Evolution of the floc size
distribution of cohesive sediments
Ontwikkeling van de vlokgrootteverdeling
van cohesief sediment
Evolution of the floc size
distribution of cohesive sediments
Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus Prof.ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,
in het openbaar te verdedigen op maandag 12 april 2010 om 15.00 uur
door
Francesca Mietta
Ingegnere per l’ambiente e il territorio, Politecnico di Milano, Itali¨e geboren te Milano, Itali¨e
Dit manuscript is goedgekeurd door de promotor: Prof. dr. ir. G. S. Stelling
Copromotor:
Dr. J. C. Winterwerp
Samenstelling promotiecommissie:
Rector Magnificus, Technische Universiteit Delft, voorzitter Prof. dr. ir. G. S. Stelling, Technische Universiteit Delft, promotor Dr. ir. J. C. Winterwerp, Technische Universiteit Delft, copromotor Prof. dr. J. Berlamont, Katholieke Universiteit Leuven, Belgi¨e Dr. A. J. Manning, University of Plymouth, Verenigd Koninkrijk Prof. dr. ir. J. B. van Lier, Technische Universiteit Delft
Prof. dr. ir. W. S. J. Uijttewaal, Technische Universiteit Delft Dr. ir. C. Chassagne, Technische Universiteit Delft
Prof. dr. ir. M. J. F. Stive, Technische Universiteit Delft, reservelid
This research has been financially supported by Technologiestichting STW and Delft University of Technology.
ISBN 978-90-8891-158-3
Copyright c 2010 by Francesca Mietta Cover design by Marta Rovatti Studihrad
All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission of the author.
Summary
This thesis focuses on the flocculation process of cohesive sediment and in particular on the time evolution of the floc size distribution. Flocculation is the combination of the processes of aggregation and breakup and is studied by analyzing both fluid-particles and particle-particles interactions.
Particle-particles interactions have been investigated in colloidal chemistry mainly for clays. The charge of the particles, strongly affected by the properties of the suspension such as pH and salt concentration, was found to have a strong influence on the probability of particles to stick together as well as on the strength of the formed flocs.
Fluid-particles interactions, which affect the rate of particles collision as well as the breakup rate, have been investigated mostly in the fields of civil and sanitary engineering. The physical properties of the system influencing flocculation are turbulence motion, quantified by the shear rate, sediment concentration and floc structure or fractal dimension. A relation between these physical properties and the flocculation behaviour of the system has been established both empirically and theoretically. This relation usually includes two parameters representing the sticking probability of particles and the strength of flocs. These parameters can be calibrated experimentally.
The setup and results of the experimental work are described in the first part of this thesis. Coupled experiments have been performed where the charge of the particles and their flocculation behaviour are observed for different suspensions. The charge of the particles is quantified by the ζ-potential, while the flocculation behaviour is studied by means of the evolution in time of the floc size distribution. We used two types of sediment: pure kaolinite and mud from the Western Scheldt. The pH and the salt type and concentration are varied. The effects of monovalent salt, divalent salt and sea salt have been investigated. For a given ζ-potential, suspensions with different salts behave in a similar way. For both types of sediment, the flocculation rate increases when the pH of the suspension decreases or the salt concentration increases. The equilibrium floc size increases with decreasing pH and increasing salt concentration. For mud suspensions, the correlation between the ζ-potential and the flocculation behaviour was found to be better than for kaolinite.
The modelling work done on flocculation is shown in the second part of this thesis. A Population Balance Equation (PBE) which takes into account the processes of aggregation and breakup is developed. The mass of flocs is discretized in size classes considering the number of primary particles they are formed of: flocs containing i primary particles belong to size class i. Such a class distribution requires a large number of classes, and therefore large computational efforts but it allows for a very straightforward implementation of the processes of aggregation and
breakup. Two parameters are considered in the PBE: the collision efficiency for the aggregation process and a parameter for the breakup rate which accounts for the strength of flocs. The effect of the flocs size distribution of daughter flocs resulting from breakup is investigated. Binary breakup is used for the comparison with the other models, the calibration of the parameters and the study on the bimodal distribution. A linear relation is found between the equilibrium mean size of flocs and the ratio between the parameters for aggregation and breakup.
It is shown that the PBE behaves similarly to two other different models which use the same equations for aggregation and breakup: the Lagrangian model developed by Winterwerp, Winterwerp [2002], and the population balance equation developed by Verney, Verney et al. [2010]. The Lagrangian model models the evolution of a characteristic floc size and does not account for the floc size distribution. In the population balance equation by Verney, classes are divided according to a logarithmic scale and mass conservation is statistically ensured. Such a PBE requires a much smaller number of classes and therefore shorter computational times than the one developed in this thesis. We show in this thesis how the PBE we developed can be used to investigate the effect of the reduction of the number of classes on the time evolution of the mean floc size on the PBEs.
The parameters for aggregation and breakup in the PBE are tuned using the experimental results shown in the first part of this thesis. Very small differences are observed comparing both observed and computed evolution in time of the mean floc size and the equilibrium floc size distributions.
For kaolinite suspensions, the collision efficiency is much larger at low pH than in saline suspensions at high pH. The same differences are not observed for the ratio between the pa-rameters for aggregation and breakup which is less sensitive to the properties of the suspension. The difference between computed and measured time evolution of the mean floc size is in most cases below 6 %.
Both the collision efficiency and the ratio between the parameters vary little for mud suspen-sions, they are both larger at low pH. While the difference for the time evolution of the mean floc size is below 4 %, the difference for the equilibrium floc size distribution is around 20 % for all mud suspensions. This may be explained considering that the equilibrium floc size distribu-tion computed with the PBE is always monomodal while in experiments with mud suspension a bimodal floc size distribution is found.
The PBE is used to investigate the stability of the bimodal floc size distribution under variable shear rate. The variation of the shear rate is imposed to reproduce a tidal cycle within the estuary of the Western Scheldt and the parameters found for the mud suspension with sea salt are used. From the simulations we conclude that a bimodal distribution is not conserved over different tidal cycles in a closed system if flocs break into two equal flocs and all differently sized particles have the same flocculation properties. As small particles are less likely to occur, the bimodal distribution is conserved for a few tidal cycles.
Overall, a step forward in establishing the relation between the small scale particle-particles interactions and the large scale flocculation behaviour is made in this thesis. This has been achieved through both experimental work and process analysis with the PBE.
Samenvatting
Dit proefschrift richt zich op flocculatie van cohesief sediment en in het bijzonder op de on-twikkeling van de vlokgrootteverdeling in de tijd. Flocculatie is de combinatie van aggregatie- en afbraakprocessen en is onderzocht door het analyseren van zowel vloeistof-deeltjes als deeltjes-deeltjes interacties. Deeltjes-deeltjes-deeltjes interacties zijn onderzocht in collo¨ıd chemie, met name voor klei. De lading van de deeltjes wordt sterk be¨ınvloed door de eigenschappen van de suspen-sie. Deze eigenschappen, zoals pH en zoutconcentratie, blijken een sterke invloed te hebben op de waarschijnlijkheid dat deeltjes aggregeren, alsmede op de sterkte van de gevormde vlokken. Vloeistof-deeltjes interacties, die zowel de botsingsfrequentie als de afbraaksnelheid van de deelt-jes bepalen, zijn voornamelijk onderzocht in de vakgebieden civiele techniek en gezondheid-stechniek. Een relatie tussen de fysische eigenschappen van het systeem (turbulente beweging, uitgedrukt in de afschuifsnelheid; sedimentconcentratie; en de structuur van de vlokken, uitge-drukt in fractale dimensie), en het flocculatiegedrag van het systeem is zowel op empirische als theoretische wijze bepaald. Deze relatie bevat meestal twee parameters: de kans dat deeltjes aggregeren als ze botsen en de sterkte van de vlokken. Deze parameters kunnen experimenteel worden gekalibreerd.
