A Study on Particle Motion and Deposition Rate:

Application in Steel Flows

Peiyuan Ni

Doctoral Thesis

Stockholm 2015

Division of Applied Process Metallurgy

Department of Materials Science and Engineering

School of Industrial Engineering and Management

KTH Royal Institute of Technology

SE-100 44 Stockholm

Sweden

Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan i Stockholm,

framlägges för offentlig granskning för avläggande av Teknologie Doktorsexamen,

fredagen den 17 April, kl. 10.00 i B2, Brinellvägen 23, Materialvetenskap, Kungliga

Tekniska Högskolan, Stockholm

ISBN 978-91-7595-448-6

ii

Peiyuan Ni A Study on Particle Motion and Deposition Rate: Application in

Steel Flows

Division of Applied Process Metallurgy

Department of Materials Science and Engineering

School of Industrial Engineering and Management

KTH Royal Institute of Technology

SE-100 44 Stockholm

Sweden

ISBN 978-91-7595-448-6

© Peiyuan Ni (倪培远), April, 2015

iii

To my beloved parents

iv

Science changes knowledge, technology/engineering changes the world.

v

Abstract

Non-metallic inclusions in molten steel have received worldwide attention due to their

serious influence on both the steel product quality and the steel production process. These

inclusions may come from the de-oxidation process, the re-oxidation by air and/or slag

due to an entrainment during steel transfer, and so on. The presence of some inclusion

types can cause a termination of a casting process by clogging a nozzle. Thus, a good

knowledge of the inclusion behavior and deposition rate in steel flows is really important

to understand phenomena such as nozzle clogging. In this thesis, inclusion behaviors and

deposition rates in steel flows were investigated by using mathematical simulations and

validation by experiments.

A ladle teeming process was simulated and Ce2O3 inclusion behavior during a teeming

stage was studied. A Lagrangian method was used to track the inclusions in a steel flow

and to compare the behaviors of inclusions of different sizes. In addition, a statistical

analysis was conducted by the use of a stochastic turbulence model to investigate the

behaviors of different-sized inclusions in different nozzle regions. The results show that

inclusions with a diameter smaller than 20 μm were found to have similar trajectories and

velocity distributions in the nozzle. The inertia force and buoyancy force were found to

play an important role for the behavior of large-size inclusions or clusters. The statistical

analysis results indicate that the region close to the connection region of the straight pipe

and the expanding part of the nozzle seems to be very sensitive for an inclusion

deposition.

In order to know the deposition rate of non-metallic inclusions, an improved Eulerian

particle deposition model was developed and subsequently used to predict the deposition

rate of inclusions. It accounts for the differences in properties between air and liquid

metals and considers Brownian and turbulent diffusion, turbophoresis and thermophoresis

as transport mechanisms. A CFD model was firstly built up to obtain the friction velocity

caused by a fluid flow. Then, the friction velocity was put into the deposition model to

calculate the deposition rate.

For the case of inclusion/particle deposition in vertical steel flows, effects on the

deposition rate of parameters such as steel flow rate, particle diameter, particle density,

vi

wall roughness and temperature gradient near a wall were investigated. The results show

that the steel flow rate/friction velocity has a very important influence on the rate of the

deposition of large particles, for which turbophoresis is the main deposition mechanism.

For small particles, both the wall roughness and thermophoresis have a significant

influence on the particle deposition rate. The extended Eulerian model was thereafter

used to predict the inclusion deposition rate in a submerged entry nozzle (SEN).

Deposition rates of different-size inclusions in the SEN were obtained. The result shows

that the steel flow is non-uniform in the SEN of the tundish. This leads to an uneven

distribution of the inclusion deposition rates at different locations of the inner wall of the

SEN. A large deposition rate was found to occur at the regions near the SEN inlet, the

SEN bottom and the upper region of two SEN ports.

For the case of an inclusion/particle deposition in horizontal straight channel flows, the

deposition rates of particles at different locations of a horizontal straight pipe cross-

section were found to be different due to the influence of gravity and buoyancy. For small

particles with a small particle relaxation time, the gravity separation is important for their

deposition behaviors at high and low parts of the horizontal pipe compared to the

turbophoresis. For large particles with a large particle relaxation time, turbophoresis is

the dominating deposition mechanism.

Key words: steel flow, ladle teeming, continuous casting, non-metallic inclusion

behavior, particle deposition, nozzle clogging, numerical simulation.

vii

Acknowledgements

First of all, I would like to express my deepest gratitude and appreciation to my excellent

supervisors Professor Lage Jonsson, Professor Pär Jönsson and Docent Mikael Ersson for

your professional guidance, great encouragement and patience.

I would like to thank Professor Sichen Du and Doctor Björn Glaser at KTH for your kind

help on my study.

I would like to thank Doctor Anders Tilliander and Doctor Andrey Karasev for your kind

help on my study at KTH.

All my colleagues in the division of TPM and all my friends in Sweden are thanked for

their kind help and good friendship. All the happy times of eating delicious food, doing

exercises and traveling with you make my life in Sweden colorful and interesting. Special

thanks to Mr. Erik Roos for your help in my work. Another special thanks to the

members in the simulation group meeting.

I would also like to thank Professor Ting-an Zhang, Professor Zhihe Dou, Doctor Guozhi

Lv and other people in the Key Laboratory for Ecological Utilization of Multimetallic

Mineral, Northeastern University in China. Thanks for your care and help on my study

and I miss the four-year study life in your team.

The financial support by the China Scholarship Council is acknowledged.

Last but not least, I would like to thank my parents and my brother who give me endless

love and support.

Peiyuan Ni

Stockholm, April 2015

viii

ix

Preface

The present thesis is based on my four-year study as a Ph. D student at the Department of

Materials Science and Engineering at KTH Royal Institute of Technology. This work is a

study mainly on the behaviors of non-metallic inclusions/particles in steel flows and the

deposition rates of non-metallic inclusions/particles onto walls.

The present thesis is based on the following supplements:

Supplement I:

“Turbulent Flow Phenomena and Ce2O3 Behavior during a Steel Teeming Process”

Peiyuan NI, Lage Tord Ingemar JONSSON, Mikael ERSSON and Pär Göran JÖNSSON

ISIJ International, 53 (2013), pp. 792-801.

Supplement II:

“On the Deposition of Particles in Liquid Metals onto Vertical Ceramic Walls”

Peiyuan NI, Lage Tord Ingemar JONSSON, Mikael ERSSON and Pär Göran JÖNSSON

International Journal of Multiphase Flow, 62(2014), pp. 152-160.

Supplement III:

“The Use of an Enhanced Eulerian Deposition Model to Investigate Nozzle Clogging

during Continuous Casting of Steel”

Peiyuan NI, Lage Tord Ingemar JONSSON, Mikael ERSSON and Pär Göran JÖNSSON

Metallurgical and Materials Transactions B, 45(2014), pp. 2414-2424.

Supplement IV:

“Deposition of Particles in Liquid Flows in Horizontal Straight Channels”

Peiyuan NI, Lage Tord Ingemar JONSSON, Mikael ERSSON and Pär Göran JÖNSSON

Submitted to the International Journal of Multiphase Flow, 2015

x

The contributions by the author to the supplements of this thesis:

I. Literature survey, numerical simulation and a major part of writing

II. Literature survey, numerical simulation and a major part of writing

III. Literature survey, numerical simulation and a major part of writing

IV. Literature survey, numerical simulation and a major part of writing

Parts of the work have been presented at the following conference:

“Numerical Study on Steel Flow and Inclusion Behavior in Nozzle during Teeming”

Peiyuan NI, Lage T.I. JONSSON, Mikael ERSSON and Pär G. JÖNSSON

The 3rd

International Symposium on Cutting Edge of Computer Simulation of

Solidification, Casting and Refining (CSSCR 2013), Stockholm, Sweden and Helsinki,

Finland, May 20-23, 2013.

xi

Contents

Chapter 1 Introduction ..................................................................................................... 1

1.1 Background ............................................................................................................... 1

1.2 Aim and Framework of the Thesis............................................................................ 3

Chapter 2 Methodology ..................................................................................................... 5

2.1 CFD Model Descriptions .......................................................................................... 5

2.1.1 Governing Equation ........................................................................................... 5

2.1.2 Turbulence Models............................................................................................. 5

2.1.3 Model Details ..................................................................................................... 6

2.2 Particle Tracking and Deposition Model .................................................................. 8

2.2.1 Lagrangian Particle Tracking Model ................................................................ 8

2.2.2 Eulerian Particle Deposition Model .................................................................. 9

2.3 Solution Methods .................................................................................................... 13

2.3.1 Solution Method of Fluid Flow Model ............................................................. 13

2.3.2 Solution Method of Particle Deposition Model ............................................... 14

Chapter 3 Results and Discussions ................................................................................ 17

3.1 Steel Flow and Inclusion Behavior in a Ladle Teeming Process ............................ 17

3.1.1 Steel Flow in a Nozzle during a Teeming Stage .............................................. 17

3.1.2 Inclusions Tracking Neglecting a Stochastic Turbulent Motion of Inclusions 18

3.1.3 Inclusion Behavior Including Stochastic Turbulent Motions .......................... 22

3.2 Inclusion Deposition Rate in Vertical Steel Flows ................................................. 25

3.2.1 Parameter Study on Inclusion Deposition Rate in Steel Flows ....................... 25

3.2.2 Inclusion Deposition Rate in a Tundish SEN ................................................... 30

3.3 Particle Deposition Rate in Horizontal Steel and Other Liquid Flows ................... 37

Chapter 4 Conclusions .................................................................................................... 41

xii

Chapter 5 Future Work................................................................................................... 43

References ........................................................................................................................ 45

1

Chapter 1 Introduction

1.1 Background

Non-metallic inclusions in molten steel have received worldwide concern due to their

serious influence on both the steel product quality and the steel production process. These

inclusions may come from the de-oxidation process, re-oxidation by air and/or slag due to

an entrainment during steel transfer, and so on. The influence of inclusions on a

continuous casting process is quite serious.

