WATER POLLUTION CONTROL RESEARCH SERIES •
16130FDQ03/71
          HEATED SURFACE
   JET DISCHARGED INTO A FLOWING
         AMBIENT STREAM
 U.S. ENVIRONMENTAL PROTECTION AGENCY

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          WATffi POLLUTION CONTROL RESEARCH SIEIES
The Water Pollution Control Research Series describes the
results and progress in the control and abatement of pollution
in our Hation's waters.  They provide a central source of
information on the research, development and demonstration
activities in the Sivironraental Protection Agency, through
inhouse research and grants and contracts with Federal,
State, and local agencies, research institutions,  and
industrial organizations.

Inquiries pertaining to Water Pollution Control Research
Reports should be directed to the Chief, Publications Branch
(Water), Research Information Division, R&M,  Efavironmental
Protection Agency, Washington, B.C. 20^60.

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    HEATED SURFACE JET DISCHARGED INTO A
           FLOWING AMBIENT STREAM

                     by

                Louis H. Motz
              Barry A. Benedict

National Center for Research and Training in
   the Hydrologic and Hydraulic Aspects of
           Water Pollution Control

            Vanderbilt University
            Nashville, Tennessee
                  for the


       Environmental Protection Agency


              Grant # 16130 FDQ
                 March, 1971

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                   EPA Review Notice
This report has been reviewed by the Environmental Protection
Agency and approved for publication.  Approval does not
signify  that the contents  necessarily reflect the views  and
policies of the Environmental Protection Agency nor does
mention  of trade names or  commercial products constitute
endorsement or recommendation for use.
c by the Siipermtcihlciit of Documents, I'.S. (iovcrnmrat Printing Ollicc, Washington. ]>.(.'. 204IK - Price *1.7~>
                           11

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                                 ABSTRACT






      The temperature distribution in the water body due to a discharge of




waste heat from a thermal-electrical plant  is a function of the hydro-




dynamic variables of the discharge and the receiving water body.  The




temperature distribution can be described in terms of a surface jet dis-




charging at some initial angle to the ambient flow and being deflected




downstream by the momentum of the ambient velocity.  It is assumed that in




the vicinity of the surface jet, heat loss to the atmosphere is negligible.




It is concluded that the application of the two dimensional surface jet




model is dependent on the velocity ratio and the initial angle of discharge,




and the value of the initial Richardson number, as low as 0.22.  Both




laboratory and field data are used for verification of the model which has




been developed.  Laboratory data is used to evaluate the two needed




coefficients, a drag coefficient and an entrainment coefficient, as well




as the length of the zone of flow establishment and the angle at the end




of that zone.  The drag coefficient and characteristics of the establish-




ment zone are found to be functions of the velocity ratio  (ambient velocity/




jet velocity), while the entrainment coefficient is primarily a function




of geometry.




      This report was submitted as a portion of the work under Grant




No. 16130 FDQ between the Federal Water Quality Administration, now in




the Environmental Protection Agency, and Vanderbilt University.




                                 KEY WORDS




Cooling Water, Thermal Pollution, Jets, Turbulent  Flow, Heated Water,




Thermal Power Plants, Thermal Stratification, Diffusion, Water Temperature,




Heat Exchange, Temperature
                                   111

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                             ACKNOWLEDGMENTS






     Special thanks are extended to Mr. E. M. Polk, Jr.,  who directed the




Vanderbilt University field surveys and made available much of the field




data used in this report.




     Major thanks are also due to Dr. F. L. Parker for general technical




report guidance, to Mmes. Beverly Laird and Peggie Bush for typing the




report, and to the laboratory staff personnel for their outstanding




cooperation.




     The investigations described herein were supported by the Federal




Water Quality Administration, of the Environmental Protection Agency,




through its establishment of the National Center for Research and Training




in the Hydrologic and Hydraulic Aspects of Water Pollution Control, con-




tract number 16130 FDQ.  This support took the form of a research assis-




tantship, equipment, and computer time.  During a portion of the work, the




senior author held a traineeship grant from the National Aeronautics and




Space Administration.  Grateful acknowledgment is made for the financial




support from these groups.




     Appreciation is also extended to the Tennessee Valley Authority for




their cooperation in conducting field surveys, supervised by Mr. Polk.

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                            TABLE OF CONTENTS

                                                                      Page

ACKNOWLEDGMENTS	    v

LIST OF TABLES	    ix

LIST OF ILLUSTRATIONS	    x

Chapter

  I.  INTRODUCTION 	     1

        Effects of Heated Discharges                                    2
        Temperature Distribution                                        2
        Description of the Problem                                      5

 II.  PREVIOUS WORK	     7

        Submerged Jets                                                  7
        Surface Jets                                                   13
        Justification for Present Study                                21

III.  ANALYTICAL DEVELOPMENT  	    24

        Assumptions                                                    24
        Shape of the Surface Jet                                       25
        Conservation Equations                                         27
        Velocity and Temperature Profiles                              33
        Solution of the Equations                                      37
        Zone of Flow Establishment                                     46
        Summary of Analytical Development                              48

 IV.  LABORATORY EXPERIMENTS  	    50

        Modeling                                                       50
        Laboratory Equipment and Procedure                             53

  V.  RESULTS OF THE LABORATORY EXPERIMENTS	    60

        Relation of Circular Jet to Half-Width bo                      60
        Laboratory Measurements                                        62
        Analysis of Data                                               64
        Presentation of Results                                        75

 VI.  FIELD SURVEYS	    87

        Description of VU Surveys                                      87
                                   VI1

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                      TABLE OF CONTENTS (Continued)

Chapter                                                               Page

         Results of the Widows Creek Surveys                           90
         Results of the New Johnsonville Survey                       114
         Results of the Waukegan Survey                               126
         Comparison of Results                                        131

 VII.  DISCUSSION	133

         Results of the Laboratory and Field Investigation            133
         Possible Sources of Error                                    140
         Application                                                  142
         Usefulness of the Proposed Model                             144

VIII.  SUMMARY AND CONCLUSIONS	145

         Analytical Development                                       145
         Laboratory Experiments                                       147
         Field Surveys                                                147
         Results of the Laboratory and Field Investigation            148
         Application                                                  149
         Future Work                                                  150

Appendix

  A.  LABORATORY LATERAL TEMPERATURE MEASUREMENTS, T'/TO 	   154

  B.  LOCATION OF LABORATORY TRAJECTORIES	159

  C.  FIGURES 46-73:  TRAJECTORIES AND TEMPERATURE AND WIDTH PLOTS
        FOR  LABORATORY EXPERIMENTS  	   164

  D.  PRACTICAL APPLICATION EXAMPLE.  	   195

  E.  LIST OF SYMBOLS	200

LIST OF REFERENCES	203
                                   vin

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                             LIST OF TABLES




Table



-  1.  Laboratory Measurements
   2.  Laboratory Results	   ™




   3.  Results of Statistical Tests on Parameters	   84




   4.  Relation of E to 6^	   86




   5.  Vertical Profile Statistics, VU Survey Number  1  	  101




   6.  Data, VU Survey Number 1	HO




   7.  Parameters, VU Survey Number  1	HI




   8.  Data, VU Survey Number 2 and  TVA  Survey	114




   9.  Parameters, VU Survey Number  2 and TVA Survey	115




   10.  Velocity  Data, New  Johnsonville Survey	120




   11.   Temperature Data, New Johnsonville Survey 	  122




   12.   Parameters,  New  Johnsonville  Survey	123




   13.   Data,  Waukegan Survey  	  129




   14.   Parameters,  Waukegan Survey 	   129




   15.   Summary of Field Results	132




   16.   Assumed Values  for  Design Problem 	   i94
                                     IX

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                          LIST OF ILLUSTRATIONS

Figure                                                                Page

   1.  Vertical Entrainment versus Richardson Number 	    15

   2.  Temperature Rise Along Jet Axis at Water Surface	    17

   3.  Temperature Concentration Along Jet Axis at Water Surface .  .    19

   4.  Definition Sketch  	    28

   5.  Volume Flux and Momentum Flux versus Non-Dimensional  Jet
       Axis Distance	    43

   6.  Temperature, Velocity, and Width Ratios versus
       Non-Dimensional Jet Axis Distance 	    44

   7.  Effect of Reduced  Drag Coefficient on Location of Jet
       Trajectory	    45

   8.  Zone of Flow Establishment	    47

   9.  Jets and Temperature Probes	    56

   10.  Surface Jet and Rotameter	    56

   11.  Temperature Probes and Other Laboratory Equipment 	    57

   12.  Observed Temperature Distribution and Fitted Gaussian Curve,
       Run 3-60-1	    66

   13.  Temperature Distribution  at  s'/bo = 35.3, Run 3-60-1	    67

   14.  Vertical Temperature Profiles,  Run  1-60  	    68

   15.  Two-Dimensional Surface Jet, Run  3-90	    69

   16.  Observed and Fitted Trajectories, Run  1-90	    76

   17.  Observed Values and Fitted Curves  for  Temperature and Width,
       Run 1-90	   77

   18.  Observed Values and Fitted Curve  for  Length of  Establishment
       Zone versus Velocity Ratio	   80

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                    LIST OF ILLUSTRATIONS (Continued)

Figure                                                                Page

   19.  Observed Values and Fitted Curve for Initial Angle versus
        Velocity Ratio 	   81

   20.  Observed Values of Reduced Drag Coefficient versus Velocity
        Ratio	   82

   21.  Observed Values and Fitted Curve for Drag Coefficient
        versus Velocity Ratio	   83

   22.  Observed Values and Fitted Curves for Drag Coefficient
        versus Velocity Ratio and Discharge Angle	   85

   23.  Temperature Distribution, °F, at 1.0-foot Depth, Widows
        Creek, VU1	   92

   24.  Temperature Rise Along Jet Axis at Cross-Sections R-l to
        R-3	   93

   25.  Temperature Rise Along Jet Axis at Cross-Sections R-4 to
        R-7	   94

   26.  Temperature Distribution, °F, in Cross-Section R-l  	   95

   27.  Temperature Distribution, °F, in Cross-Section R-2  	   96

   28.  Temperature Distribution, °F, in Cross-Section R-5  	   97

   29.  Velocity Profiles at Cross-Section R-l  	   99

   30.  Velocity Profiles at Cross-Section R-2  	  106

   31.  Observed and Fitted Trajectories, Widows Creek, VU1	  108

   32.  Observed Values and Fitted Curves for Temperature and Width,
        Widows Creek, VU1	  109

   33.  Temperature Distribution, °F, at 1.0-Foot Depth, Widows
        Creek, VU2	112

   34.  Temperature Distribution, °F, at 0.5-Foot Depth, Widows
        Creek, TVA	113

   35.  Observed and Fitted Trajectories, Widows Creek, VU2	  116

   36.  Observed Values and Fitted Curves for Temperature and Width,
        Widows Creek, VU2	  117
                                     XI

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                    LIST OF ILLUSTRATIONS (Continued)

Figure                                                                Page

   37.  Observed and Fitted Trajectories, Widows Creek,  TVA	  118

   38.  Observed Values and Fitted Curves for Temperature and
        Width, Widows Creek, TVA  	  119

   39.  Observed Temperature Distribution, °F, at 1.0-Foot Depth,
        New Johnsonville	121

   40.  Observed and Fitted Trajectories, New Johnsonville 	  124

   41.  Observed Values and Fitted Curve for Temperature, New
        Johnsonville  	  125

   42.  Temperature Distribution, °F, at 1.0-Foot Depth, Waukegan.  .  127

   43.  Observed Values and Fitted Curves for Temperature and
        Width, Waukegan	130

   44.  Observed Field Values  of  Drag Coefficient versus Velocity
        Ratio Plotted on  Laboratory  Curve	137

   45.  Observed Values of Drag Coefficient versus Reynolds Number  .  139

   46.  Observed and Fitted Trajectories, Run 2-90	165

   47.  Observed Values and Fitted Curves for Temperature and
        Width, Run 2-90	166

   48.  Observed and Fitted Trajectories, Run 3-90	167

   49.  Observed Values and Fitted Curves for Temperature and Width,
        Run  3-90	168
   50.   Observed and Fitted Trajectories,  Run 4-90
                                                                      169
   51.  Observed  Values  and Fitted Curves for Temperature and Width,
        Run  4-90	170

   52.  Observed  and  Fitted Trajectories, Run 5-90	171

   53.  Observed  Values  and Fitted Curves for Temperature and Width,
        Run  5-90	I72

   54.  Observed  and  Fitted Trajectories, Run 1-60	173

   55.  Observed  Values  and Fitted Curves for Temperature and Width,
        Run  1-60	'	174
                                      XII

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                    LIST OF ILLUSTRATIONS (Continued)

Figure                                                                Fa§e

   56.  Observed and Fitted Trajectories, Run 2-60	175

   57.  Observed Values and Fitted Curves for Temperature and Width,
        Run 2-60	176

   58.  Observed and Fitted Trajectories, Run 3-60	177

   59.  Observed Values and Fitted Curve for Temperature and Width,
        Run 3-60	178

   60.  Observed and Fitted Trajectories, Run 4-60	179

   61   Observed Values and Fitted Curves for Temperature and Width,
        Run 4-60	18°

   62.  Observed and Fitted Trajectories, Run 5-60	181

   63.  Observed Values and Fitted Curves for Temperature and Width,
        Run 5-60	182

   64.  Observed and Fitted Trajectories, Run  1-45	183

   65.  Observed Values  and Fitted  Curves for  Temperature  and Width,
         Run  1-45	184

    66.   Observed and Fitted Trajectories, Run  2-45	18S

    67   Observed Values  and Fitted  Curves for  Temperature  and Width,
         Run  2-45	186

   68.  Observed and Fitted Trajectories, Run  3-45  	

   69   Observed Values  and Fitted  Curves for  Temperature  and Width,
         Run  3-45	188
                                                                       189
    70.   Observed and Fitted Trajectories, Run  4-45 	

    71.   Observed  Values  and  Fitted  Curves  for  Temperature  and  Width,
         Run  4-45  	
                                                                       191
    72.   Observed and Fitted Trajectories,  Run 5-45  	

    73.   Observed Values and Fitted Curves for Temperature and Width,
         Run 5-45 	

    74.   Predicted Values of Temperature and Width	197

    75.   Predicted Trajectory and Width  	   198

                                    xiii

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                               CHAPTER I









                             INTRODUCTION







      The determination of the temperature distribution of power plant




condenser cooling water discharges is of immediate interest.   Almost one




half of all the water used in the United States is utilized for cooling




and condensing by the power and manufacturing industries.   In 1964,  the




cooling water intake was 50 x 1012 gallons, of which 80% was used by the




electric power generating industry, according to the Federal Water




Pollution Control Administration (FWPCA) (23).  Presently, power is




generated by hydro- and steam-electric plants, with the latter requiring




the cooling'water for dissipation of waste heat.  Since the remaining




sites which are suitable for hydro-electric plants are limited, steam-




electric plants will have to be increasingly relied on for future needs.




Of these, fossil-fueled plants operate at about 40% thermal efficiency,




and nuclear plants operate at about 33% efficiency.  These low effi-




ciencies result in a large part of the heat produced by a steam-electric




plant being wasted to the atmosphere or into water bodies.  Based on




the projected need for electric power, heat rejection from fossil- and




nuclear-fueled plants is expected to increase almost ninefold by the




year 2000.  Thus, based on the present and future requirements for




cooling water, the effects of heated discharges should be examined.

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                      Effects of Heated Discharges




      The addition of heated discharges to a water body can have  chem-




ical, biological, and physical effects.  The chemical  and biological




effects of increased temperature on water quality, aquatic life,  and




waste assimilative capacity have been noted by several authors.   Ac-




cording to the FWPCA (23), which has defined the addition of  waste heat




as thermal pollution, the dissolved oxygen may be reduced, chemical re-




actions increased, tastes and odors made more noticeable, and the rate




of  oxygen depletion by organic wastes increased.  According  to Clark (15),




fish are particularly sensitive to changes in the thermal environment be-




cause, as cold-blooded animals, they are unable to regulate their body




temperature and can be harmed by an increase or decrease in their meta-




bolic rate.  Cairns  (11) has stated that large quantities of  heat added




to a stream will cause all but the very tolerant forms of fish and other




aquatic life to disappear and may seriously impair the stream.  Krenkel,




Thackston, and Parker (32) have cited evidence that an increase in stream




temperature due to an electric generating plant's heated discharge has




the same end result in terms of reduced waste assimilative  capacity as




adding an equivalent amount of sewage or other organic waste  to the river.




Since chemical and biological effects are a function of the  temperature




distribution, the physical effects of heated discharges should be examined






                        Temperature Distribution




      The physical effect, or the temperature distribution,  is a  function




of hydrodynamic and meteorological variables.  The temperature distribu-




tion problem can be divided into two parts which  are  analyzed almost

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separately.  According to Edinger (17) ,  the first part is the initial




mixing, or dilution, of the discharge,  and the second is atmospheric




cooling.  Edinger has stated that these two parts have different length




and time scales:  mixing takes place in the immediate vicinity of the




discharge and affects a small portion of a water body, while atmospheric




cooling is relied on after mixing takes place and affects a larger por-




tion of a water body farther downstream.






Atmospheric Cooking




      Heat transfer across the air-water interface requires a large sur-




face area  and/or long time periods.  Calculations, using standard heat




transfer rates  and based on the work of Edinger and Geyer (19), indicate




that,  for  many  water bodies, the downstream distance required for cooling




is  on  the  order of miles.  The distance is particularly  large for a




relatively narrow stream whose width does not exceed 1000 feet or so,




and, thus, whose surface area is small.  Edinger and Polk (20) have




described  the initial mixing of heated discharges and have presented




data which tend to validate the assumption that atmospheric cooling can




be  neglected in the vicinity of the discharge, at least  as a first




approximation.






Mixing




       The  temperature distribution in the immediate vicinity of the dis-




charge  can take several forms.  Brooks  (9) has noted two extremes:  sur-




face spreading  of hot water with minimal mixing, and extensive jet mixing




of  the  effluent with the receiving water.  Churchill  (14) has mentioned




that at several large Tennessee Valley Authority  (TVA) steam plants, the

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stream velocity is not sufficiently high to mix the heated discharge,




and the discharge flows out across the receiving water as a density




overflow.  Harleman and Elder  (27) have noted that thermal stratification




can develop where part of the  heated water intrudes upstream as  a  heated




layer and may be recirculated  through the power plant unless a  skimmer




wall is built at the intake channel.  Bata (5) and Harleman (26) have




analytically described the formation of the upstream stratified  layer.




Harleman  (26) has also described the design of horizontal diffusers




placed across the bottom of the stream which completely mix the  heated




discharge by entraining river  water.  Edinger and Polk (20)  have studied




the lateral and vertical mixing in a uniform current from the point of




surface discharge to a completely mixed condition downstream.   In  most




literature dealing with mixing of surface discharges, it is assumed




that the mixing, if it occurs, is due to the velocity and associated




turbulence of the receiving water body.  Also, most temperature  dis-




tribution models based on the  basic conservation of heat equation   con-




sider only a uniform velocity  field in the receiving water body.






Surface Jets




      In some cases, the spatial distribution of temperature is  a  func-




tion of the velocity of the discharge as well as the velocity of the am-




bient stream.  The velocity field of the ambient stream is no longer




uniform but is influenced by the velocity field of the cooling  water in




the immediate vicinity of the  discharge.  In these cases, cooling  water




discharged at the surface can have the characteristics of a surface jet




if the discharge flow possesses sufficient momentum.  Some work on the

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problem of surface jets has already been done by Jen,  Wiegel,  and




Mobarek (29), Zeller (48),  Carter (12), and others.   However,  this




problem needs more study in order to better analyze  and predict the




spatial distribution of temperature in the vicinity  of thermal-electric




generating plants.






                       Description of the Problem




      When waste heat from a thermal-electric power  plant is discharged




into a receiving water body, the temperature distribution in the water




body is a function of the hydrodynamic variables of  the discharge and




the receiving water body.  When the velocity field of the water body is




influenced by the velocity of the heated discharge,  and the heat is




initially advected almost perpendicularly to the river flow, then the




temperature  distribution can be described in terms of a surface jet




discharging  at  some initial angle to the ambient flow and being deflected




downstream by the momentum of the ambient velocity.   The decrease in




temperature  rise along  the jet axis  is due to the entrainment of colder




ambient water into the  jet as the jet spreads laterally and, to some




extent, vertically.




      Complete  solution of the temperature distribution problem will not




be attempted in this study.  The present investigation will consider




only the problem of surface jets.  At some distance downstream from the




discharge, the  jet velocity will have been  decreased  until it is equal




to the ambient  velocity.  The decrease in the remaining temperature excess




from this point could be determined  from the temperature distribution




models which consider only the ambient, or river, velocity and ambient

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turbulence.  However, the complexity of the two phenomena,  neither of




which is at present completely understood, precludes immediate combina-




tion of the two in a model.  At some farther distance downstream,  after




mixing takes place, surface exchange of heat becomes important,  but it




is assumed that, in the vicinity of the surface jet, heat  loss to  the




atmosphere is negligible.

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                               CHAPTER II









                             PREVIOUS WORK







      A literature review by Krenkel  and Parker (31)  describing  the many




aspects of the mechanisms and modeling of heated discharges  is readily




available and will not be duplicated  here.   Instead,  the previous  work




done on submerged and surface jets will be examined in detail to provide




a basis for the development of an analytical model  describing surface




jet discharges.







                             Subm erged Jets




      Schlichting (41) has presented solutions for submerged plane and




axi-symmetric jets.   These solutions  require assumptions based on  the




mixing length theories of Prandtl, von Karman, and Taylor.




      Abramovich  (2) has discussed solutions for simple jets and for jets




in a parallel ambient stream.  For jets deflected by a cross-flow, ref-




erences are made to empirical relations, to superimposing the  stream




functions of the jet and the external flow, and to a method  which




balances the force caused by the pressure difference at the  forward and




back surface of the jet by a centrifugal force.




      Albertson, Dai, Jensen, and Rouse (4) have presented  a model which




describes the behavior of a simple air-in-air jet.   Analytic expressions




for the distributions of velocity, volume flux, and energy  flux  were




developed for the patterns of mean flow within submerged jets  from both

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slots and orifices.  A single  experimentally-determined coefficient




was evaluated.






Entrainment




      Taylor  (44) made a  simple  transfer  assumption of entrainment which




relates  the inflow  into the  edge of  a plume to a characteristic velocity




in the plume.   This assumption retains  the broad outline of the mechanics




of a rising, buoyant plume without the  necessity for understanding in




detail how the  turbulent  eddies  mix  the heated and the ambient air.  This




concept  of entrainment has made  possible  the solution to many problems




describing buoyant  and momentum  jets which are difficult,  if not impossible,




at present, to  solve by other  means.






Entrainment of  a  Buoyant  Plume




      Morton, Taylor, and Turner (34) used the concept of entrainment to




develop  relations predicting the behavior of a buoyant plume rising




through  a fluid with a linear  density gradient.  Three main assumptions




were made in deriving the prediction equations :




      (1)  The  profiles of vertical  velocity and buoyancy are similar




at all heights  and  are Gaussian,  or, as shown in Equations 1 and 2,




                             u(x,r)  = u(x) exp(-r2/b2)               (1)




and




                 g[p  - p(x,r)]         g[p  - p(x)]
                                                      exp (-rW)   (2)
where   1 u(x,r) = the vertical velocity;




            u(x) = the centerline velocity;
        Dotation used in this chapter is unique to this chapter.

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            x,r = the cylindrical  coordinates with the x axis vertical;



              b = the width of the plume;



             p  = the ambient density;



         p(x,r) = the plume density;



           p (x) = the centerline plume density; and



             Pl = a reference density.



      (2)  The rate of entrainment of fluid at any height is proportional



to a characteristic velocity at that height, or, as shown in Equation 3,




                                 -~^- - 2-rrbau                           (3)
                                 dx



where Q  = the  plume  flowrate; and a = the experimentally-determined en-



trainment coefficient.



      (3)  The fluids are  incompressible, and  local variations  in density



are small compared  to a  reference density.



      Based on the  conservation of volume,  momentum,  and heat  energy, a



system of ordinary  differential equations was  developed.  The  conservation



equations were written as  shown in Equations  4-6,



      Volume:




                             ^-  (>b2u)  =  2irbau                        (4)
                             dx    ,


      Momentum:




                         ~  (irb2u2)  =  2TTb2g(p0 -  P)                    (5)



      Density Deficiency:




                    ^—  [Trb2u(pi  - p)]  = 2irbau(p1  - p0)                 (6)



Morton  et  al. (34)  wrote the heat energy  equation in terms  of the density



difference instead of the temperature rise for a more unified treatment



of all  types of convection problems.   The temperature field is expressed

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                                                                     10
indirectly  in terms of the  equivalent density deficiency.






Jets, Plumes, and Wakes




      Morton  (33) showed  that jets, plumes, and wakes could be related by




means of  a  momentum-mass  flux diagram.  Using a simple model based on the




concept of  entrainment, the relation was found from the solution of a




single, ordinary differential equation.




      Morton's  treatment  was based on a common set of assumptions applied




to  jets,  plumes, and wakes.  Mean cross-sectional profiles of velocity




were assumed  similar along  the  axis.  Longitudinal dispersion was assumed




negligible  compared to lateral  dispersion, making possible the usual




boundary  layer  assumptions  for  free  turbulent shear flows.  The flow was




assumed to  be affected by density differences only in the form of




buoyancy  forces.  Entrainment,  or turbulent mixing of the jet and the




ambient fluid,  was represented  by an inflow velocity across the jet




boundary, and this inflow was assumed to be proportional to the dif-




ference between a characteristic velocity along the axis and the velocity




of  the ambient  fluid.




      Equations representing the conservation of mass, momentum, and den-




sity deficiency, similar  to those used by Morton et_ al. (34), were written.




The non-dimensional variables,  volume flux and momentum flux, and the den-




sity deficiency were defined, and then solutions representing a simple jet,




a jet in a  uniform current,  a buoyant jet, a simple plume in a stratified




environment, a  buoyant jet  projected along a uniform stream, a simple wake,




a forced wake,  and a buoyant forced wake were presented in which the mo-




mentum flux, M, was a function  of the volume flux, V.  Morton (33) called




this M vs.  V curve a momentum-mass diagram.

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                                                                      11
Jet Trajectory



      Bosanquet, Horn, and Thring (7)  studied a negatively buoyant,  sub-



merged jet flowing into a lighter, non-flowing ambient fluid.  The jet



trajectory was predicted in terms of the initial angle of the discharge,



the initial velocity, the density ratio, and the nozzle diameter.  The



predicted axis was shown to compare favorably with values obtained from



the water-in-water model.





