&EPA
United States
Environmental Protection
Agency
Environmental Research
Laboratory
Narragansett Rl 02882
EPA 600 3-85 073a
November 1985
Research and Development
Initial Mixing
Characteristics of
Municipal Ocean
Discharges:
Volume I.
Procedures and
Applications
EPA/600/3-85/073a
r r r
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EPA-600/3-85-073a
November 1985
INITIAL MIXING CHARACTERISTICS OF MUNICIPAL OCEAN DISCHARGES
VOLUME I - PROCEDURES AND APPLICATIONS
by
W.P. Muellenhoff, A.M. Soldate, Jr., D.J. Baumgartner
M.D. Schuldt, L.R. Davis, and W.E. Frick
PACIFIC DIVISION
ENVIRONMENTAL RESEARCH LABORATORY, NARRAGANSETT
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
NEWPORT, OREGON 97365
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The information in this document has been funded by the United States
Environmental Protection Agency Office of Research and Development, and
the Office of Marine and Estuarine Protection through contract numbers
68-01-5906 and 68-01-6922 to Tetra Tech, Inc. Agency Project Officers
for these contracts are Dr. John Pai and Mr. Barry Burgan, respectively.
The report has been subject to the Agency's peer and administrative review,
and it has been approved as an EPA document.
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FORWARD
A portion of this document is based on an earlier version by A.M. Teeter
and D.J. Baumgartner (1979), which it supersedes. The technical reviews
and resultant suggestions of A.R. Agg, W.A. Faisst, Irwin Haydock and P.J.
Roberts resulted in many improvements and are gratefully acknowledged.
We are also thankful to S.J. Wright, V.H. Chu, and other scientists who
have indirectly contributed to the report in the form of fruitful dialogue
during its development. Their continued inputs are encouraged, and will
ensure timely publication of addenda and further improvements in future
editions.
W.P. Muellenhoff is the Director, Corvallis Office, Tetra Tech, Inc.,
and A.M. Soldate, Jr. is a Senior Scientist in Environmental Systems Engineering
at Tetra Tech, Inc., Bellevue, WA. D.J. Baumgartner, M.D. Schuldt, and
W.E. Frick are with the U.S. Environmental Protection Agency, Pacific Division
(Newport, OR). L.R. Davis is Professor, Mechanical Engineering Department,
Oregon State University.
Users of this document or the models described herein are encouraged
to report any errors to enable appropriate corrections to be made. Direct
all correspondence to D.J. Baumgartner, U.S. Environmental Protection Agency,
Hatfield Marine Science Center, Newport, Oregon 93765. Holders of the
document should notify the above to receive errata or future revisions
to the document.
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ABSTRACT
This report describes the behavior of plumes generated when wastewater
is discharged at depth into waters of greater density. Volume I contains
analytical solutions and descriptions of five mathematical models that
provide the initial dilution and rise-height of the plume for a variety
of discharge, diffuser, and receiving water characteristics. Initial dilution
is defined as the flux-average dilution when the rising plume reaches an
equilibrium level or encounters the surface. Guidance is provided for
the range of values within which analytical solutions provide acceptable
estimates. Use of the models is recommended for conditions outside these
ranges and for detailed analysis. The format of model input data is the
same for all five computer programs. As an option, the user may interact
(via a terminal) with the models, changing one or more discharge parameters
while holding the others constant and rerun the model without reentering
existing ambient data. Any number of data sets may be stacked and all
the programs have a subroutine (LIMITS) to check that certain input data
are within prescribed limits. Example problem calculations are provided
for each model. Complete program listings in FORTRAN IV-PLUS are provided
in Volume II.
Volumes I and II are available in hardcopy from the National Technical
Information Service (5285 Port Royal Road, Springfield, VA, 22161; 703-
487-4650). Volume II is also available from NTIS on a 9-track tape or
diskette (703-487-4763). The IBM-PC compatible diskette has programs slightly
altered to compile using Microsoft FORTRAN (Version 3.1 or higher) or IBM
Personal Computer Professional FORTRAN (8087 or 80287 chip required).
1v
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CONTENTS
Page
Forward iii
Abstract iv
Figures vii
Tables vii
Output Exhibits viii
Glossary ix
SECTIONS
1. INTRODUCTION 1
Mixing Zone Concepts 2
Ocean Discharges 2
Buoyant Plume Models 4
Report Organization 7
2. INITIAL DILUTION 8
Methods 8
Appropriate Conditions 8
Mixing Zone Specification 11
3. MODELING PROCEDURES 13
Approach 13
Analytical Solutions 14
Dilution/Equilibrium Height Relationship 14
Single Plume, Stagnant Ambient 15
Single Plume, Flowing Ambient 18
Merging Plumes, Stagnant Ambient 22
Merging Plumes, Flowing Ambient 25
Approximations for Nonlinear Stratification 28
Other Diffuser Configurations 28
4. NUMERICAL MODEL DESCRIPTIONS 30
Introduction 30
UPLUME 30
UOUTPLM 37
UDKHDEN 38
UMERGE 43
ULINE 45
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5. MODEL EXECUTION 49
Introduction 49
Universal Data File Description 51
Discharge Port Spacing 55
Example Universal Data File 57
Resultant Model Output 58
Batch Processing 60
REFERENCES 78
APPENDICES
I. Development of Sa and h Relationship 85
II. UDF Format 89
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FIGURES
Number Page
1 Buoyant plume trajectory in an unstratified crossflow 20
2 Experimental measurements of minimum surface dilution for
a finite line source of buoyancy flux in a current 27
3 Cross section and profile along connecting line of merging
pi umes 42
TABLES
Number Page
1 Summary of Numerical Model Characteristics 31
2 Universal Data File Parameters Required by the Computer Models 62
3 Output Parameters for UPLUME 63
4 Output Parameters for UOUTPLM 67
5 Output Parameters for UDKHDEN 71
6 Output Parameters for UMER6E 73
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OUTPUT EXHIBITS
Number Page
1 UPLUME output for IPI=IPO=0 64
2 UPLUME output for IPI=IPO=1 65
3 UPLUME output for IPI=IPO=2 66
4 UOUTPLM output for 101=100=0 68
5 UOUTPLM output for 101=100=1 69
6 UOUTPLM output for 101=100=2 70
7 UDKHDEN output, no option 72
8 UMERGE output for IMI=IMO=0 74
9 UMERGE output for IMI=IMO=1 75
10 UMERGE output for IMI=IMO=2 76
11 ULINE output for INTER=1 and 1X1=1X0=2 77
vm
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model output descriptions)
y Horizontal distance 900 to x (see model output descriptions)
z Vertical distance upward from the discharge point or downward from
the surface (see particular model output descriptions)
6 Characteristic radius or half-width of the plume = W/2
A Normalized density disparity (P.-Pj/fpQ-Pd)
A0 Initial density disparity of the waste = P0-P(j
p Average density of the plume, or UMERGE element average density
PO Ambient density at the level of the discharge, or at the UMERGE
element boundary
Pd Density of the discharge
P! Centerline density at the end of the zone of flow establishment
P^ Ambient density at some level
PS Ambient density at the surface
dp/dz Ambient density gradient
XI
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SECTION 1
INTRODUCTION
Initial dilution is considered to be the rapid turbulent mixing which
occurs between wastewater discharged at depth and the surrounding seawater
resulting from the jet velocity at the point of release, driven by momentum
and buoyancy in the plume relative to the ambient. The understanding of
this mixing process can be used to predict the initial dilution attained
under given conditions. Such predictions are used to design ocean outfalls,
i.e., to select the number, spacing, size, and orientation of ports, and
the depth of the discharge so that water quality criteria are met following
initial dilution.
This report presents procedures for calculating the initial dilution
and for describing the zone of initial dilution near a discharge site.
Such calculations are required under U.S. Environmental Protection Agency
regulations effective December 27, 1982 (U.S. EPA 1982), that implement
Section 301(h) of the Clean Water Act (PL 97-117).
Two technical monographs provide detailed introductions to the general
areas of marine outfalls and environmental hydrodynamics. The book of
Grace (1978) discusses the characteristics of effluent wastewater; biological,
chemical, and physical oceanographic factors which influence the engineering
design of an outfall; and the construction and maintenance of an outfall.
The general physics of mixing in rivers, reservoirs, estuaries, and coastal
waters is explained in the work of Fischer et al. (1979). Useful discussions
relevant to the papers described in the present report include theoretical
and experimental bases of methods employed in determining the behavior
of turbulent jets and plumes, and the (hydraulic) design of ocean outfalls.
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MIXING ZONE CONCEPTS
When municipal wastewater is injected into the ocean, buoyancy and
momentum combine to form a plume of increasing size as it rises toward
the surface entraining seawater as a function of distance traveled. The
entrainment process slows markedly when plumes reach a position of neutral
buoyancy with respect to the ambient or when the plume surfaces. The dilution
process subsequently becomes more dependent on ambient oceanic processes.
Thus, the zone where rapid mixing takes place initially between the waste
stream and the ambient can be physically distinguished from the zone where
subsequent ambient conditions influence dilution. For purposes of regulating
discharges under Section 301(h), this zone of rapid mixing is approximated
by a defined "Zone of Initial Dilution". Water quality criteria should
be met and water quality standards must be met outside this zone. Concen-
trations of those pollutants identified in the waste might then exceed
water quality criteria within the initial dilution zone for a time which
varies depending on the oceanographic and discharge factors influencing
plume formation. This has been found to be on the order of several minutes
for municipal discharges in the coastal waters of the United States. Marine
organisms entrained in the plume, or passing through it, would thus be
exposed to concentrations exceeding the level outside this zone for only
a few minutes. Under these exposure conditions, the marine uses to be
protected by water quality standards based on soluble concentrations are
adequately addressed. Since some pollutants are not permanently dispersed
by the initial dilution process, e.g. accumulation in surface films or
sediments, and some adverse biological impacts may occur in spite of large
initial dilution factors, e.g. bioaccumulation in organism tissues, biological
and chemical tests of impacts within and beyond the zone of initial dilution
are employed in the 301(h) process in addition to water quality criteria
following initial dilution.
OCEAN DISCHARGES
Marine outfalls designed to discharge municipal wastes have a wide
variety of physical characteristics which can affect initial dilution.
Although single port outfalls are still used, many outfalls now have multiple
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ports to optimize the vertical rise and dilution attained. The size and
number of ports, and their orientation, can significantly influence discharge
patterns and achievable dilutions. The discharged material is relatively
uniform, having physical characteristics similar to freshwater. The marine
environment, however, varies greatly along the coastline and the physical
conditions at the discharge site have a significant effect on initial dilution.
The ambient receiving waters are of a higher density than the waste discharge
and are frequently vertically stratified. Currents are usually present,
and may have a tidal periodicity.
Discharges typically have fluxes of both momentum and buoyancy. The
densimetric Froude number describes the ratio between the inertial and
gravitational forces. The higher the initial Froude number, the more closely
the resulting plume resembles a momentum jet. The smaller the number,
the more the plume resembles a purely buoyant plume. While the differences
influence the rate of dilution and the trajectory of the forced plume which
is formed, there is not a great difference in the appearance of the plume.
Because the density difference between the waste and the ambient varies
only slightly and jet velocity is bounded by practical considerations,
Froude numbers for municipal ocean discharges generally vary between 10
and about 30 (Grace, 1978). The ends of this range are characteristic
of single-port and multi-port diffusers, respectively. In this range of
Froude numbers, buoyancy is likely to dominate the initial mixing process,
making volume rate of discharge, currents, ambient stratification, and
possibly water depth, important. Especially in a current, buoyancy dominates
since the plume's momentum becomes insignificant compared to the momentum
entrained from the ambient.
Existing coastal discharge sites vary in depth up to about 75 m (246
ft). The water columns at these sites are typically stratified due to
vertical variations in temperature and/or salinity. This leads to density
stratification that varies widely in time and space. The proximity of
large fresh water sources, surface heating, upwelling, and wind mixing
all can influence density stratification. Also, salt wedges which typically
occur near the mouths of estuaries having large freshwater flows cause
large vertical and horizontal density gradients. Advection and nonlinear
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interactions between turbulent surface waters and deeper stratified waters
also can produce pycnoclines or areas of pronounced density gradients.
Winds of sufficient strength and duration can erode the density gradient
by the turbulence they produce, leaving the ambient well mixed and allowing
buoyant plumes to rise to the surface.
Currents are very complex in coastal areas where high energy is present
from a number of processes. Nearshore currents at discharge sites may
have tidal components which can be strong. Oceanic currents impinge on
the coast and intensify on the western edges of the oceans (e.g., Florida
Current). Local forces such as wind shear and waves also generate currents.
Instantaneous current values (time scale of minutes), rather than long-term
net currents (time scale of hours, days, etc.), are important in the initial
dilution process.
BUOYANT PLUME MODELS
Ocean outfalls can be modeled by properly scaled hydraulic models,
or by mathematical models. In some cases involving complex geometry or
other conditions which cannot readily be incorporated into mathematical
models, hydraulic modeling may be appropriate. Mathematical models have
been used to study the characteristics and behavior of plumes and jets
in both marine and freshwater settings, and in the atmosphere. Only mathe-
matical models are considered in the remainder of this report.
Early work in this field was concerned with convection of purely buoyant
plumes having no initial momentum. Rouse et al. (1952) studied the buoyant
plume above a continuous heat source in an unstratified (uniform), stagnant
environment. Equations of mass continuity, momentum, and energy were integrated
by assuming Gaussian approximations for the lateral distributions of velocity
and temperature across the plume and with experimentally determined spreading
coefficients. The results were used to describe the distribution of velocity
and temperature as functions of mass flow and height of rise. Priestly
and Ball (1955) studied thermal plumes issuing into a stratified, as well
as uniform, environment. They used essentially the same equations as Rouse
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et al . (1952) for mass, momentum, and energy, but their lateral Gaussian
profiles of temperature and velocity assumed a lateral length scale.
Morton et al . (1956) also studied buoyant plume convection. They
introduced an entrainment function based on local characteristic plume
velocity and length scale to replace the spreading ratio. This approach
has been adapted by many subsequent investigators (Baumgartner and Trent
1970). Later, Morton (1959) assumed other lateral profiles of velocity
and temperature to account for the difference between eddy transport of
heat and momentum. The value of this coefficient was estimated from experi-
mental data.
Abraham (1963) investigated horizontally and vertically oriented jets
discharging into stagnant homogeneous and stratified fluids. Equations
of mass, momentum, and energy were integrated using lateral Gaussian profiles
recognizing the difference in transfer rates between momentum and heat
(or concentration). Entrainment was considered by integrating the continuity
of mass equation with appropriate spreading coefficients. Abraham's results
included analytical expressions for plume centerline velocity and concentration
as a function of initial plume characteristics, height of rise, and entrainment
coefficients for momentum and heat.
Fan (1967) reported on plumes issuing at arbitrary angles into a stagnant,
stratified environment. Velocity and buoyancy profiles were similar to
those used by Morton (1959) and included a lateral characteristic length.
Solutions were found by simultaneously solving six equations (continuity
of mass, vertical momentum, horizontal momentum, conservation of buoyancy,
vertical geometry, and horizontal geometry).
Fan (1967) also investigated buoyant plumes in a flowing, uniform-
density ambient. A theoretical solution was given as a function of plume
drag forces and an entrainment function. The entrainment function was
such that drag and entrainment coefficients varied with plume conditions,
a disadvantage in applications beyond the range of experimental validation
(Abraham 1971). Abraham introduced an entrainment function based on subdomains
influenced by initial momentum and buoyancy. With entrainment coefficients
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defined for these two subdomains and a single coefficient describing the
drag force, the theoretical solution was in good agreement with the experimental
results of Fan. More recently, Chu (1979) and Wright (1984) have discussed
buoyant plume dilutions and trajectories in cross flows in both uniform
and stratified ambients.
Plumes from multi-port discharges have been studied less extensively.
