PHYSICAL AND NUMERICAL MODELING OF THE
EXTERNAL FLUID MECHANICS OF
OTEC PILOT PLANTS
by
Paul N. Singarella and E. Eric Adams
Energy Laboratory Report No. MIT-EL
March 1982
82-018
CO0-4683-10
PHYSICAL AND NUMERICAL MODELING OF THE
EXTERNAL FLUID MECHANICS OF
OTEC PILOT PLANTS
Paul N. Singarella
and
E. Eric Adams
Energy Laboratory
and
Ralph M. Parsons Laboratory for
Water Resources and Hydrodynamics
Department of Civil Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
Prepared under the Support of
Division of Central Solar Technology
U.S. Department of Energy
Under Contract No. DE-AC0278ET20483.A004
Energy Laboratory Report No. MIT-EL 82-018
March 1982
ABSTRACT
This study examined the near field external fluid mechanics of
symmetrical OTEC pilot plant designs (20-80 MWe) under realistic deep
water conditions. The objective was to assess the environmental impact
of different plant configurations and to determine if pilot plants can
be expected to operate without degrading the thermal resource available
for power production. Physical modeling studies were conducted to investigate the variation of near field plume dynamics and the sensitivity of
recirculation to different pilot plant designs. Experiments were conducted in a thermally stratified 12m x 18m x 0.6m basin, at an undistorted length scale ratio of 1:300, which allowed the upper 170m of the
ocean to be studied. Measurements included temperature, dye concentration
and visual observation from photographs. Both mixed and non-mixed discharge concepts were investigated. Discharge port design included two,
four or eight discrete circular ports, with significant variations in the
MWe/port ratio, issuing either horizontally or vertically. A range of
ambient uniform current speeds was investigated while an ambient density
profile, representative of potential sites off of Hawaii and Puerto Rico,
was chosen.
A previously calibrated integral jet model (Hirst, 1971a) was tested
against experimental observation to develop a valid, predictive tool that
would facilitate study of conditions that were not modeled with the present
experimental set-up. The model was modified to more accurately represent
the dynamics of the OTEC discharge in the near field. Major modifications
included adjustment of the equations that characterized the starting length
(length of ZOFE); introduction of jet deflection in the ZOFE; introduction
of a lateral spreading formulation that allowed the "squeezing" effects of
the ambient stratification to be simulated; and introduction of an aspect
factor, which accounted for interaction of a number of closely spaced vertical jets issuing from a circular array. Overall agreement between prediction
and observation was quite good. The potential environmental impact of the
discharge plume from an OTEC plant over a broad range of realistic conditions
was assessed through additional sensitivity simulations.
Results indicate that little recirculation occurs for the designs considered in this study. The recirculation that does occur appears to be the
result of plume upwash in the lee of the plant and, possibly, internal wake
-2-
effects on the plant bow. Environmental impact is argued to be proportional to the degree of perturbation caused by the OTEC discharge to
the upper mixed layer. For the conditions considered in the sensitivity
study the OTEC plume remained below the upper mixed layer except for the
largest layer depths considered (H~ 100m). These larger depths are near
the maximum values reported for either Hawaii or Puerto Rico and represent the only conditions where significant perturbations may be likely.
-3-
ACKNOWLEDGEMENT
This report is part of a research program concerned with the near field
external fluid mechanics of OTEC plants. Previous reports produced at MIT
under this program include:
Adams, E., D. Fry, D. Coxe and D. Harleman, "Research on the
External Fluid Mechanics of Ocean Thermal Energy Conversion
Plants: Report Covering Experiments in Stagnant Water," Report
No. MIT-EL 79-041 Energy Laboratory, MIT, June 1979
Coxe, D., D. Dry and E. Adams, "Research on the External Fluid
Mechanics of Ocean Thermal Energy Conversion Plants: Report
Covering Experiments in a Current," Report No. MIT-EL 81-049,
Energy Laboratory, MIT, September, 1981
Fry, D. and E. Adams, "Buoyant Jet Behavior in Confined Regions,"
Report No. MIT-EL 81-050, Energy Laboratory, September, 1982
Support for the research program has been provided by the Ocean Systems
Branch, Division of Central Solar Technology of the U.S. Dept. of Energy under
Contract No. DE-ACO2-78ET20483,A004. Technical program support has also been
provided by Argonne National Laboratory. Drs. John D. Ditmars and Robert A.
Paddock of ANL are gratefully acknowledged for their cooperation and editorial
comments and for having supplied the original computer code upon which the
numerical calculations were based.
The reserach was performed by Mr. Paul Singarella in partial fulfillment
of the degree of Master of Science in Civil Engineering at MIT. Supervision
was provided by Dr. Eric Adams of the MIT Energy Laboratory and the Department of Civil Engineering. Appreciation is expressed to Dr. David Fry and
Mr. David Coxe, former students, who put together most of the experimental
set-up in connection with past studies, and to Messrs. Richard Baker, David
Kubiak and Peng-Chong Sien, students who provided assistance during the
experiments.
-4-
TABLE OF CONTENTS
Page
Abstract
2
Acknowledgement
4
Table of Contents
5
List of Figures
9
11
List of Tables
Chapter 1:
INTRODUCTION
12
1.1 Principles of Power Plant Operation
12
1.2 External Flow Considerations
15
1.3 Research Objectives
18
Chapter II: PREVIOUS AND PRESENT MODELING EFFORTS
Chapter III:
19
2.1 Background
19
2.2 Description of Previous Studies
19
2.3 Description of Present Study
22
24
ThE PHYSICAL MODEL
3.1 Modeling Considerations and Scaling Laws
24
3.1.1
Jet Reynolds Number Objective
24
3.1.2
Mixed-Unmixed Discharge Constraint
25
3.1.3
Ocean Profile Consideration
25
3.1.4
Experimental Basin Bottom Influence
Constraint
25
3.1.5
Scaling Laws
26
27
3.2 Model Design
-5-
3.3 Characterization of the Ambient Ocean
36
Chapter IV: THE EXPERIMENTS
4.1 Experimental Layout
36
4.1.1
The Model Basin
36
4.1.2
The Towing Apparatus
36
4.1.3
Discharge and Intake Water Circuits
36
4.1.4
The Stratification System
42
4.1.5
The Temperature Measurement System
43
4.1.6
The Dye Measurement System
45
4.1.7
The Photographs
47
4.2 Experimental Procedures
49
4.2.1
Procedures before and During an Experiment
49
4.2.2
Workup of Flourescent Dye Samples
52
4.2.3
Manipulation of Slide Photographs
53
4.2.4
Accuracy of Temperature Data
53
4.2.5
Temperature Data Manipulations
53
4.3 Experimental Results
4.3.1 Data Summary
Chapter V:
30
61
62
4.4 Comment on Jet Reynolds Numbers
65
THE INTEGRAL ANALYSIS
66
5.1 Justification for Use of an Integral Jet Model
66
5.2 Previous Integral Model Studies of Buoyant
Jets in a Current
66
5.3 Description of the Hirst Model
68
5.3.1
The Governing Equations
68
5.3.2
The Entrainment Function
72
-6-
Page
5.3.3
75
The ZOFE
5.4 Hirst's Verification of His Model
76
5.5 Previous OTEC Related Use of the Hirst Model
77
5.6 Adaptation of Hirst
IModel
5.6.1
Jet Interaction
79
5.6.2
Deflection in the ZOFE
94
5.6.3
Starting Length
95
5.6.4
Lateral Spreading of Plume
99
5.7 Hirst Model Simulations of Experimental Conditions
Chapter VI:
78
104
5.7.1
Vertical Discharge Experiments
104
5.7.2
Qualification of Comparison for Vertical
Discharge Experiments
111
5.8 Horizontal Discharge Experiments
114
5.9 Additional Comments on the Model Equations
115
5.9.1
An Infinite Entrainment Rate?
115
5.9.2
Boundary Layer Assumption Inconsistency
116
ADDITIONAL SIMULATIONS WITH THE INTEGRAL JET MODEL
119
6.1 Introduction
119
6.2 Selection of Base Case Plant and Ocean
119
6.3 Sensitivity to Perturbation from Base Case Conditions 121
6.3.1
Presentation of Results
121
6.3.2
Discussion of Results
126
6.4 Modeling a Separate Condenser Jet
129
6.5 Modeling Experimental Conditions from a Previous
Physical Model Study
130
6.6 Future Use of Model for Environmental Assessment
133
-7-
Page
Chapter VII:
135
RECIRCULATION
7.1 Introduction
135
7.2 Direct Recirculation in Stagnant Water Tests
135
7.3 The Upwash Effect in Vertical Experiments in
a current
136
7.4 Recirculation in Tests in a Current
138
Chapter VIII: SUMMARY AND CONCLUSIONS
142
8.1 Summary
142
8.2 Physical Modeling
142
8.2.1
Conditions Modeled
142
8.2.2
Conclusions and Recommendations for
Future Work
143
8.3 Numerical Modeling
144
8.3.1 Methodology
144
8.3.2 Conclusions and Recommendations for
Future Work
145
147
References
Appendix I: SIMULATIONS OF THE OTEC NODEL DISCHARGE IN THE
VERTICAL, Y-Z PLANE, COMPARED TO EXPERIMENT
151
Appendix II:SIMULATIONS OF THE OTEC MODEL DISCHARGE IN THE
HORIZONTAL X-Y PLANE, COMPARED TO EXPERIMENT
170
Appendix III: FINAL COPY OF INTEGRAL JET MODEL CODE
207
-8-
LIST OF FIGURES
Figure No.
Page
Title
1.1
Examples of Vertical Temperature
Profiles for the Tropical Ocean
13
1.2
OTEC Power Cycle
14
2.1
Zones of a Submerged Discharge
20
3.1
Cutaway View of M.I.T. OTEC Model
28
3.2
Photograph of Model and Table of
Model Characterization
29
3,3
Coordinate System of Physical Model
3.4
Comparison of Proposed Model and
40 MWe Gibbs & Cox Design
33
3.5
Superimposed Experimental Profile
(Scale 1:300) for comparison to Ocean
Density Profiles
34
4.1
Schematic Diagram of the Experimental
Setup
37
4.2
Schematic of the Towing Apparatus
38
4.3
Blowup Schematic of the Towing Carriage
39
4.4
Photograph of Flow Apparatus and
Model Support Frame
41
4.5
Typical Variability Between Basin
Density Profiles of Experiments
44
4.6
Flow Chart for the Temperature
Data Acquisition System
46
4.7
Cross Sectional Schematic of the Sideview
Photographic Apparatus
48
4.8
Typical Side View Cross Sectional Photograph
50
4.9
Typical Overhead Photograph
50
5.1
Natural Coordinate System
71
5.2
Example Aspect Factor Illustration
81
-9-
Page
5.3
Aspect Factor Definition Sketch for
Two Jet Experiments
87
5.4
Aspect Factor Definition Sketch for
Four Jet Experiments
89
5.5
Aspect Factor Definition Sketch for
Eight Jet Experiments
91
5.6
Crossflow Ratio Versus Normalized
Starting Length
96
5.7
Vertical Pressure Distribution of Water
Column When the Plume is at Equilibrium
100
5.8
5.9
S250 (Observed) Versus S250 (Predicted)
Versus t250 (Predicted)
t250 (bserved)
105
106
250 (Observed) Versus W 2 5 0 (Predicted)
5.10
(Observed) Versus heq (Predicted)
107
108
5.11
h
6.1
Base Case Plant Configuration and AF Sketch
122
7.1
Side View Photograph of Run 4A Illustrating
Circular Motion of Upwashed Dye Billows
137
7.2
Annular Intake Structure Showing
Positions of Intake Thermistor Probes
139
-10-
LIST OF TABLES
Table No.
Title
Page
4.1
Experimental Parameter Schematization
54
4.2
Experimental Parameters (Prototype dimensions
except for Re)
56
4.3
Experimental Results
58
4.4
Time-Variant Experimental Results
of Non-Steady State Stagnant
Experiments
64
5.1.1
Description of Simulations for
AF Sensitivity Analysis
82
5.1.2
Statistical Results of AF Sensitivity
Simulations
83
5.2
Equivalent Source, No Interaction
Compared to Individual Source
85
5.3
Simulation Versus Observation for Vertical
Discharge Experiments in a Current
198
5.4
Statistics of Simulation Versus
Observation for Vertical Discharge
Experiments in a Current
104
5.5
Assessment of Model's Prediction of
Geometry in the Y-Z Plane
110
6.1
Base Case-Conditions
120
6.2.1
Description of Simulations (Perturbations
from Base Case Conditions)
121
6.2.2
Results of Simulations Described
in Table 6.2.1
123
6.3
Simulated Condenser Jets
130
6.4.1
Independent Parameters of Coxe's
Single Jet, Horizontal Discharge
131
6.4.2
Simulation Versus Observation for
Coxe's Initial Conditions
132
-11-
1. INTRODUCTION
1.1
Principles of Power Plant Operation
Ocean thermal energy conversion (OTEC) is a proposed energy conversion
process that uses the temperature differential between upper and lower ocean
strata set up by the sun.
Only in tropicalor subtropical waters is this temAnd
perature differential large enough for the technology to be considered.
even here, the preservation of the surface thermal resource will be critical to
profitable plant operation.
OTEC is a second order solar technology in that it does not directly utilize the sun's rays but taps the solar radiation captured by
the upper ocean. Fig.
1.1 depicts representative vertical temperature
profiles for tropical oceans, each exhibiting a characteristic mixed layer
of warm water near the surface above a stably stratified density structure.
water
Since
makes
around-the-clock
an
excellent
operation
of
heat
OTEC
and
reservoir,
minimizes
this
permits
annual
output
fluctuations.
Heat
Fig. 1.2 is a simple illustration of an OTEC closed power cycle.
from the warm upper water is used in an evaporator to vaporize a working
fluid such as ammonia or freon in a pressurized vessel.
expanded through a turbo-generator
The vapor is
to produce electric power and
is
subsequently condensed using the cold water sink.
The ideal thermodynamic efficiency,
E
T -T
c
Th
(1 -
K
-12-
1)
T(oC)
100
*
/
200
300
,
III
/
400 -
600
I
I
z (m)
---
Lockheed (Dugger, 1975)
Carribean (Fuglister, 1960)
---------w---
Figure 1.1:
Florida Straits (Pub. No. 700)
Hawaii (Bathen, 1975)
Examples of Vertical Temperature Profiles
for the Tropical Ocean (From Jirka, et al,
1977).
-13-
Worm Water Seawater
Worm Water Seawater
Exhaust
Intake
22.8
Liquid
Pump
Electric
Power
Low Pressure
Ammonio Liquid
100C (50
0 F)
Cold Seowater Intake
50C (41 0 F)
Cold Seawater Exhaust
7.20C (45 0 F)
'
Figure 1.2: OTEC Power Cycle
-14-
of an OTEC plant
is 6 to 8% based on typical temperature differences
between the surface and a depth of 500 to 1500 meters of 18 to 240 C.
However, net mechanical efficiencies are estimated to be only about 2 to
3%, compared to 30 to 40% for conventional power plants.
In order to produce quantities of power comparable to conventional
power plants, an OTEC plant must utilize enormous amounts of water to
exploit the low grade energy
to produce
resource.
For example, for an OTEC plant
100 MWe at an efficiency of 3% with a realizable temperature
difference across the heat engine of 100 C, corresponding to a 200 C ocean
temperature difference (Allender et al, 1978), the warm and cold water
intake flows would each have to be 500 m3/sec (Fry, 1976).
1.2
External Flow Considerations
OTEC plants interact with the ocean environment by withdrawing water
through their intake ports and exhausting it through their discharge ports.
The
external
flow
field
this
process
generates
influences
plant
performance, as affected by any potential recirculation, and potential
environmental impacts, such as nutrient or pollutant transport and ocean
temperature modification.
Allender, et al (1978) have examined the sensitivity of OTEC power
output to a fractional loss in thermal resource.
AP
They found that:
A
n
P
n
1
E(l-a)
p
(1 - 2)
ep
where E, the fraction of the total thermal resource that exists across
the power cycle, is expected to be about 0.5 and a, the percentage of
-15-
A thermal resource,
parasitic power losses, anywhere from 0.16 to 0.40.
e , of 180 C represents the lower bound of practicality. If a 10C 66
p
P
0
occurred with an initial 6 of 20 C, the plant would experience a 13%
decrease in power output according to equation 1-2 with an a of 0.2.
Since the temperature of the deep cold water is expected to remain
will occur primarily through change in
approximately constant, change in 6
the warm water intake temperature.
Because of the enormous flows involved,
the warm water intake temperature is not only a function of ambient ocean
variability,
interactions
of potential
but
between
the
flow
fields
generated by the evaporator intake, the plant discharges and the mixed
layer depth (Ditmars
et al, 1979).
The extent of these interactions
depends upon a number of factors, including the location of the intake
and
discharge
ports
the
to
relative
mixed
layer
and
the
vertical
separation between them, the ambient ocean currents, the angle of the
discharge
with respect
the
to
horizontal
discharge flow relative to the ambient
discharges are always
and
density
thermal resource available
buoyancy of
structure.
colder than the warm water
such interactions can only serve to lower it.
recirculation.
the
the
Since the
intake temperature,
This degradation of the
to the plant is generally referred
to as
It may result from some fraction of the discharge volume
flux entering the warm water intake directly or from several indirect
causes such as recirculation of water entrained by the discharge jets,
turbulent
mixing
of
the
upper
stratified
layers
(induced
by
the
discharge jets) accompanied by a lowering of the mixed layer temperature
or selective withdrawal of the upper thermocline layers by the intake
port.
-16-
Operation of an OTEC plant may not only influence the utilization
of
the thermal
resource, but may
also have
ecological affects.
As
discussed briefly below, prevention and control of biofouling on the
heat
exchangers,
introduction
the
of
occurrence
deep,
of
working
nutrient-rich water
fluid
relatively
surface will all have environmental
impacts.
require
time-history
ccupling of
the OTEC plume
leaks,
and
close
to
the
the
Study of these phenomena
characteristics with
appropriate, biological and chemical, kinetic models.
Biofouling refers to the growth of organisms whether macroscopic,
such as barnacles and mussels, or microscopic, such as slime films, on
the heat exchanger surfaces.
Bell
(1977)
found that
thick slime layer precipitated a 15 to 25%
efficiency.
decrease in heat
formula
control
will
probably
transfer
be
a
infrequent and expensive mechanical scrubbing to remove
combination of
the growth
biofouling
The
a 50 micrometer
and
frequent
to its
and
cost
cheap
chemical
effectiveness,
to retard
it.
obvious choice
for
treatment
is the
Chlorine,
due
biocide.
The alternatives, chlorine dioxide and bromine are two to ten
times more expensive (Sands,
1980).
Chlorine however is toxic in trace
amounts.
Possible
isobutane.
working
Owens
fluids
include
ammonia,
freon,
propane
and
(1978) found that ammonia required the least amount
of surface area per kilowatt of net power produced of all these fluids.
Since
the
investment
heat
exchangers
(Sands, 1980),
working fluid.
represent
roughly
half
of
the
capital
ammonia appears to be the logical choice of
In the unlikely event of a massive rupture of the heat
-17-
exchangers, massive amounts of working fluid would be injected into the
Ammonia would cause the affected seawater to go strongly
environment.
basic
result
and
(Walsh, 1980).
in
massive
precipitation
of
hydroxides
metal
The depletion of these metals may be detrimental to the
local food chain.
The subsequent ammonium generated would act as a potent
inhibitor, uncoupling light
energy during photosynthesis
(Walsh, 1980)
Introduction of working fluid through continuous small leaks would cause
more subtle perturbations of the ambient water chemistry and ecosystem, if
any.
The OTEC plant will artificially upwell deep, nutrient-rich water.
However
the
discharge
plume will
generally
be
directed
into
the
thermocline in order to prevent recirculation and the upwelled nutrients
may remain essentially out of the area of primary productivity.
The
plume may entrain a portion of the surface zooplankton community and
relax the grazing stress of the species (Brookhaven, 1981).
1.3
Research Objectives
The objective of this study is to examine the external fluid mechanics
associated with modular, 10 to 100 MWe, OTEC pilot plant designs, under
realistic
ocean
conditions
to
facilitate
environmental impact and optimization.
assessment
of
pilot plant
This will be accomplished through a
series of physical model tests and subsequent verification of an integral
jet model with the test results.
-18-
II:
PREVIOUS AND PRESENT MODELING EFFORTS
2.1 Background
Claude built the first operational OTEC plant in
feasibility.
However
low
relatively
development.
As
cost
the
of
lack
fossil
technology has
of
sufficient
fuels
interest in OTEC has resurfaced.
technology
prevented
evolved and
1930 proving its
fuel
further
and
the
system
costs have
soared,
Consequently, numerical and physical
modeling studies have been undertaken to assess OTEC feasibility and to
evaluate how proposed power plant designs behave within the ocean
environment.
2.2
Description of Previous Studies
Most
studies pertain to specific zones of
A
the external flow.
submerged discharge can be divided into three such zones (Fig. 2.1), with
gradual transition between zones. In the near field zone, the jet dynamics
are governed by the buoyancy and momentum of the jet, the ambient current
and stratification, and any interaction with the OTEC plant structure, the
evaporator intake, the free surface or the ocean floor (in the case of a
shore-based or shelf-mounted plant).
In the intermediate zone, the jet has
arrived at an equilibrium elevation and exhibits lateral spreading due to
buoyant forces.
Jet momentum effects are relatively insignificant.
In the
far-field zone, only ambient turbulence is left to diffuse the plume.
The
following studies pertain to near field phenomena.
Several investigators
(Lockheed, 1975;
-19-
Fry,
1976;
Giannotti, 1977;
CI
Intermediate-field zone
Figure 2.1:
Zones of submerged discharge.
Ditmars et al, 1979) have used integral techniques to analyze the behavior
of individual buoyant jets representing evaporator and/or condenser flows
floating OTEC plant
from a
discharging
environment.
However proposed OTEC
into
a
stratified
site-specific
data
(Sands,
indicates that significant currents will be experienced.
Mangarella
1980)
Van Dusen and
(1974) performed a similar analysis except relatively strong
currents were studied.
the
stagnant
Straits
Florida,
of
Massachusetts.
Their investigation was essentially only germane to
While
the
these
site
studied
investigations
by
used
the
University
of
established modeling
techniques none of the results were verified against experimental data
pertaining to an OTEC application.
None explored the possibility of a
multiple port discharge and subsequent jet interaction.
Several investigators (Sundaram et al, 1977; Jirka et al, 1977; Adams
et al, 1979; Coxe et al, 1980) have performed physical model studies that
examine possible interaction between the intake and discharge.
explored
a highly
schematic
plant
configuration
with a
Sundaram
two-layered
Jirka explored a similar ambient
stratification and realistic currents.
environment with a more realistic plant.
Adams and Coxe incorporated a
realistic range of stratification into their experiments as well as a
realistic plant and range of currents. However all these investigations
focussed on horizontal discharges.
The type of recirculation observed by
Adams and Coxe is significantly different from the type observed in this
study.
In addition, Adams and Coxe considered plants in the 100 to 600
MWe range; these plants are much larger than anticipated pilot plants.
Paddock
et
al
(1981)
recently
analyzed
field
measurements
associated with the mixed discharge plume of the one MWe, OTEC-1 facility
-21-
Dye measurements were compared with analytical model
off of Hawaii.
The model included an
prediction.
integral analysis in the near-field
and a dimensional analysis after Jirka et al (1980) to analyze lateral
spreading
in
field.
the intermediate
The analysis cannot be easily
extended to predict pilot plant plume behavior due to the limited data
which were
collected
and
uncontrolled environment.
the
uncertainty
of
an
in
measurement
In addition, the flow of the OTEC-1 facility is
an order of magnitude less than that of the smallest proposed pilot plant.
2.3
Description of Present Study
The optimal OTEC plant size in terms of cost effectiveness has been
estimated
exception
studies with the
aforementioned
plants of this order of magnitude.
100
MWe
(Gibbs and Cox,
to be in the range of 400 MWe
pilot
plants
commercialilzation.
will
of Paddock
1979).
(1981),
The
address
However the deployment of many 10 to
undoubtedly
precede
large-scale
Sands (1980) projects that a total of ten modular,
10 to 100 MWe pilot plants will be operational by 1995 off Hawaii and
Puerto Rico.
Possible plant
types
include
land-based,
shelf-mounted,
asymmetric ship and floating, deep water, spar buoy design.
At least
some of these plants will probably fall into the last category with one
or two discharge ports per module.
to be about 10 MWe.
The size of a module is anticipated
It is also likely
that some of the plants will
have vertically directed discharges (Scott, 1979).
The fluid dynamics of these pilot plants will differ from those
previously
orientation.
studied
due
to
different
plant
size
and
discharge
This eliminates the possibility of directly extrapolating
-22-
the results
plants.
In
from the studies
light
of
this,
of
we
the larger plants
have
examined
to
include pilot
experimentally
and
mathematically the near field external fluid mechanics of 10 to 100 MWe,
modular pilot plants discharging both horizontally and vertically into
deep water.
This work represents the first experimental study of OTEC
pilot plants.
-23-
THE PHYSICAL MODEL
III.
Modeling Considerations and Scaling Laws
3.1
To retain dynamic similitude between the prototype and the model,
undistorted densimetric Froude scaling was used, which preserves the ratio
of buoyancy and momentum forces.
Both the momemtum and the buoyancy of the
discharge determine the external flow field surrounding an OTEC plant and
is to be considered valid.
both must be represented if a model
The
selection of a model to prototype length scale, Lr ,must satisfy several
Reynold's numbers large enough to
Obtaining jet
competing objectives.
insure turbulent flow dictates a large scale ratio as does measurement
resolution.
Modeling large ocean depths dictates a small scale ratio.
In
addition, the experimental facilities impose certain physical constraints.
The following discussion addresses these length scale considerations.
3.1.1
Jet Reynolds Number Objective
2u b
0
R
e
The
v
u
= discharge velocity
b
= port radius
v
= kinematic viscosity
jet Reynolds number must
fully
where
turbulent
jet
exceed a minimum value
flow needed
for model-prototype
to maintain
the
similarity.
The
minimum value is generally accepted to be 1500 (Ungate, 1975).
However the
transition from turbulent to laminar flow is gradual and this value is
somewhat arbitrary.
For a given plant size, this objective is
-24-
increasingly difficult to meet for low discharge velocities.
Mixed-Unmixed Discharge Constraint
3.1.2
The flow system currently available to our experimental basin only
allows the discharge of a single temperature water thus preventing us from
modeling separate evaporator and condenser discharges.
Therefore we can
only model mixed discharges (combined warm and cold water flows) or warm
water discharge flow.
It is therefore assumed that the warm and cold water
discharge flow fields, when discharged separately, are independent of each
other.
Ocean Profile Consideration
3.1.3
The
reasonable ocean density profiles
ability to produce
in
the
experimental basin with fresh water at a length scale ratio of 1:300 has
been demonstrated by previous work (Adams et al, 1979; Coxe et al, 1980).
The 1:300 ocean density profile has been shown to be reasonably stable over
the duration of an experiment
(see section 4.1.4).
