', /, s/t
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&EPA
United States
Environmental Protection
Agency
Office of Research and
Development
Washington, DC 20460
EPA/600/R-93/139
July 1993
Dilution Models for
Effluent Discharges
(Second Edition)
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EPA/600/R-93/139
July 1993
DILUTION MODELS FOR EFFLUENT DISCHARGES
(Second Edition)
D.J. Baumgartner1, W.E. Frick2, and P.J.W. Roberts3
1 Environmental Research Laboratory
University of Arizona, Tucson, AZ 98706
2 Pacific Ecosystems Branch, ERL-N
Newport, OR 97365-5260
3 Georgia Institute of Technology
Atlanta, GA 30332
July 22, 1993
Standards and Applied Science Division
Office of Science and Technology
Oceans and Coastal Protection Division
Office of Wetlands, Oceans, and Watersheds
Pacific Ecosystems Branch, ERL-N
2111 S.E. Marine Science Drive
Newport, Oregon 97365-5260
U.S. Environmental Protection Agency
Printed on Recycled Paper
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ABSTRACT
This report describes two initial dilution plume models, RSB and UM, and a model interface
and manager, PLUMES, for preparing common model input and running the models. Two
farfield algorithms are automatically initiated beyond the zone of initial dilution. In addition,
PLUMES incorporates the flow classification scheme of the Cornell Mixing Zone Models
(CORMIX), with recommendations for model usage, thereby providing a linkage between two
existing EPA systems.
The PLUMES models are intended for use with plumes discharged to marine and some
freshwater bodies. Both buoyant and dense plumes, single sources and many diffuser outfall
configurations may be modeled.
The PLUMES software accompanies this manuscript. The program, intended for an IBM
compatible PC, requires approximately 200K of memory and a color monitor. The use of the
model interface is explained in detail, including a user's guide and a detailed tutorial. Other
examples of RSB and UM usage are also provided.
This is Document No. N268 of the Environmental Research Laboratory, Narragansett. The
accompanying software also carries No. N268.
11
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DISCLAIMER
This document is intended for internal Agency use only. Mention of trade names or
commercial products does not constitute endorsement or recommendations for use.
111
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ACKNOWLEDGEMENTS
We acknowledge the leadership roles of Hiranmay Biswas, EPA Office of Science and
Technology, and Barry Burgan, Craig Vogt, and Karen Klima, EPA Office of Wetlands, Oceans
and Watersheds. They helped to formulate the concepts in the manual in broad terms, allocated
resources, and provided opportunities to increase the scope of our efforts.
Also, we appreciate and recognize the technical advice and assistance of Charles Bodeen, one
of the authors of the original edition, Bryan Coleman, Edward Dettmann, Kenwyn George, Norm
Glenn, Gerhard Jirka, and Mills Soldate. Other contributors include Gilbert Bogle, Wen-Li
Chiang, Michael Dowling, Karen Gourdine, Carlos Irizarry, Tarang Khangaonkar, George Loeb,
Ken Miller, Doug Mills, Tom Newman, George Nossa, Anna Schaffroth, John Yearsley, and
Chung Ki Yee. Their comments and suggestions contributed significantly to the content of this
work, however not all of their suggestions could be incorporated.
The support of Norbert Jaworski, Harvey Holm, David Young, and Mimi Johnson of EPA
ERL-N is also gratefully acknowledged.
IV
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CONTENTS
ABSTRACT ii
DISCLAIMER iii
ACKNOWLEDGEMENTS iv
GENERAL ASPECTS OF DILUTION MODELING 1
INTRODUCTION 1
REGULATORY ADAPTATION OF PHYSICAL PROPERTIES OF PLUME BEHAVIOR 4
Initial Dilution 4
Critical Initial Dilution 5
Mixing Zone 6
Dilution Factor 8
Effective Dilution Factor 10
Spacial and Temporal Variation of Plume Concentrations 11
The Dissolved Oxygen Problem 12
Recirculation, Quiescent Periods, and Other Temporal Variations 13
Effect of Wastewater Flow on Dilution 16
Depth as a Factor 18
Offshore Distance and Depth 19
Submerged Driftflow, Upwelling, Wind Drift 19
Dye Tracing of Plumes 19
Spatial Averages and Discrete Values 20
Regulatory Use 21
Verification Sampling 23
ENTRAINMENT FROM OTHER SOURCES AND RE-ENTRAINMENT 24
Regulatory Background 24
Significant Amounts 25
Relationship of Ambient Dilution Water to Plume Concentrations 25
Entrainment From Other Sources 28
Re-entrainment from Existing Discharge 31
Entrainment and Re-entrainment in Estuarine Discharges 32
Use of an Intrinsic Tracer 33
Salinity as a surrogate effluent tracer 33
FRESHWATER DISCHARGES OF BUOYANT EFFLUENTS 34
NEGATIVELY BUOYANT PLUMES 35
Nascent Density: Thermal Discharges in Cold Water 36
PARTICULATE DISCHARGES 37
USER'S GUIDE TO THE MODEL INTERFACE, "PLUMES" 39
SYSTEM REQUIREMENTS AND SETUP 39
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INTRODUCTION 40
PLUMES STRUCTURE 41
INTERFACE CAPABILITIES 44
COMMANDS 45
Conventions 45
The Main Menu 46
The Configuration Menu 48
The Movement Commands Menu 51
Other Useful Editing Commands 53
The Miscellany Menu 54
A TUTORIAL OF THE INTERFACE 57
EXAMPLE: PROPOSED SAND ISLAND WWTP EXPANSION 57
Introduction 57
Anaysis 58
STEP 1: Collect Pertinent Information 58
STEP 2: Input the Sand Island Information 58
STEP 3: Run Initial Dilution Models 69
STEP 4: Analyze the Model Results and Make Adjustments 73
STEP 5. Using the Results in the Decision Making Process. 79
EXAMPLE: CORMIX1 COMPARISON, DENSITY, AND STABILITY 81
INTRODUCTION 81
PROBLEM 82
ANALYSIS 83
General Considerations 83
Ambient Profile Simplification 88
Density: The Linear and Non-linear Forms of UM 91
THE ROBERTS, SNYDER, BAUMGARTNER MODEL: RSB 95
INTRODUCTION 95
DEFINITIONS 96
MODEL BASIS 97
MODEL DESCRIPTION 98
EXAMPLES 99
Introduction 99
Seattle Example: Linear Stratification - Zero Current 100
Seattle Example: Linear Stratification - Flowing Current 103
Seattle Example: Model Extrapolation 104
Seattle Example: Non-Linear Stratification. 106
Multiport Risers Example 108
DESIGN APPLICATIONS 110
VI
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UM MODEL THEORY 111
PERSPECTIVE 111
BASIC LAGRANGIAN PLUME PHYSICS 112
The Plume Element 112
Conservation Principles 115
Entrainment and Merging 116
MATHEMATICAL DEVELOPMENT 117
Basic Model Theory 117
Plume Dynamics 119
Boundary conditions and Other Pertinent Relationships 124
Merging 126
Average and Centerline Plume Properties 128
Experimental Justification of the
Projected Area Entrainment Hypothesis 130
FARFIELD ALGORITHM 133
PLUMES IMPLEMENTATION 133
REFERENCES 137
APPENDIX 1: MODEL RECOMMENDATIONS 145
JUSTIFICATION FOR USES OF PLUMES MODELS IN FRESH WATER 145
MODEL RECOMMENDATIONS TABLES 145
General Considerations 145
Caveats 147
Description and Usage 147
Single Port Diffuser Model Recommendations: Table V 148
Table V: Columns 149
Table V: Rows 150
Multiport Outfall Model Recommendations: Table VI 151
Table VI: Columns and Rows 151
SURFACE DISCHARGES 152
OTHER VIEWPOINTS AND RECOMMENDATIONS 152
APPENDIX 2: THE DIFFUSER HYDRAULICS MODEL PLUMEHYD 153
MODEL DESCRIPTION 153
MODEL USAGE 153
PLUMEHYD COMPUTER LISTINGS 154
Pascal Version of PLUMEHYD 154
Sample Input File 158
Sample Output File 159
Vll
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APPENDIX 3: SUPPORT FOR TABLE I (CHAPTER 1) 161
INPUT AND OUTPUT FOR CASE 1 161
APPENDIX 4: MESSAGES AND INTERPRETATIONS 163
CORMDC WINDOW RECOMMENDATIONS 163
DIALOGUE WINDOW MESSAGES 165
UM RUN TIME MESSAGES 171
RSB RUN TIME MESSAGES 175
FARFffiLD MODULE MESSAGES 177
APPENDIX 5: UNIVERSAL DATA FILE FORMAT (Muellenhoff et al, 1985) 179
INTRODUCTION 179
THE UNIVERSAL DATA FILE 179
Vlll
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GENERAL ASPECTS OF DILUTION MODELING
INTRODUCTION
Pollution control authorities frequently employ buoyant plume models to simulate expected
concentrations of effluent contaminants in ambient receiving waters. During the decade of the
1980s a great deal of attention was given to the subject because of the U.S. Environmental
Protection Agency's (EPA) regulation of publicly owned municipal wastewater discharges to
marine waters (USEPA, 1982). The central feature of this regulation was a modified permit
based on an applicant demonstrating the environmental acceptability of less than secondary
treatment, consistent with criteria listed in section 301(h) of the federal Clean Water Act.
A number of models and other methods, e.g., field data, were used in this context, primarily
to demonstrate compliance with a variety of applicable regulatory requirements of local, regional,
state, and federal agencies. In addition, models were used to aid in the design of marine
monitoring programs and in the design of new or modified ocean outfall pipelines and diffuser
systems. In 1985 EPA published a user's guide to five models used in these activities
(Muellenhoff et al., 1985) although the models had been distributed previously (e.g. Teeter and
Baumgartner, 1979) and used for years in many applications.
Possibly because of the popularity and the endorsement associated with the EPA user's guide,
the models were applied by regulatory agencies, designers, and dischargers to problems beyond
those for which they were originally intended. Some applications involved industrial wastes,
drilling fluids from offshore oil exploration and development projects, and effluent discharge into
freshwater systems, both lakes and rivers. Staff in the EPA offices were asked frequently to
assist with these applications, and many users requested EPA to develop a more general model,
or specific models for each situation. As a result of these requests, this user's guide and revised
computer programs are provided. With respect to the 1985 models (Muellenhoff et al., 1985),
UOUTPLM and UDKHDEN are neither reissued nor addressed herein, UPLUME is provided as
a separate file but neither recommended nor addressed, ULINE is provided as a separate file also
and was recommended in the first edition as an extension of RSB while RSB was not applicable
to unstratified conditions, which is no longer true, and UMERGE is modified, extended, and
replaced by the resident model UM. To the extent that PLUMES, described immediately below,
facilitates UDF file generation, all earlier models are supported by PLUMES.
Both RSB and UM are contained in and managed by the interface program PLUMES. In
addition, PLUMES contains two farfield algorithms and the CORMIX1 flow categorization
scheme (Jirka and Hinton, 1992). General recommendations for the use of RSB, UM, and
CORMIX are issued by PLUMES and explained further in Appendix 1.
The model UM is described subsequently in the manuscript, as is RSB, a model based on
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General aspects of dilution modeling
hydraulic model studies by Roberts (1977) and Roberts, Snyder, and Baumgartner (1989 a,b,c).
The new UM model provides essentially equivalent results as UMERGE, in fact, UM may
be interpreted to mean "Updated Merge". However, UM possesses considerably more capabilities
than its predecessor.
New subjects treated in this report include effluent material discharged at an arbitrary vertical
angle to address the cases of positively buoyant material discharged downward, and negatively
buoyant material discharged upward. These situations are handled by PLUMES. Discussion is
provided on the problem of particulates in the waste stream, as this is becoming recognized as
one of the more insidious problems of water pollution control, and on the possible use of the
models in freshwater systems. Verification based on field and laboratory data is addressed as
is information on uncertainty of predictions.
Subjects such as mixing zones and initial dilution concepts discussed in the 1985 report are
repeated, sometimes verbatim, and updated with current interpretations. Discussion of the
physical basis of models is expanded.
Readers of the earlier report (Muellenhoff et al., 1985) will also notice some deletions and
changes. The computer codes for the programs are not included in the manuscript nor in the
diskettes generally provided. (However, the RSB and UM model kernels are available on
request.) Another is that the executable models are to be provided on diskette by the EPA
marine research laboratory in Newport, Oregon, rather than by NTIS. (They will also be made
available on the CEAM, Athens Bulletin Board Service.) These procedural changes are related.
Due to user experiences as well as work conducted by EPA it is at times necessary to correct or
improve the computer codes. It now appears that changes will occur sufficiently frequently so
that it will be more effective to provide current models to users directly from EPA rather than
from NTIS. New diskettes distributed by EPA will be accompanied by brief narratives describing
the improvements in the physics or the computational routines that take place following
publication of this report. These adjustments are judged to be too difficult to arrange on a timely
basis through NTIS.
The authors assume readers will have some familiarity with terminology of buoyant plume
mechanics, either as applied in regulatory practice or in fluid mechanics generally. Terms used
in equations are defined in the text, frequently using different symbols than in original works
cited. In different parts of the document, a symbol may represent different quantities, however,
the meaning should be clear from the context. Terms and relationships are also explained in the
"Help" screens of the interface program PLUMES. General definition sketches are shown in
Figure 1.
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General aspects of dilution modeling
Cross—section
horizont*! dictate
Plan
parts
Figure 1. Definition sketch.
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General aspects of dilution modeling
REGULATORY ADAPTATION OF PHYSICAL PROPERTIES OF PLUME BEHAVIOR
Initial Dilution
Initial dilution is the dilution achieved in a plume due to the combined effects of momentum
and buoyancy of the fluid discharged from an orifice, and due to ambient turbulent mixing in the
vicinity of the plume. The rate of dilution is quite rapid in the first few minutes after exiting the
orifice and decreases markedly after the momentum and buoyancy are dissipated. Figure 2
CO
O
CO
fl
o
10000
1000
100
10
03 O
Diffuser
Length
100m
600m
•Drift Flow
0 200 400 600 800 1000
Time (minutes)
Figure 2. Plume dilution as a function of time.
schematically represents the relative dilution factors achieved in buoyant plumes and in the
subsequent drift flow region under low to moderate current conditions.
Ambient currents will also influence the rate of dilution during the buoyant rise of the plume
irrespective of jet momentum and buoyancy. As current speed increases so does initial dilution.
This is shown in Figure 3 from Baumgartner et al. (1986) for certain west coast conditions using
the models in Muellenhoff et al. (1985). UPLUME, not including current, gives constant dilution.
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General aspects of dilution modeling
Open Ocean, Unstratified
5 10 15
Current Speed (cm/sec)
Figure 3. Dilution as a function of current speed.
It is useful to compute expected
dilutions and plume locations under the
vast range of current regimes likely to be
encountered near an outfall. The
information would be useful in optimizing
monitoring programs intended to sample
the distribution of ambient values of
effluent constituents in analyzing the
effectiveness of regulatory controls.
Given sufficient data on environmental
impacts in the region and accurate
exposure data, one could imagine that
regulatory agencies might evaluate the
societal benefits derived from modifying
the definition of critical initial dilution.
For example, perhaps the twenty or thirty
percentile value of current might be
employed, rather than zero current or the
ten percentile current, if data show only
a slightly increased adverse effect! The
increased uncertainty, and risk, associated
with calculated values based on these still developing physical models of turbulent dispersion
mechanics is not always recognized. It is a cost of attempting to describe more completely the
behavior of the plume under actual conditions.
Critical Initial Dilution
The models described in this report are not constrained by any regulatory definition of
allowable current speed, although there are limiting current conditions that each model can
simulate. In relation to permit requirements of regulatory agencies it is necessary to think of
"allowable" initial dilution factors, or "critical" initial dilution factors based on conservative
values of parameters in addition to current speed. "Critical" values in terms of EPA's 301(h)
permit requirements (USEPA, 1982) include consideration of current direction as well as speed,
and other environmental and wastewater factors. The importance of current direction will be
discussed subsequently in the report.
The California Ocean Plan (State Water Resources Control Board, 1988) requires zero current
speed to be used in computing initial dilution values intended to predict compliance with permit
conditions. Whether intended or not, this regulatory approach results in a predicted initial
dilution that is less uncertain than would be obtained when the effects of current are included.
In the EPA regulations for a permit modified by section 301(h) of the Clean Water Act (USEPA,
1982), EPA allowed the lowest ten percentile current to be used in computation of the critical
initial dilution value. In many coastal settings the ten percentile value is below 5 centimeters per
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General aspects of dilution modeling
second (cm sec-1), i.e., 0.16 ft sec-1, or less than 0.1 knot. At current speeds this low there is
essentially no effect on the rate of dilution.
Other environmental and wastewater flow considerations are not discussed, primarily because
the models are generalized to the extent that any set of regulatory constraints may be handled in
use of the models. Furthermore, these parameters do not influence the physics of plume
behavior.
Mixing Zone
Permit conditions of regulatory agencies usually allow exceptions within a mixing zone
adjacent to the point of discharge. With respect to EPA's 301(h) regulations, the rationale and
the precautions associated with mixing zones and the relationships to initial dilution are described
in Muellenhoff et al. (1985). The use of the initial dilution models since 1985 in defining
mixing zones and in computing allowable discharge concentrations has suggested the need for
additional discussion.
In nature, regulatory restrictions notwithstanding, the initial dilution process occurs over a
wide spatial range compared to the length of an outfall diffuser or the depth of water at the
discharge site. The effect of current on the scale of the initial dilution process is portrayed in
Figure 4. Under low current conditions, e.g. U = 0.1 m/sec, initial dilution is virtually completed
before the plume is carried downcurrent a
distance X, equal to the water depth, for
example 30 meters when the buoyancy
frequency N, a measure of density
stratification, is 0.03 per sec. In a strong
current the process can extend
downcurrent a distance equal to multiples
of diffuser lengths (Roberts et al., 1989b).
At a current speed of 1 m/sec X, would be
300 meters.
Recognizing this, what might a
regulatory agency prescribe as a mixing
zone, that is, a zone in which water
quality criteria are permitted to be
exceeded? If a conservative posture is
adopted, the agency would allow a mixing
zone of 30 meters on both sides of the
diffuser. If a more liberal view prevails a
distance of 100 meters could be
established. With the possible exception
of riverine settings, it is necessary in most
cases to describe the zone on both sides of
10000
100
o>
o
10
X,- 8.6 U/N
0.01 0.1 1 10
Ambient Current (m/sec)
Figure 4. Length of the zone of initial dilution as
a function of current speed.
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General aspects of dilution modeling
the diffuser because coastal and estuarine currents during one part of a day are likely to be about
180 degrees opposite those six hours later.
EPA has adopted the conservative posture, at least for marine outfall problems regulated
under section 301(h). Thus a smaller area of the environment is removed from the general region
protected for unlimited use. Organisms entrained into the plume would be exposed to rapidly
decreasing concentrations of pollutants and within minutes, e.g., three, would be in an
environment containing pollutants at concentrations below the safe limit. The expectation is that
most of the time, e.g., 90% of the time or more, currents are sufficiently high to cause even a
greater rate of dilution. Under high currents the concentrations at the boundary of the mixing
zone would be expected to be less than the specified criteria values and quite possibly a good
portion of the mixing zone would actually meet the necessary criteria.
This expectation has not been rigorously tested. Hydraulic model tests conducted by Roberts
et al. (1989 a, b, c) suggested that situations might exist where the expectation is not realized.
The model UM can be used to generate simulated data that might be useful to test this
assumption. A hypothetical outfall situation is described as follows:
EXAMPLE PROBLEM
Flow: 4.47 cubic meters per second
Number of 8.5 cm ports: 143
Port spacing: 7.3 m
Discharge angle: horizontal
Water depth: 76 m
PLUMES model UM was run for a range of currents, and the plume concentrations at a
downcurrent distance of 30 m were interpolated from the output data. (The Zone of Initial
Dilution, or ZID, defined in the 301(h) regulations, would be larger but, in general, mixing zone
regulations vary from state to state.) The data shown graphically in Figure 5 demonstrate that,
as currents increase, the dilution at the boundary increases to a maximum but then begins to
decrease.
(Three noteworthy inflections appear in Figure 5. At current speeds lower than those marked
(a) the plumes reach maximum height inside the mixing zone and impinge on the surface.
Adjustments have been made to cause the simulation to reach the mixing zone boundary. All
cases with currents less than (b) encounter the overlap condition to be described subsequently.
Finally, at speeds higher than (c) the plumes no longer merge.)
Assuming this example is somewhat representative, what importance should be attached to
the concentrations above a standard level at the boundary when the currents exceed a relatively
large value? Organisms entrained into the plume will have traveled with the rapidly diluting
wastefield for only a couple of minutes before the concentration is reduced below the standard,
whereas with a small current the exposure time in the mixing zone is approximately 10 minutes.
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General aspects of dilution modeling
o
CO
p
1000
800
600
400
200
0
0 20 40 60
Current Speed (cm/sec)
80
Figure 5. Dilution at the mixing zone boundary as a function of current speed.
Organisms at and beyond the boundary will then be more greatly stressed than entrained
organisms in low current conditions. If for example the regulatory authority established the
mixing zone boundary to protect a community of benthic organisms from being exposed to
concentrations above the standard, then the standard will be abrogated when currents are large.
Even in unstratified ambients it is possible that high current speeds will cause effluent streams
to hug the seabed thus placing benthic resources at greater risk. Under low currents the plumes
will rise and be retained closer to the diffuser. Entrained organisms and near-surface resources
are more at risk under this scenario. Regulatory agencies may effectively incorporate this
knowledge into mixing zone boundaries which are narrower near the surface and wider at depth
based on these model simulations.
The term "near field" was adopted in narratives associated with the 301(h) regulations to
describe the region near the outfall inside the zone of critical initial dilution, and "farfield" was
similarly meant to apply to areas possibly impacted beyond this zone. For most cases "near
field" would be consistent with the term "mixing zone".
Dilution Factor
The average dilution factor, Sa , used in some regulatory applications, including the EPA
model UM is the reciprocal of the volume fraction of effluent, ve, contained in the diluted plume.
8
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General aspects of dilution modeling
An equivalent way of expressing this term is the ratio of effluent volume plus volume of ambient
dilution water, va , to the effluent volume, as in Equation 1.
i _ (yy«)
" "
' v. v. (1)
Thus in the region immediately outside the discharge orifice the volumetric dilution factor is very
nearly 1. In some discussions of this term in other works, e.g. the California Ocean Plan (State
Water Resources Control Board, 1988), the factor is considered to be the ratio of the volume of
ambient dilution water, va , to the volume of effluent discharged, ve. In this definition the
volumetric dilution factor approaches zero near the orifice. Above a value of 30 the difference
in the two definitions is progressively less than 3%, an inconsequential amount for most
regulatory purposes.
The former definition, i.e., Equation 1 is used in this report. This is not an arbitrary decision,
but rather is based on the general equation used to calculate the contaminant concentration in the
plume. Using the continuity equation,
CP VP = C* Ve + Ca Va (2)
where
cp = Cross sectional average concentration in the plume,
vp = Volume flux of the plume,
ce = Concentration in the effluent,
ve = Volume flux of the effluent,
ca = Concentration in the ambient dilution water, and
v = Volume flux of the ambient dilution water.
a
Substituting va + ve for vp and rearranging,
" Ve + Ca Va (3)
c -
Ve+Va
The volume fraction, Equation 1, is a useful approximation of the concentration of a pollutant
in the diluted plume only if the pollutant concentration in the ambient dilution water is very low
compared to the concentration in the effluent. Thus if Sa = 30 (which means the effluent is
diluted with 29 volumes of ambient water), the concentration of any volumetric tracer or
conservative pollutant in the effluent is one thirtieth the concentration in the effluent only if the
ambient concentration is zero. In the case of zero ambient concentration Equation 3 reduces to:
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General aspects of dilution modeling
C V
\ + Va
Dividing both sides by ce and inverting,
f* V + V
-i = -2 2 = Sa (5)
c, v«
Equation 5 demonstrates that for the special case of zero ambient concentration the volumetric
dilution factor also describes the dilution of a pollutant. In most regulatory uses of the plume
models, however, it is necessary to consider the actual, nonzero, ambient concentration of the
suite of pollutants in the effluent. In the remainder of this report the term "effective dilution
factor" (5aei) is used to describe the dilution achieved for each pollutant in a plume. That is^
c.
Sat = — (6)
1 C.
where the index, i, is used to demonstrate that in determining the final concentration of a
pollutant in the diluted effluent the effective dilution must be determined for each pollutant
individually.
Effective Dilution Factor
It is instructive to recognize that SMi is not necessarily constant for a suite of pollutants in
a discharge for any given volumetric dilution factor, Sa. This is so because the ratios cei / cai are
not necessarily constant, and the volumetric dilution factor is determined only by the density of
the plume irrespective of the contribution made by any of the pollutants individually. The
effective dilution factor, Saei, can be determined from Equation 6 for each pollutant by first
determining the concentration of each pollutant in the plume. The general solution is related to
the volumetric dilution factor, Sa, through Equation 3. First, multiply the right side of Equation
3 by ve I ve , giving
V« (7)
Next, recalling Equation 5, substitute Sa-l for va I ve , and 1 I Sa for ve I (ve+va), Equation 7
becomes
10
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_
PI 5
General aspects of dilution modeling
(8)
a
This is simplified to
c =.!>!- f* + c (9)
*' S S "'
a a
which is analogous to equation (1) given in Muellenhoff et al. (1985).
The advantage of Equation 9 is that for many situations the computer program for a plume
model needs to be run only once, that is, to obtain Sa. With Sa in hand cpi can be computed
repeatedly using paired values for cei and cai. If cai is not uniform over the depth through which
the plume rises, an average value can be used to provide an estimate of cpi. However, this is
only an estimate as entrainment is not generally a linear function of the vertical position of the
plume in the receiving water. The new model, UM, described in this report, accepts a tabular
input of the vertical distribution of ambient concentration and computes the actual, effective
diluted concentration. Since this model is quick and easy to run, there is only a modest
advantage in using Equation 9 to obtain subsequent estimates of cpi.
However with the CORMIX models and with RSB the dilution factors and plume
concentrations provided are based strictly on volumetric dilutions and must be corrected for the
ambient background. For a first order correction it is possible to assume the rate of dilution is
uniform over the rise to the trapping level so that if the ambient concentration is uniform over
that depth a simple correction can be applied using Equation 9. In the simple example problem
given above, the ambient pollutant concentration is given as 1.6 concentration units, thus a
volumetric dilution factor of 316 results in a plume concentration of 1.91, or an effective dilution
factor of only 52.4! The influence of background on effective dilution is apparent.
Spatial and Temporal Variation of Plume Concentrations
The concentrations of water quality indicators, such as contaminants and desired constituents
(e.g., dissolved oxygen) are neither uniform nor steady with respect to the space and time scales
involved in regulating the concentrations at the end of the mixing zone. The nonuniformity of
constituents in the horizontal extent of an outfall diffuser is generally not investigated and is
usually assumed to be uniform, as is the incremental volumetric flux. If nonuniformities along
the length of the diffuser are encountered the dilution model can be run for each segment of the
diffuser that may be assumed uniform. A separate hydraulic model to compute the distribution
of port flows along the length of the diffuser is described in Appendix 3, and is included in the
software. Vertical nonuniformity is more commonly encountered in design, performance analysis,
11
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General aspects of dilution modeling
and compliance monitoring.
Vertical nonuniformity is important to consider from the standpoint of the constituent
concentrations in the ambient receiving water, i.e., the dilution water mixed with the effluent
being discharged. The variations in the vertical are due to physical processes influencing the
advection of ambient water into the region of the discharge, and, for some constituents,
antecedent biological and chemical processes that have changed the form or concentration of the
constituent. Typically, field observations during synoptic surveys are relied on to provide vertical
profiles of the water quality indicators. Dissolved oxygen (DO) is an example of one water
quality indicator that exhibits vertical nonuniformity in many lake, estuarine, and coastal
situations. The concentration of DO in a plume is important to determine because of direct
biological effects, and because the strategy for effective regulation of DO at the end of the
mixing zone is strongly dependent on the relative influence of effluent constituents and the
vertical profile of receiving water constituents. The way in which the dilution models are used
to analyze the plume DO concentration illustrates a method for dealing with other ambient
nonuniformities.
The Dissolved Oxygen Problem
The DO concentration in a plume is affected by the DO in the effluent, the chemical and
biological constituents in the effluent which exert a DO demand, chemical and biological demand
factors in the seabed, and by oxygen demand in the water column carried by currents into the
region of mixing. The DO demand in the effluent is conveniently represented by the effluent
parameter called the Immediate Dissolved Oxygen Demand, IDOD. According to Standard
Methods for the Examination of Water and Wastewater (APHA, 1975), IDOD is the amount of
oxygen consumed in a 15 minute reaction time. (Later additions of Standard Methods do not
include this method because the authors were not able to interpret the significance of the
measurement in relation to total oxygen demand.) Since mixing zones established under the EPA
regulations for 301(h) permits represent travel times generally of the order of less than 10
minutes, IDOD is a conservative estimate of the mixing zone demand. On this time scale
chemical and biological demands in the ambient are inconsequential although for farfield water
quality considerations after initial dilution they are frequently decisive. Under these assumptions
the concentration of DO in the plume, CDO, is found using the equivalent of Equation 9 with an
additional term to represent the immediate demand, viz:
" - IDOD - c
LUUU C_ (1Q)
To solve this equation it is necessary to have field data on the cDOa profile; the values of cDOe
and IDOD being derived from laboratory analyses. In many cases the cDOa is low near the seabed
due to benthic demand, reaches a maximum at an intermediate depth in the water column, and
then is constant or slightly decreasing in the near surface layer of the receiving water.
12
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General aspects of dilution modeling
In some coastal regions there are deep permanently anoxic or hypoxic basins. Lakes and
reservoirs may also have such basins, perhaps only seasonally. If an outfall is placed in an
oxygen-poor basin and the vertical density structure is such that the plume rises into near surface
waters, the resulting DO in the plume will be very nearly the same as the deep water, thus quite
likely abrogating a desired DO standard irrespective of the amount of oxygen demand in the
effluent. While the violation of the standard is not due to the pollutant discharged in this case,
it is due to the discharge of effluent! If aquatic organisms in the surface layers are sensitive to
low oxygen concentrations it will matter little to them if the deficit is due to effluent or deep
oxygen-poor water forced to the surface by the buoyant effluent. The potential for "forced
upwelling" or "effluent pumping", as it has at times been labeled, should be considered in the
design of outfalls, both from a standpoint of selecting the site, and of the mechanics influencing
the height of rise of the plume. By careful balancing of those design factors which influence
final plume concentration, optimum strategies can be developed for achieving ambient standards.
Equation 10 is analogous to Equation VI-7 in EPA's Revised Section 301(h) Technical
Support Document (USEPA, 1982). However it is not stated that the tabular listing (page VI-21)
of IDOD contributions to the final plume dissolved oxygen concentration are negative
contributions.
Recirculation, Quiescent Periods, and Other Temporal Variations
The models reported in Muellenhoff et al.(1985) were steady state models, as are the models
used in this report and, as such, they do not take into account temporal variations in any of the
variables. For most applications this limitation should not be a problem. In the EPA 301(h)
regulations the effective initial dilution is determined for a set of effluent and receiving water
conditions that approaches a worst case scenario, that is, there is only a very low probability that
there would be physical circumstances under which a predicted final plume concentration would
be exceeded. The models can be used repeatedly however to generate a data set for a range of
values expected or observed in nature, as done for example to construct Figure 3 showing the
effect of different current speeds on volumetric dilution. Although this result is not a
time-variable solution to a buoyant plume problem the rate of change in dilution between two
current speeds is not an important consideration in regulatory practice, because the effect of
current on plume behavior is nearly instantaneous. Thus it is eminently satisfactory to use the
steady state model at discrete time steps.
Data sets can be generated to show the frequency distribution of currents and associated
dilutions at a discharge site, as in Figure 6 (Baumgartner et al., 1986). From an environmental
management perspective it may be important to investigate the distribution of dilutions achieved
as a result of seasonal changes. Figure 7 shows the monthly distribution of initial dilution values
calculated by UMERGE (Muellenhoff et al., 1985) for incremental changes in tidal currents
superimposed on a steady longshore current for a typical U.S. west coast discharge site.
The dramatic effect of current speed, in this case the effect of tidal current, shown in Figure
7 demonstrates that most of the time dilutions at the end of the buoyant plume phase will be
13
-------
General aspects of dilution modeling
o
o
a
co
I
§
UMERGE
Southwest Coast — Open Ocean
o
"W
_o
o
-------
General aspects of dilution modeling
3g .
Figure 7. Simulated annual variation in dilution.
15
-------
General aspects of dilution modeling
Dim
700.21 "
407. )6 '
234. 12 '
1.07
2. SO
1.33
0. 17
Locnt
Figure 8. UMERGE dilution response surface as a function of Froude number and
stratification.
Effect of Wastewater Flow on Dilution
Depending on the densimetric Froude number at the discharge port, the effect of increased
effluent flow per port on dilution can be shown to be detrimental, insignificant, or favorable.
With low Froude numbers as frequently found with municipal ocean outfalls, an increase in flow
causes a decrease in dilution, while at higher Froude numbers, as might be found with modem
power plant cooling water discharges, an increase in discharge results in an increase in dilution.
According to Rawn, Bowerman, and Brooks (1960), the 1930 data from the Los Angeles outfall
provided a guide to the conditions under which the transition occurs (see Figure 9).
If density stratification or shallow water prevents the plume from rising very far, the
transition to increased dilution is seen in this graph to occur at lower Froude numbers. This
reflects the importance of high jet-like plume velocity near the discharge causing an increased
rate of entrainment and a greater horizontal travel before reaching the trapping level or the
surface. In deep water the vertical travel of the plume and the entrainment caused by buoyancy
16
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General aspects of dilution modeling
10
20 30 40
Densimetric Froude Number
Figure 9. Examples of plume rise, dilution (SJ, and densimetric Froude number (effluent
flow) relationships.
over the major portion of the travel distance play an increasingly greater role than conditions near
the port in determining the final dilution. In deep water the transition to increased dilutions
would be seen only at very high effluent flows.
An example of the effect of wastewater discharge flow on dilution can be seen in Figure 10
also. In this graphic the negligible effect of low current speeds as simulated by the model
UMERGE (Muellenhoff et al., 1985) is shown in the Roberts' Froude number (Roberts, 1977).
Increased effluent flow causes the densimetric Froude number to increase from 0.7 to 7, resulting
in a decrease in dilution in deep water from about 250 to 150.
Table I shows the effect of increased effluent flows calculated by the UM and RSB models
described in this report for the outfall characteristics described in more detail in Appendix 2.
UM predicts a more substantial negative effect on dilution for an increase in flow from 100
MGD to 275 MOD. The volumetric dilution (Sa) at the surface at 100 MGD is 316 while at 275
MGD it is 187. The volumetric dilution is not calculated at the surface by RSB; instead the RSB
model calculates the volumetric dilution at the end of the buoyancy dominated region assuming
the water depth is sufficiently great to accommodate the complete initial dilution regime. At a
flow of 100 MGD through this outfall the water depth would have to be greater than 65 meters
to achieve a dilution of 394. At a flow of 275 MGD, the water depth would have to be greater
17
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General aspects of dilution modeling
SN • 46
SP - 0.0001
0. 1
0.01
0.001
E -4
E -5
1
10
Froude No.
E -6
E -7
"100
Roberts' Number
Figure 10. Dilution response surface as a function of Roberts Froude number and the
densimetric Froude number.
Q
(MGD)
100
137
183
228
275
Dilution
UM
316
260
222
201
187
Factor
RSB
394
387
384
383
382
90 meters to achieve a dilution of 382. Later in this Table I. Dilution factor, S,,
report example problems which are more appropriately predicted by UM and RSB vs.
defined will be provided to examine the differences effluent flow
between the two models and the dilution values they
calculate. The effects of port spacing and density
stratification will be explored in these examples also.
Depth as a Factor
Depth as a governing factor in the effective
placement of ocean outfalls has taken on significance that
is not always warranted. It is true that all other things
being equal, the greater the extent of vertical travel
experienced by the plume, the greater is the amount of
entrainment. If a location is chosen with greater depth but poorer circulation, the net result may
be less effective dilution of wastes than placement in a shallower but more open coastal area.
This is the major concern with placement of outfalls in fjords, embayments, and, in some cases,
estuaries, but this consideration must also be kept in mind when canyons, trenches, and deep
basins offshore are considered as outfall sites. The implications for seabed accumulation of
effluent paniculate matter may be more important in the long run than the water column
implications of re-entrained effluent
18
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General aspects of dilution modeling
Offshore Distance and Depth
The rationale for great depth as a factor in design of ocean outfalls seems to have been
recognized empirically as a result of observations by A. M. Rawn on the Los Angeles outfall
built in 1937 (Pomeroy, 1960). The primary consideration evidently was to reduce nearshore
pollutant (coliform) concentrations through greater travel times, and thus more die-off, associated
with outfalls further offshore. Greater depth, at least in the Southern California Bight was a
gratuitous benefit of offshore distance. Through thoughtful analysis of monitoring data Rawn and
coworkers recognized that lower beach coliform counts in the summer were in large part related
to summer density stratification at the discharge site. In designs for subsequent outfalls
submergence of the diluted sewage field was a conscious objective in addition to distance from
shore (Brooks, 1956). This dependence on depth took on unique significance in the early
legislative history of the 301(h) amendment, and was even proposed as the basis for granting
waivers in estuaries! EPA scientists suggested that physical criteria relating to effective seaward
displacement of pollutants from estuaries would be necessary in addition to depth and these were
then included in the final language.
Submerged Driftflow, Upwelling, Wind Drift
The practice of designing diffusers to retain the drift field in the pycnocline, a region of large
vertical gradient in density, below a surface layer may result in adverse implications for nearshore
water quality due to characteristic upwelling of deep water along some major continental margins.
This may not be a problem in the Southern California Bight, but needs to be considered when
exporting southern California technology to other locations. It has been mentioned as a factor
to be considered in outfall designs for the Oregon coast (Behlke and Burgess, 1964). The
concentration of contaminants carried nearshore may be higher than if the outfall had been
designed to take advantage of greater dilution offered by the full depth of water. This is a
tradeoff to be considered in light of the potential damage caused by onshore drift of surface
waters under prevailing winds in certain parts of the year.
By careful attention to wind, current and density patterns, it may be possible to design an
outfall so that the plume is submerged when there is the least chance of upwelling, and above
the pycnocline when there is the least chance of onshore winds. Most outfalls do not have the
design or operational luxury to allow for opening or closing some of the ports. For those that
do there is an additional option for adjusting the height of rise of the diluted plume.
Dye Tracing of Plumes
Dye tracing is a well known technique used in hydraulic models and prototype outfall
settings, although the cost of added tracers in prototype situations is considerable because of the
large volumetric flow rates and large dilutions usually achieved within several tidal cycles. The
rate of dye addition (Qd) to the effluent flow ve needed to provide a dye concentration of cd
following dilution of Sa is:
19
-------
General aspects of dilution modeling
Qd = Vg ** *' ' (11)
where
cca = specific gravity of diluted plume
ad = specific gravity of dye solution
W = weight fraction of dye in stock solution.
The required dye rate in gallons per hour is shown in Figure 11 for various dilution factors
and effluent flows in MOD to achieve an ambient dye concentration of 1 ppb. Rhodamine WT,
typically used in dye studies, is available as a 20% solution (ad = 1.19) in small (15 gallon)
drums.
Spatial Averages and Discrete Values
Some buoyant plume models produce dilution factors in terms of the centerline concentration,
sometimes referred to as the "minimum" dilution for the cross section of the plume at a given
distance downstream from the orifice. As the plume radius continues to expand with increasiqg
distance, the minimum dilution progressively increases. For example the centerline (minimum)
dilution at a distance of 6 meters from the diffuser port may be 6 while 10 meters from the
orifice the minimum dilution would be more like 9. Some models calculate an average dilution
for the cross section of the plume and this of course also increases downstream. The average
dilution is always larger than the minimum dilution. The appropriate average is termed the
flux-average dilution found by weighting the concentration distribution by the velocity
distribution over the cross section of the plume.
In some models the physics of the dilution process is based on the centerline mass
concentration so that the resulting calculation of average dilution is external to the physics. That
is, if a modeler assumes the effective width of a single round plume is defined by the five
percentile value of a Gaussian distribution, the average dilution will be less than if the 33
percentile value is chosen. In either case the centerline concentration would be the same. For
this reason they prefer to compare model results in terms of the centerline value rather than
average values. However, both values need to be considered in field or lab verification studies,
and both values may be useful for regulatory purposes.
In other models a uniform cross sectional or average concentration (referred to as a "top hat"
profile) equivalent to the centerline concentration is assumed. Thus, UM uses an assumed profile
to help establish minimum dilutions from predicted model average dilutions. The relationship
between the profiles is discussed further in following chapters: "Example: A CORMIX1
Comparison, Density, and Stability," and "UM Model Theory." While minimum dilutions are
often of interest to regulators, average dilutions are especially consistent with the dynamic
requirements of plume theory (Frick, 1984).
20
-------
General aspects of dilution modeling
30.
10 20 30 40 50 60 70 80 90 100
Effluent Flow Rate, Ve (MOD)
Figure 11. Dye flow rate to achieve 1 ppb in seawater with 20% Rhodamine WT.
Regulatory Use
Regulatory interest may be appropriately directed toward both average values and discrete
values. Unfortunately the state of the art of regulatory practice is not as sophisticated as plume
modeling and is generally constrained by lack of information on the temporal and spatial scales
of aquatic organisms' responses to exposure conditions in natural settings. For some parameters
California (State Water Resources Control Board, 1988) and the USEPA (1986) specify
maximum allowable instantaneous and several temporal average values. If an applicable criterion
for a certain biological resource near the outfall is an instantaneous value, a discrete value
obtained over 5 to 30 seconds, as could be achieved by sampling methods used for plume studies
in the field, would be appropriate. Many such samples would be taken to attempt to find the
highest concentration of pollutants, i.e., the centerline value.
Additionally it might be argued that a biological resource at risk at any moment is
appropriately evaluated over an expanse of space so that a spatial average is required, again
evaluated in a short time period. The time period over which this averaging would take place
is unfortunately not easily defined in relation to "instantaneous". It certainly is not seconds
because it is impractical to acquire these data synoptically across the expanse of even one plume
diameter let alone a multiport diffuser. If the data are obtained in an hour or two during slack
tide, calm seas, and low currents, it is possible that the values will not be greatly different from
21
-------
General aspects of dilution modeling
one plume to the next in the same diffuser. Depending on the biology of the resource, either the
maximum concentration (the minimum dilution) or the flux average dilution might be the
appropriate value to use in determining compliance with "instantaneous" criteria applied to a
spatial resource expanse.