Non-metallic inclusions can lead to nozzle clogging. This is due to that the particles and

the ceramic nozzle wall materials are normally not wetted by liquid steel. Furthermore,

particles tend to stick to a ceramic refractory wall driven by the decrease in interfacial

energy when they come close to the wall. [1-3]

Some studies have been carried out to

remove inclusions in molten steel during a ladle treatment as well as during a tundish

operation. [4-7]

However, it is impossible to obtain completely clean steels with the current

steel production technology. In addition, clogging is closely related to the inclusion

behavior in molten steel. Therefore, the knowledge on the steel flow and the inclusion

behavior is important for the understanding of the nozzle clogging process and for

making a prediction on clogging situations.

A great amount of mathematical simulations have propelled our understanding of the

reality of both the steel flow and inclusion behavior. [8-30]

As early as 1973, Szekely et

al.[8]

modeled the fluid flow in a mold with a straight nozzle and a bifurcated nozzle,

respectively. In addition, Thomas et al. [10-20]

carried out a systematic research on steel

flows in nozzles. They investigated the steel flow characteristics in submerged entry

nozzles (SEN) and the effects of nozzle parameters as well as the operating practice on

steel flow. In addition, steel flows in nozzles were also studied by using different

turbulence models. [25-30]

Some researchers also studied the inclusion behavior in a nozzle

during casting. Wilson et al. [9]

investigated the steel flow characteristics in a nozzle and

tracked the trajectories of inclusions. The deposition of inclusions due to a centripetal

force and turbulence was also studied. Yuan [19]

et al. predicted the fraction of inclusions

with different densities and sizes entrapped by a lining in a stopper-rod nozzle. Zhang et

2

al. [20-22]

tracked the trajectories and entrapment locations of inclusions in a sliding-gate

nozzle. Long et al. [30]

studied the Al2O3 inclusion behavior in a turbulent pipe flow. The

effects on the entrapment-probability of factors, such as release location of inclusion,

inclusion size, pipe diameter, casting speed, were investigated.

Previous studies have increased the understanding of the behavior of an individual non-

metallic inclusion in molten steels by using a Lagrangian particle tracking scheme. This

approach tracks each individual particle by considering all the forces acting on it. Also,

this kind of tracking provides a great amount of information with respect to an individual

particle’s behavior, such as the particle velocity, location, transport time before touching

a wall, and so on. In these studies, light inclusions, e.g. Al2O3, with a density smaller than

molten steel and a diameter normally larger than 10 µm were studied. However,

behaviors of different-sized heavy inclusions, e.g. Ce2O3, with a density similar with

molten steel in steel flows were not investigated. In general, Ce2O3 is formed during Rare

Earth Metal (REM)-alloyed stainless steel production, where nozzle clogging is a

common problem. It is difficult to remove Ce2O3 due to that its density is close to steel.

In order to know the nozzle clogging situation, the deposition rate of inclusions is a very

important parameter. However, none of previously presented studies enables the particle

deposition rate to be predicted. For the Lagrangian scheme, it is necessary to track a large

number of particles in a turbulent flow to obtain sufficient statistical data considering that

one kilogram of typical low-carbon aluminum-killed steel contains 107-10

9 non-metallic

particles. [31]

To track such many particles, a Lagrangian scheme is very time-consuming.

A valuable contribution towards an efficient particle deposition rate prediction was made

by Guha [32, 33]

and Young and Leeming [34]

, who developed theoretical Eulerian

deposition models. These kinds of models not only provide a good physical framework,

by solving the particle continuity equation and the particle momentum conservation

equations to determine the particle deposition rate, but can also easily be extended to

consider other effects on the particle deposition rate. In this model, a specific type of

particle is considered as a continuum. This way of describing the deposition rate for a

large number of particles is much less time-consuming than a stochastic Lagrangian

scheme. The predictions were shown to be in good agreement with the experimental data

3

over a range of particle sizes in air flows. [32]

However, this kind of method to simulate

particle depositions has not been reported for metal flows.

1.2 Aim and Framework of the Thesis

In this thesis, the behavior of heavy Ce2O3 inclusions in a steel flow was studied.

Furthermore, a statistical analysis on their deposition at different regions of a nozzle

during a steel teeming process was carried out. In addition, an Eulerian particle

deposition model was further developed to predict the deposition rate of

particles/inclusions under different conditions in liquid metal flows. An outline of the

thesis is given as follows:

Ce2O3 inclusion behavior in a steel flow,

statistical analysis of deposition possibility of

different size inclusion in different parts of

ladle nozzle and comparison with experiment

Supplement I

Prediction of particle deposition rate in metal

flows by using an extended Eulerian

deposition model under different parameters,

e.g. temperature difference near wall, wall

roughness, particle size

Supplement II

Prediction of particle deposition rate in a

tundish nozzle during steel continuous

casting

Supplement III

Individual

Inclusion

Behavior

Inclusion

Deposition

Rate

Prediction of particle deposition rate in

horizontal straight channel flows under the

influence of gravity and buoyancy on the

particle deposition

Supplement IV

4

5

Chapter 2 Methodology

2.1 CFD Model Descriptions

2.1.1 Governing Equation

The conservation of a general variable 𝜙 within a finite control volume can be expressed

as a balance among the various processes, which tends to increase or decrease the

variable. The conservation equations, continuity equation, momentum equation and

turbulence equations, can be expressed by the following general equation: [35]

𝜕

𝜕𝑡(𝜌𝜙) +

𝜕

𝜕𝑥𝑖(𝜌𝜙𝑢𝑖) =

𝜕

𝜕𝑥𝑖(𝛤𝜙

𝜕𝜙

𝜕𝑥) + 𝑆𝜙 (1)

where the first term on the left-hand side is the change of 𝜙 with time and is not

considered in a steady state calculation. Furthermore, the second term on the left-hand

side represents the transport due to convection. Also, the first term on the right-hand side

expresses the transport due to diffusion where 𝛤𝜙 is the diffusion coefficient and is

different for different turbulent models, and the second term on the right-hand side is the

source term.

2.1.2 Turbulence Models

Since the k-ε model is widely used to simulate similar processes, an improved version [36,

37] of the model, Kim-Chen k-ε turbulence model,

[38] in PHOENICS

® was used to

simulate the steel flow in a ladle, in a vertical pipe and in a tundish in supplement I-III.

The used improved model adjusts the dynamic response of the dissipation equation by

introducing an additional time scale. Furthermore, it also offers advantages in separated

flows and in other flows, where the turbulence is removed from a local equilibrium

situation. In addition, the realizable k-ε turbulent model [39]

in ANSYS FLUENT® was

used to simulate a fluid flow in a horizontal straight pipe in supplement IV. Compared to

the standard k-ɛ model, the realizable k-ɛ model contains a changed transport equation for

the dissipation rate and a changed formulation for the turbulent viscosity.

6

2.1.3 Model Details

Teeming Process of a Ladle (Supplement I)

The steel density and viscosity are 7000 kg/m3 and 0.0064 kg/(m·s), respectively. The

computational domain of a ladle teeming process can be seen in Figure 1, where

inclusions are released from the curve with a radius 5 cm. The ladle nozzle was divided

into 6 regions to enable a statistical analysis of the inclusion attachment onto the nozzle

wall. The exact release location information of the inclusions is shown in Table 1.

Table 1. Release locations of inclusions.

Locations α,defined in

Figure 1(a)

1 5º

2 25º

3 45º

4 65º

5 85º

A constant pressure boundary condition was used at the inlet and the outlet of the ladle.

The pressure is equal to be the atmospheric pressure. The boundary condition was a non-

slip wall and the logarithmic-law wall function was used to bridge the near-wall layer.

Figure 1. Schematic diagram of: (a) calculation domain and inclusion release location and

(b) nozzle regions.

(a) (b)

0.7cm

0.25cm

Release

location

Moving interface

Gas

α

Inlet

Outlet

Wall Steel

1

2

3

4 5

Sym

met

rica

l

Axis

Region 6 2cm

Region 5 2cm

Region 4 2cm

Region 3 2cm

Region 2 3cm

Region 1 1cm

Nozzle Inlet

Nozzle Outlet

3.5cm

Z

Y(X)

7

Inclusion was assumed to stick to the wall when they touch a wall and escape from the

calculation domain when they pass the outlet.

Steel Flow in a Tundish (Supplement III)

A three-dimensional model of a tundish was developed. The diagram and parameters of

the tundish are shown in Figure 2 and Table 2, respectively. Steel density and viscosity

are 7000 kg/m3 and 0.0064 kg/(m·s), respectively.

The liquid steel flow rate, 2.3 m/s, is fixed at the inlet of the tundish. The pressure at the

SEN outlet is constant and equal to the atmospheric pressure. The slag at the top surface

of tundish steel is assumed to behave as a stationary wall. The roughness of the SEN wall

is 2.7×10-4

m. The boundary condition was a non-slip wall and the logarithmic-law wall

function was used to bridge the near-wall layer.

Table 2. Parameters of the tundish

Parameter Value

Tundish width at the free surface 0.7 m

Tundish width at bottom 0.5 m

Tundish length at the free surface 2.0 m

Tundish length at bottom 1.8 m

Bath depth 0.5 m

Distance between inlet and outlet 1.27 m

Distance between inlet and dam 0.55 m

Distance between inlet and weir 0.4 m

Inlet pipe diameter 0.068 m

SEN diameter 0.07 m

SEN length 0.5 m

Inlet Weir

Dam

Stopper-rod

Outlet

Fig. 2. Diagram of a tundish.

8

Water Flow in a Horizontal Straight Pipe (Supplement IV)

A three-dimensional model of a water flow in a horizontal straight pipe was developed.

