i
CRWR Online Report 06-05 
 
 
 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 78712-4497  
 
 
 
This document is available online via the World Wide Web at  
http://www.crwr.utexas.edu/online.shtml  
 ii
Acknowledgements 
 
This research was funded by the Texas Department of Transportation (TxDOT) under 
grant number 0-4611, “Non-Proprietary, 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.  
 
 iii
 
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 SIL-CO-SIL®49 (PROVIDED BY 
US SILICA) 67 
FIGURE 4.8. PARTICLE SIZE DISTRIBUTION OF SIL-CO-SIL®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 runoff-impacted 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 non-point 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 right-of-way 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, L-W-H 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 small-footprint 
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 right-of-way 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 sub-sample 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 open-topped 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 
Small-scale modifications of the physical model, such as different orifice sizes or 
different energy dissipaters, can be easily done. However, large-scale 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 180-548 mg/L
BOD 12-19 mg/L
COD 82-178 mg/L
Total P 0.42-0.88 mg/L
Dissolved P 0.15-0.28 mg/L
TKN (Total Kjeldahl Nitrogen) 1.90-4.18 mg/L
Nitrite (NO
2
-
) and Nitrate (NO
3
-
) 0.86-2.2 mg/L
Total Cu 43-118 μg/L
Total Pb 182-443  μg/L
Total Zn 202-633 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 oxide-phosphate, 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 organic-carbon 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 (1-5 microns to 30-40 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 k-ε  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 non-ideal 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 non-ideality 
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 1-D 
advection-diffusion 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 full-scale 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/cm-s 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/cm-s, 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 (30-40 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 runoff-drainage 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/cm-s 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 
(Top-Left: Sedimentation basin set on wooden supports (view from inlet to 
outlet), Top-Right: mixing box, 3-inch pipe connecting mixing box and 
sedimentation basin, and Bottom-Left: 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, 3-inch 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 3-inch 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 water-based 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 multiple-coat 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 1-inch pipe was attached near the center at the bottom of the end piece of the 
sedimentation basin as the inflow opening. A 3-inch pipe was used in the first basin 
because a 3-inch pipe is approximately 1/5 diameter of an 18-inch 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, 1-inch 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 t-T
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 
SIL-CO-SIL®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 SIL-CO-SIL®49 were less than 50 μ m. It 
seems SIL-CO-SIL®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 SIL-CO-SIL®49 for experiments are reasonable. The other reason is the removal 
efficiency of the sedimentation basin. Removal efficiency of SIL-CO-SIL®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 SIL-CO-SIL®49 (provided by US Silica) 
 
Although the PSD of SIL-CO-SIL®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 SIL-CO-SIL®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 SIL-CO-SIL®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 4.0-600 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, 1-3400 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 5-10 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, 934-AH, Whatman) within a few hours after samples are taken. Before use in the 
measurements, stored filters were washed by reagent-grade water (milli-Q), 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
t-Ts (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 non-linear, 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
k-1
 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*(G-F)*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 1-Cell(K6)/Cell(K5). 
Cell (K9): Maximum removal ratio, calculated by 1-Cell(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
-40-200 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<T
s
) such as Run C. Outflow tracer concentration should be 1 for the time 0 to T 
since tracer is injected with a constant concentration. After the tracer injection stops, 
concentration change can be calculated by the mass balance equation (5.38). During the 
period, tracer concentration keeps declining because of the dilution by clean water. 
 
0.1*C.C.I
*C
BL
)t(gh2
e
A
dt
*dC
)t(h
s
TtTfor*C
=
⋅−=
≤≤
(5.38)
Equation (5.38) was numerically calculated by writing a code in C++. Time was 
discretized with forward differencing. Water level was explicitly calculated, followed by 
the concentration calculation so that tracer concentration can be solved explicitly. After 
 129
the runoff stops, the outflow tracer concentration becomes constant because there is no 
longer dilution.  
In the case with tracer injected from time T to the end of the runoff, T
s
, such as 
Runs A and B, outflow tracer concentration can be calculated with transforming equation 
(5.38) by changing datum concentration from 0 to inflow tracer concentration, C
in
*, 
which is 1. Therefore, the solution of outflow tracer concentration for step input can also 
be obtained from the same equation form of (5.38). 
 
