Modeling of the Surface Marks Formation in an …...Library and Archives Canada Bibliotheque et Archives Canada Published Heritage Branch 395 Wellington Street Ottawa ON K1A 0N4 Canada

Modeling of the Surface Marks Formation in an Immovable Mold during Continuous Casting of Steel

by

Serguei Mikloukhine Master of Engineering

A thesis submitted to the Faculty o f Graduate Studies and Research

in partial fulfillment o f the requirements for the degree o f

Master of Applied Science

Ottawa-Carleton Institute for Mechanical and Aerospace Engineering

Department o f Mechanical and Aerospace Engineering

Carleton University

Ottawa, Ontario

Canada

March, 2007©

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3

Abstract

Continuous casting is the main process used today for manufacturing o f billets.

During continuous casting the surface marks (SM) are forming on the surface o f

continuous casting billets. The surface marks can cause cracking and can decrease the

yield o f the casting process because the billet surface containing marks has to be grinded

away to avoid crack growth. Using a mathematical model, it is possible to understand

how to decrease the negative impact o f the SM formation process.

Most o f previously developed models o f surface mark formation had a

phenomenological, qualitative character and the major factor determining process o f

mark formation in these models was the mold oscillation. The basic mathematical model

o f SM formation in which the surface o f the liquid metal in the mold is free from

additional factors such as reciprocating mold movement does not exist. The objective o f

this work is to create the mathematical model o f the SM formation during continuous

casting in an immovable mold.

In order to numerically describe the complex process o f SM formation the two

stage approach for the description o f the cycle o f SM formation was developed. For the

thermal analysis the equation o f transient heat transfer with the source function has been

derived. An initial condition and boundary conditions were formulated based upon a

physical conception o f SM formation process. The temperature distribution on meniscus

during first stage was calculated. Adequacy testing o f the mechanical part o f model

(using mathematical statistics) demonstrates excellent correspondence between the model

and experimental data.


The developed model can be considered as a basic model for the simulation o f the

initial solidification process under different conditions during continuous casting.


Ackno wledgm ents

5

I would like to thank Doctor V. Tsukerman for his help in research and useful

discussion. I would like also to thank my Supervisor, Professor A. Artemev for his

continued guidance and support o f my work. Finally, I would like to thank Professor J.

Beddoes for support o f my research.


6

Contents

A bstract................................................................................................................................................ 3

Acknowledgments.............................................................................................................................. 5

C ontents............................................................................................................................................... 6

List o f figures...................................................................................................................................... 8

List o f symbols..................................................................................................................................10

1. Literature Review.................................................................................................................... 14

1.1. Introduction. A brief history o f the continuous casting process............................14

1.2. Factors influencing the initial process o f the shell solidification: mold slag and

mold oscillation parameters....................................................................................................... 17

1.3. Formation o f Ripples & Oscillation m arks................................................................19

1.3.1. Models o f ripple form ation................................................................................. 19

1.3.2. Models o f oscillation mark formation.............................................................. 21

1.3.3. Classification o f the oscillation m arks..............................................................35

1.4. Conclusions. Objectives o f the thesis .........................................................................39

2. Qualitative description o f the proposed m odel..................................................................40

2.1. The first stage o f the mark form ation....................................................................... 40

2.2. The second stage o f mark formation..........................................................................42

3. Mechanical model o f the first stage...................................................................................45

3.1. Basic assumptions. Forces acting on meniscus.........................................................45

3.2. Differential equation o f equilibrium and its solution in parametrical form 47

3.3. Meniscus line equation and it solution in Cartesian coordinate system ..............49

3.4. Basic parameters o f the m eniscus................................................................................52


7

3.5. Numerical simulation o f the mechanical model at the first stage........................ 59

4. The model o f the thermal processes on the meniscus at the first stage o f mark

formation process............................................................................................................................. 64

4.1. Basic assumptions. The representative element.......................................................64

4.2. Derivation o f differential equation............................................................................. 66

4.3. Equations which describe the dynamics o f R E ...................... 70

4.4. Transformation o f the differential equation..............................................................73

4.5. BCi calculation........................................... 75

4.6. Thermal boundary layer................................................................................................78

4.7. Summary o f equations..................................................................................................82

4.8. Analysis o f the model o f thermal processes on meniscus......................................83

4.9. Conclusions..................................................................................................................... 92

5. Analysis o f the adequacy o f the developed m odel........................................................... 93

5.1. Analysis o f the adequacy o f the mechanical model o f the first stage o f mark

formation....................................................................................................................................... 93

5.2. Results o f the model adequacy testing.......................................................................97

6. Conclusions. Advantages o f the developed m odel........................................................... 99

Bibliography................................................................................................................................... 101

Appendix A ..................................................................................................................................... 106

Appendix B ..................................................................................................................................... 131


8

List of figures

Figure 1.1 The twin-roll caster suggested by Bessemer [1].....................................................14

Figure 1.2 Process following rupture o f billet skin in the mold [2]....................................... 15

Figure 1.3 A vertical and radial continuous casting machine [4 ] ...........................................16

Figure 1.4 Schematic o f the meniscus reg ion .............................................................................17

Figure 1.5 Ripple formation mechanism in ingots [24]........................................................... 20

Figure 1.6 Reciprocation mark formation mechanism in steel billet [25]........................... 22

Figure 1.7 The mark formation starts with solidification o f the meniscus [3 0 ]................. 25

Figure 1.8 Takeuchi and Brimamacombe model o f O M .........................................................27

Figure 1.9 Takeuchi and Brimacombe model o f O M .............................................................. 28

Figure 1.10 Schematic drawing for ripple mark formation [34].............................................29

Figure 1.11 Delhalle mark formation mechanism [35]............................................................ 30

Figure 1.12 E.L.V. solidification m odel..................................................................................... 31

Figure 1.13 The formation o f oscillation marks in continuous casting w ith ....................... 32

Figure 1.14 M eniscus shell attachment mechanism [13].........................................................33

Figure 1.15 The concept o f electromagnetic casting o f steel [39]......................................... 34

Figure 1.16 Bending o f the originally lapped interface, x l5 [2 9 ]......................................... 35

Figure 1.17 a) Reciprocation mark showing strong outward bending o f solid shell 36

Figure 1.18 Remelting o f the reciprocation mark valley and b leed ing ................................36

Figure 1.19 Types o f oscillation marks [32]...............................................................................37

Figure 1.20 Schematics illustrating formation o f curved hook...............................................38

Figure 2.1 The scheme o f the 1st stage mark formation process............................................ 41


9

Figure 2. 2 “Lower” meniscus transformation........................................................................... 42

Figure 3.1 Forces acting on the m eniscus................................................................................... 45

Figure 3.2 Diagram o f the forces acting on an infinitesimally small section o f the

meniscus.............................................................................................................................................46

Figure 3.3 Notations used in mechanical m odel........................................................................50

Figure 3. 4 Parameters o f the equation o f meniscus lin e ......................................................... 55

Figure 3.5 Calculated intermediate positions o f the m eniscus...............................................62

Figure 4.1 The RE geometry changes as a result o f meniscus grow th................................. 65

Figure 4. 2 The RE boundary conditions.................................................................................... 67

Figure 4. 3 Distribution o f the liquid metal flows in the mold [40]....................................... 79

Figure 4.4 Variation o f RE temperature resulting from ........................................................... 86

Figure 4.5 2D views o f variation o f R E .......................................................................................87

Figure 4.6 Temperature o f RE changes resulting......................................................................88

Figure 4.7 2D views o f temperature o f R E .................................................................................89

Figure 4.8 Temperature field o f meniscus at th e .......................................................................89

Figure 4.9 Temperatures o f the meniscus at the end o f first s tag e ........................................ 90

Figure 4.10 Calculated shape o f meniscus (blue) and..............................................................91

Figure 5.1 Photo o f template with position o f ........................................................................... 93

Figure 5.2 Results o f optimization for the parameters H h and 0{)..................................... 97


List of symbols

Vm instantaneous mold velocity, mm/min;

âverage average mold velocity, mm/min;

Vc the casting velocity, mm/min;

s the stroke length, mm;

f the oscillation frequency, 1/min;

t the cycle time, s;

tc total cycle time, s;

tN negative strip time interval in each cycle o f mold , s;

a the Laplace capillary constant;

S , Rs forces acting on meniscus, N;

R radius o f curvature, mm;

<p angle o f inclination at any point on the meniscus, rad;

dl length o f the infinitely small section o f meniscus, mm;

T transverse force, N;

M bending moment, N*m;

d(p, d X , dY increments o f the angle and coordinate;

d S , d T , dM increments o f the appropriate values;


11

Q the pressure o f liquid metal (distributed load) - lfom the side o f

liquid metal, N/m2;

q the pressure o f liquid slag (distributed load) - from the side o f

liquid slag, N/m2;

R bseg longitudinal reaction at the beginning o f a meniscus section, N;

R esnd longitudinal reaction at the end o f a meniscus section, N;

Ab width o f the considered section o f meniscus in the direction,

perpendicular to the plane o f drawing, mm;

<T the coefficient o f interphase tension between liquid metal and

liquid slag, N/m;

R q the resultants o f distributed load Q , N;

Rq the resultants o f distributed load q , N;

g gravity acceleration, N/m*s2;

p m the density o f liquid metal, Kg/m3;

p sl the density o f fluid slag, Kg/m3;

0q initial angle, rad;

S x the height o f the not solidifying part o f the meniscus (initial section

o f meniscus), mm;

Ahsl slag rim thickness, mm;

Ay solid slag thickness on the mold wall (below slag rim), mm;


12

Ahsl_j- thickness o f the liquid layer between the meniscus and the solid

slag and rim, mm;

l0 initial base’s length o f RE, mm;

ST size o f prism in the direction, perpendicular to the tangent to the

meniscus (thickness o f the layer o f thermal perturbation), mm;

A/ the RE (control section o f meniscus) extension during meniscus

stretching, mm;

AQy the summary heat content o f the RE; J

AQp the RE heat content due to the substance, which left from the core

during the meniscus stretching, J;

AQa heat transfer from the RE to the mold wall, J;

c the coefficient o f the liquid metal heat capacity, kJ/(kg °C.);

X the coefficient o f the thermal conductivity, W/(m °C );

Tm the metal temperature in the core o f ingot, °C;

T temperature o f metal on the boundary RE - liquid slag °C;

T s l - f liquid slag temperature °C;

Tcr_sl temperature on the boundary liquid slag - solid slag, the

temperature o f the liquid slag crystallization °C;

Tsl temperature o f solid slag on the boundary solid slag - the mold

wall °C;


13

Tw-s temperature o f the mold wall from the solid slag side °C;

Tw-w temperature o f the mold wall from the cooling water side °C;

Twat temperature o f the cooling water °C;

a thermal diffusivity coefficient, m2/sec;

Rz total heat resistance

Woo average velocity rate out o f the boundary layer limits

P r Prandtl number;

R e the Reynolds number;

CO the speed o f the liquid metal flow;

P the kinematics viscosity coefficient o f liquid metal, poise;

r the displacement o f the starting point, mm;

RSS sum o f squares o f residuals;

N - p - l The number o f freedom;

N the number o f experimental data;

P the number o f the model parameters;

p a criterion o f Fisher;

^hgap gap thickness mm.


14

1. Literature Review

1.1. Introduction. A brief history of the continuous casting

process

There are two basic casting methods used today for billets manufacturing: ingot

casting and continuous casting. The benefits o f the latter, such as:

• Higher yield

• Smaller number o f necessary manufacturing steps

• More mechanized casting process

• More even composition

• Better surface finish.

make it the main process for billets manufacturing. The first continuous strip casting

process was proposed by Bessemer in 1856 (Figure 1.1).

Figure 1.1 The twin-roll caster suggested by Bessemer [1]


15

At that time the major problem was “breakouts”(Figure 1.2), which occur when the

solidifying steel shell sticks to the mold, tears, and allows molten steel to pour out from

the mold [2] . This problem was overcome by Junghans in 1934 by introducing a

vertically oscillating mold, utilizing the concept o f “negative strip” wherein the mold

travels downward faster than the steel shell during some portion o f the oscillation cycle to

dislodge any sticking. A similarly important development was the introduction o f

Figure 1.2 Process following rupture o f billet skin in the mold [2]

sinusoidal oscillation, first used on two Russian slab casters in 1959 [3]. Now the

sinusoidal mode is the standard regime o f oscillation worldwide. It is relatively simple to

design, provides easy implementation and features a smooth character o f movement.

Many other developments and innovations have transformed the continuous casting

process into the sophisticated process currently used to produce over 90% o f steel in the

world, including plain carbon, alloy and stainless steel grades [4],


16

The scheme o f the modem continuous casting machines is shown in Figure 1.3

[4], M elt flows through ceramic tubes (nozzles) from a ladle to a tundish and further to a

mold. The nozzles are submerged in the melt. In the mold, the mold slag (flux) covers the

melt surface. The mold slag protects the metal from contact with air and may act as a

lubricant. Once in the mold, the molten steel freezes in the water-cooled copper

oscillating mold and forms a solid shell. During solidification the defects, such as

oscillation marks (OM) are produced on the surface o f the billets.

Driving rolls continuously withdraw the shell from the mold at a rate or “casting speed”

that matches the flow o f incoming metal, so the process ideally runs in steady state.

Beneath the mold are a secondary cooling zone and a straightening zone. W ater or air

iubmerged Entry Nozzle

Meniscus

Torch Cutoff Point ,Support roll

SprayCooling

SolidifyingShell

MetallurgicalLength

Strand

Figure 1.3 A vertical and radial continuous casting machine [4]


17

mist sprays cool the surface o f the strand between the support rolls. The spray flow rates

are adjusted to control the strand surface temperature until the molten core becomes solid.

After the center o f the strand is completely solid (at the “metallurgical length” o f the

caster, which is 10 - 40m) the strand is cut with oxyacetylene torches into slabs or billets

o f any desired length.

1.2. Factors influencing the initial process of the shell

solidification: mold slag and mold oscillation

parameters.

The most critical part o f the continuous casting process is the initial solidification at

the meniscus (Figure 1.4). The meniscus is the free surface o f a liquid which is near the

walls o f vessel and which is curved because o f surface tension [5]. This is the location

where the surface o f the final product is created, and where defects such as oscillation

marks and surface cracks can form.

moldwall

Solidshell

Figure 1.4 Schematic o f the meniscus region


18

Mold flux powder, normally consisting o f silica (SiCb), lime (CaO), sodium oxide

(Na2 0 ), spar (CaF2), MgO, AI2O3, covers the surface o f liquid metal in the mold.

Graphite is also used as a flux material to control the flux melt rate. The casting flux

serves [6 - 1 2 ] to:

• Prevent oxidation

• Provide thermal insulation, prevent meniscus solidification

• Absorb inclusions

• Provide lubrication

• Determine the meniscus shape

The following major flux properties must be controlled to give optimum flux

performance: row flux insulation, rate o f inclusion absorption, flux melting rate, flux

viscosity and crystallization temperature.

The oscillation o f the mold is one o f the most important factors influencing the

OM formation process [3, 6 , 13, 14]. There are a number o f parameters that describe the

oscillation, the most important are:

Instantaneous mold velocity for sinusoidal cycle [3]:

Vm = 2 • tc • s • f • c o s (2 ; r • / - / / 6 0 ) , (mm/min) ( 1 .1 )

where s - is the stroke length, mm, f - is the oscillation frequency, 1/min

and t - is the cycle time, s.

Average mold velocity [3]:


19

V,average = 2 -s- f , (mm/min) ( 1.2)

Total cycle time [3]:

tc = 6 0 / f , (s) (1.3)

Negative strip time is the interval in each cycle o f mold oscillation when downward

velocity o f the mold exceeds the withdrawal rate o f the strand. Positive strip time is the

interval in each cycle o f mold oscillation when withdrawal rate o f the strand exceeds the

downward velocity o f the mold.

Negative strip time interval in each cycle o f mold [13]:

where Vc - is the casting velocity.

The depth o f oscillation marks decrease with increasing oscillation frequency [15 -18]

and increases with increasing oscillation stroke length [16, 18, 19] and negative-strip time

1.3. Formation of Ripples & Oscillation marks

1.3.1. Models of ripple formation

A particular group o f defects characterized by ripples and laps in the ingots (ingot

casting) and OM in strands (continuous casting) have been studied previously and a

number o f possible mechanisms were introduced to explain their formation. W aters [23]

and Thornton [24] described a qualitative model which involves meniscus solidification

during casting. Figure 1.5 illustrates the mechanism proposed by Thornton. As there is no

tN = (60 In- / ) • arccos (Vc / n ■ f - s) , (s) (1.4)

[20, 21].