Water density and viscosity are 998 kg/m3 and 0.001003 kg/(m·s), respectively. The

diameter and the length of the pipe are 0.2 m and 1.5 m, respectively. The water flow rate,

1.5 m/s, is fixed at the inlet of the pipe. The pressure at the pipe outlet is constant and

equal to the atmospheric pressure. The fluid flow and its turbulent properties in the pipe

were solved by using the commercial software ANSYS FLUENT 14.5®

. The boundary

condition was a non-slip wall and the Enhanced Wall Treatment method was used to

bridge the near-wall layer.

2.2 Particle Tracking and Deposition Model

2.2.1 Lagrangian Particle Tracking Model

A Lagrangian method[40]

was used to track an individual particle to see its behavior in a

steel flow. The position of an individual particle can be obtained by solving the following

equation:

𝑑𝑥𝑝𝑖

𝑑𝑡= 𝑢𝑝𝑖 (2)

where 𝑥𝑝𝑖 is the particle position, 𝑢𝑝𝑖 is the particle velocity in i direction .

The particle velocity is obtained after solving the following particle momentum equation

by considering drag force, buoyancy force and gravity force:

𝑚𝑝𝑑𝑢𝑝𝑖

𝑑𝑡= 𝐷𝑝(𝑢𝑖 − 𝑢𝑝𝑖) + 𝑚𝑝𝑔𝑖 (1 −

𝜌

𝜌𝑝) (3)

where 𝑚𝑝 is the mass of the particle, 𝑢𝑖 is the continuous-phase velocity, 𝜌𝑝 is the pure

particle density. Also, the drag function, 𝐷𝑝, is expressed as follows:

𝐷𝑝 =1

2𝜌𝜋𝑑𝑝

2

4𝐶𝐷1|𝑢𝑖 − 𝑢𝑝𝑖| (4)

where 𝐶𝐷1 is the drag coefficient, which according to Clift, Grace and Weber [41]

may be

expressed as follows:

𝐶𝐷1 =24

𝑅𝑒_𝑝(1 + 0.15𝑅𝑒_𝑝

0.687) +0.42

1+4.25×104𝑅𝑒_𝑝−1.16 (5)

9

where 𝑅𝑒_𝑝 is the particle Reynolds number. This drag coefficient function is valid for

spherical particles and 𝑅𝑒_𝑝 < 3 × 105.

In order to incorporate the effect of turbulent fluctuations on inclusion motion, a

stochastic turbulent model can be used. The eddy lifetime [42]

spawned the eddy-

interaction models. The model is based on an approach in which fluid velocities (eddies)

are taken to be stochastic quantities, which remains constant for the lifetime of the eddy

or, if shorter, the transit time of the particle through the eddy.[43,44]

The continuous-phase

velocity can be expressed by the following equation:

𝑢𝑖 = 𝑢�̅� + 𝑢𝑖́ (6)

where 𝑢�̅� and 𝑢𝑖́ are the continuous-phase average velocity and the fluctuating component,

respectively.

2.2.2 Eulerian Particle Deposition Model

The model is based on the following: 1) the volume fraction of the dispersed particles is

very low, and the particles are spherical and do not interact with each other; 2) a one-way

coupling is assumed; 3) particles are assumed to deposit onto a wall after they touch the

wall. The model arises from the basic conservation equations of a fluid-particle system in

an Eulerian frame of reference. In a steady state, the equations of motion can be written

as follows: [32, 33]

∇ ∙ (𝜌𝑝𝑽𝑝) = 0 (7)

𝜌𝑝(𝑽𝑝 ∙ ∇)𝑽𝑝 = −∇𝑝𝑝 + 𝜌𝑝𝑭 + 𝜌𝑝𝑮 (8)

where 𝑽𝑝 represents the velocity vector of the particles, on which a random thermal

velocity is superposed that gives rise to the partial pressure 𝑝𝑝, 𝑮 is the total external

force vector per unit mass (e.g. gravitational, electromagnetic) on the particles, 𝜌𝑝 is the

partial density of particles, 𝑭 is the force vector per unit mass on the particles due to the

fluid: [45]

𝑭 = ∅𝐷𝑽𝑓 − 𝑽𝑝

𝜏𝐶− (

𝜂

𝑚𝑝)∇ln𝑇 + 𝑭𝑉 + 𝑭𝐵 + 𝑭𝑆 + 𝑭𝐵𝑢𝑜𝑦 (9)

10

𝜏𝐶 = 𝜏𝑝 [1 +𝜆

𝑑𝑝(2.514 + 0.8 × exp (−0.55

𝑑𝑝

𝜆))] (10)

where ∅𝐷 =𝑅𝑒𝑝

24𝐶𝐷2

[34] and is equal to 1 for the case of Stokes flow assumed, 𝐶𝐷2is the

particle drag coefficient can be approximated by Clift and Gauvin [46]

, 𝑽𝑓 is the fluid

velocity vector and 𝜏𝑝 = 2𝜌𝑝𝑜𝑟2/(9𝜇) is the particle relaxation time

[32] with 𝜏𝐶

[47] as its

corrected form. Furthermore, 𝜌𝑝𝑜, 𝑑𝑝, r and 𝑚𝑝 are the mass density, diameter, radius and

mass of a pure particle, respectively. Also, µ is the dynamic viscosity of the fluid, 𝜆 is the

mean free length of the fluid, T is the temperature of the fluid, and 𝜂 is the

thermophoretic force coefficient (the thermophorectic force was defined as in the

reference of Talbot et al. [48]

). 𝑭𝑉 is the virtual mass force vector, [49]

since an accelerating

or decelerating body must move (or deflect) some volume of surrounding fluid as it

moves through it. For a single, non-deformable and spherical particle, the virtual mass

force per unit particle mass can be expressed as 𝑭𝑉 = −𝜌𝑓

2𝜌𝑝𝑜 𝒂𝑽,

[49] where 𝜌𝑓 and 𝑎𝑉 are

the density of the fluid and the virtual mass acceleration term, respectively. The

parameter 𝑭𝐵 is the Basset-Boussinesq force vector, [50]

which is difficult to implement

and is commonly neglected for practical reasons. The parameter 𝑭𝑆 is the Saffman lift

force vector.[51]

However, the lift force is questioned by many researchers with regard to

its significance, its region of validity and its formulation, and it is not clear whether it

improves the result or not. [32, 52-54]

As Guha [32]

pointed out, Saffman originally derived

his results for an unbounded shear flow whereas the deposition rate prediction applies in

the vicinity of a solid wall. This makes the use of the expression for lift force

questionable. Therefore, this force is not considered in the present work. The buoyancy

vector on the particle can be expressed as: [33]

𝑭𝐵𝑢𝑜𝑦 = − (𝜌𝑓

𝜌𝑝𝑜)𝒈 (11)

where 𝒈 is the gravity acceleration vector.

Deposition in Vertical Flows

Finally, the flux of particles in the direction perpendicular to the wall Eq. (12), the

particle momentum equations Eq. (13) in the y-direction (the direction normal to the wall)

11

and Eq. (14) in the x-direction (the direction parallel to the wall) can be obtained after

carrying out a Reynolds-averaging for a fully developed vertical pipe flow. [32, 34]

The x-

and the y-directions are illustrated in Figure 3.

𝐽 = −(𝐷𝐵 + 𝜀𝑝)𝜕�̅�𝑝

𝜕𝑦− �̅�𝑝𝐷𝑇

𝜕ln𝑇

𝜕𝑦+ �̅�𝑝�̅�𝑝𝑦

𝑐 (12)

�̅�𝑝𝑦𝑐𝜕�̅�𝑝𝑦

𝑐

𝜕𝑦+ ∅̅𝐷

�̅�𝑝𝑦𝑐

𝜏𝐶= −

𝜕𝑉𝑝𝑦′2̅̅ ̅̅

𝜕𝑦+ 𝐹𝑉𝑦 + 𝑔𝑦 + 𝐹𝐵𝑢𝑜𝑦_𝑦 (13)

�̅�𝑝𝑦𝑐𝜕�̅�𝑝𝑥

𝜕𝑦=∅̅𝐷𝜏𝐶(�̅�𝑓𝑥 − �̅�𝑝𝑥) + 𝐹𝑉𝑥 + 𝑔𝑥 + 𝐹𝐵𝑢𝑜𝑦_𝑥 (14)

where 𝐷𝐵 is the Brownian diffusivity [47]

, 𝜀𝑝 is the particle eddy diffusivity, �̅�𝑝𝑦𝑐 is the

particle convective velocity in the y direction, 𝐷𝑇 is the coefficient of a temperature-

gradient-dependent diffusion [32]

, �̅�𝑝 is the mean partial density of the particles, 𝑉𝑝𝑦′2̅̅ ̅̅ is the

particle mean-square velocity, and �̅�𝑓𝑥 and �̅�𝑝𝑥 are the mean velocities of the fluid flow

and the particles in the x direction, respectively. 𝐹𝑉𝑥 and 𝐹𝑉𝑦 are the virtual mass force in

the x and y direction, respectively. Since the y-direction velocity of the fluid is assumed

to be equal to zero in a fully developed flow and since 𝑽𝑝 is very close to 𝑽𝒑𝒄 (particle

convective velocity vector) for large particles, and since the acceleration term is not

significant for small particles, as is commented by Guha [32]

, 𝐹𝑉𝑦 can be simply expressed

as follows:

𝐹𝑉𝑦 = −𝜌𝑓

2𝜌𝑝𝑜 𝑎𝑉𝑦 = −

𝜌𝑓

2𝜌𝑝𝑜

∂�̅�𝑝𝑦𝑐

∂𝑡= −

𝜌𝑓

2𝜌𝑝𝑜 �̅�𝑝𝑦

𝑐∂�̅�𝑝𝑦

𝑐

∂𝑦 (15)

y

x

Fig. 3. Diagram of the x and y directions of the particle conservation equations.

