Numerical and Experimental Study of Deposition of...

UNIVERSITY OF BERGEN

Numerical and Experimental Study of

Deposition of Polystyrene Particles in

Multiphase Pipe Flows

by

Fasil Ayelegn TASSEW

supervisor

Alex C. HOFFMANN

A thesis submitted in partial fulfilment for the degree of

Master of science in process technology

Faculty of Mathematics and Natural Sciences

Institute for physics and technology

June 2015

http://www.uib.no

http://www.uib.no/matnat

http://www.uib.no/ift

Declaration of Authorship

I, Fasil Ayelegn Tassew, declare that this thesis titled, “Numerical and Experi-

mental Study of Deposition of Polystyrene Particles in Multiphase Pipe Flows”

and the work presented in it are my own. I confirm that:

� This work was done wholly while in candidature for a master’s degree at the

University of Bergen.

� Where any part of this thesis has previously been submitted for a degree or

any other qualification at this University or any other institution, this has

been clearly stated.

� Where I have consulted the published work of others, this is always clearly

attributed.

� Where I have quoted from the work of others, the source is always given.

With the exception of such quotations, this thesis is entirely my own work.

� I have acknowledged all main sources of help.

� Where the thesis is based on work done by myself jointly with others, I have

made clear exactly what was done by others and what I have contributed

myself.

Signed:

Date:

i

UNIVERSITY OF BERGEN

AbstractFaculty of Mathematics and Natural Sciences

Institute for physics and technology

Master of science

by Fasil Ayelegn TASSEW

Solid particle deposition on surfaces occur in various industries. Although parti-

cle deposition may be beneficial in some industries where processes such as spray

coating or filtration is essential, for many other industries particle deposition is

seen as a problem and poses a challenge. Deposited particles often block process

equipment, pipelines etc. In order to alleviate this problem it is important to

understand why and how particles deposit and what factors influence the deposi-

tion behaviour of particles. Years of research have been dedicated to accumulate

knowledge about particle deposition.

In this thesis the deposition of spherical Polystyrene particles with a diameter of

100µm was studied in a flow cell containing an obstruction. The influence of the

flow and the particle properties such as the Reynolds number, the work of cohe-

sion/adhesion, the Adhesion parameter and the Tabor parameter (determines the

particle stiffness) were investigated by numerical simulations using a commercial

computational fluid dynamics software called STAR-CCM+ as well as laboratory

experiments.

The discrete element model coupled with the Lagrangian multiphase model was

used to simulate the effects of variations of the Reynolds number, the Adhesion

parameter and the Tabor parameter and the results were analysed and discussed.

Laboratory experiments were also carried out by varying the Reynolds number val-

ues to validate the results from the simulations. The results from the simulations

were found to be in agreement with the results from the laboratory experiments.

Moreover, a literature review was carried out to validate the findings and they

were found to be in good agreement with the simulation and laboratory experi-

ment observations in this study.

http://www.uib.no

http://www.uib.no/matnat

http://www.uib.no/ift

Acknowledgements

First and foremost I would like to thank my academic supervisor Professor Alex

Christian Hoffmann for his invaluable support and supervision throughout this

study. I would also like to thank Maryam Ghaffari who contributed and provided

suggestions during meetings, laboratory experiments and simulations. Professor

Tanja Barth and Marit Bøe Vaage helped me with providing distilled water and

other apparatuses for the experimental work and I am very thankful. I also would

like to extend my thanks to Professor Pawel Kosinski and associate professor Boris

Balakin for facilitating simulation software training, installation and updates.

I would like to thank my friends at multiphase group who have been very friendly

and supportive: Arman Salimi, Ingeborg Elin Kvamme, Jaime Luis Suarez, Yuchen

Xie and Kari Halland from safety group. I would like to thank Steve R. Gunn

and Sunil Patel for providing the LaTex script that greatly helped me prepare

this document[1]. Special thanks to my family who have been understanding

and supportive: Ayelegn Tassew, Yeshi Teshager, Frehiwot Ayelegn, Tanawork

Ayelegn, Samuel Ayelegn and Kidus Ayelegn I miss you every day. Finally, I would

like to express my gratitude to Sunniva Lode Roscoe who have been supportive

and helpful.

iii

Contents

Declaration of Authorship i

Abstract ii

Acknowledgements iii

List of Figures vii

List of Tables ix

Abbreviations x

Symbols xi

1 Theory 2

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Multiphase flow modelling and particle deposition . . . . . . 4

1.2 Description of contact models . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Particle-Particle interactions . . . . . . . . . . . . . . . . . . 5

1.2.1.1 Hard sphere model . . . . . . . . . . . . . . . . . . 6

1.2.1.2 Soft sphere model . . . . . . . . . . . . . . . . . . 9

1.2.2 Particle-wall interactions . . . . . . . . . . . . . . . . . . . . 10

1.2.3 Particle aggregation and deposition . . . . . . . . . . . . . . 16

1.2.4 Origin of adhesive forces . . . . . . . . . . . . . . . . . . . . 16

1.2.5 Factors affecting particle deposition . . . . . . . . . . . . . . 18

1.2.6 Mechanisms of particle deposition . . . . . . . . . . . . . . . 25

2 Numerical methods 30

2.1 Computational fluid dynamics . . . . . . . . . . . . . . . . . . . . . 30

2.1.1 Explicit and implicit methods . . . . . . . . . . . . . . . . . 34

2.1.2 Numerical stability and convergence . . . . . . . . . . . . . . 35

iv

Contents

2.1.3 Multiphase flow simulation . . . . . . . . . . . . . . . . . . . 36

2.1.3.1 Continuous phase equations . . . . . . . . . . . . . 36

2.1.3.2 Single particle equations . . . . . . . . . . . . . . . 38

2.1.3.3 Dispersed phase equations . . . . . . . . . . . . . . 41

2.1.4 Properties of the dispersed phase . . . . . . . . . . . . . . . 42

2.1.5 Multiphase flow simulation in STAR-CCM+ . . . . . . . . . 44

3 Experimental methods 48

3.1 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.1.1 Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . 49

3.1.2 Numerical experiment set-up . . . . . . . . . . . . . . . . . . 49

3.1.3 Numerical experiment variables . . . . . . . . . . . . . . . . 49

3.2 Laboratory experiments . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2.1 Experimental set-up . . . . . . . . . . . . . . . . . . . . . . 55

3.2.2 Laboratory equipment . . . . . . . . . . . . . . . . . . . . . 56

3.2.3 Polystyrene particles . . . . . . . . . . . . . . . . . . . . . . 56

3.2.4 Re-experiments . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2.5 Ensuring correct fluid velocity . . . . . . . . . . . . . . . . . 57

4 Results and discussion 60

4.1 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.1.1 Results from Re-experiments . . . . . . . . . . . . . . . . . . 60

4.1.1.1 Results from low cohesivity experiments . . . . . . 61

4.1.1.2 Results from high cohesivity experiments . . . . . . 64

4.1.2 Adhesion parameter results . . . . . . . . . . . . . . . . . . 68

4.1.3 Tabor parameter results . . . . . . . . . . . . . . . . . . . . 71


5 Conclusions 79

6 Recommendations 81

6.1 Simulation capability . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.2 Post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82


A Dimensional analysis 84

B Grid points at the boundary: Polynomial approach 87

C Post-Processing particle tracks 89

D Extended hard sphere model equations 91

D.1 Solution manual for the extended hard sphere model particle-wallcollision equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

v

Contents

D.1.1 Case-I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

D.1.2 Case-II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

D.1.3 Case-III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Bibliography 94

vi

List of Figures

1.1 Hard sphere collision of particles. . . . . . . . . . . . . . . . . . . . 6

1.2 Geometry of hard sphere collision. . . . . . . . . . . . . . . . . . . 7

1.3 Soft sphere model analogy. . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Particle-wall collision A. . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Particle-wall collision B. . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6 Comparison of standard and extended hard sphere models. . . . . 12

1.7 Formation of liquid bridge. . . . . . . . . . . . . . . . . . . . . . . . 18

1.8 Particle-wall collision. . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.1 Discrete grid points. . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2 Illustration of volume averaging procedure. . . . . . . . . . . . . . . 37

2.3 Moving control surface enclosing a particle. . . . . . . . . . . . . . . 40

3.1 Flow cell geometry dimensions. . . . . . . . . . . . . . . . . . . . . 51

3.2 Flow cell geometry regions. . . . . . . . . . . . . . . . . . . . . . . 51

3.3 Flow cell geometry after mesh operation. . . . . . . . . . . . . . . 52

3.4 Diagram of the experimental set up. . . . . . . . . . . . . . . . . . . 55

3.5 Experimental set-up. . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.1 The effect of Reynolds number on deposition (low cohesivity). . . . 63

4.2 Velocity profile of the particle tracks. . . . . . . . . . . . . . . . . . 63

4.3 Front view of particle deposition (low cohesion). . . . . . . . . . . . 64

4.4 Bottom view of particle deposition (low cohesion). . . . . . . . . . 65

4.5 The effect of Reynolds number on deposition efficiency (high cohe-sivity). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.6 Front view of particle deposition (high cohesivity). . . . . . . . . . . 67

4.7 Bottom view of particle deposition (high cohesivity). . . . . . . . . 67

4.8 The effect of adhesion parameter on deposition efficiency (Re=333). 68

4.9 The effect of adhesion parameter on deposition efficiency (Re=1733).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.10 The effect of adhesion parameter on deposition efficiency (particleslip velocity) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.11 Front view of particle deposition (adhesion parameter). . . . . . . . 71

4.12 Bottom view of particle deposition (adhesion parameter). . . . . . . 72

4.13 Front view of particle deposition (tabor parameter). . . . . . . . . . 73

4.14 Bottom view of particle deposition (tabor parameter). . . . . . . . . 74

vii

List of Figures

4.15 Flow cell locations of interest. . . . . . . . . . . . . . . . . . . . . 75

4.16 Particle deposition in Section A. . . . . . . . . . . . . . . . . . . . . 76

4.17 Particle deposition in Section B. . . . . . . . . . . . . . . . . . . . . 77

4.18 Particle deposition in Section C. . . . . . . . . . . . . . . . . . . . . 77

B.1 Boundary grid points[5] . . . . . . . . . . . . . . . . . . . . . . . . 87

viii

List of Tables

2.1 Summary of single particle equations . . . . . . . . . . . . . . . . . 41

3.1 Experimental values for Re simulations . . . . . . . . . . . . . . . . 50

3.2 Experimental values for Adhesion parameter simulations . . . . . . 50

3.3 Experimental values for Tabor parameter simulations . . . . . . . . 50

3.4 Description of physics model . . . . . . . . . . . . . . . . . . . . . . 52

3.5 particle-particle interaction model . . . . . . . . . . . . . . . . . . . 53

3.6 particle injector settings . . . . . . . . . . . . . . . . . . . . . . . . 53

3.7 Velocity and Volumetric flow rate values for Re simulations . . . . . 58

4.1 Threshold velocity values . . . . . . . . . . . . . . . . . . . . . . . 61

4.2 High cohesivity threshold values and deposition efficiency . . . . . 65

4.3 The effect of Adhesion parameter on deposition efficiency . . . . . 68

4.4 Tabor parameter results . . . . . . . . . . . . . . . . . . . . . . . . 73

ix

Abbreviations

CFD Computational Fluid Dynamics

DE Deposition Efficiency

DEM Discrete Element Method

DMT Derjaguin Muller Toporov

DPM Discrete Parcel Method

JKR Johnson Kandall Roberts

LES Large Eddy Simulation

MD Maugis Dugdale

MYD Muller Yushchenko Derjaguin

PDE Partial Differential Equation

RANS Reynolds Averaged Navier Stokes

RTT Reynolds Transport Theorem

x

Symbols

Symbol Name Unit

a/aH/aDMT/aJKR Contact radius m

a1 Liquid curvature radius m

a2 Liquid bridge radius m

Ap Projected area m2

Ad Adhesion parameter Dimensionless

Ainterface Area of interface m2

Avdw Hamaker constant J

Bdisp Dispersion energy coefficient Dimensionless

cd Specific heat of dispersed phase J/kg −K

CD Drag coefficient Dimensionless

Cfs Static friction coefficient Dimensionless

Cvm Virtual mass coefficient Dimensionless

d Surface separation distance m

D Particle diameter m

D0 Interfacial contact separation m

Dch Characteristic channel diameter m

e Restitution coefficient Dimensionless

Eeq Equivalent Young’s modulus Pa

f Body force N

fc Collision frequency s−1

fu User defined body force per unit volume N m−3

Fadh Force of adhesion N

xi

Symbols

Fcontact Contact force between spheres N

Fcoh Force of cohesion N

Fd Particle drag force N

Fg Gravitational force N

FLiquid bridge Liquid bridge force N

Fn Normal component of contact force N

Fp Particle pressure gradient force N

Fs Particle surface force N

Ft Tangential component of contact force N

Fvdw Van Der Waals force of attraction N

Fvm Particle virtual mass force N

Fu User defined body force N

g Gravitational constant m s−2

G Total Gibb’s free energy KJ mol−1

Gb Gibb’s free energy with bulk properties KJ mol−1

Geq Equivalent shear modulus Pa

Gs Gibb’s free surface energy KJ mol−1

I Moment of inertia Ns

J Impulse force Ns

k′c Thermal conductivity of the continuous phase W/mK

Kn/Kt Normal/tangential spring stiffness coefficients N m−1

l Characteristic dimension dispersed/continuous phase m

lsd Length of interparticle spacing m

L Characteristic dimension of physical system m

m/M/mp Mass of particle kg

Meq Equivalent mass of particle kg

n Number density of particles m−3

N1/N2 Number of molecules per unit volume m−3

P Pressure Pa

Q Volumetric flow rate m3 s−1

r Distance between isolated molecules m

xii

Symbols

rm Meniscus radius m

R Particle radius m

Re Reynolds number Dimensionless

Req Equivalent radius of particle m

RCmin Minimum contact radius m

Rmin Minimum sphere radius m

Stv Stokes number for particle velocity Dimensionless

Stmass Stokes number for particle mass Dimensionless

StT Stokes number for particle energy Dimensionless

t Time s

T Temperature K

U(r) Dispersion energy J

vr Relative velocity between particles m s−1

V Particle velocity m s−1

Vave Characteristic dimension of averaging volume m3

Vc Volume of continuous phase m3

Vd Volume of dispersed phase m3

Vp Volume of a particle m3

VRayleigh Velocity of Rayleigh wave m s−1

Vs Particle slip velocity m s−1

Vthreshold Particle threshold velocity m s−1

Wadh Work of adhesion J m−2

Wcoh Work of cohesion J m−2

5PstaticStatic pressure gradient Pa m−1

α Thermal diffusivity m2 s−1

αc Continuous phase volume fraction Dimensionless

αd Dispersed phase volume fraction Dimensionless

γ Surface energy J m−2

δn/δt Normal/tangential deformation distances m

ε Interatomic spacing m

η Damping coefficient Dimensionless

xiii

Symbols

ηn/ηt Normal/tangential damping coefficients Dimensionless

θ Contact angle Degree

µc Fluid dynamic viscosity Pa s−1

µtabor Tabor parameter Dimensionless

µctabor Modified Tabor parameter Dimensionless

ν Poisson’s ratio Dimensionless

ρf Fluid density kg m−3

ρp Particle density kg m−3

σ Capillary force N m−1

τ1 DEM time step s

τc Time between collisions s

τF Time characteristic of flow field s

τM Characteristic mass transfer time s

τT Thermal response time s

τV Momentum response time s

υ Fluid velocity m s−1

υ0 Fluid inlet velocity m s−1

ω Angular momentum s−1

xiv

Dedicated toAyelegn Tassew, Yeshi Teshager, Frehiwot Ayelegn,

Tanawork Ayelegn, Samuel Ayelegn and Kidus Ayelegn

xv

Chapter 1

Theory

This chapter lays out the theoretical background for particle deposition, reviews

contact models that describe contact between solid bodies, discusses the origins of

adhesive forces as well as factors that affect particle deposition.

