MODELLING OF UNIDIRECTIONAL THERMAL
DIFFUSERS IN SHALLOW WATER
Joseph H. Lee
Gerhard H. Jirka
Donald R. F. Harleman
Energy Laboratory
Report No. MIT-EL 77-016
__________
l__--·llll---LIIII
July 1977
I
·IL·"·E)---·s·IT.II*IIIDI-^I.PIII
MODELLING OF UNIDIRECTIONAL THERMAL
DIFFUSERS IN SHALLOW WATER
by
Joseph H. Lee
Gerhard H. Jirka
and
Donald R. F. Harleman
Energy Laboratory
and
Ralph M. Parsons Laboratory
for
Water Resources and Hydrodynamics,
Department of Civil Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
sponsored by
New England Electric System
Westboro, Massachusetts
and
Northeast Utilities Service Company
Hartford, Connecticut
under the
MIT Energy Laboratory Electric Power Program
Energy Laboratory Report No. MIT-EL 77-016
July 1977
ABSTRACT
This study is an experimental and theoretical investigation of
the temperature field and velocity field induced by a unidirectional
thermal diffuser in shallow water.
A multiport thermal diffuser is essentially a pipe laid along the
bottom of the water body and discharging heated water in the form of
turbulent jets through a series of ports spaced along the pipe.
A uni-
directional diffuser inputs large momentum in one direction; it can
achieve rapid mixing within relatively small areas, and has the advantage of directing the thermal effluent away from the shoreline.
The theory considers a unidirectional diffuser discharging into
shallow water of constant depth in the presence of a coflowing ambient
current.
diffuser.
A fully mixed condition is hypothesized downstream of the
A two dimensional potential flow model is formulated and solved
in'the near field, where flow is governed by a dominant balance of pressure and inertia.
A control volume analysis gives the total induced flow,
which is used as an integral boundary condition in the potential flow
solution.
The shape of the slip streamline is solved using Kirchoff's
method; the velocity and pressure field are then computed by a finite
difference method.
are deduced.
The correct boundary conditions along the diffuser
Knowledge of the flow field defines the extent of the near
field mixing zone.
The near field solution is coupled into an inter-
mediate field theory.. In this region turbulent lateral entrainment,
inertia and bottom friction are the governing mechanisms of the flow, and
the mixed flow behaves like a two dimensional friction jet.
model is formulated and solved numerically.
An integral
The iiodel predictions of
induced temperature rises, velocities, plume widths enable comparisons of
the overall effectiveness of different heat dissipation schemes.
3
A model for calculating the near field dilution and plume trajectory
of a unidirectional diffuser discharging into a perpendicular crossflow
is formulated and solved.
The phenomenon of heat recirculation from the
far field is ascertained in the laboratory and a semi-empirical theory
is developed to evaluate the potential temperature buildup due to far
field recirculation.
A comprehensive set of laboratory experiments have been carried
out for a wide range of diffuser and ambient design conditions.
The
model is validated against the experimental results of this study as
well as those of hydraulic scale model studies by other investigators.
The analytical and experimental insights gained in this study
will aid future numerical modelling efforts and in better design of
physical scale models.
The principal results are applied to establish
general design guidelines for diffusers operating in coastal regions.
The ecological implications of the model predictions are discussed.
4
ACKNOWLEDGEMENTS
This study is funded by New England Electric System, Westboro
Massachusetts and Northeast Utility Service Company, Hartford,
Connecticut through the Electric Power Program of the M.I.T. Energy
Laboratory.
The co-operation of Dr. William Renfro of Northeast
Utilities and Mr. Bradley Schrader of New England Electric is appreciated.
The material in this study was submitted by Joseph H. Lee,
Research Assistant, to the Department of Civil Engineering in partial
fulfillment of the requirements for the degree of Doctor of Philosophy.
The research work was carried out under the technical supervision of
Professor Donald R.F. Harleman, Professor of Civil Engineering and
Director of the Ralph M. Parsons Laboratory for Water Resources and
Hydrodynamics, M.I.T. and Dr. Gerhard H. Jirka, Research Engineer in the
Energy Laboratory and Lecturer in the Department of Civil Engineering,
M.I.T.
Several discussions with Dr. E. Eric Adams, Research Engineer and
Professors Chiang C. Mei and Keith D. Stolzenbach have been very helpful.
Mr. Roy G. Milley, machinist, and students Richard Trubiano, Jeff Diercks,
Charles Almquist, and David Fry have provided assistance in the experimental program of this study.
Computational work was carried out at the
Civil Engineering-Mechanical Engineering Joint Computer Facility at M.I.T.
This manuscript was capably typed by Ms. Josephine Mettuchio.
5
TABLE OF CONTENTS
Page
TITLE PAGE
1
ABSTRACT
2
ACKNOWLEDGEMENTS
4
TABLE
OF CONTENTS
5
I.
INTRODUCTION
11
II.
1.1
Background
1.2
Multiport
1.3
Objectives
REVIEW
11
20
and Summary of this study
OF PREVIOUS
WORK
2.1
The Free Turbulent
2.2
Turbulent
2.3
14
Diffusers
Buoyant
Interaction
24
Jet
24
Jets
26
27
of Round Jets from the
Multiport Diffuser
2.4
Application
of Buoyant
Jet Models
for
30
Hydrothermal Prediction
III.
2.4.1
Unidirectional Multiport Diffusers
in a Laterally confined Environment
30
2.4.2
Unidriectional Multiport Diffusers
in a Laterally unconfined Shallow
Environment
34
Basis
37
2.5
Criterion for Unstable Near Field:
of a Two Dimensional Approach
2.6
Summary of Previous
Work
A THEORY FOR THE TEMPERATURE FIELD INDUCED BY
A UNIDIRECTIONAL DIFFUSER IN A COFLOWING CURRENT:
THE NEAR FIELD SOLUTION
Statement
of the Problem
3.1
General
3.2
The Complete Three Dimensional Formulation
38
41
41
43
6
Page
3.3
Dimensional Analysis
45
3.4
The Vertically Averaged Equations
47
3.5
A Synopsis
50
3.7
of the Flow Field
3.6.1
Control Volume Analysis
59
3.6.2
Limiting Cases
63
A Potential Flow Model for the Near Field
64
3.7.1
Basic Assumption
66
3.7.2
Derivation of the Boundary Conditions
67
3.7.3
Solution
70
3.7.4
Results
3.7.5
of the Slip Streamline
Sensitivity
75
of Results
to the
81
Polygonal Approximation
3.7.6
3.8
IV.
Solution for the Flow Field
Summary
81
87
THE INTERMEDIATE FIELD SOLUTION
89
4.1
Formulation of the Intermediate Field Theory
89
4.1.1
Model Structure
89
4.1.2
Governing Equations
92
4.1.3
Basic Assumptions
93
4.1.4
System of Integrated Equations
94
Solution of the Intermediate Field Equations
96
4.2.1
Non Dimensionalization
96
4.2.2
Numerical Solution Procedure
98
4.2.3
Initial Conditions
98
4.2.4
Entrainment Coefficients
99
4.2
7
Page
4.3
Solution
4.3.1
100
Analytical Solution in the Absence
of
4.3.2
Physical
Interpretation
Intermediate
4.4
V.
Solution
of the
103
Field Solution
for u
a
0
109
EXPERIMENTAL
INVESTIGATION
5.1
General
Experimental
5.2
Multiport Diffuser Design
118
5.3
Temperature Measurement System
121
5.4
Flow Visualisation and Velocity Measurement
121
5.5
Experimental
124
5.6
VI
Numerical
101
a Current
117
Setup
117
Runs
5.5.1
Series MJ
124
5.5.2
Series FF
125
5.5.3
Series 200
127
Data Reduction
134
COMPARISON OF THEORY WITH EXPERIMENTAL RESULTS
136
6.1
Near Field Mixing
136
6.2
Comparison of Theory and Laboratory
Series MJ and FF
Results:
140
6.3
Comparison of Theory and Laboratory
Series 200
Results:
148
6.4
Detailed Thermal Structure
154
6.5
Comparison of Theory with Perry Model Study
163
6.6
Comparison
173
of Theory
Fitzpatrick Plant
with Field Data:
James A
8
Page
VII.
UNIDIRECTIONAL DIFFUSER DISCHARGING INTO A
PERPENDICULAR CROSSFLOW
179
7.1
179
7.2
7.3
7.4
VIII.
T-Diffuser
in a Crossflow
7.1.1
Near Field Dilution of a
Unidirectional Diffuser in an
inclined Crossflow
181
7.1.2
Calculation of T-Diffuser Plume
Trajectory
184
7.1.3
Theoretical Results
187
Experimental Results
187
7.2.1
Experimental Observation:
This Study
189
7.2.2
Data Analysis of the Perry Tests
192
Comparison of Predicted and Observed
Plume Trajectory
195
7.3.1
Plume Trajectory:
Series FF
196
7.3.2
Plume Trajectory:
Perry Tests
196
7.3.3
Sensitivity of Results to'CD
202
202
Summary
FAR FIELD RECIRCULATION
205
8.1
Conceptual Framework of Heat Recirculation
205
8.1.1
Characteristic Scales of Recirculation
208
8.1.2
A Dimensionless
209
Formulation
of Heat
Recirculation
8.2
Experimental Investigation
210
8.2.1
Experimental Results
211
8.2.2
Empirical Correlation
216
8.2.3
Short vs Long Diffusers
219
9
8.3
IX.
Design Application
8.3.1
Prototype Considerations
220
8.3.2
Design of Models
226
DEISGN
9.1
9.2
220
APPLICATION
228
Diffuser Design in Shallow Water:
Discussion
A General
228
9.1.1
Design Alternatives
229
Design
Interpretation
230
of Theoretical
Predictions
9.3
9.4
9.2.1
Case A: Effect of Diffuser Length
for Fixed Discharge Velocity
232
9.2.2
Case B: Variable Discharge Velocity
and Diffuser Length with Constant
Near Field Dilution
239
Diffuser Performance and Aquatic
Environmental Impact
245
9.3.1
Comparison of Ecological Performance:
Variable Diffuser Length at Fixed
Discharge Velocity
246
9.3.2
Case B: Variable Diffuser Length
and Discharge Velocity at Constant
near Field Dilution
250
9.3.3
Concluding Remarks
252
Prototype Considerations:
Diffusers
Short vs Long
253
10
Page
X.
SUMMARY AND CONCLUSIONS
10.1
257
Objectives
257
10.2
Theory
258
10.3
Experiments
263
10.4
Results and Comparison with Theory
263
10.5
General Conclusions
265
10.6
Recommendations for Future Research
267
LIST OF REFERENCES
269
LIST OF FIGURES AND TABLE S
272
LIST OF SYM4BOLS
278
APPENDICES
A
B
CONTROL VOLUME ANALYSIS VALID FOR ALL DILUTIONS
SCHWARTZ-CHRISTOFFEL TRANSFORMATION FOR AN
ARBITRARY POLYGON WITH SIX VERTICES
285
287
C
COMPUTATION OF ISOTHERM AREAS
294
D
NUMERICAL RESULTS OF SLIP STREAMLINE COMPUTATIONS
297
11
CHAPTER
I
INTRODUCTION
1.1
Background
Technological advances in recent decades have led to man's
increasing consumption of electricity.
A by-product of this
activity is the large quantities of waste heat resulting from
steam-electric power production.
At present the thermal
efficiency
production
of this mode of energy
is about 40% for a
fossil fueled plant, and 33% for a nuclear fueled plant.
Thus
for every kilowatt of electrical energy produced, an equivalent
of 1.5 to 2 kilowatts
as waste heat.
of energy
is rejected
to the environment
The heat rejection rate is about 7x109 BTU/Hr.
for a standard 1000 MWe fossil plant.
As power plants are
usually located far from populated areas, the present state of
technology
has not found
of the low grade waste
it economical
to make beneficial
use
heat.
In many instances a "once through" condenser cooling water
system is employed.
Cooling water is withdrawn from an adjacent
water body, circulated through the plant once, and discharged
back into the water
environment
ambient water temperature.
at 20 to 30 F higher
than the
The increase in water temperature
changes our environment: the chemical composition and physical
properties of the water-dissolved oxygen levels, viscosity,
nutrient concentrations, etc. may be altered; the life processes
of many aquatic organisms may be encouraged or hampered.
The
12
waste heat input affects the ecological system.
knowledge
of interactions
of the essential
Although our
components
of an
eco-system is admittedly meagre, it is important to provide
a sound
basis on which
environment
the impact
can be assessed.
of thermal
discharges
It is of interest
on our
to understand
the various heat mixing mechanisms and induced fluid motion
that govern
the ultimate
distribution
of heat in the water body.
For additional purposes of avoiding heat recirculation through
the plant, it is desirable to predict the various temperature
configurations associated with different thermal outfall
designs.
This is the point of departure
of this study.
Different standards have been set up in the United States
to regulate thermal discharges from power plants.
Prior to the
issuance of a thermal discharge permit the applicant must per-i
form detailed engineering and ecological investigations to
demonstrate that the thermal standards will not be violated.
Depending on the aquatic environmentand a multitude of factors,
the regulatory standards are rather site specific, and vary
from state to state.
A common type of standards dictates that
a certain temperature rise above the natural background water
temperature
mixing
is not to be exceeded
zone; a high rate of mixing
the thermal outfall.
beyond
a relatively
is thus required
small
close to
Other standards may restrict the warm
effluent to an upper layer of the receiving water for protection
of benthic organisms.
Important considerations that are of
interest to biologists include the maximum temperature
13
rise near the discharge,
time to elevated
cooling water
the exposure
temperatues
flow,
of organisms
the location
entrained
in the
and extent of the plume.
In
recent years there has been a tendency to make thermal impact
assessments on a case to case basis, allowing more interdisciplinary interaction involving engineers, biologistsand
ecologists.
This also furthers the need for a more complete
picture of the induced temperature and velocity field.
For once through cooling systems there are basically two
types of waste heat discharge schemes-surface schemes discharge
heated
cooling water
at the free surface
through
number of pipes located near the shoreline.
results
in a relatively
but has the advantage
stratified
a canal or a
This method usually
large surface area of elevated temperatures,
that the heated water
surface layer
forms a stably
and the effect on the bottom
of the
receiving water is reduced.
Submerged discharge schemes consist
of single port or multiports
located near the bottom
water body.
of the
The first type of thermal standards usually precludes
the use of surface discharge or submerged single port discharges.
A submerged multiport diffuser discharging the heated water in
the form of high velocity
turbulent
jets has proved
to be an
efficient device to achieve rapid temperature reduction within
a relatively
small
area.
Condenser cooling water systems require large quantities
of cooling water.
MWe
plant
A typical condenser flow for a standard 1000
is in the order of 1000 cfs.
The projected
daily
14
condenser cooling water requirement for steam-electric power
generation for 1970 and 1980 was 113 and 193 billion gallons,
as compared to the nation's average
runoff of 1200 billion
gallons/day (National Water Commision, 1971).
Thus for once
through systems, there has been a trend to build plants adjacent
to coastal
areas
(great lakes or oceans)
where there is an ample
supply of water, the mixing capacity of thermal discharges in
rivers being limited by the ratio of the river flow to the
condenser flow.
1.2
Multiport Diffusers
A multiport diffuser is essentially a pipe laid along the
bottom of the water body and discharging waste water in the
form of turbulent jets through a series of ports spaced along
the pipe.
performance,
sufficient
The ports
can be oriented
and are usually
interaction
closely
to give optimal mixing
spaced
so that there is
among the jets to approximate
a line
source of pollutant, flow, and momentum.
Historically, submerged multiport diffusers have been used
for sewage disposal in the marine environment.
Recently they
have been used in thermal discharge applications.
There is,
however, a marked difference in the mixing mechanisms of the
two cases:
Submerged
sewage
diffusers
are characterised
by
high buoyancy of the discharge under deep water conditions.
The relative density difference between the discharge and
receiving water is about 0.025, and the typical water depth
15
about
100 ft.
equivalent
two dimensional
as an
can be represented
Thus the discharge
slot jet in an infinite
and
field,
classical theories of buoyant jet analysis are applicable to
Submerged thermal diffusers,
predict diffuser performance.
on the other hand,
by low buoyancy
are characterised
discharge under shallow water conditions.
difference
is about 0.003
of the
The relative density
and the typical water depth 20 ft.
For thermal discharge applications the buoyant jet patterns
predicted by classical theories cannot be maintained and fully
mixed conditions occur downstream of the diffuser (Fig. 1-1).
In almost all practical situations, three dimensional thermal
diffusers produce fully mixed conditions downstream; the nature
of the flow field
is altered
considerably,
and traditional
methods of predicting diffuser performance for sewage
applications cannot be carried over.
fig. 1-2 shows
A
three basic
types of multiport
diffusers.
unidirectional diffuser consists of ports all pointing in
one direction and discharging perpendicular to the diffuser
axis.
A unidirectional diffuser inputs large momentum in one
direction; it can achieve rapid mixing within a relatively
small area, and has the advantage
effluent
away from the shoreline.
of directing
the thermal
The ports in an alternating
diffuser discharge alternately in opposite directions.
net momentum
input is zero; the plume extends
The
in both directions
and flow is driven by buoyancy beyond a near field mixing zone.
16
V
f-,I-
-4
ly
tified
entrainment
X
xXX
Fig. 1-1 a)
II
X
X
x
X
x
xx
Multiport diffuser discharge in deep water:
Stable buoyant jet
V
->
discharge
induced
flow
/
./
X '
Fig.
x
1-1 b)
x
x
x
/
/C
X
x
,q;
/
JX
/
/Full
Vertical
/ixing
MX-X
~1
Multiport diffuser discharge in shallow water:
Unstable flow situation
17
ional
Alternating
Diffuser
;2:lis-~---L
............
-----. .
z-~~~~~~~~~
~
Fig. 1-2
~
Staged
Diffuser
-----
Basic types of multiport diffuser configurations
18
As the induced
diffuser
flow has no preferred
is suitable
direction,
this type of
with
for use in a tidal situation
reversing currents. In a staged diffuser the ports are oriented
in a tandom fashion:
by a narrow
diffuser
plume
axis.
this type of arrangment
and distributed
momentum
For a given heat rejection
is characterised
input along the
rate and under
stagnant conditions, the diffuser length required to achieve
a specified
degree
of initial mixing
is shortest
for the
unidirectional design (Harleman, Jirka, Stolzenbach, 1977).
The impact of a multiport diffuser operating in shallow
water can be illustrated as follows:
plant with
a condenser
Consider a hypothetical
flow of 2000 cfs and having
flow of 20,000
of 10, the total induced
low flow of the Mississippi
a dilution
cfs approaches
River in Illinois!
the
Tor a typical
discharge nozzle velocity of 20 fps, the induced velocity
around
is of the order of 2 fps-larger
the diffuser
current speeds frequently found in coastal regions.
initial mixing
also means
distance offshore
the diffuser
than
The high
has to be placed
some
so as to ensure a sufficient entrainment
flow from behind.
Fig. 1-3
shows
an example
of induced
temperature
rise at
the James A. Fitzpatrick Nuclear Power Plant on Lake Ontario.
The site is characterised
ambient
stratification,
by a relatively
small condenser
and a strong offshore
flow,
slope of 1/50.
19
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20
The isotherms
are inferred
from measurements
of dye concentration
taken in a hydrothermal field survey (Hydrothermal field studies,
Stone and Webster,
1977).
For unidirectional
discharge
dominated
diffusers
flow regimes
the travel time in the
is of the order of an hour;
for such time scales surface heatloss to the atmosphere is
negligible, and the temperature reduction is governed by
turbulent mixing with the ambient water.
1.3
Objectives
and Summary
of this study
This study is an analytical and experimental investigation
of the temperature
field and velocity
by a
field induced
unidirectional thermal diffuser in shallow water.
This type
of design provides an effective source of horizontal momentum
and is capable of meeting stringent thermal standards as
regards allowable mixing zones and temperature rises.
This study considers a unidirectional diffuser discharging
into shallow water
depth in the presence
of constant
ambient unidirectional current.
hypothesized
downstream
of an
A fully mixed condition is
of the diffuser.
The objective
is to
develop a theory that predicts quantities of environmental
interest; velocities, temperature rises, areas of surface
isotherms.
Surface temperature patterns are crucial for a
comparison of the overall effectiveness of different heat
dissipation
schemes.
Based
on an understanding
of the
diffuser induced flow field, the effect of the heat return
21
from the far field on potential
the
temperature
near field can be evaluated.
build up in
The analytical insights
gained in this study will aid future numerical modelling
efforts and in better design of physical scale models.
The
principal results are applied to establish general design
guidelines for diffusers in coastal zones.
In Chapter
II a summary
of previous
work on submerged
multiport diffusers is presented and the conceptual basis
underlying the present analytical and experimental modelling
efforts is discussed.
Chapter III presents a two dimensional potential flow
model
for the near field of a unidirectional
coflowing
current.
In this region
flow is governed
dominant balance of pressure and inertia.
analysis
gives
the total induced
diffuser
in a
by a
A control volume
flow, which is used as an
integral boundary condition in the potential flow solution.
The shape of the slip streamline
is solved using Kirchoff's
method; the velocity and pressure field are then computed by a
finite difference method.
along
the diffuser
defines
the extent
The correct boundary conditions
are deduced.
Knowledge
of the near field mixing
of the flow field
zone.
This is
important in the prediction of surface temperature configuration
and in future numerically
conditions.
modelling
for a variety
of ambient
22
In Chapter
IV the near field solution
intermediate field theory.
is coupled
into an
In this region turbulent lateral
entrainment and bottom friction are the driving mechanisms of
the flow, and the mixed
friction jet.
numerically.
flow behaves
like a two dimensional
An integral model is formulated and solved
Considerable
of no current,
insight
when an analytical
can be gained
solution
for the case
can be obtained.
The model is capable of predicting velocity,plume width,
temperature rises, and area of surface isotherms.
In Chapter V the experimental
employed
design
apparatus
in this study is described.
of each set of experiments
and run procedure
The motivation
is discussed.
for the
The run
parameters are tabulated and the experimental results presented.
In Chapter VI the predictions
of the model are compared
with laboratory experiments conducted in this study as well as
results of hydraulic scale model studies carried out by other
investigators.
In Chapter VII the effect of diffuser orientation with
respect to current is discussed and experimental results are
interpreted
in light
of the theoretical
insights
set forth in
the simpler situations.
In Chapter
the far field
VIII the phenomenon
is ascertained
of heat recirculation
in the laboratory
and a semi-
empirical theory is developed to evaluate the potential
temperature
build
from
up due to far field recirculation.
23
In Chapter IX the theoretical and experimental results
in this study are interpreted
from the view-point
of practical
design of thermal diffusers for once through cooling systems.
An
ecological interpretation of the model predictions is
advanced.
In Chapter
X the findings
in this study are summarised.
General conclusions are drawn and recommendations are made
in areas requiring further research.
24
CHAPTER
II
REVIEW OF PREVIOUS WORK
A multiport
diffuser
can be characterised
by a line of
equally spaced incompressible turbulent jets issuing into a
receiving water body of depth H.
2.1
The free turbulent
The mechanics
free turbulent
jet
of a two dimensional
fluid has been studied
jet in an infinite
extensively (Abramovich, 1963).
jet from a finite source.
or three dimensional
Fig. 2-1 shows a turbulent
Near the source the sharp discont-
inuity in velocity produces a region of high shear.
This
unstable situation gives rise to eddies, resulting in the
less turbulent ambient fluid being entrained into the turbulent fluid. The jet, or the region of high turbulent intensity,
thus grows
in width
as it entrains
more ambient
fluid.
The
mechanism of turbulent entrainment dilutes the concentration
of any tracer material
that may be introduced
at the source.
For this type of free shear flow experiments have shown
that pressure is approximately constant throughout the jet
and that the jet width is small compared with the longitudinal
length scale.
legitimate.
A boundary layer type approximation is then
Based on different hypothesis of relating the
turbulent shear terms -u'v' to the mean flow quantities, exact
similarity solutions have been obtained by Prandtl, Taylor,
25
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1
rla)
15
p4
or
fsisq
4..
W
w
I
4%,
-
-
4
I eL*
4.
r1C4d
i ?~
%t~
'r4
26
Reichardt (Abramovich).
pure source
of buoyancy
True similarity also exists for a
or a plume;
this case was treated by
Rouse, Yih and Humphreys (1952), and Morton, Taylor, and
Turner (1956).
The similarity of velocity profiles and concentration (or
buoyancy) profiles has been confirmed experimentally.
The
profiles can also be well approximated by Gaussian functions
(Albertson
where
et al, 1950)
u(x,y)
= u(x,O)
e
b
c(x,y)
= c(x,O)
e(
Xb)2
c : concentration
of tracer
: empirical constant
2.2
Turbulent buoyant jets
establishment
Fig.
2-2
Schematics
of a slot buoyant
s,n: axial co-ordinates
uc : centerline axial
velocity
I: ambient density
jet
u: axial velocity
e,:angle of discharge
B: slot jet width
27
The free jet theories have been extended by
many
investigators to study.t.hebehavior of turbulent buoyant jets
in a variety
of situations
(Abraham,
Stolzenbach and Harleman, 1971).
1963; Fan, 1967;
The turbulent buoyant jet
possesses both initial momentum and buoyancy, and as such it is
neither a pure source of momentum nor a pure
source of buoyancy.
(Fig. 2-2) A review of previous work on turbulent buoyant jets
as related to multiport diffusers has been given by Jirka and
Harleman (1973).
For purposes of predicting gross quantities of interest in
many practical applications it has been found convenient to assume
a priori cross-sectional profiles for the velocity and
concentration (buoyancy).
across
integrated
The equations of motion are then
the jet to yield a set of ordinary
differential
equations for the jet characteristics-plume position, centerline velocity, jet width, centerline concentration.
This so
called 'integral technique' has been successfully extended to
other buoyant jet flows where true similarity does not exist
(Abraham,
2.3
1963; Fan, 1967; Hoult et al, 1969).
Interaction of round jets from the multiport diffuser
The round jets issuing from the diffuser will gradually
merge into one another as they spread by turbulent entrainment
(Fig. 2-3).
turbulent
Knystautas
(1963) studied
the interaction
of
jets from a series of holes in line and discharging
into a still fluid at incompressible speeds.
Longitudinal
28
I'
i g,1
'I
a)
N-4
42
0
4d
0O
4.3
,-4
co
0
4-)
t"
p,
Q)
S
kA
Itl
A
III
4--
I
cY
I
bt
CNJ
FC'4
'a
7
-
It TM
I
.,s
Lll*
.
29
variations
in static
are assumed
pressure
thus the jet momentum
is conserved.
hypothesis
for the turbulent
linearises
the equation
to be negligible
By extending
shear stress
for the square
and
Reichardt's
in a free jet which
of the mean velocity
downstream, the momentum fluxes at each point of the interfering
The theory predicts
jet group may be found by superposition.
that fully merged two-dimensional flow occurs at approximately
12 hole spacings downstream of the source.
Cederwall
(1971) examined
the same problem
diffuser used in sewage and thermal outfalls.
momentum
and volume
slot jet for a seriesof
equivalent
TrrD
to be B =
radius
fluxes per unit width,
.
Defining
at the point where
By conserving
the width
momentum
b as the nominal
for a multiport
of an
jets can be shown
jet half-width,
the local jet velocity
equals
the
l/e of
the centerline velocity, and assuming interaction occurs when
2b=s,
it can be shown
that the jets merge
at x4.4s.
The same
line of reasoning can be worked through for a series of round
plumes, in which case interaction occurs at x5.1s.
the velocity
point of merging,
¥2
-u
M
.e
At the
at the edge of the jet is
uc =0.52 uc (obtained by summing the momentum
contributions from the individual jets); thus flow is far from
fully two dimensional.
Fully two dimensional flow is established
over a transitional region beyond the point of merging.
Cederwall
momentum
compared
jets with
comparison,
E is
the mixing
efficiency
that of an equivalent
defined
as
of a row of round
slot jet.
An index of
30
Dilution for multiple ports
Dilution
for equivalent
slot jet
At the point of merging, using well established relationships
for free jets E was shown
to be
0.95.
For the case of a
row of round plumes E0.78.
2.4
Application of buoyant jet models for hydrothermal
prediction
The considerations in the preceding sections are applicable
to multiport diffusers under deep water conditions, such as
the case for sewage applications, when the static or piezometric
head can be regarded
as constant
flow can then be modelled
in the flow field.
as an equivalent
buoyant
The diffuser
slot jet.
Predictions of dilutions of sewage pollutant or temperature
reduction of thermal discharges can then be estimated using
buoyant jet models based on integral techniques (Brooks, 1972).
Under shallow water conditions, however, the presence
of the free surface
of the flow field
regarded
and the bottom
drastically;
as constant,
boundary
pressure
and the momentum
changes
the nature
can no longer
be
flux is not conserved.
Multiport diffusers designed for use as thermal outfalls
usually fall into this category.
(Jirka, Abraham, Harleman,
1975)
2.4.1
Unidirectional multiport diffusers in a laterally
confined environment
Cederwall (1971) carried out an experimental classification
of the different
of a buoyant
types of flow regimes
slot jet in a confined,
that can occur downstream
uniformly
flowing environment
31
5.00 F
I
~ ~ ~~
~~
!
'
1
I
....
I~~~~~~~~~~~~
I
I
I
I
II
I
I
I 1111
Legend: Critical flow
I
-
I .
.I
·
__ ....
I
sf
I
I I
7
Maintained buoyant o
EI0~
.
-
jet flow pattern
.I
t,I
E
C
u
discharge in river]
1.00
-
a'8~~~~~~~~~~~~~~~~~~~~~~~~
+Zl
O..
,
0
U.
=1
0
S
Subcritical flow
0
Jet or plume-like
patterns .
.-
d--13
o3
-
-1_1%
I-
El
0.10
Occean disposal
nn
of sewvage o
r .I
I
0. .
I
I
I
0.5
I II
_
1.0
I
I
I
I
I
I
llI
5
The momentum flux ratio
I
'10
V
I
I
I
I I
50
uH
UB
-.
Observed flow regimes for a horizontal buoyant slot jet in a co-flowing stream..
Critical flow defined as the situation when the formation of a surface wedge is
incipient at the reference section.
- -
Fig. 2-4
Downstream flow regimes for a horizontal
buoyant slot jet in a coflowing stream
( from Cederwall, 1971)
U4 jet velocity
8 jet width
Uc ambient current velocity
A initial density difference
b' buoyancy flux
100
32
(Fig. 2-4). This simulates a unidirectional thermal diffuser
operating
in a river
situation.
For a horizontal
jet the
downstream flow condition is primarily governed by a source
Froude
number
defined
by F
u
3
APo
/g -
u B, a measure
of the
a
strength
of the current
relative
to the buoyancy
flux at the
source.
When F<l a subcritical flow with a surface wedge
penetrating upstream of the diffuser is observed.
When F>l
a super-critical flow occurs with the jet flow swept downstream when reaching the surface; this represents the situation
when the buoyant jet cannot entrain all the oncoming flow while
maintaining the free buoyant jet flow pattern.
The jet then
breaks up and efficient mixing takes place close to the source.
Hydraulic scale model studies and field tests done for
the thermal outfall into the Mississippi River from the Quad
Cities Nuclear Plant, Illinois, also indicate complete
vertical mixing of condenser water with the river flow downstream
of the diffuser
(Jain, Sayre,
1971; Parr, D., 1975).
Similar conclusions were also reached by Harleman, Hall, Curtis
(1968) in a study
of thermal
diffusion
of condenser-water
in
a river with application to the T.V.A. Browns Ferry Nuclear
Power Plant.
Harleman, Jirka, Stolzenbach (1971) investigated the
mixing downstream of three jets placed in a channel. Temperature measurements demonstrate that full horizontal and vertical
mixing are obtained in the downstream region.
temperature profiles are shown in Fig. 2-5.