12

Eq. (13) depends on �̅�𝑝𝑥 in Eq. (14) only through Saffman’s lift force. Since lift force is

neglected, the particle flux can be obtained simply by solving the flux Eq. (12) and the y-

direction momentum Eq. (13). These two equations can be made dimensionless as

follows:[32, 34]

𝑉𝑑𝑒𝑝+ = −(

𝐷𝐵𝜈+𝜀𝑝

𝜈)𝜕𝜌𝑝

+

𝜕𝑦+− 𝜌𝑝

+𝐷𝑇+𝜕ln𝑇

𝜕𝑦++ 𝜌𝑝

+�̅�𝑝𝑦𝑐+ (16)

(1 +𝜌𝑓

2𝜌𝑝𝑜)�̅�𝑝𝑦

𝑐+𝜕�̅�𝑝𝑦

𝑐+

𝜕𝑦++ ∅̅𝐷

�̅�𝑝𝑦𝑐+

𝜏𝐶+ = −

𝜕

𝜕𝑦+(𝑉𝑝𝑦

′+2̅̅ ̅̅ ̅̅ ) (17)

where 𝜈 is the kinematic viscosity of the fluid. The expressions for some dimensionless

parameters are shown in Table 3.

Table 3. Expressions for the dimensionless parameters.

𝑉𝑑𝑒𝑝+ �̅�𝑝𝑦

𝑐+ 𝜌𝑝+ 𝑉p𝑦

′+ 𝐷𝑇+ 𝜏𝐶

+ 𝜏𝑝+

𝐽/(𝜌𝑝𝑜𝑢∗) �̅�𝑝𝑦

𝑐 /𝑢∗ �̅�𝑝/𝜌𝑝𝑜 𝑉p𝑦′ /𝑢∗ 𝐷𝑇/𝜈 𝜏𝐶𝑢

∗2/𝜈 𝜏𝑝𝑢∗2/𝜈

Note: 𝜌𝑝𝑜 is the partial density of particles at the pipe center, 𝑢∗ is the friction velocity.

Deposition in Horizontal Straight Flows

The particle flux can be obtained by solving the flux equation Eq. (12) and the y-direction

momentum equation Eq. (13) by considering the influence of gravity and buoyancy in y-

direction momentum equation. Eq. (12) and Eq. (13) can be non-dimensionalized as

follows: [32, 34]

𝑉𝑑𝑒𝑝+ = −(

𝐷𝐵𝜈+𝜀𝑝

𝜈)𝜕𝜌𝑝

+

𝜕𝑦+− 𝜌𝑝

+𝐷𝑇+𝜕ln𝑇

𝜕𝑦++ 𝜌𝑝

+�̅�𝑝𝑦𝑐+ (18)

(1 +𝜌𝑓

2𝜌𝑝𝑜)�̅�𝑝𝑦

𝑐+𝜕�̅�𝑝𝑦

𝑐+

𝜕𝑦++ ∅̅𝐷

�̅�𝑝𝑦𝑐+

𝜏𝐶+ = −

𝜕

𝜕𝑦+(𝑉𝑝𝑦

′+2̅̅ ̅̅ ̅̅ ) − (1 −𝜌𝑓

𝜌𝑝𝑜)𝑔

+ cos 𝜃 (19)

where 𝑔+ = 𝑔𝜈

𝑢∗3 is the dimensionless form of the gravity acceleration rate. The

definition of θ in a pipe can be seen from Figure 4. In a rectangle cross-section ducts, θ is

equal to be 0 for a floor deposition, π/2 for a wall deposition and π for a roof deposition.

13

For deposition in an air-particle system where the density ratio 𝜌𝑓/𝜌𝑝𝑜 is very small, it is

reasonable to neglect the virtual mass force. Since the heavy particles may not firmly

follow the air flow fluctuation, we use the relationship between the particle eddy

diffusivity and the turbulent viscosity of the fluid, 𝜈𝑡 [55]

, which also was used by Zhao

and Wu [47]

, 𝜀𝑝

𝜈=

𝜈𝑡

𝜈(1 + 𝜏𝐶/𝑇𝐿)

−1. This, in turn, is based on Hinze’s results [56]

. The

particle mean-square velocity was estimated by using its relationship with the fluid mean-

square velocity. [32, 52, 55]

However, for a liquid-particle system, e.g. a molten steel-particle

system, the liquid density is close to or even greater than the particle density. Therefore,

the virtual mass force is considered in the particle momentum equation. In addition,

particles tend to firmly follow the steel flow because their density is close to or less than

that of steel, especially for small-size particles. Therefore, the particle eddy diffusivity

and the particle mean-square velocity are equal to the turbulent viscosity of the liquid

flow and the liquid flow mean-square velocity, respectively.

2.3 Solution Methods

2.3.1 Solution Method of Fluid Flow Model

Fluid flow field was solved by using commercial software PHOENICS®

used in

supplement I-III and commercial software ANSYS FLUENT 14.5®

used in supplement

IV.

Fig. 4. Diagram of the cross section of the horizontal pipe.

θ

Horizontal

Vertical

Pipe wall

Location 4

Location 5

Location 1

Location 3

Location 2

14

In PHOENICS, the Kim-Chen modified k-ɛ model [36]

was used to describe the turbulence

properties of steel flow. The Log-Law wall function was used to bridge the near-wall

layer and the outside fully developed turbulent flow region. The equation formulation

used was elliptic-staggered. The solution algorithm used for velocity and pressure was

the Semi-Implicit Method for Pressure-Linked Equations Shortened, abbreviated as the

SIMPLEST method, which is a modified version of the well-known SIMPLE method.

The Hybrid scheme was used as the differencing scheme. The global convergence

criterion was set to 0.01% for all variables.

In FLUENT, the realizable k-ε model [39]

including an enhanced wall treatment method

was used. The SIMPLE scheme was used for the pressure-velocity coupling. The

Standard discretization method was adopted to discretize the pressure. The governing

equations were discretized by using a second order upwind scheme. The convergence

criteria were as follows: the residuals of all dependent variables were less than 10-4

.

2.3.2 Solution Method of Particle Deposition Model

Guha [33]

recommended the use of a time-marching FDM (finite difference method) to

solve Eq. (17) and (19) by adding a time-dependent term 𝑑�̅�𝑝𝑦

𝑐+

𝑑𝑡 and a Gaussian elimination

to solve equations originating from FDM discretization of Eq. (16) and (18), writing Eq.

(16) and (18) as: 𝑑

𝑑𝑦+[− (

𝐷𝐵

𝜈+𝜀𝑝

𝜈)𝜕𝜌𝑝

+

𝜕𝑦+− 𝜌𝑝

+𝐷𝑇+ 𝜕ln𝑇

𝜕𝑦++ 𝜌𝑝

+�̅�𝑝𝑦𝑐+] = 0, to obtain a density

profile in the boundary layer. Actually, a single-pass marching FDM (without adding the

time-dependent term) on Eq. (17) and (19) can also yield an accurate solution (only one

solution is physically meaningful). In addition, in order to guarantee the numerical

stability, an artificial viscosity term, −∆𝑡

2[(1 +

𝜌𝑓

2𝜌𝑝𝑜)�̅�𝑝𝑦

𝑐 ]2𝜕2�̅�𝑝𝑦

𝑐

𝜕𝑦2, can be added into the

discrete momentum equation based on Young and Leeming [34]

. Both a time-marching

FDM and a single-pass marching FDM require a sufficient fine grid. The lower and upper

boundary conditions are listed in Table 4. The commercial MATLAB®

software was

used to solve the deposition model equations.

15

Table 4. Boundary conditions used in the deposition model equations.

Boundary y+ value Eq. (17) and (19) Eq. (16) and (18)

The lower boundary y+=F

++r

+ - 𝜌𝑝

+ = 0

The upper boundary y+=60 �̅�𝑝𝑦

𝑐+ = 0 𝜌𝑝+ = 1

Note: F+ is a hybrid parameter that combines the surface roughness and the peak-to-peak distance

[52]; r

+ is

the dimensionless particle radius with the following expression: 𝑟+ = 𝑟𝑢∗/𝜈.

According to Hussein et al. [52]

, the hybrid parameter of the roughness, F+, has the

following expressions:

𝐹+ =

{

𝑚𝑢∗

𝜈ln (

𝐾+

𝐿+) +

𝑐𝑢∗

𝜈 𝛼𝑜 ≤

𝐾+

𝐿+≤ 0.082

0 𝐾+

𝐿+< 𝛼𝑜

(20)

where 𝛼𝑜 = 0.0175 , m = 54.86 ± 10.29 µm, 𝑐 = 222.02 ± 8.94 µm, 𝐾+ = 𝐾𝑢∗/𝜈 ,

and 𝐿+ = 𝐿𝑢∗/𝜈. K is the roughness height and K+ is its dimensionless form and L is the

peak-to-peak distance between roughness elements and L+ is its dimensionless form.

Kay and Nedderman [57]

gave a temperature profile in a boundary layer, which may be

expressed as follows:

𝑇(𝑦+) − 𝑇𝑊∆𝑇

=

{

𝑃𝑟𝑦+

∆𝑇60+ 𝑦

+ < 5

5𝑃𝑟 + 5ln (0.2𝑃𝑟𝑦+ + 1 − 𝑃𝑟)

∆𝑇60+ 5 ≤ 𝑦+ ≤ 30

5𝑃𝑟 + 5 ln(1 + 5𝑃𝑟) + 2.5ln (𝑦+/30)

∆𝑇60+ 30 ≤ 𝑦+ ≤ 60

(21)

where ∆𝑇60+ = 5𝑃𝑟 + 5 ln(1 + 5𝑃𝑟) + 2.5ln (60/30), TW is the wall temperature, Pr is

the Prandtl number, ∆𝑇 is the temperature difference between the fluid at the turbulent

core and fluid at the wall.

16

17

Chapter 3 Results and Discussions

3.1 Steel Flow and Inclusion Behavior in a Ladle Teeming Process

In this part, first a ladle teeming process was simulated by using PHOENICS®. Thereafter,

the behavior of different-size Ce2O3 inclusion in a fixed steel flow field was investigated.

The number of inclusions that touched the different regions of the nozzle wall was

counted and compared with the available experimental data.