Pearson (1956) considered flow from a line diffuser in a stagnant medium
in which merging of the individual jets had occurred. Liseth (1970, 1976)
did experimental work on merging round buoyant jets from a row of ports
in a line diffuser. Koh and Fan (1970) proposed a mathematical model of
multi-port diffusers by interfacing single jet and slot jet solutions at
a transition point. Cederwall (1971) studied the flow regimes of line
sources for discharges in confined and unconfined environments. Sotil
(1971) proposed a mathematical model for slot jets or continuous line sources
in stagnant, stratified ambients. Kannberg and Davis (1976) examined the
mixing characteristics of a multiport, submerged, thermal diffuser discharging
into a uniform ambient as a function of port spacing, discharge Froude
number, discharge angle and discharge-to-ambient velocity ratio. More
recently, Roberts (1977; 1979a,b) examined line plumes discharging into
steady currents. A number of excellent plume modeling reviews have been
published (Briggs 1969; Baumgartner and Trent 1970; and more recently Davis
and Shirazi 1978) and Roberts (1983, 1984, 1985) have summarized recent
applicable modeling efforts.
Five computer models are described in this report. They include modified
versions of PLUME, OUTPLM, and DKHPLM described in Teeter and Baumgartner
(1979), and two additional models entitled UMERGE and ULINE. All of the
models accept a variety of density gradients (i.e., zero, linear, or non-
linear). The model PLUME by Baumgartner et al. (1971) simulates a solitary
plume in a stagnant environment. OUTPLM by Winiarski and Frick (1976)
also models a single plume, but in either a stagnant or uniformly flowing
ambient. A revised version of DKHPLM by Davis (1975) describes single
plumes which are allowed to merge with identical adjacent plumes in either
stagnant of flowing environments with a variety of velocity profiles (i.e.,
zero, constant, or varying with depth). UMERGE, based on the work of Frick
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(1981) also accounts for interference of adjacent plumes for a variety
of current speeds. ULINE, based on Roberts (1977; 1979b) models a line
source of finite length or closely spaced ports in a current flowing at
an arbitrary direction and speed. Complete FORTRAN IV-PLUS program listings
of each of the five models are available in Volume II of this report.
To simplify the use of these models, all have been tailored to read
a single input file termed the Universal Data File. To indicate this modifi-
cation, the names of the models have been preceded with the letter U.
Hence the model designated in earlier work as PLUME is referred to in this
report as UPLUME. In addition, DKHPLM has been changed to accept density
and current profiles, and is renamed UDKHDEN.
Simplified analytical solutions are provided as they were in Teeter
and Baumgartner (1979). However, they have been revised to reflect published
experimental data. Dilution and trajectory equations are provided for
single and merging forced plumes in both stagnant and flowing environments
which have no density gradient or are linearly stratified. Approaches
to solving equations for non-linear stratifications are cited. Example
calculations are included for most conditions to assist the user in performing
similar determinations. The analytical solutions are useful in situations
where the models are not available, or where it is impractical to run a
computer model.
REPORT ORGANIZATION
Section 2 presents general methods for determining plume initial dilutions,
approaches to defining the critical minimum initial dilution and mixing
zone concepts. Plume modeling parameters are presented in Section 3, followed
by analytical solutions for selected discharges and receiving water conditions.
Five numerical models are described in Section 4, and Section 5 is devoted
to an explanation of the Universal Data File and numerical model execution.
Required input data parameters are summarized along with tables of output
parameters and a test run printout for each. The appendices contain a
detailed description of the Universal Data File and the development of
average dilution and height of rise relationships.
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SECTION 2
INITIAL DILUTION
METHODS
The buoyant-plume phase of waste dispersion can be described by a
variety of published methods. However, to evaluate the water quality impacts
of municipal ocean discharges, the methods presented here are well suited
and reconmended for consistency. Some of these may be applicable to streams,
rivers and lakes, although emphasis herein is on ocean discharges. Application
of the models to these other environments may require additional caveats.
For most cases, either analytical methods or computer models can be used.
When there is uncertainty about the influence of simplifying assumptions,
or when more detail is required, the computer models should be used. For
unusual situations or conditions, other methods such as physical models
should be considered. Although these models are felt to be reliable, they
should be continually evaluated relative to theoretical developments and
especially, quality field measurements.
APPROPRIATE CONDITIONS
Dilution is herein defined as the total volume of a sample divided
by the volume of effluent contained in it. The dilution achieved during
the initial mixing process is dependent on ambient and discharge conditions
and is, therefore, highly variable. To prevent or minimize biological
effects, occurrences of pollutant concentrations greater than limiting
water quality criteria must be avoided. In evaluating a discharge's effect
on water quality, therefore, the appropriate conditions to consider are
those which result in the "lowest" dilution and those which occur at times
when the environment is most sensitive. For example, minimum dilution
can be predicted using a combination of maximum vertical density stratification,
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minimum initial density difference between the effluent and the ambient
seawater, maximum waste flow rate, and minimum currents for a particular
site. Other situations may be more critical depending on the ambient water
quality, and applicable criteria.
Predicting dilution reliably depends on the availability of statistically
valid data with which to estimate ambient conditions. The statistical
uncertainty in estimates of absolute worst case conditions is generally
great. Also there are inherent biases to some oceanographic measurements.
For example, current measuring instruments have finite thresholds. It
therefore becomes difficult to distinguish low values (which may be as
high as 5.0 cm/sec) from zeroes in these data sets. In estimating environmental
conditions, a more reliable estimation can be made at the lowest 10 percentile
on a cumulative frequency distribution. Data on ambient density structure
are not routinely collected. Consequently, there is not usually an existing
data set for the site under consideration. To increase the reliability
of "worst-case" estimates, data should be evaluated not only for the discharge
site but for nearby coastal areas of similar environmental setting.
Defining "worst-case" conditions as a combination of those conditions
affecting initial dilution, each taken at the worst 10 percentile on cumulative
frequency distributions, is recommended (Tetra Tech 1982). This approach
allows a reliable estimation of these conditions to be made and prevents
the unlikely occurrence of more extreme conditions from biasing the predic-
tions. The probability of these conditions occurring simultaneously is
much less than 10 percent, ensuring that the predicted dilution will be
exceeded most of the time. Application of multiple "worst case" factors
(i.e. flows, stratification and currents) to determine a minimum dilution
must be done carefully, however, and in recognition of the criteria for
which compliance is being determined. For example, although application
of an absolute "worst case" dilution may be appropriate for determining
compliance with an acute toxicity limit, it is more appropriate to identify
the lowest 6-month median dilution to determine compliance with a 6-month
median receiving water limitation.
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To determine initial dilutions it is necessary to know specific charac-
teristics of the discharge, the outfall and the receiving waters. The
discharge volumetric flow rate and density are required. Alternately the
effluent temperature and salinity (major inorganic ions contributing to
density) can be used to estimate a density,based on known relationships
for seawater (U.S. Navy Hydrographic Office 1952). Municipal effluent
densities typically range from 0.9970 to 1.0003 g/cm3, and salinities range
to 5 ppt. The highest 2-3 hour flow rate during a period of concern should
be used to calculate the minimum initial dilution for that period. Data
from the last 2 years or longer should be used to ensure that the flows
are representative. The flow from each port, which is not necessarily
uniform, can be determined from an evaluation of manifold hydraulics.
If the flow distribution among all ports is relatively uniform, the total
outfall flow divided by the number of ports can be used as the representative
per-port flow rate. Relevant diffuser characteristics include number of
ports, size, spacing, angle of discharge, and depth.
In running the models, the port diameter specified should be the effective
diameter, reflecting the effects of the orifice on the contraction of the
jet. This effective diameter can be specified in terms of an appropriate
discharge coefficient and true port diameter (Fischer et al. 1979).
The principal environmental quantities to consider in dilution prediction
are the ambient density stratification and local currents. These parameters
should be considered for periods of maximum wastewater flow, any other
periods of maximum loadings (e.g. canning seasons), times of seasonal maximum
and minimum stratification, low ambient water quality, low net circulation
or flushing and exceptional biological activity. The quantities selected
to represent these periods should reflect lowest 10 percentile conditions.
Current speed data usually consist of discrete values and can be ranked
into cumulative frequency distributions to select the 10 percentile design
current.
The worst stratification is that which results in the lowest dilution
if other conditions are constant. If the density gradients are uniform
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with depth they can be ranked numerically. If not, measured profiles can
be input to the computer models and the results used to rank the profiles.
MIXING ZONE SPECIFICATION
After initial dilution, the concentrations of waste constituents (Cf)
are a function of the average dilution achieved (Sa) and their concentrations
in the ambient (Ca) and the effluent (Ce)
°f - Ca + (Ce-Ca)/Sa (1)
If the effluent has been adequately treated and disposed of in an
environmentally appropriate area, the final concentrations of various con-
stituents should comply with applicable quality criteria.
The zone surrounding the discharge site which geometrically bounds
the critical initial dilutions is termed the zone of initial dilution (ZID)
to distinguish it from other mixing zone definitions. It defines, theo-
retically, a concentration isopleth. Thus, there would be a discrete ZID
for each density and current velocity profile at each site. In practice,
the ZID defined for Clean Water Act Section 301(h) purposes is regularly
shaped (e.g., circular, rectangular or "Y" shaped) to encompass the set
of theoretically calculated dimensions. The ZID does not attempt to describe
the area bounding the entire initial mixing process for all conditions
(e.g. high currents and low stratification) or the area impacted by the
sedimentation of particulate organic material.
Within the ZID, concentrations of pollutants in the water column may
exceed water quality criteria. There will be times when dilution will
be much higher than calculated for critical conditions, and consequently
water quality may be met within the ZID. Beyond the ZID boundaries water
quality standards are expected to be met essentially all the time. If
biological impacts are detected beyond the ZID they would not be expected
to have been due directly to water column concentrations. Since the models
do not attempt to predict physical, chemical, and biological accumulation
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of constituents following initial dilution, other methods must be included
in a complete evaluation of biological impact beyond the ZID boundary.
These methods should account for seabed accumulation of particulates, surface
film concentration, and bioconcentration in tissues of marine organisms.
If potential problems are identified with these methods, additional initial
dilution may be required although additional treatment or pre-treatment
control may be much more effective.
The ZID dimensions and location are defined to establish a sampling
perimeter at which adherence to water quality criteria is to be evaluated
through monitoring. These dimensions can be specified by analyzing model
results for a range of critical conditions. However, it can be simply
approximated using the height of rise predicted for the critical conditions
as a radial distance measured horizontally from the outfall diffuser or
port. This distance will often equal the depth of water at the discharge
site. During periods of higher currents, the plume will be carried further
horizontally and initial dilutions will be higher than predicted for the
critical current conditions. The dilution achieved over that portion of
the trajectory within the ZID, however, will be approximately equal to
the initial dilution predicted for the critical conditions. The U.S. EPA
accepts ZID dimensions equal to the water depth from any point of the diffuser,
provided these do not violate mixing zone restrictions in applicable water
quality standards (Tetra Tech 1982).
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SECTION 3
MODELING PROCEDURES
APPROACH
The plume attributes of primary interest are the average dilution,
position, and dimensions at the equilibrium level or end of convective
ascent through the water column, whichever occurs first. For evaluation
purposes, it is important to identify the lowest flux-average dilution
and trapping depth anticipated during critical periods, which in turn serve
as input to analyses of immediate and farfield impacts on water quality
and biota.
Behavior of buoyant plumes can be mathematically modeled by properly
considering mass, momentum, energy, and a scalar variable (e.g., salt)
(Hirst 1971a). A form of entrainment function must be assumed and fitted
to experimental data. Other assumptions generally made are that flows
are steady and incompressible, pressure is hydrostatic throughout, the
plume is fully turbulent and axisymmetric, and turbulent diffusion dominates
and is significant only in the radial direction. Distributions of velocity
and concentration may also be assumed. Plume solutions can be obtained
in various ways. Often, systems of differential equations are integrated
across the plume to reduce the variables to a single independent one, namely
arc length along the plume axis.
Mathematical models of jet discharge are systems with internal variables
(mass, momentum, and energy), external variables (discharge characteristics,
ambient vertical density, and currents), and boundary mechanisms (entrainment).
Equations can be used to describe the resultant plume's behavior in terms
of principal quantities. For stagnant conditions, the principal quantities
are initial density difference between the waste and ambient (AO) , ambient
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density gradient (dP/dz), and flow rate per port (Q) or flow rate per unit
length (q). Current speed (U) , if considered, is also a principal quantity.
ANALYTICAL SOLUTIONS
The remainder of this chapter is devoted to presenting equations that
can be used to approximate the plume average dilution and rise height,
knowing selected characteristics of the discharged waste, the diffuser,
and the receiving environment. Environmental conditions addressed include
single and merging plumes, in stagnant and flowing environments which are
linearly stratified or have no density gradient. Approximations for nonlinear
stratifications are also discussed. The objective of this chapter is to
provide practical equations for typical ranges of parameters of interest.
Example calculations are included to assist the user in performing similar
calculations.
The equations provide approximate solutions in lieu of running a model.
However, for more exact solutions, use of one or more of the models described
in Section 4 is encouraged. Use of the Universal Data File allows the
running of several models for a particular input data set, or multiple
runs with any one model to examine the effects of input parameter variation.
Dilution/Equilibrium Height Relationship
As developed in Appendix I, the following general relationships apply
to cross-sectional average plume dilution (Sa) and the equilibrium height
of rise (h) of a buoyant plume (i.e., negligible initial momentum) in receiving
waters with a linear density gradient.
Given that
Sa = Bz" (2)
then
h = {[(n+lJ/BKgd'/G]} (3)
14
-------
where
6 = -(g/P0)(dP/dz) (4)
and
9d' -
Although authors of reviewed works present dilution and equilibrium height
equations in a variety of forms, the above forms are generally used throughout
this section. Where another author's work is referenced, equations will
have been transformed to these forms to enable comparison of equations
for the various ambient conditions and port configurations considered.
As the density gradient decreases, so also does the parameter G, resulting
in an increasing equilibrium height of rise, h. It is at this height that
the average density of the plume is equal to the density of the surrounding
water. The value of h calculated with equation (3) cannot exceed the water
depth. For a surfacing plume, the average dilution can be estimated using
Equation (2), with the water depth H reduced to account for the thickness
of the waste field.
Single Plume, Stagnant Ambient^
The plume average dilution as a function of elevation above the discharge
point, z, for a single port discharging at an arbitrary angle into a quiescent
unstratified environment can be calculated using the following (based on
Brooks 1973)
Sa = 0.155 gd'1/3Q-2/3z5/3 (6)
From this expression, 3 is 0.155 gd'1/3Q~2/3 and n=5/3.
Substituting into Equation (3) results in
15
-------
h = 2.91 gd'1/4Q1/4G-3/8 (7)
In deriving equation (7), the entrainment given by equation (6) is assumed
to apply to both unstratified and stratified environments.
For unstratified conditions, the rising plume is deflected horizontally
upon nearing the ocean surface. Compensation for this effect should be
taken into account. Because the extent of any further dilution within
the trapped wastefield is not well documented, dilution can be estimated
by using an effective distance over which dilution is occurring equal to
the full water depth minus the vertical thickness of the surface waste
field.
Brooks (1973) suggests a correction in H of approximately one quarter
the plume diameter, or 0.07z. Lee and Jirka (1981) indicate that, for
large values of the water depth to port diameter ratio (i.e., >10), the
waste field thickness is approximately 0.08 times the water depth for a
vertical round buoyant jet. Frankel and Gumming (1965) reported a surface
field thickness of 0.25 times the water depth for a horizontal round buoyant
jet near the bottom. Fan and Brooks (1966) report that, in most cases,
the surface waste field thickness is considerably less than one-fourth
the water depth. They also report that the surface transition zone dimension
is not a simple proportion of the water depth, but also of rising plume
parameters (e.g. z/D, F), the discharge angle, and the character of the
horizontal surface flow layer. An important parameter that may be dominant,
but has not been mentioned in the papers reviewed, is the magnitude of
the residual buoyancy possessed by a plume as the surface is reached.
