i
CRWR Online Report 0605
Physical and Conceptual Modeling of Sedimentation Characteristics
in Stormwater Detention Basins
by
Masatsugu Takamatsu, Ph.D.
Michael Barrett, Ph.D.
Randall J. Charbeneau, Ph.D.
December 2006
Center for Research in Water Resources
The University of Texas at Austin
J.J. Pickle Research Campus
Austin, TX 787124497
This document is available online via the World Wide Web at
http://www.crwr.utexas.edu/online.shtml
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Acknowledgements
This research was funded by the Texas Department of Transportation (TxDOT) under
grant number 04611, “NonProprietary, Small Footprint Stormwater Quality Structures
for Use in Urban Areas”. The TxDOT Program Coordinator was David Stolpa and
TxDOT Project Director was David Zwernemann. The research project, led by Dr.
Harlow Landphair, was a joint project with Texas A&M University (TAMU). Full scale
experiments were conducted by the research team from TAMU. We appreciate the
research funding from TxDOT and the support from the TAMU research team.
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TABLE OF CONTENTS
List of Tables .......................................................................................................... v
List of Figures........................................................................................................ vi
Chapter 1 Introduction.................................................................................. 1
1.1 Background........................................................................................... 1
1.2 Motivation............................................................................................. 3
1.2.1 Need for Investigation of Patent and Proprietary Technology .... 3
1.2.2 Need for Design Development..................................................... 4
1.2.3 Need for Physical Models............................................................ 7
1.2.4 Need for a Conceptual Model...................................................... 8
1.2.5 Need for Prototype Scale Testing ................................................ 9
1.3 Research Objectives.............................................................................. 9
1.4 Dissertation Organization ................................................................... 10
Chapter 2 Literature Review....................................................................... 11
2.1 Characteristics of Stormwater............................................................. 11
2.2 Mathematical modeling of treatment systems .................................... 15
2.2.1 Modeling of particle settling velocity........................................ 15
2.2.2 Modeling of primary settling tank ............................................. 18
2.2.3 Modeling of gravitational treatment systems for stormwater runoff
........................................................................................... 20
2.3 Physical Model Scaling....................................................................... 22
Chapter 3 Conceptual Model and Application ........................................... 25
3.1 Conceptual Model............................................................................... 25
3.1.1 Model description ...................................................................... 25
3.1.2 Hydraulics of the basin .............................................................. 27
3.1.3 Critical Settling Velocity ........................................................... 29
3.1.4 Overall Removal Ratio of Particles ........................................... 31
3.2 Conceptual Model Application........................................................... 34
iv
3.2.1 First Application Example: Constant Inflow Case .................... 34
3.2.2 An Example with Triangular Hydrograph inflow...................... 46
Chapter 4 Physical Model........................................................................... 55
4.1 Physical Model Design ....................................................................... 55
4.2 Physical Model Description................................................................ 57
4.2.1 The first built sedimentation basin – an unsuccessful example. 57
4.2.2 The second sedimentation basin – a successful example........... 59
4.2.3 Suspended sediment particle...................................................... 65
4.3 Data Acquisition, Experiment Procedure, and Post Processing ......... 72
4.3.1 Data acquisition ......................................................................... 72
4.3.2 Experimental procedure............................................................. 74
4.3.3 Data Processing.......................................................................... 75
Chapter 5 Model Results, Analyses, and Uses ........................................... 80
5.1 Experimental results and analysis....................................................... 80
5.1.1 Experimental results................................................................... 80
5.1.2 Nondimensionalization of outflow SSC and its analysis........... 88
5.2 Conceptual Model and Analysis ....................................................... 107
5.2.1 Conceptual model results......................................................... 107
5.2.2 Scaling...................................................................................... 116
5.2.3 Analyses for Designing............................................................ 119
5.2.4 Nonideality............................................................................... 124
Chapter 6 Conclusions and future work ................................................... 135
6.1 Conclusions....................................................................................... 135
6.2 Future work....................................................................................... 140
Appendix A Conceptual Model Program in MATLAB .................................. 141
Appendix B Nomenclature ................................................................................. 146
References........................................................................................................... 148
v
List of Tables
TABLE 2.1. MAJOR CONSTITUENTS IN URBAN RUNOFF AND NURP
RECOMMENDATIONS FOR LOAD ESTIMATES (EPA 1983) 12
TABLE 3.1. INPUT PARAMETERS, THEIR UNITS, AND VALUES 35
TABLE 3.2. INPUT PARAMETERS, THEIR UNITS, AND VALUES 49
TABLE 3.3. INFLOW CONDITION AND CALCULATED REMOVAL RATIO 51
TABLE 3.4. INFLOW CONDITIONS AND RESULTING PARTICLE REMOVAL RATIOS
FOR FLAT HYDROGRAPHS (A’, B’, AND C’) AND PARTICLE REMOVAL RATIO OF
CORRESPONDING TRIANGULAR HYDROGRAPH (A, B, AND C) 53
TABLE 4.1. SCALING RATIO OF EACH PARAMETER 57
TABLE 5.1. SETUP CONDITIONS AND INFLOW HYDRAULIC CONDITIONS 81
TABLE 5.2. INFLOW EXPERIMENTAL CONDITIONS 83
TABLE 5.3. REMOVAL EFFICIENCY AND EMC OF EACH RUN 87
TABLE 5.4. CALCULATED AND MEASURED RATIO OF OUTFLOW SSC TO INFLOW
SSC FOR FILLING PERIOD, C
*
F
, AND REMOVAL EFFICIENCY 107
TABLE 5.5. GRADIENT AND CALCULATED INFLOW SSC DIFFERENCE BETWEEN
THE START AND THE END OF RUNOFF DURATION 113
TABLE 5.6. MEASURED AND CALCULATED REMOVAL EFFICIENCIES 114
TABLE 5.7. SCALING RATIO OF EACH VARIABLE 117
TABLE 5.8. DIMENSIONS AND INFLOW CONDITIONS OF PROTOTYPE TO HAVE
THE SAME PARTICLE REMOVAL EFFICIENCY OF RUN A IN MODEL SCALE 118
TABLE 5.9. PARTICLE REMOVAL RATIO AND CORRESPONDING MAXIMUM
WATER LEVEL FOR DIFFERENT FLOW RATE AND DIFFERENT ORIFICE SIZES.122
TABLE 5.10. EXPERIMENTAL CONDITIONS OF THREE DIFFERENT TRACER TESTS
126
TABLE 6.1. SUMMARY OF THE MODEL CHARACTERISTICS BETWEEN THE PLUG
FLOW BASE CONCEPTUAL MODEL AND THE CSTR BASE EMPIRICAL MODEL.139
vi
List of Figures
FIGURE 1.1. RESEARCH PROJECT OVERVIEW DIAGRAM 3
FIGURE 1.2. DESIGN SCHEMATIC FOR BOX CULVERT TYPE STORMWATER
SEDIMENTATION BASIN 4
FIGURE 2.1. COMPETITION OF RECOMMENDED DRAG CORRELATION (EQ. 2.16)
TO EXPERIMENTAL DATA (LAWLER 2003) 18
FIGURE 2.2. SCHEMATIC VIEW OF IDEAL HORIZONTAL FLOW REACTOR 19
FIGURE 3.1. SCHEMATIC OF RECTANGULAR STORMWATER DETENTION BASIN
26
FIGURE 3.2. TRAJECTORY OF WATER MOLECULES AND A PARTICLE WITH
CRITICAL SETTLING VELOCITY IN AN IDEAL HORIZONTAL FLOW REACTOR
(ABOVE) AND RECTANGULAR STORMWATER DETENTION BASIN (BELOW)
DURING FILLING PERIOD 27
FIGURE 3.3. LOCAL CONVECTIONAL VELOCITY AT STORM RUNOFF PERIOD 28
FIGURE 3.4. TRAJECTORY OF A PARTICLE WITH LOWER SETTLING VELOCITY
THAN CRITICAL SETTLING VELOCITY AND THE PARTICLE RATIO OF CAPTURED
AND ESCAPED 32
FIGURE 3.5. RELATIONSHIP BETWEEN PARTICLE DIAMETER AND SETTLING
VELOCITY 36
FIGURE 3.6. (A) PDF OF MASS BASE PARTICLE SIZE DISTRIBUTION, (B) CDF OF
MASS BASE PARTICLE SIZE DISTRIBUTION, (C) PDF OF THEORETICAL SETTLING
VELOCITY, AND (D) CDF OF THEORETICAL SETTLING VELOCITY 37
FIGURE 3.7. (A) CALCULATED WATER LEVEL CHANGE IN THE SEDIMENTATION
BASIN AND (B) INFLOW AND OUTFLOW RATE 39
FIGURE 3.8. THEORETICAL DETENTION TIME OF WATER COLUMNS WITH
RESPECT TO THEIR INFLOW TIME. 41
FIGURE 3.9. CRITICAL PARTICLE SIZE CHANGE WITH RESPECT TO INFLOW TIME
42
FIGURE 3.10. PARTICLE SIZE DISTRIBUTION OF INFLOW SUSPENDED SOLID AND
THE FRACTION REMOVED. 43
vii
FIGURE 3.11. REMOVAL RATIO OF PARTICLES AS A FUNCTION OF INFLOW TIME
45
FIGURE 3.12. TIME SERIES SSC CHANGE IN INFLOW AND OUTFLOW 46
FIGURE 3.13. SCS TRIANGULAR HYDROGRAPH 47
FIGURE 3.14. INFLOW AND OUTFLOW RATE 49
FIGURE 3.15. TIME SERIES OUTFLOW SSC 50
FIGURE 3.16. INFLOW HYDROGRAPHS OF THREE TESTED EXAMPLES 51
FIGURE 3.17. TIMESERIES OUTLET CONCENTRATIONS FOR THE THREE DIFFERENT
HYDROGRAPHS 52
FIGURE 3.18. TRIANGULAR AND FLAT INFLOW HYDROGRAPH 53
FIGURE 3.19. OUTFLOW SSC FOR TRIANGULAR AND FLAT HYDROGRAPH 54
FIGURE 4.1. THE FIRST SEDIMENTATION BASIN, BUILT IN 2003, CAUSED A LOT
OF PROBLEMS 58
FIGURE 4.2. WHOLE STRUCTURE OF THE PHYSICAL MODEL 60
FIGURE 4.3. PIPE NETWORK FOR THE PHYSICAL MODEL 61
FIGURE 4.4. INLET PIPE AND ENERGY DISSIPATOR 63
FIGURE 4.5. SEDIMENTATION BASIN WITH VIEW WINDOW 64
FIGURE 4.6. SETUPS AROUND OUTLET ORIFICE (LEFT: VIEW FROM TOP, RIGHT:
ORIFICE) 64
FIGURE 4.7. PARTICLE SIZE DISTRIBUTION OF SILCOSIL®49 (PROVIDED BY
US SILICA) 67
FIGURE 4.8. PARTICLE SIZE DISTRIBUTION OF SILCOSIL®49 (MANUFACTURE
PROVIDED), INFLOW SAMPLE MEASURED BY COULTER COUNTER, AND THE
BEST FIT LOGNORMAL DISTRIBUTION 68
FIGURE 4.9. PARTICLE DELIVERY SYSTEM FOR THE SECOND TRIAL 70
FIGURE 4.10. PARTICLE DELIVERY SYSTEM (LEFT) AND INSIDE OF THE MIXING
BUCKET 71
viii
FIGURE 4.11. BUBBLE FLOW METER (RIGHT) AND TUBING ATTACHED TO THE
BOTTOM (LEFT) 73
FIGURE 4.12. INLET SAMPLING NOZZLE (LEFT) AND OUTLET SAMPLING CASCADE
(RIGHT). 73
FIGURE 4.13. h (SQUARE ROOT OF WATER LEVEL) CHANGE WITH TIME 76
FIGURE 4.14. MEASURED AND CALCULATED WATER LEVEL CHANGE 77
FIGURE 4.15. CONCEPTUAL FIGURE OF MEASURED OUTFLOW SSC DATA
(DISCRETE) AND ESTIMATED OUTFLOW RATE (CONTINUOUS) 78
FIGURE 5.1. TIME SERIES INFLOW SSCS FOR ALL EIGHT RUNS 82
FIGURE 5.2. MEASURED INFLOW AND OUTFLOW SSC CHANGE 84
FIGURE 5.3. EXCEL SPREADSHEET TO CALCULATE REMOVAL EFFICIENCY 85
FIGURE 5.4. RELATIONSHIP BETWEEN REMOVAL EFFICIENCY AND THEORETICAL
OVERFLOW RATE 87
FIGURE 5.5. WATER LEVEL CHANGE WITH TIME IN NONDIMENSIONALIZED FORM
90
FIGURE 5.6. OUTFLOW SSC CHANGE WITH TIME IN NONDIMENSIONALIZED
FORM 91
FIGURE 5.7. RELATIONSHIP BETWEEN THEORETICAL OVERFLOW RATE AND
AVERAGE NONDIMENSIONALIZED SSC FOR FILLING PERIOD 92
FIGURE 5.8. EFFECTIVE PARTICLE SETTLING VELOCITY 94
FIGURE 5.9. RATE OF SSC CHANGE FOR EMPTYING PERIOD 96
FIGURE 5.10. RELATIONSHIP IN EQUATION (5.16) 98
FIGURE 5.11. MEASURED AND CALCULATED CHANGE OF
*
F
C/*C WITH RESPECT
TO NONDIMENSIONALIZED TIME 99
FIGURE 5.12. MEASURED AND EMPIRICALLY CALCULATED TIME SERIES
OUTFLOW SSC 101
FIGURE 5.13. RATIO OF MASS FLOWED OUT DURING FILLING AND EMPTYING AND
THEIR SUM WITH RESPECT TO EFFECTIVE ORIFICE AREA 106
ix
FIGURE 5.14. MEASURED AND CALCULATED WATER LEVEL CHANGE 109
FIGURE 5.15. MEASURED AND CALCULATED OUTFLOW SSC 111
FIGURE 5.16. RELATIONSHIP BETWEEN ERROR IN PEAK SSC AND INFLOW SSC
DIFFERENCE BETWEEN AT THE BEGINNING AND END OF THE RUNOFF 113
FIGURE 5.17. MEASURED AND CALCULATED (USING CONCEPTUAL MODEL)
REMOVAL EFFICIENCIES 115
FIGURE 5.18. TIME SERIES WATER LEVEL CHANGE (ABOVE) AND OUTFLOW SSC
CHANGE (BELOW) BETWEEN MODEL AND PROTOTYPE 118
FIGURE 5.19. REMOVAL RATIO CHANGE WITH DIFFERENT THEORETICAL
OVERFLOW RATES AND DIFFERENT RUNOFF VOLUMES 120
FIGURE 5.20. REMOVAL RATIO CHANGE WITH DIFFERENT EFFECTIVE ORIFICE
SIZES 122
FIGURE 5.21. REMOVAL RATIO CHANGE WITH DIFFERENT PARTICLE DENSITIES
124
FIGURE 5.22. CALIBRATION LINE BETWEEN CONDUCTIVITY AND SALT
CONCENTRATION 125
FIGURE 5.23. NONDIMENSIONALIZED TRACER CONCENTRATION AT INLET AND
OUTLET 127
FIGURE 5.24. MEASURED OUTFLOW TRACER CONCENTRATION WITH
CALCULATED RESULTS FROM PLUG FLOW AND CFSTR MODEL. 130
FIGURE 5.25. TYPICAL VELOCITY PROFILES WITH VELOCITY INTERFACES FOR THE
FILLING PERIOD (LEFT) AND THE EMPTYING PERIOD (RIGHT) 132
FIGURE 5.26. PARTICLE CONCENTRATION PROFILE IN THE SEDIMENTATION BASIN
IN THE PLUG FLOW MODEL AND THE REAL SITUATION DURING THE FILLING
PERIOD. 134
1
Chapter 1 Introduction
1.1 BACKGROUND
The importance of stormwater management has increased from controlling floods
in urban areas to include preventing the spread of polluted stormwater runoff to the
environment. According to Walesh (1987), “stormwater management is primarily
concerned with limiting future flood damages and environmental impacts due to
development, whereas flood control aims at reducing the extent of flooding that occurs
under current conditions.”
Polluted runoff from highways may cause serious water quality problems such as
excess loadings of nutrient and metals to rivers and lakes. Many of the pollutants that are
emitted by heavy traffic are accumulated on the roadway surface, where they attach to
soil particles, and are carried by storm water runoff into receiving water bodies. Thus,
there is a need for appropriate treatment of stormwater runoff. Many factors should be
considered in the treatment design process, such as the type, cost, size and detention time
of the treatment device, as well as the physical and chemical properties of stormwater
runoff.
Many treatment structures have been developed to improve the quality of
stormwater runoff. The most common treatment structures are passive processes that rely
on gravity and naturally occurring biological processes. Pollutant removal efficiencies are
different for each constituent depending on the specific technology used (TTI Research
Report 2000). It has been found that some types of pollutants (e.g., lead, zinc, oil and
grease) are well attached to particles and can be effectively removed by removing the
suspended solids in storm water runoff (Urbonas and Stahre 1993).
2
Stormwater has been cited in regulations as one of the major contributors of non
point source pollution. In response to the 1987 Amendments to the Clean Water Act
(CWA), the U.S. Environmental Protection Agency (EPA) developed the National
Pollutant Discharge Elimination System (NPDES) Stormwater Program in 1990 (US
EPA NPDES Website). The NPDES program regulates stormwater discharge from
industrial facilities, construction activities, and municipal separate storm sewer systems
(MS4s) to receiving water bodies. Many local governments intending to improve the
quality of their runoffimpacted streams are incorporating best management practices
(BMPs) into their stormwater management plans. EPA defines stormwater BMPs as
“methods that have been determined to be the most effective, practical means of
preventing or reducing pollution from nonpoint sources” (Bartosh 2004).
To utilize stormwater BMPs in a limited space, it is necessary that they be
installed underground. Several different proprietary stormwater BMPs are already being
marketed for use underground. While they are effective, these marketed BMPS are more
expensive than simple precast units such as pipe and box culverts. Texas Department of
Transportation (TxDOT) funded a research project that was launched in September 2003.
The primary objective of the project is
To develop simple effective stormwater quality treatment structures that are
composed of off the shelf materials including pipe and box culvert sections with
specialized inlet and outlet controls. These structures will be designed for low
head loss and will affect efficient, low maintenance and cost effective stormwater
quality treatment in limited rightofway conditions.
3
1.2 MOTIVATION
Several areas of research are needed to fully accomplish the project objective, as
shown in Figure 1.1, a project overview diagram. These various research needs and the
related approaches are discussed below.
Investigation of Patent and Proprietary Technology
(Completed in Bartosh (2004))
Project Start (Fall, 2003)
Develop a Design and Build a Physical Model
zInlet structure (Energy dissipator)
zSedimentation basin (Volume, LWH ratio)
zOutlet structure (Orifice size)
Prototype Scale
Testing
(at Texas A&M )
Project End (Fall, 2006)
Conceptual Model
zDevelopment of theory
zApplication
zStudy scaling issues
Physical Model
zMeasurements
•Outflow SSC
•Flow rate
zData analysis
•Particle removal ratio
Design storm (no overflow conditions)
zInflow rate
zRunoff duration
zInflow Suspended Sediment Concentration (SSC)
Verification of
Conceptual Model
Research Scope
Compare results
Figure 1.1. Research project overview diagram
1.2.1 Need for Investigation of Patent and Proprietary Technology
A survey of manufactured stormwater BMPs currently on the market was
necessary to determine how each BMP works and to avoid violating any patents that may
exist. For this purpose, Bartosh (2004) did an extensive survey on marketed small
4
footprint stormwater BMPs and established which parts of each product, if any, are
proprietary. Because of this research, the design process for new smallfootprint
stormwater BMPs, including inlet and outlet structures, became much easier. Also
according to the research, box type culverts and baffles are not allowed to be patented
because they are fundamental hydraulic structures (Bartosh, 2004). This fact is important
to consider within the context of the design process.
1.2.2 Need for Design Development
After the review of proprietary BMPs was completed, a design process for a
nonproprietary BMP was begun. There were three major components to be designed.
• Inlet structure
• Sedimentation basin
• Outlet structure
Figure 1.2 shows a schematic figure of box culvert type stormwater sedimentation basin.
Inlet structure
Sedimentation basin
Outlet structure
Energy dissipator
Outlet orifice
Overflow weir
Water level is changeable
Figure 1.2. Design schematic for box culvert type stormwater sedimentation basin
5
“Inlet structure” refers to the inlet pipe system and the energy dissipation system.
In a water treatment device, the inlet structure is important because a strong jet from the
inlet pipe can cause resuspension of previously deposited sediment near the inlet. Thus
the inclusion of an energy dissipator or a diffuser is an important design component so
that effective sedimentation is not adversely affected by a strong inflow jet. Also, the
inlet pipe system should be carefully designed such that the inlet pipe is large enough to
convey roadway runoff caused by design storm (e.g., 15 minute duration and 25 year
storm). Sometimes, excessive runoff is bypassed to the outlet in order to avoid flooding
so that the inlet pipe doesn’t need to account for all the runoff in case of large storms. The
design of the inlet structure must account for both water quality and hydraulic
considerations.
“Sedimentation basin” refers to a box type culvert section that is used for settling
and retaining sediments. Use of highway rightofway for subsurface placement of box
culvert type structures may be the most cost effective stormwater management strategy.