As L
r
decreases, the
relative perturbation to the temperature profile due to surface cooling
increases which would contribute to a general decrease of the stability
of
the
ocean density
profile
over
the
duration of
an
experiment.
Therefore to avoid further experimentation, it would be advantageous and
convenient to work with an L
r
3.1.4
of 1:300.
Experimental Basin Bottom Influence Constraint
At a scale ratio of 1:300, the experimental basin has a prototype
depth of 174 meters, much less than required for OTEC operation.
-25-
Therefore
the success of our simulations depends on the absence of significant
This consideration was most constraining to the
basin floor influences.
case of vertically directed discharges in stagnant water.
An investigation of available data (Fry,1980) suggested that bottom
basin interference could be avoided for vertical flows corresponding to
10 to
100 MWe
and
a length
scale
ratio
of
1:300 if
the
proper
combination of discharge depth, ambient current velocity, port size and
number of ports was maintained.
The investigation also indicated that a
length scale ratio of 1:400 severely limits examination of low velocity
discharges and low power
criteria.
(MWe) plants because of the Reynold's number
For example, at an L
r
of 1:300, R is approximately 2000 for
e
an evaporator discharge from a 10 MWe plant with a b
of
1:400, Re is approximately
o
of 3.1m. At an L
r
1400 for the same plant, if the same
similarity conditions are preserved.
In fact, under the same similarity
conditions, only about one third of our experiments would have had an Re
greater than
3.1.5
1500 for an L
r
of 1:400.
Scaling Laws
Due to the considerations described above, a length scale ratio of
1:300 was chosen.
This was a convenient choice since previous studies
(Adams et al, 1979; Coxe et al, 1980) had proven the viability of modeling
OTEC plants in the basin at this L .
r
The densimetric Froude number is
defined as:
IF = u(g A-- h)
where u,h , p
and Ap
density difference.
rp a characteristic velocity, length, density and
If the ratio of a characteristic quantity between the
-26-
model and protype is designated with subscript r, and if AO is assumed
r
equal to one,
then equality of densimetric Froude number
protype implies the following conditions for the ratios
in
model and
of velocity, time
and flow rate at an undistributed length scale of 1:300:
u
r
t
r
= L 1/2 = 0.058
r
= L 1/2 = 0.058
r
Qr = L
5/2
-7
/2 = 6.4 x 10
3.2 Model Design
The designs considered in
be modeled as
horizontally
intake
this study are limited to those which can
columns, discharging vertically
symmetrical vertical
from
round,
(see Fig. 3.1).
multiple
ports,
Designs with two,
with
an
annular
or eight
four
or
warm water
evenly
spaced
ports located at a single radius from the plant axis were investigated
It
Fig. 3.2 shows a photograph of the model.
model used by Coxe et al
are designed
(1980) in several ways.
differs
The discharge ports
instead
to discharge vertically downward,
from the
of horizontally.
However horizontal discharge was achieved by fitting 900 elbows into the
ports.
The cold water pipe is
now included in
possible interaction between it
and
the model so as to study
a vertical
discharge.
The Coxe
model could discharge from a radial or annular port, as well as from
discrete ports, whereas the present model can only discharge from discrete
ports.
-27-
DISCHARGE
LINES
Figure 3.1:
Cutaway View of M.I.T. OTEC Model (shown for vertical
discharge; for horizontal discharge, 90 elbows were
attached to each discharge line and were directed
radially outward)
-28-
4 1
EXPERIMENTAL MODEL (1:300),
RANGE OF PARAMETER VARIATION
Qi (m3/s)
hi(m)
100-400
4.0
Qo (m3/s)
100-400
u (m/s)
0.8-3.9
b (m)
2.8-3.1
hd(m)
39-43
01(o)
0, +45 +90
02( 0)
0 and
J
90
2,4,8
r
(m)
23
4.1
r (m)
c
38-57
u.(m/sec)
-H (m)
Notes:
38-57
u
= discharge velocity
b
= port radius
o
r
o
r
u
o
c
= plant radius
= cold water pipe radius
co = ambient current velocity
Figure 3 .2: Photograph
of Model and Table of
Model Characterization
A
3.3
Characterization of the Ambient Ocean
Figure 3.5
(Miller, 1977) shows ocean density profiles for several
Underneath a well-mixed layer near the surface lies
tropical locations.
a thermocline, where temperature drops rapidly accounting for a strong
density gradient.
important
The stable density structure in the thermocline is an
inhibitor
to
momentum
vertical
and
heat
transfer,
and
therefore, is an important consideration in the study of OTEC external
fluid mechanics.
As shown in Fig.
3.5, this
study
considers realistic continuous
density profiles which are comparable to actual ocean density profiles.
Although each density profile produced in the laboratory is different in
detail, they are characterized by their values of H and Apa*
H is the mixed
layer depth, defined as the depth where the ambient temperature differs from
is the density difference between
the surface temperature by 1C. Apa
the surface and 165 meters, far enough above the basin bottom to avoid
any potential thermal boundary layer.
As reported by
Sands
the most
(1980),
frequently observed mixed
layer depths in Puerto Rico and Hawaii are 68 and 66m respectively.
As shown in Fig. 3.2,
monthly average is less than 40m.
examines a slightly conservative
(ie
shallow) range
No
this study
of mixed
layer
depths.
In addition
modeled.
to density
stratification, ocean currents were also
Monthly mean surface currents, u.,
range from 11 to 37 cm/sec.
in Hawaii and Puerto Rico
As shown in Fig. 3.2, this study examines a
realistic range of currents.
-30-
As previously mentioned, two discharge configurations are considered
in this study.
For a horizontal discharge, the vertical angle e2 between
the horizontal plane and the direction of discharge, equals 0O
3.3).
(see Fig.
The horizontal angle 61, measured from the axis normal to the
current toward the direction of the current (see Fig. 3.3), may vary from
-900 for a counterflowing jet to +900 for a co-flowing jet.
discharge, 62 = 900 while 61 is undefined.
For a vertical
The orientations of the vertical
port arrays for tests with 2.4 and 8 jets are shown in Figs. 5.3, 5.4 and
5.5.
The table in Fig. 3.2 lists the parameters used to characterize the
OTEC plant and the ranges corresponding to their variations in the model
tests.
The evaporator intake flow, Qi, which enters the model through a
radial configuration of circular port holes, is located at a depth hi below
the surface.
The condenser intake is not modeled.
The discharge flow, Qo'
is exhausted through a total of J ports, at a depth h d .
flow is mixed, Qo = 2Qi..
hen the discharge
When it is non-mixed (referred to from here on as
an evaporator discharge), Qo =
Q..
Fig. 3.4 shows the Gibbs and Cox (Scott, 1979) 40 MNe spar OTEC plant
against the outline of our experimental model.
discharge
are
shown at
experiments were run.
the
depths at
which
The model's intake and
the vertical
discharge
The horizontal experimental discharge was a few
meters deeper (see Table 4.2).
The separation between the intake and discharge in the model is less
than half of the separation in the Gibbs and Cox design.
To some degree, this
was necessary to avoid interaction with the basin bottom; however, it also
allowed us to explore conditions under which recirculation was most likely to
occur.
-31-
01
O
Y/
Figure 3.3:
Coordinate System of Physical Model
:rodel Intake
Prototyp
-
Intake
A
A
--
'rotot' pe
1)ischarge
Model
O
S....
Discharge
5
,
. .-
O
0a . - 115
•
Figure 3.4
;
_._....._ "
Mode!
30m.
rn.Prototype
Section A-A
Comparison of Proposed ,Model and 40 M1:e
Gibbs 6 Cox Design
-33-
1:crr.
6
5
241
3
0_
D
/
/
I
I
J
D
1
200-
IT/
200
I
*II
r
/
8/
I
/
i:
6:
DENSITY
3:
Figure 3.5:
I
I
S. Atlantic-Brazil
I2 If'S, 30002'W1
Rico
Puerto
iss.-la.
'WI
04'%, 64
1 0 ()'R
8 2 '\
[8
Superimposed Experimental Profile (Scale 1:3007',) for Comparisonh
to Ocean Density Profiles
-34-
[Miller, 1977].
The currents in our model are simulated by towing the model OTEC
plant
through
a
temperature-stratified
basin.
Thus,
prototype
conditions are modeled with a uniform ambient current and a horizontally
uniform, vertically stratified environment.
-35-
THE EXPERIMENTS
IV.
Experimental Layout
4.1
4.1.1
The Model Basin
The experiments were conducted in a 12.2 m x 18.3 m x 0.58 m basin
located on the first floor of the Ralph M. Parsons laboratory for Water
Resources and Hydrodynamics at M.I.T.
are insultated to minimize heat
The floor and sides of the basin
loss to the surroundings.
Figure 4.1
presents a general layout of the basin showing the experimental setup
for the tests performed in a current.
The Towing Apparatus
4.1.2
The towing apparatus is presented schematically in Figs. 4.2 and 4.3.
A continuous belt, driven by a reversible 3 horsepower varispeed motor,
pulls the towing carriage across the basin.
guides the belt,
carriage.
forming a closed
The model intake
An overhead support rail
loop with each side of the towing
and discharge
hoses,
attached to trolley
wheels in the overhead support rail, are pulled by the towing carriage
as it
crosses the basin.
Figure 4.3 shows the OTEC model located in
stage no. 1 of the towing carriage with data acquisition equipment
located on stages 2 and 3.
4.1.3
The Discharge and Intake Water Circuits
The intake and discharge water flow circuits
for the stratified
current tests are schematically illustrated in Fig. 4.1.
-36-
K
!-8
3m (60')
12 2m
(40')
To Drain
Window
U Filter
0
Roto Meter
H Head Tank
8
Control Valve
M Manifold
* Pulley
+ Profile Probe Stations
Figure 4.1:
Schematic Diagram of the Experimental Setup
Guide Pulley
,
To/From Pumps*Drive Pulley ---Motor
,
--
Guide Track
V Towing
Carriage
Figure 4.2:
OTEI
Schematic of the Towing Apparatus
Counterweight
Liftracks
Peristaltic
Sample Pump
/Mntnr
Solenoid Valves
Stage no 3
__
_
Apparatus
Support Frame
Sample Troy
IFrame
/ i
S/Supports
__
/Woz
-=
Z-
"-
.
•
/7
"
I
.
j_
Stage no 2
Sampling Pr0ote
Support Frame
(Vorable Height'
Metal
Coaster - -
Wheels
Stage no I
Model
Support
Frame
\/
Figure 4.3:
Blow Up Schematic of the Towing Carriage
-39-
To
simulate
temperature
steady
of the
state
discharge
operation
of
an
flow must be kept
OTEC plant,
constant.
the
This
is
accomplished by mixing cold tap water with hot water that has passed
through a steam heat exchanger.
A mixing valve adjusts the relative flow
of hot and cold water to achieve the desired temperature.
The water flows
through the mixing valve to a constant head tank which provides a constant
pressure to the discharge flow and helps to damp out short term temperature
fluctuations.
The water is
filter, where it
pumped
from the head tank into a diatomaceous earth
is purified for photographic purposes.
Rhodamine B, a
flourescent dye, is introduced into the water with a peristaltic pump as
it leaves the filter.
Then the water passes through a rotameter and
control valve to the discharge hose.
The discharge hose carries the
water to a flow manifold located on stage no. 3 of the towing carriage.
The manifold
distributes
the water
through
eight
valves.
Hoses
connected to each valve lead to copper tubes in the upper portion of the
model.
The flow passes through these copper tubes and out to the
discharge ports.
The discharge temperature is monitored in the flow
lines near the model and before it enters the discharge hose.
Figure
4.4 is a photograph of the flow apparatus showing the constant head
tank, the filter, the rotameters, the intake and discharge hoses and the
towing carriage.
The intake circuit, driven by a pump, withdraws water from the
basin through the perforations in the annulus at the top of the model
(see figure 3.1).
The water flows through the intake hose, is measured
by a rotameter and controlled by a valve, before it is fed to a drain.
-40-
4-.
Figure 4.4:
Photograph of Flow Apparatus and Model Support Frame
It should be noted that for a mixed discharge flow configuration a net
flow was introduced into the basin.
No effort was made to adjust the
water level to account for this effect.
4.1.4
The Stratification System
The basin is filled with water of different temperatures, all of
which passes through the filter for photographic purposes.
Initially
the
takes
basin
is
filled
partway
approximately two hours.
with
Then,
cold
city
hot water
water,
which
from the mixing valve
is
bypassed through a hose network to a radial manifold located on a float
in the center of the basin, providing an even distribution of the hot
water over the cold water surface and minimizing mixing of the cold and
hot water.
The
period, diffusion
hot water
fill takes
takes place between
17
to
20 hours.
the warm and
resulting in a smooth temperature profile.
During this
the
cold water
Once the filling is over,
surface cooling mixes the upper layers thereby lowering the mixed layer
temperature.
water
The density difference between the entering hot and
is designed to be greater than the one desired, Apa.
cold
Thus a
cooling period of one to four hours follows the fill and precedes an
experiment.
Coxe et al, (1980) reported on the spatial and temporal variability of
typical temperature profiles obtained with this filling procedure.
found that
constant
temperature profiles throughout
at
any
given time.
(The average
They
the basin were essentially
standard
deviation
of
temperature was about 0.10 C with a maximum of 0.25*C occurring at the
thermocline.)
Temporally, for a typical experiment lasting 30 minutes,
-42-
the maximum change
surface and was
in temperature occurred near the
about 0.50 C.
These profiles
Figure 4.5 shows typical basin density profiles.
display the range of variability that can be expected with the filling
procedure.
The Temperature Measurement System
4.1.5
Temperature
thermilinear,
series 700,
0.05 0 C).
repeatability
made
measurements were
thermistor probes
to monitor the discharge
into
the
perforations
of
temperature.
the
temperature and fluctuation.
(time
constant =
Inc.,
1 sec,
the discharge
Four probes were lowered
to
intake
annular
Springs
stationed in
Two probes were
hose
Yellow
using
monitor
intake
Three probes were fixed on stage no. 1 at
an elevation that would immerse them in the mixed layer of the filled
basin.
These probes thus travelled with the model but were located in
areas that the discharge plume should
of
indication
ambient
temperature
among temperature
variance
not perturb.
(characterized by
variability
readings)
in
They provided an
the mixed
layer, which was
compared to the variance of recorded intake temperatures.
arrays of
ten
stationary probes,
designated as
located in the two far corners of the basin.
the ambient
temperature profile and
the
profile
Two vertical
probes, were
They were used to measure
its variability at various times
during the experiment.
The
(1980),
data
acquisition
system,
designed
consists of the following components:
-43-
originally
by McCaffrey
"104
10
'':/'m 3
20
30
I)
Run 16
60 -
Run 6
12()
I 1(m)
Ipt
Figure 4.5:
Typical Variability Between
Density Profiles of Experiment
-44-
Basin
A) General purpose computer; MITS, Altair, 8800B.
B) Disk storage units; MITS, Altair, 99DCDD.
C) Display terminal; Lear Siegler ADM-3A.
D) Data scanner; ADDS YModel 012130.
Fig. 4.6 shows a flow chart for these components as integrated into
a 300 channel per second thermal data acquisition system.
Temperature
information from the YSI 700 thermistor is scanned by the reed relay
which connects the temperature probe output to a YSI thermivolt signal
conditioner, which
is
scaled
to produce
signals which are directly convertible
linear DC analog millivolt
to temperature readings in °C.
After a prescribed number of scans, the digitized scaled analog voltages
for each individual probe are averaged and the average temperatures and
computed variances
are sent
to the display
screen.
During a typical
experiment approximately 600 temperature readings were made using this
system.
4.1.6 The Dye Measurement System
Flourescent dye
direct
recirculation
(Rhodamine B) measurements were used to determine
and
downstream
dilutions.
Sample
dye
concentrations were measured with a Turner Model IV fluorometer allowing
a threshold detection of 1 part per billion
(ppb).
Experiments could be
run with discharge concentrations of as much as 50,000 ppb and basin
background concentrations of less than 30 ppb.
A dye concentration of 10
ppb above background concentration was distinguishable and it was estimated
that measurement of direct recirculation down to 10 ppb/50,000 ppb = 0.0002
or 0.02% was possible (Coxe et al, 1980).
-45-
I
E
L-J
S
E
C
D
A
N
N
eF
I
I
Cormputer
Digital
9o ter
,R SConditioner
R
Figure 4.6:
Scan Interval
Digital
Real Time
Clock
Time Data
Flow Chart for the Temperature Data Acquisition System
Three types of dye samples were taken during an experiment using
the sampling apparatus shown schematically on stage no. 3 of the towing
carriage
in
Fig.
A
4.3.
peristaltic
pump
delivered
a
steady
Two sample
simultaneous flow from four sample points to a bottle rack.
probes attached to stage no. 2 of the towing carriage and located at the
same elevation were positioned
1.5m
(450m prototype)
and
O.8n
prototype) behind the OTEC model to measure field dilution.
(250m
Two more
samples were taken, one each from the intake and discharge flow lines,
and were used to measure direct recirculation.
The
field probes were
steered into
the center of
the plume by
adjusting the elevation of stage no. 2 from a switch panel located next
to the computer.
Stage no. 2 was supported by two motorized vertically
traversing lift racks.
The sample flow and bottle rack were also
controlled from the switch panel.
4.1.7
The Photographs
Injection of fluorescent dye into the discharge water also served to
tag
the
discharge
photographs of
plume
for
photographic purposes.
the power plant wake and
Both
overhead
side view photographs
of a
cross-sectional plane along the axis of the model were taken.
Figure 4.7 illustrates the apparatus used to take the
side view
pictures.
A spotlight emits a horizontal slit of light above the water
surface.
A long, narrow mirror attached to the towing carriage deflects
the light slit downward to illuminate a vertical plane, approximately 50cm
wide x 60cm high, along the axis of the model, parallel to the direction of
current.
A water tight box uses mirrors to reflect the field of vision of
-47-
Adjustable
Mirror
Spotlight
Su pport
00
35 mm
Camera
1000
Towing
Carriage
Mirror
Water Tight
Photo Sub
Watt
Light
Beam\
Light Shutter
Spotlight
_
Plane of Light
Optic Light
So urce
Figure 4.7:
Fiber
Pole
T""lJBasin
ondow
Floor
Weld
Adjustable Mirror
Vision
Cross Sectional Schematic of the Side View Photographic Apparatus
a 35 mmcamera through a front
submerged model.
glass window at
the elevation of the
As the towing carriage moves past the photo station,
pictures are taken of the 50cm x 60cm plane illuminated by the spotlight.
On the average, 5 such pictures were needed to capture the flow field
perturbations extending to 1.5m (450m prototype) downstream of the model.
Figure 4.8 shows a cross section photographed as the model moved by the
field of vision.
The water used to fill the basin was filtered to provide adequate
clarity.
In order to measure quantitatively the position and thickness
of the spreading layer, marker poles containing fiber optic strands were
mounted along the mirror attached to the towing carriage, providing a
reference grid.
Overhead pictures were taken from a balcony
basin.
located next
to the
A grid of black crosses painted on the basin floor provided a
reference grid.
Figure 4.9 is an overhead photograph taken during an
experiment.
4.2
4.2.1
Experimental Procedures
Procedures Before and During an Experiment
Once the basin had been filled, DEMO, a computer program written in
BASIC (all the programs mentioned below are also in BASIC), was used to
scan and print out the thermistor probe readings on the display terminal,
which permitted continuous monitoring of changes in the temperature profile
and the density difference between the upper and lower layers.
Twenty minutes before the start of an experiment, the water for the
-49-
Figure 4.8:
Typical Side View Cross-sectional Photograph
.
Figure 4.9:
Typical Overhead Photograph
-5^.-
discharge
flow was turned on, injected with
fluorescent dye and run
This allowed for
through a bypass loop and rotameter at 45 gal/min.
fine adjustment and stabilization of the discharge temperature and dye
concentration and purged the main sections of the discharge and intake
lines of air.
The intake and profile probes were placed in the upper 3cm of the
water column for calibration.
They and the 3 surface probes attached to
stage no. 1 were calibrated by the program CAL against the temperature
0
of the mixed layer as ascertained by a mercury thermometer to +0.05 C.
After
calibration, the profile probe
the
intake
probes
arrays were positioned
were
placed
inside
the
the
for
intake
experiment
and
structure.
At the end of an experiment, the program DISCAL calibrated
the
discharge
thermistors
the
against
temperature
discharge
as
ascertained by a mercury thermometer.
When the proper density difference,
experiment began.
Ap_,
had been reached, the
The dyed discharge flow was routed through the model,
the intake circuit was activated
and the varispeed
turned on and adjusted to the proper tow speed.
when the wake was judged
to be in steady
towing motor was
Temperature scans began
state.
The program OTEC
collected a prescribed number of scans, designated as n, calibrated the
readings by calling CAL and displayed the average and the variance of
the readings of the n scans for each thermistor probe on the display
terminal.
A photograph was taken of this result.
Typically n= 15.
Since one tow of the model across the basin significantly disrupted
the stratification, an experiment could only last one complete end to
end tow.
This was enough time to investigate two different tow speeds.
-51-
Each tow speed was given a run letter.
Thus experiment 5 consisted of
runs 5a and 5b.
Three to five sets of four dye samples were taken during each run.
As the model passed the photo-station, the photographer told the switch
panel operator in which direction to move stage no. 2 in order to place
the sampling probe in the center of the plume.
Overhead pictures of the wake and side view pictures of
discharge
jets
and flow field were
taken
during each run.
photo-stations were used when an experiment was run in a current.
the
Two
In the
stagnant experiments, the model was positioned in the center of the
basin at the start of an experiment so that the length of time before
wall effects became important could be maximized.
One photo-station was
directed at the model while the other was directed at the mirror 1.5m
(450m prototype) from the model.
4.2.2
Workup of Flourescent Dye Samples
Fluorescent dye samples taken from the intake line, discharge line
and
flow
field were
flourometer.
diluted
as
necessary
for
analysis with a
Concentrations were determined from a calibration curve.
These measurements were used to determine direct recirculation defined
as intake concentration/discharge concentration and centerline dilutions
at 250m and 450m (prototype) directly behind the power plant; dilution was
defined as the discharge concentration/maximum concentration of the
field samples.
-52-
4.2.3
Manipulation of Slide Photographs
A photo-enlarger was used to trace the visible dye boundaries, as
The side view slides were
seen in the slide photographs, onto paper.
pieced together so that
into
one
tracing.
the complete set of
Appendix
I shows
slides
the complete
could be combined
set
of
side view
tracings and compares them with computer simulation (see Section 5.7).
For the overhead pictures, parallax error was corrected by tracing the
wake and the reference grid of black crosses and then reconstructing an
undistorted representation of
the picture.
Appendix II indicates
the
width of the wake at 250m prototype as ascertained by this procedure and
compares it with computer sinulation (see Section 5.7).
4.2.4
Accuracy of Temperature Data
The horizontal uniformity of the upper layer of the water column is
about ±0.15°C.
Since the probes were located at three distant locations
in the basin, the three sets (i.e. the two profile arrays and the intake
calibrated
probes) were
against
three
Thus the overall accuracy of the probes,
locally measured temperatures.
reflecting the accuracy of the
mercury theremometer and the repeatability of the individual probes, was
about 0.1
0
C.
4.2.5 Temperature Data Manipulations
Tables
4.2 and 4.3
temperatures,
list
the
characteristic discharge and intake
the mean intake temperature
-53-
depression and
the average
Table 4.1:
RUN #
Net
Power
(MWe)
Experimental Parameter Schematization
TYPE OF DISCHARGE
DICAG
M/E
61
0
CURRENT
SPEED um
S/I/H
1
1A
40
M
8
90
I
lB
40
M
8
90
H
2A
80
E
8
I
2B
80
E
8
90
90
3A
40
E
8
90
I
3B
40
E
8
90
H
4A
20
M
8
I
4B
20
M
8
90
90
5A
20
M
4
90
I
5B
20
M
4
H
6A
40
M
4
90
90
6B
40
M
4
90
H
7A
80
E
4
90
I
7B
80
E
4
90
H
8A
40
E
4
90
I
8B
40
E
4
90
H
9
40
E
4
90
S
10
40
M
4
90
S
11A
20
M
2
90
I
11B
20
M
2
H
12A
40
E
2
12B
40
E
2
13A
80
E
8
0±45±90
90
90
90
0
13B
80
E
8
0±45±90
0
H
14A
40
M
8
0±45±90
0
I
14B
40
M
8
0±45±90
H
15A
40
E
8
0±45±90
0
0
15B
40
E
8
0±45±90
H
16A
20
M
8
0±45±90
0
0
16B
20
M
8
0±45±90
0
I
-54-
H
H
I
I
H
I
I
H
Table 4.1
RUN #
17
(Continued)
TYPE OF DISCHARGE
CURRENT
Net
Power
(MWe)
M/E
40
E
8
0±45±90
0
0±90
0
0
J
,
J
SPEED u 0
S/I/H
1
02
18
40
E
4
19A
80
E
4
0±90
19B
80
E
4
0 ±90
20A
40
E
4
0±90
0
0
20B
40
E
4
0-+90
21A
40
M
4
0±90
0
0
21B
40
M
4
0±90
22
40
M
4
0±90
23A
40
E
4
+450
23B
40
E
4
+450
0
0
0
Notes)
Type of discharge:
Current speed:
M = mixed; E = evaporator; J = number of jets.
S = stagnant;
I = intermediate; H = high.
-55-
Experimental Parameters (Prototype dimensions except for]Ree )
Table 4.2:
OCEAN
PLANT
I
Qi
(m/se3
(a /sec)
-
4
41
Qo
-
(m/sec)
-----
~b
1
hi
(m /sec)
(m/sec)
(m)
(m)
I
hd
(m)
-I
T'o
4
II
o
del)
(Model)
(oc)
Apx10
H
4
.'
U,
T'a(z hi)
Ta (z =hd
(g/cm
(m)
(m/sec)
(0C)
(oC)
3)
1A
200
400
1.67
3.1
4
17.8
7.4+
1531
24.3
54
0.28
25.2
25.1
1B
200
400
1.67
3.1
4
16.5
7.4+
1443
23.7
2A
400
400
1.67
3.1
4
22.2
12.44+
1993
24.3
57
45
0.51
0.28
25.2
25.2
25.1
24.8
2B
400
400
1.67
3.1
4
22.8
14.6+
2052
23.8
51
0.51
25.2
25.0
3A
200
200
0.83
3.1
4
22.8
1027
24.0
0.28
25.0
24.7
3B
200
200
0.83
3.1
4
22.8
7.5+
1026
23.8
50
56
0.51
25.0
24.8
S4A
100
200
0.83
3.1
4
17.2
3.8+
778
24.2
49
0.28
25.3
25.1
S4B
100
200
1.67
3.1
4
17.2
3.8+
780
24.1
50
0.51
25.3
25.'