A large difference in buoyancy between the plume and the receiving water
will force formation of a thinner layer than would a small difference.
16
-------
In the absence of a clearly definitive value for the surfaced waste
field thickness, a nominal value of one-tenth the water depth is used herein.
Therefore equation (6) becomes
Sa = 0.155 gdll/3Q~2/3(H-0.10H)5/3 (8)
or Sa = 0.130 gd'1/3(T2/3H5/3 (9)
When the height of rise computed using equation (7) reaches or exceeds
0.9H, then equation (9) should be used.
It should be noted that the equilibrium height of rise in UPLUME is
the same as h in equation (3), namely the height in the water column where
the average plume density equals the density of the surrounding ambient.
UPLUME does, however, contain an algorithm to correct for a finite plume
thickness of 0.1H at the surface. The earlier version of the model, PLUME,
did not contain this correction.
Additional references for single buoyant jets in quiescent stratified
environments include List (1982), Hofer and Mutter (1981), Henderson-Sellers
(1978), Baines (1977), Cederwall (1975, 1968), Koh and Brooks (1975), Fox
(1970), Abraham and Eysink (1969), Fan (1967), Abraham (1963), Hart (1961),
Morton (1959), and Morton et al. (1956). References for single buoyant
jets discharging into quiescent unstratified environments include Lee and
Jirka (1981), Chen and Nikitopoulos (1979), Kotsovinos (1978), Schau (1978),
Abraham (1978, 1963, 1960), Baines (1977), Cederwall (1975, 1968), Morton
(1959), Morton et al. (1956), and Rouse et al. (1952).
In the following calculations, port spacing is assumed to be sufficiently
large to preclude merging of adjacent plumes. Verification of noninterference
can be made by estimating the plume half-widths at the equilibrium point
or at the surface as appropriate, using the methods described in Fan and
Brooks (1969), or in Brooks (1973).
17
-------
Example--
A 50-port diffuser has a total flow rate of 2.19 m3/Sec (50 MGD) and
a density of 0.9995 g/on3. The ambient at the discharge site has a surface
density of 1.0246 g/cm3 and, at a depth of 30.5 m (100 ft), a density of
1.0258 g/cm3. Determine the maximum height of rise and average initial
dilution.
9d' » (9.81)[(1.0258-0.9995)71.0258] = 0.2515 m/sec?
Q = 2.19/50 = 0.0438 m3/sec
6 = -(9.81/1.0258)[(1.0246-1.0258)730.5)] = 3.763 (1Q-4) sec-2
According to equations (6) and (7),
h = 2.91(0.2515)1/4(0.0438)1/4[(3.763)(10'4)]-3/8 = 18.1 m
Sa = 0.155 (0.2515)173 (0.0438)"2/3(18.1)5/3 = 98
For this example, assume that the ambient density gradient is zero.
The surfacing plume average dilution according to Equation (9) is
Sa = 0.130(0.2515)1/3(0.0438)'2/3(30.5)5/3 = 197
Single Plume, Flowing Ambient
The appropriate form of plume initial dilution and trajectory equations
when a crossflow is present depends on whether the plume is momentum- or
buoyancy-dominated at the point of consideration, and whether it is in
the nearfield or farfield region. Wright (1984) provides minimum dilution
and trajectory equations for each of the four possible combinations of
conditions. The subsequent discussion is limited to the buoyancy-dominated
farfield condition most likely to apply to plumes at the point of equilibrium
in a flowing environment.
18
-------
The average dilution Sa can be computed with
Sa = C(U/Q)Z2 (10)
where e=C(U/Q) and n=2, which is analogous to equation (2).
Using the method of Chu (1979, 1985), the average dilution of a plume
in a crossflow, in the buoyancy dominated farfield can be expressed as:
Sa = 0.49(U/Q)z2 (11)
Substituting p=0.49(U/Q) and n=2 into equation (3) gives
h = 1.83[(Q)(p0-pd)/(U dp/dz) ]1/3 (12)
For a stratified flowing ambient, Wright (1984) provides the following
relationship for the equilibrium height of rise
h = 1.85 (U/e1/2)2/3 (B/U3)1/3 (13)
where
e - -(g/P0)dP/dz
B = gd'Q = [g(p0-Pd)/p0] Q
Substituting these identities, Equation (13) can also be written as
h = 1.85 [(Q)(Po-pd)/(u dp/dz) ]1/3 (14)
which is in close agreement with Equation (12).
In a crossflow, a blocking correction for a surfacing plume may be
appropriate (Roberts, P.J.W., 9 October 1984, personal communication).
In the absence of experimental data, it is assumed that the proper elevation
at which dilution should be determined is the plume centerline as the plume
touches the surface. As shown in Figure 1, the plume centerline lies an
19
-------
3
O
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20
-------
amount rs below the water's surface. Chu (1979) gives a linear relationship
between plume width and height z above the discharge point (based on observed
plume dye boundaries), when the exit-to-crossflow velocity ratio exceeds
four
W = 0.68z (15)
or
rs = W/2 = 0.34z (16)
Therefore
z = H - rs = H - 0.34z
or
z = H/1.34 = 0.746H (17)
Substituting this value of z into Equation (11) gives
Sa = 0.27 (U/Q)R2 (18)
Wright (8 April 1985, personal communication) suggests that blocking
effects are much smaller than previously reported. Taking this into account
(i.e., let z=H), using his 0.4 coefficient for axis-of-symmetry dilution,
and an averaging factor of 0.72 results in
Sa = (0.4)(0.72)(U/Q)z2 = 0.29 (U/Q)H2 (19)
which is within 10 percent of Equation (18).
Single buoyant jets discharging into a flowing, linearly stratified
ambient are discussed by Manins (1979), Schatzmann (1978), Henderson-Sellers
(1978), Luti and Brzustowski (1977), Hayashi (1972), and Hirst (1971b).
Additional references for single buoyant jets discharging into an unstratified
21
-------
environment with a significant current include Crabb et al. (1981), Hwang
and Fletcher (1978), Schatzman (1978), Abraham (1978), Abdelwahed and Chu
(1978), Krausche et al. (1978), Baines (1977), Jiji and Hoch (1977), Wright
(1977), Chien and Schetz (1975), Chu and Goldberg (1974), Hayashi (1972),
Hoult et al. (1969), and Fan (1967).
Example--
A 0.5 m3/sec (11.4 MGD) municipal effluent is discharged from a single
port in 50 m (164 ft) of ocean water. The discharge density is 1.000 g/cm3,
and ambient density varies linearly from 1.0260 g/cm3 at the discharge
depth to 1.0240 g/cm3 at the surface. Ambient current speed is 15 cm/sec
(0.49 ft/sec). Determine the plume equilibrium height of rise, and the
average dilution.
Calculate
dP/dz = -(1.0240-1.0260)750 = 4xlO~5 g/cm3/m
Equations (11) and (12) give
h = 1.83[(0.5)(0.026)/(0.15)(4xlO-5)]1/3 = 23.7 m
Sa = 0.49 (0.15/0.5)(23.7)2 = 83
If the receiving water was unstratified (i.e., dp/dz=0), the corresponding
average dilution upon surfacing would be according to Equation (18)
Sa = (0.27)(0.15/0.5)(50)2 = 203
Merging Plumes, Stagnant Environment
At a given height, z, above a diffuser which has ports spaced at an
interval, 1, little merging occurs provided the ratio of z/l<5. As z/1
increases beyond five, there is a significant decrease in the dilution
rate due to merging of adjacent plumes, and a 2-dimensional plume condition
22
-------
is approached. There are two approaches to expressing the results for
an unstratified environment. One explicitly considers the effects of blocking,
whereas the other includes blocking implicitly. If blocking is ignored,
then for an unstratified ambient, the average initial dilution can be expressed
as
Sa = 0.54 (g'/q2)l/3 H (20)
(Brooks 1973; Fischer et al., 1979). However, the average initial dilution
observed at the edge of the zone of initial dilution includes blocking,
which can be taken into consideration by assuming that the waste field
has a finite thickness and that dilution ceases at its lower boundary.
Koh (1983) suggests that blocking is significant and reports that
the thickness for laboratory multi-port diffusers varies from less than
30 to about 40 percent of the water depth, as reported by Liseth (1970),
Buhler (1974), Liu (1976), Koh (1976), and Roberts (1977). Roberts measured
minimum near-surface dilutions whose magnitudes correspond to those calculated
using Equation (20) at a height of 0.7 times the water depth, with the
coefficient decreased by a factor of 1.41 (to 0.38). This suggests that
dilution ceases in the upper 30 percent of the water column. In contrast,
Wright (1985) cites evidence that blocking is minimal, and suggests that
until a more complete analysis is developed, the buoyant jet formula should
be applied all the way to the surface in an unstratified environment, and
to the maximum height of rise in a stratified environment.
Pending resolution of this issue and in the interests of determining
compliance with water quality criteria, the dilutions calculated herein
are conservatively determined at the lower boundary of the waste field
in an unstratified environment, and at the equilibrium height in a stratified
environment. Therefore, a correction to equation (20) can be made as follows
Sa = 0.54 (g'/q2)l/3 (H-Q.3H)
or
23
-------
Sa = 0.38 (g'/q2)173 H (21)
The second approach, which implicitly includes the effects of blocking
(Roberts, P.J.W., 25 July 1985, personal communication), employs the experi-
mental results of Roberts (1977). The average initial dilution achieved
by a line source in a slowly moving unstratified ambient flow is expressed
as
Sa = 0.38 (g'/q2)1/3 z (22)
provided the ratio of ambient current velocity cubed to buoyancy flux per
unit length is sufficiently low, that is
F = U3/g'q £0.1 (23)
The constant in Equation (22) includes a factor of 1.41 to convert the
measured minimum near-surface dilution to an average dilution for a line
source (Fischer et al. 1979).
The entrainment parameter @ based on Equation (22) is assumed to be
valid for stratified conditions. The equations for equilibrium height
and dilution for these conditions can be deduced from the form of Equation
(22). Since 3 is 0.38(g'/q2)173 and n=l, substitution into Equation (3)
gives
h = 2.29 (g'q)1/3/61/2 (24)
Using this equilibrium height in the average dilution Equation (22) gives
Sa = 0.87 (g')2/3/(q1/3G1/2) (25)
Further references on 2-dimensional slot jets discharging into a quiescent
unstratified ambient include Fischer et al. (1979), Kotsovinos (1978),
Liseth (1976, 1970), Cederwall (1975, 1971), Abraham (1963), and Rouse
et al. (1952). Authors treating 2-dimensional buoyant jets in a quiescent
stratified environment include Wright (1982), Sorrel! and Smith (1981),
24
-------
Chen et al. (1980), Wright and Wallace (1979), Fischer et al. (1979), Cederwall
(1975), Liseth (1970), and Abraham (1963).
Example--
Estimate the average initial dilution for an outfall whose volumetric
flow rate is 4.38 m^/sec (100 MGD). The water depth is 30.5 m (100 ft),
port spacing is 1.5 m (5 ft), and the diffuser is 1,000 m (3,280 ft) long.
Surface and bottom ambient densities are 1.0240 and 1.0258 g/cm3, respectively.
Effluent density is 1.0000 g/cm3. Ambient current is constant with depth
at 4 cm/sec (0.13 ft/sec).
To determine the applicability of Equation (25), calculate 7/1=30.48/1.5
=20»5, suggesting merging. Also, F=U3/g'q is 0.06. Therefore, the equilibrium
height and associated dilution can be calculated as follows
G = -(9.81/1.0258)[(1.0240-1.0258)/30.5] = 5.64xlO'4 sec'2
Equations (24) and (25) give
h = 2.29[(0.253)(0.00438)]1/3/(5.64xlO-4)1/2 = 10.0 m
Sa = 0.87(0.253)2/3/[(0.00438)1/3 (5.64xlO'4)1/2] = 90
In the case of an unstratified environment, the near surface dilution using
Equation (21) is
Sa = (0.38)(0.253)1/3(0.00438)"2/3(30.5) =274
Merging Plumes, Flowing Environment
Roberts (1977) provides graphical solutions for vertical slot jets
oriented at angles of 0°, 45°, and 90° to the ambient current flow, for
values of F=U3/b up to 100. For increasing values of F above 0.1, the
effect of current angle becomes significant. For ambient flow perpendicular
25
-------
to the slot jet ( e=90°) and 0.2_0.1) , which is perpendicular to a vertically oriented diffuser
with merging plumes, the average initial dilution and height of rise can
be expressed as follows. Since 6 is 0.82 U/q and n=l, the rise height,
in the form of Equation (3) is
h = 1.56[(g'q)/(UG)]1/2
Substituting this value of h into Equation (26) gives
Sa = 1.28 [(g'U)/(qG)]1/2 (28)
Example--
Determine the near-surface initial dilution for the same outfall and
ambient characteristics given in the previous example, for an ambient current
of 15 cm/sec (0.49 ft/sec).
The merging condition still applies since z/1 is 20. To determine
whether F>0.1 calculate
U3 = (0.15)3 = 0.0034 m3/sec3
26
-------
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g1 = (9.81)(1.0258-1.0000)7(1.0000) = 0.253 m/sec2
q = 4.38/1000 = 0.00438 m3/sec/m
F = 0.00347(0.253x0.00438) = 3.1
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dilution can be calculated using Equations (27) and (28) as follows
h = 1.56[(0.253)(0.00438)7(0.15)(5.64xlO-4)]1/2 = 5.6 m
Sa = 1.28[(0.253)(0.15)/(0.00438)(5.64xlO-4)]1/2 = 159
Applying Equation (26) for an unstratified condition gives
Sa = (0.82)(0.15/0.00438)(30.5) = 857
APPROXIMATIONS FOR NONLINEAR STRATIFICATIONS
Brooks (1970, 1973) and Roberts (1979b) discuss general approaches
to solving buoyant jet problems, i.e., direct computer solutions by models
such as those presented herein, or by linear approximation of the measured
density profiles. For most problems an approximate solution is obtainable
by assuming an equivalent uniform density gradient over that portion of
the water column over which the plume rises. The reader may consult Brooks
(1973) for a graphical method of determining h, which can then be used
to determine the initial dilution by previously presented methods.
OTHER DIFFUSER CONFIGURATIONS
Staged diffusers with ports oriented in the general direction of the
pipe axis have been constructed. Authors who have addressed the dynamics
of discharges from such diffusers include Adams and Trowbridge (1979),
Trowbridge (1979), Chu (1977), Brocard (1977), and Almquist and Stozenbach
(1976). Adams (1982, 1972), has examined the flow produced by unidirectional
diffusers in both parallel and perpendicular ambient currents, comparing
28
-------
measured and predicted dilutions. Another form of diffuser currently receiving
more consideration is the multiport riser. Isaacson et al. (1983, 1979,
1978a,b) examined the nearfield plume dilutions from an ocean outfall diffuser
consisting of evenly spaced risers with clusters of two to eight ports.
For additional information on jets and plumes from risers or other diffuser
configurations, refer to the annual mixing and transport literature review
in each June issue of the Water Pollution Control Federation Journal.
29
-------
SECTION 4
NUMERICAL MODEL DESCRIPTIONS
INTRODUCTION
The theoretical developments of five numerical models are described
in this section. UPLUME and UOUTPLM are essentially the same numerical
models contained in Teeter and Baumgartner (1979). The model UMERGE is
a generalization of OUTPLM to take into account the effects of plume merging.
UDKHDEN is an improved version of DKHPLM described originally in 1979 (Teeter
and Baumgartner), and ULINE is a generalization of the analytical formulas
of Roberts (1977; 19795). UPLUME and UOUTPLM accept multiple port data
but do not consider the effects of merging. UOUTPLM accepts ambient current
constant with depth in a direction perpendicular to the diffuser axis.