20

mold/liquid steel contact, the meniscus tends to build up until, when the internal pressure

becomes too great, the meniscus bends to the mold wall at the limit o f the frozen surface

layer, as shown in Figure 1.5.

S 0*20

N.hollow

OZO0-20 0 0 TIME t* '/ * »

010 TIME t

D I S T A N C E F R O M M O U L D W A L L , I n .

Figure 1.5 Ripple formation mechanism in ingots [24]

The steel surface is shown by :

a b e d at time T ;

a g e ’d ’ at time T+(l/4)sec ;

a g ’I c ”d ” at time T+(l/2)sec .

The position o f the boundary between solid and liquid steel is shown as:

e f b at time T

e ’f ’g at time T+(l/4)sec

e ”f ”j k at time T+(l/2)sec


21

The precise position o f the boundary cannot be determined. The dashed lines f ’hg and

/ ’ h ’g show an estimate o f its position allowing for the reduced rate o f chilling at the

hollow (Figure 1.5 at time T + (l/2 )s )) o f the ripple where there is a gap between steel and

mold.

Saucedo I, Beech J and Davis G.J. [29] studied ripples formation in the ingots during

ingot casting . Their conclusions were:

1. The cooling factor (rate o f heat extraction) o f the mold is the major factor in the

formation o f the laps and ripples in ingot.

2. An increase o f superheat and casting speed reduce these defects.

3. All o f the observations made are consistent with solidification over the meniscus

being the main step in the formation o f surface rippling.

1.3.2. Models of oscillation mark formation

With the development o f the continuous casting o f steel, the mold reciprocation

needed to be taken into account in the OM formation models. It has been proposed that

mold reciprocation represented an additional mechanical effect. Consequently the model

proposed by Savage [25] (see Figure 1.6) and Sato [26] incorporated mechanical and

thermal mold - metal interactions.

Figure 1.6a illustrates a shell o f the solidifying ingot in the mold at the time the mold is

just beginning its upstroke. It is postulated that when the stress rate and maximum stress

are sufficiently high, the thin and weak shell is broken near the meniscus level at position

Pi.


22

POSITION AT WHICH

SUBSEQUENT “lAP* IS FORMED DURING MOULD DOWNSTROKt

m o u u d MOVEMENT

te e m in g NS .L , LAP FORMED

surface of liquid metal

4

C Xm

X

Figure 1.6 Reciprocation mark formation mechanism in steel billet [25]

The top element o f the shell is then carried upwards by the mold, and it is assumed that

during the upstroke some slipping may occur between the surface o f the mold and the

element Ej. Consequently, Ei is carried upwards by a smaller distance than the mold

stroke, and arrives at the position shown in Figure 1.6b. In this time, element C has

continued its downward movement at the casting speed. Consequently, a substantial gap


23

between Eiand C is formed, which begins to freeze as the element Bi, Figure 1.6b. The

mold downstroke then begins, and continuous to the point shown in Figure 1.6d. In this

period the stress applied by the mold to the ingot shell is either zero or slightly

compressive, depending on whether the mold moves at the same or at a slightly greater

speed than the ingot. Consequently, only a very weak mechanical link such as Bi is

required to permit the element C to move the element Ei. Now when the upper edge Ei

penetrates the liquid steel meniscus, it is suggested that a fine lap Li is produced, as

indicated in Figure 1.6c. During the reminder o f downstroke, the lap Li passes beyond the

meniscus level to reach the position shown in Figure 1.6d. When the mold is again

reversed on the subsequent cycle, it is proposed that tearing occurs again as shown at P2 ,

and the cycle o f events is repeated.

The use o f flux in the slab casting o f steel introduced a further complexity. In this

case it has been proposed that a mechanical interaction exists between flux and mold that

changes the conditions under which the solid shell was formed. The next mechanism

proposed by Davies and Sharp [28] for slabs considers that the negative strip in

combination with the encapsulating effect o f the solid flux can give rise to a pressure in

the molten slag between strand and mold. The pressure acts so as to bend the solid shell

inwards, producing an overflow o f the liquid metal so forming the reciprocation mark.

In 1979 Saucedo I, Beech J and Davis G.J. suggested that the simple

consideration o f the casting speed can be used to understand how and when reciprocation

marks are formed. [29]. The strongest rippling occurs at the point where mold and strand

speed are equal. Lesser rippling would also be expected before and after this stage.


24

Meniscus solidification can account for the formation o f reciprocation marks and the

presence o f smaller ripples in the region between them. After analyzing the model and

experiments they suggested [29]:

1. Metallographic evidence shows that the same basic meniscus solidification

mechanism is responsible for the formation o f the reciprocation marks in

continuous casting and ripples in ingots. In the former case this mechanism is

modified by the mechanical effects produced by the mold reciprocation.

2. The final shape and size o f the surface feature depends upon both the casting

conditions and the rate o f the heat transfer. For example the solidifying meniscus

shell may be bent back towards the mold walls or it may be ruptured and bleeding

can occur.

In 1980 Saucedo et al. [29] explained that the oscillation marks, or ripples, form

because the meniscus solidifies. No marks will form if the rate o f heat extraction is

lowered. In 1991 Saucedo [30] presents the theory in more detail. The oscillation marks

form when oscillation forces the liquid metal to regain contact with the mold wall. This

can happen in two ways, either an overflow occurs, or the shell is bent towards the mold

by the metallostatic pressure, or that these two processes combine. F igurel.7 shows the

different ways the liquid metal can come into contact with the mold wall above the frozen

meniscus. The left mark is usually termed the folding mark (1), and the mark in the

middle is called an overflow mark (2).


25

1 2 3

Figure 1.7 The mark formation starts with solidification o f the meniscus [30]

1. The rising liquid pushes the solid shell towards the wall

2. The liquid overflows the shell

3. The first two are combined

The following conclusions were given [30]:

1. Samples indicate that initial solidification and oscillation mark formation are

strongly influenced by meniscus effects.

2. The surface topography o f oscillation marks has dendritic features, indicating that

they were formed by meniscus solidification, without contacting the mold.

3. Oscillation marks produced in turbulent conditions can ’’split” in two, indicating

that they are not produced by mechanical deformation effects.

4. Oscillation marks can have multiple depressions, indicating that there are not

solely the products o f mechanical effects during the mold downstroke.

5. Longitudinal streak lines on the surface o f as-cast strands are produced by


26

meniscus solidification effects.

In 1980 Tomono et al [31] performed experiments with organic substances. They

observed the formation o f surface marks and concluded that the two mark types, i.e.

folding marks and overflow marks are formed differently. They suggested that oscillation

marks were formed when the meniscus was subjected to compressive force by particles

sticking to the wall, and folding marks were formed independently o f the oscillation.

They used the Bikerman equation [31] for calculating the meniscus shape and connected

the oscillation mark properties to the discrepancy between observed and calculated

meniscus shape [3, 31].

In 1984 Takeuchi and Brimacombe [22] described a mechanism in which the

meniscus is covered with a rigid solid skin (or without, depending on superheat and local

convection) and the pressure generated in the liquid slag channel varies and draws the

meniscus back towards the mold wall during the negative strip. The pressure is positive

when the mold is moving downward faster than the strand and is negative when the

strand is moving faster. The difference between marks with and without hooks is

assumed to be caused by the difference in strength o f the meniscus skin. I f the skin is

strong, an overflow will occur, and a hook forms (Figure 1.8). I f the skin is weaker, the

shell is pressed against the wall and no overflow is needed, and no hook forms (Figure

1.9). Takeuchi and Brimacombe described how the meniscus follows the oscillation and

suggested the classification o f oscillation marks [22, 32, 33].


2 7

time

meniscus

Figure 1.8 Takeuchi and Brimamacombe model o f OM

formation with subsurface hook [22]


28

meniscus

Figure 1.9 Takeuchi and Brimacombe model o f OM

formation without subsurface hook [22]


29

melt

l s \

shel

y

b)

Figure 1.10 Schematic drawing for ripple mark formation [34]

a) Balance o f forces at the tip o f solidified shell

b) Growth o f shell and movement o f melt surface

Suzuki et al [34] suggested that the meniscus in surface tension balance with the

shell moves upwards as the solid shell grows inwards (see figure 1.10). Further J.

Elfsberg [3] used this idea for her calculation.

In 1989 Delhalle et al. [35] described three different mechanisms for the

formation o f oscillation marks, as shown in Figure 1.11. The three mechanisms are based

upon meniscus solidification. The size and shape o f the oscillation marks is said to

depend on the heat extraction, the oscillation parameters and the interfacial properties

[35].

Based on the results o f three experiments Lanez and Busturia [36] suggested that

solidification does not start at the meniscus, but further down in the mold, at the lower

part o f the solid slag rim (Figure 1.12). The oscillation mark formation was connected to


30

this region and it has been noted that marks formed under “positive strip” (where the

mold downward speed is at maximum). They termed their model E.L.V. (“Extra Liquid

Volume”) solidification model, Figure 1.12.

---- - i"V/ r V

/ **

\ j r I \ 4s. .»*: i :

r ’

Ii i ■ *

1I S M H i

i 1

1 i

A B COverflew Overflew * Remitting MvnUcu* t»«nf (mm;*

Figure 1.11 Delhalle mark formation mechanism [35]

A. Mold oscillation and product withdrawal cause liquid steel

to overflow the solid hook. The overflowing liquid freezes

against the mold wall and a new shell tip forms.

B. Same as A, but here partial or total remelting o f the solid

shell tip is assumed.

C. The metallostatic pressure bends the solid shell back against the

mold [35]

In 2003 J. Elfsberg [3] proposed that there can be two types o f oscillation marks:

overflow marks and folding marks. It was suggested that they both were caused by

overflows. The difference in their appearance depends on that they formed at different

times o f the oscillation cycle. The overflow marks are formed during the down strokes,

and the folding marks during the upstrokes. The oscillation marks may form patterns.

One deeper mark is followed by some shallow marks.


31

Mould

R im

Raw powd>f> II II H i ll l|IM I« i.l .lM»l . » ^ . * «

Liquid powder

^ H U q u id s te e i:> > ; . C -2 5 H S « « -v -* -v - H H K W K K S T " “

V W V V V : V W % « « A rU 5W 4H* «m» « f c « ■ » r n * m W i T m m m

issasssfisfiS S S S H K S S S S S

Figure 1.12 E.L.V. solidification model

(’’Extra Liquid Volume”) [36]

Then there comes a deep one and so on. She suggested that the deeper marks form when

the interfacial tension and the oscillation co-operates (acting in the same direction). Also

J. Elfsberg suggested that it is possible to avoid the oscillation mark formation by

avoiding the overflows. If the oscillation frequency is chosen so that the maximum

meniscus height is never reached, the overflows will not occur. In other words, if the

mold is switching to upward movement just as the maximum meniscus height is reached,

the overflow will be suppressed. The optimal oscillation frequency was suggested to be

[3]: f = Vc / ( j2 - a ) (1.5)


32

Sato [26, 27] suggests that the marks are formed in two steps, first the solid

meniscus shell is lifted by the upward moving mold. This lift causes the formation o f two

convex surfaces, ab and be in Figure 1.13. Then, as the mold turns, the two convex

surfaces are forged together and the mark is formed. Figure 1.13 shows the conditions for

casting with the use o f mold slag. The presence o f the slag will cause a different pressure

[27].

O s c ii i t in j bhsW w all ( b )Q w ciktir,* -o ir i « »U ( # i

Md&m *!*f

P'm ^ Norma.! K ieniseiss

•s' -*— OaeiHttiBff m*rk

Figure 1.13 The formation o f oscillation marks in continuous casting with

the use o f mold slag. The mark forms during the upstroke. The solid meniscus

lifted by the mold movement [27]

Edward S. Szekeres [13] suggested “the secondary meniscus mechanism” .

According to [13] the oscillation mark forms at a secondary meniscus, not at the primary

meniscus. In Figure 1.14 section A demonstrates the state just after the start o f the cycle

and an element o f solid steel is assuming to be adhered to the mold wall and moving


33

upward with the mold. The surface o f liquid metal between the meniscus shell and

downward moving strand shell is a secondary meniscus. There is no tearing at this

junction because the meniscus shell is not yet connected to the strand shell. Section B

represents the start o f negative strip. It is possible that the secondary meniscus freezes

over as early as the start o f negative strip which establishes a solid bridge between the

meniscus shell and the strand shell.

» r a b]........ cj ~ d[ eJ “I**— Negative Strip —*

0 ^ r .....* PnAarvid ■ Eg '

■*$.00 010 030 0.90 0.40 0 60TtMC. teoond

Figure 1.14 Meniscus shell attachment mechanism [13]

As the negative strip condition intensifies, the meniscus shell breaks free from the mold

wall. Also the author pointed out that the Savage model included a tearing healing action

at the junction between upward - moving solid and the downward - moving strand.

Although the tear and heal mechanism was referred to for many years, it has been

abandoned because “Metallographic examinations o f as-cast surfaces have revealed no

evidence o f tearing” [13].

c f' Negative Strip

> v. ^Meniscus Shell 1 \* «ilW> FnAarvitf ' j H a

^ Secondary I ^ C M e n isc u s I | U q

1 1 1

| S u q j l u q

1 iI I I Strand | j | | | 1 l l r Shell l i l t 1 1 ^SsSsl.i....i...._____ j__J u s ls M ..1......1 ■ t


34

Information about SM formation in an immovable mold was presented in 1990

[38]. The mechanism o f slag film renewal during continuous casting in an immovable

mold was suggested. The new research was published in 2002. After experiments with

electro magnetic casting (EMC) o f steel (Figure 1.15) Joonpyo Park et al. [39] concluded:

in ordinary casting, the mold is oscillated to prevent sticking between the mold and the

billet. In EMC, the magnetic pressure increased a curvature form in the meniscus, thus

NozzleMold

Mold flux

4 ^

Soft-contact zone

Molteta metal

Solidified shell

Conventional continuous castin:

Soft contact EMC

Figure 1.15 The concept o f electromagnetic casting o f steel [39]

the inflow o f the mold flux was improved. Therefore, it was suggested that EMC could

be performed without mold oscillation. If the casting can be performed without mold

oscillation, the casting plant could be dramatically simplified [39].


35

1.3.3. Classification of the oscillation marks

In 1979 Saucedo I, Beech J and Davis G J . [29] described the different types o f

the OM (presented in Figures 1 .1 6 -1 .1 8 ) and their formation. It is possible to observe

that the lap can be covered by liquid overflow (Figure 1.16), also it is possible to observe

an outward bending o f an originally inward-growing shell (Figure 1.17), solid formation

outside reciprocation marks, remelting o f the reciprocation mark valley and bleeding

through (Figure 1.18).

Figure 1.16 Bending o f the originally lapped interface, x l5 [29]


36

a) b)

Figure 1.17 a) Reciprocation mark showing strong outward bending o f solid shell, x24;

b) Solidification outside the reciprocation mark, x60 [29]

Figure 1.18 Remelting o f the reciprocation mark valley and bleeding through, x60 [29]


37

Later, in 1985 Brimacombe J.K. et al. [32] suggested the following classification o f OM

(Figure 1.19):

Types ofOsc Motion

Mark

Types of Positive Segregation

Hooks in

subsurfaces tru c tu re

Typel

kjrqearea

Type 2

fee

Type 3

week segrego'entine

Hooks ab sen t in

su b su rfa c e structu re

Type 4

segregation layer

Figure 1.19 Types o f oscillation marks [32],

The OM with subsurface hooks were studied by B.G Thomas et al.[37] in details.

Hook formation is initiated by meniscus solidification. The instantaneous shape o f the

meniscus dictates the curvature o f the line o f hook origin. It has been established that

hooks formation usually occur at the beginning o f the negative strip period [37] as

schematically illustrated in Figurel.20.


38

Figure 1.20 Schematics illustrating formation o f curved hook

in an ultralow -carbon steel slab by meniscus solidification

and subsequent liquid steel overflow [37]

a) meniscus freezing;

b) meniscus overflow;

c) line o f hook origin;

d) shell growth.