Cross Flows



      Keffer and Baines  (30) studied the flow of a vertical air-in-air



jet directed normally to a uniform, steady ambient current for velocity



ratios  (jet/ambient velocity) of  2, 4, 6, 8, and 10.  The integrated
                                                   X.

equations of continuity  and motion along the deflected jet axis were made



non-dimensional after the general method of Morton  (33).  Entrainment



was defined in  terms of  an  inflow velocity, v., which was assumed pro-



portional to the difference between the centerline jet velocity, U, and



the ambient velocity, U  .  The entrainment relation was written as shown
                       a
in Equation 7,
                                v  = E(U -U )                         (7)
                                 J.         d
      Keffer and Baines observed that the ambient flow was decelerated



at the upstream surface of the jet, creating a positive pressure region,



and that separation occurred at the rear, creating a negative pressure



region.



      They also studied the effects of the velocity ratio on the zone of



flow establishment, which is defined as the region in which the velocity



distribution changes from a uniform distribution at the jet nozzle to a

-------
                                                                     12
fully developed Gaussian distribution at the beginning of the zone of




established  jet flow.  They observed that at low velocity ratios (2 and 4)




the  establishment  zone was deflected downstream by the pressure field,




and  the  beginning  of the established flow region was at an angle less




than 90°  to  the ambient current.  At these  low ratios, the jet was ob-




served to cling to the wall,  and  entrainment may have been restricted by




the  proximity  of a solid surface.   For velocity ratios greater than 4,




the  beginning  of the established  flow region was observed to be approx-




imately  above  the  center of the jet orifice.  For these larger ratios,




the  effect of  the  pressure field  was mainly to change the cross-section




from a circular shape at the  orifice to a distorted kidney shape at the




end  of the zone of flow establishment.  In all of the cases studied, a




pair of vortices was observed along the jet axis.  Keffer and Baines




felt that the  vortices were caused  by the separation of the ambient flow.




      Keffer and Baines observed  that the velocity excess for the jet in




an ambient current decreased  much more rapidly than that reported by




Albertson _et_ al. (4) for a simple jet where the ambient velocity is zero,




and  that  the rate  of decrease increased with distance from the source.




They felt that the entrainment was  augmented by the twin vortices, which




do not exist in the free jet  where  the ambient velocity is zero.  Also,




they observed  that the lateral spreading did not appear to be affected




by the vortices.   They reported that the entrainment coefficient, E,




which was  determined by fitting the observed velocities measured with a




hot-wire  anemometer to the predicted values, varied along the jet axis




from 0.3  to  1.6 and also varied as  a function of the velocity ratio.

-------
                                                                     13
Buoyant Jet in a Cross  Flow




      Fan (22) used Morton's work (33)  as a basis for studying an in-




clined, round buoyant jet in a stagnant environment with linear density-




stratification and for  studying a round buoyant jet in a uniform cross




stream of homogeneous density.  Extending the integral technique of




analysis, for the case  of a uniform cross stream, Fan included the effect




of an initial angle of discharge at the end of the zone of flow establish-




ment that was not 90° to the ambient flow,  presented empirical relations




describing the zone of flow establishment,  assumed an entrainment




mechanism based on the vector difference between a characteristic jet




velocity and  the ambient velocity, and approximated the effect of the




pressure gradient across the jet parallel to ambient current by a drag




coefficient.  From laboratory experiments in which the jet Froude Number




was varied from 10 to 80 and  the velocity ratio  (jet/current) varied




from 4 to 16, the entrainment coefficient and the drag coefficient were




found  to vary from 0.4 to 0.5 and from 1.7 to 0.1, respectively.  Fan




determined the entrainment coefficient from the  jet centerline dilution




ratios and noted that the cross-sectional concentration profiles were




horse-shoe shaped with the maximum concentration at two sides of the




plane  of symmetry.  The maximum concentrations were from 1.60 to 1.80




times  greater than the values at the jet axis, or centerline.






                              Surface Jets




       Jets discharged at the  surface of  a heavier fluid have been studied




by several authors.  The jet  fluid, usually water, was the same as the




ambient fluid except that its density was less.  This was accomplished

-------
                                                                     14
by heating the jet fluid or by increasing the density of the ambient




water by  adding salt.






Vertical  Entrainment and the Richardson Number




      Ellison and Turner  (21) studied the entrainment process in strati-




fied flows and found that vertical entrainment could be expressed as a




function  of  the Richardson Number.  In the case of a surface jet with




the ambient  velocity zero, the inflow velocity into the turbulent jet




region  was assumed to be proportional to the velocity of the surface jet.




The constant of proportionality was called the entrainment coefficient,  E.




      Ellison and Turner performed a series of laboratory experiments




using a salt solution to increase the density of the ambient fluid.   With




no density difference,  the flow formed a simple, two-dimensional half-jet,




the depth of which increased nearly linearly with distance.   When the




ambient fluid was heavier than the jet fluid, the rate of increase of




depth became smaller and at some distance downstream, depending on the




density difference and  the rate of flow, the jet depth changed very-




little  with  distance.   These experiments also demonstrated that, in most




practical cases, the surface layer will attain an equilibrium state in




which the Ri Number does not vary with distance downstream.   The observed




relation between E and  the Ri Number is shown in Figure 1.






Small Richardson Numbers




      Jen, Wiegel, and  Mobarek (29) performed laboratory studies on




the mixing of heated, buoyant jets discharging horizontally at the




surface of a large body of initially quiescent receiving water.  The

-------
                                                            15
FIGURE 1.--VERTICAL ENTRAINMENT VERSUS RICHARDSON NUMBER
            [After Ellison and Turner (21)]

-------
                                                                        16
initial Ri Number of the jets was defined as shown in Equation 8,




                                 Apgd'

                          Ri  =  - °                               (8)
where Ap = the density difference;




       g = the acceleration of gravity;




       p = the ambient density;




      d  = the jet diameter; and



      U  = the initial jet velocity.




In these studies, the Ri  Number was small, i.e., 5.0 x 10~5 <  Ri  <
                        o                                    —   o —



3.0 x 10~3.



      Jen jrt &\_. observed that the jet mixed less with depth than it did




laterally by a factor of about two.  They attributed this difference to




the buoyancy force, which they felt  inhibits the vertical component of




the turbulent velocity fluctuations  but does not inhibit the lateral




component.  An empirically-determined  equation for the temperature along




the jet axis was presented.






Larger Richardson Numbers




      Taraai, Wiegel, and Tornberg  (43) studied the same type of buoyant




jets as Jen et_ al_.  (29) except that  the Ri  Number was larger, i.e.,




1.0 x 10" 2 _<_ Ri  ^  2.0 x 10" l.  A comparison of  the data taken at these




higher Richardson Numbers with the empirical equation of Jen _e_t al. ,




plotted in Figure 2, showed that the temperature rise along the jet




centerline was generally greater than  predicted  for values of Ri




approaching 2 x lO'1.

-------
     1.0
     0.5
T/TO
     o.i
      .05
      .01
TJ
                                       o  o
                                          A   A
                 A
                 O
       0.16
       0.09
       0.15
       0.008
                                                         BEST FIT LINE
                                                          JEN, WIEGEL, & MOBAREK (29)
TAMAI, WIEGEL, & TORNBERG (43)
      RU
                                   10
                                            x/d
                                     50
100
500     1000
                                              o
                   FIGURE 2.--TEMPERATURE RISE ALONG JET AXIS AT WATER SURFACE
                             [After Tamai, Wiegel, and Tornberg (43)]

-------
                                                                     18
Approximate Theory




      Hayashi and Shuto (28) also studied the diffusion of warm water jets




discharged horizontally at the surface of an initially quiescent water




body.  The initial temperature was from 0.2°C to 28°C higher than the




receiving water, and the Ri  Number of the jets varied from 0.004 to 0.54.




An approximate theory was developed in which the non-linear inertial




terms representing the square of the jet velocity in the equation of




motion were neglected, and an approximate solution was presented for the




case of no vertical entrainment.  The experimental results were compared




with the approximate theory, and the effect of the Richardson Number on




vertical entrainment was in agreement with the work of Ellison and




Turner  (21).  Figure 3 shows a comparison between the approximate theory




and one set of experimental data of Hayashi and Shuto.






Similarity Criteria




      Stefan  (42), using the dimensionless equation of motion, continuity,




and heat transfer, presented similarity criteria for the flow of heated




water over a  stagnant and colder body of water.  He felt that turbulent




flow in the model, matching of reduced gravitational forces and surface




heat losses as well as sufficient size of the model were the most es-




sential modeling requirements.






Integral Technique




      Zeller  (48) developed a two-dimensional mathematical model based




on the work of Morton (33) and Fan (22) to describe the observed tempera-




ture distribution offshore from a steam-electric generating plant which




discharged heated water through two outfalls 640 feet apart.  According

-------
     1.0
    0.5
T/T,
   o
              o  o
     «   u
SS88
                             THEORETICAL CURVE FOR NO
                              VERTICAL ENTRAPMENT
         —   O
               8
                   o
    0.2
    o.i
          0
       1000
2000
3000
4000
5000
6000
7000
            FIGURE  3.--TEMPERATURE CONCENTRATION ALONG JET AXIS AT WATER SURFACE

                              [After Hayashi and Shuto (28)]

-------
                                                                     20
to Zeller, field surveys indicated that, beyond a region close  to  shore,




the warm water from each outfall spread as a two-dimensional  surface  jet




about 3 feet thick and could be followed for several thousand feet out




into the  lake.  Measurements of the temperature distribution  were  made




by traversing the jets in a boat, and jet velocities were determined  by




tracking  the paths of drogues.




      Zeller developed a system of two-dimensional mass,  momentum, and




heat energy conservation equations.  Based on the work of Ellison  and




Turner  (21), vertical entrainment was assumed negligible because of the




stability of the surface jet at high Richardson Numbers.   Lateral  en-




trainment was assumed to be represented by the expression shown in




Equation  9,





                              |J  =  EzU                             (9)





which equates the increase in flow rate along the jet axis to the  product




of the  entrainment coefficient, E, the depth, z, and the jet  velocity,  U.




Zeller  reported that the value of the entrainment coefficient varied from




0.127 to  0.993 for 22 field surveys.






Slot Jet




      Carter (12) developed a mathematical model to describe  the be-




havior of a two-dimensional slot jet discharged perpendicularly into a




flowing ambient stream.  Equations representing the change in momentum




along the jet axis were developed, and the decrease in temperature rise




along the axis was described in terms of an  empirically-measured dilution.




The solution to the equations predicted the  location of the jet trajectory




in terms of empirically-determined coefficients and the dilution.   The

-------
                                                                     21
zone of flow establishment and the pressure gradient that exists across




the jet parallel to the ambient flow were both considered in the mathe-




matical model.






                    Justification for Present Study




      A survey of the literature indicates that further study on surface




jets is justified.  Previous work does not contain a completely suitable




method for analyzing and predicting the spatial temperature distribution




in the vicinity of power plants located on rivers when the discharge




velocity is important.






Laboratory Studies




      The results of laboratory studies by Jen, et al.  (29), and Tamai,




et al.  (43) are applicable primarily to jets discharging into ambient




water bodies that have no appreciable velocity.  The effect of a cross




flow on a submerged jet has been discussed by Keffer and Baines (30),




and this effect must be considered in the present case of a surface jet.






Theory




      None of the theoretical models examined is completely satisfactory.




The model of Hayashi and Shuto  (28) neglects the inertial terms in the




equation of motion, which is permissible only for velocities much less




than those normally encountered at a power plant discharge.  Also, their




model is applicable only to a quiescent water body.




      Zeller's  (48) definition of entrainment, shown in Equation 9,






                              §  -  EzU                             (9)

-------
                                                                        22
is different from that used by Morton  (33), Fan  (22), and Keffer and
Baines  (30) , all of whom have defined  the  entrainment in terms of the
difference between some characteristic jet velocity  and the velocity of
the moving  ambient current.  Also,  Zeller's model does not consider the
effect  of the  pressure gradient  that exists across the surface jet
parallel to the ambient current  in  predicting the trajectory of the jet,
nor does it consider  the  zone of flow  establishment.
      The theoretical model developed  by Carter  (12) differs from others
in several  respects.  The  decrease  in  temperature along the jet axis is
described in terms of an  empirically-measured dilution, but the entrain-
ment mechanism causing the dilution is not specifically considered.  The
equation of volume continuity is not used, and its integrated form,
the volume  flux equation,  does not  appear.  Since the continuity equation
is not  used, in order to  integrate  the combined momentum equation which
contains two dependent variables and one independent variable, an
assumption has to be made  concerning one of the dependent variables.  In
order to graphically integrate the  momentum equation, it is assumed that
the component  of the jet velocity in the same direction as the ambient
velocity is equal to the ambient velocity  everywhere in the zone of
established jet flow.  The validity of this assumption, according to
Carter,  is "...it appears  from the  data that the pressure field, the
length of the  zone of flow establishment,  and the velocity field may
adjust themselves so that  at (x  ,y  ) [the  beginning  of the zone of
                               C G
established jet flow] the  x-component  of the jet velocity is equal to
the u   [the ambient velocity]."   In addition, the length of the zone of
     3.
flow establishment is assumed constant with respect  to the velocity

-------
                                                                       23
ratio, which differs from the results reported by Fan (22), Keffer and




Baines (30), Pratte and Baines (36), and others.  Finally, the increase




in the width of the jet cannot be predicted by the solution presented




by Carter.






Present Study




      Therefore, using the work of Morton (33), Fan (22), and Zeller (48)




as a base, a more refined model of surface jets has been developed which




is able to describe certain cases of heated power plant discharges.

-------
                              CHAPTER  III









                           ANALYTIC  DEVELOPMENT






       The  present  study,  which  is based on the Morton technique of




 integral analysis,  develops  a system of ordinary differential equations,




 which,  when solved numerically,  predicts the jet trajectory, width,




 velocity,  and  temperature distribution for the case of a two-dimen-




 sional  surface jet.






                              Assumptions






       Certain  assumptions are made  in  the development of the mathe-




 matical model:




       (1)   The jet is  assumed to be two-dimensional.




       (2)   Profiles of velocity and temperature normal to the jet




 axis  are similar along the length of the jet axis.




       (3)   The flow regime is completely turbulent, which means that




 molecular  diffusion can be neglected,  and that the flow is independent




 of the  Reynolds Number.




       (4)   Changes  in  density are small compared to a reference den-




 sity.   Thus, inertial  forces due to density gradients are negligible,




 and mass flux  terms  can be replaced by volume flux terms.  This is




 commonly called the Boussinesq  assumption.




       (5)   Turbulent mixing  into the jet can be represented by entrain-




ment, or an inflow  velocity  across  the jet boundary.






                                   24

-------
                                                                    25
      (6)   Separation of the ambient flow around the jet can be  repre-




sented by a drag force.




      (7)   Temperature losses in the region of the jet are small com-




pared to losses farther downstream and can be neglected.






                        Shape of the Surface Jet




      Observation of warm water surface jets indicates that a surface




jet may be either two- or three-dimensional.  For a jet to be two-




dimensional, only the width should increase, and the change in depth




along  the  axis  should be  zero.  For a jet to be three-dimensional,




both  the depth  and  the width  should increase along the  jet axis.




Primarily,  two  forces determine the extent  of vertical  spreading.  The




inertial force,  or  the  difference between the axial velocity of the jet




and the ambient velocity,  creates the shearing  force which  entrains




ambient fluid,  causing  the jet boundary  to  spread laterally  and




vertically.  The buoyancy force due to the  density difference between




the heated surface  jet  and the heavier ambient  fluid  opposes the  vertical




spreading  of the jet.   When the inertial forces dominate,  the jet will




spread equally in the vertical and  lateral  directions.   However,  when




the buoyancy is large,  vertical spreading is suppressed.






 Richardson Number



       Quantitatively, vertical spreading, or vertical entrainment, is




 an inverse function of the Richardson Number of the jet.   When  the




 ambient velocity is not zero, the Richardson Number can be defined as




 shown in Equation  10, or

-------
                                                                    26
                                      Apgd'


                             Ri  =  - ° -                       (10)
where   p = density of the jet;



       Ap = density difference between the jet and the ambient flow;



        g = acceleration of gravity;



       d  = the diameter of the jet;



        U = axial velocity of the jet; and



       U  = ambient velocity.
        3.


When the inertial force is large, or the Richardson Number much  less



than 1.0, vertical entrainment takes place.  However,  when the buoyancy



force is large, or the Richardson Number approximately 1.0 or greater,



vertical entrainment is small.






Field Cases



      In many field cases, the density difference due to a heated  dis-



charge is relatively large, and the square of the velocity difference is



relatively small, particularly when the initial jet velocity is  not much



greater than the ambient velocity.  Thus, the initial Richardson Number



of the jet is 1.0, or very close to 1.0.  Theory and laboratory  experi-



ments by Ellison and Turner  (21) and field observations by  Zeller  (48)



indicate that the Richardson Number along the jet axis changes rapidly



from an initial value less than 1.0 to a value equal to or  greater than



1.0,  and that vertical entrainment decreases rapidly almost  to  zero




along the axis.



      Thus, if the initial Richardson Number of the jet is  1.0,  or close



to 1.0, vertical entrainment can be neglected, and the decrease in

-------
                                                                     27
temperature along the jet axis can he approximately described in terms




of a two-dimensional surface jet deflected downstream by the ambient




velocity.






                         Conservation Equations




      The differential equations for conservation of volume flux, momen-




tum flux,  and heat energy can be integrated over the cross-sectional area




of a two-dimensional momentum jet to obtain integral equations based on




the method of Morton  (33).  A definition sketch is shown in Figure 4.






Conservation of Volume Flux




      Integrating the volume continuity equation over the cross-section




gives the relation between the rate of change of volume flux along the




jet axis and the flow entrained into the jet across the outer edge as




shown in Equation 11,
                                    v.dC                             (11)
where   u = the jet velocity directed along the s axis;




       v. = the inflow velocity;




       dA = the differential area; and




        C = the circumference through which entrainment takes place.




      The inflow, or entrainment, velocity is assumed in this study to




be proportional to the magnitude of the difference between a characteris-




tic velocity along the jet and the parallel component of the ambient




velocity.  If this entrainment is expressed as an equality, then the




inflow velocity can be written as shown in Equation 12,

-------
                                                           28
7
               FIGURE 4.--DEFINITION SKETCH

-------
                                                                     29
                     v.   =  E(U - U cos (3)                            (12)
                      1             3.



where     U = the centerline velocity;



    U cos B = the parallel component of the ambient velocity, U ;
     a                                                         a


          8 = the angle  between the jet and the ambient current; and



          E = the experimentally-determined entrainment coefficient.




      Substituting this  expression for entrainment, Equation 12, into



the continuity relation, Equation 11, gives the integral form of the



volume continuity equation, Equation 13,




                d
                     j u dA  =  CE(U - U^cos 3)                       (13)


                    A



      This definition of entrainment is different from the definition



used by  Zeller  (48), who assumed that entrainment was related to only



the jet  velocity, as shown in Equation 9,





                         |f  =  EzU                                  (9)




      Zeller's  definition is really applicable only when the jet is at



90° to the ambient flow, or when the jet velocity is an order of magni-



tude greater than the ambient velocity.  Realistically, as the jet is



deflected, the  parallel component of the ambient velocity becomes more



and more important.  When the jet and ambient velocities are nearly



equal and parallel, mixing due to entrainment becomes quite small, be-



cause the velocity field is nearly uniform.   Zeller's model does not



take this into  account.

-------
                                                                     30
Conservation of Momentum



      The momentum equation along the jet axis can be integrated over



the cross-section to give the relation between the rate of change of



jet momentum flux, the rate of entrainment of ambient momentum flux,



and the force exerted by the pressure gradient across the jet.   For



convenience, the resulting momentum equation is resolved into its



longitudinal and lateral components along the x and y axes,  respectively.



The pressure gradient, due to the separation of the ambient  flow, is



assumed to be represented by a drag force.



      In the x-direction, the rate of change of momentum flux is equal



to the rate of entrainment of ambient momentum flux and to the x-com-



ponent of the drag term, or as in Equation 14,
^     U2dA
                       cos
=  CE(U - U cos B)  U  + F  sin &        (14)
           3.       3-D
where the drag term, F , is assumed to be related to the ambient velocity,
U , as shown in Equation  15,
 a
                                CnU2  z sin
and where C  is defined as the  experimentally-determined drag coefficient,



      In the y-direction, since all the ambient flow is oriented along



the x axis, the rate of change  of momentum flux decreases due to the



y-component of the drag term, or as in Equation 16,




                                                                     (16)



      To compare the drag term  used in the present study with the drag



terms used by Carter (12) for a two-dimensional slot jet and by Fan (22)
d
ds
u2dA sin 6
= -FD cos

-------
                                                                     31
for a submerged ax i- symmetric jet,  Equation 15 can be written in terms



of the drag force normal to the jet axis as shown in Equation 17,



                          Cp U2(zds)  sin B

                       =     a a -                         (17)
      Carter defined the drag force,  dP,  normal to the jet axis in terms



of the ambient density, a drag coefficient,  the ambient velocity,  and



the projected area of the jet as shown in Equation 18,



                          C p U2zds

                   dp  =   U a a -                                 (18)
Upon examining the definition sketch, Figure 4, it can be seen that at



distances downstream, where 3 approaches zero, the y-component of



Equation 18, or dP cos 3, would continue to deflect the jet by con-



tinuing to change the y- mom en turn flux.  Thus, the trajectory of the jet



would be an ellipse.  This is not realistic, since all the ambient



momentum is in the x-direction, and, as 3 approaches zero, all the jet



momentum is also in the x-direction.  Thus, the y-momentum of the jet



could not continue to change, and the jet should remain parallel to the



ambient flow.



      In comparison, the y-component of Equation 17, or (p F ds) cos 3,
                                                          a L)


becomes zero as the jet becomes parallel to the ambient flow, because



the projected area normal to the ambient current, or (zds sin 6) in



Equation 17, goes to zero.  The addition of the sin 3 term makes the



drag force representation in this study more realistic, since the jet



remains parallel to the ambient flow as (3 goes to zero.



      Fan used a sin23 term in his formulation of the drag force



normal to the jet axis as shown in Equation 19,

-------
                                                                     32
                          Cnp U2sin2B (2/2 rds)
               (F.ds)  -
where r = radius of the axi- symmetric jet.  The U2sin2B  term  is the




square of the ambient velocity component normal to the projected area.




This formulation differs from Equation 17, in which the  component of




the projected area normal to the ambient velocity is considered.




According to Prandtl and Tietjens (35), pressure drag depends more on




the form of the body and on separation at the rear of the body than on




conditions at the front of the body.  Thus, it is felt that the sin 3




term used in the present study is adequate to describe the conditions at




the front of the jet, and that Fan's sin26 term may be an unnecessary




refinement .






Conservation of Temperature




      If the temperature can be treated as a conservative property,




which is reasonable considering the relatively small size of  the jet




surface area, then the integrated temperature equation can be written as




in Equation 20,





                        ±-  | | uT' dA j =  0                         (20)





where T  is the temperature rise at any point in the cross-section.  The




temperature rise is the difference between the jet and the  ambient




temperatures .




      Equations 13, 14, and 20 represent the general integrated  forms




of the equations of volume, momentum, and heat flux.

-------
                                                                    33
                   Velocity and Temperature Profiles






Profile Shape




      The Morton technique of integral analysis requires an assumption




regarding the lateral distribution of velocity and temperature.   Ac-




cording to Morton (33),  the result of assuming similar profiles  and the




form of the inflow velocity is to suppress analytic solution of  the




details of the lateral structure of the jet.   Thus, any reasonable pro-




file shape can be assumed in the theoretical  model.  One such shape




assumed by Morton is the laterally-averaged,  or top-hat, profile in which




the temperature and velocity have constant values across the entire width




of the jet.  However, the top-hat profile is  not the most realistic




assumption, since profiles almost Gaussian in shape in turbulent jets




have been reported by Wiegel, Mobarek, and Jen (46), Cederwall (13),  and




Samai  (40) among others.  Thus, the Gaussian profile seems to more




accurately describe the details of the lateral distribution and  is the




shape assumed in this study.




      This study also assumes that the lateral spreading of heat is the




same as the lateral spreading of momentum.  For some cases of jets, de-




tailed velocity and temperature measurements indicate that the assumption




is not completely valid.  Rouse, Yih, and Humphreys  (39) report  that the




spread of temperature is greater than the spread of velocity by  a factor




of 1.16 for the case of a vertical buoyant jet.  However, for other cases,




detailed velocity and temperature measurements are lacking.  For example,




Fan (22) assumes that the spreading ratio is unity for the case  of a jet




discharged into a flowing stream.  Since no detailed velocity measurements

-------
                                                                       34
were made  in  the present study,  the  same  assumption —that the spreading
ratio  is unity --is made here.

Continuity Equation
       If the  velocity  distribution of  the jet  is  approximated by a
Gaussian profile,  then the  velocity, u, at any point in the cross-
section can be related to the  centerline  velocity, U, by Equation 21, or

                        u   =   U exp(-n2/2o2)                         (21)

where  o = the standard deviation; and
       n = distance along the  axis perpendicular  to the s axis.
       Substituting Equation 21  into  the integral  on the left side of the
continuity equation, Equation  13,  gives Equation  22,
              +b                             -t-00
               U exp  (-n2/2a2)  zdn  =  Uz   exp  (-n2/2o2) dn        (22)
            -b                             -°°
where   b  = the jet half-width;
       dA  = zdn; and
        z  = jet depth.
The integral  on the right side  of  Equation 22  is  a definite integral
whose value is a/2rT .   Equating  this result to the right side of the
continuity  equation, Equation  13,  gives,  for the  continuity equation,
Equation 23,

                    |- (Uza/2T  ) = 2zE(U  - U   cos 6)                 (23)
                    G.S                      3-
where 2z = C,  the  circumference  through which  entrainment takes place.
      If the half-width b is defined as in Equation 24,
                               b = a/2                              (24)

-------
                                                                      35
and this relation, Equation 24, substituted into equation 23, then the

continuity equation can be written in terms of the centerline velocity

and jet width as in Equation 25,


            i-  (Ub)  =  —  (U - U  cos 8)                            (25)
            QS           j         a

after dividing both sides by z.


Momentum Equations

      The square of the velocity, U, is approximated using Equation  21

and is written  as  in Equation  26,

                    u2 = U2 exp(-n2/a2)                               (26)

      Upon substituting  this relation,  Equation  26,  into the general

form  of the x-momentum equation,  Equation 14,  and using the  relations

dA  =  zdn  and  C  =  2z; the definition of the drag  term,  Equation  15;  and

the definition  of  the half-width,  Equation 24; then the x-momentum

equation  can  be written  in terms  of the centerline  velocity  and the jet

width as  in Equation 27,
   ^- (U2b  cos  0)  = /2  —  (U - Ua cos 3)  Ua + —  -2-2	       (27)
                         VTT                      /IT

 after dividing  both sides by z.