The models UMERGE, UDKHDEN, and ULINE consider multiport diffusers in a
stagnant or flowing environment in which the effects of merging are considered
and the current speed, if present, is allowed to vary with depth. ULINE
is the most simple while UDKHDEN is the most complex. A brief summary
of the model characteristics is given in Table 1. The vertical extent
of the ambient is considered infinite in the theoretical development of
all the models e.g., there are no built-in plume-surface interactions,
and flow is assumed to be fully turbulent. All models provide average
dilutions, and UPLUME can optionally provide centerline dilution.
UPLUME
Theoretical Development
The computer model PLUME (Baumgartner and Trent 1970; Baumgartner
et al., 1971) considers a buoyant plume issuing at an arbitrary angle into
a stagnant, stratified environment. Two zones of plume behavior are con-
30
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31
-------
sidered. The region required to develop fully established profiles is
the zone of flow establishment. Beyond this zone, similarity profiles
are assumed for velocity and concentration. This allows the equations
for mass continuity, momentum, and density disparity to be integrated across
the plume reducing them to one independent variable, the arc length along
the axis of the plume.
Zone of Flow Establishment--
The length of the flow establishment zone, se, has been found to depend
on the initial Froude number, F (Abraham 1963). As initial Froude numbers
approach infinity, se=5.6 port diameters. At low Froude numbers, this
distance is shorter. The inclination of the plume's axis,ee, at se is
a function of se, F, and the initial discharge angle. The centerline velocity
at se depends on the Froude number and, for low Froude numbers, may be
greater than the initial velocity due to the influence of buoyancy (Abraham
1963).
Zone of Established Flow--
The following assumptions are made
• Flow is steady and incompressible
0 Turbulent diffusion is significant only in the radial direction
t Pressure is hydrostatic throughout
• Plume flow is axisymmetric
• Velocity and concentration distributions are Gaussian across
the plume, so that
V/Vm = exp [-k (r/s)2] (29)
= C/Cm = exp [-ku(r/s)2] (30)
32
-------
where V and C are the velocity and concentration respectively, at some
radial distance, r. The subscript m refers to the centerline. The distance
along the plume's trajectory is s; k and u are empirical coefficients (Abraham
1963).
The governing equations are
Conservation of mass,
d/ds / Vrdr = -rbub (31)
Conservation of vertical momentum,
d/ds J V2rdr = sine J [(P.-Pd)/Pd] Qrdr (32)
Conservation of density disparity,
d/ds /*VArdr= C [Vr/(P0-Pd)] (dP./dsJdr (33)
Conservation of pollutants,
d/ds I VCrdr = 0 (34)
Substitution of equations (29) and (30) into these four equations,
simplifying, and reducing to nondimensional form, these equations become
Momentum,
d/ds (V,,^ 53/^3/2) = [3gAmVm $3 sinQfpQ-Pd)]/^^^) (35)
Density disparity,
d/ds [(Am Vm s2/(ku + k)] = [(Vm s2)/ k(P0-Pd)]dPai,/ds (36)
33
-------
Conservation of pollutant,
vm Cm s2/(ku + k) = D2 V0 C0/4 (37)
where
s* = s/D (38)
vm* = Vm/V0 (39)
E* - (Vm*s*/k1/2)3 (40)
R* - [Vm*Ams*2]/[k(u+l)] (41)
Substitution of (40) into (41) gives
R* = [E*1/3Ams*]/[k1/2(u+l)] (42)
p.* sp./(po-pd) (43)
The angle of the plume's axis, e, from the horizontal can be evaluated
by considering conservation of horizontal momentum,
d/ds / V2 cos e r dr = 0 (44)
Substitution of equation (29) and simplifying using equation (40), results
in
e= cos-l [(Ee*/E*)2/3 cosee] (45)
where ee is the plume inclination at se*. Abraham (1963) shows that for
a horizontal discharge,
tane = [l + (l/4)(s*/se*) + (l/6)(s*/se*)2]s*/F (46)
34
-------
For other than a horizontal discharge, Rittall and Baumgartner (1972 errata
to Baumgartner et al . 1971) derived an expression for the angular orientation
of the plume's axis at the end for the zone of flow establishment using
a linear interpolation process and equation (46),
Alpha = (Beta/900) (9QO-ee) + ee (47)
where Beta is the angular orientation of the discharge port from the hori-
zontal. When Beta = 90<\ Alpha = 90°, and when Beta = 0°, Alpha = ee.
Model Description
The initial conditions for the zone of established flow are determined
by evaluating ee, E*, and R* at s*=se*. The length of the zone of flow
establishment as a function of the initial Froude number is
se* = 2.8 F2/3 F< 2 (48)
se* = 0.113 F2 + 4 2_ 3.2 (50)
Assuming that the centerline concentration at se* is equal to the initial
concentration, Cm = C0, and solving equation (37) for E* as a function
of s*, gives
E*l/3 = [k1/2(U+l)]/4s* (51)
which, when evaluated at s*=se*, gives the initial condition for Ee*.
Assuming also that the centerline density at se* is equal to the initial
density and that there is no ambient stratification along the length of
the zone of flow establishment, the centerline density disparity, am, defined
by,
Am = (P«,-Pm)/(Po-Pd) (52)
35
-------
is therefore equal to 1.
Substitution of equations (51) and (52) into equation (42) gives
Re* = 1/4
The angle ee is calculated from equation (46) with s*=se* and together
with the discharge port angle from the horizontal, the initial angle is
then calculated with equation (47).
Equations (35) and (36) are solved at steps along the plume's axis
by the Runge-Kutta approximation. E* is used to calculate the inclination
6 and concentration C. The position of the plume's axis is incremented
by
dx/ds = cos e (53)
dz/ds = sine (54)
where x and z are the horizontal and vertical coordinates respectively.
The centerline concentration is inverted to describe the centerline
dilution. To convert the centerline dilution to a flux-average dilution,
the distribution of concentration must be weighted by distribution of velocity.
With the distributions assumed for UPLUME, the flux-average dilution can
be found by
Sa = 1.77 (l/Cm) or = 1.77 Sm (55)
where ^ is the centerline dilution. The model UPLUME produces flux-average
dilutions and, for one output option, also gives centerline dilutions as
did the model PLUME contained in Teeter and Baumgartner (1979). The dilution
achieved for a plume trapped at a subsurface equilibrium level below 0.9
times the port depth is calculated from E* and S* at the elevation where
R* is estimated to be zero, i.e., where the average density of the plume
36
-------
equals the ambient density. This is normally somewhat below the maximum
height of rise, but is where similarity ends. Above this level, the plume
tends to spread and become passive, possibly interfering with further dilution.
If the plume reaches 0.9 times the port depth, then the plume is considered
to reach the depth at which no further dilution is possible (due to blocking).
The centerline velocity, averaged over each step length, is divided
into the step length to obtain time. The plume diameter is found by W=0.308s.
The calculation of these parameters also ceases when the final dilution
is calculated. The program terminates when the vertical velocity is zero,
the angle of the centerline is 0°, or the surface is reached, whichever
occurs first.
UOUTPLM
Theoretical Development
The computer model OUTPLM (Winiarski and Frick 1976, 1978) considers
a single plume element. By following the element as it gains mass due
to ambient fluid entrainment, the characteristics of a continuous plume
in a flowing ambient are described. The original cooling tower plume model
has been adapted for marine discharges (Teeter and Baumgartner 1979).
Density (or temperature and salinity) and velocity are assumed to be average
properties of the element. The sums of plume element and entrained mass,
horizontal momentum, and energy are conserved. An equation relating temper-
ature, salinity, and density (U.S. Navy Hydrographic Office 1952) is used
to calculate the density of the ambient and the plume element at each time
step.
Entrainment brings ambient mass (plus momentum, temperature, and salinity)
into the plume element. Entrainment is assumed to consist of either of
two mechanisms. One mechanism, sometimes called forced entrainment, is
due to the impingement of current on the plume. It is the mass flux through
the boundary area of the plume element projected on a plane normal to the
current. The element is usually a section of a bent cone. Therefore,
the projected area formulation contains a cylindrical term, a growth term,
37
-------
and a curvature term as described in Frick (1984). The second mechanism
is aspiration entrainment (i.e., the Taylor entrainment hypothesis discussed
in Taylor et al. 1956) which captures 0.1 times the product of the external
area of the plume element and its shear velocity. Total entrainment is
taken to be the larger of these two mechanisms.
Model Description
In the computer program, the entrained mass is added to the element's
mass to become the new mass. The new temperature and salinity of the element
are the averages of the old values and the entrained ambient values weighted
by their relative masses. The horizontal velocity is found in the same
way, thus conserving horizontal momentum. The vertical velocity depends
on buoyant force as well. The new density, and thus buoyancy, creates
a vertical acceleration on the plume segment. Since the element is considered
to be one of a train, each following the preceding element, drag is assumed
to be negligible. The segment length is changed in proportion to the total
velocity to conserve mass and pollutant. The radius is changed to correspond
to the new mass and density. Dilution is calculated by comparing the initial
volume to that of the element. The program terminates execution when the
vertical velocity reaches zero, the surface is reached, or length scales
or execution step limits are reached whichever occurs first.
UDKHDEN
Theoretical Development
UDKHDEN is a fully three-dimensional model which can be used for either
single or multiple port diffusers and is based on the technical developments
of Hirst (1971a,b), Davis (1975) and Kannberg and Davis (1976). It considers
variable profiles through the zone of flow establishment and through the
merging zone of multiple plumes. Detailed development of the plume is
considered through the zone of flow establishment rather than by approximating
it in a single step as do most other models. In addition, the changing
geometric form of merging, multiple plumes is approximated instead of sharply
38
-------
transitioning from multiple, round plumes into a two dimensional equivalent
slot plume.
UDKHDEN is easier to use than DKHPLM. Ambient conditions are entered
in tabular form, thereby allowing for variation in density and/or current
as a function of depth. The user can input either temperature and salinity,
or density, for both the effluent and the receiving water characteristics.
Entrainment is an explicit function dependent on the local Froude number,
plume spacing, excess velocity, and ambient velocity. Similar profiles
(power function form) are assumed for velocity, concentration, and tempera-
ture. These profiles are superimposed in their merging zones.
Zone of Flow Establishment
All quantities are assumed uniformly distributed in the plume at the
point of discharge. In the zone of flow establishment, these uniform profiles
change to similar profiles as the boundary layer diffuses inward to the
centerline of the jet. The rate at which the profiles of velocity, concentra-
tion, and temperature develop may vary. The governing equations are
Conservation of mass:
d/ds / Vrdr = E (56)
Conservation of energy:
00 OD
d/ds I V(T-T.)rdr = -dT./ds f Vrdr (57)
o o
Conservation of pollutant:
to oo
/*V(C-CJrdr = -dC./ds /*
d/ds V(C-CJrdr = -dC./ds Vrdr (58)
o o
Conservation of momentum in the s equation:
d/ds I V^rdr = UEsin6icos62 + / g(P<1,-P)/Pd rdr sine£ (59)
//•
V2rdr = UEsin6icos82 + I
/
39
-------
where ej is the horizontal angle between the plume centerline and the x
axis, and 62 is the vertical angle between the plume centerline and the
horizontal. Two additional integral equations have been developed from
equation (59) to describe momentum in two additional plume coordinates.
These "natural" coordinates of the plume are converted to conventional
3-dimensional Cartesian coordinates for model output. Implicit in the
derivation of these equations are the assumptions that
• Flow is steady in the mean
0 The fluid is incompressible and density variations are included
only in the buoyant terms
t All other fluid properties are constant
t No frictional heating
• Pressure variations are purely hydrostatic
t Ambient turbulence effects are included in the entrainment
function only
t Flow within the jets before merging is axisymmetric and
is free, boundary layer type flow.
If temperature and salinities are input, densities are calculated internally
using an equation relating temperature, salinity, and density (U.S. Navy
Hydrographic Office 1952) in the subroutine SIGMAT. The six governing
equations are solved simultaneously in the subroutine SIMQ and stepped
forward in space by Hamming's modified predictor-corrector generator in
subroutine HPCG. This procedure continues until velocity, temperature,
and concentration profiles become fully developed. Subroutine OUTP1 contains
the results which are stored as initial conditions for the zone of established
flow.
40
-------
Zone of Established Flow
The governing equations presented in the previous section are also
solved in the zone of established flow but are slightly different in form.
They are evaluated using a power function approximation to Gaussian profiles
leaving the centerline concentrations, temperature, width, and plume coordinates
as dependent variables. Entrainment is determined from an empirical function
which is a function of plume size, excess velocity, local Froude number,
and ambient velocity. Subroutine DERIV evaluates the derivatives of these
dependent variables which are used in subroutine HPCG to step forward in
space. Subroutine OUTP evaluates the values of the variables at each integra-
tion step and prints them out periodically.
Zone of Merging
When adjacent plumes begin to overlap, the plumes are no longer considered
axisynmetric. The distributions of plume properties are superimposed as
shown in Figure 3. The entrainment function is modified to account for
the interaction of plumes and the reduction in the entrainment surface
as the merging process proceeds. It is assumed that the plumes are equally
spaced, in a line, and that end effects are negligible.
Sample Run and Model Listing
The program is for interactive operation from a terminal. The program
contains many comments and explanations which serve as further model documen-
tation and will aid the user in operating the model. Required inputs and
resultant outputs are described in Section 5. The program is terminated
when the surface is reached, when the preprogrammed length scale (SF=1000)
is exceeded, the plume has reached its maximum height or error conditions
were detected in subroutine HPCG (see comment section of that subroutine
under IHLF) whichever occurs first.
41
-------
n
i,
WIDTH
CONNECJ1NG_1
LINE
ENTRAPMENT
SURFACE
MERGED PROFILE
Figure 3. Cross section and profile along connecting line
of merging plumes.
42
-------
UMERGE
Theoretical Development
The model UMERGE analyzes a positively buoyant discharge by tracing
a plume element through the course of its trajectory and dilution. Conditional
controls, rather than conceptual limitations, prevent analysis of negatively
buoyant discharges. UMERGE is a two-dimensional model which accounts for
adjacent plume interference and which accepts arbitrary current speed variations
with depth. Oiffuser ports are assumed to be equally spaced and may be
oriented at any common elevation angle. The current is assumed to be normal
to the diffuser axis and the discharge velocity vector is assumed to be
in the plane formed by the current direction and the vertical axis.
The basic plume equations are summarized as follows
dm/dt = entrainment (Taylor hypothesis + forced continuity) (60)
d(mu)/dt = u0(dm/dt) (conservation of horizontal momentum) (61)
d(mv)/dt = (AP/p)mg (vertical momentum) (62)
d(mT)/dt = T0(dm/dt) (conservation of temperature) (63)
d(mS)/dt = S0(dm/dt) (conservation of salinity) (64)
Ah/(u2+v2)l/2 = Ahi/(ui2+Vi2)1/2 = constant (65)
where
i = initial conditions
o = ambient conditions.
Equation (65) transforms the integral flux plume equations to their Lagrangian
counterparts. Also required is an equation for density (subroutine SIGMAT)
as a function of temperature and salinity (U.S. Navy Hydrographic Office
1952). The equations are integrated with respect to time.
Forced and aspiration entrainment (Taylor hypothesis, see Morton et
al. 1956) are handled in much the same way as in UOUTPLM. However, rather
than considering the larger of the two components as being the operative
43
-------
mechanism, they are considered additive, based on superimposed flow fields.
In the absence of a current, entrainment is due solely to aspiration.
At moderate current levels, entrainment is from both mechanisms but aspiration
is somewhat reduced in the lee of the plume. In the presence of higher
currents, entrainment is largely forced (Frick 1981, 1984).
The merging equations are based on purely geometric considerations.
The mass of overlapping portions of adjacent plumes is redistributed by
increasing the normal dimensions of the plumes, and entrainment is adjusted
accordingly.