Oscillation marks (OM) are formed by steel shell growth after overflow. The meniscus

subsequently overflows the line o f hook origin, as shown in Figure 1.20(b). The final

shape o f the hook is formed as the hook tip fractures and is carried away, as shown in

Figure 1.20(c). The hook protruding from the solidifying shell captures inclusions and

bubbles in the liquid steel until the shell finally solidifies past the hook. These events are

illustrated in Figure 1.20(d).


39

1.4. Conclusions. Objectives of the thesis

As has been demonstrated in the review, the most o f the presented OM models

have a qualitative character [13, 23, 24, 25, 27, 29, 30], i.e. there is no numerical

simulation o f the meniscus behavior. Therefore it is difficult to utilize them in specific

applications. Usually, such models are based on a particular experimental data (templates

with the sections o f the billets surface, on which the meniscus was fixed). All these

models have lack o f completeness. There is no connection o f the observed phenomenon

to the previous history o f process. Most researchers proposed models in which mold

oscillation is the major factor determining the OM process [3, 33]. A large number o f

different models exist because there is no basic model o f initial solidification wherein the

meniscus is free from additional forces. The objective o f this work is to create a

mathematical model describing the mechanical behavior o f the meniscus and the thermal

processes acting upon the meniscus during continuous casting in an immovable mold.


40

2. Qualitative description of the proposed model

The developed in this thesis model gives a picture o f the process o f initial shell

formation through a description o f the mark formation process. The model contains a

description o f meniscus movement and deformations that are used in calculations o f the

temperature distribution on meniscus. Behavior o f the meniscus was considered during

the stationary process o f ingot formation. The stationary process means that at the end o f

a mark formation cycle the next cycle started from the same point that the previous cycle

began. In the developed model the mark formation process was divided into two stages,

which followed each other successively, thus making a cycle.

2.1. The first stage of the mark formation

The scheme o f the 1st stage o f process is shown in Figure 2.1. This stage begins

when the previous cycle comes to an end and solidified shell starts to move downwards

relatively to the slag rim. At this moment solidified shell looses contact with the rim. The

point on the boundary between the solidified portion o f the meniscus and liquid (the top

o f solid mark) becomes the support for the meniscus o f the liquid metal (Figure 2.1).

Surface tension force keeps the liquid metal above the solid mark and prevents contact

with solid slag on the mold wall. The meniscus stretches and moves in accordance with

the balance o f surface tension forces and gravity. The meniscus growth arises from two

processes: the meniscus moving as a whole and the meniscus stretching. The initial

condition o f the 1st stage is shown in Figure 2.1(a). The top o f the solid mark started

moving downwards with the billet (Figure 2.1 (b). The curved liquid meniscus (CLM)


41

expanded between two points: the first point is the top o f the solid mark (T) and the

second point is point o f junction (J) between the curved meniscus and the flat portion o f

surface o f liquid metal in the mold. The CLM started stretching and moving. The gap

between the meniscus and rim increases and liquid slag flows into the gap (Figure 2.1 (b).

When the top o f the solid mark moved, the meniscus height increased. Under the action

o f the ferrostatic pressure, growing meniscus is pressed to jo in the slag rim again. In

addition, liquid slag is forced to move out from the gap (Figure 2.1 (c). Thus, at the first

stage, the billet grows due to the meniscus height increasing and overflow o f the liquid

metal over the solid mark. The first stage is divided into two phases.

d)a) b) c)

Figure 2.1 The scheme o f the 1st stage mark formation process

1- solid mark; 2 - meniscus; 3 - liquid slag; 4 - slag rim

The first one - when the meniscus moves from the slag rim, as shown in Figure 2.1(a-b)

and second one when meniscus moves to the slag rim as show in Figure 2.1(c-d). When

the meniscus contacted the slag rim the upper part o f the meniscus stops. The contact o f

liquid metal and solid slag (Figure 2.1 (d)) leads to splitting o f the meniscus into upper


42

and lover parts. We called this moment the sticking o f meniscus and this is the end o f

stage 1. The lower meniscus still continues to follow the movement o f the solid mark.

At the first stage we assume that up to the moment when meniscus stops at the mold

wall it has not been solidified: less heat escapes the meniscus than accrues to it. This

happens because new portions o f superheated liquid metal are being added to the external

surface o f meniscus.

2.2. The second stage of mark formation

The second stage begins when the meniscus touched the mold wall and two

menisci formed: upper and lower. The lower meniscus continues to follow the movement

o f the solid mark. After the meniscus had made contact with solid slag, part o f the liquid

slag becomes closed (under the point E, see Figure 2.1 (d). From this moment liquid slag

cannot move upward on the liquid metal surface in the mold from the gap.

.. 4--A ..

a) b)

Figure 2.2 “Lower” meniscus transformation

1 - solid mark; 2 - meniscus o f the liquid metal; 3 - liquid slag;

4 - slag rim; 5 - solid part o f meniscus; 6 - closed slag.


9985^42998

43

The slag closing inside liquid metal leads to a changing o f the forces which had provided

a balanced interaction on the lower meniscus at the second stage. Thus, at the moment o f

slag closing the meniscus changes its shape very rapidly i.e. there is a transition from the

form shown in Figure 2.1 (d), to that shown in Figure 2.2 (a). The lower meniscus

becomes convex on the side o f liquid metal. The top o f the solid mark continues to move,

therefore the ferrostatic pressure acting on the lower meniscus increases. The lower

meniscus moves and deforms as a result o f the increasing force. The process o f the lower

meniscus transformation is shown in Figure 2.2 (a, b). The bottom support point o f the

lower meniscus moves along the solid mark. The upper support point o f lower meniscus

moves down and its position is defined by the solidification and deformation processes.

As the solid mark moved downwards the lower meniscus leaned on it, and following the

solid mark became extended. As a result, fresh portions o f liquid metal contact the

external surface o f the lower meniscus thereby adding superheat. I f during the movement

o f the lower meniscus the heat extraction rate does not exceed the rate o f the heat supply

via fresh portions o f liquid metal, the casting process became unstable. The liquid core is

poured out under the mold through the non-solidified lower meniscus. On the contrary, if

the level o f heat withdrawal from the lower meniscus exceeds the level o f incoming

additional superheat then the gap between the sticking upper meniscus and solid mark

solidifies. In the thesis we considered only the first stage o f surface mark formation. To


44

describe the evolution o f upper meniscus shape and its heat state we need to develop the

mechanical and thermal models o f the meniscus state. Based on the model o f meniscus

mechanical behavior (Chapter 3), the model o f its thermal condition will be discussed in

Chapter 4.


45

3. Mechanical model of the first stage

The mechanical model is formulated based on the equilibrium existing at the

surface o f meniscus between surface tension forces and the pressure o f the liquid metal

and slag (Figure 3.1).

3.1. Basic assumptions. Forces acting on meniscus

Meniscus is considered to be an interphase film. The coefficient o f interphase

tension assumed to be constant along meniscus length. To evaluate the mechanical state

o f meniscus we consider the unit width element o f the meniscus along the ingot axis.

L iq u id s la g — ------------s

W a te rc o o le dmold,wall

liquid metal

solidshell

X

Figure 3.1 Forces acting on the meniscus

S , Rs , Q , q - forces acting on meniscus.


229696895952492690978251144225999^8356^6452693918057514928615642299195

46

The forces, which act on this element from the remaining part o f the meniscus,

perpendicular to the plane o f the drawing direction are neglected and only the forces,

which act in the plane o f drawing (Figure 3.1), are discussed, hence a 2-dimensional

problem is considered. We assume, that the top o f the solidified mark (point T) moves

along an axis X (Figure 3.1), without changing coordinate Y .

Such an assumption is equivalent to the assumption that within the limits o f the mark

formation cycle neither solidification, nor a melting at the top o f the solid mark occur.

To derive a mathematical model describing the mechanical behavior o f the liquid

metal meniscus, we examine the equilibrium o f the forces acting on an infinitely small

section o f meniscus (Figure 3.2).

Figure 3.2 Diagram o f the forces acting on an infinitesimally small section o f the

meniscus

Ab - unit width o f the considered section o f meniscus in the direction,


47

perpendicular to the plane o f drawing;

(p - angle between tangent to the meniscus and axis X;

dl - length o f the infinitesimally small section o f meniscus;

S - longitudinal force - is always directed tangential to the surface o f the

considered

section; at any point o f meniscus always has one and the same value (the surface

tension force);

d(p, dx , d y— increments o f the appropriate values.

Q - the pressure o f liquid metal (distributed load), acts from the side o f liquid

metal;

q - the pressure o f liquid slag (distributed load), acts from the side o f liquid slag.

Within the limits o f an infinitesimally small section o f meniscus with the length dl we

neglect the changes o f Q and q .

3.2. Differential equation of equilibrium and its solution in

parametrical form

Distributed loads Q and q can be presented by the resultant forces

Rq —Q ’dl • Ab and Rq = q ■ dl ■ Ab . Rq and Rq are applied in the middle o f the

infinitesimally section o f meniscus. The forces o f interphase tension are equal to

S = <7- Ab,

where cr - the coefficient o f interface tension between liquid metal and liquid slag.

Equation o f equilibrium forces (sum o f the forces projections on the direction o f the

resultants o f distributed forces):


48

2 5 - s \ x ^ - - { Q - q ) - A b - d l= 0

To simplify this expression we use

S = cr • Ab and Ab = 1.

For small d(p , s in d(p ~ d<p, we have:

( Q - q ) - c r ^ = 0 .V dl

Here (Q — q) is a function o f meniscus height x : Q{x) = p m ■ g ■ x ; q{x) = p sl • g ■ x .

Finally differential equation o f the force balance looks like:

(Pm -Psi ) ' g x - < 7 - ~ = ° ’ C3-1)dl

where g - the gravity acceleration ;

p m - the density o f liquid metal;

p sl - the density o f fluid slag.

At a certain point on the meniscus with the coordinates (x , >’) the projection o f

elementary section dl on axis X is equal to dx = dl- cos (p, where q> - angle o f

tangent inclination to the meniscus at point on the meniscus with coordinates (x , y ) .

The curvature o f the considered section is determined by the known expression:

d (p1 _ dcp _ dx

T~~ ~ d T ~ ~dTdx

R - radius o f curvature;


49

from which after the transformation follows:

d(p d<pdT=COS(p'S x W

then equation (3.1) becomes:

( p w - p sl )• g ■ x ■ dx — a • cos <p • dq> = 0 (3.3)

To simplify we introduce the variable: B = ^ .2 cr

Integrating (3.3) over the meniscus height x h

xh <ph

J:2B-x d x- | cos<p-d(p = 0 ;

we get: B ' xh = (s in ~ s in )# A^

where 0Q - initial angle at the point J (paragraph 2.1) where x = 0 (see Figure 3.3).

3.3. Meniscus line equation and it solution in Cartesian

coordinate system

Notations for the description o f the meniscus line are shown in Figure 3.3. An

axis Y is directed along the flat portion o f molten metal surface. An increase in the y

values occurred to the right side from the beginning o f coordinates.


50

surface of the liquid metal

meniscus

solidified—murk ~

X '1

Figure 3.3 Notations used in mechanical model

The axis X is parallel to the strand’s axis and crosses the top o f the solidified mark

(point C 0 - point where the new cycle o f the mark formation begins), directed

downward.

CH - the point corresponds to an extremum on the meniscus line;

Ch - a certain point on the meniscus at a distance h from the beginning o f coordinates;

CH - the top o f solidified mark in the end o f the first stage (height o f the meniscus =

max);

h - the current height o f meniscus 8 x < h < / / max, which corresponds

to the displacement o f the mark top C 0 on the distance hc;

H s - displacement o f the top o f solid mark;

H h - limiting height o f meniscus for concrete conditions;


51

h[ - the height o f the not solidifying part o f the meniscus (initial section o f meniscus);

H g - distance from the beginning o f coordinates (along the axis X ) to

x - coordinate o f the current point on meniscus;

y(x, h) - coordinate o f the current point on meniscus;

y hg - coordinate o f point H G ;

Ay - the thickness o f solid slag on the mold wall (out o f the slag rim zone);

0O- initial angle; theoretically this angle should be constant and equal to —it1 2 , but in

reality the surface o f the liquid metal in the mold is not absolutely plane surface (liquid

metal continuously moving) therefore we assume 0O is one o f the fitting parameters

close to — it / 2 ;

J - ju n c tio n point (Paragraph 2.1).

The solution o f the equation (3.4) we present as function y = f { x ) .

„ „ dy sine?From Figure 3.2 — = t a n (p = --------- ;dx cos cp

substituting from equation (3.4):

• 9 •dy _ sin (p _ B x + sin 0O

the point o f extremum on the meniscus ^ hg 5

(3.7)

then

(3.8)


52

or in integral form:

, in . r 2 + s i n / 9 „ )

■dx (3.9)y(x) = y M + ) (S * 2+ ^ )jc0 - \ / 1 - ( - 5 - x 2 + s i n # 0 j

3.4. Basic parameters of the meniscus

Maximum height o f the meniscus

The maximum height o f meniscus can be found from equation (3.4):

_ s in p -s in & 0B

The height o f meniscus is maximal i f the residual (sin (p — s in 6{)) is maximal. In this

case (sin (p — s in 0())m a x = 2 when <p = n ! 2 , B() = - n 12

Substituting these values and expression for B in equation (3.4):

(Pm ~ Psh ) ‘ S ' ^ max _ ^2

From last equation we get:

(.Pm P si) ' S *

Substituting the numerical values o f the parameters:

p m = 7 2 0 0 ^ / 0 w 3 ; p sl - 2 6 6 7 , 3 K g / M 3 ; g - 9 , S \ m / s 1-, a - 1 ,2 0 5 8 h / m ,

We obtain the following:


53

4 • 1,2058 . . . .Hmax = J 7--------------------r = 10.42 m m

ma \ (7 2 0 0 - 2667,3) • 9,81

We assumed that the height o f the studied meniscus is less than the maximum

possible (in this case the random fluctuations will not lead to the destruction o f the

meniscus). In the further descriptions we accept the height o f the

m eniscus//^ = 0.985 -Hmax.

Points o f the extremum on the line o f meniscus

The position o f the extremum o f function _y(jc) can be found from condition:

dy _ (b • x 2 + s in 0Q)

^ -Ji ~(b -x2 T s in ^ o )20 ,

therefore dy = ( B- x + sin 0Q ) = 0 ,

the position o f the extremum point is:

* = (3-10)B

After substitution 0O - - n 12, and finally we obtain H G = J — .V B

The J - point coordinates at the maximum height o f the meniscus

The coordinate t ( x o) ° f junction point J, (Figure 3.3) can be found from

equation (3.9):

t ( x ) = t ( xo )+ \ S B( X +Sm<9°) -dx. xoyl - \B - x 2 + sin 00 J


54

Substituting xQ - 0 , x — H max and the point on meniscus with coordinate x = H n

has the coordinate y{H max ) = 0 :

^ m a x / 7-j 2 . • r\ \

m „ ) = y ( o ) + J I * Sln o) . ^ = 0 ;0 ^ l - [ B - x 2 + sin<90)

then

V ( s - x 2 + s in 0 o) i y(o) = - J V \2 * ‘ (3‘U)

0 -^l - - x 2 + sin (90<T

The height o f the initial section o f the meniscus

The initial section o f meniscus is a part o f the meniscus (higher than point C 0 ),

which does not solidify during the period o f mark formation. At the beginning o f the new

cycle and at the end o f the stage l, the y ( h j ) is equal to 0.

From (3.9) y [ht )= y ( x Q)+ f (* * + s i n #o) . =*o )/l - (-5 • x 2 + sin 0q J

^max | p 2 , /) IFrom (3 .I I ) y (x 0) = y (o ) = - j I x sin o) . ^ , then

0 - \ J l - \ B - x 2 + s i n 0o )

) ( f l * 2 + ^ o ) . A . " J ^ (3.I2)

•*0 y 1 — • x 2 + s i n j 2 *0 ^ l - ^ - x 2 T s in ^ o )2


55

This equation 3.12 is implicit relatively to hi and solved by the numerical methods,

where y’(o ) can be found from (3.11).