Some manufactured BMPs utilize centrifugal force to promote sedimentation. These
products are effective when there is limited space, but the development of such a system
takes more time and is more expensive than that of simple box culvert type.
Although very few municipalities have set requirements for the removal of total
suspended solids (TSS) from storm water, those that have requirements commonly call
for 80% removal of TSS in influent storm water. The Edwards Aquifer Recharge Zone in
Austin, Texas, for example, is a sensitive drainage area and state regulations call for the
removal of 80% TSS in influent storm water. Practically all storm water BMP
manufacturers claim their products provide 80% removal of TSS in storm water but do
not have proper documentation (lab and field studies) to back up these claims (Bartosh
6
2004). A box type culvert structure might be more cost effective than a centrifugal storm
water treatment system to achieve the required 80% removal.
While TSS (total suspended solids) is commonly used as the measure of
suspended sediment concentration, some authors claim that SSC (suspended sediment
concentration) should be used. The difference between TSS and SSC is in the
measurement methods. For TSS, a subsample is taken from the suspension with a pipette
and then is dispensed onto filter paper to gather the sediment. For SSC, the whole sample
is poured onto the filter paper to catch the sediment (Minton, 2002). The reason that the
SSC technique is considered by many to be the preferred method for runoff samples is
that it is difficult to get a representative sample into a pipette. The stormwater runoff
samples have particles with high settling velocities. These particles settle rapidly to the
bottom of the container, so it is difficult for them to be pulled up by the pipette
(Karamalegos, 2006). Therefore, SSC was used instead of TSS in this research.
The design shape of the sedimentation basin in this research is a rectangular box
culvert. Parameters for sedimentation basin design are volume and the ratios between
width, length, and height. The physical model for the sedimentation basin was designed
as an opentopped tank. Tank performance can be measured by physical experiments;
however, it is difficult to test different size and length ratios because a new tank is
required for each scenario. A mathematical model was used to help evaluate different
sizing scenarios as part of the design process.
“Outlet structure” refers to a simple overflow weir with an orifice placed at the
bottom and the outlet pipe system. Storm runoff, once stored in the tank, slowly exits the
system through the orifice; in situations when the volume of runoff exceeds the holding
capacity of the tank, water flows over the weir to the outlet pipe system to prevent
7
surcharge of the system and roadway flooding. Drainage time of stored stormwater
through the orifice depends on the initial depth of the water and the size of the orifice.
1.2.3 Need for Physical Models
Physical models are indispensable in testing pollutant removal ratio for a given
stormwater detention basin and allow for the variation of input variables such as flow rate
and initial SSCs of stormwater runoff. Also, performance results for each physical model
should be used to check if the model design meets required criteria such as the particle
removal ratio.
In the physical model testing, two conditions can be considered; no overflow
conditions and over flow conditions. “No overflow” describes the condition that runoff
volume is within the capacity of sedimentation basin and all runoff water is drained by
the outlet orifice. Outlet SSCs are measured with time, and then event mean
concentration (EMC) and particle removal ratio are estimated for each runoff event.
“Overflow” describes the condition in which runoff volume exceeds the capacity of the
sedimentation basin and excess water overflows to the other side of the overflow weir.
Once overflow starts, flow in the tank becomes more dynamic because outflow rate is
dramatically increased. In such dynamic flow conditions, flow and the sedimentation
pattern in the sedimentation basin are expected to be very different from the condition of
no overflow. However, the overflow case was not studied in this research.
According to a simple methodology for estimating solids removal fraction in
detention ponds (Urbonas and Stahre 1993), volume, residence time, and average depth
of a detention pond are used as the major parameters when flow conditions are ideal. In a
simple box type culvert sedimentation basin, these three parameters can be translated into
detention time, depth, and the area of detention basin. Detention time, determined by
8
outlet orifice size and water depth, can be controlled by using different orifice sizes. The
relationship between detention time and particle removal ratio was studied in evaluation
of the no overflow condition.
1.2.4 Need for a Conceptual Model
Smallscale modifications of the physical model, such as different orifice sizes or
different energy dissipaters, can be easily done. However, largescale modifications to the
physical model design are more difficult; testing alternate tank sizes may require a new
tank for each scenario. A mathematical model can be a useful tool in estimating tank
performance and can help evaluate different sizing scenarios in the designing process.
According to Ford and Hamilton (1996), a mathematical model can be
categorized either as empirical or conceptual. According to the definition, an empirical
model is formulated from observation of input and output without seeking to represent
explicitly the process of conversion. A conceptual model, by definition, is formulated
from consideration of physical, chemical, and biological processes acting on the system.
The modeling for stormwater treatment performance requires a large number of
variables such as inflow rate, runoff duration, inflow SSC, particle sizes, etc. Taking this
into account, formulation of a conceptual model is more feasible than an empirical model
since there are too many variables associated with empirical modeling. Therefore, the
strategy for this research includes creating a conceptual model and verifying it through
use of physical model results.
Numerical modeling using computational fluid dynamics (CFD) has been popular
in the stormwater area. However, such a numerical model was not developed for the same
reason that the empirical model was not pursued. The system cannot be at steady state
due to the natural variation of inflow rate and the variation of outflow rate governed by
9
orifice size and water level. Parameters such as fluid velocity are a function of time and
space, which requires enormous amount of data collection to calibrate velocity profile
over time and space.
The conceptual model for this research was not designed to include the overflow
conditions since an overflow event should be less frequent than no overflow conditions in
real sites. Also it was considered that flow during overflow is more dynamic and makes
the sedimentation phenomena more complicated. Thus the conceptual model is designed
only for no overflow conditions in this research.
1.2.5 Need for Prototype Scale Testing
An operational design prototype, which can be tested with actual storm runoff, is
the final component of this project. A prototype scale stormwater sedimentation basin
was installed at College Station, Texas and has been tested by Texas A&M University
project team members. Data have been collected from synthetic storms. Their
measurement results can be compared with the physical and conceptual modeling results
of this research using scaling techniques.
1.3 RESEARCH OBJECTIVES
The main scope of this research is physical and conceptual modeling of a
rectangular stormwater sedimentation basin and evaluation of performance tests of
candidate designs. Specific objectives to achieve this overall research objective are listed
below.
1. Build a rectangular sedimentation basin as a physical model, which can
reproduce and vary input conditions such as inflow rate and suspended
10
sediment concentration (SSC), and can be used to take samples at any
time from inlet and outlet to measure SSC.
2. Build a conceptual model, which can estimate time series outflow SSC,
event mean concentration (EMC), and particle removal ratio, and verify
the conceptual model using physical model results.
3. Test different lengths of the basin and outlet orifice sizes in the physical
model to evaluate how overflow rate and detention time affect the
particle removal ratio.
4. Study how geometric and hydraulic parameters should be scaled to have
identical particle removal efficiency in the physical model and
prototype using conceptual and physical model results.
1.4 DISSERTATION ORGANIZATION
This dissertation consists of six chapters. A literature review is presented in
Chapter 2. Chapter 3 describes the details of the conceptual model and the methodology
of the model application. Chapter 4 describes the physical model including how it was
designed and built and how measurements were made. Chapter 5 includes results and
analysis of both the conceptual and the physical model. A CSTR (Continuous Stirring
Tank Reactor) model to estimate time series outflow SSC and particle removal ratio using
empirically determined parameter is also shown in this chapter. In Chapter 6, conclusions
and future work are discussed.
11
Chapter 2 Literature Review
A literature review is presented to provide a background relating to the issues of
stormwater and its treatment by gravitational separation. This chapter includes 1)
characteristics of stormwater, 2) modeling of treatment systems, and 3) physical model
scaling.
2.1 CHARACTERISTICS OF STORMWATER
Pollutants in stormwater
Many different constituents are found in stormwater runoff. The EPA
(Environmental Protection Agency) adopted for their Nationwide Urban Runoff Program
(NURP) several constituents as standard pollutants characterizing urban runoff shown.
For these constituents, the NURP evaluated data collected between 1978 and 1983 from
81 sites in 22 cities and presented the recommendations for concentration estimates.
Table 2.1 shows these major constituents in urban runoff and the NURP
recommendations for load estimates (ASCE, 1992).
12
Table 2.1. Major constituents in urban runoff and NURP recommendations for load
estimates (EPA 1983)
Constituent Concentration estimates
TSS 180548 mg/L
BOD 1219 mg/L
COD 82178 mg/L
Total P 0.420.88 mg/L
Dissolved P 0.150.28 mg/L
TKN (Total Kjeldahl Nitrogen) 1.904.18 mg/L
Nitrite (NO
2

) and Nitrate (NO
3

) 0.862.2 mg/L
Total Cu 43118 μg/L
Total Pb 182443 μg/L
Total Zn 202633 μg/L
In Texas, the Lower Colorado River Authority (LCRA) and the Texas
Commission on Environmental Quality (TCEQ) added oil and grease as important
stormwater runoff constituents (TTI Research Report, 2000). These organics usually
contain polyaromatic hydrocarbons (PAHs) such as pyrene, phyenanthrene, and chrysene.
Some pollutants are attached to particles rather than being dissolved in the runoff.
For example, phosphorus is primarily sorbed on the soil surface or precipitated through
chemical reaction, forming calcium phosphate, aluminum oxidephosphate, iron oxide
phosphate, and ferric phosphate (Minton, 2002). Metals are also sorbed on the soil
surface, and the degree of sorption depends on metal species, ionic strength, pH, redox
potential of the soil, and so on. Most organics such as PAHs and petroleum hydrocarbons
are hydrophobic and sorbed to the soil. The sorption correlates strongly with the organic
content of the soil, f
oc
, and organiccarbon partition coefficient, K
oc
. The significance of
sorption processes implies that particle removal from stormwater could realize the
removal of other pollutants from stormwater runoff to some extent.
13
Particle size distribution
Particle size distribution (PSD) is also one of the most important characteristics of
stormwater runoff since the diameter and density of a particle determines particle settling
velocity, and surface area of a particle affects the capacity of sorption in soil in addition
to surface charge, and organic contents of a particle.
There are several methodologies to measure particle size distribution, which are
dry/wet sieving, electrical sensing zone (ESZ) instrument, and a laser diffraction particle
size analyzer. Dry/wet sieving method can measure the mass distribution directly, but it is
only useful for larger particles since fine sieves are easily clogged with fine particles. The
ESZ instrument determines the number distribution of particles by measuring the voltage
flux between two electrodes caused by the particles. The laser diffraction particle
analyzer determines relative volume distribution by measuring diffraction pattern and
intensity (Shimazu 2006). Each of these methodologies determines the size distribution
by measuring either number, mass, or volume. However, there is no clear protocol for
accurately determining particle size distribution (Karamalegos, 2006).
Also, particle size distribution of stormwater runoff is very different from place to
place or time to time even in a single storm event. Minton (2002) pointed out several
possible reasons for the particle size difference.
• Particle mass distribution may be affected by the runoff rate. In general,
larger material is resuspended with increasing flow rate.
• Sampling location affects results. Samples taken immediately at the edge of
pavement may find a greater fraction of large particles than those taken
within a drainage system where the large particles may have already settled.
Furthermore, very small particles aggregate while traveling through drainage
14
pipes (15 microns to 3040 microns) by precipitation with silica leachate
from concrete.
• Site conditions: In temperate climates with winter snow and ice, the
application of sand affects the particle size. Snow causes small particles to
aggregate.
In general, street sediment is larger than stormwater sediment and its particle size ranges
from 10 μm to 10 mm, while the particle size of stormwater sediment ranges from 1 μm
to 1 mm. Many studies have found that the majority of particles are less than 50 to 75 μm
(Minton, 2002).
Density of particles
Particle density affects particle settling velocity. Frequently, 2.65 g/cm
3
, which is
equivalent to the density of quartz and sand, is used to estimate particle settling velocity.
However, the density of smaller particles is sometimes much less than the density of
sand. This is because of organic matter, which is lighter than sand, tends to be absorbed
onto the smaller particles which have larger specific surface area per mass than to the
larger particles, and a small particle is sometimes wholly or intrinsically organic as
condensed organic material or fragmentary plant material (Lee, et al. 2005). Karamalegos
(2006) summarized previous measurements of density of suspended solid particles and
performed additional measurements of stormwater runoff from highway in Austin, TX
using an ESZ instrument. Karamalegos reported that the density, from thirteen papers and
reports, ranges from 1.1 to 2.86 g/cm
3
, snowmelt density was slightly higher than the
density of road sediment, and smaller particles had a lower density compared to bulk
samples. The measurements by Karamalegos shows that the density of particles smaller
than 75 μm ranged from 1.4 to 2.4 g/cm
3
with the mean of 1.8 g/cm
3
for four different
15
storm samples from a highway, and 1.4 to 2.8 g/cm
3
with the mean of 1.8 g/cm
3
for three
different storm samples from the inlet of an extended detention basin. These
measurement data show that density of suspended solid particles from stormwater runoff
are much lighter than the density of sand. Therefore, this should be taken into account
when removal efficiency of any detention basin is designed.
2.2 MATHEMATICAL MODELING OF TREATMENT SYSTEMS
2.2.1 Modeling of particle settling velocity
Particle settling velocity has been measured, calculated, and discussed in the
environmental engineering field for a long time as the settling velocity affects gravity
separation systems. Stokes (1851) was the first to derive the terminal settling velocity of a
spherical particle. Terminal settling velocity, v
s
, occurs when gravitational force, F
g
,
minus buoyancy force, F
b
, equals drag force, F
d
. This relationship can be extended using
the spherical assumptions:
2
sf
2
pDd
3
p
fpbg
dbg
vdC
8
1
FRHS
6
d
g)(FLHS
FFF
ρπ==
ρ−ρπ==
=−
− (2.1)
Where d
p
is the diameter of a particle, ρ
p
and ρ
f
are the density of a particle and a
fluid, and C
D
is the drag coefficient of a particle. By solving Equation (2.1), the drag
coefficient C
D
can be described as follows:
2
s
p
f
fp
D
v
d
g
3
4
C
ρ
ρ−ρ
=
(2.2)
16
For creeping flow conditions where inertial effects are negligible, Stokes derived the
following equation for drag force:
spd
vd3F πμ=
(2.3)
where μ is viscosity. C
D
can be simplified by substituting Eq. (2.3) for Eq. (2.1):
Re
24
dv
24
C
psf
D
=
ρ
μ
=
(2.4)
where the Reynolds Number, Re, is defined as
μ
ρ
=
psf
dv
Re (2.5)
The terminal settling velocity of a sphere particle for the creeping flow conditions can be
finally obtained by substituting drag coefficient equation, (2.4), for the force balance
equation (2.1) as follows:
2
pfps
d)(
18
g
v ρ−ρ
μ
=
(2.6)
When inertial effects cannot be neglected, the drag coefficient cannot be predicted
theoretically. Experimental data have been correlated to estimate C
D
and v
s
to develop
predictive tools. The drag coefficient has different values depending on whether the flow
surrounding the particle is laminar or turbulent (i.e., depending on the value of Re). For
smooth, spherical particles with Re < 10
4
, the drag coefficient can be approximated by
(Fair and Geyer, 1954):
34.0
Re
3
Re
24
C
D
++=
(2.7)
17
This equation also includes Eq. (2.4) for Re<0.3. Recently, more empirical equations
have been suggested to estimate the drag coefficient C
D
more precisely (Clift et al 1978,
Turton and Levenspiel 1986). For example, Khan and Richardson (1987) suggested
45.306.031.0
D
)Re36.0Re25.2(C +=
−
(2.8)
Haider and Levenspiel (1989) proposed
Re
95.6880
1
4251.0
)Re1806.01(
Re
24
C
6459.0
D
+
++=
(2.9)
Brown and Lawler (2003) suggested revising Eq. (2.9) since this equation might include
wall effect errors. They took the general form of (2.9) with parameters determined by a
local minimization of the sum of squared errors, Q, defined in Eq. (2.10)
∑
−=
n
1
2
cal,Dexp,D
)ClogC(logQ (2.10)
They found Eq. (2.11)
Re
8710
1
407.0
)Re150.01(
Re
24
C
681.0
D
+
++=
(2.11)
Figure 2.1 shows the good fit of experimental data and Eq. (2.11).
18
Figure 2.1. Competition of recommended drag correlation (Eq. 2.16) to experimental
data (Brown and Lawler 2003)
These results show that the drag coefficient, C
D,
could be estimated satisfactorily and
implies that terminal settling velocity also could be estimated by these revised empirical
equations.
2.2.2 Modeling of primary settling tank
An ideal horizontal flow reactor, or a Class I settling tank, is one of the oldest and
most fundamental conceptual models for modeling gravitational settling, developed by
Camp (1946). The following figure is a schematic of a rectangular, continuous flow
settling tank, with horizontal flow.
19
H
v
or
v
x
L
Q
Figure 2.2. Schematic view of ideal horizontal flow reactor
Overflow rate, v
or
, is defined by
LB
Q
areaSurface
rateFlow
v
or
⋅
==
(2.12)
where B is the width and L is the length of the reactor.
Camp (1946) developed the following equation for removal efficiency, R, expressed as a
function:
∫
+−=
0
f
0
or
s
0
df
v
v
f1R
(2.13)
where v
s
is Stokes settling velocity (2.6), f is the fraction of the particles with v
s
< v
or
, and
f
0
is the fraction with v
s
= v
or
.
The ideal horizontal flow reactor model has been modified both conceptually and
numerically. Swamee and Tyagi (1996) added a scour effect for the model and developed
modified removal efficiency. The sedimentation performance was also calculated with
advection and diffusion equation with settling velocity term. Takamatsu, et al. (1974)
estimated the diffusion coefficient and scouring coefficient experimentally and
analytically solved the system. Hydrodynamics of the sedimentation tank have been
20
analyzed numerically. Zhou and McCorquodale (1993) calculated the velocity field and
suspended solid transport numerically to investigate the effects of density stratification on
the hydrodynamics. Turbulence modeling, using kε model, has been applied by several
researchers (Adams and Rodi, 1990; and Zhou, et al., 1994). Sediment concentration and
bed depth in the sedimentation basin were modeled numerically by Jin and Guo (2000).
As shown above, research in the modeling of settling basins is extensive since
such basins are common to drinking water and waste water treatment. However, these
findings may not be directly applied to the modeling of stormwater sedimentation basin.
This is because stormwater detention basins are not steady state systems; inflow rate from
runoff always fluctuates and outflow rate is not always the same as inflow rate especially
when outflow is controlled by an orifice and the shape of the detention basin, which, in
most cases, is not simply rectangular. In the following section, how sedimentation has
been modeled in the stormwater detention basins is described.
2.2.3 Modeling of gravitational treatment systems for stormwater runoff
Modeling of particle removal efficiency in gravitational treatment systems for
stormwater runoff has been more empirical and statistical rather than conceptual or
numerical because the system is far from steady state during a storm and the runoff
characteristics (e.g., inflow rate, volume, and sediment concentration) are different from
storm to storm. To deal with these nonideal problems and their effects on long term
performance, a methodology was suggested by EPA in 1986 for the analysis of detention
basins for the control of urban runoff pollution (Urbonas and Stahre, 1993). The
methodology combines probability concept with theoretical removal efficiency modeling
concepts. The calculation method is different for dynamic conditions and quiescent
conditions and the calculation should be done for different particle size fractions. The
21
bases of the probability concept using the methodology was developed by Di Toro and
Small (1979). This methodology can be applied when there are non ideal situations such
as turbulent condition or flow short circuiting in the basin by applying nonideality
parameter. However water level change is not considered in this model.
For wetland, Su and Mitchell (2006) presented a removal efficiency model using
plug flow reactor (PFR) and CSTR (continuous stirred tank reactor) model to estimate the
characteristics of first flush effect reduction. In a single storm event, initial concentrations
of most pollutants including sediment are generally much higher than the concentration
during the latter part of the storm because the initial flush of water washes off the
accumulated contaminants in air and ground. This is called first flush phenomena. The
flush effect was modeled by the first order concentration reduction using the measured
inflow concentration decline. Their model results show that PFR has more significant
effects on first flush reduction than CSTR. However, the simulation results of constituent
concentrations were not validated with measured data in Su and Mitchell’s research. For
combined sewer overflow (CSO), removal of sewer solids by treatment facilities such as
storage sedimentation tank, high side weir overflow, and vortex overflow is modeled with
Hazen number, which is the ratio of particle settling velocity to the overflow rate.
Luyckx, et al. (2005) proposed a normalized particle efficiency and showed its strong
relationship with the Hazen number for several CSO treatment facilities. This research
provides insight in terms of how the Hazen number is effective on the removal efficiency
of stormwater treatment systems. Jensen (2005) modeled the CSO system using a 1D
advectiondiffusion equation and estimated the dispersion coefficient and depositional
velocity by comparison with laboratory data. However Jensen’s model deals with only
steady state conditions.
22
For the purpose at the present work, no existing model was found which could be
applied to this research. This is mainly because existing models do not incorporate water
level change, which affects the unsteady outflow rate and therefore the particle settling
too in the proposed stormwater BMP design.
2.3 PHYSICAL MODEL SCALING
Physical models are often employed in hydraulic studies. The fullscale object of
interest is called the “prototype” while the scaled down experimental object is called the
“model”. The length ratio of the prototype to the model is called the geometric length
ratio (L
R
). This length ratio should be kept constant throughout the model except for
distorted models that use a different length ratio in the vertical compared with horizontal.