1.1 Background

Deposition of solid particles is a common problem in multiphase flow transport.

Various industries that require pipe transportation of multiphase systems or sys-

tems that involve solid particle contact with a surface often had to deal with

deposition of solid matter on surfaces at a big maintenance costs.

In oil and gas industries deposition of wax on inner subsea pipeline surfaces cause

gradual decrease in flow rate and ultimately lead to complete blockage of the pipe

unless the deposit is removed periodically[6].

Adhesion of powder particles on to solid surfaces is a challenge in pharmaceutical

and food industries[7][8]. The problem of particle adhesion/deposition on to pipe

surfaces also extends its effect on human health. Deposition of particulate matter

in human lung[9], trachea[10] and blood vessels[11] have been known to cause

significant health risks.

Understanding how particles adhere on to solid surfaces and what factors influence

their deposition behaviour as well as understanding how these factors affect the

extent of deposition is important in preventing or reducing solid matter build-up.

2

Chapter 1. Theory

1.1.1 Objectives

At the beginning of August 2014 several thesis objectives were being considered.

After consulting with my academic supervisor and reviewing available literature

the following topics were chosen as the thesis objectives:

• To understand how the deposition of Polystyrene particles is affected by

the variation in the Reynolds number within the laminar flow regime in an

obstructed flow cell.

• To assess the influence of cohesivity of the particle (Wcoh and Wadh) on the

deposition of Polystyrene particles.

• To understand how the Adhesion parameter affects the deposition of Polystyrene

particles in an obstructed flow cell.

• To understand how the Tabor parameter, Young’s modulus and Poisson’s

ratio affect the deposition of Polystyrene particles.

• To establish the capability of the DEM-Lagrangian multiphase model in

simulating Polystyrene particle deposition in an obstructed flow cell.

• To compare and contrast the results from the numerical simulations with

the results from the laboratory experiments.

3

Chapter 1. Theory

1.1.2 Multiphase flow modelling and particle deposition

Development of powerful computational fluid dynamics(CFD) tools that simulate

particle (dispersed phase) and fluid (continuous phase) flows as well as particle-

fluid, particle-particle and particle-boundary interactions provided an attractive

approach to study particle deposition. There are various commercial CFD soft-

ware that are used to simulate particle-particle and particle-fluid interactions in

multiphase flows.

In this thesis STAR-CCM+, one of these commercial CFD software, was used

to simulate particle deposition in multiphase pipe flows. STAR-CCM+ software

provide the discrete element model (DEM) and the Lagrangian multiphase models

which were extensively used to simulate the deposition of the Polystyrene particles.

Description of both DEM and Lagrangian multiphase models are given in Chapter

two Section 2.1.5.

1.2 Description of contact models

The contact of solid bodies have been studied for several decades and various

theories have emerged over the years. In his pioneering work Hertz described the

contact between spherical elastic solid bodies. His work showed that the radius

of the circle of contact between the solid bodies depends on the pressure (P),

the spherical radius(R) and the elastic properties of the bodies involved in the

contact(Eeq)[12].

a3H =PR

Eeq

(1.1)

Eeq =4

3

(1− υ21E1

+1− υ22E2

)(1.2)

Where, υ1 and υ2 are the Poisson’s ratios for the solid bodies and E1 and E2 are the

Young’s moduli. The Hertz model works well for contacts where adhesion doesn’t

play a significant roll. However, when the adhesive force become significant, such

as in cases of contacts that involve small, “soft” particles with high adhesive/cohe-

sive nature, the Hertzian model becomes less effective. Several other models have

4

Chapter 1. Theory

been proposed to accommodate the adhesivity of solid bodies in contact mechan-

ics models. One such proposal is the DMT model proposed by Derjaguin, Muller

and Toporove[13]. In their model they added an extra “load” term into the Hertz

equation for the contact radius to account for the adhesive effect.

a3DMT =(P + 2π∆γR)R

Eeq

(1.3)

Where ∆γ is the work of adhesion and the extra “load” term is PDMT = 2π∆γR

Johnson, Kendall and Roberts in their JKR theory studied the contact between

a rigid solid surface and an elastic half-space (a half space is a segment of an n-

dimensional space that remains when a segment on the side of an (n-1) dimensional

hyperplane is removed[14]) and came up with a model for the contact radius (aJKR)

a3JKR =R

Eeq

[P + 3π∆γ +

(6π∆γRP + (3π∆γR)2

) 12

](1.4)

Others such as the MD(Maugis and Dugdale) and MYD (Muller, Yushchenko

and Derjaguin) models are also proposed. Each with their own advantages and

disadvantages. Although, they are not discussed in this thesis, an interested reader

may refer to the following literature[12][15].

1.2.1 Particle-Particle interactions

In dilute particle flows particle-particle interactions are rare. As a result, the ef-

fect of the interactions on the flow behaviour can safely be ignored. However, if

the number density (the number of particles per unit volume) of the particles is

increased enough, the flow can be considered as a dense flow and the frequency

of particle-particle interactions and subsequent loss of kinetic energy can not be

ignored. In such cases particle-particle interactions have to be studied and ac-

counted for. Several models have been proposed to model particle-particle as

well as particle-wall collisions for different particle inertias and continuous phase

flow conditions[16]. For example, in 1956 Saffman and Turner[17] proposed a

model for the collision of droplets of equal sizes in turbulent flows. Later in 1975

Abrahamson[18] proposed a model for the rate of particle collisions in a high

intensity turbulent flows. Other examples of particle collision models such as

5

Chapter 1. Theory

Veeramani et al.[19], who studied collision between two non-Brownian particles in

multiphase flows and suggested a model based on stereomechanical impact model,

were also attempted. However, the two most common models that are used to

study particle-particle interactions are the hard sphere and soft sphere models. A

review of each model is presented below.

1.2.1.1 Hard sphere model

The hard sphere model is based on the assumptions that particles are rigid spheres

with no deformation during collision or contact, the Coulomb’s friction law dictates

the friction on sliding particles and once a particle stops sliding no further sliding

occurs.

Figure 1.1: Hard sphere collision of particles. Where ω1 and ω2 are angularmomentums and V1 and V2 are particle velocities of particles 1 and 2 respec-

tively

The hard sphere model is based on integrated form of the Newtonian equations

of motion for the colliding particles. The collision is not resolved in time, which

means the particles translational and rotational velocities after the collision are

determined by the integration of the conservation law. However, this restricts the

hard sphere model to be applicable only for a binary collision at a time. The

equations that govern the hard sphere collision of particles are based on impulsive

forces[3].

6

Chapter 1. Theory

m1 = (V1 −V(0)1 ) = J (1.5)

m2 = (V2 −V(0)2 ) = −J (1.6)

I1 = (ω1 − ω(0)1ω1 − ω(0)1ω1 − ω(0)1 ) = r1n× J (1.7)

I2 = (ω2 − ω(0)2ω2 − ω(0)2ω2 − ω(0)2 ) = r2n× J (1.8)

Where, n is the unit normal vector, J is the impulse force on the particles, I is the

moment of inertia, r is the particle radius, m is the particle mass and superscript

(0) indicates values before the collision.

Extended hard sphere model

Kosinski and Hoffmann[2] provided a model to account for the cohesivity of the

particles during particle-particle collision by directly incorporating a cohesive im-

pulse into the impulse based hard sphere collision model.

n

J

Jt

Jn

t

Plane of collision

Particle 1

Particle 2

Figure 1.2: Geometry of hard sphere collision[2]

Since J, the impulse that is acting on the particles during the collision, represents

particle repulsion due to elastic deformation then incorporating a factor for the

7

Chapter 1. Theory

particle cohesivity would account for particle attraction during the collision. In the

standard hard sphere model, the normal component of the impulse(Jns) is given

as:

Jns = − m1m2

m1 +m2

(1 + e)(n.G(0)) (1.9)

Jt = fJns (1.10)

Where, Jt, f, e, m1, m2, n and G(0) are the tangential impulse component, friction

coefficient, restitution coefficient, mass of particle 1, mass of particle 2, unit vector

from particle 1 to particle 2 and relative velocity before the collision. Kosinski

and Hoffman argued that the above equation should include a cohesive impulse

component. The new normal component of the impulse(Jn) would then be:

Jn = Jns + Jn,c (1.11)

Jt = f(Jns − Jn,c) (1.12)

Where, Jn,c is the cohesive impulse that accounts for particle attraction due to the

cohesivity of the particle during the collision. Consequently, equations were de-

rived for the particles post-collision velocities. For particles that slide throughout

the collision the requirement and post collision velocities are:

Requirement ⇒ n.G(0) < −m1 +m2

m1m2

Jn,c1 + e

+2 | G(0)

ct |7f(1 + e)

(1.13)

V1 = V(0)1 +

Jn,cm1

(n− ft)− m2

m1 +m2

(1 + e)n.G(0)(n + ft) (1.14)

V2 = V(0)2 +

Jn,cm2

(n− ft) +m1

m1 +m2

(1 + e)n.G(0)(n + ft) (1.15)

ω1ω1ω1 = ω(0)1ω(0)1ω(0)1 +

5

2r1(n× t)f

[−Jn,cm1

− m2

m1 +m2

(1 + e)n.G(0)

](1.16)

8

Chapter 1. Theory

ω2ω2ω2 = ω(0)2ω(0)2ω(0)2 +

5

2r2(n× t)f

[−Jn,cm2

− m1

m1 +m2

(1 + e)n.G(0)

](1.17)

Similarly equations for particles that stop sliding during the collision were also

formulated. The requirement for this condition is the same as the requirement in

the standard collision model. The equations for post-collision velocities are given

below:

Requirement ⇒ Jt = − 2m1m2

7(m1 +m2)| G(0)

ct | (1.18)

V1 = V(0)1 +

[(Jn,c

m1− m2

m1+m2(1 + e)n.G(0)

)n− 2m2

7(m1+m2)| G(0)

ct | .t]

(1.19)

V2 = V(0)2 +

[(Jn,c

m2− m1

m1+m2(1 + e)n.G(0)

)n− 2m1

7(m1+m2)| G(0)

ct | .t]

(1.20)

ω1ω1ω1 = ω(0)1ω(0)1ω(0)1 −

5

7r1(n + t) | G(0)

ct |m2

m1 +m2

(1.21)

ω2ω2ω2 = ω(0)2ω(0)2ω(0)2 −

5

7r2(n + t) | G(0)

ct |m1

m1 +m2

(1.22)

1.2.1.2 Soft sphere model

Unlike the hard sphere model, the soft sphere model is based on differential form

of the Newtonian equation of motion of the particles. The collisions are resolved in

time, hence, the collision of more than two particles at a time can be studied [20]

The soft sphere model takes particle deformation into consideration. When par-

ticles collide they exert energy on each other, this makes them to be momentarily

deformed and lose energy. Due to the loss of energy, the coefficient of restitution

is less than one. The coefficient of restitution is the ratio of pre-collisional and

post-collisional velocities. The energy loss can be modelled using a dash-pot or

viscous damper. The equation for a damped oscillator is given as follows:

md2x

dt2+ η

dx

dt+ kx = 0 (1.23)

9

Chapter 1. Theory

Where the first term is mass times acceleration, the second term is the damping

term and the third term is the spring term. The constants η and k are the damping

coefficient and the stiffness of spring, respectively.

x

k

η

m

x

Figure 1.3: Soft sphere model analogy using spring-damper system[3]

1.2.2 Particle-wall interactions

There are two types of particle-wall interactions. The first one is hydrodynamic

interactions. These interaction arise due to the close proximity of particles to a

wall such as a pipe surface. An example of this is the Saffman lift force where

velocity gradient prevents the particles from contacting the wall. The second

category is the fluid force interactions with a particle. When a particle approaches

a wall, pressure increases and prevents the particle from contacting the wall. If

the particle has a large inertial force, contact with the wall takes place regardless

of the hydrodynamic and fluid forces. A particle that has collided with a wall has

two likely outcomes, either it bounces back off the wall with a loss of some of its

kinetic energy or it remains stuck on the wall if the adhesive/cohesive forces are

dominant and can overcome the particles inertial force[3].

10

Chapter 1. Theory

Momentum and energy exchange during particle-wall inter-

action

The hard sphere model describes particle-wall collision using the impulse equa-

tions, however, the impulse equations alone do not establish the relationship be-

tween pre-collision and post-collision velocities of the particle. To achieve this,

the coefficient of restitution and the coefficient of friction must be used. There

are different ways to define the coefficient of restitution. Crowe et.al [3] provided

some of these definitions.

e =

∣∣∣∣V(2)

V(0)

∣∣∣∣ (1.24)

e =V

(2)y

V(0)y

(1.25)

ex =V

(2)x

V(0)x

= ey =V

(2)y

V(0)y

(1.26)

e =J(2)y

J(1)y

(1.27)

V(0)=(VX(0)),VY

(0),VZ(0))

V(2)=(VX(2)),VY

(2),VZ(2))

ω(0)

ω(2)

Figure 1.4: Particle-wallcollision A.[3]

Vy(0) V(0)

Vx(0)

Vx(0)

V(2) Vy(2)

wall

Figure 1.5: Particle-wallcollision B[3]

If the coefficient of restitution and the coefficient of friction are known for a spher-

ical particle colliding with a flat wall, then the impulse equations can be used to

solve the post-collision translational and angular velocities.

11

Chapter 1. Theory

During particle-wall collision a particle passes through a compression and recovery

periods as well as brief slide across the wall surface. The post-collision velocities

are dictated by how long the particle slides and at what period the particle stopped

sliding. Generally, there are three cases, the first one is when the particle stops

sliding in the compression period, the second one is when the particle stops sliding

in the recovery period and the last case is when the particle slides in both the

compression and recovery periods.

Kosinski and Hoffmann[21][4] provided an extension of the hard sphere model to

account for the adhesive force during particle-wall collisions. They introduced an

impulse term in to the impulse equations of the standard hard sphere model. Both

the extended and standard models consider the three cases.

