FLOCCULATION AND TRANSPORT OF COHESIVE SEDIMENT

Transcription

1 FLOCCULATION AND TRANSPORT OF COHESIVE SEDIMENT By MINWOO SON A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 29 1

2 29 Minwoo Son 2

3 To my parents, wife and lovely daughter, Jiwoo 3

4 ACKNOWLEDGMENTS I would like to appreciate to the academic advisor and supervisory committee chair, Dr. Tian-Jian Hsu, Assistant Professor of Civil and Environmental 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 Engineering at the University of Florida for his discerning comments and questions on study of flocculation. Sincerely, 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, Yhung-Gyung Kang, for their warm hearts from Manhattan. Finally, I would like to express my sincerest appreciation to my parents, Youngseuk Son and Yeonsuk Kang, for their presence itself, wife, Kunhwa Choi, for her love, and daughter, Jiwoo Son, for her lovely smile and kiss. 4

5 TABLE OF CONTENTS ACKNOWLEDGMENTS...4 LIST OF TABLES...7 LIST OF FIGURES...8 LIST OF ABBREVIATIONS...13 ABSTRACT...17 CHAPTER 1 INTRODUCTION...19 page 1.1 Significance of Study on Cohesive Sediment Transport Objectives of This Study Terminology Outline of Presentation LITERATURE REVIEW Studies on Flocculation and Yield Strength of Floc Sediment Transport Modeling STUDY ON PROPERTIES OF COHESIVE SEDIMENT General Properties of Cohesive Sediment Fractal Dimension Flocculation Process MODELING FLOCCULATION OF COHESIVE SEDIMENT Overview on Flocculation Modeling Lagrangian Flocculation Models Winterwerp s Flocculation Model Flocculation Model Using a Variable Fractal Dimension Flocculation Model Using a Variable Fractal Dimension and Variable Yield Strength Investigation of Flocculation Models Application of FM A and FM B Application of FM C and FM D MODELING TRANSPORT OF COHESIVE SEDIMENT Governing Equations for Flow Momentum and Concentration

6 5.2 Flow Turbulence Bottom Boundary Conditions Flow Forcing for Tidal and Unsteady Flow Condition Preliminary Tests MODEL APPLICATION TO EMS/DOLLARD ESTUARY In-situ Measurement in Ems/Dollard Estuary Calibration of Models Investigation of Sediment Transport Model SUMMARY, CONCLUSIONS AND REMARKS Summary and Conclusion Concluding Remarks for Future Study APPENDIX DERIVATION OF EQUATION LIST OF REFERENCES BIOGRAPHICAL SKETCH

7 LIST OF TABLES Table page 3-1 Intensity of cohesion according to size of sediment (Mehta and Li, 1997) Classification of sediment by size Properties of clay minerals Cation exchange capacity of clay minerals Experiment values and parameters of flocculation models Summary of flocculation models used in this study Empirical parameters of the flocculation models used for experiment of Spicer et al. (1998) Empirical parameters of the flocculation models used for experiment of Biggs and Lant (2) Experimental conditions of Burban et al. (1989) Empirical parameters of the flocculation models used for experiment of Burban et al. (1989) Numerical coefficients adopted for the eddy viscosity and k-ε equations Sediment transport models combined with or without flocculation models Assumed values and calibrated coefficients of FMs Calibrated values of empirical coefficients for the critical shear stress

8 LIST OF FIGURES Figure page 3-1 Example of definition of fractal dimension Example of two same sized aggregates having different fractal dimensions. A) F=3. and B) F= Schematic sketch of floc structure due to flocculation process Conceptual diagram for effect of turbulent shear and concentration on floc size (Dyer, 1989) Evolutions of F n (X) with X for three experiments and values of p and q. A) p = 1., q =.5, B) q =.5, and C) p = Model results with different initial floc sizes (4, 1, 2, 4, 6 and 8 μm) Experimental results of equilibrium floc size reported by Bouyer et al. (24) and modeled results of FM A and FM B for several dissipation parameters Experimental results of equilibrium floc size measured by Biggs and Lant (2) and model results of FM A and FM B for several dissipation parameters Temporal evolution of floc size measured by Biggs and Lant (2) and calculated by FM A for the case of G=19.4 s -1. Three curves represent model results using different sets of k A and k B Temporal evolution of floc size measured by Biggs and Lant (2) and calculated by FM B for the case of G=19.4 s -1. Three curves represent model results using different sets of k A and k B Temporal evolution of floc size measured by Biggs and Lant (2) and calculated by FM A for the case of G=19.4 s -1. Three curves represent model results using different sets of p and q Temporal evolution of floc size measured by Biggs and Lant (2) and calculated by FM B for the case of G=19.4 s -1. Three curves represent model results using different sets of p and q Change of the fractal dimension of FM B with time for the case of G=19.4 s Comparison of two flocculation models, FM A and FM B, for T71 experiment of Delft Hydraulics Comparison of two flocculation models, FM A and FM B, for T69 experiment of Delft Hydraulics

9 4-12 Comparison of two flocculation models, FM A and FM B, for T73 experiment of Delft Hydraulics 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=12 mg/l 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=16 mg/l Experimental result of Spicer et al. (1998) and model results of FM C Experimental result of Spicer et al. (1998) and model results of FM D Experimental result of Spicer et al. (1998) and model results of FM A and FM B Experimental result of Biggs and Lant (2) and model results of FM C Experimental result of Biggs and Lant (2) and model results of FM D Experimental result of Biggs and Lant (2) and model results of FM A and FM B Experimental results of case B12 of Burban et al. (1989) and model results of FM C and FM D Experimental results of case B12 of Burban et al. (1989) and model results FM A, FM B, and FM C Experimental results of case B4 of Burban et al. (1989) and model results of FM C and FM D Experimental results of case B4 of Burban et al. (1989) and model results FM A, FM B, and FM C Temporal evolution of floc size simulated by FM A combined with a variable yield strength Definition of coordinate system Depth-averaged flow velocity and water depth used to test the sediment transport model. A) The depth-averaged flow velocity and B) the water depth Mass concentration calculated by sediment transport model combined with FM C using two types of concentrations Volumetric concentration calculated by sediment transport model combined with FM C using two types of concentrations Velocity calculated by sediment transport model combined with FM C using two types of concentrations

10 5-6 Mass concentration calculated by sediment transport model combined with FM C using one type of concentration Volumetric concentration calculated by sediment transport model combined with FM C using one type of concentration The Ems/Dollard estuary and the measuring pole equipped with a rigid frame for insitu measurement (van der Ham et al., 21) Time evolution of floc sizes simulated by FMs combined with sediment transport model Time evolution of fractal dimensions simulated by FMs combined with sediment transport model Time evolution of densities simulated by FMs combined with sediment transport model Time evolution of settling velocities simulated by FMs combined with sediment transport model Velocities measured and calculated by CMC at 1. m Velocities measured and calculated by CMB at 1. m Velocities measured and calculated by CMA at 1. m Velocities measured and calculated by CMN at 1. m Measured mass concentrations and mass concentrations calculated by CMC using a variable fractal dimension and yield strength Measured mass concentrations and mass concentrations calculated by CMB using a variable fractal dimension and a constant yield strength Measured mass concentrations and mass concentrations calculated by CMA using a constant fractal dimension and yield strength Measured mass concentrations and mass concentrations calculated by CMN using constant floc size and density Measured and simulated mass concentration profiles Settling velocities of floc and dissipation parameter at.5 m, 1. m, and 1.5 m above the bottom calculated by CMC. A) settling velocity and B) dissipation parameter Mass concentrations at.3 m and.7 m calculated by CMC using the constant τ c and the variable τ c

11 6-17 The bottom stress (dotted lines) and the critical shear stress (solid lines) calculated by CMC The bottom stress (dotted lines) and the critical shear stress (solid lines) calculated by CMB The bottom stress (dotted lines) and the critical shear stress (solid lines) calculated by CMA The bottom stress (dotted lines) and the critical shear stress (solid lines) calculated by CMN 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 Vertical profiles of size, settling velocity, and mass concentration of floc calculated by CMC and CMB at t=18. hr. The velocity at t=18. is at the peak Simulated volumetric concentration profiles. Solid and dotted lines represent simulation results of CMC and CMN Settling velocity plotted as function of floc size Relationship between settling velocity and mass concentration calculated by CMC Relationship between settling velocity and dissipation parameter (G) calculated by CMC Relationship between floc size and mass concentration calculated by CMC Relationship between settling velocity and volumetric concentration calculated by CMC Relationship between settling velocity and density of floc calculated by CMC Relationship between settling velocity and mass concentration calculated by CMB Relationship between settling velocity and dissipation parameter (G) calculated by CMB Relationship between floc size and mass concentration calculated by CMB Relationship between settling velocity and volumetric concentration calculated by CMB Relationship between settling velocity and density of floc calculated by CMB Relationship between settling velocity and density of floc calculated by CMA

12 6-36 Mass concentration calculated by CMC without damping effect of density stratification Mass concentration calculated by CMC with σ c = A-1 Schematic description on adopting the mensuration by parts for Eq. A A-2 Schematic description on adopting the mensuration by parts for Eq. A

13 LIST OF ABBREVIATIONS a B 1 B 2 Empirical coefficient for breakup process Empirical parameter for yield stress of floc [N] Empirical parameter for yield strength of floc [N] c Mass concentration [kg/m 3 ] C ε 1, C ε 2, C ε 3 Numerical parameter C μ CMA CMB CMC CMN d D D D e D fc e b, e c, e d E f s F F c F c,p F n F y FM Numerical parameter Sediment transport model combined with FM A Sediment transport model combined with FM B Sediment transport model combined with FM C Sediment transport model without flocculation model Size of primary particle [m] Size of floc [m or μm] Initial floc size [m or μm] Equilibrium floc size [m or μm] Characteristic size of floc [m or μm] Efficiency parameter Erosion flux Shape factor Three-dimensional fractal dimension of floc Characteristic fractal dimension Cohesive force of primary particle [N] Function for equilibrium floc size Yield strength of floc [N] Flocculation model 13

14 FM A FM B FM C FM D Flocculation model using constant fractal dimension and constant yield strength Flocculation model using variable fractal dimension and constant yield strength Flocculation model using variable fractal dimension and variable yield strength theoretically derived Flocculation model using variable fractal dimension and variable yield strength empirically proposed by Sonntag and Russel (1987) G Dissipation parameter (Shear rate) [s -1 ] h Water depth [m] k Turbulent kinetic energy [m 2 /s 2 ] ' k A ' k B M n N N rup N turb p q r s s t T rel T L u u τ Empirical dimensionless coefficient for aggregation process Empirical dimensionless coefficient for breakup process Total eroded mass [kg] Number of flocs per unit volume Number of primary particles within a floc Number of primary particles in the plane of rupture Rate of collision of particles due to turbulent flow Empirical coefficient for breakup process Empirical coefficient for breakup process Empirical parameter for τ y Specific gravity of primary particle Time [s] Relaxation time [s] Turbulent eddy time scale [s] x-direction flow velocity [m/s] Total bottom friction velocity [m/s] 14