This consistency is called geometric similarity. In the same manner, to make two flow
patterns similar, velocity ratio (v
r
) and force ratio (FC
r
) must be kept the same throughout
the model. These consistencies are called kinematic similarity and dynamic similarity,
respectively. To maintain the dynamic similarity, an appropriate dimensionless number
must be given the same value in the model as in the prototype.
In hydraulic models, the Froude number is used to maintain dynamic similarity if inertial
force and gravity force are significant, such as in open channel flow. The Froude number
is specified as follows:
gL
v
Fr
2
= (2.14)
Where v is flow velocity, g is gravitational acceleration, and L is characteristic length.
23
On the other hand, if inertial force and viscous forces are significant, such as in pipe flow
the Reynolds Number (Re) may be important (Henderson, 1966):
μ
ρ
=
vL
Re
(2.15)
However, a different scaling methodology is proposed for the particle settling system.
This scaling method is called Hazen number scaling. Hazen number is the ratio of particle
settling velocity, v
s
, to overflow velocity, v
or
.
or
v
s
v
a
H =
(2.16)
Some researchers show the validity of Hazen number scaling. For primary settling
basin, Thompson (1969) proved by using two geometrically similar models that the same
removal efficiency of a rectangular sedimentation tank can be reproduced using the same
powder if the overflow rates are the equal in model and prototype even though the
removal efficiencies in the model and prototype were around 10% worse than theoretical
estimation using ideal horizontal flow reactor theory.
For CSO structures, physical scaling problems have been much discussed
(Luyckx, et al., 1998, 2002, Fenner and Tyack, 1998). Luyckx, et al. (2002) did a
comprehensive research in removal efficiency of CSOs, including high side weir
overflows, hydrodynamic separator, and vortex separator, with many physical model
results. The research indicated that the same removal ratio could be achieved with
identical Hazen number and geometrically scaled model if Reynolds number is less than
200,000.
However, not much research has been done to deal with the scaling problem in the
area of stormwater detention basin. One of the reasons is that wet basin is usually not
24
only for sediment treatment but also for other pollutants removal by biological activity
(Urbonas and Stahre, 1993), which is no longer just a physical process. In this research,
the only target pollutant was particles, and the treatment method was purely gravitational
separation. Also, the rectangular detention basin had a simple shape, which was easy to
be scaled down geometrically and reproducible in a laboratory.
25
Chapter 3 Conceptual Model and Application
The removal efficiency of particles with a known settling velocity in a primary
settling basin can be simply calculated by ideal horizontal flow reactor theory. A
conceptual model to calculate particle removal efficiency was developed in the same
manner of the ideal horizontal flow reactor theory by employing similar assumptions
used in the theory. This chapter covers 1) how the conceptual model was developed to
calculate removal efficiency of particles and 2) how the model can be applied in
example problems such as constant inflow runoff rate and a triangular hydrograph inflow.
3.1 CONCEPTUAL MODEL
3.1.1 Model description
This section describes how a rectangular detention basin was modeled and how
the modeling process was similar and/or different from the ideal horizontal flow reactor
theory. Figure 3.1 shows the schematic of the detention basin. Stormwater runoff from a
roadway surface flows into the basin through inlet pipes and drains out of the basin
through the outlet orifice. Water level increases when the inflow rate is larger than the
outflow rate and decreases when the opposite occurs. This flow process is different from
a primary settling tank because inflow and outflow rates are not always constant and they
are not equal and therefore water level varies depending on the inflow rate and the
volume of water in the basin.
26
Water level is changeable
Inlet Zone
Outlet Zone
Orifice
Figure 3.1. Schematic of rectangular stormwater detention basin
Next, assumptions used for modeling are explained here. The water level was
assumed to be zero when a runoff starts. Inflow was assumed to be well distributed over
the submerged depth of the basin. Also, flow in the basin was assumed to be plug flow
and resuspension effects were assumed to be negligible as in the ideal horizontal flow
reactor theory. Outflow rate was controlled by an orifice set at the bottom of the end wall.
Particles that did not reach the outlet zone would be removed. Water was overflowed in
the case that runoff volume was larger than the basin capacity. However, the overflow
condition was not studied in this research.
The critical settling velocity is the velocity where all the particles with higher
settling velocities will settle out in a reactor. In the ideal horizontal flow reactor, critical
settling velocity is simply the overflow velocity, which is a constant flow rate divided by
the surface area. However, in this rectangular detention basin, calculating critical settling
velocity was not as simple because of unsteady inflow and outflow rates and variable
water level. Figure 3.2 shows the schematic of trajectories of water molecules and a
particle which has a critical settling velocity in an ideal horizontal flow reactor and the
rectangular detention basin. Trajectories of a water molecule and a particle are curved
while the trajectory of a particle in an ideal horizontal flow reactor is straight. Here,
27
methodologies to calculate the curved particle trajectory and critical settling velocity for a
rectangular detention basin are developed within the stated model assumptions.
Water molecule trajectory
SS particle trajectory with critical settling velocity
Figure 3.2. Trajectory of water molecules and a particle with critical settling velocity in
an ideal horizontal flow reactor (above) and rectangular stormwater
detention basin (below) during filling period
3.1.2 Hydraulics of the basin
Water level keeps changing from the start of runoff to the end of drainage which
makes the system unsteady. Water level change can be determined as follows from mass
balance if both the inflow rate, Q
in
(t), and outflow rate, Q
out
(t), are known. h(t) is the
water level as a function of time, B is width, L is the length of the basin, and t represents
time.
())t(
out
Q)t(
in
Q
LB
1
dt
)t(dh
−
⋅
=
(3.1)
28
It was assumed that there is no horizontal or vertical mixing in the sedimentation basin.
Numerical simulation would be required if mixing is significant. This assumption implies
that longitudinal velocity is uniform over the vertical cross section and vertical velocity is
uniform over the horizontal cross section.
Mass balance was considered in a control volume at the right hand side of the
detention basin including the outlet orifice as shown in Figure 3.3. The local flow rate at
x and t, Q(x,t), is equal to the sum of the outflow rate, Q
out
(t), and upflow rate (or
downflow rate when the tank is emptying) within the control volume as shown in
Equation 3.2.
)t(
out
Q
dt
)t(dh
B)xL()t,x(Q +⋅⋅−=
(3.2)
x
Q(x,t)
H(t)
dH
L
V
x
(x,t)
Q
out
(t)
dt
)t(dH
B)xL( ⋅−
Figure 3.3. Local convectional velocity at storm runoff period
29
The corresponding local horizontal flow velocity at x and t, u(x,t), is Q(x,t) divided by the
submerged cross sectional area at t as shown in Equation (3.3).
)t(hB
)t,x(Q
)t,x(u
⋅
=
(3.3)
Vertical flow velocity at the water surface is the velocity of water level change,
dt
)t(dh
,
and 0 at bottom. Vertical velocity in between, v(z,t), is linearly distributed over depth;
)t(h
z
dt
)t(dh
dt
dz
)t,z(v ⋅==
(3.4)
3.1.3 Critical Settling Velocity
As mentioned in the model description section, the particle path is not straight as
seen in the ideal horizontal flow reactor, but is curved. Therefore, trajectories of particles
should be calculated first in order to calculate critical settling velocity. Trajectories, or
pathlines, of particles were calculated from the velocity field. A pathline is a line which is
traced out in time by a given fluid particle as it flows, while a streamline is an
instantaneous line whose tangents are everywhere parallel to the velocity vector (Currie,
1974). A pathline is calculated using the following equation.
)t,z(v
dt
dz
)t,x(u
dt
dx
=
=
(3.5)
The benefit of using the pathline concept in this study is the position of a particle can be
traced with time. In this section, both water molecules and particle pathlines are
described. Pathlines for a water molecule can be derived in (3.6) from local flow rate
(3.2), local flow velocity (3.3) and pathline equation (3.5) as follows;
)t(hB
)t(
out
Q
dt
dh
)t(h
)
w
xL(
dt
w
dx
⋅
+⋅
−
=
(3.6)
30
where x
w
means the position of the traced water molecules. Equation (3.6) was
analytically solved with the initial condition that the longitudinal position of a water
molecule at time t
in
, x
w
(t
in
; t
in
), is 0. x
w
(t; t
in
) means the longitudinal position at time t of a
traced water molecule that flowed into the tank at time t
in
. Equation (3.7) is the solution
of (3.6).
∫
⋅
=
t
in
t
dt)t(
in
Q
)t(hB
1
)
in
t;t(
w
x
(3.7)
Vertical position of water molecules can be determined by the integration of Equation
(3.4) from t
in
to t and h(t
in
) to h(t) as follows depending on initial height, z(t
in
; t
in
);
)
in
t;
in
t(
w
z
)
in
t(H
)t(h
)
in
t;t(
w
z =
(3.8)
The combination of (3.7) and (3.8) shows the position of water molecules.
Next, the pathline of a particle is derived. The velocity vector of a particle can be
determined as follows using the velocity vector of water and settling velocity of the
particle, v
s
. Settling velocity was assumed to be independent of the ambient flow field.
s
v)
in
t;t(v)
in
t;t(
p
v
)
in
t;t(u)
in
t;t(
p
u
−=
=
(3.9)
where u
p
(t;t
in
) is the x component of particle velocity and v
p
(t;t
in
) is the z component of
particle velocity. Longitudinal position of a suspended solid particle at t with given t
in
,
x
p
(t;t
in
), is the same as the position of water particle, x
w
(t;t
in
) since longitudinal velocity
of water molecules and particles are assumed to be the same;
∫
⋅
=
t
in
t
dt)t(
in
Q
)t(hB
1
)
in
t;t(
p
x
(3.10)
Substitution of the vertical flow velocity equation (3.4) and (3.9) for the pathline equation
(3.5) yields vertical velocity vector equation of a particle;
s
v
dt
)t(dh
)t(H
)
in
t;t(
p
z
dt
)
in
t;t(
p
dz
−⋅=
(3.11)
31
Equation (3.11) was analytically solved with the initial position of the particle at
t=t
in
., which is z
p
(t;t
in
)=z
p
(t
in
;t
in
).
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
∫
+−=
)
in
t(h
)
in
t;
in
t(
p
z
t
in
t
dt
)t(h
1
s
v)t(H)
in
t;t(
p
z
(3.12)
Finally, detention time of the particle is calculated here. Since the position of the
suspended solid particle, (x
p
(t;t
in
), z
p
(t;t
in
)), is known, time to reach the end of the tank,
t
out
, for the particle can be calculated by substituting x
p
(t;t
in
)=L in (3.10) and solving for t.
Detention time of a particle can be simply calculated from t
out
 t
in
. As shown in Figure
3.2, a particle with critical settling velocity, which enters the basin at the very top of the
water column, will settle to the bottom on the right hand side when reaching the outlet.
Therefore, the focus was on a particle released from the water surface at the inlet. Initial
vertical position of such a particle is the same as the water level at t
in
, or simply h(t
in
).
Therefore, vertical position of the particle is shown in equation (3.13) by substituting
z
p
(t
in
;t
in
)=h(t
in
) in (3.12), the pathline equation of a particle.
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
∫
+⋅−= 1dt
t
in
t
)t(h
1
s
v)t(h)
in
t;t(
top
,
p
z
(3.13)
Then, the minimum particle which meets 0)
in
t;
out
t(
top
,
p
z > establishes the critical
settling velocity, )
in
t(
c,s
v as shown in equation (3.14).
∫
=
out
t
t
dt
)t(h
1
1
)
in
t(
c,s
v
(3.14)
3.1.4 Overall Removal Ratio of Particles
Some particles settle out, but some don’t settle if the settling velocity of a particle
is smaller than critical velocity. Figure 3.4 shows the trajectories of a particle with lower
32
settling velocities. The figure shows a particle released from the water surface that didn’t
settle out, but the other particle, released from the middle of the water level, just settles
out at the outlet orifice. This exact particle (solid line in Figure 3.4) has the initial height
of )t;t(1z
ininp
at inlet, which is the maximum initial height for particles to settle out.
This implies that particles with higher initial position than this particle will all flow out
and those with lower initial position will all settle out. The ratio of settling is exactly
same as the height ratio,
)
in
t(h
)
in
t;
in
t(
p
z
, at inlet since it is assumed that particles are
uniformly distributed at inlet.
Ratio settled
Ratio escaped
Water molecule trajectory at surface
SS particle trajectory from surface
SS particle trajectory which runs into right bottom
h(t
in
)
)t;t(1z
ininp
Outlet orifice
Figure 3.4. Trajectory of a particle with lower settling velocity than critical settling
velocity and the particle ratio of captured and escaped
The ratio of settling can be calculated as described below. First, solve the pathline
equation for a particle which exactly runs into the bottom orifice. Mathematically, put
33
(t,z
p
)=(t
out
,0) into the particle pathline equation (3.12) and solve for the term
)
in
t(h
)
in
t;
in
t(
p
z
.
The settled mass ratio, which is the settled number ratio with the same size particles, can
be simplified as the ratio of settling velocity to the critical settling velocity.
)
in
t(
c,s
v
s
v
out
t
in
t
dt
)t(h
1
s
v
)
in
t(h
)
in
t;
in
t(
p
z
=⋅=
∫
(3.15)
The settled mass ratio, F(v
s
,t
in
), also called reactor settling potential function (Lawler,
2006), is shown in equation (3.16), and this equation suggests that all particles with
settling velocity larger than critical settling velocity will settle out and the fraction
)t(v
v
inc,s
s
will settle out for particles with settling velocity smaller than critical settling
velocity.
⎪
⎩
⎪
⎨
⎧
<
>
=
)t(vvfor
)t(v
v
)t(vvfor1
)t,v(F
inc,ss
inc,s
s
inc,ss
ins
(3.16)
Assume the settling velocity of inflow particles has its own probability density function,
E(v
s
). Then, mass removal ratio can be calculated as a function of inflow time.
∫∫
∫
⋅+⋅=
⋅=
∞
∞
)tin(c,s
)tin(c,s
v
0
ss
inc,s
s
v
ss
0
sssinout
dv)v(E
)t(v
v
dv)v(E1
dv)v(E)v(F)t(R
(3.17)
Overall mass removal ratio of suspended solids can be calculated by the total mass of
particles flowing out divided by total mass of particles flowing in the basin as follows:
∫
∫
⋅
⋅⋅
=
Ts
0
ininin
Ts
0
ininoutinin
dt)t(C)t(Q
dt)t(R)t(C)t(Q
R
(3.18)
where T
s
represents duration of storm runoff.
34
Time series outflow SSC can be estimated by equation (3.19).
( ))t(R1)t(C)t;t(C
inoutininout
−⋅=
(3.19)
3.2 CONCEPTUAL MODEL APPLICATION
Two example problems and their solutions using the developed conceptual model
are shown in this section to illustrate how the conceptual model could be applied. The
first is a simple example which has a constant inflow rate and SSC so that it is
reproducible using the physical model. The second example employs a triangular
hydrograph as inflow, which makes the inflow condition more realistic than the first
example.
3.2.1 First Application Example: Constant Inflow Case
Assume that there is constant inflow of Q
in
=0.53 L/s for the duration of T
s
=40
min and the inflow SSC is C
in
=202 mg/L. Outflow is governed by a small orifice
installed at the bottom of the end wall. The effective area of the orifice is
2
pde
cm43.0ACA =⋅= , the multiplication of orifice coefficient, C
d
, and the area of
orifice, A
p
. Assume the particle size distribution of inflow suspended solid can be
properly described by lognormal distribution and the mean of ln(d
p
), λ
p
, is 2.286 (dp
(μm)) and the standard deviation of ln(d
p
), ζ
p
, is 0.908 (d
p
(μm)) where d
p
is the diameter
of a particle. Several articles (Campbell, 1985; Buchan, 1989) suggested that soil particle
size distribution can be well described by lognormal distribution. Table 3.1 summarizes
the input parameter values used in the example.
35
Table 3.1. Input parameters, their units, and values
Parameters Unit Value
Ae
cm
2
0.43
L m 6.96
B m 0.62
Q
in
L/s 0.53
T
s
min 40
C
in
mg/L 202
λ
p
0.908
ζ
p
2.286
ρ
p g/cm
3
2.65
ρ
w g/cm
3
1
μ g/cms 0.01
Inflow suspended solid particles
Stokes law, shown in Equation (2.6), was applied to calculate particle settling
velocity, v
s
(m/hr), from the diameter of a particle, d
p
(μm).
()
()
()
2
p
3
2
p
7
wp
2
6
p
wp
s
d10*24.3
d10*6.3
18
g
hr
s
100
3600
m
m
10*d
18
g
v
⋅=
⋅⋅
μ
ρ−ρ⋅
=
⎥
⎦
⎤
⎢
⎣
⎡
⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎥
⎦
⎤
⎢
⎣
⎡
μ
⋅
μ
ρ−ρ⋅
=
−
−
−
(3.20)
For the parameter values in this equation, gravitational acceleration, g, is 981cm/s
2
,
density of particles, ρ
p
, is 2.65 g/cm
3
, density of water, ρ
w
, is 1.00 g/cm
3
, and
viscosity of water is 10
2
g/cms, which are shown in Table 3.1. Particle Reynolds
number, defined in the equation (2.5), for the particle with the diameter of 50μ m is
approximately 0.1 for the silica particle. This is within the range of particle Reynolds
36
number where the Stokes law can be applied. Figure 3.5 shows the relationship between
diameter of the silica particles and their settling velocity using the Stokes law.
Figure 3.5. Relationship between particle diameter and settling velocity
Equation form in (3.20) implies suspensions settling velocities will have a
lognormal distribution (v
s
=LN(λ
v,
ζ
v
) ) if particle size assumes to have a lognormal
distribution (d
p
=LN(λ
p,
ζ
p
)) with the random variable transformation. The validity of
the assumption will be shown in the section 4.2.3. The relationship of lognormal mean
and lognormal standard deviation of particle diameter and settling velocity is as follows
using the relationship (3.20).
37
( )
pv
3
pv
2
10*24.3ln2
ζ⋅=ζ
+λ⋅=λ
−
(3.21)
Then, the mean of lognormal settling velocity, λ
v
, is 1.186 (v
s
(m/hr)) and the standard
deviation of lognormal settling velocity, ζ
v
, is 1.816 (v
s
(m/hr)) from the calculation of
equation (3.21). Figure 3.6 shows Particle Size Distribution (PSD) and settling velocity
distribution of inflow runoff in the forms of a Probability Density Function (PDF) and
Cumulative Density Function (CDF).
Figure 3.6. (A) PDF of mass base Particle Size Distribution, (B) CDF of mass base
particle size distribution, (C) PDF of theoretical settling velocity, and (D)
CDF of theoretical settling velocity
38
The PDF of the particle size distribution graph shows the majority of the particle sizes are
less than 20 μm, and the PDF of the particle settling velocity graph shows the majority of
the particle settling velocities are less than 1 m/hr. That can be confirmed in the graphs of
CDFs with actual numbers. The CDF of particle size distribution graph shows that about
50% of particles on mass base are smaller than 10 μm which correspond to the settling
velocity smaller than 0.32 m/hr, and 80% of particles are smaller than 20 μm which
correspond to the settling velocity smaller than 1.3 m/hr. The CDF of the settling velocity
distribution graph shows that more than 70% of particles have settling velocities less than
1m/hr.
Model application
The example problem was solved by writing a code in MATLAB. How the
MATLAB code was built is shown in this section. The code is provided in Appendix A.
1) Water level and flow rate calculation
Mass balance equation (3.1) was rewritten by substituting an orifice equation for
outflow rate.
( ))t(gh2
e
A)t(
in
Q
LB
1
dt
)t(dh
−
⋅
=
(3.22)
This nonlinear differential equation was solved numerically. Equation (3.22) was
discretized as Equation (3.23) and water level was solved explicitly by using previous
time step water level values with an initial condition of water level (H
0
=0).
⎟
⎠
⎞
⎜
⎝
⎛
+
⎟
⎠
⎞
⎜
⎝
⎛
−
⋅
=
+
−
⋅
=
−
+
n
out
Q
n
in
Q
LB
dt
n1n
n
out
Q
n
in
Q
LB
1
dt
n
h
1n
h
hh
(3.23)
Where
39
nanyforgh2AQ
n
dt
T
for0Q
dt
T
n0forQQ
n
e
n
out
s
n
in
s
in
n
in
=
<=
≤<=
Calculated water level change, given inflow rate, and calculated outflow rate are shown
in the Figure 3.7. As the graph shows, it takes about 6 hours to drain all the water after
runoff stops and the drainage time is about as 9 times long as the 40 minute runoff period.
0.00
0.05
0.10
0.15
0.20
0.25
0.30
01234567
W
a
t
e
r
le
v
e
l
(
m
)
0.00
0.10
0.20
0.30
0.40
0.50
0.60
01234567
Time (hr)
Fl
o
w
r
a
t
e
(
L
/
s
)
Outflow
Inflow
(A)
(B)
Figure 3.7. (A) Calculated water level change in the sedimentation basin and (B) Inflow
and outflow rate
40
2) Critical settling velocity calculation
Think about an imaginary water column which is traveling from inlet to outlet
without exchanging water with surrounding water because no mixing is assumed. Each
imaginary water column is released from the inlet every time step, dt, from the beginning
to the end of the runoff. The water column travels through the sedimentation basin and
eventually reaches the outlet. Time to reach outlet, t
out
n
, corresponding to each inflow
time, t
in
n
, was calculated by substituting x
p
(t
out
;t
in
)=L in Equation (3.10).
