EPA-R2-72-005b
    May 1974
                     Environmental Protection Technology Series

    Workbook of Thermal  Plume
    Prediction-Vol.2-Surface Discharge
                                  I
                                  55
                                  \
SSSZ
\
 LLJ
 CD
                          National Environmental Research  Center
                          Office of Research and Development
                          U.S. Environmental Protection Agency
                          Corvallis, Oregon  97330

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            RESEARCH REPORTING SERIES
Research reports of the  Office  of  Research  and
Monitoring,  Environmental Protection Agency, have
been grouped into five series.  These  five  broad
categories  were established to facilitate further
development  and  application   of   environmental
technology.   Elimination  of traditional grouping
was  consciously  planned  to  foster   technology
transfer   and  a  maximum  interface  in  related
fields.  The five series are:

   1.  Environmental Health Effects Research
   2.  Environmental Protection Technology
   3.  Ecological Research
   4.  Environmental Monitoring
   5.  Socioeconomic Environmental Studies

This report has been assigned to the ENVIRONMENTAL
PROTECTION   TECHNOLOGY   series.    This   series
describes   research   performed  to  develop  and
demonstrate   instrumentation,    equipment    and
methodology  to  repair  or  prevent environmental
degradation from point and  .non-point  sources  of
pollution.  This work provides the new or improved
technology  required for the control and treatment
of pollution sources to meet environmental quality
standards.

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                                                      EPA-R2-72-005b
                                                      May 1974
             WORKBOOK OF  THERMAL PLUME  PREDICTION

                             Volume 2

                        SURFACE DISCHARGE
                               By

                       Mostafa A. Shirazi
                         Lorin R. Davis
                   Thermal  Pollution Branch
      Pacific Northwest Environmental Research Laboratory
                        Project 16130 FHH
                    Program Element 1BA032
            NATIONAL  ENVIRONMENTAL RESEARCH CENTER
              OFFICE  OF  RESEARCH AND DEVELOPMENT
             U.S. ENVIRONMENTAL PROTECTION AGENCY
                   CORVALLIS, OREGON   97330
For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 - Price $4. 5

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                              •PREFACE

 In a contlnulnq effort to present current knowledge on heated plume
 prediction to the public, nomograms are presented in this second volume
 that describe the behavior of surface jets for a wide range of ambient
 and initial discharae conditions encountered in practice.  As in the
 first volume, an attempt is made to present the material in a concise
 manner and in a format that is clear and accessible to a nonspecialist
 user.  Many fundamental derivations are left outside the body
 of the workbook and retained for further reading in the appendix.
 These undoubtedly would be of use to the specialist researcher who
 seeks to advance the status of knowledge.

 The nomoarams provide qualitative results describing the surface
 olume trajectory, width, temperature, depth, surface area and time
 of travel along the plume center!ine.  The nomograms are not intended
 to be used as exclusive design tools for surface discharge problems
 nor for use in a nrecise prediction of any specific surface plume
 condition.

 The nomoqrams are generated predominately from an idealized mathematical
 model of a plume.  Some field and laboratory data have been used to
 ad.iust the performance of the model so that more realistic predictions
 are obtained.  However, the class of problems that can be handled this
wav are limited due to the limitations in the model itself.  We have
made an earnest attempt to help the nonspecialist user by pointing out
the main restrictions included in the model  both in a special chapter
in the workbook as well  as in example problems.
                                 111

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                              CONTENTS

                                                             Page

Preface                                                      111
List of Symbols and Dimensionless Numbers                    v - vii
List of Figures                                              viii
List of Tables                                               ix

Sections-

I    Introduction                                            1-14

II   Nomogram Organization                                   15
     A - (TTWD) Working Nomograms                            16-1.7
     B - (TA) Working Nomograms                              18
     C - (Tt) Working Nomograms                              18-20
     D - (TTWD) Supplementary Nomograms                      20-22

III  Analytical Considerations                               23
     A - Model Description                                   23
         1.  Model Idealization                              24-25
         2.  Other Limitations                               25-28
     B - Model Applicability                                 28-29

IV   Example Problems                                        30

V    Acknowledgement                                         42

VI   References                                              43

VII  Appendices
     A - (TTWD) Working Nomograms                            44-177
     B - (TA) Working Nomograms                              178-251
     C - (Tt) Working Nomograms                              252-325
     D - (TTWD) Supplementary Nomograms                      326-354
     E - Temperature Density Nomograms and                   355-365
              Computational Aids
     F - Further Analytical Considerations                   366-430
              and Computer Program
                                 IV

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                             SYMBOLS

.   	9    °        Channel (or jet) aspect ration
    H0  =HQ
B                   Characteristic (half) plume width = *^~cr

2BQ                 Channel (or jet) full width

C                   Specific heat of water

D/H                 Dimensionless plume depth = &  H/HQ

E, or E             Jet entrainment coefficient

E.  , E               Horizontal and vertical turbulent exchange coefficients
 F                   Jet dens i metric Froude number, U / (— gH )

 g                   Acceleration of gravity

H                   Characteristic plume depth = & a

 H                   Channel (or jet) depth

 K                   Dimensionless surface heat exchange coefficient
                    KE/pCpUo

KF                  Absolute surface heat exchange coefficient commonly
 •                    -.     .   Btu
                    91Ven  in
R                   U/U  , ambient to jet velocity ratio
                     a  O

S                   Length along plume center! ine trajectory

T = AT /ATQ         Dimensionless excess center! ine temperature
                   - ratio
                                  v

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T                   Ambient temperature
 a

(TA)                (Temperature, Area) Nomogram


T  = T              Surface tmperature on centerline  (
 v*    »5

T                   Jet temperature


TH = ©Q             Initial discharge angle


(Tt)                (Temperature, time) nomogram


(TTWD)              (Temperature, Trajectory, Width, Depth) Nomogram
U  = V              Ambient velocity
 a

U                   Jet velocity

W                   Dimensionless plume full width = 2JZ B/H

WQ                  Channel full width = 2 B

X                   Longitudinal coordinate of plume centerline -
                    along the shoreline or direction of the uniform
                    ambient current
Y                   Lateral coordinate of plume centerline - perpendicular
                    to the shoreline or direction of the uniform ambient
                    current

0                   Angle of jet discharge with respect to ambient current
                    expressed in degrees and measured relative to the
                    direction of ambient current

AT                  Centerline excess temperature = T  - T  = T  - T
  c                 ,                                 c    a    s    a
                                 v1

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AT                  Discharge excess temperature = T  - T
  o                                                 o    a
v                   Kinematic viscosity





pa                  Ambient water density
p                   Jet discharge density




a                   Standard deviation of horizontal  temperature


                    distribution at surface
0                   Standard deviation of vertical temperature


                    distribution on plume centerline

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                               FIGURES
1     Plume Trajectory for Surface Jet Showing
     Effects of Ambient Current.

2    Plume Trajectory for Surface Jet Showing                  ^
     Effects of .Jet Densimetric Froude Number.

3    Plume Trajectory for Surface Jet Showing                  7
     Effects of Jet Aspect Ratio.

4    Plume Trajectory for Surface Jet Showing                  8
     Effects of Discharge Angle.

5    Plume Trajectory, Temperature, Width and         ,        10
     Depth for Surface Jet Showing Effects of
     Ambient Current.

6    Plume Trajectory, Temperature, Width and Depth           11
     for Surface Jet Showing Effects of Jet Densimetric
     Froude Number.

7    Plume Trajectory, Temperature, Width and Depth           13
     for Surface Jet Showing Effects of Jet Aspect
     Ratio.

8    Plume Trajectory, Temperature, Width and Depth           14
     for Surface Jet Showing Effects of Discharge
     Angle.
                                viii

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                              TABLES

Table                                                       Page

1    Flqure numbers for (TTWD) working Nomograms            17 & 45
     of Appendix A, GO = 90° and K  = 1Q-5

?.    Summary of Flqure Numbers for (TTWD)                   17 & 45
     Working Nomograms of Appendix A

3    Fiqure Numbers for (TA) Working Nomograms              19 & 179
     of Appendix B, 9Q  = 90°, K  = 10"5

4    Summary of Figure Numbers for (TA) Working             19 & 179
     Nomoarams of Appendix B

5    Figure Numbers for (Tt) Working Nomograms              21 & 253
     of Apoendix C, 0Q = 90°, K  = 10-5

6    Summary of Figure Numbers for (Tt) Working             21 & 253
     Nomoqrams of Appendix C

7 -   Figure Numbers for (TTWD) Supplementary                22 & 327
     Nomoqrams of Appendix D, 9rt = 90°,
 *   K = 10-", F = 4, A • 5    °

*    Summary of Figure Numbers for (TTWD)                   22 & 327
     Sunplementary Nomograms of Appendix D
                                 ix

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                           I.  INTRODUCTION

The  surface  discharge  of heated water  is often  chosen  over the
 submerged  discharge  because it affects  a greater rate  of heat
 transfer to  the  atmosphere due to  normally  higher surface temperatures
 and  it  is  more economical  due to the relatively simple discharge
 structure.

 The  specific plume behavior from a surface  discharge  is a result
 of a complex interaction of such factors as (a) the jet discharge
 characteristics, (b) the ambient water current  and turbulence  level,
 (c)  the atmospheric  conditions, and (d) the bottom and shoreline
 geometries.   Since the precise effects  of all factors  and their mutual
 interaction  on a plume in a natural environment are not well understood,
 only solutions to idealized situations  can  be obtained.  Briefly,
 solutions  presented  in this workbook are limited to a  uniform  and
 constant  (steady) surface discharge of heated water from a rectangular
 channel into a large and deep body of  water that is either at  rest
 or moving  at a uniform and constant velocity.   Details of these
 and  other  limitations  are given in Section  II and in Appendix
 F which the  reader is  urged to review  before using the results.

 Meanwhile, we will give an overview of the  material in the workbook
 with the objective to  familiarize  the  prospective user with (a)
 the  manner with  which  the plume characteristics are presented
 and  (b) the  physical interpretation of the  results.

 The  plume  characteristics in  this  book are  presented  for several
 values  each  of the initial jet discharge, ambient flow, and atmospheric
 conditions.   However,  for the sake of  convenience let  us limit our

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attention in this introductory section to only a few of the physical
factors involved.  These are:  the jet velocity, U ; the jet
                 ",   j  '                            U
temperature, T   (or density, p ) ; the jet depth, HQ; the jet half width,
B  ;  the ambient velocity, U ; and the ambient temperature T  (or density
 o                          a                               a
Since information on all four plume characteristics, namely,
trajectory, centerline temperature, plume width and plume depth is
needed, plots of these characteristics must be presented for each
variation in U  , T  , H , B  , U  , and T .  Clearly, a handbook
              o   o   o   o   a       a
containing such expanded information is difficult to use and a
modest number of parametric representations of the above information
may run into several thousand pages.  Considerations of other factors
such as angle of discharge  (9 ), ambient turbulence, etc., would add
considerably more pages to the book.  However, by using appropriate
dimensionless numbers and by combining information on all four plume
characteristics on a single plot, it becomes possible to present
the same material in less than two hundred pages.

The six parameters U , T , H , B , U  and T  can all be collected
in three dimension! ess numbers defined by R = U /U  for the velocity
                                               a  o
ratio, A =  2B /H  for the channel  aspect ratio, and F = U /VAp/p H q
              oo                                        o      o o3
for the jet densimetric Froude number.  Note that the water densities
p  and p  (with Ap' = p -p ) are related to the temperature T  and T
 a      o             a  o                                  so
respectively, and g denotes the acceleration of gravity.
The jet densimetric Froude number represents the relative strength
of two forces imparted initially to the discharged water, one due
to the combined water mass and its initial velocity (inertia forces)
and the other due to the initial density difference of the discharged
water volume with respect to the ambient (buoyancy forces).

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Consider as a first example the plots shown in Figure 1 for F=4, A=5
representing a surface discharge at ninety degrees into an ambient current
The plots are made for a typical dimensionless surface heat exchange
coefficient K.  Its value will be kept constant for all calculations
given in this chapter. The curves marked R=0.01, 0.05, 0.1, ... etc.,
are the spatial coordinates (also called trajectories) of the plume
centerlines whose initial jet discharge velocities (U )  are, respectively
one hundred times, twenty times, ten times, ... etc., the ambient
current velocities.  The trajectory of the plume initially discharging
into a zero ambient current (i.e., R=0) is coincident with the ordinate
(Y) of the plot.  Note also that both coordinates (X and Y) are presented
in terms of dimensionless numbers, consistent with other uses in
the plot.

The important physical observation drawn from Figure 1 is that plume
penetration across a weak ambient current, say on the order of 5
to 10 percent of the initial jet velocity, is substantial.  However,
plume penetration is inhibited considerably by an incremental relative
increase in the ambient current.  A more specific observation related
to those just mentioned is that the plume trajectory bends over
more sharply in a strong ambient current than in a weak current.
The cross current momentum of the jet, while initially at a maximum,
is forced by the strong ambient current to change direction rapidly thus
causing a relatively small initial plume penetration.  At some distance
from the source the plume is eventually carried along the general
direction of the ambient current even if the latter is small.

Figure 1 shows plume trajectories for only the stated initial jet
flow characteristics, namely for F=4 and A=5.  In general, the degree
of penetration depends upon the magnitude and the distribution of
the initial plume momentum in the cross current direction as well
as on the momentum of the ambient flow field.  Since the initial

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 o

I
UJ
O
Q
-1
<
or
in
           20   40   60   80  100   120   140  160   180   200
                  LONGITUDINAL DISTANCE  X/H0
        Figure  1    Plume Trajectory for Surface Jet Showing
                   Effects of Ambient Current.

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magnitude and distribution of jet momentum depend on A, F, and 0 ,
it follows, therefore, that the degree of plume penetration across
the current and thus the trend shown in the above figure is modified
with changes in these factors.  These trends are shown in Figures
2, 3 and 4 as discussed below.

In Figure 2 the Froude number is varied while other parameters are
held constant.  The trajectories show a substantial influence of
the jet discharge Froude number on plume penetration.  The plume
penetration across ambient current is greater for small Froude numbers
than with large ones.

A physical explanation for this phenomenon can be attempted by reasoning
                                             /
that (a) the buoyancy force adds cross current momentum to the initial
inertia forces thus providing a forward driving force in that direction,
and  (b) the same buoyancy forces cause the plume to float on top
of the cooler ambient water and to "thin out" vertically thereby
reducing jet-current interaction and increasing plume penetration.
In this particular case penetration was further enhanced because
the jet is initially five times as wide as it is deep.  For larger
channel aspect ratios,  we should expect even less interaction with
the cross current and a greater plume penetration, as shown by the
plots in Figure 3.  Conversely,  for a small aspect ratio where the
plume tends to block the flow of the ambient water, the interaction
is large  and the plume penetration is small.

The effects of the initial discharge angle on plume trajectory for
F = 6, A = 10, and R = 0.1 are shown in Figure 4.  The results support
the common intuition that a plume with a shallow initial discharge
angle, say 60 degrees, does not penetrate as effectively into an
ambient current as one with an angle of 90 or 120 degrees.

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  20   40  60   80   100  120   140  160  180  200
         LONGITUDINAL DISTANCE X/H0
Figure 2    Plume Trajectory for Surface Jet,Showing
           Effects of Jet Densimetric Froude Number.

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    20   40  60   80  100  120   140
                                   180   200
           LONGITUDINAL DISTANCE X/H0
Figure 3
   "Hi.
Plume Trajectory for Surface Jet Showing
Effects of Jet Aspect Ratio.

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-20
20   40   60    80   100   120   140   160   180




LONGITUDINAL DISTANCE X/HO
    Figure 4.   Plume  Trajectory for Surface Jet Showing

               Effects of Discharge Angle.
                                             ''
                       8

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In addition to the plume trajectory,  Figures  1, 2, 3, and 4 provide
information on the local plume depth  D/HQ.  Note that centerline
plume trajectories are plotted in broken lines.  The length of each
broken line segment along the plume trajectory is a measure of the
local plume depth as shown by the accompanying D/H  scale in each
Figure.  Referring now to Figures 1,  2, 3 and 4, one can see from
variations in the lengths of these segments the effects; respectively
of the ambient current,  Froude number, aspect ratio and discharge
angle on the plume depth.  These effects are  most evident in Figures
2 and 3.

The plume depth  generally increases rather rapidly at first, then
it either becomes small  and remains essentially constant because
of stratification or it  continues thickening  (at a low rate) due
to jet and ambient mixing.  Plumes with lower Froude numbers tend to
become thin and  stratified while those with higher Froude numbers tend
to thicken.

It is important  when attempting to explain these results in physical
terms to make a  mental note of the actual problem being solved.   Moving
from one trajectory to another of Figures 2 and 3, in practice, may
require a change in the  channel depth, relative to which magnitude all
plume coordinates and local depth are measured.  Consequently, a direct
comparision of the relative magnitudes may not be meaningful.

The foregoing figures contain   information on the plume trajectory and
depth.  It is possible to superimpose on these figures other plume
character!'si tics such as centerline temperature and plume width,
thereby eliminating the  need for presenting additional figures as
well as providing at once a more complete picture of the plume in
a single plot.   Examples of this manner of presentation are given-
in Figures 5, 6, 7, and  8 corresponding respectively, to Figures
1, 2, 3 and 4.
                                  9

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300
 20
         20   40   60   80   100  120   140  160   180  200
                LONGITUDINAL DISTANCE X/H0
   Figure 5  Plume Trajectory, Temperature, Width and Depth for
            Surface Jet Showing Effects of Ambient Current.


                       10

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          20   40   60   80   100   120  140   160   180  200
                 LONGITUDINAL DISTANCE X/H0
Figure 6  Plume Trajectory, Temperature, Width and Depth for Sur-
         face Jet  Showing Effects of Jet Densimetric Froude Number.


                          11

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The local excess center!ine temperature ATC, is plotted 1n Figure 5
as a fraction of the initial excess temperature ATQ.  The local
plume width W is also plotted in this figure as a dimensionless number.
The local width is taken equal to four standard deviations of the
temperature distribution when such distribution is assumed Gaussian.

The following observations are based on the analysis of Figure 5.
(1) The effect of a high ambient current is to reduce the centerline
plume temperature by causing additional entrainment.  (2)  The plume
width generally varies inversely with changes in the ambient current.
Thus when the ambient water is nearly stagnant one expects to find
a hot and wide plume.  Figures 6, 7, and.8 show that hot and wide plumes
are also caused by low discharge Froude numbers, large aspect ratio and
large discharge angle.

A general physical interpretation of the above can be presented as
follows.  We have discussed the process of interaction with respect
to plume trajectories in Figures 1 through 4.  Jet interaction is
greatest at a high Froude number and low aspect ratio.  Figures
6 and 7 are plotted to demonstrate this effect.  They show a narrow
and cool plume when there is a strong interaction and a wide and
warm plume when there is a weak interaction.  The effect of the discharge
angle is somewhat complicated as shown in Figure 8.  It shows that the
plume is wider and hotter for a discharge angle of 120° than for 90
                                                               !
or 60°.   This is true at least near the source.  If calculations are
allowed to continue, as will in the working nomograms, this trend
is shown to be reversed.

The four plume character! si tics, namely, trajectory, temperature,
width and depth, are combined in generalized nomograms similar to
Figure 5.  Their discussion and other plume characteristics such
as surface area and time of travel are presented in the next section.
                                 12

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     20   40  60   80  100   120   140  160
200
            LONGITUDINAL DISTANCE  X/H0
Figure 7  Plume Trajectory, Temperature, Width and Depth for
         Surface Get Showing Effects of Jet Aspect Ratio.

                    13

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°-20   0    20   40   60    80   100   120  140   160   180


             LONGITUDINAL DISTANCE  X/H0
 Figure 8  Plume Trajectory, Temperature, Width  and Depth for
          Surface Jet Showing Effects of Discharge Angle.
                       14

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                       II NOMOGRAM ORGANIZATION
                               . 'i.
Nomograms are presented in  two  distinct  functional groups.  The
first group given in Appendices A,  B, and C  are  representative of
typical situations encountered  in practice.  They do not  represent
extreme conditions according  to our judgment.   This group of nomograms
constitutes the main body of  the workbook.  The second group of nomograms
is given in Appendix D to show  the  variations in the results due
to changes in certain  of the  input  parameters.  Ordinarily  these
are of little concern  to the  prospective user seeking qualitative
results.

There are at least three reasons for presenting the additional nomograms
of Appendix D.  First, the  user might have  a problem that he considers
an extreme situation.  In this  case he will identify a possible variation
of one or more input parameters to  his problem  and thus use the appropriate
nomogram.  Second, a researcher seeking  information on the  sensitivity
of the model to possible variations of the  input parameters can make
direct use of the nomograms and thus spare  himself the ordeal of
obtaining an operational computer program of the model for  that purpose.
The third reason for providing  the  supplementary material in Appendix
D is for the sake of completeness and an admission of the fact on
our part that'the results presented in Appendices A, B, and C for
general use, while reasonable according  to  present information, are
                                                       V
not the final word.  This is  because the resolution of the  available
data is not sufficiently  good  to support a firm stand on details.
                                 15

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A - (TTWD) WORKING NOMOGRAMS

Parametric  information on the local plume width, depth and center!ine
temperature and trajectory is given in Appendix A.  A typical plot
of trajectory, temperature, width, and depth (TTWD) is given in
Figure 5 and discussed at considerable length in the introductory
section.

A summary of the entire set of nomograms in Appendix A is listed
in Tables 1 and 2.  The (TTWD)-plots are subgrouped conveniently
in a 4 by 4 matrix of aspect ratio and Froude number variations,
respectively,as shown in Table 1.  The matrix elements Al, A2,...,
                /•
A16 refer to the appropriate figure numbers in Appendix A.  Each
subgroup of 16 plots provides plume information for a fixed initial
discharge angle and heat exchange coefficient.  The numerical values
of these variables are <
referred to in Table 1.
of these variables are 0  = 90° and K =10~5  for all  the  16  plots
Table 2 is a 3 by 3 matrix showing variations of 0   and K, respectively.
Note that each of the entries in this matrix refer to subgroups of
plots similar to Table 1.  Note also that Table 1 is the first matrix
entry of Table 2.  Since it is likely that plots of the intermediate
values of 0  and K  receive greater use, we have placed them as the
first entry for convenience.

In order for the user to locate a specific plot, some cross-referencing
of the formats in Tables 1 and 2  is required.   It is helpful to
illustrate this by an example.  The figure that gives the plume characteristics
for F=4, A=5, 6o=60° and K =10"4 is Figure A70.  Seventy is the sum of
two numbers, namely, 64 (from Table 2) plus 6 (from Table 1) where 64
is the highest figure number in the preceding matrix element.
                                 16

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                    TABLE 1
  Figure numbers for (TTWD) Working Monograms
     of Appendix A, G= 90° and K = 10-5
F ->
+A
1
5
10
15
2
Al
A5
A9
A13
4
A2
A6
A10
A14
6
A3
A7
All
A15
10
A4
A8
A12
A16
                    TABLE 2
        Summary of Figure Numbers for (TTWD)
          Working Nomograms of Appendix A
V
+K
io-5
ID'4
io-6
90°
A1-A16
A49-A64
A97-A112
60°
A17-A32
A65-A80
A113-A128
120°
A33-A48
A81 -A96
A128-A144
NOTE:  Figures A-37, 41,  45,  46,  85, 89,  93, 94,
       133, 137, 141, 142 are not included due to
       computational difficulties.
                        17

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B -  (TA) WORKING NOMOGRAMS

The  nomograms of Appendix B contain information on water surface
area within  the plume that equals    or exceeds the designated excess
temperature  contour namely, temperature-area (TA)-plots. The areas
plotted  are  normalized by the square of the discharge depth.   There
are  nine sets of plots subgrouped as 8 plots to a set as shown in
Table  4.  The format for each subgroup is detailed in Table 3.  The
combined use of Tables 3 and 4 is identical to the previous example
for  the  use  of Tables 1 and 2.  For instance, if we require information
on area  for  Figure A70 alone, we must refer to Figure B35  (i.e.,
32 + 3).  Note also from Table 3 that Figure B35 contains information
on plumes for F=2, A=5.

C -  (Tt) WORKING NOMOGRAMS

The  nomograms of Appendix C contain information on the time of
travel of a  parcel of water along the trajectory of the plume.
Since  the plume temperature is not constant, the parcel of water
is exposed to continually varying temperature levels.  The time of
exposure has been calculated in these nomograms for the centerline
temperature  and plotted in terms of the total exposure time, t,
to a temperature excess ratio equal to or greater than AT /AT  .
                                                         t*   \J            *•
For  instance, the total time of exposure of a parcel of water  that
is entrained near the discharge (say, when AT /AT  * 1) to a point
along  the plume trajectory whose temperature ratio is, say, 0.1,
can  be read  directly from the plots.  However, since entrainment
could  take place anywhere along the plume trajectory, say at a point
where  the temperature ratio is 0.3, the total exposure time to temperature
ratios b'etween 0.3 and 0.1 is the difference between two direct
readings of  the times.  Note also that in these plots, the exposure
time t is made dimensionless by multiplying by U /H  .
                                 18

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                  TABLE 3
Figure Numbers for (TA) Working Nomograms
    of Appendix B, 6  = 90°, K = 1Q~5
F-*
+ A
1
5
10
15
2
Bl
B3
B5
B7
4
Bl
B3
B5
B7
6
B2
B4
B6
B8
10
B2
B4
B6
B8
                  TABLE 4
 Summary of Figure Numbers  for (TA)  Workii
           Nomograms  of Appendix B
8
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There are nine sets of plots subgrouped as 8 plots to a set as
shown in Table 6.  The format for each subgroup is detailed in
Table 5.  The combined use of Tables 5 and 6 is identical to the
use of  Tables 3 and 4 just explained.  Thus the (Tt)-plots corresponding
to Figure A70 are  found in Figure C35.