( )
( )
()
() 1C
*
in
*C1
BL
)t(gh2
e
A
dt
*C1d
)t(h
*
in
C*C
BL
)t(gh2
e
A
dt
*
in
C*Cd
)t(h
*C
BL
)t(gh2
e
A
*
in
C
BL
in
Q
dt
*dC
)t(h
=−⋅−=
−
−⋅−=
−
⋅−⋅=
Q
(5.39)
 
Results 
Fig 5.22 shows the measured and modeled outflow results.  
 130
Run (A)
0.0
0.2
0.4
0.6
0.8
1.0
0 20 40 60 80 100 120 140 160 180 200
Time (min)
C*
Outlet
Inlet
Plug flow
CFSTR
Run (C)
0.0
0.2
0.4
0.6
0.8
1.0
0 20 40 60 80 100 120 140 160 180 200
Time (min)
C*
Outlet
Inlet
Plug flow
CFSTR
Run (B)
0.0
0.2
0.4
0.6
0.8
1.0
0 20 40 60 80 100 120 140 160 180 200
Time (min)
C*
Outlet
Inlet
Plug flow
CFSTR
 
Figure 5.24. Measured outflow tracer concentration with calculated results from plug 
flow and CSTR model. 
 
The graphs show that the interface between tracer and water traveled much faster 
than the interface calculated with plug flow model. The measured results are also 
different from CSTR model because the measured results show that the concentration 
rises for Runs A and B or drop for Run C at certain point, which cannot be explained by a 
 131
CSTR model. Among the three experimental results, deviation between measured and 
plug flow model is the largest in Run B with the tracer injected for the last 10 minutes 
among 38 minutes runoff duration.  
1-D dispersion coefficient is sometimes employed for modeling the outflow tracer 
concentration change of column tests. However, this model would not well describe the 
situation. This is because 1-D dispersion term can only smooth out the sharp 
concentration rise of the plug flow solution.  
 
(C) Observation from view window 
The tracer tests imply that the flow is different from CSTR, plug flow, or plug 
flow with 1-D diffusion.  The flow pattern was observed visually using the view 
window. Dye was used to visualize the velocity profile in the cross section for both the 
filling and emptying period. For the filling period, the velocity pattern looked like the 
plug flow except near the very bottom of the cross section due to the friction until water 
level reached approximately 12 cm. As water level increased during the filling period, an 
interface was created approximately at the height of 12 cm. the flow was still like the 
plug flow beneath the interface, while the flow above the interface was very slow and 
went backward. The height of the interface did not change much throughout the filling 
period. The velocity profile when the water level was approximately 20 cm during the 
filling period is shown in Fig 5.23. The pictures were captured from the videos for the 
filling and emptying period. Three layers can be seen in each picture. The top layer is the 
water surface, which is covered by scum, the middle layer is the other side of the basin 
wall painted gray, and the bottom layer is the bottom of the basin covered by white silica 
particles. The profile implies that a big counterclockwise direction eddy was developed 
during the latter half of the filling period.  
 132
 
 
t=0(s)
t=3(s)
t=6(s)
For the filling period For the emptying period
t=0(s)
t=6(s)
t=15(s)
 
Figure 5.25. Typical velocity profiles with velocity interfaces for the filling period (left) 
and the emptying period (right) 
 
For the emptying period, mean flow flux across a cross section was much smaller 
than the flux for the filling period because the mean flux is determined only by outflow. 
At the beginning of the emptying period, there remained the backwater caused by the big 
slow eddy developed during the filling period. The backwater circulation in the upper 
 133
layer disappeared approximately 10-15 minutes after the runoff stopped. Figure 5.25 
shows the velocity profile observed 8 minutes after the runoff stopped. The figure shows 
that there was a weak backward flow for 1/2 depth of the basin from the top and there is 
also a weak forward flow for 1/2 depth of the basin from the bottom. After the backwater 
disappeared in the upper layer, the velocity profile becomes close to the plug flow except 
for the very top and bottom of the cross section.  
The pictures also shows the reason for the short circuiting observed in the tracer 
test results such as Runs A and B. The tracer velocity in the bottom layer is much larger 
than the calculated average velocity of a cross section, which shorten the traveling time of 
the tracer front from the inlet to the outlet. However, some of the tracer might be 
dispersed in the basin due to the shear stress created by the large velocity gradient at the 
velocity interface and some of the tracer, which happened to be in the top layer, will be 
carried backward by the large vertical counterclockwise eddy. This can be a reasonable 
explanation of the tracer concentration graph, which never reached 1 throughout runs.  
The reason why the plug flow model underestimated the results is speculated 
using the nonideality results on flow patterns. The particle sedimentation rate can be 
theoretically calculated as follows; 
 
