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PAGE 1 1 FLOCCULATION AND TRANSPORT OF COHESIVE SEDIMENT By MINWOO SON A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2009 PAGE 2 2 2009 Minwoo Son PAGE 3 3 To my parents, wife a nd lovely daughter, Jiwoo PAGE 4 4 ACKNOWLEDGMENTS I would like to appreciate to the academic advisor and supervisory committee chair, Dr. TianJian Hsu, Assistant Professor of Civil and E nvironmental Engineering at the University of Delaware, for his kind and academic guidance. I thank to the committee member, Dr. A. J. Mehta, Professor Emeritus of Civil and Coastal E ngineering at the University of Florida for his discerning comments and questions on study of flocculation. Sincer ely, thanks are also extended for academic advices and suggestions of Dr. K. Hatfield, Professor of Civil and Coastal Engineering of the University at Florida and Dr. J. S. Curtis, Professor of Chemical Engineering at the University of Florida. Special thanks go to the family of my uncle, YhungGyung Kang, for their warm hearts from Manhattan. Finally, I would like to express my sincerest appreciation to my parents, Youngseuk Son and Yeonsuk Kang, fo r their presence itself, wife, Kunhwa Choi, for her love, and daughter, Jiwoo Son, for her lovely smile and kiss. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4LIST OF TABLES................................................................................................................. ..........7LIST OF FIGURES................................................................................................................ .........8LIST OF ABBREVIATIONS........................................................................................................13ABSTRACT....................................................................................................................... ............17 CHAPTER 1 INTRODUCTION..................................................................................................................191.1 Significance of Study on Cohe sive Sediment Transport..................................................191.2 Objectives of This Study..................................................................................................201.3 Terminology................................................................................................................ .....211.4 Outline of Presentation.................................................................................................... .232 LITERATURE REVIEW.......................................................................................................252.1 Studies on Flocculation a nd Yield Strength of Floc.........................................................252.2 Sediment Transport Modeling..........................................................................................293 STUDY ON PROPERTIES OF COHESIVE SEDIMENT....................................................323.1 General Properties of Cohesive Sediment........................................................................323.2 Fractal Dimension.......................................................................................................... ...353.3 Flocculation Process....................................................................................................... ..394 MODELING FLOCCULATION OF COHESIVE SEDIMENT...........................................424.1 Overview on Flocculation Modeling................................................................................424.2 Lagrangian Flocculation Models......................................................................................444.2.1 Winterwerps Flocculation Model..........................................................................444.2.2 Flocculation Model Using a Va riable Fractal Dimension......................................484.2.3 Flocculation Model Using a Variable Fractal Dimension and Variable Yield Strength....................................................................................................................... .564.3 Investigation of Flocculation Models...............................................................................624.3.1 Application of FM A and FM B.............................................................................624.3.2 Application of FM C and FM D.............................................................................815 MODELING TRANSPORT OF COHESIVE SEDIMENT.................................................1015.1 Governing Equations for Flow Momentum and Concentration.....................................101 PAGE 6 6 5.2 Flow Turbulence............................................................................................................ .1045.3 Bottom Boundary Conditions.........................................................................................1065.4 Flow Forcing for Tidal and Unsteady Flow Condition..................................................1075.5 Preliminary Tests.......................................................................................................... ..1086 MODEL APPLICATION TO EMS/DOLLARD ESTUARY..............................................1176.1 Insitu Measurement in Ems/Dollard Estuary................................................................1176.2 Calibration of Models.....................................................................................................1196.3 Investigation of Sediment Transport Model...................................................................1277 SUMMARY, CONCLUSIONS AND REMARKS..............................................................1727.1 Summary and Conclusion...............................................................................................1727.2 Concluding Remarks for Future Study...........................................................................176APPENDIX DERIVATION OF EQUATION 424...............................................................178LIST OF REFERENCES.............................................................................................................181BIOGRAPHICAL SKETCH.......................................................................................................190 PAGE 7 7 LIST OF TABLES Table page 31 Intensity of cohesion according to size of sediment (Mehta and Li, 1997).......................3432 Classification of sediment by size......................................................................................3433 Properties of clay minerals................................................................................................ .3434 Cation exchange capacity of clay minerals........................................................................3541 Experiment values and parame ters of flocculation models...............................................5542 Summary of flocculation m odels used in this study..........................................................6143 Empirical parameters of the flocculation m odels used for experiment of Spicer et al. (1998)......................................................................................................................... ........8844 Empirical parameters of the flocculation models used for experiment of Biggs and Lant (2000).................................................................................................................... .....8845 Experimental conditions of Burban et al. (1989)...............................................................9246 Empirical parameters of the flocculation m odels used for experiment of Burban et al. (1989)......................................................................................................................... ........9251 Numerical coefficients adopt ed for the eddy viscosity and kequations.......................10661 Sediment transport models combined with or without flocculation models....................12162 Assumed values and calibr ated coefficients of FMs........................................................12663 Calibrated values of empirical coeffi cients for the critical shear stress...........................128 PAGE 8 8 LIST OF FIGURES Figure page 31 Example of definition of fractal dimension.......................................................................3732 Example of two same sized aggregates having different fractal dimensions. A) F =3.0 and B) F =2.5......................................................................................................................3833 Schematic sketch of floc stru cture due to flocculation process.........................................4034 Conceptual diagram for eff ect of turbulent shear and con centration on floc size (Dyer, 1989).......................................................................................................................... ........4141 Evolutions of Fn( X ) with X for three experiments and values of p and q A) p = 1.0, q = 0.5, B) q = 0.5, and C) p = 1.0........................................................................................5442 Model results with different in itial floc sizes (4, 10, 20, 40, 60 and 80 m).....................5543 Experimental results of equilibrium fl oc size reported by Bouyer et al. (2004) and modeled results of FM A and FM B for several dissipation parameters...........................6444 Experimental results of equilibrium floc size measured by Biggs and Lant (2000) and model results of FM A and FM B fo r several dissipation parameters...............................6545 Temporal evolution of floc size measur ed by Biggs and Lant (2000) and calculated by FM A for the case of G =19.4 s1. Three curves represent model results using different sets of k A and k B.................................................................................................6846 Temporal evolution of floc size measur ed by Biggs and Lant (2000) and calculated by FM B for the case of G =19.4 s1. Three curves represent model results using different sets of k A and k B.................................................................................................6947 Temporal evolution of floc size measur ed by Biggs and Lant (2000) and calculated by FM A for the case of G =19.4 s1. Three curves represent model results using different sets of p and q ......................................................................................................7048 Temporal evolution of floc size measur ed by Biggs and Lant (2000) and calculated by FM B for the case of G =19.4 s1. Three curves represent model results using different sets of p and q ......................................................................................................7149 Change of the fractal dimension of FM B with time for the case of G =19.4 s1................72410 Comparison of two flocculation models, FM A and FM B, for T71 experiment of Delft Hydraulics............................................................................................................... ..73411 Comparison of two flocculation models, FM A and FM B, for T69 experiment of Delft Hydraulics............................................................................................................... ..74 PAGE 9 9 412 Comparison of two flocculation models, FM A and FM B, for T73 experiment of Delft Hydraulics............................................................................................................... ..75413 Equilibrium floc sizes due to different dissipation parameters measured by Manning and Dyer (1999) and the calculated results of FM A and FM B for c =120 mg/l...............77414 Equilibrium floc sizes due to different dissipation parameters measured by Manning and Dyer (1999) and the calculated results of FM A and FM B for c =160 mg/l...............78415 Experimental result of Spicer et al. (1998) and model results of FM C............................85416 Experimental result of Spicer et al. (1998) and model results of FM D............................86417 Experimental result of Sp icer et al. (1998) and model results of FM A and FM B...........87418 Experimental result of Biggs and La nt (2000) and model results of FM C.......................89419 Experimental result of Biggs and La nt (2000) and model results of FM D......................90420 Experimental result of Biggs and Lant ( 2000) and model results of FM A and FM B.....91421 Experimental results of case B12 of Burb an et al. (1989) and model results of FM C and FM D....................................................................................................................... ....96422 Experimental results of case B12 of Bu rban et al. (1989) and model results FM A, FM B, and FM C................................................................................................................97423 Experimental results of case B4 of Burb an et al. (1989) and m odel results of FM C and FM D....................................................................................................................... ....98424 Experimental results of case B4 of Burb an et al. (1989) and m odel results FM A, FM B, and FM C.................................................................................................................... ...99425 Temporal evolution of floc size simulate d by FM A combined with a variable yield strength....................................................................................................................... ......10051 Definition of coordinate system.......................................................................................10152 Depthaveraged flow velocity and water depth used to test the sediment transport model. A) The depthaveraged flow velocity and B) the water depth.............................11153 Mass concentration calcul ated by sediment transport model combined with FM C using two types of concentrations....................................................................................11254 Volumetric concentration calculated by sediment transport model combined with FM C using two types of concentrations................................................................................11355 Velocity calculated by sediment trans port model combined with FM C using two types of concentrations....................................................................................................114 PAGE 10 10 56 Mass concentration calcul ated by sediment transport model combined with FM C using one type of concentration.......................................................................................11557 Volumetric concentration calculated by sediment transport model combined with FM C using one type of concentration...................................................................................11661 The Ems/Dollard estuary and the measuri ng pole equipped with a rigid frame for insitu measurement (van der Ham et al., 2001)..................................................................11862 Time evolution of floc sizes simulate d by FMs combined with sediment transport model.......................................................................................................................... ......12263 Time evolution of fractal dimensions simulated by FMs combined with sediment transport model................................................................................................................12364 Time evolution of densities simulated by FMs combined with sediment transport model.......................................................................................................................... ......12465 Time evolution of settling velocities simulated by FMs combined with sediment transport model................................................................................................................12566 Velocities measured and calculated by CMC at 1.0 m....................................................12967 Velocities measured and calculated by CMB at 1.0 m....................................................13068 Velocities measured and calculated by CMA at 1.0 m....................................................13169 Velocities measured and calculated by CMN at 1.0 m....................................................132610 Measured mass concentrations and mass concentrations calcu lated by CMC using a variable fractal dimens ion and yield strength..................................................................133611 Measured mass concentrations and mass concentrations calcu lated by CMB using a variable fractal dimension a nd a constant yield strength.................................................134612 Measured mass concentrations and mass concentrations calcu lated by CMA using a constant fractal dimens ion and yield strength..................................................................135613 Measured mass concentrations and ma ss concentrations cal culated by CMN using constant floc size and density..........................................................................................136614 Measured and simulated mass concentration profiles.....................................................137615 Settling velocities of floc and dissipa tion parameter at 0.5 m, 1.0 m, and 1.5 m above the bottom calculated by CMC. A) settling velocity and B) dissipation parameter........140616 Mass concentrations at 0.3 m and 0.7 m calculated by CMC using the constant c and the variable c ...........................................................................................................142 PAGE 11 11 617 The bottom stress (dotted lin es) and the critical shear stre ss (solid lines) calculated by CMC......................................................................................................................... ...143618 The bottom stress (dotted lin es) and the critical shear stre ss (solid lines) calculated by CMB......................................................................................................................... ...144619 The bottom stress (dotted lin es) and the critical shear stre ss (solid lines) calculated by CMA......................................................................................................................... ..145620 The bottom stress (dotted lin es) and the critical shear stre ss (solid lines) calculated by CMN......................................................................................................................... ..146621 Vertical profiles of size, settling velocity, and mass con centration of floc calculated by CMC and CMB at t =14.6 hr. The velocity at t =14.6 hr is around zero......................148622 Vertical profiles of size, settling velocity, and mass con centration of floc calculated by CMC and CMB at t =18.0 hr. The velocity at t =18.0 is at the peak............................149623 Simulated volumetric concentration pr ofiles. Solid and dotted lines represent simulation results of CMC and CMN..............................................................................151624 Settling velocity plotted as function of floc size..............................................................153625 Relationship between settling velocity and mass concentration calculated by CMC......154626 Relationship between settling ve locity and dissipation parameter ( G ) calculated by CMC............................................................................................................................ .....157627 Relationship between floc size and mass concentration calculated by CMC..................158628 Relationship between settling velocity and volumetric concentration calculated by CMC............................................................................................................................ .....159629 Relationship between settling velocity and density of floc calculated by CMC.............161630 Relationship between settling velocity and mass concentration calculated by CMB......162631 Relationship between settling ve locity and dissipation parameter ( G ) calculated by CMB............................................................................................................................ .....163632 Relationship between floc size and mass concentration calculated by CMB..................164633 Relationship between settling velocity and volumetric concentration calculated by CMB............................................................................................................................ .....165634 Relationship between settling velocity and density of floc calculated by CMB.............166635 Relationship between settling velocity and density of floc calculated by CMA.............168 PAGE 12 12 636 Mass concentration calculated by CM C without damping effect of density stratification................................................................................................................. ....169637 Mass concentration calculated by CMC with c =1.0.....................................................171A1 Schematic description on adopting the mensuration by parts for Eq. A1......................179A2 Schematic description on adopting the mensuration by parts for Eq. A3......................180 PAGE 13 13 LIST OF ABBREVIATIONS a Empirical coefficient for breakup process B1 Empirical parameter for yield stress of floc [N] B2 Empirical parameter for yield strength of floc [N] c Mass concentration [kg/m3] 1C 2C 3C Numerical parameter C Numerical parameter CMA Sediment transport model combined with FM A CMB Sediment transport model combined with FM B CMC Sediment transport model combined with FM C CMN Sediment transport model without flocculation model d Size of primary particle [m] D Size of floc [m or m] D0 Initial floc size [m or m] De Equilibrium floc size [m or m] f cD Characteristic size of floc [m or m] eb, ec, ed Efficiency parameter E Erosion flux fs Shape factor F Threedimensional fractal dimension of floc Fc Characteristic fractal dimension Fc,p Cohesive force of primary particle [N] Fn Function for equilibrium floc size Fy Yield strength of floc [N] FM Flocculation model PAGE 14 14 FM A Flocculation model using constant fractal dimension and constant yield strength FM B Flocculation model using variable fractal dimension and constant yield strength FM C Flocculation model using variable fractal dimension and variable yield strength theoretically derived FM D Flocculation model using variable fractal dimension and variable yield strength empirically proposed by Sonntag and Russel (1987) G Dissipation parameter (Shear rate) [s1] h Water depth [m] k Turbulent kinetic energy [m2/s2] A k Empirical dimensionless coefficient for aggregation process Bk Empirical dimensionless coefficient for breakup process M Total eroded mass [kg] n Number of flocs per unit volume N Number of primary particles within a floc Nrup Number of primary particle s in the plane of rupture Nturb Rate of collision of partic les due to turbulent flow p Empirical coefficient for breakup process q Empirical coefficient for breakup process r Empirical parameter for y ss Specific gravity of primary particle t Time [s] Trel Relaxation time [s] TL Turbulent eddy time scale [s] u xdirection flow velocity [m/s] u Total bottom friction velocity [m/s] PAGE 15 15 U Computed depthaveraged xdirection flow velocity [m/s] U0 Desired depthaveraged xdirection flow velocity [m/s] v ydirection flow velocity [m/s] V Computed depthaveraged ydirection flow velocity [m/s] V0 Desired depthaveraged ydirection flow velocity [m/s] Ws Settling velocity [m/s] X Ratio of the equilibrium floc size to primary particle size Empirical coefficient for variable fractal dimension s Slope of the bottom 1 2 3 Empirical parameter for variable critical stress Empirical coefficient for va riable fractal dimension e Empirical parameter for upward erosion flux f Immersed density of floc [kg/m3] s Immersed