Assumptions inherent in the model formulation include
• Exchange between adjacent plumes does not change the average
properties of a plume element (mirror imaging) but does
affect the plume radius
• The model calculates average plume properties
• The ambient fluid is largely undisturbed by the presence
of the plume
• No net pressure forces are exerted on the plume by the ambient
and adjacent plume elements exert no net force on each other
• Energy and salinity are conserved
• Specific heat is considered to be constant over the range
of temperatures observed in the system
• In addition to entrainment by aspiration, all fluid impinging
on the projected area of the plume is entrained
44
-------
• Current direction is assumed to be normal to the diffuser
axis
t The plume boundary encloses all the plume mass.
Model Description
Entrainment is considered as the mass flowing through the projected
plume area plus the aspirated quantity. While the concept is simple, the
computation for the projected plume area is complex and the reader is referred
to Frick (1984) for further development. The changes in mass (Am) and
time (At) are scaled internally by the model, allowing for a variable time
step. This feature shortens execution time, important when using micro-
computers or when using the program to optimize a design. The new plume
element average horizontal velocity, temperature, and salinity are calculated
using weighted averages of both the element and entrained masses. In calcu-
lating the vertical velocity, the effect of buoyancy is taken into account.
The subsequent position of the plume element is found by multiplying
the new element velocity by the time increment and adding to the previous
coordinates. The length of the plume element changes during each time
increment due to the velocity gradient between the two faces of the element.
Elongation, or contraction, can be estimated by comparing the element velocities
between iterations. The effect of merging is estimated by distributing
the overlapping mass to other portions of the plume, calculating the resulting
changes in the element radius, and by adjusting entrainment terms.
Once all plume properties have been calculated for a given time step,
the iteration process begins anew until the vertical velocity becomes negative
(maximum rise), the surface is reached, or the maximum number of specified
iterations is exceeded.
ULINE
The model ULINE is based on Roberts' (1977) uniform density flume
experiments and is a generalization of Roberts' (1979b) discussion of dilution
45
-------
achieved in an arbitrarily stratified environment. The ambient current
direction is assumed constant but no restriction is imposed on the current
direction relative to the diffuser axis. The ambient current speed and
ambient density are allowed to vary with depth.
The results of the flume experiments of Roberts (1977) are shown in
Figure 2. As indicated, the minimum surface initial dilution Sm, for a
fixed current direction relative to the diffuser axis (0, 45, or 90°) ,
is given by
STO = (UH/q) fe(F) (66)
where
fe = function dependent on e
e = current direction relative to the diffuser axis (a current flowing
perpendicularly to the diffuser axis has e= 900)
F = U3/g'q
9' = 9(P0-pd)/pd-
The model ULINE linearly interpolates the results of Roberts for an arbitrary
current angle. The average initial dilution Sa for a slot jet (Brooks
1973) is approximately
Sa • 1.41 Sm (67)
These relations are used to derive the function
a - dSa/dz - 1.41 (U/q)fe(F) (68)
On the basis of mass conservation the plume density at a height h
above the diffuser can be expressed as
(69)
46
-------
where
P"a(h) = ambient density over the height of rise
Pe = effluent density
Sa = average initial dilution
Equation (69) can be rewritten as
pj(h) = [Pe+Pa(n) (Sa-l)]/Sa (70)
where
h h
P"a(h) =
and
h
Sa = A(z) dz (71)
o
o(z) « 1.41 U(z)fe(F)/q (72)
Trapping of the plume occurs if pa(h) = Pj(h) for some h. Otherwise
the plume surfaces. The program ULINE numerically integrates the two integrals
h
J a(z)Pa(z)dz (73)
o
and
h
/«(z)dz (74)
by the trapazoidal rule, and uses their values to determine whether the
plume is trapped. As indicated earlier, the initial dilution Sa is given
by equation (71), where h is the plume height of rise.
47
-------
The program terminates when the trapping level is reached or when
the plume surfaces.
48
-------
SECTION 5
MODEL EXECUTION
INTRODUCTION
The five models are written in FORTRAN IV-PLUS and are running on
a PDF 11/70. The program listings, available in a separate volume, have
statements specific to this POP system and may need to be modified to conform
to the user's system. For example, the third read statement in UDKHDEN
is
READ(3,102,END=221,ERR=999)N11
which might have to be modified to something like
READ(3,102)N11
IF(EOF)GO TO 221 If no more cards* to read - STOP.
IF(ERR)GO TO 999 Input error, inform user and STOP.
Also, the terminology used in the following discussion is specific to the
POP 11/70 running the IAS operating system, and the user will have to make
appropriate changes. For example, the POP 11/70 system prompt is
PDS>
The models are set up to be run from a terminal and require a UNIVERSAL
DATA FILE (UDF) which is described below. Assume, for example, the user
wishes to run UPLUME (the program must be in the user's directory) and
has created a UDF named MARC.IN. The following is a step by step procedure
used herein, the word card refers to a record or line of information.
49
-------
for running this example from a terminal connected directly to the computer.
To distinguish between system or program prompts and user responses, the
latter are underlined. Each response is terminated with a carriage return
indicated by . It should be noted that the file names (MARC.IN, MARC.OUT
etc.) selected for the following examples can be replaced by names chosen
by the user.
SWITCH THE TERMINAL ON
CTRL C Type the letter C while holding
the control key down to obtain
computer recognition.
(System information will be displayed.)
PDS> LOGIN
USER NAME? Your user name.
PASSWORD? Your password (system does
not echo password).
(Messages to system users will be displayed.)
PDS> RUN UPLUME
*****PROGRAM UPLUME, AUGUST 1985*****
ENTER UDF NAME MARC.IN Your input file name.
ENTER OUTPUT FILE NAME MARC.OUT Your input file name.
15:09:00 SIZE 19K CPU: 2.68 STATUS: SUCCESS
The results of this example are in the user's directory with the file name
MARC.OUT. To display or print the results, enter the following:
50
-------
PDS> TYPE MARC.OlfT
This response will display
the results at the user's
terminal.
or,
PDS> PRINT MARC.OUT
PDS> LOGOUT
This response will send the
results to the line printer.
Ends terminal session.
(User identification, terminal number, time, date, connect and system
utilization times will be displayed.)
BYE
UNIVERSAL DATA FILE DESCRIPTION
Each of the five models described requires particular input data.
Although these data (port diameter, spacing, etc.) are similar, the previous
versions of the models (Teeter and Baumgartner 1979), had unique input
data formats. To simplify the use of the models, they have been modified
so that any one will read the same input data file which is termed the
UNIVERSAL DATA FILE (described below and in Appendix II). This file contains
all the parameters required to execute each of the models. Further, parameters
that were usually held constant and entered each time a model was run,
(e.g., printout interval and aspiration coefficient) are now preprogrammed.
However, the user may change these default values by setting the parameter
ICUTOP=1 (Card 2) and including Card 5 with the new value(s). IF ICUTOP=0,
Card 5 must be omitted. In an earlier version of one of the models, the
data could be entered in English or metric units. All of these models
now require metric units. To distinguish the new models from the old ones,
they are now named UPLUME, UOUTPLM, UDKHDEN, UMERGE and ULINE.
51
-------
A UDF consists of one or more sets of "card images" created and maintained
with any editor. See Appendix II for the data required, the units of the
variables and their limitations where appropriate.
Card 1 can be used to identify a particular data set in the UDF.
Card 2 is a control card providing the user with the following options:
INTER (Interactive Control variable)
If INTERS
The programs will process this data set and go to the next set or
exit if there are no more data sets in the UDF.
If INTER=1 (Interactive mode)
The programs will prompt the user for a run title which is useful
for identifying successive interactive runs. The user responds by
typing in a title for the run, terminated by a carriage return. The
programs will process the data set and display the following results
at the user's terminal:
If the equilibrium level was reached or that the plume reached
the surface.
Reason for terminating calculations, e.g., VERTICAL VELOCITY
went through zero.
Depth of equilibrium level if appropriate.
Average dilution.
CHANGE VARIABLES?
52
-------
The user is asked if another run is to be made with the existing ambient
data, YES or NO?
+ If NO, the programs go to the next data set or exit as the
case may be.
+ If YES, the user is prompted for a run title. After entering
this title,
1. The present values of the parameters that may be changed
are displayed. Each variable is numbered.
2. The user responds with the number of the variable to
be changed and is then prompted for the new value.
3. After entering the new value, the user is asked if
another variable is to be changed, YES or NO?
If YES, 1, 2, and 3 are repeated.
If NO, the programs compute the results using the new
value(s) and the entire sequence is repeated, i.e.,
results are displayed and the user is asked if another
run is to be made.
IDFP (Input data file)
If IDFP=0
The card images of the input data are not included as part of the
output.
If IDFP=1
The card images of the input data as they exist in the UDF will be
included in the output for that run. It will not reflect any changes
53
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made by the user in the interactive mode (INTER=1). These changes
are shown in the heading of the results.
ICUTOP (Control Parameter Change)
If ICUTOP=0
Card 5 must be omitted from the data set. The programs will use the
preprogrammed default values for those parameters defined on Card 5.
If ICUTOP=1
Card 5 must be included in the data set even if blank. If it's blank,
the default values will be used. If it's not blank, the user's values
will be used with that data set.
If ICUTOP=1 and Card 5 is omitted or if ICUTOP=fl and Card 5 is included,
an input conversion error will occur and the programs will exit even
if there are more data sets in the UDF. Correct that data set and
reenter the UDF.
Output Format Control 0, 1, or 2
IPI=IPO, 101=100, IDI=IDO, IMI=IMO, ILI=ILO
If zero, the output format is 8-1/2 inches wide by 11 inches long.
It may be longer depending on how many images of Card 7 are in the
data set.
If one, the output format is as originally programmed; varies depending
on the model.
If two, the output is condensed by omitting the results of intermediate
iterations (except UDKHDEN). When in the interactive mode (INTER=1),
the ambient data is not repeated but pertinent parameters are.
54
-------
Card 3 contains the flow rate and all but one of the required diffuser
parameters.
Card 4 contains a uniform ambient current speed (used in UOUTPLM only),
the horizontal angle of the current relative to the diffuser and
the discharge port spacing which is discussed below.
Card 5 is omitted if ICUTOP=0, included if ICUTOP=1. This card permits
the user to change the default values of the programs.
Card 6 contains the number (NPTS) of images of Card 7 (ambient data table)
included in the data set and the density of the discharge as either
g/cm3 or salinity (ppt) and temperature (°C).
Card 7 is the ambient data table, one card for each depth of ambient data.
Density may be in g/cm3 Or salinity (ppt) and temperature (°C)
All however must be in the same units. The number of cards must
be equal to NPTS or an input conversion error will occur and the
programs will exit even if there are more data sets in the UDF.
The order of the ambient data table is immaterial as all the programs
sort this table, arranging the depths in increasing order.
All of the programs have a subroutine (LIMITS) to check that certain
input data are within prescribed limits, e.g., the port depth cannot be
zero meters or deeper than the deepest depth of the ambient data table.
If INTER=1 (interactive mode), the user is prompted for corrections which
may be made from the terminal. See the comments at the beginning of SUBROUTINE
LIMITS in each of the programs for the specific data that is checked for
that program.
DISCHARGE PORT SPACING
Selecting values for port spacing, flow rate, number of ports, and
port diameter may not be that straightforward. The variety of diffuser
55
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designs requires the user of these models to exercise care in selecting
values to use as input data.
For the simple case where all the ports are on the same line, equally
spaced, discharging at the same angle and all the same diameter, the input
values may be taken directly from the design data sheet(s). However, this
is usually not the case and the data need to be modified to represent the
simple case, and multiple runs may be needed to simulate segments of the
diffuser.
One of the requirements for a well designed diffuser is that the flow
rate per port be uniform (or nearly so) and thus the data are readily reduceable
to any number of ports. Do not however reduce the data to a single port
as at least two ports are required for adjacent port merging to be detected
(see footnotes a and b to Table 2). If the port diameter is varied from
one end to the other, usually in groups, and ports are all in the same
line, the port diameter (PDIA, card 3) is a variable which may be easily
changed by running the model interactively (INTER=1, card 2). The value
to use for NP (card 3) is the number of ports in the group; the flow rate
(QT, card 3) is NP times the flow rate per port; and the spacing (SPACE,
card 4) is the distance between adjacent ports on the same side of the
diffuser.
Often, diffuser designs specify half the ports to be on one side and
the other half diametrically opposite or with staggered spacing, and all
the same diameter. For this condition, model one side of the diffuser
and verify that merging with plumes from the other side does not occur.
Then, the results represent the dilution achievable based on adjacent ports.
In this case, QT would be equal to half the total flow, NP equal to half
the total number of ports and SPACE is the distance between adjacent ports
on the same side of the diffuser.
All possible diffuser designs cannot be covered here but this should
give the user some insight into selecting input data so the models give
realistic results.
56
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EXAMPLE UNIVERSAL DATA FILE
This DDF contains three data sets for the three printout options and
is formatted using short field terminators.
UDF file name; User's choice. (For this example it is MARC.IN.)
#1 EFFLUENT & AMBIENT DENSITY AS G/CM3, ZERO CURRENT, IxI=IxO=ZERO
0,1,0 ,0,0 ,0,0,0,.
1.266,148,.0915,0.,55.2,
0.,90.,3.0,
7,.99744,0.,
00.00,1.02261,,,
20.00,1.02275,,,
45.00,1.02302,,,
50.00,1.02344,,,
55.00,1.02348,,,
60.00,1.02365,,,
60.96,1.02367,,,
#2 EFFLUENT AS G/CM3, AMBIENT AS S 8 T, 0.02 M/SEC CURRENT, 1x1=1x0=1
0,1,0,1,1,1,1,1,
1.266,148,.0915,0.,55.2,
0.02,90.,3.0,
7,.99744,0.,
00.00,34.72,26.75,0.02,
20.00,34.72,26.30,0.02,
45.00,34.66,25.30,0.02,
50.00,34.74,24.10,0.02,
55.00,34.71,23.90,0.02,
60.00,34.71,23.30,0.02,
60.96,34.71,23.23,0.02,
#3 EFFLUENT & AMBIENT DENSITY AS G/CM3, 0.04 M/SEC CURRENT, 1x1=1x0=2
0,1,0,2,2,2,2,2,
1.266,148,.0915,0.,55.2,
0.04,90.,3.0,
7,.99744,0.,
00.00,1.02261,,0.04,
20.00,1.02275,,0.04,
45.00,1.02302,,0.04,
50.00,1.02344,,0.04,
55.00,1.02348,,0.04,
60.00,1.02365,,0.04,
60.96,1.02367,,0.04,
57
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Shown below is the second data set in this UDF formatted according
to the individual card format specified in the description of the UDF.
#2 EFFLUENT AS G/CM3, AMBIENT AS S & T, 0.02 M/SEC CURRENT, 1x1=1x0=1
01011111
55.2
1.266
0.02
7
00.00
20.00
45.00
50.00
55.00
60.00
60.96
148
90.0
.99744
34.72
34.72
34.66
34.74
34.71
34.71
34.71
.0915
3.0
0.0
26.75
26.30
25.30
24.10
23.90
23.30
23.23
0.0
0.02
0.02
0.02
0.02
0.02
0.02
0.02
In this example, the total effluent flow is 1.266 m^/sec with a density
of 0.99744 g/cm3. The diffuser has 148 0.0915-m diameter ports spaced
3.0 m apart. The discharge is horizontal at a depth of 55.2 m. The ambient
current is zero for the first case, a uniform current of 2.0 cm/sec 90
degrees to the diffuser for the second case and 4.0 cm/sec also at 90 degrees
for the third case. For Cases 1 and 3, the density option is used; Case
2 uses the salinity-temperature option.
For this example the programs will print a "card image" (IDFP=1) of
the data on the first page of the output with the results on the second
page. This is shown in the following exhibits for UPLUME only, as it would
be the same for the others. The results of the first data set will be
the 8-1/2 by 11 format (1x1=1x0=0). The second will be as "originally"
programmed (1x1=1x0=1) and the third will be the condensed format (1x1=1x0=2).
UDKHDEN and ULINE do not have output format options.