∫∫ ⋅=
L
0
dx
B
0
dy)t,y,x(
B
C
s
v)t(M
(5.40)
Where M(t) is the total sedimentation rate and C
B
 is the particle concentration close to the 
bottom. The particle concentration in the bottom layer in the real situation is higher than 
that in the plug flow model since inflow was introduced to the lower part of the 
sedimentation basin when the water level is higher than a certain height. The particle 
concentration profile during the filling period was conceptually drawn in the Figure 5.26. 
Assume the settling velocity for the both situations are the same, the sedimentation rate, 
 134
M(t) in equation (5.41), for the real situation is higher than that of the plug flow model 
since the mean SSC is higher in the bottom layer.  
  
Plug flow model
Real situation
 
Figure 5.26. Particle concentration profile in the sedimentation basin in the plug flow 
model and the real situation during the filling period.  
This could be a reasonable explanation of the situation that plug flow model 
underestimated the removal ratio. In this sense, the plug flow model estimates the 
removal efficiency conservatively. However, advection velocity in the lower layer is 
much higher than that in the plug flow model as shown in the Figure 5.26. Therefore, the 
higher advection velocity might enhance the resuspension of particles sitting on the 
bottom and entraining particles which are going to settle out. Therefore, it is necessary to 
use devices such as an energy dissipator or diffuser around the inlet and a sill around the 
outlet to prevent these resuspension and entrainment effects. 
 
 135
 
Chapter 6 Conclusions and future work 
Conclusions drawn from the research and recommendations for future work are 
discussed in this chapter. 
 
6.1 CONCLUSIONS 
This research work is part of a research project funded by TxDOT. Four research 
objectives were set to fulfill the research project. How these research objectives were 
achieved and what kinds of findings that were drawn for each objective are discussed. 
 
Objective 1: Build a rectangular sedimentation basin as a physical model, which can 
reproduce and vary input conditions such as inflow rate and SSC, and can be used 
to take samples at any time from inlet and outlet to measure SSC 
Two physical model basins were built in the laboratory of CRWR (Center for 
Research in Water Resources). The first one, which unfortunately was not used, provided 
many insights about how the sedimentation basin should be built and protected from 
water leakage and how measurements should be taken. The second one had the same size 
of the first basin, but had a modified particle delivery system and better waterproofing. 
This second physical model could reproduce inflow conditions well except for controlling 
inflow SSC at a constant value. However, the particle mixing and delivery system was 
also modified and achieved 80% uniformity of inflow SSC during the filling period using 
a better peristaltic pump and mixing method. The system has an inflow nozzle right 
before the sedimentation basin and outflow nozzle right after the outlet orifice, and these 
nozzles were successfully used for taking samples. Therefore, the first research objective 
was successfully achieved.  
 
 136
 
Objective 2: Build a conceptual model which can estimate time series outflow SSC, 
event mean concentration (EMC), and particle removal efficiency; and verify the 
conceptual model using physical model results  
A conceptual model used to calculate outflow SSC and particle removal 
efficiency was theoretically developed based on the assumptions and concepts that are 
used in the ideal horizontal flow reactor theory, and a program to apply the theory to 
simple cases was developed in MATLAB. The code can handle either constant flow rates 
or SCS triangular hydrographs as inflow rates. As a result, outflow SSCs were accurately 
calculated by the conceptual model except for a few cases, which had increasing trends of 
inflow SSC. Particle removal efficiencies were underestimated by a few percent for all 
cases. That’s may be because highly concentrated inflow rather stay in the lower part of 
the tank and effectively be settled out than spread out and evenly be distributed vertically 
around at inlet, which is the assumption in the conceptual model. 
Several simulations and analyses were conducted with the conceptual model. 
When the overflow rate is increased, removal ratio will decrease but not significantly. 
This may be because no matter how strong the storm is, the detention basin will have 
almost the same or a slightly longer drainage time for emptying period. However, the 
length/width ratio of the sedimentation basin will not affect removal efficiency when 
surface area of the basin is constant. This may not be true when the length of the 
sedimentation basin is nearly equal to or shorter than the width of the sedimentation basin 
because the flow will be far from plug flow. When the drainage time is decreased, by 
increasing the area of outlet orifice, the removal efficiency will decrease. This result is 
consistent with the analysis done from physical model results. Finally, particle removal 
efficiency was calculated when particle density is lower than 2.65 (g/cm
3
). The decrease 
 137
of particle density will affect removal efficiency especially when the density is smaller 
than 2 (g/cm
3
), which is possible for smaller particles with a high organic carbon fraction. 
The tracer tests and the dye tests from the view window shows that there were 
some mixing in the basin and it may be because of a large scale vertical counterclockwise 
eddy formed in the basin, which resulted in the flow pattern different from plug flow. An 
example solution to make the flow close to the plug flow is to put a series of baffles in the 
basin to minimize the momentum created by the strong inflow. If it is necessary to 
understand the effect of the eddy on sedimentation, numerical simulations would be 
required to calculate at least 2-D (longitudinal and vertical axes) flow profile in the basin.   
  