density of primary particle [kg/m3] Turbulent dissipation rate (dis sipation rate of energy) [m2/s3] 0 Kolmogorov micro scale [m] Dynamic viscosity [N s/m2] Kinematic viscosity [m2/s] t Eddy viscosity [m2/s] f Density of floc [kg/m3] s Density of primary particle [kg/m3] w Density of water [kg/m3] c k Numerical parameter b Bottom stress [N/m2] PAGE 16 16 c Critical shear stress [N/m2] s Surface shear stress [N/m2] y Yield stress of floc [N/m2] 0 y Scaling parameter for y [N/m2] w x z xdirection fluid stress [N/m2] y zw ydirection fluid stress [N/m2] f Volumetric concentration of floc s Solid volume concentration of primary particle sf Solid volume concentration of pr imary particle within a floc / p x Pressure gradient in xdirection [N/m3] / p y Pressure gradient in ydirection [N/m3] PAGE 17 17 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy FLOCCULATION AND TRANSPORT OF COHESIVE SEDIMENT By Minwoo Son December 2009 Chair: TianJian Hsu Major: Civil Engineering An earlier model for floc dynamics utilizes a constant fractal dimension and a constant yield strength as a part of the model assumptions However, several prior studies suggest that the fractal dimension of floc change s as floc size increases or de creases. Furthermore, the yield strength of floc is observed to be proportiona l to floc size and fractal dimension during breakup process. In this research, a variable fractal dimension is adopted to improve the previous flocculation model. Moreover, an equation for yield strength of floc is theoretically and mathematically derived. The newly derived equati on is combined with flocculation models. By comparing with laboratory experiments on temp oral evolution of floc size (mixing tank and Couette flow), this research demonstrates the importance of incorpora ting a variable fractal dimension and a variable floc yield strength in to the model for floc dynamics. However, it still remains unclear as what are effects of variable fractal dimension and variable yield strength on the prediction of cohesive sediment transport dynamics. The second goal of the present study is to further investigate ro les of floc dynamics in determining the predicted sediment dynamics in a tidedominated environment. A 1DV numerical mode l for fine sediment transport is revised to incorporate four different modul es for flocculation, i.e., no floc dynamics, floc dynamics with assumptions of constant fractal dimension and yield strength, floc dynamics for variable fractal PAGE 18 18 dimensional only, and floc dynamics for consider ing both fractal dimensi on and yield strength variables. Model results are compared with meas ured sediment concentration and velocity time series at the Ems/Dollard estuary. Numerical model predicts very small (or nearly zero) sediment concentration during slack tide when floc dynamics is neglected or inco rporated incompletely. This feature is inconsis tent with the observation. When cons idering variable fractal dimension and variable yield strength in the flocculation model, numerical model predicts much smaller floc settling velocity during slack tide and hen ce is able to predict measured concentration reasonably well. Model results furt her suggest that, when sediment concentration is greater than about 0.1 g/l, there exists a power law relati onship between mass concentration and settling velocity except very near the bed where turbulen t shear is strong. This obs ervation is consistent with earlier laboratory and field experiment on fl oc settling velocity. It is concluded that a complete floc dynamics formulation is importa nt to modeling cohesive sediment transport. PAGE 19 19 CHAPTER 1 INTRODUCTION 1.1 Significance of Study on Cohesive Sediment Transport Sediment transport is an important physic al process that further controls many environmental, geomorphological, and biologica l processes and their relationship with the natural environment. Furthermore, studying sediment transport is of economi cal interest such as the maintenance of navigatable harbors and ch annels through dredgi ng. Sediment transport process is determined by hydrodynamics of carrier flow and sediment characteristics. However, these are very dynamic factors that are dete rmined in accordance with complicated fluidsediment interactions (Winterwer p and van Kesteren, 2004). Thus, to study sediment transport, it is important to understand two representative characteristics, the hydrodynamics of carrier flow and the dynamics of sediment. Hydrodynamic c onditions in fluvial, estuarine, and coastal environment are generally highly dynamic temporally and spatially and flow is often turbulent. Turbulence is the main mechanism to suspend sedi ment. Here, turbulence is one of the essential elements in the study of sediment transport. Sediment is classified into two groups, noncohe sive sediment and cohesive sediment in a broad sense. Sand and gravel are typical nonco hesive sediments. Their electrochemical or biochemical attraction is small enough to be igno red and, as a result, sediment particles are transported individually. On the other hand, Cohesive sediments, the mixture of water and finegrained sediments such as cl ay, silt, fine sand, and organi c material, have cohesive characteristics due to significant electrochemi cal or biologicalchemi cal attraction. The physics of cohesive sediment transport is more co mplicated than noncohesive sediment due to flocculation processes (e.g. Dyer, 1989; Wint erwerp and van Kester en, 2004). Cohesive sediments form floc aggregates through binding t ogether of primary particles and smaller flocs PAGE 20 20 (aggregation), and flocs can disa ggregate into smaller flocs/pa rticles due to flow shear or collision (breakup or disaggreg ation) (Dyer, 1989). The prope rties of floc aggregates continuously change with the fluid flow cond ition. The averaged size of cohesive sediment aggregate is determined by flow turbulence, co ncentration of sediment biologicalchemical properties of water, properties of primary partic le and so on (Lick et al., 1992). Thus, accurate prediction of cohesive sediment transport may requi re detailed water column models that resolve timedependent flow velocity, turbulence and sedi ment concentration (Winterwerp, 2002; Hsu et al., 2007). Moreover, the density of floc aggregates, which is of great importance to further estimate of settling velocity, has a tendency to decrease or increase as the floc size changes (Dyer, 1989; Mehta, 1987; Kranenburg, 1994) Hence, flocculation process should be appropriately investigated when studyi ng cohesive sediment transport. The earths surface is almost entirely covere d with large or small amounts of cohesive sediment (Winterwerp, 2004). In estuaries, large amount of cohesive sediment can be found near the river mouth. Studying the fate of these terrestr ial sediments in the estuary is critical because it significantly affects properties of river and sea bed, carbon sequestration and the health of riverine and coastal habitat/ecology (Goldsmith et al., 2008; Fabricius and Wolanski, 2000). Hence, understanding detailed dynamics of cohesi ve sediment transport is as important as noncohesive sediment transport process. 1.2 Objectives of This Study The major objective of this re search is to understand the dyna mics of cohesive sediment transport in tidedominated environment. The primary tasks performed to achieve this objective are to: (1) develop a flocculation mode l representing natural properti es of cohesive sediments, PAGE 21 21 (2) develop a comprehensive sediment trans port model which can describe transport of cohesive sediment under the conditio n of tide flow and river flow (3) incorporate a flocculation model into a numerical model for cohesive sediment transport, (4) apply the model to estuarie s where river input from upstream and tidal flow coexist, (5) investigate the effect of modeling floc dynamic on cohesi ve sediment transport, and (6) assess the needs for future research. 1.3 Terminology In this section, terminology adopt ed in this study is defined: Aggregate : see floc Aggregation : the process to increase floc size through binding together of primary particles and smaller flocs Breakup : the process to break floc into sma ller flocs/particles due to flow shear or collision Brownian motion : the random movement of pa rticles in fluid due to thermal molecular motion Cohesive force : a physical property of a substance, caused by the electrochemical or biochemical attraction Cohesive sediment : the mixture of finegrained sediment, such as clay particles, silt, fine sand, organic material and so on, having cohesive properties Critical shear stress : the minimum stress to cause erosion of bed Disaggregation : see breakup Dissipation parameter (shear rate) : the para meter that characterizes the effects of turbulence on the evolution of floc si ze (Winterwerp and van Kesteren, 2004) Equilibrium floc size : the size of floc when aggregation and breakup are in equilibrium state Erosion : the removal of sediment from a bed Erosion flux : the rate of erosion from a bed (m/s) Floc : an aggregated pa rticles through bi nding together of primary particles PAGE 22 22 Flocculation : a series of aggr egation and breakup due to cohe sive properties of sediment and flow turbulence Fractal dimension : a statistical quantity th at gives an indication of how completely a fractal appears to fill space Hindered settling effect : the e ffect of particles or concentr ation on settling velocity of a substance Kolmogorov micro scale : the smallest scales of turbulent eddy Lagrangian flocculation model : flocculation model of which interest is in closed system and averaged values Lutocline : a pycnocline due to sedi ment concentration stratification Mass concentration : the mass of sediment pre unit volume of fluidsediment mixture Noncohesive sediment : general coarser sedi ment, such as sand and gravel, of which attraction is not sufficient to aggregate particles Number concentration : the number of suspende d particles such as floc and particles per unit volume of fluidsediment mixture Primary particle : an individual particle not to be broken into smaller particles by general stress in nature such as turbulent sh ear and collisional stress between particles Sediment transport : the move ment of solid particles (sediment) and the processes that govern their motion Self similarity : the property of aggregate that the whole has the same shape as one or more of the parts Selfweight consolidation : Compaction of bed due to selfweight of sediment Settling velocity : the gravityi nduced terminal velocity at which particles fall through the water column Shear rate : See dissipation parameter Sizeclasses flocculation model : flocculation model of which interest is in individual particles having various sizes and which c onsiders input and output of particles Solid volume concentration : the volume of su spended primary particles per unit volume of fluidsediment mixture and the mass concentration is obtained by multiplying it with density of primary particle Total eroded mass : the mass of suspended sedi ment in the water column above unit area of bottom PAGE 23 23 Turbulent eddy time scale : characteristic timescal e of an eddy turnover Turbulent kinetic energy : the mean kinetic energy per unit mass associated with eddies in turbulent velocity fluctuations Turbulent dissipation rate : the rate of the dissipation of turbulent kinetic energy Volumetric concentration of floc : the volum e occupied by flocs per unit volume of fluidsediment mixture Yield strength of floc : the minimum force to break a floc Yield stress of floc : the yield strength divided by the ruptured area of floc 1.4 Outline of Presentation This dissertation is organized with seven chapters and one appendix. Chapter 1 (Introduction) presents the impor tance of study on cohesive sediment, the objectives of the present research, and definitions of terminology used in this dissertation. In Chapter 2 (Literature Review), pr evious studies on flocculation, floc yield strength, and sediment transport modeling are reviewed. Chapter 3 (Study on Properties of Cohesive Sediment) presents general properties of cohesive sediment, fract al theory, fractal dimension, and flocculation process, one of the most im portant properties of cohesive sediment. Chapter 4 (Modeling Flocculation of Cohesive Sediment) first disc usses the characteristic s of two types of flocculation models, sizeclasses flocculati on model and Lagrangian flocculation model. Secondly, the Lagrangian flocculation models ar e derived based on the different assumptions: a constant fractal dimension and a constant yield strength of floc, a variable fractal dimension and a constant yield strength of floc, and a variable fr actal dimension and a vari able yield strength of floc. Thirdly, these different flocculation m odels are used to model several laboratory experiments on the equilibrium floc sizes and temporal evolutions of floc size. In Chapter 5 (Modeling Transport of Cohesive Sediment), governing equations for flow momentum, concentration, turbulent closures, boundary cond itions for tidal flow forcing adopted in the PAGE 24 24 numerical model of sediment transport are presented. These proposed equations and boundary conditions are tested with idealized cond itions as preliminary study. Chapter 6 (Model Application to Ems/Dollard Estuary) discusses re sults of sediment transport models combined with three different flocculation models. The numerical models are applied to the insitu measurement conducted at the Ems/Dollard estuar y (van der Ham et al., 2001). The effects of flocculation and bed erodibility on cohesive sedi ment transport in tidedominated environment are studied in details. In Chapter 7 (Summary, Conclusions and Remarks), major findings of this study are summarized. Concluding remarks for future study is also suggested in this chapter. The appendix demonstrates the detailed deri vation of the equation for the number of particles in the plane crossing the center of a floc with schematic figures to show application of the method of mensuration by parts to this derivation. PAGE 25 25 CHAPTER 2 LITERATURE REVIEW 2.1 Studies on Flocculation and Yield Strength of Floc The flocculation process has been studied by ma ny researchers. The theoretical aspects of the flocculation process have been developed by pioneering studies su ch as Smoluchowski (1917), Camp and Stein (1943), and Ives (1978). These studies have been based on the rate of change of particle numbers due to particle aggregation after collision (Tsai and Hwang, 1995). Lick and Lick (1988) present a more general model for floc dynamics that includes the effects of disaggregation due to collision and shear. Tsai et al. (1987) investigate the effect of fluid shear with natural bottom sediments and suggest th e important factors of collision mechanism according to particle sizes. Lick et al. (1993) fu rther study the effect of differential settling on flocculation of finegrained sediments using natural sediments. McAn ally and Mehta (2000) develop a dynamical formulation for estuarine fine sediment aggregation. The spectrum of fine particle has been represented by a discrete num ber of classes and the frequency of particle collisions due to Brownian mo tion, turbulent shearing and differe ntial settling are described by statistical relationships. They conclude that it is very important to char acterize particle density and strength when flocculation approaches equilibrium state. Flocculation of finegrained particles depe nds on collisions resulted from Brownian motion, differential settling, and turbulent flow shear (Dyer, 1989; Dyer and Manning, 1999; Lick et al., 1993). According to the studies of OMelia (1980), McCa ve (1984), van Leussen (1994), and Stolzenbach and Elimelich (1994), it can be concluded that for cohesive sediment transport in rivers, estuaries and continental shel ves (or other aquatic syst em with more energetic flow) the effects of Brownian motion and differe ntial settling on the flocculation process may be less important. Hence, many studies have focu sed on understanding the effects of turbulence on PAGE 26 26 the flocculation process. Parker et al. (1972) describe the change of number of particles in a turbulent flow as a function of G, the dissipation parameter (or shear rate) defined as / Herein, is the turbulent dissipation rate and is the kinematic viscos ity of the fluid. It is important to note that G is a measure of the small scale turbulent shear. To control G, many studies use a mixing tank. Ayesa et al. (1991) develop an algorith m to calibrate the parameters proposed by Argamam and Kaufman (1970) using data obtained from mixing tank experiments. Tambo and Hozumi (1979) conclude that the maximum floc size is in proportional to the Kolmogorov turbulent length scale. However, none of these studies explicitly describes the variation of floc size with time, which may be necessary for proper understanding and modeling of cohesive sediment transport processes in dynamical environmen t, especially wavedominated condition (Hill and Newell, 1995; Hsu et al., 2007; Traykovski et al., 2000; Winterwerp, 2002). Biggs and Lant (2000) conduct expe riments in order to obtain the temporal change of floc size with respect to a prescribed constant di ssipation rate. In this experiment, samples of activated sludge are stirred in a batch mixing vessel. They conclude that the change in floc size with flow shear follows a power law relationshi p due to the breakage mechanisms. Bouyer et al. (2004) analyze the relationship betw een characteristic floc size and turbulent flow characteristics in a mixing tank. This experiment demonstrates that the average floc sizes are similar after flocculation or reflocculation steps, but the floc size distributions can be different with different impellers. Manning and Dyer (1999) investigat e the relationship between floc size and dissipation parameter under different sediment c oncentrations using an annular flume. They conclude that at low shear rate increasing turbidity encourages floc growth. However, at high shear rate, increasing turbidity in su spension may enhance breakup of floc. PAGE 27 27 Winterwerp (1998; 2002) develops a floccu lation model adopting fractal theory. The concept of fractal geometry has been widely used in order to describe fl oc geometry (see Vicsek, 1992, and Kranenburg, 1994, for a review). Winterwe rpss model describes one characteristic floc size and considers turbulence as the domin ant factor affecting flocculation processes. However, a fixed value of fractal dimension such as 2.0 and 2.2 (Winterwerp, 1998; Winterwerp et al., 2006) has been assumed in the model. Although it is practical and for the sake of simplicity to use a fixed fractal dimension, the applicability of this assumption for sediment transport in different regimes is unclear. For example, fractal dimension of floc in the water column of dilute flow is considered to be around 2.0 (Hawley, 1982; Meakin, 1988). However, large variations of fractal dimension are obtai ned based on field observed estuaries mud (Dyer and Manning, 1999). Moreover, using measured data and constitutive relations for rheology (Kranenburg, 1994), effective stress and permeability (Merckelbach and Kranenburg, 2004) in a consolidating bed, the resulting fr actal dimension is significantly larger than 2.0 (around 2.75). As illustrated by Khelifa and Hill (2006), considering a completely consolidated bed, where all the floc structure is complete ly destroyed, the fractal dimension is 3.0. Hence, a general flocculation model that is able to describe floc dynamics from consolidating bed to dilute suspension must incorporate a variable fractal dimension. Khelifa and Hill (2006) propose a model to pred ict the effective density of flocs and the resulting settling velocity using a variable fract al dimension that depe nds on floc size. They demonstrate that, by using the concept of vari able fractal dimension, the resulting settling velocity converges to Stokes law when the floc size approaches to that of the primary particle. On the other hand, as the floc size becomes ve ry large (more than about 2 mm), the settling velocity decreases as floc size increases. C onsequently, Khelifa and Hill (2006) suggest a new PAGE 28 28 settling velocity formulation that is able to predict measured settling velocity data reported previously for a much wider range of floc sizes. Maggi et al. (2007) also adopt a variable fractal dimension to develop a sizeclasse s flocculation model and conclude that the use of a variable fractal dimension results in better predictions of flocculation process. More recently, Son and Hsu (2008) further extend the floc dynamic equati ons of Winterwerp (1998) for variable fractal dimension suggested by Khelifa and Hill (2006). However, Son and Hsu (2008) show that none of the two flocculation models of Winterwerp (1998) and Son a nd Hsu (2008) is in satisfactory agreement with experimental resu lts for the temporal evolution of floc size in mixing tanks. They conjecture that a constant floc yield strength adapted by these flocculation models may be the main reason causing such deficiency. The yield strength of a floc is a very importa nt parameter in floccu lation process because it has a direct relationship with breakup proce ss during flocculation. Ma ny types of cohesive sediments and techniques have been employed to determine the yield strength of flocs (e.g. Leentvaar and Rebhun, 1983; Francois, 1987; Bache and Rasool, 2001; Wu et al., 2003; Gregory and Dupont, 2001; Wen and Lee, 1998; Yeung a nd Pelton, 1996; Zhang et al., 1999). For example, Wen and Lee (1998) apply a controllabl e ultrasonic field to a floc suspension and observe floc erosion. Zhang et al. (1999) squeez e a single floc in su spension between a glass slide and fiber optic using a for ce transducer. The values of floc yield stress estimated in these studies are in very wide ra nge between the order of 102 to 103 (N/m2) (see Javis et al., 2005, for more details). McAnally (1999) proposes an equati on for yield stress of floc. His derivation starts with the assumption that a floc yield strength is constant as Kran enburg (1994) suggests. Whereas, Tambo and Hozumi (1979) postulate that the floc yield strength is related to the net solids area at the plane of rupture. Son and Hs u (2009) have theoretically derived a constitutive PAGE 29 29 equation for floc yield strength which varies according to change of floc size and fractal dimension. When this variable yield strength is adopted in the floccula tion models of Son and Hsu (2008) and Winterwerp (1998) significant improvement on the temporal evolution of floc size is obtained. Hence, it is conc luded that the flocculation mode l, which uses both a variable fractal dimension and a variable yield strength, is the appropriate flocculation model to further predict sediment transport. 