19.3 15.1
0.051 17.1 15.5
234
-------
APPENDIX A (continued). TABULATED DATA
L/D 8 F R X/D AT0/AT0 W/D T/D Vo To Ta
10. 90 32.62 0.095 40 0.050 18.6 10.7 51.75 34.03 17.60
0.049 22.0 10.3
10. 90 31.78 0.099 40 0.048 20^6 14.1 50.80 34.30 17.74
0.052 22.3 15.0
0.051 15.0 16.5
10. 90 33.68 0.238 40 0.048 10.9 10.5 52.58 34.64 19.45
0.047 8.3 9.9
10. 90 33.47 0.237 40 0.049 11.9 11.1 52.93 34.97 19.50
0.051 8.5 10.0
10. 90 31.77 0.245 30 0.065 9.7 9.3 51.06 35.46 19.67
0.069 6.7 9.0
10. 90 33.23 0.242 30 8.056 9.9 11.0 52.38 34.97 19.63
0.059 6.8 9.3
10. 90 32.22 0.244 20 0.097 8.0 8.9 51.75 35.47 19.72
0.093 9.4 7.3
10. 90 31.83 0.247 20 0.083 9.4 9.6 51.18 35-54 19.78
0.085 7.9 8.6
10. 90 31.89 0.244 10 0.103 7.3 7.1 51.75 35.81 19.86
10. 90 30.52 0.245 10 0.126 6.7 8.5 51.50 36.81 19.89
10. 90 30.42 0.249 5 0.183 5.4 6.3 51.23 36.83 20.00
0.194- 5.7 4.4
10. 90 30.20 0.250 5 0.185 6.8 6.7 50.80 36.82 20.06
0.187 5.7 5.4
10. 90 30.39 0.244 5 0.163 4.8 6.9 51.14 36.87 20.12
0.173 5.9 5.4
10. 90 30.02 0.248 5 0.166 4.8 6.6 50.47 36.88 20.17
0.160 6.9 5.1
10. 90 29.43 0.257 5 0.163 6.0 8.5 49.48 36.91 20.23
0.168 7*0 6.6
10. 90 29.73 0.245 10 0.107 5.8 7.9 49.89 36.90 20.31
0.105 4.3 6.3
10. 90 33.13 0.255 20 0.088 8.6 9.2 49.68 34.45 20.70
0.085 8.3 7.2
235
-------
APPENDIX A (continued). TABULATED DATA
6 F R J/BT AVATo W/D T/D vo To T*
ICU90 32,22 0.252 ~200.078 6^5 9.3 50.05 35.26 20.74
0.089 7.5 6.9
10. 90 31.04 0.253 30 0.059 8.7 9.7 49.48 35.89 20.77
0.059 9.2 8.9
10. 90 30.59 0.256 30 0.059 8.4 9.7 48.00 35.53 20.80
0.058 9.1 9.5
10. 90 31.98 0.245 40 0.064 9.0 11.4 50.55 35.72 20.82
0.056 8.8 8.7
10. 90 32.30 0.244 40 0.065 9.6 9.3 50.78 35.59 20.82
0.073 6.3 10.1
10* 90 32.04 0.494 50 0.043 12.2 11.7 50.63 35.70 20.80
0.042 7.7 12.6
0.039 9.7 11.6
10. 90 32.15 0.496 50 0.046 5.9 3.5 50.53 35.57 20.81
0.054 6.3 5.4
0.048 7.2 5.7
10. 90 31.80 0.494 40 0.064 6.6 7.5 50.80 35.97 20.83
0.069 6.3 5.7
10. 90 31.49 0.497 40 0.053 9.1 7.6 50.47 36.05 20.84
0.053 7.6 5.9
10. 90 31.49 0.497 30 0.054 7.5 6.4 50.24 35.95 20.85
0.054 6.1 4.4
10. 90 32.18 0.509 30 0.065 7.9 5.5 50.05 35.36 20.86
0.068 5.7 4.6
10.. 90 30.76 0.515 20 0.090 5.9 4.7 49.56 36.20 20.88
10.. 90 30.57 0.496 20 0.093 7.8 5.0 50.55 36.86 20.92
0.104 6.2 4.2
10. 90 30.60 0.503 10 0.132 4.6 4.7 50.38 36.78 20.95
10. 90 30.16 0.507 10 €.181 5.1 4.5 49.73 36.83 21.00
0.181 5.0 3.6
10. 90 31.02 0.503 5 0.193 4.2 4.4 49.97 36.27 21.02
0.204 4.2 3.8
236
-------
APPENDIX A (continued). TABULATED DATA
L/D
10.
