micro-macro aspects of particle sedimentation …...micro-macro aspects of particle sedimentation...

Micro-Macro Aspects of Particle Sedimentation Analysis

An Application in Suspension Stability Assessment

Dissertation

zur Erlangung des akademischen Grades

Doktoringenieur

(Dr.-Ing.)

von: M.Sc. Olakunle Nosiru OLATUNJI

geb. am: 29. Februar 1976

in: Lagos, Nigeria

genehmigt durch die Fakultät für Verfahrens – und Systemtechnik der Otto-von-Guericke-Universität Magdeburg

Promotionskommission: Prof. Dr.-Ing. habil. Eckehard SPECHT (Vorsitz)

Prof. Dr.-Ing. habil. Jürgen TOMAS (Gutachter)

Prof. Dr.-Ing. habil. Hermann NIRSCHL (Gutachter)

Prof. Dr. rer. nat. Franziska SCHEFFLER (Mitglied)

eingereicht am: 27. Juni 2014

Promotionskolloquium am: 28. November 2014

Prof. Dr.-Ing. Jürgen TOMAS M.Sc. Olakunle Nosiru OLATUNJI

Micro-Macro Aspects of Particle Sedimentation Analysis – An Application in

Suspension Stability Assessment

Abstract

The dynamics of dispersion and agglomeration is one important characteristic that must be

well understood in order to ease the chemical formulation step during the synthesis of

colloidal dispersions for optimal benefits in research and commercial applications. Colloidal

dispersions are of utmost benefit when the disperse phase can be kept suspended in the

continuous phase throughout the lifetime of the product and/or should be readily redispersable

just by simple shaking or stirring should sedimentation occur. Hence, particle sedimentation

analysis is an important characterization technique that can be used to make efficient, fast and

reliable assessment of the dynamics of dispersion and agglomeration of colloidal dispersions.

It may also be suitable for investigating possible changes occuring when physicochemical and

mechanical treatments are performed on colloidal dispersions to enhance their stability.

In this work, a new numerical analytical model was derived similar to Kynch’s continuity

equation in that negligible compression effect and constant solid volume concentration ϕs are

assumed at the thickened sludge (compression) phase. However, this new model differs as it

includes new expressions for forces due to inertia and accelerated fluid eddies. Pressure drop

during flow through the particle-bed is also defined by a new expression equivalent to single

particle fluid drag plus an additional term for particle-bed porosity function. Furthermore, the

model was employed in analyzing dynamics of particles during sedimentation thereby making

it useful in assessing dispersion stability as well as predicting sedimentation profiles if

variation of ϕs with time can be accurately measured.

Particle sedimentation analysis in this study was done via an instrument, Turbiscan Lab

Expert, operating based on principle of multiple light scattering technique. The instrument

measures variations in dispersion particle size and position as a function of time and records

as backscattered (BS) or transmitted (T) light intensity which can then be converted to ϕs

expressed in terms of attenuation time and height. This makes the instrument applicable in

validating the newly derived numerical analytical model.

Prof. Dr.-Ing. Jürgen TOMAS M.Sc. Olakunle Nosiru OLATUNJI

Micro-Macro Aspects of Particle Sedimentation Analysis – An Application in

Suspension Stability Assessment

Abstrakt

Die Dynamik der Dispersion und der Agglomeration in kolloiden Dispersionen ist eine

bedeutende Eigenschaft, die gut verstanden sein muss, um die Synthese kolloidaler

Dispersionen, im Hinblick auf den wissenschaftlichen und kommerziellen Einsatz, zu

optimieren. Kolloide Dispersionen sind von größtem Nutzen, wenn die dispergierte Phase in

der kontinuierlichen Phase über die gesamte Lebenszeit des Produktes suspendiert bleibt

und/oder durch Schütteln oder Rühren leicht redispergierbar ist, falls Sedimentation eintritt.

Die Analyse der Partikelsedimentation ist ein bedeutendes Charakterisierungsverfahren für

eine effiziente, schnelle und verlässliche Bewertung der Dynamik der Dispersion und

Agglomeration kolloidaler Dispersionen. Dieses Charakterisierungsverfahren kann auch auf

die Untersuchung von möglichen Veränderungen während der physikochemischen und

mechanischen Behandlung zur Verbesserung der Stabilität kolloidaler Dispersionen

anwendbar sein.

In dieser Arbeit wurde ein neues analytisches Modell hergeleitet, welches vergleichbar mit der Kynch Kontinuitätsgleichung ist und in welchem vernachlässigbare Kompression und konstante Feststoffvolumenkonzentration ϕs in der Dickschlamm- (Kompressions-) Phase angenommen werden. Das neue Modell enthält zusätzlich neue Ausdrücke für Trägheitskräfte und Fluid-Verwirbelungen. Der Druckverlust beim Durchströmen des Partikelbetts wird durch einen neuen Ausdruck definiert, der den Strömungswiderstand am Einzelpartikel sowie die Funktion der Partikelbettporosität berücksichtigt. Weiterhin wurde das Modell auf die Analyse der Partikeldynamik während der Sedimentation angewendet um die Dispersionsstabilität zu bewerten als auch Sedimentationsprofile für definierte Zeitabhängigkeit von ϕs vorherzusagen.

Die experimentelle Sedimentationsanalyse erfolgte mit dem Messgerät Turbiscan Lab Expert,

basierend auf der Mehrfachlichtstreuung. Mit dieser Messmethode können Veränderungen

der Partikelgröße und Partikelposition als Funktion der Zeit bestimmt werden. Ferner werden

die Lichtintensität des rückgestreuten Lichts als auch des durchgelassenen Lichts

aufgezeichnet um hieraus ϕs abzuleiten, beide Parameter sind von der Sedimentationszeit und

der Position der Sedimentationsfront abhängig. Das Messgerät kann daher als geeignet für die

Validierung des neuen hergeleiteten analytischen Modells eingeschätzt werden.

Dedication

This work is dedicated, first to my creator, the giver of life and eternal wisdom, I call Him

Jehovah El-Effizzy and to the sweet loving memory of my mum, Mrs. Felicia Gbemisola

Fatimo Olatunji (Nee Shonde).

Acknowledgement

This work would be incomplete without acknowledging the direct and indirect contributions

of several friends and colleagues to its success.

My gratitude first goes to Prof. Dr.-Ing. habil. Jürgen Tomas for giving me the opportunity to

work on one of his several derived numerical analytical models at the Chair of Mechanical

Process Engineering (MPE) and for the funding through DFG-GRK1554.

I will like to acknowledge the following for their contributions to the experimental part of this

work: Du and Zhao (MSc Thesis students), Frank (Turbiscan), Antje (BET), Jakob (TGA),

Dr. Heyse (SEM) and Dr. Schwidder (Rheology).

My sincere gratitude also goes to Dr. Hintz (my MSc Thesis supervisor), Dr. Ssemangada and

Dr. Aman (for helping out with mathematical problems and MatLAB codes).

Several other old and new colleagues at MPE also made my time at the chair worthwhile and

really interesting, particularly, Sebastian, Sören, Madeleine (Finance), Katja, Zinaida, Abbas,

Talea, Alexander (Russian), Alexander (Indian), Nicole, Asim, Salman, Bernd, Andreas,

Hannes, Zheni, Peter Müller and Peter Siebert.

To all my great friends in Magdeburg – Judith, Jan, Akin, Ahams, Emmanuel (QSE, Ghana),

David, SB, Paul, the Onwutas, Lekan, Yinka, Alaba, the Spüllers and the Scala Gemeinde,

thanks so much for making my time in Magdeburg so pleasant and memorable.

I am also very grateful to members of GRK1554 (students and professors), all my Nigerian

and African friends in and out of Deutschland, so innumerable I cannot sincerely list them all.

Finally, I thank my loved ones in Nigeria for keeping faith in me throughout my nine years of

academic sojourn in Deutschland – Baba Kunle, Yinkus, Dami, Seun and Kemi, thank you all

so much for always being there for me.

Most importantly, thank you so much dear Lord, for your abundant grace and making me a

wonder even to myself. I will forever praise you my dear God.