Y

X Jys J

Jx

Jys J

Jx Jy,t

Jy

Standard hard sphere model


Figure 1.6: Comparison of standard and extended hard sphere models[4]

In the first case the particle stops sliding during the compression period, in the

second case the particle stops sliding in the recovery period and the last case

considers a particle that continues to slide throughout the collision period. All the

impulse terms in the standard model that are acting on the particle as a “push-off”

force were given an extra impulse term in the extended model to account for the

force acting on the wall surface due to adhesive force. The extra impulse force acts

in the Y-direction (i.e Jt = (0, Jy,t, 0)). In the case where particle stops sliding in

the compression period, the modified equations are:

J (s)x = m(V (s)

x − V (0)x ) (1.28)

12

Chapter 1. Theory

J (s)ys + J

(s)y,t = m(V (s)

y − V (0)y ) (1.29)

J (s)z = m(V (s)

z − V (s)z ) (1.30)

Where, the superscripts (s) and (0) indicate the sliding and the pre-collision peri-

ods. For a particle that stops sliding in the recovery period, the equations are:

J (r)x = m(V (1)

x − V (s)x ) (1.31)

J (r)ys + J

(r)y,t = m(V (1)

y − V (s)y ) (1.32)

J (r)z = m(V (1)

z − V (s)z ) (1.33)

Where, the superscripts (r) and (1) indicate the recovery and the compression

periods. For a particle that continues to slide throughout the whole collision

period, the equations are:

J (2)x = m(V (2)

x − V (1)x ) (1.34)

J (2)ys + J

(2)y,t = m(V (2)

y − V (1)y ) (1.35)

J (2)z = m(V (2)

z − V (1)z ) (1.36)

Since the standard model doesn’t contain the impulse in the Y-direction for the

equations of the particle rotation, both the extended and the standard models

have the same equations for the particle rotation. They are given as:

− aJ (s)z = I(ω(s)

x − ω(0)x ) (1.37)

13

Chapter 1. Theory

0 = I(ω(s)y − ω(0)

y ) (1.38)

aJ (s)x = I(ω(s)

z − ω(0)z ) (1.39)

− aJ (r)z = I(ω(1)

x − ω(s)x ) (1.40)

0 = I(ω(1)y − ω(s)

y ) (1.41)

aJ (r)x = I(ω(1)

z − ω(s)z ) (1.42)

− aJ (2)z = I(ω(2)

x − ω(1)x ) (1.43)

0 = I(ω(2)y − ω(1)

y ) (1.44)

aJ (2)x = I(ω(2)

z − ω(1)z ) (1.45)

Where, I(∆ω = r× J), r=(0,-a,0) and a is the particle radius. In addition the

surface velocities of the particle at the point of contact are given as:

(V (s)x + aω(s)

z )i + (V (s)z − aω(s)

x )k = 0 (1.46)

(V (1)x + aω(1)

z )i + (V (1)y )j + (V (1)

z − aω(1)x )k = 0 (1.47)

(V (2)x + aω(2)

z )i + (V (2)z − aω(2)

x )k = 0 (1.48)

The relationships between the impulses are given as:

14

Chapter 1. Theory

J (s)x = −εxf(J (s)

ys − J(s)y,t ) (1.49)

J (s)z = −εzf(J (s)

ys − J(s)y,t ) (1.50)

ε2x + ε2z = 1 (1.51)

J (2)y = (J (2)

ys + J(2)y,t ) = em(J (s)

ys + J (r)ys + (J

(2)y,t ) (1.52)

Where, εx and εz are direction cosines for the sliding particle, J(r)ys is the impulse

during the remainder of the recovery period and em is the equivalent restitution

coefficient in the extended hard sphere model. Kosinski and Hoffmann[4] also

derived estimations of the extra impulse terms, namely the impulse during com-

pression period(J(1)y,t) and the impulse during the recovery period (J

(2)y,t). In addition

they set out a particle deposition requirement:

J(1)y,t = −m

(√2Fy,t

m(D −D1) + (V (0)

y )2)

(1.53)

J(2)y,t = m

(√2Fy,t

m(D1 −Dc) + V 2

2 − V2)

(1.54)

Fy,t = − aAvdw

6DcD1

(1.55)

V1 =

√2Fy,t

m(Dc −D1) + (V (0)

y )2 (1.56)

V2 = −V1em (1.57)

Where, Avdw is the Hamaker constant, D is the surface separation distance, D1 is

surface separation distance just before the collision where the attractive interaction

is still negligible, Dc is the surface separation distance at the end of the compression

15

Chapter 1. Theory

period, V1 is the velocity of the particle at impact and V2 is the velocity of the

particle after the impact. The criteria for the deposition of the particle is given

as:

2Fy,t(D1 −Dc)

m+ V 2

2 6 0 (1.58)

A solution manual for each of the above cases is provided in Appendix D.

1.2.3 Particle aggregation and deposition

Particles dispersed in a continuous phase often exhibit aggregation and deposition

behaviours. Aggregation of particles occur when individual particles associate

to form clusters whereas deposition occur when particles are transported to a

surface where they become attached. Aggregation and deposition processes have

several similarities and each can be considered as an extreme form of the other

(eg. deposition can be considered as heteroaggregation where particles of different

type form aggregates). Both processes involve the transport and attachment steps.

Various mechanisms of particle transport, hydrodynamic and electrical forces play

part in determining the fate of the aggregated clusters and deposits [22].

Since the topic of this thesis is about particle deposition, detailed discussions were

limited to the mechanism of particle-wall interactions and particle deposition only.

Aspects related to particle aggregation and particle-particle interactions were only

discussed briefly. (A previous masters student wrote a masters thesis on the subject

of cohesive particle agglomeration[23]).

1.2.4 Origin of adhesive forces

Adhesion between particles and surfaces is thought to be a resultant force from sev-

eral contributing forces such as the electrostatic, capillary and the van der Waals

forces. Most of these contributing forces are active in short distance. The extent

of contribution of each force to the overall adhesion force depends on the environ-

mental and the experimental conditions as well as physicochemical properties of

the involved particles and surfaces[7].

16

Chapter 1. Theory

Salazar-Banda et al.[8] showed that the adherence of dry and inert particles onto

solid surfaces is dominated by the van der Waals force. However, the adhesion of

wet particles is dominated by the capillary force due to liquid bridge[3].

van der Waals force

The van der Waals force is a short range attractive force between macroscopic

surfaces due to intermolecular interactions. The pairwise summation of all inter-

molecular interactions gives the van der Waals force [22].

Hamaker(1937)[24] studied the van der Waals interactions between two spherical

particles as a function of separation distance between the particles and the diam-

eter of the particles and It was found out that van der Waals force depends on the

geometry and the molecular property of the interacting bodies.

In general, the van der Waals interaction for any given two bodies such as two

spheres, a sphere and a flat plate or two flat plates can be expressed as a product

of Avdw, the Hamaker constant and H, a factor that is dependent on the geometry

and dimension of the interacting bodies. For example, for two identical spheri-

cal particles H is, -D/24d , where D is the diameter of the sphere and d is the

separation distance between the spheres.

Capillary force

When interacting bodies are in a liquid medium, the capillary force comes into

significance. The role of the capillary force in particle adhesion/aggregation is that

when the contacting bodies are approaching each other, there will be a build-up

of pressure that acts as a buffer and prevents the contact. This can be countered

if the contacting bodies have a high enough momentum and become close enough

with each other for the capillary force to be active.

Fliquid bridge = πa22σ

(1

a1+

1

a2

)+ 2πa2σcosθ (1.59)

The above equation is used to quantify the force due to the liquid bridge and the

capillary force and it is called the Younge-Laplace equation[3]. In the equation,

17

Chapter 1. Theory

a1 is the curvature radius of the bridge, a2 is the radius of the liquid bridge, σ is

the capillary force and θ is the contact angle.

a2

d

a1

Figure 1.7: Formation of liquid bridge

Electrostatic force

Since the van der Waals interaction between two solid surfaces in a liquid medium

is always attractive[25], one would expect the particles to immediately aggregate

or deposit. However, this is not always the case and the reason is because of

the presence of the electrostatic repulsive forces. There are various mechanisms

in which the electrostatic charges form on the surfaces of interacting bodies in

liquid medium but apart from DVLO theory, which will be discussed in the next

section, it is beyond the scope of this thesis to go further into electrostatic charge

formation. However, interesting read can be found on this subject in books such as

“Intermolecular and surface forces” by Jacob N. Israelachvili and “An introduction

to interfaces & colloids: a bridge to nanoscience” by John C. Berg.

1.2.5 Factors affecting particle deposition

There are several factors that affect particle deposition on a surface. The rate

and the magnitude of deposition of particles is heavily influenced by the electrical

properties of both the particles and the continuous phase, the flow properties,

18

Chapter 1. Theory

the surface properties, the particle properties and the physical properties. Brief

overviews of the above properties are given below.

Electrical properties

The electrical interaction between particles influence the particle stability, aggre-

gation and deposition [22]. Most particles in aqueous solutions are electrically

charged but the distribution of ions around the particles are in such a way that

give rise to an electrical double layer. This means the surface charge present on

charged particles are balanced by oppositely charged counter ions. There are dif-

ferent models that describe the electrostatic interactions between charged particles

and how that affect the deposition and aggregation behaviour of particles.

One such theory is the classical DLVO theory. The DLVO theory explains the

stability of dispersed particles as a product of a balance between the attractive

van der Waals force and the repulsive electrostatic Coulomb force interactions.

Moreover, It has been shown that the electrical double layer interactions play a

significant role in both the kinetics of particle deposition and the structure of

deposited particles[26] [27].

The van der Waals force between two particles is always an attractive force and

its magnitude depends on the size of the particles involved and their shape.

Hamaker(1937) and De Boer (1936) suggested that the Van Der Waals interaction

between two macroscopic objects is the summation of energies acting between all

molecules in one macroscopic object with those in the other object.[28] For two

identical spherical particles of radius R that are separated by a distance of d, the

van der Waals attractive force, Fvdw, as a function of separation distance between

the particles is given as:

Fvdw = −AvdwR/12d (1.60)

As mentioned earlier, the interaction between macroscopic objects is a product of

the Hamaker constant and a function derived from the geometries of the objects

involved in the interaction. Hamaker constant is dependent on the nature of the

interacting materials and it is defined as: [29]

19

Chapter 1. Theory

Avdw(12) = π2N1N2B12 (1.61)

Where, N1 and N2 are the number of molecules per unit volume in material 1

and 2, respectively. B12 is a coefficient in an expression for the dispersion energy,

U(r), between isolated pairs of molecules 1 and 2 by a distance of r. Generally,

B12 may be simply referred to as Bdisp and it represents sum of the dipole-dipole,

dipole-induced dipole (debye) and the london dispersion interactions[28].

U(r) = −B12r−6 (1.62)

Flow properties

The manner and extent in which particles aggregate with each other or deposit on

a surface depends on the flow characteristics to which the particles are subjected.

There are several different flow characteristics that influence particle aggregation

and deposition. The most important flow characteristics are discussed below.

Reynolds number

The Reynolds number, Re, is defined as the ratio of inertial and viscous forces of

the fluid flow. It is an important parameter in determining the regime of a flow(i.e

whether the fluid flow is laminar or turbulent). It is a dimensionless number and

is usually defined as:

Re =ρfDchυ

µc

(1.63)

Where ρf is the density of the fluid, Dch is the characteristic length of a channel

in which the fluid is flowing (eg. Diameter in pipe flows), υ is the velocity of the

fluid and µc is the viscosity of the fluid. The effect of Reynolds number on particle

deposition and aggregation characteristics have been studied extensively. Adams

et al. [30] studied the effect of Reynolds number on the deposition and dispersion

of spherical particles in turbulent square duct flows using LES and RANS methods

and found out that generally the rate of the deposition increases with the Reynolds

number.

20

Chapter 1. Theory

Stokes number

The Stokes number is a ratio of the particles response time to that of the fluid’s

response time. It is a dimensionless parameter and it can be defined in relation to

the particles mass, velocity and energy. For example, the Stokes number related

to the particle velocity is defined as follows[3]:

Stv =τvτF

=ρdD

2/18µc

Dch/υ(1.64)

In the above equation, τv and τF are the particles velocity response time and the

time characteristic of the flow field, Dch is length or diameter in which the fluid is

passing through and υ is the flow velocity. The Stokes number can also be defined

for the particles mass and energy as follows.

Stmass =τmτF

=τmυ

Dch

(1.65)

StT =τTτF

=τTυ

Dch

(1.66)

where τm is the particles mass transfer response time and τT is the particles ther-

mal response time. When the Stokes number is much less than one, the particles

response time is small, hence any change in the flow field would not affect the par-

ticles properties significantly. At a small response time the particle takes a very

short time to adjust to the change in the flow field. On the other hand, a high

Stokes number means the response time is high, hence the particle would take a

long time to adjust to the change in the flow field. As a result, the particle prop-

erties would be affected by changes in the flow field. This will have implications

when there is a need to consider coupling between the continuous and dispersed

phases.

In multiphase flows phase coupling describes how the continuous and dispersed

phases affect one another. The effect can be one-way coupling where only the

dispersed phase is affected by the continuous phase but not the other way around

or it can be two-way coupling, where both phases are affected by one another. Flow

properties of the continuous and dispersed phases such as velocity, temperature,

particle size, density etc can be affected by phase coupling[3].

21

Chapter 1. Theory

Adhesion parameter

The Adhesion parameter, Ad, is a dimensionless parameter that is often used in

studies involving adhesive particle flows. The Adhesion parameter is defined as

the ratio of adhesive surface energy and the particle inertia[31].

Ad =2γ

ρpV 2D(1.67)

Where, γ is the adhesive surface energy, ρp is the particle density, V is the particle

velocity and D is the particle diameter. The adhesive surface energy can be related

to Hamaker constant as follows:[25].

γ =Avdw

24πD20

(1.68)

In the above equation D0 is the interfacial contact separation. Due to ambiguities

of what value to use for D0, Israelachvili (2011) suggested that a universal value

that is less than interatomic center-to-center distances be used and this yields

values of adhesive surface energies that are in good agreement with experimental

values. For large values of the Adhesion parameter particles colliding with each

other or with the wall tend to stick, forming particle agglomerates and deposits.

The adhesive force facilitates formation of agglomerates and deposits by reducing

the rebound velocity.

Tabor parameter

The Tabor parameter, µtabor, is a dimensionless number often used in particle

adhesion/cohesion studies. It is defined as[12].

µtabor =

(RW 2

adh

E2eqε

3

)1/3

(1.69)

Eeq =1

1− ν21E1

+1− ν22E2

(1.70)

22

Chapter 1. Theory

Where: Eeq is the equivalent Young’s modulus, E1 and E2 are Young’s moduli for

the contacting particles and surface material, respectively ν1 and ν2 are Poissons

coefficients for the particle and surface material respectively, ε is the interatomic

spacing (equilibrium spacing in Lennard-Jones potential), R is the equivalent ra-

dius and Wadh is the work of adhesion.

µtabor is introduced by Tabor et al. [32][33]. It is understood to be the ratio of

elastic deformation due to adhesion and the effective range of surface forces[12].

It is closely associated with adhesive contact models such as JKR and DMT mod-

els. More rigid solid particles with small radius tend to have low µtabor values

(µtabor < 0.1) in such cases DMT model provides a better alternative [33]. In the

other extreme, soft solids with large radius have high µtabor values (µtabor > 5) and

they are better suited for the JKR model. Description of the JKR contact model

is presented in Section 1.2.6.

For the adhesion of particles in the presence of liquids a modified Tabor parameter

can be used as suggested by Xu et al.,2007 [34]. In such cases, the equilibrium

separation distance in Lennard-Jones potential value, ε, should be replaced with

the mean radius of the meniscus (rm).