15 U U v V V W s X α α s α 1, α 2, α 3 β β e Computed depth-averaged x-direction flow velocity [m/s] Desired depth-averaged x-direction flow velocity [m/s] y-direction flow velocity [m/s] Computed depth-averaged y-direction flow velocity [m/s] Desired depth-averaged y-direction flow velocity [m/s] Settling velocity [m/s] Ratio of the equilibrium floc size to primary particle size Empirical coefficient for variable fractal dimension Slope of the bottom Empirical parameter for variable critical stress Empirical coefficient for variable fractal dimension Empirical parameter for upward erosion flux Δ ρ f Immersed density of floc [kg/m 3 ] Δ ρ s Immersed density of primary particle [kg/m 3 ] ε Turbulent dissipation rate (dissipation rate of energy) [m 2 /s 3 ] λ Kolmogorov micro scale [m] μ Dynamic viscosity [N s/m 2 ] ν ν t Kinematic viscosity [m 2 /s] Eddy viscosity [m 2 /s] ρ f Density of floc [kg/m 3 ] ρ s Density of primary particle [kg/m 3 ] ρ w Density of water [kg/m 3 ] σ c, σ k, σ ε, Numerical parameter τ b Bottom stress [N/m 2 ] 15

16 τ c Critical shear stress [N/m 2 ] τ s Surface shear stress [N/m 2 ] τ Yield stress of floc [N/m 2 ] y τ y Scaling parameter for τ y [N/m 2 ] w τ x-direction fluid stress [N/m 2 ] xz w τ y-direction fluid stress [N/m2 ] yz φ f φ s φ sf Volumetric concentration of floc Solid volume concentration of primary particle Solid volume concentration of primary particle within a floc p / x Pressure gradient in x-direction [N/m 3 ] p / y Pressure gradient in y-direction [N/m 3 ] 16

17 Chair: Tian-Jian Hsu Major: Civil Engineering Abstract of Dissertation Presented 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 29 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 changes as floc size increases or decreases. Furthermore, the yield strength of floc is observed to be proportional 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 equation is combined with flocculation models. By comparing with laboratory experiments on temporal evolution of floc size (mixing tank and Couette flow), this research demonstrates the importance of incorporating a variable fractal dimension and a variable floc yield strength into 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 roles of floc dynamics in determining the predicted sediment dynamics in a tide-dominated environment. A 1DV numerical model for fine sediment transport is revised to incorporate four different modules for flocculation, i.e., no floc dynamics, floc dynamics with assumptions of constant fractal dimension and yield strength, floc dynamics for variable fractal 17

18 dimensional only, and floc dynamics for considering both fractal dimension and yield strength variables. Model results are compared with measured 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 incorporated incompletely. This feature is inconsistent with the observation. When considering variable fractal dimension and variable yield strength in the flocculation model, numerical model predicts much smaller floc settling velocity during slack tide and hence is able to predict measured concentration reasonably well. Model results further suggest that, when sediment concentration is greater than about.1 g/l, there exists a power law relationship between mass concentration and settling velocity except very near the bed where turbulent shear is strong. This observation is consistent with earlier laboratory and field experiment on floc settling velocity. It is concluded that a complete floc dynamics formulation is important to modeling cohesive sediment transport. 18

19 CHAPTER 1 INTRODUCTION 1.1 Significance of Study on Cohesive Sediment Transport Sediment transport is an important physical process that further controls many environmental, geo-morphological, and biological processes and their relationship with the natural environment. Furthermore, studying sediment transport is of economical interest such as the maintenance of navigatable harbors and channels through dredging. Sediment transport process is determined by hydrodynamics of carrier flow and sediment characteristics. However, these are very dynamic factors that are determined in accordance with complicated fluidsediment interactions (Winterwerp and van Kesteren, 24). Thus, to study sediment transport, it is important to understand two representative characteristics, the hydrodynamics of carrier flow and the dynamics of sediment. Hydrodynamic conditions 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 sediment. Here, turbulence is one of the essential elements in the study of sediment transport. Sediment is classified into two groups, non-cohesive sediment and cohesive sediment in a broad sense. Sand and gravel are typical non-cohesive sediments. Their electro-chemical or biochemical attraction is small enough to be ignored and, as a result, sediment particles are transported individually. On the other hand, Cohesive sediments, the mixture of water and finegrained sediments such as clay, silt, fine sand, and organic material, have cohesive characteristics due to significant electrochemical or biological-chemical attraction. The physics of cohesive sediment transport is more complicated than non-cohesive sediment due to flocculation processes (e.g. Dyer, 1989; Winterwerp and van Kesteren, 24). Cohesive sediments form floc aggregates through binding together of primary particles and smaller flocs 19

20 (aggregation), and flocs can disaggregate into smaller flocs/particles due to flow shear or collision (breakup or disaggregation) (Dyer, 1989). The properties of floc aggregates continuously change with the fluid flow condition. The averaged size of cohesive sediment aggregate is determined by flow turbulence, concentration of sediment, biological-chemical properties of water, properties of primary particle and so on (Lick et al., 1992). Thus, accurate prediction of cohesive sediment transport may require detailed water column models that resolve time-dependent flow velocity, turbulence and sediment concentration (Winterwerp, 22; Hsu et al., 27). 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 studying cohesive sediment transport. The earth s surface is almost entirely covered with large or small amounts of cohesive sediment (Winterwerp, 24). In estuaries, large amount of cohesive sediment can be found near the river mouth. Studying the fate of these terrestrial 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., 28; Fabricius and Wolanski, 2). Hence, understanding detailed dynamics of cohesive sediment transport is as important as noncohesive sediment transport process. 1.2 Objectives of This Study The major objective of this research is to understand the dynamics of cohesive sediment transport in tide-dominated environment. The primary tasks performed to achieve this objective are to: (1) develop a flocculation model representing natural properties of cohesive sediments, 2

21 (2) develop a comprehensive sediment transport model which can describe transport of cohesive sediment under the condition of tide flow and river flow (3) incorporate a flocculation model into a numerical model for cohesive sediment transport, (4) apply the model to estuaries where river input from upstream and tidal flow coexist, (5) investigate the effect of modeling floc dynamic on cohesive sediment transport, and (6) assess the needs for future research. 1.3 Terminology In this section, terminology adopted 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 smaller flocs/particles due to flow shear or collision Brownian motion : the random movement of particles 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 fine-grained 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 parameter that characterizes the effects of turbulence on the evolution of floc size (Winterwerp and van Kesteren, 24) 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 particles through binding together of primary particles 21

22 Flocculation : a series of aggregation and breakup due to cohesive properties of sediment and flow turbulence Fractal dimension : a statistical quantity that gives an indication of how completely a fractal appears to fill space Hindered settling effect : the effect of particles or concentration 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 sediment concentration stratification Mass concentration : the mass of sediment pre unit volume of fluid-sediment mixture Non-cohesive sediment : general coarser sediment, such as sand and gravel, of which attraction is not sufficient to aggregate particles Number concentration : the number of suspended particles such as floc and particles per unit volume of fluid-sediment mixture Primary particle : an individual particle not to be broken into smaller particles by general stress in nature such as turbulent shear and collisional stress between particles Sediment transport : the movement 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 Self-weight consolidation : Compaction of bed due to self-weight of sediment Settling velocity : the gravity-induced terminal velocity at which particles fall through the water column Shear rate : See dissipation parameter Size-classes flocculation model : flocculation model of which interest is in individual particles having various sizes and which considers input and output of particles Solid volume concentration : the volume of suspended primary particles per unit volume of fluid-sediment mixture and the mass concentration is obtained by multiplying it with density of primary particle Total eroded mass : the mass of suspended sediment in the water column above unit area of bottom 22

23 Turbulent eddy time scale : characteristic timescale of an eddy turn-over 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 volume 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 importance of study on cohesive sediment, the objectives of the present research, and definitions of terminology used in this dissertation. In Chapter 2 (Literature Review), previous 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, fractal theory, fractal dimension, and flocculation process, one of the most important properties of cohesive sediment. Chapter 4 (Modeling Flocculation of Cohesive Sediment) first discusses the characteristics of two types of flocculation models, size-classes flocculation model and Lagrangian flocculation model. Secondly, the Lagrangian flocculation models are 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 fractal dimension and a variable yield strength of floc. Thirdly, these different flocculation models 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 conditions for tidal flow forcing adopted in the 23

24 numerical model of sediment transport are presented. These proposed equations and boundary conditions are tested with idealized conditions as preliminary study. Chapter 6 (Model Application to Ems/Dollard Estuary) discusses results of sediment transport models combined with three different flocculation models. The numerical models are applied to the in-situ measurement conducted at the Ems/Dollard estuary (van der Ham et al., 21). The effects of flocculation and bed erodibility on cohesive sediment transport in tide-dominated 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 derivation 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. 24

25 CHAPTER 2 LITERATURE REVIEW 2.1 Studies on Flocculation and Yield Strength of Floc The flocculation process has been studied by many researchers. The theoretical aspects of the flocculation process have been developed by pioneering studies such 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 the important factors of collision mechanism according to particle sizes. Lick et al. (1993) further study the effect of differential settling on flocculation of fine-grained sediments using natural sediments. McAnally and Mehta (2) develop a dynamical formulation for estuarine fine sediment aggregation. The spectrum of fine particle has been represented by a discrete number of classes and the frequency of particle collisions due to Brownian motion, turbulent shearing and differential settling are described by statistical relationships. They conclude that it is very important to characterize particle density and strength when flocculation approaches equilibrium state. Flocculation of fine-grained particles depends 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 O Melia (198), McCave (1984), van Leussen (1994), and Stolzenbach and Elimelich (1994), it can be concluded that for cohesive sediment transport in rivers, estuaries and continental shelves (or other aquatic system with more energetic flow) the effects of Brownian motion and differential settling on the flocculation process may be less important. Hence, many studies have focused on understanding the effects of turbulence on 25

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 ε / ν. Herein, ε is the turbulent dissipation rate and ν is the kinematic viscosity 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 algorithm to calibrate the parameters proposed by Argamam and Kaufman (197) 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 environment, especially wave-dominated condition (Hill and Newell, 1995; Hsu et al., 27; Traykovski et al., 2; Winterwerp, 22). Biggs and Lant (2) conduct experiments in order to obtain the temporal change of floc size with respect to a prescribed constant dissipation 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 relationship due to the breakage mechanisms. Bouyer et al. (24) analyze the relationship between 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) investigate the relationship between floc size and dissipation parameter under different sediment concentrations using an annular flume. They conclude that at low shear rate, increasing turbidity encourages floc growth. However, at high shear rate, increasing turbidity in suspension may enhance breakup of floc. 26