10..
10.
10.
10.
10.
10.
10.
10.
10.
10..
10.
10.
10.
e
90
90
90
90
90
90
90
90
90
90
90
90
90
90
F
31.05
30.79
31.66
31.35
31.55
31.51
30.23
31.15
30.90
31.44
33.31
31.21
31.05
29.94
R
0.501
0.504
0.496
0.489
0.494
0.490
0.503
0.492
0.499
0.490
0.502
0.494
0.497
0.516
I/D
5
5
5
5
10
10
20
20
30
30
40
40
40
50
™*Ql"*Q
0.233
0.245
0.179
0.200
0.205
0.188
0.183
0.140
0.137
0.129
0.142
0.097
0.101
0.106
0.098
0.076
0.069
0.079
0.081
0.056
0.061
0.061
0.073
0.073
0.063
0.061
W/D
4.2
3.7
2.9
4.9
5.8
4.6
5.7
5.4
5.2
5.3
5.5
6.0
4.4
6.7
5.0
6.7
6.3
6.8
5.9
6.3
7.2
8.9
8.5
5.1
9.7
8.4
T/D
3.3
3.5
1.7
4.0
3.7
4.3
3.8
3.5
3.4
3.7
4.4
5.4
4.8
4.7
4.4
4.9
4.4
5.6
5.0
6.2
5.7
5.1
5.5
4,8
6.3
5.5
*o
50.05
49.40
50.21
51.33
51.01
51.23
49.56
50.80
50.30
51.02
49.80
50.47
50.89
49.16
T°
36.32
36.22
35.97
36.75
36.47
36.63
36.85
36.74
36.72
36.67
31.37
33.41
33.80
33.88
*a
21.06
21.08
21.14
21.14
21.19
21*24
21.27
21.30
21.34
21.39
15.17
15.30
15.37
15.43
10. 90 59.70 0.050 40 0.058 32.7 41.5 80.37 31.74 19-92
0.053 30.1 39.1
10. 90 55.55 0.049 40 0.051 35.9 38.1 78.00 32.60 19.95
0.050 29.3 38.9
0.049 34.3 47.0
237
-------
L/D e
APPENDIX A (continued). TABULATED DATA
F R X/D ATC/AT0 W/D I/D
10. 90 56.33 0.048 30 0.069
0*070
0.070
10. 90 56.77 0.051 30 0.064
0.064
0.060
10. 90 56.54 0.049 20 0.066
0.063
0.068
10. 90 57.68 0.050 20 0.063
0.065
0.066
0.064
10.. 90 60.18 0.052 10 0.090
0.087
0.083
0.078
10. 90 58.05 0.053 10 0.097
0.089
0.093
10. 90 59.89 0.047 5 0.147
0.146
0.148
10. 90 58.27 0.051 5 0.141
0.141
0.141
0.138
10. 90 58.72 0.052 2.5 0.214
0.200
0.195
0.190
0.121
0.200
30.7 30.1 77.79 32.29 20.00
27.4 36.5
30.9 36.6
34.4 38.6 78.70 32.45 20.13
36.3 35.1
31.9 36.9
27.7 31.5 78.50 32.52 20.19
25.9 30.6
27.5 32.7
22.6 29.8 78.40 32.17 20.30
26.3 30.4
24.8 29.8
25.4 33.1
21.5 23.8 78.98 31.54 20.35
19.3 21.5
19.0 23.7
21.1 23.4
19.8 24.0 77*11 31.82 20.44
20.4 24,3
21.6 26.7
24.5 21.3 79.00 31.73 20.50
23.2 21.9
22.5 19.7
20.1 19.3 76.72 31.73 20.56
23.2 17.0
18.5 18.7
20.6 16.9
19.8 14.0 77.70 31.84 10.58
17.6 13.8
19.0 14.0
17.1 12.4
17.9 12.9
16.2 13.0
10. 90 60.09 0.105 50 0.044 15.1 15.9 77.63 30.06 18.47
0.053 lfc.2 15.7
10. 90 52.36 0.113 50 0.047 17.1 18.4 68.80 30.44 18.55
0.045 19.2 14.1
238
-------
APPENDIX A (continued). TABULATED DATA
L/D 6 F R I/D ATe/AT0 W/D Y/D Vo To
10..
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
58.42
57.83
57.77
57.71
58.31
56.60
56.50
53.65
54.41
53.52
58.79
55.90
55.41
55.08
55.57
56.6?