Olakunle Nosiru OLATUNJI

Magdeburg, December 10, 2014

19:45 Uhr

Table of Contents Abstract ii

Nomenclatures x

1 Introduction 1

1.1 Sedimentation as Dispersion Separation Technique 4

1.2 Sedimentation as Dispersion Instability Mechanism 8

1.2.1 Agglomeration 10

1.2.2 Ostwald Ripening 10

1.2.3 Sedimentation and Creaming 10

1.3 Motivation and Outline of Task 10

Part I Literature Review

2 Fundamental Theory of Dispersion Sedimentation 13

2.1 Forces of Particle Sedimentation 14

2.1.1 Microscopic Particle Flow-around 18

2.1.2 Macroscopic Particle-bed Flow-through 21

2.2 Sedimentation Models 24

2.3 Numerical Analysis of Particle Sedimentation 25

2.4 Zone Sedimentation 36

2.5 Graphical Illustrations of Zone Sedimentation 39

3 Fundamental Principles of Dispersion Synthesis 42

3.1 Dispersion Stability Mechanism 42

3.2 Particle-Particle Interactions 46

3.2.1 Intermolecular Forces 47

3.2.2 Interparticle Forces 48

3.2.3 Continuous Phase Effect 49

3. 2.4 Electrical Double Layer Effect (Electrostatic Forces) 49

3. 2.5 Adsorbed or Anchored Layer Effect (Steric Repulsion) 59

3. 2.6 Miscellaneous Effects 65

T a b l e o f C o n t e n t s | vii

3. 2.7 Total Interaction Potential: DLVO Theory 65

3.3 Dispersion Instability Mechanism 68

3.3.1 Agglomeration Mechanism 68

3.3.2 Kinetics of Agglomeration 74

3.3.3 Aggregation and Particle Growth (Ostwald Ripening) Mechanism 78

3.3.4 Sedimentation Mechanism 81

3.4 Definition of “Dilute”, “Concentrated” and “Solid” Dispersions 81

3.5 Structure of Agglomerates and Sediments 82

4 Particle Precipitation by Sol-Gel Synthesis and Surfactant Adsorption 86

4.1 Sol-Gel Synthesis 86

4.1.1 Process of Sol-Gel Synthesis 86

4.1.2 Applications of Sol-Gel Synthesis 90

4.2 Surfactant Adsorption 92

4.2.1 Process of Surfactant Adsorption 95

4.2.2 Applications of Surfactant Adsorption 96

Part II Experiments and Measurements

5 Experimental Processes and Dispersion Characterization Techniques 98

5.1 Description of Selected Model Materials 98

5.2 Preparation and Stabilization of Dispersions using Commercial Powders 98

5.2.1 Preparation and stabilization of alumina dispersion 98

5.2.2 Preparation and stabilization of titania dispersion 101

5.3 Sol-Gel Synthesis and Stabilization of Nanoparticles 103

5.3.1 Sol-gel Synthesis and stabilization of titania nanoparticles 104

5.3.2 Sol-gel Synthesis and stabilization of silica nanoparticles 106

5.4 Characterization of Colloidal Dispersions 108

5.4.1 Dynamic Light Scattering (DLS) 108

5.4.2 Specific Surface Area Analysis (BET Theory) 112

5.4.3 Electrokinetic Analysis 113

T a b l e o f C o n t e n t s | viii

5.4.4 Macroscopic Rheological Analysis 116

5.4.5 Scanning Electron Microscopic Analysis 126

5.4.6 Thermogravimetric Analysis (TGA) 126

5.4.7 Sedimentation Analysis 128

6 Results and Discussions of Dispersion Characterizations 142

6.1 Experiment with Sterically Stabilized Commercial Alumina Powder 142

6.1.1 Effect of varying propionic acid and solid volume concentrations 142

6.1.2 Effect of varying heptanoic acid concentrations on TSK 154

6.1.3 Effect of varying lauric acid concentrations on TSK 154

6.1.4 Effect of fatty acid molecular architecture on stability of alumina dispersion 154

6.2 Experiment with Sterically Stabilized Commercial Titania Powder 159

6.3 Experiment with Electrostatically Stabilized Commercial Titania Powder 170

6.3.1 Effect of varying electrolyte and solid volume concentrations 170

6.3.2 Effect of electrolyte on stability of titania dispersion 183

6.4 Experiment with Sol-Gel Synthesized and Sterically Stabilized Titania Nanoparticles 184

6.4.1 Effect of varying precursor concentrations with Brij 30 as stabilizer 185

6.4.2 Effect of varying Brij 76 concentrations 192

6.4.3 Effect of varying Tween 20 concentrations 197

6.4.4 Effect of surfactant molecular architecture on stability of sol-gel synthesized titania nanoparticles 200

6.4.5 Effect of mixed solvent on synthesized titania nanoparticles 201

6.5 Experiment with Sol-Gel Synthesized and Electrostatically Stabilized Titania Nanoparticles 204

6.5.1 Effect of varying electrolyte and electrolyte concentrations 204

6.5.2 Effect of electrolyte on stability of synthesized titania nanoparticles 209

6.6 Experiment with Sol-Gel Synthesized and Sterically Stabilized Silica Nanoparticles 210

6.6.1 Effect of varying solvent and concentrations of NH4OH 210

T a b l e o f C o n t e n t s | ix

6.6.2 Effect of surfactants on stability of synthesized silica nanoparticles in isopropanol 220

6.6.3 Effect of solvent on stability of sol-gel synthesized silica nanoparticles 220

6.7 Comparison between Analytical Models and Experiments 222

6.7.1 Analytical Stability Kinetics (ASK) 222

6.7.2 Analytical Sedimentation Profiles 228

6.8 Summary 233

7 Conclusions and Recommendations 237

References 240

Appendix 252

N o m e n c l a t u r e s | x

Nomenclatures Greek Characters

Character Meaning Unit

α Wetting (contact) angle; atom or molecule polarizability; polymer expansion coefficient; collision frequency -

β Collision efficiency factor -

fβ Volume ratio of fluid eddies -

Γ Molar amount of adsorbed surfactant mol/m2

γ Solid-fluid density ratio; solid-liquid interfacial tension -

γ Shear rate s-1

0γ Amplitudes of strain -

φ∆ Streaming potential mV

δ Phase angle shift rad

Dδ Polymer adsorbed layer thickness nm

ε Particle-bed porosity; permittivity of solvent -

Lε Binding energy per link J

,r oε ε Dielectric constants of solvent and vacuum -

ζ Zeta potential mV

η Fluid viscosity Pa.s

*η Complex viscosity Pa.s

'η Complex (real part) viscosity Pa.s

''η Complex (imaginary part) viscosity Pa.s

,pl rη η Plastic; relative viscosity Pa.s

κ Debye-Hückel parameter nm-1

λ Photon mean free path -

N o m e n c l a t u r e s | xi


*λ Photon transport length -

iµ Mass fraction in class i -

*iµ Contact friction coefficient -

, ol lµ µ Chemical potential in the presence and absence of free polymer -

ν Poisson’s ratio; Characteristic frequency Hertz

ξ Damping ratio -

ρ Density kg/m3

,0 ,b bρ ρ Initial, final bulk density kg/m3

oσ Particle surface charge density C.m-3

0 ,, M Stσ σ Initial, mean consolidation (steady-flow) stress Pa

τ Delay time s

, yτ τ Shear or yield stress Pa

0τ Amplitudes of stress Pa

sφ Solid volume concentration -

effφ Effective solid volume concentration -

χ Flory-Huggins interaction parameter -

, oψ ψ Surface potential mV

( )Dψ Electrical potential mV

δψ Stern potential mV

ω Frequency rad.s-1

,d oω ω Damped and undamped angular frequencies Hertz

N o m e n c l a t u r e s | xii

Latin Characters


A London dispersion constant -

effHA Effective Hamaker constant -

am Area per adsorbed molecule m2

Ap Particle cross-sectional area m2

As Specific Surface Area m2/g

B Reflection coefficient of backscattering photon; Born repulsion constant -

B(ε)B Particle-bed porosity function -

c(+), c(-) Concentration of positive, negative ions mol/l

c1 Initial concentration of surfactant in solution mol/l

c2 Equilibrium concentration of surfactant in solution mol/l

ci Number of ions m-3, mol/l

cv Contact consolidation coefficient m2/s

cW Single particle drag coefficient -

D Particle separation distance m

d Particle diameter m

D(ρf), DB(ρf) Density function -

D0 Diffusion coefficient m2/s

d10,r Particle size below which 10% of sample lies nm

d50,r Median value of size distribution nm

d90,r Particle size below which 90% of sample lies nm

dH Hydrodynamic diameter nm

di Mean class diameter nm

dST Equivalent surface (Sauter mean) diameter nm

dε Average pore diameter nm

EA Total binding energy J

N o m e n c l a t u r e s | xiii


Esep Separation energy J

EuB Particle-bed drag coefficient -

e Elementary or proton charge C

f (κr) Henry’s correction factor -

F Faraday’s constant Cmol-1

FA Accelerated fluid eddies force N

FAd Characteristic adhesion force nN

FB Buoyance force N

FG Field (gravitational) force N

FI Inertia force N

FK Particle-particle interaction forces N

FP Particle pressure force N

FS Stochastic Brownian force N

FW Drag force N

ffc Flow function of dry powder -

g Gravitational acceleration, Optical parameter m2/s, -

g(τ) Auto-correlation function -

G Shear modulus Pa

G* Complex shear modulus Pa

G’ Storage modulus Pa

G” Loss Modulus Pa

H(t) Interface height or Clarification front m

H0 Initial height of dispersion m

HTS(t) Sediment height m

Planck’s constant Js

K Kozeny constant -

k Electrical conductivity of dispersion S/m (s-1)