Criteria that are expressed in terms of temporal averages (daily to semi-annual) suggest that
plume concentrations be assessed extensively in three dimensions, both at the boundary of fhe
mixing zone and in some cases at sensitive biological resource locations down-current. Current
speed and direction play significant roles when assessing the concentrations at the boundary.
By incorporating data on the cyclical variation of effluent composition, density profiles, and
current direction it is possible to construct a running six month average (or median) for a number
of points on the mixing zone boundary. The six month average is expected to be quite variable
at these points, and the point with the highest exposure frequency may not have the highest
average concentration.
Beyond the mixing zone there may be regions where current streams of diluted effluent,
leaving the zone at different times in different directions, would converge over a reef, a kelp
Pacific
Ocean
Figure 12. Visitation frequency (percent) of effluent about the San Francisco Southwest
Ocean Outfall.
22
-------
General aspects of dilution modeling
forest, or a swimming area. Thus if frequency and duration are important exposure
characteristics in resource response, the exposure may be more critical even if the concentration
(intensity) is lower, as it almost surely will be. In this case current direction is important to
understand on a larger scale so that circulation patterns are evaluated. Some formal applications
of this "visitation frequency" approach (Figure 12) have been used in regulatory assessment of
criteria that are presumably "instantaneous" (Roberts, 1990). Depending on the size and nature
of the resource to be protected either discrete or spatially averaged values might be appropriate.
The regulatory authority may not need to prescribe specific criteria for each of several
segments along the mixing zone boundary. More likely they will be interested only in the
highest six month average concentration wherever and whenever it occurs. Thus the formal
methods for determining a relationship between frequency of occurrence, intensity of the stress
(concentration), and duration of the exposure for plume performance at the mixing zone boundary
are not rigorously established. However, designers, environmental scientists, and regulators
should assess these performance characteristics conceptually, and possibly with a well chosen
suite of model simulations, to conscientiously achieve responsible regulations and to guide
improvements in the state of the art. USEPA (1986) provides a method to evaluate the
appropriate relationship for ammonia in freshwater streams, which may be taken as an indication
that frequency, intensity, duration relationships developed for evaluating outfall performance
would be useful in improving regulatory practice.
Aside from the question of whether discrete values or cross sectional averages are used to
test compliance with criteria, the way in which field samples are used to verify or compare with
model results is an important consideration.
Verification Sampling
In laboratory or field verification studies of plume performance the average value is measured
or captured in a sample bottle only by chance. Characteristically the field value measured is
from a very small spatial region and represents a signal over a certain time span. A large number
of samples is sought from the same cross section in order to arithmetically compute an average.
In the laboratory, using a single plume, this is relatively easy to do. But in the field where
multiple plumes are usually involved, and a moving flow field too deep below the surface to see
is being sampled by a moving sampler from a moving boat, it is quite uncertain what portion of
the cross section the value represents. Attempts to acquire a large number of samples from a
different radial position of the same cross section are frustrated because of the relative horizontal
motions involved. Surface waves and possibly internal waves in the pycnocline can also cause
the sample to be obtained from a shallower or deeper cross section.
For these reasons field verification studies are best attempted for a cross section as far from
the orifice as practical as long as the region is still within the range where the buoyant plume
physics apply. Nearer to the orifice the values are changing more rapidly and the dimensions
of the plume are much smaller, making it much harder to get the sampler in the right place, or
23
-------
General aspects of dilution modeling
even in the plume. In addition it is best to conduct the study when currents are low so that the
plume rises nearest to the surface, shortening the interval between samples, as the sampling
device need not be lowered so far. Placement of the sampling device may be improved because
it may even be possible to see the plume. Aside from the use of the data for verification of the
physics, samples taken during low currents may be especially useful for verification of regulatory
compliance. Field verification data taken near the end of the initial dilution region can be
compared with controlled laboratory simulations for similar conditions, and then, if necessary,
the laboratory verification data can be relied upon for estimation of field values closer to tfie
orifice.
ENTRAINMENT FROM OTHER SOURCES AND RE-ENTRAINMENT
Regulatory Background
In drafting modifications to the Federal Water Pollution Control Act (Anon., 1982), the
United States Senate (Anon., 1983) proposed strengthening the authority of the Environmental
Protection Agency (EPA) to deny waivers from secondary treatment for publicly owned treatment
works (POTWs) discharging partially treated wastes into estuaries. Concern was expressed for
re-entrainment of contaminants discharged previously from the POTW under consideration, and
also for entrainment of contaminants discharged by other sources. Amendments to section 301(h)
of the Act appearing in section 303 of the Water Quality Act (WQA) of 1987 (Anon., 1987)
addressed these concerns:
Section 301 (h) is amended by striking out "such modified requirements will not
interfere" and inserting in lieu thereof "....will not interfere, alone or in
combination with pollutants from other sources..."
and further on:
Section 301(h) is further amended by adding "....marine waters must exhibit
characteristics assuring that water providing dilution does not contain significant
amounts of previously discharged effluent from such treatment works."
These amendments suggested that EPA would need to revise the methods used to calculate
compliance with water quality standards at and beyond the boundary of a mixing zone. Three
topics needed to be addressed:
1. Definition of "significant amounts"
2. Entrainment of contaminants from other sources
3. Re-entrainment of contaminants from the proposed discharge
The water quality standard to be met is most easily assessed if it is expressed in terms of a
concentration of a pollutant, i.e., a numerical criterion. For example, the California Ocean Plan
24
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General aspects of dilution modeling
(State Water Resources Control Board, 1988) contains such limitations, a few of which are listed
in Table II, along with background seawater concentrations. The questions raised by the 1987
WQA amendments concern the proper value to use for the ambient (background) concentration
for certain environmental settings, and how much is too much for a given discharge.
Table II. Concentrations of contaminants in coastal waters of California.
Contaminant
Allowable
Instantaneous
Maximum, C=1
Arsenic
Mercury
Silver
Zinc
Ammonia
(N)
Toxaphene
DDT and
derivatives
80
0.4
7
200
6000
0.021
0.003
ug/1
ug/1
ug/1
ug/1
ug/1
ug/1
ug/1
Background
Seawater
Concentration
3
0.0005
0.16
8
0
0
0
ug/1
ug/1
ug/1
ug/1
Significant Amounts
The definition of significant amounts is easily resolved by use of mathematical models such
as UM. That is, significance, in the sense of "importance", rather than a statistically computed
value, is eloquently expressed in the test of compliance against a numerical standard in this
model. If for a given setting Equation 9 provides values of cpi that are lower than the values of
csi, then indeed the diluting water does not contain significant amounts of previously discharged
effluent. Thus the question of how much re-entrained effluent is allowable is operationally
defined with the types of models that were already in use in 1987, and at least for this purpose
the 1987 revisions did not require a change in the models or their application. The major
question is, "What is the proper value to use for each cail"
Relationship of Ambient Dilution Water to Plume Concentrations
The following discussions is intended to show that the amount of effluent that is allowed to
be re-entrained is a variable amount depending on the value of the standard, the amount of the
contaminant in the effluent, and the volume of entrained diluting water. This can be seen by
rearranging the terms of Equation 8 as follows:
25
-------
Sa
General aspects of dilution modeling
(12)
The requirement that the plume concentration of contaminant be less than the standard for each
contaminant can be expressed in the following inequality:
cpi < c (13)
where csi is the numerical value for the ith standard. Substituting the expression for cpi from
Equation 12 into Equation 13
C_li + °a' (S" ~ 1} < c (14)
For cases where the ratio (cai/csi)/Sa is less than 0.02 there would be less than a 3% error writing
Equation 14 as:
-c
(15)
Whether Equation 14 or Equation 15 is used, it is helpful to visualize the initial dilution
requirement in this form for three reasons. First, it clearly shows that a certain standard may be
met with different sets of values for Sa, cei and cai. For example, if one effluent has an ammonia
nitrogen concentration of 120 mg/1 and the local ambient is 3.9 mg/1, the California instantaneous
allowable maximum of 6 mg/1 would be met if an Sa of 60 were achieved. Another outfall, or
the same outfall at a different time, achieving an Sa of 60 could meet the standard with an
effluent value of 305 mg/l if the local ambient were 0.9 mg/1!
Second, the value of the ambient concentration is seen to be of the same relative importance
as the designated standard value in determining compliance. Thus if one locality has a standard
one unit higher than another, but the ambient is also one higher, the necessary ratio of cei / Sa is
the same. In other words, both dischargers have theoretically the identical options of reducing
cei or building a more efficient diffuser or any favorable combination of these options. And if
one locality has a standard one unit higher, and an ambient one unit lower, the discharger at this
location would have to meet a less stringent ratio of cei / Sa, i.e., it is two units higher. This
relationship is shown in Figure 13.
Third, notice that Sa is not subscripted with an "i" meaning that Sa is not dependent on the
contaminant under consideration, as explained previously. It may be helpful to think of a
"contaminant specific effective initial dilution" as the ratio of the concentration of a specific
26
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General aspects of dilution modeling
Water Quality Standard, Cs.
Figure 13. Maximum ratio of effluent concentration to Sa for standard compliance and
dependence on ambient concentration.
"contaminant specific effective initial dilution" as the ratio of the concentration of a specific
contaminant in the effluent to the concentration resulting after the volumetric process of critical
initial dilution is achieved, i.e., cei I cpi. By rearranging Equation 12 and again accepting an error
no greater than 3% for dilution factors greater than 30, Equation 12 becomes:
(16)
Expressed in this way it is clear that the effective dilution of the specific contaminant, limited
by regulation to less than a given numerical standard, depends on both Sa and the ratio cai I cei.
Figure 14 graphically depicts that the ratio cei I cpi, the contaminant specific effective initial
dilution, is dramatically reduced below Sa as the ratio cai I cei increases.
This analysis has shown that the computational technique employed to test compliance with
27
-------
General aspects of dilution modeling
800
-4 -3 -2 -1
Relative Ambient Concentration. Log(Cai /Ccl )
numerical water quality standards does
take into consideration the entrainment
of contaminants existing in the
ambient dilution water. Thus the
Senate revisions, contrary to first
impressions, did not require a change
in the EPA evaluation procedures to
determine "significant amounts" of
previously discharged effluents.
What is yet to be shown is how
the value of cai may be determined or
estimated to reflect the influence of
other discharges nearby. The first
requirement is for regulatory
instructions to explain clearly that cai
must accurately reflect the quality of
the water entrained, i.e., the water
adjacent to the diffuser, not the water
at some remote, pristine location.
Thus, for example, the "ambient"
values in Table n are not likely to be
generally useful, and may be
inaccurate for California coastal
discharges.
Entrainment From Other Sources
In the case of existing discharges it is not necessary to employ mathematical models to assess
the amount of entrainment from other sources, and the amount of re-entrainment of previously
discharged effluent, because field monitoring data will reflect the combined result of these
factors. _A priori assessment is needed in cases where a major change in effluent quality is
proposed, or the outfall is to be modified or relocated, and models are useful for this purpose.
In the preceding sections it is shown that the effect of entrainment from other sources is
properly incorporated in mathematical models such as UM as long as a proper data set for the
ambient concentration of specific contaminants is used for input. Data available for an existing
outfall may be useful for the relocated site if it is within the region covered by sampling stations,
and sufficient vertical detail is provided in the data set. The presumption is that the new site or
the modified outfall (e.g., longer or more ports) would provide better critical initial dilution.
Since the data set would reflect both entrainment from other sources as well as re-entrainment
of effluent, the data set would provide a conservative estimate. If the new site is outside the
region sampled, new monitoring stations could be established and coastal circulation models
could be employed to assess transport of pollutants from known sources in the region.
Figure 14. Effect of ambient concentration on effective
dilution.
28
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General aspects of dilution modeling
Entrainment into the plume of an outfall from other point and nonpoint sources is not
generally a problem in the open ocean because of many factors. In most cases there is a large
distance between point sources, providing ample opportunity for diluted waste to be dispersed
and carried away from the region of entrainment of another outfall. Also, the volume of
nonpoint sources of pollutants discharged directly to the ocean is small. Greater care is now
given to locate modern ocean outfalls in well-flushed offshore environments rather than near
shore. The volume of coastal waters available for dilution of point and nonpoint sources is great.
For example, a 100 km section of coastal shelf out to a distance of 10 km with an average depth
of 25 meters contains 25 x 109 m3 of water, about 5000 times the daily effluent flow that might
be generated by a municipality of 10 million people.
For a simple generalized case of contaminants transported from a source, for example another
outfall, the concentration contributing to the ambient at the new site can be determined from
Equation 17 (Brooks, 1960):
c = c erf
Saax J
*
Ub2 (17)
16 e_ X
where
cpi = Plume concentration at the end of initial dilution
cmax = Centerline (maximum) concentration at distance X
erf() = Standard error function of ( )
U - Current speed in the X direction
b = Width in the Y direction (orthogonal to X) at the end of initial dilution
e0 = Constant Horizontal (Y direction) eddy diffusivity
X = Travel distance
Computed in this way, cmaa is a conservative estimate for open coastal environments, and an
appropriate estimate for near coastal and inshore waters. In some open coastal situations the
farfield centerline dilution, cmax, is appropriately estimated using a 4/3 power law to continuously
increase the coefficient of lateral dispersion as the width of the field increases (Okubo, 1962).
Further details, including the relationship between e.0 and the farfield diffusion coefficient input
in PLUMES, are given in the chapter entitled "Farfield Algorithm."
PLUMES automatically computes the farfield centerline dilutions according to both equations,
providing a table of output data under column headings "4/3 Power Law" and "Const Eddy Diff'.
Corresponding data columns provide the centerline farfield pollutant concentrations using the first
order decay coefficient (or T-90) provided by the user, however, the RSB model does not
calculate decay in the near field. If ambient concentrations are specified they are factored into
the mass balance as ambient fluid is entrained and they are subject to first order decay as well.
(Again, RSB does not include ambient concentration in the near field).
In other cases, for example involving ULINE predictions, these dilution factors would assume
29
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General aspects of dilution modeling
negligible contribution from contaminants in the ambient water, thus they must be reduced to
represent the effective dilution at the down-current site. Figure 14 can be used for this purpose,
substituting cpl for cei in the abscissa term and cpllcmax for cjcpi in the ordinate term. These
dilution factors are minimums, that is, a cross-field functional form such as a Gaussian curve
should be used to estimate the cross sectional average.
It must be recognized that the dispersing plume from one outfall will contaminate near
surface waters while the principal source of entrainment for another plume is the deeper waters.
Verified two-layer circulation models for the coastal segment under consideration may be useful
to estimate the vertical exchange of contaminants as well as horizontal migration, thus providing
an estimate of distant deep water quality.
Diffuse source inputs and episodic events are difficult to deal with in assessing the quality
of ambient water expected to be entrained into new outfalls. During major storms that may occur
as frequently as two or three times per year in the northeast and northwest, annually in the
southeast, and perhaps once in ten years in the southwest, (1983 in the Los Angeles Bight, 1988
in Hawaii), storm runoff flushes riverine and estuarine contaminants into the coastal waters.
Wind driven currents and waves re-suspend coastal sediments and distribute contaminants
throughout the waters of the nearshore continental shelf, in many cases causing impairment of
water quality entrained into ocean outfall plumes.
Mathematical models of coastal circulation may be able to predict dispersion of a given slug
of contaminants washed out of an estuary up the coast from an outfall. Under storm conditions
large dilution factors would be expected, however it is unlikely data are available to quantify
contaminant levels in estuarine discharges. Direct land runoff and runoff from combined and
storm sewers discharging directly to the ocean complicate both the analysis of transport and
dispersion calculations as well as specification of contaminant levels.
Single-layer circulation models are likely to be inadequate in assessing runoff related effects.
Depending on the concentration of dissolved and suspended materials, the bulk density of the
runoff-contaminated coastal waters may be sufficiently low so that a short time after subsidence
of the storm, deep denser offshore water will gradually move in toward shore and the turbid
storm water will be carried in a thinner lens on or near the surface. Since a large percentage of
the water entrained into the plume occurs at depth, there may be considerably less entrainment
of contaminated storm water into the plume than would appear to be the case as one views the
situation from the surface (or from the air). Mathematical models of coastal circulation may not
be as useful for the period just on the heels of the storm event because of the difficulty in dealing
with multi-layer flows in the high energy coastal environments. Because of the importance of
entrainment at depth in achieving the proper degree of initial dilution before reaching the level
of buoyant equilibrium, it is not appropriate to use a one-layer model which assumes the water
column is completely well mixed under conditions of low currents.
During the storm event it is reasonable to expect that water quality values related to human
use of the marine resource in the vicinity of the outfall might well be suspended de facto. For
30
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General aspects of dilution modeling
example, sport fishing and scuba diving are not likely to be engaged in near the outfall during
a coastal storm. Consequently no harm is expected to be done to this use if effective dilution
during the storm is impaired by entrainment of poor quality ambient water.
No references have been identified describing the behavior of marine organisms during storm
events and their response to the mixture of effluent and runoff constituents. Their sensitivity
must be considered irrespective of the suspension of human uses. There may be sufficient
resiliency in coastal ecosystems so that short period perturbations can be accommodated. The
incremental perturbation due to entrainment of runoff-contaminated ambient may be either small
or large compared to average shelf conditions, depending on the circumstances of each event and
each locality. It should be recognized, however, that even with entrainment of contaminated
dilution water, the amount of dilution will be significantly increased over that predicted by
conservative plume assessments specified by EPA due to the much greater energy dissipation
occurring during storms. The net effect may be that organisms will experience a much lower
concentration of pollutants during a storm than in the average case.
Given the concern over the inapplicability of models for the complex cases of shelf advection
of pollutants in a variety of conditions, monitoring data may be the best option for estimating
ambient quality under all conditions. In light of the generally poor water quality data base
available in coastal shelf areas, if there is indeed a national priority for improvement of methods
to estimate entrainment of other sources into extant outfall dilution fields, there is an opportunity
to build a monitoring network that will serve a host of other highly important coastal resource
issues. A report of a panel convened by the Marine Board (Eichbaum et al., 1990) contains
recommendations for improvements in this area.
One important advantage of the use of field data to determine the quality of dilution water
over the use of model simulations is that it is an operationally responsive approach. As new data
are obtained, management options for control of the point source or the remote source, or both,
can be balanced.
Re-entrainment from Existing Discharge
In addition to contamination of dilution water from other sources there are circumstances
under which an existing discharge can re-entrain a portion of previously discharged effluent.
However, the farther offshore an outfall is located the less this is likely to be a problem. Coastal
currents and winds, which dominate replenishment of coastal waters with relatively clean offshore
water, are not likely to be suppressed to the extent that flushing of diluted effluents is materially
impeded for long periods of time. Under critical conditions of low wind and current, diluted
effluents rise to the surface or to a level of buoyant equilibrium in the pycnocline. Water which
is entrained between the discharge on the seabed and the spreading layer is not contaminated with
previously discharged effluent due to the density stratification, thus Ca is not increasing with
time. Tidal currents typically have a rotational character so that previously discharged effluent
is carried some distance inshore on one reversal past the discharge point and offshore past the
31
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General aspects of dilution modeling
diffuser on the next reversal. Again, under stratified, low current conditions the effluent rises
nearly to the surface or at least into the upper mixed layer. It does not remain at depth where
the majority of entrainment takes place.
In shallow coastal settings where some outfalls historically had been placed, vertical
turbulence is sufficient to reduce the degree of density stratification. If the discharge site
happens to be between headlands the replenishment of shelf water by deep ocean water may be
significantly restricted. In either of these settings partially diluted effluent can be returned to the
deeper water levels and effective dilution can be substantially reduced. EPA has provided the
model DECAL (Tetra Tech, 1987) to deal with this problem in a general coastal setting, i.e., not
necessarily near shore, however it is restricted to cases where vertical turbulence is sufficient to
cause complete vertical mixing near the outfall. Coastal circulation models and monitoring data
as discussed in preceding sections may be used for these cases as well.
Relocation of the terminal end of an outfall to a site further offshore is frequently considered
among the options to reduce environmental impacts of wastewater disposal. Another possible
scenario for relocation of an outfall is lateral displacement upcoast or downcoast from the present
location at about the same distance offshore. The rationale might be to minimize distance to the
location of a new treatment plant, or any number of water and sediment quality considerations.
If topographic and bathymetric features are similar at the former and proposed site, the
circulation features will be similar. Re-entrainment could then be estimated taking into account
any differences associated with the characteristics of the new diffuser. Monitoring data on
conditions around the outfall to be replaced would be useful in estimating the degree of
re-entrainment.
Entrainment and Re-entrainment in Estuarine Discharges
The above discussion focuses on open ocean conditions. For estuarine discharges the use of
Equation 17 may not be appropriate as advection and turbulent mixing is not so conveniently
described by this simple model. Monitoring data and estuarine circulation models may be useful,
although point and diffuse sources may not be well characterized.
Compared to waste discharges along a stretch of open coastline, discharge of effluents into
an estuary almost surely guarantees recirculation to other points in the system, and the
entrainment of effluents from other sources into the plume generated by the outfall in question.
Estuarine water quality analysis techniques have improved steadily since an EPA resource
management assessment was made in 1971 (Ward and Espey, 1971). The assessment of research
needs to support a national estuarine research strategy (Menzie and Associates, 1986) cites
examples of additional model development that is still needed, but the state of the art is sufficient
already for many management purposes. It is possible to adapt available models to many if not
most estuarine problems and to conduct simulations with computers available to every modern
regulatory program.
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General aspects of dilution modeling
EPA maintains an estuarine modeling repertoire and provides computer programs and
documentation manuals to potential users. These can be used to estimate the steady state
concentration of contaminants at a variety of sites in the estuary given the mass loadings and
input locations. Some models may be able to simulate varying concentrations of pollutants
within a period of critical conditions such as portions of a tidal cycle. As water quality criteria
become sophisticated enough to address short time variations the demand for detailed data on
time varying mass inputs will begin to limit the utility of the models. Simulations conducted for
all source inputs except the extant outfall, compared to simulated water quality in the absence
of inputs, will show the effect of "other sources" on the quality of water entrained in the outfall.
Monitoring data would be useful for verification of the modeling results except for the fact
that monitoring data will include the contribution from the extant outfall. For example, if several
of the other sources contribute nitrogen, monitoring data could not partition the estuary-wide
distribution of nitrogen since a municipal outfall also contributes nitrogen. It would be rare, and
extremely valuable, if baseline monitoring data were available over long enough periods of time
to provide some verification of the pristine case.
Use of an Intrinsic Tracer
There is a possibility, though unlikely, that a surrogate approach to partitioning of
contemporary monitoring data may be useful. If the effluent were the unique (ambient effectively
zero) source of any water quality constituent whose physical, chemical and biological fate
mechanisms were known, or could reasonably be assumed to be inconsequential, the distribution
of this tracer throughout the estuary could serve as a proportional marker for any other
constituent in the outfall.
Thus if the tracer was at concentration 10 in the outfall and contaminant "X" was at
concentration 4, then at some point in the estuary where the concentration of tracer was found
to be 0.1 and the concentration of "X" was found to be 3, the amount of "X" from other sources
could be found by solving Equation 9 first for Sa using the tracer data and then solving Equation
9 for ca using the contaminant data and the value found for Sa. Of course the behavior of the
surrogate and the contaminant "X" must be the same or adjustments to the correction have to be
made to account for any differences in coagulation, adsorption, decay etc. While easily stated,
this environmental behavior question may limit the practical use of the approach. No literature
citations have been found that report use of this technique although in a practical sense it is of
the same form of approach as injecting dye or some other tracer to determine the estuarine
distribution of outfall constituents generally.
33
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General aspects of dilution modeling
Salinity as a Surrogate Effluent Tracer
Under some specialized situations the distribution of salinity, which is more easily verified
than nonconservative pollutants, can be an effective surrogate for a nominal effluent constituent
in the water column. The simplest case is when an effluent is proposed to be discharged near
the major freshwater inflow to the estuary.
In the case of a discharge near the entrance, salinity may be an approximate surrogate only
if the wastewater flow is very much smaller than the incoming seawater volumetric flux during
periods of small tidal exchange.
Unfortunately, neither case deals with the question of environmental fate factors (adsorption,
speciation, decay), and surrogate values based on salinity have to be modified to account for
evaporation, direct rainfall, and other influences on the salinity value. Nor are salinity
distribution patterns useful for estimating particulate sedimentation values, which may be the
most important consideration because the 301(h) modified permit usually results in greater
suspended solids emissions than would be achieved with full secondary treatment.
FRESHWATER DISCHARGES OF BUOYANT EFFLUENTS
The buoyant plume problems of major interest to scientists and regulators have typically
involved the discharge of lighter material into a denser environment, such as a smoke plume in
the atmosphere or freshwater sewage effluent discharged into the marine environment. The
models developed for these cases are also able to handle the discharge of heated water into a
colder lake because of the slight density difference associated with temperature differences.
The models may be employed in some riverine situations as well as in lakes. That is, if the
effluent is warmer than the river and is discharged at depth, the effluent would be expected to
behave as a buoyant plume. The relative size of the diffuser ports in relation to the depth of the
river may be important in achieving the dilution factors predicted by the models. Muellenhoff
et al. (1985) recommended the depth be greater than ten times the port diameter, although there
is no strong experimental or observational basis for this rule. Rather it is based on the
knowledge that plume models were developed for deep water discharges and modelers are not
confident in extrapolating verification data from deep water situations to shallow water
applications.
For riverine situations in which the effluent is discharged through a multiport diffuser placed
along the stream bed in the direction of flow rather than across the current, only the RSB (line
source) model in this report may be applicable for analysis of the dilution field.
Industrial wastes discharged to rivers or lakes may have bulk densities greater than the
receiving water due to high concentrations of dissolved contaminants. But if an effluent is
substantially warmer than the lake or river the net result might be a lesser density and a
34
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General aspects of dilution modeling
positively buoyant plume would develop from a discharge at depth. However, modelers should
be aware of the nonconservative nature of heat in describing the density of an effluent at the
discharge point. The wastewater temperature at the diffuser port may be significantly lower than
at the treatment plant due to heat lost as the effluent runs through an underground and underwater
sewer.
Because most rivers will not have density gradients it is likely that warm water plumes will
reach the surface of the receiving stream, and the surface plume will be subject to heat exchange
with the atmosphere. The models in this guide do not incorporate atmospheric heat transfer
functions so that any temperature output generated after the water surface is encountered must
be accepted with caution. For short time periods atmospheric heat exchange will not make a
large difference.
The subjects of subsurface and surface discharges of large heated effluent flows as for
example from thermal electric power plants are treated in many reports.
The special phenomenon of nascent dense plumes, initially buoyant thermal plumes
discharged into near-freezing freshwater, which rise briefly before becoming dense and sinking
to the bottom are discussed in the next section.
NEGATIVELY BUOYANT PLUMES
Many industrial wastes whether discharged to fresh or marine waters have sufficient
dissolved or suspended solids concentrations so that the bulk density is greater than the receiving
waters into which they are discharged. The cases can include wastes discharged horizontally or
at an angle (including 90 degrees) downward from the surface or upward from the seabed.
Simple plume models such as UPLUME (Muellenhoff et al., 1985) have been used to fashion
a surrogate solution to the problem of predicting trajectories and dilution factors for vertical
discharges of negatively buoyant wastes. This has been accomplished by recasting the problem
in terms of an analogous positively buoyant case.
It may help the reader to appreciate this approach by pointing out that many laboratory
experimental data sets, and photographs, of positively buoyant plumes rising from the bottom of
a simulated stably stratified ocean are in fact results from a negatively buoyant plume discharged
from the surface, sinking toward the bottom! The laboratory experiment is set up this way for
the physical convenience of the modelers. The photographs are typically presented in published
reports upside down so that they visually depict the conceptual problem being addressed. The
proper analogy is effected by due regard to the density differences between the plume elements
and the local ambient so that the forces acting on the plume element are the same regardless of
the direction of motion. Thus a freshwater plume rising from the seabed is simulated physically
by a heavy liquid sinking in a lighter fluid. The mathematical simulation is analogous, and the
printout from the computer program is an equivalent, surrogate solution.
35
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General aspects of dilution modeling
An example of the above approach is the simulation of dilution factors computed for near
surface, downward discharge of drilling fluids into a marine ambient by Ozretich and
Baumgartner (1990). In this example the mathematical models PLUME, OUTPLM, and
DKHPLM, which would accept only positively buoyant discharges directed up from the seabed,
were provided input for a surrogate freshwater discharge into an ambient having an initial density
difference and a density gradient equal and opposite to the prototype situation. The
mathematically simulated results were comparable to data from a physical model of heavy fluids
discharged downward from the surface, i.e., exactly as in the prototype.
Extrapolation of the usual plume model results to cases of very large solids concentrations,
and slurries or solutions with very high specific gravities compared to the ambient fluid may
violate the Boussinesq approximation which is generally assumed. This assumption, incorporated
in plume models to simplify calculations, requires that density differences between the plume and
the ambient must be small compared to the density of the fluid. For example, the specific gravity
difference between sewage and seawater compared to seawater is approximately 0.02. Sewage
sludge is about the same, whereas drilling fluids used in offshore oil exploration could have a
ratio of as high as 0.5! Clearly 0.5 is not a small difference compared to 0.02, but there has not
been a rigorous examination of the importance of the Boussinesq assumption in plume modeling,
or for that matter what a useful criterion is for judging "small." Morton (1959) pointed out that
density differences are rapidly dissipated within a short distance from the orifice, suggesting that
violation of the Boussinesq approximation is not very serious for the major flow region. Fluid
modeling studies by Roberts (1977), and by Roberts, Snyder, and Baumgartner (1989 a, b, c)
show no effect of the ratio over a wide range.
In the hydraulic model studies of drilling fluids reported by Ozretich and Baumgartner
(1990), drilling muds with specific gravities as high as 2.17 were adequately modeled by the
model PLUME (Teeter and Baumgartner, 1979) as judged by comparison to measured depth of
penetration to the level of buoyant equilibrium. The ratio of predicted to observed depths
averaged 0.93 (standard error = 0.03) for 27 trials.
The model UM described in this report will accept direct input matching all physically
observed positively or negatively buoyant plumes discharged at any angle from either the surface
or the seabed. Furthermore it does not depend on the Boussinesq assumption. Other models
accessed through the UM interface may or may not produce output for certain negatively buoyant
cases, and output which appears complete for other than positively buoyant plumes discharged
from the seabed must be considered carefully by the user.
Nascent Density: Thermal Discharges to Cold Water
A special class of negatively buoyant plumes are nascent dense plumes, plumes which begin
as buoyant plumes but reverse buoyancy, becoming dense and sinking to the bottom or to some
more deeply submerged trapping level. The best known examples are thermal freshwater plumes
discharged to freezing ambient freshwater (Frick and Winiarski, 1978; Frick, 1980). The
behavior, which can also occur in brackish water up to a salinity of approximately 14 o/oo,
36
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General aspects of dilution modeling
occurs because the plume, as its temperature cools by mixing with water near the freezing point,
becomes denser than the ambient because the maximum density of freshwater is around 4 C.
Thus, if the temperature of the ambient is less than 4 C, the potential for the nascent dense plume
phenomenon exists.
The non-linear equation of state used in UM may be used to model nascent dense plumes,
as explained in the chapter entitled: "A CORMIX1 comparison, density, stability, and profiles".
PARTICIPATE DISCHARGES
Particulates in fluid discharges may vary from 10 ppm in municipal secondary effluent to
over 100,000 ppm in drilling fluids. The mass of solids may contribute to the bulk density of
the fluid, influencing the transient behavior of the plume and its equilibrium position. For
municipal effluents this contribution is neglected because of the low concentration of particulates.
Simple plume models (e.g., UPLUME) have also been used to analyze the behavior of
municipal sewage sludge in relation to alternative discharge methods such as pumping from
barges. Comparison of the mathematically simulated results to small scale hydraulic models
results demonstrated that sewage sludges containing between 2 to 6% suspended solids have
essentially the same properties as aqueous solutions of the same bulk densities. As the buoyant
Figure 15. Separation of plume and flocculating particulates.
37
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General aspects of dilution modeling
equilibrium level is reached in a density stratified ambient fluid the particulates begin to separate
from the diluted sewage field, some rising, some settling, with or without flocculation. See
Figure 15.
The physics of plume models does not attempt to describe the behavior of particulates within
the buoyant plume region or following equilibrium, except to the extent they behave as part of
the fluid continuum. Models are available (Tetra Tech, 1987, Bodeen, et al., 1989) to simulate
the dispersion and settling of sewage effluent particulates based on pioneering work of Hendricks
(1982, 1983) in the Southern California Bight. These models may be applicable for analysis of
other types of particulates. It should be borne in mind that the equations of state used in UM,
RSB, and CORMIX are not necessarily appropriate for the fluids at hand. (Some additional
amplification on this point is found in the section entitled: "Example: A CORMIXl Comparison,
Density, Stability, and Profiles.")
It may be possible to influence the behavior of particles in relation to the physics of
sedimentation by adjusting the discharge conditions at the diffuser port, especially the exit speed.
High exit speed may break up agglomerated particles causing them to behave as discrete particles
at the equilibrium level. Low exit speeds may preserve the integrity of agglomerated particles
and enhance the flocculation of others prior to arrival at the equilibrium level. This is a separate
area of research beginning to be questioned. Attention so far has been focused primarily on the
interactions of particulates following the transition from plume mixing to ambient turbulent
transport (Hunt, 1990). Whether or not discrete or agglomerated particle are the more
environmentally benign form has not been rigorously established, although a task force report of
the Marine Board suggests dispersal is preferred to seabed accumulation (NRC, 1984). This
recommendation is based on broad physical considerations rather than detailed ecological
considerations which may be preemptory.
38
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USER'S GUIDE TO THE PLUME MODEL INTERFACE, "PLUMES"
SYSTEM REQUIREMENTS AND SETUP
PLUMES is designed to be used on IBM compatible PCs running under DOS. The program
does not make use of graphics but does require a color monitor. The memory requirements of
PLUMES are modest, less than 200K, and should not interfere with other resident programs. The
latest advisories are contained in a file called READlst.exe, which, as its name implies, should
be read first. READlstexe contains information on how to unzip the program and document
files. Information describing a few supplementary files is also found there.
PLUMES can be run from the A: prompt using the diskette provided, however, the tutorial
notation assumes that it is installed on a hard drive, generally Drive C. We suggest that you
create a new directory on which to install PLUMES. If this new directory is a sub-directory of
the root directory the following procedure could be used at the C:> prompt to create a sub-
directory called, for example, MODELS, and to change to the new directory. At the prompt type
"mkdir MODELS" followed by a carriage return (i.e. the Enter key: ). The installation
commands might look like this:
C:> mkdir MODELS
C:> chdir MODELS
C:\MODELS> a:PLUMEPRO
After each command an is implied. The mkdir (or md) command makes the
PLUMES sub-directory, the chdir (or cd) command moves you to the new PLUMES sub-
directory. The A:PLUMEPRO command executes the self-unzipping executable file by that name
on the A: drive. (Substitute the appropriate drive designation). At this point, the program,
PLUMES.exe, may be run by typing at the prompt:
C:\MODELS > plumes
The case of the command is unimportant.
For further guidance on setting up the directories consult the DOS reference manual.
39
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User's guide to the plume model interface, "PLUMES"
INTRODUCTION
PLUMES is a computer implementation for preparing input data and controlling two plume
models, RSB and UM, and two farfield algorithms. RSB and UM are relatively sophisticated
mathematical models for analyzing and predicting the initial dilution behavior of aquatic plumes
discharged from diffusers or (UM only) single ports. The farfield algorithms are relatively
simple implementations of the Brooks farfield dispersion equations.
The interface itself presents a spreadsheet environment, scoreboard-like in appearance, that
allows you to describe effluent parameters, environmental conditions, diffuser design features,
and program controls in an organized but flexible manner. The various program elements are
intended to work together to help reduce the amount of time required to analyze various plume
problems, or cases. For example, the interface provides limited control over output format to
help in writing reports. The goal is to make it easier to explore options, conduct sensitivity
analyses, and generally produce more in-depth project reviews, designs, or assessments.
In addition, PLUMES can provide the corresponding CORMIXl flow categories based on
the CORMIXl classification scheme (Doneker and Jirka, 1990). Thus, PLUMES can offer
recommendations on model usage that go beyond the built-in models — including EPA
CORMIXl, CORMIX2 (Akar and Jirka, 1990), and CORMK3 (Jones, 1990), for single port
discharges, diffusers, and surface discharges respectively, in its appraisal. More comprehensive
recommendations on model usage are provided in Appendix 1.
The software is bundled with several stand-alone models: UPLUME, ULINE, and
PLUMEHYD. UPLUME and ULINE are initial dilution models described by Muellenhoff et al.
(1985). PLUMES supports them by providing a way to create UDF files from the input data.
PLUMEHYD may be used to analyze the hydraulic performance of simple, linear diffusers; it
is described in Appendix 2.
The PLUMES interface uses several main structures to display information, activate various
functions, and control the resident models:
4 the case (or record)
^ cells
i pop-up menus
f dialogue windows
+ help windows
+ configuration string
which are described in the next section. In addition, various specialized built-in features are
included to support the analytical process. Perhaps the most unique specialized capability is the
conflict resolution feature which allows many ways of defining the problem, i.e. entering different
sets of variables, and, consequently, must be able to detect instances of conflict when they occur
and help to remedy them. The following tutorial chapter demonstrates the conflict resolution
40
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User's guide to the plume model interface, "PLUMES"
mode.
Another feature is a units conversion capability to minimize the need for a calculator.
The structure, commands, special capabilities, and the plume models themselves work
together to help you analyze initial dilution, mixing zone, and farfield dispersion problems. The
level of refinement available in each of these zones varies considerably, being relatively high in
the near field simpler and approximate in the farfield.
PLUMES STRUCTURE
When PLUMES is started, introductory information is displayed which must be
acknowledged by pressing any key. Once acknowledged, the main screen, often referred to as
the interface level or simply the interface, appears. An example of the interface is given in
Figure 16. The screen represents a single problem, or case, which, as the information in the
upper right corner implies, could be just one record in a file of many cases.
A color monitor is required. Color is used to help organize the input and enhance the
readability of the interface.
Jun 19, 1992, 11:35
Title Sand Island
tot flow # ports
4.469 285
port dep port dia
70.1 0.085
port elev ver angle
0.84
0.0
hor angle red space
90
depth
0.0
30 .48
45 .72
60.96
76.20
7.315
current
le-5
le-5
le-5
le-5
le-5
6 ERL-N PROGRAM PLUMES, Jun
validation: no blockage
port flow spacing effl sal
0.01568 7.315 0.0
plume dia total vel horiz vel
0.08500 2.763 2.763
cont coef effl den poll cone
1.0 -2.893 100
p amb den p current far dif
24.080.00001000 0.000453
salinity
34.99
35.00
35.02
35.00
35.02
10, 1992 Case:
2 of 2
non-linear
far inc far dis
500 2000
asp coeff print frq
0.10 500
Froude # Roberts F
18.40 2.044E-14
K:vel/cur stratif #
2763000.00004871
N (freq) red grav.
0.01217 0.2652
buoy flux puff-ther
0.004159 35.60
jet-plume jet-cross
1.472 20810
plu-cross jet-strat
4.159E+12 4.136
plu-strat
6.932
hor dis>=
^ORMIXl flow category algorithm is turned off.
0.0 to any range
: Fl. Quit: . Configuration:ATNOO. FILE: sandis.var;
density
22 .99
23 .18
23 .40
23 .49
24.47
temp
26.18
25.60
24.95
24.60
21.22
effl temp
25
ve rt 1 ve 1
0. 000
decay
55.26
far vel
0.15
amb cone
0
0
0
0
0
Figure 16. The PLUMES main screen, or interface level. (Software version is in color.)
The greater part of the interface is occupied by ceils. In general, each cell has a short label
and a space beneath it for numeric data (the value of the mathematical variable). The title cell,
occupying the second line, is longer and is suited to alphanumeric input. In the main body of
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User's guide to the plume model interface, "PLUMES"
green ambient cells, which define conditions in the receiving water, vertically stacked cells share
common labels.
The cells are organized into colored blocks. Outfall structure variable labels are on magenta
background; effluent characteristics, brown; miscellaneous variables, gray; ambient variables,
green; and specialized information, red. The actual colors depend on the brand and settings of
the monitor in use. There is also a multipurpose "pause" cell (identified initially by the header
"hor dis>="), near the lower right hand corner of the interface, which may be used with UM to
control output of information under specified conditions (useful for specifying dilutions at the
mixing zone boundary). The color of the numeric information in the cells is either displayed in
yellow or in white, depending on whether the information was entered manually (or selected from
a default value) or was computed by PLUMES. Yellow variables are independent variables;
white ones are dependent. Only some of the cells, which you select to suit the problem
(independent cells), need to be specified — PLUMES computes the rest (dependent cells).
This flexibility makes it possible to define problems in a variety of ways.
At the top of the interface is a clock, the PLUMES version identification, the case counter,
and the Equation-of-state-window (showing linear or non-linear). At the bottom are three lines
of data: the first is reserved for the CORMIX flow classification predictions and modeling
recommendations, the second is the dialogue line, and the third contains basic help information,
program configuration identification, and the name of the file of cases in use.
The dialogue line may be passive, displaying useful information that is relevant at various
times, or it can be active, awaiting instructions to continue. Sometimes you are alerted to new
information in the window by sound. An example of a passive message explaining how to use
the menus after the Fl key is pressed is shown in Figure 17. When action is required, the
Hit bolded letter or arrow keys and ; use control sequences for speed
Figure 17. An example of the dialogue line.
options will be displayed or a cell will be provided for inputting string information, such as a file
name. The latter often display a default string which may be accepted or simply typed over,
referred to as "typeover" input. Explanations of messages may be found in Appendix 4.
Except for some of the editing commands which are described only in the Miscellaneous
Editing Commands section, the commands can be selected from several menus, the main one
of which is shown in Figure 18 as it appears as a window on your screen. The menus are
provided mainly as a memory aid, and, in general, it is faster to use the keystroke form of the
commands at the interface level. The *• symbol after some of the commands on the main menu
indicates the presence of sub-menus. The mode of implementation is explained subsequently.
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User's guide to the plume model interface, "PLUMES"
Jun 11, 1992, 19:57:49 ERL-N PROGRAM
PLUMES , Jun
Title Sand Island validation: (no blockage) TRR
tot flo
4.46
plume de
70.
port ele
0.8
hor angl
9
dept
0.