55.*0
0.103
0.104
0.102
0.101
0.103
0.103
0.096
0.101
0.099
0.099
0.103
0.105
0.108
0.106
0.105
0.103
0.104
50
40
40
30
30
30
20
20
20
10
5
5
5
5
5
10
10
0.050
0.049
0.048
0.055
0.045
0.039
0.056
0.050
0.049
0.048
0.064
0.057
0.067
0.065
0.073
0.075
0.093
0.086
0.081
0.082
0.146
0.152
0.158
0.165
0.140
0.139
0.139
0.136
0,153
0.147
0.085
0.091
0.085
16.9
21.9
18.1
16.7
15.3
19.5
19.8
15.9
16.8
15.9
15.0
15.1
14.1
14.4
12.7
10.4
10.1
8.3
11.0
10.9
8.7
10.6
9.5
10.3
9.7
8.3
8.5
9.9
13.5
9.3
11.6
10.9
11.5
15.9
13.4
13.1
15.3
16.0
19.9
14.3
16.8
13.4
14.3
15.1
14..3
20.5
14.7
16.3
18.2
14.8
5.7
15.6
15.6
11.2
10.9
13.1
13.3
11.9
11.4
11.9
12.1
12.1
11.2
14.1
15.9
15.1
77.24
76.07
77.36
77.38
78.04
75.74
77.76
74.70
76.20
75.20
74.85
74.50
74.10
74.46
74.70
74.70
74.33
30.59
30»53
30.94
30.98
30.98
31.02
31.71
32.02
32.19
32.30
30.38
31.29
31.40
31.66
31.60
31.24
31.62
18.61
18.67
18.78
18.81
18.86
18.93
19.16
19.28
19.36
19.43
19.40
19.50
19.58
19.65
19.73
19.76
19.82
0.087 10.6 15.6
239
-------
APPENDIX A (continued). TABULATED DATA
L/D 6
F
R Z/D ATe/AT0 W/D I/D V,
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
5*.09
5*.62
58.0*
58.08
56.23
56.36
56..57
58.93
57.88
58.5*
57.2*
57.59
59.26
59.01
59.7*
57.87
60.00
58.39
59.09
58.01
59.86
58.90
0.107
0.107
0.101
0.100
0.099
0.098
0.266
0.27*
0.25*
0.257
0.25*
0.256
0.2*8
0.2*7
0.2*3
0.253
0.2**
0.2*9
0.250
0.2*9
0.2*3
0.2*8
20
20
30
30
*0
*0
*0
40
30
*0
*0
20
10
10
5
5
5
5
5
5
10
10
0.063
0.065
0.053
0.051
0.059
0.060
0.058
0.053
0.05*
0.05*
0.057
0.05*
0.060
0.061
0.067
0.059
0.063
0.097
0.096
0.112
0.179
0.196
0.173
0.190
0.18*
0.213
0.108
0.117
240
12.2
11.2
15.7
11.9
16.8
1*.3
16.6
15.1
27.1
20.0
21.8
19.3
8.*
9.6
8.*
9.7
13.9
7.5
8.3
7.3
6.1
6.2
8.*
7.0
*.8
5.8
6.3
6.7
13.3
15.9
15.*
17.2
1*.7
15.9
13.*
12.8
13.6
16.9
18.0
16.3
9.1
9.1
8.*
10.3
9.*
8.0
9.2
8.7
6.3
6.9
6.8
6.6
6.1
5.*
5.8
7.3
72.56
73.38
78.8
79.9*
77.89
78.16
76.29
77.**
75.82
78.82
76.23
75.95
77.90
77.2*
78.70
76.*6
78.16
76.*6
77.53
77.90
79.95
78.57
31.63
31.70
31.73
32.05
32.22
32.28
30.75
30.19
30.32
29.98
30.52
30.50
30.*8
30.*6
30.57
30.67
28.20
28.40
29.85
30.38
30.33
30.33
19.85
19.91
19.62
19.72
19.78
19.85
18.23
18.15
18.35
18.25
18.26
18.55
18.60
18.70
18.70
18.77
14.92
15.10
17.62
17.7*
17.8*
17.89
-------
APPENDIX A (continued). TABULATED DATA
L/D
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
e
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
F
58.87
59.04
59.99
60.48
55.57
54.30
56.69
57.12
58.84
61.58
59.60
60.72
61.97
60.59
64.07
63.15
60.42
59.19
56.97
56.90
59.06
59.74
58.80