De experimentele opstelling en de meetresultaten worden beschreven in het eerste deel van dit proefschrift. Experimenten zijn uitgevoerd waarbij zowel de lading van de deeltjes en hun flocculatiegedrag worden gemeten voor verschillende suspensies. De lading van de deeltjes wordt uitgedrukt in de ζ-potentiaal, terwijl flocculatiegedrag bepaald is aan de hand van de ontwikke-ling van de vlokgrootteverdeontwikke-ling in de tijd. Twee soorten sediment zijn gebruikt: zuiver kaolin-iet en slib uit de Westerschelde. De pH-waarde, het soort zout en de zoutconcentratie zijn gevarieerd. De effecten van monovalent zout, divalent zout en zeezout zijn onderzocht. Voor een gegeven ζ-potentiaal gedragen suspensies met verschillende zoutsoorten zich vergelijkbaar. Voor beide sedimenttypen neemt de flocculatiesnelheid toe als de pH van de suspensie afneemt of de zoutconcentratie toeneemt. De vlokgrootte in de evenwichtssituatie neemt toe met afnemende pH en toenemende zoutconcentratie. De correlatie tussen de ζ-potentiaal en het flocculatiege-drag bleek voor de slibsuspensies sterker te zijn dan voor kaoliniet.
Het modelleren van het flocculatiegedrag vormt het tweede deel van dit proefschrift. Een populatievergelijking die rekening houdt met aggregatie- en afbraakprocessen is ontwikkeld. De massa van de vlokken is gediscretiseerd in verschillende klassen, die gebaseerd zijn op het aantal primaire deeltjes waaruit zij gevormd zijn: vlokken die i primaire deeltjes behoren tot klasse i. Een dergelijke verdeling vereist een groot aantal klassen, en daarmee een grote rekeninspanning, maar het leidt wel tot een zeer eenvoudige implementatie van de aggregatie- en afbraakprocessen. Twee parameters worden beschouwd in de populatievergelijking: de botsingseffici¨entie voor het
aggregatieproces en een parameter voor de afbraakfrequentie, die gebaseerd is op de sterkte van de vlokken.
Het effect van de vlokgrootteverdeling van de dochtervlokken, die het gevolg zijn van af-braak, is onderzocht. Binaire afbraak is gebruikt voor de vergelijking met de andere modellen, de kalibratie van de parameters en voor onderzoek naar de bimodale verdeling. Een lineaire relatie is gevonden tussen de gemiddelde grootte van de vlokken in de evenwichtssituatie en de verhouding tussen de parameters voor aggregatie en afbraak. De populatievergelijking gedraagt zich vergelijkbaar als twee andere typen modellen die dezelfde vergelijkingen voor aggregatie en afbraak gebruiken: het Lagrangiaanse model ontwikkeld door Winterwerp, Winterwerp [2002] en de populatievergelijking ontwikkeld door Verney, Verney et Al. [2010]. Het Lagrangiaanse model modelleert de evolutie van een karakteristieke vlokgrootte en houdt geen rekening met de vlokgrootteverdeling. In de populatievergelijking van Verney zijn de klassen verdeeld op basis van een logaritmische schaal en massabehoud is statistisch gewaarborgd. Een dergelijke populatievergelijking vereist een veel kleiner aantal klassen en daarmee minder rekentijd. We tonen in dit proefschrift aan hoe de PBE kan worden gebruikt om het effect te onderzoeken van de afname van het aantal klassen op de ontwikkeling van de gemiddelde vlokgrootte in de tijd. De parameters voor aggregatie en afbraak in de populatievergelijking zijn gekalibreerd op basis van de experimentele resultaten uit het eerste deel van dit proefschrift. De verschillen tussen de waargenomen en berekende ontwikkeling in de tijd van de gemiddelde vlokgrootte en de verdeling van de evenwichtsvlokgrootte.
Voor kaoliniet suspensies is de botsingseffici¨entie veel groter bij lage pH dan in zoute suspen-sies bij hoge pH. Vergelijkbare verschillen zijn niet waargenomen voor de verhouding tussen de parameters voor aggregatie en afbraak. Deze verhouding is minder gevoelig voor de eigenschap-pen van de suseigenschap-pensie. Het verschil tussen de berekende en gemeten ontwikkeling in de tijd van de gemiddelde korrelgrootte is in de meeste gevallen kleiner dan 6%. Zowel de botsingseffici¨entie als de verhouding tussen de parameters vari¨eren weinig voor slibsuspensies: beide groter bij lage pH. Hoewel het verschil voor de ontwikkeling van de gemiddelde korrelgrootte in de tijd kleiner is dan 4%, is de fout voor de verdeling van de evenwichtsvlokgrootte ongeveer 20% voor alle slibsuspensies. Dit kan verklaard worden doordat de verdeling van de evenwichtsvlokgrootte zoals berekend wordt met de populatievergelijking altijd monomodaal is, terwijl in experimenten met slibsuspensies een bimodale vlokgrootteverdeling is gevonden. De populatievergelijking is tevens gebruikt om de stabiliteit te bepalen van de bimodale vlokgrootteverdeling onder vari-abele afschuifsnelheid. Een vari¨erende afschuifsnelheid is opgelegd om een getijdencyclus te reproduceren in de monding van de Westerschelde. De parameters, die voor de slibsuspensie in zeezoutoplossing zijn gecalibreerd, zijn gebruikt. Uit de simulaties concluderen we dat een bi-modale verdeling niet behouden blijft over verschillende getijdencycli in een gesloten systeem als vlokken opbreken in twee gelijke vlokken en als alle deeltjes, van verschillende grootte, dezelfde flocculatie-eigenschappen hebben. Als kleine deeltjes minder waarschijnlijk voorkomen, dan is de bimodale distributie behouden over een enkele getijdencycli.
In dit proefschrift is een stap vooruit gemaakt bij de bepaling van de relatie tussen de kleinschalige deeltje-deeltje interactie en het grootschalige flocculatiegedrag. Dit is bereikt met zowel experimenten als procesanalyse met de PBE.