3.1.1 Steel Flow in a Nozzle during a Teeming Stage

Figure 5-8. Predicted properties of the steel flow field at a time when around 300 kg steel is left in the ladle. Figure 5. Turbulent kinetic energy. Figure 6. Turbulent dissipation rate.

Figure 7. Shear stress. Figure 8. Velocity contour.

Note: Figure (b) in the Figure 5-8 is the enlarged pictures of the oval enclosed parts of the corresponding

Figure (a).

(a) (b) (b) (a)

(b) (a) (a) (b)

18

The steel flow field in the nozzle at a point when around 300 kg steel is left in the ladle is

shown in Figure 5-8. Figure 5 and Figure 6 show the turbulent kinetic energy and

turbulent dissipation rate of the steel flow field in the nozzle predicted by using the Kim-

Chen k-ɛ turbulent model, respectively. It can be seen that the turbulent properties reach

their maximum values approximately at the connection region of the straight pipe part

and the expanding part of the nozzle (region 5 in Figure 1(b)). This illustrates that the

steel flow is very chaotic in this region. The largest shear stress value also exists around

this region, as is shown in Figure 7. Figure 8 shows the velocity distribution of the steel

flow in the nozzle. The quick change of the velocity contour around region 5 indicates a

high velocity gradient, which means that a turbulent flow is developed very quickly. In

the nearby regions, where a nozzle geometry transition exists, a change of the steel flow

direction occurs. The velocity of the radical steel flow greatly decreases due to its

collision with a downwards directed steel flow at the nozzle core part. This kind of

collision also increases the chaotic characteristics of the flow, which results in an

increased turbulence level of the steel flow.

3.1.2 Inclusions Tracking Neglecting a Stochastic Turbulent Motion of Inclusions

Figure 9. Locations of inclusions at different times (Coordinate origin is located

at the center of the ladle).

Nozzle

wall

Release

location 3

Construct

object

400µm Other sizes of

inclusions

Z

X(Y)

19

Six different sizes of inclusions, 0.5 µm, 3 µm, 10 µm, 20 µm, 100 µm and 400 µm,

released from location 3 in Table 1 were tracked using a Lagrangian method under the

previously obtained fixed flow field. As mentioned before, it is reasonable to use the

fixed flow field due to that only a small change of the flow field occurs during the short

time that the inclusions pass through the nozzle. In addition, the focus is to compare the

inclusion behaviors in the same flow field. In order to obtain a clear view on the flow

abilities of different sizes of inclusions, a stochastic turbulent model for inclusion

movement is not used at first. In this way, the uncertainty that a stochastic turbulent

random motion leads to is reduced.

Figure 9 shows the locations of inclusions at different times in the nozzle. It can be seen

that inclusions with a diameter of 0.5 µm, 3 µm, 10 µm, 20 µm and 100 µm have similar

trajectories. However, for an inclusion with a diameter of 400 µm, the trajectory is

obviously different from the other inclusions. It moves closer to the nozzle center and

takes a much longer time before it reaches the nozzle region than the other inclusions.

This can be seen in Figure 10(b). The behaviors of inclusions are mainly determined by

three forces: i) an inertia force, ii) a drag force and iii) a buoyancy force due to a density

difference between an inclusion and steel. In the current situation, the angle between the

upwards buoyancy force and the downwards drag force is larger than 90°. Therefore, the

drag force in the z direction needs to combat the buoyancy force to make inclusions move

to the nozzle region. In order to explain the obviously different behaviors of 400 µm

inclusions compared to other sizes of inclusions, the buoyancy force and the drag force

for 100 µm and 400 µm inclusions at the release location 3 as well as at the straight pipe

location, around 0.026 m from the nozzle outlet, were calculated, as is shown in Table 5.

At the release location 3, it can be seen that the downwards drag force of 400 µm

inclusions has a similar magnitude to the upwards directed buoyancy force. However, a

much larger drag force than a buoyancy force was obtained for the 100 µm inclusions.

This means that 100 µm inclusions can move much faster in the downwards z direction

compared to 400 µm inclusions. The competition of these two forces in the z direction

makes 400 µm inclusions to take a longer time than smaller inclusions, like 100 µm

inclusions, to move to the nozzle region. This also gives them more time to move towards

the nozzle center, as is shown in Figure 9. The larger inertia of big inclusions than that of

20

small inclusions also causes them to take a longer time to respond under the same

conditions. At the straight pipe location, the drag forces of both 100 µm and 400 µm

inclusions are much larger than the buoyancy forces, which cause them to move fast in

the nozzle pipe region.

Table 5. Buoyancy force and drag force of inclusions in the z direction. Location Size, µm Buoyancy force, N Drag force, N Acceleration a, m2/s

Release location 3 100 -1.02×10-9 1.50×10-8 3.93

400 -6.53×10-8 6.70×10-8 7.66×10-3

Pipe location, 0.026m from nozzle outlet

100 -1.02×10-9 1.50×10-7 36.79 400 -6.53×10-8 2.09×10-5 91.70

The change of inclusion velocities in the y direction, parallel to the cross section of the

nozzle, and the z direction, vertical to the cross section of the nozzle, as a function of

time are shown in Figure 10(a) and (b), respectively. The characteristics of the inclusion

velocities in the x direction, which is not shown here, are similar as those in the y

direction, except with respect to the velocity magnitude. It can be seen that inclusions

with diameters of 0.5 µm, 3 µm, 10 µm and 20 µm have a similar velocity pattern with

minor differences of the magnitude for the same elapsed time. This means that they have

similar trajectories, as previously shown in Figure 9. With the increase of inclusion sizes,

for especially inclusions larger than 100 µm, the maximum velocities of inclusions in

both the y and z directions decrease. This can clearly be seen in Figure 10(c) and (d). In

the y direction, the main reason for that is the inertia of inclusions. For the z-direction

inclusion velocity, both the inertia force and the buoyancy force should be responsible for

a little bit smaller velocity magnitude for the larger inclusions than that for the smaller

inclusions. From Figure 10(c), it can be seen that a sharp decrease of the y-direction

inclusion velocities occurs at the locations of 3 cm and 7 cm away from the nozzle outlet,

where the connection regions of the nozzle exist. As previously mentioned, a rapid

decrease of steel velocity in the y direction should be the reason for that. In Figure 10 (d),

the inclusion velocity increases rapidly within a 2 cm distance, going from 5 cm down to

3 cm distance from the nozzle outlet. This location is situated just above the straight pipe

part of the nozzle. This also illustrates that the turbulence intensity increases quickly in

this region.

21

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0 2 4 6 8 10 12 14 16 18

y-d

irec

tio

n v

elo

city

, m

/s

Time, s

-0.5

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10 12 14 16 18

z-d

irec

tio

n v

elo

city

, m

/s

Time, s

0

0.02

0.04

0.06

0.08

0.1

0.12

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

y-d

irec

tio

n v

elo

city

, m

/s

Vertical distance from the release location 3, m

0

0.5

1

1.5

2

2.5

3

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

z-d

irec

tio

n v

elo

city

, m

/s

Vertical distance from the release location 3, m

Figure 10. Velocities of inclusions in nozzle, (a) and (b) are y-direction and z-direction velocities

of inclusions as a function of time, respectively; (c) and (d) are y-direction and z-direction velocity

distributions of inclusions at different distance from the release location 3, respectively.

0.5μm

3μm

10μm

20μm

100μm

400μm

(a)

(b)

(c)

0.5μm

3μm

10μm

20μm

100μm 400μm

(d)

0.5μm

3μm

10μm

20μm 400μm

100μm

10&20μm

100μm

0.5μm

3μm

400μm

Nozzle

outlet

Nozzle

outlet

Reaching nozzle inlet Reaching nozzle inlet

Nozzle

inlet

Nozzle

inlet

22

3.1.3 Inclusion Behavior Including Stochastic Turbulent Motions

The behaviors of inclusions in a nozzle were statistically analyzed. In order to understand

the inclusion behaviors at different nozzle regions (shown in Figure 1(b)), a statistical

analysis was also carried out to investigate the sensitivity of different nozzle regions on

the possible inclusions deposition. Considering the result of 3.1.2 and also that only a

small number of large-size inclusions, e.g. bigger than 100 µm, exist in steel, three sizes

of inclusions, 1 µm, 10 µm and 100 µm, were tracked. Twenty inclusions were released

from each location, as is shown in Table 1.

0

2

4

6

8

10

12

14

16

1µm 10µm 100µm

Nu

mb

er o

f in

clu

sio

ns

tou

ch t

he

wal

l

Particle size

0

2

4

6

8

10

12

14

Location 1 Location 2 Location 3,4 and 5

Nu

mb

er o

f in

clu

sio

ns

tou

ch t

he

wal

l

Inclusion release location

1µm

10µm

100µm

0

1

2

3

4

5

6

7

Region 1 Region 2 Region 3 Region 4 Region 5 Region 6Nu

mb

er o

f in

clu

sio

ns

tou

ch t

he

wal

l

Nozzle region

1µm

10µm

100µm

0

2

4

6

8

10

12

14

16

Region 1 Region 2 Region 3 Region 4 Region 5 Region 6Nu

mb

er o

f in

clu

sio

ns

tou

ch t

he

wal

l

Nozzle region

Location 1

Location 2

Figure 11. Number of inclusions that touch the nozzle wall, (a) influence of all 5 release

locations on the number of inclusions touching the nozzle wall, (b) number of different-size

inclusions from all release locations touching the nozzle wall, (c) and (d) influence of inclusion

sizes from all release locations and release locations for all sizes of inclusions on the number of

inclusions touching the nozzle wall at each nozzle region.

