Master´s Thesis in Scientific Computing at Stockholm University, Sweden 2007

A Solid Inclusion Separation at the Steel-Slag Interface for Tundish Conditions in the Continuous Steel Casting Process

Sami Jani

A Solid Inclusion Separation at the Steel-Slag Interface for Tundish Conditions in the Continuous

Steel Casting Process

Sami Jani

Master´s Thesis in Scientific Computing (20 credits) Single Subject Courses Stockholm University year 2007 Supervisor was Pär Jönsson, KTH Examiner was Axel Ruhe TRITA-CSC-E 2007:043 ISRN-KTH/CSC/E--07/043--SE ISSN-1653-5715 Department of Numerical Analysis and Computer Science Royal Institute of Technology SE-100 44 Stockholm, Sweden

Abstract In this thesis an attempt was carried out to solve mathematical equation of PDE type describing the phenomena of solid inclusion separation at the steel slag interface in the continuous steel casting process for tundish conditions. The separation of non-metallic inclusions from the steel to the slag phase in the ladle during secondary steel making operations and it the tundish and Mod during casting is very crucial to the production of clean steel. The work is aimed to provide a better understanding of inclusion behavior at the steel-slag interface. The model descriptions of the inclusion transfer are based on the equation of motion at the system. It is assumed that the inclusion transfer is governed by four forces acting on the inclusion as it has reached the steel-slag interface. These are the buoyancy force, the added mass force, the drag force and the rebound force. The models assume two cases of inclusion separation depending on the inclusion Reynolds number. In the case where Reynolds number is larger or equal to unity, i.e. , a steel film is formed between the inclusion and the slag. 1Re≥If Reynolds number , there will be no steel film formation and the inclusion will be in direct contact with the slag.

1Re<

A consistent study of the mathematical model based on the equation of motion with the above mentioned forces is performed the equations were solve numerically by the Runge-Kutta-Gill fourth order method, and code was written in the C++ programming language. In this thesis an attempt was carried out to develop a mathematical modeling technique for casting of steel. Emphasis was given on solidification and particle tracking. Keywords: mathematical modeling, solid inclusions, tundish, inclusion removal.

Avskiljning av fasta icke-metalliska inneslutningar vid ett stål/slagg-gränsskikt i en kontinuerlig ståltillverknings-process innehållande en gjutlåda

Sammanfattning I den här uppsatsen har jag försökt att lösa en matematisk ekvation av typen PDE som beskriver fenomenet avskiljning av fasta icke-metalliska inneslutningar vid ett stål/slagg-gränsskikt i en kontinuerlig ståltillverkningsprocess innehållande en gjutlåda. Avskiljning av en icke-metallisk inneslutning från stålfasen till slaggfasen under behandling i skänk eller gjutlåda är väldigt kritisk för tillverkning av rent stål. Arbetet har som mål att ge en bättre förståelse av inneslutningars beteende vid stål/slagg-gränskiktet. Modellens beskrivning av inneslutningars förflyttning är baserad på rörelse ekvationen för systemet. Det antas att inneslutningarnas förflyttning styrs av fyra krafter som påverkar inneslutningen när den har nått stål/slagg-gränsskiktet. Dessa är flytkraften, masskraften, dragkraften och återflyttningskraften. Modellerna föreslår att det finns två fall hur inneslutningar kan separeras till slaggfasen beroende på inneslutningarnas Reynolds tal. Vid fallet där Reynold tal Re , bildas en stål film mellan inneslutningen och slaggen. Om Reynold nummer Re<1, då blir det inget filmbildning och inneslutningen kommer att vara i direkt kontakt med slaggen.

1≥

En genomgående studie av matematiska modellen baserad på ekvationen av rörelsen med ovan nämnda krafter är gjort och ekvationerna är lösta numeriskt med Runge-Kutta-Gill metoden av ordning fyra, och koden skrevs i programmering språket C++. Nyckelord: matematisk modellering, inneslutning, gjutlåda, avskiljning.

ACKNOWLEDGEMENTS First, and most importantly, I want to thank my supervisor, Professor Pär Jönsson, Head of the Department of Materials Science and Engineering, for giving me opportunity to work in this project, and excellent guidance throughout the work. As well as I am thankful to my all friends for their understanding and support throughout my long hours of work. My dear friends! Without you there would be a lot less of good and enjoying times. A special Thank to Mr. Alexey Striey for all his help and advices when writing the C++ code. Sami October 02, 2006. Stockholm, Sweden This thesis is dedicated to my parents, Ibtisam and Tawfik, my brother Sattar, my sisters Mayssa and Lina and my wife Siba whom I am greatly indebted for their great support.

TABLE OF CONTENTS

INTRODUCTION…………………………………………………………………1 CHAPTER 1 1.1 An overview of the Fluid Dynamics of the Continuous Casting…………………... 2 1.2 The Objectives of the Thesis Work………………………………………………... 3 CHAPTER 2 2.1 Mathematical Model of Particle Behavior at the Steel-Slag Interface…………….. 5 2.2 General Equations…………………………………………………………………. 5

2.2.1 Inclusion Transport without a Steel Film…………………………………… 6 2.2.2 Inclusion Transport with a Steel Film………………………………………. 7

CHAPTER 3 3.1 The Runge-Kutta Method…………………………………………………………. 10 3.2 The Runge-Kutta-Gill Method…………………………………………………….. 11 3.3 Analytical solution………………………………………………………………… 12 3.3.1 Case without steel film formation…………………………………………... 12 3.3.2 Case with steel film formation…………………………………………….... 13 CHAPTER 4 4.1 Implementation of algorithm in C++……………………………………………… 15 4.2 The solution algorithm…………………………………………………………….. 15 4.2.1 Inclusion Transport without steel film formation……………………………16 4.2.2 Inclusion Transport with steel film formation……………………………….17 4.3 The choice of initial conditions…………………………………………………….19 CHAPTER 5 5.1 The case without steel-film formation…………………………………………...... 22 5.2 The case with steel-film formation………………………………………………... 22 5.3 Discussions…………………………………………………………………………23 5.4 Characteristics of Particle Behavior at the Interface………………………………. 24 5.5 Concluding Remarks………………………………………………………………. 24 5.6 Possibilities of coupling in the future………………………………………………25 References……………………………………………………………………………….. 26 Nomenclature……………………………………………………………………………. 28 List of Figures…………………………………………………………………………… 29 Appendix………………………………………………………………………………… 36

Introduction The demands on the steel cleanliness and the steel quality from the producers are vital to the customers, and critical conditions to the companies in the steel market. This gives a rise to an increasing interest of researches in the area of steel producing cleanliness. One of the facts that control the steel quality is the control of the inclusion characteristics throughout the steelmaking and casting process. Studies of inclusion separation during the steel making process are impossible by direct visualization, because of the high temperature (making the steel glow), and non-transparent cover powder and slag. Moreover, it is difficult to get samples representing the steel-slag interface. Solhed et al. [1] developed a sampler for this purpose called the MISS-sampler (Momentary Interfacial Solidification Sampling). But the problem with sampling at the interface in a ladle or tundish is that a cover powder or slag is covering the steel surface, protecting the steel from reoxidation. In order to take samples when a cover powder is present in a tundish, it must be removed. This is difficult to do without also removing the slag. Therefore it is better and easier to the mathematical models describing the separation and growth of inclusions during the steel casting process for the tundish conditions. So, if the desire of the steel maker is to use the tundish as an important metallurgical tool, it is essential to have a good control and a deep understanding of all aspects of the tundish operation, including for example, throughput rate of steel, chemical composition and amount of tundish slag, temperature of the steel and suitable measure to avoid ladle slag carry over. In order to be able to control and affect the separation of non metallic inclusions from the steel to the tundish covering flux it is important to understand the mechanisms involved in the transfer of inclusions across the interface between the steel and the tundish covering slag. It is assumed that the inclusion has been carried to the interface by the flow field and that it reaches the interface with the terminal velocity determined by the buoyancy force and the fluid dynamic drag. The model predicts two modes of inclusion transfer, one where a steel film is formed between the inclusion and the slag as the inclusion is crossing the interface and one without the formation of the steel film. From the research of Nakajima et al. [2], [3] and Strand et al. [4] it was assumed that the most important parameters controlling the inclusion behavior at the steel-slag interface are the slag viscosity and the interfacial tensions between the phases. This study of parameter was made for mµ20 inclusions. Also they showed that for mµ100 inclusions also the inclusion density affects the inclusion behavior. They also showed in their work that the parameters that have the largest influence on the inclusion displacement are the interfacial tensions ( MSISMI σσσ ,, ) and the slag viscosity ( Sµ ) have the largest influence on the predicted displacement. It was also concluded that the overall wettability should be positive and that the slag viscosity should be as low as possible to obtain the most favorable conditions for inclusion transfer at the steel-slag interface. In this work we will consider the mathematical model that describes the phenomena of the solid inclusion at the steel slag interface for tundish conditions in the continuous steel casting process. The model for the separation of inclusion from the steel to the slag was developed by Nakajima and Okamura et al. In this work I’ll consider the part of solid inclusion transfer at the steel-slag interface developed by Nakajima and Okamura et al.

1

CHAPTER 1 In this chapter I will give an overview of the Fluid Dynamics of the Continuous Casting. 1.1 An overview of the Fluid Dynamics of the Continuous Casting Continuous casting (CC), figure (1), has become an increasingly important step in the manufacture of steels in the last three decades. Concurrent with the increase in production levels, the quality requirements of the final steel product have also become quite stringent. High-quality steel should satisfy the customer specifications for composition and the content of non-metallic inclusions. The tolerance level for overall inclusion content and fluctuations in composition in the final steel product has been steadily decreasing over the years, and this trend is expected to continue.

Figure 1 Diagram of the continuous casting system [5]. 1.1.1 Tundish The tundish is important for regulating the flow of molten steel from individual ladles to the CC mould, see figures (2) and (3). Ideally, it is also used as a metallurgical vessel to aid in Figure 2 Diagram of the tundish [6]. Figure 3 A tundish in a steel factory [7].

2

inclusion removal and improves the quality of the cast steel product. Inclusion removal in a tundish is controlled by many factors, including the fluid flow pattern, inclusion agglomeration and flotation, and top surface condition, where inclusions may be either removed or generated, depending on the slag composition and reoxidation conditions. Furthermore, unsteady flow phenomena, such as occur during a ladle exchange, are important to inclusion removal and quality. The shape of the tundish is typically rectangular. Nozzles are located along its bottom to distribute steel to the mould. The tundish also serves several other key functions:

• Enhances oxide inclusion separation. • Provides a continuous flow of liquid steel to the mold during ladle exchanges. • Maintains a steady metal height above the nozzles to the mould, thereby keeping

steel flow uniform. • Provides more stable stream patterns to the mould.

