EPA-R2-72-005b
May 1974
Environmental Protection Technology Series
Workbook of Thermal Plume
Prediction-Vol.2-Surface Discharge
I
55
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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
-------
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
-------
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
-------
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
-------
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).
-------
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.
-------
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.
-------
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.
-------
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.
-------
-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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
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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
-------
o
I
UJ
O
CO
tr
UJ
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
-------
o
I
X
UJ
<
-i
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
-------
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
O
I-
co
Q
QC
P
<
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
-------
UJ
o
I
o
_J
a:
ui
I-
50
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
-------
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
-------
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
-------
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
-------
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
-------
X
x
iy
o
(0
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
X
UJ
w
o
o:
P
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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
-------
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
-------
X
LU
O
o:
UJ
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
-------
UJ
o
tr
UJ
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
-------
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
-------
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
-------
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
-------
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
-------
UJ
o
I
W
S
-J
o:
UJ
I-
50 100 150 200 250 300 350 400 450 500
LONGITUDINAL DISTANCE X/H0
FIG (A26) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
71
-------
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
-------
111
o
I
(0
o
o:
UJ
I-
50
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
-------
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
-------
UJ
o
I
cc
UJ
I-
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
-------
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
-------
UJ
u
CO
5
cc
111
h-
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
-------
-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
-------
o
UJ
O
1
UJ
-10 0 10 20 30 40 50 60 70 80 90
LONGITUDINAL DISTANCE X/H
FIG (A34) TEMPERATURE, TRAJECTORY.WIDTH.AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
-------
o
I
LU
O
z
o:
LU
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
-------
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
-------
-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
-------
-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
-------
LU
0
I
OT
a:
lu
I-
"-50 0 50 100 150 200 250 300 350 400 450
LONGITUDINAL DISTANCE X/H0
FIG (A42) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
85
-------
UJ
o
i
5
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o:
IU
h-
3
"-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
-------
-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
-------
I
V.
IU
u
(O
a
cc
LU
i-
-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
-------
X
X
ui
o
i
UJ
I-
-50
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
-------
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
-------
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
-------
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
-------
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
-------
x
V
X
UJ
o
I
oc
Id
I-
300 350 400 450 500
LONGITUDINAL DISTANCE X/H0
FIG (A53) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
94
-------
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
-------
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
-------
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
-------
LU
O
1
W
cc
UJ
K-
50
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
-------
x
X
UJ
(0
5
o:
ui
i-
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
-------
x
X
UJ
u
I
CO
2
UJ
I-
100
50
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
-------
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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o
X
yj
o
or
LU
0 100 200 300 400 500 600 700 800 900 1000
LONGITUDINAL DISTANCE
FIG (A6 I) TEMPERATURE, TRAJECTORY. WIDTH, AND DEPTH
(TTWDhPLOTS FOR SURFACE JET DISCHARGE
102
-------
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
-------
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
-------
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
-------
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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X
X
tu
o
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o
o:
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10 20 30 40 50 60 70 80 90 100
LONGITUDINAL DISTANCE
Fl G (A66) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
107
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X
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to
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QC
10 20 30 40 50 60 70 80 90 100
LONGITUDINAL DISTANCE X/H0
FIG(A67) TEMPERATURE.TRAJECTORY, WIDTH, ANDDEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
108
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I.