Parametric equation for the meniscus line

It is important to note that dependence (3.9) has two parameters, i.e. in addition to

the determination o f the coordinate o f the observation point x , the value o f the meniscus

height h at the given moment o f time is required (equation 3.11, Figure 3.4).

e

Figure 3. 4 Parameters o f the equation o f meniscus line

For the considered point C the dependence (3.9) can be presented as two-parametrical:

y{x,h) = y(C) = -Y(h)+ Y{x), (3.13)

where Y(h ) - depends only on the height o f the meniscus at considered moment in time


56

as:

Y(h)J t ( t £ j ^ h L . & ; (3.14)0 -Jl - ( 5 • x 2 + sin (90 j2

Function F (x ) depends only on coordinate x o f the point o f observation:

r ( x ) = ] . (f l -*2+ sin go) ,A (3.15)0 A/ l - ( 5 - x 2 +sin<90)2

The depth o f the mark (3.13):

AY = y(h,, H h ) - y{H a , H , ) = -Y (H h) + Y(h,) - [ - Y(Hh )+ Y(Ha )] =

= r(A,)-r(H0)

where Y(hi) and Y (H g ) we determine from formula (3.15).

The equation o f the slag line

Solid slag on the mold wall has two regions:

1.The slag rim (in the interval HG > x > 0 ) , i.e. coordinates o f points on this section can be

described by functional dependence on coordinate x )

2. The solid slag with a constant thickness located under the slag rim (in the interval

x > H g ).

Curve o f the slag rim in the rim limits (HG>x> 0) can be described by (3.9) when h=Hh

In parametric form the equations o f the slag rim:

y si{x >H h ) = - Y (H h ) + Y (x ) ; ( h g >x >o)


57

for x > H g , y sl(x, Hh) = const.

The generalized description o f the solid slag line jyv/(x ) , which includes rim and solid

slag with a constant thickness, can be done by symbols o f Boolean algebra:

y d ( x , H h) = y d (x ,Hh) D ( H g > x > 0 ) + y ( H G, H h) U ( x > H a ),

y,i (*, M - y(Hu)+ 1-W]n (h g > x > o)++ [ - y ( H h) + y ( H a ) ) n ( x > H G)

where Y(H„). Y ( H g ), Y(x ) can be found from formula (3.14) and (3.15) and the sign

Pi indicates the union o f sets - domain o f existence o f the functions values that

simultaneously stand to the left and to the right o f sign.

The solid slag thickness

Outside o f the slag rim we assume the solid slag thickness is constant and equals

to Ay . In the slag rim zone the thickness o f the rim is added to the thickness o f the solid

slag. The slag rim thickness depends on the coordinate x and h as:

Ah,l(x ,h )= A y + y d (Ha , H ll) - y d ( x ,H h) == Ay + [ - Y(Hh ) + 7 ( / / c )] - [ - Y(Hh ) + 7 (x )] = (3.18)

= Ay + Y(Ha ) - Y ( x )

where HG>x>0;

Ahs} (x, h) - slag rim thickness;

Ay - solid slag thickness on the mold wall (below slag rim) Ay = c o n s t;

Y(Hg ) , y ( x ) - is defined by equation (3.14) and (3.15).


58

Thickness o f the liquid slag layer

Thickness o f the liquid layer between the meniscus and solid slag on the mold

wall is the distance along an axis Y between points on the solid slag and on the meniscus

at co n s tan tx . To find this we use the equation (3.17) and (3.13). Thickness o f the liquid

layer between the meniscus and the solid slag and rim Ahsl_j- can be expressed in the

parametrical form as:

- [ (Y(x) - r (Hl,))a(Hc > x > o ) + (r(Ha ) - r ( H h))f ] (x>Hc )] = = [Y(Hh) - Y ( h \ n ( H a > x > 0 ) + + [r(*) -Y(h)+Y(H„)- Y{Hg )] n (x > Hg )

where h - current height o f meniscus;

H h - limiting height o f meniscus for given conditions;

It can be noted that in the section o f meniscus {Hg > x > 0 ) , the Ahst_j- does not

depend on x , but it depends only on the height o f the meniscus h .

The length o f the meniscus

The length o f the meniscus is required for a description o f the m eniscus’ thermal

condition during moving and lengthening. The length o f the line from a reference point

x 0 up to some point Cx is determined by the formula:


59

where x 0 and X - the beginning and the end o f the studying part o f curve y = f { x )

accordingly.

dyAfter the substitution the value o f derivative — defined by (3.8) fo rx 0 = 0 ; X - h we

dx

obtain:

1 +[b • x 2 + sin <90 )

-Ji ~(b -x 2 H -sinô)2■dx

Or for the meniscus with height h:

w , 1 +(b -x2 + sinff0)

■Ji ~(b -x2 + s in d0f■dx (3.20)

3.5. Numerical simulation of the mechanical model at the

first stage

The solution o f differential equation (3.9) was done with the help o f the

mathematical package "MathCAD". For the best illustration o f the meniscus

displacements, the result o f calculations has been represented as graphs. These graphs

determined the form and position o f solid slag, and also the end and intermediate

positions o f meniscus.

The physical values for calculation

The basic physical characteristics o f the considered materials:


60

Density o f the liquid metal 7500 kg / m3

Density o f the liquid slag 2500 kg / m3

Coefficient o f the surface tension 0.2 kg / m2

We assumed that the metal remains liquid and neglected dependence o f their basic

physical properties on temperature. Furthermore, we assumed that the height o f the

studied meniscus is less than the maximum possible (in this case the random fluctuations

will not lead to the destruction o f the meniscus). In the further descriptions we accept the

height o f the meniscus H^ = 0 .985 -Hmax.

Influence o f the basic parameters on the form and dimensions o f the

mark.

The results o f the calculations are shown in figures 3.4 (a - h), where solid slag is

marked by green line. Dotted blue line is the x coordinate o f the point o f extremum. The

meniscus and solidified mark are blue. The top o f solidified mark is located on an axis

X ( y = 0).

- 0.1 - 0.1

X, mm

3.99 3.99

8.09

Y, mm

a) b )


61

3.99

8.092 0 2 4 6 -2 0 2 4 *

b ) d )

- 0.1

I.W3.99

8.096

e) f)


62

-oi

3.99

8.09

g ) h )

Figure 3.5 Calculated intermediate positions o f the meniscus

To the figure 3.4:

a - H s = 0 - the initial position;

b - H s = 0.25 • [Hh - / 2J - 25% o f the solidified mark movement;

c - H s = 0.50 -{Hh - h i ) - 50% o f the solidified mark movement;

d - H s = 1.00 • (Hg - hi) - exactly on the line o f solid slag lower boundary;

e - H s = 0.75 • {H h - hi) - 75% o f the solidified mark movement;

f - H s - 0.85 • {li h - h / ) - 85% o f the solidified mark movement;

h - H s = 0.95 • {H h - h t ) - 95% o f the solidified mark movement;

g - H s = 1.00 • {H h - h j ) - 100% o f the solidified mark movement.


63

Two phases o f the first stage

From the previous numerical analysis o f the meniscus displacement during the

first stage o f the mark formation, it is possible to separate two phases o f meniscus

movement, such as:

1. From the beginning o f the mark formation cycle (point Q - see Figure 3.3) until the

x - H g (interval hi>x>HG). During the first phase the line o f the meniscus moves along

the axis Y from the slag rim (Figure 3.4 a-d). The maximum distance y(Ha) achieved at

x = H g (Figure 3.4 d).

2. The second phase starts when the top o f solidified mark passed through the point

x =Hg (Figure 3.4 d) and continues until the meniscus contacts the slag rim, i.e. till the

ending o f the first stage mark formation (Figure 3.4g).

During the second phase the line o f the meniscus, moving along an axis Y, comes nearer

to the slag rim. Here the distance between meniscus and slag rim in the section

Hg > x > 0 varies from y(HG) up to zero.


64

4. The model of the thermal processes on the

meniscus at the first stage of mark formation

processIn the previous chapter the model o f mechanical behavior o f meniscus was

described. In present chapter we consider how the temperature o f each selected point o f

meniscus changes during the first stage o f mark formation process.

4.1. Basic assumptions. The representative element

We can assume that it is possible to neglect the heat removal from the meniscus at

the first phase o f the first stage o f mark formation. At the first phase when the meniscus

moved away from the solid slag, the gap between meniscus and solid slag increased. The

lfesh portions o f fluid slag with the high temperature moved into the gap. The

temperature equalization occurred across the entire surface o f meniscus, and became

equal to temperature o f the liquid metal in its core TM . Reduction o f the meniscus

temperature starts only at the beginning o f second phase after meniscus height became

equal H G. In the second phase the gap between the meniscus and the solid slag

decreases. Fresh liquid slag stops coming into the gap and liquid slag starts to escape

from the gap. It leads to appearance o f a temperature gradient in the liquid slag providing

intensive heat removal from the meniscus. For description o f the meniscus thermal state

during first stage, we consider the thermal state o f n- points o f observation on the

meniscus starting from the end o f the first phase.

First, we choose some point B on the meniscus with coordinate x = x 0 and


65

y = y(x0, H G) (see Figure 3.3). At the point B(x0) we place the representative element

(RE) with infinitesimally width (see figure 4.1) and from this point we start to consider

changes o f the heat state o f RE. This RE is represented as orthogonal prism which is

created by two planes parallel and two planes perpendicular to the plane o f drawing. The

centerline o f the RE is perpendicular to tangent to the meniscus. One base o f RE is

located on the meniscus and the second base in the liquid metal (figure 4.1). The length

o f the RE is equal to the thickness o f thermal boundary layer ST (see Chapter 4.6).

L m iiu lm e in i

tolidified marl;

h

a)

Figure 4.1 The RE geometry changes as a result of meniscus growth

a) initial position (the beginning of the second phase);

b) current position.

B(xq ) - initial position of point of observation;

B(x„) - current position;

/q - initial width of RE base;

I — f (h) - current width;

b)


8921912390902323902323905323899091239048238990912390902389909123909023

66

ST - length of RE (thickness of the thermal boundary layer);

h - current height of meniscus;

S q- initial meniscus height;

Xq - X- coordinate of point B (x0 );

Xfa - X- coordinate of point i?(x /j) ;

J - junction point;

0q - initial angle.

Considering the process o f the meniscus stretching the following special features o f the

RE stretching were taken into consideration:

- the RE stretched with the meniscus, the stretching is accompanied with liquid

metal coming into RE from the core o f the billet;

- the new portions o f metal are evenly distributed throughout entire RE;

We study only 1 D transient heat transfer, i.e. we consider heat transfer only in the

direction o f the RE axis. We neglect heat exchange between the RE and the adjacent

sections o f meniscus. The heat transfer occurs only through the bases o f RE.

4.2. Derivation of differential equation

The basic principles o f the heat balance equation

The infinitesimally element du was selected inside RE (see Figure 4.2). Thermal

conditions for du determined by:

- heat transfer through the bases o f d u ;

- change o f the du heat content;

- heat comes with fresh portion o f metal from the core o f billet


67

Heat balance for the du

The change o f the du heat content during the time dr (see Figure 4.2):

dQv = dQfr + d Q f , (4-1)

where dQv - the total change o f the heat content in the du during dr ;

dQA= (-dQ]+dQ2) - change o f the heat transferred through the bases o f the du

during d r ;

dQF - change o f the heat content o f the RE which comes with the fresh portion o f metal

from the core o f billet.

B Q = * -E i(,vc)|u=D =,*ke( ^ x» >

11 =

Figure 4. 2 The RE boundary conditions

The total change o f the heat content in the du during dr

A change o f the heat content in the du is determined by the equation:

dQv W = c • p ■ b • [ /( r2 )■ T(u, r2 ) - l(rx) • T(u, r , )] • d u , (4.2)


68

c - coefficient o f heat capacity o f liquid metal;

p - density o f liquid metal;

b - width o f infinitesimally section o f meniscus;

u - the coordinate along the prism axes {u\ <u <u2).

Change o f the heat transfer through the bases o f the du

The notation o f the coordinate o f the RE bases: u = 0 on the meniscus, and

u = ST inside liquid metal area. For the du element respectively: at u = u2 the heat

enters in the d u , and at U = U \ the heat exits from the d u . According to the Fourier law

the heat transfer through the ends o f the du for the time from r , to t 2 :

dQA =b , dT / \ , dT/ \du du

where l{t) = l0 + A /(r) - is the du extension along the meniscus (tangentially to it);

A /( r ) - lengthening o f the RE along the meniscus as the function o f time r , < t < t2.

Change o f the heat content o f du due to the metal flows from the core

The meniscus stretching is accompanied by metal influx from the core o f the

strand. The temperature o f this metal is equal to the temperature o f the liquid metal,

which was not cooled. In this case the heat, which entered into du is proportional to the

extension o f the du along the meniscus A /( r ) . The contribution to the du thermal

equilibrium from the metal, can be determined as a difference in the values o f the heat

content o f outgoing metal at r 2 and at r , :

dQF(r) = c -p -b -T M • [ /(r2) - / ( T 1)]-t/w (4.4)

After substitution o f dQA, dQv and dQF into equation o f heat balance (4.1) we get:


69

c-p-b-[l{r)-T{u,r2)-l{r^)-T{u,rx)\ - du =

b- 0 dT( \ ]dT( \du du

-l(r2)-dr + c -p -b -T M - [ l ( T 2 ) - l ( T l ) \ - d u (4.5)

Equation (4.2) can be transformed:

^ u d U(T) ' T(U’T)] J JdQy = c- p - b — — *— — -du-dr =dr

= o . p . b . j[,(r)] . + T ( u , r ) . 4 W 1 1 . d u . d t( dr dr J

After substitution dQv into equation o f heat balance (4.5) and transformation:

c- p -b - \ [/(r)] • r ) \ . du- dr = b •'a d T ( \

dudu

du-dr +

+ c ■ p -b -\Tm - ^ ^ - T ( u , r ) - d- ^ ^ \ - d u - d r dr dr

After reducing on du-dr the differential equation o f heat balance takes the form

(taking into account, that the du is the uniform prism):

2

* • W ) } • 4 4 - + c • P ' [tm - T(u , r)]•dT

du dr dr

d T ^ _ _ l _ d 2T | Tm - T ( u , t ) d[/(r)]

dr c - p du2 l(r) dr

This differential equation corresponds to the classical form o f the parabolic differential

equation with the source function:

dT^__A_ d T dr c - p du2

+ F ( r ) , (4.6)


70

where F(t ) = — —J ̂ ^ • —— — - source function; (4.7)l \ t) dr

Initial conditions

At t ' , when the first phase completes, the RE has the temperature in the whole

volume:

T{u, t)\t^ = T m = const (4.8).

Boundary conditions

Boundary conditions are formulated for two RE bases: BC2 is located at the

distance ST from the surface o f meniscus:

T(u, t )\ =Tm = const (4.9)I U2 —Sj

At the surface o f the meniscus, boundary condition (BCi) is:

l d T ( \du <1r e (t ’x o ) » ( 4 - 1 0 )

u=0

where <1re{t,Xq) - heat-flux density extracted from the base o f the RE (it depends on the

time o f observation and coordinate o f initial observation point). In the process o f the

meniscus stretching, RE is moving and, therefore heat-flux density is the function o f the

RE position (initial point is the point x0 ).

4.3. Equations which describe the dynamics of RE

Functional description o f the RE position is determined by meniscus displacement

and stretching. The change o f the position o f the RE is the result o f two simultaneous

motions addition:

- displacement together with the meniscus resulting from meniscus height


71

increasing;

- displacement resulting from meniscus stretching.

The RE heat state at a given moment o f time is defined by the heat transfer from RE

depending on the RE displacement from the wall, and additional superheat from the core

o f the strand depending on the RE stretching. We assume that the start o f the change in

the RE heat state coincided with the start o f the second phase. For the solution o f the

differential equation o f heat conductivity it is necessary to know the functional

dependence o f RE location and stretching on the meniscus height.

RE position as a function o f the meniscus height

In the end o f the first phase the length o f meniscus L0 is equal (3.20):

(4.11)

The length o f the part o f meniscus L B between point J and point B (x0 ) is equal:

(4.12)

If the height o f meniscus will increase to h (Hh > h > H G), the meniscus length Lh

becomes:

(4.13)


72

Aspect ratio o f meniscus : ----------- .h

Point B(x) comes from point Z?(x0) into point B(xh) and length o f the part o f

meniscus between point J and point B(xh ) will be:

(4.14)

The coordinate xh o f the point B(xh ) can be found from implicit equation:

(4.15)

where Lh - can be found from equation (4.13);

L0 - can be found from equation (4.11);

LBa - can be found from equation (4.12);

x 0 - is known a priory;

xh - unknown.