RESULTANT MODEL OUTPUT
Four of the five models were run with this UDF. Table 2 shows the
input parameters required by each model. Tables 3 through 6 define the
output parameters for UPLUME, UOUTPLM, UDKHDEN, and UMERGE. No intermediate
results are output for ULINE, and thus no table of output variables is
58
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necessary. Exhibits 1 through 6, and 8 through 10 show the output for
both the original and new formats for the three models UPLUME, UOUTPLM,
and UMERGE. UDKHDEN has only one output format (Exhibit 7, Case 2).
Even though the printed ambient densities for Case 2 (salinity-temperature
option) are identical to the input ambient densities for Cases 1 and 3,
(density option) the results are slightly different (UPLUME, Exhibits 1
and 2). This is because the number of significant digits used in the calcu-
lations involving densities are not the same. Cases 1 and 3 use six significant
digits, but the number of digits used in Case 2 depends on the specific
computer system and whether the programs are compiled using single or double
precision. Case 2 densities are calculated values. Therefore the resulting
number of significant digits may vary. This applies to the others programs
as well but is not evident in the respective Exhibits as Cases 2 and 3
have ambient currents of 2.0 and 4.0 cm/sec respectively.
Another UDF (MARC2.IN), shown below, was run with ULINE. The inter-
active control (INTER=1) was used to show first the 8 1/2 x 11 format and
then the condensed output (Exhibit 11). A uniform current of 4.0 cm/sec
is included and the ambient current angle to the diffuser is varied; 90
45, and zero degrees.
EFFLUENT & AMBIENT DENSITY AS G/CM3, 1x1=1x0=2
1,0,0,2,2,2,2,2,
1.266,148,.0915,90,55.2,
0.04,90.,3.0,
7,.99744,0.,
00.00,1.02261,,0.04,
20.00,1.02275,,0.04,
45.00,1.02302,,0.04,
50.00,1.02344,,0.04,
55.00,1.02348,,0.04,
60.00,1.02365,,0.04,
60.96,1.02367,,0.04,
59
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BATCH PROCESSING
While these programs are designed to be run in the interactive mode
from a terminal, they may also be run in batch mode. This mode does not
require the attention of the user or tie-up a terminal while the program
is running. This is especially useful for long runs, say 12 different
density profiles with 5 different flows per profile which would tie-up
a terminal for a considerable length of time. All systems provide this
alternative but require job control language (JCL) cards, a special file
or something similar. The POP 11/70 requires a special file (filetype.BIS)
containing the necessary instructions. To run UPLUME using MARC.IN data
in batch mode on a POP 11/70, the .BIS file is shown below. Before submitting
the job, check to be sure that INTER=zero in every data set in the input
file or the job will terminate prematurely.
Name of file, MARC.BIS
$JOB MARC DON (MIN)
$RUN UPLUME
MARC.IN
MARC.OUT
SPRINT/DELETE MARC.OUT
$EOJ
The first line is the user identification; DON is to receive the output,
and (MIN) is the time in minutes which overrides system default time-cut
for BATCH jobs. The second line identifies the program to be run. The
third and forth lines identify the input and output files respectively.
The fifth line prints and then deletes the output file and the last line
terminates the job. Note that the third and forth lines do not have the
dollar sign ($). This means that information is in response to program
prompts and in the interactive mode would be entered from the terminal
during program execution.
To run the model, the user responds to PDS prompts as shown below.
60
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PDS> SUBMIT MARC (.BIS is the default filetype for the SUBMIT command.)
PDS> LOGOUT
The system puts the job in a queue and runs it along with time sharing
tasks. Alternately, it may be submitted as a night job in which case it
would not be run until after 10:00 pm. For this, the response to the PDS
prompts is shown below.
PDS> SUBMIT/NIGHT MARC
PDS> LOGOUT
61
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TABLE 2. UNIVERSAL DATA FILE PARAMETERS REQUIRED
BY THE COMPUTER MODELS
Parameter
QT
NP
PDIA
VANG
PDEP
UN
HANG
SPACE a
NPTS
S
T
DP( )
SA( )
TA( )
UA( )
UPLUME
X
Xa
X
X
X
X
X
X
X
X
X
X
UOUTPLM
X
xa
X
X
X
X
X
X
X
X
X
X
X
UMERGE
X
Xa
X
X
X
X
X
X
X
X
X
X
X
UDKHDEN
X
Xa
X
X
X
X
X
X
X
X
X
X
X
X
ULINE
X
Xb
X
X
Xb
X
X
X
X
X
X
X
a SPACE is used to determine if merging of adjacent plumes occur if NP>1.
If NP=1, then SPACE=1000 (DEFAULT) and the merging flags are inactive.
b In ULINE, the length of the diffuser is defined as the product of (NP-1)
and SPACE. NP=1 is not allowed.
Blanks (no X) indicate parameters ignored by those models.
QT Total effluent flow (m3/sec)
NP Number of ports
PDIA Port diameter (m), effective diameter if known (Fischer et al. 1979)
VANG Vertical angle of discharge (900 is vertical)
PDEP Depth of discharge (m)
UW Ambient current speed (m/sec)
HANG Horizontal angle of current relative to the diffuser
SPACE Spacing between ports on the same side of the diffuser
NPTS Number of cards in the ambient data table
S Effluent salinity (ppt) or density (gm/cm3) if T=zero
T Temperature of the effluent (°C) or zero if S is density
DP Depth (m)
SA Salinity (ppt) at DP or density (gm/cm3) if TA=zero
TA Temperature (°c) at DP or zero if SA=density
UA Current speed (m/sec) at DP
62
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TABLE 3. OUTPUT PARAMETERS FOR UPLUME
Description
Original
New
A time based on the centerline velocity T T
Distance of the plume element from the port
orifice along the centerline S S
Horizontal distance of the center of the plume
element from the port orifice X X
Depth of plume element from the surface Z Z
Diameter of plume element D DIA
Height of rise of plume element ELEV H
Angle (degrees) of the plume's velocity (or
centerline) at time T from the horizontal THETA THETA
Plume dilutions DILNa FLUX-AVG
DILUTION
a In Teeter and Baumgartner (1979), DILN is the centerline dilution. In
program UPLUME for IPI=IPO=1, both centerline dilution and flux-averaged
dilution (which is 1.77 times the centerline dilution) are printed. For
the other two print options, only flux-average dilution is printed. The
flux-averaged dilution is the appropriate dilution to use in water quality
computations.
63
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EXHIBIT 1 UPLUME OUTPUT FOR IPI=IPO*0
UNIVERSAL DATA FILE: MARC.IN
*1 EFFLUENT 8 AMBIENT DENSITY AS G/CN3, ZERO CURRENT, IxI=IxO»ZERO
0,1,0,0,0,0,0,0,
1.266,148,.0915,0.,55.2,
0.,90.,3.0,
7,.99744,0.,
00.00,1.02261,,,
20.00,1.02275,,,
45.00,1.02302,,,
50.00,1.02344,,,
55.00,1.02348,,,
60.00,1.02365,,,
60.96,1.02367,,,
UPLUME VERSION 1.0 AUGUST 1985 (BASED ON 053 VERSION 2.3 9/12/77)
UNIVERSAL DATA FILE: MARC.IN
CASE I.D. #1 EFFLUENT & AMBIENT DENSITY AS G/CM3, ZERO CURRENT, 1x1=1x0=ZERO
PRINTOUT INTERVAL
INITIAL DENSITY OF THE PLUME
DISCHARGE VELOCITY
FROUDE NUMBER
(DEFAULT)
0.99744 G/CM3
1.301 M/S
8.5
DEPTH
0.00
20.00
45.00
50.00
55.00
60.00
60.96
DENSITY
1 .02261
1 .02275
1.02302
1 .02344
1 .02348
1 .02365
1.02367
TOTAL EFFLUENT FLOW
NUMBER OF PORTS
PORT DIAMETER
PORT SPACING
VERTICAL PORT ANGLE FROM HORIZONTAL
PORT DEPTH
T
(SEC)
5.89
14.70
25.16
S
(M)
3.38
6.38
9.36
X
(M)
2.51
3.21
3.58
Z
(H)
53.32
50.42
47.46
1.2660 CMS
148
0.0915 (1
3.00 M
0.0 DEGREES
55.20 M
DIA
(N)
1.04
1.96
2.88
PLUMES MERGED, PARAMETERS AT THAT TIME WERE:
26.67 9.74 3.62 47.08 3.00
FOLLOWING CALCULATIONS DO NOT ACCOUNT FOR MERGING, A
12.33 3.89 44.50
15.32 4.68 41.71
COMPUTATIONS CEASE: PLUME TRAJECTORY IS HORIZONTAL
H
(M>
1.88
4.78
7.74
8.12
SINGLE
10.70
13.49
THETA
(DEG)
68.6
81.1
84.0
84.1
FLUX-AVE
DILUTION
18.95
54.79
98.05
102.99
PLUME IS ASSUMED.
83.0
0.0
TRAPPING LEVEL = 46.58 M BELOW WATER SURFACE.
AVERAGE DILUTION - 108.7
TIME TO TRAP: 28.48 SEC. PLUME DIA AT THE TRAPPING LEVEL: 3.16 M
64
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EXHIBIT 2 UPLUME OUTPUT FOR IPI=IPO=1
UNIVERSAL DATA FILE: MARC.IN
«2 EFFLUENT AS G/CM3, AMBIENT AS S & T, 0.02 M/SEC CURRENT, 1x1=1x0=1
0,1,0,1,1,1,1,1,
1.266,148,.0915,0.,55.2,
0.02,90.,3.0,
7,.99744,0.,
00.00,34.72,26.75,0.02,
20.00,34.72,26.30,0.02,
45.00,34.66,25.30,0.02,
50.00,34.74,24.10,0.02,
55.00,34.71,23.90,0.02,
60.00,34.71,23.30,0.02,
60.96,34.71,23.23,0.02,
OS3 PLUME VERSION 2.3 9/12/77 (MODIFIED FOR UNIVERSAL DATA FILE, AUGUST 1985.)
*****************A BUOYANT PLUME IN A DENSITY STRATIFIED MEDIA*****************
UNIVERSAL DATA FILE: MARC.IN
CASE I.D. #2 EFFLUENT AS G/CM3, AMBIENT AS S t T, 0.02 M/SEC CURRENT, 1x1=1x0=1
CASE NO. 2 WITHIN THE UDF, UNITS: NCS, INITIAL CONDITIONS.
PORT ANGLE 0.0
FROUDE NUMBER 8.5
LENGTH FOR FLOW ESTABLISHMENT. ... 0.51
INTEGRATION STEP LENGTH 0.062
PRINTOUT INTERVAL 3.00
XO 0.51
ZO 55.14
DISCHARGE DENSITY 0.99744
PORT DEPTH 55.20
FLOWRATE 1.2660
NUMBER OF PORTS 148
DISCHARGE VELOCITY 1.30
PORT DIAMETER. ........... 0.0915
PORT SPACING 3.000
DENSITY STRATIFICATION: DEPTH
0.00
20.00
45.00
50.00
55.00
60.00
60.96
T
6.00
14.94
25.54
S
3.43
6.45
9.46
X
2.53
3.22
3.59
RHO
1.02261
1.02275
1.02302
1.02344
1.02348
1.02365
1.02367
Z
53.28
50.35
47.36
D
1.06
1.99
2.91
PLUMES MERGED, PARAMETERS AT THAT TIME WERE:
26.66 9.74 3.62 47.08 3.00
ELEV
1.92
4.85
7.84
8.12
THETA
69.0
81.2
84.1
84.1
DILN(CL)
10.92
31.58
56.17
58.23
DILN(AVE)
19.33
55.89
99.42
103.07
FOLLOWING CALCULATIONS DO NOT ACCOUNT FOR MERGING, A SINGLE PLUME IS ASSUMED.
12.45 3.91 44.39 10.81 82.8
15.35 4.71 41.70 13.50 0.0
LAST LINE ABOVE IS FOR MAXIMUM HEIGHT OF RISE.
TRAPPING LEVEL IS 46.58 M WITH CL DILUTION OF 61.4 AND AVE. DILUTION OF 108.7
TIME TO TRAP: 28.46 SEC. PLUME DIA AT THE TRAPPING LEVEL: 3.16 M
HEIGHT OF RISE= 15.6 PERCENT OF DEPTH
65
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EXHIBIT 3 UPLUME OUTPUT FOR IPI=IPO=2
UNIVERSAL DATA FILE: MARC.IN
*3 EFFLUENT ft AMBIENT DENSITY AS G/CM3, 0.04 H/SEC CURRENT, 1x1=1x0=2
0,1,0,2,2,2,2,2,
1.266,148,.0915,0.,55.2,
0.04,90.,3.0,
7,.99744,0.,
00.00,1.02261 ,,0.04,
20.00,1.02275,,0.04,
45.00,1.02302,,0.04,
50 .00,1 .02344,,0.04,
55.00,1.02348,,0.04,
60.00,1.02365,,0.04,
60.96,1.02367,,0.04,
UPLUME VERSION 1.0 AUGUST 1985 (BASED ON OS3 VERSION 2.3 9/12/77)
UNIVERSAL DATA FILE: MARC.IN
CASE I.D. »3 EFFLUENT 8 AMBIENT DENSITY AS G/CM3, 0.04 M/SEC CURRENT, 1x1*1x0=2
INITIAL DENSITY OF THE PLUME » 0.99744 G/CM3
DISCHARGE VELOCITY = 1.301 M/S
FROUDE NUMBER = 8.5
DEPTH
0.00
20.00
45.00
50.00
55.00
60.00
60.96
DENSITY
1 .02261
1.02275
1.02302
1 .02344
1 .02348
1 .02365
1 .02367
TOTAL EFFLUENT FLOW
NUMBER OF PORTS
PORT DIAMETER
PORT SPACING
VERTICAL PORT ANGLE FROM HORIZONTAL
PORT DEPTH
1.2660 CMS
148
0.0915 M
3.00 M
0.0 DEGREES
55.20 M
COMPUTATIONS CEASE: PLUME TRAJECTORY IS HORIZONTAL
NOTE: AVERAGE DILUTION WAS 103.0 WHEN PLUMES
MERGED AT 47.08 M BELOW THE WATER SURFACE.
TRAPPING LEVEL NOT YET REACHED. AVE. DILUTION
SHOWN BELOW DOES NOT ACCOUNT FOR MERGING.
TRAPPING LEVEL = 46.58 M BELOW WATER SURFACE.