D - (TTWD) SUPPLEMENTARY NOMOGRAMS

The nomograms of Appendices A, B, and C are plotted for fixed values
of entrainment coefficient, horizontal and vertical eddy diffusivities,
drag and shear coefficients.  Some variations on these inputs (both
for lower and higher magnitudes) are calculated for fixed values
of typical Froude  number, F=4, aspect ratio, A=5 discharge angle,
0  = 90° and surface heat transfer coefficient K  = 10"5.   These
 o
results are given  in Appendix D in the form of (TTWD)-plots.  The
summary of the plots is given in Tables 7 and 8.
                                 20

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                 TABLE 5
   Figure Numbers for (Tt) Working Monograms
     of Appendix C, 0  = 90°, K = 10"1*
R-
+ A
1
' 5
10
15
2
Cl
' C3
C5
C7
4
Cl
C3
, C5
C7
6
C2
C4
C6
C8
10
C2
C4
C6
C8
                 TABLE 6
Summary of Figure Numbers for (Tt) Workii
         Nomograms of Appendix C
4-K
c
10'5
10"4
io-6
90°

Cl-8
C25-C32
C49-C56
60°

C9-C16
C33-C40
C57-C64
120°

C17-C24
C41-C48
C65-C72
                     21

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               TABLE 7
 Figure Numbers for (TTWD) Supplementary
  Nomograms of Appendix D, 0'  • 90°,
        K = 10'5' F = 4, A *°5
Eh*
VEh
.001
' .01
.2

.005
Dl
D4
D7

.02
D2
D5
D8

.1
D3
D6
D9
               TABLE 8
 Summary of Figure Numbers for (TTWD)
Supplementary Nomograms of Appendix D
          .01

          .05

          .15
D1-D9

D10-18

D19-D27
                   22

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                     Ill .ANALYTICAL  CONSIDERATIONS

The nomograms in this  volume were generated  predominately  from a
                 t
modification to the  surface jet model  of  Reference  1.  The reader
should consult References  1 and 3 for  the original  derivations
and Appendix F for the analysis, verification, and a Fortran listing
of the full modified model with an  example input and output.

A - MODEL  DESCRIPTION

The mathematical model describes the three dimensional behavior
of a heated jet discharged from a rectangular channel  at the  surface
of a deep  and wide body of homogeneous water that is either at
rest or moving with  a  uniform  and constant (steady)  velocity.  The magnitudes
of the discharge angle, channel dimensions,  discharge  velocity
and temperature are  arbitrary.  The jet velocity and temperature
distributions at the outlet are assumed uniform and constant.

The mathematical model is  greatly idealized, that is,  it describes
the behavior of a plume whose  discharge characteristics and ambient
environment are closely controlled.  Furthermore, the  model is subjected
to considerable limitations in the  process of arriving at  a mathematical
solution.  These two aspects of the model are discussed further.
                                  23

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1. Mode1 Ideali zat i ons

While we need to discuss model idealizations and limitations for
the benefit of tlie prospective user, we must also hasten to add    ;
that these shortcomings are not necessarily unique to this
model.  An exact simulation of a real plume in the natural
environment has not been achieved.  The difficulties confronting
this level of effort are too numerous to enable a general solution
at a reasonable cost.  Some of the more obvious difficulties can
be readily outlined.  For example, surface discharges are not confined
to open rectangular channels; the jet is usually discharged over a
sloping beach; the ambient current is neither uniform, nor
constant; a natural water body is often stratified, or nearly so;
there will always be some wind;  the discharge velocity and temperature
profiles are usually nonuniform and not always constant; the natural
water  body is not infinite giving rise to shoreline effects and
obstruction, etc.

Consequently, the nomograms do not provide a precise description
of a real plume.  They do, however, describe a plume realistically
if one or more of the real factors that have been ignored or idealized
in the model do not become dominant.  Following this line of reasoning,
the utility of the idealized model can, sometimes, be rightfully
defended.  Here are some examples:  (1) The effect of wind can enter
indirectly in the wind induced ambient current.  In rivers, its effect
may be overshadowed by the high river current; (2) The ambient water
need be homogeneous only in a thin top layer with a thickness on the
order of channel depth; (3) Slowly changing ambient conditions may
be assumed nearly constant when compared with the vplume response.
However, time dependent back and forth motion of tidal currents cannot
be handled with the steady state model used in this workbook; (4)
The effects of discharge geometry, velocity and temperature profiles

                                  24

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are felt mainly near the discharge and, in general, the model does
not treat the development portion in a precise way thus leaving some
doubts as to the need for a precise specification of these conditions;
(5) The model may still provide meaningful results if the initial
plume depth is small compared with the ambient water depth, regardless
of beach slopes.

These are only a few comments regarding the possible adequacy of
the idealized model for application to real situations.  Clearly,
there is considerable judgment involved on the part of the user.
The difficult problem is that the assumptions can be stated and
introduced in the model rather precisely, but matching these with
a specific problem cannot be done with nearly as much precision.
The degree of accuracy and successful prediction is principally
dictated by the close agreement one finds between the stated idealizations
in the model and the conditions of the real problem.  This is an
important fact not to be overlooked by the prospective user.

2 . Other Limitations

Other limitations imposed mainly in the process of arriving at an
analytical solution to the problem are important to mention though
more difficult to discuss in depth without the resort to mathematical
formalism.  The most significant among these limitations will be
discussed very briefly.

Development Calculations- The turbulent mixing of the discharged
fluid with the ambient water begins immediately at the point of
discharge.  However, turbulent mixing penetrates both the jet region
and the ambient surrounding only gradually.  Thus, the width and
depth of the turbulent mixing region are zero at the point of discharge
                                 25

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but grow gradually, downstream of this point until at a short distance
 from  it the zone occupies the entire interior of the jet.  The
 turbulent jet is then called fully developed.  Obtaining a complete
 mathematical solution to describe the characteristics of this so  called
 development portion of the jet is very difficult.  Consequently,
 a  partial solution is obtained by resorting to experimental data
 which provide estimates of the development length.  Then, following
 the procedure outlined in Reference 1 other required information
 at the end of development zone is calculated from the governing
 equations of motion.  The reader is referred  to Reference 3 for
 a  more thorough analytical treatment of this zone.

 Similarity - Once  the turbulent mixing is fully developed it is
 assumed that the mean profiles of the temperature, velocity
 and density do not change throughout the plume trajectory.  In the
 present study, these profiles are assumed to be Gaussian even when
 there is a cross current that disturbs the symmetry within the
 plume. The symmetric Gaussian profile assumption can lead to
 reasonable results for a very small or no ambient current.  Such
 a  model is inapplicable when there is a very strong ambient current.

 Additionally, in a strong current the plume may be forced against
 the shoreline thereby confining entrainment mainly to one side of
 the plume.  Again, the present analysis cannot handle this complication.
 In this Workbook nomograms are provided for a minimum discharge angle
 of 60°.  Even then, the velocity ratio  is limited to 0.7 to avoid
 conditions where the edge of the plume comes into contact with the
 shoreline.  If in  a particular application this condition is violated,
 the user should treat the results with caution.

 Entrainment - The plume expands by entraining ambient water at its
 outer boundaries, that is, along the plume edge.  The forward motion

                                  26

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of the jet establishes a lateral mean ambient fluid velocity into
the jet.  This entrainment velocity is assumed proportional to the
local mean velocity at the jet centerline relative to the ambient
fluid.  The entrainment velocity is also affected by the jet temperature
but only in the vertical direction.  The proportionality factor, known
as entrainment coefficient, must be prespecified in the model.  Its
value must be based on experimental data.

In addition to the mean convective velocity at the edge of the plume,
there are large scale turbulent fluctuations that cause the turbulent
diffusion of heat and momentum.  In the model this turbulent entrainment
is assumed to be due to the ambient contribution only.  Since the
turbulence scale in the ambient water is generally different in the
horizontal and vertical directions, separate information on diffusivities
in these directions must be provided for the model in terms of appropriate
coefficients.  These diffusivity coefficients must be prespecified.
Their values must be based on experimental data.

Buoyant Spreading - The plume from a heated jet spreads in directions
due to buoyant forces.  Ordinarily the spreading of the plume, whether
due to jet inertia forces or buoyant forces, must be predicted from
the governing equations of motion.  However, due to excessive simplification
of the equations, the mechanism for generating this information from
the original equations  is bypassed.  Consequently additional information
on buoyant spreading of the plume must be provided independently
of the original equations.  The analysis due to Reference 1 was retained
in the model but with a slight modification introduced  in the spreading
function.
                                  27

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Singularity - The original, as well as the modified model, runs into
computational difficulties (singularity) at a local plume densimetric
 i
Froude number of unity.  Once this value is reached computations
based on the model are questionable.

Singularities occur in two equations: (a) the equation for
calculating the plume depth and (b) the equation for the lateral
buoyant spreading.  The two equations are independent of one another
and in the modified model, the first singularity occurs before the
second.  Unfortunately, the model becomes inoperative due to these
singularities for many problems of practical interest.  Greatest
difficulty is noted at small values of R and F as well as large
values of A.  Since for these conditions the plume trajectory is
nearly coincident with the Y axis, little deviation of calculated
trajectory from data was noted even for calculations beyond the
first singularity.  Therefore, trajectories calculated by the program
were used in the nomograms.  However, the information on temperature and
width given in (TTWD) nomograms beyond the first singularity are
based on an extrapolation of previously calculated values of temperature
and width for higher R.  In order to clearly identify these extrapolations
the plume width and temperature curves are continued with broken lines.
These values should be used with-caution.

B - MODEL APPLICABILITY

The foregoing discussion of the idealizations and model limitations
should provide the reader with some understanding of the model,
particularly with respect to its applicability to practical problems.
In order to be useful, the model must be verified against laboratory
and field data.  If the magnitude of the plume characteristics
are in reasonable agreement with data and if the trends of these
characteristics are likewise reasonable, the model can be used to
predict similar situations with confidence.
                                 28

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It is true the present surface discharge model has many shortcomings,
but we have made a careful attempt to choose all the free input
coefficients to the model in order to produce the best fit with
the available data.  Consequently, we consider the results both
reasonable and of considerable practical utility.  The procedure
for producing the best fit to the data is contained in Appendix F.

In all cases, we have avoided presenting information on R = 0.
Instead, whenever extrapolation was meaningful, calculations were
extended to R = 0.01 which, in reality, may be interpreted as
R = 0.  For nomograms with @Q = 120°, R - 0.1.  In several instances
a complete set of nomograms was omitted because of computational
difficulties.

The words of caution given earlier concerning the use of the extra-
polated- port ion of the nomograms might have escaped the avid user.
They  should not.  The presence of the broken temperature and width
lines should signal not  only difficulties in the mathematical model
used, but also a similar difficulty in obtaining good experimental
data  in a stagnant water body.
                                  29

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                         IV.  EXAMPLE PROBLEMS
Examples are given in this section to demonstrate the use of the
nomograms.  Even though an attempt was made to develop more or less
realistic problem statements, the examples should not be construed as
representing a preferred design or recommended temperature zone.
Numbers in these examples have been conveniently rounded off.   Problems
4, 5, and 6 are presented to show example cases where using the nomograms
would lead to questionable results.

Example Problem #1

Given:

A 500 MWe nuclear power plant is located on a large freshwater lake.
The cooling water is discharged at the surface through a rectangular
open channel.  The following design data apply:
                                               g
     a.  Waste heat to cooling water « 8.1 x 10  kcal/hr
     b.  Condenser AT (AT ) • 12°C
     c.  Discharge angle (e ) = 90°
     d.  Discharge velocity (UQ) = 1 m/sec
     e.  Channel width (WQ = 2 BQ) = 10 m
     f.  Ambient water temperature (T.) * 15°C
                                     a
     g.  Offshore current of .05 m/sec
     h.  Dimensionless surface heat transfer coefficient K = 10"5

Determine:

     1.   The location along the centerline of the plume where the
          surface temperature is 4°C above ambient.

                                      30

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     2.    The width and depth of the plume at this point.




     3.    The plume area enclosed by the 4°C AT isotherm.




     4.    The time of travel for a water parcel following the plume


          centerline from the discharge structure to the 4°C AT isotherm.
Solution:


1.  Determine A (Aspect ratio) • 2BQ/H0



n,-e^hav,«Q *.a4-« n -       Heat load (kcal/hr)	~
uiscnarge rate ^ -   ^  x 36QO (se^/ftrj'x fooo (kcal/nr-0C)





                                x 1000   = 18-8 m3/sec
         BoHoUo
HQ   = q/2 BQU0 = 18.8/10 x 1
HQ   = 1.88 m
       2 BQ/HO - 10/1.88 =5.3-5
2.  Determine F  (Froude Number)
              UP£)/P)
From Figure El, Aja/p  » 0.0026




F = (1 m/sec)/[(0.0026)  (9.8 m/sec2)  (1.88m)]1* =  4.6
                                   31

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3.  Location of 4°C AT  isotherm:
                          c
     For e  = 90°, K = 10  ,  A = 5 and F = 4,  use nomogram A6.  At
     ATC/ATQ = 4/12 = 0.333 and R = 0.05, X/HQ = 40 and Y/HQ =  180.   For
     eQ, K, and A unchanged with F = 6, use nomogram A7,  where  X/HQ = 16
     and Y/HQ = 82.  Interpolating to F = 4.6  gives X/HQ  = 33 and Y/H^ =
     151.  Since H  - 1.88 m, the 4°C AT isotherm will  be located at
     (151) (1.88 m) = 283 m offshore and (33)  (1.88 m)  =  62 m downstream
     from the discharge.  It should be noted that we have used  the
     nomograms of Figure A6 in an extrapolated region.   The answer is,
     therefore, approximate.

4.   The width of the plume at the 4°C AT isotherm:
     Using nomograms A6 and A7, W/H  values of 200 and  170 are  obtained.
     Interpolation for F = 4.6, gives W/H  = 190.   Thus,  the width at
     this point is (1.88 m) (190)  = 357 m.

5.   The center!ine depth of the plume at 4°C  excess temperature:
     Using nomograms A6 and A7, D/HQ values are scaled  from the length
     of the dashes at the desired location to  be 2.0 and  3.0 for F = 4,
     and 6, respectively.  Interpolating to F  = 4.6,
     gives D/HQ = 2.3.  Thus, D = (2.3) (1.88  m) = 4.3  m.

6.   The area within the 4°C AT isotherm:
                                    2                "31
     Using nomograms B3 and B4, A/H   values of 6 x 10   and 1.1 x 10
                                                            •?
     are obtained.  Interpolating for F = 4.6  we obtain A/HQ = 4.5 x
     103.  Thus the area equals (4.5 x 103) (1.88 m)2 = 1.6 x 104 m2 or
     about 4 acres.
7.   The time for a water parcel  to travel  along the plume center!ine
     from the outlet to the 4°C AT isotherm:
                                                        2           9
     Using nomograms C3 and C4, tU/ /H  values of 7 x 10  and 2 x 10
     are obtained.  Interpolating to F = 4.6,  gives tU^/tt  = 550.  Thus,
     t = (550)  (1.88 m/1 m/sec)= 1034 sec, or about 17 minutes,

                                     32

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Example Problem #2

Given:

A 1000 MWe fossil fueled power plant is located on a large fresh water
lake.  The cooling water is discharged through a one kilometer  long
rectangular discharge canal.  The following design data apply:

     a.   Waste heat to cooling water » l.OSxlO9 kcal/hr
     b.   Condenser AT UT   ) = 16.3°C

     c.   Discharge angle (0Q) = 60°

     d.   Discharge velocity = 0.8 m/sec

     e.   Depth of water in channel at outlet = 1.25 m

     f.   Ambient water is stagnant with an ambient temperature  of  20°C.
          Assume K = 10"4.
Determine:
     1.   The location of the plume center line where the temperature  is
          23°C

     2.   The thickness of the water layer at this point that is 21 °C or
          warmer.

     3.   The plume width at this point.

     4.   The surface plume area having a temperature equal  to or
          warmer than 23°C.
                                     33

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     5.   The time required for a water parcel to travel from the
          condenser outlet to the 23°C isotherm.
Solution:
     1.   Determine the water temperature at the end of the discharge
          canal.

          The temperature distributions in a constant width canal with
          mixed flowing fluids is given by

                    T - TP
                          f  = EXP[-KF2B X/pQC ]
                          I i-       .HO     P
                    con
          where 1C is the surface heat transfer coefficient, 2BQ is the
          canal width, X is the distance from the beginning of the canal
          to the point desired, p is the density of water, Q is the
          volumetric flow rate in the canal, C  is the heat capacity of
          the water, T    is the condenser outlet temperature, and T.- is
                      c*on                                           L,
          the equilibrium temperature of the water.   Given Q = U H 2B
          and K = IC/pC U , the above equation for canal temperature can
          be simplified to yield the canal dischage temperatures.  This
          expression is:
                           = EXP[-K L/HQ]
          if T£ is taken as the ambient receiving water temperature and
          L is the total canal length.
                                   AT              .
          Thus, for this problem  -rf- = EXP[-(10"^)(1000)/(1.25)]= .923
                                     con

                                    34

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          Therefore ATQ = .923 x 16.3 = 15°C
2.   Determine A = 2BQ/H0
     Q =  (1.08xl09)/[(16.3)  (1000) (3600)] = 18.5 m2/sec
     2BQ = Q/HQU0 = 18.5/(.8)(1.25) = 18.5 m
     A = 2B0/HQ = 18.5/1.25 = 14.8 : 15
3.   Determine F
     From Figure E3, 6 = '5
                            cm
         GUo   _ (.5)x(.8)(10) = 3.6
     F = 3.6, use F = 4

4.   ATC/ATQ = (23-20)/15 =0.2
     Use R = 0.01 as an approximation for R = 0,  GQ = 60°,  K =  10~4, A =
     15, and F = 4, to enter nomogram A78. At AT r/ATn = 0.2,  X/H  =  380
                                               I*   U
     arid ;V/H  = 620.  This gives a distance S/HQ  = 727 as measured  along
     the trajectory.  Thus, 23°C occurs approximately on the plume
     center! ine at (727) (1.25 m) = 909 m from the discharge.

5.   The Gaussian distribution curve, Figure E5,  is used to find  the
     plume depth where the temperature is 21 °C at a cross  section located
     909 m offshore.  (T -Tj(T -Tj = (21-20)7(23-20) = .333.  At this
                        iia   c  a
     value on Figure E5, I/a  = 1.48.  The standard deviation az  can  be
     found from knowing  that the depth D = 2a .   Therefore,  the  depth

                                    35

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     to the 21°C point is 1.48(D)/2 from the surface.  The value of D
     for this problem is found from the length of the dash at the
     desired point on Figure A78 to be 2.8 HQ = (2.8)(1.25) = 3.5 m.
     This gives a thickness of the desired layer to be (1.48)(3.5)/(2) =
     2.6 m.
6.   Width of the plume at the 23°C center!ine temperature:
     Using nomogram A78, W/HQ is found to be greater than 450.  Thus the
     plume is over (450) (1.25m) = 562 m wide.  A gross extrapolation of
     the data on Figure A78 indicates the width to be more like 900
     meters wide.
7.   The plume area warmer than 23 C:
                                        2
     Using nomogram B39, a value of A/H   slightly greater than 1.5 x
       5
     10  is read off.  Thus, the plume area enclosed by the 23°C isotherm
     is (1.5 x 105) (1.25 m)2 = 2.34 x 105m2, or 58 acres.

8.   The time for a water parcel to travel from the condenser to the
     23°C isotherm:
     Assuming a constant cross section in the rectangular channel, the
     water parcel will take (1000m)/(0.8m/sec) = 1250 seconds to travel
     through the canal to the point of discharge.   Using nomogram C39,
  '   tUo/Ho is found to be approximately 5.2 x 103.  Thus, t = (5.2 x
     TO3) (1.25/0.8) = 8120 sec.  Adding this value to 1250 sec gives a
     total travel time of 9370 sec - 2.6 hr.

Example Problem #3

An 800 MWe nuclear power plant is located on an open coastline.  Condenser
cooling water is discharged through an open rectangular channel.  The
following design data apply:

                                   36

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     a.   Waste heat to cooling water =  1.3  x  109  kcal/hr
     b.   Discharge velocity =1.0 m/sec
     c.   Offshore current along coastline = .2 m/sec
     d.   Condenser AT = 12°C
     e.   Total width of channel = 14 m
     f.   Ambient water temperature and  salinity are respectively 10°C
     and 30 ppt.
     g.   Surface heat transfer coefficient  K = 10

Determine:

     1.   The location on the plume centerline where T  = 16°C
                                                    v

     2.   The plume width and depth at this  point.

     3.   The surface area that has a temperature  between 14 and 16°C

Solution:

1.   Determine A

     Q =  (1.3 x-109)/(103)  (1) (10) (3.6 x  103)  = 36 m3/sec

     HQ = Q/2B0UQ =  (36)/(l)(14) = 2.6 m

     A =  14/2.6 = 5.4 ~ 5

2.   From Figure E2

          = 0.0024
      o

     F =  (1)/[(0.0024)  (9.8)  (2.6)]^ = 4.04

                                    37

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3.   Location of 16°C isotherm
     ATr/AT  = (16-10)/12 = .5
       C   0

     R = U/U  = 0.2/1 = 0.2
          a  0
                                       »
     Use nomogram A6 for Q =90°, K  = 10~5, A=5 and F=4 at R = .2 and T
     = .5 to find X/HQ = 18 and Y/HQ = 38.   Thus, the 16°C isotherm is
     encountered at the centerline (18) (2.6 m) « 47 m down current and
     (38) (2.6 m) = 99 m away from shore.
4.   The width of the plume at the 16°C centerline temperature.
     Using nomogram ft
     (2.6 m) = 135 m.
Using nomogram A6, W/HQ = 52.   Thus,  the plume width  is  (52)-
5.   The plume depth at this point is scaled off Figure A6 from the
     length of dashes.  D/HQ * 2, D = 2 x 2.6 = 5.2 m

6.   The plume area between 4 and 6°C.
                                                       y
     For AT/AT. = 0.333 and 0.5, nomogram B3 gives A/H ' values of 2.2
         q    O        0                              °
     x 10  and 4.3 x 10  at R = 0.2.  The difference between the two is
     1.77 x 10^ giving the appropriate area as (1770) (2.6m)2 = 1.2 x
       4.  ?
     10  m  (3 acres).
Example Problem 4

     Repeat Problem #1 for discharge at 60° into a deep river whose
velocity is .7 m/sec.

Solution:

     From Problem fl, F = 4.5, A = 5, K = 10"5, HQ = 1.88 m.
                                   38

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1.   Location at 4°C AT isotherm:

     For 0Q = 60°, K • 10  , F = 4 and A = 5 use nomogram A22.   For R =
     .7/1.0 = 0.7 and ATC/ATQ = 0.333, X/HQ = 68 and Y/HQ = 18.   From
     Figure A23 for F = 6, X/HQ = 67 and Y/HQ = 18 (nearly the  same).
     The total width at this point is found from the same two nomograms
     to be about 30 or a half width of 15.  Since Y/HQ is only  about 18
   1  this places the edge of the plume very near the shore.  Due to
     uncertainties in the model, this plume could well be attached to
     the shore, limiting entrainment from that side.  This is especially
     true if the bottom near the shore is not deep.  The results from
     the nomograms for this case are therefore, to be used with caution.
     Hydraulic modeling or field data could bt used to obtain more
     reliable information.
Example Problem #5

     Repeat Problem 1 for a channel width of 20 m and a discharge
velocity of 0.25 m/sec.

Solution:

1.   Determine A

     Q is given Problem #1 to be 18.8 m3/sec
     HQ = 18.8/£0 x .25)= 3.76 m
     A  = 20/3.76 « 5.3                               ;

2.   Determine F
     Since the discharge ATQ is the same as Problem #1.
                                     39

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     F = 0.25/[(.0026)(9.8)(3.76)]^ = 0.81

     For a discharge Froude number less than 1.0,  a cold water wedge
     probably exists which means the actual  discharge velocity
     is greater than that indicated and the  depth  is less than estimated
     from calculations.  The nomograms cannot handle this type of problem.
     Hydraulic modeling or field measurements could be used to find the
     actual values.

Example Problem #6

     Repeat Problem #1 for the case where the bottom of the lake slopes
away from the shore with a 5 meter drop for  every  100 meters offshore.
The bottom of the discharge channel is flush with  the bottom of the
lake.

Solution:

     The problem with a sloping bottom is whether  there is sufficient
ambient water between the bottom of the plume and  the bottom of the lake
to ensure adequate entrainment as calculated by the model.  To determine
this, the depth of lake is compared with plume depth.  If the plume
depth is close to or greater than the bottom depth, the model fails and
alternate methods of analysis or hydraulic modeling would have to be
used.