Contents
Summary i
Samenvatting iii
1 Introduction 1
1.1 Background . . . 1
1.2 Aim and approach . . . 2
1.3 Outline . . . 4
2 Literature survey and definitions. 5 2.1 Overview . . . 5
2.2 Sediment characterization and particle-particle interactions . . . 6
2.2.1 Cohesive sediment . . . 6
2.2.2 Suspension . . . 8
2.2.3 Electrokinetic properties of the particles . . . 8
2.2.4 Particle-ions interactions . . . 10
2.2.5 Particle-particle interactions . . . 10
2.2.6 Ions and particles interaction: the case of kaolinite . . . 11
2.3 Water column hydrodynamics and fluid-particles interactions . . . 12
2.3.1 Water column hydrodynamics . . . 12
2.3.2 Fluid-particles interactions . . . 13 2.4 Flocs properties . . . 14 2.4.1 Structure of flocs . . . 14 2.4.2 Strength of flocs. . . 16 2.4.3 Settling velocity . . . 17 2.5 Flocculation dynamics . . . 18 2.5.1 Aggregation . . . 18 2.5.2 Breakup . . . 20
2.5.3 Equilibrium and flocculation time . . . 22
2.6 Modelling flocculation . . . 23
2.7 Experimental work . . . 25
2.7.1 In situ observations . . . 25
2.7.2 Indirect laboratory flocculation observations . . . 27
2.7.3 Direct laboratory flocculation experiments . . . 27
2.8 Conclusions . . . 28
3 Experiments: methodology 31 3.1 Sediment characterization . . . 31
3.1.1 Sample composition . . . 31
3.1.2 Organic matter content . . . 31
3.1.3 Electrokinetic properties and ζ-potential . . . 32
3.2 Small scale flocculation experiments: mixing jars . . . 33
3.2.1 Description . . . 33
3.2.2 Turbulence field . . . 33
3.2.3 Floc size distribution measurements . . . 34
3.2.4 Data representation . . . 34
3.2.5 Preliminary experiments . . . 36
3.2.6 Setup of the flocculation experiments . . . 39
3.3 Large scale flocculation experiments: settling column . . . 40
3.3.1 Description . . . 40 3.3.2 Measuring techniques . . . 41 3.3.3 Data representation . . . 44 3.3.4 Injection system . . . 44 3.3.5 Turbulence field . . . 45 3.3.6 Separation chamber. . . 45
3.3.7 Floc sample chamber . . . 45
3.3.8 Setup of flocculation experiments . . . 46
3.4 Conclusions and planning of the experimental work . . . 48
4 Kaolinite: characterization and flocculation experiments 49 4.1 Sediment characterization . . . 49
4.1.1 ζ-potential . . . 51
4.1.2 Comparison between two different types of kaolinite . . . 53
4.2 Small scale flocculation experiments . . . 54
4.2.1 Influence of the shear rate . . . 55
4.2.2 Influence of salinity and pH . . . 57
4.3 From small to large scale . . . 62
4.4 Conclusions . . . 65
5 Western Scheldt mud: characterization and flocculation experiments 67 5.1 Sediment characterization . . . 67
5.1.1 ζ-potential . . . 68
5.2 Small scale flocculation experiments . . . 70
5.2.1 Influence of the shear rate . . . 70
5.2.2 Influence of salinity and pH . . . 72
5.2.3 Influence of the organic matter content . . . 75
5.3 From small to large scale . . . 76
6 Setup of a flocculation model 81
6.1 Lagrangian model . . . 81
6.1.1 Model description . . . 81
6.1.2 Behaviour . . . 83
6.2 Population Balance Equation (PBE) . . . 84
6.2.1 Model description . . . 84
6.2.2 Numerical integration and stability . . . 86
6.2.3 Behaviour . . . 87
6.3 Simplified population balance equation . . . 94
6.3.1 Model description . . . 95
6.3.2 Semi-analytical solution and numerical integration . . . 96
6.3.3 Behaviour . . . 97
6.4 Comparison between different models . . . 99
6.4.1 Lagrangian - complete PBE . . . 100
6.4.2 Lagrangian - simplified PBE . . . 102
6.4.3 Complete PBE - simplified PBE . . . 104
6.5 Conclusions . . . 105
7 Analysis of the flocculation process 109 7.1 Calibration of the parameters . . . 109
7.1.1 Simulations at different shear rates . . . 114
7.1.2 Sensitivity of the parameters to the fractal dimension d0 . . . 116
7.2 Relation between physico-chemical properties of the suspension and the calibrated parameters . . . 116
7.3 Comparison with the parameters calibrated for the Lagrangian model by Winter-werp [1999] . . . 119
7.4 Comparison with a PBE with geometrical class division, Verney et al. [2010] . . 120
7.5 Analysis of in situ flocculation . . . 123
7.5.1 Setup of the study . . . 123
7.5.2 Results of the simulations and discussion . . . 124
7.6 Conclusions . . . 127
8 Conclusions 129 8.1 Achievements of this research . . . 129
8.1.1 Experimental work . . . 129
8.1.2 Modelling work . . . 132
8.2 Applications . . . 134
8.3 Recommendations for future research . . . 134
A Semi-analytical solution of the simplified PBE 137
List of symbols 141
Acknowledgements 155
Chapter 1
Introduction
1.1
Background
Rivers and estuaries have always been a privileged environment for human settlements thanks to the fertility of the land, the abundance of fresh water and the transport facilities. As a large part of the world population is settled along estuaries, it is important to protect these environments, and the flora and fauna within. Moreover, the major commercial harbours of the world are situated in estuaries and sediment behaviour has a strong economical impact as sediment settling induces harbour siltation and hinders fluvial transport.
The behaviour of suspended sediment affects the environment in different ways at various space and time scales. On the large scale, sediment transport and deposition induce long term morphological changes as well as harbour siltation. On the mesoscale, the transport of pollu-tants and nutrients may be enhanced or modified by suspended sediment particles to which the pollutants and nutrients can adhere. On the small scale, fine suspended particles reduce the transmission of light which is important for primary production and the flora and fauna within an estuary.
The settling velocity of sediment particles governs to a large extent the transport of fine sediment in open water systems, such as rivers, estuaries and seas. The settling velocity is therefore an important parameter in sediment transport models such as Delft 3D. For the study of sediment transport it is therefore necessary to be able to estimate the settling velocity.
Cohesive sediment flocculates and its floc size and settling velocity vary with the environ-mental conditions. Flocculation can be regarded as a competition between aggregation and breakup. The rate at which flocs grow and the size they attain depend on hydrodynamic con-ditions, residence time, sediment properties, pH and salinity. All these quantities are highly variable in estuaries, where river fresh water mixes with salt marine water, and where the tidal forcing induces variations in shear rate and sediment concentration.
If on the one hand the flow properties and the sediment concentration determine the collision rate and the shear stress applied to flocs, the probability of flocs to aggregate and the strength of the formed flocs are mainly dependent on the properties of the suspension such as sediment type, particles structure, surface charges, dissolved ions type and concentration. These quantities vary in time and space within an estuary and between different estuaries.
Studies have already been done to link the electrokinetic (i.e. measured by electrokinetic
2 Chapter 1. Introduction
studies) charge to the sedimentation behaviour of kaolinite, Wang and Siu [2006], or to its rheological properties, Melton and Rand [1977]. The influence of the particles charge on the flocculation rate of kaolinite at zero shear rate has also been investigated, Tombacz and Szekeres [2006]. While the relation between shear rate and floc size has been widely investigated, Serra and Casamitjana [1998a], Spicer et al. [1998] and Bouyer et al. [2004], few studies have been done to relate the flocculation behaviour to both the chemical properties of the suspension and the applied shear rate, Bouyer et al. [2005a] .
Besides its relevance in estuarine environments, flocculation is studied to optimize water treatment plants as the formation of large flocs enhances the removal of suspended particles [Bouyer et al., 2005a]. Furthermore, several branches of industry deal with flocculation. Paper or colour making and food industry are some examples.
The research presented in this thesis investigates the flocculation behaviour of clays and cohesive sediments in relation with the physico-chemical properties of the suspension.
1.2
Aim and approach
The aim of this thesis is to develop a tool to predict the flocculation behaviour of cohesive sediment on the basis of a few, easily measurable quantities such as sediment concentration, shear rate and ζ-potential. In this thesis we focus on the time evolution of the floc size distribution; from this, the settling velocity distribution at any time can be estimated if the density of flocs is known. We seek therefore for laboratory investigations and process modelling which can be used for the prediction of the floc size distribution.
Experimental study. We focus the experimental part of this thesis on the research of a relation between the properties of the suspension, the surface charge of the sediment and the flocculation behaviour. Two different types of experiments need to be done for this purpose: characterization of the sediment and of its surface charge, mainly through electrokinetic studies, and shear induced flocculation experiments. Both types of experiments are performed for a wide range of pH and salinity of the suspension. Flocculation experiments have been done both at small scale with 1 litre mixing jars and at large scale with a 4 m high settling column.
Mud is a complex material formed by different clays mixed with organic matter. An insight on the physics of the flocculation process can be gained studying the behaviour of single clay components. We choose to work on kaolinite, as its structure and charge properties have been widely studied in literature. Starting from the analysis of the behaviour of kaolinite, we analyze the behaviour of a complex mud system with different concentrations of organic matter.
We use the ζ-potential to quantify the charge of the particles in the electrokinetic study. Through flocculation experiments we assess the following different quantities:
• The equilibrium floc size which depends on both the strength of flocs and their collision efficiency, next to the shear rate and the sediment concentration.