5A
100
200
1.67
3.1
4
17.2
6.3+
1629
24.0
48
0.28
25.3
25.1
5B
100
200
3.35
3.1
4
17.2
6.34
1634
23.9
51
0.51
25.3
25.2
6A
200
400
3.35
3.1
4
17.2
12.6+
2893
24.7
50
0.28
25.7
6B
200
400
3.35
3.1
4
17.2
12.6+
3256
24.8
49
0.51
25.8
25.9
7A
400
400
3.35
3.1
4
23.6
31.2+
4180
24.3
39
0.28
25.3
24.4
7B
400
400
1.67
3.1
4
23.6
30.84+
4180
25.2
50
0.51
25.4
25.2
8A
200
200
1.67
3.1
4
22.S
10.8+
2056
24.6
50
0.28
25.4
25.4
8B
200
200
1.67
3.1
4
23.3
12.0+
2067
24.2
54
0.51
25.3
25.3
9
200*
200
1.67
3.1
4
22.5*
8.7+
2513*
22.9*
38*
0.00
24.6*
24.1
10
200
200
3.35
3.1
4
17.2f
9.7+
3537*
21.4*
53*
0.00
24.1*
24.1
11A
100
200
3.35
3.1
4
17.0
10.6+
3521
23.78
48
0.28
25.3
25.3
11B
100
200
3.35
3.1
4
17.1
10.8+
3546
23.51
48
0.51
25.3
25.3
12A
200
200
3.35
3.1
4
23.6
21.9+
4461
24.87
48
0.28
25.5
,,,
25.7
-----
25.4i
i
Experimental Parameters (Prototype dimensions except for P)
Table 4.2:
OCEAN
PLANT
I
U
Qo
Qi
u
o
b
hi
hd
(m)
o0
Ao x104
H
(Model)
(g/cm 3)
(m)
1R
e
IF
o
u
(m/sec)
I
I
I
RI
I
I1
bT
T'a (z-h
i )
T'(z =hD)
(oc)
(m3/sec)
(m3/sec)
(m/sec)
(m)
(m)
12B
200
200
3.35
3.1
4
23.6
21.9+
4466
24.87
48
0.51
25.5
25.4
13A
400
400
1.96
2.9
4
19.8
7.4+
2259
23.13
48
0.28
25.5
25.3
13B
400
400
1.96
2.9
4
21.1
7.7+
2278
23.85
48
0.51
25.5
24.7
14A
200
400
1.96
1.9
4
18.3
6.0+
2175
23.45
54
0.28
24.5
24.4
14B
200
400
1.96
2.9
4
18.3
6.0+
2175
23.45
54
0.51
24.5
24.4
n 15A
200
200
0.98
2.9
4
16.0+
1345
44
0.28
15B
200
200
0.98
2.9
4
17.0
16.0+
1345
23.68
43
0.51
25.3
24.2
16A
100
200
0.98
2.9
4
19.4
3.7+
1223
20.22
54
0.51
23.6
23.5
16B
100
200
0.98
2.9
4
29.4
3.9+
1227
20.07
55
0.28
23.6
23.5
17
200
200
0.98
2.9
4
23.2*
7.2+
1330*
22.9*
54*
0.00
24.6*
24.5*
18
200
200
1.93
2.9
4
23.3*
16.0+
2837*
22.5*
44*
0.00
24.9*
23.6*
19A
400
400
3.88
2.9
5
22.3
21.3+
5400
23.14
54
0.28
24.4
24.1
19B
400
400
3.88
2.9
5
23.3
20.84
5468
23.06
52
0.51
24.4
24.1
20A
200
200
1.93
2.9
5
22.5
13.84,
2757
20.9
54
0.28
24.2
24.0
20B
200
200
1.93
2.9
5
22.5
13.8+
2757
20.8
54
0.51
24.2
23.7
21A
200
400
3.88
3.9
5
18.3
12.8-
5143
22.3
48
0.28
24.4
23.9
21B
200
400
3.88
2.9
5
28.5
13.24
5205
22.3
45
0.51
24.4
23.5
22
200
400
3.88
2.9
5
17.2*
10.7+
5343*
24.5*
48*
0.00
25.4*
25.4
23A
200
200
1.93
2.9
5
22.8
12.6+
3040
23.6
52
0.28
25.3
25.0
23B
200
200
1.93
2.9
5
22.9
14.8+
3040
23.3
52
0.51
25.3
25.0
I
_______________
(OC)
I ____________________________________________
I-___________
1
4
__________-I
________________
" 4 " under IF indicates directioL of plume buoyancy
Note:
"*" connotes time averaged value.
"--" signifies that the data does not exist
(o0C)
OT '0
1600
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TWOr
TO '0
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TO'0
170*0
90
00
fl
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9010
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9L 'T
L'1 L '6T-
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900
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I
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00£
VZT
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98
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V/OT
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69
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08
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69
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05;T
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01
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(ul
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QCN
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90'0Colo-
$1'L -
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L8'T
(Ta3aS)
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6*0
89'T
0&$1
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69'T
OOOZ-
0£ '0
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8'6TT'0
S1'0-
C 06T-
Z0 '0
TO'0
ZO'0
$18
8'1:
89'T
0'0
170*0
0 0
£0'0
ZO'0
nso'o
flzoO*
8L'T:
TO'0-
u6Z'0
'V7'N
901
T'OZ-
*8 16T-
~'9
cL'T
Q
fso -
ZO'0
0
$10'0
0'0
0'0
C0'0
TO '0
£0 '0
00
Z0'0
00
0 '0
TOO0
6' -
1O'0
0'0
TO 0
C0'0
T00
TO'0
17T'0
90'0-
f60'0
(Z 00)
TO'0
(O)
xVni~TQ
(M)
OSZ3
VT
N
-'
t
RUN
S
P
I
D
h
250
"250
(m)
(m)
eq
(m)
Y
s
(m)
I
1
°250
rv
'I
D
P
P
x104
Ap
(g cm
Nxl02
(sec- 1 )
3)
ATi
T
(Oc)
i
T
1r
2
ci
o.1
I
2
2
0
MAX
a
(oC 2 )
(Oc2)
(Oc2
13A
13.7
36
844
51
126
N.D.
2.8
-12.5
1.59
-0.04
0.04
0.08U
0.01
13B
11.0
61
543
51
76
N.D.
2.6
-11.5
1.57
-0.18
0.09
0
. 16 U
0.02
14A
8.9
41
808
55
121
N.D.
0.6
-18.8
1.74
-0.01
0.03
0.05U
0.02
14B
8.9
56
543
48
77
N.D.
0.4
-19.0
1.71
-0.06
0.05
15A
15.2
45
329
43
90
N.D.
2.6
0.7
78
N.D.
1.0
0.7
1.56
-0.02
0.01
0.03
0.02U
0.01
15B
15.2
45
212
42
16A
11.9
47
300
51
58
N.D.
0.4
-12.4
1.57
-0.01
0.02
0.o3F
0.02
16B
8.5
52
426
48
78
N.D.
0.5
-11.4
1.56
-0.02
0.03
0.
0 3N
0.03
N.A.
-
3.3*
1.61*
0.01
30*
N.A
-
2.6*
1. 32*
0.01
40
271
-
5.9
1.60
17
30*
18
68
644
N.D.
0.9
0.03
0.051
0.03
-0.19
0.12
0.17
D
0.01
0.02
0.01
0.02 U
0.02
D
0.01
U
0.01
0 .1 0
U
0.01
0.02
U
-0.05
19A
13.4
19B
9.3
59
293
46
113
N.D.
0.4
-
6.2
1.62
20A
8.0
48
446
34
149
N.D.
0.5
-
3.5
1.52
20B
8.0
21A
11. 2
56
944
48
249
N.D.
0.6
-16.3
1.45
0.05
0.03
0.03
21B
8.6
66
402
62
163
N.D.
1.0
-15.3
1.46
-0.05
0.06
45
N.A.
-23.5*
1.63*
45
22
77
336
23A
30.8
40
500
31
28
23B
76.6
45
400
35
15
N.D.
N.D.
N.D.
-
0.5
-
0.1
-
0.1
i
i
1.50
3.5
1.53
4.0
1.53
3.0
-1
0.01
0.03
0.02
**
0.01
0.01
-0.05
0.02
-0.05
-
0.02
J
N
0.01
0.01
Notes:
"*"
connotes a time averaged value.
"**"
means that that quantity changes with time and is reported
as a function of time in Table 4.4
f"-"
signifies that that data does not exist
"N.A."
"U"
indicates that that parameter is not applicable to that
particular experiment
indicates that ii (MAX) occurred in the upstream half of
intake annulus
"D"
indicates that oi2 (MAX) occurred in the downstream half
of the intake annulus
"N"
indicates thatai2(MAX) occurred at least twice and in both
the upstream and downstream halfs of the intake annulus
"N.D."
means that that•v parameter was not detectable for that experiment
0
and maximum variance of the intake temperature as determined from the
intake and discharge thermistors.
Experimental Results
4.3
Table 4.1 gives
Although a range of parameter variation
conducted.
experiments were not meant
operating
the experimental series that was
of
a summary
conditions.
was examined, these
to represent a comprehensive set
Rather
towards ocean/plant operating
the
experimental
conditions most
series
likely to
was
of plant
oriented
affect power
production by inducing recirculation and those which could be performed
resulting in significant interaction between the plume and the
without
Thus, the discharge depth was relatively shallow compared to
basin bottom.
the discharge proposed by industry (Scott, 1979).
depth was somewhat
3.3).
Also, the mixed layer
shallow compared to site-specific data (see section
Currents slightly over and under those anticipated (see section 3.3)
were studied.
In addition, previous work
(Coxe et al, 1980) led us to
anticipate maximum recirculation at currents
least for the horizontal experiments.
currents
of
at
least
this
magnitude.
of approximately 50 cm/sec, at
Therefore it made sense to include
The
other objective
of
the
experimental series was to amass enough data with respect to plume geometry
and dilutions under a range of operating conditions that an integral jet
model could indeed be verified and later used with assurance.
Finally it
should be noted that the nominal plant size is based on a flow rate of
Qi
=
5 m /s-MWe.
-61-
The numerical values of the dimensional parameters resulting from
the experimental schematizations expressed in Table 4.1 are given in
Table 4.2.
The values of Apa
explanation.
and T' indicated in Table 4.2 deserve further
Because ambient temperatures used in the experiments
varied, experimentally measured temperatures (T) were cast in terms of
density differences, using the density at a depth of 165m as a reference.
Thus in general
Ap'= p ra (z = 165m),
S =0 o/oc
-p [T,S = 0 O/oo
Furthermore since one is more able to identify with ocean temperatures
than with density differences, the values of Ap
were converted to charac-
teristic tropical ocean temperatures, denoted by primes, as well.
ical ocean with uniform salinity of 35
depth of 165m was assumed.
0
A trop-
/oo and a temperature of 170C at a
Thus associated with every T and Ap',
a value
of T' was defined for which
Ap' = p
4.3.1
' = 17'C, S = 35 o/oo
-p
V ',
= 35
/oo]
Data Summary
Tables 4.2, 4.3 and 4.4 present a summary of the results which can be
expressed in parameter form.
The measurement source for finding the parameter
is denoted by a D, P or T (dye, photographic or temperature measurement).
The
listed parameters are:
Sc:
centerline dilution at y = 250m (prototype) as determined from
measurements of discharge and plume centerline dye concentrations.
-62-
t250:
thickness of the plume at y = 250m (protytype as seen in
the side view tracings.
W250:
width of the plume at y = 250m (prototype) as seen in the
overhead tracings.
h
:
eq
equilibrium depth, taken as depth of plume centerline at y = 450m
(prototype) as seen in the side view tracings.
Y :
s
distance to the stagnation point as seen in the side view
tracings.
0250:
number of oscillations that the discharge makes around an
equilibrium elevation after it has reached its maximum depth
of penetration (vertically
X:
APo:
direct recirculation (percent)
discharge density difference
=
N:
directed discharges only).
Ia(z
= hd)
-
0o
Brunt - Vaisala frequency
p0
o
where g = 9.8 m/s 2 and
p/D3z is taken as the slope of the
thermocline.
AT.:
1
The difference between the evaporator intake temperature and
the average of the temperatures in the mixed layer near the
evaporator intake.
=
'
t i
-T
(z
a
= h
i
the 4 intake probes
-63-
Table 4.4: Time-Variant Experimental Results
of Non-Steady State Stagnant Experiments
'''
Run T(mtin
9
9
9
9
9
15
AT.
1
-0.03
2
2
oa
Smax)
0.01
0.03
87
160
0.06
0.01
0.02
232
319
9
9
9
452
9
10
696
528
0.08
0.10
0.01
0.01
0.02
Run
0.00
AT '
i-0.17
ai
)X
17
377
0.3
17
468
0.2
18
16
0.0
18
52
0.5
0.0
18
81
0.4
0.0
18
117
18
183
18
241
18
290
0.0
0.10
K(min)
0.9
0.01
597
]
2
0.01
0.01
0.02
-0.11
0.02
0.02
-0.05
0.02
0.04
-0.17
-0.02
0.01
0.02
1.(,
6.3
0.3
0.1
-0.04
0.01
0.03
0.1
0.2
0.01
29
-0.10
0.01
0.02
41.6*
18
360
-0.01
0.01
10
81
-0.06
0.02
0.02
3.3
22
479
-0.02
0.01
10
10
125
-0.06
0.01
0.02
0.6
22
15
-0.22
0.02
189
-0.07
0.00
0.01
0.6
22
44
10
247
0.2
22
109
10
290
-0.08
0.01
0.02
0.4
22
148
10
10
435
-0.06
0.01
0.02
0.3
22
189
17
21
151
17
225
17
296
0.02
--
-0.16
0.00
0.01
4.6
0.4
0.4
0.2
-0.17
0.01
0.03
0.5
0.2
537
17
0.5
0.00
-0.25
-0.19
0.02
0.01
0.03
0.01
4. 1
0.4
0.4
-0.17
0.01
0.01
0.5
-64-
Notes: " -"
signifies that that
data does not exist
Time is in prototype minutes
"*"
large values of X at initial times attributed to
"jet" start-up".
2
a i (max):
a
:
:
maximum variance of the 4 intake probes.
average variance of the ambient water at the elevation
of the intake as determined from the 3 near-surface
probes in current.
4.4
Comment on Jet Reynolds
Numbers
Inspection of Table 4.2 shows that in experiments 3, 4, 15, 16 and
17, the jet Reynolds
number was less than 1500.
However as previously
mentioned (Section 3.1.1), the transition to laminar flow is gradual
and viscous effects in these experiments are undoubtedly small.
The side
view slides indicated that significant turbulent mixing did occur.
-65-
V.
THE INTEGRAL ANALYSIS
Justification for Use of an Integral Jet Model
5.1
The
physical
discharging jets
reasonable
model
experiments
indicated
that
the
formed a cohesive, well-defined plume.
vertically
It
is also
to expect the coflow to crossflow jets of the horizontal
experiments
However
to be well-behaved.
any horizontal
jet with a
discharge component into the current will experience reentrainment and its
cohesiveness will be inversely proportional to the rate of reentrainment.
The
low values of recirculation in all the experiments showed that the
Therefore it
discharge does not significantly interact with the intake.
was decided to analyze the gross behavior of the discharge independently
from the intake and address recirculation as an intermittent phenomenom
related to fluctuations that cannot be captured with a treatment of average
discharge properties.
integral jet
All the vertical discharges were analyzed with an
model as were
the horizontal
discharges
that
did not
experience significant reentrainment, which cause the integral analysis to
fail.
A verified integral jet model
represents the major
is a powerful tool.
Because
it
independent variables of a system, it allows rapid
simulation of conditions that would otherwise require numerous experiments,
some
outside
of
the
range
of
a
particular
experimental
facility.
Considering the limitations of our experimental basin, this was quite
advantageous.
5.2
Previous Integral Model Studies of Buoyant Jets in a Current
The near-field of a submerged discharge is characterized by a zone of
flow establishment (ZOFE) and a zone of established flow (ZOEF).
-66-
The ZOFE
is characterized by a potential core, which is the region that turbulent
mixing has not yet penetrated.
An integral analysis may either begin at
the physical origin (using modified parameters to treat the ZOFE) or at the
end
of the
ZOFE, using conditions at
as
this cross-section
initial
conditions.
Analysis of discharges in a continuous environment of infinite extent,
whether plumes
or jets,
using the
integral form of
the conservation
Rouse et al (1952)
equations has progressed considerably over the years.
and Albertson et al (1950) confirmed the adequacy of representing velocity
and scalar profiles in pure plumes and jets respectively, with Gaussian
distributions.
Morton et al (1956) used the conservation of mass equation
in their analysis of buoyant jets discharged to a quiescent medium, which
required specification of an
entrainment function.
The
function they
proposed depended only on the local mean velocity and width.
Several investigators
(Keefer and Baines, 1963; Priestley, 1966; Fan,
1967; Platten and Keffer, 1968; Hoult, 1969; Hirst,
1970; Winiarski and
Frick, 1978) have analyzed vertical discharges in crossflow.
Generally
these analyses have resulted in integral models that have been applied to
and verified against
either atmospheric
smoke stack or cooling tower
emissions, or wastewater discharges or ocean outfalls from power plants.
The hydrodynamic characteristics of the OTEC discharge studied in this work
are typically in between these two kinds of discharges.
For example, an
OTEC discharge would have a Froude number characteristic of a relatively
non-buoyant atmospheric emission or, conversely, a relatively buoyant ocean
outfall.
Thus the model that we choose will undoubtedly not
verified for our entire experimental range.
Respecting this conclusion,
several other criteria were important in selecting a model.
-67-
have been
Those criteria
were that the model include the effect of buoyancy on entrainment and that
it be versatile.
Fox (1970) first suggested that entrainment is indeed a function of
(1971,a) incorporated buoyancy
into his
entrainment
buoyancy.
Hirst
function.
In addition, he combined the entrainment functions of several
investigators
for
coflow
crossflow
and
situations
formulate
to
a
generalized entrainment function that performed well over a wide range of
Because it can
discharge orientations with respect to the free stream.
treat both horizontal and vertical discharges and because its entrainment
function does reflect buoyancy effects, the Hirst model was chosen to
simulate the pilot plant discharge.
5.3
Description of Hirst Model
The Hirst integral model was designed to treat three dimensional flow
of round, turbulent, buoyant jets discharged at arbitrary angles to flowing
stratified ambients.
coordinate
The integral equations are formulated in a "natural"
system that
follows
the
jet
centerline.
A
generalized
entrainment function was established that Hirst found to perform well
without calibration over a broad range
of jet-plume conditions.
The
following discussion describes the Hirst model as documented, (Hirst,
1971,a) then addresses the modifications made on it.
5.3.1
The Governing Equations
The basic conservation equations of mass, momentum, energy and a
scalar in Castesian coordinates were simplified for a steady mean, fully
turbulent, incompressible flow in which the boundary layer assumption could
be invoked and in which the pressure variation is assumed to be purely
-68-
hydrostatic (i.e. drag is neglected).
These equations were expressed in
a "natural" coordinate system to allow efficient tracking of the flow trajectory and properties.
The resulting set of non-linear, coupled,partial differential equations
includes three independent variables - r, s and
assumption of axisymmetry removes
(See Fig.
5.1).
The
dependency on the azimuthal angle ¢.
The assumption of Gaussian distributions of fluid properties allows integration over the jet cross-sectional area.
These distributions, for velocity
and density are:
-r2/b
u = (um - uSIC2)e
p
2
+
u0SIC2
= Ame(r/Xb)
(5-
1)
(5 -
2)
where b is a characteristic jet radius, the subscript infinity refers to
the ambient value of that quantity, Si = sin 0i, Ci = cos e,
e, is the local
jet angle in the horizontal plane with respect to the x-axis, 82 is the local
let angle with respect to the horizontal plane (see fig. 5.1) and X
is a
spreading factor to be addressed later.
It should be noted that the model simulates only density differences
between discharge and ambient, rather than temperature or concentration
differences. This implies a linear equation of state between temperature
(2) and concentration (C) of the form
P = Po
1 -B
(T -
To)
-
Y(C -
(5 -
C
3)
which includes a reference density po and concentration (y) and temperature
(3) coefficients of volumetric expansion.
It includes no simultaneous de-
pendency on both C and T.
The derivatives of the local jet centerline velocity (um),local jet
"radius," (b), local jet centerline density difference (A0m),
-G9
local jet
angles (01 and 62) and the local Castesian coordinates (x,y,z) with respect to s are expressed as:
du
ds m
ds
deO
=
u
(SS
1 2
1 + CCC
-
d
1
)
+
1 2 ds
ds
g(pm - pm)b2S2
4 (uS1C2 - um)E
b2(um + uS C2)
b2
E
db
~2
2
7
-du
2p
dO
U0rn(S1S
2
s
I
(5 - 4)
de
+ CC
12
*-12
ds
ds
2
1)i
(
(5- 5)
ds
(um + uCS 1 C2 )b
dAPm
p b
+
m+
ds
u
F
APmb2
)7
um)T
2
-u
-uu(uSC 1 2(1 +
C
SS C
du
-
2)
de
u2S2 ds
0
2 ds
2 ds
/
Lo
b 2 X2 K(US1C2
0I22 2
ds
do d2
ds
1
2
2
(1 + X2 )
d
2
m
22
de
S
db
ds
1C2 )
9z 2 S 2 (urnm
ds
- Um)
(i
1
+
(5 - 6)
2)
EuuSC
(5 - 7)
qC2
- (P
- P
0
2m)bC
2
EuS lS
gX
2]
(5 -
q
-70-
8)
. a
gI
Figure 5.1:
Natural Coordinate System
dx
ds
=
CIC2
(5 - 9)
_
SC2
12
(5 - 10)
S
(5 - 11)
dz
ds
where
q =
1 [bK2
Lb (uS
)2
1
C 2 + um )
- E
2
,
S1,2 = sin1,2
C1,2 = cos1,2.
(5
-
12)
The numerical solutions to these equations were obtained through a
variable order, Adams predictor - corrector method on an IBM, VM-1, 370 computer using the IMSL subroutine called DGEAR - Differential Equation Solver.
-5
The relative error bound specified to the subroutine was 1.0 x 10-5
Numerical solution yields values of the centerline velocity and density
difference, the width, orientation and position of the discharge as a
function of the centerline coordinate s.
Dependence of velocity and density
difference on rmust be found using the similarity profiles of Equations 5.1
and 5.2.
A solution can only be found after the entrainment function E and the
initial conditions at the end of ZOFE are specified.
These specifications
are now addressed.
5.3.2
The Entrainment Function
Jet entrainment is a function of the mean flow conditions, buoyancy
within the jet, jet orientation, free stream velocity and the ambient turbulence.
The effect of ambient turbulence is neglected due to a lack of per-
tinent data.
An extensive body of knowledge exists that addresses the other
-72-
functionalities.
Fox (1970) considered byoyant jets discharged vertically up to a stagUsing an integral equation of mechanical energy conserva-
nant ambient.
tion, he deduced that:
a2
) umb
E = (a1 +
TF
rL
Where F
r
(5 - 13)
the local jet Froude number,
is
ur(
um
/
a
Ap gb)
m
and a, and a
2
are
This equation provides for transition between
entrainment coefficients.
buoyancy and momentum induced entrainment.
Hirst generalized this result for non-vertical discharges to give:
E = (a
1
a
ub
+ a2 2
2 2)
b
F
(5 - 14)
rL
The term alum b represents internal jet turbulence induced entrainment.
The constant a
l
has been found experimentally to be equal to 0.057 in the
limiting case of a pure momentum jet (Albertson, 1950).
2
The term a2 S2 u b/F
The termrLrepresents buoyancy induced entrainment, which
can be
shown to be a function of the turbulent Schmidt number (Hirst,
a2 = 2
2
2
1971a):
2
3X
3X2
2+1
(5 -
For a simple momentum jet, Becker et al
For a simple plume, Rouse et al
15)
(1967) found that X= 1.11.
(1952) found that X = 1.16.
Since a2 en-
trainment is only significant for low Fr flows, Hirst set X equal to 1.16.
Thus a2 = 0.97.
Hoult (1969) developed an entrainment function for buoyant jets discharged to a cross flow:
-73-
E = a 3 blu
- umS 1 C2
+ a 4 bu1
-(S
2
(5 - 16)
The first term in Eq. 5 - 16 accounts for jet induced entrainment
while the second term accounts for ambient current induced entrainment.
To produce an expression that incorporates the effects of both
buoyancy and ambient current, Hirst combined the Fox and Hoult equations
to yield:
a
+ a22
E =(a
S 2 )bl m-uS
1 C2
+
1
rL
a6
(a5 +
1 -
2 S2 ) ub
(S1 C2 )
(5 - 17)
(5 - 17)
rL
where a l and a2 are as previously defined and a 5 and a 6 represent ambient
induced entrainment as a function of buoyancy.
Hirst assumed that an ambient current will not effect the ratio of
buoyancy induced entrainment to internal turbulence induced entrainment.
This is
a2
al
tantamount to:
a6
a5
which reduces Eq. 5 - 17 to:
E = (al
1
S2)
ebum
- uS 1 C2 1 +
rL
a3umb
0
- (SC
2
(5 - 18)
where a 3 is a new entrainment coefficient that must be empirically ascertained while a l and a2 are the same.
used in Hirst's integral analysis.
-74--
This is the final form of the equation
Hirst evaluated a 3 by calculating jet trajectories for several flows
and adjusting the value of a 3 to provide a good fit.
He recommended
a3 = 9.0 as the optimal value, admitting that this method is somewhat
subjective.
5.3.3
The ZOFE
One definition of the end of the ZOFE is the first location along the
jet centerline where u < u . Prior to this point, the velocity profile in
m
o
the potential core is top-hat.
Obviously assuming a Guassian profile in
the ZOFE is a bad approximation and the integral analysis described above
is inappropriate.
Hirst (1971,a) solves for conditions at s = se, the end of the ZOFE,
by assuming top-hat profiles at s = 0 and Gaussian profiles at s = s .
e
In
= u0 and performing mass, momentum, energy and
voking the fact that u
Se
scalar balances between s = 0 and s = s ,
e
the following conditions at s
are derived:
u = u
o
m
(5 - 19)
2u
2
b =
o
2
b
+ uSC
o
o
o 1 2
u
u
2
X +1
2
Ap
+
o
(5-
uSC
12
2
o1 2
20)
0
(5 - 21)
=
1
(5 - 22)
2 =
2
(5 - 23)
e1
o
S=X
o
Y = Y
y
(5 - 24)
+ SeC 1 C 2
o
+ s S
e 1
o
C
2
(5 - 25)
-75-
e
z = z
+
(5 - 26)
S 2S
Hirst assumed that buoyancy forces and free stream velocity have negligible
effect on trajectory in the ZOFE.
These eight conditions provide the essen-
tial boundary conditions to the integral analysis.
Hirst also formulates an
expression for the length of se which we modified considerably.
This modifica-
tion will be developed later.
It should be noted that Hirst (1971 b) formulated an integral model for
the ZOFE, consisting of two sets of equations,one each for the inner (tophat) and the outer (turbulence eroded) region.
However since this study is
not particularly concerned with parameters within the ZOFE, the algebraic
approach was used.
Hirst's Verification of His Model
5.4
Hirst (1971,a) compared theoretical prediction
to observation for 100
different flows with the entrainment coefficients held constant.
Initial con-
ditions for these flows fell in the following ranges:
2
10 <Frr <C,
0
R < 0.54,
-450
<e1
< 450
1-
00 <e2
2- < 900
where
IFr= u O (Pa/APmgbo)
and
R = 4,/u .