27 Winterwerp (1998; 22) develops a flocculation model adopting fractal theory. The concept of fractal geometry has been widely used in order to describe floc geometry (see Vicsek, 1992, and Kranenburg, 1994, for a review). Winterwerps s model describes one characteristic floc size and considers turbulence as the dominant factor affecting flocculation processes. However, a fixed value of fractal dimension such as 2. and 2.2 (Winterwerp, 1998; Winterwerp et al., 26) 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. (Hawley, 1982; Meakin, 1988). However, large variations of fractal dimension are obtained 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, 24) in a consolidating bed, the resulting fractal dimension is significantly larger than 2. (around 2.75). As illustrated by Khelifa and Hill (26), considering a completely consolidated bed, where all the floc structure is completely destroyed, the fractal dimension is 3.. 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 (26) propose a model to predict the effective density of flocs and the resulting settling velocity using a variable fractal dimension that depends on floc size. They demonstrate that, by using the concept of variable 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 very large (more than about 2 mm), the settling velocity decreases as floc size increases. Consequently, Khelifa and Hill (26) suggest a new 27

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. (27) also adopt a variable fractal dimension to develop a size-classes flocculation model and conclude that the use of a variable fractal dimension results in better predictions of flocculation process. More recently, Son and Hsu (28) further extend the floc dynamic equations of Winterwerp (1998) for variable fractal dimension suggested by Khelifa and Hill (26). However, Son and Hsu (28) show that none of the two flocculation models of Winterwerp (1998) and Son and Hsu (28) is in satisfactory agreement with experimental results 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 important parameter in flocculation process because it has a direct relationship with breakup process during flocculation. Many 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, 21; Wu et al., 23; Gregory and Dupont, 21; Wen and Lee, 1998; Yeung and Pelton, 1996; Zhang et al., 1999). For example, Wen and Lee (1998) apply a controllable ultrasonic field to a floc suspension and observe floc erosion. Zhang et al. (1999) squeeze a single floc in suspension between a glass slide and fiber optic using a force transducer. The values of floc yield stress estimated in these studies are in very wide range between the order of 1-2 to 1 3 (N/m 2 ) (see Javis et al., 25, for more details). McAnally (1999) proposes an equation for yield stress of floc. His derivation starts with the assumption that a floc yield strength is constant as Kranenburg (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 Hsu (29) have theoretically derived a constitutive 28

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 flocculation models of Son and Hsu (28) and Winterwerp (1998), significant improvement on the temporal evolution of floc size is obtained. Hence, it is concluded that the flocculation model, 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 cohesive 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 are determined by flow turbulence, sediment concentration, properties of carrier fluid, properties of primary particle and so on. Thus, to accurately predict transport of cohesive sediment, detailed water column models which resolve time-dependent flow velocity, turbulence and sediment concentration are required (Winterwerp and van Kesteren, 24; Winterwerp, 21; Hsu et al., 29). van der Ham and Winterwerp (21) analyze flow velocity, sediment concentrations and turbulence measured in the Ems/Dollard estuary with a 1-dimensional vertical (1DV) numerical model. In this study, it is concluded that the damping of flow turbulence due to sediment-induced density stratification can explain rapid settling of suspended sediment towards slack water. Li and Parchure (1998) have modeled the damping effect by stratification in the vertical suspended sediment transport equation and obtained reasonable site-specific results on predicted profiles of suspended sediment over mud bank. Hsu et al. (29) incorporate 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. 29

30 The settling velocity of sediment is a very important factor to determine downward flux of sediment transport. Furthermore, the settling velocity of cohesive sediment significantly varies with flocculation process (e.g. Mehta and Partheniades, 1975; van Leussen, 1994). Mehta (1986) proposes empirical equations for the settling velocity and shows that the settling velocity depends on sediment concentration and turbulence. Khelfa and Hill (26) suggest a model to predict settling velocity based on the concept of fractal geometry. Erosion or resuspension of sediment from the bottom is one of the most important factors governing sediment transport in natural water body (Sanford and Maa, 21) and it is highly dependent on critical shear stress of the bed. For cohesive 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 particles. 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 shear stress can be parameterized by amount of eroded sediment from the bed. Many studies propose the use of a power law relationship between amount of erosion and critical shear stress (Lick, 1982; Roberts 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 Piedr-Cueva and Mory (21) shows an important characteristic of cohesive sediment. Their flume erosion experiment shows increase of critical shear stress with depth of bed, suggesting that the consideration of variable critical shear stress of cohesive bed is of great importance. Sanford and Maa (21) propose a power law relationship between critical shear stress and total eroded mass in the water column (unit of kg/m 2 ) 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 eroded mass and the rate of increase of critical shear 3

31 stress approaches zero at non-zero values of total eroded mass. They conclude that this is evidence for the presence of a floc or fluffy layer. 31

32 CHAPTER 3 STUDY ON PROPERTIES OF COHESIVE SEDIMENT As mention in Chapter 1, cohesive characteristics of very fine sediments are due to significant electrochemical or biological-chemical attraction. Flocculation is one of the most important characteristics of cohesive sediment. Via flocculation, aggregate 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 concept of fractal geometry, floc structure is described by a fractal dimension, F. Flocculation process is governed 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 process because breakup is considerably affected by the force to break a floc. In this chapter, fundamental concepts for flocculation and yield 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, 24). 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 such as clay and silt are considered to be affected significantly by cohesion compared to coarse sediment such as sand because weight of fine sediment (that is, inertia) is not large enough to resist its electrochemical or bio-chemical cohesive force. Mehta and Li (1997) classify fine sediment into several groups by size and describe their intensity of cohesion (Table 3-1). As mentioned in the previous chapter, cohesive sediment is a mixture of non-organic solid material such as clay and silt, organic materials, water and so on. The solid materials can be classified in accordance with size by ASTM International Standards shown in Table 3-2. Though 32

33 colloid can be also classified into solid material, it is often considered 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 3-3 following studies of Mehta and Li (1997), Grim (1968), Dade and Newell (1991), and Ariathurai et al. (1977). Clay minerals are important in the sense that cohesion of sediment is significantly dependent on them. The cation exchange capacity (CEC) is the total amount of exchangeable cation. The CEC is proportional to reactivity (that is, cohesive force) of mineral. Table 3-4 provides the range of CEC for major materials of solid matter of cohesive sediment (Horowitz, 1991; Grim, 1968). The non-clay solid in estuarine environment is commonly composed of quartz and calcium carbonate. The size of non-clay solid is usually larger than 2 μm which is a criterion dividing clay and non-clay mineral. The relative composition of organic matter in estuarial cohesive sediment shows very wide variation spatially and seasonally (Kranck, 198). However, it is clear that organic matter plays an important role on cohesive properties of sediment and flocculation process. Organic matter can be composed of many kinds of material. Following Berner (198), the main organic material shown in estuarine environments are : Ploysaccharides and proteins composed of peptides and amino acid Lipides, hydrocarbons like cellulose, lignin composed of aliphatic and aromatic hydrocarbons Humic actids 33

34 Table 3-1. Intensity of cohesion according to size of sediment (Mehta and Li, 1997) Size (μm) Wentworth Scale Classification Intensity of Cohesion < 2 Very fine clay to medium clay Very important 2 2 Coarse clay to very fine silt Important 2 4 Fine silt to medium silt Increasingly important with decreasing size 4-62 Medium silt to coarse silt Partially ignorable Table 3-2. Classification of sediment by size Sediment Colloid Clay Silt Fine Sand Medium Sand Size <1 μm - 5 μm 75 μm μm - 2, μm (2 mm) Table 3-3. Properties of clay minerals Clay Mineral Grain Size (μm) Equivalent Diameter (μm) Density (kg/m 3 ) Kaolinite ,6 2,68 Illite ,6 2,96 Smectite ,2 2,7 Chlorite ,76 3, 34

35 Table 3-4. Cation exchange capacity of clay minerals Clay Mineral Cation Exchange Capacity (meq/1g) Kaolinite 3 15 Smectites 8 15 Illite 1 4 Chlorite 5 1 Halloysite (2H 2 O) 5 1 Halloysite (4H 2 O) 4 5 Vermiculites 12 2 Attapulgite-palygorskite-sepiolite Fractal Dimension Fractal theory describes the geometry of many natural structures that show a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole (Mandelbrot,1982). This theory can be ideally applied to structure of floc when a floc is composed of infinitesimally small particles. Although primary particles in nature have a size O{1 } μm, Kranenburg (1994) shows that the structure of floc can be described by fractal theory from experimental data. According to fractal theory, the structure of fractal entity is considered to follow a power-law behavior (Winterwerp and van Kesteren, 24). The fractal dimension is the value of power to indicate the structure of fractal entity and the number of smaller particles (primary particles in this study) in a larger aggregate (a floc in this study). 35

36 Figure 3-1 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 and 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: N p D = d F (3-1) where N p is the number of primary particles within one aggregate and F, the three-dimensional fractal dimension, is calculated to be 3.. Figures 3-2 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 3-2 A has 64 primary particles of size d. Whereas, the aggregate shown in Figure 3-2 B is composed of 32 primary particles of size d. According to Eq. 3-1, the fractal dimensions of Figures 3-2 A and 3-2 B are 3. and 2.5, respectively. In many studies, flocs are considered as self-similar fractal entities (e.g. Tambo and Watanabe, 1979; Krone, 1984; Logan and Kilps, 1995; Chen and Eisma, 1995, Winterwerp, 1998; Son and Hsu, 28; Son and Hsu, 29). This approach to the structure of floc is very effective way to describe a number of primary particles within a floc. It is important to know the number of primary particles because it becomes possible to calculate the density of a floc based on known information on density and size of primary 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 3-2 gives us a hint to further calculate the density of floc. In this case, the number of primary particles whose volume is d 3 and void cubes filled with water whose size is also d 3 are 36

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 D F d 3-F and D 3 - D F d 3-F. Now, the equation for the density of floc, ρ f, is obtained by the definition of density, which is the total mass divided by the total volume: F 3 D ρ f ρw = ρf = ( ρs ρw) (3-2) d where, ρ s and ρ w are densities of primary particle and water. This equation is derived by Kranenburg (1994) also (see Eq. 4-11). d D=2d Figure 3-1. Example of definition of fractal dimension 37

38 D=4d A D=4d Figure 3-2. Example of two same sized aggregates having different fractal dimensions. A) F=3. and B) F=2.5. B 38

39 3.3 Flocculation Process Due to eletrochemical or biological-chemical 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 mechanisms such as collision and turbulence shear because its strength is very small compared to a primary particle. This series of processes is the flocculation process, one of the most important characteristics of cohesive sediment. The schematic sketch of floc structure due to flocculation process is depicted in Figure 3-3. Flocculation is essentially different from coagulation. van Olphen (1977) distinguishes these two processes: Flocculation : Reversible process and the result of simultaneous aggregation and breakup Coagulation : Irreversible process to form 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 mentioned in the section 2.1. Among these mechanisms, turbulent shear mainly governs floc aggregation (O Melia, 198; McCave, 1984; van Leussen, 1994; Stolzenbach and Elimelich, 1994). Breakup is considered to be occurred by inter-particle collisions and turbulent shear (e.g. McAnally, 1999). However, the effect of inter-particle collision on breakup is often ignored as size and density of floc are small (Winterwerp, 1998;Winterwerp and van Kesteren, 24; Son and Hsu, 29). 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. The effect of turbulent shear on flocculation is 39