The equation (4.15) shows that xh = f ( h ,x 0)

Coordinate y{xh,h ) can be found from equation (3.13):

Where Y(h) and Y(xh) found from equation (3.14) and (3.15).

RE stretching as a function o f h

Let us assume that RE width equal to /0 when meniscus height equal to H G .

y(xh,h )= Y {h ) -Y {x h),


73

Then if the meniscus height equal to H G < h< H h then RE will stretched to lh .

where Lh = f ( h ) (4.13);

L0 - can be found from equation (4.11).

4.4. Transformation of the differential equation

For further calculations we need transformation from variable t to variable/?.

This single-valued conversion o f variables leads to a significant simplification in the

calculations (4.6-4.10). Let assume that during A t = T — Tl the top o f solidified mark

moved along axis X and the height o f the meniscus h :

h = H a +Vc AT = H G +Vc ( T - T l) ,

we start to consider the cooling process from the second phase when meniscus height

already equal to H G ( see Figure 3.3);

Vc = const - the velocity o f the displacement o f the top o f solidified mark.

At the time equal T the length o f meniscus equal to Lh and the width o f RE equal to lh .

1 — 1 ^ h lh ~ l0 r ’ (4.16)

For the source function the derivative dW)\ can be determined from (4.11):dr

£ M = _ ? i y _ = i ( L . ra - f 1. \ T CS t J h _ \ L0 dh

‘h

In this case from the expression (4.7) it follows, that the sources function:


74

F r _ \_ Tm ~ T (u >T) 5 [ / ( r ) ]_ T m - T ( u ,r) l0 8Lh _F[T)~ — j ^ ) F — Vc — ~d t

Tu - T ( u j V c . a i = F W

7 . L q

° Tx>n

dh

L,(h) dh

dL>dh

can be determined from equation (4.12) by differentiation o f integral:

d hdh

1 + {b • x 2 + sin 0O)

-y 1 - { b - x 2 + sin 0O Jdx

dh°r:

dLh1 +

(.5 • h2 + sin 0O)

dh ]j *\jl-(B-h2 +sin0oJ

Thus, source function became:

F(h) - T m ~ T (U ’^ ) • Vc ■ Lh

1 +{b • h2 + sin 0O)

■\Ii ~(b -h2 + sin0oJ(4.17)

where Lh = J o

1 +i(b ■ x 2 + sin 0O)

- J l - ( 5 - x 2 + s in ^ 0)^• d x ;

dT dTKnown, that — = Vr ------

dr C dh

After substitution o f this expression in (4 .6 ), we obtain:


75

dT _ X d 2T | TM -T (u ,h ) dh Vc -c-p du2 Lh i1 +

(B-h2 + sinfl0)

~ { B h 2 + s in ^ 0)2(4.18)

/ W =Tm - T(u, h)

Li1 +

{b ■ h2 + sin 0O )

+sin<90)2

4.5. BCi calculation

To determine boundary condition the estimation o f the heat-flux density from the

meniscus to the water through the slag and the mold wall is required. We examine heat

transfer from the RE at t > t ' .

Basic assumptions

Heat-flux density is determined according to the heat transfer through the

multilayered wall. The coefficient o f the liquid slag specific heat is more than 5 times

lower than the coefficient o f the specific heat o f the liquid metal. Therefore for

determination o f the heat flux density function, we neglect the quantity o f heat,

introduced by liquid slag into the gap between the meniscus metal and the solid slag on

the mold wall. We also neglect the heat, absorbed by liquid slag from the liquid metal,

(low heat capacity o f slag and its small quantity). Since boundary liquid slag - solid slag

- is the boundary o f the phase transition between liquid and solid slag, then the

temperature o f this boundary is constant and equal to the temperature o f the slag

crystallization (we neglect the interval o f crystallization o f slag). We assume the steady

configuration o f the slag rim observed during periodic displacements o f meniscus. We

consider only ID heat transfer from RE to the examined the non-uniform prism, which is


76

located at a distance x from the beginning o f coordinates. This prism is notes as the heat-

transfer prism. The centerline o f heat-transfer prism is parallel to axis Y. The heat

transfer prism consists consecutively from liquid slag, solid slag and water-cooled mold

wall. One base o f heat-transfer prism is the base o f the RE (located on meniscus o f liquid

metal). Second base o f the prism is part o f the surface o f the water-cooled mold wall. RE

and the heat-transfer prism are inclined toward each other at the angle (p .

For evaluation o f the heat exchange from the RE to the water, the law o f

stationary heat exchange was used. Since we examined the stationary heat exchange

(from the RE to the cooling water), an equal quantity o f heat passed through each part o f

the considered prism, i.e., heat-flux density in all parts (thermal resistances) is equal

(4.19)

where:

- heat-flux density according to Fourier’s law o f thermal

conductivity;

A,i - the coefficient o f the thermal conductivity o f / -thermal resistance;

(Ti+ — 7 j_) - the temperature differential in the Z -thermal resistance ;

hj - the thickness o f I -thermal resistance;

q = a,j • [Tj + —Tj_) - heat-flux density according to the Newton’s law o f heat

emission;

a j - the thermal transfer coefficient o f j -thermal resistance;


77

{Tj+ — Tj_ ) - the temperature differential on the j - boundary o f two surfaces.

Considering structure o f thermal resistance on the way from the surface o f the meniscus

to the cooling water, density o f the thermal flux determined from the equation (without

taking into account that the RE base is not perpendicular to the prism axis):

<7 = {Tm ~ Twat ) ' Rat > (4.20)

where RAT - thermal resistance on the way from the liquid metal meniscus to the

cooling water.

Heat-flux density in the gap

Taking into account that the thermal resistance o f the gap between liquid metal

and mold wall, filled with solid and liquid slag has up to 80-90% o f all thermal resistance

on the way from the meniscus to the cooling water [42], we have:

q * r b '}T" ~ Tw* \ ; <4 -2 »Ahsi_f | Ahsl

v A 's l - f ^ s i ,where b varied from 0.8 to 0.9 depending on slag thickness, slag conductivity,

mold wall material;

A hsi - f-------------heat conductivity o f fluid slag;A'sl-f

A/z— — - heat conductivity o f solid slag.

I.e. the change o f heat flux was determined by the change o f the slag thermal resistance

which resulted from a change in the thickness o f the layers o f solid slag and liquid slag (it

is determined by definition o f RE position with coordinate x):


78

1 _ AK i - f | Ahsl

& sl A 's l - f A l

Hence, heat-flux density (4.21) is a function o f the thickness o f the layers o f solid and

liquid slag.

During the estimation o f the heat-flux density from the RE we consider not only

o f the RE exothermal base on the a x i s X ). Finally, heat-flux density from the RE with

the correction on the angle between the ax o f the heat-transfer prism and RE:

4.6. Thermal boundary layer

The meniscus and meniscus area as parts o f the shell o f the billet are subject o f

the action o f the liquid metal flow. The typical distribution o f flow near the liquid metal

surface in the mold is shown on Figure 4.3 (standard technology o f continuous casting o f

steel).

RE position (coordinate x ) , but also angle between the RE and the axis X (projection

For xh definition we use equation 4.15, for cos^?(x^ j definition we use equation:


79

3. SoM Flux Layer3 . StmVFlux Interface

Shel6. GsrfHafon Mark6. Meniscus7. StMi Surface 6. Uaiid SteelD. Liquid Flux Layer 13. Cr)«teMne "Flux Layef 1 l . « a ^ “nuxLayef

4, SoMifyIng Steel

1. Air GapLiqiwS Slag

Figure 4. 3 Distribution of the liquid metal flows in the mold [40]

The layer o f liquid metal near the meniscus consists o f the thermal boundary layer (TBL)

and the velocity boundary layer (VBL). Thickness o f the TBL is functionally connected

to the thickness o f the VBL and for liquid metals thickness o f the TBL exceeds the

thickness o f the VBL [41].

Estimation o f the TBL

Thickness o f the TBL can be defined as the function o f VBL thickness S on the

basis o f criterial equation (Prandtl number). For liquid metals it is possible [42] to use

dependence:

S * S T - P r 3 (4.23)

where S - the VBL thickness;

8t - the thickness o f the TBL;

P r - Prandtl number.

where /j - coefficient o f kinematics viscosity o f liquid metal;


80

k - coefficient o f thermal conductivity o f liquid metal;

cp - the coefficient o f heat capacity o f liquid metal.

Estimation o f the thickness o f VBL is done on the basis o f the criterial equation o f

the flow (Reynolds number):

where z - distance from the end o f plate, where the flow began;

R e = 03 ^ - the Reynolds number;

p - the characteristic size, determined from the dimensions o f the liquid metal

flow;

CO - the speed o f the liquid metal flow.

Let us consider that the velocity o f the liquid metal flow CO is directly proportional to the

speed o f casting (extraction speed o f ingot from the mold is Vc ), co ~ k * V c where k

- coefficient o f proportionality. After substituting R e , p , co,k,S in (14):

wherez • jU6

(4.25)


81

In the limits o f the mark formation cycle it is possible to accept the physical

characteristics o f the liquid metal as constants: fi — const , cp = const , A = const.

Taking into account that the meniscus length is small in comparison with the flow

extension above the solidified shell, it is possible to accept that the flow parameters

above the meniscus are the same as above solidified shell. I.e. for any point o f meniscus

z — const , p = const . Thus, it is possible to assume Cs - const and

Some thermophysical parameters in expression (4.26) are unknown a priori. The

speed o f the flow o f liquid metal near meniscus for the specific case is also unknown; we

do not know exact value o f the distance from the end o f plate z , where the flow began

(equation 4.24). Hence, the direct determination o f ST from equation (4.26) is not

possible and ST was chosen as fitting parameter. The following procedure was

developed for estimation o f the thickness o f the TBL. We assume that at the end o f the 1st

stage meniscus is still liquid, i.e. its minimal temperature (T) is not less than temperature

o f liquidus (Tt,). For some assigned value o f ST at one fitting step we determine

temperature distribution on meniscus (using the thermophysical model o f meniscus). The

iteration process is organized as follows. The minimum value T o f the temperature

distribution is compared with T l . I f |T - T, |> 0.2 then we change value ST and repeat

determination o f the temperature distribution on meniscus and comparison T and TL, i.e.

do the next fitting step. The process ends w h e n |r - TL |< 0 .2 . It has been found that value

dT = 0.3 mm.

Sconst _ Cs

(4.26)


4.7. Summary of equations

Basic differential equation

dT _ A d T dh Vc -c- p du2 +/(*)

Source function

j r ^ y TM -T {u ,h ) 1 + (g • h2 + sin Q0)

s j \ - ( B ■ h 2 + sin 8 o f

where Lh - can be found from equation (4.9):

h

01 +

(g • x 2 + sin 0O)

sJi - ( b -x2 + sinB0f■dx,

xh - the coordinate determining RE position and can be found from equation:

J * h

Lb° 0 T6J,

( ( \\B ■ x 2 + sin 0O)

| [ ^ - ( g - x 2 +sin6>0)21 + ■dx,

x0

Lbo = Jl 1 +(g • x2 + sin O0 )

-^1 - (b ■ x2 + sin 0O f■dx,

Hr,4 = J 1 +

(g • x 2 + sin 6>0)

-y/l - { b -x 2 + s in g 0)2■dx,

aBJ =B̂

(Chapter 3.4)


83

Initial condition

T{u, h)\h=Hc =Tm = const

Boundary conditions

T ( u ’ t y \ u = s t = T m = const,

. 0 const C?where oT ~ -------- -- — —

A , h)ou

■ < 1 r e { K x o )

where ( 1. r ( 0 , 8 . . . 0 , 9 ) - Twat)-cos<p(xh ) qRE \n, x0)~ f — - - -=jAhsi-f{xh) | Ahsl(xh)

S l - f Si

* h A * k)= * y + Y{HG) - Y { x h),

^ f {xh) = [Y{h)-Y(Hh)][\{HG>xh > 0 ) +

+ [Y{h)-Y{xh) -Y { H h) -Y { H G)}D(x* >HG) ’

cos <p^h)=-1

1 + j B . + sin 0O1'n2

tJi - ( b -x% +sin0of

(Chapter 3.4)

(Chapter 3.4)

4.8. Analysis of the model of thermal processes on meniscus

A special program was developed for the temperature field o f meniscus

calculation (using MathCAD). This program was a continuation o f the program which

was developed for the model o f the mechanical state o f meniscus. Calculation was done

for the second phase o f the first stage o f mark formation. Twenty points o f observation

were selected on the meniscus. For each point the differential equation o f thermal


84

conductivity was solved. In addition, the displacement o f the observation point was taken

into account. Thus, the values o f temperatures o f meniscuses were determined.

Initial data for evaluating o f the meniscus thermal condition

Thermophysical characteristics published for some slag and steels are shown in

tables 4.1 and 4.2.

Table 4.1 Thermophysical characteristics o f the mold slag

Thermophysical characteristics o f the mold slag

Name o f parameters

Notation Unit

Value o f parameters

For steel 0.05%C

Forlowcarbonsteel

Formedium- carbon steels steel

Temperature o f slag on the surface in the mold

T sl °C

Temperature o f crystallization o f slag °c 950 1043 1163

Coefficient o f heat conductivity o f solid slag

A t W /(m°C) 1.5 1.73 1.83

Thickness o f the solid slag on the mold wall

d sl mm 0.3...1.0 0.4.. .0.9

C a0 /S i02 0.96 1.41


85

Table 4.2 Thermophysical characteristics o f steel

Thermophysical characteristics o f steel

Name of parameters Notation Unit

Value o f parameters

Steel10.1-.2%C

Steel20.05%C

Steel3

0.5%C

Steel4

0.2%C

TemperatureLiquidus Tl °C 1510 1529 1495 1520

Temperaturesolidus Ts °c 1475 1509 1445 1480

Density o f liquid metal Pm kg/m3 7400 7400 7500 7500

Coefficient o f thermalconductivity o f liquid metal

^M W /(m °C ) 23.26 23.2 29.1

Coefficient o f heat capacity o f liquid metal

CM kJ/(kg°C) 0.691 0.84 0.71

thermal diffusivity coefficient o f liquid metal

a - 1(T6 m2/sec 4.546 3.682 5.465

Temperature in the core (with superheat) o f strand

Tm °C 1540 1550 1515

Specific heat o f crystallization

L kJ/kg 268 268

Extractionvelocity

vc m/sec 0.2

Thermophysical characteristics o f the mold slag mixtures used in experimental castings

are shown in table 4.1. It is the medium carbon slag with CaO/SiC>2 equal to 1.41.


86

The results o f the temperatures determination

The calculations were done for the 0,5% -carbon low-alloy steel with the

thermophysical characteristics, shown in the tables 4.1. The appropriate solutions o f the

equation for the mechanical equilibrium o f meniscus are used as the initial data for the

solution o f the differential equation o f thermal conductivity. The changes o f the

temperature at the meniscus points were considered from time, when meniscus has the

height h — H G and its temperature was constant throughout all length and equal to

temperature o f metal in the ingot’s core T - Tm . While the position o f the RE point

B(x()) moved and the RE temperature field was calculated in the process o f RE

displacement. The examples o f these calculations are shown in Figures 4.5 - 4.9.

Figure 4.4 Variation o f RE temperature resulting from

the changes o f the meniscus height and determined for

h —0.\- Hq , 3D view


87

For the case shown in Figure 4.4 the casting temperature was 1515 °C and the

point B{x()) is located at the distance 0.1 • H G from the surface o f liquid metal. The axis

U in Figure 4.4 is the distance along the RE centerline, the axis H is the parameter o f the

meniscus height (H = (h — H G) / H h). In Figure 4.4 and its 2-D projection, Figure 4.5

we can see only small changes o f metal temperature because o f close location o f B{x0 )

to the point J.

T,°C

T,°C

1510

1500

1490

1510

1500

U. m m h , m m

a) graph T v s .U , ( H = 0) b) graph T vs. h , (U = 0;

Figure 4.5 2D views o f variation o f RE

temperature resulting from the changes

o f the meniscus height and determined

for h = 0.1 • H g

For the case shown in Figure 4.6 the casting temperature is equal to 1515 °C and

the point B (x 0) located at the distance 0.9 • H Cj from the surface o f liquid metal. Here

we have significant changes o f temperature because o f close location B{x{)) to the mold

wall.