• The time evolution of the mean floc size, which depends on the collision efficiency, next to the shear rate and the sediment concentration. The evolution in time of the mean floc size is important for the study of natural systems where conditions vary continuously in
1.2. Aim and approach 3
time, and a dynamic equilibrium may not be reached. It is also important for engineering applications such as water treatment, where the efficiency of the plant depends on the flocculation time and on the settling velocity of the particles.
• The time evolution of the floc size distribution. The shape of the floc size distribution and its evolution in time, depend on the composition of the sediment as well as on the ki-netics of aggregation and breakup. The different size fractions of the floc population affect the estuarine environment in different ways: the fine particles influence the transmission of light through the water column and are responsible for the transport of pollutants; large flocs mainly affect harbour siltation and sediment transport.
By combining the electrokinetic study and the flocculation experiments we show how the mean equilibrium floc size and its evolution in time vary with the composition (clay type and organic matter content) and the ζ-potential of the sediment.
Modelling work. Flocculation modelling allows for a better understanding of the effect of the physical processes and the prediction of the flocculation behaviour under specific conditions. Two different types of model have been developed:
• characteristic floc growth, modelling the evolution in time of a characteristic floc size as a function of the processes of aggregation and breakup. An example of this type of models is the Lagrangian model by Winterwerp [1998].
• population balance equation, modelling the evolution in time of the floc size distribution. The Lagrangian model consists of a single equation for which the analytical solution can be found for some specific fractal dimensions. This model is computationally very fast. The population balance equation consists of a large number of equations, each of them representing a size class to be integrated numerically. Contrary to the Lagrangian model, the population balance equation gives a complete information on the flocculation process, as it describes the evolution of the floc size distribution, and it allows for the investigation of different breakup dynamics, but it requires long computational times.
We analyze in this thesis the behaviour of both models and their sensitivity to all involved parameters. We furthermore compare the two models to find a relation between their charac-teristic parameters.
Synthesis. In the last part of this study, the experimental and modelling parts of this work are combined and the parameters of the model are calibrated. These parameters quantify the time evolution of the process as well as the equilibrium floc size. We then establish a relation between the electrokinetic properties of the suspension, namely the ζ-potential, and the parameters of aggregation and breakup.
The population balance equation Is also compared to the population balance equation de-veloped by Verney, Verney et al. [2010]. The difference between the two models lays in the discretisation in size classes.
The model and the calibrated parameters are at last used to analyze the in situ flocculation. For example the effect of variable forcing on the floc population is investigated.
4 Chapter 1. Introduction
1.3
Outline
An overview of the framework of this study is given in the literature survey in Chapter 2. Previous works which lay at the basis of this study are described in this chapter and definitions are given. The experimental work is described in Chapters 3, 4 and 5. In Chapter 3 the experimental devices and techniques used are described and discussed. This chapter also includes the preliminary experiments done for the definition of the setups of the experiments and the description of the setups used. All experimental data obtained for kaolinite are shown and discussed in Chapter 4 while Chapter 5 deals with the experiments done on the Western Scheldt mud. In Chapter 6, the Lagrangian model and the population balance equation are described and their behaviour is analyzed. A comparison between the different models is also done. Experimental and modelling work are combined in Chapter 7 where the parameters of the population balance equation are calibrated on the basis of experimental data. The relation between the parameters and the ζ-potential of the particles is then established. The population balance equation is also compared to the model by Verney, Verney et al. [2010] in this chapter. At last, the PBE is used in this chapter to observe the flocculation behaviour under more realistic conditions as for example variable shear rate. The conclusions of this work and the recommendations for further investigations in this field are given in Chapter 8.
Chapter 2
Literature survey and definitions.
Many research groups have studied flocculation of cohesive sediment, both theoretically and experimentally. Both approaches are treated in this thesis and in this chapter an overview of what has been done so far is given. Definitions are given.
The first part of this chapter focuses on the factors affecting flocculation such as particle-particle and fluid-particle-particle interactions. The survey continues with a description of the floccula-tion dynamics and of the properties of the floc populafloccula-tion. The state-of-the-art in flocculafloccula-tion modelling and experimental studies is then introduced.
2.1
Overview
The interactions between water and sediment and the sediment cycle in a natural system are sketched in Figure 2.1. The smallest flocs in suspension are the primary particles of size Lp, which
consist of many clay mineral particles, salt and organic matter. The formation of these primary particles is referred to as coagulation, and should be considered as an irreversible process under the forces at hand. Flocculation is the combination of floc breakup and aggregation. The forces acting on the flocs are turbulence which is responsible for aggregation, breakup and erosion, and gravity; this last is mainly responsible for particles settling leading to deposition. Turbulence affects flocculation in two ways:
• turbulent diffusion, at the scale of the floc sizes, induces collisions between the various flocs, and is responsible for turbulence-induced aggregation;
• turbulent shear stresses can disrupt flocs when they exceed floc strength; floc disruption is in this case defined as turbulence-induced breakup.
When flocs become large and heavy, and gravitation dominates over turbulence, they settle towards the bed. The settling velocity depends on size, shape and density of flocs. When turbulent fluctuations are sufficiently large, flocs are eroded from the bed. The eroded sediment is lifted up and resuspended in the water body, where it can take part in the flocculation process again. The rate of sedimentation represents the difference between the amount of material which settles and the one which is eroded from the bed. In this thesis, we limit ourselves to the study of flocculation and we do not consider sedimentation.
6 Chapter 2. Literature survey and definitions. Settling ws2 Settling ws1<ws2 Break-up Flocculation Erosion Aggregation Primary Particles Presence Water level Water mixing
Sedimentation Interface water-bed
Water body
Bed Water mixing
Clay mineral particles
Figure 2.1: Cycle of deposition and resuspension of suspended matter subject to aggregation and breakup, Maggi [2005]. ws1and ws2 indicate the settling velocity of differently sized flocs.
2.2
Sediment characterization and particle-particle interactions
Particle-particle interactions influence the probability of particles to stick after collision and they are dependent on the properties of both the sediment and the suspension.
2.2.1 Cohesive sediment
Cohesive sediment, or mud, consists of a mixture of inorganic and organic material, water, and sometimes gas. Its cohesive behaviour depends on the electric charge and coating of particles, the electrochemical properties of the suspension and on the organic matter content.
Inorganic matter. The inorganic fraction of cohesive sediment consists in general of a mixture of clay minerals, silt and sand. These three materials have similar chemical composition and they are classified according to their grain size distribution. According to the Dutch grain size classification standards, particles are considered as “clay” if smaller than 2 µm, as “silt” if smaller than 64 µm and as “sand” if larger.
The mineral component of cohesive sediment mainly consists of clay and silt. Clay particles are characterized by a flat shape, a large specific surface and they are negatively charged in water suspensions. The most common clay minerals in marine environment are kaolinite, illite, montmorillonite, chlorite and smectite. Clay minerals are build from two dimensional silica
2.2. Sediment characterization and particle-particle interactions 7
tetrahedra combined with aluminium-hydroxide octahedra (gibbsite) or magnesium-hydroxide octahedra (brucite). The way in which they are combined determines the different types of clay as well as their specific properties. A more detailed description of some of the most common clay minerals is given below and in Table 2.1, Winterwerp and Van Kesteren [2004].
Kaolinite. Kaolinite is a 1:1 type of clay mineral where multiple basal layers are attached by strong hydrogen bonds forming thick crystalline flake-like particles. Hydrogen bonds are strong, thus water can hardly enter between the layers, and the swelling capacity of kaolinite is very low. Kaolinite is characterized by relatively large crystalline particles resulting in a small specific area and a low cation exchange capacity cec. The cec is defined as the number of cations that can be exchanged between the surface layers of the crystalline particle and the surrounding fluid. It is measured in milli-equivalent per 100 gram of dry clay.