       Similarly,  the y-momentum equation, Equation 16,  can be written in

 terms of  U  and  b  as in Equation 28,


    ,                    /=•  CnU sin B cos 6
   |- (U2b  sin  3)  =  - —  -5-2	                           (28)
   ds                  ^        2

-------
                                                                     36
Temperature Equation

      If the temperature rise  distribution  is  approximated by a Gaussian

profile, then the temperature  rise,  T' ,  in  the cross-section is related

to the centerline temperature  rise,  T, by Equation  29,


                  T*   =  T  exp (-n2/2o2)                              (29)


      Substituting Equations  21  and  29  into the general form of the

temperature equation,  Equation 20,  gives the conservation of temperature

equation in terms of U, T,  and b,  or Equation  30,


                         (UTb)   =  0                                   (30)
Summary of  the  Equations

      The integrated mass,  momentum,  and temperature rise  equations

give a system of 4  equations and 4 unknowns:

      Four  equations:

      continuity:

             ^- (Ub)   =  — (U - U  cos 6)                            (25)
             dS          v^       a

      x- component momentum:
                                                                 U2sin23
                                    ?P                      /?
                 (U2b  cos  3)   =  /2 —  (U - U  cos 6)  U  + —
                                              a         &
                                    /TT                      /IT
                                                                      (27)

      y- component momentum:

              ,                     /=•  CU2sin 3 cos g
             ^ (U2b  sin 6)   = - ^  JLS -                 (28)
                                  /if       ^

      temperature rise:


                 (UTb)  =  0                                           (30)

-------
                                                                     37
      Four unknowns :



           U,  T,  b, 6



      The geometry of the jet trajectory as shown in Figure 4 gives two



additional equations  and unknowns,



      Two geometry equations:




               ~  =   cos B                                          (31)






               &.  =  sin 6                                          (32)
               ds




      Two unknowns :



           x, y




      Thus, there are six unknowns--U, T, b, B, x and y--and six



equations, all functions of  the jet axis distance, s.  In addition,



there are two experimentally-determined coefficients- - C  and E.






                       Solution of the Equations





No n- Dimensional Volume Flux  and Momentum Flux



      The volume continuity  equation and the two momentum equations are



ma-1' non-dimensional by defining V as the non-dimensional volume flux



and M as the non-dimensional momentum flux as shown  in Equation 33,




                                                                     (33)
                    o o
                                  o o
      Using Equation 33, then Equations 25, 27, and 28 take the re-



spective forms:

-------
                                                                    38
      volume continuity:



              f. - S - A COS „                                      (34,


      x- component of momentum flux:


              d(M If g) = /2 A  [f - A cos 8 + A C; sin2e]          (35)


      y- component of momentum flux:


              d(M ^n g) = -/2  A (A C^ sin 3 cos 6)                 (36)


 where S  =  the non-dimensional jet axis distance;


              S = (2E//i? b )s                                       (37)


      A  =  the ratio of the ambient velocity to the initial jet velocity;


              A = Ua/Uo                                             (38)

      i
 and  CL = the reduced drag coefficient;

               p = CD/4E                                            (39)
Geometry Equations


      The geometry equations, Equations 31  and  32, can be written in


non-dimensional form as shown in Equations  40 and 41,


              3! =  cos 3                                           (40)
                                                                    (41)


where
              X= (2E/bo)x      ;     Y=  (2E/vbo)y              (42)



Temperature Equation


      The temperature equation,  Equation 30,  can be  integrated immediately


to give Equation 43,
                              w0-=   l                           (43)

-------
                                                                     39
where U   T , and b  = the initial values of the velocity, temperature
       o   o       o


rise, and jet half-width, respectively.




Solution in Terras of S, and X and Y


      There are now five differential equations, Equations 34, 35, 36,



40,  and 41, and five unknowns, or V, M, B, X, and Y, all  functions of



the  jet axis distance, S.  By solving this set of differential equations



for  M  and V, then  the  velocity ratio, U/UQ,  the  temperature rise  ratio,



T/T  ,  and the  half-width  ratio, b/bQ,  can be solved  as  functions  of  S.



       Using the definition of  the volume flux and the momentum flux,



 or Equation 33,  the ratio U/U   is related to M and V as shown in



 Equation 44,


                      U/U   =  M/V                                    (44)
                         o
       Using the definition of V, Equation 34, and the integrated tempera-



 ture equation, Equation 43, the ratio T/TQ is related to V as shown in



 Equation 45,


                      T/T   =  1/V                                    (45)
                         o



       Using Equation 34, the ratio b/bQ is related to V and M as in



 Equation 46,


                      b/b   =  V2/M                                   (46)
                         o
       The ratios U/U  , T/T  , and b/b  are functions of S, since M and
                     O     O         \j


 V are functions of S.


       Next, by solving the  set of differential  equations for 3, then



 U/U  , T/T  , and b/b   can  also be expressed  as functions of the rectangular



 coordinates,  X and Y.  As shown in  Equation 47  and 48, these coordinates

-------
                                                                     40
can be  located by integrating Equations 40 and 41 along the jet axis,
S, or
                            S
                     X  =   f  cos  B dS                               (47)
                            o

                     Y  =     sin  B dS                               (48)
                            o

Numerical  Solution
      A numerical integration method  was used to solve the five equations
for the five  unknowns--V, M, B,  X, and Y -- as functions of S.  Then,
at each step  of  the  integration, values of U/U , T/T , and b/b  were
calculated from  the  values  of M  and V and expressed as functions of
S, and  X and  Y.
      The  system of  differential equations was solved using General
Electric's  FORTRAN subprogram RKPBX$, which is available in the time-
sharing service  program library.   This subprogram integrates a system
of first-order differential equations by the fourth-order Runge-Kutta
method.
      The five equations — the  volume flux equation, the two momentum
flux equations,  and  the two geometry  equations — were rewritten in a
form suitable for the subprogram.  Equations 34, 35, 36, 40, and 41
became, respectively,
      continuity:
                                1/2
               dV  _  (L2 + F2)              AL                       (49)
               dS  "       V         "

-------
                                                                41
x-component of momentum:
  dL
  dS
      (L2 + F2)
                          1/2
                                      AL
                                  (L2
\CF
ALD
                                         (L2 + F2)
            (50]
y-component of momentum:
                  ACDFL
                (L
                                                                 (51)
 geometry equations:
   dX
   dS
(L2 + F2)1/2
                                                       (52)
   dY
   dS
          (L
         T7T
 where   L  =  M cos  6  ;
         F  =  M  sin  B  ;

         M  =   (L2  +  F2)1/2;  and

         3  =   arctan (F/L)
            (53)



            (54)


            (55)


            (56)

            (57)

-------
                                                                    42
Example Solution



      The numerical solution was used to solve for values of V,  L,  F,



X,  and Y and then M, T/T  , U/U  , and b/b  at each increment of



dS  =  0.1 along the jet  axis, S, for given values of the coefficients A



and C*



      For example, consider a surface jet discharging into an ambient



stream at an initial angle of 6  = 60.0  .  The initial value of  the



volume flux is V  =1.0 from Equation 33.  The initial value of  L  =0.50
                o             M                                  o


from  Equation 54, and the initial value of F  = 0.86603 from Equation  55,



since the initial value of the momentum flux is M  =1.0 from Equation 33.
                                                 o


The initial values of the rectangular coordinates are X  =0.0 and



Y   =0.0.  For this example, the value of the initial jet velocity  is



assumed to be four times greater than the ambient velocity, or A =  0.25



from  Equation 38.  The  reduced drag coefficient, Cn,  is assigned the



values 0.0, 0.5, and 1.0 to illustrate its effect on the location of



the jet trajectory.



      After integrating the system of equations, the result can be  shown



graphically.  The volume flux, V, and the momentum flux, M, are plotted



as functions of the jet axis distance, S, in Figure 5.  The temperature



rise,  velocity, and width ratios -- T/T  , U/U , and b/b  -- are plotted



as functions of S in Figure 6.  The location of the jet trajectory  as a



function of the reduced drag coefficient, C*, is plotted in Figure  7.



The value of T/T  along the jet trajectory is not significantly changed



by the different values of C .  For example, at S = 100.0, the value of



the temperature rise is T/T  = 0.0597, 0.0582, and 0.0572 for the re-



spective values of Cl = 0.0, 0.5, and 1.0.

-------
FIGURE 5.--VOLUME FLUX AND MOMENTUM FLUX VERSUS NON-DIMENSIONAL JET AXIS DISTANCE

-------
0.01
                                                                                    50.0     100.
       FIGURE 6.--TEMPERATURE, VELOCITY, AND WIDTH RATIOS VERSUS NON-DIMENSIONAL JET AXIS DISTANCE

-------
  40
  30   -
Y
  20   _
  10   _
   0
                  10
80
                FIGURE  7.--EFFECT OF REDUCED DRAG COEFFICIENT ON  LOCATION OF JET TRAJECTORY

-------
                                                                     46
                       Zone of Flow Establishment



      In practical applications, the zone of flow establishment  of  the



jet, illustrated in Figure 8, must be determined.  This  zone  is  a mixing



region in which turbulent mixing changes the uniform temperature and



velocity profiles at the jet origin to fully-developed turbulent profiles



which are Gaussian in shape at the beginning of the established  flow



region.



      Previously, when the zone of establishment has been determined  for



jets in a flowing stream, it has been done by measuring  the length, sg,



and the initial angle, 3  , at the end of the zone of establishment.



Fan (22), for example, empirically related the length of the  establish-



ment zone and the initial angle, (3 , to the ratio of the ambient velocity



and the initial jet velocity, or A = U /U .  Based upon  Gordier's  data  (25),
                                      cl  O


the length of the establishment zone was found by Fan to be related to



A as shown in Equation 58,






                        ——  =  6.2 exp (-3.32 A)                   (58)

                         d1
                          o



The initial angle at the end of the zone, 8  , was found  to be related



to A as shown in Equation 59,




                         (3   =  90° - 110° A                         (59)
                          o



      Unfortunately, Fan's results in Equations  58  and 59 are not



directly applicable to the present study.  In cases where the initial



angle of the discharge is 60° or even 45°, Fan's results can not be used



directly since the initial angle, 8*, in Equation 59  is 90°.  In cases

-------
u
                                            U  =  U0  T  -  T0
U )
0
To
— >
(U
TF

1
0'
f f^&
J 0
/ \
i 7
\ ^.DO
                                                                                        X
                            FIGURE 8.--ZONE OF FLOW ESTABLISHMENT

-------
                                                                    48
of two-dimensional surface jets where the range of the velocity ratio,  A,



is anticipated to be 0. 20 <_ A <_ 0.80 or higher, Fan's results  should  not



be directly applied even when &  = 90° since his results were  developed



for a different range of A, or 0.125 <_ A <_ 0.25.  Therefore, it will  be



necessary to extend Fan's work by empirically determining how  the  length



of the zone of flow establishment, s', and how the initial angle at the



end of the zone of establishment, B , are related to the velocity  ratio,



A = U /U , for cases in the range of 0.20 <_ A <_ 0.80 and at the same
     a  o


time for cases where the initial angle of the discharge, 6 , is not



90° to the ambient flow.





                   Summary of Analytical Development



      By considering the basic volume, momentum, and heat equations,  a



model has been developed describing a two-dimensional surface  jet  de-



flected downstream by a flowing ambient current.  By using a numerical



solution, values of temperature rise, velocity, and width can  be pre-



dicted as functions of a non-dimensional jet axis distance. The location



of the jet trajectory can be predicted in terms of non-dimensional



rectangular coordinates.  The model contains two experimentally-determined



coefficients,  the entrainment coefficient and the drag coefficient.   Since



the non-dimensional coordinates S, X, and Y are defined as functions  of



the entrainment coefficient, E, in Equations 37 and 42, and since  the



location of the jet trajectory depends on the drag coefficient, C  ,  these



coefficients must be evaluated by laboratory experiments before the model



can be used to predict temperature distributions at field sites.  Also,



the zone of flow establishment of the jet must be determined from laboratory

-------
                                                                    49
experiments before the mathematical model  can be  used  in practical




situations.

-------
                               CHAPTER IV









                         LABORATORY EXPERIMENTS






      The laboratory experiments were designed to  functionally relate




the entrainment and drag coefficients and the zone of flow establishment




to the velocity ratio, A, and the initial angle  of discharge, 3  .






                                Modeling




      The complete modeling of a heated discharge  would require modeling




several phenomena:  the inertial mixing due to the jet velocity,  the




buoyancy due to the temperature difference between the jet and the




ambient current, the advection of the heat by the  ambient current,  the




turbulent diffusion in the ambient stream, and the evaporative heat loss




across the air-water interface.  According to Ackers  (3) and  other  in-




vestigators, it is impossible to accurately reproduce all the phenomena




in the same model.  Thus, it is necessary to identify the dominant  forces




present in a given situation.






Dominant Forces in a Surface Jet




      As assumed in the development of the mathematical model,  the  three




dominant forces involved in a surface jet are the inertial,  viscous,  and




buoyancy forces.  The ratio of the difference between the inertial  force




of the jet and the ambient current to the buoyancy force due to  the




temperature difference between the jet and the ambient current  is the




Richardson Number and can be written as  in Equation 10.   The ratio of






                                     50

-------
                                                                   51
the inertial force of the jet to the viscous  force is  the  jet  Reynolds



Number.   In addition, the ratio of the inertial  to the viscous forces



in the stream, or the ambient Reynolds Number, must be considered.






Jet Reynolds and Richardson Numbers



      Two of the criteria considered in modeling a surface jet are  the



equivalence of the model and prototype jet Reynolds and Richardson



Numbers.  Unfortunately, exact equivalence cannot be achieved in the




laboratory.



      Equivalence of the initial Richardson Numbers (RiQ Number) can be



achieved by reducing the velocity  and velocity differences in the model




This  is shown  in  Equation  60,

Kl -
O
Apgd^
p(U - U )2
^ o a

m
Apgd^
>
-------
                                                                     52
 where  v = kinematic viscosity.   This  is because the diameter, d', of the




 model is still much smaller than that  of the prototype, and the kinematic




 viscosity, v,  is approximately the  same in both the model and the proto-




 type.




       Thus,  approximate equivalence of the Ri  Numbers can be achieved




 only at the expense of very small jet Re  Numbers in the model.   Other




 investigators, such as Abraham (1) and Burdick and Krenkel (10), have




 assumed that,  if the model  jet flow is turbulent, as in the prototype,




 i.e., if the Re  Number is  greater than some critical value,  then the




 criterion for  exact equivalence between model and prototype jet Re




 Numbers can be relaxed.  Then, the criterion for similarity between




 model and prototype Ri  Numbers can be achieved.






 Ambient Reynolds Numbers




       The third criterion,  exact equivalence of prototype and model




 ambient Reynolds Numbers, also cannot be achieved in the laboratory.




 It  is  assumed  that  the requirements for exact equivalence can be relaxed




 if  the  flow  regime  in  the laboratory flume  is turbulent.






Model Surface  Jet




      The model  surface jet is thus an approximation of the prototype.




Equivalence of model and prototype Ri  Numbers  is chosen in preference




to equivalence of Re   Numbers.  Prototype Ri  Numbers can be  calculated




based on conditions representing  typical  field  sites, and the model Ri




Numbers should  be on the same order of magnitude.   Since prototype Re




Numbers indicate a turbulent flow regime  in the jet and in the ambient




stream, the model Re  Numbers should also indicate a turbulent flow regime.

-------
                                                                     53
                   Laboratory Equipment  and  Procedure




      The laboratory experiments  were performed in the hydraulics




laboratory of the Department  of Environmental  and Water Resources  En-




gineering at Vanderbilt University.






Flume



      The flow system used was a rectangular flume 60  feet long,  2.0 feet




wide, and 1.0 foot deep.  The flume has  glass sides and a painted  steel




bottom, and is mounted on a truss system so that the slope can easily




be varied.






Jets



      Circular jets were  selected for laboratory use for two reasons --




convenience and  the object of  the experiments.  Using circular jets re-




quired only a minimum  of  commercially-available  fittings, while using




square or rectangular  orifices would have required  extensive fabrication




of  special  shapes  and  equipment.  The object  of  the experiments --  re-




 lating the  entrainment and drag  coefficients  and the  zone of establish-




ment to  the velocity  ratio,  A,  and  the  initial angle,  $Q  -- was re-




 stricted to what were considered the significant factors  influencing the




 spatial  distribution  of temperature. Understanding these factors,  the




 influences  of A and g', more fully  was  felt to be necessary before   ^




 considering the many  possible variations of orifice shape.




       The design of the jets was based  on the desire  to maintain  con-




 stant Re Numbers in the jet  and in the  flume.  It was decided  to  main-




 tain a constant jet velocity and jet ReQ Number and vary the velocity




 ratio, A = U /U , by varying the ambient velocity, U  ,  in the  flume.
             a  o'

-------
                                                                    54
Since it was decided to use the maximum flowrate,  Q   - 0.15 cfs, this
                                                  d



meant that the flume, or ambient, Reynolds Number  could be kept constant




by inversely changing the ambient depth, z ,  with  changes in the flume




velocity, U, .   The ambient Reynolds Number can be  written as shown  in
           cl
Equation 62,
                             a
                                    v


where Re  = the ambient Re Number; and
        a


        z  = the ambient depth.
        a
                                  U z

                           Re  = _JLJL_                              (62)
Thus, for each desired value of the velocity ratio,  A  =  ua/uo>  a  dif~
 ferent value of U  was  indicated, since U  was constant.   For  each  value
                 a                       o


 of  U  , then, a different value of z  was indicated from Equation 62,
    3.                             **


 since Re  and Q  were  to be kept constant.  This resulted in the jets
        a      a


 being designed to enter the flume at different heights so that the  top



 of  the jet discharge for each jet when in use would coincide with the



 ambient water surface.



      The 1.0- inch diameter jets were built using PYC pipe and Plexiglass,



 One of the glass panels near the upstream end of the flume was removed,



 and a Plexiglass panel  with a circular opening approximately 0.7 feet in



 diameter was sealed in its place.  Into this opening could be placed a



 Plexiglass "plug" drilled to hold a vertical row of four jets mounted so



 thpt  they would be flush with the inside wall of the flume.  The jets



 were  designed so that  each one could be sealed when not in use.  Since



 it  was decided to study the effect of three different jet angle orienta-



 tions, or g' = 90°, 60°, and 45° to the ambient flow, three of these



 plugs were constructed, each containing a vertical row of jets oriented

-------
                                                                     55
at one of the angles,  3 ,  mentioned above.   Figure 9 shows  the three




plugs with the jets offset horizontally for clearance,  while Figure 10




shows one of the jets  in use with the other three sealed off.






Temperature Determination




      The temperature distribution was determined using probes small




enough to produce minimum disturbance in the flow pattern,  a manually-




operated switchbox, and a digital thermometer.  Point measurements were




made by using a single Cole-Parmer No. 8432-1 probe mounted on a traveling




point gauge  apparatus.  This probe had a length of 2 1/2 inches, and the




temperature-sensitive bulb at the tip of the probe had a diameter of




1/8  inch.  Lateral  temperature measurements were determined using




eleven Cole-Parmer  No.  8434  stainless steel probes with a length of




4 1/2 inches  and  a  diameter  of 5/32  inches.  These probes were spaced




laterally  on a movable  platform  which was  suspended over the water sur-




face.  Preliminary  investigation indicated that  the thermal sensitivity




of these probes was limited  to the  lower 1/2  inch or so of  the  tip.




Thus, when the tips were  immersed below the water surface,  a vertically-




averaged temperature  could be read  over this  depth.  A manually-operated




YSI  twelve-point  switchbox was used  to route  the output of  the  probes to




a Digitec  Model  1515  digital thermometer.   Preliminary  investigation also




indicated  that  the  time-varying  turbulent  fluctuations  of temperature




along the  centerline  of the  surface jets were sufficiently  small  to allow




average  centerline temperatures  to  be determined directly from  the read-




out scale  of the digital  thermometer,  precluding the necessity  of re-




 cording  apparatus.   Figure  11  shows  the temperature probes, the switch-




box, and the digital  thermometer.

-------
                                                   6
PIGURE 9.--JETS AND TEMPERATURE  PROBES
FIGURE 10.--SURFACE JET AND ROTAMETER

-------
         ill*
         ?:     •
FIGURE  11.--TEMPERATURE PROBES AND OTHER LABORATORY EQUIPMENT


-------
                                                                    58
Other Equipment



      Other laboratory equipment was also used in performing the ex-



periments.  A 60° V-notched weir installed at the upstream end of the



flume was used to measure the ambient flowrate, Q .   A 2.00-gpm capacity
                                                 cL


rotameter was used to measure the initial jet flowrate,  Q .   The



traveling point gauge was used to measure the flume depth,  z , which
                                                            3.


was  controlled by means of a perforated baffle installed at  the down-



stream  end of the flume.  The spatial location of the temperature probe



attached to the traveling point gauge apparatus relative to  the origin



of the  jet was determined by measuring vertically with the point gauge,



laterally by means of a scale mounted on the lateral arm of  the



traveling cart, and  longitudinally by means of a tape stretched along



the  upper edge of the flume wall.  The longitudinal  location of the



platform containing  the eleven laterally-spaced probes was determined



by means of the same tape.  The jet discharge was heated by  mixing



about 150 gallons of water in a tank into which steam could  be admitted



at a flowrate adjusted to maintain the desired elevated temperature.



The  heated water, dyed with Pontacyl Pink to allow visual observation of



the  jet, was continuously pumped into a 10.0-foot constant heat tank,



then allowed to flow through the rotameter and into the flume, entering



at the ambient water surface.






Procedure



      The procedure  followed in performing the laboratory experiments



was similar for all  the runs.  Selection of parameters such  as ambient



depth,  z ,  and the initial temperature of the heated discharge for each
        3.


run was based on preliminary calculations of the desired value of the

-------
                                                                    59
velocity ratio, A, and the desired values of the Ri  and Re Numbers.
                                                   o


Uniform ambient flow was obtained by adjusting the slope of the flume



before each run.  Steady-state conditions were insured by allowing the



jet to discharge into the ambient flow for at least 15 minutes before



data were taken.  Lateral temperature distributions were measured at



predetermined intervals starting at the downstream end of the flume.



Point temperature measurements were made to accurately locate the jet



centerline, to determine the zone of flow establishment, and to measure



vertical temperature distributions.  The initial temperature of the jet



was measured with the point probe in the mouth of the jet flush with



the inside wall of the flume.  The flume water was not recirculated,



since preliminary calculations had indicated that the ambient tempera-



ture would be  increased during the estimated time for each run by the



heated jet by  about 5°F, which was significant compared to the initial



temperature rises planned.

-------
                               CHAPTER V







                  RESULTS OF THE LABORATORY EXPERIMENTS





      The results of the laboratory experiments were analyzed  in terms



of the parameters in the mathematical model.





               Relation of Circular Jet to Half-Width b
               — - • -- — -- o


      The two-dimensional mathematical model  predicts the temperature



distribution along the jet axis in terms of the initial  jet  half-width,



b  ,  located at the end of the zone of flow establishment  as  seen in
 o


Equation 37,



                       S = (2E/A b )s                               (37)




In order to plot the observed trajectories and temperature data in



terms of non-dimensional coordinates, it was  necessary to calculate the


                                                                      i

value of b , which can be determined from the value of the diameter,  d ,



and  the value of the equivalent half-width at the origin of  the jet,  b .





Circular and Square Orifices



      According to Yevdjevich (47), the entrainment characteristics of a



square jet approximate the entrainment characteristics of a  circular jet



if the cross-sectional areas of the two orifices are equal.   If the



circular orifice diameter is d , and the square orifice  half-width is b ,



then equating the cross-sectional areas of the two shapes gives



Equation 63,



                          = 2.26 b^                                  (63)





                                    60


-------
                                                                    61
This equation is established by Yevdjevich for a vertical three-



dimensional jet but is assumed in this study to be applicable to a



horizontal surface jet as well.




             t

Relation of b  to b
	o	o


      The value of the initial jet half-width, b , at the end of the


                                                              !

zone of flow establishment can be related to the half-width, b , at the



origin by considering the conservation of heat flux between the two



cross-sections at 0' and 0 shown in the definition sketch of the zone



of  flow establishment, Figure 8.  Using the assumed Gaussian velocity



and temperature profiles, Equations 21 and 29, and integrating the con-



servation of heat flux relation, Equation 30, at 0' and 0, and then



equating the resulting heat flux at O1 to the flux at 0 gives Equation 64,




                      2 U T z b'  =  UTzb /n~//2                      [64)
                         O O O 0         O



Since the temperature rise, T  , the jet velocity, U  , and the depth, z ,



at  0' are equal to the respective centerline values of T and U and the



value of z at 0 in Equation 64, then b  can be related to b', as shown
                    M          '       o                    o'
in Equation 65,
                           b    =  1.60 b'                            (65)
                            oo                            •  J
Relation of b  to d
	 o	 o


      The two equations, Equations 63 and 65, express d' and b  as



functions of b .  Solving the  equations to eliminate b' relates the



initial half-width at the end  of the zone of flow establishment, b ,



to the diameter, d', of a circular jet as shown in Equation 66,




                           bo  =  0.708 d^                           (66)

-------
                                                                   62
In the laboratory experiments the diameter,  d^,  of  the jets was  equal



to 1.0 inch.  Therefore, from Equation 66, the value of  the half-width



was b  = 5.90 x 10-2 feet.
     o



                        Laboratory Measurements



      A complete summary of the data is presented in Appendices  A  and B.





Velocities



      The ambient velocity, U , and the initial  jet velocity, UQ,  for



each run were calculated from the measured flowrates, since no attempt



was made to directly measure the ambient velocities or the spatially-



varied velocity distribution in the jet flow field. The value of  the



velocity ratio, A = U /U  , was calculated from these values of U  and
                     a  o                                       £*


b'  .  Then the initial jet Reynolds Number was calculated using the
 o


diameter, d  , and Equation 61, and the ambient Reynolds  Number was cal-



culated using measured values of the ambient depth, z  ,  and Equation 62.



These values are presented in Table 1.





Temperature



      The initial jet Richardson Number was calculated  from Equation 60,



using the initial jet and ambient temperatures,  which  are presented in



Table 1, and standard tables for the density of  water.   These values are



presented in Table 1.





Geometry



      The values of the initial angle of  the discharge,  &',  are  also



presented in Table 1.