     The plume depth at various distances from discharge for Problem #1
can be found using nomogram A6 and A7 with R = .05 and interpolating
to F = 4.6.  The results are tabulated below along with the lake depth
for selected values of S/H .
                                   40

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S/HQ
10
20
30
D/HQ
1.5
2.1
2.25
Lake Bottom
5:100 slope
1.5
2.0
2.5
Since the plume depth exceeds the lake depth, the analysis fails.
                               41

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                         ACKNOWLEDGEMENT

The nomograms in this workbook were plotted by the computer.  A
program was prepared specially for this task by Mr. Kenneth V. Byram
of the National Environmental Research Center in Con/all is.  His
valuable assistance is acknowledged with appreciation.
                                42

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                             REFERENCES

.1 .    Prych, Edmund A.   "A Warm Water Effluent  Analyzed  as a Buoyant
      Surface Jet"  Swedish Meteorological  and  Hydrological Institute,
      Series Hydology Report, No.  21, 1972, Stockholm.

 2.    Shirazi, Mostafa A., "Some Results from Experimental Data on
      Surface Jet Discharge of Heated Water" Proceedings of the
      International Water Resources Association, Chicago, 1973.

 3.    Stolzenbach, K. D., Harleman, D. R. F.  "An Analytical and
      Experimental Investigation of Surface Discharges of Heated Water."
      Water Pollution Control Series 16130 DJV  02/71,  Feb. 1971.
                                43

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       APPENDIX A
(TTWD)  WORKING NOMOGRAMS
            44

-------
                     TABLE  1
   Figure numbers  for (TTWD) Working Nomograms
      of Appendix  A,  Qn = 90°  and  K =  10~5
                      o
F +
+A
1
5
10
15
2
Al
A5
A9
A13
4
A2
A6
A10
A14
6
A3
A7
All
A15
10
A4
A8
A12
A16
                     TABLE 2
         Summary  of  Figure Numbers for (TTWD)
           Working Nomograms of Appendix A
V
+K
10"5
io-4
io-6
90°
A1-A16
A49-A64
A97-A112
60°
A17-A32
A65-A80
A113-A128
120°
A33-A48
A81 -A96
A128-A144
NOTE:  Figures A-37,  41,  45,  46,  85, 89, 93, 94,
       133, 137, 141, 142 are not included due to
       computational  difficulties.
                        45

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   40
   20
          20   40  60   80  100  120   140  160  180  200
                LONGITUDINAL DISTANCE X/H0

      FIG( A I ) TEMPERATURE,TRAJECTORY,WIDTH, AND DEPTH
             (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                     46

-------
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          10   20  30   40  50   60  70  80   90  100
                LONGITUDINAL DISTANCE
      FIGC A2 ) TEMPERATURE.TRAJECTORY.WIDTH.ANDDEPTH


             (TTWD)-PLOTS FOR SURFACE JET DISCHARGE


                      47

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     10   20  30   40   50  60   70  80  90   100
          LONGITUDINAL DISTANCE X/H0

FIG(A3 ) TEMPERATURE.TRAJECTORY.WIDTH.ANDDEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

-------
X

LU
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QC
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   20
    10
          10  20   30  40   50  60   70  80   90  100
                LONGITUDINAL DISTANCE X/U0

     FIG( A4 ) TEMPERATURE,TRAJECTORY,WIDTH,ANDDEPTH
             (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                      49

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      0    50   100  150  200 250  300 350 400  450  500
                 LONGITUDINAL DISTANCE X/H0




       FIG( A5 ) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH

              (TTWDhPLOTS FOR SURFACE JET DISCHARGE


                       50

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20
       20  40   60   80  100   120  140  160  180   200
             LONGITUDINAL DISTANCE X/H0

   FIG( A6 ) TEMPERATURE,TRAJECTORY,WIDTH, AND DEPTH
       :   (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                  51

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300
280
260
        20  40   60  80   100  120  140  160  180  200

             LONGITUDINAL DISTANCE X/H0

    FIG( A7 ) TEMPERATURE,TRAJECTORY,WIDTH, AND DEPTH
           (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                   52

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    20   40  60   80  100  120  140  160  180  200
          LONGITUDINAL DISTANCE X/H0

FIG( A8 ) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
               53

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750
        50  100  ISO 200  250 300  350 400  450  500
             LONGITUDINAL DISTANCE X/H0

    FI6( A9 ) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH
           (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                   54

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X
x
iy
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5
3
            50  100   150  200  250  300  350  400 450  500
                  LONGITUDINAL DISTANCE X/H0



        FIG (AIO ) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH

              (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                       55

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    50  100  150  200  250  300  350 400 450  500
          LONGITUDINAL DISTANCE X/H0

FIG ( All ) TEMPERATURE,TRAJECTORY,WIDTH. AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

               56

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750
   0    5p  100  150  200  250  300  350  400 450  500

              LONGITUDINAL DISTANCE X/H0

    FIG ( AI2 ) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH
           (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                   57

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    100  200  300  400 500  600 700  800  900 1000
          LONGITUDINAL DISTANCE X/H0

FIG ( AI3 ) TEMPERATURE,TRAJECTORY.WIDTH,AND DEPTH
       (TTWDhPLOTS FOR SURFACE JET DISCHARGE
               58

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    50  100  150 200  250 300  350 400  450  500
          LONGITUDINAL DISTANCE X/H0

FIG (AI4 ) TEMPERATURE. TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                59

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    50  100  150  200  250 300  350 400 450  500
          LONGITUDINAL DISTANCE X/H0

FIG (AI5 ) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
               60

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    50  100  150 200  250 300  350 400 450  500

          LONGITUDINAL DISTANCE X/H0

FIG ( AI6 ) TEMPERATURE, TRAJECTORY, WIDTH. AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                61

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    20   40  60   80  100  120  140  160  180  200
          LONGITUDINAL DISTANCE X/H0

FIG (Al 7) TEMPERATURE.TRAJECTORY,WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                  62

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          10   20  30   40  50   60  70   80  90  100


                LONGITUDINAL DISTANCE X/H0


      FIG(AI8 ) TEMPERATURE.TRAJECTORY. WIDTH, AND DEPTH

             (TTWDhPLOTS FOR SURFACE JET DISCHARGE

                      63

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     10   20  30  40   50  60   70  80   90  100
          LONGITUDINAL DISTANCE X/H0

Fl G (A 19) TEMPERATURE.TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                  64

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X



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               20  30   40   50  60   70  80  90   100



                LONGITUDINAL DISTANCE X/U0
      FIG(A20) TEMPERATURE,TRAJECTORY,WIDTH,ANDDEPTH

             (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                        C5

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            50  100   150  200 250  300 350 400 450  500
                 LONGITUDINAL DISTANCE X/H
        FIG (A21 ) TEMPERATURE, TRAJECTORY. WIDTH. AND DEPTH

               (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

                         66

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    20   40  60   80  100  120   140  160   180  200

          LONGITUDINAL DISTANCE  X/H0

FIG (A22 ) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                 67

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    20   40  60   80  100  120   140  160   180  200
          LONGITUDINAL DISTANCE X/H0

FIG (A23) TEMPERATURE,TRAJECTORY,WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                 68

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20
       20  40   60   80  100  120  140 160  180   200
             LONGITUDINAL DISTANCE X/H0

   FIG (A24) TEMPERATURE,TRAJECTORY,WIDTH, AND DEPTH
          (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                    69

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    50  100  150  200  250 300  350 400 450  500
          LONGITUDINAL DISTANCE X/H,
FIG (A25 ) TEMPERATURE. TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                 70

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                  LONGITUDINAL DISTANCE X/H0



        FIG (A26) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH

               (TTWD)-PLOTS FOR SURFACE JET DISCHARGE


                         71

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    50  100  150  200  250  300  350  400 450  500
          LONGITUDINAL DISTANCE X/H,
FIG (A27 ) TEMPERATURE, TRAJECTORY,WIDTH, AND DEPTH
       (TTWDhPLOTS FOR SURFACE JET DISCHARGE

                  72

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111
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            50  100  150  200  250  300  350 400 450  500
                  LONGITUDINAL DISTANCE X/H0




        FIG (A28) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH

               (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                        73

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0   100  300  300  400 500  600 700  800  900 1000
          LONGITUDINAL DISTANCE X/H0

FIG (A29 ) TEMPERATURE,TRAJECTORY.WIDTH,AND DEPTH
       (TTWDhPLOTS FOR SURFACE JET DISCHARGE

                 74

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            50  100   150  200 250  300  350  400 450  500
                 LONGITUDINAL DISTANCE X/H0



        FIG (A30) TEMPERATURE, TRAJECTORY, WIDTH. AND DEPTH

               (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                         75

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i    50   100  150 200 250  300 350 400  450  500

          LONGITUDINAL DISTANCE X/H0

FIG (A3I ) TEMPERATURE, TRAJECTORY,WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                 76

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            50  100   150  200  250  300  350 400 450  500
                  LONGITUDINAL DISTANCE X/H,
        FIG (A32 ) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH

               (TTWD)-PLOTS FOR SURFACE JET DISCHARGE


                         77

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-20
20   40   60   80   100   120   140  160  180

LONGITUDINAL DISTANCE X/H0
FIG (A 33) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                  78

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                LONGITUDINAL DISTANCE X/H
      FIG (A34) TEMPERATURE, TRAJECTORY.WIDTH.AND DEPTH


             (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

-------
 o
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               10   20   30  40   50   60  70   80   90
               LONGITUDINAL DISTANCE X/H0




     FIG (A35) TEMPERATURE,TRAJECTORY,WIDTH,AND DEPTH

            (TTWDhPLOTS FOR SURFACE JET DISCHARGE


                      80

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         10
20   30  40   50  60   70  80
90
          LONGITUDINAL DISTANCE X/H0

FIG (A36) TEMPERATURE,TRAJECTORY,WIDTH,AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                81

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-20
20   40   60   80   100   120  140  160  180

LONGITUDINAL DISTANCE X/H0
FIG (A38) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

-------
-20
20   40   60  80   100  120   140  160  180

LONGITUDINAL DISTANCE X/H0
FIG (A39) TEMPERATURE,TRAJECTORY,WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                  83

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-20
20   40  60   80   100   120  140  160   180

LONGITUDINAL DISTANCE X/H0
FIG (A40) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                  84

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                  LONGITUDINAL DISTANCE X/H0



        FIG (A42) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH

               (TTWD)-PLOTS FOR SURFACE JET DISCHARGE


                         85

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       "-50   0   50  100  150  200  250  300 350  400  450



                  LONGITUDINAL DISTANCE X/H0




        FIG (A43) TEMPERATURE, TRAJECTORY, WIDTH. AND DEPTH

               (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                         86

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-50  0   50   100   150  200  250  300  350 400  450

          LONGITUDINAL DISTANCE X/H0

 FIG (A44) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH
        (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

                  87

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       -50   0   50  100  150   200  250  300  350  400  450


                  LONGITUDINAL DISTANCE X/H0



        FIG (A47) TEMPERATURE, TRAJECTORY, WIDTH. AND DEPTH

               (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                         88

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X

X
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50   100  150  200  250  300 350  400  450


 LONGITUDINAL DISTANCE X/HQ
        FIG (A48) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH

               (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                        89

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300
280
260
        20  40  60   80  100  120  140  160  180  200
              LONGITUDINAL DISTANCE X/H0

    FIG (A49) TEMPERATURE.TRAJECTORY, WIDTH, AND DEPTH
           (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

                    90

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     10   20  30  40   50  60   70  80   90  100
          LONGITUDINAL DISTANCE X/H0

FIG (A50) TEMPERATURE.TRAJECTORY, WIDTH, AND DEPTH
        (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                 91

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     10   20  30   40  50   60  70   80  90   100
          LONGITUDINAL DISTANCE X/H0

Fl G (A 51) TEMPERATURE.TRAJECTORY, WIDTH, AND DEPTH
       O;TWD)-PLOTS FOR SURFACE JET DISCHARGE
                 92

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     10   20  30   40   50  60   70  80  90   100
          LONGITUDINAL DISTANCE X/H0

FIG (A52) TEMPERATURE.TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                 93

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                                  300 350 400  450  500
                 LONGITUDINAL DISTANCE X/H0



        FIG (A53) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH

              (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                        94

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20
       20  40   60  80  100  120  140  160  180  200
             LONGITUDINAL DISTANCE X/H0

   FIG (A54) TEMPERATURE.TRAJECTORY, WIDTH, AND DEPTH
          (TTWO)-PLOTS FOR SURFACE JET DISCHARGE
                   95

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300
280
260 il
        20  40  60   80  100  120  140  160  180  200
              LONGITUDINAL DISTANCE X/H0
    FIG (A55) TEMPERATURE,TRAJECTORY,WIDTH, AND DEPTH
           (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

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300
280
260
 20
        2O   40  60   80  100  120  140  160  180  200
              LONGITUDINAL DISTANCE X/H0

    FIG (A56) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
           (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                     97

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            50  100   150  200  250  300  350 400  450  500
                  LONGITUDINAL DISTANCE X/H0



        FIG (A57) TEMPERATURE, TRAJECTORY, WIDTH. AND DEPTH

               (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

                        98

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            50 i 100  150  200  250 300  350 400  450  500
                  LONGITUDINAL DISTANCE X/H0



        FIG (A 58) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH

               (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                         99

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            50  100   150  200 250  300  350  400 450  500



                 LONGITUDINAL DISTANCE X/H0




        FIG (A59) TEMPERATURE, TRAJECTORY, WIDTH. AND DEPTH

               (TTWD)-PLOTS FOR SURFACE JET DISCHARGE


                        100

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    50  100  150  200  250 300  350 400 450  500

          LONGITUDINAL DISTANCE X/H0

FIG (A60) TEMPERATURE/TRAJECTORY, WIDTH. AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                  101

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      FIG (A6 I) TEMPERATURE, TRAJECTORY. WIDTH, AND DEPTH
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                        102

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    50  100  150  200  250  300  350  400 450  500

          LONGITUDINAL DISTANCE X/H0

FIG (A 62) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH
       (TTWDhPLOTS FOR SURFACE JET DISCHARGE

                   103

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    50  100  150  200  250  300  350 400 450  500
          LONGITUDINAL DISTANCE X/H,
FIG (A63) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

                   104

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    50  100  150 200  250 300  350 400 450  500
          LONGITUDINAL DISTANCE X/H0

FIG (A64) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

                    105

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    20   40  60   80  100  120   140  160   180  200

          LONGITUDINAL DISTANCE X/H0

FIG (A65) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

                      106

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                LONGITUDINAL DISTANCE
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                            107

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                      108

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      FIG(A68) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH

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                       109

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        FIG (A69) TEMPERATURE, TRAJECTORY. WIDTH, AND DEPTH

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                        no

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300
280
260
 20
        20  40  60   80  100  120
160   180  200
              LONGITUDINAL DISTANCE X/H0

    FIG (A70) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
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                    111

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      FIG (A71) TEMPERATURE,TRAJECTORY,WIDTH, AND DEPTH

             (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                      112

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    20   40  60   80  100  120  140  160  180  200
          LONGITUDINAL DISTANCE X/H0

FIG (A72) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

                113

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               (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                        114

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        FIG (A74) TEMPERATURE. TRAJECTORY,WIDTH. AND DEPTH


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                        115

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        FIG (A75) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH

               (TTWD)-PLOTS FOR SURFACE JET DISCHARGE


                        116

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750
   0    50  100  150 200  250 300  350 400 450  500

              LONGITUDINAL DISTANCE X/H0

    FIG (A76) TEMPERATURE. TRAJECTORY, WIDTH, AND DEPTH
           (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

                    117

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    100  200  300  400 500  600 700  800  900 1000
          LONGITUDINAL DISTANCE X/H,
FIG (A 77) TEMPERATURE,TRAJECTORY.WIDTH,AND DEPTH
       (TTWDhPLOTS FOR SURFACE JET DISCHARGE

                its

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                  LONGITUDINAL DISTANCE X/H0




        FIG (A78) TEMPERATURE, TRAJECTORY.WIDTH, AND DEPTH

               (TTWD)-PLOTS FOR SURFACE JET DISCHARGE


                        119

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                 LONGITUDINAL DISTANCE X/H0




        FIG (A79) TEMPERATURE, TRAJECTORY, WIDTH. AND DEPTH

               (TTWD)-PLOTS FOR SURFACE JET DISCHARGE


                        130

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            50  100   150  200  250  300  350 400 450  500




                  LONGITUDINAL DISTANCE X/H0




        FIG (A80) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH

               (TTWD)-PLOTS FOR SURFACE JET DISCHARGE


                        121

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      -20    0   20  40  60   80   100   120   140   160   180



                LONGITUDINAL DISTANCE X/H0   -



      FIG (A81) TEMPERATURE.TRAJECTORY, WIDTH, AND DEPTH

             (TTWD)-PLOTS FOR SURFACE JET DISCHARGE


                        122

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-10   0    10    20  30   40  50   60  70   80   90
           LONGITUDINAL DISTANCE X/H0

 FIG (A82) TEMPERATURE,TRAJECTORY,WIDTH,AND DEPTH
        (TTWDhPLOTS FOR SURFACE JET DISCHARGE
                  123

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               10   20   30   40  50   60  70  80   90
                LONGITUDINAL DISTANCE X/H0




      FIG (A83) TEMPERATURE,TRAJECTORY,WIDTH,AND DEPTH

             (TTWD)-PLOTS FOR SURFACE JET DISCHARGE


                       124

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-10
10    20  30   40  50   60  70  80   90

 LONGITUDINAL DISTANCE X/H0
 FIG(A84) TEMPERATURE,TRAJECTORY,WIDTH,AND DEPTH
        (TTWDJ-PLOTS FOR SURFACE JET DISCHARGE
                  125

-------
'-20   0   20   40   60   80   100  120  140   160   180

          LONGITUDINAL DISTANCE  X/H0

 FIG (A86) TEMPERATURE,TRAJECTORY,WIDTH, AND DEPTH
        (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                   126

-------
      0   20  ~40   60  80" 100   120  140  160  180

          LONGITUDINAL DISTANCE X/H0
FIG (A87) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                   127

-------
'-20
20   40   60   80   100   120   140   160  180

LONGITUDINAL DISTANCE X/H0
 FIG (A88) TEMPERATURE.TRAJECTORY, WIDTH, AND DEPTH
        (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

                   128

-------
-50
50   100  150  200  250  300 350  400 450

 LONGITUDINAL DISTANCE X/H0
 FIG (A90) TEMPERATURE, TRAJECTORY. WIDTH, AND DEPTH
        (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                 129

-------
-50
50  100  150  200  250 300  350  400  450

 LONGITUDINAL DISTANCE X/H«
 FIG (A91) TEMPERATURE, TRAJECTORY, WIDTH. AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                 130

-------
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        FIG (A92) TEMPERATURE, TRAJECTORY, WIDTH. AND DEPTH

               (TTWD)-PLOTS FOR SURFACE JET DISCHARGE


                          131

-------
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                  LONGITUDINAL DISTANCE X/H0



        FIG (A95) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH

               (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                         132

-------
-50  0    50   100  150  200  250 300  350 400  450
           LONGITUDINAL DISTANCE X/H0

 FIG (A96) TEMPERATURE, TRAJECTORY. WIDTH, AND DEPTH
        (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                   133

-------
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      FIG (A97) TEMPERATURE.TRAJECTORY,WIDTH, AND DEPTH

             (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

                       134

-------
     10   20   30  40   50  60   70  80   90  100
          LONGITUDINAL DISTANCE X/H0

FIG (A98) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
        (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                 135

-------
     10   20  30   40   50  60   70  80   90  100
          LONGITUDINAL DISTANCE X/H0

FIG (A99) TEMPERATURE.TRAJECTORY,WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                 136

-------
     10   20   30  40   50  60   70  80   90  100

          LONGITUDINAL DISTANCE  X/H0

FIG (AIOO) TEMPERATURE.TRAJECTORY, WIDTH, AND DEPTH
        (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                 137

-------
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                  LONGITUDINAL DISTANCE X/H0



        FIG (AIOI) TEMPERATURE,TRAJECTORY.WIDTH, AND DEPTH

               (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                        138

-------
    20   40  60   80  100  120   140  160   180  200
          LONGITUDINAL DISTANCE  X/H0

FIG (AI02) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

                 139

-------
20
       20  40   60   80  100  120  140  160  180  200
             LONGITUDINAL DISTANCE X/H0

   FIG (AI03) TEMPERATURE,TRAJECTORY,WIDTH, AND DEPTH
          (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
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-------
20
       20  40  60   80  100  120   140  160  180  200
             LONGITUDINAL DISTANCE  X/H0

   FIG (AI04) TEMPERATURE.TRAJECTORY, WIDTH, AND DEPTH
          (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                   141

-------
50
 0    50   100   150  200 250  300 350 400  450  500
            LONGITUDINAL DISTANCE  X/H0

  FIG (Al'05) TEMPERATURE, TRAJECTORY, WIDTH. AND DEPTH
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                  T42

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        FIG (AI06) TEMPERATURE, TRAJECTORY, WIDTH. AND DEPTH

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                         143

-------
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            50  100  150  200  250 300  350 400  450  500
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        FIG (AI07) TEMPERATURE. TRAJECTORY. WIDTH. AND DEPTH

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                         V44

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            50  100   150  200  250  300  350  400 450  500
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        FIG (AI08) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH


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                         145

-------
0   100  200  300  400 500  600  700  800  900 1000
          LONGITUDINAL DISTANCE X/H0

FIG (AIO9) TEMPERATURE,TRAJECTORY,WIDTH,AND DEPTH
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                146

-------
O    50   100   150  200 250  300 350 400  450  500
           LONGITUDINAL DISTANCE X/H0

 FIG (Al 10) TEMPERATURE, TRAJECTORY, WIDTH. AND DEPTH
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                 147

-------
    50  100  ISO  200  250  300  350 400 450  500
          LONGITUDINAL DISTANCE X/H0

FIG (A 111 ) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH
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-------
750
700
        50  100  150 200  250  300  350 400  450  500

              LONGITUDINAL DISTANCE X/H0

    FIG (Al 12) TEMPERATURE. TRAJECTORY, WIDTH, AND DEPTH
           (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                     149

-------
    20   40  60   80  100  120   140  160   180  200
          LONGITUDINAL DISTANCE X/H0

FIG (Al 13) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
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                150

-------
     10   20  30  40   50  60   70  80   90  100

          LONGITUDINAL DISTANCE X/H0

FIG(AI 14) TEMPERATURE.TRAJECTORY, WIDTH, AND DEPTH
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                        152

-------
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                       153

-------
    50  100  150  200  250 300  350 400  450  500
          LONGITUDINAL DISTANCE X/H,
FIG (A 117) TEMPERATURE. TRAJECTORY, WIDTH, AND DEPTH
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          20   40  60  80  100   120  140  160  180  200
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                        155

-------
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300
280
        20   40  60  8.0   100  120   140  160  180  200
              LONGITUDINAL DISTANCE  X/H0

    FIG (AI20) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH
           (TTWDhPLOTS FOR SURFACE JET DISCHARGE
                     157

-------
750
700
 50
        50  100   150  200 250  300 350 400  450  500
             LONGITUDINAL DISTANCE  X/H,
    FIG (A 121) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH
          (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                    158

-------
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       0    50  100  150 200  250  300  350 400  450  500




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                         V5S

-------
    50  100  150  200  250 300  350 400  450  500
          LONGITUDINAL DISTANCE X/H0

FIG (AI23) TEMPERATURE. TRAJECTORY, WIDTH, AND DEPTH
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                160

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                        161

-------
0   100  200  300  400 500  600  700 800  900  1000
          LONGITUDINAL DISTANCE X/H,
FIG (AI25) TEMPERATURE, TRAJECTORY.WIDTH,AND DEPTH
       (TTWDhPLOTS FOR SURFACE JET DISCHARGE
                 T62

-------
    50  100  150 200  250 300  350 400  450  500

          LONGITUDINAL DISTANCE X/H0

FIG (AI26) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                163

-------
    50  100  150  200  250 300  350 400  450   500
          LONGITUDINAL DISTANCE X/H0

FIG CAI27) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                164

-------
    50  100  150 200  250  300  350 400  450  500
          LONGITUDINAL DISTANCE X/H0

FIG (AI28) TEMPERATURE, TRAJECTORY,WIDTH. AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                165

-------
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                LONGITUDINAL DISTANCE X/H0



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             (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

                        166

-------
-10
10    20  30   40  50   60  70   80   90

^LONGITUDINAL DISTANCE X/H0
 FIG (AI30) TEMPERATURE, TRAJECTORY.WIDTH,AND DEPTH
        (TTWDhPLOTS FOR SURFACE JET DISCHARGE

                  H7

-------
-10
10    20  30   40  50   60  70   80   90
           LONGITUDINAL DISTANCE X/H,
 FIG(AI3I) TEMPERATURE,TRAJECTORY.WIDTH,AND DEPTH
        (TTWDhPLOTS FOR SURFACE JET DISCHARGE
                 158

-------
-10
10    20  30   40  50   60  70   80   90

 LONGITUDINAL DISTANCE X/H0
 FIG (AI32) TEMPERATURE, TRAJECTORY.WIDTH,AND DEPTH
        (TTWDhPLOTS FOR SURFACE JET DISCHARGE
                  169

-------
300
    -20
20   40   60  80   100   120   140   160   180

LONGITUDINAL DISTANCE X/H0
   FIG (AI34) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
           (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                     170

-------
 0   0   20   40   60  80   100  120   140  160  180

          LONGITUDINAL DISTANCE X/H0
FIG (AI35) TEMPERATURE,TRAJECTORY,WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                   171

-------
300
280
   '-20
20   40   60   80   100   120  140  160  180

LONGITUDINAL DISTANCE X/H0
   FIG (AI36) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
           (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                      172

-------
-50
0   50   100   150  200  250  300  350 400  450
     LONGITUDINAL DISTANCE X/H0
 FIG (AI38) TEMPERATURE, TRAJECTORY. WIDTH, AND DEPTH
        (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                  >73

