A Study on Particle Motion and Deposition Rate...
Transcript of A Study on Particle Motion and Deposition Rate...
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
References
[1] A.G. Levada, D.N. Makarov, V.N. Artyushov, A.G. Zyryanov, V.P. Sosnin: Steel in
Transl., 39(2009), 1084.
[2] N. Shinozaki, N. Echida, K. Mukai, Y. Takahashi, Y. Tanaka: Tetsu-to-Hagané
80(1994), 14.
[3] K. Mukai, R. Tsujino, I. Sawada, M. Zeze, S. Mizoguchi: Tetsu-to-Hagané, 85(1999),
307.
[4] Y. J. Kwon, J. Zhang and H. G. LEE: ISIJ Int., 48(2008), 891.
[5] L. T. Wang, Q. Y. Zhang, S. H. Peng and Z. B. Li: ISIJ Int., 45(2005), 331.
[6] H. Arai, K. Matsumoto, S. I. Shimasaki and S. Taniguchi: ISIJ Int., 49(2009), 965.
[7] G. S. Diaz, A. R. Banderas, J. j. Barreto and R. D. Morales: Steel Res. Int., 80(2009),
223.
[8] J. Szekely and R. T. Yadoya: Metall. Tran., 4(1973), 1379.
[9] F. G. Wilson, M. J. Heesom, A. Nicholson and A. W. D. Hills:Ironmak. Steelmak.,
14(1987), 296.
[10] B. G. Thomas, L. J. Mika and F. M. Najjar: Metall. Tran. B, 21B(1990), 387.
[11] D. E. Hershey, B. G. Thomas and F. M. Najjar: Int. J. Numer. Methods in Fluids,
17(1993), 23.
[12] F. M. Najjar, B. G. Thomas and D. E. Hershey: Metall. Mater. Trans. B, 26B(1995),
749.
[13] B. G. Thomas, A. Dennisov and H. Bai: Proc. of ISS 80th
Steelmak. Conf., Chicago,
(1997), 375.
[14] H. Bai and B. G. Thomas: TMS Annu. Meet., Nashville, (2000), 85.
[15] H. Bai and B. G. Thomas: Metall. Mater. Trans. B, 32B(2001), 253.
[16] H. Bai and B. G. Thomas: Metall. Mater. Trans. B, 32B(2001), 269.
[17] H. Bai and B. G. Thomas: Metall. Mater. Trans. B, 32B(2001), 707.
[18] Q. Yuan, B. G. Thomas and S. P. Vanka: Metall. Mater. Trans. B, 35(2004), 685.
[19] Q. Yuan, B. G. Thomas and S.P.Vanka: Metall. Mater. Trans. B, 35(2004), 703.
[20] L. F. Zhang, B. G. Thomas: XXIV Steelmak. Natl. Symp., Morelia, (2003), 26.
[21] L. F. Zhang: 5th Int. Conf. on CFD in the Process Ind. CSIRO, Melbourne, (2006), 1.
[22] L. F. Zhang, Y. F. Wang and X. J. Zuo: Metall. Mater. Trans. B, 39B(2008), 534.
[23] C. Pfeiler, M. Wu and A. Ludwig: Mater. Sci. Eng. A, 413-414(2005), 115.
46
[24] A. R. Banderas, R. D. Morales, R. S. Perez, L. G. Demedices, and G. S. Diaz: Int. J.
Multiph. Flow, 31(2005), 643.
[25] R. Sambasivam: Ironmak. Steelmak., 33(2006), 439.
[26] C. Real, R. Miranda, C. Vilchis, M. Barron, L. Hoyos and J. Gonzalez: ISIJ Int.,
46(2006), 1183.
[27] C. Real, L. Hoyos, F. Cervantes, R. Miranda, M. P. Pardave and J. Gonzalez: Mec.
Comput., XXVI(2007), 1292.
[28] X. W. Zhang, X. L. Jin, Y. Wang, K. Deng and Z. M. Ren: ISIJ Int., 21(2011), 581.
[29] R. Chaudhary, C. JI, B. G. Thomas, S. P. Vanka: Metall. Mater. Trans. B, 42(2011),
987.