N o m e n c l a t u r e s | xiv


kB Boltzmann constant JK-1

k Average coordination number -

Mw Molecular mass g/mol

mf,p Mass of fluid eddies g

mp Particle mass g

NA Avogadro’s number mol-1

n Coordination number or number of adsorbed polymer segments -

nf Liquid refractive index -

nL Number of links -

np Particle number , refractive index -

p, p0 Adsorbate partial pressure and saturation vapour pressure Pa

Q Scattering angles rad

q Number of molecules per unit volume of particles m-3

q1,q2 Electrical charges of equal sign C

qr (d) Particle size frequency distribution -

Qr (d) Cumulative particle size distribution -

Qs Optical parameters -

R Universal gas constant Jmol-1K-1

Rf, Rb Forward, backward reaction rate s-1

r Particle radius m

reff Effective radius m

rG Radius of gyration of free polymer m

s(t) Displacement-time function m

T Temperature K

Tm Melting temperature K

t Time s

N o m e n c l a t u r e s | xv


t1/2 Half-life -

t63 Stokes characteristic settling time s

t76 Newton characteristic settling time s

tcrit Critical time step s

to Experimental observation time s

tr System relaxation time s

u Fluid velocity m/s

ur Relative velocity between particle-bed and fluid m/s

V Equilibrium volume of gas adsorption per unit mass of adsorbent m3/g

V1,1,V2,2 Solvent-solvent, polymer segment-segment interaction energy J

Vf Volume of fluid m3

Vf,B Volume ratio of accelerated fluid eddies -

Vm Volume of gas needed for the monolayer m3

Vp Particle volume m3

Vs Molar volume of solvent mol/l

v Relative particle-fluid velocity m/s

1 2,v v Molar volumes of solvent and adsorbed polymer m3/mol

vs,N, vs,B,N Newton stationary settling velocity m/s

vs,St, vs,B,St Stokes stationary settling velocity m/s

Ws Stability ratio -

ws Dimensionless velocity -

z+, z- Valencies of positive, negative ions -

ΔG Surface free energy J

ΔH1 Heat of adsorption of first monolayer J

ΔHads Heat of adsorption J

ΔHL Heat of liquefaction of extra layers J

N o m e n c l a t u r e s | xvi


Δp Pressure difference Pa

Δp Dynamic pressure N/m2

ΔS System entropy J/K

Δt Time step s

N o m e n c l a t u r e s | xvii

Abbreviations Meaning

ASK Analytical Stability Kinetics

BET Brunauer-Emmett-Teller

BS Backscattered light

ccc Critical Coagulation Concentration

cmc Critical Micelle Concentration

DEM District (Distinct) Element Method

DLS Dynamic Light Scattering

DLVO Derjaguin-Landau-Verwey-Overbeek

DSC Differential Scanning Calorimetry

EDX Energy-dispersive X-ray

ESEM Environmental Scanning Electron Microscope

GC Gas Chromatography

HPLC High Pressure (Performance) Liquid Chromatography

IR Infra Red

ISO International Standard Organization

lft Lower flocculation temperature

NMR Nuclear Magnetic Resonance

PAA Poly (acrylic acid)

PCD Particle Charge Detector

PCS Photon Correlation Spectroscopy

PEG Polyethylene glycol

PEO Poly(ethylene oxide)

PFC Particle Flow Code

PI Polydispersity Index

PLZT Lead Lanthanum Zirconium Titanates

ppm Parts per million

PSS Poly (sodium styrene sulfonate)

N o m e n c l a t u r e s | xviii

Abbreviations Meaning

QELS Quasi Elastic Light Scattering

SAXS Small Angle X-ray Scattering

SDS Sodium dodecyl/lauryl sulphate

SEM Scanning Electron Microscopy

SSA Specific Surface Area

T Transmitted light

TEOS Tetraethyl orthosilicate

TGA Thermogravimetric Analysis

TSI Turbiscan Stability Index

TSK Turbiscan Stability Kinetics

TTIP Titanium tetraisopropoxide

uft Upper flocculation temperature

vdw Van der Waals

Chapter 1

Introduction Processing and handling of solid-liquid-systems are paramount day-to-day activities in

several chemical process industries such as mineral processing, agrochemicals, cosmetics,

paints, printing inks, energy production, crude oil exploration, pharmaceuticals, ceramics,

paper coatings, food, water and wastewater treatment, pulp and paper. Right from the mid-

1990s, about half of the capital expenditure and more than half of the operating cost of

chemical process plants is earmarked for solid-liquid-systems handling processes [1].

Importance of such processes can never be overemphasised and will continue to experience

growth due to: (a) increasing demands on product quality, especially in the pharmaceutical,

biotechnology and food industries, (b) rising demand for environmentally friendly waste

materials, and (c) decreasing quality of raw materials, e.g. in mineral processing.

A number of equipments are available in process industries which are meant solely for

processing and handling solid-liquid-systems. Such equipments are broadly classified into

two as depicted in Fig. 1.1.

The focus in this task is on dispersion, which is that wherein finely divided solid referred to as

disperse (discontinuous) phase is uniformly distributed in a continuous phase (dispersion

medium), a liquid. When dimensions of the disperse phase fall within 1 nm to 1000 nm, the

dispersion is regarded as containing colloidal particles. Dispersions containing disperse phase

larger than 1000 nm are outside the colloidal range and such solids are said to be suspended.

However, terminologies do not connote particle sizes here in this work. Particles, solids,

colloids, colloidal dispersion and colloidal suspension are used indiscriminately.

Colloidal dispersions are often characterized by the properties of the interface separating

disperse and continuous phases. Such interfacial region strongly influences significant

proportion of the dispersion because its structure determines the effect of particle-particle

interactions and the state of the dispersion, that is, whether the dispersion consists of

agglomerated or uniformly dispersed particles during long term storage.

In addition, filterability and sedimentation rate also depend on the state of the dispersion. The

properties of the dispersion may also be time dependent, such that filterability and

sedimentation rate depend on the dispersion history and conditions during preparation [3].

1 I n t r o d u c t i o n | 2


Figure 1.1: Classification of solid-liquid-systems processing equipment [2]

Dispersive and agglomerative forces present in the dispersion are functions of variables such

as pH, electrolyte concentration, surfactant concentration, temperature, agitation, pumping

conditions, etc., all of which complicate the process conditions and result in dispersion

properties that cannot be simply explained by only hydrodynamic terms [2].

Each process in the equipment shown in Fig. 1.1 is usually complex and rarely stands alone,

various pre- and post-treatment stages may be necessary to make a complete process shown in

Fig. 1.2. Typical example is the chemical or physical pre-treatment required to improve

separation in filtration or sedimentation process.

Filtration

Gravity

Vacuum

Pressure

Centrifugal

Discontinuous

Continuous

Discontinuous

Discontinuous

Discontinuous

Continuous

Continuous

Continuous

Strainer or Nutsche Sand-Charcoal Filter

Grids: Sieve Bends Rotary Screen Vibratory Screen

Nutsche Filter Candle and Cartridge

Grids: Sieve Bends Rotary Screen Vibratory Screen

Pressure Nutsche Plate & Frame Filter Tube Candle & Leaf Filter

Belt Press Screw Press

Basket: 3 Column Centrifuge Peeler Centrifuge

Pusher, Vibratory & Tumbler Centrifuge, Helical Conveyor

Discharge Typical Equipment

Gravity

Centrifugal

Hindered

Unhindered

Rotary Wall

Stationary Wall

Stationary Screens & Jigs, Tables, Lamella Settlers, Froth Flotation

Batch & Continuous Sedimentors & Thickeners

Hydrocyclone

Sedimentation

Disc or Tubular Centrifuge, Multi-Chamber Bowl

1 I n t r o d u c t i o n | 3


Figure 1.2: Treatment stages in solid-liquid-systems processing [3 – 4]

Filtration may also require a post-treatment stage like product (wet solids) deliquoring in

order to reduce liquid content of the filter cake. In some other instances, the filter cake may be

the essential product and further purification by washing with clean liquid may be necessary.

Pre-treatment

Chemical Coagulation, Flocculation

(Tests: Sedimentation Buchner Funnel, Capillary-Suction

Time, Cake moisture content)

Physical Crystallisation, Freezing,

Ageing and Filter aids addition (Tests: Bomb Filter Filtrate

Clarity)

Solids Concentration

Assisted Separations Magnetic, Dielectric/Electric,

Acoustic, Vibration

Thickening Gravity Sedimentation, Centrifugal

sedimenters, Hydrocyclones, Delayed Cake Formation with Cross-Flow

(Test: Kynch Sedimentation Analysis)

Clarification Gravity Sedimentation (Tests: Sedimentation,

turbidity of supernatant liquid)

(ii) Media plus precoat (Tests: blinding, cake

discharge, clarity)

(iii) Continuous Batch (Tests: rate of cake-thickness

buildup: filtration rate vs. pressure, cake-moisture

content vs. pressure)

Clarification Granular-bed Cartridge (Tests: pressure drop vs. time, rate vs. time, clarity

vs. time)

Precoat-drum filter (tests: clarity as a function of knife

cut)

Sedimentation

Combination

Filtering

(i) Pressure, vacuum, gravity

filters

Solids Separation

Centrifuges (Tests: rate of cake build up vs.

rpm (g’s); cake consistency; filtrate clarity)

1 . 1 S e d i m e n t a t i o n a s S e p a r a t i o n T e c h n i q u e | 4


Figure 1.2 (contd.): Treatment stages in solid-liquid-systems processing [3 – 4]

Generally, dispersions are needed in many major process industries for one or more of the

following objectives [4]:

separate valuable disperse phase (solid) while discarding the continuous phase

(liquid)

separate the liquid while discarding the solid

discard both disperse and continuous phases, for example, wastewater treatment prior

to discharge so as to prevent water pollution

keep both disperse and continuous phases, in essence, the solid remains stable and

does not settle for a considerable period

Some chemical manufacturing processes require the solid remain stable because unstable

products not only will be expensive to produce but in some cases like in medicine and

pharmaceuticals, it could be harmful to the patient. In Chapter 3, the fundamental principle of

synthesizing stable colloidal dispersions would be discussed.