55.97
R
0.247
0.241
0.237
0.238
0.259
0.266
0.255
0.273
0.251
0.482
0.501
0.490
0.494
0.490
0.507
0.496
0.499
0.487
0.499
0.511
0.494
0.490
0.507
0.503
X/D
20
20
30
30
40
50
50
40
20
50
50
50
50
40
40
40
30
30
20
20
10
10
5
5
ATC/AT0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
.076
.095
.064
.063
.066
.050
.058
.070
.091
.051
.051
.053
.056
.055
.067
.058
.072
.085
.095
.110
.159
.164
.228
0.221
W/D
9.3
7.1
9.8
11.2
9.2
11.8
10.7
10.0
8.5
6.7
6.6
11.2
8.0
6.9
6.3
8.3
7.2
7.6
7.5
6.6
6.9
6.0
5.4
6.1
Y/D
7.4
8.1
7.3
9.0
10.2
9.4
11.0
9.3
8.4
5.8
5.4
5.6
5.1
4.8
3.5
5.1
5.2
5.3
4.8
5.5
3.5
4.2
4.0
3.7
7o
78.60
80.16
82.10
82.70
75.32
73.50
76.24
76.20
77.50
77.63
76.33
78.47
77.50
76.85
76.72
78.03
76-33
78.15
75.85
75.70
78.33
78.43
76.72
77.37
TO
30.36
30.71
30.92
30.96
30.90
30.89
30.79
30.53
30.45
28.15
28.49
28.72
28.19
28.50
27.48
28.13
28.96
29.85
30.05
30.07
30.05
29.88
29.70
30.93
*a
17.91
17.93
18.01
18.12
18.30
18.34
18.39
18.22
18.50
16.02
16.10
16.17
16.34
16.42
16.47
16.63
17.33
17.41
17.47
17.53
17.60
17.64
17.55
17.67
241
-------
L/D
e
APPENDIX A (continued). TABULATED DATA
F R I/D AT0/AT0 W/D I/D
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
10.
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
54.83
55.70
53.89
53.71
57.46
59.82
59.61
62.32
60.44
60.57
60.34
63.36
60.74
59.97
59.62
59.62
0.514
0,489
0.501
0.505
0.501
0.487
0.497
0.489
0.494
0.497
0.497
0.512
0.514
0.511
0.503
0.506
5
5
5
5
10
10
20
20
20
30
30
40
40
40
50
50
0.214
0.218
0.214
0.192
0.114
0.122
0.104
0.109
0.104
0.089
0.099
0.077
0.062
0.055
0.046
0.057
5.2
5.9
4.7
5.6
5.1
6.1
6.8
8.4
9.3
8.2
6.0
8.0
7.5
7.3
6.2
8.0
3.3
3*5
3.7
3.2
3.8
4.3
4.5
4.5
3.6
3.8
5.0
4.7
5.4
4.3
5.7
5.3
75.25
78.65
76.71
76.20
76.74
78.60
77.36
78.30
77.90
77.76
77.34
77.57
77.76
77.43
77.24
77.24
30.81
31.48
31.67
31.64
30.41
30.10
29.86
29.26
29.75
29.69
29.67
28.61
29.50
29.71
29.81
29.84
17.73
17.84
17.88
17.95
17.98
18.00
29.86
18.00
18.03
18.04
18.07
17.70
17.82
17.90
17.96
18.02
242
-------
APPENDIX A (continued). TABULATED DATA
L/D 0 F R I/J> ATC/AT0 W/D I/D Vo To Ta
5. 45 9.62 0.105
5. 45 9.65 0.101
5.. 45 11.40 0.097
5.. ^5 10.10 0.106
5. 45 10.19 0.104
5. 45 52.81 0.049
5. 45 54.13 0.050
10 0.193
0.215
0.198
0.217
0.208
0.209
0.203
20 0.126
0.123
0.132
0.126
0.121
0.121
0.123
0.126
30 0.093
0.113
0.119
0.115
0.111
0.115
40 0.088
0.089
0.084
0.072
60 0.077
O.OF6
0.066
10 0.259
0.290
0.279
0.24*
0.263
20 0.153
0.148
0.155
0.144
0.146
0.146
7.3
8.5
8.2
6.0
5.8
7.7
6.8
8.6
8.8
10.4
8.6
8.8
11.1
12.0
10.6
13.4
10.1
11.7
11.9
12.6
13.7
14.3
17.9
13.3