-------
750
    -50   0   50   100  150  200 250  300 350  400  450

              LONGITUDINAL DISTANCE X/H0

    FIG (AI39) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH
           (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                    174

-------
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                          175

-------
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               (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                        176

-------
-50  0
50   100  150  200  250  300 350  400  450

 LONGITUDINAL DISTANCE X/H0
 FIG (AI44) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH
        (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

                  177

-------
       APPENDIX B
(TA)  WORKING NOMOGRAMS
            178

-------
                 TABLE  3
 Figure  Numbers  for (TA)  Working  Nomograms
    of  Appendix B, 6n  =  90°,  K =  10'5
F*
+A
1
5
10
15
2
Bl
B3
B5
B7
4
Bl
B3
B5
B7
6
B2
B4
B6
B8
10
B2
B4
B6
B8
                 TABLE 4
Summary of Figure Numbers for (TA) Working
          Nomograms of Appendix B
eo
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io-5
ID'4
ID'6
90°
B1-B8
B25-B32
B49-B56
60°
B9-B16
B33-B40
B57-64
120°
B17-B24
B41 -B48
B65-B72
                 179

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FIG(BI )  TEMPERATURE, AREA (TA)-PLOT
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                    FIG ( B 3 ) TEMPERATURE. AREA (TA)-PLOTS

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PLUME  AREA
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FIG ( B7 ) TEMPERATURE, AREA (TA)-PLOTS
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                                                        10
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FIG (B8 ) TEMPERATURE, AREA (TA)-PLOTS
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FIG (B 51) TEMPERATURE, AREA (TA)-PLOTS
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             PLUME AREA/H;
FIG (B52) TEMPERATURE. AREA (TA)-PLOTS
         FOR SURFACE  JET  DISCHARGE

-------
ro
GO
                                                            10
10
                                   PLUME AREA/H;
                     FIG (B 53) TEMPERATURE, AREA (TA)-PLOTS

                              FOR SURFACE JET DISCHARGE

-------
ro
CA>
CA>
                                    PLUME  AREA/H;
                       FIG (B54) TEMPERATURE, AREA (TA)-PLOTS

                                FOR SURFACE JET DISCHARGE

-------
ro
CO
                                           10


                                   PLUME  AREA
                     FI<3(B55) TEMPERATURE, AREA (TA)-PLOTS

                              FOR SURFACE JET DISCHARGE

-------
f«O
(M
Wt
                                  PLUME AREA/H:
                     FIG (B56) TEMPERATURE. AREA (TA)-PLOTS
                              FOR SURFACE JET DISCHARGE

-------
                     10
             PLUME  AREA/Ho

FIG(B57)  TEMPERATURE, AREA (TA)-PLOT
         FOR  SURFACE JET DISCHARGE

-------
10
                  10
        10
PLUME  AREA
10*
10
              FIG(B58)  TEMPERATURE, AREA (TA)-PLOT
                        FOR SURFACE JET DISCHARGE

-------
IN3
                                 PLUME AREA/H;
                    FIG (B 59) TEMPERATURE, AREA (TA)-PLOTS
                             FOR SURFACE JET DISCHARGE

-------
c*
    CO
        IO
                                   PLUME AREA/H;
                     FIG (B6O) TEMPERATURE, AREA (TA)-PLOTS
                              FOR SURFACE JET DISCHARGE

-------
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                                                                                10
                                     PLUME AREA/H;
                       FIG (B 61) TEMPERATURE, AREA (TA)-PLOTS


                                FOR SURFACE JET DISCHARGE

-------
             PLUME  AREA/H;
FIG (B62) TEMPERATURE, AREA (TA)-PLOTS
         FOR SURFACE  JET  DISCHARGE

-------
ro
*=.
IV
                                                                              10
                                   PLUME AREA/H;
                      FIG (B63) TEMPERATURE, AREA (TA)-PLOTS

                               FOR SURFACE JET DISCHARGE

-------
ro
-P>
CO
                                    PLUME AREA/H;
                      FIG(B64) TEMPERATURE, AREA (TA)-PLOTS

                               FOR SURFACE JET DISCHARGE

-------
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a.

ui
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                                PLUME  AREA/HJ



                   FIG(B65)  TEMPERATURE. AREA (TA)-PLOT

                            FOR  SURFACE JET  DISCHARGE

-------
en
   *°
   K°
Ul
a:
i>

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iij
a.
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IU
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3
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                           10
                                          10
                                    PLUME  AREA/H*
10*
10
                       FIG(B66)  TEMPERATURE. AREA (TA)-PLOT

                                 FOR  SURFACE JET DISCHARGE

-------
                      IO
              PLUME  AREA
FIG (B67) TEMPERATURE, AREA (TA)-PLOTS
         FOR SURFACE JET DISCHARGE

-------
             PLUME  AREA/H;
FIG (B68) TEMPERATURE. AREA (TA)-PLOTS
         FOR SURFACE  JET  DISCHARGE

-------
10
                 10
                           PLUME ARE A/H
             FIG (B69) TEMPERATURE. AREA (TA>-PLOTS
                      FOR SURFACE JET D.SCHARGE

-------
             PLUME AREA/H;
FIG (B 7O) TEMPERATURE, AREA (TA)-PLOTS
         FOR SURFACE JET  DISCHARGE

-------
    10'
        10
PLUME  AREA
10*
10
                                                           6
FIG (B7I ) TEMPERATURE. AREA (TA)-PLOTS
         FOR SURFACE  JET  DISCHARGE

-------
             PLUME  AREA/H;
FIG (B72) TEMPERATURE, AREA (TA)-PLOTS
         FOR SURFACE  JET  DISCHARGE

-------
      APPENDIX C
(Tt) WORKING NOMOGRAMS
            252

-------
                 TABLE 5
   Figure Numbers for (Tt) Working Nomograms
     of Appendix C, 0   = 90°, K = 10
F-*
+A
1
5
10
15
2
Cl
C3
C5
C7
4
Cl
C3
C5
C7
6
C2
C4
C6
C8
10
C2
C4
C6
C8
                 TABLE 6
Summary of Figure Numbers for (Tt) Working
         Nomograms of Appendix C
so*
+K
ID'5
io-4
ID'6
90°
Cl- 8
C25-C32
C49-C56
60°
C9-C16
C33-C40
C57-C64
120°
C17-C24
C41-C48
C65-C72
                     253

-------
                                        10s
10
              TIME- tU0/H0
FIG(C1  ) TEMPERATURE. TIME  (Tt)-PLOtS
         FOR SURFACE  JET DISCHARGE

-------
              TIME - tU0/H0
FIG ( C2 ) TEMPERATURE. TIME  (Tt)-PLOTS
         FOR SURFACE JET DISCHARGE

-------
re.
en
cv
         10
                                     TIME- tU0/H0



                      FIG(C3 ) TEMPERATURE, TIME (Tt)-PLOTS

                               FOR SURFACE JET DISCHARGE

-------
ro
                                                                                  \0'
                                      TIME- tU0/H0

                        FIG ( C4  ) TEMPERATURE. TIME  (Tt)-PLOTS
                                 FOR SURFACE  JET DISCHARGE

-------
01
CO
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Si
cr
UJ
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2
UJ
     UJ
     o

     £
     o:
                                        TIME-  tU0/H0



                        FIG(C5  ) TEMPERATURE, TIME  (Tt)-PLOTS

                                 FOR SURFACE JET DISCHARGE

-------
ro
8
                                      TIME-  tU0/H0


                      FIG(C6 )  TEMPERATURE, TIME (Tt)-PLOTS

                                FOR SURFACE JET DISCHARGE
                                                                                 10

-------
                                        10
10
               TIME- tU0/H0

FIG ( C7 ) TEMPERATURE,  TIME (Tt)-PLOTS
         FOR SURFACE JET  DISCHARGE

-------
                TIME-  tU0/H0

FIG(C8 ) TEMPERATURE, TIME (Tt)-PLOTS
         FOR SURFACE JET DISCHARGE

-------
              TIME- tU0/H0

FIG ( C9  ) TEMPERATURE. TIME  (Tt)-PLOTS
         FOR SURFACE JET DISCHARGE

-------
                                                         10
             TIME-
FIG (CIO) TEMPERATURE. TIME (Tt)-PLOTS
         FOR SURFACE JET DISCHARGE

-------
                                        10
10
               TIME- tU0/H0

FIG (CM  ) TEMPERATURE,  TIME  (Tt)-PLOTS
         FOR SURFACE JET DISCHARGE

-------
ro
        in
                                   TIME- tU0/H0

                      FIG(C12 ) TEMPERATURE. TIME  (Tt)-PLOTS
                               FOR SURFACE JET DISCHARGE

-------
               TIME- tU0/H0

FIG (C 13) TEMPERATURE,  TIME  (Tt)-PLOTS
         FOR SURFACE JET DISCHARGE

-------
                                         10
10
                TIME-  tU0/H0

FIG (C 14) TEMPERATURE, TIME (Tt)-PLOTS
         FOR SURFACE JET DISCHARGE

-------
UJ
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QC
3
V)
                                                           10
                                  TIME-  tU0/H0


                  FIG (CIS)  TEMPERATURE, TIME (Tt)-PLOTS

                            FOR SURFACE JET DISCHARGE
10

-------
ro
C?i
vo
                                        TIME-  tU0/H0



                         FIG (C 16) TEMPERATURE, TIME (Tt)-PLOTS

                                  FOR SURFACE JET DISCHARGE

-------
              TIME- tU0/H0
FIG(CI7) TEMPERATURE. TIME  (Tt)-PLOTS
         FOR SURFACE  JET DISCHARGE
                                                          10'

-------
                                                          10'
              TIME- tU0/H0
FIG (CIS) TEMPERATURE. TIME (Tt)-PLOTS
         FOR SURFACE  JET DISCHARGE

-------
                                                          10
                TJME-  tU0/H0

FIG (C 19) TEMPERATURE, TIME (Tt)-PLOTS
         FOR SURFACE JET DISCHARGE

-------
               TIME-  tU0/H0

FIG(C2I) TEMPERATURE, TIME  (Tt)-PLOTS
         FOR SURFACE JET DISCHARGE

-------
              TIME- tU0/H0

FIG(C20) TEMPERATURE. TIME  (Tt)-PLOTS
         FOR SURFACE  JET DISCHARGE
                                                          ICT

-------
               TIME- tU0/H0

FIG(C22) TEMPERATURE,  TIME  (Tt)-PLOTS
         FOR SURFACE JET DISCHARGE

-------
               TIME- tU0/H0

FIG(C23) TEMPERATURE, TIME  (Tt)-PLOTS
         FOR SURFACE JET DISCHARGE

-------
ro
                                                                                  10
                                       TIME-  tU0/H0

                       FIG(C24)  TEMPERATURE, TIME (Tt)-PLOTS
                                 FOR SURFACE JET DISCHARGE

-------
              TIME- tU0/H0

FIG(C25) TEMPERATURE. TIME  (Tt)-PLOTS
         FOR SURFACE JET DISCHARGE

-------
              TIME- tU0/H0
FIG(C26) TEMPERATURE, TIME  (Tt)-PLOTS
         FOR SURFACE  JET DISCHARGE

-------
rv,
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oo
   LU

   CC
   a:
   UJ
   Ul
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   2
   o:
                                     TIME- tU0/H0




                     FIG(C27) TEMPERATURE, TIME (Tt)-PLOTS


                               FOR SURFACE JET  DISCHARGE

-------
IN3
CO
                                    TIME - tU0/H0
                       FIG (C28) TEMPERATURE. TIME  (Tt)-PLOTS

                                FOR SURFACE JET DISCHARGE

-------
                                        10
               TIME- tU0/H0

FIG (C29) TEMPERATURE,  TIME (Tt)-PLOTS
         FOR SURFACE JET  DISCHARGE
IO

-------
r\3
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                                     TIME- tU0/H0



                      FIG (C3O) TEMPERATURE, TIME (Tt)-PLOTS

                               FOR SURFACE JET DISCHARGE

-------
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   cc

   1
   a:
   UJ
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   V)
                                      TIME- tU0/H0



                      FIG(C3I)  TEMPERATURE,  TIME (Tt)-PLOTS

                                FOR SURFACE JET  DISCHARGE

-------
X

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                                  TIME-  tU0/H0




                   FIG (C32) TEMPERATURE, TIME (Tt)-PLOTS


                            FOR  SURFACE JET DISCHARGE

-------
IO
                           TIME-  tU0/H0

             FIG (C 33)  TEMPERATURE, TIME (Tt)-PLOTS
                       FOR SURFACE JET DISCHARGE

-------
ro
CO
                                    TIME- tU0/H0



                       FIG (C3A) TEMPERATURE. TIME  (Tt)-PLOTS

                                FOR SURFACE JET DISCHARGE

-------
CO
CO
                                                             10
10
                                    TIME -  tU0/H0


                     FIG(C35) TEMPERATURE, TIME  (Tt)-PLOTS

                              FOR SURFACE JET DISCHARGE

-------
ro
                                   TIME- tU0/H0

                      FIG(C36) TEMPERATURE. TIME (Tt)-PLOTS
                               FOR SURFACE JET DISCHARGE

-------
                                                          10
               TIME- tU0/H0

FIG(C37) TEMPERATURE,  TIME  (Tt)-PLOTS
         FOR SURFACE JET DISCHARGE

-------
               TIME-  tU0/H0

FIG(C38) TEMPERATURE, TJME  (Tt)-PLOTS
         FOR SURFACE JET DISCHARGE

-------
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                                   TIME-  tU0/H0



                   FIG(C39) TEMPERATURE, TIME  (Tt)-PLOTS

                            FOR  SURFACE JET DISCHARGE
                                                                               IO

-------
               TIME- tU0/H0

FIG (C40) TEMPERATURE, TIME (Tt)-PLOTS
         FOR SURFACE JET  DISCHARGE

-------
ro
10
                                  TIME- tU0/H0



                     FIG (C4I ) TEMPERATURE, TIME  (Tt)-PLOTS

                              FOR SURFACE JET DISCHARGE

-------
              TIME-  tU0/H0

FIG(G42) TEMPERATURE, TIME  (Tf)-PLOTS
         FOR SURFACE JET DISCHARGE

-------
ro
£
                                                                                10
                                     TIME-  tU0/H0

                      FIG (C43) TEMPERATURE, TIME  (Tt)-PLOTS
                               FOR SURFACE JET DISCHARGE

-------
                                                          IO
              TIME- tU0/H0

FIG (C44) TEMPERATURE, TIME  (Tt)-PLOTS
         FOR SURFACE  JET DISCHARGE

-------
ro
to
CX)
                                       TIME-  tU0/H0



                       FIG (C45)  TEMPERATURE, TIME (Tt)-PLOTS

                                 FOR SURFACE JET DISCHARGE

-------
      H°
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ID
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2
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                                         TIME-  tU0/H0



                         FIG.(C46) TEMPERATURE, TIME (Tt)-PLOTS

                                   FOR SURFACE JET DISCHARGE

-------
               TIME-  tU0/H0

FIG(C47) TEMPERATURE. TIME  (Tt)-PLOTS
         FOR SURFACE JET DISCHARGE

-------
to
                                     TIME- tU0/H0

                      FIG (C48) TEMPERATURE, TIME  (Tt)-PLOTS
                               FOR SURFACE JET DISCHARGE

-------
10
                           TIME-  tU0/H0

             FIGCC49)  TEMPERATURE, TIME (Tt)-PLOTS
                       FOR SURFACE JET DISCHARGE

-------
CO
o
CO
                                    TIME- tU0/H0



                      FIG (C50) TEMPERATURE, TIME  (Tt)-PLOTS

                               FOR SURFACE  JET DISCHARGE
                                                                                10

-------
               TIME- tU0/H0

FIG(C5I) TEMPERATURE,  TIME  (Tt)-PLOTS
         FOR SURFACE JET DISCHARGE

-------
                           TIME- tU0/H0
10
10
             FIG (C 52) TEMPERATURE, TIME (Tt)-PLOTS
                       FOR SURFACE JET DISCHARGE

-------
               TIME- tU0/H0

FIG(C53) TEMPERATURE,  TIME  (Tt)-PLOTS
         FOR SURFACE JET  DISCHARGE
                                                          10

-------
               TIME- tU0/H0

FIG (C54) TEMPERATURE, TIME  (Tt)-PLOTS
         FOR SURFACE JET DISCHARGE

-------
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C3
                                                                                    10
                                         TIME-  tU0/H0



                          FIG (C 55) TEMPERATURE, TIME  (Tt)-PLOTS

                                   FOR SURFACE JET DISCHARGE

-------
CO
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Xa
                                     TIME- tU0/H0



                     FIG (C56) TEMPERATURE, TIME (Tt)-PLOTS

                              FOR SURFACE JET DISCHARGE

-------
                           TIME-  tU0/H0
IO
             FIG(C57) TEMPERATURE, TIME (Tt)-PLOTS
                       FOR SURFACE JET DISCHARGE

-------
              TIME- !U0/H0
FIG (C58) TEMPERATURE. TIME  (Tt)-PLOTS
         FOR SURFACE  JET DISCHARGE

-------
               TIME- tU0/H0

FIG(C59) TEMPERATURE,  TIME  (Tt)-PLOTS
         FOR SURFACE JET DISCHARGE

-------
CO
CO
                                     TIME-  tU0/H0

                       FIG (C6O) TEMPERATURE, TIME (Tt)-PLOTS
                                FOR SURFACE JET DISCHARGE

-------
                      10
               TIME- tU0/H0
10
10
FIG(C61) TEMPERATURE, TIME (Tt)-PLOTS
         FOR SURFACE  JET  DISCHARGE

-------
CO
*J
en
                                       TIME- tU0/H0




                        FIG (C62) TEMPERATURE,  TIME (Tt)-PLOTS

                                 FOR SURFACE JET  DISCHARGE

-------
   Ul
   a:
   cc


co  CL
_j  5
t" *S\  ^^m,

   UJ
   UJ
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   if
   o:
                                      TIME-  tU0/H0




                      FIG(C63) TEMPERATURE, TIME  (Tt)-PLOTS


                                FOR SURFACE JET DISCHARGE

-------
u»
                                    TIME- tU0/H0

                     FIG(C64) TEMPERATURE, TIME  (Tt)-PLOTS
                              FOR SURFACE JET DISCHARGE

-------
CO
                                  TIME-  tU0/H0
        10
                     F1G(C65) TEMPERATURE. TIME (TtH>LOTS
                              FOR SURFACE JET DISCHARGE

-------
CO
i««ff

to
        IOV
                          10'
                                            10s
                                         -  fUo/H0
                     FIG(C«> TEMPERATURE. TIME (TtHH.073
                               FOR SURFACE  JET DISCHARGE

-------
               TIME- tU0/H0

FIG (C67) TEMPERATURE, TIME  (Tt)-PLOTS
         FOR SURFACE JET DISCHARGE

-------
10
                           TIME-  tU0/H0

             FIG (C68)  TEMPERATURE, TIME (Tt)-PLOTS
                       FOR SURFACE JET DISCHARGE

-------
                                                          10
               TIME- tU0/H0

FIG(C69) TEMPERATURE,  TIME (Tt)-PLOTS
         FOR SURFACE JET  DISCHARGE

-------
CO
IS3
CO
                                       TIME-  tU0/H0


                        FIG (C70) TEMPERATURE, TIME (Tt)-PLOTS

                                 FOR  SURFACE JET DISCHARGE

-------
CO
        UJ

        GC

        ID
o:
UJ
Q.


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       HI
       o

       2
       cc
                                          TIME-  tU0/H0




                          FIG (C7I )  TEMPERATURE, TIME  (Tt)-PLOTS


                                    FOR SURFACE JET DISCHARGE

-------
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rs
C'l
                                                                                   10
                                       TIME-  tU0/H0


                        FIG (C72 ) TEMPERATURE, TIME (Tt)-PLOTS

                                 FOR SURFACE JET DISCHARGE

-------
          APPENDIX D
(TTWD)  SUPPLEMENTARY NOMOGRAMS
               326

-------
               TABLE 7
 Figure Numbers for (TTWD) Supplementary
  Nomograms of Appendix D, 0,  = 90°,
        K = 1(T5> F = 4, A =°5
V^h
.001
.01
.2
.005
Dl
D4
D7
.02
D2
D5
D8
.1
D3
D6
D9
               TABLE 8
 Summary of Figure Numbers for (TTWD)
Supplementary Nomograms of Appendix D

Eo

.01
.05
.15
D1-D9
D10-18
D19-D27
                   327

-------
    20   40  60   80  100  120   140  160  180  200
          LONGITUDINAL DISTANCE  X/H0

FIG ( D I ) TEMPERATURE,TRAJECTORY,WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                 32-8

-------
    20   40  60   80  100  120   140  160   180  200
          LONGITUDINAL DISTANCE X/H0

FIG( D2 ) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                  329

-------
    20   40  60   80  100  120   140  160   180  200
          LONGITUDINAL DISTANCE X/H0

FIG ( D 3 ) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                  330

-------
    20   40  60   80  100  120   140  160   180  200
          LONGITUDINAL DISTANCE  X/H0

FIG( D4 ) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                   331

-------
o

X


LtJ
O
CO
o:
UJ
          20   40  60   80  100  120   140  160  180   200
                LONGITUDINAL DISTANCE  X/H0


      FIG ( D5 ) TEMPERATURE9TRAJECTORY, WIDTH, AND DEPTH

             (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                          332

-------
300
        20   40  60   80  100  120   140  160   180  200
              LONGITUDINAL DISTANCE X/Ha

    FIG (D 6 ) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
           (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                        333

-------
300
280
260
 20
        20  40   60  80  100  120  140  160  180  200
              LONGITUDINAL DISTANCE  X/H0

    FIG (D 7 ) TEMPERATURE.TRAJECTORY,WIDTH, AND DEPTH
           (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                       334

-------
20
       20  40   60   80  100  120  140  160  180   200
             LONGITUDINAL DISTANCE X/H0

   FIG ( D8 ) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
         (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

                      335

-------
    20   40  60   80  100  120   140  160   180  200
          LONGITUDINAL DISTANCE X/H0

FIG (D9 ) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                   336

-------
20
       20  40   60   80  100  120  140  160  180  200
             LONGITUDINAL DISTANCE X/H0

   FIG ( D10) TEMPERATURE,TRAJECTORY,WIDTH, AND DEPTH
          (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                      337

-------
X


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IT
LU
          20   40  60   80   100  120  140  160  180  200
                LONGITUDINAL DISTANCE  X/H0


      FIG ( D11 ) TEMPERATURE,TRAJECTORY,WIDTH, AND DEPTH

             (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                          338

-------
20
       20  40   60   80  100  120  140  160  180  200
             LONGITUDINAL DISTANCE X/H0

   FIG (D 12 ) TEMPERATURE.TRAJECTORY, WIDTH, AND DEPTH
          (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                      339

-------
    20   40  60   80  100  120   140  160  180  200
          LONGITUDINAL DISTANCE  X/H0

FIG (Dl 3 ) TEMPERATURE,TRAJECTORY,WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                  340

-------
    20   40  60   80   100  120   140  160  180  200
          LONGITUDINAL DISTANCE  X/H0

FIG (D14 ) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                  341

-------
    20   40  60   80  100  120   140  160   180  200
          LONGITUDINAL DISTANCE X/H0

FIG (Dl 5) TEMPERATURE,TRAJECTORY,WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                   342

-------
                               140  160  180  200
          LONGITUDINAL DISTANCE X/H0

FIG (D 16) TEMPERATURE,TRAJECTORY,WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

-------
    20   40  60   80  100  120   140  160   180  200
          LONGITUDINAL DISTANCE X/H0

FIG (Dl 7) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                   344

-------
    20  40  60   80  100  120  140  160  180  200
          LONGITUDINAL DISTANCE X/H0

FIG (D18) TEMPERATURE.TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

                  345

-------
300
280
260
        20  40   60   80  100  120  140 160  180  200
              LONGITUDINAL DISTANCE X/H0

    FIG (Dl 9) TEMPERATURE.TRAJECTORY,WIDTH, AND DEPTH
           (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                      346

-------
    20   40  60   80  100  120   140  160  180  200
          LONGITUDINAL DISTANCE  X/H0

FIG (D 20) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                   347

-------
    20   40  60   80  100  120   140  160  180  200
          LONGITUDINAL DISTANCE  X/H0

FIG (D2 I ) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                    348

-------
20
       20  40   60   80  100  120  140  160  180  200
             LONGITUDINAL DISTANCE X/H0

   FIG (D 22) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH
          (TTWD)-PLOTS FOR SURFACE JET DISCHARGE
                      349

-------
20
      20   40  60  80   100  120   140  160   180  200
            LONGITUDINAL DISTANCE  X/H0

   FIG (D23) TEMPERATURE,TRAJECTORY,WIDTH, AND DEPTH
         (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

                    350

-------
    20   40  60   80  100  120  140  160  180  200
          LONGITUDINAL DISTANCE X/H0

FIG (D24) TEMPERATURE,TRAJECTORY,WIDTH, AND DEPTH
       (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

                   351

-------
300
280
        20  40   60   80  100  120  140 160  180   200
              LONGITUDINAL DISTANCE X/H0

    FIG (D25) TEMPERATURE.TRAJECTORY. WIDTH, AND DEPTH
           (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

                      352

-------
20
       20  40   60   80  100  120  140  160  180  200
             LONGITUDINAL DISTANCE X/H0

   FIG (D26) TEMPERATURE,TRAJECTORY,WIDTH, AND DEPTH
          (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

                     353

-------
300
280
260
 20
        20  40   60   80  100  120  140 160  180   200
              LONGITUDINAL DISTANCE X/H0

    FIG (D27) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
           (TTWD)-PLOTS FOR SURFACE JET DISCHARGE

                      354

-------
                    APPENDIX E
TEMPERATURE DENSITY NOMOGRAMS AND COMPUTATIONAL AIDS
                          355

-------
                         FIGURES
No.         '                                                Page
El   Temperature-density Relations for Fresh and             358
     Salt Water.
E2   Temperature-density Relations for Salt Water.           359
E3   Temperature-density Relations for Froude Number         360
     Calculations.
E4   Temperature-density Relations for Froude Number         361
     Calculations.
E5   Gaussian Distribution Curve Used for Calculating Off-   362
     Centerli ne Temperatures.
                                 356

-------
                        COMPUTATIONAL AIDS

 In order to aid  in the  calculation of discharge  Froude numbers, the
 following figures have  been  prepared.   Figures El and E2 are curves of
 Apr/p  as a function of  the temperature  difference, ambient temperature,
 and salinity.  The figures are  for salinities of S=0 (fresh water),
 10, 30, and 40 ppt.  The ambient  temperature differences and temperatures
 have been given  in both °C and  °F for convenience.