-------
        TABLE 1




LABORATORY MEASUREMENTS
Run
1-90
2-90
3-90
4-90
5-90
1-60
2-60
3-60
3-60-1
4-60
5-60
1-45
2-45
3-45
4-45
5-45
Ua
ft/sec
0.38
0.23
0.16
0.12
0.10
0.31
0.23
0.16
0.17
0.12
0.10
0.34
0.22
0.16
0.12
0.10
U0
ft/sec
0.52
0.52
0.52
0.52
0.52
0.52
0.52
0.52
0.52
0.51
0.52
0.52
0.52
0.52
0.52
0.52
A
Eq.38
0.73
0.44
0.30
0.23
0.20
0.67
0.44
0.30
0.32
0.23
0.19
0.66
0.42
0.30
0.23
0.18
Re0
Eq.61
5240
5240
5240
5240
5240
5240
5240
5240
5240
5240
5240
5240
5240
5240
5240
5240
Rea
Eq.62
5370
5340
5370
5300
4590
5300
5340
5.370
5400
5300
4300
5370
5300
5340
5370
4300
za
ft
0.20
0.33
0.48
0.63
0.62
0.21
0.33
0.49
0.46
0.64
0.63
0.22
0.34
0.49
0.63
0.62
Ambient
Temp.
°F
43.5
45.5
46.0
47.0
47.7
43.5
45.6
46.2
43.9
47.1
46.8
44.7
45.5
46.4
46.6
47.0
Initial
Jet Temp.
°F
111.0
106.0
91.5
84.2
70.5
101.0
106.0
93.0
93.0
83.0
74.5
101.0
107.0
92.5
84.0
73.0
BO
90.0
90.0
90.0
90.0
90.0
60.0
60.0
60.0
60.0
60.0
60.0
45.0
45.0
45.0
45.0
45.0
(RicOm
Eq.60
1.17
2.61X10'1
9.55xlO-2
6.90xlO-2
3.20xlO-2
6.32X10-1
2.65X10-1
l.OSxlO"1
8.74xlO"2
6.15xlO-2
3.88xlO-2
5. 92X10'1
2.48X10'1
1.04X10-1
6.11xlO-2
3.83xlO"2
(Ri0)p
Eq.60
4.12
3. 55x10' 1
l.llxlCT1
5,07xlO-2
3.57xlO-2
2.51
3.67X10-1
1.07X10'1
1.37X10-1
5.13xlO-2
3.04xlO-2
2.27
3.16X10'1
l.OSxlO'1
5.32x10-2
2.82xlO-2

-------
                                                                   64
                            Analysis  of Data



Reynolds Number



      The Re Numbers of the jet and ambient flow, shown in Table 1, were



all Re = 5,000, which indicates that  the Re Numbers were constant, and



that the flow regime was almost fully turbulent.





Comparison of Model and Prototype Ri



      The initial Richardson Number for each value of A in Table 1 can



be compared to a representative prototype RiQ Number calculated from



Equation 60 by assuming a 15° temperature rise  from 70°F to  85°F, a



10.0-foot discharge depth, and an ambient velocity of Ua =  1.0 ft/sec.



These prototype values were based upon anticipated conditions at several



Tennessee Valley Authority generating plants.   The model and prototype



Ri  Numbers for corresponding values of A  = U /U in Table  1 are seen
  o                                         a   o


to all be of the same order of magnitude.





Gaussian and Two-Dimensional Assumptions



      The temperature data were examined  to determine  the validity  of



the Gaussian and two-dimensional assumptions made  in  the mathematical



model.





      Gaussian Assumption.-- The assumption made in  Equation 29  that the



lateral temperature distribution perpendicular to  the jet  axis  is



Gaussian in shape was examined for a representative  run.   Surface



temperature measurements were made with the point probe at  a distance



far enough downstream from the origin of the jet so  that the cross-



section, measured laterally across the flume,  was perpendicular to the

-------
                                                                    65
axis of the jet.   The surface temperature measurements  were  plotted,  and




then a Gaussian curve was fitted to the distribution as shown  in




Figure 12.   The mean of this distribution is 0.56 feet, and  the standard




deviation of the sample is 0.27 feet.   The correlation  coefficient for




the fitted Gaussian distribution is equal to 0.90, which indicates that




the Gaussian distribution is adequate to describe the temperature pro-




file.  The bimodal distribution observed by Fan (22) for a submerged jet




was not observed  in  this study of a surface jet.






       Two-Dimensional Assumption. - - The assumption that the surface jet




 spreads  laterally much greater  than it does vertically was examined by




 measuring  cross-sectional  and vertical temperature distributions.  A




 cross-section  drawn from measurements made with the point probe is




 shown in Figure  13.   In  terms  of the 0.05 temperature  rise contour in




 Figure 13,  the jet  has spread  laterally  from  y'/bQ  = 1.25 to y'/bQ =  20.0,




 while vertically the jet has spread from z/bQ =  1.25 to  z/bQ =  2.75.




 Thus, for  this representative  cross-section,  the  rate  of lateral




 mixing is  7.3  times greater than the rate of  vertical  mixing,  which  is




 approximately  a full order of magnitude.




       Vertical temperature profiles were measured with the  point  probe




 at several points along  the jet centerline for each run.  The  repre-




 sentative profiles shown in Figure 14 indicate that vertical  entrainment




 is small,  since the jet does not spread vertically along the  jet axis.




 If vertical entrainment were significant, then the temperature profiles




 shown in Figure  14 would indicate more and more vertical mixing with




 distance along the  jet  axis.  Visual observation also  indicated the




 limited extent of vertical mixing, as seen in Figure 15.

-------
0.3
0.2  _
0.1  -
0.0
                                            y  ,  feet
             FIGURE 12.--OBSERVED TEMPERATURE  DISTRIBUTION AND FITTED GAUSSIAN CURVE
                                   AT s'/bQ  =  35.3,  RUN  3-60-1

-------
0
1   _
2   -
3   -
4   -
1	1	1	1	\
                FIGURE  13.--TEMPERATURE DISTRIBUTION AT s'/b  = 35.3, RUN 3-60-1, T/T

-------
                 1	1	1	T
    0
z   2
             i     I	I	I	L
                                                    s'/b0=17.0
        I      I	I	L
      .0    .1   .2   .3    .4    .5   .6
   .0
T/T0
.1   .2    .3    .4   .5.0    .1   .2
                       FIGURE 14.--VERTICAL TEMPERATURE PROFILES, RUN 1-60

-------
FIGURE 15.--TWO-DIMENSIONAL SURFACE  JET,  RUN  3-90
                                                                                      •

-------
                                                                    70
The vertical temperature profiles in Figure 14 also indicate that



temperature measurements made just below the water surface with  the



point probe should agree with measurements taken just below the  surface



with the larger set of probes.  The larger probes measure average



temperature values over a vertical distance of approximately z/b  =  0.5.



The reason for this agreement is that the slope of the vertical  profiles



over this depth is almost zero.  As a result, the centerline values  of



temperature reported  in this study were made with the larger probes



mounted on the movable platform and with the point probe.  These center-



line values, then, are the maximum temperatures, laterally and vertically,



in each of the cross-sections.






Geometry Effects



      The laboratory  flume was not wide enough for the jet to spread



completely unaffected by the sides of the flume.  At some point  down-




stream, the jet would intersect the sides of the flume, and the  location



of the trajectory and the width of the jet would no longer be functions



of only the relative momenta of the jet and the ambient current.  As had



been anticipated, this effect was particularly apparent for those  runs



where the value  of A - U /U  was relatively small and the angle g'  of the
                         a  o              '                      o


discharge equal to 90°, and for those runs where A was relatively  large



and 3'  equal  to 45°.
     o  M


      Temperature measurements taken with the larger probes mounted  on



the movable platform at x'/h>o = 33.8 (x'  = 2.0 feet), x'/bQ = 50.7



(x  = 3.0 feet),  and farther downstream showed a tendency for the



trajectories  to converge along the sides of the flume.  Also, the  exact

-------
                                                                    71
lateral location of the maximum temperature rise in the cross-sections




from the measured lateral  distributions was difficult to determine.  As




a result, data for the observed trajectories at x'/b  = 33.8 and beyond




were not used in locating  the jet centerline or in determining the drag




coefficient, Cn.  However, the decrease in the temperature rise due




to lateral mixing as far downstream as x'/b  = 84.5 (x  =5.0 feet)  seemed




to be relatively unaffected by the sides of the flume, and these data,




taken with  the  larger probes, were used, along with values taken from




x'/b  =  0.0 to  x'/b  = 16.9 with the point probe, in determining the




value of the entrainment  coefficient, E.






Width Measurements




      The width of the jet was determined  from measurements made with




 the  eleven  probes mounted on  the movable platform.  At  each of several




 points  along the jet, measurements were made  of  the  lateral temperature




 distribution, and then the standard  deviation of the  temperature rise




 in terms of l'/T  was  computed.  Then, values of b/bQ were  calculated
 using b   =0.71  inches  and Equation 24,
                                                                       (24)
 Determination of  the  Zone  of Flow  Establishment




 and the Entrainment  and Drag Coefficients




       Graphical methods were used  to determine the  parameters and co-




 efficients from the  laboratory data and  from numerical  solutions of the




 mathematical model.

-------
                                                                      72
      Trajectories and Temperature Plots.-- The jet trajectory  for  each



run was determined from the measured location of the maximum temperature



in each cross-section taken along the jet axis.  These data points  were



plotted relative to the jet origin in terms of x'/b  and y /b .   A  smooth



curve was fitted through the observed points on the jet centerline  by



using an n-th order polynomial fitting routine.  Next, the temperature



data were plotted on log-log plots in terms of T/T  vs. s'/b ,  where



s'/b  had been measured for each data point along the fitted y  /b  vs.
x /b  curves.





      Zone of Flow Establishment.-- The length of the zone of flow



establishment in terms of s'/b  for each run was measured by examining
                           e  o                            '


the data points on the T/T  vs. s /b  curve and interpolating to  the

                  i
largest value of s /b  where T/T  was still equal to 1.0.  This measured

          i
value of s /b  was the length of the zone of flow establishment,  or



s'/b .
 e  o


      The angle, 3 , at the end of the zone of flow establishment was



determined for each run bv using the value of s /b  from the T/T   vs.
                               0               e  o            o
 f
s /b  plot and measuring this distance along the fitted trajectory curve



from x /b  = y'/b  = 0.0.  The angle at this value of s /b  was measured,



giving the value of the angle at the beginning of the established jet



flow,  or 6 ,   The rectangular coordinates xe/b  and y /b  were also



measured at this value of s'/b .
                           e  o




      Entrainment Coefficient.-- To determine the value of the entrainment



coefficient,  numerical solutions of T/T  vs. S were computed and  plotted



on a log-log scale for each observed value of A and 3 .  A preliminary

-------
                                                                      73
value of E for each run was determined by over-laying the predicted



T/T  vs  S curve with the observed T/T  vs.  s'/b  data plot.   A set of
   o                                  o         o


corresponding values of S and s'/bQ was noted after sliding the T/TQ vs.



s'/b  plot horizontally until the best fit of the data and the theoreti-



cal curve was obtained.  Then, Equation 37 was solved for E,



                        E =  (^ S/2) / (s/bQ)                         (37)




      Next,  the value  of S* , which is the length of the establishment



zone  in  non-dimensional  form, was calculated from the observed value



of s'/b  ,  the preliminary  value of E, and Equation 67,
     e o     c



                                       s   .                           (67)
       The T/T   vs.  S  plot,  which is  referenced  to  the beginning of the



 jet flow region,  was  then redrawn as T/TQ vs.  S* ,  referenced  to the  jet



 origin,  using  the value of S  and Equation 68,
                         S'  = S'  + S                                   (68)
                               e
 A set of corresponding values of S*  and s /bQ was noted after sliding



 the T/T  vs.  s'/b  plot horizontally over the T/TQ vs.  S*  curve until



 the best fit of the data and the theoretical curve was  obtained.  Then,



 Equation 69 was solved for the final value of E,




                         E =  CAT s'/2) /  (s'/bj
       Drag Coefficient.-- To determine the value of the drag coefficient,



 C , for each run, numerical solutions of X and Y were computed along

-------
                                                                     74
with the T/T  solutions for each value of A and 6 •   For each run,  at
            o                                    o


least two numerical solutions for two different assumed values of the



reduced drag coefficient, CL, were computed.  Each of the predicted



trajectories was plotted in terms of x'/b  and y'/b , using Equation 42



to calculate x/b  and y/b ,




                   X  =  C2E//JT b )x  ;  Y  =  (2E//JT bQ)y           (42)
and then using Equations 70 and 71 to calculate x /b  and y /bQ,





                   *'/bo = x/bo + Xe/bo                              (70)




                   y'/b  = y/b  + v'/b                               (71)
                   J   o   '  o   • e' o




This was possible since the value of E, and x'/b  and y'/b  had already
         r                                   e  o      e  o


been determined from the temperature plots.



      The observed value of the reduced drag coefficient for each run



was determined by over-laying the predicted trajectories with the



observed jet trajectory.  The value of CL was chosen by interpolating



between the predicted trajectories to the observed trajectory.   For



example, if the values CL = 0.5 and 1.0 were assumed in obtaining two



predicted trajectories, and if the observed trajectory were located



exactly half-way between the two predicted trajectories, then the



observed value would be C  = 0.75.



      The value of the drag coefficient, C , for each run was obtained



from the observed values of E and C  and Equation 39,




                          Cp = CD/4E                                 (39)

-------
                                                                     75
h'uitli of the J_e£



      The predicted and observed values of b/b  were then plotted.   The



numerical solution obtained for each run predicted b/b  vs.  S.   Predicted

                    t
 alue? of b/b  vs.  s  /b  were calculated and plotted as smooth  curves



by using the observed values of E and S*  determined for each run from  the



centerline data and Equations 68 and 69,




                        S'  = S*  + S                                  (68)
                              "
 v
                                    bo)s                             (69)



Then, the observed values of b/b  vs. s'/b  were plotted on the same
                                o         o


figure.




                        Presentation of Results


Trajectories, and Temperature and Width Plots



      The trajectories and temperature plots, from which the values of



C  and E were determined, and the width plots are presented in Figures



16 and 17 for a representative laboratory run and in Appendix C for the



other cases.  The data points which indicate the observed trajectory



for each run are plotted on the same figure as the fitted trajectory



computed from the numerical solution.  The observed T/T  vs. s'/b  data
                                                       oo


for each run are plotted on the same figure as the fitted T/T0 curve com-



puted from the numerical solution.  The observed b/b  vs. s'/b  data for
                                                    o         o


each run are plotted on the same figure as the predicted b/b  curve also



computed from the numerical solution.

-------
                    A =0.73
                    Cn= 0.4
                 V4EV°-7
0
                  FIGURE 16.--OBSERVED AND FITTED TRAJECTORIES, RUN 1-90

-------
1.0
0.5   -
0.1   -
0.05  -
0.01
    0.1
                                                                                             100.0
                                                                                         -  50.0
                                                                                              10.0
                                                                                              5.0,
                                                                                               1.0
                        0.5     1.0
5.0    10.0
50.0    100.
                                           s'/b
                                               o
           1'lCURli 17.--OBSHRVHD VALUHS AND IMTTHi) CURVliS 1'OR TJ-MPliRATURli AND WIDTH, RUN 1-90

-------
                                                                    78
Parameters


      The values of the various parameters as determined  from  the



laboratory data are presented in tabular form.  The values  of  A, 3Q,



s'/b , x'/b , and y'/b  are presented in Table 2.   The corresponding
 e  o   e  o      -V o     ^


best-fit values of E and C^, the values of S^ calculated  from  Equation 67



and X* and Y* calculated from Equation 72,
     e      e



             X* =  (2E//rF b )x'  ;  Y* = (2E/v/if bo)/g                 (72)




and values of C  calculated from Equation 39 are also presented  in



Table 2.





Relation of Parameters to A and g


      The object of  the laboratory experiments was to determine how the



establishment  zone and the entrainment and drag coefficients are related



to the velocity ratio and to the discharge angle.   The experimentally-



determined values of s'/b  and 6/6* are related to A as  shown in
                      e  o      o  o


Figures 18 and 19, respectively.  The values of C' are plotted as a



function of A  in Figure 20, and the values of the drag coefficient, CD,



are related to A as  shown in Figure  21.  The correlation coefficients of



the relations  and the correlation coefficient at the 1% level of



significance are shown in Table 3.    These correlation coefficients were



obtained using a regression analysis routine, which gives  the correlation



coefficients and predicted values for assumed arithmetic,  semi-log, and



log-log relations between the dependent and  independent variables.  The



assumed relation which gave the highest correlation  is the relation



presented in each of Figures 18, 19, and  21.  The  regression  analysis

-------
     TABLE 2




LABORATORY RESULTS
A
0.73
0.44
0.30
0.23
0.20
0.67
0.44
0.30
0.23
0.19
0.66
0.42
0.30
0.23
0.18
>;
90.0
90.0
90.0
90.0
90.0
60.0
60.0
60.0
60.0
60.0
45,0
45.0
45.0
45.0
45.0
3
o
r 	 ~
34.5
47.5
50.0
62.0
71.5
13.5
36.5
40.0
41.0
45.0
25.0
23.0
26.5
33.5
35.0
s'/b
e o
0.9
1.4
1.3
1.7
1.9
0.7
1.3
1.5
2.0
2.3
0.7
1.2
1.7
2.1
2.3
x'/b
e o
0.6
0.6
0.6
0.4
0.1
0.6
0.9
1.2
1.4
1.4
0.6
1.1
1.5
1.6
1.8
y'/b
e o
0.8
1.3
1.3
1.7
2.0
0.3
0.8
1.0
1.3
1.9
0.4
0.6
0.8
1.1
1 .3
E
0.46
0.39
0.31
0.47
0.44
0.24
0.13
0.19
0.25
0.29
0.19
0.13
0.21
0.27
0.22
s'
e
0.47
0.62
0.46
0.90
0.94
0.24
0.19
0.32
0.56
0.75
0.15
0.18
0.40
0.64
0.57
t
X
e
0.31
0.26
0.21
0.21
0.05
0.19
0.13
0.26
0.39
0.46
0.13
0.16
0.36
0.49
0.45
Y'
e
0.41
0.57
0.45
0.90
0.99
0.16
0.12
0.21
0.37
0.62
0.08
0 . 09
0.19
0.33
0.52
c;
0.4
1 .0
2.8
2.0
2.0
0.08
1.5
3.0
2.7
3.4
1.5
1.0
1.0
1.0
2.0
cn
0.7
1.6
3.5
3.8
3.5
0.1
0.8
2.3
2.7
3.9
1.1
0.5
0.8
1.1
1 .7

-------
                                                                    80
    5.0
    2.0
Ji
 bo
    1.0
    0.5
        0.1
                                                 I    I    I   I   I
                                                O    90.00
                                                D    60.0°
                                                A   45.00
                                                         O     -
                                                      ID  \-
                  I     I    I   I   I  I
0.2
0.5
1.0
          FIGURE 18.--OBSERVED VALUES AND FITTED CURVE FOR LENGTH OF
                   ESTABLISHMENT ZONE VERSUS VELOCITY RATIO

-------
1.0
I I
'o
0.5
0.0
                                                                         O 90.0°
                                                                         D60.0°
                                                                         A45.00
                                                             A
                                                              D
                                               0.5

                                                A
                 FIGURE 19.--OBSERVED VALUES AND FITTED CURVE FOR  INITIAL
                                 ANGLE VERSUS VELOCITY RATIO
                                                                                            1.0

-------
                                                                82
 4.0,	1	1	1	1    |   |   |  |
                    D
                               D
                         D     O


 2.0  I-            AO  O
 1.0  h-                 A   A


CD


 0.5
 0.2
              0   90.0  °
              a   60.0  °
              A  45.0  °
                                 1      1     I    I    I   I   1  I
     0.1              0.2                   0.5              1.0
        FIGURE 20.--OBSERVED VALUES OF REDUCED DRAG COEFFICIENT
                         VERSUS VELOCITY RATIO

-------
                                                              83
FIGURE 21.--OBSERVED VALUES AND FITTED CURVE FOR DRAG
          COEFFICIENT VERSUS VELOCITY RATIO

-------
                                                                     84
also indicated, as shown in Table 3, that the entrainment  coefficient,


E, does not appear to be a function of A over the range  of A  investigated


in this study.



                                TABLE 3


               RESULTS OF STATISTICAL TESTS ON PARAMETERS
Observed Correlation Coefficient
Correlation at 1% Level of
Relation Coefficient Significance
s;/bo
t
CD vs
E vs.
CD)VS
CD(VS
( O —
IP ™~
f O —
vs. A 0.95
vs. A 0.87
. A 0.72
A 0.01
. A
90.0°) 0.98
. A
60.0°) 0.97
. A
45.0°) 0.99
0.64
0.64
0.64
0.64
0.96
0.96
0.96
      Examination of Figure 21 indicates that C  may be a function of


B', the initial angle of discharge, as well as a function of A.   This


possible relation was tested, and the results are shown in Figure 22.


The correlation coefficients, which are presented in Table 3, are good.


However, because of the small number of data points, the confidence


limits of each of the curves plotted in Figure 22 tend to overlap, and

                       t
the relation of C  to 8  as well as to A cannot be conclusively es-


tablished from this study.

-------
                                                              85
 0.3
                  \     \

                        °

                               o
         \             \
                               \
                                 \
                               I I   v

 2.0  f-        \            \     V

CD                 ^o =OU-U-A      V?

                        •           \    \
                        A         \      \
                           \           \      \



                                 \         \
                                   \          \       \

 0.5
                                      I     l\  I    I   K  (
    0.1              0.2                  0.5              1.0
           FIGURE 22.--OBSERVED VALUES AND FITTED CURVES

                 FOR DRAG COEFFICIENT VERSUS  VELOCITY

                      RATIO AND DISCHARGE ANGLE

-------
                                                                      86
      A dependence of E on the initial angle of discharge was  indicated
by an F test, which was used to compare the variances of the experi-
mentally-determined values of E for each of the three discharge angles.
The results of the F test are presented in Table 4, and they show that
the mean value of E at 6* = 90° is significantly different from the mean
                        o
of the values obtained at g' = 60° and 45°.  Thus, E appears to be a
                           o
function of 8*.
             o

                                TABLE 4
                          RELATION OF E TO g'
e' E
o
Level of
F Ratio Significance
      90°               0.41
                                           23.1               0.1%
      60°               0.22
                                           0.20               None
      45°               0.20

-------
                              CHAPTER VI








                            FIELD SURVEYS






      Data were obtained  from  five surveys  at  three  different  steam-




electric  generating plants.  Data from  three of the  surveys  were  col-




lected by Vanderbilt  University  (VU)  personnel (37), and data  from  the




other two surveys  were taken from reports written by Churchill (45)




and Beer  and Pipes (6) .






                       Description  of VU Surveys




Flowrates



      The field surveys were made on regulated river systems over a




period of time when plant discharge and river flowrate were maintained




very near to steady-state conditions.  Stream flowrates were not




measured in the field but were determined  from data provided by the




regulating agency.  Plant flowrates were determined from pumping records




routinely kept by plant  personnel.






Geometry



       River  and discharge channel geometries  were determined  from  field




measurements and  from maps.   Ground  control was  established using  a




 transit  and  a  stadia rod to lay out  a  base line  which  paralleled the




 river along one bank for a  distance  extending several  thousand feet




 downstream.   From this  base line,  stations were  located which were




 perpendicular to  the river  centerline and, if possible, to  the jet





                                      87

-------
                                                                     88
centerline.  At each of these stations, the cross-sectional area of the



channel was determined using a Raytheon portable  recording fathometer



carried in one of the survey boats.  From the base  line measurements,



the longitudinal location of each of the cross-sections could be plotted



on a scale drawing traced from a U. S.  Geological Survey  (USGS) 1:24000



scale map of the area.  In addition, the initial  angle, 8^, of the



discharge channel was measured from these maps.





Velocity Data



      Measurements were made to determine the magnitude and direction  of



the initial jet velocity, U , and the ambient velocity, U  .
                           o                             a


      The magnitudes of U  and U  were determined by  two methods.  By



the first method, the magnitudes of the velocities  were measured at a



cross-section in the discharge channel and at a river cross-section



just upstream from the discharge.  The measurements were made with a



Price current meter mounted on one of the survey  boats.   Lateral



stations within each cross-section were located using anchored buoys



whose lateral distance from the bank had been measured either with a



transit on the bank and a stadia rod held in the  boat or  by means of



timed runs from the bank in the boat.  Vertical distances were measured



using the cable to which the velocity meter was  attached.   Ideally, this



first method should have given detailed velocity  profiles  from which



spatially-averaged ambient and initial jet velocities could be determined.



      By the second method, magnitudes of the velocities  were cal-



culated indirectly using the cross-sectional areas  obtained from  the



depth-sounding records and the flowrates obtained from gauging  station

-------
                                                                    89
and power plant records.   This second method yielded  average values of
U  and U  which were in close agreement with the values  obtained  by the
 o      a
first method.
      The directions of U  and U  were measured from  the mar* of the
                         o      a
area.  This method was chosen because the velocity vectors  could  not  be
measured directlv.
Temperature Distribution
      Temperature measurements were made at each of the cross-sections
using Whitney temperature probes carried in the survey boats.   Lateral
stations were located using the anchored buoys, while vertical measure-
ments were made using the cable to which each of the probes was attached.
Approximately 100 measurements were made in each of the cross-sections.

Procedure
      The normal procedure for the field surveys involved most of two
days spent on preliminary work and most of a third day spent obtaining
the temperature and velocity data.  The preliminary work involved laying
out the base line, locating the cross-sections, making a depth-sounding
record across the channel at each of the cross-sections, and setting out
the buoys.  The distance from the base line to  each of the buoys was
then measured.  The temperature and velocity data were measured on the
last day during the time when steady plant and  river discharges were
maintained.  Normally,  one boat and at least three men were required for
the preliminary work, while three boats -- one  to measure velocities and
two  to measure temperatures --  and as many as  eight or nine men were re-
auired  on  the  third day.  The field surveys were quite extensive in scope

-------
                                                                     90
and recorded considerable data, which were used in other studies in




addition to this present study.






                  Results of the Widows Creek Surveys





Introduction




      Data were collected by VU and USGS personnel on November 20 and 21,




1968, and by TVA personnel on August 30, 1967, at TVA's Widows Creek




Steam Plant.  The data  for the VU surveys were obtained from a report




by Polk  (37) and for  the TVA survey from the report by Churchill (45) .




At this plant, cooling  water from the condensers is discharged into a




small coal-barge harbor and flows directly onto the river surface with




a minimum of mixing.  Velocity measurements taken at several stations




in the river indicated  that the velocity field of the receiving ambient




water body was influenced by the discharge.  Since the discharged heat




was initially advected  across the river almost perpendicularly to the




river flow, its spatial distribution can be described in terms of a jet




discharging at some initial angle to the ambient flow.  Temperature




measurements indicated  that the rate of lateral spreading for the first




5000 feet downstream  was approximately 10 times greater than the rate




of vertical spreading due to the buoyancy of the heated discharge.




Thus, the application of the two-dimensional surface jet theory to de-




scribe a large, initial portion of the mixing zone is considered




reasonable.






VU Survey Number 1, November 20




      Temperature Measurements.-- The results of the temperature




measurements are shown  in Figures 23-28.  The temperature contours at the

-------
                                                                     91
1.0-foot depth and the location of the cross-sect ions  are  shown  in


Figure 23.   The vertical temperature profiles  along the jet centerline,


which is assumed to be located at the point of the maximum temperature


in each cross-section  for a distance of about 7000 feet downstream,


are shown in Figures 24 and 25.  The temperature contours  for three of


the downstream cross-sections are shown in Figures 26-28.   The jet


spread laterally from an initial discharge width on the order of 100


feet to a width of approximately 1200 feet, while it spread vertically


from an initial depth on the order of 10 feet to almost 15 feet.  Thus,


the rate of vertical spreading was almost an order of magnitude less


than lateral spreading and can be considered negligible for the first


five downstream cross-sections, which is approximately 5000 feet.