Thirty flows experienced ambient current.
Hirst achieved what he called "good" but not "excellent" agreement with the
data.
For example, for the forty cases considered with a stably stratified,
stagnant ambient, the predicted maximum height of rise of a vertically directed
buoyant jet was within about 5%.
This was excellent agreement considering that
the average rise was about 200 b .
Trajectories for buoyant and non-buoyant
jets discharged at various angles to the free stream were typically within 15
-76-
to 20% of measured values with maximum error of about 50%.
Trajectories
for buoyant jets (IF 2 = 17 to 25) discharged to a crossflow were within 20%
r
or less.
Hirst also looked at centerline velocity and concentration decay, but
was more concerned with the prediction of trajectory, since he had more
data on trajectory and since he calibrated a 3 to trajectory.
noted that Hirst neglected drag in his analysis.
It should be
If however bending due
to drag is important, then the calibrated value of a 3 may prove to be high
since it is calibrated against data on trajectory and since, as a 3 increases, so too will bending.
5.5
Previous OTEC-Related Use of the Hirst Model
Because of its generality, the Hirst model has been used in a number
of applications.
Van Dusen and Mangarella (1974) used it to analyze the
behavior of the condenser discharge from an OTEC plant.
Because they had
no physical data to compare with model predictions, they were unable to
evaluate the performance of the model and had no justification to alter it nor did they alter it.
As their base-case, they considered a
400 MWe asymmetric plant in
which the condenser flow discharges horizontally at 80 m from one circular
conduit.
They also modeled discharges of up to 300 from a horizontal plane.
They treated crossflowing and coflowing currents with velocity of 3 m/sec,
corresponding to the Gulf Stream in the Straights of Florida.
Discharge
velocities ranged between 4.6 and 6.1 m/sec corresponding to maximum port
diameters of 17.2 and 20.0 m respectively.
In short, their conceptualiza-
tion of the plant and the ambient environment was significantly different
than the one reported here.
-77-
In a later paper addressing submerged nuclear power plant discharge,
Mangarella
(1975)
again used the Hirst model.
This time he
grouped four horizontal submerged discharges into one equivalent source.
However he did not represent the effect this grouping has on the length
of the ZOFE and the entrainment.
5.6
Adaptation of Hirst Model
The Hirst model was modified to treat multiple jets, represent the
ZOFE more accurately and include the phenomenom of plume collapse in the
near field.
These modifications, as described below, permitted more repre-
sentative simulation of OTEC plume behavior.
That the model be able to treat multiple, vertically directed
jets
was essential since an OTEC modular design with several discharge ports in
proximity was anticipated.
source.
Multiple jets were grouped into an equivalent
The introduction of an aspect factor (Winarski and Frick, 1978)
captured the difference in entrainment between the multiple and the equivalent sources.
Because the experimental range of the length of the ZOFE was approximately 40 m, comparison of different experiments demanded accurate representation of the ZOFE.
Therefore the effects of buoyancy and free stream
velocity on orientation were introduced into the description of the ZOFE.
In addition, the dependency of the length of the ZOFE on buoyancy and
crossflow velocity were reformulated from those given by Hirst (1971,a).
A model that assumes axisymmetry cannot be expected to predict the
shape of a discharge into a stratified environment.
The OTEC plume will
most likely be discharged into or near the thermocline were it will spread
laterally due to a net pressure difference with the ambient.
-78-
This phenomenom
was formulated and superimposed on the model output.
5.6.1
Jet Interaction
Interaction between closely spaced jets has been treated in the lit-
erature in two different ways.
Wu and Koh (1977) combined several plume
models to represent multiple sources in a row.
Alternatively, Winiarski
and Frick (1978) used geometrical constructs to account for entrainment
differences between an equivalent single source and the actual sources.
The process of merging in the OTEC design considered in this study is
complicated by the presence of the cold water pipe (CWP).
This precludes
explicit treatment of interaction in the context of the integral equations.
Obviously it is desirable to work with a single integral model (i.e. the
Hirst model).
Therefore the tack of Winiarski and Frick was followed.
Winiarski and Frick (1978) provide a simple method involving the concept of an aspect factor (AF) to treat an array of jets in close proximity.
They hypothesized that the main difference between multiple sources and
a hypothetical equivalent area single source with the same fluxes of mass,
momentum and energy is that the projected and peripheral areas of the two
cases differ.
This difference can be geometrically monitored along the
trajectory of a single equivalent jet by the AF, which is defined as:
actual projected cylindrical area of multiple sources
projected area of equivalent single source
Since current-induced entrainment is proportional to actual projected
cylindrical area and jet-induced entrainment is proportional to actual
peripheral area, the AF effectively expresses the entrainment velocity for the
equivalent jet divided by the entrainment velocity of the true multiple
sources.
It is used as a coefficient multiplying the nominal entrainment
-79-
velocity of the equivalent source.
Alternatively the average behavior of
one of the multiple sources could be represented by decreasing the entrainment for a single source by the appropriate factor.
The value of the AF depends on the orientation of the discharge
port array with respect to the ambient current and on position along
the plume trajectory.
Winiarski and Frick (1978) illustrate the rela-
tionship between the AF and the equivalent radius with an easily followed
example
that is restated here
with the values changed to reflect a
potential OTEC scenario.
Imagine three sources of radius 3.0 m whose centers form an equilateral triangle 15.0 m on a side (see Fig. 5.2).
An equivalent source of
radius 5.2 m (giving the same cross-sectional area as the three sources
combined) is centered at the centroid of the triangle. The AF is initially
1.73 and remains constant until shadowing begins, at which time it begins
to decrease linearly.
The slope of the decrease can be determined by
solving for the equivalent radius and the AF at the beginning of merging.
This point and the point that defines the initiation of shadowing are
connected and extrapolated to the equivalent radius for which AF = 1.0, indicating that the sources are completely merged.
Formulation of the AF as proposed by Winiarski and Frick (1978) assumes that the only difference between an equivalent area source and a group
of closely spaced multiple ports issuing in the same direction is the entrainment rate and that this difference in entrainment can be accounted for
by multiplying both current-induced and jet-induced entrainment.by the same
coefficient, that all shadowing is effective, that the completely merged
multiple sources behave as an axisymmetric equivalent source and that the
-80-
a) Initial separate sources
3.0 m
(initial radius)
-
3x2x3.0
25.2 =1.73
2x5.2
A.F.
b) Shadowing starts
c)
7.5 m
3.8 m
1.6 m
I
A.F.-
3x2x3.8
S2xb.6
1.73
16.6
(a)
5.2
(b)
6.5
(c)
13.0
14.7
2.0
1.
1.0
4--
S
-
"0
I
IIII
2.5
1
I
7.5
---
Figure 5.2:
1
I
II
10.0
Equivalent radius(m)
Example Aspect Factor Illustration
-81-
I
12.5
15.0
sources deform symmetrically.
These assumptions facilitate evaluation of
the AF and,while approximate, are justified by the fact that the AF works
reasonably well as documented in Section 5.7 and cannot readily be evaluated without them.
The validity of several of the assumptions were tested
in a series of simulations of the vertical discharge experiments in a current
in which the base case AF was modified.
See Figs. 5.3 - 5.5 for the
orientation of the port array with respect to the current.
Since use of
an AF can be expected to affect trajectory as well as dilution, we examined the behavior of these two parameters with different AF modifications
and compared this behavior to the base case results.
The results are pres-
ented in tabular form below.
Table 5.1.1
Description of Simulations for AF Sensitivity
Simulation
Case
A.F.
f(beq,s)
radius
E.
1
Base Case
2
Merged Initially
N.A.
E.
3
No Interaction
/-;
E.
4
Individual Source
N.A.
A.
5
No Shadowing
f(beq, s)
E.
Notes:
"E."
= equivalent
"A ." = actual
,, f"
= function of
= number of actual sources
"b
1
eq
"N.A."
= equivalent radius
= not applicable
-82-
Table 5.1.2
Statistical Results of
Sensitivity Simulations
X
S
1
c
1
h
eq
2
S
c
2
h
eq
S 3
c
h
eq
S 5
c
h
-0.7
-2.4m
2.7
9.7m
3
5
eq
Statistic
x. -x
-1.6
a
x -x
p o
2
.6m
-0.8
27.4m
1.5
-3.8m
2.5
11.5m
2.7
19.3m
2.4
13.8m
-0.16
0.03
-0.08
0.34
0.15
-0.05
0.26
0.14
0.28
0.24
0.25
0.17
-0.07 -0.03
x
G
0.28
0.12
x
Notes:
superscript of parameter indicates case
(see previous table)
"p" = predicted
"0" = observed
"5" = mean value of parameter x
Case four was not meant to be compared with observation and therefore does
not appear in Table 5.1.2.
Case two is an attempt to ascertain whether or not inclusion of the
AF really does improve simulation when an equivalent source is used.
De-
leting the AF but invoking an equivalent area source implies that the multiple sources behave as a completely merged equivalent source from their
origin.
As indicated by Table 5.1.2, this assumption causes a significant
reduction in the ability to predict equilibrium elevation, while affecting
dilution only slightly, compared to the base case results.
The model per-
forms much better with the AF than without it.
Case three is an attempt to determine whether or not it is important
to account for interaction at all, when using an equivalent area source.
-83-
Using a constant AF equal to
n assumes that the multiple sources do not
interact (i.e. that no shadowing nor merging occurs).
The results of
Table 5.1.2 indicate that this approach causes a 15% overprediction of
dilution.
Since predicted dilution should be approximately 15% less than
observed dilution according to the analysis of Section 5.7, it was concluded that it is important to account for interaction, which as demonstrated by the base case and case five decreases overall dilution relative
to this case.
Comparison of case three to case four indicates the validity of the
AF formulation.
Since in case three, the sources are assumed to be in-
dependent, the equivalent area source should exhibit the dilution and
elevation characteristics of any one of the individual sources it accounts
for, when that individual source is modeled separately, which is exactly
the situation in case four. These two parameters were within
9% of each other in the mean with minor standard error reflecting the consistency of the error.
Thus it is concluded that the AF theory is reason-
ably sound analytically, but does not correlate perfectly between instances
where it should.
Table 5.2 presents a summary of the comparison.
-84-
Table 5.2: Equivalent Source,
No Interaction Compared to
Individual Source
x
S
h
1.1
-6.4m
c
eq
Statistic
xi-x V
a
0.7
x -x
xiXn
1.6 m
0.09
0.08
0.06
0.02
x
.
x
Notes:
"i"
= simulation of individual jet
" 2'
= simulation of equivalent source
with AF = Vn
= mean value of x
Sc and heq as defined in section 4.3.1
-85-
The validity of the effective shadowing assumption will depend on the
distance between the sources and the free stream turbulence and uniformity.
For example, a source directly behind another source along the free stream
axis will probably experience complete shadowing.
As the distance between
the two sources increases, it is clear that this geometrical shadowing becomes ineffective.
Because the distance between ports in the vertical ex-
periments was as much as 6.2 ro, this consideration was explored in case
five.
In case five, all shadowing was considered ineffective.
initial value of AF was always
geometrical configuration.
Thus the
vn, despite the number of ports and their
Only when merging commences does the AF in case
five decrease.
The results of case five are similar to those of the base case except
that predicted dilution is slightly higher and predicted elevation is
slightly lower.
The dilution of the base case still corresponds better
with the experimental uncertainty analysis of Section 5.7.
In addition,
since dilution is increased in case five relative to the base case, mean
error now exists in the prediction of t2 5 0 and W250 when virtually none exists for the base case simulation.
Thus the base case AF does perform
better, although nothing definitive can be concluded concerning the effectiveness or lack thereof of geometrical shadowing.
Figures 5.3, 5.4 and 5.5 depict the initial jet arrangements and
illustrate the evolution of the base case AF for the three experimental
cases of two, four and eight ports.
While both current-induced and jet-
induced entrainment are multiplied by the AF, it is formulated specifically for current-induced entrainment.
These two types of entrainment are
of the same order of magnitude under the vertical experimental conditions
j yl_ __ 11 V I ___
I ~___
_ il_I
~1
(b)
(a) Initial Separate Sources
Shadowing Starts
Plant Periphery
Plant
Periphery
22.50
A.F.
=
2 x 2 x 3.1
2 x 4.4
Figure 5.3:
= 1.41
A.F.
=
2 x 2 x 7.4
2 x 10.5
Aspect factor definition sketch for two jet experiments.
= 1.41
Equivalent Source
A.F.
(2 x 19.4) + 14.8
2.0
2 x 27.4
(a)
4.4
= 1.00
(c)
(b)
10.5
27# .'t
,
-
1.5
01
U
cJ
I0
- $-4
t4
PO
6
-a
0 .0
o
0.0
11
mr
p
5.0
10.0
15.0
Equivalent Radius
Figure 5.3:
Continued.
p
20.0
(m)
25.0
30.0
a)
b)
Initial separate sources
3.Im
Shadowing starts
Plant
Periphery
Plant Periphery
22.50
A.F. = 4 x 2 x 3.1 = 2.0
A.F.=
2 x 6.2
Figure 5.4:
Aspect factor definition sketch for four jet experiments.
2 x 10.5
= 2.0
(c)
Merging Begins
Equivalent
Source
13.7m
'Plant Periphery
(2 x 2 x 13.7) + 11.4
2 x 27.4
A.F.
(a)
6.2
2.0 r
= 1.21
(b)
10.5
(c)
27.4
32.0
I
1.5
1.0
C
a.'
w
)
-o
0.5
aD
a
0.0
a
0.0
5.0
10.0
15.0
20.0
Equivalent Radius
Figure 5.4:
a
a
~i
Continued
-90-
(m)
25.0
30.0
35.0
(a)
Initial Separate Sources
Shadowing Starts
(b)
Plant Periphery
22.50
A.F.
=
4x
2x3.1
2 x 8.8
Figure 5.5:
= 1.41
A.F.
=
4 x 2 x 5.3
2 x 14.8
Aspect factor definition sketch for eight jet experiments.
= 1.41
(c)
Merging Begins
Current
Equivalent
Source
7.4 m
Plant Periphery
N
A.F.
= (3 x 2 x 7.4) + 9.2
2 x 20.9
(a)
8.8
(b)
(c)
20.9
15.0
20.0
2.0
1.5
1.0 m
0.5 1
0.0
0.0
5.0
10.0
Equivalent Radius (m)
Figure 5.5:
Continued.
-92-
25.0
30.0
35.0
in current of this study.
Exact geometrical treatment of jet-induced en-
trainment would require a peripheral factor reflecting the ratio of the
sum of the peripheries of the multiple sources divided by the periphery of
the equivalent source.
This ratio would remain constant until merging be-
gan, after which it would approach one as the multiple sources became completely merged.
Except for the condition of no geometrical shadowing, the peripheral
factor would always be greater than the base case AF.
Thus it would appear
that multiplication of jet-induced entrainment by the AF instead of a peripheral factor would result in an underestimation of entrainment.
a cluster of jets in proximity are in effect
fluid.
"competing" for the
However
local ambient
The pressure field set up around an individual jet is not as effec-
tive in aspirating ambient fluid in the presence of the pressure fields
of neighboring jets.
Thus while geometry indicates that shadowing and
proximity do not decrease jet induced entrainment, it is physically reasonable that they do.
In light of this, the base case AF was applied to jet-
induced entrainment also.
The AF was formulated for the ZOEF.
If the concept of an equivalent
source is employed from the point of discharge, it is clear that similar
logic would hold in the ZOFE.
Because entrainment in the ZOFE is repre-
sented by the starting length, se, the AF was used to correct for an overprediction of it.
For a vertically directed aggregate jet,
se = Q/Eag ' where Eag is
the entrainment rate of the aggregate source.
Now, in order to reflect
entrainment in the multiple sources, Eag must be multiplied by the
appropriate AF.
It is therefore assumed that the starting length for the
-93-
multiple sources is
s is computed based on the dimensions of
e
s /AF, where
e
the equivalent jet as discussed in the following section.
Deflection in the ZOFE
5.6.2
Simple momentum balances allowed representation of deflection in the
ZOFE.
Deflection of 81 can be ascertained through crossflow and coflow component momentum balances in the x-y plane.
Consider a jet issuing from the
origin at 610 from the x-axis in the horizontal plane (see Fig. 3.3).
If
drag is neglected, the conservation of coflow momentum can be expressed as:
d
ds
(M)
y
(5 - 27)
= Eu
while the corresponding crossflow
d (M
ds
x
balance is:
(5 - 28)
= 0
Therefore at s
e
M (se) =
Mx(S
where
it
e
M(o)
y(O)
+(Sc
- 1) u O
(5 - 29)
and
) = Mx(o) = M(o)C 1 = QoUoC 1
= Q u
and Sc = centerline dilution.
(5 - 30)
From
M (s ) and M (se )
can be shown that:
-(S) u S + (S-)UQo
(5 -
31)
This relationship is written into the model code for the various combinations of initial orientation and free stream velocity.
Deflection of 82 can be ascertained through crossflow
and coflow mo-
mentum balances in the vertical plane defined by the free stream direction.
Consider a jet issuing from the origin at e2 0 from the y-axis in
plane (see fig. 3.3).
the y-z
The coflow momentum conservation balance is still the same
-94-
(Eq. 5-27).
Conservation of vertical crossflow momentum can be expressed
as:
1
d
ds (z
p f
j-
0
(5 - 32)
A
where A is the jet cross-sectional area.
The distribution of the jet den-
sity p. will vary over the cross-sectional area and trajectory.
the approximation is made here that pj
only over the initial jet area.
However
= po and integration is performed
Thus the buoyancy contribution is evalu-
ated as if the jet were instantaneously placed in the middle of the trajectory.
M (s
z
This yields:
)
e
where P
M (o)
z
= p(z)
+ H (o
cogb
r
at z=zo +
SeS
.
o
s
(5 -33)
e
My(s
is as previously defined.
Thus:
-1
-
(s)
2 e
2
= tan
-1
Qu S
Quo S2
0
+ H (
O -
0
0
2
)gb s
oe
(5 -
34)
(S-1)Qu
Q0UoC
0 O
0 2 + c
This relationship was written into the model code for the various combinations of initial orientation, direction of buoyancy and free stream magnitude.
5.6.3
Starting Length
The relationships
which Hirst (1971,a) used for the dependence of se
on crossflow and buoyancy have been redefined.
Dependence on Crossflow
function of R.
Unfortunately little data exist concerning se as a
Fan (1967) graphically defined se for jets in a crossflow us-
ing only the point at R = 0 from Albertson (1950) and the three points from
Gordier (1959) at R = 0.250, 0.167 and 0.125 (Fig. 5.6).
-95-
Fan's linear fit to
S:
Gordier's and Albertson's data
-
:
Hirst's fit to data
---
:
Fit used in this study
s
e
O
0.71
+ 0.247 R)
S(0.057
0
0.0
0.1
0.2
0.3
0.5
0.4
0.6
0.7
0.8
R
Figure 5.6:
Crossflow Ratio Versus Normalized Starting Length
0.9
1.0
the data neglects the fact that the rate of decrease in se seems to decrease as R increases.
The fit allows se to go negative at values of
R > 0.4 which is physically unreasonable.
for Fan's line into his code.
Hirst incorporated an equation
He claimed that it was valid up to values
of R of = 0.5.
If it is assumed that the dependency of entrainment on R in the ZOFE
is proportional to the corresponding dependency in the ZOEF, then manipulation of the entrainment function together with the conditions specified
at s=s epermits formulation of a relationship that better reflects the
actual physical processes.
For the non-buoyant case in a crossflow the entrainment function
within the ZOEF is found from Eq. 5-18 using Hirst's values for a and a :
l
3
E =
(5-35)
mb (0.057 + 0.513 R)
If b = bo and it is recognized that at s = se in a crossflow,Ese
=
Q0,
then
the entrainment function for the ZOFE can be found by normalizing Eq. 5-35
for the stagnant case where it is known that se = 12.4b o.
This process
gives
Se(R)=
b
0.71
(0.057 +0.513R)
(5-36)
o
A graph of Se/bo versus R according to Eq.5-36would overpredict the
measured decrease in se /bo with increasing R
(see fig. 5.6).
This seems to indicate that Hirst overestimated a 3 , the current
induced entrainment coefficient.
This observation can be accounted for by
the fact that Hirst neglected drag force.
Jet experiments by Chan and
Kennedy (1972) indicate significant drag forces at least near the jet
origin.
Apparently in calibrating a 3 to trajectory, Hirst compensated for
the neglected beading effect of drag by inflating entrainment.
-97-
Nevertheless equation 5-36 appears to model the shape of the decay indicated by the data quite well.
If the velocity ratio coefficient, 0.513,
is changed to 0.247, excellent fit with the data is achieved (see fig. 5.6).
It is this curve that is used to express starting length as a function of R.
It is approximated by the following series of straight lines:
=
12.4 - 31.1 R
R)/ b =
Se(R)/bo
0.0_R0.167
(5-37.1)
9.1 - 11.6 R
0.167 <R<-0.4
(5-37.2)
s (R)/ b =
e
5.8 -
3.1 R
0.4<R<1.2
(5-37.3)
se (R)/b =
2.9 -
0.8R
1.2<R.3.625
(5-37.4)
s
Dependence on buoyancy
Abraham (1963) graphically defined starting
length for buoyant jets discharged vertically up into a stagnant ambient.
Abraham's graphical solution can be approximated by a series of straight lines
such that
(IF )/bo = 2.40 F
+ 3.00
= 1.28 F
S (F )/b
e
r
o
r
< 1.5
(5 - 38.1)
+ 4.68
1.5< IF < 4.0
(5 - 38.2)
+ 8.37
4.0< IF < 7.0
r
(5-
7.0< IF < 35
(5 - 38.4)
(5 - 384)
35 < 7r
< 200
(5 - 33.5)
Fr
> 200
(5 - 39.6)
s (F )/b
0
e r
- 0.34 F
Sc r)/bo
c r
= 0.014 Fr + 10.7
s
r
r)/bo = 0.0071Fr + 10.95
seo Fr)/b
= 12.40
0< IF
r
38.3)
This approximation is appreciably different than the one Hirst recommended in his code, especially for low Froude number cases where several
discontinuities in the approximation occur.
-98-
Abraham provides theoretical results for IF > 0 but there is no data
r
For IFr < Obut
to substantiate them.
IFri large (conditions representative
of OTEC operation), the approximation, seF ) = 12.4b should be reasonable.
o
e r
Dependence on both Crossflow and Buoyancy
For a buoyant jet in a
crossflow the starting length used in the model is given by combining
Eqs. 5-37 and 5-38:
s
b
5.6.4
s (R)
e
e
b
0
s OF)
er
b
.1(5-39)
12.4
Lateral Spreading of Plume
A jet discharged to a stratified medium will seek an equilibrium ele-
vation at which it will proceed to spread laterally, due to a net pressure
force difference between it and the ambient.
This lateral spreading or
collapse may be treated in an intermediate field analysis (eg. Larsen and
Sorenson, 1968; Jirka, 1980).
However, the experiments of this study in-
dicate that the collapse is significant in the near field as well.
collapse within the near field can be addressed, then
If
proper boundary con-
ditions of plume width and thickness can be input to an intermediate field
spreading model.
The assumption of axisymmetry disallows direct simulation
of this phenomenom with the Hirst model.
And, since the Hirst model has
been previously calibrated, we wanted to leave the integral equations intact.
Accordingly, a relatively simple modification was made along the
following lines.
Consider first a rectangular jet of uniform density at some equilibrium
elevation D.
As illustrated in Fig. 5.7 nominal width and thickness are
2w* and 2t* such that the area of the rectangular jet is the same as that
of a circular jet of radius b; thus
2
=Sb
4w t*
(5-40)
-99-
Pressure
--- ~
-~
01
E -w *
t
I
S--
I-
-
-
I
Figure 5.7:
Vertical pressure distribution of water column when the plume is at equilibrium
(Schematic) .
t
+
)
Note, however, that the width W and thickness
t output by the program are
each 2/rTtimes the respective values of w* and t*.
Thus they are consistent
with the jet width output by the original Hirst model and correspond to a
jet cross-sectional area evaluated at radius V2b.
In a stratified environment, a pressure difference will exist between
This difference is represented by the dashed
the jet and the ambient.
region in Fig. 5.7.
If the ambient pressure at the top of the jet is PD-t'
then the ambient pressure force acting on the jet in the negative x direction
is
-
D+t
LPDt*
z
FA =
-
(P. + (n-D) T)
+
dz
D-t
D-t
2
+ 2pgt
+
2pmgt
P Dt
= 2t S2t**
D-t
-
2 g
3
(5-41.1)
t*3
p
z
3z
= the ambient density at D and
where p
gdn
9z
= the average density gradient
The jet pressure force acting in the positive
from D - t* to D + t*.
direction is
D+t*
z
gdn
PD-t* +I
=
F
D-t*
D-t*
= 2t*P
D-t
*+
dz
i
2pgt* 2
(5-41.2)
J
The net pressure force in the x direction is thus
F
P
= F
-
F
A
= 2t*g(p-
J
) + 2 gt*3
3
9z
(5-41.3)
Spreading along the jet centerline s can be represented by two
components:
-101-
dw_*
ds
where
ds J
db
+dw
(5-42)
IB
dJs
ds
represents axisymmetric jet spreading already accounted for
dw *
by the Hirst model and -ds *is the buoyant spreading being formulated
B
presently.
In lieu of solving a lateral momentum equation for the buoyant spreading rate, the following proportionality is assumed:
udw* 2 ,
udsw*
Fp O(
(5-43)
t
This type of assumption has been used with reasonable success in analyzing
the buoyant spreading of surface jets as well as internal and surface layers
(Jirka, 1980 and Larsen and Sorenson, 1968).
dw
ds
B
c
u
Thus
(5-44)
ph
The constant of proportionality c will be evaluated below.
The discussion so far assumes that the jet has reached its equilibrium
elevation and that the pressure within the jet is hydrostatic.
these assumptions are invalid near the jet origin.
Clearly
However it is still
desireable to represent lateral spreading within the near field.
Indeed,
because of oscillation about the equilibrium elevation, true hydrostatic
conditions will not exist until well into the intermediate field.
In lieu
of using a vertical momentum equation to determine FJ under non-hydrostatic
conditions, the transition to lateral spreading provided by Eq. 5-44 for
horizontal jets is represented by multiplying Fp by C2.
is initiated at the end of the ZOFE at which point
-102-
This procedure
w
t*
(5-45)
= bT
In summary, the procedure begins by computing Fp
then
ds
*
B
from Eq 5-44 and thus w* from Eq. 5-42.
from Eq. 5-41.3,
A new value of t* is
then found from Eq. 5-40 using updated values of w* and b.
(b is computed
as part of the original integral jet calculations.)
The uncalibrated values of W generated by this method (coefficient
of proportionality in Eq. 5-44 = 1) were considerably larger than experimental ones.
This trend was anticipated since a jet of constant
density was assumed rather than a smoothly varying Gaussian distribution
Since the constant jet
that matched the ambient density at the jet edges.
density was taken as the jet centerline density, FJ was overestimated. Although Eq. 5-44 could have been derived for a Gaussian distribution, it was
not since the analysis demanded calibration anyway.