30.4
45.7
60.9
76.2
raci_LU menu
run rsB program
run Urn program
show Independents
units Konversion
List equations
get Work file
fill New file
add to Output
cell Precision
shallow/surface Z
configuRe models ••
movement commands *•
miscellanY menu >•
t;pciL:j_uy
7.315
otal vel
2.763
effl den
-2.893
current
00001000
salinity
34.99
35.00
35.02
35.00
35.02
KLL J. isdJ-
0.0
horiz vel
2.763
poll cone
6.1e8
far dif
0.000453
temp
26.18
25.60
24.95
24.60
21.22
10,
case.
c-.f fl
ei r i
vertl
0
1992
temp
25
vel
.000
decay
far
amb
vel
0 .15
cone
0
0
0
0
0
Case:
1 of 1
non-linear
far inc
500
asp coeff
0.10
Froude #
18.40
K: vel /cur
2763000
N (freq)
0.01217
buoy flux
0.004159
jet-plume
1.473
plu-cross
4.159E+12
LdL UXti
2000
print frq
500
Roberts F
2 .044E-14
Stratif #
.00004871
red grav.
0.2653
puf f-ther
35.61
jet-cross
20820
jet-strat
4.136
Figure 18. The main pop-up menu superimposed on the PLUMES interface.
The most pervasive help screens are the cell definition windows. These come up by issuing
the (AL) command on the main menu. The information provided is specific
to the cell identified by the cursor and has one of two forms. An abbreviated form is used when
the file EQNS is not in the current directory; it consists only of a definition of the cell and
descriptive notes. With the file in the current directory, a second form adds the equations that
are used by PLUMES to define dependent (white) variables. The extended form, in this example
showing the equations and terms involved in various methods for computing density, is shown
in Figure 19. If file EQNS is not in the current directory because it was not copied or was
deleted, it may be restored from the original disk.
The Configuration string, which may vary from case to case, appears in the middle of the
bottom line of the interface. Each character in the string is a mnemonic for different program
attributes. Changing the string will cause the program to work in one of several fundamentally
different ways. For example, the "O" in "ATNOO" in Figure 16 indicates that the plume model
UM, under overall control of the PLUMES interface, will terminate the initial dilution phase
(near-field) if and when the mathematical condition of element overlap is encountered.
INTERFACE CAPABILITIES
It is easy to be unaware of some of the special capabilities available in PLUMES because
all are not controlled directly. However, understanding them will enhance the use of the system.
The more notable ones are described below.
4 an unstructured data input environment
+ a conflict resolution mode for resolving many over-specified input conditions
f a configuration file
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User's guide to the plume model interface, "PLUMES"
Help for variable: den = effl den sigmat
Effluent density. When calculated from temperature and salinity,
the salinity is assumed to have the composition of sea salt. If the
density is independent, a linear equation of state is assumed (see
Example 2 in the manual for more detail).
Equations and variable definitions:
den = (dena+1000)/(1.0 +vel*vel/(g*dia*abs(Fr)*Fr) - 1000
{ note single use of abs to retain sign }
= (dena+1000)/(1.0+gp/g)-1000
= dena -SP*(dena-dal)*dia/pdep
= sigmat(s,t).
dal
dena
dia
Fr
g
gp
pdep
SP
t
vel
s
surface (level 1) density
ambient density at plume depth
plume diameter
densimetric Froude number
acceleration of gravity
reduced acceleration of gravity
plume depth
stratification parameter
plume temperature
for more [ for the continuation page below
plume vena contracta velocity
plume salinity
den, dena, etc. expressed in sigma-t units
Figure 19. Example of a "cell definition window." The help window for the plume density cell.
+ selection from multiple solutions to governing equations
4 display based on significant digits
Perhaps the most outstanding feature of the interface is its unstructured data input
environment. The user is free to move about, skipping over cells, just as in a spreadsheet
program. This facilitates "what if inquiries.
The unstructured environment would not have much purpose if all the cells had to be filled
in anyway. But, in fact, only some of the cells need ever be filled. The reason is that PLUMES
provides redundant variables as a convenience. For example, there are cells for the total flow,
number of ports, and port flow. Since it is assumed that all ports have equal flow, only the first
two cells are necessary to specify the port flow. (Given that they are specified, the port flow
should not have to be input, in fact, it would be potentially incorrect to do so because the value
could be inconsistent with the total flow, which, as is explained below, would be brought to your
attention by the conflict resolution algorithm.) In this case the total flow and number of ports
are displayed as independent variables, i.e. in yellow, while the calculated (dependent) port flow
cell is displayed in white.
For even more flexibility data can be entered into cells defined previously. This capability
facilitates sensitivity analyses. If the superseded value was yellow (independent), the affected
dependent (white) cells are simply recalculated. However, things are more complicated when a
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User's guide to the plume model interface, "PLUMES"
white cell is superseded with new information you entered. In this case the overspecification
alluded to above will, in general, cause the data set to be inconsistent. In the above example,
the product of the port flow and the number of ports would no longer equal the total flow.
PLUMES detects most such inconsistencies1 and goes into a conflict resolution mode in
which you select (space bar to move to the selected variable, followed by the "D" or delete keys)
which variable is to be calculated (dependent).
PLUMES maintains a configuration file called SETUP, an ASCII file that is created if it
is not present in the current dkectory. It is routinely updated and stores information on the last
use, including the location of the cursor, and the variables on the output table list. PLUMES
attempts to find and read the file each time it is run.
Some of the equations used to define dependent cells in the interface have more than one
solution. A good example is density as a function of temperature and salinity. It is well known
that the greatest density for fresh water at standard temperature and pressure is around 4 C.
Thus, there is a range of densities smaller than the maximum density in which temperatures both
less and greater than 4 C are compatible. Whenever this occurs, PLUMES provides for the
selection of the desired solution from the multiple solutions to the governing equations. The
same occurs when the dependent variable is the solution to a square root, in which case the
proper root, either positive or negative, must be selected.
The interface displays numbers to 3 or 4 significant digits. This capability assures that
information is not lost due to formatting deficiencies. Numbers that cannot be displayed to the
proper precision within the allotted space are converted to the "E" format of scientific notation,
e.g. 1.4xlO"8 is displayed as 1.4E-8. The "E" format may also be used to enter data. The | command may be used to show extra precision.
COMMANDS
Conventions
Control over the interface is exercised through a system of commands which may be issued
at any time. The commands are listed on a series of menus and can be implemented by
bringing up a menu or by holding the "control" key and striking an appropriate letter key. The
former is convenient for remembering the commands while the latter is faster. Thus, the "run
rsB program" command, which is listed on the Main menu, can be issued at the interface level
by simply holding down the control key and then pressing the letter B, also denoted by AB. (The
case of the letter is irrelevant, i.e., AB = Ab.)
' Strictly speaking, instances of over-specification are detected only when at least one of the defining variables
of the offending cell is independent. However, a special command is available for checking the consistency of all
variables, irrespective of their independent/dependent lineage.
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User's guide to the plume model interface, "PLUMES"
There is only one way to access the Main menu directly, which is not to say the
commands, and that is with the "Fl" key. From the main menu the commands, which include
bringing up the sub-menus, may be issued by hitting the chosen highlighted key, or, using the
arrow keys to move to the chosen command and selecting it with the enter (carriage return) key
or the space bar.
In the catalogue of commands to follow commands will be enclosed by < > brackets, to
indicate they are keystrokes. Thus or are equivalent. For commands
issued directly from the interface level without going through , sequences are harder to
represent, for example, does not convey very well the fact that the keys are to be
depressed simultaneously. For such cases the notation AB is more useful and will be used
extensively. For sub-menus, the chosen highlighted letter can be added to the key sequence. For
example, to use the command on the Miscellany menu from the interface
press AY followed by AH or ; this sequence is summarized as AYH. If a command is issued
which is invalid in context, PLUMES will send a reminder to the dialogue window. The
commands are case insensitive.
In the following listing, the name of the command as it appears on the menus is given,
followed by the interface level keystroke command sequence and a brief description of the
command itself.
The Main Menu
The Main menu (Help, ) is shown in Figure 20.
, AU:
Run the UM model. Subsequent dialogue window
prompts ask you to specify the number of cases to run
and the destination of the simulations (console, printer, or disk file). (See explanation of
the AB command above.)
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User's guide to the plume model interface. "PLUMES"
, AI:
Typically, PLUMES can use several equations to define dependent cells. AI examines
each of these and, in turn, identifies all the potential defining variable sets for the cell'in
which the cursor is located. The cell's independent variables are revealed by black
hatching of the background color of the cells' labels. AI is useful for establishing which
data (cells) will define the cell at the cursor for which data may be unavailable. For
example, you might only be using the interface to calculate salinities and wish to
determine the appropriate cells to input. (REMINDER: the variables in the defining set
can themselves be either independent or dependent.)
, AK:
Allows you to change the input units of a cell to one of the units shown in the dialogue
window. After the desired unit appears in the dialogue window, you may input the value
in its native units. Upon leaving the cell the value is automatically converted to the
system units (primarily SI, i.e. kg, m, sec, C). Subsequently, the conversions will appear
in the dialogue window whenever the cursor is moved back into the cell.
, AL:
Provides a definition of the present cell. An example is given in Figure 19. The header
name is displayed at the top of the screen in the cell's interface color. If the file EQNS
is in the current directory, the set of equations that define the cell, together with variable
explanation, is also provided.
, AW:
Used to specify a new working file of records, or cases. A typeover window is provided
for file name input. The <4> key may be used to cycle through existing .VAR filenames
in the present directory. The existing active file is stored and the new file is opened.
The new file name replaces the old one at the bottom of the interface after the word
"FILE". If the file does not exist it is created and filled with default data. The length
of the new file is checked to help ascertain that the appropriate format exists.
, AN:
Directs the interface to create a new file of records from the current file of records. You
are asked for a new file name (existing files are rejected). The filename extension, .VAR,
is recommended (see the command above). You must specify which
records are to be copied to the new file. The numbers of the cases must be separated by
blanks (spaces, not commas) but may be in any order. Sequential cases may be specified
by connecting their beginning and end members with "..", e.g. the sequence 5 3..7 1
causes the cases 5, 3, 4, 5, 6, 7, and 1, in that order, to be copied to the new file. The
command is useful for reorganizing your case files.
odd to Output>, AO:
For the UM model AO allows cells to be added to the list of cells that are displayed as
47
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User's guide to the plume model interface, "PLUMES"
output. Affected cells are highlighted by a blue rectangle in the first character of the cell
label. Certain auxilliary variables like centerline dilution may be added or removed by
using the AYS command on the Miscellany menu.
| , AP:
Increases the precision to which dependent cell values are expressed. The effect is global
to all cases but is reset at PLUMES run time. Up to six significant digits.
, AZ:
This command allows the analysis of single port plumes into very shallow water. Usage
is explained in the unofficial accompanying file called EGSFC.WP.
>, ARx:
Displays the Configuration menu. The "x" indicates another key is to follow. If AR is
pressed at the interface level, the Configuration menu will appear after a timed delay if
the "x" has not followed in the allotted time.
• >, AVx:
Displays the Movement menu. The "x" indicates another key is to follow. If AV is
pressed at the interface level, the Movement menu will appear after a timed delay if the
"x" has not followed in the allotted time. Some mnemonics of some of the editorial
commands are also displayed. Note: the AV prefix is not required.
>, AYx:
Displays the Miscellany menu. The "x" indicates another key is to follow. If AY is
pressed at the interface level, the Miscellany menu will appear after a timed delay if the
"x" has not followed in the allotted time.
The null command. Returns the interface level. At the interface level it is used to quit.
The Configuration Menu
The configuration prescribes one of several possible running modes for the interface, UM,
and RSB. The settings are identified in capital letters and numbers after the word
"Configuration". Defaults are provided if the file SETUP is missing, otherwise they are read in
from SETUP. The menu is shown in Figure 21. Unlike other menus which disappear after a
command is selected, the configuration menu remains on the screen until is hit, allowing
the entire configuration string to be edited in one pass. It can be accessed with followed
by , or AR. Once the commands are known, it is more convenient to use the command
sequences given below.
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User's guide to the plume model interface. "PLUMES"
Configuration Menu -|
Auto ambient
Brooks eqn input
Cormixl categories
Farfield start
Reversal set
Show configuration
, ARA:
Possible settings are A (on) and N (off) in the first
character of the configuration string, e.g. ATNOO or
NTNOO. In the ambient block starting with the line below
the surface ambient line, while moving from cell to cell,
Auto ambient (on) will fill the cell with the value
immediately above it if that value is independent (yellow).
This is a convenient way of filling out the ambient block
when many of the values are similar. The provided values
can be edited in the usual ways. The default is A (on).
Figure 21. The
, ARB: Configuration menu.
The command toggles between two options. Possible
settings of T or R are identified by the second character in the configuration string at the
bottom of the interface level. The R setting (reset), e.g. NRCOO, indicates that PLUMES
will prompt you to approve or change the inputs (wastefield width and origin distance).
This allows you to essentially run the farfield model independently of the initial dilution
models. The T setting (transmitted) will establish the initial dilution model results as the
farfield model inputs. The default value is T (transmitted).
, ARC:
The command toggles between two options. Possible settings of C or N are identified by
the third character in the configuration string at the bottom of the interface level. The C
setting, e.g. NTCOO, indicates that PLUMES will attempt to define CORMDC1 flow class
corresponding to the input conditions. Recommendations for model usage are also
presented. The N setting, e.g. NTNOO, specifies that no classification is attempted. The
default mode is N.
, ARF:
Used to configure the UM model, it is identified by the fourth character in the
configuration string at the bottom of the interface level. This command specifies at which
point the farfield dispersion model is initiated following the use of UM in the initial
Start far-field at Max-rise, Overlap, or Pause criterion?
Figure 22. Farfield configuration options.
dilution phase, i.e. in the near field. When the command is issued the prompt shown in
Figure 22 appears in the dialogue window. Using the M, or Max-rise option, e.g.
ATNMO, the initial dilution phase is terminated when the plume reaches maximum rise
(or the surface), after which the farfield model is initiated. The default value is O
(Overlap), e.g. ATNOO, which specifies the farfield model begins when the plume
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User's guide to the plume model interface, "PLUMES"
element can no longer be consistently defined due to geometric constraints (Frick, Fox,
and Baumgartner, 1991). This condition, sufficiently pronounced, is associated with
upstream anvil formation (Frick et al., 1990). The P (Pause criterion) option, e.g.
ATNPO, initiates the farfield model when the condition in the pause cell, set by the AYS.
command, is met.
, ARR:
Plumes rising in stratified receiving waters frequently trap at an intermediate level, a level
of zero net buoyancy. Generally, plumes will traverse, or overshoot, this level and
perform wavelike motion because they still have vertical momentum. Thus, above and
below the trapping level the buoyancy will switch from positive to negative or vice versa.
This reversal in buoyancy will ultimately slow the vertical motion to a standstill before
reversing again. Each reversal point is a crest or trough of the wave.
The setting specifies how many extrema are to be modeled before the
farfield model takes control. The farfield setting must be M or P. If the number of
reversals (the last character in the configuration string) is set to zero, e.g. AONOO,
PLUMES will determine the number of reversals to be one, 1, for buoyant plumes and
two, 2, for negatively buoyant plumes. The reason for this option is that normally rising
plumes usually entrain much more vigorously between discharge and maximum rise than
they do in the farfield, thus the initial dilution region is confined to the region between
discharge and the first reversal (i.e. maximum rise). Negatively buoyant discharges are
frequently discharged upwards and pass through maximum rise before their turbulence is
dissipated, hence it is appropriate to continue relatively active entrainment through the
subsequent sinking region. In any case, by specifying a nonzero integer between 1 and
9, the user can specify the number of oscillations which will be modeled. The 0 value
is generally recommended but may be altered for the rare instances that a different choice
would be more conservative or for special purposes.
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User's guide to the plume model interface. "PLUMES"
options are also summarized. An example configuration is shown in Figure 23.
The null command. Returns the interface level.
The Movement Commands Menu
The Movement Commands menu is shown in Figure 24.
It can be accessed with followed by AV or . Once
the commands are known, it is more convenient to use the
commands given below. Note that even though they appear
on a submenu, to use the movement commands from the
interface level it is NOT necessary to first use the AV key.
The movement keys given on the Movement
Commands menu are augmented by other editing
commands described in the next section: Other Useful
Editing Commands. They are basic and useful and should
be learned thoroughly.
, AA:
In the title cell, AA moves the cursor to the beginning
of any word in which the cursor is located. If the
cursor is at the beginning of the string, it moves the cursor to the [tot flow] cell.
In the other cells, AA moves the cursor to the beginning of the number in a cell, or, to
the previous cell if the cursor is already at the beginning.
, AS, or < <— >:
Moves the cursor one character to the left of its present position. If it is already at the
beginning of the number or string, it moves the cursor to the previous cell.
, AD, or < —> >:
Moves the cursor one character to the right of its present position. If the cursor is at the
end of the number or string, it moves the cursor to the next cell.
— Movement commands —
A cell left
S char left
D char right
F cell right
E cell up
X cell down
go to next Case
Jump cell blocks
P (return last cell)
del left
"t del word right
"ql sorry key
(more: see manual)
Figure 24. The Movement
menu.
, AF:
In the title cell, AF moves the cursor to the end of any word in which the cursor is
located. At the end of the title cell it moves the cursor to the [tot flow] cell. In all other
cells, AF moves the cursor to the right side of the value in cell or to the next cell if the
cursor is already on the right side.
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User's guide to the plume model interface, "PLUMES"
works normally in the title cell but moves the cursor to the next cell in
the rest of the interface.
, AE, or < T >:
Moves the cursor up one cell in the interface. If the cursor is in the uppermost row of
cells, the cursor is moved to one line below the deepest defined line in the ambient block
or to the bottom of the column of cells.
, AX, or < i >:
Moves the cursor down one cell. If the cursor is in the row of cells in the ambient block
one below the lowest defined depth, or is at the bottom of a column of cells, the cursor
is moved to the top of the column of cells. Affected by the command.
, AC:
Directs PLUMES to go to another case specified in response to a typeover prompt in the
dialogue window. The next case is always offered as a default and can be accepted with
or . Otherwise, the default may be overridden by typing any other
number followed by or .
If the specified case number is one greater than the number of cases that currently exist
in the file of cases, a new case, is appended and filled with the same information
contained in the case from which the AC command is issued. Any number less than one
or greater than the number of cases plus one is ignored.
See the and commands below.
, AJ:
Moves the cursor into the next colored block of the interface. AJ is a fast way to move
about in the interface and the only way to move the cursor into the [title] cell.
, AVP:
This command is useful after a variable is selected for deletion in the conflict resolution
mode. When a deletion is made the cursor normally returns to the cell in which the
cursor was located after the value that caused the conflict was entered. The AVP
command returns the cursor to the cell which was deleted. Also works after the AJ, AE,
and AX commands. NOTE: Due to the presence of the AP (cell precision) command on
the main menu, this command can only be accessed by using the AV prefix.
The null command. Returns the interface level.
Mnemonics:
The Movement commands menu lists a few editing commands which also may be issued
at the interface level. These are described in the next section.
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User's guide to the plume model interface. "PLUMES"
Other Useful Editing Commands
The following commands perform useful editing functions in the interface. Many of the
commands are similar to those in the WordStar (trademark) word processing program and in the
Borland Pascal editor. Some common WordPerfect (trademark) commands are also used.
, or :
Moves the cursor to the next cell, except in the title cell, where it works normally.
, or AH:
Erases the character or digit to the left of the cursor.
, or AG:
Erases the character or digit under the cursor.
AT:
Erases the rest of the word or number to the right of the cursor.
:
Directs PLUMES to go to the previous case of the case file. When used in Case 1, the
highest numbered case is brought into the interface.
For skipping over many cases or for creating new cases using an intermediate case as a
template, use the AC command.
:
Directs PLUMES to go to the next case of the case file. When it is the last case in the
case file, a beep is issued to alert you to the fact that a new case will be created if the
command is issued again. This is a fast way for browsing the case data file and for
creating new cases using the last case as a template.
For skipping over many cases or for creating new cases using an intermediate case as a
template, use the AC command.
AQD:
Moves the cursor to the right of the last character or digit in the cell.
AQY:
Erases everything in the cell to the right of the cursor, all of it.
AQL:
"Sorry-I-changed-it-command". Restores the original value of a cell providing the cursor
has not left the cell. Some conditions cause exceptions to this rule and data may have
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User's guide to the plume model interface. "PLUMES"
to be re-entered.
May be used in many situations to dump whatever is on the screen to an ASCII file
called DUMPALL. Subsequent uses of the command will cause the DUMP ALL file to
be appended so that occasional examination or deletion of the file may be appropriate.
Intended for debugging and documentation purposes.
The Miscellany Menu
The Miscellany Menu (AY) is shown in Figure 25.
, AYF:
If variables in an ambient column are all the same, it
is often useful to fill only the surface cell for that
column and use the to skip over
successive cells in that column. After all the depths
are entered (i.e. all the [depth] cells are filled with the
appropriate depths), move to the surface cell in the
empty column and issue the AYF command. All the
remaining cells in that column down to the deepest
depth will be filled with the same value.
The command on the Configuration
menu is useful for achieving these results on a
continuous basis.
ambient column Fill
Interpolate amb cell
Copy ambient line
Delete ambient line
Beget new cases
cHeck consistency
Notes
clear Output cells
Purge cases
construct Udf file
pauSe cell
cormiX category
Zap most variables
Figure 25. The Miscellany
menu.
, AYI:
This command is used to place depth interpolated ambient values into intermediate empty
cells in a given column in the ambient block. For example, similar to the AYF command,
you could specify a surface current of 0.10 m/sec and a bottom current of 0.20 m/sec.
Then, from the cell below the empty cell(s), issue the AYI command. The empty
intermediate cells will be filled with depth interpolated values.
, AYB:
Used to copy the cell in which the cursor is located to the same cell in a specified
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User's guide to the plume model interface, "PLUMES"
number of subsequent cases. The number of cases involved is specified in a dialogue
window which is provided.
, AYH:
Instructs PLUMES to evaluate all possible solutions for the cell from the set of equations
which may be displayed with the AL command. The results are compared and any
difference greater than a tenth of one percent is reported in the dialogue line. Not all
differences reported are cause for concern. In particular, very small values, which are for
all practical purposes identical to zero, can occasionally differ by more than the criterion.
Also, if the defining equation has more than one solution, as for example is the case when
the horizontal velocity [hor vel] is computed from the total velocity [total vel] and the
vertical velocity [ver vel], the signs of the reported values may differ. Nevertheless, any
reported differences should be contemplated.
, AYN:
Reports the previous messages, up to 20, that have displayed in the dialogue window,
, AYO:
Just as cells may be added to the list of variables to be printed or displayed by UM at run
time, cells already on the list may be cleared using AYO. The dialogue window gives a
choice for clearing all cells from the table or for returning to the default list of variables.
After the command is used the AO command may be used to establish a different list.
, AYP:
All cases after the one shown on the interface may be deleted from the case file. The
command is especially useful when terminal cases have been added to the file
inadvertently by the use of the < t > command.
construct Udf file>, AYU:
Used to translate the cases specified in the dialogue window from and into the UDF
format used in the 1985 plume models (Muellenhoff et al., 1985). See Appendix 5 for
UDF.IN file format. This makes PLUMES operationally compatible with the earlier
models. The intent is to support the 1985 models and users who may not have adopted
the resident models. The interpreted cases are read from or are appended to an ASCII
file called UDF.IN. When reading the UDF.IN file the Append option may be used to
transmit some variables found in the interface but not in the UDF.IN file, for example,
the farfield increment cell. In other words, the present case may be used as a template
for variables not included in the UDF.IN file.
, AYS:
Used to edit and set up the pause cell located near the lower right hand corner of the
interface. After typing AYS the dialogue line shown in Figure 26 is displayed. The cell
is the only way to access selected model variables not present on the interface screen, viz.
average dilution, centerline concentration, time, density difference, and horizontal
55
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User's guide to the plume model interface, "PLUMES"
Back, Inequalities, Output, Variables(space), or ,
Figure 26. The pause cell dialogue window.
distance. The program control function of the pause cell works in conjunction with the
command on the Configuration menu. (Other cells that can be controlled,
via conditions given below, include [port dep], [plume dia], [effl sal], [effl temp], [horiz
vel], [vertl vel], and [p amb den].) The capital letters in the window are highlighted; their
functions are:
or :
Moves backwards through the list of model variables, including those listed above.
or :
Selects the possible inequality conditions or criteria. The idea is to set up
conditions under which UM will be forced to output data or terminate before
initiations of the farfield algorithm. For example, if the pause cell is the
horizontal distance (travelled) [hor dis] cell, with a numeric value of 10 m, the
inequality >=, and the Farfield start character in the Configuration string is set to
"P" for Pause criterion, then UM will output a dilution immediately after 10 m is
reached and initiate the farfield algorithm. If the Configuration string is not set
to the Pause criterion, then UM will simply output a value at that point and
terminate. This is a convenient way to establish output at desired points (like the
mixing zone boundary) or criteria. The inequalities include >=, <=, =.
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A TUTORIAL OF THE INTERFACE
EXAMPLE: PROPOSED SAND ISLAND WWTP EXPANSION
Introduction
This example is a step-by-step development, or tutorial, of the kind of problems encountered
in applying 301(h) regulations. It is designed to make you familiar with the use of PLUMES and
to give you a feel for its capabilities and limitations. Several figures are given along the way
to allow you to compare your progress with a prepared example. These figures do not adequately
convey what is a full color display on the computer monitor. Consequently, the tutorial is most
effective if it is used as a guide while filling out the PLUMES interface input form.
The Sand Island example is intended to be realistic, not only as being representative of the
problems encountered in practice but in terms of how analyses are not unique. In other words
there is not a single right way, instead, an analysis is likely to be an evolutionary process. An
examination of work and simulations already completed are likely to identify other factors that
need to be considered. Thus, part of planning the analysis is to carefully examine modeling
results already in hand to guide further changes which, fortunately, with PLUMES, are easily
made. But, the greater flexibility available in PLUMES also requires vigilance on the part of the
user because it is easy to overlook cells that, no longer standing out because they are filled, need
however to be changed.
It is assumed that the installation procedures described briefly at the beginning of the
previous chapter have been completed.
The problem described here is based on a proposal by the Sand Island Waste Water
Treatment Plant (WWTP) of the City and County of Honolulu, Hawaii which seeks to increase
its permitted wet-weather flow capacity from 102 to 130 MGD. An increase in the design
capacity of 202 MGD is also under consideration.
What will be the effects of the proposed actions on initial and farfield dilution? How are
bacterial, turbidity, and other contaminant levels likely to change? Is the new discharge likely
to meet water quality standards under the proposed operating changes? How do new techniques
compare to earlier analytical procedures? These are some of the questions addressed.
The problem involves a diffuser with 285 ports located on both sides of the diffuser. Over
time, the landward ports have become clogged with sand so that the discussion changes between
285 and 142 ports, 24 foot spacing and 12 foot spacing, which is confusing. This poses the
question: "How do you compare the performance of the diffuser, now clogged, with the
previously unclogged diffuser?" Ultimate answers are not provided and this analysis is
incomplete. In fact, THIS EXAMPLE CONTAINS DELIBERATE MISTAKES.
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A tutorial of the interface
Analysis
The problem can be broken down into five different parts:
(1) Collect pertinent information.
(2) Input information into the PLUMES interface.
(3) Run the PLUMES initial dilution and farfield plume models.
(4) Analyze the model results and make adjustments, if necessary.
(5) Use the results in the decision making process.
STEP 1: Collect Pertinent Information
One way to get a feel for the information needed is to run PLUMES and work an example.
The first time you do so you may be dismayed by the number of cells displayed by the interface.
It may seem imposing at first but only some of the variables, which you may choose, need to be
defined — the interface automatically calculates the rest, as soon as sufficient data is provided.
You are free to pass over cells for which you have no data, filling those for which data is
available.
When you create subsequent cases, the data contained in an existing case may be used as a
template for the new case by using the AC command to simply move from the case to be copied
to the new case to be appended (which will have a number one greater than the number of cases).
Minor changes may then be made very quickly to only the affected variables.
STEP 2: Input the Sand Island Information
It is assumed the necessary data needed for Sand Island have been acquired; the appropriate
references are given. Begin by entering the main menu using and set up a file for the
example by pressing , the command, or, better, press AW without first
pressing . The dialogue window changes to request the work filename. Type in
(which does not exist yet) followed by or . Notice that the
default filename can be overwritten without first deleting it. The .VAR filename extension is
recommended because the command may be used to scan existing .VAR files
in the current directory by simply using the <•!> key.
It is worth noting that the default name given in the dialogue box can be edited. For
example, pressing < —> >, or some other editing key like AQD (move to the last character), before
you type an ordinary character, will move the cursor into the field of the cell where it may be
edited by adding or deleting characters.
When you are done the monitor will look somewhat like Figure 27, without the 4.469 values
or title. The other values are default values which may be accepted or rejected as appropriate.
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To start, give this case, identified as Case 1 in the upper right hand corner, a descriptive title,
e.g. "Sand Island validation". First, of course, you must move into the title cell. You could go
to the Movement menu using the key but it is faster to use the "jump" command, AJ,
several times until the cursor moves into the title cell. Go ahead and type in the title. Push
, AJ, or AX when you are finished, in either case the cursor moves to the [tot flow] cell
which is a good place to start filling out the rest of the interface.
Jun 19, 1992, 11:26:42 ERL-N PROGRAM PLUMES, Jun 10, 1992 Case: 1 of 1
Title Sand Island validation linear
tot flow # ports port flow spacing effl sal effl temp far inc far dis
4.469 1 4.469 1000 0.0
port dep port dia plume dia total vel horiz vel vertl vel asp coeff print frq
0.10 500
port elev ver angle cont coef effl den poll cone decay Froude # Roberts F
0.0 1.0 100
hor angle red space p amb den p current far dif far vel K:vel/cur Stratif #
90 1000.0 0.000453
depth current density salinity temp amb cone N (freq) red grav.
0.0
buoy flux puff-ther
jet-plume jet-cross
plu-cross jet-strat
plu-strat
hor dis>=
CORMIX1 flow category algorithm is turned off.
4.469 m3/s, 102.0 MGD, 157.8 ofs. >0.0 to 100 m3/s range
Help: Fl. Quit: . Configuration:NTNOO. FILE: sandis.var;
Figure 27 The PLUMES interface with the dialogue line showing units conversions of the total
flow cell.
The total flow corresponding to the current permit is 102 MGD. It is the appropriate value
for the [tot flow] cell in which the cursor should now be located. However, the dialogue line
informs you that the primary units in this cell are m3/s, so a conversion is required. On the
main menu we note there is a command called , the dialogue line will appear as shown at the bottom of Figure 27.
In addition to showing the total flow in m3/sec, MGD (million gallons per day), and cfs (cubic
feet per second), the dialogue line also gives the recommended range of values for the cell,
which, incidentally, is not enforced.
The cursor should now be in the [# ports] cell and the value shown in the [tot flow] cell
should be 4.469. When the cursor was moved to the number of ports [# ports] cell, the third cell,
labeled [port flow], acquired a white value equal to 4.469, even though we did not input a value
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in this cell. This is an example of how the interface is event driven, i.e. an event, your pressing
the space bar, automatically initiated an action. We will have more to say about this shortly.
The number of ports is 285, but they are not uniform and the diffuser has sections of
different diameters. Strictly speaking, a hydraulic model is needed to properly analyze the
effluent velocities from the ports, but we will assume that the flow is uniformly distributed. If
the diffuser is well designed, the deviations from this assumption will not be too great. If there
is doubt a program such as PLUMEHYD.EXE (Appendix 2) may be run to give better estimates
of the port flow distribution. In that case the total flow may not be consistent with the port flow
and we may need to do a piecewise analysis of the diffuser. Alternatively, some other more
conservative assumptions could be made. To simplify the analysis we will assume uniformity.
Do not worry about what to do about the "1" that is already in the cell, just type 285. As
explained previously, the "1" disappears when you begin to type. Again, there are a number of
editing commands explained in the previous chapter which allow you to modify the information
that is previously contained in the cell.
After typing in 285 use the space bar to move the cursor to the [port flow] cell: it has now
changed from 4.469 to 0.01568. The new value in m3/sec is consistent with the flow from 285
equal ports producing a total flow of 102 MGD (4.469 m3/sec). The [port flow] cell value is
white (when the cursor is not in the cell) instead of yellow to remind you that this is a
dependent variable which you did not input but was calculated by PLUMES from information
you did input. Which cells are independent and which cells are dependent depends entirely on
how you fill in the interface, i.e. whichever variables are most compatible with the available
information. This gives you flexibility to use the data you have, not data you wish you had.
Before going on, note also that the new value is expressed as 0.01568 and not 0.016 (three
decimal places to the right of the period) as might have been expected based on the formatting
pattern established in the total flow [tot flow] cell. PLUMES reports data to three or four
significant digits and up to six are accessible with the | command.
The spacing is 7.315 m (24 ft). It is also noted that the ports are opposed so that there are
really two ports per 24 ft section. This presents an interesting problem because, if the plumes
from both sides merge, as they would in a crosscurrent or as they might even in the absence of
current because they tend to attract each other by mutual suction, then this spacing is too large
because this kind of merging is not modeled in the UM program, only side by side merging is.
Thus, there is an intuitively appealing suggestion that we should use spacing of 12 instead of 24
feet. But for the moment we will ignore this complication. It will be easy to estimate its effect
later when we develop additional cases. Input 7.315 or use the AK command to input in feet.
In the case of Sand Island, we encounter another complication. Because the diffuser parallels
the isobaths it acts as a barrier to sand moving seaward. This has apparently clogged the ports
on one side and causes the port flow to double in the remaining ports. But, for now leave the
spacing at 7.315.
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A tutorial of the interface
It is important to note that with the spacing described in this way, the farfield predictions will
not be correct unless the Configuration menu is used to enable you to input the correct length
of the wastefield and the end of initial dilution. The reason is that, by ignoring cross diffuser
merging, we have described the diffuser as if all ports are on one side of the diffuser.
Consequently, the initial wastefield width needed for the farfield algorithm will be overestimated
by approximately a factor of two. More will be said about this subsequently.
It may seem that we are following a rather cavalier path in defining the problem. However,
in practice, it is common to first estimate parameters and play around. In effect, this represents
a screening analysis. If it is found that the initial dilution is close to being inadequate for
meeting water quality standards, then the analysis can be refined. In fact, as will be seen, the
interface is ideally suited for this purpose because it is easy to change values anywhere in the
interface without starting over. Thus, there is no disadvantage to first scoping out the problem
and becoming aware of some of the potential pitfalls in the analysis beforehand.
When you are finished with the [spacing] cell you could move to the salinity cell. But wait
a minute, we have just made an important observation about spacing so let's jump (repeated AJ's)
back to the title and reflect this fact there. This will give you a chance to practice your editing.
In the title cell you could use the arrow keys to move to the end of the string, but, for touch
typists, it is easier to use the control key movement cluster. A few AF's get you to the right
place. If there are several words to jump over, the AQD command does the move in one step.
Type in ": no blockage" or something like that. Then return to where you were using the
movement commands.
The header of the next cell shows "effl sal" and, because the cursor is now in the brown
block, you may infer, correctly, that this refers to effluent salinity. If you are unsure about the
content of any cell however, the AL command ( on the main menu) may be used
to define the cell, as shown in Figure 28.
Help for variable: s = effl sal o/oo
Effluent salinity. Sea salt composition is assumed.
Equations and variable definitions:
s = sigmasal(t,den).
den : plume density
t : plume temperature
sigmasal: Newton approximation using the SigmaT function
(if no solution to sigmasal then s is quasi-defined) .
Figure 28. A screen for the effluent salinity cell.
The information given, that s = sigmasal(t,den), is somewhat more cryptic than most cells.
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A tutorial of the interface
It is an abbreviated way of showing that salinity is derived from a complicated function, in this
case involving the Newton-Raphson method because the function cannot be solved analytically
for salinity. Also, most cells have more than just one defining equation.
Information on the salinity of municipal effluent is not always available. We suspect that
the effluent is largely fresh water and guess that it is close to 0.0. (If possible, this should be
checked later.) The default is accepted by passing over the cell using the space bar and going
to the effluent temperature cell [effl temp]. Being in Hawaii, the temperature is estimated to be
about 25 C. As soon as both the salinity and temperature are specified, PLUMES calculates a
value of -2.893 for plume density ([effl den] cell). This density is given in sigma-t units and
translates to kg/m3 when 1000 is added. Thus, the approximated density of the effluent is
997.107 kg/m3. The conversion can be verified directly by taking an excursion to the [effl den]
cell and consulting the dialogue line which gives the value in additional units. (Try the AP
command).
We can approximate for now the effluent salinity and temperature because the effluent is
discharged to sea water with a much higher salinity. Thus, the greater part of the density
difference, i.e. buoyancy, is due to salinity differences, and the temperature approximations are
unlikely to affect the outcome by more than a few percent. However, in regulatory work you
would try to define these variables more accurately. (See also the discussion in the Freshwater
Discharges of Buoyant Plumes section of Chapter 1.)
The cursor should now be in the upper right corner of the interface in the Miscellaneous
(gray) block of cells. Again using the command, it is determined that the
farfield increment cell [far inc] is the distance between points at which the farfield dilution
estimates are reported during the simulation. Notice also that the header typeface is black, which
means that cell input is not necessary to determine the initial dilution, (i.e. neither UM nor RSB
require it for input). However, as we are interested in farfield bacterial concentration predictions,
values for these should be established.
It is known that a surfing area is located within approximately 2000 m of the outfall and,
therefore, this is considered to be an appropriate value to put in the farfield distance [far dis] cell.
Since only the bacterial levels at this distance are of interest, the farfield increment, the [far inc]
cell in which the cursor is presently located, can be rather large, 500 m will do. (An
unnecessarily small value may give more output than you want causing previous information to
scroll off the screen when RSB or UM are run, necessitating a dump to a file.) Enter this value
and follow this by putting the value 2000 m in the [far dis] cell.
Now enter the plume depth measured from a standard datum such as mean lower low water
(MLLW). (This is an UM program variable which, like a few others, is initialized by the
interface.) In this case we know the depth to the center of the ports from which the plumes
emanate to be 70.1m. All ports are at essentially the same depth. The blue background before
the first letter in [port dep] indicates the variable, (centerline) plume depth, will be an output
variable when running UM. It can be turned on and off with the AO command.
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The next cell, [port dia], is the actual physical diameter of the port (as opposed to the vena
contracta plume diameter, the minimum diameter of the plume, in the following [plume dia] cell).
The Sand Island diffuser has five different diameters to choose from, so which one should be
used? Technically, a diffuser hydraulics model (Appendix 2) could be used to provide estimates
for port flow from each port. Experience shows, however, that varying port flow over a limited
range does not affect initial dilution radically. Nevertheless, it would be wise, especially for
beginners, to do a sensitivity analysis by changing some of the values somewhat. The interface
is ideal for this kind of exploration. A conservative value is always appropriate if the screening
test is ultimately passed. (See Chapter 1: Effect of Wastewater Flow on Dilution.)
For now, enter 8.5 cm. Notice that the possible input units are in meters, feet, and inches,
not centimeters. This time the proper conversion is not available through the AK command. It
is assumed that you are familiar with the fact that 8.5 cm is equal to 0.085 m. Notice that the
leading 0 does not have to be entered. Notice also that after inputting the diameter many cells
are starting to fill up with white values1.
Now keep moving the cursor until the cursor is on the [print frq] cell. The print frequency
cell [print frq] simply determines how many model steps there are between outputs. Except when
the time step becomes too large, UM is designed to double dilution every 100 program steps.
Thus a [print frq] cell value of 100 will cause UM to output dilutions of 1, 2, 4, 8...,
approximately. This can be adjusted to taste, we will accept the default value for the time being.
It is not critical in any case because the model outputs at important milestones, e.g. the trapping
level. The performance of RSB is not affected by this cell.
1 The basic idea behind filling empty cells in the interface is this: PLUMES can calculate cell values from input
you provide because it is event driven and because it normally has many ways to calculate each cell. To give an
example, move the cursor to the [total vel] cell. You may have wondered why some cell labels are displayed against
a checkered background which changes as you move from cell to cell. These checkered labels tell you which other
variables (cells) serve as independent variables for the cell in which the cursor is located. For example, right now
the [port flow] and [plume dia] labels should be on a checkered background. That means that if [port flow] and
[plume dia] are defined (either white or yellow), [total vel] will be calculated by PLUMES, as it apparently has been.
This is a basic characteristic of the PLUMES interface that makes it act like a specialized spreadsheet. Essentially,
most cells have one or several equations associated with it (cf. Figure 19), just like spreadsheets, that allows
unknown cells to be defined, providing the appropriate information is available.
But PLUMES provides more than the standard spreadsheet in this respect. If you will now push followed
by (or simply AI at the interface level) for the command, you will see that other labels
are now checkered: first [horiz vel] and [vertl vel], then [plume dia], [p amb den], [effl den], and [Froude #], etc..
Many cells have a multitude of ways of being calculated by PLUMES. The AL command will reveal just how many
there are and define them if the file EQNS resides in the current directory where it can be accessed by PLUMES.
It is this ability of PLUMES to calculate variables in many different ways that helps assure that you will have to
input only a minimum of information and that you do not have to be an expert and know how to provide specialized
information. Your job is to keep finding cells that you know something about and fill them until the interface is
completely defined. You can do this by moving directly to the cells you know, passing over the others. If at the
end some cells remain unfilled, you will need to continue the process. Remember, cells with black lettering in their
labels are not needed for initial dilution calculations, only for farfield estimates.
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The port elevation cell [port elev] is used in calculating the CORMIX flow categories, and
it also affects UM's prediction of when the plume hits bottom. Here we use the radius of the
diffuser pipe, in this case enter 0.84 m.
Accept the default value in the vertical angle cell [ver angle] by skipping over it. A value
of 0.0 indicates that the effluent is being discharged horizontally, which is the case with Sand
Island and many modern diffusers. It should be becoming apparent that filling out the interface
is not such a difficult task after all.
The contraction coefficient cell [cont coef] is normally used to compute the actual initial
plume diameter by adjusting [port dia] on the basis of the differentiation between bell shaped
ports, which have a coefficient approximately equal to 1.0, and sharp-edged ports, which have
a value near 0.61. Sometimes this information is not provided in which case the value that yields
the more conservative dilution could be used. Its value tends not to affect dilution very much.
If salinity and temperature are specified, as they are here, the [effl den] (effluent density) cell
is calculated using the non-linear equation of state found in Teeter and Baumgartner (1979).