Beyond this point, it appears that the ambient turbulence of the stream


can no longer be neglected, because vertical mixing began to take place.




      Determination of Ambient Velocity.-- The value of the ambient


velocity, U  , was  determined  from the ambient flowrate, cross-sectional
           £L

areas, and velocity measurements.  The river  flowrate  at Widows Creek


was  estimated  to be  26,170 cfs, which was  the average  flowrate during


the  period of  the  survey  at  Nickajack Dam,  located about  20 miles up-


stream.  The minimum  flowrate at  Nickajack was  26,000  cfs  and the max-


imum approximately 27,000 cfs.  The  flow at Guntersville  Dam, located


approximately 60  miles  downstream, was  also maintained steady during the


period  of  the field  survey.   Thus,  the value  of Q^ =  26,170 cfs  is con-


 sidered a  reasonable estimate of  the river flowrate.   The  value of the


 ambient flowrate  past the discharge  was  calculated to  be  Q = 23,800 cfs

-------
R-l          R-2         R-3         R-4          R-5         R-6         K-i
                                         I	I	I	I
 FIGURE 23.--TEMPERATURE DISTRIBUTION,  °F,  AT  1.0-FOOT DEPTH,  WIDOWS CREEK, VU 1
                                                                                             i

-------
       AT CROSS-SECTION R-l
                    L
                 j
I
                            AT  CROSS-SECTION R-2
J
I
1
I
                                 AT CROSS-
                                SECTION  R-3
0
  46    8   10    12    0246802

                      TEMPERATURE RISE, °F

FIGURE 24.--TEMPERATURE RISE ALONG JET AXIS AT CROSS-SECTIONS R-l TO R-3

-------
AT R-4
     1	L
       AT R-5
J	I	L
AT R-6
AT R-7
              J	L
               024602
                        TEMPERATURE RISE,  °F
                                0
 FIGURE 25.--TEMPERATURE RISE ALONG JET AXIS AT CROSS-SECTIONS R-4 TO R-7

-------
0
 200       400       600       800       1000       1200
   DISTANCE FROM LEFT BANK,  FEET, FACING DOWNSTREAM
FIGURE 26.--TEMPERATURE DISTRIBUTION, °F, IN CROSS-SECTION R-l
1400
                                                                                     •

-------
CO
LU
CO
D-
LU
Q
 0
 2
 4
 6
 8
10
12
14
16
18
20
22
24
       0         200        400       600       800       1000       1200
            DISTANCE FROM LEFT BANK,  FEET,  FACING  DOWNSTREAM
             FIGURE 27.--TEMPERATURE DISTRIBUTION, °F, IN CROSS-SECTION R-2
                                                                                     •;
                                                                                     ' i

-------
    0
    2
uj  4
U_
    6
    8
   10
   12

CO
o_
   16
   18
   20
   22
   24
                           I
I
I
1
      0         200        400        600      800        1000      1200
          DISTANCE FROM  LEFT  BANK,  FEET, FACING DOWNSTREAM
          FIGURE 28. --TEMPERATURE DISTRIBUTION, °F,  IN CROSS-SECTION R-5

-------
                                                                    98
by subtracting from Q  = 26,170 cfs the value of the diverted plant



flowrate, Q  = 2370 cfs.



      Cross-sectional areas were measured for each of the downstream



cross-sections.  The average value of the first five cross-sections  was



22,400 ft2.  This downstream reach is considered to be the extent  of



the  influence of the two-dimensional surface jet.  The value of the



ambient velocity was found to be U  =1.06 ft/sec by dividing the  am-
                                  3


bient flowrate by the average cross-sectional area.



      To check this method of obtaining U , velocity measurements  made
                                         3


during the survey by USGS personnel were used.  Velocity measurements



taken in the undisturbed ambient velocity field at cross-section R-l



are  shown in Figure 29.  The value of the velocity obtained after  ver-



tically averaging each of the profiles shown in Figure 29 using a



planimeter and then averaging these values was equal to 1.05 ft/sec  at



cross-section R-l.  Dividing the value of Q  = 23,800 cfs by the value
                                           3


of the cross-sectional area at R-l, or 23,200 ft2, gave a comparable



value of U  =1.03 ft/sec.  Thus, using the flowrate and the cross-
          3


sectional area gave the same value of ambient velocity as did detailed



velocity measurements for cross-sections as well.  Since the cross-



sectional areas varied from 23,900 ft2 to 18,800 ft2, it is felt that



using the flowrate Q  = 23,800 cfs and the average value of the cross-
                    3


sectional areas of the first five downstream stations gave the best



determination of the ambient velocity along the reach of stream affected



by the jet discharge.

-------
   0


   2


 ~- 4
        1l

LU
a:
^ «
to o


  io
214
LU
QQ

£16


£318
o


   20



   22



   24
   STATION  1

  MB





    I   I   I   I
STATION 2



i   I   L  I  I
                                   I   I
STATION 3



I  I   1  L
                                                                    i   i
                                                            STATION 4
                                                             I   I    I   I
                                                           STATION 5
.0
1.0.0
                          .5    1.0    .0
                                             .5    1.0

                                             FT/SEC
                                 .0     .5     1.0  .0    .5    1.0
                         FIGURE 29.--VELOCITY PROFILES AT CROSS-SECTION R-l
                                                                                                  to
                                                                                                  <£>

-------
                                                                    100
      Initial Jet Width.--  The value  of  the width of the discharge was



estimated to be 187 feet from the scale  drawing traced from the USGS



map of the Widows Creek area.   The value of the half-width at the origin



was half of this, or b' = 93.5 feet.   The value of bQ, the initial half-



width at the beginning of the zone of established jet flow, was found to



be b  = 150.0 feet from Equation 65,




                            b  = 1.60 b'                              (65)
                             o        o
      Depth of the Jet.-- It was originally  intended to theoretically



calculate the depth of the jet at the mouth  of the discharge channel.



According to Harleman (26),  when the density difference between the dis-



charge water and river water is sufficiently great, the colder ambient



water will intrude into the discharge channel under the lighter con-



denser water.  The interface between the  discharge water and ambient



water can be described in terms of two-layer stratified flow theory.



Upon assuming that a critical Froude Number  occurs at the junction of



the discharge channel and the river, the  depth of the upper layer, or



the depth of the jet, can be predicted based on Harleman's work in terms



of the discharge flowrate, the channel width, and the density difference.



      However, the predicted depth of the upper layer using the observed



field values for the plant flowrate, the  discharge width, and the density



difference was on the order of 15.0 feet,  or almost the entire depth of



the ambient stream.   The vertical temperature profiles in Figure  24  and



25 indicate that the average depth of the jet in the ambient stream was



considerably less than 15.0 feet.  Therefore, it was concluded that  the



conditions necessary to theoretically predict the jet depth were  not

-------
                                                                    101
met at Widows Creek.



      Therefore, the depth of the jet was empirically estimated from the



field data in a manner which is consistent with the definition of the



half-width, b, in Equation 24,



                              b = /2 G                              (24)



The jet depth, which is assumed constant compared to the jet width,  was



estimated from the vertical temperature data shown in Figures 24 and 25



by using a similar definition, shown in Equation 73,



                              z  = /2 a                              (73)
                              o       z


where o  is the standard deviation of the vertical temperature data.
       z


      The standard deviation  and sample variance for each of the first



five sets of vertical temperature data are shown in Table 5.  The es-



timated jet depth was found by calculating the  standard deviation from



the average of  the variances.  The standard deviation was 4.24 feet, and



the jet depth was found to be z  =6.0 feet by  using Equation  73.





                                 TABLE 5



                       VERTICAL PROFILE  STATISTICS

                            VU SURVEY NUMBER  1
Location
R-l
R-2
R-3
R-4
R-5
Standard
Deviation
3.71
3.07
5.07
4.31
4.75
Sample
Variance
13.7
9.4
25.7
18.6
22.6

-------
                                                                    102
      Initial Jet Velocity.-- The value of the initial jet velocity was



calculated to be U  = 2.12 ft/sec from Equation 74,



                         Q  = U z 2 b'                              (74)
                          0    0 O   0



where   Q  = 2370 cfs;



        z  =6.0 feet; and



        b' = 93.5 feet.
         o




      Velocity Ratio.-- The value of the velocity ratio was found to be



A = 0.50 using U  =1.06 ft/sec and U  =2.12 ft/sec and Equation 38,
                3.                    O


                          A = U /U                                  (38)
                               a  o
      Initial Angle of Discharge.-- The initial angle,  3 ,  of the dis-



charge was found by assuming that the centerline axis of the river



passed through the center of cross-sections R-l and R-5, which are



shown in Figure 23.  This reach of the river between R-l and R-5 was



assumed to be a straight segment, with the direction of the ambient



velocity along this axis.  The x -axis was parallel to this centerline



axis.  The origin of the x  and y  axes was assumed to be located, in



terms of x , in the middle of the discharge channel, and, in terms of



y , at the edge of the ambient stream.  After locating the x1 and y1



axes in this way, the initial angle of the discharge was measured,



giving g'  = 85.0°.





      Zone of Flow Establishment.--  From Figure 18, the length of the



zone of flow establishment was estimated to be s'/b  = 1.05, using
                                                e  o             6


A = 0.5.   From Figure 19, the initial angle at the beginning of the

-------
                                                                    103
zone of established jet flow was estimated to be 3  =46.7°,  using



A = 0.5 and B  = 85.0°.
      Jet trajectory.-- The distance along the jet trajectory was found



from Figure 23 by measuring the distance s /b  along a smooth curve from



the origin to the location of the maximum temperature in each cross-



section.  Then, the value of T/T  at each of the cross-sections was



calculated.





      Entrainment Coefficient.-- Numerical solutions for A = 0.5 and



g  = 46.7° were used to determine the value of E.  First, a preliminary
 o


value of E = 0.2 was determined by  sliding the T/T  vs. s'/b  data plot



horizontally over the  T/T  vs. S curve, which was computed from S = 0.0.



When the best fit of the data and the theoretical curve was obtained, a



corresponding set of values of s'/b and S was noted.  Equation 37 was



then solved for the preliminary value of E,



                         E  -   (v^  S/2) /  (s/bQ)                       (37)




      Next, Equation 67 was solved  for s', using  the value of sg/bo = 1.05



and E =  0.2,



                         S' =   (2E/A"b )s'                            (67)
                           Q            O   c



The T/T  vs. S plot was then redrawn as T/TQ vs.  S*, referenced  to  the



jet origin, using  the  value of  S  and Equation  68,



                         S' =   S' H- S                                  (68)
                                 e



A  set of corresponding values  of  s' and s'/bQ was noted  after sliding



the T/T  vs.  s'/b   plot horizontally over  the T/TQ  vs. s'/bQ  curve  until



the best fit  of  the data and  the  theoretical  curve  was obtained.  Then,

-------
                                                                     104
Equation 69 was solved for the final value of E,




                      E  =  (/S" s'/2) / (s'/b)                       (69)
                                             0
      Drag Coefficient.-- To determine the value of the drag  coefficient,



C , numerical solutions of X and Y were computed along  with the T/TQ



solutions with A = 0.50 and g  = 46.7°.  Two different  values of the
                             o


reduced drag coefficient, C* = 0.5 and 1.0, were assumed.   Each of the



predicted trajectories was plotted in terms of x /b  and y'/b ,




                   X - (2E//Fb )x  ;  Y = (2E//F bQ)y                 (42)



                                   ,         i
In the field cases, the values of x /b  and y /b  were  considered to be
                                   e  o      e  o


negligible compared to the values of x/b  and y/b .  Thus,  it was



assumed that x'/b  = x/b  , and y /b  = y/b .
                 o      o          o      o


      The value of the reduced drag coefficient was determined by over-



laying the predicted trajectories with the observed jet trajectory.  For



this field case, interpolation between the predicted trajectories to the



observed trajectory was not necessary, because the assumed value of



C* = 1.0 gave the best fit.



      The value of the drag coefficient was found to be CL =  0.6 from

                                     t

the observed values of E  = 0.16 and CL = 1.0 and Equation 39,




                        Cp = CD/4E                                     (39)






      Width Ratio.-- The  observed values of the width ratio b/b  were



obtained by calculating the standard deviation of the temperature values



at each of the cross-sections and then using Equation 24,

-------
                                                                   105
                                                                    (-4)
and the value of b  =150.0 feet.
      Initial  Ri  Number.--  The value of the initial  Richardson Number


of the jet was calculated to be RiQ = 0.22 using the  values


AD/P = 1.29 x  10~3, U  =2.12 ft/sec, U  =1.06 ft/sec,  ZQ =  6.0  feet,



and Equation 60.




      Check of Initial Jet Velocity.-- The value of UQ =2.12 ft/sec



was obtained by using Equation 74,



                           Q  = U  z  2 b'                            (74)
                           xo    o o   o



      To check  this method of obtaining U  , the velocity measurements


made by USGS personnel were used.  Since the velocity measurements made


at  the discharge near cross-section  R-l were inconsistent and scattered,


presumably due  to  turbulence caused  by  the presence of several large


piers used by coal barges,  it was  necessary to use the measurements


taken at R-2.   These  velocity measurements, shown in Figure  30, in-


dicated that  the jet  discharge created  a  non-uniform velocity field


when  compared to the  unaffected,  uniform  part  of the velocity field at



R-l,  shown  in Figure  29.


      The  observed velocity and  temperature rise at  R-2 was  examined to



see if  the  values  of  U   and A  used in the numerical  solution were  con-


sistent with  the  observed values.   As seen in  Figure 30,  the maximum


value of  the  velocity in cross-section R-2 was 1.7 ft/sec.   At this


 same cross-section,  the value  of the temperature rise was T/TQ =  0.49.


 The numerical solution using A = 0.50 and 6Q  = 46.7° predicted that

-------
   0
   2
B 4
l_l_
   6
        T—r
                  '   '
         1   I ~T
         I   I   I
                                                                                I  I
o
a
DC:
00
 8
10
12
14
16
18
  22
         STATION 4
  24
     I   I   I   I   I   I
                          STATION 5
                     STATION  6
I   I   I   I   I   I
                    STATION 7
I   I   I   I   I   I
                                                                       I   I   I   I   I   I
    .0    .5    1.0      .0    .5    1.0       .0    .5     1.0      .0    .5    1.0
                                            FT/ SEC
                        FIGURE 30.--VELOCITY PROFILES AT CROSS-SECTION R-2

-------
                                                                   107
T/T  = 0.49 at S* = 28.0.  At this same value of s' ,  the numerical  solu-



tion also predicted a value of U/U  =0.8.  If U = 1.7 ft/sec,  then the



initial jet velocity based on data taken at this one cross-section  should



have been U  = 2.13 ft/sec, which compares favorably with the value cal-



culated from Equation 74, or U  =2.12 ft/sec, which is based on data



taken at five cross-sections.





      Presentation of Results.-- The data for the November 20 survey are



presented in Table 6, and the parameters are presented in Table 7.   The



trajectory and temperature plots, from which the values of CL and E were



determined, are presented in Figures 31 and 32.  The data points which



indicate the observed centerline of the jet are plotted on Figure 31,



along with the fitted trajectory computed from the numerical solution.



The observed T/T  vs. s'/b  data are plotted on Figure 32, along with



the fitted T/TQ curve also computed from the numerical solution.  The



observed values of b/b   are plotted on Figure 32 along with predicted



b/b  curve computed from the numerical solution.   The observed value of



E = 0.16 determined from the centerline temperature data was used to



calculate the values of  s'/b  and from S, using Equations 68 and 69,





                         S' =   S' + S                               (68)
                                 e
                                (2E/A" b  )s'                         (69)

-------
20.0  -
0.0
                      10.0    15.0      20.0     25.0     30.0      35.0     40.0     45.0
                                            x'/b0
                   FIGURE 31.--OBSERVED AND FITTED TRAJECTORIES,  WIDOWS CREEK,  VU 1
                                                                                                         00

-------
1.0
0.5
0.1
0.05
0.01
     0.1
                         I        I
                 A = 0.50
                 E - 0.16
 I	I
0.5     1.0
5.0     10.0
50.0    100.
                                           s'/br
        FIGURE 32.--OBSERVED VALUES AND FITTED CURVES FOR TEMPERATURE AND WIDTH,  WIDOWS CREEK, VU 1

-------
                                                                  110
                                TABLE 6


                        DATA, VU SURVEY NUMBER 1
Cross-Section
R-l
R-2
R-3
R-4
R-5
R-6
R-7
VU Survey Number
Temperature
S'/bo
4.46
14.0
24.3
34.5
45.2
54.2
63.5
2 and TVA Survey
Measurements . - - The
T/TO
0.82
0.49
0.38
0.33
0.31
0.099
0.13

temperature
b/bo
1.00
2.11
2.02
2.80
3.16
--
--

measurements taken
November 21 by VU personnel and August 30 by TVA personnel at Widows


Creek showed that the temperature distribution could again be described


in terms of a two-dimensional surface jet.  The surface temperature


contours for these two surveys are shown in Figures 33 and 34.  The


location of the cross-sections was the same as for the first survey.




      Presentation of Result!;.-- The procedure for analyzing the data


was the same as that used for the first survey.  The numerical solution


was used with different values of A and 3Q since the velocity ratio,


A = U /U , was different in each case from the first survey.  The data
     a  o

-------
                                             Ill
           TABLE 7




PARAMETERS, VU SURVEY NUMBER 1
Observed Values
T , Initial Temperature Rise, °F
Q , River Flowrate, ft3/sec
Q , Plant Flowrate, ft3/sec
b', Discharge Half-Width, ft
B0, Discharge Angle, °
14.2
26,170
2,370
93.5
85.0
Calculated Values
Qa, Ambient Flowrate, ft3/sec
Rio, Initial Ri Number
Average Cross-Sectional Area, ft2
Ua, Ambient Velocity, ft/sec
b0, Initial Jet Half-Width, ft
zo, Jet Depth, ft
Uo, Initial Jet Velocity, ft/sec
A, Velocity Ratio
se/b0, Establishment Zone
3 , Initial Jet Angle, °
23,800
0.22
22,400
1.06
150
6.0
2.12
0.50
1.05
46.7
Observed Results
E, Entrainment Coefficient
Cp, Reduced Drag Coefficient
CD, Drag Coefficient
0.16
1.0
0.6

-------
FIGURE 33.--TEMPERATURE DISTRIBUTION,  °F,  AT 1.0-FOOT  DEPTH,  WIDOWS CREEK, VU  :

-------
FIGURE 34.--TEMPERATURE CONTOURS,  °F,  AT 0.5-FOOT DEPTH,  WIDOWS CREEK,  TVA


-------
                                                                    114
for VU Number 2 and TVA surveys are presented in Table 8,  and  the




parameters are presented in Table 9.   The trajectories and temperature




plots are presented in Figures 35-38.   The observed and predicted




values of b/b  are presented in Figures 36 and 38.   For the VU survey,




the observed values were E = 0.16, CD = 0.9, and CD = 0.6.  For the




TVA survey, the observed values were E = 0.16, CD = 0.4, and CD = 0.3.






                                TABLE 8




                DATA, VU SURVEY NUMBER 2 AND TVA SURVEY
Cross- t
Section s /b
R-l
R-2
R-3
R-4
R-5
R-6
R-7
4.46
14.0
24.3
34.5
45.2
54.2
63.5
VU No.
0.81
0.56
0.40
0.42
0.25
0.15
--
2
b/b
o
--
1.66
2.03
2.48
2.58
--
--
TVA
T/T
o
0.74
0.50
0.47
0.41
0.28
0.07
0.08
b/b
o
--
1.28
1.98
2.55
2.88
4.01
4.15
                 Results of the New Johnsonville Survey
 Introduction
       Data (37)  were collected by VU personnel on May 29, 1969, at




 TVA's  New Johnsonville Steam Plant  located on Kentucky Lake.  This




 plant  is  similar to Widows Creek in that  condenser water is discharged




 into a small  coal barge harbor and  then flows onto the river surface.




 The spatial distribution of the discharged heat  is again described in

-------
                                                      115
                  TABLE 9




PARAMETERS, VU SURVEY NUMBER 2 AND TVA SURVEY
              Observed Values

T0, °F
Qr, ft3/sec
Q0, ftVsec
b0, ft
30,
VU No. 2
13.0
26,170
1,840
93.5
85.0
TVA
14.2
47,000
2,200
93.5
85.0
             Calculated Values
Qa, ft3/sec
Rio
Average Cross-
Sectional Area, ft2
Ua, ft/sec
b0, ft
z0, ft
U0, ft/ sec
A
1
Bo. °
VU No. 2
24,330
0.64
22,400
1.09
150
6.0
1.64
0.67
0.84
36.5
TVA
44,800
1.21
26,250
1.71
150
5.2
2.27
0.75
0.75
32.3
               Observed  Results

E
c;
D
CD
VU No. 2
0.16
0.9

0.6
TVA
0.16
0.4

0.3

-------
                i         i         i         i          i         r
i        r
  20.0
  15.0
 o
•O
  10.0
                      A = 0.67


                         -  0.9
   5.0
   0.0
               5.0      10.0     15.0    20.0      25.0     30.0      35.0     40.0     45.0

                                             x'/bQ
                  FIGURE 35.--OBSERVED AND  FITTED TRAJECTORIES, WIDOWS CREEK, VU 2

-------
1.0
0.5
0.1
0.05
0.01
     0.1
      r

Q = 36.5°
A = 0.67
E = 0.16
0.5      1.0
                                       s'/b
                                          o
                                 5.0     10.0
                                                                    100.0
                                                                     50.0
                                                                     10.0
                                                                     5.0
                                                                      1.0
50.0    100.0
     FIGURE 36.--OBSERVED VALUES  AND FITTED CURVES FOR TEMPERATURE AND WIDTH,  WIDOWS CREEK,  VU  2

-------
20.0   -
15.0
o
10.00-
 5.0   -
 0.0
               5 0       10.0    15.0     20.0     25.0     30.0      35.0      40.0     45.0
                                              x'/D0
                   FIC.URH 37.--OBSl-;RVH1>  AND 1MTTHI) TRA.JKCTORTT.S,  W1POWS CRIiliK,  TVA

-------
1.0
0.5
0.1
0.05
0.01
     0.1
         r

ft  =36.5(
 A = 0.75
 E = 0.16
0.5      1.0
                                    5.0      10.0
                                             s'/b
                                                                                              100.0
                                                                      50.0
                                                                      10.0
                                                                      5.0
                                                                       1.0
50.0    100.0
                                                o
      Flf.URi; 38.--OBS1-RV1:!) VALUES AND FTTTHD CURVES FOR  TllMPHRATURP. AND WIDTH,  WIDOWS CRTJiK, TV A

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                                                                   120
terms of a two dimensional  surface jet  deflected  by  the  ambient  current.




At the discharge,  the width of the ambient  water  body  is only  four  times




larger than the width of the discharge.   As a result,  the  entrainment




characteristics and the spatial location of the jet  were affected by the




narrow boundaries.  The surface temperature contours are shown in




"igure 59.






Depth of the Jet




      At New Johnsonville,  detailed velocity measurements  were made at




the mouth of the discharge channel.  These velocity measurements were




used to estimate the depth of the jet at this field site.   The sample




standard deviation of the velocity data shown in Table 10  was  calculated,




and then the depth of the jet was found to be ZQ = 5.95  feet,  using




Equation 73,




                                                                     (73)
z
o

VELOCITY DATA,
Depth, ft
1.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
22.0
= v'2 a
z
TABLE 10
NEW JOHNSONVILLE SURVEY
Velocity, ft/sec
0.99
0.93
0.73
0.10
0.14
0.01
0.10
0.14
0.0
0.0

-------
                                                      0         1000       2000 FEET
                                                       i   i  i   i            ^
OLD RIVER

CHANNEL
      FIGURE 39.--OBSERVED TEMPERATURE  DISTRIBUTION,  °F, AT 1.0-FOOT  DEPTH, NEW JOHNSONVILLE

-------
                                                                   122
Presentation of Results
      Except for the depth determination discussed  above,  the  procedure
for analyzing the data was the same as for the Widows  Creek  surveys.
The numerical solution was used with A = 0.57 and 8Q = 30.0°.   The
temperature data are presented in Table 11 and the parameters  in
Table 12, while the trajectory and temperature plots are presented in
Figures 40  and 41, respectively.  The width ratio, b/bQ, is  not cal-
culated.  The  distortion  caused by the narrow boundaries precluded
measuring with any degree of  accuracy the location of the lateral
temperature rise perpendicular  to  the jet axis.  The observed values
of E  =  0.04 and  C*  =  3.0  reflect the  influence of  the geometry.

                                TABLE 11
                          TEMPERATURE  DATA, NEW
                           JOHNSONVILLE  SURVEY
                        0.63                  1.00
                        5.62                  0.91
                       12.5                   0.82
                       15.7                   0.73

-------
                                                 123
              TABLE 12




PARAMETERS,  NEW JOHNSONVILLE SURVEY
Observed Values
T , °F
Qr, ftVsec
Q0, ft3/sec
bO> ft
C °
11.0
26,500
2,180
250
60.0
Calculated Values
Qa, ft3/sec
Ri0
Average Cross-
Sectional Area, ft
Ua, ft/sec
b0, ft
z0, ft
U0, ft/sec
A
V
se/b
B0' °
24,320
3.30
58,182
0.42
400
5.95
0.73
0.57
0.93
30.0
Observed Results
E
CD
CD
0.044
3.0
0.5

-------
4o.--oBsuRvi-;i)  AND MTTI;U TRAJiiCTORii-s,  NHW

-------
 1.0
 0.5    -
T/T,
   o
 0.1
 0.05
  0.01
  3  =30.0
   o

   A = 0.57


   E = 0.04
 J	L
 J	L
                                                                                   1
      0.1
0.5      1.0
5.0     10.0
                                             s'/b
                                                o
50.0    100.
             FIGURE 41.--OBSERVED VALUES AND FITTED CURVE  FOR TEMPERATURE,  NEW JOHNSONVILLE

-------
                                                                   126
                      Results of the Waukegan Survey




Introduction



      Data obtained from a report by Beer and Pipes (6)  at Waukegan



generating station located on Lake Michigan were analyzed in terms  of a



two-dimensional surface jet.   The cooling water from the plant  was  dis-



charged into a 2000-foot long channel and then into the lake.   There was



no appreciable ambient velocity, and, therefore, the velocity  ratio was



equal to zero, or A = U /U  =0.0.  This case of a jet discharging  into
                       3.  O


a stagnant environment ordinarily could not be described as two-



dimensional, since the Richardson Number is quite low, and vertical en-



trainment would be significant.   However, the shallow depth of the  lake



near the shore inhibits vertical entrainment to some degree.  Over  a



longitudinal distance of 3600 feet, the jet spread laterally from an



initial width on the order of 250 feet to about 2500 feet and spread



vertically from about 6 feet  to a depth on the order of 15 feet. Thus,



the rate of lateral mixing was approximately five times greater than



vertical mixing.  Even though the rate of lateral mixing was not a  full



order of magnitude greater than vertical mixing, it is felt that appli-



cation of the two-dimensional model is reasonable, if only in order to



examine a case in which the two-dimensional assumption is not fully met.