(a) (b)

(c) (d)

23

The number of inclusions that touch the nozzle wall is shown in Figure 11. Figure 11(a)

shows the influence of inclusion release locations on the number of inclusions touching

the wall. It can be seen that some inclusions released from location 1 and 2 touch the

nozzle wall. However, there is no inclusion, released from location 3, 4 and 5, that

touches the nozzle. This illustrates that the deposition possibility of inclusions from

location 3, 4 and 5 is very low. The inclusions released from location 1 have the highest

possibility to touch the nozzle wall among all the release locations. One possible reason

is that location 1 is close to the nozzle wall. Therefore, an inclusion released at this

position has a short distance to pass through to the nozzle wall. Figure 11(b) shows the

number of different-size inclusions from all the release locations touching the nozzle wall.

For each size of inclusions, 10 µm inclusions seem to have a little bit higher possibility to

touch the nozzle wall than the other two sizes of inclusions.

Figure 11(c) and 11(d) show the influence of inclusion sizes and release locations on the

number of inclusions touching the wall for each nozzle region. It can be seen that for the

1 µm inclusions, the distribution of inclusions that touch the nozzle wall along the nozzle

height is more uniform than for the other two inclusion sizes. For 10 µm and 100 µm

inclusions, two nozzle regions, region 1 and region 5, have a higher number of inclusions

that touch the nozzle wall compared to the other regions. This illustrates that region 1 and

region 5 may have a higher possibility of clogging than the other regions. In region 1, the

steel flow velocity is very small. The turbulence properties in Figure 5 and 6 also show

that turbulence intensity is not high in this region. Furthermore, Figure 11(d) shows that

all the inclusions that touch the nozzle wall within regions 1 to 4 are from release location

1. Therefore, the following reasons should be responsible for a large number of

inclusions moving to the wall in the nozzle region 1: 1) the release location 1 is close to

the wall of nozzle region 1; 2) turbulence plays a positive role for the transport of

inclusions to the wall; 3) a centripetal force, which has also been investigated by Wilson

[9], should be helpful for the inclusion transport during flow direction changing in region

1. In region 5, steel velocity changes quickly, as is previously shown in Figure 8.

Therefore, the steel flow turbulence developed very fast. Furthermore, the collision

between the radial steel flow and the downwards core flow contributes to the flow

turbulence. Inclusions can obtain high momentum from turbulent fluid and move towards

24

the nozzle wall. A smaller diameter of the nozzle in this region gives inclusions a shorter

distance to pass through to reach the nozzle wall. Both the high turbulence intensity and

the short moving distance cause inclusions to easily touch the nozzle wall. For the

inclusions released from location 2, it can be seen from Figure 11(d) that inclusions only

touch the nozzle wall in region 5 and region 6. As previously mentioned, a high

turbulence as well as a short moving distance should be the reason for that.

The experimental results, as is shown in Figure 12, illustrate the reliability of this

simulation work. It can be seen that a serious clogging is found in region 5, which is in

good agreement with the present simulation results. Region 1 is located at the connection

part of the nozzle and ladle bottom. After an experiment, the steel left in ladle must be

tapped out. This makes it difficult to get a sample of solidified steel from that region.

However, the experiments reported by Kojola [58]

supports the model results, which

demonstrate that clogging also frequently occurs in the upper part of the nozzle. Both the

simulations and experiments show that the transition region of the geometry, or flow field,

is the sensitive region for an inclusion deposition as well as for clogging.

Figure 12. Pictures of a part of the clogged nozzle that correspond to region 5 and 6 in Figure

1(b). ((a)[58]

and (b): Erik Roos, personal communication, January 10, 2013)

Steel flow direction Steel

Inclusions

Inclusions

(b)

Steel flow direction Steel

Inclusions

Inclusions

(a)

25

The analysis in this study gives some information on inclusion behaviors in nozzles.

Despite that the model cannot give an estimation on the deposition rate of inclusions, it

provides information on where inclusions touch the nozzle wall and which places may be

sensitive for nozzle clogging. Thus, a more complete deposition model will be developed

to predict the deposition rate of particles/inclusions.

3.2 Inclusion Deposition Rate in Vertical Steel Flows

In this part, the deposition rate of inclusions was predicted by the use of an Eulerian

deposition model. A parameter study was carried out to investigate the influence of

particle density, particle size, steel flow rate, wall roughness and temperature difference

between wall and turbulent flow core on the particle deposition rate. Furthermore, the

deposition rate of inclusions on the SEN wall of a tundish was predicted.

3.2.1 Parameter Study on Inclusion Deposition Rate in Steel Flows

3.2.1.1 Effect of Steel Flow Rate

Figure 13 shows the influence on the deposition rate of the inlet steel flow velocity (or

friction velocity) in the vertical pipe. In Figure 13 (a), it can be seen that the deposition

rates for these three inlet velocities tend to be similar when the particle size becomes very

small. For such small particles, Brownian and turbulent diffusion rather than

turbophoresis play the dominant role in causing a particle deposition. Turbophoresis is

more important for larger particles, which normally have a larger particle relaxation time.

Brownian diffusion is primarily related to the particle diameter and turbulent diffusion is

primarily related to the distance from the wall in a turbulent flow. Consequently, the

deposition rate of very small particles is almost independent of the friction velocity.

Therefore, it is reasonable to neglect the influence of turbophoresis on these small

particles, since their relaxation time is small. This simplification will not lead to any large

errors in the predicted deposition rates for small particles, as Lai and Nazaroff [59]

did. As

the particle diameter increases, the turbophoresis becomes dominant. This leads to a

steady increase of the particle deposition rate. Large particles reach the wall mainly by

the convective velocity imparted by the turbophoresis. A higher frictional velocity will

lead to a higher dimensionless relaxation time for the same size of a particle as well as a

26

stronger effect of the turbophoresis. Therefore, the deposition rate increases with an

increased friction velocity, which follows from a higher steel flow rate. However, for

smaller particles with a shorter particle relaxation time, turbophoresis can be neglected.

In Figure 13 (b), the same value of the dimensionless particle relaxation time under

different steel flow velocities corresponds to different particle diameters. The larger the

friction velocity, the smaller is the particle diameter. Smaller particles have a larger

Brownian diffusion rate which is dominant. Therefore, a larger deposition rate for a

higher flow rate was observed for the same 𝜏𝑝+in the lower end of the graph.

(a)

(b)

10-7

10-6

10-5

10-4

10-6

10-5

10-4

10-3

10-2

10-1

dp,m

Vdep

+

A vf=0.2 m/s, N

grid=10000

B vf=0.2 m/s, N

grid=20000

C vf=1 m/s, N

grid=10000

D vf=3 m/s, N

grid=10000

10-8

10-6

10-4

10-2

100

102

10-6

10-5

10-4

10-3

10-2

10-1

p+

Vdep

+

A vf=0.2 m/s, N

grid=10000

C vf=1 m/s, N

grid=10000

D vf=3 m/s, N

grid=10000

Fig. 13 Influence of the inlet steel flow rate on the particle deposition rate: (a) deposition rate

versus the particle diameter; (b) deposition rate versus 𝜏𝑝+; 𝜌Ce2O3 = 6800 kg/m

3, ΔT=0, F+=0.

D

C

A&B

A C D

27

3.2.1.2 Effect of Particle Density on Deposition Rate

Figure 14 shows the effect of the particle density on the deposition rate for two different

steel flow rates. When the particle diameter is small, the deposition rates of different-

density particles under the same friction velocity (steel flow rate) are similar. However,

the influence of the density on the deposition rate increases with an increased particle

diameter. The relaxation time for the small particles is very small, as is shown in Figure

14. This means that the effect of turbophoresis is weak compared to diffusion. With an

increasing particle size, the relaxation time is increased by several orders of magnitude.

Therefore, the role of turbophoresis becomes increasingly important. Different densities

lead to different particle relaxation times for the same particle size. Thus, the influence of

the density on the deposition rates is greater for larger particles, i.e. when the

turbophoresis becomes important.

3.2.1.3 Effect of Wall Roughness on Deposition Rate

Figure 15 shows the effect of the wall roughness on the particle deposition rate. In Figure

15 (a), the presence of a slight roughness significantly enhances the deposition rate of

smaller particles. However, its effect is minor for large particles. Figure 15 (b) shows that

Fig. 14. Influence of particle density on the particle deposition rate for two different inlet steel

flow rates, 𝜌Ce2O3 = 6800 kg/m3, 𝜌Al2O3 = 3500 kg/m

3, ΔT=0, F+=0. (Note: The values

given in the figure are the corresponding dimensionless particle relaxation times)

10-7

10-6

10-5

10-4

10-6

10-5

10-4

10-3

10-2

10-1

dp,m

Vdep

+

C vf=1 m/s,

p=6800 kg/m3

E vf=1 m/s,

p=3500 kg/m3

D vf=3 m/s,

p=6800 kg/m3

F vf=3 m/s,

p=3500 kg/m3

𝜏𝑝+ value

D: 1.40×10-3 F: 7.06×10-4

C: 2.01×10-4

E: 1.03×10-4

𝜏𝑝+ value

D: 1.22 F: 0.63

C: 0.18

E: 0.09

28

F+ has a much larger value than the r

+ value for small particles, which means that the

roughness value is very important. The roughness of a wall reduces the extension of the

boundary layer, i.e. particles have a shorter distance to reach the wall. This has a great

influence on small particles for which diffusion is the main transport mechanism. For

large particles, the contribution of the F+

value to the decrease in the boundary layer

extension is very small compared to the r+ value, as shown in Fig. 15(b). In addition,

turbophoresis is the main transport mechanism for large particles. The transport of large

particles is governed mainly by the magnitude of the convective velocities that particles

can obtain due to turbophoresis rather than by diffusion. The large particles which can

pass through the boundary layer most probably reach the wall regardless of whether or

not there is a slight additional thickness of boundary layer near the wall. Therefore, the

reduction in boundary layer extension that the roughness represents has a small influence

on the deposition rate of larger particles.