X
Ul
o
H
c/2
o
o:
LU
10 20 30 40 50 60 70 80 90 100
LONGITUDINAL DISTANCE
FIG(A68) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
109
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yj
o
1
(0
o:
UJ
50 100 150 200 250 300 350 400 450 500
LONGITUDINAL DISTANCE X/H0
FIG (A69) TEMPERATURE, TRAJECTORY. WIDTH, AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
no
-------
300
280
260
20
20 40 60 80 100 120
160 180 200
LONGITUDINAL DISTANCE X/H0
FIG (A70) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
111
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LJ
O
Z
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(0
tr
UJ
20 40 60 80 100 120 140 160 180 200
LONGITUDINAL DISTANCE X/H0
t
FIG (A71) TEMPERATURE,TRAJECTORY,WIDTH, AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
112
-------
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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X
X
UJ
U
1
W
5
-J
£t
111
50 100 150 200 250 300 350 400 450 500
LONGITUDINAL DISTANCE X/H0
FIG (A73) TEMPERATURE, TRAJECTORY, WIDTH. AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
114
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o
I
(0
o:
UJ
I-
50 100 150 200 250 300 350 400 450 500
LONGITUDINAL DISTANCE X/H0
FIG (A74) TEMPERATURE. TRAJECTORY,WIDTH. AND DEPTH
(TTWDJ-PLOTS FOR SURFACE JET DISCHARGE
115
-------
111
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1
V)
o:
UJ
i-
i 50 100 150 200 250 300 350 400 450 500
LONGITUDINAL DISTANCE X/HQ
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
-------
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
-------
111
o
I
V)
i
UJ
H
50 100 150 200 250 300 35O 400 450 500
LONGITUDINAL DISTANCE X/H0
FIG (A78) TEMPERATURE, TRAJECTORY.WIDTH, AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
119
-------
LU
o
1
2
UJ
I-
50 100 150 200 250 300 350 400 450 500
LONGITUDINAL DISTANCE X/H0
FIG (A79) TEMPERATURE, TRAJECTORY, WIDTH. AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
130
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z
X
IU
o
I
CO
u
H
<
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
-------
o
X
UJ
o
tr
in
-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
-------
-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
-------
o
I
LU
O
o:
UJ
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
-------
-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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UJ
u
1
CO
5
-i
ct
UJ
K-
-50 0 50 100 150 200 250 300 350 400 450
LONGITUDINAL DISTANCE X/H0
FIG (A92) TEMPERATURE, TRAJECTORY, WIDTH. AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
131
-------
I
X
UJ
u
en
o
tr
UJ
-50 0 50 100 150 200 250 300 350 400 450
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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300
280
260
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o
_i
DC
LU
I-
20 40 60 80 100 120 140 160 180 200
LONGITUDINAL DISTANCE X/H0
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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X
X
LJ
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iu
H-
50 100 150 200 250 300 350 400 450 500
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
HO
-------
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
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
T42
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UJ
o
1
CO
5
-j
oc
UJ
50 100 150 200 250 300 350 400 450 500
LONGITUDINAL DISTANCE X/H0
FIG (AI06) TEMPERATURE, TRAJECTORY, WIDTH. AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
143
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I
X
UJ
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1
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0
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H-
50 100 150 200 250 300 350 400 450 500
LONGITUDINAL DISTANCE X/H0
FIG (AI07) TEMPERATURE. TRAJECTORY. WIDTH. AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
V44
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UJ
o
CO
5
2
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150
100
50
50 100 150 200 250 300 350 400 450 500
LONGITUDINAL DISTANCE X/H0
\ w
FIG (AI08) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
145
-------
0 100 200 300 400 500 600 700 800 900 1000
LONGITUDINAL DISTANCE X/H0
FIG (AIO9) TEMPERATURE,TRAJECTORY,WIDTH,AND DEPTH
(TTWDhPLOTS FOR SURFACE JET DISCHARGE
146
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O 50 100 150 200 250 300 350 400 450 500
LONGITUDINAL DISTANCE X/H0
FIG (Al 10) TEMPERATURE, TRAJECTORY, WIDTH. AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
147
-------
50 100 ISO 200 250 300 350 400 450 500
LONGITUDINAL DISTANCE X/H0
FIG (A 111 ) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
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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
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
150
-------
10 20 30 40 50 60 70 80 90 100
LONGITUDINAL DISTANCE X/H0
FIG(AI 14) TEMPERATURE.TRAJECTORY, WIDTH, AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
151
-------
150
illiliiiii iiiiiiiiiliiii
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K
(0
O
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QL
0 20 30 40 50 60 70 80 90 100
LONGITUDINAL DISTANCE X/H0
FIG (A115) TEMPERATURE.TRAJECTORY,WIDTH, AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
152
-------
X
UJ
o
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V)
a
UJ