 Figures El and E2 were  used  to  determine  values  for the curves plotted
                                                                            1 in
 on Figures E3 and E4. The latter  two figures, present values of G = (g Ap/p)  '
(such that the Froude number  F = GU / vH )  for selected values of
 S and T  as a function  of AT.   Figure E3  is for  the metric system
       a
 of units where U  is in cm/sec, H in cm,  and temperature in °C.  Figure
 E4 is for the British engineering system  of units with U  in ft/sec,
 HQ. in  ft, and temperature in °F.

 The distribution of temperature and velocity within the fully developed
 region of the plume are assumed to be Gaussian both in the lateral
 and vertical directions.  Figure  E5 is  a  Gaussian distribution
 curve with n  = 0 at the plume  centerline or water surface.  The
 variable n can be thought of as either  4n/W or   2 Z/D  depending on
 whether the lateral or  vertical direction is considered.  Also
 indicated on the curve  is the part where  the local excess temperature
 is 1/2 the centerline excess temperature, T -T,/T -T,  = 0.5.  The
                                            i  u  U  d
 value of n at this point is  1.1775.   This point is of interest since
 many investigators use  nQ 5   as  a measure of plume size.  The relationship
 between  nQ 5 and W/2 or D is W/2 = 1.7 nQ 5 or  D = 1.7 nQ 5-

 Table El in this appendix is a  comprehensive list of conversion factors
 useful in solving practical  problems.                /
                                   357

-------
   o
   o
   h-
CO
en
OS-

                                                                   .OO2
.OO4
.006
                       Figure El  Temperature-density Relations for Fresh and Salt,Water.

-------
   O
  o

  h-
to
en
                                                                    SALINITY-30%.
SALINITY-40%.
                                                                .002
                                                    .OO4
.006
                         Figure E2 Temperature-density Relations for Salt Water.

-------
CO
CT»
O
<->
o
LU
                                                 LEGEND

                                                 	Ta-20*C

                                                             IO°C
                  Figure  E3  Temperature-density Relations for Froude  Number Calculations.

-------
CO
at
                                                    LEGEND

                                                           TQ «50°F
              SALINITY

                  %0
               Figure E4  Temperature-density Relations for Froude Number Calculations.

-------
       1.0
       0«8
t*j  r-
cr>  ,
co  '.

                        ££
                              Dl/2

                              B(H)

                              W

                              D
HALF-WIDTH (-DEPTH) = 1-177 cf

CHARACTERISTIC  WIDTH(DEPTH)*1-4I4

WIDTH-4 cf

DEPTH=2tf
                                                     2-0
                    2.5
3-O
3.5
                                                   z/tf
         Figure E5  Gaussian Distribution Curve Used for Calculating Off-Centerline Temperatures.

-------
Table  El  (Continued);

Multiply                 §y_

Centimeter/sec2          .03281
Feet/sec2                30.48

gn = 980.7 Cm/sec
 0 = 32.17 Ft/sec
   = 35.30 Km/hr
   = 9.807 Meters/sec2
   = 21.94 Meters/hr

Feet /sec                30.48
Feet/sec                 1.097
Feet/sec                 .0183
Feet/sec                 .592
Feet/sec                 .3048
Feet/sec                 .6818
Centimeters/sec          1.969
Centimeters/sec          .0328
Centimeters/sec          .036
Centimeters/sec          .0194
Centimeters/sec          .0224
Meters/sec               3.281
Meter/sec                1.943
Meters/sec               2.237
Knots                    51.48
Knots                    1.689
Knots                    1.853
Knots                    .5148
Knots                    1.152

Pounds                   453.6
Pounds                   -4536
Kilogram                 2.205
 British Thermal Unit
 Btu
 Btu
 Btu
 Btu
 Btu
 Btu
 Horsepower-hours
 Horsepower-hours
 Horsepower-hours
 Hoursepower-hours
 Joules (watt-sec)
 Kilowatt-hours
(Btu) 1.076
    778.3
    3.929 x
    1054.8
    .2520
    2.93 x
    .2930
    2545
    1.98 x
    2.684 x
    .7457
    9.48 x
    3413
 x 10?

 10-4
10


10
 6

-4
             To  Obtain
                     2
             Feet/sec
             Centimeters/sec'
Centimeters/sec
 Kilometers /hour
 Ki1ometers/mi nute
 Knots (nautical  miles/hr)
 Meters/sec
 Miles/hr
 Feet/minute
 Feet/sec
 Kilometer/hr
 Knots
 Miles/hr
 Feet/sec
 Knots
 Miles/hr
 Centimeters/sec
 Feet/sec
 Kilometers/hr
 Meters/sec
 Miles/hr

 Grams
 Ki 1 ograms
 Pounds

 Centimeter-grams
 Foot-pounds
 Horsepower-hours
 Joules (Watt-seconds)
 Kilogram-calories
 Kilowatt-hours
 Watt-hours
 Btu
 Foot-Pounds
 Joules (Watt-sec)
 Kilowatt-hours
 Btu
 Btu
                                   363

-------
                             TABLE El
                        CONVERSION FACTORS
Multiply
Centimeters
Centimeters
Inches
Feet
Feet
Meters
Acres
Acres
Acres
Acres
Acres
Square Miles
Square Kilometers
Hectares
Cubic Feet
Cubic Feet
Cubic Feet
Cubic Meters
Cubic Meters
Gallons
Gallons
Gallons
Liters
Acre Feet
Cubic Feet/sec
Cubic Feet/sec
Cubic Feet/sec
Cubic Meters/sec
Cubic Meters/sec
Cubic Meters/sec
Acre Feet/year
0.3937
0.0328
2.540
30.48
.3048
3.281
43,560
, 4,047
.001562
.004047
.4047
640
247.1
2.471
.02832
7.481
28.32
35.31
264.2
.1337
.003785
3.785
.2642
1233
448.86
.0283
.6464
15850
35.31
22.82
.62
Gallons per minute       ,00223
Cubic Meters per second  .0283
Million Gallons/day      1.547
Million Gallons/day      54.63
 to  Obtain

 Inches
 Feet
 Centimeters
 Centimeters
'Meters
 Feet

 Square  feet
 Square  Meters
 Square  Miles
 Square  Kilometers
 Hectares
 Acres
 Acres
 Acres

 Cubic Meters
 Gallons
 Liters
 Cubic Feet
 Gallons
 Cubic Feet
 Cubic Meters
 Liters
 Gallons
 Cubic Meters

 Gallons per minute
 Cubic Meters  per second
 Million Gallons per day
 Gallons per minute
 Cubic Feet per second
 Million Gallons/day
 Gallons/minute

 Cubic Feet per second
 Cubic Feet per second
 Cubic Feet per second
 Cubic Meters  per second
                                  364

-------
        10
Table El (Continued):

Multiply                 By_


Kilogram-Calories        3.969
Kilogram-Calories        3089
Kilogram-Calories        1.559 x
Kilogram-Calories        1.163 x 10
Joules(Watt-seconds)     9.48 x 10
Joules(Watt-seconds)     .7376
Joules(Watt-seconds)     3.722 x 10

Btu per square Foot      .2713
Gram Calories per square
 Centimeter              3.687
                                   -3
                                   -3
          -7
To Obtain


Btu
Foot-pounds
Horsepower-hours
Kilowatt-hours
Btu
Foot-pounds
Horsepower-hours

Gram Calories per square Cm

Btu per square Foot
Degrees  Kelvin

°c = | (°F - 32)

°F = 32  + 1.8°C
1.8
                                             Degrees Rankine
         365

-------
             APPENDIX F
FURTHER ANALYTICAL CONSIDERATIONS
                AND
          COMPUTER PROGRAM
                    366

-------
                             CONTENTS
                            APPENDIX F
                                                               Page
List of Symbols                                                368-370
List of Figures                                                371-372
Sections
I.   Modifications Leading to PDS Model
     A.   Introduction                                        373-374
     B.   Buoyant Spreading                                   374-376
     C.   Development Length                                  377-378
     D.   Temperature-Time Exposure                           378-379
     E.   Temperature-Area Calculations                       379
     F.   Extrapolating the Model                             379-381
     G.   Dimensionless PDS Program                           382

II.  Fitting PDS with Data                                    383
     A.   Turbulent Exchange Coefficients Eh> Ey              384-385
     B.   Entrapment Coefficient EQ                          385-389
     C.   Drag Coefficient Cp                                 389-396
     D.   Conclusions                                         396-399

III. Theoretical Analysis                                     400-409
IV   Computer Program                                         410-428
V.   References                                               429-430
                                  367

-------
                         LIST OF SYMBOLS

B    Local characteristic width of jet = i/2 a
B    Half width of outlet
 o
B, /0 Plume half width = 1.177 a,
  \l i                           h
CD   Form drag coefficient
Cp   Interfacial shear drag coefficient
c    Celerity of a density front
D    Local plume depth = 20
E.   Dimensionless horizontal eddy diffusion coefficient e../UH
E    Entrainment coeffient
EV   Ratio of vertical to horizontal eddy, diffusion coefficients E /E.
           A    •  * •  c   A     u      TT^(U'/2 + V  COS 0
F    Gross densimetric Froude number,   — •
FD   Form drag per unit length of jet
F    Densimetric Froude number at outlet, U
 o                                         o  3  o
g    Acceleration due to gravity
g1   Reduced, gravitational acceleration, g Ap/pa
H    Local characteristic thickness of jet
HQ   Depth of outlet
KE   Atmospheric heat transfer coefficient
M    Local s-momentum flux in jet
n    Horizontal coordinate"" perpendicular to s
PDS  Prych model modified by Davis and Shirazi
P    Pressure force on jet cross section
Q    Local volume flux in jet
Ri   Richardson number
Sp   Shear forces in X-  and Y- directions per unit length
     of jet
s    Curvilinear coordinate along jet centerline
S.   Distance from outlet to end of initial zone
ATr  Local excess water surface temperature on jet center! ine
  w
AT   Difference  between outlet and ambient water temperatures
t    Time of travel along centerline trajectory
                                 368

-------
TH    Angle between positive  S-  and  X-  directions  (0)
T     Local plume temperature T(s , n, Z)
U     Local  excess jet velocity on jet centerline
U     Velocity in s -direction
U     Local plume velocity U(s,n,Z)
U     Discharge  velocity from outlet
X     Rectilinear coordinate  parallel to ambient current
Y     Rectilinear coordinate, horizontal and perpendicular to X
Z     Coordinate in vertical  direction
V     Ambient current velocity
a     Angle used in data analysis of Ref.  (2) a= tr-0
Ap    Difference between outlet  and  ambient water densities
eu.e  Ambient turbulent diffusion coefficient for horizontal and
 hi* v
      vertical directions
8     Angle  between positive s-  and x~ directions^]
0     Angle between x-axis and outlet velocity direction
 v     Kinematic  viscosity
 p      Fluid density
T „     Shear stress
 SUBSCRIPTS

 a     Ambient  conditions
 c     Centerline  value  at surface
 i     Refers to variables at  end of  development zone
 o     Discharge conditions
 r     Indicates variable is a function of  s, n, Z directions
                                   369

-------
DIMENSIONLESS VARIABLS
u1 =
B1 =

H1 =

r =

K1 =
S' =

X1 =

Y' =

V =

P' =
M1 =
SF •
FD -

Q' =
A
R

Re '
Uc/Uo
B/H
0
H/H
0
AT/AT
C 0
Wo
S/H
0
X/H
0
Y/H
0
v/un
0
9 9
P/U H
M/UoHo
SF/U0H0
VUoHo
2

2Bo/Ho
V/U
0
UoHo/v
            370

-------
                         List of figures
                            Appendix F
Figure No-                                                               Page

     Fl   Comparison of Original Model Predictions of References 1  and    375
          3 for Plume Width with a Typical Measured Plume Width from
          Reference 3.
     F2   Correlation of Measured Surface Plume Temperature Data from     386
          a Jet Initially Discharged in a Turbulent Coflowing Channel
          with Zero Relative Velocity, R=l.
     F3   Comparison of Calculated Temperatures with Mean Measured         387
          Centerline Plume Temperature Data of Figure F2.
     F4   Comparison of Calculated Widths with Mean Measured              388
          Width for a Jet Initially Discharged at the Surface in a
          Turbulent Coflowing Channel with Zero Relative Velocity,  R=l.
     F5   Correlation of Selected Field and Laboratory Surface Plume      390
          Temperature for Typical Froude Number and Aspect Ratio at
          Zero or Negligibly Small Ambient Current.
     F6   Comparison of Calculated Temperatures with Mean Measured         391
          Surface Plume Temperature Data Shown in Fig. F5
     F7   Correlation of Selected Field and Laboratory Surface Plume      392
          Width Data for Typical Jet Froude Number and Aspect Ratio
          at Zero or Negligibfy Small Ambient Current.
     F8   Comparison of Calculated Widths with Mean Measured Surface      393
          Plume Width Data of Fig. F7.
     F9   Correlation of Selected Field and Laboratory Surface Plume      394
          Trajectory Data for Typical Jet Froude Number, Aspect Ratio
          and Ambient Cross Current.
     F10  Comparison of Calculation with Mean Measured Surface Plume      395
          Trajectory Data of Fig. F9.
     Fll  Comparison of Calculated Widths with Selected Field and         397
          Laboratory Surface Plume Width Data Correlated for Typical
          Jet Froude Number, Aspect Ratio and Ambient Cross Current.
                                  371

-------
Figure No.                                                             page

     F12  Comparison of Calculated Temperatures with Selected Field     398
          and Laboratory Surface Plume Temperature Data, Correlated
          for Typical Jet Froude Number, Aspect Ratio and Ambient
          Cross Current.

     FT3  Schematics of a Surface Plume Showing the Coordinate          399
          System.
                                   372

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             I. MODIFICATIONS LEADING TO PDS MODEL

A. INTRODUCTION

Various mathematical models of heated surface jets are available
for the prediction of two and three dimensional plume configurations.
Two widely accepted methods are used for solving the equations in
these models, namely one based on the integral analysis approach
and the other based on the differential numerical analysis methods.
The latter approach, while capable of greater generality, is considerably
more costly and due to limited funds and resources was excluded
from further consideration for this work.  However, a certain degree
of generality of results is retained by considering only three
dimensional plume models for generating the nomograms in this workbook.

A comprehensive review of thermal plume models is presented in
Reference  4. Among the three dimensional surface jet models seriously
considered for generating the nomograms in this workbook is  one by
Stolzenbach and Harleman (MIT Model), another by Prych and the third
model by Stefan, et al. These are discussed in References 1, 3, 5,
and 10.   It is outside the scope of this Workbook to discuss in detail
results of all experiments  performed  on  the  three models during  our  attempt  to
provide a working program.  The MIT model, despite its many fine features,
has considerable computational difficulties.  Prych's model is the
result of reasonably successful attempt to remove from MIT's model
some of these difficulties.  Stefan's model was written for the developed
zone alone and thus can't be compared with others directly. Even
though it includes wind effects absent in the other two, it ignores
the hydrostatic pressure in the longitudinal direction.
                                   373

-------
In general, the MIT and Prych models yield comparable predictions .
The greatest deviation between the predictions of both models and
data is in plume width.  Figure Fl shows the predictions of width
from both models for a particular Froude number and aspect ratio
as compared with the mean of experimental data of Fig. F8 and the
experimental data from Run 3 of Ref. 3.  Both models overestimate
the plume width.

An effort is made here to introduce modifications in Prych's model
to make it better agree with existing data.  These modifications for
improving the model, as well as certain other additions, are discussed
below.  The result of these changes is a modified model henceforth
referred to as  PDS.

B. BUOYANT SPREADING

                        3
Stolzenbach and Harleman  present an order of magnitude analysis of
the momentum equations as applied to the jet.  They show that the
lateral acceleration of fluid particles within the plume is negligible
only when the jet is nonbuoyant.  Otherwise, the fluid particles accelerate
(spread) due to the influence of two interacting forces,  namely,
the inertia and buoyant forces.  Since the full nonlinear equations
of motion describing a buoyant plume are too difficult to solve, the
lateral spreading due to buoyant forces in the MIT, and Prych
models are calculated independently of spreading due to nonbuoyant
forces.  The two spreading rates are assumed to make additive contributions,
thereby ignoring the nonlinear interaction between the two forces.
As a consequence of the assumptions in this linearization their  analyses
overestimate the plume width when the inertia and buoyant forces are
the same order of magnitude (i.e.,  when the densimetric Froude number
is not too large).  When the plume inertia forces are dominant such
as with strong ambient current or large densimetric Froude numbers,
reasonable width predictions can be obtained.
                                 374

-------
     100
      80
      60
X
X
      40
      20
              /
             /   LEGEND
            /   	 MIT
            /    	PRYCH
           1     O   MIT DATA
           /     	MEAN DATA FIG F8
                    20
40          60

     S/HO
80
100
    Figure FT   Comparison  of Original Model Predictions of References
               1  and  3  for Plume  Width With a Typical Measured Plume
               Width  from  Reference  3.
                                375

-------
The buoyant spreading  function  used   by Prych is based on the analysis
of an immiscible film, such as oil spreading over water that ignores
the shear interaction between the fluid systems.  In this analysis,
the fluid particles are assumed to move with a velocity equal  to the
velocity caused by density waves alone.

In a separate analysis of a buoyant spreading of a pool of warm water,
Koh and Fan  accounted for the interfacial shear interaction but ignored
the actual entrainment of the cool water.  They found that near the
source the spreading velocity is in agreement with  Prych 's analysis
and the spreading velocity and the fluid velocity  are the same, i.e.,
                           ?     ?
                          v£  =  c  ~ g'H
Where H is the local depth of the buoyant pool.  However, far away
from the source where the shear forces become very important,  the
fluid front velocity is
                         V (
                              2
Where g'H is proportional to c , (e/Hp) is proportional to the shear
velocity and H/B is the ratio of the local pool depth to its width.
If interpreted in terms of plume spread, this finding implies that
spreading velocity is inversely proportional to the local aspect ratio
of the plume.
The appearance of the local aspect ratio in the expression for the
plume velocity offers an intuitively appealing ground for assuming,

                         Vjj ~ (g'H)(H/B)

This slight modification to Prych's analysis was introduced in the
model.  As a result, a satisfactory fit with data became possible.
                                  376

-------
C.    DEVELOPMENT LENGTH

Analysis of the jet development zone is complicated because of the need
to examine simultaneously the characteristics of a core region as well
as a turbulent outer jet region.  Stolzenbach and Harleman developed a
three dimensional program for this  region, but in his modification of the
program, Prych adopted a one dimensional approach in which he employed
celerity relations  for the spreading of the buoyant unmixed core region.
He then used the appropriate  conservation equations to relate the
fluid properties at four jet diameters away from the outlet to the
fluid properties at the outlet.  The fixed development length of
four diameters is based on the assumption of a semicircle with an
area 2 B H .  Prych 's development length S^ can be written as
                        HO
where A is the channel  aspect  ratio.

 Note that the above development  length  does not  change with the
 initial densimetric Froude number.  However,  calculations with the
 MIT model show that the development length does  change with initial
 densimetric  Froude number as well  as  the jet  aspect ratio.

 Since a better agreement of model  predictions with the data is expected
 if this aspect of the model is also appropriately adjusted, resort
 was made to  laboratory  experiments to obtain  this information.  Experiments
 were conducted in a still water  tank  with a heated jet at the EPA
 Pacific Northwest Environmental  Research Laboratory.   Several jet
 aspect ratios and jet densimetric  Froude numbers were tested.  A hot
 film anemometer probe was used to  traverse the jet development zone
 laterally at several stations  downstream from the outlet.  The presence
 of the core was detected from  subdued turbulent  temperature fluctuations
 as well as the temperature level.  The  coincidence of the increased
 turbulence fluctuations, the beginning  of the temperature drop, and

                                   377

-------
the disappearance of a uniform  core at a point downstream of the
outlet signaled the end of development zone.  The data for this length
was correlated to give

                    ^i            A?  i /-a
                    —      R  A  /A*- \ 1/6
                    u     -   O.f  (-g— )
                    Ho            h°
This tentative result is subject to refinement (particularly with
respect to the effect of the ambient current) when better experimental
investigations currently underway  become available.   Meanwhile,
the use of this correlation was found very helpful to fit the model
with available plume data.

It should be noted that generally the excess temperature ratio at the
end of the development zone is different from unity.  It is hoped that
future experiments will shed light on this aspect of the problem as
well.  In anticipation of the latter point, we have introduced
a slight modification to Prych's program to allow the use of
excess temperature ratios other than unity

D.  TEMPERATURE-TIME EXPOSURE

The total exposure time of organisms to a given excess temperature
within the plume can be calculated if we assume that (a) such organisms
are uniformly distributed within the natural water environment that
supply the initial jet discharge with dilution water, and (b) that the
motion of these organisms is totally governed by the motion of the
entrained fluid.  That is,  the organisms are not self propelled and
they are small enough so that they faithfully follow the motion of
the entrained fluid.
                                  378

-------
Naturally, the organisms can be entrained  at  any point s. along the
trajectory from S] to a second point s2-   Thus, the travel time
t can  be estimated from the calculated centerline plume velocity as


                f£2  ds/Uc
Where U  is the local centerline  velocity, itself a function of s and
       V*
calculated stepwise by the  program.  For each plume, unique  temperatures
 T,  and TV are calculated that correspond, respectively with s,  and  s^.

The above calculation performed for the travel along the centerline
yields a minimum time of travel between s-, and s? associated with an
exposure to a maximum elevated temperature existing in the plume
between T, and l^.  Off-center, the temperatures are lower, and
particles do not travel as  fast as along the  plume centerline.

E.    TEMPERATURE-AREA CALCULATIONS

The surface area within an  isotherm is calculated in Prych's program
for each integration step DS by using the  trapezoidal rule of
integration.  The  areas are calculated for excess temperature ratio
 increments of .05  starting  at  1.0. Near the source the excess temperature
ratio may drop more than  .05 within one integration step.  As a result,
the area calculations are in error.  To avoid this problem in the
PDS model the integration step is adjusted so that the maximum allowable
drop in excess temperature  per time step  is  .05.

F.    EXTRAPOLATING THE MODEL

Prych suggests that at a local densimetric Froude number of unity,
calculations with  the plume model be discontinued because of
mathematical singularities.  In the PDS model,  there are two  singularities
 which do not occur at the  same point along the trajectory.
                                   379

-------
Even though calculations can be made to continue beyond the first
singularity, adequate justification is missing to support the
use of information so obtained.  The problem is that the first singularity
can be bypassed almost all the time by a judicious selection of the
step size and error terms, thereby passing from regions where the
gross densimetric Froude number is slightly greater than unity to
a region where it is below unity.  Calculations continue normally until the
second singularity is reached, there the program is terminated.

The effect of passing over a Froude number of unity while allowing
calculations to continue is that the gradient of the plume depth changes
sign which initiates a small damped oscillation.  Conservation equations
are satisfied and all plume characteristics seem to change very
smoothly.  The trajectory of the plume is calculated without apparent
discontinuity,from the equations.  The trajectory curves also agree
very well with the available data.  Motivated by these observations,
we have decided to retain at least the trajectory calculations when
the solution is allowed to continue beyond a Froude number of unity.

It is important to resolve whether the singularity at a local Froude number
of unity is only a mathematical obstacle or that a discontinuity actually
exists in nature at this point.   The question arises because (a) examination
of the available three dimensional plume data does not reveal the presence
of such a discontinuity in plume characteristics for a low discharge
Froude number and a small ambient current, (b) the existence of a
discontinity for two dimensional plumes has been confirmed in
laboratory experiments only and in analyses, of References 6 and 10
and these occur at local Froude numbers greater than unity,
                                  380

-------
(c) the uncoupling of the hydrostatic pressure forces in
the longitudinal direction from the same forces in the lateral direction
retains essentially the nature of a two dimensional plume in the
model with respect to those factors that influence an internal hydraulic
jump the most, and (d) it is difficult to insist that a singularity
in the calculations occurs at a point in the plume where the internal
hydraulic jump takes place based solely on the simplified analysis
of  the plume without confirmation with data.