40 rim ary ParticleFcP locsmal(daughter)flodescribed by Dyer (1989). From this study, it is suggested that floc size first increases and then decreases as shear stress increases, and the size is proportional to sediment concentration. Figure 3-4 shows these characteristics qualitatively. The yield strength of floc is the minimum force to break a floc and yield stress is defined as the yield strength divided by the ruptured area. The yield strength depends on the inter-particle bonds between primary particles within a floc (Parker et al., 1972; Bache 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., 25). Therefore, the yield 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 flocculation process is studied because the breakup rate due to turbulent stress is significantly affected by the yield strength. erfigure 3-3. Schematic sketch of floc structure due to flocculation process 4

41 Floc Size Concentration Shear Stress Figure 3-4. Conceptual diagram for effect of turbulent shear and concentration on floc size (Dyer, 1989) 41

42 CHAPTER 4 MODELING FLOCCULATION OF COHESIVE SEDIMENT 4.1 Overview on Flocculation Modeling To quantitatively predict change of floc properties, such as density and size, many types of flocculation models (FM) have been 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 (2) develop a dynamic model for aggregation rate of cohesive sediment. This model considers both binary and multi-body 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 = ε / ν. Herein, ε is the turbulent dissipation rate and ν is the kinematic viscosity of the fluid. Ayesa et al. (1991) develop an algorithm based on data obtained from a mixing tank experiment to determine the parameters suggested by Argamam and Kaufman (197). 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 more concentrated fluid mud transport (e.g. Hill and Newell, 1995; van der Ham and Winterwerp, 21; Uncles and Stephens, 1999; Shi and Zhou, 24). Winterwerp (1998) develops a semi-empirical flocculation model which describes the rate of change of averaged floc size in turbulent 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 aggregate may be too restricted for modeling general cohesive 42

43 sediment transport that has a wide range of flow condition and sediment concentration. Khelifa and Hill (26) suggest a model for floc composed of mono-sized primary particles based on variable fractal dimension using a power law. Maggi et al. (27) also adopt a variable fractal dimension to develop a size-classes flocculation model and conclude that the use of a variable fractal dimension results in better predictions of flocculation process. A variable fractal dimension formulation used in Maggi et al. (27) is very similar to the power law of Khelifa and Hill (26). The size-classes flocculation models such as models of Maggi et al. (27) and McAnally and Mehta (2) can resolve more mechanisms causing flocculation compared to the Lagrangian flocculation models (e.g. Winterwerp, 1998; Son and Hsu, 28). 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 (28) further extend the floc dynamic equations of Winterwerp (1998) for variable fractal dimension suggested by Khelifa and Hill (26). However, Son and Hsu (28) show that none of the two flocculation models of Winterwerp (1998) and Son and Hsu (28) is in satisfactory agreement with experimental results for the temporal evolution of floc size in mixing tanks. These models show much gradual increase during the initial flocculation state and much abrupt increase of floc size when the floc size approaches its equilibrium state. Son and Hsu (28) conjecture that a constant yield strength adapted by these flocculation models may be the main reason causing such deficiency. From this, Son and Hsu (29) further improve the flocculation model adapting a variable yield strength. In this study, they show the flocculation model describes temporal evolution of floc size reasonably well only when both a variable fractal dimension and a variable yield strength are considered. 43

44 4.2 Lagrangian Flocculation Models When modeling cohesive sediment transport, it is needed to consider the change of floc size because many physical quantities, such as the settling velocity depends on floc size. On the other hand, the carrier flow turbulence can be damped due to the presence of sediment. This mechanism may directly or indirectly depend on floc size (Winterwerp, 1998; Hsu et al., 27). Thus, it is important to develop a model calculating the temporal evolution of floc size under given flow conditions. As mentioned the section 4.1, the size-classes 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 adopted and investigated in this study Winterwerp s Flocculation Model A Lagangian flocculation model is developed 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 concentration of flocs, the mass concentration of primary particles and the number of flocs per unit volume of fluid is given by: φ ρ ρ c s w f = = ρ f ρ w ρs f nd s 3 (4-1) where φ f is the volumetric concentration of flocs, c is the mass concentration of primary particles, f s is a shape factor (for sphere, f = π / 6 ) and n is the number of flocs per unit volume s of fluid (water in this study). By combining Eqs. 3-2 and 4-1, n can be represented as: 44

45 c = (4-2) ρ f F 3 F n d D s s By assuming particle diameters are much smaller than the Kolmogorov length scale ( λ ) and based on the theory of Smoluchowski (1917), Levich (1962) suggests the rate of collision of particles due to flow turbulence can be determined by integrating the diffusion equation over a finite volume: N turb = πedgd n (4-3 a) where e d is an efficiency parameter for turbulent diffusion. Winterwerp (1998) further assumes that only a certain portion of the collisions causes flocculation and proposes the equation for the rate of aggregation between the flocs in a turbulent fluid: dn dt 3 e e GD n 3 2 = π (4-3 b) 2 c d where e c is a constant efficiency parameter accounting for the fact that not all collisions result in coagulation. Although e c can be a variable as floc size and floc density change, it is assumed to be constant for simplicity because e c cannot be determined at present on the basis of theoretical arguments (Lick et al., 1992). 45

46 From Eqs. 4-1 and 4-3 b, the increase rate of flocs is obtained for the constant mass concentration: dn F c d D dd f ρ F 3 F 1 = (4-4) s s Eq. 4-4 can be expanded below: dn dn dd F c dd d D dt dd dt f ρ dt F 3 1 F = = (4-5) s s By substituting Eq. 4-5 to Eq. 4-3 b and rearranging this, the equation describing the increase rate of floc size is obtained: ' dd 3 ecπ ed c F 3 4 F ka 1 Gd D cgd F 3 D 4 F = = (4-6) dt 2 f F ρ ρ F s s s where ' k A is a dimensionless coefficient. This equation describes the aggregation process to increase the size of floc by attracting primary particles to a larger floc. Winterwerp (1998) assumes inter-particle collisions 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 function of the dissipation parameter of the disrupting turbulent eddies and proposes the following relation based on dimensional considerations: 46

47 p 1 dn D d μg Ga (4-7) ndt d F D 2 y / q where μ is the dynamic viscosity of the fluid, F y is a yield strength of flocs, and a, p, and q are the coefficients to be discussed later. Winterwerp (1998) considers that F y is dependent of the number of primary particle in a plane of failure and the number of primary particle in the plane is constant. As a result, F y is fixed to be constant in this flocculation model. By further incorporating an efficiency parameter for floc breakup, e b, the increase rate of floc due to breakup is obtained: p q q ' p aeb μ k B μ q+ 1 2q+ 1 = DG G D 2 = dt F d Fy / D F F y d dd D d G D d (4-8) where ' k B is also a dimensionless coefficient. The negative sign (-) is added because breakup of floc decreases floc size. By linearly combining Eq. 4-6 for aggregation process and Eq. 4-8 for breakup process, the complete flocculation model of Winterwerp (1998) is proposed and it is denoted as Model FM A (see Table 4-2): q ' p F 3 4 F k B μ q+ 1 2q+ 1 ' dd ka 1 D d = cgd D G D dt ρs F F F y d (4-9) 47

48 As mentioned above, this flocculation model has been derived assuming that the fractal dimension, F, is constant. Many experimental studies suggest that floc size is proportional to the Kolmogorov length scale (Bratby, 198; Akers et al., 1987; Leentvaar and Rebhun, 1983), 3 1/4 λ = ( ν / ε ), where ν is the kinematic viscosity of fluid and ε is the turbulent dissipation rate. Hence, Winterwerp (1998) assumes that the equilibrium floc size, D e, 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. and q=.5 in this flocculation model using fixed fractal dimension. With these values for empirical coefficients and F=2. which is assumed by Winterwerp (1998), Eq. 4-9 is simplified:.5 dd G ' c 2 ' μg 2 = ka D kb ( D d) D dt 2d ρ s F y (4-1) Flocculation Model Using a Variable Fractal Dimension The main concept of fractal theory is self-similarity 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 adopted in this section is based on previous two studies of Winterwerp (1998) for floc dynamics and Khelifa and Hill (26) for variable fractal dimension. For mono-sized primary particles of size d, the effective density of floc is calculated as (Kranenburg, 1994): 48

49 D ρ f = ρw + ( ρs ρw) d F 3 (4-11) where F is the three-dimensional fractal dimension of flocs. To take into account the possible variability in the structure of flocs, a variable fractal dimension depending on floc size is proposed by Khelifa and Hill (26). The fractal dimension of a floc with size closer to the size of the primary particles should approach the value of 3. (Khelifa and Hill, 26). On the other hand, the fractal dimension of large flocs should be close to the value of 2. (Dyer, 1989; Dyer and Manning, 1999; Meakin, 1988; Winterwerp, 1998). Hence, a power law is proposed by Khelifa and Hill (26) to describe variation of fractal dimension: D F = α d β (4-12) log( / 3) where α = 3 and β = Fc log( D / d). F c is a characteristic fractal dimension and D is a fc fc characteristic size of flocs. Khelifa and Hill (26) suggest the typical value of F c and F c = 2. and D fc to be D fc = 2 μ m. Eg gives a plausible description of fractal dimension such that when d<<d, F approaches 3. but for very large floc, F approaches 2.. By combining Eqs. 4-1 and 4-11, n can be represented as: c = (4-13) ρ f F 3 F n d D s s 49

50 and dn dd can be calculated as: dn 3c F 3 β F 1+ β D = d D β ln + 1 dd ρ f d s s (4-14) Utilizing dd dd dn = and Eqs. 4-3 b and 4-14, an equation representing the evolution of dt dn dt floc size due to aggregation is obtained: dd cecπ e = dt 2ρ f s d s Gd D F 3+ β F+ 4 β 1 D β ln + 1 d (4-15) As mentioned previously, the concept of fractal theory is based on self-similarity of the structure. Although this is appropriate for aggregation process, breakup process may be too abrupt to entirely adopt variable fractal dimension. Following Winterwerp (1998), it is assumed that inter-particle collisions are apt to cause aggregation of flocs rather than breakup. Hence, only the breakup by turbulent shear stress is incorporated here. Substituting Eqs. 4-2 and 4-14 into Eq. 4-7, the balance equation for the decay rate of flocs by breakup process can be written as: q dd ega b μg p 1 2q p 1 d β D β+ + = ( D d ) dt 3 F D y β ln + 1 d (4-16) 5