 
Objective 3: Test different lengths of the basin and outlet orifice sizes in the physical 
model to evaluate how the overflow rate and detention time affect particle removal 
ratio 
Eight runs were conducted with different inflow conditions and setups with two 
different orifice sizes and two different basin lengths. To compare these results, which 
have different conditions, a time nondimensionalization method was developed for the 
system such that all runs have the same length of nondimensionalized filling and 
emptying period. As a result, time series outflow SSCs of all runs were compared 
efficiently. 
For the filling period, outflow SSC was almost constant and was proportional to 
the theoretical overflow rate. The phenomenon was explained by a semi steady state 
CSTR model using effective settling velocity, which was calculated to 2.17 (m/hr). For 
the emptying period, outflow SSC decreased with time as if particles have a settling 
velocity which is the same as the speed of water level dropping except for few cases. 
These exceptional cases are runs with the two highest theoretical overflow rates and two 
 138
shortest durations, which implies almost all the bigger particles will settle out for a filling 
period if runoff duration is long enough and the overflow rate is not too large compared 
to particle settling velocity. Overall removal efficiency will decrease if the effective 
orifice area is increased; however, the mass of particles flowing out during the emptying 
period may be decreased if the effective orifice area is increased because the increased 
speed of water height drop will push particles down to settle out.  
 Table 6.1 shows the summary of the two different models developed in this 
research. One is the conceptual model with the plug flow assumption and the other is the 
empirical model developed using CSTR model concept and the physical model results. 
As shown in the table, the empirical model is easier to handle and as accurate as the 
conceptual model. The empirical model should determine the effective settling velocity, 
which would be different for different inflow particles, and sizes and shapes of the 
sedimentation basin. However, the behavior of the effective settling velocity was not 
studied yet.    
 
 
 
 
 
 
 
 
 139
Table 6.1. Summary of the model characteristics between the plug flow base 
conceptual model and the CSTR base empirical model.  
Conceptual Empirical
Model base Plug flow CFSTR + experimental data
Required inflow data Mean flow rate, duration, Mean SSC Mean flow rate, duration, Mean SSC
Water level change Time series water level change should be
calculated
Only maximum water level should be
calculated
Required software MATLAB or any programming compiler Excel spread sheet
Required inflow particle
size distribution (PSD)
Lognormal mean and standard deviation N/A
Calibration parameter N/A Effective particle settling velocity
Model accuracy Good, but model results always
underestimates for a few %
Good, but better for less intense and
longer runoffs
Nonideality Not considered Incorporated with the effective settling
velocity
Applicability with
different sizes of particles
Fine if inflow particle size distribution is
known
Effective settling velocity should be
determined experimentally
 
 
 
Objective 4: Study how geometric and hydraulic parameters should be scaled to 
have identical particle removal efficiency in physical model and prototype using 
conceptual and physical model results 
 
This scaling study was done using the conceptual model since prototype SSC data 
are not available yet. As a result, Hazen number scaling, which sets the velocity ratio 
between model and prototype equal to each other, was found to be the proper scaling 
method. With this scaling method, particles do not need to be scaled, which makes 
experiments much easier. However, the physical model has a weak but large vertical eddy 
in the basin, which made the system a little different from plug flow. This kind of 
nonideality may occur in the prototype, but it has not been studied yet.  
 