A visual best fit value
of the constant of proportionality in Eq. 5-44 was found to be 0.2.
In effect what this formulation achieved was a superimposition of buoyant spreading over entrainment-induced spreading.
More formally, the gov-
erning differential equation for the effective radius could have been replaced by two equations for thickness and width.
However in this case, the
entrainment function would also demand reformulation.
Note that, due to
density stratification, the laterally spreading jet would entrain faster on
its vertical faces than on its horizontal faces.
However Hirst had com-
pared the predictions of his entrainment function with a number of experiments with appreciable stratification.
-133-
Although the stratification in
these experiments was generally less than that expected near the thermocline at potential OTEC sites, comparison with data from the OTEC experimental dilutions demonstrated that the generalized entrainment funcThus it was not tampered with.
tion worked reasonably well.
5.7
Hirst Model Simulations of Experimental Conditions
Vertical Discharge Experiments
5.7.1
The model simulations of the vertical discharge experimental conditions in current (1A-8B, 11A-12B) were compared to experimental data.
Figs.
5.8 through 5.11 illustrate the model's performance in predicting Sc , t2 5 0 ' W 2 5 0 ,
and h
eq
Note that because the plume centerline is predicted to
.
oscillate about the equilibrium elevation, the predicted value of heq
in this and later simulations was chosen as an average of maximum and
minimum centerline elevations once the plume turned horizontal.
are presented in tabular form in Table 5.3.
Results
Table 5.4 indicates the mean
error, the standard error, the normalized mean error and the normalized
standard error of the distributions.
Table 5.4: Statistics of Simulation Versus Observation for
Vertical Discharge Experiments in a Current
x
S
t
c
50
W
250
h
eq
(h -h
eq d)
02
250
Statistic_
x -x
p o
a
(x -x
p
xo
/x
Notes:
-1.6
-3.6m
7.9m
2.6m
2.6m
-0.4
2.5
14,8m
61.9m
11.5m
11.5m
1.2
-0.16
0.05
0.03
0.03
0.06
0.24
0.26
0.22
0.21
0.14
0.28
0.80
"p"
=
predicted
"o"
=
-104-
observed
"R
=
mean value of x
,/
*
-
/
,/
/
*
/
9-
d
6-
3
9
SMOBSERVED
FIG 5.8:PREDICTED VERSUS OBSERVED.
PRRED WITR PERFECT FIT LINE, XYT.
-105-
CM-
1/
/
120
/0
I
/
/
/
-i
30-
//
3
120
0
s
'tmiBSERVED
FIG 5.9:PREDICTED VERSUS OBSERVED.
P"RED HITH PERFECT FIT LINE.
XmY.
PgREO Hull PERFECT FIT LINE. X=.
-106-
COK-
50
500
/
4
Od
C3
•*
/n/
cc* 300.
200-
100-
I0.
2oo
Ko
400
WnOBSERVED
FIG5.10:PREDICTED VERSUS OBSERVED,. CSM-PARED WITH PERFECT FIT LINE, X=Y.
-107-
5oO
150
120
90
.
60-
30
/
/
I
Il
80
30
6Q
I
90
p
L20
h.. OBSERVED
FIG5.11:PREDICTED VERSUS OBSERVED. COMPRRED WITR PERFECT FIT LINE. X=Y.
-108-
L50
Table 5.3:
Simulation Versus Observation for Vertical
Discharge Experiments in a Current
RUN
P
S
c
o
S
c
t 250
P
(m)
t 250
0
W250
P
W 250
O
P
heq
heq
(m)
0
P
25heq
0
0
250
0
(m)
(m)
(m)
(m)
100
88
1.5
1.4
1A
8.3
7.5
85
448
450
lB
6.2
7.5
78
257
330
98
85
0.9
0.7
2A
6.6
70
406
450
101
71
1.8
1.3
2B
4.7
100
229
300
96
76
1.0
1.0
3A
4.9
70
274
300
86
72
2.0
2.5
3B
4.6
7.2
59
174
240
63
64
1.0
2.0
4A
8.2
9.3
58
342
300
90
80
1.9
1.9
4B
6.9
9.3
56
203
150
71
74
0.9
0.7
5A
9.7
8.0
59
348
255
82
77
1.7
2.1
5B
7.2
8.0
42
188
140
76
70
0.8
1.1
10.1
69
500
360
93
104
1.8
1.0
10.1
86
279
300
84
94
1.0
ND
6A
6B
11.6
8.0
12.2
9.3
12.0
7A
10.9
8.7
81
473
450
85
84
1.9
1.8
7B
7.4
8.7
61
260
345
75
80
0.9
0.9
8A
7.5
9.2
50
305
300
76
75
1.9
2.0
8B
5.7
9.2
50
170
180
72
68
1.0
1.4
300
92
106
1.9
1.4
80
93
1.0
ND
11A
11B
12A
12B
13.7
12.1
72
12.1
76
220
12.8
69
387
300
85
91
1.9
1.5
8.8 12.8
58
210
225
74
75
1.0
1.5
9.7
12.5
Notes:
"0"
= observed
"P"
= predicted
"ND" = not detectable
Parameters Sc, t250' W250
-109-
heq and 0250 as defined in chapter 4.
I
Appendix I permits a visual, qualitative assessment of how well the
Table 5.5 pre-
model is predicting geometry in the vertical (y-z) plane.
sents the conclusions of this visual assessment.
Table 5.5:
Assessment of Model's Prediction
of Geometry in the Y-Z Plane
El.
Run
Thk.
El.
E
E
6A
G
G
lB
E
E
6B
E
G
2A
E
G
7A
C
E
2B
7B
E
F
3A
8A
E
E
Run
Thk.
1A
3B
F
E
8B
E
E
4A
E
G
11A
G
G
4B
E
E
11B
F
F
5A
E
E
12A
E
G
G
12B
F
E
5B
____
G
____I.
___
___
_12__
Notes:
"Thk"
= thickness
"El"
= elevation
"E"
= excellent; "G"
-110-
= good;
"F"
= fair;
"P"
= poor.
The model achieves excellent agreement simultaneously for thickness
and elevation 33% of the time and separately 61 and 50% of the time respectively.
Overall the agreement is very good.
cillation is not that accurate.
The agreement for os-
However it is consistent in that in both
experiment and simulation for a given plant configuration, the number of
oscillations is inversely proportional to the current speed.
Appendix II permits a qualitative assessment of how well the model is
predicting geometry in the x-y plane.
In no case is a glaring anomaly observed.
Furthermore, in comparison with the vertical profile an overprediction of
width is generally correlated with an underprediction of thicknes and vice
versa.
This implies that the approach used to treat lateral spreading has
introduced the error.
5.7.2
Overall the agreement is very good.
Qualification of Comparison for Vertical Discharge Experiments
The statistics and conclusions presented above must be qualified with
respect to the accuracy of the data.
Due to the uncertainty associated with placing the dye sampling probe
(referred to in Section 4.2.1) at the centerline of the plume for a given
experiment, the observed centerline dilutions will generally be greater than
the ones that actually occurred.
at y=250m (prototype) is 67m.
dilution is 9.8.
The mean value of the observed thickness
The corresponding measured mean centerline
The ability to place the sampling probe at the centerline
is estimated to be approximately + 15% of the mean thickness or ± 10m.
Now
if the sampling process is assumed to exhibit a Gaussian distribution and
if the mean plume thickness corresponds to 4z,then the expected value of
measured concentration
E(c) can be evaluated.
-111-
It would be equal to:
2
E(c) =
f (z)
c o exp ( -. z
) dz
(5 - 46)
2a
where
f(z) = the Gaussian probability distribution function of the
sampling process with a = 10m.
E(c) = 1/9.8 = reciprocal of the observed mean centerline
dilution.
az
z
= 16.75 = standard deviation of the observed plume.
z = coordinate normal to the free surface in the y-z plane
with its origin along the jet trajectory (jet assumed
approximately horizontal at y = 250m prototype)
The solution to Eq. 5-46 gives c
= 0.12.
Recalling that E(c) =0.10,
this implies that the sampling procedure will detect concentrations that
are approximately 85% of the actual centerline value.
Therefore an optimal
predicted value of Sc would be approximately 85% of 9.8 or 8.3 giving a
mean error of 1.5.
This agrees very well with the results presented in
Table 5.4 .
The observed values of t250,heq, (heq -h d ) and 0250 are determined
from side view photographs.
The major limitation in measuring the
parameters is associated with intermittent billowing at the plume boundary; it is not always clear what the average elevation of the boundary is.
The uncertainty attributed to this is estimated to be ±5 % of the mean plume
thickness or about ± 3m.
The mean error in the prediction of thickness is
-112-
less than the uncertainty of measurement while the mean error in the prediction of elevation and change in elevation is on the order of the uncertainty.
Thus, with respect to mean error, the model can be considered cal-
ibrated within the experimental uncertainty for these parameters.
The
standard deviations of these quantities are considerably larger than the experimental uncertainty.
bility of the model.
This reflects inaccuracies in the predictive capa-
The mean error in the prediction of oscillation is
particularly significant.
However, since in both prediction and observation
the amplitude of oscillation is an order of magnitude less than the mean
thickness, the importance of this phenomenom is minor.
The observed values of W250 are determined from photographs taken
above and to the side of the experimental basin; thus they must be corrected
for parallax error.
The uncertainty of this procedure is estimated to be
+ 5% of the observed mean width or about + 15m.
The mean error in predicting
W250 is near zero because the coefficient of proportionality governing
lateral spreading in Eq. 5-44 was calibrated against observations of W250.
However, the scatter indicated by the standard deviation of W250 is again
significant compared to the experimental uncertainty.
To summarize, the scatter in the prediction of W250 and t250 results
largely from the inexactness of the lateral spreading formulation rather
than from any significant experimental uncertainty or lack of reproducibility.
The scatter in the prediction of dilution arises largely from the
uncertainty of the experimental technique.
The mean error in the predic-
tion of dilution reflects the bias of this uncertainty.
The scatter and
error in the correlation of elevation between prediction and observation
arises from a combination of experimental uncertainty and error in the
-113-
predictive capacity of the model.
In light of the experimental uncertainty
and bias, the overall correlation between observation and prediction is
very good.
The model is calibrated and will predict reasonably well the
behavior of a vertical discharge from an OTEC pilot plant.
5.8 Horizontal Discharge Experiments
The orientations (e0) of the individual jets of a horizontal discharge
experiment vary about the plant periphery as indicated in Table 4.1.
More-
over, the jets interact both directly through merging and indirectly by
altering the ambient flow field.
The integral jet model can treat an in-
dividual horizontal jet, but is incapable of accounting for interaction.
Therefore, only limited integral model simulation is possible.
The crossflow jet (81 = 0 ) of a particular horizontal jet array
should exhibit the maximum lateral penetration.
Thus simulations were
obtained for a crossflow jet from experiments 13A - 16B and 19A - 21B.
The predicted lateral.penetrations of these jets (equal to x2 5 0 + W250/2)
at y = 250m (prototype) were compared to the observed composite plume
half-width at the same point.
(See Table 4.3.
Note that W250 in this
table refers to the composite plume width; one half of this value was
used for comparison.)
The mean predicted lateral penetration was 59m less
than that observed for the intermediate current experiments (u. = 0.28 m/s)
while it was only 5m less for the higher current experiments (um = 0.51 m/s).
For all experiments the average underprediction was 32m with an associated
standard error of 85m. This suggests a shielding effect associated with
adjacent jets pointing in the upstream direction which block the ambient
current making the apparent crossflow velocity lower.
Apparently this
effect is inversely proportional to the magnitude of the ambient current.
-114-
This analysis is supported by results shown in Appendix II.
Dilution associated with a horizontal experiment is spatially
complex and variable.
Since all such experiments, except for experiment 23,
had a counterflowing jet which cannot be simulated with an integral model
due to reentrainment, it was not justified to compare the observed dilutions
with simulation.
In addition, even at y = 250m (prototype; the position
of the first sampling probe), the flow field of the other jets had
significantly overlapped making it difficult to determine what percentage
of dye came from which discharge.
This problem is compounded by the fact
that each individual jet exhibited its own elevation and thickness.
As a
result comparison between observed and predicted geometry in the y-z plane
was not made.
5.9 Additional Comments on the Model Equations
5.9.1 An Infinite Entrainment Rate?
Inspection of the Hirst model entrainment function, Eq. 5-18, indi2
cated that as IF2 rL
0, E-l-,
which is physically unreasonable.
It is ob-
vious that the entrainment function must approach some asymptotic value to
prevent it from going to infinity.
fied forF 2 <
rL
2
value for IF
rL
Since the function has not been veri-
10.5, it was reasonable to establish a lower limit of the
in Eq. 5-18.
This value was chosen to be ten.
Only in experiments 4A and 4B was the value of F
at the start of the integral analysis.
2
r
less than ten
In these experiments, F
2
equalled
L
about 3.0 initially, which is about four times smaller than the next smallest
initial value which occurred in experiment IA.
Thus it would be the ex-
ception,not the norm, in simulating OTEC pilot plant conditions with the
-115-
model if the lower limit of F
rL
was invoked.
Even so, when the lower limit
is invoked, because F
rL grows rapidly as the jet penetrates the stratification and buoyancy decreases, IFrL will not remain less than the lower
limit for long.
For example, the actual local Froude numbers for simula-
tions 4A and 4B were less than the lower limit for approximately 24m and
28 m into the ZOEF respectively, which represents a small percentage of
the total trajectory studied.
5.9.2
Boundary Layer Assumption Inconsistency
As previously stated the differential equations of the Hirst model
were formulated presuming that flow in a buoyant turbulent jet is of the
boundary layer type, meaning that gradients tangent to the trajectory are
much smaller than those normal to the trajectory and that >>
. However
in approximating the equations, Hirst apparently ignored this assumption
at one juncture and carried a second order term through his analysis.
The term
v
r
(4 ar
-2
q* = u
+
(5 -- 47)
47)
(5
v
ar
appeared when Hirst introduced turbulent fluctuations into the axisummetric momentum equations.
2
2
-2
ar
ar
r
Note that since v' = v - v,
(5 - 48)
.
Therefore q* reduces to
q*
2
r
(5 - 49)
av
and since u>>v the second term on the right hand side should be neglected.
-116-
Hirst, however, retained this term, defining q* as:
Lb2
q* =
(u
+ u SC
2
)2 - E
(5 - 50)
,
2
in which the second order term is represented by E
Removal of E 2 significantly affects the performance of the Hirst
model.
Inspection of Eq. 5-8 shows that the resulting increase in q*
decreases the rate of bending which increases current induced entrainment.
Simulating the 22 experimental vertical discharge conditions with
current, using the model with the second order term removed, overpredicted
dilution and underpredicted jet bending relative to both measurement and
prediction with the term retained.
An error analysis gave the following
mean results:
Sc (predicted) - Sc (observed) = 1.6
heq (predicted) - heq (observed) = 3.5m
t2 5 0 (predicted) - t2 5 0 (observed) = 2.9m
W 2 5 0 (predicted) - W250 (observed) = 41.5m
where S c' heq
t2 50 and W250
are as defined in Section 4.3.1.
Removal of E 2
affected the simulations fairly uniformly.
In light of these results and since the Hirst model was calibrated
against a large dataset (Section 5.4), q* was left intact.
The fact
that we were able to achieve reasonable simulation with the model with
q* as defined by equation 5- 50was sufficient justification not to tamper
with it.
It is nevertheless worthwhile exploring what would have happened if
Hirst had calibrated his model without the E2 term.
Our results indicate
that with a3 = 9.0, and without the E2 term, bending would be underestimated.
However, assuming that dilution would be high if a 3 = 9.0, as
our data suggests, an increase in a3 to reflect actual bending would force
-117-
dilutions even higher.
resolve this problem.
Hirst would have had to introduce a drag force to
These trends suggest that if a true, first order
calibration of the model is pursued, drag forces will have to be accounted
for.
-118-
VI.
ADDITIONAL SIMULATIONS WITH THE INTEGRAL JET MODEL
6.1
Introduction
Perturbations of the ambient ocean, particularly of the well mixed
layer, by the OTEC plant and external flows must be understood in order to
make a valid assessment of an OTEC plant's environmental impact.
The per-
turbation will reflect the plume's position within the water column and
the concentration of chemical species (e.g. biocides, products of corrosion or possible working fluid leaks), nutrients and temperature within
the plume.
In this chapter a sensitivity study is described which ex-
plored a range of plume behavior that is anticipated.
The results
are interpreted with specific reference to the negative impact associated
with biocides and the (positive or negative) impact associated with nutrients artificially upwelled by the condenser intake.
Simulations are
also made that examine the behavior of a condenser jet discharged within
the thermocline and the response of the integral jet model to conditions
from a previous OTEC physical model study.
6.2
Selection of Base Case Plant and Ocean
Additional simulations were performed for realistic OTEC plant and
ocean conditions.
In some instances, the conditions are such that plant
performance may be adversely affected through recirculation.
stances, they may induce negative environmental impacts.
In other in-
Base case (B.C.)
conditions were selected after which a series of simulations were run, in
which one or several complementary parameters were varied at a time while
the others remained at their B.C. values.
conditions.
-119-
Table 6.1 lists the base case
Table 6.1:
Base Case Conditions
The MWe/port ratio reflects current modular design notions.
It was
practical to model a mixed discharge for a vertically discharging plant
since closely spaced but separate evaporator and condenser flows (e.g. as
envisioned by Gibbs and Cox; Scott, 1979) should mix soon after discharge.
Also, studies have indicated that internally mixing the flows may not
present serious mechanical nor economic problems.
The discharge depth
was a compromise between the hd of 80m proposed by Gibbs and
Cox (Scott, 1979) for a 40 MWe vertical plant and the hd of 40m used in the
physical model tests of this study, which demonstrated that such a shallow
discharge will not significantly deteriorate plant performance.
The base case ocean had a vertical temperature profile that was an
average between the profiles reported by Bathen
(1975) for Hawaii and
Fuglister (1960) for the Caribbean, except that the surface temperature
was taken from the Hawaiin profile.
A ep
of 180 C, which represents the
lower bound of practicality for plant operation was chosen.
be the minimum conceivable at any time during a year.
-120-
Thus T o would
Fig. 6.1 shows the base case plant in a hydrodynamically stable configuration with respect to the ambient current; note that the orientation
of the port array with respect to the current is different than in the fourjet physical model tests.
Fig. 6.1 indicates that the AF is always equal to
unity when the plant is in this configuration and that the equivalent area
radius for the base case is 6.6m.
Sensitivity to Perturbation from Base Case Conditions
6.3
Presentation of Results
6.3.1
Parameters that were varied in the sensitivity study were uo, Qo'
hd
ep , H and u .
was achieved.
Table 6.2.1 describes how variation of the parameters
Table 6.2.2, which lists the runs from the six simulation series,
indicates the values of the independent variables that have been changed
from the base case and documents the response of the major dependent variables, Sc, heq
t250 and W250 at y = 250m (prototype), to these changes.
Table 6.2.1:
Description of Simulations
(Perturbations from Base Case Conditions)
Method
Study
1
u
o
b
was varied.
o
was simultaneously adjusted to
maintain a constant Qo"
2
Q
tain a constant u
3
h d was varied.
4
6
was simultaneously adjusted to main-
b
was varied.
o
Everything else remained the same.
was varied by changing the temperature of the
upper mixed layer.
Thus T
and the temperature grad-
ient of the thermocline also changed, since Ta (165m)
was held constant.
5
H was varied.
As H increased, the slope of the thermo-
cline decreased since Ta (165m) was held constant.
6
u
was varied.
-121-
Everything else remained the same.
V4
Current
Plant Periphery
A.F.
= 2 x 2 x 3.3
2 x 6.6
6.6
27.4
2.0
1.0
5.0
10.0
15.0
20.0
25.0
30.0
Equivalent Radius (m)
Figure 6.1:
Base case configuration and aspect factor sketch.
-122-
35.0
Table 6.2.2:
Results of Simulations
Described in Table 6.2.1
Study 1:
Run
~
(n ~ ~
u~~
u o (m/sec)
bo(m)
S
(i/ec
heq
(m)
t256 m )
W2 5 0 (m)
1
1.0
5.6
7.3
147
82
387
2
2.0
4.0
9.0
142
92
407
3*
3.0
3.3
10.9
149
102
431
4
4.0
2.8
13.1
149
110
449
5
5.0
2.5
15.3
149
119
471
6
6.0
2.3
17.4
148
128
494
b o(m)
Sc
Study 2:
Run
Q (m3 /sec)
heq
(m)
t
250
(m)
W 2 5 0 (m)
1
40
1.0
18.3
98
41
186
2*
400
3.3
10.9
149
102
431
3
1000
5.2
8.4
189
154
524
4
2000
7.3
7.5
233
194
531
5
4000
10.3
6.4
276
240
730
Study 3:
-123-
Study 3:
Run
S
hd (m)
h
t
(in)
eq
c
25
0 (m)
W250(m)
6*
60
10.9
149
102
431
7
70
10.8
159
106
417
8
80
168
108
409
9
90
10.0
181
110
399
10
100
10.0
193
110
400
I
9.9
I
I
II
1
i
_
Study 4:
Run
ep ( C)
T(Cm)
Sc
heq
(m)
t
250
(m)
W2
50
1*
18
24.7
10.9
149
102
431
2
19
25.7
10.7
148
100
441
3
20
26.7
10.2
146
98
451
4
21
27.7
10.1
145
93
457
5
22
28.7
10.3
144
89
461
6
23
29.7
10.4
142
87
466
7
24
30.7
10.4
141
88
475
(m)
Study 5:
t2 5 0 (m)
W 2 5 0 (m)
Run
H(m)
c
1
10
9.9
153
107
412
2
20
10.0
152
105
416
3
30
10.2
151
104
420
4
40
10.6
150
103
425
5*
50
10.9
149
102
431
-124-
h
(m)
Study 5:
heat (m)
t2 5 0 (m)
W 2 5 0 (m)
Run
H(m)
Sc
6
60
10.9
149
10i
437
7
70
10.6
148
100
444
8
80
10.1
146
97
450
9
90
9.9
144
91
454
10.1
141
87
457
10
100
Study 6:
Run
u (m/sec)
1
0.2
2*
0.3
3
0.4
4
0.5
5
0.6
6
0.7
Note : "*"
indicates B.C.
-125-
6.3.2
Discussion of Results
The perturbation of the well mixed upper layer of the ambient ocean
caused by an OTEC discharge must be known in order to make a valid assessment of the plant's environmental impact.
One form of perturbation is the
entrainment of mixed layer water into the OTEC plume, which would occur if
the upper horizontal face of the plume was in the proximity of the bottom
of the mixed layer.
phytoplankton
This would induce an artificial downwelling of
(Walsh, 1981).
Another form of interaction is associated
with the physical presence of all or part of the OTEC plume in the mixed
layer.
This would directly introduce any biocide or nutrient load of the
plume into the mixed layer.
An explanation of the relationship between
phytoplankton and the photic zoneand the well mixed layer is necessary to
fully develop the effects of these phenomena.
A typical, subtropical, oceanic water column has a critical depth
(Gross, 1977) at which there is just enough light to support growth of the
phytoplankton population.
does not occur.
mixed layer.
Below this depth, significant photosynthesis
The phytoplankton are distributed throughout the upper
The critical depth can occur either in the upper mixed layer
or below it, and its position changes throughout the year.
When it occurs
in the upper mixed layer, OTEC induced downwelling would remove a higher
percentage of the phytoplankton population from its photosynthetic cycle.
However since downwelling will expose the phytoplankton to the nutrient
rich water of the OTEC plume, it may actually relieve the grazing stress
of the population (Brookhaven, 1981), assuming that the phytoplankton
can eventually escape the OTEC plume and float back to the surface after
having absorbed nutrients from the plume.
Since the phytoplankton will
only be exposed to biocide in the plume when the phytoplankton is below
-126-
the critical depth, the inhibiting effect of some biocides, such as chlorine, to photosynthesis, will be minimal.
However, when the critical depth
occurs below the mixed layer, while the OTEC plume may augment the nutrient supply of the phytoplankton, it will also expose the phytoplankton to
the inhibiting effects of some biocides when photosynthetic activity is
intense.
Clearly the closer the critical depth is to the bottom of the
well mixed upper layer, the less such exposure will occur.
Thus the en-
vironmental impact of downwelling depends on the depth of the well mixed
upper layer and the location of the critical depth.
Direct introduction
of biocide and/or nutrient by OTEC plume intrusion into the well mixed
upper layer, will always impact the entire phytoplankton community - although
not necessarily uniformly
- since the input will effectively become well
mixed.
In Hawaii and Puerto Rico the maximum, most probable, monthly mixed
layer depth is about 100m (Sands, 1980).
This maximum generally occurs
in the winter months when the critical depth lies in the upper mixed layer.
Clearly under these circumstances, the potential impact of any artificial dowiwelling is maximized as is the potential introduction of substances into the
waters above the critical depth.
Therefore in the following discussion
of the simulation series, only when the results indicated that the upper
horizontal face of the OTEC plume would reside at or above a depth of 100m
was that particular simulation assessed as having potential environmental
An assessment of the relative value of that impact was not
impact.
attempted in this study.
thickness
Although most of the simulations were made with a mixed layer
of 50m, the results of study five indicated that the dependent variables
Sc, heq
,
t2 5 0 and W250 are all fairly insensitive to change in H.
-127-
Thus
conclusions reached for simulations at H = 50m are approximately valid
for any other H less than 100m, and the method of environmentally assessing results as described above is also approximately valid.
The results of study one indicated that heq is relatively insensitive
to change in uo for a given plant size, discharge configuration
and ambient environment.
Since Sc is positively correlated with uo while
t250 is negatively correlated with u , selection of uo for a plant entails
choice between a relatively thin, undiluted discharge (i.e. small u ) and
one that is relatively thick and
diluted
(i.e. large u ). Note that the
ensuing plume of all the runs ih this study may cause downwelling of the
upper mixed layer.
In addition, as u0 increases it becomes increasingly more
difficult to prevent intrusion of the plume into the mixed layer.
The results of study two indicate that as Qo increases, Sc decreases
while heq increases.
Thus a tradeoff occurs between S
and h
. A
small plant will have a relatively shallow, but highly diluted discharge
that is likely to intrude into the upper mixed layer.
Obviously if the
discharge is going to reside partially within this layer, it is advantageous that rapid dilution take place.
In this way, biocide will be dis-
persed faster, mitigating any negative impact, and nutrient will be distributed faster, so that a larger percentage of the phytoplankton population may utilize it, maximizing any positive impact.
The discharge
from a large plant, although unlikely to intrude into the upper mixed
layer, may induce significant downwelling of that layer.
As indicated by study three, heq is nearly proportional to h d suggesting
that (due to high dilution) the density of the diluting rather than the discharge
flow is primarily responsible for establishing equilibrium plume elevation.
Furthermore, Sc, t250 and W250 are not sensitive to changes in h d . Thus at least
-128-
when the critical depth occurs within the mixed layer, it may be desirable
to design hd so that the OTEC plume resides just beneath the mixed layer.
This would prevent inhibitation of photosynthesis by biocide, while still
introducing any artificially upwelled nutrient to the phytoplankton community.