The discussions in this work would be limited only to sedimentation; in subsequent sections,

sedimentation would be presented as both a separation technique and an instability

mechanism.

1.1 Sedimentation as Separation Technique

Sedimentation can be described simply as the entire process through which the solid separates

out of the fluid. This description is however ambiguous in that the external field of

acceleration such as body forces (gravitational or centrifugal), magnetic field, or electrostatic

Deliquoring, Increased Filter pressure

(Tests: Moisture content vs. pressure)

Mechanical Squeezing (Test: Cake Thickness vs. time)

Post-treatment

Washing, Drying (Tests: Concentration vs. number

of displacements: moisture content vs. time during suction; volume of air during suction)

Washing, Displacement Repulping

(Tests: Concentration vs. number of Displacements)



field responsible for the separation was not specified. Also, the fluid could either be liquid or

gas. So, to be precise, sedimentation is defined as the gravitational separation of the disperse

phase from the continuous phase due to difference in phase densities. The chemical pre-

treatment stage in Fig. 1.2 essentially modifies the particle surface, enhances the interfacial

phenomena and at the end increases sedimentation rate.

As mentioned above, sedimentation rate depends on the state of the dispersion and also relates

to the effect of particle-particle interactions. The interparticle interaction can be explained

from the view point of three types of attractive forces existing between atoms or molecules at

short distances from each other: Dipole-dipole interaction (Keesom), dipole-induced-dipole

interaction (Debye) and the London dispersion force (van der Waals). These forces are

generally referred to as van der Waals attraction or London dispersion forces. They arise from

fluctuations in the electron density distribution and exist for both polar and non-polar

molecules. If the van der Waals attraction is not effectively overcome by electrostatic

repulsions (zeta potentials), steric hindrance or fluid-dynamic shear, the particles collide and

agglomerate resulting in different sedimentation regimes depending on solid volume

concentration, surface properties and relative tendency of particles to agglomerate. The

different sedimentation regimes form the basis of various sedimentation models or

mechanisms commonly available in literatures [5]. A literature review of some of these

sedimentation models is presented in Chapter 2.

Schematic illustration of the different sedimentation regimes is shown in Fig. 1.3. From left to

right, the abscissa depicts sedimentation of particles with little or no interaction (left) to

particles with strong interaction (right) during sedimentation. The ordinate depicts solid

volume concentration varying from low at the top to high at the bottom.

Sedimentation in dilute dispersion or low solids concentration system consists of a

clarification regime. In this regime, particles are far apart and free to settle independently or

as single particles while the fluid flow around individual particles. In Chapter 2, this type of

sedimentation is referred to as “microscopic particle flow-around”. With slight increase in

solid volume concentration, the interparticle contact is increased even though the dispersion is

still regarded as dilute. Depending on the nature of the particles and their relative tendency to

agglomerate, linkage into some sort of floc structure could result. The formed structure could

either be continuous [6 – 7] or a bed of closely-spaced floccules [8]. Particles contained in

such structures sediment at the same rate while fluid flow through the particle-bed. Such is

termed particle-bed/zone settling. In Chapter 2, the sedimentation behaviour is named

“macroscopic particle-bed flow-through”.



Further increase in solid volume concentration makes the floc structure so firm it develops

compressive strength and results in each layer of solids transmitting mechanical support to the

layers above. At a stage, the solids stress or squeeze ensures the structure compacts to higher

solids concentration. This is regarded as compression or compaction regime and it is mostly

evident in highly concentrated dispersions. For dispersions with intermediate concentration, a

regime termed channelling often exists. In this situation, the solid structure can be described

as a porous medium with channelling enhancing its permeability and lowering its resistance to

percolation. As such, channelling is a process that significantly increases sedimentation rate.

Phase-settling is due to channelling or local segregation of floc structure into grossly non-

homogeneous structure [5].

Figure 1.3: Typical sedimentation regimes [6]

Typical application of sedimentation as a separation technique in the industry is depicted in

Fig. 1.4. The process flow route of a continuous sedimenter/clarifier starts with the feed layer

involving settleable solids running through the supernatant layer and then spreading out at

hydrostatic equilibrium [9 – 12]. Non agglomerating fines leave the sedimenter together with

the overflow while agglomerating fines either stay or sediment through the feed layer. In

some instances, agglomerating fines may accumulate as slime layer above feed zone.

Particle flow-around

Particle-bed flow-through

Channelling

Compression

Particulate Agglomerate Interparticle cohesiveness High

Low

Solid

vol

ume

conc

entr

atio

n



Figure 1.4: Ideal-model Sedimenter [Courtesy: Paul Bungartz GmbH & Co. KG;

www.bungartz.de]

Following the feed layer is what is known as the critical zone; the existence of this zone is as

a result of the solids flux into the sedimenter exceeding what can settle through the most

limiting concentration. Excess solids add to the critical zone, increasing in depth until it fills

the sedimenter and then overflows. Critical zone is absent in sedimenters operating at optimal

conditions. It is present only when the pool solid settling capacity is exceeded. Hence, an important design criterion is to ensure there is enough pool area to prevent formation of the

critical zone at any particular maximum solids throughput.

Next to the critical zone is the compaction zone whereby sedimenting solids are subjected to

ever-increasing squeeze or solids stress. If the solids are compressible, they are compacted to

an ever increasing concentration. Though the solids seem to still be sedimenting with respect

to the liquid, most of the weight is sustained hydrodynamically. Since, it is just a fraction of

weight of the solids that is available to produce stress, the concentration and solids stress

reached at the bottom of the compaction zone is not an obvious function of solids loading and

zone depth. Mostly, an ideal model sedimenter consists of two zones: (a) clarification zone,

governed by liquid flux, overflow rate and retention of liquid, and (b) thickening or sludge

zone, governed by downward flux of solids and depth of compaction zone [5].

Clarification zone

Overflow

Compaction zone

Underflow

http://www.bungartz.de/

1 . 2 S e d i m e n t a t i o n a s I n s t a b i l i t y M e c h a n i s m | 8


1.2 Sedimentation as Instability Mechanism

A stable dispersion consists of particles that have slight or no tendency to self agglomerate

(colloidal stability) and also sediment quite slowly (mechanical stability). Such dispersions

show no signs of phase-separation over a period of time. An unstable dispersion on the other

hand, consists of particles that readily agglomerate and sediment rapidly. Fig. 1.5 shows some

extrinsic (environmental) process variables that could cause instability in colloidal dispersions

as they are inherently thermodynamically unstable.

Figure 1.5: Extrinsic process variables responsible for instability mechanisms

Dispersion instability mechanisms are processes that indicate the breakdown of stability in

colloidal dispersions. At such situations, the disperse phase is no longer uniformly dispersed

in the continuous phase during storage. Such breakdown processes are mostly driven by

particle-particle interactions occurring at the colloidal level. Examples of such mechanisms

responsible for dispersion instability are illustrated in Fig. 1.6.

In Chapter 3, a comprehensive discussion on each of these mechanisms would be presented,

but before that a short introduction is given in the next subsections.

Microorganisms

Mechanical agitation

Oxygen, electrolyte concentration, surfactant concentration, pH

Temperature

Pressure

Light Stable Dispersion

Particle-particle interactions

1 . 2 S e d i m e n t a t i o n a s I n s t a b i l i t y M e c h a n i s m | 9


Figure 1.6: Schematic representation of dispersion instability mechanisms

Agglomeration Ostwald Ripening

Stable Dispersion

Flotation

Sedimentation

1 . 3 M o t i v a t i o n a n d O u t l i n e o f T a s k | 10


1.2.1 Agglomeration

This is the process whereby more or less firmly bound primary particles are formed via

collisions between primary particles, primary particles and agglomerates, as well as pre-

existing agglomerates. It includes both destabilization and transport processes. The

destabilization process can occur as either coagulation or flocculation, or in parallel.

1.2.2 Ostwald Ripening

The phenomenon occurs when small particles in polydisperse dispersion systems dissolve and

redeposit on the surfaces of larger particles thereby making them bigger [13]. The driving

force is the difference in solubility between small and large particles. Thermodynamically,

larger particles are more energetically stable to dissolution than smaller particles.

1.2.3 Sedimentation and Flotation

These two processes result from external fields of acceleration, usually, body forces (gravity

or centrifugal). When the body force exceeds Brownian (thermal) motion of particles, a

concentration gradient develops in the dispersion such that, for flotation, large particles

migrate rapidly to the top of the container as a result of their density being less than that of the

continuous phase. On the other hand, for sedimentation, large particles migrate rapidly to the

bottom of the container due to their density being greater than that of the continuous phase.