13.2
16.3
14.9
9.6
8.0
7.8
7.1
8.4
8.5
10.0
10.4
11.6
13.8
12.1
11.1
8.9 24.03 46.85
9.2
8.2
8.2
7.9
8.2
8.6
11.9 23.88 46.40
11.7
10.2
11.0
10.4
11.0
9.8
11.4
15.1 24.83 41.02
11.7
10.3
10.5
10.5
10.8
12,2 24.29 44.83
12.0
11.4
10.5
11.2 24.22 44,47
15.9
14.7
9.4 62.86 23.91
9.5
8.8
9.4
9.4
11.8 60.97 22.94
13.7
13.3
13.2
14.0
13.6
9.41
9.19
8.96
8.60
8.35
9.66
9.91
243
-------
APPENDIX A (continued). TABULATED DATA
L/D 0 F R
5. 45 54.19 0.051
5. 45 57.51 0.051
5. 45 57.48 0.053
5. 45 58.18 0.101
5. 45 57.71 0.100
5. 45 56.91 0.101
5- 45 57.13 0.103
5. 45 58.45 0.097
5. 90 10.69 0.099
5. 90 9.90 0.095
X/D ATC/ATO
30 0.128
0.127
0.135
0.130
40 0.094
0.109
0.118
0.102
0.102
60 0.080
0.097
0.095
0.094
10 0.225
0.217
0.225
0.234
0.233
20 0.121
0.129
0.120
0.130
30 0.110
0.118
40 0.099
0.099
0.092
60 0.072
0.088
0.075
10 0.117
0.121
0.109
0.125
20 0.116
0.099
0.099
0.105
0.117
0.103
W/D
12.7
16.2
14.7
12.2
16.3
14.2
18.5
16.5
18.4
16.5
23.2
17.2
19.4
8.2
7.2
6.7
6.1
6.1
10.1
9.5
10.2
10.7
12.0
13.6
14.7
12.3
16.4
14.3
17.7
21.3
13.4
13.0
11.7
14.0
16.8
18.2
9.9
15.9
16.9
19.6
*/D «0 *o
17.2 59.87 22.67
18.9
17.6
16.5
19.9 60.89 21.98
20.1
18.7
20.3
18.1
23.4 60.29 21.90
20.6
22.1
20.8
8.1 59.78 23.71
7.6
7.9
7.4
7.3
9.9 59.39 23.71
10.6
11.5
10.6
12.0 59.77 23.96
12.6
12.0 59.23 23.74
11.6
13.1
14.0 61.69 23.99
15.5
14.6
11.4 25.20 46.47
12.2
13.3
12.7
16.4 25.03 49.49
12.7
10.2
12.1
15.3
11.3
**
10.09
10.23
10.40
14.36
14.32
14.25
14.18
14.13
16.93
17.15
244
-------
APPENDIX A (continued). TABULATED DATA
L/D
5.
5-
5.
5.
5.
5.
5..
5.
5.
5..
5.
5..
5.
5.
5.
6
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
F
9.68
9.55
9.51
10.03
10.07
10.42
10.77
11.33
12.01
10.40
10.50
10.61
10.40
10,91
11.07
R
0.101
0.104
0.107
0.504
0.503
0.499
0.493
0.503
0.482
0.512
0.503
0.500
0.504
0.503
0.495
Z/D
30
40
60
10
10
10
20
20
30
30
30
40
40
40
50
ATC/AJ0
0.090
0.076
0,074
0.072
0.071
0.069
0.077
0.064
0.052
0.050
0.051
0.052
0.156
0.155
0.164
0.148
0.158
0.140
0.087
0.091
0.095
0.091
0.083
0.094
0.080
0.085
0.081
0.067
0.071
0.063
0.067
0.064
0.058
0.064
W/D
14.6
17.5
18.9
20.7
17.0
18.8
17.5
20.1
13.9
15.9
16.7
22.1
6.4
5.5
5.7
3.9
6.8
5.5
5.6
7.3
6.3
6.9
8.0
7.7
5.8
8.8
8.5
8.5
9.6
7.1
9.0
9-1
10.5
10.1
I/D
19.2
16.7
18.2
16.1
20.4
19.5
17.1
16.5
22.4
26.3
24.4
18.5
3.9
4.6
4.2
3.8
3.9
4.5
4.1
4.1
4.3
4.7
5.0
*.3
5.7
4.8
4.2
4.8
4.8
5.0
4.3
*.9
4.1
4.9
V
*o
24.82
24.17
24.06
24.71
24.91
25.07
25.07
24.92
25.76
24.66
24.87
24.85
24.94
24.56
24.89
TO
50.13
49.61
49.63
48.27
48.41
47,21
45.87
43.66
42.72
46.53
46.48
46.00
46.92
44.34
44.29
*.