 
 
 140
6.2 FUTURE WORK 
For the conceptual model, application methods can be modified a little bit more. 
For example, the current program cannot handle discrete inflow SSC but can only handle 
constant inflow SSCs. This is good enough when outflow SSC results are compared with 
physical model results or when only the removal ratio is to be estimated. However, 
inflow SSC sometimes significantly fluctuates or has a declining trend when so called 
first flush effect is significant. In these cases, using mean inflow SSC as an input will 
result in a large deviation from the actual removal ratio. Also, it is convenient if the 
conceptual model can calculate removal ratio of particles with known size especially 
when particle size distribution of inflow is unknown.    
For the physical model, larger orifice sizes should be tested to see if the empirical 
model continues to works well and the calculated effective settling velocity has the same 
value. Also, several test results of the prototype should be acquired from the Texas A&M 
group and reproduce runs with properly scaled input to see if the Hazen value scaling 
works well. 
Finally, optimization of removal efficiency should be done. For example, 
nonideality should be studied more. How the large vertical eddy makes the sedimentation 
different from the plug flow based sedimentation should be studied. Also, ways to 
minimize the resuspension effects around inlet and outlet area should be studied because 
the resuspension effects were significant especially with a strong runoff and the effect 
may be more significant for the prototype. 
 
 
 
 141
 
Appendix A  Conceptual Model Program in MATLAB 
 
The following is the MATLAB program developed for calculating particle removal efficiency of the rectangular 
detention basin by the conceptual model. Water level, outflow rate, and outflow SSC are also calculated in the program. 
The same kinds of calculation can be done for prototype by setting the flagScale to be 1.  
 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%This is a MATLAB code to calculate Water level change, Outflow rate, outflow SSC, and SS removal ratio 
%Author: Masatsugu Takamatsu 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 
clear all 
 
%<<Define flags>> 
    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: dp-vp relationship 
    flagSSC=1;          %0 for not calculating sedimentation process 
                        %1 for calculating sedimentation process  
     
%<<Define Variables>> 
%%%%%%%%%%%%%%%%%%%%%%%%%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=1e-3;  %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*(densP-densL)*3.6e-6; %(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=1e-4;        %Water level(m) 
     
    %%%%%%%%%%%Constant Inflow%%%%%%%%%%%%%%%%%%%%%% 
     
    if flagInflow==0     %Constant flow rate case 
        QQ=0; 
        tautau=0; 
        while i<= 20*Ts/60/dt & WL>1e-5 
            time(i)=i*dt; 
            if i<(Ts/60+1)/dt 
                WL=WL+60*dt/B/L*(Qconst/1000-Ae/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/QQ-Ts/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>1e-5 
            time(i)=i*dt; 
            if i<(Tp/60+1)/dt 
                WL=WL+60*dt/B/L*(i*dt/(Tp/60)*Qp/1000-Ae/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*((1.6-0.6*i*dt/(Tp/60))*Qp/1000-Ae/10000*(2*g*WL)^0.5); 
                WLL(i)=WL; 
                Qin(i)=1000*(1.6-0.6*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:ni-1 
         time=0; 
         RHS=0;LHS=1; 
         tDrain(i)=0; 
         n=1; 
         t0(i)=i*dt; 
        while RHS<LHS %t0=TT*i and T=time(i) 
            if i+n<ni 
                iRange=[i:i+n]; 
            else 
                iRange=[i:ni]; 
            end 
            LHS=L; 
            RHS=1/B/WLL(i+n)*trapz(iRange,Qin(iRange))*60/1000; 
            RHSS(i)=RHS; 
            tDrain(i)=i+n; 
            tDetention(i)=n; 
            n=n+1; 
         end 
     end 
%         plot(t0, tDetention); 
%         title('Figure2. Detention Time') 
%         xlabel('Inflow time(min)'); 
%         ylabel('Detention time (min)') 
  
             
        %%%%%%%%%%Time series TSS Concentration Calculation%%%%%%%%%%%%%%%%    
        for i=1:ni-1 
            C00(i)=C0; 
            vpMini=(densP-densL)*(100*g)*(1.5*1e-4).^2/18/(10*viscs)/100*3600; %vp(m/hr) for 1.5 micron 
particle 
            vp=0;%vpMini; 
   %Calculating critical velocity 
            time=[t0(i):tDrain(i)]; 
            vp=60*1./trapz(time,1./WLL(time)); 
            vpp(i)=vp; 
            %Calculating critical particle size 
            cpp(i)=10000*(vp/3600*100*10*viscs*18/((densP-densL)*100*g))^0.5; 
            vp_interval=0.00001; 
            if vp> 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 1-P_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:ni-1]; 
         totalMassIn=C0/1000*60*trapz(timeIn,Qin(timeIn)); 
         totalMassOut=60/1000*trapz(timeIn,Qin(timeIn).*C(timeIn)); 
         RemovalRatio=1-totalMassOut/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*((lndp-rmdPsize)/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=(densP-densL)*(100*g)*(exp(lndp)*1e-4).^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*((lnvp-rmd)/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
 
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