The PDS model is fitted with data in a systematic way as will be
discussed. Based on the model, (TTWD) plots are made that show the
effect of ambient current.  For nearly all cases examined, the model
produces continuous data near the source as the calculations proceed
from high ambient current to low and zero ambient current.  However,
at low densimetric Froude numbers and far away from the source, the
local Froude number becomes unity  and the calculations are stopped.
A smooth extrapolation based on calculated data at higher ambient
current (usually R > 0.05) to lower currents (R < 0.01)  can be
made only if there is assurance that the plume does not go through
                                 i
an internal hydraulic jump.  Since the available data do not support
the existence of discontinuity in a three dimensional model, we
find it useful to present the said extrapolation.  In order to
avoid misunderstanding, the extrapolated curves are presented as
broken lines.  The reader is cautioned not to confuse the dashed
extrapolation lines with the trajectory lines which are also dashed.
The trajectories are completely dashed and originate at X and Y = 0.

The extrapolation of the temperature lines was obtained  according
to S = a (l-R)b where a and b are constants determined from two
consecutive neighboring points just before the occurrance of singularity
The width lines were extrapolated in the same manner.
                                  381

-------
G.    DIMENSIONLESS PDS PROGRAM

In order to make the results of the mathematical model more general
and independent of any particular system of units used, the governing
equations and all calculated plume characteristics are made dimensionless
by dividing all length character!si tics by H  , velocities by U  .
                                              22
momentum, pressure, shear and drag forces by U  H , flow rate by
O2,   diffusivity by U H , excess temperature by (T -T ) and the
 0. 0                    00                          0  a
kinematic heat transfer coefficient by U .  These quantities are
"primed" and listed under a separate heading in the list of symbols.
The choice of H  for nondimensionalizing the length scale is made
for the convenience it offers over 41 B , etc.
                                     o o

When the dimensionless terms are introduced in the governing equations
and the primes dropped, the equations for the developed region of
the jet become identical with those given by Prych if only one
interprets:
            1         1                        -
               as 91> ft  as v,  K£/UO as K and UQH0 as e
             o
                                  382

-------
                      II. FITTING PDS WITH DATA

Reference 2 provides a comprehensive set of data that is a good
representation of available experiments both in the field and
laboratory.  The data provide a wide range of plume conditions with which
one can test and accordingly adjust numerous analytical functions of the
plume model.  The plume model contains a number of free variables such
as entrapment coefficient EQ, turbulent exchange coefficients Eh> Ey,
drag coefficient CQ and shear coefficient Cp. The magnitudes of these
coefficients must be prespecified so that the model produces
the best fit with the measured plume characteristics.

In order to accomplish this task, the following procedure is adopted:
(a) Data for plume characteristics are subgrouped with a narrow range of
certain experimental parameters such as the current ratio, R, the
densimetric Froude number, F , the jet aspect ratio, A, or the angle of
discharge, 0Q.   Each subgroup consists of several experiments and
several sources, thus providing considerable degree of realism with
respect to possible experimental scatter and variations in experimental
parameter scales.  The choice of a narrow range in certain experimental
parameters was dictated by the desire to obtain as strong a correlation
of the data within a given subgroup as possible, (b) For each subgroup,
the range and the mean of all experimental parameters are determined.
(c) The data are correlated using dimensional analysis and multiple
regression methods separately for each subgroup following the procedure
outlined in Reference 2. (d) The measured plume characteristics are
plotted against dimensionless axial distance using the correlation
results, (e) A representative smooth curve is drawn through the
mean data and local  standard deviations are displayed on both
sides of the mean curve to show the scatter.   This mean curve is
a fair representation of the subgroup, and is represented by the
                                  383

-------
mean parameters obtained  in  item b  above,  (f)  Finally, the  PDS
 program  is  used to  calculate  the plume  characteristics in each
 subgroup  for  the mean of  the  experimental  parameters  R,  F,  A, and
 0  .  Agreement between  the calculated characteristics and the
 data mean is  sought by  adjusting one or more of  the model coefficients
 E  , E.,  E  , DD and  Cp.  This  process is repeated for  several
 subgroups,  adjusting in each  trial  one  or  more coefficients until
 best fits are obtained  to plume characteristics  for all  subgroups.

 It.should be  pointed out  that correlations of  each data  subgroup are
 useful mainly for the mean data in  that subgroup.  They  are not
 universal correlations  and cannot be used  outside the data  range
 they represent.

 A.  TURBULENT EXCHANGE  COEFFICIENTS, Eh, Ey

 The data  set  most suitable for determining the effects of ambient
 turbulence  on plume behavior  is provided by Weil2>8      in
 his experiments, Weil injected heated water at the surface  in
 a  turbulent channel  from  a semi-circular jet at  a relatively
 large  densimetric jet Froude  number.  The  discharge was  in  the
 direction of  the channel  current (6 =0). The jet velocity in
 all his  experiments  was held  equal  to the  local  channel  flow
 velocity.*    Since  the  relative velocity between the  plume  and ambient
 water  is  zero and since buoyancy effects are small due to a high
 Froude number, dilution is largely  due  to  turbulence  effects.

 For the  conditions  of this experiment,  the following  simplifications
 can be introduced in the  mathematical model: (a) The  entrainment
 coefficient,  E  , can be set equal to zero  because there  is  no relative
 velocity  between the jet  and  the ambient water,  (b) For  the same
 *When  retrieving Weil's  data  from  Reference  2  the  following  adjustments
 should be  made only  on that data:  multiply  F  by 1.189,  S  by 1.414 and W
 by  0.707 to  account  for  an error in presentation.
                                  384

-------
reason, the shear coefficient, Cp,is also zero,   (c) The drag coefficient
CD> is zero because the jet is parallel to the ambient current
and the pressure distributions on the left and" right hand sides
of the plume are identical,   (d) For the dimensionless surface heat
exchange coefficient, one can choose a typical value of K  = 10
without affecting the calculated plume characteristics greatly one
way or another, because we are dealing with small areas and small
temperature differences, (e)  Since the jet densimetric Froude number
is high, the influence of the buoyant forces on the plume spread
is not substantial.  The plume width grows  predominantly due to
turbulent entrainment of the  ambient water, a mechanism which the
model accounts for through E. and E  .

Figure F2 is the plot of correlated temperature data showing the
local mean and standard deviations.  Figure F3 is a replot of the
mean  temperature data together with several computer calculations
based on-the PDS model for F  = 16  , A= 2, 0Q = 0, and K = 10"5.
Calculations are made for several values of E.  and E /E,  as well
as the free factor of the spreading function,XK1.  The plots for
the calculated and measured  plume width data are  shown in Fig. F4.
The measured width data were  closely spaced with  excellent correlation.
For this reason individual data  points were not plotted.  Instead,
a  narrow band showing the spread of all experimental data are presented.

A  visual inspection of Weil's data of Figures  F3  and F4 shows that
the best fit is obtained with
       Eh = .02, Eu/Eh = .2 and XK1 = 1.4
        II       Vi,  V  I I     ,
B. ENTRAPMENT  COEFFICIENT  EQ

The next group  of data  consists  of  information  from several  sets
of laboratory and some  field experiments  for a  surface  discharge
                                   385

-------
  in
CO
co
   . o
                  L EGEND
                         LABORATORY DATA
                         REF(2).WEIL(I972)
                         LOCAL ME AN OF DATA
                         LOCAL STANDARD DEVIATION
                                                                      100
                                            X/H
                 Figure  F2   Correlation of Measured Surface  Plume Temperature Data from
                            a Jet Initially Discharged in a  Turbulent Coflowing Channel
                            with Zero Relative Velocity, R=l.

-------
    u.
    X
CO
00
                  LEGEND
                  	 MEAN DATA FROM FIG F2
                  REFER TO FIG F4  FOR A.B»'"
         id
           10°
10'
                     Figure   F3   Comparison of  Calculated Temperatures with  Mean Measured
                                   Centerline Plume  Temperature Data of Figure F2.

-------
^  10'
1°
X
CO
                 LEGEND
                      Eh
                  0   0.01
                  O   o.i
                  O   0.001
                  V   0.02
                  A   0.02
                  O   0.012
                     "0.02
                       - DATA. REFERENCE (2), WEIL (1972)
                       CORRELATION
                Figure  F4   Comparison of Calculated Widths  with Mean Measured
                             Width  for a Jet Initially Discharged at the Surface  in a
                             Turbulent Coflowing Channel with Zero Relative  Velocity, R=l

-------
in zero or negligibly small cross current.  The correlation of temperature
data are plotted in Fig. F5 and the width data in Fig. F7.

For the  conditions of this group, one can assume that the drag
coefficient is zero.  As a first approximation we also assume that
the contribution of the ambient turbulence is accounted  for by the
previously assigned values of E.  and E .  As we continue to adjust
other coefficients in the model, we may have to reevaluate E,  and E .

Figures F6 and F8 show the replots of the mean data and several
computer calculations of the PDS model with preassigned values of
entrainment and shear coefficients.  The best fit of the computer
model with the data is obtained with E  = 0.05.   Since  the shear coefficient
has negligible effect on the result, it will be set equal to zero.

 C.    DRAG COEFFICIENT, CQ

For given values of discharge angle, Froude number, aspect ratio,
and ambient current, the plume trajectory is mainly influenced by
the entrainment of ambient fluid with a minor influence due to pressure
drag.  Since the entrainment coefficient is prespecified from the
above, only the drag coefficient can be used to further adjust the
trajectory.   Consequently, we need to regroup the trajectory data
for a reasonably wide range of all plume parameters mentioned above.
Such data are plotted in Fig. F9 showing the data sources,  the  local
mean, and standard deviations.  Figure  F10 is a replot of the mean
trajectory showing the comparison with computed values.  Originally,
considerable deviation of computed vs. mean data was found near
the source.  This deviation was corrected by assuming in PDS that
                        X(o)  =  Sj  cos 0 (o)
                        Y(o)  =  S.  sin 9 (o)
                                   389

-------
   ro
   (D
CO
<£»
a
 u.
 X

 £


H°
 <
EGENO

    DATA SOURCE, SHIRAZI ( 1973)

    REF 3

    REF  I
                       O   REF 9


                       A   REF 8


                       •   LOCAL MEAN


                       •   LOCAL STANDARD  DEVIATION
                                                  X/H.
                 Figure  F5   Correlation of Selected  Field and Laboratory Surface  Plume
                             Temperature for Typical  Froude Number and Aspect Ratio  at
                             Zero or Negligibly Small Ambient Current.

-------
CO

tO

JO
 i"
CO
VO
 H°
 <3
            Hi|»MPUTED  FOR  Eh - -02. E- «2.
            - - - u o   _   .      n       v
                a   F   A
                O  2.5  2.0
                V  2.5  2.0
                A  3.0  2.5
                03  4.O  2.5

                O  4-0  25
                A  3.O  25

                   25  2.7
                    MEAN OF MEASURED DATA

                                                   10
                                                                                      100
                                                      X/HO
                      Figure F6   Comparison of Calculated Temperatures with Mean Measured
                                  Surface Plume Temperature Data Shown in Fig. F5

-------
CO
«3
ro
             ffi
                   10°
                         LEGEND
                     DATA SOURCE,  SHIRAZI (1973)
                      — REF  3
                      -* REF  8
                       V REF  9
                      • LOCAL  MEAN
                      • LOCAL  STANDARD DEVIATION
10'
I0a
I03
                                                        X/H.
                      Figure  F7   Correlation of Selected Field and Laboratory Surface Plume
                                   Width  Data for Typical Jet Froude Number and Aspect Ratio
                                   at Zero  or Negligibly Small Ambient Current.

-------
GO
U»
CO
    sr
    o
    (0
    CM
     1°
     00
LEGEND

   D  SEE FIGF6

      MEAN OF  DATA
                                                      X/H.
                     Figure  F8   Comparison of Calculated Widths  with  Mean  Measured Surface
                                  Plume Width Data of Fig. F7.

-------
CO
i£>
•Pa
                   LEGEND

                   	REFER TO FIGS F 10 and FII

                   • LOCAL MEAN

                   • LOCAL STANDARD DEVIATION
          10°
             icr
10
                   Figure   F9   Correlation of  Selected Field and Laboratory Surface Plume
                                 Trajectory Data for Typical Jet Froude  Number, Aspect Ratio
                                 and Ambient Cross  Current.

-------
   cv
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   10

   0>

   'u.


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01 "il.
I02
10'
10°
1 1 1 1 1 1 1 1 1 1
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LF


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1 1 1 1 1 1 1 1 1 	 1
•GEND
COMPUTED FOR MEAN DA
CD-0
CD»I
MEAN OF MEASURED DAI
1.2 «F< 5.6
.087« R <: .73
l.57« 04; 2.06
.5« A« 31













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                      Figure  F10  Comparison  of Calculation with Mean  Measured Surface Plume
                                   Trajectory  Data of Fig. F9.

-------
which replaces Prych's assumptions,

                                           6(0}
                        X(o) ••S1 cos -£

                        Y(o) = S, sin
   2
en + e(o)
Figure F10 shows that further improvement can be obtained by setting
cd = i.o.

In order to complete the adjustment of the PDS model to fit the
data, we need to check the model against measured plume width and
temperature for a wide range of parameters.  If agreement is obtained
with such data without the need to readjust the previously specified
coefficients EQ,..Eh, Ey, Cpand CD, then the fitting of PDS with
data is considered complete.

The raw data and calculated values based on the previously determine co-
efficients are  compared  in  Fig.  Fll  for plume width and  Fig.  F12  for plume
temperature.   The agreement obtained from the comparison of calculated
and measured plume width is excellent and the agreement for plume
temperature is reasonably good.

D. CONCLUSIONS
     >i
The PDS model is best fitted to a wide range of field and laboratory
experimental  data if we set EQ » .05, Eh = .02, Ey/E.  = 0.2, CQ
= 1.0 and CF  = 0.  It is felt that PDS can be used to predict typical
situations for surface discharge consistent with the major assumptions
made in the model.  Calculations based on extreme, but perhaps possible,
values of these coefficients are also presented in the supplementary
nomograms of  Appendix D.
                                   396

-------
                                             LEGEND

                                          DATA SOURCE, SHIRAZI (1973)

                                               REF II
                                               REF 5
                                                                     cr
                             x/a
Fi gure  Fl 1
Comparison of Calculated Widths with Selected Field and
Laboratory Surface  Plume Width Data Correlated for Typical
Jet Froude Number,  Aspect  Ratio and Ambient Cross Current.

-------
                                                                              LEGEND
                                                                          DATA SOURCE, SHIRAZI (1973)

                                                                           —  REF 3
                                                                           V  REF 9
                                                                           X  REF 6

                                                                           t  REF 5
                                                                          COMPUTED FOR MEAN DATA
                                                                              R   F   A
                                                                          Q  .39  1.4   23   SO(va-)

                                                                          O  -49  1.6   7.2   87 {X )

                                                                          A  .28  3*   1.0   90 (t )
10'
10
                                                S/H,
            Figure   F12  Comparison  of Calculated Temperatures with  Selected Field
                          and Laboratory Surface Plume Temperature  Data, Correlated
                          for Typical  Jet Froude Number, Aspect Ratio and Ambient
                          Cross Current.

-------
                 111. THEORETICAL ANALYSIS

The theoretical analysis used to develop the three-dimensional
surface plume program (PDS) is based on Prych's model which has
been non-dimensionalized and modified according to previous
sections.  The analysis is given here in an abbreviated form for
completeness.  The terms have all been non-dimensionalized as mentioned
above and primes have been omitted for convenience.

The'method of analysis is an integral approach which assumes
similarity of temperature and velocity profiles and the principle
of entrainment.  The profiles assumed are Gaussian such that

     Tr = T exp(-n2/B2) • exp(-Z2/H2)                         (1)

     Ur = U exp(-n2/B2) • exp(-Z2/H2) + V cos e               (2)

Where n and Z are distances perpendicular to the plume center!ine in
the lateral and vertical directions,respectively.  T and U are the
centerline temperature and velocity, respectively.  See Figure F13.

With the temperature and velocity profile assumed, the energy, volume,
and momentum fluxes  can be integrated across the plume at any cross
section leaving them in terms of centerline values and plume characteristic
width, B, and depth, H.  Accordingly, the volume flux

     Q " // (Uw + V  cos 0) dndZ = vBH(? + V cos 0)            (3)
         A    r                       c
where the limits of  integration for V cos 0 are taken as the bottom
half of the region.
                       2
                                399

-------
         u
o
o
          ///////////////     irr/T//•///.//
//7     £

V'
  7  U«
               Figure F13 Schematics of a Surface Plume Showing the Coordinate System,

-------
     solving (3) for U, yields

     U = 2 (- - V cos 0)                                   (5)
The heat flux, 0, is
     J • // UT dndZ =   TBH  (  + cos  0) -                    (6)
The momentum flux, M, 1s
     M - // Ur2 dndZ =TrBH  (jj-  + V cos e)2                    (7)
         M
The quantities dQ/ds, dT/ds, and dM/ds are calculated from the
conservation equations.  dQ/ds is assumed to be due to contributions
of jet entrainment and ambient turbulent mixing, thus
    dQ.= da i. + dg.i                                        (9)
    ds   ds  'j   ds  'a
The jet and ambient contributions are both divided into vertical
and horizontal components.  The horizontal jet entrained fluid is
                                401

-------
where

     AU  =  (U2 +  V2  sin2  0)1/2  exp(-Z2/H2)                      (11)

and EQ is an entrainment coefficient.  By inserting (11) into (10),
we obtain

     3s  'j h =  ^ HE0(u2 + y2  si"2  e>V2                        <12>

The vertical jet entrained fluid is

     f |jjV =  2 ABEAUvdZ                                 (13)

where
        =  Eo  f                                             (14)
 R.  is the  local Richardson number given by
      R  -  &   HTfs.n.o)
       1  "
The  function f  (R^) is a curve fit to the experimental data of
Ellison and Turner which gives
     f = [expf-SR   - .0183J/.982                             (15)
The velocity difference AU  is given by
AUV =
               exp(-2n2/B2) + V2 sin2 e]1/2
The  term T  is the surface excess temperature at the distance n from
the  plume center!ine.  The value of the integral  (13)  is determined
numerically in the  program.
                                 402

-------
The effective entrainment due to ambient turbulent mixing is due to an
analysis used by Prych1 which is

                   H  %                                        (16)
    f  la   -11.0^ ^H  f(V>                              (17)
         "             0 0
where
     Ri =  v£ HT/[Fjj(U  + V  cos  9)2]
and en and ey are the horizontal and vertical turbulent diffusion
coefficients, respectively.
The change in heat flux along the plume due to heat exchange with
the atmosphere is expressed as
      -= -2 /       Trdn =
            o
Where K is the dimension! ess heat exchange coefficient.

Substituting (18) into (11) yields

                                                              (19)
The conservation of momentum is applied in the s-direction and then
divided into X and Y components.  The net forces on the plume are
balanced by the change in momentum flux.  The forces considered
important are (a) internal pressure forces due to buoyancy, (b) form
drag due to ambient current and 
-------
The pressure forces are found by determining the excess pressure
due to buoyancy as a function of depth and then integrating the
pressure over the vertical cross section of the plume.  Thus, the
normalized pressure force is

    P = p-2 // ( /  TfdZ} dA =  v€ TH2B/2F2                       (20)
         o   A   -oo
The form drag acting normal  to the plume center!ine is assumed
similar to the drag on a solid body such that

    FD = lcD^HV |V| sin20                                   (21)

where CD is a drag coefficient

The interfacial shear forces are assumed to be similar to turbulent
flow over a flat surface with a boundary layer thickness of *2 H and
a velocity equal  to the vector velocity difference between the plume
and ambient current.  Accordingly, the X and Y components of this shear
force reduce to
                    fn n
SFX = CF(1H) 1/4 /     AU-V3/4 [V sin2 G - U cos 9 exp(-n2/B2)]dn (22)
           ,   1/4  v^B    ...                    9
SFY= •CF(RT)     /     AU-V    £v cos 9 H- U exp(-nVB2)]dn       (23)
           e       o
where Cp is a friction coefficient and Re is the jet discharge
Reynolds number.   The value of CF includes constants carried
throughout Prych's program.

The change in momentum flux includes the effects of the momentum
of the entrained ambient fluid, V ^-, which acts in the X-direction.
Equating the forces to the change in momentum flux in the X and Y
directions yields
                                404

-------
     s (M+P) cos 0 = SFX +  FD  sin  e + V                       (24)

    4-s (M+P) sin 0 = SFY -  FD  cos  0                           (25)

Using equations (8) and (20) for M and P, multiplying  (24)  by  - sin 6,
(25) by cos 6 and combining yields an expression  for the change in flow
direction
          SFy cos 0 - SFX sin  0 -  FD - V  sin  0 (dQ/ds)
             Q    . /IT  TU2D
            TrBH  + 2F2 TH B
                                                               (26)
Differentiating M and P, multiplying  (24)  by  cos 6 and  (25)  by sin G
and combining yields

    3Jr = [SFy sin G -i- SFX cos 0 +  (V  cos 0 -  2Q/TrBH) (dQ/ds)
          -(vTBH2/2F2)(dT/ds) + (Q2/TrB2H  - vfH2T/2F2)(dB/ds)]
          [*4FTHB/2F2 - Q2/^]"1                             (27)

It is noted that this expression for  change in  depth  is undefined when
the denominator is zero.  Hence, results beyond this  singularity are
questionable.

Momentum in the lateral direction  is  included only indirectly through
lateral spreading.  It is assumed  that  the contributions  to  spreading
by nonbuoyant horizontal jet mixing and buoyancy are  independent of one
another such that
     dB_ _  /dEU    .  /dBv
     ds "  Wnb   Wb

where the subscripts b and nb denote  buoyant  and nonbuoyant  terms.
                                 405

-------
The nonbuoyant spreading 1s found by writing equation (27) without
the buoyancy terms (any term containing FQ) and assuming that
where (dQ/ds)h and (dQ/ds)y are the horizontal and vertical
entrainment rates.  This yields
                                                  29.)     ,      -(QZ/7rBH-)[(dQ/4s)v/(dQ/ds)h + 1]

This equation differs from one given by Prych in that all terms
containing the local densimetric Froude number are deleted.

As discussed earlier in this appendix, the spreading due to buoyancy
is assumed to be a function of the local excess density ratio, plume
depth and aspect ratio such that
                                                                 (29)
     vds't> " (§-Fc - 1)"'

where XK1 is a constant and F is the local Froude number.  It is
noted that this also has a singularity.  But due,to the fact that
B/H  is usually large, the singularity is not encountered.

The  proceeding equations are sufficient to perform a step-wise
integration along the plume.  From the local conditions of the plume,
dQ/ds is calculated.  When this is known dT/ds, dG/ds and dB/ds are
calculated.  With these known, dH/ds can be calculated.  These
derivatives are integrated step—wise along the plume trajectory
to give local values of X, Y, T, H, B, Q, and Q.
                                406

-------
In order to start the Integration within the developed zone where
the above analysis Is valid, starting conditions must be calculated.
These are determined by a simplified analysis of the development zone.
                           *
As was mentioned earlier 1n this appendix, the length of the development
zone 1s assumed to be

    ^ • 5.4 ( 4?-)1/3                                           (30)
                0
The values of B and H at the end of the development zone are calculated
from a method that superimposes entrained fluid, and fully developed
temperature and velocity profiles at the end of development onto an
analysis which ignores entrainment.  For the no-entrainment case, the
plume remains rectangular with 2bhU0 • QQ such that b.^ = BQH0.
However", it 1s assumed that b spreads due to buoyancy such that

    f-tg'h/Cu'-g'h)]1/2                                    (31)

Since bh 1s assumed constant (31) can be Integrated from S - 0 to
S* S.j to give
                                           +1}                 (32)
          0         0
Thus
    hi  - H0BQ/b1                                                (33)
It  is  now assumed  that the actual plume including entrapment has an
aspect ratio equal  to b^/h.j  such that

       •
                                407

-------
  The actual  flow rate  at the end  of the development  zone  is
  calculated  assuming fully developed profiles  and  ignoring surface
  heat losses such that J is constant.   Using equation (6)  yields
                                                                          (35)

  Where T is the excess temperature at the end  of development  and
  Q  is the discharge flow rate.

  The X and Y component to the momentum equations are  written  as in
  the previous section over the length of the development  zone.

  (MQ + PQ) cos 9Q + V QQ (f- - 1)  + FD.  sin GO  +  SF.x  =  (M. +  P.)  cos  9.   (36)
  (MQ + P0) sin  9Q +  FD. cos 0Q + SF.Y = (M.-fP.) sin 8.

  Where the drag and shear forces  are approximated by
(37)
             Cn      1 +  h,           ,,
      FDi  =TSi  <-T-> v  IV' Sln  9o                                 (38)

                          _                9       1/4
      SF.X = CFS. (Bo+b.) |V - UQ|  (__i-_)     (V-cos  &l          (39)

      ce  -  qp    sin 0n
      bMY "  bMX  V - cos  00                                              (40)

  and U  is a unit vector in  the direction of 8  and 7 is the vector
  velocity of the ambient current.

  Multiplying (36) by sin 8.  and (37)  by cos  81 and subtracting yields

8'= arc tan {[(MQ + PQ)  sin 8Q - FD1 cos 8Q + SF1y]/[(MQ  + PQ) cos  8Q +
          Q0V(2/T- 1) ^  s1neo +  SF]>                              (41)
                                  408

-------
Multiplying (36) by cos G^ and (37) by sin e^ and combining yields

(MQ + Po) cos  (ereo)  + VQQ cos 0.  (f -1)  + (SF1X  +  FD. cos 0Q) cos 0.