Typical
33
12 FT. FROM PORTS
Ho"atvramLSLE
VECIc¢L
-a45'. 3."'
SCILE : -
3F
7.5 FT. FROMPORTS
Horizontal
Temperature
profiles
3' FROM PORTS
PORT
Q~~~BTWPOt
BETWEENPORTS
~~~~~~~~'
_
WATERSURFACE :
i
3 FT. FROMI
12 FT.FROM PORT
-"''..
.
..
-
-
.~~~
..
HORIZONTAL SCA
VERTICAL SCA
Vertic
CHANNEL
.
OTTS..
Temper
._1
Prof i
.CHANNEL BOTTOM
Fig. 2-5
Temperature profiles downstream of heated jets
placed in a channel
F
34
Jirka and Harleman (1973) gave a theoretical treatment
of the stability
stagnant
shallow water.
combination
measure
of a two dimensional
of H/B
It was
(a measure
of the buoyancy
buoyant
found that for a certain
of shallowness)
of the discharge)
flow is possible near the jet.
slot jet in
and FS ( a
no stably stratified
Intense vertical mixing is
created and consequently the temperature profile downstream
is vertically homogenous.
Whereas
the instability
in a confined
ambient
in the case of a buoyant
flowirg environment
current,
the instability
is primarily
induced by the
of a buoyant
slot jet in
stagnant water is dominated by discharge conditions:
momentum
and the shallowness
slot jet
of the receiving
the jet
water.
It should be pointed out that the rate of initial mixing
of a multiport
of the river
with length
diffuser
in a river is determined
flow to the condenser
comparable
to the width
flow.
by the ratio
Also, for diffusers
of the river
(the Quad
Cities Plant or the Browns Ferry Plant, for example), the
mixing downstream is strongly influenced by lateral boundaries.
2.4.2
Unidirectional multiport diffusers in a laterally
unconfined shallow environment
A theory
on the initial
mixing
performance
of a unidirec-
tional multiport diffuser in an unconfined environment was
first proposed by Adams (1971).
Assuming full vertical mixing
a control volume analysis indicates a contraction of the mixed
35
flow downstream.
The theory predicts the total induced flow
downstream from the diffuser in a coflowing current.
However,
the two-dimensional nature of the flow field was not discussed.
The location of the contraction, the boundary conditions along
the diffuser, the extent and nature of the flow field cannot
be obtained from the solution.
discussed
in Chapter
Adams's analysis is further
III.
Boericke (1975) studied the interaction of multiple jets
with an ambient current by solving the vertically averaged
equaions of motion numerically.
Predictions of flow patterns
surrounding the proposed thermal outfall for the D.C. Cook
Nuclear Plant on Lake Michigan were in qualitative agreement
with dye studies reported by the Alden Research Laboratories
(Fig. 2-6).
The numerical solution reveals large scale
recirculation (in which the discharge flow is re-entrained in
a circulatory flow pattern) can occur when the ambient current
is not strong enough to supply the entrainment demand.
This
would impair the dilution cooling near the diffuser as heat
is being recirculated.
Although the numerical modelling
promise
for a more complete
description
approach holds much
of the flow and temp-
erature field, there are several shortcomings;
In Boericke's
work the velocity boundary conditions around the thermal outfall are obtained
by adding
the velocities
due to each
individual slot jet at the outfall vectorially.
However, even
36
___
___
s7wit
Patterns Observed
FIG. 1.-Recirculation
ft = 0.305 m; 1 fps = 0.305 m/s)
4*
.S4
-.
_-.
c,*
e4.-
.7.
,
in Dye Studies of Hydraulic Mldel
t
o
~~
i::~
4,..*- o
e G.-o..e
-
W
-..
-"---- :.::
--
4k
t
e,
Flow Pattern for 0.5-fps Ambient Current (1 fps 0.305 ms;
I ft - 0.305 m)
dl.4h4M!*4' .~0,. r
Flow Pattern for 0.5-fps Ambient Current (1 fps = 0.305 m/s;
0.305 m)
IFIG.
ft =4.--Calculated
FIG. 4.-Calculated
Fig. 2-6
b
Numerical predictions of diffuser75
induced flow patterns ( from Boericke,19 )
37
if the water
momentum
body can be considered
fluxes can be
superposed
infinite,
only the
and not the velocities.
Furthermore, as discussed before, in shallow water pressure
can hardly be considered constant and therefore even superposition of momentum fluxes is not justified.
introduced
in the particular
The error
case cited may not be large as
there are only four slots oriented in different directions
in the thermal outfall.
In the general
case of a unidirec-
tional multiport diffuser with many closely spaced ports, the
interaction of the jets in shallow water have to be accounted
for in order
to arrive
at the correct boundary
be imposed around the diffuser.
conditions
to
In some cases, the near field
mixing of a diffuser may be strongly influenced by the presence
of an ambient
current,
i.e., the boundary
conditions
for the
vertically averaged equations may be coupled with an ambient
condition and cannot be specified independently.
2.5
Criterion
for unstable
near field:
basis of a
two dimensional approach
Fig. 2-7 shows the stability criterion given by Jirka
and Harleman
water.
for a horizontal
On the same diagram
slot buoyant
jet in stagnant
the (F , H/B) values
of
unidirectional diffuser experiments in a laterally unconfined
water body for which direct temperature measurements have
verified the vertical homogeneity of the temperature field
downstream are plotted.
From all the available data, we
38
see that the prediction
of stability
for a two dimensional
slot jet is a good indication of the near field stability
of a three dimensional unidirectional diffuser.
The foregoing
discussion
suggests-it is useful
to look
at the diffuser induced temperature field two dimensionally
and ignore the vertical variations as a valid approximation.
Except for an initial jet mixing region of limited extent,
this approach is valid up to the point where stratification
becomes
important
in determining the flow field.
In practice
the region of validity will cover the surface isotherms of
interest.
Furthermore, under prototype conditions, wind
induced mixing
and turbulence
created by heat flux out of the
water surface will enhance the vertical homogeneity of the
temperature field.
2.6
Summary of previous work
It has been recognised both experimentally and theoretically
that the heat discharged by thermal multiport diffusers in
shallow water will be vertically fully mixed.
plume
like patterns
cannot
be maintained
Buoyant jet or
and the nature of the
flow field is changed considerably.
For unidirectional multiport diffusers in an open water
body, simulating the Great Lakes or ocean coastal regions,
existing theory can only predict the total induced flow near
the diffuser.
To date no comprehensive treatment of the
temperature field induced by a unidirectional diffuser
39
i-
, Us
Fig. 2-7
us equivalent slot jet velocity
initial density difference
p$ ambient density
H water depth
8 equivalent slot jet width
A
Near Field Instability of Unidirectional thermal
diffusers in shallow water
40
Data reported by previous workers are
has been done.
limited to hydraulic scale model studies of specific sites
and as such little general insights can be extracted from
these scatteredexperimental
data.
modelling of thermal discharges in shallow
Numerical
water has been attempted.
However, previous efforts suffer
from the lack of knowledge of the correct boundary conditions
for the flow and temperature
field near the diffuser.
Areas of knowledge requiring further research include:
understanding
1.
A better
2.
A predictive
model
of the diffuser
of the temperature
flow field.
induced
field as a function
of diffuser design parameters.
3.
set of experiments
A comprehensive
that can provide
a
data base for validation of existing or future theoretical
modelling efforts, as well as providing meaningful empirical
guidelines
in situations
that are too difficult
to model
analytically.
4.
An understanding
field;
of the nature
this may be self-induced
the diffuser
situation,
over
of heat return
by the pumping
from the far
action of
prolongedperiods of slack in a tidal
or due to the interaction
of an ambient
with the diffuser flow.
All the above
points
are addressed
in this study.
current
41
CHAPTER III
A THEORY FOR THE TEMPERATURE FIELD INDUCED BY A UNIDIRECTIONAL
DIFFUSER IN A COFLOWING CURRENT: THE NEAR FIELD SOLUTION
The next
two chapters
are devoted
to a theoretical
discussion of the diffuser induced temperature field for the
case when the diffuser momentum is aligned in the same direction
as the ambient current.
in a coastal
This simulates a diffuser operating
zone and in particular,
the limiting
case, when
there
is no current, often reprsents the most severe design condition
for thermal diffusers.
this chapter.
A
near field theory is presented in
The coupling of the near
field solution with an
intermediate field theory is treated in the next chapter.
The
near field theory does not consider the question of heat return
from the far field.
3.1
General
Statement
of the Problem
Consider a unidirectional multiport diffuser consisting of
N identical ports of cross-sectional area a , equal spacing s,
and discharging heated water in the form of turbulent jets at
temperature
T
and velocity
u
in the same direction
into an
infinite receiving water body of depth H and uniform temperature
The ports are located near the bottom.
Ta.
LD
is defined
port.
as the distance
The upstream
schematized
current
between
velocity
The diffuser length,
the first and the last
is ua.
in Fig. 3-1 and the relevant
The problem
parameters
is
are defined.
42
J
-h
T(x,I gc(bs@}>
L-+ ,Ta,
,.
I
L
-- y
J~
-4
Pl
:O
IT
-§--
-40
-4
PLAN VIEW
!
j
4
4 U&
-
H
I
//
J
-
I
///
I6
. ,at///
~~ ~~~__~ ~ ~ ~
-
//
ELEVATION
/
/
//
///
/
ELEVATION
u,
a,
N
s
LD
To
c
discharge velocity
port area
number of ports
port spacing
diffuser length
discharge temperature
port clearance from
bottom
Fig. 3-1
ua
T
Ha
h(x,y)
OT=T-T
Oh=h-Ha
ambient current velocity
ambient temperature
receiving water depth
free surface elevation
temperature excess
free surface deviation
Unidirectional diffuser in a coflowing current:
Problem Definition
z
'7
43
3.2
The Complete Three-dimensional Formulation
Setting up a Cartesian co-ordinate system (x,y,z) with
the origin
at the center of the diffuser
q (x,y,z) = [u,v,w]
velocity:
and defining
(x,y,z)
T (x,y,z)
temperature:
h (x,y)
free surface:
level
A turbulent flow situation is considered and the quantities
over the scale of turbulence.
are time-averaged
Assuming
an
incompressible fluid and neglecting molecular viscosity and
transport terms, the following governing equations are obtained
for steady
flow:
Continuity:
+
+
0
az
ay
ax
(3.1)
Momentum:
au
au
au
vy a+ +w at~Z)=
pax
+ auv
ay
(u av
Bax (+
+
ap
au2
a x
ax
(3.2)
az
v + w a)
av)
ay
az
= pg -
-
ay
ax
+ avw'
+ aw'
ay
az
~
(u
+ v
w +
ay
ax
+
w w)
az
v'w'+
aw
ay
az
(3.3)
g
p
uw
az
ax
(3.4)
44
Heat conservation:
T
IT +
=y
w
v
X
UUT
ax
ay
a
av'T'
ay
u'"T'
x
w'T'
aZ
(3.5)
For small temperature differences the equation of state can be
written
as:
P -Paa =
where
e
(3.6)
(T-T a)
a
e aa
is the coefficient of thermal expansion of water at T .
a
The Boundary conditions are:
z = h (x,y)
On the free surface,
ah
ah
ax
ay
u ax + v
= w
-y
p=o
At large distances from the diffuser,
(x,y) +
(U,V,W) * ua
h(x,y)
At the bottom,
+
H
T(x,y,z)+ Ta
a
z=0
aT c
az
At each discharge port
(u,v,w)
T =
=
(u
T
o
,o o)
(3.7)
45
The system of governing equations (3.1 - 3.6) with the
associated boundary conditions presents a formidable task in
arriving at a complete solution valid for the entire region.
Instead a zone modelling approach is taken.
Physical insight
gained from experimental experience allows the division of the
entire solution domain into several regions, each with distinct
hydrodynamic properties.
In the following sections, the
approximate governing equations for each region are presented;
the solution
is sought
in each region
and the solutions
are
coupled to give a global solution for the entire field.
3.3
Dimensional Analysis
As a prelude
to the ensuing
theoretical
discussion,
a
dimensional analysis is presented in this section to single
out the important
Letting
dimensionless
AT = T - T
a
of the problem.
parameters
excess
be the temperature
above
the
ambient temperature, we can write the functional relationship
, u
f(AT,AT ,x,y,z, a ,N,H,LD,U,V,w,c,u,g,T
Choosing
as the fundamental
AT ,H,u
O
variables,
a)
= 0
(3.8)
the following
o
parameter groupings can be obtained,
AT
(u,vw))
u
F(
c
H
(xy,z)
H
s
H
uO
0-
LDH
D
'Na.
u
u
a
u
_
AT
'N
(3.9)
46
-
-
-
.
l
-
Dimensionless
Parameter
-
Nomenclature
Typical
in this
study
range of
- -
-
|
LDH
Na
50-500
O
or
sH
a
physical interpretation and
comments
values
--
H/B
|--
--
-
--
-
_
ratio of intercepted crosssectional area per
port to port area:
relative submergence
o
of equivalent
jet-B
For
-
=
a
/s
slot
is
the
width of
the slot
jet having the same
discharge flow and
momentum per unit
length
N
large
LD
H
10-50
relative diffuser
length
U
u
a
o
u
< 0.1
strength
current
Fc
50C-500
densimetric Froude
S°
number of the equivalent slot jet:
measure of the buoyancy of the discharge
<
H
of ambient
o
,/gBeAT
S
Y
3.0
relative spacing:
important only in
initial jet mixing
zone where temperature
field is three
dimensional
C
< 0.2
H
relative clearance
of the ports
U
0gH
s/gH
0(1)
O
jet Froude
number based on the
water depth
m
Table 3-1
Summary of Governing Dimensionless Parameters
47
The physical interpretation of each parameter is given in Table 3-1;
typical values are also indicated.
hider vertically fully mixed conditions variations in the
z direction
can be neglected
and the two dimensional
problem
ceases to depend on the details of the diffuser design described
by
c
s
H ' H.
LDH
D
AT v)(u F((xLy)
AT
u
H
o
'Na
o
LD
D
H
u
' u
u
a
u
o
o
Na
/gB AT
3.4
L
o
o
gH
LD
The vertically Averaged Equations
In this section,
integrated
the equations
in the z direction
of motion
and energy are
to yield a set of equations
for the depth averaged quantities.
Assuming u,v>>w, so that pressure is hydrostatic we
integrate
eq. 3.1, 3.2, 3.3, and 3.5 vertically
z = h(x,y).
Invoking
free surface
and the no slip condition
repeated
the kinematic
use of Leibnitz's
boundary
from z = 0 to
condition
at the bottom,
rule, we arrive
at the
and with
at
Continuity
a fh(xy)
u dz +
o
x
y
(xy)
v
dz = 0
(3.10)
48
M omentum
h
P ( (!_
a I
y
d
udz
uvdz)
+
h
x
h
h
(ph) dz
h
2
P
u'
a
dz
1P uTdz
+ ay
J p u'v' dz
2
-
(3.11)
T
vTd =z-
u
3h
T-h
ax
I
p uv'
-ay
uTdz
a
a
ay
0v 2 dz
vTdz
-
(3.12)
T'
a
3'T'dz
aV'T' dz - w'T'
dz
v
(3.13)
-
49
Defining the vertically averaged quantities
h
u
=
u dz
(3.14)
v dz
(3.15)
Jo
h
V=
J
we have
h22dh
o
udz
=
(
)
22
v2dz
2
v dz = (
v
) h -2
(3.17)
hv
o
h
fh
o
(3.16)
hu
hu
uvdz
uvdz (- 5
h- uv
) huv
(3.18)
The following assumptions are made:
i)
The coefficients
resulting
3.18 are close to unity.
from the averaging
in eq. 3.16-
For turbulent open channel flows,
for example, the vertical variation of velocity can be assumed
to obey a 1/7 power
law; the coefficient
computed
for this
profile is 1.015.
ii)
Changes
in free surface
level are small,Ah/H,<<
1.
50
iii)
Density
differences
are small, AP
<<1, and can be
a
neglected except when multiplied by the gravity term in the
equations of motion (The Boussinesq approximation).
iv)
A quadratic friction law is assumed for the bottom shear
Pu2
o
zX = 8°
8
2
vu2+v
XT:
u
(3.19)
f
:
0
T
= -
zy
2
of
U +v
8
2
Friction
Factor
v
transfers
(3.20)
v)
Turbulent
vi)
Surface heat transfers are neglected.
Incorporating
in the x direction
these assumptions
is neglected.
into eq. (3.10 - 3.13) we
obtain the vertically averaged equations. For simplicity the
depth averaged quantities will henceforth appear without bars.
au + av
v = 0
ay
ax
au
ax
au
ay
(3.21)
h
ah _
aAT
eg
ax
dz
ax
u'v'
f
ay
8H
2+v2
(3.22)
av
av
ax
va ay
Uay 5dz
ay-o
9ay
aTdz
ah
f
8H
/u2+v
v
2
2
(3.23)
u
3.5
aT
aT
ax + v ay
-
A synopsis
A brief
_v'T'
(3.24)
ay
(3.24)
of the flow field
description
of a physical
picture
of the diffuser
induced flow field is given in this section.
Turbulent mixing is created by the high velocity shear near
each discharge port.
The jets merge with one another and the
51
spreading reaches the water surface at approximately
a predominantly
A negative
two dimensional
flow field as shown in
pressture zone is created
a fluid particle
at infinity
jets produces
of the multiple
'he interaction
away.
5 to 10
behind
the diffuser,
would tend to converge;
Fig.3-2a.
to which
the flow
pattern behind the diffuser resembles that of a sink flow.
The
momentum input riises the pressure across the diffuser, creating
a positive
behind
pressure
'separates'
diffuser
from theambient
and is confined
Fig. 3.2b illustrates
field
3.5.1
fluid at the ends of the
stream
contracting
in a
flow from
The induced
in front of it.
an instantaneous
snapshot
tube.
:
of the flow
in an experiment.
Scaling
Through a proper choice of characteristic scales for the
physical variables we can estimate the relative importance of
the terms
in eq. (3.21-3.23)
for each region.
Let
(U,V) be characteristic
velocities
(L,6) be characteristic
lengths
in the x,y direction
in the x,y direction
pagAH be characteristic pressure deviation
Ap be a characteristic
density
difference
52
>
-4I
I
Ua,
/ Late alr
Ent
MV0,ctf
inmenK
Ent
It
p<o
-40
2
-
Fig.
3-2 a)
?I
l
--- >svmmetrv
line X
x
z~-
Schem
.Li
1L6
- I
Fig. 3-2 b), An instantaneous snapshot of the diffuser
induced flow field in slack water
( photo by C. Almquist )
53
Thus in eq. 3.22,
inertia
3.23
U2/L
pressure
gAH/L
buoyancy
The magnitude
p gH/L
of the lateral
turbulent
shear stress
terms can
be estimated and related to mean flow quantities by mixing
length concepts commonly used in free shear flows (Abramovich)
-u'v'
where
2 (.u)2
-
: mixing length
The mixing
length
can be assumed
transverse length scale
to be proportional
= c'6
where c' is an empirical constant.
a)
The Near
the following
L
Field:
In a region
scales apply:
LD
UsingV
6
LD
AH
U2/2g
Using typical values of
LD =
1000
ft.
H
=
20
ft.
U
=
2
fps
to the
For free jets
c ' 101
close to the diffuser
L
LD
54
temperature
f
=
o
to an induced
(corresponding
= 0.0004
Ap/p
of 4 0 F)
rise
0.02
The relative importance of each term can be estimated:
H 0(1)
pressure %
U2
inertia
,2U2L2
2
6
shear
inertia
inertia
lateral
U
2
2
X
/L
inertia
bottom shear .
L
-2
3
( 6 )3
0
(10-2)
0 (10-2
g
2
buoyancy
c'
fL
o D
0 (
8H
inertia
We therefore conclude that pressure must balance inertia in
this region.
Lateral shear, buoyancy and bottom friction are
not important
in the near
for the near field
au + av
ay
ax
au
ah
ay
ax
av
ay
g ah
ax
u
aT + v aT.
ax
equations
0
ax
v
The approximate
are then
au
u av
field.
ay
Oy
_ av'T
ay
(3.25a)
55
Field
The Intermediate
b)
the near field,
In a region beyond
takes on a boundary
6
layer character,
10LD, the flow
L
so that
<<L
V <<U
The scale of free surface deviation is then due to turbulent
side entrainment
(U) U
AH
where
2g
a = entrainment
= 0.06
coefficient
Then
pressure
a2
103
inertia
so that the dominant
balance
is among the convective
This region of extent
lateral shear and bottom friction.
103H
is called
field'.
the 'intermediate
terms,
Beyond
a certain
point, when friction has reduced the velocity to a scale
that
Ap
g AP
H
P2 -
0(1), buoyancy effects will have to be accounted.
U
The governing equations for the intermediate field are
then
au
av =O
_
+
ay
ax
au +v
ax
aau+ _
ay
_-
au'v
ay
fo
8Hu
ulul
(3.25b)
56
Cont.
(3.25b)
av
3.6
2
av
av'
aT
aT
av'T'
ax
ay
ax
ay
fo
ay
u
8H
ay
Outline of Near Field Theory
A two dimensional
model
for the flow field downstream
of
the undirectional diffuser is presented in the following sections.
A vertically fully mixed condition is hypothesized, the equations
of motion
are then independent
of the temperature
field.
The
total induced flow near the diffuser is first derived by a
control volume analysis.
This is then used as an integral
boundary condition to obtain a potential flow solution of the
near field.
In Chapter IV the near field solution is coupled
into an intermediate field theory that predicts the global
temperature field.
A definition sketch of the near field problem is illustrated
in Fig. 3-3.
1)
The following
As a first approximation,
neglected
hold.
in this region
assumptions
turbulent
and Laplace's
As far as the depth-averaged
are made:
side entrainment
equation
motion
is
is assumed
is concerned,
to
in the
region beyond the initial jet mixing zone, vorticity can be
generated only by turbulent lateral shear.
that if turbulent
entrainment
It thenfollows
has a negligible
effect
on the
fluid motion in this region (as shown by the scaling arguments),
--
-------
57
f~~~~~I
57Ar
t
I -f
A
\ t2
\:4
X
-P
.\
.e
tL
'.~~~~~~~~
L
()
1D
--)
-?
I\
I,I
,II
of)
1
I
;
21
W
~
4~ 4'1
'~-7
I1----.1
II
6L,
I~~~~~~~~~~~~~~~~~~~~
lI
~I
I
I
41 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I
A.
-:>
I
I
.
I
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1
s1,
I
--30!
I
I I
I
i
I
1'
_~j
p,4
(~3
ELV
Fig. 3-3
nON
Control volume analysis of a unidirectional
diffuser in a coflowing current
58
the two dimensional fluid motion can be considered irrotational.
Furthermore, a solution of Laplace's equation also satifies
eq. (3.25).
2)
The induced flow from behind the diffuser 'separates'
from the ambient
confined
at the ends of the diffuser,
fluid
A A A' A'.
in a streamtube
The slip streamline
as a locus of velocity
can then be regarded
and is
discontinuity
separating the flow within the streamtube from the ambient
Pressure is continuous across the slip streamline.
fluid.
In a real fluid
the discontinuity
in velocity
is of course
unstable and generates eddies that will draw the ambient fluid
into the streamtube.
included
This correction
as a perturbation
is small and can be
in the development
of the inter-
mediate field theory in Chapter IV.
3)
A
high
rate of mixing
is assumed;
i.e. the induced
flow
is large compared with the discharge flow.
The momentum
4)
of the diffuser
input
creates a pressure
discontinuity across the diffuser; but velocities are
continuous across it.
AH/H << 1.
Alternatively, depth changes are small
In principle,
this amounts
to a "rigid lid"
assumption.
Bottom friction is neglected.
5)
_
L_
I_
59
3.6.1
Control Volume Analysis
In this section
an integral
quantity
of interest,
the
total induced flow QN' is derived using a control volume
analysis of the two dimensional flow field.
the same as given
by Adams
(1972).
The approach is
The analysis
is included
here for completeness; unnecessary assumptions are removed and
issues relating
to the formulation
of a complete
two dimentional
problem are discussed in greater detail.
The following quantities are defined:
QN= SQ
S
~qD(y)
total induced flow from behind
near field dilution
velocity distribution along the diffuser
(x=0,-LD/
qD ( Y ) =I
D
(Y )
2
< Y < LD/
2
)
I
As shown in Fig. 3-3, two control volumes are chosen:
Bounded by the free surface
and the bottom
in the vertical
direction, the outer control volume A 1 A 2 A 3 A 4 is chosen such that
A1A4 is sufficiently far upstream from x=O, A2A 3 at a downstream
position where h=H.
The four corners are at y = + a.
The
inner control volume, B1AArB2, is chosen to just enclose the
diffuser.
The
then:
boundary conditions along the control surfaces are
60
Along
A1A 2
q
=
A4 A 3
h'n=
(u a,O)
(3.26)
H
A 1A 4
A2 A
A A'
3
X
Along
A co A' oo
=
(uNO)
h= H
Along
B1B 2
q
AA'
= qD (
)
h = h
(y)
q = qD
(Y )
h
(y)
= h
Applying Bernoulli's theorem along any streamline that
extends from A1A4 to B1B2, we have
Ua2
a
2g
Similarly
U 2
2
2g
_____111_____1_11___.__11.1·1111111
qD 2 (y)
H =(y)
2g
+
for the same streamline
q
(3.27)
leading
from AA' to A A'
2
2(y)
H = +2g
+ h+(y)
_11_1^1_1__
.1^^__
.1_1____.·._._
.·
(3.28)
61
Subtracting eq. 3.27 from eq. 3.28, we have an expression for
the head rise Ah along the diffuser:
2
Ah(y) = + h (y) - h(y)
we see that regardless
2
UN
of the velocity
the head rise is constant
(3.29)
2g
distribution
qD(y),
along the diffuser,
Ah = constant
Applying
(3.30)
the momentum
theorem
in the x-direction
to the
outer control volume described by A1A2A3A 4 we obtain
SQu a
+ PQoo
=
SQ
(3.31)
U
As there is no net contribution
difference
in the x-momentum
A 1 A 4 must be equal
is equal
of pressure
fluxes
the
out of A2A 3 and into
to (pSQou N-SQoUa),
to the momentum
forces,
assuming
S
S-1; this
input pSQo u
Similarly for the inner control volume B1AA'B2,
PLD
Qu
0
gJ
L
1/2
h+
[ h+(y)
2
2
-
- h
(y)
] dy
(3.32)
- LD
D
2
The momentum input is balanced by an increase in pressure
force across
the diffuser.
For a small deviation
in the
62
free surface
level
(h
+ h-)x H, and by virture
of eq. 3.30
eq. 3.32 then becomes
Qou
Recapitulating
= gLDHAh
(3.33)
the complete
set of equations
for the control
volume analysis:
2
u
Ah
=
2
-U
Naa
N
(3.34)
2g
Qouo = gLDHAh
SQua
+ Q u
oa
Noting
Q
0
=
(3.35)
=
o o
(3.36)
SQuN
oUN
Na u , and rearranging, an equation for S, the near
field dilution can be obtained
L IIH
( D)(
S
0
The solution
u
ua
)
LD)
S
Na
0
=
(3.37)
0
is
S
-
P
+
V6
2
2y +
2B
(3.38)
wlhere
LDH
=
The width
a
of
u
a
0o
0
00
A 00 ' can be obtained
SQ° o = UNaLDH
D
by equating
(3.39)
63
from which we get the contraction coefficient
a = Sy + 0.5
3.6.2
(3.40)
Limiting Cases
Considerable insight can be gained by examining the limiting
case of no ambient
current,
u
a
= 0.
Setting y = 0 in eqs.
3.36, 3.38, 3.40 gives
LDH
S=
2Na
o
a = 1/2
UN= u /S
Thus,
(3.41)
the contraction
of the mixed
regardless of the diffuser length.
be evaluated as
Ah
u N
H
2H
S = 10
u
uN
gives
H = 0.003,
indeed small;
=
2
= 20 fps
that the pressure
deviation
is
Ah is less than an inch in 20 feet of water.
In the limit of high crossflows
eqs. 3.38,
H = 20 ft
fps
confirming
LDHu >>Na u , or
y>>l
3.40 gives
S = By
and
a
=
1
1/2
The pressure deviation can
Using typical values of
u
/S
flow away is always
(3.42)
64
The dilution
crossflow
is then equal
to the ratio of the intercepted
flow.
and the discharge
be pointed
It should
of high dilutions,
out the assumption
S>>1, is not a restrictive
one.
A similar
analysis
valid for all dilutions is included in Appendix A.
even for S=3, the deviation
owever,
for the near
is only a few per cent.
field dilution
3.7
in the prediction
that is
A Potential
Flow Miodel for the
ear Field
From the control volume analysis we have derived the total
induced
flow confined
in a streamtube AA A' A'.
the shape of the streamtube,
of the contraction,
location
However,
the
and
are unknown.
the extent of the flowfield
In this section, the boundary conditions along the diffuser
The position
are derived.
of the slip streamline is solved by
free streamline theory and a potential flow solution is obtained
in the near field by the method
of finite
differences.
By symmetry we need consider only one half of the streamtube
(Fig. 3-4).
potential
variable
Assuming
potential
flow we can define
W(z) of the flow field as a function
z = x + iy
corresponding
to
a point
in
a complex
of a complex
the
physical
plane.
2
a=u
=y
ax
-v
W(z) -
(x,y)+ i(x,y)
(3.43)
q(x,y)
(-----
---- ·- · r
I·ll----p-·-r.r.
·LI
·I(-----··II·-·I1·-·--··111_-.1
=
dz
= qe
·LI·li·L-·I·II··I-·l*··--lll·_L-·
----
--------
-I---
65
II
4
*/,
0.56s UNL
Ad
O.5
U,
A potential flow model for the near field
Fig. 3-4
where
= velocity potential
= stream function
q
= complex velocity u(x,y) + iv(x,y)
dW
o
= inclination of velocity vector from x-axis
qD=
velocity along diffuser line
qA = velocity
at point of separation
W A = complex potential
at A
WB = complex potential
at diffuser
A
center B
66
3.7.1
Basic Assumption
Consider a general case where the induced flow from
behind the diffuser is given by 0. 5 oLDuN
where
uN
: velocity
at A
a : contraction coefficient of streamtube
uN,a are given by eqs. 3.38-3.40.
This may be conceived as
an integral boundary condition.
We suppose that locally near x=O, each section of the diffuser
induces
the same amount
of flow: i.e., each individual
jet
entrains the same amount of ambient flow.
O)
0: angle of separation of
It
-- 0---t
lfin trPamlinp
---
,--1
4
L((
-*.;+wYU.
I+
I 1:4
ox
t 1,&.X
Fig. 3-5
Boundary conditions along the diffuser line
67
This assumption implies that locally near x=O, the
horizontal velocity gradient, au , is also
not a function of y.
Applying conservation of mass to an infinitesimal portion of
the streamtube
near x=O, as shown in Fig. 3-5, we have
u(x=O)
= (1 - Ax tan
A
) u(x = Ax)
(3.44)
= (1 -
Ax tan eA)(u(x=O) +
A
tx
Ax)
~x0
Equating terms of order Ax
aux |
u(O) tan
ax
a
A
constant
(3.45)
Thus
u X=O
= constant
au
=
ax
3.7.2
XO
constant
Derivation of the Boundary Conditions
By symmetry
BB. is a streamline.
If we define
= 0 on BB
it follows
= 0.5LDuN
Along
(3.46)
the slip streamline
P
= P.
on AA.
AA.
(3.48)
68
2
=2
p
+
1
pu
2
N
(3.48
N
of the velocity
The magnitude
and equal
on the slip streamline
at the contraction,
to the speed
the diffuser
Along
Pq
u
q =
therefore
i)
1
+
P1
to any point on the slip streamline
theorem
Bernoulli's
Applying
is constant
u N.
axis AB
Since u=constant by continuity we have
(3.49)
U = au N
N
q
and
D
=
cos
(3.50)
, by definition
'
ii) Since au= constant
invoking continuity again we have
av = constant
-
By symmetry
v = 0 at
diffuser
v(y) = vA
therefore
where
the
the x-component
Whilst
B
fY
y component
vA
center
of velocity
(3.51)
at A
of q (y) is constant
along the
diffuser axis, the y-component varies linearly.