[30] M. J. Long, X. J. Zuo, L. F. Zhang and D. F. Chen: ISIJ Int., 50(2010), 712.
[31] Kiessling, R., 1980. Clean Steel- a debatable concept. Met. Sci. 15, 161-172.
[32] A. Guha: J. Aerosol Sci., 28(1997), 1517.
[33] A. Guha: Annu. Rev. Fluid Mech., 40(2008), 311.
[34] J. Young, A. Leeming: J. Fluid Mech. 340(1997), 129.
[35] S.V. Patankar: Numerical Heat Transfer and Fluid Flow, Hemispere Publishing
Corp., New York, 1980.
[36] Y.S. Chen, S.W. Kim: Computation of turbulent flows using an extended k-ɛ
turbulence closure model, NASA CR-179204, 1987.
[37] D.J. Monson, H.L. Seegmiller: Comparison of experiment with calculations using
curvature- corrected zero and two-equation turbulence models for a two-dimensional U-
duct, AIAA, 90-1484, (1990).
[38] B.E. Launder: Mathematical Models of Turbulence. In: D.B. Spalding (Editor).
Academic Press, London, 1972.
[39] T.-H. Shih, W.W. Liou, A. Shabbir, Z. Yang, J. Zhu: Comp. Fluids, 24(1995), 227.
[40] J. C. Ludwig, N. Fueyo and M. R. Malin: CHAM/TR211, 2006.
[41] R. Clift, J. R. Grace and M. E. Weber: Drops and Particles, Academic Press, New
York, (1978).
[42] A. D. Gosman, E. Ioannides: AIAA 19th
Aerospace Sciences Meeting, Paper 81-
0323, St. Louis, (1981).
[43] A. ORMANCEY: “Simulation du Comportement de Particules dans des
Ecoulements Turbulents”, Ph.D. thesis, Ecole Nationale Supérieure des Mines, Paris,
1984.
[44] D. I. Graham: Int. J. Multiph. Flow, 24(1998), 335.
47
[45] J.D. Ramshaw: Phys. Fluids 22(1979), 1595.
[46] R. Clift, W.H. Gauvin: Proc. Chemeca 70(1970), 14.
[47] B. Zhao, J. Wu: Atmos. Environ., 40(2006), 457.
[48] L. Talbot, R.K. Cheng, R.W. Schefer, D.R. Willis: Fluid Mech., 101(1980), 737.
[49] D.A. Drew, L.Y. Cheng, R.T. Jr. Lahey: Int. J. Multiph. Flow, 5(1979), 233.
[50] C.T. Crowe, J.D. Schwarzkopf, M. Sommerfeld, Y. Tsuji: Multiphase flows with
droplets and particles, CRC Press, 1998.
[51] P.G. Saffman: J. Fluid Mech., 22(1965), 385.
[52] T. Hussein, J. Smolik, V. M. Kerminen, M. Kulmala: Aerosol Sci. Technol.,
46(2012), 44.
[53] M.R. Sippola, W.W. Nazaroff: Lawrence Berkeley National Laboratory Report
LBNL-51432, Lawrence Berkeley National Laboratory, Berkeley, California, 2002.
[54] M.J. Zufall, W.P. Dai, C.I. Davidson, V. Etyemezian: Atmos. Environ., 33(1999),
4273.
[55] S.T. Johansen: Int. J. Multiph. flow, 17(1991), 355.
[56] J.O. Hinze: Turbulence, second ed. McGraw-Hill, New York, 1975, 466.
[57] J.M. Kay, R.M. Nedderman: Fluid Mechanics and Transfer Processes, Cambridge
University Press, Cambridge, 1985.
[58] N. Kojola, S. Ekerot, M. Andersson and P. G. Jönsson: Ironmak. Steelmak.,
38(2011), 1.
[59] A.C.K. Lai, W.W. Nazaroff: J. Aerosol Sci., 31(2000), 463.
[60] M. Tsuda, H. Shinagawa, R. Kamata, Y. Hiraga and T. Hara: J. Tech. Assoc.
Refract., 20(2000), 138.
[61] L.M. Aksel'rod, M.R. Baranovskii and G.G. Mel'nikova: Refract. Ind. Ceram.,
32(1991), 661.