17.25
17.37
17.49
16083
16.75
16.66
16.58
16.55
16.51
16.44
16.38
16.28
16.16
16*03
15.97
245
-------
L/D
5.
5..
5-
5*
e
90
90
90
90
F
10.51
10.96
10.92
10.79
I^V^BM •. * \^*
R
0.521
0.508
0.510
0.540
W**V^bM*
I/D
50
60
60
60
ATC/AT0
0.065
0.071
0.066
0.061
0.054
0.055
0.050
0.048
tf/D
7.7
6.6
6.5
10.1
8.7
8.5
8.3
10.2
•• ••^tOT
r/D
4)7
5.0
4.4
4.2
3.5
3.1
3.0
V0
23.78
24.64
24.52
23.20
TO
44.51
44.22
44.15
42.46
*.
15.89
15.79
15.71
15.61
5. 90 55.45 0.051 10 0.128
0.115
0.120
5- 90 55.80 0.050 20 0.132
0.100
0.105
0.095
5. 90 55.26 0.049 30 0.098
0.104
0.092
0.111
5. 90 55.25 0.052 40 0.086
0.078
0.086
0.099
5. 90 55.47 0.051 60 0.055
0.075
0.071
0.071
5. 90 59.84 0.098 10 0.129
0.150
0.132
5.. 90 59.92 0.100 10 0.128
0.137
0.133
5. 90 59.19 0.100 20 0.101
0.110
0.103
17.1 21*1 59.10 24.40 14.54
16.0 22.0
21.7 19.8
21.2 27.7 60.20 24.61 14.60
21.9 25.*
16.6 31.0
19.4 28.9
19.6 34.3 59.57 24,65 14.68
23.3 33.8
24.9 36.8
24.5 37.9
14.3 33.6 59.57 24.70 14.76
25.0 30.9
24.9 33.6
20.8 34.8
17.6 44.7 59.60 24.67 14.81
31.9 45.1
17.3 *3.7
23.6 45.7
11.9 10.9 60.49 24.15 15.^5
13.0 12.1
15.3
13.7 11.4 60.78 24.17 15.39
12.9 11.0
12.9 10.9
13.5 14.0 60.27 24.19 15.35
14.2 13.9
19.2 15.3
246
-------
APPENDIX A (continued). TABULATED DATA
L/D 6 F R X/D ATC/AT0 W/D Y/D Vo To Ta
5. 90 59.65 0.098 20 0.152 15*9 14.0 60.9 24.21 15.29
0.145 16.4 15.8
5- 90 58.84 0.098 30 0.135 16.5 15.6 60.49 24.26 15.24
0.135 15.6 17.5
5. 90 56.32 0.106 30 0.109 17.8 15.9 58.83 24.46 15.17
0.109 15.7 15.0
5.. 90 51.65 0.106 40 0.119 16.0 16.8 56.10 25.02 15.12
0.115 16.6 15.5
5- 90 52.58 0.105 40 0.098 18.3 16.3 57.25 25.03 15.05
0.103 19.8 16.9
5. 90 55.08 0.100 50 0.082 16.1 17.0 59.68 24.94 15.02
0.086 15.7 18.2
5- 90 56.59 0.097 60 0.093 20.1 19.8 60.83 24.73 14.89
0.078 22.6 17.9
0.077 20.8 20.6
5. 90 56.64 0.102 60 0.073 20.0 17.0 59.90 24.44 14.84
0.080 22.7 16.3
5. 90 57.81 0.503 10 0.134 6.3 4.0 59.70 24.22 15.06
5. 90 57.57 0.503 10 0.144 6.3 3.5 59.59 24.23 15.01
0.131 4.1 3.4
5. 90 59.90 0.478 20 0.097 6.1 5.3 62.21 24.25 14.98
0.092 4.6 4.9
5. 90 56.98 0.508 20 0.086 7.0 , 4.8 59.22 24.23 14.93
5. 90 58.46 0.498 20 0.119 7.3 ^.8 60.00 24.03 14.90
0.084 3.7 4.8
5. 90 57.51 0.486 30 0.083 8.7 5*0 61.51 26.11 17.21
5. 90 58.50 0.476 30 0.067 6.9 4.9 62.64 26.08 17.13
0.059 8.0 4.5
5. 90 54.29 0.522 40 0.066 8.4 3.4 57.97 26.00 17.08
0.050 7.9 3.9
247
-------
L/D e
APPENDIX A (continued). TABULATED DATA
P R I/D ATC/AT0 W/D T/D
2.5 45 11.46 0.098
2.5 45 11.45 0.095
.