At
the
point
qA
and
-----
--
-----
aUA
of separation
=
A
uN
U
auN
~-~~~1-^11-"-"1P---·Llil·IBII·ll
--
(3.52)
69
since
qACosOA
= uA
cosA
= a
we have
8
or
where
A
is the angle
As vA = qAsinSA,
=
cos
(3.53)
o
of separation
by virtue
of eq. 3.49-3.51
(3.54)
vA - UNSinA
A LD
N
A
Recapitulating the boundary conditions, and defining the
velocity potential at the diffuser center
B
0, we have
Along AA
-
=
0.5 LDuN
4
= varies
q =
uN
a
varies
from
at A
A at A to
from eA at A to 0 at A
Along BB
=
4
0
varies
from
0 at
B to
0
at
v =0
B
0
v
Along AB
dJ =
au
u
N
dy
d~ = uNSinGALD2y
v=
dy
O(o)
= 0
v
'p(0)= O
a UN
q
0
=
cosE
varies
from 0 at B to
A
at A
(3.55)
70
3.7.3
Solution of the slip streamline
The position
of the slip streamline
is solved by Kirchoff's
method (alternatively called the Hodograph plane method) which
has been commonly used in classical free streamline theory
(Milne-Thomson,
the complex
1968; Gurevich,
potential
intermediate
1965).
W and the complex
common variable
i.
The idea is to map
dW
velocity
d-z onto an
If each mapping
is kept one-
to-one, we will have obtained a one-to-one correspondence
between W and dW , thus solving the problem.
The complex
illustrated
potential
in Fig. 3-6.
W diagram
The shaded
for the case when u =0 is
a
portion
within A ABB
corresponds to the flow within the streamtube. The positions
of the points
A,A
,B,B
on the W plane,
values at the locations A,A ,B,B
The boundaries
of the shaded
Let the hodograph
Q = log(u
=
The hodograph
n
plane
N
( -)
the
(,')
in the physical plane (x,y).
portion
is derived
plane Q be defined
N
represent
using eq. 3.55.
as
dW
- i
(3.56)
Q is illustrated
in Fig. 3-7.
Again
the shaded portion describes the flow field within the streamtube.
uN
The points
A,A ,B,B
on the Q plane represent
values at the locations
A,A.,B,B.
the (ln
in the physical
-
plane
,0 )
(x,y).
The boundaries of the shaded portion is derived using eq.3.55
71
If we can find intermediate
transformations
W
0. 5 oL
()
fl
u
D N
Q =
(3.57)
f2 ()
then
dz
dW
-
1
exp (Q)
UN
-
d[0. 5a LDUN]
d(
5L ) =
di;
dr
exp [Q(%)]
D
dC
(3.58)
Replacing z,W by its non-dimensionalized counterparts:
df
z =z +
Given
df
will
give the position
(3.59)
di
the path in the C plane corresponding
eq. 3.59
the
o exp(f2())
to the slip streamline,
of z by intergrating
along
C plane.
Both the W-diagram
and the Q diagram
are approximated
by
polygons and the interior of these polygons are mapped onto the
upper half plane
by the Schwartz-Christoffel Transformation,
with the following correspondences specified, as shown in
Fig. 3-6, 2-7.
C
W,Q plane
A
<
>
(-,0)
A
<
>
(-1,0)
B
<
>
(+1,0)
plane
72
our*
L,
.·-
--&'
i * -
/
./
I
-0-6
i"/
°
..5
-W0.
0.2twa
-
__
lis~am \
~~I
-}oo - .9 -0-6
-o-s -0
f
r_
I
I
.
I
I
i
I
I
Q.2
..
04,
I
I
0.
!-o
·
P
·
-
H
w/
8
6s.
A,
W(S)
Fig. 3-6
= CW
)
(-
J
b0rcCL
eA=.
~ . ·
3
@4_
·
o.6
a( I41-t
tef
aI,) ( ( - ?z3 (4_1) den +
_I
Conformal mapping of the complex potential plane
W onto the upper half plane
o
73
oZ
6'
o4
o6
t( LN)
o.
I
sr
=;=//
7
-t o
I.0
(IQ) =a f(-+I(.g-A
-I - -I
) C-c-I)
AS
I
Fig. 3-7
Conformal mapping of the hodograph plane
onto the upper half plane
+
74
The points
onto three
shown
B
A,A,B
for the W and Q diagrams
arbitrary
choices
is then mapped
The curvilinear
on the real axis.
onto (+
are mapped
It can be
,0)
boundary A ABB ,
in the W plane is
approximated by a six sided polygon A AW1W2BB.
By the Schwartz-Christoffel
w()
= C
f4
j
ca
-1
(C+1)A
-
a
(C-a )
Transformation,
1
1
a-l
(_-a2 ) 2
-
we have
B-1
1) d
+ WA
(3.60)
-1
where
a
:
interior
WA
:
co-ordinate
CW
: constant
angles of the W polygon
of
point
A in
It can be shown
that W 1 ,W 2 will be mapped
(a 2,0)
al
where
-1
< a2
in units of
the W plane
onto points
(a 1,0),
< 1
The computational details required to calculate Cw,ala
are included
in Appendix
In an entirely
2
B.
analogous
manner
by a six-sided polygon A AQ 1 Q 2 BB .
the Q diagram
is approximated
The Q-C transformation
is given by
(r+l)
Q() = Cq-1
A
Q((+) = CQ
-1
(C
(2-1B-1 +
b
) 1
(b2
(-1)
d
+ QA
(3.61)
75
where
8i
: interior
(b,0),(b2,0)
angles
: corresponding
QA : co-ordinate
Substituting
the expression
of the Q-polygon
points
in units of
of Q1'Q 2 on the real axis
of the point A in the Q plane
for W()
and Q(c) in eq. 3.59
we obtain
z
+
=
-
exp [C
AC (+1)
z
QJ
(C+l)
(-b
l)
1
(-b
Q 22
a2-1
1
(C-al
B
(-)
dC+ QA ]
B-1
(-L1)
('-a 2
2
d
=(0,1)
(3.62)
Since the slip streamline AA corresponds to the strip
(-1,0) to (-o,0) in the
plane,
the (x,y) position
of the slip
streamline can then be evaluated using eq.(3.62) by integrating
from i = - 1 to r = -X.
3.7.4
Results
For a given diffuser design and ambient current, the
contraction
coefficient
The angle of separation
W and Q diagrams
finding
can be calculated
A
is then obtained
are generated
accordingly
the W-C, Q-c transformations,
slip streamlines
by eq. 3.38, 3.40.
can be evaluated
by eq. 3.53.
by eq. 3.55.
the positions
The
After
of the
by eq. 3.62.
In table .3-2 computed values of the near field
dilution
and the contraction coefficient are presented for typical
76
L H
u
Na
u
o
a
S
a
o
50.0
0.0
5.0
0.5
150.0
0.0
8.7
0.5
300.0
0.0
12.3
0.5
50.0
0.025
5.7
0.56
150.0
0.025
10.7
0.60
300.0
0.025
16.6
0.65
50.0
0.04
6.1
0.60
150.0
0.04
12.2
0.67
300.0
0.04
19.6
0.72
50.0
0.067
6.9
0.65
150.0
0.067
15.0
0.75
300.0
0.067
25.9
0.82
Table 3-2
Near Field Dilution and
Contraction coefficient
for typical
LDH
B= Na
Na
0
u
and y= u
0
values
of
77
values
of
diffuser
and y.
designs
It should be pointed
(differentcombinations
lead to the same near field dilution
out that different
of
and y) can
and contraction
coefficient.
The predicted slip streamline is computed for different
values
of ,
eA
the angle of separation,
range of practical interest.
in Fig. 3-8.
i)
The results
that encompass
the
The results are illustrated
show that
The slip streamline reaches an asymptotic value at x0.5L
This means
the extent
D.
of the near field is 0. 5 LD; this result
coincides with visual observation of experiments performed
over a wide range of diffuser lengths, and is illustrated in
Fig. 3-2b.
ii) The slip streamline matches asymptotically the contraction
coefficient
given by eq. 3.40 for all
0
.
A
This is considered
sufficient proof that the basic assumption stated in section
3.7.1 (from which the boundary conditions along the diffuser
are derived)
is correct.
It is emphasized
that an incorrect
specification of the velocity distribution along the diffuser
will lead to an erroneous asymptotic value that does not agree
with the contraction coefficient given by eq. 3.40.
Details of the computations are given in Appendix D.
78
1.0
A
predicted slip streamline (eq.3.62)
aA
contraction
j
y/0.5 LD o.6
…
s
-
coefficient (eq.3.40)
_contraction.
4
40
a4
6,CX6-
o.766
6: C=
0.766
o.4
0.2
x/O. 5 LD
i
0.2
I
I
a.
0.6
I
I
o.
1.o0
1 2.
I.0
0.8
y/O.5LD
, -
o.6
'o
6. 70o7
o.+
0.2
x/O.
I
0.2
Fig.
3-8
0.4
I
0.6
o.g
I
.0
5
LD
I_
2
Theoretical solution of slip streamline
for different angles of separation
79
I.o
y/0.5L
D
0.8
0.6
466- 50
6
o 643
0.4
0.2
o.Z
0.4
0.6
Os
1.0
LD
1.Z
-L
.0
y/O.5LD
5
x/O.
0.$
__O.-_
--_A__ Coo
0.6
t:-
53.
6
0
5
o.4
0.2
x/O. 5LD
0.2
0.+
Fig. 3-8 cont'd
0-6
0.9
r.a
1.z
80
1.0
y/O.
5
LD
0.g
0.6
z4,-6o
o.4
6' 05
((4- o)
0.2
x/0. 5 LD
Fig.
3-8 cont'd
81
3.7.5
of results
Sensitivity
It was shown
to the polygonal
approximation
in Fig. 3-7 that the curvilinear
boundaries
along AB in both the W plane and the Q plane are approximated
by polygonal boundaries A
W2B and AQ1 Q2 B respectively.
The
straight boundaries AW1,W2B and AQ ,Q2B are chosen to closely
approximate
the slopes of the curvilinear
The exact positions
of W1,W
2
boundary
near A and B.
or QIQ2 are chosen such that the
polygon closely resembles the W, Q diagrams.
Fig. 3-9 illustrates
two quite different approximations of the W,Q diagrams for
6A=
A
-
600, a= 0.5.
Approximation
A is indicated
by the solid
lines AW1 ,W2 B,AQ1 Q2 B.
B is indicated
Approximation
Fig. 3-10 illustrates
AWVlW: 2 B,AQ'lQ' 2 B.
the computed
streamlines based on the two approximations.
in the predictions
the limited
lines
by the dotted
The difference
for the two cases is very small,
sensitivity
of the results
slip
demonstrating
to the polygonal
approximations.
3.7.6
Solution
It is found
can be well
for the flow field
that for
0
L
- 0.5L
g(x*) = x*4-(tanlOeA
x*
-
by a 4th order polynomial
approximated
tanle
< 1.0 the slip streamline
D
A*
x
0. 5LD
+ 2)x*
+ 1
3
+ (2tanleAI
+ 3
-2)x* 2
(3.63)
82
.g'
UiLL.
i
A_
a-
-/-O
--0.
-6
-04
-
B
a
ie
-4
-c
Fig. 3-9
Two different approximations of the W and Q diagram
ApproximationA
Approximation B
83
O'
It
I
I
0
I
I
I
I
o
(n
o)
I
CI
I
4, I
zz
I
-71
r-
I
I
Q)r4 c=
L4
I
rH
p
oI
(iI,
t
C
a lo
o
0
0
ND
C)
84
Given
solved
a numerical
g(x*),
in
solution
terms of the stream
of the flow field can be
function
finite differences in the region A ABBO.
are indicated
by the method
of
The boundary conditions
in Fig. 3-11
I'
/.0
¥/,'=
.0
=0
0.5aLDU
x' ,y
N
Boundary
Fig. 3-11
1.0
0.5
0.5LD
conditions
for the stream
function in the near field
A uniform
system for
grid system is used.
=
-
600, a=0.5.
Fig. 3-12 shows the grid
A second order finite
difference approximation is employed.
1.0
0.
0.6
. +
0 2
0.2
Fig. 3-12
o.+
0o.6
o.
1.o
Grid system used for finite difference solution;
step size=0.1
At an interior
node,
'~(i,)=-(iA,JA)=[(i-l,)+*(i+l,j)+*(i,j-l)+i(i,j+l)]
(3.64)
where A= step size
Details for treating grid points near the boundaries can be
9 ) and will be omitted here.
found in Ref. (
Fig. 3.-13 shows
tide case (A
=
-
the computed
streamlines
for the slack
600 a= 0.5), and the transverse
variations
of pressure at different longitudinal positions x.
The near
86
r O-
o .7Wiv
0.5 _
-
-,
-
.
014
--
3o-2
Fig. 3-13 a)
o.5 a6
;s
.3
~--
07
'
S oq
._
;-
1
Finite difference solution for the potential
flow model. Illustrated streamlines ( Y= 0)
"'
/,O
o.q
l
~
0.
-
.X
cr
,
I
0-3 .
0.2
.
o.
i
Fig.
3-13
b)
Fig.
3-13
b)
0o.1
04
05
o
o.
1.cl
O.
Transverse pressure distribution at different
longitudinal positions in the near field (Y = 0)
87
field
in front of the diffuser
high pressure, h(x,y)>H.
is marked
by a region
of
The excess potential energy gained
from the diffuser momentum input is redirected into longitudinal
momentum as the flow contracts and accelerates downstream.
The finite difference approximations are checked using
two step sizes A=0.1 and
A=0.05.
The solutions at the common
nodal points agree to within three decimal places; thus
of the numerical
convergence
size A=0.1
3.8
is judged
scheme
sufficiently
is assured,
and a step
small for the calculations.
SUMMARY
A two-dimensional model for the flow field near a uni-
directional diffuser in a coflowing current has been presented
in this chapter.
The vertically averaged equations are derived; it is shown
that in the 'near field', pressure and inertial effects dominate.
In the 'intermediate field' beyond the region, turbulent
entrainment and bottom friction become important determinants
of the flow.
A control volume analysis yields two important
integral quantities (the total induced flow and the contraction
coefficient
flow away) as a function
of the mixed
parameters
LDH
u
Na
u
D
o
a
o
of two physical
88
A potential flow model for the near field is formulated
and solved.
The position of the slip streamline bounding the induced
flow from behind is computed by classical free streamline theory.
The
theory indicates the extent of the near field is 0.5LD.
The velocity distribution along the diffuser can also be
inferred
as:
u =
SQ /LD
uSin6A 2y
v(y)
=LD
oL D
89
CHAPTER
IV
THE INTERMEDIATE FIELD SOLUTION
In chapter
formulated
III a two dimensional
and solved
In this chapter
potential
for the diffuser
an integral
flow model
flow in the near
model for the temperature
is
field
and velocity
field downstream of a unidirectional diffuser in stagnant water
or a coflowing current is derived from the vertically averaged
equations of motion.
the model
rises.
The near field solution is coupled into
to give a global
solution
of the induced
temperature
The characteristic features of the solution are discussed
and physical interpretations are given.
4.1 Formulation of the Intermediate Field Theory
4.1.1
Model Structure
The postulated
Fig. 4-1.
x=O.
structure
of the model
The flow is assumed
The diffuser
extends
is illustrated
to be vertically
in
fully mixed
from (0,-L /2 ) to (0, L
2
).
at
By
symmetry only one half of the flow and temperature field have
to be considered.
i)
Region
refers
I
The flow field is divided into two regions:
(0 < x
x),
to the zone where
penetrated
the region of flow establishment,
turbulent
side entrainment
to the axis of the diffuser
flow characteristics
can be described
plume.
has not
At any x, the
as consisting
of a
90
i
f
c)
P
I
I
I~~~~
i
I
I
0
P.i
H-
o
-f i
/7--t; tAit
Ili
t
I
--
is
1-1
iw
-,
i
.L
AN
tI
IY
III
Q)
I:t
I,a
a
o1
IH
JLt
k4
q
·rH
1
i
4
i
L
fc J. Y
%,
t
IC
II-
I
\
*
-I
!.li
1t
g r.Ii
-14
,,
I
91
potential'
width
layer
L(x).
to be equal
(x),
The temperature
and a diffusion
layer of
in this potential
to the near field temperature
being predicted
profiles
of width
by eq. 3.38.
The velocity
layer is assumed
AT
riseATN =
S
and temperature
take on the form:
u(x,y)
-c
AT(x,y)
u(x,y)
= AT
- u = {u
a
N
c
(x)
=
0
S
(x)
a
)2
L
=
AT
N
<6(x)
- u ) e
_(Y-
AT(x,y)
< y
e
6(x)
< y
< 6(x)
+
L(x)
The unknowns in this region are u (x),6(x),L(x).
extent of region I,xI, is defined by 6(xI)=O,
as
ii)
and is obtained
of the problem.
a solution
Region
The
II( x > XI), the region
of established
flow, refers
to the zone where turbulent side entrainment has reached the
center of the plume. The velocity and temperature profiles are
assumed
to be:
(y)2
u(x,y)
AT(x,y)
- u
a(
= {u (x)-u } e
= AT
sy)c
C
a
(x) eL
In this region, the unknowns
are u (x),L(x),AT (x).
92
The implicit assumption involved in the foregoing is
that the width
of the plume
is small compared
with
the
longitudinal extent, and that the velocity and temperature
are similar
profiles
at all x.
in transverse
Differences
heat and momentum transfer are neglected.
4.1.2
overning Equations
Letting b(x) = 6(x) + L(x), the vertically averaged
eq. 3.25b,
chapter,
in the previous
equations
can be integrated
across the diffuser plume (in the y direction) from y=O to y=b,
The resulting
i.e. to the edge of the plume.
equations
for
conservation of mass, momentum and heat are:
b
Jf
d
d
dx
Continuity
u dy
= u
o
db
db
a dx
v
(4.1)
e
Momentum
b-b
u
{
dx
o
dy =
(u v
a dx
a
f
J
8
8H
Heat
u
_
)
e
dx
gAh dy
0
(4.2)
dy
o
b
d J
uAT
dy
=
(4.3)
0
o
where
u
: ambient current velocity
v
: entrainment
a
e
velocity
at plume edge
93
Basic Assumptions
4.1.3
assumptions
Two basic
in the subsequent
are employed
analysis:
The entrainment assumption, first proposed by Morton,
a)
(1956) relates
Taylor and Turner
in
velocity
a characteristic
to
entrainment
the rate of turbulent
It
a plume.
transfer
recognised that in turbulent mixing the eddies causing
of heat between
two parallel
the volume
db
flux of the plume, u a db
dx
-
by the
The rate of increase
of the two streams.
velocity
relative
are characterised
streams
is
v
e
is assumed
of
to be
proportional to the centerline velocity excess:
u
u
e
current;
v
(4.4)
= a(u -u )
c
is the incremental
-
a dx
plume
v
db
a dx
a
flow addition
is the incremental
inflow
axis. a is the entrainment
different
values
by the coflowing
perpendicular
coefficient
in the two regions.
to the
that takes on
Its value is determined
from well established data on classical free jets.
b)
As shown
in Chapter
III, in the near field, 0 < x < 0.5LD
pressure and inertia effects dominate and the longitudinal
gradient of the pressure force acting over the plume crossd b
section
dx
gAh dy, therefore
does not vanish.
As the
diffusion layer in this region is expectedly small, the
pressure
force is approximated
by the solution
of the potential
flow model that neglects turbulent side entrainment.
94
g(x*)0.
_
dP
d
dx
where
P
g (x*)
u
=
(x,y) dy
(4.5)
pressure force over plume cross-section
given by eq. 3.63
slip streamline
= dimensionless
=
(x,y)
u2
d
gAh dy
J
dx
5LD
induced velocity as given by the potential
flow solution, equals the total induced flow
per unit depth divided by the width of the
bounding slip streamline, or
SQ
u p (x,y)
Beyond
the pressure
the near field,
(4.6)
X*=-0.5L
=
g(x*)LDH
is due to the
deviation
turbulent side entrainment and can be neglected as was shown
in section
3.5.1.
Thus the pressure
is evaluated
gradient
dP
d
SQO
dx
dx
LDH
D
2 0.5LD
D
g(x*)
= 0
4.1.4
as
x
> 0. 5L D
(4.7)
System of Integrated Equations
Invoking the previous assumptions, the postulated forms
of the velocity
Eq. 4.1-4.3.
and temperature
profiles
are substituted
out the integrations,
Carrying
we obtain
into
the
following system of equations:
Region
I
d=
dx
I(Uc-u
I
c
a)
a
(4.8)
95
dM
dQ
dP
dx
a dx
dx
o
(4 . 9)
8H
0.5QoAT
H
=ATN [U 6 + L{Ua
Lc (u
(-u) -U
+
}]
(4. 10)
where
udy = volume
Q =
flux across
plume
o
= u
c
6 +
(b
L [ C1 (u
1
c
-ua)
+
a
u
(4.11)
]
a
2
u dy = momentum
M =
u
6+ L
= U
+ L [c 2u2
across
flux
plume
+ 22+
2u u (Cl-C ) + u
2
(1-2c+c2
)
]
(4.12)
c1 ,c2 are integration constants
2
C
1
=
e
-k 2
dk =
(4.13)
/!2
o
c
I
e
2
dk =12 -
k
o
Regior
/2
II
(4.14)
dxdx aI11 (u c -u a )
dM
dQ
dx
a dx
ffM
o
(4.15)
8H
O.5Q AT
= AT L [ c2(c-ua)
where
Q =
L [ c
M= L [
(Uc-u a)
c
c
22
a
+
+ C1
a
(4.17)
a
a
ua
(4.16)
1
2
(1-2c
+ ua (-c)
+ c2)
(4.18)
96
aI a H
in region I, II
coefficient
: entrainment
respectively.
Solution
4. 2
4.2.1
of the Intermediate
Field Equations
Non-dimensionalization
The system
can be cast in dimension-
(4.8-4.18)
of equations
less form by defining the following dimensionless varibles
(x*,6 *,L*) = (x,6 ,L)/0.5LD
AT * = AT /AT
c
c
U /U
U * =
C
c
O
Q*=
Q/(u 0o5LD
M*
M/(uo 0.5LD )
The length
)
2
D
o
2
P* =
P/(uo 0.5LDD )
is chosen
scale
0. 5 LD, the velocity
temperature
o
to be the half-diffuser
temperature
rise by AT , the condenser
a
,
Q
o
Region
u , the
rise.
LH
u
Recalling that y=
set of dimensionless
velocity
is scaled by the nozzle
length
equations
LDH
a o uo ,
Na
the following
o
in stantardform
can be obtained:
I
dx*
dx*
dM*
dM*
a(uc*
I
C
dQ*
Ydx*
Ydx*
-)
(4.19)
dP*
f LD
+ dx*_
16HD
dx*16H
(4.20)
where
2 d(x)
dP*
dP* =S)2
dx* Bdx*
0
g(x*)0
0
x*
<
x*
<
<
1.0
< 1.0
> 1.0
(4.21)
(4.21)
97
and
from heat
+ L* [ c 2 (Uc*-y) + cl Y1
S = u *
c
(4.22)
conservation:
from integration
of
the profiles:
Q*
=
*6
c
2
= U,*c
+ L*
C*-y)
+
c
+y
eq. 4.22-4.24
2
(1-2c
1
u *,L*,6
+c
(4.23)
+ y]
L* [ c 2 u* 2
6* +
M
By rearranging
[Cl(u
2u
2 )
c)
*Y(c
c
1
2
(4.24)
can be expressed
in terms of M*,Q*:
U c * = Y+
Q*u *
L*
(4.25)
(M* -yQ*)
-
M*
=
(u *-y)[(c-c2 )uc* + y(1-2c 1+c 2)]
* Q*-L*
6=
-
C
6*(XI*)
= 0 defines
(4.26)
[ cl (u c *- Y) +y]
*
the length
(4.27)
of the region
of flow establish-
ment.
Region
II
dx*
dM*
dx*
= a(U
dQ*
Y dx*
(4.28)
* -Y)
f LD
oD
16H
(4.29)
where
1 = AT * L* [c 2 (u * -y)
c
2
c
a
+
C 1Y
]
(4.31)
Q* = L* [ c 1 (u * - y) + y]
M* = L* [c2u
2 c *2 + 2u c * y(c 1 -
(4.30)
) +Y
2(cc2
2(1-2c +c 2 )]
(4.32)
98
By rearranging
eq. 4.30-4.32,
u *,L*,AT
* can be related
to M*,Q*:
C1
rcnM*
)(2y) +
2
Q*
c
C1
2
(1)
(2y
c
2
M*
2
)2 +
Q*
4 (M*¥
2
Y)
2
2
(4.33)
L*
AT
c
c
*
4.2.2
=
Q*
(u c-* y)
=
+ Y
(4.34)
L*[ c2 (uc
*-Y)+cY]
(4.35)
Numerical Solution Procedure
The set of first order ordinary differential equations
(4.19-4.20,
numerically
initial
in x* with
characteristics
be related
be solved
using a fourth-order Runge-Kutta scheme by marching
out the solution
The plume
in M* and Q* can in general
4.28-4.29)
to (Q*, M*)
values
u *(x*),L*(x*),
c
at any x* by virtue
specified
6*(x*),
at x*=0.
AT (x*) can
c
of eq. 4.25-4.27,
4.33-4.35, thus enabling the derivatives dx*
dx* ' d*to
dx* to be
be
computed.
4.2.3
Initial Conditions
From section
3.7, we have obtained
SQo
u(x=0,y)
= L
LDH
Hence,
the initial
momentum
flux are:
conditions
for the volume
flux and the
99
Q*(o)
a)
=
u (0)
0.5L
o
D
D
0.5LD
Q
dy
o
i
u 0.5L
b)
M*(0)
S
--O
(4.36)
D
M(0)
2
u
0.5L
o
D
0.5L
D
-O.
o
dy
=
S )2
(4.37)
d
LD
0.5LDD u2
o
4.2.4
Entrainment Coefficients
The values
obtained
from the experimental
free jet published
flow across
u
c
coefficients
for the entrainment
by Albertson
aIaII
are
data for a two-dimensional
et al (1950).
the jet can be related
The rate of
to the centerline
velocity
(x).
i)
In
the
region
of
where
B
o
Q
B
establishment of
7(V2-1) 0.109 xB
Q(x) = 1 +
Q
flow
o
: initial
jet flow
: initial
jet width
can be combined
The two equations
o
a free
jet,
uc=uo
'
c
o
to give, for the half-diffuser
plume,
('2-1) 0.109
dQ
dx
2
a
= 0.040
c
(4.38)
100
ii)
In the region
of established
1 B
x
Qo
The two equations
dQ
o
°o
can be combined
_ 0.109
dx
u
flow,
1
x
to give
(x)
4
c
therefore
aII
4.3
= 0.068
(4.39)
Solution
u
The entire
problem
can be seen to be dependent
on
=
o
the relative strength of ambient to nozzle velocity;
LDH
= Na , a near field design parameter which together with y
o
determines
fL
the near field dilution
A (eq. 3.38); andB
2
=
0oD
16H
an intermediate field parameter measuring the momentum
dissipation of the diffuser plume.
In the limiting
case of no ambient
current,
= 0, the
system of equations (4.19-4.20, 4.28-4.29) can be solved
analytically.
Considerable insight can be gained by examining
the characteristics of the analytical solution which elucidates
the essential feature of the diffuser plume.
Hencforth all starred subscripts will be dropped and
all variables encountered later on are dimensionless variables.
101
4.3.1
Analytical Solution in the absence of a current
Setting
Region
y=O in eq. 4.19, 4.20, 4.22 gives
I
u c
dx
dx [ c (6+cL)]
1
d
c
(4.40)
I c
d
=82
[u 2(6+c L)]
dx
a u
=
1
2
1
g(x) )
dx
-
(2
2
[
(6+
c
(6+c 2
L)
(4.41)
u c ( 6 +c 2 L)
(4.42)
1
where
f L
S
1
B
Eliminating
du
82
6 +c
2L
16H
from eq. 4.41 and eq. 4.42 gives
c
d
d
Uc
Using an integrating
and invoking
g(x)
factor
e
Uc(O) =
uc(x) = e
1
1 dx
2X1
1
(4.43)
aX
, eq. 4.43 can be solved
[1.0 +
X e
d
(
to give
) dx
0
(4.44)
The entrainment relationship eq. 4.40 gives
r,x
a
uc(6+c1L)=
Io
u(X) dx +
eq. 4.45 from eq. 4.42, the solution
Substracting
layer L(x) can be obtained
=
of the diffusion
as
ouc (x)dx
(c -c 2 )
and the core thickness
(X)=u
(4.45)
fx
aI
L(x)
1
1
u c(x)
6(x) as
(x) - c2L(X)
u x
(4.46)
(4.47)
102
Region
II:
Setting
y= 0 in Eq. 4.28, 4.29, we have
(4.48)
(ucL) = a uc
C1 d
(ud
-
(u
Eq. 4.49 can be solved
m(x)
(4=
.49)
to give
L(x)
= u
L)
2
L
exp
I
-2(x-)
(4.50)
exponentially
by bottom
XI
i.e. the momentum
flux is dissipated
in a classical
friction;
free jet the momentum
flux is
constant.
Substituting
the expression
d
dx
(u c
equation
2
c
+
2
2
Since m(x) is known
solution
can
u
in u
2
c
c
the following
eq. 4.48 and expanding,
differential
for u 2L(x) in eq. 4.50 into
+
2
type
can be derived:
II
by virtue
be obtained:
Bernoulli
m(x)(u
22
c
)
=
0
4.51)
of eq. 4.50, the Region
II
f L
oD
o
8H
= u
(x)
u cW=
103
(x-x8H)
e
e
I {
I
1/2
L
f L
-f
+
(1-3)
oD
3e 16H
(x-xI)
f0L D
L(x) = L I {B3 + (1-
3 )e
(x-xI
16H
}
(4.52)
foLD
AT
=
S {(1-8
(x)
3 e
)
16H (xI
)-1/2
where
-32aIIH
3
I
LI
4.3..2
1L Ifo LD
Uc(
=
(4.53)
I)
= L(x I)
Interpretation
Physical
The features
Field Solution
of the Intermediate
of this intermediate
field solution
is best
demonstrated by plotting the variables relative to their respective
u
the beginning
valuesat
the distance
of region
from the beginning
II,
c
(x)
n
(x)
U,
L
of region
II,
T c S, against
'
Using
L
D
typical values of
LD
= 100,50,25,5
f
o
= 0.02
the solution
is plotted
that as the variables
in Fig. 4-2.
are plotted
It should
against
(x-xI)
be pointed
out
the figure
L
does
D
not illustrate the tradeoff among different diffuser designs
(e.g. a long diffuser
vs a short diffuser).
The implication
of
the theory on diffuser design will be treated in a later chapter.
104
n ,-
0
Co-
C1
c0
*,
ie,
0
C)
r
Il
Cm
)i
I
cO
IA
I
.H
P
0
,0
ci
_s
(S
105
qiZ
a)
6
C)
(3
0
$o
r-.
4.-i
a
4-i
-l
4-4
0
U)
a)
r
-4
.4
P:4
O
0
10
la
106
.
,r
a
0
o
tn
cj1
0
,0
C
C06
C
a)
a)
4.)
0)
.D
0
a)
0
ai
C1
(H
w
1 1,
.H
P..,
:
c;
Ci
C
1C5
107
It is interesting
to note that in contrast
to a classical
two dimensional jet in which the centerline velocity decays as
x
-1/2 , the velocity
in a two dimensional
friction
jet decays
exponentially with x, due to turbulent side entrainment and
momentum dissipation.
exponentially.