In general, dispersion stability and the morphology of agglomerates formed depend on [14]:

magnitude and balance of the various interaction forces,

particle size distribution and density difference between disperse and continuous

phases,

conditions and prehistory of the dispersion, such as agitation, needed to determine the

structure of agglomerates formed (chain agglomerates, compact clusters, etc.),

presence of additives in dispersion, for instance, high-molecular weight polymers

may cause bridging or depletion flocculation.

All these listed factors can also influence dispersion flow characteristics or rheology.

1.3 Motivation and Outline of Task

As mentioned above, colloidal dispersions are employed in many process industries for

various objectives. In most industrial applications, colloidal dispersions are formulated to

ensure long-term physical stability during storage and application. Hence, when formulating

colloidal dispersions, it is important to understand how to control certain process variables so



as to achieve desired product. More so, at the early stage of production, a reliable rapid

dispersion stability characterization technique is necessary to predict changes that might occur

during storage. Some stability tests currently in use take quite a long time while in some

cases, the results are inconclusive unless accelerated aging tests (e.g. centrifugation) are

applied and which often may be unreliable.

Therefore, the first main motivation of this task is to simulate the dynamics of dispersion and

agglomeration of ultrafine (nano) particles in wet environment (colloidal dispersion) using

newly derived numerical analytical models that can also accurately predict sedimentation

dynamics of colloidal dispersions. Second motivation comes from answer to the question

“when is the dispersion dispersed or agglomerated?” In order to answer this question several

dispersion stability characterization techniques were compared and the most reliable which

can give current degree of dispersion stability at any particular time is presented.

The newly derived numerical analytical models are applicable in analyzing the dynamics of

particles during sedimentation either independently (particle sedimentation) or as particle-bed

(zone sedimentation) and the result used in dispersion stability assessment. Consequently, the

dispersion stability is characterized according to sedimentation rate determined by numerical

analysis. For instance, the faster the sedimentation rate the poorer the ability of the colloidal

dispersion to maintain long-term physical stability during storage.

Furthermore, the numerical analytical models are derived by resolving several forces acting

around smooth spheres during sedimentation in a stagnant fluid according to Newton’s

second law of motion resulting in analytical differential equations of motion that gives the

possibility of estimating settling velocity and displacement of the smooth spheres as a

function of time.

The analytical differential equations of motion developed are then solved by assuming two

sedimentation models: steady (stationary) and uniformly accelerated. The fluid drag around

single particle (microscopic particle flow-around) is defined according to a three-term model

drag coefficient which varies with prevailing flow regime. Likewise, the pressure drop during

flow through particle-bed (macroscopic particle-bed flow-through) is defined in relation to the

single particle fluid drag plus an additional term for particle-bed porosity function.

This study is also meant to serve as a basic understanding of an aspect of computational

simulation pioneered by Dong et al. (2009) [15], involving discrete or distinct element

methods (DEM), fluid dynamics and their coupling. Experimental results from this work can

be used to validate such computational simulations.



Outline of this task is in parts. Starting with Chapter 1, the introductory part, the importance

of processing and handling of solid-liquid-systems in process industries is extensively

discussed. In addition, sedimentation is described as both a separation technique and a

dispersion instability mechanism. Also in Chapter 1, the different sedimentation regimes are

highlighted because in order to derive new numerical analytical models for the dynamics of

particle sedimentation some assumptions based on sedimentation regimes are required.

This was followed by Part I, titled literature review, consisting of three chapters, two to four.

Chapter 2 provides fundamental theory of dispersion sedimentation and the relevant

assumptions for mathematical developments of numerical analytical models. To appreciate

the synthesis, properties and characterization of colloidal dispersions, Chapter 3 describes the

effects of particle-particle interactions in dispersions. Also included in Chapter 3 are

extensive discussions on dispersion instability mechanisms, definitions of “dilute”,

“concentrated” and “solid” dispersions, and the structure of agglomerates and sediments.

Chapter 4 discusses the principles and applications of sol-gel synthesis and surfactant

adsorption.

Part II includes discussions on experiments and measurements performed. Beginning from

Chapter 5 where detail description of selected model materials, preparation and stabilization

of the colloidal dispersions are presented. Several characterization techniques employed in

order to fully investigate characteristics of dispersion nanoparticles and their sol-gel synthesis

are also highlighted in Chapter 5. In Chapter 6, results and discussions on all experiments and

measurements carried out are presented. Also included in this chapter is the comparison

between analytical and experimental stability assessments. Analytically predicted and

experimentally observed sedimentation profiles are also compared. Conclusions and

recommendations for future studies are presented in Chapter 7.

PART I

Literature Review Historical Perspective

Effective handling of colloidal dispersions regularly encountered in process industries is one of their paramount concerns. Sedimentation process provides a means of separation as well as a stability assessment for the colloidal dispersions.

These possibilities make sedimentation an important topic of research both in academics and chemical process industries. Some process industries where sedimentation is applied and regularly investigated as earlier mentioned are mineral, petroleum, agrochemicals, paints, ceramics, pharmaceuticals, food, pulp and paper, water and wastewater treatments.

Documented research on sedimentation began well as far back as in the 16th century with the Saxons in Germany and Cornishmen in England who developed mineral processing from mere unskilled labour and craftsmanship to a scientifically governed process industry with an international exchange of technology between the two countries. Agricola, then in Saxony published De Re Metallica first in Latin in 1556. The publication which was later translated to German and Italian was the first major contribution to the development and understanding of the mining industry.

Washing and settling tanks by Agricola (1556) [Courtesy: Concha and Bürger, 2002]

With the advent of the Dorr thickener in 1905 and its application in mineral processing, there was a rapid increase in the intensity of experimental and theoretical studies on sedimentation at the beginning of the 21st century. It is evident in all the references that the early researchers knew the practical effect of the difference in specific gravity of the various components of the ore and applied sedimentation operations, now identified in modern day technology as classification, clarification and thickening. In earlier applications, the three operations were not distinguished.

This part of the dissertation is presented chronologically, starting with fundamental theory of particle sedimentation, a theory pioneered by the works of Newton (1687) and Stokes (1844) on flow around single particle. Subsequent chapters like fundamental principles of colloidal dispersion synthesis, describes the principles behind the forces resulting from particle-particle and particle-fluid interactions in colloidal dispersions.

Chapter 2

Fundamental Theory of Dispersion

Sedimentation Basic underlying principle of particle sedimentation is the relative flow of particles and fluids.

In low solid concentration systems, the particles are far apart and move through the fluid as

single entities or in isolation while on the other hand, in high solid concentration systems,

particles move in rather consolidated network through the fluid. All numerical models of

particle sedimentation fundamentally assume particles in a defined zone sediment at the same

rate.

Several studies in the field of particle technology have been devoted to deriving simple

analytical equations relating the settling velocity of particles to their size, shape and

concentration. The pioneering work was in part due to the works of Newton (1687) [16] and

Stokes (1844) [17] which describes spherical particles settling in the fluid in isolation. More

recent experimental and theoretical studies published on sedimentation as a separation

technique include works by Nichols (1908a; 1908b) [18 – 19], Mischler (1912; 1918) [20 –

21], Coe and Clevenger (1916) [22], Lapple and Shepherd (1940) [23], Kynch (1952) [24],

Heywood (1962) [25], Fitch (1962, 1972, 1979, 1983) [26 – 29], Michaels and Bolger (1962)

[30], Shannon et al. (1963, 1964) [31 – 32], Shannon and Tory (1965) [33], Dixon (1977a;

1977b) [34 – 35], Font (1988) [36], Holdich and Butt (1997) [37], Bustos et al. (1999) [38]

and Concha (2001) [39].

Bürger (2000) [40] pioneered the phenomenological theory of sedimentation, a well received

mathematical framework for the description of sedimentation-consolidation processes of

flocculated suspensions based on the theory of mixtures. Concha and several of his co-

workers in the series publication “Settling velocities of particulate systems 1 – 13” defined

the fundamentals of sedimentation-consolidation processes. They also introduced a new

method for particle sedimentation analysis based on fundamental principles of particle

mechanics termed discrete sedimentation or discrete approach to sedimentation [41]. Further

publication in the series was done by Bürger and Concha (2001) [42]. The method starts by

first analyzing the dynamics of an isolated particle as it settles in the fluid followed by

introducing corrections for particle-particle interactions which results in an appreciable

decrease of dispersion settling rate. The method has been quite useful in deriving analytical

2 . 1 F o r c e s o f P a r t i c l e S e d i m e n t a t i o n | 14


equations for various sedimentation systems and establishing sedimentation properties for

certain material in given fluid. A century of research in sedimentation and thickening [43]

gives a concise historical perspective of research contributions to sedimentation and

consolidation made during the 20th century starting from the invention of Dorr thickener in

1905.

Broader review on dispersion hydrodynamics without the emphasis on sedimentation as a

separation technique has been done by Batchelor (1982) [44], Batchelor and Wen (1982) [45],

Tiller and Khatib (1983) [46], Buscall and White (1987) [47], Buscall (1990) [48], Auzerais

et al. (1988; 1990) [49 – 50], Soo (1989) [51], Russel et al. (1989) [52], Kim and Karrila

(1991) [53], Ungarish (1993) [54], Landman and White (1994) [55], Tory (1996) [56],

Holdich and Butt (1997) [57] and Drew et al. (1998) [58].