LLAADDLLEE

MMOOLLDD

TTUUNNDDIISSHH

SSUUBBMMEERRGGEEDD NNOOZZZZLLEE

IINNGGOOTT

Figure 4 Schematic of the continuous casting system [8]. 1.1.2 The Nozzle A nozzle is a mechanical device designed to control the characteristics of a fluid flow as it exits from an enclosed chamber into some medium. A nozzle is often a pipe or tube of varying cross sectional area, see figure (4), and it can be used to direct or modify the flow of a fluid (liquid or gas). Nozzles are frequently used to control the rate of flow, speed, direction, mass, and/or the pressure of the stream that emerges from them. Tundish nozzle geometry is one of the few variables that are both very influential on the casting process and relatively inexpensive to change. Designing an effective nozzle requires quantitative knowledge of the relationship between nozzle geometry and other process variables on the influential characteristics of the flow exiting the nozzle. This relationship depends on the flow pattern within the nozzle components. Argon injection into the nozzle is an efficient and widely employed method to reduce nozzle clogging, even though the real working mechanism(s) are still not fully understood [9]. Argon injection may greatly affect the flow pattern in the nozzle, and subsequently in the mould.

3

1.1.3 The Mould The main function of the mould is to establish a solid shell sufficient in strength to support its liquid core upon entry into the secondary spray cooling zone. The mould is basically an open-ended box structure, containing a water-cooled inner lining fabricated from a high purity copper. The inner face of the copper mould is often plated with chromium or nickel to provide a harder working surface, and to avoid copper pickup on the surface of the cast strand, which otherwise can facilitate surface cracks on the product [10]. 1.1.4 Heat Transfer in Continuous Casting

Because heat transfer is the major phenomenon occurring in CC, it is also the limiting factor in the operation of a casting machine. The distance from the meniscus to the cut-off stand should be greater than the metallurgical length, which is dependent on the rate of heat conduction through the solid shell and of heat extraction from the outside surface, in order to avoid cutting into a liquid core. Thus, the casting speed must be limited to allow sufficient time for the heat of solidification to be extracted from the strand. 1.2 The Objectives of the Thesis Work The present thesis is aimed at studying the behavior of solid inclusions at a steel-slag interface with focus on tundish process. The parameters introduced by Nakajima et al and Strand et al have determined which of the model variables that have the largest influence on the predicted inclusion displacement at the interface. The Runge-Kutta-Gill method was used for numerical solution of the problem, and the coding was done in the C++ programming language. The code is proposed to be a stand-alone program, which can be applied into some other appropriate CFD software by coupling, in order to analyze the displacement of the solid inclusion and also the different forces that act on the inclusion as it approaches the steel-slag interface. I hope that this work will be a step in the work of improving the quality of the steel as well as any future work aimed in this area.

4

CHAPTER 2 In this chapter we discuss the formulation of the problem and the mathematical equations for the two different cases of the particle transport across the steel slag interface. 2.1 Particle Behavior at the Steel-Slag Interface The mathematical model of the inclusion transfer at a steel-slag interface has been presented originally at a conference by Nakajima and Okamura. The model describes two types of inclusion transfer behavior: inclusion transport without steel film and, inclusion transport with steel film; please see Figures (6) and (7). The two cases differ in the inclusion transfer displacement and the velocity of the transfer and the bounded forces. Also, in the film case the phenomena of film rupture will arise, it occurs when the film thickness reaches to 0.001mm. In this case one should take care of the calculation of the four different forces acting on the inclusion as it approaches the steel-slag interface before the film rapture and after the film rapture: the buoyant force, the round force, the added mass force, and the drag force. However in the case, when the inclusions are always accompanied with steel around them, so that the steel is not pushed away rapidly. Thus, a thin steel film will still exist between the inclusion and the interface when the inclusion reaches the steel-slag interface. In case of , no steel film will be formed and the inclusions will be in direct contact with the slag at the interface.

1Re≥

1Re≥

To formulate a proper mathematical model of the inclusion transfer across the steel-slag interface, we need to consider some important assumptions made by Strand et al. These assumptions are:

• The inclusion is solid and spherical with constant volume. • No chemical reactions between the phases. • All fluids are incompressible and isothermal. • The slag phase is assumed to be liquid. • The interface between steel and slag is flat. • The inclusion transfer depends on the buoyancy, added mass, drag and rebound

force. • The interfacial tension is uniform along the interface.

When the inclusion has traveled one diameter form its original position, it is assumed to be completely absorbed in the slag.

2.2 General Equations

To carry out a proper calculation, we start by making the center of the inclusion is defined as the initial position, i.e. Z=0, situated one inclusion radius from the interface, see Figures (6), (7). Thus the surface of the inclusion is in contact with the interface 1. Although the inclusion is generally carried by the steel flow to the interface, here it is assumed to have an initial velocity equal to its terminal velocity, which is given by

RZ =

MIMI

gRuµ

ρρ )(92 2 −=∞ (1)

Where ∞ - initial velocity; Mu ρ - density of the metal; Iρ - density of the inclusion; Mµ -viscosity of the metal phase; g - gravitational acceleration; - inclusion radius. IR

5

As the inclusion approaches the steel-slag interface, there are four different forces acting on it, these forces are the buoyant force ( b ), the rebound force ( r ), the added mass force ( f ) and the drag force ( ). And the general formulation for the equation of motion over the system is

F F FdF

rdbfII FFFFdt

ZdR −−=+2

3

34 ρπ (2)

Those forces are acting differently on the inclusion transfer:

• The buoyant force acts always upwards. b• The added mass force r and the drag force d can act both upwards and

downwards depending on the behavior of the inclusion at the interface.

FF F

For simplicity, the equations will be expressed in dimensionless variables, though it is easer to solve them. In dimensionless variables, it is assumed that the inclusion is completely transferred to the slag when the dimensionless displacement . Then for making the displacement and the time and the velocity is made as the following:

2* =Z

• The displacement Z is made dimensionless by multiplying it withIR

1 .

• The time t is made dimensionless by multiplying it withIR

g .

• The velocity dtdZ is made dimensionless by multiplying it with

IgR1 .

Now by utilizing the equations above, we can derive the equations necessary for the two different forms of the inclusion transfer: inclusion without and with steel film. 2.2.1 Inclusion transport without a steel film First, we need to determine the inclusion displacement from its original location, to do this we need to consider the equation of motion, Equation (2), and using the properties of the acting forces mentioned above. For when there is no steel film formation, the buoyant force b , is positive and hence having its direction upward, then it is described by the equation below:

F

))((34 *3

ISIb ZAgRF ρρπ −×= (3)

Where sρ is the density of the slag and is defined as )( *ZA

( )S

M

S

M ZZZAρρ

ρρ

+−×⎟⎟⎠

⎞⎜⎜⎝

⎛−= 3)(1

41)( *2** (4)

For the case of inclusion transfer without steel film the added mass force is positive in the downward direction, is given by the following equation

fF

2

*2*3 )(

32

dtZdgZARF SIf ××= ρπ (5)

The rebound force is positive in the downward direction, and is given by the following equation:

)(32 *ZBRF MSIr σπ= (6)

Where MSσ is the interfacial tension between the metal and the slag phase and is defined as

)( *ZB

6

IMSZZB θcos1)( ** −−= (7)

The overall wettabilility IMSθcos is defined as

MS

ISIMIMS σ

σσθ

−=cos (8)

• IMσ = The interfacial tension between the inclusion and the metal. • ISσ = The interfacial tension between the inclusion and the slag.

In the work of Nakajima et al. it was concluded that if IMS then the liquid steel is said to wet the solid inclusion, and if cos then the steel is said to be non-wetting.

o0cos <θ0>θ o

IMSThe drag force d which is acting on the inclusion, positive in the downward direction, is given by the following equation

F

*

** )(6

dtdZgRZCRF ISId ××= µπ , (9)

Where Sµ is the viscosity of the slag and is defined as )( *ZC

S

M

S

M

S

M ZZZCµµ

µµ

µµ

+⎟⎟⎠

⎞⎜⎜⎝

⎛−−⎟⎟

⎠

⎞⎜⎜⎝

⎛−= *2** 12)(1)( (10)

Now it is possible to formulate the equation of the dimensionless displacement of the inclusion from its initial position, it is by using the above defined equations of buoyant force, Equation (3), the added mass force, Equation (5), the rebound force, Equation (6) and the drag force, Equation (9), by inserting these equations in the equation of the motion, Equation (2), to obtain the following equation of the dimensionless displacement:

*

**

***

*

*

2*

*2

)()(

9)()(32)(

)(2dtdZZC

ZEZBZD

ZAZA

dtZd

IS

IS ××−×−+×−×

=ρρρρ (11)

Where and are given by: )( *ZD )( *ZE

gZARZD

ISI

MS

)2)(()( *2

*

ρρσ

+×= (12)

)2)(()( *3

*IS

S

I ZAgR

ZE ρρµ

+×= (13)

By solving Equations (11) numerically, by using the Runge-Kutta-Gill method outlined in section (3.2) and implemented in a C++ code, the displacement of the inclusion transport with steel film formation from its initial position and the velocity of the inclusion and the four acting forces on the inclusion can be calculated, see Figures (9) – (11). 2.2.2 Inclusion Transport with a Steel Film When , a steel film is formed between the rising inclusion and the slag interface, as mentioned earlier. But in this case need to consider some additional assumptions mentioned in section 2.1 in the former case of inclusion with no film. Particularly, the steel film is assumed to be uniform along the surface of the inclusion also very thin. It is also assumed that the following stream function in spherical coordinates can be used to describe the flow around the inclusion:

1Re ≥

7

θψ 23

sin21

23

21

⎟⎟⎠

⎞⎜⎜⎝

⎛+−−=

rRrR

dtdZ I

I , (14)

Where r is the distance in the r direction from the center of the inclusion. Figure (8) describes the streamlines around a moving sphere in a liquid at rest. This stream function is valid for , andIRr≥ πθ ≤≤0 . Inclusion is normally relatively small, and as a result the deformation of the interface is small. Therefore, the slag pressure is approximately equal to the metal pressure . SP MP Now we could formulate the continuity equation of normal stress across the steel film-slag interface by using Equation (14) together with the slag pressure and the metal pressure , the equation is expressed as the following

SP MP

θµσ cos

41

21

2322

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−+

++

=−≅−SRSRdt

dZSR

PPPPII

SI

MSMFSF (15)

Where F the steel film is pressure and is the steel film thickness. The rebound force is obtained by integrating Equation (15) as follows:

P S

θθπθθ

dRRPPF IIMFr

c

×××−= ∫ sin2cos)(0

=

= ⎥⎦

⎤⎢⎣

⎡−×⎟⎟

⎠

⎞⎜⎜⎝

⎛+

−+

××++

)cos1(4

12

123

3sin

)(24 322

CII

S

I

MSI SRSRdt

dZSR

R θµ

θσ

π (16)

Where the following expressions for the angles are obtained form the geometry in figure (6) as

22

)())(2(sin

SRZSZSR

I

IC +

+−+=θ (17)

SRZR

I

IC +

−=θcos (18)

Because of the accelerated liquid motion, the buoyancy force is written as

gRF IMIb )(34 3 ρρπ −= (19)

The drag force is described by

dtdZRF MId µπ6= (20)

The added mass force is expressed as

2

23

32

dtZdRF MIf ρπ= (21)

The equation of motion, Equation (2), for the model including a steel film can now be expressed as follows for the dimensionless displacement of the inclusion:

8

*

*

*

*****

2*

*2 9),(2),(32

2dtdZ

IdtdZSZK

JSZHG

dtZd

IM

IM −−×−+−ρρρρ (22)

Where G, H, I, J and K are defined as:

gRG

IMI

MS

)2(2 ρρσ+

= (23)

3*

******

)1())(2(),(

SZSZSSZH

++−+

= (24)

M

IMI gRI

µρρ )2(3 +

= (25)

S

IMI gRJ

µρρ )2(3 +

= (26)

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−

−×⎟⎠⎞

⎜⎝⎛

+−

+=

3

*

*

****

111

411

211

23),(

SZ

SSSZK (27)

Based on the equation for the steel flow around the inclusion (Equation (14)) the steel film flow-out velocity is given by

drd

rU ψ

θ×−=

sin1 | =

CRr θθ == , CdtdZ θsin− (28)

The film surface areaδ is described as

SRZSRdRR

IIII

C

++

×=×= ∫ 2

02sin2 πθθπδ

θ (29)

And the continuity of the film flow is expressed as

))(()sin2( δδθπδ ddSSSdtRUS CI ++=− (30)

Where , dZ and dS δd are small variations of , S Z , δ corresponding to a small time step . By substituting Equations (28) and (29) into Equation (30) and after some rearrangement, assuming that I and neglecting the second order dS and (since they are so small) the following expression is obtained for the dimensionless steel film thickness:

dt

SR >> dZ

**

*

******

*

*

*

*

2

))(2(

ZSdtdZSSZSZ

dtdZ

dtdS

+

−+−−= (31)

By solving Equations (22) and (31) numerically, by using the Runge-Kutta-Gill method outlined in the next chapter and implemented in a C++ code, the displacement of the inclusion transport with steel film formation from its initial position and the velocity of the inclusion and the film thickness and the four acting forces on the inclusion can be calculated, see Figures (12) – (15).

9

CHAPTER 3 In this chapter we will present the analytical solutions and the numerical solutions of the equations presented in chapter 2. 3

.1 The Runge-Kutta Method

Runge-Kutta (RK) methods [11] are single-step methods that are similar in motivation to Taylor series methods, but do not involve explicit computation of higher derivatives. Instead, RK methods replace higher derivatives by finite difference approximations based on values of a given function at points between kt and 1+k . Alternatively, RK methods can be derived by applying numerical quadrature rules [12] to evaluate the integral:

f t

∫+=−+

1 .))(,(1k

k

t

tkk dttytfyy (32)

In either case, some bootstrapping1 will be required to obtain the necessary values of , since we do not know the second argument of , namely the solution , for t between and .

ff )(ty kt 1+kt

To demonstrate the derivation of a Runge-Kutta method, recall the Taylor series method [14]:

...)(6

)(2

)()()( '''3

''2

' ++++=+ tyhtyhthytyhty (33)

Note, however, that this approach requires the computation of higher derivatives of y . These can be obtained by differentiating using the chain rule2 e.g.: ),(' ytfy =

),,(),(),(),(),( ''' ytfytfytfyytfytfy ytyt +=+= (34) Where, each function is evaluated at , and the subscripts indicate partial derivatives with respect to the given variable.

),( yt

We can approximate the term on the right by expanding in a Taylor series in two variables f

)(),( 2hOfhfhffhfyhtf yt +++=++ , (35) From which we obtain

)(),(),( hOh

ytfhfyhtffff yt +−++

=+ (36)

1 In statistics bootstrapping is a method for estimating the sampling distribution of an estimator by resembling with replacement from the original sample [13] . 2 If is differentiable at the point )(xg x and is differentiable at the point , then

is differentiable at)(xf )(xg

gf o x . Furthermore, if we let ))(( xgfy = and )(xgu = , then

dxdu

dudy

dxdy

×= [15]

10

With this approximation to the second derivative, the second-order Taylor series method shown in (33) becomes:

=−++

×++=+k

kkkkkkkkkkkkkk h

ytfytfhyhtfhytfhyy

),()),(,(2

),(2

1

= ))),(,(),((2 kkkkkkkkk

k ytfhyhtfytfh

y ++++ (37)

The term in (37) can be implemented in the form

)(2 211 kkh

yy kkk ++=+ (38)

Where, and ),(1 kk ytfk = ),( 12 khyhtfk kkkk ++= . The best-known Runge-Kutta method is the classical fourth-order scheme

)22(6 43211 kkkkh

yy kkk ++++=+ , (39)

Where

),(1 kk ytfk = , )2/,2/( 12 khyhtfk kkkk ++= ,

)2/,2/( 23 khyhtfk kkkk ++= , ),( 34 khyhtfk kkkk ++= .

In order to solve a given PDE with this method we need to know the following parameters:

== 0ttstart initial time;

== tendtend end value of the time interval;

== dth The step size. And also we need to know the initial conditions and ),0(y ).0('y The RK methods have a number of virtues. To proceed to time 1+kt , they require no history of the solution prior to time kt , which makes them self-starting at the beginning of the integration, and also makes it easy to change the step size during the integration. These features also make Runge-Kutta methods provide no error estimate on which to base the choice of step size. More recently, however, embedded Runge-Kutta methods have been developed in which an error estimate is based on the difference between a pair of methods of different order, but which share function evaluations for efficiency. 3.2 The Runge-Kutta-Gill Method As mentioned in Chapter 3.1 the embedded Runge-Kutta methods have been developed in which an error estimate is based on the difference between a pair of methods of different order, but which share function evaluations for efficiency. One of these methods is the Runge-Kutta-Gill (RKG) method.

11

A RKG [16], [17] scheme in common use is based on an incomplete adaptation for floating point operations of Gill's method. An improved version reduces round-of error significantly. To illustrate this method we need first to consider the following. A formula for numerical solution of differential equations is given by the RKG method:

f

),)22()22((61

43211 kkkkyy kk +++−++=+ (40)

Where

),(1 kk ythfk =

)21,

21( 12 kyhthfk kk ++=

))2211()21(

21,

21( 213 kkyhthfk kk −++−++=

))2211(2

21,( 324 kkyhthfk kk ++−+=

We see that the scheme of the RKG method is similar to the one of the common RK method, but here in (40) it multiplies with 2 by k )22( − instead of (2) in (39) and 3 by k )22( + instead of (2) in (39), this is meant to make a better error estimate in a solution of a PDE.

In order to solve a given PDE with this method we need to know the following parameters:

== 0ttstart initial time;

== tendtend end value of the time interval;

== dth step size.

And also we need to know the initial conditions and ),0(y ).0('y 3.3 Analytical solution Here, we solve the equations described in chapter 2 analytically. 3.3.1 Case without steel film formation Given the equation for dimensionless displacement for the case of no steel film formation, Equation (11):

*

**

***

*

*

2*

*2

)()(

9)()(32)(

)(2

dtdZZC

ZEZBZD

ZAZA

dtZd

IS

IS ××−×−+×−×

=ρρρρ

And the equations of the four acting forces on the inclusion for the case without steel-film formation

))((34 *3

ISIb ZAgRF ρρπ −×=

12

2

*2*3 )(

32

dtZdgZARF SIf ××= ρπ

)(32 *ZBRF MSIr σπ=

*

**)(6

dtdZgRZCRF ISId ××= µπ

Equation (11) is autonomous differential equations3. To solve them, first, we have to rewrite them in order. We let and then equation (11) can be written as, *,1 Zy = ,*'

2 Zy =

2'1 yy = and 21

111

1

1'2 )(

)(9)()(3

2)()(

2 yyCyE

yByDyAyA

yIS

IS ××−×−+×−×

=ρρρρ

(41)

And the equations of the four acting forces as

))((34

13

ISIb yAgRF ρρπ −×=

'21

3 )(32 ygyARF SIf ×××= ρπ

)(32

1yBRF MSIr σπ=

21 )(6 ygRyCRF ISId ×××= µπ

3.3.2 Case with steel film formation Equation (22):

*

*

*

*****

2*

*2 9),(2),(32

2dtdZ

IdtdZSZK

JSZHG

dtZd

IM

IM −−×−+−

=ρρρρ

And the four acting forces on the inclusion:

⎥⎦

⎤⎢⎣

⎡−×⎟⎟

⎠

⎞⎜⎜⎝

⎛+

−+

××++

= )cos1(4

12

123

3sin

)(24 322

CII

S

I

MSIr SRSRdt

dZSR

RF θµ

θσ

π

gRF IMIb )(34 3 ρρπ −=

dtdZRF MId µπ6=

3 A differential equation is said to be autonomous if it does not explicitly contain the independent variable (usually denoted t ). A second-order autonomous differential equation is of the form , where

By the chain rule, can be expressed as 0),,( ''' =yyyF

./' vdtdyy ≡≡ ''y

vdydv

dtdy

dydv

dtdvvy ==== *''' [18].

13

2

23

32

dtZdRF MIf ρπ=

Here, Equation (22) is also an autonomous differential equation. Therefore, we rewrite it in order to make them possible to solve. We let and then equation (22) can be written as: *,1 Zy = ,*'

2 Zy =

2'1 yy = , and

22*

1*

1'2

9),(2),(32

2 yI

ySyKJ

SyHGyIM

IM ×−−×−+−

=ρρρρ

(42)

And the equations of the four acting forces on the inclusion for the case with steel-film formation can be written as

⎥⎦

⎤⎢⎣

⎡−×⎟⎟

⎠

⎞⎜⎜⎝

⎛+

−+

××++

= )cos1(4

12

123

3sin

)(24 3

222

CII

S

I

MSIr SRSR

ySR

RF θµ

θσ

π

gRF IMIb )(34 3 ρρπ −=

26 yRF MId µπ= '2

3

32 yRF MIf ρπ=

And Equation (31)

1*

2**

1*

12*

*

2))(2(

ySySSySyy

dtdS

+−+−−

= (43)

Now in order to solve above equations by the RKG we need to know the following parameters: , , and . startt endt dt And we need also to know the initial conditions, i.e.