Also, the width of the plume L(x) increases
This diverging behavior is distinctly different
from the two dimensional free jet
(U x
-1/2 ,
Lox).
It can be seen from eq. 4.52, that as x-
AT c approaches an asymptotic value.
lim
AT
This is again
jet, which
This
=
AT (x)
1
in sharp contrast
predicts
(1-03
-1/2
to the behavior
(4.54)
of the free
that AT X x -1/2 , and hence AT +0 as x+.
behavior stems from the strong exponential decay in
velocity and the exponential spread of the diffuser plume, so
that at large x there is virtually
due to the relative
velocity
This feature also places
no turbulent
entrainment
of the plume and the ambient
a limit on the mixing
capacity
fluid.
of any
unidirectional diffuser.
The above features together imply that
a)
At some distance from the diffuser, due to the divergent
nature of the plume spreading, the lateral scale of the plume
is no
longer small compared with the longitudinal scale, and
the boundary layer asumption ceases to be valid.
108
b)
At any x, a measure
of the degree
of stratification
in the flow is given by a characteristic
densimetric
Froude
number based on the water depth defined as:
u
F
(x)
xc
e
The densimetric
as
F
c
oude number decreases with x, so that buoyancy
effects will become
diffuser,
(4.55)
important at large distances from the
+0.
Both a) and b) suggest
prone
to heat recirculation
that the diffuser
mixed
into the near field.
flow is
This phenomenon
is experimentally confirmed and discussed in detail in Chapter VIII
As will be seen later, the intermediate field solution is valid
for a region
of extent
103H, a sufficiently
large area over
which the importance of the thermal impact of diffuser discharges
can be assessed.
In the limiting
case of f
0, the solution
eq. 4.52 can be shown to possess
dimensional
free jet.
given by
the same behavior
as a two
At each x, as f0 + 0,
f L
oD
e
16H
16
(x-x )
I
Thus u (x) can be written
c
u Cx)
1
CLi3
(1+
f L
1
2
(
I)
as
fL
oD
8H
-
(x-x
16H
a 8HH
(x-xI
1/2
I32a
D
H
I6H
1 I f0LDD ClI
1LI
(4.56)
f L
I
109
therefore
lim
rn uc(X)
c( )
f +0 u I
o
I
_=_1
(1+
2l
L
(x-x )
C1L
cl II
1/2
1/2
(4.57)
I
In an entirely analogous manner,
lim
L(x) = 1 +
f +
L
o
(x-x
(4.58)
cL
I
1
revealing the linear spread of thefree jet.
4.4
Numerical
Solution
In the presence
for u
~ 0
acurrent,
of a coflowing
yi
0, the
theoretical solution to the system of equations (4.19-4.20,
has to be obtained
4.28-4.29)
section 4.2.
as discussed
numerically
In Fig. 4-3 the numerical
results
and contrasted
for y=0.04
Two interesting
(
are illustrated
L
L
for two sets of parameters
= 10,B =50; HD = 30,
= 150)
with the slack tide solution
observations
in
for y =0
can be made:
AT
a)
diagram shows the near field temperature rise
The AT
o
is less than the slack
tide case.
This is evident
as the near field dilution is greater in a
from Eq. 3.38,
coflowing current.
This leads to a more pronounced velocity field; however, the
u
velocity excess (c
-y) is smaller implying a lesser mixing
o
capacity in the intermediate field.
of Region
I is larger,
The results show the extent
and that difference
after several diffuser lengths.
in AT (x) is small
110
b)
slack
The plume width
tide case.
is much narrower
This is consistent
in extent
with
than the
the behavior
of a
constant pressure free jet in a coflowing stream and suggest
a lesser
tendency
for the heat in the diffuser
plume to
recirculate.
It should
be noted
when u (x) =u ,
relative
velocity
that the calculations
as further mixing
are discontinued
may not be governed
(u -u ), but by the turbulence
present in the ambient stream.
by the
initially
111
a
a4
eq
aJ
4
0
N
0
a)
a)o
u
a)
c:
9
' l
0
a)
C
Ti,
a
n
r4
0
It
1;
\D
0
o,4
a
ci
1o
0
I
zCY
rl1
0
4:
a
0
4
ua n
%
ci
119
0
N
N
#
It
N
z
16
0
II
I!
Q)
-C,Iz
4'J
.l
0
To~~~~~~~~~~~~~~r
-1
T/
O0
0
0
O
lif
U
0C
4
CY
113
I
0
4-
II
a
14!
mn
0
In
-H
0
II
I
U,1
4-i
0r
a
*4-.
a
"i
I-
0
114
w
Ut
.H
ra)
-·I
0C"
14~~~rDX
i3
&
sw
a,
o
U
k~
0
a
o.
xlX
X
..
a
o
o
~~k
>~~~~~~~-
.
c%
.Qj:i
o~~~ UX
C"
.xU
<~~E
O
I
O
0
o~~~
e3
C
U
0)
C
o
.,
ed
o
q
a
a
0
I
%
6
It
Id
d
l
115
O
C
r
4J
o
0
U
C_)
0
Or
.z
',
6
d
d
4
116
0
o
N4
03
0
eI
a
0
0
0o
I
4J
-W
:3
>0
w
C"
tor
I,
0
aL:
a)
cn
4 i
4-
0a
0
c
Q
-o
-,
pq
O
0a
0
117
CHAPTER V
EXPERIMENTAL INVESTIGATION
The objectives
1)
of the experimental
program
are:
To provide experimental insights into the global
circulation patterns and temperature configurations associated
with a wide range of diffuser designs and ambient conditions.
2)
To establish
a comprehensive
set of laboratory
data of
sufficient spatial resolution for comparison with theoretical
model predictions.
Details of the experimental setup and procedure are
described in the following sections.
5.1
General experimental setup
Three sets of experiments (Series MJ, Series FF, Series
200) were carried out in a 40'x60'xl.5' model basin.
artificial
floor was constructed
An
in the basin using 16"x8"x2"
concrete blocks so as to insure a completely horizontal bottom.
Discharge water at a desired temperature is obtained from a
heat exchanger and constant head tank.
water
level, water
is withdrawn
To ensure a constant
at the same rate at one end
of the model.
Currents can be generated by a crossflow system installed
at the two basin
ends.
Water can be pumped
from either
end of
the basin through PVC manifolds into a large reservoir and vice
versa.
The lateral uniformity of the crossflow is improved
118
by horsehair matting and vertical slotted weirs at the basin
The crossflow
ends.
system
can be used to mix the water in the
uniform ambient temperature before the start
basin to ensure a
All flows are monitored
of each experiment.
by calibrated
Brooks rotameter flowmeters.
A schematic
shown
in Fig. 5-1.
placed
in position
the basin.
Ibr full diffuser
tests the diffuser
B or C roughly midway
For half-diffuser
edge of one side of the basin
a plane
setup is
of the plan view of the experimental
the sides of
tests the diffuser
is placed
(position
then simulates
for the induced
of symmetry
between
is
A) which
temperature
at the
field.
5.2 Multiport diffuer design
A special multiport diffuser injection device was designed
and constructed to allow maximum flexibility in the operation
of the experimental
program.
The entire diffuser discharge system is mounted on an
6' x 2.5' aluminum
steel
threaded
framethat
rods
is supported
(Fig.5-2).
on four galvanized
To ensure uniformity
of the
nozzle discharge, discharge water is fed from a 15" long 3"
PVC pipe manifold
into one side of a large 6' long 3" diffuser
manifold through four 3/4" hoses, valves and couplings
uniformly spaced apart.
side of the manifold
Discharge water is fed on the other
to a system
tygon tubings and brass fittings.
air bleed valves
are installed
of copper nozzles
via flexible
Ior priming purposes, two
at either
end of the diffuser manifold.
119
0'
I.
I
I
=:X
-
-
X-
-
II-
-
.51*4M4, 4v"
.
a'ya
Zi
II
r.
.
.
a
.
.
I
-
ASUILAAA;11,ttt~h
A
B
TIT4
-
-
-
-
-
tlA
- -
I
LI
60'
I
t Ic It
4tsn~bc~ci~nf;
4., CCCJL~tLALR;
I CI'~
C~r/
off
fvLdf
fo say-
- -- -
,-
-.
.
-
P*
~motsd
p~cttfn
CoAcng
-;"dtet~LO
zsd/L
-
i
f
i
J
-
II
P,4
11I ,
cc
r
,4.,O
Q*RL
¢
O', rkeVA&II,
&+,CtVdAIC Grt&IrUh
TO
r POVKP
( 'O
T
tr
1.5
._
X
/
I
,
I
/
l
/
I
,
Al
/
2" Cv 1'%4
g.ea
f&
-
/
4w>n41K
I
/
/-! /1/-
Bumst1
fv,-
-FViL
S CT,
o)
Fig. 5-1
I--
Schematic
4-V,
PP
of Experimental
Setup
3" PVC
4nan4foaUd
120
3" PC
Aea
cowttcl~-m
Side View
Top View
Fig. 5-2
Multiport diffuser injection device
121
The copper tubes can be fitted between two aluminum
plates with indentations made to accomodate the tubes.
Tightening
the plates
together
fixes the position
of the
copper tubes.
5.3
Temperature measurement system
A temperature measurement system consisting of 90 Yellow
Springs thermistor probes (Series 701, time constant = 9 sec)
is mounted
at the same level
on a motorized
aluminum
platform
(vertical speed 1.5"/min.),thusenabling vertical temperature
profiles to be taken.
Nearly instantaneous temperature readings
are recorded by an electronic scanner manufactured by Data
Entry Systems (Scanning rate 100 records/min.) and printed on
paper as well as punched on tape.
converted
to computer
tape-to-card reader.
The data
cards by passing
on paper tape was
the tape through
a
The thermistor probes are individually
calibrated against a mercury-in-glass thermometer in a steam
bath at several temperatures within an expected range of temperatures.
The measurement
5.4
accuracy
of the calibrated
probes
is + 0.1
F.
Flow visualisation and velocity measurement
Rhodamine
blue
visualize the flow.
of travel method:
dye is injected
into the discharge
line to
Velocity measurements were made using a time
light
styrofoam
particles
are dropped
onto the
water surface and their pathlines are recorded either photographically or visually.
The concrete blocks that constitute
the floor form a clear grid system that greatly facilitates
122
locating
marked
the particle
at about
at any given time.
6 foot intervals
Crosses are also
on the floor with
orange
fluorescent dye.
Surface velocity measurements are made by taking time-lapse
pictures of the trajectory of a light floating particle with
a Topcon Super Dm autowinder camera mounted on a tripod
on a walkway above
the model.
placed
The pictures are taken using
high speed Ektachrome film with shutter speeds in excess of
1/30 sec.
When steady conditions are assumed to prevail (usually
about
30 minutes
from the start of the experiment)
a styrofoam
particle is carefully dropped onto the water surface at the
center of the diffuser.
When the flow starts to drag the
particle along the first shot is taken.
Subsequent photographs
were taken at timed intervals, the length of which are chosen
to reflect the magnitude of the observed velocities (smaller
At for larger velocities
and vice versa).
The distance
of the
particle from the diffuser are then measured off from the
projections
of the photos
the form of (t
Usually
(xt
or slides.
),x
x) 2 t 2 ), (xm tm)
two sets of measurements
are made
assure repeatability of the results.
great care is taken
Thus a set of data in
is obtained.
for each run to
For half-diffuer tests
to make sure the particle
stays out of
the boundary layer near the basin wall that simulates a plane
of symmetry.
123
Typical raw data recorded
Run MJ
Set
are illustrated below:
06
1
Time
(sec.)
ti
0
5
10
20
30
40
Xi (# of blocks)
0.9
3.2
5.5
10.6
15.1
19.1
Xi.(ft.)
0.57
2.04
3.49
6.72
9.6
12.1
0
5
10
20
40
60
90
1.0
3.0
5.3
10.5
18.8
25.7
33.2
0.67
1.87
3.39
6.66
11.97
16.32
1
1
Set
Time
x.(
1
2
(sec.)
t.
X
# of blocks)
xi(ft.)
1
21.08
124
5.5
Experimental
Runs
In this section
and experimental
the motivation
procedure for each series of runs will be described.
Before
the start of each experiment
the water in the
basin is thoroughly mixed to attain a uniform ambient
temperature by operating the crossflow system.
The discharge
system is primed under high pressure to drive away any air
that might be present in the diffuser system.
at a desired
temperature
is allowed
Discharge water
to run through
the bypass
system until the temperature of the discharge water is steady.
The discharge temperature is monitored throughout the experiment
by a mercury-in-glass
thermometer
inserted
into the PVC manifold
feeding into the copper nozzles.
5.5.1
Series MJ
Due to the limited size of usual experimental setups
previous studies have been unable to study the behaviour of the
It is desirable to
diffuser plume beyond the near field.
investigate under what conditions a diffuser will induce
recirculation patterns that carry heat back into the near
field.
Series MJ was performed to study the circulation patterns
induced by a thermal diffuser discharging under shallow water
conditions.
As discussed
in section
3.5, it was judged
that
for diffusers of practical interest, the velocity field would
be independent
of the temperature field.
As such, experiments
125
series were done by discharging
in the MJ
at about the same
temperature as the ambient water.
A typical
one-minute
run usually
dye injection
start of the experiment
lasts for about 45 minutes.
is made at 15 minutes
A
from the
the flow pattern.
to visualize
Subsequent dye patterns at different times are recorded.
measurements
Two sets of velocity
are made at t=30 minutes,
when steady state of the near field is usually attained.
Only slack water conditions were tested in this series.
The diffuser length as well as the nozzle spacing were kept
Only the water depth and the jet velocity were
constant.
18 runs were per-
varied to produce different flow patterns.
formed for a half-diffuser with the basin wall approximating a
plane of symmetry. Effectively
doubling
minimize model boundary effects.
with a full diffuser
diffuser
tests.
listed
in Table.
5.5.2
Series
Six additional runs were done
to corroborate
the results
The run parameters
of the half-
for the MJ series
are
5-1.
FF
In this series
discharged
the basin size helps to
of tests, heated
multiple
at about 25 to 30°0F above ambient
jets are
temperature.
Tests were made to investigate how the recirculation changes
the temperature configuration
The effect
respect
of the diffuser
to a cross current
in the near field of the diffuser.
length
and its orientation
on the overall
performance
with
of the
126
MOMENTUM JET EXPERIMENT
Series
RUN
Q
NO.
(EPM)
N
H
(in)
T
(F)
( F)
MJ
U
o
RD
0.5LD/H
1.22
1980
32.9
05
2
ft /sec
MJ04
2.32
30
0.91
59.5
(FPS)
2.66
MJ05
2.88
30
0.66
59.0
3.29
1.22
2450
45.5
M.106
2.3
30
1.21
56.0
2.63
1.28
1866
24.8
MJ07
2.88
30
1.21
61.0
3.29
1.18
2533
24.8
MJ08
1.73
30
1.21
61.0
1.98
1.18
1524
24.8
MJ09
2.88
30
0.98
61.0
3.29
1.18
2533
30.5
MJ10
1.73
30
0.97
60.0
1.98
1.20
1499
30.9
MJll
2.88
30
0.77
57.0
3.29
1.26
2372
39.1
MJ12
1.73
30
0.73
59.0
1.98
1.22
1474
41.0
MJ13
1.73
30
0.68
58.0
1. 98
1.24
1450
43.9
MJ14
1.0
30
1.57
64.0
1.14
1.14
908
19.1
MJ15
.0
(
30
1.33
67.0
1.14
1.10
941
22.5
MJ16
1.0
30
1.07
66.0
1.14
1.1
941
28.1
M.;17
2 .0
30
1.08
58.0
2.28
1.24
1670
27.8
MJ18
2.0
30
0.66
61.0
2.28
1.18
1755
45.5
MJ19
2.0
30
0.98
65.0
2.28
1.12
1849
30.5
MJ20
1.0
30
0.98
62.0
1.14
1.16
893
30.5
MJ2]
2.0
30
0.6
60.0
2.28
1.20
1726
50.0
*MJ22
0.8
30
1.43
62.0
0.91
713
10.5
*MJ23
1.0
30
0.66
62.0
1.14
1.16
1.16
893
22.7
*MJ24
1.0
30
0.84
69.0
1.14
1. 06
2.0
30
0.72
66.2
2.28
*MJ26
4.0
40
1.13
67.1
3.42
1.10
1.10
977
2060
17.9
*MJ25
3091
17.7
*MJ27
1.5
40
1.13
67.1
1.28
1.10
1.157
17.7
Table
5-1 Run
s, nozzle
parameters
spacing
xlO
for Series
20.8
MJ experiments
1.C"
D, nozzle Diameter = 0.109"
N, No. of nozzles
* RUNS WITH FULL DIFFUSER IN MIDDLE OF BASIN
kinematic viscosity of water at discharge temperature
v0
=
R = Reynolds number based on jet diameter
D
uo D
V
0
127
heat
dissipation scheme was also studied.
Runs FF02-29 were half-diffuser tests.
of the temperature
the probe
sensors
spacing
The arrangement
for these runs is shown in Fig.5-3a,
is of the order of the diffuser
F1 30-36 were done with a full diffuser
placed
length.
Runs
in the middle
of
the model and discharging perpendicularly with respect to a
current
( Fig. 5 -3b)
As these
tests are designed
to study the recirculation
of
heat into the near field, the experiments usually took a little
over two hours, when it was judged that the model boundaries
had significantly
distorted the test results.
Temperature
scans
were made every ten minutes from the start of the experiment.
Dye patterns and path lines are recorded and velocity measurements are made.
Fr
crossflow runs, steady state is usually
achieved after about forty minutes, when temperature rises
change by only fractions of a degree.
listed
in Table
5.5.3
Series 200
The run parameters are
5-2.
The objective
of the Series
200 experiments
is to provide
a data base for comparison with the theory presented in
ChapterIII and IV.
dimensional
All previous experiments on the three-
temperature
field of a unidirectional diffuser
in an
open, shallow receiving waterbody employ a temperature measurement grid much larger than the nozzle spacing.
Because these
tests are mostly either aimed at testing plume behavior on a
large scale (e.g. far field recirculation) or providing global
128
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Run Parameters for Series FF experiments
0
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131
information
basic
(e.g. maximum
structure
temperature
of the temperature
rise)they
do not reveal the
field to a degree
that can
be used for quantitative comparison with theoretical
predictions.
resolution
In this series
of experiments,
of the temperature
sensors
spacing close to the diffuser.
region,
the spatial
is equal to the nozzle
Beyond the near field mixing
the spread of the plume is taken into account,
the sensor spacing is adjusted accordingly.
measurement
grid is shown in Fig. 5-4.
and
The temperature
Vertical
temperature
profiles are recorded along the centerline and across six
transverse
sections
of the diffuser
plume.
Altogether
10 full
diffuser tests were performed under stagnant ambient conditions.
The diffuser
No velocity
presented
is placed
in position
measurement
is made.
in Table.
As the limiting
B as is shown in Fig. 5-1.
The run parameters
are
5-3.
case of long diffusers
with
strong
potential for heat recirculation has been treated previously,
this program concentrates on relatively short diffuser (LD/H<10).
This enables the study of the temperature field for many diffuser
lengths in theexperimental setup.
Furthermore, as will be seen
later, these short diffusers are characterized by large scales
of recirculation (weak recirculation effects).
detailed
temperature
field can be observed
Hence, the
in the absence
of
heat recirculation
A typical
result
begins
run usually
takes about 30 minutes
to be distorted
by heat recirculated
when the
due to the
132
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134
the model boundaries.
Steady state is usually attained in
less than 10 minutes.
5.6
Data reduction
Temperature Data
For each run the ambient
the average
before
temperature
of the recordings
excess
is defined
of each probe is defined
(xi,Y i) is the co-ordinate
as
sensors
At each time instant
AT(xi,Yi,t) = T(xi,Yi,t) where
a
of all the temperature
the start of the experiment.
the temperature
T
as
T(xiYi,0)
of the i th probe.
The near field dilution S is inferred from the temperature
data as
S
AT
AT
where
T
T -T
o
o
a
T
max -T a
max
is the maximum
surface
max
absence of any heat recirculation.
temperature
recorded
in the
Velocity Data
The reduced velocity data are obtained from the photographic measurements
in the form of
(x ,to), (xl t1)...(xnt
where
x
is the distance
)
of the particle
from the diffuser
time ti
time
t..
1
_
___
__
at
135
We assume
that the centerline
can be inferred
velocity
from the trajectory
variation
of a straight
u (x)
c
pathline
in
the following manner.
An average velocity is computed for the interval
(xi,xi+l)
u(-xi+xi+l
c
X i+l-Xi
t
2
i+l-t i
The error in locating
be
x = 0.4x
width
a particle
of a concrete
The error in recording
t
1
can be estimated
block
=0.25'.
can be regarded
for the range of velocities being measured.
error
in measuring
the velocity
xi+l -xi+6x
i
6u
as negligible
Thus, the relative
can be estimated
as
x i+l-x
i+l i
At
u
to
6
At
x
xi+x
X i+lAt
For the record
measurement
illustrated
in section
can be estimated
in the first interval
5.4, the error of
to be 18% for the average
and 5% for the last interval.
velocity
136
CHAPTER
VI
COMPARISON OF THEORY WITH EXPERIMENTAL RESULTS
In this chapter the theoretical predictions presented
in Chapter
III and IV are compared
with experimental
results
of this study and results reported by other investigators.
Experimental observations are also described.
are made for runs that do not exhibit
the near field.
This
The comparisons
heat recirculations
topic will be separately
discussed
into
in
a later chapter.
6.1
Near Field Mixing
As indicated in section
3.5 , vertical temperature
measurements near the diffuser show that the temperature is
vertically homogenous, demonstrating the flow is vertically
fully
lengths
behind
mixed.
LD
(
Surface pathlines for a wide range of diffuser
LD
3
is concentrated
the toal induced
The diffuser
concerned.
100) show that the induced
flow from
in a contracting
that
flow from the ambient
stramtube
fluid.
(Fig. 6-la).
acts as a sink as far as the external
The contraction
occurs
'separates'
flow is
at about 0.5 L D from the
diffuser and the contraction coefficient is observed visually
to be about 0.5 for the slack tide case
(cf. Fig. 3-2b.) Dye
injection patterns usually indicate a contraction coefficient
greater
than predicted
by eq. 3.40 as shown in Fig. 6-lb.
experimentally observed value of the near field dilution
The
137
u =0
Run FF13
a
surface
pathlines
diffuser
Run FF19
-30
Ua
scale:
.5 feet
2.5 feet
/
/
/
Fig. 6-1 a)
/
/, ,
/
/V
Recorded surface flow patterns for a unidirectional
diffuser in shallow water; with and without current
/ff
138
I
.
ST"Ctgl-1z.r
Fig. 6-1 b)
is
taken
Dye injection pattern in the near field
of a unidirectional diffuser
to be
AT
S
=
(observed)
-
ATMax
is the maximum surface temperature rise recorded
where ATMAX
MAX
in the near field
i.e.,
AT
max
in the absence of any heat recirculation.
= max AT.
1
AT.
being the temperature excess recorded by the ith probe .
1
In Fig. 6-2, we compare
given by eq. 3.38 with
is obtained
for a wide
the theoretical
the experimental
range of S.
predictions
results.
of S
Good agreement
It should be noted the
relative error in S (observed) introduced by temperature
measurement errors is greater for larger dilutions.
139
-.
r-.
i o,i
a)
CZ
(D
CJ .H
co C_
M C4 ,1
mn
r--r-- 4
~,,
L)
9n
G
U) U]
m
on r
'4
"U
cn
0
4-i
5
4
"U
.H
r
0a)
0U
'U
4
0
14
Q
a)
z
O 0
4
'U
"u
"U
1-a)
a)
4J
O
0
4
Oj)
U1
W
zrl
4-0
'H
a)
Ga)
c\
0 A4
Q)
u
'IOa)
a) 'H
z
co w;
am
40
riI
;0
140
6.2
Comparison
of theory with data:
In Fig. 6-3, the predicted
the observed
surface
Series
isotherms
temperature
FF, MJ
are compared
with
field for the FF Runs of a
unidirectional diffuser in a coflowing current with no heat
recirculation into the near field.
are shown
to within
The recorded temperatures
0.1 F on the diagram;
the decimal
point
locates the temperature probe relative to the diffuser and
the figure
is drawn
to scale.
For each run, the system of
equations 4.19-4.20, 4.28, 4.29 are solved numerically for
the set of governing
parameters
friction
is assumed
coefficient
laboratory experiments.
LDH
B=N
u
foLD
Y= a and
H
The
~~Na
'
and
H
.The
o ~
o
to be f = 0.035 for all the
0o
The solution yields the centerline
velocity u (x),the centerline temperature excessAT (x), and
c
c
the plume width
L(x).
At each x, knowing
AT
and L(x), the
c
position of a particular isotherm can be calculated using
the assumed temperature profiles given in section 4.1.1.
Both the quantitative and qualitative features of the model
are well
plume
slight
supported
is larger
by the data.
In general
than predicted;
stratification
observed
the spread
of the
this can be attributed
to the
at the edge of the plume in
these experiments.
In Fig.
6-4, the prediction
of centerline
velocities
are compared with measured surface velocities for runs in the
FF and MJ series.
described
The method of determinirg the velocity was
in section.
5.6.
The agreement between theory and data is
141
'N
FI1l)
v:o.oo76
/ =77
4r-I
4
3 .6
4F
svrnmmt r
9-
3
2
4.'
a r-_ .2. 5 1~--'t'
,51:
,F--- 3 Z
2 2
3 4
-,
I-
3- S
3 4
-
2-
0.0
0.1
o .
o0.0o
1i n
-
o .3
o '3
o --
o4.
0I
o-2
o-.3
I
3
1-I
2.5 Fr.
5 cJ.
iUN
0
,. =!1o7 ,I---o.o7oO
.1(
ar,3..
2.6
.
F
-2.
o
1.4
(.6
0.0
24 - 5 ,
o.66
a-I
*.1
r,, o.oq
1/1
6
.6
o0
f
4
_--
.i
syvmmet rv
8
2.3
Z..
I .5
22
21-
o.I
0.0
.
0.
0.1
0 *C
0.0o
o
o.)
I in
.
2.0
3
-
o.I
-
2.4-
2 3
.7
f4T
2
o oo
0.0
0.1
0.0
RI'N I FO)8 /9
line
2.
2.7
-s 2
2
±.-o
2.7
2.7
.
symmetry
o
00 .1
.
0 1
0.0
O.2
o 2
6
0.1
o.
.
o .5.o
-e
0 1
Of
.o
Fig.6-3: Comparison of predicted isotherms
with temperature
measurements in Series FF ( eq. 4.19-4.20, 4.28-4.29 )
142
Rt'N
1.()
-
921/I
a- 0.oog7
L9
-
E- z Y-1°F
--'
IF _-2 1. o.I
O-1
O.1
.S
=3
svmmetr' i i nc
1'
2.=
o.
O.-
o.
0o-2
o.
o0.
0.
OZ
o.
2.5
F.
Sc.*J.L
r o
/ -
RtI ' 1
2.7
:=
26
symmetry line
2.3
-5
.3
.5
'1 2
07 . o-6
14
0.5
o.3
oC.
o.-
1.0
Oq
0-7
1.2
1-1
c.q
o .
0-
0.3
1
-
- _ -- tr
_
L,
RlUN
60G
=
i;",.
mL
=
C.FO
=_
o0.1
I · 6,
I.Z
I
I/ L
0-6
°
0.7
0.2
0.2
0.2.
0./
0 2
o 4
o.2
o 3
0.2
O-Z
.
0-
o.
0o2z
o- 2
o.o
72Z
0.1
Fig. 6-3 cont 'd:
/L
11_
_
_I
I
I
_
1X
143
I-11
/4: 161
0
_0TTZ-o F
vnmeetrv
2-4
2 5
2-S
0
t.o
0.7
04
*.
o -3
o -3
o .4
o .2
0o3
.
.
_.
o 3
ITINFF 14
23 2-2 -. .
/-5
O -5 05
o
o
0.4-
o-S
ro.oo75
j/I,
4=
2.. 12
f'AT
5 3D/2
~~
4T~<.
0..
JI NF 1p:jc
/A x q
FTr.
Scae,
symmetry
-
2-1
line
.-.........
T- I\
o4
o-3
0-1
0.1
o.2
o.3
o.3
O .
O -2
o-I
O.
O
o.z-3
o .2
O.2
o0-2
0.2
0 .2
O.
o -I
o.
o -Z
o- 2.
LK
t
2.5
z
k - O.oo6f
L
*
0.1
= 26.6
2.3 F
I-~./1-
3
linc
o-J
. 72.4
symmetry
3
in
3.2
l
0.2.
2 2arr
-6
o.-g
o6
O.-
o.o0
00
6.1
O.2
o.o
0.o
o. i
-
o .2
O-Z
o 4-
I
Fig. 6-3 cont'd:
o-3
-02
.3
o. 3
144
fairly good.
For some runs there is scatter
in the data and
the recorded velocities are lower than predicted values;
this may be due to surface
tension
effects,
limitation in the time-of-travel velocity
an
intrinsic
measurement
technique at low velocities.
RUNFF 2
Y.o3
0-0
Z1
-=z
~r,~
1~
'F ~~I/I
I.q
Ab0
I.
1
FT~~~~~~ ~~~
.- / 1sqf
0
6-2
0.a
Fig.6-3 cont'd:
a.2
symmetry
line
/.q /.q
,.q
. 'w
2. o
2.'°
2,4
o-3
0-3
O2
*.Z
o.3
0~~~~3*3
0.3
916
o.
s
l0
2.0
\
_
2. 5' Fr
Sca.e,
145
I-.
CU
cU
4-)
0
-.
rdq
04
!> E
d
O-
eB
r.
C
_
"io
r1
_,
o HU
o!
.R*
__
E
a
0
I
o
0
I
9
1
.I
60 ° 0 I
t
*
o
o
o
o
0
o
I
e
e
6
I
°
to
0
a)
rli
4)
zI
0
O '
aU)
*r
!
'o
mt
w~a
O
4 la0
- ql '-1:1
rHtO
15
m
la
_
!d .-z.
l/
U)CD
t
o
ZoQ
O
a
II.
.p
^l
IAt
/
ot
:
N
i
H
0-
i°
i
._
-.
I Q
*
o
1_
r,
6
C1
6
LI
V
0
o
6
I
C)
I -L
a
0
i
o
Cs
.rd
146
I
lo
I
-
1
z"II
a
R
/
~~~~~~~~~~~~~~~~~~~~~~I
i
~~~~~~~i e
4,
on
/
Z,
.4
Q
..
1
,.
4
i
19
4
/
6
*
I
0 *
o
.
0
4
~~4~~~~~4
0
06
0
40
44
4
I
s
0~d
ok
6
Id
0
44 -
~w
4"-
V0
I
Iz
w~~~~~~
I
It
0.
o
1°
A
i~~~~~~~~
Q~~~~~
<
X
I
d.
'0
Q 44 -
_a
/4
/.
0
0~~~~~~~~~~~
/
as
2:It:
a
e
I
t
6
4
tb
_~~~~~~
~~~~~~
_
-;
6
Z4
0
0
Z
6
V
1
*
I$
-H
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
___
-,q
j
_1111^1_-·-·11111s---I
147
?
I
I?
a Io
"
*t.
.
.
s
8IqI
z
B
0-
°
0
6
e6
i6
I
0
6o
o
0
a
T
v
t
16
6
6
d
,,
.4
Al¢
ip
.
1*
a-
'i 1
I 11.
.
Ijam.
11. &
4
I
I
4)
4
0
4
:
0
0
a
_
a
°'
t
Is
o
Ii
:
'u l
01
I
6
16
.
-rI
148
6.3
Comparison of theory and laboratory results:
For this set of experiments
the plume
is symmetrical
(Fig. 6-5).