)
n
out
t(hB
in
Q
n
in
t
n
out
t
n,out
t
n,in
t
dt
in
Q
)
n
out
t(hB
1
L
⋅
⋅−
=
⋅
=
⎟
⎠
⎞
⎜
⎝
⎛
∫
(3.24)
End of the travel time, t
out
n
, of a water column which started traveling at t
in
n
, can
be calculated iteratively from the equation (3.24). Figure 3.8 shows the calculated
detention time, t
out
n
t
in
n
of water columns released at t
in
n
. In the graph, traveling time, or
detention time, of the first quarter (0 to 10 minutes) is less than an hour while detention
time of the last quarter (3040 minutes) is around 4 hours. Therefore, initial part of a
runoff, sometimes it is the dirtiest, won’t stay in the basin longer compared to the latter
part of the runoff.
41
0
50
100
150
200
250
300
350
010203040
Inflow time (min)
T
h
e
o
r
e
t
i
c
a
l
de
t
e
nt
i
o
n t
i
m
e
(
m
i
n
)
Figure 3.8. Theoretical detention time of water columns with respect to their inflow
time.
Critical settling velocity, v
p,c
(t
in
), was calculated from equation (3.14). The
integral, in the denominator of equation (3.14), was calculated using the MATLAB
function “trapz”, which conducts a trapezoidal numerical integration. Figure 3.9 shows
the critical particle size change with respect to the inflow time, t
in
. This implies that
particles larger than 8 μm would all settle throughout the runoffdrainage process.
42
0
2
4
6
8
10
0 10203040
Inflow time (min)
C
r
i
t
i
c
a
l
p
a
rt
i
c
l
e
si
ze (
μ
m)
Figure 3.9. Critical particle size change with respect to inflow time
3) Overall particle removal ratio calculation
As shown in Equation (3.17), PDF of settling velocity, E(v
s
), and potential
settling function, F(v
s
), should be known to calculate the overall particle removal ratio.
PDF of settling velocity, E(v
s
), shown in Figure 3.6 (b) can be described as follows using
the calculated mean and standard deviation value of natural logarithm of particle settling
velocity, ln(v
s
):
2
v
vs
)vln(
2
1
vs
s
e
2v
1
)v(E
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
ζ
λ−−
πζ⋅
= (3.25)
Settling potential function, F(v
s
), can be calculated from the calculated critical particle
velocity. For this application case, Stokes velocity, equation (3.20), can be put into (3.16)
and can be further transformed as follows:
43
⎪
⎪
⎩
⎪
⎪
⎨
⎧
<
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
>
=
c,ss
2
c,p
p
c,s
s
c,ss
s
vvfor
d
d
v
v
vvfor1
)v(F
(3.26)
This settling potential function varies depending on inflow time. This function can be
visually understood by Figure 3.10. This graph shows the particle size distribution of
inflow and the fraction removed in the form of Probability Density Function (PDF). The
group of particles shown in the graph flowed into the detention basin at t
in
=5 min and
flowed out at t
out
=40 min. Calculated critical particle settling velocity, v
s,c
, is 0.195 m/hr
and the corresponding critical particle size, d
p,c
, is 7.8 μm. The area surrounded by the
solid and dotted lines corresponds to the fraction flowing out, and the area under the
dotted line is the fraction settled out.
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 102030405060
Diameter (micron)
PD
F
Original size distribution
Removed fraction
Fraction in outflow
Figure 3.10. Particle size distribution of inflow suspended solid and the fraction
removed.
44
Given the particle settling velocity distribution, E(v
s
) in Equation (3.25), and
calculated reactor potential settling function, (3.26), these can be plugged into (3.17) to
calculate the removal ratio of particles flowing into the tank at t
in
. The first term, called
R
1
(t
in
), of the right hand side of (3.17) is the ratio of particles larger than critical particle
size and the second term, R
2
(t
in
), is the ratio of captured particles smaller than critical
particle size. Each term was transformed as follows and calculated separately using the
numerical integration function “trapz” in MATLAB. Figure 3.11 shows the calculation
results of R
1
(t
in
) and R
2
(t
in
). The graph shows the R
1
(t
in
) is much greater than the R
2
(t
in
),
which is surrounded by the R
1
(t
in
) lines and R
1
(t
in
)+ R
2
(t
in
) lines. This implies most of
inflow particles will be removed because the particle size is larger than the critical
particle size.
∫
∫
∫
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
ζ
λ−−
∞
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
ζ
λ−−
∞
πζ⋅
−=
πζ⋅
=
⋅=
)tin(c,vs
0
s
)vln(
2
1
vs
)tin(c,vs
s
)vln(
2
1
vs
)tin(c,vs
ssin1
dve
2v
1
1
dve
2v
1
dv)v(E1)t(R
2
v
vs
2
v
vs
∫
∫
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
ζ
λ−−
πζ
=
⋅=
)tin(c,vs
0
s
)vln(
2
1
vinc,s
s
)tin(c,vs
0
s
inc,s
s
in2
dve
2
1
)t(v
1
dv)v(E
)t(v
v
)t(R
2
v
vs
(3.27)
45
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
10203040
Inflow time (min)
Re
mo
v
a
l
r
a
t
i
o
R1(tin)
R1(tin)+R2(tin)
Figure 3.11. Removal ratio of Particles as a function of inflow time
Finally, total removal ratio of suspended solid and time series outlet SSC were
calculated by substituting the calculated (3.17) for (3.18) and (3.19). Figure 3.12 shows
the calculated time series outflow SSC. This graph shows the outflow SSC gradually
increased until approximately 50 min, which is 10 minutes after runoff stopped. Then, the
concentration dropped steadily with increasing time until water was completely drained.
The calculated total particle removal ratio was 0.875.
46
0
50
100
150
200
250
0 30 60 90 120 150 180 210 240 270 300 330
Time (min)
SSC
(
m
g
/
L
) Outflow
Inflow
Figure 3.12. Time series SSC change in inflow and outflow
Time series outflow SSC from the outlet may not be necessary to be estimated for
stormwater management purpose since the total particle removal ratio is more important
in the general case. However this model can determine the particle size distribution
change in the outflow. This implies that this study could be extended for estimating
removal ratio of other important pollutants in stormwater such as nutrients, heavy metals,
and organic materials in particulate form since hydrophobic pollutants are likely to attach
to the surface of suspended solids. Adsorption is usually highly dependent on the surface
area of sediment, which can be estimated from the particle size distribution.
3.2.2 An Example with Triangular Hydrograph inflow
The conceptual model was applied to deal with the SCS triangular hydrograph as
an inflow rate. The SCS triangular hydrograph is a synthetic unit hydrograph, which
relates direct runoff to a unit depth of excess rainfall. The benefit of applying a synthetic
hydrograph, such as this triangular hydrograph, is that it is more realistic than a constant
47
runoff (inflow) rate since a hydrograph tends to have a skewed bell shape rather than a
rectangular shape. A drawback, on the other hand, is these bell shaped hydrographs are
very difficult to reproduce experimentally. The triangular hydrograph can be determined
by specifying the peak flow rate, Q
p
, and peak time, T
p
, as shown in Figure 3.13.
T
p
8/3T
p
Q
Q
p
Figure 3.13. SCS triangular hydrograph
Water level and flow rates calculation
Water level in the detention basin and corresponding outflow rate were calculated
based on the mass balance equation (3.28).
()
())t(gh2
e
A)t(
in
Q
LB
1
)t(
out
Q)t(
in
Q
LB
1
dt
)t(dh
−
⋅
=
−
⋅
=
(3.28)
48
Water level was solved numerically since mass balance is a nonlinear differential
equation. Equation (3.28) was discretized as follows. Water level was solved explicitly
using the previous time step water level value.
⎟
⎠
⎞
⎜
⎝
⎛
+
⎟
⎠
⎞
⎜
⎝
⎛
−
⋅
=
−
⋅
=
−
+
+ n
gh2
e
A)
n
t(
in
Q
LB
dt
n
gh2
e
A)
n
t(
in
Q
LB
1
dt
n
h
1n
h
n1n
hh
(3.29)
where
tT
3
8
for0)t(Q
T
3
8
tTforQ
T
t
5
3
5
8
)t(Q
Tt0forQ
T
t
)t(Q
pin
ppp
p
in
pp
p
in
<=
<<
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−=
<<=
An example problem
The MATLAB program was modified to have an unsteady state inflow and run
with a triangular hydrograph. As an example, the inflow condition was set with Q
p
=1 L/s
and T
p
=30 min at the model scale. Inflow and outflow rate are snown in Figure 3.14 and
the calculated time series SSC is shown in Figure 3.15. The calculated removal ratio was
84.1%. The outflow peak occured at 70 min and the outlet SSC peak was at 77 min.
49
Table 3.2. Input parameters, their units, and values
Parameters Unit Value
Ae
cm
2
0.43
L m 6.96
B m 0.62
Q
p
L/s 1
T
p
min 30
C
in
mg/L 202
λ
p
0.908
ζ
p
2.286
ρ
p g/cm
3
2.65
ρ
w g/cm
3
1
μ g/cms 0.01
Figure 3.14. Inflow and outflow rate
50
0
10
20
30
40
50
60
70
0 60 120 180 240 300 360 420 480 540 600
Time (min)
SS
C
(
m
g
/
L
)
Concentration peak at 77 (min)
Figure 3.15. Time series outflow SSC
Application and comparison
A) Three different triangular hydrographs
The conceptual model was run in the model scale with three different triangular
hydrographs (Figure 3.16) and the same runoff volume of 2400 L to see how removal
ratio will be different for different runoff intensity. Table 3.2 shows the inflow conditions
and resulting particle removal ratio. This table shows that the removal ratio for the
slowest runoff, C, is approximately 5% higher than the most intense runoff, A. Figure
3.17 shows outflow SSC change for run A, B, and C. This graph shows that the peak SSC
in a short and intense storm is higher than that in a slower storm.
51
Table 3.3. Inflow condition and calculated removal ratio
Run Peak time (min) Duration (min) Peak flow rate (L/s) Runoff Volume (L) Removal ratio
A 20 53 1.5 2400 0.828
B 40 107 0.75 2400 0.852
C 80 213 0.375 2400 0.879
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0 50 100 150 200 250
Time (min)
In
flo
w
r
a
te
(L
/s)
Run A
Run B
Run C
Figure 3.16. Inflow hydrographs of three tested examples
52
0
10
20
30
40
50
60
70
80
0 100 200 300 400 500 600
Time (min)
SSC
(
m
g
/
L
)
Run A
Run B
Run C
Figure 3.17. Timeseries outlet concentrations for the three different hydrographs
B) Triangular hydrograph vs flat hydrograph
It is of interest to see how performance would be different between a triangular
hydrograph and a flat hydrograph. Three runoff values with flat hydrographs, each of
which corresponds to the triangular hydrograph show in Table 3.2, were tested to
compare their removal ratios. These three flat hydrographs were made to have the same
durations and runoff volumes from their corresponding triangular hydrographs. Table 3.4
shows the inflow conditions and Figure 3.18 shows the tested inflow hydrographs of both
the triangular and flat hydrographs.
53
Table 3.4. Inflow conditions and resulting particle removal ratios for flat hydrographs
(A’, B’, and C’) and particle removal ratio of corresponding triangular
hydrograph (A, B, and C)
Run Duration (min) Inflow rate (L/s) Runoff Volume (L) Removal ratio
Removal ratio of corresponding
triangular hydrograph (A, B, and C)
A' 53 0.750 2400 0.837 0.828
B' 107 0.375 2400 0.863 0.852
C' 213 0.188 2400 0.893 0.879
0.0
0.5
1.0
1.5
2.0
0 50 100 150 200 250
Time (min)
Inflow
rate (L/s)
Run A
Run B
Run C
Run A'
Run B'
Run C'
Figure 3.18. Triangular and flat inflow hydrograph
Table 3.4 also shows the results of the removal ratio calculation for both
triangular and flat hydrographs. The removal ratio results from corresponding
hydrographs are very close (around 1% difference), while removal ratio of triangular
hydrograph is always a little less than that of flat hydrograph. However, the peak SSCs
54
are different. Figure 3.19 shows the calculated SSC change of runoff B for both triangular
and flat hydrograph. The graphs shows that the peak SSC of the triangular hydrograph is
much higher than that of the flat hydrograph. These simulation results imply that particle
removal efficiency results from physical models with constant inflow would not be very
different from the results with the corresponding actual stormwater runoffs such as bell
shape inflows.
0
10
20
30
40
50
60
0 100 200 300 400 500 600
Time (min)
SS
C
(
m
g
/
L
)
Triangular hydrograph
Flat hydrograph
Figure 3.19. Outflow SSC for triangular and flat hydrograph
55
Chapter 4 Physical Model
A physical model was built in a smaller scale of the prototype rectangular
detention basin to conduct performance tests of particle removal efficiency. By using a
physical model, it is possible to see the validity of the developed conceptual model and
see what parameters will strongly affect particle removal ratio. To cover these subjects,
this chapter includes 1) physical model design; 2) detailed description about physical
model; and 3) data acquisition and processing.
4.1 PHYSICAL MODEL DESIGN
The physical model was designed based on the design of a prototype scale
rectangular stormwater detention basin. The prototype is assumed to have a 1 acre (4047
m
2
) watershed. The basin volume is equivalent to the water quality volume, where the
water quality volume is typically taken as the first 25.4 mm (=1 inch) of runoff from the
impervious areas of the watershed (Young, et al., 1996). Based on the criteria, the
dimensions were determined to be 34.8 m long, 3.1 m wide, and 1.7 m high to be able to
fit in the right of way area along a highway. Drainage time was assumed to be 24 hours.
A physical model, geometrically similar to the prototype, was built with length
scaling ratio, L
R
, of 1:5. However, issues were raised when kinematic similarity was set
up as to whether Froude number scaling, Hazen number scaling, or other scaling methods
should be considered. If Froude number scaling was chosen, which is used when
gravitational force and inertial force are dominant such as in open channel flow problems
(Hwang and Houghtalen, 1996), both the time ratio and velocity ratios are
R
L .
However, this doesn’t take sedimentation processes into account. If Hazen number
56
scaling is chosen, the ratio of settling velocity to overflow velocity should be identical for
both model and prototype and overflow velocity should be identical if the same particles
are used for both. Then the time ratio is set to
R
L since the length ratio is L
R
and velocity
ratio is 1.
Here, Hazen value scaling was chosen for kinematic similarity since it was
experimentally proved by Thompson (1969) that Hazen value scaling would work for an
ideal horizontal tank, which is similar to the rectangular stormwater detention basin in the
aspect of shape and function. Accordingly, the drainage time was determined to be 4.8
hours, or 24 hours divided by time ratio of 5. Then, time is nondimensionalized by runoff
duration, T
s
, as follows;
s
T
t
*t =
(4.1)
This is necessary because the water level and inflow and outflow rates are a function of
time and they should be scaled on the same time frame. With the nondimensionalized
time, the scaling ratio of water level,
M
P
*)t(h
*)t(h
, should be equal to L
R
because water level
has the length scale. The ratio of flow rate,
M
P
*)t(Q
*)t(Q
, should be the multiplication of
velocity ratio and area ratio which is equal to L
R
2
. Orifice area ratio was calculated to be
L
R
3/2
to have a properly scaled outflow rate at the orifice as follows:
2/3
R
M
P
M
out
P
out
M
M
out
P
P
out
M
e
P
e
L
*)t(H
*)t(H
*)t(Q
*)t(Q
*)t(H*g*2
*)t(Q
*)t(H*g*2
*)t(Q
A
A
==
=
(4.2)
57
This assumption on scaling is verified in chapter 5 using the conceptual model. Table 4.1
has the summary of scaling ratio of each parameter. The physical model dimensions were
calculated based on design dimensions of the prototype and scaling down using this table.
Table 4.1. Scaling ratio of each parameter
Variable Description Ratio
L Length of the detention basin L
R
B Width of the detention basin L
R
h Water level of the detention basin L
R
Ts Storm duration L
R
v Velocity 1
Q
in
and Q
out
(t*) Flow rate L
R
2
A
e
Effective Area of orifice L
R
3/2
4.2 PHYSICAL MODEL DESCRIPTION
Two similar (L=20 ft) sedimentation basins were built at Center for Research in
Water Resources (CRWR). The first, built in fall 2003, did not provide consistency of
experimental conditions while the second, built in summer 2004, worked better since it
was modified to overcome drawbacks of the first. How the first sedimentation basin was
built and why it did not work well are explained below, followed by the details of the
successful second basin.
4.2.1 The first built sedimentation basin – an unsuccessful example
The first sedimentation basin, shown in Figure 4.1, was built with several 3/4 inch
plywood sheets and was set on several wooden supports.
58
Figure 4.1. The first sedimentation basin, built in 2003, caused a lot of problems
(TopLeft: Sedimentation basin set on wooden supports (view from inlet to
outlet), TopRight: mixing box, 3inch pipe connecting mixing box and
sedimentation basin, and BottomLeft: mixing box)
Water, taken from the outside reservoir through a pump, was conveyed to a
wooden mixing box by flowing on a roadway surface. Then the water was mixed in the
box, shown in the bottom left of Figure 4.1, with an introduced mixture of silica particles.
Silica particles were introduced to the mixing box with a peristaltic pump with a flow rate
controller. The water with silica particles was conveyed to the sedimentation basin
through 10 ft long, 3inch diameter pipe. Then the water was drained to the return
59
channel which brought water back to the reservoir. This basin was not successful, and the
data taken from the structure were not reliable enough to be used for several reasons:
• Mixing of water and particles in a rectangular mixing box by a propeller type mixer
was neither efficient nor constant and resulted in too much sedimentation in the
mixing box. Therefore, the box couldn’t achieve a constant inflow SSC.
• The 3inch diameter pipe connecting the mixing box to the basin for approximately
10ft was too long and too wide. This caused sedimentation and resuspension to occur
in the pipe, which made experimental conditions more uncontrolled.
• Water proofing which was done by silicone caulking couldn’t hold water well;
leakage from the basin was significant, especially from the seams of plywood and the
connection between the pipe and the basin.
4.2.2 The second sedimentation basin – a successful example
The second basin was built after the careful consideration of solutions for the
problems encountered in the first sedimentation basin. This section covers details of the
sedimentation basin including construction, water proofing, pipe networking,
characteristics of structures, and particle mixing and the delivery system.
• Construction and water proofing
The sedimentation basin was made of 3/4 inch plywood. Prior to construction,
nine cutout boards, three each for both sides and bottom, were painted with a polycrylic
waterproofing medium, followed by primer paint and a tinted waterbased latex paint.
The nine boards were carefully connected to each other with screws and put on cinder
blocks. Pieces of wood boards were attached to support connections between
longitudinally connected boards, and 1 by 1 inch boards surround several cross sections
of the basin to firmly support the hydraulic pressure (Figure 4.2). The seams were then
sealed with silicone caulking. To complete the water proofing of the model basin, resin
60
was applied in a multiplecoat thickness throughout the whole inside surface of the model
basin. This resulted in excellent water proofing.
Figure 4.2. Whole structure of the physical model
• Pipe network
Inflow to the model basin was taken either from the indoor reservoir via a pump
system or from tap water. Figure 4.3 shows the schematic of the pipe network used for
the physical model. Tap water was preferred to pump water because the reservoir water
contains a few mg/L of SSC, and this might affect the particle size distribution of inflow
particles. Therefore, the pump system was used only when tap water couldn’t achieve the
required flow rate. Water, taken either from pump or tap water, was circulated back to the
indoor reservoir when the water was clean and was drained out when the water contained
a certain SSC. Once the pump was switched on, the flow rate was regulated by valve V
B
,
while valve V
C
is open for bypass and valve V
D
is closed, blocking inflow to the model
61
basin. V
A
was always widely open when water was taken from the indoor reservoir
during an experimental run to prevent pump impairment because the required inflow to
the model basin is very small relative to the pump capacity and too little flow through a
large pump can easily impair the pump.
The flow rate was roughly measured at the outfall, following valve V
C
, by
measuring the time needed to fill a bucket of known volume. When multiple
measurements showed the flow rate to be steady, this rate was noted, then V
C
was closed
and V
D
was opened. Flow was lead to bypass through V
E
by closing V
F
until the silica
slurry was prepared in the mixing tank, delivered via the peristaltic pump to the pipe
system and stabilized. Finally, runoff with a certain SSC was lead to the model basin.
VA
VB
VC
VD
Model basin
Drain to outside
Pump
Reservoir
P
VE
VF
Tap water
Inflow measurment
Pipe for stabilization
Mixing bucket
Figure 4.3. Pipe network for the physical model
62
• Inlet structure
A 1inch pipe was attached near the center at the bottom of the end piece of the
sedimentation basin as the inflow opening. A 3inch pipe was used in the first basin
because a 3inch pipe is approximately 1/5 diameter of an 18inch pipe, which is the
minimum diameter of stormwater pipes. However, it caused particle accumulation in the
pipe due to small flow velocity in the pipe. Therefore, 1inch pipe was used to pursue
experimental consistency and prevent the influence of previous runs.
A small wooden block with a 9 cm by 9 cm square face and streamline tail was
attached to the bottom, 13 cm behind the inlet opening as an energy dissipator. The
purpose of the energy dissipator is to disperse the strong momentum of the inflow jet and
prevent resuspension of sediment around the inlet area. The position and shape of the
energy dissipator determine the effect it will have on the flow and sedimentation pattern
around it. These effects were not studied intensively in this research because resuspention
effect was out of our modeling focus and the existing energy dissipator did effectively
disperse the strong momentum of inflow jet to minimize the resuspension effect. Figure
4.4 shows the picture around the inlet including the wooden block used as an energy
dissipator.