2.2 Sediment Transport Modeling By flocculation process, properties of cohesi ve sediment such as size and density of cohesive sediment aggregates (so called, floc) are variables corresponding to the flow condition. The averaged values of floc properties ar e determined by flow turbulence, sediment concentration, properties of carrier fluid, propert ies of primary particle and so on. Thus, to accurately predict transport of cohesive sediment detailed water column models which resolve timedependent flow velocity, turbulence and se diment concentration are required (Winterwerp and van Kesteren, 2004; Winterwerp, 2001; Hsu et al., 2009). van der Ham and Winterwerp (2001) analyze flow velocity, sediment con centrations and turbul ence measured in the Ems/Dollard estuary with a 1dimensional vertic al (1DV) numerical model. In this study, it is concluded that the damping of fl ow turbulence due to sedimenti nduced density stratification can explain rapid settling of suspe nded sediment towards slack wate r. Li and Parchure (1998) have modeled the damping effect by stratification in the vertical suspende d sediment transport equation and obtained reasonable sitespecific results on predic ted profiles of suspended sediment over mud bank. Hsu et al. (2009) inco rporate turbulence modulation by sediment and the mud rheological stress in their 1DV numerical model. In this study, it is concluded that the damping effect due to sediment induced density stratification and rheological stress play a key role in determining the fluid mud behavior and hydrodynamic dissipation. PAGE 30 30 The settling velocity of sediment is a very important factor to determine downward flux of sediment transport. Furthermore, the settling velo city of cohesive sediment significantly varies with flocculation process (e.g. Mehta and Part heniades, 1975; van Leussen, 1994). Mehta (1986) proposes empirical equations for the settling ve locity and shows that the settling velocity depends on sediment concentration and turbulence. Khelfa and Hill (2006) suggest a model to predict settling velocity based on the concept of fractal geometry. Erosion or resuspension of sedi ment from the bottom is one of the most important factors governing sediment transport in natural water bo dy (Sanford and Maa, 2001) and it is highly dependent on critical shear stress of the bed. For c ohesive sediment, the effect of consolidation is of great importance because critical shear stress of a bed becomes a variable according to the stage of consolidation even if it is composed of the same kind of primary pa rticles. That is, under the condition of consolidated bed, the bed critical shear stress is not constant but a function of various factors. In an idealized 1DV condition, the bed critical sh ear stress can be parameterized by amount of eroded sediment from the bed. Ma ny studies propose the use of a power law relationship between amount of eros ion and critical shear stress (Lick, 1982; R oberts et al., 1998), an exponential form (Mehta, 1988; Chapalain et al., 1994), and a linear relationship (Mclean, 1985; Mei et al., 1997). The work of PiedrCue va and Mory (2001) shows an important characteristic of cohesive sediment. Their flume erosion experiment shows increase of critical shear stress with depth of bed, s uggesting that the consideration of variable critical shear stress of cohesive bed is of great importance. Sanfor d and Maa (2001) propose a power law relationship between critical shear stress and total erode d mass in the water column (unit of kg/m2) from insitu erosion rate measurements in Baltimore Harbor, MD. The rate of increase in critical shear stress is most rapid at low values of total erode d mass and the rate of in crease of critical shear PAGE 31 31 stress approaches zero at nonzero values of total eroded mass. They conclude that this is evidence for the presence of a floc or fluffy layer. PAGE 32 32 CHAPTER 3 STUDY ON PROPERTIES OF COHESIVE SEDIMENT As mention in Chapter 1, cohesive characte ristics of very fine sediments are due to significant electrochemical or biologicalchemica l attraction. Flocculation is one of the most important characteristics of cohe sive sediment. Via flocculation, a ggregate of cohesive sediment (so called, floc) changes their size To study flocculation process, it is of great necessity to understand the structure of flocs. Based on the co ncept of fractal geometry, floc structure is described by a fractal dimension, F. Flocculation process is g overned by aggregation and breakup. The floc yield strength is the force to break a floc into two parts. Thus, it also has an important effect on flocculation pr ocess because breakup is consider ably affected by the force to break a floc. In this chapter, fundamental concep ts for flocculation and yi eld strength of floc are discussed extensively. 3.1 General Properties of Cohesive Sediment The term, cohesive sediment, is generally associated with sediments that are sticky, muddy, and gassy (Winterwerp and van Kesteren, 2004). These properties are also closely to cohesion of sediment. Cohesion is the tendency of particles to bind together under certain conditions (McAnally, 1999). The fine sediments su ch as clay and silt are considered to be affected significantly by cohesion compared to co arse sediment such as sand because weight of fine sediment (that is, inertia) is not large enough to resist its electrochemical or biochemical cohesive force. Mehta and Li (1997) classify fine sediment into several groups by size and describe their intensity of cohesion (Table 31). As mentioned in the previous ch apter, cohesive sediment is a mixture of nonorganic solid material such as clay and silt, organic materi als, water and so on. The solid materials can be classified in accordance with size by ASTM In ternational Standards shown in Table 32. Though PAGE 33 33 colloid can be also classified into solid materi al, it is often consider ed as dissolved matter because it does not settle in water due to Brownian motion. Clay is composed of kaolinite, illite, smectite (or montmorillonite), chlorite and so on and properties of these four clay minerals are listed in Table 33 following studies of Mehta and Li (1997), Grim (1968), Dade and Newe ll (1991), and Ariathurai et al (1977). Clay minerals are important in the sense that cohe sion of sediment is significan tly dependent on them. The cation exchange capacity (CEC) is th e total amount of exchangeable cation. The CEC is proportional to reactivity (that is, cohesive force) of mineral. Table 34 provides the range of CEC for major materials of solid matter of cohesive sediment (Horowitz, 1991; Grim, 1968). The nonclay solid in estuarine environment is commonly composed of quartz and calcium carbonate. The size of nonclay solid is usua lly larger than 2 m which is a criterion dividing clay and nonclay mineral. The relative composition of organic matter in es tuarial cohesive sediment shows very wide variation spatially and seasonally (Kranck, 1980). However, it is clear that organic matter plays an important role on cohesive properties of se diment and flocculation process. Organic matter can be composed of many kinds of material. Fo llowing Berner (1980), the main organic material shown in estuarine environments are : Ploysaccharides and proteins composed of peptides and amino acid Lipides, hydrocarbons like cellulose, ligni n composed of aliphatic and aromatic hydrocarbons Humic actids PAGE 34 34 Table 31. Intensity of cohe sion according to size of sediment (Mehta and Li, 1997) Size (m) Wentworth Scale Classifi cation Intensity of Cohesion < 2 Very fine clay to medium clay Very important 2 20 Coarse clay to very fine silt Important 20 40 Fine silt to medium silt Increasingly importa nt with decreasing size 40 62 Medium silt to coarse silt Partially ignorable Table 32. Classificatio n of sediment by size Sediment Colloid Clay Silt Fine Sand Medium Sand Size <1 m 5 m 75 m 425 m 2,000 m (2 mm) Table 33. Properties of clay minerals Clay Mineral Grain Size (m) Equivalent Diameter (m) Density (kg/m3) Kaolinite 0.1 1 0.36 2,600 2,680 Illite 0.01 0.3 0.062 2,600 2,960 Smectite 0.001 0.1 0.011 2,200 2,700 Chlorite 0.01 0.3 0.062 2,760 3,000 PAGE 35 35 Table 34. Cation exchange capacity of clay minerals Clay Mineral Cation Exchange Capacity (mEq/100g) Kaolinite 3 15 Smectites 80 150 Illite 10 40 Chlorite 5 10 Halloysite (2H2O) 5 10 Halloysite (4H2O) 40 50 Vermiculites 120 200 Attapulgitepalygorskite sepiolite 3 15 3.2 Fractal Dimension Fractal theory describes the geometry of ma ny natural structures that show a rough or fragmented geometric shape that can be split into part s, each of which is a reducedsize copy of the whole (Mandelbrot,1982). This theory can be idea lly applied to structure of floc when a floc is composed of infinitesimally small particle s. Although primary particles in nature have a size O{100} m, Kranenburg (1994) shows that the struct ure of floc can be described by fractal theory from experimental data. According to fr actal theory, the structur e of fractal entity is considered to follow a powerlaw behavior (W interwerp and van Kest eren, 2004). The fractal dimension is the value of power to indicate the structure of fr actal entity and the number of smaller particles (primary particles in this study) in a larger aggregate (a floc in this study). PAGE 36 36 Figure 31 shows an example of fractal dimension as an indicator of number of daughter entities. In this figure, d and D represent size of primary particle an d aggregate of primary particles. The larger aggregate of D=2d is composed of 8 primary particles of size=d and this is described by the equation below: F pD N d (31) where Np is the number of primary part icles within one aggregate and F, the threedimensional fractal dimension, is calculated to be 3.0. Figur es 32 A and B show two different aggregates having the same size, D, and composed of the same primary particle but with different numbers of primary particles. The aggregate shown in Figure 32 A has 64 primary particles of size d. Whereas, the aggregate shown in Figure 32 B is composed of 32 primar y particles of size d. According to Eq. 31, the fractal dimensions of Figures 32 A and 32 B are 3.0 and 2.5, respectively. In many studies, flocs are considered as se lfsimilar fractal ent ities (e.g. Tambo and Watanabe, 1979; Krone, 1984; Logan and Kilps, 1995; Chen and Eisma, 1995, Winterwerp, 1998; Son and Hsu, 2008; Son and Hsu, 2009). This appr oach to the structure of floc is very effective way to describe a number of primary par ticles within a floc. It is important to know the number of primary particles because it becomes po ssible to calculate the density of a floc based on known information on density and size of prim ary particle. Under the assumption of monosized primary particles of size d and when the void within a floc is assumed to be filled with water, Figure 32 gives us a hint to further calculate the density of floc. In this case, the number of primary particles whose volume is d3 and void cubes filled with water whose size is also d3 are PAGE 37 37 (D/d)F and (D/d)3(D/d)F, respectively. Herein, D is the size of floc and d is the size of primary particle. Thus, the total volumes of primary particles and water within a floc are DFd3F and D3DFd3F. Now, the equation for the density of floc, f is obtained by the definition of density, which is the total mass divided by the total volume: 3()F fwfswD d (32) where, s and w are densities of primar y particle and water. Th is equation is derived by Kranenburg (1994) also (see Eq. 411). d D =2 d Figure 31. Example of defi nition of fractal dimension PAGE 38 38 D =4 d A D =4 d B Figure 32. Example of two same sized aggreg ates having different fractal dimensions. A) F =3.0 and B) F =2.5. PAGE 39 39 3.3 Flocculation Process Due to eletrochemical or biologicalchemical properties of cohesive sediment mentioned in Chapter 1, a floc aggregate increases its size by attracting primary particles or smaller flocs. Whereas, a floc can be broken up by many mechanis ms such as collision and turbulence shear because its strength is very small compared to a primary particle. This seri es of processes is the flocculation process, one of the most important characteristics of c ohesive sediment. The schematic sketch of floc structure due to fl occulation process is de picted in Figure 33. Flocculation is essentially different from coagul ation. van Olphen (1977) distinguishes these two processes: Flocculation : Reversible process and the re sult of simultaneous aggregation and breakup Coagulation : Irreversible process to fo rm the primary particles known as flocculi Flocculation process is composed of two main mechanisms, aggregation and breakup. Floc aggregation is the process to increase floc size by gathering primary particles or smaller flocs. It can occur through collisions of particle. This collision is due to Brownian motions, differential settling velocity, and turbulent shear, as mentione d in the section 2.1. Among these mechanisms, turbulent shear mainly governs floc aggregati on (OMelia, 1980; McCave, 1984; van Leussen, 1994; Stolzenbach and Elimelich, 1994). Breakup is considered to be oc curred by interparticle collisions and turbulent shear (e.g. McAnally, 1999). However, the effect of interparticle collision on breakup is often i gnored as size and density of floc are small (Winterwerp, 1998;Winterwerp and van Kesteren, 2004; Son and Hsu, 2009). Turbulent shear exerts fluctuating stress on a floc or the surface of floc. As a result, a floc can be broken into several smaller flocs or peeled off by turbulent shear. Th e effect of turbulent shear on flocculation is PAGE 40 40 described by Dyer (1989). From this study, it is s uggested that floc size first increases and then decreases as shear stress increa ses, and the size is proportional to sediment concentration. Figure 34 shows these characteristics qualitatively. The yield strength of floc is the minimum force to break a fl oc and yield stress is defined as the yield strength divided by the ruptured area. The yield stre ngth depends on th e interparticle bonds between primary particles within a floc (P arker et al., 1972; Bach e et al., 1997). This means that the yield strength is determined by the intensity of each bond and number of primary particles within the floc (Jarvis et al., 2005). Therefore, the yiel d strength and stress have a close relationship with the structure of floc represented by the fractal dimension. The yield strength of floc is an important parameter when flocculati on process is studied because the breakup rate due to turbulent stress is significan tly affected by the yield strength. Primary Particle Figure 33. Schematic sketch of floc structure due to flocculation process PAGE 41 41 Floc Size Shear Stress Concentration Figure 34. Conceptual diagram for effect of turbulent shear and concentration on floc size (Dyer, 1989) PAGE 42 42 CHAPTER 4 MODELING FLOCCULATION OF COHESIVE SEDIMENT 4.1 Overview on Flocculation Modeling To quantitatively predict change of floc proper ties, such as density and size, many types of flocculation models (FM) have b een developed. The first type of flocculation models is based on the rate of change of particle numbers due to particle aggregation by collision (Tsai and Hwang, 1995). McAnally and Mehta (2000) develop a dynamic model for a ggregation rate of cohesive sediment. This model considers both binary and multibody collisions. Parker et al. (1972) consider the change of number concentration as a function of turbulent shear quantified by the dissipation parameter (or shear rate), / G Herein, is the turbulent dissipation rate and is the kinematic viscosity of the fluid. Ayesa et al. (1991) develop an algorithm based on data obtained from a mixing tank experi ment to determine the parameters suggested by Argamam and Kaufman (1970). However, these studies do not provide explicit information on the temporal evolution of floc size which is needed for modeling cohesive sediment transport in dynamic environment such as that under tidal forcing and mo re concentrated fluid m ud transport (e.g. Hill and Newell, 1995; van der Ham and Winterwerp 2001; Uncles and Stephens, 1999; Shi and Zhou, 2004). Winterwerp (1998) develops a semiempirical flocculation model which describes the rate of change of averaged floc size in turbulen t flow. This model is based on the collisional frequency derived by Levich ( 1962), dimensional analysis and assuming flocs are of fractal structure with constant fractal dimension. Flocs have been considered as fractal entities (Tambo and Watanabe, 1979; Huang, 1994; Logan and Kilp 1995). However, the assumption of constant fractal dimension for floc aggr egate may be too restricted for modeling general cohesive PAGE 43 43 sediment transport that has a wide range of fl ow condition and sediment concentration. Khelifa and Hill (2006) suggest a model for floc com posed of monosized primary particles based on variable fractal dimension using a power law. Maggi et al. (2007) also adopt a variable fractal dimension to develop a sizeclasse s flocculation model and conclude that the use of a variable fractal dimension results in be tter predictions of flocculation process. A variable fractal dimension formulation used in Maggi et al. (2007) is very similar to the power law of Khelifa and Hill (2006). The sizeclasses flocculation models such as models of Maggi et al. (2007) and McAnally and Mehta (2000) can resolve more m echanisms causing flocculation compared to the Lagrangian flocculation models (e.g. Winterwerp, 1998; Son and Hsu, 2008). However, the sizeclasses flocculation models are too numerically expensive to be combined with numerical sediment transport models (Lick et al., 1992) and demand many empirical coefficients. More recently, Son and Hsu (2008) further extend the floc dynamic equations of Winterwerp (1998) for variable fractal dime nsion suggested by Khelifa and Hill (2006). However, Son and Hsu (2008) show that none of the two flocculation models of Winterwerp (1998) and Son and Hsu (2008) is in satisfactory agreement with experimental results for the temporal evolution of floc size in mixing ta nks. These models show much gradual increase during the initial flocculation state and much ab rupt increase of floc size when the floc size approaches its equilibrium state. Son and Hsu ( 2008) conjecture that a co nstant yield strength adapted by these flocculation models may be the ma in reason causing such deficiency. From this, Son and Hsu (2009) further improve the flocculati on model adapting a variable yield strength. In this study, they show the flocculation model descri bes temporal evolution of floc size reasonably well only when both a variable fractal dimension and a variable yield strength are considered. PAGE 44 44 4.2 Lagrangian Flocculation Models When modeling cohesive sediment transport, it is needed to consider the change of floc size because many physical qua ntities, such as the settling velocity depends on floc size. On the other hand, the carrier flow turbulence can be da mped due to the presence of sediment. This mechanism may directly or indirectly depend on floc size (Winterwerp, 1998; Hsu et al., 2007). Thus, it is important to develop a model calcul ating the temporal evolution of floc size under given flow conditions. As mentioned the section 4.1, the sizeclasses model is numerically too expensive to be combined with detailed turbulence sediment transport model although it can describe many mechanisms of flocculation process and size distribution of flocs. Therefore, the Lagrangian flocculation models are adopt ed and investigated in this study. 4.2.1 Winterwerps Flocculation Model A Lagangian flocculation model is devel oped by Winterwerp (1998) based on previous studies (e.g. Levich, 1962; van Leussen, 1994) and under the assumption of a constant fractal dimension. The relationship between the volumetric concentr ation of flocs, the mass concentration of primary particles and the number of floc s per unit volume of fluid is given by: 3 sw fs fwsc f nD (41) where f is the volumetric con centration of flocs, c is the mass concentration of primary particles, fs is a shape factor (for sphere, /6sf ) and n is the number of flocs per unit volume of fluid (water in this study). By combining Eqs. 32 and 41, n can be represented as: PAGE 45 45 3 FF ssc ndD f (42) By assuming particle diameters are much smaller than the Kolmogorov length scale (0 ) and based on the theory of Smoluchowski (1917), Levi ch (1962) suggests the ra te of collision of particles due to flow turbulen ce can be determined by integrating the diffusion equation over a finite volume: 323 2turbdNeGDn (43 a) where ed is an efficiency parameter for turbulent diffusion. Winterwerp (1998) further assumes that only a certain portion of the collisions cau ses flocculation and proposes the equation for the rate of aggregation between the flocs in a turbulent fluid: 323 2cddn eeGDn dt (43 b) where ec is a constant efficiency parameter accounting for the fact that not all collisions result in coagulation. Although ec can be a variable as floc size and floc density change, it is assumed to be constant for simplicity because ec cannot be determined at pres ent on the basis of theoretical arguments (Lick et al., 1992). PAGE 46 46 From Eqs. 41 and 43 b, the increase rate of flocs is obtained for the constant mass concentration: 31 FF ssdnFc dD dDf (44) Eq. 44 can be expanded below: 31 FF ssdndndDFcdD dD dtdDdtfdt (45) By substituting Eq. 45 to Eq. 43 b and re arranging this, the equation describing the increase rate of floc size is obtained: 343431 2FFFF cd A sssee k dDc GdDcGdD dtfFF (46) where 'Ak is a dimensionless coefficient. This e quation describes the aggregation process to increase the size of floc by attracting primary particles to a larger floc. Winterwerp (1998) assumes interparticle collis ions are apt to cause aggregation of flocs rather than breakup. Hence, only the breakup by turbulent shear stress is incorporated here. Winterwerp (1998) suggests that the breakup rate is a f unction of the dissipat ion parameter of the disrupting turbulent eddies and proposes the following relation based on dimensional considerations: PAGE 47 47 21 /q p ydnDdG Ga ndtdFD (47) where is the dynamic viscosity of the fluid, Fy is a yield strength of flocs, and a, p, and q are the coefficients to be discussed later. Winterwerp (1998) considers that Fy is dependent of the number of primary particle in a plane of failu re and the number of primary particle in the plane is constant. As a result, Fy is fixed to be constant in this flocculation model. By further incorporating an efficiency parameter for floc breakup, be, the increase rate of floc due to breakup is obtained: 121 2/qq pp qq b B yyae k dDDdGDd DGGD dtFdFDFFd (48) where Bk is also a dimensionless coefficient. The negative sign () is a dded because breakup of floc decreases floc size. By linearly combining Eq. 46 for aggregation process and Eq. 48 for breakup process, the complete flocculation model of Winterwerp (199 8) is proposed and it is denoted as Model FM A (see Table 42): '' 341211q p FFqq AB sykk dDDd cGdDGD dtFFFd (49) PAGE 48 48 As mentioned above, this flocculation model has been derived assuming that the fractal dimension, F is constant. Many experimental studies sugge st that floc size is proportional to the Kolmogorov length scale (Bratby, 1980; Aker s et al., 1987; Leentvaar and Rebhun, 1983), 0 =31/4(/) where is the kinematic viscosity of fluid and is the turbulent dissipation rate. Hence, Winterwerp (1998) assumes that the equilibrium floc size, De, is in proportion to 1/ G Further assuming that the equilibrium floc size is much larger than primary particle size, Winterwerp (1998) suggests p =1.0 and q =0.5 in this flocculation model using fixed fractal dimension. With these values for empirical coefficients and F =2.0 which is assumed by Winterwerp (1998), Eq. 49 is simplified: 0.5 '2'22AB sydDGcG kDkDdD dtdF (410) 4.2.2 Flocculation Model Using a Variable Fractal Dimension The main concept of fractal theory is selfs imilarity of the floc structure. Under this assumption, the concept of fractal theory can be used to develop a model describing the floc aggregation process. The model development adopt ed in this section is based on previous two studies of Winterwerp (1998) for floc dynamics a nd Khelifa and Hill (2006) for variable fractal dimension. For monosized primary particles of size d the effective density of floc is calculated as (Kranenburg, 1994): PAGE 49 49 3()F fwswD d (411) where F is the threedimensional fractal dimension of flocs. To take into account the possible variability in the structure of flocs, a variab le fractal dimension depending on floc size is proposed by Khelifa and Hill (2006). The fractal dime nsion of a floc with size closer to the size of the primary particles should approach the value of 3.0 (Khe lifa and Hill, 2006). On the other hand, the fractal dimension of large flocs should be close to the value of 2.0 (Dyer, 1989; Dyer and Manning, 1999; Meakin, 1988; Winterwerp, 1998). Hence, a power law is proposed by Khelifa and Hill (2006) to describe variation of fractal dimension: D F d (412) where 3 and log(/3) log(/)cc fF Dd Fc is a characteristic fractal dimension and f cD is a characteristic size of flocs. Khelifa and Hill (2006) suggest the typical value of Fc and f cD to be Fc = 2.0 and f cD = 2000 m. Eg. 412 gives a plausible descri ption of fractal dimension such that when d< PAGE 50 50 and dn dD can be calculated as: 313 ln1FF ssdncD dD dDfd (414) Utilizing dDdDdn dtdndt and Eqs. 43 b and 414, an equa tion representing the evolution of floc size due to aggregation is obtained: 341 2 ln1FF cd sscee dD GdD D dtf d (415) As mentioned previously, the c oncept of fractal theory is ba sed on selfsimilarity of the structure. Although this is appropriate for a ggregation process, breakup process may be too abrupt to entirely adopt variab le fractal dimension. Following Winterwerp (1998), it is assumed that interparticle collis ions are apt to cause aggregation of flocs rather than breakup. Hence, only the breakup by turbulent shear stress is inco rporated here. Substituting Eqs. 42 and 414 into Eq. 47, the balance equation for the decay rate of flocs by breakup process can be written as: 121 () 3 ln1q pqp b yeGa dDG dDDd D dtF d (416) PAGE 51 51 In the present study using vari able fractal dimension, it is necessary to check the robustness and the sensitivity of p and q in the context of variable fractal dimension. This issue shall be discussed next. Using a linear combination of aggregation a nd breakup processes (Winterwerp, 1998), i.e., Eqs. 415 and 416, a complete flocculation mode l using a variable fract al dimension can be obtained and it is denoted as Mode l FM B (see also Table 42): '' 3421() 33 ln1q FFpqp AB sykk dDGdcG dDdDDd D dtF d (417) where '3 2cd A see k f and Bbkae are empirical dimensionless coefficients. For equilibrium condition, i.e. dD/dt=0, and using the assumption that De is much larger than d, Eq. 417 can be simplified as: 2 3(23) '()0q Xpq A n sByk cGd FXX kF (418) where X=/eDd. Eq. 418 is a nonlinear algebraic equation of De. A numerical solution for equilibrium floc size ca n be obtained by setting Fn zero. In this study, a numerical solution of the time evolution equation of floc si ze, i.e. 417, will be directly calculated using a RungeKutta method. However, it is important to first examine the effects of p and q on the flocculation model using Eq. 418 because their behavior affects th e nonlinearity and hence numerical stability of PAGE 52 52 time evolution of Eq. 417. Figur e 41 A shows the variation of Fn(X) with X for three flocculation experiments: T69, T 71, and T73 carried out in Delf t Hydraulics (see Van Leussen, 1994). Flow conditions and coefficients for thes e simulations are shown in Table 41. Using values suggested by Winterwerp (1998), p=1.0, q=0.5 (and with Ak=0.15 and Bk=105), Eq. 418 has solution (i.e., for the range X > 1, Fn(X)=0 exists). Figure 41 B fu rther presents the evolution of Fn(X) with several p values for test T71 but with q remains 0.5. When p is as large as 1.3, Fn(X) approaches zero rapidly and the resulting De is very close to d (i.e., De/d=3.15). Moreover, when setting p to be 0.7, Fn(X) has no root and the equilibrium fl oc size does not exist. That is, Eqs. 418 and the flocculation model, equation 417, become unrealistic when p=0.7. Figure 41 C further shows the evolution of Fn(X) with several q values for test T71 with p=1.0. When q is 0.7, Fn(X) increases rapidly with respect to X. In contrast, Fn(X) increases very slowly when q is 0.3. Hence, when q is smaller than 0.3, a highly accurate an d stable numerical solver is necessary in order to obtain a solution for Eq. 417. Fr om these observations, it can be concluded that values of p=1.0 and q=0.5 originally suggested by Wi nterwerp (1998) based on physical arguments are also rather robust numerically fo r the present flocculati on model using variable fractal dimension. When the values suggested by Winterwerp (1998) for p=1.0 and q=0.5 are adopted for FM B using variable fractal dimension, Eq. 417 can be rewritten as: 0.5 '3412() 33 ln1FF B A syk dDGdcG kdDdDDd D dtF d (419) Figure 42 shows the dependence of modeled time evolution of floc si ze on the initial floc size with other parameters kept the same: G=7.3 s1, c=0.65 kg/m3, '0.98Ak '53.310Bk PAGE 53 53 and Fy=1010 N. It is observed that the initial floc si ze affects the time to reach the equilibrium state, but not on the final (equilibrium) floc size. Notice that most of the field or laboratory experiments cannot start with primary particles becau se it is difficult to keep all primary particles completely separated before each experiment. Model results presented here are insensitive to this uncertainty as far as the final floc size is concerned. This section presents a semiempirical model to describe flocculation process of cohesive sediment in turbulent flow. For aggregation pr ocess, a variable fractal dimension is adopted under the assumption that a floc has the character istic of selfsimilarity the main concept of fractal theory. The model for breakup mechanism is based on studies of Winterwerp (1998) and Kranenburg (1994), which are semi empirical and requires determination of several empirical coefficients. By a linear combination of the formulations for aggregation and breakup processes, a flocculation model which can desc ribe the evolution of floc size with time is obtained. The values of the exponent p and q for breakup process suggested by Winterwerp (1998) are shown to be also appropriate here for model based on variable fractal dimension (see Figure 41). However, a yield strength of floc is still kept to be constant although it is considered to be variable based on previous studies (e.g. Ye ung and Pelton, 1996; Tambo and Hozumi, 1979). A variable yield strength is further incorporated into floccula tion modeling in the next section. PAGE 54 54 0 50 100 100 80 60 40 20 0 20 40 60 80 100 XFn(X) T73 T71 T69 A 0 50 100 150 200 100 80 60 40 20 0 20 40 60 80 100 XFn(X) p=1.3 p=1.0 p=0.7 0 50 100 100 80 60 40 20 0 20 40 60 80 100 XFn(X) q=0.7 q=0.5 q=0.3 B C Figure 41. Evolutions of Fn(X) with X for three experiments and values of p and q. A) p = 1.0, q = 0.5, B) q = 0.5, and C) p = 1.0. PAGE 55 55 Table 41. Experiment values and parameters of flocculation models A k 'Bk Test NO. c (kg/m3) G (s1) d ( m) Fy ( N ) s (kg/m3)FM B FM A FM B FM A T71 0.65 7.3 4 10102650 0.98 0.3095 3.3 105 3.54 105 T69 1.17 28.9 4 10102650 0.98 0.3095 3.3 105 3.54 105 T73 1.21 81.7 4 10102650 0.98 0.3095 3.3 105 3.54 105 Figure 42. Model results with different initial floc sizes (4, 10, 20, 40, 60 and 80 m) PAGE 56 56 4.2.3 Flocculation Model Using a Variable Frac tal Dimension and Variable Yield Strength As presented in Eq. 411, the fractal dimension is an indicator to describe how dense a floc is for a given size and density of the primary pa rticles. In other words, the number of primary particles within a floc and in the ruptured plane of a floc is also a function of the fractal dimension. By the definition, the floc density can be calculated as: (1) f sssw (420) where s f is the solid volume concentration of primary particles within a floc. Rearranging Eq. 420, s f is expressed as: f sf s (421) where f (= f w ) is the immersed density of floc and s (= s w ) is the immersed density of primary particle. By substitu ting Eq. 411 to Eq. 421, the equation for s f is rewritten as: 3 F sfD d (422) PAGE 57 57 Under the assumption that the fl oc and the primary particles ar e spherical for the sake of simplicity, the number of primary particles within a floc, N, is derived from the definition, 3 3 sfNd D, and Eq. 422: FD N d (423) In addition, the number of prim ary particles within a floc is assumed to be sufficient to adopt mensuration by parts. Using mensuration by parts, the averag e distance between two neighboring primary particles within a floc is determined. From this, one can further determine the averaged area occupied by one primary partic le in the ruptured plane. Consequently, an equation for the number of primary particles rupN in the plane crossing the center of a floc, whose size is D, can be derived as (see Appendix for more details): 2 2/3 346F rupD N d (424) In this study, the plane crossing the center of a floc is assume d to be the ruptured plane of the floc due to the action of turbulent shear. This shall be discussed more later. FM A and FM B adopt a consta nt yield strength of floc, Fy, under the assumption that the number of bonds (or primary partic les) in a ruptured plane is inde pendent of the size of the floc (e.g. Kranenburg, 1994). However, Boadway ( 1978) and Tsai and Hwang (1995) have observed floc breakup process and conclude d that a floc often disaggregat es into two roughly equalsized PAGE 58 58 flocs. Hence, it is assumed here that during floc br eakup, a floc is divided by the plane which contains the center of floc as the two daughter flocs have the same size after breakup. The number of primary particles in this plane shou ld be a function of floc size and its fractal dimension. The yield strength of floc depends on the strength of interp article bonds between the primary particles and the number of these bonds within a floc (Parker et al., 1972; Boller and Blaser, 1998). Thus, the yield stre ngth is also a function of floc size and its structure (e.g. Yeung and Pelton, 1996). Sonntag and Russ el (1987) suggest an empirical equation for the yield stress (units, Pa), which is the yiel d strength divided by the cr osssectional ar ea of floc ( 24D), based on a power law: (3) 00((/))rrF yysfyDd (425) where 0y is a scaling parameter and r is an empirical coefficient. According to a later study by Bache (2004), the value of r is usually in the range between 0.5 and 1.5. Following the definition of the solids volume concentration within a floc, s f (see Eq. 422), it can be concluded that the yield strength is not constant but a func tion of fractal dimension and floc size. This discussion on a yield strength is cons istent with earlier st udy by Tambo and Hozumi (1979) who postulate that the yield strength of fl oc is proportional to th e net solid area in the ruptured plane. In addition, it is clear that the yield stre ngth of floc has a direct relationship with the cohesive force of each primary particle. Thus, the magnitude of the yield strength is considered as the sum of cohesive force of all primary particles in the ruptured plane and, in this section, a new equation for floc yield st rength is proposed based on Eq. 424: PAGE 59 59 2 3 2 F yD FB d (426) where 2/3 2,46cp B F and Fc,p is the cohesive force of primary particle. Fc,p is considered as an empirical parameter because it depends on the properties of sediment and chemicalbiological effects. Further dividing Eq. 426 by th e area of ruptured plane, an equation for the yield stress of a floc, y is obtained: 2 3 2 1 2(/4)F y yF D B D Dd (427) where 2/3 1,6cp B F It can be seen that in this formulation, the only parameter that is difficult to obtain and needs to be empirically determined is Fc,p. Utilizing chain rule, Eqs. 414, 47 and 27 for floc yield stress, the equation for floc breakup due to turbul ent shear is obtained: 22 12 33 11 3 ln1q qq pFqF p beGa dDG dDDd D dtB d (428 a) The above equation adopts the yield stress equa tion theoretically derive d in this study (Eq. 427). Similarly, utilizing the yield stress equa tion suggested empirica lly by Sonntag and Russel (1987) (see Eq. 425), the following equa tion for floc breakup is obtained: PAGE 60 60 (3)1(3) 01 3 ln1q p prqFrqF b yeGa dDG dDDd D dt d (428 b) The complete flocculation models are obt ained by linearly co mbining flocculation processes due to collisions and turbulent shear in this section also. For the flocculation model that utilizes theoretical derivation of floc yield stress developed in this study (Eq. 427), it is denoted as Model FM C (see also Table 42) and is written as: 22 '' 1(3) 34 33 133 ln1q qq pFF p FF AB skk dDGdcG dDdDDd D dtB d (429 a) The flocculation model utilizes floc yield stre ss of Sonntag and Russel (1987) is denoted as FM D (see Table 42) and is given as: '' 34(3)1(3) 033 ln1q p FFprqFrqF AB sykk dDGdcG dDdDDd D dt d (429 b) Essentially, p and q are empirical coefficients as men tioned in the previous section. The values of p =1 and q =0.5 based on several additional constr ains are also used FM C and FM D. PAGE 61 61 Table 42. Summary of flocculation models used in this study Model name Characteristic Reference FM A Constant fractal dimension Constant Fy Eq. 49 FM B Variable fractal dimension Constant Fy Eq. 417 FM C Variable fractal dimension Variable yield strength theoretically derived in Section 4.2.3 Eq. 429 a FM D Variable fractal dimension Empirical variable yield stress of Sonntag and Russel (1987) Eq. 429 b PAGE 62 62 4.3 Investigation of Flocculation Models 4.3.1 Application of FM A and FM B As shown in Eq. 419, the fl occulation model using a variab le fractal dimension (FM B) depends primarily on five parameters: the floc size d and density s of the primary particle, the yield strength of flocs Fy, and empirical parameters Ak and Bk For test of FM A and FM B, this study follows Winterwerp (1998) where d Fy, and s are assumed to be 4 m O{1010} N, and 2,650 kg/m3. Winterwerp (1998) specifies these va lues based on experimental data and information adopted by previous literatures (Matsuo and Unno, 1981; van Leussen, 1994). Specifically he estimates th e yield strength of floc Fy, to be about O{1010} N, but also acknowledges that for natural mud Fy may change by several order of magnitude depending on the chemicalbiological properties of the floc. Finally, Ak and Bk are empirical coefficients that may vary with fluid/sediment pr operties and possibly sediment c oncentration. Hence, these two empirical coefficients are calibrated for each experiment. Bouyer et al. (2004) carry out experiments on floc size distribution in a mixing tank. In these experiments, a synthetic suspension of bentoni te is used to mimic the behavior of particles in natural water. The concentration of bentonite is fixed at 0.03 kg/m3. The dissipation parameter, G varies from 5 to 300 s1 and the mean floc sizes of floc are measured for each value of G It is assumed that the equilibrium floc size is close to the measured mean floc size of flocs. To simulate these experiments, 'Ak and 'Bk for FM A and FM B are determined to be 1.82 and 1.9106 and 1.02 and 3.8106 by matching the model results with measured data. The results for all 9 tests reported by Bouyer et al. ( 2004) are plotted in Figure 43, which shows the variation of the equilibrium floc size with the dissipation pa rameter. Solid line is a regression of measured results and dotted lines are those of the flocculation models. In this figure, the floc size predicted PAGE 63 63 by the FM B shows good agreement with the expe rimental data. The model is capable of predicting the equilibrium fl oc size at different levels of homogeneous turbulence. Biggs and Lant (2000) report th e measured equilibrium floc sizes of activated sludge for various magnitudes of dissipation parameter. 60 ml of activated sludge is added with 1.135 liter of filtered effluent (0.45 m Millipore filters) to a 1.2 l baffled batch vessel and mixed with a flat six blade impeller. Because the mass of total slud ge diluted with effluent is not reported, the mass concentration is derived here from the volumetric concentration based on the assumption that the density of sludge to be 1,300 kg/m3 and the density of primary particle to be 2,650 kg/m3. The calculated mass concentration is 24.19 kg/m3 and is rather concentrated. Using the impeller, 4 dissipation parameters are tested: 19.4, 37.0, 113, and 346 s1. To model these experiments, sets of values, Ak = 0.017 and Bk = 2.4105 for FM B and Ak = 0.008 and Bk = 4.4105 for FM A, are used based on bestfit of the model results with the case of G = 19.4 s1. The results for all test cases are shown in Figure 44. The solid line is a regressi on of measured results and the dotted lines are for results of the flocculati on models. The floc sizes in the range of G = 19.4 and G = 113.0 s1 are in good agreement with experimental results. However, it is evident that the equilibrium floc sizes calculated by both FM A and FM B show a m ilder slope in the loglog plot than the experimental results. Comparing the me asured results between Bouyer et al. (2004) and Biggs and Lant (2000), it appears that the decay of equilibrium floc size (or breakup process) is enhanced when concentration is higher (see also Figure 414). PAGE 64 64 1101001000G (s 1 ) 10 100 1000Floc size (micron ) Measured result FM A FM B Figure 43. Experimental results of equilibrium floc size reported by Bouyer et al. (2004) and modeled results of FM A and FM B for several dissipation parameters PAGE 65 65 101001000G (s 1 ) 10 100Floc size (micron ) Measured result FM A FM B Figure 44. Experimental results of equilibri um floc size measured by Biggs and Lant (2000) and model results of FM A and FM B for several dissip ation parameters PAGE 66 66 Figures 45 and 46 present the temporal ev olutions of floc size for the case of G = 19.4 s1. Because Ak and Bk are chosen to bestfit the calculated equilibrium floc size with the measured data according to this case, it allows us to ev aluate the model capability on the timedependent floc evolution. According to Biggs and Lant (2000), the initial floc size is about 15 m for this experiment. It appears that this experiment is not started with completely deflocculated primary particles because 15 m appears to be too large for typical size of primary particles. Thus, the model calculation is conducted based on the assu mption that the initial condition of cohesive sediment in the vessel is not primary particles but micro flocs having larger size. Under this assumption, the initial floc size is set to be 15 m and the primary particle size is assumed to be 4 m. The measured and modeled tem poral evolutions of floc size ar e plotted in Figures 45, 46, 47, and 48. The dotted curves of Figures 46 and 48 represent results of FM B using Ak =0.017, 'Bk =2. 4105, p =1.0, and q =0.5 (i.e., parameters that are iden tical to that shown in Figure 44). Overall, the shapes of the fl oc size evolution are not predic ted well by the models although the final equilibrium floc size is pr edicted. The measured floc evolution shows a less apparent Scurve shape. The floc size has a more rapid initi al increase with time but shows a more gradual increase of floc size when appr oaching equilibrium. On the cont rary, the model results predict a more gradual increase during the initial stage and a pproach to the equilibrium state more rapidly. In Figure 46, the dasheddot curve and da shed curve represent model results with Ak =0.020 and Bk =2.84105 and Ak =0.015 and Bk =2.13105, respectively. All of them use p =1.0 and q =0.5. The purpose of these tests is to evaluate the sensitivity of model results on 'Ak but 'Bk need to be changed slightly in order to match the given equi librium floc size. It can be concluded that the shape of curve is only slightly affected by Ak and Bk In order to furthe r study the effects of p PAGE 67 67 and q three sets of p and q are tested (with Ak = 0.017 and Bk = 2.4105) and the model results are shown in Figures 47 and 48. The dasheddot curve and dashed cu rve represent the model results with p = 1.05 and q = 0.34 and p = 0.95 and q = 0.65 with fixed kA =0.008 and kB =4.4105 for FM A and kA =0.017 and kB =2.4105 for FM B. Apparently, changes of p and q also do not have significant effect on the shape of the time evolution of floc size. Figure 49 represents the temporal change of fractal di mensions calculated by FM B. Three lines have different sets of kA and kB with fixed p =1.0 and q =0.5. As the floc sizes approach the equilibrium, the values of fractal dimensions approach 2.4. Because the initial floc size is assumed to be 15 m the initial value of fractal dimension is not 3.0 but 2.75. Winterwerp (1998) develops FM A based on fi xed fractal dimension and his research is one of the bases of FM B. In Winterwerp ( 1998), the model coefficien ts are calibrated or estimated using experimental data measured in Delft Hydraulics (see van Leussen, 1994). It is assumed that at t = 0 the initial particle size equals to the size of the primary particles, i.e. D0 = d = 4 m and the maximum floc size measured equals the equilibrium value. Other values required by the flocculation model are given in Table 41 and all these values are determined from the measured results of test T73. PAGE 68 68 0 20 40 60 80 100 0 20 40 60 80 100 120 140 160 Measured result kA ` = 0.008, kB ` = 4.40e5 kA ` = 0.009, kB ` = 4.95e5 kA ` = 0.007, kB ` = 3.85e5 Figure 45. Temporal evolution of floc size measured by Biggs and Lant (2000) and calculated by FM A for the case of G =19.4 s1. Three curves represent model results using different sets of k A and k B. PAGE 69 69 0 20 40 60 80 100 0 20 40 60 80 100 120 140 160 Measured result kA ` = 0.017, kB ` = 2.40e5 kA ` = 0.020, kB ` = 2.84e5 kA ` = 0.015, kB ` = 2.13e5 Figure 46. Temporal evolution of floc size measured by Biggs and Lant (2000) and calculated by FM B for the case of G =19.4 s1. Three curves represent model results using different sets of k A and k B. PAGE 70 70 0 20 40 60 80 100 0 20 40 60 80 100 120 140 160 Measured result p = 1.00, q = 0.50 p = 1.05, q = 0.34 p = 0.95, q = 0.65 Figure 47. Temporal evolution of floc size measured by Biggs and Lant (2000) and calculated by FM A for the case of G =19.4 s1. Three curves represent model results using different sets of p and q PAGE 71 71 0 20 40 60 80 100 0 20 40 60 80 100 120 140 160 Measured result p = 1.00, q = 0.50 p = 1.05, q = 0.34 p = 0.95, q = 0.65 Figure 48. Temporal evolution of floc size measured by Biggs and Lant (2000) and calculated by FM B for the case of G =19.4 s1. Three curves represent model results using different sets of p and q PAGE 72 72 0 5 10 15 20 25 30 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 Time (min)Fractal dimension kA ` = 0.017, kB ` = 2.40e5 kA ` = 0.020, kB ` = 2.84e5 kA ` = 0.015, kB ` = 2.13e5 Figure 49. Change of the fractal dime nsion of FM B with time for the case of G =19.4 s1 Figures 410, 411, and 412 present the result s of two flocculation models, FM A and FM B. The solid curves are the results of FM A, th e dotted curves are results of FM B, and circles are experimental data. It is can be observed th at the flocculation model using a variable fractal dimension, FM B, has a slightly more smoot h Scurve than the flocculation model using a constant fractal dimension, FM A. Results calcu lated by both models are in fair agreement with experimental results in terms of the equilibrium floc size. Considering all three test cases for equilibrium floc size, model resu lts using a variable fractal dime nsion, FM B, appear to agree with the experimental data slightly better than results of flocculation model using a constant fractal dimension, FM A. This is qualitatively consistent with the conclusion made by Khelifa and Hill (2006) on settling velocity us ing a variable fractal dimension. PAGE 73 73 102 104 100 101 102 103 FM B FM A Measured result Figure 410. Comparison of two flocculation models, FM A and FM B, for T71 experiment of Delft Hydraulics PAGE 74 74 102 104 100 101 102 103 FM B FM A Measured result Figure 411. Comparison of two flocculation models, FM A and FM B, for T69 experiment of Delft Hydraulics PAGE 75 75 102 104 100 101 102 103 FM B FM A Measured result Figure 412. Comparison of two flocculation models, FM A and FM B, for T73 experiment of Delft Hydraulics PAGE 76 76 Manning and Dyer (1999) examine the rela tionship between