Beyond
dye injection
for a distance
Series 200
patterns
show
of approximately
this point the plume
meanders
40 H
and some-
what loses its shear structure.
In Fig. 6-6 the theoretical
4.52 and 4.53 are compared
data.
with
predictions
the series
of eqs. 4.44,4.47,
200 experimental
Both centerline temperatures and areas within surface
isotherms
are illustrated.
(Details
of the computation
isotherm areas are included in Appendix C).
of
The centerline
temperature rise normalised with the predicted near field
temperature rise, AT/ATN, is plotted against distance from
the diffuser x/L D .
At every x, the different symbols
represent temperatures recorded at six vertical positions
from just beneath the free surface to near the bottom.
the vertical
spread
of these points
is a measure
vertical homogeneity of the thermal structure.
Thus
of the
For a region
close to the diffuser, where the temperature field is strongly
three
dimensional
(the
individual
submerged jets have not
completely merged with each other and the flow not yet
vertically fully mixed), the
model is not valid.
prediction of the two dimensional
Beyond the initial jet mixing zone,
however, good agreement is obtained.
The data clearly shows
a shorter diffuser mixes more efficiently in the intermediate
field.
In general the vertical scatter of temperature data at
any point
is less than 10% of the mean, suggesting
_
__
a 2D
__
149
approximation is sufficient for practical purposes.
plume
-- diffuser
Fig.
6-5:
Dye Injection
Fig. 6-7 shows
Pattern
of Diffuser
the comparison
Plume:
of predicted
isotherm areas with the series 200 results.
is obtained.
induced
nature
close
rises.
of the model.
to ATN
surface
Good agreement
This can be attributed
The area of the initial
AT <ATN ) is of the order of LH,
N
D.
a substantial
200
Ingeneral the model overpredicts for the higher
temperature
(in which
Series
portion
of the predicted
in the short diffuser
jet mixing
zone
and can account
isotherm
tests.
to the 2D
for
areas for AT
__
Cl
150
L-D,'H=
.4
In
4
.
AITC
AT,
I
I
I2
L
4
I
Fig.6-6: Comparison of predicted centeyline temperature rises
with eries 200 experiments
eq. 4.44-4.47 , 4.5 2-4.53 )
_
III
I_
_·_
____I
_·
151
. Li
;D H
BI. '3
'Z/H
I.
)
0.89
0.76
3.64
0.52
ATC
0.40
0.28
ATN
0.
Jj
LI
+4
u1
*
<n
<.
C,
>
A
0.
flY
-A
IL
w
A
w
*
0
0. 4
H
-J
iof
*
o
A
0
PI
0
*
o
*
w
F-
rn
0
z
i._i
u
I
C
*
o
a
+
I
10
i5
DISTANCE FROM DIFFUSER
C
RUN 200.C
20
X/LD
I
25
,
58
,D/H=
O
0
H
9
7
5
3
9
*
+
*
w
>
0
i.-Un
4
C
H
.a.I; 0.
+
4.
:
*_
O
C
~~~
N
*
C
;
z
-J n
z
Urjc"
U.C
RUN 2d04,
Fig.6-6 cont'd
10
DISTANCE FROM DIFFUSE.R
15
K/LD
.
152
.
D/H--
I
Z/H
ATc
A AA
·.TN
[1
1.
4
7
9
A
4
~n
X
©
L'-
0.
.
A
u
Li
4
4
4
Z.
4
-J
o
,J
2'
u
7
L
L
o
+
L
.
2
I
iD
15
DISTANCE' FRM
RUN 205,0
25
3D
K/LD
DIFFUSER
,_D/H= 8.
0
0
§
cC
&TN
0o
O.B
Ld
ul
A *4
A4
F-. 0. G
.J
-I~~~
+
*
+
o
1.0
X
"-4
--0 0
~-
0.89
0.76
0.64
0.51
0039
0.26
Q
10
-x
T---
0
4
A
C>
+
+
4 _
Z /H
*
w
I--
7
w
IJ n
u.
I
-O
A
n e
IC
1
RIJ
"Iu_
Fig. 6-6 cont'd
I
4
1
h
S
DISTANCE FROM DtFFUS'R
h
1
10
X/LD
14
o
153
I
* 0.89
+ 0.77
El 0.64
& 0.52
O 0.40
o 0.27
H
89
76
64
51
25
Fig.6-6 cont'd
.
I
-
-
E D)IH=
2
C
Z/H
0.89
0.76
0.64
0.51
0.38
0.25
T.
u
L
L~
Fig.6-6 cont'd
6.4
Detailed thermal structure
In Fig. 6-8 typical transverse surface temperature
profiles at different longitudinal positions are illustrated
along with the theoretical predictions.
The surface centerline
temperatures indicate a constant temperature rise plateau
region
of % 3 LD,
confirming
of a zone of flow
the assumption
In Fig. 6-9 the cross-sectional
establishment.
The temperature
temperature profiles are also illustrated.
measurements
mixing
show that beyond
the details
the diffuser
of the diffuser
length L
3D temperature
the region
is the correct
distribution,
of
design
transverse
near field
are not important
length scale.
as influenced
and
The exact
by the merging
and
155
-
I
AT
50
5.0
5 0 As
1.0
4.0 I~)
4.0
O.8
o61-
3.0
04 _
aT 0.6
G.2
...
10
RUN 201
__
_
100
AREA WITHIN SURFACE ISOTHERM
..
2
7
0.4
10
5
30
AT.
20
2.0
AT
3.FI
O
0.
F
., ,I..
500
5
A}H
.
I
RUN
202
RUN 202
00
AREA WITHIN SURFACE
~
AREA WITHIN SURFACE ISO3THEI,
.
.
2
500
A/H
5.0
I
--
I
SO
50
_
0.6
40
0.6
3.0 Ar
40
0.8t
I:
AT 0 6
3D
-
AT
AT
AT04-
20(F
04._
0.4
0.2
O..
02
_10
,_ . ..
5
. _1_.._
..1 ._...---L.._-.
_i .._,._ .__. 1 . ....
10
100
RUN 203RN23 AREA
AEA WITHIN
ITHIN SUAFACE
SUFACE ISOTHERM
ISOT-ER-
A H
Al/H
E
10
80
.-..
5
.
.
0
RUN 204
..
I
SURF00ACE
WITHIN ISOERM
00AREA
A
AREA WIITHIN
SURFACE
ISOTHERM
l
L I.._l.,..1.
5
A04
1.01
-40
4.0
1.0
,T-0.46
0.8
3.0
AT 0.6_
A7e
1.0
0.2i
II
R0
RUN 205
.
I
,
I
100
AREA WITHIN SURFACE ISOTHERM
Big. 6-7:
....
I
A/H 2
500
J
1
I
.6
2.0
ATO.
.
0.4
.
02
,
.
...
10
RUN 20T
I0
. .
.
.
100
AREA WITHIN SURFACE ISOTHERM
.
11.0
.........
50
A/H
Conmparisonof predicted isotherm areas with
Series
200 experiments ( eq. 4.44-4.47, 4.52-4.53 )
156
50
5.0
4.0
4.0
'.C
$.0 I,T
3.0
AT
{*F)
zo
0.1I
3.0(ITAT
I12.0
0.G
A
T
0.4i
I
0.84
_1.0
AT 02
1.0
0.24
0.2Z
_
5
10
RUN 20O
100
AREA WITHIN SURFACE ISOTHIrfM
500
A/H
i
5.0
0 ,-\
40
53.0
A
AT
AT
1.0F)
.0
0.2
I
5
2 _
RUN 210.0
S. C
A.
.
Fig.6-7 cont'd
50
I00
ISOTHERM
AREA WITHIN SURFACE
A /H
0
20
O
RUN 208
,
,
I
.0_.
00
.
E00
AREA WITHIN SURFACE 'OTHERM
A/H2
50_,0.............
L-__
0__
500
157
A
1.0
A
A
A
a
"
' '
0.t
arc
srl .6
..
-
_.
A
&4
A
p
o.
2.0
3.o
*o
I
p
~~~~~~111
f.
.o
8.o0
o
?.o
10.o
Fig. 6-8a) Typical centerline surface temperature profile
along diffuser axis ( RM
M
ef)
and bending of the individual jets, surface and bottom
The temperature is more
interaction, is very complicated.
homogenous
vertically
at -the center
than at the edge of the
The flow becomes more stratified with increasing x.
plume.
that the low Reynolds
setup
the degree of stratification
enhances
in the experimental
number
It is expected
and that in the
prototype the flow will have a stronger tendency to stay
fully mixed.
It cannot be inferred from the temperature data alone
that the temperature profile is well predicted by the standard
profiles
(top hat or gaussian).
However,
for
158
I,,
I
-Ip
r-4A
Co
CM
X
4-,
-
4
pd
QI)
HI)
r
0
1
I
41!
Q
-0
o
o
U)
H
a) r.
a)
,4
LV
.
w
<M
q X
_
I
Q
<3
C.
r1'
*_
j b
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Fig. 6-9: Cross-sectional temperature transects at various
longitudinal positions (
predicted profiles )
_
160
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Fig. 6-9 cont'd
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Fig. 6-9
C0
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TR,NSVJP
cont'd
L' L'?.
it
PRDFILE'
2 0O
V/LD
2.9
162
purposes of thermal prediction, the assumption of standard
profiles appears to be a valid way of describing the essential
features of the field variables.
The temperature
recordings
in the series
200 experiments
are taken with a Digitec (United Systems) scanning and printing
unit;
each scan of 90 probes
takes about three minutes.
Thus
the cross-sectional temperature profiles at six vertical
positions are measured over an interval of about 20 minutes and
as such these measurements
do not represent
a truly instantaneous
snapshot or time-averaged values (over a long time interval)
of the thermal field.
However, the cross-sectional measurements
are taken after the surface temperature rises indicate steady
state conditions, and duplication of several runs demonstrate
the experimental results can be reproduced.
163
6.5
Comparison of Theory with Perry model study
In this section the theory is compared with the results
of hydraulic scale model studies for a proposed diffuser
discharge for the Perry Nuclear Power Plant on Lake Erie.
An extensive
diffusers
for this specific
site has been carried
in this report
isotherms
are presented
as illustrated
the temperature
out and
The experimental
reported by Acres American Corp. (1974).
results
of unidirectional
set of tests on the performance
in the form of surface
All
in Fig. 6-10.
field used for comparison
on
information
are reduced
from
The relevant data are the unidirectional
these diagrams.
diffuser tests covered in plates 10-63 in Ref.3.
The centerline temperature rise along the plume and
isotherm areas are compared with the slack tide theoretical
predictions
in Fig. 6-11, 6-12.
The diffuser
design
and
ambient conditions are indicated on each diagram. For each
LDH
design, the near field parameter = Na , and the intermediate
field parameter
Eq. 3.41;
4.44-4.47,
foLD
H
are computed;
the intermediate
4.52-4.53.
0
AT N is then given by
field solution
It can be seen that the data shows
the same type of trend as observed
in this study,
comparison shows satisfactory agreement.
jet mixing
region
of
is given by eqs.
10 H is indicated
and the
An estimated initial
on each plot.
It
should be pointed out that many of the Perry tests were
done for rather
short diffusers
and as such the initial
jet
164
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___
165
mixing region can extend for several diffuser lengths.
Furthermore,
the assessment
the nature
of the data at hand do not permit
of the spatial
resolution
of the temperature
measurements and hence any maximum temperature rises that may
be missed
by the sensors.
Only tests in which the near field mixing is not severely
affected
by the cross-current
are included.
In some test
with strong crossflows the near field temperature rise is
considerably higher than predicted; the effect of an ambient
crossflow on diffuser performance is separately discussed
in the next chapter.
From eq. 4.52 it can be seen the
2
for isotherm
areas A/L D vs AT/AT
intermediate
field parameter
D
N
functionalrelationship
is dependent
f LD/H.
only on the
In Fig. 6-13 all the
Series 200 experimental data and Perry test data are condensed
on the same diagram
predictions.
and compared
with the theoretical
166
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173
6,6
Comparison of theory with field data:
James A.
Fitzpatick Plant
In this section the theory is compared with field data
on the induced
temperature
field by the thermal diffuser
-:
discharge of the James A. Fitzpatrick Nuclear Power Plant on
Lake Ontario.
The
hydrothermal survey was carried out and
reported by Stone and Webster Engineering Corporation (Ref. 29).
Dye at a monitored
tracer.
rate is injected
into the discharge
flow as a
Meterological and hydrographic conditions are recorded;
both temperature and dye concentrations are measured at various
levels below the free surface.
The plant is located 3500 feet eastward of Nine Mile
Point Nuclear Station, another station employing once through
cooling.
in about
The diffuser discharge is located 1200 feet offshore
30 feet of water
direction.
and discharging
(Fig. 6-15) The diffuser
in the offshore
consists
of 6 double
ports
(included angle 420) each spaced 150 feet apart; the design
values are
L D = 750 feet
u
=
14
fps
H
=
30 feet
Q
=825
D
= 2.5 feet
cfs
The site is characterised by a relatively strong offshore
slope of 1/50.
174
As an ambient stratification of 3-5°F was always present
during the temperature survey, it is difficult to extract
information about diffuser performance from the temperature
pattern, which is also affected by the discharge from the
Nine Mile Point Station.
The induced excess temperature AT
is, therefore, inferred from measurements of dye concentration.
For this diffuser,
submergence
the densimetric
of the equivalent
Froude number
and
slot jet can be calculated
by
conserving momentum and volume flux per unit length along the
diffuser.
Accounting
for the angle of inclination
of the
double ports to the offshore direction, we have
B.
s = u
B
giving
cos
()
= 13.1
fps
o
= 0.0903
ft.
F S = 128, H/B = 332
where 8o is the included angle between the double ports.
1
The stability
criterion
shown in Fig. 2-7 indicates
the
design falls marginallyin the vertically fully mixed region.
In fact vertical
beyond
transects
a small region
near
of dye concentration
the diffuser
show that
the flow is always
stratified.
The predicted near field dilution
LDH
Bi
S=
2Na
well with
0o
cos (
-) = 13.3.
the observed
concentrations.
as given by eq. 3.41 is
This value compares reasonably
value of 10, as inferred
from dye
175
In Fig. 6-14, the stagnant
prediction
for isotherm
area (eq. 4.44-4.47, 4.52-4.53) is compared with the field
data for June 13, 1976, a day marked
(X0.05 fps).
The data is recorded
(8 a.m. to 4 p.m.).
The different
by very weak currents
over a period-of
symbols
8 hours
refer to measurements
at different levels below the free surface.
The discrepancy
between theory and data, especially for the lower temperatures,
is about
the same as the scatter
in the data.
In Fig. 6-15 a,b the predicted
isotherms
are compared
with the observed excess temperature contours for two different
times; the lateral plume spread is larger than the predicted
cigar shaped isotherms.
were carried
Unfortunately, no plume measurements
out at a distance
beyond
2000 ft. from the diffuser
line.
It should be pointed
by a low momentum
input
into a strong
such it does not strictly
outlined
in Chapter
out that this design
meet
is characterised
sloping bottom,
the assumptions
III and IV; the comparison
and as
of the model
above, however,
still shows fair agreement regarding the approximate position of
the diffuser plume and the isotherm areas.
1 7-
i1
-
C
Cl
-1
I
'"C
ct
0
C
H )>b
~
~~~~~
Q
"0
.. Q
,4Z
0~~
I)
H
-i
U
OU
oCD
CJ
Ho N
0 I;
~D
g3
6
o
0
aC
E-d
H1
0
177
I
I
A
II
I
I
I
I
I
I
II
I
'd
of
I'
.
1,
II t
Condenser flow 835 efs
Discharge temp.rise 30 F
30 feet
Water depth
Discharge velocity 14 fps
II '. s
t
i
,
I
,
4I
I I,
I
I'
I
II
I
II
I
I
II
1(
I II
I
I
i
LAKE
ONTARIO
/
current
0.05-0.1
shorelii
ps
depth
Fig. 6-15 a)I Comparison of predicted isotherms
: James A. Fitzpatrick
with field measurements
Power Plant
178
0\
//
i
II
eVf
' ~~k~~~~~~~~'.
\
,k,
,,
\
I
r,
,
K'
\ ,
t
,~~~~"
.i
Condenser
f low
li.scharge temp. rise
Wattr depth
I)ischarge v lot ity
835 cIs
/.
30 '
30 ft.
14 fps
0.0
LAKE ONTARIO
<0.05fps
shiorel ine
power
pth
Fig.6-15
b)
-
r---------
-
with field measurements
Power Plant
- YV
-
I ..-
- -
-
-
-
: James A. Fitzpatrick
179
CHAPTER VII
UNIDIRECTIONAL DIFFUSER DISCHARGING INTO A PERPENDICULAR CROSSFLOW
The temperature field induced by a unidirectional diffuser
discharging
chapter.
into a perpendicular
crossflow
is treated
This situation is often found in coastal
predominantly alongshore currents.
in this
regions with
A theory is developed for
calculating the total induced flow from behind the diffuser and
the plume trajectory.
and the Perry model
of the theoretical
7.1
T-diffuser
The experimental results of this study
tests are analysed
and interpreted
in light
prediction.
in a crossflow
Fig. 7-1 shows the diffuser
plume in the presence
of a
perpendicular crossflow in an open waterbody; a unidirectional
diffuser discharging into the ambient current at right angles
is often called
a Tee-diffuser.
The plume is deflected
by the
dynamic pressure of the intercepting stream; eventually the
plume will be swept in the direction
In contrast
of the ambient
to a round jet which allows
the ambient
current.
flow to get
around it, the vertically fully mixed plume forms an obstruction
to the crossflow;
the blockage
deprives
the lee of the plume
of entrainment water from the ambient crossflow - this can, in
some instances, cause recirculation ofheated water(Boericke,
1975).
180
-l,
1
a
4
(x,y) : Cartesian co-ordinates
(s,n) : natural
co-ordinates
: inclination of plume trajectory
u
: mean velocity inside plume
Fig. 7-I
Unidirectional
diffuser
to x-axis
in a perpendicular
crossf low
In the following
field dilution
sections
a prediction
and plume trajectory
T-diffuser is presented.
fully mixed conditions.
for the near
of a trajectory
of a
Both analyses assume vertically
181
A
I
I
_
-
l
IIl
II
I
-
i
I
I
zke
9.
i
I~~~~~~~~~1
.
_~~~~~~,
I
I
I
%
I
I
I
I
I
\
\
\
Fig. 7-2
7.1.1
'C
Near field dilution induced by a unidirectional
diffuser in an inclined crossflow
Near field dilution
of a unidirectional
diffuser
in an
inclined crossflow
Fig. 7-2 schematizes a unidirectional diffuser aligned
at an angle
of
a
a to the ambient
=0 corresponds
By carrying
a control
current
u .
The special
to the case of a perpendicular
over the same assumptions
volume analysis
stated
is used to derive
crossflow.
in section
S
case
3.6,
the near field
dilution.
As in the case of a coflowing
is negligible
section
turbulent
mixing
i be the section where
current,
we assume
in the near field,
the free surface
there
and let
returns
to the
182
undisturbed
level; u
the flow at section
is the velocity
1
i;
in the direction
a. is the inclination
1
direction to the horizontal.
ABCD is an
volumejust enclosing the diffuser.
clature
as presented
in Cahpter
of
of the flow
outer control
Using the same nomen-
III, we have for energy
conservation along a streamline,
2
2
u
2g
+ H = h
2g
+
q D
2
2
u
+
qD
+ H =h
+ 2g
-2g
where
(7.1)
.
2g
qD is the magnitude
(7.2)
of the velocity
along the diffuser
A momentum balance for the inner control volume A'B'C'C' gives
gLDH(h
gL
H h
D
Equating
+
~~~~0
h-) = Qo
ou
h
-
momentum
fluxes
(7.3)
in the x and y direction
for the
outer control volume ABCD, we have
SQ u.sina
o
SQ
o
u.cosa.
1
= Q
1
=
i
o
+ SQ u sina
o
0
SQ u cosa
o
a
Combining
eq. 7.1 through
equations
and an equation
obtained
S
a
(7.4)
(7.5)
a
7.5, ai. can be eliminated
1
in
the near field dilution
from the
can be
as:
ysina
-
a
S
where
LDH
B
a
Na
Na
=
0
u /u
ao
2
-
0
(7.6)
183
the solution
is
Y sin
S
2+
=a
+
2.
a
(7.7)
2
Using a different argument, Adams (1972) obtained the
same result by directly
replacing
y by ysinaa
a
For the special
case of a perpendicular
in eq. 3.38.
crossflow,a
a
=0,
eq. 7.7 then gives
s =/
(7.8)
2
The induced near field dilution in a perpendicular crossflow
is then the same as the case of no ambient
current.
In the2foregoing analysis, a crucial implicit assumption
Uu
U.
u.
i
a
is is
thatthat2g
2 << 2gI
-1,
so that any correction due to the ambient
current in the energy equation or the momentum balance equation
can be
u
AH=
a
2
29
eglected as a first approximation.
due to the blocking
The head difference
of the ambient
stream by the
diffuser plume, provides the necessary centrifugal force for
the bending
of the plume,
and hence it cannot
the calculation of theplume trajectory.
in the equations necessary
small.
For example,
written
as:
for calculating
2
2
qD
-'- + H +e
2g
where
2
u
a
2g
=
2g
-
+ h
+
in
However, its contribution
S can be shown
the more exact version
U.
be neglected
to be
of eq. 7.2 can be
184
importance
The relative
of
in equation
7.2 can be estimated
as
_o_
( _a
_
ui
2
u
U.i
)22
i
2g
u.
% u 0 /S
1
As
U
(--a
U
2
U.
S)2
0
Using typical values of
= 0.5
fps
u
=
fps
S
=
u
The error
a
25
5
introduced
2
U.
by neglecting
c is
% 0.01
(2g
7.1.2
Calculation of T-diffuser
The situation
definition
iss2hematised
of symbols.
lume trajectory
in Fig. 7-1 along with the
s is the distance
u is the mean plume velocity
in the stream
For simplicity
the analysis,
and the plume cross-section
(i.e. extending
direction;
top hat profiles
plume width.
rectangular
along plume centerline;
L is the
are assumed
is assumed
over the entire depth.)
to be
in
185
Integrating the continuity and momentum equations in
co-ordinates across the plume in the normal direction
natural
yields the following conservation eqations:
Continuity
dQ
ds
d(uL)
ds
= 2a
(7.9)
a
As in the coflowing
relates
cose)
(u-u
current
case, the entrainment
the rate of increase
of volume
flux,' d
ds
assumption
to a
characteristic velocity difference in the plume direction.
Axial Momentum
2
d (u2L)
-fM
=d
1dM L)-+u
ds
ds
The momentum
dM
cose ds
8H
a
dM
ds
variation,
(7.10)
ds
is governed
by the exchange
momentum with the crossflow and bottom dissipation.
variations
in the s direction
are neglected;
that this approximation is only valid
(cf. coflowing
realistic
Pressure
it is recognised
beyond the near field,
case in Chapter
current
of
IV) and is judged
to bea
3implificationas far as calculating the plume
trajectory is concerned.
Normal Momentum
The momentum equation in the normal direction is given
by
u LH dO
ds
-
1
2 CDp(ua
sine)(si02
(7.11)
The main factor in determining the plume trajectory is assumed
to be the centrifugal
force provided
by the dynamic
pressure
of the crossflow component perpendicular to the plume trajectory.
186
force per unit length along the plume is
The bending
2
1
CDP(u
Fb =
(7.12)
sin )
where CD: blocking coefficient.
Non-dimensionalising eq. 7.9.11 with u , 0. 5 LD, the following
system of governing equations can be obtained.
dQ
*
(u
= 2
- ycosO
)
ds
L
ffoD
dM
D ) M*dQ*+ ycosO
16H
dS
dO
*
*
(s
x
)
_
dS
sin
2
D
= _2
*
-2¥ **
M
cos ds*
ds
cosO
=
o
*
(s
)
oy sinO
0
=
(7.13)
ds
where
*
.*
uL
u (0.5L D
, M
)
D
u
*
-
u
u
,
(x
*
*
u uL
22L
u
o
(0.5L )
D
*
,y ,s ) = (x,y,s)/0.5LD
o
u
a
y
o
that
such
at
s =
0
= 90°
Q (O) = 2(-)
M
(0)
=
2
()2
where S is the near field dilution.
187
Given the initial conditions and the values of the coefficients
a, f , CD , the system of equations
o
7.12 can be solved
numerically using a fourth order Runge-Kutta scheme.
7.1.3
Theoretical Results
It can be seen from the system of equations
the governing
parameters
of the T-diffuser
7.13 that
problem
are
,y and
fLD
H
H
,
as in the coflowing
current
case (cf. section
4.2).
The theoretical solution forthe plume trajectory is plotted in
Fig. 7.2.1 for two diffuser
lengths
(LD/H = 5,20) and different
D
combinations of8 and y; the origin is chosen to coincide with
the diffuser
center.
The comparison
for the different
input
dimensionless parameters demonstrates the plume trajectory is
more sensitive to ythan to
.
A parameter measuring the ratio
of ambient momentum to diffuser momentum can be defined as
M
2
u
F=a
LDH
D
Q
y
u
2
o0
For a given
plume
relative
trajectories
diffuser
length,
there is an ordering
in terms of FM; the significance
of the
of this
parameter is further discussed in later sections.
7.2
Experimental Results
In this section
tests
with
the experimental
in this study are reported
the results
experimentally
results
and analysed
of the Perry model
that for low momentum
study.
of T-diffuser
in conjunction
It is found
input in a strong
crossflow, the near field temperature rise can be considerably
higher
than AT /S, S being given by eq. 7.8.
0
An empirical
188
fo=0.02
LD/H
D
=
5
20
/3-5$0O, rgO.OZ
5
o31
*,
Y.
LD
/0
- o.lz5
/o
20
30
x/LD
fo=0.02
LD/H = 20
,63
Ia
- !r,
LD
Y
O -0-1r
FFIz 0·12
5
/O
Sv
x/LD
Fig. 7-2.1
Predicted plume trajectory as a function of
governing parameters
189
correlation
is developed
give estimates
7.2.1
to delimit
this condition,
of the near field dilution
and to
when this occurs.
Experimental observation: this study
The T-diffuser
The gross featuresof
Fig. 7-3.
tests in this study are runs FF30-36.
these experiments
A high pressure
diffuser;
the diffuser
is created
plume, as depicted
by the ambient current.
penetrate upstream
region
can be summarised
in front of the
by 1, is deflected
Some of the heat is observed
and is swept downstream
in
further
to
away.
A stagnant region is found slightly upstream of the
diffuser
where the maximum surface temperature rise is usually observed.
Some of the heat in this region
entrainment
advected
is recirculated
flow at the back of the diffuser
downstream,
is well-stratified,
as depicted
by 2.
into the
and is also
Whereas
the flow near the diffuser
the flow in 2
in 1 is well-
mixed, indicating that unstable conditions are present.
from the diffuser,
the flow in 1 is also stratified.
Away
Fig. 7-4
illustrates a typical surface and bottom scan of the temperature
field.
It can be seen the temperature
field is highly
non-
uniform near the diffuser.
The highest temperature is found
near the stagnation
As the heat load is uniform
the diffuser,
induced
point.
this non-uniformity
flow is not uniform
of high local temperature
a diffuser
length
of temperature
along the diffuser
rises,
along
suggests
axis.
the
In spite
the temperature rise
beyond
AT
°- than the
away is much closer to S-
maximumtemperature
surface
observed.
maximum surface
temperature
observed.
190
u
-
a
4
Fig.
7-3
Experimental observation of a T-diffuser
FF
test: Series
arm ient
current
direction
la
,"%'T
t
wI
1.iz1
Il
diffusr Runreflections
diffuser
Run FF33
I
of ceiling lights
diffuser
plume (dye
injection pattern)
191
Rasin hnundary
.1
o.1
0*
0.
*,0+~
e.g
0.2
e.1 o.a f ;' 3
PA~~~~~~~~~~~~~~~~~~~~~~~~~~~~.I
~~~~
~ .,02
03
, 0202
0I
*.,
2
*.0
. 1.
.s..o~~IJ'f
,ts-i
q
2
i.g.t
e.,
use r
4.&
di
0.1
.''3
0.
.1
e.
*.t
0.0
.
3*..
2
1.5
.
2.4
*.
0.-2
*.2
0.5
F
2
1.6
2
I.o
basi
boun(ary
Surface temperaturefield
Run FF31 tlme=50min.
T"/s
pF
3.1
z Pr
basin boundary
.3
0o3
0.2
0.3
*o
0.2 o-j o.a
.3)
ko.
U.
O0.2 FPS
Is.T
.
2
o-04.3 02 oo.f
.
0.3
·2 -,.e 2'
0.3
e-
-2
.3
0'3
0.3
_rF
o.a o.3*
.
o.~
0-.
*.J
,.31
0-o
1.
10
I'3 0.1 /
/,,
7
0.r
.6_e7.S.2I.
8
g40
-
1.7
5.3-2
,,-.. },,.,., :.f,
. -.
:.
-
-0-.
diffuser
0.-3
.4
07
*-a
0.
o0-
0.7
.0.0
o6
basii
bounlary
Temperaturefield . H below water surface
Run FF31 time=50min.
47s
3 'F
sf'a
Fffrr
Fig.7-4
Typical recorded surface and bottom
temperature
uSeries FF
field in a T-diffuser
test;
192
7.2.2
Data analysis
of the Perry tests
A comprehensive
the unidirectional
effect
analysis
tests
isotherms
in the Perry study indicates
of a perpendicular
distinct
of the surface
crossflow
types of temperature
the
can be discerned
fields:
of
in two
1) Fig. 7-5 a) illustrates
a temperature field little affected by the ambient current.
The near field mixing
is dominated
by the diffuser
design.
The normal temperature gradient is symmetrical with respect to
the centerline.
2)
Fig.
7-5 b) illustrates
a temperature
field
heavily influenced by the crossflow, resulting in temperature
rises higher
than AT /S extending
for large distances
0
gradient
an lee and the normal temperature
(--n)Tside
(n acurrent
is not symmetrical
with respect
to the centerline.
In Fig. 7-6 the experimentally
the near field
are plotted
FM
the momentum
temperature
versus
is a measure
downstream.
observed
values
of
rise in this study and {erry tests
a momentum
flux ratio
of the bending
F
M
ua LDH
= --Q
QoUo
0 0
force on the plume relative
input of the diffuser.
ATMA X is defined
to
to be
the maximum surface temperature rise recorded beyond a distance
LD from the diffuser; defining ATMAX in this manner eliminates
isolated
heated
spots
due to local stagnation
that are not indicative
is the near field
of the diffuser
temperature
and stratification
induced flow. AT /S
0
rise that would be obtained
the fully mixed assumption is met.
if
In general
deviates
In gnerlATMAX
AMAX dvae
u
from AT /S for larger values
o
of F M .
MN
For a given
u
,
0
91
a longer
U)
0
F4.
01)
4-4-4
'4
Do
00
'-LO
ON~
rC*'J
J
.-
0
*o,
21)
U) t~
(a) 0
uZ
Cd~
14-44-4
-T1
44
co
'4-4
4-J
3-.
Cl) C4IFI
CN
rIIi
H.A
Le) E-
II- .H
0
03
*-4 )
U')
,4
4
0
r
II
(Cl
:::1
.
4.
4.
4.
4
4.
LI)
4%
U
4.4
3tj
/
I
ci)
r
(
4J
3 oI-
0
rZ 4
co 0o
0 4-4U)
U)
14-4
C-
4-44-
0
4-1
0
.t-I<t
0
.,-4
III
I
in
0
4- a;
co
C. 4
C,,
~4J
4i
I
00
Cll
3-N1
II
II II
bI
3-4
U)
S~
u'
C
I 44 *-
G-0)
O CQ.IJ=
o00C.')
U)
N1 U)
l4 j
-i
;o
a) 0 Cl.