63
Figure 4.4. Inlet pipe and energy dissipator
• Sedimentation basin
The sedimentation basin has the length of 6.96 m and width of 0.62 m. The
bottom of the basin was painted dark blue and the sides were painted gray. At the middle
of the basin, a Plexiglas window was attached to view sedimentation pattern or flow
pattern using dye. Particles precipitated from previous runs were not removed as is the
case for real detention basin. Therefore, each experimental result has some extent of
influences such as resuspension from previous runs. Figure 4.5 shows the sedimentation
basin with Plexiglas window. After several measurements were taken at the original
length basin, the basin was shortened to have 2/3 length of the original length by sliding
the inlet wall to the 1/3 of the original sedimentation basin.
64
Figure 4.5. Sedimentation basin with view window
• Outlet structure
At the far end of the sedimentation basin, plywood and a small section of metal
sheeting made up a composite 0.37 m high overflow weir and outlet orifice (Figure 4.6).
The effective area of the orifice, A
e
, was calculated as follows to take 4.8 hours to drain
completely from the very top of the basin.
Figure 4.6. Setups around outlet orifice (Left: view from top, Right: orifice)
65
Mass balance for the emptying period can be expressed using the orifice equation,
dtg2
BL
A
h
dh
gh2A
dt
dh
BL
e
e
−=
−=
(4.3)
This can be integrated as
[] []
()
s
e
max
t
Ts
e
h
h
Tt
BL
A
2
g
hh
tg2
BL
A
h2
max
−−=
⋅−=
(4.4)
Put h
max
=0.37 m, h= 0 m, and tT
s
=4.8 hr, then A
e
is calculated as 0.69 cm
2
.
To have this effective orifice area, the diameter of orifice, D
o
, was calculated as 1.1 cm
assuming the orifice coefficient, C
d
, is 0.8.
cm1.1
C
A4
D
4
D
CACA
d
e
o
2
o
dode
=
⋅π
=
π
⋅=⋅=
(4.5)
Then the circular orifice with a diameter of 1.1 cm was cut out from metal sheet by metal
scissors.
4.2.3 Suspended sediment particle
Synthetic silica was used as the suspended sediment for the physical model
instead of using real suspended sediment from the natural environment because the
particle size distribution of this commercial synthetic sediment is known. This
characteristic made experiments reproducible and made the verification process of
conceptual modeling easier. In this section, how the particle size distribution of inflow
particles was determined and how particles were introduced to the system are discussed.
66
• Particle size distribution determination
SILCOSIL®49, a product of US Silica Company, was used for this experiment.
According to the information that the manufacturer provides, density of the product is
2.65 g/cm
3
, which is very close to the density of quartz, and the particle size distribution
was also given. Figure 4.7 shows the provided particle size distribution of the product. As
shown in this figure, almost all the particles of SILCOSIL®49 were less than 50 μ m. It
seems SILCOSIL®49 is too small to be a representative of suspended solid particles in
stormwater since stormwater particle size ranges up to 500 to 1,000 μ m (Minton, 2003).
Andral et al (1999) reported that the particles with a diameter larger than 100 μm settle
out of the suspension easily; however, the particles with a diameter less than 100 μm
remain in suspension. Thus, the investigators concluded that particles with a diameter less
than 50 μm must be studied in order to effectively treat runoff. That is one reason that
using SILCOSIL®49 for experiments are reasonable. The other reason is the removal
efficiency of the sedimentation basin. Removal efficiency of SILCOSIL®49 was
around 85% and there were small fluctuations with different experimental conditions.
This implies removal efficiency would be higher than 85% with bigger particles and this
would make comparison of experimental data more difficult.
67
Figure 4.7. Particle size distribution of SILCOSIL®49 (provided by US Silica)
Although the PSD of SILCOSIL®49 is given by the manufacturer, particle size
of an inflow sample was also measured. The inflow sample was taken from a nozzle right
before the sedimentation basin. The measurement was done using a Coulter Counter
operated by an experienced graduate student. Figure 4.8 shows the mass base particle size
distribution of SILCOSIL®49 read from the manufacturer data and the PSD of an
inflow sample measured by the Coulter Counter in the form of a cumulative density
function. A gap is observed between the two, and the gap is approximately 20 % for
particles smaller than 15 μm. This gap comes from mixing and the sediment delivery
process, which will be explained in the next section. Some or most of the bigger particles
might settle in the mixing tank, or they may not be pumped by the peristaltic pump.
Therefore, the measured PSD by the coulter counter was used as the PSD of the inflow
sample.
68
The next step is fitting the PSD using a lognormal distribution. This fitting
process was done using the solver function in Excel to find the best fit parameters, which
are the mean and standard deviation of the logarithm of particle diameter, ln(d
p
), in order
to minimize the summation of square errors of percentage for each particle size range.
The calculated mean of ln(d
p
), λ, is 2.286 and standard deviation of ln(d
p
), ζ , is 0.908.
The equivalent mean particle size is m9.14)2/exp(
2
μ=ζ+λ . These values were used
for the PSD of inflow in the conceptual model. Figure 4.8 also shows the fitted lognormal
curve. The graph shows the error is very small for particles less than 25 μm and the error
is less than 10% for the entire particle size range. As shown in the conceptual model,
bigger particle would all settle out in the basin and the interest is more on smaller
particles. Therefore, this lognormal fit, better for smaller particles, is useful for this aspect
also.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
11010
dp (micron)
CDF
Manufacture's data
Coulter counter data
Lognormal fit
Figure 4.8. Particle size distribution of SILCOSIL®49 (manufacture provided), inflow
sample measured by coulter counter, and the best fit lognormal distribution
69
• Sediment delivery system
Three mixing systems were tried in this research. The first is the mixing box tried
in the first built basin, which resulted in failure. The second and the third are explained
and compared here. Figure 4.9 shows the second trial system. This system consists of a
4L mixing box, a variable flow peristaltic pump (VWR Vera 4.0600 mL/min), and
tubing. This system worked simply. First, a mixture of particles was prepared in the 4L
mixing box and mixed by a propeller. Then, the mixture was sucked from bottom of the
box by a peristaltic pump with variable flow rate and pushed into the inflow pipe through
the tubing. Unfortunately, this system couldn’t achieve a constant inflow SSC due to the
following reasons:
• The flow rate of the peristaltic pump was significantly affected by the head drop of
the mixing tank.
• The propeller type stirrer created stratification in the mixing box and it prevented
uniform mixing.
• Since the volume of the mixing box was too small for a run, a mixture was added to
the box 2 or 3 times during a run which made the consistency even worse.
70
Figure 4.9. Particle delivery system for the second trial
Fortunately, several ideas for mixing were given by professors and a successful
system was built. The final design of the sediment delivery system and the mixing box
are shown in Figure 4.10. The particle concentrate was prepared in a 22 L bucket and
mixed by circulation created by a submersible pump (Rule 1800) placed at the bottom of
the mixing bucket. The submersible pump drew water from the bottom of the bucket and
pushed water out through 1 inch diameter opening at the rate of 1.8 L/s. The opening
angle was set almost parallel to the bottom to enhance scouring and resuspention of silica
particles. A large stirrer was also tried for the mixing purpose with the same 22 L bucket,
but a constant SSC couldn’t be achieved at inlet.
Mixed particle concentrate was delivered using a peristaltic pump (Manostat
Vera, 13400 mL/s), the flow rate of which can be kept constant. The flow rate of the
pump was controlled by a dial, so the relationship between dial indicates and the flow
71
rates was calibrated beforehand to control inlet SSC. The mixed sediment was introduced
with the constant flow rate to the pipe network at right before the inlet opening of the
model basin.
Figure 4.10. Particle delivery system (Left) and inside of the mixing bucket
Next, the experimental process for sediment delivery is shown below:
1. Determine the duration and flow rate of a model runoff
2. Determine the inflow SSC, approximately 200 mg/L is used.
3. Calculate the SSC in slurry, C
slurry
, and volume of slurry in the mixing basin,
V
slurry
, to be delivered, using the following mass balance;
( )
()
stubeslurry
tube
intubepipe
slurry
slurrytubeintubepipe
TQV
Q
CQQ
C
CQCQQ
⋅≥⇒
⋅+
=⇒
⋅=⋅+
(4.6)
where Q(L/min) is flow rate, C(mg/L) is concentration, and Ts(min) is runoff duration.
Even though the particle delivery system was modified, the inlet SSC was still
variable and difficult to control. Therefore a uniformity coefficient parameter was
introduced to determine whether inflow SSC was kept constant enough. Uniformity
coefficient of inflow SSC was determined by using the following equation.
72
100*
C*N
CC
1(%)CU
in
N
1i
ini,in
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎝
⎛
−
−=
∑
=
(4.7)
where N = total number of samples,
i,in
C represents inflow SSC of i
th
sample, and
in
C
represents average inflow SSC among total samples. A criterion was set at a uniformity
of 80%. Then eight runs with the inflow SSC uniformity coefficient larger than the
criteria were selected.
4.3 DATA ACQUISITION, EXPERIMENT PROCEDURE, AND POST PROCESSING
Measurable parameters in the physical model are water level of the sedimentation
basin and SSC. Inflow and outflow rate, event mean concentration (EMC), and removal
ratio are the parameters to be calculated. How values of these parameters were acquired
and calculated and how a set of experiments was conducted are explained here.
4.3.1 Data acquisition
A) Water level
Water level in the model basin was measured by an automatic bubbler flow meter
(ISCO 3230). A flexible polyurethane tube is fixed to the model basin bottom and then
connects to the bubble meter, which sends a slow air stream through the tube allowing the
release of air bubbles from the bottom of the model basin. Figure 4.11 shows the pictures
of the bubble flow meter and tubing attached to the bottom of the basin adjacent to the
orifice. The bubble meter continuously measures the hydrostatic pressure due to the depth
of water above the bubbling end of the tube. The meter automatically transfers the
pressure into a water height reading. The measurement interval was set for 1 minute for
this research.
73
Figure 4.11. Bubble flow meter (right) and tubing attached to the bottom (Left)
B) Suspended sediment concentration (SSC)
Water samples were taken from the inlet and outlet periodically. Inlet samples
were taken through a nozzle attached before the sedimentation basin and outlet samples
were taken directly from a cascade installed at the bottom of the outlet area. Inflow
samples were taken around 510 seconds after opening the nozzle to prevent taking
particles accumulated in the nozzle. Occasionally, samples were taken from the
lengthwise midpoint of the sedimentation basin when necessary. Turkey basters are used
to take samples instead of actual pipets since this is much quicker in taking a sample of
approximately 200 (mL). Figure 4.12 shows both the inlet sampling nozzle and outlet
sampling cascade.
Figure 4.12. Inlet sampling nozzle (Left) and outlet sampling cascade (Right).
74
Samples were stored in 250 mL polyethylene bottles. Each sample was filtered
through a 1.5 μm of particle retention size glass fiber filter (Glass Microfiber Filters
47mm, 934AH, Whatman) within a few hours after samples are taken. Before use in the
measurements, stored filters were washed by reagentgrade water (milliQ), dried in an
oven at 105 °C for 1 hour, stored in a desiccator for at least half an hour, and weighed
just before the filtration process. All filters with solid residues were again dried by an
oven at 105 °C for 1 hour, kept in the desiccator at least half hour, and weighed. At the
time of filtration, sample volume is also measured using a 250 mL graduated cylinder.
Then suspended solid concentration (SSC) can be estimated as
mL,volumesample
1000*)BA(
)L/mg(SSC
−
=
(4.8)
where A is the weight of the dry filter and dried residue in mg, and B is the weight of the
dried clean filter in mg.
4.3.2 Experimental procedure
Here is the experimental procedure that took place at each run.
1. Target inflow rate, runoff duration, and inflow SSC were determined prior to each run.
Inflow was bypassed to drainage and inflow rate was roughly measured using a
stopwatch and bucket of known volume.
2. Detemine SSC in the mixing bucket, C
slerry
, and the speed of peristaltic pump, Q
slerry
.
3. Inflow runoff with a given SSC was introduced to the sedimentation basin and
maintained for a predetermined duration.
4. Samples were taken periodically from the inlet and outlet by using 250 mL
polyethylene bottles. Inflow samples were taken periodically so that at least 4 or 5
samples can be taken. Outflow samples were taken periodically (5 to 10 minutes
interval) from the time runoff started until 10 to 20 minutes after the runoff stopped.
After that, intervals of taking samples became longer, up to 30 minutes, since the SSC
change was small.
75
5. Water level in the basin was recorded every minute for the entire runoff and drain
process using a bubble flow meter (ISCO 3230). Acquired data were used to calculate
inflow and outflow rates.
6. SSC of each sample was measured by the filtration method after the entire process
was finished.
4.3.3 Data Processing
Variable water level and resulting outflow change made the system unsteady even
though inflow rate and inflow SSC were kept constant. However, the hydraulics part
concerning the water level and both inflow and outflow rates are fairly easy to estimate,
which is a big advantage in consideration of this unsteady system.
(A) Inflow and outflow rate estimation
Effective orifice area
Effective orifice area was measured using time series water level data for the
emptying period. This calculation was done using water level change. Equation (4.9)
shows that water level change is only a function of time for emptying period and h is
a linear function of time. Figure 4.13 shows the h change with time for two different
orifices which were actually used in experiments. Effective orifice area can be calculated
from the gradient of the graph of h with respect to time.
76
()
()
()
()
4
2
2
12
122
e
12
12
e
s2
e
max2
s1
e
max1
10
2/)s/m(g
)m(BL
min)/s(60*(min)t(min)t
)m(h)m(h
cmA
,unitswithor
2/g
BL
tt
hh
A
Tt
BL
A
2
g
hh
Tt
BL
A
2
g
hh
⋅⋅
−
−
−=
−
−
−=
−−=
−−=
(4.9)
Where (t
1
, h
1
) and (t
2
, h
2
) are two combinations of a time during the emptying
period and the corresponding measured water level.
0
0.1
0.2
0.3
0.4
0.5
0.6
50 0 50 100 150 200
tTs (min)
h^
0.
5
Orifice 1
Orifice 2
Figure 4.13. h (square root of water level) change with time
In the experiment, two different size orifices were used and their effective areas were
calculated using equation (4.3’) as 0.43 cm
2
for orifice 1 and 0.49 cm
2
for orifice 2.
77
Outflow rate
Outflow rate was calculated by the orifice equation (the RHS of equation (4.3)) with
the calculated effective orifice sizes.
Inflow rate
Since the mass balance equation for filling period is nonlinear, as shown in
(3.22), the inflow rate was adjusted by trial and error until the numerically calculated
water level change, fitted with the measured water level change. Figure 4.14 shows an
example of water level change comparison between measured and calculated. The figure
shows a good fit although there are some errors equal to or less than a centimeter between
these two. The errors are mainly because the water level was measured near the outlet
and travel time was not considered.
0.0
0.1
0.1
0.2
0.2
0.3
0.3
0 50 100 150 200
Time (min)
W
a
t
er l
e
v
e
l
(
m
)
Measured
Calculated
Figure 4.14. Measured and calculated water level change
78
(B) Event mean concentration, EMC, and particle removal ratio
Samples for measuring SSC can be taken during a storm runoff for only limited
times. The interval of taking samples may not be the same through each storm runoff.
Specifically, samples should be taken more frequently during a high flow period than
during a low flow period because the mass flow rate of particles is greater during a high
flow period due to both high flow rate and high particle concentration. The discharged
mass of particles in outflow can be estimated by the measured outflow SSC and the
estimated outflow rate, as described in the previous section. Figure 4.15 shows a
conceptual figure of measured discrete outflow SSC data and continuous estimated
outflow data.
Q, C
Time
Outlet TSS concentration
1
23 4 5 6
Outflow rate
Figure 4.15. Conceptual figure of measured outflow SSC data (discrete) and estimated
outflow rate (continuous)
Event Mean Concentration (EMC)
EMC is defined in the following equation;
∫
∫
⋅
==
dtQ
dtCQ
V
M
EMC
out
outout
out
out
(4.10)
79
In the case of discrete sampling, the k
th
value of outflow SSC taken at T
k
minutes is the
representative value of a group of time which is closer to T
k
than T
k1
or T
k+1
. Then, EMC
should be estimated as follows:
{ }
∫
∑
∫
=
⋅
=
dtQ
dtQC
EMC
k
out
n
1k
k
out
k
out
(4.11)
In the case of Figure 4.15, n=6.
Particle removal ratio
Mass removal ratio of TSS, or simply particle removal ratio, can be defined as
follows:
∫
∫
⋅
⋅
−=−=
dtCQ
dtCQ
1
M
M
1R
inin
outout
in
out
(4.12)
In this physical model study, total mass of inflow particles are simply defined as
sinin
TCQ . Therefore, the mass removal ratio of particle can be estimated as follows:
{ }
sinin
n
1k
k
out
k
out
TCQ
dtQC
1R
∑
∫
=
⋅
−=
(4.13)
The relationship between EMC and removal ratio is
( )
()R1
V
M
EMC
R1MEMCVM
in
in
ininout
−=⇒
−=⋅=
(4.14)
80
Chapter 5 Model Results, Analyses, and Uses
The conceptual model, described in Chapter 3, was developed to explain how
sedimentation would occur in a rectangular detention basin under ideal conditions and to
estimate removal efficiency of suspended sediment particles. The physical model,
explained in Chapter 4, was built to measure the sedimentation performance of a scaled
down basin. In this chapter, results from both the physical model and conceptual model
are described and compared.
5.1 EXPERIMENTAL RESULTS AND ANALYSIS
5.1.1 Experimental results
(A) Inflow conditions
Hydraulic and setup conditions
Eight experimental runs were conducted with different inflow rates, runoff
durations, and inflow SSCs. Table 5.1 shows the summary of inflow conditions including
the experimental setups and given hydraulic conditions. Three combinations of setup
conditions were tested. Two runs (A and B) were conducted with a smaller orifice area
and full length of the basin. The next three (C, D, E) were conducted with a larger orifice
area and full length of the basin. The remainder (F, G, H) were conducted with the larger
orifice area and 2/3 length of the original sedimentation basin.
Inflow rate, runoff duration, and measured maximum water level are also shown
in Table 5.1. Among them, only the inflow rates were calculated values which were fitted
to the measured time series water level change based on the method described in section
4.3.3. Inflow volume, V
in
, and theoretical overflow rate, Q
in
/(BL), in the table were the
calculated values. Theoretical overflow rate is distinguished from overflow rate in this
81
research. The theoretical overflow rate, the rate of water level increase by ignoring the
drainage from outflow is slightly larger than the overflow rate, the rate of water level
increase for the filling period. The theoretical overflow rate is often used in this research
since outflow rates are negligible compared to inflow rates and easier to calculate.
Theoretical overflow rates range from 0.27 m/hr to 1.16 m/hr, which made these eight
inflow conditions very different.
Table 5.1. Setup conditions and inflow hydraulic conditions
Effective area of
orifice, A
e
(cm
2
)
Basin length,
L (m)
Inflow rate,
Q
in
(L/s)
Runoff duration,
T
s
(min)
Maximum water
level, h
max
(m)
Inflow volume,
V
in
(L)
Theoretical overflow
rate, Q
in
/BL (m/hr)
A 0.43 6.96 0.53 40 0.257 1272 0.44
B 0.43 6.96 0.32 80 0.283 1536 0.27
C 0.48 6.96 0.9 20 0.283 1080 0.75
D 0.48 6.96 1.17 15 0.228 1053 0.98
E 0.48 6.96 0.73 30 0.276 1314 0.61
F 0.48 4.64 0.93 15 0.269 837 1.16
G 0.48 4.64 0.51 30 0.274 918 0.64
H 0.48 4.64 0.32 40 0.212 768 0.40
Setup conditions Inflow hydraulic conditions
Inflow SSC
To calculate mean inflow SSC and to check whether or not the inflow particle
concentration was uniform enough for each run, 5 to 10 inflow samples were taken from
the inlet pipe through the nozzle and SSCs were measured. Figure 5.1 shows measured
SSC data with respect to sampling time. The graph shows that all of the inflow SSCs are
greater than 150 mg/L and less than approximately 350 mg/L. This inflow SSC range is
within the range of TSS in urban runoff recommended by NURP (Table 2.1).
82
0
50
100
150
200
250
300
350
20406080
Time (min)
SSC
(
m
g/
L
)
A C
D E
F G
H B
Figure 5.1. Time series inflow SSCs for all eight runs
Mean inflow SSC for each run was calculated by taking an arithmetic average of
all the inflow samples SSC because inflow SSC is designed to be constant and samples
were taken uniformly over time. Then to check the uniformity of inflow SSCs, a
uniformity coefficient was calculated using equation (4.7). Table 5.2 shows mean SSC,
total mass of inflow particles, and uniformity coefficient of each run in addition to inflow
rate and runoff duration. Overall, uniformity coefficient values are very high, and all of
them are much greater than 0.8, which was set as the criteria. Any strong trend such as
long term increase or decrease cannot be found from the graph. The table implies that it is
difficult to hold inflow SSC uniform when the inflow is very low (e.g., B and H) or
duration is very long (e.g., B). Actually, it was very difficult to conduct run B in terms of
the uniformity since the flow rate of the peristaltic pump was very low in order to
83
maintain a certain SSC value for a long time. This might have resulted in sedimentation
in the tube.