=0V Initial velocity of the inclusion; =0Z Initial position of the inclusion

By choosing appropriate values for the parameters above, it is possible to solve the problem numerically by using Equation (40).

14

CHAPTER 4 In this chapter we discuss the implementations of the numerical algorithms presented in chapter 3. 4.1 Implementation of algorithm in C++ Here, description of the implementation of algorithm in C++ is given. The results can also be implemented in the same way in any other programming language. However, one have to be careful when using arrays in other programming languages, since in low level languages4, such as Pascal, C, C++, where one has to allocate memory for the arrays, and then destroy the array in the end of the program for not occupying the memory address and then causing a memory lack. In the high level languages5 such as Java, Delphi, or in MatLab, since it would destroy the array automatically after running the program and executing the results. 4.2. The solution algorithm To implement the method described in section 3.2, the algorithm presented by Nakajima et al. [23] is used. This algorithm has better numeric stability6, rather than the method described in 3.2. Nakajima et al rewrite Equation (40) in section 3.2 as:

443322111 kRkRkRkRyy kk ++++=+ (44) Where

).,( 41 ytfhk kk= ).,

2( 12 yhtfhk kk +=

).,2

( 23 yh

tfhk kkk +=

4 kk

).,( 3yhtfhk k+=

And

)2(21

411 QkR −=

))(211( 122 QkR −−=

4 Low-level programming languages are sometimes divided into two categories: first generation, and second generation [19], [20].

5 The term "high-level language" does not imply that the language is always superior to low-level programming languages – in fact, in terms of depth of knowledge of how computers operate the inverse may in fact be true. Rather, "high-level language" refers to the higher level of abstraction from machine language. Rather than dealing with registers, memory addresses and call stacks, high-level languages deal with variables, arrays and complex arithmetic or boolean expressions. In addition, they have no codes that can directly compile the language into machine code [21],[22]. 6 Numerical stability refers to how a malformed input affects the execution of an algorithm. In a numerically stable algorithm, errors in the input lessen in significance as the algorithm executes, having little effect on the final output. On the other hand, in a numerically unstable algorithm, errors in the input cause a considerably larger error in the final output [24].

15

))(211( 233 QkR −+= )2(

61

344 QkR −=

And

1141 213 kRQQ −+=

1212 )211(3 kRQQ −−+=

3323 )211(3 kRQQ +−+=

4434 213 kRQQ −+=

As mentioned earlier, for solving above, we need also to know the following parameters:

startt , and . endt dt As well:

• for the case of inclusion transport without steel film: =0V the initial velocity of the inclusion; =0Z the initial position of the inclusion;

y is a vector containing the initial conditions above, i.e. y = [Z0 V0].

• for the case of inclusion transport with a steel film: =0V the initial velocity of the inclusion; =0Z the initial position of the inclusion;

S0= the initial film thickness; y is a vector containing the initial conditions above, i.e. y = [Z0 V0 S0].

4.2.1 Inclusion Transport without steel film formation

end value of the time interval, step size, == dthWe let: initial time, == 0start tt == tendtendinitial velocity of the inclusion, =0V =0Z initial position of the inclusion, and y is a vector

containing the initial conditions above, i.e. y = [Z0 V0]. Then, initialize , and 0While (t < tend – dt,)

tt =

for i = 1 to the number of the initial conditions (in this case 2, since we have V0 and Z0)

).,(*)( 41 ytfdtik k=

))(2)((21)( 411 iQikiR −=

)()()( 141 iRiyiy +=

)(21)(3)()( 1141 ikiRiQiQ −+=

end for for i = 1 to the number of the initial conditions (in this case 2, since we have V0 and Z0)

),2

(*)( 12 yhtfdtik k +=

16

))()()(211()( 122 iQikiR −−=

)()()( 212 iRiyiy +=

)()211()(3)()( 1212 ikiRiQiQ −−+=

end for for i=1 to the number of the initial conditions (in this case (2), since we have V0 and Z0)

),2

(*)( 23 yh

tfdtik kk +=

))()()(211()( 233 iQikiR −+=

)()()( 323 iRiyiy +=

)()211()(3)()( 3323 ikiRiQiQ +−+=

end for for i=1 to the number of the initial conditions (in this case 2, since we have V0 and Z0)

),(*)( 34 yhtfdtik kk +=

))(2)((61)( 344 iQikiR −=

)()()( 434 iRiyiy +=

)(21)(3)()( 4434 ikiRiQiQ −+=

end for Then, to obtain the dimensionless displacement and the velocity of the inclusion for the case without steel film formation we write Z1(t) = , where Z1 is the dimensionless displacement )1(4)1(3 Ry +V1(t) = , where V1 is the velocity of the inclusion )2(4)2(3 Ry + t = t + 1; make an iteration for the values of t. End While 4.2.2 Inclusion Transport with steel film formation: Implement equation (48) from 2.8.1, then let

== 0ttstart The initial time. == tendtend The end value of the time interval.

== dth The step size.

=0V the initial velocity of the inclusion. =0Z the initial position of the inclusion.

S0 = The initial film thickness. y is a vector containing the initial conditions above, i.e. y = [Z0 V0 S0].

17

Initialize and then 0tt = While (t < tend – dt, and (Reynolds number) > 1 ) for i = 1 to the number of the initial conditions (in this case 3, since we have V0 and Z0, S0)

).,(*)( 41 ytfdtik k=

))(2)((21)( 411 iQikiR −=

)()()( 141 iRiyiy +=

)(21)(3)()( 1141 ikiRiQiQ −+=

end for for i = 1 to the number of the initial conditions (in this case 3, since we have V0 and Z0, S0)

),2

(*)( 12 yhtfdtik k +=

))()()(211()( 122 iQikiR −−=

)()()( 212 iRiyiy +=

)()211()(3)()( 1212 ikiRiQiQ −−+=

end for for i = 1 to the number of the initial conditions (in this case 3, since we have V0 and Z0, S0)

),2

(*)( 23 yh

tfdtik kk +=

))()()(211()( 233 iQikiR −+=

)()()( 323 iRiyiy +=

)()211()(3)()( 3323 ikiRiQiQ +−+=

end for for i = 1 to the number of the initial conditions (in this case 3, since we have V0 and Z0, S0)

),(*)( 34 yhtfdtik kk +=

))(2)((61)( 344 iQikiR −=

)()()( 434 iRiyiy +=

)(21)(3)()( 4434 ikiRiQiQ −+=

end for Then to obtain the dimensionless displacement and the velocity and the dimensionless steel film thickness of the inclusion for the case with steel film formation we write

18

Z2(t) = , where Z2 is the dimensionless displacement )1(4)1(3 Ry +V2(t) = , where V2 is the dimensionless velocity of the inclusion )2(4)2(3 Ry +S(t) = , where S is the dimensionless film thickness. )3(4)3(3 Ry + t = t + 1; make an iteration for the values of t. End While 4.3 The choice of initial conditions As described in Chapter 3.2, the choice of the initial conditions is critical for calculation efficiency. For implementation of algorithm in Chapter 3.2, the initial conditions suggested by K. Nakajima et al were used. These parameters where chosen after a careful study and shown to be the most appropriate for treating the both cases of the inclusion transport with and without a steel film formation. The following tables outline the values of the initial conditions I used when implementing the algorithms described in chapter 4.

Table 1 Inclusion Transport without Steel-Film Formation.

0t 0 tend 0.1 dt 0.0002

0V 0,011040Z 0

Table 2 Inclusion Transport with Steel-Film Formation.

0t 0 tend 0.1 dt 0.0002

0V 0,349110Z 0

S0 0,002

In the above tables, the values of initial conditions are provided. Initial conditions that affect the numerical stability are 0t , and . If end or is changed, it will effect the numerical stability of the algorithm, i.e. it will make the calculation longer if we choose a longer interval from 0 to end or if we choose a smaller time-step And, the calculation would be shorter if we choose a smaller calculation interval from t to t or a bigger time-step [21]

tend dt t ,dt

t t .dt0 end .dt

tend

T

tstart

Figure 5 Time discritization.

19

However, the choice of the step-size or the calculation interval length should be done after a proper consideration of the physical problem given, so if the choice of the step-size and the interval length results in a satisfactory solutions of the physical problem given, then there is no need to try to seek a smaller step-size or a longer calculation interval length, though the smaller step-size length we choose the smaller error between the time steps we obtain and hence a better calculation, see figure (5), but it would affect the numerical stability of the algorithm and resulting in a longer calculation and execution time of the algorithm. Some programming languages are quicker in execution time than other, for instance Fortran 90, C and C++ have a shorter execution time than Java and Delphi, so the choice of the programming language should also be taken in consideration, also the choice of the numerical method, since some numerical methods gives a better error estimation than other numerical methods, for instance the Finite Element Method and the Finite Volume Method give a better error estimation for the solutions than the Runge-Kutta Method. So, when deciding to solve a mathematical problem, if very small time-step and a longer calculation interval are chosen, then one has to choose the programming language and the numerical method in order to carry out the calculations; sometimes it is even necessary to use super-computers with multiple processors in order to carry out proper calculations. But as I mentioned one has really be sure whether this is necessary to do or not to solve a given problem, since it is a time consuming possess and even highly in costs. The other initial conditions i.e. the initial velocity V0 and the initial position of the inclusion Z0 and the initial film thickness S0 from tables 1 and 2 are presented in Nakajima et al .Those parameters where chosen after a careful study and they should be implemented as they are given, since any small change in them would be critical for the calculation of the solid inclusion. Finally all the parameters used, in this work, are the one presented by Nakajima et al, as follows.

20

Table 3 Numerical parameters.

Parameter Name Description Value Unit MSσ The surface tension

of the metal-slag interface

1.375 N/m

MIσ The surface tension of the metal-inclusion interface

1.518 N/m

ISσ The surface tension of the inclusion-slag interface

0.44 N/m

Mµ The viscosity of the metal

0.006 Pa*s

Sµ The viscosity of the slag

0.1998 Pa*s

Id (NO FILM) Inclusion diameter 0.00002 m

Id (WITH FILM) Inclusion diameter 0.0002 m R (NO FILM) The radius of the

inclusion 0.00001 m

R (WITH FILM) The radius of the inclusion

0.0001 m

IMS)cos(θ The overall wettability

0.784 N/m

RI (NO FILM) Reynolds number 0.0025509 - RI (WITH FILM) Reynolds number 2.5509 - G The gravitational

force 9.80665 2/ sm

Iρ The density of the inclusion

3990 3/ mkg

Mρ The density of the metal

7000 3/ mkg

Sρ The density of the slag

2543 3/ cmkg

21

Chapter 5 In this chapter we discuss the results obtained after solving the algorithms in chapter 4. 5.1 The case without steel-film formation Figures 9–11 outline the results of the implementation of Chapter 3.2. Three plots are obtained, for the dimensionless displacement and the velocity of the inclusion and the four forces acting on the inclusion. Figure (9) shows the displacement of the inclusion. We see from that figure that the displacement of the inclusion particle starts just when the time is zero we have the displacement Z (m) = 0.039590m, then it will continue developing up for different time steps until it reaches its highest point at 1.7790m at time s, and then the inclusion particle displacement continues in steady state mode.

