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

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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-

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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

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                              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

-------
           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

-------
                            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-
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12.   Iwasa, Yoshiaki, and Mashio  Yatsuzuka,  Spread of  Heated  Waters
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     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
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     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,
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17.  Davis,  L.  R. , Analysis  of Multiple Cell Mechanical Draft Cool-
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     Ecological Res. Series,  EPA-660/3-75-039, June 1975.

18.  Hirst,  E.  A.  Analysis of Buoyant Jets Within the Zone of
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     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
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     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

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            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

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                                   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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