floc size and dissipation parameters (12.8 45.2 s1) under the condition of increasing concentrati on (80 200 mg/l). The experiment is carried out in a laboratory flume with a nonintrusive macrolens miniature video camera. The sediments used for the experiment have been collected from an intertidal mudflat. Figures 413 and 414 show the results calcul ated by FM A and FM B. To simulate these experiments, the initial floc size is assumed to be 15 m and kA and kB for FM B are 0.55 and 4.8106 when c = 120 mg/l and 0.50 and 5.8106 when c = 160 mg/l. Using FM A, empirical coefficients for kA and kB are 0.33 and 1.15105 when c = 120 mg/l and for higher concentration c = 160 mg/l condition, kA and kB are specified as 0.30 and 1.40105, respectively. In this section, di fferent types of sediments of vari ous concentrations are tested and the resulting empirical coefficients kA and kB are quite different. It can be concluded here that for the same sediment source considered in this case, the variation of calibrated coefficients are significantly smaller. However, it appears that these empirical coefficients may still depend on sediment concentrations despite concentration is al ready a variable in the aggregation term of the flocculation models. In practical applications, when the variation of sediment concentration is significant, it may be necessary to calibrate the empirical coefficients acc ording to the magnitude of concentration. PAGE 77 77 1020304050G (s 1 ) 100200 90 80 70 60 50Floc size (micron ) Measured result FM A FM B Figure 413. Equilibrium floc sizes due to different dissipation parameters measured by Manning and Dyer (1999) and the calcul ated results of FM A and FM B for c =120 mg/l PAGE 78 78 1020304050G (s 1 ) 100200 90 80 70 60 50Floc size (micron ) Measured result FM A FM B B Figure 414. Equilibrium floc sizes due to different dissipation parameters measured by Manning and Dyer (1999) and the calcul ated results of FM A and FM B for c =160 mg/l PAGE 79 79 Such a weak point of the flocculation models is emphasized by further considering the decay rate of equilibrium floc size with resp ect to the dissipation parameter for different sediment concentrations. Floccu lation models show good agreements with experimental results when the mass concentration is 120 mg/l. Howeve r, when the mass concen tration is 160 mg/l, it is evident that the regression curves of the mode l results for both FM A and FM B show different slopes compared to the experimental results. Th erefore, according to measured data reported by Manning and Dyer (1999), the decay rate of equilib rium floc size with respect to the dissipation parameter also depends on the mass concentrat ion. However, following Winterwerps study (1998), the equilibrium floc size of FM A depends on the dissipation parameter G to the power of 2/( p +2 q + F 3) and the value of p and q are further chosen such that it is 0.5. This strategy is also similarly adopted by FM B. As a result, mo del results can only produce more or less a single value of slope under the condition of different concentrations when the relationship between equilibrium floc sizes an d dissipation parameters, G are plotted. Future work is necessary to further study this issue. More experimental data is necessary to fully unde rstand the decay rate of floc size with respect to the dissipation paramete r for various concentrations. In addition, it is also possible to propose an empirical relation of p and q that depends on concentration. In this section, the capability and limitati on of FM A and FM B are validated by four experimental data sets. In terms of equilibri um floc size, model results agree reasonably well with the measured data (see Figures 43, 44, 410, 411, 412, and 413) provided that empirical coefficients are calibrated. This is partially because of the variation of chemicalbiological properties of the cohesive sediment tested. However, through modeldata comparisons with Manning and Dyer (1999) (Figur e 414), it becomes clear that the empirical coefficients, specifically q also depends on sediment concentration. Qualitatively, the model predicts PAGE 80 80 equilibrium floc size decreases as dissipation parameter increa ses, suggesting that the model captures observed floc dynamics that strong tu rbulence has a tendency to break the floc and reduce the floc size. As shown in Figures 45, 46, 47, and 48, when comparing model results with measured time evolution of floc size by Biggs and Lant (2 000), the performances of FM A and FM B are limited. Model results show a gradual increase during the initial flocculation stage and after larger aggregates are created, the floc size a ppears to increase too ra pidly as the floc size approaching the equilibrium condition. This weak point related to time evolution of floc size cannot be improved by adjusting model coefficient such as p q kA and kB (see Figure 45, 46, 47, and 48). Because FM A using fixed fractal dimension also shows similar Sshaped curve, it is concluded that the existing de scription for floc dynamic may n eed to be revised for a more accurate description on the timedependent behavior of floc size. It is likely that other term representing additional physics of floc aggregation and breakup need to be incorporated. On the other hand, if the fractal di mension is deemed to be a variable, the floc strength Fy, which is shown to be a constant under the assumption of fixed fractal dime nsion (Kranenburg, 1994), shall also be a variable (Khelifa and Hill, 2006). This aspect is further investigated in the next section. Although incorporating va riable fractal dimension base d on empirical relationship of Khelifa and Hill (2006) dose not improve FM A signi ficantly, it is believed that using variable fractal dimension remains to be more physica lly reasonable for a more extensive study of cohesive sediment transport processes, such as a unified model for sedimentation and consolidation. Jackson (1998) and Thomas et al. (1999) propose the model of the equivalent spherical diameter of floc c onsidering size distribution of pr imary particles. Using their PAGE 81 81 approaches, it is possible to deve lop a flocculation model of polysi zed particles. In this study, only monosized primary particles are considered. However, sediments in nature are the mixture of primary particles having various sizes. In order to simulate the natural phenomenon more completely, it is necessary to consider polysized primary particles. 4.3.2 Application of FM C and FM D In this section, the new flocculation models ba sed on variable yield strength, i.e. Eqs. 429 a (FM C) and 429 b (FM D), are validated with several existing expe rimental data sets. Comparisons with FM A and FM B based on constant yield strength are also carried out in order to demonstrate the effect of variable yield stre ngth. A summary on the flocculation models tested in this study in given in Table 42. Numerical solutions of flo cculation models are obtained using an explicit RungeKutta method (ODE45 functi on of MATLAB is used in this study). Spicer et al. (1998) carry out an experime nt on flocculation and measure the temporal evolution of floc size in a mixi ng tank. In this experiment, polys tyrene particles, whose primary particle diameter and density are 0.87 m and 1,050 kg/m3 (Spicer and Pratsinis, 1996), are mixed in a 2.8 liter, baffled, stirred tank using a Rushton impeller. The volumetric concentration of primary particle is set to be 1.45. From the volumetric concentration and the density of primary particle, the mass concentrati on is calculated to be 0.0147 kg/m3. The average dissipation parameter, G in the tank is 50 s1. Sampling the floc is a crucial step to accurately characterize flocculation. Three kinds of sampling techniques ar e used: (1) withdrawal of a sample into the sample cell of light scattering instrument using a hand pipette; (2) withdrawal of a sample into the flowthrough sample cell using a syringe pump; (3) continuous recycle of the suspension through the sample cell using a perist altic pump which is a class of mechanical pumps with relatively simple structure and is suitable for miniaturiza tion (Xie et al., 2004). Among three techniques, the results obtained using peristaltic pump show the largest number of PAGE 82 82 samples and the most stable shape of evolution cu rve in the experiment of Spicer et al. (1998). Thus, the result with peristaltic pum p is selected and used to validate flocculation models in this study. Figures 415, 416, and 417 present the experi mental results of Spi cer et al. (1998) and model results of different flocculation models. The initial floc diameter is set to be 10 m for all the model simulations presented in Figures 415, 416, and 417. For numerical stability, the size of primary particle is assumed to be 1 m instead of 0.87 m. The values of empirical coefficients used to generate these model resu lts are summarized in Table 43. Following prior section, the criterion of specifying these empirica l coefficients is to match the equilibrium floc size (except r in FM D, which is used to evaluate tem poral evolution of floc size). FM C (Eq. 429 a) using a variable yield stre ngth derived theoretically in th is study and a variable fractal dimension in modeling flocculation processes shows good agreement with the experimental results (Figure 415). Figure 416 presents the re sults for FM D using the empirical yield strength equation of Sonntag and Russel (1987) and a variable fractal dimension in fl occulation processes. Three values of r have been tested: r= 0.7; r= 1.0; r= 1.3. The model results are sensitive to the choice of r Specifically, the case of r= 0.7 gives the best result among three values of r FM D shows numerical instability around r= 0.66. The results given by FM A and FM B based on a constant yield streng th are shown in Fig 417. These two models use a constant yield strength, which is set to be 1010 N (van Leussen, 1994; Matsuo and Unno, 1981). As mentioned in the previous section, the model of Son and Hsu (2008) uses a variable fractal di mension whereas FM D is based on a constant fractal dimension ( F =2.0). However, two models predict similar re sults on temporal evolutions of floc size and they both do not agree with measured data. Wh en comparing FM C and FM D using a variable PAGE 83 83 yield strength to FM A and FM B using a consta nt yield strength, it is notable that results obtained with a variable yield st rength (FM C and FM D) are cl early better than those of a constant yield strength (Between 0 and 11 min, the values of root mean square error of FM A, FM B, FM C, and FM D are 17.4, 24.7, 60.3, and 58.7 m). Overall, a variable yield strength has significant effect on the temporal evolution of floc size. FM C and FM D adopting a variable yield strength improve the prediction of timedependent behavior of flocculation. It is also emphasize here that FM D adopting Sonntag and Russel (1987) with r =0.7 show very similar results with FM C. This is not surprising if Eqs. 425, 426 and 427 are further compared. The powers of D in Eqs. 425 and 426 are r ( F 3) and 2 F /32, respectively. It can be seen that these two power become identical when r =2/3=0.67. In other words, the new formulation for variable yield st rength provides a theoretical appr oach to determine an empirical coefficient r required in the formation of Sonntag a nd Russel (1987). Using FM C with a yield strength derived theoretically in Section 4.2.3, the floccula tion model has one less empirical parameter. Biggs and Lant (2000) report th e temporal evolution of floc size of activated sludge under the conditions of G =19.4 s1 and with sludge volumetric con centration of 0.05. For this experiment, a baffled batch vessel and a flat six blade impeller are used. Because the total diluted mass concentration is not reported, the mass c oncentration is estimated from the volumetric concentration under the assumption that the density of sludge is 1,300 kg/m3 and the density of primary particle is 2,650 kg/m3. Hence, the calculated mass concentration is 24.19 kg/m3. In addition, the size of primary particle is also assumed to be 4 m. Empirical coefficients used to model this experiment are shown in Table 44. Th e experimental result of Biggs and Lant (2000) and the model results are presented in Figures 418, 419, and 420. Similar to those shown in PAGE 84 84 Figures 415, 416, and 417, FM C and FM D usi ng a variable yield strength show more smooth Scurves and are in better agreement with the experimental data. Overa ll, results predicted by FM C and D are quite similar. Comparing the mode l performance with the case of Spicer et al. (1998) (Figures 415, 416, and 417), it can be no ted that the predicted temporal evolution of floc size in this case agrees le ss favorably with experimental da ta. However, the adoption of a variable yield strength al lows the prediction of floc size that increases more rapidly in the initial stage of flocculation and, after larger aggreg ates are created, the floc size increases more gradually as it eventually appro aches the equilibrium value. Burban et al. (1989) perform experiments w ith Detroit River sediment in a Couette chamber. The experiments have been reproduced by McAnally (1999) and two cases of temporal evolution of floc size (Case B12 and Case B4) are shown in McAnally (1999). The mass concentrations are 0.05 kg/m3 for Case B12 and 0.80 kg/m3 for Case B4. The dissipation parameter for both cases is set to be G=200 s1. To simulate this experi ment, the size and density of primary particle are assumed to be 4 m and 2,650 kg/m3 due to absence of more information. In addition, McAnally (1999) provides information on the time required for the floc size to reach equilibrium state. Hence, empirical paramete rs of the flocculation models are calibrated according to equilibrium time determined by McAn ally (1999). More details about experiment conditions and model coefficients are shown in Table 45 and Table 46. PAGE 85 85 0 10 20 30 0 50 100 150 200 250 300 Measured result FM C Figure 415. Experimental re sult of Spicer et al. (1998) and model results of FM C PAGE 86 86 0 10 20 30 0 50 100 150 200 250 300 Measured result FM D, r=0.7 FM D, r=1.0 FM D, r=1.3 Figure 416. Experimental re sult of Spicer et al. (1998) and model results of FM D PAGE 87 87 0 10 20 30 0 50 100 150 200 250 300 Measured result FM B FM A Figure 417. Experimental result of Spicer et al. (1998) and m odel results of FM A and FM B PAGE 88 88 Table 43. Empirical parameters of the flocculati on models used for experiment of Spicer et al. (1998) FM A k 'Bk B1 r 0 y Fy FM A 0.44 1.106N.A. N.A.N.A. 1010 FM B 2.50 1.726N.A. N.A.N.A. 1010 FM C 6.74 4.5962.6314N.A.N.A. N.A. FM D 5.99 4.086N.A. 0.7 3.022 N.A. FM D 3.74 2.556N.A. 1.0 1.071 N.A. FM D 2.99 2.046N.A. 1.3 3.811 N.A. Table 44. Empirical parameters of the floccu lation models used for experiment of Biggs and Lant (2000) FM A k 'Bk B1 r 0 y Fy FM A 0.004 2.235N.A. N.A.N.A. 1010 FM B 0.02 2.405N.A. N.A.N.A. 1010 FM C 0.02 2.8854.2013N.A.N.A. N.A. FM D 0.02 2.895N.A. 0.7 2.822 N.A. FM D 0.02 2.415N.A. 1.0 5.242 N.A. FM D 0.01 1.925N.A. 1.3 9.652 N.A. PAGE 89 89 0 20 40 60 80 100 0 20 40 60 80 100 120 140 160 Measured result FM C Figure 418. Experimental re sult of Biggs and Lant (2000) and model results of FM C PAGE 90 90 0 20 40 60 80 100 0 20 40 60 80 100 120 140 160 Measured result FM D, r=0.7 FM D, r=1.0 FM D, r=1.3 Figure 419. Experimental re sult of Biggs and Lant (2000) and model results of FM D PAGE 91 91 0 20 40 60 80 100 0 20 40 60 80 100 120 140 160 Measured result FM B FM A Figure 420. Experimental result of Biggs and Lant (2000) and model results of FM A and FM B PAGE 92 92 Table 45. Experimental condi tions of Burban et al. (1989) Case G (s1) Mass conc. (kg/m3) Equilibrium time (min) Equilibrium floc size ( m) B12 200 0.05 45 8089 B4 200 0.80 10 25 Table 46. Empirical parameters of the flocculati on models used for experiment of Burban et al. (1989) Case FM A k 'Bk B1 r 0 y Fy FM A 0.18 9.607N.A. N.A. N.A. 1010 FM B 0.58 7.567N.A. N.A. N.A. 1010 FM C 1.05 1.3866.9213N.A. N.A. N.A. FM D 1.05 1.386N.A. 0.7 4.552 N.A. FM D 0.88 1.166N.A. 1.0 7.452 N.A. B12 FM D 0.77 1.026N.A. 1.3 1.211 N.A. FM A 0.07 2.095N.A. N.A. N.A. 1010 FM B 0.18 1.645N.A. N.A. N.A. 1010 FM C 0.30 2.9554.1012N.A. N.A. N.A. FM D 0.30 9.236N.A. 0.7 2.602 N.A. FM D 0.29 2.755N.A. 1.0 3.101 N.A. B4 FM D 0.29 2.715N.A. 1.3 3.601 N.A. PAGE 93 93 Figures 421, 422, 423, and 424 present the ex perimental results of Burban et al. (1989) and model results for Case B12 and Case B4. Consistent previous modeldata comparisons presented in Figures 415, 416, 417, 418, 419, and 420, flocculation models combined with a variable yield strength predict be tter temporal evolution of floc size than that using a constant yield strength in Case B12 (see Figures 421 and 422). However, there is less significant difference among the model results fo r Case B4 (see Figure 423 and 424) and in fact all models predict temporal evolution of fl oc size that agree reasonably well with measured data. Similar to the previous simulations, results predicted by FM C and FM D with r =0.7 are almost identical and show the best agreement with experimental data. In this study, the equations for a variable yield strength are combined with FM B. As mentioned previously, FM B uses a variable fr actal dimension whereas FM A uses the fixed fractal dimension of 2.0. According to results presented in Figure 415 to Figure 424, it have established that it is necessary to utilize flo cculation models based on va riable fractal dimension and using variable yield strength in order to predict the temporal evolution of floc size. However, it is not yet clear if one can simply implement a variable yield strength in a flocculation model based on fixed fractal dimension and obtain similar model performance. To examine the effect of variable fractal dimension, Eq. 426 is combined with FM A with a fixed fractal dimension, F=2.0 Figure 425 shows the results of experiment of Spicer et al. (1998), FM C, FM A combined with variable yield st rength of Eq. 426, and the original model of FM A. Although FM A combined with Eq. 426 is slightly better than FM A using a constant yield strength, FM C based on variable fractal dimensi on and variable yield strength re mains to be superior. Hence, it can be concluded that floccula tion model based on variable fractal dimension is a more PAGE 94 94 physicallybased mathematical fo rmulation while variab le yield strength is critical process during floc breakup that needs to be carefully parameterized. It can be concluded that a variable yield strength is a more reasonable approach to flocculation modeling than a cons tant yield strength. Although in corporating sole ly a variable fractal dimension in the floccula tion models may not predict the te mporal evolution of floc size well, it gives good agreement with measured data wh en it is further combined with variable yield stress formulations. However, it shall be also em phasized here that when simply using a variable yield strength in a flocculation model based on the fixed fractal dimension, the results for temporal evolution of floc size remains unsatis factory (Figure 425). He nce, it is recommended in this study that both variable yield strength an d variable fractal dimension are critical to predict flocculation processes. The empirical yield stress proposed by Sonnt ag and Russel (1987) (see Eq. 425) shows the best agreement with measured data when using r =0.7. It is also demonstrated that, when specifying r =2/3 in Sonntag and Russel (1987), it redu ces to theoretical m odel for yield stress developed in this study. Hence, it is suggested that the theoretical model proposed in this study is robust and it may be appropriate to specify the empirical parameter r in Sonntag and Russel (1987) to be around 0.7. There still remain several weak points in th e present model of flocculation process. To describe flocculation due to collision, the constant efficiency parameter, ec, has been adopted in this study and it is assumed that collisions cause only aggreg ation. However, it has been observed that collisions can make both aggr egation and breakup (McAnally, 1999). When the collisional stress is larger than the yield stress of floc, the breakup due to collision is expected rather than aggregation. It is not easy to adopt a variable effici ency parameter because it can be PAGE 95 95 highly empirical and explicit formulation has not been proposed at present although it is clear that ec is a function of potential and yield stress. In this study, it ha s been assumed that a floc is simply disaggregated into two roughly equalsized flocs (Boadway, 1978; Tsai and Hwang, 1995) due to lack of more detailed evidence. However, it is possible that a floc can fragment into a number of particles having a range of sizes (e.g. Srivas tava, 1971). More complicated flocculation model assuming more general types of breakup process is warranted. In addition, more studies are needed to understand parameters p and q (in Eq. 47) because they are currently highly empirical. Winterwerp (1998) uses the as sumption that the equilib rium floc size is independent of primary particle size and fractal dimension is 2.0 (see Eq. 25 of Winterwerp (1998). If p + nf 3 equals to zero, De is not a function of primary pa rticle size) to determine their values. In the context of variab le fractal dimension, more phys icalbased criterion shall be incorporated to determine p and q PAGE 96 96 0 10 20 30 40 50 60 70 80 0 20 40 60 80 100 120 Measured result FM C FM D, r = 0.7 Figure 421. Experimental result s of case B12 of Burban et al. (1989) and model results of FM C and FM D PAGE 97 97 0 10 20 30 40 50 60 70 80 0 20 40 60 80 100 120 Measured result FM C FM B FM A Figure 422. Experimental result s of case B12 of Burban et al. (1989) and model results FM A, FM B, and FM C PAGE 98 98 0 5 10 15 20 0 5 10 15 20 25 30 35 Measured result FM C FM D, r = 0.7 Figure 423. Experimental result s of case B4 of Burban et al. (1989) and model results of FM C and FM D PAGE 99 99 0 5 10 15 20 0 5 10 15 20 25 30 35 Measured result FM C FM B FM A Figure 424. Experimental result s of case B4 of Burban et al. (1989) and model results FM A, FM B, and FM C PAGE 100 100 0 5 10 15 20 25 30 0 50 100 150 200 250 300 Time (min)Floc size (micronmeter) Measured result FM C FM A with variable Fy FM A Figure 425. Temporal evolution of floc size si mulated by FM A combined with a variable yield strength PAGE 101 101 CHAPTER 5 MODELING TRANSPORT OF COHESIVE SEDIMENT 5.1 Governing Equations for Flow Momentum and Concentration The present model formulations for cohesive se diment transport with consideration of floc dynamics are revised from an earlier model of Hsu et al. (2009) in which constant floc size and floc density are assumed. The governing equati ons are obtained by simplifying EulerianEulerian twophase equations for the sediment limit, i.e., small particle response time. The xand y directions represent the directions of main flow (streamwise dire ction in rivers and crossshelf direction in coastal zones) and the di rection horizontally perpendicular to x direction (spanwise direction in rivers and alongshel f direction in coastal zones). Figure 51 shows the coordinate system used in this study. The governing equations for the x and y direction flow momentums are: Figure 51. Definition of coordinate system s x Streamwise Crossshelf y Spanwise Alongshelf z PAGE 102 102 (1) 11 sin (1)1w xzss s wwsss up g txz (51 a) 11 (1)w yz wwsvp tyz (51 b) where ss is the specific gravity of primary particle, s is the solid volume concentration of primary particles per unit volum e of fluidsediment mixture, s is the slope of the bottom, and g is the gravitational accelerat ion. In this equation, s is used instead of the floc volumetric concentration, f because the momentum transport of flow is considered to be affected