'H
PO
rX4
c -4 J aJ=I
U)
N)
(Hr Clj
H
1
II
w
14-4
4
0
4. t4.4 4%1
11co
I
194
O
tn
O
o
0
NO
0
0
0
0
:r
-
iO,
0
<
It
,.
0
(D1
I
I
a
4O w
0'
H
I
~-H
0
<
H
o
0
·
1jf
OI
<<tD
0
In
0
O
0
I
U4
I
O
I
4 z
W
,,J
I
I
0
0 0
I
I
I
I
-
-
I
|
0
N
I
I
I
|L
0
In
-0
·
0
O
O
0
195
diffuser is mre prone to adverse effects of an ambient current.
For a given
diffuser
design,
an increase
of the ambient
current velocity would likewise worsen the diffuser performance.
The correlation
is not affected
the near field
AT /S.
0
shows
that for FM<0.1
by the ambient
temperature
The implication
more concentrated
current.
the near field mixing
Whereas
for
rise can be considerably
is that if a diffuser
momentum
input (low values
FM> 0.1
higher
is designed
than
with
of FM), the
adverse effect of an ambient current on near field mixing
can be eliminated.
With the limited amount of available data,
it is our speculation
that for low momentum
input
(high F M )
the ambient current induces strong stratification near the
diffuser; the fully mixed assumption breaks down, and the
prediction
7.3
of S ceases
to be valid.
Comparison of predicted and observed plume trajectory
In this section
the computed
plume
trajectories
are
compared with experimental results of this study and the Perry
model
studies.
In all the calculations,
are used for the physical
the following
values
coefficients.
= 0.068
f =
0.035
0o
CD= 4.0
For each experiment,
,
are determined from the
diffuser design and ambient conditions.
by eq. 7.8.
For FM>0.1,
For FM<0O.1 S is given
S is experimentally
determined;
196
AT
o
S
=
oBs
ATMAX
7.3.1
,
ATMAX being defined in section 7.2.
MAX
Series FF
Plume trajectory:
Fig. 7.7 shows
the comparison
the predicted
between
plume trajectory and the observed plume boundaries.
The
observed plume boundaries are visually recorded from dye
injection patterns during the experiment.
agreement
more the plume
the value of F
The larger
is reasonable.
is deflected.
In general, the
the
In some runs the predicted
plume trajectory (e.g. FF30, 34) extends beyond the model
limit:
distortion of results is expected in regions close
to the boundary.
7.3.2
Plume trajectory:
Perry tests
Fig. 7-8 shows the predicted plume trajectory along
with the observed
surface
isotherms
for the Perry
same trend is seen in these comparisons;
to stronger plume deflection.
tests.
The
a larger F M leads
The agreement between the
predicted and observed plume trajectory is satisfactory.
The
predicted plume width, however, is considerably smaller than
the lateral
extent
of the surface
isotherms.
As the theory
is based on a vertically fully mixed assumption, this suggests
the plume
is stratified
beyond
the near field.
Only tets with distinct input parameters to eq. 7.3
are included;
some Perry
tests
differ
only in design
parameters that do not affect the global temperature field
(e.g. clearance of the discharge ports from the bottom).
197
o
0
,.0
,--I
I
.IU)
I
II
':
-- 4
CM
. 1- -
.
, r.
C-
.
c
*
I.
*
2
I
U)
(.°
0
. .
.
U
;
04
e-
I
V
0
I
a
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V
o
oI
.
_-
H-
+
+
+
t
t
+
I
9
9,
4
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I
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a
t
i
C-
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E.I
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I..
9
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C.
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4-i
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201
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ill
t
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u
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lb
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202
For tests characterized
by large FM>O.1
the calculated
trajectory indicates the plume rapidly approaches the ambient
The calculations
direction.
current
point where
u(x) =u
a
.
cos
at a
are discontinued
As the induced
in these
velocity
cases (with impaired near field dilution) is usually small,
this occurs
compared
at a distance
with
the scale
from the diffuser
of the isotherm
plot
that is small
(1000 ft.).
Consequently it is judged sufficient to illustrate only several
comparisons
in other
of this type (Runs in Fig.
tests belonging
Sensitivity
7.3.3
blocking
to this category
of results
The sensitivity
coefficient
7-8 b); the agreement
to CD
predictions
of the theoretical
CD
is similar.
is illustrated
in Fig. 7-9.
to the
The
major determinant of the plume trajectory, CD, is varied
+ 40% from the assumed value in all the previous calculations
for a typical
run with FM <0.1.
The results
show that a
larger assumed value of CD leads to more pronounced plume
deflection; the ultimate lateral position of the plume is
changed
7.4
by only about
10 % from the CD
=
4 prediction.
Summary
This chapter concerns the induced temperature field of
a unidirectional diffuser in a perpendicular crossflow.
203
Perry:2M2S09
y
2000
ft.
1000
x
IUUU
Fig.7-9
JUUU
ZUUU
it.
Sensitivity of predicted plume trajectory
to blocking coefficient CD
Based on a vertically fully mixed assumption, a theory
for calculating
is developed
the near field dilution
and
plume trajectory.
Data analysis
of tests in this study and the Perry model
that for a low momentum
study shows
u 2
flow,
characterised
o
An empirical
correlation
Tee-diffuser
is given.
T-diffuser
>0.1, the near field
o
rise can be considerably
temperature
higher
than predicted.
of near field temperature
The correlation
can be designed
in the FM
adverse effect of the crossflow.
cross-
H
L
D
by F
input in a strong
rise for a
also suggests
that a
<0.1 range to prevent
204
The computed plume trajectory is compared with
observed plume boundaries in this study and surface isotherm
plots in the Perry Study.
in all cases.
situation
Satisfactory agreement is obtained
Eq. 7.13 can be used to estimate
the distance
of the diffuser
in a prototype
plume from the shore.
205
CHAPTER VIII
FAR FIELD RECIRCULATION
In the previous
a theoretical model for a uni-
chapters
directional diffuser in shallow water has been formulated,
solved, and validated against experimental data.
In this
chapter we investigate a phenomenon that has hiterto been set
aside;
the potential
in the vicinity
heat build-up
of the
diffuser.
Because of its large entrainment demand, a diffuser
represents
a strong
sink.
of interest
The question
is,
therefore, under what conditions will the diffuser mixed flow
recirculate, carrying heat back into the near field?
In the following sections an experimental confirmation
for far field recirculation
is given.
The effect
of the return
flow on the temperature field will be discussed and semiempirical correlations are presented.
8.1
Conceptual framework of heat recirculation
The conceptual basis upon which the experimental
investigation of far field recirculation has been conducted can
be schematized in Fig. 8-1.
Under slack water conditions, we
have seen from eq. 4.50 that the momentum
is
dissipated
exponentially
as
e
-f
o
x/8H
of the diffuser
; thus
at
plume
a
-8H
characteristic distance of L rec
=8H
f
away, the diffuser plume
0o
will have lost most of its initial momentum; a stagnant pool
of water at this distance would, by flow continuity, serve
206
II
!
__________
diffuser
Fig. 8-1:
as a source
symmetry
line
LRC
Schematic of diffuser induced recirculation
for the back entrainment
thus creating
sink
>|___
demand
a far field recirculation
of the diffuser,
resembling
a source-
type of flow.
Fig. 8-2 illustrates
an experiment.
Twenty
the recirculation
minutes
a 30 sec. dye injection
pattern
observed
in
after the start of the experiment,
is made;
Fig. 8-2a shows the diffuser
plume immediately following dye injection, the plume edge being
blurred
by dye diffusion.
Fig. 8-2b is a snapshot
minutes after the dye injection has stopped.
taken five
The clear discharge
water and the dyed recirculation patterns reveal clearly the
shape
of the diffuser
flow is laminar.
plume.
The blue streak
lines indicate
the
207
Ifrc
&.,
Fg. -2 2-w
.
Fe.?-2.
mswa
ZaL
,~a4
( Mj' 23 )
:
RaREMZ3
2./
,,6_
5 *4A"~
a
e SdOffuetXZ-ft
&ttfewest e,,
..-
aMA. dre
84±-
L
208
Characteristic scales of recirculation
8.1.1
From the discussion
of the previous
we see that
scale of recirculation,
of the characteristic
an estimate
section,
LRE C is
L
LREC
REC
(
(8.1)
8H
f8
o
=
of the diffuser
is
Thus the time scale of recirculation
is
(per unit depth)
The sink strength
SQ
measured
by
H-
characterised
.
by a parameter
HL 2
REC
S REC
T RE
TREC
Q-
(8.2)
o
for runs that exhibit
In the laboratory,
strong
far field
reciculation, the flow beyond the near field would be laminar
Using
a pipe flow analogy,
the friction
factor f
0
can be
as
estimated
64
f=
o
R
(8.3)
e
where
R
= characteristic Reynolds number of mixed flow
uN(4H)
e
induced velocity
uN
in the near field
LREC
Thus the scale of recirculation,
out in this study can be estimated
carried
experiments
u
L
REC
% 1
2
, for the laboratory
as
H2
N
(8.4)
-V
From eq. 3.41 we see that uN =u
0
/s
u H2
L
REC
2
2
0
S
(8.5)
209
Thus L
RECC
is dependent
The constant
on a characteristic
of proportionality
able less than unity
parameter
can be expected
as the actual
jet velocity
u 0H2
vS
v9S
to be considerrapidly
decreases below the characteristic velocity uN as the diffuser
plume spreads in the lateral direction
8.1.2
A dimensionless formulation of heat recirculation
As heat gets recirculated
into the near field due to
diffuser induced recirculation, the maximum temperature rise
in the near field would
entrained
be higher
flow is at a temperature
than ATN, as part of the
higher
than the ambient
temperature Ta . The transient temperature build-up in the
near field can b correlated with the diffuser design variables.
The previous considerations suggest it is reasonable to assume
a physical
dependence
of the following
form for the far field
recirculation effects.
AT
= f
Thus the maximum
induced
(ATN'
SQ
H
temperature
temperature
field
LREC,
rise AT
is assumed
L
max
t)
(8.6)
in the diffuser
to depend
on ATN, the
near field temperature rise in the absence of heat recirculation;
SQ0
H ' the 'sink strength' of the diffuser; LREC, the scale of
recirculation;
LD, the diffuser
length;
and
t, the time from
the start of the experiment.
Choosing
ATNLREC,
t as the fundamental
variables,
following dimensionless grouping can be obtained:
the
210
AT MA X
ATN
8.2
D
HL2
The correlation
results
LD
SQt
= ' L
M
REC
REC
)
(8.7)
8.7 can be obtained
of this study,
from the experimental
as shown in a later section.
Experimental Investigation
All the slack
tide runs in Series MJ and FF are performed
to test the phenomenon
of far field recirculation.
these tests were done with a half-diffuser;
Most of
a few full diffuer
tests in series MJ were made to corroborate the findings of the
half diffuser
confirmed
tests.
The phenomenon
in these runs.
recirculation
of recirculation
It was possible
that is significantly
to produce
was
a scale of
less than the basin extent,
thus minimizing model boundary effects.
The experiments in the
MJ series were designed to investigate how the scale of
recirculation varies as the diffuser design parameters; no heat
is introduced.
series
Each run lasted
the slack
phenomenon
about 45 minutes.
tide runs were performed
of recirculation
In the FF
to confirm
in the presence
the
of heat input and
investigate how the recirculation changes the temperature configuration in the near field of the diffuser.
Each run usually
lasted a little over two hours, when it was judged the model
boundaries have significantly distorted the results.
runs a steady state
accumulating
is not approached
in the basin.
It was
For these
as heat is continually
found that small scales of
recirculation (strong far field recirculation) are in general
211
characterised by long diffusers, low discharge velocities,
and shallow water
8.2.1
depths
(cf. eq. 8.5).
Experimental Results
Fig. 8-3 (Run FF04)
times after
illustrates
the beginning
the dye pattern
of an experiment.
at various
Also shown is a
surface pathline which indicates a scale of recirculation that
is significantly
smaller
than the basin size.
the far field recirculation
illustrated
in Fig. 8-4.
normalized by ATN,
The effect
on the temperature
The excess
AT~~y =AT(x,y)
AT(xy))
of
field is
temperature
field
ispote-ear
is plotted n
the
N
beginning
and an hour from the start of the experiment.
value of AT greater
recirculation
than 1.0 is then indicative
effects.
Any
of heat
Characterised by a small time scale
of recirculation (TREc= 3.9 min), the run shows strong heat
REC
recirculation leading to temperatures much higher than ATN within
60 minutes
from the start of the experiment,
T
t
=
15.4.
REC
As a
contrasting
the surface
scale
example,
Fig. 8-5 (Run FF13) illustrates
flow field of a run that is characterised
of recirculation;
thus the recirculation
is clearly
influenced by the presence of the model boundaries.
TRe c is large
for this run (TRE C
51 min.),
by a large
Because
we see from Fig.8-6
that no heat recirculation in the near field is visible even
after
70 minutes
from the start of the experiment,-t
TRec
= 1.4.
212
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predicted near field temp. rise=4.9 F
slack tide
I2 6
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bas I
he. ,
time=60 in.
predicted near field temp. rise=4.9 F
slack tide
r,,*
15.4-
a.24
0.1
o-o8
0o.0
scal e:
2 feet
Fig. 8-4
Effect of far field recirculation on
the normalised excess temperature field:
AT = _7
strong recirculation
4
N
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214
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Run FF13 time=1O min
predicted near field temp. rise=2.7 F
slack tide
00
scale:
*--e
2 feet
~~~~~~~~diffuser
14o.1
0.s1
0
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0.04
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0. 04
00
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0-0
0.0o
o°°8
Run FFI13 time=70 min.
basi
predicted near field temp. rise=2.7 F
slack tide
Fig. 8-6
arv
boudare
Effect of far field recirculation on the
normalised excess temperature field:
weak recirculation
.
2 feet
216
8.2.2
A.
Empirical Correlation
The scale of recirculation,
In the experiment
to the point
plume
to recirculate
Fig. 8-2).
By defining
LRE C can be measured
experimentally
-REC
LRE C is taken to be the distance
the diffuser
begins
LC
where the center
(as indicated
of the diffuser
in Fig. 8-1
the scale of recirculation
to within
observed
values
about 2 feet.
be well borne
correlation
and
in this manner,
In Fig. 8-7 the
for LE C in the MJ and2 FF series
REC
u H
are plotted against the characteristic parameter
as it is, the theoretical
from
consideration
out by the experimental
in section
data.
S--.
Vs
Rough
8.1 seems to
The empirical
is:
u H2
LC=
REC
B.
0.088 - Vs
(8.8)
Maximum temperature rise
In Fig. 8-8 the normalised
maximum
temperature
rise in the
diffuser induced temperature field recorded at any time during
AT
(t)
an experiment,
MA
an expAT(obseriment
t
less time
d) , is plotted against a dimension-
N
,where
REC
TREC = characteristic time of recirculation
REC
HL 2 RE (Obs.)
__REC
SQ o
In general
it can be seen that for small values
of LD/LREC;
i.e. large scale of recirculation, the heat build up is much
more gradual whereas for large values of LD/LREC, the maximum
temperature
rise in the near field can be as high as two to
217
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0o
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QJ
En
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C)
000
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010
1-1 0
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T-co
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04
W~
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N
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218
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____11__11_1111111____1__11_·____1_11_
11--·--11_
219
three
rise without
times ATN, the near field temperature
heat recirculation as given in section 4.1.1.
The temperature
data of three FF slack tide runs have
been excluded in these correlations for experimental reasons;
Runs FF09 and FF25 are characterised
jet mixing.
The diffuser
with the sensor
compared
jet laminarity may affect
so that potential
RDn 1500,
numbers,
the
length LD of Run FF27 is short
spacing,
not sufficiently close to resolve
8.2.3
by low jet Reynolds
so that the measurements
are
ATMAX.
Short vs. long diffuser
All the previous discussion and experimental results have
the far field heat recirculation
suggested
parameters
tion LREC.
- the sink strength
this notion
With
is dependent
on two
SQ /H, and the scale of recircula-
in mind we can compare
the effect
of diffuser length on far field recirculation.
Let us consider
two diffuser
designs.
The design
parameters are indicated below:
Design
Design
A
Qo,2LD,2s,uo,H a ,N
QOLDS,uOH,aON
Thus design
B differs
B
from design A in that only
the port
spacing (and hence the diffuser length) is doubled, everything
else being kept the same.
slack
From Eq. 3.41 we see that for the
tide case the near field dilution
S is proportional
to
L H
/Na
o
thus design B would have a near field dilution
i2 times
220
that of A.
For these
two cases the sink strength
and LREC
are compared:
SQ /H
L
%
REC
(f2 S)Q
uH
2
O
O
H2
L
vS
The sink strength
with a scale
/H
of design
REC
v(2
S)
B is /2 times that of design A,
of recirculation
1//2 times.
These elementary considerations are demonstrated by Runs
FF28,
29.
Fig. 8-9 compares
the surface
flow field recorded
for the two runs; it can be sei that a longer
diffuser
smaller
of the dyed
scale of recirculation.
The movement
has a
diffuser plume at various stages after dye injection for these
two runs is also shown
normalised
excess
in Fig. 8-10.
temperature
Fig. 8-11 compares
the
field for the same two runs;
a
longer diffuser induces more far field heat recirculation in
the same real time t (cf. t= 120 minutes)
after the beginning
of the experiment, which corresponds to
different normalised
times
diffuser).
8.3
t/TREC
10.2 (short) vs.
30(0ng
Design Application
8.3.1
Prototype Considerations
In the prototype flow is turbulent;
of recirculation
values
are hence
of
H
f
= 20
0
=
feet
0.02
characteristic scales
given by eq. 8.1, 8.2.
Using typical
221
Q0 = 0.8 GPM
L,= 2.5 FT.
A = 1.0 IN.
H =0.73 IN.
RUN F '7-9
o=1.89 FPS
Fig. 8-9
Recorded surface flow field
( short vs long diffuser)
222.
r-
r=
CD
C)
rS
C)-
cl)
C'
-11c+
-W
0
C
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~~~~~~~~~~~~~~~~~~~~basi
Iar
bou
0.06
Run FF28 time=120 min.
predicted near field temp. rise= 3.2 F
slack tide
/0o.2
0-03
0.0
0o
scale:
e'J
2 feet
svmmer-
diffuser
lina
0*0
0. J
o 33
0.0
0.0
o 33
0ol
O,/4.
o.o
0.0
basi
0'0
0.
0'O
0.0
bo U
arv
0.0
Run FF29 time=120 min.
predicted near field temp. rise= 2.1 F
slack tide
scale:
- 0
Fig. 8-11
_______________I_·_1__1__11___11_1111
2 feet
Recorded normalised excess temperature
( short vs long diffuser )
field,
_ I·IIXLLI__II._-(-_·--_I
-
-C
L_·-
I
225
we have
LRE C = 1.5 miles
REC
Also
the time scale of recirculation,
by eq. 8.2.
For small diffuser
TREC, can be computed
flows and small initial
near field dilution, for example.
S =
Q
Qo
=
5
1000 cfs
we have
TRE
= 71 hour
REC
For large diffuser flows and large initial near field dilution,
for example,
S =
Q
0
10
= 3000
cfs
we have
TRE C = 11.8 hr.
Thus at a given site, a larger diffuser flow and larger
initial back entrainment is characterised by a smaller time
scale of recirculation i.e. relatively stronger heat recirculation effects.
Also for a given discharge flow, a longer
diffuser (everything else kept constant) would have a shorter
time scale of recirculation
T
REC
1
S
-
as
(cf. eq.8.2,
3.41);
ED
these considerations were demonstrated with relation to the
laborabory tests as discussed in section 8.2.3.
226
In general,
for any given
LRE C and TRE C
situation,
estimated using eq. 8.1, 8.2 respectively.
frequency
and duration
importance
can also be used to estimate
8.3.2
Based upon the
of slack in a prototype
of heat recirculation
can be
situation,
can then be assessed.
the heat build-up
the
Fig. 8-8
in the near field.
Design of Models
for undistorted models
In the laboratory,
that preserve
dynamic similitude of the near field, the friction factor is
not scaled
properly.
of flow in the model tends to
Laminarity
pronounce frictional effects.
Thus stronger recirculation is
observed in the physical scale model
where
than in the prototype,i.e.
(LREC) prototype
r(f ) prototype
(LREC) model
(f ) model
; length scale ratio of prototype to model.
For example, using typical values of
(f0 ) model = 0.035
(f)
prototype
= 0.02
the scale of recirulation
LRE C is overestimated
by 50% in the
laboratory.
For a given
site,
heat recirculation
used to establish
if it is desired
in a hydraulic
the minimum
to test diffuser
scale model,
induced
eq. 8.8 can be
model scale ratio to be used.
By preserving Froude similitude in the prototype and model
we have
(V) model
1
(v) prototype
_______1_1________________1_1111
_·
227
where V: characteristic velocity
Then for the hydraulic scale model
U.
o. 088( -
°-
/Z-XQ
L
r
=
REC
)( H) 2
rr
vS
= 0.088
u H2
( 0 -)prototype
VS
prototype
For any given diffuser design and L
-5/2
(8.9)
r
del' the appropriate
length scale ratio can be chosen using eq. 8.9.
Using typical values of
i
u
= 20
fps
H
= 20
ft.
S
= 10
r
=
100
we have
LREC
= 70
At this ratio,
ft.
only by using model basins
200 feet or longer
can diffuser induced heat recirculation be tested.
228
CHAPTER IX
DESIGN APPLICATION
In this chapter, the theoretical results presented in the previous
chapters are interpreted from the viewpoint of practical design of
thermal diffusers for once through cooling systems.
overview of diffuser design is given.
First, a brief
Second, general features of
diffuser performance for various designs are discussed for two cases of
fundamental interest.
Finally, an ecological interpretation of the
model predictions is advanced, and prototype considerations are made.
9.1
Diffuser design in shallow water:
a general discussion
At a particular site and for a given plant capacity, the design
objective is often to meet state or Federal thermal standards.
These
standards can be either quantitative (e.g., the allowable extent of
an isotherm area) or qualitative (e.g., siting to locate the discharge
away from fish spawning areas, making provisions for zones of passage
for marine organisms) in nature.
evaluation
of the impact
The final design is based on the
of the complicated
interaction
of a multitude
of factors--frequency, magnitude and direction of the prevailing
currents, topography and hydrography of the site, the type of organisms
that are to be protected, and economic considerations.
It was pointed out in Chapter II there are three basic types of
multiport thermal diffusers:
and unidirectional diffusers.
the nature
of the
ambient
alternating diffusers, staged diffusers,
Depending on the plant design and
current,
one type of
229
diffuser
may be
preferable
a tidal situation,
to the other.
it may be desirable
For example,
in
to use an alternating
diffuser with no directional momentum input, whereas in the
case of currents
predominantly
in one direction,
either
a
staged or unidirectional diffuser may be more preferable.
The
details of the general design philosophy can be found in Ref.15
In many instances, and especially for well designed
diffusers,
ambient
the temperature
current
is greater
reduction
in the presence
of an
than the case of no current.
Thus
the stagnant water condition is often chosen as the worst or
design condition upon which general design principles and
first estimates
of design values
The comments
in the ensuing
can be drawn.
discussion
in this chapter
are limited to unidrectional diffusers, which are particularly
applicable in shallow coastal waters.
A unidirectional diffuser inputs large momentum in one
direction;
it is effective
in minimizing
the area of mixing
zones of elevated temperatues, and directingthe thermal effluent
offshore.
9.1.1 Design Alternatives
For a given plant capacity, the heat rejection rate is
fixed;
thus Q0 AT is constant.
The design
variables
are
230
Q
AT
condenser flow
0
condenser temperature rise
0
LD
diffuser length
u
nozzle velocity
N
number
a0
port area
H
water depth at discharge
0
of nozzles
Many design alternatives can be explored.
increase Q
number of ports
2) increase
and hence
u , keeping
constant,
increase
and hence
the diffuser
flow Q
can mean either 1) increasing the
Increasing Q
.
being
We may reduce AT 0 to
LD, keeping
s and u
the same number
the port area a
length
constant,
of ports and hence LD
so that the number
0
or
L D is decreased
of ports
- the condenser
fixed.
In the next section
are discussed.
Design
two design
alternatives
cases of fundamental
can be understood
interest
in terms
of these two cases.
Design interpretation of theoretical predictions
9.2
General features of diffuser plume characteristics can be
extracted by examining the theoretical solution given in ChapterIV.
It is possible
view.
to interpret
eq. 4.52, 4.53 from adesign
We have seen from eq. 4.19-4.20,
4.28-4.29,
point of
that for
=0,
LDH
the solution
L
L
HD
____
sh
is dependent
eltv
the relative
ifue
diffuser
on N
-
a near field parameter,
lnt.
Fo
length.
For a specific
peii
and
itwNa
site, we can
231
think of Q
ATo, H as fixed.
The near field temperature
rise
ATN is given by
AT
AT
-=
=
N
AT
-(9.1)
S
LH
*D
2Na
Since Q
0
= Na u , by definition,
eq. 9.1 can be written
as
AT
o
Thus for a specific
site and plant design,
only on the diffuser
Given L
determined
length L
ATN is dependent
and the nozzle velocity
D--~~~~~--
u , the details of the design
D'
~~~~~~~~o0
- s,N,a
u.
can be
by
(N-1)
s = L
(9.3)
D
N
a
o
=
Qo
u
(9.4)
0
Given eq. 9.3 and 9.4, an arbitrary
of the three variables
choose
s sufficiently
momentum,
and then a
For purposes
0
s,N, and a .
0
choice can be made
Thus in principle
for any
we can
small to approximate
a line source
and N are determined
accordingly.
of discussion,
the design
values
of
for a 2410
MWe power plant in shallow water - the Perry Nuclear Power Plant
on Lake Erie are chosen
Q
= 2780
H
=
19 ft.
=
28
0
AT
f
o0
o
= 0.02
cfs
F
232
Two cases of fundamental interest are considered:
9.2.1
Case A:
Effect of diffuser length for fixed discharge
velocity
LD
The nozzle
velocity
u
o
shown by Eq. 9.2, a longer
field mixing.
is fixed and
diffuser
D
H
is varied.
is more efficient
As
in near
The intermediate field performance, however, can
be contrasted to elucidate the tradeoff between a longer diffuser
and a shorter
one in the entirely
fully mixed region.
Fig. 9-1 illustrates comparisons of variables of interest
for four cases with
u
= 20
fps
LD
H
The nozzle
-
10,
diameter
20,
D
0o
40,
60
is fixed at 2 feet.
Fig. 9-1 a) illustrates
temperature rise with x.
the variation
in centerline
It can be inferred from the temperature
drop that a longer diffuser induces more back entrainment, whereas
the shorter diffuser entrains more flow in the intermediate field
The ultimate
AT at large distances
Fig. 9-1 b) shows
the variation
with distance from the diffuser.
due to inertial
is similar.
acceleration
u
c
of centerline
u (x)
increases in the near field
and then decreases
entrainment and momentum loss.
velocity
due to side
A shorter diffuser induces less
near field mixing and hence leads to a more pronounced velocity
field.
233
0
%3
0
tG
W
0
1 rzP.
X,
Cd
)u
v4
)
D0
C4
O
w
lb
li
WI
234
I
0
04
En
04
(D
rH
U
00m
00
owH
rH-H
H
UO
-41Oc
N
i
4-) -H
Uo
Ow
-H
0
\t
V'4~I~
L
X
-
4E
C4
235
Fig. 9-1 c) shows the shape of the diffuser
longer
diffuser
starts
out a wider plume,
plume.
The
but the rate of
increase of the shorter diffuser plume is greater by virture
of its larger
capacity
to entrain.
Fig. 9-1 d) shows the areas within
different diffuser designs.
surface
isotherms
for
For a given diffuser design the
area within a AT0 F surface isothermis meaningfully defined
only for
AT
< AT
X
< AT
-
N
whereAT N is the near field temperature rise as given in Section
AT
4.1.1; ATN = -S
S being given by Eq. 3.41. AT
is the asymptotic
temperature
4.3.2,
rise given by eq. 4.54.
this is a measure
of the mixing
flow) of any unidirectional
tures when the different
As pointed
diffuser.
diffuser
capacity
out in Section
(total induced
For the range of tempera-
designs
overlap,
functions are virtually indistinguishable.
the A
s
vs AT
This means that if
the criterion of interest is a relatively low temperature isotherm
(e.g. the 3F
isotherm),
as a diffuser
twice as long.
a shorter diffuser
Fig. 9-1 e) shows the travel
a function
entrained
time along the centerline
of AT ; this is a characteristic
c
history of
might be as satisfactory
measure
as
of the time
exposure to elevated temperatures for aquatic organisms
into the diffuser
induced
flow.
It can be seen that the
thermal impact of short vs long diffusers on an entrained organism
is intimately
related
to the ability
of the organism
236
N
0
trt
Nh¢
In
0 :1:
Q~~~~~Q
4i
H C
-
3 'H
Q)
-I
o
.
C)h
:n E
QH
rz s
.n
0
I
0
C
An
-
237
-
..
I
4J
U
o c
fU0
.1i
*H
D~
0)02
u, En
44
p
Ch
.H
o0f
¢z 0
.H
c H
0
'.6
o0
F<
La
o0
*
-
9
~
0
O
:)
cn
uC)
0
m4
E
J
V0
V) a1)
)
N
-4
's
4O
a)
44>
pi.-
.-
H
~
0
$-0cd
)
4U
:~I--
·...
0
0 .H
~4 ~4
0-C,
O
W
4 JQ0
CO
aH Q)
0
r O~a
a)
OII
Al
-J :c
I
0
0E
0
w
t(1
0
+,,d
O
.
239
to withstand high temperature rises for small durations.
is discussed
in greater
detail
As a check against
This
in the next section.
the assumption
of a two dimensional
friction jet, the densimetric Froude number based on the water
u
depth
F
=
H
(x)
x
is plotted
VgfAT H
be seen FH>1 up to x/H = 600.
by this model
effects.
vs x/H in Fig. 9-1 f).
The essential
are thus consistent
features
with the exclusion
It can
revealed
of buoyancy
(see section 4.3.2).
In summary,
the physical
one striking
variables
~10 H the effect
feature
of interest
of the diffuser
that is common
is that beyond
length
to all
a distance
is no longer
felt.
of
Short
diffusers with its more concentrated momentum flux, make up for
its lesser near field mixing capacity by entraining more ambient
flow in the intermediate
field.
The results
also support
the
two dimensional boundary layer type approximation for a region
large enoughto reveal the essential characteristics of the
diffuser plume.
9.2.2
Case B:
Variable discharge velocity and diffuser length
with constant near field dilution
In this case we study
and decreasing
five diffuser
L D /H is
D
LDu
L.
stays constant.
changed
Fig. 9-2 compares
designs
proportionally
the tradeoff
in which u
decreased
between
the thermal
increasing
impact
u
of
varies from 15 fps to 45 fps.
LD
LD
from
H = 60 to
H
Thus the near field dilution
(cf. eq. 9.2) and the global
performance
=
20
so that
H
is not
can be compared.
240
To
..
E
-4
4.0
U.
C<
mo
0
s
Wn
D
-W
rzH
81 rz) Q)
4-J
0
-4
o'
o
0
0
-1
%3
CI
241
zo
0
o
-4
,,
0
41-
o
~4J
o
X
co
4 .,
0U
r-4
J -4CD
0 4Jc
dU
o
C4X en
fX
a0
Q
o
a
4oii
I-L)
a
CO
Q
o..L
0
c
C
a)
c
o3
-I
(.)
C
,.o4 0U
U)
X
1,
c4
r4
s~i2
0
4U
a
CI
N
If
,O"
tCI
r
t
40
o
I<3
I
40
(3
e
I
0
(:
LL
I
0
(
<
N
I
Or
243
,
.c.
Co
C
,-4
-W
aNO
:I0
.0
r)
"U
CJ0
4-C.