Temperature Measurements



      The temperature contours at the 1.0-foot depth are shown  in



Figure 42.

-------
FIGURE 42.--TEMPERATURE DISTRIBUTION, °F, AT 1.0-FOOT DEPTH, WAUKEGAN
                     [AFTER BEER AND PIPES (6)]

-------
Zone of Establishment



      The length of the zone of establishment  is  not  estimated  from



Figure 18 because the laboratory experiments did  not  include  the  case  of



A = 0.0.   Instead, the length was taken to be  s /d  = 6.2  from  the work



of Albertson ejt al_. (4).   Expressed in terms of b ,  the length  is found



to be s'/b  = 4.4 bv using Equation 66,
       e  o        '      '


                              b  = 0.708d'                            (66)
                               o         o
Results of Analysis



      The procedure for analyzing the data was the same as for the other



surveys.  The numerical solution was used with A = 0.0.  The temperature



data are presented in Table 15 and the parameters in Table 14, while the



temperature plot is presented in Figure 43.  Values of b/bQ estimated



from Figure 42 are also presented in Figure 43 along with the predicted



b/b  curve.  The value of the entrainment coefficient was found to be
   o


E = 0.44.



      The temperature and width plots, Figure 43, indicate that the two-



dimensional model can be matched to the observed data reasonably well in



the zone of established flow.  However, in the zone of establishment,



the predicted values are greater than the observed data.  For a distance



of s'/b  =4.4, the curve predicts T/TQ =1.0, while over the same



distance, the values of the data decrease from T/TQ = 0.8 to 0.7.



      The initial temperature rise, T  = 15.0°F, was computed from the



difference of the reported condenser temperature, 60°F, and the lowest



ambient  temperature in the region of the jet, or 45°F.  Examination of



the surface contour plot, Figure 42, indicates that the temperature at

-------
                                               129
            TABLE 15
     DATA,  WAUKECAX SURVEY
>'»„
0.
0.
0.
1.
1.
3 .
4.
5 .
6.
7
9.
11.
14.
16.
0
39
48
29
95
41
34
60
89
99
65
6
7
8
T/T
o
1
0
0
0
0
0
0
0
0
0
0
0
0
0
.0
.80
.80
.80
.6~
.6~
.67
.60
.55
.40
.40
. 55
. 53
.20
b/b
o
--
--
--
--
--
--
2.0
2.75
3.5
4.75
5.25
5.0
--
--
            TABLE 14
  PARAMETERS, IVAUKEGAN SURVEY
        Observed Values
T0, °F                    15.0
Q0, ft3/sec            1,690
b'Q, ft                   125
       Calculated Values
Ri0                        0.01
b0, ft                   200
U0, ft/sec                 2.6
A                          0.0
        Observed Results
                           0.44

-------
1.0
0.5
0.1
0.05
0.01
     0.1
                     •  90.0
                  A =0.0
                  E = 0.44
 I	I
                                         o    o  o
0.5     1.0
                              T/l
                                                     b/b
                                                        o
                                      D
 I        I
5.0    10.0
                                          s'/b
                                             o
  I
                                                                  100.0
                                                                   50.0
                                                                   10.0
                                        5.0
1.0
50.0    100.0
                                                                                                        O
           FIGURE 43.--OBSERVED VALUES AND FITTED  CURVES FOR TEMPERATURE AND WIDTH, WAUKEGAN

-------
                                                                   131
the origin of the jet ma}' have been on the order  of 58°F,  and  that the



initial temperature rise may have been 13°F,  instead of 15°F.   If an



initial temperature rise of T  = 15.0°F could be  assumed,  then all the
po
  ints in the zone of establishment would be closer to the predicted
value of T/T  =1.0.
            o
                          Comparison of Results



      The results from the three Widows Creek surveys, the New Johnsonville



survey, and the Waukegan survey are summarized in Table 15.  The values



of the entrainment coefficient at Widows Creek were consistently E = 0.16



for three different field surveys with different values of A = ua/l]Q-



The small value of E = 0.04 at New Johnsonville indicates the influence



of the relatively narrow boundaries, which apparently greatly inhibited



lateral entrainment.  The value of E = 0.4 at Waukegan seems reasonable,



since  there were no boundaries to inhibit the lateral entrainment of



colder water  into the jet.  However, this value may indicate the effects



of vertical entrainment, since the rate of lateral mixing was not a full



order  of magnitude greater than vertical mixing at the Waukegan site.



Because of the scatter of the data, it is difficult to determine the



significance  of  this  effect.



       The values of the  drag coefficient, C  ,  ranged  from  0.3 to 0.6, as



A varied from 0.75 to 0.50.  Over this narrow  range of A,  at  least, it



appears  that  C   is a  function of A under field  conditions  as well as



under  laboratory conditions.

-------
                                           132
        TABLE 15




SUMMARY OF FIELD RESULTS

Widows Creek
VU No. 1 0.
VU No. 2 0.
TV A 0 .
New Johnsonville 0.
U'aukegan 0.
A

50
67
75
57
00
;' Ri
0 0

85.0 0
85.0 0
85.0 1
60.0 5
0

T 1
.64
.21
.30
.01
E

0.
0.
0.
0.
0.
CD

16 1.0
16 0.9
16 0.4
04 5.0
44
Cn

0.
0.
0.
0.
	


6
6
5
5


-------
                               CHAPTER VII






                                DISCUSSION




               Results  of Laboratory and Field Investigations



 Establishment  Zone



      Laboratory  results describing how the zone of flow establishment


 is related  to  the velocity  ratio are presented in Figures 18 and 19.


 These relations were used in analyzing the field data in order to obtain


 initial values of the  jet axis distance and the initial angle, -   from
                                                                o

 which to obtain the numerical solution.  The laboratory results were


 used because detailed  field measurements of the establishment zone could


 not be obtained due to time and equipment limitations.  It is felt that


 the empirical  curves in Figures 18 and 19 are adequate to describe the


 observed field conditions, as can be seen by examining the temperature


 and trajectory plots of the field data.


      The length  of the zone of flow establishment is, in general,


 slightly less than values found in the literature.  A possible explana-


 tion is that the  relations in this study were derived for the cases  of


 the jet origin coinciding with the flume wall,  while the relation used


by Fan (22), for  instance,  was developed for a jet placed away from  the


boundary into a region of more nearly uniform flow.   The effect  of the


velocity gradient near the boundary,  it is felt,  is  to shorten the


 length of the establishment  zone.   Since the discharge coincides  with
                                    133

-------
                                                                    134
the boundary at most field  sites,  the  relations  derived  in  the present




study are felt to be more realistic  when applied to  field sites.







Entrainment Coefficient




      The entrainment mechanism,  or  the inflow velocity,  is assumed  to




represent the turbulent mixing of the  jet.   The  laboratory  results




presented in Tables 2, 3, and 4 indicate that  E  = 0.4 when  PQ =  90.0°,




and that E = 0.2 when t'  = 60.0°  and 45.0°.   Thus,  E appears to  be a
                       o



function of the initial angle of discharge of the jet.   Apparently,  as




;;'  is decreased, the volume of colder water available for inflow and

 o


dilution of the heated jet on the side of the jet near the  boundary  is




also decreased.  Thus, the smaller values of E at PQ = 60.0° and 45.0°




indicate the effect of the near boundary on entrainment.




      The values of E at the two  field sites located on rivers  are  lower




than the laboratory results for comparable values of j3\  These  lower




values apparently indicate the effect of the narrow boundaries  on lateral




inflow on both sides of the jet,  since the ratios of the ambient width




to the discharge width of 7.5 at Widows Creek and 3.5 at New Johnsonville




are  lower than the  ratio of 24.0 used in the laboratory experiments.




      The values of E at Widows Creek are consistently equal to 0.16 for




three different field survevs with different values of A = U /U .  Thus,
                                                            d  O



it appears  that, for  a given site, the  entrainment coefficient  is




reasonably  constant.




      For  cases where A  = 0.0, other  investigators, such as Fan (22),




have reported  laboratory values of the  entrainment coefficient on the




order of 0.10.  The value of E = 0.44 at the Waukegan field site is

-------
                                                                   135
areater and possibly indicates the effects of ambient  turbulence  and
£>


vertical mixing, the effects of which are not considered in the



analytical model.


      The entrainment coefficients determined in this study differ



slightly from values found in the literature.  Fan (221 has reported



values  on the order of 0.45 determined from measurements along the axis



of  a  buoyant jet discharged vertically into a uniform,  ambient current.



However, his values would be  smaller  if  a more realistic, bimodal pro-



 file  had been  assumed,  since  the  maximum cross-sectional concentrations



 on  both sides  of the  jet axis were  60 to 80%  greater  than  concentrations



 along the  jet  axis.   Keffer and Baines (30)  have  reported  values  of  the



 entrainment coefficient which varied along  the jet axis from 0.5  to



 1.6 for air-in-air jets subject to  a steady cross-wind.  Zeller  (48)



 presented values which varied from  0.127 to 0.993, not along the jet



 axis, but as determined from 22 sets of data for two surface jets at a



 field  site.  The values of E determined  in the present study are not



 unreasonable, however, when  the variation in the values of E reported by



 these  other investigators  is considered.



        The  entrainment  coefficient  appears to be  a function of the



 assumed profile shape  and  other  factors, the  effects  of which are not



 yet  completely understood.   For  example, although some investigators



  suggest  that  E is  a  function of  A, the  results of the present study



  indicate  that boundary geometry  may be  more significant when relatively



  narrow channels and discharge angles less  than 90°  are considered.

-------
                                                                    156
Drag Coefficient



      The laboratory results presented in Tables  2 and 3,  and in



Figures 21 and 22 indicate that C  varies with the velocity  ratio  and,



possibly, with the angle of discharge.  Values of C_.  vs. A have  also



been reported by Fan (22) and Carter (12).



      The range and magnitude of C  from C.I  to 3.8 are greater  than



would be expected based on the work of Fan,  who reported a range of C



from 0.1 to 1.7, and Carter, who reported values  of CD ranging from 0.74



to 1.36.  However, as shown in Equations 17-19, the analytical form of



the drag force used in this study is different from the forms used by



Fan and Carter.  Thus, a direct comparison of C  values cannot be  made.



      The possible dependence of C  on 3  is indicated in  Figure 22.



The values of C  are seen to decrease as S*  decreases from 90° to  60°
               D                          o


and 45°.  In the analytical model, the drag  force is assumed to  represent



the effects of the pressure gradient caused  by the separation of the



ambient flow behind the jet.  At o'  = 60.0°  and 45.0°, the jet is  closer



and closer to the near boundary, the presence of which may inhibit the



ambient flow and reduce the separation of the flow.  Thus, the drag force



mav be reduced at 6' = 60.0° and 45.0°, which would account  for  the
                   o


smaller values of C  observed at 3'  = 60.0°  and 45.0°.



      The field values of C  are very close to the values  that would



have been predicted based on the laboratory results, as shown in



Figure 44.  In this figure, the observed field values of C  vs.  A, de-



termined over the range of A from 0.5 to 0.75, are plotted along with



the empirical laboratory curve, which was shown in Figure  21. Unfor-



tunately, the lack of field values of CQ determined in the range of

-------
                                                           137
1.0 -
0.5
 'D
0.2
 0.1
                                       i—rr
               LABORATORY CU
                                             D
   WIDOWS  CREEK

O  VU1
D  VU  2
0  TVA	
    NEW JOHNSONVILLE
                                      o  V
                                   I     I    I    I   I  I
                   0.2
                        0.5
1.0
           FIGURE 44.--OBSERVED FIELD VALUES OF DRAG COEFFICIENT
            VERSUS VELOCITY RATIO PLOTTED ON LABORATORY CURVE
                           (From Figure 21)

-------
                                                                    138
A from 0.5 to 0.2 or lower restricts the usefulness  of Figure  44  at  the



present time.



      The comparison of the observed field values  of C  with the  laboratory-



determined curve in Figure 44 implies that C  is not a function of the



Reynolds Number but only a function of A.  However,  since Reynolds Num-



bers in the field may be two or more orders of magnitude greater  than in



the laboratory, it may not be possible to merely extrapolate the  C  vs.  A



relation from the laboratory to field sites as is  done in Figure  44.



      Literature references, such as Prandtl and Tietjens (35), present



C  values for solid objects in a moving stream as  functions of a  Reynolds



Number defined in terms of the ambient velocity and viscosity and some



characteristic length, or diameter, of the solid body.  In the present



study, the jet is treated as a solid body in terms of the drag force.



Therefore, it is possible to define a Reynolds Number, ReD, in terms of



the ambient velocity and viscosity and the depth of the jet as shown in



Equation 75,


                                  U z

                           Re  = __§_                                (75)
                             D     v
                                    a




      A plot of Cn vs. Re , shown in Figure 45, indicates that C   may



have the same characteristics as the drag coefficient usually associated



with flow around solid bodies.  The laboratory values of CD decrease as



the Re  Number increases  in the range of  102 < ReQ < I0k, and the field



values of C  become relatively constant  as the Re  Number approaches



values of Re  - 106.  A quantitative relation between CD and ReD   in



Figure 45 cannot be derived, since there  are not enough data points, par-




 ticularly  in the range  of 104 < ReQ  <  105.  However,  a  possible relation

-------
10


 5
. J
           i—r
             D

            A

             A
                D
                            LABORATQRY  RESULTS
                      A
                                  90^
                                      o
O
D  60.0
    45.0°
i     i	1—r
          FIELD RESULTS
             WIDOWS CREEK
             O  VU 1
             D  VU 2
             Q  TVA	
             A  NEW  JOHN-
                SONVILLE
.1
                     D
                                                                      o
           1      1
                             J	L
                  J	L
                103
                                       Re
                                         D
                                                    10
                                         10
       FIGURE 45.--OBSERVED VALUES OF DRAG COEFFICIENT VERSUS REYNOLDS NUMBER

-------
                                                                    140
between C  and Re  is suggested by Figure 45,  and  this  relation  should




be studied further.






                        Possible Sources  of Error





Determination of E and Cn




      Values of E and C  were obtained by fitting  the theoretical  curves




to the observed data.  Since the solutions were in numerical   rather




than analytical  form, best-fit values were obtained by inspection.   Some




of the scatter in the reported results may be due  to the graphical method




of solution, but it  is felt that this source of error is negligible  when




reasonable care is taken in determining E and Cp.






Ambient Turbulence




      The effects of mixing due to ambient turbulence are not considered




in the analytical model.  It is assumed that the decrease in the tempera-




ture is due to the entrainment of ambient fluid, which is caused by the




difference between the jet and the ambient velocities.  As long as the




jet velocity  is greater  than the parallel component of the ambient




velocity, the decrease in temperature can be described by the model.




However,  as the jet  velocity approaches  the same direction and magnitude




as  the  ambient velocity, the model predicts that entrainment and dilution




decrease  as the velocity difference  decreases.  Beyond this point, the




observed  decrease  in the temperature would have to  be  accounted for by




 turbulent diffusion  in the ambient stream.




       Temperature  measurements smaller than T/TQ  =0.10 were not used in




 determining  the  values of  the  entrainment  coefficient.  It was  felt that




 the let velocities beyond  this point had approached the same magnitude

-------
                                                                    141
and direction as the ambient velocity, and that the temperature decreases



beyond this point were caused by the ambient turbulent diffusion.





Density Differences



      Changes in density along the jet axis could, in some cases,  affect



the characteristics of the jet.  However, since the maximum density dif-



ference was approximately A,?/r = 0.9°o in the laboratory and approximately



Ar/o = 0.1°! in the field, changes in density are felt to have been



negligible compared to a reference density, c .




Temperature Losses



      Heat losses across the jet surface can be considered negligible,  as



demonstrated by the following calculations based on the work of Edinger



and ("ever (19) .   The decrease in temperature rise along a stratified



stream can be approximated by Equation 76,



                        T/T  = expC-Kx/c C U z )                     ("61
                           o     J v     o p o o


where K = the thermal exchange coefficient; and



      C = the heat capacity of water.
       P             ^

Applying this equation to a surface jet actually exaggerates the de-



crease in temperature rise due to surface heat exchange, since lateral



mixing of the jet will also reduce the temperature rise and, consequently,



reduce the driving force for surface heat exchange.



      Selecting representative values of K = 200 Btu/ft-i-day-°F



(=2.32 x 10'3 Btu/ft2-sec-°F), c  = 62.4 lb/ft3, and C  = 1.0 Btu/lb °F,



then values of T/T  versus x/z  can be computed for laboratory and field
                  o           o           ^


conditions from Equation 76.  For example, using the laboratory value of



U  = 0.5 ft/sec, Equation 76 predicts T/T  = 0.98 at x/z  = 200.  In the
 a                                       o              o

-------
                                                                    14;
laboratory,  values on the order of T/TQ  =0.10 were measured  at X/ZQ  =  200,



The observed decrease in T/T  due to both lateral mixing  and  surface  heat
                            o

exchange was therefore on the order of 0.90,  while  the predicted  decrease



in T/T  due to surface heat exchange alone from  Equation  76 is  only about
      o

0.02.  Thus, the decrease in T/T  due to lateral mixing and surface heat



exchange is more than an order of magnitude greater than  the  decrease due



to surface heat exchange alone.


      Using the field value of U  =1.06 ft/sec, Equation 76  predicts
                                3.


T/T  = 0.965 at x/z  = 1000, or x/b  = 40.  In the field, values  on  the
   o               o               o


order of T/T  = 0.32 were measured at x/b  = 40.  The observed decrease
            o                            o


in T/T  due to lateral mixing and surface heat exchange was therefore on
      o

the  order of 0.7, while the predicted decrease due to surface heat ex-



change alone from Equation 76 is about 0.04.  Therefore,  since the de-



crease in T/T  due to lateral mixing and surface heat exchange is more
             o

than an order of magnitude greater than the decrease due to surface heat



exchange alone in the laboratory and in the field, then the effects of



surface heat  exchange can reasonably be neglected in the present study.




                               Applicat ion



       The usefulness  of  the  two-dimensional surface jet model in



analyzing field data  appears  to  be  dependent on the velocity ratio and



the  discharge angle,  the value of  the initial Richardson Number, and the



intensity of  the  ambient turbulence.




Velocity Ratio  and  Discharge Angle


       For  the ranges  of A and s'  considered  in  the  laboratory  and in the



 field, the  results  of the present study indicate  that, even  when the

-------
                                                                    143
discharge velocity is on the same order of magnitude as the ambient veloc-
ity, the effects of the discharge velocity should be considered,  as long
as the initial direction of the discharge velocity is significantly dif-
ferent from the direction of the ambient current.  The effect of  changes
in the discharge velocity on the spatial temperature distribution can be
seen by examining the observed laboratory and field trajectories.   As the
velocity ratio is changed, the location of the trajectory is also changed.
The analytical model, which considers the non-uniform velocity field
caused by the surface jet, can be used to describe how these changes in
the velocity ratio affect the temperature distribution.

Initial Richardson Number
      The value of the initial Ri Number can be used to indicate  whether
or not the jet will be two-dimensional.  Jen et_ al_. (29)  report that the
rate of lateral mixing is only twice the vertical rate for the range of
5.0 x 1CT5 < Ri  < 3.0 x 10"3, while the results of Tamai et al.  (43)
               o
indicate that the lateral rate is somewhat greater when the Ri Number
                                           &                  o
is in the range of 1.0 x 10~2 < Ri  < 2.0 x 10'l.  The results of the
                              —   o —
present study indicate that the two-dimensional assumption is valid when
the Ri  Number is as low as Ri  =0.22, which is the lowest of the values
      o                       o
determined at Widows Creek, where the rate of lateral mixing was  found
to be about 8.0 times the vertical rate.  This is reasonable, since
Ellison and Turner (21) and Zeller (48) have all reported for the case of
a surface jet that the value of the Ri Number along the jet axis  rapidly
increases to 1.0 or greater, and that vertical mixing becomes negligible.

-------
                                                                  144
The two-dimensional assumption also  appears  to  be  valid at  even  lower




values of the Ri  Number in cases where vertical  entrainment  is  inhibited




by shallow depths.






Ambient Turbulence



      The effects of ambient turbulence have not  been investigated in




the present study.   Based on the laboratory and field results,  the




analytical model seems adequate to describe the decrease in the tempera-




ture rise to values as low as T/TQ •=• 0.10 as far downstream as  x/bQ =  100.




In highly turbulent streams, the temperature decrease may be even more




dependent on the intensity of the ambient turbulence.






                     Usefulness of the Proposed Model




      When a heated discharge has the characteristics  of a two-dimensional




surface  jet, then  the location of the jet trajectory and the changes in




temperature  and width along the  trajectory  can be determined by the




model developed in the present study.  The  model developed by Zeller  (48)




does  not consider  the pressure gradient  across the jet.  The importance




of  the pressure gradient on the  location of the jet  can be seen by con-




sidering the magnitude  of  the  CD values  determined in  the present  study.




The model  developed by  Carter  (12)  cannot predict the  change in the jet




width along  the jet trajectory.



       It is  felt  that  the  present study  represents an  advance  in  the




 analysis and prediction of temperature distributions.   The usefulness




 of the proposed model  should  increase as more  cases  of field data are




 analyzed to  determine the  values of the entrainment  and drag coefficients




 over a wide  range of  field conditions.

-------
                               CHAPTER VIII









                         SUMMARY  AND  CONCLUSIONS






      A survey of the literature  indicated  that  further study  of surface




lets was justified.  Previous work did  not  contain a completely suitable




method for quantitatively describing the spatial  temperature distribution




in the vicinity of power plants located on  rivers where the discharge and




the ambient velocities should both be considered.  The analytical and




experimental studies performed by Jen,  Wiegel, and Mobarek (29), Tamai,




Wiegel, and Tornberg  (43), and Hayashi and Shuto  (28) are applicable




primarily to jets  discharging  into ambient water bodies that have no




appreciable velocity.  The two-dimensional surface  jet model developed




by  Zeller  (48)  does  not  consider  the pressure gradient that exists across




the surface jet parallel to  the  ambient  current,  while the model developed




by  Carter  (12)  cannot  predict  the change in  the  width  of  the  jet along




the axis.  Therefore,  a  model  is  developed which is able  to better de-




scribe certain cases of  heated power plant discharges.






                           Analytical Development




       The present study, which is based on the  previous work  of Morton




 (54), Fan (22), and Zeller (48), develops  a  system of ordinary differen-




 tial equations, Equations 25,  27, 28,  30,  31, and 32,  which,  when solved




 numerically,  predicts the jet trajectory,  width, velocity, and temperature




 distribution for the case of a two-dimensional surface jet.
                                       145

-------
                                                                     146
      The entrainment mechanism is assumed to represent the turbulent




mixing of the jet.  The inflow velocity, which is assumed to be propor-




tional to the difference between the centerline velocity of the jet and




the parallel component of the ambient velocity, is written as in




Equation 12,




                         v  = E(U - U cos 3)                          (
                          i          a





The  coefficient  E is defined  as  the  experimentally-determined entrain-




ment coefficient.



       The  pressure gradient,  due to  the separation  of  the ambient  flow,




 is assumed to  be represented  by  the  drag force normal  to the jet axis,




 as shown in Equation 17,


                                C n U2(z ds)  sin 6
                                 Da  a
 The coefficient,  C ,  is the experimentally-determined drag coefficient.




 As the jet becomes parallel to the ambient flow,  the drag force goes  to




 zero, because the projected area normal to the ambient current, or




 (z ds sin i?) , goes to zero.



       In practical applications, the zone of flow establishment of the




 jet, illustrated in Figure 8, must be determined.  As in previous work on




 jets in a non-parallel stream,  it was necessary to empirically determine




 how  the length of the  zone of flow establishment and how the initial




 angle at  the end of the  zone are related  to the velocity ratio and to




 the  initial  angle of the discharge.

-------
                                                                    147
                          Laboratory Experiments



      The laboratory experiments were designed to study the entrainment



and drag coefficients and the zone of flow establishment.   The object  was



to functionally relate these coefficients and the zone of establishment



to the velocity ratio and to the initial angle of discharge.   The velocity



ratio, A. = U /U , was varied from 0.180 to 0.727, and lets with discharge
            a  o


angles of a' = 45.0°, 60.0°, and 90.0° were used.



      The values of the entrainment coefficient are presented in



Tables 2, 5, and 4.  The results indicate that E = 0.4 when |3^ = 90.0°,

                        i
and that E = 0.2 when [3  =  60.0° and 45.0°.  Thus, it appears that E is



a function of boundary geometry.  For the range of A used in the present



study, E was not found to be a  function of A.



      The results presented  in  Tables 2 and 5 and Figures 21 and 22



show that C  ranged from 0.1 to 3.8.  It appears that CQ varies with the



velocity ratio, as shown in  Figure  21, and, possibly, with the initial



angle of discharge, as shown in Figure 22.



      The length of the  zone of establishment and the initial angle at



the end of  the  zone were related to the velocity ratio.  The empirically-



determined  relations describing s'/bo vs. A and  PO/BO vs. A are shown



in Figures  18 and  19, respectively.




                              Field Surveys



      Data  obtained  from five field surveys at three  different  steam-



electric generating plants  were analyzed  to see  how well  the two-dimen-



sional surface  jet model describes  the observed  temperature distribu-



tions .

-------
                                                                    148
      The values of E and C   from  the  field  surveys  are  presented  in



Table 15.  The values of E at Widows  Creek are consistently  equal  to



0.16 for three different field surveys with  different  values of



\ = U /U .   The snail value  of E = 0.04 at New Johnsonville  reflects
     a  o


the influence of the relatively narrow boundaries, which apparently



greatly inhibit lateral entrainment.   The value of E = 0.4  at Waukegan



seems reasonable, since there are no  boundaries to inhibit  the lateral



entrainment of cooler water into the  jet. However,  this last value may



indicate the effects of ambient turbulence and vertical  entrainnent,  the



effects of which the analytical model does not consider.  The scatter



of the data at this site makes it difficult  to determine how great these



effects are.



      The values of C  are very close to the values  that would have been



predicted based on the laboratory results, as shown  in Figure 44.



      The laboratory-derived relations describing the zone  of flow



establishment were used in analysing  the field data.  The laboratory



results were used because detailed field measurements of the establish-



ment zone could not be obtained due to time  and equipment limitations.





             Results of the  Laboratorv and Field Investigations
      Results indicate that the value of the entrainment coefficient is



a function of boundary geometry.  The results of the laboratory studies



indicate that lateral inflow, or dilution, is inhibited along the near



boundary when the discharge angle is decreased:   the value of E decreased



from 0.4 to 0.2 when •'  was decreased from 90° to 60° and 45°.
                      o

-------
                                                                    149
      The results of the field studies  indicate that  lateral  inflow on




both sides of the jet is  inhibited by  a narrow channel.   It  appears




that E decreases as the ratio of the channel  width  to the  discharge




width decreases, because values of E =  0.16 were determined  at  Widows




Creek, where the ratio is about 7.5, and a smaller  value of  E = 0.04




was determined at New Johnsonville, where the ratio is about  3.5.