51004.2 −×

Figure (10) shows the velocity of the inclusion particle. Here, the particle has its lowest velocity equal to 283.7 m/s at time s, and it reaches its highest velocity 461.4 m/s at time s, and then after this point the velocity of the inclusion particle will start decreasing.

7102.2 −×61001.1 −×

Figure (11) shows the plot of the four acting forces on the inclusion for the case of inclusion transport without steel film formation, from that plot we conclude that the rebound force r it reaches its highest point 0.00015071 N at time 0s and after that time it will continue decreasing downwards across the time axis.

F

The added mass force f , it reaches its highest point for N at time

s, and then it will continue decreasing. F 6108180.1 −×

5104680.4 −× The drag force d , it reaches its lowest point for N at time -0.0001378 s, and then it will continue growing up across the time axis.

F 6106160.1 −×

Finally the buoyancy force b , which always acts upwards, is rather growing up nor decreasing down, i.e. it is continuing across the time axis.

F

5.2 The case with steel-film formation Figures 12–15 outline the results of the steel film case. We obtained three plots for the dimensionless displacement and the velocity of the inclusion and the dimensionless steel-film thickness and the four forces acting on the inclusion. Figure (12) shows the displacement of the inclusion, we see from that figure that the displacement of the inclusion particle starts just when the time is zero we have the displacement Z(m) = m, then it will continue developing up for different time steps until it reaches its highest point at 0.00020m at time 0.00016990s.

8103870.1 −×

Figure (13) shows the velocity of the inclusion particle, here the particle has its lowest velocity equal to 0.01072 m/s at time s, and it reaches its highest velocity 2.04 m/s at time 0.0001009 s, and then after this point the velocity of the inclusion particle will start decreasing.

7103870.6 −×

Figure (14) shows the dimensionless steel film thickness of the inclusion, it has its highest point for dS/dt = 0.001932 m/s at the start time 0 s, then it decreases until it reaches its lowest point for dS/dt = 0.000995 m/s at time s. 5105130.3 −×

22

Figure (15) shows the four acting forces on the inclusion, here we see that the rebound force r it reaches its highest point 0.0015390 N at time s and after that time it will

continue decreasing downwards. F 5105130.3 −×

The added mass force f , it reaches its lowest point for -0.0015090 N at time s, and then it will continue growing up.

F 5100890.7 −×

The drag force d , it reaches its lowest point for -0.000754 N at time 0.0001003 s, and then it will continue growing up.

F

Finally the buoyancy force b , which always acts upwards, here this force is rather growing up nor decreasing down, it’s continuing across the time axis.

F

5.3 Discussions For the results obtained for the both cases of inclusion transports above, we can compare the results of the two cases by taking the average value of the results of the dimensionless displacement and the velocity. To calculate the average value, its simplest done by the following formula:

Average Value =2

valuelowestvaluehigest + .

In Table (4) the values for both cases are summarized

Table 4.

Variable Unit Average Value (without film)

Average Value (with film)

Average Time (without film)

Average Time (with film)

Displacement M 0.90930 41000010.1 −× -5101.0200 × s s108.4950 -5× Velocity m/s 372.5500 1.0254 s106.1500 -7× s-5105.07690 ×

We see from Table 4 that the average displacement of the inclusion particle transport is greater for the case of without steel film formation than for the case with steel film formation. Also, average time of the displacement is shorter for the case without steel film than the one with steel. The velocity is greater for the case of inclusion transport without steel film formation than the one with steel film formation, but the average time of the velocity for the case with film is shorter for the case with film than the case without film. Those measurements above are somehow showing the different behavior the inclusion particle takes in the two different cases.

23

5.4 Characteristics of Particle Behavior at the Interface By analyzing the mathematical model above, we can conclude that the inclusion can adopt three driftnet types of behavior at the interface depending on the interfacial properties of the system, and the size and initial velocity of the inclusion. These types of the behavior of the inclusion are:

• Pass. • Remain. • Oscillate.

These three modes of inclusion behavior are schematically outlined in Figure (16). As mentioned before the center of the inclusion will have to be displaced one inclusion diameter for its original position in order to pass from the steel to the slag. In the case of remain, the inclusion stays at the interface and is not completely transferred to the slag. This is the case occur in the case of inclusion transfer without steel film. This happens when the viscosity of the slag Sµ is (0.1199) (Pa*s). In the case of oscillate, the inclusion initially rises upwards to a maximum position from where it descends to a position slightly above the original location, this happens when the viscosity of the slag Sµ (Pa*s) is between (0.0399) and (0.1199) (Pa*s). From this position the inclusion rises once again. This continues for some time while the oscillations are gradually dampened out and the inclusion finally comes to rest at an equilibrium position, with the center of the inclusion located beneath the interface. It is apparent that both in the remain and in the oscillate mode the inclusion might be washed back into the steel, by a steel flow that is parallel to the interface. These two modes should therefore be avoided if an increased steel cleanliness is desired. The pass case occurs in the case of inclusion transfer with steel film, in this case we have the inclusion diameter is roughly 150 mµ . 5.5 Concluding Remarks A mathematical model has been used to predict how inclusions are transferred across a steel-slag interface. The inclusion can behave in three ways as it reaches the steel-slag interface. The inclusion can behave in three ways as it reaches the steel-slag interface. It can pass and enter the slag almost instantaneously, remain at the interface where it may act as a build up place for clusters that can re-enter the steel bath with the steel flow and case nozzle clogging in the end, or oscillate and follow the flow to the steel causing product defects. There also exist two situations when an inclusion passes the interface, one in which a steel film is formed between the inclusion and the slag and another where no steel film is formed. However, it has been concluded in this work that the steel film formation occurs when inclusion diameter is roughly mµ150 . Since most non-metallic inclusion found in modern steel grades in a tundish far smaller that that diameter, the non-film case of the model is most relevant to use. The main conclusion of this work is that the proposed mathematical model can be used to predict and to determine the critical parameters governing the separation of non-metallic inclusions at the interface between the steel and the slag. Strand et al. [4] has showed that the most important parameters on the inclusions transfer have been found to be the interfacial tensions ( MSISMI σσσ ,, ) and the slag viscosity ( Sµ ) for a 20 Mµ inclusions. In the case of the100 Mµ inclusion also the slag and inclusion density plays a role. The combined effect of these parameters showed that the overall wettability should be positive and that the slag entrainment to the steel, for most favorable inclusion transfer conditions.

24

5.6 Possibilities of coupling in the future The code is a stand alone program, which can be coupled with appropriate CFD software, using appropriate conditions. Therefore, it would be possible, for instance, to model the displacement and the velocity of the inclusion particle for the cases with and without steel film in the tundish. Here, we briefly outline this process in FLUENT® solver. The solver provides an option of defining UDF (User Defined Functions), with which a stand-alone code can be coupled to the solver at some boundary. This means that at specified boundary the solver will use the UDF to calculate specified values. . To do this, some steps are needed to be completed in the solver:

1. Define the problem. 2. Create a C source code file. 3. Start FLUENT and read in (or stet up) the case file. 4. Interpret the source file. 5. Hook the UDF to FLUENT. 6. Run the calculation. 7. Analyze the numerical solution and compare it to expected results.

25

References

1. H. Solhed, L. Jonsson, P. Jönsson. A theoretical and Experimental Study of Continuous Casting Tundish Focusing on Slag-Steel Interaction, Metallurgical and Materials Transactions B, Volume 33, Number 2, 1 April 2002, pp. 173–185 (13).

2. K. Nakajima and K. Okamura. Proc. 4th International Conference on Molten Slag and

Fluxes, ISIJ, Sendai, Japan (1992), p. 505.

3. K. Nakajima. PhD. Thesis, Osaka University, Osaka, Japan (1993).

4. J. Strand. A Study of Solid and Liquid Inclusion Separation at the Steel-Slag Interface. April 2005.

5. Report: Continuously Cast Steel Output, 1999. International Iron Steel Institute,

Brussels, Belgium, 2000, source: www.worldsteel.org.

6. A diagram of the Tundish, source: www.fibretech.com.

7. Tundish ( Stahlindustrie–Flow Control ), source: www.rhi.at.

8. S. Kholmatov. Mathematical Modeling of Particle Inclusion Removals during Continuous Casting of Steel, MSc. Thesis, KTH-MSE, May 2005.

9. F. D. Najjar. Finite-Element Modeling of Turbulent Fluid Flow and Heat Transfer

through Bifurcated Nozzles in Continuous Steel Slab Casters, MSc. Thesis, University of Illinois at Urbana-Champaign, 1990.

10. B. Kiflie and Dr-Ing D. Alemu. Thermal analysis of Continuous Casting, ESME 5th

Annual Conference on Manufacturing and Process Industry, September 2000.

11. Michael T. Health. Scientific Computing, an Introductory Survey, 2nd Edition, 2002, page 405.

12. Michael T. Health. Scientific Computing, an Introductory Survey, 2nd Edition, 2002,

page 342–359.

13. http://en.wikipedia.org/wiki/Bootstrapping.

14. Michael T. Health. Scientific Computing, an Introductory Survey, 2nd Edition, 2002, page 404.

15. Kaplan, W. Derivatives and Differentials of Composite Functions and the General

Chain Rule. §2.8 and 2.9 in Advanced Calculus, 3rd edition. Reading, MA: Addison-Wesley, pp. 101–105 and 106–110, 1984.

16. H. M. Anita. Numerical Methods for Scientists and Engineers, 2002.

17. Abramowitz, M. and Stegun, I. A. (Eds.) Handbook of Mathematical Functions with

Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover p. 896, 1972.

26

18. Weisstein, Eric W. Autonomous. From MathWorld–A wolfram Web Resource.

http://mathworld.wolfram.com/Autonomous.html.

19. Alan Perlis Humorous epigram from Epigrams on Programming, 2001.

20. Murdocca, Miles J., Vincent P. Heuring. Principles of Computer Architecture. Prentice-Hall, 2000.

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3rd Edition, 1972.