Computed values vary slightly from published values (see Table III in the next chapter). The
equation of state used at run time in UM is indicated in the linear/non-linear window below
the case counter. In running UM, if suspended or dissolved substances factor prominently into
determining density it may be better to use a linear equation of state, invoked by defining the
density cells while leaving the temperature and salinity cells empty. Any such empty cells
(providing, in the ambient block, the layer is defined) will cause the linear mode of UM to run.
Now move the cursor to the pollutant concentration [poll cone] cell. This cell is used to
specify the concentration of a specific pollutant in the effluent and, in combination with the
ambient concentration cell [amb cone], to help determine the effective dilution achieved by the
diffuser (see Chapter 1: Dilution Factor, Effective Dilution Factor, and Relationship of Ambient
Dilution Water to Plume Concentrations). For example, if the ambient concentration is
everywhere zero then the effective dilution is identical to the effluent dilution. However, suppose
we accept the default value of 100 (i.e. thinking in terms of percentage) given in the [poll cone]
cell and all the ambient concentration cells have a concentration of 1.0. Then, no matter how
great the volume dilution is, the effective dilution can never exceed 100.
Any consistent units of concentration may be used, which means that the units in the
pollutant and ambient concentration cells must match. We will use a value of
6.1xl08(colonies)/100ml for the bacterial concentration. In PLUMES format, scientific notation
is input in "e" format, for example as 6.1e8. Note again, that to replace the default value we
simply start typing in the value of 6.1e8 and when done.
The cursor should now be on the decay cell [decay]. This is the simple first order decay
constant, k, used in the equation
c = c^-* <18)
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where c is the concentration time t after a concentration of cmax is measured. For convenience,
the primary unit is inverse days. Often, however, decay is expressed in terms of T90 values,
which specifies how much time is required for 90 percent of the pollutant to decay, or how much
time is required for 90 % of the bacteria to die. The T90 time must be input in hours; for Sand
Island we use 1 hr. Thus, after one hour of exposure to daylight in surface waters, 90 % of the
bacteria have died. This unit is available by using the command; when t90hr
is indicated in the dialogue window enter the value 1.
As you move to the next cell you will notice that the space bar movement command
bypasses the densimetric Froude number [Froude #] and Roberts Froude number [Roberts F]
cells; the red block parameters are normally of interest only to researchers and designers. (When
it is convenient to use them the AJ command may be used to get into this block.) These numbers
will be calculated by the interface when all necessary input is entered.
The cursor should be in the horizontal diffuser angle cell [hor angle]. The outfall structure
variables, effluent characteristics, and miscellaneous blocks are complete. The interface screen
should now look like Figure 29.
Jun 19, 1992, 11:32: 1 ERL-N PROGRAM PLUMES, Jun 10, 1992 Case: 1 of 1
Title Sand Island validation: no blockage linear
tot flow # ports port flow spacing effl sal effl temp far inc far dis
4.469 285 0.01568 7.315 0.0 25 500 2000
port dep port dia plume dia total vel horiz vel vertl vel asp coeff print frq
70.1 0.085 0.08500 2.763 2.763 0.000 0.10 500
port elev ver angle cont coef effl den poll cone decay Froude # Roberts F
0.84 0.0 1.0 -2.893 6.1e8 55.26
hor angle red space p amb den p current far dif far vel K:vel/cur Stratif #
90 7.315 0.000453
depth current density salinity temp amb cone N (freq) red grav.
0 .0
buoy flux puff-ther
jet-plume jet-cross
plu-cross jet-strat
plu-strat
hor dis>=
CORMIX1 flow category algorithm is turned off.
90 deg 45 to 135 deg range
Help: Fl. Quit: . Configuration:ATNOO. FILE: sandis.var;
Figure 29 A partially completed interface.
The cursor is now in the green ambient block, specifically, in the horizontal diffuser angle
cell. An angle of 90 degrees (the default value) indicates that the current is perpendicular to the
axis of the diffuser, i.e. it is flowing across the pipe and parallel (co-flowing) to the effluent
plume. Notice that if 45 degrees were entered the value in the following reduced spacing cell
[red space] would change from 7.315 m (the physical port spacing) to 5.172m, the geometrically
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projected spacing. In UM, the effect of changing the direction of the current simply changes
the reduced spacing. The justification for this procedure is derived from Roberts (1977) and
is valid over angles ranging from 45 to 135 degrees. While 90 degrees is the desired angle for
now you may change it temporarily to see how it works. Values of 0 to 44 and 36 to 180
degrees, which are outside the range shown in the dialogue window, would produce reduced
spacings of 0 to 5.1m and should not be used for UM (but are appropriate for RSB). Similarly,
values of 181 to 360 produce negative reduced spacings and should not be used.
Now skip the reduced spacing cell [red space] and move to the port ambient density [p amb
den] cell. Notice that it is not one of the cells preferred for input (it has a white header) and we
will not enter a value into this cell, even though we could, or the following port ambient current
[p current] cell. Both cells will be calculated by the interface when the ambient depth, density
(or temperature and salinity), and current are completed2.
Now move to the farfield diffusion coefficient cell [far dif] and use the AL command to get
an explanation of this parameter. While the value of the coefficient is not known accurately, it
is considered to have the properties of a universal constant. The value, 0.000453, used in this
chapter corresponds closely to the 0.01 crn2/3/sec found in Fischer et al. (1979), however, a more
conservative one, 0.0003, has been adopted as a default value in PLUMES.
The next cell is the farfield velocity [far vel]. The cell label is black, to indicate it is not
required for initial dilution estimates. However, it is our goal to estimate farfield dispersion in
order to determine maximum bacterial levels in areas where water contact activities occur.
Although the dilution of contaminants in the near field would be enhanced by greater current
speeds we recognize that high current speed will also result in shorter travel times for the diluted
wastes that are carried to the protected zone, thus resulting in less die-off of bacteria. However,
the current speed should be realistic and take into consideration not only consistency with the
near field current but also factors such as tidal reversals and the likelihood that high currents will
2. This is a good place to point out something you may have already noticed, some of the labels have yellow
letters (yellow lettering on a colored field like the [tot flow] cell) while others have white ones (white lettering like
the port ambient density [p amb den] cell). In general, the yellow labels mark the variables that are recommended
for input, in a sense, they are preferred variables. There are a variety of reasons why they are preferred which are
rather technical and have to do with the math of the equations. For example, the program may need additional
information about the sign of a calculated number if one of the secondary variables is input (e.g. if it is a solution
of a square root). There is even a possibility of inconsistencies in the input (refer to the manual for an
explanation). The miscellanY submenu has a command that can be issued when it is suspected
that there is an inconsistency. Normally, inconsistencies will not develop unless the user overrides a cell containing
a white (not to be confused with the header lettering color described above) numeric dependent value with a yellow
independent input value, a topic that has not been covered yet. Even under these circumstances, inconsistencies (or
conflicts) will not usually arise. Also, to avoid alarm, in some cases the command will report
values of the same magnitude but different sign; this does not necessarily indicate the case is inconsistent. Finally,
the check is based on a comparison of values of the same parameter calculated from each of the different equations
that can be seen when issuing the AL command. Sometimes it will report two very small values, both essentially
equal to zero, which nevertheless differ by more than the fractional criterion.
66
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A tutorial of the interface
persist for long periods of time. In the case of Sand Island, a current of 15 cm/sec is used
corresponding to a travel time to affected areas 2000 m away of 3.7 hrs. Input 0.15 m/sec.
The cursor is now in the main ambient block [depth]. This is where information on various
layers of the ambient receiving water is input. The first depth cell [depth] should normally
contain the default value of 0.0 m (water surface), so move to the ambient current [current] cell
of the surface layer. We will input depth, salinity, and temperature data shown in Figure 30.
Jim 19, 1992, 11:36
Title Sand Island
tot flow # ports
4.469 285
port dep port dia
70.1 0.085
port elev ver angle
0.84 0.0
nor angle red space
90 7.315
depth current
: 5 ERL-N PROGRAM PLUMES, Jun 10, 1992 Case:
1 of 1
non-linear
far inc far dis
500 2000
asp coeff print frq
0.10 500
Froude # Roberts F
18.40 2.044E-14
K:vel/cur Stratif #
2763000.00004871
N (freq) red grav.
0.01217 0.2653
buoy flux puff-ther
0.004159 35.61
jet-plume jet-cross
1.473 20820
plu-cross jet-strat
4.159E+12 4.136
plu-strat
6.933
nor dis>=
CORMIX1 flow category algorithm is turned off.
deg C, deg F -2.0 to 50 deg C range
Help: Fl. Quit: . Configuration:ATNOO. FILE: sandis.var;
0.0
30.48
45.72
60.96
76.20
le-5
le-5
le-5
le-5
le-5
validation: no blockage
port flow spacing effl sal
0.01568 7.315 0.0
plume dia total vel horiz vel
0.08500 2.763 2.763
cont coef effl den poll cone
1.0 -2.893 6.1e8
p arnb den p current far dif
24.080.00001000 0.000453
salinity
34.99
35.00
35.02
35.00
35.02
density
22.99
23 .18
23.40
23.49
24.47
temp
26.18
25.60
24.95
24.60
21.22
effl temp
25
vertl vel
0.000
decay
55.26
far vel
0.15
amb cone
0
0
0
0
0
Figure 30 Completed interface.
Zero current is often chosen to estimate minimum dilution, which we input in the surface
ambient current [current] cell. Note that upon moving to the next cell, the 0 is replaced by a
small, near-zero value of le-5, which is the e-form scientific notation for 0.00001 m/s. This is
done to avoid a mathematical singularity elsewhere in the interface3. The value of le-5 is
practically equivalent to zero but can be input as a smaller value still if necessary.
Note: Other quasi-defined cells can still be generated, if they are, usually the last cell
entered caused the condition and can be changed to resolve it. In the case of the [Stratif #] cell,
3. Originally a 0.0 value was allowed but resulted in the creation of quasi-defined cells (identified by the
background color of the cell turning cyan) which made this capability inconvenient. For example, a zero current
throughout the ambient block would make it impossible to define a value for the effluent to current ratio cell
[K:vel/cur] because the ratio would involve a division by zero. Thus, a quasi-defined cell is one which would
normally be defined (all the independent variables that are needed exist), however a singularity (division by zero,
negative square root, etc.) keeps that from happening. This is now avoided.
67
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A tutorial of the interface
a non-zero density gradient in the ambient density cells will keep it from being quasi-defined.
To establish the minimum dilution it is necessary to also use the maximum density gradient.
The appropriate values, as shown in Figure 30 in terms of the depth and density columns, are
established by filling in the salinity and temperature columns for the depths shown.
For now, go to the surface ambient salinity cell [salinity], and then the surface ambient
temperature cell [temp] and type in the appropriate values shown in the figure. As soon as you
do, and follow it with , the ambient density [density] value at the surface of 22.99
sigma-t units is computed. The cursor should now be in the ambient concentration [amb cone]
cell. Here it is safe to input 0.0 since we expect the receiving water to be generally very pristine,
the ambient currents carry the effluent out of the region of the diffuser, and, most importantly,
the die-off is generally sufficiently rapid that even recirculated water is likely to contain
negligible bacterial concentrations, but that should not generally be assumed. If background were
specified the analysis would be correspondingly more conservative because the pollutant
concentrations are assumed to be horizontally homogeneous, i.e. constant, even though they
would be expected to decrease away from the source.
The cursor should now be in the next depth cell [depth]. Since data are given at 100 feet
and every 50 feet thereafter, use the AK command to bring up the ft units in the dialogue line and
enter 100 ft. Move to the salinity and temperature cells and continue to fill in the ambient block
as shown (the remaining depths are 150, 200, and 250). Because the Configuration string shows
a leading "A" the auto-ambient mode is on, which means that default values are taken from the
line above. Thus, none of the ambient current speeds or ambient concentrations below the
surface need to be typed.
As the last cell in the ambient temperature [temp] column was filled the remaining red cells
were automatically calculated by PLUMES and also filled in. The stratification parameter [Stratif
#] characterizes the degree to which the ambient is stratified between the surface and seabed
when a linear approximation is appropriate. Some technical references (e.g. Fisher et al., 1979)
use the linear approach in estimating dilution factors and trapping levels. Like the Froude
number, the stratification number is also used to determine similitude between prototype and
hydraulic model representations of plume behavior. While useful especially for laboratory
experiments, most environmental problems involve complex nonlinear density profiles. The RSB
and UM models calculate plume variables, such as dilution and rise, based on the density
gradients established by the inputted ambient salinities, temperatures, or densities, rather than the
overall average represented by the stratification parameter4. You can demonstrate that the
stratification parameter does not change when intermediate lines of ambient data are added,
deleted, or changed, as long as the data that determine the average parameters are not changed.
However, by running successive cases you will see that dilutions and geometric variables
4. Actually, the RSB model uses a stepwise series of linear gradients. It starts with an overall gradient and steps
down until the dilution is no longer reduced by more than an arbitrary small amount.
68
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A tutorial of the interface
calculated by RSB and UM do change appropriately.
STEP 3: Run Initial Dilution Models
The fact that all the cells (except for the elective Pause cell which is presently showing the
horizontal distance [hor dis] cell, its default value) are filled is a sign that the plume model can
now be run. Issue the AU, or , command. The dialogue line will then query
"From this case on, run how many cases" and offer a default of 1 in the dialogue window. Since
we still have only one case we can simply use the space bar to accept the default value. A
second query asks "Write to ("prn" for printer, "console", or disk file name):" with a default
value of "console". Accepting the default with a routes the output to the monitor.
The result is shown in Figure 31. Note the use of the non-linear equation of state is indicated.
UM INITIAL DILUTION SIMULATION (non-linear mode)
plume dep plume dia poll cone dilution hor dis
mm m
70 10 0.08500 610000000 1.000 0.000
67
61
27 1.822 18800000 31.18 4.417
20 3.682 6751000 84.58 6.463< trap level
54 65 7.502 3744000 144.2 8.460< merging
53 04 10.75 3406000 153.5 9.365< begin overlap
Farfield calculations based on Brooks (I960), see guide for details:
Farfield dispersion based on wastefield width of 2088m
--4/3 Power Law— -Const Eddy Diff-
conc dilution cone dilution distance Time
m sec hrs
420200 153.6 420200 153.6 500.0 3271 0.9
49130 155.9 49430 154.9 1000 6604 1.8
5594 162.6 5723 158.8 1500 9938 2.8
626.7 172.3 656.8 164.2 2000 13270 3.7
Figure 31 UM simulation of the first Sand Island case. Note that the initial wastefield width
of 2088m is too large by a factor of two and the farfield predictions should be ignored.
The trapping level dilution is 84.58 which corresponds almost exactly to the dilution found
by UMERGE (84.48) and UPLUME (Muellenhoff et al., 1985), and the earlier reported value of
84. Experience shows that under a large range of conditions (without current) UPLUME and
UMERGE agree very closely (Baumgartner et al., 1986). Therefore, it is not surprising that we
obtain close agreement with UM. It gives us some confidence in the new methodology.
Nevertheless, this degree of agreement should not be expected in general. For one thing, in
comparing UM and UMERGE, the definition of the aspiration velocity has been simplified which
can cause small differences depending on the relative importance of forced and aspiration
entrainment. Also, some of the input was approximated and the values are subject to some
adjustment. Later, you can make some of these adjustments.
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The farfield bacteria concentration based on the open water diffusion equation described in
the final chapter — Farfield Algorithm — which uses the less conservative eddy diffusivity factor
appropriate to coastal waters (the 4/3 power law), is 626.7, above the water quality standard of
400 colonies/100ml. However, this estimate may be too liberal, in other words, since the
wastefield is deeply submerged the survival of the bacteria may be much higher. As a result of
this run we could now adjust the T90 time to a value more appropriate for a submerged flow
field, such as 10 hours. We would then see a bacterial concentration l.SxlO6 colonies per 100
ml.
The message "plume element overlap", which is discussed further in the sections on model
theory, means that dilution predictions beyond this point would degrade increasingly if UM (not
the farfield algorithm) were continued to be used. It may not be significant if dilution increases
little in the overlapped region, which can be established by running the simulation to maximum
rise using the AR command.
The UM simulation can be interrupted at any time, execution is then suspended until another
keypress restarts or terminates it. After it is finished running, any key will reestablish the input
screen, i.e. the interface. The same procedure can be used to run the program again. If we
override the word "console" with "prn" (do not enter the quotation marks) on the dialogue line
the output will go to the printer (be sure that it is properly connected). Given any other name,
PLUMES will attempt to send the output to a disk file (created or appended). Notice that the
output contains a copy of the interface screen so that there is an exact record of the input.
As has been indicated, the farfield predictions shown in Figure 31 are not correct because
the length of the wastefield is overestimated owing to the assumption that all ports are on one
side of the diffuser and are spaced 7.315m apart. The farfield simulation could be "corrected"
without changing the near-field predictions by accessing the Configuration menu (AR) and
toggling the option. The Configuration string will then change from, for
example, "ATNOO" to "ARNOO", where the R stands for "reset" the farfield algorithm initial
conditions. Then run UM or RSB as you normally would. After the initial dilution phase is
completed PLUMES will prompt "Input wastefield width:" in the dialogue window. Enter an
approximate width of 1040 m to override the default value of 2088. PLUMES then prompts
"Input starting longitudinal coordinate", i.e., the horizontal travel distance. Here we will accept
9.36m which is the horizontal distance between the source and the end of the initial dilution
zone.
The results are shown in Figure 32. As was anticipated, the farfield concentration is now
somewhat lower: 536.9. We hasten to add however that this underpredicts farfield concentration
because the effect of cross diffuser merging is ignored. At least we have had the opportunity to
demonstrate the Configuration menu, and, in any case, we now feel more certain that the 400
colonies/lOOml standard will be exceeded. A better estimate of farfield concentration awaits a
more complete analysis.
70
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UM INITIAL DILUTION SIMULATION (non-linear mode)
plume dep plume dia poll cone dilution hor dis
m
70.10
67.27
61.20
54.65
53.04
m
0.08500 610000000
1.822 18800000
6751000
3744000
3406000
Farfield calculations based on Brooks (1960)
Input starting longitudinal coordinate:
Farfield dispersion based on wastefield width of
3 .682
7.502
10.75
1.000
31.18
84.58
144.2
153.5
m
0.000
4.417
6.463< trap level
8 .460< merging
9.365< begin overlap
see guide for details:
--4/3 Power Law--
cone dilution
418200
46750
5033
536.9
154.4
164.0
181.1
201.8
-Const Eddy Diff-
conc dilution distance
m
419100 154.0 500.0
48000 159.6 1000
5387 169.0 1500
602.7 179.4 2000
1040m
Time
sec hrs
3271
6604
9938
13270
0.9
1.8
2 .8
3.7
Figure 32 Using the Configuration menu to gain control over farfield input and output.
Go ahead and change the Configuration string back to "ATNOO" and run RSB by using the
AB command. The results are given in Figure 33. For those who also run the non-PLUMES
version of RSB, it is important to note that for equivalence the PLUMES RSB version must
use a spacing value half as large as the original model since the latter assumes two ports
per spacing distance while PLUMES RSB assumes only one. This is done to be consistent
with the UM convention.
Notice that RSB does not report a trapping level or intermediate dilution. However, we may
compare the average volume flux dilutions at the plume element overlap level: they are 153.6
and 182 for UM and RSB respectively. The corresponding wastefield thicknesses are 10.75 (see
[plume dia]) and 12.2 meters respectively, varying by a similar amount. Finally, the respective
centerline rises are 17.06 and 10.9 meters.
Once again, if the analysis allows the luxury, it is convincing to present the results of the
most conservative conditions likely to be encountered for the variables even if they are unlikely
to occur simultaneously.
(Note that if the simulated plume is allowed to develop to maximum rise, which is possible
when the Configuration string is changed to, for example, "ATNMO" ("M" is maximum rise), the
corresponding far-field dilution, diameter, and rise are 156.2, 16.5, and 17.23 respectively. This
is characteristic of the overlap problem under which plume diameter is overestimated, which, if
prolonged, feeds back and increases the initial dilution. Frick, Baumgartner, and Fox (1992)
show this problem is shared by Lagrangian and Eulerian integral flux plume models generally,
due to inadequacies of the standard round plume assumption. It is unimportant in this case, the
dilution increasing from only 153 to 156 in the overlapped region.)
71
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A tutorial of the interface
RSB
Written by Philip J. W. Roberts (12/12/89)
(Adapted by Walter E. Frick (1/12/92))
Case: 2: Sand Island validation: no blockage
Lengthscale ratios are: s/lb = 3.42 1m/Ib = 0.21
Froude number, u3/b = 0.00
Jet Froude number, Fj = 18.6
Rise height to top of wastefield, ze = 16.3
Wastefield submergence below surface = 53.8
Wastefield thickness, he = 12.2 m
Height to level of cmax, zm = 10.9 m
Length of initial mixing region, xi = 8.6m
Minimum dilution, Sm = 158
Flux-average dilution, Sfa = 182 ( 1.15 x Sm)
Results extrapolated beyond their experimental values, may be unreliable
Wastefield submerged
Interpolation count: 10
Roberts Fr. # < 0.01 (aspiration dominated), no avg. flux dilution formed
for farfield prediction
Farfield calculations based on Brooks (1960), see guide for details:
Farfield dispersion based on wastefield width of 2085m
--4/3 Power Law— -Const Eddy Diff-
conc dilution cone dilution distance Time
m sec hrs
411800 182.2 411800 182.2 500.0 3276 0.9
48130 184.9 48420 183.8 1000 6610 1.8
5473 192.9 5603 188.4 1500 9943 2.8
612.2 204.5 642.4 194.9 2000 13280 3.7
Farfield result will not reflect decay in the near-field.
Figure 33 The RSB simulation of the first Sand Island case. (Note the excessive estimate of
the wastefield width.)
PLUMES links the same Brooks farfield model to RSB as it does to UM. It may seem odd
then that RSB predicts a farfield concentration almost equal to that of UM (612.2 vs. 626.7) even
though the dilution is substantially higher (204.5 vs. 172.3). One reason is the small T90 time:
in UM the decay mechanism is functional from discharge, while the pollutant is assumed to be
conservative (non-decaying) in the initial dilution region in RSB.
As was suggested previously, it is perhaps appropriate to consider a weakly stratified case,
as shown in Figure 34, in order to simulate a surfacing waste field that might impact recreational
waters. Notice that this case is Case 2, as is shown in the upper right corner of Figure 34. To
create a new case use the AC command, on the Movement menu. The new
case will use the information contained in the present case from which the AC command is given
as a template. Once in the new case, it may be edited. In Figure 34, some of the ambient data
has been changed: the case title, one line of ambient has been removed using the AYD command,
and changes shown in bold.
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Jun 19, 1992, 12:11:52 ERL-N PROGRAM PLUMES, Jun 10, 1992
Title Sand Island validation, no blockage, min strat.
tot flow # ports port flow spacing effl sal effl temp
4.469 285 0.01568 7.315 0.0 25
port dep port dia plume dia total vel horiz vel vertl vel
70.1 0.085 0.08500
port elev ver angle cont coef
2.763 2.763
effl den poll cone
0.84
0.0
1.0
-2 .893
nor angle red space p amb den p current
Case: 2 of 2
non-linear
far inc far dis
500 2000
asp coeff print frq
0.10 500
Froude # Roberts F
18.68 2.105E-14
K:vel/cur Stratif #
2763000.00001395
N (freq) red grav.
0.006417 0.2576
buoy flux puff-ther
0.004039 35.96
jet-plume jet-cross
1.494 20820
plu-cross jet-strat
4.039E+12 5.696
plu-strat
11.12
nor dis>=
CORMIX1 flow category algorithm is turned off.
0.0 to any range
Help: Fl. Quit: . Configuration:ATNOO. FILE: sandis.var;
90
depth
0.0
30
61
76
7.315
current
le-5
le-5
le-5
le-5
23.290.00001000
density salinity
22.99
23.23
23.31
23.28
34.99
35.11
35.16
35.15
6.1e8
far dif
0.000453
temp
26.18
25.71
25.56
25.64
0.000
decay
55.26
far vel
0.15
amb cone
0
0
0
0
Figure 34 Case 2: a weakly stratified Sand Island case.
As before, you can now run UM and RSB, the results are given in Figure 35. Again, the
predicted UM and RSB dilutions compare well, being 601.6 and 671 respectively. This time the
UM plume diameter and the RSB wastefield thickness, which are not totally comparable
quantities, also agree closely, being 37.57 and 36.6 meters respectively. The message warning
plume element overlap, indicates upstream intrusion of the wastefield is possible (Frick et al.,
1989). The rises are considerably different, being 50.53 ( 70.10 - 19.57 ) and 32.7 meters
respectively. The UM farfield concentration is now 134.2 colonies/lOOml, which is much less
than the previous farfield concentration and would meet the water quality standard of 400
colonies/lOOml.
STEP 4: Analyze the Model Results and Make Adjustments
In the previous section the RSB, UM, and historical model results were compared. Now we
will delve into the implications of some of the findings and question some of the assumptions
that were made. In doing so, we will change the program configuration to make it possible to
find the CORMDC flow categories for the cases in question. We will also illustrate the PLUMES
conflict resolution capability.
From the standpoint of assumptions made earlier, in Sand Island Case 3 we will first examine
the implications of sand blockage of half of the diffuser ports. In Case 4 the focus shifts to the
sensitivity of the models to the magnitude of the decay coefficient, to other assumptions and
input data. Finally, in Case 5, we examine the effect of current on predictions.
73
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A tutorial of the interface
UM INITIAL DILUTION SIMULATION (non-linear mode)
plume dep plume dia poll cone dilution hor dis
m m
70.10 0.08500 610000000
67.28 1.810 18800000
44.72 7.382 1603000
29.51 11.73 936100
19.57 37.57 727000
m
1.000 0.000
31.21 4.432
331.5 8.008< merging
520.1 8.868< trap level
601.6 9.652< begin overlap
Farfield calculations based on Brooks (1960), see guide for details:
Farfield dispersion based on wastefield width of
--4/3 Power Law-- -Const Eddy Diff-
2115m
cone dilution
89800
10510
1197
134.2
602 .0
610.5
636.2
673 .9
cone dilution distance
89810
10570
1224
140 .5
601.9
606.9
621.8
642 .9
m
500.0
1000
1500
2000
Time
sec
3269
6602
9936
13270
hrs
0.9
1.8
2.8
3.7
RSB
Written by Philip J. W. Roberts (12/12/89)
(Adapted by Walter E. Frick (1/12/92))
Case: 2: Sand Island validation, no blockage, min strat.
Lengthscale ratios are: s/lb = 0.92
Froude number, u3/b = 0.00
Jet Froude number, Fj = 18.9
1m/Ib =
0.06
Rise height to top of wastefield, ze = 48.7
Wastefield submergence below surface = 21.4
Wastefield thickness, he = 36.6 m
Height to level of cmax, zm = 32.7 m
Length of initial mixing region, xi = 31.9 m
584
671 ( 1.15 x Sm)
Minimum dilution, Sm =
Flux-average dilution, Sfa =
Wastefield submerged
Interpolation count: 3
Roberts Fr. # < 0.01 (aspiration dominated), no avg. flux dilution formed
for farfield prediction
Figure 35 UM and RSB predictions for Sand Island Case 2.
Instead of using the AC command, in going from one case to the next it is easier to use the
key. Use it to create Case 3. Now make the changes indicated in Figure 36 to the
ambient block (remember to delete the middle lines using AYD), the title, and the [# ports] cell.
To change the PLUMES configuration use the AR command to obtain the Configuration menu,
then toggle the CORMIX flow categorization feature — simply press . Notice that the
configuration string at the bottom of the interface changes from ATNOO to ATCOO after which
the flow category is given above the dialogue line: "CORMDC1 one port flow s5 unattached".
74
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A tutorial of the interface
Jun 19, 1992, 12:17:37 ERL-N PROGRAM PLUMES, Jun 10, 1992 Case: 3 of 3
Title Sand Island validation, blockage, mln strat. non-linear
tot flow # ports port flow spacing effl sal effl temp far inc far dis
4.459 142 0.03147 7.315 0.0 25 500 2000
port dep port dia plume dia total vel horiz vel vertl vel asp coeff print frq
70.1 0.085 0.08500 5.546 5.546 0.000 0.10 500
port elev ver angle cont coef effl den poll cone decay Froude # Roberts F
0.84 0.0 1.0 -2.893 6.Ie8 55.26 37.50 1.049E-14
hor angle red space p amb den p current far dif far vel K:vel/cur Stratif #
90 7.315 23.270.00001000 0.000453 0.15 554600 4.089E-06
depth current density salinity temp amb cone N (freq) red grav.
0.0 le-5 23.19 35.13 25.90 0 0.003473 0.2574
76 le-5 23.28 35.15 25.64 0 buoy flux puff-ther
0.008100 72.19
jet-plume jet-cross
3.001 41780
plu-cross jet-strat
8.100E+12 10.97
plu-strat
20.97
hor dis>=
CORMIX1 one port flow s5 unattached. Use UH to overlap point. (See manual)
0 0.0 to any range
Help: Fl. Quit: . Configuration:ATCOO. FILE: sandis.var;
Figure 36 Sand Island blocked ports case.
The PLUMES CORMIX classification algorithm is presently limited to single ports, thus the
classification applies only to the unmerged region of the plume. Also, CORMIX is limited to
predicting plume behavior in, at most, two layer systems. Consequently, the interface will not
predict the flow category unless there are at least two and not more than three lines of ambient
input information. This is one reason why the middle lines in the ambient block in Figure 34
have been deleted. (Also, the surface salinity and temperature cells have been arbitrarily adjusted
to give about the same density gradient found between the 30 and 76 m depths, ignoring the
measured values at 61 m.) In this case this is not a significant simplification, especially since
the original density structure is not entirely self-consistent as is evidenced by the unstable layer
in the third line of ambient stratification in Figure 34 (denser fluid of 23.31 sigma-t units would
appear to lie over less dense fluid of 23.28 sigma-t units). This could be the result of
measurement anomalies or a real transient condition.
Run Case 3 — Figure 37. The initial dilutions do not change very much: 601.6 to 577.1 and
671 to 658.7 for UM and RSB respectively. The farfield concentrations also change little: from
134.2 to 124.8 and 184.0 to 174.0 for UM and RSB respectively. The changes would be greater
except for the fact that the surface is reached in Figure 37. Under these conditions involving
light currents, the two models, very different in formulation, are in close agreement.
While you are still in Case 3, use the command to create Case 4. Earlier it was
assumed that the effluent temperature was 25 C, its more "correct" value is 25.1 C. While this
is a trivial change, go ahead and enter it anyway. Also, an effluent of 0.99979 gm/cc is reported;
do not enter it just yet. While the differences are seemingly trivial, it does provide an
75
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UM INITIAL DILUTION SIMULATION (non-linear mode)
plume dep plume dia poll cone dilution hor dis
m m
70.10 0.08500 610000000 1.000
68.53 2.388 18860000 31.21
49.04 7.377 2578000 211.2
2.752 19.19 790700 561.3
0.1349 20.25 758700 577.1
m
0.000
5.957
13.38< merging
17.97< trap level
18.17< surface hit
Farfield calculations based on Brooks (1960) , see guide for details:
Farfield dispersion based on wastefield width of
--4/3 Power Law-- -Const Eddy Diff-
1052m
cone dilution
96780
10840
1169
124.8
579.9
614.9
678.0
754.9
cone dilution distance
96960
11120
1249
139.8
578.8
599.2
633 .6
672.4
m
500.0
1000
1500
2000
Time
sec
3212
6546
9879
13210
hrs
0.9
1.8
2 .7
3.7
RSB
Written by Philip J. W. Roberts (12/12/89, 4/22/93)
(Adapted by Walter E. Frick (1/12/92, 5/6/93))
Case: 2: Sand Island validation: blockage, min strat.
Lengthscale ratios are: s/lb =
Froude number, u3/b =
Jet Froude number, Fj =
0.59 1m/Ib = 0.09
0.00
38.0
PLUME SURFACES
Rise height to top of wastefield, ze = 70.1 m
Wastefield submergence below surface = 0.0 m
Wastefield thickness, he = 52.6 m
Height to level of cmax, zm = 47.0 m
Length of initial mixing region, xi = 50.0 m
Minimum dilution, Sm = 572.8
Flux-average dilution, Sfa = 658.7 ( 1.15 x Sm)
Interpolation count: 0
Roberts Fr. # < 0.01 (aspiration dominated), no avg. flux dilution formed
for farfield prediction
FARFIELD CALCULATION (based on Brooks, 1960, see guide)
Farfield dispersion based on wastefield width of 1039m
--4/3 Power Law-- -Const Eddy Diff-
conc dilution
135500
15190
1634
174.0
661.1
699.1
770.7
858.7
cone dilution distance Time
m sec hrs
135600 660.2 500.0 3000 0.8
15570 682.3 1000 6333 1.8
1746 721.5 1500 9667 2.7
195.0 766.1 2000 13000 3.6
Farfield result will not reflect decay in the near-field.
Figure 37 UM and RSB predictions for Case 3.
76
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A tutorial of the interface
opportunity to demonstrate the conflict resolution capability.
By now you are familiar with the fact that PLUMES differentiates between independent
(yellow) variables, or values, that you provide (or accept by default) and dependent (white)
variables that PLUMES can create on its own from the information you type into the spreadsheet
interface. You may have wondered, "What would happen if I move to a cell which contains a
white value and I input a new value, thus overriding the old value?" This is the primary reason
why other programs do not allow the input of redundant variables. The danger is that you will
either create an inconsistency or, as it is called in mathematics, overspecify the system.
PLUMES has the capability to resolve many of these conflicts.
Go ahead and move the cursor in the effluent density [effl den] cell and, ignoring the
dependent value it contains, use the AK command to obtain the units of gm/cm3 and enter
0.99997 kg/m3.
As soon as you are finished entering the data the background in the effluent salinity and
temperature cells ([effl sal] and [effl temp]), the plume (port) depth cell [port dep] and the brown
effluent plume density [effl den] cells acquire a magenta background color and the 70.1 value
in [port dep] should begin to blink. PLUMES has detected the conflict that your overriding of
the density value has caused. You are now confined to the conflict resolution mode until you
complete the actions shown in the dialogue window. The will move you from cell
to cell, showing its location by blinking the value in each in turn. You must determine which
of the conflicting independent variables you wish to delete and then do so. That is the only
normal way to leave the mode. In this case, move the cursor to the [effl sal] and press or
the delete key. Immediately, PLUMES replaces it with the value of 3.600 o/oo.
This new value has interesting implications. The question might be asked whether the
effluent is indeed so saline, or is it more likely that suspended or other dissolved contaminants
contribute to the density or is it a case of analytical measurement errors? This question will not
be resolved here but may be important to pursue if the reduction in dilution caused by reduced
buoyancy results in a standards violation. In any case, by running UM you find that the farfield
bacterial count has been raised only a few percent and is still below the critical value.
Now we will create Case 5 to provide another variation of Case 2, the case with minimum
stratification and no blockage. Use AC and <2> to return to Case 2 followed by AC and <5> to
establish the new case. Our main objective is to analyze the effect of current, but first we will
look at another bacterial contaminant that is regulated, Enterococcus, which is found in the
effluent at 6.3x106 colonies/lOOml. When you make just this change in the [poll cone] cell and
run UM and RSB you see that UM provides a farfield dilution of 673.9 which is exactly the
same as Case 2 and a plume concentration of 1.386 which is 1/96.6 of 139, the concentration
found with Case 2. Of course this is the same fraction as 6.3xl06 is of 6.1xl08.
RSB provides a greater farfield dilution (993.9) than UM because the initial dilution is
higher. However, the 1.901 colonies/100ml is greater than the UM concentration because UM
77
-------
A tutorial of the interface
includes die-off in the initial dilution region while RSB does so only in the farfield. However,
both RSB and UM plume concentrations are proportional to the effluent concentrations.
Both RSB and UM are now in agreement: the discharge will meet the Enterococcus water
quality criterion of 7 colonies/lOOml, predicting 1.901 and 1.386 colonies/100ml respectively.
However, predicted concentrations are very sensitive to the value of the decay constant. For
example, if the T90 time for Enterococcus is 1 hour and 15 minutes, rather than one hour, an
increase of only 25%, the Enterococcus bacteria concentration predicted by UM increases more
than six-fold, to 8.044 colonies/100ml, a value close to the numerical value of the Enterococcus
standard. The corresponding RSB concentration is 10.18.
Before wrapping this example up, we will make one more change. It may be argued that it
is unrealistic to subject the plume to zero current in the initial dilution (or rise) region and then
assume that the subsequent current is 15 cm/sec. We will now examine the effect of current on
predictions of the two models.
From Case 3 create Case 6 using the AC command. Add to the title the word "current".
Move to the ambient current cell [current]. Now type .1, i.e. 10 cm/sec, over each of the
currents. Also, change the concentration in the [poll cone] cell to 6.3e6. The interface for Case
6 should agree with Figure 38.
The predictions for both UM and RSB are shown in Figure 39. The UM average dilutions
at the end of initial mixing (overlap is no longer a problem) and at 2000 meters are substantially
higher than RSB, 1667 compared to 1111.8 and 2131.3 versus 1400.9. The UM farfield
Enterococcus concentration is disproportionally lower than the RSB value, 0.5343 compared to
2.144 colonies/100ml, reflecting again the treatment of die-off in the initial dilution phase by UM
but not by RSB. Since the initial dilution region is 275 m long, the effluent takes the better part
of an hour to traverse this distance. Nevertheless, both RSB and UM predict that the standard
would be met under these conditions. However, at 1500m the models would be in disagreement
on the standard being met.
Some of the differences in the models can be attributed to the fact that RSB uses a constant
peak-to-mean ratio of 1.15. The average flux dilution calculation of 1440.7 also given in Figure
39 suggests a higher ratio would agree more closely with the average flux dilution calculation
and with UM. It is difficult to describe the relationship between average and centerline values
based on empirical measurements because it is necessary to define the plume boundary. Thus,
it is possible that the RSB average predictions are overly conservative.
All along we have been finessing the issue of port blockage and using the spacing on one
side of the diffuser (half the number of ports) versus using half the spacing (all ports). Of
course, PLUMES can be easily set up to do either. When the number of ports in Case 6 is
restored to 285 and the spacing is reduced to 3.858m (12ft) the initial dilution for UM increases
from 1667 to 1774, a relatively small change. Increased dilution from more ports more than
offsets decreases due to smaller spacing. For RSB it increases comparably from 1111.8 to
78
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A tutorial of the interface
1268.9. Neither change is really significant, although it may be in other circumstances. If the
most conservative analysis still shows the standards will be met, the "right" answer is really
irrelevant. However, if standards are not met then refinements are in order.
Jun 23, 1992, 12:50:31 ERL-N PROGRAM PLUMES, Jun 10, 1992 Case: 6 of 6
Title Sand Island validation: blockage, min strat., current non-linear
tot flow # ports port flow spacing effl sal effl temp far inc far dis
4.469 142 0.03147 7.315 0 25 500 2000
port dep port dia plume dia total vel horiz vel vertl vel asp coeff print frq
70.1 0.085 0.08500 5.546 5.546 0.000 0.10 500
port elev ver angle cont coef effl den poll cone decay Froude # Roberts F
0.84 0.0 1.0 -2.893 6.3e6 55.26 37.50 0.01049
hor angle red space p amb den p current far dif far vel K:vel/cur Stratif #
90 7.315 23.27 0.1000 0.000453 0.15 55.46 4.089E-06
depth current density salinity temp amb cone N (freq) red grav.
0.0 0.1 23.19 35.13 25.90 0 0.003473 0.2574
76 0.1 23.28 35.15 25.64 0 buoy flux puff-ther
0.008100 3.351
jet-plume jet-cross
3.001 4.178
plu-cross jet-strat
8.100 10.97
plu-strat
20.97
hor dis>=
CORMIX1 one port flow s4 attached. al. Use UM. (See manual)
76 m, 249.3 ft 0.0 to -200 m range
Help: Fl. Quit: . Configuration:ATCOO. FILE: sandis.var;
Figure 38 Case 6, with current.
STEP 5. Using the Results in the Decision Making Process.
As we said it might, the analysis evolved along several paths and examined several issues.
Yet, the higher flow cases are still not analyzed and the analysis remains incomplete.
Completing the job is left an as exercise. However, given that the data was reliable and
appropriate, that our conclusions about the proper use of UM and RSB are correct, and that this
is the only contaminant of concern, it seems that the proposed plant expansion should meet the
state's water quality criterion for Enterococcus. Thus, even doubling the flow rate would allow
the standard to be met according to the UM predictions.
But how good are the input data? In the case of the decay constant, we have observed
extreme sensitivity of the bacterial concentration to minor changes in the decay constant.
Bacterial survival in ocean water is known to depend strongly on solar insolation, protozoan
predation, and other factors. We also saw that dilutions and concentrations were sensitive to
ambient current speeds and ambient density, and, in this case, less sensitive to port spacing.
With these considerations in mind, it is important for the analyst to obtain the best data
possible and to encourage regulatory agencies to acquire environmental data over a wide range
of conditions. Even then, it is apparent that judgment is also likely to play a role in the decision
making process.
79
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UM INITIAL DILUTION SIMULATION (non-linear mode)
plume dep plume dia poll cone dilution hor dis
mm m
70.10 0.08500 6300000 1.000 0.000
69.22 2.082 195300 31.21 5.830
62.52 7.347 28050 208.3 16.46< merging
42.47 34.54 4950 997.8 45.46
35.52 48.89 3433 1335 57.70< trap level
30.02 64.53 2546 1667 70.17< surface hit
Farfield calculations based on Brooks (1960), see guide for details:
Farfield dispersion based on wastefield width of
--4/3 Power Law--
conc dilution
-Const Eddy Diff-
conc dilution
406.3
45.97
4.987
0.5343
1670.4
1753.4
1921.2
2131.3
406.7
46.91
5.285
0.5924
1669.0
1717.4
1810.3
1917 .9
distance
m
500.0
1000
1500
2000
1096m
Time
sec hrs
2866 0.8
6199
9532
12870
1.7
2 .6
3 .6
RSB
Written by Philip J. W. Roberts (12/12/89, 4/22/93)
(Adapted by Walter E. Frick (1/12/92, 5/6/93))
Case: 6: Sand Island validation: blockage, min strat., current
Lengthscale ratios are: s/lb =
Froude number, u3/b =
Jet Froude number, Fj =
0.59 1m/Ib =
0.91
38.0
0.09
Rise height to top of wastefield, ze = 70.1 m
Wastefield submergence below surface = 0.0m
Wastefield thickness, he = 62.4 m
Height to level of cmax, zm = 47.0 m
Length of initial mixing region, xi = 206.2 m
PLUME SURFACES
Minimum dilution, Sm =
Flux-average dilution, Sfa =
Interpolation count: 0
Wastefield width: 1031.50m
for farfield prediction
966.8
1111.8
( 1.15 x Sm)
Avg. flux dilution (width*he*u/Q)
FARFIELD CALCULATION (based on Brooks, 1960, see guide)
Farfield dispersion based on wastefield width of 1039m
--4/3 Power Law-- -Const Eddy Diff-
conc dilution
1618
185.4
20.11
2.144
1112.4
1151.7
1259.2
1400.9
cone dilution distance Time
m sec hrs
1618 1112.4 500.0 1959 0.5
188.0 1135.4 1000 5292 1.5
21.18 1195.6 1500 8625 2.4
2.367 1269.1 2000 11960 3.3
Farfield result will not reflect decay in the near-field.