Table 5.2. Inflow experimental conditions
Inflow rate,
Q
in
(L/s)
duration,
T
s
(min)
Mean inflow SSC,
C
in
(mg/L)
Total inflow mass,
M
in
(g)
Uniformity coefficient,
CU ()
A 0.53 40 202 257 0.97
B 0.32 80 267 410 0.85
C 0.9 20 246 266 0.97
D 1.17 15 258 271 0.97
E 0.73 30 212 279 0.95
F 0.93 15 192 161 0.97
G 0.51 30 255 234 0.91
H 0.32 40 277 213 0.89
(B) Outflow SSC and particle removal efficiency
Outflow SSC
Outflow samples were taken periodically for both the filling and emptying periods
and their SSCs were measured. Figure 5.2 shows time series SSC change of both inflow
and outflow for all the eight runs. Actually, these time series data are difficult to compare
since multiple inflow parameters such as inflow rate, runoff duration, and initial SSC are
different between runs. Therefore, comparison of time series outflow SSC is done in the
next section using the nondimensionalized technique. However, one thing to be noted
from the graph is that outflow SSC changes are all very smooth even when the inflow
SSC fluctuates significantly such as for run B, G, and H, which implies that the system
has the effect of equalization to eliminate the fluctuation of inflow SSC.
84
A: Qin=0.53 (L/s), Ts=40 (min)
0
50
100
150
200
250
300
350
0 50 100 150 200 250
Time (min)
SS
C
(
m
g
/
L
)
Inflow
Outflow
B: Qin=0.32 (L/s), Ts=80 (min)
0
50
100
150
200
250
300
350
0 50 100 150 200 250
Time (min)
SS
C
(
m
g
/
L
)
Inflow
Outflow
C: Qin=0.9 (L/s), Ts=20 (min)
0
50
100
150
200
250
300
350
0 50 100 150 200 250
Time (min)
SS
C
(
m
g
/
L
)
Inflow
Outflow
D: Qin=1.17 (L/s), Ts=15 (min)
0
50
100
150
200
250
300
350
0 50 100 150 200 250
Time (min)
SS
C
(
m
g
/
L
)
Inflow
Outflow
E: Qin=0.73 (L/s), Ts=30 (min)
0
50
100
150
200
250
300
350
0 50 100 150 200 250
Time (min)
SS
C
(
m
g
/
L
)
Inflow
Outflow
F: Qin=0.93 (L/s), Ts=15 (min)
0
50
100
150
200
250
300
350
0 50 100 150 200 250
Time (min)
SSC
(
m
g
/
L
)
Inflow
Outflow
G: Qin=0.51 (L/s) and Ts=30 (min)
0
50
100
150
200
250
300
350
0 50 100 150 200 250
Time (min)
SS
C
(
m
g
/
L
)
Inflow
Outflow
H: Qin=0.32 (L/s), Ts=40 (min)
0
50
100
150
200
250
300
350
0 50 100 150 200 250
Time (min)
SS
C
(
m
g
/
L
)
Inflow
Outflow
Figure 5.2. Measured inflow and outflow SSC change
85
Particle Removal efficiency
The calculation method for particle mass removal efficiency was conceptually shown in
Chapter 4. Here, the calculation for removal efficiency of each run is shown. Figure 5.3 is
a screen shot of the Excel spreadsheet used in the process of removal efficiency
calculation for run A.
Figure 5.3. Excel spreadsheet to calculate removal efficiency
Column A: The inflow conditions consisting of runoff duration, Ts (min), mean
inflow SSC, Cin (mg/L), and inflow rate, Qin (L/s), are listed here.
Column B: Time of each sampling (min).
Column C: Measured water level (m) at the time of sampling.
Column D: Calculated outflow rate (L/s) at the time of sampling using the orifice
equation.
Column E: Measured SSC (mg/L) at the time of sampling.
Column F and G:
Start time (min) and end time (min) during the sampling period. The
sampling time is assumed to be in the middle of the range.
86
Column H: Volume (L) of water discharged from time F to time G, which is
calculated by D*(GF)*60.
Column I: Mass (g) of particles discharged from time F to time G, which is
calculated by E*H/1000.
Cell (K2): Total volume of inflow water (L), calculated by A3*A7*60.
Cell (K3): Total outflow volume from the beginning of runoff to the end time
represented by the last sampling, calculated by the sum of column H.
Cell (K4): Volume of water that remained in the tank after the end of the last
sampling, calculated by Cell (K3) Cell (K2).
Cell (K5): Total mass of inflow particles (g), calculated by total inflow volume
multiplied by mean inflow SSC.
Cell (K6): Minimum mass of discharged particles (g), calculated by the sum of
column I.
Cell (K7): Maximum mass of discharged particles (g) assuming all the residual
water drained out with the SSC at the last sampling, calculated by
Cell(K6)+Cell(K4)*Cell(E16)/1000.
Cell (K8): Minimum removal ratio, calculated by 1Cell(K6)/Cell(K5).
Cell (K9): Maximum removal ratio, calculated by 1Cell(K7)/Cell(K5).
Cell(K10): Removal ratio, average value of Cell(K8) and Cell(K9).
Maximum and minimum calculated values of total discharged particle mass, M
in
,
removal efficiency, and EMC are shown in Table 5.3 with the inflow mean SSC, C
in
, and
the total inflow mass, M
in
. For practical reasons, the average of maximum and minimum
removal efficiencies are considered as the removal efficiency of each run, which was
calculated in Cell (K10). This is because maximum and minimum values of removal
efficiencies are all very close to each other. The table shows removal efficiencies are over
86% except in run F, which was conducted with the highest theoretical overflow rate
(1.16 m/hr see Table 5.1) and had the removal ratio of only 82%. Figure 5.4 shows the
87
relationship between removal ratio and theoretical overflow rate. This figure shows that
the larger the overflow rate is, the worse the removal efficiency is, but it is not a very
strong relationship. These data are further analyzed in the next section with the assistance
of nondimensionalization.
Table 5.3. Removal efficiency and EMC of each run
Min Max Max Min Min Max
A 202 257 26 31 0.90 0.88 20 24
B 267 410 27 30 0.93 0.93 17 19
C 246 266 29 30 0.89 0.89 27 28
D 258 271 34 36 0.87 0.87 33 34
E 212 279 32 34 0.88 0.88 25 26
F 192 161 28 29 0.82 0.82 34 35
G 255 234 31 32 0.87 0.86 34 35
H 277 213 26 27 0.88 0.88 34 35
Removal efficiency EMC (mg/L)
Inflow mean SSC,
C
in
(mg/L)
Total mass of inflow
particle, M
in
(g)
Total mass of discharged
particle, M
out
(g)
R
2
= 0.6692
0.80
0.85
0.90
0.95
1.00
0.00 0.50 1.00 1.50
Theoretical overflow rate (m/hr)
R
e
moval efficiency
Figure 5.4. Relationship between removal efficiency and theoretical overflow rate
88
5.1.2 Nondimensionalization of outflow SSC and its analysis
Although time series outflow SSCs and removal efficiencies of all runs were
shown in the previous section, they are difficult to compare. For example, all the graphs
in Figure 5.2 shouldn’t be overlapped since both the concentration and time scale is
different. Differences in the concentration scale can be treated by nondimensionalizing
the outflow SSC by mean inflow SSC for each run.
in
out
C
C
*C =
(5.1)
However, differences in time scale are not compared as easily since the time scale for the
filling period and emptying period is different. The filling period ends at the end of the
runoff duration which is different for each run. The end of the emptying period is
governed by the orifice size and maximum water level at the end of filling period, both of
which are different for each run. In this section, time nondimensionalization is considered
first, and then outflow SSC for each run is compared.
(A) Nondimensionalization of time and outflow SSC
Nondimensionalization of time was done such that all runs have the same filling
and emptying period in the nondimensionalized time scale. This allows all the
experimental data can be compared in a single graph. First, water level was
nondimensionalized simply by the maximum water level. This is because the time scale is
dominated by the maximum water level and orifice size during the emptying period.
max
*
h
h
h =
(5.2)
89
The nondimensionalization of time is shown here. For the filling period, time was simply
nondimensionalized by runoff duration, which is the end time of the filling period so that
filling periods of all runs end at the same nondimensionalized time.
s
*
F
T
t
t =
(5.3)
For emptying period, water level change is controlled by the outlet orifice. The
relationship between water level and time was defined as equation (4.4), and this equation
can be modified by nondimensionalized h* as
()
()
*
Es
max
e
s
e
max
tTt
2
g
hLB
A
*h1
Tt
2
g
LB
A
hh
=−
⋅
=−⇒
−
⋅
=−
(5.4)
The right hand side of this equation can be defined as the nondimensionalized time for
emptying period,
*
E
t, so that
*
E
t is equal to one when the water is completely drained
(h*=0) at the end of the emptying period. Finally, overall nondimensionalized time can
be expressed as
()
⎪
⎪
⎩
⎪
⎪
⎨
⎧
>−
⋅
+=+
≤=
=
ss
max
e*
E
s
s
*
F
TtforTt
2
g
hLB
A
1t1
Ttfor
T
t
t
*t
(5.5)
The water level change with the defined nondimensionalized time is plotted in Figure 5.5.
The nondimensionalized filling period starts from t*=0 and ends at t*=1, then
nondimensionalized emptying period starts from t*=1 and ends at t*=2. As shown herein,
the scaling method is different between the filling period and the emptying period. While
the length of nondimensionalized filling period and the nondimensionalized emptying
period are the same, actual filling period is much shorter than the emptying period.
90
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0 0.5 1.0 1.5 2.0
Nondimensionalized time, t*
N
o
ndim
e
n
s
io
na
liz
e
d
w
a
t
e
r
le
v
e
l,
h
A
B
C
D
E
F
G
H
Figure 5.5. Water level change with time in nondimensionalized form
The change in water level can be simplified using the redefined time and water level;
2*t1for*)t2(*h
1*tfor*t*h
2
≤≤−=
≤≈
(5.6)
h* cannot be exactly the same as t* for the filling period because there were outflows
during this period. h* can be equal to t* only when there is no outflow in the filling
period. Next, nondimensionalized outflow SSC, defined by equation (5.1), was plotted on
the nondimensionalized scale as shown in Figure 5.6.
91
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.0 0.5 1.0 1.5 2.0
Nondimensionalized time, t*
N
o
ndi
m
e
ns
i
o
n
a
l
i
z
e
d out
f
l
ow
SSC
,
C
A
B
C
D
E
F
G
H
Figure 5.6. Outflow SSC change with time in nondimensionalized form
The graph shows the following three characteristics:
• Some runs have strong initial drops of outflow SSC (e.g., run F and G)
• After the initial drops, SSCs are nearly constant throughout filling period.
• SSCs drop exponentially during the emptying period
The initial SSC drops were considered as the resuspension effect from previous runs and
were not considered in depth for this research. The change in SSC data is further
analyzed in the next section.
92
(B) Analysis of nondimensionalized outflow SSC
Filling period
Now it is appropriate to analyze the time series outflow SSC data. Assume outflow SSC
is a constant throughout the filling period as seen in Figure 5.6, excluding the initial drop
of SSC. The SSC for the filling period,
*
F
C , was compared with the theoretical overflow
rate, BL/Q
in
in Figure 5.6.
*
F
C was calculated from the arithmetic average of two C*
from before t*=1.0 and one from after or exactly at t*=1.0. Figure 5.7 suggests that SSC
for the filling period has a strong relationship with theoretical overflow rate.
Interestingly, this relationship (R
2
=0.94) is slightly better than the relationship between
*
F
C and real overflow rate, h
max
/Ts, which has the R
2
=0.92.
R
2
= 0.9346
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.00 0.50 1.00 1.50
Theoretical overflow rate, Q
in
/BL
N
o
n
d
i
m
en
si
o
n
a
l
i
zed
S
S
C
f
o
r f
i
l
l
i
n
g
p
eri
o
d
,
C
F
*
Figure 5.7. Relationship between theoretical overflow rate and average
nondimensionalized SSC for filling period
93
The CSTR (Continuous Stirred Tank Reactor) model with effective particle
settling velocity was introduced to understand the relationship between theoretical
overflow rate and average nondimensionalized SSC for filling period. The mass balance
of water for the filling period is
BL
Q
BL
Q
dt
dh
outin
−=
(5.7)
The mass balance of particle, on the other hand, is
CQCBLvCQ
dt
dM
outsinin
−−=
(5.8)
Where M is the mass of particles suspended in the basin and v
s
is the effective settling
velocity and C is the SSC in the basin that is equal to the outflow SSC. LHS of equation
(5.8) can be decomposed as follows by setting the concentration in the tank constant
throughout the filling period.
dt
dh
BLC
dt
)hC(d
BL
dt
)hBLC(d
dt
dM
≈==
(5.9)
Then the mass balance equation for particles can be simplified as
outs
in
in
ininouts
outsinin
QBLv
dt
dh
BL
Q
C
C
CQQBLv
dt
dh
BLC
CQCBLvCQ
dt
dh
BLC
++
=
=⎟
⎠
⎞
⎜
⎝
⎛
++
−−=
(5.10)
This can be simplified using the mass balance of water, equation (5.7), as
in
s
*
F
in
s
sin
in
*
F
in
Q
BL
v1
C
1
Q
BLv
1
1
BLvQ
Q
C
C
C
=−⇒
+
=
+
==
(5.11)
Figure 5.8 plots this relationship. This figure shows the effective settling velocity, v
s
, is
constant and the value is approximately 2.17 m/hr. In Figure 5.8, the far right point has a
94
big impact to determine the gradient of the trend line since the gradient will be 1.98 m/hr
and the R
2
value is 0.78 without the far right point. Therefore, the accuracy of the
gradient should be enhanced by taking more data in the future. However 2.17 m/hr will
be used as the gradient value in the dissertation because there is no reason to exclude this
far right point.
R
2
= 0.9186
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
0.00 1.00 2.00 3.00 4.00
(Q
in
/BL)
1
1/
C
F
*
1
BL/Q
17.2
1
*C
1
inF
=−
Figure 5.8. Effective particle settling velocity
Here the calculated effective settling velocity, 2.17 m/hr is much higher than
mean settling velocity of inflow particles 0.5 m/hr. This difference may be mainly from
the assumption that SSC in the basin is equal to the outlet SSC. In reality, the basin
cannot be a CSTR and the concentration in the basin is a function of time and location.
Therefore, average SSC in the basin should be higher than outlet SSC. However, this
CSTR model works well with the calibrated effective settling velocity to estimate the
95
outflow SSC for the filling period. Finally, nondimensionalized SSC for filling period,
*
F
C , can be calculated as follows.
in
*
F
Q
BL
17.21
1
C
+
=
(5.12)
This equation form would work for other rectangular settling basins, although the
effective settling velocity, v
s
, may change when particle is different or the basin size is
very different.
Emptying period
To see the rate of outflow SSC decline, outflow SSC was divided by the SSC at
the end of filling period, and the ratio of outflow SSC for the emptying to the average
outflow SSC for the filling,
*
F
*
E
C/C , was plotted with respect to time in Figure 5.9. This
figure shows an interesting feature. There exist two curves; one for A and B, and the
other for C through H. The only difference between runs A and B and runs C through H
is the effective orifice size as shown in Table 5.1. This suggests sedimentation during the
emptying period is highly dominated by the rate of drop in water level.
96
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 50 100 150 200 250
Time from the end of filling period (min)
C*
E
/C
*
F
A
B
C
D
E
F
G
H
Figure 5.9. Rate of SSC change for emptying period
Mass balance of water in emptying period is
BL
Q
dt
dh
out
−=
(5.13)
Mass balance for particles can be described in the CSTR model with a new effective
settling velocity, v
s
, for the emptying period as follows and simplified with (5.13).
dt
h
v
C
dC
C
dt
dh
v
dt
dC
h
dt
dh
C
C
dt
dh
vC
BL
Q
Cv
dt
)Ch(d
CQCBLv
dt
)Ch(d
BL
dt
dM
s
s
s
out
s
outs
−
=
⎟
⎠
⎞
⎜
⎝
⎛
+−=+
⎟
⎠
⎞
⎜
⎝
⎛
+−=−−=
−−==
(5.14)
97
Here, assume the effective settling velocity is proportional to the change in water level,
dh/dt.
dt
dh
kv
s
−=
(5.15)
Equation (5.14) can be analytically integrated after substituting (5.15) for (5.14) and
integrating as follows:
[] []
()
k
*
F
*
E
max
*
F
*
E
h
h
C
C
*h
C
C
*)hln(k
h
h
lnk
C
C
ln
)hln(k)Cln(
h
dh
k
C
dC
max
*
E
*
F
=⇒
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⇒
=
=
(5.16)
Figure 5.10 shows the data follow this relationship when k is 0.98, which is very close to
1.0. k=1.0 in equation (5.15) means the particle settling velocity is the same as the rate of
drop in water level for emptying period and the driving force of particle settling is not
particle’s own settling velocity but the water level drop.
98
R
2
= 0.9033
0.0
0.5
1.0
1.5
2.0
2.5
0 0.5 1 1.5 2 2.5
ln(h*)

l
n
(
CE
*
/
CF
*
)
()*)hln(98.0
*C
*C
ln
F
E
−⋅=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
Figure 5.10. Relationship in equation (5.16)
Therefore, nondimensionalized outflow SSC during the emptying period can be
simplified by setting k equal to 1.0.
2*
F
*
E
2
*
F
*
E
*)t2(CC
*)t2(*h
C
C
−⋅=⇒
−==
(5.17)
Nondimensionalized outflow SSC for the entire period, C*, can be summarized as
follows;
()
⎪
⎪
⎩
⎪
⎪
⎨
⎧
≤≤−=
≤≤
+
=
=
)periodemptying(2*t1for*t2CC
)periodfilling(1*t0for
Q
BL
17.21
1
C
*C
2*
F
*
E
in
*
F
(5.18)
99
To see how this relationship fits with the measured time series outflow SSC,
*
F
C/*C
was calculated as follows and the relationship was compared with measured data in
Figure 5.11. The graph shows the equation fits well with measured data.
()⎪
⎩
⎪
⎨
⎧
≤≤−
≤≤
=
)periodemptying(2*t1for*t2
)periodfilling(1*t0for1
C
*C
2*
F
(5.19)
0.0
0.5
1.0
1.5
2.0
0.0 0.5 1.0 1.5 2.0
t*
C*
/
C
F
*
Measured
Model
Figure 5.11. Measured and calculated change of
*
F
C/*C with respect to
nondimensionalized time
Also, the measured outflow SSC and calculated outflow SSC using equation
(5.18) were plotted in Figure 5.12 for the normalized time scale. The figure shows that
the decline of measured outflow SSC for the emptying period of run C, D, and F are
much steeper than the estimation curve. Actually, Run C, D, and F have the top three
highest theoretical overflow rates (see Table 5.1) and three shortest runoff durations,
100
which implies that the larger overflow rate kept the basin more turbulent during the
filling period and didn’t allow bigger particles to settle out. However, runs with moderate
overflow rates and longer durations had enough time for larger particles to settle out for
filling period, and the settling velocity is almost same as the change in water level
velocity for the emptying period. So, this empirical model, developed from experimental
results, should work better for slower runoff events when compared to rapid runoffs.
101
(A)Qin=0.53(L/s),Ts=40(min),C0=202(mg/L)
0.0
0.1
0.2
0.3
0 50 100 150 200 250 300
Time (min)
SS
C
*
Measured
Empirical
(B)Qin=0.32(L/s),Ts=80(min),C0=267(mg/L)
0.0
0.1
0.2
0.3
0 50 100 150 200 250 300
Time (min)
SS
C
*
Measured
Empirical
(C)Qin=0.90(L/s),Ts=20(min),C0=246(mg/L)
0.0
0.1
0.2
0.3
0 50 100 150 200 250 300
Time (min)
SS
C
*
Measured
Empirical
(D)Qin=1.17(L/s),Ts=15(min),C0=258(mg/L)
0.0
0.1
0.2
0.3
0.4
0 50 100 150 200 250 300
Time (min)
SS
C
*
Measured
Empirical
(E)Qin=0.73(L/s),Ts=30(min),C0=212(mg/L)
0.0
0.1
0.2
0.3
0 50 100 150 200 250 300
Time (min)
SS
C
*
Measured
Empirical
(F)Qin=0.93(L/s),Ts=15(min),C0=192(mg/L)
0.0
0.1
0.2
0.3
0.4
0.5
0 50 100 150 200 250 300
Time (min)
SS
C
*
Measured
Empirical
(G)Qin=0.51(L/s),Ts=30(min),C0=255(mg/L)
0.0
0.1
0.2
0.3
0 50 100 150 200 250 300
Time (min)
SSC
*
Measured
Empirical
(H)Qin=0.32(L/s),Ts=40(min),C0=277(mg/L)
0.0
0.1
0.2
0.3
0.4
0 50 100 150 200 250 300
Time (min)
SS
C
*
Measured
Empirical
Figure 5.12. Measured and empirically calculated time series outflow SSC
102
(C) Empirical Model for calculating removal efficiency
Removal ratio can be calculated using the empirically developed CSTR model in
the previous sections. The process to calculate removal efficiency from the CSTR model
is shown below.