        + (SFTY  '  FD1  Sln  0o)  Sln-el  =Mi  + pi                      (42)
Using equations (8),  (20),  (32) and 2B0/HQ = A, equation (42)
after some manipulation becomes
  LHS =       ()  +      ()                                    (43)
        T IT    °1           Di    F'
Where LHS is the left  hand side of  (42).  This equation is solved
numerically in  the  program for B. .  Equation  (33) is used to find
H. .  The value  of T at the end of the development zone is retained
as a variable depending on the expression for S.. .  Equation 30 is
used for development length when T  = 1.0.
                                 409

-------
                   IV.COMPUTER PROGRAM

The computer program used to generate the nomograms  for this
workbook is presented in this section.  The program  is  written in
FORTRAN 4 and consists of a main program entitled PDS and five subroutines
KHPCG, AREA, FCT, RED, and OUTP,

The main program PDS reads the input variables .initializes constants,
and calls subroutine KHPCG which performs the actual  calculation!
Subroutine KHPCG is a standard IBM scientific subroutine which performs
the stepwise integration of differential equations by the Hamming
Predictor-Corrector Method.  It has been modified slightly for compatability
of common statements.  This method was found to be faster than the
fourth order Range-Kutta solution  used by Prych.

Subroutine AREA is a step-wise integration of the area  enclosed
by surface isotherms.  Subroutine FCT calculates the derivatives
of the program variables which are used in KHPCG. Subroutine RED
calculates the reduction in the vertical entrainment coefficient
as a function of local Richardson's number.

Subroutine OUTP prints out the input parameters followed by desired
dimensionless variables at each integration step along  the trajectory
of the plume.  The variables printed out are  S1, X1, Y1, TH, T1,
U', t1, Q/Q0, HT, H1, B1, F, RI, and IHLF.  See the  list of symbols
in this appendix for definition of these terms.

Input to the PDS program consists of one card giving the number
of cases to be calculated followed by a set of three cards for each
case.  The variables on each card and the required format are as
follows:
                                 410

-------
Card 1
     Format (13)
     Number of cases to be calculated

Card 2
     Format (20A4)
     Any information the user wishes to have printed out
     relating to this case.  This information is printed out at
     the top of each output page.

Card 3
     Format (7F10.5)
     FQ   =    Discharge densimetric Froude  number
     A    =    Aspect ratio
     V   =    Ambient current ,  v/U
     RE   =    Reynolds number
               (only used when CF f 0)
     0    =    Discharge angle (degrees)  6  = 0 is in the
               direction of the ambient current
     CD   =    Drag coefficient (1.0 is suggested)
     CF   =    Interfacial shear (friction)  coefficient
               (0.0 is suggested)

Card 4
     Format (7F10.5)
     E    =    Entrapment coefficient  (0.05 suggested)
     K    =    Surface heat exchange coefficient K£/pC UQ
     SLIM =    Value  of S1 at which integration is to stop
     DS   =;    Largest integration step to be used (DS =
               is  reasonable)
                                  411

-------
     EV   =    The ratio of vertical  to  horizontal  ambient  turbulence
               diffusion coefficients(0.2  is  suggested).
     EH   -    Dimensionless horizontal  ambient  turbulent diffusion
               coefficient (0.02  is suggested)
     XK1   =    Spreading coefficient  (1.4  is  suggested)

Cards 2*4 are repeated for each case  to  be run.

Following are (a)  listings of sample  input, (b)  sample output and
(c) complete program listing.
                                 412

-------
                 SAMPLE  INPUT  DATA
                  i
                SAMPLE RUN OF PDS PROGRAM
                4.        5.        .1                90.       1.
                .05       .00001    500.      5.       .2        .02          1.4
CO

-------
                                         OUTPUT
FLOATING WARM WATER JETS —  SAMPLE RUN OF PDS PROGRAM
FRO «   0.0    2BO/HO =  5.0    V/UO •   .100
E * .0=100   CD s 1.0000   CF =     0   »E =

    S        X        Y     TH     T     U
  ANGLE * 90.0     SURFACE H
OE 00   EV • 2.000E-01   EH =
TIME
              HT
                                                                  H
« l.OOOE-05
 2.000E-02

   8       F
RI
                                                  IHLF
9.6
9.9
10.?
10.5
10. 8
11.1
11.4
12.1
12.7
13.3
13.9
15.2
16.4
17.7
18.9
21.4
23.9
26.4
28.9
31.4
33.9
38.9
43.9
48.9
53.9
58.9
63.9
68. 9
73.9
7B.9
A3. 9
ftfl.9
93.9
9ft. 9
103.9
108. 9
1)3.9
118.9
1?3.9
128.9
133.9
138.9
143.9
148.9
153.9
1.0
1.0
1.1
1.1
1.1
1.2
1.2
1.3
1.4
1.5
1.6
l.R
2.0
?.2
2.4
3.0
3.5
4.1
4. A
5.5
6.2
7.7
9.3
11.1
12.9
14.8
16.9
19.0
21.1
23.4
?5.7
28.0
30.5
32.9
15.5
38.1
40.7
43.4
46.1
48.9
51.8
54.6
57.5
60.5
63.5
9.5
9.8
10.1
10.5
10. S
11.1
11.4
12.0
12.6
13.2
13.9
15.1
16.3
17.6
18.8
21.2
23.7
26.1
29.5
30.9
33,3
38.1
42.8
47.5
52.1
56.7
61.3
65.9
70.4
74.8
79.3
83.7
88.0
92.4
96.7
101.0
105.?
109.4
113.6
117.8
121.9
126.0
130.1
134.1
138.1
84.1
83.8
83.6
83.3
83.1
(32.9
82.7
82.3
81.9
81.5
81.1
80.4
79.8
79.2
78.6
77.5
76.5
75.5
74.6
73.8
73.0
71.6
70.2
69.0
67.8
66.7
65.7
64.8
63.8
63.0
62.1
61.3
60.6
59.8
59.1
58.4
57. ft
57.1
56.5
55.9
55.3
54.7
54.2
53.6
53.1
1.000
.980
.962
.944
.928
.913
.899
.873
.849
.828
.809
.775
.746
.721
.699
.662
.632
.607
.585
.567
.550
.522
.500
.481
.465
.451
.438
.427
.417
.408
.400
.392
.385
.379
.373
.367
.362
.356
.351
.347
.342
.338
.334
.331
.327
.951
.928
.906
.886
,«67
.849
.832
.801
.773
.748
.725
.684
.649
.619
.592
.547
.511
.481
.455
.433
.413
.381
.355
.334
.315
.300
.286
.274
.264
.254
.245
.237
.230
.223
,?17
.211
.206
.200
.196
.191
.187
.182
.179
.175
.171
9.8E 00
.OE 01
.OE 01
.IE 01
.IE 01
.IE 01
.2E 01
.3E 01
.3E 01
.4E 01
.5E 01
.7E 01
.9E 01
2.0E 01
2.2E 01
2.7E 01
3. IE 01
3.6E 01
4. IE 01
4.6E 01
5.2E 01
6.4E 01
7.6E 01
8.9E 01
l.OE 02
1.2E 02
1.3E 02
USE 0?
1.6E 02
1.8E 02
2.0E 02
2. IF 02
2.3E 0?
2.SE 02
2.7E 02
2.9E 02
3. If 02
3.3E 02
3.5E 02
3.7E 02
3.9E 02
4. IE 02
4.3E 02
4.5E 02
4.7E 02
2.000
2.041
2.080
2.118
2.155
2.190
2.225
2.291
2.354
2.414
2.472
2.579
2.679
2.772
2.859
3.019
3.163
3.295
3.416
3.529
3.634
3.826
3.998
4.154
4.298
4.431
4.555
4.672
4.783
4.887
4.987
5.083
5.174
5.263
5.348
5.430
5.510
5.588
5.664
5.738
5.809
5.879
5.94ft
6.014
6.079
.000
.000
.000
.000
.000
.000
.000
.000
.000
1.000
1.000
1.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.999
.999
.999
.999
.999
.998 .
.998
.998
.998
.997
.997
.997
.997
.996
.996
.996
.995
.995
.995
.995
.994
.994 1
.994 1
.88
.91
.94
.96
.98
t.OO
L.03
1.06
1. 10
1.13
1.16
L.21
1.24
1.28
1.30
1.34
1.37
1.39
.40
.41
.41
.41
.41
.40
.39
.38
.37.
.36
.34
.33
.32
.31
.30
.29
.28
.28
.27
.26
.25
.25
.24
.23
.23
1.22
1.21
7.4
7.5
7.6
7.7
7.8
7.9
8.1
8.3
8.5
8.8
9.0
9.5
10.0
10.5
11.0
12.1
13.2
14.2
15.3
16.3
17.4
19.4
21.4
23.3
25.2
27.0
28.8
30.5
32.2
33.8
35.4
37.0
38.5
40.0
41.5
42.9
44.3
45.7
47.0
48.4
49.7
51.0
52.2
53.5
54. 7
2.751
2.675
2.606
2.542
2.483
2.429
2.379
2.289
2.210
2.141
a. 080
1.976
1.892
1.822
1.762
1.667
1.595
1.539
.493
.456
.425
.376
.341
.316
.295
.279
.267
1.257
1.250
1.244
1.239
1.236
1.234
1.231
1.230
1.229
1.228
1.228
1.227
1.227
1.228
1.228
1.229
1.230
1.231
.086
.091
.096
.101
.106
.111
.116
.126
.136
.145
.155
.173
.190
.207
.224
.254
.283
.310
.335
.359
.381
.424
.461
.496
.529
.559
.587
.615
.640
.664
.689
.711
.733
.756
.777
.798
.819
.841
.862
.882
.903
.923
.942
.962
.981
0
4
4
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
                                              414

-------
FLOATING HARM MATER JETS —  SAMPLE RUN OF DOS PROGRAM
FRO *   4.0     2RO/HO a  5.0
E »  .0500   CD = 1.0000   CF
                             TH
V/UO •   .100
   0   HE «.

   T      U
  ANGLE =90.0     SURFACE  H
OE 00   EV « 2.000E-01    EH =
* l.OOOE-OS
 2.000E-02
                                                  TIME
                                                                  HT
                                                                                                 RI
                                                                                                        IHLF
158.9
163.9
168.9
173.9
17B.9
183.9
1BH.9
193.9
1919.9
203.9
?OR.9
213.9
218.9
2?3.9
228.9
233.9
238.9
243.9
248.9
253.9
258.9
243.9
768.9
273.9
278.9
283.9
288.9
293.9
298.9
303.9
308.9
313.9
318.9
323.9
328.9
333.9
338.9
343.9
348.9
353.9
35H.9
363.9
368.9
373.9
378.9
66.5
69.5
72.6
75.7
78.9
82.1
R5.3
88.5
91. H
95.1
98.4
101. R
105.1
108.5
112.0
115.4
118.9
122.4
125.9
129.4
133.0
136.5
140.1
143.8
147.4
151.0
154.7
158.4
162.1
165.8
169.6
173.3
177.1
180.9
184.7
188.5
192.3
196.1
200.0
203.9
P07.7
?n.6
215.5
?19.5
223.4
142.1
146.1
150.0
153.9
157.8
161.6
165.5
169.3
173.0
176.8
180.5
184.3
187.9
191.6
19S.3
198.9
202.5
206.1
209.6
213.1
216.7
220.2
223.6
227.1
230.5
233.9
237.3
240.7
244.1
247.4
250.7
254.1
257.3
260.6
263.9
267.1
270.3
273.5
276.7
279.9
283.0
286.2
289.1
292.4
295.5
52.6
52.1
51.6
51.?
SO. 7
50.2
49.8
49.4
48.9
48.5
48.1
47.7
47.3
47.0
46.6
46.2
45.9
45.5
45.2
44.9
44.5
44.2
43.9
43.6
43.3
43.0
42.7
42.4
42.1
41.8
41.5
41.2
41.0
40.7
40.4
40.2
39.9
39.7
39.4
39.2
39.0
36.7
38.5
38.3
38.0
.323
.320
.317
.314
.311
.30H
.305
.302
.300
.297
.295
.29?
.290
.288
.286
.283
.281
.279
.277
.275
.274
.272
.270
.268
.266
.265
.263
.261
.260
.258
.257
.255
.254
.252
.251
.250
.248
.247
.245
.244
.243
.242
.240
.239
.238
.168 4.9F 02 6.143
.165 . 5. IE 02 6.206
.162 S.4E 02 6.267
.159 5.6E 02 6.327
.156 5.6E 02 6.386
.153 6. IE 0? 6.443
.151 6.3F 02 6.500
.148 6.5E 02 6.556
.146 6.8F 02 6.611
.144 7.0E 02 6.665
.142 7.2F 02 6.718
.140 7.5E 0? 6.770
.137 7.7E 02 6.822
.135 8.0E 02 6.873
.134 8.2E 02 6.923
.132 8.4E 02 6.973
.130 8.7F 02 7.022
.128 9.0E 02 7.070
.127 9.2E 02 7.118
.125 9.5E 02 7.165
.123 9.7E 02 7.212
.122 .OE 03 7.258
.120 .Of 03 7.304
.119 .OE 03 7.349
.117 .IE 03 7.393
.116 .IE 03 7.438
.115 .IF 03 7.481
.113 .2E 03 7.525
.112 .2E 03 7.568
.111 .2E 03 7.611
.110 .2E 03 7.653
.108 .3E 03 7.695
.107 .3E 03 7.736
.106 .3E 03 7.778
.105 .3E 03 7.818
.104 .4E 03 7.859
.103 .4E 03 7.899
.102 .4€ 03 7.939
.101 .5E 03 7.979
.100 .5E 03 8.019
.099 .5E 03 8.058
.098 .5E 03 8.097
.097 .6F Q3 8.135
.096 1.6E 03 8.174
.095 1.6E 03 8.212
.993
.993
.993
.992
.992
.992
.991
.991
.991
.990
.990
.990
.989
.989
.989
.988
.988
.987
.987
.987
.986
.986
.986
.985
.985
.984
.984
.984
.983
.983
.982
.982
.982
.981
.981
.981
.980
.980
.979
.979
.979
.978
.978
.977
.977
.21
.20
.20
.19
.19
.18
.18
.17
.17
.16
.16
.16
.15
.15
.14
.14
.14
.13
.13
.13
.12
.12
.12
.11
.11
.11
.10
.10
.10
.10
.09
.09
.09
.09
.09
.08
.08
.08
.08
.07
.07
.07
.07
.07
.07
55.9
57.1
58.3
59.4
60.5
61.7
62.8
63.9
64.9
66.0
67.1
68.
69.
70.
71.
72.
73.
74.
75.0
76.0
76.9
77.9
78.8
79.7
80.6
81.5
82.4
83.3
84.1
85.0
85.8
86.7
87.5
88.4
89.2
90.0
90.8
91.6
92.4
93.2
94.0
94.8
95.6
96.4
97.1
1.233
.235
.236
.238
.240
.242
.244
1.246
1.249
1.251
1.253
1.256
1.258
1.260
1.263
1.265
1.268
1.270
1.273
1.275
1.278
1.280
.283
.285
.288
.290
.293
.295
.298
1.300
1.302
1.305
1.307
1.310
.312
.314
.317
.319
.322
.324
.326
.328
.331
.333
.335
.000
.019
.037
.056
.074
.092
.109
.127
.145
.162
.180
.197
.214
.231
.248
.265
.282
.299
.315
.332
.349
.366
.382
.399
.415
.432
.449
.465
.482
.498
.515
.532
.548
.565
.581
.598
.615
1.631
1.648
1.665
1.681
1.698
1.715
1.732
1.748
0
0
0
0
0
0
0
0
fl
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
                                                     415

-------
FLOATING WARM WATER JETS ~  SAMPLE RUN OF POS PROGRAM
FRO a   4.0     ?BO/HO =  5.0     V/IJO =   .100




E »  .0500   CD = 1.0000   CF *      0   HE "






    S        X        Y      TH      T      U
                                                    ANGLE =90.0      SURFACE  H



                                                  OE 00   EV =  2.OOOE-01    EH =
                                                  TIME
                                                                   HT
=  l.OOOE-05



2.000E-02





   B        F
                                                                                                   RI
                                                                                                          IHLF
383.9
388.9
393.9
398.9
403.9
408.9
413.9
418.9
423.9
428.9
433.9
438.9
443.9
448.9
453.9
458.9
463.9
468.9
473.9
478.9
483.9
488.9
493.9
498.9
503.9
508.9
5)3.9
518.9
523.9
528.9
533.9
538.9
543.9
548.9
553.9
558.9
563.9
568.9
573.9
578.9
583.9
588.9
593.9
598.9
603.9
?27.3
231.3
235.3
239.?
243.?
247.?
251.?
255.3
259.3
263.3
267.4
271.5
275.5
?79.6
283.7
287.8
291.9
296.0
300.?
304.3
308.5
312.6
316.8
320.9
325.1
3?9.3
333.5
337.7
341.9
346.1
350.4
354.6
358.8
363.1
367.3
371.6
375.9
380.1
384.4
388.7
393.0
397.3
401.6
405.9
410.2
298.5
301.6
304.6
307.7
310.7
313.7
316.7
319.6
322.6
325.5
328.5
331.4
334.3
337.2
340.0
342.9
345.7
348.6
351.4
354.2
357.0
359.8
362.5
365.3
368.0
370.8
373.5
376.2
378.9
381.6
384.3
3H6.9
389.6
392.2
394.8
397.5
400.1
402.7
405.2
407.8
410.4
412.9
415.5
418.0
420.5
37.8
37.6
37.4
37.2
36.9
36.7
36.5
36.3
36.1
35.*
35.7
35.5
35.3
35.?
35.0
34.8
34.6
34.4
34.2
34.1
33.9
33.7
33.5
33.4
33.2
33.0
32.9
32.7
3?. 6
J2.4
32.?
32.1
31.9
31.8
31.6
31.5
31.3
31.2
31.0
30.9
30.8
30.6
30.5
30.3
30.2
.237
.236
.234
.233
.232
.231
.230
.229
.228
.227
.226
.225
.224
.223
.222
.221
.220
.219
.218
.217
.216
.215
.214
.213
.21?
.211
.210
.?10
.209
.208
.207
.206
.205
.205
.204
.203
.202
.201
.201
.200
.199
.198
.197
.197
.196
.094 ,7F 03
.093 ,7F 03
.092 .7F 03
.092 .7F. 03
.091 .8F 03
.090 .8F 03
.089 .8E 03
.088 ,9E 03
.088 .9E 03
.087 .9E 03
.086 2.0E 03
.085 2.0E 03
.085 2.0E 03
.084 2.0E 03
.083 2. IE 03
.082 2. IF 03
.082 2. IE 03
.081 2.2E 03
.080 2.2E 03
.080 2.2F 03
.079 2.3E 03
.079 2.3E 03
.078 2.3E 03
.077 2.3E 03
.077 2.4E 03
.076 2.4E 03
.076' 2.4F 03
.075 2.5E 03
.074 2.SE 03
.074 2.5E 03
.073 2.6F 03
.073 2.6F 03
.072 2.6E 03
.072 2.7f 03
.071 2.7E 03
.071 2.7E 03
.070 2.8E 03
.070 2.8E 03
.069 2.8E 03
.069 2.9E 03
.068 2.9E 03
.068 2.9F 03
.067 3.0E 03
.067 3.0E 03
.066 3.0E 03
8.250
8.288
8.325
8.363
8.400
8.437
8.473
8.510
8.546
8.563
8.619
8.654
8.690
8.726
8.761
8.796
8.832
8.867
8.901
8.936
8.971
9.005
9.040
9.074
9.108
9.142
9.176
9.210
9.244
9.277
9.311
9.344
9.378
9.411
9.444
9.477
9.510
9.543
9.576
9.609
9.642
9.674
9.707
9.740
9.772
.976
.976
.976
.975
.975
.974
.974
.974
.973
.973
.972
.972
.971
.971
.971
.970
.970
.969
.969
.968
.968
.968
.967
.967
.966
.966
.965
.965
.965
.964
.964
.963
.963
.962
.962
.962
.961 1
.961 1
.960 1
.960 1
.959 1
.959 ]
.958 1
.958 1
.958 ]
.06
.06
.06
.06
.06
.06
.05
.05
.05
.05
.05
.05
.05
.05
.04
.04
.04
.04
.04
.04
.04
.04
.04
.04
.03
.03
.03
.03
.03
.03
.03
.03
.03
.03
.03
.03
L.03
1.03
1.02
1.02
L.02
1.02
1.02
1.02
1.02
97.9 i .337
98.6 .340
99.4 .342
100.1 .344
100.9 .346
101.6 .348
102.3 .351
103.1 .353
103.8 .355
104.5 .357
105.2 .359
105.9 .361
106.6 .363
107.3 .365
108.0 .367
108.7 .369
109.4 .371
110.1 .373
110.8 .375
111.5 .377
112.1 .379
112.8 1.381
113.5 1.383
114.1 1.385
114.8 1.387
115.4 1.389
116.1 1.391
116.7 1.393
117.4 1.394
118.0 1.396
118.7 1.398
119.3 1.400
119.9 .402
120.6 .404
121.2 .405
121.8 .407
122.4 .409
123.0 .411
123.7 .412
124.3 .414
124.9 .416
125.5 .418
126.1 .419
126.7 .421
127.3 .423
1.765
1.782
1.T99
1.816
1.833
1.850
1.867
1 . 884
1.901
1.919
1.936
1.953
1.970
1.988
2.005
2.023
2.040
2.058
2.076
2.093
2.111
2.129
2.147
2.165
2.183
2.201
2.219
2.237
2.255
2.274
2.292
2.310
2.329
2.347
2.366
2.385
2.404
2.422
2.441
2.460
2.479
2.499
2.518
2.537
2.556
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

-------
AREAS  OF  EXCESS
 SAMPLE RUN OF POS PROGRAM
                 TEMPERATURE   FOR
REU TEMP.