3.2.1.4 Effect of Temperature Gradient near the Wall on Deposition Rate

Figure 16 shows the influence of the deposition rate on the temperature gradient near the

wall. It can be seen that thermophoresis can greatly enhance the deposition rate of small

particles, even if the temperature of the turbulent core is only 5 K higher than that of the

pipe wall. It also shows a much higher contribution to the deposition rate of small

Fig. 15. (a) Influence of wall roughness on the particle deposition rate, (b) Change of r+/F

+ on the

particle diameter. (𝜌Ce2O3 = 6800 kg/m3, ΔT=0, 𝑣𝑓 = 1 m/s)

(a) (b)

10-7

10-6

10-5

10-4

10-6

10-5

10-4

10-3

10-2

dp,m

Vdep

+

F+=0.0

F+=0.25

F+=0.5

10-7

10-6

10-5

10-4

0

0.5

1

1.5

2

2.5

3

3.5

4

dp,m

r+/F

+

F+=0.25

F+=0.5

F+=0

F+=0.25

F+=0.5

29

10-7

10-6

10-5

10-4

10-6

10-5

10-4

10-3

10-2

dp,m

Vdep

+

T=0 K

T=5 K

T=10 K

T=20 K

particles than that of either turbulent or Brownian diffusion. In steel flows, there certainly

exists a temperature difference between the wall and the fluid. Thus, thermophoresis is a

vital mechanism for the deposition of small particles in liquid metals. However, for larger

particles (e.g. particle diameters greater than 10 µm for a steel flow rate of 1 m/s),

turbophoresis is still the dominant contribution to the particle deposition rate.

Figure 17 shows the effects of the temperature gradient and wall roughness on the

deposition rate. It can be seen that the influence of the temperature gradient is obviously

different from the influence of the wall roughness. For the influence of a wall roughness,

a gradual change in the deposition rate is observed with a gradually decreasing boundary

layer extension. This is because of both the roughness and particle radius. The roughness

reduces the distance which the particles need to travel to reach the wall and to thereby

increase the probability of deposition of particles. This results in an increase in the

deposition rate, even for a little bit larger particles. However, the curve of deposition

under a temperature gradient has an inflexion point. For particles with a diameter smaller

than that of the inflexion points, the deposition rate remains almost constant, whereas for

particles with a diameter larger than that of the inflexion point, a temperature gradient has

only a very small effect on the deposition rate. It can also be seen that the inflexion

region in the deposition-rate curves depends on the magnitude of the temperature gradient.

Fig. 16. Influence of the temperature gradient on the particle deposition rate, 𝜌Ce2O3 =

6800 kg/m3, F+=0, 𝑣𝑓 = 1 m/s.(A positive value of ΔT means that the fluid temperature

is higher than the wall temperature)

ΔT=0 K

ΔT=5 K

ΔT=20 K

ΔT=10 K

30

10-7

10-6

10-5

10-4

10-6

10-5

10-4

10-3

10-2

10-1

dp,m

Vdep

+

A: F+=0, T=0

B: F+=0.25, T=0

C: F+=0.5, T=0

D: T=10, F+=0

E: T=20, F+=0

3.2.2 Inclusion Deposition Rate in a Tundish SEN

3.2.2.1 Steel Flow in Tundish and SEN

Figure 18 shows the steel flow field in the tundish, and Figure 19 shows the steel flow

properties on the longitudinal sections of the middle SEN plane. It can be seen from these

figures that the steel flow in each longitudinal section of the SEN is almost symmetric.

On the longitudinal section of Y-Z middle SEN plane, two swirls exist at the bottom of

the SEN where a maximum turbulent kinetic energy can be found. This is in good

Fig. 17. Influence of the temperature gradient and the wall roughness on the particle deposition rate,

𝜌Ce2O3 = 6800 kg/m3, 𝑣𝑓 = 1 m/s.

(b) (a)

right view

Figure 18. Velocity of steel flow field in tundish, (a) front view of the X-Z middle SEN

plane, (b) right view of the Y-Z middle SEN plane.

X

Z

Y

Z

A

B D

C E

31

agreement with previous research result [15, 16]

. The steel flows in the Y-Z middle SEN

plane and in the X-Z middle SEN plane are very different in magnitude and distribution.

Figure 19. Properties of steel flow field in the middle sections of the SEN. X-Z Plane Y-Z Plane

Velocity Contour Velocity

Vector KE

Velocity

Contour

Velocity

Vector KE

Figure 20 shows the turbulent kinetic energy at different cross sections of the tundish

SEN. Obviously, this figure shows that the steel flow along the near-wall region in the

cross sections of SEN is non-uniform. The high turbulent kinetic energy occurs at the

near wall cross regions with the X-Z and Y-Z middle plane of the SEN. The predicted

shear stress on the inner vertical wall surface of the SEN is shown in Figure 21. A high

shear stress appears at the regions around the inlet of the SEN, the outlet regions of the

SEN, and the near-wall region of middle Y-Z section of the SEN. The obtained shear

stress was used for calculating the friction velocity. Thereafter, the friction velocities

were used to evaluate the deposition rate of inclusions in the steel flow at different

locations of the inner wall of the SEN.

32

Figure 21. Shear stress distribution on the vertical wall surface of the SEN. X-Z Plane

(front view) Y-Z Plane

(right view) X-Z Plane

(rear view) Y-Z Plane

(left view)

Figure 20. Turbulent kinetic energy of steel flow at different cross-sections of the SEN: (a) 0.06

m from the SEN bottom, (b) 0.2 m from the SEN bottom, (c) 0.4 m from the SEN bottom.

(b) (c)

X

Y

X

Y

(a)

X

Y

right SEN

port side left SEN

port side

33

3.2.2.2 Ce2O3 Inclusion Deposition in SEN

Figure 22. Deposition rates of inclusions of different diameters at the inner surface of the SEN X-Z Plane

(front view)

Y-Z Plane

(right view)

X-Z Plane

(rear view)

Y-Z Plane

(left view)

X-Z Plane

(front view)

Y-Z Plane

(right view)

X-Z Plane

(rear view)

Y-Z Plane

(left view)

1 µm 5 µm

10 µm 20 µm

Figure 22 shows the contours of the deposition rates of inclusions of different sizes at the

inner SEN wall surface. It can be seen that the locations around the SEN inlet, the SEN

bottom and the upper region of the two SEN ports have a large deposition rate. This

means that clogging may be serious in these regions. This is in agreement with previous

experimental observations. [60, 61]

The distribution of the deposition rate at the inner SEN

wall surface observed from the front view of a X-Z plane is similar to that observed from

the rear view of a X-Z plane. This is due to the symmetric flow properties based on the

X-Z and the Y-Z middle planes of SEN. This is also the situation for the observations

from the right and left views of the Y-Z planes. These distribution characteristics of the

deposition rate are generally the same as the distribution of shear stress, as shown

previously in Figure 21. The reason for the distribution similarity between the shear stress

and the deposition rate is that a larger shear stress represents a larger friction velocity.

34

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Distance from top to bottom along SEN, m

Vdep

+

1 m

5 m

10 m

20 m

This leads to a larger dimensionless inclusion relaxation time for the same size inclusion,

which normally indicates a stronger influence of turbophoresis on the deposition rate of

inclusions.

From Figure 22, it can also be seen that the deposition rate of inclusions increases

significantly with an increased inclusion size for the same flow conditions. This can

clearly be observed from the data in Figure 23, which shows the deposition rate of

inclusions of different sizes at the locations along line 3 in Figure 23(a). The reason for

this increase of the deposition rate with an increased inclusion size is due to that large

size inclusions have a larger inclusion relaxation time than that of small inclusions for the

same flow condition. This leads to a strong effect of turbophoresis on the deposition rate.

Another obvious observation from the surface contour of the deposition rate is that the

deposition rates are non-uniform at the inner surface of the SEN. This is true both in the

vertical direction and in the circular direction of a cross section. The uneven distribution

of the deposition rates in the circular direction near the SEN wall can clearly be observed

in Figure 24.

Cross-section of SEN

Line 1 Line 2

Line 3

Line 4

SEN bottom

SEN top

X

Y

left SEN

port side

(a) (b)

Figure 23. Deposition rates of inclusions of different sizes along line 3 at the inner

surface of the SEN.

20 µm

10 µm 5 µm

35

Figure 24 shows the contours of deposition rate of 10 µm inclusions along the circular

wall of different cross sections of the SEN. Figure 24 (a) gives the distribution of the

deposition rate at the cross section just above the SEN port. Furthermore, Figures 24 (b)

and (c) show that in the cross sections 0.2 m and 0.4 m above the SEN bottom,

respectively. The degree of an uneven distribution generally increases from the top to the

bottom of the SEN cross section. This results in that Figure 24 (a) has the largest range of

deposition rates among the three cross sections. The uneven distribution of the deposition

rate in the vertical direction can clearly be seen from Figure 23 (b), as was shown

previously. Furthermore, this can also be seen from Figure 25 where deposition rates of

10 µm inclusions along different vertical lines at an inner surface of the SEN were

presented. Furthermore, it can be seen that the deposition rates of inclusions along line 1

(b) 0.2 m (c) 0.4 m

X

Y

X

Y

(a) 0.06 m

X

Y

left SEN

port side

right SEN

port side

Figure 24. Deposition rates of 10 µm inclusions at different cross-sections of the SEN: (a) 0.06

m from the SEN bottom, (b) 0.2 m from the SEN bottom, (c) 0.4 m from the SEN bottom.

36

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

1

2

3

4

5

6

7

8x 10

-3

Distance from top to bottom along SEN, m

Vdep

+

Line 1

Line 2

Line 3

Line 4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

2

4

6

8x 10

-3

Vdep

+

Distance from top to bottom along SEN, m

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

KE

m2/s

2

Deposition rate

KE

and line 2 are similar. This is also the true for line 3 and line 4. The reason is its

symmetric location between line 1 and 2, and line 3 and 4. For these positions, similar

steel flow phenomena were observed, as is shown in Figure 19. The largest deposition

rate appears at the region where a swirl flow exists, at the bottom of the Y-Z middle plane

section of the SEN in Figure 19.

Figure 25. Deposition rates of 10 µm inclusions along different vertical

lines (shown in Figure 23(a)) at the inner surface of the SEN.