by mass of sediment rather than its volume. / p x and / p y represent the pressure gradients in the x and y direction which are implemented as flow forc ing in the present model (see section 5.4). w x z and w yz are fluid stresses: ()w xzwtu z (52 a) ()w yzwtv z (52 b) where t is the eddy viscosity. The closure of t is discussed in section 5.2. Rheological stress is neglected in this study due to low sediment concentration considered. PAGE 103 103 The governing equation for solid volume con centration (volumetric concentration of primary particles) is: s ts ss cW tzz (53) where Ws is the settling velocity of floc and c is the Schmidt number set to be 0.5 in this study. The first term on the righthandside of Eq. 53 repr esents settling of floc and the second term is turbulent and molecular suspensi on. The mass concentration of noncohesive sediment is in direct proportion to the volumetri c concentration because its density is always constant. However, the mass concentration of cohesive sediment (floc), c is a function of both the volumetric concentration, f and the density of floc which usually varies according to floc size and its fractal dimension following fractal theory (Vicsek, 1992; Kranenburg, 1994). As a result, it is theoretically possible that the mass concentration of floc does not exactly follow the change rate of the floc volumetric concentration. For example, the fixed volumetric concentration of floc can be in wide range of the mass concentration depe nding on the average density of flocs. Thus, it is reasonable to select one concentration among the mass and volumetric concentrations for each governing equation considering phys ical meaning of a term wher e a concentration is needed although the equations for noncohesi ve sediment use any of them. In Eq. 53, the solid volume concentration (that is, the concept of mass conc entration) is used in stead of the volumetric concentration of floc because the mass conservati on is not to be satisfied when the volumetric condition is used for Eq. 53. The volumetric condition is mainly governed by not only settling PAGE 104 104 and turbulent suspension (the fi rst and second terms in the righthandside of Eq. 53) but also flocculation process. The settling velocity is calculated by (Richa rdson and Zaki, 1954): 241 ()(1) 18sfwfWDg (54) This equation is a function of f because the hindered settling of particle is considered to be affected by the volume of suspended flocs (eff ects of pore flow in the floc is neglected). f and D are the variable parameters in this equa tion because the density and size of cohesive sediment continuously change according to the conditions of fluid and sediment. D is determined by a flocculation model in this study (see Chapter 4). Following the fractal theory, f is calculated by Eq. 411. Once s is obtained by the above equations, f can be calculated by the equations shown below: 3F swsw f ss fwsfwcD d (55) 5.2 Flow Turbulence Solutions for Eqs. 51 and 53 are obtained by incorporating the eddy viscosity and a kclosure in this study. The balance equations of the turbulent kinetic energy, k and dissipation rate, have been adopted in many studies (e .g. Rodi, 1980; Elghobashi and AbouArab, 1982). Herein, the kequations for suspended sediment transpor t, which have been reduced from the full twophase kequations by Hsu and Liu (2004), are re vised to consider the feature of PAGE 105 105 cohesive sediment that the density and size of floc are not constant. The eddy viscosity, t is calculated by: 2(1)tfk C (56) In this equation, f is used in the sense that the turbul ence exists only in the carrier fluid. C is a numerical coefficient. k and are solved by their balance equations under the assumption of local equilibrium: 22(1) (1)(1)(1)f tts ft fs kck kuv sg tzzzzz (57) 22 1(1) (1)f t ftuu C tkzzzz 2 23(1)(1)ts fs cCCsg kkz (58) where c k 1C 2C ,and 3C are numerical parameters. Standard values of these numerical parameters including C are shown in Table 51 (see Hsu et al., 2007, for more details and references). Basically, f is used in kequations because it is assu med in this study that the turbulence in water within a floc is out of interest in this study. The last terms of above equations represent the effects of sediment on turbulence of carrier fluid (mostly damping effect) due to PAGE 106 106 density stratification. Density stratification is expected to be governed mainly by mass of suspended sediment. Thus, ss and s are adopted here instead of the specific gravity of floc, sf, and f Table 51. Numerical coefficients adopted for the eddy viscosity and kequations Coefficient C k c 1C 2C 3C Value 0.09 1.00 1.30 0.50 1.44 1.92 0.00 5.3 Bottom Boundary Conditions The continuous erosion formulation is a dopted for the bottom boundary condition (e.g. Sanford and Halka, 1993) and the upward erosion flux, E is: () 1 ()b e ct E M (59) where E is the erosion flux, e is an empirical erosion flux coefficient and M is the total eroded mass above unit area of bed. The total bottom st ress is calculated at every time step by the equation below: 2()()bwtut (510) PAGE 107 107 Herein, () ut is the total bottom friction velocity (see Hsu et al., 2007, and Mellor, 2002, for more details). The critical shear stress,c is assumed to be a function of M in this study. Sanford and Maa (2001) have suggested a si mple power law relationship between c and M : 312()cM (511) where 1 2 and 3 are empirical coefficients. From co mparison and calibra tion with field measurement data, it has been c oncluded that the coefficients in the equation are very sitespecific. This power law relationship for the critical shear stress is adopted here and the coefficients are calibrate d for each simulation case. Under the conditions of tide, uns teady river flow, tsunami and so on, the change of water depth is often seriously large, especially, in estuaries. Thus the fixed calculation domain for numerical model becomes inappropriate. Furthe rmore, it is more physically reasonable to consider variable water depth instead of fixed on e because the scale of large eddies and hence the turbulent mixing process scale with the water de pth. For this study, the variable calculation domain height is used with a moving top bounda ry condition implementation. The domain height is increased or decreased according to the pres cribed water depth and the top boundary condition is applied to the top cell of calculation domain. 5.4 Flow Forcing for Tidal and Unsteady Flow Condition The proposed model is a timedependent unst eady flow model which can be driven by arbitrary oscillations, such as tide and wave, a nd current such as river flow. To fulfill such objective for tidal and river flow, the pressure gradient terms in flow momentum equations are PAGE 108 108 calculated by the approach developed by Uitte nbogaard et al. (1996) and Delft Hydraulics (Winterwerp, 2002): 0()() 1sb wwrelUtUt p xhT (512 a) 0()() 1sb wwrelVtVt p yhT (512 b) where h is water depth, Trel is a relaxation time, U(t) and V(t) are actual computed depthaveraged xand ydirection flow velocities, U0(t) and V0(t) are desired depthaveraged xand ydirection flow velocities, and s is the surface shear stress. s is set to be zero because the free surface condition is calculated in this study. 5.5 Preliminary Tests In the sections 5.1 and 5.2, the governing e quations of the sediment transport model for this research use both the volum etric concentration of floc, f and the solid volume concentration of pr imary particles, s which is linearly proporti onal to the mass concentration of sediment, c. The reasons for such choice are also expl ained. When flocculation is considered, the density of floc becomes vari able and solid volume concentrat ion (or mass concentration) is the physical variable that quantifies the am ount of suspended mass. Hence, solid volume concentration (instead of floc volumetric concentration) is used in this study as the primary variable. However, many transport properties, such as settling velocity and turbulencesediment interactions, are better represented by floc volumetric con centration. Hence, floc volumetric concentration is used for many constitutive rela tionships and closures in the present model. PAGE 109 109 Figure 52 represents the depthaveraged flow velocity and wa ter depth used to test the sediment transport model. These conditions are generated from a simple sine function having 30,000 s of period and 0.5 m of tidal elevation amp litude and 0.5 m/s of tidal velocity amplitude. The average water depth and flow veloci ty are set to be 2 m and 0 m/s. The results of the sediment transport model de scribed in Chapter 5 are shown in Figures 53, 54, and 55. Figure 53 represents the mass c oncentrations at 0.2 m, 0.5 m, and 1.0 m above the bottom calculated by the sediment transport model. The mass concentrations in Figure 53 shows smooth increase and decrea se according to flow velocity, which also affects the bottom stress. The second peak of concen tration is larger than the first peak because the lowest water depth occurs at the second peak whereas the high est water depth occurs at the first peak. Figure 54 represents the floc volumetric concentrations at 0.2 m, 0.5 m, and 1.0 m above the bottom. The floc volumetric concentrations also show smooth changes according to flow velocity. The velocities calculated by the sediment transport m odel described in Chapter 5 are represented in Figure 55. The calculated velocities follow the tendency of the depthav eraged velocity in Figure 52 A without numeri cal instability such as irregular fl uctuation. The velocities calculated by model considering the floc volumetric c oncentration also show reasonable result. It is worthy to mention that if the floc vol umetric concentration is considered as the primary variable, similar to some earlier c ohesive sediment transport models without consideration on floc dynamics (e.g. Hsu et al., 2009), model results for sediment concentration are unreasonable (see Figure 56 and Figure 57). This reinforces the previous discussion that when floc dynamics is considered in sediment transport calculation, solid volume concentration, s (or mass concentration) needs to be sued as the primary variable. PAGE 110 110 From Figure 53 to Figure 57, it can be concluded that the sediment transport model composed of equations discussed in Chapter 5 is a physically and math ematically reasonable approach to model transport of cohesive sediment with the cons ideration on flocculation process. PAGE 111 111 0 1 2 3 4 5 6 7 8 9 1 0.5 0 0.5 1 Time (hr)Velocity (m/s) A 0 1 2 3 4 5 6 7 8 9 1 1.5 2 2.5 3 Time (hr)Water Depth (m) B Figure 52. Depthav eraged flow velocity and water depth used to test the sediment transport model. A) The depthaveraged flow velocity and B) the water depth. PAGE 112 112 0 1 2 3 4 5 6 7 8 9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Time (hr)Mass Conc.n (g/l) At 0.2 m At 0.5 m At 1.0 m Figure 53. Mass concentration calculated by sediment transport model combined with FM C using two types of concentrations PAGE 113 113 0 1 2 3 4 5 6 7 8 9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Time (hr)Volumetric Conc. (%) At 0.2 m At 0.5 m At 1.0 m Figure 54. Volumetric concentration calculate d by sediment transport model combined with FM C using two types of concentrations PAGE 114 114 0 1 2 3 4 5 6 7 8 9 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 Time (hr)Velocity (m/s) At 0.2 m At 0.5 m At 1.0 m Figure 55. Velocity calculate d by sediment transport model combined with FM C using two types of concentrations PAGE 115 115 0 1 2 3 4 5 6 7 8 9 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Time (hr)Mass Conc.n (g/l) At 0.2 m At 0.5 m At 1.0 m Figure 56. Mass concentration calculated by sediment transport model combined with FM C using one type of concentration PAGE 116 116 0 1 2 3 4 5 6 7 8 9 0 0.01 0.02 0.03 0.04 0.05 0.06 Time (hr)Volumetric Conc. (%) At 0.2 m At 0.5 m At 1.0 m Figure 57. Volumetric concentration calculate d by sediment transport model combined with FM C using one type of concentration PAGE 117 117 CHAPTER 6 MODEL APPLICATION TO EMS/DOLLARD ESTUARY 6.1 Insitu Measurement in Ems/Dollard Estuary van der Ham et al. (2001) carri ed out insitu highfrequency measurements of velocities and suspended sediment concentrations (SSC) in the Ems/Dollard estu ary to understand the behavior of cohesive sediment in a tidal cha nnel. The insitu measurem ent was conducted in a straight homogenous reach of the tidal channel Groote Gat, approximately 30 m to the east of the bank of the adjacent flat Herigsplaat at 7 09'43"E, 53'15"N in 1996. Figure 61 shows the Ems/Dollard estuary and the meas uring pole equipped with a rigid frame for insitu measurement. From sediment samples taken from positions directly adjacent to the measuring pole, it is known that the bed sediment was composed of silt and clay. The channel width was 600 m and the average elevation of the bed was 3.3 m below Amsterdam ordnance datum. The condition of bed was considered to be very smooth. Only sm all undulations of typically 0.05 m over a 10 m distance were found in the bed. The location where the measurements were conducted shows small horizontal SSC gradients in longitudinal dir ection. Thus, the effect of horizontal advection on suspended sediment transport is assumed to be negligible (Dorrrestei n, 1960). Furthermore, it has been known from horizontal flow measurements conducted at the same location that the direction of the flow was very pa rallel to the adjacent tidal flat. Only during slake water and part of the ebb, a small spanwise velocity perpendicula r to the border of the tidal flat was measured (van der Ham et al., 2001). These characteristics of the field site are appropriate for testing 1DV model. PAGE 118 118 Figure 61. The Ems/Dollard estuary and the m easuring pole equipped with a rigid frame for insitu measurement (van der Ham et al., 2001) PAGE 119 119 6.2 Calibration of Models In this study, four types of sediment transport models combined with three types of flocculation models (FM A, FM B, and FM C) and without consideration on flocculation are utilized to model high concentration mud suspension observe d by van der Ham et al. (2001) in the Ems/Dollard estuary in order to evaluate the effect of floccula tion on modeling cohesive sediment transport. The first type is the sedime nt transport model combined with FM C (Eq. 429 a) and denoted CMC in this study. The second and third types of models are combined with FM B and FM A (Eqs. 417 and 49), called CMB a nd CMA, respectively. The forth model (CMN) assumes constant floc density and floc size. That is, CMN does not consider floc dynamics. Instead, a low density and repres entative size of floc are used (e.g. Hsu et al., 2009). Table 61 summarizes sediment transport models combin ed with flocculation models. The empirical coefficients of each flocculation model are to be calibrated. Properties of floc determine the settling velocity and have very important effects on sediment transport. Moreover, floc properties highly depend on floc dynamics. The study of van de r Ham et al. (2001) doe s not provide insitu measurement on properties of primary particle. As a result, the properties of sediment are to be assumed in this study. In addition, floc size and density had not been measured during the experiment. Hence, modeldata comparison based on sediment mass concentration can be mainly carried out. For this reason, em pirical coefficients of FMs are calibrated to have the same equilibrium floc size under the condition of c=0.5 kg/m3 and G=3 s1, which are representative values during the field experiment (see Figures 610, 611, 612, 613, and 616). Figures 62, 63, 64, and 65 show time evolut ions of floc size, fractal dime nsion, floc density, and settling velocity calculated by th ree different FMs used for sediment tr ansport model in this study. While FM C and FM B calculate the variable fractal dimension, F, according to floc size, FM A assumes F to be constant. Thus, the value of F for FM A should be empirically determined. PAGE 120 120 Winterwerp (1998) assumed F=2.0 to model flocculation experiments in the settling column and developed FM A (see van Leussen, 1994, for more details). Following this assumption, the value of F for FM A is also fixed to be 2.0 in this study. The calibrated coefficients and assumed values for each FM used in this study are shown in Table 62. In Chapter 4, it is concluded that FM C is a more physicallybased mathematical formulation compared with FM A and FM B based on validations w ith several laboratory experimental results (e.g. Spi cer et al., 1998; Biggs and Lant, 2000). Only FM C can predict the gradual increase of floc size that is similar to measured data. This feature can also be observed in Figure 62. It is notable that th e characteristic shape of the tem poral evolution of floc size is insensitive to calibrati ng the coefficients in FM A and FM B (Son and Hsu, 2008). Hence, coefficients for FM A and FM B are calibrated to match the same equilibrium floc sizes of 440 m. The assumption of equilibrium floc size of 440 m is base on previous studies (e.g. Dyer and Manning, 1999; Milligan, 1996). From previous studies, it is known that floc sizes measured by insitu experiments are D=O{102} m in many cases. CMN assumes the constant size and density of floc. To determine these values, the result of CMC has been referred to. Although it is impossible to exactly calculate the mean size of floc over time and space, the mean size of floc obtained by CMC is consid ered to be about 90 m. The density of floc ha s been calculated by the size of floc and Eqs. 41 and 412. The floc density for CMN is determined to be 1,300 kg/m3 under the assumption of Dfc=2,000 m, Fc=2.0, d=4 m, and s =2,650 kg/m3. The resulting settling velocity is about 1.3 mm/s. PAGE 121 121 Table 61. Sediment transport models comb ined with or without flocculation models Model name Flocculation model Assumptions CMC FM C Variable fractal dimension Variable yield st rength of floc CMB FM B Variable fractal dimension Constant yield strength of floc CMA FM A Constant fractal dimension Constant yield strength of floc CMN N.A. Constant size and density of floc PAGE 122 122 0 100 200 300 400 500 600 700 800 900 1000 50 100 150 200 250 300 350 400 450 500 Time (second)Floc Size (micron meter) FM C FM B FM A Figure 62. Time evolution of floc sizes simulated by FMs comb ined with sediment transport model PAGE 123 123 0 100 200 300 400 500 600 700 800 900 1000 1.5 2 2.5 3 Time (second)Fractal Dimension FM C FM B FM A Figure 63. Time evolution of fractal dimensions simulated by FMs combined with sediment transport model PAGE 124 124 0 100 200 300 400 500 600 700 800 900 1000 500 1000 1500 2000 2500 Time (second)Density of Floc (kg/m3) FM C FM B FM A Figure 64. Time evolution of densities simulated by FMs combined with sediment transport model PAGE 125 125 0 100 200 300 400 500 600 700 800 900 1000 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (second)Settling Velocity (mm/s) FM C FM B FM A Figure 65. Time evolution of settling velocities simulated by FMs combined with sediment transport model PAGE 126 126 Table 62. Assumed values and calibrated coefficients of FMs d (m) s (kg/m3) 'Ak 'Bk Fc F Dfc (m) B1 Fy (N) FM C 4 2,650 31.50 4.824 2.0 N.A.2,000 2.6314 N.A. FM B 4 2,650 32.42 2.254 2.0 N.A.2,000 N.A. 1010 FM A 4 2,650 8.65 1.594 N.A.2.0 N.A. N.A. 1010 To calculate the flow forcing terms (Eq. 512) in flow momentum equation, the desired depthaveraged flow velocity has to be prescribed and given as an input data. van der Ham et al. (2001) measured streamwise velocity (U0) at only three elevations (0 .1, 0.4, and 1.0 m above the bottom) and assumed the spanwise velocity co mponent to be neglig ible. Thus, the depth averaged streamwise velocity is assumed to be close to the velocity at 1.0 m considering a logarithmic profile with the aver age depth of the tidal channel of about 3.3 m. The elevation of water surface also has to be give n as an input data. Measured tem poral variation of flow depth is utilized and the flow depth variation in th e numerical model is calculated with a moving boundary scheme. The empirical erosion flux coefficient, e, is set to be 3105 m/s while empirical coefficients determining the critical sh ear stress are calibrated to match measured data (see the next section). The re laxation time is set to be 2relTt Herein, t is the time step of calculation. Due to empirical coefficients involved in the FMs and limited information from measured data, our goal is to utilize ex isting FMs of different complexity in order to evaluate their importance on sediment transport. To achieve this goal, it is of great importance to keep one main criterion same to determine sediment properties for all FMs. Herein, the criterion is the PAGE 127 127 same kind of primary particle and the equa l equilibrium floc size under the same mass concentration and the dissipation parameter. 6.3 Investigation of Sediment Transport Model Figures 66, 67, 68, and 69 present the measured and mode led streamwise velocities (u). All models predict streamwise velocity at various elevations reasonably well by using the measured velocity at 1.0 m above the bottom as the depth averaged velocity to drive the numerical model. Model results al so suggest that flocculation pro cess plays a minor role in the resulting flow velocity due to relatively low to moderate concentration considered here. Figures 610, 611, 612, and 613 presen t measured and predicted mass c oncentrations at 0.3 m and 0.7 m above the bottom. The empirical coefficients associated with c have been calibrated to match measured concentration at 0.3 m above the bottom around t=5 hr except for CMB. Empirical coefficients of c of Sanford and Maa (2001) for each mode l are summarized in Table 63. Thus, concentrations at 0.7 m and at 0.3 m after t=6 hr are of interest here. Due to limited model capability of CMB (reasons given in a later section) it is difficult to calibrate coefficients for CMB to match measured concentra tion at 0.3 m above the bottom around t=5 hr. Thus, CMB is calibrated to match overall te ndency of measured mass c oncentration at 0.3 m. By and large, it can be observed that CMC, CMA, and CMN can predict qualitatively the observed magnitude and tendency of sediment concentration. Suspension and settling of sediment calculated by these models reasonably follo w the observed data in slack water, ebb, and flood. These three models also capture a ma in observed feature that the magnitudes of concentration at these two vertical locations are similar, suggesting a more or less wellmixed condition (van der Ham et al., 2001). It is of great necessity to discuss more detailed results in order to understand the effects of flocculation on sediment dyna mics and to possibly improve PAGE 128 128 model capability in a more quant itative manner. Among all models CMC predicts the smallest concentration gradient between 0. 