U, mm

Figure 4.6 Temperature o f RE changes resulting

from the changes o f the meniscus height and

determined for h = 0 .9 • H G , 3D view

Below 2-D images o f the 3-D graph for better illustration o f the dependences T vs. U and

T vs. h are shown.


T, °C

89

1510

1500

14900.1 0.2

U, mm

1500

1480

1460

108.5 9 9.5

a) graph T v s .U , (H = 0j b) graph T vs. h , (U = 0)

Figure 4.7 2D views o f temperature o f RE

changes resulting from the changes o f the

meniscus height and determined for h - 0 .9 • H G

Representation and analysis o f the meniscus temperatures

h, m m

Figure 4.8 Temperature field o f meniscus at the

end o f first stage( h = H h), 3D view

Resulting temperature field o f the meniscus at the end o f the first stage is shown

on Figure 4.8. The set o f the temperature values for this graph inferred from the sets o f


90

the twenty computed values o f temperatures for different position o f RE. Below are 2-D

image o f the 3-D graph (with the liquidus temperature) is shown.

T,°CTemperature o f meniscus

1510

1500

Line o f liquidus

1490

h, mm

Figure 4.9 Temperatures o f the meniscus at the end o f first stage

As is evident, to the end o f first stage o f surface mark formation, curve for the

temperature o f meniscus is located above the temperature o f liquidus.


91

Y , m m0 2 4 6

4.34

mi

Lipe of extremum

>.67

Top of solid mark

13.011500 1505 1510 1515

Figure 4.10 Calculated shape o f meniscus (blue) and

calculated meniscus temperature field (red)

In the Figure 4.10 the changing o f the temperature o f the meniscus along the

meniscus height is shown. The blue dashed line is the calculated shape o f meniscus, red

line is the temperature distribution along the meniscus height.

After obtained meniscus temperature field at the end o f first stage we can verify

that ID heat transfer equation can be used in proposed thermal model. Let the qt is the

heat flux density in the direction tangent to meniscus and qn is the heat flux density in

horizontal direction. The heat-flux density according to Fourier’s law o f thermal

conductivity: q = A ■ AT / h ;


92

From Figures 4.9 and 4.10 and table 4.2 we have: AT - 20° C ; h = 10m m ;

A = 23.2W/m-° C.

qt = 4 6 ,4 0 0 W / m 2;

For qn we have: AT = 1 4 7 5 ° C ; h - 4m m ; X —1.9 W/m-° C .

qn = 700,600 W / m 2;

hence qt / qn = 7% . It confirms validity o f our assumption about ID heat transfer from

the m eniscus.

4.9. Conclusions.

• A mathematical model o f the thermal state o f meniscus was developed. Meniscus

displacement and stretching were taken into consideration. In the estimation o f the

meniscus thermal state, the source function was introduced.

• Developed mechanical and thermal model o f the meniscus can also be used for

the mark formation study in the case o f mold oscillation.

• The developed model can be further supplemented by the meniscus solidification

calculation. It will permit to use the model for description o f the meniscus thermal

conditions at the second stage o f mark formation.


93

5. Analysis of the adequacy of the developed model

Approbation o f the developed mechanical model was done for the purpose o f

testing the model by experimental data and possible correction model parameters. The

level o f correspondence between model and data was evaluated using some methods o f

mathematical statistics.

5.1. Analysis of the adequacy of the mechanical model o f the first

stage of mark formation

Experimental data

The data were obtained from the surface marks on real billets were used for

adequacy test o f the model. These data were represented by measurements o f fixed

SCALE 1 : 0.5 mm

Figure 5.1 Photo o f template with position o f meniscus marked as yellow dots


94

position o f the meniscus, and were obtained by measurements o f the templates images o f

surface marks. Figure 5.1 gives the sample o f measurements for the medium-carbon low-

alloyed steel. The experimental data values (Appendix B) represent a two-dimensional

numerical file x ; , y t . Quantity o f pair’s numbers we denote as N .

Optimization o f the modelfitting parameters

The developed model required a set o f parameters that do not have an exact a

priori determination. These fittings parameters are:

H h - height o f the meniscus at the end o f the first stage o f mark formation, thus

H g - H h < H max;

Oq - initial angle o f the meniscus ( Chapter 3.3).

Since the parameters H h and 0{) cannot be exactly assigned a priori, it is necessary to

find them first.

The coordinate system o f the theoretical model does not obviously coincide with

coordinate system o f measurements. The parameter, which corrects the value x ( was

introduced:

Zi =xt + r ,

where r - correction o f initial point, which corrects the parameter x ; r does not

depend on the number o f observation i .

Now equation 3.13 can be re-written as:

y (x i ,H h ,0 O’r) = - Y ( H h,0o)+Y(x i ,0o,r ) , (5.1)


95

where Y(Hh,0o)= J (B x + smgo) ,dx) o y i - ( s - x 2 + s i n 0 of

/ v ,

o - ^ l - ( i ? - x 2 + s i n # 0j

Thus, during adequacy testing the following parameters were corrected: H h, 6{), r .

The level o f correspondence was evaluated using the residual sum o f squares

( R SS ). RSS o f the calculated and measured values:

RSS = £ [ y , - y ( x „ H h,e0,r )Y , (5.2)/=1

where x , , y, - pairs o f numbers, which determine the position o f the point on the

meniscus, these pairs were found using experimental measurements;

y(x i, H h,9q,r) - predicted by the model.

The best approximation is obtained if we will get the minimum value o f expression (5.2).

From this condition fitting parameters were found (Table 5.1). Minimization was done

using mathematical software written for "MathCAD" (Appendix B, “Adequacy test”).

Test o f adequacy between experimental data and model

The conclusion that the model is adequate to the experimental data was done on

the basis o f Fisher test. Fisher's test is just the determination o f the ratio o f the estimated

variances o f two distributions. I f the following inequality (5.3) is satisfied, then the model

is adequate to the experimental data.

(5-3)V

r e p


96

RSSwhere s 2d = —--------- - - the variance o f adequacy;

N — p — \ - the number o f degree o f freedom for variance o f adequacy;

N - the number o f experimental data;

p = 3 - the number o f the param eters, determined by selection o f their values for

the best correspondence between experimental data and model;

2s rep ‘ variance ° f reproducibility;

1 - the number o f the denominator degrees o f freedom (for the variance o f

reproducibility);

F{xN_p_l - Fisher ratio, the value o f Fisher ratio we can find from the table o f F-

distributions [43]. In this case the level o f significance accepted as

a = 0,05 (probability that the null hypothesis would be reject even if null

hypothesis is true [44]).

The dispersion o f reproducibility was evaluated as follows:

Within the present research it was found that the maximum error o f positioning o f the

point on the mark surface is + 0 ,2 5 m m (systematic error, i.e. half o f scale, see Figure

5.1). Hence, the length o f interval is A / = 0 , 5 0 m m . It is known that for the normal

distribution 95% confidence interval corresponds to ± 2 standard deviation. From here

we can write: Al = 4 ■ Srep. Then the dispersion o f reproducibility can be determined:

2 fAnerep


97

By the fact that the image o f template was increased by 20 times, the value o f the

dispersion o f reproducibility needs to be corrected:

2

3 r e pf A / 1 f 0,5 1U - 2 0 j U - 2 0 j = 0 . 0 0 0 0 3 9

5.2. Results of the model adequacy testing.

Adequacy testing o f the model was done for four samples, obtained from the surface o f

the different ingots. An example o f the parameter H h and 0Q optimization results is

given in Figure 5.2:

2.5

X, mm

15

- 1 .5 5 0 Y, mm

Figure 5.2 Results o f optimization for the parameters H h and 0(),

number o f points N = 15 ; 0O = 0,05°

green wide line - solid slag;

black line - meniscus;

blue crosses - experimental points

The results o f calculations for four samples are given in the Table 5.1.


98

Table 5.1 The results o f calculations

SaITiples

Numbero f

points

Calculated parameters of the

model

RSS N -p-1 s hS 2rep

F%n-p- iCorresponden

ceHh/

/ T-f/ ma 0o+9O

1 15 0.963 0.05 0.02033 12 43.37 243.91 yes2 9 0.798 0.5 0.008617 6 36.76 233.99 yes3 5 0.877 0.05 0.005893 2 75.43 199.50 yes4 5 0.937 0.05 0.0095 2 121.60 199.50 yes

Table 5.1 data shows, that the model corresponds well to the experimental data, obtained

in varied conditions. The Proposed model has quantitative and qualitative

correspondence. Qualitative correspondence can be seen in Figure 5.2. This figure

demonstrates that the points, obtained on the solidified part o f the meniscus, do not

exceed the top o f the solidified mark, what corresponds to the prediction o f the model.

The points correspond to the meniscus extremum (model presentation) also coincides

well with the experimental data.


99

6. Conclusions. Advantages of the developed model

The SM formation process occurs at the initial stage o f solidification o f the

billet’s shell and defines stability o f continuous casting and quality o f the billets surface.

In order to describe the complex process o f SM formation a two stage approach to the lull

cycle o f SM formation was suggested. The first stage describes the changes o f the

meniscus position before touching the wall o f the mold, after which the second stage

begins. A mathematical model describing the first stage o f the SM formation process was

developed in this thesis. Oscillation factor which considerably complicates the modeling

process was eliminated. Basic results o f the modeling were tested during the continuous

casting process in an immovable mold.

Novel features o f the mechanical model

The developed mechanical model describes meniscus displacement and stretching

during the first stage o f SM formation process.

• The numerical analysis o f the meniscus displacement and stretching during the

first stage shows the existence o f the two separate phases o f meniscus movement.

During the first phase meniscus moves away from the mold wall and during

second phase meniscus comes to the mold wall.

• The top o f the solidified mark is located at the same distance from the mold wall

during the whole cycle o f the mark formation;

• The upper meniscus after the first stage o f mark formation is immovable.


100

• The developed numerical model has been tested against the experimental data.

The level o f correspondence between simulated and measured data has been

evaluated and shows that the model corresponds well to the experimental data.

• The model was used to completely describe the process o f displacement and

deformation o f meniscus. Results generated by the mechanical model were further

used in the simulation o f the thermal processes on meniscus.

Distinctive novel features o f thermal model

• For purpose o f thermal analysis the equation o f 1 D transient heat transfer has

been derived and the source function was obtained. Initial condition and boundary

conditions were formulated on the basis o f physical conception o f mechanical

model o f SM formation process.

• The source function was obtained on the basis o f detailed description o f the

meniscus stretching and considers the superheat which comes with portions o f

liquid metal from the core o f billet to the meniscus.

• The numerical model o f the temperature distribution on meniscus during its

evolution was developed.

• The thermal model was used to predict o f the thickness o f the thermal boundary

layer ( dT ) accepting that at the end o f the first phase the minimal temperature o f

the meniscus equals to the temperature o f liquidus.


101

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106

Appendix A

Thermophysical model of meniscus for 1st stage formation


107

Parametric repres. o f solution

Parameter o f limiting value o f meniscus dbh

Parameter o f top o f solidified mark displacement dd

Liquid steel density, Kg/M^

Liquid slag density, K g / M 3

Gravity acceleration, m/s^

Coefficient o f interphase tension, N/m

g iB := (RoM - RoS)-

RoM := 7200

RoS := 2667.3

g l := 9.807

SmS:= 1.2058

B = 0.0182SmS(10)

Upper meniscus Overflow process

Initial angle, degree Om := 5

Calculated initial angle, radian ^ OmitOml(Om) := — + -------2 180

sin o f initial angle pp(Om) := sin(Oml(Om)) pp(Om) = -0 .996

Hl(Om) :Coordinate x= H q along meniscus height, (point o f

extremum on meniscus) y=ymax Hl(Om) = 10.407

Hmax, initial angle Omega > - 90°

-f1~ PP(Om)

B

hl(Om) :=

First derivative

fl(x , Om)

-10

X

-pp(Om)

B

fl(x , Om) :=

hl(Om) = 7.352

(pp(Om) + B x 2)

J l ~ ( pp(Om) + BJ


108

Assign limiting height o f meniscus H^—bb

Parameter of limiting height of meniscus 0<dbh<l xHG< Hh<Hma

dbh := 0.9

bb(dbh, Om) = 10.101

[bb(dbh, Om) := hl(Om) + dbh(Hl(Om) - hl(Om))

Hl(Om) - bb(dbh ,Om) = 0.306

Junction point(meniscus and liquid surface o f metal coord.

Bh0(dbh, Om) :=bb(dbh,Om)

Profile o f rim

fl[x,(O m )] dx

Bh0(dbh, Om) = 14.314

Vertical coordinate axis located on the top o f solidified mark

BhO(dbh,Om) + fl[x ,(O m )]dx 0

(hl(Om ) > x > 0)Fb2(x, dbh, Om)

Rim line equation.Fb(x,dbh,Om) := Fb2(x,dbh, Om) + Fb2(hl(Om),dbh ,O m )(x > hl(Om ))

Solid slag line equation Initial location o f the top o f solidified mark Sx (along Fb2(20, dbh, Om) = o axis X) zz l Fb2(hl(Om ), dbh, Om) = -1 .5 5 Fb(hl(O m ), dbh, Om) = -1 .55

Giveraa := .0001 z := 0 + aa

Liquid meniscus lengthIn the end o f 1 phase Fb(z, dbh, Om) = 0 zzl:=F ind(z) zz l = 3.979

Parameter o f displacement o f solidified mark ddfhl(Om )

Value o f displacement db(bb)LLQ(Om) :=

Coordinate o f the top o f solidified mark X after its displacement (bbl) = current height o f meniscus hCoordinate o f the extremum point along X, ( y=ymax) on solidified mark

Coordinate o f junction point if h

1 + fl[x,(O m )] d>dd := .80

db(dd, dbh, Om) = 4 .898

db(dd, dbh, Om) := dd-(bb(dbh,Om) - zzl)

bbl(dd,dbh,O m ) := db(dd,dbh,O m ) + zzl

bbl (dd, dbh, Om) = 8 .8 7 7

hh3(dd,Om) :=hl(O m ) + db(dd,dbh,O m )

hh3(dd,Om) = 12.249


109

Bh(dd,dbh,O m ) :=■/•bbl(dd, dbh, Om)

B h(dd, dbh, Om) = 15.508

Meniscus line equation, meniscus leaned on solidif. mark

Fb3(x, dd , dbh, Om) := Bh(dd,dbh,Om ) + fl[x ,(O m )]dx 0

(x < bbl(dd,dbh,O m ))

x:= 5 Fb3(x, dd , dbh, Om) = 0.373 Fb3(zzl, 1, dbh, Om) = -2 .555 x 10~

M eniscus line equation, meniscus leaned on solidif. mark + section o f solid, mark + solid slag.