      The field values of C  closely agree with the empirically-determined




laboratory curve, as shown in Figure 44.  However,  the possible dependence




of C  on the Reynolds Number is also considered.  A plot of  CD vs.  the




Reynolds Number, shown  in Figure 45, indicates that C  may have the same




characteristics as the drag coefficient usually associated with flow




around solid bodies.  The laboratory values of C  decrease as the ReQ




Number increases in the range of 102 < Re  <  10\ and the  field values




of C  become relatively constant and equal to about 0.5 at values of




Re  - 106.  It  is felt  that this possible relation between CD and the




Re  Number should be studied further.






                               Application




      The results of the laboratory and field  investigations indicate




that  the application of the two-dimensional surface jet model  is dependent




on the velocity ratio and the  initial angle of discharge,  the value of




the initial Richardson  Number,  and, possibly,  the  intensity of the am-




bient turbulence.




      Even when the  jet velocity  is on  the same order of magnitude as




the ambient velocity,  it appears  that the  temperature distribution can




still be  described  in  terms  of  a  surface  jet,  as long as the initial

-------
                                                                    150
direction of the discharge velocity is significantly  different  from  the




direction of the ambient current.




      The two-dimensional assumption appears  valid when the initial




Richardson Number is as low as 0.22, which is the lowest of the values




determined at Widows Creek, where the rate of lateral mixing was found




to be about 8.0 times greater than the vertical  rate.




      The effect of ambient turbulence is not considered in the proposed




model.  However, the model seems to adequately describe the decrease in




the temperature rise as far downstream as s/b  - 100, unless the ambient




stream is highly turbulent.






                               Future Work





Entrainment and Drag Coefficients




      The accurate prediction of values of E and C  for design  purposes




depends upon additional work.  More field data should be obtained and




compared to the proposed model in order to determine the values of E and




C  at different field sites under widely varying conditions.




      The effects of boundary geometry on the temperature distribution




are reflected in the empirically-determined values of E.  According to




Rouse (58), at some distance downstream in a narrow channel, lateral




inflow becomes negligible, and the volume flux of the jet becomes con-




stant.  Thus, boundary  effects could be treated theoretically by




modifying the system of equations so that lateral entrainraent goes to




zero  at some point downstream.




      The effects of the ratio of the channel width  to  the discharge




width could be studied  in  the laboratory by varying  the width of the

-------
                                                                    151
channel or the size of the jet.   Also,  these effects would have to be




considered in modeling a specific field site.






Vertical Entrainment




      Future work should consider the effects of vertical  entrainment,




which causes a greater decrease in the temperature rise that  the  two-




dimensional model predicts.  Vertical entrairunent occurs when the




shearing force of the jet velocity is greater than the opposing buoyancy




force, which is due to the density difference between  the  jet and the




heavier ambient fluid.  When the initial Richardson Number is very  small,




the rate of vertical spreading is no longer negligible compared to  the




lateral rate and must be considered.




      It appears that the rate of vertical spreading could be related




to the lateral rate in terms of the Richardson Number, which, according




to Ellison and Turner (21), increases with distance along  the jet axis.




Along the jet axis, the vertical rate of spreading would decrease as a




function of the increasing Richardson Number.  A three-dimensional




analytical model would have to be formulated in which  the  decrease in




temperature rise would be greater than that predicted  by a two-dimensional




model but less than the decrease predicted by an axi-symmetric model.




Fietz and Wood (24) have studied the case of a three-dimensional  density




current along a sloping floor.  However, considerable work remains to be




done  in order to develop a useful model for cases of surface jets.






Ambient Turbulence




      Two approaches to the problem of temperature prediction in  the




mixing  zone downstream  from a power plant are currently in use.   The

-------
                                                                     152
first approach,  illustrated by the present  study,  is  to  assume  that  the




heated discharge can be described in terms  of a momentum jet.   The




second approach, illustrated by the work of Edinger  and  Polk  (20), is  to




assume that mixing can he determined by considering  the  diffusion due  to




ambient turbulence.   However, in some cases, as noted by Csanady  (16)  and




others, both jet momentum and ambient turbulence should  be considered.




      Future investigators should consider  combining  the two  approaches.




Pratte and Baines (36) note the importance  of the relative size of the




turbulent eddies compared to the size of the jet cross-section.   Some




suggestions are offered by Briggs  (8), who  has studied plume  rise and




dispersion in the atmosphere.  He  feels that the point at which the  plume




velocity becomes small compared to the ambient velocity can be  charac-




terized by the eddy dissipation, which is a measure of the ambient  tur-




bulence, and some characteristic radius of  the plume.  Such an  approach




might be applied to the case of a  surface jet.






Establishment Zone




      The present derivation of the establishment zone assumed  a  uniform




temperature and velocity distribution at the point of discharge.   However,




at many field sites, the temperature and velocity profiles are  likely to




be non-uniform at the mouth of the discharge channel, and the establishment




zone  is likely  to be shorter than  would be predicted by the present




relations.  More consideration should also be given to predicting the




initial jet depth in terms of the  cold water wedge that, according to




Harleman  (26), may  intrude  into the discharge channel.

-------
                                                                     155
Surface Heat Exchange




      The effects of surface heat exchange should be considered  for  cases




where a large surface area is available for cooling.  To  more  accurately




describe conditions on lakes and wide rivers,  the equation for the  con-




servation of heat could be rewritten to include a "sink"  term, which




would represent the loss of heat across the air-water interface.  Based




on the work of Rdinger and Geyer (19) and Edinger, Puttwciler, and




Geyer  (18), the heat loss term would be a function of the equilibrium




temperature and the thermal  exchange coefficient.







Analytical  Solution




       The present  study solves  a system of equations by numerical inte-




gration  and then  evaluates  the  entrainment and drag  coefficients by in-




spection.   If  an  analytical  solution describing  the location  of the jet




trajectory  and the decrease in  the  temperature rise could be  obtained,




then the coefficients  could be  evaluated,  for  instance, by a  least-




squares  technique.   This  would  eliminate  some  of the subjectivity inherent




in the present method.  At  present,  however, analytical  solutions for




all  but  the simplest cases  of jets  are limited to cases  of irrotational




flow.   Gordier (25),  for  instance,  studied a two-dimensional  slot jet




 in terms of free streamline analysis and  conformal mapping.   The useful-




 ness of ideal flow theory would depend on realistically  approximating




 the turbulent entrainment of the jet.




       These suggestions for future work should help to further refine the




 analysis and prediction of the temperature distribution  in the vicinity  of




 thermal-electric power plants.  It is hoped that the present  study  is




 adequate to serve as a base for future research.

-------
              APPENDIX A
              LABORATORY
LATERAL TEMPERATURE MEASUREMENTS,  T/T
Run
1-90
1-90
1-90
1-90
1-90
2-90
2-90
2-90
2-90
2-90
3-90
3- 90
3-90
3-90
3-90
\^bo
xl/v\
84.5
67.7
SO. 7
33.8
16.9
84.5
67.7
50.7
33.8
16.9
84 . 5
67.7
50 . 7
33.8
16.9
1.41
0.08
0.08
0.10
0.11
0.19
0.04
0.04
0.03
0.03
0.06
0.03
0.02
0.02
0 . 03
0.05
4.51
0.08
0.09
0.12
0.16
0.26
0.04
0.05
0.06
0.11
0.23
0.03
0.03
0.04
0.07
0.15
7.61
0.09
0.10
0.11
0.14
0.07
0.07
0.07
0.10
0.13
0.24
0.05
0.05
0.09
0.13
0.24
10.7
0.10
0.09
0.07
0.05
0.01
0.09
0.10
0.10
0.15
0.16
0.08
0.10
0.12
0.13
0 . 1 9
13.8
0.04
0.04
0.04
0.00
0.00
0.08
0.11
0.12
0. 17
0.10
0.12
0.12
0.13
0.14
0.16
16.9
0.04
0.04
0.01
0.00
0.00
0.09
0.10
0.02
0.09
0.03
0.09
0.08
0.10
0.11
0.07
20.0
O.OJ
0.01
0.00
0.00
0.00
0.06
0.07
0.10
0.05
0.00
0.07
0.07
0.05
0.11
0.00
23.1
0.00
0.00
0.00
0.00
0.00
0.06
0.07
0.04
0.02
0.00
0.07
0.07
0.05
0.08
0.00
26.2
0.00
0.00
0.00
0.00
0.00
0.0.3
0.03
0.01
0.01
0.00
0.05
0.07
0.07
0.08
0.00
29.3
0.00
0.00
0.00
0.00
0.00
0.04
0.03
0.01
0.00
0.00
0.05
0.05
0.08
0.01
0 . 00
32.4
0.00
0.00
0.00
0.00
0.00
0.06
0.04
0.01
0.01
0.00
0.08
0.03
0.01
0.01
0.00
h/bo
7.9
7.4
6.7
5.2
4.2
12.6
11.7
9.8
8.1
6.5
12.5
11.5
11.4
5.0
3.5

-------
APPENDIX A -- Continued
Run
4-90
4-90
4-90
4-90
4-90
5-90
5-90
5-90
5-90
5-90
1 - 60
1-60
1-60
1-60
1-60
1-60
\^bo
x'/bX
o N^
84.5
50.7
33.8
16.9
8.45
84.5
50.7
33.8
16.9
8.45
84.5
67.7
50.7
33.8
16.9
8.45
1.41
0.03
0.01
0.01
0.01
0.02
0.02
0.02
0.01
0.01
0.02
0.09
0.10
0.12
0.18
0.34
0.53
4.51
0.03
0.02
0.04
0.09
0.22
0.02
0.03
0.02
0.04
0.14
0.13
0.18
0.23
0.28
0.20
0.01
7.61
0.03
0.05
0.07
0.17
0.30
0.02
0.07
0.07
0.14
0.25
0.17
0.20
0.17
0.09
0.02
0.00
10.7
0.04
0.06
0.13
0.21
0.24
0.07
0.06
0.11
0.24
0.30
0.09
0.09
0.07
0.03
0.01
0.01
13.8
0.06
0.10
0.15
0.08
0.16
0.07
0.10
0.15
0.20
0.06
0.04
0.04
0.06
0.00
0.00
0.00
16.9
0.09
0.13
0.17
0.02
0.02
0.10
0.10
0.19
0.08
0.01
0.04
0.02
0.01
0.01
0.01
0.01
20.0
0.09
0.11
0.13
0.00
0.01
0.09
0.14
0.17
0.02
0.00
0.01
0.01
0.00
0.00
0.00
0 . 00
23.1
0.13
0.08
0.11
0.00
0.01
0.10
0.14
0.03
0.00
0.00
0.01
0.00
0.00
0 . 00
0.00
0.00
26.2
0.12
0.06
0,05
0.00
0.00
0.11
0.13
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
29.3
0.15
0.06
0.03
0.00
0.00
0.11
0.06
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
32.4
0.13
0.06
0.01
0.00
0.00
0.11
0.00
0.00
o.oc
0.00
0.00
0.00
0.00
0.00
0.00
0.00
b/h0
12.2
11.1
9.4
5.8
6.5
11.6
10.2
7.3
5.5
4.4
7.6
6.9
6.9
6.1
5.7
6.1

-------
APPHNIHX A -- Continued
Run
2- 60
2-60
2-60
2-60
2-60
3-60
3-60
3- 60
3-60
3-60
4-60
4-60
4-60
4-60
4-60
Q\*
84 . 5
67.7
50.7
33.8
16.9
84.5
67.7
50 . 7
33.8
16.9
84.5
50. 7
33.8
16.9
8.45
1.41
0.06
0.06
0.06
0 . 04
0.07
0.05
0.04
0.04
0.04
0.03
0.07
0.05
0 . 04
0 . 03
0.12
4.51
0.04
0.04
0.06
0.13
0.37
0.03
0.04
0 . 06
0.09
0.25
0.05
0 . 0-1
0.05
0. 22
0.40
7.61
0.06
0.07
0. 14
0 . 30
0.24
0.04
0.05
0.11
0.22
0.22
0.04
O.OS
0.15
0.27
0.04
10.7
0.13
0.17
0.23
0.24
0.06
0.09
0.15
0.19
0.21
0.05
0.07
0.14
0.18
0.08
0.02
13.8
0 . 1 5
0 . 20
0.21
0.15
0.01
0 . 1 7
0.21
0.25
0.19
0.04
0.08
0.18
0.18
0.04
0.01
16.9
0.15
0. 15
0.12
0.06
0.01
0 . 1 5
0.17
0.10
0.08
0.01
0.11
0.15
0.12
0 . 03
0.01
20.0
0.10
0.07
0.06
0.01
0.00
0.15
0.11
0.05
0.04
0.00
0.11
0.15
0.04
0.00
0.00
23.1
0 . 0 6
0.07
0.05
0.01
0.00
0.13
0.10
0.03
0.17
0.00
0.11
0.07
0.04
0.00
0.00
26.2
0.06
0.04
0.02
0 . 00
0.00
0.06
0.05
0.01
0.00
0.00
0.08
0.05
0.01
0.00
0.00
29.3
0.04
0.04
0.01
0.00
0.00
0.06
0.03
0.00
0.00
0.00
0.07
0.04
0.01
0.01
0.01
32.4
0.06
0.01
0.01
0 . 00
0 . 00
0.08
0.02
0.01
0.01
0.00
0.14
0.03
0.00
0.01
0.01
h/b
o
11.5
9.8
8.6
6.7
6.1
1 I . 3
9.6
7.8
7.5
	
13.4
10.4
8.2
6.8
6.8

-------
APPF.NinX  A  --  Continued
Run
5-60
5-60
5-60
5-60
5-60
1-45
1-45
1-45
1-45
1-45
1-45
2-45
2-45
2-45
2-45
2-45
\^bo
X'/V,
84.5
50,7
33 . 8
16.9
8.45
84.5
67.7
50.7
33.8
16.9
8.45
84.5
67.7
50 . 7
33.8
16.9
1.41
0.05
0.03
0.03
0.02
0.03
0.09
0.09
0.11
0.13
0.38
0.56
0.10
0.12
0.11
0. 12
0.15
4.51
0.02
0.04
0.04
0.15
0.22
0.13
0.17
0. 25
0.38
0.17
0.02
0.11
0.09
0.11
0. 14
0.43
7.61
0.04
0.08
0.08
0.22
0 . 35
0.18
0.23
0 . 23
0.09
0.01
0.01
0.09
0.11
0.18
0 . 34
0.21
10.7
0.05
0.10
0.15
0.19
0.08
0.17
0.16
0.09
0.04
0.01
0.01
0.15
0.20
0.30
0. 29
0.07
13.8
0.04
0.15
0.19
0.11
0.01
0.06
0.06
0.04
0.01
0.01
0.01
0.18
0 . 22
0.25
0.13
0.01
16.9
0.08
0.15
0.19
0.01
0.01
0.07
0.04
0.02
0.01
0.01
0.01
0.]9
0.21
0.17
0.07
0.01
20.0
0.11
0.11
0.04
0.01
0.00
0 . 04
0.01
0.01
0.01
0.01
0.01
0.13
0.13
0.08
0.02
0 . 00
23.1
0.11
0.08
0.04
0.01
0.00
0.01
0.01
0.01
0.01
0.01
0.01
0.11
0.10
0.06
0.01
0 . 00
26.2
0.11
0.04
0.01
0.00
0.00
0.01
0.01
0.01
0.01
0.01
0.0]
0.08
0.06
0.02
0.00
0 . 00
29.3
0.11
0.04
0.01
0.00
0 . 00
0.01
0.01
0.01
0.01
0.01
0.01
0.06
0.06
0.01
0.00
0 . 00
32.4
0.15
0.03
0.01
0.01
0.0.1
0.01
0.01
0.01
0.01
0.01
0.01
0.09
0.02
0 . 0 1
0.00
0 . 00
b/h0
12.6
10.4
8.1
7.4
6.2
9.2
8.6
8.5
8.6
9.4
9.4
12.3
10.8
8.8
7.0
6.1

-------
APPHNDIX A -- Continued

Run
3-45
3-45
3-45
3-45
3-45
4-45
4-45
4-45
4-45
4-45
5-45-
5-45
5-45
5-45
5-45
X^o
x^v\
84.5
67.7
50.7
33.8
16.9
84.5
50 . 7
33.8
16.9
8.45
84.5
50.7
33.8
16.9
8.45
1.41

0.06
0.07
0.07
0.07
0.11
0.06
0.05
0.05
0.04
0.10
0,06
0.04
0.03
0.01
0.04
4.51

0.03
0.06
0.07
0.11
0.27
0.05
0.05
0 . 09
0.21
0.51
0.06
0.08
0.06
0.19
0.42
7.61

0.05
0.07
0.12
0.25
0.08
0.05
0.08
0.16
0.33
0.02
0.04
0.10
0.17
0.29
0.06
10.7

0.10
0.14
0.20
0.17
0.05
0.09
0.13
0.24
0.05
0.01
0.08
0.11
0.21
0.21
0.02
13.8

0.09
0.17
0.16
0.05
0.00
0.09
0.17
0.21
0.01
0.00
0.08
0.15
0.13
0.06
0.00
16.9

0.14
0.15
0.08
0.03
0.00
0.12
0.17
0.14
0.01
0.00
0.11
0.17
0.15
0.08
0.00
20.0

0.13
0.12
0.05
0.00
0.00
0.09
0.12
0.05
0.00
0 . 00
0.10
0.11
0.10
0.02
0.00
23.1

0.07
0.06
0.02
0.00
0.00
0.09
0.06
0.04
0.00
0.00
0.10
0.10
0.08
0.00
0.00
26 . 2

0.06
0.05
0.00
0.00
0.00
0.08
0.02
0.01
0.00
0.00
0.10
0.06
0.02
0.00
0.00
29.3

0.06
0.03
0.00
0.00
0.00
0.08
0.02
0.00
0.00
0.00
0.08
0.04
0.00
0.00
0.00
32.4

0.03
0.01
0.01
0.01
0.00
0.10
0.02
0.01
0.00
0.00
0.11
0.01
0.01
0.00
0.00
b/b0

11.5
10.4
8.3
6.9
6.6
13.2
10.0
7.8
5.8
5.8
13.2
10.3
8.7
6.6
4.9

-------
                                                     APPENDIX  R
                                        '.OCATION Ol;  LABORATORY TRAJIiCTORIKS
•-D
                 Run 1-1)0
x'/b
o
0.07
0.49
0. 71
1.41
2.82
4.23
5 . 64
7.05
8.46
8.46
12.7
16.9
16.9
33.8
50.7
67.7
84.5
v'/b
0
0.28
0.63
0 . 99
1.41
1.97
2.26
2.26
2.93
3.03
3.03
3.81
4.09
4.09
4.51
4.51
7.61
10.7
*'"•„
0.29
0.80
1.21
1 .96
3.41
4.81
6.2
7.6
9. 1
9.1
13.3
17.6
17.6
34.5
51.4
68.4
85.2
1 .00
1 .00
0.97
0.85
0.75
0.63
0.54
0.48
0.4:
0.47
0.33
0.26
0.26
0.16
0.1?
0. 10
0.10








)


7
7

)
)
)
)
)
)



x'/b
o
0.01
0.01
0.14
0.28
0.71
0.71
1 .41
2. 82
4.2.3
5 . 64
7.05
8.46
12.7
16.9
16.9
33.8
50.7
67.7
84.5
Run
v'/b
0
0.14
0.71
0.71
0.99
1.41
1.55
2.11
3.17
3.67
3.88
4.72
4.44
5.64
5.99
5.99
13.8
13.8
13.8
10.7
2-90
s'/b
o
0.14
0.71
0.72
1 .03
1 .58
1 . 70
2.71
4.15
5 . 79
7 . 09
8.40
9.81
14.2
18.4
18.4
35.3
52.2
69.2
86.0

[ T/T
o
0.97
0.97
1 .00
0.98
0.97
0.97
0.83
0.67
0.57
0.47
0.41
0.37
0.27
0.22
0.24
0.17
0.12
0.11
0.09

x'/h
0
0.01
0.35
0.42
0.35
0.71
0.49
1.41
2.82
4.23
5.64
7.05
8.46
12.7
16.9
16.9
33.8
50.7
67.7
84.5
Run
v'/b
o
0.13
0.78
0.99
1 .06
1.20
1 .41
2.33
2.95
3.95
4.37
4.93
4.79
5.77
6.35
6.34
13.8
13.8
13.8
13.8
3-90
s'/b
o
0.13
0.85
1.07
0. 12
1 . 39
1 .49
2.65
4 . 1 5
5.85
7.25
8.65
10.1
14.4
18.5
18.5
35.3
52.3
69 . 3
86.1

T/T
o
0.99
1 . 00
0.98
1 . 00
0.96
0.97
0.79
0 . 64
0.57
0.53
0.47
0.44
0 . 29
0.25
0.24
0.14
0.13
0.12
0.12

-------
APPENDIX B -- Continued

x'/b
o
0.01
0.01
0.01
0.28
0.71
0.71
1.41

2.82
4.23
5.64
7.05
8.45
8.45
16.9
33.8
50.7
84.5
Run ^
y'/bo
0.28
0.85
1.13
1.55
1.69
2.11
3.17

4.79
4.79
5.50
6.35
7 . 33
7.61
10.0
16.9
16.9
29.3
1-90
s'/b
o
0.28
0.85
1.13
1.57
1.83
2.22
3.35

5.10
6.55
8.09
9.4
11.4
11.4
20.2
37.1
54.0
87.8

T/T
o
1.00
1.00
0.99
0.99
0.96
0.87
0.70

0.52
0.43
0.40
0.28
0.30
0.30
0.21
0.17
0.13
0.16

x'/bo
0.01
0.01
0.01
0.01
0.01
0.71
1.41

2.82
4.23
5 . 64
7.05
8.46
12.7
16.9
33.8
50.7
84.5
Run .
y'/bo
0.07
0.56
0.99
1.13
1.55
1.97
3.38

5.78
5.99
6.91
8.11
9.16
11.1
14.2
16.9
23.1
29.3
5-90
''/"o
0.07
0.56
0.99
1.13
1.55
2.09
3.64

6.30
7.35
9.05
10.4
12.2
16.8
22.0
38.9
55.8
89.6

T/T
o
1.00
1.00
0.98
0.98
0.95
0.85
0.60

0.50
0.40
0.36
0.32
0.31
0.23
0.18
0.19
0.15
0.11

x'/bo
0.07
0.42
0.71
1.41
2.82
4.23
5 . 64
I
7.05
8.46
8.46
12.7
16.9
16.9
33.8
50.7
67.7
84.5
	
Run
y'/bo
0.13
0.25
0.35
0.49
0.77
0.83
1.20

1.41
1.62
1.76
2.26
2.54
2.54
4.51
4.51
7.61
7.61
1-60
''">,,
0.15
0.49
0.79
1.45
2.86
4.30
5.71

7.15
8.55
8.56
12.8
17.0
17.0
33.9
50.8
67.8
84.6

T/T
o
1 .00
0.98
0.96
0.85
0.67
0.59
0.57

0.57
0.55
0.55
0.51
0.41
0.41
0.28
0.23
0.20
0.17

-------
APPHNDIX B -- Continued
Run 2-60
x'/b
o
0.71
1.20
1.41
2.82
4 . 23
5.64
7. OS
8.46
12.7
16.9
16.9
35. 8
SO . 7
67.7
84. S



_Z^
0.56
0.85
1.06
1.90
2.68
2.82
3.24
3.67
5.36
6.35
4.51
7.61
10.7
13.8
13.8



**o
0.90
1.65
1.95
3.60
5.15
6 . 60
8.05
9 . 50
13.9
18.3
18.3
35.2
52.1
69 . 1
85.9



T/T
o
1.00
0.99
0.90
0.84
0.80
0.73
0.69
0.65
0.57
0 . 50
0.37
0.31
0. 23
0. 21
0.16



Run 3-60
x'/b y/b s'/b
o J o o
0.28 0.13 0.31
0.85 0.44 0.96
0.49 0.40 0.63
0.71 0.80 1.07
1 .41 1.2 1 .79
2.82 2.1 3.55
4.23 2.8 5.20
5.64 3.5 6.65
7.05 3.9 8.15
8.46 4.0 9.51
12.7 5.5 13.9
16.9 6.3 18.4
16.9 6.3 18.4
16.9 4.5 18.4
53.8 7.6 35.3
50.7 13.8 52.2
67.7 13.8 69.2
8-1.5 15.8 86.0
T/TO
1.00
0.98
0.99
0.94
0.91
0.77
0.66
0.53
0.49
0.38
0 . 34
0.27
0.31
0.25
0.22
0.25
0.21
0.17
Run 4-60
x'/b
o
0.28
0.71
1.41
1 .69
2.82
4.23
5.6'l
7.05
8.45
8.45
16.9
1 6 . 9
33. 8
50 . 7
84 . S



y'/bo
0.23
0.69
1.10
1.49
2.61
3.07
3.98
4 . 09
5.23
4.51
7.47
7.61
12.3
15.8
20.0



^0
0 . 36
0.99
1.79
2.25
3.80
5.30
6 . 90
8.25
9.90
9 . 90
18.7
18.7
3 5 . 6
52.5
86.3



T/TO
1 .00
1 .00
0.99
0.97
0.75
0.55
0.53
0.43
0.46
0.40
0.30
0.28
0.18
0.18
0 . 1 4




-------
APPENDIX B -- Continued

Run 5-60
x'/bo
0 01
0 71
0.71
141
2.82
4.25
5.64
7 05
8.46
8 45
12 7
16 9
169
^3 8
50.7
84.5


y'/bn
0.13
1.01
1.16
1.96
3.10
4.02
4.51
5.64
5.91
7.61
7.47
8.32
7.61
15.5
15.5
20.0


s''bo
0.13
1.23
1.36
2.35
4.10
5.90
7.35
9.10
10.5
10.5
15.0
19.2
19.2
36.1
53.0
86.8


T/To
0.98
1.00
0.99
1.00
0.60
0.45
0.44
0 . 39
0.37
0.35
0.29
0.24
0.22
0.19
0.15
0.15