µctabor =

(RW 2

adh

E2eqr

3m

)1/3

(1.71)

Surface properties

Properties of the wall surface such as surface energy , surface roughness and contact

area play a role in the mechanism for, and extent of, particle deposition. Due to

asymmetry of intermolecular force field near interfaces, the density of molecules

near an interface is different from that of the bulk density. Hence, real interfaces

always have a concentration gradient[35].The asymmetry of intermolecular forces

gives rise to excess energy at the interface. This excess energy is defined in terms

of Gibb’s free surface energy:

Gs =G−Gb

Ainterface

(1.72)

23

Chapter 1. Theory

Where G is total Gibbs free energy, Gb is Gibbs free energy if the interface prop-

erties were the same as that of the bulk and Ainterface is area of the interface. For

practical purposes Gibbs free surface energy, Gs, is assumed to be equal to surface

energy, γ

Gs = γ =

(∂G

∂A

)p,V,ni

(1.73)

Surface energy, γ, is related to work of cohesion and work of adhesion. For phases

A and B, Work of cohesion, Wcoh, and work of adhesion, Wadh, are defined as:

Wcoh = 2γA (1.74)

Wadh = γA + γB − γAB (1.75)

STAR-CCM+ software provides a model for work of cohesion in DEM simulations

that can be defined both for particle-particle as well as particle-wall multiphase

interactions.

Since surface energy and work of cohesion have units of energy per unit interfacial

area, it is then imperative to consider what the interfacial area is and what role

it has in particle deposition. In ideal cases where there are only perfectly smooth

surfaces, it is easy to estimate the interfacial area, in practice surfaces have some

degree of roughness. Tabor et al. (1977) performed an experiment to study the

effect of surface roughness on adhesion and they observed that adhesion decreases

with increasing surface roughness[32]. In addition, they concluded that higher

roughness prevents effective adhesion between surfaces by forcing the surfaces

apart.

Particle properties

Properties of the dispersed particles have various effects on their agglomeration and

deposition behaviours. Particle size and shape have been known to have influences

on the contact area and the surface energy and ultimately on the magnitude of

adhesive/cohesive forces[36].

24

Chapter 1. Theory

Ermis et al. [37] studied the effect of size and shape of salt and glass particles on

particle adhesion strength to a flat surface. They observed that the adhesive force

between the particles and the surface increased with the increase of the particle

size. However, when the particle size was increased even further the adhesive

force weakened. They ascribed this to the increase in the impact force, which is

directly related to the detachment force, with the increasing particle size. Overall,

they observed the strongest adhesive force in medium sized particles. They also

studied the effect of the particle shape on adhesion by using particles of different

geometries such as sphere, cube, tetrahedron and octahedron. They observed that

spherical particles show the lowest contact area whereas cubical particles show the

highest.

1.2.6 Mechanisms of particle deposition

The Hertz contact model doesn’t account for the adhesive force during particle

contact, however presence of the adhesive force during contact results in a small

but observable increase in the contact area[38]. The JKR theory takes the effect of

adhesion force on the elastic deformation into consideration [39]. Most of present

understanding of particle adhesion stems from the JKR theory[40]. It is composed

of three distinct terms; mechanical energy, elastic energy and surface energy terms.

Assume an elastic particle thrown to a flat wall, since the particle is elastic the

collision with the wall results in a temporary deformation. The extent of the de-

formation is dependent on how fast the particle is thrown, how elastic the particle

is and how strongly the wall surface attracts the particle. The deformation can be

measured as a contact radius. The contact radius in the JKR theory can be given

in terms of Wadh as follows:

a3JKR =D

2Eeq

P +3

2wadhπD +

√3πWadhDP +

(3πWadhD

2

)2 (1.76)

If the surface energy is not accounted for the JKR model will be reduced to the

Hertzian contact model. Since the surface energy is zero then, Wadh = 0, as a

result the contact radius becomes:

a3JKR = a3H =DP

2Eeq

(1.77)

25

Chapter 1. Theory

There are also other contact models that take the contribution of adhesive forces

into account such as the DMT model, the MD model and the MYD model. The

DMT model, unlike the JKR model, considers a long range surface force, whereas

the MD model uses a “square-well” potential for the model. The MYD model, on

the other hand, uses both short and long range surface forces for the model[12].

wall

Surface energy term

D

a

Figure 1.8: Particle-wall collision

In STAR-CCM+ three types of DEM contact models are provided to model DEM

phase interactions. They are Hertz-Mindlin, Linear spring and Walton Braun

models. The Hertz-Mindlin model is the standard contact model in STAR-CCM+

particle contact simulations. It is a variant of the spring-damper model and it

assumes particles as elastic, perfectly smooth and with a small contact area during

contact [41]. The contact force between two spheres is given as a sum of the normal

and tangential components [42].

Fcontact = Fn + Ft = (−Knδn − ηnVn) +|Knδn|Cfsδt|δt|

(1.78)

26

Chapter 1. Theory

The tangential term in the above equation becomes −Ktδt − ηtVt if |Ktδt| is less

than |Knδn|Cfs. Moreover, the normal and tangential spring stiffness terms, Kn

and Kt as well as the normal and tangential damping terms, ηn and ηt are calcu-

lated as:

Kn =4

3Eeq

√δnReq (1.79)

Kt = 8Geq

√δtReq (1.80)

ηn =√

5KnMeqη (1.81)

ηt =√

5KtMeqη (1.82)

where Req is the equivalent radius, Meq is the equivalent particle mass, Eeq is the

equivalent Young’s modulus, Geq is the equivalent shear modulus,Cfs is the static

friction coefficient, δn and δt are overlaps in the normal and tangential directions

at the contact point and η is the damping coefficient. Subscripts A and B denote

particle A and particle B. The equivalent values are calculated as follows:

Req =1

1

RA

+1

RB

(1.83)

Meq =1

1

MA

+1

MB

(1.84)

Geq =1

2(2− νA)(1 + νA)

EA

+2(2− νB)(1 + νB)

EB

(1.85)

The above model is formulated for contact of two spheres(particle-particle interac-

tion) but it can be applied for a contact between a sphere and a wall (particle-wall

27

Chapter 1. Theory

interaction). In a case where particle-wall interaction is needed the formulas to be

used are the same as above except that the radius and the mass of the wall are as-

sumed to be infinite. This results the equivalent radius and the equivalent mass to

be equal to the radius of the particle and the mass of the particle, respectively[42].

Adhesion/cohesion modelling in STAR-CCM+ is provided for the JKR and DMT

models. In both cases the force of cohesion/adhesion is calculated as:

Fcoh/adh = RCminπWcoh/adhF (1.86)

Where RCmin is the minimum contact radius and F is a factor of 1.5 for the JKR

model and 2 for the DMT model.

28

Chapter 2

Numerical methods

This chapter is devoted to the discussion of numerical methods used in multiphase

flow simulations. The basic principles of computational fluid dynamics as well

as the techniques used to simulate fluid flows in general and multiphase flows in

particular are discussed. Common techniques used in computational fluid dynamics

such as discretization, finite difference method, implicit and explicit approaches

are summarized. Brief reviews of the continuous and dispersed phase equations

are also given from the viewpoint of multiphase flow simulation.

2.1 Computational fluid dynamics

Traditionally, fluid dynamics problems have been solved using combinations of

theoretical and experimental approaches. However, the development of modern

computers and algorithms allowed inclusion of a third approach into solving fluid

dynamics problems, that is a computational approach[5]. The CFD approach lies

in between the theoretical and experimental approaches and it simplifies problems

and reduces costs associated with experiments.

Fluid flows are governed by three fundamental principles: the conservation of

mass, the conservation of momentum and the conservation of energy. These prin-

ciples can be expressed in mathematical equations and they are generally referred

to as the governing equations. An equation derived by applying the mass conser-

vation principle is called the continuity equation. Similarly the momentum and

30

Chapter 2. Numerical methods

energy equations are derived by applying the principles of momentum and energy

conservation on a fluid system, respectively.

The governing equations can be derived in different ways and they can have dif-

ferent mathematical expressions depending on the type of flow system involved.

For example, the governing equations for the continuity and momentum equations

(only in x direction) for an unsteady, three-dimensional, compressible, viscous

flows in a conservation form are given below [5]:

∂ρ

∂t+5(ρV ) = 0 (2.1)

∂(ρu)

∂t+5(ρuV ) = −∂P

∂x+∂τxx∂x

+∂τyx∂y

+∂τzx∂z

+ ρfx (2.2)

τxx = λ(5.V ) + 2µ∂u

∂x(2.3)

τyx = µ

[∂v

∂x+∂u

∂y

](2.4)

τzx = µ

[∂u

∂z+∂w

∂x

](2.5)

λ = −2

3µ (2.6)

Where ρ is density, u,v and w are velocity components in the x, y and z directions,

V is velocity, P is pressure, fx is body force per unit mass acting on the fluid in the

x-direction, τxx, τyx and τzx are shear stress on different sides of the fluid element

and λ is the second viscosity coefficient.

The governing equations of fluid dynamics are a collection of partial differential and

integral equations that must be solved to get any meaningful result. However, the

process of solving these equations is not straightforward. There are several ways

in which the equations can be solved. In CFD, partial differential equations and

integral components of the governing equations are replaced with an equivalent and

approximated discrete algebraic equations through a process called discretization.

Discretization is a process in which a closed form mathematical equations such as

partial differential equations (PDE) are approximated into a discrete and finite

values[5]. Closed form equations such as partial differential equations that govern

flow fields have dependent variables that continuously vary throughout the domain.

31


The purpose of the discretization process is to replace these continuously varying

variables with an approximate and finite values at specified grid points in the

domain. There are different methods of discretization such as the finite difference,

finite volume and finite element methods. An example of how a discretization is

carried out is given below using finite difference method[5].

In the finite difference method a numerical grid is defined for a given geometric

domain[43] as shown below and the finite values for each grid point is calculated

using an approximation of the Tyler’s serious expansion or other alternatives such

as polynomial fitting.

P2

Y

X

∆X

∆Y

i-2,j i-1,j i,j i+1,j i+2,j

i,j-1

i,j+1

P1

Figure 2.1: Discrete grid points[5]

If one has ui,j at point p1, then according to the Tyler’s polynomial expansion, at

point p2, ui+1,j is given as:

ui+1,j = ui,j +

(∂u

∂x

)i,j

∆x+

(∂2u

∂x2

)i,j

(∆x)2

2+

(∂3u

∂x3

)i,j

(∆x)3

6+ ... (2.7)

Getting the exact value of ui+1,j, requires inclusion of an infinite number of deriva-

tives. However, a reasonable accuracy can be achieved by using only the first few

derivatives and truncating the rest. This, of course, introduces a truncation error.

For applications where more accurate results are required, one can include the

second and third even the fourth derivatives to obtain finite values at each grid

32


points and then truncate the rest of the derivative terms. But, this will increase

the computational time and cost dramatically. If the above equation is rearranged

one can have the following expression:

(∂u

∂x

)i,j

=ui+1,j − ui,j

∆x︸︷︷︸finite difference representation

−(∂2u

∂x2

)i,j

(∆x)2

2−(∂3u

∂x3

)i,j

(∆x)3

6− ...︸︷︷︸

truncation error

(2.8)

The above equation is a first order forward difference equation with respect to x.

This is because in order to solve the term,(∂u

∂x i,j

), the algorithm uses the value

of u at a grid point one step forward (i+ 1, j) from the starting grid point, (i, j).

The term can also be solved using the first order rearward difference and the first

order central difference methods. In such cases, the algorithm uses the value of u

at a grid point one step backward at (i, j − 1) for the rearward difference method

and uses two grid points (one grid point forward and one grid point backward(i+

1, j and i, j − 1)) for the central difference method. A similar approach can be

used to evaluate second order partial differential equations such as(∂2u∂x2

)and( ∂2u

∂x∂y

)that are often found in the Navier-Stokes equations. Once the partial

differential equations are replaced with the finite difference representations, the

resulting equation is called a difference equation[5]. Second order central second

difference equation with respect to x and second order central mixed difference

equation with respect to x and y without the truncation errors are given below[5].

(∂2u

∂x2

)i,j

=ui+1,j − 2ui,j + ui−1,j

(∆x)2(2.9)

(∂2u

∂x∂y

)i,j

=ui+1,j+1 + ui−1,j−1 − ui−1,j+1 − ui+1,j−1

4∆x∆y(2.10)

At the boundaries of the grid points the value of u or any other variable can

be determined using different methods. One of such methods is the polynomial

approach[5]. An example of a polynomial evaluation of u at the boundary of the

grid points is shown in Appendix B [5].

33


The variables in the governing fluid dynamics equations are calculated based on

the grid points that are discretized at specified locations. This is known as “space

marching”. However, governing equations also require “time marching”, which

is a discretization where an unknown value of a variable at time step n + 1 is

solved from known values at time step n[5]. The solutions obtained using time

discretization account for the transient effects of the fluid flow[44].

2.1.1 Explicit and implicit methods

Algorithms that are used to solve the governing equations of fluid dynamics use

two different methods to achieve the solution. These are, the explicit and the

implicit methods. In the explicit method, the value of an unknown variable is

calculated exclusively from a single equation where the values of all the other

variables are known. John D. Anderson[5] illustrated the difference between the

two methods using the following one dimensional heat conduction equation.

∂T

∂t= α

∂2T

∂x2(2.11)

first order forward difference:∂T

∂t=T n+1i − T n

i

∆t(2.12)

central second difference:∂2T

∂x2= α

(T ni+1 − 2T n

i + T ni−1

(∆x)2

)(2.13)

Where n is the time step, α is a constant and T is temperature. Rearranging the

above equation gives:

T n+1i =

∆tα(T ni+1 − 2T n

i + T ni−1)

(∆x)2+ T n

i (2.14)

In the above equation there is only one unknown variable, T n+1i , and the equation

can readily be solved.

An alternative method of solving equation 2.11 is to use the implicit method. The

Crank-Nicolson implicit method for example uses the average values of variables

Ti+1, Ti and Ti−1 at time steps n and n+ 1 to solve for T n+1i .

34


T n+1i − T n

i

∆t= α

12(T n+1

i+1 + T ni+1) + 1

2(−2T n+1

i − 2T ni ) + 1

2(T n+1

i−1 + T ni−1)

(∆x)2(2.15)

In the above equation new unknowns are introduced, namely T n+1i+1 , T n+1

i and

T n+1i−1 . As a result, the equation cannot be solved on its own. In order to solve

it, the values for all equations at all grid points must be solved concurrently. The

implicit method has an advantage of maintaining the stability of the solution even

at a relatively large ∆t values but it is also more difficult to program and it is

computationally more demanding than the explicit method.

2.1.2 Numerical stability and convergence

The representation of a partial differential equation with a difference equation

introduces a truncation error. In addition, a round-off error can become signif-

icant. These errors will propagate and amplify as the calculation is performed

while marching in space and time. The amplified errors might become too big and

ultimately makes the numerical method unstable. Due to this, there are restric-

tions placed on the values of ∆t and ∆x in order to keep the numerical method

stable[5]. When the numerical calculation progresses from a time step n to n+ 1

the error, εi, should shrink or stay unchanged for the solution to be stable.

| εn+1i

εni|6 1 (2.16)

For example, the application of the above requirement on the one dimensional heat

conduction equation (Von Neumann method) reveals that the stability criteria

should be as follows[5]:α∆t

(∆x)26

1

2(2.17)

In order to get an accurate numerical solution, a numerical algorithm usually

involves an iterative process, where the value of a variable is calculated repetitively

while the accuracy of the value become progressively better. After a number of

repetitions the difference between values from any two consecutive calculations

become close to zero (this difference value is referred to as, residual). When the

35


residual becomes lower than a pre-defined value the numerical solution is said to

be convergent[45].