2.5 *5 11.28 0.097
2.5 *5 11.59 0.100
2.5 *5 10.93 0.108
2.5 90 10.70 0.101
2.5 90 10.29 0.099
2.5 90 10.19 0.976
10 0.280
0.265
0.282
0.286
0.279
0.305
20 0.198
0.195
0.219
0.200
0.187
0.191
30 0.171
0.171
0.169
0.156
0.156
0.167
40 0.123
0.125
0.133
0.132
0.132
0.130
60 0.105
0.104
0.100
0.112
10 0.221
0.180
0.189
20 0.140
0.149
0.134
0.128
0.120
0.138
30 0.098
0.102
0.096
0.108
4.9
7.0
5.7
6.7
6.5
6.6
7.4
8.1
11.4
10.6
9.9
10.3
9.8
11.6
12.4
12.1
11.6
12.1
8.4
14.6
14.9
13.8
13.8
12.9
16.3
10.4
15.8
11.5
11.3
13.9
12.8
12.8
14.1
12.2
16.3
11.4
12.4
21.5
22.6
19.5
23.3
4.8 27.03 44.40
5.3
5.8
4.9
5.7
6.1
12.6 27.03 44.50
13.3
12.7
13.2
10.4
13.0
16.4 26.53 44.40
15.0
13.8
14.5
13.6
15.5
19.1 25.93 *2.39
19.1
19.1
16.9
16.9
17.4
23.1 23.84 41.46
22.6
22.6
20>
18.9 25.15 45.51
18.6
18.4
24.6 25.10 47.02
22.7
24.6
22.4
24.4
23.7
29.4 25.10 47.30
28.2
27.6
30.2
9.67
9.91
10.17
10.41
10.63
14.38
14.21
14.04
248
-------
APPENDIX A (continued). TABULATED DATA
L/D 0 F R X/D ATC/AT0 W/J> I/D Vo To Ta
2.5 90 10.92 0.097 **0 0.092 28.5 33.1 25.04 43.43 10.54
0.081 27.3 37.2
0.064 27.7 35.3
0.080 36.5 30.2
0.096 25.3 31.9
2.5 90 10.75 0.105 60 0.080 55.9 36.9 24.62 43.31 10.25
0.087 53.1 30.3
0.057 46.3 ^3.4
0.065 40.1 35.8
0.066 37.6 42,9
2.5 90 10,24 0.499 10 0.179 5.1 5.8 24.91 47.00 14.56
0.178 4.7 5.3
2.5 90 9.33 0.491 10 0.162 6.0 5.5 25.24 51.82 14.69
0.147 6.2 5.6
2.5 90 10.42 0.496 20 0.141 6.8 7.5 25.21 46.85 14.77
0.129 7.0 7.0
2.5 90 11.79 0.498 20 0.135 7.4 7.8 25.16 41.82 14.58
0.140 5.9 7.8
2.5 90 13.00 0.487 30 0.119 8.3 8.6 25.50 39.11 15.07
0.113 8.4 8.7
2.5 90 12.11 0.490 30 0.118 9.9 9.4 25.27 41.07 14.72
0.118 10.3 9.3
2.5 90 11.85 0.504 40 0.096 10.8 8.8 24.82 41.23 14.85
0.103 7.3 7.7
2.5 90 12.12 0.493 40 0.112 10.1 9.1 25.27 41.06 14.79
0.116 10.7 9.1
2.5 90 12.26 0.504 60 0.086 11.7 8.5 24.9 40.29 14.99
0.075 11.2 9.7
2.5 90 12.11 0.499 60 0.100 10.0 8.6 25.00 40.73 14.92
0.101 12.1 8.4
2.5 90 51.70 0.050 10 0.168 23.6 31.9 49.12 23.02 15.02
0.158 23.1 33.8
0.169 29.5 29.9
0.140 22.8 38.5
0.147 30.1 33.9
249
-------
APPENDIX A (continued). TABULATED DATA
L/D
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
Q
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
90
F
48.51
47.71
52.31
50.95
51.80
51.33
55.02
56.70
56.91
53-52
57.66
56.61
63.96
62.95
61.02
61.22
61.3*
60.3*
R
0.0*9
0.0*9
0.096
0.10*
0.102
0.102
0.50*
O.*91
0.*96
0.492
0.*9*
0.503
0.*93
0.*99
0.*97
0.*99
0.501
0.509
X/D
20
30
10
20
30
*0
10
10
10
20
20
20
30
30
30
*0
40
40
AT0/AT0
0.128
0.112
0.091
0.1*3
0.229
0.252
0.233
0.235
0.209
0.197
0.13*
0.137
0.101
0.120
0.129
0.131
0.170
0.200
0.180
0.196
0.122
0.129
0.1*6
0.1*5
0.116
0.126
0.118
0.106
0.107
0.113
tf/D
3*.*
3*.5
33.5
*5.2*
12.0
13.6
11.9
1*.6
11.*
13.9
1*.3
17.9
15.9
15.5
17.8
21.3
6.7
6.0
5.*
6.7
9-*
8.1
7.3
9.1
9.5
10.7
7.8
10.8
12.1
13.5
I/O
*OJ3
*3.9
53.5
7*.9*
16.0
18.7
18.7
18.2
21.3
20.*
21.8
22.1
21.8
25.*
25.2
31.1
*.o
5.0
*.o
*.o
8.2
5.6
5.7
6.2
10.2
6.3
6.1
5.8
7.0
3.9
V0
50.20
*9.86
50.60