Smectite. Smectite is a 2:1 type of clay in which silica tetrahedra are attached to a gibbsite octahedra. Layers are attached by K+-ions forming flake-like minerals with low charge density between particles. This enables water to enter between basal layers allowing swelling of the interlayer. The large surface area implies an increase in the capability of ion exchange and therefore a cec much higher than in kaolinite and illite. Examples of the smectite group are bentonite and montmorillonite.
Illite. Illite is a 2:1 type of clay with strong ionic bonds which do not allow molecules of water between layers. This results in low swelling capacity of the material. Primary particles are flake like and they are much thinner than the kaolinite ones. This implies a larger specific area of illite flocs. When the amount of potassium is not sufficient, the number of basal layers is limited. A smaller number of basal layers implies that the specific surface as well as the cec are higher than for kaolinite.
More focus is given in this work to kaolinite as it is the clay we use for flocculation experi-ments.
Property Kaolinite Smectite Illite
Structure 1:1 2:1 2:1
Shape platelets flakes flakes
Specific surface [m2/g] 10-20 50-120 60-100
cec[meq/100 g] 3-15 80-150 10-40
Thickness [µm] 0.05-2 0.001-0.01 0.003
Length [µm] 0.1-4 1-10 0.1-10
Type of bonding hydrogen van der Waals K+ ions
Magnitude of bonding strong weak strong
Cohesiveness low high medium
Swelling capacity low high low
Specific weight [g/l] 2650 2530 2800
Table 2.1: Physico-chemical properties of three main clay minerals.
Organic matter. Organic matter consists primarily of organic polymers which can occur as charged or neutral particles [Winterwerp and Van Kesteren, 2004]. Organic matter strongly
8 Chapter 2. Literature survey and definitions.
influences the flocculation behaviour of cohesive sediments and its production depends on the amount of light, the water temperature, and on the ongoing biological processes. The amount of organic matter can be measured for example by loss-on-ignition or estimated from the amount of chlorophyll-a. The organic polymers can adsorb to the flocs and modify their surface charge, coating and structure and therefore the interactions between particles.
Few qualitative studies on different types of organic matter exist and no systematic study has been done, to our knowledge, to evaluate the effect on flocculation of the different components of organic matter.
2.2.2 Suspension
Electrolyte. An electrolyte consists of water and dissolved ions. The properties of the elec-trolyte influence both particle-particle interactions and floc strength. An elecelec-trolyte can be characterized by two quantities: pH and ionic strength (salinity). pH can be defined by pH ≈ − log10H+where H+is the concentration of H+ions in solution in mol/L. The ionic strength
is proportional to the amount of salt dissolved in water and is also referred to as salinity. Salinity is usually expressed in weight ratio (grams of salt per kilogram of solution), part per thousands (ppt), in civil engineering and marine applications, and in moles per litre (M = mol/L) in chemical and colloidal applications.
Sediment concentration. The sediment concentration can be expressed in three different ways:
• The mass concentration c [kg/m3] is the mass of flocs/particles per unit volume; • The volume fraction φ [-] is the volume of flocs/particles per unit volume;
• The number concentration N [m−3] is the number of flocs/particles per unit volume. Assuming flocs as fractal objects, a relation between N , c and φ can be established, Winterwerp [1999]: N = 1 fs c ρs Ld0−3 p L −d0 (2.1) and φ = ρs− ρw ρf − ρw c ρs = c ρs L Lp 3−d0 , (2.2)
where fs is a shape factor (fs= π/6 for spheres), ρs, ρf and ρw are, respectively, the density of
sediment, flocs and water and Lp and L are the sizes of primary particles and of the flocs. The
fractal dimension of the flocs is d0.
2.2.3 Electrokinetic properties of the particles
When a charged particle is in suspension, ions with a charge opposite to the one of the particles (counterions) will accumulate near the particle surface. Counterions interact by electrostatic forces with the particle’s surface. If the ions are bound by electrostatic attraction alone, they are defined as indifferent ions. Note that ions (of any charge) can also be chemically bound to the surface.
2.2. Sediment characterization and particle-particle interactions 9
Electric double layer. The electric double layer around a charged colloidal particle consists of the ions at the surface of the particle at x = 0 in Figure 2.2, and the ions (mostly counteri-ons) electrostatically interacting with the particle. The electric double layer extends from the particle’s surface into the surrounding suspension over a distance called the Debye length. The Debye length, κ−1, is defined as:
κ−1 = s ǫ0ǫ1kT e2n ∞P νizi2 (2.3)
where e is the absolute value of the electron charge, n∞is the ionic density, νi and zi are
respec-tively the stoichiometric coefficient and the valence of the ion i, and ǫ0 and ǫ1 are respectively
the dielectric permittivity of vacuum and the relative dielectric permittivity of the electrolyte. We have furthermore n∞= CsNa where Cs is the salt concentration in millimol/liter [mM] and
Na [mol−1] is Avogadro’s number. For x >> 0 the standard electrokinetic equations can be
applied, Kruyt [1983]. For x ≈ 0 the finite size of ions or ion adsorption can play a role and the standard theory cannot be used. The region where this occurs, 0 < x < xStern, is defined as the
Stern layer.
ζ-potential. In electrokinetic studies, such as electrophoresis, the accessible quantity to esti-mate the surface charge of a particle is the ζ-potential. The ζ-potential is defined as the electric potential at the surface of shear x = xshear of the particles. The surface of shear can be seen
as the surface such that the ions between the particle surface and the plane of shear act as if they were “attached” to the particle; the ions beyond the plane of shear are free to move in response to any applied field. For more details see Kruyt [1983]. A schematic representation of the ions distribution around the particles surface, the electrical double layer and the location of the ζ-potential as a function of the distance from the particle’s surface are given in Figure 2.2.
+ + + + + -+ -+ -- -+ + + + + -+ + + + Particle surface
Electrical double layer Shear plane P ot ent ial x -+ + -ȗ-potential Ȍ0surface potential -+ + + + + + 0 xStern xshear ț -1
Figure 2.2: Schematic representation of the electrical double layer and ζ-potential of particles, Kruyt [1983]. x represents the distance from the particle’s surface located at x = 0.
Although the ζ-potential represents an average value which does not include the charge distribution at the particle surface, it gives an indication of the interactions between the dissolved
10 Chapter 2. Literature survey and definitions.
ions and the particles. The ζ-potential and the surface charge of kaolinite have been widely studied in literature [Rand and Melton, 1977; Williams and Williams, 1978; Vane and Zhang, 1997; Chassagne et al., 2009; Tombacz and Szekeres, 2006]. These authors found for example that both a decrease in pH and an increase in ionic strength yield a drop in the absolute value of the ζ-potential.
The ζ-potential of illite varies with pH and salt concentration similarly to that of kaolinite, Sondi et al. [1996].
2.2.4 Particle-ions interactions
Dissolved ions can be classified in two groups depending on the way they interact with the suspended particles:
• Indifferent electrolytes. The ions of indifferent electrolytes are assumed not to chemi-cally react with the particles surface groups. They only contribute to the compression of the electric double layer, the length of which is proportional to 1/√Cs, where Cs is the
salt concentration. When the salt concentration is high, the double layer is very small and the surface charge of the particles is nearly completely screened by counterions very close to the surface. In this case, the ζ-potential should be close to 0 mV according to the standard theory provided that no Stern Layer is considered.
• Other electrolytes. The ions of these electrolytes can react with the particle surface and modify its charge, which may go through zero and even reverse sign upon addition of salt, Sposito [1989]. The ζ-potential will reach zero and change sign accordingly. As shown in Sposito [1989], ions with higher valence are more likely to react with the particle surface groups.