2.1.3 Multiphase flow simulation

Multiphase flows involve mixtures of macroscopically distinct phases known as dis-

persed phases and continuous phases. Flow of gases in liquids and solid particles

in liquids are examples of such flows[42][46]. These phases have clearly defined

interfaces and they often interact with each other as well as with the boundary of

the flow system (nature of the particle-particle and particle-wall interactions are

described in the first chapter). These interactions are often sources of complexi-

ties and due to this it is difficult to use experimental and analytical approaches

to investigate multiphase flows[47]. Numerical simulations give an attractive al-

ternative to investigate the behaviour of multiphase flows.

However, numerical simulation of multiphase flows come with complexity of their

own. An accurate and detailed description of multiphase flows are often heavy

on computational time and cost. In addition, industrial applications, that often

consist of trillions of particles, are restricted by computational capability[3].

2.1.3.1 Continuous phase equations

The presence of dispersed phase particles in the continuous phase complicates

the numerical solution. Hence, for practical applications the continuous phase

equations are based on averaging procedures where the average property of a flow

over time, volume/space or ensemble is used to formulate the continuous phase

equation[48]. A detailed description of each averaging procedure is presented by

Crowe et al. [3] but here, only the volume averaging procedure is discussed.

The Volume averaging procedure is the most commonly used method since com-

putational models are themselves based on averaged discretized cells of a flow

domain. If B is a flow variable of the continuous phase such as fluid density or

velocity and if Vave is the averaging volume, then the volume average of B in the

continuous phase is defined as:

〈B〉 =1

Vave

∫Vc

BdV (2.18)

36


L

l

Vd

Vc

Figure 2.2: Illustration of volume averaging procedure.

Where 〈B〉 is the volume averaged property of the continuous phase and Vc is the

volume of the continuous phase. In order to apply the volume averaging technique,

the characteristic dimension of the averaging volume(Vave) must be much smaller

than the characteristic dimension of the physical system over which the flow vari-

ables change significantly (L), and much larger than the characteristic dimension

of the continuous phase (usually the average inter-particle spacing)(l)[49]. This

requirement is needed to ensure that microscopic variations in the flow variables

(B) are levelled off and the volume averaged properties are continuous. When

the averaging volume is too small, microscopic fluctuations of the flow variables

affect the averaged properties significantly. On the other hand, when the volume

is too large the averaged properties will be affected by macroscopic variations of

the variables and the dimensions of the physical system. At the right averaging

volume, the average properties become independent of the average volume. The

local volume average of Equation 2.18 is given as:

B =VcVave

1

Vc

∫Vc

BdV (2.19)

VcVave

= αc (2.20)

⇒ B = αc〈B〉 (2.21)

37


Where, B is the local volume average of the flow variable, αc, is the volume fraction.

When the average properties of the flow become independent of the averaging

volume, the local and global average properties become equal.

B = 〈B〉 (2.22)

The mass, momentum and energy conservation equations of the continuous phase

can then be formulated based on the above averaging technique. The conservation

equations can be formulated as quasi-one-dimensional form or three-dimensional

form.

2.1.3.2 Single particle equations

The basis for the formulation of single particle equations is the application of the

Reynolds transport theorem (RTT) on a control volume. A control volume is an

arbitrary section of the flow field in which the conservation laws such as mass

conservation are applied. The resulting equations of the control volume are then

related to the whole flow system using the Reynolds transport theorem. For a

control volume in a flow field with a defined control surface at its boundary, the

Reynolds transport theorem states that the rate of change of an extensive property

of the system is equal to the time rate of change of the extensive property in the

control volume and the net rate of flux of the extensive property across the control

surface. In other words, the total change in the property of a system in the control

volume is the sum of the property of the control volume at time t and the difference

in the property that flows in to the control volume and the property that flows out

of the control volume across the control surface. Mathematically, the Reynolds

transport theorem is given as:

dBsys

dt=

∫cv

∂(ρβ)

∂tdV +

∫cs

ρβuini dS (2.23)

Where, ui is the velocity of the fluid at the control surface with respect to the

coordinate reference frame, ni is a unit vector normal to the control surface, ρ

is the fluid density, S is the surface area of the control surface, Bsys is the ex-

tensive property of the system and β is the corresponding intensive property of

the system. The subscripts cv and cs refer to control volume and control surface,

38


respectively. The Reynolds transport theorem equation can now be applied to for-

mulate the single particle equations and the dispersed phase equations in general.

For example, the mass conservation principle states that, dMdt

= 0. In this case

the extensive property is mass and since the intensive property of mass is one (i.e

M/M), application of the Reynolds transport theorem on the mass conservation

principle gives:dM

dt=

∫cv

ρddtdV +

∫cs

ρswini dS = 0 (2.24)

⇒ dm

dt= −

∫cs

ρswini dS (2.25)

If the efflux velocity(wi) and the fluid density are uniform across the control sur-

face, then the above equation can be written as:

dm

dt= −ρswSd (2.26)

Where ρd is the particle density, ρs is the density of the fluid at the control surface,

Sd is the surface area of the particle, wi is the velocity across the control surface

with respect to the control surface and w is the magnitude of efflux velocity vec-

tor. The above strategy can be applied on the momentum as well as the energy

conservation principles to formulate their respective equations.

Control surface

r1

v1

w1

x3

x1

x2

Figure 2.3: Moving control surface enclosing a particle[3]

39


In the momentum conservation, the Newton’s second law of motion states that:

Fi =d(MUi)

dt(2.27)

The extensive variable is linear momentum (MUi) and the intensive variable is

velocity (Ui). When the Reynolds transports theorem is applied on the above

principle, it gives:

Fi =d

dt=

∫cv

ρdUi dV +

∫cs

ρsUi,swini dS (2.28)

Where, Ui,s is the velocity of the fluid at the control surface with respect to the

inertial reference frame. Similarly from the moment of momentum conservation

equation:

Ti =d(MHi)

dt(2.29)

Where, Ti is the applied torque vector and Hi is the moment of momentum per

unit mass. In this case the extensive property is the moment of momentum (MHi)

and it’s intensive counterpart is the moment of momentum per unit mass(Hi).

Hence, applying the Reynolds transport theorem gives:

Ti =d

dt

∫cv

ρdhi dV +

∫cs

ρshiwini dS (2.30)

The Reynolds transport theorem can also be applied on the first law of thermo-

dynamics (principle of energy conservation) in a control volume. The first law of

thermodynamics states that:

dE

dt= Q−W (2.31)

E = M(i+UiUi

2) + Sσ = Me+ Sσ (2.32)

Where, E is the sum of internal, external and surface energies, i is internal energy

per unit mass, Ui is velocity, Q is the rate of heat transfer and W is the rate of work

done. When the Reynolds transport theorem is applied on the above equation it

40


gives:dMe

dt=

d

dt

∫cv

ρde dV +

∫cs

ρsewini dS (2.33)

⇒ Q−W =d

dt

∫cv

ρde dV +

∫cs

ρseswini dS +d(Sσ)

dt(2.34)

Detailed derivations for linear momentum, moment of momentum and energy

equations for a dispersed phase particle can be found in Crowe et al.[3] and Jakob-

sen et al.[49]. However, the results from the derivations are summarized in the

following table.

Conservation principle Equation Single particle equation

Mass conservation dMdt = 0 dM

dt = −ρsωSd

Momentum conservation Fi = d(MUi)dt Fi = mdV

dt +∫cs ρ(rn + w)w.n dS

Angular momentum conservation Ti = d(MHi)dt Ti = Idωdt

Energy conservation dEdt = Q−W mcd

dTddt = Q + m(hcs − hd + ω′ω′

2 )

Table 2.1: Summary of single particle equations

2.1.3.3 Dispersed phase equations

The dispersed phase equations are classified into two groups.The Lagrangian and

Eulerian approaches. The Lagrangian approach is further classified into two groups

as the discrete element method(DEM) and the discrete parcel methods(DPM). The

difference between the DEM and the DPM methods is that DEM simulates the flow

properties of every single dispersed phase particle whereas in DPM the smallest

unit of simulation is a parcel of the dispersed phase particles. In both DEM and

DPM, the dispersed phase particles are assumed as a separate entities from the

continuous phase. Here, an important distinction between the Lagrangian and

Eulerian approaches arise. In Eulerian approach the dispersed phase particles are

assumed to posses properties of the continuous phase. Due to this the Eulerian

approach is also known as the two fluid model.

The choice of what approach to use is dependent on whether the fluid flow is dilute

or dense with the dispersed phase particles. If the flow is dilute, the dispersed

41


phase particles are sparsely present and they can’t be considered continuous. As

a result the Eulerian approach can’t be used. The only choice will then be the

Lagrangian approach. In dense flows where the dispersed particles are abundant,

contact between the dispersed particles will also be abundant. The dispersed

phase starts to show behaviours of continuity, the abundance of particle-particle

contact would enable information to travel in all directions[3]. In this case the

the dispersed phase can be assumed to be a continuous phase and one can use the

Eulerian approach.

Crowe et al. presented a dimensionless number to determine whether a dispersed

phase flow is dilute or dense. It is the ratio of the momentum response time(τv)

and the average time between collisions(τc). It is given as:

τvτc

=τv

1/fc=

ρD2

18µc

1

nπD2vr

=nπρD4vr

18µc

(2.35)

Where, fc is the collision frequency, n is the number density of the particles,

vr is the relative velocity of one particle to other particles. If the value of the

dimensionless number is less than one, then the flow is dilute. If it is more than

one the flow is dense.

2.1.4 Properties of the dispersed phase

Multiphase flows involve a continuous phase and dispersed phase particles. One

of the main difference between the two phases is that unlike the continuous phase,

the dispersed phase particles are not in continuum. The properties of the dispersed

phase particles affect the overall property of the flow. In this section some of the

most important properties of the dispersed phase are discussed.

Particle spacing

The average distance between the dispersed particles is an important property of

the dispersed phase. It’s importance lies in determining whether a particle should

be considered isolated from other particles or not. An isolated particle exerts little

42


or no influence on other particles. The ratio of the average interparticle distance

and the particle diameter is used to quantify particle spacing. If two particles are

at the centres of two adjacent cubes of length, lsd then the ratio of the interparticle

spacing and the particle diameter is given as:

lsdD

=

[π

6αd

] 13

(2.36)

Where, αd is the volume fraction of the dispersed phase. If the value oflsdD

exceeds

a certain limit, particles of the dispersed phase are considered isolated. The limit

depends on the nature of the continuous and dispersed phases. For example, for

systems involving gas-particle flows anlsdD

value of 10 or above indicates particle

isolation[3].

Particle response times

How fast particles respond to changes in the flow characteristics give an important

information about the overall flow. Two important response times are usually

encountered. These are the momentum and the thermal response times. The

momentum response time is the time it takes for the dispersed phase particle to

respond to changes in the fluid velocity. Whereas the thermal response time is the

time it takes for the dispersed phase particles to respond to changes in the fluid

temperature.

τV =ρdD

2

18µc

(2.37)

τT =ρdcdD

2

12k′c(2.38)

Where, τV is the momentum response time, τT is the thermal response time, cd is

the specific heat of the particle material and k′c is the thermal conductivity of the

continuous phase. Another property that determine the interaction between the

continuous and dispersed phases is the the Stokes number. The Stokes number is

a dimensionless quantity that is usually defined as a ratio of time characteristic

43


of the dispersed phase to that of the continuous phase. Some examples of Stokes

number are given in Chapter 1 Section 1.2.5

How the dispersed particles are packed in the flow is also an important property.

It is used to classify flows as dense or dilute flows. In dense flows the particles

are densely packed and particle-particle collisions are frequent. As discussed in

section 2.1.3.3, this affects the flow behaviour in a fundamental way in part because

when particles are highly densely packed the limit of dispersion will be reached

and the particles can be considered to be in a continuum.

The continuous and dispersed phases also exert influences on each other. As

discussed in Chapter 1 Section 1.2.5, these influences are called phase coupling.

In one-way coupling the continuous phase influences properties of the dispersed

phase but not the other way around. However in two-way coupling the dispersed

phase also influences the continuous phase properties. When the two phases are

in a dynamic and thermal equilibrium the flow can be considered as a single phase

flow.

2.1.5 Multiphase flow simulation in STAR-CCM+

In general, multiphase flows can be classified into two broad categories as dispersed

flow and stratified or separated flows[42][50]. Dispersed flows are flows where the

flow consists of distinct particles such as bubbles and solid particles along with the

continuous phase, whereas separated flows are flows where the flow is consisted of

multiple continuous phases separated by interfaces. There are six models for the

simulation of multiphase flows in STAR-CCM+. They are:

• Lagrangian multiphase

• Discrete element method

• Multiphase segregated flow

• Dispersed multiphase model

• Fluid film

• Volume of fluid

44


Since the later four models are not the focus of this paper, only the description of

the Lagrangian multiphase and the discrete element method models are presented.

Lagrangian multiphase model

In Lagrangian multiphase model the basic unit for calculation is a parcel. A

parcel is composed of elements of the dispersed phase. The model selects a sta-

tistically representative number of parcels instead of considering a large number

of dispersed phase elements. This enables faster simulation. For each parcel, the

Lagrangian multiphase model solves the equation of motion[42]. The Lagrangian

multiphase model is provided in STAR-CCM+ software and it is suited for flows

where particle-wall interactions are important.

The Lagrangian multiphase model is able to model several properties of the dis-

persed phase and record the state of the parcels as a separate track file from which

post processing of the results could be carried out. A detailed formulation of the

model is found in the user guide manual for the simulation software[42].

Discrete element model

The discrete element method is an extension of the Lagrangian multiphase model.

It is used for the simulation of several interacting, usually solid particles. The

DEM method is suited for flows where particles are densely packed, collide fre-

quently or when flow behaviour is dependent on particle size and shape as well

as contact mechanics. Hence, the DEM is a good alternative to study the effect

of particle-particle and particle-wall interactions. However, the DEM requires a

small integration time steps for an adequate resolution of particle-particle and

particle-wall contact properties , this makes the DEM simulations time and cost

intensive[51].

DEM formulation

The momentum balance for the material particle is generally given as the sum of

particles surface and body forces:

mpdVpdt

= (Fs) + (f) = (Fd + Fp + Fvm) + (Fg + Fu) (2.39)

45


Where, Fs ,f,Fd,Fp,Fvm, Fg, and Fu are the particles surface, body, drag, pressure

gradient, virtual mass, gravity and user defined body forces respectively.

Fd =1

2CDApVs. | Vs | (2.40)

Fp = Vp5Pstatic(2.41)

Fvm = CvmρVp

(DV

Dt− DVp

Dt

)(2.42)

Fg = mpg (2.43)

Fu = Vpfu (2.44)

Where CD is the particle drag coefficient, Vs is the particle slip velocity, Ap is the

particle projected area, Vp is the particle volume, 5Pstaticis the gradient of static

pressure in the continuous phase and Cvm is the virtual mass coefficient.