Study four shows that as 6
increases, t2 50 decreases because the
steepness of the thermocline increases with 0
also increases with it.
6 .
Sc and heq
and thus lateral spreading
are fairly insensitive to change in
For a constant u , um and heq , it might be expected that environ-
mental impact would be maximized for small
however that despite the effect 6
P
the OTEC plant, the largest
. It should be recognized
has on the external fluid mechanics of
0p available will always be selected in order
to maximize plant output.
The results of study six indicate that since both h
eq
and S
c
vary
inversely with u , the elevation of the upper horizontal face of the OTEC
plume does not change significantly as u. is varied.
To illustrate, note
that this elevation (heq - t2 5 0 /2) is 99m and 105m for runs one
(u
= 0.2m/sec) and six (u
ever since S
=
0.7m/sec),respectively,of study six.
How-
varies inversely with u , the discharge fluid within the
near field under high u, conditions will be characterized by higher concentrations of biocide or nutrient.
6.4
Modeling
a Separate Condenser Jet
It is possible that the evaporator and condenser will be discharged
separately and that the orientation of the two discharges will prevent
interaction.
For example, consider the case where the evaporator dis-
charges horizontally from the side of the plant while the condenser discharges vertically from its bottom, or the case where both flows discharge
horizontally at a large separation.
-129-
Table 6.3 lists the independent
Both these cases were simulated.
variables that differ from the B.C. and the response of the dependent
Note that for the horizontal condenser dis-
variables to these changes.
charge (Run 2) a co-flowing orientation (e1=900) was assumed.
)
Run
1 (vertl
-
2 (hor)
90
_
_
e2 (0)
_
'
hcq(
250(n
>250(m
T' (OC)
hd(m)
b (m)
90
8.0
80
2.3
14.7
154
83
368
0
[I_
2.3
5.6
132
78
144
80
8.0
_
_
_
S
___
It is obvious that a vertical discharge is superior in diluting the
condenser flow.
Surprisingly, the relatively large dilution of the vertical
discharge does not result in a significantly larger thickness, but rather
translates into more lateral spreading.
(Note the t
term in
expression for Fp in Eq. 5-41 governing lateral spreading.)
the
Thus while
both a horizontal and a vertical discharge will probably prevent the discharge
from penetrating the photic zone, the vertical discharge is superior in
dispersing any toxic biocides and in distributing any beneficial nutrient
load into the local environment.
6.5 Modeling Experimental Conditions from a Previous Physical Model Study
Coxe et al (1980) ran several experiments in which a 100 MWe,
evaporator flow was discharged horizontally through one rectangular port
at either cross-flowing or co-flowing orientation with respect to the free
stream.
Table 6.4.1 lists the independent parameters of these experiments,
along with the run number assigned by Coxe.
Using an equivalent area circular port, these experiments were
simulated with the integral jet model.
-130-
Table 6.4.2 documents the results of
%
I
.
a
Independent Parameters of Coxe's
Table 6.4.1
Single Jet, Horizontal Discharge Experiments
-I
Qo
Orientation
RUN
33B
34A
34B
(01 = 900
I
(m/sec)
(m /sec)
crossflow
(o1 = 0)
crossflow
(01 = 0)
coflow
(61 = 900)
coflow
33A
h
U
W
hd
(m)
(m)
(m)
H
0
(oC)
(m)
u
(m/sec)
(m/sec)
T'
0.51
24.5
21.2
55
0.28
24.5
21.2
55
0.51
24.5
21.2
11.98
10.74
75
23.4
55
500
3.9
11.98
10.74
75
23.7
55
500
3.9
11.98
10.74
75
22.2
500
3.9
11.98 110.74
75
22.3
I.
I
4
(oC)
21.2
3.9
I
T" (z=hd)
24.5
500
)
I
(z=h.)
( oC)
a
4
Simulation versus Observation ,or Coxe's Initial Condition
-I
P
h (m)p
t 4 5 0 (m)
t 4 5 0 (m)
x450(m)
x450(m)o
450(m)
W450(m)
Table 6.4.2:
Sp
Run
S
S
I
h (m)0
p
C
33A
14.4
68
93
638
825
33B
11. 3
68
77
405
508
213
261
54
248
147
0
0
46
195
116
0
0
34A
8.5
4.5
70.4
68
34B
6.6
3.8
74.7
69
__i
Notes:
I
62
£J
Superscript "o" = observed
Superscript "p"
=
predicted
All variables measured or predicted at y = 450 m (prototype)
385
the simulations and compares them with experimental observations.
For the co-flowing jets (Runs 34A and B) the integral model
significantly underpredicts jet dilution.
rise is a little high (h eq
< h
0)
As a result, predicted plume
and both predicted width and thickness
are too small, though the later is reasonably close.
Turbulent mixing
associated with the plant wake (and which is not simulated in the
integral model) may be responsible for the observed dilution being larger
than predicted.
For the cross-flowing jets (Runs 33A and B),
the predicted
trajectories and widths are somewhat higher than observed.
This may
reflect inaccuracy in the lateral spreading formulation and/or the
entrainment function.
Since no experimental values of t
4 50
450
for the cross-flowing cases, these cannot be distinguished.
or S exist
c
However,
since the lateral spreading formulation was calibrated to the vertical
experiments, where spreading occurred well into the thermocline, it is
possible that the calibration would be different for the horizontal
experiments, where spreading takes place typically in the lower portion
of the well mixed layer.
6.6
Future Use of Model for Environmental Assessment
It should be noted that the plant and ocean conditions presented in
this chapter are by no means exhaustive.
Rather a reasonable methodology
has been presented and a sensitivity analysis has been undertaken to explore the response of a particular base case to individual variation response of its independent variables.
The trends identified in this
study will help facilitate the inclusion of environmental impact considerations into generic OTEC design.
The versatility of the model will also
allow detailed studies to be made of any particular design.
-133-
More detailed environmental assessment could be attained if the integral model were coupled to chemical and biological kinetic models.
could be accomplished by writing
This
integral equations expressing conserva-
tion of the appropriate chemical or biological species analagous to
the conservation of thermal energy (heat) equation presently included in
If coupling is pursued, it is important to identify the appro-
the model.
priate rate constants and to guarantee that the step size used in the
numerical
model is consistent with the fastest reaction rate.
For ex-
ample, consider the pulsing of the discharge water with chlorine to retard biofouling.
As the chlorine is introduced to the seawater, it will
almost instantaneously hydrolyze to hypochlorite by the following reaction:
+
C12 + H20 -+ OCIl + Cl- + 2H
(6 - 1)
The degradation of the hypochlorite follows several pathways that differ
in rate.
The step size of the numerical model must accomodate the fastest
of these rates to prevent predicted concentrations of some chemical
species from decreasing too slow with respect to position along the
trajectory.
-134-
RECIRCULATION
VII.
7.1
Introduction
Determination of recirculation was one of the major objectives of
the physical model experiments.
small.
Fortunately, recirculation was generally
Mean direct recirculation (as determined from dye measurements) for
the vertical experiments in crossflow was 0.6% with a standard deviation
of 0.7%.
For the horizontal experiments in a current mean direct recir-
culation was 0.9% with a standard deviation of 0.9%.
stagnant water tests was even less.
Recirculation in
These mean values and standard devia-
tions are relatively small compared to the mean values of direct recirculation (-5%) and standard deviation (-5 - 10%) that Coxe et al (1980)
observed in their physical model study of 200 to 600 MWe plants.
Attempts to correlate direct recirculation to the independent variables in this study were generally unsuccessful due to the extremely low
values of observed recirculation and, thus, the relatively high level of
uncertainty in the measurements.
(This is in contrast to the reasonable
success which Coxe et al (1980) achieved in their analysis of recirculation.)
Nevertheless it is worthwhile to qualitatively discuss the me-
chanisms thought to lie behind the recirculation which was observed.
7.2
Direct Recirculation in Stagnant Water Tests
Direct recirculation in the stagnant water experiments displayed
significant temporal variability (see Table 4.4), with highest levels of
direct recirculation occurring at the beginning of an experiment due to
the evolution of the discharge plume (plant start-up).
These high levels
quickly diminished, until direct recirculation became relatively steady,
at levels that were insignificant (mean steady value typically about
-135-
0.2 %).
In the context of actual OTEC operation, the recirculation at startup is not important and, for practical purposes, only insignificant levels
of recirculation existed for the stagnant water pilot plant tests.
It
must be noted however that the duration of the stagnant water experiments
was typically of order 10 hours (prototype).
It is possible that an ex-
tended period of ambient stagnation, a period of ambient current following stagnation or a period of current reversal could result in recirculation due to unsteady build-up of the discharge plume.
While not expected
to be significant, such phenomena would require a transient intermediate
field spreading analysis in addition to a near field analysis.
7.3
The Upwash Effect in Vertical Experiments in a Current
Experimental observation during tests with a current indicated that a
significant captive eddy of low pressure often formed in the lee of the
plant.
Fig. 7.1 is a side view photograph illustrating this phenomenom.
It is characterized by a clockwise circular gyre (as viewed in Fig. 7.1) of
swirls of dye that intermittently billow from the plume.
It is believed
that the intake flow coupled with intense turbulent mixing within the
wake combine to produce this intermittent swirling.
This phenomenom is hydrodynamically similar to that of downwash which
occurs in mechanical draft cooling towers (Chan and Kennedy, 1980).
Both
the initial Froude numbers and crossflow ratios are in the same range.
The
major differences between the two situations are that the OTEC orientation
is inverted compared to that of the cooling tower and the stable stratification of the ocean profile at the level of the OTEC discharge is more
significant than the corresponding atmospheric stratification.
The phen-
omenon is henceforth referred to as upwash when addressing OTEC plants.
-136-
Figure 7.1:
Side View Photograph of Run 4A Illustrating
Circular Motion of Upwashed Dye Billows
-137--
cause
In addition to intermittent swirling, the upwash effect can also
deflection of the plume towards the plant.
This is probably the primary
reason that integral jet simulation (see Appendix I) did not accurately
capture the shape of the top of the discharge at the lee side of the
plant.
7.4
Recirculation in Tests in a Current
Recirculation is manifested by an intake temperature depression,
ATi.,
between the intake and the ambient at the level of the intake, which,
0
0
for tests in a current, was of order -0.05 C ±0.1 C.
the intake water temperature itself, T i, was
The variability of
±0.4 0 C with the greatest
variability occuring in the upstream half of the intake structure.
(Fig.
7.2 depicts the position of the four intake thermister probes.)
The vertical variation in ambient temperature measured over the
upper 30m (prototype) of the water column was typically 0.2
0
C +0.1 0 C.
Thus
since T' ( z= hi) is generally the maximum temperature of the water column,
a
i
the mean intake temperature depression could be accounted for by assuming
the intake withdraws from throughout the upper 30m (prototype) of the water
column.
However the spatial and temporal variability of T.1 strongly sug-
gests that cooler water from below this upper layer occasionally reaches
the intake.
One mechanism for recirculation,
is
the occurrence of internal waves,
consistent with the above observations,
which result from the perturbation of
the ambient water column as it moves past the plant.
These waves
could occasionally cause water from near the bottom of the mixed layer,
or below,
to be drawn to the intake.
internal waves on the
upstream
The greater height expected for
side of the plant - correlated with the
-138-
= Location of thermistor probe
L
Current
II
271
o
64
0
03
O
OO
Figure 7.2:
Annular intake structure showing positions of intake
thermistor probes (only one half of structure depicted).
-139-
increased variability of intake temperature on the upstream side - promotes
this possibility.
Another mechanism is upwash, which has been previously described.
Due
to the intermittency of the dye billows that escape the plume, upwash recirculation would be expected to cause significant temporal variability of
the intake water.
Unfortunately the time scale of upwash recirculation,
as visually estimated from the experiments, is small compared to the time
it takes to collect water sample during an experiment (=10 sec).
Thus while
direct recirculation is measured from the water samples, its anticipated
variability is
not resolved.
However other indirect evidence suggests the existence of upwash recirculation.
Because upwash recirculation consists of the intermittent
intake of highly diluted billows of discharge water, it would not be expected to cause significant mean recirculation, which is consistent with
the observed low values of direct recirculation and intake temperature
depression.
Also the relatively high variability associated with T. sug-
gests that the mean T i represents an average of frequent readings with a
temperature near T'a ( z= hi) and infrequent readings of cooler temperature.
The notion of upwash recirculation is
ing heat budget analysis for the intake.
also consistent with the followAssume that the intake tempera-
ture depression is caused (at least in part) by direct recirculation of
the discharge fluid,(upwash induced recirculation) and that the discharge
has been mixed with ambient water of lower temperature relative to
T' (zmh ).
a
i
QiiT'
Then the heat balance of the intake flow can be expressed as:
+ (Sa= XQT'
iohere
1)
T'e Q + (1 - SavX ) T' Qi
where
-140-
(7- 1)
Sa
= a characteristic average jet dilution
V'.
= a characteristic average jet temperature
T'
= The average temperature of the ambient fluid that enters
je
the intake without having been entrained by the jet
If S
and T' are set equal to the average dilution of the jet and the
je
av
average temperature of the jet respectively at its maximum point of penetration ( zma)
then a solution for T' can be
and X are already kno .n.
obtained since Qi. Ti, T
When solutions for T' were obtained, it was
(z = h.).
discovered that T' was always approximately equal to T'
a
I
Fur-
thermore, this trend was fairly insensitive to changing the assumptions of
how values of T
je
were obtained.
and S
av
These results suggest that direct
recirculation (occuring through an upwash mechanism) is the primary
cause of intake temperature depression.
-141-
VIII.
8.1
SUMMARY AND CONCLUSIONS
Summary
Ocean thermal energy conversion plants are being considered to
produce power based on the thermal difference between the upper and
lower temperature strata in a tropical or subtropical ocean.
This study
has examined the behavior of the near field external fluid mechanics of
generic pilot plant OTEC designs under realistic deep water operating
conditions to assess the environmental impact of different plant
configurations and to determine if pilot plants can be expected to
operate without degrading the thermal resource available for power
production.
8.2
8.2.1
Physical Modeling
Conditions Modeled
Physical model studies investigated variation of near field plume
dynamics and sensitivity of recirculation to different pilot plant
designs.
Both mixed and non-mixed (evaporator) discharge concepts were
examined for power plant sizes ranging from 20 MWe to 80 MWe with
nominal discharge flow rates of 5 m 3 /sec-MWe for an evaporator discharge
and 10m3 /sec-MWe for a mixed discharge.
Discharge port designs included
two, four and eight discrete circular ports, with significant variations
in the MWe/port ratio, issuing either horizontally or vertically.
An
axisymmetric, annular intake structure which promoted vertical downward
inflow was located near the surface, in the mixed upper layer, for all
of the experiments.
-142-
Prototype ambient ocean conditions were modeled by towing the
model pilot plant through a temperature-stratified basin.
Uniform
current speeds of up to 0.5 m/sec (prototype) were studied for oceans
with continuously stratified density profiles characterized by a mixed
layer depth of order 50m (prototype).
8.2.2
Conclusions and Recommendations for Future Work
As anticipated, little recirculation was observed in the
experiments.
Values of direct recirculation as determined from
discharge and intake dye samples were on the order of 0.7% ± 0.8%.
The
0
corresponding intake temperature depression was on the order of -0.05 C
± 0.09 0 C.
Two mechanisms (upwash and internal waves) were proposed to
explain the intermittent recirculation which was observed.
Certain
experimental adjustments could be made to allow more definitive
characterization of these mechanisms.
A continuous flow-through
fluorometer could be used so that fluctuations in direct recircultion
are resolved.
The discrete values of the intake temperature readings,
in addition to the mean and standard deviation, could also be recorded
so that maximum intake temperature depressions could be documented.
Finally, the time histories of the dye sampling and temperature
recording procedures could be coupled.
These modifications would permit
virtually complete charactertization of the mechanisms of intake
temperature depression.
The environmental parameters that were measured include the
dilution, elevation, width, thickness and trajectory of the OTEC
discharge plume.
The latter four were ascertained to within a few
-143-
percent from photographic data with the most uncertainty associated with
width (due to parallax error).
Dilution was measured by a water
sampling probe that moved vertically in the water column.
Measured
values of centerline dilution generally fell in the range of 10 to 15
with an uncertainty of about 15% due to estimated measurement bias.Use of
a continuous flow through fluorometer over the entire water column
would eliminate this bias.
Because numerical simulations suggested that plume dynamics are
fairly insensitive to the mixed layer depth as long as the ZOEF begins
near the thermocline, it seems reasonable to conduct some future physical
model tests with both a shallower discharge and mixed layer depth.
This would test conditions of maximum recirculation and would increase
the effective vertical dimensions of the experimental basin.
Finally, the modeling in this study has been limited to those
designs which can be represented as symmetrical, vertical columns,
discharging in deepwater.
into this category.
However many pilot plant designs may not fall
For instance, shelf-mounted and shore-based plants
can be expected to interact with the ocean bottom or shoreline.
The
present experimental set-up could be modified to represent discharge
interaction with the ocean floor.
8.3
8.3.1
Numerical Modeling
Methodology
A previously calibrated integral jet model (Hirst, 1971a) was used
to represent the dynamics of the OTEC discharge in the near field.
Major modifications included adjustment of the equations that
characterized the starting length (ZOFE); introduction of jet deflection
-144-
in the ZOFE; introduction of a lateral spreading formulation that
allowed the "squeezing" effects of the ambient stratification to be
modeled; and introduction of an aspect factor, which accounted for
interaction of a number of ports issuing from a circular array (i.e. the
vertical discharge experiments).
Comparison of simulation to
observation indicated that overall agreement over the range of vertical,
experimental conditions was quite good.
In light of this, additional
simulations were made to characterize the environmental impact of the
discharge plume from an OTEC pilot plant over a broad range of realistic
conditions.
8.3.2
Conclusions and Recommendations for Future Work
Although the integral jet model performed well and should be useful
for design purposes, it is believed that a reformulation of the integral
analysis and a subsequent recalibration of the model would produce an
even better predictive tool.
This conclusion is based on the fact that
Hirst (1971,a) included a second order term in his integral equations
which was shown to affect the results of the simulations significantly.
It is believed that the inclusion of this second order term, which
increases the rate of bending of the plume, compensates for the omission
of drag forces in the integral formulation and results in an
overprediction of the calibrated jet entrainment rate.
The mechanism of
jet spreading (collapse) due to ambient stratification should be
reformulated to include separate differential equations and entrainment
-145-
relationships governing jet thickness and jet width.
The integral model could be coupled with chemical and biological
kinetic models to help assess environmental impacts associated with such
effects as the introduction of biocides or artificial nutrient upwelling
by the condenser intake.
-146-
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Boundary Source," Tellus, 4, 201-210 (1952).
Sands, M.D., "Ocean Thermal Energy Conversion Programmatic Environmental
Analysis," DOE Publication LBL-10511, Vol.1 (January, 1980).
Scott, R.J., "Conceptual Design and Costs of OTEC 10/40 MW Spar
Platforms," proc. Sixth OTEC Conference, paper 5.6 (June, 1979).
Sundaram, T.R., S.K. Kapur, A.M. Sinnarwalla, and E. Sambuco, "The
External Flow Field Associated with the Operation of an Ocean
Thermal Power Plant," Hydronautics, Inc., Report No. C00-2348-1
(December, 1979).
Ungate, C.D., D.R.F. Harleman and G. H. Jirka, "Mixing of Submerged
Turbulent Jets at Low Reynolds Numbers," R.M. Parsons Lab. for
Water Resources and Hydrodynamics, M.I.T. Technical Report No. 197
(February, 1975).
Van Dusen, E., and P. A. Mangarella, "An Analysis of the Thermal and
Nutrient Properties of the Condenser Discharge Plume Created by an
Ocean Thermal Difference Power Plant," NSF Report No.
NSF/RANN/SE/GI-34979/TR/74/2, Univ. of Massachusetts, Amherst,
Massachusetts (October, 1974).
Walsh, J.J., "The Potential Environmental Consequences of Ocean
Thermal Energy Conversion (OTEC) Plants," Proc. of Workshop,
Brookhaven National Laboratory, Upton, New York (January 1980)
Winiarski, L.D., W.E. Frick, "A Simple Method of Predicting Plume
Behavior from Multiple Sources," Corvallis Environmental Research
Lab., Corvallis, Oregon (September, 1978).
Wolf, A.W., "OTEC Thermal Resource Report," Ocean Data Systems, Inc.,
Monterey, CA, Contract No. ET-78-C-01-2893, (May, 1979).
Wu, F.HI.Y., "A Mathematical Model for Multiple Cooling Tower Plumes."
W.M. Keck Lab. of Hydraulics and Water Resources, Cal. Inst. of
Tech., Pasadena, California, Publication No., KH-R-37 (July, 1977).
-150-
APPENDIX I
SIMULATIONS OF THE OTEC MODEL
DISCHARGE IN THE VERTICAL, Y-Z
PLANE, COMPARED TO EXPERIMENT
Legend
(A)
Dotted line indicates centerline of simulation.
(B)
Solid lines indicate simulated boundaries.
(C)
Dashed line indicates experimentally observed boundaries.
-151-
Exp IA: 40 MWe plant; vertical, 8-port, mixed
discharge; current speed = 0.3 m/s.
DIST14NCE. POSITIVE IN DIRECTION OF CURRENT
200
tOO
(M)
-tOa
O
PLANT
/
*
I
0 ..
I
-
'
f
'
LaO
L.-
200
200
C3
m
I
.
I
Exp IB: 40 MWe plant; vertical, 8-port, mixed
discharge; current speed = 0.5 m/s.
DISTRNCE, POSITIVE IN DIRECTION OF CURRENT
20Q
a
OC
-Oo
I-3
PLANT
•
200
j
(M)
.
2z1
I
Exp 2A: 80 Me plant; vertical, 8-port, evaporator discharge;
current speed = 0.3 m/s.
DISTNCE, POSITIVE IN DIRECTION OF CURRENT
Loo
2(1
(M)
-L4AQ
P
PLANT
X-.
C
200
I
2W
40 MWe plant; vertical, 8-port, evaporator
Exp 3B:
discharge; current speed = 0.5 m/s.
DISTANCE,.
POSITIVE IN DIRECTION OF CURRENT
200
(M)
0
100
0
-
0L0
m
,--M
-200
200
-
__
L
--
20 MWe plant; vertical, 8-port, mixed
Exp 4A:
discharge; current speed = 0.3 m/s.
DOISTNCE,
200
POSITIVE
IN DIRECTION OF CURRENT
0
t00
(M)
-t00
PLANT
I
,I
29I
Loa
Loa
I
.
M
I I
I
Exp 4B: 20 MWe plant; vertical, 8-port, mixed
discharge; current speed = 0.5 m/s.
DISTANCE, POSITIVE IN DIRECTION OF CURRENT
2Q0
-0a
O
tOO
(M)
PLANT
I
OO
/
'Na-
-*
/I
20Q
z20
20 MWe plant; vertical, 4-port, mixed
Exp 5A:
discharge; current speed = 0.3 m/s.
DISTRNCE, POSITIVE IN DIRECTION OF CURRENT
LQQ
2aQ
(M)
-LOQ
O
PLANT
* * *....*
I
00
,
'-M
200.
20
Exp 5B:
20 MWe plant; vertical, 4-port, mixed
discharge; current speed = 0.5 m/s.
OISTRNCE, POSITIVE
ZQ
Luu
IN DIRECTION OF CURRENT
1!3.
(M)
0
L(Mrn
-10
.--I
2z
Man
Exp 6A: 40 MWe plant; vertical, 4-port, mixed
discharge; current speed = 0.3 m/s.
DISTFINCE, POSITIVE IN DIRECTION OF CURRENT
MI
I
i2
(M)
i
PLANT
C3
1
1L..**
t
.O
MC
..
..
-U
)
z20
M
*
9-4
20a
Exp 6B: 40 MWe plant; vertical, 4-port, mixed
discharge; current speed = 0.5 m/s.
DISTANCE, POSITIVE IN DIRECTION OF CURRENT
200
-tOO
O
LOQ
(M)
PLANT
-
1I
C-'
I -
/
wI
20
20
-LOu
M
Exp 7A:
80 MWe plant; vertical, 4-port, evaporator
discharge; current speed = 0.3 m/s.
DISTFNCE. POSITIVE IN DIRECTION OF CURRENT
200
-,
I
-..
.
PLANT
-
.
"
1
.-
,
.
2a
.
*
-taO
€
tO
(M)
. . .
•
•
.
7
o
200
80 MWe plant; vertical, 4-port, evaporator
Exp 7B:
discharge; current speed = 0.5 m/s-
DISTRNCE, POSITIVE IN DIRECTION OF CURRENT
200
t
LOO
(M)
-Lta
PLANT
-.
100
2s
I
N.
_
4'
_
-~
- -
-D
"-
/
0
0n0
m
80 MWe plant; vertical, 4-port, evaporator
Exp 8A:
discharge; current speed = 0.3 m/s.
DISTRNCE, POSITIVE IN DIRECTION OF CURRENT
1
I
LOQ
I
I
P
ON
(M)
ol
PLANT
I
Y
200
-
Man
Exp 8B: 80 MWe plant; vertical, 4 -port, evaporator
discharge; current speed = 0.5 m/s.
DISTNCE,
POSITIVE IN DIRECTION OF CURRENT
200
LOQ
O
(M)
-LOC
PLANT
Ln
20.0
_
200
20 MWe plant; vertical, 2-port, mixed
Exp 11A:
discharge; current speed = 0.3 m/s.
DISTNCE, POSITIVE IN DIRECTION OF CURRENT
20
SP
100
(M)
-tLOO
O
L A NIT
PLANT
I
LOM
L(M
1..0n
//
20a2
M
-o
-I
I
2011
Exp 11B: 20 MWe plant; vertical, 2-port, mixed
discharge; current speed = 0.5 m/s.
DISTRNCE.
POSITIVE IN DIRECTION OF CURRENT
(M)
LQQ
LQ
200
I
200.
I
m:
Exp 12A: 40 MWe plant; vertical, 2-port, evaporator
discharge; current speed = 0.3 m/s.
DISTRNCE, POSITIVE IN DIRECTION OF CURRENT
-Q
LtOO
200
CM)
PLANT
*I
--
La
2aa
!l
m
2za
,.
40 MWe plant; vertical, 2-port evaporator
Exp 12B:
discharge; current speed = 0.5 m/s.
DISTANCE, POSITIVE IN DIRECTION OF CURRENT
-400
0
100
200
(M)
PLANT
PN
I
L
-N
C3
2
200
111~-
I
_
I
J
r
____ 1
~_ _~
I
,
APPENDIX II
SIMULATIONS OF THE OTEC MODEL DISCHARGE
IN THE HORIZONTAL, X-Y PLANE,
COMPARED TO EXPERIMENT
Legend
(A) Dashed line indicates centerline of simulation.
(B)
Solid lines indicate simulated boundaries.
(C)
" 04" or "4
"1 indicates
left and right experimentally
observed boundaries respectively.
(D)
"..."
indicates observed wake too wide to be indicated
on plot.