   .05
   .10
   .15
   .20
   .?5
   .30
   .35
   .40,
   .45
   .50
   .55
   .60
   .6,5
   .70
   .7S
   .80
   .85
   .90
   .95
  1.00
1
9
6.
3
t
5.
2.
1
7.
4
3
?
1.
1
9.
8.
6.
5.
4
2.
 AREA

205E 05
037E 04
637E 04
848E 04
449E 04
897E 03
676E 03
336E 03
493E 02
S18E 02
049E 02
1B2E 02
627E 02
236E 02
819E 01
039E 01
470E 01
280E 01
186E 0)
393E 01
PARTIAL AREA - CALCULATION VALID TO T « .196
PARTIAL AREA - CALCULATION VALID TO T = .196
PARTIAL AREA - CALCULATION VALID TO T « .196
                                          417

-------
                      PROGRAM    LISTING
c
c
c
c
c
c
r*
  PROGRAM PDS
  COMMON E»CO»CF«EH,EVtGH»DR»AK.V»A»OS»P.SP»S2»*E
  COMMON! THO»FRO»FS» INFO (20) .NLINE.MP AGE*XKl
  COMMON  Y<7).  0(7) »  »PMT<5>«  AUX  »YY (ff) * IHLFtS
  COMMON Ul»Sl«TtME
  COMMON/ ID ASH/ Y?L AST
  EQUIVALENCE (Y(D,Q),  (Y(?).T)»  
  DO 100 K=1»KK

  REAO ALL DATA  FW  ONE  JET
  Rfc"AO(<5.2>  (INFO(I) »I = 1«?0)
                  A»Vtf*t»THtCD«CF
  W^ITE (6»6)  (IMFO(I) tl=l«
2 FORMAT 3*rH/MO.O
  M=A*( l.+.S^GP)
  STH = SIN(THO)
  CTH = COS(THO)
  N=0
  OELH=.?
  9M=1.0
  CONTINUE
  FH = 0.5*CO*V*AHS(V)*STh*STH*SI«(l.*Hl)/2.0
00001
0000?
00003
00004
00005
00006
00007
00008
00009
00010
00011
00012
00013
00014
00015
0001ft
00017
0001*
00019
00020
00021
00022
00023
00024
00025
00026
00027
00028
00029
00030
00031
00032
00033
00034
00035
00036
00037
00038
00039
00040
00041
00042
00043
00044
00045
00046
00047
0004ft
00049
00050
00051
00052
00053
00054
00055
00056
                                   4-18

-------
   17

   19
C
C
C
C
32
C
C
C
C
C
      RV = SURT(U«U  -  e
      IF (RV.EO.0.0) GO  TO  17
      SF = RV«O.S»(1.*H1)
      SF = SORT(SORT(l./(RF»SF))
      Sf = 0.0?06»SF»HV*RV
      SF = CF»SI*(.S*A+H1)*SF
      SFX = SF*(V-U*CTH)/RV

      GO TO 19
      SFX = 0.0
      SFY = 0.0
      CONTINUE

      CALCULATE  INITIAL  ANGLE
      TH = M»STh - FO*CTH +  SFY
      TH = ATAN(FH/*SIM(TH)
      OIF=RHS-LHS
      IF(M.GT.O)GO  TO  3?
      N=l
      IF(ARS(OIF1/LHS) .LT.. 001)60  TO  80
      gM=BM+OELH
      GO TO 31
      OIF? =  ABS(OIF)
      IF(A8S(OIF2/LHS)
      IF(OIF?.GT.OIF1)
      BM=BM*DELH
      OIF1=I)IF2
      N=N+1
      IF(N.LT.500  XiO
      WRITF.(6»5)
      FORMAT (/" ASSUMING UPSTRfA^  wFOGt  PRESENT - USING F =  1.02")
      8M = 1.
                       .LT.. 001)60 TO  fcO
                            = -.5*OFLH
                      TO
      CC=1.-1./C
      Bl=ft.*0**4
      H1=SP»LHS**?*CC/ 63


        ASSIGN VALUES TO PARAMETERS  IN SUBROUTINES ODTP AND KHPCG
      NLINE = 50
00057
00058
00059
00060
00061
00062
00063
00064
00065
00066
00067
00068
00069
00070
00071
00072
00073
00074
00075
00076
00077
00078
00079
00080
00081
00062
OOOP3
00084
OOOS5
00086
00087
00088
00089
00090
00091
00092
00093
00094
00095
00096
00097
00096
00099
00100
00101
00102
00103
00104
00105
00106
00107
00108
00109
00110
00111
OOU2
                                   419

-------
   85
      NPAGE = 0
      PRMTU)=SI
      »RMT(?) = SLIM
      PRMTI3) = OS
      PRMT<4) = 0.01
      00 fl5 1 = 1.7
      0(1)  » 0.1
      D(4>  = 0.?5
      0(5)  = 0.25
      IST=0

      CALL  KNPCG
                                        OF  EXCESS   TEMPERA
c
C     WKITE AREAS vi/ITHlN ISOTHERMS
      WRITE (6»?00) (INFO(I).1=1,20)
  200 FORMAT (1H1,//////,bX»"A R E A S
     IT U R E   F 0 *     ",/,6X,20A4l
      in/RITE (6.201)
  201 FORMAT (//,5X»»PFL. TEMP.»,5X."AREA"./)
      DO 90 1=1.20
      TI = I
      TI = TI/20.0
         IF(T.GT.TI) GO TO 301
         WRITE(6»20?)  TI.AR(I)
         GO TO 90
301      «/RITE(6,302)TI.AH(I),T
302      FORMAT(6X,F5.2,5X»F10.3»2X»"PARTIAL ARFA - CALCULATION VALID TO
     1 T ="F5.3)
90       CONTINUE
  202 FORMAT (6X,F5.2,5X,E10.3)
  100 CONTINUE
      STOP
      END
      SUBROUTINE AREA(J)
      COMMON E»CO»CF»EH»EV,GP»DR»AK,V,AM,DSS,P,SP,S2»RE
      COMMON THO,FRO,FS»INFO(20).NLINE»NPAGE»XK1
      COMMON  Y(7)« U(7), PWMT(5). AUX(20.7).A(20).YY(8)»IHLF.S
      COMMON Ul,SI,TIME
      EQUIVALENCE (Y(l),0), (Y(2),T), (Y(3),TH),  (Y(4),rt), (Y(5),H)
      EQUIVALENCE (Y(6),XP), (Y(7)»YP)
      DIMENSION  rfl(20), W2(20>
C
      IF(J)GO TO 20

    ,  INITIALIZE VARIABLES A AND Wl
      00 15 1=1,20
      A(I)  = 0.0
   15 WKI) = H
      SLAST=0.
      GO TO 150

      CALCULATE AREAS
   20 OS=S-SLAST
      DO 100 1=1,'0
      TI = 1/20.0
      IF(TI.LE.T)GO TO 40
C
C
C
C
00113
00114
00115
00116
00117
0011H
00119
00120
00121
00122
00123
00124
00125
00126
00127
00128
00129
00130
00131
00132
00133
00134
00135
00136
00137
00138
00139
00140
00141
00142
00143
00144
00145
00146
00147
00148
00149
00150
00151
00152
00153
00154
00155
00156
00157
00158
00159
00160
00161
00162
00163
00164
00165
00166
00167
00168
                                   420

-------
'40
 SO
100
              «*SQRT(ALOr,(T/TI>)
                  + (W2(I> » Wl(I))*DS
              *2(l>
c
c
        )  = 0.0
    GO TO ^0
    W2(I>
    A(D  =
    WKI)  =
    SLAST=S
150 CONTINUE
    RETURN
    END
    SUBROUTINE FCT
    COMMON E»CO»CF,EH,EV»GP,DR»AK,V,A»DS,P,SP»S2»R£
    COMMON THO»FRO»FS,INFO(20>,NLINE»NPAGE»XK1
    COMMON  Y<7>, 0(7), PRMT(5), AUX **?)

    IF  Q,T,TH,B, OR M «RF NEGATIVE, RETURN
    DO 5 1=1,5
    IF (Yd)) 50,5,5
  5 CONTINUE
C
C
C
C
C
C
C
    CALC HORIZONTAL AMBIENT AND JET ENT9AINMENT
    EA = EH
    OA = 3.5*P*EA*H/8
    RV = U*tl + VST»VST
    QJ s SP*E*SQRT(RV)*H
    QH = QA + QJ

    CALC VERTICAL AMBIENT tNTRAINMENT
    RI = GP*S2»T*H/(U+VCT)»»?
    CALL REO(RI,RF)
    QA = 3.5*P*b»EV*EA*ftF/H

    CALC VERTICAL JET ENTRAlNMENT AND INTERFACIAL SHEAR
    QJ = 0.0
    SFX = 0.0
    SFY = 0.0
    DO ?5 1=1,10
    YI = I
    YI = 0.1*YI - 0.05
    YI - F.XP(-2.0»YI»YI)
    RV = U*U*YI*YI + VST»VST
    RI s. f-iP*S?*T*YI*H/RV
00169
00170
00171
00172
00173
00174
00175
00176
00177
00178
00179
00180.
00181
00182
00183
00184
00185
00186
00187
00188
00189
00190
00191
00192
00193
00194
00195
00196
00197
00198
00199
00200
00201
00202
00203
00204
00205
00206
00207
00208
00209
00210
00211
00212
00213
00214
00215
00216
00217
00218
00219
00220
00221
00222
00223
00224
                                   421

-------
   25
c
c
CALL
DO =
QJ =
RV =
SFX
SFY
QJ =
0V -
ON =
SFX
SFX
SFY
SFY

CALC
0(1)
0(2)
DTH
DTH
D(3)
IF(F
D(4)
Dri =
OB =
OH =
08 =
   50
   10
   20
     RED(RI»«F)
     2.0»RF*E»SQKT(RV)*S2*H/10.0
     QJ «• OQ
     RV**0.375
    = SFX » RV*(VST*STH - U*YI*CTH)
    s SFY * RV»(VCT + U»YI>
     OJ * 1.77*00
     QA + OJ
     SQRT*0. 0206*52*3/10.0
    s SFX»GiM»?.0
    = CF*SFX
    = SFY*GN*2.0
    = -SFY*CF*STH

     DERIVATIVES X3F O.T.THt B»H»X»  AND Y
     = QM + QV
     = -T*(2.n#SP*AK»a * 0(1) )/0
    = -FD + SFY*CTH - SFX*STH
    = DTH - VST«0(1)
     = DTH/(0*Q/(P*B*H) + rjP*T*:-'*H*8*SFJ/2.0)
    .LT.l.O) F=1.001
    =XK1/(SQWT(B*F/H-1.0»
     SFY*P*STH + 5FX»°*CTH
     08 + (P*VCT-?.0»()/(H*H))*n(l)
     OB - FA*rt*0(2)
     -D8/((QV/OH+1.0)«HE*H/P)
c
c
DH=DH+0*Q«D (4) / (8*B*H)
OH = OH - FA*T»0(4)
0(5) = OH/f D(7)t PRMT(S)t AUX (20.7) » AR (20) » YY  , IHLFtS
COMMON U1.S1«TIN>E
COMMON/IDASH/Y2LAST
EQUIVALENCE  (Y(1),Q), (Y(2)»T)« 

IS NUMBER OF LINES 50.
Y?LAST=T
IF (NLINE-45) 11.10*10
 00225
 00226
 00227
 00228
 002P9
 00230
 00231
 00232
 00233
 00234
 00235
 00236
'00237
 00238
 00239
 00240
 00241
 00242
 00243
 00244
 00245
 00246
 00247
 00248
 G0249
 00250
 00251
 00252
 00253
 00254
 00255
 00256
 00257
 00258
 00259
 00260
 00261
 00262
 00263
 00264
 00265
 00266
 00267
 00268
 00269
 00270
 00271
 00272
 00273
 00274
 00275
 00276
 00277
 00278
 00279
 00280
                                   422

-------
c
c
          OF LINES IS
10 NPAGE » NPAGE *1
   NLINE = 0
   WRITE(6,1) (INFO(I),1*1,20),NPAGE
 I FORMATdHl,///," FLOATING WARM WATER
   50 OR MORE, SKIP TO NEXT PAGE AND WRITE HEADING
C
C
                                           JETS —",2X»20A4»3X,"PAGE»»

      WRITE(6.2)FRO»A.ViTHO.AK
      FORMAT(IX."FRO = "»F5.1»5X»"2BO/HO  =",F5.1»5X,"V/UO =»»F7.3»
     15X."ANGLE a"»FS.l»5X»»SURFACE H  =",E10.3/>
      WRITE(6»3)E»CD»CF»RE,EV,EH
    . FORMAT(1X»"E =",F7.4.3X,"CD ="»F7.4»3X»»CF ="»F7.4»3X»
     1"RE ="»E10.3,3X,»EV =»,E10.3»3X»"EH =»»E10.3//)
      WRITE  (6,4)
    4 FORMAT  (5X.»SI'»8X»»X"»8X,"Y",6X»"TH"»6X»"T"»6X,"U»»5X,"TIME"*4X,
     1»Q",7X,»HT",6X,»H",7X,"3",7X,"F  "»6X,"RI»,SX,"lHLF»,//>
   11
C
C
LESS
1
                        THAN 50 , CALCULATE ANO WRITE OUTPUT DATA
 NUMBER  OF LINES
 NLINE = NLINE *
 CALL AREA(l)
 U  - 2.0*(0/(P*a«H)  - V«COS(TH»
 F  = (U*0.5 +  V»COS(TH))*SORT(SP/(GP*T*H»
 RV a U»U + (V*SIN(TH))*»2
 RI r 6P»T*H*S2/RV
 UVCT =  U + V#COS(TH)
 TIME=TIME+?.»(S-S1)/(U1*UVCT>
 VC=TIME
 U1=UVCT

 THOUT=  Y(3)*90.0/1.5707963
 OOUT= 0/A
 HT = OOUT*T*0.5
 WRITE  (6fS)  S,XP,YP,THOUT»T,UfVC,OOUT»HT,H»S,F  ,RI,IHLF
 FORMAT  (3(lX,F7.1,lX),F6.1,2X,2(F6.3,lX)»lXfE7.3»lX»2(F6.3,2X)
1 *F5.2,1X,F7.1«2(2X*F6.3>«3X«I3,3X,I3)

 PUT PRMT(5)=  1.0  IF NUMERICAL INTEGRATION SHALL  STOP
 IF (T  .LT. 0.01) PRMT(5)«1.0
 IF (NPAGE .GT.  5) PRMT(5)=1.0
 RETURN
 END
 SUBROUTINE KHPCG
 COMMON  E.CD»CF,EH,EV,GP,DR»AK,V,VV,AM,DS»P,SPtS2,RE»GPP»TPRIM
 COMMON  THO»FRO,C<5»6)»FS,INFO<20>»NLINE»NPAGEtNOPT,IOR»XKl
 COMMON   Y(7)« OERY(7)» PHMT<5),  AUX(20»7)»A(20),YY(8)»IHLF,X
    COMMON/DASH/Y2LAST

 NDIM=7
 IHLF=0
 X=PRMT(1)
 H=PRMT(3)
 PRMT(5)aO.
 DO 1  I=1.NOIM
 AUX(16,I)-0.
 AUX(15,n=DERY(I)
 AUX(1,I)=Y(I)
00281
00282
00283
00284
00285
00286
00287
00288
00289
00290
00291
00292
00293
00294
00295
00296
00297
00298
00299
00300
00301
00302
00303
00304
00305
00306
00307
00308
00309
00310
00311
00312
00313
00314
00315
00316
00317
00318
00319
00320
00321
00322
00323
00324
00325
00326
0032?
00328
00329
00330
00331
00332
00333
00334
00335
00336
                                    423

-------
c
c
c
c
c
c
c
c
c
c
c
c
   IF(H»3»2.4

ERROR RETURNS
 2 IHLF=12
   GO TO 4
 3 IHLF=13

COMPUTATION OF OERY FOR STARTING VALUES
 4 CALL FCT

RECORDING OF STARTING VftLUES
   CALL OUTP
   IF)6.5,6
 S IF(IHLF>7,7»6
 6 RETURN
 7 DO 8 I=ltNDIM
 8 AUX<8»I)=OERY(I>

COMPUTATION OF AUX(?,I)
   ISW=1
   GO TO 100
 9 X=X+H

   oo 10 I=I»NDIM
10 AUX(2.I)=Y(I)

INCREMENT IS TESTED BY MEANS OF BISECTION
11 IHLF=IHLF+1
   X=X-H
   DO 12 I=1.NDIM
12 AUX(4,I)=AUX(2»I)
   H=.5*H
   N=l
   ISW=2
   GO TO 100

13 X=X+H
   CALL FCT
   N=2
   DO 14 I=1»NDIM
   AUX
-------
c
c
c
c
c
c
    GO TO 4

 THERE IS SATISFACTORY ACCURACY AFTER  LESS  THAN  11  BISECTIONS
 19 X=X + H
    CALL FCT
    DO 20 I=1,NOIM
    AUX<3»I)=Yd)
 20 AUX.dO»I)=DERYd>
    N=3
    ISW=4
    GO TO 100

 21 N=l
   . X=X+H
  .; CALL FCT
    X=PRMTd>
    00 22 I=1»NUIM
    AUX(11»I)=DERY(I)
 22 Yd) =AUX ( 1 , I ) »H* ( . 375»AUX (8,1) + .7916666667«AUX (9, I )
   X-. 2083333333* AUX dO» I) +.04166666667»JERY d ) )
 23 X=X»H
    N=N+1
    CALL FCT
    CALL OUTP
    IF(PRMT(5)>6.24,6
 2* IF
 26 AUX(N + 7,I)=DERYd)
    IF +DELT + AIM «10, I ) )
    GO TO 23

 29 DO 30 I=1,NDIM
    DELT = AUX (9, I ) +AUX dO» I )
    DELT=DELT+DELT*OELT
 30 Yd) =AUX ( 1 » I ) + . 37S*H» ( AUX ( 8« I > +DELT* AUX (11,1))
    GO TO 23

 THE FOLLOWING PART OF SUBROUTINE HPCG COMPUTES BY MEANS  OF
 RUNGE-KUTTA METHOD STARTING VALUES FOR THE NOT SELF-STARTING
 PREDICTOR-CORRECTOR METHOD
100 DO 101 I=1»NDIM
    Z=H»AUX(N+7»I)
    AUX(5»I)=Z
 Z IS AN 'AUXILIARY STORAGE LOCATION
  101 Yd) =AUX(N,I) *
      CALL FCT
      DO 102 I=1,NDIM
 00393
 00394
 00395'
 00396
 00397
 00398
 00399
 00400
 00401
 00402
 00403
 00404
 00405
 00406
 00407
 0040ft
 00409
 00410
 00411
 00412
 00413
 00414
 00415
 00416_
 00417
 00418
 Ob419"
 00420
 00421
 0042,2
 60423"
 004^4
 00425
 00426
 0042t
 0042ft
 004^9
 00430
 00431
 00432
 00433
..00434
 (J0435
 004^6
 00437
 00433
 00439
 00440
 00441
 00442
 00443
 00444
 00445
 00446
 00447
 00448
                                   425

-------
  102
cc
    Z=H»DE»Y(I)
    AUX(6,I)=Z
    Yd) = AUX  -3. 0509651486* AUX (6. I )
      »3.8328647604*Z
c
c
c
c
c
c
c
  103
    CALL FCT
    00  10*  I=1»NOIM
 104 Y(I>=AUX
    GOTO  (9.13»lS,<>l>ISw
 ','i
 POSSIBLE BREAK-POINT FOR  LINKAGE

 , STARTING VALUES  ARE COMPUTED.
- NOW START  HAMMINGS MODIFIED PREDICTOR-CORRECTOR METHOD
 200 ISTEP=3
 201 IF (N-8) 204, 202,204
 N=R CAUSES THE *o»/s
202 DO 203 N=?»7
    DO 203 I=1»NOIM
  203
                     OF  AUX TO CHANGE THEIR STORAGE LOCATIONS
c
c.
c
c
   LESS/ THAN
    N=N*1
              H  CAUSES N*l  TO GET N
c
c
c
c
c
  COMPUTATION OF  NFXT  VECTO* Y
     DO  205  I=1.NOIM
     AUX(N-1»I)=Y(I>
 205  AUX(N+6»n=DERY(I)
     X=X + H
 206  ISTEP=IST£P+1
     DO  207  I=1»NQIM
     DELT=AUX (N-4» I ) + 1 . 333333333»H* ( AUX (N+6» I ) +AUX (N+6. I ) -
   * AUX(N+5«I) +AUX(M+4,I) + AUX (N* 4,1) )
     Y ( I ) sDELT- . 9256 1 98* AUX ( 1 6, 1 )
 207  AUX(lft,I)=OELT
  PREDICTOR  IS  NOW  GENERATED IN  POw 16  OF  AUX» MODIFIED  PREDICTOR
,  IS  GENERATED  IN Y.   DELT MEANS  AN AUXILIARY  STORAGE
    CALL FCT
 DERIVATIVE OF
                MODIFIED  PREDICTOR  IS  GENERATED  IN  DERY
      DO 208 I=1»NDIM
      OELT=.125*(9.»AUX»3.»H*+AUX0463
00464
00465
00466
00467
00468
00469
00470
00471
00472
00473
00474
00475
00476
00477
00478
00479
00480
00481
00482
00483
00484
004P5
00486
00487
00488
00489
00490
00491
00492
00493
00494
00495
00496
00497
00498
00499
00500
00501
00502
00503
00504
                                   426

-------
208
    AUX < 16, I )
c
c
                AUX (16.1) -DELT
                .07438017»AUXU6»!)
c
c
c
c
 TEST WHETHER H MUST BE HALVED  OR  DOUBLED
    DELT=0.
    DO 209 I=1»NOIM
209 DELT=OELT* AUX (15,1) *ABS < AUX ( 1 6« I »
       IF(DELT.GT.PRMT<4>. OR. Y2LAST-Y(2) .GT. .05)00  TO  222

 H MUST NOT BE HALVED.  THAT MEANS Y (I)  ARE  GOOD
210 CALL FCT
  ,  CALL OUTP
  " IF(PRMT<5»21?,ZH«21?
211 IF(IHLF-11>213»212»212
212 RETURN
213 IF(H*(X-PR'4T<2>»214,212,212
214 IF(ABS201»220«201
220 H=H+H
    IHLF=IHLF-1
    ISTEP=0
    DO 221 I=1»NOIM
    AUX(N-1»I)=AUX(N-2»I)
    AUX(N-?»IJ=AUX
    AUX(N-3.I)=AUX(N-6»I)
    AUX(N»6.I)=AUX(N*5.I)
    AUX(N+5»I)=AUX
    AUX(N*4»I)=AUX(N+1.I)
    DELT=AUX 223»223,210
223 H=.5»H
    ISTEP=0

    ??lfr!o0390625»(80.»AUX(N-l,I)*135.*AUX(N-2,I)+40.»AUX(N-3,I)*
   XAUX(N-4,I))-.1171875«(AUX(N»6,I)-6.*AUX(N*5,I)-AUX(N*4.I)>*H
    . . .^ .*.  .  .»_  A ^ -,rtrt /L r*e- & * »*a  «• AI IV ' NJ* ^ t I) *135«*AUX(N"*2f I) *
     X9.*AUX(N*4.I> >*H
      AUX(N-3»I)=AUX(N-?,I)
00505
00506
00507
00508
00509
00510
00511
00512
00513
00514
00515
00516
00517
00518
00519
00520
00521
00522
00523
00524
00525
00526
00527
00528
00529
00530
00531
00532
00533
00534
00535
00536
00537
00538
00539
00540
00541
00542
00543
00544
00545
00546
00547
00548
00549
00550
00551
00552
00553
00554
00555
00556
00557
00558
00559
00560
                                    427

-------
224 AUX(N+4,I)=AUX(N+S»I)                                                 00561
    X=X-H                                                                 0056?
    DELTsX-                                                    00567
225 Y                                                       00568
    DELT=DELT-(H+H>                                                       00569
    CALL FCT                                                              00570
    00 ?2ft IsltNOI'M                                                       00571
    OELT=AUX(N + 5t D+AUX (N + 4.I)                                            00572
    OELT=OELT+OELT*D£:uT.                                                   00573
    AUX(16,'l) =8. 9*296 J*-3. 36111 l*H*
-------
                            REFERENCES

1.   Prych, Edmund A. "A Warm Water Effluent Analysis as a Buoyant
     Surface Jet"  Swedish Meteorological and Hydrological Institute,
     Series Hydroli, Nr 21, 1972.

2.   Shirazi, Mostafa A., "Some Results from Experimental Data  on
     Surface Jet Discharge of Heated Water" Proceeding of the
     International Water Resources Association, Chicago, 1973.

3.   Stolzenbach, K. D., Harlemann, D. R. F. "An Analytical and
     Experimental Investigation of Surface Discharges of Heated Water."
     Water Pollution Control Series 16130 DJV 02/71, Feb. 1971.

4.   Policastro, A. J. and Tokar, J. V.  "Heated Effluent Dispension
     in large Lakes: State-of-the-art of Analytical Modeling Part  I,
     Critique of Model Formulations? Argonne National Laboratory
     ANL/ES-11 Jan. 1972.

5.   Stolzenbach, K. D., Adams, E. E. and Harleman, D. F.
     "A User's Manual for Three-Dimensional Heated Surface Discharge
     Computations" Environmental Protection Technology Series EPA-
     R2-73-133, Jan. 1973.

6.   Koh, R. C. Y., Fan, L. N. "Mathematical Models for the
     prediction of Temperature Distributions Resulting from the
     Discharge of Heated Water into Large Bodies of Water"  Water
     Pollution Control Series 16130 DWO  10/70, Oct. 1970.

7.   Stefan, Heinz. Personal Communication.
                                     429

-------
8.   Well, J.  "Mixing  of  a  Heated Surface Jet 1n Turbulent Channel
     Flow"  Report No.  WHM-1,  Department of Civil Engineering,
     University of California, Berkeley, June 1972.

9.   Ellison, T. H., and Turner,  J.  S.,  "Turbulent Entrainment in
     Stratified Flows," Jour, of Fluid  Mechanics, Vol 6, Part 3
     p. 423-448.
         \
10.  Stefan, Heinz, Hayakawa N.,  and Schiebe, F.  R. " Surface
     Discharge of Heated Water"  Water Pollution  Control Research
     Series 16130 FSU 12171, Dec.  1971.
                                     430
                                           *OA GOVERNMENT PRINTING OFFICE: 1974 546-319/442 4-3

-------
 SELECTED WATER
 RESOURCES ABSTRACTS

 INPUT TRANSACTION FORM

 *  Title    Workbook of Thermal  Plume  Prediction, Volume 2,
            Surface Discharges     ,
                                                             3.  Accession No,
                                                             w
  7.  Author^) pp. Mostafa A> shlrazl
            Dr. Lorln R. Davis
  9. Q
organization
Thermal  Pollution Branch, Pacific Northwest Environmental
Research Laboratory, NERC-CorvalUs
                                                            10.  Project No.

                                                                 16130 HE
                                                                  11,  Contract/Grant No,
  IS.  Supplementary Notes

      Environmental Protection Agency report number, EPA-B2-72-005T), May 19jk
  16. Abstract  In a continuing effort  to present current knowledge on heated plume pre-
 diction to the public, nomograms are  presented in this second volume that describe the
 behavior of surface jets for a wide range of ambient and Initial  discharge conditions
 encountered In practice.  An attempt  1s made to present the material in a concise manner
 and In a format that Is clear and accessible to a nonspeclallst user.  Many fundamental
 derivations are left outside the body of the workbook and retained for further reading
 in the appendix.  These undoubtedly would be of use to the specialist researcher who
 seeks to advance the status of knowledge.

 The nomograms provide qualitative results describing the surface plume trajectory, width,
 temperature, depth, surface area and  time of travel  along the plume center!ine.   The
 nomograms are not Intended to be used as exclusive design tools for surface discharge
 problems nor for use in a precise prediction of any specific surface plume condition.

 The nomograms are generated predominately from an idealized mathematical  model  of a
 plume.  Some field and laboratory data have been used to adjust the performance  of the
 model so that more realistic preddtlons are obtained.  However,  the class of  problems
 that can be handled this way are limited due to the limitations 1n the model Itself.
 We have made an earnest attempt to  help the nonspeclallst user by pointing out the main
 restrictions Included 1n the model  both 1n a special  chapter 1n the workbook as  well as
 In example problems.	
  17s. Descriptors
 Thermal Pollution, Discharge  (Water),  Water Quality,  Water Quality  Control
 Pollution, Water Pollution Control,  Surface Waters
                                                                        Water
  IJb. Identifiers
I Surf ace Discharge, Thermal Plumes
  17c. COWRR Field & Group  F16l d 05    fifOUp
  •MMMMMMMMMW^MI^M

  18. Availability
  Abstractor
                                                 Send To:


                                                 WATER RESOURCES SCIENTIFIC INFORMATION CENTER
                                                 U.S. DEPARTMENT OF THE INTERIOR
                                                 WASHINGTON. O. C. 2O24O
WRSIC 102 (REV. JUNE 1870

-------