3 m and 0.7 m that matches with observed data. In general, a lutocline formation can divide a water column into high and low concentration regimes (Ross and Mehta, 1989; Kineke et al., 1996). The data measured by van der Ham et al. (2001) in the Ems/Dollard estuary is considered not to have a clear lutocline in the re gime between 0.3 m to 0.7 m. To further illustrate this, the vertical profiles of mass concentrations calculated by CMC and CMN are plotted in Figure 614 along with measured data at two discrete locations. As the elevation of water surface osci llates between 2 m and 4.5 m above the bottom, the calculated concentration is usually very low in the regime higher than 3.5 m. Thus, profiles from 0 m to 3.5 m are plotted here for convenience. It can be obs erved that, as the elevation increases, the mass concentration usually decreases gradually until it reaches the free surface. The lack of lutoline feature is consistent with the relatively low to intermediate concentration in the case (maximum near bed concentration calculated by the model is about 1.0 g/l). Table 63. Calibrated values of empirical coefficients for the critical shear stress CMC CMB CMA CMN 1 0.30 0.40 0.50 0.45 2 0.05 0.05 0.05 0.05 3 1.00 1.00 1.00 1.00 PAGE 129 129 5 10 15 20 25 1 0.5 0 0.5 1 Velocity (m/s)Time (hr) Calculated Vel. Measured Vel. 5 10 15 20 25 1 2 3 4 5 Elevation (m) Tidal Ele. 1 0.5 0 0 0.5 1 1.5 2 Velocity (m/s)Elevation (m)At 12 hr 0 0.5 1 0 0.5 1 1.5 2 Velocity (m/s)Elevation (m)At 15 hr 0 0.5 1 0 0.5 1 1.5 2 Velocity (m/s)Elevation (m)At 18 hr Figure 66. Velocities measured and calculated by CMC at 1.0 m PAGE 130 130 5 10 15 20 25 1 0.5 0 0.5 1 Velocity (m/s)Time (hr) Calculated Vel. Measured Vel. 5 10 15 20 25 1 2 3 4 5 Elevation (m) Tidal Ele. 1 0.5 0 0 0.5 1 1.5 2 Velocity (m/s)Elevation (m)At 12 hr 0 0.5 1 0 0.5 1 1.5 2 Velocity (m/s)Elevation (m)At 15 hr 0 0.5 1 0 0.5 1 1.5 2 Velocity (m/s)Elevation (m)At 18 hr Figure 67. Velocities measured and calculated by CMB at 1.0 m PAGE 131 131 5 10 15 20 25 1 0.5 0 0.5 1 Velocity (m/s)Time (hr) Calculated Vel. Measured Vel. 5 10 15 20 25 1 2 3 4 5 Elevation (m) Tidal Ele. 1 0.5 0 0 0.5 1 1.5 2 Velocity (m/s)Elevation (m)At 12 hr 0 0.5 1 0 0.5 1 1.5 2 Velocity (m/s)Elevation (m)At 15 hr 0 0.5 1 0 0.5 1 1.5 2 Velocity (m/s)Elevation (m)At 18 hr Figure 68. Velocities measured and calculated by CMA at 1.0 m PAGE 132 132 5 10 15 20 25 1 0.5 0 0.5 1 Velocity (m/s)Time (hr) Calculated Vel. Measured Vel. 5 10 15 20 25 1 2 3 4 5 Elevation (m) Tidal Ele. 1 0.5 0 0 0.5 1 1.5 2 Velocity (m/s)Elevation (m)At 12 hr 0 0.5 1 0 0.5 1 1.5 2 Velocity (m/s)Elevation (m)At 15 hr 0 0.5 1 0 0.5 1 1.5 2 Velocity (m/s)Elevation (m)At 18 hr Figure 69. Velocities measured and calculated by CMN at 1.0 m PAGE 133 133 5 10 15 20 25 0 0.5 1 1.5 Mass Conc. (g/l)Time (hr) 5 10 15 20 25 0 0.5 1 Mass Conc. (g/l)Time (hr) 5 10 15 20 25 1 0 1 Velocity (m/s)Time (hr) Velocity at 1.0 m 5 10 15 20 25 0 5 Elevation (m) Elevation Calculated at 0.3 m Measured at 0.3 m Calculated at 0.7 m Measured at 0.7 m Figure 610. Measured mass concentrations and mass concentrations calculated by CMC using a variable fractal dimens ion and yield strength PAGE 134 134 5 10 15 20 25 0 0.5 1 1.5 Mass Conc. (g/l)Time (hr) 5 10 15 20 25 0 0.5 1 Mass Conc. (g/l)Time (hr) 5 10 15 20 25 1 0 1 Velocity (m/s)Time (hr) Velocity at 1.0 m 5 10 15 20 25 0 5 Elevation (m) Elevation Calculated at 0.3 m Measured at 0.3 m Calculated at 0.7 m Measured at 0.7 m Figure 611. Measured mass concentrations and mass concentrations calculated by CMB using a variable fractal dimension a nd a constant yield strength PAGE 135 135 5 10 15 20 25 1 0 1 Velocity (m/s)Time (hr) Velocity at 1.0 m 5 10 15 20 25 0 5 Elevation (m) Elevation 5 10 15 20 25 0 0.5 1 1.5 Mass Conc. (g/l)Time (hr) Calculated at 0.3 m Measured at 0.3 m 5 10 15 20 25 0 0.5 1 Mass Conc. (g/l)Time (hr) Calculated at 0.7 m Measured at 0.7 m Figure 612. Measured mass concentrations and mass concentrations calculated by CMA using a constant fractal dimens ion and yield strength PAGE 136 136 5 10 15 20 25 1 0 1 Velocity (m/s)Time (hr) Velocity at 1.0 m 5 10 15 20 25 0 5 Elevation (m) Elevation 5 10 15 20 25 0 0.5 1 1.5 Mass Conc. (g/l)Time (hr) Calculated at 0.3 m Measured at 0.3 m 5 10 15 20 25 0 0.5 1 Mass Conc. (g/l)Time (hr) Calculated at 0.7 m Measured at 0.7 m Figure 613. Measured mass concentrations a nd mass concentrations calculated by CMN using constant floc size and density PAGE 137 137 5 10 15 20 25 1 0 1 Velocity (m/s)Time (hr) Velocity at 1.0 m 5 10 15 20 25 0 5 Elevation (m) Elevation 0 1 2 0 1 2 3 At 5 hr Mass Concentration (g/l)Elevation (m) 0 1 2 0 1 2 3 At 8 hr 0 1 2 0 1 2 3 At 11 hr 0 1 2 0 1 2 3 At 14 hr 0 1 2 0 1 2 3 At 17 hr Mass Concentration (g/l)Elevation (m) 0 1 2 0 1 2 3 At 20 hr 0 1 2 0 1 2 3 At 23 hr 0 1 2 0 1 2 3 At 26 hr CMC CMN Measured Figure 614. Measured and simu lated mass concentration profiles PAGE 138 138 Another critical feature of se diment concentration captured only by CMC is the noticeable amount of concentration during slack water ( tidal current reversal ). Specifically, at t=14 hr and t=26 hr, the hydrodynamic and sediment transport processes are confined by small flow depth (see Figure 614). The tidal flow ve locities at these two periods b ecome very low (close to zero) and flow turbulence, a main mechanism of sedime nt suspension, becomes also weak (not shown here). Thus, it can be argued that notabl e concentration up to the water surface at t=14 hr and t=26 hr may not be due to strong upward suspen sion but related to sma ll settling ve locity of suspended flocs compared to the descending ra te of water surface during ebb. The settling velocity of CMC, CMB, and CMA are variable at any given instant and lo cation because it is a function of floc size, D, and floc density, f that are determined by flocculation process. Figure 615 represents the settling velocity and the di ssipation parameter at 0.5 m, 1.0 m, and 1.5 m calculated by CMC. It is seen that the settling velocity is very small (around 0.2 to 0.4 mm/s) when the flow velocity is close to zero such as at t=9 hr, t=14 hr, t=20 hr, and t=26 hr. On the other hand, because CMN uses constant floc size and density, the settli ng velocity is almost constant around 1.3 mm/s (the fl oc volumetric concentration is not significant to cause notable hindered effect). Carefully exam ining the results of CMC, CMB and CMA further suggests that the floc sizes predicted by mode ls without consideri ng variable yield stre ngth, i.e., CMB and CMA, are generally larger, which result in larger settling velocity (see, for example, Figure 621). This observation illustrates the importance of va riable yield strength in predicting cohesive sediment transport. Moreover, model results further suggest although fl oc sizes predicted by CMB and CMA are qualitatively similar, utilizin g variable fractal di mension gives larger calculated settling velocity (see Figure 624). This provides an explanation why CMB utilizing variable fractal dimension (but c onstant yield strength) predicts negligible sediments remain in PAGE 139 139 the water column when flow is weak. The time series of G (Figure 615 B) is very similar to that of settling velocity. The settling velocity and G seem to have a clos e positive relationship. However, it can be referred that they show a similar tendency because floc size and density, which govern the settling velocity of floc, are seriously affected by G. Small but noticeable amount of sediment con centration measured during slack water may be also related to bed erodibil ity. The present models rely on Eq 59 to supply upward sediment flux, E, from the bed into the water column. The magnitude of E is determined by variable critical shear stress, c which is further parameterized as a function of total eroded mass (see Eq. 511). To evaluate the effect of variable critical shear stress on the predicted sediment transport, Figure 616 represents predicted co ncentration of CMC but using a constant critical shear stress of 0.17 N/m2 (solidcurves). This value for a constant cr itical shear stress is calibrated to match measured concentration based on the same criteria mentioned before (i.e., results in Figure 610). Comparing with the concentration calculated by variable critical shea r stress of CMC (dashed curve), concentration predicted by constant critical sh ear stress is much la rger than measured data when flow is energetic (e.g. maximum ebb around t=12 hr and t=23 hr). In addition, during tidal flow reversal (e .g. around slack water at t=6~9 hr and t=19~22 hr), CMC with constant critical shear stress predict zero concentration even at locations ve ry close to the bed due to large (constant) critical shear stress used. Also notic e in Figure 616, whenever sediments remain suspended, sediment concentration at 0.7 m calcu lated by CMC with constant critical shear stress is similar to the concentration at 0.3 m. This is consistent with previous observation in Figures 610 and 614 that CMC predicts more wellmixed c oncentration due to sma ller settling velocity via flocculation. PAGE 140 140 5 10 15 20 25 0 1 2 3 4 Settling Vel. (mm/s)Time (hr) At 0.5 m At 1.0 m At 1.5 m 5 10 15 20 25 1 0.5 0 0.5 1 Velocity (m/s) Measured Vel. at 1.0 m A 5 10 15 20 25 0 2 4 6 8 10 Time (hr)Dissipation Parameter (m/s)0.5 At 0.5 m At 1.0 m At 1.5 m B Figure 615. Settling velocities of floc and dissipation paramete r at 0.5 m, 1.0 m, and 1.5 m above the bottom calculated by CMC. A) settling velocity and B) dissipation parameter. PAGE 141 141 Figure 617 shows the time series of c and b calculated CMC. From Eq. 511, it is known that c is determined by the total eroded mass, M, which is also highly correlated with bottom stress, b The inset in Figure 617 shows more detailed c and b time series around t=13~15 hr during tidal velocity re versal. It can be seen that c calculated by Eq. 511 is usually smaller than b even when b is close to zero whereas the constant c =0.17 N/m2 is much larger than b during such weak flow condition. This characteristic of the variable c allows sediment to be suspended from the bottom such as a fluffy layer under the condition of low b The variable c of CMB, CMA, and CMN also s how similar relationships with b (Figures 617, 618, and 619). In summary, long duration of zer oconcentration of Figure 616 (s olid curve) is due to lack of sedimentsupply from the bottom. But as long as sediment suppl y is available from the bed, CMC allows a more wellmixed concentration which is more consistent with measured data, especially for concentration far from the bed (e.g. 0.7 m above the bed). The variable c is a reasonable approach to model sediment transpor t in the Ems/Dollard estuary where cohesive sediment is expected to make up fluffy layer in the upper bed re gime and consolidated layer in the lower regime (van der Ham and Winterwerp, 2001). PAGE 142 142 5 10 15 20 25 0 0.5 1 1.5 2 Mass Conc. (g/l)Time (hr) 5 10 15 20 25 0 0.2 0.4 0.6 0.8 1 Mass Conc. (g/l)Time (hr) Measured at 0.3 m Constant c Variable c Measured at 0.7 m Constant c Variable c Figure 616. Mass concentrati ons at 0.3 m and 0.7 m calcula ted by CMC using the constant c and the variable c PAGE 143 143 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time (hr)Stress (N/m2) Critical Shear Stress Bottom Stress 13 13.5 14 14.5 15 0 0.02 0.04 0.06 Time (hr)Stress (N/m2) Figure 617. The bottom stress (dot ted lines) and the critical shear stress (solid lines) calculated by CMC PAGE 144 144 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time (hr)Stress (N/m2) Critical Shear Stress Bottom Stress Figure 618. The bottom stress (dot ted lines) and the critical shear stress (solid lines) calculated by CMB PAGE 145 145 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time (hr)Stress (N/m2) Critical Shear Stress Bottom Stress Figure 619. The bottom stress (dot ted lines) and the critical shear stress (solid lines) calculated by CMA PAGE 146 146 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time (hr)Stress (N/m2) Critical Shear Stress Bottom Stress Figure 620. The bottom stress (dot ted lines) and the critical shear stress (solid lines) calculated by CMN PAGE 147 147 Figure 621 and Figure 622 show the vertical profiles of floc size, settling velocity and sediment concentration calculate d by CMC and CMB during the phase of tidal velocity reversal (Figure 621) and peak ti dal velocity (Figure 622). It can be observed that CMB predicts very large flocs near the bottom during tidal velocity re versal. In addition, duri ng energetic tidal phase (Figure 8b), CMB predicts sudden increase of floc size around z=1.0 m. These sharp increases of predicted floc size coincide with vertical locations where sediment concentration starts to become large (see dashed curves in the predicted concentration profiles). On the other hand, floc size profiles predicted by CMC have a much mild er variation (so is settling velocity). This feature can be realized from the temporal evolu tion of floc size approaching the equilibrium state shown in Figure 62. The equilibr ium state is achieved when th e magnitude of the breakup term catches up with that of aggrega tion term during the later stage of the flocculation process. The decrease rate of floc yield stress due to increase of floc size calc ulated by variable yield strength formulation (Eq. 426 and Eq. 427) is smaller than that obtained by the co nstant yield strength assumption. Hence, the breakup term of FM C increases more gradually compared to breakup terms of FM B. As a result, FM C shows more gr adual increase of floc size when approaching the equilibrium state. From this, it can be inferr ed that CMB combined with FM B is likely to have abrupt changes of floc size and density when a certain conditi on is met. Abrupt increase of floc size predicted by CMB further enhances more rapid settling that fu rther gives more sudden increase of predicted concentration as shown in Figure 621 and Figure 622. PAGE 148 148 0 100 200 300 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Floc Size (micron meter)Elevation (m)At 14.6 hr CMC CMB 0 1 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Settling Velocity (mm/s) 0 2 4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Mass Conc. (g/l) Figure 621. Vertical profiles of size, settling velocity, and mass concentration of floc calculated by CMC and CMB at t=14.6 hr. The velocity at t=14.6 hr is around zero. PAGE 149 149 0 500 0 0.5 1 1.5 2 2.5 3 3.5 4 Floc Size (micron meter)Elevation (m)At 18.0 hr CMC CMB 0 5 0 0.5 1 1.5 2 2.5 3 3.5 4 Settling Velocity (mm/s) 0 2 4 0 0.5 1 1.5 2 2.5 3 3.5 4 Mass Conc. (g/l) Figure 622. Vertical profiles of size, settling velocity, and mass concentration of floc calculated by CMC and CMB at t=18.0 hr. The velocity at t=18.0 is at the peak. PAGE 150 150 Discussions made so far are mostly based on sediment mass concentration. The number of floc or floc volumetric concentra tion may be of interest when one is interested in applications related to chemicalbiological processes. When the size and density of floc are set to be constant, the volumetric concentration of floc is in direct proportion to the mass con centration. As a result, model results from CMN do not need to be sepa rately investigated fo r mass concentration and floc volumetric concentration. However, thes e two physical quantities are different when flocculation process is considered. Both CMC and CMN calculate the solid volume concentration as primary variables via Eq. 53. Then, the floc volumetric concentration is calculated using the mass concentration and floc de nsity via Eq. 55. As a result, the volumetric concentration of CMC also depends on the density of floc (or floc size). It is clear that the volumetric concentrations calculated from these two models show qualitative difference. As expected, the shape of the floc volumetric concen tration profile of CMN is exactly same as its mass concentration profiles. Every floc volumet ric concentration profile of CMN shown in Figure 623 has the highes t concentration at the bed and decreas ing concentration away from bed. However, most of floc volumetric concentr ation calculated by CMC though has a peak around the lower portion of the water column, further decreases approaching the bed. For example, during energetic flow at t=11 hr and t=23 hr, the peaks of volumetric concentration are located at around 0.5 m above the bed. In these cases, floc size is smaller near the bed (see Figure 622) and floc volume concentration also become smalle r. Data on floc volumetric concentration in the Ems/Dollard estuary is not availabl e (also in most of the other fiel d experiments). At present, it is not possible to compare the present numerical m odel results with insitu measurement data of volumetric concentration. However, it is believe d that this phenomenon is probable in nature because turbulent shear near the bed in a boundary layer is very strong co mpared to that away PAGE 151 151 from the bed. This can lead to significant floc breakup and as a result, small volumetric concentration of suspended floc near the bed. 0 0.2 0.4 0 0.5 1 1.5 2 2.5 3 3.5 At 5 hr Vol. Concentration (%)Elevation (m) CMC CMN 0 0.05 0.1 0 0.5 1 1.5 2 2.5 3 3.5 At 8 hr 0 0.5 1 0 0.5 1 1.5 2 2.5 3 3.5 At 11 hr 0 0.05 0.1 0 0.5 1 1.5 2 2.5 3 3.5 At 14 hr 0 0.2 0.4 0 0.5 1 1.5 2 2.5 3 3.5 At 17 hr Vol. Concentration (%)Elevation (m) 0 0.05 0.1 0 0.5 1 1.5 2 2.5 3 3.5 At 20 hr 0 0.2 0.4 0 0.5 1 1.5 2 2.5 3 3.5 At 23 hr 0 0.05 0.1 0 0.5 1 1.5 2 2.5 3 3.5 At 26 hr Figure 623. Simulated volumetric concentratio n profiles. Solid and dotted lines represent simulation results of CMC and CMN. PAGE 152 152 For a given floc size, models using variab le fractal dimension give larger calculated settling velocity than models using constant fractal dimensi on (see Figure 624). The primary particle size is assumed to be 4 m in this figure. This relationship is obtained by substituting Eq. 411 into Eq. 54 with hinde red settling effect neglected for simplicity. Although CMB calculates similar magnitude of floc size with CM A, a variable fractal dimension formulation in CMB gives larger settling velocity. This provide s the explanation why CMB predicts the most unsatisfactory results on sediment concentration shown in Figure 611. It is possible to further infer that good model performance of CMC, which predicts smaller settling velocity, is due to its capability to predict smaller floc size under the same flow condition as compared to CMA, CMB and CMN. Many prior field or laboratory studies of cohesive sediment transport focus on investigating the relationship between settling ve locity and mass concentration in order to better predict sediment transport with the effect of flocculation (e.g Ross 1987; Mehta 1988, Wolanski et al., 1995; Dyer and Manning, 1999). Present model results pr ovide detailed information on floc properties, sediment concen tration and hydrodynamics and it is of interest here to investigate the relation between floc settling velocity and se diment mass concentration using results of CMC for the Ems/Dollard case discussed later (see Fi gure 625). Each symbol in Figure 625 shows model results at a given time and data has been collected from 10 cm to 2.0 m above the bottom with an interval of 5 cm. A clear relationship between c and the settling velocity is observed during strong tidal velocity (maxim um flood and ebb velocities during t=5hr, t=11hr, t=17hr, and t=23 hr) where sediment concentration exceed 0.1 g/l in most of the water column. The settling velocity is loglinearly proportional to c between c=O{101} kg/m3 and c=O{100} and the slope, log(SV)/log()c in the loglog plot is slightly larg er than 1.0. Mehta (1988) proposes the PAGE 153 153 power law relationship between the settling velocity and c when c is in the intermediate regime (flocculation settling regime). In this study, the settling velocity is proportional to c4/3, which is consistent with theoretical study of Krone (1962). 100 101 102 103 102 101 100 101 FM C & B FM A Figure 624. Settling velocity pl otted as function of floc size PAGE 154 154 104 103 102 101 100 101 102 101 100 101 Mass Conc. (g/l)Settling Velocity (mm/s) At 11 hr At 13 hr At 17 hr At 20 hr At 23 hr At 26.8 hr Figure 625. Relationship between settling velo city and mass concentra tion calculated by CMC PAGE 155 155 It is important to further note that, for example, at t=23 hr the settling velocity in such energetic tidal phase starts to decrease when concentration exceed s 0.8 g/l. Carefully examining the model results suggests these data points th at show decrease of se ttling velocity when concentration becomes larger are rather close to the bed (within 0.5 m above the bed). Hence, the reduced floc settling velocity is a direct result of reduced fl oc size (see Figure 622) caused by strong boundary layer turbulence variation (large variation of G) near the bed. In boundary layer theory, it is wellknown that, the turbulence quantities, such as Re ynolds stress, are more or less uniform in the logarithmic regime. However, very near the bed, turbulent dissipation increases dramatically approaching the bed. Therefore, it is inferred that model results shown here do not suggest the upper concentration limit for the wellknown power law (Krone 1962, Mehta 1988) is only within 1 g/l. In fact, this power law shal l be applicable to few g /l before hindered settling effect become important. Model results shown here simply suggest when using this power law to locations where variation of G is large (e.g. near the bottom of boundary layer), the effect of G shall be further considered. It is observed that dur ing the phases of weak ti dal flow velocity (e.g. t=13 hr, t=20 hr and t=26.8 hr), the settling ve locity does not have appa rent relationship with c. Due to weak tidal velocity, sedime nt concentration rarely exceed c=O{101} g/l. The settling velocity (and floc size) under th ese weaker tidal phases are results of more complicated sediment transport and floc dynamics evolving from previ ous strong tidal velocity phase (i.e., strong history effect). Based on laboratory experiments, Mehta ( 1988) concludes that the settling velocity of floc is independent of c when c is smaller than c1 which is in the range of 0.1 to 0.3 kg/m3 (free settling regime). Thus, it can be concluded that the se ttling velocity shown in Figure 625 is in qualitative agreement with earlier laboratory experimental results carried out in annular flume and mix tanks. PAGE 156 156 Figure 626 shows the relationship between G and settling velocity calculated by CMC. It is clearly known that these two values do not have a strong relationship. The settling velocity is determined by the density and size of floc. Although G is one of parameters to govern the density and size of floc through flocculation process, its effect on settling velocity is not clear. This is considered to be due to the effect of mass con centration. As shown in Figure 625, the settling velocity of floc has the strong relationship with mass concentration which is also one parameter of flocculation process modeling. Because the size and density of floc seriously depend on mass concentration, the effect of G is expected to be relatively weak. Figure 627 shows the relationship between floc size a nd mass concentration calculated by CMC. As expected, they are highly correlated except t= 13 hr and t=26.8 when tidal flow velocity is very small. Flocculation process is not active when flow turbulence is we ak because both aggregation due to particle collisions and breakup due to tu rbulent shear depend on turbulent intensity. Thus, floc size is kept to be almost constant wh en turbulence, which has a relati onship with flow velocity, is negligibly low. Figure 628 represents the re lationship between floc set tling velocity and volumetric concentration of floc calculated by CMC. Even though the volumetric concentration of floc is not linearly proportional to mass concentration of fl oc, they are considered to have relationship. As a result, the settling velocity of floc has the strong relationship with volumetric concentration of floc also. When floc size is constant as noncohesive sedi ment is simulated, th e settling velocity of sediment is linearly proportional to density of sediment (see Eq. 54). However, cohesive sediment changes both its size and density simu ltaneously through flocculation process. Thus it is necessary to investigate the relationship between density of floc and floc settling velocity. PAGE 157 157 101 100 101 102 102 101 100 101 G (s1)Settling Velocity (mm/s) At 11 hr At 13 hr At 17 hr At 20 hr At 23 hr At 26.8 hr Figure 626. Relationship between settli ng velocity and dissipation parameter (G) calculated by CMC PAGE 158 158 104 103 102 101 100 101 100 101 102 103 Mass Concentration (g/l)Floc size (micronmeter) At 11 hr At 13 hr At 17 hr At 20 hr At 23 hr At 