0
J lt
-ii
0
U: 4 U
4
0 -'
0)
i)u
-H
OCsom
XoU'
H 0)
C
N
0
o4o
C)
°
'3~ ~
q
)
0
244
..
X
.0
t4
.0
a)
4J
O 0
-'0).)'I'
I
O
-4 0
0)
o,
©
H
0)02
0) X
G
CI.
I
C,
.H
Ic
a
0
0
0
ch
-4
X
245
The five designs
LD
u
D
correspond
to
fps
o
H
60
15
50
18
40
22.5
30
30
20
45
It can be seen from Fig. 9-2 a,b,c,d
that by increasing
the discharge velocity (keeping a constant and just decreasing
0
N and hence L),
D
a significant decrease is obtained both in
terms of centerline temperature rises and areas of different
surface
isotherms.
Comparing
the
LD
LD
H = 60 case and the
- =30
case, for example, an order of magnitude decreasein the area
within
the 3F isotherm
is achieved
by increasing
u
from
15 fps to 30 fps.
Fig. 9-2 d. shows,
on the other hand,
the increasing
velocities which are induced by the shorter diffusers with
higher discharge velocities.
9.3
Diffuser performance and aquatic environmental impact
Thus far, the comparison of diffuser designs have been
presented in terms of centerline temperature rises, centerline
velocities, and different isotherm areas in the previous sections.
In addition
to this type of Eulerian
description,
it may be
desirable for purposes of quantifying the environmental impacts
246
of different diffuser designs to display the model predictions
in an alternative manner that may provide more direct answers
to question
of interest
What do these results
to the aquatic
biologist,
such as:
mean in terms of the organisms
entrained into the diffuser plume?
that are
Is therea quantitative
measure for the total number of organisms that will be affected?
What is the distribution
the aquatic
population
of heat to which various portions
will be exposed
of
, and for how long?
(Fig. 9-3)
In this section
we attempt
to translate
to a form that could be of more potential
and decision makers.
the model results
use for biologists
Diffuser performance is compared on the
basis of "ecological histograms" which capture information from
both the temperature and velocity field on one diagram.
To fix ideas,
the concept
to the same two design
In the following
is illustrated
cases presented
with reference
in Section
the basic assumption
9.2.
is made that the
number of aquatic organisms entrained into the diffuser plume
is proportional
to the induced
9.3.1
Comparison of ecological performance:variable
diffuser length at fixed discharge velocity_
Case A:
We recall
u
flow.
in this case the nozzle
= 20 fps and only the diffuser
LD
compare
dilution
the
H
=~~~~~DD
60 and
achieved
velocity
length
is fixed at
L D is varied.
Let us
L
= 20 designs.
The asymptotic
for the two cases is about
the same
247
I
4'
aq
en
di
Fig. 9-3
An ecological interpretation of diffuser
performance
(cf.Fig. 9-1 a): thus the total number
be entrained
into the diffuser
of organisms
that will
plume will be the same.
However,
different proportions of this number of organisms will experience
different temperature rises for different durations.
In Fig. 9-4 we plot a histogram
distance/total
designs.
of induced
induced flow (in increments
It can be seen, as in section
flow per unit
of 50 H) for the two
9.1, the shorter
produces a more distributed kind of flow pattern.
diffuser
The back
entrainment is larger for the longer diffuser, whereas the
shorter diffuser entrains more flow in the intermediate field.
In Fig. 9-5 we plot a histogram
of the fraction
of the
total number of organisms experiencing a maximum temperature
rise in a certain
range and its duration.
Thus for the design
248
.
1.c
as
,
LIf
Fraction of
total
a6 _
entrainment
flow
6
'2
Z
100
Fig.
9-4
200
1T
-.
t
l
I
l
I1
r
300
U
4W
1
5oO
Distributionof induced flow along diffuser plume
1.0 p.
Fraction of
total
entrainment
flow
_o0
l.,20
H
_I
p-
a,
@2-
I
I
0
200oo
t
r~~~~~~~~~~~~~~~~~~~~~
·
I
I
X
!
·
300
400
Soo
a2W
249
ILo
__
..
0%
O
'A
t
00
N
I
a
j-J:
C,
Cil
rfi
0Q
co
!
I
0
,,
0
\
6
m
o
C
-n
,,
C
C
E
-E
"L
C
L
$-
.
-C
o
o
N
5
1
4O
O
-o
N
-
0U00
w
w
m
C
-
-
w
r_
2
E-
-,
, C, C
I-
*T-I
01
0
00
Z7
t,
0
! i^
0aU)I
_
o
o
0
r4
ICt
,q
r
0
.C
C
,-
-9 I
4-4
Lb
0
0l
0
U)
'-I
0C
rl
1
0
C.4
'AL
I-
U
o
N4
0
_ O
I
0
|
.
I'D
oo
O
-
b
.
.. : 6O
O
u,2
E
C
C
..
._4
U.
. i
C
u
J
CWi.O
L 0C
C
I
.~~~~~~~~~
I
.
1+
N
dO
~S
O
I
I
I
CCC
C
r:
C-
I1
rl
250
LD
= 20 fps, 70% of the entrained
- = 60, u
H
will be
organisms
o
exposed
rise in the range of 3-3.5 F
temperature
to a maximum
for 38 minutes, 15% will experience a maximum temperature rise
in the range
than one hour.
three
rise in the range 2-2.5 F for longer
temperature
a maximum
and the rest will experience
F for 40 minutes,
2.5-3
tradeoff"
The "ecological
( and perhaps
times as short
a diffuser
three times less costly) can
of the entrained
a fraction
as diverting
then be interpreted
of building
organisms (45%) to be exposed to higher maximum temperature rises
9.3.2
(less than 10 minutes)
for short durations
(3.5-5.5F)
Case B:
Variable diffuser length and discharge velocity
at constant near field dilution
In this case when L D is decreased
D
the near field dilution
i)
L
H
ii)
the same.
= 60
u
=
15
fps
30
u
=
30
fps
LD
H
The total induced
is increased
u
to keep
~~o
We compare
flow for these two designs
the two designs
is not the same as
the asymptotic dilution is greater for the shorter diffuser
(cf.
the
LD
H
Fig.
9-2a);
thus
a greater
number
of organisms.
number
of entrained
ecological
organisms
histograms
be seen for the higher
30
=
=
30 fps
for design
the total
i), we plot the same
for the two designs.
velocity
entrains
design
Using as base value
(Fig. 9-6).
case the duration
temperature rise is reduced greatly
ten minutes
u
It can
for the maximum
from e.g. 50_minutes to
for 3.5<AT<4 0 F) at the expense
number of organisms at lower temperatures.
of entraining
a larger
251
U.
Sf
k0
it
r
I
sr
E
[,
oI
7,
H.
0 O
.)
·tI
0
-
na
Q,!
.,
N
U)
4 JJE
- I
o 0
4 -
C)
-
-4
0
C
C
(4:
Q *.4
C1
I.,z
I-
.-4
V
4C
,,-
0
C
4.
*r
W8 -
C
C
g
cU
)
_
C
s
c3
c~
c
%
0C
c;
OC(C
C -
o
4-
4-'
i-
I.
a
0p
--Z
d
ci
-4
^~~~~~~~~~~~~~~~~~~~~~~-
o
0
.,4
It
C
0
4
L
0i
I
I
I
o0
ti
)
o
E
o
sC
(
U)
4-4
ar4
0
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I-,
o0
.H1
o
o
a)
-
0
-4
o
l c
I
o
L
.,..
U,
0
*
I0
C-
I
o
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44
(',1
I
0C.-1
L
0n~
;
cr
.H
tc
M-.4-- >
V
·~
C
E
0CO
C
-C
,
0
II
il
CC
cO
C
o
C U:
C CC%
CC
VO
0
a)
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V4
m
:
c-
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V~
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E
4- C,,
C
-0X
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am
\0
+
I
o
252
9.3.3
Concluding Remarks
By presenting
the results
in this discretised
fashion
more information pertaining to possible ecological impact
can be extracted
in addition
to other quantities
previously
adhered to: induced velocities, maximum temperature rise in
the field, plume
areas.
It should be pointed
out the assumption
of the quantity
of entrained organisms being proportional to induced flow is
only strictly valid for weak swimmers and drifting organisms,
such as embryos
circulated
and
through
larvae of fish species.
Aquatic
the power plant are not accounted
organisms
for in
the ecological histograms, as their inclusion would involve
the consideration of many site specific conditions:
intake
design, condenser flow, length of connecting piping to the
discharge structure, etc.
Furthermore, the number of organisms
actually pumped through the plant at much higher temperatures
than AT
affected,
is a small percentage
the ratio being
for example,
S
types
characterised
by 1/S
.
of organisms
In case A,
= 15, and the ratio is 0.067.
The concept
to other
of the total number
outlined
in this section
of heat dissipation
schemes
can be extended
(e.g. surface
discharges) as well as other sources of pollutant (e.g. chlorine
and copper), provided the initial concentration at the discharge
point
can be estimated.
253
9.4
Prototype Considerations:
All the model
point
Short vs long diffusers
interpretations
in sections
to say that in general by increasing
velocity u
9.2 and 9.3
the discharge
(short diffusers), the diffuser performance is
greatly enhanced by a large reduction in
i)
isotherm areas
ii)
the
duration
in the high temperature ranges for
entrained organisms.
The advantages offered by a short diffuser discharging
at relatively higher velocities than conventional designs can
be further
a)
substantiated in a prototype
setting:
In coastal regions, thermal diffusers are usually
located close to the shoreline and oriented to direct the
thermal
piping
effluent
required
offshore,
as shown in Fig. 9-7.
for a diffuser
is then the sum of the diffuser
length LD
and the length of piping
structure
and the power plant, L .
Because
The total
connecting
of the two dimensional
nature
the discharge
of the fluid
motion induced by the shallow water discharge, the diffuser
induced flow effectively blocks the oncoming current for some
distance offshore.
is greater
If the total induced back entrainment, SQo,
than u L H, the amount
ap
of flow
that normally
passes
the region between the diffuser and the shoreline, possible
local
re-entrainment of warm mixed water can occur.
Thus
L , the length of connecting piping required to avoid local
P
254
Long diffuser
Low velocity disc]
ambient
current
A1
back
entral
power plant
Short diffuser
High velocity discharge
-
I
lateral
entrainment
ambient
current
I
-;O
ting piping
shoreline
li
Fig. 9-7
power plant
Prototype considerations of diffuser design in
coastal region: short vs long diffusers
255
re-entrainment
LD.
As S fL,
D'
expect
L
p
/L
D
is also a function
(for constant
D
u
of the diffuser
length
cf e.g. Eq. 9.2) we would
o
.
In order to avoid local recirculation leading to undersirable heat build-up, therefore, a longer diffuser has to be
located further offshore.
proportional
Assuming the cost of piping is
to its length,
diffusers of length LD1
L
L
. would
pl' p2
the economic
tradeoff
of two
LD2, and connecting pipes of length
be characterised
by
LD1 + LP
LD2 + Lp2
(cost of piping)l
(cost of piping)2
LDI
(yLP)
L
L
L
L
2
(L1) + ( PI___P-
LD2
p2
1 +
Supposing
the connecting
L
L
p2
Thus building
LDILD+
= (LD1)
D1
D2
D2
a diffuser
As pointed
we have
twice as long would bear an
2= 3.4 times higher.
out in Chaster VII,
u
momentum
/ L
D2
cost
economic cost 2 +
b)
)
D2
pipe is as long as the diffuser
length for design 2; since
ratio of pipes
(L
D2
flux ratio
FM =
for low values
of the
LDH
U
(small L D , large u ), the
0 0
near
field mixing
is not affected
by the ambient
current;
whereas for FM> 0.1 (large LD, small u ), plumes of temperature
rises
two or three times greater than ATN areobserved for
rises two or three times greater than AT N are observed for
256
large distances downstream.
pointed
out in Chapter
The empirical correlations
VII can be used to design
diffusers
with sufficiently strong momentum flux, so that the
temperature reduction is improved by the crossflow.
In periods of prolonged slack, long diffusers tend
to induce stronger
was discussed
far field recirculation;
in detail
In summary,
in Chapter
VIII.
it should be pointed
on the geomorphological
this topic
and hydrologic
out that depending
conditions
at a
specific site, the economic and ecological advantages offered
by short diffusers have to be weighed against higher pumping
costs, possible navigation hazards caused by higher induced
velocities, and sedimentation problems.
discharge
velocity
u
In general, the
should be set to the highest value that
isacceptable
by other considerations of importance.
is acceptable by other considerations of importance.
257
CHAPTER X
SUMMARY AND CONCLUSIONS
Objectives
10.1
This study is concerned with the understanding and
prediction of the velocity and excess temperature field
induced by a unidirectional thermal diffuser in shallow
water.
A unidirectional thermal diffuser is a submerged pipeline discharging waste heat from steam-electric power
in the form of a line of high velocity
generation
heated
All the discharge
jets.
direction.
ports are pointing
turbulent
in one
This type of once-through condenser cooling
water discharge is effective in mixing the heat with the
ambient water within a small area, and is hence capable of
meeting stringent thermal standards.
momentum
in one direction,
It provides large
and can direct
the thermal
effluent
offshore.
The objective is to understand induced flow and temperature
patterns as a function of diffuser design and ambient conditions
(discharge velocity, diffuser length,strength of ambient current)
The ultimate
goal is to establish
a framework
upon which
the
design of unidirectional thermal diffusers in coastal regions
can be based.
Three
cases of fundamental
1) diffuser operating in
interest
are studied:
stagnant conditions or a coflowing
258
current, 2) diffuser discharging into a perpendicular crossflow, and 3) potential heat build-up due to recirculation
induced by the pumping
Prior
action
of the diffuser.
to this study, knowledge
of a unidirectional
diffuser in shallow water was primarily limited to a onedimensional analysis of the total induced flow in the near
field
(Adams, 1972),
and scattered
experimental
results of
site specific hydraulic scale model studies of limited
general value.
No comprehensive treatment of the excess
temperature field induced by the diffuser in an open water
body had previously been attempted.
10.2
a)
Theory
The theory for the diffuser operating in stagnant conditions
or a coflowing
current,
forms the analytical
as outlined
foundation
in Chapters
for the entire
III and IV,
study.
A
vertically fully mixed condition is hypothesized downstream of
the diffuser.
Vertical variations of temperature are neglected
and a two dimensional
theory
is based
model
is formulated
on the coupling
and solved.
of a potential
The
flow model in the
near field with a friction jet model in the intermediate field.
The predictions
do not involve
any fitting
of theory to
experimental data; all physical coefficients are taken from well
established
diffuser
data in the literature.
plume
As the travel
time in the
is of the order of an hour, surface heat loss
effects are neglected.
259
By neglecting turbulent side entrainment and bottom friction,
the total induced flow near the diffuser is first derived from
a control volume
analysis.
This is then used as an integral
boundary condition to obtain a potential flow
the near field.
The near field solution
solution of
is coupled
into an
intermediate field model that predicts the global temperature
field, by incorporating lateral entrainment and bottom friction.
Scaling arguments show that Laplace's equation satisfies
the equationsof motion in the near field, where pressure and
inertial effects dominate.
A control volume analysis yields
two gross parameters - the induced flow from behind SQ , and
the contraction
ratio
a of the mixed
flow (Eq. 3.38, 3.40)
These provide the integral boundary conditions for the potential
flow model.
By assuming uniform induced flow locally along
the diffuser line, the complex potential plane and the hodograph
plane
corresponding
of a
slip streamline
to the flow can be derived.
which models
discontinuity is hypothesized.
The existence
the locus of velocity
The position of the slip stream-
line can be calculated using classical free streamline theory
(eq. 3.62) by mapping
plane.
the complex
Once the slip streamline
potential
is known,
onto the hdograph
the flow and
pressure field can be calculated by solving Laplace's equation
in a region with known boundaries and boundary conditions.
260
is a function
The near field model
=
LDH
the ratio of the intercepted cross-sectional area
,
NaO '
to toal discharge
u
a
port area, and y= ua , the ratio of the
o
ambient current to discharge velocity.
results
of two parameters
show that the extent
The theoretical
of the near field is 0.5L D and
the computed slip streamlines always match asymptotically the
contraction ratio predicted independently by the control volume
The solution reveals a region of high pressure in
analysis.
The potential
front of the diffuser.
energy
gained
from the
diffuser momentum is re-directed into longitudinal momentum
as the flow contracts
distribution
along
and accelerates
the diffuser
downstream.
The velocity
line can also be inferred
as:
SQ
a
=
(10.1)
LD
u sinO
v(y)
A
2y
A
(10.2)
LD
In the intermediate
field,
an integral
friction
jet model
is formulated for the diffuser plume; inertia, turbulent side
entrainment, and momentum dissipation due to bottom friction
are included.
potential
Pressure variations are computed using the
flow solution
beyond the near field.
in the near field, and are neglected
The entrainment assumption relates
the increase in volume flux across the plume to a characteristic
centerline velocity excess.
The global velocity and temperature
261
field is assumed to possess a boundary layer type structure
(section 4.1) and is divided
of
flow establishment
lateral entrainment
temperature
refers
into two regions:
The region
to the zone where
has not penetrated
turbulent
to the core.
rise in the core is assumed
The
to be AT IS.
The
0
unknowns in this region are u , the centerline velocity,S(x)
the core width,
and L(x) the diffuser
length of this region
is obtained
layer thickness.
as a solution
The
to the problem.
In the region of established flow gaussian profiles are assumed
for the velocity
region are u,
and temperature
AT , and
Cc
rise.
H
.
in this
L(x).
The entire model is a function
foLD
field parameter
The unknowns
of
,y and an intermediate
Integrating the governing equations
across the plume yields a system of first order ordinary
differential equations in terms of the unknowns in each region.
An analytical
solution
of the friction
jet model
in the
special case of no ambient current yields the following
interesting
features
free turbulent
(eq. 4.52):
In contrast
to a 2D or 3D
jet, the momentum flux of a friction
foX
dissipated exponentially as e
8H
.
jet is
The solution possesses
an exponential type of behavior, leading to a divergent plume
with rapid decrease in velocity.
of heat recirculation
predicts
plume.
into the near field.
an asymptotic
This places
This suggests the possibility
temperature
rise AT
a limit to the mixing
The model also
(eq. 4.54) in the
capacity
of any
262
unidirectional diffuser.
In the presence
of a coflowing
current,
the numerically
computed results indicate that the plume is much narrower
and that the difference
in AT
c
small
b)
in the intermediate
For a diffuser
from the stagnant
case is
field.
discharging
into a perpendicular
crossflow,
(Tee-Diffuser) a control volume analysis predicts that the
near field
dilution
is independent
of the current,
a
S =
2
(eq. 7.8).
Given
for calculating
the total induced
the plume
formulated (eq. 7.13).
in natural
flow from behind,
trajectory
in a crossflow
a model
is
The conservation equations are written
co-ordinates.
The model uses top hat profiles
and
incorporates the same entrainment assumption and frictional
dissipation as in the coflowing case.
Pressure variations
in the stream direction is neglected.
The major factor in
determining the plume trajectory is the dynamic pressure
created
by the blocking
is assumed
to take on the form
is a blocking
deflection
of the crossflow.
coefficient.
is dependent
on
Fb
=
The bending
CDp(u
The results
a parameter
sine)2 where CD
show that the plume
u 2LDH
=
F
M
Q0
y
c)
the value of F M , the greater
The phenomenon
by the analytical
The
the plume deflection.
of far field recirculation
results
2
40
measuring the ratio of ambient to diffuser momentum.
larger
force
of the coflowing
is suggested
current
case. Under
slack water conditions, the momentum flux of the diffuser plume
263
is dissipated
at a characteristic
distance
of
A
f
0
stagnant pool of water at this distance would, by flow
continuity, serve as a source for the back entrainment demand
of the diffuser,
resembling
thus creating
a source-sink
type of flow.
and time scale of recirculation
Heuristic
arguments
a far field recirculation
suggest
Characteristic
can be estimated
length
(eq. 8.1, 8.2).
that the heat recirculation
can be
SQ
regarded
as a function
of the 'sink strength'
of the diffuser
H
and the scale of recirculation LREC .
Semi-empirical correlations
can be made based on these considerations.
10.3
Experiments
A comprehensive
set of laboratory
tests were
carried
out in
a 60'x40'xl.5' model basin for a wide range of diffuser lengths
LD
D
D
L
3
~D
D = 100) and ambient
conditions.
Velocities
are
measured using time-of-travel method and detailed temperature
measurements are made.
Several series of experiments with
different objectives were conducted as described in Chapter V.
10.4
Results and Comparison with Theory
For the case of a coflowing
current
the experimental
data of this study validates the theory both qualitatively and
quantitatively.:
of the diffuser
surface pathlines always indicate a contraction
induced
of diffuser lengths.
flow at about
0.5 L D for a wide
range
The observed flow patterns are in
qualitative
agreement with the predictions of the potential
flow model.
A core region of roughly constant temperature
264
is confirmed, and the predicted isotherms compare favorably
with the observed temperature field.
The centerline temperature
rise, centerline velocities, and isotherm areas agree well
The theory
with data.
an extensive
out by Acres American
set of tests carried
(Ref. 3)
Good agreement is obtained.
structure
also demonstrates
in the discharge
with the results of
is also compared
dominated
The detailed thermal
the validity
regime.
Corp.
of the 2D approxmiation
The results
show the exact
temperature distribution,as influenced by the detailed diffuser
design, surface and bottom conditions, is very complicated.
good agreement
interest,
of
of theory with data for all the parameters
however,
justifies
the use of the model
The
as a valid
way of describing the essential features of the complicated
The observed plume width is usually greater than
phenomenon.
predicted,
due to stratification
tests it is found experimentally
For Tee-diffuser
FM
at the edge of the plume.
2.1, the observed
near field temperature
considerably higher than ATo /S.
ATMAX is correlated with FM .
The maximum
that for
rise can be
temperature rse
The predicted plume trajectory
compares favorably both with results of this study and the Perry
tests.
A CD
4 is assumed
The phenomenon
in the laboratory.
in all the calculations.
of far field recirculation
is ascertained
It is found that for long diffusers
and
strong momentum dissipation, heat recirculation into the near
field can cause
temperature
rises two to three times higher
265
than predicted.
Empirical correlations of the scale of
temperature
(Fig. 8-7) and maximum
recirculation
rise
(Fig. 8-8) are presented.
10.5
General Conclusions
This study provides a basis for comparing the effectiveness
of different
heat dissipation
schemes.
The model can be used to
calculate induced velocities and temperature rises for a
particular diffuser'design.
for estimating
isotherm
Simple, engineering formulae
areas of interest
are given in Appendix
C
The design implications of the model predictions are interpreted
in Chapter
IX.
A novel
interpretation
of the results
for purposes
of environmental aquatic impact assessment is advanced.It is
proposed to compare designs on the basis of 'ecological
histograms' which embody information from both the velocity
field and the temperature
field.
The general
design
implications
are:
i)
For fixed discharge velocity, a longer diffuser induces more
back entrainment, but less lateral entrainment in the intermediate
field.
The striking feature is that beyond a certain distance
(about 200 water
reflected
depths)
the effect
in the field variables,
show that if thecriterion
of the diffuser
is not felt.
of interest
length,
as
The results
is the area within
a low
temperature isotherm, a short diffuser may be as satisfactory
as one that is two or three times as long.
266
The 'ecological tradeoff' of building a shorter diffuser
for this case can be interpreted
of the entrained
organisms
as diverting
to be exposed
a fraction
to higher maximum
temperature rises for very short durations.
ii)
For variable diffuser length and discharge velocity at
constant near field dilution, it is shown that by increasing
the discharge velocity while proportionally decreasing the
diffuser length,a large reduction in centerline temperature
rises and isotherm areas is obtained.
interpretation
case the
greatly
The ecological
for this case is that for the high velocity
duration
for the maximum
at the expense
temperature
of entraining
rise is reduced
a larger number
of
organisms at lower temperatues.
iii)
The empirical
correlation
of Chapter VII shows that
by designing diffusers with more concentrated momentum input
(short diffusers with velocities higher than conventionally
accepted values), so that FM<O.1, the temperature reduction
can be improved
iv)
by the crossflow.
The experimental results of Chapter VIII demonstrates
that a longer diffuser is more prone to heat recirculation
into the near
field.
In summary, by designing shorter diffusers with higher
velocities, the diffuser performance is greatly enhanced
by a large reduction
in isotherm
areas and the duration
of
267
high temperature ranges for entrained organisms.
These
advantages, however, have to be weighed against higher
pumping costs, possible navigation hazards caused by
higher induced velocities, and sedimentation problems.
10.6
Recommendations for Future Research
More studies, both experimental and theoretical,
on the mechanics
of a Tee-diffuser
discharging
perpendicular crossflow should be pursued.
limited amount of available
that for a low diffuser
crossflow,
into a
Based on the
data, it is our speculation
momentum
input into a strong
F
> 0.l,stratification
is induced in the near
M
field and the mixing capacity of the unidirectional
diffuser is impaired.
A more satisfactory explanation of
more general applicability is needed.
Based on the insights gained in this study, numerical
modelling of unidirectional diffusers can be attempted.
Numerical models can offer more information in a specific
prototype situation with complicated boundary geometries
and conditions.
and distance
Effects of a sloping bottom, ambient current,
from the shoreline
can then be studied.
Our results indicate that beyond a large distance
(X1000 water depths) the flow will be stratified for most
designs without affecting the results in the discharge
dominated fully mixed regime.
More work should be
directed to
268
investigate
flow with
the coupling
of this far field buoyant
the near and intermediate
this study.
field model of
269
BIBLIOGRAPHY
1.
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Hydraulics Laboratory, Publication No. 29, (1963).
2.
Abramovich,
G.N.,
"The Theory
of Turbulent
Jets".,
The
M.I.T. Press, M.I.T. Cambridge, Massachusetts (1963).
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4.
Adams, E.E., "Submerged Multiport Diffusers In Shallow
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5.
Albertson,
Dai, Y.B., Jensen,
M.L.,
R.A., and Rouse, H.,
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6.
Almquist, C.W., and Stolzenbach, K.D., "Staged Diffusers
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Boericke, R.R., "Interaction of multiple Jets with Ambient
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8.
Brooks, N.H., "Dispersion in Hydrologic and Coastal
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Carnahan, B., Luther, H., Wilkes, J., Applied Numerical
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10.
Dec. 1972.
Cederwall,
Wiley and Sons, 1969.
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Gurevich,
M.I.,
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Harleman,
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D.R.F.,
Jirka,
G. and Stolzenbach,
K.D.,
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the Shoreham Nuclear Power Station", M.I.T.
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Technical
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272
LIST OF FIGURES AND TABLES
Page
Figure
1-1 a) Multiport Diffuser Discharge in Deep Water:
Stable Buoyant Jet
16
b) Multiport Diffuser Discharge in Shallow Water:
Unstable Flow Situation
1-2
Basic Types of Multiport Diffuser Configurations
1-3
An Example
of Temperature
Patterns
Induced by a
17
19
Unidirection Thermal Multiport Diffuser
Two-Dimensional Free Turbulent Jet
25
2-1
The
2-2
Schematics
2-4
Downstream Flow Regimes for a Horizontal Buoyant
of a Slot Buoyant
Slot Jet in a Coflowing
1971)
2-5
Jet
Stream
26
Temperature Profiles Downstream of Heated Jets
placed
31
(From Cederwall,
33
in a Channel.
2-6
Numerical Predictions of Diffuser-Induced Flow
Patterns (From Boericke, 1975)
36
2-7
Near Field Instability of Unidirectional Diffusers
in Shallow Water
39
3-1
Unidirectional
Diffuser
in a Coflowing
Current:
42
Problem Definition
3-2 a) Schematic of Surface Flow Field
b) An Instantaneous
Snapshot
of the Diffuser
Induced
52
Flow Field in Slack Water
3-3
Control
Volume
Diffuser in a
Analysis
of a Unidirectional
57
Coflowing Current
Flow Model
for the Near Field
65
3-4
A Potential
3-5
Boundary Conditions along the Diffuser Line
66
3-6
Conformal Mapping of the Complex Potential Plane
W onto the Upper Half Plane
72
273
Figure
3-7
Page
Conformal Mapping of the Hodograph Plane
/
onto the Upper Half Plane
3-8
Theoretical
Solution
Different Angles of
3-9
Two Different
of Slip Streamline
for
78
of the W and Q
82
.
Approximations
Diagram
3-10
Comparison of
Approximation
3-11
Boundary
Predicted
A and B
Conditions
Slip Streamlines
of
for the Stream Function
83
in
84
the Near Field
3-12
Grid System used for Finite Difference
Step Size = 0.1
Solution;
3-13 a)Finite Difference Solution for the Potential
Flow Model. Illustrated Streamlines
85
86
b)Transverse Pressure Distribution at Different
Longitudinal Positions in the Near Field
4-1
4-2
A Unidirectional
a)Centerline
Diffuser
Velocity
in a Coflowing
Current
Decay in a Two Dimensional
90
104
Friction Jet
b)Plume Width
c)Centerline Temperature Decay
4-3
Numerical
Solution
5-1
Schematic
of Experimental
5-2
Multiport Injection Device
120
5-3
Temperature Probe arrangementfor Series FF
128
Temperature
132
5-4
in a Coflowing
Measurement
Current
Setup
Grid for Series
111
119
200
Experiments
6-1
a)Recorded Surface Flow Patterns for a Unidirectional 137
Diffuser in Shallow Water: with and without
Current
274
Page
Figure
6-1 b) Dye Injection Pattern in the Near Field
Diffuser
of a Unidirectional
6-2
Comparison of Predicted Near Field Dilution S
Eq. 3.38 with Experimental Results
139
6-3
Comparison of Predicted Isotherms with Temperature
141
Measurments
in
Series
FF
6-4
Comparison of Predicted Centerline Velocities with
Experimental Data
6-5
Dye Injection
Series 200
6-6
Comparison of Predicted Centerline Temperature
rises with Series 200 Experiments
150
6-7
Comparison of Predicted Isotherm Areas with
Series 200 Experiments
155
Pattern
of Diffuser
Plume:
6-8 a) Typical Centerline Surface Temperature Profile
along Diffuser Axis (Run 201)
145
149
157
b) Transverse Surface Temperature Profiles at
Different
6-9
6-10
x (Run 201)
Cross-Sectional Temperature Transects at Various
Longitudinal Positions
159
Isotherm Plot of Model Study by Acres
164
American,
Inc.