According to Fitch (1979) [28], most analytical derivations for numerical sedimentation

analysis start by taking more physical effects into consideration and making fewer

assumptions. Almost all the authors cited above have different assumptions and/or analytical

derivations for the underlying physics of particle sedimentation. The assumptions correspond

to the different particle sedimentation models.

Here in this work, physical effects not accounted for in previous analytical derivations for

dispersion sedimentation analysis (specifically, accelerated fluid eddies) were taken into

consideration. There also exist superficial variations as the derivations here include only

different but equivalent way of expressing same physical effects accounted for in previous

derivations.

In order to compare analytical equations derived here with previous ones, some general

differential equations are developed which are then solved by neglecting some terms and/or

making assumptions about relationship between variables. The differential equations include

descriptive specific terms for build-up and compression of porous particle-bed as well as main

contact and adhesive forces at the microscopic level during the sedimentation.

The differential equations are derived by summing the forces acting on single particle or a

particle-bed settling under gravity at a critical time-step and then resolving according to the

Newton’s second law of motion. Some of these essential forces are described below.

2.1 Forces of Particle Sedimentation

List of essential forces that could act on sedimenting particle or particle-bed is typically

inexhaustible. However, some commonly listed in literatures are inertia force FI, drag force

FW, force accounting for accelerated fluid eddies (that is, added (or virtual) mass and the



‘Basset history integral’) FA, field force FG, buoyancy force FB, the stochastic Brownian force

from particle thermal diffusion FS, particle-particle interaction forces FK and the transmitted

stress or ‘particle pressure’ force FP. Resolution of these forces via the Newton’s second law

of motion is termed Langevin equation (Eq. 2.1) in some references [59 – 64].

0I W A G B S K PF F F F F F F F+ + − + + + + = (2.1)

Inertia Force FI

As the particle or particle-bed accelerates from a rest position (t = 0, v = 0) to a stationary

(settling or terminal) velocity (t = ∞, v = vs) in an unbounded fluid, it develops the inertia

force FI.

Drag Force (or Dynamic Pressure) FW

Drag force or dynamic pressure FW is the hydrodynamic force exerted by the fluid on cross

sectional area of the particle surface. It is a linear function of the relative particle-fluid

velocity. It is regarded as a frictional term, corresponding to dissipation of mechanical energy

as the particle accelerates in the fluid. For an incompressible and Newtonian fluid, the drag

force FW depends on the particle size, relative particle-fluid velocity, density and viscosity of

the fluid. Hence, the Reynolds number Re is a very important parameter to characterize FW.

Re is the ratio between inertia force and viscous force. For small Re, a much cited expression

for the drag force around a rigid sphere undergoing uniformly accelerated motion in stagnant

fluid is given by Clift et al. (1978) [60] in Eq. (2.2). The expression includes terms for Stokes

drag (3πηdv), added (virtual) mass and the ‘Basset history integral’ which accounts for past

acceleration history from particle diffusion eddies, respectively.

2332 2

i

tp i

W f fit t

V dtdv dvF dv ddt dt t t

pη ρ pρ η−∞ =

− = + + − ∫ (2.2)

where ƞ and ρf are the fluid viscosity and density respectively, v is the relative particle-fluid

velocity, d and Vp are the diameter and volume of spherical particle respectively while ti and t

stand for the initial and final times respectively.

For relatively high solid concentration (concentrated dispersion systems), particles sediment

relative to other particles in their close proximity (zone or bed), thereby the effect of particle-

particle interactions is enhanced. Consequences of these interparticle interactions make the

spatial distribution of the particles in a specific zone or bed and the elastic energy stored

within the zone to be affected. As particle-bed or zone accumulates in layers from the bottom

of the container, a ‘dynamic pressure gradient’ is propagated as such that it opposes gravity.



This pressure gradient is responsible for the flow of fluid through the particle-bed and should

be accounted for in the force balance or Langevin Equation for the particle-bed. In the case of

stable colloidal dispersions, the pressure gradient is accounted for just by particles osmotic

pressure. However, for flocculated systems where particle networks exist as a result of

particle-particle interactions or at gel-point, the pressure gradient is accounted for by elastic

stress in the particulate network [47]. Obviously, the numerical analytical derivations for real

dispersion systems will require a force balance over cross sectional area element of particle-

bed and not on single particle.

Accelerated Fluid Eddies Force FA

During particle or particle-bed sedimentation, streamlines or fluid eddies develop as depicted

in Fig. 2.1. Depending on flow regime or Re, the developed fluid eddies, attach and accelerate

with the particle(s). FA corresponds to mass of fluid eddies mf,p accelerating with the particle

or particle-bed during sedimentation. In Clift et al. (1978) [60], this force is represented by

two terms: added (virtual) mass and ‘Basset history integral’ term as given in Eq. (2.2). In this

work, new numerical analytical models are derived by considering only the added (virtual)

mass. The justification for this is that the ‘Basset force’ describes only the history (time lag)

during acceleration of ‘virtual’ fluid mass of eddy backlash, hence it is more significant for

creep flow analysis in solids rather than in fluid [65]. Also, according to Hjelmfelt and

Mockros (1966) [66], the ‘Basset history integral’ term is significant only for particle settling

within turbulent eddies.

Figure 2.1: Streamlines around a spherical particle settling in fluid

u

Vp vs

Accelerated ‘virtual’ fluid mass of eddies

, 2f

f p pm Vρ

=



Field Force FG

The most readily available and commonly applied field force in particle sedimentation is the

gravitational force FG.

Buoyancy Force FB

Particle settling under gravity in fluid at rest experiences hydrostatic pressure at several points

on its surface. Sum of this hydrostatic pressure over its entire surface is the so-called static

buoyancy force FB. It acts in opposite direction to gravity and is equivalent to weight of the

fluid displaced [67].

Stochastic Brownian Force FS

The stochastic Brownian force FS arises when particles diffuse relative to each other as a

result of Brownian thermal motion. The magnitude of the effect of the field force FG

(gravitational force) relative to the effect of the stochastic Brownian force FS is determined by

particle Péclet number PéP. PéP is defined as the ratio of directional convective particle

transport to non-directional random particle transport by Brownian motion (Eq. 2.3). FS

decreases with increasing particle size, hence, the larger the particles, the higher the

possibility of sedimentation under gravitational force [44]. PéP less than unity indicate

dominance of FS while on the other-hand PéP greater than unity indicates dominance of FG

[68].

Einstein-Stokes equation in Eq. 2.4 gives an estimate of the particle diffusion coefficient (D0)

or the significance of Brownian diffusion motion on the particle.

( ), ,0

1s f ps B st STpB

g d Vv dPe

D k Tρ ρ− ⋅ ⋅ ⋅⋅

′ = = <⋅

(2.3)

0 3Bk TD

dpη= (2.4)

where vs,B,St is relative velocity of a particle-bed (see Table 2.2), dST is equivalent surface

(Sauter mean) diameter, ρs is particle density, g is due to gravity, T is absolute temperature in

Kelvin, kB is Boltzmann constant, Vp and d are given for single particle volume and diameter.

Later in Chapter 3, it would be seen that when 0 , ,s B St STD v d⋅ , it implies the particle are

quite small (submicron range) and stability of the dispersion could be due to net repulsion

from the: (i) presence of low electrolyte concentration (i.e. extended electrical double layer),

(ii) steric repulsion produced by adsorption of non-ionic surfactants or polymers, (iii) the



combination of both electrical double layer and steric repulsions, known as electrosteric

repulsion. Such dispersions are both colloidally and mechanically stable thereby no phase

separation or sedimentation would be observed on storage even for long periods. On the other

hand, when , . 0s B St STv d D⋅ , the particles can still be colloidally stable as a result of the net

repulsion described above, but mechanically unstable because the particles size range is no

longer within the colloidal range (> 1 µm). Hence, gravitational force will overcome

stochastic Brownian force leading to rapid gravitational sedimentation. Also, the particle size

distribution could either be uniform or widely distributed. If the particles are uniform in size,

the presence of repulsive forces causes them to slide past each other to form hard sediment

known technically as “clay” or “cake” which could be very difficult to redisperse just by

simple shaking. If the particles are widely distributed, the sediment may contain larger

proportions of bigger particles and would still be difficult to redisperse, due to the presence of

repulsive forces.

Particle-Particle Interaction Forces FK

These forces are described extensively in Chapter 3. Here, it is sufficient just to state that they

include van der Waals attractive force and the repulsive forces due to electrical double layer

or steric repulsions. These forces are responsible for the transmission of solids stress among

the particles and sometimes regarded as the particle pressure force described next.

Transmitted stress or ‘Particle pressure’ Force FP

The transmitted stress or ‘particle pressure’ force FP includes the solids stress transmitted

mechanically via particle-particle interactions and hydrodynamic stress transmitted as

particles approach each other. When two particles approach each other, the fluid flow

between them is resisted leading to increased transmitted fluid pressure through the particle-

bed (zone). However, it is a fundamental assumption in the theory of particle sedimentation

that no stress is transmitted within the particles.

Neglecting FS, FK and FP, the next two subsections present expressions and sum of forces

acting around single particle (microscopic particle flow-around) and through particle-bed

(macroscopic particle-bed flow-through) during sedimentation.