• H+ ions. A specific behaviour has been observed in the interaction between H+ ions and
kaolinite particles [Tombacz and Szekeres, 2004; 2006]. H+ ions adhere to the edges of the platelets, which charge reverses sign, from negative to positive, at pH = pH∗. At pH =
pH∗ the average charge of the edges is zero. The edges are charged negatively at pH > pH∗ and positively at pH < pH∗. The point of zero charge of the edges of kaolinite has
been estimated at pH∗
∈ [5 − 7] by Rand and Melton [1977], pH∗ = 7.2 by [Williams and
Williams, 1978] and pH∗ = 5 by Wang and Siu [2006]. pH∗ depends on the origin of the kaolinite and on the sample preparation Melton and Rand [1977].
The edges of montmorillonite platelets have been observed to interact with H+ ions in the same way as the ones of kaolinite [Tombacz and Szekeres, 2004; Lagaly and Ziesmer, 2002] but the ζ-potential of montmorillonite varies little with pH [Vane and Zhang, 1997].
2.2.5 Particle-particle interactions
The likelihood of particles to aggregate depends on the particles’ charge, charge distribution and coating. The particles surface charge can be estimated on average by the ζ-potential. As a rule of thumb, if all the other conditions are equal, particles with a higher ζ-potential (in absolute value) will have less tendency to aggregate than those with lower ζ-potential, Kruyt [1983].
2.2. Sediment characterization and particle-particle interactions 11
Three different mechanisms may lead to particle aggregation:
• Charge screening. Indifferent electrolytes or charged polymers cumulate in the proxim-ity of the particle surface, screening its charge. The ζ-potential of the particle therefore decreases in magnitude and the suspension becomes unstable. Aggregation is in this case due to Van der Waals forces, Kruyt [1983]. Large flocs can be formed, Tombacz and Szekeres [2006].
• Coulombic attraction. Coulombic attraction between two oppositely charged particles induces fast aggregation. This is the case for kaolinite at low pH when the edges and faces have opposite charges. Edge-face flocculation is fast and leads to the formation of large and strong flocs, Tombacz and Szekeres [2006].
• Bridging. This happens when a polymer string adheres to two different clay particles creating an aggregate. This aggregation mechanism can generate large, strong and very porous flocs, Li et al. [2006].
2.2.6 Ions and particles interaction: the case of kaolinite
The effect of indifferent electrolytes and pH on the flocculation behavior of kaolinite with no organic matter is schematically represented in Figure 2.3, Tombacz and Szekeres [2006]. Four different regimes can be distinguished in the sketch.
+ Ͳ + ͲͲ ͲͲ Ͳ Ͳ Ͳ Ͳ ͲͲ ͲͲͲ Ͳ Ͳ Ͳ ͲͲ ͲͲͲ pH Ionic strength + Ͳ + ͲͲͲͲͲ 2 4 3 1 aggregation mode particlescharges anddoublelayer contour aggregation mode particlescharges anddoublelayer contour
Figure 2.3: Sketch of the interactions between the kaolinite particles and the dissolved ions; more probable modes of aggregation. The dashed line represents the contours of the electric double layer at a distance κ−1 of the particle’s surface.
• Regime 1 and 2. When the ionic strength is increased, the double layer is compressed until a point where the ζ-potential is low enough so that the particles can flocculate easily. Aggregation occurs because of charge screening. The most probable contact between particles is therefore face-face.
12 Chapter 2. Literature survey and definitions.
• Regime 3. When the pH is decreased well beyond the point of zero charge of the edges, Coulombic attraction between the positively charged edges and the negatively charged faces induces edge-face flocculation. Large, strong and porous flocs may form rapidly.
• Regime 4. At high pH and low ionic strength, the double layer is large and particles are unlikely to aggregate. The suspension is in this case rather stable.
2.3
Water column hydrodynamics and fluid-particles
interac-tions
Water column hydrodynamics affect both aggregation and breakup as the fluid can entrain particles and induce shear stresses, see Section 2.1.
2.3.1 Water column hydrodynamics
Instantaneous velocity u in a turbulent field is characterized by a mean component U and a fluctuating component u′. The fluctuating velocity can be isolated by means of a Reynolds decomposition u′
= u − U and it can be characterized by a normalized autocorrelation function: ρ(τ′) = ρ(−τ′) = u(t)u(t
′)
u′2 , (2.4)
where u(t) and u(t′) are the velocities separated by a time lag τ′
= t − t′.
The area defined by the autocorrelation function ρ in the (τ′ρ)-plane is the integral time scale ℑ:
ℑ = Z ∞
0
ρ(τ′)dτ′. (2.5)
The integral scale is a measure of the time interval beyond which u(t) is not self-correlated anymore and it can be interpreted as the largest time scale in the turbulent flow: it is related to the size of the largest eddies, equivalent to the depth or width of the flow for shallow water conditions. The largest eddies do not play a role in flocculation as at these length scales the sediment particles are only transported by the flow and no relative transport between particles or shear stresses are induced.
The Taylor microscale λ is the zero-centred second derivative of ρ(τ′), Tennekes and Lumley
[1972]: ∂2ρ ∂τ′2 τ′=0= − 2 λ2. (2.6)
The Taylor microscale is representative of the energy transfer from large to small scales. The process of energy transfer is represented by the energy dissipation per unit mass ǫ. If the Reynolds number is high, the structure of the turbulence can be considered isotropic and the energy dissipation can be approximated by [Tennekes and Lumley, 1972]:
ǫ ≈ 15νu
′2
λ2 ≈ 15
νu∗2
λ2 , (2.7)
where u∗ is the isotropic fluctuating velocity scale and ν is the kinematic viscosity. The
2.3. Water column hydrodynamics and fluid-particles interactions 13
The Taylor microscale is not the smallest scale in turbulence: at very small scales, viscosity is responsible for dissipating the velocity fluctuations. The smallest scale of motion adjusts itself to the viscosity of the fluid. By relating the viscosity ν and the energy dissipation rate ǫ, the Kolmogorov length η, time τ and velocity υ microscale can be defined as:
η =νǫ3 1 4
, τ = νǫ12 , υ = (νǫ)14. (2.8)
The shear rate G at scale η, commonly used to describe the kinetics of flocculation and to model aggregation and breakup, can be defined as:
G =r ǫ
ν (2.9)
The shear rate can also be expressed as a function of the Kolmogorov microscale as G = ν/η2.
At this small length scales, turbulence influences flocculation by inducing collision between particles (turbulence diffusion) and stresses on the flocs which may lead to breakup.
2.3.2 Fluid-particles interactions
Turbulent motion, at the scale of the floc sizes, governs the collisions frequency between the various flocs. However, turbulence also induces pressure differences and differential velocities. These may result in shear and normal stresses, which may disrupt the flocs when they exceed floc strength. Both turbulent motion and stresses can be conveniently quantified by the shear rate G at these small scales, e.g. Levich [1962]. We will therefore use the shear rate G for flocculation modelling.
.
.
L
iL
jL
cFigure 2.4: Sketch of the collision sphere centred in particle i.
Collision frequency. The collision diameter Lc between two particles sized Li and Lj can
be computed as the sum of the two diameters Lc = Li+ Lj, Saffman and Turner [1956]. The
collision sphere is a surface centred in the centre of one of the particles such that, if the centre of the other particle is on that surface, the two particles collide, see sketch in Figure 2.4. If
14 Chapter 2. Literature survey and definitions.
the small eddies are isotropic, the particles are moving with the fluid and they are randomly distributed, the collision frequency can be defined as the product between the flux through the collision sphere and the number concentration of the particles. The above defined collision frequency reads as:
βi,j = G/6 (Li+ Lj)3NiNj (2.10)
where Ni and Nj are the number concentrations of particles of size Li and Lj. The coefficient
1/6 is a commonly used approximation of the value derived by Saffman and Turner [1956].
Breakup. The turbulent eddy velocities uλe scale with the eddy size λe as [Levich, 1962]: uλe ≈
υ
ηλe. (2.11)
in which υ and η are respectively the velocity and size of the Kolmogorov microscale. As a consequence, the turbulent stresses τt in the viscous regime at the scale of the floc λe = L are
given by, Winterwerp [1998]:
τt= µ
∂u ∂z ≈ µ
υL/η
L ≈ µG, (2.12)
in which µ is the dynamic viscosity of the fluid. Shear stresses increase rapidly at length scales larger than the Kolmogorov microscale η, and the floc size is usually smaller than η.