1440.7
Figure 39 UM and RSB predictions for Case 6.
80
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Example: CORMIX1 comparison, density, and stability
EXAMPLE: CORMIX1 COMPARISON, DENSITY, AND STABILITY
INTRODUCTION
Beginning in 1973, the U.S. EPA sponsored research which ultimately led to a succinct,
untuned statement of forced entrainment, the Projected Area Entrainment (PAE) hypothesis (Prick
and Winiarski, 1975; Winiarski and Frick, 1976, 1978; Teeter and Baumgartner, 1979; Frick,
1984; Frick, Baumgartner, and Fox, 1993). Models using PAE, all currently expressed in the
Lagrangian formulation, include OUTPLM (Teeter and Baumgartner, 1979), UMERGE
(Muellenhoff et al., 1985), UM, and JETLAG (Lee and Cheung, 1990). Sometimes criticized,
the work was recently verified and justified by Lee and Cheung (1990) and Cheung (1991).
Cheung (1991) shows that JETLAG, a three-dimensional plume model, clearly outperforms the
highly regarded Chu (1975) and Schatzmann (1979) models in predicting trajectory and dilution
constants in asymptotic flow regimes. It also indicates the correct power law dependence of the
trajectory in different flow regimes. Frick, Baumgartner, and Fox (1993) demonstrate the
similarity between UM and JETLAG for two-dimensional flow, showing that Cheung's
conclusions concerning JETLAG apply to UM as well.
Nevertheless, while it should be possible to apply the PAE hypothesis to plume behavior in
general, the EPA UM model is presently limited to simple merging geometries and surface
interaction phenomena. Thus, it performs best when plumes are discharged in deep water. It is
also a two-dimensional model, though the merging version is pseudo-three-dimensional and an
experimental single-port three-dimensional vector version comparable to JETLAG exists.
The RSB model overcomes some of these limitations. In addition, they played a role in
EPA's decision to develop the EPA CORMIX models, or expert systems (summarized by Jirka
and Hinton, 1992). CORMIX stands for CORnell MIXing zone models. The idea was to exploit
accumulated laboratory and field experience to compile a set of methods and empirical models
to bridge the gaps evident in theoretical modeling. The Cornell initiative resulted in the
development of CORMIX1, CORMIX2, and CORMIX3 for the analysis of submerged single port
discharges, submerged diffusers, and surface discharges, respectively. About 80 different diffuser
and ambient profile combinations, or flow classifications are represented.
But, while theoretical models are subject to assumptions, their behavior is fairly predictable
when those assumptions are met. On the other hand, empirical models are most effective when
prototype and model variables and conditions match closely. When they do not the predictions
can degrade substantially. This is a real, if fine, distinction. In other words, it is often difficult
to extrapolate to conditions which were not included in the experimental design on which the
models are based. Since it is often not clear to the user when extrapolation occurs, this can be
a real problem. The example in this chapter demonstrates some of the pitfalls.
An example comparing the UM and CORMIX models is presented to give you an
81
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Example: CORMIX1 comparison, density, and stability
appreciation of how PLUMES may be used to assess the appropriate uses for CORMIX, RSB,
and UM. At the same time it will help you understand the differences between the plume
models, their strengths and weaknesses. The example chosen is from Appendix B of Jirka and
Hinton (1992), in which a full statement of the problem and a description of the CORMIX 1
solution is found.
In addition to the references to CORMIX, this example provides an opportunity to explore
the very important roles of density and stability in plume behavior and modeling. They are the
sources of some of the pitfalls alluded to above. It also addresses the relationship between
average and centerline plume properties.
PROBLEM
A manufacturing plant is discharging effluent into a reservoir. The effluent of 3.5 MOD
contains chlorides at a concentration of 3500 ppm (3.5 o/oo) and is released at a temperature of
20 C. The reservoir is large and deep, a cross-section is shown in Figure 40(a). The discharge
is at a depth of 29.9 m, 0.6 m off the bottom, and is directed upward at an angle of 10 degrees,
whose horizontal projection is perpendicular to the current. The port diameter is 25.4 cm. In
summer, the temperature profile (CORMIX approximation) of the lake is 29 C at the surface and
28.1, 19.1, and 11 C at depths of 15.5, 15.5, and 35 m respectively as shown in Figure 40(b).
The current in the bottom layer is small: 0.015 m/s.
The maximum allowable concentration is 1200 ppm of chloride and the allowable continuous
concentration is 600 ppm. The mixing zone boundary is 60 m away from the port. CORMIX 1
calculates an effluent density of 998.3872 kg/m3 and an ambient density of 999.6476 kg/m3.
Using a layer boundary depth of 15.5 m, CORMIX predicts the flow class S3 for this
example. No bottom attachment is indicated. The dilution at the boundary of the specified
regulatory mixing zone is predicted to be 11.9 at a depth of 27.5 m. This is a centerline dilution
— the average dilution would be significantly greater. The dilution is sufficient to meet both the
maximum and continuous criteria.
Because the plume is expected to trap in the stable bottom layer, we expect the PLUMES
UM model to simulate this case well, even though some of the underlying assumptions are not
met. RSB, as a multiple port diffuser model, is not applicable. If issued, the RSB command will
cause the message "Use RSB for multiple port diffusers" to appear in the dialogue window.
82
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Example: CORMIX1 comparison, density, and stability
(a)
(b)
View Looking Downstream
Field Measurements
Cormiz Approximation
UM3 Case 2
0.61m
\- Discharge Pipe
600 600 400 300 200 100
Distance From Shore (ft)
10 -|
20 -_
30 -
C- 40 -
Depth :
(m)
Discharge
Level
, , I , , . . I , . . . I . , , , I . , . . I . . , ,
10 20 30
Temperature (C)
Figure 40. (a) Reservoir cross-section, (b) Temperature profile.
S3
NH2
Figure 41. Schematics of flow classifications S3 and NH2 (Hinton and Jirka, 1992).
ANALYSIS
General Considerations
To begin this exploration of the relationship of the UM model to CORMIX and the issues
of density, stability, and plume profiles, start PLUMES and type in the data as shown in Figure
42. (Since the PLUMES interpolation capability will be demonstrated, leave the blank cells in
the ambient block as shown. Since the Configuration string shows that the auto ambient option
is on, which will normally provide a default value for these cells, you can turn it off.
83
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Example: CORMIX1 comparison, density, and stability
Jun
23, 1992, 20:
6:45
Title CORMIX1 example,
tot
0
flow
.1533
port dep
port
hor
29.9
elev
0.6
angle
90
depth
0.0
12.5
18.0
29.9
35 .0
t ports
1
port dia
0.2540
ver angle
10
red space
1000.0
current
0.015
0.015
0.015
0.015
Q .015
port
0.
plume
0.
cont
p amb
ERL-N PROGRAM
H&J
flow
1533
dia
2540
coef
1.0
den
density
-3
-3
_^
-0.
.993
.733
.250
3299
1992
spacing
1000
total vel
3.026
effl den
0.9296
p current
0.01500
salinity
0
0
0
0
PLUMES, July 1, 1992
effl sal
3.5
horiz vel
2.980
poll cone
3500
far dif
0.000453
temp
29.0
28.1
17 .5
11
effl temp
20
vertl vel
0.5255
decay
0
far vel
0.015
amb cone
0
0
0
0
0
Case:
far inc
20
asp coeff
0.10
Froude #
K: vel /cur
201.8
N (freq)
buoy flux
jet -plume
plu-cross
plu-strat
1 of 1
linear
far dis
60
print frq
500
Roberts F
Stratif #
red grav.
puf f-ther
jet-cross
45.41
jet-strat
hor dis>=
CORMIX1 algorithm limited
to three lines
of ambient
0.0 to any range
Help:
Fl.
Quit: . Configuration :ATCOO . FILE
: cormixl
var ;
Figure 42. First draft input for the CORMIX1 example.
Alternatively, you could move around the cells or delete the default values using AT.)
Several assumptions and statements concerning the input should be clarified: The default
port spacing [spacing] of 1000m is acceptable. It means that merging will not occur because
the plumes will never grow that large and thus UM will run as a point source model. Also,
as a first cut, the effluent salinity cell is input as 3.5 o/oo (3500 ppm) even though, since the
effluent is neither fresh nor sea water, the PLUMES equation of state is not valid for accurately
estimating the density of the effluent. The AK (units conversion) command has been used to
convert units in several cells, e.g. to input 3.5 MOD in the total flow cell. For purposes of
demonstration, the ambient depth of 29.9m has been entered into the ambient block while the
density, salinity, and temperature cells have been left blank.
It is assumed the effluent is co-flowing, i.e. discharged in the direction of the current, even
though it contradicts the actual geometric flow configuration. This is necessary because for
single ports UM is a two-dimensional model. A horizontal angle [hor angle] of 90 degrees
indicates the plume will be modeled as a co-flowing case. It is a justifiable assumption because
the current is small. Furthermore, it is a conservative assumption because entrainment will be
underestimated and, therefore, dilution will be less than it would be otherwise. This is due to
the fact that the plume will project less area to the current and therefore forced entrainment will
be reduced. Thus, a solution combining near and far field solutions may be patched together.
CORMIX1 works in a somewhat similar fashion, patching together different modules valid in
different parts of the plume's trajectory.
84
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Example: CORMIX1 comparison, density, and stability
PLUMES Configuration
A: Automatic ambient fill is on
R: Brooks equation input deleted
C: The CORMIX flow categorization algorithm is inactive
0: UM farfield predictions begin at element overlap
Farfield model initiation choices are:
M: maximum rise; 0: element overlap; P: pause criterion.
Other criteria, such as surface interaction, will override these choices.
0: Brunt-Vaisala reversals determined by UM as 1 or 2
Figure 43. The PLUMES configuration.
Before proceeding, it is good practice to assure that the model configuration options are set
appropriately. Use the command, ARS, to show the current settings
(Figure 43). It shows that the CORMIX 1 classification algorithm is currently inactive. Since
we want to illustrate the association between PLUMES and CORMIX, use the command, or , to activate the option. The third letter in the configuration string
at the bottom of the screen will change from "N" to "C" (e.g., ATNOO to ATCOO). The new
configuration string is shown in Figure 44.
To establish the proper, interpolated temperature at the 29.9 m depth in the ambient block,
Jun 23, 1992, 20:10:37 ERL-
Title CORMIX1 example, H&J
tot flow # ports port flow
0.1533 1 0.1533
port dep port dia plume dia
29.9 0.2540 0.2540
port elev ver angle cont coef
0.6 10 1.0
hor angle red space p amb den
90 1000.0 -0.5584
depth current density
0 .015
0 .015
0.015
0.015
0.015
N PROGRAM PLUMES, July 1, 1992 Case: 1 of 1
1992 non-linear
spacing effl sal effl temp far inc far dis
1000 3.5 20 20 60
total vel horiz vel vertl vel asp coeff print frq
3.026 2.980 0.5255 0.10 500
effl den poll cone decay Froude # Roberts F
0.9296 3500 0 -49.73 0.0003835
p current far dif far vel K:vel/cur Stratif #
0.01500 0.000453 0.015 201.8 -0.01961
salinity temp amb cone N (freq) red grav.
0 29.0 0 0.03357 -0.01458
0 28.1 0 buoy flux puff-ther
0 17.5 0 -0.002236 18.59
0.00 13.0 0 jet-plume jet-cross
0 11 0 11.89 45.41
plu-cross jet-strat
662.4 4.504
plu-strat
2 .772
hor dis>=
CORMIX1 algorithm limited to three lines of ambient
o/oo 0.0 to -200 o/oo range
Help: Fl. Quit: . Configuration:ATCOO. FILE: cormixl.var;
0 .0
12.5
18.0
29.9
35.0
-3 .993
-3 .733
-1.250
-0 .5584
-0.3299
Figure 44. Interface with CORMIX flow category and interpolated temperature.
85
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Example: CORMIX1 comparison, density, and stability
put the cursor in the temperature cell at 35 m depth. Since the automatic ambient fill option is
on you may have to use the AT command to keep the 29.9 m temperature cell empty after the
cursor is moved through it. From the 35 m temperature cell issue the command AYI; the correct
interpolated temperature, 13.0 C, appears immediately as shown in Figure 44. The same could
be done for the salinity cell although it will be easier to simply move the cursor through the cell
(or input 0 if the automatic ambient fill option is off). The ambient density will be calculated
automatically upon moving from the salinity cell. While density issues will be discussed further,
it is worth noting here that interpolating temperature and salinity value, will not result in the
same density as interpolating on density directly. The interpolated values are also shown in
Figure 44. The inclusion of the 29.9 m ambient line is not a requirement.
Notice that the CORMIX window near the bottom of the screen states: "CORMIX1
algorithm limited to three lines of ambient". This is a limitation of the PLUMES interface which
does not yet implement the full CORMIX classification algorithm. (CORMIX provides also for
two layers with a discontinuity, requiring four lines of ambient data. Also, the abridged version
that is implemented has not been reviewed by authors of the CORMIX models.) In some cases
it is possible to circumvent this limitation. For example, if the plume remains in the bottom layer
plume dep plume dia poll cone dilution nor dis
mm m
29.90 0.2540 3500 1.000 0.000
28.82 4.045 212.8 16.47 9.574
< only growth and aspiration entrainment after this point
< local maximum rise or fall
30.50 7.458 109.4 32.05 19.49
30.55 7.507 108.6 32.27 19.64< trap level
31.37 8.660 93.91 37.32 22.92< bottom hit
31.72 10.00 81.19 43.17 26.55< bottom hit
< only growth and aspiration entrainment after this point
< local maximum rise or fall< plume element overlap.
Farfield calculations based on Brooks (1960), see guide for details:
Farfield dispersion based on wastefield width of 10.00m
--4/3 Power Law-- -Const Eddy Diff-
conc dilution cone dilution distance Time
m sec hrs
50.19 69.8 62.32 56.2 40.00 897.0 0.2
24.69 141.8 44.74 78.3 60.00 2230 0.6
RSB not compatible with input conditions: negative buoyancy
Figure 45. UM and RSB output for the draft case (Case 1).
the details of the ambient temperature near the surface will be superfluous, making it possible
to simplify the ambient profile in order to obtain the CORMIX flow class. (In other cases, the
one or two layer approximation used in CORMIX may be inadequate.) Thus UM may be used
to predict the rise of the plume which, after the fact, shows that the simplification is appropriate
(i.e. the rise is limited to the bottom layer). The predictions are given in Figure 45.
86
-------
Example: CORMIX1 comparison, density, and stability
UM predicts a plume concentration of 212.8 at maximum rise at a downstream distance of
9.574 m and a depth of 28.82 m. This is clearly in the bottom stratified layer and within the
specified mixing zone of 60 m. Thus, the simplification of the ambient data to two lines of data,
as is done in Case 2 (which is developed in Figure 48), is appropriate. Consistent with the
predictive strategy for negatively buoyant plumes indicated by the configuration string, the UM
prediction continues past the point of maximum rise. With the option set to
0, UM determines the number of reversals, i.e. levels of maximum rise and fall, to be two
if the effluent is negatively buoyant. The average concentration at maximum fall is 81.19.
UM may also be used to estimate plume centerline concentrations. For this purpose there
is a centerline concentration [CL cone] cell which does not normally appear on the interface,
because, unlike the average properties which are fundamental model variables, it is an
approximated value. It can be added by manipulating the Pause Cell in the lower right hand
corner of the interface. To get the centerline concentration into the Pause Cell use the ), or
Figure 46. The Pause cell dialogue window.
cell> command, AYS, on the Miscellany menu. When you do the dialogue window shown in
Figure 46 appears. Press space bar to move through the list of available cells until the centerline
concentration [CL cone] cell appears. If you go too fast and pass it you can return to it by
pressing , then press to put it on the output table. The left byte of the cell turns blue
to indicate the variable will be output. It is worthwhile becoming familiar with this procedure.
Run UM again. The results are shown in Figure 47. We see that the maximum rise
centerline and average concentrations are 424.7 and 212.8 ppm respectively. At maximum fall,
the corresponding concentrations are 171.1 and 81.19 ppm. Note that the ratio of the centerline
to average concentration is not constant but increases from 1.0 at the source to 2.1 (171.1/81.19)
at the end of the initial dilution zone. This is a typical range although its theoretical limit for
single round plumes is 3.89. All concentrations are well below the 600 ppm standard.
The farfield concentrations are centerline concentrations. However, with merged plumes,
between the near and farfield, the concept of pollutant profiles shifts its orientation from vertical
to horizontal. Thus, since the initial wastefield is assumed to be well mixed, there is a region
in which the horizontal Gaussian profile is established whose length is difficult to determine.
If we were confident about the assumptions the analysis would be complete; after all, the
standards are met under relatively conservative conditions (e.g. at the first maximum rise for a
co-flowing plume). The same basic conclusion that the criteria will be met has been reached by
87
-------
Example: CORMIX1 comparison, density, and stability
plume dep plume dia poll cone dilution CL cone hor dis
mm rn
29.90 0.2540 3500 1.000 3500 0.000
28.82 4.045 212.8 16.47 424.7 9.574
< only growth and aspiration entrainment after this point
< local maximum rise or fall
30.50 7.458 109.4 32.05 225.6 19.49
30.55 7.507 108.6 32.27 224.1 19.64< trap level
31.37 8.660 93.91 37.32 195.7 22.92< bottom hit
31.72 10.00 81.19 43.17 171.1 26.55< bottom hit
< only growth and aspiration entrainment after this point
< local maximum rise or fall< begin overlap
Farfield calculations based on Brooks (1960), see guide for details:
Farfield dispersion based on wastefield width of 10.00m
--4/3 Power Law-- -Const Eddy Diff-
cone dilution
50.19
24. 69
69.8
141.8
cone dilution distance
62.32
44.74
56.2
78.3
m
40.00
60.00
Time
sec
897 .0
2230
hrs
0.2
0.6
Figure 47. UM output with centerline concentrations.
the PLUMES and CORMIX1 analyses. However, it is instructive to continue with the analysis,
especially as it serves the purpose of further illustrating the subtleties of the modeling process.
Ambient Profile Simplification
If the assumption that the effluent brine density obeys the PLUMES equation of state were
valid, there would be no reason to continue the analysis. However, since one is buoyant (rises)
while the other is negatively buoyant (sinks), it is clear that CORMDC uses another relationship
and therefore the assumption is questionable. Further examination shows that the effluent density
calculated by CORMIX1 corresponds closely to fresh water. Of course, the freshwater
assumption is also tenuous because chloride is a major constituent of sea water and the effluent
should probably exhibit a greater density. In any case, it is sobering to see how little it takes to
switch from one flow pattern shown in Figure 41 (S3) to another (NH2). Thus, understanding
the role that density plays in plume behavior and ambient stability is very important.
Continue with the analysis by forming a new case, Case 2, using the AC command or key. Then, delete the three intermediate lines from the ambient block using the AYD
command. Finally, move to the surface ambient temperature cell and type in 25.5 (the extension
of the bottom temperature gradient shown as a dotted line in Figure 40). When you are done the
interface screen should look like the upper part of Figure 48.
PLUMES predicts a negatively buoyant flow classification type: NH3 with [bottom]
attachment a5 (see Doneker and Jirka, 1990 for schematic descriptions of these classes). That
the plume is negatively buoyant is also apparent from the fact that the effluent density (0.9296
88
-------
Example: CORMIX1 comparison, density, and stability
Jun 24, 1992, 12:49:36 ERL-N
Title CORMIX1 example, H&J
tot flow # ports port flow
0.1533 1 0.1533
port dep port dia plume dia
29.9 0.2540 0.2540
port elev ver angle cont coef
0.6 10 1.0
hor angle red space p amb den
90 1000.0 -0.7222
depth current density
0.0 0.015 -3.022
35.0 0.015 -0.3299
PROGRAM PLUMES, July 1, 1992
1992, reduced ambient lines
spacing effl sal effl temp
1000 3.5 20
total vel horiz vel vertl vel
3.025 2.979
effl den poll cone
0.9296 3500
p current far dif
0.01500 0.000453
salinity temp
0 25.5
0 11
CORMIX1 one port flow nh3 attached. a5. Use CORMIX.
11 deg C, 51.80 deg F
Help: Fl. Quit: . Configuration:ATCOO.
Case: 2 of 2
non-linear
far inc far dis
20 60
asp coeff print frq
0.10 500
Froude # Roberts F
-47.19 0.0003455
K.-vel/cur Stratif #
201.7 -0.01183
N (freq) red grav.
0.02748 -0.01618
buoy flux puff-ther
-0.002481 17.95
jet-plume jet-cross
11.28 45.40
plu-cross jet-strat
735.1 4.979
plu-strat
3.307
CL conc>=
(See manual)
-2.0 to 50 deg C range
FILE: cormixl.var;
0.5254
decay
0
far vel
0.015
amb cone
0
0
plume dep plume dia poll cone dilution CL cone hor dis
mm m
29.90 0.2540 3500 1.000 3500 0.000
28.83 4.019 214.3 16.36 427.5 9.505
< only growth and aspiration entrainment after this point
< local maximum rise or fall
30.44 7.370 110.9 31.61 228.5
30.52 7.467 109.4 32.05 225.6
31.34 8.741 93.26 37.58 194.4
31.58 9.814 82.89 42.29 174.4
< only growth and aspiration entrainment after this point
< local maximum rise or fall
Farfield calculations based on Brooks (1960), see guide for details:
19.21< trap level
19.49
23.07< bottom hit
25.97< bottom hit
Farfield dispersion based on wastefield width of
--4/3 Power Law-- -Const Eddy Diff-
9.814m
cone dilution
49.57
24.50
70.7
142 .9
cone dilution distance
62.57
45.12
56.0
77.6
m
40.00
60.00
Time
sec
935 .4
2269
hrs
0.3
0.6
Figure 48.
PLUMES.
Simplified Case 1 input to enable the CORMIX flow classification algorithm in
sigma-t) in the brown block is greater than the ambient density (-0.7222 sigma-t) in the green
block and the Froude # is negative. The NH3 is a classification similar to the NH2 classification
given if Figure 41. It differs significantly in character from the S3 classification.
It is gratifying that the overall flow characterization is essentially equivalent to the one
analyzed in Case 1, as it should be since the plume is negatively buoyant. Specific numerical
differences with the previous case may be attributed to small inaccuracies in specifying the
surface temperature, which was estimated graphically.
89
-------
Example: CORMIX1 comparison, density, and stability
Case: 3 of 3
non-linear
far inc far dis
20 60
asp coeff print frq
0.10 500
Froude # Roberts F
64.17 0.0006391
K:vel/cur Stratif #
201.7 0.02193
N (freq) red grav.
0.02748 0.008750
buoy flux puff-ther
0.001341 22.03
jet-plume jet-cross
15.35 45.40
plu-cross jet-strat
397.5 4.979
plu-strat
2 .836
CL conc>=
CORMIX1 one port flow s3 unattached. Use UM. (See manual)
11 deg C, 51.80 deg F -2.0 to 50 deg C range
Help: Fl. Quit: . Configuration:ATCOO. FILE: cormixl.var;
Jun 24, 1992, 9: 6:50 ERL-N
Title CORMIX1 example, H&J
tot flow # ports port flow
0.1533 1 0.1533
port dep port dia plume dia
29.9 0.2540 0.2540
port elev ver angle cont coef
0.6 10 1.0
hor angle red space p amb den
90 1000.0 -0.7222
depth current density
0.0 0.015
35.0 0.015
-3.022
-0.3299
PROGRAM PLUMES, July
1992, effl den = 998.
spacing effl sal
1000 0.1573
total vel horiz vel
3.025 2.979
effl den poll cone
-1.613 3500
p current far dif
0.01500 0.000453
salinity temp
0 25.5
0 11
1, 1992
3872
effl temp
20
vertl vel
0.5254
decay
0
far vel
0.015
amb cone
0
0
dilution CL cone
plume dep plume dia poll cone
m m
29.90 0.2540 3500 1.000 3500
28.19 3.897 218.8 15.98 436.1
27.00 7.494 110.9 31.53 228.5
< only growth and aspiration entrainment after this point
< local maximum rise or fall
Farfield calculations based on Brooks (1960), see guide for details:
hor dis
m
0.000
9.160<
18.44
trap level
Farfield dispersion based on wastefield width of
--4/3 Power Law-- -Const Eddy Diff-
7 .494m
cone dilution
110.6
42.26
22.25
31.6
82 .8
157 .3
cone dilution distance
110 .7
67.54
51.39
31.6
51.8
68.1
m
20.00
40.00
60.00
Time
sec
103 .7
1437
2770
hrs
0.0
0 .4
0.8
Figure 49. The interface screen after correction of CORMDC effluent density, with output.
For the sake of comparison, we will attempt to correct the discrepancy between the two
models by revising the assumption that the discharged chloride brine has the same equation of
state as that built into the interface. To do so, make a new case, Case 3. Then move the cursor
to the effluent plume density [effl den] cell, invoke the AK command, and input the effluent
density in kg/m3 given in the CORMIX1 analysis: 998.3872. After attempting to move from the
cell, the conflict resolution mode will trap the overspecification. Press to move to
the effluent salinity cell and delete its value. The interface screen should now look like that in
Figure 49. The effluent salinity now indicates 0.1573 o/oo which supports the conclusion that
the density of fresh water is used in the CORMIX example with chloride being treated as a
noncontributing component to density. With this assumption, the flow classification now agrees
with the CORMIX1 prediction — both are S3, with no bottom attachment in the initial dilution
region.
90
-------
Example: CORMIX1 comparison, density, and stability
The corresponding simulation is also shown in Figure 49. The predicted dilution is now
31.53 at the end of the initial dilution zone, i.e. at maximum rise. This is almost twice large as
the dilution found in Case 1 and consequently, if the density assumption were valid, which it is
not, the criterion for chloride will be easily met. (Note the inverse relationship between
concentrations and dilutions.) Consistent with the fact that the plume is now said to be buoyant
(-1.613 sigma-t < -0.7222 sigma-t), the farfield model begins at maximum rise and the advisory
message about growth and aspiration entrainment may be ignored.
Density: The Linear and Nonlinear Forms of UM
The CORMIX equation of state applies only to fresh water. For sea water, the user is
required to input density values directly. The PLUMES equation applies strictly only to both
fresh and sea water. However, there is another option in UM — a linear form of the equation
of state. In this form, as in the CORMIX sea water equation, the density is assumed to be linear
function, i.e. to have a constant coefficient of bulk expansion. Essentially the density is a
weighted average of the densities of the plume and ambient fluids. It is a useful approximation
in many cases where the non-linear form is inappropriate. However, it does not account for non-
linearities and is totally inadequate for predicting nascent density effects.
The linear equation of state is invoked simply by entering densities instead of salinity
and temperature, which are left undefined. In this mode the complex equation of state built
into PLUMES is ignored in favor of the simple linear equation of state. To illustrate, create Case
4 pressing the key in Case 3. Then delete ambient temperatures and salinities and
override the values in the ambient density cells. See Figure 50 and note the linear designation.
In this case the differences with the previous run in Case 3 are relatively small. The
predicted dilution at maximum rise for the linear form is 29.02 compared to 31.53 for the
nonlinear. The differences in rise are correspondingly small: 2.63 m (29.90 - 27.27) for the
linear form compared to 2.90 m (29.90 - 27.00) for the nonlinear form.
While the linear form is appropriate here, in most cases involving fresh or sea water,
without significant dissolved or suspended species, it is best to use the nonlinear form of
UM, i.e. to specify salinity and temperature rather than only density as input. It is recommended
because the equation of state of water, especially fresh, cold water, is significantly nonlinear.
For plumes discharged to very cold, fresh water, the linear form of the model can lead to
significant errors in the predictions, in extreme cases predicting monotonically rising plumes
where, in fact, real plumes will rise briefly before sinking to the bottom (Frick and Winiarski,
1978). This is the nascent density effect described in the first chapter.
To illustrate this very interesting behavior, consider the case of a highly buoyant plume
discharged to near freezing, fresh water. This is a common occurence in cold climates with
thermal discharges to fresh water bodies. From Case 4 press to create Case 5. Now
enter the temperatures shown in Figure 51. Then, after you are finished, run this nonlinear
91
-------
Example: CORMIX1 comparison, density, and stability
Jun 24, 1992, 9: 7:56 ERL-N PROGRAM PLUMES, July 1, 1992
Title CORMIX1 example, H&J 1992, linear equation of state
tot flow # ports port flow spacing effl sal effl temp
0.1533 1 0.1533 1000
port dep port dia plume dia total vel horiz vel vertl
29.9 0.2540 0.2540 3.025 2.979
port elev ver angle cont coef effl den poll cone
Case: 4 of 4
linear
far inc far dis
20 60
vel asp coeff print frq
0.5254 0.10 500
decay Froude # Roberts F
0.6 10 1.0 -1.613 3500 0 64.17 0.0006391
angle red space p amb den p current far dif far vel K:vel/cur Stratif #
90 1000.0 -0.7222 0.01500 0.000453 0.015 201.7 0.02193
depth current density salinity temp amb cone N (freq) red grav.
0.0 0.015 -3.022 0 0.02747 0.008750
35.0 0.015 -0.3299 0 buoy flux puff-ther
0.001341 22.03
jet-plume jet-cross
15.34 45.40
plu-cross jet-strat
397.5 4.979
plu-strat
2.836
CL conc>=
(See manual)
-2.0 to 50 deg C range
FILE: cormixl.var;
CORMIX1 one port flow s3 unattached. Use UM.
deg C, deg F
Help: Fl. Quit: . Configuration:ATCOO.
UM INITIAL DILUTION SIMULATION (linear mode)
plume dep plume dia poll cone dilution CL cone
m m
29.90 0.2540 3500 1.000 3500
28.38 3.527 242.7 14.41 482.2
27.27 6.934 120.5 29.02 247.0
hor dis
m
0.000
8.212< trap level
16.98
< only growth and aspiration entrainment after this point
< local maximum rise or fall
Farfield calculations based on Brooks (1960), see guide for details:
Farfield dispersion based on wastefield width of
--4/3 Power Law-- -Const Eddy Diff-
6.934m
cone dilution
115.3
41.39
21.99
30.3
84.5
159.2
cone dilution distance
117.4
70.06
53 .75
29.8
49.9
65.1
m
20.00
40.00
60.00
Time
sec
201.5
1635
2868
hrs
0.1
0.4
0.8
Figure 50. The linear equation of state mode of UM. Note the empty cells.
form of UM. The predicted plume reaches a false trapping level at the 29.46 m level and,
expending its vertical momentum, rises to a depth of 28.37 m. At this point the plume is, and
remains, negatively buoyant and, therefore, descends back to the bottom.
The reason for this behavior is due to the fact that water has its maximum density around
4C. Initially the plume is very buoyant (-7.724 sigma-t < 0.09290 sigma-t), but, as the plume
ascends in the water column, it rapidly cools through entrainment and becomes more dense than
the ambient fluid as the average density of the plume element approaches 4 C. At that
temperature it is considerably more dense than its surroundings which has a temperature
somewhere between 0 and 4 C at this point. Consequently, the upward ascent of the plume is
first inhibited and finally reversed due to the negative buoyancy.
92
-------
Example: CORMIX1 comparison, density, and stability
Jun 24, 1992, 9:57:22 ERL-N
Title CORMIX1 example, non-
tot flow # ports port flow
0.1533 1 0.1533
port dep port dia plume dia
29.9 0.2540 0.2540
port elev ver angle cont coef
0.6 10 1.0
nor angle red space p amb den
90 1000.0 -0.09290
depth current density
0.0 0.015 -0.09295
35.0 0.015 -0.09289
Case: 5 of 5
non-linear
far inc far dis
20 60
asp coeff print frq
0.10 500
Froude # Roberts F
21.860.00007415
K:vel/cur Stratif #
201.7 6.449E-08
N (freq) red grav.
0.0001378 0.07542
buoy flux puff-ther
0.01156 10.74
jet-plume jet-cross
5.227 45.40
plu-cross jet-strat
3426 70.29
plu-strat
257.7
CL conc>=
CORMIX1 one port flow h4-0 unattached. Use UM until near surface. (See manual)
o/oo 0.0 to -200 o/oo range
Help: Fl. Quit: . Configuration:ATC02. FILE: cormixl.var;
PROGRAM PLUMES, July 1, 1992
linear mode, very cold ambient
spacing effl sal effl temp
1000 0 40
total vel horiz vel vertl vel
3.025 2.979 0.5254
effl den poll cone decay
-7.724 3500
p current far dif
0.01500 0.000453
temp
0
0.001
salinity
0
0
0
far vel
0.015
amb cone
0
0
UM INITIAL DILUTION SIMULATION (non-linear mode)
plume dep plume dia poll cone dilution CL cone
m m
29.90
29.46
28.37
0.2540
1.237
4.850
3500
705 .8
175.2
1.000
4.921
19.82
3500
1372
352.3
nor dis
m
0.000
2.441< trap level
11.61
< only growth and aspiration entrainment after this point
< local maximum rise or fall
29.70 7.340 109.4 31.75 225.4 19.27
32.14 8.340 89.46 38.82 186.9 23.99< bottom hit
35.13 8.940 77.34 44.91 163.5 27.74< bottom hit
Leaving defined depth range
Figure 51. Discharge of a highly buoyant plume to very cold water; nonlinear form of UM.
With output.
To compare this simulation to one with the linear model, form Case 6 starting from Case 5
and override all the dependent (white) densities with equivalent independent densities. First erase
any salinity or temperature values. (Or, if the conflict resolution capability is used, the AQD
command is handy for moving to the end of the cell where you can add an extra zero to the
replacement string so that PLUMES knows that the densities are to become independent). When
you are done the interface should look like that in Figure 52.
In this, a case superficially identical to Case 5, the plume rises to the surface. Clearly it is
important to be aware of these extreme differences in model behavior. They are not both right.
Depending on the analysis, in one case one would conclude that benthic organisms will be
affected, in the other, surface organisms. Thus, whenever the data are available and suspended
and dissolved, foreign solids are not an important factor, the nonlinear equation of state should
be considered.
93
-------
Example: CORMIXl comparison, density, and stability
Jun 24, 1992, 10: 1: 5 ERL-N PROGRAM PLUMES, July 1, 1992
Title CORMIXl example, linear mode, very cold ambient
tot flow # ports port flow spacing effl sal effl temp
0.1533 1 0.1533 1000
port dep port dia plume dia total vel horiz vel vertl vel
29.9 0.2540 0.2540
port elev ver angle cont coef
1.0
Case: 6 of 6
linear
far inc far dis
20 60
asp coeff print frq
0.5254 0.10 500
decay Froude # Roberts F
0 21.860.00007414
far vel K:vel/cur Stratif #
0.015 201.7 5.706E-08
amb cone N (freq) red grav.
0 0.0001297 0.07542
0 buoy flux puff-ther
0.01156 10.74
jet-plume jet-cross
5.227 45.40
plu-cross jet-strat
3426 72.47
plu-strat
269.9
CL conc>=
CORMIXl one port flow h4-0 unattached. Use UM until near surface. (See manual)
0 o/oo 0.0 to -200 o/oo range
Help: Fl. Quit: . Configuration:ATCO2. FILE: cormixl.var;
3.025 2.979
effl den poll cone
-7.724 3500
hor angle red space p arab den p current far dif
90 1000.0 -0.09290 0.01500 0.000453
depth current density salinity temp
-0.09295
-0.09289
0.6
0.0
35.0
10
0.015
0.015
UM INITIAL DILUTION SIMULATION (linear mode)
plume dep plume dia poll cone dilution CL cone
m m
29.90 0.2540 3500 1.000 3500
20.93 5.665 109.4 31.76 225.4
1.529 10.48 38.14 91.08 86.82
hor dis
m
0 .000
14.08
22.36< surface hit
Farfield calculations based on Brooks (1960), see guide for details:
Farfield dispersion based on wastefield width of
--4/3 Power Law-- -Const Eddy Diff-
10 .48m
cone dilution
cone dilution
distance
m
Time
sec hrs
20.28
10.73
171.9
325 .5
27.17
20.36
128 .1
171.2
40.00
60.00
1176
2510
0.3
0.7
Figure 52. Discharge of a highly buoyant plume to very cold water; linear form of UM. With
output.
Some densities, including ones pertinent to this problem, are compared in Table III.
Table III. PLUMES and CORMIXl densities compared with published values (Weast, 1977).
Temperature
(C)
0.0
4.0
13.0
28.1
20.0
20 .0
20.0
20.0
Salinity
0
0
0
0
0
34
79
149
.0
.0
.0
.0
.0
.84
.69
.5
DGRS itiss (K.cj/m. ,
UM CORMIXl
998.267
998. 691
999.442 999.648
996.267 996.341
998.267
1024.66
1060.04
1126.3
Weast
999 .842
999.975
999.380
996.208
998.207
1024.5
1058.5
1112.2
94
-------
THE ROBERTS, SNYDER, BAUMGARTNER MODEL: RSB
INTRODUCTION
RSB is based on the experimental studies on multiport diffusers in stratified currents
described in Roberts, Snyder, and Baumgartner (1989a,b,c), which should be consulted for
detailed explanations. These studies were conducted with an experimental configuration shown
in Figure 53. The diffuser is straight and consists of horizontally discharging round nozzles
which are uniformly spaced. The ports discharge from both sides of the diffuser, which is similar
to most prototype applications. This configuration would include diffusers consisting of pipes
with ports which are holes along each side or T-shaped risers each containing two ports as shown
in Figure 53.
The receiving water is linearly density-stratified, and flows at a steady speed at an arbitrary
angle relative to the diffuser axis. RSB is intended for stratified conditions producing a fully
submerged wastefield; other models should be used for surfacing wastefields, for example
ULINE (Muellenhoff et al., 1985).
u
Wastefteld
Figure 53. Diffuser configuration considered by RSB.
As discussed later, RSB is also a good approximation for diffusers in which the ports are
clustered in multiport risers, at least up to 8 ports per riser. The range of the experimental
parameters (port spacing, port diameter, jet exit velocity, current speed, current direction, and
95
-------
The Roberts, Snyder, Baumgartner model: RSB
density stratification) was chosen to be representative of highly buoyant discharges such as
domestic sewage and some industrial wastes into coastal and estuarine waters. When RSB is
used outside the parameter range for which these experiments were conducted, it extrapolates the
results to obtain a solution and gives a warning that the answers are extrapolated.
The model can be thought of as a replacement for and a significant update of ULINE
(Muellenhoff et al., 1985). Whereas ULINE was based on experiments in unstratified
environments, RSB is based on experiments in stratified environments, and so is therefore more
reliable in this situation. Also, ULINE applies only to single line plumes whereas RSB is based
on experiments with multiport diffusers. It therefore includes the effects of port spacing and
source momentum flux, and is more realistic in that it includes discharges from both sides of the
diffuser.
DEFINITIONS
The definitions used in RSB in relation to the geometry of the initial mixing region are
shown in Figure 54 and described below. At the end of this region the dilution is called the
initial dilution and the wastefield is said to be established. The established wastefield then drifts
with the ocean currents and is diffused by oceanic turbulence.
V
Established wastefield — ,
Side view
(Fore -OCP)
Initial mixing region
Figure 54. Definition of Wastefield Geometry.
In RSB this "initial mixing region" or "hydrodynamic mixing zone" is defined to end where
the self-induced turbulence collapses under the influence of the ambient stratification and initial
dilution reaches its limiting value. The length of the initial mixing region is denoted by x,. The
96
-------
The Roberts, Snyder, Baumgartner model: RSB
geometrical wastefield characteristics (see Figure 53) at this point are thickness he height to top
z£ and height to level of maximum concentration (or minimum dilution) zn. The minimum initial
dilution Sm is defined as the smallest value of dilution (corresponding to maximum concentration)
observed in a vertical plane through the wastefield at the end of the initial mixing region.
MODEL BASIS
The initial mixing of wastewater discharged from a multiport diffuser depends on diffuser
design and receiving water characteristics. The diffuser can be characterized by fluxes of
volume, momentum, and buoyancy per unit diffuser length:
q = 9. m = uq b=g'0q (19)
Lt
where Q is the total discharge, L the diffuser length, «, the jet exit velocity, and g0' =
g(Pa' Po)/Pa is tne reduced gravitational acceleration due, g is the acceleration due to gravity, pa
is the ambient density at the level of the ports and p0 the effluent density. A linear density
stratification can be characterized by the buoyancy frequency, N, also referred to as the Brunt-
Vaisala frequency, usually expressed in units of sec"1:
(20)
We define three length scales:
= i i = i = J!L (21)
* IT •-
Note that these length scales are defined based on the total fluxes, rather than the flux from each
side of the diffuser. The geometrical characteristics defined in Figures 53 and 54 can then be
expressed as:
ze, he, zm = f(q,b,m,s,u,N,&) (22)
Which, by means of dimensional analysis, becomes:
/
j_ < m = f
m
s
(23)
Where F = i/lb is a dynamic variable which is a type of Froude number. In Equation 5, the
effect of the source volume flux q is neglected as an independent variable. This is because /y/6
97
-------
The Roberts, Snyder, Baumgartner model: RSB
is usually much less than one and therefore has little dynamic effect except very near to the ports.
The corresponding normalized expression for dilution is:
= /
n S F
-T—> —» " •>
(24)
where Sm is the minimum initial dilution, as previously defined. An average dilution Sa is
computed as 1.15 Sm based on hydraulic model tests by Roberts (1989).