2.1.1 Microscopic Particle Flow-around

The term microscopic particle flow-around describes the sedimentation analysis of low solid

volume concentration (or dilute) systems. In this situation, particles sedimentation in the fluid

takes place singly or individually. Fig. 2.2 depicts some forces acting on a smooth sphere

settling as the liquid flow around it. For particle sizes greater than 1 µm, FS is negligible.



Figure 2.2: Microscopic Particle Flow-around

Expressions for the forces in Fig 2.2 are as follows:

Inertia force I s pdvF Vdt

ρ= ⋅ ⋅ (2.5)

Drag force ( )( ) ( )2

Re2

rW W r f p

u tF c u Aρ= ⋅ ⋅ ⋅ (2.6)

Accelerated fluid eddies A f fdvF Vdt

ρ= ⋅ ⋅ (2.7)

Gravitational force G s pF V gρ= ⋅ ⋅ (2.8)

Buoyancy force B f pF V gρ= ⋅ ⋅ (2.9)

The volume ratio of accelerated fluid eddies Vf to particle Vp is given by

ff

p

VV

β = (2.10)

Eq. (2.10) is based on prevailing flow condition (i.e. Re) and shape of the settling particle. It

varies from 0 to 1. For a spherical particle βf = 0.5 and for a cylindrical object settling across

the flow stream βf = 1.

Resolving the forces around the spherical particle in Fig. 2.2 gives

X

Y FG

FI

FB FW FA

u = 0

∑Fi



( )2

12

fp s f p s f W p f

s

dv vV V g c Adt

ρρ β ρ ρ ρ

ρ

⋅ ⋅ + ⋅ ⋅ = ⋅ − ⋅ − ⋅ ⋅ ⋅

(2.11)

If the volume Vp and area Ap of the particle is expressed in terms of the diameter d then

( )( ) ( ) ( )

23 1 04

s f f W

s f f s f f

cdv g v tdt d

ρ ρ ρρ β ρ ρ β ρ

− ⋅− ⋅ + ⋅ ⋅ ⋅ =

+ ⋅ + ⋅ (2.12)

Further simplification of Eq. (2.12) requires an expression to be given for the drag coefficient

cW which relates settling velocity v to diameter d of the spherical particle at the prevailing

flow regime (or Re). Various expressions for cW of spheres are available, ranging from very

low (Stokes regime) to medium (transition regime) and high (turbulent or Newton’s regime)

Re. Some common expressions for cW published in literatures are given in Appendix A.

From the various expressions given for cW by Kaskas (1964) [69] in Table 2.1, it is clear they

are based on prevailing flow regimes and Re.

Theoretically, every flow regime is possible but for colloidal dispersions, small particle sizes

and low density difference between continuous and disperse phases limit sedimentation

analysis only to the laminar (viscous) flow regime.

Table 2.1: Drag coefficient cW expressions and flow regimes according to Kaskas (1964) [69] Flow regimes

Re range 24 4 0.4Re ReW

c = + +

Viscous/Laminar

Re < 0.25 24ReW

c = (Stokes)

Transition

0.25 < Re < 103 cW (trial-and-error)

Square range of inertia 103 < Re < 2 ⋅ 105

cW = 0.44 (Newton)

Turbulent boundary layer 2 ⋅ 105 < Re < 4 ⋅ 105 cW = 0.07 to 0.3

2.1.2 Macroscopic Particle-bed Flow-through

In the case of high solid concentration systems, the force balance is usually performed over a

cross sectional area element of the particle-bed rather than on single or isolated particles. The

term macroscopic particle-bed flow-through is based on the assumption that particles in a

defined zone or same vicinity sediment at same rate and as the particles sediment, fluid flow



through the particle-bed. The forces depicted in Fig. 2.3 are expressed per unit area A of the

particle-bed in Eq. (2.13) to (2.17).

Figure 2.3: Macroscopic Particle-bed Flow-through

Inertia I s sF dvdyA dt

ρ φ= ⋅ ⋅ ⋅ (2.13)

Dynamic pressure ( )( ) ( )2

Re , ,2

rWr f

u tFp Eu u dA ε

ε ρ∆ = = ⋅ ⋅ (2.14)

Accelerated fluid eddies ,A

f f BF dvdyA dt

ρ β= ⋅ ⋅ ⋅ (2.15)

Gravitational G s sF dy gA

ρ φ= ⋅ ⋅ ⋅ (2.16)

Buoyancy B f sF dy gA

ρ φ= ⋅ ⋅ ⋅ (2.17)

Combining Eq. (2.13) to (2.17) gives

( ) ( ) ( ), rs s f f B s s f sdp udv g

dt dyρ φ ρ β ρ φ ρ φ⋅ + ⋅ ⋅ = ⋅ − ⋅ ⋅ − (2.18)

Y

X

dy

u = 0

A

FG

FI

FB

FW

FA

∑Fi

Compression

Tension



As earlier mentioned, the dynamic pressure gradient ( )rdp udy

responsible for the fluid flow

appears in the resolution of forces through the particle-bed in Eq. (2.18).

If the average pore diameter within the particle-bed is given by

( )23 1

STddεε

ε⋅ ⋅

=⋅ −

(2.19)

Fluid drag coefficient within the particle-bed EuB can then be expressed as

( )( )

( )( )

2

2 2

2 43 1

STB

f rf r

dp dy d dp dy dEu

uuε ε

ρ ερ ε ε

⋅ ⋅ ⋅ ⋅ ⋅= =

⋅ ⋅ ⋅ −⋅ ⋅ (2.20)

Rearranging Eq. (2.20) gives the dynamic pressure gradient in terms of fluid drag coefficient

within the particle-bed

( ) ( )22

3 14

f rrB

ST

udp uEu

dy dρ ε

ε⋅ ⋅ ⋅ −

= ⋅⋅ ⋅

(2.21)

where ur is the relative velocity between the particle-bed and fluid while ɛ is the bed porosity.

According to Tomas (2011) [64], the drag coefficient within the particle-bed EuB can also be

expressed in terms of the drag coefficient around single particle cW by the following:

Eq. (2.22) is based on Molerus (1993) [70] numerical approximation deduced from several

experimental observations for Re range 0 < Re < Recrit = 2 ⋅ 105 and expressed as

23 3

3 3

1.53 3

0.13 3

24 1 1 11 0.692Re 20.95 1 0.95 1

4 1 1 0.8911 0.12 0.4ReRe 0.95 1 0.95 1

BEuε ε

ε ε

ε εε ε

− − = ⋅ + ⋅ + ⋅ + − − − − − − ⋅ + ⋅ + + ⋅ − − − −

(2.23)

It shows in the limit ɛ → 1, cW proposed by Kaskas (1964) [69] is obtained

( )1ε B W

= clim Eu→

(2.24)

Macroscopic particle-bed flow-through resistance

= Microscopic particle flow-around resistance +

Characteristic resistance of the particle-bed (2.22)



For laminar flow regime (Stokes range), the last three terms in Eq. (2.23) are neglected to

give

( )2

3 3

3 3

24 1 1 1 241 0.692Re 2 Re0.95 1 0.95 1B B

Eu Bε ε εε ε

− − = ⋅ + ⋅ + ⋅ ≡ ⋅ − − − − (2.25)

Substituting Eq. (2.25) into (2.21) and then (2.18) simplifies to Eq. (2.26) which can be

employed for numerical sedimentation analysis of particle-bed within laminar flow.

( )( )

( )( ) ( )2,

181 0s f B

s f f B s s f ST

Bdv g v tdt d g

ρ ρ η ε

ρ ρ β φ ρ ρ ε

− ⋅ ⋅ − ⋅ ⋅ − ⋅ =

+ ⋅ − ⋅ ⋅ ⋅ (2.26)

Similarly, a differential equation for numerical sedimentation analysis of particle-bed within

all other flow regimes can be obtained by substituting Eq. (2.23) into (2.21) and then (2.18) to

give

( )( ) ( ) ( )

22

,

31 0

4s f f B

s f f B s s f ST

Eudv g v tdt d g

ρ ρ ρρ ρ β φ ρ ρ ε

− ⋅ ⋅ − ⋅ ⋅ − ⋅ =

+ ⋅ ⋅ − ⋅ ⋅ ⋅ (2.27)

Volume ratio of accelerated fluid eddies Vf,B to particle-bed Vp can be expressed in terms of

the angle at which there is maximum compression and tension between particles in contact

within the particle-bed (see Fig. 2.3) which is usually given as α = 45o.

But before that, assuming volume of cylindrical liquid bridge formed between two particles in

contact is given by

3 4, 0.264 sin4f B

V dp α≈ ⋅ ⋅ ⋅ (2.28)

For particle-bed consisting of spherical particles with wetting (contact) angle α and average

coordination number 6k p ε= ≈ (coordination number per particle), the volume ratio of

accelerated fluid eddies to particle-bed is

3 4, , , 4

3 3

36 0.264 sin 1.2 sin2 6 8

f B f B f B

s p

V V dkV d d

β α αφ p

⋅ ⋅ ⋅= = ⋅ ≈ ≈ ⋅

⋅ ⋅ ⋅ (2.29)

In the case of homogeneous particle-bed packing or colloidal dispersions where wetting

(contact) angle between particles in contact is α ≤ 45o, the volume ratio of accelerated fluid

eddies to particle-bed would be βf,B/ϕs ≤ 0.3 which can now be substituted into Eq. (2.27).