6-11
Comparison of Theory with Perry Model Tests:
Plume Centerline Temperature Rises
166
6-12
Comparison of Theory with Perry Model Tests:
Isotherm area
169
6-13
Comparison
of Theoretical
Predictions
of
172
Isotherm Areas with Perry Tests and Series 200
Experiments
6-14
Comparison of Predicted Isotherm area
with Field Data: James A. Fitzpatrick Nuclear
Power Plant, New York
176
275
Figure
6-15
Page
Comparison
of Predicted
Isotherms-----
with Field Measurements:
Fitzpatrick Power Plant
7-1
Unidirectional
Diffuser
177
James A.
in a Perpendicular
180
Crossflow
7-2
Near Field Dilution induced by a Unidirectional
Diffuser in an Inclined Cross Flow
187
7-2.1
Predicted
of
188
Tests:
190
Plume Trajectory
as a Function
Governing Parameters
7-3
Experimental
Series FF
7-4
Typical Recorded Surface and Bottom Temperature
Field
7-5
Observation
in a T-Diffuser
of a T-Diffuser
Effect of Ambient Crossflow on Near Field
Mixing:
Surface
191
Test; Series FF
193
Isotherms for Two Diffuser
Designs
7-6
Empirical Correlation of Near Field
Temperature rise for T-Diffuser
194
7-7
Comparison of Predicta Plume Trajectory
and Observed Plume Boundary -----
195
Series FF
7-8
Comparison of Predicted Plume Trajectory
and Observed Surface Isotherms in Perry Model
Study
199
7-9
Sensitivity
203
of Predicted
Plume Trajectory
to
Blocking Coefficient CD
8-1
Schematic of Diffuser Induced Recirculation
206
8-2
Recirculation
Test
Demonstrated
207
8-3
Recirculation
Dye Front
Illustrated
8-4
Effect
in a Full Diffuser
By Movement
of Far Field Recirculation
of
212
on the
213
Normalised Excess Temperature Field:
Recirculation
Strong
276
Page
Figure
8-5
214
Surface Flow Field of an Experiment
Characterised by Weak Recirculation Effects:
LREC ' model dimension
8-6
of Far Field
Effect
Recirculation
on the
normalised Excess Temperature Field:
recirculation effects
8-7
Correlation
Empirical
215
Weak
of the Scale of
217
Recirculation
8-8
Empirical Correlation of Observed Maximum
Temperature
8-9
Recorded
218
rise due to Heat Recirculation
Surface
(Short vs
221
(Short Diffuser)
222
Flow Field
Long Diffuser)
8-10 a)Recorded
Dye Patterns
b)Recorded Dye Patterns (Long Diffuser)
8-11
Recorded Normalised Excess Temperature Field
224
(Short vs Long Diffuser)
9-1
a)Centerline Temperature Rise along the
Diffuser Plume: Design Comparisons (Case A)
233
b)Centerline Velocity along the Diffuser
Design
Plume:
Comparisons
(Case A)
c)Diffuser Plume Width: Design
Comparison
(Case A)
d)Area within Surface Isotherms:
Comparisons
Design
(Case A)
e)Time History of Exposure to Elevated Temperatures
for Entrained Organisms: Design Comparison
(Case A)
f)Densimetric Froude number along the Diffuser Plume:
Design Comparisons (Case A)
9-2
a)Centerline Temperature Rise along the Diffuser
Plume:
Design
Comparisons
(Case B)
b)Area within Surface Isotherms:
(Case
B)
Design Comparisons
241
277
Figure
Page
c)Time History of Exposure to Elevated
Temperatures for Entrained Organisms:
9-2
Design Comparisons
(Case B)
d)Centerline Velocity along the Diffuser Plume:
Design Comparisons (Case B)
9-3
An Ecological Interpretation of Diffuser Performance 247
9-4
Distribution of Induced Flow along Diffuser
Plume
248
9-5
Comparison
of "Ecological"
Diffuser Design:
9-6
Comparison
Performance
of
249
Performance
of
251
Case A
of "Ecological"
Diffuser Design (Case B)
Prototype Considerations of Diffuser Design
in Coastal Region: Short vs Long Diffusers
254
3-1
Summary of Governing Dimensionaless Parameters
46
3-2
Near Field Dilution and Cenfiraction Coefficient
76
9-7
Table
for Typical values
of
D
o
Na
5-1
Run Parameters
for Series
0
and y= u
0
126
5-2
Run Parameters for Series FF Experiments
130
5-3
Run Parameters
133
A-1
Comparison of Theory assuming High Dilutions
with Exact Theory
MJ Experiments
for Series
200 Experiments
286
278
LIST OF SYMBOLS
Superscripts
turbulent fluctuating quantities
time-mean quantity, or quantity
normalised with ATN, or depth averaged
quantities
dimensionless quantity
*e
in front of and behind
the diffuser
Subscripts
a
ambient value
A
value at point of separation
0
initial or discharge value
c
centerline value
D
quantity at diffuser line
e
entrainment quantity
i ,j
indices
cx0
value at large distances from diffuser
r
ratio of prototype
1,2
indices
I,
quantities
II
to model
in Region
I and II
N
near field quantity
a
diffuser
al,a 2
constants determined by conformal mapping
iq slip streamline calculations
0
ao a 2
A,
As,
A AT
port area
isotherm area
279
b
bi
edge of plume
determined
constant
b
in conformal
transformation in slip streamline
calculations
B
slot jet width
b0 ,BO
initial jet width
c
concentration, clearance of diffuser
port from bottom
~c'l~ ~empirical
coefficient relating mixing
length
c 1,
c2
CD
CQ CW
to jet width
of
constants
integration
blocking coefficient
scale factor in conformal mapping
diameter
D ,D
port or nozzle
E
index of dilution comparison of multiple
jets with an equivalent
F
slot jet
source densimetric Froude number of
slot jet in confined
buoyant
flow
Fb
plume deflecting force
FD
densimetric Froude number based on
nozzle diameter
u
o
Ap
/ g-° D0
a
FH
local densimetric Froude number based
on water depth
.U
c
.
.
/r e AT cH
280
ratio of ambient to diffuser momentum
FM
u
a
2L
H
LII
D
QQoU0o
u
F
densimetric Froude number of equivalent
s
slot jet
u
_
_
_ _ .--
/co
B
aa
f
friction factor
0
acceleration
y coordinates
g
y-coordinate
due to gravity
of given isotherm
at any x
of slip streamline
at any x
H
water depth
AH
characteristic free surface deviation
h
local free surface elevation
h+ , h
free surface
level in front of and behind
diffuser
Ah
,,
hi, h 2
I
I
1
AT'
head rise across diffuser, free surface
deviation
steps towards convergence in the determination of ala2,blb
2
12
I3
integrals in conformal mapping
I
I
integrals in isotherm area computation
1'
.2
i ,
imaginary
number
= /-1
i,j
indices
k
variable of integration
281
L
characteristic length scale
L(x)
diffusion thickness in diffuser plume
LD
diffuser length
LI
diffuser layer thickness at x
LREC
scale of recirculation
L
length of piping connecting power
plant and diffuser
p
mixing length scale
length cale ratio of prototype to model
r
momentum flux across plume
M
=u 2
m
cC
L
N
number of diffuser ports
n
coordinate in direction normal to plume
trajectory, index
p
pressure
P
pressure force over plume cross-section
Qo
power plant condenser flow
Q
hodograph plane, volume flux across plume
QA,
Q1,1Q29QB
points on Q plane
diffuser induced flow from behind
q
velocity vector
qD
induced velocity along diffuser line
magnitude
of velocity
=
qD
i
282
R
Reynolds number of ambient current
based on water depth
u H
a
v
RD
discharge Reynolds number based on
nozzle diameter
u D
00
S
near field dilution
s
port spacing, distance along plume
centerline
T
temperature
AT
temperature
T
ambient temperature
a
T
o
excess
= T-T
a
discharge temperature
ATMAX
maximum temperature rise
t
time, variable
TREC
TREC
time scale of recirculation
(u,v,w)
velocity components in (x,y,z) direction
U,V
characteristic
of integration
velocities
in x and y
direction
ua
ambient current velocity
u
discharge velocity
ud
x-component
uAvA
velocity components at diffuser end
ve
entrainment velocity
u
centerline velocity
uN
velocity
c
of
qD
in induced
near field
283
u
induced.velocity given by potential
flow solution
P
velocity
at x
UI
centerline
u.
1
mean plume velocity in the near field
W
complex potential plane
WAWl,
W 2 ,WB
points on W plane
(x,y,z)
caresian co-ordinate directions
xI
location
of end of region
uM
velocity
at point of merging
I
of
interacting jets
XAT
centerline position of isotherm
_____________________________________________________________
aI' II
entrainment coefficients
aAala2,aB
interior angles of polygons approximating
curvilinear
figures
near field parameter
=
LDH
Na
0
81
o D
=S./a
momentum
dissipation parameter =16H
-32 aIiH
cL foLD
coefficient of thermal expansion of water
e
u
to discharge
velocity
y
ratio of ambient
p
density
Ap
density difference
6
potential core width, characteristic
transverse length scale
=
u
a
0
284
6
angle of inclination
aA
angle of separation
ratio
1' 2
of
sides
of
at diffuser
end
polygons
in limits of integration,
small quantity
E
to x-axis
error of approximation
a
~a
~inclination
of ambient current to diffuser
jet spreading
X
ratio between mass and
momentum
v
kinematic molecular viscosity
a
contraction coefficient
~~~T
~bottom
shear
velocity potential
'p+~~ ~stream
function
upper half plane, variable of
integration
285
APPENDI
A
CONTROL VOLUME ANALYSIS VALID FOR ALL DILUTIONS
Following the same general assumptions outlined in
Chapter
III, the more exact version
the case of arbitrary
dilution
of eq. 3.34-3.36
for
values of the induced near field
S is
a + Q ou
(S-1) Q
= SQ ouN
2
2
uN -u
Ah
+ Qo u
(S-1) Qud
N
a
2g
= SQ OUd + gHLDAh
(s-1)Q
where
ud
=
x component
of
q
d
Combining
D
D
and rearranging,
H
~~~~LDH
a cubic equation
in S can be
obtained:
S
3
($+l)S
2
+2
2
0
2
02(1-
(A.1)
0
of A.1 are compared
for a range of dilutions;
1-
ua
u
LDH
Na
The predictions
a2
(1-y)S +
with that of eq. 3.38
the comparisons
show that even for dilutions
in Table A-1
as small as 3, the error
involved in using the thory for high dilutions is only a
few percent.
286
Predicted
dilution
assuming
S>>1
Predicted
dilution
using
Eq. A.1
Ratio of
predicted
values
Eq. 3.38
10.0
2.24
2.41
1.08
15.0
2.74
2.93
1.07
20.0
3.16
3.37
1.07
25.0
3.54
3.75
1.06
40.0
4.47
4.69
1.05
50.0
5.00
5.22
1.05
60.0
5.48
5.71
1.04
85.0
6.52
6.75
1.04
100.0
7.07
7.31
1.03
y
Table A-1:
= 0O
Comparison of theory assuming high
dilutions with exact theory
287
APPENDIX
B
SCHWARTZ-CHRISTOFFEL TRANSFORMATION FOR AN ARBITRARYPOLYGON
WITH SIX VERTICES
A
A..
W A"Ui
kw
Be
AO
I
;
A
-I.o
The W-C
I1
ihA
.
.~~~~~~~~
(,ao)
.
.
(at,o) B
/wo
transformation
is illustrated
transformation is entirely analogous.
below.
The Q-4
288
W()
= Cw
aA-1
(+i)
(4-a1)
1
a
t
-1
-1
(4-a2)
(~-i )
LB-1
dC + W A
-1
=
W(1)
B
-
W
>
1
Ia
CW
W
-1
a2-1
-1
-1
A_ i (_-ai)
(4+)
(4-1)
(4-a2)2
B
-1
2:
To find a,a
ratio of sides, for 'conformal' transformation;
preserve
I2W1
BW
I w2wli
it
w l!
1
IA
W lI
A2
Let
a
I! (a,
2
)
=11
aA
1
(4+1)
a22
I2 (a 1 ,a 2 ) =
(C+1)
)-la
(
1 -1
(a2 -C
(ai-4)
2
-1
A1
A-1
)
2
(-a
(a2 -)
a 2-1
(1-) aB-1
(i-~)
a B-1
(1-C)
a
1
+A
I3(al a 2)
a2
(4+1)
-1
(c-a 1)
a B-1
1
a2-1
-1
(C-a2)
Then
(*)
i22 = A 1I1
I
3=
(ala
Solution
X
2I1
2)
for a
is the solution
a2:
Start with an initial
that satisfies
Newton-Fourier
(*)
Method
guess a(O) = (a (0) , a2 (o)))
289
T.nen, a Taylor's series expansion gives
I'
a )
(ka)
2
aa
(a 2
( ° ) ) ) ++ h 1
1 (a
(a)
13 ah
a
= Il
(a) =
_
(a(°))
hl
1
1
a
(~~~~~~o)
+
2 a 21-
II 1( a()
(2(a )
+ r2
(0))
()
+ 2(a
Da 2
i2( a ) +a
2
2
1
I3(a) = i3(a()
1 (a (o) )+
(O)
2
+ hl a
+
I (a())
h2
I(a
) +
Substituting these expressions into (*)
(a())
I2-a(°))
++ h 1aa I2(a(o))
-
+
h2 a a
(
2
1
1(
1
+ h
3 (- )
(a(o
h 11
1
() + h
(a ())+
I
aaa 1
3(
1
a
31
I (a())+
(a())
1
a
I
a
h2
I1
h 1 [
1 3T 1
-->hi
[D~a
h2D
a
)
2h2
2
t
I1
2Daaa22
2
3
a 2
1
a
1
Da
(0
)
]
1
(a-)=%2
(°)
1
)
1
I2
2
1+*
aa
2
(B. 1)
aI
22
a
2
Solve B.1 for (h1 ,h2 )
set -a(1) = -a(°) -+ ( 2 )
and iterate
[I 1-(a(
~(o)
I3
Da1
(a(0)
%2 [2I(a(°))
1
1
h1
)=
(a(0)
)
-
__
h
h
until convergence
i.e.
h I = h2 = 0
2
1
3
290
The partial
derivatives
in B.1 have to be evaluated
iteration:
Their expressions are given below
at each
a
a -1
a l
a
l
2
a
2
-
-I
+a
a2 -2
(a2- C)
I1-
(
1
1
a -1
(a_1
-(
(1-cD)
a
1
i+ A
(C+1) a
Ia
B-)
l+al
a
,1-1
-1
(C+I)
cA
el-1
(al-C)
(a B-1)
2
12
a -a
(a2 -C)
cx2-2
2
-
(1-)
previously
a
1
d
(B.2)
(l+ai)-1
2
-2
(1-C)
a
(a2-1)
I 1 is defined
c
c 2-l
(a 2 -%
C)
a 2-a
2
1
UA(C+1)
(C-a1 )
-1
a2
(a2 -C) 2(1-)
aB -2
1
(aA
1
a2
)
tA-
2
( +1)
(a -a )
a1-1
(c-a)
2
a2
OB-1
)i-
drC
a1
I2
a2
l +a 2-1
a2-a
(a Bi)
2
a2-al
[
(C+1)
a
(C-al 1 )
(a 2-1)
a -1
)-1
a
( 1-B)
2
1
a
(a - 1 )
( a -a
a
aA -2
)
(;+1)
Sa
(C-al )
(a2 -)
(B. 3)
-)
1
3-
I2 is defined
(
previously
- )
C A- 1
[1
a2
1 -2
(
a 2- (
B-1
291
3
aa 2
=
i
-B I
2
1-a 2
+
3
(al-l)-i
aA-1
I
Ja (1+0)
1-a 2
a-2CA-~2
(a -i) (1
a
A (C-a
- a J (+C)
+ +1-a2
)
1
(-a
a2-1
al-2
(t-a2)
(C-a 1 )
-1
a -1
(r-a 2 )
a
(C-t)
(l-_a) dt
(B.4)
2 a2
I3 is defined previously
Remarks
a)
out the partial
In carrying
the variable
the integrals,
itself
differentiation
of some of
in the limits
appears
of
integration
a2
a
aa
_
edge.aa
3aa = Ha
aa 1
e~~g.
A
in such cases a change
details
of variable
(a2
t
aB-
d~
(1- )
)
t
is needed.
We present
The others
as an example.
As the integrand
intergration,
a2
1
1)
for this derivative
similarly.
1
i al((-a
the
follow
does not exist at the limits
of
Leibtnitzrule cannot be used.
C -al
Let
t =
=
a2-a1
d
This transforms
a(l 1-t) + a2t
= (a2-a 1 ) dt
(al,a 2 ) to (0,1)
Let the integrand
be I2*(C )
Then it can be shown by rearranging,
(1 - t ) ]
a2-1
al-1
aA-1
I2*(t) = [l+a2t+a l
[t(a 2-a1)]
[(a 2-a 1 ) (l-t)]
a* aB-1
[1l-a 2 t-a
l ( l-t)]
B
d{
292
aa
Then
1 I*
a
1 (al
) 11
_. ~
J
(t) dt
(a
(a2 -a )
b)
1
1-t
1
B
l1-a2 t-a1 (l-t)]
2
(aA-1) (l-t)
from tis
+
(a
) 2 -a 1
Ll+a2t+a
)
(a
o
2
(a2-
2
1
]2
we have what is given in B.3
Every
integral
that does not exist
in these computations
at the limits
As such the singularity
has an integrand
of integration.
has to be subtracted
off first before
the integrals can be evaluated numerically.
off part is evaluated
As an example,
analytically.
consider
ral
I
(4+1
(a-) A
the integrand
F1 ( {) -(+a)
a1
a-
a-i
-I
Near a,
The subtracted
1
behaves
2-a2-1
(a2-
(
(1-
B-)
like
aA-1
al-1
(a1 -O)
(a2 -a1)
a2 - 1
aB-1
(1-a
1)
Near -1, the integrand behaves like
aAF2( =
(1+)
Thereforeal - c
I
1I(4C)
= j
a1 -B-1
(l+al)
(l+a
al
I-F
1
1
-+E:
1
2
al
I
(C) - F(
1
2)
2
4
)]d
+
1
F (d
~~-1
1
well behaved:
evaluated numerically
+
F ()d
-12
293
=
1~
Jfa[Ii-F
-F ]
d
+A a
+ (1+a )
a l+aA-1
+ (+a 1)
(1+a )
a1
(a2 -a 1 )
cA1
1
2 B
a
o 2-1
aB-1
(1-a
i )
294
APPENDIX
C
COMPUTATION OF ISOTHERM AREAS
y
-L
LD
2
x
Atz
T
Region
I-
x
x
AT
Fig. C-1
Computation
: length of Region
: centerline position of
of isotherm
area
of heat recirculation,
In the absence
Ii
given isotherm
: diffuser length
D
Region
)l<
I
(x) : lateral position of isotherm
5(x) : potential core width
LD
YXA
the two dimensional
model outlined in ChapterIII and IV predicts the excess
temperature field AT(x,y) induced by a unidirectional thermal
such that
diffuser
AT
AT
For any diffuser
isotherms
(C.1)
< AT < design
it is only meaningful
in this range using this model.
S is the near field dilution
AT
to calculate
is given by Eq. 4.54;
given by Eq. 3.41
it characterises
the ulitimate
temperature reduction in the discharge dominated region.
295
A typical
C-1.
isotherm
is illustrated
are the same as used in section
The notations
the figure shows
AT = constant
the area within
the isotherm,
in Fig.
4.1.
AT - ATN
Thus
AT
0
=
S'
is given by
AT N
(C.2)
6(x) dx
= 2
N~~
xI,6(x) are given by the Region I solution in section 4.3.1.
In general,
a given AT isotherm,
the area within
AT
is
given by
xA T
AAT
where
=
Jo gs(x)
(C.3)
dx
g (x) is the y-co-ordinate
the position
of the isotherm
of the isotherm
at any x,xAT ,'
along the centerline,
is given by
eq. 4.52.
rearranging
1
1
~~1
AT
x + -in
AT = XI
ai- in {{-[-C1-8.)]
63
[
(SAT)2
~~~~~3
(C.4)
3
2
AT
where
0
f LD.
= 16H
3 =C
The computation
-32a IIH
L f
as given by eq. 4.53
of AAT has to be split into two parts-
the
contributions from region I and II respectively.
[x I
x
AAT = 2 [
gs(X) dx +
AT .0
AT
gs(x) dx ]
x
~~~~~~I
x
(C.5)
I
The contribution
=
from region
[XI
xXI
r XI
J
I is
g(x)
dx =
6(x) dx +
[ gs(x)
6(x)
(C.6)
dx
296
of the profile
By virtue
stated
assumptions
in section 4.3.1
we have
(XI
(XI
6 (x)
~
~~~~~~~~~
0
dx
(C. 7)
TS
II is
from Region
the contribution
f~~~~~~~~~~~~~
AT
A
XA
gs
2 =
(s)
XII
where
S (x)/Q,1 (--)
+
o
0
Similarly,
dx
dx =
AT
(
L(x)/Zn
(x)
AT)
dx
(C.8)
XI
xI
AT (x) is given by eq. 4.52.
C
The isotherm
area can then be obtained
numerical
necessary
integrations
by performing
the
in Eq. C.7 and C.8.
Design formula for isotherm areas
For estimating isotherm areas of interest it is desirable
to derive
simple,
engineering
formulae
that can give close
approximations very quickly instead of performing numerical
computations
for the solution
given in section
4.3.1.
From numerical experience it is found that the isotherm
areas
AT
toAT-
corresponding
approximated
ATN
= 0.8
and AT
AT-
0.5 can be well
ATN
by
f LD
A
= 1.8 exp [ 0.534 (
(T)T
AT
D
=
0.8
H
1.437
-)
ATN
and
A
LD
f LD
9.5 exp [ 3.22 (--)
2
AT
ATN
0.5H
=0.5
183
]
297
Appendix D
Numerical Results of Slip Streamline Computations
The numerical values used in the computations of the slip streamlines (Eq. 3.62) and an abridged version of the results are given for
the five cases illustrated in Fig. 3-8.
The integrations for Eq. 3.62
are carried out using Simpson's rule with step sizes AC = 0.0001 close
=
to
= 0.01 further
(-1, 0) and
1.
C=.5
WA
=
A
away.
-60°
-1.0472
= (-0.866, 1.0), W1 = (-0.4, 0.73), W 2
AA = 0.167
,
= 1.118
c
|W2W1i
IA W11
,
1.125
2
(0.0, 0.23), WB
, aB
5
IBW2
1.187
21
1
a1 = -0.9693
QA = (0,-1.0472) ,
AW
,
1=
2=
=0.427
2
1
a2 = 0.487
, CW = 0.323
(0.4,-0.77)
,Q2
= (0.693,-0.16) ,
QB = (0.693,0)
A
= 0.3071
1 = 0.8354
,
IQ 1 Q 2 1
IA Qll
,
2
0.8575
=
,
IBQ21
1
b 1 = -0.818
1.39
,
,
AQ
=
b 2 = 0.8769
2
,
=
C
0.3285
= 0.268
B
0.5
=
(0,0)
298
x
Y
0. LD
0. 5LD
0.0000
0.0605
0.0686
0.0738
0.0778
0.0811
0.0863
0.0905
0.0956
0.1010
0.1054
1. 0000
0. 1101
0.1141
0.1202
0.1247
0. 1860
0.8977
0.8855
0.8777
0.8719
0.8673
0.8599
0.8542
0.8473
0. 8402
0.8344
0.8285
0.8235
0.8160
0. 8105
0. 4628
0.7468
0.7221
0.6937
0.6629
0.6406
0.6207
0.5911
0.5141
0.5754
0.2149
0.2529
0.3016
0.3430
0.3845
0.5526
0.6095
0.6514
0.6989
0.7560
0. 8053
0.8485
0.9029
0.9524
1.0011
1.0299
1.0466
0.5652
0.5523
0.5442
0.5362
0.5280
0.5221
0.5176
0.5124
0.5087
0. 5056
0.5040
0.5032
Q
-1.0472
-0.9927
-0.9798
-0.9709
-0.9638
-0.9579
-0.9482
-0.9403
-0.9305
-0.9197
-0.9107
-0.9010
-0.8927
-0.8798
-0.8700
-0. 7376
-0.6786
-0.6069
-0.5248
-0.4634
-0.4070
-0.3206
-0.2739
-0.2433
-0.2041
-0.1793
-0.1548
-0.1296
-0.1112
-0.0972
-0.0800
-0.0682
-0.0583
-0.0530
-0.0502
299
2.
a = 0.5907
aA =0.2156
X
1
,
=
,
QA = (0, -0.9389),
A
= 0.2811
,
,
a 2 = 0.7498
1 = 0.8761
A = 2.099
C
Q1 = (0.23,
,
WB
=
(0)
0.5
= 0.319
-0.75),
Q2 =(0.526,
2 = 0.8428 ,
0.2),
8B = 0.5
x2 = 0.672
bl = -0.9739 , b2 = 0.672 ,
CQ= 0.2143
x
Y
0. 5LD
0.5L D
0.0000
0.0481
0.0621
0.0700
0.0804
0.0909
0.1008
0.1100
0.1201
1.0000
0.9366
0.1932
0.8053
0.7831
0.7578
0.7418
0.7253
0.2277
0.2727
0.3047
0.3415
0.3951
0.4284
0.5207
0.5811
=
, aB
0.15),
0. 3134
A2
1 = 1.318
a 1 = -0.919
a2 = 1.1634
,
1.121
, W2 = (0.0,
0.70)
, W1 = (-0.31,
1.0)
WA = (-0.683,
= -0.9389
-53.8
A
-0.9389
0.8901
-0. 8676
-0. 8418
-0. 8269
-0. 8069
-0. 7867
0.8804
-0.7675
0.8716
0.8625
-0. 7496
0.9206
0. 9119
0.9008
-0.7303
-0.5992
-0.5454
-0.4825
-0.4425
-0. 4027
0.6519
-0.3468
-0.3153
-0.2450
-0.2073
0.6265
0.6430
-0. 1826
0.6933
0.6317
-0. 1511
0.7043
0.6922
0.6658
QB = (0.526,
0)
300
2.
(Continued)
0.5 LD
0.7424
0.7981
0.8533
0.9053
0.9460
0.9602
0.9854
1.0370
0.6248
0.6180
0.6123
0.6077
0.6047
0.6037
0.6015
0.5986
3.
x
0. 5LD
Y
= 0.6428
aA
=
0.2418,
a
=
1.096
X=
1.3154
1
a
= -0.9105 ,
, Q1
b
UI____II___U__PYP____IP___1PI·III··
1
= 2.017
= -0.9836,
=
1.163
, WB = (0,)
0.5
XB
cB
=
0.5
2 = 0.4591
a1 -0.9105
,' 2=07,C
039
=
,
X
'2
b 2 = 0.4626
0.3193
C =
, Q2 = (0.442,-0.2),
(0.2,-0.69)
61 = 0.8816 ,
A = 0.2644,
X
,
a2 = 0.6227
QA = (0,-0.8727)
8A1'
2163 ,
=1.096
=021,
,
1
A
-0. 1115
-0.0946
-0.0808
-0.0712
-0.0681
-0.0611
-0.0514
W2 = (0, 0.2)
0.7),
= (-0.28,
W
-0.1312
-50. = -0.8727
A
1.0),
WA = (-0.5959,
Q
2 = 0.854 ,
B
QB = (0.442,0)
= 0.5
= 0.738
CQ = 0.1875
y
x
0.5LD
0. 5LD
0.0000
0.0432
0.0517
0.0574
0.0656
0.0742
1.0000
0.9507
0.9423
0.9367
0.9290
0.9212
I1---I_-- _-YII*---YY·II-*Y·--
6
-0.8727
-0.7917
-0.7755
-0.7645
-0.7489
-0.7327
301
3.
(Continued)
x
0.5LD
y
0. 9122
0.9069
0.9006
0.8932
0.8852
0.8825
0. 8307
0.8099
0. 7862
0.7713
0.7605
0.7520
0.0844
0.0906
0.0982
0.1073
0.1177
0.1210
0.1995
0.2375
0.2872
0.3225
0.3506
0.3743
0.4043
0.4297
0.4582
0.5590
0.6250
0.7141
0.7758
0.8616
0.9219
0.9883
1.0064
1.0309
0.7013
0.6889
0.6753
0.6678
0.6594
0.6546
0.6504
0.6494
0.6481
=-450
= -0.7859
WA = (-0.5, 1.0), W 1 = (-0.2, 0.65),
tA =0.2744
,
a
= 1.1045 ,
QA-
,
(0, -0.7859)
QB = (0.347,
0)
,
W2 = (0.0, 0.15),
1.121
,
WB = (0, 0)
aB = 0.5
2 = 0.3254
= 0.7857
a2
=
2
A = 1.168
a1 = -0.804
-0.7137
-0.7022
-0.6884
-0.6720
-0.6538
-0.6479
-0.5255
-0.4754
-0.4176
-0.3810
-0.3543
-0.3332
-0.3082
-0.2885
-0.2664
-0.2037
-0.1703
-0.1327
-0.1111
-0.0859
-0.0710
-0.0570
-0.0536
-0.0491
0. 7420
0. 7342
0. 7254
a= 0.7068
4.
9
0. 5LD
=
(19,
,
CW
= 0.3185
-57)
= (0.347,
0.15) ,
302
(Continued)
4.
A
= 0.2298
X
,
= 0.8841
B
= 1.559
1 '2
bI = -0.9844 ,
Y
0.5LD
0. 0000
0. 0368
1.0000
0.9654
0.9587
0.9509
0.9457
0.9400
0.9356
0.9300
0.9263
0.9204
0.9157
0. 1067
0.1146
0.2011
0.2423
0.2962
0.3344
0.3906
0.4231
0. 9109
0.8654
0.8476
0.8274
0.8147
0.7983
0.7899
0.7853
0.7813
0.7755
0.7553
0.7448
0.7382
0.7336
0.7252
0.7219
0.7206
0.7185
0.7176
0.7169
0.7168
0. 4420
0.4591
0.4818
0.5916
0.6635
0.7175
0.7609
0.8556
0.9020
0.9220
0.9574
0. 9 731
0.9879
1.0017
· Y---
-·l--llp----l
,
= 0.5
= 0.1499
0. 5LD
0.0992
I
C
x
0. 0901
_
2 = 0.8861
2 = 0.5216
b 2 = 0.4763 ,
0.0451
0.0552
0.0622
0.0701
0.0763
0.0846
_
,
P·I--·_I-F__·_L-ZIL^--I_·Y_··_II-
0
-0.7859
-0.6851
-0.6678
-0.6475
-0.6340
-0.6194
-0.6081
-0.5936
-0.5842
-0.5691
-0.5570
-0.5447
-0.4296
-0.3855
-0.3357
-0.3046
-0.2644
-0.2437
-0.2324
-0.2226
-0.2091
-0.1579
-0.1308
-0.1132
-0.1005
-0.0767
-0.0668
-0.0628
-0.0562
-0.0534
-0.0509
-0.0486
I^---·DIIICI11·-
· IL
-
_-_l.l_s
·.111 I1
-
pi
303
c = 0.766
5.
0A = -40
= -0.6981
WA = (-0.4195, 1.0) , W 1 = (-0.19, 0.7) , W 2 = (0, 0.2) , W B = (0, 0)
aA =.2921
a1 = 1.0923 ,
,
A
a
= -0.8558
QA
=
= 1.416
,
'
aB
=
0.5
A2 = 0.5295
a2 = 0.6747
,
= 0.3185
C
(0,-0.6981) , Q1 = (0.15, -0.49) , Q2 = (0.2666, -0.1) ,
QB = (0.2666,
A
= 1.1156 ,
2
= 0.1988
,
1
O)
0.8937
'
A1 = 1.587
b 1 = -0.995
,
2 = 0.9075
,
B = 0.5
2 = 0.3899
= 0.5356 ,
CQ = 0.1113
x
y
0. 5LD
0. LD
0.0000
0.0343
0.0426
0.0483
0.0528
0.0628
0.0723
0.0827
0.0842
0.0897
0.0977
0.1097
0.1158
0.2040
0.2475
0.2791
0.3263
1.0000
0.9735
0.9682
0.9647
0.9620
0.9562
0.9510
0.9453
0.9446
0.9418
0.9377
0.9319
0.9289
0. 8930
0.8783
0.8687
0.8556
0
-0.6981
-0.5755
-0.5575
-0.5457
-0.5369
-0.5183
-0.5018
-0.4848
-0.4825
-0.4741
-0.4623
-0.4456
-0.4375
-0.3438
-0.3085
-0.2860
-0.2562
304
5.
x
0.O.SLD
5L D
0
0. 5LD
0.3624
0.3922
0.4403
0.4957
0.6214
0.6988
0.8038
0.9325
0.9779
1.0333
1.0642
0.8465
0.8396
0.8294
0.8189
0.7987
0.7891
0.7783
0.7681
0.7651
0.7618
0.7601
-0.2361
-0.2209
-0.1987
-0.1761
-0.1338
-0.1135
-0.0908
-0.0691
-0.0627
-0.0557
-0.0522
Copies of a listing of computer programs can be obtained
by request to the R.M. Parsons Laboratory.
j_
I
I_
P
111
^111__1__111111_1__1____I-··
1
· ·- ·- l-L···-·----··---------·