10 20 30 40 50 60 70 80 90 100
LONGITUDINAL DISTANCE
FIG(AII6) TEMPERATURE.TRAJECTORY, WIDTH, AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
153
-------
50 100 150 200 250 300 350 400 450 500
LONGITUDINAL DISTANCE X/H,
FIG (A 117) TEMPERATURE. TRAJECTORY, WIDTH, AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
V54
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tfl
cr
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20
20 40 60 80 100 120 140 160 180 200
LONGITUDINAL DISTANCE X/H0
FIG (Al 18) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
155
-------
o
X
o
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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
-------
X
X
Ul
o
1
V)
5
0 50 100 150 200 250 300 350 400 450 500
LONGITUDINAL DISTANCE X/H0
FIG (AI22) TEMPERATURE, TRAJECTORY, WIDTH. AND DEPTH
(TTWDhPLOTS FOR SURFACE JET DISCHARGE
V5S
-------
50 100 150 200 250 300 350 400 450 500
LONGITUDINAL DISTANCE X/H0
FIG (AI23) TEMPERATURE. TRAJECTORY, WIDTH, AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
160
-------
x
V
UJ
o
I
0)
2
UJ
50 100 150 200 250 300 350 400 450 500
LONGITUDINAL DISTANCE X/H0
FIG (AI24) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
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
-------
300
UJ
o
(O
Q
DC
IU
-20 0 20 40 60 80 100 120 140 160 180
LONGITUDINAL DISTANCE X/H0
FIG (AI29) TEMPERATURE,TRAJECTORY, WIDTH, AND DEPTH
(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
-------
750
UJ
o
1
CO
§
ui
-50 0 50 100 150 200 250 300 350 400 450
LONGITUDINAL DISTANCE X/H0
FIG (AI40) TEMPERATURE, TRAJECTORY, WIDTH, AND DEPTH
(TTWD)-PLOTS FOR SURFACE JET DISCHARGE
175
-------
Ul
o
i
o
QC
UJ
I-
-50 0 50 100 150 200 250 300 350 400 450
LONGITUDINAL DISTANCE X/H0
FIG (AI43) TEMPERATURE, TRAJECTORY, WIDTH. AND DEPTH
(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
•IK
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
-------
PLUME AREA/HQ
FIG(BI ) TEMPERATURE, AREA (TA)-PLOT
FOR SURFACE JET DISCHARGE
-------
CD LU
PLUME AREA/HJ
FIG(B2 ) TEMPERATURE, AREA (TA)-PLOT
FOR SURFACE JET DISCHARGE
-------
CO
IM
10
PLUME AREA/H;
FIG ( B 3 ) TEMPERATURE. AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
CO
10
10
PLUME AREA
FIG ( B4 ) TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
CO
PLUME AREA/H;
FIG ( B5 ) TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
o
01
PLUME AREA/H;
FIG (B6 ) TEMPERATURE. AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
10'
PLUME
FIG ( B7 ) TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
10
PLUME AREA/H;
FIG (B8 ) TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
oo
PLUME AREA/H*
FIG(B9 ) TEMPERATURE, AREA (TA)-PLOT
FOR SURFACE JET DISCHARGE
-------
CD
VO
UJ
(T
cr
LU
Q.
Ul
I-
Ul
o
tr
-^
CO
PLUME AREA/H^
FIG(BIO) TEMPERATURE, AREA (TA)-PLOT
FOR SURFACE JET DISCHARGE
-------
10
10
PLUME AREA
10
10
$
FIG (Bll ) TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
IO
PLUME AREA/H;
FIG (B 12 > TEMPERATURE. AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
ro
PLUME AREA/H;
FIG(BI3) TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
iO
CO
10
10
PLUME AREA/H;
FIG (BI4) TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
PLUME AREA/H:
FIG ( BI5) TEMPERATURE. AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
ID
(71
PLUME AREA/H;
FIG (
TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
Ot
liJ
tr.
tr
iij
a.
UJ
o
tr
PLUME AREA/HQ
FIG© 17) TEMPERATURE, AREA (TA)-PLOT
FOR SURFACE JET DISCHARGE
-------
vo
HI
oc
cr
UJ
a.
2
UJ
t-
LJ
O
(0
PLUME
FIG (B18) TEMPERATURE, AREA (TA)-PLOT
FOR SURFACE JET DISCHARGE
-------
PLUME AREA/H;
FIG (B 19) TEMPERATURE. AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
10
PLUME AREA/H;
FIG (B2O) TEMPERATURE. AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
O
(O
PLUME AREA/H;
FIG (B2I ) TEMPERATURE. AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
PLUME AREA/H;
FIG (B22) TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
I.Or-
rsj
10
PLUME AREA
FIG (B 23) TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
10*
10*
-------
10
PLUME AREA/H;
FIG (B24) TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
ro
10
PLUME AREA/HO
FIG(B25) TEMPERATURE, AREA (TA)-PLOT
FOR SURFACE JET DISCHARGE
-------
o
10
10
FIG(B26)
10
PLUME AREA/H*
TEMPERATURE, AREA (TA)-PLOT
FOR SURFACE JET DISCHARGE
10
10
-------
ro
Q
CT»
IO
PLUME AREA/H;
FIG (B27) TEMPERATURE. AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
PLUME AREA/H;
FIG (B 28) TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
ro
o
co
10
PLUME AREA
FIG (B29) TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
10
10
-------
to
PLUME AREA/H;
FIG (B3O) TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
PLUME AREA/H;
FIG (B3I ) TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
10
10
PLUME AREA
10"
FIG(B32) TEMPERATURE. AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
10
6
-------
INJ
ro
PLUME AREA/HQ
FIG (B 33) TEMPERATURE, AREA (TA)-PLOT
FOR SURFACE JET DISCHARGE
-------
CA>
PLUME AREA/H*
FIG (B 34) TEMPERATURE, AREA (TA)-PLOT
FOR SURFACE JET DISCHARGE
-------
PLUME AREA/H;
FIG (B35) TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
ro
in
\o
PLUME AREA/H;
FIG (B36) TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
IV)
cr»
PLUME AREA/H;
FIG (B 37) TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
ro
PLUME AREA/H;
FIG (B 38) TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
ro
CD
PLUME AREA/H:
FIG(B39) TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
PLUME AREA/H!