51 In the present study using variable 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 and breakup processes (Winterwerp, 1998), i.e., Eqs and 4-16, a complete flocculation model using a variable fractal dimension can be obtained and it is denoted as Model FM B (see also Table 4-2): q β ' ' dd Gd ka c F 3 F 4 k β B μg + p β+ 2q+ 1 p = d D d D ( D d) dt D β ln 1 3 ρs 3 F + y d (4-17) where k ecπ ed = and 2 f ' 3 A s k ' B = ae are empirical dimensionless coefficients. b For equilibrium condition, i.e. dd/dt=, and using the assumption that D e is much larger than d, Eq can be simplified as: q ' 2 β 3 X + ( p+ 2q 3) c k A μgd Fn ( X) = ( X) ' = ρs k B F y (4-18) where X= De / d. Eq is a nonlinear algebraic equation of D e. A numerical solution for equilibrium floc size can be obtained by setting F n zero. In this study, a numerical solution of the time evolution equation of floc size, i.e. 4-17, will be directly calculated using a Runge-Kutta method. However, it is important to first examine the effects of p and q on the flocculation model using Eq because their behavior affects the nonlinearity and hence numerical stability of 51

52 time evolution of Eq Figure 4-1 A shows the variation of F n (X) with X for three flocculation experiments: T69, T71, and T73 carried out in Delft Hydraulics (see Van Leussen, 1994). Flow conditions and coefficients for these simulations are shown in Table 4-1. Using values suggested by Winterwerp (1998), p=1., q=.5 (and with ' k A =.15 and ' k B =1-5 ), Eq has solution (i.e., for the range X > 1, F n (X)= exists). Figure 4-1 B further presents the evolution of F n (X) with several p values for test T71 but with q remains.5. When p is as large as 1.3, F n (X) approaches zero rapidly and the resulting D e is very close to d (i.e., D e /d=3.15). Moreover, when setting p to be.7, F n (X) has no root and the equilibrium floc size does not exist. That is, Eqs and the flocculation model, equation 4-17, become unrealistic when p=.7. Figure 4-1 C further shows the evolution of F n (X) with several q values for test T71 with p=1.. When q is.7, F n (X) increases rapidly with respect to X. In contrast, F n (X) increases very slowly when q is.3. Hence, when q is smaller than.3, a highly accurate and stable numerical solver is necessary in order to obtain a solution for Eq From these observations, it can be concluded that values of p=1. and q=.5 originally suggested by Winterwerp (1998) based on physical arguments are also rather robust numerically for the present flocculation model using variable fractal dimension. When the values suggested by Winterwerp (1998) for p=1. and q=.5 are adopted for FM B using variable fractal dimension, Eq can be rewritten as:.5 β ' dd Gd c ' F 3 F 4 k β B μg + 1 β+ = kd 2 A D d D ( D d) dt D β ln 1 3ρs 3 F + y d (4-19) Figure 4-2 shows the dependence of modeled time evolution of floc size on the initial floc size with other parameters kept the same: G=7.3 s -1, c=.65 kg/m 3, k =.98, ' A = 3.3 1, ' 5 k B 52

53 and F y =1-1 N. It is observed that the initial floc size 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 because 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 semi-empirical model to describe flocculation process of cohesive sediment in turbulent flow. For aggregation process, a variable fractal dimension is adopted under the assumption that a floc has the characteristic of self-similarity, 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 describe 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 4-1). 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. Yeung and Pelton, 1996; Tambo and Hozumi, 1979). A variable yield strength is further incorporated into flocculation modeling in the next section. 53

54 F n (X) T73-8 T71 T X A p=1.3 p=1. p= q=.7 q=.5 q= F n (X) F n (X) X X B C Figure 4-1. Evolutions of F n (X) with X for three experiments and values of p and q. A) p = 1., q =.5, B) q =.5, and C) p =

55 Table 4-1. Experiment values and parameters of flocculation models Test NO. c (kg/m 3 ) G (s -1 ) d (μm) F y (N) ρ s (kg/m 3 ) ' k A ' k B FM B FM A FM B FM A T T T Figure 4-2. Model results with different initial floc sizes (4, 1, 2, 4, 6 and 8 μm) 55

56 4.2.3 Flocculation Model Using a Variable Fractal Dimension and Variable Yield Strength As presented in Eq. 4-11, the fractal dimension is an indicator to describe how dense a floc is for a given size and density of the primary particles. 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: ρ = φρ+ (1 φ) ρ f s s s w (4-2) where φ sf is the solid volume concentration of primary particles within a floc. Rearranging Eq. 4-2, φ sf is expressed as: φ sf Δρ f = (4-21) Δ ρ s where Δ ρ f (= ρ f ρ w ) is the immersed density of floc and Δ ρs (= ρ s ρ w ) is the immersed density of primary particle. By substituting Eq to Eq. 4-21, the equation for φ sf is rewritten as: φ sf D = d F 3 (4-22) 56

57 Under the assumption that the floc and the primary particles are spherical for the sake of simplicity, the number of primary particles within a floc, N, is derived from the definition, 3 Nd φ sf =, and Eq. 4-22: 3 D F D N = d (4-23) In addition, the number of primary particles within a floc is assumed to be sufficient to adopt mensuration by parts. Using mensuration by parts, the average distance between two neighboring primary particles within a floc is determined. From this, one can further determine the averaged area occupied by one primary particle in the ruptured plane. Consequently, an equation for the number of primary particles whose size is D, can be derived as (see Appendix for more details): N rup in the plane crossing the center of a floc, N rup 2/3 π 2F 3 π D = 4 6 d (4-24) In this study, the plane crossing the center of a floc is assumed 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 constant yield strength of floc, F y, under the assumption that the number of bonds (or primary particles) in a ruptured plane is independent of the size of the floc (e.g. Kranenburg, 1994). However, Boadway (1978) and Tsai and Hwang (1995) have observed floc breakup process and concluded that a floc often disaggregates into two roughly equal-sized 57

58 flocs. Hence, it is assumed here that during floc breakup, 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 should be a function of floc size and its fractal dimension. The yield strength of floc depends on the strength of inter-particle 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 strength is also a function of floc size and its structure (e.g. Yeung and Pelton, 1996). Sonntag and Russel (1987) suggest an empirical equation for the yield stress π D (units, Pa), which is the yield strength divided by the cross-sectional area of floc ( 4 2 ), based on a power law: r r( F 3) τ y = τ yφsf ( = τ y( D/ d) ) (4-25) where τ y is a scaling parameter and r is an empirical coefficient. According to a later study by Bache (24), the value of r is usually in the range between.5 and 1.5. Following the definition of the solids volume concentration within a floc, φ sf (see Eq. 4-22), it can be concluded that the yield strength is not constant but a function of fractal dimension and floc size. This discussion on a yield strength is consistent with earlier study by Tambo and Hozumi (1979) who postulate that the yield strength of floc is proportional to the net solid area in the ruptured plane. In addition, it is clear that the yield strength 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 strength is proposed based on Eq. 4-24: 58

59 D = Fy B2 d 2F 3 (4-26) 2/3 π π where B2 = F cp,, and F c,p is the cohesive force of primary particle. F c,p is considered 4 6 as an empirical parameter because it depends on the properties of sediment and chemicalbiological effects. Further dividing Eq by the area of ruptured plane, an equation for the yield stress of a floc, τ y, is obtained: τ y 2F Fy D 3 = = B 2 1 ( π /4) D 2 D d (4-27) where 2/3 π 1 =, B F cp. It can be seen that in this formulation, the only parameter that is 6 difficult to obtain and needs to be empirically determined is F c,p. Utilizing chain rule, Eqs. 4-14, 4-7 and 27 for floc yield stress, the equation for floc breakup due to turbulent shear is obtained: ega μ 3 B q 2q 2q dd 1 2 b G β p + 3 F β+ d D q 3 = F ( D d ) p D dt β ln 1 d (4-28 a) The above equation adopts the yield stress equation theoretically derived in this study (Eq. 4-27). Similarly, utilizing the yield stress equation suggested empirically by Sonntag and Russel (1987) (see Eq. 4-25), the following equation for floc breakup is obtained: 59

60 ega 3 τ q dd b μg p rq(3 F) 1 rq(3 F) p 1 d β D β+ = ( D d ) D dt y β + ln 1 d (4-28 b) The complete flocculation models are obtained by linearly combining 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. 4-27), it is denoted as Model FM C (see also Table 4-2) and is written as: β ' dd Gd ka c G = d D d D D d dt D β ln 1 3 ρ 3 B d q ' 2q 2q p F 1 (3 F) F 3 F 4 k β β B μ p ( ) + s 1 (4-29 a) The flocculation model utilizes floc yield stress of Sonntag and Russel (1987) is denoted as FM D (see Table 4-2) and is given as: q β ' ' dd Gd ka c F 3 F 4 k β B μg + p rq(3 F) 1 β+ rq(3 F) = d D d D D d dt D β ln 1 3 ρ 3 τ d ( ) + s y p (4-29 b) Essentially, p and q are empirical coefficients as mentioned in the previous section. The values of p=1 and q=.5 based on several additional constrains are also used FM C and FM D. 6

61 Table 4-2. Summary of flocculation models used in this study Model name Characteristic Reference FM A Constant fractal dimension Constant F y Eq. 4-9 FM B Variable fractal dimension Constant F y Eq FM C Variable fractal dimension Variable yield strength theoretically derived in Section Eq a FM D Variable fractal dimension Empirical variable yield stress of Sonntag and Russel (1987) Eq b 61

62 4.3.1 Application of FM A and FM B 4.3 Investigation of Flocculation Models As shown in Eq. 4-19, the flocculation model using a variable fractal dimension (FM B) depends primarily on five parameters: the floc size d and density ρ s of the primary particle, the yield strength of flocs F y, and empirical parameters ' k A and ' k B. For test of FM A and FM B, this study follows Winterwerp (1998) where d, F y, and ρ s are assumed to be 4 μ m, O{1-1 } N, and 2,65 kg/m 3. Winterwerp (1998) specifies these values based on experimental data and information adopted by previous literatures (Matsuo and Unno, 1981; van Leussen, 1994). Specifically he estimates the yield strength of floc F y, to be about O{1-1 } N, but also acknowledges that for natural mud F y may change by several order of magnitude depending on the chemical-biological properties of the floc. Finally, ' k A and ' k B are empirical coefficients that may vary with fluid/sediment properties and possibly sediment concentration. Hence, these two empirical coefficients are calibrated for each experiment. Bouyer et al. (24) carry out experiments on floc size distribution in a mixing tank. In these experiments, a synthetic suspension of bentonite is used to mimic the behavior of particles in natural water. The concentration of bentonite is fixed at.3 kg/m 3. The dissipation parameter, G, varies from 5 to 3 s -1 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, ' k A and ' k B for FM A and FM B are determined to be 1.82 and and 1.2 and by matching the model results with measured data. The results for all 9 tests reported by Bouyer et al. (24) are plotted in Figure 4-3, which shows the variation of the equilibrium floc size with the dissipation parameter. Solid line is a regression of measured results and dotted lines are those of the flocculation models. In this figure, the floc size predicted 62