In DEM an extra body force is introduced to account for the particle-particle and

the particle-wall contacts. Hence, the body force in DEM becomes the sum of

gravity force, user defined body force and contact force:

f = Fg + Fu + Fcontact (2.45)

DEM time step

The maximum time-step that is allowed for a DEM particle is limited due to the

assumption that the force acting on a particle is only affected by the particle’s

immediate neighbours during a single time-step. Hence, the time-step is limited

by the time it takes for the Rayleigh wave to propagate across the surface of the

sphere to the opposite pole and the minimal sphere radius (Rmin)[42].

τ1 = πRmin

VRayleigh

(2.46)

46

Chapter 3

Experimental methods

This chapter provides descriptions of the experimental techniques used for the nu-

merical simulations and the laboratory experiments employed in this thesis. The

experimental set-ups are discussed and the values for the selected experimental

variables are given along with the justifications for the selections.

3.1 Numerical experiments

Several numerical experiments were carried out during the course of this thesis. In

order to accomplish the best possible accuracy for the simulations while keeping

simulation time as shorter as possible, some considerations were implemented. For

example, the number of injected particles was chosen to be 1000 after trial simu-

lations showed that it was possible to assess particle deposition with a reasonably

shorter time (about 12 hrs) than the time it would take if higher number of in-

jected particles were used. Other considerations were made so that the simulation

parameters are not over complicated and over specified hence, parameters such

as “coefficient of restitution” and “coefficient of friction” as well as some of the

solver properties were left at default values. To identify parameters that affect

particle deposition, dimensional analysis and literature review were carried out.

In the following sections the experimental parameters and their values as well as

the justifications for their choice will be presented.

48

Chapter 3. Experimental methods

3.1.1 Dimensional analysis

Dimensional analysis was carried out to identify the most important variables

that influence particle deposition. Ultimately it was decided to use variables such

as the Reynolds number, the Adhesion parameter and the Tabor parameter as

the variables to investigate particle deposition. However the dimensional analysis

gave an interesting insight into other variables that may also influence particle

deposition. The dimensional analysis can be seen in Appendix A

3.1.2 Numerical experiment set-up

Numerical experiments were carried out using a commercial CFD software STAR-

CCM+ 9.06 model. The numerical experiments involve the following steps: Ge-

ometric preparations, defining surface and boundary types, meshing, defining the

physics models and interactions between the boundary, the fluid and the particles,

set-up and define the injector settings, set-up the numerical solver settings. After

these steps were completed the simulations were run until converging solutions

were obtained. Most of the simulations took about 12 hours to complete. The

simulation time depends on factors discussed in chapter 2 Section 2.1.5.

After solutions were obtained, post-processing of the solutions were performed.

The track files method provided in the STAR-CCM+ software were deemed to be

enough to extract data about the particle deposition.

3.1.3 Numerical experiment variables

The variables for the numerical experiments were the Reynolds number, the Ad-

hesion parameter and the Tabor parameter. All of the experiments were carried

out within the laminar flow regime(below Re value of 2300). The experimental

parameters for each variable are presented in the following tables:

49


Experiment name Re333 Re800 Re1266 Re1733 Re2200

Viscosity of the fluid(Pa.S) 0.001002 0.001002 0.001002 0.001002 0.001002Density of the fluid(kg/m3) 1000 1000 1000 1000 1000Diameter of flow cell pipe(m) 0.01 0.01 0.01 0.01 0.01Velocity of the fluid(m/s) 0.033 0.08 0.1266 0.1733 0.22Reynolds number 333 800 1266 1733 2200

Table 3.1: Experimental values for Re simulations

Experiment name Ad1 Ad2 Ad3 Ad4

Density of the particle(kg/m3) 1050 1050 1050 1050Particle diameter(m) 0.0001 0.0001 0.0001 0.0001Particle velocity(m/s) 0.03 0.03 0.03 0.03Work of cohesion(J/m2) 0.0814 0.1628 0.407 0.814Adhesion parameter 1722 3445 8613 17227

Table 3.2: Experimental values for Adhesion parameter simulations

Experiment name Tab1 Tab2 Tab3 Tab4

Young’s modulus(Pa) 3.0× 108 1.0× 106 1.0× 106 3.0× 108

Poisson’s ratio 0.35 0.35 0.1 0.1Separation distance (m) 5× 10−5 5× 10−5 5× 10−5 5× 10−5

Work of cohesion (J/m2) 0.0814 0.0814 0.0814 0.0814Particle radious (m) 5× 10−5 5× 10−5 5× 10−5 5× 10−5

Tabor parameter 5.78× 10−10 1.16× 10−6 1.36× 10−6 6.79× 10−10

Table 3.3: Experimental values for Tabor parameter simulations

Geometry, surface preparation and boundary types

This thesis focuses on study of particle deposition on a pipe containing an ob-

struction. The shape and dimensions of the pipe is given in Figure 3.1 and 3.2:

This geometry was made using STAR-CCM+ 3D CAD design tool. After the

geometry was made, the inlet, the outlet and the wall of the pipe were assigned a

specific region and each region was assigned an appropriate boundary type. For

the inlet the boundary type was “velocity inlet” and for the outlet it was “flow

split outlet” boundary type. The wall region was assigned “wall” boundary type.

50


20 cm

9 cm 9.6 cm

0.7 cm 1 cm

1.4 cm

Figure 3.1: Flow cell geometry dimensions

Pipe inlet

Pipe outlet

Wall Wall

Figure 3.2: Flow cell geometry regions

Meshing

Meshing is a step in which the designed geometry is discretized to represent the

computational domain [42] in which the physics solver finds a solution. In this

thesis the “polyhedral mesher” with a base size of 0.0015 was selected along with

the “surface remesher” and the “surface wrapper”. The “Polyhedral mesher”

generates volume meshes of polyhedral shapes that have the advantage of being

easy and efficient to build as compared to other mesh types (such as tetrahedral

meshes). The “Surface remesher” was used to refine and improve the geometry

surfaces. The “Surface wrapper” was used to remove any problem that may arise

due to the complexity of the geometry, intersection of parts or sharp edges.

51


Figure 3.3: flow cell geometry after mesh operation

Physics models

The physics models provides settings, variables and constants that will be used

for the calculation of the numerical solutions within the computational domain.

In this thesis the aim is to study the deposition of Polystyrene particles using the

DEM and The Lagrangian multiphase models. In general, the following models

were used:

Setting ModelEquation of state Constant densityTime Implicit unsteadySpace 3DGradient metrics GradientsFlow Segregated flowViscous regime LaminarMulti-phase interaction Multiphase interactionOptional Gravity, passive scalar,

DEM, Lagrangian multiphase

Table 3.4: Description of physics model

The Lagrangian multiphase model provides a model that helps define the La-

grangian phase. In this paper, the Lagrangian phase consists of solid Polystyrene

particles whose properties and interactions are defined as constant density, DEM

particle type, drag force, residence time, solid particle, spherical particle and track

file. The multiphase interaction model lets the interaction between phases to be

52


defined. Polystyrene-Polystyrene and Polystyrene-wall interactions were defined

using the physics model. The values for each interactions are listed below:

DEM phase interactionsFirst phase PolystyreneSecond phase Polystyrene

Hertz MindlinStatic frictional coefficient 0.61Normal restitution coefficient 0.6Tangential restitution coefficient 0.6

Linear cohesionWork of cohesion(J/m2) 0.0814 - 0.814Factor 1.5

Rolling resistance method Coefficient of rolling resistance 0.001

Table 3.5: particle-particle interaction model

Particle injector

The particle injectors provides a way to introduce the Lagrangian particles into

the computational domain. There are different types of particle injectors in STAR-

CCM+. The “Part injector”, the “Point injector” and the “Surface injector” are

some examples. However, the “Random injector” was selected because it is best

suited for DEM simulations and it represents the random fashion in which particles

are distributed in the domain better than the other injectors. The conditions and

values of the “Random injector” are given as follows:

condition setting valueParticle amount specification number of particles 1000Particle size specification diameter 0.0001 mParticle velocity specification “absolute” various

Table 3.6: particle injector settings

53


Solver settings and stopping criteria

Unlike the steady state solvers, the implicit unsteady solver offers information

about the particles at each specified time step. For this reason the implicit un-

steady solver is favoured over the steady solver. The implicit unsteady solver

requires a time step specification and a time step of 0.01s was selected for all

simulations. Other properties of the solver were left at default values.

When the Lagrangian multiphase model is selected for the physics model, as is the

case in the thesis, a separate solver called Lagrangian multiphase implicit unsteady

solver is also activated. This solver offers controls for the maximum “substeps”

and “verbosity” (level of detail of information required) among other things. All

of these properties were again left at the default values or settings.

The stopping criteria is used to tell the solver how long the computation runs and

how and when the solver should stop it. Once the stopping criterion are specified,

they are evaluated after each iteration or time-step. When the criterion are met

the simulation ends. For the unsteady solvers there are three important stopping

criterion that must be specified [42]. They are the maximum inner iterations,

the maximum physical time, and the maximum steps. Generally, the choice of

the stopping criteria depends on how much detailed information is required, how

much computational time and cost is available and other related aspects. In this

study, the maximum inner iteration was set to 2, the maximum physical time was

2 s and the maximum steps were 10,000.

3.2 Laboratory experiments

Laboratory experiments were carried out with the objective of comparing results

with the results suggested by the low cohesion Re-simulations. In order to ac-

complish this, a laboratory experiment was designed. In the next sections, the

laboratory set-up, the experimental procedure, the equipments and the apparatus

used for the experiments are discussed.

54


3.2.1 Experimental set-up

The experimental observations were carried out based on the following procedure:

Two litres of distilled water was added into tank-1 (see Figure 3.4) followed by

two drops of concentrated polystyrene particle suspension (this corresponds to

nearly a thousand polystyrene particles). The tank was continuously stirred using

a magnetic stirrer to ensure uniform particle distribution.

Contents from tank-1 were then allowed to pass through connecting tubes into

the flow cell. A microscope connected to a computer was mounted in such a way

to monitor particle deposition behaviour on a particular sections of the flow cell.

The microscope was used to take pictures every 5 minutes. The pictures were

then transferred to a computer where a software was used to analyse the extent of

deposition of particles. The undeposited particles and the fluid were then allowed

to pass to tank-2 from which a pump transported the contents back into tank-1.

The recirculation process was continued for 15 minutes.

The velocity of carrier fluid was controlled by using the pump and the inlet valve

at tank-1. The experiments were carried out at various Reynolds number values.

The results were then compared with results from the numerical experiments.

Figure 3.4: Diagram of the experimental set up

55


3.2.2 Laboratory equipment

The experimental set-up involved the use of the following laboratory apparatus

and equipment:

Two reservoir tanks

Connecting tubes

A light source

A magnetic stirrer

Polystyrene particles

KnF lab liquiport pump

A flow cell with an “obstruction”

A Nikon SMZ800 microscope with Q-imaging camera and software

Figure 3.5: Experimental set-up

3.2.3 Polystyrene particles

Dark red micro-particles made from polystyrene were purchased from SIGMA-

ALDRICH. The particles were spherical with a diameter of 100µm. The Polystyrene

particles were in a 10 ml aqueous suspension form with a solid content of 5 wt%.

56


3.2.4 Re-experiments

Laboratory experiments were carried out for different values of the Reynolds num-

ber. All of which were within the laminar flow regime. The value of the Reynolds

number for each experiment were controlled by using the velocity of the fluid as a

variable. For polystyrene particles, the diameter and the density are known and

constant(10−4m and 1050kg/m3, respectively), in addition, the continuous phase

(water) has a viscosity of 0.001 Pa.s at 20 ◦C, this leaves the fluid velocity as the

only variable that can be used for the Reynolds number variation.

Re =ρυD

µc

(3.1)

For example for the Reynolds number value of 1000, the corresponding fluid ve-

locity will be:

υ =Reµc

ρD=

(1000)(0.001 Pa.s)

(1050 kg/m3)(10−4 m)= 9.55 m/s (3.2)

3.2.5 Ensuring correct fluid velocity

In order to ensure a precise fluid velocity, a pump and a value located at the

outlet of tank-1 were used in combination. The pump used in the experiments,

KnF lab liquiport pump, has a volumetric flow rate capability ranging from 0.2

lt/min to 3.0 lt/min. Based on a starting 2 lt volume of water in tank-1 and flow

cell diameter of 0.01 m, the volumetric flow rate required to ensure 0.0333 m/s

flow velocity (corresponding to Re of 333) will be:

Q333 = υ333.D2flowcell = (0.0333m/s)(0.001m2) = 0.0000333m/s = 0.2lt/min (3.3)

Once the flow rate for a given velocity of the fluid was known, the next step was to

turn the pump to that flow rate value, then the outlet value on tank-1 was adjusted

so that the volume of water in the tank remains constant. The recirculation of the

water was continued with out adding the polystyrene particles into the tank for

57


up to 10 minutes until the volume of water in tank-1 was completely stabilized.

When there was no more variation of the water volume in tank-1, the Polystyrene

particles were added into tank-1. Images of the particle deposition were then

taken at 5, 10 and 15 minutes after the particles were added. This procedure was

repeated for different Reynolds number values.

Re Q(lt/min) υ(m/s)333 0.2 0.0333800 0.48 0.081266 0.76 0.012661733 1.04 0.17332200 1.32 0.22

Table 3.7: Velocity and Volumetric flow rate values for Re simulations

58

Chapter 4

Results and discussion

This chapter presents the findings from the numerical and the laboratory experi-

ments. Comparisons of the simulations with the laboratory experiments as well as

with the results from published literature are discussed.

4.1 Numerical experiments

Simulation of deposition of particles using the DEM-Lagrangian multiphase model

was performed. The Reynolds number, the Adhesion parameter and the Tabor

parameter were varied to understand how they affect the deposition. In the next

sections the results are presented and discussed in detail.

4.1.1 Results from Re-experiments

The effect of the Reynolds number on the particle deposition was investigated using

two sets of numerical experiments. Each set consisted of five experiments for the

Reynolds number values of Re=333, Re=800, Re=1266, Re=1733 and Re=2200.

All the other parameters and model specifications were kept constant. The first

set of experiments were assigned a low cohesivity (Wcoh = 0.0814J/m2), while the

second set of experiments were assigned a high cohesivity (Wcoh = 0.814J/m2)

60

Chapter 4. Results and discussion

4.1.1.1 Results from low cohesivity experiments

The amount of deposition of Polystyrene particles was determined based on the

particle velocity near the wall surfaces. A threshold “derived part” was created

using STAR-CCM+ and particles with velocities below the threshold velocity were

recorded as deposited particles. The threshold velocity of the particles depends

on the inlet velocity of the fluid and was defined in such a way that made compar-

isons of particle depositions from different Reynolds number experiments possible.

Initial post-processing of the numerical experiments showed that using a constant

threshold velocity for all Reynolds number experiments led to a false increase in

the deposition of particles. This effect was specifically observed for Re=333 and

Re=800 experiments. Lower inlet fluid velocities were found to increase this ef-

fect. Hence, it was deemed necessary to define threshold velocities in comparison

with the inlet fluid velocities of each numerical experiment. Threshold velocities

calculated based on this definition are shown below:

Vthreshold =υ0

1.0× 104 (4.1)

Where, Vthreshold is particle threshold velocity and υ0 is fluid inlet velocity.