26.8 hr Figure 627. Relationship between floc si ze and mass concentration calculated by CMC PAGE 159 159 105 104 103 102 101 100 102 101 100 101 Vol. Conc. (%)Settling Velocity (mm/s) At 11 hr At 13 hr At 17 hr At 20 hr At 23 hr At 26.8 hr Figure 628. Relationship between settling veloc ity and volumetric conc entration calculated by CMC PAGE 160 160 Figure 629 shows the relationshi p between floc settling velo city and density of floc calculated by CMC. In loglog axes, the settling velo city of floc is nearly linearly proportional to floc density. It means that the settling velocity follows a power law of floc density. It is an obvious result because the density of floc also has a power law relationship with floc size (see Eq. 411). However, it is notable that the settling velocity of floc can be determined when both floc size and density are known and the settling velocity is expected to be proportional to a certain power of density. Figures 630, 631, 632, 633, and 634 repres ent the relationships calculated by CMB. By and large, it can be concluded that CMB has a tendency similar to CMC. Specifically, the relationship between floc settling velocity and density of floc of CMB has a same tendency as the result of CMC. As shown in Figure 624, FM B and FM C have the same density theoretically and mathematically when floc size is same because these two models adopt the same approach to fractal dimension which dete rmines the density of floc (see Eq. 412). However, it is of interest that CMB shows abrupt changes of floc settling velocity and size of floc when concentrations become higher than certain values. These abrupt changes are considered to be due to the characteristic FM B. As shown in Figures 415 and 417, FM B shows abrupt changes of floc size evolution compared to FM C which shows more smooth Sshaped curves of temporal evoluti on of floc size. From this charac teristic of FM B, it is inferred that CMB combined with FM B calculates abrupt ch ange of floc properties such as floc size and density as concentrations approach certain magnit udes. As a result, the settling velocity, which is a function of floc size and density, of CM B suddenly change under the condition of concentration. PAGE 161 161 103.1 103.2 103.3 103.4 102 101 100 101 Density of Floc (kg/m3)Settling Velocity (mm/s) At 11 hr At 13 hr At 17 hr At 20 hr At 23 hr At 26.8 hr Figure 629. Relationship between settling velo city and density of floc calculated by CMC PAGE 162 162 104 103 102 101 100 101 100 Mass Conc. (g/l)Settling Velocity (mm/s) At 11 hr At 13 hr At 17 hr At 20 hr At 23 hr At 26.8 hr Figure 630. Relationship between settling velo city and mass concentration calculated by CMB PAGE 163 163 101 100 101 102 102 101 100 101 G (s1)Settling Velocity (mm/s) At 11 hr At 13 hr At 17 hr At 20 hr At 23 hr At 26.8 hr Figure 631. Relationship between settli ng velocity and dissipation parameter (G) calculated by CMB PAGE 164 164 104 103 102 101 100 101 101 102 103 Mass Concentration (g/l)Floc size (micronmeter) At 11 hr At 13 hr At 17 hr At 20 hr At 23 hr At 26.8 hr Figure 632. Relationship between floc si ze and mass concentration calculated by CMB PAGE 165 165 104 103 102 101 100 101 100 Vol. Conc. (%)Settling Velocity (mm/s) At 11 hr At 13 hr At 17 hr At 20 hr At 23 hr At 26.8 hr Figure 633. Relationship between settling veloc ity and volumetric conc entration calculated by CMB PAGE 166 166 103.1 103.2 103.3 102 101 100 101 Density of Floc (kg/m3)Settling Velocity (mm/s) At 11 hr At 13 hr At 17 hr At 20 hr At 23 hr At 26.8 hr Figure 634. Relationship between settling velo city and density of floc calculated by CMB PAGE 167 167 Over all, it is difficult to figure out the rela tionships between floc settling velocity and concentrations calculated by CMA incorporated w ith FM A using a constant fractal dimension. Thus, the results are not shown in this dissertation. Howeve r, the relationship between floc settling velocity and density of fl oc (Figure 635) is of interest in the sense that it is different from CMC and CMB combined with FM C and FM B using a variable fractal dimension. In this research, the fractal dimension of FM A is fixed to be 2.0 following the previous study of Winterwerp (1998). As a result, the density of floc calculated by FM A is proportional to D1 (see Eq. 411) and the settling velo city of floc mathematically becomes a function of only D without any relationship with floc density. Because th e density of floc calculated by FM A is also determined by floc size even though the constant variable fractal dimension is used, the settling velocity of floc shows a kind of relationship with density of floc and this relationship is different from relationships of CMC and CMB of which the settling velocity of floc is a function of both floc size and density. Figure 636 represents the result s of CMC without damping eff ect of density stratification on turbulence. The solid curves show the result with the damping effect (Figure 610). The dotted curves show the results i gnoring the damping effect of de nsity stratification. The last terms of Eq. 57 and Eq. 58 are set to be zero in this case. Compared to result with damping effect, the result without damping effect show s remarkable increase of sediment suspension during strong tidal flow velocity (around t=11 hr and t=24 hr). It is noticeable that the effect of damping is more significant at 0.3 m above the bed. It is considered because the vertical gradient of concentration is la rge near the bottom and, as a resu lt, the damping effect of density stratification becomes more significant in this regime. Over all, CMC without damping effect is considered to overestimate the mass concentration compared to measured results. PAGE 168 168 103.1 103.2 103.3 103.4 102 101 100 101 Density of Floc (kg/m3)Settling Velocity (mm/s) At 11 hr At 13 hr At 17 hr At 20 hr At 23 hr At 26.8 hr Figure 635. Relationship between settling velo city and density of floc calculated by CMA PAGE 169 169 5 10 15 20 25 1 0 1 Velocity (m/s)Time (hr) Velocity at 1.0 m 5 10 15 20 25 0 5 Elevation (m) Elevation 5 10 15 20 25 0 0.5 1 1.5 Mass Conc. (g/l)Time (hr) At 0.3 m With Damping Without Damping Measured 5 10 15 20 25 0 0.5 1 Mass Conc. (g/l)Time (hr) At 0.7 m Figure 636. Mass concentration calculated by CMC without damping effect of density stratification PAGE 170 170 Figure 637 shows the results of CMC with 1.0 of the Schmidt number (c ). The solid curves show the result with c =0.5 (Figure 610). The dotted cu rves show the results with c =1.0. Compared to the results with c =0.5, the increased c has a tendency to increase the mass concentration at 0.3 m above the bottom a nd decrease the mass concentration at 0.7 m above the bottom. c is included in the terms for tur bulent and molecular suspension of sediment (Eq. 53) and the damping effect due to density stratification (E q. 57 and Eq. 58). As c increases, turbulent and molecular suspen sion of sediment and damping effects on the turbulent kinetic energy (k) and the turbulent dissipation rate () are decreased. Thus, it is known that the effect of k and on sediment suspensi on is more significant than and the turbulent and molecular suspension (the last term of Eq. 53) near the bottom where tur bulence is expected to be strong and the effect of tur bulent and molecular suspension b ecomes significant in the regime far from the bottom. PAGE 171 171 5 10 15 20 25 1 0 1 Velocity (m/s)Time (hr) Velocity at 1.0 m 5 10 15 20 25 0 5 Elevation (m) Elevation 5 10 15 20 25 0 0.5 1 1.5 Mass Conc. (g/l)Time (hr) At 0.3 m 5 10 15 20 25 0 0.5 1 Mass Conc. (g/l)Time (hr) At 0.7 m 0.5 of Schmidt # 1.0 of Schmidt # Measured Figure 637. Mass concentration calculated by CMC with c =1.0 PAGE 172 172 CHAPTER 7 SUMMARY, CONCLUSIONS AND REMARKS 7.1 Summary and Conclusion The objective of this research is to understand of the dyna mics of cohesive sediment transport. To achieve this objec tive, two research directions are pursued. The first research emphasis is the development of flocculation mode l representing natural behaviors of cohesive sediment. The second research task is the de velopment of a comprehensive numerical model describing transport of cohesive se diment. The main results of this research are a semiempirical flocculation models to describe fl oc dynamics of cohesive sedime nt in turbulent flow and a 1 DV numerical model for transport of cohesive sedi ment driven by tidal flow. The flocculation models are tested with several laboratory experiments such as Biggs Land (2000), Burban et al. (1989), and Spicer et al. (1998) and the numerical models for sediment transport combined with flocculation models are compared with insitu measurement of van der Ham et al. (2001). The flocculation model (FM A) using a consta nt fractal dimension and a constant floc yield strength is introduced in this study. This m odel is developed in stud y of Winterwerp (1998). Based on FM A, the flocculation model using a va riable fractal dimensi on (FM B) is proposed (Son and Hsu, 2008). For aggregatio n process of this model, a variable fractal dimension is adopted under the assumption that floc has the characteristic of selfsimilarity, the main concept of fractal theory, and the previ ous research of Khelifa and Hi ll (2006). The model for breakup mechanism is based on studies of Winterwerp (1998) and Kranenburg (1996), which is semiempirical and requires determination of several empirical coefficients. By linearly combining models for aggregation and br eakup processes, a Lagrangian flocculation model which can describe the evolution of floc size with time is obt ained. In terms of equilibrium floc size, results of FM B agree reasonably well with the measured data (see Figures 43 and 44) provided that PAGE 173 173 empirical coefficients are calibrated. This is partially because of the variation of chemicalbiological properties of the c ohesive sediment tested. However, through modeldata comparisons with Manning and Dyer (1999) (see Fi gures 413 and 414), it is clear that the empirical coefficients, specifically q, also depend on sediment concen tration. By and large, the model qualitatively predicts equilibrium floc size decreases as dissipation parameter increases, suggesting that FM B can capture observed floc dynamics that strong tu rbulence has a tendency to break the floc and reduce the floc size. As shown in Figures 45, 46, 47, and 48, when comparing model results with measured temporal evolutions of floc size by Biggs and Lant (2000), the performances of FM A and FM B are uns atisfactory. Model results of FM A and FM B show a gradual increase during the initial flocculation stage a nd after larger aggregates are created, the floc size appears to increase mu ch rapidly as the floc size approaching the equilibrium condition. This weak point related to time evolution of floc size cannot be improved by adjusting model coefficient such as p, q, kA and kB (see Figures 45, 46, 47, and 48). Because FM A using fixed fractal dimension also shows similar Ssh aped curve, it is clear that the description for floc dynamic may need to be further revised for a mo re accurate description on the timedependent behavior of floc size. If the fractal dimension is deemed to be a variable, the floc yield strength Fy, which is shown to be a constant under the assumption of fixed fractal dime nsion (Kranenburg, 1994), shall also be a variable. From this idea, further study on floc yiel d strength starts. The equations for the yield strength and yield stress (Eqs 426 and 427) are deri ved theoretically and combined with FM B. The yield strength a nd yield stress equations are derived under the assumptions that (1) the shapes of the floc a nd primary particle are spherical, (2) a floc is composed of monosized primary particles, (3) the number of primary particles is large enough PAGE 174 174 to adopt mensuration by parts, (4) the yields strength of floc is determined by the number of primary particles in the ruptured plane and cohesi ve force of each primary particle, and (5) a floc is disaggregated into two roughly equalsized flocs (Boadway, 1978; Tsai and Hwang, 1995). Using a variable fractal dimension suggested by Khelifa and Hill (2006) (see Eq. 412), the number of primary particles in th e ruptured plane crossing the cente r of a floc is calculated. The cohesive force of primary partic le is considered as an empirical parameter because it depends on the properties of sediment and ch emicalbiological effects. The re sulting equations for the yield strength and yield stress play a key role in the flocculation pro cess driven by turbulent flow. The effect of a variable yiel d strength on flocculation is tested with four laboratory experiments. FM A and FM B assuming constant yield strength cannot predict the temporal evolution of floc size well as s hown in Figures 45, 46, 47, a nd 48. A much milder increase of floc size approaching the equili brium state observed in the experiment is not captured (see Figures 415, 416, and 417). On the other hand, adopting va riable yield strength based on variable fractal dimension (FM C) shows clear improvement on th e evolution of floc size that better agrees with experimental re sults (see Figures. 415 to 424). Thus, it can be concluded that a variable yield strength is a mo re reasonable approach to floccu lation modeling than a constant yield strength previously adopted for FM A and FM B. The empi rical yield stress proposed by Sonntag and Russel (1987) (see Eq. 425) shows th e best agreement with measured data when using r=0.7. It is also demonstr ated that when specifying r=2/3 in Sonntag and Russel (1987), it reduces to theoretical model for yield stress deve loped in this study. Henc e, it is suggested that the theoretical model proposed in this study is robust and it may be appropriate to specify the empirical parameter r in Sonntag and Russel (1987) to be around 0.7. PAGE 175 175 Although incorporating solely a variable fractal dimension in the flocculation models may not predict the temporal evoluti on of floc size well, it gives good agreement with measured data when it is further combined with variable yield stress formulations. However, it shall be also emphasize here that when simply using variable yield stress in a flocculation model based on fixed fractal dimension, the resu lts for temporal evolution of floc size remains unsatisfactory (Figure 623). Hence, it is recommended in this study that both variable yield stress and variable fractal dimension are critical to predict flocculation processes. Sediment transport model combined with a flocculation model based on variable yield strength and variable fractal dimension (CMC) is shown to predict essentia l features of cohesive sediment transport measured in Ems/Dollard estu ary. Flocculation model based on variable yield strength and variable fractal dimension predict a smaller settling velocity and hence more wellmixed sediment concentration profiles as compared to results of other flocculation models or model results without considering flocculation process. Moreover, in corporating variable critical shear stress, which is parameterized as a function of total eroded se diment mass, is also shown to be effective in modeling cohesive sediment tran sport, especially for near bed time series of concentration. Without incorporatin g variable critical sh ear stress, the temporal variation of near bed concentration becomes too large, which is no t consistent with measured data. Essentially, flocculation controls the settling velocity and be d erodibility (parameterized by critical shear stress here) controls the botto m supply of cohesive sediment. Both processes need to be incorporated appropriately in or der to predict cohesive sediment transport with satisfactory results. CMC shows a clear re lationship between c and the settling velocity during strong tidal velocity where sediment concentration exceed 0.1 g/l in most of the water column. The settling PAGE 176 176 velocity is loglinearly proportional to c. It is also observed that during the phases of weak tidal flow velocity, the settling velocity does not have apparent relationship with c. Due to weak tidal velocity, sediment concentration is usually less than c=O{101} g/l. Mehta (1988) concludes that the settling velocity of floc is independent of c when c is smaller than about 0.1 to 0.3 kg/m3. Thus, this relationship between c and the settling velocity calculat ed by CMC is considered to be consistent with studies of Mehta (1988). 7.2 Concluding Remarks for Future Study In this study, only monosized primary particle s are considered. However, sediments in nature are the mixture of primary particles having various sizes. Jackson (1998) and Thomas et al. (1999) propose the model of the equivalent s pherical diameter of floc considering size distribution of primary particle s. Using their approaches, it may be possible to develop a flocculation model of polysized particles. In order to simulate the natural phenomenon more completely, it is necessary to consider polysized primary particles. To describe flocculation due to collisi on, the constant efficiency parameter, ec, has been adopted in this study and it is a ssumed that collisions cause only aggregation. However, it is observed that collisions can make both aggr egation and breakup (McA nally, 1999). When the collisional stress is larger than the yield stress of floc, the breakup due to collision is expected rather than aggregation. It is not easy to adopt a variable effici ency parameter because it can be highly empirical and an explicit formulation has not been propos ed at present although it is possible that ec is a function of potential a nd yield stress. In addition, more studies are needed to understand parameters p and q because they are currently highly empirical. FM A uses the assumption that the equilibrium floc size is i ndependent of primary pa rticle size and fractal dimension is 2.0 (see Eq. 25 of Winterwerp, 1998). If p+nf3 equals to zero, De is not a function PAGE 177 177 of primary particle size) to dete rmine their values. In the context of variable fractal dimension, more physicalbased criterion sha ll be incorporated to determine p and q. At present, there is few available laboratory or field experimental data of the floc volumetric concentration with high spatial and temporal resolutions to investigate interrelationship between cohesive sedi ment and carrier fluid. The fl oc volumetric concentration is sometimes a more important quantity than mass concentration for environmental and biological applications such as the transillumination. The volumetric co ncentration of cohesive sediment and density of floc is consider ed to be determined by measur ing floc size and the settling velocity of floc. Thus, it is encouraged to car ry out insitu or laboratory experiments on floc properties to further underst and floc dynamics and transp ort of cohesive sediment. One of outstanding characteristics of cohesive se diment is the selfwei ght consolidation of the bed. As flocs accumulate in the bed, the fl ocs that have already ar rived are squeezed by the ones on top (Winterwerp and van Kesteren, 2004). This process is consolidation of cohesive sediment. consolidation is important for its effect on critical shear stress of the bed surface. In the whole column of cohesive sediment bed, the upper part such as fluffy layer has smaller critical shear stress than the lower part because the upper part has been less consolidated and density and strength are smaller than that of the lower part. Although th e power law relationship of critical shear stress with the total eroded mass, M, (Eq. 511) is adopted and used to consider variable critical shear stress due to conso lidation effect of cohesive sedime nt, it is determined empirically with insitu measurement at pr esent. Thus, more comprehensiv e study on consolidation process of cohesive sediment is needed to understand various processes of cohesive sediment transport as a single dynamic system. PAGE 178 178 APPENDIX DERIVATION OF EQUATION 424 Here the derivation of equation for the number of primary particles in the plane crossing the center of a floc assumed to be spherical (Eq. 424). From Eq. 423, it has been known that the num ber of primary partic les within a floc, N, is /FDd Under the assumption that the number of primar y particles within a floc is large enough to adopt the mensuration by parts, the volume of a floc of which size is D is calculated by the mensuration by parts: 336 f isVDND (A1) where Vf is the volume of a floc of which size is D and Dis is the average distance between two neighbor primary particles. Figure A1 illustrate s Eq. A1 schematically. From Eq. 423 and Eq. A1, Dis is obtained: 1/3 1/3/36FF isDDd (A2) The area of the plane crossing the center of a floc, Af, is also derived by the mensuration by parts: 224 f rupisADND (A3) PAGE 179 179 where Nrup is the number of primary particles in the plane crossing the center of a floc. Figure A2 describes Eq. A3 briefly. By substituting Eq. A2 to Eq. A3, the equation for Nrup (Eq. 424) is derived. Figure A1. Schematic description on adop ting the mensuration by parts for Eq. A1 D Dis Floc with Size= D Dis d Primary Particle with Size= d PAGE 180 180 Figure A2. Schematic description on a dopting the mensuration by parts for Eq. A3 D Plane with Area= Af d Primary Particle with Size= d Dis Dis PAGE 181 181 LIST OF REFERENCES Akers, R.J., Rushton, A.G., Stenhous, J.I.T., 1987. Floc breakage: th e dynamic response of particle size distribution in a flocculated susp ension to a step change in turbulent energy dissipation. Chemical Eng. Sci. 42, 787. Allen, J.R.L., 1985. 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Zhang, Z., Sisk, M.L., Mashmoushy, H., Thomas C.R., 1999. Characterisation of the breaking force of latex particle aggregates by micromanipulation. Part Part System Characterisation 16, 278. PAGE 190 190 BIOGRAPHICAL SKETCH Minwoo Son was born in Jinju, South Korea. With parents who were educators, he had developed a passion for study and a love of learning. He went to the Civil Engineering Department of Seoul National University (SNU) Seoul, South Korea in March 1995. In Feb. 1999, he completed his undergraduate degree in Ci vil Engineering department of SNU. Then, he continued his higher study in the graduate school of SNU. During his Mast er of Science courses, his academic interest was prediction of water dept h in narrow and steep natural streams under the condition of tropical shower using the artifi cial neural network theory. After Minwoo Son received MS degree, he worked for a consultant company (Samho Consultant Inc) and a national research institute (Korea Instit ute of Construction Technology) as a hydraulics engineer and a researcher from Jan. 2001 to July 2006. He got married to Kunhwa Choi in May 2002 and begot a daughter, Jiwoo, on June 13th 2006. A life as a researcher of nati onal research institute had insp ired him to further study his research topics, river mechanics and estuarine hydr aulics. Furthermore, he had realized that he needed to go to a higher stage of education and have enough time to concentrate on his own study and research. So, he moved to the United St ates. with his wife and 8 week old daughter and started his doctorate study in Civil and Coastal Engineering Depa rtment of the University of Florida (UF). The department offered him a prestigious and competitive Alumni Graduate Award which was awarded only to outstanding candidates for the Ph.D. He had spent three and half years as a graduate research assistant and received his Ph.D degree from UF in 2009. His research topic was the development of theoretica l and numerical models for cohesive sediment transport. He had been at the University of Delaware (UD) for one and half years as a visiting scholar of Civil and Environment department. At UD, he continues this study with his academic advisor and colleague, Dr. TianJian Hsu. 