63 by the FM B shows good agreement with the experimental data. The model is capable of predicting the equilibrium floc size at different levels of homogeneous turbulence. Biggs and Lant (2) report the measured equilibrium floc sizes of activated sludge for various magnitudes of dissipation parameter. 6 ml of activated sludge is added with liter of filtered effluent (.45 μm Millipore filters) to a 1.2 l baffled batch vessel and mixed with a flat six blade impeller. Because the mass of total sludge 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 1,3 kg/m 3 and the density of primary particle to be 2,65 kg/m 3. The calculated mass concentration is kg/m 3 and is rather concentrated. Using the impeller, 4 dissipation parameters are tested: 19.4, 37., 113, and 346 s -1. To model these experiments, sets of values, ' k A =.17 and ' k B = for FM B and ' k A =.8 and ' k B = for FM A, are used based on best-fit of the model results with the case of G = 19.4 s -1. The results for all test cases are shown in Figure 4-4. The solid line is a regression of measured results and the dotted lines are for results of the flocculation models. The floc sizes in the range of G = 19.4 and G = 113. s -1 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 milder slope in the log-log plot than the experimental results. Comparing the measured results between Bouyer et al. (24) and Biggs and Lant (2), it appears that the decay of equilibrium floc size (or breakup process) is enhanced when concentration is higher (see also Figure 4-14). 63

64 1 Floc size (micron mr) ete1 Measured result FM A FM B G (s -1 ) Figure 4-3. Experimental results of equilibrium floc size reported by Bouyer et al. (24) and modeled results of FM A and FM B for several dissipation parameters 64

65 Floc size (micron mr) ete1 Measured result FM A FM B G (s -1 ) Figure 4-4. Experimental results of equilibrium floc size measured by Biggs and Lant (2) and model results of FM A and FM B for several dissipation parameters 65

66 Because Figures 4-5 and 4-6 present the temporal evolutions of floc size for the case of G = 19.4 s -1. ' k A and ' k B are chosen to best-fit the calculated equilibrium floc size with the measured data according to this case, it allows us to evaluate the model capability on the time-dependent floc evolution. According to Biggs and Lant (2), the initial floc size is about 15 μm for this experiment. It appears that this experiment is not started with completely deflocculated primary particles because 15 μm appears to be too large for typical size of primary particles. Thus, the model calculation is conducted based on the assumption 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 μm and the primary particle size is assumed to be 4 μm. The measured and modeled temporal evolutions of floc size are plotted in Figures 4-5, 4-6, 4-7, and 4-8. The dotted curves of Figures 4-6 and 4-8 represent results of FM B using ' k A =.17, ' k B = , p=1., and q=.5 (i.e., parameters that are identical to that shown in Figure 4-4). Overall, the shapes of the floc size evolution are not predicted well by the models although the final equilibrium floc size is predicted. The measured floc evolution shows a less apparent Scurve shape. The floc size has a more rapid initial increase with time but shows a more gradual increase of floc size when approaching equilibrium. On the contrary, the model results predict a more gradual increase during the initial stage and approach to the equilibrium state more rapidly. In Figure 4-6, the dashed-dot curve and dashed curve represent model results with ' k B = and ' k A =.15 and ' k A =.2 and ' k B = , respectively. All of them use p=1. and q=.5. The purpose of these tests is to evaluate the sensitivity of model results on ' k A but ' k B need to be changed slightly in order to match the given equilibrium floc size. It can be concluded that the shape of curve is only slightly affected by ' k A and ' k B. In order to further study the effects of p 66

67 and q, three sets of p and q are tested (with ' k A =.17 and ' k B = ) and the model results are shown in Figures 4-7 and 4-8. The dashed-dot curve and dashed curve represent the model results with p = 1.5 and q =.34 and p =.95 and q =.65 with fixed k A =.8 and k B = for FM A and k A =.17 and k B = 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 4-9 represents the temporal change of fractal dimensions calculated by FM B. Three lines have different sets of k A and k B with fixed p=1. and q=.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 μm the initial value of fractal dimension is not 3. but Winterwerp (1998) develops FM A based on fixed fractal dimension and his research is one of the bases of FM B. In Winterwerp (1998), the model coefficients are calibrated or estimated using experimental data measured in Delft Hydraulics (see van Leussen, 1994). It is assumed that at t = the initial particle size equals to the size of the primary particles, i.e. D = d = 4 μm and the maximum floc size measured equals the equilibrium value. Other values required by the flocculation model are given in Table 4-1 and all these values are determined from the measured results of test T73. 67

68 Measured result k À =.8, k B` = 4.4e-5 2 k À =.9, k B` k À =.7, k B` = 4.95e-5 = 3.85e Figure 4-5. Temporal evolution of floc size measured by Biggs and Lant (2) and calculated by FM A for the case of G=19.4 s -1. Three curves represent model results using different sets of k A and k B. 68

69 Measured result k À =.17, k B` = 2.4e-5 2 k À =.2, k B` k À =.15, k B` = 2.84e-5 = 2.13e Figure 4-6. Temporal evolution of floc size measured by Biggs and Lant (2) and calculated by FM B for the case of G=19.4 s -1. Three curves represent model results using different sets of k A and k B. 69

70 Measured result p = 1., q =.5 2 p = 1.5, q =.34 p =.95, q = Figure 4-7. Temporal evolution of floc size measured by Biggs and Lant (2) and calculated by FM A for the case of G=19.4 s -1. Three curves represent model results using different sets of p and q. 7

71 Measured result p = 1., q =.5 2 p = 1.5, q =.34 p =.95, q = Figure 4-8. Temporal evolution of floc size measured by Biggs and Lant (2) and calculated by FM B for the case of G=19.4 s -1. Three curves represent model results using different sets of p and q. 71

72 Fractal dimension k À =.17, k B` = 2.4e-5 k À =.2, k B` = 2.84e-5 k À =.15, k B` = 2.13e Time (min) Figure 4-9. Change of the fractal dimension of FM B with time for the case of G=19.4 s -1 Figures 4-1, 4-11, and 4-12 present the results of two flocculation models, FM A and FM B.. The solid curves are the results of FM A, the dotted curves are results of FM B, and circles are experimental data. It is can be observed that the flocculation model using a variable fractal dimension, FM B, has a slightly more smooth S-curve than the flocculation model using a constant fractal dimension, FM A. Results calculated 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 results using a variable fractal dimension, 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 (26) on settling velocity using a variable fractal dimension. 72

73 FM B FM A Measured result Figure 4-1. Comparison of two flocculation models, FM A and FM B, for T71 experiment of Delft Hydraulics 73

74 FM B FM A Measured result Figure Comparison of two flocculation models, FM A and FM B, for T69 experiment of Delft Hydraulics 74

75 FM B FM A Measured result Figure Comparison of two flocculation models, FM A and FM B, for T73 experiment of Delft Hydraulics 75

76 Manning and Dyer (1999) examine the relationship between floc size and dissipation parameters ( s -1 ) under the condition of increasing concentration (8 2 mg/l). The experiment is carried out in a laboratory flume with a non-intrusive macro-lens miniature video camera. The sediments used for the experiment have been collected from an inter-tidal mudflat. Figures 4-13 and 4-14 show the results calculated by FM A and FM B. To simulate these experiments, the initial floc size is assumed to be 15 μm and k A and k B for FM B are.55 and when c = 12 mg/l and.5 and when c = 16 mg/l. Using FM A, empirical coefficients for k A and k B are.33 and when c = 12 mg/l and for higher concentration c = 16 mg/l condition, k A and k B are specified as.3 and , respectively. In this section, different types of sediments of various concentrations are tested and the resulting empirical coefficients k A and k B 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 already 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 according to the magnitude of concentration. 76

77 2 Floc size (micron metemeasured result FM A FM B r) G (s -1 ) Figure 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=12 mg/l 77

78 2 Floc size (micron metemeasured result FM A FM B r) G (s -1 ) Figure 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=16 mg/l B 78

79 Such a weak point of the flocculation models is emphasized by further considering the decay rate of equilibrium floc size with respect to the dissipation parameter for different sediment concentrations. Flocculation models show good agreements with experimental results when the mass concentration is 12 mg/l. However, when the mass concentration is 16 mg/l, it is evident that the regression curves of the model results for both FM A and FM B show different slopes compared to the experimental results. Therefore, according to measured data reported by Manning and Dyer (1999), the decay rate of equilibrium floc size with respect to the dissipation parameter also depends on the mass concentration. However, following Winterwerp s study (1998), the equilibrium floc size of FM A depends on the dissipation parameter G to the power of -2/(p+2q+F-3) and the value of p and q are further chosen such that it is -.5. This strategy is also similarly adopted by FM B. As a result, model 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 and dissipation parameters, G, are plotted. Future work is necessary to further study this issue. More experimental data is necessary to fully understand the decay rate of floc size with respect to the dissipation parameter 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 limitation of FM A and FM B are validated by four experimental data sets. In terms of equilibrium floc size, model results agree reasonably well with the measured data (see Figures 4-3, 4-4, 4-1, 4-11, 4-12, and 4-13) provided that empirical coefficients are calibrated. This is partially because of the variation of chemical-biological properties of the cohesive sediment tested. However, through model-data comparisons with Manning and Dyer (1999) (Figure 4-14), it becomes clear that the empirical coefficients, specifically q also depends on sediment concentration. Qualitatively, the model predicts 79

80 equilibrium floc size decreases as dissipation parameter increases, suggesting that the model captures observed floc dynamics that strong turbulence has a tendency to break the floc and reduce the floc size. As shown in Figures 4-5, 4-6, 4-7, and 4-8, when comparing model results with measured time evolution of floc size by Biggs and Lant (2), 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 appears to increase too 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, k A, and k B (see Figure 4-5, 4-6, 4-7, and 4-8). Because FM A using fixed fractal dimension also shows similar S-shaped curve, it is concluded that the existing description for floc dynamic may need to be revised for a more accurate description on the time-dependent 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 dimension is deemed to be a variable, the floc strength F y, which is shown to be a constant under the assumption of fixed fractal dimension (Kranenburg, 1994), shall also be a variable (Khelifa and Hill, 26). This aspect is further investigated in the next section. Although incorporating variable fractal dimension based on empirical relationship of Khelifa and Hill (26) dose not improve FM A significantly, it is believed that using variable fractal dimension remains to be more physically 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 considering size distribution of primary particles. Using their 8

81 approaches, it is possible to develop a flocculation model of poly-sized 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 poly-sized primary particles Application of FM C and FM D In this section, the new flocculation models based on variable yield strength, i.e. Eqs a (FM C) and 4-29 b (FM D), are validated with several existing experimental 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 strength. A summary on the flocculation models tested in this study in given in Table 4-2. Numerical solutions of flocculation models are obtained using an explicit Runge-Kutta method (ODE45 function of MATLAB is used in this study). Spicer et al. (1998) carry out an experiment on flocculation and measure the temporal evolution of floc size in a mixing tank. In this experiment, polystyrene particles, whose primary particle diameter and density are.87 μm and 1,5 kg/m 3 (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 From the volumetric concentration and the density of primary particle, the mass concentration is calculated to be.147 kg/m 3. The average dissipation parameter, G, in the tank is 5 s -1. Sampling the floc is a crucial step to accurately characterize flocculation. Three kinds of sampling techniques are 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 flow-through sample cell using a syringe pump; (3) continuous recycle of the suspension through the sample cell using a peristaltic pump which is a class of mechanical pumps with relatively simple structure and is suitable for miniaturization (Xie et al., 24). Among three techniques, the results obtained using peristaltic pump show the largest number of 81