22. Nicklaus Wirth. Hardware Architectures for Programming Languages and Programming Languages for Hardware Architectures, 1987.

23. A part of Nakamima’s Fortran programs on Inclusion Transfer Behavior Across a

Molten Steel-Slag Interface and its memorandums. May 14, 2003.

24. Nicholas J. Higham. Accuracy and Stability of Numerical Algorithms, Society of Industrial and Applied Mathematics, 1996.

27

Nomenclature Symbol Description Unit g gravity 2/ smt time s

*t dimensionless time - ∞

Inclusion at 0u terminal/initial velocity of the sm /

=t

ID diameter of the inclusion mIR radius of the inclusion m

M

IMe

DuRµ

ρ ∞= Reynolds number -

S steel film thickness m)0(S initial steel film thickness m

Z displacement of the inclusion m)0(Z initial position of the inclusion m

xρ density 3/ mkgxσ surface tension mN /

interfacial tension mN /xσ y

xµ viscosity sPa. Superscript * dimensionless

28

List of Figures

Figure 1 Diagram of the continuous casting system.

Figure 2 Diagram of the tundish.

Figure 3 A tundish in a steel factory.

Figure 4 Schematic of the continuous casting system.

Figure 5 Time discritization.

Figure 6 Schematic diagram showing the inclusion transfer to the slag without the film formation between the inclusion and the slag.

Figure 7 Schematic diagram showing the inclusion transfer to the slag with

the formation of a thin film between the inclusion and the slag.

Figure 8 Streamlines around a solid moving sphere in a liquid at rest.

Figure 9 The displacement of the inclusion at time t (s) for the case with no steel film.

Figure 10 The velocity of the inclusion at time t (s) for the case with no steel film.

Figure 11 Four different forces acting on the inclusion at time t (s) for the case with no steel film.

Figure 12 The displacement of the inclusion at time t (s) for the case with steel film.

Figure13 Velocity of the inclusion at time t (s) for the case with steel film.

Figure 14 The film thickness at time t(s) for the case with steel film formation.

Figure 15 Four different forces acting on the inclusion at time t (s)

for the case with no steel film.

Figure 16 The three types of behavior (remain, oscillate and pass) for the inclusion at the steel-slag interface.

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Figure 6 Schematic diagram showing the inclusion transfer to the slag without the film formation between the inclusion and the slag [4].

Figure 7 Schematic diagram showing the inclusion transfer to the slag with

the formation of a thin film between the inclusion and the slag [4].

30

Figure 8 Streamlines around a solid moving sphere in a liquid at rest [4]. Figure 9 The displacement of the inclusion at time t (s) for the case with no steel film.

31

Figure 10 The velocity of the inclusion at time t (s) for the case with no steel film.

Figure 11 Four different forces acting on the inclusion at time t (s) for the case with no steel film.

32

Figure 12 The displacement of the inclusion at time t (s) for the case with steel film.

Figure13 Velocity of the inclusion at time t (s) for the case with steel film.

33

Figure 14 The film thickness at time t(s) for the case with steel film formation. Figure 15 Four different forces acting on the inclusion at time t (s)for the case with no steel film.

34

Remain Oscillate Pass Figure 16 The three types of behavior (remain, oscillate and pass) for the inclusion at the steel-slag interface.

35

Appendix

CODE LISTING

36

The code for the files constants.h, functionsNofilm.cpp, mainNofilm.cpp, functionsWithfilm.cpp and mainwithfilm.cpp are given below. Constants.h #ifndef CONSTANTS_H #define CONSTANTS_H #define PI 3.14159 #include <math.h> //Some constants and initial values //double pi = 3.1416; double g=9.80665; //The gravitational force (m/s) double roI=3990; //The density of the inclusion (kg/m3) double roM=7000; //The density of the metal (kg/m3) double roS=2543; //The density of the slag (kg/cm3) double sigmaMS=1375e-3; //The surface tension of the metal-slag interface (N/m) double sigmaMI=1518e-3; //The surface tension of the metal-inclusion interface (N/m) double sigmaIS=440e-3; //The surface tension of the inlucion-slag interface (N/m) double myM=0.006; //The viscosity of the metal (Pa*s) double myS=0.1998; //The viscosity of the slag (Pa*s) //double myS=0.0399; double AITA=1.0; //Constant double XAITA=1.0; //Constant double dI=20E-6; //The inclusion diameter double RI=dI/2; //The radius of the inclusion double alfar1=(int)ceil(sigmaMS/((pow(RI,2))*g*(AITA*roM+2*roI))); //Constant double betad1=(sqrt(g*pow(RI,3)))*(AITA*roM+2*roI)/myM; //Constant double gammar1=(sqrt(g*pow(RI,3)))*(AITA*roM+2*roI)/myS; //Constant double costhetaIMS=(sigmaMI-sigmaIS)/sigmaMS; //The overall wettability //double costhetaIMS=-0.6909; double NT=501; //Constant double DT=0.0002; //Time step double tEND=(NT-1)*DT; double MN=3; double MT=NT; double JTIME=1; double JJ=1; double t0=0.0; //The initial time double Z0=0.0; //The initial position of the inclusion double V0=((2.0/9.0)*(pow(RI,2))*(roM-roI)*g)/(myM*sqrt(g*RI)); //The initial velocity of the inclusion double S0=2e-3; //The initial film thickness double S1=1e-3; //The film thickness at rupture double ReI=roM*V0*sqrt(g*RI)*2*RI/myM; //Reynolds number double Jflg=0; double JJflg=0; #endif

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functionsNofilm.cpp #include <iostream> #include <math.h> using namespace std; #include "constants.h" void CopyColumn(int n, int ColumnNumber, double **x, double *y) // n - Array lenght // ColumnNumber - number of the column to copy // x - 2d (n*m) input array // y - result array { for (int i=0; i < n; i++) y[i] = x[i][ColumnNumber]; // 2d array memory allocation /* int n,m; double **a; n=5; // rows m=6; // col a = new double *[n]; for (int i=0; i<n; i++) a[i] = new double[m]; */ } double z2(int k, double t, double *Y, int n) { // n must be initialized with some value // n=50; // n=3; int F = 1; // k=0; double AS, BS, C1, C2; double Phai1 = 0.0; double wmemb = 0.0; double P = 0.0; double Phai2 = 0.0; double Phai3 = 0.0; // Y = new double[n]; // Z=Y(:,1); double Z = Y[0]; // V=Y(:,2); double V = Y[1]; double PP = myM/myS; double xmemb = 0.0; if (Z < 1) { Phai1 = (PP-1)*pow(Z,2)-2*(PP-1)*Z+PP; } else if (Z >=1) { Phai1 = 1;

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} P = roM/roS; if (Z < 2) { Phai2 = 0.25*(P-1)*pow(Z,3)-0.75*(P-1)*pow(Z,2)+P; } else { Phai2 = 1; } Phai3 = Z-1-costhetaIMS; AS=sigmaMS/((g*pow(RI,2))*(XAITA*roS*Phai2+2*roI)); BS=(sqrt(g*pow(RI,3))/myS)*(XAITA*roS*Phai2+2*roI); C1=2*(roS*Phai2-roI)/(XAITA*roS*Phai2+2*roI); C2=3*AS*Phai3; if (k==1) { xmemb = V; } else if(k==2) { xmemb=C1-C2-(9*F*Phai1/BS)*V; } return xmemb; } void out1(double *arr, int n) { for (int i=0; i<n;i++) { // cout<<"{"<<arr[i]<<","<<" "<<arr[i+1]<<"}"<<endl; cout<<"{"<<arr[i]<<","<<" "<<arr[i+1]<<","<<" "<<arr[i+2]<<"}"<<endl; } } void out2(double **arr, int n) { //int i = 1; int i; int j; for (i=0; i<n; i++) { // int j=2; i = 0; for(j=0;j<30;j++) cout<<"i0 ="<<"{"<<arr[i][j]<<"}"<<endl; cout<<"BRAKE1"<<endl; cout<<"BRAKE1"<<endl; cout<<"BRAKE1"<<endl; cout<<"BRAKE1"<<endl; i = 1;

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for(j=0;j<30;j++) cout<<"i1 ="<<"{"<<arr[i][j]<<"}"<<endl; } // for (i=1; i<n; i++) // { // for(j=0;j<4;j++) // cout<<"i1 ="<<"{"<<arr[i][j]<<"}"<<endl; // } } void out3(double *arr, int n) { for (int i=1; i<n; i++) cout<<"res["<<i<<"] ="<<arr[i]<<" "; cout<<endl; } int SaveToFile(int i0, int n, double *a, char *FileName) { FILE *f; int m = n - i0; if ((f = fopen(FileName, "wb")) == NULL) return -1; fwrite(&m, sizeof(int), 1, f); for (int i=i0; i<n; i++) fwrite(&(a[i]), sizeof(double), 1, f); fclose(f); }

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mainNofilm.cpp #include <iostream> #include <math.h> using namespace std; #include "constants.h" #include "functionsNofilm.cpp" int main(int argc, char *argv[]) { double *Y0; Y0 = new double[3]; Y0[0] = Z0; Y0[1] = V0; Y0[2] = S0; /* CALCULATION WITHOUT STEEL FILM */ int t = (int)ceil(t0+(JTIME-JJ)*DT); int MXN=50; double HROOT1=1.0-sqrt(0.5); //Constant double HROOT2=1.0+sqrt(0.5); int N = 3; int i; int x; double *YW = new double[3]; YW = Y0; double *Q4 = new double[N]; for (i=0; i<N; i++) { Q4[i]=0; } int m = 0; int p = 0; int L = 2; int k; double t1 = 0.0; double t2 = 0.0; double VF = 1.0; double VPP = myM/myS; double VP = roM/roS; double Vphai1 = 0.0; double Vphai2 = 0.0; double Vphai3 = 0.0; double *K0 = new double[L]; double *R1 = new double[L]; double *Y1 = new double[L]; double *Q1 = new double[L]; double *K1 = new double[L]; double *R2 = new double[L]; double *Y2 = new double[L]; double *Q2 = new double[L]; double *K2 = new double[L]; double *R3 = new double[L];