FIG (B40) TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
CO
ro
o
UJ
o:
or
uj
a.
UJ
UJ
o
0
10
10
10
PLUME AREA
10*
FIG(B4I ) TEMPERATURE, AREA (TA)-PLOT
FOR SURFACE JET DISCHARGE
10
-------
10
PLUME AREA/HO
FIG (B 42) TEMPERATURE, AREA (TA)-PLOT
FOR SURFACE JET DISCHARGE
-------
IV
IM
PLUME AREA/H;
FIG (B43) TEMPERATURE. AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
ro
PLUME AREA/H;
FIG (B44) TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
ro
ro
PLUME AREA/H;
FIG (B45) TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
re
iv:
01
PLUME AREA/H;
FIG (B46) TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
ro
10
PLUME AREA
FIG (B47) TEMPERATURE. AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
IVJ
ro
10
PLUME AREA
FIG (B48) TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
IV3
o:
o:
oc
ui
Q.
LJ
UJ
O
if
QL
3
V)
10
10
PLUME AREA/HQ
10
10
FIG(B49) TEMPERATURE. AREA (TA)-PLOT
FOR SURFACE JET DISCHARGE
-------
ro
ro
10
10
PLUME AREA/H^
10*
FIG(B50) TEMPERATURE, AREA (TA)-PLOT
FOR SURFACE JET DISCHARGE
10
-------
PLUME AREA/H;
FIG (B 51) TEMPERATURE, AREA (TA)-PLOTS
FOR SURFACE JET DISCHARGE
-------
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
-------
ro
-P*
o
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
-------
UJ
o:
tr
UJ
a.
ui
UJ
o
PLUME AREA/HJ
FIG(B65) TEMPERATURE. AREA (TA)-PLOT
FOR SURFACE JET DISCHARGE
-------
en
*°
K°
Ul
a:
i>
s
o:
iij
a.
2
IU
H
UJ
O
if
cc
3
o>
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
UJ
oc
^
Si
cr
UJ
o.
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
QL
UJ
o
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,
ro
oo
LU
CC
a:
UJ
Ul
O
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
oo
CO
TIME- tU0/H0
FIG (C3O) TEMPERATURE, TIME (Tt)-PLOTS
FOR SURFACE JET DISCHARGE
-------
IV)
oo
tu
cc
1
a:
UJ
UJ
o
cr
i>
V)
TIME- tU0/H0
FIG(C3I) TEMPERATURE, TIME (Tt)-PLOTS
FOR SURFACE JET DISCHARGE
-------
X
o
UJ
a:
a:
UJ
a.
UJ
UJ
o
s
cr
z>
(O
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
-------
UJ
QL
ID
a:
UJ
a.
2
UJ
UJ
o
if
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°
ro
vo
to
UJ
o:
ID
i-
oc
UJ
Q.
UJ
H
UJ
O
2
cr
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
-------
u>
o
C3
10
TIME- tU0/H0
FIG (C 55) TEMPERATURE, TIME (Tt)-PLOTS
FOR SURFACE JET DISCHARGE
-------
CO
>o
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
O
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.
UJ
HI
o
2
cc
TIME- tU0/H0
FIG (C7I ) TEMPERATURE, TIME (Tt)-PLOTS
FOR SURFACE JET DISCHARGE
-------
CJ
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
UJ
o
OT
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
-------
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
'0
10
0>
'u.
u> evi
01 "il.
I02
10'
10°
1 1 1 1 1 1 1 1 1 1
1 1
LF
— ID
—
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
/"
/
X
r
/
s>
4
O
X
/
X
f
\
FA —
rA
<
?
>E
X
X
X
~
x"
C
4
J*
V
1
O
Q ^x"
"""
)
n
X^
S
O
•1
^
(
c
id1 10° id
.,
i-""""^
r
*"^E
^-^
^
I02
«^
I
s
i
i
ICf
ij
I04
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
------- |