82 samples and the most stable shape of evolution curve in the experiment of Spicer et al. (1998). Thus, the result with peristaltic pump is selected and used to validate flocculation models in this study. Figures 4-15, 4-16, and 4-17 present the experimental results of Spicer et al. (1998) and model results of different flocculation models. The initial floc diameter is set to be 1 μm for all the model simulations presented in Figures 4-15, 4-16, and For numerical stability, the size of primary particle is assumed to be 1 μm instead of.87 μm. The values of empirical coefficients used to generate these model results are summarized in Table 4-3. Following prior section, the criterion of specifying these empirical coefficients is to match the equilibrium floc size (except r in FM D, which is used to evaluate temporal evolution of floc size). FM C (Eq a) using a variable yield strength derived theoretically in this study and a variable fractal dimension in modeling flocculation processes shows good agreement with the experimental results (Figure 4-15). Figure 4-16 presents the results for FM D using the empirical yield strength equation of Sonntag and Russel (1987) and a variable fractal dimension in flocculation processes. Three values of r have been tested: r=.7; r=1.; r=1.3. The model results are sensitive to the choice of r. Specifically, the case of r=.7 gives the best result among three values of r. FM D shows numerical instability around r=.66. The results given by FM A and FM B based on a constant yield strength are shown in Fig These two models use a constant yield strength, which is set to be 1-1 N (van Leussen, 1994; Matsuo and Unno, 1981). As mentioned in the previous section, the model of Son and Hsu (28) uses a variable fractal dimension whereas FM D is based on a constant fractal dimension (F=2.). However, two models predict similar results on temporal evolutions of floc size and they both do not agree with measured data. When comparing FM C and FM D using a variable 82

83 yield strength to FM A and FM B using a constant yield strength, it is notable that results obtained with a variable yield strength (FM C and FM D) are clearly better than those of a constant yield strength (Between 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, 6.3, and 58.7 μm). 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 time-dependent behavior of flocculation. It is also emphasize here that FM D adopting Sonntag and Russel (1987) with r=.7 show very similar results with FM C. This is not surprising if Eqs. 4-25, 4-26 and 4-27 are further compared. The powers of D in Eqs and 4-26 are r(f-3) and 2F/3-2, respectively. It can be seen that these two power become identical when r=2/3=.67. In other words, the new formulation for variable yield strength provides a theoretical approach to determine an empirical coefficient r required in the formation of Sonntag and Russel (1987). Using FM C with a yield strength derived theoretically in Section 4.2.3, the flocculation model has one less empirical parameter. Biggs and Lant (2) report the temporal evolution of floc size of activated sludge under the conditions of G=19.4 s -1 and with sludge volumetric concentration of.5. 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 concentration is estimated from the volumetric concentration under the assumption that the density of sludge is 1,3 kg/m 3 and the density of primary particle is 2,65 kg/m 3. Hence, the calculated mass concentration is kg/m 3. In addition, the size of primary particle is also assumed to be 4 μm. Empirical coefficients used to model this experiment are shown in Table 4-4. The experimental result of Biggs and Lant (2) and the model results are presented in Figures 4-18, 4-19, and 4-2. Similar to those shown in 83

84 Figures 4-15, 4-16, and 4-17, FM C and FM D using a variable yield strength show more smooth S-curves and are in better agreement with the experimental data. Overall, results predicted by FM C and D are quite similar. Comparing the model performance with the case of Spicer et al. (1998) (Figures 4-15, 4-16, and 4-17), it can be noted that the predicted temporal evolution of floc size in this case agrees less favorably with experimental data. However, the adoption of a variable yield strength allows the prediction of floc size that increases more rapidly in the initial stage of flocculation and, after larger aggregates are created, the floc size increases more gradually as it eventually approaches the equilibrium value. Burban et al. (1989) perform experiments with 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.5 kg/m 3 for Case B12 and.8 kg/m 3 for Case B4. The dissipation parameter for both cases is set to be G=2 s -1. To simulate this experiment, the size and density of primary particle are assumed to be 4 μm and 2,65 kg/m 3 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 parameters of the flocculation models are calibrated according to equilibrium time determined by McAnally (1999). More details about experiment conditions and model coefficients are shown in Table 4-5 and Table

85 Measured result FM C Figure Experimental result of Spicer et al. (1998) and model results of FM C 85

86 Measured result 5 FM D, r=.7 FM D, r=1. FM D, r= Figure Experimental result of Spicer et al. (1998) and model results of FM D 86

87 Measured result FM B FM A Figure Experimental result of Spicer et al. (1998) and model results of FM A and FM B 87

88 Table 4-3. Empirical parameters of the flocculation models used for experiment of Spicer et al. (1998) FM ' k A k B 1 r τ y F y ' B FM A N.A. N.A. N.A. 1-1 FM B N.A. N.A. N.A. 1-1 FM C N.A. N.A. N.A. FM D N.A N.A. FM D N.A N.A. FM D N.A N.A. Table 4-4. Empirical parameters of the flocculation models used for experiment of Biggs and Lant (2) FM ' k A k B 1 r τ y F y ' B FM A N.A. N.A. N.A. 1-1 FM B N.A. N.A. N.A. 1-1 FM C N.A. N.A. N.A. FM D N.A N.A. FM D N.A N.A. FM D N.A N.A. 88

89 Measured result FM C Figure Experimental result of Biggs and Lant (2) and model results of FM C 89

90 Measured result FM D, r=.7 2 FM D, r=1. FM D, r= Figure Experimental result of Biggs and Lant (2) and model results of FM D 9

91 Measured result 2 FM B FM A Figure 4-2. Experimental result of Biggs and Lant (2) and model results of FM A and FM B 91

92 Table 4-5. Experimental conditions of Burban et al. (1989) Case G (s -1 ) Mass conc. (kg/m 3 ) Equilibrium time (min) Equilibrium floc size (μm) B B Table 4-6. Empirical parameters of the flocculation models used for experiment of Burban et al. (1989) Case FM ' k A k B 1 r τ y F y ' B FM A N.A. N.A. N.A. 1-1 FM B N.A. N.A. N.A. 1-1 B12 FM C N.A. N.A. N.A. FM D N.A N.A. FM D N.A N.A. FM D N.A N.A. FM A N.A. N.A. N.A. 1-1 FM B N.A. N.A. N.A. 1-1 B4 FM C N.A. N.A. N.A. FM D N.A N.A. FM D N.A N.A. FM D N.A N.A. 92

93 Figures 4-21, 4-22, 4-23, and 4-24 present the experimental results of Burban et al. (1989) and model results for Case B12 and Case B4. Consistent previous model-data comparisons presented in Figures 4-15, 4-16, 4-17, 4-18, 4-19, and 4-2, flocculation models combined with a variable yield strength predict better temporal evolution of floc size than that using a constant yield strength in Case B12 (see Figures 4-21 and 4-22). However, there is less significant difference among the model results for Case B4 (see Figure 4-23 and 4-24) and in fact all models predict temporal evolution of floc size that agree reasonably well with measured data. Similar to the previous simulations, results predicted by FM C and FM D with r=.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 fractal dimension whereas FM A uses the fixed fractal dimension of 2.. According to results presented in Figure 4-15 to Figure 4-24, it have established that it is necessary to utilize flocculation models based on variable 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 is combined with FM A with a fixed fractal dimension, F=2.. Figure 4-25 shows the results of experiment of Spicer et al. (1998), FM C, FM A combined with variable yield strength of Eq. 4-26, and the original model of FM A. Although FM A combined with Eq is slightly better than FM A using a constant yield strength, FM C based on variable fractal dimension and variable yield strength remains to be superior. Hence, it can be concluded that flocculation model based on variable fractal dimension is a more 93

94 physically-based mathematical formulation while variable 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 constant yield strength. Although incorporating solely a variable fractal dimension in the flocculation models may not predict the temporal evolution 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 emphasized 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 unsatisfactory (Figure 4-25). Hence, it is recommended in this study that both variable yield strength and variable fractal dimension are critical to predict flocculation processes. The empirical yield stress proposed by Sonntag and Russel (1987) (see Eq. 4-25) shows the best agreement with measured data when using r=.7. It is also demonstrated that, when specifying r=2/3 in Sonntag and Russel (1987), it reduces to theoretical model 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.7. There still remain several weak points in the present model of flocculation process. To describe flocculation due to collision, the constant efficiency parameter, e c, has been adopted in this study and it is assumed that collisions cause only aggregation. However, it has been observed that collisions can make both aggregation 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 efficiency parameter because it can be 94

95 highly empirical and explicit formulation has not been proposed at present although it is clear that e c is a function of potential and yield stress. In this study, it has been assumed that a floc is simply disaggregated into two roughly equal-sized 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. Srivastava, 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. 4-7) because they are currently highly empirical. Winterwerp (1998) uses the assumption that the equilibrium floc size is independent of primary particle size and fractal dimension is 2. (see Eq. 25 of Winterwerp (1998). If p+nf-3 equals to zero, D e is not a function of primary particle size) to determine their values. In the context of variable fractal dimension, more physical-based criterion shall be incorporated to determine p and q. 95

96 Measured result FM C FM D, r = Figure Experimental results of case B12 of Burban et al. (1989) and model results of FM C and FM D 96

97 Measured result 2 FM C FM B FM A Figure Experimental results of case B12 of Burban et al. (1989) and model results FM A, FM B, and FM C 97

98 Measured result FM C FM D, r = Figure Experimental results of case B4 of Burban et al. (1989) and model results of FM C and FM D 98

99 Measured result 5 FM C FM B FM A Figure Experimental results of case B4 of Burban et al. (1989) and model results FM A, FM B, and FM C 99

100 3 25 Floc size (micronmeter) Measured result 5 FM C FM A with variable F y FM A Time (min) Figure Temporal evolution of floc size simulated by FM A combined with a variable yield strength 1

101 CHAPTER 5 MODELING TRANSPORT OF COHESIVE SEDIMENT 5.1 Governing Equations for Flow Momentum and Concentration The present model formulations for cohesive sediment transport with consideration of floc dynamics are revised from an earlier model of Hsu et al. (29) in which constant floc size and floc density are assumed. The governing equations are obtained by simplifying Eulerian-Eulerian two-phase equations for the sediment limit, i.e., small particle response time. The x- and y- directions represent the directions of main flow (stream-wise direction in rivers and cross-shelf direction in coastal zones) and the direction horizontally perpendicular to x direction (span-wise direction in rivers and along-shelf direction in coastal zones). Figure 5-1 shows the coordinate system used in this study. The governing equations for the x- and y-direction flow momentums are: z y Spanwise Along-shelf αs x Streamwise Cross-shelf Figure 5-1. Definition of coordinate system 11