SEN top SEN bottom

Up of

SEN port

Figure 26. Distribution of deposition rate of 10 µm inclusion and steel

flow turbulent kinetic energy along line 3 in Figure 23(a).

Line 4 Line 3

Line 1

Line 2

KE

Deposition rate

37

The present thesis shows that the locations around the SEN inlet, the SEN bottom and the

upper region of the two SEN ports have a large deposition rate of inclusions as well as a

high shear stress and a high turbulent kinetic energy. The similar characteristics of the

distribution of the deposition rates of inclusions, shear stress and turbulent kinetic energy

reflect the close relationship between the steel flow properties and the deposition rate of

inclusions. This can clearly be seen from Figure 26, which shows the distribution of the

deposition rate of inclusions and the turbulent kinetic energy of the steel flow along line 3

in Figure 23 (a). Particles can obtain a convective velocity towards the SEN wall

imparted from the turbulent eddies due to their inertia. This convective velocity, caused

by turbophoresis, steadily increases as the increase of the particle relaxation time. This, in

turn, increases with an increased shear stress and inclusion diameter for the current steel

flow and inclusion size range. Therefore, inclusions with a diameter 20 µm are found to

have the largest deposition rates on the inner surfaces of the SEN wall among the

investigated inclusion sizes.

Due to the non-uniform distribution of the steel flow in the SEN, the distribution of the

deposition rate of inclusions are also non-uniform. A non-uniform deposition rate of

inclusions may lead to a non-uniform clogging. This, in turn, may lead to an uneven flow

and an uneven distribution of the temperature in a mold.

3.3 Particle Deposition Rate in Horizontal Steel and Other Liquid Flows

The deposition rate of particles in a horizontal straight channel may differ along the

channel wall of a cross section, due to the influences of gravity and buoyancy. In order to

show the characteristics of a horizontal straight channel deposition and to show the model

performance, the particle deposition in the circulating cooling water (ρf=998 kg/m3,

ρp=2710 kg/m3) in a horizontal straight pipe was taken as an example and investigated

more in depth.

Figure 27 shows the deposition rates of particles of different sizes at different locations

(these locations are shown in Figure 4) of the pipe wall. It can be seen from Figures 27 (a)

and (b) that the deposition rate increases with an increased particle size. This is due to the

function of the turbophoresis, which is very important for particles with a large inertia. At

location 3 (θ=π/2) of a pipe cross section, the gravity and buoyancy have no effect on the

38

0 20 40 60 80 100 120 140 16010

-6

10-5

10-4

10-3

10-2

10-1

100

dp, m

Vdep

+

Location 1

Location 2

Location 3

Location 4

Location 5

0 20 40 60 80 100 120 140 16010

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

dp, m

Vdep

+

Location 1

Location 2

Location 3

Location 4

Location 5

deposition rate. This is the same situation as for the deposition onto a vertical wall. At

location 1 (θ=0) and 2 (θ=π/4), the net influence of buoyancy and gravity increases the

particle deposition rate. This leads to a larger deposition rate for the particles of the same

size compared to the value at location 3. At locations 4 (θ=3π/4) and 5 (θ=π), the net

influence of the buoyancy and gravity leads to a decreased particle deposition rate.

Therefore, the deposition rates at these two locations are smaller compared to the values

at location 3 for the same particle sizes. At location 5, it can be seen that the deposition

rates of particles in Figure 27 (b) (with a friction velocity 0.064 m/s) are somewhat

smaller than that in Figure 27 (a) (with a friction velocity 0.08 m/s), especially for small

particles. For the same particle size, e.g. 10 µm, the net influence of gravity and

buoyancy is the same. In addition, a higher friction velocity results in a larger particle

relaxation time. This, in turn, represents a larger influence of the turbophoresis. Therefore,

a higher value of the deposition rate is observed in the case of a larger friction velocity.

However, this is not obvious at location 1 due to the important influence of gravity and

buoyancy compared to the influence of a small difference in turbophoresis between that

in Fig. 27 (a) and (b).

In a fluid flow with a different density ratio of a particle and the fluid, the deposition rates

of particles are expected to be different. This is due to the net influence of buoyancy and

gravity, −(1 −𝜌𝑓

𝜌𝑝𝑜)𝑔 cos 𝜃.

u*=0.08 m/s u

*=0.064 m/s

Figure 27. Deposition rates of different-sized particles at different locations of the pipe

wall, (a) cross section I, 0.1 m from inlet; (b) cross section II, 1.3 m from inlet.

(a) (b)

39

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.510

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

u*, m/s

Vdep

+

Water Flow, =0

Water Flow, =

Petroleum Flow, =0

Petroleum Flow, =

Steel Flow, =0

Steel Flow, =

0 20 40 60 80 100 120 140 16010

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

dp, m

Vdep

+

p/

f=1.5, =0

p/

f=2.7, =0

p/

f=4.0, =0

p/

f=1.5, =

p/

f=2.7, =

p/

f=4.0, =

Figure 28 shows the deposition rates of particles of different densities at location 1 (θ=0)

and at location 5 (θ=π) in a water flow. It can be seen that the deposition rates of

different-density particles can vary a lot for the same water flow. This is especially

obvious for small size particles, for which the turbophoresis is weak, compared to the

influence of gravity and buoyancy. Figure 29 shows the deposition rates of particles at

location 1 (θ=0) and at location 5 (θ=π) in different particle-liquid systems in horizontal

straight channels, for a variety of friction velocities. The values of deposition rates show

a large difference in different systems. When the friction velocity is large enough, the

deposition rates of 20 µm particles with θ=0 and θ=π show almost no difference for all

three studied systems for the current conditions. This illustrates that the net influence of

buoyancy and gravity is no longer important compared to the effect of turbophoresis. In

addition, the value of the lower boundary location r+ (r

+ is the dimensionless particle

radius, 𝑟+ = 𝑟𝑢∗/𝜈) should be very large.

Figure 28. Deposition rates of particles with

different density ratios of water at location 1

(θ=0) and at location 5 (θ=π).

u*=0.08 m/s

Figure 29. Deposition rates of particles in

different fluid-particle systems as a function of

the friction velocity at location 1 (θ=0) and at

location 5 (θ=π): water-particle system

ρp/ρf=2.7, petroleum-particle system ρp/ρf=1.4,

steel-particle system ρp/ρf=0.5.

dp = 20 µm

40

41

Chapter 4 Conclusions

The efforts of this thesis have been dedicated to increase the understanding of high-

density inclusion behavior in steel flows and to predict the deposition rate of non-metallic

inclusions in steel flows. The focus of the present thesis includes: (I) A simulation of

Ce2O3 inclusion behavior during a ladle teeming process and a comparison of predictions

with experimental data; (II) An up-to-date Eulerian deposition model was enhanced and

for the first time introduced in a particle-molten-metal system. Furthermore, this model

enabled the prediction of the deposition rates of non-metallic inclusions in a Submerged

Entry Nozzle of a tundish. The main conclusions can be summarized as follows:

A ladle teeming process was simulated and the inclusion behavior at a stage of

teeming was investigated. The study shows that 0.5 μm, 3 μm, 10 μm and 20 μm

inclusions had similar trajectories and velocity distributions in the nozzle.

However, the trajectories of larger inclusions (400μm) were quite different from

the smaller ones. Both the inertia force and the buoyancy force play a very

important role for the behaviors of large-size inclusions. The statistical analysis

shows that inclusions that enter the nozzle inlet from a close-wall location have a

high probability of touching the nozzle wall. The nozzle inlet region and the

connection region of the straight pipe and the expanding part of the nozzle were

found to be the sensitive regions for the inclusion deposition as well as for nozzle

clogging.

An Eulerian deposition model was developed and used to predict the particle

deposition rate. For the case of deposition in vertical flows, effects of different

parameters on the deposition rates of particles in a turbulent steel flow were

investigated. The friction velocity (steel flow rate) was found to have a large

influence on the deposition rate caused by turbophoresis for large particles.

However, it had a small influence on smaller particles for which diffusion is the

main deposition mechanism. The density of the particles has some influence on

only large particles, for which the turbophoresis is significant. Both the wall

roughness and the temperature gradient in the boundary layer can greatly enhance

the deposition rate of small particles. A temperature gradient over the boundary

42

layer leads to a sharp inflexion region in the graph of the deposition rate versus

the particle size. The deposition rate of particles with a diameter smaller than that

of the inflexion region remains almost constant under a temperature gradient,

while the thermophoresis has a minor effect for larger particles for which

turbophoresis is the dominating mechanism.

A non-uniform distribution of the deposition rate was observed both at the cross

sections of the SEN and at the vertical direction of the SEN. A large deposition

rate was found at the regions near the SEN inlet, the SEN bottom and the upper

region of two SEN ports, where high turbulent properties also exist, e.g. turbulent

kinetic energy. 20 µm inclusions have the largest deposition rate among the sizes

of inclusions considered in present study (inclusions diameter of 1 µm, 5 µm, 10

µm, 20 µm) due to the strong effect of turbophoresis.

For the case of particle deposition in horizontal straight channel flows, the

deposition rates of particles/inclusions at different locations of a horizontal

straight pipe cross-section were found different due to the influence of gravity and

buoyancy. For small particles with a small particle relaxation time, the gravity

separation is important for their deposition behaviors at high and low parts of the

horizontal pipe compared to the turbophoresis. For large particles with a large

particle relaxation time, turbophoresis is the dominate deposition mechanism.

43

Chapter 5 Future Work

In order to achieve an even better understanding of the inclusion behavior and its

deposition in steel flows, the following future work is proposed.

Water model experiments and/or high-temperature experiments should be carried

out to investigate the particle deposition rate in a solid-liquid system to validate

the Eulerian deposition model. Such a comparison has not been reported in

literature.

The motion of non-spherical inclusions, e.g. clusters, in steel flows should be

investigated, since the behavior of these during ladle refining and casting are also

important in order to improve production of clean steel.

Parameters related to particle and eddy interaction are required to be further

investigated.

44

45

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