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double *Y3 = new double[L]; double *Q3 = new double[L]; double *K3 = new double[L]; double *R4 = new double[L]; double *Q5 = new double[3]; double **Y4 = new double*[L]; for(i=0;i<L;i++) { Y4[i] = new double[502]; for(int j=0; j<502;j++) Y4[i][j] = 0.0; } double *Z2 = new double[501]; double *V2 = new double[501]; double *alfa = new double[501]; double *AA2 = new double[501]; double *BB2 = new double[501]; double *CC2 = new double[501]; double *Ff2 = new double[501]; double *Fd2 = new double[501]; double *Fr2 = new double[501]; double *Fb2 = new double[501]; double *TT2 = new double[501]; while((JJflg == 0) && (p < MT-m)) { p; if (m == 0) t2 = t0+(p-1)*DT; else t2 = t1+p*DT; TT2[p] = t2; //out1(TT2, p); for (k=0; k<L; k++) { K0[k] = DT*z2(k+1,t2, YW, MXN); R1[k] = 0.5*(K0[k]-2*Q4[k]); Y1[k] = YW[k]+R1[k]; Q1[k] = Q4[k]+3*R1[k]-0.5*K0[k]; } for (k=0; k<L; k++) { K1[k] = DT*z2(k+1, (t2+0.5*DT), Y1, MXN); R2[k] = HROOT1*(K1[k]-Q1[k]); Y2[k] = Y1[k]+R2[k]; Q2[k] = Q1[k]+3*R2[k]-HROOT1*K1[k]; } for (k=0; k<L; k++) { K2[k] = DT*z2(k+1, (t2+0.5*DT), Y2, MXN); R3[k] = HROOT2*(K2[k]-Q2[k]); Y3[k] = Y2[k]+R3[k]; Q3[k] = Q2[k]+3*R3[k]-HROOT2*K2[k]; }

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for (k=0; k<L; k++) { K3[k] = DT*z2(k+1, (t2+DT), Y3, MXN); R4[k] = (K3[k]-2*Q3[k])/6; Y4[k][p+1] = Y3[k]+R4[k]; Q5[k] = Q3[k]+3*R4[k]-0.5*K3[k]; YW[k] = Y3[k]+R4[k]; } Z2[p] = Y3[0]+R4[0]; V2[p] = Y3[1]+R4[1]; p=p+1; } for (k=0; k<(m+p); k++) { if (k == (m+p)) alfa[k] = (V2[k]-V2[k-1])/DT; else alfa[k] = (V2[k+1]-V2[k])/DT; } for (x=0; x<p; x++) { if (Z2[x] < 1) Vphai1 = (VPP-1.0)*pow(Z2[x],2)-2*(VPP-1)*Z2[x]+VPP; else Vphai1 = 1.0; AA2[x] = Vphai1; if (Z2[2] < 2.0) Vphai2 =0.25*(VP-1)*pow(Z2[x],3)-0.75*(VP-1)*pow(Z2[x],2)+VP; else Vphai2 = 1.0; BB2[x] = Vphai2; Vphai3=Z2[x]-1.0-costhetaIMS; CC2[x] = Vphai3; Ff2[x] = -(2.0/3.0)*PI*pow(RI,3)*((XAITA*roS*Vphai2+2*roI)*g*alfa[x]); //KOLLA HÄR Fd2[x] = -6*PI*RI*myS*VF*Vphai1*V2[x]*sqrt(RI*g); Fr2[x] = -2*PI*RI*sigmaMS*Vphai3; Fb2[x] = (4.0/3.0)*PI*pow(RI,3)*g*(roS*Vphai2-roI); } for (int i=0;i<501; i++) { TT2[i] *= sqrt(RI/g); } int size = 501; SaveToFile(0, size, V2, "V2.bin"); SaveToFile(0, size, Z2, "Z2.bin"); SaveToFile(0, size, Fr2, "Fr2.bin"); SaveToFile(0, size, Ff2, "Ff2.bin"); SaveToFile(0, size, Fb2, "Fb2.bin");

43

SaveToFile(0, size, Fd2, "Fd2.bin"); SaveToFile(0, size, TT2, "TT2.bin"); delete []YW; delete []Q4; delete []K0; delete []R1; delete []Y1; delete []Q1; delete []K1; delete []R2; delete []Q2; delete []K2; delete []R3; delete []Y3; delete []Q3; delete []K3; delete []R4; delete []Y4; delete []Z2; delete []V2; delete []alfa; delete []AA2; delete []BB2; delete []CC2; delete []Ff2; delete []Fd2; delete []Fr2; delete []Fb2; delete []TT2; system("PAUSE"); return EXIT_SUCCESS; }

44

functionsWithfilm.cpp #include <iostream> #include <math.h> using namespace std; #include "constants.h" double z1(int k, double t, double *Y, int n) { int F = 1; double C1 = 0.0; double C2 = 0.0; double C3 = 0.0; double wmemb = 0.0; double Z = Y[0]; double V = Y[1]; double S = Y[2]; double D1 = pow((1+S), 3); C1 = 2*(roM-roI)/(AITA*roM+2*roI); C2 = (3*alfar1*(2+S-Z)*(S+Z))/D1; C3 = (2/gammar1)*(3/2)*((1/(1+2*S))-(1/(1+4*S)))*(1-pow(((1-Z)/(1+S)),3)); if (k == 1) { wmemb = V; } else if (k == 2) { wmemb=C1-C2-(C3+9*F/betad1)*V; } else if (k == 3) { wmemb=(-F*V*(2-Z)*(S+Z)*S-S*V)/(2*S+Z); } return wmemb; } double z2(int k, double t, double *Y, int n) { int F = 1; double AS, BS, C1, C2; double Phai1 = 0.0; double wmemb = 0.0; double P = 0.0; double Phai2 = 0.0; double Phai3 = 0.0; double Z = Y[0]; double V = Y[1]; double PP = myM/myS; double xmemb = 0.0; if (Z < 1) { Phai1 = (PP-1)*pow(Z,2)-2*(PP-1)*Z+PP; } else if (Z >=1)

45

{ Phai1 = 1; } P = roM/roS; if (Z < 2) { Phai2 = 0.25*(P-1)*pow(Z,3)-0.75*(P-1)*pow(Z,2)+P; } else { Phai2 = 1; } Phai3 = Z-1-costhetaIMS; AS=sigmaMS/((g*pow(RI,2))*(XAITA*roS*Phai2+2*roI)); BS=(sqrt(g*pow(RI,3))/myS)*(XAITA*roS*Phai2+2*roI); C1=2*(roS*Phai2-roI)/(XAITA*roS*Phai2+2*roI); C2=3*AS*Phai3; if (k==1) { xmemb = V; } else if(k==2) { xmemb=C1-C2-(9*F*Phai1/BS)*V; } return xmemb; } void out3(double *arr, int n) { for (int i=1; i<n; i++) cout<<"res["<<i<<"] ="<<arr[i]<<" "; cout<<endl; } int SaveToFile(int i0, int n, double *a, char *FileName) { FILE *f; int m = n - i0; if ((f = fopen(FileName, "wb")) == NULL) return -1; fwrite(&m, sizeof(int), 1, f); for (int i=i0; i<n; i++) fwrite(&(a[i]), sizeof(double), 1, f); fclose(f); }

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mainwithfilm.cpp #include <iostream> #include <math.h> using namespace std; #include "constants.h" #include "functionsWithfilm.cpp" int main(int argc, char *argv[]) { double *Y0; Y0 = new double[3]; Y0[0] = Z0; Y0[1] = V0; Y0[2] = S0; /* CALCULATION WITHOUT STEEL FILM */ int t = (int)ceil(t0+(JTIME-JJ)*DT); int MXN=50; double HROOT1=1.0-sqrt(0.5); //Constant double HROOT2=1.0+sqrt(0.5); int N = 3; int i; int x; double *YW = new double[N]; YW = Y0; double *Q4 = new double[N]; for (i=0; i<N; i++) { Q4[i]=0; } int m = 0; int p = 0; int L = 2; int k; double t1 = 0.0; double t2 = 0.0; double VF = 1.0; double VPP = myM/myS; double VP = roM/roS; double Vphai1 = 0.0; double Vphai2 = 0.0; double Vphai3 = 0.0; double *K0 = new double[N]; double *R1 = new double[N]; double *Y1 = new double[N]; double *Q1 = new double[N]; double *K1 = new double[N]; double *R2 = new double[N]; double *Y2 = new double[N]; double *Q2 = new double[N]; double *K2 = new double[N]; double *R3 = new double[N];

47

double *Y3 = new double[N]; double *Q3 = new double[N]; double *K3 = new double[N]; double *R4 = new double[N]; double *Q5 = new double[3]; double **Y4 = new double*[N]; for(i=0;i<N;i++) { Y4[i] = new double[502]; for(int j=0; j<502;j++) Y4[i][j] = 0.0; } double *Z2 = new double[501]; double *V2 = new double[501]; double *alfa = new double[501]; double *AA2 = new double[501]; double *BB2 = new double[501]; double *CC2 = new double[501]; double *Ff2 = new double[501]; double *Fd2 = new double[501]; double *Fr2 = new double[501]; double *Fb2 = new double[501]; double *TT2 = new double[501]; double *TT1 = new double[501]; double *S = new double[501]; double *V1 = new double[501]; double *Z1 = new double[501]; double *V = new double[501]; double *Fr1 = new double[501]; double *Ff1 = new double[501]; double *Fb1 = new double[501]; double *Fd1 = new double[501]; double *Z = new double[501]; double *TT = new double[501]; double *Fr = new double[501]; double *Fd = new double[501]; double *Ff = new double[501]; double *Fb = new double[501]; //Starts the Runge-Kutta-Gill for the film case // for (m=0; m<MT; m++) // { // if ((Jflg == 0) && (ReI > 1)) // { while((Jflg == 0) && (m < MT) && (ReI > 1)) { m; t1 = t0+(m-1)*DT; TT1[m] = t1; for (k=0; k<N; k++) { K0[k] = DT*z1(k+1,t1, YW, MXN); R1[k] = 0.5*(K0[k]-2*Q4[k]); Y1[k] = YW[k]+R1[k]; Q1[k] = Q4[k]+3*R1[k]-0.5*K0[k]; }

48

for (k=0; k<N; k++) { K1[k] = DT*z1(k+1, (t1+0.5*DT), Y1, MXN); R2[k] = HROOT1*(K1[k]-Q1[k]); Y2[k] = Y1[k]+R2[k]; Q2[k] = Q1[k]+3*R2[k]-HROOT1*K1[k]; } for (k=0; k<N; k++) { K2[k] = DT*z1(k+1, (t1+0.5*DT), Y2, MXN); R3[k] = HROOT2*(K2[k]-Q2[k]); Y3[k] = Y2[k]+R3[k]; Q3[k] = Q2[k]+3*R3[k]-HROOT2*K2[k]; } for (k=0; k<N; k++) { K3[k] = DT*z1(k+1, (t1+DT), Y3, MXN); R4[k] = (K3[k]-2*Q3[k])/6; Y4[k][m+1] = Y3[k]+R4[k]; Q4[k] = Q3[k]+3*R4[k]-0.5*K3[k]; YW[k] = Y3[k]+R4[k]; } Z1[m] = Y3[0] + R4[0]; V1[m] = Y3[1] + R4[1]; S[m] = Y3[2] + R4[2]; if (Z1[m] >= 2) { Jflg = 1; } if (S[m] <= S1) { Jflg = 2; } m=m+1; } //} if (Jflg == 0) { if (ReI < 1) { cout<<"No film. Change equation and continue"<<endl; } else { cout<<"Reaches 2.0 ... Normal end=too short calculation time to desc ribe the phenomena"<<endl; cout<<"Jflg=0, The film is still on"<<endl; } } if (Jflg == 1) { cout<<"Reaches 2.0 ... Normal end"<<endl; }