Run 1-45
x'/bo
0.01
0.14
0.71
1.41
2.82
4.23
5.64
7.05
8.45
8.46
12.7
16.9
16.9
33.8
50.7
67.7
84.5
y'/b0
0.13
0.17
0.42
0.63
0.85
1.13
1.27
1.34
1.41
1.41
1.83
2.68
1.41
4.51
4.51
7.61
7.61
s'/bo
0.13
0.20
0.82
1.45
2.85
4.30
5.72
7.10
8.49
8.50
12.8
17.1
17.1
34.0
50.9
67.9
84.7
T/To
1.00
0.98
0.88
0.76
0.67
0.65
0.61
0.57
0.56
0.52
0.50
0.49
0.38
0.38
0.25
0.23
0.18
Run 2-45
xVb
o
0.28
0.71
1.41
2.82
4.23
5.64
7.05
8.46
12.7
16.9
16.9
16.9
33.8
50.7
67.7


y'/bo
0.14
0.42
0.85
1.55
1.83
2.40
2.68
3.10
3.95
4.79
4.79
4.51
7.61
10.7
13.8


s'/bo
0.31
0.82
1.60
3.15
4.55
6.10
7.50
8.95
13.2
17.5
17.5
17.5
34.4
51.3
68.3



T/To
1.00
1.00
0.90
0.82
0.78
0.72
0.69
0.63
0.53
0.46
0.49
0.43
0.34
0.30
0.22




-------
APPENDIX B -- Continued
Run 3-45
x'/bo
0.71
1.41
1.62
2.82
4.23
5.64
7.05
8.46
12.7
16.9
16.9
16.9
33.8
50 . 7
67.7
84.5
yVb0
0.30
0.90
0.70
1.41
2.0
2.7
3.1
3.4
5.1
5.5
5.5
4.5
7.6
10.7
13.8
16.9
s'/bo
0.77
1.69
1 . 80
3.25
4.75
6.30
7.75
9.20
13.7
17.9
17.9
17.9
3.48
53.7
68 . 7
85.5

T
00

92
98
90
66
58
51
46
40
34
33
27
25
20
17
14


x'/bo
0.71

1.41
1.41
2.12
2.82
4.23
5.64
7.05
8.45
8.46
12.7
16.9
16.9
33.8
50 . 7
94 . 5

Run
yVbo
0.41

0.97
1 . 00
1.25
1 ,83
2.95
5.52
3.95
4.51
4.65
6 . 49
6 . 63
6.63
35.7
52.6
96 . 4

4-45
s'/bo
0.81

1.75
1.75
2.75
3.65
5.35
6.90
8.40
9 . 90
9.80
14.5
18.8
]8.8
35.7
52.6
96 . 4


T/T
o
1.00

0.99
0.97
0.97
0.92
0.72
0.57
0.51
0.51
0.44
0 . 36
0.31
0.33
0.24
0.17
0.12


x'/b
o
0.21

0 . 49
0.71
0.63
1.41
2.82
4.23
5.64
7. 05
8.45
8.46
12.7
16.9
1 6 . 9
33.8
50.7
84.5
Run
v'/b
o
0.07

0.13
0.44
0.63
I . 11
2.11
2.82
3.38
4.51
4.94
4.51
6.63
7.61
7.61
10.7
16.9
16.9
5-45
o
0.22

0.51
0.83
0.89
1.85
3 . 60
5.25
6.70
8 . 50
9.90
9.90
14.4
18.7
.18.7
35.6
52.5
86.5

_^o___
0.98

1 . 00
1 .00
0.94
0.98
0.91
0.78
0 . 63
0,56
0.42
0.49
0 . 4 2
0 . 34
0 . 29
0.21
0. 15
0. 11

-------
                APPENDIX C






              FIGURES 46 - 73










TRAJECTORIES AND TEMPERATURE AND WIDTH PLOTS




         FOR LABORATORY EXPERIMENTS
                    164

-------
FlGURli 46.--OBSHRVKD  AND FITTliU TRAJfcCTORH-S, RUN  2-90

-------
1.0
0.5
0.1
0.05
0.01
    0.1
                          - 47.5
                        A  -  0.44
                        E = 0.39
0.5      1.0
5.0      10.0
                                         s'/b
                                                                   100.0
                                                                    50.0
                                                                    10.0
                                                                     5.0
                                                                     1.0
50.0     100.0
                                            o
          FIGURE 47.--OBSERVED VALUES AND FITTED CURVES FOR  TEMPERATURE AND WIDTH, RUN 2-90

-------
FIGURE 48.--OBSERVED AND FITTED TRAJECTORIES,  RUN 3-90

-------
          A - 0.30
          E = 0.31
                                                                                  100.0
                                                                               -  50.0
                                                                               -=   10.0
                                                                               -   5.0
                                                                                     1.0
                                                                        50.0   100.0
FIGURE  49.--OBSERVED VALUES  AND FITTED CURVES FOR  TEMPERATURE AND WIDTH, RUN 3-90

-------
10
 8
         r~

    V62.0
      A =0.23
     (V2.0
 0
2
4
8        10
  x'/b0
12
14      16
18      20
                   PIGURH 50.--OBSKRVHU ANU FITTHU TRAJKCTORIHS,  RUN  4-90

-------
1.0
0.5
0.1
0.05
0.01
    0.1
                 A =0.23
                 E = 0.47
                              i        i
0.5      1.0
5.0    10.0
                                                                   100.
                                                                    50.0
                                                                    10.0
                                                                     5.0
                                        1.0
50.0   100.
                                       s'/br
          FIGURE 51.--OBSERVED VALUES AND FITTED CURVES FOR TEMPERATURE AND WIDTH, RUN 4-90

-------
0
                                       8
10
                                         x'/b
12
14
16
18
20
                                             o
                   FIGURE 52.--OBSERVED AND  FITTED  TRAJECTORIES,  RUN  5-90

-------
1.0
jo—oo-o
                A- 0.20
                E= 0.44
                                                                                          0
                                                 5.0    10.0
                                                   50.0    100.
                                         s'/b,
         FIGURE 53.--OBSERVED VALUES AND FITTED CURVES FOR TEMPERATURE AND WIDTH,  RUN 5-90

-------
  10
            i       r
                         i       i        i        r
   8
   6
 o

"V,

-------
1.0
                      A  -0.67
                      E  = 0.24
0.01
                                                                                    50.0     100.
           FJGURIi 55.--OBSJ;RVHD VALUHS AND F ITT1-D  CURVHS  FOR THMPHRATIIRH AND WIDTH,  RUN 1-60

-------
lOi—
  0
                                                                                           18       20
                     •IGURIi 56.—OBSI-RVI-D  AND FITTI-U TRAJECTORIHS,  RUN 2-60

-------
 1.0
 0.5
0.1
0.05
0.01
    0.1
                 V36-5
                  A =0.44
                  E-0.13
 I
       Oi	O
I
0.5    1.0
                                                          100.
                                                               -  50.0
                                           s'/b,
                   5.0    10.0
                                                                   10.0
                                                               -    5.0
                                                                    1.0
                                                                              50.0    100.
         FIGURE 57.--OBSERVED VALUES AND  FITTED CURVES FOR TEMPERATURE AND WIDTH, RUN 2-60

-------
  10
   8
   6
 O
.0
V40'0'
 A =0.30

CD-3.o
   0
                              6
8
                            10
14
16
18
20
                    FIGURE 58.--OBSERVED AND FITTED TRAJECTORIES, RUN 3-60

-------
1.0
0.5
0.1
0.05
0.01
     0.1
                     A =0.30
                      E=0.19
0.5     1.0
                             I	I
                                           s'/b
5.0    10.0
                                                                                I
                                                                 100.
                                                                  50.0
                                                                  10.0
                                                                   5.0
                                        1.0
50.0   100.
                                                                                                       00
          FIGURE 59.--OBSERVED VALUES AND FITTED CURVES FOR TEMPERATURE AND WIDTH, RUN 3-60

-------
  10
  8
 O
_0
T	r
i	r
     V41-0
       A =0.23

     CD-2J
i	r
   0
                       8
        10      12      14      16      18     20
                                    x'/b
                                       o
                  FIGURE 60.--OBSERVED AND FITTED TRAJECTORIES, RUN 4-60
                                                                             '.£>

-------
1.0
0.5
0.1   -
0.05
0.01
     0.1
0.5      1.0
                                               s'/b
5.0      10.0
                                                                                                   1.0
50.0     100.
                                                   o
           FIGURE 6l.--OBSliRVIiU VALUI-S AND FITTliD CURVIiS FOR TBlPIiKATURli AND WIDTH, RUN 4-60

-------
 A = 0.19
Cn =3.4
  FiCURli  62.--OBSI:;RVL;i) AND  H1TTLD TRAJECTORILiS, RUN 5-60
                                                                                         CO

-------
      A =  0.19
      E -  0.29
                                                                                50.0     100.
•JGURl- 03.--OBSURV1-L) VAl.Ul-S  AND !•'ITTIiD  CURVliS FOR Tl;Ml'l:RATURli AND WIDTH, RUN  5-bO

-------
  10
   8
 O
.O
1        I         I        I
                      A =0.66

                    Cp-1.5
                                                                 I         I
                                                                                  18      20
                    FIGURE  64.--OBSERVED AND FITTED TRAJECTORIES,  RUN 1-45
                                                                                                    c»

-------
1.0
0.5
0.1
0.05
0.01
    0.1
                   V25'0'
                    A =0.66

                    E = 0.19
0.5
50.0    100.
                                                                                                          00
         FIGURE 65.--OBSERVED VALUES AND  FITTED CURVES FOR TEMPERATURE AND WIDTH, RUN 1-45

-------
 A-0.42
cIT i.o
 I;ir,URK bb.--OBSl:RVLiU AND FITTED TRAJliCTOR ILiS, RUN  2-45
                                                                                           CO
                                                                                           Ui

-------
              A =0.42

              E - 0.13
                                                                                  50.0    100.
                                                                                                              co
                                                                                                              
-------
10
  8   -
  6   -
o
  4   _
  2   -
  0
16
18      20
                        HGUKI- b8.--OBSl:RVm) AND l:l'lTi;L) TRAJLCTORlliS,  RUN 3-45

-------
1.0
0.5   -
0.1   -
0.05  -
0.01
     0.1
0.5      1.0
5.0     10.0
50.0    100.
                                              s'/b
                                                  o
          FJGURl: 69.--OBSLiRVI;D VALUHS AND F1TTHD CURVliS FOR TliMPLiRATURIi  AND  WIDTH,  RUN  3-45

-------
  10
   8   -
   6  -
 o
.a
   0
ft-33-5
 A =0.23
   2  _
                    FIGURE  70.--OBSERVED AND FITTED TRAJECTORIES, RUN 4-45

-------
1.0
0.5
0.1
0.05
0.01
     0.1
                  $,-33.5
                   A-0.23
                   E-0.27
0.5      1.0
5.0     10.0
                                           s'/b
                                                                   100.
                                                                   50.0
                                                                    10.0
                                                                     5.0
                                                                     1.0
50.0    100.
                                              o
          FIGUKi;  71 .--OBSI'KViiU  VALLIliS AND FiTTFU CURVIiS FOR TBllMiRATURH  A.N'D IV1DT1I,  RUN

-------
  10   _
   8   -
    6  _
  o
JQ
    0
    2   -
18     20
                         FIGURE 72.--OBS1-RV1-U  AND FITTliU TRAJliCTORIl-S,  RUN 5-45

-------
                                                                           50.0    100.
FIGURE 73.--OBSliRVliD VALUES AND FITTED CURVES FOR TEMPERATURE AND WIDTH,  RUN 5-45

-------
                                APPENDIX D



                      PRACTICAL APPLICATION EXAMPLE







       Representative  values  of field parameters have been  chosen  to  il-



 lustrate the practical  application  of  the model developed  in the  present



 study.



       The assumed values  of  the parameters  are summarized  in Table 16.



 The  assumed plant flowrate,  QQ =  2600  ft3/sec, is subtracted from the



 assumed  river  flowrate, OF = 31,900 ft3/sec, to obtain the ambient flow-



 rate,  0   = 29.300 ft3/sec.   The ambient width is assumed to be 2000 feet,
       d                                                               *


 and  the  ambient depth is  assumed  to be  15  feet, giving a cross-sectional



 area of  30,000 ft2.  Dividing  the ambient  flowrate by the area gives



 the  ambient velocity, U   =0.98 ft/sec.
                       3.


      The jet discharge is assumed to have a flowrate of Q  = 2600 ft3/sec,



 a half-width of b  = 100  feet,   and a depth of 10 feet.   Dividing the value



 of Q  by the cross-sectional area, or 2000 ft2, gives the initial jet



 velocity, U  =1.3 ft/sec.  The value of b , the initial half-width at
           o                              o


 the beginning of the zone of established jet flow,  is calculated to be



b  = 160 feet from Equation 65,





                         b0 =  1.60 t/                                 (65)




      The value of the velocity ratio is found to be A = 0.75,  using



U  =0.98 ft/sec and U  =1.3 ft/sec and Equation 38,
 a                    o




                          A - Ua/UQ                                  (38)
                                   193

-------
                                                             194
                          TABLE 16



             ASSUMED VALUES FOR DESIGN PROBLEM
            Parameter                        Value




Q , River Flowrate, ft3/sec               31,900



Q , Plant Flowrate, ft3/sec                2,600



Q , Ambient Flowrate, ft3/sec             29,300
 3


Ambient Width, feet                        2,000



Ambient Depth, feet                           15.0



Ambient Cross-sectional Area, ft2         30,000



U , Ambient Velocity, ft/sec                   0.98
 3.


b', Discharge Half-Width, feet               100.0



z , Discharge Depth, feet                     10.0



Discharge Cross-sectional Area, ft2        2,000



U , Initial Jet Velocity, ft/sec               1.3



b , Initial Jet Half-Width, feet             160.0
 o


A, Velocity Ratio                              0.75



B , Initial Discharge Angle, °                90.0



E, Entrainment Coefficient                     0.25



CD, Drag Coefficient                           0.5



C , Reduced Drag Coefficient                   0.5



T , Initial Temperature Rise, °F              15.0



Ri , Initial Richardson Number                >1.0

-------
                                                                  195
      The initial angle of discharge is  assumed to be  6   =90.0°.   From



Figure 18, the length of the zone of flow establishment  is  estimated



to be s'/b  = 0.75,  using A = 0.75.   From Figure 19,  the initial  angle
       e  o

at the beginning of the zone of established jet flow  is  estimated to be



B  = 34.2°, using A = 0.75 and p1 =  90.0°.
 Q                              Q


       Values  of  the  entrainment  and drag coefficients are  chosen based



 on the laboratory and the limited field results of the  present study.



 The entrainment  coefficient is assumed  to be  E = 0.25,  since  the dis-



 charge width is  one- tenth as  large  as the ambient width, while the drag



 coefficient  is  assumed to be  C  = 0.5,  based  upon the field results


 shown in Figures 44  and 45.   The value  of the reduced drag coefficient


 is found to  be  C'  =  0.5 from  Equation 39,
       A  numerical  solution  for T/T   and b/b  vs. S and for X and Y can
                                  o        o


be  computed, using the values of A = 0.75, (3  -  34.2°, and C* = 0.5.



       Using  the  values s'/b  = 0.75  and E = 0.25, the value of S? is
                         e   o                                    e


found  to be  equal  to  0.2 from Equation 67,





                         S; = C2E//F bo) S;                           (67)





       The values of T/T  and b/b  are then referenced to the discharge



in  terms of  S?,  using the value of s' and Equation 68,
                                     G




                            s' = S* + S                                (68)
                                 e




       Using  the  value of E  = 0.25 and solving Equation 69 for s'/b ,

-------
                                                                    196
                           s'/b  = (vV S?/2E)                        (69)





then, the predicted values of T/T  and b/b  vs. s /b  can be plotted



as shown in Figure 74.



      Using the value of E = 0.25 and solving Equation 42 for x/b  and
                  x/bQ = (vV X/2E)  ;   y/bo = (A~ Y/2E)             (42)





then, the predicted location of the trajectory can be plotted as shown



in Figure 75.  The width of the surface jet in terms of b/b  is also



shown in Figure 75.  It is assumed that x /b  = x/b , and y'/b  = y/b ,
                                            o      o          o      o


since the values of x'/b  and y'/b  were considered to be negligible



when compared to the values of x/b  and v/b  in the field cases of the
                                  o     '   o


present study.



      The application of the model depends on the magnitude of the



initial Richardson Number, accurate prediction of the initial jet depth,



and the intensity of the ambient turbulence.



      Assuming a 15°F initial temperature rise from 70°F to 85°F, the



initial Richardson Number is found to be greater than 1.0, using the



values Ap/p = 2.6 x 10~3, U  = 1.3 ft/sec, LI  = 0.98 ft/sec, and
                           O                3.


z  = 10.0 feet.  Thus, the two-dimensional model should be applicable,



since the value of the Ri  Number > 1.0 indicates a lack of vertical
                         o


mixing.



      Since the initial jet velocity is calculated from the value of



the initial jet depth, accurate prediction of the depth is necessary



before the proposed model can be used.   At present, this depth is

-------
1.0
0.5   -
0.1    -
0.05   -
0.01
     0.1
0.5      1.0
5.0     10.0
                                            s/b
50.0     100.
                                               o
                         FIGURE- 74.-- PREDICTED VALUES OF TEMPERATURE AND WIDTH

-------
20.0   -
15.0   _
10.0   ~
 5.0
    0
5.0     10.0      15.0    20.00    25.0     30.0     35.0    40.0     45.0    50.0
                            x'/b,
                                               o
                           FIGURE 75.-- PRF.niCTF.n TRAJECTORY  AND WIDTH
                                                                                                          '30

-------
                                                                  199
difficult to predict, particularly at field sites where a cold water




wedge would be expected to intrude into the discharge channel.   Harleman's




work (26) could be used as a guide,  hut more research must be done con-




cerning those field sites where the  conditions necessary to apply




Harleman's two-layer stratified flow theory are not met.




      In some cases, ambient turbulence can be expected to cause a




greater decrease in the temperature  rise than the two-dimensional jet




model predicts.  Vertical mixing due to ambient turbulence was  noted at




some of the field sites in the present study.   The effects of ambient




turbulence must be studied further before they can be quantitatively




included in the surface jet model.

-------
                              APPENDIX  R

                           LIST OF  SYMBOLS
a  -  Subscript pertaining to ambient fluid

A  -  Ratio of the ambient velocity to the initial jet velocity,
      U /U  ,   M°L°T0
       a  o
b  -  Half-width of jet, L

C  -  Circumference through which entrainment takes place,  L

C  -  Drag coefficient, M°L°T°

C' -  Reduced  drag coefficient, MCL°T°

dA  -  Differential area of  jet cross-section, L-

d'  -   Initial  let diameter,  L
  o
 e  -  Subscript pertaining  to  establishment  zone

 E  -   Entrainment coefficient, M°L°T°

 exp   -   Exponential  function
 F   -   Drag  term,  L3T~-

 Fr -   Froude  Number, M°L°T°

 a   -   Acceleration of  gravity,  LT~^

 M   -   Non-dimensional  momentum  flux

 o   -   Subscript denoting initial  values

 0   -   Origin  of the coordinate  system (x,y), beginning of zone of
       established flow

 O' -   Origin  of the coordinate  system (x',v'), point of jet discharge

 Q  -   Flowrate, L:'T~l

 T  -   Subscript pertaining to river flow

 Re -   Revnolds Number, M°L°T°
                                  200

-------
                                                                 201
                        APPENDIX E--Continued
Ri -  Richardson Number,  M°L°T°

s  -  Coordinate axis  of  jet referenced to beginning of zone of
      established flow,  L

s  -  Coordinate axis  of  jet referenced to point of jet discharge,  L

s  -  Length of zone of  flow establishment, L

S  -  Non-dimensional  distance along jet axis referenced to beginning
      of zone of established flow

S! -  Non-dimensional  distance along axis referenced to point of jet
      discharge

S  -  Non-dimensional  length of zone of flow establishment
 e
I  -  Centerline temperature rise, °F

T' -  Temperature rise at any point in the jet cross-section, °F

u  -  Velocity along the jet axis, LT~-

U  -  Centerline jet velocity, LT~:

U  -  Ambient velocity,  LT~L
 3-
U0 -  Initial jet velocity, LT"2

Y  -  Non-dimensional x'olume flux

v. -  Inflow velocity, LT~ ^

x  -  Longitudinal axis referenced to beginning of zone of established
      flow, L

x  -  Longitudinal axis referenced to point of discharge, L

x' -  Longitudinal length of establishment zone, L
 e
X  -  Non-dimensional longitudinal axis referenced to beginning of
      zone of established flow

X1 -  Non-dimensional longitudinal axis referenced to point of
      discharge

x' -  Non-dimensional longitudinal length  of establishment zone

-------
                                                                 202
                        APPENDIX E--Continued


y  -  Lateral axis referenced to beginning of zone of established
      flow, L

y' -  Lateral axis referenced to point of discharge,  L

y' -  Lateral length of establishment zone, L
 c
Y  -  Non-dimensional lateral axis referenced to beginning of zone
      of established flow

Y  -  Non-dimensional lateral axis referenced to point of discharge

Y* -  Non-dimensional lateral length of establishment zone

z  -  Depth of jet, L, subscript pertaining to vertical coordinate

B  -  Angle between jet and ambient current, degrees

n  -  Distance along axis perpendicular to s axis, L

v  -  Kinematic viscosity, L2!'1

p  -  Density, FT2L-k

a  -  Standard deviation, L

-------
                           LIST OF REFERENCES
 1.  Abraham, G., "Horizontal Jets in Stagnant Fluid of Other Density,"
        Journal of the Hydraulics Division, Am. Soc. of Civil Eng.,
        Vol. 91, No. HY4, Proc. Paper 4411, July, 1965, pp. 139-154.

 2.  Abramovich, G. N., The Theory of Turbulent Jets, M.I.T. Press,
        Cambridge, Mass., 1963.

 3.  Ackers, P., "Modeling of Heated-Water Discharges," Engineering
        Aspects of Thermal Pollution, edited by F. L. Parker and
        P. A. Krenkel, Vanderbilt Univ. Press, Nashville, Tenn.,
        1969, pp. 172-212.

 4.  Albertson, M. L., Dai, Y. B., Jensen, R. A., and Rouse, H.,
        "Diffusion of Submerged Jets," Transactions, Am. Soc. of
        Civil Eng., Vol. 115, 1950, pp. 639-697.

 5.  Bata, G. L., "Recirculation of Cooling Water in Rivers and Canals,"
        Journal of the Hydraulics Division, Am. Soc. of Civil Eng.,
        Vol. 93, No. HY3, Proc. Paper 1265, June, 1967.

 6.  Beer, L. P., and Pipes, W. 0., "Environmental Effects of Condenser
        Water Discharge in Southwest Lake Michigan," Consulting En-
        gineering Report, Environmental Sciences  Industrial Bio-Test
        Laboratory, Inc., 106 pp.

 7.  Bosanquet, C. H., Horn, G.,  and Thring, M. W.,  "The Effect of
        Density Differences on the Paths of Jets," Proceedings, Royal
        Soc. of London,  Vol. 263A, No. 263, September,  1961, pp. 340-352.

 8.  Briggs, G. A., "Plume Rise," Air Resources Atmospheric Turbulence
        and Diffusion Laboratory, Environmental Science Services
        Administration,  Oak Ridge, Tenn.,  September, 1969, 81 pp.

 9.  Brooks, N. H., Discussion of "Mechanics of Condenser-Water Dis-
        charge from Thermal-Power Plants,  " by D. R. F. Harleman,
        Engineering Aspects of Thermal Pollution, edited by F.  L.
        Parker and P.  A.  Krenkel, Vanderbilt Univ. Press, Nashville,
        Tenn.,  1969,  pp.  165-172.

10.  Burdick, J.  C.  Ill,  and Krenkel,  P.  A., "Jet Diffusion Under
        Stratified Flow Conditions,"  Tech.  Report No.  11, Sanitary
        and Water Resources Engineering, Vanderbilt  Univ., Nashville,
        Tenn.,  1967,  100 pp.
                                 203

-------
                                                                   204
11.   Cairns,  John,  Jr.,  "Effects of Heat on Fish,"  Industrial  Wastes,
        Vol.  1,  No. 5,  May-June, 1956,  pp.  180-183.

12.   Carter,  H.  H., "A Preliminary Report on the Characteristics  of a
      .  Heated Jet  Discharged Horizontally into a Transverse Current,
        Part  I - Constant Depth," Tech.  Report No.  61,  Chesapeake Bay
        Institute,  The Johns Hopkins Univ., Baltimore,  Md.,  November,
        1969, 38 pp.
              ;  i
13.   Cederwall,  K., "Jet Diffusion: Review of Model  Testing  and Com-
        parison with Theory," Hydraulics Division,  Chalmers  Institute
        of Technology,  Goteborg, Sweden, February,  1967,  28  pp.

14.   Churchill,  M.  A.,  "Effects of Density Currents  in  Reservoirs on
        Water Quality," Water and Sewage Works, Reference No.  1965,
        November, 1965.

15.   Clark, J. R.,  "Thermal Pollution and Aquatic Life,"  Scientific
        American, Vol.  220, No. 3, March, 1969, pp.  19-26.

16.   Csanady, G. T., "The Buoyant Motion Within a Hot Gas Plume in a
        Horizontal  Wind," Journal of Fluid Mechanics, Vol.  22, 1965,
        pp. 225-239.

17.   Edinger, J. E., Discussion of "The Cooling of Riverside Thermal-
        Power Plants," by Andre Goubet, Engineering  Aspects  of Thermal
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                                                                  205
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                                                                  206
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                                                                   207
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1 Accession Number
w
n 1 Subject Field & Group
05D, 02E
SELECTED WATER RESOURCES ABSTRACTS
INPUT TRANSACTION FORM
     Organization
               Department of  Environmental and Water Resources Engineering,
            Vanderbilt University,  School of Engineering, Nashville, Tennessee

               HEATED SURFACE JET DISCHARGED INTO A FLOWING AMBIENT STREAM
 10
  Aathorfs)


    Motz,  Louis H,

    Benedict,  Barry A.
|£ I Project Destination

         FWQA Contract  No.  16130FDQ
                                      21J
    Note
 221
2 I Citation
          Federal  Water Quality Administration Water Pollution Control Research Series,
    Office  of Research and Development, Report based  on Contract No. 16130 FDQ, March,1971
 23
  Descriptors (Starred First)
    *Cooling Water,  *Thermal Pollution, *Jets, Turbulent  flow,  *Heated water
    *Thermal power plants, Thermal stratification, Diffusion,  Water temperature,
    Heat  exchange,  Temperature
 25
  Identifiers (Starred first)
      *Heat  discharge, Density differences, Buoyant Jets,  Ambient Fluids,
       Temperature profiles, Widows Creek, New Johnsonville,  Waukegan Surveys
 27
  Abstract
           The temperature distribution in the water body due to a discharge of waste
  heat  from a thermal-electrical plant, is a function of the hydrodynamic variables
  of  the  discharge and the receiving water body.  The temperature distribution can be
  described in terms of a surface jet discharging at  some initial angle to the ambient
  flow  and being deflected downstream by the momentum of the ambient velocity.  It is
  assumed that in the vicinity of the surface jet, heat  loss to the atmosphere is
  negligible.  It is concluded that the application of the two dimensional surface jet
  model is dependent on the velocity ratio and the initial angle of discharge, and the
  value of the initial Richardson number, as low as 0.22.  Both laboratory and field
  data  are used for verification of the model which has  been developed.  Laboratory
  data  is used to evaluate the two needed coefficients,  a drag coefficient and an
  entrainment coefficient, as well as the length of the  zone of flow establishment and
  the angle at the end of that zone.
Abstractor
                               Institution
       
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