Nondimensionalization of outflow rate and particle mass rate in outflow
Outflow rate can be calculated as a function of nondimensionalized time as follows:
*)t(gh2A*)t(Q
eout
=
(5.20)
This outflow rate can be nondimensionalized by the maximum outflow rate as
*h
h
*)t(h
gh2A
*)t(gh2A
*)t(Q
*)t(Q
*)t(Q
maxmaxe
e
max,out
out
*
out
====
(5.21)
h* in equation (5.21) can be replaced by t* using (5.6) as
)periodemptying(2*t1for*t2
)periodfilling(1*t0for*t*)t(Q
*
out
≤≤−=
≤≤≈
(5.22)
Then, the nondimensionalized outflow mass rate of particles can be defined as the
multiplication of nondimensionalized SSC and outflow rate as
() )periodemptying(2*t1for*t2C
)periodfilling(1*t0for*tC
*)t(*Q*)t(*C*)t(*M
3*
F
*
F
≤≤−=
≤≤≈
⋅=
(5.23)
Removal efficiency calculation
Here, the process for calculating removal efficiency is presented.
Total particle mass in inflow can be calculated as
sininin
TCQM ⋅⋅=
(5.24)
Total particle mass in outflow is the sum of particle mass for the filling and the emptying
period.
For the filling period, particle mass in outflow,
F,out
M , can be calculated as
103
∫
∫
⋅
=
=
s
s
T
0
*
out
*
outinmax,out
T
0
outoutF,out
dt)t(C)t(QCQ
dt)t(C)t(QM
(5.25)
This equation can be further transformed by replacing t with t* as follows:
10:*t
T0:t
ofrangethefor
*dtTdt
*tTt
s
s
s
→
→
=
=
(5.26)
Therefore, total mass of particles flowing out during the filling period is
3
TC2
CQ
*dt*tTCCQ
*dt*hTCCQ
*dt*)t(C*)t(QTCCQM
s
*
F
inmax,out
1
0
s
*
Finmax,out
1
0
s
*
Finmax,out
1
0
*
out
*
outs
*
Finmax,outF,out
⋅
⋅=
⋅⋅≈
⋅⋅=
⋅⋅=
⋅
⋅
⋅
⋅
∫
∫
∫
(5.27)
In this equation, *dt*h
1
0
∫
is approximated by *dt*t
1
0
∫
using *t*h ≈ for the filling
period. In order to check how much error this approximation has, the most deviated case,
which is run H in Figure 5.5 was analyzed. As a result, the real value of *dt*h
1
0
∫
was
0.70, while *dt*t
1
0
∫
was 0.67, which has approximately 4% error for the worst case,
run H. This means equation (5.27) would slightly underestimate particle mass in outflow
for the filling period, but with no greater than 4% error.
For the emptying period, in the same manner, total mass of discharged particles is
104
max
e
sout
T
T
*
out
*
outinmax,out
T
T
outoutE,out
h2
g
BL
A
TT
where
dt)t(C)t(QCQ
dt)t(C)t(QM
out
s
out
s
+=
⋅=
=
∫
∫
⋅
(5.28)
Here, t was replaced by t* using equation (5.5) as follows:
()
21:*t
TT:t
ofrangethefor
dt*dt
g
h2
A
LB
Tt
2
g
hLB
A
1*t
outs
max
e
s
max
e
→
→
=
⋅
−
⋅
+=
(5.29)
The particle mass in outflow for the emptying period can be calculated by replacing t
with t* using equation (5.29) as follows:
maxFin
max
e
*
F
inmax,out
2
1
3
max
e
*
F
inmax,outE,out
BLh*CC
2
1
g
h2
A4
BLC
CQ
*dt*)t2(
g
h2
A
BLC
CQM
=
⋅
⋅=
−
⋅
⋅=
⋅
⋅
∫
(5.30)
Finally, the total particle mass in outflow can be calculated for the sum of equation (5.27)
and (5.30).
⎭
⎬
⎫
⎩
⎨
⎧
+≈
+=
maxmax,outs
*
Fin
E,inF,outout
BLh
2
1
QT
3
2
CC
MMM
(5.31)
105
The removal efficiency can be calculated from the particle mass in inflow, equation
(5.24), and in outlflow, equation (5.31), as follows:
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+⋅
+
−=
⋅
⎭
⎬
⎫
⎩
⎨
⎧
+
−≈
−=
sin
max
in
maxe
in
sin
maxmax,outs
*
F
in
out
TQ
BLh
2
1
Q
gh2A
3
2
BL/Q
17.2
1
1
1
TQ
BLh
2
1
QT
3
2
C
1
M
M
1R
(5.32)
This equation was used to examine how drainage time affects removal ratio.
Inflow and setup conditions of Run A were used for an example. Several different
effective orifice areas were used to evaluate the relationship between effective orifice
area, A
e
, and removal ratio, R. The ratio of particle mass released during the filling period
to the total mass is called η
F
and the same ratio for emptying period is called η
E
. They
can be expressed as
sin
max
in
E
in
maxe
in
F
EF
TQ
BLh
2
1
BL/Q
17.2
1
1
Q
gh2A
3
2
BL/Q
17.2
1
1
where
1R
⋅
+
=η
⋅
+
=η
η−η−=
(5.33)
The particle mass ratios released during the filling, η
F
, and during the emptying, η
E
,
were calculated and the results were plotted in Figure 5.13. This figure shows that overall
removal efficiency,
EF
1R η−η−= , will decrease if drainage time is longer, but the ratio
of released mass during the emptying period is decreased if the drainage time is shorter.
106
0.0
0.0
0.0
0.1
0.1
0.1
0.1
0.1
0.0 0.5 1.0 1.5 2.0
Effective orifice area (cm
2
)
R
a
t
i
o of
m
a
ss f
l
ow
e
d
ou
t
ηF+ηE ηF
ηE
Figure 5.13. Ratio of mass flowed out during filling and emptying and their sum with
respect to effective orifice area
Table 5.4 shows the measured and calculated removal ratio using equation (5.32)
for all eight runs, as well as the calculated error, which is the calculated removal
efficiency minus measured removal efficiency. Calculated removal efficiencies should be
greater than the measured values since the calculated values don’t take resuspension into
account, thereby overestimating the results. However, runs C, D, and F have large
negative errors, which means the deviation of outflow SSC calculated for the emptying
period, shown in Figure 5.12, affected the removal efficiency calculation by this amount.
107
Table 5.4. Calculated and measured ratio of outflow SSC to inflow SSC for filling
period, C
*
F
, and removal efficiency
Measured CF* Modeled CF* Calculated R
Measured removal
efficiency
Error (calculated
measured) (%)
A 0.18 0.17 0.91 0.89 1.6
B 0.10 0.11 0.93 0.93 0.2
C 0.23 0.26 0.83 0.89 5.5
D 0.29 0.31 0.84 0.87 3.4
E 0.24 0.22 0.88 0.88 0.3
F 0.36 0.35 0.81 0.82 1.1
G 0.22 0.23 0.87 0.87 0.4
H 0.19 0.16 0.91 0.88 3.0
The CSTR model was developed to pursue simplicity and make the best use of
experimental results. The CSTR model is accurate in estimating both outflow SSC and
removal efficiency, especially for slower storms. Also, the particle removal efficiency
can be easily calculated if effective settling velocity is known for the filling period. The
effective settling velocity might be a function of both the size and shape of the detention
basin, and inflow particle settling velocities. Therefore, more data should be taken with
different shapes and sizes of basins including prototype to see if the model equation
works well. If so, methods to determine the effective settling velocity should be
determined and studied more.
5.2 CONCEPTUAL MODEL AND ANALYSIS
5.2.1 Conceptual model results
In the previous section, experimental results were nondimensionalized so that they
could be easily compared, and the behavior of time series outflow SSC was explained
using the CSTR model with effective particle settling velocity. In this section, the
developed conceptual model is examined by comparison with experimental results.
108
Conceptual model results were compared with experimental results using three
parameters: change in water level, outflow SSC, and particle removal efficiency.
Water level change
Water level change was measured every minute during the run. The water level
change was used for the conceptual model to determine water balance in the basin. Water
level change was explicitly calculated numerically using equation (3.23) for the entire
period. Figure 5.14 shows experimental and calculated results of water level change. This
figure shows that the water level was accurately calculated by the numerical calculation.
109
(A) Qin=0.53(L/s), Ts=40(min), long basin
0
0.05
0.1
0.15
0.2
0.25
0.3
0 50 100 150 200 250
Time (min)
W
a
t
e
r
le
v
e
l (
m
)
Measured
Calculated
(B) Qin=0.32(L/s), Ts=80(min), long basin
0
0.05
0.1
0.15
0.2
0.25
0.3
0 50 100 150 200 250
Time (min)
W
a
t
e
r
le
v
e
l (
m
)
Measured
Calculated
(C) Qin=0.90(L/s), Ts=20(min), long basin
0
0.05
0.1
0.15
0.2
0.25
0.3
0 50 100 150 200 250
Time (min)
W
a
t
e
r
le
v
e
l (
m
)
Meas ur ed
Calculated
(D) Qin=1.17(L/s), Ts=15(min), long basin
0
0.05
0.1
0.15
0.2
0.25
0.3
0 50 100 150 200 250
Time (min)
W
a
t
e
r
le
v
e
l (
m
)
Measured
Calculated
(E) Qin=0.73(L/s), Ts=30(min), long basin
0
0.05
0.1
0.15
0.2
0.25
0.3
0 50 100 150 200 250
Time (min)
W
a
te
r
l
e
v
e
l
(m
)
Measured
Calculated
(F) Qin=0.93(L/s), Ts=15(min), short basin
0
0.05
0.1
0.15
0.2
0.25
0.3
0 50 100 150 200 250
Time (min)
W
a
t
e
r le
v
e
l
(
m
)
Measured
Calculated
(G) Qin=0.51(L/s), Ts=30(min), short basin
0
0.05
0.1
0.15
0.2
0.25
0.3
0 50 100 150 200 250
Time (min)
W
a
t
e
r
le
v
e
l
(
m
)
Measured
Calculated
(H) Qin=0.32(L/s), Ts=40(min), long basin
0
0.05
0.1
0.15
0.2
0.25
0.3
0 50 100 150 200 250
Time (min)
W
a
t
e
r
le
v
e
l
(
m
)
Measured
Calculated
Figure 5.14. Measured and calculated water level change
110
Outflow SSC change
The outflow SSC was calculated using the calculated water level change. Figure
5.14 shows the calculated and measured outflow SSC for all eight runs. Calculation error
for peak concentration is smaller than 15 (mg/L) except for Run B and G shown in Table
5.5. Interestingly, inflow conditions of run B and G are quite different since run B has the
longest duration and the slowest theoretical overflow rate, but run G has medium duration
and medium overflow rate.
111
(A)Qin=0.53(L/s),Ts=40(min),C0=202(mg/L)
0
20
40
60
80
100
0 100 200 300
Time (min)
SS
C
(
m
g
/
L
)
Measured
Calculated
(B)Qin=0.32(L/s),Ts=80(min),C0=267(mg/L)
0
20
40
60
80
100
0 100 200 300
Time (min)
SS
C
(
m
g/
L
)
Measured
Calculated
(C)Qin=0.90(L/s),Ts=20(min),C0=246(mg/L)
0
20
40
60
80
100
0 100 200 300
Time (min)
SS
C
(
m
g/
L
)
Measured
Calculated
(D)Qin=1.17(L/s),Ts=15(min),C0=258(mg/L)
0
20
40
60
80
100
0 100 200 300
Time (min)
SS
C
(
m
g
/
L
)
Measured
Calculated
(E)Qin=0.73(L/s),Ts=30(min),C0=212(mg/L)
0
20
40
60
80
100
0 100 200 300
Time (min)
SS
C
(
m
g/
L
)
Measured
Calculated
(F)Qin=0.93(L/s),Ts=15(min),C0=192(mg/L)
0
20
40
60
80
100
0 100 200 300
Time (min)
SSC
(
m
g
/
L
)
Measured
Calculated
(G)Qin=0.51(L/s),Ts=30(min),C0=255(mg/L)
0
20
40
60
80
100
0 100 200 300
Time (min)
SS
C
(
m
g
/
L
)
Measured
Calculated
(H)Qin=0.32(L/s),Ts=40(min),C0=277(mg/L)
0
20
40
60
80
100
0 100 200 300
Time (min)
SS
C
(
m
g
/
L
)
Measured
Calculated
Figure 5.15. Measured and calculated outflow SSC
112
One possible reason to guess why these two estimations are worse than others is
seen in the inflow SSC graph, shown in Figure 5.1. To determine if the inflow SSC
increases or decreases with time, a trend line was drawn for each run to check if the
gradient of trend lines are significant. In Table 5.5, the gradient of the trend line for time
series inflow SSC shown in Figure 5.1 is shown. Then, runoff duration was multiplied by
the gradient in order to know how much the difference in inflow SSC for each run is
between the beginning and the end of each runoff. This table shows that runs B and G
had the greatest inflow SSC differences, which could result in the large deviations of
these two cases. Figure 5.16 supports the speculation. In Figure 5.16, the relationship
between error in peak outflow SSC and inflow SSC difference at the beginning and
ending, which corresponds the 3
rd
and 4
th
column in Table 5.5, is plotted. The figure
shows the peak concentration error depends on the magnitude of outflow SSC difference
at the beginning and ending. Less inflow SSC at the initial stage of a runoff would make
peak outflow SSC lower, and a greater inflow SSC at end of runoff would not reach the
outlet until a later time when the basin is considered as a plug flow reactor. This implies
that fluctuation and stability of inflow SSC are one of the error sources for the conceptual
model prediction. If more accurate results are necessary, discrete SSC should be used for
input data instead of using mean inflow SSC.
113
Table 5.5. Gradient and calculated inflow SSC difference between the start and the end
of runoff duration
Run
Gradient of inflow
SSC (mg/min)
Calculated difference of inflow
SSC between at the beginning
and end of the runoff (mg)
Peak concentration difference between
conceptual model prediction and
experimental results (mg/L)
A0.2 8.1 7
B10 74 21
C1.4 2.1 8
D 1.9 28.3 2
E1.1 34.0 5
F 0.1 1.9 3
G2.4 71.5 16
H12 42 2
R2 = 0.84
0
5
10
15
20
25
40200 2040608010
Difference in inflow SSC between at the beginning and end
of the runoff (mg/L)
P
e
a
k
o
u
tflo
w
S
S
C
e
rro
rs
(m
g
/
L
Figure 5.16. Relationship between error in peak SSC and inflow SSC difference between
at the beginning and end of the runoff
114
Particle removal ratio
Table 5.6 shows the calculated and measured particle removal efficiency, as well
as the error which is the calculated minus measured removal efficiency for each run.
Figure 5.17 shows the resulting and measured removal efficiencies. The figure shows that
calculated results underestimated the removal efficiency for a few %, but the trend is
consistent.
Table 5.6. Measured and calculated removal efficiencies
Run
Calculated removal
efficiency
Measured removal
efficiency
Error (calculated
measured) (%)
A 0.88 0.89 1.4
B 0.89 0.93 4.6
C 0.86 0.89 3.2
D 0.85 0.87 1.8
E 0.85 0.88 2.8
F 0.80 0.82 2.4
G 0.81 0.87 5.2
H 0.84 0.88 3.6
115
0.70
0.75
0.80
0.85
0.90
0.95
1.00
ABCDEFGH
Measured
Conceptual
Figure 5.17. Measured and calculated (using conceptual model) removal efficiencies
Here, the conceptual model results are briefly summarized. The conceptual model
was developed to treat the sedimentation basin as a plug flow reactor since the basin has a
long path and narrow cross sectional area, which is similar to channel flow. The model
works well to estimate time series outflow SSC, although the model always
underestimated the removal efficiency by a few percent. If a storm runoff is evaluated by
only average inflow rate and SSC, the conceptual model may overestimate the removal
efficiency because the high inflow SSC, called first flush, will result in higher outflow
SSC and lower removal efficiency.
116
The conceptual model has no calibration parameter, which means the model
results cannot be modified by experimental results. This is a desirable characteristic when
designing because the conceptual model requires basin dimensions (e.g., Length, width,
height, and drainage time) and inflow particle characteristics (e.g., particle size and
density).
5.2.2 Scaling
The next objective is to see whether or not Hazen number scaling works to
accomplish the same sedimentation performance between the physical model and the
prototype. The conceptual model was used for analysis since the prototype scale
experimental results were not available. As shown in the physical model design section,
length ratio,
mp
L/L , time ratio,
mp
t/t , and velocity ratio, )t(u/)t(u
mmpp
,were set to L
r
,
L
r
, and 1 respectively. The orifice size ratio,
m,ep,e
A/A , was
2/3
r
L to keep the water
level change between the model and the prototype consistent with the time and the length
scales. The maximum water level of the prototype should be L
r
times larger than that of
the model, and the time required to drain water from the maximum water level to zero for
the prototype should be L
r
times longer than that of the model. Particle size distribution
should be the same between model and prototype so that settling velocity ratio can be
kept at 1. This is a big advantage for scaling because to prepare particles, which has
exactly scaled up (or down) size distribution as particle size distribution in prototype, is
very difficult. Inflow SSCs are not necessary to scale up since the outflow SSC is always
normalized to the inflow SSC and the ratio is considered to be the same for any inflow
SSC in the conceptual model. Table 5.7 summarizes the scaling ratio for parameters used
for the physical model scaling.
117
Table 5.7. Scaling ratio of each variable
Variable Description
Ratio
mp
X/X
L Length of the detention basin L
R
B Width of the detention basin L
R
h(t) Height of the water level L
R
Ts Storm duration L
R
v Velocity 1
Q
in
and Q
out
(t) Flow rate L
R
2
A
e
Effective Area of orifice L
R
3/2
C
in
and C
out
(t*) Concentration 1
λ and ζ
Parameters of particle size distribution 1
Rd Removal ratio 1
To see the validity of these scaling assumptions, Run A, shown in the physical
model section, was scaled up to the prototype with a length ratio, L
R
, of 5. Inflow
conditions for the model and prototype are shown in Table 5.8. The conceptual model
was run for both the model and the prototype scale and time series outlet SSC and
particle removal efficiency were calculated. Figure 5.18 shows water level change and
outflow SSC change between the model and prototype.
118
Table 5.8. Dimensions and inflow conditions of prototype to have the same particle
removal efficiency of run A in model scale
Inflow rate,
Qin (L/s)
Runoff duration,
Ts (min)
Effective area of
orifice, Ae (cm
2
)
Basin length, L
(m)
Basin width,
B (m)
Inflow mean SSC
(mg/L)
Model 0.53 40 0.43 6.96 0.62 202
Prototype 2.65 200 4.81 34.8 3.1 202
Model
Prototype
Model
Prototype
Figure 5.18. Time series water level change (above) and outflow SSC change (below)
between model and prototype
119
The figure shows that both water level and SSC were properly scaled. The calculated
removal efficiencies were 88% for both. This result implies that Hazen number scaling
would work for the physical scaling of the sedimentation process in the rectangular
detention basin since the conceptual model explained the sedimentation process, at least
for the model scale. However, this comparison was done by using the conceptual model
and not by using physical models. Therefore, Hazen number scaling should be verified
using at least two similar physical models. This is not covered by this paper.
5.2.3 Analyses for Designing
Several typical design questions are presented and are attempted to be answered
by either the conceptual model results or the experimental results. All the following
analyses were done with the particles which have the same size distribution as the ones
used for the physical models.
(A) Which storm has worse removal efficiency of suspended sediment: a strong but
short storm, or a weak but long storm?
Experimental results can answer this question. Figure 5.4 shows the removal
efficiency with respect to the theoretical overflow rate, which is a good index of how
storm strength influences removal efficiency. The figure shows removal efficiency is
worse if the storm strength is greater. However, this is not a good comparison because the
storm volume is different for each run. Therefore, several simulations were done with the
conceptual model keeping the runoff volume constant. Figure 5.17 shows the relationship
between removal efficiency and theoretical overflow rates for three different storm
volumes (0.82, 0.41, and 0.20 of the basin volume). As shown in the graph, storms with
larger runoff volumes have lower suspended solid removal efficiency, although 80%
120
removal was calculated even for the worst case. This graph suggests that intense storms
would result in a lower removal ratio than less intense storms. The removal efficiency
doesn’t drop significantly when the theoretical overflow rate is very strong. It is because
no matter how large the storm is, the detention basin will have almost the same or even
slightly longer drainage time for emptying period.
0.0
0.2
0.4
0.6
0.8
1.0
0.0 1.0 2.0 3.0 4.0
Theoretical overflow rate(m/hr)
R
e
m
oval
rat
i
o
0.82*V
0.41*V
0.20*V
Figure 5.19. Removal ratio change with different theoretical overflow rates and different
runoff volumes
(B) How does the length/width ratio affect removal efficiency?
In the conceptual model, which employs the plug flow concept, the calculated
removal efficiency didn’t change when only length/width ratio, L/B, changed. This is
because a narrower basin results in a faster advection velocity. However, this may not be
true in an extreme case such as for the length/width ratio less than 1. For this case, plug
121
flow cannot exist and the short cutting and large dead space may result in a lower
removal efficiency.
(C) How does drainage time affects removal efficiency?
The process to determine the drainage time is equivalent to the process used to
determine the outflow orifice size. The theoretical drainage time, t
d
(hr), is defined as the
time to drain all the water from full (h=H(m)) to empty.