AVERAGE DILUTION - 108.7
TIME TO TRAP: 28.48 SEC. PLUME DIA AT THE TRAPPING LEVEL: 3.16 M
66
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TABLE 4. OUTPUT PARAMETERS FOR UOUTPLM
Description
Horizontal distance of plume from port orifice
Depth of plume from the surface
Plume radius
Thickness of plume element in the numerical
integration scheme
Mass of plume element
Entrainment due to impingement of the ambient
current on the plume
Aspiration entrainment
Plume dilution
Density of plume minus ambient density
expressed in sigma units
Horizontal component of the plume's velocity
Vertical component of the plume's velocity
Magnitude of the plume's velocity
Original
X
Z
B
THICK
MASS
EINS
ZWEI
DILUTION
DENDIFF
HOR VEL
VER VEL
TOT VEL
New
X
Z
PLUME RADIUS
DILUTION
DENDIFF
HORIZ VEL
VERT VEL
TOTAL VEL
Temperature of plume minus ambient temperature
(No heading printed if density option used)
TEMPDIF
67
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EXHIBIT 4 UOUTPLM OUTPUT FOR 101=100=0
UOUTPLM VERSION 1.0 AUGUST 1985 (BASED ON OS3 VERSION 2.3 5-16-79)
UNIVERSAL DATA FILE: NARC.IN
CASE I.D. #1 EFFLUENT & AMBIENT DENSITY AS G/CM3, ZERO CURRENT,
INITIAL THICKNESS OF PLUME ELEMENT > PORT RADIUS (DEFAULT)
IMPINGEMENT ENTRAINMENT COEFFICIENT - 1.00 (DEFAULT)
ASPIRATION ENTRAINNENT COEFFICIENT = 0.10 (DEFAULT)
NUMBER OF STEPS ALLOWED = 5000 (DEFAULT)
PRINTOUT INTERVAL = 50 (DEFAULT)
AMBIENT CURRENT SPEED * 0.00 M/S
INITIAL DENSITY OF THE PLUME * -2.5600 SIGMAT UNITS
FROUDE NUMBER * 8.5
DEPTH SIGMAT
(M)
0.00 22.61
20.00 22.75
45.00 23.02
50.00 23.44
55.00 23.48
60.00 23.65
60.96 23.67
TOTAL EFFLUENT FLOW - 1.2660 CMS
NUMBER OF PORTS = 148
PORT DIAMETER = 0.0915 M
PORT SPACING = 3.00 M
VERTICAL PORT ANGLE FROM HORIZONTAL - 0.0 DEGREES
PORT DEPTH
X Z
(M) (M)
0.00 55.20
0.00 55.20
0.09 55.20
0.22 55.19
0.41 55.18
0.66 55.13
1.00 55.01
1.40 54.74
1.79 54.29
2.15 53.64
2.47 52.81
2.76 51.76
3.03 50.45
3.28 48.82
3.53 46.76
*****PI_UMES MERGE
PLUME
RADIUS
(M)
0.05
0.05
0.06
0.09
0.13
0.18
0.24
0.31
0.39
0.47
0.57
0.69
0.85
1.05
1.37
AT 46.04
***** FOLLOWING CALCULATIONS
*****NORMAL TRAPPING LEVEL
3.86 43.93
4.46 41.21
2.05
4.69
DILU-
TION
1.0
1.0
1.4
2.0
2.8
3.9
5.5
7.8
11.1
15.6
22.1
31.2
44.1
62.4
88.2
M BELOW
= 55.20
DENDIFF
(SIGMA)
26.05
25.87
18.42
13.02
9.21
6.51
4.60
3.25
2.30
1.62
1.14
0.80
0.56
0.31
0.07
THE SURFACE
M
HORIZ
VEL
(M/S)
1.30
1.29
0.92
0.65
0.46
0.33
0.23
0.16
0.11
0.08
0.06
0.04
0.03
0.02
0.01
WITH AN
VERT
VEL
(M/S)
0.00
0.00
0.02
0.03
0.05
0.08
0.11
0.14
0.17
0.18
0.18
0.17
0.16
0.15
0.13
IxI*IxO=ZERO
TOTAL
VEL
(N/S)
1.30
1.29
0.92
0.65
0.46
0.34
0.26
0.22
0.20
0.19
0.19
0.18
0.17
0.15
0.13
AVE. DILUTION OF 97.6
DO NOT ACCOUNT FOR MERGING
REACHED
124.8
154.7
-0.08
-0.09
0.01
0.01
0.08
-0.01
0.08
0.01
NUMBER OF STEPS= 732
COMPUTATIONS CEASE: VERTICAL PLUME VELOCITY WENT THRU ZERO
PLUMES MERGED BEFORE TRAPPING LEVEL REACHED
TRAPPING LEVEL* 45.97 M BELOW WATER SURFACE. DIUJTION= 98.52
68
-------
EXHIBIT 5 UOUTPLM OUTPUT FOR 101=100=1
OUTFALL BUOYANT (JET) PLUME IN FLOWING, STRATIFIED AMBIENT
UOUTPLM VERSION 1.0 AUGUST 1985 (BASED ON OS3 VERSION 2.3 5-16-79)
UNIVERSAL DATA FILE: MARC.IN
CASE I.D. *2 EFFLUENT AS G/CM3, AMBIENT AS S 8 T, 0.02 M/SEC CURRENT, 1x1=1x0=1
E= 1.00 A= 0.10 ITERB* 5000 IR* 50
PORT SPACING (M) = 3.00, PORT DIA (M)
AMBIENT STRATIFICATION
0.0915, PORT ANGLE (DEG) = 0.0
DEPTH, M
0.00
20.00
45.00
50.00
55.00
60.00
60.96
SALIN
34.72
34.72
34.66
34.74
34.71
34.71
34.71
TEMP,C
26.75
26.30
25.30
24.10
23.90
23.30
23.23
SIGMAT
22.61
22.75
23.02
23.44
23.48
23.65
23.67
K FROUDE Q CURRENT
6.50E+01 8.50E+00 8.55E-03 2.00E-02
MODEL INPUT (LINE 1) AND MODEL OUTPUT
X Z B THICK MASS EINS ZWEI
O.OOE-01 5.52E+01 4.57E-02 4.57E-02 3.00E-01 1.63E-04 1.72E+00
1.56E-03 5.52E+01 4.59E-02 4.54E-02 3.02E-01 1.63E-04 1.72E+00
9.38E-02 5.52E+01 6.40E-02 3.26E-02 4.24E-01 6.61E-05 2.89E-03
2.27E-01 5.52E+01 8.99E-02 2.33E-02 6.00E-01 1.43E-04 4.09E-03
4.17E-01 5.52E+01 1.26E-01 1.67E-02 8.49E-01 3.24E-04 5.78E-03
6.82E-01 5.51E+01 1.74E-01 1.23E-02 1.20E+00 7.62E-04 8.19E-03
1.04E+00 5.50E+01 2.36E-01 9.52E-03 1.70E+00 1.73E-03 1.16E-02
1.46E+00 5.47E+01 3.04E-01 8.08E-03 2.40E+00 3.48E-03 1.65E-02
1.90E+00 5.43E+01 3.77E-01 7.44E-03 3.39E+00 6.15E-03 2.33E-02
2.33E+00 5.37E+01 4.60E-01 7.06E-03 4.80E+00 9.96E-03 3.30E-02
2.74E+00 5.28E+01 5.61E-01 6.71E-03 6.79E+00 1.55E-02 4.67E-02
3.15E+00 5.18E+01 6.86E-01 6.35E-03 9.60E+00 2.37E-02 6.60E-02
3.56E+00 5.05E+01 8.41E-01 5.96E-03 1.36E+01 3.61E-02 9.33E-02
4.01E+00 4.89E+01 1.04E+00 5.49E-03 1 .92E+01 5.64E-02 1.32E-01
4.55E+00 4.68E+01 1.35E+00 4.64E-03 2.72E+01 9.76E-02 1.86E-01
*****NORMAL TRAPPING LEVEL REACHED
*****PLUMES MERGE, WHICH IS NOT ACCOUNTED FOR IN THE FOLLOWING
5.42E+00 4.40E+01 1.99E+00 2.99E-03 3.84E+01 2.50E-01 2.58E-01
6.63E+00 4.18E+01 3.50E+00 1.36E-03 5.43E+01 3.75E-01 4.74E-02
7.33E+00 4.14E+01. 5.16E+00 8.93E-04 7.68E*01 5.31E-01 1 .37E-02
7.52E+00 4.14E+01 5.34E+00 8.72E-04 8.06E+01 5.57E-01 2.82E-02
NUMBER OF STEPS= 807
DILUTION
1.OOE+00
1.01E+00
1.40E+00
1.97E+00
2.78E+00
3.92E+00
5.54E+00
7.82E+00
1.11E+01
1.56E+01
2.21E+01
3.12E+01
4.41E+01
6.24E*01
8.82E+01
DENDIFF
2.60E+01
2.59E+01
1.84E+01
1 .30E+01
9.21E+00
6.51E+00
4.60E+00
3.25E+00
2.30E+00
1.62E+00
1 .14E+00
7.99E-01
5.57E-01
3.11E-01
7.35E-02
HOR VEL
1.30E+00
1.29E+00
9.26E-01
6.60E-01
4.73E-01
3.40E-01
2.46E-01
1.80E-01
1.33E-01
1 .OOE-01
7.66E-02
6.00E-02
4.83 E-02
4.00E-02
3.42E-02
VER VEL
O.OOE-01
3.08E-04
1.57E-02
3.31E-02
5.42E-02
8.06 E-02
1.12E-01
1.43E-01
1.64E-01
1.74E-01
1.75E-01
1.70E-01
1.62E-01
1.51E-01
1.27E-01
TOT VEL
1.30E+00
1.29E+00
9.26E-01
6.61E-01
4.76E-01
3.50E-01
2.71E-01
2.30E-01
2.12E-01
2.01E-01
1.91E-01
1 .80E-01
1.69 E-01
1.56 E-01
1.32 E-01
CALCULATIONS
1.25E+02-8.55E-02 3.00E-02 7.95E-02 8.50E-02
1.76E+02-7.92E-02 2.71E-02 2.76E-02 3.86E-02
2.50E+02-5.93E-02 2.50E-02 4.46E-03 2.54E-02
2.60E*02-5.66E-02 2.48E-02-1.59E-03 2.48E-02
COMPUTATIONS CEASE: VERTICAL PLUME VELOCITY WENT THRU ZERO
PLUMES MERGED AFTER TRAPPING LEVEL REACHED
TRAPPING LEVEL- 46.00 M BELOW WATER SURFACE, DILUTION 99.06
69
-------
EXHIBIT 6 UOUTPLM OUTPUT FOR 101=100=2
UOUTPLM VERSION 1.0 AUGUST 1985 (BASED ON OS3 VERSION 2.3 5-16-79)
UNIVERSAL DATA FILE: MARC.IN
CASE I.D. #3 EFFLUENT S AMBIENT DENSITY AS G/CN3, 0.04 N/SEC CURRENT, 1x1=1x0=2
INITIAL THICKNESS OF PLUME ELEMENT
IMPINGEMENT ENTRAINMENT COEFFICIENT
ASPIRATION ENTRAINMENT COEFFICIENT
NUMBER OF STEPS ALLOWED
AMBIENT CURRENT SPEED
INITIAL DENSITY OF THE PLUME
FROUDE NUMBER
PORT RADIUS (DEFAULT)
1.00 (DEFAULT)
0.10 (DEFAULT)
5000 (DEFAULT)
DEPTH
(M)
0.00
20.00
45.00
50.00
55.00
60.00
60.96
SIGNAT
22.61
22.75
23.02
23.44
23.48
23.65
23.67
TOTAL EFFLUENT FLOW
NUMBER OF PORTS
PORT DIAMETER
PORT SPACING
VERTICAL PORT ANGLE FROM HORIZONTAL
PORT DEPTH
NUMBER OF STEPS* 843
0.04
-2.5600
8.5
M/S
SIGMAT UNITS
1.2660 CMS
148
0.0915 H
3.00 M
0.0 DEGREES
55.20 M
COMPUTATIONS CEASE: VERTICAL PLUME VELOCITY WENT THRU ZERO
PLUMES MERGED AFTER TRAPPING LEVEL REACHED
TRAPPING LEVELS 46.03 M BELOW WATER SURFACE, DILUTION" 100.69
70
-------
TABLE 5. OUTPUT PARAMETERS FOR UDKHDEN
Parameter
Description
X
Y
Z
TH1
Note:
TH2
WIDTH
DUCL
DRHO
DCCL
TIME
DILUTION
And not
and salinity
DTCL
DSCL
Horizontal distance perpendicular to the ambient current.
Horizontal distance parallel to the ambient current.
Vertical distance from the discharge port.
Local horizontal flow angle relative to the X coordinate.
TH1 tends to approach 90 degrees in all cases. Initially,
the horizontal angle of the ambient current with respect
to the diffuser (90 degrees is perpendicular).
If the ambient current is zero, set TH1=90, then X will
be parallel and Y will be perpendicular to the diffuser.
Initially, angle of the discharge port with respect to the
horizontal (0 degrees is horizontal). Thereafter, its the
angle of the plume's center line with respect to the horizontal.
Initially the plume diameter. If merging does not occur,
WIDTH is plume diameter. If merging occurs, WIDTH is the
width of the plume.
Excess velocity: (U(cl)-U(a))/(U(o)-U(ao))
U(cl) Instantaneous plume centerline velocity.
U(a) Ambient current velocity at U(cl) depth.
U(o) Initial discharge velocity.
U(ao) Ambient current velocity at the depth of discharge.
Excess density, defined the same ways as for DUCL except
densities instead of velocities.
Ratio of instantaneous centerline concentration of a tracer
to the discharge concentration of that tracer, assuming
an ambient concentration of 0.0.
Time in seconds.
Average dilution.
shown here but will replace DUCL and DCCL when the temperature
option is used:
Excess temperature, defined the same was as for DUCL except
temperatures instead of velocities.
Excess salinity, defined the same way as for DUCL except
salinities instead of velocities.
71
-------
EXHIBIT 7 UDKHDEN OUTPUT
-------
TABLE 6. OUTPUT PARAMETERS FOR UMERGE
Description
Iteration step number
Horizontal distance of plume from port
orifice
Depth of plume from the surface
Plume diameter
Plume dilution
Horizontal component of the plume's
velocity
Vertical component of the plume's
velocity
Magnitude of the plume's velocity
Density of plume minus ambient density
expressed in sigma units
Ambient current (horizontal)
Original
J
HOR COR (X)
DEPTH (Z)
DIAMETER
VOL DIL
HOR-VEL (V)
VER-VEL (V)
TOTAL VEL
DEN-DIFF
CURRENT
New
X
Z
PLUME DIAMETER
DILUTION
HORIZ VEL
VERT VEL
TOTAL VEL
DENDIFF
AMBIENT CURRENT
Time (seconds) of plume from the port
orifice
TIME
73
-------
EXHIBIT 8 UMERGE OUTPUT FOR IMI=IMO=0
UNER6E VERSION 1.0 AUGUST 1985.
UNIVERSAL DATA FILE: MARC.IN
CASE I.D. #1 EFFLUENT 8 AMBIENT DENSITY AS G/CM3, ZERO CURRENT, IxI=IxO=ZERO
ASPIRATION ENTRAINMENT COEFFICIENT
NUMBER OF STEPS ALLOWED
ITERATION PRINTOUT FREQUENCY
PRINT ARRAY AA (0=NO, 1»YES)
PRINT ARRAY AB (0=NO, 1»YES)
PRINT ARRAY AC (0=NO, 1=YES)
INITIAL DENSITY OF THE PLUME
FROUDE NUMBER
DEPTH SIGMAT U
(M) CM/S)
s
•
s
3
S
*
S
s
0.10
5000
150
0
0
0
-2.5600
8.5
(DEFAULT)
(DEFAULT)
(DEFAULT)
(DEFAULT)
(DEFAULT)
(DEFAULT)
SIGMAT UNITS
0.00
20.00
45.00
50.00
55.00
60.00
60.96
22.61
22.75
23.02
23.44
23.48
23.65
23.67
0.000
0.000
0.000
0.000
0.000
0.000
0.000
TOTAL EFFLUENT FLOW
NUMBER OF PORTS
PORT DIAMETER
PORT SPACING
VERTICAL PORT ANGLE FROM HORIZONTAL
PORT DEPTH
1.2660 CMS
148
0.0915 M
3.00 M
0.0 DEGREES
55.20 M
FIRST LINE OF OUTPUT ARE INITIAL CONDITIONS
(M)
(M)
PLUME
DIAMETER
(M)
DILU-
TION
DENDIFF
(SIGMAT)
HORIZ VERT
VEL VEL
(M/S) (M/S)
0.00
0.00
0.41
1.40
2.47
3.28
55.20
55.20
55.18
54.74
52.81
48.82
0.091
0.092
0.255
0.625
1.139
2.110
1.00
1.01
2.78
7.82
22.08
62.40
26.05
25.87
9.21
3.25
1.14
0.30
*****MERGING BEGINS
*****NOMINAL TRAPPING LEVEL REACHED
3.62 45.97 3.029 98.59
0.00
1.30
1.29
0.46
0.16
0.06
0.02
0.01
0.00
0.00
0.05
0.14
0.18
0.15
0.12
COMPUTATIONS CEASE: VERTICAL PLUME VELOCITY IS LESS THAN 0
PLUMES MERGED AND TRAPPED AT THE SAME TIME.