48.88
*9.*1
48.96
*9.58
50.89
50.40
51.22
50.65
*9.55
50.18
*9.8l
*9.98
*9.98
*9.90
48.85
TO
2*. 17
24.30
23.32
23.26
23.21
23.26
23.55
23.48
23.29
23.2*
23.15
23.05
21.21
21.28
21.61
21.56
21.50
21.*3
*.
1*.93
1*.85
15.11
15.19
15.23
15.30
16.76
16.72
16.68
15.20
16.61
16.52
15.57
15.55
15.52
15.*9
15.46
15.*3
250
-------
APPENDIX A (continued). TABULATED DATA
L/D 0 F R X/D ATC/AT0 W/D I/D VQ To Ta
2^590 60.14 0.509 60 0.08? 12^25.7 49.08 21.68 15.6?
2.5 90 59.32 0.518 60 0.110 11.2 7.7 48.27 21.70 15.74
2.5 90 59.77 0.512 60 0.086 10.4 7.4 43.86 21.79 15.79
2.5 90 61.43 0.499 60 0.110 9.8 6.8 50.37 21.85 15.83
251
-------
TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO.
EPA-600/3-76-101'
3. RECIPIENT'S ACCESSION>NO.
4. TITLE AND SUBTITLE
An Experimental/Analytical Investigation of Deep
Submerged Multiple Buoyant Jets
5. REPORT DATE
September 1976
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
8. PERFORMING ORGANIZATION REPORT NO.
L. U. Kannberg and L. K. Davis
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Oregon State University
Corvallis, Oregon 97330
10. PROGRAM ELEMENT NO.
1BA032
11. CONTRACT/GRANT NO.
Grant No. R-800818
12. SPONSORING AGENCY NAME AND ADDRESS
U.S. Environmental Protection Agency
Office of Research and Development
Corvallis, Environmental Research Laboratory
Corvallis, Oregon 97330
13. TYPE OF REPORT AND PERIOD COVERED
Final
14. SPONSORING AGENCY CODE
EPA-ORU
15. SUPPLEMENTARY NOTES
16.ABSTRACTTf)e resui £5- of an experimental and analytical study of deep submerged multiple
port tnennal discharges are presented. The experimental results include the measured
downstream thermal dilution, width, and center!ine trajectory of the buoyant thermal
plume from multiple port discharges consisting of a row of equally spaced discharge
ports. Independent parameters for which measurements were obtained include port
spacing, discharge Froude Number, discharge angle, and discharge to ambient velocity
ratio. Results indicate that decreasing port spacing greatly decreases thermal
dilution.
The analytical portion of this report presents a modified version of the
Hirst lumped differential plume model. It has been extensively modified to include
multiple plume effects including gradual transition of the plume profiles from simple
axisymmetric profiles to merging profiles and finally to fully merged profiles and
entrainment oased on the variable available entrainment surface of merging plumes.
The results of the tuned model agree well with abailable experimental data.
7.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.IDENTIFIERS/OPEN ENDED TERMS C. COSATI Field/Group
Thermal Pollution*, Jet Flow
Submerged Multiple Jets
20U
8. DISTRIBUTION STATEMENT
Release to Public
19. SECURITY CLASS (ThisReport)
Unclassified
21. NO. OF PAGES
266
20. SECURITY CLASS (Thispage)
Unclassified
22. PRICE
EPA Form 2220-1 (9-73)
252
ft U.S. GOVERNMENT PRINTING OFFICE: 1976— 796-31*1 12 REGION 10
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