Thomas et al. [1999] show that if flocs are smaller than the Kolmogorov microscale, they are more likely to break by surface erosion, while larger flocs are more exposed to breakage by fracture.
2.4
Flocs properties
Fluid-particle interactions and particle-particle interactions lead to the formation of a population of flocs characterized by different structure, strength and settling velocity. These properties are described in more detail in this section.
2.4.1 Structure of flocs
Shape. The shape of flocs can vary widely with the environmental conditions and the shear stresses. At high shear, flocs are mainly spherical and rather compact in all cases [Eisma, 1986] while at low shear, the structure of flocs is influenced by the environment. Long chain-like flocs as well as spherical flocs have been observed in marine environment, Manning [2001]. The irregularity of the floc shape increases also with the amount of organic matter. In most of the modelling and laboratory works, flocs are assumed to be spherical.
Fractal dimension d0. It is convenient to describe flocs as fractal objects which porosity can
be expressed as a function of their size, Kranenburg [1994]. The mass m of a fractal object scales as a power law of the dimensionless size l = LL
p as follows:
m ∝ ld0 (2.13)
where the exponent d0 is the fractal dimension (or Housdorff dimension), L is the size of the
2.4. Flocs properties 15
d0 is a measure of how the constituent particles fill the space within an aggregate [Kranenburg,
1994], and is expressed as follows [Vicsek, 1992]:
d0 =
ln(n) lnLL
p
, (2.14)
where n is the number of primary particles in the floc. The number n of primary particles in a floc of size L can therefore be computed as:
n = L Lp
d0
. (2.15)
The fractal dimension of a floc may depend on the process of aggregation. In particular two different processes can be distinguished [Tang et al., 2000]: Diffusion Limited Cluster Ag-gregation (DLCA), when every collision between particles results in agAg-gregation and Reaction Limited Cluster Aggregation (RLCA) when only few collisions result in aggregation. In the first case, flocs are formed very rapidly and their fractal dimension derived from analytical work is d0 = 1.8 while in the second case flocculation takes long time and compact flocs are formed.
The derived fractal dimension in this case is d0 = 2.2.
Kranenburg [1994] observed that the large variability of mud implies that flocs can vary largely in structure. On the basis of experimental results on kaolinite, Maggi [2005] suggests that flocs may be described as statistically self-similar objects where the structure is not exactly repeated at different length scales and the fractal dimension is a function of the size of flocs. Maggi proposed a power law relation between fractal dimension and floc size L which reads as follows:
d0 = δ L
Lp
ξ
, (2.16)
where δ = 3 is the fractal dimension of primary particles, and ξ = −0.1 gives an indication of the distance from fully self-similar structure.
The value of the fractal dimension measured in the water column can vary from d0= 1.4 for
very fragile flocs, like marine snow, up to d0 = 2.2 for strong estuarine flocs [Winterwerp, 1999;
McCave, 1984]. Typical values within estuaries and coastal waters range from 1.7 to 2.2 with an average value of d0 = 2.
SizeL. The 3D size of a fractal object can be defined in different ways. The gyration diameter is twice the average distance of particles composing the floc from its centre of mass. The hydraulic diameter is the diameter of a sphere which volume is equivalent to the cluster’s volume. The box length is the size of the minimum cube circumscribing the floc. The collision diameter is the maximum distance necessary for two flocs to get in contact. Different authors and measuring instruments make use of the different definitions to compute the floc size from the experimental data. For example, laser diffraction techniques ( i.e. Malvern mastersizer) make use of the hydraulic diameter while the box length may be used to process the data of a camera system, Maggi [2005].
16 Chapter 2. Literature survey and definitions.
Relative floc density ∆ρf. The relative floc density can be written as follows [Winterwerp,
1998; Serra and Casamitjana, 1998c]:
∆ρf = (ρs− ρw) L
Lp
3−d0
, (2.17)
where ρs and ρw are, respectively, primary particles and water densities.
The dynamics of aggregation and breakup are influenced by the shape, the size, and the structure of flocs while the relative density, together with the other properties, is important for the computation of the settling velocity of flocs.
2.4.2 Strength of flocs.
The strength of flocs is the main parameter for the determination of the breakup rate and it depends on the composition of flocs (for example the organic matter content) as well as on the type of bonds, see Section 2.2.5.
It is very difficult to experimentally determine the strength of flocs and both direct and indirect evaluations have been done. Jarvis et al. [2005] give an overview of the different exper-imental techniques used.
Direct measurements. Two examples of direct floc strength measurements are the work by Yeung and Pelton [1996], and the one by Ani et al. [1991]. The strength of flocs in these works is defined as the critical force needed to break a floc. Yeung and Pelton [1996] measured the force needed to break a floc by pulling on the two sides of the flocs with a micropipette. They carried out a large number of experiments on flocs of different structure and they found only a very poor correlation between the floc size and the floc strength.
Ani et al. [1991] measured the critical shear rate at which a floc breaks while falling in a column with increasing G. They observed that the floc strength increases linearly with the floc diameter.
Indirect estimations. Kranenburg [1999] estimates the floc strength as the critical force needed to break a floc by the combined measure of the maximum floc size and the settling velocity. The stress on a floc of size L with a settling velocity ω can be computed from the drag force as:
Fy =
3
4πµωsmLm, (2.18)
where µ is the dynamic viscosity of water and ωm and Lm are the settling velocity and size of
the largest floc. For estuarine macroflocs he estimates Fy ≈ (10−8) N.
It has been observed that the maximum flocs size can be related to the shear rate as:
Lm = CG−γ (2.19)
where C is the floc strength coefficient and γ is the stable floc size exponent, Jarvis et al. [2005]. Observed values of γ range from 0.29 up to 0.81 and in most cases γ ≈ 0.5. γ is rather constant with the properties of the suspension and with the experimental setup while the floc
2.4. Flocs properties 17
strength coefficient (C ∈ [1.9 − 6]) is strongly dependent on the amount of flocculant added to the suspension.
Parker et al. [1972] consider in detail the forces on a floc due to the turbulent flow. From a force balance, they derive the maximum stable floc size in the case of erosion or binary breakup as a function of the shear rate. They find that in the viscous subrange, where the length scales are smaller than the Kolmogorov microscale, the maximum stable floc size follows Eq. 2.19 with γ = 1 if erosion takes place and γ = 0.5 if floc fragmentation takes place. These results are obtained without taking into account the aggregation process.
A similar relation was also found through both the theoretical study by Sonntag and Russel [1987] and the numerical simulations by Higashitani and Iimura [2001] who found, in our nota-tion, the relation n ∝ G−γ where n is the number of particles in a floc, proportional to the floc size L. Sonntag and Russel [1987] found γ = 0.879 and Higashitani and Iimura [2001] found γ to vary between 0.75 and 1 when the fractal dimension of flocs is varied.
2.4.3 Settling velocity
The settling velocity can be computed from the balance of the three forces acting on a suspended floc: gravitational force, buoyant force and drag force. The drag force opposes to the motion of an object in a fluid and is computed as the product of the object velocity and surface, the viscosity of the fluid and a non dimensional drag coefficient cd, which depends on the shape of
the object. Stokes’ relation for the settling velocity is defined for Euclidean (non-fractal) spheres and is valid in the Stokes regime (Re < 1):
ωs=
g(ρs− ρw)
18ν L
2, (2.20)
where g is the gravity acceleration, ρs is the sediment density, ρw is the density of water and ν
its viscosity. In Eq. 2.20, the drag coefficient as a function of the Reynolds number cd= 24/Re
has been used.