TRAPPING LEVEL = 45.97 M BELOW SURFACE; DILUTION
TOTAL
VEL
(M/S)
1.30
1.29
0.46
0.22
0.19
0.15
0.12
AMBIENT
CURRENT
(M/S)
0.000
0.000
0.000
0.000
0.000
0.000
0.000
98.52
74
-------
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75
-------
EXHIBIT 10 UMERGE OUTPUT FOR IMI=IMO=2
UMERGE VERSION 1.0 AUGUST 1985.
UNIVERSAL DATA FILE: MARC.IN
CASE I.D. #3 EFFLUENT 8 AMBIENT DENSITY AS G/CM3, 0.04 M/SEC CURRENT, 1x1=1x0=2
ASPIRATION ENTRAPMENT COEFFICIENT
NUMBER OF STEPS ALLOWED
PRINT ARRAY AA (0=NO/ 1=YES)
PRINT ARRAY AB <0=NO, 1=YES)
PRINT ARRAY AC (0=NO, 1=YES)
INITIAL DENSITY OF THE PLUME
FROUDE NUMBER
s
s
s
s
s
-
s
0.10
5000
0
0
0
-2.5600
8.5
(DEFAULT)
(DEFAULT)
(DEFAULT)
(DEFAULT)
(DEFAULT)
SIGMAT UNITS
DEPTH
(M)
SIGMAT
U
(M/S)
0.00
20.00
45.00
50.00
55.00
60.00
60.96
22.61
22.75
23.02
23.44
23.48
23.65
23.67
0.040
0.040
0.040
0.040
0.040
0.040
0.040
TOTAL EFFLUENT FLOU
NUMBER OF PORTS
PORT DIAMETER
PORT SPACING
VERTICAL PORT ANGLE FROM HORIZONTAL
PORT DEPTH
1.2660 CMS
148
0.0915 M
3.00 M
0.0 DEGREES
55.20 M
COMPUTATIONS CEASE: VERTICAL PLUME VELOCITY IS LESS THAN 0
PLUMES MERGED BEFORE TRAPPING LEVEL REACHED
TRAPPING LEVEL = 47.01 M BELOW SURFACE; DILUTION = 130.35
76
-------
EXHIBIT 11 ULINE OUTPUT FOR INTER=1 AND 1X1=1X0=2
ULINE VERSION 2.0 AUGUST 1985 A LINE SOURCE OF BUOYANCY FLUX ONLY
UNIVERSAL DATA FILE: MARC2.IN
CASE I.D. EFFLUENT & AMBIENT DENSITY AS G/CM3, 1x1=1x0=2
RUN TITLE: CURRENT ANGLE PERPENDICULAR CHANG=90) TO THE DIFFUSER
ROBERTS FACTOR SA/SM = 1.41 (DEFAULT)
INTEGRATION STEP SIZE = 0.10 (DEFAULT)
INITIAL DENSITY OF THE PLUME = 0.99744 G/CM3
ROBERTS FROUDE NUMBER = 0.09
DEPTH
(M)
0.00
20.00
45.00
50.00
55.00
60.00
60.96
DENSITY
(G/CM3)
1.02261
1 .02275
1 .02302
1 .02344
1 .02348
1 .02365
1 .02367
U
(M/S)
0.040
0.040
0.040
0.040
0.040
0.040
0.040
TOTAL EFFLUENT FLOW = 1.2660 CMS
NUMBER OF PORTS = 148
PORT SPACING = 3.000 M
HORIZONTAL ANGLE - 90.0 DEGREES
PORT DEPTH = 55.20 M
TRAPPING LEVEL = 46.60 M BELOW WATER SURFACE, DILUTION = 107.12
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
RUN TITLE: CURENT ANGLE 45 DEGREES (HANG=45) TO THE DIFFUSER
ROBERTS FROUDE NUMBER = 0.09
TOTAL EFFLUENT FLOW . * 1.2660 CMS
NUMBER OF PORTS = 148
PORT SPACING = 3.000 M
HORIZONTAL ANGLE = 45.0 DEGREES
PORT DEPTH = 55.20 M
TRAPPING LEVEL = 46.48 M BELOW WATER SURFACE, DILUTION = 104.36
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
RUN TITLE: CURRENT ANGLE PARALLEL (HANG=0) TO THE DIFFUSER
ROBERTS FROUDE NUMBER = 0.09
TOTAL EFFLUENT FLOW = 1.2660 CMS
NUMBER OF PORTS = 148
PORT SPACING = 3.000 M
HORIZONTAL ANGLE = 0.0 DEGREES
PORT DEPTH = 55.20 M
TRAPPING LEVEL = 46.48 M BELOW WATER SURFACE, DILUTION = 104.36
77
-------
REFERENCES
Abdelwahed, M.S.T., and V.H. Chu. 1978. Bifurcation of buoyant jets in
crossflows. Tech. Rep. 78-1. Fluid Mechanics Lab., McGill University,
Montreal, Canada. 130 pp.
Abraham, G. 1960. Jet diffusion in liquid of greater density. J. Hydraul.
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84
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APPENDIX I
Development of Sa and h Relationship
A general relationship between plume average initial dilution and
height of rise in a linearly stratified ocean can be developed as follows:
Let
Sa = ez" (75)
where
Sa = flux average initial dilution
3 = coefficient
z = vertical coordinate, positive upward with origin at the discharge
depth
n = constant greater than 0.
Also, describe a linear density profile as
Pa(z) = PO + az (76)
where
PO = ambient density at the discharge depth
Pa = constant ambient density gradient from the discharge depth to
the water surface
Define
o(z) = dSa/dz = 3nzn-1 (77)
85
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£.
J o(z)dz =
- Sa (78)
Then
az) = P0gzn + [apnzn+1/(n-H)] (79)
Therefore
n+1
n + aenzn/(n+l)]/ezn » PO + [anz/(n+lj] (80)
Assume the plume is trapped at z=h below the water surface.
At z = h
Pj(h) - Pa(h) (81)
where
Pj(h) = average density of the plume
Pa(h) = ambient density at the equilibrium height
But the plume average density can be expressed as
Pj(h) =7a(h) + [Pa - Tfe(h)]/Sa (82)
Therefore, from Equations (75) and (80),
+ {(Pd - PO) - [anh/(n+l)]}/Bhn (83)
Equating this to equation (76) at z=h results in
86
-------
- anh/(n+l) = (Bhn) (ah)/(n+l) = Bahn+1/(n+l) (84)
In a "typical" ocean
Pd = 1 g/cm3
PO = 1.025 g/cm3
Surface = l'02* 9/cm3
Therefore
kd - PO| = +0.025 g/cm3
|anh/(n+l)|<]ah| - 0.001 g/cm3
|anh/(n+l)|«|Pd - PO|
Neglecting this term in (84) leads to
(n+l)(Pd -P0) =
or
h =
Rewriting
h = {[(n+l)/0][g(P0 - Pd)/P0][P0/(-gdP/dz)]}1/(n+1) (87)
(88)
where
G = -(g/P0) dP/dz
87
-------
9d' = Q(PO -
It should be noted that the neglect of the second term in Equation (84)
is not necessary for the formulas of Roberts.
Substituting n=l into Equation (84) gives
= eah2/2 (89)
or
h2 + (h/3) + 2(P0 - Pd)/ a = 0 (90)
or
h2 + (h/e) - (2gd'/BG) = 0 (91)
which has a solution
h = (l/2e)[-l+(H-2gd'e/G)1/2] (92)
-------
APPENDIX II
UNIVERSAL DATA FILE (UDF) "CARD" DECK
THE DATA ENTERED ON CARDS 2 THROUGH 7 MAY BE EITHER IN THE
FORMAT REQUIRED BY EACH CARD OR EACH VALUE ON THE CARD MAY
BE SEPARATED BY A COMMA (SHORT FIELD TERMINATION).
AN EXPLICIT DECIMAL POINT OVERRIDES THE FIELD DESCRIPTOR.
CARD 1 FORMATdOAS)
IDENTIFICATION OF A DATA SET WITHIN THE UDF.
CARD
2 FORMAT(8I2)
INTER =1 INTERACTIVE CONTROL OF CARDS 3 AND 4 PARAMETERS.
=0 "SINGLE" RUN USING PARAMETERS IN DATA SET ONLY.
IDFP =1 PRINT "CARD IMAGE" OF DATA SET.
=0 DO NOT PRINT CARD IMAGE OF DATA SET.
ICUTOP =1 USE OPTIONAL CARD 5 TO CHANGE CONTROL PARAMETERS
THE DEFAULT VALUES.
=0 DO NOT READ A CARD 5 (THUS CARD 5 MUST BE OMITTED).
FROM
IPI
101
IDI
IMI
ILI
IPO=IPI
100=101
IDO=IDI
IMO=IMI
ILO=ILI
INPUT PRINTOUT CONTROL FOR
OUTPUT PRINTOUT CONTROL
UPLUME
UOUTPLM
UDKHDEN (SEE NOTE 1)
UMERGE
ULINE
FOR UPLUME
UOUTPLM
UDKHDEN (SEE NOTE 1)
UMERGE
ULINE
FOR EACH OF THE PARAMETERS IPI TO ILI
=0 USE NEW (8.5 X 11) FORMAT.
=1 USE ORIGINAL FORMAT.
=2 USE CONDENSED FORMAT (USEFUL IN INTERACTIVE MODE).
NOTE! 1) IDI AND IDO ALLOWED FOR BUT PRESENTLY NOT USED
IN UDKHDEN, ENTER THE SAME VALUE AS THE OTHERS.
CARD 3 FORMAT(F10.0,I10,3F10.0)
QT TOTAL EFFLUENT FLOW (CUBIC METERS PER SEC).
NP NUMBER OF PORTS (SEE NOTE 2).
PDIA PORT DIAMETER (M), EFFECTIVE DIAMETER IF KNOWN.
VANG VERTICAL ANGLE (DEG) OF PORT RELATIVE TO THE
HORIZONTAL (90 DEGREES FOR A VERTICAL PORT).
ULINE ASSUMES VANG=90 DEG.
PDEP PORT DEPTH (M) MUST BE GREATER THAN 0.0 AND
LESS THAN OR EQUAL TO THE DEEPEST DEPTH OF THE
AMBIENT DENSITY PROFILE.
NOTE! 2) ULINE REQUIRES TWO OR MORE PORTS, FOR THE
OTHERS, IF NP=1 SPACE=1000.0 (DEFAULT) MAKING
THE MERGING FLAGS INACTIVE.
CARD 4 FORMAT(3F10.0)
UW HORIZONTAL CURRENT SPEED (M/S) (USED IN UOUTPLM ONLY).
HANG ANGLE (DEG) OF CURRENT DIRECTION WITH RESPECT TO DIFFUSER
AXIS (90 DEGREES CORRESPONDS TO A CURRENT DIRECTION
PERPENDICULAR TO THE DIFFUSER AXIS AND IF VANG=0, BOTH
THE CURRENT AND THE DISCHARGE ARE IN THE SAME DIRECTION)
(SEE NOTE 3).
SPACE DISTANCE (M) BETWEEN ADJACENT PORTS (SEE NOTE 2).
NOTE! 3) HANG NOT USED IN UPLUME. UOUTPLM AND UMERGE
ASSUME 90 DEG. UDKHDEN RANGE 45 - 135 DEG FOR
MORE THAN ONE PORT AND 0 - 180 DEG FOR A SINGLE
PORT (NOTE, SINGLE PORT ONLY: FOR VALUES GREATER
89
-------
APPENDIX II
THAN 90 DEG BUT LESS THAN OR EQUAL TO 180 DEC, THE
PROGRAM SETS HANG EQUAL TO THE SUPPLEMENTARY ANGLE).
ULINE RANGE 0 - 180 DEG.
CARD 5 OPTIONAL (INCLUDE THIS CARD ONLY IF ICUTOP
FORMAT(F5.0,2I5,3I2,6F5.0,2I5)
=1)
USED IN UNERGE
A ASPIRATION COEFFICIENT 0.1 BY DEFAULT
ITER MAXIMUM NUMBER OF ITERATIONS 5000 BY DEFAULT
IFRQ ITERATION PRINTOUT FREQUENCY 150 BY DEFAULT
NAA PRINT ARRAY AA IF =1, DO NOT IF =0 0 BY DEFAULT
NAB PRINT ARRAY AB IF *1, DO NOT IF =0 0 BY DEFAULT
NAC PRINT ARRAY AC IF =1, DO NOT IF -0 0 BY DEFAULT
(SEE LISTING OF PROGRAM UMERGE FOR CONTENTS Of ARRAYS
AA, AB, AC WHICH ARE MAINLY DEBUGGING AIDS.)
USED IN UPLUME
PS PRINTOUT "INTERVAL"
USED IN ULINE
RK RATIO OF SA/SM IN ROBERTS' EXPERIMENTS
DH INTEGRATION STEP SIZE (M)
USED IN UOUTPLM
H
E
A
ITERB
IR
INITIAL THICKNESS OF PLUME ELEMENT
IMPINGEMENT ENTRAINMENT COEFFICIENT
ASPIRATION ENTRAINMENT COEFFICIENT
NUMBER OF INTEGRATION STEPS ALLOWED
PRINTOUT INTERVAL
3. BY DEFAULT
1.41 BY DEFAULT
0.1 BY DEFAULT
.5*PDIA BY DEFAULT
1.0 BY DEFAULT
0.1 BY DEFAULT
5000 BY DEFAULT
50 BY DEFAULT
NOTE!
WHEN CARD 5 IS USED, ALL OF THE PARAMETERS NEED NOT BE
GIVEN A NEW VALUE, ONLY THE ONES TO BE CHANGED. ENTER ZERO
FOR THE OTHERS AND THERE DEFAULT VALUES WILL BE USED.
ITER, IFRQ, ITERB AND IR NOT TO EXCEED FOUR DIGITS.
NO OPTIONS AVAILABLE FOR UDKHDEN.
CARD 6 FORMAT(I10,2F10.0)
NPTS NUMBER OF DEPTHS WHERE AMBIENT TEMPERATURE, SALINITY, AND
HORIZONTAL CURRENT SPEED ARE KNOWN (NPTS MUST BE AT LEAST
EQUAL TO 2 AND NOT MORE THAN 30).
S EFFLUENT SALINITY (PPT) IF T NOT EQUAL TO ZERO
EFFLUENT DENSITY (G/CM3) IF T=0
T EFFLUENT TEMPERATURE (DEGREES CELSIUS).
IF T=0 PROGRAMS ASSUME S IS EFFLUENT DENSITY IN
G/CM3, SEE NOTE 4.
CARD 7 FORMATUF10.0)
DP( ) DEPTH IN METERS, MUST HAVE DATA FOR DP( )=0.0
SA( ) AMBIENT SALINITY (PPT) IF TA( ) NOT EQUAL TO ZERO
AMBIENT DENSITY (G/CM3) IF TA( )=0
TA( ) AMBIENT TEMPERATURE (DEGREES CELSIUS).
IF TA( )=0 PROGRAMS ASSUME SAC ) IS AMBIENT DENSITY
IN G/CM3, SEE NOTE 4.
UA( ) HORIZONTAL AMBIENT CURRENT SPEED (M/S) (USED IN UMERGE,
UDKHDEN, AND ULINE).
NOTE! 4) THERE MUST BE NPTS IMAGES OF CARD 7. ALSO, EITHER
ALL TA(I) MUST BE ZERO OR ALL NOT ZERO, OR ERRORS
IN THE INTERPRETATION OF SA( ) AND TA( ) WILL OCCUR.
IF, FOR SOME I, SA(I) IS DESIRED TO REPRESENT
AMBIENT SALINITY AND TA(I) SHOULD BE EXACTLY 0, SET
TA(I) EQUAL TO A SMALL NUMBER INSTEAD (0.000001 FOR
INSTANCE). THIS APPLIES TO S AND T AS WELL.
90
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