-170-
Run 1A:
-25G
40 MWe plant; vertical, eight port, mixed discharge, current speed = 0.3 m/s.
-20(
-15Q
-1 O
DISTFRtC'CE NORMtGL T
-SQ
Q
50
N
N,
CURRENT
10
-T
(MI
1.5(Q
--I
2(1
2 O
3G0
I
Ic
on
p
C
z
Fp
0U-,
o, -
P-
FFR
Q
F
P>
ca
n
+
+
-rn:
+
+
+
+
+
+
+
+
t+
2t
0
-
40 MWe plant; vertical, eight port, mixed discharge, current speed
Run iB:
-250
-'200
-O0
-450
Ccco
pw
DISTRNCE NORHAL TO CURRENT (MI
1.50
Loo
50
0
-50
•0
±
=
0.5 m/s.
200
250
300
zC-,
,,-4
f•
e
r
ra
C3
p
rt--A
p1
+
+
+
'-4
a
+
z
+++
0.-
'.
c~l
l+
p2
80 MWe plant; vertical, eight port, evaporator discharge, current speed = 0.3 m/s.
Run 2A:
-250
-20
-450
OISTRNCE NORKRL TO CURRENT (M)
.OO
1.50
-50
O
50
-100
200
25(0
300
GCm
F(A
CC7
e -r-
en
0
rt
°
.
,C
p
+z
t
t
-I
t
-'-4
p-
0
Run 2B:
80 MWe plant; vertical, eight port evaporator discharge, current speed = 0.5 m/s.
DISTANCE NORKFIL TO CURRENT tMI
300
O
-
,
0
IV
9-4
0
m
-2
a
ra
znr
a
m
Z
fu,
C")
Q
'I,,-4
2
z
'4l
Run 3A:
-25O
40 MWe plant; vertical, eight port evaporator discharge, current speed = 0.3 m/s.
-2(0
-LOO
-t50
OJSTRNCE NORMAIL TO CURRENT (M)
1.S0
100
50
o
-50
200
250
300
u
p
on,
FC,
N
M
+P
C3
p
0-4
0
ca
~n
+
+
+
U'-
r0
a
-n
fl
Run 3B:
-250
40 MWe plant; vertical, eight port, evaporator discharge, current speed = 0.5 m/s.
-200
-1 0
-150(
DISTRNCE NORMFL TO CURRENT (MI
150
100
50
0
-50
200
250(1
300
:)
0
CC-
0
en
p
+
o
th
+
m
a
z
w
a
r
p+
+
+
pg
+
+
+
+
+
+
+
+
2-Iz
0 1
Run 4A:
N
Un
in
-250
-200
I
20 MWe plant; vertical, eight port, mixed discharge, current speed = 0.3 m/s.
-150
-100
DISTRNCE NORMAL TO CURRENT
100
5s0
0
-50
I
I
I
I
(M)
15
200
I
251
I
1
Run 4B:
20 MWe plant; vertical, eight port, mixed discharge, current speed = 0.5 m/s.
DISTRNCE NORIKRL TO CURRENT
-250
-200
-150
- 00
-50
0o
50
100
(MI
150
200
250
300
N-
I+
00
6-P
0
z
-
+
+
r
+
+
0
+
I
0I
I
I.o
I-'
pC+t
t
C1
~
+
+
+
+
+
0
Run 5A:
20 MWe plant; vertical, four port mixed discharge, current speed = 0.3 m/s.
DISTFNCE NORKRL TO CURRENT
(Ml
Run 5B:
20 MWe plant; vertical, four port mixed discharge, current speed = 0. 5 m/s.
DISTANCE NORKRL TO CURRENT (MI
3OO
300
J)
cIV
IV
ch
en
m
00
U)
z
0D
CD
wP-
2
6O
a
2
I:
Run 6A:
-250
40 MWe plant; vertical, four port, mixed discharge, current speed = 0.3 m/s.
-200
-50so
-00
DISTRNCE NORMRL TO CURRENT (M)
-50
0
50
100
50
2(00
250
300
CD
ca
'-4
p
Z
m
-
+
+
+
+
+
+
+
+
+.
-U
z
l
p~
CR
+
mr
0-
Run 6B:
40 MWe plant; vertical, four port, mixed discharge, current speed = 0.5 m/s.
DISTINCE NORKHIL TO CURRENT t(M
!00
Is
n
CA
w.z
m
a
U)
2
0
n1
pl-
a
2
2
a-n
2
p-I
Run 7A:
80 MWe plant; vertical, four port, evaporator discharge, current speed = 0.3 m/s.
OISTRNCE NORKRL TO CURRENT
(MI
gOO
0
um
-t
II(
Q
zU
tm
p.4
z
I
rn
30
C)
'-4
'-4
n-
2
z
p.'
Run 7B:
80 MWe plant; vertical, four port evaporator discharge, current speed = 0.5 m/s.
DISTANCE NORKRL TO CURRENT
(H)
Run 8A:
-'250
40 MWe plant; vertical, four port evaporator discharge, current speed = 0.3 m/s.
-200
-150
-100
DISTINCE NORHFPL TO CURRENT (M)
-50
Q
s5
L00
1.50
200
250
300
CJ'D
cR
N-
+fl+
+
0
m
Rn
s-a
CR
+flu
T
-~o
+r
Pr
5-
Run 8B:
40 MWe plant; vertical, four port, evaporator discharge, current speed
1OISTRNCE NOR1AL TO CURRENT
-25a
-200
-100
-150
-'50
50
0
oo
=
0.5 m/s.
[M)
t50
200
300
250
SC7
in
aa
+
S:
p
00
Iz
$.I
0"a
ca
-n
a
z)
0)a
++++
p
~
rt-
-+
+.++
+1
pg
I
E
'.4
+
-+
-+
+
Run IIA:
-250
20 MWe plant; vertical, two port, mixed discharge, current speed = 0.3 m/s.
-20
-150
-400
DISTRNCE NORHRL TO CURRENT
0
50
l00
-5a
(MI
L15
2a0
250
300
,-6
CD
00-
SC3
p
n'
0
0n
'-4
Ln
ca
S
pI
+
+
+
+
+
+
+
+
+
2
ra
+W-
-
Run 11B:
20 MWe plant; vertical, two port, mixed discharge, current speed = 0.5 m/s.
DISTRNCE NORKRL TO CURRENT (Ml
Run 12A:
-250
I
p
-200
40 MWe plant; vertical, two port evaporator discharge, current speed = 0.3 m/s.
-15Q
-1 00
DISTRNCE NORMAL TO CURRENT (M)
150
50
100
-50
200
250
Run 12B:
40 MWe plant; vertical, two port evaporator discharge, current speed = 0.5 m/s.
DISTINCE NORMFIL TO CURRENT
-250
-200
-400
-150
-50
0
100
50
(MI
150
250
200
300
UI
+
++
-
+
+-
+
p
-I
N
I
Run 13A:
80 MWe plant; horizontal, eight port, evaporator discharge at crossflow;
current speed = 0.3 m/s.
a
6
.
Run 13B:
80 MWe plant; horizontal, eight port, evaporator discharge at crossflow;
current speed = 0.5 m/s.
DISTFINCE NORhFIL TO CURRENT
(.MI
3SQ
3
0
C2
-u
P
wCm
Cm
P
2
r-0
'.4
1
i
Run 14A:
40 MWe plant; horizontal, eight port, mixed discharge at crossflow;
current speed = 0.3 m/s.
DISTRNCE NORRAL TO CURRENT (MI
100
5
\0
-50
Run 14B:
40 MWe plant; horizontal, eight port, mixed discharge at crossflow;
current speed = 0.5 m/s.
DISTRNCE NORKFIL TO CURRENT (M)
-200
-50
-50
-LO
0
S
l00
1,50
200
250
300
350
P
0
u)
+
p
C,
rn
++'.0
FFR
+
I"R
CA
p++
+
+.
+
+
+
+
+
20
2
tvl
1
I A
Run 15A:
40 Me plant; horizontal, eight port, evaporator discharge at crossflow;
current speed = 0.3 m/s.
DISTONCE NORMAL TO CURRENT
50
0
-50
(MI
Run 15B:
40 MWe plant; horizontal, eight port, evaporator discharge at crossflow;
current speed = 0.5 m/s.
DISTANCE NORKMRL TO CURRENT
(M)
Run 16A:
-2GQ
20 MWe plant; horizontal, eight port, mixed discharge at crossflow;
current speed - 0.5 m/s.
DISTI NCE NORMAL TO CURRENT
LOQ
5Q
QI
-5
-- QQ
-L5Q(
IMI
LSQ
zQQ
251
35(1
3QQ
PC3
+a
S+
+
+
+
+l
I
I
On
pt-
•.
"
p
+
C?
M
ID
,
"4
(A
z
+
++
•
--
+
c,
,-4
""-
FR
z
1 ra
'.0
p
p~
+
\o
+
1
c,2
I
0
r'
+++++'
-t-4
c-
I
,t -.
+
t
p2
t
t
"4
Run 16B:
20 MWe plant; horizontal, eight port, mixed discharge at crossflow;
current speed = 0.3 m/s.
DISTRNCE NORPRL TO CURRENT (M)
I5Q
0
u)
M-
z
WE
M
-In
2
A
rWE
C?
u-I
*.6
2
I
'-4
0
Run 19A:
80 MWe plant; horizontal, four port, evaporator discharge at crossflow;
t
g
Run 19B:
-200
-450
80 MWe plant; horizontal, four port, evaporator discharge at crossflow;
current speed = 0.5 m/s.
-L00
DISTRNCE NORMAL TO CURRENT (MI
-50
O0
54
100
LSQ
wN
+
,.-
+
U
p
200
250
+
.
i
*
Run 20A:
.
40 MWe plnat; horizontal, four port, evaporator discharge at crossflow;
current speed = 0.3 m/s.
350
,-4
U)
- U
IV
a
m
C3
9-4
"?
-4
0C1
Ca
n
2I
'-I
Run 20B:
-150
40 MWe plant; horizontal, four port, evaporator discharge at crossflow;
curent speed = 0.5 m/s.
-100
DISTANCE NORTL TO CURRENT (M)
LO
-50
0
\ 0
200
Run 21A:
-200
N
I +
-50
I
40 MWe plant; horizontal, four port, mixed discharge at crossflow;
current speed = 0.3 m/s.
-1Q0
I
DISTRNCE NOR ~L TO CURRENT CMI
OQ
150
0
50
-50
I
I
+
.
1
I
200
-1
250
i
Run 21B:
40 MWe plant; horizontal, four port, mixed discharge at crossflow;
current speed = 0.5 m/s.
DISTFANCE NORP~1L Ti
a40
-150
-40
-50
a
0
(/l
10
I
r.
0
Run 23A:
-200
-150
40 MWe plant; horizontal, four port, evaporator discharge, 450 into current;
current speed = 0.3 m/s.
-1 00
___
_ __
$
Run 23B:
MWe plant; horizontal, four port, evaporator discharge, 450 into current;
current speed = 0.5 m/s.
DISTFINCE NORKAL TO CURRENT
-200
-150
-O00
-50
0
10
50
I(M
150
200
250
300
350
I)
+U
-
2
+
+
+
+
+
+
zr
+
U-'
APPENDIX III
FINAL COPY OF INTEGRAL
JET MODEL CODE
Note:
This code was originally written at Argonne National
Laboratory by Tony C. Ku. It was written according
to a publication by E.A. Hirst (1971a).
-207-
C*************************************
C
C
C
PROGRAM INTEGRATES THE INTEGRAL MODEL EQUATIONS (BASED
C
ON THE HIRST ANALYSIS) FOR AN INCLINED, ROUND, BUOYANT
JET INTO AN ARBITRARILY STRATIFIED FLOWING ENVIRONMENT.
C
C
THE PROGRAM CAN ACCOUNT FOR A CIRCULAR MULTIPLE ARRAY
C
BY USING THE ASPACT FACTOR FUNCTION.
C
ALL REFERENCES TO PAGE NUMBERS, EQUATION NUMBERS, AND
C
C
SECTION NUMBERS REFER TO:
C
E. A. HIRST, 'ANALYSIS OF ROUND, TURBULENT, BUOYANT
C
JETS DISCHARGED TO FLOWING STRATIFIED AMBIENTS,'
C
C
OAK RIDGE NATIONAL LABORATORY, ORNL-4685, OAK RIDGE,
C
TENNESSEE (JUNE 1971).
C
ADDITIONAL MODIFICATIONS ARE DOCUMENTED IN THE
C
C
M.S. THESIS OF PAUL SINGARELLA AT M.I.T. IN
CIVIL ENGINEERING.
C
C
C
C***************************************
C
C
WRITTEN BY:
C
C
ROBERT A. PADDOCK
C
ENERGY AND ENVIRONMENTAL
C
SYSTEMS DIVISION
C
ARGONNE NATIONAL LABORATORY
C
ARGONNE, ILLINOIS 60439
C
(JULY 1981)
C
C
C
ADAPTED FROM A PROGRAM WRITTEN BY:
C
TONY C. KU
C
ENERGY AND ENVIRONMENTAL
C
SYSTEMS DIVISION
C
ARGONNE NATIONAL LABORATORY
C
ARGONNE, ILLINOIS
60439
(1976)
C
C
C
C
C
ADAPTED FOR AN OTEC PILOT PLANT
C
ANALYSIS BY PAUL SINGARELLA
C
OF THE RALPH M. PARSONS LAB
C
OF WATER RESOURCES AND HYC
DRODYNAMICS, M.I.T.
C
C
C***********************************************************
C
C
C
INPUT VARIABLES:
-208-
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
TITLE = ANY ALPHANUMERIC TITLE
XD = DISCHARGE POSITION ALONG X AXIS
YD = DISCHARGE POSITION ALONG Y AXIS
ZD = DISCHARGE DEPTH (POSITIVE DOWNWARD WITH ZERO AT
THE SURFACE)
THETID = DISCHARGE ANGLE IN HORIZONTAL PLANE WITH
RESPECT TO THE X AXIS (DEGREES, 0 IS ALONG
X AXIS, 90 IS ALONG Y AXIS)
THET2D = DISCHARGE ANGLE WITH RESPECT TO THE HORIZOTAL
PLANE (DEGREES, DOWNWARD ANGLE IS POSITIVE)
UD = DISCHARGE VELOCITY
DD = DISCHARGE DIAMETER
SD=DISCHARGE SALINITY IN PARTS PER THOUSAND
TD=DISCHARGE TEMPERATURE, DEGREES C
UAMB = AMBIENT CURRENT SPEED (ALONG POSITIVE Y AXIS)
GRAV = ACCELERATION DUE TO GRAVITY
SCHMD = SQUARE ROOT OF THE TURBULENT SCHMIDT NUMBER
(1.16 IS RECOMMENDED)
Ai = ENTRAINMENT COEFFICIENT FOR A SIMPLE MOMENTUM JET
(0.057 IS RECOMMENDED)
A3 = THIRD COEFFICIENT IN THE ENTRAINMENT FUNCTION
(9.0 IS RECOMMENDED)
VISC = KINEMATIC VISCOSITY
NPTS=NUMBER OF AMBIENT PROFILE POINTS THAT
CHARACTERIZE THE STRATIFICATION
DELS = STEP SIZE ALONG JET CENTERLINE
XLIMIT = LIMIT ON INTEGRATION IN X DIRECTION FROM
ORIGIN
YLIMIT = LIMIT ON INTEGRATION IN Y DIRECTION FROM
ORIGIN
ZLIMIT = LIMIT ON INTEGRATION IN Z DIRECTION FROM
ORIGIN (SURFACE)
NLIMIT=MAXIMUM NUMBER OF STEPS ALLOWED
IR=PRINTOUT FREQUENCY
AFI=INITIAL ASPACT FACTOR
AF2=ASPACT FACTOR WHEN MERGING IS COMPLETE
RDI=EQUIVALENT RADIUS WHEN AFI BEGINS TO CHANGE
RD2=EQUIVALENT RADIUS WHEN ASPACT FACTOR
BECOMES EQUAL TO ONE
MULTOW=I WHEN THE RUN IS FOR A MULTIPLE TOWER
GEOMETRY, AND IT EQUALS ZERO FOR A
SINGLE TOWER RUN
NARRAY EQUALS UNITY WHEN THE COORDINATES OF
THE DISCHARGE, CENTERLINE AND PERIPHERY,
ARE TO BE SENT TO DATA FILES.
DP(I) = VERTICAL COORDINATE OF AMBIENT PROFILE
SA(I) = AMBIENT SALINITY AT Z=DP(I)
TA(I) =AMBIENT TEMPERATURE AT Z = DP(I)
-209-
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
INTERNAL VARIABLES:
AMBDEN = LOCAL AMBIENT DENSITY
DENGRD = LOCAL VERTICAL AMBIENT DENSITY GRADIENT
DELDND = AMBIENT DENSITY MINUS JET DENSITY AT POINT
OF DISCHARGE
RHOO = AMBIENT DENSITY AT POINT OF DISCHARGE (USED
AS 'REFERENCE' DENSITY)
FD = DENSIMETRIC FROUDE NUMBER AT POINT OF DISCHARGE
INBUOY = PLUS ONE FOR A NEUTRAL OR POSITIVELY BUOYANT
JET WITH AN UPWARD COMPONENT OF MOMENTUM.
PLUS ONE FOR A NEUTRAL OR NEGATIVELY BUOYANT
JET WITH A DOWNWARD COMPONENT OF MOMENTUM. NEGATIVE ONE FOR A NEGATIVELY BUOYANT JET WITH
AN UPWARD COMPONENT OF MOMENTUM. NEGATIVE ONE
FOR A POSITIVELY BUOYANT JET WITH A DOWNWARD
COMPONENT OF MOMENTUM. ZERO FOR A NEUTRALLY BUOYANT JET AND NO VERTICAL MOMENTUM COMPONENT.
SDD = JET CENTERLINE DISTANCE TO POINT OF DISCHARGE
(ALWAYS ZERO)
BD = JET RADIUS AT POINT OF DISCHARGE
WD = JET DIAMETER AT POINT OF DISCHARGE
REYND = REYNOLDS NUMBER AT POINT OF DISCHARGE
SO = CENTERLINE DISTANCE FROM POINT OF DISCHARGE TO
THE END OF THE 'ZOFE'
THETIO = JET ANGLE WITH RESPECT TO THE X AXIS AT
THE END OF 'ZOFE' (DEGREES)
THET20 = JET ANGLE WITH RESPECT TO THE HORIZONTAL
AT THE END OF 'ZOFE' (DEGREES)
ZO = JET DEPTH AT END OF 'ZOFE'
XO = POSITION ALONG X AXIS AT END OF 'ZOFE'
YO = POSITION ALONG Y AXIS AT END OF 'ZOFE'
UO = JET CENTERLINE VELOCITY AT END OF 'ZOFE'
BO = JET 'RADIUS' AT END OF 'ZOFE'
WO = MEASURE OF JET 'DIAMETER' AT END OF 'ZOFE'
DELDNO = CENTERLINE DENSITY DIFFERENCE AT END OF
'ZOFE'
FO = DENSIMETRIC FROUDE NUMBER AT END OF 'ZOFE'
REYNO = REYNOLDS NUMBER AT END OF 'ZOFE'
Y(1) = LOCAL JET CENTERLINE VELOCITY
Y(2) = B = LOCAL JET 'RADIUS'
Y(3) = DELDN = LOCAL CENTERLINE DENSITY DIFFERENCE
Y(4) = THETI = LOCAL ANGLE WITH RESPECT TO X AXIS
(RADIANS)
Y(5) = THET2 = LOCAL ANGLE WITH RESPECT TO
HORIZONTAL (RADIANS)
Y(6) = X = LOCAL COORDINATE ALONG X AXIS
Y(7) = Y = LOCAL COORDINATE ALONG Y AXIS
Y(8) = Z = LOCAL JET CENTERLINE VERTICAL COORDINATE
(POSITIVE DOWNWARD)
SO = INITIAL DISTANCE ALONG JET CENTERLINE FROM
POINT OF DISCHARGE TO START OF CALCULATION
(END OF 'ZOFE')
S = DISTANCE ALONG JET CENTERLINE FROM POINT OF
-210-
DISCHARGE
C
N, EPS, ETA, INDEX, METH AND MITER ARE VARIABLES NEEDED
C
TO ALLOW SUBROUTINE 'DGEAR' TO INTEGRATE THE
C
JET EQUATIONS.
C
THET1 = LOCAL JET CENTERLINE ANGLE WITH RESPECT
C
TO X AXIS (DEGEES)
C
THET2 = LOCAL JET CENTERLINE ANGLE WITH RESPECT
C
THE HORIZONTAL (DEGREES)
C
Z = Y(8) = LOCAL JET CENTERLINE VERTICAL COORDINATE
C
W = 2.0*SQRT(2.0)*B = MEASURE OF JET 'DIAMETER'
C
AVGDIL = LOCAL AVERAGE DILUTION BY VOLUME
C
CLDIL = LOCAL CENTERLINE DILUTION FOR SUBSTANCE WHICH
C
SPREADS LIKE DENSITY (OR TEMPERATURE)
C
C
C
C*********************************************************
C
C
IMPLICIT REAL*8 (A-H,O-Z)
DIMENSION Y(8),TITLE(10),SA(30),TA(30)
DIMENSION IWK(8),WK(208)
EXTERNAL DIFFUN
EXTERNAL PEDERV
COMMON/BLKI/UAMB,GRAV,SCHMD,A1,A2,A3,RHOO,
RDI,RD2,AF1,AF2,MULTOW
&
),NPTS
COMMON/BLK2/DP(100),DENPRO(100)
NDIM=100
FMAX=10000.ODO
ZERO=O.ODO
PAI=3.1415927DO
GRAV=9.80DO
DTOR=PAI/180.ODO
RTOD=180.ODO/PAI
SR2=DSQRT(2.0DO)
IPASS=0O
C
C
C*************************************
C
C
C
READ INPUT VARIABLES.
10 IPASS=IPASS+1
C
C
C
C
THE PROGRAM HAS BEEN MODIFIED TO HANDLE
UP TO 99 CASES DURING ONE EXECUTION
84
86
88
90
NCASEN=O
NOCASE=O
READ(15,84) NOCASE
FORMAT(I2)
NCASEN=NCASEN+1
WRITE(16,88) NCASEN,NOCASE
FORMAT(48X,'CASE NUMBER',I3,' OF',I3,/)
READ(15,90,END=80) TITLE
FORMAT(IOA)
-211-
READ(15,92,END=80) XD.YD,ZD,THETIDTHET2D
92 FORMAT(5D10.0)
IF(THETID.GT.180.ODO.OR.THETID.LT.-180.ODO) GO TO 100
IF(THET2D.GT.180.ODO.OR.THET2D.LT.-180.ODO) GO TO 100
READ(15,94,END=80) UD,DD,TD,SD
94 FORMAT(4D10.0)
READ(15,95,END=80) UAMB,SCHMD,A1,A3,VISC
95 FORMAT(5D10.0)
READ(15,96,END=80) NPTS,NLIMIT,DELS,XLIMIT,YLIMIT,ZLIMIT
96 FORMAT(2I5,5X,4D10.0)
READ(15,97,END=80) AFI,AF2,RD1,RD2,MULTOW,IR,NARRAY
97 FORMAT(4DI0.0,315)
C
C
C
READ IN AMBIENT STRATIFICATION.
98
8
24
26
C
C
C
C
IF(NPTS.GT.NDIM) GO TO 100
IF(NPTS.LE.O.AND.IPASS.GE.2) GO TO 22
IF(NPTS.LE.O.AND.IPASS.LT.2) GO TO 100
DO 8 I=I,NPTS
READ(15,98,END=80)DP(I),SAI),A(I)
FORMAT(3D10.0)
DENPRO(I)=DENSIT(SA(I),TA(I))
CONTINUE
IF(NPTS.GE.2) GO TO 24
NPTS=2
DP(2)=DP(I)+IOO.ODO
DENPRO(2)=DENPRO(1)
DO 26 I=2,NPTS
IF(DP(I).LE.DP(I-1)) GO TO 82
IF(DENPRO(I).LT.DENPRO(I-1)) GO TO 82
CONTINUE
NOLD=NPTS
CALCULATE DISCHARGE FROUDE NUMBER AND OTHER NEEDED
PARAMETERS.
22 NPTS=NOLD
CALL GETAMB(ZD,AMBDEN,DENGRD)
DEND=DENSIT(SD,TD)
DELDND=AMBDEN-DEND
RHOO=AMBDEN
DDD=DSQRT((GRAV*DABS(DELDND)*DD)/RHOO)
FD=FMAX
IF(UD.LT.DDD*FMAX) FD=UD/DDD
SDD=O.ODO
BD=DD/2.0DO
WD=DD
IF (NARRAY.EQ.O) GO TO 28
CALL ARRAY (WD,WD,THET1D,THET2D,XD,YD,ZD)
28 REYND=UD*DD/VISC
C
SCHMD2=SCHMD*SCHMD
COEFF=SCHMD2/(I.ODO+SCHMD2)
A2=2.ODO*SCHMD2-3.ODO*COEFF
Ul2D=UAMB*DSIN(THETID*DTOR)*DCOS(THET2D*DTOR)
-212-
C
C
C
C
C
CALCULATE THE LENGTH OF THE 'ZOFE' AND PARAMETERS AT
THE END OF THE 'ZOFE' USING EXPRESSIONS IN APPENDIX D,
PAGES 34-35.
RATIO=UAMB/UD
R12=Ul2D/UD
C
C
C
C
C
C
C
C
THE PROGRAM HAS BEEN MODIFIED TO HANDLE ALL
POSSIBLE COMBINATIONS OF BUOYANCY AND
DISCHARGE ORIENTATION
INBUOY=O
IF (DELDND.GT.ZERO.AND.THET2D.GT.ZERO) INBUOY=-1
IF (DELDND.GE.ZERO.AND.THET2D.LE.ZERO) INBUOY=I
IF (DELDND.LE.ZERO.AND.THET2D.GE.ZERO) INBUOY=1
IF (DELDND.LT.ZERO.AND.THET2D.LT.ZERO) INBUOY=-1
FD=FD*DFLOAT(INBUOY)
FACTI AND FACT2 HAVE BEEN MODIFIED TO MORE
ACCURATELY REFLECT THE STARTING LENGTH.