The two length scale ratios ljlb and s/lb are diffuser parameters which characterize the
significance of source momentum flux and port spacing respectively. Note that these length scale
ratios encompass the jet exit velocity, port diameter, port spacing, effluent density, and ambient
stratification. Based on consideration of actual operating conditions, the range of experiments
was chosen to be 0.31 < sllb < 1.92 and 0.078 < ljlb < 0.5. For sllb < 0.3 and ljlb < 0.2, the
discharge approximates a line plume, i.e. the individual plumes rapidly merge and the effect of
source momentum flux is negligible, many ocean outfalls operate in the regime in which
momentum is negligible (Roberts et al., 1989a). Therefore the range of diffuser parameters can
be considered to be sllb < 1.92 and ljlb < 0.5
A more important parameter is F, which characterizes the importance of the current speed
relative to the buoyancy flux of the source. Small values of F signify little effect of current;
according to Roberts et al. (1989a) the current exerts no effect on dilution if F < 0.1. Larger
values of F denote situations where the plumes are rapidly swept downstream by the current;
dilutions are always increased by increased current speeds, although not always at the regulatory
(critical) mixing zone boundary, as shown in Figure 5. (See Figures 4 and 6 in Roberts, Snyder,
and Baumgartner, 1989a for photographs of plumes at various Froude numbers, F). The tests
were run at differing current speeds to obtain F = u3lb in the range 0 (zero current speed) to 100.
The effect of the current also depends on the direction of the current relative to the diffuser
0. For a line diffuser 0 < 0 < 90°. Tests were run with 0 = 90° (diffuser oriented perpendicular
to the current), 45°, and 0° (parallel to the current). In general, diffusers oriented perpendicular
to the current result in highest initial dilutions and lowest rise heights.
MODEL DESCRIPTION
Results for wastefield geometry and initial dilution were presented graphically (Figures 8,
10-12 of Roberts et al. 1989a) in the dimensionless form of Equations 5 and 6 for line plume
conditions (sllb < 0.3 and ljlb < 0.2). Results to predict the length of the initial mixing zone xt
are in Figures 4 and 8 of Roberts et al., 1989b. For higher port spacings and higher momentum
fluxes the results are given in Figures 5 and 6, and 7 and 8 of Roberts et al., 1989c.
For some of these results, semi-empirical equations are given. These equations are semi-
98
-------
The Roberts. Snyder. Baumgartner model: RSB
empirical because they are physically based, but the coefficients must be obtained from the
experiments. Examples are the dilution and rise height of line plumes in perpendicular currents
(Equations 14 and 17 of Roberts et al., 1989a):
= 2.19 F1/6 -0.52, ll. = 2.5F-1'6 (25)
^b
In other cases, for example, high momentum jets in a parallel current, only graphical solutions
are available. In these cases, purely empirical equations are fitted to the curves, and the results
interpolated as appropriate. RSB can therefore be thought of as a coding of the graphs and
equations in the original papers. For linear stratifications, the model should give exactly the
same results as obtaining the solution graphically.
For non-linear stratifications, RSB assumes that the density profile is linearized over the
rise height. In RSB, the solution procedure is iterative, solving automatically for the rise height
ze. This method, which is similar to that used by Brooks (1973) is shown in Figure 59. As
discussed later, this approximation works very well, even for very non-linear stratifications. In
fact, this is a conservative assumption, as linear stratifications lead to less rapid spreading, thinner
wastefield, less subsequent mixing, and therefore less dilution than in a wastefield at the same
rise height in a non-linear stratification (Roberts, 1993).
EXAMPLES
Introduction
RSB can be run either as a stand alone program or from PLUMES. When run in stand
alone mode, RSB uses the same UDF input file format as previous EPA models (Muellenhoff
et al., 1985). This file can be created using the AYU command in PLUMES, with any ASCII
text-editor, or interactively by following prompts within RSB. Note, however, that RSB assumes
discharges from both sides of the diffuser, whereas the original EPA models implicitly assume
discharge only from one side of the diffuser, so the data may be different for different models.
In UM this requirement is accommodated by running the cross-diffuser merging configuration,
i.e. by specifying half spacing between ports. For example, if ports are staggered every two
meters with adjacent ports on one side of the diffuser four meters apart, then the appropriate
spacing is two meters. Whether the model is run stand alone or from PLUMES, the solution
procedure is the same, so the results should be practically identical.
Recommendations on usage are given in Appendix 1. The ambient density must be stable,
i.e. density must not decrease downwards, however, under some circumstances RSB will produce
valid results if intermediate levels are specified as unstable due to the method used in RSB to
calculate a linear gradient. The total number of ports n and spacing s are inputted to determine
the diffuser length L which is then used to compute q and the length scales.
99
-------
The Roberts. Snyder. Baumgartner model: RSB
(26)
Seattle Example: Linear Stratification - Zero Current
The following example follows that given in Roberts et al., 1989a,b,c. The parameters are
taken from the Metropolitan Seattle outfall discharging into Puget Sound (Fischer et al., 1979):
Design average flow, Q = 194 ftVs (5.49 m3/s)
Number of ports = 202
Port spacing (on each side of the diffuser), s = 6 ft (1.83 m)
Port diameters, d = 4.5 to 5.75 inches (0.114 to 0.146 m)
Assume d = 5.0 inches (0.127 m)
Effluent density, p0 = 1.000 g/cm3
The port depth is about 70 m, and density stratifications at nearby Alki Point vary
between 0.002 and 0.025 at-units per meter. Taking the strongest stratification (0.025 ot-
units per meter) yields, for example, a density of 1.02425 g/cm3 at the surface and
1.02600 g/cm3 at 70 m depth. The pipe diameter is 96 inches (2.44 m) so the port
elevation is 1.22 m and the total depth is set at 71.22 m.
The input and output files of the original RSB (Basic) model for zero current are shown in
Figure 55. The computed length scales ratios are sllh = 0.14 and ljlb = 0.13 which suggests no
effect of the source momentum flux and port spacing so we expect the behavior of this discharge
to approximate a line plume. The predicted minimum initial dilution Sm for this case is 80, and
rise height ze is 32.9 m. No farfield calculation is provided.
The corresponding PLUMES RSB and UM runs are given in Figure 56 without farfield
calculations. Notice the close agreement between Basic RSB and PLUMES RSB; maximum
difference are less than one percent. Also, notice the approximate agreement between the
models, e.g. average dilutions of 92 and 82 for RSB and UM respectively. In the remainder of
this chapter only the PLUMES RSB runs will be displayed. The corresponding UM run is given
in Figure 57.
The Basic RSB program is not bundled with the plumes package.
100
-------
The Roberts, Snyder. Baumgartner model: RSB
Port spacing = 1.83 m
Input flie:
Seattle Example
5.490 202 0.127 0.00 70.0
0.000 90.000 1.830
2 1.0000 0
0.00 1.02425 0.0 0.000
70.00 1.02600 0.0 0.000
Output file:
Input data:
Seattle Example
Flowrate = 5.49 m3/s
Effluent density = 1 g/cm3
Number of ports = 202
Port diameter = .127 m;
Discharge depth = 70 m
Current speed = 0 m/s; Angle of current to diffuser = 90 degrees
Computed diffuser length = 183.0 m
Density profile:
Depth (m) Density (g/cm3)
0.0 1.02425
70.0 1.02600
Results:
Length scale ratios are: s/lb = 0.14, Im/lb = 0.13
Froude number, u3/b = 0.00; Jet Froude number, Fj = 12.1
Rise height to top of wastefield, ze = 32.9 m
Wastefield submergence below surface = 37.1 m
Wastefield thickness, he = 22.7 m
Height to level of cmax, zm = 21.5 m
Length of initial mixing region, xi = 25.3 m
Minimum dilution, Sm = 80; Flux-average dilution, Sfa = 92 (1.15 x Sm)
Figure 55. Input and output of the original RSB program (Roberts, 1991).
101
-------
The Roberts. Snvder, Baumgartner model: RSB
Jun 28, 1992, 11:23:13 ERL-N PROGRAM PLUMES, July 1, 1992
Title Seattle Example
tot flow # ports port flow spacing effl sal effl temp
202 0.9144
port dep port dia plume dia total vel horiz vel vertl vel
70 0.127 0.1270
port elev ver angle cont coef
0.0
1.0
2.145
effl den poll cone
0 100
hor angle red space p amb den p current far dif
90 0.9144 26.000.00001000 0.000453
density salinity temp
24.25
26.00
depth
0.0
70
current
le-5
le-5
Case: 1 of 8
linear
far inc far dis
asp coeff print frq
0.000 0.10 500
decay Froude # Roberts F
0 11.92 1.833E-14
far vel K:vel/cur Stratif #
214500 0.0001221
amb cone N (freq) red grav.
0 0.01546 0.2550
0 buoy flux puff-ther
0.006930 36.61
jet-plume jet-cross
1.425 24150
plu-cross jet-strat
6.930E+12 3.952
plu-strat
6 .581
CL conc>=
CORMIX1 flow category algorithm is turned off.
5.49 m3/s, 125.3 MGD, 193.9 cfs. >0.0 to 100 m3/s range
Help: Fl. Quit: . Configuration:ATNOO. FILE: rsbeg.var;
RSB
Written by Philip J. W. Roberts (12/12/89)
(Adapted by Walter E. Frick (1/12/92))
Case: 1: Seattle Example
Length scale ratios are: s/lb =
Froude number, u3/b =
Jet Froude number, Fj =
0.14 Im/lb =
0.00
12.1
0.13
Rise height to top of wastefield, ze = 32.9
Wastefield submergence below surface = 37.1
Wastefield thickness, he = 22.8 m
Height to level of cmax, zm = 21.5 m
Length of initial mixing region, xi = 25.3 m
Minimum dilution, Sm = 80
Flux-average dilution, Sfa = 92 ( 1.15 x Sm)
Wastefield submerged
Interpolation count: 1
Roberts Fr. # < 0.01 (aspiration dominated), no avg. flux dilution formed
for farfield prediction
.. UM Simulation ...
plume dep plume dia poll cone dilution CL cone
m
70.00
69.64
59.89
42.82
29.89
m
0.1270
0.9207
3.047
7.686
27.12
100.0
12.94
3 .125
1.509
1.192
1.000
7.556
31.22
64.62
81.79
100.0
24.10
4.529
2 .159
1.702
hor dis
m
0 .000
2 .019< merging
6.675
9.387< trap level
11.98
< plume element overlap.
Figure 56. PLUMES RSB run for Seattle example.
102
-------
The Roberts. Snyder, Baumgartner model: RSB
Seattle Example: Linear Stratification - Flowing Current
Consider now an ambient flowing current of 0.30 m/s perpendicular to the diffuser. The new
input and output data files are shown in Figure 57.
The minimum dilution is now increased by the current to 181, and the rise height (to the
top of the wastefield) reduced from 32.9 m to 26.5 m. This process can be continued for other
current speeds to generate the results shown as Table 2 in Roberts et al., 1989a. Note that
numbers may differ slightly from this table due to slightly differing interpolation procedures.
Jun 28, 1992, 11:27:44 ERL-N PROGRAM PLUMES, July 1, 1992 Case: 3 of 8
Title Seattle Example; with current linear
tot flow # ports port flow spacing effl sal effl temp far inc far dis
202 0.02718 0.9144
port dep port dia plume dia total vel horiz vel vertl vel asp coeff print frq
70 0.127 0.1270 2.145 2.145 0.000 0.10 500
port elev ver angle cont coef effl den poll cone decay Froude # Roberts F
1 0.0 1.0 0 100 11.92 0.4948
nor angle red space p amb den p current far dif far vel K:vel/cur Stratif #
90 0.9144 26.00 0.3000 0.000453 7.152 0.0001221
depth current density salinity temp amb cone N (freq) red grav.
0.0 0.3 24.25 0 0.01546 0.2550
70 0.3 26.00 0 buoy flux puff-ther
0.006930 1.178
jet-plume jet-cross
1.425 0.8049
plu-cross jet-strat
0.2567 3.952
plu-strat
6.581
hor dis>=
CORMIX1 flow category algorithm is turned off.
5.49 m3/s, 125.3 MGD, 193.9 cfs. >0.0 to 100 m3/s range
Help: Fl. Quit: . Configuration:ATNOO. FILE: rsbeg.var;
Case: 3: Seattle Example; with current
Length scale ratios are: s/lb = 0.14 Im/lb = 0.13
Froude number, u3/b = 3.62
Jet Froude number, Fj = 12.1
Rise height to top of wastefield, ze = 26.5
Wastefield submergence below surface = 43.5
Wastefield thickness, he = 21.5 m
Height to level of cmax, zm = 17.4 m
Length of initial mixing region, xi = 164.9 m
Minimum dilution, Sm = 180
Flux-average dilution, Sfa = 208 ( 1.15 x Sm)
Wastefield submerged
Interpolation count: 1
Wastefield width: 183.92m Avg. flux dilution (width*he*u/Q): 216.3
Figure 57. RSB Seattle example, with current.
103
-------
The Roberts, Snyder. Baumgartner model: RSB
Seattle Example: Model Extrapolation
This example illustrates the effect of running RSB outside the range of values on which
it is based. The port diameter is reduced to 60 mm (0.06 m); the new data files are shown in
Figure 58.
In this case the decrease in nozzle size causes an increase in nozzle exit velocity and an
increase in momentum flux. The length scale ratio ljlb becomes equal to 0.60, which exceeds
the experimental range. Note that RSB still gives answers in these situations and gives a warning
message that the predicted results are extrapolated and therefore may be unreliable; the
interpretation of these results is at the discretion of the model user. The primary predicted effect
of the increased momentum flux is a decrease in rise height; the dilution is unchanged. The
reasons for this type of behavior are discussed in Roberts et al., 1989c.
104
-------
The Roberts. Snyder, Baumgartner model: RSB
Jun 28, 1992, 11:28:14 ERL-N PROGRAM PLUMES, July 1, 1992
Title Seattle Example; extrapolated
tot flow # ports port flow spacing effl sal effl temp
202 0.02718 0.9144
port dep port dia plume dia total vel horiz vel vertl vel
70 0.06 0.06000 9.612 9.612 0.000
port elev ver angle cont coef effl den poll cone decay
1 0.0 1.0 0 100
nor angle red space p amb den p current far dif far vel
90 0.9144 26.000.00001000 0.000453
depth current density salinity temp amb cone
0.0 le-5 24.25 0
70 le-5 26.00 0
. . . RSB . . .
Case: 4: Seattle Example; extrapolated
Length scale ratios are: s/lb = 0.14 1m/ Ib = 0.60
Froude number, u3/b = 0.00
Jet Froude number, Fj = 78.7
Rise height to top of wastefield, ze = 26.5
Wastefield submergence below surface = 43.5
Wastefield thickness, he = 19.9 m
Height to level of cmax, zm = 17.8 m
Length of initial mixing region, xi = 25.3 m
Minimum dilution, Sm = 80
Flux-average dilution, Sfa = 92 ( 1.15 x Sm)
Results extrapolated beyond their experimental values, may
Wastefield submerged
Interpolation count: 1
Case: 4 of 8
linear
far inc far dis
asp coeff print frq
0.10 500
Froude # Roberts F
77.71 8.658E-15
K:vel/cur Stratif #
9612000.00005769
N (freq) red grav.
0.01546 0.2550
buoy flux puff-ther
0.006930 99.49
jet-plume jet-cross
4.390 51110
plu-cross jet-strat
6.930E+12 5.750
plu-strat
6.581
hor dis>=
be unreliable
Roberts Fr. # < 0.01 (aspiration dominated), no avg . flux dilution formed
... UM ...
plume dep plume dia poll cone dilution hor dis
mm m
70.00 0.06000 100.0 1.000 0.000
69.96 0.9254 6.381 15.30 2 . 160< merging
68.90 2.882 3.125 31.21 7.323
Figure 58. Seattle example, reduced port size, RSB model extrapolation.
105
-------
The Roberts. Snyder. Baumgartner model: RSB
Seattle Example: Non-Linear Stratification
In this example the non-linear ambient density profile shown in Figure 59 is used. The
density profile is the one used in the Boston Harbor Diffuser model tests. It consists of a
uniform, well-mixed surface layer about 8 m thick, followed by a sharp change in density
through the thermocline, which is about 13 m thick, then a uniform density down to the bottom.
The port depth in this case is 31.3 m below the water surface. The diffuser of the Seattle
example is used and the new data files are given in Figure 60.
Wetter surface
5-
10
Depth 15 ,
(m)
20
25
30
35
Port depth
1.020 1.021 1.022 1.023 1.024 1.025 1.026
Density (Q/cc)
Figure 58. Density Profile used in Non-Linear Example.
RSB predicts a rise height of 17.4 m, which is in the pycnocline. The solution procedure,
which is transparent to the user, is to linearize the density profile over this 17.4 m.
106
-------
The Roberts, Snyder. Baumgartner model: RSB
PLUMES, July 1, 1992
profile
effl sal effl temp
Jun 28, 1992, 11:29:16 ERL-N PROGRAM
Title Seattle example; Boston density
tot flow # ports port flow spacing
202 0.02718 0.9144
port dep port dia plume dia total vel horiz vel vertl vel
31.3 0.127 0.1270 2.145
port elev ver angle cont coef effl den
0
0
2.145
poll cone
100
far dif
0.000453
temp
hor angle red space p amb den p current
90 0.9144 25.200.00001000
depth current density salinity
0.0 le-5 21.4
5 le-5 21.4
7.3 le-5 21.5
10 le-5 22.2
15 le-5 24.2
17.3 le-5 24.9
20 le-5 25.1
25 le-5 25.2
35 le-5 25.2
CORMIX1 flow category algorithm is turned off.
5.49 m3/s, 125.3 MGD, 193.9 cfs.
Help: Fl. Quit: . Configuration-.ATNOO .
0 .000
decay
0
far vel
amb cone
0
0
0
0
0
0
0
0
0
Case: 6 of 8
linear
far inc far dis
asp coeff print frq
0.10 500
Froude # Roberts F
12.11 1.891E-14
K:vel/cur Stratif #
214500 0.0006118
N (freq) red grav.
0.03408 0.2471
buoy flux puff-ther
0.006717 36.99
jet-plume jet-cross
1.448 24150
plu-cross jet-strat
6.717E+12 2.662
plu-strat
3.609
hor dis>=
>0.0 to 100 m3/s range
FILE: rsbeg.var;
Case: 6: Seattle example; Boston density profile
Length scale ratios are: s/lb =
Froude number, u3/b =
Jet Froude number, Fj =
0.26 Im/lb =
0.00
12.3
0.25
Rise height to top of wastefield, ze = 17.4
Wastefield submergence below surface = 13.9
Wastefield thickness, he = 13.1 m
Height to level of cmax, zm = 11.7 m
Length of initial mixing region, xi = 13.9 m
Minimum dilution, Sm = 43
Flux-average dilution, Sfa = 50 ( 1.15 x Sm)
Wastefield submerged
Interpolation count: 8
Roberts Fr. # < 0.01 (aspiration dominated), no avg. flux dilution formed
Figure 59. Seattle example, non-linear density profile.
107
-------
The Roberts, Snyder. Baumgartner model: RSB
Multiport Risers Example
Many outfalls with multiport risers are now operating (San Francisco and Sydney), under
construction (Boston), or proposed (Hong Kong). Except for San Francisco, these are tunneled
outfalls for which the cost of the risers is very high, of the order of several million dollars each.
It is therefore necessary to minimize the number of risers without unduly impairing dilution.
This is different from a pipe diffuser in which, for a given diffuser length, the number of ports
in the pipe wall and their spacing is not a significant cost consideration.
The following example is for the Boston outfall. This is a convenient example as
experimental results from the hydraulic model tests done for this diffuser are available (Roberts,
1989). The example also illustrates the effects of non-linear stratifications.
The basic assumption is that the behavior of the wastefield is the same as if the ports were
uniformly distributed along both sides of the diffuser, rather than clustered in multiport risers.
This was originally demonstrated by Isaacson et al. (1978, 1983) to be a good assumption for
certain limited conditions. The caveat to this assumption is that entraining water must be
available to the plumes. This implies that not more than 8 ports per riser be used, otherwise the
flow collapses to a rising ring with reduced dilution.
The following examples are of the final design, which has 55 risers spaced a distance of 122
ft (37.2 m) apart. Each riser has 8 ports with nominal diameters of 6.2 inches (0.157 m). Tested
flowrates were 390 mgd (17.08 m3/s), 620 mgd (27.16 m3/s), and 1270 mgd (55.63 m3/s). If the
ports were uniformly distributed along the diffuser, the port spacing s would be 122/4 = 30.5 ft
(9.30 m). A typical data file for 390 mgd, zero current speed, with a density profile as shown
in Figure 59 (this is referred to as the Late Summer Profile in Roberts, 1989), is given in Figure
61. Table IV gives more comparisons between measured and predicted dilutions.
Table IV. Measured and predicted wastefield characteristics for Boston Harbor Outfall.
Current
speed
(cm/s)
0
25
0
0
Flowrate,
Q (mgd)
390
390
620
1270
Minimum initial
dilution, Sm
Measure
d
81
223
70
56
Predicted
67
215
59
46
Rise height to top of
wastefield, ze (m)
Measure
d
16.3
16.3
17.8
17.8
Predicted
17.1
15.8
16.9
16.9
Wastefield thickness,
Mm)
Measure
d
7.5
14.5
10.5
14.5
Predicted
12.8
14.1
12.7
12.7
108
-------
The Roberts, Snyder, Baumgartner model: RSB
Jun 28, 1992, 11:29:45 ERL-N PROGRAM PLUMES, July 1, 1992
Title Boston, multiport risers
tot flow # ports port flow spacing effl sal effl temp
440 0.03882 4.15
port dep port dia plume dia total vel horiz vel vertl vel
Case: 7 of 8
linear
far inc far dis
31.3 0.157 0.1570
port elev ver angle cont coef
101
2.005 2.005
effl den poll cone
0
nor angle red space p amb den p current
90
depth
0.0
5
7 .3
10
15
17.3
20
25
35
RSB . . .
4.150 25.200.00001000
current density salinity
le-5 21.4
le-5 21.4
le-5 21.5
le-5 22.2
le-5 24.2
le-5 24.9
le-5 25.1
le-5 25.2
le-5 25.2
100
far dif
0.000453
temp
0 .000
decay
0
far vel
amb cone
0
0
0
0
0
0
0
0
0
asp coeff
0.10
Froude #
10.18
K:vel/cur
200500
N (freq)
0.03408
buoy flux
0.009593
jet-plume
1.505
plu-cross
9.593E+12
plu-strat
3 .946
hor dis>
print frq
500
Roberts F
1.637E-14
Stratif #
0.0007564
red grav.
0.2471
puff-ther
39.82
jet-cross
27900
jet-strat
2.861
Case: 7: Boston, multiport risers
Length scale ratios are: s/lb =
Froude number, u3/b =
Jet Froude number, Fj =
1.70 1m/Ib = 0.22
0.00
10.3
Rise height to top of wastefield, ze = 17.2
Wastefield submergence below surface = 14.1
Wastefield thickness, he = 12.9 m
Height to level of cmax, zm = 11.5 m
Length of initial mixing region, xi = 9.7 m
Minimum dilution, Sm = 66
Flux-average dilution, Sfa = 76 ( 1.15 x Sm)
Wastefield submerged
Interpolation count: 5
Roberts Fr. # < 0.01 (aspiration dominated), no avg. flux dilution formed
UM
plume dep plume dia poll cone dilution
m
31.30
25.39
18.96
16.92
14.16
m
0.1570
2 .569
4.183
4.802
9 .749
100.0
3 .125
1.408
1 .217
1.075
1.000
31.24
69.29
80 .15
90.80
No farfield prediction; cause not known.
hor dis
m
0.000
5 .444
6 .703< merging
6.989< trap level
7.539< plume element overlap.
Figure 60. Boston example, multiport risers; RSB and UM simulations.
It can be seen that, despite the very large difference between the conditions on which RSB
is based (paired ports, linear stratification) and the Boston tests (ports clustered 8 per riser, very
non-linear stratification), the predictions are very good. Dilutions are generally underestimated,
i.e. the model is conservative. This is most probably due to the additional mixing which occurs
in the horizontally spreading layer in the non-linear profile compared to that in the linear profile.
109
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The Roberts. Snyder. Baumgartner model: RSB
DESIGN APPLICATIONS
RSB is a useful tool for the design of outfall diffusers. Time can be saved when doing this
by keeping in mind the following guidelines:
The most important parameter for an ocean outfall diffuser for a fairly large flow is the
length L. This can be chosen first, and the details, i.e. port spacing and diameter chosen
later.
The flow approximates a line source for sllb < 0.3. At this point the dilution is a
maximum (for fixed diffuser length) and adding more ports so that the spacing is less will
have no effect on dilution or rise height. Also, there is little point in making the port
diameter smaller than the value which results in ljlb = 0.2, as this will result in increased
head losses. The only constraints are internal hydraulics (which may be complex for
tunneled outfalls) and that the ports flow full, i.e. Fj > 1.
Momentum only affects dilution when ljlb > 0.2. Therefore decreasing the port diameter
to increase momentum will only affect dilution if it results in ljlb > 0.2. Even then the
primary effect on the wastefield is reduced rise height (in a linear stratification), and
dilution is only slightly affected.
110
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UM MODEL THEORY
PERSPECTIVE
UM is the newest of a series of models first developed for atmospheric and freshwater
applications by Winiarski and Frick (1976) and for marine applications by Teeter and
Baumgartner (1979). The marine version, known as OUTPLM, became the basis of the MERGE
model (Frick, 1980). Both underwent modifications to become the UOUTPLM and UMERGE
models (Muellenhoff et al., 1985). Since 1985 the UMERGE model has been further generalized
and enhanced; including treatments of negatively buoyant plumes and background pollution.
These improvements are included in UM, one of two resident initial dilution models in PLUMES.
Other active research focusing on the generalization to three dimensions and to geothermal
applications continues (e.g. Frick, Baumgartner, and Fox, 1993).
Outstanding UM features are the Lagrangian formulation and the projected area entrainment
(PAE) hypothesis. The Lagrangian formulation offers comparative simplicity that is useful in
developing PAE.
The projected area entrainment hypothesis is a statement of forced entrainment — the rate at
which mass is incorporated into the plume in the presence of current. As a general statement it
was articulated at least as early as 1960 (Rawn, Bowerman, and Brooks). However, Frick (1984),
Lee, Cheung, and Cheung (1987), and Cheung (1991) find that most implementations (e.g. Hoult,
Fay, and Forney, 1969) of the hypothesis are incomplete. They typically include only one or two
of the terms that have been identified, which are then tuned for best fit. For two-dimensional
flow, UM and JETLAG (Lee and Cheung, 1990) use all three terms, thereby eliminating the need
for tuning. In addition to PAE, the traditional Taylor entrainment hypothesis (Morton, Taylor,
and Turner, 1956) is also used.
It is not in the scope of this work to present extensive verification of the UM model, however,
Figures 61 and 62 do give a general indication of the quality of prediction. The superiority of
the PAE hypothesis is demonstrated by Lee and Cheung (1990) and Cheung (1991) who adapt
the approach to three dimensions in the JETLAG model and show that the Lagrangian plume
models using PAE predict observed asymptotic behavior in a number of flow regimes. Frick,
Baumgartner, and Fox show example comparisons between UM and JETLAG.
In Figure 62 the densimetric Froude number of the effluent is given by F-. a measure of the
ratio of momentum to buoyancy in the plume, with large Froude numbers indicating relatively
high momentum and small Froude numbers indicating strong buoyancy. The ratio of efflux
velocity to current is given by k, a high value indicates a relatively strong effluent velocity or
low current speed.
The Lagrangian model and its entrainment hypotheses are described below in some detail.
To understand the model it is necessary to first have an appreciation of the basic model building
111
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UM model theory
60r
(a)
Z/D
0
X/D
40
100
Z/D
(c)
UmrtratMed
100
X/D
200
lOOr
Z/D
100
800
100
Z/D
(d)
100
X/D
800
Figure 61. UM centerline and boundary predictions in stagnant ambient compared to Fan
(1967). (a) Jet No. 10, (b) Jet No. 16, (c) Jet No. 22, unstratified, and (d) Jet No. 32.
block — the plume element. On that basis, the plurne element dynamics, conservation principles,
entrainment, and merging are more easily understood. Simultaneously, a detailed mathematical
description of the model is given.
BASIC LAGRANGIAN PLUME PHYSICS
The Plume Element
The shape of the element is very important to plume modeling because it determines the
projected area, to which forced entrainment is directly proportional, at least in the initial dilution
region. In UM the constant of proportionality is simply unity — 1. Forced entrainment and
Taylor entrainment determine the growth of the element and play a key role in the dynamics of
the element center-of-mass, i.e. the particle.
In terms of the dynamics of the plume element, shown at three stages of development in
Figure 63, simple models like the Lagrangian or Eulerian integral flux models provide only an
estimate of the element trajectory, i.e., s, the path of the center-of-mass of the plume element.
It is shown as a solid line passing through the centers of the elements as if all the mass of the
112
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UM model theory
100
(a)
100
(c)
too
Z/D
(b)
0 100
X/D
100 r
(d)
Figure 62. UM predictions in flowing ambient compared to Fan (1967). (a) F~10, £=8, (b)
Fj=20, £=12, (d) Fj=40, k=l6, and (d) F-80, k=16.
plume element were concentrated there.
In Lagrangian and comparable integral flux models, that is the only coordinate variable that
is predicted by the plume model. Other variables characterizing the distribution of mass are
inferred or assumed. The shape of the element is established arbitrarily before the growth of the
particle can be determined. In other words, the modeler determines how the shape of the plume
is specified. Normally, a particular interpretation of the round plume assumption is used to
establish the distribution of mass about the trajectory of the plume element; it holds that the
plume element is basically cylindrical in shape.
But, if it is assumed, as it generally is, that the element is defined by a smooth surface on
the exterior of the plume and by interior planes, or faces, that are perpendicular to the particle
trajectory, and that the plume trajectory is curved, then this definition results in an element that
is not cylindrical but has the shape of a section of bent cone. Because the length of the element
along the trajectory must be small for mathematical reasons, it is better to conceive of the
element as a thin round wedge with a blunt or sharp edge. This is the element form assumed
in UM.
In special cases of plume trajectory of smaller radius-of-curvature than the plume radius
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-------
UM model theory
(a)
(b)
line of
overlap
Figure 63. Plume trajectory, the element at three stages of development, and selected plume
variables.
itself, the element faces would intersect, or overlap, a physically impossible situation.
complication is considered further subsequently.
This
Furthermore, the asymmetry in shape is not consistent with the conventional practice of
constructing equal plume element radii symmetrically about the trajectory. The plume trajectory
represents the center-of-mass of the plume element which is generally not at the center of the
circular cross-section and therefore the lengths of the "radii" are directionally dependent.
The rigorous treatment of these complications is beyond the scope of the UM model.
However, UM does issue a warning when overlap begins and, in its the default mode, terminates
the initial dilution computation. In other models of the same class, both Lagrangian and Eulerian
integral flux, the condition is not identified, or even recognized, and results in the over-prediction
of plume radius and entrainment unless the increase has been effectively tuned out, a practice that
would introduce spurious behavior elsewhere. Empirical models are not subject to the error.
The plume is assumed to be in steady state. In the Lagrangian formulation that implies that
114
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UM model theory
successive elements follow the same trajectory. The plume envelope remains invariant while
elements moving through it change their shape and position with time. However, conditions can
change as long as they do so over time scales which are long compared to the time in which a
discharged element reaches the end of the initial dilution phase, usually at maximum rise. The
steady state assumption is used to derive the length of the plume element as a function of the
instantaneous average velocity, its initial length, and the initial effluent velocity.
Thus, the length of the element does not in general remain constant but changes with time
due to the different velocities of the leading and trailing faces. It follows that the radius of the
element must respond to this velocity convergence or divergence, as well as to entrainment,
because the fluid is practically incompressible, though incompressibility and the limiting
Boussinesq approximations (Spiegel and Veronis, 1960) are not incorporated in UM.
The exterior boundary of the plume element coincides initially with the edge of the orifice
from which it issues (or the vena contracta diameter). By integrating from this known initial and
boundary condition the plume volume is calculated based on the entrained mass and the assumed
element shape. It is assumed that the properties of the plume at the boundary are
indistinguishable from those in the adjacent ambient fluid. This has important implications, one
being that drag is not an important force in plume dynamics. It also implies that mass crosses
the projected area of the element at the speed of the ambient current.
Conservation Principles
The model includes statements of conservation of mass (continuity), momenta, and energy.
Conservation of mass states that the initial mass of the element and that added, or entrained, over
time is conserved. In modeling terms the element mass is incremented by the amount of fluid
that flows over the outside boundary of the plume element in a given amount of time. Given that
mathematical artifacts like overlap do not occur, the PAE assures that excessive or inadequate
amounts of entrainment are not inadvertently incorporated, i.e. entrained, into the plume.
Similarly, horizontal momentum is conserved. The horizontal momentum, the product of the
element mass and horizontal velocity, is increased by the horizontal momentum of the entrained
fluid in the same time step. Vertical momentum is not generally conserved because it is usually
changed by buoyancy, a body force arising from the density difference between the element and
the ambient fluid.
Finally, energy is conserved, similarly incremented by adding an amount of energy equal to
the product of a constant specific heat, the entrained mass, and the ambient temperature. It
provides the means for estimating the average temperature of the element which is used in the
equation of state to obtain the densities of fresh and sea water in salinity and temperature ranges
that are representative of terrestrial and coastal waters.
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UM model theory
Entrainment and Merging
Entrainment is the process by which the plume incorporates ambient material into itself. It
may be thought of as a process in which fluid flows into the plume interior through the exterior
surface. Alternatively, it may be considered to be a process of accretion followed by the
redistribution of material. The former model is used here and is consistent with the projected
area entrainment hypothesis.
Several mechanisms of entrainment are considered: aspirated, forced, and turbulent, or eddy,
diffusion. Aspirated entrainment is shear (or Taylor) entrainment which is present even in the
absence of current. It is due to the fact that high velocity regions are regions of relative low
pressure which causes inflow of material into the plume. Thus the plume induces a flow field
in the surrounding ambient fluid. Forced entrainment is due to the presence of current that
advects mass into the plume. Diffusion is assumed always to be present but is only important
beyond the zone of initial dilution. It becomes dominant after the other two entrainment
mechanism die off due to the steady reduction in shear between the plume and the ambient. The
transition separates the near-field from the farfield. Strictly speaking, the latter dilution is not
a part of the UM theory because UM is still primarily a near-field model. Instead, farfield
diffusion is parameterized, for example, by the "4/3 law" (Tetra Tech, 1982).
Entrainment through the projected area of the plume is composed of three terms. The first
term is proportional to the length and radius of the element (the cylinder component), the second
to the growth in diameter of the plume, and the third to the curvature of the plume trajectory that
opens or closes area on the element surface. All are simply mathematical parts of the overall
projected area that contribute to forced entrainment. A fourth term, encompassing the entire
peripheral area, accounts for aspiration entrainment.
When adjacent plumes grow sufficiently they begin to merge and entrain each other. Merging
of plumes has the immediate effect of reducing entrainment by reducing the contact area between
the plume and its environs. Each of the four entrainment terms is decremented to a different
degree as merging proceeds. In essence, merging simply necessitates some geometric corrections.
Surface and bottom effects as demonstrated by Wood (1990), or Coanda attachment (Akar and
Jirka, 1990), are not modeled.
Only the merging of adjacent plumes discharging from linear diffusers (pipes) are considered
here. This simplification helps to reduce the problem to two dimensions. Diffusers are assumed
to be long so that end effects can be ignored and unbalanced internal diffusion is neglected.
Variations in the angle between the diffuser and the current are accommodated by
mathematically reducing the spacing distance between adjacent ports by the appropriate
trigonometric factor. Currents between 90 and 45 degrees may be handled in this way and lead
to reductions of entrainment in agreement with measurements made by Roberts (1977).
Typically diffusers are perforated on both sides. In a current the upstream plumes will then
116
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UM model theory
frequently bend over and merge with downstream plumes. This cross-diffuser merging is not
simulated explicitly. In UM there are three ways to estimate the reduction in dilution due to
cross-diffuser merging. The simplest way is to reduce the spacing between ports by a factor of
two (i.e. spacing is equal to the diffuser length divided by the total number of ports). This
method is justified by experience but it is not known with certainty how accurate it is. The effect
may also be estimated by specifying the "background" concentration generated by the upstream
plume, which results in the prediction of a reduced effective dilution. A third method involves
doubling the flow per port and increasing the diameter of the port to maintain approximately the
same densimetric Froude number. None of the methods account for the changes in density
profile that the upstream plume effects on the downstream plume.
MATHEMATICAL DEVELOPMENT
Basic Model Theory
With respect to the foregoing discussion, it is emphasized that the element in Figure 63 is
not a cylinder but is in general a section of a bent cone. The consequences of this fact cannot
be overstated because the shape of the element determines the projected area which in turn
determines forced entrainment, frequently the dominant source of entrainment. In general, a bent
cone plume element has a projected area that differs substantially from the projected area of a
simple cylinder. Thus, the growth and curvature terms are required to accurately describe the
projected area of the plume element.
As has been stated, the principle of superposition allows the entrainment terms to be
described separately. The projected area entrainment hypothesis states that
f - PA« <27)
where dm is the incremental amount of mass entrained in the time increment dt, Ap is the
projected area, u is the ambient current speed normal to the projected area, and pa is the local
ambient density. This hypothesis, neglecting Taylor entrainment for a moment, makes it possible
to explain observed plume behavior in simple terms.
Equation 27 can be written in vector terms
f - -P. 4,u on
where the underline notation is used to indicate vectors. A^ lies in a vertical plane containing the
current vector and points generally upstream out of the element. U_ is the average velocity of the
ambient flow through the projected area. A^ and U_ point in opposite directions so that their dot
product is intrinsically negative.
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UM model theory
To estimate the projected area it is necessary to express mathematically how the length of
the element, h, changes in response to changes in other plume properties. The reason h changes
is due to the difference in velocity of the leading and trailing faces of the element which causes
the faces to converge or diverge with time. Just how much their separation changes depends on
how much the local current velocity differs from the element velocity. Because mass is
conserved, changes in h result in changes to the radius. The effect is substantiated by dilution
and radii data tabulated by Fan, 1967.
Referring to Figure 64, A/K/ is seen
to be the difference in velocity at two
opposing faces of the semi-infinitesimal
element. (The velocity vectors are
proportional to the displacement vectors
shown. Also, in both formulations the
element is infinitesimal only along the
trajectory, thus it is a hybrid integrating
volume which is treated differently from
truly infinitesimal volume elements.)
Since the Lagrangian formulation deals
with material elements and it is assumed
the velocity is uniform, the faces separate
or converge, proportional to A|V|, i.e.,
6f
(29)
displacement
of leading lace
displacement
of trailing face
where 8f is an arbitrary, but constant, time Figure 64. Convergence of element faces due to
increment. Integrating Equation 29 and differences in face velocities.
noting that the corresponding speeds and
lengths are A | .£, | and h0, and, A|V| and h yields
[Hdh = 5r f'du,
Jh* Ju- *
(30)
where us = |V| and usa = |Vo|. Equation 30 can be integrated to yield
(31)
Finally, since 8t can be chosen to be hjust
h ^s
(32)
and \V\ and h change proportionally.
118
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UM model theory
Plume Dynamics
It is convenient to begin a discussion of the Lagrangian plume equations with the equation
of continuity, in other words, the entrainment equation. Equations 27 or 28 is a partial
expression for entrainment; it states that the "forced" part of the amount of mass added to the
element in time dt is equal to the total mass flux through the element surface. The complete
entrainment equation is a sum of the forced and Taylor induced entrainment terms
= -P A.U - PAT v, (33)
at f
where AT is the area of the plume element in contact with the ambient fluid and VT is the Taylor
aspiration speed. Since, in the absence of merging, AT wraps completely around the element it
is not expressed as a vector. VT is often related to an average plume velocity through a
proportionality coefficient, a:
VT = a \Y\ (34)
where \V_\ is the average, or top hat, plume element velocity (but in other formulations it could
be the centerline velocity with a scaled accordingly).
For plumes (jets with buoyancy) adequately described by a Gaussian profile (see a
subsequent section entitled "Average and Centerline Plume Properties") a value of 0.082 is often
attached to a. However, this is based on a nominal plume boundary which encompasses only
the central portion of the plume. The corresponding value for jets in stagnant ambient is 0.057.
However, Frick (1984) makes arguments for a constant a. The conversion from nominal
Gaussian plumes to a "top hat", or average, description of the plume element yields
corresponding values of 0.116 and 0.081. According to Frick (1984), the latter is underestimated
so that an average value for a of 0.1 is thought to be slightly conservative in terms of describing
aspiration entrainment. A comparison with JETLAG supports this conclusion (Frick,
Baumgartner, and Fox, 1993).
Strictly speaking, the areas are infinitesimal areas which might be indicated with the
differential d prefix. This is because h is ideally an infinitesimal distance. However, the model
equations are approximations in which small algebraic values substitute for infinitesimal ones.
Both entrainment areas need further elaboration. The Taylor aspiration area in the absence
of merging, dynamic collapse, and element facial overlap (sharp trajectory curvature) is simply
AT = 2nbh (35)
where b is the element radius. The reduction in this area due to merging is described in a later
section. Dynamic collapse (Frick et al., 1990) is not included in UM.
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UM model theory
Deriving the projected area is more difficult than deriving the Taylor entrainment area. An
approach that applies to three-dimensional plumes is useful. It holds that, since the current, U_,
Figure 65. The local coordinate system.
is a vector field it may be transformed into a useful coordinate system by well established rules
of vector rotation. A particularly useful coordinate system is the local coordinate system shown
in Figure 65. The ambient velocity vector, i.e., the current, can be expressed as the sum of
components in each of the local coordinate system directions
U = «!«! + «A + «A (36)
where et, e2, and e3 are the unit vectors in the direction of the trajectory, the horizontal normal
to the trajectory, and in a vertical plane respectively. The vector e3 can be expressed in terms
of the cross-product of e, and e2:
& = e x £ (37)
e3 - elx ez v /
The unit vectors are derived by constructing a rotation matrix that transforms between the
coordinate systems.
As far as each velocity component is concerned the corresponding projected areas are
particularly simple, see Figure 66. Again ignoring merging, collapse, and overlap, the projected
area associated with ut, i.e., Ah is simply an annulus that wraps around the plume
120
-------
UM model theory
growth term
curvature
term
cylinder
term
Figure 66. The projected area entrainment components: a) the growth area, b) side view of
the element, and c) the cylinder and curvature area.
Al =
(38)
where Ab is the difference between the radius of the leading and trailing faces of the plume
element. This is the "growth" contribution to the projected area (see Figure 66a). The
assumption is made that only the upstream portion of the area, half the circumference, has flow
going through it. The flow in the wake is altered and is assumed to flow parallel to the plume
surface.
The difference in radius over the length of the element is
a?
(39)
where s is the distance along the centerline. The derivative is estimated from the difference in
121
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UM model theory
radius in successive program steps divided by the distance traversed.