102 w u 1 p 1 τxz ( ss 1) φs = + + g sinαs t ρ x ρ (1 φ ) z 1 φ w w s s (5-1 a) w v 1 p 1 τ yz = + t ρ y ρ (1 φ ) z w w s (5-1 b) where s s is the specific gravity of primary particle, φ s is the solid volume concentration of primary particles per unit volume of fluid-sediment mixture, α s is the slope of the bottom, and g is the gravitational acceleration. In this equation, φ s is used instead of the floc volumetric concentration, φ 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 forcing in the present model (see section 5.4). w and τ yz are fluid stresses: w τ xz w u τxz = ρw( ν + νt ) z (5-2 a) w v τ yz = ρw( ν + νt ) z (5-2 b) where ν t is the eddy viscosity. The closure of ν t is discussed in section 5.2. Rheological stress is neglected in this study due to low sediment concentration considered. 12

103 The governing equation for solid volume concentration (volumetric concentration of primary particles) is: φ s t s sw ν + ν φ φ = s + t z σ c z (5-3) where W s is the settling velocity of floc and σ c is the Schmidt number set to be.5 in this study. The first term on the right-hand-side of Eq. 5-3 represents settling of floc and the second term is turbulent and molecular suspension. The mass concentration of non-cohesive sediment is in direct proportion to the volumetric concentration because its density is always constant. However, the mass concentration of cohesive sediment (floc), c, is a function of both the volumetric concentration, φ 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 depending 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 physical meaning of a term where a concentration is needed although the equations for non-cohesive sediment use any of them. In Eq. 5-3, the solid volume concentration (that is, the concept of mass concentration) is used instead of the volumetric concentration of floc because the mass conservation is not to be satisfied when the volumetric condition is used for Eq The volumetric condition is mainly governed by not only settling 13

104 and turbulent suspension (the first and second terms in the right-hand-side of Eq. 5-3) but also flocculation process. The settling velocity is calculated by (Richardson and Zaki, 1954): W = D g( ρ ρ )(1 φ ) (5-4) s 18μ f w f This equation is a function of φ f because the hindered settling of particle is considered to be affected by the volume of suspended flocs (effects of pore flow in the floc is neglected). and D are the variable parameters in this equation 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, ρ f is calculated by Eq Once φ s is obtained by the above equations, φ f can be calculated by the equations shown below: ρ f 3 F ρs ρ w c ρs ρ w D φ f = = φs = φ s ρf ρ w ρ s ρ f ρ w d (5-5) 5.2 Flow Turbulence Solutions for Eqs. 5-1 and 5-3 are obtained by incorporating the eddy viscosity and a k-ε closure in this study. The balance equations of the turbulent kinetic energy, k, and dissipation rate, ε, have been adopted in many studies (e.g. Rodi, 198; Elghobashi and Abou-Arab, 1982). Herein, the k-ε equations for suspended sediment transport, which have been reduced from the full two-phase k-ε equations by Hsu and Liu (24), are revised to consider the feature of 14

105 cohesive sediment that the density and size of floc are not constant. The eddy viscosity, ν t, is calculated by: 2 k ν t = C μ (1 φ f) (5-6) ε In this equation, φ f is used in the sense that the turbulence exists only in the carrier fluid. C μ is a numerical coefficient. k and ε are solved by their balance equations under the assumption of local equilibrium: 2 2 k u v ν (1 φ f ) k t νt φs f t f ss g t z z z σk z σc z (1 φ ) = ν + + ν + (1 φ ) ε + ( 1) (5-7) 2 2 u u ν (1 φ f ) ε t t σ ε ε ε (1 φf ) = Cε 1 ν + + ν + t k z z z z 2 ε ε ν t φs Cε2 (1 φ f ) + Cε3 ( ss 1) g k k σ z c (5-8) where σ c, σ k, σ ε, C ε 1, C ε 2,and C ε 3 are numerical parameters. Standard values of these numerical parameters including C μ are shown in Table 5-1 (see Hsu et al., 27, for more details and references). Basically, φ f is used in k-ε equations because it is assumed 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 15

106 density stratification. Density stratification is expected to be governed mainly by mass of suspended sediment. Thus, s s and φ s are adopted here instead of the specific gravity of floc, s f, and φ f. Table 5-1. Numerical coefficients adopted for the eddy viscosity and k-ε equations Coefficient C μ σ k σ ε σ c C ε 1 C ε 2 C ε 3 Value Bottom Boundary Conditions The continuous erosion formulation is adopted for the bottom boundary condition (e.g. Sanford and Halka, 1993) and the upward erosion flux, E, is: τ b() t E = βe 1 τ c ( M ) (5-9) where E is the erosion flux, β e is an empirical erosion flux coefficient and M is the total eroded mass above unit area of bed. The total bottom stress is calculated at every time step by the equation below: τ () t () b 2 = ρwuτ t (5-1) 16

107 Herein, uτ ( t) is the total bottom friction velocity (see Hsu et al., 27, and Mellor, 22, for more details). The critical shear stress, τ c, is assumed to be a function of M in this study. Sanford and Maa (21) have suggested a simple power law relationship between τ c and M: α3 c = 1( M 2) (5-11) τ α α where α 1, α 2, and α 3 are empirical coefficients. From comparison and calibration with field measurement data, it has been concluded 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 calibrated for each simulation case. Under the conditions of tide, unsteady 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. Furthermore, it is more physically reasonable to consider variable water depth instead of fixed one because the scale of large eddies and hence the turbulent mixing process scale with the water depth. For this study, the variable calculation domain height is used with a moving top boundary condition implementation. The domain height is increased or decreased according to the prescribed 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 time-dependent unsteady flow model which can be driven by arbitrary oscillations, such as tide and wave, and current such as river flow. To fulfill such objective for tidal and river flow, the pressure gradient terms in flow momentum equations are 17

108 calculated by the approach developed by Uittenbogaard et al. (1996) and Delft Hydraulics (Winterwerp, 22): 1 p τ s τ b Ut () U() t = + ρ x ρ h T w w rel (5-12 a) 1 p τ s τ b Vt () V () t = + ρ y ρ h T w w rel (5-12 b) where h is water depth, T rel is a relaxation time, U(t) and V(t) are actual computed depthaveraged x- and y-direction flow velocities, U (t) and V (t) are desired depth-averaged x- and y- direction flow velocities, and τ s is the surface shear stress. τ 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 equations of the sediment transport model for this research use both the volumetric concentration of floc, φ f, and the solid volume concentration of primary particles, φ, which is linearly proportional to the mass concentration s of sediment, c. The reasons for such choice are also explained. When flocculation is considered, the density of floc becomes variable and solid volume concentration (or mass concentration) is the physical variable that quantifies the amount 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 turbulence-sediment interactions, are better represented by floc volumetric concentration. Hence, floc volumetric concentration is used for many constitutive relationships and closures in the present model. 18

109 Figure 5-2 represents the depth-averaged flow velocity and water depth used to test the sediment transport model. These conditions are generated from a simple sine function having 3, s of period and.5 m of tidal elevation amplitude and.5 m/s of tidal velocity amplitude. The average water depth and flow velocity are set to be 2 m and m/s. The results of the sediment transport model described in Chapter 5 are shown in Figures 5-3, 5-4, and 5-5. Figure 5-3 represents the mass concentrations at.2 m,.5 m, and 1. m above the bottom calculated by the sediment transport model. The mass concentrations in Figure 5-3 shows smooth increase and decrease according to flow velocity, which also affects the bottom stress. The second peak of concentration is larger than the first peak because the lowest water depth occurs at the second peak whereas the highest water depth occurs at the first peak. Figure 5-4 represents the floc volumetric concentrations at.2 m,.5 m, and 1. m above the bottom. The floc volumetric concentrations also show smooth changes according to flow velocity. The velocities calculated by the sediment transport model described in Chapter 5 are represented in Figure 5-5. The calculated velocities follow the tendency of the depth-averaged velocity in Figure 5-2 A without numerical instability such as irregular fluctuation. The velocities calculated by model considering the floc volumetric concentration also show reasonable result. It is worthy to mention that if the floc volumetric concentration is considered as the primary variable, similar to some earlier cohesive sediment transport models without consideration on floc dynamics (e.g. Hsu et al., 29), model results for sediment concentration are unreasonable (see Figure 5-6 and Figure 5-7). This reinforces the previous discussion that when floc dynamics is considered in sediment transport calculation, solid volume concentration, φ, (or mass concentration) needs to be sued as the primary variable. s 19

110 From Figure 5-3 to Figure 5-7, it can be concluded that the sediment transport model composed of equations discussed in Chapter 5 is a physically and mathematically reasonable approach to model transport of cohesive sediment with the consideration on flocculation process. 11

111 1.5 Velocity (m/s) Time (hr) A 3 Water Depth (m) Time (hr) B Figure 5-2. Depth-averaged flow velocity and water depth used to test the sediment transport model. A) The depth-averaged flow velocity and B) the water depth. 111

112 .7.6 At.2 m At.5 m At 1. m.5 Mass Conc.n (g/l) Time (hr) Figure 5-3. Mass concentration calculated by sediment transport model combined with FM C using two types of concentrations 112

113 .7.6 At.2 m At.5 m At 1. m.5 Volumetric Conc. (%) Time (hr) Figure 5-4. Volumetric concentration calculated by sediment transport model combined with FM C using two types of concentrations 113

114 .6.4 At.2 m At.5 m At 1. m.2 Velocity (m/s) Time (hr) Figure 5-5. Velocity calculated by sediment transport model combined with FM C using two types of concentrations 114

115 At.2 m At.5 m At 1. m.1 Mass Conc.n (g/l) Time (hr) Figure 5-6. Mass concentration calculated by sediment transport model combined with FM C using one type of concentration 115

116 .6.5 At.2 m At.5 m At 1. m Volumetric Conc. (%) Time (hr) Figure 5-7. Volumetric concentration calculated by sediment transport model combined with FM C using one type of concentration 116

117 CHAPTER 6 MODEL APPLICATION TO EMS/DOLLARD ESTUARY 6.1 In-situ Measurement in Ems/Dollard Estuary van der Ham et al. (21) carried out in-situ high-frequency measurements of velocities and suspended sediment concentrations (SSC) in the Ems/Dollard estuary to understand the behavior of cohesive sediment in a tidal channel. The in-situ measurement was conducted in a straight homogenous reach of the tidal channel Groote Gat, approximately 3 m to the east of the bank of the adjacent flat Herigsplaat at 7 9'43"E, 53 17'15"N in Figure 6-1 shows the Ems/Dollard estuary and the measuring pole equipped with a rigid frame for in-situ 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 6 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 small undulations of typically.5 m over a 1 m distance were found in the bed. The location where the measurements were conducted shows small horizontal SSC gradients in longitudinal direction. Thus, the effect of horizontal advection on suspended sediment transport is assumed to be negligible (Dorrrestein, 196). Furthermore, it has been known from horizontal flow measurements conducted at the same location that the direction of the flow was very parallel to the adjacent tidal flat. Only during slake water and part of the ebb, a small span-wise velocity perpendicular to the border of the tidal flat was measured (van der Ham et al., 21). These characteristics of the field site are appropriate for testing 1DV model. 117

118 Figure 6-1. The Ems/Dollard estuary and the measuring pole equipped with a rigid frame for insitu measurement (van der Ham et al., 21) 118