Various authors derive the settling velocity taking into account the fractal nature of flocs which affects the density of flocs and the drag coefficient. Their findings are based on different experimental observations.
• Johnson et al. [1996] express the drag coefficient as a power law of the Reynolds number as suggested by experimental observations and define the floc porosity as pr= 1 − nVp/V
where n is the number of primary particles of volume Vp and V is the volume of the floc:
ωs=
g(ρs− ρw)nVp
3πνL (2.21)
• Winterwerp [1999] uses cd= Re24(1 + 0.15Re0.687) as experimentally determined by Vanoni
[1977] and obtains: ωs = aw 18bw (ρs− ρw)g µ L 3−d0 p Ld0−1 1 + 0.15Re0.687 (2.22)
where aw and bw are two shape parameters, Re = Lωνs is the particle Reynolds number
18 Chapter 2. Literature survey and definitions.
The settling velocities computed with the different equations for flocs with fractal dimension d0 = 2 are plotted in Figure 2.5 together with the settling velocity computed with Stokes’ law.
For small flocs, the two settling velocities computed considering flocs as fractal objects (Eq. 2.21
10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1 100 Floc size [m] Settling velocity [m/s] Stokes (Eq. 2.20) Johnson et al. (Eq. 2.21) Winterwerp (Eq. 2.22)
Figure 2.5: Comparison between the different formulations for the settling velocity. The Winterwerp settling velocity (Eq.2.22) has been computed using aw= bw= 1.
and Eq. 2.22) are the same if d0 = 2 and they are both smaller than the Stokes’ settling velocity.
For large flocs, the Winterwerp settling velocity varies very little with the floc size while the Johnson et al. settling velocity increases with floc size in the same way over the complete size range.
2.5
Flocculation dynamics
Flocculation is the result of the simultaneous processes of aggregation and breakup. A dynamic equilibrium is reached when these two processes balance each other. The time needed to reach equilibrium is defined as the flocculation time. Knowing the value of the flocculation time is important for the study of natural systems where conditions are continuously varying in time and the dynamic equilibrium may not be reached. The flocculation time is also important for engineering applications such as water treatment where the efficiency of the plant depends on the flocculation time and on the settling velocity of the particles.
2.5.1 Aggregation
Aggregation can be assessed either from the so called Smoluchowski growth rate or from calcu-lations based on a statistical approach which involve the use of a collision frequency βi,j and a
collision efficiency α. The collision frequency represents the number of collisions per unit vol-ume per unit time while the collision efficiency represents the probability of collisions to result in aggregation.
Smoluchowski growth rate. The rate of variation in time t of the number concentration N , dN/dt, of flocs of size L due to aggregation can be derived from the Smoluchowski formulation
2.5. Flocculation dynamics 19 as [Winterwerp, 1998]: dN dt = −k ′ aGL3N2, (2.23) where k′
a is a coefficient which includes the efficiencies for collision and diffusion. As introduced
in Section 2.2.5 the efficiency for collision depends on the physico-chemical properties of the suspension. Equation 2.23 accounts for particles collisions due to shear rate solely. Brownian motion and differential settling are not taken into account (see below).
Collision frequency βi,j. Aggregation kinetics have been studied in detail as a function of
shear rate, temperature, concentration and relative density of aggregates [Hunt, 1980]. Collision between particles can be considered as the combination three main (f)actors, βi,j = β(BM )i,j +
βi,j(DS)+ βi,j(SH) where:
• Brownian motion βi,j(BM ). Particles collide by thermal motion. The collision frequency
is a function of the absolute temperature T , the dynamic viscosity of the suspension µ and the Boltzmann constant k. If flocs are spheres with diameter Li and Lj the collision
efficiency reads as [Hunt, 1980]:
βi,j(BM ) = 2 3 kT µ (Li+ Lj)2 LiLj . (2.24)
This collision frequency can be neglected if the size of the particles is Li > 1 µm [Hunt,
1980]. This is the case in this study.
• Differential settling βi,j(DS). Differently sized flocs settle with different velocities and a
faster floc may collide with a slower one if they fall along the same trajectory. The collision frequency in this case can be computed as a function of the settling velocity of the flocs as [Hunt, 1980]:
βi,j(DS) = π
4(Li+ Lj)
2|w
s,i− ws,j| , (2.25)
where the settling velocities ws,i and ws,j of flocs of size Li and Lj respectively are
com-puted for instance with Stokes’ law. Stolzenbach and Elimelech [1993] investigated the relevance of differential settling both using a theoretical approach and with experiments in a settling column. They showed that the chance that a large, but less dense, rapidly falling particle will collide with a small, but denser, slowly falling particle is very small. • Shear rate βi,j(SH). The collision frequency due to turbulence diffusion can be computed
as shown in Section 2.10 [Saffman and Turner, 1956; Hunt, 1980]:
βi,j(SH)= G
6(Li+ Lj)
3. (2.26)
Collision between particles in estuarine environment is mainly due to turbulent diffusion, Hunt [1980].
All the three collision frequencies defined above can decrease due to hydrodynamic inter-actions, as the flow lines around a particle are affected by the particle motion. Approaching particles may therefore have less chances to collide. This effect is stronger when the ratio be-tween the sizes of the interacting particles is large as the trajectory of the smaller particle will
20 Chapter 2. Literature survey and definitions.
strongly be affected by the flow lines around the larger particle. Friedlander [1957] and McCave [1984] modify the collision frequency βi,j with a function of the ratio between the sizes of the
interacting particles. Han and Lawler [1992] and Li et al. [2004] account also for the ratio of hydrodynamic shear forces to Van der Waals forces between the colliding particles.
Collision efficiency αi,j. The collision efficiency is mainly responsible for the stability of a
suspension. If the collision efficiency is very small, flocs can hardly aggregate and the suspension is stable. If the collision efficiency is close to unity all collisions result in aggregation and the suspension is unstable.
The collision efficiency is mainly influenced by the short range forces (DLVO), Van de Ven and Mason [1977]. These forces depend on the charge and coating of the particles which are affected by the physico-chemical properties of the suspension as explained in Section 2.2.5. Van de Ven and Mason [1977] derived a semi-empirical formulation for α which takes into account the effect of both hydrodynamics and DLVO forces:
α = f (λv/L) A/4πµGL30.18, (2.27)
where f (λv/L) is a function of the dispersion wave length and A is the Hamaker constant,
see Van de Ven and Mason [1977] for more details. In this equation, the collision efficiency decreases with increasing shear rate and floc size. On the basis of experimental work, Serra and Casamitjana [1998b] suggest that the collision efficiency is independent of the shear rate and of the floc size and it depends on the short range interactions between particles solely. In this work we also consider the collision efficiency to vary only with the properties of the suspension. It is very difficult to measure the collision efficiency in a direct way or to estimate it through force balance, and in most cases α is estimated indirectly from flocculation experiments.
2.5.2 Breakup
Breakup of flocs is due to shear stresses and inter-particle collisions, and can be characterized by a breakup rate and a breakup distribution function. The breakup rate represents the number of flocs which break in each class per unit volume per unit time. The breakup distribution function, on the other hand, indicates the gain in each class derived from the breakup of flocs in a larger class.
Breakup ratesi. The breakup rate can be computed as the sum of breakup induced by shear
rate and breakup induced by inter-particle collisions, Serra and Casamitjana [1998a]:
si = si,lin+ ∞
X
j=1
α′βi,jNj, (2.28)
where βi,j is the collision frequency between particles in class i and j (defined in Section 2.5.1),
α′ is the probability of breakup after collision and N
j is the number of particles in class j per
unit volume. The first term of the right hand side represents the linear breakup due to shear rate while the second, non linear term represents the breakup due to inter-particle collisions. This can be neglected if the sediment concentration considered is low (volume concentration φ < 5 · 10−5) , Serra and Casamitjana [1998a] and Winterwerp [1999].