Re υ0(m/s) Vthreshold(m/s)2200 2.2× 10−1 2.2× 10−5

1733 1.7× 10−1 1.7× 10−5

1266 1.3× 10−1 1.3× 10−5

800 8.0× 10−2 8.0× 10−6

333 3.3× 10−2 3.3× 10−6

Table 4.1: Threshold velocity values

Deposition efficiency (DE) was then defined for the flow and plotted against the

Reynolds number. Deposition efficiency was defined as the ratio of the number

of particles deposited and the number of particles that were injected into the

simulation domain.

In all the numerical experiments the number of Lagrangian phase particles (Polystyrene)

to be injected into the domain was specified to be 1000. However, after each sim-

ulation the total number of particles in the domain was found to be slightly less

61


than 1000. The reason for this is, in “Random injectors” the number of injected

particles depend on the “number of seeds” and the “injection cycle limit”, both

of which can be specified to keep the total number of injected particles within an

acceptable range.

The “number of seeds” is defined as the number of particles that are injected

at the same time. Generally, it’s value is set at a tenth of the total number of

particles that can be packed inside the volume of the injector region. The “injection

cycle limit” controls the number of times the injector injects new particles[42].

Both the “number of seeds” and the “injection cycle limit” introduce a small

degree of uncertainty to the injector and this leads to discrepancies between the

specified number of particles to be injected and the actual number of particles in

the domain[23] [52]. Nevertheless, it was found necessary to use the total number

of particles in the domain after the simulations were completed as the number of

injected particles for the calculation of deposition efficiencies(DE).

DE =

(Deposited particles

Injected particles

)× 100% (4.2)

Numerical experiments were carried out to study the effect of Reynolds number

variation on the deposition efficiency of Polystyrene particles. Reynolds number

values ranging from 333 to 2200 were used. Deposition efficiency of each Reynolds

number experiment was then calculated.

Overall, it was found that deposition efficiency of Polystyrene particles increased

with increasing Reynolds number. A sharp increase was observed in the lower

range of Re-values. An increase from Re=333 to Re=800 resulted an increase in

deposition efficiency from 13% to 35.9%. The increase became less sharp at higher

Re-values. An increase in Reynolds number from 1266 to 2200 resulted an increase

from 62.5% to 68.3% in deposition efficiency.

In addition to deposition efficiency, the location in which particles deposited were

visually investigated. It was found that more particles were deposited after the

obstructed section of the flow cell (for nomenclature of the flow cell sections,

please refer to Figure 4.15). The particles also deposited on the edges of the

obstructed section. However, no significant deposition was observed on the surface

the obstruction itself. It was also found that the location of particle deposition

did not vary significantly from one Reynolds number to the other.

62


0

10

20

30

40

50

60

70

80

0 500 1000 1500 2000 2500

Reynolds number

DE

Figure 4.1: The effect of Reynolds number on deposition efficiency (low cohe-sivity, Wcoh = 0.0814 J/m2)

Figure 4.2: Velocity profile of the particle tracks (Re=333)

63


The type of velocity profile for particle tracks shown in Figure 4.2 can be converted

into a more realistic representation of the particle flow and deposition. STAR-

CCM+ provides a tool to convert particle tracks into animation and record them.

All particle tracks from experimental simulations were converted into animation

for a better visual analysis. The procedure to convert particle tracks can be found

in Appendix C.

Re=333

Re=1266

Re=800

Re=1733

Re=2200

Flow direction

Figure 4.3: Front view of particle deposition (low cohesion simulation, bluedots represent deposited particles)

4.1.1.2 Results from high cohesivity experiments

In order to investigate and compare how Reynolds number variation affects particle

deposition at higher cohesivity as compared to the low cohesivity experiments

discussed above, numerical experiments were carried out for Reynolds number

values ranging from 333 to 2200 with a high particle cohesivity. The parameters

and procedures in the simulations were the same as the low cohesivity experiment

counterparts. However, in the high cohesivity experiments the work of cohesion

for each experiment was assigned to be 0.814J/m2. The same method was also

used to calculate deposition efficiency.

The threshold velocity for high cohesivity experiments was defined in a slightly

different way. Initially, the definition used in the low cohesivity experiments was

64


Re=333

Re=1266

Re=800

Re=1733

Re=2200

Flow direction

Figure 4.4: Bottom view of particle deposition (low cohesion simulation)

used but the results were found to be unrealistic. Hence, a new definition of

threshold velocity had to used. In the new definition the velocity of the particles

still had to be reasonably near zero so that particles with velocities below the

threshold velocity could safely be considered deposited while obtaining a realistic

result. Several trial simulations and threshold values were used to obtain more

accurate results and ultimately the following definition of threshold velocity was

chosen.

Vthreshold =υ0

1.0× 103 (4.3)

Re Vthreshold(m/s) Deposition efficiency(%)2200 2.2× 10−4 72.11733 1.7× 10−4 89.11266 1.3× 10−4 65.1800 8× 10−5 23.8333 3.3× 10−5 21.9

Table 4.2: High cohesivity threshold values and deposition efficiency

65


Results from the high cohesivity experiments indicated that, overall, as the Reynolds

number increased deposition efficiency also increased. This trend was found both

in the low and high cohesivity experiments. Another similar trend observed was

the fact that increase in the deposition efficiency was more pronounced when the

Reynolds number was increased from Re=333 to Re=1266 than from Re=1266 to

Re=2200.

Adams et al.[53] studied the effect of Reynolds number on the deposition of spheri-

cal particles of different sizes using LES-RANS approach coupled with Lagrangian

particle tracking method and found out that overall, increase in Reynolds number

led to increase in particle deposition. However, their study also concluded that the

increase in the deposition was more pronounced for particles with larger diameter

(500µm) than particles with diameters from 50µm-100µm. Similarly, Afkhami et

al.[54] studied the Reynolds number effect on particle agglomeration in turbulent

channel flows using LES-DEM method. They found out that increased Reynolds

number led to increased agglomeration and observed that particle agglomeration

often happened near channel walls.

0

10

20

30

40

50

60

70

80

90

100

0 500 1000 1500 2000 2500

Reynolds number

DE

Figure 4.5: The effect of Reynolds number on deposition efficiency (highcohesivity, Wcoh = 0.814 J/m2)

66


Re=333

Re=1266

Re=800

Re=1733

Re=2200

Flow direction

Figure 4.6: Front view of particle deposition (high cohesivity)

Re=333

Re=1266

Re=800

Re=1733

Re=2200

Flow direction

Figure 4.7: Bottom view of particle deposition (high cohesivity)

67


4.1.2 Adhesion parameter results

Four numerical experiments were carried out to study the effect of the Adhesion pa-

rameter on the particle deposition. Adhesion parameter values ranging from 1722

to 17227 were used for the experiments. In each experiment the fluid velocity was

set to 0.0333m/s (Re=333) and the threshold velocity was set to 3.33× 10−6m/s.

Deposition efficiency was then calculated and the following results were obtained.

Ad υ0(m/s) Vthreshold(m/s) Deposition efficiency(%)Ad1 3.3× 10−2 3.3× 10−5 14.6Ad2 3.3× 10−2 3.3× 10−5 86.2Ad3 3.3× 10−2 3.3× 10−5 86.8Ad4 3.3× 10−2 3.3× 10−5 22.5

Table 4.3: The effect of Adhesion parameter on deposition efficiency

0

10

20

30

40

50

60

70

80

90

100

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

DE

Adhesion parameter (x104)

Figure 4.8: The effect of Adhesion parameter on deposition efficiency(Re=333)

The results showed that as Adhesion parameter increased the deposition efficiency

also increased. From Ad1 to Ad2 there was a sharp increase in deposition efficiency,

however the increase from Ad2 to Ad3 was minimal. At the highest Adhesion pa-

rameter (i.e Ad4) the deposition efficiency decreased sharply. The decrease was

not expected since, theoretically, increase in Adhesion parameter should also lead

68


to increase in deposition efficiency. Since deposition efficiency showed dependency

on the Reynolds number, a new set of Adhesion parameter experiments were car-

ried out at a different Reynolds number to check if the decrease in the deposition

efficiency could still be observed. In the new experiments, the Reynolds number

was set to Re=1733 and the threshold velocity was 1.7× 10−5m/s. Then the de-

position efficiencies were calculated for each experiment. The results are shown in

Figure 4.9 and they indicated that the deposition efficiency did not decrease from

Ad3 to Ad4.

0

10

20

30

40

50

60

70

80

90

100

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

DE


Figure 4.9: The effect of Adhesion parameter on deposition efficiency(Re=1733)

In addition to the analysis based on the particle velocity, the deposition efficiency

was analysed based on the particle slip velocity(Vs). The slip velocity of the

particle is generally defined as the difference in the velocity of the particle and the

continuous phase [42]. Since, the new Adhesion parameter experiments were all

assigned an inlet fluid velocity of 0.1733 m/s, ideally the particle slip velocity would

be close to the fluid velocity if the particles are deposited (i.e since Vs = υ0 − V

then a near zero value of V would indicate deposition of a particle). However,

particle slip velocity is known to vary from one section of the flow to another[55].

For example, the particles that were deposited on the obstruction of the flow cell

generally showed high slip velocity. This was expected because of the fact that

the fluid attained high velocity as it passed through the narrow section around the

69


obstruction. On the other hand, in the section after the obstruction (Figure 4.15

section C) there were eddy formations and the fluid velocity was much lower than

the section containing the obstruction. Visual inspection of the particles deposited

in Section C showed that the slip velocity was generally around 1.1 m/s. Hence, it

was decided to use a slip velocity value of 1.1 m/s or above as a basis to determine

deposition efficiency (the reasoning here is that, if the slip velocity of a particle

deposited on a section where the fluid velocity is the lowest is 1.1m/s, then it is

safe to assume that the particles deposited on other sections would have higher

slip velocities than 1.1m/s). A new threshold “derived part” was created using

STAR-CCM+ and the deposition efficiencies were calculated.

0

10

20

30

40

50

60

70

80

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

DE


Figure 4.10: The effect of Adhesion parameter on deposition efficiency (par-ticle slip velocity)

The result showed that as the Adhesion parameter was increased the particle

deposition efficiency also increased. The increase was observed from Ad=1722 to

Ad=8613 but a slight decrease was observed from Ad=8613 to Ad=17227. In

addition to the investigation of the amount of deposited particles, the location

of deposition of particles in the flow cell was also investigated visually for each

70


experiment. These investigations revealed an increasing particle deposition with

increasing Adhesion parameter in most sections of the flow cell. In particular,

Section A showed linear increase in particle deposition with Adhesion parameter.

Once again particle deposition in Section B was minimal in all experiments apart

from the few deposited particles at the front edge of the section. Most of the

deposited particles were observed in Section C. A review of scientific literature was

carried out to assess the validity of these experimental observations. The review

revealed that particle deposition/adhesion increases with increasing surface energy

[56] [57] [58]. Since the only Adhesion parameter variable in the experiments was

Wcoh and since it is the sum of surface energies of particle-particle or particle-wall

interactions then it is imperative to say that the experimental observations were

in agreement with prior studies.

Ad1

Ad3

Ad2

Ad4

Flow direction

Figure 4.11: Front view of particle deposition (Adhesion parameter)

4.1.3 Tabor parameter results

In order to understand the effect of the Tabor parameter on the deposition of parti-

cles, four numerical simulations with different Tabor parameter values were carried

out. In these experiments Young’s modulus and Poisson ratio were independently

varied. Simulations with high values of Young’s modulus and Poisson’s ratio val-

ues were compared with simulations with low values of these two parameters. The

71


Ad1

Ad3

Ad2

Ad4

Flow direction

Figure 4.12: Bottom view of particle deposition (Adhesion parameter)

values of these parameters for each simulation are presented in Table 3.3. The

Tabor parameter experiments revealed lower deposition efficiency for experiments

with high Young’s modulus values than for those with lower Young’s modulus.

Experiments Tab1 and Tab4 both were assigned a Young’s modulus of 3× 108 Pa

while experiments Tab2 and Tab3 were assigned 1× 106 Pa. Tab1 and Tab4 had

deposition efficiencies of 13.7% and 13.2%, respectively which contrasted with the

deposition efficiencies of Tab2 and Tab3, which were 69.1% and 67% . In addition

to the variation of deposition efficiency with Young’s modulus, variation of depo-

sition efficiency with Poisson’s ratio was also investigated. Experiments Tab1 and

Tab4 were assigned identical parameters except Poisson’s ratio values. In Tab1

the value was 0.35 while in Tab4 it was 0.1. The simulation results revealed that

although there was no significant difference in the deposition efficiency between

the two experiments, it was apparent that a small increase can be observed in the

deposition efficiency as Poisson’s ratio increased. The same trend was observed in

experiments Tab2 and Tab3.

Since Young’s modulus is a measure of “stiffness” of a particle, large value of

Young’s moduli (rigid particles) lead to increase in “tear-off” force. Hence, it is

expected that large Young’s modulus values lead to lower deposition. Literature

review on the subject matter revealed a similar trend with the observations found

72


Name Tabor parameter Threshold velocity(m/s) DE(%)Tab1 5.8× 10−10 1.7× 10−5 13.7Tab2 1.2× 10−6 1.7× 10−5 69.1Tab3 1.4× 10−6 1.7× 10−5 67.0Tab4 6.8× 10−10 1.7× 10−5 13.2

Table 4.4: Tabor parameter results

in this study. A pioneer research done by Muller et al.[33] showed that the “tear-

off” force is entirely dependent on the Tabor parameter and it decreased when the

Tabor parameter was increased. This suggested that the deposition of particles is

facilitated by low “tear-off” force and hence high Tabor parameter. Similar conclu-

sions were made by papers written by Cheng et al.[59], Maguis[60], Greenwald[61]

and Feng[62].

Tab1

Tab3

Tab2

Tab4

Flow direction

Figure 4.13: Front view of particle deposition (Tabor parameter)

73


Tab1

Tab3

Tab2

Tab4

Flow direction

Figure 4.14: Bottom view of particle deposition (Tabor parameter)


Laboratory experiments were carried out for various Reynold’s number values and

influence of the Reynolds number on the particle deposition behaviour in a flow

cell was investigated. Polystyrene particles with the work of cohesion value of

0.0814J/m2 were used during the experiments. Hence, comparison of results from

laboratory experiments and numerical experiments (Low cohesion simulations)

were carried out.

The results from the laboratory experiments were images taken by a microscope

during different stages of the experiment. Due to limited capacity, the microscope

can only take images at specified locations in the flow cell. Three separate locations

were chosen. These were, Section A, Section B and Section C.

During each laboratory experiment images of the particle depositions at the above

three specified locations were taken using a microscope. Each experiment was

carried out for 15 minutes and images of deposition of the particles were taken at

5, 10 and 15 minute marks. However, it was observed that there was no appreciable

difference both in the number of particles deposited and the location of deposition

between images taken for each experiment. Due to this, it was decided that only

74


Section A

Inlet

Section C

front edge

back edge

Section B

Outlet

Figure 4.15: Flow cell locations of interest

a single representative image for each experiment at each specified location to be

presented.

Due to lack of appropriate measurement technique, the deposition efficiency of

particles in the laboratory experiments was not possible to estimate. But it was

possible to visually observe the increase in deposition of particles with the Reynolds

number. Analysis of images from Section A showed that the number of particles

deposited increased when the Reynolds number was increased. This trend was

also observed in the simulation results from low cohesion experiments although

the increase was less prominent in Section A than Section C.