Fb4(x,dd,dbh ,Om) := F b 2(x-d b (d d ,d b h ,O m ),d b h ,O m )(x -d b (d d ,d b h ,Om) > zzl) + Fb2(hl(Om),dbh ,O m )-(x -d b (d d ,d b h ,Om) > h l(O m )) + Fb

Fb4(hl(O m ),dd, dbh, Om) = -0 .356Fb4(hh3(dd, Om), d d , dbh, Om) = -3 .101

DFb(x, dd , dbh, Om) := (Fb4(x, d d , dbh, Om) - Fb(x, dbh, Om))Gap between solid slagAnd meniscus D Fb(0,dd,dbh,O m ) = 1.194

dbh = 0.9 DFb(h 1 (Om), dd, dbh, Om) = 1.194

x:= 0 ,-1 .. 2 Hl(Om) dd = 0.8

O m = 5

6.94

20.81

dd := .95

13.88 “


25

Meniscus height parameter dd ifThe volume=max(0<dd<l)

db (0.9 )

ddO = 0.66

Length o f meniscus when the volume=max - base length

Initial position o f observation point on the bases part o f meniscus for Temperature changes

Assign B(xq) with coordinate x=Bx (0<Bx<bbl(ddO,dbh))

Parameter o f position xx (0<xx<l)Find coordinate x B(xq) along X (0<Bx<hh(ddO,dbh)

r bb1(ddO, dbh, Om)

LLBCfddO, dbh, Om) :=

LLBQ[ ddO, dbh, Om) = 19.62


I l l

BxG[xx) := xxbbl(ddO, dbh, Om) YxQ xx) := Fb3(BxfJxx), ddO, dbh, Om)

xx:= .5 BxC(x^ = 4.011 Yxdxx) = 1.456

Parameter o f displacemento f solid mark from the base position I>dd01>dd0

M eniscus length up to the pointB0(x)

L L B X x ? ) :=

■Bx0(xx)

1 + fl[x,(O m )] d>

Liquid meniscus length after top displ.to ddOl

Meniscus lengthening after top displ. from base position

Meniscus length, from point B(xq) to B(x)

ddOl := .8

bbl(dd01, dbh, Om) = 8 .877

LLBQddOl ,dbh , Om) = 20.527

DLL(dd01) = 1.046LLBQfddO 1, dbh, Om)

DLL(dd01) :=LLBtfddO, dbh, Om)

Coordinate x for B(x): x=Bxl LL^xx,ddOl) := DLI(ddOI) LLBXxx)

LLKxxddOl) = 15.885

LL?(xx,dd01) = 15.885

LL(Bbx) :=

Bbx

■I1 + fl[x,(O m )] d>0̂

ddOl := 1 xx:= 1

B bxl(xxdd01) = 10.101

Tangent in point B(x): cos

DL(xx,dd01,Bbx) := LIYxx,ddOI) - IJ(Bbx)

Bbx:= .lG iven DL( xx, ddO 1 ,Bbx) = 0

COS(xx,dd01) :=1 + (fl(B bxl(xx, dd01),Om))

Thickness o f solid slag in point B(x): DY0

COS(xxdd01) = 0.467

x:= hl(Om)

DYO(x,dbh,Om) = 0 dd := 0.8


112

Finding the point B(c), which cam to the top

Gap between meniscus and solidslag in point B(x): DFb DFb(x,dd,dbh ,Om) = 1.194

Meniscus length after top displ.dd := 1

Meniscus length to the final position B(c) bbl (ddO, dbh, Om) = 8.022

Length o f the base part o f meniscusLLBG(0, dbh, Om) = 15.138

LLB((ddO, dbh, Om) = 19.62Meniscus length to B(c)

* tt T , xx x LLBG(dd, dbh, Om) IJLIfdbh ,Om) = 1.135ULI/dbh.Om ) := -----------------------------

LLBG(ddO, dbh, Om)Liquid meniscus lengthening inrespect to base LLBC(0,dbh,Om)

LLQdbh, Om) := -------------------------LLQ dbh, Om) = 13.34 U L l/dbh , Om)

/•Bx

LL(Bx) := J] „ x21 + fl(x,Om ) d>■'0

Coordinate x finding for point B (c): x = B cx DLC(dbh, Om,Bx) = 0

B x := .l GivenDLC(dbh, Om, Bx) := LLQdbh, Om) - LL(Bx)

Bcx(dbh,Om) = 2.864

Coordinate y finding for B (c): x=Y c01 By(dbh,Om) = 2.893

By(dbh, Om) := Fb3(Bcx(dbh, Om), ddO, dbh, Om)

Parameter o f coordinate x for Bcxfdbh, Om) xc = 0.357x c *= .................

p o in t B q( c): XC=BCX bbl(ddO,dbh,Om)

Coordinate 6 ^(0) after m eniscus stoppedLL(hl(Om)) = 18.946

C Bbx2

LL(Bbx2) := y j l + fl[x ,(O m )]2 d>•'O

The liquid m eniscus length i f the liq. slag vol.=m ax

LLBC(ddO, dbh, Om) = 19.62

Initial position o f the point o f observation (for temperature calc.)


113

Meniscus length till the point B q (c^x:= xc ddOl := 1

LLC(dbh,Om) = 13.34

Meniscus length after top displacement to 1

Meniscus lengthening after the top DLL(ddOl) := ’ ^bh ’ ° m^displacement

Liquid meniscus length till point B(c) after top displacement from the base position till the end o f first stage

Coordinate x for point B(c) finding: x=Bxl

LL>(xx,dd01) = 15.138

LLBC(ddO, dbh, Om)

DLL(dd01) = 1.135

LL;(xx,dd01) := DLfrdd01)-LLBXx^

LL?(xx,dd01) = 15.138

rBbx

LL(Bbx) := 1 + fl[x,(O m )] dx0

DL(xx,ddOl, Bbx) := LL>(xx, ddOl) - LL(Bbx)

B b x := .l Given Dl/xx,dd01 ,Bbx) = 0 xx:= xc

Bbxl(xx,ddOl) = 3.979 Bby3 := Fb3(Bbxl(xx,ddOl), ddOl, dbh, Om)

x := 0 ,.2 ..2 H l(O m ) dbh = 0.9 O m = 5 dd:=dd0

6.94

13.88

20.81


114

Heat constants

Coef. o f solid slag heat conductivity Ksl s 1.83

Coef. o f liquid slag heat conductivity Ksl _ f := 2

Solid slag thickness outside rim, mm Dsl = 1.5

Extraction speed, MM/sec V V = 3.33

Thermal diffusivity RE MM^/sec Am = 3.68

Coef. o f liquid metal heat conductivity Ktm = 23.2

Thickness o f disturbance layer (RE length) Lm := .3

Water temperature T w = 15

Liquid metal temperature (with superheat) Tm = 1515

Temperature solidus Tsoi = 1445

Temperature liquidus Tlik= 1495

Differential equation o f unsteady state heat transfer

ddO = 0.66 xx := xc dd := 1 bbl (dd, dbh, Om) = 10.101

T1 := 1 — ddO - .000001 bbl(ddO,dbh,Om) = 8.022Dc = 6.766

Am (bb(dbh, Om) - zzl)Dc := --------------- — --------------- LLBffddOl, dbh, Om) = 22.265

ddOl := 1 fl(Bbxl(xx,dd01),O m ) = -0 .992 COS(xx,dd01) = 0.71

DYQ(xx,dd01,Om) = 12.119 ddO = 0.66 dbh = 0.9 O m = 5 Bbxl(xx,dd(>) = 2.864

DFb(Bbxl(xx, ddOl), ddOl, dbh, Om) = 0 (fi(Bbxl(xx,dd01),O m ))2 = 0.985

Given

(T m —u(x,t))(bb(dbh ,O m ) — zzl) [~ ~ 2ut(x,t) = Dc uvJ x ,t) H----------------------------------------------------- J 1 + fl(B bxl(xx,t + ddO),Om)

1 x* LLBQ(t + ddO, dbh, Om)

u(x, 0) = Tm spacepts = 5 timepts = 10

______________________________[(Tm - Tw)-0.8-CQS(xx,t + ddO)]_______________________

’ ~ [TTDYO(Bbxl(xx,t + ddO), dbh, Om) + P sQ DFb(Bbxl(xx,t + ddO),t + ddO,dbh,Om)

\ x Ksl ) + K s l f■Ktm


115

u := Pdesolveu(Lm,t) = Tm U'X’1 L n ,/ '

° 'i. r° 'iT 1)

, spacepts ,timepts

HHh(t) := bbl(ddO,dbh,Om) + (b b l(l,d b h ,O m ) - bbl(0 ,dbh ,O m ))t

A := C reateM esh(u ,0,L m ,0,T l)

u (0 ,T l) = 1.50856 x 103 XX 0-357

n := 2 0 i : = 0 . .n j : = 0 . .n ^ _ l£n . ^ T l-i1 n * n

HT. := bbl (ddO, dbh, Om) + (b b l(l,d b h ,O m ) - bbl(0,dbh ,Orn))-^xT.j

UU<,,i:- " K T1) UU0 i , = 1.509 * 1 0 3

UUft „ = 1.509 X 103( \ u ,uxL,xT.)

i j;


1500

UU0 i!480

1460

0 0.1 0.2

xLj

1500

1460

8.5 9 9.5 10


117

xx:= 0

Given

u ,(x ,t) = D c „ xJx,t> + (T m - U (».t» (bb(dbh.Om) - zzl) J + n , Bbxl( + dd0) 0m )3 1 LLB0(t + ddO, dbh, Om)

u(x, 0) = Tm

ux(0 , t) =[(Tm - Tw)-0.8-CQS(xx,t + ddO)]

7 DY(XBbxl(xx,t + ddO), dbh, Om) + Pslî Di'b(Bbxl(xx,t + ddO),t + ddO,dbh,Om)~

Ksl J Ksl f■Ktm

u(Lm,t) = Tm

u := Pdesolve u ,x ,L m J’^ T l J

.,spacepts ,timepts

UU,UUQ = 1.515 x 10

UU,

xx:= 0.05 Given

ut(x ,t) = Dc-ux^x,t)(Tm - u(x,t))-(bb(dbh,O m ) - zzl) / ~ 2 ---------- v ■■■— ------: ---d l + fl(Bbxl(xx,t + ddO),Om)

LLBQ(t + ddO, dbh, Om)

u (x ,0) = Tm

ux(0,t) =[(Tm - Tw) 0.8-CQS(xx,t + ddO)]

f DYO(Bbxl(xx,t + ddO),dbh,Om) + Dsl DFb(Bbx l(xx,t + ddO),t + ddO,dbh,Om)

\ Ksl ) K s l f■Ktm


118

u(Lm,t) = Tm

u := Pdesolve <n r <nu ,x ,| , ,t , | , spacepts ,timeptsLmy v T ly

UU., . := u |x L ,T l) rlTT t - t -l , i V i / UUj Q = 1.515 x 10

3 UHj . := uÔ, xT.j

xx:= O.l Giver

ut(x ,t) = Dc-ux>(x, t) +

u(x ,0) = Tm

(T m - u(x,t))-(bb(dbh,O m ) - zzl) I ~ " " ~ 2- --------------------- :----- --------- --Jl + fl(B bxl(xx,t + ddO), Om)

LLBQ[t + ddO, dbh, Om)

[(Tm - Tw)-0.8-COS(xx,t + ddO)]

DYOf Bbxl(xx,t + ddO),dbh,Om) + Dsl ^ DFb(Bbx l(xx,t + ddO),t + ddO,dbh,Om)

Ksl J K sl_f■Ktm

u(Lm,t) = Tm u := Pdesolve“ ■ w '

0 ) - Ispacepts ,timepts

UU.2 , i :=uK TI) U lij = 1.515 x 10


119

1510

1500

14900.1 0.2

xLj

1510 -

UH2>i

1500

UU,.2>i := u (x L ,T l)

xx:= 0.15 Given

U U , n. = 1.514 x 102,0i

(Tm - u (x ,t))(bb(dbh ,Om) - zzl) f ~ 2v v ’ -— ------! L-yj 1 + fl(B bxl(xx,t + ddO),Om)

LLBQ[t + ddO, dbh, Om)

u (x,0) = Tm


120

ux(0,t) =[(Tm - Tw) 0.8 COS(xx,t + ddO)]

( DYG(Bbxl(xx,t + ddO),dbh,Om) + Dsl DFb(Bbxl(xx,t + dd0),t + ddO,dbh,Om)

\ Ksl J K sl_f•Ktm

u(Lm,t) = Tm

UU3 .:= u (x L ,T l) UU3 0

Given xx:= 0.2

o W o û := Pdesolve u ,x ,| , ,t , , , spacepts ,timepts

Lmy v T iy

1.514 x 10^ UP 5 .i - " ( 0 xTi)

ut(x,t) = Dc-uXJJx,t) +(T m - u(x ,t)) (bb(dbh,Om) - zzl) L ~ 2 ---------- - v— ------!----- --------- - >/1 + fl(B bxl(xx,t + ddO), Om)

LLBlft + ddO,dbh,Om)

u(x, 0) = Tm

u ^ 0 , t ) :[(Tm - Tw)-0.8-CQS(xx,t + ddO)]

u(Lm,t) = Tm

( DYC(Bbxl(xX|t + ddO),dbh,Om) + Dsl ^ DFb(Bbxl(xx,t + ddO),t + ddO,dbh,Om)

Ksl ) + K s l f•Ktm

u := Pdesolve < 0 . 'u,x,[ . ,t , spacepts ,timeptsLm J v T ly

UU4 i := u (x L ,T l) UU4 , o = L :

Given xx:= 0.25

513 x 10UH4>.:=u(0 ,xT j)

ut(x ,t) = D c u xx(x ,t) +(T m -u (x ,t ) ) (bb(dbh,Om) - zzl) f " n2 ---------- — i---------------------- ' | + fl(B bxl(xx,t + ddO), Om)

LLBCft + ddO, dbh, Om)

u(x,0) = Tm

uÔ .t) =[(Tm - Tw)-0.8-CQS(xx,t + ddO)]

DYO(Bbxl(xx,t + dd0),dbh,Q m ) + D sl^ DFb(Bbx l(xx,t + ddO),t + ddO,dbh,Om)

Ksl ) K s l f•Ktm


121

u(Lm,t) = Tin

u := Pdesolveo W o ^

u ,x ,| , ,t , ., spacepts ,timeptsLm) l^T l)

UU5 .:= u (x L ,T l) UU5 ,0 = Li * , , m 3 U H . . :=u(0,xT .l = 1.512 x 10 5 ,i V 1/

Given xx:= 0.3

, n n n . (Tm - u (x,t)) (bb(dbh,Om) - zzl) 7 . „ , . , ^ ,2ut(x ,t) = Dc uvTx,t) H------------------------------------------------------\11 + fl(B bxl(xx,t + ddO),Om)

1 x* LLB((t + ddO, dbh, Om) V

u(x, 0) = Tm

LScCO.t) =[(Tm - Tw)0.8-CQ S(xx,t + ddO)]

7 DYO(Bbxl(xx,t + ddO), dbh, Om) + Dsl ̂ DFb(Bbxl(xx,t + ddO),t + ddO,dbh,Om)~

V Ksl J + K s l f

u(Lm,t) = Tm u := Pdesolve 0 W » û ,x ,| , ,t , ..spacepts ,timeptsL m ; V T i;

U U , . := u (x L ,T ll TTTT ln 3 U H , . := u |0 ,xT .)6 ,i V i / UU6 0 = 1.511 x 10 6 ,i ( if

Given xx:= 0.35

W = D , M i . 1 ) + (T m - u(x,t))-(bb(dbh,O m ) - zzl) J fl(B b x |( t d d 0 )0 m )i 1 X)f LLB0(t + ddO, dbh, Om)

u (x,0) = Tm

ux(0,t) =[(Tm - Tw)0.8-CQ S(xx,t + ddO)]

( DYO(Bbxl(xx,t + ddO),dbh,Om) + Dsl ̂ DI’b(Bbxl(xx,t + ddO),t + ddO,dbh,Om)

Ksl Ksl f

u(Lm,t) = Tm u := Pdesolve u ,x , 0 "l, , t , spacepts ,timeptsLmJ 1,11 J

•Ktm

■Ktm


122

uu, u (x L ,T l )

UU? = 1.509 X 103 UH7>. : = u ( 0 , xT.)

xx:= 0.4 Given

ut(x ,t) = D c u x>(x, t) +(Tm - u(x,t))-(bb(dbh ,0m ) - zzl) f ~ 2-----------------------------------------------------J 1 + fl(B bxl(xx,t + ddO),Om)

LLBCft + ddO, dbh, Om)

u(x, 0) = Tm

Lix(0,t) = [(Tm - Tw)0.8-CQ S(xx,t + ddO)]

f DYC(Bbxl(xx,t + ddO),dbh,Om) + D sl^ DF'b(Bbxl(xx,t + ddO),t + ddO,dbh,Om)

ly Ksl ) + K s l f

u(Lm,t) = Tm

u := Pdesolve 0 ^ ( 0 ̂u ,x ,l , ,t , , , spacepts , timepts' LmJ V T1)

■Ktm

UU8,i;-" K T1) LJLL . = 1.507 x 10o , U

3 UH, , i := u ( ° ’xTi)

Given xx:= 0.45

ut(x,t) = Dc-uxx(x, t) +(T m - u(x ,t))(bb(dbh ,O m ) - zzl) I ~ ' ~ ~~ 2 --------------- :----- --------- - J l + fl(B bxl(xx,t + ddO),Om)

LLB(f t + ddO, dbh, Om)

u (x ,0) = Tm

ux(0 ,t) :[(Tm - Tw)-0.8-COS(x*t + ddO)]

( DY()Bbxl(xx,t + dd0),dbh,Q m ) + Dsl ̂ DFb(Bbxl(xx,t + ddO),t + ddO,dbh,Om)

Ksl ) + K s l f■Ktm

u(Lm,t) = Tm u := Pdesolve' . - u ,x , , ,t , ,, spacepts , timepts. {Lm) ^ T lJ

UU,9 ,i u(x lf TI) UU9 0 = 1.504 x 103 U H ,, ,1 = 8 (0 , xT)

xx:= 0.5


123

Given

ut(x ,t) = D c - u j x ,t ) +(Tm - u (x ,t))(bb(dbh ,O m ) - zzl) f 2 ---------- -■-■■■ -— ------:----- -■'J 1 + fl(B bxl(xx,t + ddO), Om)

LLBQft + ddO,dbh,Om)

u (x ,0) = Tm

ux(0 ,t) =[(T m -T w ) • 0.8-COS(xx,t + ddO)]

f DYC(Bbxl(xx,t + ddO),dbh,Om) + D sl^ DFb(Bbxl(xx,t + ddO),t + ddO,dbh,Om)

i Ksl ) K s l f•Ktm

u(Lm,t) = Tm u := Pdesolve u ,x ,L m / \ T 1 J

1.501 x 103 UH1 0 ,i := u ( ° ’xTi)UUi0>i:=u(xL,T1)

UU,10,0

.spacepts .timepts

Given xx:= 0.55

ut(x ,t) = D c u x;|x , t ) +(T m - u (x .l)) (bb(dbh.Om) - z z l ) ^ ^ fl t 1

L LB ((t+ ddO.dbh.Om)

u(x,0) = Tm

[(Tm - Tw)-0.8 CQS(xx,t + ddO)]

7 DYOfBbxl(xx,t + ddO), dbh, Om) + D sl^ DFb(Bbxl(xx,t + ddO),t + ddO.dbh.Om)'

\ Ksl J K s l fKtm

u(Lm,t) = Tm u := Pdesolve' fo W o ^ . ■u .x , , ,t , ..spacepts .timepts

UU.[ll .i-ufxL.Tl) UUn ,0 == 1.499 x 10 UK,! ^ (O .z T .)