Each one of the velocity components u2 and u3 has two projected area terms associated with
it, one which is due to the curvature of the plume trajectory, the other simply being the projection
of a cylinder (see Figure 66b and 66c respectively). Since only the two-dimensional problem is
considered the u3 component is ignored; its cylinder and curvature contributions are due to
current flowing into the side of the plume element caused by directional changes with depth in
the ambient flow.
The cylinder projected area is simply
(40)
The change in direction of the average plume element velocity, V, which is parallel to et,
over the length of the plume element h, in other words the curvature of the centerline s, produces
the "curvature" component to the projected area. Since the faces defining the element are normal
to 5, in regions of strong trajectory curvature the element is deformed into a wedge shape. A
(a)
(b)
negative
volume
Figure 67. a) The plume element in a region of weak trajectory curvature and b) strong
trajectory curvature (showing overlap).
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-------
UM model theory
depiction is given in Figure 67.
The curvature component of the projected area is
A = -^b2^h (41)
w 2 &
where 0 is the elevation angle of s. This area can be positive or negative depending of the sign
of 90/95 which is determined with reference to successive values of U_. Positive curvature has
the effect of reducing the total projected area.
Historically the growth and curvature terms have either not been recognized or have been
thought to be small compared to the cylinder term (Schatzmann, 1979). However, in general,
it can be shown that all three contributions to the total projected area are important. Any earlier
perceived inadequacies in the projected area entrainment hypothesis can be attributed to the
omission of the growth and curvature terms. Further details are available in Lee, Cheung, and
Cheung (1987), Cheung (1991), and Frick (1984).
Conservation of momentum is given by
dm
- m
dt dt p
where m is the mass of the plume element (m = pntfh), pa and p are the ambient and average
element densities respectively, and & is the gravity vector. Ideally U_ represents the average
ambient velocity over the exposed plume surface. This point is worth emphasizing since the
surface area is infinitesimal only along the centerline and can be extensive in the two dimensions
orthogonal to the centerline, over which, therefore, the ambient velocity can vary significantly.
In UM it is approximated by the ambient velocity at the level of the particle, i.e., the center of
the cross-section.
Equation 42 states that the change in momentum in the element is due to the amount of
momentum introduced by the entrained mass dm and the change in vertical momentum generated
by the buoyant force. The implicit assumption is that drag effects are absent. This is consistent
with the conception of the element having the same properties as the ambient on the outside
surfaces of the element. Effectively, there are no shears that can generate drag.
While interactions with solid and free surfaces are not modelled, UM gives warning when
some of them occur. The warnings, which are not exhaustive, are explained in Appendix 4. The
bottom is assumed to be flat. In Muellenhoff et al. (1985) predicted dilutions were reduced by
10% when the sea surface was encountered. Generally, plumes rise in a matter of minutes so
that the Coriolis force is safely ignored.
To evaluate the buoyancy term in the conservation of momentum equation, it is necessary
123
-------
UM model theory
to define the conservation of energy equation, approximated by
tojF-TJ =C(T_T )^ (43)
dt >( * «' dt
where cp is the specific heat at constant pressure. T, Ta, and Tref are the average element
temperature, the ambient temperature, and an arbitrary reference temperature, respectively. More
correctly, the terms in Equation 43 should be represented by integrals. However, it is assumed
that cp is constant over the range of interest permitting Equation 43 to be simplified,
j, dm /44^
adt
Radiation, conduction, and diffusion are assumed to be small. Like salinity, temperature is
assumed to be a conservative property.
Several other relationships are necessary. Conservation of salinity is expressed by
(45)
dt *dt
where S and Sa are the average element salinity and the ambient salinity respectively. The
symbol for ambient salinity should not be confused with average dilution of the plume.
Conservative pollutants would be expressed similarly, however, since important pollutants, such
as coliform bacteria, are subject to decay, a first order decay term is included.
(46)
where % and %a are the concentrations of the species of interest in the element and ambient
respectively and k is a first order decay constant, which is zero for conservative pollutants. Non-
conservative pollutants are also assumed to be subject to decay in the farfield.
The momentum equation includes the reduced gravity, ((pa-p)/p)£, which must be determined.
Densities are derived from the equation of state (Sigmat function) used by Teeter and
Baumgartner (1979). It is independent of pressure, limiting UM to shallow water, by deep ocean
standards. It is also limited to ordinary temperatures. At 150 o/oo the error in density in sigma-t
units is about 10 percent.
Boundary Conditions and Other Pertinent Relationships
To completely describe the problem, the boundary and initial conditions must also be
specified. The main boundary condition is the location of the source from which the subsequent
124
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UM model theory
position of the element may be determined by integrating the trivial relationship
Y. (47)
dt
where /? is the radius vector of the particle, i.e., the center-of-mass of the element. To give an
example of how the equations are solved in a finite difference model, the new R_ is
*u - Rt ^dt (48)
Another boundary condition is the initial plume radius. Initial conditions include the efflux
velocity, the effluent temperature, etc..
Various auxiliary equations are also required. They include linear interpolations that
determine ambient conditions at the level of the particle. Also, because the Lagrangian plume
equations require a very small time step initially, but not later in the simulation, a method of
varying the size of the time step is used to control the relative amount of mass that is entrained
during any one single step. This is done in the interest of computational efficiency.
The general computational procedure followed in the model is: 1) a time step is provided
(guessed), 2) the entrainment equations are then used to determine the amount of mass that will
be added given this time step, 3) this increase is then compared with the target mass increase and
the appropriate adjustments are made to the time step and the entrainment components to meet
the appropriate doubling criterion, 4) the equations of motion and other model equations are
solved, and 5) the new time step is established and the cycle is repeated.
It is important to recognize that some of the above equations are not always solved for the
quantity on the left hand side of the equal sign. In other words, the dependent variable may be
some other variable besides the one on the left hand side of the equal sign. For example consider
Equation 49 which expresses the mass of the element in terms of its dimensions and the density:
m = pnb2h
(49)
For modeling purposes the radius, b, is not an independent variable, rather it is a dependent
variable. Since mass is computed by integrating from its initial value using the entrainment, or
continuity, equation, it is effectively an independent variable in Equation 49. Equation 49 is
inverted and used to solve for the radius:
>
(50)
t+dt
When overlap occurs Equation 50 gives anomalous results (Frick, Baumgartner, and Fox,
1993.) This is the source of the overestimation of radius and entrainment described previously.
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UM model theory
Merging
The basic approach to handling plume merging is to 1) reduce the entrainment areas, both
Taylor and forced, to account for the loss of exposed surface area that occurs when neighboring
plumes interfere with each other, and, 2) to confine the plume mass from each plume to the space
between them that is known to be available from symmetry considerations. It is assumed that
the plumes are identical and any interaction between them is mutual, i.e. gains equal losses.
Considering Taylor entrainment first, the conditions of merging are depicted in Figure 68.
h- L—I
Reflection planes
Figure 68. Merging geometry and reflection planes.
It is seen that the uncorrected Taylor entrainment area can be multiplied by a factor equal to the
ratio of the exposed circumference to the total circumference to reduce it to the actual exposed
area. Assuming no overlap, the side of the plume element that is longer and larger in area due
to trajectory curvature nearly compensates for the opposite side that is shorter and smaller.
The appropriate ratio of correction is
(51)
7T
where
126
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arctan
N
UM model theory
(52)
where ty is defined in Figure 68 and L is the spacing between adjacent ports. The same
correction factor applies to the growth entrainment term.
While it is assumed that the current is perpendicular to the diffuser axis, the method may be
used for angles between 45 and 135 degrees (90 degrees being equivalent to a current
perpendicular to the diffuser) by multiplying L by the factor sin \|/ where \j/ is the angle between
U_ and the diffuser axis. This method is justified by measurements of dilution of merging plumes
(Roberts, 1977).
The correction factor for the cylinder projected area is simply
a , = — (53)
^ 2b
Finally, the correction term for the curvature projected area entrainment contribution is
2d> sin2d> /c/it
a = 1 - —— + — (.54;
cur
Equations 49 and 50 must also be modified when merging occurs. As was pointed out in
the previous section, the mass of the plume element is obtained by knowing the initial mass and
integrating the entrainment equation. Given that the mass, average plume density, and element
length are known, the element volume can be determined. Upon merging, the transverse
dimension of the plume element (i.e. along e2~) is assumed to be limited to a maximum length of
L, the spacing distance. Effectively, a vertical plane half way between the ports acts as a wall
or reflecting plane. This technique is common in air pollution modeling (Turner, 1970) where
a fictitious mirror source is used to estimated dispersion in the presence of an actual physical
barrier. With plume merging the sources are real.
Thus, the volume of the plume element can be thought to be the product of h and the area
of a rounded rectangle, see Figure 69. This area is the quotient of the element volume and the
length which, after simplification, becomes
1 - ^) + 2Z>2sin<|>cos4> (55)
where br is the unmerged round element radius and b is now the radius of the element in the
vertical plane. In other words, b describes the plume element parallel to £3. Solving for b
127
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UM model theory
Figure 69. Derivation of dimensions under merging: a) the merged element with volume
confined between reflection planes, and b) the corresponding unmerged element of equal
volume.
b =
Tib.
TT - 2 + 2sin<|>cos<|>
(56)
the subscript t+$t has been left off for simplicity. Since (p is larger than sin (p cos (p, b is larger
than br.
Average and Centerline Plume Properties
The previous discussion is in terms of average plume properties because average plume
properties are physically compatible with the average motion of the plume element. We do not
expect that centerline buoyancy can accurately describe, via vertical acceleration, the plume
trajectory traced by the center-of-mass of the plume element. After all, the element is an entity
which stretches from one boundary with the ambient flow to the other, with widely varying
properties in between.
On the other hand, centerline concentrations often concern environmentalists because they
128
-------
UM model theory
have the potential for acutely affecting organisms. Fortunately, plumes are often found to possess
predictable patterns of cross-sectional properties. For example, plumes discharged into quiescent
fluid tend to display the Gaussian profile, very dilute at the edges and concentrated at the center.
However, the Gaussian profile is not very compatible with the plume element described above
because it extends to infinity whereas we have described an element with definite boundaries.
Consequently, another profile, the 3/2 power profile (Kannberg and Davis, 1976), which closely
matches the Gaussian profile over the concentrated portion of its range, is used to determine the
centerline concentration as a function of the average concentration, or dilution, that UM predicts.
The 3/2 power profile is expressed by
(57)
where is instantaneous scaling factor relating differences between the plume and the ambient
of an appropriate property, such as the concentration of some pollutant or velocity, b is the plume
radius, and r is the distance from the center of the plume to the point within the plume at which
is measured.
The peak-to-mean ratio is simply the ratio of the centerline to the average concentration, it
is obtained from a flux integral. We start with the relationship for the average concentration
[ CvdA
C = ^ _ (58)
avg f
f vdA
JA
where Cmg is equivalent to the average concentration obtained from UM, C and v are the
instantaneous concentration and velocity in the plume element, A is the cross-sectional area, and
dA is the corresponding infinitesimal area. The peak-to-mean ratio is defined to be CmjCavg
(59,
CvdA
A
where Cmax is the centerline concentration. The integrals in this quotient are not easy to solve
analytically and, therefore, are estimated numerically in UM.
It is illuminating to define limiting values of the coefficient. When dilutions and currents
are large a simplification is possible. In this case the velocity can be considered constant and
can be factored from the integrals, giving
(60)
f CdA
JA
129
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UM model theory
Using this approximation and assuming the 3/2 power profile a peak-to-mean ratio of 3.89 is
found for round plumes. The corresponding ratio for a fully merged line plume is 2.22.
However, the ratios vary and in much of the plume the peak-to-mean ratios are considerably
smaller than these limiting values, in fact, near the source they often approach 1.0, depending on
the uniformity of the source. The centerline concentration prediction is approximate and
occasionally deviates from the expected trend when vertically varying background pollutant
concentrations are present.
Experimental Justification of the Projected Area Entrainment Hypothesis
In 1989, Roberts, Snyder, and Baumgartner published three papers in ASCE (1989a,b,c)
which record the behavior of merging laboratory plumes in flowing, stratified environments.
Although they did not set out to do so, their findings directly corroborate PAE, as shown below:
Starting with Equation 13a of Roberts, Snyder, and Baumgartner (1989a)
= 1.08F1/6 (61)
where Sm is the centerline dilution in the plume, q is the diffuser volume flux per unit length, b
is the buoyancy flux per unit length (i.e. the product of the reduced gravitational acceleration and
the volume flux per unit length), F is a type of Froude number (t//b, where u is the current
speed), and N is the buoyancy (Brunt-Vaisala) frequency
(62)
and dp/dz is the ambient density gradient. Their Equation 13b states
^ = 1.85F-1/6 (63)
where ze is the rise above the port datum of the top of the fully merged wastefield and lb is a
buoyant length scale defined by Roberts et al., 1989a Equation 4
/ - (64)
* AT
Combining, noting that q = Q/L, where L is the length of the diffuser and Q is the diffuser
total volume flux, and making the appropriate substitutions yields
130
-------
UM model theory
= L08« (65)
m 1.86 Q
The quantity Lzeu is, of course, just the flux through the projected area, which is the
integrated form of PAE! The coefficient is within the general range described in the previous
section, however, it differs markedly from the factor of 1.15 used in RSB.
This derivation proves, at least in an overall sense, that, in sufficiently high current, initial
dilution is given simply by the quotient of the flux through the projected area of the wastefield
divided by the source flux, multiplied by a constant factor. In lieu of convincing evidence to the
contrary, it is eminently reasonable to assume that such an integrated outcome is the result of
adding the individual projected area fluxes throughout the plume trajectory. In other words, it
is not reasonable to assume, a priori, that the plume entrains differentially over its projected area,
perhaps at twice the rate at one point and half the rate at another. Any such deviations are
thought to be due to the aspiration effect of the Taylor entrainment coefficient which can be
treated separately. In other words, the two entrainment mechanisms act independently, are
mathematically linear, and may be added.
131
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FARFIELD ALGORITHM
PLUMES IMPLEMENTATION
Equation 17, developed by Brooks (1960), has been transformed into Equation 66 for near
shore coastal waters, confined channels, and wherever a conservative analysis is desired
.
(66)
16 a
where erf is the error function, S is the centerline dilution in the farfield plume, Sa is the initial
dilution (at maximum rise, overlap, or other special condition), a is a dispersion coefficient
(Fischer, 1979; Okubo, 1962), b is the width of the plume field at the end of initial dilution, and
t is the time of travel from the point of the end of initial dilution to the point of interest.
The relationship between a (in Equation 66) and e0 (in Equation 17) is simply
a = eb4* <67)
For example, for e0 = 4 m2/sec and b = 900 m, then
a = 4/9004/3 = 0.00046 mV3/sec
The value for a is entered into the farfield diffusion [far diff] cell of the interface. PLUMES
uses the value in the farfield increment [far inc] cell divided by the farfield velocity [far vel] cell
to compute the travel time, t.
The corresponding equation for open coastal waters, where the dispersion coefficient is
continuously increased according to the 4/3 power of the local plume field width is:
S*
1.5 , (69)
(1 + Sab413— )3 - 1
For coastal areas of known high energy dissipation features, or in many geographical areas
at certain times of the year, a may have a value as high as 0.0005 m2/3/sec. In less turbulent
situations a may be as low as 0.0001 m2/3/sec, thus the user has many options to employ in
generating more or less conservative estimates of farfield dilution. Small values of a yield the
most conservative estimates of farfield dilution.
133
-------
Farfield algorithms
In these Equations the width, b, is the horizontal width of the wastefield measured
perpendicular to the current:
b = (N - 1)5^ + d (70)
where N is the number of ports, seff is the effective spacing (spacing multiplied by sin\|/), and d
is the diameter of the plume at the end of initial dilution. Equation 70 is simply the physical
projection of the diffuser plus the additional growth of the plumes outside of this region. It is
an approximation which does not account for the "attraction" of the plumes to each other or other
mechanisms which can affect the width of the wastefield, including upstream intrusion.
Equations 66 and 69 only provide estimates of volume dilution, which is appropriate for
conservative pollutants (decay = 0) and unpolluted ambient receiving water. PLUMES uses
additional equations to estimate the effect of first order decay and ambient background
concentrations. The sequence in each time step is as follows.
First a distance (path), presumed to be along ambient streamlines, is established. It is
computed by repeatedly adding the value in the [far inc] cell to the distance of the element in
the present time step. When the sum is greater than the value found in the [far dis] cell then it
is set to that value and the program is terminated. The time elapsed in traversing the distance
between successive values is found by solving the distance-is-rate-times-time formula. The total
time is also incremented and Equations 66 and 69 are solved. The incremental mass gained by
the element during the time step is determined by
Am = CVA, - 5>0 (71)
where Am is the mass entrained during the time step and m0 is the plume element mass at the
port. The total pollutant in the element is given by
a
-------
Sm
Farfield algorithms
(73)
o
where % with the bar is the average pollutant concentration in the element and ^ is the pollutant
concentration in the effluent.
The farfield algorithm is much simpler than the initial dilution part of UM. The quality of
the estimates should not, in general, be expected to be as high as the initial dilution model.
Consequently, if better methods for estimating the farfield concentration are available they should
be considered.
135
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APPENDIX 1: MODEL RECOMMENDATIONS
JUSTIFICATION FOR USES OF PLUMES MODELS IN FRESH WATER
The title of this work "Dilution models for effluent discharges" signifies that this report
encompasses a broader scope than Muellenhoff et al. (1985) which addressed primarily ocean
discharges. The reasons are many but most importantly, users of Muellenhoff et al. (1985) often
applied the plume models to freshwater outfalls because experience showed that some of the
models, UMERGE included, worked well in that setting.
However, since 1985 the CORnell MIXing zone models (Hinton and Jirka, 1992), CORMIX,
have been developed, supported in part by EPA, for the express purpose of addressing the
problem of discharges to shallow and confined water bodies. CORMIX uses a classification
scheme based on length scales to associate a number of formulae and methods appropriate for
each sub-category, linking together several discrete plume behaviors into an estimate of overall
behavior, much like PLUMES links RSB and UM to a farfield algorithm. This is done for a
broad range of conditions, including single ports, merging plumes, and surface discharges,
covering many conditions encountered in practice.
In addition to this practical reason for addressing the freshwater uses of our models, there
are valid reasons for occasionally recommending them, even for those categories for which
CORMIX was expressly developed. Speed of analysis is one reason. Suppose, for example, that
it is to be established what percentage of time annually a plume surfaces and that this estimate
is to be based on available hourly data collected during a monitoring study. This may require
hundreds of simulations, which might be developed relatively easily with PLUMES.
MODEL RECOMMENDATION TABLES
General Considerations
Recommendations for use of the models UM and RSB are based on the experience of the
authors who have contributed to the formulation of the models and the interface, PLUMES, and
have gained experience with the models in a large number of design and analysis applications.
Our experience with CORMIX is much less extensive and we have not contributed directly to
its formulation. Furthermore, CORMIX is only recently available for multiport and surface
discharges and we have seen few results of its application to actual cases.
The basic responsibility for choice of a model lies with the user, especially in relation to
application for regulatory permits, which may carry important legal implications in addition to
professional responsibility. There are many models and other approaches than can be used to
estimate initial dilution that may be acceptable to regulatory agencies. By presenting the
following recommendations we do not claim that any others should not be used. We do not
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Appendix 1: Model recommendations
provide recommendations for ULINE, UPLUME, and UOUTPLM because wherever they may
have been used appropriately in the past we now believe UM or RSB is used more effectively,
even in the case where the regulatory agency requires use of zero ambient current. We do not
include UDKHDEN (Muellenhoff et al., 1985) in our recommendations because we have not
followed its use since 1985 and we believe Dr. Lorin Davis has made further improvements to
his models.
The extensive verification of the Projected Area Entrainment (PAE) hypothesis given by Lee
and Cheung (1990) and Cheung (1991) supports our recommendations. As has been shown, UM
uses the PAE hypothesis which is further supported by the experimental data on which RSB is
based. Thus the RSB and UM models support each other, though they are certainly not identical.
In general we believe RSB (indicated in Table VI by "R," when well suited, or "r," when less
suitable) is applicable to any case that matches closely the experimental conditions used in its
development, which were limited to multiple port discharges. Figure 2 of Roberts, Snyder, and
Baumgartner (1989a) may be used as a guide — a complete list of experimental parameters is
included as Appendix 1 (Table 5) of Roberts, Snyder, and Baumgartner (1989c). Other cases in
which the density gradient over the height of rise can be represented by a linear gradient may
be effectively modeled by RSB. However, the model does accept non-linear density gradients.
Submerged diffusers with fairly closely spaced multiport risers may be modeled (Roberts, 1989).
The model UM (indicated in Tables V and VI by "U" or "u") is useful for a similar range
of conditions for both single port and multiple port discharges. Again, a lower case "u" is used
to indicate where UM is less useful, such as in the case of parallel currents and in shallow water
discharges. In addition to coastal applications, UM may be used for freshwater discharges and
provides exceptional capability in nascent density cases, where discharge is to cold, fresh water
(less than 4 C), owing to a robust and rigorously defined equation of state. Vertical non-
uniformities in current speed and direction (primarily merging plumes), as well as non-uniform
density and ambient contaminant concentrations are handled directly by UM (however,
approximate corrections can be made to RSB dilution predictions for vertically uniform ambient
concentrations of contaminants too). UM is well suited for dense seawater brines because the
model is not constrained by the Boussinesq approximations and in addition can handle negatively
buoyant flows. For very high density discharges the error in the calculated density increases and
the linear equation of state may be more appropriate. While not frequently encountered, UM is
appropriate for analysis of diffusers with ports along only one side.
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Appendix 1: Model recommendations
Caveats
The recommendations given in the following tables are intended for general guidance
purposes and to emphasize the complementary capabilities of the RSB and UM models and the
CORMIX expert system. No attempt is made to define a rigorous classification system as
defined in CORMIX, which, between CORMIX 1, 2, and 3, classifies perhaps 90% of common
plume problems. The CORMIX classification system is made possible by adopting assumptions
which, while making it possible to analyze a majority of freshwater and seawater outfall problems
objectively, does not define the remainder. Some of the latter are important in certain regions
of the country and/or under special circumstances. Hence, a different, somewhat complementary
system was devised, albeit one which must appeal to the user for help in assuring that the models
are appropriately implemented. However, cases may arise which even this generalized system
does not include. The user must be the ultimate judge of the applicability of any given model
under the circumstances at hand.
Description and Usage
Table V specifies the applicability of the CORMIX 1 (single port CORMIX) and UM models
to single port submerged discharge problems. Similarly, Table VI addresses multiport submerged
diffusers. General applicability is indicated by the placement, in alphabetical order, of either a
C for CORMIX 1 or R, r, U, or u, for RSB or UM. Because we are more knowledgeable with
our own models than with CORMIX, we indicate a general quality of our models with an upper
case letter, e.g. U, signifying that we think the model generally performs well in this category,
or lower case letter, e.g. u, suggesting that the user may wish, depending on the sensitivity of the
project and other considerations, to seek other models, like CORMIX, if they apply.
An italicized C, i.e. C, for CORMIX conveys the fact that we are not experts in CORMIX
usage and do not feel justified in assigning a measure of quality it. We simply include it to
indicate the general domain of applicability of the CORMIX models, bearing in mind that the
importance of a particular category is not necessarily represented by the relative size of the box.
In its domain CORMIX can be used in analysis and generally be accepted by the authors and
regulators in regulatory situations, providing that some special circumstances, some of which are
identified below, do not invalidate such usage.
Each table classifies conditions and effluent types in an array in which the categories are not
exclusive, but rather assimilative. Guidance is derived from the tables by identifying the
appropriate effluent type (row) and then examining the applicability ratings in that row. The row
can be likened to a chain in which each condition relevant to the problem is a link. The weakest
link determines the strength of the chain.
For example, with respect to Table V, if there exists a deeply submerged outfall (i.e.
boundary conditions, BCs, are unimportant), discharging effluent which is moderately buoyant,
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Appendix 1: Model recommendations
into a lake which is stratified into two layers, with co-flowing current (directed in the same
general direction as the effluent), and no background pollution, decay, or upstream intrusion (the
presence of which would be indicated by UM with an overlap message), then both CORMIX and
UM would be applicable. In this case, the chain would consist of the "1,2 Stratification" and "2-
D Current" categories (columns) which show U's in both instances, i.e. strong links.
If the current were not co-flowing but directionally stratified, implying need for a 3-D current
modeling capability (link), then the UM link would be relatively weak, and, given that all
CORMIX simulation modules use formulae and coefficients that are uniformly appropriate,
CORMIX would be the model of choice. On the other hand, going back to the original case, if
background pollution is present then the CORMIX chain would contain a weak link.
It should be noted that CORMIX does not explicitly include background in its simulations,
but a C followed by the word decay is entered in that column to indicate that decay has been
added since the first edition of this manual was published. Calculations could be made separately
to estimate the consequences of background concentrations on predictions.
The meaning of the table columns and rows and other comments are given in the following
sections.
Single Port Diffuser Model Recommendations: Table V
Table V: Columns
Table V sub-divides the Stratification column into three sub-columns, one each for
unstratified, singly or doubly stratified, or multiply stratified water bodies. Length scale analysis
may be used, as it is in CORMIX, to define these categories more precisely. Whether
stratification is important depends on the strength of stratification as well as the buoyancy flux
of the source, however, an unstratified system is one in which truly buoyant discharges
(possessing no nascent density) reach the surface, which, if there is doubt, can be established
quickly simply by running UM. In stratified systems the density varies with depth and the plume
will trap (come to equilibrium) at some intermediate depth.
For cases with current, the 2-D sub-column is restricted to effluents and conditions where
the current is either substantially co-flowing or counter-flowing, or, the current is sufficiently
weak and does not affect trajectory plume direction significantly in the initial dilution region, i.e.
before attaining maximum rise, overlap, or trapping. The latter condition, i.e. weak current,
justifies the use of UM in the example given in the CORMIX1 Comparison Chapter even though
the problem is three-dimensional (the fact that the analysis was conservative further justifying
its use). Three dimensional current (3-D) means there is a significant component of current
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Table V. Single port discharge model recommendations.
Appendix 1: Model recommendations
Conditions
Effluent Types
Buoyant discharges: sewage,
industrial waste especially to
saline waters
Slightly buoyant discharges,
signif. momentum:
thermal discharges
Dense discharges:
light brine.
R.O. discharge, industrial waste
Discharges with nascent or non-
linear density effects: thermal
discharge to cold water
Stratification
none
C
U
C
U
C
U
U
1,2
C
U
C
U
C
U
U
3+
U
U
U
U
Current
2-D
C
U
C
U
C
U
U
3-D
C
u
C
u
C
u
u
Other
sources,
decay
C (decay)
U
C (decay)
U
C (decay)
U
U
BCs
C
u
C
u
C
u
u
Intrusion
C
u
C
u
C
u
u
VSW
u
u
u
u
perpendicular to the flow of the effluent or the current direction varies with depth and
significantly affects the trajectory.
The Other sources, decay column indicates that there are significant levels of uniform
horizontally distributed background pollution (ambient pollution concentration) in the water body,
or that there is a nearby source which creates a localized background pollution field in the
vicinity of the outfall, and/or the pollutant in the effluent is subject to first order decay. Note,
while the effect of uniform horizontally distributed background is well simulated by UM, nearby
sources may create fields with large horizontal gradients which may make farfield estimates
questionable. For example, can the user establish that spatially separated plumes actually
interact? Also note, that UM assumes background fluid is entrained at the level of the center-of-
mass of the plume element so that pollution profiles may need to be adjusted to compensate for
the effect of this assumption. For example, given a body of water stratified with high pollution
near the surface and low pollution near the bottom, the plume pollutant concentration would tend
to be underestimated.
The boundary conditions (BCs) column indicates that boundaries, bottom, surface, and/or
sides, play an important role in the plume problem. The concern here is whether the models
appropriately limit entrainment due to the interference of the boundary. If side boundaries are
important then CORMIX should be used exclusively, given there are no missing or weak links.
However, if only surface boundaries are important, then UM can generally be used up to the
point where it indicates the surface is hit. In general, the UM message indicating that the bottom
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Appendix 1: Model recommendations
is contacted is less important because the interaction is along the weakly entraining side of the
plume. However, for negatively buoyant plumes, the bottom boundary condition is as important
as the surface boundary condition is to truly buoyant plumes.
The Intrusion column indicates that portions of the plume will flow upstream and form either
stable or unstable upstream protrusions. If an estimate of the length of the effect is wanted, it
is usually appropriate to use CORMIX. However, for estimating the dilution in the wastefield
UM will provide estimates which are consistent with the amount of dilution water available for
entrainment due to current or aspiration and can be considered to be reliable. As in Muellenhoff
et al. (1985), the dilution could be reduced by ten percent to assure the analysis is conservative.
The final column, VSW, or very shallow water, defined to be water less than three plume
diameters deep, was built into UM to take advantage of its merging algorithm (reflection
technique) to estimate initial dilution in cases in which CORMIX provides no estimates, an
excluded category brought to our attention by one of our reviewers. While such outfalls are not
recommended, where they exist they sometimes need to be analyzed. UM can be applied using
the command. (Run the READlst.exe file for the latest developments on
this topic.) In such cases the surface or bottom are encountered almost immediately and no
criterion is known to establish an appropriate beginning of the farfield. As a result, widely
varying estimates of plume spreading are given, depending on where the farfield zone is initiated
using the Pause Cell capability in the Configuration menu for the farfield start. Our
recommendation is that the VSW capability be used only for screening purposes. If it needs to
be established that a migration path exists for various fish, then the solution giving the greatest
spread might be used as a conservative indicator of wastefield width. If maximum concentration
at a mixing zone are of concern, the solution giving the highest concentration might be used.
Table V: Rows
The first three rows in Table V are self-explanatory. Additional information is available in
other parts of this manual, especially the introductory chapter. The CORMIX manuals (Doneker
and Jirka, 1990; Jirka and Hinton, 1992) may also be consulted. The term "R.O. discharge"
refers to brine plumes created by a reverse osmosis desalination process.
The nascent density row is important, even though the effect is not widely recognized. At
low ambient temperatures the non-linearities in the equation of state for fresh or low salinity
water, particularly in the 0 to approximately 4 C range, cause initially buoyant thermal plumes
to become negatively buoyant as they cool by mixing. The effect, described in the first chapter,
is important in cold climate regions. As explained in the CORMIX example chapter, existing
versions of CORMIX do not address the problem.
As was pointed out, the problem causes some models to fail completely (one could say
catastrophically), by predicting that the effluent will rise to the surface instead of sinking to the
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Appendix 1: Model recommendations
bottom. The ramifications could be serious, causing, for example, a monitoring program to be
designed to study healthy surface biota while the benthic community is actually at risk.
Multiport Outfall Model Recommendations: Table VI
Table VI. Model recommendations for multiport diffusers.
Conditions
Effluent Types
Buoyant discharges:
sewage, industrial
waste especially to
saline waters
Slightly buoyant
discharges, signif.
momentum: thennal
discharges
Dense discharges:
light brine,
R.O. discharge,
industrial waste
Discharges with
nascent or non-linear
density effects:
thermal discharge to
cold water
Stratification
no
C
R
U
C
r
U
C
R
U
U
1,2
C
R
U
C
r
U
C
R
U
U
3+
R
U
r
U
r
U
U
Current
cross
C
R
U
C
R
U
C
T
U
U
par'l
C
R
C
R
C
r
u
Merging
part
R
U
C
R
U
C
R
U
U
full
C
R
U
C
R
V
C
R
U
u
Other
sources &
decay
C (decay)
U
C (decay)
U
C (decay)
U
U
BCs
C
u
C
u
C
a
u
Intru
sion
C
R
u
C
R
u
C
R
U
u
Stage
C
C
C
Table VI: Columns and Rows
The multiport discharge model recommendations are given in Table VI. In general, the same
comments that apply to Table V apply to Table VI as well. Notable differences are the addition
of the models RSB (denoted by R or r) and columns for degree of merging and staged diffusers.
The Current category sub-columns have been changed to indicate the importance of diffuser
alignment on plume behavior. Generally, cross-diffuser flow is from perpendicular to 45 degrees
off perpendicular, other cases falling in the parallel sub-column.
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Appendix 1: Model recommendations
The Merging column indicates the degree of merging, either partial or full. It is worth noting
that RSB is considered to be particularly appropriate to tunneled outfalls with multiport risers.
With respect to the Intrusion column, only CORMIX provides an estimate of the length of
penetration upstream. However, RSB and UM do provide estimates of the dilution in the
wastefield. RSB is considered to be especially applicable for making dilution estimates and
provides other information lacking with CORMIX. If the surface is hit, UM predictions should
be interpreted at that point, that dilution being consistent with the amount of dilution water
available for entrainment due to current or aspiration. Again, the dilution could be reduced by
ten percent to assure the analysis is conservative.
The Stage column refers to staged diffusers, diffuser pipes with ports not perpendicular to
the diffuser axis. Such diffusers are staged to use the momentum in the effluent to carry effluent
farther from shore. Of the models under consideration, only CORMIX applies to this diffuser
configuration.
SURFACE DISCHARGES
CORMIX (CORMDC3) is recommended for modeling surface discharges.
OTHER VIEWPOINTS AND RECOMMENDATIONS
As described previously, the plume classification scheme presented in this appendix differs
from the CORMIX classification scheme. Within the CORMIX classification scheme UM is
thought apply to the near-field of the following classes (Jirka, 1992).
Single ports: SI, S2, S3, S4, S5, VI, V2, V3, V5, HI, H2, H3, H4, NV1, NV2, NH1, NH2, and
NH4, provided they are not associated with an attachment suffix (A..).
Multiport diffusers: MSI, MS2, MS3, MS4, MS5, MS6, MS7, MS8, MU1V, MU1H, and
MNU2.
These recommendations do not necessarily correspond to the ones described in Appendix 4.
Also, no attempt has been made to define the applicability of the RSB model in the above
context.
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APPENDIX 2: THE DIFFUSER HYDRAULICS MODEL PLUMEHYD
MODEL DESCRIPTION
The model PLUMEHYD is based on the hydraulics model DPHYDR used by Tetra Tech in
the early 1980's to help assess 301(h) applications (Gremse, 1980), and, based on a limited
number of trials, gives approximately identical results. It is appropriate for use with multiport
diffusers with bell shaped or sharp-edged ports. It also considers multi-segmented diffusers of
varying diameter. The program uses metric (SI) units and works in batch mode. A discussion
of diffuser hydraulics is available in Grace (1978).
MODEL USAGE
At this time PLUMEHYD.exe works only in the
batch mode, which means you must construct the input
file in an ASCII editor, like the built-in Turbo Pascal
editor. Sample input is shown in Figure 78.
Honouliuli diffuser hydraulics
74 4 0.0267
bell
1 1 1.22 7.315 0.0 0.215
2 22 1.22 7.315 0.0 0.134
23 47 1.677 7.325 0.0 0.129
48 74 1.982 7 .315 0.00 .123
0.014 0.1818
The first line of input is a title. It is followed by a
line containing the number of ports, number of diffuser
sections, and the ratio of the density difference between Figure ?8 pLUMEHYD batch input
the ambient and effluent fluids to the effluent density, ^e
(Pa • P*)/P«- ^e individual values must be separated by
blanks.
The third line should contain the words "bell" or "sharp", for bell shaped or sharp edged
ports. Sharp edged ports cause a dynamic constriction in the plume diameter within a short
distance of the port and increase the effective densimetric Froude number of the discharge.
There follow a variable number of lines defined by the number of diffuser sections on the
second line of input, in this case, 4. Each line, starting from the end of the diffuser, specifies
the number of the first port in the section, the last port, the pipe diameter, the port spacing, the
rise, and finally the port diameter. The spacing is the distance between adjacent ports on
opposite sides (staggered ports). If there are two ports at the same point but on opposite sides
of the pipe, half the spacing between pairs of ports should be used. Note that in this case the
diffuser has a large port at the end of the diffuser described in the line immediately below the
word "bell". Its purpose may be to maintain a high flow velocity in the end of the diffuser to
prevent sedimentation within the line.
The last line of input specifies the Mannings number and the total flow rate. The units are
SI (MKS). Estimates of the Mannings number may be obtained from Brater and King (1976)
or other engineering texts.
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Appendix 2: The diffuser hydraulics model PLUMEHYD
PLUMEHYD COMPUTER LISTINGS
Pascal Version of PLUMEHYD
{$r+}
{
Program PLUMEHYD.pas
Metric system (SI) units assumed
}
const
g = 9.807;
criterion = le-6;
type
porttype = (bell,sharp);
st80 = string[80];
var
piped,dxpipe,dzpipe,ff,portd: array[1..20] of real;
fm,fxn,title: st80;
nf,nl: array[1..20] of integer;
qq,ee: array[1..50] of real;
e,cd,pipev,portfn,portv,q: array[1..400] of real;
ab,al,al,cdc,dr,dx,dz,error,eorg,eO,f,fnf,gprime,hlf,hlz,
mann,pd,pid4,pod,qc,qorg,qsum,qt,qO,v,vnew,vorg,zman: real;
i,iter,np,ns,ans: integer;
ptype: porttype;
fi,fx: text;
{
dr = drho/rho
dxpipe = horizontal length of the section
dzpipe = vertical rise of the section
mann = Manning's n
nf = number of the first port in a given section
nl = number of the last port in a given section
np = number of ports
ns = number of diffuser sections
piped = pipe diameter of the section
portd = port diameter
ptype = port type, bell or sharp
qt = total discharge
154
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Appendix 2: The diffuser hydraulics model PLUMEHYD
function pwr(a,b: real):real; var sign: integer;
{ an exponentiation function }
begin
if a < 0 then begin sign:--l; a:=-a; end else sign:=l;
a:=exp(b*ln(a)); if sign = -1 then pwr:=-a else pwr:=a; end;
function strip(s:st80): st80;
{ strips blanks out of a string of characters }
begin while s[l] = ' 'do delete(s,l,l); strip:=s; end;
procedure cvnew(var enew,vold,cd,vnew: real);
{ sets up PLUMEHYD for analyzing diffusers with bell or sharp-edged ports }
var dv,fl,f2,v,v2: real;
begin
v:=0;
fl:=0.5/g/enew;
f2:=al/ab*sqrt(2*g*enew);
if ptype = bell then begin
v:=vold;
repeat
v:=vnew;
v2:=sqr(v);
cd:=0.975*pwr((l-v2*fl),0.375);
vnew:=vold+cd*f2;
dv:=v-vnew;
v:=vnew;
until abs(dv) - criterion < 0;
end
else
begin { sharp }
v:=vold;
repeat
v2:=sqr(v);
cd:=0.63-0.58*v2*fl;
vnew:=vold+cd*f2;
dv:=v-vnew;
v:=vnew;
until abs(dv) - criterion < 0;
end;
end;
procedure loop; var j,k,nl,n2: integer;
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Appendix 2: The diffuser hydraulics model PLUMEHYD
{ main program element }
begin
vorg:=0; eorg:=eO; k:=0; qsum:=0;
for j:=l to ns do begin
pd:=piped[j];
ab:=pid4*sqr(pd);
dx:=dxpipe[j];
dz:=dzpipe[j];
f:=ff|j];
pod:=portd[j];
al:=pid4*sqr(pod);
fnf:=4/al/sqrt(gprime*pod);
nl:=nf[j];
n2:=nl[j];
hlz:=dz*dr;
hlf:=f*dx/pd/2/g;
for i:=nl to n2 do begin
cvnew(eorg,vorg,cdc,vnew);
k:=k+l;
e[k]:=eorg;
qc:=(vnew-vorg)*ab;
q[k]:=qc;
cd[k]:=cdc;
pipev[k]:=vnew;
portv[k]:=qc/al;
portfn[k] :=qc*fnf;
eorg:=hlz+eorg+vnew*vnew*hlf;
qorg:=qc;
qsum :=qsum+qc;
vorg:=vnew;
end;
{}if j-ns < 0 then begin
v:=vorg*sqr(piped[j]/piped[j+l]);
eorg:=eorg+0.7*sqr(v-vorg)/2/g;
vorg:=v;
end;
end;
iter:=iter+l; ee[iter]:=eO; qq[iter]:=qt-qsum; end;
procedure input; var portst: st80;
begin
write('Input file (CR for default name of "HYD.IN": '); readln(fin);
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Appendix 2: The diffuser hydraulics model PLUMEHYD
if fin = " then fin:='hyd.in';
assign(fi,fin); reset(fi);
writeCOutput file (CR for default name of "HYD.EX": '); readln(fxn);
if fxn = " then fxn:='hyd.ex';
assign(fx,fxn); rewrite(fx);
readln(fi,title); readln(fi,np,ns,dr);
readln(fi,portst); portst:=strip(portst);
if upcase(portst[l]) = 'B' then ptype:=bell else ptype:=sharp;
for i:= 1 to ns do
readln(fi,nf[i],nl[i],piped[i],dxpipe[i],dzpipe[i],portd[i]);
{ write('Input Mannings n, q (mA3/sec)'); } readln(fi,mann,qt);
end;
procedure initialize;
{ initializes program variables }
begin
error:=0.001; pid4:=pi/4;
zman:=124.58*mann*mann;
for i:=l to ns do ff[i]:=zman/pwr(piped[i],0.33333);
qO:=qt/np;
al :=pid4*sqr(portd[l]);
eorg:=sqr(qO/al)/2/g;
ee[l]:=eorg; eO:=eorg;
iter:=0; gprime:=dr*g; end;
procedure outputit; var j,k: integer; begin
writeln(fx,title); writeln(fx);
writeln(fx,'Number of ports =',np:4);
writeln(fx,'drho/rho = ',dr:9:4);
writeln(fx,'Number of sections = ',ns:4);
if ptype = bell then writeln(fx,'bell')
else
writeln(fx,'sharp');
writeln(fx);
writeln(fx,'Mannings N = ',mann:9:4);
writeln(fx,'Desired Q = ',qt:9:4);
writeln(fx,'Calculated Q = ',qc:9:4); writeln(fx);
for k:= 1 to ns do begin
writeln(fx,
'Friction factor F = ',ff[k]:9:4,' ':9,
'Pipe diameter =',piped[k]:9:4);
writeln(fx,
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Appendix 2: The diffuser hydraulics model PLUMEHYD
'Length between ports = ',dxpipe[k]:9:4,' ':9,
'dz between ports =',dzpipe[k]:9:4);
writeln(fx,'Port diameter = ',portd[k]:9:4);
writeln(fx);
writeln(fx,
'Port Specific Coeff Pipe Port Port Port');
writeln(fx,
'number energy cd velocity velocity discharge Froude #');
writeln(fx,
(m) (m/sec) (m/sec) (mA3/sec)');
writeln(fx);
for j:=nf[k] to nl[k] do
writeln(fitj:6,e[j]:10:4,cd|j]:10:4,pipev|j]:10:4,
portv[j]: 10:4,q|j]: 10:4,portfn[j]: 10:4);
writeln(fx); end;
end;
{ main program element }
begin
input; initialize;
repeat
loop;
if iter = 1 then
eO:=ee[ l]*sqr(qt/qsum)
else
eO:=(ee[iter-l]*qq[iter]-ee[iter]*qq[iter-l])/(qq[iter]-qq[iter-l]);
until abs(qq[iter]) < error;
qc:=qsum;
outputit;
close(fi); close(fx); end.