IF (RATIO.LE.O.1670DO)
& FACT1=6.2DO*(1.ODO+R12)/(1.ODO-Ri2)/DSQRT(1.ODO+1.18DO*RI2)
-15.55DO*RATIO*DSQRT(1.ODO-DSIN(THETID*DTOR)**2
&
*DCOS(THET2D*DTOR)**2)
&
IF (RATIO.GT.O.167DO.AND.RATIO.LE.O.4DO)
8
& FACTI=4.55DO*(1.ODO+RI2)/(1.ODO-R12)/DSORT(1.ODO+1.1 DO*Ri2)
-5.8DO*RARIO*DSQRT(1.ODO-DSIN(THETID*DTOR)**2
&
*DCOS(THET2D*DTOR)**2)
&
IF (RATIO.GT.O.4DO.AND.RATIO.LE.1.2DO)
2
& FACTi=2.9DO*(1.ODO+Ri2)/(1.ODO-R12)/DSQRT(1.ODO+1.18DO*R1 )
-1.55DO*RATIO*DSQRT(1.ODO-DSIN(THETID*DTOR)**2
&
*DCOS(THET2D*DTOR)**2)
&
IF (RATIO.GT.1.2DO)
& FACTI=i.45*(I.ODO+Ri2)/(1.ODO-RI2)/DSORT(1.ODO+1.18DO*R12)
-0.4DO*RATIO*DSQRT(I.ODO-DSIN(THET1D*DTOR)**2
&
*DCOS(THET2D*DTOR)**2)
&
FACT2=6.2DO
IF (FD.LE.1.500O.AND.FD.GE.O.ODO)
FACT2=1.2DO*FD+1.5DO
&
IF (FD.GT.I.5DO.AND.FD.LE.4.ODO)
FACT2=0.64DO*FD+2.34DO
&
IF (FD.GT.4.ODO.AND.FD.LE.7.ODO)
FACT2=0.17DO*FD+4.185DO
&
IF (FD.GT.7.ODO.AND.FD.LE.35.ODO)
FACT2=0.007DO*FD+5.35DO
&
IF (FD.GT.35.ODO.AND.FD.LE.200.ODO)
FACT2=0.0035DO*FDD+5.475DO
&
BO=(DD/SR2)*DSQRT(UD/(UD+DABS(U12D)))
C
C
C
C
C
THE STARTING LENGTH HAS BEEN ADJUSTED TO
BE CONSISTENT WITH THE USE. OF AN ASPACT FACTOR.
ENCYL=I.ODO
-213-
SO=(FACTI*FACT2*DD)/6.2DO/AF1
UO=UD
AVDILO=(BO*BO*UO)/(BD*BD*UD)
CLDILO=COEFF*AVDILO
WF=AVDILO-1.ODO
SEDP=ZD+SO/2.0DO
CALL GETAMB (SEDP,AMBDEN,DENGRD)
BV=((DEND-AMBDEN)*GRAV*SO)/(AMBDEN*UO)
C
C
C
C
C
C
C
DEFLECTION IN THE ZOFE HAS BEEN ADDED SO
THAT INITIAL CONDITIONS TO THE ZOEF
ARE MORE ACCURATE
CALCULATE DEFLECTION OF THET2D IN THE ZOFE
IF (THET2D.LT.-90.ODO.AND.WF*UAMB.GT.-UO*DSIN((THET2D+90.ODO)*DTOR
))THET20=DATAN((UO*DCOS((THET2D+90.ODO)*DTOR)-BV)/
(-UO*DSIN((THET2D+90.ODO)*DTOR)-WF*UAMB))/DTOR
IF (THET2D.LT.-90.ODO.AND.WF*UAMB.LE.-UO*DSIN((THET2D+90.ODO)*DTOR
))THET20=DATAN((WF*UAMB+UO*DSIN((THET2D+90.ODO)
&
*DTOR))/(UO*DCOS((THET2D+90.ODO)*DTOR)-BV))-90.ODO
&
/DTOR
&
IF (THET2D.GT.90.ODO.AND.WF*UAMB.GT.UO*DSIN((THET2D-90.ODO)*DTOR))
THET20=DATAN((UO*DCOS((THET2D-90.ODO)*DTOR)+BV)/(WF
&
*UAMB-UO*DSIN((THET2D-90.ODO)*DTOR)))/DTOR
&
IF (THET2D.GT.90.ODO.AND.WF*UAMB.LE.UO*DSIN((THET2D-90.ODO)*DTOR))
THET20=DATAN((UO*DSIN((THET2D-90.ODO)*DTOR)-WF*UAMB)
&
/(UO*DCOS((THET2D-90.ODO)*DTOR)+BV))+90.ODO/DTOR
&
IF (UAMB.EQ.O.ODO.AND.DABS(THET2D).EQ.90.ODO) GO TO 38
IF (DABS(THET2D).LE.90.ODO)
THET20=(DATAN((UO*DSIN(THET2D*DTOR)+BV)/
&
(UO*DCOS(THET2D*DTOR)+WF*UAMB)))/DTOR
&
38 IF (DABS(THET2D).EO.90.ODO.AND.UAMB.EQ.O.ODO)
&
THET20=THET2D
&
&
C
C
C
CALCULATE DEFLECTION OF THETID IN ZOFE
THETIO=THET1D
IF (DABS(THETID).LT.90.ODO)
THETIO=(DATAN((UO*DSIN(THETID*DTOR)+WF*UAMB)/
&
(UO*DCOS(THET1D*DTOR))))/DTOR
&
IF (THET1D.GT.90.ODO)
THETIO=(DATAN((UO*DSIN((THETID-90.ODO)*DTOR))/
&
(UO*DCOS((THETID-90.ODO)*DTOR)+WF*UAMB))+90.ODO)
&
/DTOR
&
IF (THETID.EQ.-90.ODO.AND.WF*UAMB.LE.UO)
THETIO=THET1D
&
IF (THET1D.EQ.-90.ODO.AND.WF*UAMB.GT.UO)
THET1O=-THET1D
&
IF (THETID.LT.-90.ODO.AND.UO*DCOS((THET1D+90.ODO)
*DTOR).GT.WF*UAMB)
&
THETIO=(DATAN((UO*DSIN((THETID+90)*DTOR))/(UO*
&
DCOS((THET1D+90.ODO)*DTOR)-WF*UAMB))-90.ODO)/
&
&
DTOR
IF (THET1D.LT.-90.ODO.AND.UO*DCOS((THET1D+90.ODO)
-214-
*DTOR).LT.WF*UAMB)
THET10=(DATAN((UO*DSIN((THETID+90.ODO)*DTOR))/(UO
*DCOS((THETID+90.ODO)*DTOR)-WF*UAMB))-90.ODO)/
DTOR
IF (THETID.LT.-90.ODO.AND.UO*DCOS((THETID+90.ODO)*DTOR).EQ.WF*UAMB
)THET10=180.ODO
&
ZO=ZD+SO*DSIN(THET20*DTOR)
YO=YD+SO*DSIN(THETIO*DTOR)*DCOS(THET20-DTOR)
XO=XD+SO*DCOS(THET1O*DTOR)*DCOS(THET20*DTOR)
WO=2.0DO*SR2*BO
DELDNO=DELDND*(UD+U12D)/(2.ODO*COEFF*(UD+SCHMD2*Ul2D))
IF (NARRAY.EQ.O) GO TO 106
CALL ARRAY (WO,WO,THETIO,THET20,XO,YO,ZO)
106 CALL GETAMB (ZO,AMBDEN,DENGRD)
DDD=DSQRT((GRAV*DABS(DELDNO)*WO)/AMBDEN)
FO=FMAX
IF (UO.LT.DDD*FMAX) FO=UO/DDD
REYNO=UO*WO/VISC
&
&
&
&
C
C
C
PRINT DISCHARGE PARAMETERS AND PARAMETERS AT END OF 'ZOFE'.
104 WRITE(16,900) TITLE
900 FORMAT(IOX,'INTEGRAL JET MODEL FOR MULTIPLE PORT DIS',
'CHARGE ',/,15X,'INTO A STRATIFIED FLOWING EN',
&
'VIRONMENT',///,5X,10A8,//)
&
WRITE(16,902) SDD,SO,XD,XO,YD,YO,ZD,ZO,THETID,THETIO,
THET2D,THET20
$
902 FORMAT(5X,'PARAMETER',3X,'DISCHARGE',4X,'END-ZOFE',//,
13X,'S',2X,F10.3,2X,F10.3,/,
$
13X,'X',2X,FIO.3.2X,F1O.3,/,
$
13X,'Y',2X,F10.3,2X,F1O.3,/,
$
13X,'Z',2X,F10.3,2X,F10.3,/,
$
9X,'THETI',2X,F10.2,2X,FiO.2,/,
$
9X,'THET2',2X,F1O.2,2X,FIO.2)
$
WRITE(16,904) UD,UO,BD.BO,WD,WO,DELDND,DELDNO,FD,FO,
REYND,REYNO
$
904 FORMAT(13X,'U',2X,F1O.3,2X,FiO.3,/,
13X,'B',2X,F1O.3,2X,FIO.3,/,
$
13X,'W',2X,F10.3,2X,FIO.3,/,
$
9X. 'DELDN',2X,FIO.7,2X,FIO.7,/,
$
13X,'F',2X,FIO.3,2X,FiO.3./.
$
10X,'REYN',2X,IPE10O.3,2X,1PEIO.3)
$
WRITE(16,906) DEND,RHOO,TD,SD
906 FORMAT(//,17X,'DISCHARGE',5X,'AMBIENT',//,
7X,'DENSITY',2X,FIO.7,2X,FIO.7.//,7X,
$
'TD=',FIO.3,7X,'SD=',FIO.3)
&
WRITE(16,908) UAMB,A1,XLIMIT,AF1,RATIO,A2,YLIMIT,
AF2,GRAV,A3,ZLIMIT,RD1,SCHMD,NPTS,NLIMIT
&
,RD2,VISC,DELS,IR,MULTOW
&
='
908 FORMAT(//,6X,'UAMB=',F10.3,8X,'Ai=',F10.5,4X,'XLIMIT
10
.5,
,FIO.3,7X,'AFI=',FIO.3,/,3X,'UAMB/UD=',F
&
8X,'A2=',F1O.5,4X,'YLIMIT=',FIO.3,7X,'AF2=',
&
=
F1O.3,/,6X,'GRAV=',F1O.3,8X,'A3=',FIO.5,4X,'ZLIMIT ',
&
F1O.3,7X,'RDI=',FIO.3,/,5X,'SCHMD=',FIO.5,6X,'NDEN=',
&
=
IIO,4X,'NLIMIT=',IO0,7X,'RD2=',FIO.3,/,6X,'VISC ',
&
-215-
&
1PE10.3,6X,'DELS=',F10.3,8X,'IR=',I10,4X,'MULTOW='
IF(NPTS.GT.9) WRITE(16,900) TITLE
WRITE(16,911)
911 FORMAT(5X,'AMBIENT STRATIFICATION',//,
11X,'DP',10X.'SALINITY',IOX,'TEMP',7X,'DENSITY'./)
$
DO 34 I=1I,NPTS
34 WRITE(16,912) DP(I),SA(I),TA(I),DENPRO(I)
912 FORMAT(5X,F10.3,5X,F1O.3,5X,FIO.3,5X,FIO.7)
C
C
C
WRITE HEADER AND PREPARE FOR INTEGRATING EQUATIONS.
WRITE(16,914)
914 FORMAT(/,'I',BX,'S',4X,'THETI',4X,'THET2',9X,'W',8X,'U',
7X,'DELDN',3X,'AVGDIL',4X,'CLDIL',IOX,'X',10X,
$
'Y',IOX,'Z',6X,'AMBDEN',/)
$
NPRINT=1
AVGDIL=1.0
CLDIL=1.0
WRITE(16,916) SDD,THETID,THET2D,WD,UD,DELDND,AVGDIL,
CLDIL,XD,YD,ZD,RHOO
$
CALL GETAMB(ZO,AMBDEN,DENGRD)
NPRINT=NPRINT+I
WRITE(16,916) SO,THETIO,THET20,WO,UO,DELDNO,AVDILO,
CLDILO,XO,YO,ZO,AMBDEN
$
916 FORMAT(1X,F9.3,F9.2,F9.2,FIO.3,F9.3,FI2.7,F9.3,F9.3,
F11.3,F11.3,Fil.3,F12.7)
$
C
918 Y(1)=UO
Y(2)=BO
Y(3)=DELDNO
Y(4)=THET10*DTOR
Y(5)=THET20*DTOR
Y(6)=XO
Y(7)=YO
Y(8)=ZO
N=8
EPS=0.00OOO1DO
ETA=O.O00OO0DO
INDEX=I
MITER=2
METH=1
C
C
C
C
C
C
C
C
MOST OF THE NEXT SECTION HAS BEEN
MODIFIED TO REFLECT LATERAL
SPREADING IN THE NEAR FIELD.
CARRY OUT THE CALCULATIONS.
CALL GETAMB (ZO,AMBDEN,DENGRD)
WSDELS=DSQRT(PAI)*Y(2)/2.ODO
HSDELS=WSDELS
PI=AMBDEN-DENGRD*HSDELS
A
-216-
PJ=AMBDEN-Y(3)
40 NPRINT=NPRINT+1
IF (NPRINT.EQ.3) GO TO 886
CALL GETAMB (Z,AMBDEN,DENGRD)
PI=AMBDEN-DENGRD*HSDELS
PJ=AMBDEN-Y(3)
FP=DCOS(Y(5))*DABS((2.0DO*HSDELS**2*GRAV*(PJ-PI))-GRAV
*DENGRD*((2.ODO*Z**2*HSDELS)-(2.0DO
&
*HSDELS*HSDELS**2/3.ODO)))
&
886 DWDS=(0.103DO/Y(1))*DSQRT((2.ODO*WSDELS*FP)/
(PAI*Y(2)**2))
&
S=SO+DELS
CALL DGEAR(N,DIFFUN,PEDERV,SO,ETA,Y,S,EPS,METH,MITER,
INDEX,IWK,WK,IER)
$
THETI=Y(4)*RTOD
THET2=Y(5)*RTOD
Z=Y(8)
+DELS*DWDS
WSDELS=WSDELS+(Y(2)-BO)
HSDELS=(PAI*Y(2)**2)/(4.ODO*WSDELS)
WG=2.ODO*SR2*WSDELS
HG=2.ODO*SR2*HSDELS
BO=Y(2)
AVGDIL=(Y(2)*Y(2)*Y(1))/(BD*BD*UD)
CLDIL=COEFF*AVGDIL
IF(NPRINT/IR-(NPRINT-1)/IR.NE.1)GO TO 888
WRITE(16,916) S,THETI,THET2,WG,Y(1),Y(3),AVGDIL,CLDIL,
Y(6),Y(7),Y(8),AMBDEN
$
IF (NARRAY.EQ.O) GO TO 888
X=Y(6)
YY=Y(7)
CALL ARRAY (WG,HG,THET1,THET2,X,YY,Z)
IF(INDEX.NE.0) GO TO 10
888
IF(Y(I).LT.I.OD-3.AND.Y(3).LT..0OD-5) GO TO 80
IF(DABS(Y(6)).GT.XLIMIT) GO TO 80
IF(DABS(Y(7)).GT.YLIMIT) GO TO 80
IF(Y(8).GT.ZLIMIT) GO TO 80
IF(Y(8).LT.ZERO) GO TO 80
IF(NPRINT.GE.NLIMIT) GO TO 80
GO TO 40
C
C
80 WRITE (16,5555) Y(8),HG
5555 FORMAT (' PLUME THICKNESS AT ',F10.1,
&
'
Z,
=
',FIO.1)
WRITE(16,6666)NPRINT
6666 FORMAT(' NUMBER OF STEPS = ',14)
WRITE(6,7777)NCASEN
IS OVER ')
7777 FORMAT(' CASE NUMBER ',12,'
IF(NOCASE-NCASEN.GT.0) GO TO 86
100 STOP
C
C
82 WRITE(16,950)
950 FORMAT(///,' *****AMBIENT PROFILE DATA OUT OF ORDER*****',///)
STOP
-217-
C
C
C***************************************
END
C**********
*******************************
C
SUBROUTINE DIFFUN(N,S,Y,YDOT)
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
ROUTINE SUPPLIES THE DERIVATIVES OF THE EIGHT VARIABLES
OF THE HIRST MODEL FOR AN INCLINED, ROUND, BUOYANT JET
INTO AN ARBITRARILY STRATIFIED FLOWING ENVIRONMENT.
ALL REFERENCES TO PAGE NUMBERS, EQUATION NUMBERS, AND
SECTION NUMBERS REFER TO:
E. A. HIRST, 'ANALYSIS OF ROUND, TURBULENT, BUOYANT
JETS DISCHARGED TO FLOWING STRATIFIED AMBIENTS,'
OAK RIDGE NATIONAL LABORATORY, ORNL-4685, OAK RIDGE,
TENNESSEE (JUNE 1971).
VARIABLES:
N = NUMBER OF SIMILTANEOUS ORDINARY DIFFERENTIAL
EQUATIONS TO BE SOLVED
S = DISTANCE ALONG JET CENTERLINE
Y(i) = U = LOCAL JET CENTERLINE VELOCITY
Y(2) = B = LOCAL JET 'RADIUS'
Y(3) = DELDN = LOCAL JET CENTERLINE DENSITY DIFFERENCE
Y(4) = THET1 = LOCAL ANGLE WITH RESPECT TO X AXIS
Y(5) = THET2 = LOCAL ANGLE WITH RESPECT TO HORIZONTAL
PLANE
Y(6) = X
Y(7) = Y (DIRECTION OF AMBIENT CURRENT)
Y(8) = Z (POSITIVE DOWNWARD)
YDOT(1)-YDOT(8) = DERIVATIVES OF JET PARAMETERS WITH
RESPECT TO S
UAMB = AMBIENT CURRENT VELOCITY (ALONG POSITIVE Y AXIS)
GRAV = ACCELERATION DUE TO GRAVITY
SCHMD = SQUARE ROOT OF TURBULENT SCHMIDT NUMBER
A1,A2,A3 = COEFFICIENT IN ENTRAINMENT FUNCTION
RHOO = AMBIENT DENSITY AT DEPTH OF DISCHARGE (USED AS
REFFERENCE DENSITY)
AMBDEN = LOCAL AMBIENT DENSITY
DENGRD = LOCAL VERTICAL AMBIENT DENSITY GRADIENT
FRLINV = INVERSE OF LOCAL FROUDE NUMBER BASED ON JET
'RADIUS', B (HIRST'S DEFINITION)
ENTRAN = LOCAL ENTRAINMENT RATE
U12 = COMPONENT OF AMBIENT VELOCITY ALONG LOCAL JET
AXIS
-218-
C
C
C*******************************
C
C
IMPLICIT REAL*8 (A-H,O-Z)
DIMENSION Y(8),YDOT(8)
COMMON/BLKI/UAMB,GRAV,SCHMD,AI,A2,A3,RHOO,
&
RD1,RD2,AF1,AF2,MULTOW
C
C
C***********************************
C
C
C
C
GET LOCAL DENSITY GRADIENT.
Z=Y(8)
CALL GETAMB(Z,AMBDEN,DENGRD)
C
C
C
CALCULATE NEEDED COEFFICIENTS AND PARAMETERS.
SCHMD2=SCHMD*SCHMD
CONi=(-GRAV*SCHMD2)/(2.ODO*RHOO)
CON2=I.ODO+SCHMD2
SI=DSIN(Y(4))
S2=DSIN(Y(5))
CI=DCOS(Y(4))
C2=DCOS(Y(5))
BSQ=Y(2)*Y(2)
U12=UAMB*S1*C2
U12MUM=UI2-Y(1)
VSTAR=Y(I)+UI2
C
C
C
C
CALCULATE INVERSE OF LOCL FROUDE NUMBER BASED ON HIRST'S
DEFINITION.
FRLINV=(Y(3)*Y(2)*GRAV)/(Y(1)*Y(1)*RHOO)
C
IF (FRLINV.LT.O.1DO) FRLINV=O.IDO
C
C
C
C
C
C
C
C
C
C
CALCULATE ENTRAINMENT FROM EQUATION 78, PAGE 21.
THE ASPACT FACTOR HAS BEEN INTRODUCED TO
ACCOUNT FOR DIFFERENTIAL ENTRAINMENT
BETWEEN AN EQUIVALENT AREA SOURCE AND
ACTUAL MULTIPLE SOURCES.
ENCYL=I.ODO
IF (DFLOAT(MULTOW).EQ.I.ODO) ENCYL=ASPFAC(Y(2),RD1,RD2,AFI,AF2)
ENTRAN=ENCYL*(AI+A2*DABS(S2)*DABS(FRLINV))*(Y(2)*DABS(UI2MUM)
&
+A3*UAMB*Y(2)*DSQRT(1.ODO-SI*SI*C2*C2))
C
O=0.25DO*(BSQ*VSTAR*VSTAR-ENTRAN*ENTRAN)
-219-
C
C
C
C
C
CALCULATE DERIVATIVES OF JET PARAMETERS.
FROM EQUATION 62, PAGE 13.
YDOT(4)=(ENTRAN*UAMB*CI)/(Q*C2)
C
C
C
FROM EQUATION 63, PAGE 13.
YDOT(5)=(Y(3)*BSQ*CONI*C2-ENTRAN*UAMB*SI*S2)/Q
C
C
DUI2DS=UAMB*(-S1*S2*YDOT(5)+C1*C2*YDOT(4))
C
C
C
FROM EQUATION 61 USIING EQUATION 58, PAGE
$
C
C
C
13.
YDOT(1)=-DU12DS+(4.ODO/(BSQ*VSTAR))*(O.5DO*Ul2MUM*ENTRAN
+Y(3)*BSQ*CON1*S2)
FROM EQUATION 58, PAGE 13.
YDOT(2)=(ENTRAN-O.5DO*BSQ*(YDOT(1)+DU12DS))/(VSTAR*Y(2))
C
C
C
FROM EQUATION 59, PAGE
$
$
$
C
C
C
13.
YDOT(3)=(DENGRD*O.5DO*VSTAR*BSQ*S2
-Y(2)*YDOT(2)*Y(3)*SCHMD2*(U12-U12MUM/CON2)
-Y(3)*BSQ*SCHMD2*O.SDO*(DU12DS+(YDOT(1)-DU12DS)
/CON2))/(BSQ*SCHMD2*O.5*(U12-Ul2MUM/CON2))
FROM EQUATIONS 54, PAGE
12.
YDOT(6)=CI*C2
C
YDOT(7)=S1*C2
C
YDOT(8)=S2
C
RETURN
C
C
END
C***************************************************
C
SUBROUTINE PEDERV(N,S,Y,PD)
C
C****************
C
C
C
C
C
C
********************
DUMMY SUBROUTINE TO SATISFY THE EXTERNAL REFERENCE
GENERATED BY SUBROUTINE 'DGEAR'.
C******************t********************
-220-
C
C
REAL*8 Y(N),PD(N,N)
RETURN
C
C
C****************
**
END
***
C***
***
**
*******************
C
SUBROUTINE GETAMB(Z,AMBDEN,DENGRD)
C
C
ROUTINE RETURNS THE LOCAL AMBIENT DENSITY (AMBDEN) AND
DENSITY GRADIENT (DENGRD) AT ELEVATION Z. THE RESULTS
ARE FOUND BY LINEAR INTERPOLATION/EXTRAPOLATION OF THE
TABLE OF PROFILE DATA TRANSMITTED IN COMMON/BLK2/.
NOTE THAT NPTS MUST BE GREATER THAN OR EQUAL TO 2.
C
C
C
C
C
C
C
C**
**
********
*******
*******
C
c
IMPLICIT REAL*8 (A-H,O-Z)
COMMON/BLK2/DP(OO1),DENPRO(100),NPTS
C
C
IF(Z.LE.DP(1)) GO
IF(Z.GE.DP(NPTS))
DO 10 I=2,NPTS
IF(DP(I).EQ.Z) GO
IF(DP(I).GT.Z) GO
10 CONTINUE
GO TO 30
TO 20
GO TO 30
TO 40
TO 50
C
20 DENGRD=(DENPRO(2)-DENPRO(1))/(DP(2)-DP(1))
AMBDEN=DENPRO(I)+DENGRD*(Z-DP(1))
RETURN
C
30 DENGRD=(DENPRO(NPTS)-DENPRO(NPTS-1))/(DP(NPTS)-DP(NPTS-1))
AMBDEN=DENPRO(NPTS)+DENGRD*(Z-DP(NPTS))
RETURN
C
40 IF(I.EQ.NPTS) GO TO 30
AI=(DENPRO(I+1)-DENPRO(I))/(DP(I+I)-DP(l))
A2=(DENPRO(I)-DENPRO(I-1))/(DP(I)-DP(I-1))
DENGRD=0.5DO*(AI+A2)
AMBDEN=DENPRO(I)
RETURN
C
50 DENGRD=(DENPRO(I)-DENPRO(I-1))/(DP(I)-DP(I-1))
AMBDEN=DENPRO(I-I)+DENGRD*(Z-DP(I-1))
RETURN
-221-
C
C
C**********************************
***
END
C***************************************
C
C
C
C
C
C
THE ASPACT FACTOR HAS BEEN INTRODUCED TO
ACCURATELY PREDICT ENTRAINMENT WHEN AN
EQUIVALENT SOURCE IS USED TO REPRESENT
MULTIPLE SOURCES.
10
20
FUNCTION ASPFAC(Y2,RD1,RD2,AF1,AF2)
IMPLICIT REAL*8 (A-H,O-Z)
IF (Y2.GT.RDI) GO TO 10
ASPFAC=AF1
RETURN
IF (Y2.GE.RD2) GO TO 20
S=(AF2-AFI)/(RD2-RDI)
ASPFAC=AF2+(Y2-RD2)*S
RETURN
ASPFAC=I.ODO
RETURN
END
C
C***************************************
C
C
C
C
C
C
THE DENSIT SUBROUTINE ALLOWS TEMPERATURE
AND SALINITY DATA TO BE INPUT TO THE
PROGRAM INSTEAD OF DENSITY. A MATTER
OF CONVENIENCE FOR THE USER.
FUNCTION DENSIT(SAL,T)
SIGO=(((6.8E-6*SAL)-4.82E-4)*SAL+.8149)*SAL-.093
C=1.E-6*T*((.01667*T-.8164)*T+18.03)
D=.001*T*((.0010843*T-.09818)*T+4.7867)
SUMT=(T-3.98)*(T-3.98)*(T+283.)/(503.57*(T+67.26))
SIGMAT=(SIGO+.1324)*(1.-D+C*(SIGO-.1324))-SUMT
DENSIT=1.ODO+(SIGMAT*O.OO1DO)
RETURN
END
C
C
C
C
*******************************
SUBROUTINE ARRAY (W.H,THET1,THET2,X,Y,Z)
IMPLICIT REAL*8 (A-H,O-Z)
PAI=3.1415927DO
DTOR=PAI/180.ODO
C
C
C
C
C
C
ARRAY GENERATES THE DATA POINT ARRAYS
USED TO PLOT THE PLUME PROFILES
AND CENTERLINES IN BOTH THE Z-Y
AND X-Y PLANES.
CLX=X
I,
-222-
CLY=Y
CLZ=Z
ALPHA =90.ODO-THET2
GAMMA=90.ODO-THET1
DELY=H*DCOS(ALPHA*DTOR)/2.ODO
DELZ=H*DSIN(ALPHA*DTOR)/2.ODO
PYI=CLY-DELY
PZ1=CLZ+DELZ
PY2=CLY+DELY
PZ2=CLZ-DELZ
WRITE (17.1) CLY,CLZ,PY1,PZI,PY2,PZ2
C
C
DELY=W*DSIN(GAMMA*DTOR)/2.ODO
DELX=W*DCOS(GAMMA*DTOR)/2.ODO
PXI=CLX+DELX
PYI=CLY-DELY
PX2=CLX-DELX
PY2=CLY+DELY
WRITE (18,1) CLX,CLY,PXI,PYI,PX2,PY2
1 FORMAT (6F10.3)
RETURN
END
-223-