( )
( )
)hr(t
16.3
)hr(t
1
81.9
34.0*2
*
3600
10*96.6*62.0
cmA
)s/m(g
)m(h2
)hr(t)hr/s(3600
)m/cm(10)m(L)m(B
cmA
g
h2
t
LB
A
t
2
g
LB
A
h
dd
4
2
e
2
d
224
2
e
d
e
d
e
=⋅=
⋅
⋅⋅
=
⋅
=
⋅
⋅
=
(5.34)
Then, removal efficiency of suspended solid was calculated for four different orifice sizes
(0.41, 0.82, 1.23, and 1.64 (cm
2
)), which correspond to theoretical drainage times of 7.7,
3.9, 2.6, and 1.9 hours for three different flow rates (Q
in
=0.5, 1.0, and 2.0 (L/s)).
However, the runoff volume is the same for all cases. The calculation results are shown in
Figure 5.18. Removal efficiency decreased significantly with increasing orifice size when
compared to increasing flow rate. Table 5.9 shows the maximum water level and removal
efficiency for the same cases shown in Figure 5.20.
122
Table 5.9. Particle removal ratio and corresponding maximum water level for different
flow rate and different orifice sizes.
Effective area of orifice (cm
2
)
0.41 0.82 1.23 1.64
Particle removal efficiency 0.88 0.82 0.78 0.76
Maximum water level (m) 0.25 0.21 0.19 0.16
Particle removal efficiency 0.86 0.79 0.75 0.71
Maximum water level (m) 0.26 0.25 0.23 0.21
Particle removal efficiency 0.85 0.77 0.72 0.68
Maximum water level (m) 0.27 0.26 0.25 0.25
Q
in
=2.0
(L/s)
Q
in
=1.0
(L/s)
Q
in
=0.5
(L/s)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Effective orifice area (cm2)
R
e
m
oval
e
f
f
i
c
i
e
n
c
y
Q=0.5 (L/s)
Q=1 (L/s)
Q=2 (L/s)
Figure 5.20. Removal ratio change with different effective orifice sizes
The table shows for a short but strong storm, a smaller orifice works much better
than a larger orifice and the maximum water level is almost the same. However, for a
weak but long storm, the difference in removal efficiency for different orifice sizes is not
as big as the difference for a short but strong storm. For a flow rate of 0.5 L/s, the
123
maximum water level was 16 cm, or 47% of the overflow weir height, for an orifice area
of 1.64 cm
2
. However, for the same storm event (0.5 L/s), the maximum water level was
25 cm, or 74 % of the overflow weir height, for the orifice area of 0.41 cm
2
. Therefore a
smaller orifice opening is preferred if the design is for an intense short duration storm.
However, a larger orifice opening can be effective when the design is for less intense
storms since the volume which can be captured by the basin before overflow will occur
will increase.
(D) How does the density of suspended solids affect removal efficiency?
To answer this question, particle removal efficiency was calculated for particles
with different densities, and simulations with different flow rates were compared while
keeping the runoff volume the same. Of primary interest are particles with densities less
than 2.65 g/cm
3
. Figure 5.21 shows the removal efficiency of suspended solids for
particles with densities ranging from 1.2 to 2.65 (g/cm
3
), using the assumption that
density is uniform for all the particle sizes. Removal efficiency is lower if density is
lower, and the decrease is more significant for lighter particles. The density of smaller
particles in stormwater runoff can range from 1.1 (g/cm
3
) (Cristina, et al., 2001) for
typical highway organic soil to around 3.0 (g/cm
3
) for larger particles (Zanders, 2005).
When combined with the influence of density on removal efficiency, these results suggest
that characteristics of particle density should be considered in analysis and design of
stormwater treatment sysmtems.
124
0.0
0.2
0.4
0.6
0.8
1.0
1.01.52.02.53.0
Density (g/cm3)
Re
m
o
v
a
l
r
a
t
i
o
Q=0.5 (L/s)
Q=1 (L/s)
Q=2 (L/s)
Figure 5.21. Removal ratio change with different particle densities
5.2.4 Nonideality
The ideal plug flow solution always underestimates the removal efficiency to
some extent although the errors were not significant. To understand the reason and to see
how the flow in the basin is different from the plug flow, conductivity tracer tests were
done with salt as a tracer. Also, dye tests were conducted using the view window at the
side of the tank to see the velocity profile.
(A) Conductivity test
Three conductivity tests were done using salt adding to the water as a tracer. A
conductivity meter (Check mate 90, Corning) was used for detecting salt concentration at
inlet and outlet.
125
Calibration test
To know the relationship between salt concentration and conductivity, a
calibration test was done. Fig.5.20 shows the calibration line obtained from 6 different
samples, salt concentrations of which were known.
R2 = 0.9996
0
200
400
600
800
1000
1200
0 100 200 300 400
Salt Concentration (mg/L)
C
o
n
d
u
c
tiv
i
ty
(
μ
S
)
Figure 5.22. Calibration line between conductivity and salt concentration
Salt concentration can be obtained from the measured conductivity as follows:
93.1
341)S(Cond
)L/mg(C
−μ
=
(5.35)
341 (μS) is the tap water conductivity, which was used as the background concentration.
Therefore, nondimensionalized conductivity was defined subtracting background
concentration as follows:
126
341Cond
341Cond
*C
in
−
−
=
(5.36)
Where Cond is the measured outflow conductivity and
in
Cond is the mean inflow
conductivity.
Methodology
Three tracer tests were conducted with the same detention time of approximately 35
minutes and inflow rates of approximately 0.4 L/s. These experiments were done without
injecting sediment. Therefore, salt was put into the inflow system using the mixing tank
which was used to mix suspended sediemnt particles. Table 5.10 shows the experimental
conditions of these three tracer tests. Salt was introduced as a step function for runs A and
B while salt was injected as an inverse step function for run C. For example in the run A,
the basin was continuously filled with water for 35 minutes but the tracer was injected
only from 15 to 35 minutes during the filling period. These experiments were conducted
with low salt concentration, which was always lower than 150 (mg/L), so that the density
effect on the flow pattern was negligible.
Table 5.10. Experimental conditions of three different tracer tests
Qin (L/s) Ts (min) Total Q (L) Tracer from (min) Tracer to (min)
A 0.42 35 882 15 35
B 0.38 38 866 28 38
C 0.4 35 840 0 10
Test results
Fig 5.23 shows the test results. Each graph has nondimensionalized inlet and
outlet tracer concentration. In Run A, tracer was first detected at 26 minutes, which is 11
127
minutes after the tracer was added at 15 minutes. The traveling time of the tracer front
was 12 minutes for Run B and the traveling time of clean water front was 11 minutes for
Run C. Another interesting point is that tap water (runs A and B) or tracer (run C) was
never depleted. If this detention basin is a plug flow reactor, the outlet tracer
concentration should become 1 for Runs A and B, and 0 for Run C eventually. Therefore,
these conductivity test results show that there is some sort of mixing or dispersion
process in the basin and these processes prevent the detention basin from being a plug
flow reactor.
Run (A)
0.0
0.2
0.4
0.6
0.8
1.0
0 102030405060708090100
Time (min)
C* Outlet
Inlet
Run (B)
0.0
0.2
0.4
0.6
0.8
1.0
0 102030405060708090100
Time (min)
C*
Outlet
Inlet
Run (C)
0.0
0.2
0.4
0.6
0.8
1.0
0 102030405060708090100
Time (min)
C*
Outlet
Inlet
Figure 5.23. Nondimensionalized tracer concentration at inlet and outlet
128
(B) Model comparisons
Outflow tracer concentrations were calculated using plug flow and CSTR model.
These model results were compared with the measured outlet tracer concentrations in
order to see if the settling basin works more like plug flow or CSTR.
Plug flow model
Outflow tracer concentration was calculated using the conceptual model. In the
case of step function input, such as Runs A or B, the outflow tracer concentration should
jump from 0 to 1 when the tracer front reaches to the outlet. The time when the tracer
front reaches to the outlet was calculated using the conceptual model since the conceptual
model employed the plug flow model. Specifically, t
out
in equation (3.10) was solved
numerically as follows with the tracer input time and the shortened basin length of L
3
2
:
dt
tout
tin
)t(
in
Q
)t(hB
1
L
3
2
∫
⋅
=
(5.37)
CSTR model
Consider the case with the tracer injected from time 0 to T and the runoff duration
is T
s
(T>
flagScale=0; %Model or Prototype (0 for model scale and 1 for scaled model)
flagInflow=0; %Inflow hydrograph type (0 for constant inflow, 1 for SCS triangular inflow)
flagInput=1; %Among eight data, A=1, B=2, C=3, D=4, E=5, F=6, G=7, H=8
flagGraph=0; %Graph options
%0 for no graph
%1 for water level change
%2 for flow rate
%3 for PDF of inflow PSD
%4 for CDF of inflow PSD
%5 for PDF of particle settling velocity of inflow
%6 for CDF of particle settling velocity of inflow
%7: dpvp relationship
flagSSC=1; %0 for not calculating sedimentation process
%1 for calculating sedimentation process
%<>
%%%%%%%%%%%%%%%%%%%%%%%%%coeff for model%%%%%%%%%%%%%%
g=9.81; % Gravity acceleration, g (m/s^2)
B=0.62; % Width of the tank, B (m)
if flagInflow==0 %parameter for constnat Qin
if flagInput==1
Ae=0.43*4;L=6.96;C0=202;Qconst=0.53;Ts=40*60;%runA
elseif flagInput==2
Ae=0.43;L=6.96;C0=267;Qconst=0.32;Ts=80*60;%runB
elseif flagInput==3
Ae=0.48;L=6.96;C0=246;Qconst=0.9;Ts=20*60;%runC
elseif flagInput==4
Ae=0.48;L=6.96;C0=258;Qconst=1.17;Ts=15*60;%runD
elseif flagInput==5
Ae=0.48;L=6.96;C0=212;Qconst=0.73;Ts=30*60;%runE
elseif flagInput==6
Ae=0.48;L=6.96*2/3;C0=192;Qconst=0.93;Ts=15*60;%runF
elseif flagInput==7
Ae=0.48;L=6.96*2/3;C0=255;Qconst=0.51;Ts=30*60;%runG
elseif flagInput==8
Ae=0.48;L=6.96*2/3;C0=277;Qconst=0.32;Ts=40*60;%runH
end
142
K2=(Ae/10000*(2*g)^0.5/2/B/L)^2; %K2 from (3.1.5)
elseif flagInflow==1 %parameters for SCS triangular hydrograph
Ae=0.43; % Effective area of orifice
L=6.96; % Length of the tank, L=6.96 (m)
C0=202; %C0=constant inlet SSC (Suspended Solid Concentration)
K2=(Ae/10000*(2*g)^0.5/2/B/L)^2; %K2 from (3.1.5)
Qp=1.5/4; %Peak flow rate (L/s)
Tp=20*4*60; %Peak time (s)
end
%%%%% Variables on particles %%%%%%%%%%%%
densP=2.65; %density of silica particle (g/cm^3)
densL=1.00; %density of water (g/cm^3)
viscs=1e3; %viscs (viscosity of water) (Pa*s)=0.1*viscosity of water(=0.01)(g/cm/s)
rmdPsize=2.286; %mean of ln(particle size) (particle size (micron)) 2.252 from Ana's data and 2.651 for best
fit, 2.642 for manufacture data, (2.287, 0.906) is Jeff's data
tyetaPsize=0.908; %standard deviation of ln(particle size) 0.876 from Ana's data and 1.102 for best fit,
and 0.588 for manufacture data
k=g/18/viscs*(densPdensL)*3.6e6; %(vs(m/hr)=k*dp(micron)^2)
rmd=2*rmdPsize+log(k);%rmd is mean of ln(vp) and tyeta is std dev of ln(vp)
tyeta=2*tyetaPsize; %tyeta is std dev of ln(vp)
%%%%%%%%%%%%%%%%%%%%%%Coeff for Scaled up model%%%%%%%%%%%%%%%%%
if flagScale==1
Lr=5; %Length ratio
Ae=Ae*Lr^1.5; % Orifice area, Ap (cm^2)
L=L*Lr; % Length of the tank, L (m)
B=B*Lr; % Width of the tank, B (m)
%parameters for SCS triangular hydrograph
if flagInflow==0
Qconst=Qconst*Lr^2;
Ts=Ts*Lr;
elseif flagInflow==1
Qp=Qp*Lr^2;%Constant inflow rate, Qin (L/s), 0.53(L/s) and Ts=40 (min) is default value
Tp=Tp*Lr;% TS is the time when the runoff stops
end
end
%%%%%%%%%%%%%%%%%%%%%%%% Program for hydraulics %%%%%
i=1;
dt=1; %dt: time interval (min)
WL=1e4; %Water level(m)
%%%%%%%%%%%Constant Inflow%%%%%%%%%%%%%%%%%%%%%%
if flagInflow==0 %Constant flow rate case
QQ=0;
tautau=0;
while i<= 20*Ts/60/dt & WL>1e5
time(i)=i*dt;
if i<(Ts/60+1)/dt
WL=WL+60*dt/B/L*(Qconst/1000Ae/10000*(2*g*WL)^0.5); %WL (m)
WLL(i)=WL;
Qin(i)=Qconst; %Qinn (m^3/s)
Q(i)=1000*Ae/10000*(2*g*WL)^0.5; %Q (L/s)
tau(i)=Q(i)*i*dt;
else
WL=WL+60*dt/B/L*(Ae/10000*(2*g*WL)^0.5);
143
WLL(i)=WL;
Qin(i)=0;
Q(i)=1000*Ae/10000*(2*g*WL)^0.5;
tau(i)=Q(i)*i*dt;
end
QQ=QQ+Q(i);
tautau=tautau+tau(i);
i=i+1;
end
tmax=Ts/60/dt;
detention=tautau/QQTs/2/60
WL_max=WLL(tmax);
ni=Ts/60/dt; %ni:Number of i
%%%%%%%%%%%SCS Triangular inflow%%%%%%%%%%%%%%%%%
elseif flagInflow==1 %SCS triangular hydrograph case
while i<= 30*Tp/60/dt & WL>1e5
time(i)=i*dt;
if i<(Tp/60+1)/dt
WL=WL+60*dt/B/L*(i*dt/(Tp/60)*Qp/1000Ae/10000*(2*g*WL)^0.5);
WLL(i)=WL;
Qin(i)=1000*i*dt/(Tp/60)*Qp/1000;
Q(i)=1000*Ae/10000*(2*g*WL)^0.5;
elseif i<(8/3*Tp/60+1)/dt
WL=WL+60*dt/B/L*((1.60.6*i*dt/(Tp/60))*Qp/1000Ae/10000*(2*g*WL)^0.5);
WLL(i)=WL;
Qin(i)=1000*(1.60.6*i*dt/(Tp/60))*Qp/1000;
Q(i)=1000*Ae/10000*(2*g*WL)^0.5;
else
WL=WL+60*dt/B/L*(Ae/10000*(2*g*WL)^0.5);
WLL(i)=WL;
Qin(i)=0;
Q(i)=1000*Ae/10000*(2*g*WL)^0.5;
end
i=i+1;
end
ni=8/3*Tp/60/dt; %ni:Number of iteration during runoff period
end
%Water level (m) change with time
if flagGraph==1
plot(time/60, WLL);
hold on
title('Water level change with time')
xlabel('time(hr)');
ylabel('Water Level (m)')
elseif flagGraph==2
plot(time/60, Q);
hold on
plot(time/60, Qin);
title('In and outflow change')
xlabel('time(hr)');
ylabel('Flow Rate (L/s)')
end
%%%%%%%%%%%%%%%%%%%%%%%Program for calculating TSS outlet
concentration %%%%%%%%%%%%%%%%
144
if flagSSC==1
tic
%%Calculating time to reach outlet zone
for i=1:ni1
time=0;
RHS=0;LHS=1;
tDrain(i)=0;
n=1;
t0(i)=i*dt;
while RHS vpMini
criticalVP=vpMini:vp_interval:vp;%0.0073(m/hr) is settling velocity for dp=1.5(micron)
%P_Conc is the ratio of particles smaller than critical particle size, which means 1P_Conc is the
portion all of which settles down
P_Conc(i)=1/(tyeta*(2*pi)^0.5)*trapz(criticalVP, exp(0.5*((log(criticalVP)
rmd)./tyeta).^2)./criticalVP);
%E2_out is the portion that settles down among whole PSD
E2_out(i)=60*trapz(time,1./WLL(time))*1/(tyeta*(2*pi)^0.5)*trapz(criticalVP, 1/3600*exp(
(log(criticalVP)rmd).^2/(2*tyeta^2)));
E_out(i)=P_Conc(i)E2_out(i);
C(i)=C0*E_out(i);
else
C(i)=0;
end
end
plot(tDrain/60, C);
145
hold on
title('Figure2. Time series TSS concentration')
xlabel('Time (hr)');
ylabel('Concentration (mg/L)')
%%%%%%%%%%%%%5%Total removal ratio calculation%%%%%%%%%%%%%
timeIn=[1:ni1];
totalMassIn=C0/1000*60*trapz(timeIn,Qin(timeIn));
totalMassOut=60/1000*trapz(timeIn,Qin(timeIn).*C(timeIn));
RemovalRatio=1totalMassOut/totalMassIn
end
%%%%%%%%%%%%%%%%%%%%%% PSD graphs %%%%%%%%%%%%%%%%%%%%%%%
lndp=[5:0.1:4];
x_dp=exp(lndp);
lnvp=[8:0.1:2];
x_vp=exp(lnvp);
if flagGraph==3  flagGraph==4
pPDF=1./((2*pi)^0.5*tyetaPsize.*exp(lndp)).*exp(0.5*((lndprmdPsize)/tyetaPsize).^2);
pCDF=cumtrapz(exp(lndp),pPDF);
if flagGraph==3
plot(x_dp,pPDF);
title('PDF of Particle Size Distribution')
xlabel('Particle Size (micron)');
ylabel('PDF')
else
plot(x_dp,pCDF);
title('CDF of Particle Size Distribution')
xlabel('Particle Size (micron)');
ylabel('CDF')
end
end
if flagGraph==7
vp=(densPdensL)*(100*g)*(exp(lndp)*1e4).^2/18/(10*viscs)/100*3600; %vp(m/hr)
plot(x_dp,vp)
title('Relationship between particle size and settlign velocity')
xlabel('Particle Size (micron)');
ylabel('Particle Settling velocity (m/hr)')
end
if flagGraph==5  flagGraph==6
pPDF=1./((2*pi)^0.5*tyeta.*exp(lnvp)).*exp(0.5*((lnvprmd)/tyeta).^2);
pCDF=cumtrapz(exp(lnvp),pPDF);
if flagGraph==5
plot(x_vp,pPDF);
title('PDF of Particle Settling Velocity')
xlabel('Particle settling velocity (m/hr)');
ylabel('PDF')
else
plot(x_vp,pCDF);
title('CDF of Particle Settling Velocity')
xlabel('Particle settling velocity (m/hr)');
ylabel('CDF')
end
end
146
Appendix B Nomenclature
Symbols Descriptions
A
0
Actual orifice area
A
e
Effective area of an orifice
B Width of the sedimentation basin
C Concentration
C
D
Drag coefficient
Cd Orifice coefficient
C
in
Inflow SSC
C
out
Outflow SSC
CU Uniformity coefficient
d
p
Diameter of a particle
d
p,c
Critical particle diameter
EMC Event Mean Concentration of particles
F Force (in Chapter 2)
F Potential settling function (except Chapter2)
F
b
Body force
F
d
Drag force
F
g
Gravitational force
F
r
Froude number
g Gravitational acceleration
h Water level
H
a
Hazen number
L Length of the sedimentation basin
L
R
Length ratio of prototype to model
MMas
M
in
Total mass of inflow particles
M
out
Total mass of outflow particles
Q Flow rate
Q
in
Inflow rate
Q
out
Outflow rate
Q
p
Peak flow of a triangular hydrograph
R Removal efficiency
R
e
Reynolds number
R
out
(t
in
) Removal ratio of particles which flows into the basin at t
in
tTime
tin Time when a particle flows into the basin
147
t
out
Time when a particle flows out of the basin
T
p
Peak time of a triangular hydrograph
T
s
Duration of a runoff
u Longitudinal fluid velocity
u
p
Longitudinal particle velocity
v Velocity
v Vertical fluid velocity
V
in
Total inflow volume
v
or
Overlflow velocity
V
out
Total outflow volume
v
p
Vertical particle velocity
v
s
Particle settling velocity
v
s,c
Critical particle settling velocity
x Longitudinal position
x
p
Longitudinal position of a particle
x
w
Longitudinal position of water molecules
ζ Lognormal standard deviation
ζ
p
Lognormal standard deviation of particle diameter
ζ
v
Lognormal standard deviation of particle settling velocity
λ Lognormal mean
λ
p
Lognormal mean of particle diameter
λ
v
Lognormal mean of particle settling velocity
μ Viscosity of a fluid
ν Kinematic viscosity of a fluid
ρ Density
ρ
f
Density of a fluid
t
d
Drainage time
ρ
p
Density of a particle
Superscript
MModel
PPrtyp
* Nondimensionalized
k Time step
Subscript
E Emptying period
F Filling period
148
References
Adams, E.W., and Rodi. (1990). “W. Modeling Flow and Mixing in Sedimentation
Tanks.” Journal of Hydraulic Engineering. 116(7), 895913.
Andral, M. C., Roger, S., MontrejaudVignoles, M., and Herremans, L. (1999). “Particle
Size Distribution and Hydrodynamic Characteristics of Solid Matter Carried by Runoff
from Motorways.” Water Environment Research, 71 (4), 398407.
ASCE. (1992). “Design and Construction of Urban Stormwater Management Systems.”
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