Sample Input File
Honouliuli diffuser hydraulics
74 4 0.0267
bell
1 1 1.22 7.315 0.0 0.215
2 22 1.22 7.315 0.0 0.134
23 47 1.677 7.325 0.0 0.129
48 74 1.982 7.315 0.0 0.123
0.014 0.1818
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Appendix 2: The diffuscr hydraulics model PLUMEHYD
Sample Output File
Honouliuli diffuser hydraulics
Number of ports = 74
drho/rho = 0.02*
Number of sections = 4
bell
Mannings N = O.OK
Desired Q =0.18]
Calculated Q = 0.18]
Friction factor F = 0.02^
Length between ports = 7.31!:
Port diameter = 0.21?
Pipe diameter = 1.2200
dz between ports = 0.0000
Port
number
Specific
energy
(m)
0.0017
Port Port
velocity discharge
(m/sec) (m^S/sec)
0.1763
0.0064
Port
Froude #
0.7429
Friction factor F
Length between ports
Port diameter
Port Specific
number energy
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
Friction
(m)
0.0017
0.0017
0.0017
0.0017
0.0017
0.0017
0.0017
0.0017
0.0017
0.0017
0.0017
0.0017
0.0017
0.0017
0.0017
0.0017
0.0017
0-.0018
0.0018
0.0018
0.0018
factor F
Length between ports
Port diameter
0
7
0
Coeff
cd
0.9744
0.9739
0.9734
0.9728
0.9721
0.9713
0.9704
0.9694
0.9683
0.9671
0.9658
0.9644
0.9629
0.9613
0.9597
0.9580
0.9562
0.9543
0.9524
0.9504
0.9483
0
7
0
.0229
.3150
.1340
Pipe
velocity
(m/sec)
0.0076
0.0097
0.0119
0.0140
0.0161
0.0182
0.0203
0.0225
0.0246
0.0267
0.0288
0.0310
0.0331
0.0352
0.0373
0.0395
0.0416
0.0437
0.0459
0.0480
0.0501
.0206
.3250
.1290
Pipe diameter
dz between ports
Port Port
velocity discharge
(m/sec)
0.1762
0.1762
0.1761
0.1761
0.1760
0.1759
0.1759
0.1759
0.1759
0.1759
0.1759
0.1759
0.1760
0.1761
0.1763
0.1764
0.1767
0.1769
0.1772
0.1776
0.1780
Pipe
(mA3/sec)
0.0025
0.0025
0.0025
0.0025
0.0025
0.0025
0.0025
0.0025
0.0025
0.0025
0.0025
0.0025
0.0025
0.0025
0.0025
0.0025
0.0025
0.0025
0.0025
0.0025
0.0025
diameter
dz between ports
1.2200
0.0000
Port
Froude #
0.9408
0.9405
0.9402
0.9399
0.9396
0.9393
0.9391
0.9389
0.9388
0.9388
0.9389
0.9392
0.9396
0.9402
0.9410
0.9419
0.9431
0.9445
0.9462
0.9481
0.9503
1.6770
0.0000
159
-------
Appendix 2: The diffuser hydraulics model PLUMEHYD
Port Specific
number energy
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
Friction
(m)
0.0018
0.0018
0.0018
0.0018
0.0018
0.0019
0.0019
0.0019
0.0019
0.0019
0.0019
0.0019
0.0019
0.0019
0.0019
0.0019
0.0019
0.0019
0.0019
0.0020
0.0020
0.0020
0.0020
0.0020
0.0020
factor F
Coeff
cd
Pipe
velocity
(m/sec)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Length between ports
Port diameter
Port
number
48
49
50
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
Specific
energy
(m)
0 .0020
0.0020
0.0021
0.0021
0.0021
0.0021
0.0021
0.0021
0.0022
0.0022
0.0022
0.0022
0.0022
0.0022
0.0022
0.0022
0.0022
0.0023
0.0023
.9672
.9666
.9659
.9653
.9646
.9639
.9632
.9625
.9617
.9609
.9601
.9593
.9585
.9576
.9568
.9559
.9550
.9541
.9532
.9523
.9513
.9504
.9494
.9484
.9475
0
= 7
0
Coeff
cd
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
.0194
.3150
.1230
0276
0287
0298
0309
0320
0331
0341
0352
0363
0374
0385
0396
0407
0418
0429
0440
0451
0462
0473
0484
0495
0506
0517
0528
0539
Port
velocity
(m/sec)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
.1834
.1835
.1836
.1836
.1837
.1838
.1839
.1841
.1842
.1843
.1845
.1847
.1849
.1851
.1853
.1855
.1858
.1861
.1864
.1867
.1870
.1873
.1877
.1881
.1885
Port
discharge
Port
Froude #
(mA3/sec)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
.0024
.0024
.0024
.0024
.0024
.0024
.0024
.0024
.0024
.0024
.0024
.0024
.0024
.0024
.0024
.0024
.0024
.0024
.0024
.0024
.0024
.0024
.0025
.0025
.0025
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Pipe diameter =
dz between
Pipe
velocity
(m/sec)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
.9607
.9602
.9596
.9547
.9541
.9536
.9530
.9524
.9518
.9512
.9506
.9500
.9494
.9488
.9482
.9476
.9470
.9464
.9457
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
039-4
0401
0408
0475
0483
0491
0498
0506
0513
0521
0528
0536
0543
0551
0559
0566
0574
0582
0589
Port
velocity
(m/sec)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
.1921
.1923
.1925
.1944
.1947
.1950
.1952
.1955
.1958
.1961
.1965
.1968
.1972
.1975
.1979
.1983
.1986
.1991
.1995
ports
Port
discharge
(mA3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
/sec)
.0023
.0023
.0023
.0023
.0023
.0023
.0023
.0023
.0023
.0023
.0023
.0023
.0023
.0023
.0024
.0024
.0024
.0024
.0024
=
.9981
.9984
.9988
.9992
.9997
.0002
.0008
.0015
.0022
.0030
.0039
.0049
.0059
.0070
.0082
.0095
.0109
.0124
.0140
.0156
.0174
.0193
.0213
.0233
.0255
1.9820
0.0000
Port
Froude #
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
.0706
.0715
.0725
.0833
.0848
.0863
.0879
.0895
.0912
.0929
.0948
.0966
.0986
.1005
.1026
.1047
.1069
.1091
.1115
160
-------
APPENDIX 3: SUPPORT FOR TABLE I (CHAPTER 1)
Input and Output for Case 1
Two examples given in Table I corresponding to flows of 100 and 137 MOD are presented
below:
Apr 25, 1993, 13:
Title EXAMPLE 1,
tot flow # ports
4.381 100
port dep port dia
29.87 0.1524
port elev ver angle
0.6096 0
hor angle red space
90 10.000
depth current
0.0 0.05
30.48 0.05
8:30 ERL-N PROGRAM PLUMES, Mar 3, 1993
5 cm/sec
port flow spacing effl sal effl temp
0.04381 10 0.0 15
plume dia total vel horiz vel vertl vel
0.1446 2.669 2.669 0.000
cont coef effl den poll cone decay
0.9 -0.8363 100 1.157e-8
p amb den p current far dif far vel
24.28 0.05000 0.000453
density salinity temp amb cone
24.10 32 13 1.6
24.29 32 12 1.6
Case: 1 of 2
non-linear
far inc far dis
30 100
asp coeff print frq
0.10 150
Froude # Roberts F
14.13 0.001673
K:vel/cur Stratif #
53.370.00003593
N (freq) red grav.
0.007731 0.2466
buoy flux puff-ther
0.01080 2.936
jet-plume jet-cross
1.924 6.838
plu-cross jet-strat
86.41 6.650
plu-strat
12 .36
hor dis>=
CORMIX1 flow category algorithm is turned off.
0.10
Help: Fl. Quit: . Configuration:ATNOO. FILE: DB JMANUL.VAR;
UM INITIAL DILUTION CALCULATION (non-linear mode)
plume dep plume dia poll cone dilution CL cone
0.0 to 0.5 range
m
29.87
29.86
29.59
27.13
21.25
11.41
7.872
3.406
m
0.1446
0.3974
1.051
2.191
4.144
8.168
10.05
13 .60
100.0
36.39
13.90
5.949
3.138
2.144
2.006
1.904
1.000
2.783
7.828
22.10
62.45
17S.6
236.3
316.1
100.0
69.34
26.01
10.44
4.613
2.488
2.207
1.982
hor dis
m
0.000
0.6407
2.386
5.819
9.418
14.10
15.89
18.42
-> merging
-> surface hit
161
-------
Appendix 3: Support for Tables I and II (Chapter 1)
Apr 25, 1993, 13 :
Title EXAMPLE 1,
tot flow # ports
6.002 100
port dep port dia
29.87 0.1524
port elev ver angle
0.6096 0
hor angle red space
90 10.000
depth current
0.0 0.05
30.48 0.05
8:33 ERL-N
5 cm/sec
port flow
0.06002
plume dia
0.1446
cont coef
0.9
p amb den
24.28
density
24.10
24.29
PROGRAM PLUMES, Mar 3, 1993
spacing
10
total vel
3.656
effl den
-0.8363
p current
0.05000
salinity
32
32
effl sal
0.0
horiz vel
3.656
poll cone
100
far dif
0.000453
temp
13
12
effl temp
15
vertl vel
0.000
decay
1.157e-8
far vel
amb cone
1. 6
1.6
CORMIX1 flow category algorithm is turned off.
0.10
Help: Fl. Quit: . Configuration .-ATNOO . FILE:
UM INITIAL DILUTION CALCULATION (non-linear mode)
plume dep plume dia poll cone dilution CL cone
m
29.87
29 .86
29 .71
27.67
21.03
9.464
7.812
3.045
m
0 .1446
0.3993
1.082
2.469
4.686
9.270
10.09
13.24
100.0
36.39
13 .90
5.949
3.138
2.144
2.083
1.969
1.000
2 .783
7.828
22.10
62.45
176.6
198.7
260.4
100.0
68.90
25 .63
10.18
4.520
2.475
2.337
2.078
Case: 2 of 2
non-linear
far inc far dis
30 100
asp coeff print frq
0.10 150
Froude # Roberts F
19.36 0.001221
K:vel/cur Stratif #
73 .120.00003593
N (freq) red grav.
0.007731 0.2466
buoy flux puff-ther
0.01480 4.022
jet-plume jet-cross
2.636 9.369
plu-cross jet-strat
118.4 7.784
plu-strat
13 .38
hor dis>=
0.0 to 0.5 range
DBJMANUL.VAR;
hor dis
m
0.000
0.6421
2.420
6.595
11.51
17 .34
18 .18 -> merging
20.71 -> surface hit
162
-------
APPENDIX 4: MESSAGES AND INTERPRETATIONS
CORMIX WINDOW RECOMMENDATIONS
Historically, work culminating in this manuscript and corresponding software and the EPA
sponsored work on CORMIX proceeded independently. Since about 1990, efforts have been
made to integrate the two approaches to take advantage of their complementary capabilities, as
explained in Appendix 1. For example, a CORMIX work element exists to in some way include
the traditional EPA models within its framework. The CORMIX window of the PLUMES
interface, implemented for CORMIX 1, integrates the CORMIX categorization schemes into
PLUMES. See Hinton and Jirka (1992) for a graphic description of the flow categories.
Providing there are no limitations to its use as described in Table V of Appendix 1,
CORMIX 1 is considered to be an appropriate solution to the plume problem under consideration
in the PLUMES interface. It is assumed that the Configuration menu has been used to turn the
CORMIX 1 algorithm on.
Note, since RSB is exclusively designed for merging plumes, only CORMIX 1 and UM are
applicable to this discussion. However, in some cases the CORMDC1 categories have a clear
relationship to CORMIX2 categories. Also, in questionable cases, a few runs using both*
CORMIX1 and UM may be helpful, either corroborating each other or suggesting caution.
Single: use CORMIX1; merging: UM ok
Displayed in cases in which PLUMES predicts flow categories v4 and v6: The use of
CORMIX is definitely recommended for single plumes, but only in cases in which
nascent density effects are absent and other weak links in the CORMIX chain (see
Appendix 1) do not exist. Excluded cases must be handled on a case-by-case basis.
To the extent that some CORMIX1 flow categorize have obvious CORMIX2 counterparts,
the appropriate use of the models for merging plumes may be apparent. Mutual
validation and the use of the more conservative analysis are recommended in questionable
cases.
Use CORMIX
Displayed in cases in which PLUMES predicts flow categories h4-90, h5-90, nv5, nh3:
The use of CORMIX 1 is definitely recommended, but only in cases in which nascent
density effects are absent and other weak links in the CORMIX chain (see Appendix 1)
do not exist. Excluded cases must be handled on a case-by-case basis.
Use CORMIX or UM to surface hit
Displayed in cases in which PLUMES predicts flow categories nv3, nv4, and nh5: It is
appropriate to continue the analysis with UM until the surface is hit. The use of
163
-------
Appendix 4: Messages and interpretations
CORMIX is appropriate and possibly preferred, but only in cases in which nascent density
effects are absent and other weak links in the CORMIX chain (see Appendix 1) do not
exist. Mutual validation with CORMIX and the use of the more conservative analysis is
recommended in questionable cases.
UseUM
Displayed in cases in which PLUMES predicts no CORMIX1 category or flow categories
si, s3, s4: It is appropriate to continue the analysis with UM. The use of CORMIX is
appropriate too, but only in cases in which nascent density effects are absent and other
weak links in the CORMIX chain (see Appendix 1) do not exist.
Use UM to bottom hit
Displayed in cases in which PLUMES predicts flow categories nvl, nv2, nhl, nh2, and
nh4: It is appropriate to continue the analysis with UM until the bottom is hit. The use
of CORMIX is also appropriate, but only in cases in which nascent density effects are
absent and other weak links in the CORMIX chain (see Appendix 1) do not exist.
Because two of the entrainment terms are disabled after plume vertical directional
reversal, the UM analysis is thought to be conservative. Mutual validation with CORMIX
and the use of the more conservative analysis is recommended.
Use UM to overlap point
Displayed in cases in which PLUMES predicts flow categories s2, s5, h4-180, h5-180:
UM is considered appropriate to the point of overlap, with the farfield model being
initiated at that point. The use of CORMIX is appropriate, but only in cases in which
nascent density effects are absent and other weak links in the CORMIX chain (see
Appendix 1) do not exist.
Use UM until near surface
Displayed in cases in which PLUMES predicts flow categories v3, v5, h3, h40: UM is
weaker and CORMIX is correspondingly stronger in these categories. The ten percent
prohibition suggested by Muellenhoff et al. (1985) may be appropriate and can be
implemented using the Pause criterion in the Farfield configuration of PLUMES. The use
of CORMIX is appropriate, but only in cases in which nascent density effects are absent
and other weak links in the CORMIX chain (see Appendix 1) do not exist.
Use UM until surface hit
Displayed in cases in which PLUMES predicts flow categories vl, v2, hi, h2, and h5-0:
UM is considered appropriate to the point of the surface being hit, with the farfield model
being initiated at that point. The use of CORMIX is appropriate, but only in cases in
which nascent density effects are absent and other weak links in the CORMIX chain (see
Appendix 1) do not exist.
164
-------
Appendix 4: Messages and interpretations
DIALOGUE WINDOW MESSAGES
The following messages appearing in alphabetical order are more or less frequently displayed
by the PLUMES interface. Here they are explained in more detail. Some have subsidiary
messages, shown here below the main message. Content that depends on context is contained
in brackets []. Some of the notes may be revealed only briefly, the AYN command may be used
to examine them at your leisure.
Absolute value of decay too large, reduce value.
Warns of a value for decay that does not convert into the correct units and can cause
program crashes.
A descriptive title.
Used to describe the title cell, it is issued by the command.
At [variable] Change sign or to accept [default]
This message usually indicates that PLUMES is trying to define the identified cell from
an equation involving a square root for which both positive and negative roots are valid.
You have to make the appropriate choice.
Back, Inequalities, Output, Variables (space), or to quit
Used to manipulate data in the Pause (or stop) cell. Typing "V" or the spacebar brings
the various available cells into the window, "B" doing so in the reverse manner. "I"
selects the appropriate pause inequality. The "O" option only installs the hidden
variables, e.g. the centerline concentration, on the output table. The cell is filled with a
numeric value in the usual manner, e.g. by using the AJ command to enter the cell.
Bad file name, old or default file restored
Indicates a non-existent case file, normally one with a .VAR extension, was specified for
opening. Usually this happens when you have forgotten the name of the case files and
inadvertantly specify a non-existent file name. Exit to DOS and use the DIR command
to refresh yourself on the appropriate names. The <-l> may be used in the command to refresh yourself on existing case files.
Note: [message] [equation number] of [variable]
Appears when a potential data inconsistency is detected. This can be automatic or happen
when the command is used. The AYN command may be
used to check for their occurence. The [equation number] refers to the cardinal number
of the equations listed for the cell [variable] when the AL command is
used. While efforts should be made to resolve inconsistencies, they do not always
indicate incompatible input data.
165
-------
Appendix 4: Messages and interpretations
Default table, or New table?
Asks you whether to include the default output variables when running UM or to clear
the table (New) for the addition of variables of your choosing using the
command.
Discharge in Middle or Surface/bottom of water column?
Appears when the command is used. You must choose or
(or ) to specify your choice, which establishes the proper spacing for the reflection
surfaces and other parameters.
Error detected in case range
Appears after invoking the (Miscellany Menu) to indicate that an error
in specifying the number of cases to which to copy the current cell has been made.
Farfield result will not reflect decay in the near field
This is a reminder that RSB, as an initial dilution model, does not include decay.
Consequently, if decay is fast or rise times are long, the pollutant concentration can be
significantly over-predicted.
File access denied, directory name?
The inputted file name is not valid because it already specifies a sub-directory.
File [filename] exists or name illegal, must be new'
Issued while using the AN command when an existing file name or an
illegal name, such as a sub-directory name, is specified. You are asked to provide unique
case file name.
Go to case ( for default): [default case number]
Used to specify how many cases to run or translate into Universal Data File (UDF)
format or to which case to move using the AC command. For the first two functions all
cases between the present and the indicated case, including the present case, will be
processed.
Hit bold letters or arrow keys and ; use control sequences for speed
Issued when accessing the main menu to remind you that the control key sequence for
issuing commands is faster than using the menus.
Inconsistency at [variable name 1]: [value 1] vs. [variable name 2]: [value 2]
These messages may appear when using the command if tolerances
are not met. In other words, if two different equations of the same dependent variable
yield values which differ by more than 1 part per thousand, then this message is issued.
166
-------
Appendix 4: Messages and interpretations
Inputted case # invalid, reset
The case number input is invalid. For example, when running UM, specifying a negative
case number will cause this message to be issued. In this case the input is changed to
the present case number and a single case is run.
Input file name (or <-i> to select .VAR file):
Requests you to enter the name of the case file, i.e. the non-ASCII file used to store the
input screen data, such as PLMSTUFF.VAR, or to select the appropriate file with <-l>
followed by . These files cannot be edited by an ASCII editor.
Input starting longitudinal coordinate:
When the Brooks-equation-width-input-toggle in the Configuration String is set to "user",
PLUMES prompts for the initial width of the wastefield and the initial starting distance,
thus allowing for the override of these two parameters. This allows runs of the Brooks
equation which are essentially independent of the initial dilution estimates.
Input wastefield width:
See related message, "Input starting longitudinal coordinate:", above.
Invalid file name
An illegal file name was specified while using the AW command.
for far field prediction
RSB output is displayed on two screens, the near field output and the far field output.
once again to start PLUMES
While using the command, some condition needing your attention
in the initialization phase has been identified. Make tot flow, spacing, plume dep, port
dia, port eleve cells independent, and, a non-surf ace independent ambient depth cell must
be defined, which must satisfy: ambient depth >= plume dep. A message appears on
three separate dialogue windows when some or all cell values needed to complete the
command are missing.
New file name (or to cancel command>
An empty line will appear to use to enter the string of case numbers
Enter the record numbers of the records to keep (followed by )
Use spaces as separator, .. to indicate a range, e.g. 12 3..9 14
A tutorial on using the command. The command is used to fill a
previously non-existent file with cases from the file in the interface. Cases may be
specified in any order and repetition is allowed.
167
-------
Appendix 4: Messages and interpretations
No changes made
Appears if a choice other than Middle or Surface/bottom, i.e. no choice, is made after
issuing the command.
No direct independents to hilite for [variable], remove others.
Issued when the problem is overspecified and a confict arises. This happens when a
dependent (white) value is replaced by an independent (yellow) value but no immediate
independent values for the cell can be identified, i.e. the cell is totally defined by other
dependent (white) values. YOU SHOULD IMMEDIATELY REMOVE THE LAST
VALUE YOU INPUT OR FIND OTHER INDEPENDENT VALUES TO REMOVE.
USE THE COMMAND TO ASSURE CONSISTENCY.
NO GO, incomplete effluent/ambient blocs.
Advises you that the data necessary for running UM are not complete. Return to the
input screen and check for missing cells.
Not a number: [string], correction attempted.
You tried to input non-numerical information in a numerical cell. PLUMES removes the
non-numeric characters from the input data and tries to convert the remaining string to
numeric data. Other conditions, such as multiple decimal points, will also cause this
message to be issued. The value should be checked and corrected if necessary.
Only for adding hidden variables to the table.
Variables explicitly displayed on the interface screen are put on or removed from the
output table with the AO command.
Overwrite existing cases or Append (default)?
Issued by the AYU command when the read option is chosen. The
overwrite option erases the case in which the cursor is located and all subsequent cases.
Plumes not merged, Brooks method may be invalid.
The Brooks equations are based on a continuous wastefield, an assumption which is not
valid when the plumes are not merged. However, the equations are probably valid if the
unmerged distance is small.
Probable corrupted data file, check SETUP, and files.
SETUP should be deleted; program to terminate!
An error has been identified in the case file. Possibly you asked for a file that is not in
the binary case file format, you have moved your files to some new directory and
PLUMES is unable to find the files, or some other terminal condition exists. Check the
SETUP file for clues, delete it, and start over (or shift attention to other case files).
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Appendix 4: Messages and interpretations
Probably not a .VAR file
A file was specified while using the AW command which does not contain the correct
number of bytes to be a .VAR file.
Quit (or ), all others to continue
Message appears when execution of UM has been interrupted. or will cause
the current run to be abandoned.
Replicate this cell to case ( to accept default):
Issued by the AYB command. A value in a particular cell in the
present case may be copied to the corresponding cell in a specified number of additional
cases starting with the next case.
Save (also ), Discard work, to return to PLUMES
Message appears when existing PLUMES. or causes the old case file
to be updated. restores the previously existing file i.e. all the work done in the
current session is discarded. and other keystrokes cancel the command.
See guidance material for explanation
Appears when the Miscellany Menu is accessed. Guidance may be found in the section
entitled "User's guide to the model interface, PLUMES" in the manual.
See users' guide for details
Appears when the Configuration Menu is accessed.
Specify max reversals; 0: PLUMES chooses (see manual: configuration):
You are asked how many vertical velocity reversals UM should use before giving control
over to the far field model. Reversals occur in stable ambient at the top of rise or when
the plume sinks to a maximum depth (fall). If the trajectory is plotted out, these points
are the crests and troughs of the resulting waveform.
Start far-field at Max-rise, Overlap, or Pause criterion?
Issued when invoking on the Configuration Menu for control of the UM
model. You are to specify at which point the initial dilution model should end and the
far field model begin. The overlap condition is recommended.
Sure you want to zap variables? (y/n):
Reminder after issuing the command on the Miscellany Menu, that
all variables except the aspiration coefficient, output frequency, decay, far field dispersion
coefficient, and surface ambient depth cell will be blanked out.
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Appendix 4: Messages and interpretations
Temperature A) [temperature 1] or [temperature 2]?
This message appears when temperature is the dependent variable (defined by density and
salinity). In this case an approximation technique is used to solve the density function
for temperature. This choice is presented when, starting at different initial guesses, two
solutions to the problem converge on separate values.
To use command, number of ports must = 1
Reminder that the command can only be used for single port outfalls.
To use put cursor in the filled cell below cells to be interpolated
Instructs you how to fill embedded empty cells in the ambient block. You must move
the cursor to a filled cell below the embedded empty cells. The corresponding top
bounding cell must also exist. The cells in between will be interpolated on the values of
the depths in the depth column.
UM running, to interrupt
A "Please wait" message. UM can be interrupted and stopped at anytime.
Use RSB for multiple port diffusers
This is a reminder that RSB is a multiport, not a single port, model.
Use control key sequences or see the Guide for better movement and control
Appears when the Movement menu is accessed, reminds you that better movement
controls are available by consulting the manual.
With regard to [variable name] resolve conflicts:
Issued when the problem is overspecified and a confict arises. This happens when a
dependent variable is replaced by an independent variable, i.e. one you input. You are
forced to move between the highlighted cells until you delete one of them, by pressing
or the on the flashing (chosen) cell.
Work will be lost with , to cancel
Issued when the Discard option is chosen when quitting PLUMES. It provides additional
protection from accidentally discarding changes made in the current work session.
Write to ("prn" for printer, "console", or disk file name): [default name]
Appears after specifying the number of cases to run after issuing the AB or AU commands
(see "From this case on..."). You are asked to specify the output device which can be the
printer (type in the letters prn), monitor (type in console), or disk file (any legal DOS file
name). The spacebar may be used to accept the default value.
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Appendix 4: Messages and interpretations
xx = current variable, x2 = 1st argument in PRECEDING ns.
Provides definitions of xx and x2. Used for programming purposes. Please report such
occurrences.
UM RUN TIME MESSAGES
UM issues various standard text and messages which are useful for interpreting numerical
output. They are given here in alphabetical order and explained in detail.
Before running, UM saves the case in which the cursor is located and copies the input to the
output file. Thus, even if a run time error were to cause a crash, the input is safely stored away.
Then, the screen is copied exactly the way it appears, except for the color, to the output file,
which may be a disk file, the printer, or, the console itself.
Immediately below the output, on three separate lines, UM prints the message "UM INITIAL
DILUTION SIMULATION" and either "linear mode" or "non-linear mode", followed by a
numerical tabulation of variables on the output table (the results of the simulation) headed by the
cell names and their corresponding units.
Pertinent output messages are issued when certain criteria are met. They are displayed after
the numerical data to which they apply, the association being indicated by an arrow that points
to the message. If there is sufficient space it appears on the same line, otherwise it appears on
succeeding lines. UM also prints output at the beginning of the simulation and at intervals
specified by the [print frq] (print frequency) cell, which specifies the number of program steps
between output. Such output is not followed by any message. Messages include:
absolute value Froude # < 1, potential diffuser intrusion
When the absolute value of port Froude number is less than one (1) the plume is so
buoyant (or negatively buoyant) that it separates from the bottom (or top) of the port
orifice allowing ambient water to flow into the diffuser.
bank(s) reached
Message used only when the AZ command, for very shallow water,
has been used and the AZ flag has been placed by UM at the beginning of the title cell.
It indicates that the width of the plume equals or exceeds the implied distance to the
bank.
begin overlap
Indicates that the definition of the UM plume element is not geometrically and physically
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Appendix 4: Messages and interpretations
self-consistent, viz. part of the element is composed of physically unreal negative volume
and negative mass. Note: this condition is an artifact of the uncorrected round plume
assumption which is commonly used in Lagrangian and Eulerian integral flux plume
models. The problem occurs when trajectory curvature is great and will produce errors
unless the model is specifically modified to correctly handle the problem. It significance
derives from the fact that the radius is over-estimated when overlap occurs. Since
entrainment is proportional to the radius, it is also over-estimated. For further detail see
the UM Model Theory Chapter, "The Plume Element."
The "end overlap" message, described below, indicates the cessation of the condition,
overlap, causing the error. If the dilution changes relatively little in this region the
message may be safely ignored. Otherwise, the dilution given at the beginning of overlap
may be used to give a conservative estimate of dilution or another model may be used.
bottom geometry consistent? Try increasing port elev and/or ambient depth
Issued only if the bottom is encountered in the first two program steps, i.e. at the source.
This advisory frequently has minor significance because it usually relates to the non or
weakly entraining side of the plume. In such cases the port elevation or ambient depth
cells may be increased, as appropriate, to prevent this condition from terminating the run.
However, negatively buoyant plumes are likely to be significantly affected. See the
related message "bottom hit".
bottom hit
This message is issued when the extremities of the plume element intersect the bottom,
which is assumed to be at a distance of [port elev] below the port depth or the deepest
ambient layer, whichever is greater. Because the bottom is often hit by the downstream
portion of the plume, which is not the primary entraining surface, the condition can
sometimes be ignored, at least as long as it is not violated excessively. However, it
should be recognized that the presence of the condition implies considerations of mass
continuity and, indirectly, the dimensions of the plume which affect entrainment.
dilution overestimated
Associated with the message "begin overlap" explained above.
end overlap
The overlap condition ceases. See the "begin overlap" message above.
end curvature, cylinder entrainment
Associated with the "local maximum rise or fall" message below. The entrainment terms
are desribed in detail in the UM Model Theory Chapter, "Plume Dynamics."
Although no message is issued to that effect, the assumption tends to cause UM to under-
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Appendix 4: Messages and interpretations
predict dilution, i.e. conservatively, in the affected region. However, a general statement
cannot be made because the vertical penetration of the plume element is also affected.
Circumstances can be imagined where the assumption would actually over-estimate
dilution, e.g. if the assumption allows the plume to penetrate into an unstratified layer
which it would otherwise not reach.
leaving defined depth range
Occurs if the extremities of the plume penetrate to a depth below the tenth ambient line
allowed, if defined. See the "bottom hit" message.
local maximum rise or fall
When moving through the trap level, see below, the plume element reverses its buoyancy,
becoming negatively buoyant if initially positively buoyant and vice versa. The vertical
accelerating force then opposes the direction of motion and the plume element ultimately
reaches maximum rise or fall, unless some other condition, such as surface interaction
intervenes. This message indicates the reversal in vertical motion occurred during the
previous time step.
In many applications, the first maximum rise is an appropriate point for determining the
initial dilution achieved and for initiating the farfield diffusion algorithm.
merging
Indicates that neighboring plume elements, assumed to be uniformly spaced and identical,
have grown sufficiently to merge. Merging occurs when the plume diameter is equal to
the reduced spacing which is a function of the physical spacing and the horizontal angle
of discharge. The effect of the condition is to reduce the surface area of the plume
element and the entrainment.
End effects are not modelled by UM, in other words, it is assumed that the diffuser is
infinitely long, the fewer the number of ports, the more important end effects become.
Also, the ports are assumed to be on one side. Cross-diffuser merging can be simulated
by using half the port spacing or by specifying background pollutant concentration in the
ambient pollutant concentration [amb cone] cells.
Quit (or ), all others to continue
Issued when UM has been interrupted while running. Execution may be continued with
any keystrokes except and which terminate the run and return to the interface.
surface hit
The extremities of the plume element have intersected the surface. Since the intersection
generally occurs at the upstream, i.e. entraining, portion of the plume, this is an important
criterion. Generally, the dilution process should be assumed to stop here and the
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Appendix 4: Messages and interpretations
PLUMES configuration string and the pause cell should be manipulated accordingly if the
farfield algorithm is used. The details of mass continuity are not properly estimated by
models of the UM class beyond this point.
In certain special cases the criterion is unreasonably conservative. This is generally true
in shallow water where the surface is intersected by the plume element soon after
discharge but in fact it retains substantial kinetic energy to drive the entrainment process.
surface reflection begins
Message used only when the AZ command, for very shallow water,
has been used and the AZ flag has been placed by UM at the beginning of the title cell.
It indicates that the plume has reached the surface implied, in this case, by the port
spacing.
trap level
This message indicates that the plume element has acquired, if only momentarily, an
average density that is equal to that of the surrounding ambient fluid at the same depth.
If the plume element where at rest it would remain at rest at this level. However,
normally the plume element has a vertical velocity when this level is reached and will
traverse the level. If the ambient is density stratified, and normally it will be, multiple
trap levels are possible. Thus, in a current, the plume element will trace a wavy path
which is sometimes observed in nature.
Historically, the initial trap level has been used as a cut-off point for the initial dilution
process. This cut-off is often applied rather arbitrarily. In many cases, the newer models,
such as UM, provide reasonable estimates of dilution beyond this point. Generally, unless
there is significant overlap, UM is believed to provide good estimates through the level
of maximum rise. In negatively buoyant cases UM is sometimes run past the second trap
level because such plumes are frequently discharged upward and the plume often has
considerable potential energy when reaching maximum rise. However, to be conservative,
the cylinder and curvature terms of forced entrainment are arbitrarily set to zero beyond
this point.
UM running, to interrupt
A "Please wait" message. UM is running but may be interrupted at any time.
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Appendix 4: Messages and interpretations
RSB RUN TIME MESSAGES
Avg. flux dilution (width*he*u/Q): [value]
Estimated volume flux through a cross-section in the ambient flow comparable to the
wastefield cross-section at the end of initial dilution.
for farfield prediction
Strike any key to continue the simulation.
No farfield prediction when far vel = 0.
Gives the reason for no farfield simulation when using RSB; the far field velocity cannot
be equal to zero.
No farfield prediction; far dif, far inc, far dis, or far vel defined?
No farfield simulation is attempted because the farfield diffusion coefficient, increment,
maximum distance, or farfield velocity are not defined.
Results extrapolated beyond their experimental values, may be unreliable
s/lb > 1.92, \J\b > 0.5, or f > 100. These parameters define experimental ranges beyond
which the quality of the empirical model is increasingly unknown. See the RSB chapter
for further details.
Roberts Fr. # < 0.01 (aspiration dominated), no avg. flux dilution formed
Avg. flux dilution not calculated because forced entrainment is small or zero.
Entrainment flow is primarily induced by the plume and not very much by the current.
RSB not compatible with input conditions: [reason]
This advisory states that RSB be cannot be run for one of the following ([reason]): 1)
stratification not defined, information to complete the stratification is missing from the
ambient block; 2) effluent density or current not defined, these cells or cells that are
needed to define them are undefined; and 3) negative buoyancy, RSB is restricted to
cases with positive buoyancy.
Wastefield plume surfaces
Warns that a basic assumption of the model, i.e. that the water is infinitely deep, is not
met. If rise above the surface is significant the dilution will be substantially over-
estimated.
Wastefield submerged
RSB ran as intended.
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Appendix 4: Messages and interpretations
In addition to these messages, RSB always displays the following text:
Written by Philip J. W. Roberts (12/12/89)
Credit
Adapted by Walter E. Frick (1/12/92))
Credit
Case: [case number]: [title]
Case identification
Lengthscale ratios are: s/lb = [value] Im/Ib = [value]
See the RSB Chapter for these variables and the "Results extrapolated..." message above.
Froude number, u3/b = [value]
A measure of current strength. When this value is large ( > 0.1) forced entrainment
dominates.
Jet Froude number, Fj = [value]
A small value indicates a buoyancy dominated plume, a large value a momentum
dominated one. A value of 1.0 is a cut-off value for intrusion of ambient fluid into the
diffuser.
Rise height to top of wastefield, ze = [value]
See Figure 54.
Wastefield submergence below surface = [value]
A negative value indicates overprediction of dilution resulting from the fact that water is
assumed to be infinitely deep.
Wastefield thickness, he = [value]
See Figure 54.
Height to level of cmax, zm = [value]
See Figure 54.
Length of initial mixing region, xi = [value]
See Figure 54.
Minimum dilution, Sm = [value]
The minimum, i.e. centerline, dilution at xt.
Flux-average dilution, Sfa = [average dilution value] ([ratio value] x Sm)
The average dilution value equals the minimum dilution value times the peak-to-mean
ratio; also defined at x,.
Interpolation count: [value]
Information on the running status of RSB. A value of 10 or more indicates that the
solution did not fall within convergence tolerances and the solution should be viewed with
caution.
Wastefield width [value]
Width of the wastefield in meters; measured at x;.
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Appendix 4: Messages and interpretations
FARFIELD MODULE RUN TIME MESSAGES
TEMPORARY NOTE The farfield algorithm is under review. The purpose of the review is
to ascertain its consistency with the proper relationship between average and centerline
concentrations.
dilution overestimated
Issued when overlap occurs in the initial dilution region and the maximum rise or the
pause cell criterion over-ride the overlap criterion as the initial dilution stopping criterion.
Dilution is likely to be overestimated.
Input starting longitudinal coordinate: [default value]
This message appears if the PLUMES configuration string ARB
command has been toggled to 'R' (or reset). The user may accept the default width by
by pressing or or type in a new value. This capability allows the
farfield algorithm to be run essentially independently of the initial dilution model.
Input wastefield width: [default value]
This message appears if the PLUMES configuration string ARB
command has been toggled to 'R' (or reset). The user may accept the default width by
by pressing or or type in a new value. This capability allows the
farfield algorithm to be run essentially independently of the initial dilution model.
No farfield prediction, check input
No farfield simulation is attempted because the farfield diffusion coefficient, increment,
or maximum distance are not defined.
No farfield prediction when far vel = 0.
Gives the reason for no farfield simulation when using RSB; the far field velocity cannot
be equal to zero.
In addition to these messages, RSB always displays the following text:
FARFIELD CALCULATION (based on Brooks, 1960, see guide)
Indicates the farfield algorithm follows.
Farfield dispersion based on wastefield width of [width]
Indicates the initial width (an initial condition) used by the farfield algorithm.
-4/3 Power Law- -Const Eddy Diff-
Headers for the farfield concentration columns that follow. The 4/3 Power Law results
are appropriate for open water while the Const Eddy Diff results are appropriate for
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Appendix 4: Messages and interpretations
channels.
cone dilution cone dilution distance Time
Column headers followed by units. The peak-to-mean ratio established at the end of the
initial dilution region may be used to estimate corresponding average dilutions in the
farfield region.
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APPENDIX 5: UNIVERSAL DATA FILE FORMAT (Muellenhoff et al, 1985)
INTRODUCTION
The Universal Data File (UDF) was introduced by Muellenhoff et al. (1985) to serve as a
common data file for the five 1985 EPA plume models: UPLUME, UOUTPLM, UKHDEN,
UMERGE, and ULINE. ULINE and UPLUME are bundled with the PLUMES software,
although we believe they are completely superseded by the new models. UMERGE has also
been completely updated in the PLUMES UM model. UOUTPLM is largely obsolete.
Experience shows that UMERGE (and also UM) and UDKHDEN have similar capabilities and
give similar results, although UMERGE is found to be slightly more conservative than
UDKHDEN (Baumgartner et al., 1986).
THE UNIVERSAL DATA FILE
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 FORMAT(10A8)
IDENTIFICATION OF A DATA SET WITHIN THE UDF.
CARD 2 FORMAT(812)
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 FROM
THE DEFAULT VALUES
= DO NOT READ A CARD 5 (THUS CARD 5 MUST BE OMITTED).
IPI INPUT PRINTOUT CONTROL FOR UPLUME
10I " UOUTPLM
IDI " UDKHDEN (SEE NOTE 1)
IMI " UMERGE
ILI " ULINE
IPO=IPI OUTPUT PRINTOUT CONTROL FOR UPLUME
IOO=IOI " UOUTPLM
IDO=IDI " UDKHDEN (SEE NOTE 1)
IMO=IMI " UMERGE
ILO=ILI " 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(F10.0,110,3F10.0)
QT TOTAL EFFLUENT FLOW (CUBIC METERS PER SEC).
NP NUMBER OF PORTS (SEE NOTE 2).
PDIA PORT DIAMETER (M), EFFECTIVE DIAMETER IF KNOWN.
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Appendix 5: Universal Data File Format (Muellenhoff et al. 1985)
CARD 4
CARD 5
CARD 6
VANG VERTICAL ANGLE (DEC) OF PORT RELATIVE TO THE
HORIZONTAL (90 DEGREES FOR A VERTICAL PORT).
ULINE ASSUMES VANG=90 DEC.
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.
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
THAN 90 DEG BUT LESS THAN OR EQUAL TO 180 DEG, THE
PROGRAM SETS HANG EQUAL TO THE SUPPLEMENTARY ANGLE).
ULINE RANGE 0 - 180 DEG.
OPTIONAL (INCLUDE THIS CARD ONLY IF ICUTOP =1)
FORMAT(F5.0,215,312,6F5.0,215)
USED IN UMERGE
A
ITER
IFRQ
NAA
NAB
NAC
ASPIRATION COEFFICIENT
MAXIMUM NUMBER OF ITERATIONS
ITERATION PRINTOUT FREQUENCY
PRINT ARRAY AA IF =1, DO NOT IF =0
PRINT ARRAY AB IF =1, DO NOT IF =0
PRINT ARRAY AC IF =1, DO NOT IF =0
0 .
5000
150
0
0
0
BY DEFAULT
BY DEFAULT
BY DEFAULT
BY DEFAULT
BY DEFAULT
BY DEFAULT
(SEE LISTING OF PROGRAM UMERGE FOR CONTENTS OF ARRAYS
AA, AB, AC WHICH ARE MAINLY DEBUGGING AIDS.)
USED IN UPLUME
PS PRINTOUT
1 INTERVAL"
USED IN ULINE
RK RATIO OF SA/SM IN ROBERTS'
DH INTEGRATION STEP SIZE(M)
EXPERIMENTS
USED IN UOUTPLM
H INITIAL THICKNESS OF PLUME ELEMENT
E IMPINGEMENT ENTRAINMENT COEFFICIENT
A ASPIRATION ENTRAINMENT COEFFICIENT
ITERB NUMBER OF INTEGRATION STEPS ALLOWED
IR PRINTOUT INTERVAL
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 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 THEIR DEFAULT VALUES WILL BE USED.
ITER, IFRQ, ITERB AND IR NOT TO EXCEED FOUR DIGITS.
NO OPTIONS AVAILABLE FOR UDKHDEN.
FORMAT(110,2F10.0)
NPTS NUMBER OF DEPTHS WHERE AMBIENT TEMPERATURE, SALINITY, AND
HORIZONTAL CURRENT SPEED ARE KNOWN (NPTS MUST BE A LEAST
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Appendix 5: Universal Data File Format (Muellenhoff et al. 1985)
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 FORMAT(4F10.0)
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 SA( ) 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.
AMBIENT DENSITY (G/CM3) IF TA( )=0
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