Section B did not show appreciable particle deposition in the laboratory experi-

ments. The few particles that were deposited did so usually at the back edge of

the section and this could well be considered as part of Section A. The results from

the simulations suggested that particles would adhere at the front edge of Section

B but it was not observed appreciably during the laboratory experiments. Several

factors might have contributed to this difference. The possible explanations are

75


(a) Re-333 (b) Re-800 (c) Re-1266

(d) Re-1733 (e) Re-2200

Figure 4.16: Particle deposition in Section A.

the presence of recurring bubbles near this section and the susceptibility of exper-

imental set-up to flow and mechanical disturbances. Analysis of Section C showed

that particle deposition increased with increased Reynolds number. Generally,

most particles deposited on the left and right corners of this section. This was in

agreement with simulation results. One interesting thing that emerged when ex-

perimental results were compared with simulation results was that, the simulations

predicted that majority of the particles would deposit in Section C. However, the

laboratory experiments revealed that Polystyrene particles deposited on Section

A just as much as in Section C. In addition, Section A seemed to show a more

linear increase in the deposition of particles with increasing Reynolds number than

Section C or B. In both the simulations and the laboratory experiments, Section

B experienced the least deposition of all sections.

76


(a) Re-333 (b) Re-800 (c) Re-1266

(d) Re-1733 (e) Re-2200

Figure 4.17: Particle deposition in Section B.

(a) Re-333 (b) Re-800 (c) Re-1266

(d) Re-1733 (e) Re-2200

Figure 4.18: Particle deposition in Section C.

77

Chapter 5

Conclusions

This chapter briefly presents the concluding remarks that were deduced based on

the results from the numerical and the laboratory experiments.

The effects of the Reynolds number, the Adhesion parameter and the Tabor param-

eter on the deposition behaviour of spherical Polystyrene particles in the laminar

flow regime in an obstructed flow cell were studied numerically. Laboratory ex-

periments were also conducted to verify results from the numerical simulations.

Various post-processing tools provided by the simulation software STAR-CCM+

were used to analyse the simulation results. Images of Particle depositions from

selected sections of the flow cell were also recorded and analysed. Deposition effi-

ciency was then defined for each simulation and used as a means to assess what the

above mentioned factors influences were. After conducting the numerical and the

laboratory experiments and analysing the results the following conclusions were

made.

The simulation results on both the low and the high cohesion experiments showed

that deposition of Polystyrene particles increased as the Reynolds number was

increased. The increase in the deposition was supported by results from the lab-

oratory experiments. Furthermore, results from literature on the dependency of

particle deposition on the Reynolds number showed that increase in particle de-

position was observed with increasing Reynolds number [53][54].

79

Chapter 5. Conclusions

Comparison of effects of the Reynolds number variation on the depositions of parti-

cles with low cohesion (Wcoh = 0.0814J/m2) and high cohesion (Wcoh = 0.814J/m2)

showed that although the overall pattern of increasing particle deposition with

increasing Reynolds number holds in both cases, particles with a high work of

cohesion values showed a higher deposition efficiency when they were compared

with particles with a low work of cohesion at the same Reynolds number. This

observation suggested that the deposition of particles was affected not only by the

flow properties such as the Reynolds number but also by particle properties such

as surface energy and work of cohesion.

The effect of Adhesion parameter variation on the particle deposition was studied.

The result suggested that the particle deposition increased with the increase of the

Adhesion parameter. Literature on the effect of Adhesion parameter were reviewed

and trends similar with the trends in this study were reported [56][57][58].

The effects of the Young’s modulus and the Poisson’s ratio on the particle deposi-

tion were studied in the Tabor parameter simulations. The results indicated that

the particle deposition was lower at a high young’s modulus value than at a low

Young’s modulus value. However, the Poisson’s ratio variation showed a slight

increase in the particle deposition. Literature review on the subject matter also

revealed that similar effects were observed in particle depositions [59] [60] [61] [62].

The results from the laboratory experiments on the deposition of Polystyrene par-

ticles at different Reynolds numbers showed increased particle deposition with

increasing Reynolds number as predicted by the simulations. Moreover, the lo-

cations of particle deposition predicted by the simulations results were for the

most part observed in the laboratory experiments as well. Finally, after conduct-

ing several simulations and comparing results with laboratory experiments it was

demonstrated that the discrete element method coupled with the Lagrangian mul-

tiphase model was capable of simulating the deposition of Polystyrene particles in

an obstructed flow cell.

80

Chapter 6

Recommendations

During the course of this study several numerical and laboratory experiments were

carried out. At this point it is worth mentioning that several challenges were met

both during simulations and experiments. Consequently, areas of improvements

were identified and recommendations for future studies on similar subject matter

were laid down. As a result, the following improvements, recommendations and

suggestions were put forward. Within the scope of this thesis there were areas

in which improvements can be made. Improvements in the simulations and post-

processing capability, as well as laboratory experiment set up are suggested.

6.1 Simulation capability

The discrete element method simulations are notorious for being time and compu-

tationally intensive. Although, most of the simulations carried out in this study

had a running time of around 12 hours (due to a limited number of injected

particles: 1000 as well as a limited simulation physical time of 2 seconds) other

simulations such as those that require high Young’s modulus values like some of

the Tabor parameter simulations took up to four times longer time to complete. If

one wants to undergo DEM simulations with more injected particles than used in

this study, as it is likely the case in real world multiphase problems, the simulation

run time would be even longer. Run time can of course be reduced using methods

such as increasing DEM particle time step by changing values for the particle den-

sity, size and Young’s modulus or using a reduced substeps and inner iterations.

However, these methods will have ill effects on the accuracy of simulation results.

81

Chapter 6. Recommendations

This problem can be solved by having a licence for a parallel processors instead of

the serial processor used in this study. The serial processor, as its name implies, is

a single processor that processes steps in the numerical simulations serially. This

obviously takes long time. A parallel processor on the other hand uses several

processors and distribute numerical calculations across all processors. This would

enable a faster DEM simulation while keeping accuracy of results.

6.2 Post-processing

STAR-CCM+ software is a multi-purpose simulation software with good post-

processing tools. However, it was found that post-processing of DEM simulations

for particle deposition would be better carried out with specialized external post-

processing software such as EnSight 10.1 software. STAR-CCM+ supports export

of particle track files to files that can be processed by external softwares and

EnSight 10.1 (or later updated versions) is one of the supported softwares. Due to

time restrictions and licence requirements post-processing using EnSight software

was not included in this study.


During the laboratory experiments different areas of improvements have been iden-

tified. The microscope used to take images of particle deposition can only focus on

a limited section of the flow cell. It is recommended that a better way be devised

to fully take images of particle deposition in the whole flow cell or maximize the

area of focus.

The Polystyrene particles were not only depositing in the flow cell but also on

the walls of the reservoir tanks and pipes. It would increase the confidence of the

result obtained from the laboratory experiments if there is a way to limit particle

depositions outside the flow cell.

It was also observed that the laboratory set-up was susceptible to mechanical and

flow disturbances. This led to repeated experimental trials and took time and

resource. It would be ideal if the laboratory set-up was made more robust.

82

Appendix A

Dimensional analysis

Dimensional analysis was carried out to identify the parameters that are important

in determining the amount of deposited particles. The Buckingham’s π-theorem

method was used[63]. The first step in dimensional analysis is to carefully se-

lect the independent variables that determine the dependent variable we want to

investigate.

In this paper the dependent variable X, is mass of the deposited particles per unit

area and the following independent variables were selected:

V, Particle velocity, [kg/m3]

D, Particle diameter, [m]

ρp, Particle density, [kg/m3]

µf , Fluid viscosity, [Pa.s]

Wcoh, Work of cohesion, [J/m2]

ym, Young’s modulus, [Pa]

The next step is to change the units of the dependent and independent variables

into a dimensional expressions based on mass(M), length(L) and time(T) as fol-

lows:

X, [kg/m2] = M.L−2

84

Appendix A. Dimensional analysis

V, [m/s] = L.T−1

D, [m] = L

ρp, [kg/m3] = M.L−3

µf , [Pa.s] = M.T−1.L−1

Wcoh, [J/m2] = M.T−2

ym, [Pa] = M.L−1.T−2

In the next step the dimensional expressions M, L and T are themselves expressed

in terms of the other variables :

M = [m]

L = [(m/ρp)1/3]

T = [(m/Wcoh)1/2]

At this step all the variables will be expressed as dimensionless variables by mul-

tiplying them with inverses of their dimensional expressions as follows:

Π0 = X.L2.[1/M] = X.[(m/ρp)1/3].[1/M] = X.m1/3

ρ1/3p

Π1 = V.T.[1/L] = V.[(m/Wcoh)1/2].[(ρp/m)1/3] = V.m1/6ρ

1/3p

W1/2coh

Π2 = ρp.L3.[1/M] = ρp.[m/ρp].[1/m] = 1

Π3 = µf .[1/M].L.T = µf .[1/m].[(m/ρp)1/3].[(m/Wcoh)1/2] = µf .m−1/6

ρ1/3p Wcoh

Π4 = Wcoh.[1/M].T2 = Wcoh.[1/m].[m/Wcoh] = 1

Π5 = ym.[1/M].L.T2 = ym.[1/m].[(m/ρp)1/3].[m/Wcoh] = ym.m1/3Wcoh

ρ1/3p

85

Appendix A. Dimensional analysis

According to the Buckingham’s theorem we can then conclude that Π0 does not

depend on Π2 and Π4. Hence we reduced the independent variables from the

starting six to three and write the relationship as follows:

Π0 = f(Π1,Π3,Π5) (A.1)

Xm1/3

ρ1/3p

= f

[V m1/6ρ

1/3p

W1/2coh

,µf

m1/6ρ1/3p Wcoh

,ymWcohm

1/3

ρ1/3p

](A.2)

86

Appendix B

Grid points at the boundary:

Polynomial approach

The usual central difference equations such as the one in Equation B.1 are not

applicable when the variable to be calculated is at a grid point on a boundary.

This is because the central difference method would require a grid point out of the

computational domain. This can be solved by using a polynomial approach.

∂u

∂y i,j

=ui,j+1 − ui,j−1

2∆y(B.1)

Y

X

∆Y

u3

u1

u2

∆Y

Figure B.1: Boundary grid points[5]

87

Appendix B. Grid points at the boundary: Polynomial approach

In terms of a polynomial serious, u can be expressed as :

u = a+ by + cy2 + ...⇒ ∂u

∂y= b+ 2cy (B.2)

at u1, y = 0⇒ u1 = a (B.3)

at u2, y = ∆y ⇒ u2 = a+ b∆y + c(∆y)2 + ... (B.4)

at u3, y = 2∆y ⇒ u3 = a+ 2b∆y + c(2∆y)2 + ... (B.5)

⇒ b =−3u1 + 4u2 − u3

2∆y(B.6)

since∂u

∂y= b+ 2cy ⇒

(∂u

∂y

)1

= b+ 2(c)(y) = b (because at point 1 y=0) (B.7)

Hence:

(∂u

∂y

)1

=−3u1 + 4u2 − u3

2∆y(B.8)

88

Appendix C

Post-Processing particle tracks

There are different ways one can convert particle tracks into animation in STAR-

CCM+. The step-by-step procedure described below is applicable for the La-

grangian particle tracks and it is available in STAR-CCM+ tutorial guide in more

detail. [42].

• First step: After simulation is finished and saved, go to “scenes” → “new

scene” → “geometry” and click to create a new geometry scene (In addi-

tion make sure you clicked the “make scene transparent” tab for a better

visibility)

• Second step: Go to “tools” → “track files” then right click and select the

correct track file (track file name will be the name of the simulation file

followed by .trk). Now a new node for Particle tracks appears.

• Third step: Go to the new “particle tracks” → click on the track file name.

The particle tracks will be highlighted on the geometry scene. Right click

on the track file name then drag and drop in the geometry scene. When a

pop-up window appears select “use representation volume mesh”.

• Fourth step: Go to scene/plot → right click “displayer” then chose “new

displayer” → “streamline”.

– In “streamline” right click “parts”→ “edit” and select “particle tracks”

then click “ok”.

89

Appendix C. Post-processing particle tracks

– Click “scalar field” then go to properties window and change function

to “track:particle velocity-magnitude”.(or any other function of choice)

– Go to “animation” and in the properties window change the “anima-

tion” mode from “off” to “tracers”. A new “streamline settings” tab

appears.

– Go to “streamline settings” and adjust values in the property window

as follows(or other appropriate settings):

Delay between tracers(sec): 5.0

Head size: 0.00045

Tail length(sec): 0.0001

cycle time(sec): 25.0

• Fifth step: Now the animation is ready to be played. Simply click “play/-

pause” button. If recording of animation is required, click “write a movie of

animation” in the current scene button and adjust the frame rate to 25.0,

start time to 0.0, animation length to 10.0. Then give file name and save.

This will record a 10 second animation of particle tracks with a total of 250

frames.

90

Appendix D


equations

D.1 Solution manual for the extended hard sphere

model particle-wall collision equations

These equations relate the translational and angular velocities of the particle before

and after the collision with a wall. For further detail about the derivation of these

equations one can refer to a work done by Kosinski and Hoffmann[4].

D.1.1 Case-I

For a particle that stops sliding in the compression period the requirement is:

27m

√(V

(0)x + aω

(0)z )2 + (V

(0)z − aω(0)

x )2 6[−mV (0)

y (1 + em) + emJ(1)y,t + J

(2)y,t

]f (D.1)

V (2)x =

[5

7(V (0)

x − 2a

5ω(0)x )

](D.2)

V (2)y =

[em(− V (0)

y −J(1)y,t

m

)+J(2)y,t

m

](D.3)

91

Appendix D. Extended hard sphere model equations

V (2)z =

[5

7(V (0)

z +2a

5ω(0)x )

](D.4)

ω(2)x =

V(2)z

a(D.5)

ω(2)y = ω(0)

y (D.6)

ω(2)z =

−V (2)x

a(D.7)

D.1.2 Case-II

For a particle that stop sliding in the recovery period the requirement as well as

the equations for the translational and angular velocities are the same as with

case-I.

D.1.3 Case-III

For a particle that slides throughout the whole collision period. The requirement

is:

27m

√(V

(0)x + aω

(0)z )2 + (V

(0)z − aω(0)

x )2 >[−mV (0)

y (1 + em) + emJ(1)y,t + J

(2)y,t

]f (D.8)

V (2)x = (V (0)

x + εxfV(0)y (1 + em) + εxf

[(2 + em)

J(1)y,t

m+J(2)y,t

m

](D.9)

V (2)y =

[em(− V (0)

y −J(1)y,t

m

)+J(2)y,t

m

](D.10)

V (2)z = (V (0)

z + εzfV(0)y (1 + em) + εzf

[(2 + em)

J(1)y,t

m+J(2)y,t

m

](D.11)

92

Appendix D. Extended hard sphere model equations

ω(2)x = ω(0)

x −5

2aεzfV

(0)y (1 + em)− 5

2aεzf

[(2 + em)

J(1)y,t

m+J(2)y,t

m

](D.12)

ω(2)y = ω(0)

y (D.13)

ω(2)z = ω(0)

z −5

2aεxfV

(0)y (1 + em) +

5

2aεxf

[(2 + em)

J(1)y,t

m+J(2)y,t

m

](D.14)

93

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