49

if (Jflg == 2) { cout<<"Film rupture, Reaches S1. Change equation and continue"<<endl; } // for (p=0; p<(MT-m); p++) // { cout<<"ffff"<<endl; // if (JJflg == 0) while ((JJflg == 0) && (p < MT-m)) { p; if (m == 0) { t2=t0+(p-1)*DT; } else t2 = t1+p*DT; TT2[p] = t2; for (k=0; k<L; k++) { K0[k] = DT*z2(k+1,t2, YW, MXN); R1[k] = 0.5*(K0[k]-2*Q4[k]); Y1[k] = YW[k]+R1[k]; Q1[k] = Q4[k]+3*R1[k]-0.5*K0[k]; } for (k=0; k<L; k++) { K1[k] = DT*z2(k+1, (t2+0.5*DT), Y1, MXN); R2[k] = HROOT1*(K1[k]-Q1[k]); Y2[k] = Y1[k]+R2[k]; Q2[k] = Q1[k]+3*R2[k]-HROOT1*K1[k]; } for (k=0; k<L; k++) { K2[k] = DT*z2(k+1, (t2+0.5*DT), Y2, MXN); R3[k] = HROOT2*(K2[k]-Q2[k]); Y3[k] = Y2[k]+R3[k]; Q3[k] = Q2[k]+3*R3[k]-HROOT2*K2[k]; } for (k=0; k<L; k++) { K3[k] = DT*z2(k+1, (t2+DT), Y3, MXN); R4[k] = (K3[k]-2*Q3[k])/6; Y4[k][p+1] = Y3[k]+R4[k]; Q5[k] = Q3[k]+3*R4[k]-0.5*K3[k]; YW[k] = Y3[k]+R4[k]; } Z2[p] = Y3[0]+R4[0]; V2[p] = Y3[1]+R4[1]; if (Z2[p] >= 2.0) { JJflg = 1; cout<<"Reaches 2.0 ... Normal end"<<endl;

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} p=p+1; } if (m == 0) { int a; for(int i=0;i<212;i++) { V[i]=V2[i]; } int q = 269; for (a=q-1;a>1;a--) { V[a] = V[a-1]; } V[0] = V0; } else { for (int i=0; i<57; i++) { V[i] = V1[i]; for (int j=0; j<212; j++) V[57+j] = V2[j];} } for (k=0; k<(m+p); k++) { if (k == (m+p)) alfa[k] = (V[k]-V[k-1])/DT; else alfa[k] = (V[k+1]-V[k])/DT; } for (x=0; x<m; x++) { double thetaIMS=acos(costhetaIMS); Fr1[x] = -4*pi*pow(RI,2)*((sigmaMS/(2*(RI+RI*S[x])))*pow((sin(thetaIMS)),2)+V[x]*sqrt(RI*g)*(myS/3)*(3/2)*((1/(RI+2*RI*S[x]))-(1/(RI+4*RI*S[x])))* (1-pow(cos(thetaIMS),3))); Ff1[x] = -(2.0/3.0)*pi*pow(RI,3)*(roM+2*roI)*g*alfa[x]; Fb1[x] = (4.0/3.0)*pi*pow(RI,3)*(roM-roI)*g; Fd1[x] = -6*pi*RI*myM*V[x]*sqrt(RI*g); } for (x=0; x<p; x++) { if (Z2[x] < 1) Vphai1 = (VPP-1.0)*pow(Z2[x],2)-2*(VPP-1)*Z2[x]+VPP; // equation (10) else Vphai1 = 1.0; AA2[x] = Vphai1; if (Z2[2] < 2.0) Vphai2 =0.25*(VP-1)*pow(Z2[x],3)-0.75*(VP-1)*pow(Z2[x],2)+VP; // equation (4) else Vphai2 = 1.0; BB2[x] = Vphai2;

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Vphai3=Z2[x]-1.0-costhetaIMS; // equation (7) CC2[x] = Vphai3; /*FOUR DIFFRENT FORCES ARE ACTING ON THE INCLUSION AS IT APPROACHES THE STEEL-SLAG INTERFACE */ /* THE REBOUND FORCE, ADDED NASS FIRCE AND THE DRAG FORCE CAN ACT BOTH UPWARDS AND DOWNWARDS DEPENDING ON THE BEHAVOIR OF THE INCLUSION AT THE INTERFACE*/ Ff2[x] = -(2.0/3.0)*pi*pow(RI,3)*((XAITA*roS*Vphai2+2*roI)*g*alfa[x]); /*THE ADDED MASS FORCE ACTS BOTH UPPWARDS AND DOWNWARDS */ Fd2[x] = -6*pi*RI*myS*VF*Vphai1*V2[x]*sqrt(RI*g); /*THE DRAG FORCE ACTS BOTH UPPWARDS AND DOWNWARDS */ Fr2[x] = -2*3.1416*RI*sigmaMS*Vphai3; /*THE REBOUND FORCE ACTS BOTH UPPWARDS AND DOWNWARDS */ Fb2[x] = (4.0/3.0)*pi*pow(RI,3)*g*(roS*Vphai2-roI); /*THE BUOYANT FORCE ALLWAYS ACTS UPWARDS*/ } if (JJflg == 1)//%Normal end=the inclusion is in the slag and the film is off { cout<<"JJflg = 1"<<endl; for (int i=0; i<57; i++) Z[i] = Z1[i]; for (int j=0; j<212; j++) Z[57+j] = Z2[j]; for (int i=0; i<269; i++) { Z[i] *= RI; } for (int i =0; i<57; i++) { Z1[i] *= RI; } for (int i =0; i<212; i++) { Z2[i] *= RI; } for (int i=0;i<269; i++) { V[i] *= sqrt(RI*g); } for (int i=0;i<57; i++) { TT1[i] *= sqrt(RI/g); }

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for (int i=0;i<212; i++) { TT2[i] *= sqrt(RI/g); } for (int i=0; i<57; i++) TT[i] = TT1[i]; for (int j=0; j<212; j++) TT[57+j] = TT2[j]; for (int i=0; i<57; i++) Fr[i] = Fr1[i]; for (int j=0; j<212; j++) Fr[57+j] = Fr2[j]; for (int i=0; i<57; i++) Fd[i] = Fd1[i]; for (int j=0; j<212; j++) Fd[57+j] = Fd2[j]; for (int i=0; i<57; i++) Ff[i] = Ff1[i]; for (int j=0; j<212; j++) Ff[57+j] = Ff2[j]; for (int i=0; i<57; i++) Fb[i] = Fb1[i]; for (int j=0; j<212; j++) Fb[57+j] = Fb2[j]; } SaveToFile(0, 57, TT1, "TT1.bin"); SaveToFile(0, 2, K0, "K0.bin"); SaveToFile(0, 2, R1, "R1.bin"); SaveToFile(0, 2, Y1, "Y1.bin"); SaveToFile(0, 2, Q1, "Q1.bin"); SaveToFile(0, 2, K1, "K1.bin"); SaveToFile(0, 2, R2, "R2.bin"); SaveToFile(0, 2, Y2, "Y2.bin"); SaveToFile(0, 2, Q2, "Q2.bin"); SaveToFile(0, 2, K2, "K2.bin"); SaveToFile(0, 2, R3, "R3.bin"); SaveToFile(0, 2, Y3, "Y3.bin"); SaveToFile(0, 2, Q3, "Q3.bin"); SaveToFile(0, 2, K3, "K3.bin"); SaveToFile(0, 2, R4, "R4.bin"); // SaveToFile(0, 3, Y4, "Y4.bin"); SaveToFile(0, 2, Q4, "Q4.bin"); SaveToFile(0, 2, YW, "YW.bin"); SaveToFile(0, 57, Z1, "Z1.bin"); SaveToFile(0, 57, V1, "V1.bin");

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SaveToFile(0, 57, S, "S.bin"); SaveToFile(0, 212, Z2, "Z2.bin"); SaveToFile(0, 212, V2, "V2.bin"); SaveToFile(0, 269, V, "V.bin"); SaveToFile(0, 268, alfa, "alfa.bin"); SaveToFile(0, 57, Fr1, "Fr1.bin"); SaveToFile(0, 57, Ff1, "Ff1.bin"); SaveToFile(0, 57, Fb1, "Fb1.bin"); SaveToFile(0, 57, Fd1, "Fd1.bin"); SaveToFile(0, 269, Z, "Z.bin"); SaveToFile(0, 212, Fr2, "Fr2.bin"); SaveToFile(0, 212, Ff2, "Ff2.bin"); SaveToFile(0, 212, Fb2, "Fb2.bin"); SaveToFile(0, 212, Fd2, "Fd2.bin"); //SaveToFile(0,N*213,Y4, "Y4.bin"); SaveToFile(0, 212, TT2, "TT2.bin"); SaveToFile(0, 269, TT, "TT.bin"); SaveToFile(0, 269, V, "V.bin"); SaveToFile(0, 269, Fr, "Fr.bin"); SaveToFile(0, 269, Ff, "Ff.bin"); SaveToFile(0, 269, Fb, "Fb.bin"); SaveToFile(0, 269, Fd, "Fd.bin"); cout<<"V0 ="<<V0<<endl; delete []YW; delete []Q4; delete []K0; delete []R1; delete []Y1; delete []Q1; delete []K1; delete []R2; delete []Q2; delete []K2; delete []R3; delete []Y3; delete []Q3; delete []K3; delete []R4; // // delete []Q5; delete []Y4; delete []Z2; delete []V2; delete []alfa; delete []AA2; delete []BB2; delete []CC2; delete []Ff2;

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delete []Fd2; delete []Fr2; delete []Fb2; delete []TT2; delete []S; delete []V1; delete []Z1; delete []Fr1; delete []Ff1; delete []Fb1; delete []Fd1; delete []TT1; delete []V; delete []Z; delete []TT; delete []Fr; delete []Fd; delete []Ff; delete []Fb; system("PAUSE"); return EXIT_SUCCESS; }

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TRITA-CSC-E 2007:043 ISRN-KTH/CSC/E--07/043--SE

ISSN-1653-5715

www.csc.kth.se