2 . 2 S e d i m e n t a t i o n M o d e l s | 24


Eq. (2.12), (2.26) and (2.27) so far developed are quite complex but can be easily solved by

neglecting certain terms and/or making assumptions about relationship between variables

based on two sedimentation models discussed in section 2.2.

2.2 Sedimentation Models

Apart from the most common assumption of spherical particles, two fundamental restrictive

assumptions give rise to the following sedimentation models:

Stationary Sedimentation Model (dv/dt = 0)

If the particle (particle-bed) motion relative to the fluid at rest is assumed negligible, then Eq.

(2.12), (2.26), (2.27) could be solved for both microscopic particle flow-around and

macroscopic particle-bed flow-through respectively by substituting the drag coefficient

expressions appropriate for prevailing flow regime. Solutions to these differential equations

using the stationary sedimentation model assumption are termed stationary (steady) or

terminal settling velocities for the particle and particle-bed at specific flow regimes as listed

in Table 2.2 and Table 2.3. For viscous (laminar) flow regime, it is commonly referred to as

‘Stokes’ terminal settling velocity and for turbulent flow regime, it is known as ‘Newton’

terminal settling velocity.

Uniformly Accelerated Sedimentation Model (dv/dt ≠ 0)

In certain scenarios whereby the particle (particle-bed) is uniformly accelerated from its rest

position (t = 0, v = 0) in the fluid at rest to its terminal settling velocity of fall (t = ∞, v = vs)

under gravity, then several time-dependent relations could be developed from Eq. (2.12),

(2.26) and (2.27) for both microscopic particle flow-around and macroscopic particle-bed

flow-through respectively by substituting appropriate drag coefficients for the flow regime.

Such relations are also presented in Table 2.2 and Table 2.3. Unlike stationary sedimentation

model, forces due to both inertia and accelerated fluid eddies are taken into consideration.

Settling velocity-time relations listed in Table 2.4A and Table 2.4B represent force balance

around the particle (particle-bed) excluding the force due to accelerated fluid eddies for

laminar and turbulent flow regimes respectively. Section 2.3 presents some numerical results

to illustrate consequences of applying one sedimentation model while ignoring the other.

Comparison with Previously Derived Analytical Equations

Obviously, many researchers have previously derived various analytical equations for the

numerical analysis of particle-bed sedimentation based on explicit functional relations

between bed porosity (solid volume concentration) and relative velocity of sedimentation R.

2 . 3 N u m e r i c a l A n a l y s i s o f P a r t i c l e S e d i m e n t a t i o n | 25


Table 2.5 lists some of these derivations which are commonly available in literatures. Fig. 2.4

depicts the graphical comparison of some these derivations for laminar flow regime. The

uniqueness of this work is that while in previous derivations where assumptions was simply

based on stationary sedimentation model, both sedimentation models highlighted above were

treated. Also, new expression was derived for the dynamic pressure gradient of fluid flow

through the particle-bed (Eq. (2.21)) using Molerus (1993) [70] numerical approximations.

In addition, displacement-time relation y (t) during particle free fall in the fluid is derived by

( ) dyv t g tdt

= = ⋅ 0 0

.y t

dy g tdt=∫ ∫ ( )2

2ty t g= ⋅ (2.30)

Figure 2.4: Comparison of derived relative velocity R = vs,B,St/vs,St in laminar flow regime

2.3 Numerical Analysis of Particle Sedimentation

Consequences of applying either stationary or uniformly accelerated sedimentation model can

only be properly understood by numerical results illustrated in the following examples.

Examples of Numerical Results

According to Hjelmfelt and Mockros (1966) [66], assuming certain conditions (such as

particle sedimentation within a turbulent eddy) are met, then Eq. (2.12) will include the

‘Basset history integral’ term to give Eq. (2.31).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

Steinour (1944)Richardson and Zaki (1954)Brauer et al. (1973)Concha and Almendra (1979)Holdich and Butt (1997)Olatunji and Tomas (2011)

Rel

ativ

e ve

loci

ty R

= v

s,B,S

t/vs,S

t

Solid volume concentration ϕs = 1 - ɛ

26

Table 2.2: Overview of Sedimentation Numerical Analytical Relations for Laminar Flow [64] Microprocess variables Laminar Particle Flow-around Laminar Particle-bed Flow-through

Reynolds Number Re, cW, EuB Re < ReSt = 0.25 ... 1,

24ReW

c = ( )24ReB B

Eu B ε= ⋅

Porosity function

ɛ = 1 and B(ɛ)B = 1 ( )

23 3

3 3

1 1 11 0.69220.95 1 0.95 1B

B ε εεε ε

− − = + ⋅ + ⋅ − − − −

Stokes stationary (terminal) settling velocity

( ) 2, 18

s fs St

d gv

ρ ρ

η

− ⋅ ⋅=

⋅

( )( )

2

, , 18s f ST

Bs B St

d gv

B ερ

η

ερ− ⋅ ⋅ ⋅=

⋅ ⋅

Density function ( ) ( )( )

s ff

s f f

Dρ ρ

ρρ ρ β

−=

+ ⋅ ( ) ( )( ),

s fB f

s f f B s

Dρ ρ

ρρ ρ β φ

−=

+ ⋅

Velocity differential equation ( ) ( ) ( ),

1fs St

dv t v tD g

dt vρ

= ⋅ ⋅ −

( ) ( ) ( ), ,

1B fs B St

dv t v tD g

dt vρ

= ⋅ ⋅ −

Velocity-time function ( ) ,

63,

1 exps

s Stv

tv t vt

= ⋅ − −

( ) , ,63,

1 exps B StB

tv t vt

= ⋅ − −

Characteristic settling time

( )( ) 2,

63, 18ss f fs St

vf

dvt

D g

ρ ρ β

ηρ

+ ⋅ ⋅= =

⋅⋅

( )( )

( )

2,, ,

63, 18s f f B s STs B St

B f BB B

dvt

D g

ρ ρ εβ φ

η ερ

+ ⋅ ⋅ ⋅= =

⋅ ⋅⋅

Displacement differential equation

( ),

63,

1 exps

s Stv

ds t tvdt t

= ⋅ − −

( )

, ,63,

1 exps B StB

ds t tvdt t

= ⋅ − −

Displacement-time function ( ) , 63,

63,

1 exps

s

s St vv

ts t v t tt

= ⋅ − ⋅ − − ( ) , , 63,

63,

1 exps B St BB

ts t v t tt

= ⋅ − ⋅ − −

Velocity-displacement function ( ) ,

, 63,

1 exp 1s

s Sts St v

sv s vv t

≈ ⋅ − − − ⋅

( ) , ,, , 63,

1 exp 1s B Sts B St B

sv s vv t

≈ ⋅ − − − ⋅

27

Table 2.3: Overview of Sedimentation Numerical Analytical Relations for Turbulent Flow [64] Microprocess variables Turbulent Particle Flow-around Turbulent Particle-bed Flow-through

Reynolds Number, Re 3 510 Re Re 2 10r f cu d ρ η< = ⋅ ⋅ < = ⋅ ( ) 4,Re Re 10r f c Bu dε ρ η= ⋅ ⋅ < = Porosity function

ɛ = 1 and EuB (ɛ = 1) = cW ≈ 0.44 23 33 3

1.53 3

0.13 3

24 1 1 11 0.692Re 20.95 1 0.95 1

4 1 1 0.8911 0.12 0.4ReRe 0.95 1 0.95 1

BEuε ε

ε ε

ε εε ε

− − = ⋅ + ⋅ + ⋅ + − − − − − − ⋅ + ⋅ + + ⋅ − − − −

Newton stationary (terminal) settling velocity ( )

,

43

s fs N

f W

d gv

cρ ρ

ρ

⋅ − ⋅ ⋅=

⋅ ⋅

( ),

2

,

43

s f STs B N

f B

d gEu

vρ ρ

ρ

ε⋅ − ⋅ ⋅ ⋅=

⋅ ⋅

Velocity differential equation

( ) ( ) ( )2

2,

1s St

dv t v tD gfdt v

ρ= ⋅ ⋅ −

( ) ( ) ( )

2

2, ,

1B fs B N

dv t v tD g

dt vρ

= ⋅ ⋅ −

Velocity-time function ( ) ,

76,

tanhs

s Nv

tv t vt

= ⋅

( ) , ,

76,

tanhs B NB

tv t vt

= ⋅

Characteristic settling time

( )( ),

76,

43s

s fs f fs Nv

s f f Wf

dvt

g cD g

ρ ρρ ρ βρ ρ ρρ

⋅ − ⋅+ ⋅= =

− ⋅ ⋅ ⋅⋅

( )( ),, ,

6,

2

7

43

s f STs f f B ss B NB

s f fB f BEdv

tgD ug

ρ ρρ ρ β

micro-macro aspects of particle sedimentation …...micro-macro aspects of particle sedimentation...

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