Given xx:= 0.6

Ut(x ,t) = D c u j x , t) +d m " u(x.t»-(bb(dbh.O m ) - zzl) ^ + + 2

L LB(ft+ddO .dbh.O m )


124

u(x,0) = Tm

u x( 0 , t ) ^[(Tm - T w )0 .8C Q S(xx ,t + ddO)]

( DY(fBbxl(xx, t + ddO),dbh,Om) + D sl^ DFb(Bbxl(xx,t + ddO),t + ddO,dbh,Om)

Lj_v Ksl J + K s l f■Ktm

u(Lm,t) = Tm

UU.12,i:=U(XV n)

i 0 ^ ( 0u := Pdesolve u ,x , . , t, , spacepts , timepts 1 Lmy y j l )

UUn Q = 1.497 x 103 UH1 2 ,i := u ( ° ’xTi)

xx:= 0.65 Given

ut(x,t) = Dc-uX!(x ,t) +(Tm - u(x,t))-(bb(dbh ,Om) - zzl) [~ ~ " ~ , 2

7 ■yjl + fl(Bbxl(xx,t + ddO),Om)LLBQft + ddO, dbh, Om)

u(x,0) = Tm

Jx( 0 , t ) :[ (T m - Tw)-0.8-COS(xx,t + ddO)]

u(Lm,t) = Tm

f DYC(Bbxl(xx>t + ddO),dbh,Om) + Dsl ̂ D l’b(Bbxl(xx,t + ddO),t + ddO,dbh,Om)

U\ Ksl J + K s l f

0 V f o

■Ktm

u := Pdesolve u ,x ,| :,t , ,, spacepts , timeptsLm J \T 1 J

UU.I3,r-"KTI) “V o " 1'= 1.497 x 10

Giver xx:= 0.7

, . _ . . (Tm - u (x ,t)) (bb(dbh ,Om) - zzl) I " ~ 2ut(x,t) = Dc u T x,t) H-----------------------------------------------------J 1 + fl(Bbxl(xx,t + ddO),Om)

1 x* LLBQ[t + ddO, dbh, Om) V

u(x,0) = Tm

u*(0,t) =7 DYO(Bbxl(xx,t + ddO), dbh, Om) + P sQ i Ksl J

[(Tm - Tw)-0.8-CQS(xx,t + ddO)]

DFb(Bbxl(x?^t + ddO),t + ddO,dbh,Om)"

Ksl f■Ktm


125

u(Lm,t) = Tm

u := Pdesolve u ,x , ° v r ° ^L m / ^ T l J

, .spacepts .timepts

u V r - K 71)

Given xx:= 0.75

UU14,0 = L497 x 10 UH14 i := u(o,xT .)

ut(x,t) = Dc-ux)|x , t) +(Tm - u(x,t))-(bb(dbh ,0m ) - zzl) / , „ /r,, „ ~2--------------------------------------------------- -Jl + fl(B b x l(xx t + ddO),Om)

LLBCft + ddO.dbh.Om)

u (x,0) = Tm

ax(0 ,t ) =[(Tm - Tw) 0.8-CQS(xx,t + ddO)]

f DYO(Bbxl(xx,t + ddO),dbh,Om) + Dsl ̂ DFb(Bbx If xx, t + ddO),t + ddO.dbh.Om)

\ Ksl ) + K s l f■Ktm

u(Lm,t) = Tmu := Pdesolve r 0 û ,x ,| , ,t , ..spacepts .timepts

LmJ v T ly

UU,, .:= u (x L ,T l) 3 U H ,_ . := u(0,xT.)15,i V i / UUlg Q = 1.499 x 10 15,i ( l)

Given xx:= 0.8

ut(x,t) = I)c-ux)x ,t ) +(T m - u(x,t))-(bb(dbh,O m ) - zzl) I ~ ~ , , ~2-------------------------------------------- -yjl + fl(B bxl(xx,t + ddO),Om)

LLBC(t+ ddO.dbh.Om)

u (x ,0) = Tm

ux(0 ,t ) =[(Tm - Tw)0.8-CO S(xx,t + ddO)]

f DYQfBbxl(xx,t + ddO),dbh,Om) + Dsl^ DFb(Bbxl(xx,t + dd0),t + ddO.dbh.Om)

\ Ksl ) + K s l f•Ktm


126

u(Lm,t) = Tm

u := Pdesolve 0 W o û,x , | ^ j >spacePts ,timepts

u U16, i :=»(xL,T |)

U U „ „ = 1.501 »103 UHl«,i:- u(°-xTi) lo,U

Giver xx:= 0.85

ut(x,t) = Dc-ux)|> , t) +(Tm - u(x,t))-(bb(dbh,O m ) - zzl) / . , ,2 -y] 1 + fl(B b x l(x x t + ddO),Om)

LLBQT + ddO, dbh, Om)

u(x ,0 ) = Tm

[(Tm - Tw)0.8-CO S(xx,t + ddO)]

f DYO(Bbxl(xx,t + ddO),dbh,Om) + D sl^ DFb(Bbxl(xx,t + ddO),t + ddO,dbh,Om)

\ Ksl ) + K s l f•Ktm

u(Lm,t) = Tm

u := Pdesolveo W o ^

u,x,I , , t , . ,spacepts ,timeptsLmJ VTly

UU.17, i : u(xIl ’T1) UUJ7 0 = 1.503 x 10 3 UH1 7 .:= u (0 ,xT .)

Given xx:= 0.9

ut(x ,t) = Dc-ux^ x ,t) +(T m - u (x ,t))• (bb(dbh, Om) - zzl) / , „ , . , ... .2--------------------------------------------------- 1 + fl(B bxl(xx,t + ddO),Om)

LLBQT + ddO, dbh, Om)

u(x ,0) = Tm

Jx(0,() =[(Tm - Tw) 0.8-CQS(xx,t + ddO)]

DYO(Bbxl(xx,t + ddO), dbh, Om) + Dsl ̂ DFb(Bbxl(xx,t + ddO),t + ddO,dbh,Om)~

Ksl ) K s l f■Ktm


127

u := Pdesolve u ,x , 0 W 0 "l, , t , ,,spacepts ,timeptsU m J i T l )

A := CreateMesh (u , 0 , Lm, 0 , T 1)

UU1Q .:= u (x L ,T ll 3 UH1fi . := u(o,xT .)18,i V i / 18 0 = X

15101500

1500

1460

14900.1 0.28.5 9.5

HT; xl


1 2 8

xx:= 0.95

Given

u .(x ,0 = D c u .J x , , ) + (T — »(x .l)H bb(dbh .O m ) - ? 1) J , + n(Bbxl( jd 0 )' xX LLBOft + ddO, dbh, Om)

u (x,0) = Tm

ux(0 ,t)[(Tm - Tw)-0.8CQ S(xx,t + ddO)]

f DYO(Bbxl(xx,t + ddO),dbh,Om) + D sl^ DFb(Bbxl(xx, t + ddO),t + ddO,dbh,Om)

U_V Ksl J + K s l f•Ktm

u(Lm,t) = Tm

u := Pdesolve ° W Ou ,x ,| , , t , pspacepts ,timeptsLm) \T 1 J

UU19, i := u (xLl’T1) UU19 = 1.508 x 10 UH19,i:=U(“ -xTi)

xx:= 1

Given

u .(x ,t) = Dc-u J x , t) + <Tm - u (x ..)).(bb (dbh ,0m ) + n (B fa K ^ , + dd0)-Om)^1 LLBQ(t + ddO, dbh, Om)

u(x,0) = Tm

Ux(0 ,t)[(Tm - Tw)-0.8CQ S(xx,t + ddO)]

^ DYOfBbx l(xx,t ddO), dbh, Om) + Dsl ̂ DFb(Bbxl(xx,t

Ksl

ddO), t + ddO, dbh, Om)

Ksl f■Ktm


129

u(Lm,t) = Tm

r r<n mu := Pdesolve u ,x , . ,t , . ,spacepts ,timepts

. I W \ T 1 )

XX = 1 ddO = 0.66 t := T it + ddO = 1 Bbxl(xx,t + ddO) = 10.101

COS(xx,t + ddO) = 0.467

fl(B bx 1( xx, t + ddO), Om)2 = 3.595 LLB(,t + dd0 ’dbh ’0m ) = 22265

UU2 # j , u ( x L , T1) U U m # = , .5 1 , | 0 3 U H ^ f . X r . )

bb(dbh,Om)-jj := 0.. n UT. .:= 0.9975Tlil UK. . := TliT xH. := ---------------------J J,i J,i J n

1500-

uu

1510

1490 0 2 4 6 8 1210

x H j, b b l(d d O , d b h , O m )


130

x := 0 ,.2 ..2 H l(O m )

dbh = 0.9

O m = 5

dd := .9995

kk:= 5

■ kkkk := FRAMET dd := ddO + (1 - ddO)—

n

TTIT b b l(d d ,d b h ,O m )jxHH. := ----------------------------

J n

O

4.34

8.67

13.01


131

Appendix B

Adequacy test


132

Physical characteristics o f materials:

Density o f liquid slag, Kg/M^ r0m := 7200

Density o f liquid steel, Kg/M^ RoS := 2667.3

Gravity acceleration, m/s^ 81 := 9 807

Coefficient o f interphase tension, h / m SmS := 1:2058

Experimental data

0 0 1.95 .11

1.14 .15

1.34 .21

1.56 .28

1.76 .37

1.93 .47

2.13 .58

2.33 .71

2.51 .84

2.67 .97

2.84 1.11

3.03 1.29

3.19 1.47

3.37 1.68

3.57 1.93 )


133

B := (R o M -R o S ) —------

2-Sm S106B = 0.018

Liquid metal overflow

Omegal := .07

Omega = -1 .57

tan o f initial angle

tt := tan (Omega)

tt = -818.511

sin o f initial angle

Omega :=■7i Omegal-Tt

2 180

PP :=1 + (tt)

2

PP = 1


134

M ax meniscus height, i f initial angle equal 90^

HiH = 10.416

M ax meniscus height, i f initial angle equal j < 90® (pp = tan j)

PP = 1

- FHI := >-‘ +-PP

B

HI = 10.416

Coordinate (x) along meniscus height in which y=ymax

-JSh = 7.366

h i := E\| B

h i = 7.366

First derivative as a function f(x)

-(pp - B-x2)fl(x) :=

J l - ( p p - B x 2)


Meniscus line y=F(x)

o

•X

m dx*• ' 0.1

Define meniscus height using parameters dbh

dbh := 0.9

ms := 2

|bb(dbh) := h i + dbh(H l - h i)

I f we place coordinate origin into point o f junction o f meniscus and surface o f liquid metal (ext. system), that coordinate o f current location o f the top o f solidified mark Bh(dbh) - displacement o f the coordinate system along X

/•bb(dbh)Bh(dbh) := fl(x) ch

Bh(dbh) = -36 .57

Profile o f rim.


136

Equation o f line o f rim.

fx 'iFb2(x,dbh) : = - Bh(dbh) - fl(x) dx (bb(dbh) > x > 0)

0 ))

Equation o f solid slag line

Fb(x,dbh) := Fb2(x,dbh)-(x< h i) + F b 2(h l,d b h )-(x> h i)

Fb(6, dbh) = -1 .3 0 8

Initial position o f the top o f solid mark on the rim (before displacement) zzl

aa := .0001

z := 0 + aa

Given

Fb(z,dbh) = 0

zzl(z,dbh) := Find(z)

zzl(z,dbh) = 3.996

rO := rows(M AS) - 1

rO = 15

Base for counting X

M<0> . z z l(z , d b h ) + m s + M A S ^q q — MAS <0>

i := 0.. rows(M ) - 1


137

M. . := zzl(z,dbh) + ms + MAS _ n - MAS. „ i , 0 ’ j rO , 0 i ,0

Base for counting Y

M ^ := M AS(l) - MASr 0 1 zzl(z,dbh) = 3.996

J ®M =

0

0 9.566

1 8.616

2 8.426

3 8.226

4 8.006

5 7.806

6 7.636

7 7.436

8 7.236

9 7.056

10 6.896

11 6.726

12 6.536

13 6.376

14 6.196

15 5.996

M(l> =

0

0 -1.93

1 -1.82

2 -1.78

3 -1.72

4 -1.65

x fc -1.56

6 -1.46

7 -1.35

8 -1.22

9 -1.09

10 -0.96

11 -0.82

12 -0.64

13 -0.46

14 -0.25

15 0

o(m ^ )meanlM / = -1 .1 6 9

row s(M )-l rrSM M (dbh,m s) := ^ U Fb(zzI(z,dbh) + ms + M ASf0 () - MAS. Q,dbhj - M; ^

i = 0

.. . . . Given -1 < ms < 1dbh := 0.8

P :=M m im iz<SM M ,dbh,m s) dbhl := PfJ dbhl = 0.963

ms := -2 ,-1 .6 .. 2 dbh := 0.7,0.75.. 1 dbh := PQ ms := Pj


138

x =

M. . := zz l(z ,P nl + P, + MAS _ n - MAS. . i ,0 V 0/ 1 rO,0 i,0i := 0.. rows(M ) - 1

b := lOroundf 2 — 'lI 10)

0.963) SMM(P0 ,P 1j = 0.02260957

x:= 0,.1..2H 1

P0.059J

rO = 15

F b(h l,d bh) = -2 .0 1 7Omegal = 0.07

d (z ,p0) = 3.57

0

0 -1.93

1 -1.82

2 -1.78

3 -1.72

4 -1.65

5 -1.56

6 -1.46

7 -1.35

8 -1.22

9 -1.09

10 -0.96

11 -0.82

12 -0.64

13 -0.46

14 -0.25

15 0

HI

(M)+ + •+

<0>

0

5

10

15

200•5 5 10

Fb(x, dbh), Fb(x, dbh), Fb2(x, dbh), Fb(x, dbh), (M)

FM. := Fb^zzl(z, dbh) + ms + M A S ^ ̂- M AS. Q,dbhj

i := 0.. rows(M ) - 1

<i>

X := M<1) Y := FM Y =

0

0 -2.013

1 -1.852

2 -1.792

3 -1.718

4 -1.598

5 -1.528

6 -1.437

7 -1.32

8 -1.19

9 -1.062

10 -0.939

11 -0.799

12 -0.63

13 -0.476

14 -0.291

15 -0.069


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